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1995-12-05T06:20:27 | 9512 | alg-geom/9512003 | en | https://arxiv.org/abs/alg-geom/9512003 | [
"alg-geom",
"math.AG"
] | alg-geom/9512003 | Bumsig Kim | Bumsig Kim | Quot schemes for flags and Gromov invariants for flag varieties | 16 pages, wrtten by latex 209, compile twice | null | null | null | null | Using Quot schemes and a localization theorem we study Gromov-Witten
invariants for partial flag varieties. The strategy is to extend A. Bertram's
result of Gromov-Witten invariants for special Schubert varieties of
Grassmannians to the case of partial flag varieties. To do so a Grothendieck's
Quot scheme is generalized for flags and proven to be an irreducible, rational,
smooth, projective variety following Str\o mme \cite{St}.
| [
{
"version": "v1",
"created": "Mon, 4 Dec 1995 21:58:00 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Kim",
"Bumsig",
""
]
] | alg-geom | \section{Introduction}
To define Gromov-Witten invariants arise in mirror symmetry, there are two
general rigorous methods so far \cite{RT}\cite{Ko}. In particular Kontsevich
introduced the notion of stable maps for a compactification of moduli spaces.
For Grassmannians, however, there is a natural compactification of the space $%
Mor_d({\PP}^1,Gr(n,r))$ of all holomorphic maps from ${\PP}^1$ to
Grassmannians with a given degree $d$ where $Gr(n,r)$ is the Grassmannian of
all rank $r$ quotient vector spaces of ${\CC}^n.$ We may see $Mor_d({\PP}%
^1,Gr(n,r))$ as the set of all rank $r,$ degree $d,$ quotient bundles of $%
{\CC}^n\otimes {\cal O}_{{\PP}^1}.$ It is not a compact space.
Hence we come to a
Grothendieck's Quot scheme $Quot_d(Gr(n,r)),$ the set of all rank $r,$
degree $d,$ quotient sheaves of ${\CC}^n\otimes {\cal O}_{{\PP}^1}$ \cite
{Gr}$.$ It is proven to be a smooth projective variety by S. A. Str\o mme
\cite{St}. Bertram and Franco-Reina used Grothendieck's Quot schemes for
Gromov-Witten invariants and quantum cohomology of Grassmannians
respectively \cite{Be}\cite{FR}. In this paper using Quot schemes and
a localization theorem we study Gromov-Witten invariants for partial flag
varieties. The strategy is the following. We extend A. Bertram's result of
Gromov-Witten invariants for special Schubert varieties of Grassmannians to
the case of partial flag varieties. To do so a Grothendieck's Quot scheme
is generalized for flags and proven to be an irreducible, rational, smooth,
projective variety following Str\o mme \cite{St}. The Hilbert schemes for
flags have already studied \cite{Se}. On a partial
flag manifold there is an action by a special linear group. It induces an
action on the Quot scheme for flags. There is another action on it by the
multiplicative group ${\CC}^{\times }.$ It is induced from the action $%
\CC ^{\times }$ on ${\PP}^1.$ The analogous action by ${\CC}^{\times }$
does not exist on the Kontsevich moduli space. These two actions are
commutative. These together give isolated fixed points. Using a localization
by action an explicit formula of Gromov invariant for special Schubert
classes with a certain condition is given.
Note that Kontsevich also uses torus actions on his moduli
space of stable maps \cite{Ko}.
In his case he has to deal with summation over trees
since the fixed subsets are rather complicated. For projective spaces the
formula derived in the sequel is shown to agree to the residue formula \cite
{Ki}. The author does not know how to directly
relate with the result of Givental and
Astashkevich-Sadov's computation \cite{GK}\cite{AS}. Now we state our
main results.
\bigskip
For given integers $s_0=0<s_1=n-r_1<\cdots <s_l=n-r_l<n=s_{l+1},$ a flag
variety $Fl:=F(s_1,s_2,...,s_l;n)$ is, by definition, the set of all flags
of complex subspaces $V_1\subseteq V_2\subseteq \cdots \subseteq V={\Bbb C}^n,
\;\dim V_i=s_i$. There are universal vector bundles $S_i$ and
universal quotient bundles $Q_i$ over $Fl$
with fibers ${\Bbb C}^{s_i}$ and $\CC ^n/\CC ^{s_i}$ respectively.
We are interested in a moduli space, the set
$Mor_d({\Bbb P}^1,Fl)$ of all morphisms $%
\varphi $ from ${\Bbb P}^1$ to $Fl$ with $<{\Bbb P}^1,c_1(\varphi
^{*}Q_k)>=d_k$, $d=(d_1,d_2,...,d_l)$.
Since $Fl$ is the fine moduli space such that the associated flag functor
is equivalent to $Mor(\cdot ,Fl)$ where the image of the functor at a scheme
$S$ is the set of all flag quotient bundles of $V\otimes\cal{O}_S$ with
ranks $r_i$. From the point of view as above, $Mor_d(\PP ^1 ,Fl)$ is
realized as the set of
all flag quotient bundles $(F_1,...,F_l)$ of $V\otimes\cal{O}_{\PP ^1}$ with
rank $r_i$ and degrees $d_i$ and hence it can be compactifyed by collecting
flag quotient sheaves. More precisely,
\bigskip
\begin{theorem}\label{thm1}
There is a smooth compactification $fQuot_d(Fl)$
of $Mor_d({\Bbb P}^1,Fl)$.
The underlying set of $fQuot_d(Fl)$
is the set of all flag quotient sheaves $(F_1,...,F_l)$
of $V\bigotimes {\cal O}_{{\Bbb P}^1}$ over ${\Bbb P}^1$ where
rank of $F_i$ is $r_i$ and its degree is
$d_i.$ Over the irreducible, rational, projective variety
$fQuot_d(Fl)\times \PP ^1$ there are tautological bundles $\cal {E}_i$
and sheaves $\cal {Q}_i$, $i=1,...,l$. They form exact sequences
$$ 0\rightarrow \cal {E}_i\rightarrow V\otimes O_{\PP ^1\times fQuot_d(Fl)}
\rightarrow \cal {Q}_i\rightarrow 0.$$
The induced sheaf morphisms $\cal {E}_i\rightarrow \cal{Q}_{i+1} $
are identically zero.
\end{theorem}
This fine moduli space $fQuot_d(Fl)$ will give the Gromov-Witten invariants
defined in \cite{KM}\cite{Ko}.
\begin{theorem}\label{thm2}
Let $p_1,...,p_N$ be fixed N distinct points in
$\PP ^1$. For i=1,...,N, let $\alpha _i$
be integers in $\{s_1,...,s_{l+1}\}$ and
let $\beta _i$
be positive integers less than $s_{\alpha _i+1}-s_{\alpha _i-1}$.
Then the number of
morphism $\varphi $ from $\PP ^1$ to $Fl$
such that each $\varphi (p_i)$ is in each the Poinc\'{a}re
dual Schubert subvariety to the classes $c_{\beta _i}(S_{\alpha _i})$
and $<\PP ^1 ,c_1(\varphi ^*Q_k)>=d_k,$ $%
k=1,...,l,$ is well-defined and it is
\[ \int _{fQuot_d(Fl)}\wedge _i c_{\beta _i} (\cal {E}_{p_i}^i). \]
The integration is not depend on the choices of the
point $p_1,...,p_N$ in $\PP ^1$.
\end{theorem}
By the torus action on $fQuot_d(fl)$ induced from the standard
$\CC ^{\times }$-action on $\PP ^1$ and the
standard $(\CC ^{\times})^n$ on $Fl$ one
can apply Bott's residue formula to the above integration to get
\begin{theorem}\label{thm3}
The integration in the theorem \ref{thm2} is
\[
\sum_{\text{all integers as in (\ref{integer})}}\frac{\prod _i
(\sigma _{\alpha _i}^{\beta _i})
\prod (\text{characters as in (\ref{tang1}))}}
{\prod (\text{characters as in (\ref{tang2}))}}.
\]
The notations will be explained as follows.
\end{theorem}
Consider a sequence of data by nonnegative integers $d_{i,j}$ and $a_{i,j}$:
\begin{equation}
(d_{1,1},a_{1,1};...;d_{1,s_1},a_{1,s_1})\cdots
(d_{l,1},a_{l,1};...;d_{l,s_l},a_{l,s_l}) \label{integer}
\end{equation}
such that $d_{i,j}-a_{i,j}\geq d_{i+1,j}-a_{i+1,j}\geq 0,\;a_{i,j}\geq
a_{i+1,j}$ and $\sum_{j=1}^{r_i}d_{i,j}=d_i.$
Set $b_{i,j}:=d_{i,j}-a_{i,j}.$ Then we consider
\begin{eqnarray}
(p-a_{i,j})\hbar +\lambda _{j^{\prime }}-\lambda _{j},\;\text{for }%
0\leq p\leq a_{i,j^{\prime }}-1,\;1\leq j,j^{\prime }\leq s_i, \nonumber
\label{tang1} \\
(b_{i,j}-p)\hbar +\lambda _{j^{\prime }}-\lambda _{j},\;\text{for }%
0\leq p\leq b_{i,j^{\prime }}-1,\;1\leq j,j^{\prime }\leq s_i, \label{tang1}
\\
(p-a_{i,j})\hbar +\lambda _{m}-\lambda _{j}.\;\text{for }0\leq p\leq
d_{i,j},\;1\leq j\leq s_i,\;s_i+1\leq m\leq n, \nonumber
\end{eqnarray}
and
\begin{eqnarray} \label{tang2} \\
(p-a_{i,j})\hbar +\lambda _{j^{\prime }}-\lambda _{j},\;
\text{for }0
&\leq &p\leq a_{i+1,j^{\prime }}-1,\;1\leq j,\leq s_i,\;1\leq j^{\prime
}\leq s_{I+1}, \nonumber \\
(b_{i,j}-p)\hbar +\lambda _{j^{\prime }}-\lambda _{j},\;
\text{for }0
&\leq &p\leq b_{i+1,j^{\prime }}-1,\;1\leq j,\leq s_i,\;1\leq j^{\prime
}\leq s_{i+1}, \nonumber \\
(p-a_{i,j})\hbar +\lambda _{m}-\lambda _{j},\;
\text{for }0 &\leq &p\leq
d_{i,j},\;1\leq j\leq s_i,\;\;s_{i+1}+1\leq m\leq n. \nonumber
\end{eqnarray}
Finally let
$\sigma ^k_i$ be the $k$-th elementary symmetry function of
$a_{i,j}\hbar +\lambda _{j}$, $j=1,...,s_i$.
\bigskip
{\bf Acknowledgments and remarks: }My special thanks goes to my advisor A.
Givental for his wonderful guide. I would also like to thank A. Bertram, R.
Hartshorne, M. Kontsevich, and S. A. Str\o mme for answering my questions.
After finishing a preliminary version of the paper
I learned there are more advanced
results by A. Bertram \cite{Be1} and I. Ciocan-Fontanine \cite{CF} in some
cases. But works seem complementary. I would like to express my thanks to
I. Ciocan-Fontanine for pointing out an error
in the preliminary version of the paper. By \cite{Be1}\cite{CF} the condition
$\beta _i < s_{\alpha _i+1}-s_{\alpha _i-1}$ could be omitted.
\section{flag-Quot schemes}
All schemes will be assumed to be algebraic schemes over
an algebraically closed field ${\bf k}$ of characteristic $0\;$and all
sheaves will be quasi-coherent.
The space of all rank $r,$ degree $d,$ quotients of trivial sheaf $V\otimes
{\cal O}_{{\Bbb P}^1}$---or equivalently, all subsheaves of rank $s=n-r$,
degree $-d$---is a smooth, rational, irreducible, projective variety $%
Qout(V\otimes {\cal O}_{{\Bbb P}^1}$, Hilbert polynomial $rx+r+d)$ \cite{Gr}%
\cite{St}. Let us denote by $Quot_d(s,V)$ the Quot scheme. It could be
considered as a compactification of the space $Mor_d({\Bbb P}^1,Gr(n,r))$ of
all degree $d$ holomorphic maps from the projective line ${\Bbb P}^1$ to the
Grassmannian $Gr(n,r)$ which is the set of all rank $r$ quotient spaces of $V$
\cite{FR}. It is equipped with a universal locally free sheaf ${\cal E}$ and
a universal quotient sheaf ${\cal Q}$ over ${\Bbb P}^1\times Quot$ with an
exact sequence $0\rightarrow {\cal E}\rightarrow V\bigotimes {\cal O}_{{\Bbb P%
}^1\times Quot}\rightarrow {\cal Q}\rightarrow 0.$ They are flat over $Quot.$
For the special Schubert varieties, Gromov invariants can be defined via
Quot schemes as enumerative invariants \cite{Be}\cite{FR}. To extend their
results to flag varieties (not necessary complete flag varieties), we will
construct Quot schemes for flags, a compactification of $Mor_d(\PP ^1, Fl)$.
\bigskip
Let $s=(s_1,...,s_l),\;s_{i+1}>s_i>0,$ and $d=(d_{1,}...,d_l)$ be
multi-indices of nonnegative integers. For this moduli problem,
first we introduce a moduli functor $F^s_d$.
A contravariant functor $F_d^s$ from the category of schemes to the
category of sets is defined to be: for a scheme $S$, $F_d^s(S):=$the set
of all flag subsheaves $(E_1,E_2,...,E_l,E%
_n=V\bigotimes {\cal O}_{{\Bbb P}^1\times S})$ over ${\Bbb P}^1\times S$ where
sheaf ${\cal E}_i$ is subsheaf of ${\cal E}_j$ if $i<j,$
sheaves are flat over $S$, and
the rank of ${\cal E}_i$ is $s_i$ and its degree is $-d\;$over ${\Bbb P%
}^1$.
The functor $F_d^s\,$can be defined by quotients in a more transparent way,
for different data $0\rightarrow E\rightarrow V\otimes\cal{O}_{\PP ^1\times S}$
could give the same data
$V\otimes \cal{O}_{\PP ^1\times S}\rightarrow F\rightarrow 0$
\cite{FR}.
Now we will show the functor is representable. First some notations;
for a sheaf $\cal F$
over $\PP ^1 \times S$, denote by $\cal {F}(m)$
$\cal{F}\otimes \pi ^*\cal {O}(m)$, where $\pi$
is the projection from $\PP ^1\times S$ to $\PP ^1$. For the
second projection from
$\PP ^1\times S$ to $S$ we will use $\pi _S$.
Taking advantage of the existence of Quot schemes,
the obvious candidate for the scheme representing the functor $F^s_d$ is
the appropriate subscheme of
$Quot_{d_1}(s_1,n)\times ...\times Quot_{d_l}(s_l,n)$.
Over the product of Quot schemes there are universal subsheaves $\cal {E}_i$
and quotient sheaves $\cal {Q}_i$ induced each from $Quot_{d_i}(s_i,n)$.
Define $fQuot_d(Fl) $ as the degeneracy loci of
\[
\bigoplus_{i=1}^{l-1}
\left( (\pi _{\Pi Quot_{d_i}(s_i,n)}){*}{\cal E}_i(m)\rightarrow V/%
(\pi _{\Pi Quot_{d_i}(s_i,n)})_{*}({\cal E}_{i+1}(m))\right)
\]
for any $m\geq \max_i\{d_i\}-1$. Note that
$(\pi _{\Pi Quot_{d_i}(s_i,n)})_{*}{\cal E}_i(m)$ and $%
V/(\pi _{\Pi Quot_{d_i}(s_i,n)})_{*}({\cal E}_{i+1}(m))$ are locally free
so that there is no problem giving a scheme structure on $fQuot_d(Fl)$.
For shorthand, we will write $Quot_{d_i}(s_i,n)$ by $Quot_i$ and $%
fQuot_d(Fl)$ by $fQuot$, then now we are ready for
\begin{proposition}
The functor $F_d^s$ is representable by the (unique) projective scheme, $%
fQuot_d(Fl)$.
\end{proposition}
\proof
The statement means that for any scheme $S$, $F_d^s(S)=Mor(S,fQuot)$
in a functorial way. Let $(E_1,...,E_l)\in F_d^s(S).$ Then we have the
morphism $g:S\rightarrow Quot_1\times ...\times Quot_l$ from
the fine moduli property of
the Quot schemes.
We see that
$g^{*}(\pi _{\Pi Quot_i})_{*}{\cal E}_i(m)
\cong (\pi _S)_{*}E_i(m)$ naturally \cite{Mu}. The
fact that $E_i\subset E_{i+1}$ implies that $(\pi _S)_{*}E_i(m)\subset (\pi
_S)_{*}E_{i+1}(m).$ Hence
$g^{*}(\pi _{\Pi Quot_i})_{*}{\cal E}_i(m)\subset g^{*}
(\pi _{\Pi Quot_i})_{*}%
{\cal E}_{i+1}(m)$ and $g$ factor through $fQuot$.
$\Box $
\bigskip
Over $fQuot$ there are exact sequences of universal sheaves
\[
0\rightarrow {\cal E}_i\rightarrow V_{{\Bbb P}^1\times fQuot}\rightarrow
{\cal Q}_i\rightarrow 0
\]
and surjections ${\cal Q}_i\rightarrow {\cal Q}_{i+1}.$ Each ${\cal E}_i$
are locally free since it is a subsheaf of a locally free sheaf over ${\Bbb P}%
^1\times fQuot$ and it is flat over $fQuot.$ For given $f\in Mor(S,fQuot),$
the corresponding quotient sheaves over ${\Bbb P}^1\times S$ is just the pull
back $(id\times f)^{*}({\cal Q}_i).$
\subsection{Irreducibility and Smoothness}
To show the Quot scheme for flags is irreducible and smooth, one can simply
adapt Str\o mme's proof \cite{St}.
\begin{theorem}\label{thm4}
$fQuot$ is an irreducible, rational, nonsingular, projective
variety.
\end{theorem}
\proof
We will work by the language of quotients rather than subsheaves.
For $m=0,-1$ and $i=1,...,l,$ let
${\cal Q}_m^i$ be $(\pi _{fQuot})_{*}{\cal Q}_i(m)$,
a locally free sheaf over $%
fQuot$ of rank $(m+1)r_i+d_i$, and let
$X_m^i\rightarrow fQuot$ be the associated
principal $GL((m+1)r_i+d_i)$-bundle. One has a smooth morphism $\rho :$%
\[
\begin{tabular}{l}
$\prod_{fQuot,i=1}^l(X_{-1}^i\times _{fQuot}X_0^i)=:Y$ \\
$\;\;\;\;\;\;\;\;\downarrow \rho $ \\
$\;\;\;\;\;\;\;fQuot.$%
\end{tabular}
\]
We will show that $Y$ is an irreducible and smooth variety after finding an
isomorphism to a smooth irreducible affine quasi-variety. Since $\rho $ is
smooth, we conclude $fQuot$ is a smooth, irreducible, projective variety. Let
$%
N_m^i:=V^{(m+1)r_i+d_i}$ for $m=0,-1$ and$\;i=1,...,l.$ Here $V^r$ is, by
definition, the $r$-dimensional vector space over the ground field $k.$ Let $%
W:=Hom(V,V^{r_1+d_1})\times Hom_{{\Bbb P}^1}(\pi ^{*}V^{d_1}(-1),\pi
^{*}V^{r_1+d_1})$
$\times Hom(V^{d_1},V^{d_2})\times Hom(V^{r_1},V^{r_2+d_2})\times Hom_{{\Bbb P%
}^1}(\pi ^{*}V^{d_2}(-1),\pi ^{*}V^{r_2+d_2})$
$\times \cdots \times Hom(V^{d_{l-1}},V^{d_l})\times
Hom(V^{r_l},V^{r_l+d_l})\times Hom_{{\Bbb P}^1}(\pi ^{*}V^{d_l}(-1),\pi
^{*}V^{r_l+d_l}).$ Let $\overline{X}:=$associated affine space. On ${\Bbb P}^1%
\times \overline{X},$ there is a tautological diagram:
\begin{eqnarray*}
&&
\begin{array}{ccc}
\pi _{\overline{X}}^{*}V^{d_l}(-1) & {\rightarrow }_{v_1} & \pi _{%
\overline{X}}^{*}V^{r_l+d_l} \\
& & \uparrow \mu _l
\end{array}
\\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,%
\,\,\,\,\,\,\,\,\vdots \\
&&
\begin{array}{ccc}
\pi _{\bar{X}}^{*}V^{d_2}(-1) & {\rightarrow }_{v_2} & \pi _{\bar{X}%
}^{*}V^{r_2+d_2} \\
& & \uparrow \mu _2
\end{array}
\\
&&
\begin{array}{ccc}
\pi _{\overline{X}}^{*}V^{d_1}(-1) & {\rightarrow }_{v_1} & \pi _{%
\overline{X}}^{*}V^{r_1+d_1} \\
& & \uparrow \mu _1 \\
& & \pi _{\overline{X}}^{*}V.
\end{array}
\end{eqnarray*}
\
Let $Z\subset \overline{X}$ be the nonsingular irreducible quasi-variety
defined by the conditions
i) $v_i$ is injective on each fiber over $X$,
ii) the induced map $\overline{\mu }_i:\pi _Z^{*}V\rightarrow $coker ($%
v_i)$ is surjective, and
iii) coker$(v_i)$ is flat over $Z$ with rank $r_i$ and degree $d_i$ on the
fibers of $\pi _Z.$
The above conditions are open conditions in algebraic geometry. Let $%
W^{\prime }:=Hom(V^{d_1},V^{r_2+d_2})\times \cdots \times
Hom(V^{d_{l-1}},V^{r_l+d_l}).$ Let $X$ be a closed subvariety of $Z$ $\,$ of
codimension $\sum_{i=1}^ld_i(r_{i+1}+d_{i+1})$ defined by the inverse image
of a morphism
\begin{equation}
W\rightarrow W^{\prime } \label{ww}
\end{equation}
measuring commutativity of the diagram
\[
\begin{array}{ccc}
\pi ^{*}_ZV^{d_{i+1}}(-1) & \rightarrow _d & \pi ^{*}_ZV^{r_{i+1}+d_{i+1}} \\
\uparrow a & & \uparrow b \\
\pi ^{*}_ZV^{d_i}(-1) & \rightarrow _c & \pi ^{*}_ZV^{r_i+d_i}
\end{array}
\]
by $d\circ a-b\circ c.$
$X$ is a product of hypersurfaces defined by irreducible quadratic
polynomials. It is smooth away where the right vertical arrow is zero map.
And so $X$ is smooth and irreducible. There is a natural morphism $%
g:X\rightarrow fQuot$ by the construction of $X.$
By the construction,
$g^{*}{\cal Q}_m^i=(N_m^i)\otimes \cal {O}_X=((m+1)r_i+d_i){\cal O}_X.$
Therefore we have a morphism $s$ in the diagram
\[
\begin{tabular}{lll}
$X\ $ \ & ${\rightarrow }_s$ & ${%
\prod_{fQuot,i}(X_{-1}^i}\times _{fQuot}X_0^i)=:Y$ \\
$g\searrow $ & & $\swarrow \rho $ \\
& $fQuot$ &.
\end{tabular}
\]
We will show $X$ and $Y$ are isomorphic finding the inverse of $s$ which
complete the proof.
We are given
isomorphisms on $Y$
\[
\lambda _m:\rho ^{*}{\cal Q}_m^i\rightarrow (N_m^i)_Y.
\]
By a proposition (1,1) in \cite{St},
we get a diagram on ${\Bbb P}^1\times Y$ except $\Uparrow$
\[
\begin{array}{ccc}
0\rightarrow \pi _Y^{*}N_{-1}^i(-1)\rightarrow & \pi _Y^{*}N_0^i &
\rightarrow (1\times \rho )^{*}{\cal Q}_i\rightarrow 0 \\
& \Uparrow & \uparrow \\
0\rightarrow \pi _Y^{*}N_{-1}^{i-1}(-1)\rightarrow & \pi _Y^{*}N_0^{i-1} &
\rightarrow (1\times \rho )^{*}{\cal Q}_{i-1}\rightarrow 0.
\end{array}
\]
Note here that$\;\pi _Y^{*}(\pi _Y)_{*}(1\times \rho )^{*}{\cal Q}_{i-1}=\pi
_Y^{*}$ $\rho ^{*}{\cal Q}_0^{i-1}\cong \pi _Y^{*}(N_0^{i-1})_Y.$
Since a morphism from free sheaf is determined and can be defined by a
morphism between the space of global sections, there exists a unique lifting
as indicated by the vertical arrow $\Uparrow $. By the defining property of $%
X$, there is an induced morphism which is the inverse of $s$.
Since $X$ is an irreducible smooth affine quasi-variety and $g$ is smooth of
relative dimension $\sum_{i=1}^l(d_i^2+(r_i+d_i)^2),$ $R$ is smooth and
irreducible. It's dimensions is $d_1(n-r_2)+d_2(r_1-r_3)+\cdots
+d_{l-1}(r_{l-2}-r_l)+d_lr_{l-1}$
$+nr_1+r_1r_2+r_2r_3+\cdots +r_{l-1}r_l-r_1^2-r_2^2-\cdots -r_l^2.$ The
rationality will be from Bialynski-Birula's theorem after considering an
action \cite{St}. The action will be studied in the following section 3.
$\Box $
\subsection{Gromov-Witten invariants and flag-Quot schemes}
From now on we will work over the complex number field ${\Bbb C}$ to
consider complex manifolds.
We shall recall the definition of Gromov-Witten invariants for homogeneous
projective variety $X$.
The variety is always smooth. Denote by $%
\overline{{\cal M}}_N(X,d)$ the moduli stack of stable maps of degree $d$
and genus $0$. The stack is represented by
a smooth compact oriented orbifold. The same notations will be taken
for the stack and the coarse moduli orbifold.
It has morphisms, contraction $\pi ^X$ and
evaluations $ev_i$ at the $i$-th marked point:
\[
\begin{array}{ccc}
\overline{{\cal M}}_N(X,d) & {\rightarrow }_{ev_i}X & \\
\downarrow _{\pi ^X} & & \\
\overline{{\cal M}}_N & &
\end{array}
\]
where $\overline{{\cal M}}_N$ is the coarse moduli space of stable $n$
marked points of genus zero. $\overline{{\cal M}}_N$ is a smooth compact
oriented manifold.
The (tree level) Gromov-Witten classes $I_{N,d}^X:H^{*}(X, \QQ)^{\otimes
N}\rightarrow H^{*}(\overline{{\cal M}}_N,\QQ )$ are defined as follows:
\[
I_{N,d}^X(a_1\otimes \cdots \otimes a_N):=(\pi ^X)_{!}(ev_1^{*}(a_1)\otimes
\cdots \otimes ev_N^{*}(a_N)).
\]
In the sequel
we are interested in $I_{N,d}^X(a_1\otimes \cdots \otimes a_N)[p]
\in \QQ$ where $[p]$ is the homology class defined by a point $p$ in
$\overline{{\cal M}}_N$. If we choose any $N$ ordered distinct points $p_i$ in
$\PP ^1$, we can naturally embed $Mor_d(\PP ^1, X)$ into $(\pi ^X)^{-1}(p)$
for any generic point $p$. The boundary $(\pi ^X)^{-1}(p)\smallsetminus
Mor_d(\PP ^1, X)$
does not matter much, namely
\begin{proposition}
Let $Y_i$ be Schubert subvarieties of $X.$ Then $%
\bigcap_{i=1}^Nev_i^{-1}(g_iY_i)=\bigcap_{i=1}^Nev_i^{-1}(g_iY_i)\cap Mor_d(%
{\Bbb P}^1,X)$ for generic $g_i.$
\end{proposition}
\proof
The proof follows from the following general setting.%
$\Box$
\begin{proposition}
$M$ be open subvariety of a variety $\bar{M}.$
Suppose a connected algebraic group $G$ acts
transitively on another variety $X$.
Given a morphism $f:\bar{M}\rightarrow X$
subvarieties $Y_i$ of pure dimension, for generic $g_i\in G,$
\[
\bigcap_if^{-1}(g_iY_i)=(\bigcap_if^{-1}(g_iY_i))\cap M
\]
provided dimensional condition $\sum_i\codim Y_i=\dim\bar{M}$.
\end{proposition}
\proof
Apply Kleiman's theorem in Fulton's Book \cite{Fu}.
For generic $g_i$, $(\bar{M}\backslash M)\cap
\bigcap_if^{-1}(g_iY_i)=\emptyset $ and for generic $g_i,$ $%
(\bigcap_if^{-1}(g_iY_i))\cap M$ is proper. Hence for generic $g_i,$ $%
(\bigcap_if^{-1}(g_iY_i))\cap M$ is proper and ($\bar{M}\backslash M)\cap
\bigcap_if^{-1}(g_iY_i)=\emptyset $.
$\Box$
Since $Mor_d(\PP ^1, X) $ is a nonsingular quasi-projective variety, we can
consider its Chow group $A_*(Mor_d(\PP ^1,X))$ with products.
Let us use the same notation $ev_i$ for the restriction of the evaluation
map to $Mor_d(\PP ^1 ,X)$. For $[ev^{-1}_1(Y_1)]\cdot ...\cdot [ev^{-1}_1(Y_1)]
\in A_0(Mor_d(\PP ^1 ,X)$, in $\ZZ $ is
\[ \int _{Mor_d(\PP ^1 ,X)} [ev^{-1}_1(Y_1)]\cdot ...\cdot [ev^{-1}_1(Y_1)]
\] after summing up the coefficients of cycles of
points in $Mor_d(\PP ^1 ,X)$, which is well-defined in these intersections.
It is equal to
$I_{N,d}^X(a_1\otimes \cdots \otimes a_N)$ for
the Poincare dual classes $a_i$ of $Y_i$ because
$(\pi ^X)^{-1}(p)$ is a projective variety
and has a resolution of singularities to avoid the intersection theory of
algebraic (smooth) stacks.
We would like to do a similar thing in $Quot$ schemes following Bertram
\cite{Be}.
\begin{proposition}
(c.f. Bertram) Suppose $Y\subset X\,$ is an irreducible subvariety of
codimension $c$ and suppose $Z\subset Mor_d({\Bbb P}^1,X)$ is an irreducible
subvariety. Then for any $p\in {\Bbb P}^1$ and a generic translate $g,$ the
intersection $Z\cap ev_p^{-1}(gY)$ is either empty or has codimension $c$
in $Z$ where $ev_p$ denoted the evaluation map at $p$.
\end{proposition}
\proof
Apply Kleiman's theorem in Fulton's Book \cite{Fu}. $\Box$
\begin{corollary}
\label{indexco}Let $c_i=$co$\dim _XY_i$ in the setting of the above
definition. If $\sum_{i=1}^Nc_i>\dim (Mor_d(C,X)),$ then, for generic
elements $g_1,...,g_N$ , $\bigcap_{i=1}^Nev_{p_i}^{-1}(g_iY_i)=\emptyset .$
If $\sum_{i=1}^Nc_i=\dim (Mor_d(C,X)),$ then, for generic elements $%
g_1,...,g_N$ , $\bigcap_{i=1}^Nev_{p_i}^{-1}(g_iY_i)$ is isolated or empty.
\end{corollary}
The points $p_i$ in the above could not be distinct.
\bigskip
Let $({\cal E}_i)_p$ be the restriction of the sheaf ${\cal E}_i$ at $p$ in $%
{\Bbb P}^1.$ Consider a commutative diagram
\[
\begin{array}{ccc}
Mor({\Bbb P}^1{\bf ,}Fl) & \rightarrow & Mor({\Bbb P}^1,Gr(n,r_i)) \\
\downarrow ev_p & & \downarrow ev_p \\
Fl & \rightarrow & Gr(n,r_i)
\end{array}
\]
where $ev_p$ is the evaluation map at $p\in {\Bbb P}^1.$ Let $W$ be the
subspace of $V^{*}$ used for defining $Z,$ i.e., the special Schubert
varieties associated to $W$. Let $V_d(p,Z)$ be the degenerate locus of the
sheaf homomorphism
$W\bigotimes {\cal O}_{Quot}\rightarrow ({\cal E}_i)_p^{*}.$
In this setting we have
\begin{proposition}
$V_d(p,Z)$ represents the ($s_i+1-\dim (W))$-th Chern class of
$({\cal E}_i)_p^*$ over $fQuot$.
\end{proposition}
\proof
When the flag variety $Fl$ is a Grassmannian, it is
proven by A. Bertram \cite{Be}. For the general case, just consider the
morphism $fQuot\rightarrow Quot_i$ from the embedding $fQuot\rightarrow \prod
Quot_i$ followed by the projection $\prod Quot_i\rightarrow Quot_i$. It is
smooth since both schemes are smooth and the induced homomorphism between
tangent spaces is surjective after looking at \ref{ww}
in the proof of the theorem \ref{thm4}. This implies the degeneracy locus has
the expected dimension and $[V_d(p,Z)]$ in the Chow ring of the smooth
projective variety $fQuot$ is the $(\codim V_d(p,Z)$)-th Chern class
of $({\cal E}_i)_p^*$ which complete the proof.
$\Box$
Using a Pl\"{u}cker embedding and a stratification $Quot_d({\Bbb P}^1,{\Bbb P}%
^n)=\coprod_{0\leq m\leq d}C_m\times Mor_{d-m}({\Bbb P}^1,{\Bbb P}^n)$ (by
locally closed schemes) where $C_m$ is the $m$-th symmetric product of ${\Bbb %
P}^1,$ one can extend Bertram's result for flag varieties.
\begin{proposition}
\label{Ber}Let $Z_i$ be a special Schubert variety with $c_i$ codimension $%
\leq s_{k_{i+1}}-s_{k_{i-1}}-1$ representing a Chern class of $S_{k_i}^{*}.$
Suppose $\sum_{i=1}^Nc_i\geq \dim (fQuot),$ then $%
\bigcap_{i=1}^Nev_{d,p_i}^{-1}(g_iZ_i)=\bigcap_{i=1}^NV_d(p_i,g_iZ_i)$ for
distinct points $p_i\in {\Bbb P}^1$.
\end{proposition}
\proof
We will use induction on the total degree $%
|d|=d_1+\cdots +d_l.$ When $|d|=0,$ it can be done by Kleiman's theorem. Let
$\widetilde{C}_m=\prod_{i=1}^lC_{m_i},$ where $m=(m_1,...,m_l)$ is a
multi-index$.$ Using the Pl\"{u}cker morphisms on Quot schemes, let us
consider the morphism
\begin{eqnarray*}
J &:&fQuot\hookrightarrow \prod_{i=1}^lQuot_{d_i}(s_i,n)\rightarrow
\prod_{i=1}^lQuot_{d_i}({\Bbb P}^1{\Bbb ,P}^{M_i}) \\
&=&Mor_d({\Bbb P}^1 ,\prod_{i=1}^l{\Bbb P}^{M_i})\cup
\bigsqcup_{|m|=1}^{|d|}\widetilde{C}_m\times Mor_{d-m}({\Bbb P}^1 ,%
\prod_{i=1}^l{\Bbb P}^{M_i}).
\end{eqnarray*}
We would like to show that
\[
\bigcap_{i=1}^NV_d(p_i,g_iZ_i)\cap J^{-1}\left( \bigsqcup_{|m|=1}^{|d|}%
\widetilde{C}_m\times Mor_{d-m}({\Bbb P}^1 ,\prod_{i=1}^l{\Bbb P}%
^{M_i})\right) =\emptyset .
\]
Then we are done. To do so one has to show, for each $m$, $|m|>0$,
\begin{equation}
\emptyset =\bigcap_{i=1}^NV_d(p_i,g_iZ_i)\cap J^{-1}\left( \widetilde{C}%
_m\times Mor_{d-m}({\Bbb P}^1 ,\prod_{i=1}^l{\Bbb P}^{M_i})\right).
\label{inter}
\end{equation}
For any subset $P$ of $\{p_1,...,p_N\}$ let
\begin{eqnarray*}
A_P &=&\{\text{quotient sheaves }Q=(Q_1,...,Q_l)\in \text{LHS of (\ref{inter})
} \\
|\text{ }\dim _{k(p_i)}(Q_{k_i})_{p_i}\otimes k(p_i) &<&r_{k_i}\text{ iff }%
p_i\in P\}
\end{eqnarray*}
\[
A_P\subset \bigcap_{p_i\notin P}J^{-1}(\widetilde{C}_m\times
ev_{d-m,p_i}^{-1}(g_iZ_i)).
\]
But for generic $g_i,\;\bigcap_{p_i\notin P}J^{-1}(\widetilde{C}_m\times
ev_{d-m,p_i}^{-1}(g_iZ_i))=\emptyset $ since $\bigcap_{p_i\notin
P}ev_{d-m,p_i}^{-1}(g_iZ_i)=\emptyset $ by dimension counting in $%
fQuot(d-m;s_1,...,s_l;n)$:
\begin{eqnarray*}
\sum_{p_i\notin P}\codim (ev_{d-m,p_i}^{-1}(g_iZ_i)) &\geq &\dim
(fQuot_d(Fl)) \\
&&-\sum_{p_i\in P}(s_{k_{i+1}}-s_{k_{i-1}}-1) \\
&&\dim (fQuot_d(Fl)) \\
&&-\sum_{p_i\in P}(s_{k_{i+1}}-s_{k_{i-1}}). \\
&\geq &\dim (fQuot_{d-m}(Fl)).
\end{eqnarray*}
Since $p_i$ are distinct, we conclude the last inequality above.
$\Box$
The proof of the theorem \ref{thm2} follows from what are done.
\section{A Formula by Localization}
\subsection{Equivariant action on ${\cal E}_i\rightarrow fQuot\times {\PP}%
^1$}
By the standard action of $SL(n)\times PGL(2)$ on $V\times \PP ^1$,
the group acts on the space of stalks of $V\otimes \cal {O}_{fQuot\times
\PP ^1}$ and hence on the subsheaves $\cal {E}_i$ and $fQuot$. The action
on the sheaves is equivariant. In particular the maximal complex torus action
of $T\times \CC ^{\times}$ will formulate integrations of wedges
products of Chern classes of $(\cal {E}_i)_p$ as certain finite
sums of characters using the localization theorem \cite{AB}.
For simplicity of notations let us do it for the Quot schemes $Quot$.
Consider the action by $T\times\CC ^{\times}$
on $Quot\times {\PP}^1$, then the action
has a lift on the total space of the vector bundle ${\cal E}$. Let ${\cal E}%
_p$ be the restriction of the sheaf ${\cal E}$ at $p$ in ${\PP}^1$.
The action has the lifting to vector bundles $\cal{E}_0$ and $\cal{E}_\infty$.
It means $\cal{E}_{0(\infty )}$ is an equivariant vector bundle and its
equivariant Chern classes can be considered.
For other points, say $p$, transitive $%
PSL(2)$-action on ${\PP}^1$ will show $\cal{E}_0$, $\cal{E}_\infty$, and
$\cal{E}_p$ are isotropic:
\[
\begin{array}{ccc}
{\cal E} & & {\cal E} \\
\downarrow & & \downarrow \\
Quot\times {\PP}^1 & \rightarrow _g & Quot\times {\PP}^1
\end{array}
\begin{array}{ccc}
{\cal E}_0 & & {\cal E}_p \\
\downarrow & & \downarrow \\
Quot & \rightarrow _g & Quot
\end{array}
\]
where $g\cdot 0=p.$ In particular the Chern classes of ${\cal E}_p\,$ are
independent to $p$ since the map induced by $g$ is homotopic to identity.
Let $\frac 12\hbar \;($resp. $\lambda _i)$ is (are) the ${\CC}^{\times
}\;( $resp. $T)\;$characteristic classes. Then,
\begin{eqnarray}
\int_{Quot}\phi (c_{i_1}({\cal E}_{p_1}),...,c_{i_m}({\cal E}_{p_m}))
&=&\int_{Quot}\phi (c_{i_1}({\cal E}_0),...,c_{i_m}({\cal E}_0)) \nonumber
\\
&=&[\text{push forward of }\phi \text{ of equivariant classes } \nonumber \\
&&\text{of }(c_1,...,c_r)\text{ at }{\cal E}_0]_{\hbar =\lambda _i=0}
\nonumber \\
&=&[\text{localization into components $P$ of the fixed subset},
\nonumber \label{local} \\
&&\text{ i.e.,}\;\sum_P\text {push forward}
\frac{i_{*}^P\phi }{E(v_P)}]_{\hbar =\lambda
_i=0} \label{local},
\end{eqnarray}
where $i^P$ is the inclusion $P\subset Quot$ and
$E(v_P) $ is the equivariant Euler class of
the normal bundle of P in $Quot$.
The last expression is independent to $\hbar $ and $\lambda _i$, without
letting them zeros, if the quasi-homogeneous degree of $\phi $ given
by degrees of the Chern classes agrees the
dimension of $Quot$. The complete analog hold for $fQuot$.
It is easy to see that the fixed subset consists of finite points. Therefore
$E(v_P)$ in (\ref{local})
is the equivariant Euler class of the normal space over the point
$P\in Quot.$ It is
the product of the complex characters of the representation of
$T\times\CC ^{\times}$
in the irreducible complex one dimensional subspaces of the tangent space of $%
Quot$ at $p.$ In the following subsection,
we devote ourselves to spell out the all fixed points
and all characteristics of the representation
to finish the proof of the theorem \ref{thm3}.
\subsection{Computation}
Let us use the standard maximal torus $T\times \CC ^{\times}$ in the
picture of flag manifolds $Fl$ and the projective line ${\PP}^1.$
Fix a sequence of $(e_{k_1},e_{k_2},...,e_{k_l})$ where $\{e_i\}_{i=1}^n$ is
the standard basis of $V.$ Then, for data (\ref{integer})
in introduction, one may associate
a flag of subsheaves
\[
{\cal O}(-d_{i,j})\longrightarrow {\cal O}(-d_{i+1,j})
\]
by the global section
\[
x^{(a_{i,j}-a_{i+1,j})}y^{(b_{i,j}-b_{i+1,j})}.
\]
It is a fixed point by the action. For such a (\ref{integer}) and a
sequence, we can associate any fixed point in $fQuot$. We have found all
fixed points.
Note that the tangent space at $x$ of a scheme $X$ is the first order
infinitesimal deformation $Mor_x(D,X),$ the set of all morphisms sending the
closed point of Spectrum of the ring $D$ of dual numbers
to $x$. Therefore, at a subsheaf ${\cal S}$ over ${\PP}$ of $%
V\otimes\cal{O}_{{\PP}^1},$ the tangent space of Quot schemes is the set of
flat
families of quotient sheaves over the Spec$D$ whose fiber over the closed
point of Spec$D$ is ${\cal S}$. It is
$Hom({\cal S},V\otimes\cal{O}_{{\PP}^1}/{\cal S}).$
For the flag-Quot scheme consider the following equivariant short exact
sequence at a fixed point ${\cal S}$ of $fQuot$
\begin{eqnarray*}
0 &\rightarrow &T_{{\cal S}}fQuot\left( d_1,...,d_l;s_1,...,s_l;n\right) \\
&\rightarrow &T_{{\cal S}}\{Quot_{d_1}(s_1,n)\times \cdots \times
Quot_{d_l}(s_l,n)\}\rightarrow \prod_{i=1}^lHom({\cal S}_i,{\cal Q}%
_{i+1})\rightarrow 0.
\end{eqnarray*}
At the fixed point associated to \ref{tang1} we find all characters of
irreducible subspace of $T_{{\cal S}}Quot$ by the torus action. They are,
for all $1\leq i\leq l,$
\begin{eqnarray*}
(p-a_{i,j})\hbar +\lambda _{j^{\prime }}-\lambda _{j},\;\text{for }0
&\leq &p\leq a_{i,j^{\prime }}-1,\;1\leq j,j^{\prime }\leq s_i, \\
(b_{i,j}-p)\hbar +\lambda _{j^{\prime }}-\lambda _{j},\;\text{for }0
&\leq &p\leq b_{i,j^{\prime }}-1,\;1\leq j,j^{\prime }\leq s_i, \\
(p-a_{i,j})\hbar +\lambda _{m}-\lambda _{j}.\;\text{for }0 &\leq &p\leq
d_{i,j},\;1\leq j\leq s_i,\;s_i+1\leq m\leq n,
\end{eqnarray*}
from
\begin{eqnarray*}
&&\bigoplus_{i=1}^lHom({\cal S}_i,{\cal Q}_i) \\
&=&\bigoplus_{i=1}^l[\bigoplus_{j=1,j^{\prime
}=1}^{s_i,s_i}Hom(.x^{a_{i,j}}y^{b_{i,j}}{\cal O}_{j}(-d_{i,j}),{\cal O}%
_{j^{\prime }}/x^{a_{i,j^{\prime }}}y^{b_{i,j^{\prime }}}\cal{O}_j) \\
&&\bigoplus \bigoplus_{j=1,m=s_i+1}^{s_j,n}Hom(x^{a_{i,j}}y^{b_{i,j}}{\cal O%
}_{j}(-d_{i,j}),{\cal O}_{m})].
\end{eqnarray*}
Characters from $\prod_{i=1}^{l-1}Hom({\cal S}_i,{\cal Q}_{i+1})$ are, for $%
1\leq i\leq l-1,$%
\begin{eqnarray*}
(p-a_{i,j})\hbar +\lambda _{j^{\prime }}-\lambda _{j},\;\text{for }0
&\leq &p\leq a_{i+1,j^{\prime }}-1,\;1\leq j,\leq s_i,\;1\leq j^{\prime
}\leq s_{i+1}, \\
(b_{i,j}-p)\hbar +\lambda _{j^{\prime }}-\lambda _{j},\;\text{for }0
&\leq &p\leq b_{i+1,j^{\prime }}-1,\;1\leq j,\leq s_i,\;1\leq j^{\prime
}\leq s_{i+1}, \\
(p-a_{i,j})\hbar +\lambda _{m}-\lambda _{j},\;\text{for }0 &\leq &p\leq
d_{i,j},\;1\leq j\leq s_i,\;\;s_{i+1}+1\leq m\leq n.
\end{eqnarray*}
The fiber space of ${\cal S}_i$ at the point $0$ has characters
\[
a_{i,j}\hbar +\lambda _{j}
\]
for $0\leq j\leq s_i.$ Therefore the $k$-th Chern character is the $k$-th
symmetric function in those characters. Let us denote it by $\sigma _i^k$.
The proof of theorem \ref{thm3} follows from the proposition 5.
\subsection{Projective spaces}
In this section we will relate the our result to the residue formula of
intersection pairing in \cite{Ki} for projective spaces. The
author does not know for the other cases.
Let $x$ be the Chern class of ${\cal O}_{{\PP}^n}(-1).\,$Then the
Gromov-Witten invariant $I^{\PP ^n}_{N,d}(x^{\otimes (n+1)d+n})$ is
\begin{equation}
\sum\Sb 0\leq i\leq n \\ 0\leq k\leq d \endSb \frac{(\lambda _i+k\hbar
)^{(n+1)d+n}}{\prod\Sb 0\leq p\leq d \\ p\ne k \endSb ((p-k)\hbar )\prod\Sb
0\leq q\leq d \\ 0\leq j\ne i\leq n \endSb ((q-k)\hbar +\lambda _j-\lambda
_i)} \label{proj}
\end{equation}
\begin{proposition}
$\sum_{N=0}^{N=\infty }
\frac {1}{N!}q^d I^{\PP ^n}_{N.d}(x^{\otimes N})$ is a global residue
\[
\frac 1{2\pi }\oint \frac{f(x)dx}{x^{n+1}-q}
\] where $q$ is a formal variable.
\end{proposition}
\proof
The identity
\begin{eqnarray*}
\frac 1{2\pi }\oint \frac{x^{(n+1)d+n}dx}{x^{(n+1)(d+1)}} &=&\left[ \frac
1{2\pi }\oint \frac{x^{(n+1)d+n}}{\prod\Sb 0\leq i\leq n \\ 0\leq k\leq d
\endSb (x-\lambda _i-k\hbar )}\right] _{\lambda _i=\hbar =0} \\
&=&(\text{\ref{proj}})
\end{eqnarray*}
implies the proof.
$\Box$
|
1996-02-05T06:20:18 | 9512 | alg-geom/9512016 | en | https://arxiv.org/abs/alg-geom/9512016 | [
"alg-geom",
"math.AG"
] | alg-geom/9512016 | Goncharov | Alexander Goncharov | Deninger's conjecture on $L$-function of elliptic curves at $s=3$ | LaTeX | null | null | null | null | I compute explicitly the regulator map on $K_4(X)$ for an arbitrary curve $X$
over a number field. Using this and Beilinson's theorem about regulators for
modular curves ([B2]) I prove a formula expressing the value of the
$L$-function $L(E,s)$ of a modular elliptic curve $E$ over $\Bbb Q$ at $s=3$ by
the double Eisenstein-Kronecker series.
| [
{
"version": "v1",
"created": "Mon, 25 Dec 1995 20:28:39 GMT"
},
{
"version": "v2",
"created": "Fri, 2 Feb 1996 18:04:47 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Goncharov",
"Alexander",
""
]
] | alg-geom | \section{Appendix}
{\bf 1. Proof of theorem \ref{z2}b)}. Let me remind the formulation of
this theorem
{\bf Theorem \ref{z2}}
a) $f_{4}(3)$ and $f_{5}(3)$ {\it do not depend on the choice
of $\omega$.
b) The homomorphisms $f_*(3)$ provide a morphism of
complexes. }
Proof. a) See the proof of similar results in chapter 3
of [G2].
b) We have to prove that $f_4(3) \circ d = \delta \circ f_5(3)$ and
$f_5(3) \circ d = \delta \circ f_6(3)$. For the first result see chapter 3 in
[G2].
The second one is much more subtle. As pointed out H.Gangl, the geometric
proof given in [G2] (see theorem 3.10 there) has some errors.
Namely, in lemma 3.8
$r = -r_3$ but not $r=r_3$ as clamed, and as a result the proof
of theorem 3.10 become more involved; further, the correct
statement in theorem 3.10 is $f_5(3) \circ d = \delta \circ 1/15 \cdot
f_6(3)$
(the coefficient $1/15$ in the definition of $f_6(3)$ was missed).
Another proof was
given in [G1]. It was actually the first proof of the statement b).
However in this proof we
used a different definition for homomorphism $f_6(3)$
(the map $M_3$ in [G1]). Moreover the proof was rather complicated and the
relation between the homomorphisms $f_6(3)$ and $M_3$ not easy to see.
Therefore I will present in detail a completely different proof togerther
with some corrections to chapter 3 in [G2].
Let us suppose that in a three dimensional vector space $V_3$ we choose a
volume form $\omega$. Then for any two vectors $a,b$
one can define the cross product $a \times b \in V_3^*$ as follows:
$<a \times b, c>: = \Delta(a,b,c)$. The volume form $\omega$ defines the
dual volume form in $V_3^*$, so we can define $\Delta(x,y,z)$ for
any three vectors in $V_3^*$.
\begin{lemma} \label{gz2}
For any $6$ vectors in generic position $a_1,a_2,a_3,b_1,b_2,b_3$ in $V_3$
$$
\Delta(a_1,a_2,b_1) \cdot \Delta(a_2,a_3,b_2) \cdot \Delta(a_3,a_1,b_3) -
\Delta(a_1,a_2,b_2) \cdot \Delta(a_2,a_3,b_3) \cdot \Delta(a_3,a_1,b_1) =
$$
$$
\Delta(a_1,a_2,a_3) \cdot \Delta(a_1 \times b_1,a_2 \times b_2,a_3 \times b_3)
$$
\end{lemma}
{\bf Proof}. The left hand side is zero if the
vectors $a_1,a_2,a_3$ are linearly dependent. So $\Delta(a_1,a_2,a_3)$
divides it. Similarly the left hand side is zero if $a_i$ is
collinear to $b_i$ or $\alpha_1 a_1 + \beta_1 b_1 = \alpha_2 a_2 + \beta_2 b_2=
\alpha_3 a_3 + \beta_3 b_3$ for some numbers $\alpha_k, \beta_k$. This
implies that $\Delta(a_1 \times b_1,a_2 \times b_2,a_3 \times b_3)$ also
divides the left hand side. It is easy to deduce the formula from this.
However it perhaps easier to check the formula directly.
Consider the following special configuration of
vectors:
$$
\begin{array} {cccccc}
a_1&a_2&a_3&b_1&b_2&b_3\\
-&-&-&-&-&-\\
1&0&0&x_1&y_1&z_1\\
0&1&0&x_2&y_2&z_2\\
0&0&1&x_3&y_3&z_3
\end{array}
$$
Then the left hand side is equal to
$x_3 y_1 z_2 - y_3 z_1 x_2$, and the computation of the right hand side
gives the same result. The lemma is proved.
{\bf Remark}. Let $a_1,...,a_n,b_1,...,b_n$ be a configuration of $2n$
vectors in
an $n$-dimensional vector space $V_n$. Set $\Delta({\hat a}_n,b_1):=
\Delta(a_1,...,a_{n-1},b_1)$ and so on. Then
$$
\Delta({\hat a}_1,b_1) \cdot ... \cdot \Delta({\hat a}_{n},b_n) -
\Delta({\hat a}_1,b_2) \cdot ... \cdot \Delta({\hat a}_n,b_1) =
$$
$$
\Delta(a_1,...,a_n) \cdot \Delta(a_1 \times ... \times a_{n-2} \times b_n,
... ,a_n \times ... \times a_{n-3} \times b_{n-1})
$$
Notice that $f_5(3) \circ d - \delta \circ f_6(3) \in B_2(F) \otimes F^*$.
There is a homomorphism
$$
\delta \otimes id: B_2(F) \otimes F^* \longrightarrow \wedge^2F^*\otimes F^*,
\qquad \{x\}_2
\otimes y \longmapsto (1-x) \wedge x \otimes y
$$
The crucial step of the proof is the following
\begin{proposition} \label{gz3}
$$
(\delta \otimes id) \circ \Bigl
(f_5(3) \circ d - \delta \circ f_6(3)\Bigr)(v_1,...,v_6) =0
\quad \mbox{in} \quad \wedge^2 F^* \otimes F^*
$$
\end{proposition}
{\bf Proof}. We will use notation
$\Delta ( i,j,k) $ for $\Delta ( v_i,
v_j, v_k)$. According to lemma (\ref{gz2})
$$
1 - \frac{\Delta ( 1, 2, 4)\Delta (2,3, 5)
\Delta(3,1,6)}
{\Delta(1,2,5 )\Delta (2,3,6)\Delta
(3,1,4)} =
\frac{\Delta (1,2,3) \Delta (v_1 \times v_4, v_2\times v_5,
v_3\times v_6)}
{\Delta(1,2,5 )\Delta (2,3,6)\Delta
(3,1,4)}
$$
Using the cyclic permutation
$1-> 2->3 ->1, 4->5->6->4$ we see that one has to calculate the element
$$
3 \cdot {\rm Alt}_{6}\left\{ \frac{\Delta (1,2,4)
\Delta (2,3, 5)
\Delta(3,1,6)}
{\Delta(1,2,5 )\Delta (2,3,6)\Delta
(3,1,4)} \wedge \frac{\Delta (1,2,3) \Delta (v_1 \times v_4, v_2\times v_5,
v_3\times v_6)}
{\Delta(1,2,5 )\Delta (2,3,6)\Delta
(3,1,4)} \otimes \frac{\Delta(1,2,4)}{\Delta(1,2,5)}\right\}
$$
in $\wedge^2 F^* \otimes F^*$.
Let us do this. We will compute first
the contribution of the factor $\otimes \Delta(1,2,4)$.
What we need to find is
$$
{\rm Alt}_{(1,2,4);(3,5,6)}\left\{\frac{\Delta (1,2,4)\Delta (2,3, 5)
\Delta(3,1,6)}
{\Delta(1,2, 5)\Delta (2,3,6)\Delta
(3,1,4)} \wedge \frac{\Delta (1,2,3)
\Delta (v_1 \times v_4, v_2\times v_5,
v_3\times v_6)}
{\Delta(1,2,5 )\Delta (2,3,6)\Delta
(3,1,4)}\right\}
$$
in $\wedge^2 F^*$.
Here ${\rm Alt}_{(1,2,4);(3,5,6)}$
is the skewsymmetrization with respect to the
group $S_3 \times S_3$ which permutes the indices $(1,2,4)$ and $(3,5,6)$.
i) Consider
$$
{\rm Alt}_{(1,2,4);(3,5,6)}\left\{\frac{\Delta (1,2,4)\Delta (2,3, 5)
\Delta(3,1,6)} {\Delta(1,2,5 )\Delta (2,3,6)\Delta
(3,1,4)} \wedge \Delta (v_1 \times v_4,v_2\times v_5,
v_3\times v_6)\right\}
$$
Using the skewsymmetry with respect to the permutation exchanging
$1$ with $3$ as well as $4$ with $ 6$ (notation: $: (13)(46)$) we see that
this expression is zero.
ii)Look at
$$
- {\rm Alt}_{(1,2,4);(3,5,6)}\left\{\frac{\Delta (1,2,4)\Delta (2,3,5)
\Delta(3,1,6)} {\Delta(1,2,5)\Delta (2,3,6)\Delta
(3,1,4)} \wedge \Delta (2,3,6) \otimes \Delta(1,2,4)\right\}
$$
The skewsymmetry with respect to $(14)$ or with respect to
$(36)$ imply that it is also zero.
iii) Consider
$$
- {\rm Alt}_{(1,2,4);(3,5,6)}\left\{\frac{\Delta (1,2,4)\Delta (2,3, 5)
\Delta(3,1,6)} {\Delta(1,2, 5)\Delta (2,3,6)\Delta
(3,1,4)} \wedge \Delta (1,2,5) \otimes \Delta(1,2,4)\right\}
$$
The skewsymmetry with respect to the permutations $(12)$ as well as $(36)$
leads to
$$
-{\rm Alt}_{(1,2,4);(3,5,6)}\left\{\frac{\Delta(2,3,5)}{\Delta(1,3,4)}\wedge
\Delta(1,2,5)
\otimes \Delta(1 ,2,4)\right\}
$$
iv) Look at the term with $\Delta(3,1,4)$:
$$
- {\rm Alt}_{(1,2,4);(3,5,6)}\left\{\frac{\Delta (1,2,4)\Delta (2,3, 5)
\Delta(3,1,6)} {\Delta(1,2, 5)\Delta (2,3,6)\Delta
(3,1,4)} \wedge \Delta (3,1,4) \otimes \Delta(1,2,4)\right\}
$$
Using the permutation $(14)$ we get
$$
-{\rm Alt}_{(1,2,4);(3,5,6)}\left\{\frac{\Delta(3,1,6)}{\Delta(1,2,5)}\wedge
\Delta(1,3,4)
\otimes \Delta(1 ,2,4)\right\}
$$
v) Finally, using $(12)$ and $(56)$
we see that
$$
{\rm Alt}_{(1,2,4);(3,5,6)}\left\{\frac{\Delta (1,2,4)\Delta (2,3, 5)
\Delta(3,1,6)} {\Delta(1,2, 5)\Delta (2,3,6)\Delta
(3,1,4)} \wedge \Delta (1,2,3) \otimes \Delta(1,2,4)\right\} =
$$
$$
- 3 \cdot {\rm Alt}_{(1,2,4);(3,5,6)}
\left\{\Delta(1,3,5) \wedge \Delta(1,2,3)
\otimes \Delta(1 ,2,4)\right\}
$$
Therefore we get
$$
{\rm Alt}_{(1,2,4);(3,5,6)}\Bigl(
\Delta(1,2,5) \wedge \frac{\Delta(2,3,5)}{\Delta(1,3,4)} + \Delta(1,3,4) \wedge
\frac{\Delta(1,3,6)}{\Delta(1,2,5)} +
$$
$$
3 \cdot \Delta(1,2,3) \wedge \Delta(1,3,5)
\Bigr)\otimes \Delta(1,2,4) =
$$
$$
{\rm Alt}_{(1,2,4);(3,5,6)} \Bigl(\Delta(1,2,5) \wedge \Delta(2,3,5) +
\Delta(1,3,4) \wedge \Delta(1,3,6) +
$$
$$
3 \cdot \Delta(1,2,3) \wedge \Delta(1,3,5)
\Bigr)\otimes \Delta(1,2,4) =
$$
$$
5 \cdot {\rm Alt}_{(1,2,4);(3,5,6)} 1759 \Delta(1,2,3) \wedge
\Delta(1,3,5) \otimes \Delta(1,2,4)
$$
The computation of the contribution of $\Delta(1,2,5)$ goes
similarly and gives the same answer.
{\it So the total result of our computation is}
\begin{equation} \label{su2}
-30 \cdot {\rm Alt}_{6} \left\{ \Delta(1,2,4) \wedge \Delta(1,4,5)
\otimes \Delta(1,2,3) \right\}
\end{equation}
Here we get the coefficient $-30$ taking into account the action of the cyclic
group of order $3$ generated by $1->2->3->1, 4->5->6->3$.
Now let us compute $f_5(3) \circ d(v_1,...,v_6)$. We will use the formula
\begin{equation} \label{su1}
\delta \{r(v_1,v_2,v_3,v_4)\}_2 = 1/2\cdot {\rm Alt}_{4}\left\{\Delta(v_1,v_2)
\wedge \Delta(v_1,v_3)\right\}
\end{equation}
Neglecting for a moment
the constant $c, c'$ we get
$$
(\delta \otimes id) \Bigl(f_5(3) \circ d(v_1,...,v_6)\Bigr) =
c \cdot {\rm Alt}_{6} \{r(v_1|v_2,v_3,v_4,v_5\}_2 \otimes \Delta(1,2,3) =
$$
$$
c' \cdot {\rm Alt}_{6} \Delta(1,2,4) \wedge \Delta(1,4,5)\otimes \Delta(1,2,3)
$$
To justify this we used here formula (\ref{su1}) and the symmetry
considerations
for transpositions $i<->j$ where $1 \leq i<j\leq 3$. More careful
consideration
shows $c' =-2$. It remains to compare it with (\ref{su2}).
That's why we need in the definition of $f_6(3)$
the coefficient $1/15$.
We have proved that
$$
(f_5(3) \circ d - \delta \circ f_6(3)\Bigr)(v_1,...,v_6) =
\sum_{1 \leq i<j<k \leq 6 } \gamma_{i,j,k} \otimes \Delta(i,j,k)
$$
where $\gamma_{i,j,k} \in B_2(F)$ and moreover $\delta(\gamma_{i,j,k}) =0$ in
$\wedge^2F^*$. According to [S2]
\begin{equation} \label{susl}
Ker\Bigl( B_2(F)
\stackrel{\delta}{\longrightarrow} \wedge^2F^*\Bigr)\otimes \Bbb Q =
K_3^{ind}(F)\otimes \Bbb Q
\end{equation}
One knows that $K_3^{ind}(F(t)) \otimes \Bbb Q = K_3^{ind}(F)\otimes \Bbb Q$.
Therefore the left hand side of (\ref{susl}) is rationaly invariant.
On the other hand one can connect by a
rational curve the configurations
$(v_1,v_2,...,v_6)$ and $(v_2,v_1,...,v_6)$ (interchanging $v_1$ with $v_2$)
in the space of all generic configurations.
This implies that $\gamma(1,2,3) = \gamma(2,1,3)$ modulo torsion.
But
$\gamma(1,2,3) = - \gamma(2,1,3)$ modulo torsion by the skewsymmetry.
So $\gamma(1,2,3) =0$ modulo torsion, and the same conclusion is valid
for $\gamma(i,j,k)$. With more work one can show that
$f_5(3) \circ d - \delta \circ f_6(3) = 0$ at least modulo 6-torsion,
but we do not need this. Theorem is proved.
{\bf 2. The geometrical definition of the homomorphism $f_6(3)$} Let
$(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3})$ be a configuration of
6 distinct points in $P^{2}$ as on
fig.\ 1. Let $P^{2}=P(V_{3})$. Choose vectors in $V_{3}$
such that they are projected to points $a_{i},b_{i}$.
We denote them by the same letters.
Choose $f_{i} \in V_{3}^{\ast}$ such that $f_{i}(a_{i}) =
f_{i}(a_{i+1}) = 0$. Put
\begin{equation}
r'_{3}(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3}) =
\frac{f_{1}(b_{2}) \cdot f_{2}(b_{3})\cdot f_{3}(b_{1})}
{f_{1}(b_{3})\cdot f_{2}(b_{1})\cdot f_{3}(b_{2})}\; .
\end{equation}
The right-hand side of (3.10) does not depend on the choice of
vectors $f_{i},b_{j}$.
\begin{center}
\begin{picture}(100,80)
\put(47,78){$a_{2}$}
\put(47,68){$\bullet$}
\put(50,70){\vector(2,-3){40}}
\put(50,70){\vector(-2,-3){40}}
\put(23,31){$\bullet$}%
\put(12,33){$b_{1}$}
\put(91,10){\vector(-1,0){80}}
\put(60,47){$\bullet$} %
\put(66,50){$b_{2}$}
\put(8,7){$\bullet$} %
\put(87,7){$\bullet$}%
\put(93,0){$a_{3}$}
\put(0,0){$a_{1}$}
\put(36,7){$\bullet$}
\put(33,0){$c_3$}
\put(63,7){$\bullet$}
\put(60,0){$b_{3}$}
\put(33,-20){(fig. 1)}
\end{picture}
\end{center}
\vskip 1cm
\vskip 3mm \noindent
{\bf Lemma 3.8} $-r(b_{1}\vert a_{2},a_{3},b_{2},b_{3}) =
r'_{3}(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3})$.
\vskip 3mm \noindent
{\bf Proof.} The same as the one of lemma 3.8 in [G2]
Now let $\hat b_3$ be the of the line $b_1b_2$ with the line
$a_1a_3$. Further, let $x$ be the intersection point of the lines $a_1b_2$
and $a_3b_1$. Let us denote by $c_3$
the intersection point of the line $a_2 x$ with the line $a_1a_3$. Then
\begin{equation}
r'_{3}(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3}) = r(a_1,a_3,c_3,b_3)
\end{equation}
Indeed, by the well known theorem $r(a_1,a_3,\hat b_3,b_3) =-1$.
Now returning to a configuration $(v_1,...,v_6)$ (see fig 2)
\begin{center}
\begin{picture}(100,80)
\put(47,78){$a_{2}$}
\put(47,68){$\bullet$}
\put(50,70){\vector(2,-3){40}}
\put(50,70){\vector(-2,-3){40}}
\put(23,31){$\bullet$}
\put(12,33){$v_{1}$}
\put(71,31){$\bullet$}
\put(80,33){$v_{5}$}
\put(91,10){\vector(-1,0){80}}
\put(60,47){$\bullet$}
\put(66,50){$v_{2}$}
\put(25,50){$v_{4}$}
\put(35,47){$\bullet$}
\put(8,7){$\bullet$}
\put(87,7){$\bullet$}
\put(93,0){$a_{3}$}
\put(0,0){$a_{1}$}
\put(36,7){$\bullet$}
\put(33,0){$v_{6}$}
\put(63,7){$\bullet$}
\put(60,0){$v_{3}$}
\put(33,-20){(fig. 2)}
\end{picture}
\end{center}
\vskip 1cm
we see that one has proceed as follows: Put $b_1:=v_1,b_2:=v_2,b_3:=v_3$ and
apply the given above definition to the configuration
$(a_1,a_2,a_3,b_1,b_2,b_3)$
and then alternate. Notice that the configuration $(a_1,a_2,a_3,b_1,b_2,b_3)$
is defined by three flags $(v_1,v_1v_4),(v_2,v_2v_5),(v_3,v_3v_6)$.
\vskip 3mm \noindent
{\bf REFERENCES}
\begin{itemize}
\item[{[B1]}] Beilinson A.A.: {\it Higher regulators and values of
$L$-functions}, VINITI, 24 (1984), 181--238 (in Russian);
English translation: J. Soviet Math. 30 (1985), 2036--2070.
\item[{[B2]}] Beilinson A.A.: {\it Higher regulators for modular curves}
Contemporary Mathematics, vol. 55, 1987, 1-35.
\item[{[BL]}] Beilinson A.A., Levin A.M.: {\it Elliptic polylogarithms}.
Symposium in pure mathematics, 1994, vol 55, part 2, 101-156.
\item[{[Bl1]}] Bloch S.: {\it Higher regulators, algebraic $K$-
theory and zeta functions of elliptic curves}, Lect. Notes U.C.
Irvine, 1977.
\item[{[Bl2]}] Bloch S.: {\it 2 letters to Deninger
regarding [D1]} Fall 1990.
\item[{[Bl3]}] Bloch S.: {\it Lectures on algebraic cycles}, Duke Math.
Lect. Series, 1980.
\item[{[BMS]}] Beilinson A.A., MacPherson R.D. Schechtman V.V: {\it Notes
on motivic cohomology}. Duke Math. J., 1987 vol 55 p. 679-710
\item[{[GGL]}] Gabrielov A.M., Gelfand I.M., Losic M.V.:
{\it Combinatorial computation of characteristic classes}, Funct.\
Analysis and its Applications V. 9 No. 2 (1975) p. 103--115
and No. 3 (1975) p. 5--26 (in Russian).
\item[{[G1]}] Goncharov A.B.:{\it Geometry of configurations,
polylogarithms and motivic cohomology}.
Advances in Mathematics, 1995. 197 - 318.
\item[{[G2]}] Goncharov A.B., {\it Polylogarithms and motivic Galois
groups}, Symposium in pure mathematics, 1994, vol 55, part 2, p. 43 - 96.
\item[{[G3]}] Goncharov A.B., {\it Explicit construction of
characteristic classes}. Advances in Soviet mathematics, 1993,
vol 16, p. 169-210 (Special issue dedicated to I.M.Gelfand 80-th
birthday)
\item[{[G4]}] Goncharov A.B., {\it Special values of Hasse-Weil
$L$-functions and generalized Eisenstein-Kronecker
series}. To appear.
\item[{[GL]}] Goncharov A.B., Levin A.M. {\it Zagier's conjecture
on $L(E,2)$}. Preprint IHES 1995.
\item[{[Del]}] Deligne P.: {\it Symbole modere} Publ. Math. IHES 1992
\item[{[D1]}] Deninger C.: {\it Higher order operations in Deligne
cohomology}. Inventiones Math. 122 N1 (1995).
\item[{[D2]}] Deninger C.: {\it Higher regulators and Hecke L-series of
imaginary quadratic fields I} Invent. Math. 96 (1989), 1-69.
\item[{[J]}] De Jeu R. {\it On $K_4^{(3)}$ of curves over number
fields} Preprint 1995.
\item[{[S1]}] Suslin A.A.: {\it Homology of $GL_{n}$,
characteristic classes and Milnor's $K$-theory}. Springer Lecture Notes
in Math. 1046 (1989), 357--375.
\item[{[S2]}] Suslin A.A.: {\it $K_{3}$ of a field and Bloch's
group}, Proceedings of the Steklov Institute of Mathematics 1991, Issue 4.
\item[{[Z1]}] Zagier D.:{\it Polylogarithms, Dedekind zeta
functions and the algebraic $K$-theory of fields}. Arithmetic
Algebraic Geometry (G.v.d.Geer, F.Oort, J.Steenbrink, eds.),
Prog. Math., Vol 89, Birkhauser, Boston, 1991, pp. 391--430.
\item[{[Z2]}] Zagier D.:{\it The Bloch - Wigner -Ramakrishnan
polylogarithm function} Math. Ann. 286 (1990), 613-624
\end{itemize}
\end{document}
{\bf Proof}. Let us use induction on $n-k$. One has
$$
\sum_i \int_{X(\Bbb C)}d{\cal
L}_{2}(f_i)\log^{n-3}|f_i|\log|g_i|\wedge \omega = -i \sum_i \int_{X(\Bbb
C)}\alpha(1-f_i,
f_i)\log^{n-3}|f_i|\log|g_i|\wedge \omega
$$
Further,
$$
\sum_i \int_{X(\Bbb C)}d{\cal
L}_{3}(f_i)\log^{n-4}|f_i|\log|g_i|\wedge \omega =
$$
$$
- \sum_i \int_{X(\Bbb C)}{\cal
L}_{2}(f_i)d\arg|f_i|\log^{n-4}|f_i|\log|g_i|\wedge \omega +\frac{1}{3}\sum_i
\int_{X(\Bbb C)}\alpha(1-f_i,f_i)\log^{n-3}|f_i|\log|g_i|\wedge \omega
$$
The second term is already in the desired form. The first one can be
written as follows:
$$
-i\cdot \sum_i \int_{X(\Bbb C)}{\cal
L}_{2}(f_i)d\log|f_i|\log^{n-4}|f_i|\log|g_i|\wedge \omega =
$$
$$
-\frac{i}{n-2}\cdot \sum_i \int_{X(\Bbb C)}{\cal
L}_{2}(f_i)d(\log^{n-3}|f_i|\log|g_i|)\wedge \omega =
$$
$$
\frac{i}{n-2}\cdot \sum_i \int_{X(\Bbb C)}d{\cal
L}_{2}(f_i)\log^{n-3}|f_i|\log|g_i|\wedge \omega
$$
After this we apply tinduction. The general case is in
complete analogy with this one: we use formula (\ref{ss}) for $d{\cal
L}_{n-k}(f_i)$ to rewrite
$$
\sum_i \int_{X(\Bbb C)} d{\cal
L}_{n-k}(f_i)\log^{k-1}|f_i|\log|g_i|\wedge \omega
$$
and then (\ref{3.11}), (\ref{masha}) and induction. Proposition is proved.
{\bf Example 2: n=4}. Set
\begin{eqnarray*}
& & r_{4}(1): \{ f\}_{4}\mapsto {\cal L}_{4}(f )\\
& &r_{4}(2) : \{ f \}_{3} \otimes g \mapsto {\cal
L}_{3} (f ) d \arg g -\frac{1}{3} {\cal L}_{2} (f )\log \vert g \vert \cdot
d \log \vert f\vert\\
& & r_{4}(3) : \{ f \}_{2} \otimes g_1 \wedge g_2 \mapsto - {\cal
L}_{2} (f ) d \arg g_1 \wedge d \arg g_2 +
\frac{1}{3} \alpha(1-f,f) \cdot \\
& &\Bigl(\log \vert g_1 \vert d \arg \vert g_2\vert -
\log \vert g_2 \vert d \arg \vert g_1\vert\Bigr) \quad
+ \frac{1}{3} {\cal
L}_{2} (f ) d \log \vert g_1 \vert \wedge d \log \vert g_2 \vert\\
& &r_{4}(4) : g_{1}\wedge ... \wedge g_{4}\mapsto
{\rm Alt}
( \frac{1}{6} \cdot \log \vert g_{1} \vert d \arg
g_{2}\wedge d \arg g_{3}\wedge d \arg g_{4} - \\
& & \qquad -\frac{1}{6} \log \vert g_{1} \vert
d\log \vert g_{2}\vert d\log \vert g_3 \vert d\arg g_4 )\in {\cal
A}^{2}_{X};\\
& & d\log^{\wedge^{4}} : g_{1}\wedge ... \wedge g_{4}\mapsto d\log
g_{1} \wedge ... \wedge d\log g_{4}\in \Omega^{4}_{X}
\end{eqnarray*}
|
1995-12-14T06:20:18 | 9512 | alg-geom/9512009 | en | https://arxiv.org/abs/alg-geom/9512009 | [
"alg-geom",
"math.AC",
"math.AG"
] | alg-geom/9512009 | null | Takesi Kawasaki | On Macaulayfication of certain quasi-projective schemes | AMSLaTeX v 1.1 with amsart.sty, amscd.sty, 25 pages | null | null | null | null | The notion of Macaulayfication, which is analogous of the desingularization,
was introduced by Faltings in 1978 and he construct a Macaulayfication of
quasi-projective scheme whose non-Cohen-Macaulay locus is of dimension 0 or 1
by a characteristic free method. In this paper, we gave a Macaulayfication of a
quasi-projective scheme whose non-Cohen-Macaulay locus is of dimension 2. Of
course out method is independent of the characteristic.
| [
{
"version": "v1",
"created": "Thu, 14 Dec 1995 04:45:18 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Kawasaki",
"Takesi",
""
]
] | alg-geom | \section{Introduction}
Let $X$ be a Noetherian scheme.
A birational proper morphism~$Y \rightarrow X$ of schemes
is said to be
a {\em Macaulayfication\/} of~$X$
if $Y$ is a Cohen-Macaulay scheme.
This notion was introduced by Faltings~\cite{Faltings:78:Macaulay}
and he established that
there exists a Macaulayfication of a quasi-projective scheme
over a Noetherian ring possessing a dualizing complex
if its non-Cohen-Macaulay locus
is of dimension $0$~or~$1$.
Of course,
a desingularization is a Macaulayfication
and Hironaka gave a desingularization
of arbitrary algebraic variety over a field of characteristic~$0$.
But Faltings' method to construct a Macaulayfication
is independent of the characteristic of a scheme.
Furthermore,
several authors are interested in a Macaulayfication.
For example,
Goto and Schenzel independently showed
the converse of Faltings' result in a sense.
Let $A$ be a Noetherian local ring possessing a dualizing complex,
hence its non-Cohen-Macaulay locus is closed,
and assume that $\dim A / {\frak p} = \dim A$
for any associated prime ideal~${\frak p}$ of~$A$.
Then the non-Cohen-Macaulay locus of~$A$ consists of
only the maximal ideal
if and only if
$A$ is a generalized Cohen-Macaulay ring
but not a Cohen-Macaulay ring~\cite{Schenzel:75:einige}.
When this is the case,
Faltings~\cite[Satz 2]{Faltings:78:Macaulay} showed that
there exists a parameter ideal~${\frak q}$ of~$A$
such that the blowing-up~$\operatorname{Proj} A[{\frak q} t]$ of~$\operatorname{Spec} A$
with center~${\frak q}$ is Cohen-Macaulay,
where $t$ denotes an indeterminate.
Conversely,
Goto~\cite{Goto:82:blowing} proved that
if there is a parameter ideal~${\frak q}$ of~$A$
such that $\operatorname{Proj} A[{\frak q} t]$ is Cohen-Macaulay,
then $A$ is a generalized Cohen-Macaulay ring.
Moreover,
he showed that $A$ is Buchsbaum if and only if
$\operatorname{Proj} A[{\frak q} t]$ is Cohen-Macaulay
for every parameter ideal~${\frak q}$ of~$A$:
see also \cite{Schenzel:83:standard}.
Brodmann~\cite{Brodmann:83:local} also studied
the blowing-up of a generalized Cohen-Macaulay ring
with center a parameter ideal.
Furthermore,
he constructed Macaulayfications
in a quite different way from Faltings.
Let $A$ be a Noetherian local ring possessing a dualizing complex.
We let $d= \dim A$
and $s$ be the dimension of its non-Cohen-Macaulay locus.
If $s=0$,
then Brodmann~\cite[Proposition 2.13]{Brodmann:83:two}
gave an ideal~${\frak b}$ of height~$d-1$
such that $\operatorname{Proj} A[{\frak b} t]$ is Cohen-Macaulay.
If $s=1$, then
Faltings' Macaulayfication~\cite[Satz 3]{Faltings:78:Macaulay}
of~$\operatorname{Spec} A$ consists of
two consecutive blowing-ups $Y \rightarrow X \rightarrow \operatorname{Spec} A$
where the center of the first blowing-up
is an ideal of height~$d-1$.
In this case,
Brodmann gave two other Macaulayfications of~$\operatorname{Spec} A$:
the first one~\cite{Brodmann:80:Kohomologische}
is the composite of a blowing-up $X \rightarrow \operatorname{Spec} A$
with center an ideal of height~$d-1$
and a finite morphism~$Y \rightarrow X$;
the second one~\cite[Corollary 3.11]{Brodmann:83:two}
consists of
two consecutive blowing-ups $Y \rightarrow X \rightarrow \operatorname{Spec} A$
where the center of the first blowing-up is
an ideal of height~$d-2$.
In this article,
we are interested in a Macaulayfication of
the Noetherian scheme
whose non-Cohen-Macaulay locus is of dimension~$2$.
Let $A$ be a Noetherian ring possessing a dualizing complex
and $X$ a quasi-projective scheme over~$A$.
Then $X$ has a dualizing complex with codimension function~$v$.
Furthermore the non-Cohen-Macaulay locus~$V$ of~$X$
is closed.
We define a function $u \colon X \to {\Bbb Z}$
to be $u(p) = v(p) + \dim \overline{\{p\}}$.
We will establish the following theorem:
\begin{thm} \label{mthm}
If $\dim V \leq 2$ and $u$ is locally constant on~$V$,
then $X$ has a Macaulayfication.
\end{thm}
If $\dim V \leq 1$,
then $u$ is always locally constant on~$V$.
Therefore, this theorem contains Faltings' result.
Furthermore,
we note if $X$ is a projective scheme
over a Gorenstein local ring,
then $u$ is constant on~$X$.
We agree that
$A$ denotes a Noetherian local ring with maximal ideal~${\frak m}$
except for Section~\ref{sec:6}.
Assume that $d = \dim A > 0$.
We refer the reader to~%
\cite{Hartshorne:66:residue,%
Hartshorne:77:algebraic,%
Matsumura:89:commutative,%
Stuckrad-Vogel:86:Buchsbaum}
for unexplained terminology.
\section{Preliminaries}
In this section,
we state some definitions and properties
of a local cohomology and an ideal transform.
Let ${\frak b}$ be an ideal of~$A$.
\begin{dfn}
The local cohomology functor~$H_{\frak b}^p(-)$ and
the ideal transform functor~$D_{\frak b}^p(-)$
with respect to~${\frak b}$
are defined to be
$$
H_{\frak b}^p(-) = \mathop{\varinjlim}_m \operatorname{Ext}_A^p(A/{\frak b}^m, -)
\quad
\text{and}
\quad
D_{\frak b}^p(-) = \mathop{\varinjlim}_m \operatorname{Ext}_A^p({\frak b}^m, -),
$$
respectively.
\end{dfn}
For an $A$-module $M$,
there exists an exact sequence
\begin{equation} \label{eqn:2.1.1}
0 @>>>
H_{\frak b}^0(M) @>>>
M @>\iota>>
D_{\frak b}^0(M) @>>>
H_{\frak b}^1(M) @>>>
0
\end{equation}
and isomorphisms
$$
D_{\frak b}^p(M) \cong H_{\frak b}^{p+1}(M)
\quad
\text{for all $p>0$}.
$$
They induces that
\begin{equation} \label{eqn:2.1.2}
H_{\frak b}^p D_{\frak b}^0(M) =
\begin{cases}
0, & p=0, 1;
\\
H_{\frak b}^p(M), & \text{otherwise}.
\end{cases}
\end{equation}
If ${\frak b}$ contains an $M$-regular element~$a$,
then we can regard $D_{\frak b}^0(M)$ as a submodule
of the localization~$M_a$ with respect to~$a$
and $\iota$ is the inclusion.
It is well-known that $H_{\frak b}^p(-)$ is naturally isomorphic to
the direct limit of Koszul cohomology.
In particular,
let ${\frak b} = (f_1, \dots, f_h)$ and
$M$ be an $A$-module.
Then
$$
H_{\frak b}^h(M) = \mathop{\varinjlim}_m M/(f_1^m, \dots, f_h^m)M
\quad
\text{and}
\quad
H_{\frak b}^0(M) = \bigcap_{i=1}^h 0 \qtn_M \angled{f_i},
$$
where $0 \qtn \angled{f_i}$ denotes
$\bigcup_{m = 1}^\infty 0 \qtn f_i^m$.
Furthermore,
let $A \rightarrow B$ be a ring homomorphism.
Then there exists a natural isomorphism
$H_{\frak b}^p(M) \cong H_{{\frak b} B}^p(M)$
for a $B$-module~$M$.
The following lemma is frequently used in this article.
\begin{lem}[Brodmann \cite{Brodmann:83:Einige}]
\label{lem:2.2}
Let ${\frak b} = (f_1, \dots, f_h)$ and
${\frak c} = (f_1, \dots, f_{h-1})$
be two ideals.
Then there exists a natural long exact sequence
$$
\cdots @>>>
[H_{\frak c}^{p-1}(-)]_{f_h} @>>>
H_{\frak b}^p(-) @>>>
H_{\frak c}^p(-) @>>>
[H_{\frak c}^p(-)]_{f_h} @>>>
\cdots.
$$
\end{lem}
Next we state on the annihilator of local cohomology modules.
\begin{dfn}
For any finitely generated $A$-module~$M$,
we define an ideal~${\frak a}_A(M)$ to be
$$
{\frak a}_A(M) = \prod_{p=0}^{\dim M -1}
\operatorname{ann} H_{\frak m}^p(M).
$$
\end{dfn}
We note that a finitely generated $A$-module~$M$ is Cohen-Macaulay
if and only if ${\frak a}_A(M) = A$,
and that $M$ is generalized Cohen-Macaulay
if and only if ${\frak a}_A(M)$ is an ${\frak m}$-primary ideal.
The notion of~${\frak a}_A(-)$ plays a key role in this article.
In fact,
Schenzel~\cite{Schenzel:79:dualizing} showed that
$V({\frak a}_A(A))$ coincides with the non-Cohen-Macaulay locus of~$A$
if it possesses a dualizing complex and is equidimensional.
He also gave the following lemma~%
\cite{Schenzel:79:dualizing,Schenzel:82:cohomological}:
\begin{lem} \label{lem:2.4}
Let $M$ be a finitely generated $A$-module and
$x_1$,~\dots, $x_n$ a system of parameters for~$M$.
Then
$
(x_1, \dots, x_{i-1})M \qtn x_i \subseteq
(x_1, \dots, x_{i-1})M \qtn {\frak a}_A(M)
$
for any $1 \leq i \leq n$.
In particular,
if $x_i \in {\frak a}_A(M)$,
then the equality holds.
\end{lem}
Let $R = \bigoplus_{n \geq 0} R_n$ be a Noetherian graded ring
where $R_0 = A$.
A graded module~$M = \bigoplus_n M_n$ is said to be
{\em finitely graded\/}
if $M_n=0$ for all but finitely many~$n$.
The following lemma is an easy consequence of~\cite{Faltings:78:uber}.
\begin{lem} \label{lem:2.5}
Let ${\frak b}$ be a homogeneous ideal of~$R$
containing~$R_+ = \bigoplus_{n>0} R_n$
and $M$ a finitely generated graded $R$-module.
We assume that $A$ possesses a dualizing complex.
Let $p$ be the largest integer
such that, for all~$q \leq p$,
$H_{\frak b}^q(M)$ is finitely graded.
Then $\operatorname{depth} M\hlz{\frak p} \geq p$
for any closed point~${\frak p}$ of~$\operatorname{Proj} R$,
that is, ${\frak p}$ is a homogeneous prime ideal
such that $\dim R/ {\frak p} = 1$
and $R_+ \not\subseteq {\frak p}$.
\end{lem}
\section{A Rees algebra obtained by an ideal transform}
\label{sec:3}
\begin{dfn}
A sequence $f_1$,~\dots, $f_h$ of elements of~$A$
is said to be a d-sequence on an $A$-module~$M$ if
$
(f_1, \dots, f_{i-1})M \qtn f_i f_j
= (f_1, \dots, f_{i-1})M \qtn f_j
$
for any $1 \leq i \leq j \leq h$.
We shall say that $f_1$,~\dots, $f_h$ is
an unconditioned strong d-sequence
(for short, {\em u.s.d-sequence\/})
on~$M$
if $f_1^{n_1}$,~\dots, $f_h^{n_h}$ is a d-sequence on~$M$
in any order and
for arbitrary positive integers $n_1$,~\dots, $n_h$.
\end{dfn}
The notion of u.s.d-sequences was introduced
by Goto and Yamagishi~\cite{Goto-Yamagishi::theory}
to refine arguments on Buchsbaum rings
and generalized Cohen-Macaulay rings.
Their theory contains
Brodmann's study on the Rees algebra with respect to an ideal
generated by a pS-sequences~\cite{Brodmann:83:local}.
But Brodmann~\cite{Brodmann:84:local} also studied
the ideal transform of such a Rees algebra.
The purpose of this section is to study
an ideal transform of the Rees algebra with respect to an ideal
generated by a u.s.d-sequence.
Let $f_0$,~\dots, $f_h$ be a sequence of elements of~$A$
where $h \geq 1$ and
${\frak q} = (f_1, \dots, f_h)$.
\begin{lem} \label{lem:3.2}
If $f_1$,~\dots, $f_h$ be a d-sequence on~$A/ f_0A$,
then
$$
[(f_1, \dots, f_k) {\frak q}^n] \qtn f_0 =
(f_1, \dots, f_k) [{\frak q}^n \qtn f_0] +
0 \qtn f_0
$$
for any $1 \leq k \leq h$ and $n>0$.
\end{lem}
\begin{pf}
It is obvious that
the left hand side contains the right one.
We shall prove the inverse inclusion
by induction on~$k$.
Let $a$ be an element of the left hand side.
When $k=1$,
we put $f_0 a = f_1 b$ where $b \in {\frak q}^n$.
By using \cite[Theorem 1.3]{Goto-Yamagishi::theory},
we obtain
$b \in (f_0) \qtn f_1 \cap {\frak q}^n \subseteq (f_0)$.
If we put $b = f_0 a'$,
then $a' \in {\frak q}^n \qtn f_0$ and
$f_0 (a - f_1 a') =0$.
Thus we get
$a \in f_1 [{\frak q}^n \qtn f_0] + 0 \qtn f_0$.
When $k > 1$,
we put $f_0 a = b + f_k c$
where
$
b \in (f_1, \dots, f_{k-1}) {\frak q}^n
$
and $c \in {\frak q}^n$.
Then we obtain
\begin{align*}
c & \in (f_0, \dots, f_{k-1}) \qtn f_k \cap {\frak q}^n
\\
& \subseteq (f_0) + (f_1, \dots, f_{k-1}) {\frak q}^{n-1}
\end{align*}
by using~\cite[Theorem 1.3]{Goto-Yamagishi::theory} again.
If we put $c = f_0 a' + b'$
where
$$
b' \in (f_1, \dots, f_{k-1}) {\frak q}^{n-1},
$$
then $a' \in {\frak q}^n \qtn f_0$.
Thus we get
\begin{align*}
a - f_k a' & \in [(f_1, \dots, f_{k-1}) {\frak q}^n] \qtn f_0
\\
& = (f_1, \dots, f_{k-1}) [{\frak q}^n \qtn f_0] + 0 \qtn f_0
\end{align*}
by induction hypothesis.
The proof is completed.
\end{pf}
Let $\trans{\frak q} = {\frak q} \qtn \angled{f_0}$.
If $f_0$ is $A$-regular
and $f_1$,~\dots, $f_h$ is a d-sequence
on~$A/ f_0^lA$ for all~$l > 0$,
then Lemma~\ref{lem:3.2} assures us that
\begin{equation} \label{eqn:3.2.1}
{\frak q}^{n-1} \trans{\frak q} =
\trans{\frak q}^n =
{\frak q}^n \qtn \angled{f_0}
\quad
\text{for all $n>0$}.
\end{equation}
Therefore the Rees algebra $\trans R = A[\trans{\frak q} t]$
is finitely generated over $R = A [{\frak q} t]$.
The following is an analogue of~\cite[Lemma 3.4]{Goto:82:blowing}.
\begin{thm} \label{thm:3.3}
Let $B = A[\trans{\frak q}/f_h] = \trans R\hlz{f_ht}$.
If $f_0$ is $A$-regular
and $f_1$,~\dots, $f_h$ is a d-sequence
on~$A/ f_0^lA$ for all~$l > 0$,
then $f_h$, $f_1/ f_h$,~\dots, $f_{h-1}/f_h$, $f_0$
is a regular sequence on~$B$.
\end{thm}
\begin{pf}
First we note that
$f_1$,~\dots, $f_h$ is a d-sequence on~$A$.
In fact,
by using Krull's intersection theorem,
we obtain
\begin{align*}
(f_1, \dots, f_{i-1}) \qtn f_i f_j
& = \bigcap_{l =1}^\infty
(f_0^l, f_1, \dots, f_{i-1}) \qtn f_i f_j
\\
& = \bigcap_{l =1}^\infty
(f_0^l, f_1, \dots, f_{i-1}) \qtn f_j
\\
& =
(f_1, \dots, f_{i-1}) \qtn f_j
\end{align*}
for any $1 \leq i \leq j \leq h$.
Next we show that
\begin{equation} \label{eqn:3.3.1}
(f_1, \dots, f_{k-1}) \qtn f_k \cap \trans {\frak q}^n =
(f_1, \dots, f_{k-1}) \trans {\frak q}^{n-1},
\end{equation}
for any $1 \leq k \leq h+1$ and $n>1$,
where $f_{h+1} = 1$.
If $a$ is an element of the left hand side,
then $f_0^l a \in {\frak q}^n$
for a sufficiently large~$l$.
By~\cite[Theorem 1.3]{Goto-Yamagishi::theory},
we have
\begin{align*}
f_0^l a &
\in (f_1, \dots, f_{k-1}) \qtn f_k \cap {\frak q}^n
\\
&
= (f_1, \dots, f_{k-1}) {\frak q}^{n-1}.
\end{align*}
Lemma~\ref{lem:3.2} says
$$
a \in [(f_1, \dots, f_{k-1}) {\frak q}^{n-1}] \qtn \angled{f_0}
= (f_1, \dots, f_{k-1}) \trans {\frak q}^{n-1}.
$$
The inverse inclusion is clear.
By~\eqref{eqn:3.3.1} and \cite[Theorem 1.7]{Goto-Yamagishi::theory},
we obtain that
$$
f_h,
\frac{f_1}{f_h},
\dots,
\frac{f_{h-1}}{f_h}
$$
is a regular sequence on~$B$.
Finally we shall show that
$f_0$ is regular on
$B/(f_h, f_1/f_h, \dots, f_{h-1}/f_h)B$.
Let $\alpha \in (f_h, f_1/f_h, \dots, f_{h-1}/f_h)B \qtn f_0$.
For a sufficiently large~$n>1$,
we may assume $\alpha = a_0/f_h^n$ and
$$
f_0 \frac{a_0}{f_h^n} =
f_h \frac{a_h}{f_h^n} +
\frac{f_1}{f_h} \frac{a_1}{f_h^n} +
\dots +
\frac{f_{h-1}}{f_h} \frac{a_{h-1}}{f_h^n}
$$
where $a_0$,~\dots, $a_h \in \trans{\frak q}^n$.
Therefore
$$
f_h^{m+1} f_0 a_0 =
f_h^m(f_h^2 a_h + f_1 a_1 + \dots + f_{h-1} a_{h-1})
$$
in~$A$ for some~$m>0$.
Take an integer~$l$
such that $f_0^l a_h \in {\frak q}^n$.
Then
\begin{align*}
f_h^{m+2} f_0^l a_h &
\in (f_0^{l+1}, f_1, \dots, f_{h-1}) \cap {\frak q}^{n+m+2}
\\
& = (f_0^{l+1}) \cap {\frak q}^{n+m+2}
+ (f_1, \dots, f_{h-1}) {\frak q}^{n+m+1}
\\
& \subseteq f_0^{l+1} \trans{\frak q}^{n+m+2}
+ (f_1, \dots, f_{h-1}) {\frak q}^{n+m+1}.
\end{align*}
If we put
$$
f_h^{m+2} f_0^l a_h =
f_0^{l+1} b_0 + f_1 b_1 + \dots + f_{h-1} b_{h-1}
$$
where $b_0 \in \trans{\frak q}^{n+m+2}$ and
$b_1$,~\dots, $b_{h-1} \in {\frak q}^{n+m+1}$,
then
\begin{align*}
f_h^{m+2} a_h - f_0 b_0 & \in
[(f_1, \dots, f_{h-1}) {\frak q}^{n+m+1}] \qtn \angled{f_0}
\\
& = (f_1, \dots, f_{h-1}) \trans{\frak q}^{n+m+1}.
\end{align*}
Let
$$
f_h^{m+2} a_h - f_0 b_0 =
f_1 c_1 + \dots + f_{h-1} c_{h-1}
$$
where $c_1$,~\dots, $c_{h-1} \in \trans{\frak q}^{n+m+1}$.
Then
$$
f_0(f_h^{m+1} a_0 - b_0) \in
(f_1, \dots, f_{h-1}) {\frak q}^{n+m}.
$$
Therefore
$$
f_h^{m+1} a_0 - b_0 \in
(f_1, \dots, f_{h-1}) \trans{\frak q}^{n+m},
$$
that is,
$$
\alpha - f_h \frac{b_0}{f_h^{n+m+2}}
\in \left(
\frac{f_1}{f_h},
\dots,
\frac{f_{h-1}}{f_h}
\right)B.
$$
The proof is completed.
\end{pf}
In the rest of this section,
we assume
that $f_0$ is $A$-regular
and that $f_1$,~\dots, $f_h$ is a u.s.d-sequence
on~$A/ f_0^l A$ for all~$l>0$.
Let $G = \bigoplus_{n \geq 0} {\frak q}^n/ {\frak q}^{n+1}$ and
$
\trans G = \bigoplus_{n \geq 0} \trans{\frak q}^n/
\trans{\frak q}^{n+1}
$
be associated graded rings
with respect to~${\frak q}$
and $\trans{\frak q}$,
respectively.
We shall compute local cohomology modules
of~$\trans G$ and~$\trans R$
with respect to~${\frak N} = (f_0, \dots, f_h)R + R_+$.
\begin{thm}
If $p < h+1$, then
$$
[H_{\frak N}^p(\trans G)]_n = 0
\quad
\text{for $n \ne 1-p$}.
$$
Furthermore
$$
[H_{\frak N}^{h+1}(\trans G)]_n = 0
\quad
\text{for $n > -h$}.
$$
\end{thm}
\begin{pf}
We shall prove that
\begin{equation} \label{eqn:3.4.1}
[H_{(f_0, f_1t, \dots, f_kt)}^p(\trans G)]_n = 0
\quad
\text{for $n \ne 1-p$}
\end{equation}
if $p < k+1$
by induction on~$k$.
It is obvious that $f_0$ is $\trans G$-regular.
Therefore $H_{(f_0)}^0(\trans G) = 0$.
Suppose $k > 0$.
Then $H_{(f_0, f_1t, \dots, f_{k-1}t)}^p(\trans G)_{f_kt} = 0$
for $p < k$ by induction hypothesis.
By Lemma~\ref{lem:2.2},
we obtain isomorphisms
$$
H_{(f_0, f_1t, \dots, f_kt)}^p(\trans G) \cong
H_{(f_0, f_1t, \dots, f_{k-1}t)}^p (\trans G)
\quad
\text{for $p<k$}.
$$
Therefore \eqref{eqn:3.4.1} is proved if $p < k$.
We also obtain an exact sequence
$$
0 @>>>
H_{(f_0, f_1t, \dots, f_kt)}^k(\trans G) @>>>
H_{(f_0, f_1t, \dots, f_{k-1}t)}^k(\trans G) @>>>
H_{(f_0, f_1t, \dots, f_{k-1}t)}^k(\trans G)_{f_kt}
$$
from Lemma~\ref{lem:2.2}.
Hence $H_{(f_0, f_1t, \dots, f_kt)}^k(\trans G)$ is
the limit of the direct system~$\{K_m\}_{m>0}$
such that
$$
K_m =
\frac
{(f_0^m, (f_1t)^m, \dots, (f_{k-1}t)^m) \trans G
\qtn \angled{f_kt}}
{(f_0^m, (f_1t)^m, \dots, (f_{k-1}t)^m) \trans G}
\, (m(k-1))
\quad
\text{for $m > 0$}
$$
and the homomorphism~$K_m \to K_{m'}$ is induced
from the multiplication of
$(f_0 \cdot f_1t \cdots f_{k-1}t)^{m' - m}$
for any $m' > m$.
We shall show that it is the zero map
except for degree~$1-k$
if $m'$ is sufficiently larger than~$m$.
Let $\alpha$ be a homogeneous element of~$K_m$
of degree~$n$
and $a$ its representative.
That is,
$a \in \trans{\frak q}^{n+m(k-1)}$ and
$$
f_k^l a \in
f_0^m \trans{\frak q}^{n+m(k-1)+l}
+ (f_1^m, \dots, f_{k-1}^m)
\trans{\frak q}^{n+m(k-2)+l}
+ \trans{\frak q}^{n+m(k-1)+l+1}
$$
for some~$l>0$.
Take an integer~$m' > m$
such that $f_0^{m' - m}\trans {\frak q} \subseteq {\frak q}$.
Then $f_0^{m' - m}\trans {\frak q}^n \subseteq {\frak q}^n$
for any~$n>0$ by~\eqref{eqn:3.2.1}.
By replacing $\alpha$ by its image in~$K_{m'}$,
we may assume that $a \in {\frak q}^{n+m(k-1)}$ and
$$
f_k^l a \in
f_0^m \trans{\frak q}^{n+m(k-1)+l}
+ (f_1^m, \dots, f_{k-1}^m) {\frak q}^{n+m(k-2) + l}
+ {\frak q}^{n+m(k-1) + l+1}.
$$
We put $f_k^l a = b + c$
where
$
b \in f_0^m \trans{\frak q}^{n+m(k-1)+l}
+ (f_1^m, \dots, f_{k-1}^m) {\frak q}^{n+m(k-2) + l}
$
and $c \in {\frak q}^{n+m(k-1)+l+1}$.
Then, by using~\cite[Theorem 2.6]{Goto-Yamagishi::theory},
we obtain
\begin{align*}
c & \in (f_0^m, \dots, f_{k-1}^m, f_k^l)
\cap {\frak q}^{n+m(k-1) + l+1}
\\
& \subseteq f_0^m \trans{\frak q}^{n+m(k-1) + l+1}
+ (f_1^m, \dots, f_{k-1}^m) {\frak q}^{n+m(k-2) + l+1}
+ f_k^l {\frak q}^{n+m(k-1) + 1}.
\end{align*}
If we put $c = b' + f_k^l a'$
where
$
b' \in f_0^m \trans{\frak q}^{n+m(k-1) + l+1}
+ (f_1^m, \dots, f_{k-1}^m) {\frak q}^{n+m(k-2) + l+1}
$
and $a' \in {\frak q}^{n+m(k-1)+1}$,
then $a-a'$ is also a representative of~$\alpha$.
Therefore we may assume that $c=0$.
By using~\cite[Theorem 2.8]{Goto-Yamagishi::theory},
we obtain
\begin{align*}
a & \in (f_0^m, \dots, f_{k-1}^m) \qtn f_k \cap
{\frak q}^{n+m(k-1)}
\\
& = (f_0^m) \cap {\frak q}^{n+m(k-1)}
+ (f_1^m, \dots, f_{k-1}^m) {\frak q}^{n+m(k-2)}
\\
& \quad +
\sum
\begin{Sb}
I \subseteq \{1, \dots, k-1\}
\\
\sharp I \cdot (m-1) \geq n + m(k-1)
\end{Sb}
\left\{
\prod_{i \in I}
f_i^{m-1}
\right\}
\{
[(f_0^m) + (f_i \mid i \in I)] \qtn f_k
\}
\\
& \subseteq
f_0^m \trans{\frak q}^{n+m(k-1)}
+ (f_1^m, \dots, f_{k-1}^m) {\frak q}^{n+m(k-2)}
+ {\frak q}^{n+m(k-1) + 1}
\\
& \quad +
\sum
\begin{Sb}
I \subseteq \{1, \dots, k-1\}
\\
\sharp I \cdot (m-1) = n + m(k-1)
\end{Sb}
\left\{
\prod_{i \in I}
f_i^{m-1}
\right\}
\{
[(f_0^m) + (f_i \mid i \in I)] \qtn f_k
\}
\end{align*}
Here $\sharp I$ denotes the number of elements in~$I$.
If $n > 1-k$,
then there is no subset~$I$ of~$\{1, \dots, k-1\}$
such that $\sharp I \cdot (m-1) = n+m(k-1)$.
If $n < 1-k$,
then such $I$ is a proper subset.
Let $j \in \{1, \dots, k-1\} \setminus I$ and
$$
d \in [(f_0^m) + (f_i \mid i \in I)] \qtn f_k
= [(f_0^m) + (f_i \mid i \in I)] \qtn f_j.
$$
Then
$$
(f_0 \cdots f_{k-1})
\left\{ \prod_{i \in I} f_i^{m-1} \right\}
d
\in
f_0^{m+1} \trans{\frak q}^{n+(m+1)(k-1)}
+ (f_1^{m+1}, \dots, f_{k-1}^{m+1}) {\frak q}^{n+(m+1)(k-2)}.
$$
In fact,
if we put $f_j d = f_0^m e + g$
where $g \in (f_i \mid i \in I)$,
then $e \in \trans{\frak q}$.
Thus the image of~$\alpha$ in~$K_{m+1}$ is zero
if $n \ne 1-k$.
Put $k=h$.
Then
$$
[H_{\frak N}^p(\trans G)]_n =
[H_{(f_0, f_1t, \dots, f_ht)}^p(\trans G)]_n =
0
\quad
\text{for $n \ne 1 - p$}
$$
if $p < h+1$.
The first assertion is proved.
Next we compute $H_{(f_0, f_1t, \dots, f_ht)}^{h+1}(\trans G)$.
It is the limit of the direct system~$\{K'_m\}_{m>0}$
such that
$$
K'_m =
\trans G/ (f_0^m, (f_1t)^m, \dots, (f_ht)^m) \trans G \, (mh)
\quad
\text{for $m>0$}
$$
and the homomorphism $K'_m \to K'_{m'}$ is induced
from the multiplication of
$(f_0 \cdot f_1t \cdots f_ht)^{m' - m}$
for any $m' > m$.
We shall show that it is the zero map for degree $n > -h$
if $m'$ is sufficiently larger than~$m$.
Let $\alpha$ be a homogeneous element of~$K'_m$ of degree~$n$
and $a$ its representative.
That is, $a \in \trans{\frak q}^{n+mh}$.
If $n > -h$, then
$$
(f_0 \cdots f_h)^{m' - m} a \in
{\frak q}^{n+m'h} =
(f_1^{m'}, \dots, f_h^{m'}) {\frak q}^{n+m'(h-1)}
$$
for a sufficiently larger~$m'$ than~$m$.
Thus the image of~$\alpha$ in~$K'_{m'}$ is zero
if $n> -h$.
Therefore $[H_{\frak N}^{h+1}(\trans G)]_n = 0$ for $n > -h$.
\end{pf}
By this theorem,
we can compute local cohomology of~$\trans R$.
\begin{cor} \label{cor:3.5}
If $h=1$, $2$,
then
$$
H_{\frak N}^p(\trans R) = 0
\quad
\text{for $p\ne 1$, $h+2$}
$$
and
$
H_{\frak N}^1(\trans R) =
[H_{\frak N}^1(\trans R)]_0 =
H_{(f_0, \dots, f_h)}^1(A)
$.
If $h \geq 3$,
then
$$
H_{\frak N}^p(\trans R) = 0
\quad
\text{for $p=0$, $2$, $3$}
$$
and
$
H_{\frak N}^1(\trans R) =
[H_{\frak N}^1(\trans R)]_0 =
H_{(f_0, \dots, f_h)}^1(A)
$.
Furthermore,
if $ 4 \leq p \leq h+1$,
then
$$
[H_{\frak N}^p(\trans R)]_n =
\begin{cases}
H_{(f_0, \dots, f_h)}^{p-1}(A),
& \text{for $-1 \geq n \geq 3-p$};
\\
0, & \text{otherwise}.
\end{cases}
$$
\end{cor}
\begin{pf}
Passing through the completion,
we may assume that $A$ possesses a dualizing complex.
Since $H_{\frak N}^p(\trans G)$ is finitely graded for~$p < h+1$,
$H_{\frak N}^p(\trans R)$ is finitely graded for~$p \leq h+1$
\cite[Proposition~3]{Marley:94:finitely}.
Considering the following two exact sequences
$$
0 @>>> \trans R_+ @>>> \trans R @>>> A @>>> 0
\quad
\text{and}
\quad
0 @>>> \trans R_+(1) @>>> \trans R @>>> \trans G @>>> 0,
$$
we obtain the assertion:
see the proof of~\cite[Theorem 4.1]{Brodmann:84:local}.
\end{pf}
Let $S = \trans R/ R$, that is,
$S = \bigoplus_{n>0} \trans{\frak q}^n / {\frak q}^n$.
The following proposition shall play an important role
in the next section.
\begin{prop} \label{prop:3.6}
If $p < h$, then
$$
[H_{\frak N}^p(S)]_n =0
\quad
\text{for $n \ne 1-p$.}
$$
Moreover,
$$
[H_{\frak N}^h(S)]_n = 0
\quad
\text{for $n > 1-h$}.
$$
\end{prop}
\begin{pf}
In the same way as the proof of Theorem~\ref{thm:3.3},
we find that $f_1$,~\dots, $f_h$ is a u.s.d-sequence on~$A$.
Hence, by using \cite[Theorem 4.2]{Goto-Yamagishi::theory},
$$
[H_{(f_1t, \dots, f_ht)}^p(G)]_n = 0
\quad
\text{for $n \ne -p$}
$$
if $p<h$.
Furthermore,
$$
[H_{(f_1t, \dots, f_ht)}^h(G)]_n = 0
\quad
\text{for $n> -h$}.
$$
By using Lemma~\ref{lem:2.2},
we obtain
$$
[H_{\frak N}^p(G)]_n = 0
\quad
\text{for $n \ne 1-p$, $-p$}
$$
if $p < h$ and
$$
[H_{\frak N}^p(G)]_n = 0
\quad
\text{for $n>1-p$}
$$
if $p = h$, $h+1$.
Since
$
\trans{\frak q}^2 =
{\frak q} \trans{\frak q}
$,
there exists an exact sequence
$$
0 @>>> S(1) @>>> G @>\phi>> \trans G @>>> S @>>> 0.
$$
Let $T$ be the image of~$\phi$.
We shall show
$$
[H_{\frak N}^p(S)]_n =
[H_{\frak N}^p(T)]_n = 0
\quad
\text{for $n> 1-p$}
$$
by induction on~$h-p$.
If $p > h+1$, then the assertion is obvious.
Let $p \leq h+1$.
Then following two exact sequences
\begin{gather*}
H_{\frak N}^p(\trans G) @>>>
H_{\frak N}^p(S) @>>>
H_{\frak N}^{p+1}(T) @>>>
H_{\frak N}^{p+1}(\trans G),
\\
H_{\frak N}^p(G) @>>>
H_{\frak N}^p(T) @>>>
H_{\frak N}^{p+1}(S)(1) @>>>
H_{\frak N}^{p+1}(G)
\end{gather*}
and the induction hypothesis imply
$$
[H_{\frak N}^p(S)]_n =
[H_{\frak N}^p(T)]_n = 0
\quad
\text{for $n>1-p$}.
$$
In the same way,
we can prove that
$$
[H_{\frak N}^p(S)]_n =
[H_{\frak N}^p(T)]_n = 0
\quad
\text{for $n< 1-p$}
$$
if $p< h$
by induction on~$p$.
\end{pf}
Finally we show that
$\trans R$ is an ideal transform of~$R$
in a sense.
\begin{prop}
$\trans R_+ = D_{(f_0, \dots, f_h)}^0(R_+)$.
\end{prop}
\begin{pf}
We first show that
$f_0$, $f_1$ is a regular sequence on~$\trans R_+$.
Let $n>0$.
Since $f_0$ is $A$-regular,
it is also $\trans{\frak q}^n$-regular.
Let
$
a \in
[f_0 \trans{\frak q}^n] \qtn f_1 \cap
\trans {\frak q}^n
$.
Then $f_0^l a \in {\frak q}^n$ for a sufficiently large~$l$.
Since $f_1 a \in (f_0)$,
we have
$
f_0^l a \in
(f_0^{l+1}) \qtn f_1 \cap {\frak q}^n \subseteq
f_0^{l+1} \trans{\frak q}^n
$,
that is, $a \in f_0 \trans{\frak q}^n$.
Thus we have shown that
$f_1$ is $\trans R_+ / f_0 \trans R_+$-regular.
By this and \eqref{eqn:2.1.1},
we obtain
\begin{equation} \label{eqn:3.7.1}
D_{(f_0, \dots, f_h)}^0(R_+) \subseteq
D_{(f_0, \dots, f_h)}^0(\trans R_+) =
\trans R_+.
\end{equation}
Since
$
\trans{\frak q}^n =
{\frak q}^{n-1} \trans{\frak q}
$
for $n \geq 2$,
$(f_0^l, f_1, \dots, f_h) \trans R_+ \subseteq R_+$
for a sufficiently large~$l$.
Hence, we obtain the inverse inclusion of~\eqref{eqn:3.7.1}.
The proof is completed.
\end{pf}
\setcounter{equation}{0}
\section{%
A blowing-up with respect to a certain subsystem of parameters}
\label{sec:4}
In this section,
we assume that $A$ possesses a dualizing complex.
We fix an integer $s \geq \dim A/ {\frak a}_A(A)$.
Since $\dim A/ {\frak a}_A(M) < \dim M$
for any finitely generated $A$-module~$M$
\cite[Korollar 2.2.4]{Schenzel:82:dualisierende},
there exists a system of parameters
$x_1$,~\dots, $x_d$ for~$A$ such that
\begin{equation} \label{eqn:4.0.1}
\begin{cases}
x_{s+1}, \dots, x_d \in {\frak a}_A(A);
\\
x_i \in {\frak a}_A(A/(x_{i+1}, \dots, x_d)),
& \text{for $i \leq s$}.
\end{cases}
\end{equation}
This notion is a slight improvement of
a p-standard system of parameters,
which was introduced by Cuong~\cite{Cuong:91:dimension}.
He also gave the statement~(1) of Theorem~\ref{thm:4.2}.
The author was personally taught it by him.
\begin{lem} \label{lem:4.1}
Let $n_1$,~\dots, $n_i$ be arbitrary positive integers.
Then
\begin{multline*}
(x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}},
x_{k+1}, \dots, x_d) \qtn x_i^{n_i} \cap
(x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}}, x_k, \dots, x_d)
\\
= (x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}}, x_{k+1}, \dots, x_d)
\end{multline*}
for any $1 \leq i \leq k \leq d$.
\end{lem}
\begin{pf}
It is obvious that the left hand side
contains the right one.
Let $a$ be an element of the left hand side
and $a = b + x_k c$ where
$
b \in (x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}},
x_{k+1}, \dots, x_d)
$.
Then
\begin{align*}
c & \in (x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}},
x_{k+1}, \dots, x_d) \qtn x_i^{n_i} x_k
\\
& = (x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}},
x_{k+1}, \dots, x_d) \qtn x_k
\end{align*}
by Lemma~\ref{lem:2.4}.
Therefore
$
x_k c, a \in (x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}},
x_{k+1}, \dots, x_d)
$.
The proof is completed.
\end{pf}
Let ${\frak q} = (x_{s+1}, \dots, x_d)$.
Lemma~\ref{lem:2.2} assures us
that $x_{s+1}$, \dots, $x_d$ is a u.s.d-sequence on~$A$.
Furthermore, we have the following theorem:
\begin{thm} \label{thm:4.2}
\rom{(1)}
The sequences $x_1^{n_1}$,~\dots, $x_s^{n_s}$,
$x_{\sigma(s+1)}^{n_{s+1}}$,~\dots, $x_{\sigma(d)}^{n_d}$ is
a d-sequence on~$A$
for any positive integers $n_1$,~\dots, $n_d$
and for any permutation~$\sigma$ on $s+1$,~\dots, $d$.
\rom{(2)}
If $s>0$, then $x_1^{n_1}$,~\dots, $x_s^{n_s}$ is
a d-sequence on~$A/ {\frak q}^n$
for any positive integers $n_1$,~\dots, $n_s$
and~$n$.
\end{thm}
\begin{pf}
(1):~Let $1 \leq i \leq j \leq d$.
We have only to prove that
$$
(x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}}) \qtn x_i^{n_i} x_j^{n_j}
=
(x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}}) \qtn x_j^{n_j}
$$
for any positive integers $n_1$,~\dots, $n_d$.
If $j>s$, then the both sides are equal to
$(x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}}) \qtn {\frak a}_A(A)$.
Assume that $j \leq s$
and take an element~$a$ of the left hand side.
By using Lemma~\ref{lem:2.4}, we get
\begin{align*}
a & \in (x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}},
x_{j+1}, \dots, x_d) \qtn x_i^{n_i} x_j^{n_j}
\\
& = (x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}},
x_{j+1}, \dots, x_d) \qtn x_j^{n_j}.
\end{align*}
Hence we have
\begin{align*}
x_j^{n_j} a & \in (x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}})
\qtn x_i^{n_i} \cap
(x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}},
x_{j+1}, \dots, x_d)
\\
& = (x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}})
\end{align*}
by repeating to use Lemma~\ref{lem:4.1}.
(2):~If $n=1$,
then the assertion is proved in the same way as above.
Let $1 \leq i \leq j \leq s$ and $n>1$.
Then $x_{s+1}$,~\dots, $x_d$ is a d-sequence
on~$A/(x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}}, x_i^{n_i} x_j^{n_j})$.
By using Lemma~\ref{lem:3.2},
we obtain
\begin{align*}
\lefteqn{[(x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}}) + {\frak q}^n]
\qtn x_i^{n_i} x_j^{n_j}}
\qquad
\\
& = (x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}})
\qtn x_i^{n_i} x_j^{n_j}
+ {\frak q}^{n-1} [(x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}},
x_{s+1}, \dots, x_d) \qtn x_i^{n_i} x_j^{n_j}]
\\
& = (x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}})
\qtn x_j^{n_j}
+ {\frak q}^{n-1} [(x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}},
x_{s+1}, \dots, x_d) \qtn x_j^{n_j}]
\\
& \subseteq
[(x_1^{n_1}, \dots, x_{i-1}^{n_{i-1}}) + {\frak q}^n]
\qtn x_j^{n_j}.
\end{align*}
Here the second equality follows from the case of~$n=1$.
Thus the proof is completed.
\end{pf}
In the same way as the proof of Theorem~\ref{thm:3.3},
we find that any subsequence of $x_1^{n_1}$,~\dots, $x_d^{n_d}$
is a d-sequence on~$A$
and any subsequence of $x_1^{n_1}$,~\dots, $x_s^{n_s}$
is a d-sequence on~$A/ {\frak q}^n$
for arbitrary positive integers $n_1$,~\dots, $n_d$ and~$n$.
\begin{cor}
Fix an integer~$k$ such that $1 \leq k \leq d$.
Then
$$
H_{(x_k, \dots, x_d)}^p(A) =
\mathop{\varinjlim}_m \frac{(x_k^m, \dots, x_{k+p-1}^m) \qtn x_{k+p}}
{(x_k^m, \dots, x_{k+p-1}^m)}
\quad
\text{for $p< d-k+1$}.
$$
\end{cor}
\begin{pf}
We shall prove that
$$
H_{(x_k, \dots, x_l)}^p(A) =
\mathop{\varinjlim}_m \frac{(x_k^m, \dots, x_{k+p-1}^m) \qtn x_{k+p}}
{(x_k^m, \dots, x_{k+p-1}^m)}
\quad
\text{for $p< l-k+1$}
$$
by induction on~$l \geq k$.
If $l=k$, then $H_{(x_k)}^0(A) = 0 \qtn_A x_k$.
Suppose $l > k$.
Then $x_k$,~\dots, $x_{l-1}$ is a regular sequence on~$A_{x_l}$
because $x_k$,~\dots, $x_l$ is a d-sequence on~$A$.
Hence we obtain isomorphisms
$$
H_{(x_k, \dots, x_l)}^p(A) \cong
H_{(x_k, \dots, x_{l-1})}^p(A)
\quad
\text{for all $p<l-k$}
$$
and an exact sequence
$$
0 @>>>
H_{(x_k, \dots, x_l)}^{l-k}(A) @>>>
H_{(x_k, \dots, x_{l-1})}^{l-k}(A) @>>>
H_{(x_k, \dots, x_{l-1})}^{l-k}(A)_{x_l}
$$
by Lemma~\ref{lem:2.2}.
This exact sequence
is the direct limit of the exact sequence
$$
0 @>>>
\frac{(x_k^m, \dots, x_{l-1}^m) \qtn x_l}
{(x_k^m, \dots, x_{l-1}^m)} @>>>
A/(x_k^m, \dots, x_{l-1}^m) @>>>
[A/(x_k^m, \dots, x_{l-1}^m)]_{x_l}
$$
Thus the proof is completed.
\end{pf}
If $s=0$,
then $\operatorname{Proj} A[{\frak q} t] \rightarrow \operatorname{Spec} A$ is
a Macaulayfication of~$\operatorname{Spec} A$:
see Theorem~\ref{thm:5.1} for details.
In the rest of this section,
we shall observe $\operatorname{Proj} A[{\frak q} t]$
when $s>0$.
Assume that $s>0$ and fix an integer~$k$
such that $1 \leq k \leq s$.
We shall compute local cohomology modules of $R = A[{\frak q} t]$
with respect to~$(x_k, \dots, x_{s+1})$.
Let ${\frak M} = {\frak m} R + R_+$.
\begin{thm}
$H_{(x_k, \dots, x_{s+1})}^0(R) = 0 \qtn_A x_k$.
\end{thm}
\begin{pf}
Since $x_k$, $x_{s+1}$,~\dots, $x_d$ is a d-sequence on~$A$,
$0 \qtn_A x_k \cap {\frak q}^n = 0$ for~$n>0$
by~\cite[Theorem 1.3]{Goto-Yamagishi::theory}.
That is,
$$
H_{(x_k, \dots, x_{s+1})}^0({\frak q}^n) =
\begin{cases}
0 \qtn_A x_k, & \text{if $n=0$};
\\
0, & \text{otherwise}.
\end{cases}
$$
Therefore,
$
H_{(x_k, \dots, x_{s+1})}^0(R) =
\bigoplus_{n \geq 0} H_{(x_k, \dots, x_{s+1})}^0({\frak q}^n)
= 0 \qtn_A x_k
$.
\end{pf}
Let $C = A[t]/R$,
that is,
$C = \bigoplus_{n>0} A/ {\frak q}^n$.
\begin{lem}
For $k \leq l \leq s+1$ and $p \leq l-k$,
the natural homomorphism
$$
\alpha_l^p \colon
H_{(x_k, \dots, x_l)}^p(A[t]) @>>>
H_{(x_k, \dots, x_l)}^p(C)
$$
is a monomorphism except for degree~$0$.
\end{lem}
\begin{pf}
We shall work by induction on~$l$.
If $l=k$,
then $0 \qtn_A x_k \cap {\frak q}^n = 0$
for $n>0$.
Therefore
$$
\alpha_k^0 \colon
0 \qtn_{A[t]} x_k @>>>
\bigoplus_{n>0} {\frak q}^n \qtn x_k / {\frak q}^n
$$
is a monomorphism except for degree~$0$.
Let $k < l \leq s$.
Then $x_k$,~\dots, $x_{l-1}$ is a regular sequence
on~$A_{x_l}$ and on~$C_{x_l}$ by Theorem~\ref{thm:4.2}.
By using Lemma~\ref{lem:2.2},
we obtain commutative diagrams
$$
\begin{CD}
H_{(x_k, \dots, x_l)}^p(A[t])
@>\sim>>
H_{(x_k, \dots, x_{l-1})}^p(A[t])
\\
@V{\alpha_l^p}VV @V{\alpha_{l-1}^p}VV
\\
H_{(x_k, \dots, x_l)}^p(C)
@>\sim>>
H_{(x_k, \dots, x_{l-1})}^p(C)
\end{CD}
\quad
\text{for $p<l-k$}
$$
and
$$
\begin{CD}
0 @>>>
H_{(x_k, \dots, x_l)}^{l-k} (A[t]) @>>>
H_{(x_k, \dots, x_{l-1})}^{l-k} (A[t]) @>>>
H_{(x_k, \dots, x_{l-1})}^{l-k} (A[t])_{x_l}
\\
@. @V{\alpha_l^{l-k}}VV @VVV @VVV
\\
0 @>>>
H_{(x_k, \dots, x_l)}^{l-k} (C) @>>>
H_{(x_k, \dots, x_{l-1})}^{l-k} (C) @>>>
H_{(x_k, \dots, x_{l-1})}^{l-k} (C)_{x_l}
\end{CD}
$$
whose rows are exact.
Therefore the assertion is true for $p<l-k$
and we find that
$\alpha_l^{l-k}$ is the direct limit of
$$
\alpha_{l,m} \colon
\frac{(x_k^m, \dots, x_{l-1}^m) A[t] \qtn x_l}
{(x_k^m, \dots, x_{l-1}^m) A[t]} @>>>
\bigoplus_{n>0}
\frac{[(x_k^m, \dots, x_{l-1}^m) + {\frak q}^n] \qtn x_l}
{(x_k^m, \dots, x_{l-1}^m) + {\frak q}^n}.
$$
Since $x_l$, $x_{s+1}$,~\dots, $x_d$ is a d-sequence
on~$A/(x_k^m, \dots, x_{l-1}^m)$,
$$
(x_k^m, \dots, x_{l-1}^m) \qtn x_l \cap
[(x_k^m, \dots, x_{l-1}^m) + {\frak q}^n] =
(x_k^m, \dots, x_{l-1}^m)
\quad
\text{for $n>0$}.
$$
Therefore $\alpha_{l,m}$ is a monomorphism except for degree~$0$
and $\alpha_l^{l-k}$ is also.
If $l=s+1$,
then $x_k$,~\dots, $x_s$ is a regular sequence on~$A_{x_{s+1}}$
and $C_{x_{s+1}} = 0$.
The assertion is proved in the same way as above.
\end{pf}
Of course, $\alpha_{s+1}^p$ is the zero map
in degree~$0$.
Therefore there exists an exact sequence
\begin{equation} \label{eqn:4.5.1}
0 @>>>
\operatorname{Coker} \alpha_{s+1}^{p-1} @>>>
H_{(x_k, \dots, x_{s+1})}^p(R) @>>>
H_{(x_k, \dots, x_{s+1})}^p(A) @>>>
0
\end{equation}
for $0 < p \leq s-k+1$.
\begin{thm} \label{thm:4.6}
Let $0 \leq q \leq s-k$.
Then
$$
(x_{k+q}, \dots, x_d) \operatorname{Coker} \alpha^q_{s+1} = 0
$$
and $H_{\frak M}^p(\operatorname{Coker} \alpha^q_{s+1})$ is finitely graded
for~$p< d-s$.
\end{thm}
\begin{pf}
We know that
$
\operatorname{Coker} \alpha_{s+1}^q =
\operatorname{Coker} \alpha_{k+q}^q =
\mathop{\varinjlim}_m \operatorname{Coker} \alpha_{k+q,m}
$
and
$$
\operatorname{Coker} \alpha_{k+q,m} =
\bigoplus_{n>0}
\frac{[(x_k^m, \dots, x_{k+q-1}^m) + {\frak q}^n] \qtn x_{k+q}}
{(x_k^m, \dots, x_{k+q-1}^m) \qtn x_{k+q} + {\frak q}^n}.
$$
By using~Theorem~\ref{thm:4.2} and Lemma~\ref{lem:3.2},
we obtain
\begin{multline} \label{eqn:4.6.1}
[(x_k^m, \dots, x_{k+q-1}^m) + {\frak q}^n] \qtn x_{k+q}
\\
= (x_k^m, \dots, x_{k+q-1}^m) \qtn x_{k+q} +
{\frak q}^{n-1} [(x_k^m, \dots, x_{k+q-1}^m,
x_{s+1}, \dots, x_d) \qtn x_{k+q}].
\end{multline}
Therefore $\operatorname{Coker} \alpha_{k+q,m}$ is annihilated
by~$(x_{k+q}, \dots, x_d)$
and $\operatorname{Coker} \alpha_{s+1}^q$ is also.
Next we compute local cohomology modules
of~$\operatorname{Coker} \alpha_{s+1}^q$.
We note that $x_{k+q}$ is a regular element
on~$A/(x_k^m, \dots, x_{k+q-1}^m) \qtn x_{k+q}$
and that $x_{s+1}$,~\dots, $x_d$ is a u.s.d-sequence
on~$A/ (x_k^m, \dots, x_{k+q-1}^m) \qtn x_{k+q} + (x_{k+q}^l)$
for any $l>0$:
see \cite[Proposition 2.2]{Huneke:82:theory}.
Therefore, by Proposition~\ref{prop:3.6},
\begin{equation} \label{eqn:4.6.2}
\text{
$
H_{(x_{k+q}, x_{s+1}, \dots, x_d)R + R_+}^p
(\operatorname{Coker} \alpha_{k+q, m})
$
is concentrated in degree~$1-p$}
\end{equation}
if $p<d-s$.
Hence
$
H_{(x_{k+q}, x_{s+1}, \dots, x_d)R + R_+}^p
(\operatorname{Coker} \alpha_{s+1}^q)
$
is also.
By the spectral sequence
$
E_2^{pq} =
H_{\frak M}^p H_{(x_{k+q}, x_{s+1}, \dots, x_d)R + R_+}^q (-)
\Rightarrow
H_{\frak M}^{p+q}(-)
$,
we obtain the second assertion.
\end{pf}
Next we compute $H_{(x_k, \dots, x_{s+1})}^{s-k+2}(R)$.
\begin{thm}
Let $A_m = A/(x_k^m, \dots, x_s^m)$ and
${\frak q}_m = {\frak q} A_m$
for any positive integer~$m$.
Then
$$
H_{(x_k, \dots, x_{s+1})}^{s-k+2}(R) =
\mathop{\varinjlim}_{m,l} A_m[{\frak q}_m t]/ x_{s+1}^l A_m[{\frak q}_m t].
$$
In particular,
$H_{\frak M}^p H_{(x_k, \dots, x_{s+1})}^{s-k+2}(R)$ is
finitely graded for~$p<d-s$.
\end{thm}
\begin{pf}
We consider the exact sequence
\begin{multline*}
H_{(x_k, \dots, x_s)}^{s-k}(A[t]) @>\alpha_s^{s-k}>>
H_{(x_k, \dots, x_s)}^{s-k}(C) @>>>
H_{(x_k, \dots, x_s)}^{s-k+1}(R) @>>>
\\
@>>>
H_{(x_k, \dots, x_s)}^{s-k+1}(A[t]) @>\beta>>
H_{(x_k, \dots, x_s)}^{s-k+1}(C).
\end{multline*}
Since $\beta$ is the direct limit of
$$
A[t]/(x_k^m, \dots, x_s^m) A[t] @>>>
C/ (x_k^m, \dots, x_s^m) C,
$$
we have $\ker \beta = \mathop{\varinjlim}_m A_m [{\frak q}_m t]$.
Taking local cohomology modules of a short exact sequence
$$
0 @>>>
\operatorname{Coker} \alpha_s^{s-k} @>>>
H_{(x_k, \dots, x_s)}^{s-k+1}(R) @>>>
\ker \beta @>>>
0
$$
with respect to $(x_{s+1})$,
we obtain
\begin{equation} \label{eqn:4.7.1}
H_{(x_{s+1})}^1 H_{(x_k, \dots, x_s)}^{s-k+1}(R) =
H_{(x_{s+1})}^1 (\ker \beta),
\end{equation}
because $\operatorname{Coker} \alpha_s^{s-k} = \operatorname{Coker} \alpha_{s+1}^{s-k}$
is annihilated by~$x_{s+1}$.
The left hand side of~\eqref{eqn:4.7.1} coincides
with~$H_{(x_k, \dots, x_{s+1})}^{s-k+2}(R)$
by Lemma~\ref{lem:2.2}.
Thus the first assertion is proved.
Since $x_{s+1}$,~\dots, $x_d$ is a u.s.d-sequence on~$A_m$,
$H_{(x_{s+1}, \dots, x_d)R + R_+}^p(A_m [{\frak q}_m t])$ is
concentrated in degree $0 \geq n \geq s-d+2$
if $p \leq d-s$:
see \cite[Theorem 4.1]{Goto-Yamagishi::theory}.
From the exact sequence
$$
0 @>>>
0 \qtn_{A_m} x_{s+1} @>>>
A_m [{\frak q}_m t] @>x_{s+1}^l>>
A_m [{\frak q}_m t] @>>>
A_m[{\frak q}_m t] / x_{s+1}^l A_m [{\frak q}_m t] @>>> 0
$$
and the spectral sequence
$
E_2^{pq} =
H_{\frak M}^p H_{(x_{s+1}, \dots, x_d)R + R_+}^q(-)
\Rightarrow
H_{\frak M}^{p+q}(-)
$,
we find that
\begin{equation} \label{eqn:4.7.2}
\text{
$H_{\frak M}^p(A_m[{\frak q}_m t]/ x_{s+1}^l A_m [{\frak q}_m t])$
is concentrated in degree $0 \geq n \geq s-d+2$}
\end{equation}
if $p< d-s$.
Taking the direct limit of it,
we obtain the second assertion.
\end{pf}
Finally we compute
local cohomology modules of
$B = A[{\frak q}/ x_{s+1}] = R\hlz{x_{s+1}t}$.
\begin{thm} \label{thm:4.8}
Let ${\frak n}$ be a maximal ideal of~$B$.
Then
$$
H_{\frak n}^p H_{(x_k, \dots, x_{s+1})}^q(B) = 0
\quad
\text{if $q=0$ or $p<d-s-1$.}
$$
Furthermore
$(x_{k+q-1}, \dots, x_{s+1}) H_{(x_k, \dots, x_{s+1})}^q(B) =0$
for~$q < s-k+2$.
\end{thm}
\begin{pf}
Since the blowing-up $\operatorname{Proj} R \to \operatorname{Spec} A$ is a closed map,
there exists a homogeneous prime ideal~${\frak p}$ of~$R$
such that $x_{s+1}t \notin {\frak p}$,
$\dim R/ {\frak p} = 1$ and
${\frak n} = [{\frak p} R_{x_{s+1}t}]_0$.
Since $x_{s+1}$ is $B$-regular,
$H_{(x_k, \dots, x_{s+1})}^0(B) =0$.
Let $1 \leq q \leq s-k+1$.
By applying Lemma~\ref{lem:2.5}
to~\eqref{eqn:4.6.2},
we obtain
$$
H_{\frak n}^p((\operatorname{Coker} \alpha_{k+q-1,m})\hlz{x_{s+1}t}) =0
\quad
\text{for $p<d-s-1$}.
$$
By taking the direct limit of it
and using~\eqref{eqn:4.5.1},
we have
$$
H_{\frak n}^p H_{(x_k, \dots, x_{s+1})}^q(B) = 0
\quad
\text{for $p<d-s-1$}.
$$
Moreover Theorem~\ref{thm:4.6}
also assures us
$
(x_{k+q-1}, \dots, x_{s+1}) H_{(x_k, \dots, x_{s+1})}^q(B) =0
$.
Next we consider $H_{(x_k, \dots, x_{s+1})}^{s-k+2}(B)$.
By applying Lemma~\ref{lem:2.5} to~\eqref{eqn:4.7.2} and
by taking direct limit,
we have
$$
H_{\frak n}^p H_{(x_k, \dots, x_{s+1})}^{s-k+2}(B) = 0
\quad
\text{for $p<d-s-1$}.
$$
Thus the proof is completed.
\end{pf}
\section{Macaulayfications of local rings}
\label{sec:5}
In this section,
we shall construct a Macaulayfication of
the affine scheme~$\operatorname{Spec} A$
if its non-Cohen-Macaulay locus is of dimension~$2$.
Assume that $A$ possesses a dualizing complex
and $\dim A/ {\frak p} = d$
for any associated prime ideal~${\frak p}$ of~$A$.
Then $V({\frak a}_A(A))$ coincides with
the non-Cohen-Macaulay locus of~$A$.
We fix an integer $s \geq \dim A/ {\frak a}_A(A)$
and
let $x_1$,~\dots, $x_d$ be a system of parameters for~$A$
satisfying~\eqref{eqn:4.0.1}.
First we refine Faltings' results~%
\cite[Satz 2, 3]{Faltings:78:Macaulay}.
Let ${\frak q} = (x_{s+1}, \dots, x_d)$,
$R = A[{\frak q} t]$ and
$X = \operatorname{Proj} R$.
\begin{thm} \label{thm:5.1}
With notation as above,
$$
\operatorname{depth} {\cal O}_{X,p} \geq d-s
\quad
\text{for any closed point~$p$ of~$X$.}
$$
If $s=0$ or $A/ {\frak q}$ is Cohen-Macaulay,
then $X$ is Cohen-Macaulay.
\end{thm}
\begin{pf}
Since $x_{s+1}$,~\dots, $x_d$ is a u.s.d-sequence on~$A$,
$H_{(x_{s+1}, \dots, x_d)R + R_+}^p(R)$
is finitely graded for $p \leq d-s$:
see \cite[Theorem 4.1]{Goto-Yamagishi::theory}.
By using Lemma~\ref{lem:2.5},
we obtain the first assertion.
Furthermore
since $\dim {\cal O}_{X,p} =d$ for any closed point~$p$ of~$X$,
$X$ is Cohen-Macaulay if $s=0$.
Assume that $s>0$ and $A / {\frak q}$ is Cohen-Macaulay.
Then $x_1$,~\dots, $x_s$ is a regular sequence on~$A/ {\frak q}$.
We use theorems in Section~\ref{sec:4}
as $k=1$.
From~\eqref{eqn:4.6.1},
we find that $\operatorname{Coker} \alpha_{s+1}^q =0$ for all~$q \leq s-1$.
That is,
$H_{\frak M}^p H_{(x_1, \dots, x_{s+1})}^q(R)$ is
finitely graded
if $p<d-s$ or $q<s+1$.
By the spectral sequence~%
$
E_2^{pq} =
H_{\frak M}^p H_{(x_1, \dots, x_{s+1})}^q(-) \Rightarrow
H_{\frak M}^{p+q} (-)
$,
we find that $H_{\frak M}^p(R)$ is
finitely graded for $p<d+1$.
Lemma~\ref{lem:2.5} assures us
$$
\operatorname{depth} {\cal O}_{X,p} \geq d
\quad
\text{for any closed point~$p$ of~$X$}.
$$
The proof is completed.
\end{pf}
From now on,
we assume that $s > 0$.
Since $x_s$ is $A$-regular,
${\frak q}$ is a reduction of~$\trans{\frak q} = {\frak q} \qtn x_s$
by~\eqref{eqn:3.2.1}.
We put $\trans R = A[\trans{\frak q} t]$ and
$\trans X = \operatorname{Proj} \trans R$.
Then $\trans X \rightarrow X$ is a finite morphism.
\begin{thm}
With notation as above,
$$
\operatorname{depth} {\cal O}_{\trans X, \trans p} \geq d-s+1
\quad
\text{for any closed point~$\trans p$ of~$\trans X$}
$$
In particular, if $s=1$,
then $\trans X$ is Cohen-Macaulay.
\end{thm}
\begin{pf}
By Corollary~\ref{cor:3.5},
$H_{(x_s, \dots, x_d)R + R_+}^p(\trans R)$ is finitely graded
for $p \leq d-s+1$.
By using Lemma~\ref{lem:2.5},
we obtain the assertion.
\end{pf}
Next we consider an ideal~%
$
{\frak b} =
{\frak q}^2 + x_s {\frak q} = (x_s, \dots, x_d) {\frak q}
$.
We put $S = A[{\frak b} t]$ and $Y = \operatorname{Proj} S$.
Then $Y$ is the blowing-up of~$X$
with center~$(x_s, \dots, x_d) {\cal O}_X$.
\begin{thm} \label{thm:5.3}
With notation as above,
$$
\operatorname{depth} {\cal O}_{Y,q} \geq d-s+1
\quad
\text{for any closed point~$q$ of~$Y$}.
$$
Furthermore,
if $s=1$ or $A$ is Cohen-Macaulay,
then $Y$ is Cohen-Macaulay.
\end{thm}
\begin{pf}
Since
$
(x_s x_{s+1}, \dots, x_s x_d, x_{s+1}^2, \dots, x_d^2)
{\frak b}^{d-s-1}
= {\frak b}^{d-s}
$,
we have only to compute the depth of
$C_0 = A[{\frak b}/ x_s x_{s+1}]$ and $C_1 = A[{\frak b}/ x_{s+1}^2]$.
If we put $B = A[{\frak q}/ x_{s+1}]$,
then
\begin{align*}
C_0 & = B[x_{s+1}/ x_s]
\cong B[T]/(x_sT - x_{s+1}) \qtn \angled{x_s},
\\
C_1 & = B[x_s/ x_{s+1}]
\cong B[T]/(x_{s+1}T - x_s) \qtn \angled{x_{s+1}},
\end{align*}
where $T$ denotes an indeterminate.
We note that $B$, $C_0$, $C_1$ are
subrings of the total quotient ring of~$A$
because $x_1$,~\dots, $x_d$ are $A$-regular elements.
First we consider $C_0$.
We regard it as a homomorphic image of~$B[T]$.
Let ${\frak l}_0$ be a maximal ideal of~$C_0$ and
${\frak n} = {\frak l}_0 \cap B$.
Then ${\frak n}$ is a maximal ideal of~$B$
because $\operatorname{Spec} C_0 \cup \operatorname{Spec} C_1 \rightarrow \operatorname{Spec} B$ is
a blowing-up with center $(x_s, x_{s+1})B$,
hence a closed map.
There exists a polynomial~$f$ over~$B$
such that ${\frak l}_0 = {\frak n} C_0 + f C_0$
and the leading coefficient of~$f$ is not contained in~${\frak n}$.
By Lemma~\ref{lem:2.2} and Theorem~\ref{thm:4.8},
we have, for any $1 \leq k \leq s$,
\begin{equation} \label{eqn:5.3.1}
H_{{\frak n} B[T] + fB[T]}^p
H_{(x_k, \dots, x_{s+1})}^q(B[T]) = 0
\quad
\text{if $p<d-s$ or $q=0$.}
\end{equation}
In fact,
the leading coefficient of~$f$ is a regular element
on $H_{\frak n}^{d-s} H_{(x_k, \dots, x_{s+1})}^q(B[T])$
because it acts on
the injective envelope of~$B/ {\frak n}$
as isomorphism.
Taking the local cohomology of
a short exact sequence
$$
0 @>>>
B[T] @>x_sT - x_{s+1}>>
B[T] @>>>
B[T]/ (x_sT - x_{s+1}) @>>> 0
$$
with respect to~%
$
(x_k, \dots, x_{s+1})
= (x_k, \dots, x_s, x_sT - x_{s+1})
$,
we obtain an exact sequence
\begin{multline*}
0 @>>>
H_{(x_k, \dots, x_{s+1})}^{s-k+1}(B[T]) @>>>
H_{(x_k, \dots, x_{s+1})}^{s-k+1}(B[T]/(x_sT - x_{s+1})) @>>>
\\
@>>>
H_{(x_k, \dots, x_{s+1})}^{s-k+2}(B[T]) @>>>
H_{(x_k, \dots, x_{s+1})}^{s-k+2}(B[T]) @>>> 0,
\end{multline*}
because
$(x_s, x_{s+1}) H_{(x_k, \dots, x_{s+1})}^{s-k+1}(B)=0$
by Theorem~\ref{thm:4.8}.
This and \eqref{eqn:5.3.1} show that
$$
H_{{\frak n} B[T] + fB[T]}^p H_{(x_k, \dots, x_{s+1})}^{s-k+1}
(B[T]/ (x_s T - x_{s+1})) = 0
\quad
\text{for $p<d-s$}.
$$
Taking the local cohomology of an exact sequence
$$
0 @>>>
\frac{(x_sT - x_{s+1}) \qtn \angled{x_s}}{(x_s T - x_{s+1})}
@>>>
B[T]/ (x_sT - x_{s+1}) @>>>
C_0 @>>> 0
$$
with respect to $(x_k, \dots, x_{s+1})$,
we obtain
$$
H_{(x_k, \dots, x_{s+1})}^{s-k+1}(C_0) =
H_{(x_k, \dots, x_{s+1})}^{s-k+1}(B[T]/(x_s T - x_{s+1})),
$$
that is,
\begin{equation} \label{eqn:5.3.2}
H_{{\frak l}_0}^p H_{(x_k, \dots, x_{s+1})}^{s-k+1}(C_0) = 0
\quad
\text{for $p<d-s$}.
\end{equation}
We note that $x_s$ is $C_0$-regular.
Put $k=s$.
Then we have
$$
H_{{\frak l}_0}^p H_{(x_s, x_{s+1})}^q(C_0) = 0
\quad
\text{if $p<d-s$ or $q<1$}.
$$
By the spectral sequence
$
E_2^{pq} =
H_{{\frak l}_0}^p H_{(x_s, x_{s+1})}^q(-) \Rightarrow
H_{{\frak l}_0}^{p+q}(-),
$
we obtain
\begin{equation} \label{eqn:5.3.3}
H_{{\frak l}_0}^p(C_0) = 0
\quad
\text{for $p<d-s+1$,}
\end{equation}
that is,
$\operatorname{depth} (C_0)_{{\frak l}_0} \geq d-s+1$.
In the same way,
we can show that
$\operatorname{depth} (C_1)_{{\frak l}_1} \geq d-s+1$
for any maximal ideal~${\frak l}_1$
of~$C_1$.
Thus the first assertion is proved.
In particular,
$Y$ is Cohen-Macaulay if $s=1$.
Assume that $A$ is Cohen-Macaulay.
Using \cite[Lemma 1]{Faltings:78:Macaulay} twice,
we find that
$$
x_{s+1} T_{s+2} - x_{s+2},
\dots,
x_{s+1} T_d - x_d,
x_s T_{s+1} - x_{s+1}
$$
is a regular sequence on~$A[T_{s+1}, \dots, T_d]$.
Therefore
$$
C_0 \cong
A[T_{s+1}, \dots, T_d] /
(x_{s+1} T_{s+2} - x_{s+2}, \dots, x_{s+1} T_d - x_d,
x_s T_{s+1} - x_{s+1})
$$
is Cohen-Macaulay.
In the same way, we can show that $C_1$ is Cohen-Macaulay.
The proof is completed.
\end{pf}
In the rest of this section,
we assume that $s \geq 2$ and
let $\trans {\frak b} = {\frak b} \qtn \angled{x_{s-1}}$.
\begin{lem} \label{lem:5.4}
For any positive integer~$n$,
$$
\trans {\frak b}^n =
{\frak b}^n \qtn \angled{x_{s-1}} =
{\frak q} {\frak b}^{n-1} [(x_s, \dots, x_d) \qtn x_{s-1}]
+ x_s^n {\frak q}^{n-1} [{\frak q} \qtn x_{s-1}].
$$
In particular, $\trans {\frak b}^2 = {\frak b} \trans {\frak b}$.
\end{lem}
\begin{pf}
It is sufficient to prove
$$
{\frak b}^n \qtn \angled{x_{s-1}} \subseteq
{\frak q} {\frak b}^{n-1} [(x_s, \dots, x_d) \qtn x_{s-1}]
+ x_s^n {\frak q}^{n-1} [{\frak q} \qtn x_{s-1}].
$$
Take $a \in {\frak b}^n \qtn \angled{x_{s-1}}$.
Then, by Lemma~\ref{lem:2.4}, Lemma~\ref{lem:3.2}
and Theorem~\ref{thm:4.2},
we have
\begin{align*}
a & \in (x_s, \dots, x_d)^{2n} \qtn \angled{x_{s-1}}
\\
& = (x_s, \dots, x_d)^{2n-1} [(x_s, \dots, x_d) \qtn x_{s-1}]
\\
& =
[{\frak q}^{2n-1} + x_s {\frak q}^{2n-2} + \dots + (x_s^{2n-1})]
[(x_s, \dots, x_d) \qtn x_{s-1}]
\\
& \subseteq {\frak q} {\frak b}^{n-1} [(x_s, \dots, x_d) \qtn x_{s-1}]
+ (x_s^n).
\end{align*}
If we put $a = b + x_s^n a'$
where
$b \in {\frak q} {\frak b}^{n-1} [(x_s, \dots ,x_d) \qtn x_{s-1}]$,
then $x_s^n a' \in {\frak b}^n \qtn \angled{x_{s-1}}$.
Since
$x_{s-1}^l x_s^n a' \in {\frak b}^n$
for a sufficiently large~$l$,
we can put $x_{s-1}^l x_s^n a' = c + x_s^n d$
where $c \in {\frak q}^{2n} + \dots + x_s^{n-1} {\frak q}^{n+1}$
and $d \in {\frak q}^n$.
Then
$
x_{s-1}^l a' - d \in
{\frak q}^{n+1} \qtn \angled{x_s} =
{\frak q}^n [{\frak q} \qtn x_s]
$.
Hence,
$x_{s-1}^l a' \in {\frak q}^n$ and
$
a' \in
{\frak q}^n \qtn \angled{x_{s-1}} =
{\frak q}^{n-1} [{\frak q} \qtn x_{s-1}]
$.
The proof is completed.
\end{pf}
Therefore the Rees algebra~$\trans S = A[\trans {\frak b} t]$
is finitely generated over~$S$.
Let $\trans Y = \operatorname{Proj} \trans S$.
\begin{prop} \label{prop:5.5}
$D_{(x_{s-1}, x_s, x_{s+1})}^0(S_+) = \trans S_+$.
\end{prop}
\begin{pf}
First show that $x_{s-1}$, $x_s$ is
an $\trans S_+$-regular sequence.
Let $n > 0$.
It is clear that $x_{s-1}$ is $\trans {\frak b}^n$-regular
because it is $A$-regular.
Let $a \in (x_{s-1} \trans {\frak b}^n \qtn x_s) \cap \trans {\frak b}^n$.
Then $x_{s-1}^l a \in {\frak b}^n$
for a sufficiently large~$l$.
Since $x_s a \in (x_{s-1})$ and
$x_s$,~\dots, $x_d$ is a d-sequence on~$A/ x_{s-1}^{l+1}A$,
\begin{align*}
x_{s-1}^l a & \in (x_{s-1}^{l+1}) \qtn x_s \cap {\frak b}^n
\\
& \subseteq (x_{s-1}^{l+1}) \qtn x_s \cap
(x_{s-1}^{l+1}, x_s, \dots, x_d)
\\
& = (x_{s-1}^{l+1}).
\end{align*}
Hence $a \in (x_{s-1})$.
If we put $a = x_{s-1} a'$,
then
$
a' \in {\frak b}^n \qtn x_{s-1}^{l+1} \subseteq \trans {\frak b}^n
$,
that is, $a \in x_{s+1}\trans {\frak b}^n$.
Thus we have proved that $x_s$ is
$\trans S_+ / x_{s-1} \trans S_+$-regular.
By~\eqref{eqn:2.1.1},
we have
\begin{equation} \label{eqn:5.5.1}
D_{(x_{s-1}, x_s, x_{s+1})}^0(S_+) \subseteq
D_{(x_{s-1}, x_s, x_{s+1})}^0(\trans S_+) = \trans S_+.
\end{equation}
Since ${\frak q} \qtn x_{s-1} \subseteq {\frak q} \qtn x_s$
by Theorem~\ref{thm:4.2},
$
(x_{s-1}, \dots, x_d) \trans {\frak b}^n
\subseteq {\frak b}^n
$
for all~$n>0$ by Lemma~\ref{lem:5.4},
that is,
$(x_{s-1}, \dots, x_d) \trans S_+ \subseteq S_+$.
We have shown the inverse inclusion of~\eqref{eqn:5.5.1}.
\end{pf}
The following theorem is one of main aims of this section.
\begin{thm}
With notation as above,
$$
\operatorname{depth} {\cal O}_{\trans Y, \trans q} \geq d-s+2
\quad
\text{for any closed point~$\trans q$ of~$\trans Y$}.
$$
In particular,
if $s=2$,
then $\trans Y$ is Cohen-Macaulay.
\end{thm}
\begin{pf}
We have only to compute the depth of
$$
\trans C_0 = A[\trans {\frak b}/x_s x_{s+1}]
\quad
\text{and}
\quad
\trans C_1 = A[\trans {\frak b}/ x_{s+1}^2].
$$
Proposition~\ref{prop:5.5} says that
$\trans C_i = D_{(x_{s-1}, x_s, x_{s+1})}^0(C_i)$
and it is a finitely generated $C_i$-module
for $i=0$,~$1$.
Let $\trans{\frak l}_i$ be a maximal ideal of~$\trans C_i$
and ${\frak l}_i = \trans{\frak l}_i \cap C_i$.
Then ${\frak l}_i$ is a maximal ideal of~$C_i$
because $\trans C_i$ is integral over~$C_i$.
We use \eqref{eqn:5.3.2} as $k=s-1$,
that is,
\begin{equation} \label{eqn:5.6.1}
H_{{\frak l}_i}^p H_{(x_{s-1}, x_s, x_{s+1})}^2(C_i) =0
\quad
\text{for $p<d-s$}.
\end{equation}
By using~\eqref{eqn:2.1.2},
we obtain
$$
H_{{\frak l}_i}^p
H_{(x_{s-1}, x_s, x_{s+1})}^q(\trans C_i) = 0
\quad
\text{if $p<d-s$ or $q<2$}.
$$
By the spectral sequence
$
E_2^{pq} =
H_{{\frak l}_i}^p H_{(x_{s-1}, x_s, x_{s+1})}^q(-)
\Rightarrow
H_{{\frak l}_i}^{p+q}(-)
$,
we find
\begin{equation} \label{eqn:5.6.2}
H_{{\frak l}_i}^p(\trans C_i) = 0
\quad
\text{for $p<d-s+2$,}
\end{equation}
that is,
$\operatorname{depth} (\trans C_i)_{\trans{\frak l}_i} \geq d-s+2$.
Thus the proof is completed.
\end{pf}
The following corollary shall be used in the next section.
\begin{cor} \label{cor:5.7}
If $A/(x_s, \dots, x_d)$ is Cohen-Macaulay, then
$$
\operatorname{depth} {\cal O}_{Y,q} \geq d-s+2
\quad
\text{for any closed point~$q$ of~$Y$.}
$$
\end{cor}
\begin{pf}
It is sufficient to prove $\trans {\frak b}={\frak b}$.
Let $a \in \trans {\frak b}$ and
$l$ be an integer such that $x_{s-1}^l a \in {\frak b}$.
Then we have
\begin{align*}
a & \in (x_s, \dots, x_d)^2 \qtn x_{s-1}^l
\\
& = (x_s, \dots, x_d) [(x_s, \dots, x_d) \qtn x_{s-1}^l]
\\
& = (x_s, \dots, x_d)^2 = {\frak b} + (x_s^2)
\end{align*}
by Lemma~\ref{lem:3.2}.
Hence,
we may assume that $a \in (x_s^2)$.
Let $a = x_s^2 a'$.
Since $x_{s-1}^l a \in {\frak b} \subseteq {\frak q}$,
$a' \in {\frak q} \qtn x_{s-1}^l x_s^2 = {\frak q} \qtn x_s$
by Theorem~\ref{thm:4.2}.
Hence $a = x_s^2 a' \in x_s {\frak q} \subset {\frak b}$.
\end{pf}
We shall give another Macaulayfication of~$\operatorname{Spec} A$
by considering an ideal
${\frak c} = (x_{s-1}, \dots, x_d) {\frak b}$.
Let $Z = \operatorname{Proj} A[{\frak c} t]$,
which is the blowing-up of~$Y$
with center $(x_{s-1}, \dots, x_d) {\cal O}_Y$.
\begin{thm} \label{thm:5.8}
With notation as above,
$$
\operatorname{depth} {\cal O}_{Z,r} \geq d-s+2
\quad
\text{for any closed point~$r$ of~$Z$}.
$$
Furthermore,
if $s=2$ or $A$ is Cohen-Macaulay,
then $Z$ is Cohen-Macaulay.
\end{thm}
\begin{pf}
Since
$
(x_{s-1}x_s, x_s^2) {\frak q} +
x_{s-1} (x_{s+1}^2, \dots, x_d^2) +
(x_{s+1}^3, \dots, x_d^3)
$
is a reduction of~${\frak c}$,
we have only to compute the depth of
\begin{align*}
D_0 & = A[{\frak c}/ x_{s-1} x_s x_{s+1}] = C_0[x_s/ x_{s-1}],
\\
D_1 & = A[{\frak c}/ x_s^2 x_{s+1}] = C_0[x_{s-1}/ x_s],
\\
D_2 & = A[{\frak c}/ x_{s-1} x_{s+1}^2] = C_1[x_{s+1}/ x_{s-1}],
\\
\intertext{and}
D_3 & = A[{\frak c}/ x_{s+1}^3] = C_1[x_{s-1}/ x_{s+1}].
\end{align*}
For $i=0$ or~$1$,
let ${\frak l}_i$ be a maximal ideal of~$C_i$.
By~\eqref{eqn:2.1.1},
there exists an exact sequence
$$
0 @>>>
C_i @>>>
\trans C_i @>>>
H_{(x_{s-1}, x_s, x_{s+1})}^1(C_i) @>>> 0.
$$
By using~\eqref{eqn:5.3.3}
and \eqref{eqn:5.6.2},
we obtain
$$
H_{{\frak l}_i}^p
H_{(x_{s-1}, x_s, x_{s+1})}^1(C_i) = 0
\quad
\text{for all $p<d-s$.}
$$
Furthermore,
$(x_{s-1}, \dots, x_d) \trans C_i \subseteq C_i$:
see the proof of Proposition~\ref{prop:5.5}.
Therefore, by~\eqref{eqn:5.6.1}, we have
\begin{equation} \label{eqn:5.8.1}
H_{{\frak l}_i}^p H_{(x_{s-1}, x_s, x_{s+1})}^q(C_i) = 0
\quad
\text{if $p<d-s$ or $q = 0$}
\end{equation}
and
\begin{equation}
\label{eqn:5.8.2}
(x_{s-1}, \dots, x_d)
H_{(x_{s-1}, x_s, x_{s+1})}^1(C_i) =0.
\end{equation}
Therefore we can prove
$$
\operatorname{depth} (D_i)_{{\frak r}_i} \geq d-s+2
$$
for any maximal ideal~${\frak r}_i$
of~$D_i$ and $i=0$,~\dots, $3$
in the same way as Theorem~\ref{thm:5.3}.
To make sure,
we compute the depth of~%
$
D_0 \cong C_0[T]/(x_{s-1}T- x_s) \qtn \angled{x_{s-1}}
$.
First we note that $x_{s+1} \in x_s C_0$
and $x_{s+1} \in x_s D_0$.
Let ${\frak r}_0$ be a maximal ideal of~$D_0$
and ${\frak l}_0 = {\frak r}_0 \cap C_0$.
Then ${\frak l}_0$ is a maximal ideal of~$C_0$
and there exists a polynomial~$f$
over~$C_0$
such that ${\frak r}_0 = {\frak l}_0 D_0 + f D_0$
and the leading coefficient of~$f$
is not contained in~${\frak l}_0$.
We obtain
$$
H_{{\frak l}_0 C_0[T] + f C_0[T]}^p
H_{(x_{s-1}, x_s)}^q (C_0[T]) = 0
\quad
\text{if $p<d-s+1$ or $q = 0$}
$$
from~\eqref{eqn:5.8.1}.
Taking the local cohomology of an exact sequence
$$
0 @>>>
C_0[T] @>x_{s-1}T - x_s>>
C_0[T] @>>>
C_0[T]/(x_{s-1}T - x_s) @>>>
0,
$$
we have an exact sequence
\begin{multline*}
0 @>>>
H_{(x_{s-1}, x_s)}^1(C_0[T]) @>>>
H_{(x_{s-1}, x_s)}^1(C_0[T]/ (x_{s-1}T - x_s)) @>>>
\\
@>>>
H_{(x_{s-1}, x_s)}^2(C_0[T]) @>>>
H_{(x_{s-1}, x_s)}^2(C_0[T]) @>>>
0
\end{multline*}
because of~\eqref{eqn:5.8.2}.
This says that
$$
H_{{\frak l}_0 C_0[T] + f C_0[T]}^p
H_{(x_{s-1}, x_s)}^1
(C_0[T]/ (x_{s-1}T - x_s)) = 0
\quad
\text{for $p<d-s+1$}.
$$
Taking the local cohomology of an exact sequence
$$
0 @>>>
\frac{(x_{s-1}T-x_s) \qtn \angled{x_{s-1}}}
{(x_{s-1}T - x_s)} @>>>
C_0[T]/(x_{s-1}T - x_s) @>>>
D_0 @>>>
0
$$
with respect to~$(x_{s-1}, x_s)$,
we obtain
$$
H_{{\frak r}_0}^p H_{(x_{s-1}, x_s)}^1(D_0) = 0
\quad
\text{for $p<d-s+1$}.
$$
Of course,
$H_{(x_{s-1}, x_s)}^0(D_0) = 0$.
By the spectral sequence
$$
E_2^{pq} =
H_{{\frak r}_0}^p H_{(x_{s-1}, x_s)}^q(-) \Rightarrow
H_{{\frak r}_0}^{p+q}(-),
$$
we get $H_{{\frak r}_0}^p(D_0) = 0$
for any $p<d-s+2$.
That is, $\operatorname{depth} (D_0)_{{\frak r}_0} \geq d-s+2$.
The last assertion is also proved
in the same way as Theorem~\ref{thm:5.3}.
\end{pf}
\setcounter{equation}{0}
\section{The proof of Theorem~\ref{mthm}}
\label{sec:6}
This section is devoted to the proof of Theorem~\ref{mthm}.
Let $A$ be a Noetherian ring possessing a dualizing complex
and $X$ a quasi-projective scheme over~$A$.
That is, $X$ is a dense open subscheme
of $X\closure = \operatorname{Proj} R$
where $R = \bigoplus_{n \geq 0} R_n$ is a Noetherian graded ring
such that $R_0$ is a homomorphic image of~$A$ and
$R$ is generated by~$R_1$ as an $R_0$-algebra.
Let $V\closure$ be the non-Cohen-Macaulay locus of~$X\closure$
and $U\closure = X\closure \setminus V\closure$.
Of course $V = V\closure \cap X$ is
the non-Cohen-Macaulay locus of~$X$.
Let $\dc$ be a dualizing complex of~$R$
with codimension function~$v$.
Assume that $X$ satisfies the assumption of Theorem~\ref{mthm}.
Without loss of generality,
we may assume that
\begin{equation} \label{eqn:6.0.1}
v({\frak p}) = 0
\quad
\text{for all associated prime ideal~${\frak p}$ of~$R$:}
\end{equation}
see \cite[p.~191]{Faltings:78:Macaulay}.
Then the local ring~${\cal O}_{X,p}$ of~$p \in X$ satisfies
the assumption of Section~\ref{sec:5},
that is,
$\dim {\cal O}_{X,p}/ {\frak p} = \dim {\cal O}_{X,p}$
for any associated prime ideal~${\frak p}$ of~${\cal O}_{X,p}$.
For the sake of completeness,
we sketch out the proof.
Let ${\frak a}$ be a homogeneous ideal of~$R$
such that $V\closure = V({\frak a})$.
Then the closed immersion $\operatorname{Proj} R/H_{\frak a}^0(R) \to X\closure$
is birational as follows.
For any minimal prime ideal~${\frak p}$ of~$R$,
${\frak a} \not\subset {\frak p}$ and
$H_{\frak a}^0(R) \subseteq {\frak p}$
because $R_{\frak p}$ is Cohen-Macaulay.
Hence the underlying set of~$\operatorname{Proj} R/H_{\frak a}^0(R)$
coincides with the one of~$X\closure$.
Furthermore,
$f^{-1}(U\closure) \to U\closure$ is an isomorphism and
$U\closure$ is dense in~$X\closure$.
By replacing $R$ by~$R/H_{\frak a}^0(R)$,
we may assume that
\begin{equation} \label{eqn:6.0.2}
\text{
every associated prime ideal of~$R$
is minimal.
}
\end{equation}
Next we fix a primary decomposition of~$(0)$ in~$R$.
For all integer~$i$,
let ${\frak q}_i$ be the intersection of
all primary component~${\frak q}$ of~$(0)$
such that $v(\sqrt{\frak q}) = i$.
Then $g \colon \coprod_i \operatorname{Proj} R/{\frak q}_i \to X\closure$
is a finite morphism
and $g^{-1}(U\closure) \to U\closure$ is an isomorphism
as follows.
Note that ${\frak q}_i = R$ for all but finitely many~$i$.
Furthermore, for any ${\frak p} \in U\closure$,
${\frak p} \supseteq {\frak q}_i$ if and only if
$v({\frak p}) - \dim R_{\frak p} = i$
because $R_{\frak p}$ is Cohen-Macaulay, hence equidimensional.
Therefore $U\closure$ is
the disjoint union of $U\closure \cap V({\frak q}_i)$.
Moreover $R\hlz{\frak p} = [R/{\frak q}_i]\hlz{\frak p}$
if ${\frak p} \in U\closure \cap V({\frak q}_i)$.
Because of~\eqref{eqn:6.0.2},
$g^{-1}(U\closure)$ and $U\closure$ are dense
in $\operatorname{Proj} R/{\frak q}_i$ and $X\closure$,
respectively.
Thus $g^{-1}(X) \to X$ is birational proper
and the connected components of~$g^{-1}(X)$
satisfy the assumption of Theorem~\ref{mthm}.
Since $u$ is locally constant,
$V_i = u^{-1}(i) \cap V$ is
closed for any positive integer~$i$.
We put $s_i = \dim V_i$.
By~\eqref{eqn:6.0.1},
we find that $V_1 = \emptyset$, $s_2 \leq 0$ and $s_3 \leq 1$.
Let $d$ be the largest integer such that $V_d \ne \emptyset$
and $s = s_d$.
We shall give a closed subscheme~$W$ of~$X$
such that $V_d = V \cap W$ and
${\cal O}_{Y,q}$ is Cohen-Macaulay
for all $q \in \pi^{-1}(W)$
where $\pi \colon Y \to X$ is the blowing-up
of~$X$ with center~$W$.
Let ${\frak a} = \prod_{i>0} \operatorname{ann} H^i(\dc)$,
which is finite product.
Then it is obvious that $V\closure = V({\frak a})$.
Fix a primary decomposition of~${\frak a}$
and let ${\frak a}_d$ be the intersection of
all primary component~${\frak q}$ of~${\frak a}$
such that $\sqrt{\frak q} \in V_d$.
Then we can take homogeneous elements $z_1$,~\dots, $z_d \in R$
such that
\begin{gather} \label{eqn:6.0.3}
V_i \cap V((z_{d-s_i}, \dots, z_d)) = \emptyset
\quad \text{for $i < d$}
\\
d({\frak p}) = d
\quad
\text{for all minimal prime ideal~${\frak p}$
of $R/(z_1, \dots, z_d) \qtn \angled{R_+}$}
\\
\begin{cases}
z_{s+1}, \dots, z_d \in {\frak a}_d;
\\
\label{eqn:6.0.5}
z_i \in
\prod_{j>d-i} \operatorname{ann} H^j(\hom(R/(z_{i+1}, \dots, z_d), \dc)), &
\text{for $i \leq s$}
\end{cases}
\end{gather}
in the same way as Section~\ref{sec:4}.
We put
$$
{\frak b} =
\begin{cases}
(z_1, \dots, z_d), & \text{if $s=0$};
\\
(z_1, \dots, z_d) (z_2, \dots, z_d), & \text{if $s=1$};
\\
(z_1, \dots, z_d) (z_2, \dots, z_d) (z_3, \dots, z_d),
& \text{if $s=2$}
\end{cases}
$$
and prove that $W = V({\frak b}) \cap X$ satisfies
the required properties.
Because of~\eqref{eqn:6.0.3},
$V_i \cap W = \emptyset$ for~$i < d$.
Let $\pi \colon Y \to X$ be the blowing-up of~$X$
with center~$W$,
$q$ a closed point of~$\pi^{-1}(W)$ and
${\frak p} \subseteq R$ the image of~$q$.
Take an element $y \in R_1 \setminus {\frak p}$
and put $x_i = z_i / y^{\deg z_i}$ for all~$i$.
Since $(\dc)\hlz{\frak p}$ is a dualizing complex of~$R\hlz{\frak p}$,
we obtain
$$
\begin{cases}
x_{s+1}, \dots, x_d \in {\frak a}_{R\hlz{\frak p}}(R\hlz{\frak p});
\\
x_i \in {\frak a}_{R\hlz{\frak p}}(
R\hlz{\frak p}/(x_{i+1}, \dots, x_d)),
& \text{for $i \leq s$}.
\end{cases}
$$
from~\eqref{eqn:6.0.5}.
When $s=2$, there exist three cases:
If $z_1$, $z_2 \in {\frak p}$,
then $x_1$,~\dots, $x_d$ is a system of parameters
for~$R\hlz{\frak p}$ satisfying~\eqref{eqn:4.0.1}
or a regular sequence
on the Cohen-Macaulay ring~$R\hlz{\frak p}$.
Since
$
{\frak b}\hlz{\frak p} =
(x_1, \dots, x_d) (x_2, \dots, x_d) (x_3, \dots, x_d)
$,
${\cal O}_{Y,q}$ is Cohen-Macaulay by Theorem~\ref{thm:5.8}.
If $z_2 \in {\frak p}$ but $z_1 \notin {\frak p}$,
then $x_2$,~\dots, $x_d$ is a subsystem of parameters
for~$R\hlz{\frak p}$ satisfying~\eqref{eqn:4.0.1}
or a regular sequence
on the Cohen-Macaulay ring~$R\hlz{\frak p}$.
Furthermore
${\frak b}\hlz{\frak p} = (x_2, \dots, x_d)(x_2, \dots, x_d)$
and $R\hlz{\frak p}/(x_2, \dots, x_d)$ is Cohen-Macaulay
because
$x_1 \in {\frak a}_{R\hlz{\frak p}}(R\hlz{\frak p}/(x_2, \dots, x_d))$
is a unit.
Hence ${\cal O}_{Y,q}$ is Cohen-Macaulay by Corollary~\ref{cor:5.7}.
If $z_1$, $z_2 \notin {\frak p}$, then
$x_3$,~\dots, $x_d \in {\frak a}_{R\hlz{\frak p}}(R\hlz{\frak p})$
is a subsystem of parameters for~$R\hlz{\frak p}$
and $R\hlz{\frak p}/(x_3, \dots, x_d)$ is Cohen-Macaulay.
Since ${\frak b}\hlz{\frak p} = (x_3, \dots, x_d)$,
${\cal O}_{Y,q}$ is Cohen-Macaulay by Theorem~\ref{thm:5.1}.
When $s=0$ or $1$,
we can prove the assertion
in the same way as above.
By repeating this procedure,
we obtain a Macaulayfication of~$X$.
We complete the proof of Theorem~\ref{mthm}.
\subsection*{Acknowledgment}
The author is grateful
to Professor S.~Goto
for his helpful discussions
and to Professor K.~Kurano
for careful reading of the draft.
\ifx\undefined\bysame
\newcommand{\bysame}{%
\leavevmode\hbox to3em{\hrulefill}\,}
\fi
|
1996-07-19T01:09:21 | 9512 | alg-geom/9512001 | en | https://arxiv.org/abs/alg-geom/9512001 | [
"alg-geom",
"math.AG"
] | alg-geom/9512001 | Richard Hain | Richard Hain | Infinitesimal presentations of the Torelli groups | 55 pages, LaTeX2e, amsart.cls. Author supplied dvi file available
from http://www.math.duke.edu/faculty/hain/ | null | null | null | null | This is a significant revision of the early version of this paper which was
posted last December. The speculative section has been removed in light of some
recent results of Morita and Kawazumi. Numerous typos have been fixed. The
companion paper "The Hodge de Rham Theory of Relative Malcev Completion" has
just been posted.
| [
{
"version": "v1",
"created": "Fri, 1 Dec 1995 21:49:54 GMT"
},
{
"version": "v2",
"created": "Thu, 18 Jul 1996 23:03:13 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Hain",
"Richard",
""
]
] | alg-geom | \section{Introduction}
By a theorem of Malcev \cite{malcev}, every torsion free nilpotent
group can be imbedded canonically as a discrete, cocompact subgroup
of a real nilpotent Lie group. One can therefore associate to a
finitely generated group $\pi$ a tower of nilpotent Lie groups
\begin{equation}\label{tower}
\cdots \to G_3 \to G_2 \to G_1 = H_1(\pi,{\mathbb R})
\end{equation}
by taking $G_k$ to be the nilpotent Lie group associated to the
maximal torsion free quotient of $\pi$ of length $k$. Since each
nilpotent Lie group is simply connected, the tower (\ref{tower})
is determined by the corresponding tower
$$
\cdots \to {\mathfrak g}_3 \to {\mathfrak g}_2 \to {\mathfrak g}_1 = H_1(\pi,{\mathbb R})
$$
of nilpotent Lie algebras. The inverse limit ${\mathfrak g}$ of this
tower is a pronilpotent Lie algebra, called the
{\it Malcev Lie algebra associated to $\pi$.} This Lie algebra
has the property that the graded Lie algebra $\Gr {\mathfrak g}$
associated with its lower central series is isomorphic to
$\left(\Gr \pi\right) \otimes {\mathbb R}$, where $\Gr \pi$ is the
graded ${\mathbb Z}$-Lie algebra associated to the filtration of $\pi$ by
its lower central series.
In this paper we give a presentation of the Malcev Lie algebra
${\mathfrak t}_{g,r}^n$ associated to the Torelli group $T_{g,r}^n$ for all
$g\ge 6$. Since each Torelli group injects into its unipotent
completion (at least when $n+r>0$), the corresponding Malcev
Lie algebra should contain significant information about the group.
Recall that the mapping class group $\Gamma_{g,r}^n$ \label{group_def}
is defined as follows. Fix a compact orientable surface $S$ of
genus $g$, together with $n+r$ distinct points
\begin{equation}\label{points}
x_1,\dots,x_n; y_1,\dots,y_r
\end{equation}
of $S$ and $r$ non-zero tangent vectors $v_1,\dots,v_r$, where
$v_j$ is tangent to $S$ at $y_j$. The group $\Gamma_{g,r}^n$
is the group of isotopy classes of orientation preserving
diffeomorphisms of $S$ that fix each of the points (\ref{points})
and each of the tangent vectors $v_j$.%
\footnote{One can replace each tangent vector in the definition
by a boundary component --- with this change, the diffeomorphisms
are required to be the identity on each boundary component.}
The Torelli group $T_{g,r}^n$ \label{torelli_def}is defined to be
the kernel of the natural homomorphism
\begin{equation}\label{rho}
\Gamma_{g,r}^n \to \Aut H_1(S,{\mathbb Z}).
\end{equation}
Observe that the classical pure braid group $P_n$ is $T_{0,1}^n$.
Our presentation of ${\mathfrak t}_{g,r}^n$ generalizes the well-known
presentation of ${\mathfrak p}_n$, the Malcev Lie algebra of the pure braid
group $P_n$, which is
of importance in the theory of Vassiliev invariants (cf.\
\cite{kontsevich:vass,cartier:knots,bar-natan}) and was first
written down by Kohno \cite{kohno:braids}. Denote the free Lie
algebra generated by indeterminates $X_1,\dots, X_m$ by
${\mathbb L}(X_1,\dots,X_m)$, and that generated by a vector space $V$
by \label{lie_def} ${\mathbb L}(V)$. Then ${\mathfrak p}_n$ is the completion of the
graded Lie algebra
$$
{\mathbb L}(X_{ij} : ij\text{ is a two element subset of }\{1,\dots,n\})/R
$$
where $R$ is the ideal generated by the quadratic relations
\begin{align*}\label{braid_relns}
[X_{ij},X_{kl}]&\text{ when $i,j,k$ and $l$ are distinct;}\cr
[X_{ij},X_{ik} + X_{jk}]& \text{ when $i,j$ and $k$ are distinct.}
\end{align*}
The property that ${\mathfrak p}_n$ is the completion of the associated graded
Lie algebra $\Gr {\mathfrak p}_n$ does not hold for the generic group, but does
hold for all Torelli groups as we shall see.
It is easiest to first state the result for the absolute Torelli
group, $T_g := T_{g,0}^0$. It follows from Dennis Johnson's computation
of the first homology of $T_g$ \cite{johnson:h1} that each graded
quotient of the lower central series of ${\mathfrak t}_g$ is a representation
of the algebraic group $Sp_g$. We will give a presentation of
$\Gr {\mathfrak t}_g$ in the category of representations of $Sp_g$. Chose a set
$\lambda_1,\dots, \lambda_g$ of fundamental weights of $Sp_g$.
Denote the representation of $Sp_g$ with highest weight
$\lambda = \sum n_j \lambda_j$ by $V(\lambda)$. Johnson's
fundamental computation is that there is a natural $Sp_g({\mathbb Z})$
equivariant isomorphism between $H_1(T_g,{\mathbb Q})$ and $V(\lambda_3)$.
For all $g \ge 3$, the representation $\Lambda^2 V(\lambda_3)$
contains a unique copy of $V(2\lambda_2) + V(0)$. Denote the
$Sp_g$ invariant complement of this by $R_g$. Since the quadratic
part of the free Lie algebra ${\mathbb L}(V)$ is $\Lambda^2 V$, we can
view $R_g$ as being a subspace of the quadratic elements of
${\mathbb L}(V(\lambda_3))$.
\begin{theorem}
For all $g\neq 2$, ${\mathfrak t}_g$ is the completion of its
associated graded $\Gr{\mathfrak t}_g$. When $g\ge 6$, this has presentation
$$
\Gr {\mathfrak t}_g = {\mathbb L}(V(\lambda_3))/(R_g),
$$
where $R_g$ is the set of quadratic relations defined above.
When $3 \le g < 6$, the relations in $\Gr {\mathfrak t}_g$ are generated by
the quadratic relations $R_g$, and possibly some cubic relations.
\end{theorem}
In fact, we will show that in genus 3 there are no quadratic relations
and the cubic relations contain a copy of $V(\lambda_3)$.
Dennis Johnson has proved that $T_g$ is finitely generated for
all $g\ge 3$, but it is not known for any $g\ge 3$ whether or not
$T_g$ is finitely presented. Geoff Mess \cite{mess} proved that
$T_2$ is a countably generated free group. (Note that when
$g = 0,1$, $T_g$ is trivial.)
\begin{corollary}
For all $g\neq 2$, and for all $r,n \ge 0$, $t_{g,r}^n$ is
finitely presented. \qed
\end{corollary}
In the decorated case, we have the extension
\begin{equation}\label{extension}
1 \to \pi_{g,r}^n \to T_{g,r}^n \to T_g \to 1,
\end{equation}
where $\pi_{g,r}^n$ denotes the fundamental group of the
configuration space of $n$ points and $r$ tangent vectors
in $S$. After applying the Malcev Lie algebra functor, we
obtain an extension
$$
0 \to {\mathfrak p}_{g,r}^n \to {\mathfrak t}_{g,r}^n \to {\mathfrak t}_g \to 0,
$$
where ${\mathfrak p}_{g,r}^n$ denotes the Malcev Lie algebra of
$\pi_{g,r}^n$.
\begin{theorem}
For all $g \ge 0$, and all $r,n \ge 0$, the Lie algebra
${\mathfrak p}_{g,r}^n$ is the completion of its associated graded
$\Gr {\mathfrak p}_{g,r}^n$. The associated graded has a presentation
with only quadratic relations.
\end{theorem}
The explicit presentation is given in Section \ref{braids2}.
In order to give the presentation for ${\mathfrak t}_{g,r}^n$, we
prove that (\ref{extension}) remains exact after taking
graded quotients. Thus, in order to give a presentation
of ${\mathfrak t}_{g,r}^n$, it suffices to determine the map
$$
[\phantom{x},\phantom{x}] :
\left(\Gr^1 {\mathfrak t}_g \otimes \Gr^1 {\mathfrak p}_{g,r}^n\right)
\oplus \left(\Gr^1 {\mathfrak t}_g \otimes \Gr^1 {\mathfrak t}_g\right)
\to \Gr^2 {\mathfrak t}_{g,r}^n.
$$
determined by the bracket.
We do this in Section \ref{decorated} to obtain the presentation
of ${\mathfrak t}_{g,r}^n$ in general.
Our results complement, and sometimes overlap with, the beautiful
work \cite{morita:casson,morita:cocycles,morita:trace,morita:conj} of
Shigeyuki
Morita who began the study of the ``higher Johnson homomorphisms''
studied in this paper. Our main theorem allows us to answer several
questions about Torelli groups, and to prove a conjecture of Morita.
These and other applications are discussed in Section~\ref{applications}.
Another feature of the classical case is the existence of a
canonical universal integrable connection. Denote the
classifying space
$$
{\mathbb C}^n - \left\{(z_1,\dots,z_n) :
\text{ the $z_i$ are not distinct}\right\}
$$
of $P_n$ by $X_n$. Denote the complex of global meromorphic
$k$-forms on a
complex manifold $Y$ by $\Omega^k(Y)$. The universal integrable
connection on $X_n$ is given by the ${\mathfrak p}_g$ valued 1-form
$$
\sum_{ij} d\log(z_i - z_j)\, X_{ij} \in \Omega^1(X_n) \otimes {\mathfrak p}_n.
$$
It plays a central role in the theory of Vassiliev invariants
(cf. \cite{kohno:KZ}, \cite{cartier:knots}, \cite{kassel}.)
We are able to prove that there is a canonical universal
connection form with ``scalar curvature'' for each $T_{g,r}^n$,
provided $g\neq 2$, although, to date, we have not been able to
give an explicit formula for it. The universal connection is
discussed in Section \ref{applications}.
The basic approach in this paper is to use Hodge
theory. The main technical theorem of the paper is:
\begin{theorem}
Suppose that $g\neq 2$ and that $r,n \ge 0$. For each choice of
a complex structure on the decorated reference surface
$$
\left(S;x_1,\dots,x_n;y_1,\dots,y_r;v_1,\dots,v_r\right)
$$
there is a mixed Hodge structure on ${\mathfrak t}_{g,r}^n$ for
which the bracket is a morphism of mixed Hodge structures.
\end{theorem}
This mixed Hodge structure is canonical once one fixes
an isomorphism of $\Gamma_{g,r}^n$ with the (orbifold) fundamental
group of the moduli space of smooth projective curves of genus
$g$ with $n$ marked points, and $r$ non-zero tangent vectors.
The theorem is proved using the mixed Hodge structure on
the completion of the mapping class group $\Gamma_{g,r}^n$
relative to the homomorphism $\Gamma_{g,r}^n \to Sp_g({\mathbb Q})$
induced by (\ref{rho}), the existence of
which follows from \cite{hain:derham}. The mixed Hodge
structure on the Lie algebra ${\mathfrak u}_{g,r}^n$ of the prounipotent
radical of the relative completion is lifted to ${\mathfrak t}_{g,r}^n$
using two results from \cite{hain:comp}. The first states that
we have a central extension
$$
0 \to \Ga \to {\mathfrak t}_{g,r}^n \to {\mathfrak u}_{g,r}^n \to 0
$$
when $g\ge 3$.
The second gives an explicit relationship between this extension
and the algebraic 1-cycle $C - C^-$ in the jacobian of an
algebraic curve $C$. The theory of relative completion is
reviewed in Section \ref{rel_comp}.
When $g \ge 3$, the weight filtration of ${\mathfrak t}_{g,r}^n$ is its
lower central series. The fact that the weight graded functors
are exact on the category of mixed Hodge structures then
allows the reduction to the associated graded with impunity
when studying ${\mathfrak t}_{g,r}^n$, ${\mathfrak u}_{g,r}^n$, ${\mathfrak p}_{g,r}^n$ and
maps between them.
In order to bound the degrees of relations in ${\mathfrak t}_g$ by $N$,
we need to know that if ${\mathbb V}$ is a variation of Hodge structure
of weight $n$ over ${\mathcal M}_g$ (the moduli space of curves) that comes
from a rational representation of $Sp_g$, then the weights on
$$
H^2({\mathcal M}_g,{\mathbb V})
$$
are bounded between $2+n$ and $n+N$ --- see Section \ref{cts_coho_tor}.
There is no {\it a priori} uniform bound on the weights of $H^k(X,{\mathbb V})$,
where $X$ is a smooth variety and ${\mathbb V}$ is a variation of Hodge structure
over $X$ of weight $l$, as there is in the case of ${\mathbb Q}$ coefficients
where the weights are bounded between $k$ and $2k$. For example, if
$\Gamma$ is a finite index subgroup of $SL_2({\mathbb Z})$ and $X$ the quotient
of the upper half plane by $\Gamma$, then the non-trivial weights on
$H^1(X,S^n{\mathbb V})$ are $n+1$ and $2n+2$ for infinitely many $n$, as can
be seen from results in \cite{zucker}. Here ${\mathbb V}$ denotes the
fundamental representation of $SL_2$ viewed as a variation of
Hodge structure over $X$ of weight 1 and $S^n{\mathbb V}$ its $n$th symmetric
power. Thus, one of the main technical ingredients in the paper is the
result of Kabanov \cite{kabanov} (see also \cite{kabanov:purity} which
states that one can take $N$ to be 2 when $g\ge 6$, and 3 when
$3 \le g < 6$.
The existence of the mixed Hodge structure on the Malcev Lie algebra
associated to the Torelli group was obtained several years ago. The
quadratic relations (proved in Section \ref{quadratic_relns}) were
derived in \cite{hain:letter}. Subsequently Morita (unpublished)
proved that when the genus is sufficiently large there are no cubic
or quartic relations in ${\mathfrak t}_g$. Kabanov's purity theorem allows us to
avoid Morita's involved computations and to show there are no
higher order relations.
\medskip
\noindent{\it Acknowledgements.} It is a pleasure to thank all those
with whom I have had useful discussions, especially A.~Borel, P.~Deligne,
Alexander Kabanov, Eduard Looijenga, Shigeyuki Morita, and Steven Zucker.
I would also like to thank Hiroaki Nakamura for his numerous comments
on the manuscript.
I would like to thank the Institute for Advanced Study, the Institut des
Hautes \'Etudes Scientifiques, the Institut Henri Poincar\'e, and the
Universit\"at Essen, each of which supported me during my sabbatical
during which this paper was written.
\section{Braid Groups in Positive Genus}
\label{braids1}
Throughout this section $g$ will be positive. Suppose that
$S$ is a compact oriented surface of genus $g$, and that
$r$ and $n$ are integers $\ge 0$. The configuration space of
$m\ge 1$ points on $S$ is
$$
F^m(S) = S^m - \Delta,
$$
where $\Delta$ is the union of the various diagonals $x_i = x_j$.
Denote the tangent bundle of $S$ by $TS$, and the bundle of non-%
zero tangent vectors by $V$. The pullback of $V$ to $F^m(S)$ along
the $j$th projection $p_j : F^m(S) \to S$ will be denoted by $V_j$.
For a subset $A$ of $\{1,\dots,m\}$ denote the fibered product
of the $V_j$, where $j \in A$, by $V_A$.
The {\it configuration space \label{config_def} $F_{g,r}^n$ of $n$
points and $r$ non-zero tangent vectors of $S$} is defined to be the
total space of the bundle
$$
V_A \to F^{r+n}(S)
$$
where $A = \{n+1,\dots,n+r\}$. Fix a base point $f_o$ of $F_{g,r}^n$.
Define \label{fund_def}
$$
\pi_{g,r}^n = \pi_1(F_{g,r}^n,f_o).
$$
When $r=0$ this is just the group of pure braids with $n$ strings
on the surface $S$. In general, this group can be thought of as the
group of pure braids on $S$ with $r+n$ strings where $r$ of the
strings are framed. It is a standard fact that the space $F_{g,r}^n$
is an Eilenberg-MacLane space of type $K(\pi,1)$ \cite[\S 1.2]{birman}.
In contrast with the genus 0 case, we have:
\begin{proposition}\label{h1_braid}
For each $g\ge 0$, there is a short exact sequence
$$
0 \to \left({\mathbb Z}/(2g-2){\mathbb Z}\right)^r \to H_1(\pi_{g,r}^n,{\mathbb Z})
\stackrel{p}{\to} H_1(S^{n+r},{\mathbb Z}) \to 0,
$$
where $p$ is induced by the natural map $F_{g,r}^n \to S^{n+r}$.
\end{proposition}
\begin{proof}
We first consider the case when $r=0$. In this case $F_{g,r}^n$
is $S^n - \Delta$. The divisor $\Delta$ is the
union of the diagonals $\Delta_{ij}$ where the $i$th and $j$th
point of $S^n$ are equal. We therefore have a Gysin
sequence
$$
\cdots \to H_2(S^n) \stackrel{\gamma}{\to} \bigoplus_{i<j}{\mathbb Z}
\stackrel{t}{\to} H_1(S^n - \Delta) \to H_1(S^n) \to 0.
$$
The map $\gamma$ takes a cycle $z$ to the element of
$\oplus_{i<j}{\mathbb Z}$ whose $ij$th term is the intersection
number $z\cdot\Delta_{ij}$. The map $t$ takes the
generator of the $ij$th factor to the homology class of a
small circle which winds about about $\Delta_{ij}$ in the
positive direction.
Fix a base point $x_o$ of $S$.
For $u\in H_k(S)$ and $i\in \{1,\dots,n\}$, denote by
$u^i$ the element
$$
x_o \times \dots \times x_o \times \stackrel{i}{u}
\times x_o \times \dots \times x_o
$$
of $H_k(S^n,{\mathbb Z})$, where $u$ is placed in the $i$th factor.
For elements $u$ and $v$ of $H_1(S)$ and $i,j\in\{1,\dots,n\}$,
denote the element
$$
x_o \times \dots \times x_o \times \stackrel{i}{u}
\times x_o \times \dots \times x_o \times \stackrel{j}{v}
\times x_o \times \dots \times x_o
$$
of $H_2(S,{\mathbb Z})$ by $u^i \times v^j$, where $u$ is placed in
the $i$th factor and $v$ in the $j$th.
By choosing representatives of $u$ and $v$ which do not
pass through the base point, one sees immediately that
$$
\gamma : u^i \times u^j \mapsto (u\cdot v)\Delta_{ij}
$$
from which it follows that $\gamma$ is surjective and that
$t$ is trivial. This proves the result when $r=0$.
Observe that
$$
\gamma : S^i \mapsto \sum_{j\neq i} \Delta_{ij}.
$$
If $a$ and $b$ are elements of $H_1(S,{\mathbb Z})$ with intersection
number 1, then
$$
S^i - \sum_{j\neq i} a^i \times b^j
$$
is in the kernel of $\gamma$, and therefore lifts to
an element $\sigma_i$ of $H_2(S^n-\Delta,{\mathbb Z})$.
We prove the general case by induction on $r$. Our
inductive hypothesis is that the result has been proven
for $F_{g,s}^m$ when $s<r$, and that there are classes
$$
\sigma_1,\dots, \sigma_m \in H_2(F_{g,s}^m,{\mathbb Z})
$$
whose images under the the maps
$$
{p_j}_\ast : H_2(F_{g,s}^m,{\mathbb Z}) \to H_2(S,{\mathbb Z})
$$
induced by the various projections $p_j : F_{g,s}^m \to S$
satisfy $p_j(\sigma_i) = \delta_{ij}[S]$. We have proved this
when $r=0$.
Suppose that $r>0$. We have the projection
$$
q: F_{g,r}^n \to F_{g,r-1}^{n+1},
$$
which replaces the first tangent vector by its anchor point.
This is a principal ${\mathbb C}^\ast$ bundle. It fits into a commutative
square
$$
\begin{CD}
F_{g,r}^n @>>> V \cr
@VqVV @VVV \cr
F_{g,r-1}^{n+1} @>>> S \cr
\end{CD}
$$
One therefore has a map
$$
\begin{CD}
H_2(F_{g,r-1}^{n+1}) @>\gamma_F>> H_0(F_{g,r}^n) @>>>
H_1(F_{g,r}^n) @>>> H_1(F_{g,r-1}^{n+1}) @>>> 0 \cr
@VVV @VVV
@VVV
@VVV \cr
H_2(S) @>\gamma_S>> H_0(S) @>>>
H_1(V) @>>> H_1(S) @>>> 0 \cr
\end{CD}
$$
of Gysin sequences. The map $\gamma_S$ is simply multiplication
by the euler characteristic.
Since $\sigma_{n+1} \mapsto [S]$, we see that $\gamma_F$ takes
$\sigma_{n+1}$ to $2 - 2g$. The result follows by induction.
\end{proof}
\begin{remark}\label{error}
This result (with $r=0$) can also be proved by considering the
natural fibrations $F_g^{n+1} \to F_g^n$ obtained by forgetting
the last point. The fiber is an $n$ punctured copy of $S$, and
its homology therefore fits into an exact sequence
$$
0 \to {\mathbb Z}^n/\text{diagonal} \to
H_1(S - n\text{ points}) \to H_1(S) \to 0.
$$
One has to be a little careful as the monodromy action is not
trivial. A simple geometric argument shows that the monodromy
acts trivially on the kernel and quotient in the sequence above,
and therefore is given by a homomorphism
$$
\pi_g^n \to \Hom(H_1(S),{\mathbb Z}^n/\text{diagonal})
\cong H^1(S^n)/\text{diagonal}.
$$
By induction on $n$, $H_1(\pi_g^n)$ is isomorphic to $H_1(S^n)$.
Since the monodromy is abelian, it factors through the quotient
map
$$
\pi_g^n \to H_1(S)^{\oplus n}.
$$
A straightforward geometric argument shows that the action of the
latter is given by the map
$$
H_1(S)^{\oplus n} \stackrel{PD^{\oplus n}}{\longrightarrow}
H^1(S)^{\oplus n}/\text{diagonal},
$$
where $PD$ denotes Poincar\'e duality.
The coinvariants are therefore given by
$$
H_0(F_g^n,H_1(\text{fiber})) = H_1(S).
$$
An elementary spectral sequence argument completes the inductive
step.
Kohno and Oda \cite[p.~208]{kohno-oda} use this method, but their
result contradicts ours as they mistakenly assume that the
monodromy representation is trivial.
\end{remark}
\section{Relative Completion of Mapping Class Groups}
\label{rel_comp}
In this section we recall the main theorem of \cite{hain:comp}
which makes precise the relationship between the Malcev completion
of $T_{g,r}^n$ and the unipotent radical of the relative Malcev
completion of $\Gamma_{g,r}^n$. We first recall the definition
of relative Malcev completion, which is due to Deligne. A reference
for this material is \cite[\S\S 2--4]{hain:comp}.
Suppose that $\Gamma$ is a discrete group, $S$ a reductive linear
algebraic group over a field $F$ of characteristic zero, and that
$\rho : \Gamma \to S(F)$ is a representation whose image is Zariski
dense. The {\it Malcev completion of $\Gamma$ over $F$ relative to
$\rho$} is a homomorphism $\tilde{\rho} : \Gamma \to {\mathcal G}$ of $\Gamma$ into
a proalgebraic group ${\mathcal G}$, defined over $F$, which is an extension
$$
1 \to {\mathcal U} \to {\mathcal G} \stackrel{p}{\to} S \to 1
$$
of $S$ by a prounipotent group ${\mathcal U}$ such that the diagram
$$
\begin{CD}
\Gamma @>{\rho}>> S \cr
@V{\tilde{\rho}}VV @| \cr
{\mathcal G} @>>p> S
\end{CD}
$$
commutes. It is characterized by a universal mapping property: If
$G$ is a linear (pro)algebraic group over $F$ which is an extension
$$
1 \to U \to G \to S \to 1
$$
of $S$ by a (pro)unipotent group, and if $\tau : \Gamma \to G$
is a homomorphism whose composition with $G \to S$ is $\rho$, then
there is a unique homomorphism ${\mathcal G} \to G$ such that the diagram
$$
\begin{CD}
\Gamma @>{\tilde{\rho}}>> {\mathcal G} \cr
@V{\tau}VV @VpVV\cr
G @>>> S
\end{CD}
$$
commutes.
When $S$ is the trivial group, the relative completion of $\Gamma$
coincides with the classical Malcev (or unipotent) completion of
$\Gamma$.
Suppose that $K/F$ is an extension of fields of characteristic
zero. When $S$ is defined over $F$ and $\rho : \Gamma \to S(F)$, one can
ask if the $K$-form of the completion of $\Gamma$ relative to $\rho$ is
obtained from the $F$-form by extension of scalars. If this is the case
for all such field extensions, we will say that the relative completion
of $\Gamma$ relative to $\rho$ can be defined over $F$.
The action of the mapping class group on $S$ preserves the intersection
pairing $q : H_1(S,{\mathbb Z})^{\otimes 2} \to Z$. We therefore have a
homomorphism
\begin{equation}\label{map}
\rho : \Gamma_{g,r}^n \to \Aut (H_1(S,{\mathbb Z}),q) \cong Sp_g({\mathbb Z}).
\end{equation}
For a positive integer $l$, we define the {\it level $l$ subgroup}
$\Gamma_{g,r}^n[l]$ \label{level_def} to be the kernel of the induced
map
$$
\Gamma_{g,r}^n \to \Aut (H_1(S,{\mathbb Z}/l{\mathbb Z}),q) \cong Sp_g({\mathbb Z}/l{\mathbb Z}).
$$
Here we interpret $Sp_g({\mathbb Z}/l{\mathbb Z})$ as the trivial group when $l=1$.
\begin{theorem}\label{rational}
For all $g\ge 3$ and all $l \ge 1$, the completion of the mapping
class group $\Gamma_{g,r}^n[l]$ relative to the homomorphism
$\rho : \Gamma_{g,r}^n[l] \to Sp_g({\mathbb Q})$ induced by (\ref{map})
is defined over ${\mathbb Q}$. \qed
\end{theorem}
This result was proved in \cite[(4.14)]{hain:comp} under the
assumption that $g \ge 8$ and that $l=1$. That the stronger
result is true follows from the strengthening \cite{borel:improved}
of Borel's stability theorem \cite{borel:triv,borel:twisted} for the
symplectic group, stated below, which ensures that
the hypothesis \cite[(4.10)]{hain:comp} is satisfied when $l\ge 1$
and $g \ge 3$.
\begin{theorem}\label{imp_borel}
Suppose that $V$ is an irreducible rational representation of
the algebraic group $Sp_g$ and that $\Gamma$ is a finite index subgroup
of $Sp_g({\mathbb Z})$. If $k < g$, then $H^k(\Gamma,V)$ vanishes when $V$ is
non-trivial, and agrees with the stable cohomology of $Sp_g({\mathbb Z})$ when
$V$ is the trivial representation. \qed
\end{theorem}
Denote the completion of $\Gamma_{g,r}^n$ relative to $\rho$ by
$\tilde{\rho} : \Gamma_{g,r}^n \to {\mathcal G}_{g,r}^n$. \label{comp_def} Denote the
prounipotent radical of ${\mathcal G}_{g,r}^n$ by \label{ugp_def} ${\mathcal U}_{g,r}^n$, and
its Lie algebra by \label{ulie_def} ${\mathfrak u}_{g,r}^n$.
\begin{proposition}\label{level}
If $g\ge 3$, then for all $l\ge 1$, the composite
$$
\Gamma_{g,r}^n[l] \hookrightarrow \Gamma_{g,r}^n \to {\mathcal G}_{g,r}^n
$$
is the completion of $\Gamma_{g,r}^n[l]$ relative to the restriction
of $\rho$ to $\Gamma_{g,r}^n[l]$.
\end{proposition}
\begin{proof}
This follows directly from results in \cite[\S4]{hain:comp} as
we shall explain. Denote the relative completion of $\Gamma_{g,r}^n[l]$
by ${\mathcal G}_{g,r}^n[l]$ and its prounipotent radical
by ${\mathcal U}_{g,r}^n[l]$. There is a natural map
${\mathcal U}_{g,r}^n[l] \to {\mathcal U}_{g,r}^n$, the surjectivity of which follows from
(\ref{imp_borel}) and \cite[(4.6)]{hain:comp}. Injectivity follows
directly from (\ref{imp_borel}) and \cite[(4.13)]{hain:comp}.
\end{proof}
We have an extension
$$
1 \to {\mathcal U}_{g,r}^n \to {\mathcal G}_{g,r}^n \to Sp_g \to 1
$$
of proalgebraic groups over ${\mathbb Q}$. The homomorphism $\tilde{\rho}$
induces a map $T_{g,r}^n \to {\mathcal U}_{g,r}^n$. Denote the classical
Malcev completion of \label{comptor_def} $T_{g,r}^n$ by ${\mathcal T}_{g,r}^n$,
and its Lie algebra by \label{lietor_def} ${\mathfrak t}_{g,r}^n$.
Since ${\mathcal U}_{g,r}^n$ is prounipotent, $\tilde{\rho}$ induces a homomorphism
$$
\theta : {\mathcal T}_{g,r}^n \to {\mathcal U}_{g,r}^n
$$
of prounipotent groups.
The following theorem is the main result of \cite{hain:comp}.%
\footnote{There is a minor error in proof of the case ``$A_{g,r}^n$
implies $A_{h,r}^n$'' of the proof of \cite[(7.4)]{hain:comp}. It is
easily fixed.}
There it is proved for all $g\ge 8$, but in view of
(\ref{imp_borel}), it holds for all $g\ge 3$. (Cf.\ the third
footnote on page~76 of \cite{hain:comp}.)
\begin{theorem}\label{central_ext}
For all $g\ge 3$, the homomorphism $\theta$ is surjective and has
a one dimensional kernel isomorphic to $\Ga$ which is central in
${\mathcal T}_{g,r}^n$ and is trivial as an $Sp_g({\mathbb Z})$ module. Moreover,
the extensions are all pulled back from that of ${\mathcal T}_g$; that is,
the diagram
$$
\begin{CD}
0 @>>> \Ga @>>> {\mathcal T}_{g,r}^n @>>> {\mathcal U}_{g,r}^n @>>> 1 \cr
@. @| @VVV @VVV \cr
0 @>>> \Ga @>>> {\mathcal T}_g @>>> {\mathcal U}_g @>>> 1
\end{CD}
$$
commutes. \qed
\end{theorem}
It is a standard fact that the sequence
$$
1 \to \pi_{g,r}^n \to \Gamma_{g,r}^n \to \Gamma_g \to 1
$$
is exact. Restricting to $T_g$, we obtain an extension
\begin{equation}
\label{exten}
1 \to \pi_{g,r}^n \to T_{g,r}^n \to T_g \to 1.
\end{equation}
\begin{proposition}\label{h1_tor}
If $g\ge 3$, then the extension (\ref{exten}) induces an
exact sequence
$$
0 \to H_1(\pi_{g,r}^n,{\mathbb Q}) \to H_1(T_{g,r}^n,{\mathbb Q})
\to H_1(T_g,{\mathbb Q}) \to 0.
$$
\end{proposition}
\begin{proof}
It follows from (\ref{h1_braid}) that the natural map
$\pi_{g,r}^n \to \pi_g^{n+r}$ induces an isomorphism on
$H_1$ with rational coefficients. The corresponding
surjection $T_{g,r}^n \to T_g^{n+r}$ induces a map
$$
\begin{CD}
H_1(\pi_{g,r}^n,{\mathbb Q}) @>>> H_1(T_{g,r}^n,{\mathbb Q}) @>>> H_1(T_g,{\mathbb Q}) @>>> 0\cr
@VVV @VVV @| \cr
H_1(\pi_g^{n+r},{\mathbb Q}) @>>> H_1(T_g^{n+r},{\mathbb Q}) @>>> H_1(T_g,{\mathbb Q}) @>>> 0\cr
\end{CD}
$$
of exact sequences. Since the middle vertical map is a surjection,
and the two outside maps are isomorphisms, it follows that the middle
map is an isomorphism. It therefore suffices to prove the result when
$r=0$.
To prove this, we need to prove that the map
\begin{equation}\label{red}
H_1(\pi_g^n,{\mathbb Q}) \to H_1(T_g^n,{\mathbb Q})
\end{equation}
is injective. We first remark that this is easily proved when
$n=1$ using the Johnson homomorphism
$$
\tau_g^1 : H_1(T_g^1) \to H_3(\Jac C).
$$
(Cf.\ \cite{johnson:def} and \cite[\S 3]{hain:normal}.)
The composition of $\tau_g^1$ with the map
$$
H_1(\pi_g^1,{\mathbb Q}) \to H_3(\Jac C,{\mathbb Q})
$$
is easily seen to be
the map $H_1(C) \to H_3(\Jac C)$ which takes a class in
$H_1$ to its Pontrjagin product with the class of $C$ in
$H_2(\Jac C)$. Since this map is injective, it follows that
(\ref{red}) is injective when $n=1$.
The general case follows from this by considering the
maps $p_j : H_1(T_g^n) \to H_1(T_g^1)$ induced by the
$n$ forgetful maps $T_g^n \to T_g^1$.
\end{proof}
Define ${\mathcal P}_{g,r}^n$ \label{comppi_def} to be the Malcev completion
of $\pi_{g,r}^n$ and \label{liepi_def} ${\mathfrak p}_{g,r}^n$ to be the
corresponding Malcev Lie algebra.
Applying Malcev completion to (\ref{exten}) we obtain a sequence
$$
{\mathfrak p}_{g,r}^n \to {\mathfrak t}_{g,r}^n \to {\mathfrak t}_g
$$
of Malcev Lie algebras.
\begin{proposition}\label{seq}
If $g\neq 2$, then the sequence
$$
0 \to {\mathfrak p}_{g,r}^n \to {\mathfrak t}_{g,r}^n \to {\mathfrak t}_g \to 0
$$
associated to (\ref{exten}) is exact.
\end{proposition}
\begin{proof}
By \cite[(5.6)]{hain:cycles} it suffices to verify two conditions.
First, that $T_g$ acts unipotently on $H^1(\pi_{g,r}^n,{\mathbb Q})$; this
follows from (\ref{h1_braid}). The second condition there is satisfied
if, for example, the extension
$$
0 \to H_1(\pi_{g,r}^n,{\mathbb Q}) \to G \to T_g \to 1
$$
obtained by pushing (\ref{exten}) out along
$\pi_{g,r}^n \to H_1(\pi_{g,r}^n,{\mathbb Q})$ is split. In our case this
follows from (\ref{h1_tor}) as the extension
above can be pulled back from the extension
$$
0 \to H_1(\pi_{g,r}^n,{\mathbb Q}) \to H_1(T_{g,r}^n,{\mathbb Q})
\to H_1(T_g,{\mathbb Q}) \to 0
$$
which is split for trivial reasons.
\end{proof}
The standard homomorphism $\Gamma_{g,r}^n \to \Gamma_g$ induces a
homomorphism ${\mathcal G}_{g,r}^n \to {\mathcal G}_g$ of relative completions. The
inclusion ${\mathcal P}_{g,r}^n \to {\mathcal T}_{g,r}^n$ induces a homomorphism
${\mathcal P}_{g,r}^n \to {\mathcal U}_{g,r}^n$. We therefore
have a sequence
$$
{\mathcal P}_{g,r}^n \to {\mathcal G}_{g,r}^n \to {\mathcal G}_g \to 1
$$
of proalgebraic groups.
\begin{lemma}\label{exactness}
If $g\ge 3$, then the sequence
$$
1 \to {\mathcal P}_{g,r}^n \to {\mathcal G}_{g,r}^n \to {\mathcal G}_g \to 1
$$
is exact. In particular, the sequence
$$
0 \to {\mathfrak p}_{g,r}^n \to {\mathfrak u}_{g,r}^n \to {\mathfrak u}_g \to 0
$$
is exact.
\end{lemma}
\begin{proof}
To prove the result, it suffices to prove that the sequence
$$
1 \to {\mathcal P}_{g,r}^n \to {\mathcal U}_{g,r}^n \to {\mathcal U}_g \to 1
$$
is exact. But this follows immediately from
(\ref{central_ext}) and (\ref{seq}).
\end{proof}
Suppose that ${\mathfrak g}$ is a finitely generated pronilpotent Lie algebra.
Denote the group of automorphisms of ${\mathfrak g}$ by $\Aut{\mathfrak g}$. Denote the
subgroup of $\Aut{\mathfrak g}$ consisting of the elements that act trivially
on $H_1({\mathfrak g})$ by $L^1\Aut {\mathfrak g}$. Since the
action of an automorphism on the graded quotients of the lower central
series is determined by the action on the first graded quotient,
$\Aut {\mathfrak g}$ is a proalgebraic group which is an extension
$$
1 \to L^1\Aut {\mathfrak g} \to \Aut {\mathfrak g} \to S \to 1
$$
of a closed subgroup $S$ of $\Aut H_1({\mathfrak g})$ by the prounipotent group
consisting of those automorphisms of ${\mathfrak g}$ that act trivially on the
graded quotients of the lower central series. Its Lie algebra is the
Lie algebra $\Der {\mathfrak g}$ of derivations of ${\mathfrak g}$. This is an extension
$$
0 \to L^1\Der {\mathfrak g} \to \Der {\mathfrak g} \to {\mathfrak s} \to 0
$$
of the Lie algebra ${\mathfrak s}$ of $S$ by the pronilpotent Lie algebra
of derivations of ${\mathfrak g}$ that act trivially on the graded quotients
of the lower central series of ${\mathfrak g}$.
If ${\mathcal G}$ is the prounipotent group corresponding to ${\mathfrak g}$, then
$\Aut {\mathcal G}$ and $\Aut {\mathfrak g}$ are isomorphic, as can be seen using the
Baker-Campbell-Hausdorff formula.
\begin{lemma}\label{rep}
For all $g\ge 0$ the natural action of $\Gamma_{g,r}^n$ on
$\pi_{g,r}^n$ induces a representation
$$
{\mathcal G}_{g,r}^n \to \Aut {\mathfrak p}_{g,r}^n.
$$
\end{lemma}
\begin{proof}
Suppose that $g \ge 0$. The mapping class group $\Gamma_{g,r}^n$
acts on ${\mathfrak p}_{g,r}^n$. We therefore have a homomorphism
\begin{equation}\label{act}
\Gamma_{g,r}^n \to \Aut {\mathfrak p}_{g,r}^n.
\end{equation}
By (\ref{h1_braid}) we know that
$$
\Aut H_1({\mathfrak p}_{g,r}^n) = \Aut H_1(S)^{\oplus(n+r)}.
$$
There is a diagonal copy of $Sp_g$ contained in this group,
and it is easy to see that this is the Zariski closure of the
image of $\Gamma_{g,r}^n$ in $\Aut H_1({\mathfrak p}_{g,r}^n)$. It follows
that the Zariski closure of the image of (\ref{act}) is an
extension of this diagonal $Sp_g$ by a prounipotent group. Since
the homomorphism from $\Gamma_{g,r}^n$ to this copy of $Sp_g$ is
the standard representation, the universal mapping property of the
relative completion implies that (\ref{act}) induces a
homomorphism ${\mathcal G}_{g,r}^n \to \Aut{\mathfrak p}_{g,r}^n$.
\end{proof}
\begin{remark}\label{sl2}
When $g=1$, the results (\ref{level}) and (\ref{central_ext})
are false.
That (\ref{level}) and (\ref{central_ext}) fail can be deduced
from \cite[(10.3)]{hain:derham}, a special case of which states that
there is a natural isomorphism
$$
H^1({\mathcal M}_1[l],S^n{\mathbb V}) \cong
\left(H^1({\mathcal U}_{1}[l])\otimes S^n{\mathbb V}\right)^{SL_2}.
$$
Here ${\mathcal M}_1[l]$ denotes the moduli space of elliptic curves with a
level $l$ structure, ${\mathbb V}$ denotes the variation of Hodge structure
over ${\mathcal M}_1[l]$ of weight 1 corresponding to $H^1$ of the universal
elliptic curve, and $S^n {\mathbb V}$ denotes its $n$th symmetric power.
Since the level $l$ congruence subgroup of $SL_2({\mathbb Z})$ is free when
$g\ge 4$, it follows by an Euler characteristic argument that
$H^1({\mathcal M}_1[l],S^k{\mathbb V})$ is non-zero whenever $g\ge 4$. Since ${\mathcal T}_1$
is trivial, it cannot surject onto ${\mathcal U}_1[l]$. Since the rank $r_l$
of the level $l$ subgroup of $SL_2({\mathbb Z})$ depends on $l$, and since
$$
\dim H^1({\mathcal M}_1[l],S^n{\mathbb V}) = (r_l-1)\dim S^k{\mathbb V},
$$
it follows that the rank of the $S^n{\mathbb V}$ isotypical part of
$H_1({\mathcal U}_1[l])$ depends on $l$. So (\ref{level}) does not hold.
\end{remark}
\section{Mixed Hodge Structures on Torelli Groups}
Denote by ${\mathcal M}_{g,r}^n[l]$ \label{mod_def} the moduli space of
ordered $(n+r+1)$-tuples
$$
(C;x_1,\dots,x_n;v_1,\dots,v_r)
$$
where $C$ is a smooth complex projective curve with a level
$l$ structure, the $x_j$ are distinct points of $C$, and the
$v_j$ are non-zero holomorphic tangent vectors of $C$ which
are anchored at $r$ distinct points of $C$ which are also distinct
from the $x_j$. We shall omit the $l$ when it is 1, and $r$ and
$n$ when they are zero. So, for example, ${\mathcal M}_g$ denotes the moduli
space of smooth projective curves of genus $g$.
For each point $x$ of ${\mathcal M}_{g,r}^n[l]$, there is a natural isomorphism
of $\Gamma_{g,r}^n[l]$ with the (orbifold) fundamental group
$\pi_1({\mathcal M}_{g,r}^n[l],x)$ of ${\mathcal M}_{g,r}^n[l]$. We will denote the
latter by $\Gamma_{g,r}^n[l](x)$. We shall denote the subgroup
of $\Gamma_{g,r}^n(x)$ corresponding to $T_{g,r}^n$ by $T_{g,r}^n(x)$.
Denote the relative Malcev completion of $\Gamma_{g,r}^n(x)$ by
${\mathcal G}_{g,r}^n(x)$, its prounipotent radical by ${\mathcal U}_{g,r}^n(x)$, etc.
The Lie algebras corresponding to $T_{g,r}^n(x)$ and $U_{g,r}^n(x)$
will be denoted by ${\mathfrak t}_{g,r}^n(x)$ and ${\mathfrak u}_{g,r}^n(x)$, respectively.
In this section we prove that for each choice of a point $x$ in
${\mathcal M}_{g,r}^n$, there is a canonical ${\mathbb Q}$ mixed Hodge structure
(MHS) on ${\mathfrak t}_{g,r}^n(x)$. The first ingredient in the construction
of this MHS is the following theorem, which is proved in
\cite[(13.1)]{hain:derham}.
\begin{theorem}\label{mhs_gen}
Suppose that $X$ is a smooth quasi-projective algebraic variety
and $({\mathbb V},\langle\phantom{x},\phantom{x}\rangle)$ is a polarized variation
of Hodge structure over $X$ of geometric origin whose monodromy
representation
$$
\rho : \pi_1(X,x_o) \to \Aut_{\mathbb R}(V_o,\langle\phantom{x},\phantom{x}\rangle)
$$
has Zariski dense image. Then the coordinate ring of the
completion of $\pi_1(X,x_o)$ relative to $\rho$ and its unipotent
radical both have natural real MHSs such that the product, coproduct,
and antipode of each are morphisms of MHSs.
\qed
\end{theorem}
We will say that a homomorphism ${\mathcal G} \to{\mathcal H}$ between proalgebraic groups,
each of whose coordinate rings is a Hopf algebra in the category of
mixed Hodge structures, is a morphism of MHSs if the corresponding map
on coordinate rings is.
Since $\Gamma_{g,r}^n(x)$ is the orbifold fundamental group of
$({\mathcal M}_{g,r}^n,x)$, the following result is not unexpected.
\begin{theorem}
For all $g,r,n \ge 0$, and for each choice of a point
$$
x = [C;x_1,\dots,x_n;v_1,\dots,v_r]
$$
of ${\mathcal M}_{g,r}^n$, there is a canonical real MHS on the coordinate
ring of ${\mathcal G}_{g,r}^n(x)$ for which
the product, coproduct and antipode are morphisms of MHS. Moreover,
the homomorphisms ${\mathcal G}_{g,r}^n(x) \to {\mathcal G}_{g,r}^{n-1}(x')$ and
${\mathcal G}_{g,r}^n(x) \to {\mathcal G}_{g,r-1}^{n+1}(x'')$, induced by forgetting a point or
by replacing a tangent vector by its anchor point, are morphisms
of mixed Hodge structure.
\end{theorem}
\begin{proof}
Since the mapping class group is not, in general, the fundamental group
of ${\mathcal M}_{g,r}^n$, we need to pass to a level.
Choose an integer $l$ such that $\Gamma_{g,r}^n[l]$ is torsion free. In
this case, the moduli space ${\mathcal M}_{g,r}^n[l]$ is smooth and has
fundamental group isomorphic to $\Gamma_{g,r}^n[l]$. Since
$\Gamma_{g,r}^n[l]$ is torsion free, there is a universal curve
$$
\pi : {\mathcal C} \to {\mathcal M}_{g,r}^n[l].
$$
Take ${\mathbb V}$ to be the dual of the local system $R^1\pi_\ast {\mathbb Z}$. This is
a polarized variation of Hodge structure of weight $-1$ and is clearly
of geometric origin. Its monodromy representation is
$$
\rho : \Gamma_{g,r}^n[l] \to Sp_g({\mathbb Z}).
$$
So by (\ref{level}) and (\ref{mhs_gen}), there is a canonical real
MHS on the coordinate ring of the relative completion ${\mathcal G}_{g,r}^n(x)$
of $\Gamma_{g,r}^n(x)$ for each choice of a point of ${\mathcal M}_{g,r}^n[l]$
that lies over $x$.
Denote the projection ${\mathcal M}_{g,r}^n[l] \to {\mathcal M}_{g,r}^n$ by $p$. The
set of lifts of a point $x$ of ${\mathcal M}_{g,r}^n$ to ${\mathcal M}_{g,r}^n[l]$ is
permuted transitively by the Galois group $Sp_g({\mathbb Z}/l{\mathbb Z})$. It follows
from the naturality of the MHS on the relative completion that the
MHSs on ${\mathcal G}_{g,r}^n(x)$ with respect any two points of $p^{-1}(x)$ are
canonically isomorphic. The MHS on $\Gamma_{g,r}^n(x)$ is therefore
indepenent of the choice of a point of $p^{-1}(x)$, and is therefore
canonical.
To show that the MHS on $\Gamma_{g,r}^n(x)$ constructed above is
independent of the choice of the level $l$, suppose that $l_1$ and
$l_2$ are two levels for which the mapping class group is torsion free.
One can then compare the corresponding MHSs by passing to the level
corresponding to the least common multiple of $l_1$ and $l_2$.
The naturality statement follows directly from
\cite[(13.12)]{hain:derham}.
\end{proof}
\begin{corollary}
For all $g\ge 0$, and for each choice of a point
$$
x = [C;x_1,\dots,x_n;v_1,\dots,v_r]
$$
of ${\mathcal M}_{g,r}^n$, the pronilpotent Lie algebra ${\mathfrak u}_{g,r}^n(x)$ of the
prounipotent radical of ${\mathcal G}_{g,r}^n(x)$ has a canonical real MHS for
which the bracket is a morphism of MHS. Moreover, the morphisms
${\mathfrak u}_{g,r}^n(x) \to {\mathfrak u}_{g,r}^{n-1}(x')$ and
${\mathfrak u}_{g,r}^n(x) \to {\mathfrak u}_{g,r-1}^{n+1}(x")$,
obtained by forgetting a point or replacing a tangent vector by its
anchor point, are morphisms of mixed Hodge structure. \qed
\end{corollary}
Given a point $x$ of ${\mathcal M}_{g,r}^n$, there are two {\it a priori} different
MHSs on ${\mathfrak p}_{g,r}^n(x)$. The first is the one obtained from the
construction given in \cite{hain:dht}. The second arises as
${\mathfrak p}_{g,r}^n(x)$ is the kernel of the natural surjection
${\mathfrak u}_{g,r}^n(x) \to {\mathfrak u}_g(x)$. The following assertion follows directly
from the naturality properties \cite[(13.12)]{hain:derham} of the mixed
Hodge structure relative completions.
\begin{proposition}
These two MHSs are identical.\qed
\end{proposition}
Fix a point $x$ of ${\mathcal M}_{g,r}^n$. Then both of ${\mathcal G}_{g,r}^n(x)$ and
${\mathfrak p}_{g,r}^n(x)$ have canonical MHSs. It is natural to expect that the
natural action
\begin{equation}\label{action}
{\mathcal G}_{g,r}^n(x) \to \Aut {\mathfrak p}_{g,r}^n(x)
\end{equation}
constructed in (\ref{rep}) is compatible with these.
\begin{lemma}\label{action_morph}
The action (\ref{action}) is a morphism of MHS. Consequently, the
morphism
$$
{\mathfrak u}_{g,r}^n(x) \to \Der {\mathfrak p}_{g,r}^n(x)
$$
is also morphism of MHS with respect to the canonical MHSs determined
by $x\in {\mathcal M}_{g,r}^n$.
\end{lemma}
\begin{proof}
It follows immediately from (\ref{exactness}) that ${\mathcal P}_{g,r}^n(x)$
is a normal subgroup of ${\mathcal G}_{g,r}^n(x)$. Since the coordinate ring of
${\mathcal G}_{g,r}^n(x)$ has a natural mixed Hodge structure compatible with its
operations, the action of ${\mathcal G}_{g,r}^n(x)$ on ${\mathcal P}_{g,r}^n(x)$ via
conjugation is a morphism of MHS. But this action is easily seen to
coincide with the canonical action of ${\mathcal G}_{g,r}^n(x)$ on ${\mathcal P}_{g,r}^n(x)$.
\end{proof}
For a curve $C$ of genus $g\ge 3$, denote by $PH_3(\Jac C,{\mathbb Q})$ the
{\it primitive three dimensional homology} of its jacobian $\Jac C$
--- that is, the subspace of $H_3(\Jac C,{\mathbb Q})$ corresponding to
$PH^{2g-3}(\Jac C,{\mathbb Q})$ under Poincar\'e duality. It has a natural
Hodge structure of weight $-3$.
\begin{proposition}\label{purity}
If $g \ge 3$, then for each
$x = [C;x_1,\dots,x_n;v_1,\dots,v_r] \in {\mathcal M}_{g,r}^n$
the canonical real mixed Hodge structure on $H_1({\mathfrak u}_{g,r}^n(x))$ is of
weight $-1$ and is canonically isomorphic to
$$
PH_3(\Jac C,{\mathbb R}(-1)) \oplus H_1(C,{\mathbb R})^{\oplus(r+n)}.
$$
\end{proposition}
\begin{proof}
As in the proof of (\ref{h1_tor}), we reduce to proof to showing
that it is true for ${\mathfrak u}_g^1$. Then, by (\ref{action_morph}), the
composite
\begin{equation}\label{comp}
H_1({\mathfrak u}_g^1(x)) \to W_{-1}H_1(\Der {\mathfrak p}_g^1(x)) \to \Gr^W_{-1}\Der{\mathfrak p}_g^1(x)
\end{equation}
is a morphism of MHS. Observe that
$$
\Gr^W_{-1}\Der{\mathfrak p}_g^1(x) \subset
\Hom(\Gr^W_{-1}{\mathfrak p}_g^1(x),\Gr^W_{-2}{\mathfrak p}_g^1(x)).
$$
{}From the work of Johnson \cite{johnson:def} (see also
\cite[\S4]{hain:normal}), it follows that (\ref{comp})
is injective, from which the result follows for ${\mathfrak u}_g^1$.
\end{proof}
The fact that $H_1({\mathfrak u}_{g,r}^n)$ is pure of weight $-1$ allows us to
conclude that the weight filtration of ${\mathfrak u}_{g,r}^n$ is essentially
its lower central series. This follows from the following general
fact.
\begin{lemma}\label{wt=lcs}
Suppose that ${\mathfrak g}$ is a pronilpotent Lie algebra in the category of
mixed Hodge structures with finite dimensional $H_1$. If the induced
MHS on $H_1({\mathfrak g})$ is pure of weight $-1$, then $W_{-l}{\mathfrak g}$ is the $l$th
term of the lower central series of ${\mathfrak g}$.
\end{lemma}
\begin{proof}
Denote the $l$th term of the lower central series of ${\mathfrak g}$ by ${\mathfrak g}^{(l)}$.
Since $H_1({\mathfrak g})$ is pure of weight $-1$, since ${\mathfrak g}$ is pronilpotent, and
since the bracket is a morphism of MHS, it follows that ${\mathfrak g} = W_{-1}{\mathfrak g}$.
An elementary argument using the Jacobi identity shows that the bracket
\begin{equation}\label{bra}
{\mathfrak g} \otimes {\mathfrak g}^{(l)} \to {\mathfrak g}^{(l+1)}
\end{equation}
is surjective. Since the bracket is a morphism of MHS, it follows that
${\mathfrak g}^{(l)} \subseteq W_{-l}{\mathfrak g}$. The fact that $H_1({\mathfrak g})$ is pure of weight
$-1$, forces ${\mathfrak g}^{(2)} = W_{-2}\,{\mathfrak g}$. The result now follows by an induction
argument (induct on $l$) using the fact that (\ref{bra}), being a morphism
of MHS, is strict with respect to the weight filtration.
\end{proof}
\begin{corollary}\label{lcs}
The $l$th term of the lower central series of ${\mathfrak u}_{g,r}^n$
is $W_{-l}{\mathfrak u}_{g,r}^n$. \qed
\end{corollary}
This result implies that the weight filtration on ${\mathcal G}_{g,r}^n$
is defined over ${\mathbb Q}$ which implies that this MHS is really defined
over ${\mathbb Q}$.
\begin{corollary}
The weight filtration of the canonical MHSs on ${\mathcal G}_{g,r}^n(x)$ and
${\mathfrak u}_{g,r}^n(x)$ associated to a point of ${\mathcal M}_{g,r}^n$ are topologically
determined and therefore defined over ${\mathbb Q}$. Consequently, the MHSs on
${\mathcal G}_{g,r}^n(x)$ and ${\mathfrak u}_{g,r}^n(x)$ each have a canonical lift to
${\mathbb Q}$-MHSs.
\qed
\end{corollary}
We are now ready to lift the MHS from ${\mathfrak u}_{g,r}^n(x)$ to ${\mathfrak t}_{g,r}^n(x)$:
\begin{theorem}\label{mhs_torelli}
Suppose that $g\neq 2$ and that $r,n \ge 0$. For each choice of
a base point
$$
x = \left[C;x_1,\dots,x_n;v_1,\dots,v_r\right]
$$
of ${\mathcal M}_{g,r}^n$ there is a canonical ${\mathbb Q}$-MHS on ${\mathfrak t}_{g,r}^n(x)$
for which the bracket and the quotient map ${\mathfrak t}_{g,r}^n(x) \to u_{g,r}^n(x)$
are morphisms of MHS. Moreover, $W_{-l}{\mathfrak t}_{g,r}^n(x)$ is the $l$th term
of the lower central series of ${\mathfrak t}_{g,r}^n(x)$ and the central $\Ga$ is
isomorphic to ${\mathbb Q}(1)$.
\end{theorem}
\begin{proof}
For all $g \ge 0$, we have the exact sequence
$$
1 \to \pi_{g,r}^n \to T_{g,r}^n \to T_g \to 1.
$$
When $g =0,1$, $T_g$ is the trivial group, so that $T_{g,r}^n$ is
isomorphic to $\pi_{g,r}^n$. It follows that in these cases
${\mathfrak t}_{g,r}^n$ is isomorphic to the Malcev Lie algebra ${\mathfrak p}_{g,r}^n$
associated to $\pi_{g,r}^n$. The choice of the base point of
${\mathcal M}_{g,r}^n$ gives the configuration space $(F_{g,r}^n,f_o)$ the
structure of a pointed smooth complex algebraic variety. Since
$\pi_{g,r}^n$ is the fundamental group of $(F_{g,r}^n,f_o)$, the
existence of the MHS on ${\mathfrak t}_{g,r}^n(x)$ when $g=0,1$ follows from
\cite[(6.3.1)]{hain:dht}.
Now suppose that $g\ge 3$.
To construct a MHS on ${\mathfrak t}_{g,r}^n(x)$, it suffices to show that ${\mathfrak t}_g(x)$
has a MHS such that ${\mathfrak t}_g(x) \to {\mathfrak u}_g(x)$ is a morphism as it follows from
(\ref{central_ext}), (\ref{seq}) and (\ref{exactness}) that the diagram
$$
\begin{CD}
{\mathfrak t}_{g,r}^n(x) @>>> {\mathfrak t}_g(x) \cr
@VVV @VVV \cr
{\mathfrak u}_{g,r}^n(x) @>>> {\mathfrak u}_g(x) \cr
\end{CD}
$$
is a pullback square in the category of pronilpotent Lie algebras.
It is useful to begin by explaining the philosophy behind the proof.
The essential point is that $\Gamma_g(x)$ acts on ${\mathfrak t}_g(x)$ and on
${\mathfrak u}_g(x)$ --- the action is induced by the action of $\Gamma_g(x)$
on $T_g(x)$ by conjugation. The central extension
\begin{equation}\label{extn}
0 \to \Ga \to {\mathfrak t}_g(x) \to {\mathfrak u}_g(x) \to 0.
\end{equation}
given by (\ref{central_ext}) can be viewed as an extension of
local systems over ${\mathcal M}_g$ (in the orbifold sense, of course)
where $\Ga$ is a trivial local system. Although we have not proved
it yet, ${\mathfrak u}_g(x)$ should be a variation of MHS over ${\mathcal M}_g$. So we should
try to construct the MHS on ${\mathfrak t}_g(x)$ so that (\ref{extn}) is both an
extension of local systems and an extension of mixed Hodge structures.
This, and the fact that the bracket has to be a morphism of MHS, gives
us no choice. We now carry this out this program.
The first point is that we know that, since $H_1({\mathfrak t}_g)$
does not contain any copies of the trivial representation,
the central $\Ga$ is contained in $[{\mathfrak t}_g,{\mathfrak t}_g]$.
The second is that by the computations in \S8 of
\cite{hain:comp}
we know that the central $\Ga$ lies in the image of the map
$$
\Lambda^2 H_1({\mathfrak t}_g) \to \Gr^{\mathrm{lcs}}_{-2} {\mathfrak t}_g
$$
induced by the bracket. So in order that the bracket be a morphism
of MHS, the central $\Ga$ must be of weight $-2$. Since it is one
dimensional, it has to be isomorphic to ${\mathbb Q}(1)$.
Now fix a base point $x$ of ${\mathcal M}_g$. There is a corresponding ${\mathbb Q}$-MHS
on ${\mathfrak u}_g(x)$. To lift this MHS to ${\mathfrak t}_g(x)$, we have to give an element of
$$
\Ext^1_{\mathcal H}({\mathfrak u}_g(x),{\mathbb Q}(1)),
$$
where $\Ext_{\mathcal H}$ denote the Ext group in the category ${\mathcal H}$ of ${\mathbb Q}$
mixed Hodge structures. Applying the functor $\Ext_{\mathcal H}^{\bullet}$ to the
sequence
$$
0 \to W_{-2} {\mathfrak u}_g(x) \to {\mathfrak u}_g(x) \to H_1({\mathfrak u}_g(x)) \to 0,
$$
we see that the natural map
$$
\Ext^1_{\mathcal H}(H_1({\mathfrak u}_g(x)),{\mathbb Q}(1)) \to \Ext^1_{\mathcal H}({\mathfrak u}_g(x),{\mathbb Q}(1))
$$
is an isomorphism.
Now let the base point vary. By (\ref{purity}), $H_1({\mathfrak u}_g)$
is a variation of ${\mathbb Q}$-MHS over ${\mathcal M}_g$ (in the orbifold sense)
of weight $-1$ --- cf.\ \cite[(9.1)]{hain:normal}. Denote the category
of admissible variations of ${\mathbb Q}$-MHS over a smooth variety $X$ by
${\mathcal H}(X)$ and the category of ${\mathbb Q}$ local systems over $X$ by ${\mathcal L}(X)$.
Then, by \cite[(8.4)]{hain:normal}, the the forgetful map
\begin{equation}\label{ext_iso}
\Ext^1_{{\mathcal H}({\mathcal M}_g)}(H_1({\mathfrak u}_g),{\mathbb Q}(1))
\to \Ext^1_{{\mathcal L}({\mathcal M}_g)}(H_1({\mathfrak u}_g),{\mathbb Q}(1))
\cong H^1(\Gamma_g,H^1({\mathfrak u}_g))
\end{equation}
is an isomorphism.\footnote{This group is one dimensional and
generated by the Johnson homomorphism, although we do not really
need to know this here --- see \cite[(5.2)]{hain:normal}.}
We will lift the MHS on ${\mathfrak u}_g(x)$ to ${\mathfrak t}_g(x)$ using an element of the
right hand group.\footnote{The class we seek, not surprisingly, is the
one corresponding to the Johnson homomorphism, and is half the class
associated to the algebraic cycle $C-C^-$ in $\Jac C$ --- see
\cite[\S8]{hain:normal}.} We do this by producing an element of the
right hand group which corresponds to the central extension (\ref{extn}).
Denote the $k$th term of the lower central series of ${\mathfrak t}_g(x)$
by ${\mathfrak t}_g(x)^{(k)}$. We can form the extension
\begin{equation}\label{extn2}
0 \to {\mathfrak t}_g(x)^{(2)}/{\mathfrak t}_g(x)^{(3)} \to
{\mathfrak t}_g(x)/{\mathfrak t}_g(x)^{(3)} \to H_1({\mathfrak t}_g(x)) \to 0
\end{equation}
As has been pointed out above, the image of ${\mathfrak t}_g(x)^{(2)}/{\mathfrak t}_g(x)^{(3)}$
in the central $\Ga$ in ${\mathcal T}_{g,r}^n(x)$ is non-trivial. The kernel of
the extension (\ref{extn2}) is a rational
representation of $Sp_g$. Since $H_1({\mathfrak t}_g(x))$ is irreducible, its
second exterior power contains exactly one copy of the trivial
representation. There is therefore a unique non-zero $Sp_g$-invariant
projection
$$
{\mathfrak t}_g(x)^{(2)}/{\mathfrak t}_g(x)^{(3)} \to \Ga.
$$
If we push the extension (\ref{extn2}) out along this map,
we obtain a canonical element of
$$
\Ext^1_{{\mathcal L}({\mathcal M}_g)}(H_1({\mathfrak u}_g),{\mathbb Q}(1)).
$$
By the isomorphism (\ref{ext_iso}), we obtain an element
of
$$
\Ext^1_{{\mathcal H}({\mathcal M}_g)}(H_1({\mathfrak u}_g),{\mathbb Q}(1))
$$
which allows us, for each point $x$ of ${\mathcal M}_g$, to lift the MHS
on ${\mathfrak u}_g(x)$ associated to $x$ to a MHS on ${\mathfrak t}_g(x)$.
Our last task is to show that the bracket is a morphism of MHS.
We only need show that the bracket preserves the Hodge and
weight filtrations.
First observe that since $\Ga$ is central and contained in $W_{-2}$,
the bracket preserves the weight filtration, and its restriction to
$W_{-2}{\mathfrak t}_g(x)$ is a morphism of MHS. It remains to show that the
the bracket preserves the Hodge filtration. In view of these facts,
it suffices to prove that
$$
\left[F^p{\mathfrak t}_g(x),F^q{\mathfrak t}_g(x)\right] = 0
$$
when $p+q > -1$. This is easily deduced from the the
fact that the map
$$
[\phantom{x},\phantom{x}] : \Lambda^2 H_1({\mathfrak t}_g)
\to \Gr^W_{-2} {\mathfrak t}_g(x) \stackrel{\text{proj}}{\to} \Ga\cong {\mathbb Q}(1)
$$
induced by the bracket is a polarization of $H_1({\mathfrak t}_g(x))$ as it is
$Sp_g$ equivariant and non-zero by results in \cite[\S7]{hain:comp}.
\end{proof}
\begin{remark}
It follows immediately from (\ref{action_morph}) and
(\ref{mhs_torelli}) that for each choice of base point in
${\mathcal M}_{g,r}^n$, the canonical morphism
${\mathfrak t}_{g,r}^n(x) \to \Der {\mathfrak p}_{g,r}^n(x)$ is a morphism of MHS.
\end{remark}
\section{Review of Continuous Cohomology}
\label{cts_coho}
In this section, we briefly review the theory of continuous cohomology
of discrete groups, which is mainly developed in \cite{hain:cycles}.
It will be our principal tool in proving that Torelli has a
presentation with only quadratic relations. As a warm up, we show
how it can be used to give a new and simpler proof of Morgan's theorem
that the complex form of the Lie algebra associated to the fundamental
group of a smooth variety has a weighted homogenous presentation with
generators of weights equal to those occurring in $H_1(X)$, and relations
of weight contained in those of $H_2(X)$.
Define the continuous cohomology of a discrete group $\pi$ to be
the direct limit of the rational cohomology of its finitely
generated nilpotent quotients:
$$
H_{\mathrm{cts}}^{\bullet}(\pi,{\mathbb Q}) := \lim_\to H^{\bullet}(N,{\mathbb Q}) \label{ctsgp_def}
$$
where $N$ ranges over the finitely generated nilpotent quotients
of $\pi$. There is an obvious natural homomorphism
\begin{equation}\label{natural_homom}
H_{\mathrm{cts}}^{\bullet}(\pi,{\mathbb Q}) \to H^{\bullet}(\pi,{\mathbb Q}).
\end{equation}
If $X$ is a topological space with fundamental group $\pi$, then
we also have a natural homomorphism
$$
H_{\mathrm{cts}}^{\bullet}(\pi,{\mathbb Q}) \to H^{\bullet}(X,{\mathbb Q})
$$
as there is a canonical map $H^{\bullet}(\pi) \to H^{\bullet}(X)$.
\begin{proposition}\label{cts_ord}
If $H_1(\pi,{\mathbb Q})$ is finite dimensional, then the natural homomorphism
(\ref{natural_homom}) is an isomorphism in degree 1 and injective in
degree 2. \qed
\end{proposition}
This is really a restatement of the result of Dennis Sullivan which
asserts that the Lie algebra of the 1-minimal model of a space is
the Malcev Lie algebra of the fundamental group. A more direct proof
can be found, for example, in \cite[(5.1)]{hain:cycles}. We present a
new proof because it is elementary.
\begin{proof}
The group $H^2(G,{\mathbb Q})$ parameterizes central extensions of $G$ by ${\mathbb Q}$.
Suppose that $\alpha \in H_{\mathrm{cts}}^2(\pi,{\mathbb Q})$ whose image in $H^2(\pi,{\mathbb Q})$
is trivial. Then there is a nilpotent quotient $N$ of $\pi$ and an element
$\tilde{\alpha}$ of $H^2(N,{\mathbb Q})$ that is a lift of $\alpha$. There is a
central extension
\begin{equation}\label{alpha}
1 \to {\mathbb Q} \to E \to N \to 1
\end{equation}
corresponding to by $\tilde{\alpha}$. The key point to note is that $E$ is
nilpotent. To say that the image of $\alpha$ is trivial in $H^2(\pi,{\mathbb Q})$
is to say that the pullback of the extension (\ref{alpha}) to $\pi$ is
split. Composing a splitting of this projection with the projection of
$\pi \to N$ gives a homomorphism $\pi \to E$ which lifts $\pi \to N$.
Denote the image of $\pi$ in $E$ by $\tilde{N}$. It is easy to see that
the pullback of the extension (\ref{alpha}) to $\tilde{N}$ splits. Since
$\tilde{N}$ is a nilpotent quotient of $\pi$, the class $\alpha$ vanishes.
\end{proof}
Similarly, we can define the continuous cohomology of a pronilpotent
Lie algebra ${\mathfrak g}$ to be the direct limit of the
cohomology of its finite dimensional nilpotent quotients:
$$
H_{\mathrm{cts}}^{\bullet}({\mathfrak g}) := \lim_\to H^{\bullet}({\mathfrak n}) \label{ctslie_def}
$$
where ${\mathfrak n}$ ranges over the finite dimensional nilpotent quotients
of ${\mathfrak g}$.
A mild generalization of a theorem of Nomizu \cite{nomizu} states
that for each finitely generated nilpotent group $N$ there is a natural
isomorphism
$$
H^{\bullet}({\mathfrak n}) \cong H^{\bullet}(N,{\mathbb Q})
$$
where ${\mathfrak n}$ is the Lie algebra of the Malcev completion of $N$. It
follows immediately from the definitions that if $\pi$ is a finitely
generated group and ${\mathfrak p}$ the associated Malcev Lie algebra, then there
is a natural isomorphism
$$
H_{\mathrm{cts}}^{\bullet}({\mathfrak p}) \cong H_{\mathrm{cts}}^{\bullet}(\pi,{\mathbb Q}).
$$
The continuous cohomology of a pronilpotent Lie algebra ${\mathfrak g}$ can
be computed using the standard complex ${\mathcal C}^{\bullet}({\mathfrak g})$ of continuous
cochains of ${\mathfrak g}$. This is defined to be the direct limit of
the Chevalley-Eilenberg cochains of the finite dimensional nilpotent
quotients of ${\mathfrak g}$. Denote the continuous dual of ${\mathfrak g}$ by ${\mathfrak g}^\ast$.
Then we have a d.g.a.\ isomorphism
$$
{\mathcal C}^{\bullet}({\mathfrak g}) = \Lambda^{\bullet} \left({\mathfrak g}^\ast[-1]\right);
$$
the differential is derivation of degree 1 whose restriction
to ${\mathfrak g}^\ast$ is minus the dual of the bracket.
The definition of continuous cohomology can be extended to the case
where the coefficients are ${\mathbb Q}$ modules on which ${\mathfrak g}$ acts via
a representation of one of its nilpotent quotients --- cf.\
\cite{hain:cycles}.
Suppose now that $H_1({\mathfrak g})$ is finite dimensional. If ${\mathfrak g}$ has a MHS,
then, by linear algebra, so do ${\mathcal C}^{\bullet}({\mathfrak g})$ and $H^{\bullet}({\mathfrak g})$. We will
call such a pronilpotent Lie algebra a {\it Hodge Lie algebra}. It
follows that if $X$ is an algebraic variety, $x\in X$, then
$H_{\mathrm{cts}}^{\bullet}(\pi_1(X,x),{\mathbb Q})$ has a canonical MHS. One can show that
this MHS does not depend on the base point $x$ of $X$ ---
\cite{hain:cycles}.
Since the weight filtration of a MHS splits canonically over ${\mathbb C}$,
each finite dimensional Hodge Lie algebra ${\mathfrak g}$ is canonically
isomorphic to the graded Lie algebra $\Gr^W_{\bullet}{\mathfrak g}$ after tensoring
with ${\mathbb C}$. The following result therefore follows by taking inverse
limits.
\begin{proposition}\label{canon_split}
If ${\mathfrak g}$ is a Hodge Lie algebra, all of whose weights are negative,
then there is a canonical Lie algebra isomorphism
$$
{\mathfrak g}_{\mathbb C} \cong \prod_{l\ge 1} \Gr^W_{-l}{\mathfrak g}_{\mathbb C}. \qed
$$
\end{proposition}
Since each choice of a base point of ${\mathcal M}_{g,r}^n$ determines a
canonical MHS on ${\mathfrak t}_{g,r}^n$, we have:
\begin{corollary}
For each choice of a base point of ${\mathcal M}_{g,r}^n$, there is a
canonical isomorphism
$$
{\mathfrak t}_{g,r}^n\otimes {\mathbb C} \cong
\prod_{l\ge 1}\left(\Gr^W_{-l} {\mathfrak t}_{g,r}^n \otimes {\mathbb C}\right). \qed
$$
\end{corollary}
The following result is proved, for example, in
\cite[(11.7)]{carlson-hain}. In Section \ref{cts_coho_tor} we will
prove a generalization needed for studying the relations in ${\mathfrak t}_g$.
\begin{theorem}\label{cts_morph}
If $X$ is a smooth algebraic variety, then the natural homomorphism
$$
H_{\mathrm{cts}}^{\bullet}(\pi_1(X),{\mathbb Q}) \to H^{\bullet}(X,{\mathbb Q})
$$
is a morphism of mixed Hodge structures.
\end{theorem}
The final two results in this section together will allow us to
use continuous cohomology as an effective tool for studying relations
in Hodge Lie algebras in general, and ${\mathfrak t}_g$ in particular.
The cochains, and therefore the cohomology, of a graded Lie algebra
both have an extra grading, and are therefore bigraded algebras. If
${\mathfrak g}$ is a Hodge Lie algebra, then $\Gr^W_{\bullet}{\mathfrak g}$ has an extra grading
is by weight. Since the functor $\Gr^W$ is exact on the category of
MHS, we have:
\begin{proposition}
If ${\mathfrak g}$ is a Hodge Lie algebra, then there is a
canonical bigraded algebra isomorphism
$$
\Gr^W_{\bullet} H_{\mathrm{cts}}^{\bullet}({\mathfrak g}) \cong H^{\bullet}(\Gr^W_{\bullet} {\mathfrak g}). \qed
$$
\end{proposition}
If ${\mathfrak g}$ is a graded Lie algebra with negative weights, then we can write
${\mathfrak g}$ as a quotient of the free graded Lie algebra ${\mathfrak f}$ generated by
$H_1({\mathfrak g})$ modulo a homogeneous ideal ${\mathfrak r}$. Note that we are not assuming
that $H_1({\mathfrak g})$ is pure --- in general it will be graded. The group
$$
H_0({\mathfrak f}/{\mathfrak r}) = {\mathfrak r}/[{\mathfrak f},{\mathfrak r}]
$$
is graded. One can obtain a minimal set of relations of ${\mathfrak g}$ by
taking the image of any splitting of the projection
$$
{\mathfrak r} \to H_0({\mathfrak f}/{\mathfrak r}).
$$
The following result is an analogue of Hopf's description
of the second homology of a group in terms of a presentation.
\begin{proposition}
If ${\mathfrak g}$ is a graded Lie algebra with negative weights,
then there is a canonical isomorphism of graded vector spaces
$$
H_0({\mathfrak f}/{\mathfrak r}) \cong H_2({\mathfrak g}).
$$
\end{proposition}
\begin{proof}
There are several ways to see this. One is look closely at the
Chevalley-Eilenberg cochains of ${\mathfrak g}$. The second is the use the
fact that a sub-Lie algebra of a free Lie algebra is free \cite%
[(2.5)]{reutenauer} to deduce that, as a Lie algebra, ${\mathfrak r}$ is free.
Then apply the Lie algebra analogue of the Hochschild-Serre spectral
sequence to the extension
$$
0 \to {\mathfrak r} \to {\mathfrak f} \to {\mathfrak g} \to 0.
$$
The details are standard and are omitted.
\end{proof}
\begin{corollary}\label{gr_presentn}
If ${\mathfrak g}$ is a graded Lie algebra with negative weights, then
there is an injective linear map
$$
\delta : H_2({\mathfrak g}) \hookrightarrow {\mathbb L}(H_1({\mathfrak g}))
$$
of graded vector spaces such that ${\mathfrak g}$ has presentation
$$
{\mathbb L}(H_1({\mathfrak g}))/(\im \delta)
$$
in the category of graded Lie algebras.
\end{corollary}
Combining (\ref{canon_split}), (\ref{gr_presentn}) and the
existence of a canonical MHS on the Malcev Lie algebra ${\mathfrak g}(X,x)$
associated to a pointed variety, we obtain
one of Morgan's theorems \cite[(10.3)]{morgan}.
\begin{theorem}\label{morgan}
If $X$ is a smooth complex algebraic variety and $x\in X$, then
then the complex Malcev Lie algebra ${\mathfrak g}(X,x)_{\mathbb C}$ associated to
$\pi_1(X,x)$ has the property that
$$
{\mathfrak g}(X,x)_{\mathbb C} \cong \prod_{l \ge 1}\Gr^W_{-l}{\mathfrak g}_C
$$
and there is a homomorphism of graded vector spaces
$$
\delta : H_2(X,{\mathbb C}) \to {\mathbb L}(\Gr^W_{\bullet} H_1(X))
$$
such that
$$
\Gr^W_{\bullet} {\mathfrak g}_{\mathbb C} \cong
{\mathbb L}(\Gr^W_{\bullet} H_1(X,{\mathbb C}))/(\delta(\Gr^W_{\bullet} H_2(X,{\mathbb C})))
$$
in the category of graded Lie algebras. \qed
\end{theorem}
\section{Remarks on the Representations of $\sp_g$}
\label{reps}
In this section we review some basic facts from the representation
theory that we shall need in subsequent sections. A
good reference is \cite{fulton-harris}.
Denote the Lie algebra of $Sp_g$ by \label{symp_def} $\sp_g$. The
representation theory of the group and the Lie algebra are the same.
Denote their common representation ring by \label{rep_def} $R(\sp_g)$.
Choose a symplectic basis $a_1,\dots,a_g,b_1,\dots,b_g$ of the
fundamental representation of $\sp_g$. Denote by ${\mathfrak h}$ the torus in
$\sp_g$ consisting of matrices that are diagonal with respect to this
basis. Choose coordinates $t=(t_1,\ldots,t_g)$ on ${\mathfrak h}$ so that
$$
t\cdot a_i = t_ia_i \text{ and } t\cdot b_i = - t_i b_i.
$$
The subalgebra of positive nilpotents ${\mathfrak n}$ has basis the elements
$S_{i,j}, (i<j)$, $T_i$, and $F_{i,j}, (i\neq j)$ of $\sp_g$,
where
$$
S_{i,j} (a_j) = a_i, \quad S_{i,j}(b_i) = - b_j, \quad
S_{i,j}(\text{other basis vectors}) = 0,
$$
$$
T_i(b_i) = a_i, \quad T_i(\text{other basis vectors}) = 0,
$$
$$
F_{i,j}(b_i) = a_j, \quad F_{i,j} (b_j) = a_i, \quad
F_{i,j}(\text{other basis vectors}) = 0.
$$
A fundamental set of weights of $\sp_g$ is $\lambda_j:{\mathfrak h} \to {\mathbb R}$,
$1\le j \le g$, where $\lambda_j$ is defined by
$$
\lambda_j(t) = t_1 + t_2 + \cdots + t_j. \label{wt_def}
$$
The irreducible representations of $\sp_g$ correspond to
positive integral linear combinations $\lambda$ of the $\lambda_j$.
Denote the irreducible representation of $\sp_g$ with highest
weight $\lambda$ by \label{module_def} $V(\lambda)$.
The irreducible representations of $\sp_g$ can also be indexed
by partitions $\alpha$ of an integer $n$ into $\le g$ parts:
$$
n = \alpha_1 + \alpha_2 + \dots + \alpha_g
$$
where
$$
\alpha_1 \ge \alpha_2 \ge \dots \ge \alpha_g \ge 0.
$$
The irreducible representation corresponding to $\alpha$
has highest weight
$$
t \mapsto \sum_j \alpha_j t_j.
$$
We shall denote the integer
$$
\sum_{k=1}^g \alpha_k = \sum_{k=1}^g k\, n_k \label{size_def}
$$
by $|\alpha|$ or by $|\lambda|$ according to whether we are using
partitions or highest weights. This can be considered as a measure
of the size of the corresponding irreducible representation; it is the
smallest positive integer $d$ such that $V(\lambda)$ occurs in the
$d$th tensor power of the fundamental representation.
There is a notion of stability of the decomposition of tensor
products and Schur functors of representations of symplectic groups.
In order to state the result, we need to first define the {\it depth},
$\delta(V)$, of a representation $V$ of $\sp_g$. If the module is
irreducible with highest weight $\sum n_k\,\lambda_k$, define
$\delta(V)$ to be the largest $d$ such that $n_d\neq 0$ --- or
equivalently, it is the number of rows in the corresponding Young
diagram. Define the depth of an arbitrary representation to be the
maximum of the depths of its irreducible components.
In order to discuss stability, we will need a stabilization map.
When $h \ge g$, define a group homomorphism
$$
R(\sp_g) \hookrightarrow R(\sp_h)
$$
by taking the irreducible representation of $\sp_g$ corresponding to the
partition $\alpha$ to the representation of $\sp_h$ corresponding to the
same partition. Equivalently, take the representation of $\sp_g$ with
highest weight $\sum n_k\, \lambda_k$ to the representation of $\sp_h$
with the same highest weight decomposition.
Recall that to each partition $\beta$ of a positive integer $n$,
one has a Schur functor $\Schur_\beta$ defined on the category of
representations of each group. For example, if $\beta = [n]$,
then $\Schur_\beta$ is the $n$th symmetric power, if $\beta = [1^n]$,
then $\Schur_\beta$ is the $n$th exterior power. We shall denote the
integer $n$ by $|\beta|$.
The second assertion of the following stability result appears to be
folklore --- the only proof I know of is in Kabanov's thesis.
\begin{theorem}\label{kab_stab}
\begin{enumerate}
\item (\cite[p.~424]{fulton-harris}) If $V$ and $W$ are representations
of $\sp_g$ and $\delta(V)
+ \delta(W) \le g$, then the irreducible representations and their
multiplicities occurring in the decomposition of $V\otimes W$ is
independent of $g$.
\item (\cite{kabanov} --- see also \cite{kabanov:stab}) If $V$ is a
representation of $\sp_g$ and $\beta$ is a partition
with $|\beta|\delta(V) \le g$, then the decomposition of
$\Schur_\beta V$ into irreducible components is independent of $g$.
\end{enumerate} \qed
\end{theorem}
\begin{remark}\label{method_comp}
Some of the computations of highest weight decompositions in this
paper have been made using the computer program {\textsf{LiE}}\ from the
University of Amsterdam. The computations were performed for a
particular $g$ in the stable range. The stability theorem was then
used to deduce the decomposition for all $g$ in the stable range. Note
that all such computations were checked using several values of $g$ in
the stable range. In addition, many of the unstable computations were
done using {\textsf{LiE}}.
\end{remark}
By composition with the canonical homomorphism
$$
\Gamma_{g,r}^n \to Sp_g({\mathbb Q})
$$
we see that each representation $V$ of $\sp_g$ gives rise to a local
system over ${\mathcal M}_{g,r}^n$, at least in the orbifold sense.
It is a standard fact that each such local system arising from an
irreducible representation of $\sp_g$ is an admissible variation
of Hodge structure over ${\mathcal M}_{g,r}^n$ in a unique way up to Tate
twist --- cf.\ \cite[(9.1)]{hain:normal}. It can always be realized
as a variation of weight $|\lambda|$, and we shall take this as the
default weight.
We would like to discuss the cohomology of ${\mathcal M}_{g,r}^n$ with
coefficients in such a local system. To do this, first choose
a level $l$ such that $\Gamma_{g,r}^n$ is torsion free. Then
${\mathcal M}_{g,r}^n[l]$ is smooth and the variation of Hodge structure ${\mathbb V}$
corresponding to an irreducible representation $V$ of $Sp_g$ is defined
over ${\mathcal M}_{g,r}^n[l]$ and has a natural $Sp_g({\mathbb Z}/l{\mathbb Z})$ action. From the
work of M.~Saito \cite{saito}, we know that $H^k({\mathcal M}_{g,r}^n[l],{\mathbb V})$ has
a canonical mixed Hodge structure with weights
$\ge k + \text{weight}(V)$. The action of $Sp_g({\mathbb Z}/l{\mathbb Z})$ preserves
this MHS. So we can define
$$
H^{\bullet}({\mathcal M}_{g,r}^n,{\mathbb V}) = H^0(Sp_g({\mathbb Z}/l{\mathbb Z}),H^{\bullet}({\mathcal M}_{g,r}^n[l],{\mathbb V})).
$$
as a MHS. Note that the underlying group is canonically isomorphic to
$H^{\bullet}(\Gamma_{g,r}^n,V)$.
\section{Continuous Cohomology of Torelli Groups}
\label{cts_coho_tor}
The next step in finding a presentation of ${\mathfrak t}_{g,r}^n$ is to determine
the relations in $\Gr^W_{\bullet} {\mathfrak t}_{g,r}^n$. Since this is a graded Lie
algebra generated in degree $-1$, the generators of the ideal of
relations is homogeneous. In this section we will use a result of
Kabanov \cite{kabanov} (see also \cite{kabanov:purity}) about the second
cohomology of $\Gamma_{g,r}^n$
to show that the ideal of relations in $\Gr{\mathfrak t}_{g,r}^n$ is generated
by quadratic and cubic generators when $g\ge 3$, and quadratic
relations alone when $g\ge 6$.
Our principal tool will be the continuous cohomology defined in
Section \ref{cts_coho}.
First some notation. Take $X$ and ${\mathbb V}$ as in the
statement of (\ref{mhs_gen}). Denote the Lie algebra associated to
the prounipotent radical of the completion of $\pi_1(X,x)$ relative
to $\rho$ by ${\mathfrak u}(x)$. This is a Hodge Lie algebra. The next result is
a generalization of (\ref{cts_morph}).
\begin{proposition}\label{morphism}
Suppose that ${\mathbb W}$ is an admissible variation of Hodge structure over
$X$ which is a subquotient of a tensor power of ${\mathbb V}$. Then for all
$k\ge 0$ and each $x\in X$, there is a natural homomorphism
$$
H^0(X,H_{\mathrm{cts}}^k({\mathfrak u}(x))\otimes W_x) \to H^k(X,{\mathbb W})
$$
which is a morphism of MHS. It is an isomorphism when $k=1$ and
injective when $k=2$.
\end{proposition}
\begin{proof}
The case $k=1$ is proved in \cite[(10.3),(13.8)]{hain:derham}.
We will prove the result when $k>1$ by induction. The most important
case for us is when $k=2$, so we will give that argument
in more detail and briefly sketch the remaining cases. We will assume
throughout that the reader is familiar with \cite{hain:derham}. A
convenient auxiliary reference for rational homotopy theory is
\cite[\S2]{hain:dht}.
We begin by recalling some well known facts from rational
homotopy theory. The base point $x\in X$ determines an
augmentation
$$
\epsilon_x : \Efin^{\bullet}(X,{\mathcal O}(P)) \to {\mathbb R},
$$
where $P$ is the principal bundle defined in \cite[\S 4]{hain:derham}.
We shall write ${\mathcal O}$ instead of ${\mathcal O}(P)$.
We can form the bar construction
$$
B(\Efin^{\bullet}(X,{\mathcal O})) := B({\mathbb R},\Efin^{\bullet}(X,{\mathcal O}),{\mathbb R})
$$
where both copies of ${\mathbb R}$ are regarded as $\Efin^{\bullet}(X,{\mathcal O})$
modules via $\epsilon_x$. The Lie algebra ${\mathfrak u}(x)$ is determined
by $B(\Efin^{\bullet}(X,{\mathcal O}))$ as follows: the dual
$$
H^0(B(\Efin^{\bullet}(X,{\mathcal O})))
$$
is a complete Hopf algebra, ${\mathfrak u}(x)$ is its set of primitive elements.
(See, for example, \cite[(2.4.5) and \S 2.6]{hain:dht}.) There is an
augmentation preserving d.g.a.\ homomorphism
\begin{equation}\label{min_model}
{\mathcal C}^{\bullet}({\mathfrak u}(x)) \to \Efin^{\bullet}(X,{\mathcal O}),
\end{equation}
unique up to homotopy, which induces the map
$$
\theta : H_{\mathrm{cts}}^{\bullet}({\mathfrak u}(x)) \to H^{\bullet}(\Efin^{\bullet}(X,{\mathcal O}))
$$
on homology. The map $\theta$ is an isomorphism in degree 1 and injective
in degree 2.\footnote{In the language of Sullivan \cite{sullivan}, the
map (\ref{min_model}) is the 1-minimal model of $\Efin^{\bullet}(X,{\mathcal O})$.}
There is a canonical isomorphism
$$
H^k(X,{\mathbb W}) \cong H^0(X,H^k(\Efin^{\bullet}(X,{\mathcal O}))\otimes{\mathbb W})
$$
of MHSs for each VHS ${\mathbb W}$ over $X$ whose monodromy representation
is the pullback of a rational representation of $\Aut(V_o,q)$ via
the representation $\rho$. Since ${\mathfrak u}(x)$ has a canonical MHS, and
since $\Efin^{\bullet}(X,{\mathcal O})$ is a mixed Hodge complex, each of the domain
and target of $\theta$ have a canonical MHS. To prove
the result, we need only prove that $\theta$ is a morphism
of MHS.
First we give an intuitive proof. The image of the map
$$
\theta^2 : H_{\mathrm{cts}}^2({\mathfrak u}) \to H^2(\Efin^{\bullet}(X,{\mathcal O}))
$$
is the subspace of the right hand side generated by the cup product
$H^1\otimes H^1 \to H^2$, all Massey triple products of 1-forms,
all Massey quadruple products of 1-forms, etc. Since the cup product
and all Massey $k$-fold products have domain which is a sub-MHS of
$\otimes^k H^1$ and are themselves
morphisms, it follows that the image of $\theta^2$ is a MHS. That
$\theta^2$ is a morphism follows as $\theta^1$ is an isomorphism of MHS.
One can continue in an analogous fashion to prove that each $\theta^k$
is a morphism.
We now make this argument precise. The spectral sequence
associated to the standard filtration of the bar construction
is called the Eilenberg-Moore spectral sequence (EMss): for an
augmented d.g.a.\ $A^{\bullet}$ with connected homology, it takes the
form
$$
E_1^{-s,t} = \left[\otimes^s H^+(A)\right]^t \implies
H^{t-s}(B(A)).
$$
Denote the EMss associated to ${\mathcal C}({\mathfrak u})^{\bullet}$ by $\{E_r({\mathfrak u})\}$ and the
EMss associated to $\Efin^{\bullet}(X,{\mathcal O})$ by $\{E_r(X)\}$.
The map (\ref{min_model}) induces a morphism of Eilenberg-Moore
spectral sequences. Each of these is a spectral sequence of MHSs
as both the domain and target of (\ref{min_model}) are mixed Hodge
complexes, but we have to prove that the map between them is a morphism
of MHSs. This is the case in total degree 0 as $E_1^{-s,s}$ is
$\otimes^s H^1$ and $\theta^1$ is an isomorphism of MHS.
It is a standard fact that
$$
H^k(B({\mathcal C}^{\bullet}({\mathfrak u}))) = 0
$$
when $k>0$; cf.\ \cite[(2.6.2)]{hain:dht} and \cite{bloch-kriz}.
Therefore, the $E^{-1,2}_\infty$ term of the associated EMss vanishes.
(This is a precise way to say that $H_{\mathrm{cts}}^2({\mathfrak u})$ is generated by Massey
products.) The edge homomorphisms
$$
H_{\mathrm{cts}}^k({\mathfrak u}) = E_1^{-1,k}({\mathfrak u}) \to E_r^{-1,k}({\mathfrak u})
$$
are all surjective. Let $M_r^k$ be the inverse image in $H_{\mathrm{cts}}^k({\mathfrak u})$
of the image of
$$
d_{r-1} : E_{r-1}^{-r,k+r-2}({\mathfrak u}) \to E_{r-1}^{-1,k}({\mathfrak u}).
$$
Then the fact that the higher cohomology of $B({\mathcal C}^{\bullet}({\mathfrak u}))$ vanishes
implies that whenever $k\ge 2$
$$
H_{\mathrm{cts}}^k({\mathfrak u}) = \bigcup_r M_r^k.
$$
Since the spectral sequence is a spectral sequence of MHS, each
$M_r^k$ is a sub-MHS of $H_{\mathrm{cts}}^k({\mathfrak u})$.
Since both spectral sequences are spectral sequences of MHSs, it
follows that the image of
$$
H_{\mathrm{cts}}^2({\mathfrak u})=E_1^{-1,2}({\mathfrak u}) \to E_1^{-1,2}(X) = H^2(\Efin^{\bullet}(X,{\mathcal O}))
$$
is a sub-MHS and that $\theta^2$ is a morphism of MHS.
If $k>2$, one can assume by induction that $\theta^m$ is a morphism
whenever $m<k$. It follows easily that the natural map
$$
E_1^{-s,t}({\mathfrak u}) \to E_1^{-s,t}(X)
$$
is a morphism of MHS whenever $-s+t < k-1$, and therefore that its
image is a sub-MHS of $E_1^{-s+t}(X)$. But since these spectral
sequences are spectral sequences in the category of MHSs, and since
$E_\infty^{-1,k}({\mathfrak u})$ vanishes, it follows that $\theta^k$ is a
morphism.
\end{proof}
\begin{remark}
This is a continuation of Remark~\ref{sl2}. The previous result implies
that ${\mathcal U}_1[l]$ is a free pronilpotent group as $SL_2({\mathbb Z})[l]$ has a free
subgroup of finite index, which implies that $H^2({\mathcal M}_1[l],S^n V)$
vanishes for all $n$. It follows that $H^2({\mathfrak u}_1[l])$ vanishes and from
(\ref{gr_presentn}) that ${\mathfrak u}_1[l]$ is free.
\end{remark}
We thus have the following version of (\ref{morphism}) for moduli
spaces of curves.
\begin{proposition}\label{morph_u}
If $g\ge 3$ and ${\mathbb V}$ is a variation of Hodge structure over ${\mathcal M}_{g,r}^n$
whose monodromy representation comes from a rational representation
of $Sp_g$, then for all $k$, there is a natural map
$$
H^0(Sp_g,H_{\mathrm{cts}}^k({\mathfrak u}_{g,r}^n(x))\otimes V_x) \to
H^k({\mathcal M}_{g,r}^n,{\mathbb V})
$$
which is a morphism of MHS. Here $V_x$ denotes the fiber of ${\mathbb V}$
over $x$. \qed
\end{proposition}
This yields the following useful result about
differentials in the Hochschild-Serre spectral sequence associated
to the group extension
\begin{equation}\label{std}
1 \to T_{g,r}^n \to \Gamma_{g,r}^n \to Sp_g({\mathbb Z}) \to 1.
\end{equation}
If we take coefficients in the irreducible representation $V(\lambda)$
of $Sp_g$, this spectral sequence takes the form
$$
E_2^{s,t} = H^0(Sp_g({\mathbb Z}),H^t(T_{g,r}^n)\otimes V(\lambda))
\implies H^{s+t}(\Gamma_{g,r}^n,V(\lambda)).
$$
\begin{corollary}\label{vanishing}
For each $\lambda$, the image of the composite
$$
H^0(Sp_g,H_{\mathrm{cts}}^2({\mathfrak t}_{g,r}^n)\otimes V(\lambda))
\to H^0(Sp_g({\mathbb Z}),H^2(T_{g,r}^n)\otimes V(\lambda)) = E_2^{0,2}
$$
is contained in $E_\infty^{0,2}$.
\end{corollary}
\begin{proof}
The result follows immediately from the fact that the diagram
$$
\begin{CD}
H^0(Sp_g,H_{\mathrm{cts}}^2({\mathfrak u}_{g,r}^n)\otimes V(\lambda)) @>>>
H^0(Sp_g,H_{\mathrm{cts}}^2({\mathfrak t}_{g,r}^n)\otimes V(\lambda)) \cr
@VVV @VVV \cr
H^2(\Gamma_{g,r}^n,V(\lambda)) @>>>
H^0(Sp_g({\mathbb Z}),H^2(T_{g,r}^n)\otimes V(\lambda))
\end{CD}
$$
commutes, where the top map is the surjection induced by the projection
of $t_{g,r}^n$ onto ${\mathfrak u}_{g,r}^n$, the right hand vertical map by the
canonical map
$$
H_{\mathrm{cts}}^{\bullet}({\mathfrak t}_{g,r}^n) \to H^{\bullet}(T_{g,r}^n)
$$
described in Section \ref{cts_coho}, the bottom map is the canonical
restriction map, and the left hand vertical map is the one given by
(\ref{morph_u}). This assertion can be proved using the constructions
in \cite[\S4]{hain:derham} by restricting to a leaf in the
principal bundle $P \to {\mathcal M}_{g,r}^n$ associated to the representation
(\ref{map}). In this case, each leaf is a copy of the classifying
space of $T_{g,r}^n$.
\end{proof}
Actually, we have proved a stronger statement than asserted. The
stronger claim will be stated in \S\ref{applications}.
Denote the fiber over $x\in {\mathcal M}_{g,r}^n$ of the variation of Hodge
structure ${\mathbb V}(\lambda)$ corresponding to the irreducible representation
$V(\lambda)$ of $Sp_g$ by $V(\lambda)_x$.
\begin{corollary}
If $g\ge 3$, then for each irreducible representation $V(\lambda)$
of $Sp_g$, there is a canonical monomorphism of MHS
$$
H^0(Sp_g,H_{\mathrm{cts}}^2({\mathfrak t}_{g,r}^n(x))\otimes V(\lambda)_x)
\hookrightarrow H^2({\mathcal M}_{g,r}^n,{\mathbb V}(\lambda)).
$$
\end{corollary}
\begin{proof}
We first prove the existence of the homomorphism. Injectivity will
then follow directly from (\ref{morphism}). It follows from
(\ref{central_ext}) and \cite[(5.5)]{hain:cycles} that the sequence
of $Sp_g$ modules
$$
0 \to {\mathbb Q}(1) \to H_{\mathrm{cts}}^2({\mathfrak u}_{g,r}^n(x)) \to H_{\mathrm{cts}}^2({\mathfrak t}_{g,r}^n(x)) \to 0
$$
is an exact sequence of MHSs. It follows that the
natural map $H_{\mathrm{cts}}^2({\mathfrak u}_{g,r}^n(x)) \to H_{\mathrm{cts}}^2({\mathfrak t}_{g,r}^n(x))$ is a
surjective morphism of MHS. According to (\ref{morph_u}), the map
$$
\left[H_{\mathrm{cts}}^2({\mathfrak u}_{g,r}^n(x))\otimes V(\lambda)_x\right]^{Sp_g}
\to H^2({\mathcal M}_{g,r}^n,{\mathbb V}(\lambda))
$$
is a morphism of MHS. So to construct the homomorphism, it suffices to
show that this map factors through the quotient map
$$
H_{\mathrm{cts}}^2({\mathfrak u}_{g,r}^n) \to H_{\mathrm{cts}}^2({\mathfrak t}_{g,r}^n).
$$
We first assume that $\lambda \neq 0,\lambda_1,\lambda_3$. Consider
the Hochschild-Serre spectral sequence
$$
E_2^{s,t} = H^s(Sp_g({\mathbb Z}),H^t(T_{g,r}^n)\otimes V(\lambda))
\implies H^{s+t}(\Gamma_{g,r}^n,V(\lambda)).
$$
By (\ref{imp_borel}), $E_2^{s,t}$ vanishes when $s\le 1$ and
$t\le 2$ provided $g \ge 3$ (cf.\ \cite[(5.2)]{hain:normal}). It follows
that
$$
H^2(\Gamma_{g,r}^n,{\mathbb V}(\lambda)) =
H^0(Sp_g({\mathbb Z}),H^2(T_{g,r}^n)\otimes V(\lambda)).
$$
The result now follows from (\ref{cts_ord}).
When $\lambda=\lambda_3$, we have $E_2^{2,1} \cong {\mathbb Q}$ (cf.\
\cite[(5.2)]{hain:normal}), so there is a possibility of having
a non-trivial differential $d_2 : E_2^{0,2} \to E_2^{2,1}$. But
by (\ref{vanishing}) this cannot occur. The argument is completed
as in the previous case. The case of $\lambda_1$ is proved in the
same way.
Finally, we consider the case of the trivial representation. In this
case, we have $E_2^{2,1}=E_2^{3,0}=0$, but $E_2^{2,0}={\mathbb Q}$. It
follows that we have an exact sequence
$$
0 \to {\mathbb Q} \to H^2(\Gamma_{g,r}^n,{\mathbb Q}) \to
H^0(Sp_g({\mathbb Z}),H^2(T_{g,r}^n)) \to 0.
$$
The result in this case now follows using the exact sequence in
the first paragraph of this proof.
\end{proof}
Denote the $\lambda$ isotypical part of an $Sp_g$ module $V$
by \label{iso_def} $V_\lambda$.
\begin{corollary} If $g\ge 3$ and $\lambda$ is a dominant integral
weight of $Sp_g$, then
$$
\dim \Gr^W_l H_{\mathrm{cts}}^2({\mathfrak t}_{g,r}^n)_\lambda \le
\dim Gr^W_{l+|\lambda|} H^2({\mathcal M}_{g,r}^n,{\mathbb V}(\lambda)). \qed
$$
\end{corollary}
So, in order to bound the degrees of the relations in ${\mathfrak t}_{g,r}^n$,
it suffices to give a bound the weights on $H^2({\mathcal M}_{g,r}^n,{\mathbb V}(\lambda))$.
In the absolute case we have the following result of Kabanov
\cite{kabanov,kabanov:purity} which is proved using intersection homology.
\begin{theorem}[Kabanov]
For each irreducible rational representation $V(\lambda)$ of $Sp_g$,
we have
$$
\Gr^W_{k + |\lambda|}H^2({\mathcal M}_g,{\mathbb V}(\lambda)) = 0
$$
when
$$
\begin{cases}
k \neq 2 & \text{ when $g\ge 6$;}\cr
k \neq 2,3 & \text{ when $3 \le g < 6$.}
\end{cases}
$$
\end{theorem}
Combining Kabanov's result with the previous results, we obtain:
\begin{corollary}
If $g\ge 3$, then $\Gr^W_{\bullet}{\mathfrak t}_g$ has a presentation with only
quadratic and cubic relations, and only quadratic relations when
$g\ge 6$. \qed
\end{corollary}
It is now an easy matter to insert the decorations:
\begin{corollary}
If $g\ge 3$, then $\Gr^W_{\bullet}{\mathfrak t}_{g,r}^n$ has a presentation with only
quadratic and cubic relations, and only quadratic relations when
$g\ge 6$. \qed
\end{corollary}
\section{The Lower Central Series Quotients of a Surface Group}
In this section we gather some information about $\Gr{\mathfrak p}_g^1$
that will be useful when computing relations in $\Gr{\mathfrak t}_g$ and
$\Gr{\mathfrak t}_g^1$. Our basic tool, once again, is continuous cohomology.
A group is called {\it pseudo-nilpotent} if $\theta$ is an
isomorphism. A proof of the following result is sketched
by Kohno and Oda in \cite[(4.1)]{kohno-oda}.
\begin{theorem}\label{curve}
If $g \ge 1$, then $\pi_g^1$ is pseudo-nilpotent. \qed
\end{theorem}
Even though we will not be needing it, we record the following
result which is stated by Kohno and Oda \cite[(4.1)]{kohno-oda}.
Their proof is incorrect --- cf.\ (\ref{error}). Nonetheless, the
result follows directly from (\ref{curve}) and \cite[(5.7)]{hain:cycles}.
\begin{corollary}\label{kohno-oda}
If $g=0$ and $r\ge 1$, or if $g \ge 1$, then, for all $n\ge 0$,
each of the decorated pure braid groups $F_{g,r}^n$ is
pseudo-nilpotent. \qed
\end{corollary}
Since $H_1({\mathfrak p}_g^1)$ is the fundamental representation of $\sp_g$,
$\Gr^W_{\bullet} {\mathfrak p}_g^1$ is a graded Lie algebra in $R(\sp_g)$, and its
complex of chains $\Lambda^{\bullet} \Gr^W_{\bullet}{\mathfrak p}_g^1$ is a complex in
$R(\sp_g)$.
We shall write ${\mathfrak p}_g$ \label{p_def} instead of ${\mathfrak p}_g^1$, and $\pi_g$
\label{pi_def} instead of $\pi_g^1$. We shall denote the $l$th weight
graded quotient of a Hodge Lie algebra ${\mathfrak g}$ by \label{gr_def}
${\mathfrak g}(l)$. In particular, we shall denote $\Gr^W_{-l}{\mathfrak p}_g^1$ by
\label{pgr_def} ${\mathfrak p}_g(l)$.
\begin{corollary}\label{complex}
If $g\ge 1$, then, for each $l\ge 3$, the complex
$$
\Gr^W_{-l} \Lambda^{\bullet}\Gr^W_{\bullet}{\mathfrak p}_g
$$
is an acyclic complex of $\sp_g$ modules. When $l=2$, we have an
exact sequence
$$
0 \to {\mathbb Q}(1) \to \Lambda^2 {\mathfrak p}_g(1) \to {\mathfrak p}_g(2) \to 0
$$
of $\sp_g$ modules. \qed
\end{corollary}
This result allows us to compute the ${\mathfrak p}_g(l)$ inductively
as elements of $R(\sp_g)$. As before, we fix a set
$\lambda_1,\dots,\lambda_g$ of fundamental weights of $\sp_g$.
\begin{proposition}\label{lcs_quots}
For all $g\ge 3$, the highest weight decomposition of ${\mathfrak p}_g(l)$
when $1\le l \le 4$ is given by
$$
{\mathfrak p}_g(l) =
\begin{cases}
V(\lambda_1) & \text{ when $l=1$}; \cr
V(\lambda_2) & \text{ when $l=2$}; \cr
V(\lambda_1 + \lambda_2) & \text{ when $l=3$}; \cr
V(2\lambda_1) + V(2\lambda_1+\lambda_2) +
V(\lambda_1 + \lambda_3) & \text{ when $l=4$}.
\end{cases}
$$
\end{proposition}
\begin{proof}
This is a straightforward consequence of (\ref{complex}). To show
how this works, we prove the case where $l=3$. In this case, we
have the exact sequence
$$
0 \to \Lambda^3 {\mathfrak p}_g(1) \to {\mathfrak p}_g(1)\otimes {\mathfrak p}_g(2) \to {\mathfrak p}_g(3) \to 0
$$
in $R(\sp_g)$. Taking euler characteristics and applying the result
for $l=2$ and $g\ge 3$, we see that
$$
{\mathfrak p}_g(3) =
V(\lambda_1)\otimes V(\lambda_2) - \Lambda^3 V(\lambda_1)
= V(\lambda_1 + \lambda_2).
$$
\end{proof}
Since the $k$th exterior power is the Schur functor corresponding
to the Young diagram with $k$ rows and one box in each row, and
since ${\mathfrak p}_g(1) = H_1(\pi_g)$ is the fundamental representation
of $\sp_g$, we obtain the following stability result for the
graded quotients of the lower central series of $\pi_g$.
\begin{corollary}
The highest weight decomposition of ${\mathfrak p}_g(l)$ is independent of
$g$ when $l\ge g$. \qed
\end{corollary}
\section{The Action of ${\mathfrak t}_g^1$ on ${\mathfrak p}_g$}
\label{inf_action}
In this section we obtain a lower bound for the size of
$\Gr^W_l{\mathfrak t}_g$ when $l=2,3$ and $g\ge 3$ by studying the action of
${\mathfrak t}_g^1$ on ${\mathfrak p}_g$. This will provide an upper bound on the size of
the quadratic and cubic relations of $\Gr^W_{\bullet}{\mathfrak t}_g$. Related
results have been obtained Asada-Kaneko \cite{japanese}, Morita
\cite{morita:trace} and Asada-Nakamura \cite{asada-nakamura}. We also
use a result \cite{asada-nakamura} of Asada and Nakamura to prove
that ${\mathfrak t}_g$ is infinite dimensional. (This fact also follows from
a recent result of Oda \cite{oda}.)
First, some notation. Denote the pronilpotent Lie algebra
$W_{-1}\Der {\mathfrak p}_g$ by \label{der_def} ${\mathfrak d}_g$, and the quotient of this
by inner automorphisms by \label{out_def} ${\mathfrak o}_g$. Once a base point $x$
of ${\mathcal M}_g^1$ has been chosen, each of these acquires the structure
of a Hodge Lie algebra.
We have natural representations
$$
{\mathfrak u}_g^1 \to {\mathfrak d}_g \text{ and } {\mathfrak u}_g \to {\mathfrak o}_g.
$$
These induce homomorphisms of their associated graded Lie algebras.
It is clear that there is an injective homomorphism
$$
{\mathfrak d}_g(l) \hookrightarrow \Hom({\mathfrak p}_g(1),{\mathfrak p}_g(l+1)).
$$
Each element $\delta : {\mathfrak p}_g(1) \to {\mathfrak p}_g(l+1)$ determines a derivation
$\tilde{\delta}$ of the free Lie algebra ${\mathbb L}({\mathfrak p}_g(1))$, the second
graded quotient of which is isomorphic to ${\mathfrak p}_g(2) \oplus {\mathbb Q}\omega$,
where ${\mathbb Q} \omega$ is the unique copy of the trivial representation
in $\Lambda^2 {\mathfrak p}_g(1)$. By taking the image of $\tilde{\delta}(\omega)$
under the projection
$$
{\mathbb L}({\mathfrak p}_g(1)) \to \Gr^W_{\bullet} {\mathfrak p}_g,
$$
we obtain an element $\sigma_g(\delta)$ of ${\mathfrak p}_g(l+1)$. Observe
that $\delta$ induces a derivation of $\Gr{\mathfrak p}_g$ if and only if
$\sigma_g(\delta)$ vanishes. We therefore have a surjection
$$
{\mathfrak d}_g \to \ker\left\{\Hom({\mathfrak p}_g(1),{\mathfrak p}_g(l+1))
\stackrel{\sigma}{\to} {\mathfrak p}_g(l+2)\right\}.
$$
\begin{proposition}\label{gr_der}
The map $\sigma$ is surjective. Consequently,
$$
{\mathfrak d}_g(l) = {\mathfrak p}_g(1)\otimes{\mathfrak p}_g(l+1) - {\mathfrak p}_g(l+2)
$$
in $R(\sp_g)$.
\end{proposition}
\begin{proof}
Consider the diagram
$$
\begin{CD}
\Hom({\mathfrak p}_g(1),{\mathfrak p}_g(l+1)) @>{\sigma_g}>> {\mathfrak p}_g(l+2)\cr
@VVV @| \cr
{\mathfrak p}_g(1)\otimes {\mathfrak p}_g(l+1) @>{[\phantom{x},\phantom{x}]}>> {\mathfrak p}_g(l+2)
\end{CD}
$$
where the left hand vertical map is induced by the quadratic
form $\omega = \sum a_i\wedge b_i$. This diagram commutes as the
left hand map satisfies
$$
\delta \mapsto
\sum_{i=1^g} a_i\otimes\delta(b_i) - b_i\otimes\delta(a_i),
$$
which goes to
$$
\sigma_g(q) =
\sum_{i=1}^g\left([\delta(a_i),b_i] + [a_i,\delta(b_i)]\right)
$$
under the bracket. Since the bottom map is surjective and all
maps are $\sp_g$ equivariant, the result follows.
\end{proof}
Combining this with the computation of the first few graded
quotients of ${\mathfrak p}_g$ given in (\ref{lcs_quots}), we obtain the following
result.
\begin{corollary}\label{der_quots}
For all $g\ge 3$, we have
$$
{\mathfrak d}_g(l) =
\begin{cases}
V(\lambda_3)+V(\lambda_1) &\text{ when }l=1;\cr
V(2\lambda_2)+V(\lambda_2)&\text{ when }l=2;\cr
V(2\lambda_1+\lambda_3)+V(\lambda_1+\lambda_2)+V(3\lambda_1)&
\text{ when } l=3.
\end{cases}
$$ \qed
\end{corollary}
The computation of ${\mathfrak d}_g(1)$ is simply another formulation of the
Johnson homomorphism.
It is proven in \cite[$A^\prime$, p.~149]{japanese} that the center
of $\Gr{\mathfrak p}_g$ is trivial, so that the inclusion ${\mathfrak p}_g \to {\mathfrak d}_g$ of
the inner automorphisms is injective.
\begin{proposition}
For all $g\ge 3$ and all $l\ge 1$,
${\mathfrak o}_g(l) = {\mathfrak d}_g(l) - {\mathfrak p}_g(l)$. \qed
\end{proposition}
Combining (\ref{lcs_quots}) and (\ref{der_quots}), we obtain
the following computation.
\begin{corollary}\label{out_quots}
For all $g\ge 3$, we have
$$
{\mathfrak o}_g(l) =
\begin{cases}
V(\lambda_3) &\text{ when }l=1;\cr
V(2\lambda_2)&\text{ when }l=2;\cr
V(2\lambda_1+\lambda_3)+V(3\lambda_1)&
\text{ when } l=3.
\end{cases}
$$ \qed
\end{corollary}
It does not seem obvious {\it a priori}, that ${\mathfrak t}_g$ is infinite
dimensional.\footnote{This result also follows quite directly from
a result of Oda \cite{oda}.}
\begin{proposition}\label{quotients}
For all $g\ge 3$, the image of ${\mathfrak t}_g$ in ${\mathfrak o}_g$ is infinite dimensional.
\end{proposition}
\begin{proof}
Since ${\mathfrak t}_g \to {\mathfrak o}_g$ is a morphism of MHS, the image ${\mathfrak g}$ has a MHS.
Since $\Gr^W_{\bullet}$ is an exact functor,
$$
\Gr^W_{\bullet}{\mathfrak g} = \text{ image of }\{\Gr^W_{\bullet}{\mathfrak t}_g \to \Gr^W_{\bullet}{\mathfrak o}_g\}.
$$
So it suffices to show that each graded quotient of ${\mathfrak g}$ is non-trivial.
It follows from the result Asada and Nakamura
\cite[Theorem~B]{asada-nakamura} that the image of
$$
{\mathfrak t}_g^1(l) \to {\mathfrak d}_g(l)
$$
contains the representation $V(2m\lambda_1 + \lambda_3)$ when
$l = 2m+1$, and $V(2m\lambda_1 + 2\lambda_2)$ when $l = 2m+2$.
These representations both have the maximal possible depth, $l+2$.
But the inner automorphisms ${\mathfrak p}_g(l)$ in ${\mathfrak d}_g(l)$
have depth at most $l$. The result follows.
\end{proof}
We can now bound below the low degree relations in ${\mathfrak t}_g$.
\begin{proposition}\label{upper}
For all $g\ge 3$, the image of ${\mathfrak u}_g(l)$ in ${\mathfrak o}_g(l)$ is
$$
\begin{cases}
V(\lambda_3) & \text{ when } l=1;\cr
V(2\lambda_2) & \text{ when } l=2;\cr
V(2\lambda_1+\lambda_3) & \text{ when } l=3.
\end{cases}
$$
\end{proposition}
\begin{proof}
It follows from (\ref{quotients}) that when $g\ge 3$, the image of
${\mathfrak u}_g(l) \to {\mathfrak o}_g(l)$ is non-trivial for all $l$. Since this map is
$\sp_g$ equivariant, the image of ${\mathfrak u}_g(2)$ must be all of ${\mathfrak o}_g(2)$.
Since $\Gr^W_{\bullet} {\mathfrak u}_g$ is generated by ${\mathfrak u}_g(1)$, and since
$V(3\lambda_1)$ does not appear in ${\mathfrak u}_g(1)\otimes {\mathfrak u}_g(2)$, the
assertion for $l=3$ follows.
\end{proof}
Note that the copy of $V(3\lambda_1)$ is detected by Morita's
trace \cite{morita:trace}.
\section{Quadratic Relations}
\label{quadratic_relns}
In this section, we find some obvious quadratic relations in ${\mathfrak t}_g$
for each $g\ge 4$. These give a lower bound for the relations in ${\mathfrak t}_g$.
Serendipitously, this coincides with the upper bound (\ref{upper})
derived in the previous section, thus yielding all the quadratic
relations.
\begin{theorem}\label{lower}
For all $g \ge 3$, we have
$$
\Gr^W_{-2} {\mathfrak t}_g = \Gr^W_{-2} {\mathfrak u}_g = V(2\lambda_2) + V(0).
$$
\end{theorem}
The proof occupies the rest of this section. We prove the result by
finding a pair of commuting elements $\phi$ and $\psi$ of the $T_g$
whose class
$$
\tau(\phi)\wedge\tau(\phi) \in \Lambda^2 V(\lambda_3)
$$
generates the $Sp_g$ complement of $V(2\lambda_2) + V(0)$ for all
$g \ge 3$. Since we know, by (\ref{upper}), that the quadratic relations
are contained in the complement of $V(2\lambda_2) + V(0)$, we have found
all quadratic relations.
\begin{lemma}\label{computation}
If $g\ge 3$, then
\begin{multline*}
\Lambda^2 V(\lambda_3) = \\
\begin{cases}
V(\lambda_6) + V(\lambda_4) + V(\lambda_2) + V(\lambda_2 + \lambda_4)
+ V(2\lambda_2) + V(0) & \text{when $g\ge 6$;}\cr
V(\lambda_4) + V(\lambda_2) + V(\lambda_2 + \lambda_4)
+ V(2\lambda_2) + V(0) & \text{when $g=5$;}\cr
V(\lambda_2) + V(\lambda_2 + \lambda_4) + V(2\lambda_2) + V(0) &
\text{when $g = 4$;}\cr
V(2\lambda_2) + V(0) & \text{when $g=3$.}
\end{cases}
\end{multline*}
\qed
\end{lemma}
{}From (\ref{mhs_torelli}), we know that $V(0)$ occurs in ${\mathfrak t}_g(2)$.
By (\ref{upper}) and the previous proposition, there is nothing to
prove when $g=3$. So we suppose that $g\ge 4$.
We use the notation introduced in Section \ref{reps}. Set
$$
\omega = a_1\wedge b_1 + \cdots + a_g\wedge b_g.
$$
\begin{proposition}\label{construction}
When $g\ge 3$, there are elements $\phi_{i,j}$, $1\le i < j \le g$
of the Torelli group whose image under the Johnson homomorphism
$$
\tau_g : H_1(T_g) \to V(\lambda_3)
$$
is given by
$$
(g-1)\tau_g(\phi_{i,j}) = (g-1) a_i\wedge a_j
\wedge b_j - a_i \wedge \omega
$$
Here we are viewing $V(\lambda_3)$ as a submodule of
$\Lambda^3 V(\lambda_1)$. Moreover, we can choose them such that
$\phi_{1,2}$ and $\phi_{3,4}$ commute when $g\ge 4$.
\end{proposition}
\begin{proof}
For $1 \le i < j \le g$ is easy to construct elements $\phi_{i,j}$
of the pointed Torelli group $T^1_g$ with
$$
\tau_g^1(\phi_{i,j}) = a_i\wedge a_j \wedge b_j \in \Lambda^3
V(\lambda_1).
$$ (To compute $\tau_g^1: H_1(T_g^1)\to \Lambda^3V(\lambda_1)$, use
Johnson's original
definition in terms of the action of $T_g^1$ on $\pi_g$.)
\vspace*{1.5in}\\
It is also easy to arrange for $\phi_{1,2}$ and $\phi_{3,4}$ to have
disjoint supports, and therefore commute.
To compute $\tau_g(\phi_{i,j}) \in V(\lambda_3)$, we just use the fact
that the maps
$$
\underline{\phantom{x}}\wedge\omega : V(\lambda_1)
\to \Lambda^3 V(\lambda_1)
$$
and
$$
p: \Lambda^3 V(\lambda_1) \to V(\lambda_1)
$$
defined by $p(x\wedge y\wedge z) = q(x,y)z + q(y,z) x + q(z,x) y$ are
$\sp_g$-equivariant and satisfy
$$
p\circ (\underline{\phantom{x}}\wedge\omega) = (g-1)\id.
$$
It follows that $V(\lambda_3)$ is the kernel of $p$ and that
$$
(g-1) \tau_g(\phi_{i,j}) = (g-1) a_i\wedge a_j\wedge b_j - a_i \wedge
\omega.
$$
\end{proof}
Take $\phi = \phi_{1,2}$ and $\psi = \phi_{3,4}$. Since these commute,
$$
v :=\tau_g(\phi)\wedge\tau_g(\psi) \in \Lambda^2 V(\lambda_3)
$$
will lie inside the $\sp_g$ module of quadratic relations. Denote
the $\sp_g$ submodule of $\Lambda^2 V(\lambda_3)$ generated by $v$
by $V$. By (\ref{construction}),
$$
v =
[(g-1)\, a_1\wedge a_2 \wedge b_2 - a_1\wedge \omega]
\wedge
[(g-1)\, a_3\wedge a_4 \wedge b_4 - a_3\wedge \omega].
$$
Recall that elements of $\sp_g$ act on exterior powers as derivations.
Note also that for all $X \in \sp_g$, $X\omega = 0$. We now compute
the highest weight decomposition of $V$.
\smallskip
\noindent{$\mathbf \lambda_2 + \lambda_4$:} Apply $F_{2,3}$, then
$F_{1,4}$, then $T_{2,3}$ to $v$ to get
$$
(g-1)^2
[a_1\wedge a_2 \wedge a_3 ] \wedge [a_1\wedge a_2 \wedge a_4]
\in \Lambda^2\Lambda^3 V(\lambda_1)
$$
which is a highest weight vector on which ${\mathfrak h}$ acts via the character
$$
\lambda_2 + \lambda_4 = (t_1 + t_2) + ( t_1 + t_2 + t_3 + t_4).
$$
To decompose the rest of $V$, consider the $\sp_g$-equivariant map
$$
\Lambda^2 V(\lambda_3) \hookrightarrow \Lambda^2\Lambda^3
V(\lambda_1) \to \Lambda^6 V(\lambda_1).
$$
Denote the image of $V$ under this map by $W$. It is spanned by
the image of $v$ in $\Lambda^6 V(\lambda_1)$. For the time being,
we suppose that $g\ge 6$.
\smallskip
\noindent{$\mathbf \lambda_6$:} In this case, the image of $v$ in $W$
is
$$
w := ((g-1)\, a_1\wedge a_2 \wedge b_2 - a_1\wedge \omega)
\wedge
((g-1)\, a_3\wedge a_4 \wedge b_4 - a_3\wedge \omega).
$$
To find a highest weight vector for the representation it generates,
first apply $F_{2,5}$, then $F_{4,6}$ to this vector to get the highest
weight vector
$$
(g-1)^2
a_1\wedge a_2 \wedge a_3 \wedge a_4 \wedge a_5 \wedge a_6
$$
of $W$ on which ${\mathfrak h}$ acts via the character
$$
\lambda_6 = t_1 + t_2 + t_3 + t_4 + t_5 + t_ 6.
$$
To show that the weights $\lambda_2$ and $\lambda_4$ occur in
$V$, it is useful to recall that for all $k\ge 2$, there is an $\sp_g$
equivariant projection
\begin{equation}\label{contraction}
\theta_k : \Lambda^k V(\lambda_1) \to \Lambda^{k-2}
V(\lambda_1)
\end{equation}
which is defined by
$$
x_1 \wedge \ldots \wedge x_k \mapsto \sum_{1 \le i < j \le k}
(-1)^{i+j+1} q(x_i,x_j)\, x_1 \wedge \ldots \wedge
\hat{x_i}\wedge \ldots \wedge \hat{x_j} \wedge \ldots
\wedge x_k.
$$
\noindent{$\mathbf \lambda_4$: } The image of $V$ in $\Lambda^4
V(\lambda_1)$ is generated by $\theta_6(w)$ which is
$$
(g-1)^2
a_1 \wedge a_3 \wedge (a_2 \wedge b_2 + a_4 \wedge b_4)
- (g-1)(g-3)\,
a_1 \wedge a_3 \wedge(a_2 \wedge b_2 + a_4 \wedge b_4)
$$ $$
- 2(g-1)\, a_1 \wedge a_3 \wedge \omega
- 2(g-2)\, a_1 \wedge a_3 \wedge \omega
$$ $$
= 2(g-1)\, a_1 \wedge a_2 \wedge (a_3 \wedge b_3 + a_4 \wedge
b_4) -2\, a_1 \wedge a_2 \wedge \omega.
$$
Applying $F_{3,6}$, then $S_{4,6}$, one gets the highest weight
vector
$$
2(g-1)\, a_1\wedge a_2\wedge a_3\wedge a_4
$$
on which ${\mathfrak h}$ acts via the character
$\lambda_4 = t_1 + t_2 + t_3 + t_4$.
\smallskip
\noindent{ $\mathbf \lambda_2$:} The image of $V$ in $\Lambda^2
V(\lambda_1)$ is generated by the image under $\theta_4$ of
$\theta_6(w)$. This is
$$
4(g-1)\, a_1 \wedge a_3 - 2(g-2)\, a_1 \wedge a_3 = 2g\, a_1 \wedge
a_3.
$$
Apply $S_{2,3}$ to this to get $2g\, a_1 \wedge a_2$ upon which
${\mathfrak h}$ acts with highest weight $\lambda_2 = t_1 + t_2$.
We sketch the remaining cases $g=4,5$. When $g=5$, $W$ is generated
by the vector
$$
(g-1)\left(a_1\wedge a_2\wedge a_3\wedge b_2 -
a_1\wedge a_3\wedge a_4\wedge b_4\right)\wedge \omega.
$$
By contracting with $q$ as above, it is easy to see that this
vector generates a submodule of
$$
\omega \wedge \Lambda^4 V(\lambda_1) \cong \Lambda^6 V(\lambda_1)
$$
isomorphic to $V(\lambda_4) + V(\lambda_2)$.
When $g=4$, $W$ is generated by $a_1\wedge a_3 \wedge \omega^2$.
Again, by contracting with $q$, it is easy to see that this vector
generates a submodule of
$$
w^2\wedge \Lambda^2 V(\lambda_1)
\subset \Lambda^6 V(\lambda_1)
$$
isomorphic to $V(\lambda_2)$.
\begin{remark}
Note that we have determined the quadratic relations for all
$g\ge 3$. One should be able to determine the cubic relations when
$3 \le g \le 5$ by applying similar methods and the fact that
the Dehn twist about the separating curve $C$ below commutes
with the bounding pair map associated to the curves $C'$ and
$C''$. The Dehn twist about $C$ is in the kernel of the Johnson
homomorphism, but has non-trivial image in the second graded
quotient of ${\mathfrak t}_g$.
\vspace*{2in}
Note that there have to be cubic relations in genus 3
as there are no quadratic relations, and there has to be one copy
of $V(\lambda_3)$ in the cubic relations to ensure the existence of
the central $\Ga$.
\end{remark}
\section{Presentations of ${\mathfrak t}_g$, $t_{g,1}$ and ${\mathfrak t}_g^1$}
\label{special}
Recall that ${\mathbb L}(V)$ denotes the free Lie algebra generated by the
vector space $V$. In this section we shall give presentations of
${\mathfrak t}_g$, ${\mathfrak t}_g^1$ and ${\mathfrak t}_{g,1}$ when $g\ge 6$. First ${\mathfrak t}_g$ ---
combining (\ref{upper}), (\ref{lower}) and (\ref{construction}), we
have:
\begin{theorem}
For all $g\ge 6$, $\Gr^W_{\bullet} {\mathfrak t}_g$ is isomorphic to
$$
{\mathbb L}(V(\lambda_3))/R_g
$$
as a graded Lie algebra in $R(\sp_g)$, where $R_g$ is the ideal
generated by the quadratic relations
$$
V(\lambda_6) + V(\lambda_4) + V(\lambda_2) + V(\lambda_2 +\lambda_4)
\subseteq \Lambda^2 V(\lambda_3). \qed
$$
\end{theorem}
Since ${\mathfrak t}_g\otimes{\mathbb C} \cong \prod_l \Gr^W_l {\mathfrak t}_g\otimes{\mathbb C}$, this gives
the desired presentation of ${\mathfrak t}_g$ for $g\ge 6$.
We next consider ${\mathfrak t}_g^1$.
Fix a point $[C;x]$ of ${\mathcal M}_g^1$ so that ${\mathfrak t}_g^1$ and ${\mathfrak p}_g$
have canonical MHSs. Then the sequence
$$
0 \to {\mathfrak p}_g^1 \to {\mathfrak t}_g^1 \to {\mathfrak t}_g \to 0
$$
is an exact sequence of MHSs. Since $\Gr^W_{\bullet}$ is an exact functor,
the sequence
$$
0 \to \Gr^W_{\bullet} {\mathfrak p}_g^1 \to \Gr^W_{\bullet} {\mathfrak t}_g^1 \to \Gr^W_{\bullet} {\mathfrak t}_g \to 0
$$
is exact in $R(\sp_g)$. Since the sequence of $H_1$'s is
canonically split in $R(\sp_g)$, there is a canonical lift
$$
{\mathbb L}(H_1({\mathfrak t}_g)) \to \Gr^W_{\bullet}{\mathfrak t}_g^1
$$
of the natural homomorphism ${\mathbb L}(H_1({\mathfrak t}_g)) \to \Gr^W_{\bullet}{\mathfrak t}_g$. Since
${\mathfrak p}_g(2)$ is isomorphic to $V(\lambda_2)$, and since this representation
does not occur in ${\mathfrak t}_g(2)$, we can take the $\lambda_2$ component of the
bracket
$$
\Lambda^2 H_1({\mathfrak t}_g) \hookrightarrow \Lambda^2 H_1(t_g^1) \to {\mathfrak t}_g^1(2)
$$
to obtain an $\sp_g$ module map
\begin{equation}\label{bracket}
c : \Lambda^2 H_1({\mathfrak t}_g) \to {\mathfrak p}_g(2) \cong V(\lambda_2).
\end{equation}
This and the map
\begin{equation}\label{action2}
H_1({\mathfrak t}_g) \otimes H_1({\mathfrak p}_g) \to {\mathfrak p}_g(2)
\end{equation}
induced by the bracket completely determine $\Gr^W_{\bullet} {\mathfrak t}_g^1$ given
$\Gr^W_{\bullet} {\mathfrak t}_g$ and $\Gr^W_{\bullet} {\mathfrak p}_g$. The map (\ref{action2})
is simply the adjoint of the Johnson homomorphism
$$
\tau_g^1 : H_1({\mathfrak t}_g) \to \Lambda^3 V \subset \Hom(H_1({\mathfrak p}_g),{\mathfrak p}_g(2)).
$$
So, to give a presentation of ${\mathfrak t}_g^1$, we have to determine the map
(\ref{bracket}). We do this by studying the action of ${\mathbb L}(V(\lambda_3))$
on ${\mathbb L}(V(\lambda_1))$.
Set $V=V(\lambda_1)$. We identify $V$ with $H_1(C)$ and $H_1(T_g^1)$
with $\Lambda^3 V$ via the Johnson homomorphism. Recall that $V(\lambda_3)$
can be realized as the kernel of the map $p:\Lambda^3 V \to V$ defined by
\begin{equation}\label{projn}
p : v_1\wedge v_2 \wedge v_3 \mapsto (v_1\cdot v_2)v_3 +
(v_2\cdot v_3) v_1 + (v_3\cdot v_1) v_2.
\end{equation}
We identify $H_1(T_g) \cong \Lambda^3 V/V$ with $V(\lambda_3)$
via the map
$$
V(\lambda_3) = \ker p \hookrightarrow \Lambda^3 V \to \Lambda^3 V/V.
$$
The natural action of $\Lambda^3 V$ on ${\mathbb L}(V)$ is defined by
\begin{multline*}
e_1\wedge e_2 \wedge e_3 \mapsto -\left\{v \mapsto (e_1\cdot v)
[e_2,e_3] + (e_2\cdot v)[e_3,e_1] + (e_3\cdot v)[e_1,e_2]\right\} \cr
\in \Hom(V,\Lambda^2 V) \subseteq \Der {\mathbb L}(V).
\end{multline*}
(With this choice of sign, $\sum x\wedge a_j\wedge b_j\mapsto\ad(x)$.)
It follows from the definition of the Johnson homomorphism that the
composite
$$
H_1(T_{g,1}) \stackrel{\tau_{g,1}}{\longrightarrow}
\Lambda^3 V \hookrightarrow \Hom(V,\Lambda^2 V)
$$
is the map induced by the action of $T_{g,1}$ on $\pi_{g,1}$.
The action descends to the action of $\Gr^W_{\bullet}{\mathfrak t}_g^1$ on
$\Gr^W_{\bullet} {\mathfrak p}_g$.
Define a projection $r : \Lambda^2 V(\lambda_3) \to V(\lambda_2)$
to be the composite
$$
\Lambda^2 V(\lambda_3) \hookrightarrow \Lambda^2 \Lambda^3 V
\stackrel{\text{mult}}{\longrightarrow} \Lambda^6 V
\stackrel{\theta_4\theta_6}{\longrightarrow} \Lambda^2 V
\to V(\lambda_2)
$$
where $\theta_k$ is the contraction (\ref{contraction}) defined in
the previous section, and the last map is the standard projection
$$
u\wedge v \mapsto u\wedge v - (u\cdot v)\,\omega/(g-1).
$$
Since there is only one copy of $V(\lambda_2)$
in $\Lambda^2 V(\lambda_3)$, this projection is unique up to a scalar.
\begin{proposition}\label{bra_const}
The map (\ref{bracket}) is given by
$$
c[u,v] = -\frac{1}{2g+2} \ad(r(u\wedge v)) \in \Hom(H_1(p_g),{\mathfrak p}_g(3)).
$$
In particular, this map is non-zero, and the extensions
$$
0 \to {\mathfrak p}_g \to {\mathfrak t}_g^1 \to {\mathfrak t}_g \to 0 \text{ and }
0 \to {\mathfrak p}_g \to {\mathfrak u}_g^1 \to {\mathfrak u}_g \to 0
$$
are not split.
\end{proposition}
We now sketch the proof.
Denote the degree $k$ part of the free Lie algebra ${\mathbb L}(V)$ by
${\mathbb L}(V)(k)$.
Recall that there is a standard exact sequence
$$
0 \to \Lambda^3 V \stackrel{j}{\to} V\otimes \Lambda^2 V
\stackrel{b}{\to} {\mathbb L}(V)(3) \to 0.
$$
The first map is the ``Jacobi identity'' map
$$
j : v_1\wedge v_2 \wedge v_3 \mapsto v_1\otimes v_2\wedge v_3
+ v_2 \otimes v_3\wedge v_1 + v_3 \otimes v_1 \wedge v_2,
$$
and the second map is the bracket. (We are identifying ${\mathbb L}(V)(2)$
with $\Lambda^2 V$ in the standard way.)
\begin{lemma}\label{compn}
The bracket $[e_1\wedge e_2 \wedge e_3, f_1\wedge f_2 \wedge f_3]$
of two elements of $\Lambda^3 V$ as derivations of ${\mathbb L}(V)$ is obtained
by summing the expression
\begin{multline*}
(e_1\cdot f_1)
\big(
e_2\otimes [e_3,[f_2,f_3]] - e_3\otimes [e_2,[f_2,f_3]]
+ f_2\otimes [f_3,[e_2,e_3]] - f_3\otimes [f_2,[e_2,e_3]]
\big) \cr
\quad \in V\otimes {\mathbb L}(V)(3) \cong \Hom(V,{\mathbb L}(V)(3))
\end{multline*}
cyclically in $(e_1,e_2,e_3)$ and in $(f_1,f_2,f_3)$. \qed
\end{lemma}
We shall view this expression as an element of
$\left(V\otimes V \otimes \Lambda^2 V\right)
/\left(V \otimes \Lambda^3 V\right)$.
The next step is to write down the projections of this group onto
$V(\lambda_2)$.
There are four copies of $V(\lambda_2)$ in
$V\otimes V \otimes \Lambda^2 V$. These are detected by the
following four projections onto $\Lambda^2 V$:
\begin{align*}
p_1 : u_1\otimes u_2 \otimes u_3 \wedge u_4
&\mapsto (u_1\cdot u_2)u_3\wedge u_4 \cr
p_2 : u_1\otimes u_2 \otimes u_3 \wedge u_4
&\mapsto (u_3\cdot u_4) u_1\wedge u_2 \cr
p_3 : u_1\otimes u_2 \otimes u_3 \wedge u_4
&\mapsto
\big((u_1\cdot u_4)u_2\wedge u_3 - (u_1\cdot u_3)u_2\wedge u_4\big)/2\cr
p_4 : u_1\otimes u_2 \otimes u_3 \wedge u_4
&\mapsto
\big((u_2\cdot u_3)u_1\wedge u_4 - (u_2\cdot u_4)u_1\wedge u_3\big)/2.
\end{align*}
One can easily check that there are two copies of $V(\lambda_2)$ in
$V\otimes \Lambda^3 V$ and that the projections $p_1 - p_3$ and
$p_2 - p_4$ vanish on these. This leaves two copies of $V(\lambda_2)$
in $\left(V\otimes V \otimes \Lambda^2 V\right)
/\left(V \otimes \Lambda^3 V\right)$. One of these vanishes in
$\Hom(V,{\mathfrak p}_g(3))$ as $V\otimes V \otimes \omega$ projects to zero there.
We are now ready to compute.
Since
$$
u_j = a_j\wedge a_3 \wedge b_3 - a_j\wedge a_4 \wedge b_4
$$
lies in the kernel of the projection $p$ above when $j=1,2$,
$u_1\wedge u_2$ is an element of $\Lambda^2 V(\lambda_3)$. The
projection $r$ takes $u_1\wedge u_2$ to $-4\, a_1\wedge a_2$.
On the other hand, by straightforward computations using
(\ref{compn}), we have
$$
p_1([u_1,u_2]) = p_2([u_1,u_2]) = 0,\text{ and }
p_3([u_1,u_2]) = p_4([u_1,u_2]) = -4\, a_1\wedge a_2.
$$
Consequently,
$$
(p_1 - p_3)([u_1,u_2]) = (p_2 - p_4)([u_1,u_2]) = 4\, a_1\wedge a_2.
$$
Next observe that $\ad [a_1,a_2]$ corresponds to the element
$$
- \sum_{j=1}^g \left( a_j\otimes b_j \otimes a_1\wedge a_2
- b_j \otimes a_j \otimes a_1 \wedge a_2\right).
$$
of $\left(V\otimes V \otimes \Lambda^2 V\right)
/\left(V \otimes \Lambda^3 V\right)$.
Since $\sum\, [a_j,b_j] = 0$, $\ad [a_1,a_2]$ is also represented
by
$$
z := - \sum_{j=1}^g \left( a_j\otimes b_j \otimes a_1\wedge a_2
- b_j \otimes a_j \otimes a_1 \wedge a_2\right)
- 2 \sum_{j=1}^g a_1\otimes a_2\otimes a_j\wedge b_j.
$$
By direct computation, we have
$$
(p_1-p_3)(z) = (p_2-p_4)(z) = -(2g+2)\, a_1\wedge a_2.
$$\,
This concludes the proof of Proposition \ref{bra_const}. \qed
Next we consider the case of ${\mathfrak t}_{g,1}$. Fix a point $(C;x,v)$ of
${\mathcal M}_{g,1}$ so that ${\mathfrak t}_{g,1}$, ${\mathfrak p}_{g,1}$, etc.\ all have compatible
MHSs. By strictness, the sequence
$$
0 \to \Gr^W_{\bullet} {\mathfrak p}_{g,1} \to \Gr^W_{\bullet} {\mathfrak t}_{g,1}
\to \Gr^W_{\bullet} {\mathfrak t}_g \to 0
$$
is exact in $R(\sp_g)$. Since the sequence
$$
0 \to {\mathbb Q}(1) \to {\mathfrak p}_{g,1} \to {\mathfrak p}_g \to 0
$$
is exact, it follows that ${\mathfrak p}_{g,1}(2)$ is isomorphic to $\Lambda^2 V$
via the bracket.
As in the case of ${\mathfrak t}_g^1$, there is a canonical lifting
${\mathbb L}(H_1({\mathfrak t}_g)) \to \Gr^W_{\bullet}{\mathfrak t}_{g,1}$
of the natural surjection ${\mathbb L}(H_1({\mathfrak t}_g)) \to \Gr^W_{\bullet} {\mathfrak t}_g$.
It follows that to give a presentation of $\Gr^W_{\bullet} {\mathfrak t}_{g,1}$ given
presentations of ${\mathfrak t}_g$ and ${\mathfrak p}_{g,1}$, it suffices to give the map
\begin{equation}\label{first}
H_1({\mathfrak t}_g)\otimes H_1({\mathfrak p}_{g,1}) \longrightarrow {\mathfrak p}_{g,1}(2)
\end{equation}
induced by the bracket, together with the $\lambda_2$ component
\begin{equation}\label{second}
\Lambda^2 H_1({\mathfrak t}_g) \longrightarrow V(\lambda_2) \subset {\mathfrak p}_{g,2}
\end{equation}
and the invariant part
\begin{equation}\label{third}
c_0 : \Lambda^2 H_1({\mathfrak t}_g) \longrightarrow
{\mathfrak t}_{g,1}(2)^{Sp_g} \cong {\mathbb Q}(1)^2
\end{equation}
of the bracket. As in the case of ${\mathfrak t}_g^1$, the first map (\ref{first})
is the adjoint of the Johnson homomorphism and the second (\ref{second}),
by naturality with respect to the projection ${\mathfrak t}_{g,1} \to {\mathfrak t}_g^1$, is
the map $c$ determined in (\ref{bra_const}). It remains to determine the
map (\ref{third}). Observe that the sequence
$$
0 \to {\mathfrak p}_{g,1}(2)^{Sp_g} \to {\mathfrak t}_{g,1}(2)^{Sp_g} \to {\mathfrak t}_g(2)^{Sp_g} \to 0
$$
splits canonically as the canonical central $\Ga$ in ${\mathfrak t}_{g,1}$
projects to the canonical central $\Ga$ in ${\mathfrak t}_g$ by (\ref{central_ext}),
and because $\Ga = {\mathfrak t}_g(2)^{Sp_g}$. As a generator of ${\mathfrak p}_{g,1}(2)^{Sp_g}$
we take $\sum\, [a_j,b_j]$.
Fix an invariant bilinear form $\bil \phantom{x} \phantom{x}$ on $V(\lambda_3)$ by
insisting that
$$
\bil {a_1\wedge a_2 \wedge a_3} {b_1 \wedge b_2 \wedge b_3} = 1.
$$
We can therefore choose a generator $\gamma$ of $\Ga$ such that
if $u,v\in H_1({\mathfrak t}_g)$, then the invariant component of $[u,v]$
in ${\mathfrak t}_g(2)$ is $\bil u v \, \gamma$.
\begin{proposition}\label{triv_cpt}
If $u,v \in H_1({\mathfrak t}_g)$, then
$$
c_0[u,v] =
\bil u v \,\gamma -
\frac{6\bil u v}{g(2g+1)}\sum_{j=1}^g\, [a_j,b_j].
$$
\end{proposition}
As in the previous case, we determine the coefficient by studying
the action of ${\mathbb L}(V(\lambda_3))$ on ${\mathbb L}(V)$. Note that $\Gamma_{g,1}$
acts on the free group $\pi_1(C - \{x\},v)$.%
\footnote{This notation denotes Deligne's fundamental group
of $C - \{x\}$ with base point the tangent vector $v \in T_x C$.} We
therefore have a representation ${\mathfrak t}_{g,1} \to {\mathfrak p}(C-\{x\},v) \cong
{\mathbb L}(V)$.\footnote{If we put the limit
MHS on $\pi_1(C - \{x\},v)$ associated with the tangent vector $v$,
then this action is a morphism of MHS.} We continue with the notation
in the proof of (\ref{bra_const}).
There are two copies of the trivial representation in
$V\otimes V \otimes \Lambda^2 V$. The corresponding projections
to ${\mathbb Q}$ are:
\begin{align*}
q_1 : u_1\otimes u_2 \otimes u_3 \wedge u_4
&\mapsto (u_1\cdot u_2)(u_3\cdot u_4) \cr
q_2 : u_1\otimes u_2 \otimes u_3 \wedge u_4
&\mapsto \bigl((u_1\cdot u_4)(u_2\cdot u_3)
- (u_1\cdot u_3) (u_2\cdot u_4)\bigr)/2.
\end{align*}
There is one copy of the trivial representation in $V\otimes
\Lambda^3 V$ and $q_1 - q_2$ vanishes on it. The vectors
$$
u_1 = a_1\wedge a_2 \wedge a_3
\text{ and } u_2 = b_1\wedge b_2 \wedge b_3
$$
both lie in $V(\lambda_3)$ and $\bil {u_1} {u_2} = 1$. It follows
from the formula (\ref{bracket}) that $[u_1,u_2]$ is obtained by
summing the expression
$$
a_2 \otimes [a_3,[b_2,b_3]] - a_3\otimes [a_2,[b_2,b_3]]
+ b_2 \otimes [b_3,[a_2,a_3]] - b_3 \otimes [b_2,[a_2,a_3]]
$$
over the cyclic group generated by $(1,2,3)$. We have
$(q_1 - q_2) ([u_1,u_2]) = 6$.
On the other hand, $\ad \sum[a_j,b_j]$ is represented by
$$
\sum_{j=1}^g \sum_{k=1}^g
(b_j\otimes a_j - a_j \otimes b_j)\otimes a_k\wedge b_k
$$
The projection $q_1 - q_2$ takes the value $-g(2g +1)$ on
this. The result follows.
\begin{remark}
The formulas in (\ref{bra_const}) and (\ref{triv_cpt}) are closely
related to those in Theorem~3.1 of Morita's paper
\cite{morita:cocycles}.
\end{remark}
\section{A Presentation of ${\mathfrak p}_{g,r}^n$}
\label{braids2}
In this section we give an explicit quadratic presentation of the
pure braid Lie algebras ${\mathfrak p}_{g,r}^n$ for all $g > 0$. We continue with
the notation of Section~\ref{braids1}. We fix a complex structure on
and a base point of $F_{g,r}^n$ by choosing a point
$$
[C;x_1,\dots,x_n;v_1,\dots,v_r]
$$
of ${\mathcal M}_{g,r}^n$.
In Section~\ref{braids1} we showed that $H^1(F_{g,r}^n(C))$ is pure of
weight 1. We will show that $H^2(F_{g,r}^n(C))$ is pure of weight 2,
from which the existence of a quadratic presentation of ${\mathfrak p}_{g,r}^n$
will follow via Morgan's Theorem (\ref{morgan}).
First, some notation. Denote the projection
of $C^n$ onto its $i$th factor by $p_i$. Denote the image of the
inclusion
$$
p_i^\ast : H^{\bullet}(C) \hookrightarrow H^{\bullet}(C^n)
$$
by $H^{\bullet}(C_i)$. For $x\in H^{\bullet}(C)$, denote $p_i^\ast x$ by $x\sup{i}$.
Denote the component of $\Delta$ where the $i$th and $j$th coordinates
are equal by $\Delta_{ij}$. Fix a
symplectic basis $a_1, \dots, a_g,b_1,\dots, b_g$ of $H_1(C)$,
and let $\alpha_1,\dots, \alpha_g,\beta_1,\dots,\beta_g$ be the
dual basis of $H^1(C)$. Denote the positive integral generator of
$H^2(C)$ by $\zeta$, and the intersection form
$$
\sum_{r=1}^g \alpha_r \wedge \beta_r
$$
by $q$. When $i\neq j$, set
$$
q_{ij} = \sum_{r=1}^g \alpha_r\sup{i} \wedge \beta_r\sup{j}
+ \alpha_r\sup{j} \wedge \beta_r\sup{i}.
$$
\begin{lemma}
The Poincar\'e dual $PD(\Delta_{ij})$ of $\Delta_{ij}$ is
$\zeta\sup{i} + \zeta\sup{j} - q_{ij}$. \qed
\end{lemma}
This is elementary. Another elementary fact we shall need is the
following statement. It is easily proved using a Mayer-Vietoris
argument.
\begin{lemma}\label{isom}
The natural map
$$
\bigoplus_{i<j} H_{2n-3}(\Delta_{ij}) \to H_{2n-3}(\Delta)
$$
is an isomorphism. \qed
\end{lemma}
We can therefore write the Gysin map
$\gamma : H_{2n-3}(\Delta) \to H^3(C^n)$
as the sum of the Gysin maps
$\gamma_{ij} : H^1(\Delta_{ij}) \to H^3(C^n)$;
the map $\gamma_{ij}$ being given by cup product with $PD(\Delta_{ij})$.
\begin{lemma}\label{formula}
The composite
$H^1(C) \stackrel{p_k^\ast}{\longrightarrow} H^1(\Delta_{ij})
\stackrel{\gamma_{ij}}{\longrightarrow} H^3(C^n)$
is given by
$$
x \mapsto
\begin{cases}
\zeta\sup{i}\wedge x\sup{j} + \zeta\sup{j}\wedge x\sup{i} &
\text{ if $k \in \{i,j\}$;}\cr
\zeta\sup{i}\wedge x\sup{k} + \zeta\sup{j}\wedge x\sup{k}
- q_{ij}\wedge x\sup{k} & \text{ if $k\not\in \{i,j\}$. \qed}
\end{cases}
$$
\end{lemma}
It follows from
(\ref{h1_braid}) that the part
$$
0 \to \bigoplus_{i<j} {\mathbb Z} \to H^2(C^n) \to H^2(C^n - \Delta)
\to H_{2n-3}(\Delta) \to H^3(C^n)
$$
of the Gysin sequence is exact. Purity of $H^2(F_g^n(C))$ therefore
follows from the following proposition.
\begin{proposition}
The Gysin map $\gamma : H_{2n-3}(\Delta) \to H^3(C^n)$ is injective.
\end{proposition}
\begin{proof}
The Gysin sequence can be viewed as the fiber over $[C] \in {\mathcal M}_g$
of an exact sequence of (orbifold) local systems over ${\mathcal M}_g$. It
follows from (\ref{isom}) that the the monodromy actions of the last
two terms of the Gysin sequence above factor through the symplectic
group. It is convenient, though not necessary, to decompose these
groups under the action of $Sp_g$.
First note that $H_{2n-3}(\Delta_{ij})$ is isomorphic to
$H^1(\Delta_{ij})$, which is isomorphic to $n-1$ copies of the
fundamental representation $V$. Next,
$$
H^3(C^n) = \bigoplus_{i\neq j} \left(H^2(C_i)\otimes H^1(C_j)\right)
\oplus \bigoplus_{i<j<k} H^1(C_i)\otimes H^1(C_j)\otimes H^1(C_k).
$$
Each of the terms $H^2(C_i)\otimes H^1(C_j)$ is a copy of the
fundamental representation that we shall denote by $V^i_j$. The
term $H^1(C_i)\otimes H^1(C_j)\otimes H^1(C_k)$ is isomorphic to
$V^{\otimes 3}$. It contains 3 copies of $V$. If, for $i,j,k$ distinct,
we set
$$
V_{ij}^k =
\text{ the image of }
\left\{H^1(C_k) \stackrel{\wedge q_{ij}}{\to} H^3(C^n)\right\},
$$
then
$$
\left[H^1(C_i)\otimes H^1(C_j)\otimes H^1(C_k)\right]_{\lambda_1}
= V_{ij}^k \oplus V_{jk}^i \oplus V_{ki}^j.
$$
It is now easy to see that $\gamma$ is injective. Indeed, by
(\ref{formula}), we see that the images of the maps
$$
H^1(C) \stackrel{p_i^\ast}{\to} H^1(\Delta_{ij})
\stackrel{\gamma_{ij}}{\to} H^3(C^n)
$$
are independent copies of $V$, and also, when $k \not\in \{i,j\}$,
that the image of
$$
H^1(C) \stackrel{p_k^\ast}{\to} H^1(\Delta_{ij})
\stackrel{\gamma_{ij}}{\to} H^3(C^n)
$$
is congruent to $V_{ij}^k$ modulo the sum of the $V_b^a$.
\end{proof}
Similarly, one can show that the $r$ Chern classes of the central
extensions
$$
0 \to {\mathbb Z}^r \to \pi_{g,r}^n \to \pi_g^{r+n} \to 1
$$
are linearly independent in $H^2(\pi_g^{r+n})$ as they correspond to
independent copies of the trivial representation in $H^2(F_g^{r+n})$.
It follows that $H^2(F_{g,r}^n)$ is also pure of weight 2.
Assembling all this, we obtain:
\begin{proposition}
For each choice of a base point $[C]$ of ${\mathcal M}_g$
and for all $g\ge 1$ and $n,r \ge 0$, the natural MHS on
$H^1(F_{g,r}^n)$ is pure of weight 1 and that on $H^2(F_{g,r}^n)$
is pure of weight 2. In addition, the cup product
$$
\Lambda^2 H^1(F_{g,r}^n,{\mathbb Q}) \to H^2(F_{g,r}^n)
$$
is surjective. \qed
\end{proposition}
{}From Morgan's Theorem we deduce that ${\mathfrak p}_{g,r}^n$ has a quadratic
presentation for all non-negative $g$, $r$ and $n$.\footnote{In the
genus zero case, it is well known that $H^1$ has weight 2 and $H^2$
weight 4 as the corresponding classifying spaces are complements
of hyperplanes in affine space.}
Our final task is to determine the relations explicitly. First some
notation. The Lie algebra ${\mathfrak p}_{g,r}^n$ is a quotient of the free
Lie algebra generated by
$$
H_1({\mathfrak p}_{g,r}^n) \cong H_1(C^{n+r})
\cong \bigoplus_{i=1}^{n+r} H_1(C_i).
$$
We shall think of elements of $H_1(C^{n+r})$ as indeterminates, and write
them as upper case letters. If $X\in H_1(C)$, we shall denote the
corresponding element of $H_1(C_i)$ by $X\sup{i}$. Fix a symplectic
basis $A_1,\dots,A_g,B_1,\dots, B_g$ of $H_1(C)$. Denote the intersection
number of $X$ and $Y \in H_1(C)$ by $(X\cdot Y)$.
\begin{theorem}
For all $g\ge 1$ and all $r,n\ge 0$,
$$
\Gr^W_{\bullet} {\mathfrak p}_{g,r}^n \cong {\mathbb L}(H_1(C)^{\oplus(n+r)})/R
$$
where $R$ is the ideal generated by the relations
\begin{xalignat*}{2}
[X\sup{i},Y\sup{j}] & = [X\sup{j},Y\sup{i}] & \text{all $i$ and $j$;} \cr
[X\sup{i},Y\sup{j}] & =
\frac{(X\cdot Y)}{g} \sum_{k=1}^g\,
[{\mathcal A}{i},\B{j}] & \text{all $i$ and $j$;} \cr
\sum_{k=1}^g\, [{\mathcal A}{i},\B{i}] & = %
\frac{1}{g} \sum_{j\neq i} \sum_{k=1}^g\,
[{\mathcal A}{i},\B{j}] & 1 \le i \le n.
\end{xalignat*}
where $X$ and $Y$ are arbitrary elements of $H_1(C)$.
\end{theorem}
Note that the last relation holds only for those factors corresponding
to a marked point, and not those corresponding to a marked tangent vector.
\begin{proof}
If ${\mathfrak g}$ is a graded Lie algebra generated in weight
$-1$ and $H_2({\mathfrak g})$ of weight 2, then we have an exact sequence
$$
0 \to H_2({\mathfrak g}) \stackrel{\text{cup}^\ast}{\longrightarrow}
\Lambda^2 H_1({\mathfrak g}) \stackrel{\text{bracket}}{\longrightarrow}
Gr^W_{-2}{\mathfrak g} \to 0,
$$
where the first map is the dual of the cup product.\footnote{There are
many ways to see this --- the easiest being from the standard complex of
Lie algebra cochains. However, the statement holds in greater generality
--- cf.\ \cite{sullivan:les}.} In our case, the natural injection
$$
H^2({\mathfrak p}_{g,r}^n) \to H^2(F_{g,r}^n,{\mathbb Q})
$$
is an isomorphism because the cup product is surjective. The
coproduct is the obvious inclusion of $H_2(F_{g,r}^n,{\mathbb Q})$ into
$\Lambda^2 H_1(C^n,{\mathbb Q})$, and the sequence is a sequence of $Sp_g$
modules:
$$
0 \to H_2(F_{g,r}^n,{\mathbb Q}) \to \Lambda^2 H_1(C^n,{\mathbb Q}) \to \Gr^W_{-2}
{\mathfrak p}_{g,r}^n \to 0.
$$
We will consider one weight at a time. Note that the three weights
occurring in $\Lambda^2 H_1(C^n)$ are 0, $\lambda_2$ and $2\lambda_1$
--- the last being the symmetric square of $H_1(C)$ and second being
the primitive part of $H_2(\Jac C)$. We also have the exact sequence
$$
0 \to H_2(F_{g,r}^n) \to H_2(C^n)
\stackrel{\gamma^\ast}{\to} \bigoplus_{i<j}{\mathbb Q} \to 0
$$
of $Sp_g$ modules. The last map is the dual of the Gysin map.
It follows that
$$
H_2(F_{g,r}^n,{\mathbb Q})_{\lambda} = H_2(C^n,{\mathbb Q})_{\lambda}
$$
when $\lambda$ is $2\lambda_1$ or $\lambda_2$.
The $2\lambda_1$ isotypical component is spanned by elements of the
form
$$
X\sup{i}\times Y\sup{j} + Y\sup{i}\times X\sup{j}.
$$
This gives the first relation:
\begin{equation}\label{comm}
[X\sup{i},Y\sup{j}] = [X\sup{j},Y\sup{i}].
\end{equation}
Since $V(\lambda_2)$ is the kernel of the symplectic form
$\Lambda^2 V(\lambda_1) \to {\mathbb Q}$,
the $\lambda_2$ isotypical component of $H_2(C^n)$ is spanned by
vectors of the form
$$
X\sup{i}\times Y\sup{j} - Y\sup{i}\times X\sup{j} -
\frac{(X\cdot Y)}{g} \sum_{k=1}^g
\left( {\mathcal A}{i}\times \B{j} - \B{i}\times {\mathcal A}{j} \right) .
$$
This gives relations of the form
$$
[X\sup{i}, Y\sup{j}] + [X\sup{j}, Y\sup{i}] =
\frac{(X\cdot Y)}{g} \sum_{k=1}^g
\left([{\mathcal A}{i}, \B{j}] + [{\mathcal A}{j}, \B{i}] \right)
$$
which simplifies to the second relation after applying (\ref{comm}).
For the time being, we assume that $r=0$.
The trivial isotypical component lies in an exact sequence
$$
0 \to H_2(F_{g,r}^n)^{Sp_g} \to H_2(C^n)^{Sp_g}
\stackrel{\gamma^\ast}{\to} H^{2g-2}(\Delta) \to 0.
$$
The map $\gamma^\ast$ takes $W\in H_2(C^n)$ to the functional
$$
\{\Delta_{ij} \mapsto W\cdot\Delta_{ij}\}.
$$
Note that
$$
H_2(C^n)^{Sp_g} = \bigoplus_{i=1}^n H_2(C_i) \oplus
\bigoplus_{i<j} \left[H_1(C_i)\otimes H_1(C_j)\right]^{Sp_g}.
$$
The first terms has basis the $Z\sup{i}$, where $Z$ denotes the
integral generator of $H_2(C)$. The second term has basis consisting
of the
$$
Q_{ij} :=
\sum_{k=1}^n \left({\mathcal A}{i}\times \B{j} - \B{i}\times {\mathcal A}{j}\right).
$$
We next determine a basis of $\ker \gamma^\ast$.
Choose $n$ distinct points $u_1,\dots,u_n$ of $C$. For $i<j$, let
$C_{ij}$ be the image of the map $C\hookrightarrow C^n$ defined by
$$
x \mapsto
(u_1,\dots, u_{i-1},x,u_{i+1},\dots,u_{j-1},x,u_{j+1},\dots,u_n).
$$
Denote its homology class by $Z_{ij}$. It is easily seen that
$$
Z_{ij} = Z_i + Z_j + Q_{ij}.
$$
Observe that
$$
Z_{ij}\cdot \Delta_{kl} =
\begin{cases}
0 & \text{$i$, $j$, $k$, $l$ distinct};\cr
1 & \#\{i,j,k,l\} = 3;\cr
2 - 2g & ij = kl.
\end{cases}
$$
The first two assertions are clear, the second follows from the
projection formula applied to the projection of $C^n$ onto $C^2$
along the $i$th and $j$th factors and the fact that the self
intersection of the diagonal in $C^2$ is the Euler number of $C$.
It follows immediately that a basis of
$$
H_2(F_g^n)^{Sp_g} = \ker\{H_2(C^n)^{Sp_g} \to H^2(\Delta)\}
$$
consists of the
$$
Z_i - \frac{1}{2g}\sum_{j\neq i} Q_{ij}.
$$
These give the relations
$$
\sum_{k=1}^g\, [{\mathcal A}{i},\B{i}] =
\frac{1}{2g} \sum_{j\neq i}
\sum_{k=1}^g\, \left([{\mathcal A}{i},\B{j}] + [{\mathcal A}{j},\B{i}]\right)
$$
which becomes the third relation after an application of (\ref{comm}).
Finally, the third relation is dual to the first Chern class of the
pullback of the tangent bundle of $C$ along $p_i : C^n \to C$. It
follows that in the general case, we do not get any relations coming
from the trivial representation associated to an index corresponding
to a tangent vector.
\end{proof}
We conclude this section with a computation of the generating function
of the lower central series of $\pi_{g,r}^n$. This corrects the formula
in \cite{kohno-oda}. (The galois analogue of this corrected formula is
also stated in \cite[(2.14)]{nak-tak-ueno}.)
\begin{theorem}
For all $g\ge 1$ and all $n\ge 1$, we have
$$
\prod_{k=1}^n \left(1 - 2g\,t -(k-2)t^2\right)
= \prod_{l=1}^\infty\left(1 - t^l\right)^{r_l}
$$
where $r_l$ is the rank of the $l$th graded quotient of the lower
central series of $\pi_g^n$.
\end{theorem}
\begin{proof}
Since $F_g^n$ is a smooth variety and a rational $K(\pi,1)$ by
(\ref{kohno-oda}), we can apply (\ref{wt=lcs}) and the formula
\cite[(9.7)]{hain:cycles} to deduce that
$$
W_{F_g^n}(t) = \prod_{l=1}^\infty\left(1 - t^l\right)^{r_l},
$$
where, for a graded (variation of) MHS $H$
$$
W_H(t) = \sum_{k\ge 0} \chi(\Gr^W_k H)\, t^k
$$
and, for an algebraic variety $X$, $W_X(t) = W_{H^{\bullet}(X)}(t)$.
(Here $\chi$ denotes Euler characteristic.) The result now
follows by induction on $n$ using the fact that
$$
W_{C - \{x_1,\dots,x_n\}}(t) = 1 - 2g\, t + (n-1)t^2
$$
and the following lemma which is proved by induction on the length
of the weight filtration of $V$.
\end{proof}
\begin{lemma}
If ${\mathbb V}$ is an admissible unipotent variation of MHS over a smooth
variety $X$, then
$$
W_{H^{\bullet}(X,{\mathbb V})}(t) = W_X(t)W_{\mathbb V}(t). \qed
$$
\end{lemma}
\section{The General Case}
\label{decorated}
In this section we assemble results from Sections \ref{special} and
\ref{braids2} to obtain a presentation of ${\mathfrak t}_{g,r}^n$ for all $g\ge 6$
and all $r$ and $n \ge 0$. Fix a base point
$$
[C;x_1,\dots,x_n;v_1,\dots,v_n]
$$
of ${\mathcal M}_{g,r}^n$ so that ${\mathfrak t}_{g,r}^n$ and ${\mathfrak p}_{g,r}^n$, etc.\ all have
mixed Hodge structures. The sequence of Lie algebras
$$
0 \to {\mathfrak p}_{g,r}^n \to {\mathfrak t}_{g,r}^n \to {\mathfrak t}_g \to 0
$$
is exact in the category of MHSs, and therefore remains exact after
applying $Gr^W_{\bullet}$:
$$
0 \to \Gr^W_{\bullet} {\mathfrak p}_{g,r}^n \to
\Gr^W_{\bullet}{\mathfrak t}_{g,r}^n \to \Gr^W_{\bullet}{\mathfrak t}_g \to 0.
$$
By (\ref{canon_split}), ${\mathfrak t}_{g,r}^n \otimes {\mathbb C}$ is isomorphic to
the completion of $\Gr^W_{\bullet}{\mathfrak t}_{g,r}^n\otimes {\mathbb C}$. So, to find a
presentation of ${\mathfrak t}_{g,r}^n$, it suffices to find a presentation of
its associated graded. As in \S\ref{special}, there is a lift of the
canonical homomorphism ${\mathbb L}(H_1({\mathfrak t}_g)) \to \Gr^W_{\bullet} {\mathfrak t}_g$ to
a homomorphism ${\mathbb L}(H_1({\mathfrak t}_g)) \to \Gr^W_{\bullet} {\mathfrak t}_{g,r}^n$. Given
presentations of $\Gr^W_{\bullet} {\mathfrak t}_g$ and $\Gr^W_{\bullet} {\mathfrak p}_{g,r}^n$,
a presentation of $\Gr^W_{\bullet} {\mathfrak t}_{g,r}^n$ is determined by maps
\begin{gather*}
a : H_1({\mathfrak t}_g)\otimes H_1({\mathfrak p}_{g,r}^n) \to {\mathfrak p}_{g,r}^n(2)\\
c : \Lambda^2 H_1({\mathfrak t}_g) \to {\mathfrak p}_{g,r}^n(2)_{\lambda_2}\\
c_0 : \Lambda^2 H_1({\mathfrak t}_g) \to {\mathfrak t}_{g,r}^n(2)^{Sp_g}
\end{gather*}
induced by the bracket.
Observe that the homomorphism
$$
{\mathfrak p}_{g,r}^n \to {\mathfrak p}_g^{\oplus(n+r)}
$$
induced by the inclusion $F_{g,r}^n(C) \hookrightarrow C^{n+r}$
induces isomorphisms
$$
H_1({\mathfrak p}_{g,r}^n) \cong H_1({\mathfrak p}_g)^{\oplus(n+r)}\text{ and }
{\mathfrak p}_{g,r}^n(2)_{\lambda_2} \to {\mathfrak p}_g(2)^{\oplus(n+r)}.
$$
By a naturality argument, the map $a$ is easily seen to be the adjoint
of the map
$$
H_1({\mathfrak t}_g) \to \Hom\left(\bigoplus_{j=1}^{r+n} H_1({\mathfrak p}_g),
\bigoplus_{j=1}^{r+n} {\mathfrak p}_g(2)_{\lambda_2}\right)
$$
which is the direct sum of $n+r$ copies of the Johnson homomorphism.
The map $c$ is simply the sum over all the marked points and tangent
vectors
$$
\Lambda^2 H_2({\mathfrak t}_g) \to
\bigoplus_{j=1}^{r+n}{\mathfrak p}_g(2) \cong {\mathfrak p}_{g,r}^n(2)_{\lambda_2}.
$$
of the maps (\ref{bracket}) which is determined in (\ref{bra_const}).
In remains to determine $c_0$. As in the case of ${\mathfrak t}_{g,1}$
considered in Section~\ref{special}, there is a canonical
splitting of the sequence
$$
0 \to {\mathfrak p}_{g,r}^n(2)^{Sp_g} \to {\mathfrak t}_{g,r}^n(2)^{Sp_g}
\to {\mathfrak t}_g^{Sp_g}(2) \to 0
$$
from which we obtain a canonical decomposition
$$
{\mathfrak t}_{g,r}^n(2)^{Sp_g} = {\mathfrak p}_{g,r}^n(2)^{Sp_g} \oplus \Ga.
$$
As in \S\ref{special}, we identify $H_1({\mathfrak t}_g)$ with the subspace of
$\Lambda^3 V$ which is the kernel of the projection (\ref{projn}),
denote by $\bil \phantom{x} \phantom{x}$ the unique $Sp_g$ invariant bilinear
form on $H_1({\mathfrak t}_g)$ such that
$$
\bil {a_1\wedge a_2 \wedge a_3} {b_1\wedge b_2 \wedge b_3} = 1,
$$
and choose a generator $\gamma$ of $\Ga$ such that if $u,v\in H_1({\mathfrak t}_g)$,
then the invariant part of $[u,v]$ in ${\mathfrak t}_g(2)$ is $\bil u v \,\gamma$.
Observe that there is an exact sequence
$$
0 \to \bigoplus_{1\le i<j\le r+n} {\mathbb Q}(1) \oplus \bigoplus_{i=1}^r {\mathbb Q}(1)
\to {\mathfrak p}_{g,r}^n(2) \to \bigoplus_{j=1}^{r+n} {\mathfrak p}_g(2) \to 0
$$
of $Sp_g$ modules. It follows that
$$
{\mathfrak p}_{g,r}^n(2)^{Sp_g} \cong
\bigoplus_{1\le i<j\le r+n} {\mathbb Q}(1) \oplus \bigoplus_{i=1}^r {\mathbb Q}(1).
$$
The terms indexed by $1\le i \le r$ correspond to the $r$ marked
tangent vectors; those indexed by $1\le i < j \le r+n$
to the diagonals $\Delta_{ij}$. It is easy to see that the composition
$$
\Lambda^2 H_1({\mathfrak t}_g) \to \bigoplus_{i=1}^r {\mathbb Q}(1)
$$
of $c_0$ with the projection
$$
{\mathfrak t}_{g,r}^n(2)^{Sp_g} \to \bigoplus_{i=1}^r {\mathbb Q}(1)
$$
is the sum of the maps $c_0$ associated to ${\mathfrak t}_{g,1}$ computed
in (\ref{triv_cpt}). So it remains to determine the composition
$$
\Lambda^2 H_1({\mathfrak t}_g) \to \bigoplus_{1\le i<j\le r+n} {\mathbb Q}(1)
$$
of $c_0$ with the projection
$$
e^{r+n} : {\mathfrak t}_{g,r}^n(2)^{Sp_g} \to \bigoplus_{1\le i<j\le r+n} {\mathbb Q}(1).
$$
To do this, it suffices to compute $e^2$ in the case of ${\mathfrak t}_g^2$,
for then the map $e^{r+n}$ is simply the sum of the $e^2$s over all
diagonals.
In order to compute
$$
e^2 : {\mathfrak t}_{g,r}^n(2)^{Sp_g} \to {\mathbb Q}(1)
$$
we use the fact that a punctured tubular neighbourhood of the diagonal
$\Delta$ in $C\times C$ is homeomorphic to the frame bundle of the
tangent bundle of $C$. In this way, we obtain homomorphisms
$$
{\mathfrak t}_{g,1} \to {\mathfrak t}_g^2 \text{ and } {\mathfrak p}_{g,1} \to {\mathfrak p}_g^2.
$$
In particular, we have a map
\begin{equation}\label{rest}
{\mathfrak p}_{g,1}(2)^{Sp_g} \to {\mathfrak p}_g^2(2)^{Sp_g}.
\end{equation}
Using the fact that $\Gr^W_{\bullet} {\mathfrak p}_{g,1} \to \Gr^W_{\bullet} {\mathfrak p}_g^2$
is a homomorphism and that on $H_1$, it is the diagonal map
$V\to V\oplus V$, we see that the map (\ref{rest}) takes the
generator $\sum_k [A_k,B_k]$ of ${\mathfrak p}_{g,1}(2)^{Sp_g}$ to
\begin{multline*}
\sum_{k=1}^g\, \left(
[{\mathcal A} 1, \B 1] + [{\mathcal A} 1, \B 2] + [{\mathcal A} 2, \B 1] + [{\mathcal A} 2, \B 2]\right) \\
= 2 \sum_{k=1}^g\,\left( [{\mathcal A} 1, \B 2] + [{\mathcal A} 2, \B 1]\right).
\end{multline*}
It follows that in ${\mathfrak t}_g^2$, the map $c_0$ is given by
$$
c_0[u,v] = \bil u v \,\gamma - \frac{12\bil u v}{g(2g+1)}
\sum_{k=1}^g\, \left([{\mathcal A} 1, \B 2] + [{\mathcal A} 2, \B 1]\right).
$$
This completes the determination of $c_0$ in general and, with it,
the descriptions of the ${\mathfrak t}_{g,r}^n$.
\section{Applications}
\label{applications}
\subsection{Cup products and Massey products}
\label{cup}
We have shown that for all $g \ge 6$ and all $r,n\ge 0$, the
Lie algebra $\Gr^W_{\bullet} {\mathfrak t}_{g,r}^n$ has a presentation with
only quadratic relations. This implies, using the short exact
sequence in \cite{sullivan:les} for example, that the cup product
$$
\Lambda^2 \Gr^W_{\bullet} H_{\mathrm{cts}}^1({\mathfrak t}_{g,r}^n)
\to \Gr^W_{\bullet} H_{\mathrm{cts}}^2({\mathfrak t}_{g,r}^n)
$$
is surjective. It follows that the cup product
$$
\Lambda^2 H_{\mathrm{cts}}^1({\mathfrak t}_{g,r}^n) \to H_{\mathrm{cts}}^2({\mathfrak t}_{g,r}^n)
$$
is also surjective.
Recall that the $l$-fold Massey products constructed from $H^1(A^{\bullet})$,
where $A^{\bullet}$ is a d.g.a., are defined on a subspace $D_l$ of
$H^1(A^{\bullet})^{\otimes l}$ and take values in $H^2(A^{\bullet})/I_{l-1}$,
where $I_{l-1}$ denotes the lift to $H^2(A^{\bullet})$ of the image of the
Massey products of order $< l$.
It follows that all Massey products in $H_{\mathrm{cts}}^2({\mathfrak t}_{g,r}^n)$ of order
$\ge 3$ vanish when $g\ge 6$ as the cup product (Massey products of
order 2) map is surjective.
Since the natural map $H_{\mathrm{cts}}^2({\mathfrak t}_{g,r}^n) \to H^2(T_{g,r}^n,{\mathbb Q})$
is injective (cf.\ (\ref{cts_ord})) and preserves Massey products,
we have:
\begin{theorem}
For all $g \ge 6$, all Massey products of order $\ge 3$ in
$H^2(T_{g,r},{\mathbb Q})$ vanish. \qed.
\end{theorem}
\begin{remark}
It follows from the fact that there are non-trivial cubic relations
and no quadratic relations in a minimal presentation of ${\mathfrak t}_3$ that
the cup product
$$
\Lambda^2 H^1(T_3,{\mathbb Q}) \to H^2(T_3,{\mathbb Q})
$$
vanishes, and that the Massey triple product map
$$
H^1(T_3,{\mathbb Q})^{\otimes 3} \to H^2(T_3,{\mathbb Q})
$$
is non-trivial.
\end{remark}
It follows from (\ref{lower}) and (\ref{computation}) that for all
$g \ge 6$, $H_{\mathrm{cts}}^2({\mathfrak t}_g)$ has highest weight decomposition
$$
H_{\mathrm{cts}}^2({\mathfrak t}_g) \cong V(\lambda_6) + V(\lambda_4) + V(\lambda_2)
+ V(\lambda_2 + \lambda_4).
$$
\begin{theorem}
For all $g \ge 3$, there is an (unnatural) isomorphism
$$
H_{\mathrm{cts}}^2({\mathfrak t}_{g,r}^n) \cong H_{\mathrm{cts}}^2({\mathfrak t}_g) \oplus
\left(H_{\mathrm{cts}}^1({\mathfrak p}_{g,r}^n)\otimes H_{\mathrm{cts}}^1({\mathfrak t}_g)\right)
\oplus H_{\mathrm{cts}}^2({\mathfrak p}_{g,r}^n)
$$
of $Sp_g$ modules.
\end{theorem}
\begin{proof}
Chose a base point of ${\mathcal M}_{g,r}^n$. Then
$$
0 \to {\mathfrak p}_{g,r}^n \to {\mathfrak t}_{g,r}^n \to {\mathfrak t}_g \to 0
$$
is an exact sequence of mixed Hodge structures, and the
corresponding spectral sequence
$$
E_2^{s,t} = H_{\mathrm{cts}}^s({\mathfrak t}_g,H_{\mathrm{cts}}^t({\mathfrak p}_{g,r}^n)) \implies
H_{\mathrm{cts}}^{s+t}({\mathfrak t}_{g,r}^n)
$$
is a spectral sequence in the category of mixed Hodge structures.
Since ${\mathfrak t}_g$ has negative
weights, the weights on $H_{\mathrm{cts}}^k({\mathfrak t}_g)$ are $\ge k$. This
and the fact that $H^k({\mathfrak p}_{g,r}^n)$ is a trivial ${\mathfrak t}_g$ module
when $k = 1$ and 2 imply that $E_\infty^{s,t} = E_2^{s,t}$ when
$s+t=2$. The result follows.
\end{proof}
\subsection{Period space is not contractible when $g\ge 4$}
Denote by ${\mathfrak h}_g$ the Siegel upper half space; that is, the space
of symmetric $g\times g$ complex matrices with positive definite
imaginary part. Denote the image of the period map
$$
\text{Teichm\"uller space } \to {\mathfrak h}_g
$$
by ${\mathcal J}_g$, and its closure by $\overline{{\mathcal J}}_g$. When $g \le 3$,
$\overline{\J}_g={\mathfrak h}_g$, so that $\overline{\J}_g$ is contractible in these cases.
\begin{theorem}
For each $g\ge 4$, $H^2(\overline{\J}_g,{\mathbb Q})$ is non-trivial. Consequently,
$\overline{\J}_g$ is not contractible.
\end{theorem}
The proof proceeds in two steps. We begin by making a definition.
\begin{definition}
The {\it extended Torelli group} $\widehat{T}_g$ is the subgroup of $\Gamma_g$
consisting of those mapping classes which act as $\pm$ the identity
on the first homology of the reference surface.
\end{definition}
We have group extensions
\begin{equation}\label{extensions}
1 \to T_g \to \widehat{T}_g \to {\mathbb Z}/2{\mathbb Z} \to 0\text{ and }
1 \to \widehat{T}_g \to \Gamma_g \to PSp_g({\mathbb Z}) \to 1
\end{equation}
where $PSp_g({\mathbb Z})$ denotes the quotient of $Sp_g({\mathbb Z})$ by $\pm I$.
Note that the first sequence gives rise to a natural action of
${\mathbb Z}/2{\mathbb Z}$ on $H^{\bullet}(T_g)$.
The first step is:
\begin{proposition}
For all $g \ge 3$, there are natural isomorphisms
$$
H^{\bullet}({\mathcal J}_g,{\mathbb Q}) \cong H^{\bullet}(\widehat{T}_g,{\mathbb Q}) \cong H^{\bullet}(T_g,{\mathbb Q})^{{\mathbb Z}/2{\mathbb Z}}.
$$
Moreover, when $g\ge 4$, $H^2(\widehat{T}_g,{\mathbb Q})$ is non-trivial.
\end{proposition}
\begin{proof}
Since $g\ge 3$, ${\mathcal J}_g$ is the quotient of Teichm\"uller space by
$\widehat{T}_g$. Since the mapping class group acts on Teichm\"uller space
virtually freely, this implies (via standard arguments) that
there is a natural isomorphism
$$
H^{\bullet}(\widehat{T},{\mathbb Q}) \cong H^{\bullet}({\mathcal J}_g,{\mathbb Q}).
$$
Applying the Hochschild-Serre spectral sequence to the first
of the extensions (\ref{extensions}) above, we see that
$$
H^k(\widehat{T}_g,{\mathbb Q}) \cong H^k(T_g,{\mathbb Q})^{{\mathbb Z}/2}.
$$
But $-I \in Sp_g({\mathbb Z})$ acts trivially on $H_{\mathrm{cts}}^2({\mathfrak t}_g)$, which
implies that
$$
H_{\mathrm{cts}}^2({\mathfrak t}_g) \subseteq H^2(\widehat{T}_g,{\mathbb Q}).
$$
The result follows.
\end{proof}
\begin{remark}
This argument also shows that the image of the cup product
$$
\Lambda^2 H^1(T_3,{\mathbb Z}) \to H^2(T_3,{\mathbb Z})
$$
is 2-torsion.
\end{remark}
To complete the proof of the theorem, note that ${\mathcal J}_g = \overline{\J}_g - {\mathcal R}$
where ${\mathcal R}$ is the locus of reducible jacobians. By standard arguments,
each component of ${\mathcal R}$ has complex codimension $\ge 2$ in $\overline{\J}_g$.
Combining Lefschetz duality and the Gysin sequence, we obtain an exact
sequence
$$
H^{BM}_{6g-k-6}({\mathcal R}) \to H^k(\overline{\J}_g) \to H^k({\mathcal J}_g)
\to H^{BM}_{6g-k-7}({\mathcal R}),
$$
where $H_{\bullet}^{BM}$ denotes Borel-Moore homology. Since ${\mathcal R}$ has real
codimension 4, it follows that $H^2(\overline{\J}_g) \cong H^2({\mathcal J}_g)$. The
theorem follows as $H^2({\mathcal J}_g)$ is non-trivial.
\subsection{Johnson's conjecture}
In \cite{johnson:survey}, Johnson constructed maps
$$
\phi_k : H_k(T_g) \to H_{k+2}(\Jac S)/[S]\times H_k(\Jac S),
$$
which generalize the classical Johnson homomorphism, which is the
case $k=1$. He conjectured that these homomorphisms are isomorphisms
for all $k$ and sufficiently large $g$.
The following result is an improvement of some unpublished computations
of Morita (cf.\ \cite[\S4]{morita:jap_acad}).
\begin{theorem}
For all $g \ge 3$, the map $\phi_2$ is not injective.
\end{theorem}
\begin{proof}
It is not difficult to see that each $\phi_k$ is $Sp_g({\mathbb Z})$
equivariant. Consider its adjoint
$$
\phi_k^t : H^{k+2}(\Jac S,{\mathbb Q})/\omega\wedge H^k(\Jac S,{\mathbb Q})
\to H^k(T_g,{\mathbb Q}).
$$
This is also $Sp_g({\mathbb Z})$ equivariant. The domain of $\phi_2^t$ is
the primitive cohomology group $PH^4(\Jac S,{\mathbb Q})$. This is the restriction
to $Sp_g({\mathbb Z})$ of the rational representation of $Sp_g$ with highest
weight $\lambda_4$. Since this is an irreducible $Sp_g({\mathbb Z})$ module,
the image of $\phi_2^t$ is either isomorphic to $V(\lambda_4)$ or
trivial. But $H^2(T_g,{\mathbb Q})$ contains the rational representation
$H_{\mathrm{cts}}^2({\mathfrak t}_g)$. It follows from the results in \S\ref{cup} that
$$
H_{\mathrm{cts}}^2({\mathfrak t}_g)/\im \phi_2^t \cap H_{\mathrm{cts}}^2({\mathfrak t}_g)
$$
is non-trivial as it contains
$V(\lambda_6) + V(\lambda_2 + \lambda_4)$ when $g\ge 6$;
$V(\lambda_2 + \lambda_4)$ when $g = 4,5$; and $V(\lambda_3)$
when $g = 3$. The result follows.
\end{proof}
\subsection{Filtrations of $T_g^1$}\label{filtn}
Define a filtration
\begin{equation}\label{filtration}
T_g =
L^1 T_g^1 \supseteq L^2 T_g^1 \supseteq L^3 T_g^1 \supseteq \cdots
\end{equation}
of $T_g^1$ by
$$
L^k T_g^1 =
\left\{\phi \in T_g^1 : \phi_\ast : \pi_1(S,x) \to \pi_1(S,x)
\text{ is congruent to the identity mod } \Gamma^{k+1}\right\}.
$$
It is quite common in the literature for this filtration to be called
the {\it relative weight filtration}, as it is in \cite{asada-nakamura}
and \cite{oda}. In view of (\ref{mhs_torelli}) and (\ref{unequal}), I
feel that this terminology is likely to result in confusion.
\begin{proposition}
This filtration is a descending central series of $T_g^1$ with
torsion free quotients and has the property that
$$
\bigcap_{k=1}^\infty L^k T_g^1
$$
is trivial.
\end{proposition}
\begin{proof}
This follows directly from the fact that the fundamental group of
a compact Riemann surface is residually nilpotent \cite{baumslag},
and the fact that the graded quotients of the lower central series
of a surface group are torsion free \cite{labute}.
\end{proof}
The most rapidly descending series with torsion free quotients
of a group $G$ is the series
$$
G = D^1 G \supseteq D^2 G \supseteq D^3 G \supseteq \cdots
$$
where
$$
D^k G = \{g\in G :
\text{ there is an integer $n>0$ such that } g^n \in \Gamma^k G\}.
$$
This filtration has the property that $D^kG/D^{k+1}$ is the
$k$th term of the lower central series of $G$ mod torsion.
Proofs of these assertions can be found in \cite{passman}.
In the current situation, we have
$$
D^k T_g^1 \subseteq L^k T_g^1.
$$
Johnson's Theorem \cite{johnson:h1} implies that $D^2 T_g^1 =
L^2 T_g^1$ when $g \ge 3$. The computations (\ref{upper}) and
(\ref{lower}) imply that the kernel of $D^2 T_g^1 / D^3 \to
L^2 T_g^1 /L^3$ is isomorphic to ${\mathbb Z}$. Morita was aware of the
fact that the kernel was at least this big --- cf.\ his work on
the Casson invariant \cite{morita:casson}, and asked whether there
is a $k$ such that $D^3 T_g^1 \supseteq L^k T_g^1$. That is, whether
the kernel of $D^2 T_g^1 / D^3 \to L^2 T_g^1 /L^3$ can be detected by
the action of $T_g^1$ on the quotients of $\pi_g$ by the terms of its
lower central series.
More generally, one can ask if the topologies on $T_g^1$ determined by
the filtrations $D^{\bullet}$ and $L^{\bullet}$ are equivalent. (Both are
separated.) That is, for each $k\in {\mathbb N}$, can one find a positive integer
$n(k)$ such that $L^{n(k)} T_g^1 \subseteq D^k T_g^1$~?
Since the groups $T_g^1/D^k$ and $T_g^1/L^k$ are torsion free nilpotent,
they imbed as a Zariski dense subgroup of a unipotent group defined over
${\mathbb Q}$. One obtains two inverse systems of unipotent groups. It is clear
that the first prounipotent group is the Malcev completion ${\mathcal T}_g^1$ of
$T_g^1$, and the second is the prounipotent group associated to the
pronilpotent Lie algebra ${\mathfrak h}_g:=\im\{{\mathfrak t}_g^1 \to {\mathfrak d}_g\}$, where ${\mathfrak d}_g$
is the pronilpotent Lie algebra defined in \S \ref{inf_action}. The two
topologies on $T_g^1$ are equivalent if and only if the natural map
${\mathfrak t}_g^1 \to {\mathfrak h}_g$ is an isomorphism. Equivalently, they are equivalent
if and only if ${\mathfrak t}_g^1 \to {\mathfrak d}_g$ is injective. It is also clear that
the filtration $L^{\bullet}$ of ${\mathfrak t}_g^1$ induced from that of $T_g^1$ is
the pullback of the weight filtration of ${\mathfrak h}_g^1$, so that
$$
\left(L^kT_g^1/L^{k+1}\right) \otimes{\mathbb Q} \cong \Gr^k_L {\mathfrak t}_g^1
\cong \Gr^W_{-k} {\mathfrak h}_g^1
$$
and that $L^k{\mathfrak t}_g^1 \supseteq \Ga$ for all $k\ge 1$ --- cf.\
(\ref{central_ext}).
\begin{theorem}\label{unequal}
For all $g \ge 3$, and all $k\ge 1$, $L^k{\mathfrak t}_g^1 \supseteq \Ga$
so that the natural representation
${\mathfrak t}_g^1 \to {\mathfrak d}_g$ is not injective as its kernel contains $\Ga$.
In particular, there is no $k\ge 1$ such that
$W_{-3}{\mathfrak t}_g^1 \supseteq L^k{\mathfrak t}_g^1$. \qed
\end{theorem}
One can define a filtration $L^{\bullet}$ of $T_g$ by defining $L^k T_g$ to
be the image of $L^k T_g^1$. Using similar arguments, one can prove that
the filtrations $L^{\bullet}$ and $D^{\bullet}$ of $T_g$ do not define equivalent
topologies.
\subsection{A question of Asada and Nakamura}
There is an issue raised by Asada and Nakamura in
\cite[(4.5)]{asada-nakamura} which is closely related to Morita's
question. Denote by $\pi_{g,1}$ the fundamental group $\pi_1(S,v)$ of
$S$ with respect to the tangent vector $v$. It is naturally isomorphic
to the fundamental group of the punctured surface $S$ minus the anchor
point $x$ of $v$. Note that $T_{g,1}$ acts on $\pi_{g,1}$. They define
a filtration $M^{\bullet}$ of $T_g^1$ as follows: First define a filtration
$L^{\bullet}$ of $T_{g,1}$ as in the previous section: $\phi$ is in
$L^kT_{g,1}$ if and only if $\phi$ induces the identity on $\pi_{g,1}$
modulo the $(k+1)$st term of its lower central series. Define $M^kT_g^1$
to be the image of $L^kT_{g,1}$ in $T_g^1$. They then ask whether, after
tensoring with ${\mathbb Q}$, the sequence
$$
0 \to \Gr^W_{\bullet} \pi_g \to \Gr^M_{\bullet} T_g^1
\to \Gr^L_{\bullet} T_g \to 0
$$
is exact. (Recall from (\ref{wt=lcs}) that the lower central series of
${\mathfrak p}_g$ agrees with its weight filtration.) We now give a proof that this
is indeed the case. We continue with the notation of the previous section.
The filtration $M^{\bullet}$ induces a filtration of ${\mathfrak t}_g^1$.
Their question then beocmes: is the sequence
$$
0 \to \Gr^W_{\bullet} {\mathfrak p}_g \to \Gr^M_{\bullet} {\mathfrak t}_g^1
\to \Gr^L_{\bullet} {\mathfrak t}_g \to 0
$$
exact? Fix a base point of ${\mathcal M}_{g,1}$ so that ${\mathfrak t}_{g,1}$, ${\mathfrak t}_g^1$,
$\pi_{g,1}$, ${\mathfrak d}_g$, ${\mathfrak p}_g$, etc.\ all have compatible MHSs; the MHS
on ${\mathfrak p}_{g,1}$ is the limit MHS on $\pi_1(S-\{x\},x_o)$ associated to
the ``degeneration'' where $x_o$ approaches $x$ from the direction of
$v$. Denote the image
of ${\mathfrak t}_g^1$ in ${\mathfrak d}_g$ by ${\mathfrak h}_g^1$, and the image of ${\mathfrak t}_g$ in ${\mathfrak o}_g$ by
${\mathfrak h}_g$. These have canonical mixed Hodge structures determined by the
choice of the base point. Since the diagram
$$
\begin{CD}
0 @>>> {\mathfrak p}_g @>>> {\mathfrak t}_g^1 @>>> {\mathfrak t}_g @>>> 0\\
@. @| @VVV @VVV \\
0 @>>> {\mathfrak p}_g @>>> {\mathfrak h}_g^1 @>>> {\mathfrak h}_g @>>> 0\\
\end{CD}
$$
commutes and since the top row is exact, it follows that
the bottom row is exact. Since $\Gr^W_{\bullet}$ is an exact functor,
and since $\Gr^W_k {\mathfrak h}_g^n \cong \Gr^L_k{\mathfrak t}_g^n$ when
$n = 0,1$, this implies that the sequence
$$
0 \to \Gr^W_{\bullet} {\mathfrak p}_g \to \Gr^L_{\bullet} {\mathfrak t}_g^1
\to \Gr^L_{\bullet} {\mathfrak t}_g \to 0
$$
is exact. To complete the proof, we show that the filtrations $L^{\bullet}$
and $M^{\bullet}$ of ${\mathfrak t}_g^1$ are equal.
Denote by $b_o$ the element of $\pi_{g,1}$ that corresponds to rotating
the tangent vector once about $x$ --- this is a ``Dehn twist about the
boundary of $S-\{x\}$.'' The action of $T_{g,1}$ on $\pi_{g,1}$ fixes
$b_o$, and therefore induces a homomorphism
$T_{g,1} \to \Aut(\pi_{g,1},b_o)$ into the automorphisms of $\pi_{g,1}$
that fix $b_o$. Set $w_o = \log b_o$. This we interpret
as an element of ${\mathfrak p}_{g,1}$. The homomorphism above induces
a homomorphism $T_{g,1} \to \Aut({\mathfrak p}_{g,1},w_o)$, and therefore a Lie
algebra homomorphism
$$
{\mathfrak t}_{g,1} \to \Der({\mathfrak p}_{g,1},w_o)
$$
into the derivations of ${\mathfrak p}_{g,1}$ that annhilate $w_o$. It follows
from standard properties of limit MHSs that $w_o$ spans a copy of
${\mathbb Q}(1)$ in $\Der{\mathfrak p}_{g,1}$. But $\Der({\mathfrak p}_{g,1},w_o)$ is the kernel
of the map $\Der {\mathfrak p}_{g,1} \to {\mathfrak p}_{g,1}$
that takes $\phi$ to $\phi(w_o) - w_o$. Since $w_o$ is a Hodge class,
this is a morphism of MHS. It follows that $\Der({\mathfrak p}_{g,1},w_o)$ has
a natural MHS.
Since ${\mathfrak p}_g$ is the quotient of ${\mathfrak p}_{g,1}$ by the ideal generated by
$w_o$ as MHS, the homomorphism
$$
\Der({\mathfrak p}_{g,1},w_o) \to \Der {\mathfrak p}_g
$$
is a morphism of MHS. The filtration $M^{\bullet}$ of ${\mathfrak t}_{g,1}$ is the
inverse image of the weight filtration under the homomorphism
${\mathfrak t}_{g,1}\to \Der({\mathfrak p}_{g,1},w_o)$. The equality of the filtrations
$L^{\bullet}$ and $M^{\bullet}$ of ${\mathfrak t}_g^1$ now follows from the strictness
properties of the weight filtration as the diagram
$$
\begin{CD}
{\mathfrak t}_{g,1} @>>> {\mathfrak t}_g^1 \cr
@VVV @VVV \cr
\Der({\mathfrak p}_{g,1},w_o) @>>> \Der {\mathfrak p}_g \cr
\end{CD}
$$
commutes and all arrows are morphisms of MHS.
\subsection{Cohomology of ${\mathfrak t}_g$ and vanishing differentials}
Since ${\mathfrak t}_{g,r}^n = W_{-1}{\mathfrak t}_{g,r}^n$, it follows that
$$
W_{k-1}H_{\mathrm{cts}}^k({\mathfrak t}_{g,r}^n) = 0
$$
for all $k\ge 0$. The {\it lowest weight subring of}
$H_{\mathrm{cts}}^{\bullet}({\mathfrak t}_{g,r}^n)$ is defined to be the subring
$$
\bigoplus_{k\ge 0} W_kH_{\mathrm{cts}}^k({\mathfrak t}_{g,r}^n).
$$
By \cite[(9.2)]{hain:cycles}, this is a quadratic algebra generated
by $H_{\mathrm{cts}}^1({\mathfrak t}_{g,r}^n)$ and where the relations are dual to the
second graded quotient of the lower central series of $T_{g,r}^n$.
The following result is a refinement of (\ref{vanishing}).
\begin{theorem}\label{van_diffls}
For each irreducible representation $V(\lambda)$ of $Sp_g$, the
image of the natural homomorphism
$$
\left[W_kH_{\mathrm{cts}}^k({\mathfrak t}_{g,r}^n) \otimes V(\lambda)\right]^{Sp_g} \to
H^0(Sp_g({\mathbb Z}),H^k(T_{g,r}^n)\otimes V(\lambda)) = E_2^{0,k}
$$
is contained in
$$
E_\infty^{0,k} = \im\left\{
H^k(\Gamma_{g,r}^n,V(\lambda)) \to H^k(T_{g,r}^n)\otimes V(\lambda)
\right\}.
$$
\end{theorem}
\begin{proof}
Fix a base point of ${\mathcal M}_{g,r}^n$ so that ${\mathfrak t}_{g,r}^n$, ${\mathfrak u}_{g,r}^n$, etc.\
all have compatible MHSs. Since the extension
$$
0 \to \Ga \to {\mathfrak t}_{g,r}^n \to {\mathfrak u}_{g,r}^n \to 0
$$
is central with kernel isomorphic to ${\mathbb Q}(1)$, it follows from the
Gysin sequence that the induced map
$$
\bigoplus_{k\ge 0} W_kH_{\mathrm{cts}}^k({\mathfrak u}_{g,r}^n) \to
\bigoplus_{k\ge 0} W_kH_{\mathrm{cts}}^k({\mathfrak t}_{g,r}^n).
$$
is surjective, with kernel the ideal generated by the cohomology class
in $W_2H_{\mathrm{cts}}^2({\mathfrak u}_{g,r}^n)$ corresponding to the extension above.
By (\ref{morph_u}), there is a canonical map
$$
\left[H_{\mathrm{cts}}^k({\mathfrak u}_{g,r}^n)\otimes V(\lambda)\right]^{Sp_g}
\longrightarrow H^k(\Gamma_{g,r}^n,{\mathbb V}(\lambda)).
$$
The result follows because the diagram
$$
\begin{CD}
\left[W_kH_{\mathrm{cts}}^k({\mathfrak u}_{g,r}^n)\otimes V(\lambda)\right]^{Sp_g}
@>>> H^{\bullet}(\Gamma_{g,r}^n,{\mathbb V}(\lambda)) \\
@VVV @VVV \\
\left[W_kH_{\mathrm{cts}}^k({\mathfrak t}_{g,r}^n)\otimes V(\lambda)\right]^{Sp_g}
@>>> H^0(Sp_g({\mathbb Z}),H^k(T_{g,r}^n)\otimes V(\lambda))
\end{CD}
$$
commutes, and because the left hand vertical map is surjective.
\end{proof}
\subsection{Morita's Conjecture}
We now prove a result which is, in some sense, an affirmation of Morita's
conjecture \cite[2.7]{morita:conj}. Our result is an analogue of his
theorem \cite[6.2]{morita:conj} which is a solution to the conjecture in
the first non-trivial case. He also informs me that he has proved the
second non-trivial case of the conjecture over ${\frac{1}{24}}{\mathbb Z}$.
Suppose that $g\ge 3$. Denote the $k$th term of the lower central
series of $\pi_g$ by $\pi^{(k)}$. Set
$$
\pi_{(k)} = \pi_g/\pi^{(k+1)}.
$$
We know from Labute's theorem \cite{labute} that this is a torsion free
nilpotent group. For each $k\ge 1$, there is a natural representation
$$
\rho_k : \Gamma_g^1 \to \Aut \pi_{(k)}.
$$
The first is simply the standard representation $\Gamma_g^1 \to Sp_g({\mathbb Z})$.
Denote the ${\mathbb Q}$ form of the unipotent completion of $\pi_{(k)}$ by ${\mathcal P}_{(k)}$.
Since $\pi_{(k)}$ is torsion free, the canonical map $\pi_{(k)} \to {\mathcal P}_{(k)}$
is injective. By the universal mapping property of unipotent completion,
we see that each $\rho_k$ extends to a homomorphism
$$
\tilde{\rho}_k : \Gamma_g^1 \to \Aut {\mathcal P}_{(k)}.
$$
Denote the Lie algebra of ${\mathcal P}_{(k)}$ by ${\mathfrak p}_{(k)}$. Then
$\Aut {\mathcal P}_{(k)} \cong \Aut {\mathfrak p}_{(k)}$.
It follows that $\Aut {\mathcal P}_{(k)}$ is a linear algebraic group. Denote
the Zariski closure of the image of $\tilde{\rho}_k$ in this by $G_k$.
It is easy to see that $G_k$ is an extension of $Sp_g({\mathbb Q})$ by a
unipotent group:
$$
1 \to U_k \to G_k \to Sp_g({\mathbb Q}) \to 1
$$
This extension is split exact, so that
$$
G_k \cong Sp_g({\mathbb Q}) \ltimes U_k.
$$
By the universal mapping property of the relative completion of
$\Gamma_g^1$, there is a homomorphism ${\mathcal G}_g^1 \to G_k$ which
commutes with the projections to $Sp_g$. The following result follows
directly from the fact (\ref{central_ext}) that the natural map
${\mathcal T}_g^1 \to {\mathcal U}_g^1$ is surjective.\footnote{A direct proof of the
lemma can be given --- cf.\ the proof of \cite[(4.6)]{hain:comp}.}
\begin{lemma}
For each $k\ge 2$, $\tilde{\rho}(T_g^1)$ is Zariski dense in $U_k$. \qed
\end{lemma}
\begin{proposition}
For each $k \ge 2$, the image of $\tilde{\rho}_k$ is a discrete subgroup of
$G_k({\mathbb R})$, and the quotient $\im\rho_k\backslash G_k({\mathbb R})$ has finite
volume with respect to any left invariant metric on $G_k({\mathbb R})$.
\end{proposition}
\begin{proof}
Since every finitely generated subgroup of the ${\mathbb Q}$ points of a unipotent
group $U$ is discrete in $U({\mathbb R})$, it follows that $\tilde{\rho}_k(T_g^1)$
is a discrete subgroup of $U_k({\mathbb R})$. Since it is also Zariski dense, it
is cocompact. The result now follows as the image of $\Gamma_g^1$ in
$Sp_g({\mathbb R})$ is $Sp_g({\mathbb Z})$, which is discrete and of finite covolume.
\end{proof}
We should note that Morita works with $\Gamma_{g,1}$ rather than with
$\Gamma_g^1$ as we do. Our arguments work equally well in his case;
we chose to work with $\Gamma_g^1$ as it seems more natural.
In conclusion, we remark that the Lie algebra of $U_k$ is simply the
image ${\mathfrak h}_g^1/W_{-k-1}$ of ${\mathfrak u}_g^1$ in $\Der {\mathfrak p}_{(k)}$. It follows
that the Lie algebra of $U_k$ has a MHS, and is therefore isomorphic
to its associated graded after tensoring with ${\mathbb C}$.
\section{The Universal Connection}
\label{connection}
In this section we construct a universal connection form
$$
\widetilde{\omega} \in
E^{\bullet}(\text{Torelli space})\comptensor \Gr^{\bullet} {\mathfrak t}_{g,r}^n
$$
with ``scalar curvature''
on Torelli space when $g\ge 3$. Here $E^{\bullet}(X)$ denotes the $C^\infty$
de~Rham complex of a smooth manifold $X$, and $\comptensor$ the completed
tensor product.%
\footnote{The completed tensor product $E^{\bullet}(X)\comptensor\Gr^{\bullet}{\mathfrak g}$
is defined to be the inverse limit
$$
\lim_\leftarrow E^{\bullet}(X)\otimes \Gr^{\bullet}{\mathfrak g} /\oplus_{l\ge m} \Gr^l{\mathfrak g}.
$$
}
This is the analogue of the universal connection
$$
\sum_{ij} d\log(z_i - z_j)\, X_{ij} \in E^{\bullet}(X_n) \otimes \Gr^{\bullet}{\mathfrak p}_n
$$
for the braid group $P_n$. Here $X_n$ denotes the classifying space
$$
{\mathbb C}^n - \left\{\text{fat diagonal}\right\}
$$
of the pure braid group, $(z_1,\dots,z_n)$ its coordinates, and ${\mathfrak p}_n$ the
Malcev Lie algebra associated to $P_n$. A reasonably precise dictionary
between the case of braid groups and the absolute mapping class groups
$\Gamma_g$ is given in the table.
\begin{table}
{\footnotesize
\begin{tabular}{c|c|p{1.2in}}
\hline
Braid Groups & Mapping Class Groups & \quad Comments \\
\hline
&&\\
$B_n$ & $\Gamma_g$ & {\raggedright the group of interest}\\
&&\\
$\Sigma_n$ & $Sp_g({\mathbb C})$ & {\raggedright a semi-simple algebraic group $G$}\\
&&\\
$\rho:B_n \to \Sigma_n$ & $\rho:\Gamma_g \to Sp_g({\mathbb C})$ & {\raggedright homomorphism
to $G$ with dense image}\\
&&\\
$\Sigma_n$ & $Sp_g({\mathbb Z})$ & {\raggedright the image of $\rho$, an arithmetic group}\\
&&\\
$P_n$ & $T_n$ & {\raggedright the kernel of $\rho$, a residually torsion free
nilpotent group}\\
&&\\
${\mathcal P}_n$ & ${\mathcal U}_g$ & {\raggedright prounipotent radical of the relative completion}\\
&&\\
$B_n \to \Sigma_n \ltimes {\mathcal P}_n$ &
$\Gamma_g \to {\mathcal G}_g \cong Sp_g({\mathbb C})\ltimes {\mathcal U}_g$ &
{\raggedright the relative completion}\\
&&\\
${\mathcal P}_n$ & ${\mathcal T}_g$ & {\raggedright the unipotent completion of the kernel of $\rho$}\\
&&\\
$\id: {\mathcal P}_n \to {\mathcal P}_n$ & ${\mathcal T}_g \to {\mathcal U}_g$ & {\raggedright the homomorphism to the
prounipotent radical}\\
&&\\
${\mathfrak p}_n$ & ${\mathfrak t}_g$ & {\raggedright the pronilpotent Lie algebra corresponding to
$\ker\rho$}\\
&&\\
$\Gr^{\bullet} {\mathfrak p}_n = {\mathbb L}(H_1(P_n))/R$ &
$\Gr^{\bullet} {\mathfrak t}_g = {\mathbb L}(H_1(T_g))/R$ &
{\raggedright quadratic presentations as graded Lie algebras in the category of
representations of $G$}\\
&&\\
$\left\{[X_{ij},X_{kl}],[X_{ij},X_{ik}+X_{jk}]\right\}$ &
$V(\lambda_6), V(\lambda_4), V(\lambda_2), V(\lambda_2 + \lambda_4)$ &
{\raggedright the quadratic relations}\\
&&\\
$X_n := {\mathbb C}^n - \{\text{fat diagonal}\}$ & ${\mathcal H}_g := \text{ Torelli space}$ &
{\raggedright the classifying space of the kernel of $\ker\rho$}\\
&&\\
$Y_n := \Sigma_n\backslash X_n$ & ${\mathcal M}_g = Sp_g({\mathbb Z})\backslash {\mathcal H}_g$ &
{\raggedright the classifying space of the group of interest}\\
&&\\
$\sum_{ij} w_{ij}\, X_{ij}
\in E^{\bullet}(X_n)\otimes \Gr^{\bullet}{\mathfrak p}_n$ &
$\omega \in E^{\bullet}({\mathcal H}_g)\comptensor \Gr^{\bullet}{\mathfrak t}_g$ &
{\raggedright the ``universal (projectively) flat connection'' on the classifying
space of $\ker\rho$}
\end{tabular}
}
\end{table}
\begin{question}
The Lie algebra $\Gr^{\bullet} {\mathfrak p}_n$ has interesting finite dimensional
representations; namely those associated to Hecke algebras.
Are there analogous representations of $\Gr^{\bullet} {\mathfrak t}_{g,r}^n$ where the
canonical central $\Ga$ acts via scalar transformations? These should
lead to interesting projective representations of $\Gamma_{g,r}^n$.
\end{question}
We now give the construction of the connection. First recall that
Torelli space ${\mathcal H}_{g,r}^n$ is the quotient of the Teichm\"uller
space associated to $\Gamma_{g,r}^n$ by $T_{g,r}^n$. It is the
moduli space of isomorphism classes of $(n+r+2g+1)$-tuples
$$
(C;x_1,\dots,x_n;v_1,\dots,v_r;a_1,\dots,a_g,b_1,\dots,b_g)
$$
where $C$ is a compact Riemann surface of genus $g$, $x_1,\dots,x_n$
are $n$ marked points, $v_1,\dots,v_r$ are $r$ marked tangent vectors,
and $a_1,\dots, b_g$ is a symplectic basis of $H_1(C,{\mathbb Z})$. Since
$T_{g,r}^n$ is torsion free and Teichm\"uller space is contractible,
${\mathcal H}_{g,r}^n$ is the classifying space of $T_{g,r}^n$.
The bulk of the work needed for the construction of the connection
has already been done in \cite[\S14]{hain:derham}.
Fix a point $x \in {\mathcal M}_{g,r}^n$. It follows from
\cite[\S14.2]{hain:derham} that there is a 1-form
$$
\omega \in E^{\bullet}({\mathcal H}_{g,r}^n)\comptensor \Gr^W_{\bullet} {\mathfrak u}_{g,r}^n
$$
which is integrable and is $Sp_g({\mathbb Z})$ invariant in that
\begin{equation}\label{invariance}
s^\ast \omega = Ad(s)\, \omega
\end{equation}
for all $s\in Sp_g({\mathbb Z})$. That is, if
$$
\omega = \sum_I w_I X_I, \text{ where } w_I \in E^1({\mathcal H}_{g,r}^n)
\text{ and } X_I \in \Gr^W_{\bullet} {\mathfrak u}_{g,r}^n,
$$
then for all $s\in Sp_g({\mathbb Z})$,
$$
\sum_I (s^\ast w_I) X_I = \sum_I w_I (X_I\cdot s^{-1})
$$
where $X\cdot s$ denotes the canonical action of $s\in S$ on
$X\in {\mathfrak u}_{g,r}^n$.
This should be compared with the case of braids where the corresponding
formula is easily verified --- cf. \cite[(14.6)]{hain:derham}.
Since there is a canonical isomorphism
$$
{\mathfrak u}_{g,r}^n(x) \cong \prod_{l\ge 1}\Gr^W_{-l}{\mathfrak u}_{g,r}^n(x)
$$
this form gives rise to a flat connection on the trivial right
${\mathcal U}_{g,r}^n$ principal bundle
$$
{\mathcal H}_{g,r}^n \times {\mathcal U}_{g,r}^n \to {\mathcal H}_{g,r}^n.
$$
Note that $Sp_g({\mathbb Z})$ acts on this bundle via the diagonal action.
The composite
$$
{\mathcal U}_{g,r}^n \to {\mathcal H}_{g,r}^n \to {\mathcal M}_{g,r}^n
$$
is a left principal $Sp_g({\mathbb Z})\ltimes {\mathcal U}_{g,r}^n$ bundle (in the
orbifold sense.) The invariance condition (\ref{invariance}) means
that the connection defined by $\omega$ is invariant under the
$Sp_g({\mathbb Z})\ltimes {\mathcal U}_{g,r}^n$ action. The monodromy yields a
representation
$$
\Gamma_{g,r}^n \to Sp_g({\mathbb C})\ltimes {\mathcal U}_{g,r}^n(x)
$$
As proved in \cite[\S14.2]{hain:derham}, this is the ${\mathbb C}$ form of
the completion of $\Gamma_{g,r}^n$ with respect to the canonical
homomorphism $\Gamma_{g,r}^n \to Sp_g({\mathbb C})$.
Since the sequence
$$
0 \to \Ga \to {\mathfrak t}_{g,r}^n \to {\mathfrak u}_{g,r}^n \to 0
$$
splits canonically over ${\mathbb C}$ (given the choice of the base point
$x$), $\omega$ has a canonical lift
$$
\widetilde{\omega} \in E^{\bullet}({\mathcal H}_{g,r}^n)\comptensor \Gr^W_{\bullet} {\mathfrak t}_{g,r}^n
$$
to $\Gr^W_{\bullet}{\mathfrak t}_{g,r}^n$.
This form is not integrable, but since $\omega$ is integrable, the
curvature of $\widetilde{\omega}$ takes values in the central $\Ga$. It also
has the invariance property (\ref{invariance}).
We will say that a representation
$\phi : \Gr^W_{\bullet}{\mathfrak t}_{g,r}^n \to \End(V)$
is {\it projective} if the image of $\Ga$ consists of scalar
matrices. If $V$ is an $Sp_g$ module and $\phi$ is $Sp_g$
equivariant, then $\phi$ should integrate to a homomorphism
$$
\Gamma_{g,r}^n \to PGL(V),
$$
at least when $\phi$ is ``sufficiently small,'' since, in this case,
the composite
$$
\omega_\phi \in E^{\bullet}(X_n)\otimes \End(V)/\text{scalars},
$$
an infinite sum, should converge to an integrable 1-form. The equivariance
of $\phi$ implies that $\omega_\phi$ has the invariance property
(\ref{invariance}), leading to a projectively flat bundle over ${\mathcal M}_{g,r}^n$
with fiber $V$ over the base point $x$.
|
1995-12-20T06:20:20 | 9512 | alg-geom/9512012 | en | https://arxiv.org/abs/alg-geom/9512012 | [
"alg-geom",
"math.AG"
] | alg-geom/9512012 | Fernando Torres | Fernando Torres | Remarks on numerical semigroups | Latex2e, ICTP preprint | null | null | null | null | We extend results on Weierstrass semigroups at ramified points of double
covering of curves to any numerical semigroup whose genus is large enough. As
an application we strengthen the properties concerning Weierstrass weights in
\cited{To}.
| [
{
"version": "v1",
"created": "Tue, 19 Dec 1995 06:21:54 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Torres",
"Fernando",
""
]
] | alg-geom | \section{Introduction}
Let $H$ be a numerical semigroup, that is, a subsemigroup of $(\mathbb N,
+)$ whose complement is finite. Examples of such
semigroups are the Weierstrass semigroups at non-singular points of
algebraic curves.
In this paper we deal with the following type of semigroups:
\begin{definition}\label{def} Let $\gamma\ge 0$ an integer. $H$ is
called $\gamma$-hyperelliptic if the following conditions hold:
\begin{itemize}
\item[($E_1$)] $H$ has $\gamma$ even elements in $[2,4\gamma]$.
\item[($E_2$)] The $(\gamma+1)$th positive element of $H$ is $4\gamma+2$.
\end{itemize}
A 0-hyperelliptic semigroup is usually called hyperelliptic.
\end{definition}
The motivation for study of such semigroups comes from the study of
Weierstrass semigroups at ramified points of double coverings
of curves. Let $\pi: X\to \tilde X$ be a double covering of projective,
irreducible, non-singular algebraic curves over an algebraically closed
field $k$. Let $g$ and $\gamma$ be the genus of $X$ and
$\tilde X$ respectively. Assume that there exists $P\in X$ which is ramified
for $\pi$, and denote by
$m_i$ the $i$th non-gap at $P$. Then the
Weierstrass semigroup $H(P)$ at $P$ satisfies the
following properties (cf. \cite{To}, \cite[Lemma 3.4]{To1}):
\begin{itemize}
\item[($P_1$)] $H(P)$ is $\gamma$-hyperelliptic, provided $g\ge
4\gamma+1$ if ${\rm char}(k)\neq 2$, and $g\ge 6\gamma-3$ otherwise.
\item[($P_2$)] $m_{2\gamma+1}=6\gamma+2$, provided $g\ge 5\gamma+1$.
\item[($P_3$)] $m_{\frac{g}{2}-\gamma-1}=g-2$ or
$m_{\frac{g-1}{2}-\gamma}=g-1$, provided $g\ge 4\gamma+2$.
\item[($P_4$)] The weight $w(P)$ of $H(P)$ satisfies
$$
\binom{g-2\gamma}{2}\le w(P)< \binom{g-2\gamma+2}{2}.
$$
\end{itemize}
Conversely if $g$ is large enough and if any of the above
properties is satisfied, then $X$ is a double covering of a curve of
genus $\gamma$. Aposteriori the
four above properties become equivalent whenever $g$ is large enough.
The goal of this paper is to extend these results for any
semigroup $H$ such that $g(H):= \#(\mathbb N\setminus H)$ is large enough.
We remark that there exist semigroups of genus large enough that cannot be
realized as
Weierstrass semigroups (see \cite{Buch1}, \cite[Scholium 3.5]{To}).
The key tool used to prove these equivalences is Theorem 1.10 in Freiman's
book \cite{Fre} which have to do with addition of finite sets. From this
theorem we deduce Corollary \ref{cor-cast} which can be considered as
analogous to Castelnuovo's genus bound for
curves in projective spaces (\cite{C}, \cite[p.116]{ACGH},
\cite[Corollary 2.8]{R}). Castelnuovo's result is
the key tool to deal with Weierstrass semigroups. This Corollary can also be
proved by means of properties of addition of residue classes (see
Remark \ref{cauchy}).
In \S2 we prove the equivalences $(P_1)\Leftrightarrow
(P_2)\Leftrightarrow (P_3)$. The equivalence $(P_1)\Leftrightarrow
(P_2)$ is proved under
the hypothesis $g(H)\ge 6\gamma+4$, while $(P_1)\Leftrightarrow (P_3)$
is proved under $g(H)=6\gamma+5$ or $g(H)\ge 6\gamma+8$. In both cases
the bounds on $g(H)$ are sharp (Remark \ref{sharp}). We mention that the
cases $\gamma\in\{1,2\}$ of $(P_1)\Leftrightarrow (P_3)$ were fixed by
Kato \cite[Lemmas 4,5,6,7]{K2}.
In \S3 we deal with the equivalence $(P_1)\Leftrightarrow (P_4)$. To
this purpose we determine bounds for the weight $w(H)$ of
the semigroup $H$, which is defined by
$$
w(H):= \sum_{i=1}^{g}(\ell_i -i),
$$
where $g:=g(H)$ and $\mathbb N\setminus H = \{\ell_1,\ldots,\ell_g\}$. It is
well
known that $0\le w(H)\le \binom{g}{2}$; clearly $w(H)=0\Leftrightarrow
H=\{g+i:i\in \mathbb Z^+\}$, and one can show that $w(H)=\binom{g}{2}
\Leftrightarrow H$ is $\mathbb N$, or $g(H)\ge 1$ and $H$ is hyperelliptic
(see e.g. \cite[Corollary III.5.7]{F-K}).
Associated to $H$ we have the number
\begin{equation}\label{even-gap}
\rho=\rho(H):=\{\ell \in \mathbb N\setminus H: \ell\ {\rm even}\}.
\end{equation}
In \cite[Lemma 2.3]{To} it has been shown that $\rho(H)$ is the unique
number $\gamma$ satisfying $(E_1)$ of Definition \ref{def}, and
\begin{itemize}
\item[($E_2'$)] $4\gamma+2 \in H$.
\end{itemize}
Thus we observe the following:
\begin{lemma}\label{feto0} Let $H$ be a $\gamma$-hyperelliptic
semigroup. Then
$$
\rho(H)=\gamma.
$$
\end{lemma}
We also observe that if $g(H)\ge 1$, then $H$ is hyperelliptic if and
only if $\rho(H)=0$. In general, $\rho(H)$ affects
the values of $w(H)$. Let us assume that $\rho(H)\ge
1$ (hence $w(H)<\binom{g}{2}$); then we find
$$
\binom{g-2\rho}{2}\le w(H)\le \left\{
\begin{array}{ll}
\binom{g-2\rho}{2}+2\rho^2 & {\rm if\ } g\ge 6\rho+5 \\
\frac{g(g-1)}{3} & {\rm otherwise}
\end{array}
\right.
$$
(see Lemmas \ref{bo-weight} and \ref{opt-weight}). These bounds
strengthen results of Kato \cite[Thm.1]{K1} and Oliveira
\cite[p.435]{Oliv} (see Remark \ref{oliv}). From this
result we prove $(P_1)\Leftrightarrow (P_4)$ (Theorem
\ref{char-weight1}) under the hypothesis
\begin{equation}\label{bound-g}
g(H)\ge \left\{
\begin{array}{ll}
{\rm max}\{12\gamma-1,1\} & {\rm if\ } \gamma\in\{0,1,2\}, \\
11\gamma+1 & {\rm if\ } \gamma \in\{3,5\}, \\
\frac{21(\gamma-4)+88}{2} & {\rm if\ }\gamma \in \{4,6\}, \\
\gamma^2+4\gamma+3 & {\rm if\ } \gamma\ge 7.
\end{array}
\right.
\end{equation}
The cases $\gamma \in \{1,2\}$ of that
equivalence was fixed by Garcia (see
\cite{G}). In this section we use ideas from Garcia's \cite[Proof of
Lemma 8]{G} and Kato's \cite[p. 144]{K1}.
In \S1 we recollet some arithmetical properties of numerical semigroups.
We mainly remark the influence of $\rho(H)$ on $H$.
It is a pleasure to thank Pablo Azcue and Gustavo T. de A. Moreira for
discussions about \S2.
\section{Preliminaries}
Throughout this paper we use the following notation
\begin{itemize}
\itemsep=0.5pt
\item semigroup:\quad numerical semigroup.
\item Let $H$ be a semigroup. The {\it genus} of $H$ is the number
$g(H):= \#(\mathbb N\setminus H)$, which throughout this article will be
assumed bigger than 0. The positive elements of $H$ will be
called the {\it non-gaps} of $H$, and those of $G(H):= \mathbb N\setminus H$
will be called the {\it gaps} of $H$. We denote by $m_i(H)$ the $i$th
non-gap of $H$. If a semigroup is generated by $m,n,\ldots $ we denote
$H=\langle m,n,\ldots \rangle$.
\item $[x]$ stands for the integer part of $x\in \mathbb R$.
\end{itemize}
In this section we recall some arithmetical properties of semigroups. Let
$H$ be a semigroup of genus $g$. Set $m_j:= m_j(H)$ for each $j$. If
$m_1=2$ then $m_i=2i$ for $i=1,\ldots,g$. Let $m_i\ge 3$. By
the semigroup property of $H$ the first $g$ non-gaps satisfy the
following inequalities:
\begin{equation}\label{prop-sem}
m_i\ge 2i+1\ \ {\rm for}\ i=1,\ldots,g-2,\ \ m_{g-1}\ge 2g-2,\ \ m_g=2g
\end{equation}
(see \cite{Buch}, \cite[Thm.1.1]{Oliv}).
\medskip
Let $\rho$ be as in (\ref{even-gap}). From \cite[Lemma
2.3]{To1} we have that
\begin{equation}\label{feto}
\{4\rho+2i: i\in \mathbb N\} \subseteq H.
\end{equation}
{}From the definition of $\rho$, $H$ has
$\rho$ odd non-gaps in $[1,2g-1]$. Let denote
by
$$
u_{\rho} <\ldots < u_1
$$
such non-gaps.
\begin{lemma}\label{feto1} Let $H$ be a semigroup of genus $g$, and
$\rho$ the number of even gaps of $H$. Then
$$
2g \ge 3\rho.
$$
\end{lemma}
\begin{proof}
Let us assume that $g\le 2\rho -1$. From $u_1\le 2g-1$ we have
$u_{2\rho-g+1}\le 4g-4\rho-1$. Let $\ell$ be the biggest even
gap of $H$. Then $\ell \le 4g-4\rho$. For suppose that $\ell \ge
4g-4\rho+2$. Thus $\ell-u_{2\rho-g+1}\ge 3$,
and then $H$ would has $g-\rho+1$ odd gaps, namely $1,
\ell-u_{2\rho-g+1},\ldots, \ell-u_{\rho}$, a contradicition. Now
since in $[2,4g-4\rho]$ there are $2g-2\rho$ even numbers such that
$\rho$ of them are gaps, the lemma follows.
\end{proof}
Denote by $f_i:=f_i(H)$ the $i$th even non-gap of $H$.
Hence by (\ref{feto}) we have
\begin{equation}\label{gene}
H=\langle f_1,\ldots,
f_{\rho},4\rho+2,u_{\rho},\ldots,u_1\rangle.
\end{equation}
Observe that $f_{\rho}=4\rho$, and
\begin{equation}\label{even}
f_{g-\rho}=2g.
\end{equation}
By \cite[Lemma 2.1]{To1} and since $g\ge 1$ we have
\begin{equation}\label{des-odd-1}
u_{\rho} \ge\ {\rm max}\{2g-4\rho +1, 3\}.
\end{equation}
In particular, if $g\ge 4\rho$ we obtain
\begin{equation}\label{first=even}
m_1=f_1, \ldots, m_{\rho}=f_{\rho}.
\end{equation}
Note that (\ref{des-odd-1}) is only meanful for $g\ge 2\rho$. For
$g\le2\rho-1$ we have:
\begin{lemma}\label{feto2}
Let $H$ be a semigroup of genus $g$, and $\rho$ the number of even
gaps of $H$. If $g\le 2\rho-1$, then
$$
u_{\rho}\ge 4\rho -2g+1.
$$
\end{lemma}
\begin{proof}
{}From the proof of Lemma \ref{feto1} we have that $H$ has $2g-3\rho$ even
non-gaps in $[2,4g-4\rho]$. Consider the following sequence of even
non-gaps:
$$
2u_{\rho}<\ldots< u_{\rho}+u_{4\rho-2g}.
$$
Since in this sequence we have $2g-3\rho+1$ even non-gaps, then
$$
u_{\rho}+u_{4\rho-2g}\ge 4g-4\rho+2.
$$
Now, since $u_{4\rho-2g}\le 6g-8\rho+1$ the proof follows.
\end{proof}
\section{$\gamma$-hyperelliptic semigroups}
In this section we deal with properties $(P_1)$, $(P_2)$ and
$(P_3)$ stated in \S0. For $i\in \mathbb Z^+$ set
$$
d_i(H):= \gcd(m_1(H),\ldots,m_i(H)).
$$
\begin{theorem}\label{char1} Let $\gamma \in \mathbb N$, $H$
a semigroup of genus $g \ge 6\gamma +4$ if $\gamma\ge 1$. Then the
following statements are equivalent:
\begin{itemize}
\item[(i)] $H$ is $\gamma$-hyperelliptic.
\item[(ii)] $m_{2\gamma+1}(H)= 6\gamma +2$.
\end{itemize}
\end{theorem}
\begin{theorem}\label{char2} Let $\gamma$ and $H$ be as in Theorem
\ref{char1}, and assume that $g\ge
1$ if $\gamma=0$. Then the following statements are equivalent:
\begin{itemize}
\item[(i)] $H$ is $t$-hyperelliptic for some $t\in \{0,\ldots,\gamma\}$.
\item[(ii)] $m_{2\gamma+1}(H)\le 6\gamma+2$.
\item[(iii)] $\rho(H)\le \gamma$.
\end{itemize}
\end{theorem}
\begin{theorem}\label{char3} Let $\gamma \in \mathbb N$, $H$ a semigroup of
genus $g=6\gamma+5$ or
$g\ge 6\gamma+7$. Set $r:= \[x]-\gamma-1$. Then the following statements
are equivalent:
\begin{itemize}
\item[(i)] $H$ is $\gamma$-hyperelliptic.
\item[(ii)] $m_r(H)=g-2$ if $g$ is even; $m_r(H)=g-1$ if $g$ is odd.
\item[(iii)] $m_r(H)\le g-1 < m_{r+1}(H)$.
\end{itemize}
\end{theorem}
\begin{theorem}\label{char4} Let $\gamma$, $H$ and $r$ be as in Theorem
\ref{char3}. Then the following statements are equivalent:
\begin{itemize}
\item[(i)] $H$ is $t$-hyperelliptic for some $t\in\{0,\ldots,\gamma\}$.
\item[(ii)] $m_r(H)\le g-2$ if $g$ is even;
$m_r(H)\le g-1$ if $g$ is odd.
\item[(iii)] $m_r(H)\le g-1$.
\item[(iv)] $\rho(H)\le \gamma$.
\end{itemize}
\end{theorem}
To prove these results we need a particular case of the result below.
For $K$ a subset of a group we set $2K:=\{a+b: a, b \in K\}$.
\begin{lemma}[[Fre, Thm. 1.10 {]}]\label{thm-fre} Let
$K=\{0<m_1<\ldots<m_i\}
\subseteq \mathbb Z$ be such that $\gcd(m_1,\ldots,m_i)=1$. If $m_i\ge
i+1+b$, where $b$ is an integer satisfying $0\le b<i-1$, then
$$
\# 2K \ge 2i+2+b.
$$
\hfill $\Box$
\end{lemma}
\begin{corollary}\label{cor-cast} Let $H$ be a semigroup of genus
$g$, and $i\in \mathbb Z^+$. If
$$
d_i(H)= 1\qquad{\rm and}\qquad i\le g+1,
$$
then we have
$$
2m_i(H) \ge m_{3i-1}(H).
$$
\end{corollary}
\begin{proof} Let $K:=\{0,m_1(H),\ldots,m_i(H)\}$. Then by
(\ref{prop-sem}), we can apply Lemma
\ref{thm-fre} to $K$ with $b=i-2$.
\end{proof}
\begin{remark}\label{rem-cast} Both the hypothesis $d_i(H)=1$ and $i\le
g+1$ of the corollary above are necessaries. Moreover the conclusion of
that result is sharp:
\begin{itemize}
\item[(i)] Let $i=g+h$, $h\ge 2$. Then $2m_{g+h}=m_{3i-h}$.
\item[(ii)] Let $m_1=4$, $m_2=6$ and $m_3=8$. Then $d_3=2$ and $2m_3=m_7$.
\item[(iii)] Let $m_1=5$, $m_2=10$, $m_3=15$, $m_4=18$,
$m_5=20$. Then $2m_6=m_{14}$.
\end{itemize}
\end{remark}
\begin{remark}\label{cauchy} (i) The Corollary above can also be proved by
using
results on the addition of residue classes: let $H$ and $i$ be as in
\ref{cor-cast}; assume further that $2\le i\le g-2$ (the remaining cases
are easy to prove), and consider
$\tilde K:=\{m_1,\ldots,m_i\}\subseteq \mathbb Z_{m_i}$ (i.e. a subset of
the
integers modulus $m_i$). Let $N:= \# 2\tilde K$. Then it is easy
to see that
$$
2m_i\ge m_{i+N}.
$$
Consequently we have a proof of the above Corollary provided $N\ge
2i-1\ (*)$. Since $m_i\ge 2i+1$ (see (\ref{prop-sem})), we get
$(*)$ provided $m_i$ prime (Cauchy \cite{Dav1}, Davenport \cite{Dav},
\cite[Corollary 1.2.3]{M}), or provided
$\gcd(m_j,m_i)=1$ for $j=1,\ldots,i-1$ (Chowla \cite[Satz 114]{Lan},
\cite[Corollary 1.2.4]{M}). In
general by using the hypothesis $d_i(H)=1$ we can reduce the proof of the
Corollary to the case $\gcd(m_{i-1},m_i)=1$. Then we apply Pillai's
\cite[Thm 1]{Pi} generalization of Davenport and Chowla results (or
Mann's result \cite[Corollary 1.2.2]{M}).
(ii) Let $H$ and $i$ be as above and assume that
$2m_i=m_{3i-1}$. Then from (i) we have $N=\# 2\tilde K=2i-1$. Thus by
Kemperman \cite[Thm 2.1]{Kem} (or by \cite[Thm. 1.11]{Fre}) $2\tilde K$
satisfies one of the following conditions: (1) there exist $m, d\in \mathbb
Z_{m_i}$, such that $2\tilde K=\{m+dj:j=0,1,\ldots,N-1\}$, or (2) there
exists
a subgroup $F$ of $\mathbb Z_{m_i}$ of order $\ge 2$, such that $2\tilde K$
is the disjoint union of a non-empty set $C$ satisfying $C+F=C$, and a
set $C'$ satisfying $C'\subseteq c+F$ for some $c\in C'$. For instance
example (iii) of \ref{rem-cast} satisfies property (2).
\end{remark}
Set $m_j:= m_j(H)$ for each $j$.
\begin{proof} {\it (Theorem \ref{char1}).} By definition
$H$ is hyperelliptic if and only if $m_1=2$.
So let us assume that $\gamma\ge 1$.
(i) $\Rightarrow$ (ii): From Lemma \ref{feto0} and (\ref{des-odd-1}) we
find that $u_{\gamma}\ge 6\gamma+3$ if
$g\ge 5\gamma+1$. Then (ii) follows from (\ref{first=even}) and (\ref{feto}).
(ii) $\Rightarrow$ (i): We claim that $d_{2\gamma+1}(H)=2$. For
suppose that $d_{2\gamma+1}(H) \ge 3$. Then $6\gamma+2= m_{2\gamma+1}
\ge m_1+ 6\gamma$ and so $m_1\le 2$, a contradiction. Hence
$d_{2\gamma+1}(H)\le
2$. Now suppose that $d_{2\gamma+1}(H) =1$. Then Corollary \ref{cor-cast}
implies $$
2(6\gamma+2) = 2m_{2\gamma+1} \ge m_{6\gamma +2}.
$$
But, since $g-2 \ge 6\gamma +2$, by (\ref{prop-sem}) we
would have
$$
m_{6\gamma +2} \ge 2(6\gamma+2) +1
$$
which leads to a contradiction. This shows that $d_{2\gamma+1}(H)=2$. Now
since $m_{2\gamma+1}=6\gamma+2$ we have that $m_\gamma \le 4\gamma$.
Moreover, there exist $\gamma$ even gaps of
$H$ in $[2, 6\gamma+2]$. Let $\ell$ be an even gap of $H$. The proof
follows from the following claim:
\begin{claim*}
$\ell <m_\gamma$.
\end{claim*}
\begin{proof} {\it (Claim).} Suppose that there exists an even gap
$\ell$ such that $\ell>m_\gamma$. Take the smallest $\ell$ with such a
property; then the following $\gamma$ even gaps: $\ell-m_\gamma<
\ldots,
\ell -m_1$ belong to $[2,m_\gamma]$. Thus, since $m_1>2$, we must have
$\ell-m_\gamma=2$. This implies
that $H$ has $\gamma+1$ even
non-gaps in $[2,6\gamma+2]$, namely $\ell-m_\gamma,\ldots,\ell-m_1,\ell$,
a contradiction.
\end{proof}
This finish the proof of Theorem \ref{char1}.
\end{proof}
\begin{proof} {\it (Theorem \ref{char2}).} The case $\gamma=0$
is trivial; so let assume $\gamma\ge 1$.
(i) $\Rightarrow$ (ii): Since $g\ge
5\gamma+1\ge 5t+1$ by Theorem \ref{char1} we have $m_{2t+1}=6t+2$.
Thus (ii) follows from Lemma \ref{feto0} and (\ref{feto}).
(ii) $\Rightarrow$ (iii): From the proof of (ii) $\Rightarrow$
(i) of Theorem \ref{char1} it follows that $d_{2\gamma+1}(H)=2$.
Consequently by using the hypothesis on $m_{2\gamma+1}$, and again from
the mentioned proof we have that all the gaps of $H$ belong to
$[2,m_\gamma]$.
Since $m_\gamma\le 4\gamma$ then we have $\rho(H)\le \gamma$
(iii) $\Rightarrow$ (i) Since $g\ge 4\gamma+1\ge 4\rho(H)+1$, the
proof follows from $(E_1)$ and $(E_2')$ (see \S0).
\end{proof}
\begin{proof} {\it (Theorem \ref{char3}).} (i) $\Rightarrow$
(ii): Similar to the proof of (i)
$\Rightarrow$ (ii) of Theorem \ref{char1} (here we need $g\ge
4\gamma+3$ (resp. $g\ge 4\gamma+4$) if $g$ is odd (resp. even)).
\smallskip
Before proving the other implications we remark that $g\le 3r-1$: in
fact, if $g\ge 3r$ we would have $g\le 6\gamma+6$ (resp. $g\ge
6\gamma+3$) provided $g$ even (resp. odd) - a contradiction.
\smallskip
(ii) $\Rightarrow$ (iii): Let $g$ even and suppose that $m_{r+1}=g-1$.
Then by
Corollary \ref{cor-cast} we would have $2g-2=2m_{r+1}\ge m_{3r+2}$ and
hence $g-1\ge 3r+2$. This contradicts the previous remark.
(iii) $\Rightarrow$ (i): We claim that $d_r(H)= 2$. Suppose that
$d_r(H)\ge
3$. Then we would have $g-1\ge m_r \ge m_1+3(r-1) \ge 3r-1$, which
contradicts the previous remark. Now suppose that
$d_r(H)=1$. Then by Corollary \ref{cor-cast} we would have
$$
2g-2\ge m_r \ge m_{3r-1},
$$
which again contradicts the previous remark.
Thus the number of even gaps of $H$ in $[2,g-1]$ is $\gamma$, and
$m_\gamma\le 4\gamma$. Let
$\ell$ be an even gap of $H$. As in the proof of the Claim in Theorem
\ref{char1} here we also have that $\ell<m_\gamma$. Now the proof
follows.
\end{proof}
\begin{proof} {\it (Theorem \ref{char4}).} (i) $\Rightarrow$
(ii): By Theorem \ref{char3} and since $t\le \gamma$ we have $g-2
=m_{g/2-t-1}$ or $g-1=m_{(g-1)/2-t}$. This implies (ii). The
implication (ii) $\Rightarrow$ (iii) is trivial.
(iii) $\Rightarrow$ (iv): As in the proof of Theorem \ref{char3} we
obtain $d_r(H)=2$. Then the number of even gaps of $H$ in $[2,g-1]$ is at
most $\gamma$. We claim that all the even gaps of
$H$ belong to that interval. For suppose there exists an even gap
$\ell>g-1$. Choose $\ell$
the smallest one and consider the even gaps $\ell-m_1<\ldots<\ell-m_r\le
g-1$. Then $r\le \gamma$ which yields to $g\le 4\gamma+2$,
a contradiction. Consequently $\rho(H)\le \gamma$.
The implication (iv) $\Rightarrow$ (i) follows from Theorem
\ref{char2}.
\end{proof}
\begin{remark}\label{sharp} The hypothesis on the genus in the
above theorems is sharp. To see this let $\gamma\ge 0$ an
integer, and let $X$ be the curve defined by the equation
$$
y^4=\mathop{\prod}\limits^{I}_{j=1}(x-a_j),
$$
where the $a_j's$ are pairwise distinct elements of a field $k$,
$I=4\gamma+3$ if
$\gamma$ is odd; $I=4\gamma+5$ otherwise. Let $P$ be the unique point over
$x=\infty$. Then $H(P)=\langle 4, I\rangle$ and so
$g(H(P))=6\gamma+3$ (resp. $6\gamma+6$), $m_{2\gamma+1}(H(P))=
6\gamma+2$ (resp. $m_{2\gamma+2}(H(P))=6\gamma+5$), and
$\rho(H(P))=2\gamma+1$ (resp. $\rho(H(P))=2\gamma+2$) provided
$\gamma$ odd (resp. $\gamma$ even).
\end{remark}
\section{Weight of semigroups}
\subsection{Bounding the weight.}
Let $H$ be a semigroup of genus $g$. Set $m_j=m_j(H)$ for each $j$ and
$\rho=\rho(H)$ (see (\ref{even-gap}). Due to $m_g=2g$ (see
(\ref{prop-sem})), the weight $w(H)$ of $H$ can be
computed by
\begin{equation}\label{weight}
w(H)=\frac{3g^2+g}{2}-\mathop{\sum}\limits^{g}_{j=1} m_j.
\end{equation}
Consequently the problem of bounding $w(H)$ is equivalent to
the problem of bounding
$$
S(H):= \sum_{j=1}^{g} m_g.
$$
If $\rho=0$, then we have $m_i=2i$ for each $i=1,\ldots,g$. In
particular we have
$w(H)=\binom{g}{2}$. Let $\rho\ge 1$ (or
equivalently $f_1\ge 4$). Then by (\ref{gene}) we have
\begin{equation}\label{weight1}
S(H)=\sum_{f\in \tilde H,\ f\le g} 2f + \sum_{i=1}^{\rho} u_i,
\end{equation}
where
$$
\tilde H:= \{f/2 : f\in H,\ f\ {\rm even}\}.
$$
\begin{lemma'}\label{bounds} With the notation of \S1 we have:
\begin{itemize}
\item[(i)] If $f_1=4$, then $f_i=4i$ for $i=1,\ldots,\rho$.
\item[(ii)] If $f_1\ge 6$, then
$$
f_i\ge 4i+2\ \ {\rm for}\ i=1, \ldots,\rho-2,\ \ f_{\rho-1}\ge
4\rho-4,\ \ f_{\rho}=4\rho.
$$
\item[(iii)] $f_i\le 2\rho+2i$ for each $i$.
\item[(iv)] $2g-4j+1 \le u_j \le 2g-2j+1$, for $j=1,\ldots,\rho$.
\end{itemize}
\end{lemma'}
\begin{proof} By (\ref{feto}), we have
$$
\tilde H= \{\frac{f_1}{2},\ldots, \frac{f_\rho}{2}\}\cup \{4\rho +i:
i\in \mathbb N \}.
$$
Thus $\tilde H$ is a semigroup of genus $\rho$. Then (i) is due to the
fact that $f_1/2=2$ and (ii) follows from (\ref{prop-sem}). Statement (iii)
follows from (\ref{even}).
\smallskip
(iv) The upper bound follows from $u_1\le 2g-1$. To
prove
the lower bound we procced by induction on $i$. The case $i=\rho$ follows
from (\ref{des-odd-1}). Suppose that $u_i\ge 2g-4i+1$ but
$u_{i-1} < 2g-4(i-1)+1$, for $1<i\le \rho$. Then $u_i=2g-4i+1$,
$u_{i-1}= 2g-4i+3$, and
there exists an odd gap $\ell$ of $H$ such that
$\ell>u_{i-1}$. Take the smallest $\ell$ with such a property.
Set $I:=[\ell-u_{i-1}, \ell-u_\rho]\subseteq [2,4\rho-2]$ and let
$t$ be the number of non-gaps of $H$ belonging to $I$. By the election
of $\ell$ we have that $\ell-u_{i-1}<f_1$. Now, since $\ell-u_j \in I$ for
$j=i-1,\ldots,\rho$ we also have that
$$
\frac{u_{i-1}-u_\rho}{2}+1 \ge t + \rho -(i-1)+1.
$$
Thus $u_\rho \le 2g-2\rho-2i-2t+1$. Now, since
$u_\rho+f_{t+1}>u_{i-1}$ and since by statement (iii) $f_{t+i-1}\le
2\rho + 2t +2i-2$, we have that the odd non-gaps $u_\rho
+f_{t+1}, \ldots, u_\rho +f_{t+i-1}$ belong to $[\ell+2,2g-1]$.
This is a contradiction because $H$ would have $(\rho -i+2)+(i-1) = \rho
+1$ odd non-gaps.
\end{proof}
\begin{lemma'}\label{bo-weight} Let $H$ be a semigroup of genus
$g$. With notation as in \S1, we have
\begin{itemize}
\item[(i)] $w(H)\ge \binom{g-2\rho}{2}$. Equality holds if and
only if $f_1=2\rho+2$ and $u_{\rho}=2g-2\rho+1$.
\item[(ii)] If $g\ge 2\rho$, then $w(H)\le \binom{g-2\rho}{
2}+2\rho^2 $. Equality holds if and only if
$H=\langle 4, 4\rho,2g-4\rho+1\rangle$.
\item[(iii)] If $g\le 2\rho-1$, then $w(H)\le \binom{g+2\rho}{
2}-4g-6\rho^2+8\rho $.
\end{itemize}
\end{lemma'}
\begin{proof} (i) By (\ref{weight}) we have to show that
$$
S(H) \le
g^2+(2\rho+1)g-2\rho^2-\rho,
$$
and that the equality holds if and only if $f_1=2\rho+2$ and
$u_{\rho}=2g-2\rho+1$. Both the above statements follow from Lemma
\ref{bounds} (i), (iv).
\smallskip
(ii) Here we have to show that
\begin{equation*}
S(H)\ge g^2 + (2\rho+1)g-4\rho^2-\rho,\tag{$\dag$}
\end{equation*}
and that equality holds if and only $H=\langle
4,4\rho+2,2g-4\rho+1\rangle$.
Since $g\ge 2\rho$ by (\ref{feto}) we obtain
\begin{equation}\label{sum1}
S(H)=\sum_{i=1}^{\rho}(f_i(H)+u_i(H))+ g^2+g-4\rho^2-2\rho.
\end{equation}
Thus we obtain $(\dag)$ by means of Lemma \ref{bounds} (ii),
(iii) and (iv). Moreover the equality in $(\dag)$ holds if and only
if $\sum_{i=1}^{\rho}(f_i+u_i)=2\rho g +\rho$. Then the
second part of (ii) also follows from the above mentioned results.
\smallskip
(iii) In this case, due to the proof of Lemma \ref{feto1}, instead
of equation (\ref{sum1}) we have
\begin{equation}\label{sum2}
S(H) = \sum_{i=1}^{2g-3\rho}f_i +
\sum_{i=2g-3\rho+1}^{g-\rho}(2i+2\rho) +
\sum_{i=1}^{\rho}u_i.
\end{equation}
We will see in the next remark that in this case we have $f_1\ge 6$.
Thus by using Lemmas \ref{feto2} and \ref{bounds} (iii), (iv) we
obtain
$$
S(H)\ge 4\rho^2-(2g+7)\rho +g^2 +5g,
$$
from where it follows the proof.
\end{proof}
\begin{remark'}\label{remark3} (i) If $f_1=4$, then $g\ge 2\rho$.
This follows from
the fact the biggest even gap of $H$ is $4(\rho-1)+2$. Moreover,
one can determinate $u_{\rho},\ldots,u_1$ as
follows: let $J\in \mathbb N$ satisfying the inequalities below
$$
{\rm max}\{1, \frac{3\rho+2-g}{2}\}\le J\le {\rm
min}\{\rho+1,\frac{g-\rho+3}{2}\},
$$
provided $g$ even, otherwise replace $J$ by $\rho-J+2$; then
\begin{eqnarray*}
\{u_{\rho},\ldots,u_1\} & = &
\{ 2g-4\rho+4J-7+4i: i=1,\ldots,\rho-J+1\}\\
& &\mbox{}\cup\{2g-4J+3+4i: i=1,\ldots,J-1\}.
\end{eqnarray*}
(see \cite[\S3]{Ko}, \cite[Remarks 2.5]{To}). Consequently from
(\ref{sum1}) and (\ref{weight}) we obtain
$$
w(H)=\binom{g-2\rho}{2}+ 2\rho^2+4\rho+6+4J^2-(4\rho+10)J.
$$
In particular we have
$$
\binom{g-2\rho}{2}+\rho^2-\rho\le w(H)\le
\binom{g-2\rho}{2}+2\rho^2.
$$
Let $C$ be an integer such that $0\le 2C\le \rho^2+\rho$. Then
$w(H)=\binom{g-2\rho}{2}+\rho^2-\rho +2C$
if and only if $4+32C$ is a square. The lower bound is attained if
and only if $H=\langle
4,4\rho+2, 2g-2\rho+1,2g-2\rho+3\rangle$.
\smallskip
\noindent (ii) Let $u_{\rho}=3$. Them from (\ref{des-odd-1}) and
Lemma \ref{feto2} we find that $g\in
\{2\rho-1,2\rho,2\rho+1\}$. Moreover, in this case one
can also obtain a explicit formula for $w(H)$ (\cite[Lemma 6]{K1}). Let
$g\equiv r \pmod{3}$, $r=0,1,2$ and let $s$ be an integer such that $0\le
s\le (g-r)/3$. If $r=0,1$ (resp. $r=2$), then
\begin{align*}
w(H) & =\frac{g(g-1)}{3}+3s^2-gs-s\le \frac{g(g-1)}{3} \\
\intertext{resp.}
w(H) & =\frac{g(g-2)}{3}+3s^2-gs+s\le \frac{g(g-2)}{3}.
\end{align*}
If $r=0,1$ (resp. $r=2$), equality occurs if and only if $H=\langle 3,
g+1\rangle$ (resp. $H=\langle 3, g+2,2g+1\rangle$).
\end{remark'}
Let $g\le 2\rho-1$. The way how we bound from below
equation (\ref{sum2}) is far away from being sharp. We do
not know an analogous to the lower bound of Lemma \ref{bounds} (iv) in
this case. However, for certain range of $g$ the bounds in \ref{remark3}
(ii) are the best possible:
\begin{lemma'}\label{opt-weight} Let $H$ be a semigroup of genus
$g\ge 11$, $r\in \{1,2,3,4,5,6\}$ such that $g\equiv r \pmod{6}$. Let
$\rho$ be the number of even gaps of $H$.
If
$$
\rho>\left\{
\begin{array}{ll}
\frac{g-5}{6} & {\rm if\ } r=5 \\
\frac{g-r}{6}-1 & {\rm if\ } r\neq 5,
\end{array}
\right.
$$
then
$$
w(H)\le \left\{
\begin{array}{ll}
\frac{g(g-2)}{3} & {\rm if\ } r= 2,5 \\
\frac{g(g-1)}{3} & {\rm if\ } r=1,3,4,6.
\end{array}
\right.
$$
If $r=2,5$ (resp. $r\not\in\{2,5\}$) equality above holds if and only if
$H=\langle 3,g+2,2g+1\rangle$ (resp. $H=\langle 3,g+1\rangle$).
\end{lemma'}
\begin{proof} We assume $g\equiv 5 \pmod{6}$; the other cases can be
proven in a similar way. By Remark \ref{remark3} (ii) we can assume
$u_1>3$, and then by (\ref{weight}) we have to prove that
\begin{equation*}
S(H) > \frac{7g^2+7g}{6}.\tag{$*$}
\end{equation*}
Now, since $\rho>(g-5)/6$, by Theorem \ref{char4} and Lemma
\ref{feto0} we must have
$$
m_{\frac{g+1}{3}}=m_{\frac{g-1}{2}-\frac{g-5}{6}}\ge g.
$$
(A) Let $S':= \sum_{i} m_i$, $(g+1)/3\le i \le g$:\quad Define
$$
F:= \{ i\in \mathbb N: \frac{g+1}{3}\le i\le g,\ m_i\le 2i+\frac{g-5}{3}\},
$$
and let $f:= {\rm min}(F)$. Then $f\ge (g+4)/3$, $m_f=2f+\frac{g-5}{3},
m_{f-1}=
2f+\frac{g-8}{3}$. Thus for $g\ge i\ge f$, $d_i=1$ and hence by Corollary
\ref{cor-cast}, $2m_i\ge m_{3i-1}=g+3i-1$. In particular, $f\ge (g+7)/3$.
Now we bound $S'$ in three steps:
\smallskip
\noindent Step (i). $(g+1)/3\le i\le f-1$: By definiton of $f$ we have that
$m_i\ge 2i+\frac{g-2}{3}$ and hence
\begin{equation}\label{aux0}
\sum_{i} m_i \ge f^2+\frac{g-5}{3}f -\frac{2g^2-2g-4}{9}.
\end{equation}
\noindent Step (ii). $f\le i\le (6f-g-7)/3$: Here we have that $m_i\ge
m_f+i-f=i+f+\frac{g-5}{3}$. Hence $$
\sum_{i} m_i \ge \frac{5}{2}f^2-\frac{4g+37}{6}f -\frac{g^2-13g-68}{18}.
$$
\noindent Step (iii). $(6f-g-4)/3\le i\le g$: Here we have $m_i+m_{i+1}\ge
g+3i+1$ for $i$ odd, $6f-g-4\le i\le g-2$. Since $m_g=2g$ then we have
$$
\sum_{i} m_i \ge -3f^2+6f+\frac{4g^2+2g-8}{3}.
$$
(B) Let $S'':= \sum_{i} m_i$, $1\le i\le (g-2)/3$:\quad By Theorem
\ref{char2} and Lemma \ref{feto0} we have that $m_i\ge 3i$ for
$i$ odd, $i=3,\ldots, (g-2)/3$. First we notice that for $i$ odd and
$3\le i\le (g-8)/3$ we must have $m_{i+1}\ge 3i+3$. Otherwise we would
have $d_{i+1}=1$ and hence by Corollary \ref{cor-cast} and
(\ref{prop-sem}) we would have $2m_{i+1}\ge m_{3i+2}\ge 6i+5$, a
contradiction.
\begin{claim*} Let $i$ odd and $3\le i\le (g-8)/3$. If
$m_i=3i$ or $m_{i+1}=3i+3$, then $m_1=3$.
\end{claim*}
\begin{proof} {\it (Claim).} It is enough to show that
$d_i=3$ or $d_{i+1}=3$. Suppose that $m_i=3i$. Since
$i$ is odd, $d_i$ is one or three. Suppose $d_i=1$. Then by Corollary
\ref{cor-cast} we have $6i=2m_i\ge m_{3i-1}$ and hence
$6i=2m_i=m_{3i-1}$. Let $\ell \in G(H)$. Then $\ell \ge m_{3i-1}+3$. In
fact if $\ell>m_{3i-1}+3$, by choosing the smallest $\ell$ with such a
property we would have
$3i+2$ gaps in $[1, 6i]$ namely,
$1,2,3,\ell-m_{3i-1},\ldots,\ell-m_1$, a contradiction.
Then it follows that $g\le 3i+1+3=3i+4$ or $g+2\le 3i+4$.
\smallskip
Now suppose that $m_{i+1}=3i+3$; as in the previous proof here we also have
that $d_{i+1}>1$. Suppose that $d_{i+1}=2$. Then $m_1>3$ and hence
$m_i=3i+1$. Since we know that $m_{i+2}\ge 3i+6$, then the even number
$\ell=3i+5$ is a gap of $H$. Then we would fine $2i+2$ even numbers in
$[2,3i+3]$, namely $m_1,\ldots,m_{i+1}$, and $\ell-m_{i+1},\ell-m_1$, a
contradiction. Hence $d_{i+1}=3$ and then $m_1=3$.
\end{proof}
Then, since we assume $u_1>3$, we have $m_i+m_{i+1}\ge 6i+5$ for $i$ odd
$3\le i\le (g-8)/3$, $m_{\frac{g-2}{3}}\ge g-2$, and so
\begin{equation}\label{aux}
\begin{split}
\sum_{i=1}^{(g-2)/3} m_i & \ge \sum_{j=1}^{(g-11)/6} (12j+10)
+m_1+m_2+m_{\frac{g-2}{3}} \\
& \ge \frac{g^2+g-78}{6} + m_1+m_2.\\
\end{split}
\end{equation}
Summing up (i), (ii), (iii) and (B) we get
$$
S(H)\ge \frac{3f^2-(2g+11)f}{6}+\frac{22g^2+32g-206}{18}+m_1+m_2.
$$
The function $\Gamma(x):= 3x^2-(2g+11)x$ attains its minimum for
$x=(2g+11)/6<(g+7)/3\le f$. Suppose that $f\ge (g+13)/3$. Then we find
$$
S(H)\ge \frac{7g^2+7g-60}{6}+m_1+m_2.
$$
We claim that $m_1+m_2>11$. Otherwise we would have $m_3=m_1+m_2=10$
which is impossible. From the claim we get $(*)$.
In all the computations below we use the fact that $2g\le (m-1)(n-1)$
whenever $m,n \in H$ with $\gcd(m,n)=1$ (see e.g. Jenkins \cite{J}).
Now suppose that $f=(g+10)/3$. Here we find
$$
S(H)\ge \frac{7g^2+7g-72}{6}+m_1+m_2.
$$
Suppose that $m_1+m_2\le 12$ (otherwise the above computation imply
$(*)$.). If $g>11$,
then $m_4\ge 13$ and so $m_3=m_1+m_2\in \{9,11,12\}$. If $m_1+m_2=9$,
then $g\le 6$; if $m_1+m_2=11$ then $g\le 10$; if $m_1+m_2=12$ then $g\le
11$ or $m_1=4$, $m_2=8$. Let
$s$ denote the first odd non-gap of $H$. Then $2g\le 3(s-1)$ and so
$s>(2g+2)/3$. In the interval $[4,(2g+2)/3]$ does not exist $h\in H$ such
that $h\equiv 2 \pmod{4}$: In fact if such a $h$ exists then we would
have $4\rho+2\le (2g-4)/3$ or $ \rho\le (g-5)/6$. Consequently
$m_3=12,\ldots,m_{(g+1)/6}=(2g+2)/3$. Thus we can improve the
computation in (\ref{aux}) by summing it up $\sum_{i=1}^{j}(4i+1)$, where
$j=(g-5)/12$ or $j=(g-11)/12$. Then we get $(*)$. If $g=11$, the first
seven non-gaps are $\{4,8,10,12,14,15,16\}$ or $\{5,7,10,12,14,15,16\}$.
In both cases the computation in (\ref{aux0}) increases at least by one,
and so we obtain $(*)$.
Finally let $f=(g+7)/3$. Here we find
$$
S(H)\ge \frac{7g^2+7g-78}{6}+m_1+m_2,
$$
and we have to analize the cases $m_1+m_2=\{9, 11, 12, 13\}$. This can be
done as in the previous case. This finish the proof of Lemma
\ref{opt-weight}.
\end{proof}
\subsection{The equivalence $(P_1)\Leftrightarrow (P_4)$.}
We are going to characterize $\gamma$-hyperelliptic semigroups by means
of weights of semigroups. We begin with the following result, which
has been proved by Garcia for $\gamma\in\{1,2\}$ \cite[Lemmas 8 and 10]{G}.
\begin{theorem'}\label{char-weight} Let $\gamma\in \mathbb N$ and $H$ a
semigroup whose genus $g$ satisfies
(\ref{bound-g}). Then the following statements are equivalent:
\begin{itemize}
\item[(i)] $H$ is $t$-hyperelliptic for some $t\in \{0,\ldots,\gamma\}$.
\item[(ii)] $w(H)\ge \binom{g-2\gamma}{2}$.
\end{itemize}
\end{theorem'}
\begin{theorem'}\label{char-weight1} Let $\gamma$, $H$ and $g$ be
as in
Theorem \ref{char-weight}. The following statements are equivalent:
\begin{itemize}
\item[(i)] $H$ is $\gamma$-hyperelliptic.
\item[(ii)] $\binom{g-2\gamma}{2}\le w(H)\le \binom{g-2\gamma}{2}+2\gamma^2$.
\item[(iii)] $\binom{g-2\gamma}{2}\le w(H)<\binom{g-2\gamma+2}{2}$.
\end{itemize}
\end{theorem'}
\begin{proof} {\it (Theorem \ref{char-weight}).} (i) $\Rightarrow$ (ii):
By Lemma \ref{feto0} and
Lemma \ref{bo-weight} (i) we have $w(H)\ge \binom{g-2t}{2}$. This implies
(ii).
(ii) $\Rightarrow$ (i): Suppose that $H$ is not $t$-hyperelliptic for any
$t\in \{0,\ldots\gamma\}$. We are going to prove that
$w(H)<\binom{g-2\gamma}{2}$, which
by (\ref{weight}) is equivalent to prove that:
\medskip
$(*)$\hfill $\sum_{i=1}^{g} m_i >
g^2+(2\gamma+1)g-2\gamma^2-\gamma.$\hfill
\medskip
\noindent We notice that by Lemma \ref{feto0} we
must have $\rho\ge \gamma+1$.
\smallskip
\noindent Case 1: $g$ satisfies the hypothesis of Lemma
\ref{opt-weight}.\quad From that lemma we have $ S(H)\ge (7g^2+5g)/3$
and then we get $(*)$ provided
$$
g>\bar\gamma:= 12\gamma+1+\sqrt{96\gamma^2+1}.
$$
We notice that
$\gamma^2+4\gamma+3\ge \bar\gamma$ if $\gamma\ge 7$. For $\gamma=1,4,6$ we
need respectively $g>11$, $g>44$ and $g>65$. By noticing that 11, 44 and
65 are
congruent to 2 modulus 3, we can use $g=11$, $g=44$ and $g=65$ because in
these cases
$S(H)\ge (7g^2+7g)/3$. For the other values of $\gamma$ we obtain the
bounds of (\ref{bound-g}).
\smallskip
\noindent Case 2: $g$ does not satisfy the hypothesis of Lemma
\ref{opt-weight}.\quad Here we have $g\ge 6\rho+5$. From (\ref{sum1})
and Lemma \ref{bounds} we have $S(H)\ge
g^2+(2\rho+1)g-4\rho^2-\rho$. The function
$\Gamma(\rho):= (2g-1)\rho-4\rho^2$ satisfies
$$
\Gamma(\rho)\ge \Gamma(\gamma+1)=(2\gamma+2)g-4\gamma^2-9\gamma-5,
$$
for $\gamma+1 \le \rho\le [(2g-1)/4]-\gamma-1$. Thus we obtain
condition $(*)$ provided $g\ge \gamma^2 + 4\gamma+3$.
\end{proof}
\begin{remark'}\label{oliv} Let $H$ be a semigroup of genus $g$, $r$ the
number
defined in Lemma \ref{opt-weight}. Put $c:= (g-5)/6$ if $r=5$, and $c:=
(g-r)/6-1$ otherwise.
{}From the proof of Case 2 of the above result we see that $S(H)\ge
g^2+3g-5$ whenever $1\le \rho(H)\le (g-3)/2$. Hence this result and Lemma
\ref{opt-weight} imply
$$
w(H)\le\left\{
\begin{array}{ll}
(g^2-5g+10/2 & {\rm if\ } \rho(H)\le c\\
{\rm min}\{(g^2-5g+10)/2, (g-1)g/3\} & {\rm if\ }
c<\rho(H)\le (g-3)/2\\
(g-1)g/3 & {\rm if\ } \rho(H)>(g-3)/2.
\end{array}
\right.
$$
\end{remark'}
\begin{proof} {\it (Theorem \ref{char-weight1}).} (i)
$\Rightarrow$ (ii) follows from Lemma \ref{bo-weight}. (ii) $\Rightarrow$
(iii) follows from the hypothesis on $g$.
(iii) $\Rightarrow$ (i): By Theorem \ref{char-weight} we have that
$H(P)$ is $t$-hyperelliptic for some $t\in \{0,\ldots,\gamma\}$. Then by
Lemma \ref{bo-weight} and hypothesis we have
$$
\binom{g-2\gamma+2}{2}>w(H)\ge \binom{g-2t}{2},
$$
from where it follows that $t=\gamma$.
\end{proof}
\begin{remark'}\label{sharp-weight} The hypothesis on $g$ in the
above two theorems is sharp:
\smallskip
(i) Let $\gamma\ge 7$ and considerer $H:=\langle 4, 4(\gamma+1),
2g-4(\gamma+1)+1\rangle$ where $g$ is an integer satisfying ${\rm
max}\{4\gamma+4, \frac{\gamma^2+6\gamma-3}{2}\}<g\le \gamma^2+4\gamma+2$.
Then $H$ has genus $g$ and $\rho(H)=\gamma+1$. In particular $H$
is not $\gamma$-hyperelliptic. By Lemma \ref{bo-weight} (ii) we have
$w(H)=\binom{g-2(\gamma+1)}{2}+2(\gamma+1)^2$. Now
it is easy to check that $w(H)$ satisfies Theorem \ref{char-weight} (ii)
and Theorem \ref{char-weight1} (iii).
\smallskip
(ii) Let $\gamma\le 6$ and consider $H=\langle 3,g+1\rangle$,
where $g=10, 22, 33, 43, 55, 64$ if $\gamma=1,2,3,4,5,6$ respectively.
$H$ has genus $g$ and it can be easily checked that $H$ is not
$\gamma$-hyperelliptic by means of inequality (\ref{des-odd-1}) and Lemma
\ref{feto1}. Moreover
$w(H)=g(g-1)/3$ (see Remark \ref{remark3} (ii)). Now it is easy to check
that $w(H)$ satisfies Theorem \ref{char-weight} (ii) and Theorem
\ref{char-weight} (iii).
\smallskip
(iii) The semigroups considered in (i) and (ii) are also Weierstrass
semigroups (see Komeda \cite{Ko}, Maclachlan \cite[Thm. 4]{Mac}).
\end{remark'}
\subsection{Weierstrass weights}
In this section we apply Theorem \ref{char-weight1} in order to
characterize double coverings of curves by means of Weierstrass weights.
Specifically we strengthen \cite[Theorem B]{To} and hence all its
corollaries. The basic references for the discussion below are Farkas-Kra
\cite[III.5]{F-K}
and St\"ohr-Voloch \cite[\S1]{S-V}.
Let $X$ be a non-singular,
irreducible and projective algebraic curve of
genus $g$ over an algebraically closed field $k$ of characteristic $p$.
Let $\pi:X\to \mathbb P^{g-1}$ be the morphism induced by the canonical
linear system on $X$. To any $P\in X$ we can associate the sequence
$j_i(P)$ ($i=0,\ldots,g-1$) of intersection multiplicities at $\pi(P)$ of
$\pi(X)$ with hyperplanes of $\mathbb P^{g-1}$. This sequence is the same
for all but finitely many points (the so called Weierstrass points of
$X$). These points are supported by a divisor $\mathcal{W}$ in such a way
that the Weierstrass weight at $P$, $v_P(\mathcal{W})$, satisfies
$$
v_P({\mathcal{W}})\ge w(P):= \sum_{i=1}^{g-1}(j_i(P)-\epsilon_i),
$$
where $0=\epsilon_0<\ldots<\epsilon_{g-1}$ is the sequence at a generic
point. One has $j_i(P)\ge \epsilon_i$ for each $i$, and from the
Riemann-Roch theorem follows that $G(P):=\{j_i(P)+1:i=0,\ldots,g-1\}$ is
the set of gaps of a semigroup $H(P)$ of genus $g$ (the so called
Weierstrass
semigroup at $P$). $X$ is called {\it classical} if $\epsilon_i=i$ for
each $i$ (e.g. if $p=0$ or $p>2g-2$). In this case we have
$v_P({\mathcal{W}})=w_P(R)$
for each $P$, and the number $w(P)$ is just the weight of the
semigroup $H(P)$ defined in
\S0. The following result strengthen \cite[Thm.B]{To}. The proof follows
from \cite[Thm.A]{To}, \cite[Thm.A]{To1}, and Theorem \ref{char-weight1}.
\begin{theorem'} Let $X$ be a classical curve, and assume that $g$
satisfies (\ref{bound-g}). Then the following statements are equivalent:
\begin{itemize}
\item[(i)] $X$ is a double covering of a curve of genus $\gamma$.
\item[(ii)] There exists $P\in X$ such that
$$
\binom{g-2\gamma}{2}\le w(P)\le \binom{g-2\gamma}{2}+2\gamma^2.
$$
\item[(iii)] There exists $P\in X$ such that
$$
\binom{g-2\gamma}{2}\le w(P)< \binom{g-2\gamma+2}{2}.
$$
\end{itemize}
\end{theorem'}
Remark \ref{sharp-weight} says that the bound for $g$ above is the best
possible. Further applications of \S3.1 and \S3.2 will be published
elsewhere \cite{To2}.
|
1995-12-07T06:20:24 | 9512 | alg-geom/9512004 | en | https://arxiv.org/abs/alg-geom/9512004 | [
"alg-geom",
"math.AG"
] | alg-geom/9512004 | Torsten Ekedahl | Torsten Ekedahl | Varieties of CM-type | plain TeX | null | null | null | null | We introduce the notion of a variety (or more generally a motive) of CM-type
which generalises the well known notion of abelian variety of CM-type. Just as
in that particular case it will turn out that the cohomology of the variety is
determined by purely combinatorial data; the type of the variety. As
applications we will show that the \l-adic representations are given by
algebraic Hecke characters whose algebraic parts are determined by the type and
give a method for computing the discriminant of the N\'eron-Severi group of
super-singular Fermat surfaces.
| [
{
"version": "v1",
"created": "Tue, 5 Dec 1995 13:08:10 GMT"
},
{
"version": "v2",
"created": "Wed, 6 Dec 1995 08:42:58 GMT"
}
] | 2008-02-03T00:00:00 | [
[
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"Torsten",
""
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\def\symb{Tr}{\symb{Tr}}
\def\symb{Irr}{\symb{Irr}}
\def\symb{Frac}{\symb{Frac}}
\mdef\coh{H^*(X,r)}
\mdef{\hodg#1.#2.}{H^{#1}(X,{\gOm}^{#2}_{X/k})}
\candef W
\head varieties of CM-type
\begin{document}
\begin{start} Varieties of CM-type \egroup\setbox1=\hbox\bgroup\smc Torsten Ekedahl
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{start}
\begin{introduction}
We will introduce the notion of a variety (or more generally a motive) of
CM-type which generalises the well known notion of abelian variety of
CM-type. Just as in that particular case it will turn out that the
cohomology of the variety is determined by purely combinatorial data; the
type of the variety. As applications we will show that the \l-adic
representations are given by algebraic Hecke characters whose algebraic
parts are determined by the type and give a method for computing the
discriminant of the N{\'e}ron-Severi group of super-singular Fermat surfaces.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{introduction}
\begin{section}Preliminaries.
To begin with let us recall the following facts from category theory. If
\Cal A is an additive category all of whose idempotents have kernels and
$R$ is a ring, then for a finitely right resp.~left projective $R$-module
$P$ resp.~$Q$ and a left $R$-object $M$ in \Cal A we can define objects
$P\bigotimess _RM$ resp.~$Hom_R(Q,M)$ of \Cal A characterised by
$$
\leqalignno{Hom_{\Cal A}(P\bigotimess _RM,N)&=Hom_R(P,Hom_{\Cal A}(M,N))\cr
\noalign{\leftline{resp. }}
Hom_{\cal A}(N,Hom_R(Q,M))&=Hom_R(Q,Hom_{\cal A}(N,M)).\cr}
$$
We always have a natural $R$-morphism $ev\co P\to Hom_{End_R(P)-\Cal
A}(Hom_R(P,M),M)$, the evaluation map, obtained by interpreting an element
$p\rum\@a\rum P$ as an $R$-morphism $R\to P$ and using $M=Hom_R(R,M)$. If
R=$\bigoplus P_i^{n_i}$ and $Hom_R(P_ i,P_j)=0$ for $i\ne j$ then for any
$R$-object $M$ in \Cal A we have
$$
M=\bigoplus P_i{\textstyle\bigotimess }_{S_i}Hom_R(P_i,M),\tag 1.1
$$
where $S_i:=End_R(P_i)$ and the map is defined using the evaluation maps.
To see this we first note that $P_i=Hom_R(R,P_i)\cong(S_i)^{n_i}$ so that
$P_i$ is $S_i$-projective and then the desired equivalence follows by
decomposing the two $R$-factors of $M=R\bigotimess _RHom_R(R,M)$.
The following setup will be with us during the rest of the paper: We let
\k\ be a perfect field of characteristic $p\ge 0$, $X$ a proper, smooth
variety over \k\ (alternatively a motive) and $S$ a set of
\k-correspondences of $X$. Furthermore, \coh, $r$ prime, will denote the
\l-adic cohomology of $X_{\k}$, where \bk\ is a fixed algebraic closure of
\k, when $r\ne p$ and the crystalline cohomology of $X/\k$ when
$r=p$. Recall that when $r\ne p$ \coh\ is a graded $\Z_r$-algebra, finitely
generated as $\Z_r$-module, having a continuous action of $Gal(\bk/\k)$ and
that when $r=p$, \coh\ is a graded $\W(\k)$-algebra, finitely generated as
$\W(\k)$-module having a {\gsi}-linear endomorphism $F$. Here $\W(\k)$ is the
ring of Witt vectors of \k\ and {\gsi} sends a Witt vector $(x_i)$ to
$(x^p_i)$. We let $L_r$ be an algebraically closed field containing $\Z_r$
resp.~$\W(\k)$. Furthermore we will denote by $K$ the
fraction field of $\W(\k)$.
Finally, if $p>0$ we will have need of the following technical condition.
There is a scheme $T$ of finite type over $\F_p$, a smooth and
proper morphism $\Cal X\to T$ and a cartesian diagram
$$
\diagram{ X&\mapright{}&\cal {X}\cr
\mapdown{}&&\mapdown{}\cr
\Sp\;k&\mapright{}&T\cr}
$$
such that for every closed point $t\rum\@a\rum T$ the eigenvalues of the Frobenius
with respect to $\k(t)$ on $H^i(X_t,p)$ are algebraic integers all of
whose archimedean absolute values are $|k(t)|^{i/2}$.
This condition is fulfilled when $X$ (possibly over \bk) is the image of a
smooth and projective variety (\[K-M]) and that this is always the case has
recently been verified by J.~de Jong (\[Jo]).
For a field $L$ and a set $R$ let $K(R,L)$ be the Grothendieck group of the
category of finite dimensional representations (i.e.~maps of $R$ into the
set of endomorphisms) of $R$. Then $K(R,L)$ is a functor in $R$ and $L$;
contravariant in $R$ and covariant in $L$. If $M(R)$ is the free monoid
generated by $R$ then the trace map gives an additive map $\symb{Tr}: K(R,L)\to
L^{M(R)}$.
\begin{lemma}1.2.
Let $L$ be algebraically closed of characteristic 0.
i) $\symb{Tr}: K(R,L)\to L^{M(R)}$ is injective.
ii) Let $L'\subseteq L$ be a subfield of $L$ and $N$ a semi-simple
$L$-representation of $R$ s.t.~for all $r\rum\@a\rum M(R)$ $\symb{Tr}_N(r)\rum\@a\rum L'$.
If $L'\langle R\rangle$ is the free associative $L'$-algebra
on $R$ then
$I :=\ker (L'\langle R\rangle\to End_L(N))$ depends only on the function
$\symb{Tr}_N: M(R)\to L'$ and $L'\langle R\rangle/I\bigotimess
_{L'}L\to End_L(N)$ is
an injection. In particular, if $L'$ is algebraically closed
$N$ is isomorphic to the
scalar extension of some $L'$-representation of $R$.
iii) If $L'$ is an algebraically closed field and $L'\to L$
a field homomorphism,
then the following diagram
$$
\diagram{
K(R,L')&\mapright{}&K(R,L)\cr
\mapdown{}&&\mapdown{}\cr
L'{}^{M(R)}&\mapright{}&L^{M(R)}\cr}
$$
is cartesian.
\pro Let us begin with ii). Note first that as
$\mathop{\rm im}\nolimits(L\langle R\rangle\to End_L(N))$ is semi-simple and that for a finite
dimensional semi-simple L-algebra $M$ and a faithful finite dimensional
$L$-representation $V$, the linear form
$$
\eqalign{ M \times\;M&\to L\cr
(m,m')\mapsto Tr_V(mm')\cr}
$$
is non-degenerate. Hence $t\rum\@a\rum L\langle R\rangle$ acts as zero on $N$ iff
$\symb{Tr}_N(rt)=0$
for all $r\rum\@a\rum M(R)$. If $t=\sum _{r\rum\@a\rum M(R)}{\gla}_rr$ then this is
a set of linear conditions on the ${\gla}_r$ with coefficients in $L'$
depending only on $\symb{Tr}_N: M(R)\to L'$. Furthermore,
$L'\langle
R\rangle/I\bigotimess _{L'}L\to End_L(N)$ is injective iff whenever
there is an $L$-linear relation in $End_L(N)$ between elements in
$L'\langle R\rangle$ there is also an $L'$-linear relation. This also follows
from the fact that the above conditions have $L'$-coefficients. Now iii)
follows immediately from i) and ii) whereas i) is well known
(cf.~\[C-R:Thm. 30.12]).
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{lemma}
If $L$ is a field of characteristic zero and $L'$ an
algebraic closure of $L$, we
put $\ovl K(R,L):=K(R,L')\bigcap L^{M(R)}$. This is clearly independent of the
choice of $L'$ and we have $K(R,L)\subseteq \ovl K(R,L)$.
If $N$ is a representative over $L'$ of an $n\rum\@a\rum \ovl K(R,L)$ then we can
construct the $L\langle R\rangle/I$ of lemma \ref{1.1}, which
depends only on $n$. It is a semi-simple $L$-algebra as its
scalar extension to $L'$ is, and we will denote it $A_L(n)$. In case $L=\Q$
then we put $A(n):=\mathop{\rm im}\nolimits(\Z\langle R\rangle\to A_\Q(n)$, where $\Z\langle
R\rangle$ denotes the free associative algebra on $R$.
If $M$ is an over-field of $L$ then we say that $n$ is
realisable over $M$ if the
induced element in $\ovl K(R,M)$ belongs to $K(R,M)$. This is equivalent to $N$
being realisable by an $A_L(n)\bigotimess _LM$-representation. For any
$n\rum\@a\rum K(R,L')$ we let $\symb{Irr}(n)\subset K(R,L')$ be the set of irreducible
constituents of n. If now $n\rum\@a\rum\ovl K(R,L)$ then $\symb{Irr}(n)$
is a finite set stable
under the action of $Gal(\bar\Q/\Q)$. Under the correspondence of Galois
theory, $\symb{Irr}(n)$ then corresponds the {\'e}tale $L$-algebra $Z(A_L(n))$.
If we return to the situation at hand we have elements
$[H^i(X,r)\bigotimess _{\Z_r}L_r]$
(resp.~$[H^i(X,p)\bigotimess _{\W(k)}L_p]$) in $K(S,L_r)$. Let
$\bar \Q$ be an algebraic closure of \Q.
\begin{lemma}1.3. Let $K_r$ denote $\Q_r$ when $r\ne p$ and
$K_r$ when it isn't. There
exists a unique element $[H^i(X)]\rum\@a\rum \ovl K(S,\Q)$ whose image in
$K(S,K_r)$ coincides with $[H^i(X,r)]$.
Furthermore, $A([H_i(X)])$ is finitely generated as \Z-module and
$A\bigotimess K_r$ equals $A_r/rad(A_r)$ where
$$
A_r:=Im(\Q_r\langle S\rangle\to
End_{\Q_r}(H^i(X,r))\;\;\;\;\;\;(resp.\;\ldots).
$$
\pro I first claim that, for every $s\rum\@a\rum M(S)$, $Tr(s,H_i(X,r))$ is a rational
number independent of $r$. By standard specialisation arguments we reduce to
\k\ being a finite field where it is \[K-M:Thm 2]
(supplemented by \[Gr] for the
definition of the cycle map in crystalline cohomology).
Note that if $p=0$, using Chow's lemma and resolution of singularities
we can get a reduction for which our technical condition is fulfilled. In this
case, a transcendental argument can also be used. This already,
using \(1.2:ii),
gives the existence of $[H_i(X)]$ and that $A\bigotimess L_r
= A_r/rad(A_r)$. Hence $A\bigotimess \Q$ is a finite dimensional
semi-simple \Q-algebra. For any $t\rum\@a\rum \Z\langle S\rangle$ the
characteristic polynomial of $t$ on $H_i(X,r)\bigotimess L_r$ is independent
of $r$ and has rational coefficients by [loc.~cit.]. As $t$
stabilises a $\Z_{\ell}$-
(resp.~$\W(\k)$-) lattice in $H_i(X,r)\bigotimess L_r$, those coefficients are
$r$-integral for all $r$ and so integral. By the Cayley-Hamilton theorem the
image of $t$ in $A\bigotimess \Q$ is integral over
\Z\ and so $A$,
being equal to $\mathop{\rm im}\nolimits:\Z\langle S\rangle\to A\bigotimes \Q$, is finitely
generated as it is contained in the different ideal of any order
containing it.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{lemma}
If still $L$ is algebraically closed of characteristic 0 and
$R$ and $T$ are two
sets, then we let $K(R,T,L)$ denote the Grothendieck group of the category of
finite dimensional $L$-representations of
$R \disjunion T$ such that every
element of $R$ commutes with every element of $T$. It is easy to see that every
simple object of this category is a tensor product of an irreducible
representation of $R$ and one of $T$ and that the two factors are
well-determined up to isomorphism. Hence $K(R,T,L)=K(R,L)\bigotimess K(T,L)$.
Furthermore, as $K(R,L)$ has a canonical base consisting of irreducible
representations we get a canonical pairing
$$
K(R,L)\bigotimess K(R,L)\to \Z
$$
where the the canonical base is orthonormal. Using this we get a mapping
$$
K(R,L)\bigotimess K(R,T,L)=K(R,L)\bigotimess K(R,L)\bigotimess K(T,L)\to K(T,L)
$$
and so for each $N\rum\@a\rum K(R,T,L)$ a mapping
$$
N\cap \co K(R,L)\to K(T,L).\tag1.4
$$
This mapping is compatible, in the obvious way, with the mappings obtained
from homomorphisms $L\to L'$ of algebraically closed fields. Hence we get
\begin{corollary}1.5.
Let $S'$ be a set of \k-correspondences of $X$ and suppose that
every element of $S$ commutes up to homological equivalence with $S'$. Then
\(1.4) gives a $Gal(\bQ/\Q)$-equivariant homomorphism
$$
[H^i(X)]\cap : K(S,\bQ)\to K(S',\bQ)\tag(1.6)
$$
which equals, for each $r$, the restriction of
$[H^i(X,r)\bigotimess L_r]\cap $ to K(S,$\bQ$).
\pro
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{corollary}
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{section}
\begin{section}Varieties of CM-type.
Let us fix an $n\rum\@a\rum \N$ and assume, for simplicity, that
$$b_n(X) =\sum _{i+j=n}\dim_kH^i(X,{\gOm}^j_{X/k}),$$
where $b_n(X) :=
\dim_{ L{_r}} H^n(X,r)$ for any $r$.
(This is of course always true
when $p=0$.) If $p>0$ this implies (cf.~\[Ek: IV, 1.2] or \[B-Og:\S8]) that
$H^n_{DR}(X/k)=H^n(X,p)/pH^n(X,p)$ and that if
$$
M^i:= \mathop{\rm im}\nolimits(F^{-1}p^iH^n(X,p)\to H^n(X,p)/pH^n(X,p))
$$
then
$$
M^i/M^{i+1}=H^{n-i}(X,{\gOm}^i_{X/k}).
$$
Hence, no matter the value of $p$, $H^n_{DR}(X/k)$
has a Hodge filtration with
the $H^{n-i}(X,{\gOm}^i_{X/k})$ as successive quotients.
\begin{definition}2.1.
$(X,S)$ is said to be of {\deffont separable CM-type in degree $n$} if the
$A_i$ of
\(1.3) has the property that $A\bigotimess \Z_{(p)}$ is a
separable (cf.~\[D-I:II,1])
$\Z_{(p)}$-algebra and for every $0\le i\ne j\le n$,
$H^{n-i}(X,{\gOm}^i_{X/k})$ and $H^{n-j}(X,{\gOm}^j_{X/k})$ are disjoint
$S$-modules (i.e.~they have no common composition factors).
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{definition}
\begin{remark}
As the quotient of a separable algebra is separable it suffices to
verify that
$\Z_{(p)}\langle S\rangle$ factored by some known relations is separable.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{remark}
\begin{example}i) If $X$ is an abelian variety of CM-type in
the usual sense and
$End_k(X)$ is separable at $p$, which is always true if $p=0$, then
$(X,End_k(X))$ is of CM-type in degree 1 (and in fact in all other degrees).
ii) Kummer surfaces associated to abelian surfaces of CM-type are of
CM-type in degree 2. Hence, by \[S-I], K3-surfaces in characteristic 0 for
which the rank of the N{\'e}ron-Severi group is 20 are of CM-type in degree 2.
iii) (Fermat hyper-surfaces, diagonal automorphisms). This is well known
(cf.~e.g.~\[Ka:Sect.~6]).
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{example}
\begin{lemma}2.2.
Suppose $(X,S)$ is of separable CM-type. If $p=0$ then $[H^n(X)]$ is
realisable over \Q\ and if $p>0$ then $A^i\bigotimess _\Z\Z_p$
is unramified (i.e.~a product of matrix algebras over unramified
extensions of $\Z_p$) and in particular $[H^n(X)]$
is realisable over $\Q_p$.
\pro The case $p=0$ follows by transcendental methods, in fact $[H_n(X)]$ is
realised by singular cohomology, and the $p>0$ is well-known (use
the fact that the Brauer group of a finite field is trivial and lift an
idempotent).
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{lemma}
\begin{remark}
Is it possible to give an algebraic proof of the first part of the
lemma? The existence of \l-adic cohomology implies that it suffices to
prove realisability over \R.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{remark}
Suppose now that $(X,S)$ is of separable CM-type and let $M$ be an
irreducible component of $[H_n(X)]$. If $p=0$ there is then an irreducible
$S$-module $N$ such that $ M\bigotimess _\bQ\bk$ is a factor of
$M\bigotimess _\k \bk$ for an embedding of $\bQ$ in \bk\ and $N$ is a
sub-quotient of $H^n_{DR}(X/k)$. (Note that the base extension of
$[H^n(X)]$ to \bk\ equals the extension of $[H^n_{DR}(X/k)]\rum\@a\rum K(S,\k)$ to
\bk, which is seen by either using a constructibility argument to reduce to
\(1.3) or a transcendental argument.) By assumption there is a unique $i$,
$0\le i\le n$, such that $N$ occurs as a sub-quotient of \hodg n-i.i. and
this $i$ depends only on $M$. If $p>0$ we get in the same way an
irreducible $K\langle S\rangle$-module $N$, $K:=\W(\k)\bigotimess \Q$, such
that $M\bigotimess _\bQ\bar K$ occurs in $N\bigotimess _{K}\bar K$ for an
algebraic closure $\bar K$ of $K$ and an embedding of \bQ\ in $\bar K$ and
$N$ occurs in $H^ n(X,p)\bigotimess _{\W(k)}K$. As $A\bigotimess \W(\k)$ is
separable, there is a unique, up to isomorphism, $A\bigotimess
\W(\k)$-lattice $N'$ with $N'\bigotimess _\W k=N$ (this follows from
\(2.2)) and $N'\bigotimess \k$ is an irreducible $A\bigotimess \k$-module
and we see that $N'\bigotimess \k$ is an irreducible sub-quotient of of
$H^n_{DR}(X/\k)$. By assumption there is then a unique $i$, $0\le i\le n$,
such that $N\bigotimess \k$ occurs in a \hodg n-i.i. and this $i$ depends
only on $M$. In both cases we put ${\gta}(M) := i$.
In conclusion we have obtained a mapping
$$
{\gta}: \symb{Irr}([H^n(X)])\to n+1\;(:=\{0,1,\relax\ifmmode \ldots\else \dots\fi ,n\}).
$$
Note that the action of $Gal$(\bk/\k) on $\ovl \symb{Irr}([H^n(X)]$) obtained through
the action of $Gal(\bQ/\Q)$ on it and the induced map $Gal(\bk/\k)\to
Gal(\bar\Q/\Q)$ (resp.~$Gal(\bk/k)\to Gal(\W(\bk)/\W(k))$)
preserves the fibers of {\gta} by construction.
\begin{definition}2.3.
Under the assumption of \(2.1) the type of $(X,S)$ is the
pair ($[H^n(X)]$,{\gta}).
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{definition}
Finally, when $p>0$ the condition that $A$ is separable at $p$ implies that the
action of $Gal(\bar \Q/\Q)$ on $\symb{Irr}([H^n(X)])$ is unramified at $p$ so that we
may unambiguously speak about the action of the Frobenius morphism on
$\symb{Irr}([H^n(X)])$ having once and for all chosen an embedding of \bQ\
in $L_p$.
This permutation of $Irr([H^n(X)])$ we will denote {\gsi}.
We have now come to the main result of the present paper. Before we
formulate it we will need to introduce some constructions. Let $T$ be a set,
$M\rum\@a\rum \ovl K(T,Q)$ and ${\gta}\co \symb{Irr}(M)\to n+1$ a function.
Let us choose an embedding of \bQ\ into \C\ and let ${\gio}\rum\@a\rum Gal(\bQ/\Q)$ be
the element corresponding through this embedding to complex
conjugation. Suppose
that for every ${\grh}\rum\@a\rum \symb{Irr}(M)$, ${\gta}({\gio}({\grh}))=n-{\gta}({\grh})$
and also that $M$ is realisable over \Q\ by a module $V$.
For each simple factor
$A_r$ of $A_\Q$ we let $V_r$ be an irreducible $A_r$-module. We then put a
rational Hodge structure on $V_r$, of weight $n$, as follows: For ${\grh}\rum\@a\rum
\symb{Irr}(M)\bigcap \symb{Irr}(V_r\bigotimess \C)$ we let $V_{r,{\grh}}$ be the
{\grh}-isotypical component of $V_r\bigotimess \C$ and then we put
$(V_r\bigotimess \C)^{i,n-i}:=\sum _{{\gta}({\grh})=i}V_{r,{\grh}}$. We then
put a Hodge structure on $V$ by forcing $V=\bigoplus V_i\bigotimess
_{End(V_i)}Hom(V_i,V)$ to be an isomorphism of Hodge
structures. By construction
$T$ acts as morphisms of Hodge structures. However, the Hodge structure itself
depends only on the action of $Z(A_\Q(M))$ on $V$ so that an alternative method
of construction is to start with the set $\symb{Irr}(M)$ with its action of
$Gal(\bar\Q/\Q)$, let $K$ be the associated {\'e}tale
\Q-algebra, let $V$ be the
$K$-module of dimension specified by $M$ and then let $(V\bigotimess
\C)^{i,n-i}:=\sum _{{\gta}({\grh})=i}V_{\grh}$, where {\grh} runs over the
\Q-algebra homomorphisms $K\to \C$. In this way it is seen that $V$ as a
rational Hodge structure depends only on the
$Gal(\bQ/\Q)$-set $\symb{Irr}(M)$ and two
functions ${\gta}\co \symb{Irr}(M)\to n+1$ and $\dim\co \symb{Irr}(M)\to \N$, where $\dim$
is defined by $\dim(n)=dim_{Z(A_\C(n))}N$ for a representative of $N$ (\C\ can
of course be replaced by any algebraically closed field). If \k\ is a subfield
of \C\ such that the action of $Gal(\bk/\k)$ on $\symb{Irr}(M)$ preserves the fibers
of {\gta} then for a choice of descent of the Hodge filtration on each
$V_r\bigotimess \C$ to $V_r\bigotimess k$ for each we get a
descent of the Hodge
filtration on $V\bigotimess \C$ again by forcing the isomorphism above to
preserve the descent.
If $p$ is a prime such that $A(M)\bigotimess \Z_{(p)}$ is
finitely generated as
$\Z_{(p)}$-module and separable
we associate, in a similar way, an $F$-crystal to $(M,{\gta})$:
Suppose that the action of $Gal(\bk/\k)$ on $\symb{Irr}(M)$ preserves
the fibers of {\gta}. We know that $M$ is realisable
over $\Q_p$ and there is, up to isomorphism, a unique
$A(M)\bigotimess \Z_p$-lattice $V$ such that $V\bigotimess \Q$ is such a
realisation. We also get analogous $V_r$. Further,
$V_r\bigotimess _{\Z_p}\W(k)$ is the sum of its isotypical components
$(V_r\bigotimess _{\Z_p}\W( k))_{\grh}$. The {\gsi}-linear
isomorphism $1\bigotimess {\gsi}$ takes
$(V_r\bigotimess _{\Z_p}\W(k))_{\grh}$ to $(V_r\bigotimess
_{\Z_p}\W( k))_{{\gsi}({\grh})}$
and we define the structure of an F-crystal on $V_r\bigotimess _{\Z_p}\W(k)$
by $F=p^{{\gta}(t)}(1\bigotimess {\gsi})\co (V_r\bigotimess
_{\Z_p}\W(k))_{\grh}\to (V_r\bigotimess _{\Z_p}\W(k))_{{\gsi}({\grh})}$.
The $F$-crystal structure on $V\bigotimess \W(k)$ is constructed as before.
Again $T$ acts by endomorphisms and there is an alternative way of
constructing the $F$-crystal if one is prepared to forget
the $T$-action. Indeed,
consider the set $\symb{Irr}(M)$ with its action of {\gsi} and the two functions
{\gta} and $\dim$.
We let $R$ be the set containing for each $n\rum\@a\rum \symb{Irr}(M)\dim(n)$ copies of
$n$ with {\gsi} and {\gta} extended in the obvious way. We then consider
$\W(\bk)[R]$, the free $\W(\bk)$-module on $R$, and define the Frobenius map
by $F{\ninerm[} r{\ninerm]}=p^{{\gta}(r)}{\ninerm[} {\gsi}(r){\ninerm]}$. Also if we
have for each $V_i$
an $End(V_i)$-representation {\grh} of $Gal(\bk/\k)$ on $V_i$, then we can
twist by this by letting $F$ act by $p^{{\gta}(t)}({\grh}\bigotimess {\gsi})$.
Finally, we would like to associate to $(M,{\gta})$ the \l-adic
analogue of this, that is a Hecke character.
We will be able to associate to our data the algebraic part of a Hecke
character but a problem arises as there is no canonical choice for a Hecke
character with a given algebraic part, indeed such a character may exist
only after an extension of the coefficient field. This will have as a
consequence that our description of the \l-adic cohomology will not be as
satisfactory as the description of the Hodge structure or $F$-crystal of a
variety of CM-type. In case $[H_n(X)]$ is multiplicity free we will be able
to do better however.
In any case the algebraic part (cf.~\[De:5.3]) can be associated
to our data as follows. Assume that for any embedding of \bQ\ in
\C\ we have ${\gta}({\gio}({\grh}))=n-{\gta}({\grh})$
as above for the corresponding {\gio}. Assume also that \k\ is a number field
for which the action of $Gal(\bQ/K)$ on $\symb{Irr}(M)$ stabilises
the fibers of {\gta}.
If again $K$ is the \Q-algebra
corresponding to $\symb{Irr}(M)$, then we can define a multiplicative map
$$
\eqalign{
k^\times &\;\longrightarrow\;\; K^\times \cr
{\gla}&\mapsto \prod _{{\grh}\rum\@a\rum \symb{Irr}(M)}{\grh}(N_{k/\Q}({\gla}))
^{{\gta}({\grh})}.\cr}
$$
By the assumption on \k\ this is well-defined and by the assumption on
{\gta} the projection onto each simple factor of $K$
fulfills the conditions for being the algebraic part
of a Hecke character of weight $n$ so we obtain in this way a set of algebraic
parts of Hecke characters.
\begin{theorem}2.4. Let $(X,S)$ be of separable CM-type in degree $n$.
i) If $k\subseteq \C$
then $H^n_{sing}(X(k),\Q)$ is isomorphic as a Hodge structure
with $S$-action to the one associated to the type of $(X,S)$ with a descent of
the Hodge filtration to \k\ of the sort described.
ii) If $p>0$ then $H^n(X,p)$ is isomorphic as $F$-crystal to the $F$-crystal
associated to the type of $(X,S)$ and a representation of $Gal(\bk/\k)$.
iii) After a finite extension of \k\ the
$Gal(\bk/\k)$-representation on $H_n(X,r)$, $(r\ne p)$
factors through the Galois group of the algebraic closure
$K$ of the prime field
in (the finite extension of) \k. If $p=0$ this representation is given, on
the Galois group of a finite extension of $K$, by a direct sum of
algebraic Hecke characters whose algebraic
parts are the ones associated to the type of $(X,S)$ and with multiplicities
given by $\dim$.
iv) If $\symb{Irr}(M)$ is multiplicity free (i.e.~every
irreducible representation of $S$
occurs at most once in $\symb{Irr}(M)$) and \k\ is a number field then the
$Gal$(\bk/\k)-representation on $H_n(X,r)$ is given by a direct sum of
algebraic Hecke characters with values in the simple components of
$Z(A_\Q(\symb{Irr}(M)))$.
\pro To begin with let $R$ be \Q, $\Z_p$ resp.~$\Q_r$. Then $S$ generates an
$R$-subalgebra $B$ of the algebra of endomorphisms of $H^n_{sing}(X(\k),\Q)$,
$H^n(X,p)$ resp.~$H^n(X,r)\bigotimess _{\Z_r}\Q_r$ such that
$$
B/(\hbox{\rm maximal
nilpotent ideal})=A^n\bigotimess _\Z R
$$
(the $A^n$ being that of \(2.1)). By \[C-R:Thm. 72.19]
$B\to A^n\bigotimess R$ splits as an algebra map and so we can assume that
$A\bigotimess R$ acts on $H^n_{sing}(X(\k),\Q)$, $H^n(X,p)$ resp.
$H^n(X,r)\bigotimess _{\Z_r}\Q_r$. Using \(1.1) we reduce to the case when
$A\bigotimess R$ is a division algebra or, in the case of ii), isomorphic to
$W(\F)$ for a finite field \F. The proof of i) is then easy: By the comparison
theorem $H^n_{sing}(X(\k),\Q)$ is a representative of $[H^n(X)]$ and as
the Hodge decomposition on $H^n_{sing}(X(\k),\Q)\bigotimess _\Q\C$ is
stable under $A\bigotimess \C$, the assumption of CM-type forces the Hodge
decomposition to be obtained as the lumping together of isotypical
components.
As for ii), the fact that $A\bigotimess \W(k)$ is separable
implies that we have a
unique isotypical decomposition $H^n(X,p)=\bigoplus M_{\grh}$, where {\grh}\
runs over the irreducible $A\bigotimess K$-modules
($K:=\W(\k)\bigotimess _\Z\Q$)
occurring in $H^n(X,p)\bigotimess \Q$. As $F$ is \gSi-linear
and commutes with $A$ it
maps $M_{\grh}$
to $M_{\gSi({\grh})}$. If $(p^{n_1},p^{n_2},\ldots,p^{n_k})$
are the elementary divisors of the linear mapping $F\co
M_{\grh}\to\gSi_* M_{\gSi({\grh}))}$, the characterisation of the Hodge
filtration recalled at the beginning of this section shows that
$M_{\grh}/pM_{\grh}$ is non-disjoint from \hod
X{n-i}ik exactly when $i$ equals
some $n_j$. Hence, by assumption, all the $n_j$ equal ${\gta}({\gLa})$ where
{\gLa} is any component of $[H_n(X)]$ which occurs in
${\grh}\bigotimess _K\bar K$.
We will denote, by abuse, this common value ${\gta}({\grh})$. Thus $F\co
M_{\grh}\to\gSi_* M_{\gSi({\grh})}$ is p$^{{\gta}({\gLa})}$ times an
isomorphism. Hence if we define $F'\co H^n(X,p)\to H^n(X,p)$ as being
$p^{-{\gta}({\grh})}F$ on $M_{\grh}$, $H_n(X,p)$ becomes a unit root crystal
and is hence described by the $Gal$(\bk/\k)-representation on the fixed
points of $F$ (over \bk). This action commutes with
$A\bigotimess _\Z{\Z_p}$ and
so gives the desired description.
Let us now turn to iii) and let us begin with the case $p=0$. We may assume
that \k\ is a finitely generated field. We have a representation ${\gph}\co
Gal(\bar k/k)\to Aut_{Z\bigotimess \Q_r}(H^n(X,r)\bigotimess
\Q)$ and after possibly
enlarging $Z$ ($:=Z(\symb{Irr}(M))$) and \k\ we may assume that there exists an
algebraic Hecke character $I_m(k')\to Z^\times $, where $k'$ is the
algebraic closure of \Q\ in \k, whose algebraic part is the one coming from
the type of $(X,S)$ (using that the condition on {\gta} is fulfilled by the
transcendental theory). Twist {\gph} by this character, considered as a
character of $Gal(\bk/\k)$ through the
morphism $Gal(\bk/\k)\to Gal(\bk'/\k')$ and (\[Se:II,2.7]). What we now need
to prove is that this twist ${\gph}'$ has finite image (cf.~\[De:Thm.~5.10]).
Let us first show that if $\gSi\rum\@a\rum Gal(\bar k/k)$ then ${\gph}'(\gSi)$ is
quasi-unipotent. The possible orders for the eigenvalues of a
quasi-unipotent matrix over $\Q_r$ of given order is bounded as the
degrees of the extensions of $\Q_r$ obtained by adjoining an $m$th root of
unity goes to infinity with $m$. It is
therefore enough to verify the quasi-unipotence on
a dense set of Frobenius elements. By the \v Ceboratev density theorem it
suffices to check quasi-unipotence for the Frobenius elements corresponding
to maximal ideals for some thickening of \Sp\k\ over which $A_n$ is
separable, $X$ is smooth and
$$
b_n(X)=\sum _{i+j=n}dim_h\hod Xijh,
$$
where $h$ is the residue field. If $F_m$ is the Frobenius element of
$Gal(\bk/\k)$ we then want to show that all the eigenvalues of
${\gph}'(F_m)$ are roots of unity or, as they are all algebraic numbers, that
all their absolute values are equal to 1. For the infinite primes we use the
Riemann hypothesis for $X$. At finite places away from $q:=\symb{char} h$
there is no
problem. Let us therefore consider the places over $q$. Pick
a place $v$ of $Z$ lying
over $q$ normalised so that $v(|h|)=1$. By definition $v({\gph}(F_ m))$ equals
the average of {\gta} over the orbit of $v$ of the action of \gSi\ on
$\symb{Irr}([H^n(X)])$, where $v$ is seen as a homomorphism $Z\to L_q$ and
thus giving an element of $\symb{Irr}([H^n(X)])$ (recall that $L_q$ is an algebraic
closure of $\Q_q$). Hence we want to show that any eigenvalue of the
action of $F_m$ on the $v$-isotypical part of $H^n(X,q)$ has the same
valuation. By construction $(X_h,S)$ is of separable CM-type in degree $n$ and
the eigenvalues of $F_m$ are of course the same for $X_k$ and
$X_h$ so we may replace $k$ by $h$. Applying \(1.5) to $S$
and $\{F_h\}$ we see, as
$F_m=F^*_h$ on $H^n(X,r)$, that the eigenvalues of $F_m$ on the
$v$-isotypical part of of $H^n(X,r)$ are the same as the the eigenvalues of
$F^*_h$ on $H^n(X,q)$. By ii), if $u$ is the length of the \gSi-orbit of $v$
then $F^u$ is divisible exactly by $p^t$ on the $v$-isotypical component of
$H^n(X,q)$, where $t$ is the sum of the values of {\gta} over the \gSi-orbit
of $v$. As $F^*_h=F^r$, where $|h|=p^r$, we immediately get what
we want.
Now again as the orders of the eigenvalues of the elements of
$Gal$(\bk/\k) are bounded after replacing \k\ by a finite extension we
may assume that the image of ${\gph}'$ consists entirely of unipotent
matrices and so by Engel's theorem ${\gph}'$ is a unipotent representation.
We aim to show that it is in fact trivial. As
$G:={\gph}'(Gal(\bar k/k))$ is a compact $r$-adic Lie group the closed subgroup
of $G$ generated by $r$th powers is of finite index in $G$ and by the Frattini
lemma any closed subgroup mapping surjectively onto the quotient of $G$ by
this subgroup equals all of G. We may then apply the Hilbert irreducibility
theorem to get a number field specialisation
$\k''$ of \k\ such that $X$ has good reduction at $\k''$ and that the composed
map $Gal(\bk''/\k'')\to Gal(\bk/\k)\to G$ is surjective. Hence we may
assume that \k\ is a number field. I claim that for each prime of \k\ over
$r$, the inertia group of that prime has finite image in
$G$. Indeed, ${\gph}'$ is
Hodge-Tate as an algebraic Hecke character is always Hodge-Tate
and by \[Fa]. The finiteness then follows from
(\[Se1:1.4,Cor.~3]) as the unipotence
implies that the Hodge-Tate weight is zero. As ${\gph}'$ is unipotent this
implies that ${\gph}'$ is unramified over $r$. The other monodromy groups
automatically have finite, and therefore trivial, images. The triviality of $G$
then follows from the finiteness of the Hilbert class field of \k.
We have therefore proved iii) when $p=0$. The case $p>0$ is similar up to
the point where we have arrived at a unipotent representation. Any
homomorphism $Gal(S,\bar s)\to \Z_r$ for a finitely generated
$\F_p$-scheme $S$ is geometrically trivial, by
\[K-L:Thm~1], so by thickening \k\ we finish.
As for iv) we start as above so that we have an action of
$A:=A^n\bigotimess \Q$ on $H^n(X)$ by correspondences. Note
that the assumption of multiplicity freeness implies
that the commutant of $A$
in $End(H^n(X,r))$ equals $Z\bigotimess \Q_r$. Let $v$ be a
place of \k\ at which
$X$ has good reduction with fiber $X_v$ over the residue field $\F_v$. Apply
the construction of a semi-simple algebra of correspondences to all
correspondences so as to get $B$. Then $B$ contains the Frobenius
correspondence in its center as well as the subalgebra $A$. Let $C$ be the
commutant of $A$ in $B$ so that $B=AC$. By the observation
just made $C\bigotimess
\Q_r=Z\bigotimess \Q_r\subseteq B\bigotimess \Q_r$ and so $C\subseteq B$ and
therefore $A=B$. Hence the Frobenius correspondence $F_v$ lies in $Z$.
Therefore we have associated to every place $v$ of \k\ outside a finite set an
element $F_v$ of $Z$. Extending by multiplicativity we get a homomorphism
$I_m(k)\to Z^\times$ for a suitable $m$. As in the proof of iii) we show that
the projections onto the simple factors of $Z$
are algebraic Hecke characters with algebraic parts given by the type of
$(X,S)$.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{theorem}
\begin{remark}
i) It is probably true that in ii) we also get the conclusion
that $[H^n(X,p)]$ is geometrically constant. This would
follow from a good theory of
over-convergent $F$-crystals.
ii) Can one find a good extension of iii) that would contain iv) as a special
case?
iii) An example showing that there are problems in the \l-adic case is
obtained as follows. Pick an imaginary quadratic field $K$ with class number
greater than one. There is an elliptic curve $E$ with complex multiplication by
the ring of integers $R$ of $K$ defined over the Hilbert
class field $H$ of K. The pair
($R_{H/K}E$,R) is of CM-type in degree 1 over $K$ yet there is no algebraic
Hecke character whose algebraic part is that obtained from the type of
($R_{H/K}E$,R).
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{remark}
As will come as no particular surprise, for abelian varieties our notion
coincides with the traditional one.
\begin{proposition}2.4.
Suppose $(X,S)$ is of separable CM-type in degree 1. Then its
Albanese variety is of CM-type in the usual sense possibly after a finite
extension of \k.
\pro This follows from \(2.4) and the Tate conjecture for homomorphisms
between abelian varieties. Another proof is for $p=0$ to
note that \(2.3:i) says
that the degree 1 Hodge structure of $X$ is visibly of CM-type and for $p>0$
that $(Alb X,S)$ is rigid as by definition $Hom_S(\hod
X01k,\hod X10k)$ is equal
to 0 and so after a finite extension of \k, $Alb\, X$ can be defined over a
finite field and is hence of CM-type.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{proposition}
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{section}
\begin{section}Hereditary CM-type
\mdef\End{\symb{End}\,}
Theorem 2.3 suffers somewhat on the $p$-adic side as the very natural
example of $(E,\End E)$ where $E$ is a supersingular elliptic curve is not
of separable CM-type; $\End E$ is not separable at $p$. It is possible to
give a result which in that case specialises to a satisfactory answer. In
this section we will give a generalisation of the previous results that
will cover this case. The maximal possible generality would seem to be to
assume that $A^n\bigotimess \Z_{(p)}$ should be a {\deffont hereditary}
order which means that any $A^n$-splitting of crystalline cohomology
tensored with \Q\ comes from an $A^n$-splitting of crystalline cohomology
itself. Let us recall that an order is hereditary if each lattice over it
is projective.
\begin{remark}
The meaning of the term differs somewhat in various areas of the literature
as hereditary sometimes means just that a submodule of a projective module
is projective. The definition used here means that the base extension of
the order to the fraction field of its base ring is semi-simple together
with the fact that every sub-module of a projective module is
projective. We will want this extra condition and hence adopt the current
definition (which is to be found for instance in \[Re]).
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{remark}
On the other hand, the example of an automorphism of order $p$ acting
(non-trivially) on a curve of genus $(p-1)/2$ shows that the condition that
different Hodge pieces be disjoint is not reasonable as the cyclic group of
order $p$ has only one irreducible representation mod $p$. The situation will
no longer be as simple as in the separable case. It is still true that one to
any irreducible $A^n\bigotimess \k$-module can associate an irreducible
$A^n\bigotimess K$-module but this map is no longer injective (though
surjective). We will use \[Re:Ch.~9] as a general reference to the theory of
hereditary orders. For the reader's convenience we repeat the salient facts in
the following proposition as well as adding a result -- a weak version of the
elementary divisor theorem -- which is not to be found in
\[loc.~cit.] (but no doubt is
not new).
\begin{proposition}eldiv
Let $A$ be a hereditary order over a henselian discrete valuation ring $R$
with fraction field $K$.
i) Any submodule of an $A$-lattice is a submodule of finite colength of a
direct factor of the lattice.
ii) In every indecomposable $A$-lattice there is exactly one submodule of a
given colength.
iii) If $M$ is an indecomposable $A$-lattice and $M\hookrightarrow N_i$ two
inclusions of finite colength. Then one of these inclusions is contained in
the other.
iv) Every indecomposable finitely generated torsion $A$-module is a quotient
of an indecomposable $A$-lattice.
v) Let $M$ be an $A$-lattice and $N$ a sub-lattice of it. Then there is a
decomposition of $M$ as a direct sum of indecomposable submodules whose
intersection with $N$ also gives a decomposition of $N$ into a direct sum of
indecomposable submodules.
\pro For i) we take the saturation of the submodule. The quotient of the
lattice by that saturation is torsion-free and hence projective and the
saturation is therefore a direct factor. For ii) we notice that by i) any
submodule of the lattice is also indecomposable so we may assume by
induction that the given colength is 1. However, the lattice being
projective is the projective hull of its co-socle (the maximal semi-simple
quotient) and so being indecomposable the co-socle is simple which means
that there is a unique submodule of colength 1; the radical. For iii) we
note that $M\hookrightarrow N_i$ are included in a common inclusion of
finite colength (being of finite colength). We then apply ii).
As for iv) we use induction on the length of the module $M$. We therefore
find a simple quotient $S$ of $M$ and apply the induction hypothesis to the
kernel $M'$ of this map. We will temporarily (and improperly) call a
torsion quotient of an indecomposable lattice a {\it cyclic} module. Thus
we may assume that $M$ is an extension of a sum of cyclic modules by the
simple module $S$. This extension is the sum, as extension, of the
extension of the cyclic summands by $S$. Let us first study the latter
extensions and let us denote by $P$ the projective hull of $S$, by $Q$ its
radical, by $V$ the cyclic summand and by $R$ its projective hull. Then
every extension of $V$ by $S$ comes from pushout by a map from $Q$ to $V$,
the same extensions being obtained if the difference of two morphism
extends to a map from $P$ to $S$. Now, I claim that all non-surjective maps
$Q\to S$ so extend. In fact the map lifts to a map $Q\to R$ which
necessarily is injective as $Q$ is indecomposable. If the map $Q\to S$ is
not surjective then the map $Q\to R$ is neither. By applying iii) we see that
$P$ must be isomorphic to the unique submodule of $R$ containing $Q$ as a
submodule of colength 1 and thus the original map lifts to $P$. This result
shows that if $V$ is not a quotient of $Q$ then any extension of $V$ by $S$
is trivial and if it is, then the group of extensions can be identified
with maps from the co-socle of $Q$ to the co-socle of $V$ which are
isomorphic simple modules. We will now show that, after possibly changing
the direct sum decomposition of $M'$ we may assume that all but one of the
extension classes of direct summands by $S$ are trivial. This will clearly
show iv). For this we may immediately discard summands of $M'$ which are
not quotients of $Q$ as their extension classes have just been shown to be
trivial. Furthermore, we may use induction on the number of non-trivial
extension classes. Note now that if $V_1$ and $V_2$ are summands then for
any map \pil\phi{V_1}{V_2} we may consider the automorphism of $V$ which
maps $v\rum\@a\rum V_1$ to $v+\phi v$ and acts as the identity on all other
factors. If $e_i$ are the extension classes then all of them but $e_2$ are
unchanged and $e_2$ is changed into $e_2+\phi_*e_1$. If we identify
extension classes of $V_i$ with homomorphisms from the co-socle of $Q$ to
that of $V_i$ then $\phi_*$ is just composition by the map on co-socles
induced by $\phi$. As $V_1$ and $V_2$ are both both quotients of $Q$, by
ii) on is a quotient by the other and we may assume that $V_2$ is a
quotient of $V_1$. In that case, any endomorphism of $Q$ induces a morphism
$V_1\to V_2$ and by the projetivity of $Q$, any map from the co-socle of
$V_1$ to that of $V_2$ is induced by an endomorphism of $Q$. Putting this
together we see that any extension class is of the form $\phi_*e_1$ so that
the we may choose $\phi$ so that $\phi_*e_1=-e_2$ which allows us to
decrease the number of non-zero extension classes.
To finally prove v) we consider the module $M/N$. This is a direct sum of
a lattice and a torsion module and the lattice may be split off from $M$
without changing $N$. Thus we may assume that $M/N$ is torsion. We then use
iv) to write that quotient as a direct sum of quotients of indecomposable
projective modules. The sum of the projective hulls of each summand is a
projective hull of the sum. That projective hull is a direct summand of the
map $M\to M/N$. This immediately gives the pair $(M,N)$ as a direct sum of
of pairs $(P_i,P'_i)$, where $P_i$ is indecomposable and a factor
$(M',M')$. As $M'$ is a sum of indecomposables, v) follows.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{proposition}
We will need a definition which is very special to the situation at hand.
\begin{definition}
Let $A$ be a hereditary order over a henselian mixed characteristic discrete
valuation ring $R$ with positive residue field
characteristic $p$ and let $M$ be
a finitely generated torsion module killed by $p$. By the
\definition{complementary module} to $M$ we mean the torsion module (defined up
to isomorphism) obtained as $P/pP'$, where $P$ is a projective hull of $M$ and
$P'$ is the kernel of the natural map $P\to M$.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{definition}
As there can be, as opposed to the separable case, non-trivial extensions of
modules we also will need to recall the definition of block, well-known in the
theory of general orders,
\begin{definition}
Let $A$ be a hereditary order over a henselian discrete valuation ring $R$
with fraction field $K$. Two indecomposable (finitely generated) $A$-modules
belong to the same block if there is a non-zero morphism from the projective
hull of one to the other. This is equivalent to the two hulls tensored with $K$
being isomorphic. If $M$ is a finitely generated $A$-module then the
\definition{$B$-component}of $M$ is the sum of all indecomposable
factors belonging to the block $B$. (It is clear that any finitely generated
$A$-module is the direct sum of its components associated to different blocks.)
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{definition}
What is different with the hereditary case as opposed to the separable case is
that we may have non-semisimple (f.g.) modules killed by $p$. This will imply
that to define CM-type it is not enough to look at what simple modules occur in
which Hodge piece; the more precise module structure needs to be taken into
account. As we will see this forces certain relations between Hodge pieces.
Our results will be purely algebraic so we will, rather than sticking
to the notation of this article as a whole, use the
following notation: $A$ will be a
hereditary $\Z_p$-order and $M$ will be an $F$-crystal with
an action of $A$. We
define the Hodge filtration on $M/pM$ by $M^i:=F^{-1}p^iM/pM$ and the Hodge
modules $H^i:=M^i/M^{i+1}$ (which may be considered as
$A\bigotimes \W$-modules).
\begin{definition-lemma}
We define the \definition{$A$-primitive part} of $H^i$ as the direct factor
(defined up to isomorphism only) by induction on $i$. For $i=0$ we let the
primitive part be all of $H^i$. For $i>0$ the complement of the primitive part
of $H^{i-1}$ is a direct factor of $H^i$ and we let the primitive part be a
complementary factor of it.
\pro What is to be proven is the statement about the
complement of the primitive
part being a direct summand. We will give another description of the primitive
part which will make this obvious. Consider therefore the Frobenius map as a
$\W$-linear map $\sigma^*M\to M$, which then also is a
$A\bigotimess \W$-linear. This is a hereditary order so we may by \(eldiv:v)
split this map up in indecomposable factors. Using Mazur-Ogus' characterisation
(\[B-Og]) of the Hodge filtration and the fact that submodules are linearly
ordered we immediately see that each indecomposable factor will contribute a
cyclic module to one Hodge piece and its complement to the next.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{definition-lemma}
We are now ready to define what we
mean by CM-type in the context of actions of hereditary orders.
\begin{definition}
The pair $(M,A)$ is of \definition{hereditary CM-type} if $A\bigotimess
\Z_p$ is a hereditary order and for each block $B$, the
$B$-component of the primitive part of $H^i$ is non-zero for at
most one $i$ and all indecomposable factors of that $B$-component have the same
length.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{definition}
We have now set up our definitions so that we may carry through the same
analysis as in the separable case (it should be noted that in the case that
$A^n\bigotimess \Z_{(p)}$ is actually separable this definition coincides with
the previous one).
\begin{lemma}galois
Let $B$ be a $\W$-algebra, finitely generated and free as a
$\W$-module. Suppose
$n$ is a positive integer and $T$ a $\sigma^n$-linear
automorphism of $B$. Using
$T$ to get a \Z-action on $B$ we have that $H^1(\Z,B^\times)=*$.
\pro If $B'$ is the $\Z_p$-algebra of $T$-fixpoints then we have that
$B=B'\bigotimes \W$ and an element of $H^1(\Z,B^*)$ is
given by an automorphism
class of a finitely generated right $B'$-module whose extension of scalars to
$\W$ is isomorphic as $B$-module to $B$ itself. As the
extension $\Z_p\to \W$ is
faithfully flat this means that such a $B'$-module is projective. Hence it is
determined up to isomorphism by its co-socle and to prove
the lemma it is enough
to show that if we have two semi-simple $B'$-modules which become isomorphic
under extension of scalars to $\W$ are isomorphic. This
however is obvious (using
for instance the independence of central characters of a semi-simple algebra).
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{lemma}
\begin{theorem}
Suppose $(M,A)$ is of hereditary CM-type and that {\k} is algebraically
closed. Then it is determined up to isomorphism by which blocks appear in the
primitive part of which Hodge-modules and the common length of indecomposable
factors of each such block.
\pro Note first that we can make {\gsi} act on the blocks of
$A\bigotimes \W$ by
the condition that $N$ belongs to the block $B$ iff $\sigma^*N$ belongs to
$\sigma^*B$. If we now split up $M$ in blocks, $M=\bigoplus_B M_B$, then it
is clear that $F$, considered as a map $\sigma^*M\to M$ is a sum of maps
$\sigma^*M_B\to M_{\sigma^*B}$. Now, for an indecomposable $B\bigotimes
W$-lattice $N$ the length of $N/pN$ only depends on which block $N$ belongs
to. Indeed, any two indecomposable $B\bigotimes W$-lattices in the same block
are contained in each other with quotient of finite length, the kernel and
cokernel of the map induced by reduction modulo $p$ then has the same length.
We now consider the component of $F$, $\sigma^*M_B\to M_{\sigma^*B}$, as a
$W$-linear map and split it up into indecomposable pieces according to lemma
\ref{eldiv}. Looking at each indecomposable piece we see that if $B$ appears in
the $i$'th Hodge piece of $M$ then $F$ maps $\sigma^*M_B$ into
$p^iM_{\sigma^*B}$, the image contains $p^{i+1}M_{\sigma^*B}$ and the length of
each indecomposable factor of $p^iM_B/M'$, $M'$ being the image, has the same
length (as each such length added to the common length of
the indecomposables of
the primitive $B$-part of the Hodge piece adds up to the common length of an
indecomposable lattice in $B$ modulo $p$). This means that any indecomposable
factor of $M_{\sigma^*B}/M'$ has the same length, which is the same as saying
that $M'=\symb{rad}^mM_{\sigma^*B}$ for a suitable $m$,
where $\symb{rad}(-)$ is
the radical functor. We may thus use $F$ to identify $\sigma^*M_B$ with
$\symb{rad}M_{\sigma^*B}$, where $m$ is determined by $i$ and the common length
of indecomposables of the primitive part belonging to the block $B$. If $n$ is
the smallest positive integer for which $\sigma^{n*}B=B$, then $F^n$ maps $M_B$
onto $\symb{rad}^kM_B$ for a suitable $k$. It is then enough to show that
all such maps are conjugate under automorphisms of $M_B$. Fix one such map
{\gph}. Now, I claim that the relation $\phi\circ f^\sigma=g\circ\phi$ defines
an automorphism $f\mapsto g$ of $\End(M_{\sigma^*B})$.
Indeed, for any $g$ there
is an $f$ fulfilling that relation as the image of {\gph} is equal to
$\symb{rad}^mM_{\sigma^*B}$. Conversely, the inverse image of $M_{\sigma^*B}$
in $\sigma^*M_B\bigotimes K$ under {\gph} is equal to its
largest sub-lattice for
which the quotient by $\sigma^*M_B$ has all its indecomposable components of
length less than or equal to $m$ which shows that to any $f$ there is a $g$. As
any map fulfilling the conditions imposed on {\gph} differs from it by an
automorphism of $\sigma^*M_B$ we can apply lemma \ref{galois} to include that
there is, up to isomorphism, only one $F$.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{theorem}
I would also like to record that, just as in the separable case, multiplicity
freeness implies CM-type.
\begin{proposition}
Suppose that $(M,A)$ is multiplicity free in the sense that an irreducible
$A\bigotimes K$-module appears at most once in $M\bigotimes K$. Then $(M,A)$ is
of hereditary CM-type.
\pro The condition implies that for any block, the component of $M$ in that
block is indecomposable. That immediately implies that a given block appears in
the primitive part of just a single Hodge piece and that part is indecomposable
so the condition on length is fulfilled.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{proposition}
We finish this section with some examples.
\begin{example}
i) Consider a supersingular elliptic curve $E$ and its ring
of endomorphisms $A$,
which is a maximal order in a division ring and hence hereditary. The
action of $A$ on the first crystalline cohomology group is
multiplicity free and
hence of hereditary CM-type. More precisely, $H^1(E,p)$ is an indecomposable
$A\bigotimes W$-lattice and $\sigma^*H^1(E,p)$ is the other indecomposable
$A\bigotimes W$-lattice -- both are in the same block. Hence, $H^0(E,\Omega^1)$
is one irreducible $A\bigotimes W$-module and $H^1(E,\Cal O_E)$ the other. The
image of $\sigma^*H^1(E,p)$ under $F$ is the maximal proper submodule of
$H^1(E,p)$.
ii) Let $C$ be the projective, smooth completion of the curve $y^p-y=x^2$,
$p\ne2$, and consider the action of $\Z/p$ given by $y\mapsto y+\alpha$. The
action of the group algebra of $\Z/p$ on $H^1(C,p)$ factors
through the quotient
$A$ that is the ring of $p$'th roots of unity. $H^1(C,p)$ is then a free
$A\bigotimes \W$-module of rank 1 and hence $(C,A)$ is of hereditary
CM-type. This time the situation is simpler as the ring is commutative and to
prove the classification theorem we could simply have divided $F$ by
$(\zeta-1)^{(p-1)/2}$ to obtain a unit root crystal.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{example}
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{section}
\begin{section}Applications to the N{\'e}ron-Severi group.
In this section we will suppose that $X$ is a surface and that $(X,S)$ is of
separable type in degree 2. Then $S$ acts on the N{\'e}ron-Severi group $NS$ of
$X_{\bar k}$. If $p=0$, \(2.4) and the Lefschetz theorem on $(1,1)$-classes
show that $[NS\bigotimess \bar\Q]\rum\@a\rum K(S,\bQ)$ equals the sum of all
irreducible {\grh}\ in $[H^2(X)]$ such that ${\gta}({\gsi}({\grh}))=1$ for all
${\gsi}\rum\@a\rum Gal(\bar\Q/\Q)$ and the Tate conjectures implies this in all
characteristics. However, \(2.3) can be used to obtain further
information on $NS$. To illustrate this let us suppose that $p>0$ and that
$\mathop{\rm rk}\nolimits NS=b_2(X)$. By possibly extending \k\ we may assume that $NS$ is
defined over \k. As $c_1\co NS\bigotimess \Z_\ell\to H^2(X_{\bk},\Z_\ell)$
($\ell\ne p$) has torsion free cokernel (cf.~\[Gro:8.7]) and
the two modules have
the same rank, $c_1$ is an isomorphism. By Poincar{\'e} duality the
intersection pairing is perfect at \l. By \[Ill:II,5.8.5,5.20] the image of
$c_1\co NS\bigotimess \W(k)\to H^2(X,p)$ is the largest
sub-$F$-crystal in which $F$ is
divisible by $p$ and, again by Poincar{\'e} duality, if ${\gsi}_0$ is the
$\W(\k)$-length of the cokernel, then $p^{2{\gsi}_0}$ is the exact power of $p$
dividing $disc(NS)$. Hence by the Hodge index theorem
$disc(NS)=(-1)^{b_2-1}p^{2{\gsi}_0}$.
As the whole $H^2(X,p)$ is determined by the type of $(X,S)$, ${\gsi}_0$ is as
well and we will now see how this can be done explicitly. By \(2.3)
$M:=H^2(X,p)=\bigoplus _{{\grh}\rum\@a\rum \symb{Irr}([H^2(X)])}M_{\grh}$
and $F\co M_{\grh}\to
M_{{\gsi}({\grh})}$ is $p^{{\gta}({\grh})}$ times an isomorphism.
Let $N\subseteq M$ be the maximal sub-$F$-crystal on which $F$
is divisible by $p$. Consider
$T:=\symb{Irr}([H^2(X)])$ with the functions {\gta} and $\dim$ and the action of
{\gsi}. We shall now describe an algorithm for computing
${\gsi}_0$. To do this we
start by by considering $M$ with $F':=p^{-1}F$ as a virtual $F$-crystal
i.e.~$p^{-1}F$ takes $M$
into $M\bigotimess \Q$ rather than into $M$ itself. Now $N$ can then be
characterised as the maximal sub-$F$-virtual-crystal which is actually a
crystal. As it is unique it is a sub-representation and so it is the direct
sum of the $N_{\grh}$. We will now concentrate on one specific {\gsi}-orbit
on $T$ and assume that $M$ is in fact the $F$-crystal associated to
it. Pick one {\grh} in this orbit. As $M_{\grh}$ is of rank 1 $N_{\grh}$ is
equal to $p_nM_{\grh}$ for some $n$. All powers of $F'$ must take
$N_{\grh}$ to $M$ which means that $n+\sum_{j=0}^k({\gta}({\gsi}^j{\grh})-1)$
is greater than or equal to 0 for all $k$. Hence if we put $n$ equal to
$-\min_k\sum _{j=0}^k({\gta}({\gsi}^j{\grh})-1)$ and define $N'$ as
$\bigoplus p^{n_{{\grh}'}}M$, where $m_{{\grh}'}:=n+\sum
_{j=0}^k({\gta}({\gsi}^j{\grh})-1)$ with ${\grh}'={\gsi}^k{\grh}$ we have a
sub-$F$-virtual-crystal of $M$ which clearly is an actual $F$-crystal (here we
use the fact that the sum of ${\gta}-1$ over the orbit is 0). We have
also seen that $N_{\grh}\subseteq N'_{\grh}$ and as $N$ is the maximal
sub-$F$-crystal we have equality. Finally, again using that the sum of
${\gta}-1$ over the orbit is 0 it is immediately realised that $N'$ is
independent of the choice {\grh} and so has to be equal to $N$. In
particular we see that the contribution of this orbit to ${\gsi}_0$ equals the
multiplicity of the orbit times the sum of the $m_{{\grh}'}$.
\begin{example}
We consider one orbit for {\gsi} and describe such an orbit by
$({\gta}(t),{\gta}({\gsi}(t)),\ldots,{\gta}({\gsi}^{h-1}(t)))$,
where $h$ is the
length of the orbit. We also assume that the starting point {\grh} is the first
element of this list.
\noindent i)\ (0,2) gives partial sums $(-1,0)$ and so $n=1$
and the list of the
$m_{{\grh}'}$ is $(0,1)$ and finally the contribution to ${\gsi}_0$ is 1.
\noindent ii)\ $(0,2,1)$ gives partial sums $(-1,0,0)$, $m$s $(0,1,1)$ and
${\gsi}_0=2$.
\noindent iii)\ $(2,1,1,1,1,0)$ gives partial sums $(1,1,1,1,1,0)$, $m$s
$(1,1,1,1,1,0)$ and so ${\gsi}_0=5$.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{example}
As a geometric example let us first consider the Fermat surface $X_m=
\{X^m_0+X^m_1+X^m_2+X^m_3=0\}$ and the group of diagonal automorphisms
$G_m=\mu^4_m/(scalars)$.
The irreducible representations of this group are the
elements of the dual group
$$
\check G_m:=\{(b_0,b_1,b_2,b_3)\rum\@a\rum(\Z/m)^4:\sum _{i=0}^3b_i=0\}
$$
and it is well known (cf.~\[Ka:Sect.~6]) that each character
occurs at most once in
$H^2(X_m)$ and those that occur are exactly those in the set
$T:=\{(b_0,b_1,b_2,b_3)\rum\@a\rum G:\forall i:i\ne0\}\cup\{(0,0,0,0)\}$. Furthermore,
if we for $b\rum\@a\rum\Z/m$ let $\langle b\rangle$ be the unique integer s.t.~$\langle
b\rangle\rum\@a\rum b$ and $0\le \langle b\rangle< m$ then ${\gta}((\b
b))=1/m\sum _{i=0}^3\langle b_i\rangle$ if $(\undl b)\ne
(\undl 0)$ and 1 if not.
Finally, the action by $Gal$(\bQ/\Q) is given by $F_p((\undl
b))=(p\undl b)$ for a
prime $p\not|m$.
\begin{remark}
The proof of this in \[Ka:Sect.~6] uses transcendental methods. A purely
algebraic proof can be given by tracing the action
of $G$ through the calculations of \[SGA7:Exp.~XI].
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{remark}
The Fermat surfaces verify the Tate conjecture (\[S-K]) so $\mathop{\rm rk}\nolimits\,NS=b_2$
over a field of positive characteristic iff the average of {\gta}
over any {\gsi}-orbit equals 1. Now complex conjugations in $Gal(\bQ/\Q)$
exchanges the values 0 and 2 and fixes 1 so we see that if the subgroup
generated by {\gsi} contains complex conjugation this is always the case.
Hence if $-1\rum\@a\rum\langle p\rangle\subseteq(\Z/m)^\times$ then
$rk~NS(X_m)=b_2(X_m)$ in characteristic p.
\begin{example}
i) $p\equiv -1 \pmod m$. Then all the orbits are of type $(1,1,\ldots,1)$
or (0,2) giving a contribution of 0 resp.~1 to ${\gsi}_0$. It is a general fact
(true whenever $rk\,NS=b_2$ and $b_2=\sum _{i+j=2}dim\;\hod Xijk$) that
$p^{2p_g}|disc\ NS$ as the morphism $H^2(X,p)\to \hod X20k$ is surjective
and vanishes on $NS\bigotimess \W$.
ii) m=5, $p\equiv 2,3 \pmod5$. Then there are four orbits of
type $(0,1,2,1)$ and
the rest are of type $(1,1,\ldots,1)$. Now the algorithm applied to $(0,1,2,1)$
gives partial sums $(-1,-1,0,0)$ and a contribution of $2$
to ${\gsi}_0$ for each copy
of this orbit and hence $disc\,NS=p^{16}$,
whereas $p_g=4$ so that we get a higher power of $p$ than is guaranteed by
$p^{2p_g}|disc\ NS$.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{example}
\mdef\tx{\tilde X} \def\tilde X{\tilde X}
\begin{proposition}3.1.
Suppose that $X$ is a smooth surface over \k\ and that
$p>0$. Suppose that $G$ is a finite group of order prime to
$p$ acting on $X$. Let
\tx be a minimal resolution of $X/G$. Then
$H^2(\tilde X,p)=H^2(X,p)^G\perp E$, where $E$ is the \W-module
spanned by the Chern classes of the exceptional curves of $\tilde X\to X/G$ and
orthogonality is wrt the cup product. Furthermore, the cup product pairing
restricted to $E$ is perfect.
\pro Let ${\gpi}\co X'\to X$ be a $G$-equivariant
blowing up of $X$ such that we have a map
${\grh}\co X'\to\tilde X$ covering the quotient map $X\to X/G$. The cup product
pairing on $E$ is perfect because the cokernel of $E\to \check E$ equals \W
tensored with the sum of the local Picard groups of the singularities of $X/G$
(cf.~\[Li:14.4]) and these are killed by the order of $G$ by the
existence of a norm map. Hence we may write
$H^2(\tilde X,p)=V\perp E$. Now ${\grh}_*{\grh}^*=|G|$ so ${\grh}^*$
is injective on $H^2(\tx,p)$
and the image is contained in the $G$-invariants and
is a direct factor.
Furthermore, by the projection formula, ${\grh}^* V$ is orthogonal to the
submodule of $H^2(X',p)$
spanned by the curves exceptional for {\gpi}. Therefore
${\grh}^* V\subseteq \pi^* H^2(X,p)^G$ and we are finished if we can
show that this is an equality. First, we show
this for the $p$-torsion. Indeed, consider the slope
spectral sequence for $X'$
and \tx\ (cf.~\[Ill:II,3]). By duality
(cf.~\[Ek1]) ${\grh}_*$ is defined as a map of
spectral sequences and we still have ${\grh}_*{\grh}^*=|G|$.
Furthermore, ${\grh}^*$ is an isomorphism on
$H^*(\tx,\W\ko \tx)\to H^*(X',\W\ko {X'})^G$ as
$H^*(X',\W\ko {X'})^G=H^*(X,\W\ko X)^G= H^*(X^G,\W\ko {X^G})
=H^*(\tx,\W\ko \tx)$ the last as the singularities are rational. Hence as
$H^0(-,\W{\gOm}_X^2)$ is torsion free as $W{\gOm}_X^2$ is we see that we
have equality on torsion groups if we have equality for the torsion of
$H^1(-,\W{\gOm}^1_-)$. The nilpotent torsion
(cf.~\[Ek1:IV,3.3.13]) of it is dual to the
nilpotent torsion of $H^2(-,\W\ko -)$
(loc.~cit.) and is hence taken care of, whereas the
semi-simple torsion (loc.~cit.:IV,3.4)
comes from $H^2(-,\Z_p(1))$ which in turn
comes from the N{\'e}ron-Severi group \[Ill:II,5.8.5] which is taken care of by
noting that
$\undl {Pic}^{\gta}(\tilde X)=\undl {Pic}^{\gta}(X/G)$ as the
singularities are rational and
$\undl {Pic}^{\gta}(X/G)=\undl {Pic}^{\gta}(X)^G$ outside of the order
of $G$. Hence, as ${\grh}^* V$ is a direct factor of $\pi^* H^2(X,p)^G$,
it suffices to show that that they have the same rank. As
the rank of $V$ is the
rank of $H^2(\tx,p)$
minus the number of exceptional curves we may replace $p$ by
\l\ and then $H^2(X,\ell)^G=H^2(X/G,\ell)$
and the latter space is isomorphic to the
orthogonal complement of the exceptional curves of $\tilde X\to X/G$ by the
Leray spectral sequence.
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{proposition}
Using the proposition we get a description of the crystalline cohomology of
the minimal resolution of the quotient of $X_m$ by any subgroup of $G$.
We can also compute other invariants of Fermat surfaces and their
quotients. Consider for instance the formal Brauer group of a surface $X$
which is Mazur-Ogus (cf.~\[Ek:IV,1.1]) (in positive characteristic) or
rather $H^2(X,\W\ko X)$ the knowledge of which is equivalent to knowing the
formal Brauer group. It follows from \[loc.~cit.:III,Thm~4.3] that
$H^2(X,W\ko X)$ is the quotient of $H^2_{cris}(X/W)\bigotimess _{W[F]}D$,
where $D$ is the Dieudonn{\'e}-ring (with power series in $V$), by the
submodule generated by $m\bigotimess 1-V(n\bigotimess 1)$ for all $m,n\rum\@a\rum
H^2_{cris}(X/W)$ for which $Fm=pn$. Hence if $X$ is of CM-type in degree 2
we get a description of $H^2(X,W\ko X)$.
\begin{example}
i) $(0,1,0,2,1,2)$ gives a $D$-module with generators $a$ and $b$ and relations
$Fa=Vb$ and $Fb=0$. This is the Dieudonn{\'e}-module of a 2-dimensional formal
group isogenous but not isomorphic to $W_2$. For $p\equiv3\pmod7$ this
appears in the cohomology of the Fermat surface of degree 7 (the orbit of
$(1,1,1,4)\rum\@a\rum (\Z/7)^4$).
ii) $(0,0,2,2)$ gives a $D$-module with generator a and
relation $F^2a=0$. This
is the Dieudonn{\'e}-module of $W_2$. For $p\equiv5\pmod7$ this
appears in the cohomology of the Fermat surface of degree 13 (the orbit of
$(3,3,3,4)\rum\@a\rum (\Z/7)^4$).
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{example}
\begin{bibliography}
\[B-Og]:\egroup\setbox1=\hbox\bgroup\smc P. Berthelot, A. Ogus \egroup\book@true\setbox2=\hbox\bgroup\sl Notes on crystalline cohomology\egroup\setbox3=\hbox\bgroup
Princeton Univ. Press \egroup\setbox4=\hbox\bgroup Princeton\yr1978
\[C-R]: \egroup\setbox1=\hbox\bgroup\smc C. W. Curtis, I. Reiner\egroup\book@true\setbox2=\hbox\bgroup\sl Representation theory of finite
groups \yr1962\egroup\setbox3=\hbox\bgroup Interscience publishers\egroup\setbox4=\hbox\bgroup New York
\[De]:\egroup\setbox1=\hbox\bgroup\smc P. Deligne\egroup\setbox2=\hbox\bgroup Applications de la formule des traces aux
sommes trigonom{\'e}triques \egroup\sln@true\setbox4=\hbox\bgroup 569 \egroup\pages@true\setbox6=\hbox\bgroup 168--232
\[D-I]: \egroup\setbox1=\hbox\bgroup\smc F. DeMeyer, E. Ingraham \egroup\setbox2=\hbox\bgroup Separable algebras over
commutative rings\sln181
\[Ek]:\egroup\setbox1=\hbox\bgroup\smc T. Ekedahl\egroup\book@true\setbox2=\hbox\bgroup\sl Diagonal complexes and F-gauge structures\egroup\setbox3=\hbox\bgroup
Hermann \egroup\setbox4=\hbox\bgroup Paris \yr1986
\[Ek1]:\egroup\setbox1=\hbox\bgroup\smc T. Ekedahl \egroup\setbox2=\hbox\bgroup On the multiplicative properties of the
de Rham-Witt complex I\egroup\jour@true\setbox3=\hbox\bgroup\sl Arkiv f{\"o}r mate\-ma\-tik \vol22
\yr1984\pages185--239
\[Fa]:\egroup\setbox1=\hbox\bgroup\smc G. Faltings \egroup\setbox2=\hbox\bgroup p-adic Hodge theory \yr1988 \egroup\jour@true\setbox3=\hbox\bgroup\sl Journal
of the AMS \vol1\pages255--299
\[Gr]: \egroup\setbox1=\hbox\bgroup\smc M. Gros \egroup\setbox2=\hbox\bgroup Classes de Chern et classes de cycles en
cohomologie de Hodge-Witt logarithmique\egroup\jour@true\setbox3=\hbox\bgroup\sl M{\'e}m. de la Soc. Math. de
France\vol21\pages1--87\yr1985
\[Gro]:\egroup\setbox1=\hbox\bgroup\smc A. Grothendieck \egroup\setbox2=\hbox\bgroup Le groupe de Brauer III \egroup\inbook@true\setbox7=\hbox\bgroup Dix
expos{\'e}s sur la cohomologie des sch{\'e}mas \egroup\setbox3=\hbox\bgroup North-Holland \egroup\setbox4=\hbox\bgroup
Amsterdam \yr1968 \pages88--188
\[Ill]:\egroup\setbox1=\hbox\bgroup\smc L. Illusie \egroup\setbox2=\hbox\bgroup Complexe de de Rham-Witt et cohomologie
cristalline \egroup\jour@true\setbox3=\hbox\bgroup\sl Ann. scient. {\'E}c. Norm. Sup \egroup\setbox4=\hbox\bgroup\bf 12 \yr1979
\pages501--661
\[Jo]:\egroup\setbox1=\hbox\bgroup\smc J. de Jong\egroup\spec@true\setbox8=\hbox\bgroup Article to appear
\[Ka]:\egroup\setbox1=\hbox\bgroup\smc N. M. Katz\egroup\setbox2=\hbox\bgroup On the intersection matrix of a hypersurface
\egroup\jour@true\setbox3=\hbox\bgroup\sl Ann. scient. {\'E}c. Norm. Sup\vol2\yr1969\egroup\pages@true\setbox6=\hbox\bgroup 583--589
\[K-L]:\egroup\setbox1=\hbox\bgroup\smc N. M. Katz, S. Lang \egroup\setbox2=\hbox\bgroup Finiteness theorems in geometric
class field theory \egroup\jour@true\setbox3=\hbox\bgroup\sl L'Ens. Math \vol27 \yr1981\pages286--314
\[K-M]: \egroup\setbox1=\hbox\bgroup\smc N. M. Katz, W. Messing \egroup\setbox2=\hbox\bgroup Some consequences of the Riemann
hypothesis for varieties over finite fields \egroup\jour@true\setbox3=\hbox\bgroup\sl Invent. Math. \yr1974
\vol23\pages73--77
\[Li]:\egroup\setbox1=\hbox\bgroup\smc J. Lipman\egroup\setbox2=\hbox\bgroup Rational singularities with applications to
algebraic surfaces and unique factorization \egroup\jour@true\setbox3=\hbox\bgroup\sl Publ. IHES
\vol36\yr1969\pages195--280
\[Re]:\egroup\setbox1=\hbox\bgroup\smc I. Reiner\egroup\book@true\setbox2=\hbox\bgroup\sl Maximal orders\egroup\setbox3=\hbox\bgroup Academic Press\egroup\setbox4=\hbox\bgroup London
\egroup\setbox5=\hbox\bgroup 1975
\[Se]:\egroup\setbox1=\hbox\bgroup\smc J.-P. Serre \egroup\book@true\setbox2=\hbox\bgroup\sl Abelian \l-adic representations and elliptic
curves \egroup\setbox3=\hbox\bgroup W. A. Benjamin\egroup\setbox4=\hbox\bgroup New York\yr1968
\[Se1]:\egroup\setbox1=\hbox\bgroup\smc J.-P. Serre \egroup\setbox2=\hbox\bgroup Groupes alg{\'e}briques associ{\'e}s aux modules
de Hodge-Tate \egroup\jour@true\setbox3=\hbox\bgroup\sl Ast{\'e}rix 65 \egroup\setbox5=\hbox\bgroup 1979\pages155--188
\[SGA7]: \egroup\setbox1=\hbox\bgroup\smc P. Deligne, N. M. Katz \egroup\setbox2=\hbox\bgroup Groupes de monodromie en
g{\'e}om{\'e}trie alg{\'e}brique \sln340
\[S-K]:\egroup\jour@true\setbox3=\hbox\bgroup\sl Toh{\^o}ku Math. Jour. \egroup\setbox1=\hbox\bgroup\smc T. Shioda, T. Katsura \egroup\setbox5=\hbox\bgroup 1979
\vol31\egroup\setbox2=\hbox\bgroup On Fermat varieties \egroup\pages@true\setbox6=\hbox\bgroup 97-115
\[S-I]: \egroup\setbox1=\hbox\bgroup\smc T. Shioda, H. Inose \egroup\setbox2=\hbox\bgroup On singular K3-surfaces \egroup\inbook@true\setbox7=\hbox\bgroup
Complex analysis \& algebraic geometry \egroup\setbox3=\hbox\bgroup Cambridge Univ. Press
\egroup\setbox4=\hbox\bgroup Cambridge \yr1977\egroup\pages@true\setbox6=\hbox\bgroup 117--136
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{bibliography}
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{section}
\egroup\end}\catcode`\^^M=\active\catcode`\ =\active\tt{document}
\vfill \supereject \@nd
|
1996-03-05T06:22:21 | 9512 | alg-geom/9512013 | fr | https://arxiv.org/abs/alg-geom/9512013 | [
"alg-geom",
"math.AG"
] | alg-geom/9512013 | Laurent Bonavero | Laurent Bonavero | In\'egalit\'es de Morse et vari\'et\'es de Moishezon | Latex (plain). PhD-thesis of the author, defended in Grenoble
(France) 18-12-1995, hard copies available on request at
[email protected] http://www-fourier.ujf-grenoble.fr/THESE/ | null | null | null | null | The central topic of this thesis is the study of some properties of a class
of complex compact manifolds~: Moishezon manifolds. In the first part, we
generalize J.-P. Demailly's holomorphic Morse inequalities to the case of a
line bundle equipped with a metric with analytic singularities on an arbitrary
compact complex manifold. Our inequalities give an estimate of the cohomology
groups with values in the line bundle tensor powers twisted by the
corresponding sequence of multiplier ideal sheaves introduced by Nadel. As a
consequence, we obtain a necessary and sufficient analytic condition, invariant
by bimeromorphism, for a manifold to be Moishezon. In the second part, we use
Mori theory to analyze the structure of Moishezon manifolds with infinite
cyclic Picard group, with big canonical bundle, and which become projective
after one single blow-up with smooth center. We study the dimension and the
structure of the center of the blow-up. In dimension four, we show that this
locus is always a surface, and when the canonical bundle
| [
{
"version": "v1",
"created": "Wed, 20 Dec 1995 10:49:04 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Bonavero",
"Laurent",
""
]
] | alg-geom | \chapter*{Introduction}
Le sujet central de cette th\`ese est l'\'etude
de certaines propri\'et\'es d'une classe
de vari\'et\'es analytiques complexes compactes~: les
vari\'et\'es de Moishezon. Ces derni\`eres sont particuli\`erement
int\'eressantes car elles forment la plus petite classe de vari\'et\'es
complexes stable par application bim\'eromorphe et contenant les
vari\'et\'es projectives. Il est bien connu depuis K.\ Kodaira
que les vari\'et\'es projectives sont caract\'eris\'ees par
l'existence d'un fibr\'e en droites ample ou de
fa\c con \'equivalente d'un fibr\'e en droites
muni d'une m\'etrique hermitienne \`a courbure
strictement positive. Le fil conducteur de cette th\`ese
est l'\'etude de l'existence ou de
l'inexistence de fibr\'es en droites v\'erifiant
des propri\'et\'es de positivit\'e faible sur les vari\'et\'es de
Moishezon.
L'\'etude que nous
avons faite est divis\'ee en deux parties~: un point de vue
analytique suivant et g\'en\'eralisant une d\'emarche pr\'esente dans
certains travaux de J.-P.\ Demailly et Y.-T.\ Siu, et
un point de vue plus alg\'ebrique reposant sur l'utilisation de
la th\'eorie de Mori, d\'emarche pr\'esente dans certains travaux
de J.\ Koll\'ar et T.\ Peternell.
\bigskip
\bigskip
\noindent{\bf \'Etude analytique}
\medskip
Cette \'etude a d\'emarr\'e avec les travaux de J.-P.\ Demailly et
Y.-T.\ Siu qui,
r\'epondant \`a une conjecture de H.\ Grauert
et O.\ Riemenschneider, ont donn\'e ind\'ependamment des conditions analytiques
suffisantes (existence
de fibr\'es en droites \`a courbure
semi-positive et g\'en\'eriquement positive)
pour qu'une vari\'et\'e complexe compacte soit de Moishezon.
Une de nos motivations vient du fait qu'aucune de ces conditions n'est
n\'ecessaire, comme le montre l'\'etude de constructions r\'ecentes.
L'un des premiers r\'esultats de cette th\`ese est de donner une
caract\'erisation analytique des vari\'et\'es de Moishezon.
Pour cela, nous montrons, et c'est le th\'eor\`eme principal
de la premi\`ere partie de notre travail, que les
in\'egalit\'es de Morse holomorphes de J.-P.\ Demailly
se g\'en\'eralisent au cas d'un fibr\'e en droites $E$ muni
d'une m\'etrique singuli\`ere $h$ au dessus d'une
vari\'et\'e complexe compacte $X$.
Nos in\'egalit\'es donnent une estimation
asymptotique de la dimension des groupes de cohomologie
\`a valeurs dans les puissances tensorielles $E^{\otimes k}$,
tordues
par une suite de faisceaux d'id\'eaux $ {\cal I}_{k} (h)$
naturellement associ\'ee
aux singularit\'es de la m\'etrique $h$~: la suite
des faisceaux d'id\'eaux multiplicateurs de Nadel.
La pr\'esence de ces faisceaux d'id\'eaux constitue le ph\'enom\`ene
nouveau par rapport au cas o\`u la m\'etrique est lisse.
Comme dans ce dernier cas, l'estimation fait intervenir des
int\'egrales de
la courbure $\Theta(E)$.
Notre r\'esultat est le suivant~:
\bigskip
\noindent {\bf Th\'eor\`eme } {\em Si la m\'etrique $h$
a des singularit\'es analytiques, alors
pour tout fibr\'e $F$ de rang $r$ et pour
tout $q$ compris entre
$0$ et $n = \dim (X)$, on a~:
$$ \sum _{j=0}^{q}(-1)^{q-j} \dim H^{j}(X,{\cal O}(E^{k}\otimes F) \otimes
{\cal I}_{k}(h)) \leq r\frac{k^{n}}{n!} \int _{X(\leq q,E)} (-1)^{q} \Theta
(E)^{n} + o(k^{n}) $$
(avec \'egalit\'e si $q=n$), o\`u $X(\leq q,E)$ d\'esigne l'ouvert
de $X$ des points lisses de la m\'etrique d'indice inf\'erieur \`a $q$.
}
\bigskip
Ce r\'esultat est \`a mettre en parall\`ele avec la g\'en\'eralisation
du th\'eor\`eme de Ka\-wa\-ma\-ta-Viehweg donn\'ee par A.\ Nadel.
Nous montrons ensuite, g\'en\'eralisant un r\'esultat de
S.\ Ji et B.\ Shiffman obtenu ind\'ependamment
et simultan\'ement au n\^otre, que les crit\`eres de J.-P.\ Demailly et
Y.-T.\ Siu deviennent,
dans ce cadre plus souple, n\'ecessaires et
suffisants. Donnons par exemple le~:
\bigskip
\noindent {\bf Th\'eor\`eme } {\em Une vari\'et\'e compacte $X$ de
dimension $n$
est de Moishezon si et
seulement s'il existe sur $X$ un courant ferm\'e $T$ de bi-degr\'e $(1,1)$
tel que~:
(i) $ \displaystyle{\{ T \} \in H^{2}(X,{\Bbb Z}) }$,
(ii) $\displaystyle{ T= \frac{i}{\pi} \partial \overline{\partial} \varphi
+ \alpha }$,
o\`u $\varphi$ est une fonction r\'eelle \`a singularit\'es analytiques
et o\`u
$\alpha$ est un repr\'esentant
$ {\cal C}^{\infty}$ de $\{ T \}$,
(iii) $\displaystyle{\int_{X(\leq 1,T)} T^{n} > 0}$ o\`u
l'int\'egrale est prise
sur les points lisses du courant $T$.
}
\bigskip
Comme nous l'avons d\'ej\`a mentionn\'e, ce type de crit\`ere
a la propri\'et\'e d'\^etre invariant par morphisme bim\'eromorphe.
\bigskip
\bigskip
\noindent{\bf \'Etude alg\'ebrique}
\medskip
La deuxi\`eme partie de cette th\`ese consiste \`a \'etudier
en d\'etail la classe des vari\'et\'es de Moishezon dont le groupe
de Picard est ${\Bbb Z}$, et dont le fibr\'e canonique $K_X$
est gros (``big").
Une de nos motivations provient d'un r\'esultat de J.\ Koll\'ar affirmant
qu'en dimension $3$, et sous les hypoth\`eses pr\'ec\'edentes, le
fibr\'e canonique est alors num\'eriquement effectif (nef).
Il n'est donc pas possible de trouver dans cette classe des
vari\'et\'es de Moishezon de dimension $3$ ne v\'erifiant pas
les crit\`eres de J.-P.\ Demailly et Y.-T.\ Siu.
Nous montrons que ceci n'est plus vrai en dimension sup\'erieure
ou \'egale \`a $4$ en construisant explicitement des exemples~:
\bigskip
\noindent {\bf Th\'eor\`eme }
{\em Pour tout entier $n$ sup\'erieur
ou \'egal \`a $4$, il existe des vari\'et\'es de Moishezon $X$,
non projectives,
de dimension $n$ v\'erifiant~:
(i) $\operatorname{Pic} (X) = {\Bbb Z}$, (ii) $K_X$ est gros, (iii) $K_X$ n'est pas nef.}
\bigskip
La construction donnant ce r\'esultat montre que les vari\'et\'es
$X$ obtenues deviennent projectives apr\`es un \'eclatement
le long d'une sous-vari\'et\'e isomorphe \`a $\displaystyle{{\Bbb P} ^{n-2}}$.
Plus g\'en\'eralement, un r\'esultat fondamental de B.\ Moishezon
affirme qu'une vari\'et\'e
de Moishezon peut \^etre rendue projective apr\`es une succession
finie d'\'eclatements le long de sous-vari\'et\'es lisses. Ce r\'esultat
difficile ne donne malheureusement aucune indication quant au choix explicite
des sous-vari\'et\'es en question.
Gr\^ace \`a l'utilisation de la c\'el\`ebre th\'eorie de Mori
sur un mod\`ele projectif, nous avons \'etudi\'e le centre
de l'\'eclatement en toutes dimensions~:
\bigskip
\noindent {\bf Th\'eor\`eme }
{\em Soit $X$ une vari\'et\'e
de Moishezon (non projective) de dimension $n$, avec $\operatorname{Pic} (X) = {\Bbb Z}$
et $K_X$ gros.
Supposons de plus que $X$ est rendue projective apr\`es \'eclatement
le long d'une sous-vari\'et\'e $Y$ lisse.
\noindent Alors, si $K_X$ n'est pas nef, on a
$\displaystyle{ \dim Y > \frac{n-1}{2}.}$
}
\bigskip
En dimension
$4$, ce r\'esultat peut \^etre pr\'ecis\'e, y compris dans
le cas o\`u le fibr\'e canonique est nef~:
\bigskip
\noindent {\bf Th\'eor\`eme }
{\em Soit $X$ une vari\'et\'e
de Moishezon (non projective) de dimension $4$, avec $\operatorname{Pic} (X) = {\Bbb Z}$
et $K_X$ gros.
Supposons de plus que $X$ est rendue projective apr\`es \'eclatement
le long d'une sous-vari\'et\'e $Y$ lisse.
\noindent Alors $Y$ est n\'ecessairement
une surface.
Autrement dit, et dans cette situation particuli\`ere,
il ne suffit pas d'\'eclater une courbe pour ``rentrer dans
le monde projectif".}
\bigskip
Nous avons vu pr\'ec\'edemment que $K_X$ n'est pas n\'ecessairement
nef \`a partir de la dimension $4$. Le r\'esultat suivant montre
que l'exemple que nous avons construit
est, en un sens, le seul possible en dimension $4$ dans le cas
o\`u $K_X$ n'est pas nef.
\bigskip
\noindent {\bf Th\'eor\`eme }
{\em Sous les hypoth\`eses pr\'ec\'edentes et si $K_{X}$ n'est
pas nef, alors le couple $\displaystyle{(Y,N_{Y/X})}$ est isomorphe \`a
$\displaystyle{({\Bbb P} ^2, {\cal O}_{{\Bbb P} ^{2}}(-1)^{\oplus 2})}$.}
\bigskip
Ces r\'esultats pr\'ecis sont accessibles en dimension $4$ car les
contractions de Mori ont \'et\'e \'etudi\'ees par T.\ Ando, Y.\ Kawamata
et M.\ Beltrametti.
Nos r\'esultats peuvent \^etre vus comme un premier pas vers
une analyse du caract\`ere non projectif des vari\'et\'es de Moishezon
de dimension sup\'erieure ou \'egale \`a $4$~; la situation
en dimension $3$ \'etant maintenant assez bien comprise suite aux
travaux de J.\ Koll\'ar et T.\ Peternell.
\bigskip
\bigskip
\noindent{\bf Plan du texte}
\medskip
- le chapitre 1 est un chapitre de pr\'eliminaires~; il contient
une description pr\'ecise des motivations et des objets utilis\'es
dans le reste de la th\`ese. Nous y d\'etaillons en particulier
une d\'emonstration du th\'eor\`eme de Siegel et donnons
les constructions de I.\ Nakamura et K.\ Oguiso montrant que les
crit\`eres analytiques de J.-P.\ Demailly et Y.-T.\ Siu
ne sont pas n\'ecessaires en g\'en\'eral.
\medskip
- le chapitre 2 est consacr\'e \`a l'\'etude analytique. Nous rappelons
les premiers r\'esultats li\'es aux m\'etriques singuli\`eres,
\'enon\c cons et d\'emontrons notre version des in\'egalit\'es
de Morse holomorphes. On en d\'eduit les caract\'erisations
analytiques
des vari\'et\'es de Moishezon.
Enfin, nous donnons une version alg\'ebrique singuli\`ere
des in\'egalit\'es de Morse.
\medskip
- le chapitre 3 est consacr\'e \`a l'\'etude alg\'ebrique.
Nous commen\c cons par rappeler le r\'esultat de J.\ Koll\'ar
en dimension $3$, puis nous faisons une \'etude des vari\'et\'es
de Moishezon \`a groupe de Picard ${\Bbb Z}$, \`a fibr\'e canonique
gros et devenant projectives apr\`es un seul \'eclatement de
centre lisse et projectif. Nous obtenons
une restriction sur la dimension du centre de l'\'eclatement.
En dimension $4$, cette restriction implique que ce dernier
est n\'ecessairement une surface.
Nous d\'ecrivons alors notre exemple et montrons qu'en dimension
$4$, cette construction est essentiellement la seule dans le cas
o\`u le fibr\'e canonique n'est pas nef.
\medskip
Mentionnons que les chapitres 2 et 3 sont dans une large
mesure ind\'ependants et peuvent \^etre lus dans un ordre
quelconque.
\chapter{Pr\'eliminaires}
Ce chapitre a pour but d'introduire les principales
notions utilis\'ees par la suite, de pr\'esenter
les premi\`eres motivations en d\'etail et de rappeler
un certain nombre de r\'esultats auxquels nous nous r\'ef\'erons
dans les chapitres suivants.
\section{\! Quelques rappels de g\'eom\'etrie
analytique complexe}
\subsection{Vari\'et\'es, fibr\'es vectoriels}
Pr\'ecisons tout d'abord que dans toute cette th\`ese,
et sauf mention explicite
du contraire, le mot {\bf vari\'et\'e} sera utilis\'e pour d\'esigner
une {\bf vari\'et\'e analytique
complexe {\em non singuli\`ere}} suppos\'ee de plus connexe.
Pour toutes les notions introduites ici, nous renvoyons
de fa\c con g\'en\'erale \`a \cite{G-H78}.
\medskip
Un {\bf fibr\'e vectoriel} complexe $F$ au dessus d'une vari\'et\'e
$X$ est la donn\'ee d'une vari\'et\'e $F$
et d'une application $\displaystyle{\pi : F \to X}$ de sorte
qu'il existe un recouvrement de $X$ par des
ouverts trivialisants $U_{\alpha}$ et des isomorphismes
(appel\'es trivialisations)
$$\theta _{\alpha} : \pi ^{-1}(U_{\alpha}) \to U_{\alpha} \times {\Bbb C} ^r$$
respectant la structure d'espace vectoriel des fibres, i.e
$$\theta _{\alpha \beta}(x,\xi)
:= \theta _{\alpha} \circ \theta _{\beta} ^{-1}
(x,\xi) = (x, g_{\alpha \beta}(x)\xi )$$ o\`u
$g_{\alpha \beta}$ est une application holomorphe
sur $U_{\alpha} \cap U_{\beta}$ \`a valeurs dans le
groupe des matrices complexes inversibles de taille $r$.
Nous notons $\xi _x$ un point de $F$ au dessus
du point $x$ de $X$ (i.e tel que $\pi (\xi _x) =x$).
L'entier $r$ est le {\bf rang} du fibr\'e $F$.
Si $r = 1$, on parle de {\bf fibr\'e en droites}.
Un exemple important de fibr\'e en droites
est le fibr\'e ${\cal O}(D)$
associ\'e \`a un diviseur $D$ sur $X$~: si $D$ est
un diviseur irr\'eductible donn\'e sur $U_{\alpha}$
par l'\'equation $f_{\alpha}=0$, le fibr\'e
${\cal O}(D)$ est le fibr\'e associ\'e au cocycle
$$g_{\alpha \beta} = \frac{f_{\alpha}}{f_{\beta}} \ .$$
\medskip
Toutes les constructions d'alg\`ebre lin\'eaire
s'\'etendent aux fibr\'es~: dual,
produit tensoriel, produit ext\'erieur.
Ainsi, un exemple fondamental de fibr\'e
en droites
sur une vari\'et\'e $X$ de dimension $n$ est le
{\bf fibr\'e canonique} d\'efini par
$$K_X := \det (T^{\ast}X) = \bigwedge ^n T^{\ast}X,$$ o\`u $T^{\ast}X$
d\'esigne le fibr\'e cotangent, dual du fibr\'e
tangent holomorphe $TX$ de $X$.
Un autre exemple important de fibr\'e en droites
est le fibr\'e ${\cal O}_{{\Bbb P} ^n}(1)$ sur l'espace
projectif ${\Bbb P} ^n$~: en associant \`a un point $[x]$
de ${\Bbb P} ^n$ la droite ${\Bbb C} x$, on construit
un sous-fibr\'e en droites du fibr\'e trivial ${\Bbb P} ^n \times {\Bbb C} ^{n+1}$~;
le dual de ce fibr\'e en droites est par d\'efinition
le fibr\'e ${\cal O}_{{\Bbb P} ^n}(1)$.
L'ensemble des
fibr\'es en droites, modulo isomorphisme, sur
une vari\'et\'e $X$ est naturellement muni
d'une structure de groupe pour le produit
tensoriel~: on l'appelle {\bf groupe de Picard}
de $X$ et on le note $\operatorname{Pic} (X)$. Mentionnons ici
que nous identifions suivant l'usage un fibr\'e
en droites $E$ au faisceau inversible
${\cal O}(E)$ des germes de sections holomorphes de $E$.
La $k$-i\`eme puissance tensorielle d'un fibr\'e
en droites $E$ sera not\'ee indiff\'eremment
$E^{\otimes k}$, $E^k$,${\cal O}(kE)$ ou m\^eme $kE$.
\medskip
Un fibr\'e vectoriel $E$ peut \^etre muni d'une m\'etrique
hermitienne ${\cal C}^{\infty}$, on parle
alors de fibr\'e vectoriel {\bf hermitien} et on note
g\'en\'eralement $h$ une telle m\'etrique~: elle correspond
\`a la donn\'ee d'une forme hermitienne sur chaque
fibre $E_x$ de $E$, d\'ependant de fa\c con ${\cal C}^{\infty}$
de $x$.
Dans le cas particulier d'un fibr\'e en droites,
une m\'etrique hermitienne est
donn\'ee localement
sur un ouvert trivialisant $U_{\alpha}$
par
$$h(\xi _{x}) = ||\xi _x||_h := |\xi|\exp(-\varphi _{\alpha} (x))$$
(la fonction $\exp(-\varphi _{\alpha})$
est appel\'ee {\bf poids} de la m\'etrique $h$ dans la trivialisation
$\theta _{\alpha}$)
o\`u la fonction r\'eelle $\varphi _{\alpha}$ est
de classe ${\cal C}^{\infty}$ sur $U_{\alpha}$.
Lorsque le fibr\'e tangent $TX$ est muni d'une m\'etrique hermitienne,
on dit que la vari\'et\'e $X$ est hermitienne.
Comme il est d'usage, nous identifions toujours
la donn\'ee d'une m\'etrique hermitienne sur une vari\'et\'e
$X$ \`a celle de la $(1,1)$ forme r\'eelle, g\'en\'eralement
not\'ee $\omega$, qui lui est naturellement associ\'ee
($\omega$ est \`a un facteur $-2$ pr\`es la partie imaginaire de la
m\'etrique). Ainsi, une vari\'et\'e est {\bf k\"ahl\'erienne}
si elle poss\`ede une m\'etrique hermitienne pour laquelle $\omega$
est une forme ferm\'ee.
Une vari\'et\'e {\bf projective} est une vari\'et\'e isomorphe
\`a une sous-vari\'et\'e ferm\'ee d'un espace projectif ${\Bbb P} ^N$.
\medskip
Pour un fibr\'e en droites hermitien $(E,h)$, on note $\Theta (E)$
la $(1,1)$ {\bf forme de courbure} de $(E,h)$~: c'est la forme
r\'eelle d\'efinie globalement sur $X$ et
donn\'ee localement par
$$\displaystyle{ \Theta (E)=
\frac{i}{\pi} \partial \overline{\partial} \varphi _{\alpha}};$$
c'est aussi la courbure de
la connexion de Chern du fibr\'e hermitien $E$.
La classe de cohomologie de $\Theta (E)$
appartient \`a $H^2(X,{\Bbb Z})$ et ne d\'epend
pas de la m\'etrique $h$~; c'est la premi\`ere classe
de Chern de $E$ et elle est not\'ee $c_1(E)$. Remarquons
qu'il n'y a pas de sens \`a parler des valeurs propres
de la forme de courbure, mais que la {\bf signature} de la courbure
(i.e le nombre de ``valeurs propres" nulles, strictement positives
et strictement n\'egatives) est une notion bien d\'efinie sans donn\'ee
suppl\'ementaire.
Par exemple, le fibr\'e ${\cal O}_{{\Bbb P} ^n}(1)$
muni de la m\'etrique induite de celle de ${\Bbb C} ^{n+1}$
est \`a courbure strictement positive~: la forme de courbure
est la m\'etrique de Fubini-Study de ${\Bbb P} ^n$.
\subsection{Th\'eor\`eme de Kodaira}
Nous sommes en mesure d'\'enoncer maintenant le c\'el\`ebre th\'eor\`eme
de plongement de Kodaira \cite{Kod54}~:
\bigskip
\noindent{\bf Th\'eor\`eme (K.\ Kodaira, 1954)}
{\em Une vari\'et\'e compacte $X$ est projective si et seulement si
elle poss\`ede un fibr\'e en droites hermitien $E$ \`a courbure
strictement positive.}
\bigskip
Signalons \'evidemment qu'un sens est ais\'e~: si une
vari\'et\'e est projective, la restriction \`a $X$ du fibr\'e
$\displaystyle{ {\cal O}_{{\Bbb P} ^N} (1) }$ muni
de sa m\'etrique naturelle ayant pour courbure la forme
de Fubini-Study de ${\Bbb P} ^N$ donne
le fibr\'e souhait\'e. L'autre sens consiste \`a montrer
que pour $k$ entier assez grand, il est possible de plonger
$X$ dans l'espace projectif des hyperplans de $H^0(X,E^{\otimes k})$,
o\`u $H^0(X,E^{\otimes k})$
d\'esigne l'espace vectoriel des sections holomorphes globales
de $E^{\otimes k}$.
Rappelons ici qu'un fibr\'e en droites pouvant \^etre
muni d'une m\'etrique \`a courbure strictement positive
est dit {\bf ample}.
C'est le th\'eor\`eme de Kodaira que H.\ Grauert et
O.\ Riemenschneider \cite{GrR70} se
proposaient de g\'en\'eraliser aux vari\'et\'es de Moishezon,
vari\'et\'es que nous introduisons dans le paragraphe suivant.
\section{Quelques rappels sur les vari\'et\'es de Moishezon}
Les vari\'et\'es de Moishezon sont, parmi les vari\'et\'es
compactes,
celles qui poss\`edent le
``plus" de fonctions m\'eromorphes alg\'ebriquement
ind\'ependantes. Cette d\'efinition heuristique est justifi\'ee
par le th\'eor\`eme de Siegel que nous rappelons maintenant.
\subsection{Th\'eor\`eme de Siegel}
En 1955, C.L.\ Siegel d\'emontre le r\'esultat suivant \cite{Sie55}~:
\bigskip
\noindent{\bf Th\'eor\`eme (C.L.\ Siegel, 1955) }
{\em Si $X$ est une vari\'et\'e
compacte
de dimension $n$, alors $X$ poss\`ede au plus $n$ fonctions
m\'eromorphes alg\'ebriquement ind\'ependantes.}
\bigskip
La d\'emonstration originale de C.L.\ Siegel repose sur une
application \'el\'ementaire du
lemme de Schwarz. Il existe maintenant plusieurs d\'emonstrations
diff\'erentes, certaines g\'en\'eralisant cet \'enonc\'e
aux espaces complexes compacts.
Nous en donnons ici une preuve ``moderne" dans le cas
non singulier. Pour cela, nous utilisons un
r\'esultat de P.\ Gauduchon \cite{Gau77}~: {\em toute vari\'et\'e
analytique
complexe compacte
de dimension $n$ poss\`ede une m\'etrique hermitienne
$\omega$ de classe
${\cal C}^{\infty}$ et d'excentricit\'e nulle, i.e
telle que $\displaystyle{ \partial \overline{\partial} (\omega ^{n-1}) =0}$.}
Commen\c cons par montrer le lemme suivant~:
\medskip
\noindent{\bf Lemme} {\em Soit $X$ une vari\'et\'e compacte
que l'on munit d'une m\'etrique de Gauduchon $\omega$
et soit $x_0$ un point de $X$.
Alors, il
existe une constante $C := C(X,x_0,\omega )$ telle que
pour tout fibr\'e en droites hermitien $(E,h)$ au dessus de $X$
et pour toute section holomorphe $s$ de $E$
non identiquement nulle, on ait~:
$$ \operatorname{mult} \, (s,x_0) \leq C \int _X \omega ^{n-1} \wedge \Theta (E).$$}
\medskip
\noindent {\bf D\'emonstration}
Soit $r$ un r\'eel strictement positif fix\'e ``petit"
(de sorte qu'il existe une carte centr\'ee en $x_0$
et contenant la boule $B(x_o,r)$).
Alors, la multiplicit\'e de $s$ en $x_0$ v\'erifie~:
$$ \operatorname{mult} \, (s,x_0) \leq
\frac{ \operatorname{vol} _{n-1}(Z_s \cap B(x_o,r))}{\operatorname{vol} _{n-1}(B_{n-1}(x_o,r))} +o(r) $$
o\`u $\operatorname{vol} _{n-1}$ est le volume $(n-1)$-dimensionnel
mesur\'e avec la m\'etrique $\omega$, et o\`u
$Z_s$ d\'esigne le lieu des z\'eros de $s$. En effet,
la multiplicit\'e de $s$ en $x_0$ est en fait \'egale
\`a la limite d\'ecroissante lorsque $r$ tend vers $0$
de la quantit\'e du membre de droite de l'in\'egalit\'e.
Comme $X$ est compacte, on a {\em a fortiori}~:
$$ \operatorname{mult} \, (s,x_0) \leq C \operatorname{vol} _{n-1}(Z_s)
= C \int_{Z_s} \omega ^{n-1}.$$
Mais l'\'equation de Lelong-Poincar\'e affirme que~:
$$ \frac{i}{\pi}\partial \overline{\partial}\log ||s||
= [ Z_s ] - \Theta (E),$$
o\`u $[ Z_s ]$ d\'esigne le courant d'int\'egration sur
l'ensemble $Z_s$.
Comme $\omega ^{n-1}$ est $\partial \overline{\partial}$-ferm\'ee,
la formule de Stokes donne de suite~:
$$ \int_{Z_s} \omega ^{n-1} = \int _X \omega ^{n-1} \wedge \Theta (E),$$
ce qui prouve le lemme.\hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
Ce lemme implique le r\'esultat suivant~:
\medskip
\noindent{\bf Proposition} {\em Soient $X$ une vari\'et\'e compacte
de dimension $n$, $\omega$ une m\'etrique de Gauduchon sur $X$
et $E$ un fibr\'e en droites hermitien au dessus de $X$.
Alors~:
(i) $\dim H^0(X,E) \leq {n+p \choose n}$ o\`u $p$
est la partie enti\`ere
de $\displaystyle{C \int _X \omega ^{n-1} \wedge \Theta (E)}$
et o\`u $C$ est la constante du lemme pr\'ec\'edent,
(ii) $\displaystyle{\dim H^0(X,E^{\otimes k}) \leq
A \left(\int _X \omega ^{n-1} \wedge \Theta (E)\right)^n k^n + o(k^n)}$,
o\`u $A$ est une constante ind\'ependante de $E$ et $k$.
}
\medskip
\noindent {\bf D\'emonstration}
Il suffit de remarquer que
${n+p \choose n}$ est la dimension de l'espace vectoriel
des polyn\^omes
de $n$ variables et
de degr\'e inf\'erieur ou \'egal \`a $p$~: le lemme implique
en effet que l'application lin\'eaire qui \`a une section
holomorphe de $E$ associe son $p$-i\`eme jet en $x_0$
est injective, d'o\`u le point (i). Le point (ii) est cons\'equence du
fait que la forme de courbure
du fibr\'e $(E^{\otimes k},h^k)$ est donn\'ee par
$\displaystyle{ \Theta (E^{\otimes k}) = k \Theta (E)}$.
On applique alors (i) au fibr\'e $(E^{\otimes k},h^k)$.\hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
Remarquons que la proposition
pr\'ec\'edente affirme que la dimension de
l'espace vectoriel des sections holomorphes des
puissances $E^{\otimes k}$ d'un fibr\'e en droites sur une vari\'et\'e
compacte de dimension $n$ cro\^\i t au plus comme $k^n$.
Ce fait est bien classique et nous avons estim\'e
la dimension en fonction d'int\'egrales de courbure. Des estimations
bien plus pr\'ecises, valables pour la dimension de tous les groupes de
cohomologie seront donn\'ees par les in\'egalit\'es de Morse holomorphes
de J.-P.\ Demailly dans le paragraphe suivant.
\medskip
\noindent {\bf D\'emonstration du th\'eor\`eme de Siegel}
L'argument est standard~: soient
$\displaystyle{(f_{i})_{1\leq i\leq N}}$ $N$ fonctions m\'eromorphes
alg\'ebriquement
ind\'ependantes sur $X$.
Notons $D$ la somme des diviseurs des p\^oles des $f_{i}$ et
${\cal O}(D)$ le fibr\'e en droites associ\'e.
Rappelons que
$H^{0}(X,{\cal O}(D))$ est isomorphe \`a l'espace vectoriel
des fonctions m\'eromorphes sur $X$ v\'erifiant
$\operatorname{div} (f) + D \geq 0$. Alors,
si $P$ est un polyn\^ome \`a coefficients complexes
en $N$ variables, de degr\'e total inf\'erieur ou \'egal
\`a $k$, la fonction m\'eromorphe $P(f_{1},\ldots,f_{N})$ est une
section holomorphe
de ${\cal O}(kD)$, et comme les $f_{i}$ sont
alg\'ebriquement ind\'ependantes, on a ${N+k \choose N}$
telles sections
lin\'eairement ind\'ependantes. De l\`a~:
$$ \dim H^{0}(X,{\cal O}(D)^{\otimes k})
\geq {N+k \choose N} \sim_{k \to +\infty} \frac{k^N}{N!}.$$
Avec la proposition, il vient $N \leq n$.\hskip 3pt \vrule height6pt width6pt depth 0pt
\subsection{\'Eclatements et vari\'et\'es de Moishezon}
Nous commen\c cons ce paragraphe par quelques rappels
sur les \'eclatements. Ces derniers jouent un r\^ole important
dans la th\'eorie des vari\'et\'es de Moishezon et la
construction d'exemples explicites.
\subsub{\'Eclatements}
Une r\'ef\'erence standard est \`a nouveau \cite{G-H78}.
\noindent Si $X$ est une vari\'et\'e, et $Y$ une sous-vari\'et\'e
de $X$ de codimension sup\'erieure ou
\'egale \`a $2$, on construit une vari\'et\'e $\tilde{X}$
appel\'ee {\bf \'eclatement de $X$ le long de $Y$}
en rempla\c cant les points de $Y$
par l'espace des directions normales \`a $Y$ dans $X$.
On note g\'en\'eralement $\pi : \tilde{X} \to X$
l'\'eclatement et $E := \pi ^{-1}(Y)$ le
{\bf diviseur exceptionnel}. La sous-vari\'et\'e $Y$
est appel\'ee {\bf centre de l'\'eclatement}. Par construction,
$\pi$ induit un isomorphisme
$$\pi _{| \tilde{X} \backslash E } : \tilde{X} \backslash E
\to X \backslash Y.$$
\noindent Signalons que le centre de l'\'eclatement
$Y$ peut \^etre
r\'eduit \`a un point.
La restriction de $\pi$ au diviseur exceptionnel $E$
munit $E$ d'une structure
de fibr\'e en espaces projectifs au dessus de $Y$~:
plus pr\'ecis\'ement, $E$ est isomorphe \`a
$\displaystyle{{\Bbb P} (N^{\ast}_{Y/X})}$ (projectivis\'e
en droites du fibr\'e normal $N_{Y/X}$
suivant la convention de Grothendieck). De plus, le
fibr\'e normal
$\displaystyle{N_{E/\tilde{X}} = {\cal O}(E)_{| E}}$ est
isomorphe au fibr\'e
${\cal O}_{{\Bbb P} (N_{Y/X}^{\ast})}(-1)$.
Mentionnons aussi que le groupe de Picard
de $\tilde{X}$ est \'egal \`a
$\pi ^*\operatorname{Pic} (X) \oplus {\Bbb Z} \cdot {\cal O}(E)$. Par exemple,
le fibr\'e canonique est donn\'e par
$$K_{\tilde{X}} = \pi ^{\ast}K_X + (r-1){\cal O}(E)$$ o\`u
$r$ est la codimension du centre de l'\'eclatement.
Enfin, si $Z$ est une sous-vari\'et\'e de $X$,
non incluse dans le centre de l'\'eclatement $Y$,
alors l'adh\'erence de $\pi ^{-1}(Z \cap (X \backslash Y))$
dans $\tilde{X}$ est appel\'ee {\bf transform\'ee stricte}
de $Z$. Si $Z'$ est la transform\'ee stricte de $Z$, alors
$$\pi _{|Z'}~: Z' \to Z$$
\noindent est l'\'eclatement de $Z \cap Y$ dans $Y$.
\medskip
Le r\'esultat suivant, d\^u \`a A.\ Fujiki et S.\ Nakano \cite{FuN72}
donne un crit\`ere pour qu'un diviseur soit
le diviseur exceptionnel d'un \'eclatement~; ce
crit\`ere est une extension du crit\`ere
de Castelnuovo sur les surfaces. Nous l'utiliserons
souvent lors de la construction d'exemples explicites de
vari\'et\'es de Moishezon non projectives.
\bigskip
{\bf Th\'eor\`eme (A.\ Fujiki, S.\ Nakano, 1972) }
{\em Soit $Z$ une vari\'et\'e et $D$ une sous-vari\'et\'e
de $Z$ de codimension $1$. On suppose que
$D$ est isomorphe \`a ${\Bbb P} (G)$
o\`u $G$ est un fibr\'e vectoriel sur une vari\'et\'e
$Y$ ; on note $p : {\Bbb P} (G) \to Y$ la projection.
On suppose enfin que $N_{D/Z}$ est
isomorphe \`a ${\cal O}_{{\Bbb P} (G)}(-1)$.
Alors, il existe une vari\'et\'e $Z'$ contenant
$Y$ comme sous-vari\'et\'e et une application $\pi : Z \to Z'$
de sorte
que $\pi$ soit l'\'eclatement de $Z'$ le long de $Y$ et
que la restriction de $\pi$ \`a $D$ soit \'egale
\`a $p$.}
\bigskip
Notons qu'au vu de ce qui pr\'ec\`ede, les hypoth\`eses
faites sur le diviseur $D$ dans cet \'enonc\'e sont
\'evidemment n\'ecessaires~: ce crit\`ere remarquable
montre qu'elles sont suffisantes.
\subsub{Vari\'et\'es de Moishezon}
Le th\'eor\`eme de Siegel
motive les d\'efinitions suivantes~:
\medskip
\noindent {\bf D\'efinition } {\em
Un fibr\'e en droites sur une vari\'et\'e
compacte de dimension $n$ est dit {\bf gros} ({\bf big}
en anglais) si la dimension de
l'espace vectoriel des sections holomorphes globales de ses
puissances $E^{\otimes k}$ cro\^\i t exactement comme $k^n$.}
\medskip
\noindent {\bf Remarque }
De fa\c con g\'en\'erale, pour un fibr\'e en droites
$E$ sur une vari\'et\'e compacte $X$, il existe
un entier $\kappa (E)$ tel que la dimension de
$H^0(X,E^k)$ cro\^\i t comme $\displaystyle{k^{ \kappa (E)}}$.
Cet entier (\'egal \`a $-\infty$ si tous les $H^0(X,E^k)$ sont nuls)
est appel\'e {\bf dimension de Kodaira-Iitaka}. Cet entier
est inf\'erieur ou \'egal \`a la dimension de $X$, et selon
ce qui pr\'ec\`ede, le fibr\'e $E$ est gros si et seulement si
$\kappa (E) = \dim X$.
\medskip
\noindent {\bf D\'efinition } {\em Une vari\'et\'e
compacte de dimension $n$ est {\bf de Moishezon} si elle
poss\`ede exactement $n$ fonctions m\'eromorphes alg\'ebriquement
ind\'ependantes ou, de fa\c con \'equivalente, si elle
poss\`ede un fibr\'e en droites gros.}
\medskip
Les premiers exemples de vari\'et\'es de Moishezon sont les
vari\'et\'es projectives. En particulier, toutes les courbes
sont de Moishezon. En fait, le th\'eor\`eme suivant,
difficile et fondamental, montre
qu'une vari\'et\'e de Moishezon n'est pas tr\`es loin d'\^etre
projective. Ce r\'esultat est d\^u \`a B.\ Moishezon \cite{Moi67}~:
\bigskip
\noindent{\bf Th\'eor\`eme (B.\ Moishezon, 1967)}
{\em Une vari\'et\'e compacte est de Moishezon si
et seulement si elle peut
\^etre rendue projective apr\`es un nombre
fini d'\'eclatements de centres
lisses. On peut m\^eme choisir les centres projectifs.}
\bigskip
\noindent {\bf Remarque }
Mentionnons d\`es \`a pr\'esent que ce th\'eor\`eme ne
donne aucune m\'ethode pour choisir les sous-vari\'et\'es
le long desquelles il faut \'eclater. Ce probl\`eme figure
dans une liste de 100 probl\`emes ouverts en g\'eom\'etrie
\'etablie par S.\ T.\ Yau \cite{Yau93}. Nous donnerons
(modestement) quelques r\'eponses dans cette direction
dans la deuxi\`eme partie de cette th\`ese.
\medskip
Une cons\'equence du th\'eor\`eme pr\'ec\'edent est le
th\'eor\`eme de Chow-Kodaira~: {\em une surface
complexe compacte lisse est de Moishezon si et seulement
si elle est projective}. En effet, on ne peut qu'\'eclater
des points en dimension $2$. Or, de fa\c con g\'en\'erale,
si $X$ est une vari\'et\'e compacte et si $\tilde{X}$
est la vari\'et\'e $X$ \'eclat\'ee au point $x$, alors
$X$ est projective si et seulement si $\tilde{X}$ l'est
(voir par exemple \cite{Kle66}).
Ceci explique le fait que tous les exemples
de vari\'et\'es de Moishezon qui figurent dans notre travail
sont de dimension sup\'erieure ou
\'egale \`a $3$. Le premier exemple de vari\'et\'e de
Moishezon non projective a \'et\'e construit par
H.\ Hironaka (voir par exemple \cite{Har77}). Nous donnons deux constructions
dues \`a I.\ Nakamura et K.\ Oguiso \`a la fin de ces
pr\'eliminaires.
\section{Les in\'egalit\'es de Morse de J.-P.\ Demailly}
Nous rappelons ici les in\'egalit\'es de Morse
holomorphes de J.-P.\ Demailly~: on renvoie \`a \cite{Dem85}
pour la d\'emonstration de ce r\'esultat.
Dans ce qui suit, $X$ d\'esigne une vari\'et\'e
compacte
de dimension $n$ et $(E,h)$ un fibr\'e en droites hermitien sur $X$.
Au couple $(E,h)$,
on associe pour tout entier $q$ compris entre $0$ et $n$
l'ouvert $X(q,E)$
form\'e des points
$x$ de $X$ pour lesquels $\Theta (E)(x)$ poss\`ede
exactement $q$ valeurs propres strictement n\'egatives et $n-q$ valeurs
propres strictement positives ;
finalement on pose $X(\leq q,E) = X(0,E)\cup \cdots \cup X(q,E)$.
Les in\'egalit\'es
de Morse holomorphes donnent une estimation des groupes de cohomologie
\`a valeurs dans les puissances tensorielles $E^{\otimes k}$
en fonction d'int\'egrales de courbure sur $X$.
L'\'enonc\'e pr\'ecis est le suivant~:
\bigskip
\noindent{\bf Th\'eor\`eme (J.-P.\ Demailly, 1985)}
{\em Pour tout $q$ compris entre
$0$ et $n$, et si $F$ est un fibr\'e vectoriel holomorphe
de rang $r$ sur $X$, on a~:
(i) $\displaystyle{ \sum _{j=0}^{q}(-1)^{q-j} \dim H^{j}(X,E^{k}\otimes F)
\leq r\frac{k^{n}}{n!} \int _{X(\leq q,E)} (-1)^{q} \Theta
(E)^{n} + o(k^{n})}$
(avec \'e\-ga\-li\-t\'e si $q=n$),
(ii) $ \displaystyle{ \dim H^{q}(X,E^{k}\otimes F)
\leq r\frac{k^{n}}{n!} \int _{X(q,E)} (-1)^{q} \Theta (E)^{n}
+ o(k^{n}).} $}
\bigskip
Le point (i) est d\'esign\'e sous le nom plus pr\'ecis
d'in\'egalit\'es de Morse fortes, alors que le point (ii),
qui est une cons\'equence imm\'ediate de (i), est d\'esign\'e
sous le nom
d'in\'egalit\'es de Morse faibles.
Ces in\'egalit\'es ont de nombreuses applications dont certaines
tr\`es r\'ecentes~: elles sont utilis\'ees dans les travaux
de J.-P.\ Demailly et Y.-T.\ Siu en direction de la conjecture de Fujita
\cite{Dem94}, \cite{Siu94}.
En g\'en\'eral, ces in\'egalit\'es peuvent se substituer
aux th\'eor\`emes
d'annulation lorsque la signature de la forme de courbure
n'est pas constante. En particulier, J.-P.\ Demailly
les a utilis\'ees initialement pour renforcer
un r\'esultat de Y.T.\ Siu donnant un crit\`ere analytique
suffisant
pour qu'une vari\'et\'e soit de Moishezon. Ces r\'esultats
fournissent la solution \`a la conjecture de H.\ Grauert
et O.\ Riemenschneider \cite{GrR70}~:
\bigskip
\noindent{\bf Th\'eor\`eme (J.-P.\ Demailly, Y.T.\ Siu, 1985)}
{\em Une vari\'et\'e compacte $X$ est de Moishezon
d\`es que $X$ poss\`ede
un fibr\'e $E$ en droites
muni d'une m\'etrique hermitienne lisse dont la forme de courbure
$\Theta(E)$ v\'erifie l'une des conditions suivantes :
(i) $\Theta(E)$ est partout semi-positive et d\'efinie positive en au moins
un point (``crit\`ere de Siu"),
(ii) $\displaystyle{ \int_{X(\leq 1,E)} \Theta(E)^{n} > 0}$
(``crit\`ere de Demailly").
}
\bigskip
Les deux \'enonc\'es d\'ecoulent des in\'egalit\'es de Morse~:
elles impliquent dans les deux cas que le fibr\'e $E$ est gros.
Avant de commenter ce r\'esultat,
rappelons qu'un fibr\'e en droites
$E$
sur une vari\'et\'e compacte $X$ est dit {\bf num\'eriquement effectif}
(en abr\'eg\'e {\bf nef}) si pour toute m\'etrique hermitienne
$\omega$ sur
$X$ et pour tout $\varepsilon > 0$, le fibr\'e
$E$ poss\`ede une m\'etrique lisse $h_{\varepsilon}$ telle que
$\Theta_{h_{\varepsilon}}(E) \geq -\varepsilon\omega$. Cette notion
a \'et\'e introduite par J.-P.\ Demailly, T.\ Peternell et
M.\ Schneider \cite{DPS94} et admet une formulation
\'equivalente sur les vari\'et\'es projectives~: sur une vari\'et\'e
projective $X$,
un fibr\'e en droites est nef si et seulement si
son intersection avec toute courbe de $X$ est semi-positive.
Sur une vari\'et\'e quelconque, il peut ne pas y avoir
de courbes, cependant il y en a ``suffisamment" sur une vari\'et\'e
de Moishezon et
Mihai Paun a \'etendu le r\'esultat pr\'ec\'edent \`a
ces derni\`eres~: {\em sur une vari\'et\'e de Moishezon,
un fibr\'e en droites est nef si et seulement si
son intersection avec toute courbe est semi-positive} \cite{Pau95}.
\medskip
\noindent {\bf Remarque-exemple }
Un fibr\'e en droites satisfaisant
au crit\`ere de Siu est simultan\'ement gros et nef.
Par ailleurs, des exemples de vari\'et\'es de Moishezon
satisfaisant les crit\`eres (i) et (ii) sont donn\'es
par les vari\'et\'es de Moishezon qui admettent
un morphisme g\'en\'eriquement fini vers une vari\'et\'e
projective.
\section{Exemples explicites}
Nous donnons dans ce paragraphe deux constructions,
l'une utilis\'ee par I.\ Nakamura \cite{Nak87}
et J.\ Koll\'ar \cite{Kol91} et l'autre due \`a K.\ Oguiso.
Nous \'etudions en d\'etail la premi\`ere
et expliquons plus bri\`evement celle de K.\ Oguiso.
Ces constructions nous permettent de montrer
qu'aucun des deux crit\`eres analytiques
pr\'ec\'edents pour qu'une vari\'et\'e soit
de Moishezon n'est n\'ecessaire~: ces \'enonc\'es n'admettent
donc
pas de r\'eciproque dans le cadre des fibr\'es hermitiens
\`a m\'etrique lisse.
\subsection{La premi\`ere construction}
La construction qui suit exhibe une famille de
vari\'et\'es de dimension
$3$ complexe d\'epen\-dant d'un param\`etre entier $m$.
L'origine de cette construction n'est pas tr\`es claire~;
elle est utilis\'ee dans \cite{Nak87} et
mentionn\'ee
dans \cite{Kol91} \S5.
Une des motivations de J.\ Koll\'ar, lorsqu'il mentionne
cet exemple, est de construire une vari\'et\'e
de Moishezon, dont le groupe de Picard est ${\Bbb Z}$
et dont le g\'en\'erateur gros du groupe de Picard
est d'auto-intersection n\'egative. En particulier,
ce g\'en\'erateur n'est pas nef.
\subsub{Construction explicite}
La construction est tr\`es simple~: elle
consiste \`a \'eclater ${\Bbb P} ^3$ le long d'une
courbe contenue dans une quadrique, et \`a
contracter lorsque ceci est possible la transform\'ee
stricte de la quadrique sur une courbe rationnelle lisse.
Soit donc ${\cal Q} \subset {\Bbb P} ^{3}$ une quadrique lisse,
donn\'ee par exemple par
l'\'equation homog\`ene $xy=zt$, o\`u $\lbrack x:y:z:t \rbrack$ sont les
coordonn\'ees homog\`enes sur ${\Bbb P} ^{3}$.
La quadrique ${\cal Q}$ est isomorphe \`a ${\Bbb P}^{1} \times {\Bbb P}^{1}$
et nous
notons $$L_{1} = \{ \ast \} \times {\Bbb P} ^{1} \ \mbox{et} \
L_{2} = {\Bbb P} ^{1} \times \{ \ast \}$$ les g\'en\'erateurs
de $H_2(\cal Q,{\Bbb Z}) \simeq {\Bbb Z} ^2$.
On a \'evidemment
$$L_{i} \cdot L_{i}=0 \ \mbox{et} \ L_{1} \cdot L_{2}=1.$$
De plus, tout diviseur $D$ de ${\cal Q}$ est num\'eriquement
caract\'eris\'e par un couple d'entiers $(a,b)$ donn\'e par
l'intersection de $D$ avec $L_{1}$ et $L_{2}$~: $$(a,b)
=(D \cdot L_{1},D \cdot L_{2})
\in {\Bbb Z}^{2}.$$ Ce couple est appel\'e le type de $D$.
Par exemple, le diviseur canonique $K_{{\cal Q}}$ est de
type $(-2,-2)$.
\medskip
\noindent {\bf Affirmation } {\em Pour tous $n$ et $m$ entiers positifs,
il existe
une courbe lisse $C_{n,m}$ incluse dans ${\cal Q}$ et
de type $(n,m)$.
Une telle courbe est de genre $g_{n,m}=(n-1)(m-1)$
et de degr\'e $n+m$.}
\medskip
Soit $C_{n,m}$ une telle courbe.
Nous \'eclatons alors ${\Bbb P} ^{3}$ le long
de $C_{n,m}$~: on obtient une vari\'et\'e
projective $\tilde{X}$, et un morphisme
$$\pi_{1} : \tilde{X} \to {\Bbb P} ^{3}.$$
Notons $E_{n,m}$ le diviseur exceptionnel de l'\'eclatement~;
il est isomorphe \`a ${\Bbb P} (N^{\ast}_{C_{n,m}/{\Bbb P} ^{3}})$.
Comme le groupe de Picard de ${\Bbb P} ^3$ est ${\Bbb Z}$,
celui de $\tilde{X}$ est ${\Bbb Z} ^{2}$.
Si $\tilde{{\cal Q}}$ d\'esigne la transform\'ee stricte
de ${\cal Q}$ et
$\tilde{L_{i}}$
celle de $L_{i}$, alors $\tilde{{\cal Q}}$ et $\tilde{L_{i}}$
sont respectivement isomorphes \`a ${\cal Q}$
et $L_{i}$ car $C_{n,m}$ est incluse dans ${\cal Q}$.
De plus, le type du fibr\'e normal
$N_{\tilde{{\cal Q}}/\tilde{X}}$ de $\tilde{{\cal Q}}$
dans $\tilde{X}$ est donn\'e par l'affirmation suivante que
nous d\'emontrons plus loin~:
\medskip
\noindent {\bf Affirmation } {\em On a
$N_{\tilde{{\cal Q}}/\tilde{X}} \cdot
\tilde{L_{1}}
= 2-n$ et $N_{\tilde{{\cal Q}}/\tilde{X}} \cdot \tilde{L_{2}} = 2-m$.
}
\medskip
Comme cas particulier de l'affirmation pr\'ec\'edente,
consid\'erons le
cas o\`u $n=3$. La restriction \`a $\tilde{L_{1}}$ du fibr\'e
$N_{\tilde{{\cal Q}}/\tilde{X}}$ est alors isomorphe au fibr\'e
${\cal O}_{{\Bbb P}^{1}}(-1)$.
Par le crit\`ere de contraction de Fujiki-Nakano,
il existe donc une vari\'et\'e $X_{m}$ et une application
$$\pi_{2} : \tilde{X} \to X_{m}$$ de sorte que $\pi_{2}$ soit
l'\'eclatement d'une courbe lisse rationnelle $C_{m}$, de fibr\'e normal
projectivement trivial (\'egal \`a ${\cal O}_{{\Bbb P} ^1}(-m)^{\oplus 2}$)
tel que le diviseur exceptionnel de $\pi_{2}$ est
exactement
$\tilde{{\cal Q}}$.
Evidemment, $X_{m}$ est bim\'eromorphiquement \'equivalente \`a ${\Bbb P}^{3}$
donc est de Moishezon. De plus, le groupe de Picard
de $X_{m}$ est ${\Bbb Z}$.
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\medskip
Avant d'\'etudier plus en d\'etail la vari\'et\'e
$X_m$, d\'emontrons les deux affirmations n\'ecessaires
\`a sa construction.
\medskip
\noindent {\bf D\'emonstration des affirmations}
L'existence de $C_{n,m}$ r\'esulte du fait que
$\displaystyle{{\cal O}(n,m)=
\mathop{\rm pr}\nolimits_{1}^{\ast}{\cal O}(n)\otimes \mathop{\rm pr}\nolimits_{2}^{\ast}{\cal O}(m)}$
est tr\`es ample sur
${\Bbb P} ^{1} \times {\Bbb P} ^{1}$. Enfin, le calcul du genre
est donn\'e par la formule classique
$$\displaystyle{2g_{n,m}-2 = C_{n,m} \cdot (C_{n,m}+K_{{\cal Q}})}.$$
Ici, $$C_{n,m} \cdot C_{n,m}=2nm
\ \mbox{et} \ C_{n,m} \cdot K_{{\cal Q}}= -2(n+m).$$ Ceci d\'emontre
la premi\`ere affirmation.\hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
Pour la deuxi\`eme, la suite exacte
$$ 0 \rightarrow T \tilde{{\cal Q}} \to T\tilde{X}_{| \tilde{{\cal Q}}}
\to
N_{\tilde{{\cal Q}}/\tilde{X}} \to 0$$
donne $N_{\tilde{{\cal Q}}/\tilde{X}}=K_{\tilde{{\cal Q}}}-
K_{\tilde{X}|\tilde{{\cal Q}}}$ o\`u $K_{\tilde{X}}=\pi_{1}^{\ast}K_{{\Bbb P}^{3}}+
{\cal O}(E_{n,m})$.
De l\`a~:
\vspace{-3mm}
\begin{eqnarray*}
N_{\tilde{{\cal Q}}/\tilde{X}} \cdot \tilde{L_{i}} & = &
K_{\tilde{{\cal Q}}} \cdot \tilde{L_{i}}-
\pi_{1}^{\ast}K_{{\Bbb P}^{3}} \cdot \tilde{L_{i}}-
{\cal O}(E_{n,m}) \cdot \tilde{L_{i}}\\
& = & K_{{\cal Q}} \cdot L_{i}-
K_{{\Bbb P}^{3}} \cdot L_{i}-C_{n,m} \cdot L_{i}.
\end{eqnarray*}
Or, $K_{{\cal Q}}={\cal O}_{{\Bbb P}^{3}}(-2)_{|{\cal Q}}$ et $K_{{\Bbb P}^{3}}=
{\cal O}_{{\Bbb P}^{3}}(-4)$. Ceci conclut le calcul.\hskip 3pt \vrule height6pt width6pt depth 0pt
\subsub{Deux propri\'et\'es de $X_m$}
Nous montrons ici que les crit\`eres de Demailly et Siu
ne sont pas satisfaits pour la vari\'et\'e de Moishezon $X_m$.
\bigskip
\noindent {\bf Th\'eor\`eme A}
{\em La vari\'et\'e $X_m$ v\'erifie les deux propri\'et\'es
suivantes~:
(i) si $m$ est strictement plus grand que $3$,
$X_{m}$ ne poss\`ede pas de fibr\'e en droites \`a la fois gros et nef,
et donc ne satisfait pas au crit\`ere de Siu,
(ii) si $m$ est strictement plus grand que $5$, $X_{m}$
ne poss\`ede pas de fibr\'e en droites $E$
muni d'une m\'etrique
hermitienne lisse $h$ telle
que la forme de courbure $\Theta (E)$ v\'erifie :
$$ \int_{X(\leq 1,E)}\Theta (E)^{3} > 0.$$
}
\medskip
L'affirmation (i) est due \`a J.\ Koll\'ar, nous en donnons
une preuve \'el\'ementaire, tandis que (ii) est nouveau \`a notre
connaissance.
\medskip
\noindent {\bf D\'emonstration de (i)}
Soit $E$ un fibr\'e holomorphe de rang $1$
sur $X_{m}$, que l'on suppose non trivial (le fibr\'e trivial, bien que nef,
n'est pas gros ! ).
Il existe alors des entiers $k$ et $l$ tels que~:
$$\pi_{2}^{\ast}E = \pi_{1}^{\ast}{\cal O}_{{\Bbb P}^{3}}(l)-{\cal O}(kE_{3,m}).$$
\noindent Comme $\tilde{L_{1}}$ est une fibre de $\pi_{2}$, on a~:
$\pi_{2}^{*}E \cdot \tilde{L_{1}} = 0$. On en d\'eduit la relation $l=3k$
et donc
$$\pi_{2}^{*}E = k(3\pi_{1}^{\ast}{\cal O}_{{\Bbb P}^{3}}(1)-{\cal O}(E_{3,m})),$$
o\`u $k$ est un entier non nul.
En particulier, si $\tilde{F}$ est une fibre non triviale de $\pi_{1}$
dans $\tilde{X}$,
on a les
nombres d'intersection suivants~:
$$\left\{
\begin{array}{l}
\pi_{2}^{\ast}E \cdot \tilde{L_{2}}=k(3-m) \\
\pi_{2}^{\ast}E \cdot \tilde{F}=k
\end{array}
\right.
$$
\noindent On en d\'eduit que pour $m>3$, le fibr\'e
$\pi_{2}^{\ast}E$ n'est pas nef (sinon son
intersection avec toute courbe serait positive ou nulle), et donc
$E$ n'est pas nef.\hskip 3pt \vrule height6pt width6pt depth 0pt
\noindent {\bf D\'emonstration de (ii)}
Notons dans la suite ${\cal O}_{X_m}(1)$ le g\'en\'erateur du
groupe de Picard de
$X_{m}$
tel que $\pi_{2}^{\ast}{\cal O}_{X_m}(1) =
\pi_{1}^{\ast}{\cal O}_{{\Bbb P}^{3}}(3)-{\cal O}(E_{3,m})$.
\bigskip
\noindent {\bf Affirmation } {\em Le fibr\'e canonique
$K_{X_{m}}$ est \'egal \`a ${\cal O}_{X_m}(-2)$.}
\medskip
En effet, on a
$$K_{\tilde{X}}=
\pi_{2}^{\ast}K_{X_{m}}+{\cal O}(\tilde{{\cal Q}}) =
\pi_1^*{\cal O}_{{\Bbb P}^{3}}(-4)+ {\cal O}(E_{3,m})$$
\noindent par
construction. Or,
$$ {\cal O}(\tilde{{\cal Q}}) = \pi_1^*{\cal O}_{{\Bbb P}^{3}}(2) - {\cal
O}(E_{3,m}),$$
d'o\`u l'affirmation.
\bigskip
\noindent {\bf Affirmation } {\em Les espaces de sections
holomorphes
$H^{0}(X_m,{\cal O}_{X_m}(k))$ sont nuls pour tout entier
$k < 0$ et le fibr\'e ${\cal O}_{X_m}(1)$ est gros.}
\medskip
Par invariance bim\'eromorphe des plurigenres, les groupes
$H^{0}(X_m,{\cal O}_{X_m}(k))$ sont nuls pour tout entier
$k$ strictement n\'egatif et pair. Comme
$X_{m}$ est de Moishezon de groupe de Picard ${\Bbb Z}$, $X_{m}$ poss\`ede
un fibr\'e gros qui n'est donc pas ${\cal O}_{X_m}(-1)$, c'est donc que
${\cal O}_{X_m}(1)$
est gros et que les $H^{0}(X_m,{\cal O}_{X_m}(k))$ sont nuls pour tout entier
$k$ strictement n\'egatif. Ceci d\'emontre l'affirmation.
\bigskip
Par dualit\'e de Serre, on d\'eduit des deux affirmations
pr\'ec\'edentes que {\em les groupes de cohomologie
$H^{3}(X_{m},{\cal O}_{X_m}(k))$ sont nuls pour tout
entier
$k > -2$.}
\medskip
Nous sommes maintenant en mesure de passer \`a la preuve
de (ii) proprement dit~:
raisonnons par l'absurde et supposons que ${\cal O}_{X_m}(1)$ poss\`ede
une telle
m\'etrique.
D'apr\`es les in\'egalit\'es de Morse holomorphes de J.-P.\ Demailly
appliqu\'ees pour $q=1$ au fibr\'e hermitien $({\cal O}_{X_m}(1),h)$, on a~:
$$\dim H^{0}(X_{m},{\cal O}_{X_m}(k))-
\dim H^{1}(X_{m},{\cal O}_{X_m}(k)) \geq
\frac{1}{6} \left(\int_{X(\leq 1,E)}\Theta (E)^{3}\right)k^{3}+
o(k^{3}).$$
\noindent Comme les groupes de cohomologie
$H^{3}(X_{m},{\cal O}_{X_m}(k))$ sont nuls pour tout
entier
$k > -2$, on a successivement~:
\vspace{-3mm}
\begin{eqnarray*}
c_{1}({\cal O}_{X_m}(1))^{3}\frac{k^{3}}{6} + o(k^{3})
& = &
\sum _{i=0}^3 (-1)^i \dim H^{i}(X_{m},{\cal O}_{X_m}(k)) \\
& = & \sum _{i=0}^2 (-1)^i \dim H^{i}(X_{m},{\cal O}_{X_m}(k)) \\
& \geq & \dim H^{0}(X_{m},{\cal O}_{X_m}(k))-
\dim H^{1}(X_{m},{\cal O}_{X_m}(k)) \\
& \geq & \left(\int_{X(\leq 1,E)}\Theta (E)^{3}\right)\frac{k^{3}}{6}+
o(k^{3}).
\end{eqnarray*}
\noindent On en d\'eduit que
$\displaystyle{c_{1}({\cal O}_{X_m}(1))^{3} \geq
\int_{X(\leq 1,E)}\Theta (E)^{3} >0}$.
Il suffit donc de montrer pour obtenir la contradiction cherch\'ee
que pour $m > 5$, on a $c_{1}({\cal O}_{X_m}(1))^{3} \leq 0$.
Or, cette derni\`ere quantit\'e est ais\'ement calculable~:
\medskip
\noindent {\bf Affirmation } {\em
La quantit\'e $c_{1}({\cal O}_{X_m}(1))^{3}$ est \'egale \`a $6-m$.}
\medskip
En effet
\vspace{-3mm}
\begin{eqnarray*}
c_{1}({\cal O}_{X_m}(1))^{3} & = &
c_{1}(\pi_{1}^{*}{\cal O}_{{\Bbb P}^{3}}(3)-{\cal O}(E_{3,m}))^{3} \\
& = &
c_{1}({\cal O}_{{\Bbb P}^{3}}(3))^{3}
-3\int_{E_{3,m}} c_{1}(\pi_{1}^{*}{\cal O}_{{\Bbb P}^{3}}(3))^{2} \\
& & +3\int_{\tilde{X}}c_{1}(\pi_{1}^{*}{\cal O}_{{\Bbb P}^{3}}(3))
\wedge c_{1}({\cal O}(E_{3,m}))^{2}
-E_{3,m}^{3}.
\end{eqnarray*}
\noindent On a \'evidemment $c_{1}({\cal O}_{{\Bbb P}^{3}}(3))^{3}=27$ et
$\displaystyle{\int_{E_{3,m}} c_{1}(\pi_{1}^{*}{\cal O}_{{\Bbb P}^{3}}(3))^{2}= 0.}$
\noindent Pour les deux termes restants, on commence par remarquer que
$\displaystyle{c_{1}({\cal O}(E_{3,m}))_{|E_{3,m}}}$ est \'egale \`a $-\xi$
o\`u
$\displaystyle{\xi=c_{1}({\cal O}_{{\Bbb P} (N_{C_{3,m}/{\Bbb P} ^{3}}^{*})}(1))}$
d\'esigne la
classe fondamentale de l'\'eclatement.
On en d\'eduit que~:
\vspace{-3mm}
\begin{eqnarray*}
\int_{\tilde{X}}c_{1}(\pi_{1}^{*}{\cal O}_{{\Bbb P}^{3}}(3))
\wedge c_{1}({\cal O}(E_{3,m}))^{2} & = &
-\int_{E_{3,m}} \pi_{1}^{*}c_{1}({\cal O}_{{\Bbb P}^{3}}(3))
\wedge \xi \\
& = & -\int_{C_{3,m}}c_{1}({\cal O}_{{\Bbb P}^{3}}(3)) \\
& = & -3(3+m),
\end{eqnarray*}
\noindent la derni\`ere \'egalit\'e venant du fait
que $C_{3,m}$ est de degr\'e
$3+m$ dans ${\Bbb P} ^{3}$ (rappelons que
$\displaystyle{{\cal O}_{{\Bbb P} ^{3}}(1)_{|Q}=
{\cal O}(1,1)}$).
Finalement, il reste \`a calculer $E_{3,m}^{3}$. Pour cela, rappelons
que $\xi$ v\'erifie la formule fondamentale suivante~:
$$ \xi^{2} - \pi_{1}^{*}c_{1}(N_{C_{3,m}/{\Bbb P}^{3}}^{*})\xi +
\pi_{1}^{*}c_{2}(N_{C_{3,m}/{\Bbb P}^{3}}^{*})=0 $$ qui se r\'eduit ici \`a
$\displaystyle{\xi^{2} - \pi_{1}^{*}c_{1}(N_{C_{3,m}/{\Bbb P}^{3}}^{*})\xi =0}$.
\noindent Il vient alors
$\displaystyle{E_{3,m}^{3}= \int_{E_{3,m}}\xi^{2}=\int_{E_{3,m}}
\pi_{1}^{*}c_{1}(N_{C_{3,m}/{\Bbb P}^{3}}^{*})\xi =
\int_{C_{3,m}}c_{1}(N_{C_{3,m}/{\Bbb P}^{3}}^{*}).}$
\noindent Or la suite exacte~:
$$ 0 \rightarrow TC_{3,m} \rightarrow T{\Bbb P}^{3}_{| C_{3,m}} \rightarrow
N_{C_{3,m}/{\Bbb P}^{3}} \rightarrow 0$$
donne de suite :
$$ \int_{C_{3,m}}c_{1}(N_{C_{3,m}/{\Bbb P}^{3}}^{*})=
\int_{C_{3,m}}c_{1}({\cal O}_{{\Bbb P}^{3}}(-4))-2g_{3,m}+2
=-6-8m .$$
\noindent Il reste finalement~:
$$c_{1}({\cal O}_{X_m}(1))^{3} = 27 - 27 - 9m + 6 + 8m = 6-m.$$
\noindent Ceci ach\`eve la preuve de l'affirmation et
par suite celle du th\'eor\`eme.\hskip 3pt \vrule height6pt width6pt depth 0pt
\subsection{La construction de K.\ Oguiso}
Pour ce paragraphe, la r\'ef\'erence est \cite{Ogu94}.
Dans cet article, K.\ Oguiso construit une vari\'et\'e
de Moishezon non projective, de dimension $3$ et qui est de plus
de Calabi-Yau~: ceci signifie que cette vari\'et\'e
est simplement connexe et \`a fibr\'e canonique trivial.
Mettons en garde le lecteur sur l'usage fait ici de l'expression
Calabi-Yau. En effet, la vari\'et\'e en question
n'est pas k\"ahl\'erienne. Rappelons plus g\'en\'eralement
qu'un th\'eor\`eme de B.\ Moishezon affirme que {\em toute vari\'et\'e
de Moishezon k\"ahl\'erienne est projective} (voir
le ``survey" de T.\ Peternell \cite{Pet95} pour une d\'emonstration
rapide de ce r\'esultat).
Le r\'esultat de K.\ Oguiso est le suivant~:
\bigskip
\noindent {\bf Th\'eor\`eme (K.\ Oguiso, 1994)}
{\em Il existe une vari\'et\'e de Moishezon $Y$, de dimension $3$
et de Calabi-Yau telle que
$\displaystyle{H^2(Y,{\Bbb Z}) = \operatorname{Pic} (Y) = {\Bbb Z} \cdot L}$
o\`u $L$ satisfait $L^3 := c_1(L)^3 =0$. Autrement dit,
la forme cubique d'intersection sur $H^2(Y,{\Bbb Z})$
est identiquement nulle.}
\bigskip
Nous donnons dans la derni\`ere partie de cette th\`ese
une construction, diff\'erente de celle d'Oguiso, permettant
de retrouver ce r\'esultat. Dans \cite{Ogu94},
K.\ Oguiso obtient ce th\'eor\`eme
comme cons\'equence du r\'esultat suivant~:
\bigskip
\noindent {\bf Th\'eor\`eme (K.\ Oguiso, 1994)}
{\em Soit $d$ un entier positif. Alors il existe
une vari\'et\'e projective $X_d$ de dimension $3$,
intersection compl\`ete d'une quadrique et d'une
quartique dans ${\Bbb P} ^5$ et contenant une courbe rationnelle
lisse $C_d$
de degr\'e $d$
dont le fibr\'e normal $N_{C_d/X_d}$ est ${\cal O}_{C_d}(-1)^{\oplus 2}$.}
\bigskip
Montrons que ce dernier r\'esultat implique le
premier.
\noindent Soit donc $X_d$ comme ci dessus et soit
$\tilde{X}_d$ la vari\'et\'e projective obtenue
en \'eclatant $X_d$ le long de la courbe $C_d$. Un argument
identique \`a celui d\'evelopp\'e dans la construction
pr\'ec\'edente assure que l'on peut contracter
le diviseur exceptionnel $E \simeq C_d \times {\Bbb P} ^1
= {\Bbb P} ^1 \times {\Bbb P} ^1$ dans l'autre direction~:
on note $Y_d$ la vari\'et\'e obtenue. Par construction,
$Y_d$ est de Moishezon, de Calabi-Yau et son groupe de
Picard est ${\Bbb Z}$.
Enfin, si $\displaystyle{{\cal O}_{Y_d}(1)}$ est le g\'en\'erateur
gros de $\displaystyle{\operatorname{Pic} (Y_d)}$, on montre comme pr\'ec\'edemment
que $\displaystyle{c_1({\cal O}_{Y_d}(1))^3 = 8 - d^3}$.
Le premier r\'esultat est d\'emontr\'e
en choisissant $d=2$ i.e $Y := Y_2$.
\noindent Comme pour les vari\'et\'es $X_m$
pr\'ec\'edemment construites, {\em les vari\'et\'es $Y_d$ pour
$d$ sup\'erieur ou \'egal \`a $2$ ne satisfont pas
les crit\`eres de Siu et Demailly.}
\medskip
\noindent {\bf Remarque } Les vari\'et\'es $Y_d$ ont \'et\'e
obtenues apr\`es
une transformation birationnelle classique en th\'eorie
de Mori appel\'ee ``flop". Nous revenons sur ce type de
construction dans la deuxi\`eme partie de cette th\`ese.
\subsection{Quelques commentaires}
Ces pr\'eliminaires illustrent une partie
des motivations de cette th\`ese. En effet, alors que
les deux constructions pr\'ec\'edentes montrent
que les crit\`eres de J.-P.\ Demailly et Y.-T.\ Siu ne sont
pas n\'ecessaires dans le cadre des fibr\'es hermitiens
munis d'une m\'etrique ${\cal C}^{\infty}$, nous montrons
dans le deuxi\`eme chapitre de cette th\`ese que ces crit\`eres
deviennent n\'ecessaires et suffisants dans le cadre
plus souple des m\'etriques singuli\`eres. Pour cela, nous
\'etendons les in\'egalit\'es de Morse en autorisant
un certain type de singularit\'es aux m\'etriques des fibr\'es
consid\'er\'es.
Remarquons ensuite que les constructions pr\'ec\'edentes
donnent des exem\-ples de vari\'et\'es de Moishezon de dimension $3$,
de groupe de Picard ${\Bbb Z}$ avec respectivement $-K_X$ gros et
$K_X$ trivial. Une des motivations du dernier chapitre de cette
th\`ese est de r\'epondre \`a la question suivante~: peut-on construire
des exemples analogues avec $K_X$ gros ?
Un r\'esultat de J.\ Koll\'ar montre
que ce n'est pas possible en dimension $3$~:
nous montrons en revanche que de tels exemples
existent en dimension sup\'erieure.
Ceci nous conduira naturellement \`a \'etudier
la structure du centre de l'\'eclatement projectif donn\'e
abstraitement par le th\'eor\`eme de Moishezon lorsque
la vari\'et\'e devient projective apr\`es un \'eclatement
seulement.
\chapter{In\'egalit\'es de Morse singuli\`eres}
Le but central de ce chapitre est d'\'etendre les in\'egalit\'es
de Morse
holomorphes de J.-P.\ Demailly au cas d'un
fibr\'e en droites $E$ muni d'une m\'etrique singuli\`ere au dessus
d'une vari\'et\'e complexe compacte $X$.
Ces in\'egalit\'es nous permettent ensuite de caract\'eriser
analytiquement les vari\'et\'es de Moishezon.
Enfin, nous donnons une version alg\'ebrique
singuli\`ere de nos in\'egalit\'es de Morse.
\section{M\'etriques singuli\`eres}
La notion de m\'etrique singuli\`ere pour des fibr\'es
en droites
a \'et\'e introduite par
J.-P.\ Demailly, A.\ Nadel et H.\ Tsuji. Nous commen\c cons
ce paragraphe en rappelant les premi\`eres
d\'efinitions et les exemples classiques.
Une r\'ef\'erence est \cite{Dem90}.
\subsection{Premi\`eres d\'efinitions}
Soit $X$ une vari\'et\'e et $E$ un fibr\'e en droites
sur $X$. Une m\'etrique hermitienne {\bf singuli\`ere}
sur $E$ est donn\'ee localement
sur un ouvert trivialisant $U_{\alpha}$ par
$$h(\xi _{x}) = ||\xi _x||_h := |\xi|\exp(-\varphi _{\alpha} (x))$$ o\`u la
fonction r\'eelle $\varphi _{\alpha}$ est
seulement suppos\'ee {\bf localement int\'egrable}.
Cette derni\`ere hypoth\`ese suffit \`a donner
encore un sens \`a la notion de courbure
d'un tel fibr\'e~: en effet, on pose
toujours $\displaystyle{\Theta(E) :=
\frac{i}{\pi} \partial \overline{\partial} \varphi _{\alpha}}$
o\`u le $\partial \overline{\partial}$ est pris au sens
des distributions. L'objet ainsi d\'efini
n'est plus une forme ${\cal C}^{\infty}$ mais
un {\bf courant} (appel\'e courant de courbure)
de bi-degr\'e $(1,1)$. Le lemme de Dolbeault-Grothendieck
\'etant vrai pour les courants, la cohomologie de
De Rham est calculable aussi bien avec les courants
qu'avec les formes et la classe de cohomologie
du courant de courbure $\Theta(E)$ est \'egale
comme dans le cas lisse \`a la premi\`ere classe de Chern
du fibr\'e $E$.
\medskip
\noindent {\bf Exemples } Les deux exemples suivants
jouent un r\^ole essentiel.
(i) Si $\displaystyle{D = \sum \alpha _j D_j}$ est un diviseur
de $X$ et si $g_j$ est l'\'equation locale de $D_j$
sur un ouvert $U_{\alpha}$, alors la fonction
$\displaystyle{\varphi _{\alpha} = \sum \alpha _j \log |g_j|}$
d\'efinit une m\'etrique singuli\`ere naturelle sur
le fibr\'e en droites ${\cal O}(D)$ associ\'e au
diviseur $D$.
\noindent Pour cette m\'etrique, l'\'equation de
Lelong-Poincar\'e est $\displaystyle{\Theta({\cal O}(D)) = [D]}$
o\`u $[D]$ d\'esigne le courant d'int\'egration
sur le diviseur $D$.
\noindent Remarquons que dans le cas o\`u $X$ est une courbe complexe
et $E$ le fibr\'e tangent de $X$, la construction pr\'ec\'edente
donne une m\'etrique plate avec des masses de Dirac de courbure~:
ce type de m\'etrique est bien connu en g\'eom\'etrie
riemannienne sous le nom de {\bf m\'etrique plate
\`a singularit\'es coniques}.
(ii) Soit $E$ un fibr\'e en droites sur une
vari\'et\'e $X$ et soient $s_1,\ldots \!,s_p$
des sections de $E^{\otimes k}$.
Alors,
$$||\xi_x||^2 :=
\left( \frac{|\theta _{\alpha}(\xi)|^2}{|\theta _{\alpha}(s_1(x))|^2+
\cdots+|\theta _{\alpha}(s_p(x))|^2} \right) ^{1/k}$$ o\`u
$\theta _{\alpha}$ est une trivialisation locale de $E$ et $E^{\otimes k}$,
d\'efinit une m\'etrique singuli\`ere sur le fibr\'e $E$.
La fonction $\varphi _{\alpha}$ est ici $\displaystyle{\varphi _{\alpha}(x) =
\frac{1}{2k} \log(|\theta _{\alpha}(s_1(x))|^2+
\cdots+|\theta _{\alpha}(s_p(x))|^2)}$.
\medskip
Comme dans le cas ${\cal C}^{\infty}$, nous d\'esirons
pouvoir donner un sens \`a l'expression ``\^etre
\`a courbure positive ou strictement positive"
pour un fibr\'e en droites muni d'une m\'etrique
singuli\`ere.
Pour cela, rappelons qu'un courant
$T$ de bi-degr\'e $(1,1)$ est {\bf positif}
si pour toutes formes $\alpha_1,\ldots \!,\alpha_{n-1}$ de type
$(1,0)$ et
${\cal C}^{\infty}$ \`a support compact, on a~:
$$ < T , i \alpha_1 \wedge \overline{\alpha}_1\wedge \ldots \wedge
i \alpha_{n-1} \wedge \overline{\alpha}_{n-1} > \ \ \geq \ \ 0,$$
o\`u $<\cdot, \cdot >$ est la dualit\'e entre les courants et les formes.
On note alors $T \geq 0$.
De m\^eme, on dira qu'un courant
$T$ de bi-degr\'e $(1,1)$ est {\bf strictement positif}
s'il existe une m\'etrique $\omega$ hermitienne ${\cal C}^{\infty}$
sur $X$ telle que $T - \omega$ est un courant positif.
On note dans ce cas $T >0$.
Nous introduisons alors la d\'efinition suivante~:
\medskip
\noindent {\bf D\'efinition } {\em Un fibr\'e en droites muni
d'une m\'etrique singuli\`ere est positif (respectivement
strictement positif) si le courant de courbure associ\'e
est positif (respectivement strictement positif).}
\medskip
En reprenant les notations de l'exemple ci-dessus,
le fibr\'e en droites ${\cal O}(D)$ muni de sa
m\'etrique singuli\`ere est positif si et seulement
si le diviseur $D$ est effectif (i.e tous les coefficients
$\alpha _j$ sont positifs ou nuls). L'exemple (ii)
d\'efinit toujours un courant de courbure
positif car une fonction de la forme
$\varphi = \log \sum |f_j|^2$ o\`u les
$f_j$ sont des fonctions holomorphes est plurisousharmonique
(en abr\'eg\'e {\bf psh}).
\subsection{Faisceaux d'id\'eaux multiplicateurs de Nadel}
L'\'etude d'un fibr\'e en droites muni d'une
m\'etrique singuli\`ere est facilit\'ee par l'utilisation
d'un outil pertinent associ\'e aux singularit\'es de
la m\'etrique~: il s'agit d'un faisceau d'id\'eaux
introduit par A.\ Nadel.
\medskip
\noindent {\bf D\'efinition } {\em
Soit $(E,h)$ un fibr\'e en droites sur une vari\'et\'e $X$.
On appelle ``faisceau multiplicateur de Nadel" le faisceau
d'id\'eaux ${\cal I}(h)$ des germes de fonctions holomorphes
$L^2$ par rapport au poids de la m\'etrique singuli\`ere,
i.e l'ensemble des germes $f \in {\cal O}_{X,x}$ tels
que $|f|^2 \exp (-2\varphi)$ est int\'egrable par rapport
\`a la mesure de Lebesgue dans des coordonn\'ees
locales au voisinage de $x$.
Plus g\'en\'eralement, si $\varphi$ est une fonction
r\'eelle
sur un ouvert $\Omega$, nous notons aussi
${\cal I}(\varphi)$ le faisceau des germes
$f \in {\cal O}_{\Omega}$ tels
que $|f|^2 \exp (-2\varphi)$ est localement int\'egrable.}
\medskip
La propri\'et\'e essentielle satisfaite par ce faisceau
d'id\'eaux, due \`a Nadel, est que
{\em si $\varphi$ est une fonction plurisousharmonique,
alors ${\cal I}(\varphi)$ est un faisceau coh\'erent.}
\medskip
Avant de donner des exemples, mentionnons
le r\'esultat suivant qui g\'en\'eralise
le th\'eor\`eme de Kawamata-Viehweg~: il s'agit du
th\'eor\`eme d'annulation de Nadel \cite{Nad89} \cite{Dem89}.
\bigskip
\noindent {\bf Th\'eor\`eme d'annulation de Nadel}
{\em Soit $X$ une vari\'et\'e k\"ahl\'erienne compacte,
et $E$ un fibr\'e en droites muni d'une m\'etrique
singuli\`ere \`a courbure
strictement positive. Alors, pour tout $q \geq 1$, on a~:
$$ H^q (X,{\cal O}(E+K_X) \otimes {\cal I}(h)) = 0.$$
}
\medskip
\vspace{-2mm}
Ce r\'esultat montre que pour g\'en\'eraliser un th\'eor\`eme
d'annulation
dans le contexte des fibr\'es en droites munis
de m\'etrique singuli\`ere, une bonne m\'ethode
consiste \`a ``tordre" la cohomologie
du fibr\'e par le faisceau de Nadel.
\medskip
Donnons quelques exemples de faisceaux multiplicateurs,
qui bien que tr\`es simples jouent un r\^ole important.
\medskip
\noindent {\bf Exemples}
(i) Soit $\varphi$ une fonction r\'eelle sur un ouvert $\Omega$
de ${\Bbb C} ^n$
contenant l'origine. Si $\varphi$ est minor\'ee au voisinage
de l'origine, alors pour tout $x$ proche de $0$, on
a clairement ${\cal I}(\varphi)_x = {\cal O}_{\Omega,x}$.
En particulier, si $\varphi$ est continue sur $\Omega$,
alors ${\cal I}(\varphi) = {\cal O}_{\Omega}$.
(ii) Pla\c cons nous dans $\displaystyle{{\Bbb C} ^n}$ au voisinage de
l'origine et, pour $\alpha_{1},\ldots \!,\alpha_{p}$ des
r\'eels positifs et $k$ un entier naturel, posons~:
$$ \varphi _k (z)=
\frac{k}{2}\log(|z_{1}|^{2\alpha_{1}}+ \cdots +|z_{p}|^{2\alpha_{p}}).$$
Alors, ${\cal I}(\varphi _k)_{{\Bbb C} ^n,0}$ est
${\cal O}_{{\Bbb C} ^n,0}$-engendr\'e
par les $\displaystyle{\prod _{j=1}^{p}z_{j}^{\beta_{j}}}$ tels que~:
$$ \sum_{j=1}^{p}\frac{\beta_{j}+1}{\alpha_{j}} > k .$$
\indent En particulier, si tous les $\alpha_{i}$ sont \'egaux
\`a $\alpha$, alors ${\cal I}(\varphi _k)_{{\Bbb C} ^n,0}$ est
${\cal O}_{{\Bbb C} ^n,0}$-engendr\'e
par les $\displaystyle{\prod _{j=1}^{p}z_{j}^{\beta_{j}}}$ o\`u
$$\displaystyle{\sum_{j=1}^{p}\beta_{j} \geq \lfloor k\alpha - p \rfloor +1}$$
($\lfloor x \rfloor$ d\'esigne la partie enti\`ere du r\'eel $x$).
Autrement dit,
$\displaystyle{{\cal I}(\varphi _k) = {\cal I}_{Y}^{
\lfloor k\alpha \rfloor -p+1}}$
o\`u $Y$ est la sous-vari\'et\'e $\{ z_{1}= \cdots =z_{p}=0 \}$
et ${\cal I}_{Y}$ son id\'eal annulateur.
(iii) Soit $\varphi$ une fonction de la forme
$\sum \alpha _j \log |g_j|$ o\`u les $\alpha _j$
sont des r\'eels positifs et o\`u les fonctions
holomorphes $g_j$ sont telles que les $D_j := g_j^{-1}(0)$
soient des diviseurs irr\'eductibles lisses
se coupant transversalement (on dit alors
que $\displaystyle{D = \sum \alpha _j D_j}$ est un diviseur
{\bf lisse \`a croisements normaux}).
Dans ce cas, le faisceau multiplicateur
${\cal I}(\varphi)$
s'identifie au faisceau inversible de rang un
${\cal O}(-\sum_{j} \lfloor \alpha _j \rfloor D_{j})$.
\medskip
\noindent {\bf D\'emonstration}
Le point (i) est trivial.
Pour (ii), il s'agit d'estimer l'int\'egrale suivante
sur un voisinage de $0$~:
$$ I := \int_{D(0,\varepsilon) ^n}
\frac{|\sum a_{\beta}z_1 ^{\beta _1} \ldots z_n ^{\beta _n}|^2}{
(|z_{1}|^{2\alpha_{1}}+ \cdots +|z_{p}|^{2\alpha_{p}})^k} \ d\lambda(z).$$
Le passage en coordonn\'ees polaires $z_j = \rho _j e^{i\theta _j}$
donne
$$ I = (2\pi)^n \sum |a_{\beta}|^2
\int_{[0,\varepsilon]^n}
\frac{\rho_1^{2\beta_{1}+1}\cdots\rho_n^{2\beta_{n}+1}}
{(|\rho_{1}|^{2\alpha_{1}}+
\cdots +|\rho_{p}|^{2\alpha_{p}})^k}\ d\rho_1 \ldots d\rho_n,$$
puis
$$ I = (2\pi)^n \sum |a_{\beta}|^2
\frac{\varepsilon^{2\beta_{p+1}+2}\ldots \varepsilon^{2\beta_{n}+2}}
{(2\beta_{p+1}+2)\ldots (2\beta_{n}+2)}
\int_{[0,\varepsilon]^p}
\frac{\rho_1^{2\beta_{1}+1}\cdots\rho_p^{2\beta_{p}+1}}
{(\rho_{1}^{2\alpha_{1}}+
\cdots +\rho_{p}^{2\alpha_{p}})^k}\ d\rho_1 \ldots d\rho_p.$$
Par le changement de variables $u_j = \rho_j^{\alpha_j}$, l'int\'egrale
ci-dessus est \'egale \`a
$$ \int_{[0,\varepsilon]^p}
\frac{u_1^{((2\beta _1 +2)/\alpha_1) -1}\ldots u_p^{((2\beta _p
+2)/\alpha_p)-1}}
{(u_1^2+ \cdots + u_p^2)^k} \ du_1 \ldots du_p,$$
Par homog\'en\'eit\'e, cette
derni\`ere int\'egrale converge si et seulement si
l'int\'egrale
$$\displaystyle{ \int_0^{\varepsilon}
\frac{ t^{ 2\sum_{j=1}^p ((\beta _j +1)/\alpha_j) - p}}{t^{2k}}t^{p-1} \ dt}$$
converge, soit
$\displaystyle{2\sum_{j=1}^p \frac{\beta _1 +1}{\alpha_1} - p -2k + p - 1 >
-1}$.
Ceci donne bien la condition annonc\'ee.
Pour le point (iii), il s'agit de d\'eterminer
le crit\`ere pour qu'une fonction de la forme
$$\displaystyle{
\frac{|f|^{2}}{|g_1|^{2\alpha_1}\ldots |g_n|^{2\alpha_n}}}$$
\noindent soit dans $L_{\mbox{\scriptsize loc}}^{1}$.
Comme les $g_j$ fournissent
des coordonn\'ees locales, et si $p_{j}$ d\'esigne l'ordre d'annulation de
$f$ le long
de $D_j = \{ g_{j}=0 \}$, la condition n\'ecessaire
et suffisante est que $2p_{j}-2\alpha _j > -2$, soit
$p_{j} \geq \lfloor \alpha _j \rfloor$. Comme les sections du fibr\'e
${\cal O}(-D_j)$ s'identifient aux fonctions
holomorphes s'annulant \`a un ordre sup\'erieur ou
\'egal \`a un le long de $D_j$, le r\'esultat en d\'ecoule.\hskip 3pt \vrule height6pt width6pt depth 0pt
\section{In\'egalit\'es de Morse singuli\`eres}
Dans cette partie, nous \'enon\c cons et d\'emontrons notre
version singuli\`ere des in\'e\-galit\'es de Morse
holomorphes.
\subsection{\'Enonc\'e du r\'esultat principal}
Dans tout ce qui suit, $X$ d\'esigne une vari\'et\'e
compacte et $(E,h)$ un fibr\'e en droites sur $X$
muni d'une m\'etrique singuli\`ere.
A $(E,h)$, on associe pour tout entier $k$ positif le
faisceau d'id\'eaux de Nadel donn\'e par la
m\'etrique singuli\`ere $h^k$ du fibr\'e $E^k$~;
il s'agit ici du faisceau
des germes de fonctions holomorphes $f$ telles que
$|f|^{2}\exp(-2k\varphi)$
est $L_{\mbox{\scriptsize loc}}^{1}$ (o\`u $\exp (-\varphi)$ est
le poids local de $h$). Nous notons ce faisceau ${\cal I}_{k} (h)$.
Pour des raisons qui apparaitront plus loin, nous sommes
contraints de n'accepter qu'un certain type
de singularit\'es, que nous appelons
{\bf ``singularit\'es analytiques"}. Plus pr\'ecis\'ement, nous
faisons l'hypoth\`ese suivante sur $h$ (c'est-\`a-dire localement
sur $\varphi$ )~:
\bigskip
\noindent {\bf Hypoth\`ese ({\cal S}) :}
{\em la fonction $\varphi$ s'\'ecrit localement~:
$$ \varphi=\frac{c}{2}\log(\sum \lambda_{j}|f_{j}|^{2})+\psi,$$
o\`u les $f_{j}$ sont holomorphes,
les $\lambda_{j}$ sont des fonctions r\'eelles positives ${\cal C}^{\infty}$
sans z\'eros communs,
$\psi$ est ${\cal C}^{\infty}$
et $c$ est un rationnel positif ou nul. Mentionnons que
la somme $\sum \lambda_{j}|f_{j}|^{2}$ peut poss\'eder une
infinit\'e de termes.}
\medskip
Cette hypoth\`ese implique que la fonction $\varphi$ est {\bf quasi
plurisousharmonique}
(ce qui signifie que son hessien complexe est minor\'e par une
$(1,1)$-forme \`a coefficients continus) et donc en particulier que le
faisceau $ {\cal I}_{k} (h)$
est un faisceau coh\'erent d'apr\`es le r\'esultat de A.\ Nadel
pr\'ec\'edemment rencontr\'e.
En terme de courbure, l'hypoth\`ese ({\cal S}) implique en particulier
que le
courant de courbure
$\displaystyle{ \Theta (E)=
\frac{i}{\pi} \partial \overline{\partial} \varphi}$
est quasi positif (c'est-\`a-dire minor\'e
par une $(1,1)$-forme \`a coefficients continus).
\bigskip
\noindent {\bf Remarque }
Signalons d\`es maintenant que cette hypoth\`ese est
``raisonnable" car c'est pr\'ecis\'ement par des fonctions
ayant ce type de
singularit\'es que J.-P.\ Demailly approche une fonction
quasi plurisousharmonique
quelconque \cite{Dem92}.
Ceci aura une importance capitale dans les applications.
Remarquons aussi que tous les exemples rencontr\'es
jusqu'\`a pr\'esent satisfont l'hypoth\`ese ({\cal S}).
Nous renvoyons ici au paragraphe 2.3.1 pour une discussion
plus compl\`ete de l'origine de ce type de
singularit\'es autoris\'ees.
\medskip
Dans notre contexte, nous introduisons les notations
suivantes~:
\medskip
\noindent {\bf Notations }
Le symbole $X(q,E)$ d\'esigne l'ouvert de $X$ form\'e des points
$x$ au voisinage desquels $\varphi$ est born\'ee et $\Theta (E) (x)$
poss\`ede
exactement $q$ valeurs propres strictement n\'egatives et $n-q$ valeurs
propres strictement positives~;
finalement et comme dans le cas lisse $X(\leq q,E)
= X(0,E)\cup \cdots \cup X(q,E)$.
\medskip
Notre r\'esultat est le suivant~:
\bigskip
\noindent {\bf Th\'eor\`eme B } {\em
Soit $(E,h)$ \`a singularit\'es analytiques sur
une vari\'et\'e compacte $X$ de dimension $n$ et soit $F$
un fibr\'e
de rang $r$ sur $X$. Alors pour tout $q$ compris entre
$0$ et $n$~:
\noindent \ (i) $\displaystyle{\
\sum _{j=0}^{q}(-1)^{q-j} \dim H^{j}(X,{\cal O}(E^{k}\otimes F) \otimes
{\cal I}_{k}(h)) \leq r\frac{k^{n}}{n!} \int _{X(\leq q,E)} (-1)^{q} \Theta
(E)^{n} + o(k^{n}) }$,\\
\noindent (a\-vec \'egalit\'e si $q=n$),
\noindent \ (ii) $\displaystyle{ \ \dim H^q(X,{\cal O}(E^{k}\otimes F) \otimes
{\cal I}_{k}(h)) \leq r\frac{k^{n}}{n!} \int _{X(q,E)} (-1)^{q} \Theta
(E)^{n} + o(k^{n}).}$}
\bigskip
Comme dans le cas o\`u la m\'etrique est lisse, le point
(ii) est cons\'equence imm\'ediate du point (i).
\medskip
Remarquons que comme dans le th\'eor\`eme d'annulation de Nadel,
le ph\'enom\`ene nouveau par rapport au cas ${\cal C}^{\infty}$
est la pr\'esence des faisceaux d'id\'eaux~; les groupes de
cohomologie que nous estimons sont les groupes de cohomologie
\`a valeurs dans les grandes puissances de $E$, tordues
par la suite des faisceaux multiplicateurs naturellement
associ\'ee aux singularit\'es de la m\'etrique.
Evidemment, si la m\'etrique est lisse, nous retrouvons les
in\'egalit\'es de J.-P.\ Demailly car tous
les ${\cal I}_{k}(h)$ sont \'egaux au faisceau structural ${\cal O}_X$.
Cependant, mentionnons d\`es maintenant que notre d\'emonstration
repose sur les in\'egalit\'es dans le cas ${\cal C}^{\infty}$ !
\subsection{D\'emonstration du th\'eor\`eme B}
\subsub{Plan de la preuve}
La d\'emarche suivie pour d\'emontrer nos
in\'egalit\'es est la suivante~:
a) apr\`es \'eclatement de $X$ le long de sous-vari\'et\'es d\'efinies par les
singularit\'es de $h$, on se ram\`ene \`a un fibr\'e muni d'une m\'etrique
lisse~; on peut alors appliquer les in\'egalit\'es de Morse holomorphes
dans le cas $h$ lisse \`a la vari\'et\'e obtenue $\tilde{X}$,
b) on relie les groupes de cohomologie sur $\tilde{X}$
\`a ceux de $X$. Pour cela, nous \'etudions
le comportement des faisceaux multiplicateurs
par rapport aux \'eclatements en montrant que
les dimensions des groupes de cohomologie
associ\'es sont asymptotiquement de m\^eme
dimension.
\subsub{R\'eduction au cas lisse}
Commen\c cons par expliquer la premi\`ere partie
de la preuve.
Pour cela, il est bon de remarquer que l'hypoth\`ese ({\cal S})
implique
que les singularit\'es de la m\'etrique
$h$ sont localis\'ees le long d'un ensemble analytique, d\'efini
localement par $ \{ x|\ \forall j,\ f_{j}(x)=0 \}$.
Comme nous l'avons vu dans les exemples, cet ensemble analytique
n'est pas n\'ecessairement irr\'eductible et ses composantes irr\'eductibles
sont en g\'en\'eral
de dimension quelconque.
La r\'eduction au cas lisse consiste dans un premier temps \`a se ramener
\`a une vari\'et\'e $\tilde{X}$, obtenue en \'eclatant $X$ le long de centres
lisses
$\pi : \tilde{X} \to X$ de telle sorte que la m\'etrique
$\tilde{h}=\pi^{*}h$ sur le fibr\'e $\tilde{E}=\pi^{*}E$ n'ait ses
singularit\'es qu'en codimension $1$ (ou, de fa\c con
\'equivalente, de telle
sorte que le faisceau $\tilde{{\cal I}}_{k} (\tilde{h})$ soit inversible).
Dans un deuxi\`eme temps, nous appliquerons les in\'egalit\'es de Morse
dans le cas lisse.
\bigskip
\noindent {\bf a) D\'esingularisation de ${\cal I}_{k} (h)$}
\medskip
Pour rendre les faisceaux ${\cal I}_{k} (h)$ localement libres,
l'id\'ee (classique) est d'\'eclater l'id\'eal
``engendr\'e par les $f_j$". Cependant, une difficult\'e
appara\^{\i}t ici car la donn\'ee des $f_j$ est locale sur $X$
et la notion d'id\'eal engendr\'e par ces fonctions
n'a donc pas de sens {\em a priori}. Nous avons cependant la proposition
suivante~:
\medskip
\noindent {\bf Proposition } {\em
Il existe un faisceau d'id\'eaux global
${\cal J}$, qui co\"{\i}ncide avec la cl\^oture int\'egrale
du faisceau
d'id\'eaux engendr\'e par les $f_{j}$ sur chaque ouvert
o\`u $\varphi$ s'\'ecrit
comme dans l'hypoth\`ese ({\cal S}).
}
\medskip
\noindent {\bf D\'emonstration}
En effet, c'est un fait bien connu (r\'esultant par
exemple du th\'eor\`eme de
Brian\c con-Skoda \cite{BSk74}, voir aussi \cite{L-T74}) que la cl\^oture
int\'egrale du faisceau
d'id\'eaux engendr\'e par les $f_{j}$ est donn\'ee par~:
$${\cal J}_{x} = \{ f \in {\cal O}_{X,x} \ ; \ \exists C >0 \ ,
\ |f(z)| \leq C\exp(\frac{1}{c}\varphi(z)) \ \mbox{au voisinage de}\ x \}.$$
Si maintenant $\varphi _{\alpha}$ et $\varphi _{\beta}$
d\'esignent les poids de la m\'etrique sur des ouverts
trivialisants $U_{\alpha}$ et $U_{\beta}$, alors
sur l'intersection $U_{\alpha} \cap U_{\beta}$,
on a $\varphi _{\alpha} = \varphi _{\beta} + O(1)$.
La caract\'erisation ci-dessus implique donc bien que
${\cal J}$ est d\'efini globalement sur $X$.\hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
Nous sommes en mesure de d\'emontrer la proposition
suivante~:
\medskip
\noindent {\bf Proposition } {\em
Sous les hypoth\`eses pr\'ec\'edentes, il existe une vari\'et\'e
$\tilde{X}$
et $\pi : \tilde{X} \rightarrow X$ une compos\'ee d'un nombre fini
d'\'eclate\-ments de centres lisses tels que le
fibr\'e $\tilde{E} :=\pi^{*}E$ muni
de la m\'etrique singuli\`ere $\tilde{h}=\pi^{*}h$
de poids local $\exp(-\tilde{\varphi})$
v\'erifie la propri\'et\'e suivante :
pour tout $x_{0} \in \tilde{X}$, il existe des coordonn\'ees holomorphes
$w_{1},\ldots \!,w_{n}$ centr\'ees en
$x_{0}$ et une fonction $\tilde{\psi}$ de classe ${\cal C}^{\infty}$
telles que~:
$$ \tilde{\varphi} (w) = c \sum_{j}a_{j}\log|g_{j}(w)|+\tilde{\psi}(w)$$
o\`u les $a_{j}$ sont des entiers positifs ou nuls et o\`u
les $g_{j}$ sont
irr\'eductibles dans ${\cal O}_{\tilde{X},x_{0}}$
et d\'efinissent un diviseur lisse \`a croisements normaux.}
\medskip
\noindent {\bf D\'emonstration}
\'Eclatons l'id\'eal ${\cal J}$ de sorte que l'image inverse
$\pi^{-1}{\cal J}.{\cal O}_{X'}$ soit un faisceau inversible.
Par le th\'eor\`eme d'aplatissement d'Hironaka \cite{Hir75},
on peut dominer
cet \'eclatement par
une vari\'et\'e $\tilde{X}$ obtenue par une suite d'\'eclatements de
centres lisses dans $X$, et l'image inverse de ${\cal J}$ est
toujours inversible !
\noindent Mais alors, la m\'etrique image r\'eciproque sur $\pi^{*}E$ est
donn\'ee
par
\begin{eqnarray*}
\ \tilde{\varphi} & = &
\frac{c}{2}\log(\sum (\lambda_{j}\circ\pi)|f_{j}\circ\pi|^{2})+
\psi\circ\pi \\
& = & \frac{c}{2}\log(|g|^{2}) +
\frac{c}{2}\log(\sum (\lambda_{j}\circ\pi)|h_{j}|^{2})+
\psi\circ\pi \\
& = & \frac{c}{2}\log(|g|^{2}) + \tilde{\psi},
\end{eqnarray*}
\noindent o\`u on a not\'e $g$ le g\'en\'erateur local du faisceau d'id\'eaux
engendr\'e
par les $f_{j}\circ\pi$. Si la d\'ecomposition de $g$ en facteurs
irr\'eductibles
dans $\displaystyle{{\cal O}_{\tilde{X},x_{0}}}$ s'\'ecrit~:
$\displaystyle{g = \prod_{j}g_{j}^{a_{j}}}$,
on a le r\'esultat apr\`es
application du th\'eor\`eme de d\'esingularisation
d'Hironaka \cite{Hir64}
pour rendre le diviseur d\'efini par les $g_{j}$ \`a croisements
normaux.\hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
Dans tout ce qui suit, les objets utilis\'es sont ceux obtenus
apr\`es application de la proposition pr\'ec\'edente.
Notons $\displaystyle{\tilde{{\cal I}}_{k} (\tilde{h})}$ le faisceau
d'id\'eaux des germes
de fonctions holomorphes sur $\tilde{X}$
telles que $\displaystyle{|f|^{2}e^{-2k\tilde{\varphi}}}$
est $L_{\mbox{\scriptsize loc}}^{1}$ (c'est le faisceau multiplicateur
de Nadel associ\'e au fibr\'e $\pi^*E$).
Si nous notons $b_{j,k} = \lfloor ca_{j}k \rfloor$ et $\tilde{D}_{j}$
le diviseur
d\'efini localement par $g_{j}=0$, alors la d\'etermination
du faisceau $\displaystyle{\tilde{{\cal I}}_{k} (\tilde{h})}$
rel\`eve des exemples
pr\'ec\'edents si bien que le lemme suivant en d\'ecoule~:
\medskip
\noindent {\bf Lemme }{\em Sous les conditions pr\'ec\'edentes,
le faisceau d'id\'eaux
$\tilde{{\cal I}}_{k} (\tilde{h})$
s'identifie au faisceau inversible de rang un
${\cal O}(-\sum_{j}b_{j,k}\tilde{D}_{j})$.}
\medskip
\noindent {\bf b) Exemples }
\medskip
Illustrons ce qui pr\'ec\`ede en reprenant les notations
de l'exemple (ii) du 2.1.2.
Dans ces cas \'evidemment simples, il n'est nul besoin d'appliquer les
th\'eor\`emes d'Hironaka~: on explicite directement le choix des
\'eclatements !
\medskip
Si on suppose que tous les $\alpha_{i}$ sont \'egaux
\`a $\alpha$, nous avons vu alors que
${\cal I}(\varphi_{k} )$ est \'egal \`a
${\cal I}_{Y}^{ \lfloor k\alpha \rfloor -p+1}$
o\`u $Y$ est la sous-vari\'et\'e de codimension
$p$ donn\'ee par $\{ z_{1}= \cdots =z_{p}=0 \}$.
\noindent Si $p=1$, le faisceau d'id\'eaux est d\'ej\`a inversible, sinon
\'eclatons ${\Bbb C} ^n$ le long de $Y$. L'expression
de la nouvelle m\'etrique est donn\'ee dans la premi\`ere carte
par $$\displaystyle{\tilde{\varphi}(w)= \alpha\log(|w_{1}|) +
\frac{1}{2}\log(1+|w_{2}|^{2\alpha}+\cdots +|w_{p}|^{2\alpha})}$$ si bien que
$\tilde{{\cal I}}(\varphi_{k}) = {\cal O}(- \lfloor k\alpha \rfloor D)$ o\`u
$D$
est le diviseur exceptionnel de l'\'eclatement.
Il suffit donc dans ce cas d'un \'eclatement en codimension
$p$ pour obtenir le r\'esultat
souhait\'e.
\medskip
De m\^eme, si $\alpha$ est un entier positif et si
$\displaystyle{\varphi(z) =
\frac{1}{2}\log(|z_{1}|^{2}+|z_{2}|^{2\alpha})}$
dans
${\Bbb C} ^n$, il faut cette fois $\alpha$ \'eclatements en codimension~2.
\noindent En effet, \'eclatons ${\Bbb C} ^n$ le long de $\{ z_{1}=z_{2}=0 \}$.
L'expression
de la nouvelle m\'etrique est donn\'ee dans la premi\`ere carte
par $$\displaystyle{\tilde{\varphi}(w)= \log(|w_{1}|) +
\frac{1}{2}\log(1+|w_{1}w_{2}|^{2\alpha})}$$ qui est de la forme voulue alors
qu'on obtient dans la deuxi\`eme carte
$$\displaystyle{\tilde{\varphi}(w)= \log(|w_{2}|) +
\frac{1}{2}\log(|w_{1}|^{2}+|w_{2}|^{2(\alpha-1)})}.$$
On \'eclate alors dans la deuxi\`eme carte le long de
$\{ w_{1}=w_{2}=0 \}$. En r\'ep\'etant ce proc\'ed\'e $\alpha$ fois,
on obtient une m\'etrique de la forme souhait\'ee en tout point.
\noindent D\'ecrivons le faisceau d'id\'eaux obtenu~: pour
tout $j$ compris entre $1$ et $\alpha$, notons $D_{j}$ la transform\'ee
stricte dans $\tilde{X}$ du diviseur exceptionnel du $j$-i\`eme \'eclatement.
Alors $D_{j}$ et $D_{j+1}$ se coupent transversalement et on a
$\tilde{{\cal I}}_{k} (\tilde{\varphi}) = {\cal O}(-kD)$ o\`u $D$
d\'esigne le diviseur \`a croisements normaux
$\displaystyle{D=\sum_{j=1}^{\alpha}jD_{j}}$.
\medskip
\noindent {\bf c) In\'egalit\'es de Morse sur $\tilde{X}$ }
\medskip
On montre maintenant comment appliquer
les in\'egalit\'es de Morse holomorphes
dans le cas ${\cal C}^{\infty}$ sur $\tilde{X}$ au
fibr\'e en droites $\tilde{E} = \pi ^{\ast}E$ muni de
la m\'etrique image r\'eciproque. Pour cela, nous devons
montrer que $(\tilde{E})^k \otimes \tilde{{\cal I}}_{k} (\tilde{h})$
peut essentiellement s'\'ecrire comme la $k$-i\`eme
puissance tensorielle d'un fibr\'e en droites hermitien fixe.
Notons $\displaystyle{c=\frac{u}{v}}$ et supposons que $k = vk'$ est
un multiple du
d\'enomina\-teur de $c$. Comme $b_{j,k}=ca_{j}k$,
le faisceau multiplicateur $\displaystyle{\tilde{{\cal I}}_k (\tilde{h})}$
est \'egal au faisceau inversible
$\displaystyle{{\cal O}(-k'\tilde{D})}$
o\`u $\displaystyle{\tilde{D}=u\sum_{j}a_{j}\tilde{D}_{j}}$.
Avec ces notations, on a la~:
\medskip
\noindent {\bf Proposition } {\em Notons $\hat{E}$
le fibr\'e $\tilde{E}^{v} \otimes {\cal O}(-\tilde{D})$. Alors~:
(i) pour tout $k = vk'$, on a
$(\hat{E})^{k'} = \tilde{E}^{k} \otimes \tilde{{\cal I}}_{k} (\tilde{h}),$
(ii) la m\'etrique hermitienne sur $\hat{E}$, produit
de la m\'etrique $\tilde{h} ^v$ et de la m\'etrique singuli\`ere
naturelle sur ${\cal O}(-\tilde{D})$, est une m\'etrique
hermitienne ${\cal C}^{\infty}$. De plus, et en
dehors des singularit\'es de la m\'etrique $\tilde{h}$,
on a l'\'egalit\'e $\Theta(\hat{E}) =
v\Theta(\tilde{E})=\pi^{\ast}\Theta(E)$.
}
\medskip
\noindent {\bf D\'emonstration }
Le point (i) est \'evident. Pour le point (ii), la m\'etrique
produit naturelle est donn\'ee localement
par le poids
$\displaystyle{\tilde{\chi}(z)=
v \tilde{\phi} (z) -\sum_{j=1}^{n}ua_{j}\log|g_{j}|
= v \tilde{\psi}(z).}$ Ainsi, cette m\'etrique est lisse
et l'\'egalit\'e de courbure d\'ecoule de suite du fait
que $\displaystyle{\partial \overline{\partial} \log|g_j|
= 0}$ l\`a o\`u $g_j$ ne s'annule pas. \hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
Cette proposition nous permet d'estimer les groupes de cohomologie qui nous
int\'eressent~:
\medskip
\noindent {\bf Proposition }{\em
On a pour tout $k$ :
$$\sum_{j=0}^{q}(-1)^{q-j}\dim
H^{j}(\tilde{X},{\cal O}(\tilde{E}^{k} \otimes \tilde{F}) \otimes
\tilde{{\cal I}}_{k} (\tilde{h})) \leq
r\frac{k^{n}}{n!} \int _{X(\leq q,E)} (-1)^{q} \Theta
(E)^{n} + o(k^{n}) $$
o\`u l'int\'egrale est prise sur les points lisses de
la m\'etrique de $E$.
}
\medskip
\newpage
\noindent {\bf D\'emonstration}
D'apr\`es la proposition pr\'ec\'edente, on peut
appliquer les in\'egalit\'es de Morse de Demailly au
fibr\'e $\hat{E}$,
si bien que pour $k=k'v$, on a~:
$$\sum_{j=0}^{q}(-1)^{q-j}\dim
H^{j}(\tilde{X},{\cal O}(\tilde{E}^{k} \otimes \tilde{F}) \otimes
\tilde{{\cal I}}_{k} (\tilde{h})) \leq
r\frac{k'{}^{n}}{n!} \int _{\tilde{X}(\leq q,\hat{E})} (-1)^{q} \Theta
(\hat{E})^{n} + o(k'{}^{n}).$$
\noindent Relions alors l'int\'egrale de courbure sur $\tilde{X}$ \`a une
int\'egrale
de courbure sur $X$.
\noindent Comme $\Theta(\hat{E})=v\Theta(\tilde{E})$ sur
les points lisses de la m\'etrique de $\tilde{E}$ si $k=k'v$,
on a
$$k'{}^{n} \int _{\tilde{X}(\leq q,\hat{E})} \Theta (\hat{E})^{n}
+ o(k'{}^{n})= k^{n} \int _{\tilde{X}(\leq q,\tilde{E})}
\Theta (\tilde{E})^{n} + o(k^{n}).$$
\noindent Notons alors $S$ la r\'eunion des diviseurs exceptionnels de
$\pi : \tilde{X} \rightarrow X $~; $S$ est n\'egligeable pour la
mesure de Lebesgue et on a
\begin{eqnarray*}
\int _{\tilde{X}(\leq q,\tilde{E})} \Theta
(\tilde{E})^{n} & = & \int _{\tilde{X}(\leq q,\tilde{E}) \setminus S} \Theta
(\tilde{E})^{n} = \int _{\tilde{X}(\leq q,\tilde{E}) \setminus S} \Theta
(\pi ^{*}E)^{n} \\
& = & \int _{\tilde{X}(\leq q,\tilde{E}) \setminus S} \pi^{*}(\Theta
(E)^{n}) = \int _{X(\leq q,E) \setminus \pi(S)} \Theta
(E)^{n} \\
& = & \int _{X(\leq q,E)} \Theta
(E)^{n}.
\end{eqnarray*}
\noindent On a donc pour les entiers $k$ multiples d'un entier fixe~:
$$\sum_{j=0}^{q}(-1)^{q-j}\dim
H^{j}(\tilde{X},{\cal O}(\tilde{E}^{k} \otimes \tilde{F}) \otimes
\tilde{{\cal I}}_{k} (\tilde{h})) \leq
r\frac{k^{n}}{n!} \int _{X(\leq q,E)} (-1)^{q} \Theta
(E)^{n} + o(k^{n}) $$
o\`u l'int\'egrale est prise sur les points lisses de
la m\'etrique de $E$.
\medskip
Pour terminer cette preuve, il suffit de montrer que l'estimation
pr\'ec\'edente est valable sans restriction sur $k$.
\noindent En reprenant les notations du lemme et en
posant $\displaystyle{c=\frac{u}{v}}$,
on a, pour $k=k'v+r$
$$ b_{j,k} = ua_{j}k'+r' $$
o\`u $r'$ ne prend qu'un nombre fini de valeurs enti\`eres. On en d\'eduit
alors
\begin{eqnarray*}
\! \tilde{E} ^{k} \otimes \tilde{F} \otimes
{\cal O}(-\sum_{j}b_{j,k}D_{j}) & = & (\tilde{E}^{v})^{k'}\otimes
\tilde{E}^{r'} \otimes \tilde{F} \otimes {\cal O}(-k'u\sum_{j}a_{j}D_{j})
\otimes {\cal O}(-r'\sum_{j}D_{j})\\
& = &
(\tilde{E}^{v})^{k'}\otimes
{\cal O}(-k'u\sum_{j}a_{j}D_{j}) \otimes \hat{F}_{r'}.
\end{eqnarray*}
On raisonne alors comme pr\'ec\'edemment~: on munit
$\hat{F}_{r'}$ d'une m\'etrique lisse quelconque tandis que
$(\tilde{E}^{v})^{k'}\otimes {\cal O}(-k'u\sum_{j}a_{j}D_{j})$
est muni de la m\'etrique lisse naturelle donn\'ee localement
par $v\tilde{\psi} (z)$.
Ceci d\'emontre la proposition.\hskip 3pt \vrule height6pt width6pt depth 0pt
\subsub{Lien entre cohomologie sur $\tilde{X}$ et cohomologie sur $X$}
Pour achever la preuve du th\'eor\`eme B, il reste \`a relier les
groupes $$H^{q}(\tilde{X},{\cal O}(\tilde{E}^{k} \otimes \tilde{F}) \otimes
\tilde{{\cal I}}_{k} (\tilde{h}))$$ de la proposition pr\'ec\'edente et les
$$H^{q}(X,{\cal O}(E^{k}\otimes F) \otimes
{\cal I}_{k}(h))$$ qui nous int\'eressent directement.
Nous adoptons ici une d\'emarche un peu diff\'erente de celle
de notre travail \cite{Bo93b}.
Soient $X$ une vari\'et\'e compacte et $\mu : X' \to X$
l'\'eclatement de $X$ le long d'une sous-vari\'et\'e lisse $Y$.
Soit $E$ un fibr\'e en droites sur $X$, muni d'une
m\'etrique hermitienne singuli\`ere $h$ de poids local
$\exp (-\varphi)$. On note ${\cal I}_k(\varphi)$ le
faisceau multiplicateur associ\'e \`a la m\'etrique
sur $E^k$. Soit enfin $F$ un fibr\'e
vectoriel sur $X$. Nous montrons dans ce paragraphe la proposition
suivante~:
\bigskip
\noindent {\bf Proposition } {\em Supposons qu'au voisinage
de tout point du diviseur exceptionnel de $\mu$, la fonction
$\varphi \circ \mu$ s'\'ecrive~:
$$ \varphi \circ \mu = \alpha \log |f| + \psi,$$
o\`u $\alpha $ est strictement positif, $f$ d\'esigne
une \'equation locale du diviseur exceptionnel de $\mu$ et
$\psi$ est une fonction psh.
\noindent Alors, si $k \alpha >1$, on a pour tout entier
$q \geq 0$~:
$$ H^q(X',K_{X'} \otimes \mu^*(E^k \otimes F) \otimes {\cal I}_k(\varphi \circ
\mu))
\simeq H^q(X,K_X \otimes E^k \otimes F \otimes {\cal I}_k(\varphi)) .$$}
\vspace*{-3mm}
\noindent {\bf Commentaires } Nous ne supposons plus dans ce dernier
r\'esultat que la m\'etrique est \`a singularit\'es analytiques,
ni que les faisceaux ${\cal I}_k(\varphi \circ \mu)$ sont inversibles.
Le point important est que l'hypoth\`ese faite sur
l'\'ecriture de $\varphi \circ \mu$ est automatiquement satisfaite
si $\varphi$ est \`a singularit\'es analytiques et le centre
de l'\'eclatement $Y$ est inclus dans le lieu singulier de la m\'etrique.
La fin de la d\'emonstration
de nos in\'egalit\'es de Morse se conclut par application
r\'ep\'et\'ee de cette proposition aux \'eclatements
successifs dont $\pi : \tilde{X} \to X$ est la compos\'ee.\hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
Avant de d\'emontrer la proposition, mentionnons le
probl\`eme suivant~:
\medskip
\noindent {\bf Question } {\em Si les singularit\'es de $\varphi$ sont
quelconques,
a-t-on toujours
$$ \dim H^q(X',K_{X'} \otimes \mu^*(E^k \otimes F)
\otimes {\cal I}_k(\varphi \circ \mu))
= \dim H^q(X,K_X \otimes E^k \otimes F \otimes {\cal I}_k(\varphi))
+ o(k^n) ?$$}
\vspace*{-3mm}
\noindent {\bf D\'emonstration de la proposition }
Elle repose tout d'abord sur le fait que les faisceaux
de Nadel se comportent bien par image directe. De fa\c con pr\'ecise,
si $(E,h)$ est un fibr\'e en droites muni d'une
m\'etrique singuli\`ere au dessus d'une vari\'et\'e
$X$, et si $\mu : \tilde{X} \to X$ est une modification,
alors \cite{Dem94}
$$\mu _*(K_{\tilde{X}}\otimes {\cal I}(\mu^*h))
= K_X \otimes {\cal I}(h).$$
\noindent \'Evidemment, ceci ne suffit pas pour d\'emontrer qu'il y a
isomorphisme en cohomologie~: l'obstruction est mesur\'ee
par les images directes sup\'erieures. Ainsi, il suffit
de montrer, gr\^ace au
th\'eor\`eme de Leray, que pour tout entier
$q \geq 1$ et pour tout $k$ assez grand,
$$R^{q}\mu_{*}(K_{X'} \otimes {\cal I}_k(\varphi \circ \mu))
= 0.$$
\noindent On note $r$ la codimension de $Y$, centre de l'\'eclatement.
\noindent Dans ce cas, le faisceau $q$-i\`eme image directe sup\'erieure
$R^{q}\mu_{*}(K_{X'} \otimes {\cal I}_k(\varphi \circ \mu))$
est un faisceau \`a support dans $Y$,
la fibre au dessus d'un point $y$ de $Y$ \'etant \'egale \`a~:
$$ F_{k,y} = \limind_{y \in U}
H^{q}(\mu^{-1}(U),(K_{X'}
\otimes {\cal I}_k(\varphi \circ \mu))_{|\mu^{-1}(U)}),$$
o\`u la limite porte sur les voisinages $U$ de $y$
dans $X$.
\noindent Soit donc $U$ un ouvert de Stein voisinage de $y$
et soit $\omega$ une m\'etrique hermitienne
sur $X'$. Il s'agit de montrer
que pour toute forme $u$ de type $(n,q)$ sur $\mu^{-1}(U)$,
\`a coefficients localement $L^2$ satisfaisant~:
(i) $\displaystyle{\overline{\partial}u = 0}$,
(ii) $\displaystyle{
I := \int _{\mu^{-1}(U)}|u|^2 \exp(-2k \varphi \circ \mu) dV_{\omega} < +\infty
,}$
\noindent il existe une forme $v$ de type $(n,q-1)$
\`a coefficients localement $L^2$ satisfaisant~:
(i)' $\displaystyle{\overline{\partial}v = u}$,
(ii)' $\displaystyle{
\int _{\mu^{-1}(U)}|v|^2 \exp(-2k \varphi \circ \mu) dV_{\omega} < +\infty .}$
\noindent R\'esoudre un tel probl\`eme est en g\'en\'eral possible
gr\^ace aux estimations $L^2$ de H\"ormander. La difficult\'e
ici est que la fonction
$k \varphi \circ \mu$
n'est pas strictement psh au voisinage du diviseur
exceptionnel $D$ de l'\'eclatement. C'est exactement ici que nous utilisons
l'hypoth\`ese faite sur $\varphi$.
\noindent En effet, l'\'egalit\'e
$$ \varphi \circ \mu = \alpha \log |f| + \psi$$
se traduit par~:
$$ {\cal I}_k(\varphi \circ \mu)
= {\cal O}(- \lfloor k\alpha \rfloor D) \otimes {\cal I}
\left( k\psi + (k\alpha - \lfloor k\alpha \rfloor)\log |f|) \right).$$
\noindent Il s'agit alors de montrer l'annulation des groupes
$$H^{n,q}\left (\mu^{-1}(U),{\cal O}(- \lfloor k\alpha \rfloor D)
\otimes {\cal I}(k\psi + (k\alpha - \lfloor k\alpha \rfloor)\log |f|\right ),$$
autrement dit de r\'esoudre le probl\`eme du
$\overline{\partial}$~:
(i)' $\displaystyle{\overline{\partial}v = u}$,
(ii)' $\displaystyle{
\int _{\mu^{-1}(U)}|v|^2 \exp(-2k \psi) dV_{\omega} < +\infty ,}$
\noindent pour des $(n,q)$ formes {\em \`a valeurs dans le fibr\'e
${\cal O}(- \lfloor k\alpha \rfloor D)$}. Or, le fibr\'e
${\cal O}(-D)_{|D} = {\cal O}_D (1)$ est strictement positif sur
les fibres de l'\'eclatement. On peut donc munir
${\cal O}(-\lfloor k\alpha \rfloor D)$
d'une m\'etrique \`a courbure strictement positive au voisinage du
diviseur exceptionnel, et comme $\psi$ peut \^etre suppos\'ee
strictement psh en dehors de $D$, nous sommes maintenant
dans les hypoth\`eses d'application des estimations $L^2$ de
H\"ormander.\hskip 3pt \vrule height6pt width6pt depth 0pt
\section{Caract\'erisations analytiques des vari\'et\'es \- de \-
Moishezon}
Dans la lign\'ee des conditions suffisantes donn\'ees par Y.-T.\ Siu
et J.-P.\ Demailly pour caract\'eriser les vari\'et\'es de Moishezon,
nous montrons la caract\'erisation suivante~:
\bigskip
\noindent {\bf Th\'eor\`eme C} {\em Une vari\'et\'e compacte $X$ de
dimension $n$
est de Moishezon si et
seulement si il existe sur $X$ un courant $T$ ferm\'e
de bidegr\'e $(1,1)$
tel que~:
(i) $ \displaystyle{\{ T \} \in H^{2}(X,{\Bbb Z}) }$,
(ii) $\displaystyle{ T= \frac{i}{\pi} \partial \overline{\partial} \varphi
+ \alpha }$,
o\`u $\varphi$ est une fonction r\'eelle \`a singularit\'es analytiques
et o\`u
$\alpha$ est un repr\'esentant
$ {\cal C}^{\infty}$ de $\{ T \}$,
(iii) $\displaystyle{\int_{X(\leq 1,T)} T^{n} > 0}$ o\`u
l'int\'egrale est prise
sur les points lisses du courant $T$.
}
\bigskip
L'id\'ee de donner une caract\'erisation analytique
des vari\'et\'es de Moishezon en terme de courant de
courbure est aussi pr\'esente dans
un travail de S.\ Ji et B.\ Shiffman \cite{JiS93} simultan\'e
au n\^otre. Nos in\'egalit\'es permettent ainsi de d\'emontrer
le r\'esultat suivant~:
\bigskip
\noindent {\bf Th\'eor\`eme D \cite{JiS93}, \cite{Bo93b}} {\em
Une vari\'et\'e compacte $X$ de
dimension $n$
est de Moishezon si et
seulement si il existe sur $X$ un courant $T$ ferm\'e de bidegr\'e $(1,1)$
tel que~:
(i) $ \displaystyle{\{ T \} \in H^{2}(X,{\Bbb Z}) }$,
(ii) le courant $T$ est strictement positif (i.e minor\'e
par une $(1,1)$-forme ${\cal C}^{\infty}$ hermitienne).
}
\bigskip
Ces deux \'enonc\'es fournissent l'extension naturelle aux vari\'et\'es
de Moishezon du th\'eor\`eme de plongement de Kodaira pour les vari\'et\'es
projectives.
Avant de d\'emontrer les r\'esultats ci-dessus,
nous rappelons deux r\'esultats sur les courants.
\subsection{Deux rappels}
Il est bien connu \cite{G-H78} que la seule obstruction
pour qu'une $(1,1)$-forme ferm\'ee r\'eelle ${\cal C}^{\infty}$
soit la forme de courbure d'un fibr\'e en droites hermitien
est que sa classe de cohomologie soit enti\`ere, i.e
appartienne \`a $H^2(X,{\Bbb Z})$. Nous montrons dans la proposition suivante
que ce r\'esultat persiste pour les courants quasi positifs~:
\medskip
\noindent {\bf Proposition }
{\em Soit $T$ un courant quasi-positif ferm\'e, de bi-degr\'e
$(1,1)$ dont la classe de cohomologie
$\{ T \}$ est dans $H^2(X,{\Bbb Z})$.
Alors, il existe un fibr\'e en droites $E$
muni d'une m\'etrique singuli\`ere dont
le courant de courbure est \'egal \`a $T$.}
\medskip
\noindent {\bf D\'emonstration }
Elle suit la d\'emonstration du cas ${\cal C}^{\infty}$ \cite{S-S85}.
On recouvre la vari\'et\'e $X$ par des ouverts de Stein
contractiles dont les intersections mutuelles
sont elles aussi contractiles. Sur chaque
ouvert $U_{\alpha}$, on \'ecrit
$\displaystyle{T = \frac{i}{\pi}\partial \overline{\partial}
\varphi_{\alpha} }$. Comme $T$ est quasi positif, les fonctions
$\varphi_{\alpha}$ sont quasi plurisousharmoniques, donc
localement int\'egrables (c'est le seul point qui
diff\`ere du cas ${\cal C}^{\infty}$).
\noindent De l\`a, on \'ecrit successivement
$\varphi_{\alpha \beta} := \varphi_{\beta}-\varphi_{\alpha}$
sur l'intersection $U_{\alpha}\cap U_{\beta}$,
puis $$ i(\overline{\partial} - \partial)\varphi_{\alpha \beta}
= 2\pi d u_{\alpha \beta}.$$
Les fonctions $c_{\alpha \beta \gamma} := u_{\beta \gamma} - u_{\alpha \gamma}
+ u_{\alpha \beta}$ sont constantes. Comme $\{ T \}$ est
enti\`ere, la classe $\{ c_{\alpha \beta \gamma} \}$ l'est aussi.
Il existe donc une $1$-c\^ochaine \`a coefficients r\'eels
$\{ b_{\alpha \beta} \}$ telle que~:
$$ c_{\alpha \beta \gamma} + b_{\beta \gamma} - b_{\alpha \gamma} + b_{\alpha
\beta}
= m_{\alpha \beta \gamma} \in {\Bbb Z}.$$
On pose alors~:
$$ g_{\alpha \beta} :=
\exp (\varphi_{\alpha \beta} + 2i\pi(u_{\alpha \beta}
+ b_{\alpha \beta})).$$
Les $g_{\alpha \beta}$ sont holomorphes sans z\'ero et
forment un cocycle~: on note $E$ le fibr\'e en droites
associ\'e. Comme
$$ |g_{\alpha \beta}|\exp(-\varphi_{\alpha})=\exp(-\varphi_{\beta}),$$
les poids $\exp(-\varphi_{\alpha})$ d\'efinissent une
m\'etrique singuli\`ere sur $E$ dont le courant de courbure
est $T$.\hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
Comme nos in\'egalit\'es de Morse supposent que la
m\'etrique est \`a singularit\'es analytiques, nous avons besoin
d'un r\'esultat d'approximation permettant de
s'y ramener. De fa\c con g\'en\'erale, \'etant donn\'e
un courant ferm\'e $T$ sur une vari\'et\'e compacte,
c'est un probl\`eme classique que de vouloir le
r\'egulariser dans la m\^eme classe de cohomologie.
Malheureusement, il n'est pas possible en g\'en\'eral
de r\'egulariser ${\cal C}^{\infty}$ tout en perdant
aussi peu de positivit\'e que souhait\'e. L'obstruction
\`a le faire est mesur\'ee par les nombres de Lelong du
courant. Pour ne pas perdre de positivit\'e,
on r\'egularise en autorisant des singularit\'es
analytiques. Le r\'esultat que nous utilisons est le th\'eor\`eme
d'approximation des courants de J.-P.\ Demailly \cite{Dem92}~:
\bigskip
\noindent {\bf Th\'eor\`eme (J.-P.\ Demailly, 1992)}
{\em Soit $T$ un courant ferm\'e de bi-degr\'e $(1,1)$
de sorte que $T \geq \alpha$ o\`u $\alpha$ est une $(1,1)$
forme ${\cal C}^{\infty}$.
Alors pour toute m\'etrique hermitienne $\omega$
de classe ${\cal C}^{\infty}$ sur $X$, il existe
une suite de courants $T_{\varepsilon}$ telle que~:
(i) $\{ T_{\varepsilon} \} = \{ T \}$,
(ii) $T_{\varepsilon}$ tend (faiblement) vers $T$ lorsque
$\varepsilon$ tend vers $0$,
(iii) $T_{\varepsilon} \geq \alpha - \varepsilon \omega$,
(iv) $\displaystyle{ T_{\varepsilon}
= i \partial \overline{\partial} \varphi _{\varepsilon}
+ \beta }$,
o\`u $\varphi _{\varepsilon}$ est une fonction r\'eelle
\`a singularit\'es analytiques
et o\`u
$\beta$ est un repr\'esentant
$ {\cal C}^{\infty}$ de $\{ T \}$.
}
\bigskip
Il n'est pas question ici de donner la d\'emonstration
de ce r\'esultat, mais pour \'eclairer le lecteur,
mentionnons la premi\`ere \'etape de la d\'emonstration,
qui est une version locale du r\'esultat~:
\medskip
\noindent {\bf Proposition }
{\em Soit $\varphi$ une fonction psh dans
la boule unit\'e $B$ de ${\Bbb C} ^n$. Pour $k$ entier
positif, notons ${\cal H}(k \varphi)$ l'espace
de Hilbert d\'efini de la fa\c con suivante~:
$$ {\cal H}(k \varphi) = \{ f \in {\cal O}(B) \ | \
\int _B |f|^2 \exp(-2k\varphi) d\lambda < +\infty \}.$$
Soit enfin $(\sigma_{j,k})_j$ une base orthonorm\'ee de
${\cal H}(k \varphi)$.
Alors la suite de fonctions
$$ \varphi _k := \frac{1}{2k} \log (\sum_j |\sigma_{j,k}|^2) $$
converge vers $\varphi$ simplement et dans $L_{\mbox{\scriptsize loc}}^1$
lorsque $k$ tend vers $+\infty$.}
\medskip
La difficult\'e du th\'eor\`eme r\'eside dans le recollement
des diverses approximations locales donn\'ees par la proposition
pr\'ec\'edente. Les fonctions $\lambda _j$ figurant dans
la d\'efinition de l'hypoth\`ese {\cal S} proviennent
essentiellement de partition de l'unit\'e. En particulier,
elles ne s'annulent pas toutes simultan\'ement si bien que les
singularit\'es sont vraiment donn\'ees par les z\'eros communs
d'une famille de fonctions holomorphes.
\subsection{D\'emonstration des th\'eor\`emes C et D}
Nous d\'emontrons ici les deux caract\'erisations
analytiques. Pour cela, on commence par remarquer
que le sens faux dans le cadre des m\'etriques lisses
est v\'erifi\'e de fa\c con presque imm\'ediate dans le cadre plus souple
des vari\'et\'es de Moishezon.
Soit en effet $X$ une vari\'et\'e de Moishezon
et $\pi : \hat{X} \to X$ une modification projective.
Si $\hat{\omega}$ est une $(1,1)$ forme ${\cal C}^{\infty}$
d\'efinie positive sur $\hat{X}$ telle que $\{ \hat{\omega} \} \in
H^{2}(\hat{X},{\Bbb Z})$, et si $\omega$ est une $(1,1)$ forme ${\cal C}^{\infty}$
d\'efinie positive sur $X$, alors la forme $\pi^{*}\omega$
est ${\cal C}^{\infty}$ et semi-positive. Il
existe donc une constante $A > 0$ telle que
$\hat{\omega} \geq A\pi^{*}\omega$.
Par cons\'equent,
le courant $T=\pi_{*} \hat{\omega}$ v\'erifie $T \geq A\omega$~:
en effet, si $\alpha$ est une $(n-1,n-1)$ forme positive
${\cal C}^{\infty}$ sur $X$, on a
\vspace{-3mm}
\begin{eqnarray*}
\ <T,\alpha> & = & \int_{\hat{X}} \hat{\omega} \wedge \pi^{*} \alpha \\
& \geq & \int_{\hat{X}} A \pi^{*}\omega \wedge \pi^{*} \alpha \\
& = & A \int_{X} \omega \wedge \alpha \\
& = & <A\omega,\alpha>.
\end{eqnarray*}
Le courant $T$ satisfait les points (i) et (ii) du th\'eor\`eme D,
et le th\'eor\`eme d'approximation des courants de Demailly
rappel\'e pr\'ec\'edemment implique qu'il existe
un courant $T' \in \{ T \}$
ayant localement les singularit\'es de l'hypoth\`ese {\cal S}
et
tel que $T' \geq (A/2) \omega$. Ainsi, $T'$ v\'erifie la conclusion
du th\'eor\`eme C.
\medskip
Pour la r\'eciproque dans le th\'eor\`eme C, soit $T$
satisfaisant (i), (ii) et (iii). Alors, il existe un fibr\'e
en droites sur $X$ muni d'une m\'etrique
hermitienne singuli\`ere dont le courant de courbure est \'egal \`a
$T$. Les in\'egalit\'es de Morse singuli\`eres impliquent
que~:
$$ \dim H^0(X,E^k \otimes {\cal I}_k(h))
- \dim H^1(X,E^k \otimes {\cal I}_k(h)) \sim_{k \to +\infty} k^n.$$
{\em A fortiori},
$$ \dim H^0(X,E^k) \sim_{k \to +\infty} k^n ,$$
i.e le fibr\'e $E$ est gros et $X$ est de Moishezon.
Pour le th\'eor\`eme D, si $T$ est strictement positif,
on peut supposer par le th\'eor\`eme d'approximation des
courants que $T$ a localement les singularit\'es de l'hypoth\`ese {\cal S}.
On conclut alors comme ci-dessus.\hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
\noindent {\bf Remarque } La preuve du th\'eor\`eme D
donn\'ee par S.\ Ji et B.\ Shiffman suit la
d\'emarche suivante~: on approche, comme pr\'ec\'edemment,
le courant $T$ par un courant strictement positif
et singulier sur un ensemble analytique $S$. Puis
on montre directement gr\^ace aux estimations
$L^2$ de H\"ormander appliqu\'ees \`a la vari\'et\'e
k\"ahl\'erienne compl\`ete $X \backslash S$ que les grandes
puissances de $E$ engendrent les $1$-jets en dehors de $S$.
\subsection{Quelques commentaires}
Les th\'eor\`emes C et D sont satisfaisants car ils donnent
une caract\'erisation analytique des vari\'et\'es de Moishezon.
Cependant, on souhaiterait pouvoir se dispenser de
l'hypoth\`ese faite sur les singularit\'es
dans le th\'eor\`eme C ou affaiblir l'hypoth\`ese de stricte
positivit\'e du th\'eor\`eme D. On conjecture le
r\'esultat suivant~:
\medskip
\noindent {\bf Conjecture }
{\em Une vari\'et\'e compacte $X$
est de Moishezon si et
seulement si il existe sur $X$ un courant $T$ positif ferm\'e
de bi-degr\'e $(1,1)$
tel que~:
(i) $ \displaystyle{\{ T \} \in H^{2}(X,{\Bbb Z}) }$,
(ii) il existe un ouvert $U$ sur lequel $T$ est
strictement positif.}
\medskip
La condition (ii) est une fa\c con d'assurer que le support
de $T$ n'est pas n\'egligeable pour la mesure de Lebesgue.
Evidemment, l'hypoth\`ese {\cal S} nous a permis de
d\'efinir les int\'egrales de courbure en int\'egrant
simplement sur la partie lisse de la m\'etrique. Dans le cas
g\'en\'eral, d\'efinir un produit de courants $T^n$ est un probl\`eme
de type {\bf Monge-Amp\`ere}.
\medskip
Le r\'esultat suivant va dans le sens de la
conjecture~:
\medskip
\noindent {\bf Proposition }
{\em La conjecture est vraie dans le cas des
surfaces complexes (et dans ce cas, $X$
\'etant de Moishezon est donc projective).}
\medskip
\noindent {\bf D\'emonstration }
Soit $\omega$ une m\'etrique hermitienne sur $X$,
et pour $\varepsilon > 0$, soit $T_{\varepsilon}$
un courant donn\'e par le th\'eor\`eme d'approximation
des courants v\'erifiant $T_{\varepsilon} \geq - \varepsilon \omega$.
Par le th\'eor\`eme C, il suffit de montrer
que pour $\varepsilon$ assez petit, on a
$$ \displaystyle{\int_{X(\leq 1,T_{\varepsilon})} T_{\varepsilon}^{n} > 0}.$$
Comme il existe un ouvert $U$ sur lequel $T$ est
strictement positif (disons sup\'erieur \`a $C_U \omega _{| U}$
o\`u $C_U$ est une constante strictement positive),
il existe un ouvert plus petit $U'$ ind\'ependant de $\varepsilon$
sur lequel $T_{\varepsilon}$ est sup\'erieur \`a $(C_{U}/2) \omega _{| U'}$.
On en d\'eduit qu'il existe une constante $C>0$, ind\'ependante de
$\varepsilon$
telle que pour tout $\varepsilon$ petit, on ait
$$\displaystyle{\int_{X(0,T_{\varepsilon})} T_{\varepsilon}^{n}
\geq \int_{U'} T_{\varepsilon}^{n} > C}.$$
Il suffit donc de montrer que
$$ \lim _{\varepsilon \to 0}\int_{X(1,T_{\varepsilon})} T_{\varepsilon}^{n}
=0.$$
Or, sur l'ouvert $X(q,T_{\varepsilon})$, on a
$$ 0 \leq (-1)^q T_{\varepsilon}^{n} \leq \frac{n!}{q! (n-q)!}
\varepsilon ^q \omega ^q \wedge
(T_{\varepsilon} + \varepsilon \omega )^{n-q}.$$
Il suffit donc de contr\^oler les produits de
Monge-Amp\`ere et plus pr\'ecis\'ement de montrer que pour $q >0$
(et dans la perspective de la conjecture, $q=1$ suffit), on a~:
$$ \lim _{\varepsilon \to 0} \varepsilon ^q \int_X \omega ^q \wedge
(T_{\varepsilon} + \varepsilon \omega )^{n-q} =0.$$
C'est \'evidemment vrai pour $q=n$ et, si on suppose
de plus que la m\'etrique $\omega$ est une m\'etrique de Gauduchon,
alors la formule de Stokes donne
$$ \int_X \omega ^{n-1} \wedge
(T_{\varepsilon} + \varepsilon \omega ) = \mbox{Cste} + O(\varepsilon),$$
et donc c'est aussi vrai pour $q = n-1$.
Dans le cas des surfaces, on a l'estimation souhait\'ee
pour $q= n-1 =1$, donc $X$ est de Moishezon par le
th\'eor\`eme C.\hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
Comme le montre la preuve ci-dessus, le cas g\'en\'eral
pourrait d\'ecouler du contr\^ole des masses de Monge-Amp\`ere
des approximations $\varphi _{\varepsilon}$ utilis\'ees par
J.-P.\ Demailly dans la d\'emonstration de son th\'eor\`eme
d'approximation des courants.
\section{Une version alg\'ebrique singuli\`ere des in\'egalit\'es
de Morse}
Dans ce paragraphe, nous donnons quelques exemples ``alg\'e\-bri\-ques"
de faisceaux d'id\'eaux de Nadel. Leur origine en g\'eom\'etrie
alg\'ebrique se situe dans la version du th\'eor\`eme de Kawamata-Viehweg
pour les diviseurs \`a coefficients rationnels. Comme dans \cite{Dem90} ou
\cite{EsV92}, ce faisceau d'id\'eaux sert de terme correctif dans le cas o\`u
les diviseurs consid\'er\'es ne sont pas \`a croisements normaux.
Par ailleurs, il existe une version alg\'ebrique
des in\'egalit\'es
de Morse holomorphes de Demailly \cite{Dem94} dans le cas d'un
fibr\'e en droites
diff\'erence de deux fibr\'es amples. Il est alors naturel de donner
une version analogue dans le cadre singulier. \'Evidemment, il s'agit
essentiellement d'une reformulation dans un cas particulier de
notre theor\`eme~B.
\subsection{Th\'eor\`eme de Kawamata-Viehweg}
Soient $X$ une vari\'et\'e projective, et $M$ un diviseur rationnel
effectif de $X$. On note $M = \sum a_i D_i$ o\`u les
$a_i$ sont des rationnels positifs et les $D_i$ sont des diviseurs
irr\'eductibles. Nous notons ${\cal I}(M)$
le faisceau d'id\'eaux de Nadel associ\'e \`a la m\'etrique singuli\`ere
$\phi = \sum a_i \log|g_i|$
o\`u $g_i$ est un g\'en\'erateur local de $D_i$.
On rappelle que si $M$ est
\`a croisements normaux, alors ${\cal I}(M)$ n'est rien d'autre
que le faisceau
inversible ${\cal O}(- \lfloor M \rfloor)$ o\`u
$\lfloor M \rfloor := \sum \lfloor a_i \rfloor D_i$.
Avec ces notations, rappelons le th\'eor\`eme
de Kawamata-Viehweg \cite{Kaw82}, \cite{Vie82}~:
\medskip
\noindent {\bf Th\'eor\`eme d'annulation de Kawamata-Viehweg } {\em
Soient $X$ une vari\'et\'e projective et
$L$ un fibr\'e en droites sur $X$.
On suppose que $L = M + \sum_{j} \alpha _jE_j$
o\`u~:
(i) $M$ est un ${\Bbb Q}$-diviseur effectif gros et nef,
(ii) les $\alpha _j$ sont des r\'eels v\'erifiant
$0 \leq \alpha _j <1$,
(iii) le diviseur $\sum_{j} \alpha _jE_j$ est
\`a croisements normaux.
\noindent Alors
$$ H^q(X,K_X + L)=0 \ \mbox{pour tout}\ q \geq 1 .$$}
Dans le cas o\`u $\sum_{j} \alpha _jE_j$
n'est pas \`a croisements
normaux dans l'\'enonc\'e du th\'eor\`eme de Kawamata-Viehweg,
le faisceau multiplicateur de Nadel sert de terme correctif.
Nous allons d\'etailler un exemple utilis\'e par
L.\ Ein et R.\ Lazarsfeld dans l'\'etude
des diviseurs \`a singularit\'e presque isol\'ee (voir
le papier de R.\ Lazarsfeld \cite{Laz93} pour plus de d\'etails).
Le contexte est le suivant : soient $X$ une vari\'et\'e projective,
$A$ un fibr\'e gros et nef sur $X$ et $D$ un diviseur dans le syst\`eme
lin\'eaire $|kA|$.
Choisissons $\pi : \tilde{X} \rightarrow X$ une compos\'ee d'un nombre fini
d'\'eclate\-ments de centres lisses de sorte que $\pi ^* D$ soit
un diviseur
\`a croisements normaux et soit $K_{\tilde{X}/X}$ la diff\'erence
des fibr\'es canoniques~:
$$K_{\tilde{X}/X} = K_{\tilde{X}} - \pi^*K_{X}.$$
Alors, pour $\lambda$ rationnel, $0\leq \lambda <1$, on pose~:
$${\cal I}_{\lambda}=
\pi_*\left(K_{\tilde{X}/X}-
\lfloor \pi^*(\lambda \frac{D}{k}) \rfloor \right).$$
La proposition suivante donne les propri\'et\'es de ${\cal I}_{\lambda}$~:
\medskip
\noindent {\bf Proposition }{\em On a
(i) le faisceau d'id\'eaux $\displaystyle{{\cal I}_{\lambda}}$
est \'egal au faisceau multiplicateur
$\displaystyle{{\cal I}(\lambda \frac{D}{k})}$, en particulier
${\cal I}_{\lambda}$ est ind\'ependant de la r\'esolution $\pi$ choisie,
(ii) les images directes sup\'erieures
$\displaystyle{R^q \pi_*\left(K_{\tilde{X}/X}-
\lfloor \pi^*(\lambda \frac{D}{k}) \rfloor \right)}$ sont
nulles pour tout
$q \geq 1$,
(iii) les groupes de cohomologie
$\displaystyle{H^q(X,(K_X + A)\otimes{\cal I}_{\lambda})}$ sont nuls pour tout
$q \geq 1$.
}
\medskip
\noindent {\bf D\'emonstration}
Pour le point (i), nous utilisons \`a nouveau l'identit\'e
$$\mu _*(K_{\tilde{X}}\otimes {\cal I}(\mu^*h))
= K_X \otimes {\cal I}(h).$$
Dans notre situation, on a
\vspace{-2mm}
$$ \pi_*\left(K_{\tilde{X}}\otimes
{\cal I}(\pi^*(\lambda \frac{D}{k}))\right)
= K_X \otimes {\cal I}(\lambda \frac{D}{k}).$$
\vspace{-3mm}
Comme $\displaystyle{\pi^*(\lambda \frac{D}{k})}$ est \`a croisements normaux,
on a
\vspace{-2mm}
$$ {\cal I}(\pi^*(\lambda \frac{D}{k}))=
{\cal O}\left(- \lfloor \pi^*(\lambda \frac{D}{k}) \rfloor \right).$$
De l\`a, on d\'eduit successivement
\vspace{-3mm}
\begin{eqnarray*}
{\cal I}_{\lambda} & = &
\pi_*\left(K_{\tilde{X}/X}-
\lfloor \pi^*(\lambda \frac{D}{k}) \rfloor \right)\\
& = & \pi_*\left(K_{\tilde{X}}-\pi^*K_X -
\lfloor \pi^*(\lambda \frac{D}{k}) \rfloor \right)\\
& = & -K_X \otimes K_X \otimes {\cal I}(\lambda \frac{D}{k}) \\
& = & {\cal I}(\lambda \frac{D}{k}).
\end{eqnarray*}
\noindent Ceci d\'emontre le point (i).
Pour le point (ii), la d\'emonstration repose sur l'observation
classique suivante (d\'ej\`a observ\'ee par H.\ Grauert et
O.\ Riemenschneider \cite{GrR70})~: les faisceaux $R^q\pi_*{\cal F}$ sont nuls
si et
seulement si pour tout fibr\'e $L$ sur $X$ suffisamment ample
et tout $q \geq 1$,
on a $H^q(\tilde{X},{\cal F}\otimes \pi^*L) = 0$.
Comme
$$ R^q \pi_* \left(\pi^*(K_X + \pi^*A) +
K_{\tilde{X}/X}- \lfloor \pi^*(\lambda \frac{D}{k}) \rfloor \right) =$$
$$(K_X + \pi^*A)\otimes R^q \pi_* \left(
K_{\tilde{X}/X}- \lfloor \pi^*(\lambda \frac{D}{k}) \rfloor \right),$$
il suffit de montrer que pour tout fibr\'e $L$ sur $X$ suffisamment ample
et tout $q \geq 1$, on a
$$ H^q\left(\tilde{X},\pi^*L + \pi^*A + K_{\tilde{X}}-
\lfloor \pi^*(\lambda \frac{D}{k})\rfloor \right) = 0. $$
Or ce dernier groupe est \'egal \`a
$$ H^q\left(\tilde{X}, K_{\tilde{X}} + \pi^*L + \left(1-\lambda\right)\pi^*A
+ \left(\lambda\pi^*A - \lfloor \lambda\pi^*A\right)\rfloor \right)$$
qui est nul gr\^ace au th\'eor\`eme de Kawamata-Viehweg
appliqu\'e au ${\Bbb Q}$-diviseur effectif gros et nef
$$ M:= \pi^*L + (1-\lambda)\pi^*A .$$
Pour le point (iii), on a
$$ H^q\left(X,\left(K_X + A\right)\otimes{\cal I}_{\lambda}\right)=
H^q\left(\tilde{X}, K_{\tilde{X}} + \pi^*A-\lfloor \pi^*(\lambda
\frac{D}{k})\rfloor \right).$$
Ce dernier groupe s'\'ecrit encore
$\displaystyle{H^q\left(\tilde{X}, K_{\tilde{X}} +\left(1-\lambda\right)\pi^*A
+ \lambda\pi^*A - \lfloor \lambda\pi^*A \rfloor \right)}$
qui est nul \`a nouveau par le th\'eor\`eme de Kawamata-Viehweg.\hskip 3pt \vrule height6pt width6pt depth 0pt
\subsection{In\'egalit\'es de Morse alg\'ebriques singuli\`eres}
En nous inspirant de l'exemple pr\'ec\'edent, nous sommes en mesure
de donner une version alg\'ebrique des in\'egalit\'es de Morse
holomorphes singuli\`eres. Pour cela, rappelons au pr\'ea\-la\-ble la version
suivante des in\'egalit\'es de Morse
holomorphes de J.-P.\ Demailly \cite{Dem94}~:
\bigskip
\noindent {\bf Th\'eor\`eme }
{\em Soit $X$ une vari\'et\'e k\"ahl\'erienne de dimension $n$ et soient
$F$ et $G$ deux fibr\'es en droites nef sur $X$.
Alors, on a~:
$$ \sum _{j=0}^{q}(-1)^{q-j} \dim H^{j}(X,k(F-G))
\leq \frac{k^{n}}{n!}\sum _{j=0}^{q}(-1)^{q-j}{n \choose j} F^{n-j}\cdot G^j
+ o(k^{n}).$$
}
\bigskip
Ce r\'esultat a \'et\'e obtenu dans un premier temps par
J.-P.\ Demailly comme
cons\'equence des in\'egalit\'es de Morse, et plus r\'ecemment,
F.\ Angelini en a donn\'e une d\'emonstration purement alg\'ebrique
\cite{Ang95}. Auparavant, S.\ Trapani \cite{Tra91} et Y.-T.\ Siu
\cite{Siu93} avaient d\'emontr\'e le cas particulier
$q = 1$ en vue d'obtenir des crit\`eres num\'eriques pour
l'existence de sections. Le terme alg\'ebrique a ici une double origine~:
les estimations font intervenir des nombres d'intersection
\`a la place d'int\'egrales de courbure, et un cas particulier
du th\'eor\`eme est celui d'une vari\'et\'e projective et
d'un fibr\'e \'ecrit comme diff\'erence de deux fibr\'es amples.
Nous montrons le r\'esultat suivant~:
\bigskip
\noindent {\bf Th\'eor\`eme E } {\em
Soit $X$ une vari\'et\'e k\"ahl\'erienne de dimension $n$ et soient
$F$ et $G$ deux fibr\'es en droites sur $X$.
On suppose que $G$ est nef, et qu'il existe un entier
positif $m$, un fibr\'e en droites nef $A$ et un diviseur effectif
$D$ de sorte que~: $mF = A + D.$
Alors, on a~:
$$ \sum _{j=0}^{q}(-1)^{q-j}
\dim H^{j}((X,k(F-G)\otimes {\cal I}_{k}(m^{-1}D)) $$
$$ \leq
\frac{k^{n}}{n!}
\sum _{j=0}^{q}(-1)^{q-j}{n \choose j} m^{-n+j}A^{n-j}\cdot G^j
+ o(k^{n}).$$
}
\bigskip
\noindent {\bf D\'emonstration}
Soit $\pi : \tilde{X} \rightarrow X$ une
compos\'ee d'un nombre fini
d'\'eclate\-ments de centres lisses de sorte que $\pi ^* D$ soit un diviseur
\`a croisements normaux.
\noindent Comme dans le cadre purement analytique de nos in\'egalit\'es
de Morse singuli\`eres, on travaille sur
$\tilde{X}$ o\`u l'on applique simplement
l'\'enonc\'e pr\'ec\'edent.
\noindent D\'etaillons bri\`evement~: le faisceau d'id\'eaux
${\cal I}_{k}(m^{-1}D)$
est \'egal
\`a l'image directe $$\pi_*\left(K_{\tilde{X}/X}-\lfloor
\pi^*(km^{-1}D)\rfloor \right),$$
si bien qu'il s'agit d'estimer les dimensions
$$\dim H^{q}\left(\tilde{X},k\pi^*(F-G)-\lfloor
\pi^*(km^{-1}D)\rfloor\right).$$
\noindent Or pour $k$ multiple de $m$, on a
$$k\pi^*(F-G)-\lfloor \pi^*(km^{-1}D)\rfloor = k\pi^*(m^{-1}A-G),$$
\noindent et il suffit d'appliquer le th\'eor\`eme pr\'ec\'edent
aux fibr\'es nef $\pi^*A$ et $\pi^*G$.\hskip 3pt \vrule height6pt width6pt depth 0pt
\chapter{\'Etude de certaines vari\'et\'es
de Moishezon dont le groupe de Picard est
infini cyclique}
Le th\`eme central de ce chapitre
est l'\'etude d'une classe particuli\`ere
de vari\'et\'es de Moishezon~: celles dont
le groupe de Picard est ${\Bbb Z}$ et dont le fibr\'e
canonique est gros. En faisant l'hypoth\`ese suppl\'ementaire
que la vari\'et\'e $X$ devient projective apr\`es un seul
\'eclatement de centre lisse et projectif, nous \'etudions ce centre
{\em via} la th\'eorie de Mori
sur le mod\`ele projectif. Nous obtenons alors
une restriction sur la dimension du centre de l'\'eclatement
dans le cas o\`u le fibr\'e canonique n'est pas nef.
Apr\`es avoir donn\'e une nouvelle famille de vari\'et\'es
de Moishezon ne poss\'edant pas de fibr\'e en droites
gros et nef et s'inscrivant
dans ce cadre d'\'etude, nous nous restreignons \`a
la dimension $4$. Nous obtenons alors une description
pr\'ecise du centre de l'\'eclatement et montrons que
notre construction est essentiellement unique dans le cas
o\`u le fibr\'e canonique n'est pas nef. Enfin, nous
obtenons aussi des restrictions partielles en dimension $4$
dans le cas o\`u le fibr\'e canonique est nef.
\section{Un th\'eor\`eme de J.\ Koll\'ar}
\subsection{\'Enonc\'e du r\'esultat}
Nous avons vu dans les pr\'eliminaires de cette
th\`ese qu'il existe des vari\'et\'es de Moishezon
ne poss\'edant pas de fibr\'e en droites
simultan\'ement gros et num\'eriquement effectif.
Plus pr\'ecis\'ement, nous avons rencontr\'e le
r\'esultat suivant~:
\bigskip
\noindent {\bf Th\'eor\`eme (J.\ Koll\'ar, K.\ Oguiso) } {\em
(i) Il existe des vari\'et\'es de Moishezon $X$ de
dimension
$3$ dont le groupe de Picard est \'egal \`a ${\Bbb Z}$,
avec $-K_X$ gros et ne poss\'edant
pas de fibr\'e gros et nef,
(ii) il existe des vari\'et\'es de Moishezon $X$ de dimension
$3$ dont le groupe de Picard est \'egal \`a ${\Bbb Z}$,
dont le fibr\'e canonique est trivial et
ne poss\'edant pas de fibr\'e gros et nef.
}
\bigskip
Remarquons que pour une vari\'et\'e de Moishezon dont le
groupe de Picard est ${\Bbb Z}$,
un et un seul g\'en\'erateur de $\operatorname{Pic}(X)$ est gros. Suivant
J.\ Koll\'ar \cite{Kol91}, nous notons ce g\'en\'erateur
${\cal O}_X(1)$ et nous \'ecrivons $\operatorname{Pic}(X)
= {\Bbb Z} \cdot {\cal O}_X(1)$.
Notons aussi $m_X$ l'entier d\'efini par la relation $K_X = {\cal O}_X(m_X)$.
Remarquons ici que les trois cas $m_X < 0$
(respectivement $m_X=0$ et $m_X >0$)
correspondent aux trois possibilit\'es $\kappa(X) = -\infty$ (respectivement
$\kappa(X) = 0$ et $\kappa(X) = \dim X$), o\`u $\kappa(X)$ d\'esigne la
dimension de Kodaira de $X$.
\medskip
Evidemment, il reste un cas non couvert par l'\'enonc\'e pr\'ec\'edent
et la question suivante est naturelle~:
\medskip
\noindent {\bf Question } {\em Existe-t-il des vari\'et\'es de Moishezon $X$,
avec $\operatorname{Pic}(X) = {\Bbb Z} \cdot {\cal O}_X(1)$ et $m_X > 0$ ne poss\'edant pas
de fibr\'e gros et nef \ ?
}
\medskip
En dimension $3$, la r\'eponse \`a cette question est n\'egative
comme le montre le r\'esultat suivant~:
\bigskip
\noindent {\bf Th\'eor\`eme (J.\ Koll\'ar, 1991) } {\em
Soit $X$ une vari\'et\'e de Moishezon
de dimension $3$. On suppose que le groupe
de Picard $\operatorname{Pic}(X)$ est ${\Bbb Z}$ et que le
fibr\'e canonique $K_X$ est gros.
Alors $K_X$ est nef.}
\bigskip
\noindent {\bf Remarque } Dans le cas o\`u le fibr\'e canonique
est gros et nef, l'un de ses multiples est globalement engendr\'e
par le ``base-point free theorem".
Au vu du r\'esultat pr\'ec\'edent, il n'existe
donc pas d'exemple
de vari\'et\'e de Moishezon de dimension $3$,
de groupe de Picard ${\Bbb Z}$ et \`a fibr\'e canonique gros
ne satisfaisant pas aux crit\`eres de J.-P.\ Demailly et Y.-T.\ Siu.
\medskip
La suite de ce paragraphe consiste \`a rappeler
la d\'emonstration de ce r\'esultat car les id\'ees
qu'elle contient seront pr\'esentes dans tout
le chapitre.
\subsection{D\'emonstration}
La r\'ef\'erence pr\'ecise est \cite{Kol91},
page 170 et suivantes.
\medskip
Le lemme suivant, bien qu'\'el\'ementaire est essentiel~:
\medskip
\noindent {\bf Lemme } {\em
Soit $X$ une vari\'et\'e de Moishezon, de dimension quelconque,
avec $\operatorname{Pic}(X) = {\Bbb Z} \cdot {\cal O}_X(1)$. Soit
$\pi : \tilde{X} \rightarrow X$ une modification projective,
de
lieu exceptionnel
$\tilde{S} \subset \tilde{X} \stackrel{\pi}{\rightarrow} S \subset X$.
\noindent Alors pour toute courbe $C$ de $X$, non incluse dans $S$, on a
${\cal O}_X(1) \cdot C > 0$.}
\bigskip
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\bigskip
\noindent {\bf Corollaire } {\em
Sous les hypoth\`eses du lemme, et si de plus $K_X$ est gros,
alors pour toute courbe $C$ (respectivement $\tilde{C}$)
de $X$ (respectivement de $\tilde{X}$) non incluse dans $S$
(respectivement $\tilde{S}$), on a $K_X\cdot C > 0$ (respectivement
$K_{\tilde{X}} \cdot \tilde{C} > 0$).}
\medskip
\noindent {\bf D\'emonstration du lemme }
Soit $\tilde{H}$ un diviseur ample dans $\tilde{X}$, et
$H := \pi _{*}(\tilde{H})$. Alors $H$ est gros, donc s'\'ecrit
${\cal O}_X(p)$, o\`u $p$ est un entier strictement positif.
Comme
$${\cal O}_X(1) \cdot C = \frac{1}{p}H \cdot C ,$$
il suffit de montrer que pour toute courbe $C$ non incluse
dans $S$, on a $H \cdot C > 0$.
\noindent Comme $C$ n'est pas incluse dans $S$, si $\tilde{C}$ d\'esigne
la transform\'ee stricte de $C$, l'\'egalit\'e suivante est v\'erifi\'ee~:
$$ H \cdot C = \pi^{*}(H) \cdot \tilde{C} =
(\tilde{H} + \sum a_iE_i) \cdot \tilde{C},$$
o\`u les $a_i$ sont des entiers positifs ou nuls, et les $E_i$ les composantes
irr\'eductibles de $\tilde{S}$. De l\`a, $H \cdot C > 0$
car $\tilde{H} \cdot \tilde{C} > 0$ et pour tout $i$, on a $E_i \cdot \tilde{C}
\geq 0$.
\hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
\noindent {\bf Remarque } Ce lemme affirme en particulier que les courbes
sur lesquelles ${\cal O}_X(1)$ est
n\'egatif ou nul sont incluses dans un ensemble analytique
de codimension sup\'erieure ou \'egale \`a $2$.
\medskip
\noindent {\bf D\'emonstration du th\'eor\`eme }
En dimension $3$, on d\'eduit du lemme pr\'ec\'edent qu'il n'y
a qu'un nombre fini de courbes sur lesquelles ${\cal O}_X(1)$ est
n\'egatif. Si $K_X$ est gros, et si $C$ est une courbe telle que
$K_X \cdot C < 0$, alors une telle courbe se d\'eforme dans $X$ car la formule
de Riemann-Roch donne
$$ \chi (N_{C/X}) = -K_X \cdot C + (n-3)(1-g) = -K_X \cdot C > 0,$$
o\`u $N_{C/X}$ est le fibr\'e normal de $C$ dans $X$. Mentionnons
que si la courbe $C$ est singuli\`ere, on d\'efinit
$N_{C/X}$ comme \'etant \'egal \`a $\displaystyle{ \nu ^{\ast}T X / T \tilde{C}
}$
o\`u $\nu : \tilde{C} \to C$ est la normalis\'ee de la courbe $C$.
Ceci donne bien la contradiction.\hskip 3pt \vrule height6pt width6pt depth 0pt
\subsection{Commentaires}
La d\'emonstration ci-dessus repose sur un argument
de d\'eformation. Comme nous utilisons dans la suite
ce type d'argument, il est sans doute bon de faire ici
un bref rappel.
\'Etant donn\'ees une vari\'et\'e $X$ et une sous-vari\'et\'e
$Y$ de $X$, c'est un probl\`eme classique et important de
d\'eterminer les d\'eformations de $Y$ dans $X$. Dans le
cadre analytique, ce probl\`eme a \'et\'e consid\'er\'e par
K.\ Kodaira \cite{Kod62} et par A.\ Grothendieck et
D.\ Mumford dans le cadre
alg\'ebrique o\`u la notion de {\bf sch\'ema de Hilbert}
joue un r\^ole essentiel~: une r\'ef\'erence importante est
le travail r\'ecent de J.\ Koll\'ar \cite{Kol94}.
La ``solution" au probl\`eme est donn\'ee par le~:
\bigskip
\noindent {\bf Th\'eor\`eme \cite{Gro62}, \cite{Kod62} }{\em
Soit $Y$ une sous-vari\'et\'e d'une vari\'et\'e $X$.
Alors le sch\'ema de Hilbert $\operatorname{Hilb} (X)$ des sous-ensembles
analytiques de $X$ admet $H^0(Y,N_{Y/X})$ comme espace tangent
de Zariski en $[Y]$. La dimension de $\operatorname{Hilb} (X)$ en $[Y]$ satisfait~:
$$ \dim H^0(Y,N_{Y/X}) - \dim H^1(Y,N_{Y/X})
\leq \dim_{[Y]} \operatorname{Hilb} (X) \leq \dim H^0(Y,N_{Y/X}).$$
En particulier, si $H^1(Y,N_{Y/X}) =0$, alors $\operatorname{Hilb} (X)$ est lisse
au voisinage de $[Y]$.}
\bigskip
Dans le cadre analytique, la construction de K.\ Kodaira
consiste \`a trouver explicitement en coordonn\'ees
locales les s\'eries enti\`eres d\'efinissant les sous-ensembles
proches de $Y$, tandis que dans le cadre alg\'ebrique, l'id\'ee
est qu'une vari\'et\'e projective $Z$ est d\'etermin\'ee par
le sous-espace vectoriel des polyn\^omes de degr\'e suffisamment
grand qui s'annulent sur $Z$.
\section{Quelques rappels sur la th\'eorie de Mori}
La th\'eorie de Mori, n\'ee dans les ann\'ees 80,
consiste \`a \'etendre et approfondir en dimension sup\'erieure
ou \'egale \`a $3$ la classification des surfaces
complexes et l'\'etude des applications bim\'eromorphes
entre surfaces complexes.
Cependant, cette
th\'eorie n'est valable que sur les vari\'et\'es
projectives. Comme une vari\'et\'e de Moishezon
est domin\'ee par une vari\'et\'e projective, une
id\'ee naturelle est d'appliquer certains r\'esultats
de la th\'eorie de Mori pour obtenir des renseignements
concernant la structure des vari\'et\'es de Moishezon
ou de leur caract\`ere non projectif.
Nous rappelons dans ce paragraphe les r\'esultats
essentiels utilis\'es dans ce chapitre. Deux excellentes
r\'ef\'erences sont \cite{CKM88} et \cite{KMM87}.
Mentionnons aussi qu'une des grandes id\'ees de S.\ Mori
est, m\^eme pour l'\'etude des vari\'et\'es
non singuli\`eres, de quitter le monde lisse pour autoriser
certains types de singularit\'es~; cette analyse
fut en particulier mise en oeuvre par M.\ Reid. Il est cependant
bon de pr\'eciser ici que la plupart
des \'enonc\'es que nous utilisons sont d'une difficult\'e
moindre dans le cas lisse, cadre dans lequel nous les
appliquons.
\subsection{C\^one des courbes effectives}
Dans tout ce paragraphe, $X$ est une vari\'et\'e projective.
\subsub{Notations}
Rappelons tout d'abord que la notation
$N_1(X,{\Bbb R})$ d\'esigne l'espace vectoriel
des combinaisons lin\'eaires finies (\`a coefficients
r\'eels) de courbes (irr\'eductibles et
\'eventuellement singuli\`eres) de $X$, modulo l'\'equivalence
num\'erique~: deux courbes sont \'equivalentes si et
seulement si leurs intersections avec tout diviseur
sont \'egales.
\noindent Pour une vari\'et\'e projective (et m\^eme de
Moishezon), cet espace vectoriel est de dimension finie et est
en dualit\'e naturelle ({\em via} la forme d'intersection)
avec le groupe de N\'eron-Severi
$\displaystyle{(\operatorname{Pic} (X)/ \operatorname{Pic} ^0(X))\otimes _{{\Bbb Z}} {\Bbb R}}$~;
la dimension de $N_1(X,{\Bbb R})$
est appel\'ee {\bf nombre de Picard de $X$}.
L'espace vectoriel
$N_1(X,{\Bbb R})$ est naturellement un sous-espace vectoriel
de $H_2(X,{\Bbb R})$.
Enfin, nous notons suivant l'usage $\operatorname{NE} (X)$ le
sous-c\^one convexe de $N_1(X,{\Bbb R})$ engendr\'e par les classes
d'homologie des courbes
effectives. L'adh\'erence de ce c\^one est not\'ee
$\overline{\operatorname{NE} }(X)$.
\subsub{Th\'eor\`eme du c\^one}
L'un des premiers succ\`es de la th\'eorie de Mori est de
montrer que {\em si le fibr\'e canonique $K_X$
n'est pas nef, alors il existe une courbe rationnelle
sur laquelle il est strictement n\'egatif}.
L'\'enonc\'e pr\'ecis est le suivant~:
\medskip
\noindent {\bf Th\'eor\`eme du c\^one } {\em
Soit $X$ une vari\'et\'e projective. Alors il existe
un ensemble minimal (fini ou d\'enombrable) de courbes
rationnelles $C_i$ dans $X$ de sorte que~:
\vspace{+1mm}
(i) pour tout $i$, on a $\displaystyle{0 < -K_X \cdot C_i \leq \dim X +1}$,
\vspace{+2mm}
(ii) $\displaystyle{
\overline{\operatorname{NE} }(X) = \overline{\operatorname{NE} }(X)_{K_X \geq 0} + \sum_i {\Bbb R}_+ \ [C_i]}$,
\noindent o\`u
$\displaystyle{
\overline{\operatorname{NE} }(X)_{K_X \geq 0} := \{ [C] \in N_1(X,{\Bbb R}) \ |
\ K_X \cdot C \geq 0 \} }$.
}
\medskip
Les courbes rationnelles $C_i$ sont appel\'es
{\bf courbes rationnelles extr\^emales} et les ${\Bbb R}_+ \ [C_i]$
sont appel\'ees {\bf rayons extr\^emaux}.
On ne connait pas de version analogue de ce th\'eor\`eme
pour les vari\'et\'es de Moishezon. \`A notre connaissance,
la question suivante est ouverte~:
\medskip
\noindent {\bf Question } {\em Soit $X$ une
vari\'et\'e de Moishezon, dont le fibr\'e canonique
n'est pas nef. Existe-t-il alors une courbe
rationnelle $C$ telle que $K_X \cdot C < 0$ \ ?
}
\subsection{Contraction de Mori}
Les rayons extr\^emaux jouent le m\^eme r\^ole
dans la th\'eorie de Mori que les courbes rationnelles
lisses d'auto-intersection n\'egative dans la th\'eorie
des surfaces complexes~: ils peuvent \^etre contract\'es.
Le th\'eor\`eme suivant d\'ecrit les diff\'erents types
de contraction obtenus en contractant un rayon extr\^emal.
\bigskip
\noindent {\bf Th\'eor\`eme de contraction \cite{CKM88} }
{\em Soit $X$ une vari\'et\'e projective
dont le fibr\'e canonique n'est pas nef. Soient
$C$ une courbe rationnelle extr\^emale et $R := {\Bbb R}_+ \ [C]$
le rayon extr\^emal engendr\'e par $C$.
Alors, il existe une vari\'et\'e (\'eventuellement singuli\`ere)
projective,
normale, et une application $$f : X \to Y ,$$ not\'ee
aussi $\operatorname{cont}_R$, de sorte que~:
(i) une courbe de $X$ est contract\'ee
par $f$ si et seulement si sa classe d'homologie
appartient au rayon $R$,
(ii) le fibr\'e $-K_X$ est $f$-ample (i.e la restriction
de $-K_X$ \`a toute fibre de $f$ est ample).
\noindent De plus, on distingue trois types de contractions~:
(a) $\dim X > \dim Y$ et $f$ est une fibration
Fano (i.e la fibre g\'en\'erique de $f$ est une
vari\'et\'e lisse dont le fibr\'e anti-canonique est ample),
(b) $\dim X = \dim Y$ et $f$ est une contraction divisorielle
(i.e $f$ est birationnelle et contracte un diviseur),
(c) $\dim X = \dim Y$ et $f$ est une petite contraction
(i.e $f$ est birationnelle et contracte un sous-ensemble alg\'ebrique
de codimension sup\'erieure ou \'egale \`a $2$).}
\medskip
Les cas (a) et (b) sont les ``bons" cas~: dans le cas (a),
on r\'eduit la compr\'ehension de la vari\'et\'e $X$
\`a celle d'une vari\'et\'e de dimension plus petite
et \`a la structure des fibres qui sont des vari\'et\'es
de Fano. Dans le cas (b), la vari\'et\'e singuli\`ere $Y$ est
${\Bbb Q}$-factorielle, \`a singularit\'es
terminales (ce sont les singularit\'es qui permettent de donner
encore un sens \`a l'expression ``$K_Y$ est ou n'est
pas nef") et le nombre de Picard de $Y$ est strictement
plus petit que celui de $X$. Le cas (c) est le
``mauvais" cas~: les singularit\'es de $Y$ sont telles
que $Y$ ne poss\`ede pas de fibr\'e canonique et il n'est pas clair
que $Y$ soit ``plus simple" que $X$.
\subsection{Contractions divisorielles}
Nous utilisons dans la suite un r\'esultat plus
pr\'ecis que le th\'eor\`eme de contraction dans
le cas d'une contraction divisorielle. Sous cette
forme, il appara\^{\i}t dans les travaux de T.\ Ando \cite{And85}
et M.\ Beltrametti \cite{Bel86}~:
\bigskip
\noindent {\bf Th\'eor\`eme (T.\ Ando, M.\ Beltrametti, 1985) }
{\em Soient $X$ une vari\'et\'e projective et
$f : X \to Y$ une contraction divisorielle d'un rayon
extr\^emal. Soit enfin $F$ une fibre g\'en\'erale
de $f_E : E \to f(E)$ o\`u $E$ est le diviseur exceptionnel
de $f$.
\noindent Alors il existe un fibr\'e en droites $L$ sur $X$ tel que~:
(i) $\operatorname{Im} (\operatorname{Pic} (X) \to \operatorname{Pic} (F)) = {\Bbb Z} \cdot L_{|F}$ o\`u $L_{|F}$
est ample sur $F$,
(ii) ${\cal O}_F(-K_X) \simeq {\cal O}_F(pL)$ et
${\cal O}_F(-E) \simeq {\cal O}_F(qL)$ o\`u $p$ et $q$ sont
deux entiers positifs.
\noindent Enfin, si $F$ est de dimension
$2$, alors $F$ est isomorphe \`a ${\Bbb P} ^2$ ou \`a la
quadrique ${\cal Q}_2$.
}
\medskip
Nous utiliserons ce r\'esultat lorsque $X$ est de dimension $4$.
\subsection{L'in\'egalit\'e de Wi\'sniewski}
Avant d'\'enoncer cette in\'egalit\'e, nous avons besoin de deux notations~:
soient $X$ une vari\'et\'e projective, $R$ un rayon extr\^emal
de $\overline{\operatorname{NE} } (X)$ et $f$ la contraction de Mori associ\'ee.
On note
$\displaystyle{ l(R) = \min \{ -K_X \cdot C \ | \
C \mbox{ est une courbe rationnelle telle que } [C] \in R \} }.$
Le nombre $l(R)$ est la {\bf longueur} du rayon $R$.
On note aussi $A(R)$ le lieu de $X$ couvert par les courbes
dont la classe appartient \`a $R$.
Dans \cite{Wis91}, J.\ Wi\'sniewski d\'emontre l'in\'egalit\'e
fondamentale suivante~:
\bigskip
\noindent {\bf Th\'eor\`eme (J.\ Wi\'sniewski, 1991) }
{\em Pour toute fibre non triviale $F$ de $f$, on a~:
$$ \dim F + \dim A(R) \geq \dim X + l(R) - 1.$$}
Cette in\'egalit\'e, dont une forme faible
est due \`a P.\ Ionescu \cite{Ion86}, a de nombreuses cons\'equences
dans la classification des vari\'et\'es projectives
de dimension sup\'erieure ou \'egale \`a $3$.
\section{Un premier r\'esultat}
Rappelons pour commencer ce paragraphe que les exemples de J.\ Koll\'ar
et K.\ O\-gui\-so d\'ej\`a cit\'es v\'erifient la propri\'et\'e
suppl\'ementaire
suivante~: les vari\'et\'es $X$ cons\-trui\-tes ne sont, bien s\^ur,
pas projectives, mais le deviennent apr\`es exactement un \'eclatement
le long d'une sous-vari\'et\'e $Y \subset X$.
Dans toute cette partie, nous nous pla\c cons dans la situation
analogue suivante~:
\medskip
\noindent {\bf Hypoth\`ese }
{\em $X$ est une vari\'et\'e de Moishezon non projective de dimension
$n$ dont le groupe de Picard
est \'egal \`a ${\Bbb Z}$, dont
le fibr\'e canonique est gros et telle qu'il existe
une sous-vari\'et\'e $Y \subset X$ de sorte que l'\'eclatement
$\pi : \tilde{X} \to X$ de $X$ le
long de $Y$ d\'efinisse une vari\'et\'e projective $\tilde{X}$.
On note
$E$ le diviseur exceptionnel de l'\'eclatement.
}
\medskip
La remarque suivante est tr\`es importante~:
\medskip
\noindent {\bf Remarque }
D'apr\`es le corollaire 3.1.2, on sait
alors que $K_X$ (respectivement $K_{\tilde{X}}$) est
strictement positif sur les courbes
non incluses dans $Y$ (respectivement dans $E$). La cons\'equence suivante
sera utilis\'ee dans la suite~: si $C$ est une courbe
de $Y$ sur laquelle $K_X$ est strictement n\'egatif, cette courbe
ne peut pas se d\'eformer (dans $X$) hors de $Y$.
\medskip
La m\'ethode que nous adoptons pour \'etudier
$X$ et $Y$ est d'appliquer la th\'eorie de Mori
\`a la vari\'et\'e projective $\tilde{X}$.
\subsection{C\^one des courbes sur $\tilde{X}$}
Dans le cadre de notre \'etude, comme $\operatorname{Pic} (X) = {\Bbb Z}$,
l'espace vectoriel
$N_1(\tilde{X},{\Bbb R})$ est isomorphe \`a ${\Bbb R} ^2$.
Dans la suite, nous repr\'esentons dans $N_1(\tilde{X},{\Bbb R}) \simeq {\Bbb R} ^2$
le c\^one ferm\'e $\overline{\operatorname{NE} }(\tilde{X})$~;
dans les figures ci-dessous,
ce dernier correspond \`a la
partie hachur\'ee. Si $D$ est un \'el\'ement de $\operatorname{Pic} (\tilde{X})$,
nous notons $D > 0$ (respectivement $D = 0$, respectivement $D < 0$)
les ensembles
$$\{ [C] \in \overline{\operatorname{NE} }(\tilde{X}) \ | \ D \cdot C > 0 \}$$
(respectivement $D \cdot C = 0$, respectivement $D \cdot C < 0$).
Deux cas se pr\'esentent suivant que $K_X$ est nef ou non.
Ces deux cas se distinguent naturellement~; ils correspondent,
comme nous le verrons plus loin,
au fait que $X$ admet ou non un morphisme vers une vari\'et\'e
(\'eventuellement singuli\`ere) projective de m\^eme dimension,
\medskip
(i) soit $K_X$ est nef et le dessin est le suivant~:
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\noindent {\bf Quelques commentaires sur ces diagrammes}
- dans les deux cas, le fait que la droite $\{ K_{\tilde{X}} = 0 \}$
coupe le c\^one effectif vient du fait qu'il y a \`a la fois des
courbes sur lesquelles $ K_{\tilde{X}}$ est strictement positif
(celles non contenues dans le diviseur exceptionnel) et
des
courbes sur lesquelles $ K_{\tilde{X}}$ est strictement
n\'egatif (toute courbe incluse dans les fibres de l'\'eclatement),
- dans le deuxi\`eme cas, la position relative de
$\{ \pi ^{\ast} K_{X} = 0 \}$ est justifi\'ee par le fait qu'il
y a des courbes sur lesquelles $\pi ^{\ast} K_{X}$ et $K_{\tilde{X}}$
sont strictement positifs (celles non contenues dans le diviseur
exceptionnel d'apr\`es 3.1.2) et que $\pi ^{\ast} K_{X}$ est nul sur
toute courbe incluse dans les fibres de l'\'eclatement.
\subsub{Quelques cons\'equences de ces diagrammes}
Nous regroupons ici les renseignements provenant
directement
de la description de $\overline{\operatorname{NE} }(\tilde{X})$.
Pour cela, appliquons le th\'eor\`eme du c\^one
\`a la vari\'et\'e projective $\tilde{X}$. On en
d\'eduit que le rayon extr\^emal du c\^ot\'e $K_{\tilde{X}} < 0$ est
engendr\'e par la classe d'une courbe rationnelle $\tilde{C}$
dans $\tilde{X}$. Alors,
le th\'eor\`eme de contraction assure
l'existence d'une
vari\'et\'e (en g\'en\'eral singuli\`ere)
projective $Z$ et d'un morphisme $f$ associ\'es
\`a la courbe extr\^emale rationnelle
$\tilde{C}$ de sorte
que la situation suivante ait lieu~:
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\medskip
\noindent (i) Cas o\`u $K_X$ est nef.
Alors le rayon extr\^emal est engendr\'e
par la classe d'une courbe rationnelle incluse
dans une fibre non triviale de $\pi$. Toutes les
fibres de $\pi$ sont donc contract\'ees par $f$ si bien
que $f$ se factorise en une application $g : X \to Z$
$$ \tilde{X} \stackrel{\pi}{\to} X \stackrel{g}{\to} Z \
\mbox{et} \ \ f = g \circ \pi.$$
\medskip
\noindent (ii) Cas o\`u $K_X$ n'est pas nef.
Alors les fibres de $f$ et les fibres de $\pi$
ne se coupent que sur un nombre fini de points~: en effet,
il n'existe pas de courbes simultan\'ement contract\'ees
par $\pi$ et $f$ car les rayons engendr\'es par $[\tilde{C}]$
et la classe d'une courbe rationnelle incluse
dans une fibre non triviale de $\pi$ sont distincts.
\subsub{Une application imm\'ediate}
Dans le cas o\`u $K_X$ n'est pas nef, la courbe
rationnelle $\tilde{C}$
n'\'etant pas contract\'ee par
$\pi$, la courbe rationnelle
$C = \pi (\tilde{C})$ v\'erifie $K_X \cdot C < 0$. Le r\'esultat suivant
en d\'ecoule~:
\medskip
\noindent {\bf Proposition }{\em
Sous les hypoth\`eses pr\'ec\'edentes, si $K_X$ n'est pas nef,
il existe une courbe rationnelle $C \subset Y $
sur laquelle $K_X$ est strictement n\'egatif.
}
\subsection{Contraction de Mori de $\tilde{X}$}
Nous \'etudions ici plus en d\'etail la contraction de Mori $f$
associ\'ee \`a la courbe extr\^emale rationnelle
$\tilde{C} \subset
\tilde{X}$ obtenue pr\'ec\'edemment pour en d\'eduire
une estimation de la dimension de $Y$ en toute dimension.
\subsub{\'Enonc\'e du r\'esultat}
Nous avons vu qu'il y a trois types de contractions
extr\^emales. Le th\'eor\`eme suivant restreint
les possibilit\'es dans notre situation~:
\bigskip
\noindent {\bf Th\'eor\`eme F } {\em
Soit $X$ de Moishezon avec $\textstyle{\operatorname{Pic} (X)} = {\Bbb Z}$ et $K_X$
gros. Si $X$ est rendue projective apr\`es \'eclatement $\pi : \tilde{X} \to X$
le
long de $Y$, alors~:
(i) on a $\dim \tilde{X} = \dim Z$, autrement dit
$f$ est une application birationnelle,
(ii) si $f$ est une contraction
divisorielle, son diviseur
exceptionnel est \'egal \`a celui de $\pi$
(not\'e $E$ pr\'ec\'edemment)~; ce cas est le seul possible
lorsque $K_X$ est nef,
(iii) si $f$ est une contraction
divisorielle et si
$K_X$ n'est pas nef, les in\'egalit\'es
suivantes sont satisfaites~:
$$ \operatorname{codim} Y -1 \leq \dim f(E) < \dim Y \ \ \mbox{\rm et} \ \
\dim Y > \frac{n-1}{2}, $$
(iv) si $f$ est une petite contraction et si
$K_X$ n'est pas nef, l'in\'egalit\'e
suivante est satisfaite~:
$$ \dim Y \geq \frac{n+1}{2}.$$
}
\smallskip
On peut reformuler ce r\'esultat sans faire intervenir
la contraction de Mori~:
\medskip
\noindent {\bf Th\'eor\`eme F' } {\em
Soit $X$ une vari\'et\'e de Moishezon de dimension
$n$ avec $\textstyle{\operatorname{Pic} (X)} = {\Bbb Z}$
et $K_X$ gros. Supposons que $X$ est rendue projective
apr\`es \'eclatement le long d'une sous-vari\'et\'e lisse $Y$.
\noindent Alors, si $K_X$ n'est pas nef, on a
$\displaystyle{ \dim Y > \frac{n-1}{2}}$.}
\bigskip
\noindent {\bf Remarque }
Le fait que $K_X$ soit gros est ici essentiel.
En effet, les constructions de J.\ Koll\'ar
et K.\ Oguiso montrent que les in\'egalit\'es
du point (iii) ne sont pas vraies en g\'en\'eral.
La construction de J.\ Koll\'ar donne aussi un exemple
o\`u les diviseurs exceptionnels de $\pi$ et $f$ ne
sont pas \'egaux.
\subsub{D\'emonstration du th\'eor\`eme F-F'}
Les points (i) et (ii) du th\'eor\`eme~F sont faciles~:
par hypoth\`ese, $f$ ne contracte que des courbes sur lesquelles
$K_{\tilde{X}}$ est strictement n\'egatif, donc incluses dans $E$
d'apr\`es le corollaire 3.2.1~; en particulier le point (i) est
d\'emontr\'e.
\noindent Ceci montre aussi que si $f$ est une contraction
divisorielle, son diviseur exceptionnel \'etant
inclus dans $E$ est donc \'egal \`a $E$. R\'eciproquement,
si $K_X$ est nef, $f$ se factorise \`a travers $\pi$ et donc
est une contraction divisorielle. Le point (ii) est d\'emontr\'e.
\bigskip
Montrons le point (iii) du th\'eor\`eme F~:
$f$ est une contraction divisorielle
et $K_X$ n'est pas nef. La situation
est r\'esum\'ee par le diagramme suivant~:
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\noindent Dans cette situation, on \'ecrit
$$ K_{\tilde{X}}=f^{\ast}K_Z + aE =
\pi^{\ast}K_X + (r-1)E $$
o\`u $r= \operatorname{codim} Y$ et o\`u $a$ est un nombre rationnel.
Cette \'egalit\'e est une \'egalit\'e
de ${\Bbb Q}$-diviseurs de Cartier~: dans le cas
d'une contraction divisorielle, le diviseur
canonique de $Z$ n'est pas de Cartier en g\'en\'eral
mais l'un de ses multiples entiers l'est.
Rappelons que toutes les notions de positivit\'e
(telle que par exemple \^etre gros, nef ou ample)
s'\'etendent naturellement aux ${\Bbb Q}$-diviseurs
de Cartier.
\noindent Les in\'egalit\'es cherch\'ees d\'ecoulent
imm\'ediatement
du lemme suivant~:
\medskip
\noindent {\bf Lemme } {\em
Si $K_X$ n'est pas nef, les nombres $a$ et $r$ v\'erifient
les deux in\'egalit\'es suivantes~:
(i) $a > r-1$,
(ii) $ \operatorname{codim} f(E) + r \leq n + 1$.
\noindent L'in\'egalit\'e suivante est vraie en toute
g\'en\'eralit\'e
pour une contraction divisorielle~:
(iii) $a \leq \operatorname{codim} f(E) - 1$.
}
\bigskip
\noindent {\bf D\'emonstration du lemme}
\medskip
\noindent {\bf In\'egalit\'e (i)~:}
Comme $Z$ est projective avec $\operatorname{Pic} (Z) = {\Bbb Z}$ et $K_Z$ gros,
on en d\'eduit que $K_Z$ est ample et donc
que $f^{\ast}K_Z$ est nef, et strictement
positif sur les courbes de $\tilde{X}$ non
contract\'ees par $f$. Choisissons
alors une courbe rationnelle $R$ incluse dans une fibre
non triviale
de $\pi$ (ces derni\`eres sont des ${\Bbb P} ^{r-1}$, on
prend pour $R$ une droite ${\Bbb P} ^{1}$).
\noindent L'\'egalit\'e
$$f^{\ast}K_Z \cdot R + a E \cdot R =
\pi^{\ast}K_X \cdot R + (r-1)E \cdot R$$
donne alors~:
$$a-(r-1) = f^{\ast}K_Z \cdot R > 0$$
car, $K_X$ n'etant pas nef,
$R$ n'est pas contract\'ee par $f$. \hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
\noindent {\bf In\'egalit\'e (ii)~:}
Cette in\'egalit\'e d\'ecoule de suite du fait
que les fibres de $f$ et $\pi$ dans $E$ ne peuvent se couper
qu'en un nombre fini de points. De l\`a ~:
$$ (n-1 - \dim f(E) ) + (r-1) \leq n-1. \hskip 3pt \vrule height6pt width6pt depth 0pt$$
\medskip
\noindent {\bf In\'egalit\'e (iii)~:}
Soit $F$ une fibre g\'en\'erique de la restriction de $f$
\`a $E$, et soit $\tilde{C}$ une courbe dans $F$.
\noindent Alors, on a
$$aE \cdot \tilde{C} = K_{\tilde{X}} \cdot \tilde{C}$$
et par la formule d'adjonction~:
$$ K_{\tilde{X} | E} = K_E - E_{| E}.$$
\noindent De l\`a, on en
d\'eduit~:
$$a+1 = \frac{K_E \cdot \tilde{C}}{E \cdot \tilde{C}}.$$
\noindent Comme le fibr\'e canonique $K_F$ est simplement la restriction
de $K_{E}$ \`a $F$, on obtient~:
$$a = \frac{K_F \cdot \tilde{C}}{E \cdot \tilde{C}} - 1.$$
\noindent Or, la vari\'et\'e $F$ est Fano, et par le th\'eor\`eme
du c\^one appliqu\'e \`a $F$, on peut supposer
que $\tilde{C}$ est une courbe (rationnelle) satisfaisant~:
$$ 0 < -K_F \cdot \tilde{C} \leq \dim F + 1 = n-1 -\dim f(E) +1 = \operatorname{codim} f(E)
.$$
\noindent De l\`a, comme $E \cdot \tilde{C}$ est un entier strictement
n\'egatif, il vient $a \leq \operatorname{codim} f(E) - 1$. Ceci termine la
preuve du lemme.\hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
\noindent {\bf Remarque } L'in\'egalit\'e (iii) peut aussi se
d\'eduire de l'in\'egalit\'e de Wi\'sniewski~: en effet $f$
\'etant divisorielle, on a $\dim A(R) = n-1$ et la dimension
de la fibre g\'en\'erique non triviale est $n-1-\dim f(E)$.
De l\`a, l'in\'egalit\'e de Wi\'sniewski donne $$\operatorname{codim} f(E) -1 \geq l(R).$$
Or, comme
$K_{\tilde{X}} = f^*K_Z + a E$, on a $-K_{\tilde{X}}\cdot C \geq a$
pour toute courbe contract\'ee, d'o\`u $l(R) \geq a$ comme souhait\'e.
\bigskip
Montrons maintenant le point (iv) du th\'eor\`eme~F.
Pour cela, on applique l'in\'egalit\'e de Wi\'sniewski~:
comme $f$ est une petite contraction, on a
\'evidemment $$\dim A(R) \leq n-2,$$ et si $F$ est une fibre
non triviale de $f$ cette derni\`ere est incluse dans $E$
et ne coupe les fibres de $\pi$ que sur un ensemble
fini. On en d\'eduit que $\dim F \leq \dim Y$ d'o\`u~:
$$ n-2 + \dim Y \geq n + l(R) -1,$$
soit $$ \dim Y \geq l(R) +1.$$
Il suffit alors d'estimer $l(R)$. Or, on a
$K_{\tilde{X}} = \pi^*K_X + (r-1) E$ et comme
$K_X$ n'est pas nef, $ \pi^*K_X$ est strictement n\'egatif sur les
courbes contract\'ees par $f$. On en d\'eduit que
$$ -K_{\tilde{X}} \cdot C \geq r$$
pour toute courbe contract\'ee par $f$. De l\`a,
$ l(R) \geq r$
et en reportant
$$ 2 \dim Y \geq n+1$$
qui est l'in\'egalit\'e souhait\'ee.\hskip 3pt \vrule height6pt width6pt depth 0pt
\subsection{Application \`a la dimension $3$}
On d\'eduit du th\'eor\`eme F un r\'esultat pr\'ecisant celui
de J.\ Koll\'ar dans notre situation~:
\medskip
\noindent {\bf Corollaire } {\em
Soit $X$ une vari\'et\'e de Moishezon non projective
de dimension $3$,
avec $\operatorname{Pic}(X) = {\Bbb Z}$ et $K_X$ gros.
\noindent Si $X$ peut \^etre rendue projective apr\`es
un \'eclatement seulement,
alors $X$ est une petite modification d'une vari\'et\'e
singuli\`ere projective ayant une unique singularit\'e
nodale ordinaire (dont le mod\`ele local est $xy-zt = 0$ dans
$({\Bbb C} ^4,0)$). En particulier, le fibr\'e canonique $K_X$
est nef.}
\medskip
\noindent {\bf D\'emonstration du corollaire}
On note toujours $\pi$ l'\'eclatement
rendant $X$ projective et $f$ la contraction
de Mori d\'efinie sur la vari\'et\'e projective
$\tilde{X}$. D'apr\`es le th\'eor\`eme F, $f$ est
birationnelle, et comme il n'y a pas de petites
contractions en dimension $3$ d'une vari\'et\'e
non singuli\`ere, c'est que $f$ est une contraction
divisorielle.
De plus, les in\'egalit\'es (iii) du th\'eor\`eme F
ne peuvent \^etre v\'erifi\'ees ici car elles
impliquent $\operatorname{codim} Y = 1$.
C'est donc que $K_X$ est nef (on retrouve ainsi le r\'esultat
de J.\ Koll\'ar), et que le rayon extr\^emal
du c\^ot\'e $K_{\tilde{X}} < 0$ est engendr\'e par la classe d'homologie
des fibres de $\pi$. Il y a alors exactement deux possibilit\'es~:
- les fibres de la contraction de Mori (restreinte au diviseur
exceptionnel $E$) sont de dimension $1$ et alors cette derni\`ere
co\"{\i}ncide avec $\pi$. Dans ce cas, $X$ est projective, ce que
l'on a exclu,
- la contraction de Mori contracte le diviseur exceptionnel $E$
sur un point. Dans ce cas, nous appliquons le r\'esultat
de S.\ Mori \cite{Mor82} qui donne la liste de toutes les contractions
extr\^emales d'une vari\'et\'e non singuli\`ere de dimension $3$.
On en d\'eduit que le diviseur exceptionnel de $\pi$ (\'egal \`a celui
de $f$)
est isomorphe \`a
${\Bbb P} ^1 \times {\Bbb P} ^1$ et que
$\displaystyle{{\cal O}_{E} (E) = N_{E/\tilde{X}}}$ est de type
$(-1,-1)$. La situation est alors la suivante~:
$$ \tilde{X} \stackrel{\pi}{\to} X \stackrel{g}{\to} Z \
\mbox{et} \ \ f = g \circ \pi,$$
o\`u $Z$ est une vari\'et\'e
singuli\`ere projective ayant une unique singularit\'e
nodale ordinaire (dont le mod\`ele local est $xy-zt = 0$ dans
$({\Bbb C} ^4,0)$).
Dans ce cas, la contraction de Mori est alors $g \circ \pi$
et correspond \`a l'\'eclatement du point singulier~: le centre $Y$
de l'\'eclatement $\pi$ est une courbe rationnelle lisse.\hskip 3pt \vrule height6pt width6pt depth 0pt
\bigskip
\noindent {\bf Exemple }
La situation pr\'ec\'edente peut effectivement \^etre
r\'ealis\'ee~: soit $Z$ une hypersurface de ${\Bbb P} ^4$
d'\'equation
$$h_0x_0^2 + h_1x_1^2 + h_2x_2^2 + h_3x_3^2 = 0,$$
o\`u $[x_0 : \cdots : x_4]$ sont les coordonn\'ees homog\`enes
dans ${\Bbb P}^4$ et o\`u les $h_i$ sont quatre polyn\^omes
homog\`enes de degr\'e $d$ sup\'erieur ou
\'egal \`a $4$ ne s'annulant pas en $[0:0:0:0:1]$ et g\'en\'eriques
parmi les polyn\^omes ayant ces propri\'et\'es.
Alors l'hypersurface $Z$ est lisse except\'e au point
$[0:0:0:0:1]$ o\`u elle poss\`ede une singularit\'e
nodale ordinaire. On obtient $X$ comme d\'ecrit pr\'ec\'edemment
en r\'esolvant la singularit\'e puis en contractant dans une direction
de la quadrique exceptionnelle.
\medskip
Nous donnons dans la suite d'autres applications du th\'eor\`eme F
mais nous commen\c cons par montrer dans le paragraphe suivant
que le r\'esultat de J.\ Koll\'ar ne s'\'etend pas en dimension
sup\'erieure ou \'egale \`a $4$.
\section{Une famille de vari\'et\'es de Moishezon}
Le but de cette partie est de montrer le r\'esultat suivant~:
\medskip
\noindent {\bf Th\'eor\`eme G} {\em
Pour tout entier $n$ sup\'erieur
ou \'egal \`a $4$, il existe des vari\'et\'es de Moishezon $X$
de dimension $n$ v\'erifiant~:
(i) $\operatorname{Pic}(X) = {\Bbb Z}$, (ii) $K_X$ est gros, (iii) $K_X$ n'est pas nef.
}
\medskip
Ainsi, le r\'esultat de J.\ Koll\'ar est propre \`a la dimension
$3$.
\medskip
\noindent {\bf Remarque } Les vari\'et\'es obtenues dans la
construction qui suit
rel\`event toutes du cas ``contraction divisorielle"
\'evoqu\'e dans le paragraphe pr\'ec\'edent.
Il serait bien s\^ur int\'eressant de cons\-truire de telles
vari\'et\'es relevant du cas ``petite contraction".
Cependant, nous verrons plus loin que ce cas ne peut pas
se produire en dimension~$4$.
\subsection{Un r\'esultat interm\'ediaire}
La d\'emonstration du th\'eor\`eme G repose
sur la proposition suivante, que nous prouvons plus loin.
Mentionnons qu'il nous a \'et\'e signal\'e par un rapporteur
anonyme que cette proposition se trouve dans \cite{BVV78}.
Pour $n$ entier, nous notons $[x_0: \cdots :x_{n+1}]$
les coordonn\'ees homog\`enes dans ${\Bbb P} ^{n+1}$. On d\'esigne
par ${\Bbb P} _{x_n} ^{1}$ la droite
$\{ x_0 = \dots = x_{n-1} = 0 \}$.
Choisissons alors $n$ polyn\^omes
homog\`enes $h_0,\ldots \!,h_{n-1}$ de degr\'e $2n-2$ et
consid\'erons
l'hypersurface $Z$ de degr\'e $2n-1$ dans ${\Bbb P} ^{n+1}$
et d'\'equation
$$\sigma = x_0h_0 +\cdots+ x_{n-1}h_{n-1} = 0.$$
Cette hypersurface contient ${\Bbb P} _{x_n} ^{1}$ et peut \^etre
singuli\`ere. On a cependant le r\'esultat suivant~:
\medskip
\noindent {\bf Proposition }{\em
Si $n$ est sup\'erieur ou \'egal \`a $3$ et si les $h_{i}$
sont choisis g\'en\'eriquement, alors~:
\smallskip
\!(i) l'hypersurface $Z$ est non singuli\`ere,
\smallskip
\!(ii) le fibr\'e normal $N_{{\Bbb P} ^{1}/ Z}$ est \'egal \`a
${\cal O}_{{\Bbb P} ^{1}}(-1)^{\oplus n-1}$,
\smallskip
\!(iii) $K_Z$ est \'egal \`a ${\cal O}_{{\Bbb P} ^{n+1}}(n-3)_{| Z}$,
\smallskip
\!(iv) $\operatorname{Pic} (Z) = {\Bbb Z}$.
}
\subsection{D\'emonstration du th\'eor\`eme G }
La construction qui suit nous a \'evidemment
\'et\'e inspir\'ee par l'analyse du paragraphe pr\'ec\'edent, dans
le cas o\`u la contraction de Mori est une contraction
divisorielle~: si une vari\'et\'e de dimension $4$
de Moishezon satisfait le point (iii) du th\'eor\`eme~F,
c'est en \'eclatant une surface, puis en contractant sur une
courbe rationnelle que l'on obtient un mod\`ele projectif. Nous
donnons cependant la construction g\'en\'erale en toute dimension.
\medskip
\noindent {\bf Construction explicite : }
On se fixe dor\'enavant une hypersurface $Z$ donn\'ee par la proposition
pr\'ec\'edente. La vari\'et\'e $X$ cherch\'ee va \^etre obtenue
en effectuant un ``flip" (plus exactement l'inverse
d'un flip) \`a partir de $Z$.
\medskip
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Suivant la figure ci-dessus, notons $\tilde{X}$ la vari\'et\'e
projective obtenue en \'eclatant $Z$ le long de ${\Bbb P} ^{1}$.
Le diviseur exceptionnel de l'\'eclatement est alors
$E = {\Bbb P} ^{1} \times {\Bbb P} ^{n-2}$, et pour pouvoir
contracter dans l'autre direction, il s'agit de montrer,
d'apr\`es le crit\`ere de contraction de Fujiki-Nakano,
que $\displaystyle{ {\cal O}(E)_{| {\Bbb P} ^1} = {\cal O}_{{\Bbb P} ^1}(-1) }$.
Pour cela, les deux suites exactes suivantes~:
$$ 0 \to N_{{\Bbb P}^1 /E} = {\cal O}_{{\Bbb P} ^1}^{\oplus n-2}
\to N_{{\Bbb P}^1/\tilde{X}} \to N_{E/ \tilde{X} | {\Bbb P} ^1} =
{\cal O}(E)_{| {\Bbb P} ^1} \to 0 ,$$
$$ 0 \to T{\Bbb P} ^1 \to T\tilde{X}_{| {\Bbb P} ^1}
\to N_{{\Bbb P}^1/\tilde{X}} \to 0 $$
\noindent donnent successivement~:
$$\deg (N_{{\Bbb P}^1/\tilde{X}})=
\deg ({\cal O}(E)_{| {\Bbb P} ^1}) \ ,\
\deg (K_{\tilde{X} | {\Bbb P} ^1}) =
-2 -\deg (N_{{\Bbb P}^1/\tilde{X}}).$$
\noindent Comme $K_{\tilde{X}} = f^{\ast}K_Z + (n-2){\cal O}(E)$
et $K_Z = {\cal O}_{{\Bbb P} ^{n+1}}(n-3)_{| Z}$, on en
d\'eduit bien que ${\cal O}(E)_{| {\Bbb P} ^1} = {\cal O}_{{\Bbb P} ^1}(-1)$.
\noindent La contraction de ${\Bbb P} ^{1}$ d\'efinit donc une
vari\'et\'e de Moishezon,
contenant un ${\Bbb P} ^{n-2}$ et telle
que $\operatorname{Pic} (X) = {\Bbb Z}$.
Montrons maintenant que $N_{{\Bbb P} ^{n-2} / \tilde{X}} =
{\cal O}_{{\Bbb P} ^{n-2}}(-1) \oplus {\cal O}_{{\Bbb P} ^{n-2}}(-1)$.
Comme $$E =
{\Bbb P} ^{1} \times {\Bbb P} ^{n-2} = {\Bbb P}(N^{\ast}_{{\Bbb P} ^{n-2} / \tilde{X}}),$$
le fibr\'e normal $N_{{\Bbb P} ^{n-2} / \tilde{X}}$ est de la forme
${\cal O}_{{\Bbb P} ^{n-2}}(a) \oplus {\cal O}_{{\Bbb P} ^{n-2}}(a)$.
Comme pr\'ec\'edemment, la suite exacte~:
$$ 0 \to T{\Bbb P} ^{n-2} \to TX_{| {\Bbb P} ^{n-2}}
\to N_{{\Bbb P}^{n-2}/X} \to 0 $$
\noindent donne $2a = -\deg (K_{X | {\Bbb P} ^{n-2}}) -n+1$,
puis
$$\deg (K_{X | {\Bbb P} ^{n-2}}) = \deg (K_{\tilde{X} | {\Bbb P} ^{n-2}}
- {\cal O}(E)_{| {\Bbb P} ^{n-2}}) = -(n-2) + 1,$$
\noindent d'o\`u finalement $a=-1$.
\noindent Par ailleurs, nous venons de montrer que
$$ K_{X | {\Bbb P} ^{n-2}} = {\cal O}_{{\Bbb P} ^{n-2}}(3-n).$$
\noindent Ainsi, si $n$ est sup\'erieur ou \'egal \`a $4$, $-K_X$
est ample sur ${\Bbb P} ^{n-2}$. Finalement, le fibr\'e $K_X$ bien
que gros n'est pas nef et le th\'eor\`eme est d\'emontr\'e.
\hskip 3pt \vrule height6pt width6pt depth 0pt
\bigskip
\noindent {\bf Remarque }
La construction pr\'ec\'edente, en dimension $3$, donne un nouvel
exemple de vari\'et\'e de Moishezon, de ``Calabi-Yau"
satisfaisant
$\displaystyle{\operatorname{Pic} (X) = {\Bbb Z} \cdot {\cal O}_X (1) ,}$
(voir aussi \cite{Ogu94}).
\subsection{D\'emonstration de la proposition}
On d\'emontre (i) et (ii) simultan\'ement.
Les points singuliers de $Z$ sont des z\'eros communs des
\'equations
$$ x_0h_0 +\cdots+ x_{n-1}h_{n-1} = 0$$ et
$$x_0 \frac{\partial h_0}{\partial x_i} +\cdots+
x_{n-1} \frac{\partial h_{n-1}}{\partial x_i} + h_i = 0
\ , \ i=0,\ldots \!,n-1.$$
En particulier, $Z$ est lisse au voisinage de
${\Bbb P} ^{1} = \{ x_0 = \dots = x_{n-1} = 0 \}$ d\`es que les
$h_i$ ne s'annulent pas simultan\'ement sur ${\Bbb P} ^{1}$. Ceci
est vrai pour un choix g\'en\'erique des $h_i$ d\`es que
$n$ est sup\'erieur ou \'egal \`a $2$.
On d\'eduit alors
du th\'eor\`eme de Bertini \cite{G-H78}
que si les $h_i$ sont \`a
nouveau g\'en\'eriques, l'hypersurface $Z$ est non singuli\`ere
partout~: en effet, de fa\c{c}on g\'en\'erale, une relation
$$\displaystyle{ \sum_i s_i f_i = 0 }$$
d\'efinit une vari\'et\'e
non singuli\`ere en dehors des z\'eros communs des $s_i$
d\`es que les $f_i$ sont g\'en\'eriques dans l'espace des
sections holomorphes d'un fibr\'e engendrant en tout point
les jets
d'ordre inf\'erieur ou \'egal \`a $1$.
\medskip
D\'eterminons ensuite le fibr\'e normal $N_{{\Bbb P} ^{1}/ Z}$, et pour cela,
consid\'erons la suite exacte des fibr\'es normaux :
$$ 0 \to N_{{\Bbb P} ^{1}/ Z} \to N_{{\Bbb P} ^{1}/ {\Bbb P} ^{n+1}} =
{\cal O}_{{\Bbb P} ^{1}}(1)^{\oplus n} \stackrel{d \sigma}{\to}
{\cal O}_{{\Bbb P} ^{n+1}}(2n-1)_{| {\Bbb P} ^1} \to 0 .$$
Il est alors clair
que $N_{{\Bbb P} ^{1}/ Z}$ est de degr\'e $-(n-1)$.
Pour montrer qu'il est \'egal \`a ${\cal O}_{{\Bbb P} ^{1}}(-1)^{\oplus n-1}$,
il suffit donc de montrer qu'il n'a pas de sections (rappelons
en effet qu'un th\'eor\`eme d'A.\ Grothendieck affirme que
tout fibr\'e vectoriel sur ${\Bbb P} ^1$ est scind\'e).
Par la suite exacte pr\'ec\'edente, une section de $N_{{\Bbb P} ^{1}/ Z}$
peut \^etre vue
comme une section de ${\cal O}_{{\Bbb P} ^{1}}(1)^{\oplus n}$,
annul\'ee par $d \sigma$. Une telle section
correspond \`a la donn\'ee d'un $n$-uplet
$(s_0,\ldots \!,s_{n-1})$ o\`u les $s_i$ sont des polyn\^omes
homog\`enes de degr\'e $1$ en les variables $x_n , x_{n+1}$,
que l'on \'ecrit $s_i (x) = s_{i,n}x_n + s_{i,n+1}x_{n+1}$.
Dans
$N_{{\Bbb P} ^{1}/ {\Bbb P} ^{n+1}}$, on a alors~:
$$\displaystyle{s = \sum_{i=0}^{n-1} s_i \frac{\partial}{\partial x_i}}.$$
De m\^eme, notons
$$\displaystyle{h_i(x) = \sum_{p=o}^{2n-2} h_{i,p}x_{n}^{p}x_{n+1}^{2n-2-p}}$$
la restriction de $h_i$ \`a ${\Bbb P} ^1$.
La relation $d \sigma (s) = 0$ donne ici~:
$$\displaystyle{\sum_{i=0}^{n-1} s_i h_i = 0}.$$
Comme $$\displaystyle{ d \sigma = \sum_{i=0}^{n-1} h_idx_i }$$
le long de ${\Bbb P} ^1$, cette relation
se traduit par un syst\`eme lin\'eaire \`a $2n$ \'equations en les $2n$
inconnues
$s_{i,n},s_{i,n+1}$. Il s'agit de montrer que pour un choix
g\'en\'erique des $h_i$, le d\'eterminant de la matrice suivante~:
$$
\left(
\begin{matrix}
h_{0,0}&h_{1,0}&\dots&h_{n-1,0}&0&0&\dots&0\\
h_{0,1}&h_{1,1}&\dots&h_{n-1,1}&h_{0,0}&h_{1,0}&\dots&h_{n-1,0}\\
\vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots\\
h_{0,2n-2}&h_{1,2n-2}&\dots&h_{n-1,2n-2}&h_{0,2n-3}&h_{1,2n-3}&\dots&h_{n-1,2n-3}\\
0&0&\dots&0&h_{0,2n-2}&h_{1,2n-2}&\dots&h_{n-1,2n-2}
\end{matrix}
\right)
$$
n'est pas nul, ce qui est clair
en prenant par exemple $$h_{0,0}= \lambda _0, \ldots \!,
h_{n-1,n-1}= \lambda _{n-1},
h_{0,n-1}= \mu _{0}, \ldots \!, h_{n-1,2n-2}= \mu _{n-1}$$
avec $\lambda _i \neq 0$, $\mu_i \neq 0$.
\noindent Ainsi, il existe un choix des $h_i$ de sorte que
l'hypersurface $Z$ (\'eventuellement singuli\`ere) est
lisse au voisinage de ${\Bbb P} ^1$, avec $N_{{\Bbb P} ^{1}/ Z} =
{\cal O}_{{\Bbb P} ^{1}}(-1)^{\oplus n-1}$.
\medskip
Si maintenant les $h_i$ sont choisis de sorte que (i) et (ii) soient
satisfaits, alors (iii) est imm\'ediat par adjonction et
(iv) d\'ecoule du th\'eor\`eme de Lefschetz~:
l'hypersurface $Z$ est le diviseur d'une section
d'un fibr\'e ample et les fibr\'es en droites
sur $Z$ sont restriction de fibr\'es en droites
sur ${\Bbb P} ^{n+1}$.\hskip 3pt \vrule height6pt width6pt depth 0pt
\bigskip
\noindent {\bf Remarque }
Le cas $n=3$ de la proposition pr\'ec\'edente correspond \`a celui des
quintiques dans ${\Bbb P} ^4$. Il a \'et\'e consid\'er\'e par S.\ Katz
dans \cite{Kat86}. Dans ce cadre, S.\ Katz d\'etermine le fibr\'e
normal $N_{{\Bbb P} ^{1}/ Z}$ dans le cas o\`u les $h_i$ sont
g\'en\'eriques, mais analyse aussi la situation non
g\'en\'erique. Signalons aussi \cite{Cle83}, o\`u H.\ Clemens
consid\`ere des questions analogues, toujours en dimension $3$.
\medskip
\noindent {\bf Remarque }
On peut reprendre plus g\'en\'eralement la construction
pr\'ec\'edente
pour les hypersurfaces de ${\Bbb P} ^{n+1}$ de degr\'e $2n-2k+1$
passant par un ${\Bbb P} ^{k}$ lin\'eaire. On peut alors
\`a nouveau montrer que (g\'en\'eriquement)
$N_{{\Bbb P} ^{k}/ Z}$ (qui est de degr\'e $k-n$) n'a pas de sections.
Cependant, le fibr\'e $N_{{\Bbb P} ^{k}/ Z}$ n'est pas
scind\'e si $k \geq 2$.
\medskip
D\'emontrons ce dernier point en consid\'erant \`a nouveau
la suite exacte des fibr\'es
normaux~:
$$ 0 \to N_{{\Bbb P} ^{k}/ Z} \to N_{{\Bbb P} ^{k}/ {\Bbb P} ^{n+1}} =
{\cal O}_{{\Bbb P} ^{k}}(1)^{\oplus n+1-k} \stackrel{d \sigma}{\to}
{\cal O}_{{\Bbb P} ^{n+1}}(2n-2k+1)_{| {\Bbb P} ^k} \to 0 .$$
En dualisant cette suite, il vient~:
$$ 0 \to {\cal O}_{{\Bbb P} ^{n+1}}(-2n+2k-1)_{| {\Bbb P} ^k} \to
{\cal O}_{{\Bbb P} ^{k}}(-1)^{\oplus n+1-k} \to N_{{\Bbb P} ^{k}/ Z}^{\ast}
\to 0 .$$
Si $k \geq 2$, comme le fibr\'e
${\cal O}_{{\Bbb P} ^{n+1}}(-2n+2k-1)_{| {\Bbb P} ^k}$ est n\'egatif,
la suite exacte longue de cohomologie donne
$H^0 ({\Bbb P} ^k, N_{{\Bbb P} ^{k}/ Z}^{\ast}) = 0$. Ceci exclut de suite
le fait que $N_{{\Bbb P} ^{k}/ Z}$ soit scind\'e car il serait
alors \'egal \`a ${\cal O}_{{\Bbb P} ^{k}}(-1)^{\oplus n-k}$.~\hskip 3pt \vrule height6pt width6pt depth 0pt
\section{Une classification en dimension $4$}
Dans ce paragraphe, nous consid\'erons des vari\'et\'es
de Moishezon (non projectives) $X$ de dimension $4$.
Comme pr\'ec\'edemment, nous
supposons que $\operatorname{Pic} (X) = {\Bbb Z}$, que $K_X$ est gros
et que $X$ est rendue projective apr\`es \'eclatement
le long d'une sous-vari\'et\'e lisse $Y$.
\subsection{\'Enonc\'e des r\'esultats}
Nous montrons les deux r\'esultats suivants~:
\medskip
\noindent {\bf Th\'eor\`eme H }{\em
Sous les hypoth\`eses pr\'ec\'edentes, $Y$ est n\'ecessairement
une surface.
\noindent Autrement dit, et dans cette situation particuli\`ere,
il ne suffit pas d'\'eclater une courbe pour rentrer dans
le monde projectif.
}
\medskip
Nous avons vu pr\'ec\'edemment que $K_X$ n'est pas n\'ecessairement
nef \`a partir de la dimension $4$. Le r\'esultat suivant montre
que l'exemple construit dans le paragraphe pr\'ec\'edent
est le ``seul possible" dans le cas
o\`u $K_X$ n'est pas nef.
Nous reprenons les notations des paragraphes
pr\'ec\'edents,
$\displaystyle{ \pi : \tilde{X} \to X}$
d\'esigne
l'\'eclatement de $X$ le
long de $Y$ et
$\displaystyle{ f : \tilde{X} \to Z}$
d\'esigne la contraction extr\^emale de Mori sur $\tilde{X}$.
\medskip
\noindent {\bf Th\'eor\`eme I }{\em
Sous les hypoth\`eses pr\'ec\'edentes et si $K_{X}$ n'est
pas nef, alors~:
(i) le couple $(Y,N_{Y/X})$ est \'egal \`a
$({\Bbb P} ^2, {\cal O}_{{\Bbb P} ^{2}}(-1)^{\oplus 2})$,
(ii) $f$ contracte le
diviseur exceptionnel de $\pi$ sur une courbe rationnelle
lisse \`a fibr\'e normal ${\cal O}_{{\Bbb P} ^{1}}(-1)^{\oplus 3}$
dans une vari\'et\'e projective lisse $Z$.
En particulier, $f$ est une contraction divisorielle.}
\bigskip
Ces r\'esultats sont accessibles en dimension $4$ car les
contractions de Mori sont ``bien comprises"
gr\^ace aux r\'esultats, rappel\'es pr\'ec\'edemment
de T.\ Ando \cite{And85}
et M.\ Beltrametti \cite{Bel86} pour les contractions divisorielles.
\bigskip
\subsection{D\'emonstration du th\'eor\`eme H}
\medskip
Nous traitons s\'epar\'ement les cas $K_X$ nef
et $K_X$ non nef.
\noindent (i) Le cas $K_X$ non nef~.
Il d\'ecoule directement du th\'eor\`eme~F~: en effet,
si la contraction de Mori $f$
est une contraction divisorielle, le point (iii)
assure
que $Y$ est une surface et que $f(E)$ est une courbe.
Par ailleurs, le point (iv)
exclut la possibilit\'e que $f$ soit une petite contraction
(car sinon $\dim Y \geq 3$ !).
Mentionnons qu'une premi\`ere version de ce travail
\cite{Bo95b} excluait ce cas en utilisant le
difficile th\'eor\`eme de structure des petites contractions de
Kawamata \cite{Kaw89}. Ceci ach\`eve le cas $K_X$ non nef.
\bigskip
\noindent (ii) Le cas $K_X$ nef.
Dans ce cas, rappelons que la contraction de Mori se factorise
par $\pi$. Notons $\displaystyle{ g : X \to Z}$ de sorte
que $f = g \circ \pi$.
Raisonnons alors par l'absurde en supposant que $Y$ est une courbe.
Dans ce cas, il est clair que $f(E)$ (\'egal \`a $g(Y)$) est un point.
En effet, dans le cas contraire, $f(E)$ est une courbe et l'application
$g$ est finie. Comme $Z$ est projective, on en d\'eduit que $X$ est
projective, ce que l'on a rejet\'e.
Ainsi $f(E)$ est un point et le diviseur $E$ est une vari\'et\'e
de Fano.
\noindent Nous allons montrer que $E$ est en fait isomorphe \`a
la quadrique de dimension $3$, ce qui fournira la contradiction~;
une quadrique de dimension $3$, dont le nombre de Picard est $1$,
ne pouvant \^etre \'egale au
projectivis\'e d'un fibr\'e de rang $3$ sur une courbe,
pour lequel le nombre de Picard est $2$ !
\noindent Pour cela, remarquons que $Z$ \'etant
${\Bbb Q}$-factorielle \`a singularit\'es
terminales, il existe un entier $m$ non nul tel que $mK_Z$ est de Cartier.
Alors~:
$$mK_X = g^{\ast}(mK_Z).$$
En particulier, la restriction de
$K_X$ \`a $Y$ est triviale. Il en d\'ecoule que $K_Y = \det N_{Y/X}$,
et par cons\'equent,
$$K_E = \pi ^{\ast}(K_Y - \det N_{Y/X} ) +
3{\cal O}_{E}(-1) = 3{\cal O}_{E}(-1).$$
On en d\'eduit que
${\cal O}_{E}(1)$ est ample et que $E$ est une vari\'et\'e
(de dimension $3$) d'indice $3$~; rappelons que l'indice
d'une vari\'et\'e de Fano $V$ est le plus grand entier
$r >0$ tel qu'il existe un fibr\'e en droites
$L$ avec $-K_V = rL$.
Or, le th\'eor\`eme de Kobayashi-Ochiai \cite{KoO73} affirme
qu'{\em une vari\'et\'e de Fano de dimension $n$ et
d'indice $n$ est isomorphe \`a la quadrique ${\cal Q}_n$}. On en
d\'eduit ici que $E$ est
la quadrique ${\cal Q}_3$ comme annonc\'e.\hskip 3pt \vrule height6pt width6pt depth 0pt
\bigskip
Dans la situation du th\'eor\`eme H et lorsque
$K_X$ est nef, nous avons
vu que la contraction de Mori $f$ sur $\tilde{X}$
se factorise en une application birationnelle
$g : X \to Z$ qui contracte la surface $Y$.
La proposition suivante pr\'ecise le cas o\`u
$g(Y)$ est r\'eduit \`a un point~:
\bigskip
\noindent {\bf Proposition } {\em Si $K_X$ est nef et si
$f(E)$ (\'egal \`a $g(Y)$) est un point, alors
le couple
$(Y,N_{Y/X})$ est \'egal \`a $({\Bbb P} ^2, T^{\ast}{\Bbb P} ^2)$,
$({\Bbb P} ^2, {\cal O}_{{\Bbb P} ^2}(-1) \oplus {\cal O}_{{\Bbb P} ^2}(-2) )$
ou $({\cal Q}_2, {\cal O}_{{\cal Q}_2}(-1,-1)^{\oplus 2} )$.}
\bigskip
Nous ne connaissons pas d'exemples explicites o\`u
ces possibilit\'es sont effectivement r\'ealis\'ees, mais
nous pouvons remarquer qu'aucune n'est exclue {\em a priori}
par les r\'esultats de T.\ Ando et M.\ Beltrametti.
\bigskip
La d\'emonstration de la proposition d\'ecoule directement
du th\'eor\`eme suivant de T.\ Peternell \cite{Pet91}~:
\medskip
\noindent {\bf Th\'eor\`eme (T.\ Peternell, 1991) } {\em
Soit $V$ une vari\'et\'e projective de dimension $n$
et soit $E$ un fibr\'e vectoriel de rang $n$ sur $V$
de sorte que $c_1(E) = c_1(X)$. Alors, le couple
$(V,E)$ est \'egal \`a
$({\Bbb P} ^n, {\cal O}_{{\Bbb P} ^n}(2) \oplus {\cal O}_{{\Bbb P} ^n}(1)^{\oplus n+1})$,
$({\Bbb P} ^n, T {\Bbb P} ^n)$ ou $({\cal Q}_n, {\cal O}_{{\cal Q}_n}(1)^{\oplus n})$.}
\bigskip
\noindent {\bf D\'emonstration de la proposition}
La d\'emonstration du th\'eor\`eme H montre que
$$K_Y = \det N_{Y/X}$$
et que
$${\cal O}_E(1) = {\cal O}_{{\Bbb P} (N_{Y/X}^{\ast})}(1)$$
est ample,
donc que $N_{Y/X}^{\ast}$ est aussi ample.
Le r\'esultat d\'ecoule
du th\'eor\`eme de T.\ Peternell appliqu\'e au couple
$(Y, N_{Y/X}^{\ast})$. \hskip 3pt \vrule height6pt width6pt depth 0pt
\subsection{D\'emonstration du th\'eor\`eme I}
Notons $F$ la fibre g\'en\'erale de $f$ restreinte
au diviseur exceptionnel $E$. Comme $f(E)$ est
une courbe, $F$ est de dimension $2$. D'apr\`es le
th\'eor\`eme de T.\ Ando et M.\ Beltrametti,
$F$ est \'egal \`a ${\Bbb P} ^2$ ou \`a
la quadrique ${\cal Q}_2$. De plus, il a \'et\'e vu
pr\'ec\'edemment que $F$ coupe les fibres de $\pi$
sur des points. On en d\'eduit que $\pi _{| F} : F \to Y$
est une application surjective finie. Deux cas sont \`a distinguer~:
\medskip
- $F$ est \'egal \`a ${\Bbb P} ^2$.
\noindent Dans ce cas, $Y$ est aussi \'egal
\`a
${\Bbb P} ^2$. En effet, un r\'esultat de R.\ Lazarsfeld \cite{Laz84}
affirme que si {\em $h : {\Bbb P} ^n \to V$ est une application holomorphe
surjective finie sur une vari\'et\'e de dimension $n$,
alors $V$ est isomorphe \`a ${\Bbb P} ^n$}~; en dimension $2$,
on peut trouver une d\'emonstration \'el\'ementaire
dans \cite{BPV84}.
\noindent Montrons alors que
$$\displaystyle{ \pi _{| F} : F \simeq {\Bbb P} ^2 \to Y \simeq {\Bbb P} ^2 }$$
est un isomorphisme. Pour cela, il suffit de montrer que
$\pi _{| F}$ est un isomorphisme local, car alors $\pi _{| F}$
est un rev\^etement donc le rev\^etement trivial.
Soient donc $x$ dans $F$ et $L$ un ${\Bbb P} ^1$ quelconque passant par $\pi (x)$.
Sa pr\'e-image $\pi ^{-1}(L)$ est une surface d'Hirzebruch bi-r\'egl\'ee
donc ${\Bbb P} ^1 \times {\Bbb P} ^1$. L'intersection
$F \cap \pi ^{-1}(L)$ est alors une r\'eunion
de ${\Bbb P} ^1$ ``horizontaux". La restriction de $\pi$
au ${\Bbb P} ^1$ horizontal passant par $x$ est donc un
isomorphisme sur son image. Ceci \'etant vrai pour tout
${\Bbb P} ^1$ passant par $\pi (x)$, ceci montre bien
que $d \pi_{| F}(x)$ est surjective, donc inversible,
et que $\pi _{| F}$ est un isomorphisme
local.
Ainsi,
$$\displaystyle{ \pi _{| F} : F \simeq {\Bbb P} ^2 \to Y \simeq {\Bbb P} ^2 }$$
est un isomorphisme.
On en d\'eduit que le fibr\'e normal $N_{Y/X}$
est scind\'e~; on d\'efinit alors $a$ et $b$ en posant~:
$$N_{Y/X} = {\cal O}_{{\Bbb P} ^2}(a) \oplus {\cal O}_{{\Bbb P} ^2}(b).$$
Le fait que $\pi ^{-1}(L) \simeq {\Bbb P} ^1 \times {\Bbb P} ^1$
montre m\^eme que $a=b$.
\noindent Comme $K_X$
n'est pas nef, $K_X$ est n\'egatif sur $Y$. Il vient alors~:
$$\displaystyle{\deg (K_{X | Y}) = -3-2a < 0}$$ d'o\`u
$a \geq -1$. L'affirmation suivante permet de conclure~:
\medskip
\noindent {\bf Affirmation }
{\em L'entier $a$ est strictement n\'egatif.}
\medskip
\noindent {\bf D\'emonstration }
Par l'absurde, supposons que $a \geq 0$. Alors, si $C$
d\'esigne un ${\Bbb P} ^1$ de $Y = {\Bbb P} ^2$, on a~:
$$ H^1(C,N_{C/X}) =
H^1({\Bbb P}^1, {\cal O}_{{\Bbb P} ^1}(1) \oplus {\cal O}_{{\Bbb P} ^1}(a)^{\oplus 2}) = 0,$$
d'o\`u~:
$$ \dim_{[C]} \operatorname{Hilb} (X) =
\dim H^0({\Bbb P}^1, {\cal O}_{{\Bbb P} ^1}(1) \oplus {\cal O}_{{\Bbb P} ^1}(a)^{\oplus 2})
= 2a + 4.$$
Or, $$\dim_{[C]} \operatorname{Hilb} (Y) = \dim_{[{\Bbb P} ^1]} \operatorname{Hilb} ({\Bbb P} ^2) = 2.$$
Comme $a \geq 0$, on en d\'eduit que~:
$$ \dim_{[C]} \operatorname{Hilb} (X) > \dim_{[C]} \operatorname{Hilb} (Y)$$
si bien que $C$ se d\'eforme dans $X$ hors de $Y$. Ceci n'est
pas possible comme nous l'avons d\'ej\`a rencontr\'e car
$K_X$ est positif sur les courbes non incluses dans $Y$.
Ici, $K_X$, n'\'etant pas nef, est n\'egatif sur $C$. Contradiction !
\hskip 3pt \vrule height6pt width6pt depth 0pt
\medskip
Ainsi, $a=-1$ et
$f$ contracte $E$ sur une courbe
rationnelle lisse \`a fibr\'e normal ${\cal O}_{{\Bbb P} ^1}(-1)^{\oplus 3}$
dans la vari\'et\'e projective lisse $Z$.
\medskip
- $F$ est \'egal \`a la quadrique ${\cal Q}_2$.
\noindent Nous montrons
que ce cas ne peut pas arriver.
En effet, $Y$ est alors isomorphe \`a ${\Bbb P} ^2$ ou ${\cal Q}_2$. Le cas
$Y \simeq {\Bbb P} ^2$ s'exclut exactement comme pr\'ec\'edemment~: $\pi _{| F}$
r\'ealise un isomorphisme entre la quadrique et ${\Bbb P} ^2$ !
\noindent Si $Y \simeq {\cal Q}_2$, le raisonnement est plus simple et il
est inutile de montrer que
$$\pi _{| F} : F \simeq {\cal Q}_2 \to Y \simeq {\cal Q}_2$$
est un isomorphisme. Choisissons en effet un ${\Bbb P} ^1$ dans $Y$,
\`a savoir un des
g\'en\'erateurs de $H_2({\cal Q}_2,{\Bbb Z})$, sur lequel
$K_X$ est strictement n\'egatif (il en existe car $K_X$
n'est pas nef). Alors $N_{Y/X}$ restreint \`a
${\Bbb P} ^1$ est de la forme $${\cal O}_{{\Bbb P} ^1}(a) \oplus {\cal O}_{{\Bbb P} ^1}(a)$$
(ceci comme pr\'ec\'edemment car
$\pi ^{-1}({\Bbb P} ^1) \simeq {\Bbb P} ^1 \times {\Bbb P} ^1$).
La suite exacte~:
$$ 0 \to T {\Bbb P} ^1 \to TX_{| {\Bbb P} ^1} \to N_{{\Bbb P} ^1/X} \to 0,$$
et le fait que $N_{{\Bbb P} ^1/{\cal Q}_2}$ est trivial entrainent
que $$\deg (-K_{X | {\Bbb P} ^1}) = 2 + 2a > 0$$ et donc que
$a \geq 0$. Ceci est, comme dans
le cas pr\'ec\'edent, absurde
car ce ${\Bbb P} ^1$ se d\'eformerait
alors dans $X$ hors de $Y$ ! \hskip 3pt \vrule height6pt width6pt depth 0pt
\subsection{Quelques commentaires}
Comme nous venons de le voir, la situation en dimension
$4$ est tr\`es satisfaisante lorsque $K_X$ n'est pas nef.
Dans le cas o\`u $K_X$ est nef, nous avons obtenu
une restriction sur le centre de l'\'eclatement seulement
lorsque le diviseur exceptionnel $E$ est contract\'e sur un point.
Au moment o\`u nous finissions la r\'edaction de cette th\`ese,
nous avons appris que M.\ Andreatta et J.A.\ Wi\'sniewski
terminent la r\'edaction d'un travail consistant \`a classifier
les contractions extr\^emales divisorielles en dimension $4$
sur une vari\'et\'e non-singuli\`ere,
\'etendant ainsi les r\'esultats de M.\ Beltrametti au cas
o\`u le diviseur est contract\'e sur une courbe
ou sur une surface. Nous sommes en mesure d'appliquer leurs
r\'esultats dans notre situation pour obtenir la
proposition suivante. Pr\'ecisons cependant que nous n'avons
pas encore une version \'ecrite du travail en question
mais que notre seule r\'ef\'erence est une s\'erie
de discussions informelles avec M.\ Andreatta, M.\ Mella et J.A.\ Wi\'sniewski.
\medskip
\noindent {\bf Proposition }
{\em Soit $X$ comme dans le th\'eor\`eme H. On suppose
que $K_X$ est nef. Si $f$ est la contraction de Mori
d\'efinie sur $\tilde{X}$, alors~:
(i) le diviseur exceptionnel $E$ est contract\'e
sur une courbe ou un point. Autrement dit, $f(E)$
n'est pas une surface,
(ii) si $f(E)$ est une courbe, cette derni\`ere est une
courbe lisse de singularit\'es nodales ordinaires
$3$-dimensionelles et le centre $Y$
de l'\'eclatement $\pi$ est une surface r\'egl\'ee
dont les fibres ${\Bbb P} ^1$ ont pour fibr\'e
normal ${\cal O}_{{\Bbb P} ^1}\oplus {\cal O}_{{\Bbb P} ^1}(-1)^{\oplus 2}$.
Autrement dit, la situation est localement le
produit d'une courbe par le mod\`ele analogue en dimension~$3$.}
\medskip
Cette proposition termine la description des
situations possibles~; cependant nous ne connaissons
pas \`a l'heure actuelle d'exemple explicite o\`u le
point (ii) est r\'ealis\'e.
\medskip
\noindent {\bf ``D\'emonstration"}
Tout d'abord, mentionnons que la contraction divisorielle que
nous \'etudions est tr\`es
particuli\`ere car nous savons {\em a priori} que le diviseur
exceptionnel a une structure de fibration en espaces
projectifs sur une base lisse.
Pour le point (i), supposons par l'absurde que $f(E)$ est une
surface. Dans ce cas, la fibre g\'en\'erale est un ${\Bbb P} ^1$
et M.\ Andreatta et J.A.\ Wi\'sniewski montrent qu'une
\'eventuelle fibre particuli\`ere est soit ${\Bbb P} ^2$, soit
la quadrique ${\cal Q}_2$, soit la quadrique
singuli\`ere ${\cal Q}_2^0$. Dans notre situation,
une \'eventuelle fibre particuli\`ere est donc ${\cal Q}_2$
et l'image $\pi ({\cal Q}_2)$ dans $Y$ est une courbe
rationnelle $C$ d'auto-intersection $-1$.
\noindent Montrons que ceci n'est pas possible, \`a
nouveau par un argument de d\'eformation. En effet,
$K_X$ est trivial sur $C$, donc
$$ N_{C/X} = {\cal O}_{{\Bbb P} ^1}(-1) \oplus {\cal O}_{{\Bbb P} ^1}(a) \oplus
{\cal O}_{{\Bbb P} ^1}(b)$$
o\`u $a$ et $b$ sont deux entiers satisfaisant la relation $a + b = -1$.
De l\`a
$$ \dim \operatorname{Hilb} _{[C]}(X) \geq \dim H^0(C,N_{C/X}) - \dim H^1(C,N_{C/X}) =
a+b+2 = 1 > 0$$
d'o\`u l'on d\'eduit que $C$ se d\'eforme dans $X$ et ce hors de $Y$.
Pour le point (ii), nous sommes dans la situation ``facile"
du travail de M.\ Andreatta et J.A.\ Wi\'sniewski car les fibres
de $f$ restreinte \`a $E$ sont \'equi-dimensionnelles.
Dans notre situation, la fibre g\'en\'erale est une quadrique
${\cal Q}_2$ et il n'y a pas de fibres particuli\`eres~: la
situtation est, transversalement \`a $f(E)$, la r\'esolution
d'une singularit\'e nodale $3$-dimensionnelle.\hskip 3pt \vrule height6pt width6pt depth 0pt
\newpage
|
1992-10-14T18:03:43 | 9209 | alg-geom/9209001 | en | https://arxiv.org/abs/alg-geom/9209001 | [
"alg-geom",
"math.AG"
] | alg-geom/9209001 | Dr Roger Brussee | Rogier Brussee | On the $(-1)$-curve conjecture of Friedman and Morgan | 13 pages, LaTeX 2.09 | null | null | null | null | Main difference with previous version: we prove that every differentiably
embedded sphere with self intersection $-1$ in a simply connected algebraic
surface with $p_g >0$ is homologous to a $(-1)$-curve if $|K_{\min}|$ contains
a smooth irreducible curve of genus at least 2 and $p_g$ is even or $K_{\min}^2
\not\equiv 7 \pmod8$ (here $K_{\min}$ is the canonical class of the minimal
model).
| [
{
"version": "v1",
"created": "Sat, 12 Sep 1992 12:01:29 GMT"
},
{
"version": "v2",
"created": "Wed, 14 Oct 1992 16:46:38 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Brussee",
"Rogier",
""
]
] | alg-geom | \section{Introduction.}
It is now well known that the deformation type of an algebraic
surface is determined by its oriented diffeomorphism type up to a
finite number of choices \cite{F&M:ellipticI}\,\cite[theorem S.2]{F&M}.
It is therefore natural to ask if a deformation invariant is in fact an
invariant of the underlying oriented differentiable manifold. For
example, Van de Ven conjectured that this is true for the Kodaira
dimension
\cite{O&V:overview}\,\cite{F&M}\,\cite{Pidstrigach&Tyurin:specialinst}.
In this paper we study whether the deformation invariant
decomposition
\[
{alggeodecomp} H_2(X) = H_2(X_{\min}) \oplus^\perp \mathop\oplus {\Bbb Z} E_i,
\]
in the homology of the minimal model and the span of the $(-1)$-curves
is invariant under orientation preserving diffeomorphisms (cf.
\cite[conj. 2]{F&M:BAMS})
A {\sl $(-1)$-curve} on a complex surface is a smooth holomorphically
embedded 2-sphere with self-intersection $-1$. A $(-1)$-curve can be
blown down to obtain a new smooth complex surface. Successively
contracting all $(-1)$-curves gives the minimal model~$X_{\min}$ which
is unique if $p_g >0$. More generally we will call the total transform
of a $(-1)$-curve on some intermediate blow-down a $(-1)$-curve as
well. It is in this sense that the decomposition~\ref{alggeodecomp} is
a deformation invariant.
A {\sl $(-1)$-sphere} on a $4$-manifold is a smooth differentiably
embedded 2 sphere with self-intersection~$-1$. A classical $(-1)$-curve
is obviously a $(-1)$-sphere, a reducible one can be deformed to a
$(-1)$-sphere by smoothing out the double points. Moreover if two
$(-1)$-curves are orthogonal, they can be deformed in disjoint
$(-1)$-spheres. Friedman and Morgan conjectured that if a surface has a
unique minimal model, then modulo homological equivalence the relation
between its $(-1)$-spheres and its $(-1)$-curves is the strongest
possible.
\pr@claim{\bf}{ \thetheorem}{\sl} Conjecture (-1)conj. (Friedman and Morgan \cite[conj. 2,3 prop.
4]{F&M:BAMS}) Let $X$ be a simply connected algebraic surface with
Kodaira dimension~$\kappa \ge 0$. Then every $(-1)$-sphere is
homologous to a $(-1)$-curve up to orientation. In particular the
decomposition~(\ref{alggeodecomp}) is invariant under orientation
preserving diffeomorphisms.
(I have slightly reformulated the conjecture, and added the simply
connectedness hypothesis). Now, $X$ contains $n$ disjoint
$(-1)$-spheres if and only if there is a differentiable connected sum
decomposition $X \buildrel\scriptscriptstyle \rm diff\over\iso X' \#n{\bar\P}$. The decomposition
\ref{alggeodecomp} can be thought of as being induced by this
special connected sum decomposition.
Friedman and Morgan also made a conjecture about more general connected sum
decompositions, which would imply conjecture~\ref{(-1)conj} above.
\pr@claim{\bf}{ \thetheorem}{\sl} Conjecture consumconj. (Friedman and Morgan \cite[conj.
9]{F&M:BAMS}) Let $X$ be a simply connected algebraic surface with
$\kappa \ge 0$. Suppose $X$ admits a connected sum decomposition $X
\buildrel\scriptscriptstyle \rm diff\over\iso X' \# N$ for a negative definite manifold~$N$, then $H_2(N,{\Bbb Z})$
is generated by $(-1)$-curves.
Note that by theorems of Donaldson (\cite[th. 1.3.1, 9.3.4,
10.1.1]{D&K}, $N$ has automatically a standard negative definite
intersection form if $p_g(X)>0$. Conjecture \ref{consumconj} (and hence
conjecture \ref{(-1)conj}) has been proved for blow-ups of simply
connected surfaces with $p_g >0$ and big monodromy (like elliptic
surfaces or complete intersections), and simply connected surfaces with
$p_g >0$ whose minimal model admits a spin structure (i.e. $K_{\min}
\equiv 0(2)$) \cite[cor. 4.5.4]{F&M}. Conjecture~\ref{(-1)conj} has been
proved for the Dolgachev surfaces (i.e. $\kappa=p_g = 0$).
For minimal surfaces, conjecture \ref{(-1)conj} would imply strong
minimality. A $4$-manifold is called {\sl strongly minimal} if for
every diffeomorphism $X \# N_1 \buildrel\scriptscriptstyle \rm diff\over\iso Y \# N_2$ with $N_i \buildrel\scriptscriptstyle \rm diff\over\iso n_i
{\bar\P}^2$, we have $H_2(N_2) \subset H_2(N_1)$ (c.f. \cite[def.
IV.4.6]{F&M}). Conjecture \ref{(-1)conj} would also imply that the canonical
class of the minimal model $K_{\min}$ is invariant mod 2 under orientation
preserving diffeomorphisms.
Conjecture \ref{consumconj} would imply that a minimal
surface is {\sl irreducible} i.e. for every decomposition $X \buildrel\scriptscriptstyle \rm diff\over\iso X'
\# N$, say $N$ is {\relax homeomorphic} to $S^4$, thereby avoiding the
Poincar\'e conjecture.
In this paper we will show that under the stronger assumption $p_g >0$,
$(-1)$-spheres must give rise, if not to $(-1)$-curves then in any
case to special algebraic 1-cycles. Indeed, the main theorem~\ref{main}
below leaves little room for $(-1)$-spheres not homologous to a
$(-1)$-curve. Furthermore we will reduce a similar statement for
general connected sum decompositions, to a technical problem in gauge
theory.
To state the theorem we need some notation. Let $N_1(X)_{\Bbb Z} \subset
H_2(X,{\Bbb Z})$ be the preferred subgroup of algebraic classes i.e. the
subgroup generated by algebraic curves. Its rank $\rho$ is the Picard
number. The effective cone $\rmmath{NE}(X) \subset N_1(X)_{\Bbb Q}$ is the cone
spanned by positive rational multiples of algebraic curves. The subcone
$\rmmath{NE}(X_{\min}) = \rmmath{NE}(X) \cap H_2(X_{\min},{\Bbb Q})$ is the cone spanned by the
pullbacks of rational curves on the minimal model, i.e. the effective
cone in $H_2(X_{\min})$. Finally we note that since $N_1(X)_{\Bbb Q}$ is a
finite dimensional vector space, the closure of the effective cone
$\overline{\rmmath{NE}}(X)$ is well defined.
\pr@claim{\bf}{ \thetheorem}{\sl} Theorem main. Let $X$ be a simply connected algebraic
surface with $p_g >0$ and let~$K$ be its canonical divisor. Then for
every $(-1)$-sphere in $X$, there is an orientation such that $e$ is
either represented by a $(-1)$-curve or $e \in
\overline{\rmmath{NE}}(X_{\min})$, depending on whether $K\cdot e$ is negative
or positive respectively.
Note that $K\cdot e \ne 0$ since $K\cdot e \equiv e^2 \!\!\pmod 2$.
I have no examples where $K\cdot e$ is positive (i.e. a counter-example to
conjecture~\ref{(-1)conj}) but without further assumptions neither can I
exclude this case.
\pr@claim{\bf}{ \thetheorem}{\sl} Corollary maincor. In addition to the assumptions of the
theorem suppose that the minimal model $X_{\min}$ has Picard number $1$
or that the linear system $|K_{\min}|$ contains a smooth irreducible
curve of genus at least two, and that $p_g$ is even or $K_{\min}^2
\not\equiv 7 \pmod 8$,
then every $(-1)$-sphere is homologous to a $(-1)$-curve (i.e.
conjecture~\ref{(-1)conj} is true for $X$).
we will use this corollary to prove conjecture~\ref{(-1)conj} for
blow-\-ups of Horikawa surfaces with $K_{\min}^2$ even and zero-sets of
general sections in sufficiently ample $n-2$-bundles on $n$-folds with
$\rho =1$ generalising Friedman and Morgan's result for complete
intersections in ${\Bbb P}^n$.
The proof of theorem~\ref{main} is based on two very general properties
of the ${\rm SO}(3)$ Donaldson-Kotschick invariant~$\phi_k$. Kotschick
observed that it follows from the invariance properties of the
$\phi_k$~polynomial, that it is divisible by the Poincare dual of a
$(-1)$-sphere. On the other hand, using Morgan's algebro geometric
description of the Donaldson polynomials we show that $\phi_k$ has
pure Hodge type for $k \gg 0$. The theorem and the corollary then follow
by using Donaldson's and O'Grady's non-triviality results.
\sloppy
\pr@claim{\sl}{\relax}{\relax}Acknowledgement *. I have greatly benefitted from discussions with
Stephan Bauer, Chris Peters, Victor Pidstrigach, Jeroen Spandaw,
Kieran O'Grady, Gang Xiao and Ping Zhang, who I would all
like to thank heartily. Special thanks for Simon Donaldson for
helping me with gauge theory, and for inviting me to Oxford
university. Its mathematics institute has proved to be a very friendly
and stimulating environment. This paper grew out of work in my thesis
\cite{RB:thesis}. It is a pleasure to thank my thesis advisors Martin
L\"ubke and Van de Ven for their help and insight. (here they
can not remove such words !)
\fussy
\section{The $\phi_k$~polynomials.}
We will need ${\rm SO}(3)$ Donaldson polynomials $q_{L,k,\Omega}$ and in
particular the $\phi_k$~invariant introduced by Kotschick
\cite{Kotschick:SO(3)}. Let $X$ be a simply connected 4-manifold with
odd $b_+ \ge 3$. The polynomial $q_{L,k,\Omega}$ on $H_2(X)$
corresponds to the moduli space $\M^{\rm asd}(L,k)$ of ASD
${\rm SO}(3)$-connections on the ${\rm SO}(3)$-bundle~$P_k$ with $w_2(P_k)
\equiv L \pmod 2$, and $p_1(P_k) = -4k$, oriented by the
choice of the lift~$L$ of $w_2(P_k)$, and an
orientation $\Omega$ of a maximal positive subspace in $H^2(X,{\Bbb R})$
\cite[\S 9.2]{D&K}. We choose $\Omega$ once and for all (e.g. using
a complex structure if present), and we will suppress it in the notation.
$q_{L,k}$ has degree
$
d = 4k - \numfrac32 (1 + b_+)
$.
Note that the ${\rm SO}(3)$ bundle~$P_k$ exists if and only if $p_1 \equiv K^2
\pmod 4$, and that $k \in {\numfrac14} {\Bbb Z}$.
To define $\phi_k(X)$ we lift the second
Stiefel Whitney class~$w_2(X)$ of the manifold to an integral class~$K$.
For complex surfaces, the canonical divisor is such a lift. Now define
$\phi_k = q_{K,k} $. $\phi_k(X)$ is invariant under
orientation preserving diffeomorphisms up to sign.
Now suppose that $X$ has a decomposition $X \buildrel\scriptscriptstyle \rm diff\over\iso X' \# N$ for a
negative definite manifold $N$, necessarily with standard intersection
form. Then we have a decomposition $H_2(X) = H_2(X') \mathop\oplus
H_2(N)$. Choose generators $e_1,\ldots,e_n$ of $H_2(N)$, such that $K\cdot
e_i \equiv -1 \pmod 4$. This fixes the generators up to permutation.
By Poincar\'e duality we can consider the generators~$e_i$ as
linear forms on $H_2(X)$. Any polynomial $Q$ on $H_2(X)$ can be uniquely
written as a polynomial in the dual classes of $e_i$ with polynomials
on $H_2(X')$ as coefficients ({\sl the $e$-expansion}).
\pr@claim{\bf}{ \thetheorem}{\relax} Definition good. A $4$-manifold is said to have a {\sl good}
connected sum decomposition $X \buildrel\scriptscriptstyle \rm diff\over\iso X' \# N$ if for every generator
$e_i$ of $H_2(N)$, $e_i$ divides $q_{L,k}(X)$ for all $L$ with
$L\cdot e_i$ odd and all $k \gg 0$.
\pr@claim{\bf}{ \thetheorem}{\sl} Proposition {or.princip}. (Kotschick \cite[prop.
8.1]{Kotschick:SO(3)}) A connected sum decomposition $X \buildrel\scriptscriptstyle \rm diff\over\iso X' \#
n{\bar\P}^2$ is a good connected sum decomposition.
\proof\ (sketch). For notational simplicity we only prove divisibility
for the $\phi_k$ invariant. The reflection~$R_e$ in the hyperplane
defined by a generator~$e$ of $H_2(n{\bar\P}^2)$ can be represented by an
orientation preserving diffeomorphism, for example ${\rm id} \#
{\Bbb C}$-conjugation (c.f. \cite[prop 2.4]{F&M:ellipticI}). Then it follows
from the general invariance properties of the ${\rm SO}(3)$-polynomials
\cite[9.2.2]{D&K} that $R_e^*\phi_k(X) = - \phi_k(X)$. Hence
$\phi_k(X)$ is an odd polynomial in the dual class of $e$, in
particular $\phi_k$ is divisible by $e$.
\ifcomment\bgroup\par\medskip\noindent\small
$$
R_e^* \phi_k = R_e^*q_{\Omega,K,k} = q_{R_e^*\Omega,R_e^*K,k}
= (-1)^{\(K-R_e^*K \over 2\)^2}q_{\Omega,K,k} = -\phi_k.
$$
\par\medskip\noindent\egroup\fi
\endproof
One should expect that any connected sum decomposition is good. This
is because the coefficients $q_{L,k,N,I}$ of the $e$-expansion have the
same invariance properties as $q_{L,k}$ under orientation preserving
diffeomorphisms of $X'$. Conjecturally, these invariants depend only on
the homotopy type of $N$, (c.f. \cite[conjecture above lemma
4.5.6]{F&M}) and so the argument for $N = n{\bar\P}^2$ would give the
divisibility by the generators in general. A naive gauge theoretic
analysis seems to confirm this conjecture, but some technical
difficulties remain to be overcome. In any case we will state and prove
our results for surfaces admitting a good connected sum decomposition.
\section{Pureness of the Donaldson polynomials.}
Now we come to the algebraic geometric part of the proof. The Hodge
structure on $H^2(X,{\Bbb Z})$ induces a natural Hodge structure on
$S^d H^2(X)$. Let
$$
S^d H^2(X) \lhook\nobreak\joinrel\nobreak\m@p--\rightarrow{j} H^{2d}(X\times\cdots\times X)
$$
be the natural injection in the cohomology of the $d$ fold product
of $X$. Then $j$ is a map of Hodge structures. Hence a polynomial $q\in
S^d H^2(X)$ is of pure Hodge type~$(d,d)$ if and only if $j(q)$ is pure
of Hodge type~$(d,d)$. Now clearly a sufficient condition for $j(q)$ to
be of type~$(d,d)$ is that it is represented by an algebraic cycle. We
will prove the rather natural statement that those Donaldson
polynomials that can be computed completely by algebraic geometry
give rise to algebraic cycles. However to make this statement precise
requires serious work (as so often in mathematics). Fortunately almost
all of the work has already been done by J. Morgan \cite{Morgan}.
\pr@claim{\bf}{ \thetheorem}{\sl} Proposition (d,d). Let $X$ be a simply connected algebraic
surface with $p_g >0$. Then if $L \in \rmmath{NS}(X)$, there is constant $k_0
>0$ such that all Donaldson polynomials $q_{L,k}$ with $k >
k_0$ and the integer ${\numfrac12} (L^2 -L\cdot K) - {\numfrac14}(L^2 + 4k)$ odd
are represented by algebraic cycles. In particular these polynomials
are of Hodge type $(d,d)$, where $d = \deg(q_{L,k}) = 4k - 3(1 + p_g)$.
\proof. First suppose that $L \equiv 0 \pmod 2$, then $q_{L,k}$ is up
to sign just the ${\rm SU}(2)$ polynomial $q_k$. Now the lemma follows
directly from recent results of Morgan \cite{Morgan}. He shows that for
odd $k\gg 0$, $q_k$ can be computed as follows.
Let ${\mkern4mu\overline{\mkern-4mu\M}}^G_k = {\mkern4mu\overline{\mkern-4mu\M}}^G_k(H,0,k)$ be the
closure of the moduli space of $H$-slope stable bundles in the moduli space
of Gieseker $H$-stable sheaves with $c_1 = 0$, $c_2 = k$. For odd~$c_2$,
there exists a universal sheaf~$\xi$ on ${\mkern4mu\overline{\mkern-4mu\M}}^G_k$ (cf.
\cite[Remark A7]{Mukai:K3I} and \cite[prop. 2.2]{OGrady}) which
determines a correspondence
$$
\eqalign{
\nu:H_2(X) &\to H^2({\mkern4mu\overline{\mkern-4mu\M}}^G_k)
\\
\Sigma &\to c_2(\xi) \slant \Sigma.
}
$$
Then if $H$ is $k$-generic (in a sense to be made more precise below)
and $k \gg 0$, we have \cite[theorem 1]{Morgan}
\[
{nupol} q_k(\Sigma) = \<\nu(\Sigma)^d,[{\mkern4mu\overline{\mkern-4mu\M}}^G_k]>.
\]
Now since the universal sheaf~$\xi$ is algebraic, it actually determines
Chow cohomology classes
$c_2(\xi) \in A^2(X\times{\mkern4mu\overline{\mkern-4mu\M}}^G_k)$ (cf. \cite[Definition
17.3]{Fulton}). Consider the diagram
$$
\cdalign{
X^d\times{\mkern4mu\overline{\mkern-4mu\M}}^G_k
\\
\llap{$\scriptstyle \pi_{X^d}$} \swarrow \quad
\searrow\rlap{$\scriptstyle\pi_i$}
\\
X^d \hskip 5em (X\times{\mkern4mu\overline{\mkern-4mu\M}}^G_k)_i. \hskip -3em
}
$$
Then by equation~(\ref{nupol}), the algebraic cycle
$$
j(q_k) = \int_{[{\mkern4mu\overline{\mkern-4mu\M}}^G_k]} \pi_1^*c_2(\xi)\cdots \pi_d^*c_2(\xi)
\in A^d(X^d) \iso A_d(X^d),
$$
represents the image of the Donaldson polynomial~$q_k$ on the level of
Chow groups.
(Integration over the fibre $\int_{[{\mkern4mu\overline{\mkern-4mu\M}}^G_k]}$ is
defined formally as the composition
$$
A^i (X^d \times {\mkern4mu\overline{\mkern-4mu\M}}^G_k) \m@p--\rightarrow{[\pi_{X^d}]} A^{i-d}(X^d \times
{\mkern4mu\overline{\mkern-4mu\M}}^G_k \to X^d) \m@p--\rightarrow{\pi_*} A^{i-d}(X^d),
$$
where $[\pi_{X^d}]$ is the orientation class of the flat map
$\pi_{X^d}$ (cf. \cite[section 17.4]{Fulton}).
This proves the lemma if $L \equiv 0 \pmod 2$.
In case $L \not \equiv 0 \pmod 2$, the results of Morgan carry over
virtually unchanged, in fact the corresponding results are rather
easier. To be more precise, for a Hodge metric~$g_H$, the moduli space
of irreducible ASD ${\rm SO}(3)$-connections with $w_2 \equiv L \pmod 2$ and
$-p_1 = 4k$ can be identified with the moduli space ${\cal M}_k$ of $H$-slope
stable bundles with $c_1 = L$ and $4c_2 - c_1^2 = 4k$. For $k \gg 0$,
the closure of the moduli space of $H$-stable bundles in the moduli
space of Gieseker stable sheaves ${\mkern4mu\overline{\mkern-4mu\M}}^G_k$ has the proper complex
dimension $d=4k - 3(1+p_g)$ and is generically smooth. Moreover
${\mkern4mu\overline{\mkern-4mu\M}}_k^G$ carries a universal sheaf~$\xi$ \cite[prop. 2.2]{OGrady},
and the class $c_2(\xi) - {\numfrac14} c_1^2(\xi)$ defines a $\nu$
correspondence and Chow cohomology classes just as in the discussion
above. Finally, we choose a polarisation~$H$ which is $k$-generic in
the sense that $H\cdot(L-2N) \ne 0$ for all $N \in \rmmath{NS}(X)$ with $-4k
\le (L -2N)^2 < 0$ (i.e. $H$ is not on a wall). Then since $L \ne 0
\pmod 2$ every Gieseker $H$-semistable sheaf is actually slope
$H$-stable. Now all of the discussion in \cite{Morgan} to relate the
Gieseker and the Uhlenbeck compactification, and the $\mu$ and $\nu$
correspondence as far as it is concerned with slope stable sheaves and
bundles carries over.
\ifcomment\bgroup\par\medskip\noindent\small
Actually the discussion of relative
classes in 6.4.2 and 6.4.3 does not carry over, but is only needed to
deal with the trivial connection anyway.
\par\medskip\noindent\egroup\fi
\endproof
\pr@claim{\sl}{ \thetheorem}{\relax}Remark (d,d)diff. The pureness of the polynomials
$q_{L,k}$ is also suggested by a differential geometric argument, which
seems to be the point of view taken by Tyurin
\cite[\S 4.22]{Tyurin:algaspect}. The
complex structure on $X$ induces the complex structure
$$
T^{10}{\cal B}^*_X = \{a \in A^{10}(\mathop{{\cal E}\mkern-3mu{\it nd}}\nolimits_0(V)),\ d^* a =0\}
$$
on the space of irreducible connections
modulo gauge ${\cal B}^*_X$. The space of irreducible ASD connections with respect
to a K\"ahler metric is then an analytic subspace. Now the explicit
formulas in \cite[Proposition 5.2.18]{D&K}
for the forms representing $\mu({\rm Pd}(\omega))$ for an harmonic form
$\omega \in H^2(X,{\Bbb C})$, show that $\mu$
preserves the Hodge structure. If we write formally
$$
\phi_k({\rm Pd}(\omega_1), \ldots, {\rm Pd}(\omega_d)) =
\int_{\M^{\rm asd}_k}\mu({\rm Pd}(\omega_1))\cdots \mu({\rm Pd}(\omega_d)),
$$
then it is clear that $\phi_k(X) \ne 0$ only if the total Hodge type
of $\omega_1,\ldots,\omega_d$ is $(d,d)$. The (probably inessential)
problem is that it is not {\relax a priori} obvious whether integrating
the form representatives over the non compact manifold ${\cal M}_k^{\rm asd}$
gives a valid way of computing $\phi_k$.
\section{Proof of theorem~\protect\ref{main}.}
We can now give proofs of the results stated in the introduction.
The main theorem \ref{main} is the special case of theorem \ref{main'}
below for $N = n{\bar\P}^2$
(see definition~\ref{good} for the definition of good connected sum
decomposition).
\pr@claim{\bf}{ \thetheorem}{\sl} Theorem main'. Let $X$ be a simply connected algebraic
surface with $p_g >0$. Let $X_{\min}$ be its minimal model and let
$K$ be its canonical divisor. Suppose $X$ admits a good smooth
connected sum decomposition $X \buildrel\scriptscriptstyle \rm diff\over\iso X' \# N$ for a negative definite
manifold~$N$. Then $H_2(N,{\Bbb Z})$ is generated by classes
$e_1,\ldots,e_n$, with $e_i^2 = -1$ such that either $e_i$ is
represented by a $(-1)$-curve, or $e_i \in \overline{\rmmath{NE}}(X_{\min})$
depending on whether $K\cdot e_i$ is negative or positive
respectively.
Here $\overline{\rmmath{NE}}(X_{\min})$ is the closure of the cone spanned by
positive rational multiples of algebraic curves on the minimal model
\proof. Choose a generator~$e$ of~$H_2(N)$. We
first prove that $e$ is homologous to an algebraic cycle. Since $e$ is
certainly integral, it suffices by the Lefschetz~$(1,1)$ theorem \cite[p.
163]{G&H} to prove that its Poincar\'e dual is of
pure type~$(1,1)$.
Choose $k$ sufficiently large as in proposition~\ref{(d,d)}
(with $L= K$), and the definition \ref{good} of good. Then $\phi_k(X)$
is non trivial and has pure Hodge type~$(d,d)$. On the other hand we
have $\phi_k = e \psi$. Since the number of
Hodge types of~$e\psi$ is at least the number of Hodge types of~$e$,
$e$ has to be of pure type as well. Since $e$ is a real class, it is
then of type~$(1,1)$.
To show that for the proper orientation $e$ lies on the closure of the
full effective cone $\overline{\rmmath{NE}}(X)$, it is enough to show that
$e\cdot H \ne 0$ for all ample divisors $H$. In fact, since the closure
of the effective cone and the nef cone are in duality \cite[proposition
2.3]{Wilson:birational}, $e \in \pm \overline{\rmmath{NE}}(X)$ if and only if $e$
defines a strictly positive or strictly negative form on the ample
cone. But since the ample cone is connected, it suffices to show that
the form~$e$ has no sign change i.e. does not vanish on the ample cone.
Since $e$ is a rational class we need to check this only for integral
ample classes. Now for a fixed ample divisor~$H$ there is a $k_0 =
k_0(H)$ such that $\phi_k(X)(H) \ne 0$ for $k > k_0$ \cite[th.
10.1.1]{D&K}. Since $e$ divides $\phi_k(X)$, it follows that $e\cdot
H\ne 0$.
By the orthogonality result \cite[th. 4.5.3]{F&M}, for every
$(-1)$-sphere~$S$ in $X$ we have either $e\cdot S = 0$ or $e = \pm
[S]$. Hence for the ``effective orientation'' of $e$ found above, $e$
is either homologous to a $(-1)$-curve, or $e$ is orthogonal to all
$(-1)$-curves, i.e. $e \in H_2(X_{\min})$. In the first case $e\cdot K
= -1 < 0$, in the latter case we have $e\cdot K = e\cdot K_{\min}> 0$
for as $p_g$ is positive, $K_{\min}$ is nef, and $K\cdot e \equiv e^2
\equiv 1 \pmod 2$. Since a divisor on the minimal model is effective if
and only if its pullback to $X$ is effective, we have
$\overline{\rmmath{NE}}(X_{\min})= \overline{\rmmath{NE}}(X) \cap H_2(X_{\min})$, and the
result follows
\endproof
Theorem \ref{main'} gives the following technical refinement of
corollary~\ref{maincor}, proving conjecture~\ref{consumconj} for
the pair $(X,N)$ under an additional hypothesis.
\pr@claim{\bf}{ \thetheorem}{\sl} Corollary maincor'. In addition to the assumptions of theorem
\ref{main'}, suppose $X$ has a deformation~$Y$ with a minimal
model~$Y_{\min}$ such that there are no classes
$C\in\overline{\rmmath{NE}}(Y_{\min})$ with $C^2 = -1$ dividing all Donaldson
polynomials $q_{L,k}$ with $L\cdot C$ odd and $k \gg 0$. Then $H^2(N)$
is generated by $(-1)$-curves. In particular this is true if
\itm{(a)}
the linear system $|K_{Y_{\min}}|$ contains a smooth irreducible curve
of genus at least 2, and either $p_g$ is even or $K_{\min}^2
\not\equiv 7 \pmod 8$, or
\itm{(b)}
the Picard number $\rho(Y_{\min}) = 1$.
As mentioned in the introduction this corollary has already been proved
without the goodness condition under the assumptions $X_{\min}$ is spin
or $X_{\min}$ has big monodromy \cite[cor 5.4]{F&M}.
\proof. Since the deformations of a surface are all oriented
diffeomorphic, we conclude that if $X$ admits a good connected sum
decomposition $X\buildrel\scriptscriptstyle \rm diff\over\iso X' \# N$, so does its deformation~$Y$.
Moreover, the subgroup generated by $(-1)$-curves is stable under
deformation by \cite[IV.3.1]{BPV}. Hence if $H_2(N)\subset H_2(Y)$ is
generated by $(-1)$-curves, so is $H_2(N) \subset H_2(X)$. Thus we can
assume $X = Y$. By definition~\ref{good} of a good decomposition, it is
clear that no generator of $H_2(N)$ can be in
$\overline{\rmmath{NE}}(X_{\min})$. Hence by theorem~\ref{main'}, $H^2(N)$ is
generated by $(-1)$-curves. It remains to see that the extra condition
is satisfied in the given special cases.
In case (b), $\rmmath{NS}(X_{\min})$ is positive definite. For case (a)
we argue by contradiction. Suppose there is a class
$C\in\overline{\rmmath{NE}}(X_{\min})$, $C^2 = -1$ and $C$ divides $q_{C,k}$
for all $k \gg 0$.
Since $C$ is orthogonal to all $(-1)$-curves,
the proof of theorem \cite[th. 4.8]{Donaldson:pol},\,\cite[th. 9.3.14]{D&K}
gives that
$$
q_{C,k}(X)|_{H_2(X_{\min})} = \pm q_{C,k}(X_{\min}).
$$
Since $C \in \rmmath{NS}(X_{\min})$, Morgan's comparison formula~\ref{nupol},
and O'Grady's non triviality result \cite[cor. 2.4, th.2.4]{OGrady},
give that for every $\omega \in H^0(K_{\min})$, which vanishes on a smooth
irreducible curve of genus $g \ge 2$ we have
$$
q_{C,k}(X_{\min})({\rm Pd}(\omega + \bar \omega)) \ne 0
$$
if $4k - 3(1+p_g)$ is even and $k\gg 0$ with ${\numfrac12}(C^2-C\cdot K) -
{\numfrac14}(C^2 + 4k)$ odd. (Strictly speaking O'Grady uses a slightly
different polynomial defined on $C^\perp \subset H_2(X)$, but it is
easy to see that on $C^\perp$, $q_{C,k}$ coincides with his polynomial).
Since $4k \equiv -C^2 \pmod 4$, and $\<C,\omega> =0$ this contradicts
the divisibility of $q_{C,k}$ by $C$ if $p_g$ is even. If $p_g$ is odd,
the same argument gives a contradiction if there is a polynomial
$q_{L,k}$ with $L\in \rmmath{NS}(X_{\min})$, $L \cdot C \equiv 1$, and
$L^2\equiv K_{\min}\cdot L \equiv 0$. This an affine equation for $L
\pmod 2$ in $\rmmath{NS}(X_{\min}) \tensor {\Bbb Z}/2{\Bbb Z}$, so it has a solution if $C
\not\equiv K_{\min} \pmod 2$. But if $C \equiv K_{\min}$, then
$C^2 = -1 \equiv K_{\min}^2 \pmod 8$ contrary to assumption.
\endproof
\pr@claim{\sl}{ \thetheorem}{\relax}Remark *. If $|K_{\min}|$ contains a smooth
irreducible curve but $p_g$ is odd and $K_{\min}^2\equiv 7 \pmod 8$ the
proof above shows that there is up to orientation at most one
generator $e_0$ of $H^2(N)$ which is not homologous to a $(-1)$-curve.
Hence all $(-2)$-spheres in $H_2(X_{\min})$ are orthogonal
to $e_0$, because the reflections they generate are represented by
diffeomorphisms. We also get that if $e_0$ exists,
$w_2(X)$ is represented by the sum of the generators of $H_2(N)$, hence
$X'$ is spin.
\pr@claim{\sl}{ \thetheorem}{\relax}Remark *. It follows from results in \cite{RB:e-exp} that if
$C$ is orthogonal to all $(-1)$-curves, then the divisibility of say
$\phi_k(X)$ by $C$ implies the divisibility of $\phi_k(X')$ by $C$ for
all blow-downs of $X'$ intermediate between $X$ and $X_{\min}$. Hence
the extra condition in corollary \ref{maincor'} gets stronger as we
blow-up more points.
It is interesting to see how the minimality of a surface plays a role
here. For a non minimal surface every curve in $|K|$ is reducible since
it contains an exceptional divisor, unless $X$ is a K3-surface blown-up
once. Indeed in the non minimal case $q_{E,k}$ is divisible by the
exceptional divisor $E$, and so O'Grady's theorem could not possibly be
true. It would be very interesting if O'Grady's results would be true
assuming the existence of an irreducible curve in $|K|$, or stretching
things even further, a smooth irreducible curve in a pluricanonical
system $|nK|$.
\ifcomment\bgroup\par\medskip\noindent\small
Suppose that $X$ is of general type and
minimal. Then say $|13K|$ contains a smooth curve of genus at least
192, and we would still have a $\theta$-characteristic that extends
over the whole surface, which seems to be one of the main points of
O'Grady's construction. Since for elliptic surfaces the $(-1)$-curve
conjecture~\ref{(-1)conj} is already known, such a result would
confirm the conjecture for all simply connected surfaces with $p_g >0$
except those of general type with $p_g$ odd and $K_{\min}^2\equiv 7
\pmod 8$.
\par\medskip\noindent\egroup\fi
\section{Examples.}
We give two examples of the use of corollary~\ref{maincor'}. The
first example follows basically by leafing through \cite{BPV}. For the
other we use Noether-Lefschetz theory to reprove and generalise
Friedman and Morgan's result that for complete intersections with $p_g
>0$ conjecture~\ref{(-1)conj} is true. Since one approach to
Noether-Lefschetz theory is through monodromy groups, it is not
surprising that this approach gives results similar to those using big
monodromy. However the proof may be interesting because the Noether
Lefschetz theorems we will use are proved using the ``infinitesimal
method'', based on Hodge and deformation theory.
\pr@claim{\bf}{ \thetheorem}{\sl} Proposition examples.
Suppose $X$ is a smooth simply connected surface admitting a good
connected sum decomposition $X \buildrel\scriptscriptstyle \rm diff\over\iso X' \# N$. Then $H_2(N)$ is
generated by $(-1)$-curves if
\itm(a) $X$ is the blow-up of a Horikawa surface with $K^2$
even (see \cite[table~10]{BPV}), or
\itm(b)
$(Y,{\cal O}(1))$ is a simply connected projective local complete
intersection of dimension $r+2$ with Picard number $\rho =1$, $V$ is an
ample $r$-bundle, and $X$ is the blow up of the smooth zero locus $X_a$
of a section in $V(a)$ with $a \gg 0$.
\proof.
Case (a). By \cite[th. 10.1, remark VII.10.1]{BPV} and Bertini's
theorem, a Horikawa surface with $K^2$ even is simply connected and the
linear system $|K|$ contains a smooth curve.
\smallskip\noindent
Case (b). For
$r=0$ the proposition is a special case of corollary~\ref{maincor'} so
we assume $r >0$.
Choose $a$ so large that $V(a)$ is globally generated.
By an application of the vector bundle version of the Lefschetz hyperplane
theorem \cite{Sommese&VdV}, \cite[cor. 22]{Okonek:Barth-Lefschetz},
$X_a$ is simply connected \cite[p.158]{G&H}. To
prove that $p_g(X_a) >0$ consider the sequence
$$
0 \to I_X \det(V(a))\tensor {\cal O}_Y(K_Y) \to
\det( V(a))\tensor {\cal O}_Y(K_Y) \to {\cal O}_X(K_X) \to 0.
$$
Now choose $a$ so large that $\det V\tensor {\cal O}_Y(K_Y) \tensor {\cal O}_Y(ra)$
has a section non-vanishing on~$X$.
Finally, if $a$ is sufficiently large then $\rho(X_a) = 1$ for the
general section by Ein's generalization of the Noether-Lefschetz theorem
in ${\Bbb P}^3$ to ample vector bundles on projective varieties \cite[th.
2.4]{Ein}.
\endproof
\pr@claim{\sl}{ \thetheorem}{\relax}Remark *. Suppose that in case (b), $Y$
is smooth, and $V = \mathop\oplus_{i=1}^r {\cal O}(d_i)$. We can then be more
precise since there is no need to twist up. It suffices that $V$ and
$\det(V)\tensor {\cal O}_Y(K_Y)$ are spanned by global sections,
$H^{1,1}(\det(V))= H^{1,1}(V\tensor \det(V)) = 0$, and
$$
H^0(V) \tensor H^0(\det(V)\tensor {\cal O}_Y(K_Y)) \to
H^0(V\tensor \det(V)\tensor{\cal O}_Y(K_Y)))
$$
is surjective (e.g. if the $d_i \gg 0$ or $Y ={\Bbb P}^n$, with the
exception of $n=3$, $V = {\cal O}(2)$ or ${\cal O}(3)$, and $n=4$, $V ={\cal O}(2)
\mathop\oplus {\cal O}(2)$). This follows from judicially checking the cohomological
conditions \cite[lemma 3.2.1,3.2.2,3.2.3]{Spandaw:thesis} using
Kodaira-Nakano vanishing. For ${\Bbb P}^n$ the statement follows from the
classical Noether Lefschetz theorem. Also note that by choosing $a$
sufficiently large we can make $K_{X_a}$ very ample, and so we can
find a smooth irreducible curve of genus at least 2 in its linear
system. Hence case (b) follows directly from corollary~\ref{maincor'}
if $p_g(X_a)$ or $K^2_{X_a}$ are even.
|
1995-03-23T06:20:39 | 9503 | alg-geom/9503014 | en | https://arxiv.org/abs/alg-geom/9503014 | [
"alg-geom",
"math.AG"
] | alg-geom/9503014 | Christian Gantz | Christian Gantz and Brian Steer | Stable Parabolic Bundles over Elliptic Surfaces and over Orbifold
Riemann Surfaces | 12 pages, LaTeX | null | null | null | null | For an elliptic surface $q:Y \to \Sigma$, with prescribed singular fibres,
Stefan Bauer proved directly via algebraic geometry that the stable bundles
over $Y$, whose chern classes are pull backs from $\Sigma$, correspond to the
stable (V-)bundles over $\Sigma$.
We show, via a short proof in differential geometry, a generalisation to
stable parabolic bundles. This uses extensions of Donaldson's deep result,
giving the existence of Hermitian-Yang-Mills (or anti-self-dual) connections on
stable parabolic bundles. In our cases these connections are flat and hence,
correspond to representations of certain fundamental groups, which in turn are
isomorphic, by Ue's work.
To generalize Bauer's equivalence of the corresponding moduli spaces of
stable bundles, we combine his arguments with Kronheimer & Mrowka's
construction of the moduli spaces of stable parabolic bundles. Finally, we
consider the pulling back of smooth parabolic bundles via $q$.
| [
{
"version": "v1",
"created": "Wed, 22 Mar 1995 15:21:07 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Gantz",
"Christian",
""
],
[
"Steer",
"Brian",
""
]
] | alg-geom | \section{Introduction}
If $q:Y \rightarrow \Sigma$ is an elliptic surface (to be made precise)
then the induced map of fundamental groups is an isomorphism if we consider
$\Sigma$ as an orbifold, \cite{ue}, \cite{dol}.
Hence, we obtain a correspondence
of flat bundles (by bundles we always mean complex vector bundles).
Donaldson showed that each stable degree zero bundle over
$Y$ or $\Sigma$ admits a unique
Hermitian-Yang-Mills (or anti-self-dual) connection.
In fact, the obvious conditions on a bundle $E' \rightarrow Y$ to come
from $\Sigma$, namely $\mbox{$\cal C$}_{1}(E') \in q^{*}\mbox{$\Lambda$}^{2}(\Sigma)$ and
$\mbox{$\cal C$}_{2}(E')=0$, imply that this H.Y.M. connection
is flat. So, there is a correspondence of stable degree zero
bundles over $Y$ and $\Sigma$.
The generalisation to all degrees
has been shown by Bauer, \cite{bau}, via a direct proof in
algebraic geometry.
Donaldson's result has been
extended by several authors to parabolic bundles,
theorem \ref{b}. This and the use of
extension results for flat bundles, theorem \ref{y},
gives an identification of stable parabolic
degree zero bundles over $(\Sigma,P)$ and $(Y,P')$,
where $P$ is a finite collection of generic points
in $\Sigma$ and $P'=q^{-1}(P)$,
with representations of the fundamental groups of
$\Sigma-P$ and $Y-P'$, respectively.
These groups are again isomorphic and so,
we have a correspondence of stable parabolic degree
zero bundles over $(Y,P')$ and over $(\Sigma,P)$.
This extends readily to all degrees by tensoring with
a parabolic line bundle.
Another proof for genuine bundles of any degree,
not using parabolic bundles at all,
relies on the correspondence between
stable bundles over $\Sigma$ ($Y$) and
representations of the fundamental group of circle
bundles (i.e. Seifert fibred spaces) over $\Sigma$ ($Y$), \cite[p 63]{fas},
\cite{bao}. These groups are also isomorphic,
see the proof of proposition \ref{v}.
Combining Bauer's arguments with Kronheimer \&
Mrowka's description of the moduli
spaces of stable parabolic bundles, we show
that these are complex manifolds if we fix
determinants and if they are pull backs from $\Sigma$.
Finally, we consider smooth parabolic bundles and produce some
details about the correspondence of stable ones.
After recalling elliptic surfaces and parabolic bundles in
the following two sections
we take one section to state the results.
More details of our work can be found in \cite{gan}.
\hfill
\noindent
{\bf Acknowledgements}
\vspace{0.2cm}
We are grateful to P. Kronheimer,
S. Bauer, M. L\"ubke, T. Peternell,
C. Okonek, S. Agnihotri and R. Plantiko for helpful remarks.
\section{Elliptic surfaces}
Throughout, let $q:Y \rightarrow \Sigma$ be an elliptic surface,
i.e. $Y$ is a compact complex
surface, $\Sigma$ a compact Riemann
surface and $q^{-1}(\sigma)$ an elliptic curve for generic,
i.e. all but finitely many, $\sigma \in \Sigma$, c.f.
\cite{gri}, \cite{bpv}.
We will assume that any non-generic
fibre is either a rational curve of multiplicity one
with one self-intersection (called singular fibre)
or a multiple elliptic curve and furthermore, that
there is at least one singular fibre.
Moishezon shows that all elliptic surfaces are deformation
equivalent to the ones we consider, \cite{moi}.
\begin{theo}[Ue, Dolgachev]
\showlabel{k}
If $q:Y \rightarrow \Sigma$ has singular fibres
and if $U_{0} \subseteq \Sigma$ is an open ball such that
$\pi^{-1}(U_{0})$ contains all singular
fibres but no multiple elliptic curves then
$\pi^{-1}(U_{0})$ is simply connected.
\end{theo}
If $Y_{\sigma}:=q^{-1}(\sigma)$ has
multiplicity $m>1$, let $\tilde{U}$ and $B$ be open discs in $\C$,
$\phi:B \rightarrow U$ a chart with $U \subseteq \Sigma$ and
$\phi(0)=\sigma$ and construct a uniformization of $U$ by
\[ \threehorss{\tilde{U}}{B}{U}{z^{m}}{\phi} \]
where $\langle \eta=e^{2 \pi i/m} \rangle = \Z_{m} \subseteq \C$
acts on $\tilde{U}$ in the standard way.
After this construction is done for
all multiple points $\sigma \in \Sigma$ we
think of $\Sigma$ as an orbifold, cf. \cite{fas}.
Bauer points out that for all orbifold Riemann surfaces there exists
an elliptic surface over it.
The natural gauge-theoretic objects on orbifolds are V-bundles:
a (local, complex) rank $r$ V-bundle $E|_{U}$ is isomorphic to
\[ (\tilde{U} \times \C^{r},\Z_{m})
\rightarrow (\tilde{U},\Z_{m}) \,\,\,\,\,\,\, \mbox{with}\]
\[ \eta (\tilde{u},z_{1},...,z_{r})=( \eta \tilde{u},
\eta^{a_{1}} z_{1},...,
\eta^{ a_{r}} z_{r}) \]
for some isotropies $(a_{1},...,a_{r}) \in \{ 0,...,m-1 \}^{r}$.
\begin{theo}[Furuta \& Steer, Seifert]
\showlabel{i}
Smooth V-bundles over $\Sigma$ are classified by rank,
degree (which is rational) and isotropies.
\end{theo}
For any $y \in Y_{\sigma}$ we can choose coordinates
$(z_{1},z_{2})$ on $U' \ni y$ such that
$\phi^{-1} \circ q (z_{1},z_{2})=z^{m}_{2}$.
Hence we can lift $\phi^{-1} \circ q$ locally to a regular
map \[ \sqrt[m]{\phi^{-1} \circ q}=z_{2}:U' \rightarrow \tilde{U} \]
uniquely up to the action of $\Z_{m}$.
In particular, $q$ is a map of orbifolds.
A divisor on $\Sigma$ can be represented by a finite sum
$D=\sum_{i \in I} \sigma_{i}n_{i}/m_{i}$ where
$n_{i} \in \Z$ and $m_{i}$ the multiplicity
of $\sigma_{i} \in \Sigma$.
The vertical divisors on $Y$ are precisely the
pull backs of divisors on $\Sigma$.
Hence, \cite{nas}, the line bundles $\mbox{$\cal O$} (D')$
over $Y$ with vertical divisor
correspond to the line V-bundles over $\Sigma$.
As $q$ is regular away from finitely many points theorem
\ref{k} implies that
$q_{*}:\pi_{1}(Y) \rightarrow \pi_{1}^{V}(\Sigma)$ is an isomorphism.
Here, the orbifold fundamental group $\pi_{1}^{V}(\Sigma)$ is
an extension of $\pi_{1}(\Sigma)$ by a
torsion group with
unipotent elements corresponding to the multiple points.
The isomorphism implies that
the first betti number of $Y$ is even and therefore
$Y$ is Kahler, by \cite{miy}.
The correspondence between
representations of the fundamental group and flat bundles on manifolds
extends to orbifolds via the construction
with simply connected coverings.
Hence, the flat bundles over $Y$
correspond to the flat V-bundles over $\Sigma$.
\section{Parabolic bundles}
Let $P:=\{ p_{1},...,p_{n} \}$ be a collection of
generic points in $\Sigma$ and put $P':=
\{ P_{j}':=q^{-1}(p_{j}) \}_{j=1...n} \subseteq Y$.
A parabolic V-bundle $E$ over the
pair $(\Sigma,P)$ is a V-bundle $E \rightarrow \Sigma$ with
proper filtrations
\[ E|_{P_{j}}=E_{j,1} \supset E_{j,2} \supset ...
\supset E_{j,l_{j}} \neq 0 \]
and weights
\[ 0 \leq \alpha_{j,1} < \alpha_{j,2} < ... < \alpha_{j,l_{j}} <1 \]
for all $j=1,...,n$.
We call $\mu_{j,k}:=\mbox{rank}\, (E_{j,k}/E_{j,k+1})$ the
multiplicity of $\alpha_{j,k}$. Let $\alpha_{j}$ be
the diagonal matrix of rank equal to $\mbox{rank}\, E$
and with entries $\alpha_{j,k}$ ($k=1,...,l_{j}$)
with multiplicities.
Then put $\alpha(E):=\{ \alpha_{1},...,\alpha_{n} \}$.
We will sometimes write $|E|$ for the underlying genuine V-bundle.
Correspondingly, one defines parabolic
bundles $E'\rightarrow (Y,P')$ where the $\{E'_{j,k} \}$
are bundles over $P_{j}'$ with
weights $\{ \alpha'_{j,k}\}$ of multiplicities
$\{ \mu'_{j,k} \}$, encoded in $\alpha'(E')$.
If the bundle and the filtrations are holomorphic we
speak of a holomorphic parabolic bundle.
Thinking of vectors in $E|_{P}$ as
having (the obvious) weights ($+\infty$ being the weight of
zero vectors), a morphism of parabolic bundles
is a bundle map not decreasing the weight of any
vector. In a direct sum of parabolic bundles the
weight of a vector is the minimum of the weights of it's
projections.
Let us fix a Kahler metric on $Y$
with (1,1)-form $\omega'$. When we use ordinary bundle invariants
and operators (like $\mbox{$\cal C$}_{i}$, $\det$ or $\deg$) on parabolic bundles
we mean to apply them to the underlying bundles.
Then we have
\[ \mbox{par}\, \mbox{$\cal C$}_{1} (E):= \mbox{$\cal C$}_{1}(E) + \sum_{j=1}^{n} \mbox{Tr\,} (\alpha_{j})
\mbox{PD}\, (p_{j} ) \in \mbox{$\Lambda$}^{2}(\Sigma,\R) \]
and similarly, $\mbox{par}\, \mbox{$\cal C$}_{1}(E') \in \mbox{$\Lambda$}^{2}(Y,\R)$.
Let $\mbox{par}\, \deg E:= \langle \mbox{par}\, \mbox{$\cal C$}_{1}(E), \Sigma \rangle \in \R$ and
$\mbox{par}\, \deg E':=\langle \mbox{par}\, \mbox{$\cal C$}_{1}(E') \cup \omega',Y \rangle \in \R$.
We also have (since $P \cdot P=0$)
\[ \mbox{par}\, \mbox{$\cal C$}_{2}(E'):=\mbox{$\cal C$}_{2}(E') + 2 \sum_{j=1}^{n}
\sum_{k=1}^{l_{j}} \alpha'_{j,k} \mbox{PD}\,(d_{j,k}')
+\frac{1}{2} \mbox{par}\, \mbox{$\cal C$}_{1}^{2}(E') \in \mbox{$\Lambda$}^{4}(Y,\R) \]
where $d'_{j,k}:= \deg (E'_{j,k}/E'_{j,k+1})$.
\begin{defi}
A holomorphic parabolic V-bundle $\mbox{$\cal E$} \rightarrow (\Sigma,P)$
is called {\bf stable}
if for all non-zero
parabolic maps $\mbox{$\cal F$} \rightarrow \mbox{$\cal E$}$,
injective over some point in $\Sigma$ and with
$\mbox{rank}\, \mbox{$\cal F$} < \mbox{rank}\, \mbox{$\cal E$}$ we have
\[\mbox{deg}\, \mbox{$\cal F$} / \mbox{rank}\, \mbox{$\cal F$} < \mbox{deg}\, \mbox{$\cal E$} / \mbox{rank}\, \mbox{$\cal E$}. \]
Similarly for $\mbox{$\cal E$}' \rightarrow (Y,P')$.
\end{defi}
\section{Statement of results}
Being a map of orbifolds,
$q:Y \rightarrow \Sigma$ induces pull backs of
holomorphic parabolic V-bundles over
$\Sigma$ to holomorphic parabolic bundles over $Y$.
Our main result is:
\begin{theo}
\showlabel{z}
Pulling back induces a correspondence between stable
parabo\-lic V-bundles $\mbox{$\cal E$} \rightarrow (\Sigma,P)$
and those stable parabolic bundles $\mbox{$\cal E$}' \rightarrow (Y,P')$
with $\mbox{par}\, \mbox{$\cal C$}_{2}(\mbox{$\cal E$}')=0$ and
$\mbox{par}\, \mbox{$\cal C$}_{1}(\mbox{$\cal E$}') \in q^{*} \mbox{$\Lambda$}^{2}(\Sigma,\R)$.
\end{theo}
In particular:
\begin{theo}
\showlabel{a}
Pulling back induces a bijection between
stable V-bundles $\mbox{$\cal E$}$ over $\Sigma$ and those stable
bundles $\mbox{$\cal E$}'$ over $Y$ with $\mbox{$\cal C$}_{2}(\mbox{$\cal E$}')=0$
and $\det \mbox{$\cal E$}' = \mbox{$\cal O$} (D')$ for a vertical divisor $D'$.
\end{theo}
Theorem \ref{a} has been shown
(under some assumptions on the Kahler metric of $Y$,
\cite[p 511]{bau}) by Bauer first,
using algebraic geometry. Our goal is a differential-geometric proof of
theorem \ref{z} using the correspondence of stable parabolic
bundles and parabolic H.Y.M. connections.
Let $\mbox{$\cal E$} \rightarrow (\Sigma,P)$ be a
stable parabolic bundle, $\mbox{$\cal E$}':=q^{*} \mbox{$\cal E$}$,
$\mbox{$\cal T$}:=\,\, \mbox{Par End}_{0} \mbox{$\cal E$}$
and $\mbox{$\cal T$}':=q^{*} \mbox{$\cal T$} = \,\, \mbox{Par End}_{0} \mbox{$\cal E$}'$.
The important thing to note is that $\mbox{$\cal T$}$
and $\mbox{$\cal T$}'$ are holomorphic (V-)bundles.
We have the deformation complexes (of smooth sections)
\begin{center}
\setlength{\unitlength}{1.5mm}
\begin{picture}(0,30)(0,-15)
\putcc{-26}{10}{\Omega_{\Sigma}^{0} (\mbox{$\cal T$})}
\putcc{0}{10}{\Omega_{\Sigma}^{0,1} (\mbox{$\cal T$})}
\putcc{26}{10}{0}
\putcc{-26}{-10}{\Omega_{Y}^{0} (\mbox{$\cal T$}')}
\putcc{0}{-10}{\Omega_{Y}^{0,1} (\mbox{$\cal T$}')}
\putcc{26}{-10}{\Omega_{Y}^{0,2} (\mbox{$\cal T$}')}
\put(-26,6){\vector(0,-1){12}}
\put(0,6){\vector(0,-1){12}}
\put(26,6){\vector(0,-1){12}}
\put(-18,10){\vector(1,0){9}}
\putbc{-15}{12}{\mbox{$\bar{\partial}$}_{\cal{T}}}
\put(-18,-10){\vector(1,0){8}}
\putbc{-13}{-8}{\mbox{$\bar{\partial}$}_{\cal{T}'}}
\put(9,10){\vector(1,0){10}}
\putbc{13}{12}{\mbox{$\bar{\partial}$}_{\cal{T}}}
\put(8,-10){\vector(1,0){9}}
\putbc{13}{-8}{\mbox{$\bar{\partial}$}_{\cal{T}'}}
\putcl{2}{0}{q^{*}}
\end{picture}
\end{center}
Let $E:=_{C^{\infty}} \mbox{$\cal E$}$, $\mbox{$\cal O$}(D)=\det \mbox{$\cal E$}$
and $\mbox{$\cal M$}(E,D)$ be the space of stable
structures on $E$ with determinant $\mbox{$\cal O$}(D)$.
Similarly, define $\mbox{$\cal M$}(E', D')$ for $E':=q^{*}E$
and $D':=q^{*}D$.
Using the extension of standard deformation theory,
\cite{dak}, to parabolic bundles makes
$\mbox{$\cal M$}(E',D')$ into a Hausdorff complex space,
see \cite[p 83]{km2} and \cite{mun}, with a description
near $\mbox{$\cal E$}'$ given by the zero set of a holomorphic map
\[ \mbox{$\Lambda$}^{0,1}_{\bar{\partial}}(Y,\mbox{$\cal T$}')
\rightarrow \mbox{$\Lambda$}^{0,2}_{\bar{\partial}}(Y,\mbox{$\cal T$}'). \]
By the vanishing of the second cohomology
over $\Sigma$, $\mbox{$\cal M$}(E,D)$ is even a complex manifold
with a chart near $\mbox{$\cal E$}$ given by $\mbox{$\Lambda$}^{0,1}_{\bar{\partial}}(\Sigma,\mbox{$\cal T$})$.
\begin{prop}[Bauer]
\showlabel{t}
Stability of $\mbox{$\cal E$}$ implies that
\[ q^{*}:\mbox{$\Lambda$}^{0,1}_{\bar{\partial}}(\Sigma,\mbox{$\cal T$})
\rightarrow \mbox{$\Lambda$}^{0,1}_{\bar{\partial}}(Y,\mbox{$\cal T$}') \]
is an isomorphism.
\end{prop}
This proposition and theorem \ref{z} imply:
\begin{coro}
If $\mbox{$\cal M$}_{\cal{E}}(E,D)$ is the connected component
of $\mbox{$\cal E$}$, correspondingly for $\mbox{$\cal E$}'$,
then \[ q^{*}:\mbox{$\cal M$}_{\cal{E}}(E,D) \rightarrow \mbox{$\cal M$}_{\cal{E}'}(E',D') \]
is an isomorphism of complex manifolds.
\end{coro}
\begin{prop}
\showlabel{u}
If $E' \rightarrow (Y,P')$ is a rank $r$ parabolic bundle satisfying
$\mbox{par}\, \mbox{$\cal C$}_{2} (E')=0$, $\mbox{par}\, \mbox{$\cal C$}_{1} (E')
\in q^{*} \mbox{$\Lambda$}^{2}(\Sigma,\R)$ and if the space
$\mbox{$\cal M$}(E')$ of stable structures on $E'$ is non-empty then
\begin{description}
\item[(i)] there exists a unique
line V-bundle $L \rightarrow \Sigma$ with $q^{*} L=\det E'$.
Also, $E'$ is uniquely determined by $L$ and it's weights $\alpha'$;
\item[(ii)] we have
\[ \mbox{$\cal M$}(E') = \bigsqcup_{a} \mbox{$\cal M$} (L,r,a,\alpha') \]
where the union is over
isotropies $a \in \Z_{m_{1}} \times ... \times \Z_{m_{k}}$
($k=\sharp \{ $ marked points on $\Sigma \}$) compatible with $L$ and
$\mbox{$\cal M$}(L,r,a,\alpha')$ is the space of stable
structures on the unique rank $r$ parabolic V-bundle over
$(\Sigma,P)$ with determinant $L$, isotropies $a$ and weights $\alpha'$.
\end{description}
\end{prop}
\section{Stable parabolic bundles}
\begin{defi}
A {\bf (parabolic) hermitian metric} $h'$ on
a parabolic bundle $E'\rightarrow (Y,P')$
(similarly for $E\rightarrow (\Sigma,P)$) is
a hermitian metric on $E'|_{Y-P'}$ such that
\[ \forall y \in P'_{j} \,\,\,\,\,\,\,\,
E'_{j,k}(y)=\{ s(y) \,\, | \,\, s
\in \Gamma_{loc}(E') \mbox{ s.t. }\, h'(s(-))=O(\mbox{d} (P'_{j},-)^{\alpha'_{j,k}})
\}. \]
\end{defi}
For a holomorphic parabolic bundle $\mbox{$\cal E$}'$, $h'$ induces
a Chern connection on $\mbox{$\cal E$}'|_{Y=P'}$.
So, one can talk of H.Y.M.-connections.
Around $P'_{j}$, a parabolic connection has holonomy
conjugated to $\alpha'_{j}$, \cite{kro1},
and can therefore not be extended over $P'_{j}$, in general.
A parabolic connection is called reducible if the
parabolic bundle decomposes together with the
connection.
The bridge between algebraic and differential geometry is
\begin{theo}[\cite{mas}, \cite{saw}, \cite{mun}, \cite{b93}, \cite{nas}]
\showlabel{b}
Any degree zero
holomorphic parabolic bundle over a
complex Kahler surface or orbifold of dimension one
is stable if and only if it admits an
irreducible H.Y.M. metric, unique up to isomorphism.
Furthermore, an H.Y.M. connection has finite action.
\end{theo}
The primary result, for genuine bundles, is due to Donaldson;
Munari's proof relies on Simpson's
work.
A H.Y.M. connection over $\Sigma$ is obviously flat.
\begin{lemm}
\showlabel{c}
If $\mbox{$\cal E$}' \rightarrow (Y,P')$ has
$\mbox{par}\, \mbox{$\cal C$}_{2}(\mbox{$\cal E$}')=0$ and $\mbox{par}\, \mbox{$\cal C$}_{1}(\mbox{$\cal E$}') \in
q^{*} \mbox{$\Lambda$}^{2}(\Sigma,\R)$ then a H.Y.M. connection is neccessarily flat.
\end{lemm}
\noindent{\em Proof:}
By \cite[p 100]{mun} or \cite{gan}
we obtain the second (in particular the existence of) and, from \cite{dak},
the third equality in
\[0=8 \pi^{2} (\mbox{par}\, \mbox{$\cal C$}_{2}(\mbox{$\cal E$}') - \frac{1}{2}
\mbox{par}\, \mbox{$\cal C$}_{1}^{2}(\mbox{$\cal E$}'))=
\int_{Y-P'} \mbox{Tr\,} (F^{2}) = ||F^{-}||^{2} - ||F^{+}||^{2}. \]
$\Box$ \hfill\vspace{0.5cm}
\begin{theo}[Munari, Biquard, Simpson]
\showlabel{y}
A holomorphic bundle over
$Y-P'$ ($\Sigma-P$) with a flat hermitian metric extends uniquely
(up to isomorphism) to a holomorphic
parabolic bundle over $(Y,P)$ ($(\Sigma,P)$) such that
the hermitian metric becomes a parabolic metric.
\end{theo}
Now we prove theorem \ref{z}:
\noindent{\em Proof:}
At first, we treat the special case of degree zero bundles.
If $\mbox{$\cal E$} \rightarrow (\Sigma,P)$ is stable and of
degree zero, let $A$ be the unique irreducible flat parabolic
connection on $\mbox{$\cal E$}$. By the uniqueness
in theorem \ref{y}, $A|_{\Sigma-P}$ is still irreducible.
The regularity of $q|_{Y-P'}$ away from finitely many
points and theorem \ref{k}
imply that $(q|_{Y-P'})_{*}:\pi_{1}(Y-P')
\rightarrow \pi_{1}^{V}(\Sigma-P)$
is an isomorphism.
Hence, if $\mbox{$\cal E$}':=q^{*} \mbox{$\cal E$}$ and $A':=q^{*}A$
then $A'$ is an irreducible flat parabolic connection
and hence $\mbox{$\cal E$}'$ is stable.
Conversely, if $\mbox{$\cal E$}' \rightarrow (Y,P')$
is stable and of degree zero, let $A'$ be the unique
parabolic H.Y.M. connection, which is
flat by lemma \ref{c} and $A'|_{Y-P'}$ is irreducible.
We push forward
$(\mbox{$\cal E$}',A')|_{Y-P'}$ to $\Sigma-P$.
By theorem \ref{y}, this push forward extends uniquely
to a holomorphic parabolic bundle with irreducible parabolic connection
$(\mbox{$\cal E$},A) \rightarrow (\Sigma,P)$.
As $q^{*}(\mbox{$\cal E$},A)$ and $(\mbox{$\cal E$}',A')$
are isomorphic over $Y-P'$ the uniqueness of theorem \ref{y}
implies that they are isomorphic over $(Y,P')$.
We are finished with the degree zero case.
To show the general case, fix a generic point $p \in \Sigma-P$ and let
$\Delta:=\int_{q^{-1}(p)} \omega'$.
For each $d \in \R$ let $[d]$ be it's
integer part, let $[p]$ be the holomorphic line bundle with
divisor $p$ and define the holomorphic parabolic line bundle
$\mbox{$\cal L$}_{d} \rightarrow (\Sigma,p)$
to be $\mbox{$\cal L$}_{d}:=[p]^{[d]}$ with weight $d-[d]$ over $p$.
Let $\mbox{$\cal L$}'_{d}:=q^{*} \mbox{$\cal L$}_{d}$ be the
parabolic pull back. We have $\mbox{par}\, \deg \mbox{$\cal L$}_{d}=d$ and
$\mbox{par}\, \deg \mbox{$\cal L$}'_{d}=\Delta d$.
Tensoring any rank $r$ stable
parabolic bundle $\mbox{$\cal E$} \rightarrow (\Sigma,P)$ ($\mbox{$\cal E$}' \rightarrow
(Y,P')$) of degree
$-dr$ ($-\Delta d r$) with $\mbox{$\cal L$}_{d}$ ($\mbox{$\cal L$}'_{d}$) gives a stable
parabolic bundle of
degree zero over
$(\Sigma,P \cup \{ p\} )$ ($(Y,P' \cup \{ q^{-1}(p) \} )$).
Furthermore, we have $\mbox{par}\, \mbox{$\cal C$}_{2}(\mbox{$\cal E$}' \otimes \mbox{$\cal L$}'_{d})=0$.
So we are reduced to the degree zero case.
$\Box$ \hfill\vspace{0.5cm}
We use Bauer's arguments, \cite[p 514]{bau}, to prove proposition \ref{t}:
\noindent{\em Proof:}
The sheaf $\mbox{$\cal T$}$ can be considered as a sheaf
on $\Sigma$ or on $|\Sigma|$, the underlying Riemann
surface of the orbifold $\Sigma$, see the correspondence
between V-bundles and parabolic
bundles in section 5 of \cite{fas}.
As sheaves, $q_{*} \mbox{$\cal T$}' = \mbox{$\cal T$}$.
The Leray spectral sequence induces an exact sequence, \cite[p 10]{bpv},
\[ 0 \rightarrow \threehorsb{\mbox{$\Lambda$}^{1}(\Sigma,\mbox{$\cal T$})}%
{\mbox{$\Lambda$}^{1}(Y,\mbox{$\cal T$}')}{\mbox{$\Lambda$}^{0}(\Sigma,q_{*1}\mbox{$\cal T$}')}{q^{*}}
\rightarrow ... \]
where $q_{*1} \mbox{$\cal T$}'$ is the first direct image sheaf of $\mbox{$\cal T$}'$.
It suffices to see that $\mbox{$\Lambda$}^{0}(\Sigma,q_{*1}\mbox{$\cal T$}')=0$.
Relative duality, \cite[p 99]{bpv}, gives
\[ q_{*1}\mbox{$\cal T$}' = q_{*}(\mbox{$\cal T$}'^{*}
\otimes \mbox{$\cal K$}_{Y} \otimes q^{*}\mbox{$\cal K$}^{*}_{|\Sigma |})^{*}=
\mbox{$\cal T$} \otimes q_{*}(\mbox{$\cal K$}_{Y}
\otimes q^{*} \mbox{$\cal K$}^{*}_{|\Sigma |})^{*} \]
where $\mbox{$\cal K$}_{|\Sigma |}$ is the canonical bundle.
By \cite[p 98, 161-162]{bpv},
\[ \mbox{$\cal K$}_{Y} \otimes q^{*} \mbox{$\cal K$}^{*}_{|\Sigma |} =
q^{*} q_{*1} \mbox{$\cal O$}_{Y}^{*} \otimes
\mbox{$\cal O$}_{Y}(\sum (m_{i}-1)Y_{\sigma_{i}}) \]
where the sum is over the singular points
$\sigma_{i}$ of $\Sigma$, $q_{*1}\mbox{$\cal O$}_{Y}$ is locally free of rank one
since
all the other sheaves in this identity are, and
\[ \deg (q_{*1} \mbox{$\cal O$}_{Y})=- \chi (\mbox{$\cal O$}_{Y}). \]
In particular, $\mbox{$\cal K$}_{Y} \cdot \mbox{$\cal K$}_{Y}=0$.
Hence, $\chi (Y)=12 \chi(\mbox{$\cal O$}_{Y})$, \cite[p 472]{gri},
which is equal to the positive number of singular fibres, cf. \cite{ue}.
Now, $\mbox{$\cal O$}_{Y}((m_{i}-1)Y_{\sigma_{i}})=
q^{*}\mbox{$\cal O$}_{\Sigma}(\frac{m_{i}-1}{m_{i}}\sigma_{i})$.
(This is in fact a trivial sheaf over $\Sigma$.)
We obtain
\[ q_{*1}\mbox{$\cal T$}'=\mbox{$\cal T$} \otimes q_{*1} \mbox{$\cal O$}_{Y}
\otimes \mbox{$\cal O$}_{\Sigma}(\sum \frac{1-m_{i}}{m_{i}} \sigma_{i})\]
and any non zero section of this induces a non zero map
$\mbox{$\cal E$} \rightarrow \mbox{$\cal E$} \otimes \mbox{$\cal L$}$
for a negative line V-bundle $\mbox{$\cal L$}$.
This is ruled out by stability of $\mbox{$\cal E$}$.
$\Box$ \hfill\vspace{0.5cm}
\section{Smooth parabolic bundles}
\begin{prop}
\showlabel{v}
Two smooth line V-bundles
over $\Sigma$ are isomorphic if their pull backs to $Y$ are isomorphic.
\end{prop}
There is an equivalence between smooth line V-bundles
over $\Sigma$ and $\mbox{$\Lambda$}^{2}_{V}(\Sigma,\Z)$, \cite{fas}.
As we don't have a sufficient theory of V-cohomology however,
our proof is not by showing
injectivity of $\mbox{$\Lambda$}_{V}^{2}(\Sigma,\Z) \rightarrow \mbox{$\Lambda$}^{2}(Y,\Z)$.
\noindent{\em Proof:}
It suffices to show that a smooth line V-bundle
$L \rightarrow \Sigma$ is trivial if $L':=q^{*}L \rightarrow Y$ is trivial.
Let us write $q':SL' \rightarrow SL$ for the induced map of circle bundles.
We use the isomorphism
$q_{*}:\pi_{1}(Y) \rightarrow \pi_{1}^{V}(\Sigma)$ and
the commutative diagram
\setlength{\unitlength}{1.0mm}
\begin{center}
\begin{picture}(0,40)(0,-20)
\putcc{-52}{13}{0}
\putcc{-26}{13}{K'}
\putcc{0}{13}{\pi_{1}(SL')}
\putcc{26}{13}{\pi_{1}(Y)}
\putcc{52}{13}{0}
\putcc{-52}{-13}{0}
\putcc{-26}{-13}{K}
\putcc{0}{-13}{\pi_{1}^{V}(SL)}
\putcc{26}{-13}{\pi_{1}^{V}(\Sigma)}
\putcc{52}{-13}{0}
\put(-45,13){\vector(1,0){12}}
\put(-19,13){\vector(1,0){10}}
\put(9,13){\vector(1,0){10}}
\put(33,13){\vector(1,0){12}}
\put(-45,-13){\vector(1,0){12}}
\put(-19,-13){\vector(1,0){10}}
\put(9,-13){\vector(1,0){10}}
\put(33,-13){\vector(1,0){12}}
\put(-52,8){\vector(0,-1){16}}
\put(-26,8){\vector(0,-1){16}}
\put(0,8){\vector(0,-1){16}}
\put(52,8){\vector(0,-1){16}}
\put(26,8){\vector(0,-1){16}}
\putbc{13}{15}{\pi'_{*}}
\putbc{13}{-11}{\pi_{*}}
\putcl{2}{0}{q'_{*}}
\putcl{28}{0}{q_{*}}
\end{picture}
\end{center}
where $K=\langle k \rangle$, $K'=\langle k' \rangle$
for regular fibres $k$ of $SL$ and $k'$ of $SL'$.
The rows are exact because bundles are always regular maps.
By the five-lemma, $q'_{*}$ is an isomorphism
if $(k' \mapsto k):K' \rightarrow K$ is an
isomorphism. Certainly, it is surjective.
Assume there is a V-homotopy $H: [0,1] \times [0,1] \rightarrow SL$
with boundary $k^{n}$.
Since $q$ is regular away from finitely many points, there exists
$H':[0,1] \times [0,1] \rightarrow Y$ lifting $\pi \circ H$.
Hence there exists
$\tilde{H}:[0,1] \times [0,1] \rightarrow SL'$ lifting $H$ and $H'$.
If $k \in S(L_{x})$ then
$\mbox{$\partial$} H' = \pi' \circ \mbox{$\partial$} \tilde{H} \subseteq Y_{x}$
and $\mbox{$\partial$} H =q' \circ \mbox{$\partial$} \tilde{H} =k^{n}$.
W.l.o.g. $x \in U_{0}$, where $U_{0}$ is as in theorem \ref{k},
and we are working on $SL'|_{q^{-1}(U_{0})}=S^{1} \times q^{-1}(U_{0})$.
So we can lift some homotopy (inside $q^{-1}(U_{0})$)
with boundary $\mbox{$\partial$} H'$ to one relating
$\mbox{$\partial$} \tilde{H}$ to $(k')^{n}$.
Hence, $q'_{*}$ is an isomorphism.
Seifert proved that $\pi_{1}^{V}(SL)=$
\[ \langle a_{j}, b_{j},g_{i},k \,\, : \,\, [a_{j},k]=
[b_{j},k]=[g_{i},k]=1=%
g_{i}^{m_{i}}k^{\beta_{i}}=k^{-b}
\prod_{j=1}^{g} [a_{j},b_{j}] \prod_{i=1}^{n} g_{i} \rangle \]
where $g$ is the genus of $\Sigma$ and
$m_{i}$ is the multiplicity of $\sigma_{i}$.
Furthermore, the isotropy
$\beta_{i} \,\,\, \mbox{mod}\,\,\, m_{i}$ of $L$ at $\sigma_{i}$ and
$\deg L=b+\sum_{1}^{n} \beta_{i}/m_{i}$ are
independent of the choices of lifts $g_{i}, a_{j}$ and $b_{j}$
of the generators of $\pi_{1}^{V} (\Sigma)$.
By the isomorphism of the above extensions
and if $SL'$ is trivial, we can choose lifts such that
all $\beta_{i}$'s are zero as well as $b$.
By theorem \ref{i}, this implies that $L$ is trivial.
$\Box$ \hfill\vspace{0.5cm}
Now we prove proposition \ref{u}:
\noindent{\em Proof:}
Theorem \ref{z} implies the
existence of some $L \rightarrow \Sigma$ with $\det E'=q^{*}L$ and that the
parabolic filtration of $E'$ along $P'$ is
by trivial bundles; in particluar $\mbox{$\cal C$}_{2} (E')=0$.
Proposition \ref{v} gives the uniqueness of $L$.
Now, $|E'|$ is uniquely determined by $\det E'$. Two different
filtrations of $E'|_{P'_{j}}$ by trivial
subbundles are related by a map $P_{j}' \rightarrow \mbox{Sl}(r,\C)$
which can be extended to an isomorphism
of $E'$ being the identity outside a tubular neighbourhood
of $P'_{j}$ since $\mbox{Sl}(r,\C)$
is simply connected. This shows (i).
After the last argument and by theorem \ref{i},
part (ii) follows from proposition \ref{v} and theorem \ref{z}.
$\Box$ \hfill\vspace{0.5cm}
|
1996-08-13T11:05:17 | 9503 | alg-geom/9503023 | en | https://arxiv.org/abs/alg-geom/9503023 | [
"alg-geom",
"math.AG"
] | alg-geom/9503023 | Richard Earl | Richard Earl | The Mumford relations and the moduli of rank three stable bundles | 35 Pages, no figures LaTeX v 2.09 | null | null | null | null | We find a complete set of relations for the rational cohomology ring of the
moduli space of rank three stable bundles over a Riemann surface of genus g and
also show that the Pontryagin ring vanishes in degree 12g-8 and greater. The
results are obtained by introducing some 'dual' Mumford relations and
generalising Kirwan's proofs of the Mumford and Newstead conjectures in the
rank two case. (In this revised version of the paper the vanishing degree of
the Pontryagin ring of the moduli space has been improved from `in and above
degree 12g-4' to `in and above degree 12g-8'. This degree is now known to be
sharp.)
| [
{
"version": "v1",
"created": "Wed, 29 Mar 1995 18:43:54 GMT"
},
{
"version": "v2",
"created": "Thu, 14 Dec 1995 13:13:43 GMT"
},
{
"version": "v3",
"created": "Tue, 13 Aug 1996 09:00:19 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Earl",
"Richard",
""
]
] | alg-geom | \section{Introduction.}
Let ${\cal M}(n,d)$ denote the moduli space of semistable holomorphic
vector bundles of coprime rank $n$ and degree $d$ over a Riemann
surface $M$ of genus $g \geq 2.$ Throughout this article we will write
\[
\bar{g} = g-1.
\]
Recall that a holomorphic vector
bundle $E$ over $M$ is said to be semistable (resp. stable) if every proper
subbundle $F$ of $E$ satisfies
\[
\mu(F) \leq \mu(E) \indent (\mbox{resp. } \mu(F) < \mu(E))
\]
where $\mu(F) = \mbox{degree}(F)/\mbox{rank}(F)$ is the slope of $F$.
Non-semistable bundles are said to be unstable. When $n$ and $d$ are
coprime the stable and semistable bundles coincide.\\
\indent Let ${\cal E}$ be a fixed $C^{\infty}$ complex vector bundle
of rank $n$ and degree $d$ over $M$. Let ${\cal C}$ be the space of all holomorphic
structures on ${\cal E}$ and let ${\cal G}_{c}$ denote the group of all
$C^{\infty}$ complex automorphisms of ${\cal E}$. Atiyah and Bott
\cite{AB} identify the moduli space
${\cal M}(n,d)$ with the quotient ${\cal C}^{ss}/{\cal G}_{c}$ where ${\cal C}^{ss}$ is the open
subset of ${\cal C}$ consisting of all semistable holomorphic structures on
${\cal E}$. In this construction both ${\cal C}$
and ${\cal G}_{c}$ are infinite dimensional; there
exist other constructions \cite{K3} of the moduli space ${\cal M}(n,d)$ as
genuine geometric invariant theoretic quotients which are in a sense
finite dimensional approximations of Atiyah and Bott's construction.\\
\indent There is a known set of generators \cite{N2,AB} for the
rational cohomology ring of ${\cal M}(n,d)$ as follows. Let $V$ denote a
universal bundle over ${\cal M}(n,d) \times M$. Atiyah and Bott then define
elements
\begin{equation}
a_{r} \in H^{2r}({\cal M}(n,d);{\bf Q}), \quad b_{r}^{s} \in
H^{2r-1}({\cal M}(n,d);{\bf Q}), \quad f_{r} \in H^{2r-2}({\cal M}(n,d);{\bf Q}) \label{000}
\end{equation}
where $1 \leq r \leq n,1 \leq s \leq 2g$ by writing
\begin{equation}
c_{r}(V) = a_{r} \otimes 1 + \sum_{s=1}^{2g} b_{r}^{s} \otimes
\alpha_{s} + f_{r} \otimes \omega \indent 1 \leq r \leq n \label{0}
\end{equation}
where $\omega$ is the standard generator of $H^{2}(M;{\bf Q})$ and
$\alpha_{1},...,\alpha_{2g}$ form a fixed canonical cohomology basis for
$H^{1}(M;{\bf Q})$. The ring
$H^{*}({\cal M}(n,d);{\bf Q})$ is freely generated as a graded algebra over ${\bf Q}$ by
the elements (\ref{000}). Notice from the definition that $f_{1}=d$. We further introduce the notation
\[
\xi_{i,j} = \sum_{s=1}^{g} b_{i}^{s} b_{j}^{s+g}.
\]
\indent The universal bundle $V$ is not unique, although its
projective class is. We may tensor $V$ by the pullback to ${\cal M}(n,d) \times
M$ of any holomorphic line bundle $K$ over ${\cal M}(n,d)$ to give another
bundle with the same universal property. This process changes the
generators of $H^{*}({\cal M}(n,d);{\bf Q})$. In particular it changes $a_{1}$ by $n
c_{1}(K)$ and $c_{1}(\pi_{!}V)$ by $(d-n\bar{g})c_{1}(K)$ where $\pi:{\cal M}(n,d)
\times M \rightarrow {\cal M}(n,d)$ is the first projection and $\pi_{!}$ is
the direct image map from K-theory \cite[p.436]{H}. Since $n$ and $d$
are coprime there exist integers $u$ and $v$ such that
\[
u n + v (d-n\bar{g}) =1.
\]
Thus if we take $K$ to be
\[
\left. \mbox{det}(V \right|_{{\cal M}(n,d)}) ^{u} \otimes (\mbox{det}
\pi_{!}V)^{v}
\]
then $V \otimes \pi^{*}(K^{-1})$ is a new universal bundle such that
\begin{equation}
u a_{1} + v c_{1}(\pi_{!}V) = 0. \label{NORM}
\end{equation}
Following Atiyah and Bott \cite[p.582]{AB} we replace $V$ by this
normalised universal bundle.\\
\indent The normalised bundle $V$ is universal in
the sense that its
restriction to $\{[E]\} \times M$ is isomorphic to $E$ for each
semistable holomorphic bundle $E$ over $M$ of rank $n$ and degree $d$
and where $[E]$ is the class of $E$ in ${\cal M}(n,d).$ Then the stalk of the $i$th higher
direct image sheaf $R^{i}\pi_{*}V$ (see \cite[$\S 3.8$]{H}) at $[E]$
is
\[
H^{i}(\pi^{-1}([E]),V_{|\pi^{-1}([E])}) = H^{i}(M,V_{|[E] \times M}) \cong
H^{i}(M,E).
\]
\indent Tensoring $E$ with a holomorphic line bundle over $M$ of
degree $D$ gives an isomorphism between ${\cal M}(n,d)$ and ${\cal M}(n,d+nD)$.
Since $n$ and $d$ are coprime we may assume without any loss of
generality that $2\bar{g} n<d<(2\bar{g}+1)n$ and so we will write
\[
d=2n\bar{g}+\delta \indent (0 < \delta < n)
\]
from now on. From \cite[lemma 5.2]{N} we know that $H^{1}(M,E)=0$
for any semistable holomorphic bundle $E$ of slope greater than
$2\bar{g}$. Thus $\pi_{!}V$ is in fact a vector bundle over ${\cal M}(n,d)$ with fibre
$H^{0}(M,E)$ over $[E] \in {\cal M}(n,d)$ and, by the Riemann-Roch theorem, of
rank $d-n\bar{g}=n\bar{g}+\delta$.\\
\indent In particular if we express the Chern classes
$c_{r}(\pi_{!}V)$ in terms of the generators $a_{r},b_{r}^{s}$ and
$f_{r}$ of $H^{*}({\cal M}(n,d);{\bf Q})$ then knowing the images of the $r$th Chern
classes in
$H^{*}({\cal M}(n,d);{\bf Q})$ vanish for $r>n\bar{g}+\delta$ gives us relations in terms of
the images of the generators in $H^{*}({\cal M}(n,d);{\bf Q}).$ Now from
\cite[prop. 9.7]{AB} we know that
\begin{equation}
H^{*}({\cal M}(n,d);{\bf Q}) \cong H^{*}({\cal M}_{0}(n,d);{\bf Q}) \otimes
H^{*}(\mbox{Jac}(M);{\bf Q}) \label{21}
\end{equation}
where $\mbox{Jac}(M)$ is the Jacobian of the Riemann surface $M$ and
${\cal M}_{0}(n,d)$ is the moduli space of rank $n$ bundles with
degree $d$ and fixed determinant line bundle.
$H^{*}(\mbox{Jac}(M);{\bf Q})$ is an exterior algebra on $2g$ generators
and we can choose the isomorphism (\ref{21}) so that these generators
correspond to $b_{1}^{1},...,b_{1}^{2g}$ and the elements
$a_{2},...,a_{n},b_{2}^{1},...,b_{n}^{2g},f_{2},...,f_{n}$ correspond
to the generators of $H^{*}({\cal M}_{0}(n,d);{\bf Q})$. So we can find
relations in terms of
$a_{2},...,a_{n},b_{2}^{1},...,b_{n}^{2g},$ and $f_{2},...,f_{n}$ by equating
to zero the coefficients of $\prod_{s \in S}b_{1}^{s}$ in the Chern
classes $c_{r}(\pi_{!}V)$ for $r>n\bar{g}+\delta$ and for every subset $S
\subseteq \{1,...,2g\}.$\\
\indent Mumford's conjecture, as proven by Kirwan \cite[$\S$2]{K2}, was that when the rank $n$ is two then these
relations
together with the relation (\ref{NORM}) from
normalising the universal bundle $V$ provide a complete set of relations in
$H^{*}({\cal M}_{0}(2,d);{\bf Q})$. Subsequently a stronger version of
Mumford's conjecture has been proven \cite{E} showing the relations
coming from the first vanishing Chern class $c_{2g}(\pi_{!}V)$ generate the
relation ideal of $H^{*}({\cal M}_{0}(2,d))$ as a ${\bf Q}[a_{2},f_{2}]$-module.\\
\begin{rem}
In the rank two case the Mumford relations above
differ somewhat from the relations $\xi_{r}$ introduced by Zagier and
studied in \cite{B,KN,ST,Z}. In the notation of \cite{Z}
\[
\Psi_{\{1,...,2g\}} \left( \frac{-t-a_{1}}{2} \right) =
\frac{(-1)^{g\bar{g}/2 + g}}{2^{2g-1}} t^{\bar{g}} F_{0}(t^{-1})
\]
where $\Psi_{\{1,...2g\}}(x)$ denotes the coefficient of
$\prod_{s=1}^{2g} b_{1}^{s}$ in $\Psi(x) =\sum_{r \geq 0}
c_{r}(\pi_{!}V) x^{2g-1-r}$ and $F_{0}(t) =
\sum_{r=0}^{\infty} \xi_{r} t^{r}$. In the notation of \cite{KN}
$\xi_{r}$ appears as $\zeta_{r}/r!$ and in \cite{ST} as
$\Phi^{(r)}/r!.$
\end{rem}
\indent We will demonstrate later (remark \ref{inadequacy}) that the Mumford relations are
not complete when the rank $n$ is greater than two. For now we
introduce a new set of relations. Let $L$ be a fixed line bundle over
$M$ of degree $4\bar{g}+1$ and
let $\phi:{\cal M}(n,d) \times M \rightarrow M$ be the second
projection. Then $\pi_{!}(V^{*} \otimes
\phi^{*}L)$ is a vector bundle over ${\cal M}(n,d)$ of rank $(3\bar{g}+1)n-d
=ng-\delta$ with fibre $H^{0}(M,E^{*} \otimes L)$ over $[E]$. By
equating to zero the coefficients of $\prod_{s \in S}b_{1}^{s}$ in the Chern
classes $c_{r}(\pi_{!}(V^{*} \otimes \phi^{*}L))$ for $r>ng- \delta$
and for every subset $S \subseteq \{1,...,2g\}$ we may find
relations in terms of the generators
$a_{2},...,a_{n},b_{2}^{1},...,b_{n}^{2g},$ and $f_{2},...,f_{n}$. We
will refer to these new relations as the dual Mumford relations.
\begin{rem}
\label{dualise}
The map $E
\mapsto E^{*} \otimes L$ induces an automorphism of $H^{*}({\cal
M}(2,d);{\bf Q})$ mapping the Mumford relations to the dual Mumford
relations and vice versa. Hence we can deduce that the dual Mumford relations
are complete when the rank is two from Kirwan's proof of Mumford's
conjecture \cite[$\S$ 2]{K2}.
\end{rem}
\indent Our first result (to be proved in $\S 4$) now reads
as:\\[\baselineskip]
{\bf THEOREM 1.} {\em The Mumford and dual Mumford relations together with the
relation (\ref{NORM}) due to the normalisation of the
universal bundle $V$ form a complete set of relations for $H^{*}({\cal
M}(3,d);{\bf Q}).$}\\[\baselineskip]
\indent The Newstead-Ramanan conjecture states \cite[$\S$5a]{N2} that the
Pontryagin ring of the tangent bundle to ${\cal M}(2,d)$
vanishes in degrees $4g$ and higher. The conjecture was proven
independently by Thaddeus \cite{T} and Kirwan \cite[$\S$4]{K2}, and
has been proven more recently by King and Newstead \cite{KN} and Weitsman \cite{W}. In $\S
5$ we will use a similar method to Kirwan's but now also involving the
dual Mumford relations to prove:\\[\baselineskip]
{\bf THEOREM 2.} {\em The Pontryagin ring of the moduli space ${\cal
M}(3,d)$ vanishes in degrees $12g-8$ and above.}
\section{Kirwan's Approach.}
\indent The group ${\cal G}_{c}$ is the
complexification of the gauge group ${\cal G}$ of all smooth automorphisms
of ${\cal E}$ which are unitary with respect to a fixed Hermitian
structure on ${\cal E}$ \cite[p.570]{AB}. We shall write
$\overline{{\cal G}}$ for the quotient of ${\cal G}$ by its $U(1)$-centre and
$\overline{{\cal G}}_{c}$ for the quotient of ${\cal G}_{c}$ by its ${\bf
C}^{*}$-centre. \\
\indent There are natural isomorphisms \cite[9.1]{AB}
\[
H^{*}({\cal C}^{ss}/{\cal G}_{c};{\bf Q}) = H^{*}({\cal C}^{ss}/\overline{{\cal G}}_{c};{\bf Q}) \cong
H^{*}_{\overline{{\cal G}}_{c}}({\cal C}^{ss};{\bf Q}) \cong H^{*}_{\overline{{\cal G}}}({\cal C}^{ss};{\bf Q})
\]
since the ${\bf C}^{*}$-centre of ${\cal G}_{c}$ acts trivially on ${\cal C}^{ss}$,
$\overline{{\cal G}}_{c}$ acts freely on ${\cal C}^{ss}$ and $\overline{{\cal G}}_{c}$ is
the complexification of $\overline{{\cal G}}$. Atiyah and Bott \cite[thm.
7.14]{AB} show that the restriction map $H^{*}_{\overline{{\cal G}}}({\cal C};{\bf Q})
\rightarrow H^{*}_{\overline{{\cal G}}}({\cal C}^{ss};{\bf Q})$ is surjective. Further
$H^{*}_{\overline{{\cal G}}}({\cal C};{\bf Q}) \cong
H^{*}(B\overline{{\cal G}};{\bf Q})$ since ${\cal C}$ is an affine space \cite[p.565]{AB}. So
putting this all together we have
\begin{equation}
H^{*}(B\overline{{\cal G}};{\bf Q}) \cong H^{*}_{\overline{{\cal G}}}({\cal C};{\bf Q})
\rightarrow H^{*}_{\overline{{\cal G}}}({\cal C}^{ss};{\bf Q}) \cong H^{*}({\cal M}(n,d);{\bf Q}) \label{14}
\end{equation}
is a surjection.\\
\indent As shown in \cite[prop. 2.4]{AB} the classifying space $B{\cal G}$
can be identified with the space $\mbox{Map}_{d}(M,BU(n))$ of all
smooth maps $f:M \rightarrow BU(n)$ such that the pullback to $M$ of
the universal vector bundle over $BU(n)$ has degree $d$. If we
pull back this universal bundle using the evaluation map
\[
\mbox{Map}_{d}(M,BU(n)) \times M \rightarrow BU(n): (f,m) \mapsto f(m)
\]
then we obtain a rank $n$ vector bundle ${\cal V}$ over $B{\cal G} \times
M$. If we restrict the pullback bundle induced by the maps
\[
{\cal C}^{ss} \times E{\cal G} \times M \rightarrow {\cal C} \times E{\cal G} \times M \rightarrow {\cal C}
\times_{{\cal G}} E{\cal G} \times M \stackrel{\simeq}{\rightarrow} B{\cal G} \times M
\]
to ${\cal C}^{ss} \times \{e\} \times M$ for some $e \in E{\cal G}$ then we obtain
a ${\cal G}$-equivariant holomorphic bundle on ${\cal C}^{ss} \times M$. The $U(1)$-centre of
${\cal G}$ acts as scalar multiplication on the fibres,
and the associated projective bundle descends to a holomorphic
projective bundle over ${\cal M}(n,d) \times M$ which is in fact the projective
bundle of $V$ \cite[pp.579-580]{AB}.\\
\indent By a slight abuse of notation we define elements $a_{r}, b_{r}^{s}, f_{r}$
in $H^{*}(B{\cal G};{\bf Q})$ by writing
\[
c_{r}({\cal V}) = a_{r} \otimes 1 + \sum_{s=1}^{2g} b_{r}^{s} \otimes
\alpha_{s} + f_{r} \otimes \omega \indent 1 \leq r \leq n.
\]
Atiyah and Bott show \cite[prop. 2.20]{AB} that the ring
$H^{*}(B{\cal G};{\bf Q})$ is freely generated
as a graded algebra over ${\bf Q}$ by the elements $a_{r}, b_{r}^{s}, f_{r}$. The
only relations amongst these generators are that the
$a_{r}$ and $f_{r}$ commute with everything else and that the
$b_{r}^{s}$ anticommute with each other.\\
\indent The fibration $BU(1) \rightarrow B{\cal G} \rightarrow
B\overline{\cal G}$ induces an isomorphism \cite[p.577]{AB}
\[
H^{*}(B{\cal G};{\bf Q}) \cong H^{*}(B\overline{{\cal G}};{\bf Q}) \otimes H^{*}(BU(1);{\bf Q}).
\]
The generators $a_{r},b_{r}^{s}$ and $f_{r}$ of $H^{*}(B{\cal G};{\bf Q})$ can be pulled
back via a section of this
fibration to give rational generators of the cohomology ring of
$B\overline{\cal G}$. We may if we wish omit $a_{1}$ since its image
in $H^{*}(B\overline{{\cal G}};{\bf Q})$ can be expressed in terms of the other
generators. The only other relations are again the commuting of the $a_{r}$ and
$f_{r}$, and the anticommuting of the $b_{r}^{s}$. We may then normalise ${\cal V}$
suitably so that these generators for $H^{*}(B\overline{{\cal G}};{\bf Q})$
restrict to the generators $a_{r}, b_{r}^{s}, f_{r}$ for $H^{*}({\cal M}(n,d);{\bf Q})$ under
the surjection (\ref{14}).\\
\indent The relations amongst these generators for $H^{*}({\cal M}(n,d);{\bf Q})$ are
then given by the kernel of the restriction map (\ref{14}) which in
turn is determined by the map
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \cong H^{*}_{\overline{{\cal G}}}({\cal C};{\bf Q}) \otimes
H^{*}(BU(1);{\bf Q}) \rightarrow \indent\indent\indent\indent\indent\indent
\]
\[
\indent\indent\indent\indent\indent
H^{*}_{\overline{{\cal G}}}({\cal C}^{ss};{\bf Q}) \otimes
H^{*}(BU(1);{\bf Q}) \cong H^{*}_{\cal G}({\cal C}^{ss};{\bf Q}).
\]
In order to describe this kernel we consider Shatz's stratification of
${\cal C}$, the space of holomorphic structures on ${\cal E}$
\cite{Sh}. The stratification $\{{\cal C}_{\mu} : \mu \in {\cal M} \}$ is
indexed by the partially ordered set ${\cal M}$, consisting of all the
types of holomorphic bundles of rank $n$ and degree $d$, as follows.\\
\indent Any holomorphic bundle $E$ over $M$ of rank $n$ and degree $d$ has a
canonical filtration (or flag) \cite[p.221]{HN}
\[
0 = E_{0} \subset E_{1} \subset \cdot \cdot \cdot \subset E_{P} = E
\]
of sub-bundles such that the quotient bundles $Q_{p}= E_{p}/E_{p-1}$
are semi-stable and $\mu(Q_{p}) > \mu(Q_{p+1})$. We
will write $d_{p}$ and $n_{p}$ respectively for the degree and rank of
$Q_{p}$. Given such a filtration we define the type of $E$ to be
\[
\mu = (\mu(Q_{1}),...,\mu(Q_{P})) \in {\bf Q}^{n}
\]
where the entry $\mu(Q_{p})$ is repeated $n_{p}$ times. When
there is no chance of confusion we will
also refer collectively to the strata of type $(n_{1},...,n_{s})$ and
we will write $\Delta$ for the collection of strata with $n_{p}=1$ for each
$p$. The
semistable bundles have type $\mu_{0} = (d/n,...,d/n)$ and form the
unique open stratum. The set ${\cal M}$ of all possible types of holomorphic
vector bundles over $M$ will provide our indexing set. A partial order on
${\cal M}$ is defined as follows. Let $\sigma=(\sigma_{1},...,\sigma_{n})$ and
$\tau=(\tau_{1},...,\tau_
{n})$ be two types; we say that $\sigma \geq \tau$ if and only if
\[
\sum_{j \leq i} \sigma_{j} \geq \sum_{j \leq i} \tau_{j} \mbox{ for } 1 \leq i
\leq n-1.
\]
The set ${\cal C}_{\mu} \subseteq {\cal C},$ $\mu \in {\cal M}$, is
defined to be the set of all holomorphic vector bundles of type $\mu$.\\
\indent The stratification also has the following properties:-\\
\indent (i) The stratification is smooth. That is each stratum ${\cal C}_{\mu}$ is a
locally closed ${\cal G}_{c}$-invariant
submanifold. Further for any $\mu \in {\cal M}$ \cite[7.8]{AB}\\
\begin{equation}
\overline{{\cal C}_{\mu}} \subseteq \bigcup_{\nu \geq \mu} {\cal C}_{\nu}. \label{11}
\end{equation}
\indent (ii) Each stratum ${\cal C}_{\mu}$ is connected and has finite (complex)
codimension $d_{\mu}$ in ${\cal C}$. Moreover given any integer $N$ there are only
finitely many $\mu \in {\cal M}$ such that $d_{\mu} \leq N$. Further $d_{\mu}$
is given by the formula \cite[7.16]{AB}
\begin{equation}
d_{\mu}= \sum_{i>j} (n_{i}d_{j}-n_{j}d_{i}+n_{i}n_{j}\bar{g}) \label{12}
\end{equation}
where $d_{k}$ and $n_{k}$ are the degree and rank, respectively, of $Q_{k}$.\\
\indent (iii) The gauge group ${\cal G}$ acts on ${\cal C}$ preserving the
stratification which is equivariantly
perfect with respect to this action \cite[thm. 7.14]{AB}. In
particular there is an isomorphism of vector spaces
\[
H^{k}_{{\cal G}}({\cal C};{\bf Q}) \cong \bigoplus_{\mu \in {\cal M}}
H^{k-2d_{\mu}}_{{\cal G}}({\cal C}_{\mu};{\bf Q}) = H^{k}_{{\cal G}}({\cal C}^{ss};{\bf Q}) \oplus \bigoplus_{\mu \neq
\mu_{0}} H^{k-2d_{\mu}}_{{\cal G}} ({\cal C}_{\mu};{\bf Q}).
\]
The restriction map $H^{*}_{{\cal G}}({\cal C};{\bf Q}) \rightarrow H^{*}_{{\cal G}}({\cal C}^{ss};{\bf Q})$ is the
projection onto the summand $H^{*}_{{\cal G}}({\cal C}^{ss};{\bf Q})$ and so the kernel is
isomorphic as a vector space to
\begin{equation}
\bigoplus_{k \geq 0} \bigoplus_{\mu \neq \mu_{0}} H^{k-2d_{\mu}}_{{\cal G}}
({\cal C}_{\mu};{\bf Q}). \label{13}
\end{equation}
\begin{rem}
\label{inadequacy}
We can at this point use a dimension argument to
show that the Mumford relations are generally not complete when the
rank $n$ is greater than two. From the isomorphism (\ref{13}) we can see that
for the Mumford relations to be complete it is necessary that the
least degree of a Mumford relation must be less than or equal to the
smallest real codimension of an unstable stratum. The degree of
$\sigma_{r,S}^{k}$ equals $2(n\bar{g}+\delta -nr -k) -|S|$ which is least
when $r=-1,k=n-1,$ and $S=\{1,...,2g\}.$ So the smallest degree of a
Mumford relation is $2(\delta + (n-1)\bar{g}).$ However a simple
calculation minimising the codimension formula (\ref{12}) shows that the
least real codimension of an unstable stratum is $2(\delta + (n-1)\bar{g})$
when $\delta < n/2$ and is $2(n-\delta +(n-1)\bar{g})$ when $\delta>n/2$.
Hence the Mumford relations are not complete when $n \geq 3$ and
$\delta>n/2.$ A similar argument shows that
the dual Mumford relations are not complete when $\delta<n/2$ since
the smallest degree of a dual Mumford relation
is $2(n-\delta +(n-1)\bar{g}).$ Clearly however this simple argument does
not tell us anything concerning the union of the Mumford and dual
Mumford relations.
\end{rem}
\indent To conclude this section we will describe a set of criteria for the
completeness of a set of relations in $H^{*}({\cal M}(n,d);{\bf Q})$ and reformulate
the Mumford and dual Mumford relations in a way more suited to these
criteria. Consider the formal power series
\[
c(\pi_{!}{\cal V})(t) = \sum_{r \geq 0} c_{r}(\pi_{!}{\cal V}) \cdot
t^{r} \in H^{*}_{\cal G}({\cal C};{\bf Q})[[t]].
\]
The vanishing of the image of $c_{r}(\pi_{!}{\cal V})$ in $H^{*}({\cal M}(n,d);{\bf Q})$
for $r > n\bar{g}+\delta$ is equivalent to the image of $c(\pi_{!}{\cal V})(t)$ being a
polynomial of degree at most $n\bar{g}+\delta$ or equally to the image of
\[
\Psi(t) = t^{n\bar{g}+\delta}c(\pi_{!}{\cal V})(t^{-1})
\]
being a polynomial of degree at most $n\bar{g}+\delta$ in
$H^{*}({\cal M}(n,d);{\bf Q})[t].$ If we write $\Psi(t)$ as the series
\[
\Psi(t)=\sum_{r=-\infty}^{\bar{g}} (\sigma_{r}^{0} + \sigma_{r}^{1}t +
\cdot \cdot \cdot + \sigma_{r}^{n-1}t^{n-1} ) (\tilde{\Omega}(t))^{r}
\]
where $\tilde{\Omega}(t)= t^{n}+a_{1}t^{n-1}+ \cdot \cdot \cdot +a_{n}$ then the
Mumford relations are equivalent to the vanishing of the images of
$\sigma_{r,S}^{k} (r<0,0 \leq k \leq n-1,S \subseteq \{1,...,2g\})$ in
$H^{*}({\cal M}_{0}(n,d);{\bf Q})$ when we write
\begin{equation}
\sigma_{r}^{k} = \sum_{S \subseteq \{1,...,2g\}} \sigma_{r,S}^{k}
\prod_{s \in S} b_{1}^{s}. \label{998}
\end{equation}
We will refer to $\sigma_{r,S}^{k} (r<0,0 \leq k \leq n-1,S \subseteq
\{1,...,2g\})$ as the Mumford relations.\\
\indent Similarly we know that the restriction of
\[
\Psi^{*}(t) = t^{ng - \delta}c(\pi_{!}({\cal V}^{*} \otimes
\phi^{*}L))(-t^{-1})
\]
to $H^{*}({\cal M}(n,d);{\bf Q})$ is a polynomial. As before we may put $\Psi^{*}(t)$ in
the form
\[
\Psi^{*}(t) = \sum_{r=-\infty}^{\bar{g}}(\tau_{r}^{0}+\tau_{r}^{1}t+ \cdot
\cdot \cdot + \tau_{r}^{n-1}t^{n-1})(\tilde{\Omega}(t))^{r}
\]
where $\tilde{\Omega}(t) = t^{n}+a_{1}t^{n-1}+ \cdot \cdot \cdot +a_{n}$ and
similarly we write
\begin{equation}
\tau_{r}^{k}=\sum_{S \subseteq \{1,...,2g\}} \tau_{r,S}^{k} \prod_{s
\in S} b_{1}^{s}. \label{999}
\end{equation}
We will refer to $\tau_{r,S}^{k} (r<0,0 \leq k \leq n-1,S \subseteq
\{1,...,2g\})$ as the dual Mumford relations.\\
\indent The motivation for this is that the restrictions of
$\sigma_{r,S}^{k}$ and $\tau_{r,S}^{k}$ to the strata ${\cal C}_{\mu}$ are
easier to calculate in this form. This is a crucial step in applying
the following completeness criteria.\\
\indent Given $\mu=(\mu_{1},...,\mu_{n}),\nu=(\nu_{1},...,\nu_{n}) \in
{\cal M}$ then we write $\nu \prec \mu$ if there exists $T$, $1 \leq T
\leq n$, such that
\[
\nu_{i} = \mu_{i} \mbox{ for } T < i \leq n \mbox{ and } \nu_{T} >
\mu_{T}.
\]
We write $\nu \preceq \mu$ if $\nu \prec \mu$ or $\nu = \mu.$ A few
easy calculations verify that $\preceq$ is a total order on ${\cal M}$
with minimal element $\mu_{0}$, the semistable type. For an unstable
type $\mu$ we will write $\mu-1$ for the type previous to $\mu$ with
respect to $\preceq.$
\begin{prop}
\label{KCC}
(Completeness Criteria) Let ${\cal R}$ be a
subset of the kernel of the restriction map
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}^{ss};{\bf Q}).
\]
Suppose that for each unstable type $\mu$ there is a subset ${\cal
R}_{\mu}$ of the ideal generated by ${\cal R}$ such that the image of
${\cal R}_{\mu}$ under the restriction map
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\nu};{\bf Q})
\]
is zero when $\nu \prec \mu$ and when $\nu = \mu$ contains the ideal of
$H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ generated by $e_{\mu}$, the equivariant
Euler class of ${\cal N}_{\mu}$, the normal bundle to the stratum ${\cal C}_{\mu}$ in
${\cal C}.$ Then ${\cal R}$ generates the kernel of the restriction map
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}^{ss};{\bf Q})
\]
as an ideal of $H^{*}_{\cal G}({\cal C};{\bf Q}).$
\end{prop}
\begin{rem}
The proof of proposition \ref{KCC} below follows similar
lines to the proof of \cite[prop.1]{K2}. However there are some
differences -- the order $\preceq$ does not generally coincide with
$\leq$ -- and further the proof of \cite[p.867]{K2} as given is true
only for the rank two case. For these reasons we include a proof of
proposition \ref{KCC} below although it clearly owes many of its origins to
\cite{K2}.
\end{rem}
{\bf Proof} Let $\mu \in {\cal M}$ and define
\[
V_{\mu} = \bigcup_{\nu \preceq \mu} {\cal C}_{\nu}.
\]
We will firstly show that $V_{\mu}$ is an open subset of ${\cal C}$
containing ${\cal C}_{\mu}$ as a closed submanifold. Note that if $\nu \leq \mu$
then $\nu \preceq \mu$ and thus by property (\ref{11}) if $\nu
\succ \mu$ then $\overline{{\cal C}}_{\nu} \subseteq {\cal C} - V_{\mu}$. The
stratification is locally finite and hence $V_{\mu}$ is open. Further
note that the closure of ${\cal C}_{\mu}$ in $V_{\mu}$ equals
\[
V_{\mu} \cap \bigcup_{\nu \geq \mu} {\cal C}_{\nu} = {\cal C}_{\mu}
\]
as required.\\
\indent Recall now that the composition of the Thom-Gysin map
\[
H_{{\cal G}}^{*-2d_{\mu}}({\cal C}_{\mu};{\bf Q}) \rightarrow H^{*}_{\cal G}(V_{\mu};{\bf Q})
\]
with the restriction map
\[
H^{*}_{\cal G}(V_{\mu};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})
\]
is given by multiplication by the Euler class $e_{\mu}$ which is
not a zero-divisor in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ \cite[p.569]{AB}.
It follows from the exactness of the Thom-Gysin sequence
\[
\cdot \cdot \cdot \rightarrow H_{{\cal G}}^{*-2d_{\mu}}({\cal C}_{\mu};{\bf Q})
\rightarrow H^{*}_{\cal G}(V_{\mu};{\bf Q})
\rightarrow H^{*}_{\cal G}(V_{\mu-1};{\bf Q}) \rightarrow \cdot \cdot \cdot
\]
that the direct sum of the restriction maps
\[
H^{*}_{\cal G}(V_{\mu};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q}) \oplus H^{*}_{\cal G}(V_{\mu-1};{\bf Q})
\]
is injective. Hence inductively the direct sum of restriction maps
\[
H^{*}_{\cal G}(V_{\mu-1};{\bf Q}) \rightarrow \bigoplus_{\nu \prec \mu} H^{*}_{\cal G}({\cal C}_{\nu};{\bf Q})
\]
is injective and in particular the image of any element of ${\cal R}_{\mu}$
under the restriction map
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}(V_{\mu-1};{\bf Q})
\]
is zero.\\
\indent For any given $i \geq 0$ there are only finitely many $\nu \in
{\cal M}$ such that $2d_{\nu} \leq i$ and so for each $i \geq 0$ there
exists some $\mu$ such that
\[
H_{{\cal G}}^{i}({\cal C};{\bf Q}) = H_{{\cal G}}^{i}(V_{\mu};{\bf Q}).
\]
Hence it is enough to show that for each $\mu$ the image in
$H^{*}_{\cal G}(V_{\mu};{\bf Q})$ of the ideal generated by ${\cal R}$ contains the
image in $H^{*}_{\cal G}(V_{\mu};{\bf Q})$ of the kernel of the restriction map
\begin{equation}
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}^{ss};{\bf Q}). \label{QQQ}
\end{equation}
Note that the above is clearly true for $\mu=\mu_{0}$ as
$V_{\mu_{0}}={\cal C}^{ss}.$ We will proceed by induction with respect to $\preceq$.\\
\indent Assume now that $\mu \neq \mu_{0}$ and that $\alpha \in
H^{*}_{\cal G}({\cal C};{\bf Q})$ lies in the kernel of (\ref{QQQ}).
Suppose that the image of $\alpha$ in $H^{*}_{\cal G}(V_{\mu-1};{\bf Q})$ is in the
image of the ideal generated by ${\cal R}.$ We may, without any loss of
generality, assume that the image of $\alpha$ in $H^{*}_{\cal G}(V_{\mu-1};{\bf Q})$ is
zero. Thus by the exactness of the Thom-Gysin sequence there exists an
element $\beta \in H_{{\cal G}}^{*-2d_{\mu}}({\cal C}_{\mu};{\bf Q})$ which is
mapped to the image of $\alpha$ in $H^{*}_{\cal G}(V_{\mu};{\bf Q})$ by the Thom-Gysin
map
\[
H_{{\cal G}}^{*-2d_{\mu}}({\cal C}_{\mu};{\bf Q}) \rightarrow H^{*}_{\cal G}(V_{\mu};{\bf Q}).
\]
Hence the image of $\alpha$ under the restriction map
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})
\]
is $\beta e_{\mu}$, and by hypothesis there is an element $\gamma$ of ${\cal
R}_{\mu}$ which maps under the
restriction map
\[
H^{*}_{\cal G}(V_{\mu};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})
\]
to $\beta e_{\mu}.$ Now the images of $\gamma$ and $\alpha$ in
$H^{*}_{\cal G}(V_{\mu-1};{\bf Q})$ are both zero and we also know the direct sum of the
restriction maps
\[
H^{*}_{\cal G}(V_{\mu};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q}) \oplus H^{*}_{\cal G}(V_{\mu-1};{\bf Q})
\]
to be injective. Thus the images of $\gamma$ and $\alpha$ in
$H^{*}_{\cal G}(V_{\mu};{\bf Q})$ are the same, completing the proof. $\indent \Box$
\begin{rem}
Kirwan's completeness criteria follow from the
above criteria since for each $\mu$
\[
V_{\mu-1} \subseteq {\cal C} - \bigcup_{\nu \geq \mu} {\cal C}_{\nu}.
\]
So if the restriction of a relation to $H^{*}_{\cal G}({\cal C}_{\nu};{\bf Q})$ vanishes for
every $\nu \not \geq \mu$ then certainly the same relation restricts
to zero in $H^{*}_{\cal G}({\cal C}_{\nu};{\bf Q})$ for any $\nu \prec \mu.$
\end{rem}
\begin{rem}
Kirwan's proof of Mumford's
conjecture \cite[$\S$ 2]{K2} amounts to showing that for each unstable
type $\mu=(d_{1},d_{2})$ the set
\[
{\cal R}_{\mu} = \bigcup \{
\sigma_{d_{2}-2g+1,S}^{0},\sigma_{d_{2}-2g+1,S}^{1} \},
\]
where the union is taken over all subsets $S \subseteq \{1,...,2g\}$,
satisfies the above criteria. In the rank two case the criteria of
proposition \ref{KCC} are in fact equivalent to Kirwan's completeness
criteria since $\preceq$ and $\leq$ coincide.
\end{rem}
\section{Chern Class Computations.}
\indent We first describe the restriction maps $H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow
H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ and our preferred generators for
$H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$. Let $\mu=(d_{1}/n_{1},...,d_{P}/n_{P})$. Let
${\cal C}(n_{p},d_{p})^{ss}$ denote the space of all semistable holomorphic
structures on a fixed Hermitian vector bundle of rank $n_{p}$ and
degree $d_{p}$ and let ${\cal G}(n_{p},d_{p})$ be the gauge group of that
bundle. Atiyah and
Bott \cite[prop. 7.12]{AB} show that the map
\[
\prod_{p=1}^{P} {\cal C}(n_{p},d_{p})^{ss} \rightarrow {\cal C}_{\mu},
\]
which sends a sequence of semistable bundles $(F_{1},...,F_{P})$ to
the direct sum $F_{1} \oplus \cdot \cdot \cdot \oplus F_{P}$, induces an
isomorphism
\[
H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q}) \cong \bigotimes_{1 \leq p \leq P} H^{*}_{{\cal
G}(n_{p},d_{p})}({\cal C}(n_{p},d_{p})^{ss};{\bf Q}).
\]
Thus we can find generators
\begin{equation}
\bigcup_{p=1}^{P} \left( \{a_{r}^{p} | 1 \leq r \leq n_{p} \}
\cup \{ b_{r}^{p,s} | 1 \leq r \leq n_{p},1 \leq s \leq 2g \} \cup \{
f_{r}^{p} | 2 \leq r \leq n_{p} \} \right) \label{A}
\end{equation}
corresponding to the generators of $H^{*}_{\cal G}({\cal C}^{ss};{\bf Q})$ described earlier in
(\ref{0}). As before we also define
\[
\xi_{i,j}^{p,q} = \sum_{s=1}^{g} b_{i}^{p,s} b_{j}^{q, s+g}.
\]
To explicitly describe the restriction map note that
$c_{r}({\cal V})$ restricts to $c_{r}(\bigoplus_{p=1}^{P} {\cal
V}_{p})$ where ${\cal V}_{p}$ is the universal bundle on ${\cal C}(n_{p},d_{p})$. The
restrictions of the generators of $H^{*}_{\cal G}({\cal C};{\bf Q})$ can be written in terms of
the generators of $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ by taking the appropriate
coefficients in the K\"{u}nneth decomposition.\\
\indent One problem that we will be faced with in due course is how to
calculate the coefficients of $\prod_{s \in S} b_{1}^{s}$ once we have
restricted to a stratum. Suppose first that the stratum concerned is of type
$\mu=(d_{1},...,d_{n}) \in \Delta$ and take
$\zeta \in H^{*}_{\cal G}({\cal C};{\bf Q}).$ We can express $\zeta$ in terms of the generators
\[
\{a_{r} | 1 \leq r \leq n \} \cup \{ b_{r}^{s} | 1 \leq r \leq n,1
\leq s \leq 2g \} \cup \{ f_{r} | 2 \leq r \leq n \}
\]
but equally we could write $\zeta$ in terms of
\[
\{a_{r} | 1 \leq r \leq n \} \cup \{ n b_{r}^{s}
-(n-r+1)a_{r-1}b_{1}^{s} | 2 \leq r \leq n,1
\leq s \leq 2g \}
\]
\begin{equation}
\cup \{ n^{2} f_{r} - n(n-r+1)(\xi_{r-1,1} + \xi_{1,r-1}) + (n-r+1)(n-r+2) a_{r-2} \xi_{1,1} | 2 \leq r
\leq n \} \label{1000}
\end{equation}
and $\{b_{1}^{s} | 1 \leq s \leq 2g\}.$ We shall take the coefficients
of $\prod_{s \in S}b_{1}^{s}$ when $\zeta$ is expressed in this latter
form. The reason for this is that
the restrictions of the elements (\ref{1000}) in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ can then
be written in terms of
\begin{equation}
\{a_{1}^{r} | 1 \leq r \leq n \} \cup \{b_{1}^{p,s}-b_{1}^{n,s} | 1
\leq p \leq n-1, 1 \leq s \leq 2g \} \label{1001}
\end{equation}
(see remark \ref{tedious}.) We can uniquely write the restriction of $\zeta$ in terms of the
elements (\ref{1001}) and the restrictions of $b_{1}^{s},(1 \leq s
\leq 2g)$. Hence we may calculate the restrictions of the coefficients of
$\prod_{s \in S} b_{1}^{s}$ in $\zeta$ by taking the coefficients of
\[
\prod_{s \in S} (b_{1}^{1,s} + \cdots + b_{1}^{n,s})
\]
in the restriction of $\zeta$.\\
\indent We deal with a general type stratum in a similar way. Let
$\mu = (d_{1}/n_{1},...,d_{P}/n_{P})$. We define formal symbols
$a^{p,k},b^{p,k,s}$ and $d^{p,k}$ such that the $r$th Chern class
$c_{r}({\cal V}_{p})$ is given by the $r$th elementary symmetric
polynomial in
\begin{equation}
a^{p,k} + \sum_{s=1}^{2g} b^{p,k,s} \otimes \alpha_{s} + d^{p,k}
\otimes \omega \indent (1 \leq k \leq n_{p}) \label{QQ}
\end{equation}
when $1 \leq r \leq n_{p}$ and $1 \leq p \leq P$. In terms of
$a^{p,k},b^{p,k,s}$ and $d^{p,k}$ the restriction map to
$H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ is formally the same as the restriction map when $\mu \in
\Delta$. Again we may uniquely write the restriction of
$\zeta$ in terms of
\begin{equation}
\bigcup_{p=1}^{P} \bigcup_{k=1}^{n_{p}} \{a^{p,k}, d^{p,k} \}
\cup \bigcup_{p=1}^{P-1} \bigcup_{k= 1}^{n_{p}} \bigcup_{s=1}^{2g} \{
b^{p,k,s} - b^{P,n_{P},s} \} \cup \bigcup_{k=
1}^{n_{P}-1} \bigcup_{s=1}^{2g} \{ b^{P,k,s} - b^{P,n_{P},s} \} \label{NEW1}
\end{equation}
and the restrictions of $b_{1}^{s}, (1 \leq s \leq 2g)$, and we take
the coefficients of
\[
\prod_{s \in S}(b_{1}^{1,s} + \cdots + b_{1}^{P,s})
\]
as before.\\
\indent So in our definitions of the Mumford and dual Mumford
relations, (\ref{998}) and (\ref{999}), we assume first that
$\sigma_{r}^{k}$ and $\tau_{r}^{k}$ have first been written in terms
of the elements (\ref{1000}) before taking the appropriate
coefficient.
\begin{rem}
\label{tedious}
It is a trivial but tedious calculation to show that the restrictions
of the elements (\ref{1000}) in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ for $\mu \in \Delta$ can
indeed be written in terms of the elements (\ref{1001}). Let
$a_{r}^{\mu}$ denote the restriction of $a_{r}$ to $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$;
this equals the $r$th elementary symmetric product in
$a_{1}^{1},...,a_{1}^{n}$. The restrictions of $b_{r}^{s}$ and $f_{r}$
in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ equal
\[
\sum_{i=1}^{n} b_{1}^{i,s} \frac{\partial a_{r}^{\mu}}{\partial
a_{1}^{i}}, \indent \sum_{i=1}^{n} d_{i} \frac{\partial
a_{r}^{\mu}}{\partial a_{1}^{i}} + \sum_{i=1}^{n} \sum_{j=1}^{n}
\xi_{1,1}^{i,j} \frac{\partial^{2} a_{r}^{\mu}}{\partial a_{1}^{i}
\partial a_{1}^{j}}.
\]
The restrictions of the elements (\ref{1000}) can then be seen to
equal
\[
a_{r}^{\mu}, \indent \sum_{i=1}^{n-1} \left( n \frac{\partial
a_{r}^{\mu}}{\partial a_{1}^{i}} - (n-r+1) a_{r-1}^{\mu} \right) (
b_{1}^{i,s} - b_{1}^{n,s}),
\]
and
\[
n^{2} \sum_{i=1}^{n} d_{i} \frac{\partial a_{r}^{\mu}}{\partial
a_{1}^{i}} + \sum_{i=1}^{n-1} \sum_{j=1}^{n-1} \sum_{s=1}^{g}
(b_{1}^{i,s} - b_{1}^{n,s})(b_{1}^{j,s+g} - b_{1}^{n,s+g}) \left(
n^{2} \frac{\partial^{2}a_{r}^{\mu}}{\partial a_{1}^{i} \partial
a_{1}^{j}} \right.
\]
\[
\left. - n(n-r+1) \left(\frac{\partial a_{r-1}^{\mu}}{\partial
a_{1}^{i}} + \frac{\partial a_{r-1}^{\mu}}{\partial a_{1}^{j}} \right) +
(n-r+1)(n-r+2)a_{r-2}^{\mu} \right).
\]
\end{rem}
\indent The remains of this section are given over to calculating the
Mumford and dual Mumford relations. Our first problem is to obtain
their generating functions from their respective Chern characters which we can
evaluate using the Grothendieck-Riemann-Roch theorem (GRR).
\begin{lem}
\label{Chernlemma}
Suppose that
\begin{equation}
{\rm ch}(E) = \sum_{i=1}^{M} \alpha_{i} e^{\delta_{i}} + \sum_{i=1}^{N}
\beta_{i} e^{\epsilon_{i}} \label{230}
\end{equation}
where the $\beta_{i},\delta_{i}$ and the $\epsilon_{i}$ are formal
degree two classes and the $\alpha_{i}$ are formal degree zero
classes. Then as a formal power series
\begin{equation}
c(E)(t) = \sum_{r=0}^{\infty} c_{r}(E) \cdot t^{r} = \prod_{i=1}^{M}
(1+\delta_{i}t)^{\alpha_{i}} \prod_{i=1}^{N} \exp \left\{
\frac{\beta_{i}t}{1+\epsilon_{i}t} \right\}. \label{231}
\end{equation}
\end{lem}
{\bf Proof}
The relationship between the Chern character and Chern polynomial
is as follows. If $\mbox{ch}(E)= \sum_{i=1}^{K} e^{\gamma_{i}}$ where $\gamma_{i}$
are formal degree two classes then
\[
c(E)(t) = \prod_{i=1}^{K} (1+\gamma_{i}t).
\]
If $\mbox{ch}(E)$ is in the form of (\ref{230}) then by comparing
degrees we find that
\[
\sum_{i=1}^{M} \alpha_{i}(\delta_{i})^{n} + \sum_{i=1}^{N} n \beta_{i}
(\epsilon_{i})^{n-1} = \sum_{i=1}^{K} (\gamma_{i})^{n}
\]
for each $n \geq 0$. Thus on
the level of formal power series $\log c(E)(t)$ equals
\[
\sum_{i=1}^{K} \sum_{r=1}^{\infty} (-1)^{r+1}
\frac{(\gamma_{i}t)^{r}}{r} = \sum_{i=1}^{M} \alpha_{i} \log(1+\delta_{i}t) +
\sum_{i=1}^{N} \frac{\beta_{i}t}{1+\epsilon_{i}t}
\]
and hence the result (\ref{231}). $\indent \Box$.\\[\baselineskip]
\indent Armed with the above lemma we are now in a position to
determine the Chern polynomials $c(\pi_{!}{\cal V})(t)$ and
$c(\pi_{!}({\cal V}^{*} \otimes \phi^{*}L))(-t)$. We can, and
will, calculate these Chern polynomials in terms of the generators
$a_{r},b_{r}^{s}$ and $f_{r}$ of $H^{*}_{\cal G}({\cal C};{\bf Q})$ (see (\ref{26})
and (\ref{210})). However the expressions obtained are
somewhat cumbersome and for ease of calculation we
will find the formal expressions, (\ref{25}) and (\ref{29}),
calculated directly from the above lemma of more use.
\begin{prop}
\label{Chernprop}
The Chern polynomial $c(\pi_{!}{\cal V})(t)$ equals
\begin{equation}
\Omega(t)^{-\bar{g}} \prod_{k=1}^{n} (1+\delta_{k}t)^{W_{k}} \exp \left\{
\frac{X_{k}t}{1+\delta_{k}t} \right\} \label{25}
\end{equation}
and $c(\pi_{!}({\cal
V}^{*} \otimes \phi^{*}L))(-t)$ equals
\begin{equation}
\Omega(t)^{3\bar{g}+1} \prod_{k=1}^{n} (1+\delta_{k}t)^{-W_{k}} \exp
\left\{ \frac{-X_{k}t}{1+\delta_{k}t} \right\}, \label{29}
\end{equation}
where $\delta_{1},...,\delta_{n}$ are formal degree two classes such
that their $r$th elementary symmetric polynomial equals $a_{r},$ and
\[
\Omega(t)= \prod_{k=1}^{n}(1+\delta_{k}t) = 1 + a_{1}t + \cdot \cdot
\cdot + a_{n}t^{n}, \indent \xi_{i,j}= \sum_{s=1}^{g} b_{i}^{s}
b_{j}^{s+g},
\]
\[
W_{k}= \sum_{i=1}^{n} f_{i} \frac{\partial \delta_{k}}{\partial a_{i}} +
\sum_{i=1}^{n} \sum_{j=1}^{n} \xi_{i,j} \frac{\partial^{2} \delta_{k}}{\partial
a_{i} \partial a_{j}}, \indent X_{k} = \sum_{i=1}^{n} \sum_{j=1}^{n} \xi_{i,j}
\frac{\partial \delta_{
k}}{\partial a_{i}} \frac{\partial \delta_{k}}{\partial a_{j}}.
\]
In terms of the generators $a_{r},b_{r}^{s}$ and $f_{r}$ for
$H^{*}_{\cal G}({\cal C};{\bf Q})$ then $c(\pi_{!}{\cal V})(t)$ equals
\begin{equation}
\Omega(t)^{-\bar{g}} \exp \left\{ \int_{0}^{t} \left( \frac{d}{u} - \sum_{i=1}^{n}
\frac{f_{i} u^{i-2}}{\Omega(u)} + \sum_{i=1}^{n} \sum_{j=1}^{n}
\frac{\xi_{i,j}u^{i+j-2}}{\Omega(u)^{2}} \right) {\rm d}u \right\} \label{26}
\end{equation}
and $c(\pi_{!}({\cal V}^{*} \otimes \phi^{*}L))(-t)$ equals
\begin{equation}
\Omega(t)^{3\bar{g}+1} \exp \left\{ \int_{0}^{t} \left( -\frac{d}{u} + \sum_{i=1}^{n}
\frac{f_{i} u^{i-2}}{\Omega(u)} - \sum_{i=1}^{n} \sum_{j=1}^{n}
\frac{\xi_{i,j}u^{i+j-2}}{\Omega(u)^{2}} \right) {\rm d}u \right\}. \label{210}
\end{equation}
\end{prop}
{\bf Proof}
Now $\mbox{ch}({\cal V}) = e^{\gamma_{1}} + \cdot \cdot \cdot +
e^{\gamma_{n}}$ where $\gamma_{1},...,\gamma_{n}$ are formal degree two classes
such that their $r$th elementary symmetric polynomial equals
\[
c_{r}({\cal V}) = a_{r} \otimes 1 + \sum_{s=1}^{2g} b_{r}^{s} \otimes
\alpha_{s} + f_{r} \otimes \omega \indent (1 \leq r \leq n) .
\]
For each $k \geq 0$ there exist coefficients $\rho_{r_{1},...,r_{n}}^{(k)}$
such that
\[
(\gamma_{1})^{k} + \cdot \cdot \cdot + (\gamma_{n})^{k} = \sum
\rho_{r_{1},...,r_{n}}^{(k)} (c_{1}({\cal V}))^{r_{1}} \cdot \cdot \cdot
(c_{n}({\cal V}))^{r_{n}}
\]
where the sum is taken over all non-negative $r_{1},...,r_{n}$ such that
$r_{1}+2r_{2}+ \cdots +nr_{n} = k$. Now
\[
(a_{1} \otimes 1 + \sum_{s=1}^{2g} b_{1}^{s} \otimes \alpha_{s} + f_{1} \otimes
\omega)^{r_{1}} \cdot \cdot \cdot (a_{n} \otimes 1 + \sum_{s=1}^{2g} b_{n}^{s}
\otimes \alpha_{s} + f_{n} \otimes \omega)^{r_{n}}
\]
equals
\[
(a_{1})^{r_{1}} \cdot \cdot \cdot (a_{n})^{r_{n}} \otimes 1 + \sum_{i=1}^{n}
\sum_{s=1}^{2g} b_{i}^{s} \frac{\partial}{\partial a_{i}} (a_{1})^{r_{1}} \cdot \cdot \cdot
(a_{n})^{r_{n}} \otimes \alpha_{s}
\]
\[
+\sum_{i=1}^{n} f_{i} \frac{\partial}{\partial a_{i}} (a_{1})^{r_{1}} \cdot \cdot \cdot (a_{n})^{r_{n}}
\otimes \omega + \sum_{i=1}^{n} \sum_{j=1}^{n} \xi_{i,j} \frac{\partial^{2}}{\partial a_{i} \partial a_{j}} (a_{1})^{r_{1}}
\cdot \cdot \cdot (a_{n})^{r_{n}} \otimes \omega.
\]
Since
\[
\sum \rho_{r_{1},...,r_{n}}^{(k)} (a_{1})^{r_{1}} \cdot \cdot \cdot
(a_{n})^{r_{n}} = (\delta_{1})^{k} + \cdot \cdot \cdot + (\delta_{n})^{k}
\]
we find that $\mbox{ch}({\cal V})$ equals
\[
\sum_{k=1}^{n} e^{\delta_{k}} \otimes 1 + \sum_{i=1}^{n} \sum_{s=1}^{2g}
\sum_{k=1}^{n} b_{i}^{s} \frac{\partial}{\partial a_{i}} e^{\delta_{k}} \otimes \alpha_{s}
\]
\begin{equation}
+\sum_{i=1}^{n} \sum_{k=1}^{n} f_{i} \frac{\partial}{\partial a_{i}} e^{\delta_{k}} \otimes
\omega + \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1}^{n} \xi_{i,j} \frac{\partial^{2}}{\partial a_{i} \partial a_{j}}
e^{\delta_{k}} \otimes \omega. \label{212}
\end{equation}
{}From GRR we have $\mbox{ch}(\pi_{!}{\cal V}) = \pi_{*} ( \mbox{ch}({\cal V}) \cdot
1 \otimes (1-\bar{g} \omega))$ and hence $\mbox{ch}(\pi_{!}{\cal V})$ equals
\[
\sum_{i=1}^{n} \sum_{k=1}^{n} f_{i} \frac{\partial}{\partial a_{i}} e^{\delta_{k}} + \sum_{i=1}^{n}
\sum_{j=1}^{n} \sum_{k=1}^{n} \xi_{i,j} \frac{\partial^{2}}{\partial a_{i} \partial a_{j}} e^{\delta_{k}} - \bar{g}
\sum_{k=1}^{n} e^{\delta_{k}}
= \sum_{k=1}^{n} (-\bar{g} + W_{k} + X_{k}) e^{\delta_{k}}.
\]
Note that $W_{k}$ has degree zero and $X_{k}$ has degree two. Hence by
lemma \ref{Chernlemma} we see that $c(\pi_{!}{\cal V})(t)$ equals
\[
(\Omega(t))^{-\bar{g}} \prod_{k=1}^{n} (1+\delta_{k}t)^{W_{k}} \exp \left\{
\frac{X_{k}t}{1+\delta_{k}t} \right\}
\]
to give equation (\ref{25}).\\
\indent Now $\frac{{\rm d}}{{\rm d}t} \log ( \Omega(t)^{\bar{g}}
c(\pi_{!}{\cal V})(t))$ equals
\[
\sum_{i=1}^{n} \sum_{k=1}^{n} f_{i} \frac{\partial \delta_{k}}{\partial a_{i}}
\frac{\delta_{k}}{1+\delta_{k}t} + \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1}^{n}
\xi_{i,j} \left( \frac{\partial^{2} \delta_{k}}{\partial a_{i} \partial a_{j}}
\frac{\delta_{k}}{1+\delta_{k} t} + \frac{\partial \delta_{k}}{\partial a_{i}}
\frac{\partial \delta_{k}}{\partial a_{j}} \frac{1}{(1+ \delta_{k} t)^{2}}
\right)
\]
\[
= \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{\xi_{i,j}}{t^{2}} \left( \sum_{k=1}^{n} t
\frac{\partial^{2} \delta_{k}}{\partial a_{i} \partial a_{j}} - \sum_{k=1}^{n}
\left( \frac{t}{1+ \delta_{k}t} \frac{\partial^{2} \delta_{k}}{\partial a_{i}
\partial a_{j}} -
\frac{t^{2}}{(1+\delta_{k} t)^{2}} \frac{\partial \delta_{k}}{\partial a_{i}}
\frac{\partial \delta_{k}}{\partial a_{j}} \right) \right)
\]
\begin{equation}
+ \sum_{i=1}^{n} \frac{f_{i}}{t} \left( \sum_{k=1}^{n} \frac{\partial
\delta_{k}}{\partial a_{i}} - \sum_{k=1}^{n} \frac{\partial
\delta_{k}}{\partial a_{i}} \frac{1}{1+ \delta_{k} t} \right) \label{219}
\end{equation}
Since $\sum_{k=1}^{n} \frac{\partial \delta_{k}}{\partial a_{i}} =
\frac{ \partial a_{1}}{\partial a_{i}}$, $f_{1}=d$, and
$\sum_{k=1}^{n} \frac{\partial^{2} \delta_{k}}{\partial a_{i} \partial
a_{j}} = \frac{\partial^{2} a_{1}}{\partial a_{i} \partial a_{j}} = 0$
then (\ref{219}) reduces to
\[
\frac{d}{t} - \sum_{i=1}^{n} \sum_{k=1}^{n} f_{i} \frac{\partial}{\partial a_{i}} \frac{\log (1+
\delta_{k} t)}{t^{2}} - \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1}^{n} \xi_{i,j}
\frac{\partial^{2}}{\partial a_{i} \partial a_{j}} \frac{\log (1 + \delta_{k}t)}{t^{2}}
\]
\[
= \frac{d}{t} - \left(
\sum_{i=1}^{n} f_{i} \frac{\partial}{\partial a_{i}} + \sum_{i=1}^{n} \sum_{j=1}^{n} \xi_{i,j}
\frac{\partial^{2}}{\partial a_{i} \partial a_{j}} \right) \frac{\log \Omega(t)}{t^{2}}
\]
to give equality (\ref{26}).\\
\indent The calculations for the dual case follow in a similar
fashion. We have that $\mbox{ch}({\cal V}^{*}) = e^{-\gamma_{1}} + \cdot
\cdot \cdot + e^{-\gamma_{n}}$ with $\gamma_{1},...,\gamma_{n}$ as
before and arguing as in the calculation of (\ref{212}) we determine that
$\mbox{ch}({\cal V}^{*})$ equals
\[
\sum_{k=1}^{n} e^{-\delta_{k}} \otimes 1 + \sum_{i=1}^{n} \sum_{s=1}^{2g}
\sum_{k=1}^{n} b_{i}^{s} \frac{\partial}{\partial a_{i}} e^{-\delta_{k}} \otimes \alpha_{s}
\]
\begin{equation}
+\sum_{i=1}^{n} \sum_{k=1}^{n} f_{i} \frac{\partial}{\partial a_{i}} e^{-\delta_{k}} \otimes
\omega + \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1}^{n} \xi_{i,j} \frac{\partial^{2}}{\partial a_{i} \partial a_{j}}
e^{-\delta_{k}} \otimes \omega. \label{213}
\end{equation}
We know that $\mbox{ch}(\phi^{*}L) = \phi^{*}(e^{(4\bar{g}+1)\omega}) = 1
\otimes (1+ (4\bar{g}+1)\omega)$ and GRR shows that $\mbox{ch}(\pi_{!}({\cal
V}^{*} \otimes \phi^{*}L))$ equals
\[ \pi_{*}(\mbox{ch}({\cal
V}^{*}) \cdot \mbox{ch}(\phi^{*}L) \cdot 1 \otimes (1 - \bar{g} \omega)) =
\pi_{*}(\mbox{ch}({\cal
V}^{*}) \cdot 1 \otimes (1 + (3\bar{g}+1) \omega))
\]
which gives
\begin{equation}
\mbox{ch}(\pi_{!}({\cal V}^{*} \otimes \phi^{*}L)) = \sum_{k=1}^{n} (
(3\bar{g}+1) -W_{k} +X_{k} ) e^{-\delta_{k}}. \label{215}
\end{equation}
Applying lemma \ref{Chernlemma} to expression (\ref{215}) gives equation
(\ref{29}). Expression (\ref{210}) is arrived at by calculating $\frac{{\rm
d}}{{\rm d}t} \log ((\Omega(t))^{-3\bar{g}-1}c(\pi_{!}({\cal V}^{*} \otimes
\phi^{*}L))(t))$ and grouping the terms in a similar manner to
expression (\ref{219}). $\indent \Box$
\begin{rem}
Note that $\delta_{k}, W_{k}$ and $X_{k}$ are not
elements of $H^{*}_{\cal G}({\cal C};{\bf Q})$. However the direct sum of the restriction
maps
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow \bigoplus_{\mu \in \Delta} H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})
\]
is injective and so we may consider $\delta_{k},W_{k}$ and $X_{k}$ as
elements of $\bigoplus_{\mu \in \Delta} H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ corresponding
respectively to $a_{1}^{k},d_{k}$ and $\xi_{1,1}^{k,k}$ in each
summand $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q}).$
\end{rem}
\begin{rem}
From (\ref{26}) we can find an expression for
\[
\frac{\Psi'(t)}{\Psi(t)} = \frac{d-n\bar{g}}{t} - \frac{c(\pi_{!}{\cal
V})'(t^{-1})}{t^{2} c(\pi_{!}{\cal V})(t^{-1})}.
\]
In fact we may write $\Psi'(t)/\Psi(t)$ as a rational function with
denominator $(\tilde{\Omega}(t))^{2}$ and a numerator of degree at most $2n-1$.
By multiplying by $\Psi(t)$ and comparing coefficients of
$t^{k} (\tilde{\Omega}(t))^{r}, (r \leq \bar{g}, 0 \leq k <n)$ we may derive
recurrence relations amongst the Mumford relations which determine
$\{\sigma_{r}^{k} : 0 \leq k < n\}$ in terms of
$\{\sigma_{r+1}^{k},\sigma_{r+2}^{k} : 0 \leq k < n \}$. Similar
recurrence relations exist among the dual Mumford relations which
determine $\{\tau_{r}^{k} : 0 \leq k < n\}$ in terms of
$\{\tau_{r+1}^{k},\tau_{r+2}^{k} : 0 \leq k < n \}$.
\end{rem}
\indent The calculation of the restriction of $c(\pi_{!}{\cal V})(t)$ to
$H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})[[t]]$ follows easily from the previous proposition. As in
\cite[prop. 2]{K2} this restriction can be expressed in terms of
elementary functions of the generators of $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ when $\mu \in \Delta$.
However for a general type $\mu$
this restriction cannot be expressed so easily and we will find
formal expressions similar to (\ref{25}) of more use.
\begin{cor}
\label{Chernresn}
Let $\mu= (d_{1}/n_{1},...,d_{P}/n_{P})$. The restriction to $H_{\cal G}^{*}({\cal C}_{\mu};{\bf Q})[[t]]$ of
$c(\pi_{!}{\cal V})(t)$ equals the formal power series
\begin{equation}
\Omega_{\mu}(t)^{-\bar{g}} \prod_{p=1}^{P} \prod_{k=1}^{n_{p}}
(1+\delta_{k}^{p}t)^{W_{k}^{p}} \exp \left\{
\frac{X_{k}^{p}t}{1+\delta_{k}^{p}t} \right\} \label{27}
\end{equation}
and similarly the restriction of $c(\pi_{!}({\cal V}^{*} \otimes
\phi^{*}L))(-t)$ to $H_{\cal G}^{*}({\cal C}_{\mu};{\bf Q})[[t]]$ equals
\begin{equation}
\Omega_{\mu}(t)^{3\bar{g}+1} \prod_{p=1}^{P} \prod_{k=1}^{n_{p}}
(1+\delta_{k}^{p}t)^{-W_{k}^{p}} \exp \left\{
\frac{-X_{k}^{p}t}{1+\delta_{k}^{p}t} \right\}, \label{211}
\end{equation}
where $\delta_{1}^{p},...,\delta_{n_{p}}^{p}$ are formal degree two
classes such that their $r$th elementary symmetric polynomial equals
$a_{r}^{p}$, where $\Omega_{\mu}(t)= \prod_{p=1}^{P} \prod_{k=1}^{n_{p}} (1+
\delta_{k}^{p}t)$ is the restriction of $\Omega(t)$ to $H^{*}_{\cal G}({\cal C}_{\mu};{\bf
Q})[t],$ and where $\xi_{i,j}^{p,p},W_{k}^{p}$ and $X_{k}^{p}$
correspond to the expressions defined in the statement of proposition
\ref{Chernprop}.
\end{cor}
{\bf Proof}
Expression (\ref{27}) is immediate from the previous proposition
once we note that the restriction of $\mbox{ch}(\pi_{!}{\cal V})$ to $H^{*}_{\cal G}
( {\cal
C}_{\mu};{\bf Q})$ equals
\[
\sum_{p=1}^{P} \pi_{*}(\mbox{ch}({\cal V}_{p}) \cdot 1 \otimes (1-\bar{g} \omega))
\]
and recall that the Chern polynomial is
multiplicative. The dual expression (\ref{211}) follows in a similar
fashion. $\Box$
\begin{cor}
\label{Deltaresn}
Let $\mu=(d_{1},...,d_{n}) \in \Delta$. Then the restriction
of $c(\pi_{!}{\cal V})(t)$ to
$H^{*}_{\cal G} ({\cal C}_{\mu};{\bf Q})[[t]]$ equals
\[
\prod_{p=1}^{n} (1+ a_{1}^{p} t)^{d_{p} -\bar{g}} \exp \left\{ \frac{\xi_{1,1}^{p,p}
t}{1 + a_{1}^{p} t} \right\}.
\]
Also the restriction of $c(\pi_{!}({\cal V}^{*}
\otimes \phi^{*}L))(-t)$ to
$H^{*}_{\cal G} ({\cal C}_{\mu};{\bf Q})[[t]]$ equals
\[
\prod_{p=1}^{n} (1+ a_{1}^{p} t)^{3\bar{g}+1-d_{p}} \exp \left\{
\frac{-\xi_{1,1}^{p,p} t}{1 + a_{1}^{p} t} \right\}.
\]
\end{cor}
{\bf Proof}
Simply note that in this case $\delta_{1}^{p} =
a_{1}^{p},W_{1}^{p} = d_{p}$ and $X_{1}^{p}=\xi_{1,1}^{p,p}. \indent
\Box$
\begin{rem}
Let $\mu=(d_{1}/n_{1},...,d_{P}/n_{P})$. From the calculation
(\ref{212}) and since the Chern
character is additive we know that the restriction of $\mbox{ch}({\cal V})$
to $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ equals
\[
\sum_{p=1}^{P} \sum_{k=1}^{n_{p}} \exp \left\{ \delta_{k}^{p} +
\sum_{s=1}^{2g} \left( \sum_{i=1}^{n_{p}} b_{i}^{p,s}\frac{\partial
\delta_{k}^{p}}{\partial a_{i}^{p}} \right) \otimes \alpha_{s} + W_{k}^{p}
\otimes \omega \right\}.
\]
Thus in terms of our earlier notation (\ref{QQ}) we have
\[
a^{p,k} = \delta_{k}^{p}, \indent b^{p,k,s} = \sum_{i=1}^{n_{p}}
b_{i}^{p,s} \frac{\partial
\delta_{k}^{p}}{\partial a_{i}^{p}}, \indent d^{p,k}=W_{k}^{p}.
\]
\end{rem}
\indent We end this section with two further calculations, namely the Chern
polynomials of the normal bundle ${\cal N}_{\mu}$ to the stratum ${\cal C}_{\mu}$ in ${\cal C}$
(necessary to the completeness criteria) and of the tangent bundle $T$
to the moduli space ${\cal M}(n,d)$ (needed for generalising the proof of the Newstead-Ramanan conjecture).
\begin{lem}
\label{normal}
Let $\mu=(d_{1}/n_{1},...,d_{P}/n_{P})$. Then the
Chern polynomial $c({\cal N}_{\mu})(t)$ of the normal bundle in ${\cal
C}$ to the stratum ${\cal C}_{\mu}$ equals
\begin{equation}
{\cal P}_{\mu}(t)^{\bar{g}} \prod_{I<J} \prod_{k=1}^{n_{I}}
\prod_{l=1}^{n_{J}}
(1+(\delta_{l}^{J}-\delta_{k}^{I})t)^{W_{k}^{I}-W_{l}^{J}} \exp
\left\{ \frac{-\Xi_{k,l}^{I,J}t}{1+(\delta_{l}^{J}-\delta_{k}^{I})t}
\right\} \label{216}
\end{equation}
where
\[
\Xi_{k,l}^{I,J}= \sum_{s=1}^{g} \left( \sum_{i=1}^{n_{I}} b_{i}^{I,s}
\frac{\partial \delta_{k}^{I}}{\partial a_{i}^{I}} -\sum_{j=1}^{n_{J}}
b_{j}^{J,s} \frac{\partial \delta_{l}^{J}}{\partial a_{j}^{J}} \right) \left(
\sum_{i=1}^{n_{I}} b_{i}^{I,s+g} \frac{\partial \delta_{k}^{I}}{\partial a_{i}^{I}} -\sum_{j=1}^{n_{J}}
b_{j}^{J,s+g} \frac{\partial \delta_{l}^{J}}{\partial a_{j}^{J}} \right)
\]
and
\[
{\cal P}_{\mu}(t)= \prod_{I<J} \prod_{k=1}^{n_{I}} \prod_{l=1}^{n_{J}}
(1+(\delta_{l}^{J} - \delta_{k}^{I})t).
\]
\end{lem}
{\bf Proof}
Kirwan \cite[lemma 2]{K2} showed that the normal bundle
${\cal N}_{\mu}$ to ${\cal C}_{\mu}$ in ${\cal C}$, equals
\[
-\pi_{!} \left( \bigoplus_{I < J} {\cal V}^{*}_{I} \otimes {\cal
V}_{J} \right).
\]
{}From the proof of the proposition \ref{Chernprop} we can find expressions for
$\mbox{ch}({\cal V}_{J})$ and $\mbox{ch}({\cal V}_{I}^{*})$ corresponding to (\ref{212})
and
(\ref{213}). The GRR implies that
\[
\mbox{ch}({\cal N}_{\mu}) = \sum_{I<J} \pi_{*} ( \mbox{ch}({\cal V}_{I}^{*}) \cdot
\mbox{ch}({\cal V}_{J}) \cdot 1 \otimes (\bar{g} \omega -1)).
\]
Substituting in these expressions for $\mbox{ch}({\cal V}_{J})$ and $\mbox{ch}({\cal
V}_{I}^{*})$ we find that $\mbox{ch}({\cal N}_{\mu})$ equals
\[
\sum_{I<J} \left\{ \sum_{k=1}^{n_{I}} \sum_{l=1}^{n_{J}}
(\bar{g}+W_{k}^{I}-W_{l}^{J}-\Xi_{k,l}^{I,J})e^{\delta_{l}^{J}-\delta_{k}^{I}}
\right\}.
\]
Applying lemma \ref{Chernlemma} produces the required result (\ref{216}). $\indent
\Box$
\begin{lem}
\label{Pont}
The total Pontryagin class of ${\cal M}(n,d)$ equals
\[
\prod_{1 \leq k < l \leq n} (1 + (\delta_{k} - \delta_{l})^{2})^{2\bar{g}}.
\]
In particular the Pontryagin ring of ${\cal M}(n,d)$ is generated by the elementary
symmetric polynomials in
\[
\{ ( \delta_{k} -\delta_{l}) ^{2} : 1 \leq k < l \leq n \}.
\]
\end{lem}
{\bf Proof}
Let $T$ denote the tangent bundle of ${\cal M}(n,d)$. From \cite[p.582]{AB} we
know that
\[
T + T^{*} -2 = \pi_{!}({\rm End} V \otimes (\Omega_{M}^{1}-1)).
\]
Applying GRR we find
\[
\mbox{ch} T + \mbox{ch} T^{*} - 2 = 2 \bar{g} \mbox{ch} ({\rm End V}|{\cal M}(n,d))
\]
which we know to equal
\[
2 \bar{g} \left(\sum_{k=1}^{n} e^{\delta_{k}} \right) \left( \sum_{l=1}^{n}
e^{-\delta_{l}} \right)
\]
from expressions (\ref{212}) and (\ref{213}).\\
\indent Now let $p(T)(t) = \sum_{r \geq 0} p_{r}(T) t^{r}$ denote the
Pontryagin polynomial. The relationship between the Pontryagin classes
and the Chern classes is given by
\[
p(T)(-1) = c(T)(1) \cdot c(T)(-1) \indent \cite[\mbox{Cor. } 15.5]{MS}.
\]
Hence $p(T)(-1)$ equals
\[
\prod_{k \neq l} (1+ \delta_{k} -\delta_{l})^{2\bar{g}} =
\prod_{k<l} (1 - (\delta_{k} - \delta_{l})^{2})^{2\bar{g}}.
\]
The total Pontryagin class of ${\cal M}(n,d)$ then equals $p(T)(1)$ and hence
the result. $\Box$
\section{A Complete Set of Relations.}
Whilst we observed in remark \ref{inadequacy} that neither the Mumford relations nor the
dual
Mumford relations are in themselves a complete set of relations when
the rank is greater than two, it is still
possible to put these relations into the context of the
completeness criteria. In terms of
these criteria we will show how
the Mumford relations contain subsets corresponding to all strata of
the form
\[
\mu= (d_{1}/n_{1},...,d_{P}/n_{P})
\]
where $n_{P}=1.$ Similarly the dual Mumford relations contain subsets
corresponding to all
those strata with $n_{1}=1.$ From this we shall deduce that in the
rank three case the Mumford and dual Mumford relations form a complete
set.\\
\indent Before we continue with the main proposition we need a lemma on the
vanishing of the Mumford and dual Mumford
relations on restriction to a stratum.
\begin{lem}
\label{vanishing}
Let $\mu=(d_{1}/n_{1},...,d_{P}/n_{P})$. The image of
the Mumford relation $\sigma_{r,S}^{k}$ under the restriction map
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\mu};{\bf
Q})
\]
vanishes when $r < d_{P}/n_{P} -2g +1$. The image of the dual Mumford
relation $\tau_{r,S}^{k}$ under the restriction map vanishes when $r <
2\bar{g}-d_{1}/n_{1}.$
\end{lem}
{\bf Proof}
Recall that the Mumford relations are given by
$\sigma_{r,S}^{k} (r<0,0 \leq k \leq n-1,S \subseteq \{1,...,2g\})$
when $\Psi(t)= t^{d-n\bar{g}}c(\pi_{!}{\cal V})(t^{-1})$ is written in the form
\[
\sum_{r=-\infty}^{\bar{g}} (\sigma_{r}^{0} + \sigma_{r}^{1} t + \cdot \cdot
\cdot + \sigma_{r}^{n-1} t^{n-1})(\tilde{\Omega}(t))^{r}, \indent
\sigma_{r}^{k}=\sum_{S \subseteq \{1,...,2g\}} \sigma_{r,S}^{k}
\prod_{s \in S} b_{1}^{s}.
\]
For $1 \leq k \leq n$ and any fixed integer $R$ the power $t^{-k}$ appears in
\[
\sum_{r=-\infty}^{\bar{g}} (\sigma_{r}^{0} + \sigma_{r}^{1} t + \cdot \cdot \cdot +
\sigma_{r}^{n-1} t^{n-1})(\tilde{\Omega}(t))^{r-R-1}
\]
only when $r=R.$ Let $C_{r}^{i}$ denote the
coefficient of $t^{-i}$ in $\Psi(t)(\tilde{\Omega}(t))^{-r-1}$. Then
\[
(\sigma_{r}^{0} + \sigma_{r}^{1} t + \cdot \cdot \cdot +
\sigma_{r}^{n-1} t^{n-1}) = (t^{n} + a_{1} t^{n-1} + \cdot \cdot \cdot
+ a_{n}) \sum_{i=1}^{n} C_{r}^{i} t^{-i}
\]
modulo negative powers of $t$ and hence
\begin{equation}
\sigma_{r}^{n-k} = \sum_{i=1}^{k} a_{k-i} C_{r}^{i} \indent (r<0,1
\leq k \leq n) \label{31}.
\end{equation}
\indent Now let $K$ be a fixed line bundle over $M$ of degree $D$ where $D$ is
the smallest integer such that
\[
\mu(Q_{P} \otimes K) = \frac{d_{P}}{n_{P}} + D > 2\bar{g}
\]
where $Q_{P}=E_{P}/E_{P-1}.$ Since $\mu(Q_{p} \otimes K) \geq
\mu(Q_{P} \otimes K) > 2\bar{g}$ then $\pi_{!}({\cal V}_{p} \otimes
\phi^{*}K)$ is a bundle over ${\cal C}(n_{p},d_{p})^{ss}$ of rank
$d_{p} +(D-\bar{g})n_{p}$ for each $1 \leq p \leq P.$ In particular
\[
\Psi(\pi_{!}({\cal V}_{p} \otimes \phi^{*}K))(t) =
t^{d_{p}+n_{p}(D-\bar{g})} c (\pi_{!}({\cal V}_{p} \otimes \phi^{*}K))(t^{-1})
\]
is a polynomial modulo relations in $H^{*}_{{\cal G}(n_{p},d_{p})}
({\cal C}(n_{p},d_{p})^{ss};{\bf Q})$. From GRR we have that
$\mbox{ch}(\pi_{!}({\cal V}_{p} \otimes \phi^{*}K))$ equals
\begin{equation}
\mbox{ch}(\pi_{!}{\cal V}_{p}) + \pi_{*}(\mbox{ch} {\cal V}_{p} \cdot 1 \otimes
D\omega) = \mbox{ch}(\pi_{!}{\cal V}_{p}) +D \sum_{k=1}^{n_{p}}
e^{\delta_{k}^{p}}. \label{32}
\end{equation}
In terms of Chern polynomials (\ref{32}) gives
\[
c(\pi_{!}({\cal V}_{p} \otimes \phi^{*}K))(t) =
(\Omega_{p}(t))^{D} c(\pi_{!}{\cal V}_{p})(t)
\]
where $\Omega_{p}(t) = \prod_{k=1}^{n_{p}} ( 1 + \delta_{k}^{p}t)$. Hence
\begin{equation}
\prod_{p=1}^{P} \Psi(\pi_{!}({\cal V}_{p} \otimes \phi^{*}K))(t)
= (\tilde{\Omega}_{\mu}(t))^{D} \Psi_{\mu}(t) \label{51}
\end{equation}
is a polynomial modulo relations in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ where
$\Psi_{\mu}(t),$ and $\tilde{\Omega}_{\mu}(t)$ are respectively the restrictions
to $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ of $\Psi(t)$ and $\tilde{\Omega}(t)$. Thus the coefficient of $
t^{-k}$ in
$\Psi_{\mu}(t)\tilde{\Omega}_{\mu}(t)^{-r-1}$ is a relation
when $r \leq -1 -D$. So by (\ref{31}) the restriction of $\sigma_{r}^{k}$ to
$H^{*}_{\cal
G}({\cal C}_{\mu};{\bf Q})$ vanishes when $r \leq d_{P}/n_{P} - 2g.$ The dual
calculation follows by a similar argument. $\indent \Box$\\[\baselineskip]
\indent Thus finally we come to
\begin{prop}
\label{biggy}
Let $\mu = (d_{1}/n_{1},...,d_{P}/n_{P})$ with
$n_{P}=1$. Then there is a subset ${\cal R}_{\mu}$ of the ideal
generated by the Mumford relations such that the image of the ideal
generated by ${\cal R}_{\mu}$ under the restriction map
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\nu};{\bf Q}) \indent
\nu=(\tilde{d}_{1}/\tilde{n}_{1},...,\tilde{d}_{T}/\tilde{n}_{T})
\]
is zero when either
\[
\mbox{(i) } \tilde{d}_{T}/\tilde{n}_{T}> d_{P} \indent \mbox{or}
\indent \mbox{(ii) } \tilde{n}_{T}=1, \tilde{d}_{T} = d_{P}, \mbox{
and } \nu \not \geq \mu
\]
and contains the ideal of $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ generated by $e_{\mu}$ when
$\nu=\mu.$\\
\indent Let $\mu = (d_{1}/n_{1},...,d_{P}/n_{P})$ with
$n_{1}=1$. Then there is a subset ${\cal R}_{\mu}$ of the ideal
generated by the dual Mumford relations such that the image of the ideal
generated by ${\cal R}_{\mu}$ under the restriction map
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\nu};{\bf Q}) \indent
\nu=(\tilde{d}_{1}/\tilde{n}_{1},...,\tilde{d}_{T}/\tilde{n}_{T})
\]
is zero when either
\[
\mbox{(i) } \tilde{d}_{1}/\tilde{n}_{1}< d_{1}/n_{1} \indent \mbox{or}
\indent \mbox{(ii) } \tilde{n}_{1}=1, \tilde{d}_{1} = d_{1} \mbox{
and } \nu \not \geq \mu
\]
and contains the ideal of $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ generated by $e_{\mu}$ when
$\nu=\mu.$
\end{prop}
{\bf Proof}
Let $\Psi(t)= t^{d-n\bar{g}}c(\pi_{!}{\cal V})(t^{-1})$ and
let $C^{R}_{K}, (R<0,1 \leq K \leq n)$ denote the coefficient of
$t^{-K}$ in $\Psi(t)(\tilde{\Omega}(t))^{-R-1}$. Let
\[
\mu=(d_{1}/n_{1},...,d_{P-1}/n_{P-1},d_{P})
\]
so that $n_{P}=1.$\\
\indent Since the Chern polynomial is multiplicative the restriction
in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ of $C_{R}^{K}$, which we will write
as $C_{R}^{K,\mu}$, equals the coefficient of $t^{-1}$ in
\begin{equation}
t^{K-1} \prod_{p=1}^{P} \Psi_{p}(t)(\tilde{\Omega}_{p}(t))^{-R-1} \label{50}
\end{equation}
where
\[
\Psi_{p}(t)= t^{d_{p}-n_{p}\bar{g}}c(\pi_{!}{\cal V}_{p})(t^{-1}), \indent
\tilde{\Omega}_{p}(t)=t^{n_{p}}+a_{1}^{p} t^{n_{p}-1}+ \cdot \cdot \cdot +
a_{n_{p}}^{p}
\]
for $1 \leq p \leq P$. Further from the previous lemma we know that
$C_{R}^{K,\mu}$ vanishes when $R<-D=d_{P}-2g+1.$\\
\indent We facilitate the proof of proposition \ref{biggy} with the following
lemma and corollaries
\begin{lem}
Let $\theta(t)$ equal
\begin{equation}
t^{d-nd_{P}+(n-1)\bar{g}} \prod_{p=1}^{P-1} \prod_{k=1}^{n_{p}} ( 1
+(\delta_{k}^{p}-a_{1}^{P})/t)^{W_{k}^{p}+\bar{g}-d_{P}} \exp \left\{
\frac{\Xi_{k,1}^{p,P}}{t+\delta_{k}^{p}-a_{1}^{P}} \right\}. \label{52}
\end{equation}
Then modulo relations in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$,
\[
C_{-D}^{K,\mu}= (-a_{1}^{P})^{K-1} (\xi_{1,1}^{P,P})^{g} \Theta
\]
where $\Theta$ is the constant coefficient of $\theta(t).$
\end{lem}
{\bf Proof}
From corollary \ref{Deltaresn} we know that
\[
\Psi_{P}(t) (\tilde{\Omega}_{P}(t))^{D-1} = (t+a_{1}^{P})^{\bar{g}} \exp \left\{
\frac{ \xi_{1,1}^{P,P}}{t+a_{1}^{P}} \right\}
\]
where $\xi_{1,1}^{P,P} = \sum_{s=1}^{g} b_{1}^{P,s} b_{1}^{P,s+g}.$
Also in a Laurent series the coefficient of $t^{-1}$ is invariant under
transformations such as $t \mapsto t-a_{1}^{P}.$ So from (\ref{50})
$C_{-D}^{K,\mu}$ equals the coefficient of $t^{-1}$ in
\begin{equation}
(t-a_{1}^{P})^{K-1} t^{\bar{g}} \exp (\xi_{1,1}^{P,P}/t) \prod_{p=1}^{P-1}
\Psi_{p}(t-a_{1}^{P})(\tilde{\Omega}_{p}(t-a_{1}^{P}))^{D-1}. \label{D}
\end{equation}
\indent From the proof of lemma \ref{vanishing} (\ref{51}) we know that
\[
\Psi_{p}(t)(\tilde{\Omega}_{p}(t))^{D-1} = \Psi(\pi_{!}({\cal V}_{p} \otimes
\phi^{*}{\cal L}))(t)
\]
where ${\cal L}$ is a fixed line bundle over $M$ of degree $D-1.$ For each
$p \neq P$, $Q_{p} \otimes {\cal L}$ is a semistable bundle of
slope
\[
\frac{d_{p}}{n_{p}} - d_{P} + 2\bar{g} > 2\bar{g}.
\]
Hence $\pi_{!}({\cal
V}_{p} \otimes \phi^{*}{\cal L})$ is a bundle over
${\cal C}(n_{p},d_{p})^{ss}$ and $\Psi_{p}(t)(\tilde{\Omega}_{p}(t))^{D-1}$ is a polynomial
modulo relations in
$H^{*}_{{\cal G} (n_{p},d_{p})}({\cal C}(n_{p},d_{p})^{ss};{\bf Q}).$ As
$(\xi_{1,1}^{P.P})^{g+1} = 0$ it follows from (\ref{D}) that
$C_{-D}^{K,\mu}$ equals the constant coefficient of
\begin{equation}
(\xi_{1,1}^{P,P})^{g} (t-a_{1}^{P})^{K-1} \prod_{p=1}^{P-1}
\Psi_{p}(t-a_{1}^{P}) (\tilde{\Omega}_{p}(t-a_{1}^{P}))^{D-1} \label{E}
\end{equation}
modulo relations in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$.\\
\indent Since $\sum_{k=1}^{n_{p}}
W_{k}^{p} =d_{p}$ then we know from corollary \ref{Chernresn} that $\Psi_{p}(t-a_{1}^{P})$
equals
\[
(\tilde{\Omega}_{p}(t-a_{1}^{P}))^{-\bar{g}} t^{d_{p}} \prod_{k=1}^{n_{p}} ( 1 +(
\delta_{k}^{p} -a_{1}^{P})/t)^{W_{k}^{p}} \exp \left\{
\frac{X_{k}^{p}}{t+\delta_{k}^{p} -a_{1}^{P}} \right\}.
\]
Recall from lemma \ref{normal} that
\[
\Xi_{k,1}^{p,P} = \sum_{s=1}^{g} \left( \sum_{i=1}^{n_{p}} b_{i}^{p,s}
\frac{\partial \delta_{k}^{p}}{\partial a_{i}^{p}} - b_{1}^{P,s}
\right) \left( \sum_{i=1}^{n_{p}} b_{i}^{p,s+g}
\frac{\partial \delta_{k}^{p}}{\partial a_{i}^{p}} - b_{1}^{P,s+g}
\right)
\]
and we also have that
\[
X_{k}^{p} = \sum_{s=1}^{g} \left( \sum_{i=1}^{n_{p}} b_{i}^{p,s}
\frac{\partial \delta_{k}^{p}}{\partial a_{i}^{p}}
\right) \left( \sum_{i=1}^{n_{p}} b_{i}^{p,s+g}
\frac{\partial \delta_{k}^{p}}{\partial a_{i}^{p}}
\right).
\]
Since
\[
(\xi_{1,1}^{P,P})^{g} = (-1)^{g\bar{g}/2} g! \prod_{s=1}^{2g}
b_{1}^{P,s}
\]
then
\[
(\xi_{1,1}^{P,P})^{g}(\Xi_{k,1}^{p,P})^{q} =
(\xi_{1,1}^{P,P})^{g}(X_{k}^{p})^{q} \indent (q \geq 0).
\]
Thus by (\ref{E}) and the identity $\tilde{\Omega}_{p}(t-a_{1}^{P}) =
t^{n_{p}}\prod_{k=1}^{n_{p}}(1+(\delta_{k}^{p}-a_{1}^{P})/t),$ we have
that $C_{-D}^{K,\mu}$ equals the constant coefficient of
\[
(\xi_{1,1}^{P,P})^{g} (t-a_{1}^{P})^{K-1} \theta(t).
\]
Since $(\xi_{1,1}^{P,P})^{g}\theta(t)$ is a polynomial modulo
relations in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ then the lemma follows. $\indent
\Box.$
\begin{cor}
Define $C_{R,S}^{K} (R<0,1 \leq K \leq n, S \subseteq
\{1,...,2g\})$ by
\[
C_{R}^{K} = \sum_{S \subseteq \{1,...,2g\}} C_{R,S}^{K} \prod_{s \in
S} b_{1}^{s}
\]
writing $C_{R,S}^{K}$ in terms of the elements (\ref{1000})
and also define $\tilde{a}_{r},\tilde{b}_{r}^{s}$ and $\tilde{f}_{r}$ by
\[
c_{r}(\bigoplus_{p=1}^{P-1} {\cal V}_{p}) = \tilde{a}_{r} \otimes 1 +
\sum_{s=1}^{2g} \tilde{b}_{r}^{s} \otimes \alpha_{s} + \tilde{f}_{r} \otimes \omega.
\]
Then the restriction of $C_{-D,S}^{K}$ to $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ equals a
non-zero constant multiple of
\begin{equation}
(a_{1}^{P})^{K-1} \prod_{s \not \in S} (\tilde{b}_{1}^{s}-(n-1) b_{1}^{P,s})
\Theta \label{M}
\end{equation}
for any subset $S \subseteq \{1,...,2g\}.$
\end{cor}
{\bf Proof}
We know that $(\xi_{1,1}^{P,P})^{g}$ equals
\[
(-1)^{g\bar{g}/2} g! \prod_{s=1}^{2g} b_{1}^{P,s} = (-1)^{g\bar{g}/2}
n^{-2g} g! \prod_{s=1}^{2g} ((\tilde{b}_{1}^{s} +
b_{1}^{P,s})-(\tilde{b}_{1}^{s}-(n-1) b_{1}^{P,s}))
\]
and also that the restriction of $b_{1}^{s}$ in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ equals
$\tilde{b}_{1}^{s}+ b_{1}^{P,s}.$ Further
\[
\tilde{b}_{1}^{s} - (n-1)b_{1}^{P,s} = \sum_{p=1}^{P-1} \sum_{k=1}^{n_{p}} \left(
\sum_{i=1}^{n_{p}} b_{1}^{p,s} \frac{\partial \delta_{k}^{p}}{\partial
a_{i}^{p}} - b_{1}^{P,s} \right).
\]
So the corollary follows once we note from
(\ref{52}) that $\theta(t)$, and hence $\Theta$, can be written in
terms of the elements (\ref{NEW1}). $\indent \Box$.
\begin{cor}
Let $\Lambda$ equal
\begin{equation}
\bigcup \{ \sigma_{-D,S}^{n-1},...,\sigma_{-D,S}^{0} \} \label{54}
\end{equation}
where the union varies over all subsets $S \subseteq \{1,...,2g\}$.
Then all elements of the form
\begin{equation}
\prod_{k=2}^{n-1} (\tilde{f}_{k})^{m_{k}} \prod_{k=1}^{n-1} \prod_{s \in
S_{k}} \tilde{b}_{k}^{s} \prod_{k=1}^{n-1} (\tilde{a}_{k})^{r_{k}} (a_{1}^{P})^{r}
\prod_{s \in S} b_{1}^{P,s} \Theta \label{81}
\end{equation}
lie in the restriction of the ideal generated by $\Lambda$, where
$r,r_{1},...,r_{n-1},m_{2},...,m_{n-1}$ are arbitrary non-negative
integers and $S,S_{1},...,S_{n-1}$ are subsets of
$\{1,...,2g\}$.
\end{cor}
{\bf Proof}
Let $(\Lambda)$ denote the ideal of $H^{*}_{\cal G}({\cal C};{\bf Q})$
generated by $\Lambda$. Using induction on (\ref{31}) we know that the
restriction of $C_{-D,S}^{K}$ lies in the image of
$(\Lambda)$. From (\ref{M}) and since $b_{1}^{s}$ restricts to $\tilde{b}_{1}^{s}
+ b_{1}^{P,s}$ it follows that all elements of the form
\[
(a_{1}^{P})^{K-1} \prod_{s \in S_{1}} \tilde{b}_{1}^{s} \prod_{s \in S_{2}}
b_{1}^{P,s} \Theta
\]
for arbitrary $S_{1},S_{2} \subseteq \{1,...,2g\}$ and $1 \leq K \leq
n$, lie in the restriction of $(\Lambda).$ The restriction of
$a_{k}$ in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ equals
$\tilde{a}_{k}+\tilde{a}_{k-1}a_{1}^{P}$. By noting that $(a_{1}^{P})^{r}$ equals
\[
(\tilde{a}_{1}+a_{1}^{P})(a_{1}^{P})^{r-1} -(\tilde{a}_{2} +
\tilde{a}_{1} a_{1}^{P})(a_{1}^{P})^{r-2} + \cdot \cdot \cdot +
(-1)^{n-1}(\tilde{a}_{n-1}a_{1}^{P})(a_{1}^{P})^{r-n}
\]
for $r \geq n$ we see that all elements of the form
\[
(a_{1}^{P})^{r} \prod_{s \in S_{1}} \tilde{b}_{1}^{s} \prod_{s \in S_{2}}
b_{1}^{P,s} \cdot \Theta \indent (r \geq 0)
\]
lie in the restriction of $(\Lambda)$. Finally working inductively on
the variables $r_{1},...,r_{n-1},$ $S_{2},S_{3},...,S_{n-1}$ and
$m_{2},m_{3},...m_{n-1}$ in that order we find that all elements of
the form (\ref{81}) lie in the image of $(\Lambda)$ since under the
restriction map $H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$
\[
a_{k} \mapsto \tilde{a}_{k}+\tilde{a}_{k-1}a_{1}^{P} \indent b_{k}^{s} \mapsto
\tilde{b}_{k}^{s}+ a_{1}^{P} \tilde{b}_{k-1}^{s} + \tilde{a}_{k-1}b_{1}^{P,s}
\]
and
\begin{equation}
f_{k} \mapsto \tilde{f}_{k} + d_{P}\tilde{a}_{k-1} + a_{1}^{P} \tilde{f}_{k-1} +
\sum_{s=1}^{g} ( \tilde{b}_{k-1}^{s} b_{1}^{P,s+g} + b_{1}^{P,s}
\tilde{b}_{k-1}^{s+g}). \quad \Box \label{60}
\end{equation}
\indent We now continue with the proof of proposition \ref{biggy}. Let
${\cal C}'={\cal C}(n-1,d-d_{P})$ and let ${\cal G}'= {\cal G}(n-1,d-d_{P}).$
Let $\mu' = (d_{1}/n_{1},...,d_{P-1}/n_{P-1})$ and let $e_{\mu'}$ denote the
equivariant Euler class of the normal bundle to ${\cal C}'_{\mu'}$ in ${\cal C}'.$ Let
\[
U_{\mu'} = {\cal C}' - \bigcup_{\nu'>\mu'} {\cal C}'_{\nu'}.
\]
Then $U_{\mu'}$ is an open subset of ${\cal C}'$ which contains
${\cal C}'_{\mu'}$ as a closed submanifold. So we have the maps\\
\begin{picture}(400,120)
\put(190,60){\makebox(0,0){$H^{*-2d_{\mu'}}_{{\cal G}'}({\cal C}'_{\mu'};{\bf Q}) \rightarrow
H^{*}_{{\cal G}'}(U_{\mu'};{\bf Q}) \rightarrow
H^{*}_{{\cal G}'}(U_{\mu'}-{\cal C}'_{\mu'};{\bf Q})$}}
\put(190,110){\makebox(0,0){$H^{*}_{{\cal G}'}({\cal C}';{\bf Q})$}}
\put(190,10){\makebox(0,0){$H^{*}_{{\cal G}'}({\cal C}'_{\mu'};{\bf Q})$}}
\put(190,102){\vector(0,-1){34}}
\put(190,50){\vector(0,-1){32}}
\put(100,50){\vector(2,-1){65}}
\put(150,30){\makebox(0,0)[tr]{multiplication by $e_{\mu'}$}}
\end{picture}
\\
Let $a'_{r},{b^{s}_{r}}'$ and $f'_{r}$ denote the generators of
$H^{*}_{{\cal G}'}({\cal C}';{\bf Q})$. Also take $\nu' \not \geq \mu'$ and let
$\hat{a}_{r},\hat{b}_{r}^{s},\hat{f}_{r}$ denote the restrictions of
$a'_{r},{b^{s}_{r}}',f'_{r}$ in $H^{*}_{{\cal G}'}({\cal C}'_{\nu'};{\bf Q})$. Since the
stratification is equivariantly perfect then the restriction map
\[
H^{*}_{{\cal G}'}({\cal C}';{\bf Q}) \rightarrow H^{*}_{{\cal G}'}(U_{\mu'};{\bf Q})
\]
is surjective \cite[p.859]{K2}. From the exactness of the Thom-Gysin
sequence we have that for every element of
the form $\alpha e_{\mu'}$ in $H^{*}_{{\cal G}'}({\cal C}'_{\mu'};{\bf Q})e_{\mu'}$ there is
some $\beta
(a'_{r},{b_{r}^{s}}' ,f'_{r})$ in $H^{*}_{{\cal G}'}({\cal C}';{\bf Q})$ such that
\[
\beta(\tilde{a}_{r},\tilde{b}_{r}^{s},\tilde{f}_{r}) = \alpha e_{\mu'} \mbox{ and }
\beta(\hat{a}_{r},\hat{b}_{r}^{s},\hat{f}_{r}) = 0.
\]
Since every element of the form (\ref{81}) lies in the restriction of
$(\Lambda)$ to $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ then every element of the form
\begin{equation}
\beta(\tilde{a}_{r},\tilde{b}_{r}^{s},\tilde{f}_{r}) (a_{1}^{P})^{r} \prod_{s \in S}
b_{1}^{P,s} \Theta \indent (r \geq 0,S \subseteq \{1,...,2g\}) \label{z1}
\end{equation}
similarly lies in the restriction of $(\Lambda)$. Now let
$\nu=(\nu',d_{P})$ with $\nu' \not \geq \mu'$. Note that the
restriction map
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\nu};{\bf Q})
\]
is formally the same as (\ref{60}) but with
$\hat{a}_{r},\hat{b}_{r}^{s},\hat{f}_{r}$ replacing
$\tilde{a}_{r},\tilde{b}_{r}^{s},\tilde{f}_{r}$. Thus there are elements of $(\Lambda)$
which restrict to (\ref{z1}) under (\ref{60}) and have restriction
\[
\beta(\hat{a}_{r},\hat{b}_{r}^{s},\hat{f}_{r}) (a_{1}^{P})^{r} \prod_{s \in S}
b_{1}^{P,s} \hat{\Theta} = 0
\]
in $H^{*}_{\cal G}({\cal C}_{\nu};{\bf Q})$.\\[\baselineskip]
\indent Define ${\cal R}_{\mu}$ to be all those elements
of $(\Lambda)$ which restrict to an element of the form
\[
\alpha e_{\mu'} (a_{1}^{P})^{r} \prod_{s \in S} b_{1}^{P,s} \Theta
\indent (r \geq 0, S \subseteq \{1,...,2g\}, \alpha \in
H^{*}_{{\cal G}'}({\cal C}'_{\mu'};{\bf Q}))
\]
in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ and which restrict to zero in $H^{*}_{\cal G}({\cal C}_{\nu};{\bf Q})$ for any
$\nu = (\nu',d_{P})$ with $\nu' \not \geq \mu'$.\\[\baselineskip]
\indent From the definition of $\Theta$ (\ref{52}) we know that
$e_{\mu'} \Theta$ is the constant coefficient of
\begin{equation}
(-1)^{d_{\mu'}} t^{d_{\mu'}} c({\cal N}_{\mu'})(-t^{-1}) \theta(t) \label{53}
\end{equation}
where ${\cal N}_{\mu'}$ is the normal bundle to ${\cal C}'_{\mu'}$ in
${\cal C}'$ and $d_{\mu'}$ is the codimension of ${\cal C}'_{\mu'}$ in
${\cal C}'$. From lemma \ref{normal} and the fact that
\[
d_{\mu'} + d - nd_{P} + (n-1)\bar{g} = d_{\mu}
\]
we know (\ref{53}) equals
\[
(-1)^{d_{\mu'}} t^{d_{\mu}} c({\cal N}_{\mu})(-t^{-1})
\]
which has constant coefficient $(-1)^{d_{\mu'}+d_{\mu}}e_{\mu}.$ Hence the
ideal
\[
H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})e_{\mu}
\]
lies in the restriction of ${\cal R}_{\mu}$ to
$H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q}).$\\
\indent Finally from lemma \ref{vanishing} and the definition of $\Lambda$ (\ref{54})
we know that the image of ${\cal R}_{\mu}$ under the restriction map
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}_{\nu};{\bf Q}) \indent \nu =
(\tilde{d}_{1}/\tilde{n}_{1},...,\tilde{d}_{T}/\tilde{n}_{T})
\]
vanishes when $\tilde{d}_{T}/\tilde{n}_{T} > d_{P}/n_{P}$ proving the
first half of proposition \ref{biggy}.\\
\indent The proof of the dual case follows in a similar fashion.$
\indent \Box$\\[\baselineskip]
\indent In the general rank case there are strata of types not covered in the
previous proposition. Moreover the strata on which the restrictions of
the relations have been demonstrated to vanish do not generally coincide with
the strata mentioned
in the hypotheses of the completeness criteria. However in the
rank two and rank three cases all unstable strata are covered by the above
proposition. In the rank two case proposition \ref{biggy} shows that the Mumford
relations and the dual Mumford relations both form complete sets,
simply duplicating Kirwan's work \cite{K2} and remark \ref{dualise}. In the
rank three case we have the following:\\[\baselineskip]
{\bf THEOREM 1.} {\em The Mumford and dual Mumford relations together with the
relation (\ref{NORM}) due to the normalisation of the
universal bundle $V$ form a complete set of relations for $H^{*}({\cal
M}(3,d);{\bf Q}).$}
{\bf Proof}
The unstable strata are now of types (2,1),(1,1,1) and
(1,2). From the previous proposition we may meet the completeness
criteria for the (2,1) and (1,1,1) strata using the Mumford relations.
In these cases those strata where the
restriction of ${\cal R}_{\mu}$ have been shown to vanish are those strata
${\cal C}_{\nu}$ such that $\nu \prec \mu$. The criteria for the (1,2) types may be met
using
the dual Mumford relations. In this case those strata where the
restriction of ${\cal R}_{\mu}$ vanishes (according to proposition \ref{biggy}) are those strata ${\cal C}_{\nu}$ such that $\nu \not \geq \mu$ which certainly
includes those strata such that $\nu \prec \mu.
\indent \Box$
\begin{rem}
As remarked earlier it was shown in \cite[thm.4]{E} that the Mumford
relations $\sigma_{-1,S}^{1}$ for $S \subseteq \{1,...,2g\}$ generate
the relation ideal of $H^{*}({\cal M}_{0}(2,1);{\bf Q})$ as a
${\bf Q}[a_{2},f_{2}]$-module. Evidence for this theorem appears in the
Poincar\'{e} polynomial of the relation ideal which equals \cite[p.593]{AB}
\[
\frac{t^{2g}(1+t)^{2g}}{(1-t^{2})(1-t^{4})}.
\]
\indent Similarly in the rank three case the Poincar\'{e} polynomial
of the ideal of relations among our generators for $H^{*}({\cal
M}_{0}(3,1);{\bf Q})$ equals
\[
\frac{(1+t^{2})^{2} t^{4g-2}(1+t)^{2g}(1+t^{3})^{2g} - (1+t^{2}+t^{4})
t^{6g-2}(1+t)^{4g}}{(1-t^{2})(1-t^{4})^{2}(1-t^{6})},
\]
The first Mumford relation $\sigma_{-1,\{1,...,2g\}}^{2}$ has degree $4g-2$
and the first dual Mumford relation $\tau_{-1,\{1,...,2g\}]}^{2}$ has degree
$4g$. This strongly suggests that the relations
\[
\{ \sigma_{-1,S}^{i}, \tau_{-1,S}^{i} : i=1,2, S \subseteq \{1,...,2g\} \}
\]
generate the relation ideal of $H^{*}({\cal N}(3,d);{\bf Q})$ as a
\[
{\bf Q}[a_{2},a_{3},f_{2},f_{3}] \otimes
\Lambda^{*}\{b_{2}^{1},...,b_{2}^{2g}\}
\]
module.
\end{rem}
\section{On the Vanishing of the Pontryagin Ring.}
\indent We now move on to discuss the Pontryagin ring of the moduli
space in the rank three case. For each $S \subseteq \{1,...,2g\}$ we
define $\Psi_{S}(t)$ and $\Psi^{*}_{S}(t)$ by writing
\[
\Psi(t) = \sum_{S \subseteq \{1,...,2g\}} \Psi_{S}(t) \prod_{s \in S}
b_{1}^{s}, \indent \Psi^{*}(t) = \sum_{S \subseteq \{1,...,2g\}}
\Psi^{*}_{S}(t) \prod_{s \in S} b_{1}^{s}.
\]
Kirwan proved the Newstead-Ramanan conjecture \cite[$\S$ 4]{K2} by considering
relations derived from the expression
\[
\Psi_{\{1,...,2g\}}(t)\Psi_{\{1,...,2g\}}(-t-a_{1}).\]
Arguing along similar lines but now considering the expression
\[
\Phi(t) = \Psi_{\{1,...,2g\}}(t)\Psi^{*}_{\{1,...,2g\}}(t)
\]
we will show that in the rank three case the Pontryagin ring vanishes
in degree $12g-8$ and above -- theorem 2 below.
\begin{lem}
\label{Pontlemma}
Let $\mu = (d_{1}, d_{2},...,d_{n}) \in \Delta$. The
restriction of $\Phi(t)$ to $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ equals
\[
(-1)^{g} \frac{A(t)^{2g}}{n^{4g}\tilde{\Omega}_{\mu}(t)}
\]
where
\[
\tilde{\Omega}_{\mu}(t) = \prod_{p=1}^{n} (t+a_{1}^{p}), \indent A(t) =
\sum_{p=1}^{n} \prod_{q \neq p} (t+a_{1}^{q}).
\]
\end{lem}
{\bf Proof}
From corollary \ref{Deltaresn} we know that the restriction of
$\Psi(t)$ to $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ equals
\[
\prod_{p=1}^{n} (t+a_{1}^{p})^{d_{p}-\bar{g}} \exp \left\{
\frac{\xi_{p}}{t+a_{1}^{p}} \right\}
\]
where $\xi_{p} = \xi_{1,1}^{p,p} = \sum_{s=1}^{g} b_{1}^{p,s} b_{1}^{p,s+g}.$
Let $v_{s} = b_{1}^{1,s} + \cdot
\cdot \cdot + b_{1}^{n,s}$ denote the restriction of $b_{1}^{s}$ to
$H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ and let $w_{i,j}^{s} = b_{1}^{i,s} - b_{1}^{j,s}$ (see
(\ref{1001})). Then
$nb_{1}^{i,s} = v_{s} + \sum_{j=1}^{n} w_{i,j}^{s}$ and hence
\[
n^{2} \xi_{i} = \sum_{s=1}^{g} v_{s}v_{s+g} + \sum_{s=1}^{g} \left(
v_{s} \sum_{j=1}^{n} w_{i,j}^{s+g} + \sum_{j=1}^{n} w_{i,j}^{s}
v_{s+g} \right) + \sum_{s=1}^{g} \sum_{j=1}^{n} \sum_{k=1}^{n} w_{i,j}^{s}
w_{i,k}^{s+g}.
\]
Note that
\begin{equation}
\sum_{p=1}^{n} \frac{\xi_{p}}{t+a_{1}^{p}} = \frac{1}{\tilde{\Omega}_{\mu}(t)}
\sum_{i=1}^{n} \sum_{q \neq i} \xi_{i} (t+a_{1}^{q}) \label{N}
\end{equation} Thus (\ref{N}) equals
\[
\frac{1}{n^{2} \tilde{\Omega}_{\mu}(t)} \left\{ A(t) \sum_{s=1}^{g} v_{s}v_{s+g}
+ \sum_{s=1}^{g} (B_{s}(t) v_{s+g} + v_{s}B_{s+g}(t)) + \Gamma(t)
\right\}
\]
where
\[
A(t) = \sum_{i=1}^{n} \prod_{q \neq i} (t+a_{1}^{q}) , \indent B_{s}(t)
= \sum_{i=1}^{n} \sum_{j=1}^{n} w_{i,j}^{s} \prod_{q \neq i}
(t+a_{1}^{q}),
\]
\[
\Gamma(t) = \sum_{i=1}^{n} \sum_{j=1}^{n}
\sum_{k=1}^{n} \sum_{s=1}^{g} w_{i,j}^{s} w_{i,k}^{s+g} \prod_{q \neq i}(t+a_{1}^{q}).
\]
The exponential of (\ref{N}) equals
\[
\exp \left\{ \frac{\Gamma(t)}{n^{2}\tilde{\Omega}_{\mu}(t)} \right\}
\prod_{s=1}^{g} \left[ 1+ \frac{B_{s}(t)v_{s+g} + v_{s}
B_{s+g}}{n^{2}\tilde{\Omega}_{\mu}(t)} + \left( \frac{A(t)}{n^{2}\tilde{\Omega}_{\mu}(t)} -
\frac{B_{s}B_{s+g}}{n^{4}\tilde{\Omega}_{\mu}(t)^{2}}\right) v_{s}v_{s+g} \right].
\]
The coefficient of $\prod_{s=1}^{2g} v_{s}$ in the above then equals
\[
(-1)^{g\bar{g}/2} \exp \left\{ \frac{\Gamma(t)}{n^{2}\tilde{\Omega}_{\mu}(t)} \right\}
\prod_{s=1}^{g} \left( \frac{A(t)}{n^{2}\tilde{\Omega}_{\mu}(t)} -
\frac{B_{s}B_{s+g}}{n^{4}\tilde{\Omega}_{\mu}(t)^{2}} \right)
\]
or equivalently
\[
(-1)^{g\bar{g}/2} \exp \left\{ \frac{\Gamma(t)}{n^{2}\tilde{\Omega}_{\mu}(t)} \right\}
\left( \frac{A(t)}{n^{2}\tilde{\Omega}_{\mu}(t)} \right)^{g}
\exp\left\{\frac{-\xi(t)}{n^{2}A(t)\tilde{\Omega}_{\mu}(t)} \right\}
\]
where $\xi(t) = \sum_{s=1}^{g} B_{s}(t)B_{s+g}(t)$. Thus the
restriction of $\Psi_{\{1,...,2g\}}(t)$ to $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ equals
\[
(-1)^{g\bar{g}/2}\left(\prod_{p=1}^{n} (t+a_{1}^{p})^{d_{p}-\bar{g}} \right)
\exp \left\{ \frac{\Gamma(t)}{n^{2}\tilde{\Omega}_{\mu}(t)} \right\}
\left( \frac{A(t)}{n^{2}\tilde{\Omega}_{\mu}(t)} \right)^{g}
\exp\left\{\frac{-\xi(t)}{n^{2}A(t)\tilde{\Omega}_{\mu}(t)} \right\}
\]
and similarly the restriction of $\Psi^{*}_{\{1,...,2g\}}(t)$ to
$H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ equals
\[
(-1)^{g\bar{g}/2} \left(\prod_{p=1}^{n} (t+a_{1}^{p})^{3\bar{g}+1-d_{p}} \right)
\exp \left\{ \frac{- \Gamma(t)}{n^{2}\tilde{\Omega}_{\mu}(t)} \right\}
\left(\frac{-A(t)}{n^{2}\tilde{\Omega}_{\mu}(t)} \right)^{g}
\exp\left\{\frac{\xi(t)}{n^{2}A(t)\tilde{\Omega}_{\mu}(t)} \right\}.
\]
The result then follows. $\indent \Box$\\[\baselineskip]
\indent Now if we write $\Phi(t)$ in the form
\[
\sum_{r = -\infty}^{2g-1} ( \rho_{r}^{0} + \rho_{r}^{1} t + \cdot
\cdot \cdot + \rho_{r}^{n-1} t^{n-1}) ( \tilde{\Omega}(t))^{r}
\]
where $\tilde{\Omega}(t) = t^{n} + a_{1}t^{n-1} + \cdot \cdot \cdot +a_{n}$ then
we know that the elements $\rho_{r}^{k},(r < 0 ,0 \leq k \leq n-1)$
lie in the kernel of the restriction map
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow H^{*}_{\cal G}({\cal C}^{ss};{\bf Q}).
\]
\indent From lemma \ref{Pontlemma} we know that the restriction of $\Phi(t)$ to
$H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})$ equals
\[
(-1)^{g} \frac{A(t)^{2g}}{n^{4g}\tilde{\Omega}_{\mu}(t)}
\]
for any $\mu \in \Delta.$ Let $\rho_{r}^{k,\mu}$ denote the
restriction of $\rho_{r}^{k}$ in $H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q}).$ Thus we have that
\[
\frac{(-1)^{g}}{n^{4g}} A(t)^{2g} = \sum_{k=0}^{n-1} \rho_{-1}^{k,\mu}
t^{k} \indent \mbox{mod } \tilde{\Omega}_{\mu}(t).
\]
Hence by substituting $t = -a_{1}^{i}$ for each $i$ we obtain
\[
\frac{(-1)^{g}}{n^{4g}} \left( \prod_{p=1,p \neq i}^{n} (a_{1}^{i} -
a_{1}^{p}) \right)^{2g} = \sum_{k=0}^{n-1}
\rho_{-1}^{k,\mu}(-a_{1}^{i})^{k}.
\]
Since the direct sum of restriction maps
\[
H^{*}_{\cal G}({\cal C};{\bf Q}) \rightarrow \bigoplus_{\mu \in \Delta} H^{*}_{\cal G}({\cal C}_{\mu};{\bf Q})
\]
is injective \cite[prop. 3]{K2} we have that
\begin{equation}
\frac{(-1)^{g}}{n^{4g}} \left( \prod_{p=1,p \neq i}^{n} (\delta_{i} -
\delta_{p}) \right)^{2g} = \sum_{k=0}^{n-1}
\rho_{-1}^{k}(-\delta_{i})^{k} \label{321}.
\end{equation}
Solving the equations (\ref{321}) we obtain
\begin{equation}
\rho_{-1}^{k} = \frac{(-1)^{g+n}}{n^{4g}} \sum_{i=1}^{n} S_{i}^{k}
\left( \prod_{p=1,p \neq i}^{n} (\delta_{i} - \delta_{p})
\right)^{2g-1} \label{rev5}
\end{equation}
where $S_{i}^{k}$ equals the $k$th elementary symmetric polynomial in
$\{\delta_{p}: p \neq i\}$.\\
\indent We will show later, in proposition \ref{Pontinad}, that the
relations $\rho_{-1}^{k}$ above are insufficient to prove any
vanishing of the Pontryagin ring in ranks greater than three. For now
consider the rank three case. We write
\[
\alpha = \delta_{1}-\delta_{2},\indent \beta = \delta_{2}-\delta_{3},
\indent \gamma = \delta_{3} - \delta_{1}.
\]
We know from lemma \ref{Pont} that the Pontryagin ring is generated by the
elementary symmetric polynomials in $\alpha^{2},\beta^{2}$ and
$\gamma^{2}.$ The relations $\rho_{-1}^{0},
\rho_{-1}^{1},\rho_{-1}^{2}$ read as
\begin{eqnarray}
(\alpha \beta)^{2g-1} + (\beta \gamma)^{2g-1} + (\gamma \alpha)^{2g-1}
= 0, \label{reva}\\
(\delta_{1}+\delta_{3}) (\alpha \beta)^{2g-1} +(\delta_{2}+\delta_{1})
(\beta \gamma)^{2g-1} + (\delta_{3}+\delta_{2}) (\gamma \alpha)^{2g-1}
= 0, \label{revb}\\
(\delta_{1} \delta_{3})(\alpha \beta)^{2g-1} +(\delta_{2} \delta_{1})
(\beta \gamma)^{2g-1} + (\delta_{3} \delta_{2}) (\gamma \alpha)^{2g-1}
= 0. \label{revc}
\end{eqnarray}
The equations (\ref{reva}), $a_{1} \times$ (\ref{reva}) --
(\ref{revb}), and (\ref{revc}) $+ a_{1} \times$ (\ref{revb}) $-a_{2}
\times$ (\ref{reva}) then show
\begin{equation}
(\delta_{2})^{k} (\alpha \beta)^{2g-1} + (\delta_{3})^{k} (\beta
\gamma)^{2g-1} +(\delta_{1})^{k} (\gamma \alpha)^{2g-1} = 0, \label{rev2}
\end{equation}
for $k = 0,1,2$. Note that
\[
(\delta_{i})^{r+3} = a_{1} (\delta_{i})^{r+2} - a_{2}
(\delta_{i})^{r+1} + a_{3} (\delta_{i})^{r}
\]
and hence equation (\ref{rev2}) holds for all non-negative
$k$. Further note that
\begin{equation}
\gamma^{2} = (a_{1})^{2} - 4 a_{2} + 2 a_{1} \delta_{2} - 3
(\delta_{2})^{2} \label{rev1}
\end{equation}
and so combining equation (\ref{rev2}) with equation (\ref{rev1}) and
two similar equations for $\alpha^{2}$ and $\beta^{2}$ we see that
\[
\gamma^{2l} (\delta_{2})^{k} (\alpha \beta)^{2g-1} + \alpha^{2l}
(\delta_{3})^{k} (\beta \gamma)^{2g-1} + \beta^{2l} (\delta_{1})^{k}
(\gamma \alpha)^{2g-1} = 0,
\]
for any non-negative $k,l$. Let $r,s,t$ be three non-negative integers
with an even sum. Note
\[
2 \alpha = (a_{1} - 3 \delta_{2}) - \gamma, \indent
2 \beta = (3 \delta_{2} - a_{1}) - \gamma,
\]
and hence $(\alpha^{r} \beta^{s} + \alpha^{s} \beta^{r}) \gamma^{t},$
when written in terms of $a_{1}, \delta_{2}$ and $\gamma$ is an even
function in $\gamma$.\\
\indent Now any element of the Pontryagin ring can be written
as a sum of elements of the form
\[
F(u,v,w) = \alpha^{u}\beta^{v}\gamma^{w} + \alpha^{v}\beta^{w}\gamma^{u} +
\alpha^{w}\beta^{u}\gamma^{v} + \alpha^{u}\beta^{w}\gamma^{v} +
\alpha^{v}\beta^{u}\gamma^{w} + \alpha^{w}\beta^{v}\gamma^{u},
\]
where $u+v+w$ is even. From the argument above we know that
\begin{equation}
F(2g-1+r,2g-1+s,t) = 0 \label{rev4}
\end{equation}
for $r,s,t \geq 0$ and $r+s+t$ even. If $u \geq 1$ then we have
\begin{equation}
F(u,v,w) = -F(u-1,v,w+1) - F(u-1,v+1,w) \label{rev3}
\end{equation}
since $\alpha+\beta+\gamma=0$.\\
\indent Suppose now that $ u \geq v \geq w$. We
claim $F(u,v,w)=0$ if $u+v+w \geq 6g-4.$ Note that
\[
\mbox{max}\{u,v,w\} > \mbox{max}\{u-1,v+1,w+1\}
\]
unless $u-v$ equals zero or one. In either case we find that $u
\geq v \geq 2g-1$ and hence $F(u,v,w) = 0$ by (\ref{rev4}). Hence by
repeated applications of identity (\ref{rev3}) we see
that $F(u,v,w) = 0$ when $u+v+w \geq 6g-4$ and so we have:\\[\baselineskip]
{\bf THEOREM 2.} {\em The Pontryagin ring of the moduli space ${\cal
M}(3,d)$ vanishes in degrees $12g-8$ and above.}
\begin{rem}
Theorem 2 falls short of Neeman's conjecture
\cite{NE} which states that the Pontryagin ring of ${\cal M}(n,d)$
should vanish in degrees above $2gn^{2}-4g(n-1)+2$. When $n=3$ this gives
$10g+2$.
\end{rem}
\begin{rem}
In the rank two case the relations (\ref{rev5}) show
that
\[
((a_{1})^{2} - 4a_{2})^{g}=0
\]
and that the Pontryagin ring of ${\cal M}(2,d)$ vanishes in degrees
greater than or equal to $4g$, duplicating Kirwan's proof of the
Newstead-Ramanan conjecture.
\end{rem}
\indent To conclude we show now that the relations $\rho_{-1}^{k}$ are
inadequate to show any vanishing of the Pontryagin ring when $n \geq 4$. From equation
(\ref{rev5}) we see that the ideal of the Pontryagin ring is contained
in the ideal generated by the formal expressions
\begin{equation}
\left( \prod_{p=1,p \neq i}^{n} (\delta_{i} - \delta_{p})
\right)^{2g-1} \label{rev6}.
\end{equation}
Let $I$ denote the ideal generated by
the relations (\ref{rev6}) and consider this as an ideal of ${\bf C}[\delta_{1},...,\delta_{n}]$. By Hilbert's Nullstellensatz the radical
$\sqrt{I}$ of $I$ consists of those elements of the Pontryagin ring which vanish on
the intersection of the subspaces given by
\begin{equation}
\prod_{p \neq i} (\delta_{i} - \delta_{p}) = 0, \indent i=1,...,n. \label{Q16}
\end{equation}
We shall consider the even and odd cases for $n$ separately.\\
\indent (i) $n$ is even -- write $n=2m.$ The intersection of the
subspaces (\ref{Q16}) consists of $(2m)!/(2^{m}m!)$ distinct
$m$-dimensional subspaces of ${\bf C}^{n}.$ One of these subspaces is
given by the equations
\begin{equation}
\delta_{2k-1} = \delta_{2k}, \indent k=1,...,m. \label{plane}
\end{equation}
We know from lemma \ref{Pont} that the total Pontryagin class $p(T)$ of
${\cal M}(n,d)$ equals
\[
\prod_{1 \leq k < l \leq n} (1 + (\delta_{k}-\delta_{l})^{2})^{2 \bar{g}}
\]
and in the subspace (\ref{plane}) $p(T)$ then equals
\[
\prod_{1 \leq k < l \leq m} (1 + (\delta_{2 k -1} - \delta_{2 l
-1})^{2})^{8 \bar{g}}.
\]
In particular we see that none of the Pontryagin classes of ${\cal M}(n,d)$
vanish on the subspace (\ref{plane}).\\
\indent (ii) $n$ is odd -- write $n=2m+1$. The intersection of the
subspaces (\ref{Q16}) consists of $(2k+1)!/(3 \cdot 2^{k}(k-1)!)$
distinct $k$-dimensional subspaces of ${\bf C}^{n}$. One of these
subspaces is given by the equations
\begin{equation}
\delta_{1} = \delta_{2} = \delta_{3}, \quad \delta_{2k} = \delta_{2k+1}, \quad k=2,...,m. \label{Plane}
\end{equation}
In the subspace (\ref{Plane}) the total Pontryagin class of ${\cal M}(n,d)$
equals
\[
\left( \prod_{2 \leq k \leq m} (1+(\delta_{1}-\delta_{2k})^{2})^{12\bar{g}}
\right) \left( \prod_{2 \leq k < l \leq m}
(1+(\delta_{2k}-\delta_{2l})^{2})^{8\bar{g}} \right).
\]
In particular we see that none of the Pontryagin classes of ${\cal M}(n,d)$
vanish on the subspace (\ref{Plane}).\\
\indent Thus we see that none of the Pontryagin classes $p_{r}(T)$ are
nilpotent modulo the formal relations (\ref{rev6}). Hence:
\begin{prop}
\label{Pontinad}
For $n \geq 4$ the Pontryagin classes $p_{r}(T) \in
H^{4r}({\cal M}(n,d);{\bf Q})$ are not nilpotent modulo $\rho_{-1}^{k}$ for $0 \leq
k \leq n-1$. In particular these relations are inadequate to
prove any non-trivial vanishing of the Pontryagin ring.
\end{prop}
|
1995-03-30T07:20:33 | 9503 | alg-geom/9503022 | en | https://arxiv.org/abs/alg-geom/9503022 | [
"alg-geom",
"math.AG"
] | alg-geom/9503022 | Luca Barbieri-Viale | L. Barbieri-Viale, C. Pedrini and C. Weibel | Roitman's theorem for singular complex projective surfaces | 36 pages, LaTeX | Duke Math. J. 84 (1996), 155-190 | null | null | null | Let $X$ be a complex projective surface with arbitrary singularities. We
construct a generalized Abel--Jacobi map $A_0(X)\to J^2(X)$ and show that it is
an isomorphism on torsion subgroups. Here $A_0(X)$ is the appropriate Chow
group of smooth 0-cycles of degree 0 on $X$, and $J^2(X)$ is the intermediate
Jacobian associated with the mixed Hodge structure on $H^3(X)$. Our result
generalizes a theorem of Roitman for smooth surfaces: if $X$ is smooth then the
torsion in the usual Chow group $A_0(X)$ is isomorphic to the torsion in the
usual Albanese variety $J^2(X)\cong Alb(X)$ by the classical Abel-Jacobi map.
| [
{
"version": "v1",
"created": "Wed, 29 Mar 1995 10:13:35 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Barbieri-Viale",
"L.",
""
],
[
"Pedrini",
"C.",
""
],
[
"Weibel",
"C.",
""
]
] | alg-geom | \section*{Introduction}
If $X$ is a smooth projective surface over the
complex numbers $\C$, the classical Abel--Jacobi map
goes from the Chow group $A_0(X)$ of cycles of degree 0
to the (group underlying the) Albanese Variety $Alb(X)$.
Roitman's Theorem \cite{Roit} states that this map induces
an isomorphism on torsion subgroups.
(See \cite{CT} for a nice compendium).
The goal of this paper is to remove the word ``smooth''
from Roitman's theorem. For this we shall modify
the definition of $A_0(X)$, replace $Alb(X)$ with Griffiths'
intermediate Jacobian $J^2(X)$, and construct a
generalization of the Abel--Jacobi map.
\medskip
\noindent{\bf Main Theorem. }\
{\it Let $X$ be a reduced projective
surface over $\C$. Then there is a natural map from $A_0(X)$
to $J^2(X)$ inducing an isomorphism on torsion:
$$A_0(X)_{tors}\cong J^2(X)_{tors}.$$
In particular, the torsion subgroup is a finite direct sum
of copies of $\Q/\Z$.}
\smallskip
If $X$ is a normal surface, this theorem is a reformulation of
a theorem of Collino and Levine \cite{C2} \cite{L-Alb}, because
(as we will show in Corollary~\ref{abnorm}), $J^2(X)$
is isomorphic to the Albanese of any desingularization of $X$.
Gillet studied the Abel--Jacobi map in \cite{GDuke} when
$X$ is a singular surface with ``ordinary multiple curves''
({\it e.g.\/}\ a seminormal surface with smooth normalization $\tilde X$).
He proved in \cite[Theorem B]{GDuke} that if $\tilde X$ satisfied
some extra hypotheses ($p_g=0$, etc.) then the Abel--Jacobi map
is surjective with finite kernel. Thus we deduce:
\medskip
\noindent{\bf Corollary. }\ {\it
Let $X$ be a surface with ordinary multiple curves
such that $H^2(X,{\cal O}_X)=0$. Assume that Bloch's conjecture
holds for the normalization $\tilde X$ of $X$.
Then the Abel--Jacobi map is an isomorphism.
$$A_0(X) \cong J^2(X)$$}
We now describe the ingredients in our main theorem.
If $X$ is a proper surface over $\C$, the intermediate
Jacobian $J^2(X)$ is defined to be
$$J^2(X) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\,\frac{H^{3}(X,\C)}{F^2H^{3}+ H^{3}(X,\Z(2))}.$$
Here $F^2H^3$ refers to the Hodge filtration of \cite{D}
and the coefficients $\Z(2)$ refer to the
embedding of $\Z$ in $\C$ sending 1 to $(2\pi i)^2$.
When $X$ is a smooth surface, it is well known that
$J^2(X)$ is isomorphic to the Albanese Variety $Alb(X)$.
Now suppose that $X$ is a singular surface.
We will show in Corollary~\ref{1-motive} that $J^2(X)$
is a complex torus, and that
if $X'$ is a resolution of singularities for $X$
then $J^2(X)$ is an extension of $Alb(X')$ by a torus.
That is, the map $J^2(X)\to Alb(X')$ forms a 1--motive
in the sense of Deligne \cite{D}; we call it the
{\it Albanese 1--motive} of $X$.
Given this, the torsion subgroup of $J^2(X)$
is a finite direct sum of copies of $\Q/\Z$.
The modified version of $A_0(X)$ is defined as a subgroup
of the Levine--Weibel Chow group $CH_0(X)$ of zero-cycles
on $X$ \cite{LW}. By definition, $CH_0(X)$ is the
abelian group generated by the smooth closed points on $X$,
modulo the subgroup generated by all terms $D=\sum n_iP_i$
(with $P_i$ smooth on $X$) such that
$D = (f)$ for some rational function $f$ on some
curve $C$, the curve being locally defined by a single equation
on the surface $X$.
If $X$ is a surface with $c$ proper components, there is a
natural surjection $CH_0(X)\to\Z^c$, called the degree map.
By definition, $A_0(X)$ is the kernel of the degree map.
In order to prove our Main Theorem, we need to reinterpret
$CH_0(X)$ in terms of algebraic $K$-theory.
Let $SK_0(X)$ denote the subgroup of $K_0(X)$
generated by the classes of smooth points on the surface $X$.
Then $CH_0(X)$ is isomorphic to $SK_0(X)$, by the map sending
a smooth point to its class in $K_0(X)$. This is the
Riemann--Roch Theorem if $X$ is smooth. It was proven for
affine surfaces in \cite[Theorem 2.3]{LW}.
For arbitrary quasi-projective surfaces it is due to
Levine \cite{L1}, who proved that both groups are isomorphic
to $H^2(X,{\cal K}_2)$ ({\it cf.\/}\ \cite{PW1}, \cite{C1}).
The isomorphism
$$CH_0(X) \cong H^2(X,{\cal K}_2) \cong SK_0(X)$$
is often called ``Bloch's formula'' for surfaces.
We have laid this paper out as follows.
In \S1 we present some basic facts about Deligne cohomology
of a proper but singular scheme. The corresponding Deligne
Chern classes which will be used in later sections is
introduced in \S2.
In \S3 we construct and compare the Mayer-Vietoris sequences
for $K$-theory and Deligne cohomology that we shall need.
In \S4 we compute $J^2(X)$ for any proper surface $X$.
Our computation shows that $J^2(X)$ is part of a 1-motive
$Alb(X)$ which we call the {\it Albanese 1-motive of $X$}.
In \S5 we describe the structure of $SK_1$ of any curve over
any algebraically closed field.
In \S6 and \S7 we establish some technical results about
${\cal K}_2$-cohomology, ending with the exact sequence of
Theorem~\ref{NH3}
for a normal surface $X$ over any field containing $\frac1n$.
$$0\to H^1(X,{\cal K}_2)/n\to N\Het3(X)\to{}_nCH_0(X)\to 0.$$
Finally we prove the Main Theorem in \S8.
\medskip
\section*{Notation}
All schemes we consider will be separated and
of finite type over a field $k$.
We call such a scheme a {\it curve} if it is 1-dimensional,
and a {\it surface} if it is 2-dimensional.
When $X$ is an algebraic scheme over $\C$, we will write
$H^*(X,\Z)$ and $H^*(X,\C)$ for the singular cohomology of the
associated analytic space $X_{an}$ as well as for
the mixed Hodge structure on it, given by Deligne \cite{D}.
The weight filtration on $H^*(X,\Z)$ will be written as
$W_iH^*$, and the Hodge filtration on $H^*(X,\C)$
will be written as $F^iH^*$.
The notation $\Z(r)$ denotes the subgroup $(2\pi i)^r\Z$ of $\C$.
Unless we wish to call attention to the relation with $H^*(X,\C)$,
we will write $H^*(X,\Z)$ instead of $H^*(X,\Z(r))$.
The notation $\Z(r)_{\cal D}$ denotes the Deligne complex
on a smooth scheme $X$ over $\C$ (see \S1).
We will use the Deligne complex to define the Deligne cohomology of
proper schemes; in the affine case the definition of
Deligne-Beilinson cohomology is different (one needs to consider
logarithmic poles), and we remain silent about this.
Similarly, the Zariski sheaves
${\cal H}_{\cal D}^*(r)$ (defined as the higher direct images of $\Z(r)_{\cal D}$)
are used only for proper schemes, as a technical device.
(See \S1, (2.4), \ref{square} and \ref{crux}.)
The Zariski sheaf ${\cal K}_q$ on $X$ is obtained by sheafifying
the Quillen higher $K$-theory functor $U\mapsto K_q(U)$. The
${\cal K}$-cohomology groups $H^p(X,{\cal K}_q)$ are just the
Zariski cohomology of these sheaves. As indicated in the
introduction, when $X$ is a surface the most important
${\cal K}$-cohomology group is $H^2(X,{\cal K}_2)\cong CH_0(X)$.
Similarly, we shall write ${\cal K}_q(\Z/n)$ and
${\cal H}^q(\mu_n^{\otimes i})$ for the Zariski sheaves associated
to the presheaves sending $U$ to $K_q(U;\Z/n)$ and
$\Het{q}(U,\mu_n^{\otimes i})$, respectively. In general,
we will always use calligraphic letters for Zariski sheaves.
Finally, we will use some standard notation.
Let $H$ be an abelian group or sheaf of abelian groups.
Then $H_{tors}$ will denote its torsion subgroup.
For each integer $n$ we will write $H/n$ for $H/nH$,
and ${}_nH$ for the subgroup $\{ x\in H\colon nx=0\}$ of $H$.
\goodbreak
\section{Deligne cohomology groups}
For $X$ smooth (possibly affine) over $\C$ we let
$\Z(r)_{{\cal D}}$ denote the ``Deligne complex''
$$0\to \Z(r)\to {\cal O}_{X_{an}}\by{d} \cdots \by{d}
\Omega^{r-1}_{X_{an}} \to0$$
of sheaves on the complex analytic manifold $X_{an}$,
where $\Z(r)\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, (2\pi i)^r\Z$ is in degree 0.
The {\it analytic} Deligne cohomology groups of the smooth
scheme $X$ are defined to be
$$H_{\cal D}^q(X,\Z(r))=H^q(X,\Z(r)_{{\cal D}})
\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\,\H_{an}^q(X,\Z(r)_{{\cal D}}).$$
We then have exact sequences of complexes of sheaves
on $X_{an}$:
\B{equation}\label{augment}
0\to \Omega^{<r}_{X_{an}}[-1]\to
\Z(r)_{{\cal D}}\by{\varepsilon_X} \Z\to 0.
\E{equation}
We can also define the Deligne cohomology groups of a smooth
simplicial scheme $X_{\bul}$ by considering
$\Z(r)_{{\cal D}}$ as a complex of analytic sheaves on $X_{\bul}$.
This yields an exact sequence of complexes parallel to
(\ref{augment}) by \cite[5.1.9.(II)]{D}.
Now let $X$ be a singular scheme.
A {\it smooth proper hypercovering} $X_{\bul}\to X$ of
$X$ ({\it cf.\/}\ \cite[6.2.5--6.2.8]{D}) is a simplicial scheme
$X_{\bul}$ with smooth components $X_i$, each proper over $X$,
together with a morphism to $X$ satisfying ``universal
cohomological descent.''
We define the Deligne cohomology of $X$ to be:
$$H_{{\cal D}}^q(X,\Z(r))\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\,\H_{an}^q(X_{\bul},\Z(r)_{{\cal D}}).$$
This definition is independent of the choice of smooth proper
hypercovering by \cite[Expos\'e V{\it bis},
5.1.7 and 5.2.4]{SGA4}.
There is a canonical descent isomorphism
$H^*(X,\Z)\cong H^*(X_{\bul},\Z)$, so the map $\varepsilon$ in
(\ref{augment}) induces a natural map
$\varepsilon_X\colon H_{\cal D}^*(X,\Z(r))\to H_{an}^*(X,\Z)$.
It is well-known (see \cite[1.6.4]{Bei}{})
that $\varepsilon_X$ preserves products.
For $X$ proper with arbitrary singularities
we have a standard long exact sequence
\B{equation}\label{modf}
\cdots\by{\varepsilon}\kern-2pt H^q(X,\Z)\to\kern-2pt
H^q(X,\C)/F^r\kern-2pt\to\kern-2pt H_{{\cal D}}^{q+1}(X,\Z(r))
\by{\varepsilon}\kern-2pt H^{q+1}(X,\Z)\to\cdots
\E{equation}
induced by (\ref{augment}) and $\Z\cong\Z(r)\subset\C$,
as well as
\B{equation}\label{star}
\cdots\to F^rH^q(X,\C)\to H^q(X,\C/\Z(r)) \to
H_{{\cal D}}^{q+1}(X,\Z(r)) \to F^rH^{q+1}(X,\C)\to\cdots
\E{equation}
If $X$ is a proper surface then from (\ref{modf})
we have an exact sequence
\B{equation}\label{extJZ}
0\to J^2(X) \to H_{{\cal D}}^{4}(X,\Z(2))
\by{\varepsilon} H^{4}(X,\Z(2))\to0.
\E{equation}
Any map $i\colon Y\to X$ lifts to a morphism
$i\colon Y_{\mbox{\Large $\cdot $}}\to X_{\bul}$ between hypercoverings;
see \cite[Expos\'e V{\it bis}, 5.1.7 and 5.2.4]{SGA4}
or \cite[6.2.8]{D}.
The {\it relative} Deligne cohomology of this map is
defined in the notation of \cite[6.3.3]{D} to be:
$$H_{{\cal D}}^q(X{\rm mod\,}Y,\Z(r))\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\,
\H_{an}^q(X_{\bul}{\rm\,mod\,}Y_{\mbox{\Large $\cdot $}},\ \Z(r)_{{\cal D}}
{\rm\, mod\,}\Z(r)_{{\cal D}})
$$
By \cite[6.3.2.2]{D} we have a functorial long exact sequence
\B{equation} \cdots\to H_{{\cal D}}^q(X{\rm\,mod\,}Y,\Z(r))\to
H_{{\cal D}}^q(X,\Z(r))\to H_{{\cal D}}^q(Y,\Z(r)) \to\cdots
\E{equation}
and of course there are relative versions
of (\ref{modf}) and (\ref{star})
which depend functorially on the pair $(X,Y)$, such as
\B{equation}\label{modf-rel}
\cdots\by\varepsilon\kern-3pt H^q(X{\rm mod\,}Y,\Z)\to\kern-3pt
H^q(X{\rm mod\,}Y )/F^r\to\kern-3pt
H_{{\cal D}}^{q+1}(X{\rm mod\,}Y,\Z(r))\by\varepsilon\cdots.
\E{equation}
\subsection*{Low degree Deligne cohomology}
We will need the following calculation of $H_{\cal D}^q(X,\Z(2))$
for $q\le2$. Given a proper scheme $X$ over $\C$, we fix a smooth
proper hypercovering $X_{\bul}\by\pi X$. By abuse of notation,
we shall write ${\cal H}_{\cal D}^q(2)$ for the complexes $R^q\omega_*\Z(2)_{\cal D}$
of Zariski sheaves on either $X_{\bul}$ or $X$,
$\omega$ denoting either
$\omega_{\mbox{\Large $\cdot $}}\colon X_{\mbox{\Large $\cdot $} an}\to X_{\mbox{\Large $\cdot $} zar}$ or
$\omega=\pi\omega_{\mbox{\Large $\cdot $}}\colon X_{\mbox{\Large $\cdot $} an}\to X_{zar}$.
\B{prop}\label{sheafHD}
For $X$ proper and connected over $\C$ we have:
\kern-8pt
\B{description}
\item[{\it (i)}] $H_{\cal D}^0(X,\Z(2)) = 0$;
\item[{\it (ii)}] $H_{\cal D}^1(X,\Z(2))\cong\C/\Z(2) \cong\C^*$;
\item[{\it (iii)}] $H_{\cal D}^2(X,\Z(2))_{tors}\cong H^1(X,\Q/\Z)$.
\E{description}\kern-8pt
Moreover, if $X$ is irreducible then we have
\B{description}
\item[{\it (iv)}] $H_{\cal D}^2(X,\Z(2)) = H^0_{zar}(X,{\cal H}_{\cal D}^2(2))
=H^0_{zar}(X_{\bul},{\cal H}_{\cal D}^2(2))$
\E{description}
and there are edge homomorphisms:
\B{description}
\item[{\it (v)}] $H^1_{zar}(X,{\cal H}_{\cal D}^2(2)) \hookrightarrow
H^1_{Zar}(X_{\bul},{\cal H}_{\cal D}^2(2))\hookrightarrow H_{\cal D}^3(X,\Z(2))$
\quad (these are injections);
\item[{\it (vi)}] $H^2_{zar}(X,{\cal H}_{\cal D}^2(2)) \longby{}
H^2_{Zar}(X_{\bul},{\cal H}_{\cal D}^2(2))\to H_{\cal D}^4(X,\Z(2))$.
\E{description}
\E{prop}
\B{proof}It is well-known that $H_{an}^1(X,\Z)$ is torsion-free.
Hence $(i)$ and $(ii)$ follow immediately from (\ref{modf}).
(Cf. the proof of Lemma~2.17 in \cite{GDuke}.)
Part $(iii)$ follows from this and (\ref{star}),
since $H^1(X,\C/\Z(2))_{tors}\cong H^1(X,\Q/\Z)$.
In order to prove parts $(iv)$, $(v)$ and $(vi)$
we use the Leray spectral sequences
for $\omega$ and $\omega_{\mbox{\Large $\cdot $}}$:
\B{equation} \label{simps}
\B{array}{ccccc}
E^{p,q}_2 &=& H^p_{zar}(X,\,{\cal H}_{\cal D}^q(i))
&\mbox{$\Rightarrow$}& H_{\cal D}^{p+q}(X,\Z(i))_{\strut} \\
E^{p,q}_2 &=& H^p_{zar}(X_{\bul},{\cal H}_{\cal D}^q(i))
&\mbox{$\Rightarrow$}& H_{\cal D}^{p+q}(X,\Z(i))
\E{array}\E{equation}
with $i=2$ ({\it cf.\/}\ \cite[(2.13)]{GDuke}).
For this, we need to compute ${\cal H}_{\cal D}^0(2)$ and ${\cal H}_{\cal D}^1(2)$.
When $U$ is smooth we may identify the analytic sheaf
${\cal O}_U/\Z(2)$ with ${\cal O}_U^*$ and
obtain a quasi-isomorphism between $\Z(2)_{\cal D}$ and the complex
$0\to{\cal O}_U^*\longby{dlog}\Omega_U^1$.
It follows that there is a distinguished triangle
of complexes of analytic sheaves on $X_{\bul}$
$$\C^*[-1] \to \Z(2)_{\cal D} \to
\Omega_{X_{\bul}}^1/dlog({\cal O}_{X_{\bul}}^*)[-2]
\to \C^*.$$
Applying $\omega_*$ and $R^1\omega_*$ immediately yields
${\cal H}_{\cal D}^0(2)=0$ and ${\cal H}_{\cal D}^1(2)=\omega_*\C^*=\C^*$ on both
$X_{\mbox{\Large $\cdot $} zar}$ and $X_{zar}$. Therefore
in either spectral sequence (\ref{simps})
the row $q=0$ vanishes and in row $q=1$ we have
$H_{zar}^p(X,\C^*)=H_{zar}^p(X_{\bul},\C^*)$.
The exact sequences of low degree terms in (\ref{simps}) become:
$$\B{array}{c}
0\to H_{zar}^1(X,\C^*)\to H_{\cal D}^2(X,\Z(2)) \to
H^0(X,{\cal H}_{\cal D}^2) \longby{d_2} H_{zar}^2(X,\C^*)
\to H_{\cal D}^3(X,\Z(2))\\
0\to H_{zar}^1(X,\C^*)\to H_{\cal D}^2(X,\Z(2)) \to
H^0(X_{\bul},{\cal H}_{\cal D}^2) \longby{d_2} H_{zar}^2(X,\C^*)
\to H_{\cal D}^3(X,\Z(2)).
\E{array}$$
The map between these sequences identifies them, and
$H^0(X,{\cal H}_{\cal D}^2)\cong H^0(X_{\bul},{\cal H}_{\cal D}^2)$ by the 5-lemma.
If $X$ is irreducible then $H_{zar}^p(X,\C^*)=0$ for $p\ne0$.
Hence $H_{\cal D}^2(X,\Z(2))$ is isomorphic to
$H^0(X,{\cal H}_{\cal D}^2)$.
Parts $(v)$ and $(vi)$ follows similarly.
\E{proof}
For each $n$ there is a distinguished triangle of
complexes of analytic sheaves on $X_{\bul}$:
\B{equation}\label{triangle}
\Z/n[-1]\by\delta\Z(2)_{\cal D}\by{n}\Z(2)_{\cal D}
\by{\bar\varepsilon}\Z/n \longby{\delta[1]}\Z(2)_{\cal D}[1].
\E{equation}
The comparison theorem between the analytic and \'etale sites,
together with universal cohomological descent, yields
$H^q(X_{\bul},\Z/n)\cong H^q(X,\Z/n)\cong\Het{q}(X,\Z/n)$.
Fixing an $n^{th}$ root of unity allows us to identify
$\mu_n$, $\mu_n^{\otimes2}$ and $\Z/n$ on $\eX$.
If $X$ is proper, the cohomology of the triangle
(\ref{triangle}) yields ``Kummer sequences''
\B{equation}\label{Kum-etHD}
0\to H_{\cal D}^{q}(X,\Z(2))/n \longby{\bar\varepsilon}
\Het{q}(X,\mu_n^{\otimes2})\longby\delta H_{\cal D}^{q+1}(X,\Z(2))_{n-tors} \to0.
\E{equation}
By \ref{sheafHD} this identifies $\mu_n\cong\Het0(X,\Z/n)$
with the $n$-torsion in $\C^*\cong H_{\cal D}^1(X,\Z(2))$,
and identifies $\Het1(X,\mu_n^{\otimes2})\cong\Het1(X,\Z/n)$
with the $n$-torsion in $H_{\cal D}^2(X,\Z(2))$.
\smallskip
Now consider the morphism $\omega\colon X_{\mbox{\Large $\cdot $} an}\to X_{zar}$.
Applying the higher direct image $R^q\omega_*$ to
(\ref{triangle}) yields an exact sequence of Zariski sheaves:
\B{equation}\label{Rq}
\longby{\bar\varepsilon} {\cal H}^{q-1}(\mu_n^{\otimes2})
\longby\delta {\cal H}_{\cal D}^q(2) \longby{n} {\cal H}_{\cal D}^q(2)
\longby{\bar\varepsilon} {\cal H}^q(\mu_n^{\otimes2})\longby\delta.
\E{equation}
In particular, $\delta$ identifies ${\cal H}^1(\mu_n^{\otimes2})$ with the
$n$-torsion subsheaf of ${\cal H}_{\cal D}^2(2)$.
The map $\Het{*-1}(X,\mu_n^{\otimes2})\by\delta H_{\cal D}^*(X,\Z(2))$
is also the abutment of a morphism of Leray spectral sequences.
At the $E_2$-level it is
$H_{zar}^p(X,{\cal H}^{q-1}(\mu_n^{\otimes2})) \by\delta H_{zar}^p(X,{\cal H}_{\cal D}^q(2))$.
If $X$ is proper and irreducible then the bottom row of both
spectral sequences degenerates
({\it e.g.\/}\ $H_{zar}^p(X,\Z/n)=0$ for $p\ne0$)
and we obtain the following result.
\B{cor}\label{morph}If $X$ is proper and irreducible,
there is a commutative diagram whose rows are the
exact sequences of low degree terms of Leray spectral sequences:
$$\B{array}{ccccccccc}
0&\to& \kern-5pt H^1(X,{\cal H}^1(\mu_n^{\otimes2}))&\to&\kern-3pt\Het2(X,\mu_n^{\otimes2})
&\to& \kern-3pt H^0(X,{\cal H}^2(\mu_n^{\otimes2}))
&\kern-4pt\longby{d_2}&\kern-8pt H^2(X,{\cal H}^1(\mu_n^{\otimes2}))\\
&&\delta\downarrow &&\delta\downarrow&&\delta\downarrow&&
\delta\downarrow\\
0&\to& \kern-5pt H^1(X,{\cal H}_{\cal D}^2(2))&\to&\kern-3pt
H_{\cal D}^3(X,\Z(2)) &\to& \kern-3pt H^0(X,{\cal H}_{\cal D}^3(2))
&\kern-4pt\longby{d_2}& \kern-8pt H^2(X,{\cal H}_{\cal D}^2(2)).
\E{array}$$
\E{cor}
\section{Chern classes in Deligne cohomology}
For each scheme $X$ of finite type over $\C$ the
exponential map ${\cal O}_{X_{an}}\to{\cal O}_{X_{an}}^*$ induces a
quasi-isomorphism between $\Z(1)_{{\cal D}}=(\Z\to{\cal O}_{X_{an}})$ and
${\cal O}_{X_{an}}^*[-1]$. This quasi-isomorphism also holds over a
simplicial scheme $X_{\bul}$ by naturality, so
$\H^q(X_{\bul},\Z(1)_{\cal D})\cong H^{q-1}(X_{\bul},{\cal O}^*_{X_{\bul}})$.
This gives a natural map from $H^1_{an}(X,{\cal O}_X^*)$ to
$\H^2(X_{\bul},\Z(1)_{\cal D})\cong H_{an}^1(X_{\bul},{\cal O}^*_{X_{\bul}})$
for every smooth proper hypercovering $X_{\bul}\to X$.
Composing with the determinant map $K_0(X)\to \mbox{Pic}(X)$
and the natural map $\mbox{Pic}(X)\to H_{an}^1(X,{\cal O}_X^*)$
yields a map $c_1\colon K_0(X) \to \H^2(X_{\bul},\Z(1)_{\cal D})$.
Now the splitting principle holds for Deligne cohomology by
\cite[5.2]{GDuke}. (Warning: if $X$ is not proper this differs
slightly from the splitting principle proven in \cite[1.7.2]{Bei}!)
Thus the map $c_1$ extends to Chern classes
$c_i\colon K_0(X) \to \H^{2i}(X_{\bul},\Z(i)_{\cal D})$
for vector bundles.
When $X$ is proper, these are the Deligne-Beilinson Chern classes
$$c_i\colon K_0(X) \to H_{\cal D}^{2i}(X,\Z(i)).$$
Recall from (\ref{augment}) that there is a map
$\varepsilon_X\colon \H^{2i}(X_{\bul},\Z(i)_{\cal D})\to
H_{an}^{2i}(X,\Z)$, and that it is product-preserving.
\B{lemma}\label{c2-an} ({\it cf.\/}\ Beilinson \cite[1.7]{Bei})
The composition of $c_i$ with the map $\varepsilon_X$
is the classical Chern class
of the associated topological vector bundle \cite{MCC}:
$$c_i^{an}\colon
K_0(X)\to H_{an}^{2i}(X,\Z)=H_{top}^{2i}(X,\Z).$$
\E{lemma}
\B{proof} Since $\varepsilon_X$ preserves cup products, the
splitting principle shows that it suffices to establish the
result for $c_1$.
If $X$ is smooth then $c_1$ is the analytic determinant map,
and $\varepsilon_X$ is just the usual map
$\partial_X\colon H_{an}^1(X,{\cal O}_X^*)\to H_{an}^2(X,\Z)$
used to define $c_1^{an}$ on analytic vector bundles,
so it is clear that $c_1^{an}=\varepsilon_X\circ c_1$.
To deduce the result for general $X$, choose a smooth proper
hypercover $u\colon X_{\bul}\to X$. Composing $\partial_X$
(which is $c_1^{an}$) with the descent isomorphism
$H_{an}^{2i}(X,\Z)\cong H_{an}^{2i}(X_{\bul},\Z)$ is the
descent map
$H^1(X,{\cal O}_X^*)\to H^1(X_{\bul},{\cal O}_{X_{\bul}}^*)$
(which is $c_1$) composed with $\partial_{X_{\bul}}$,
{\it i.e.,\/}\ with $\varepsilon_{X_{\bul}}$.
\E{proof}
Reduction of $\varepsilon_X$ mod $n$ yields a map
$\bar\varepsilon_X\colon H_{\cal D}^{2i}(X,\Z(i))\to
H_{an}^{2i}(X,\Z/n)$. Since reduction mod $n$ is
product-preserving and sends $c_1^{an}$ to the
\'etale Chern class $c_1^{et}$, we deduce the
\B{cor}\label{c-etale}
The composition of $c_i$ with $\bar\varepsilon_X$
is the \'etale Chern class
$$c_i^{et}\colon K_0(X) \to H_{an}^{2i}(X,\Z/n)
\cong\Het{2i}(X,\mu_n^{\otimes i}).$$
\E{cor}
In this paper we shall be mostly concerned with the class
$c_2\colon K_0(X)\to H^4_{{\cal D}}(X,\Z(2))$ when $X$ is a
projective surface. Recall from the introduction (or \cite{LW})
that the Chow group $CH_0(X)$ of zero-cycles on $X$ is
isomorphic to the subgroup $SK_0(X)$ of $K_0(X)$.
If $X$ has $c$ irreducible components then there is a natural
degree map $CH_0(X)\to\Z^c$, and $A_0(X)$ is defined to be
the kernel of this map.
The following cohomological interpretation
of the degree map will be useful.
\B{lemma}\label{degree} (Beilinson \cite[1.9]{Bei})
If $X$ is a projective surface, the
degree map is the same (up to sign) as the classical Chern class
$$ CH_0(X) \hookrightarrow K_0(X) \by{c_2^{an}} H_{an}^4(X,\Z)
\cong \Z^c.$$
By (\ref{extJZ}), the Deligne Chern class $c_2$ induces a natural map
$\rho\colon A_0(X)\to J^2(X)$
fitting into the diagram
$$ \begin{array}{ccccccc}
0 \to & A_0(X) &\to& CH_0(X) &\by{deg}& \Z^c &\to0 \\
&\rho\downarrow& &c_2\downarrow& & \mbox{\large $\parallel$} & \\
0 \to & J^2(X) &\to& H_{\cal D}^4(X,\Z(2)) &\by{\varepsilon}&
H^4_{an}(X,\Z) &\to0 \\
\end{array}$$
\E{lemma}
\smallskip\noindent{\bf Definition: }
We shall refer to the map $\rho$ as the {\it Abel--Jacobi map},
because if $X$ is a smooth surface then
$J^2(X)$ is the usual Albanese variety and the map
$\rho$ coincides with the classical Abel--Jacobi map
by \cite[1.9.1]{Bei} or \cite[2.24]{GDuke}.
\medskip
\B{proof} Observe that if $X$ has $c$ proper irreducible
components then $H^4(X,\Z)\cong\Z^c$, because
the singular locus of $X$ has real analytic dimension $\le2$.
Given Lemma~\ref{c2-an}, the second assertion follows from
the first. If $X$ is a smooth projective surface the result is
classical; one way to see it is to use the product formula
for two divisors on $X$:
$$c_2^{an}(D\otimes E) = -c_1^{an}(D)\cup c_1^{an}(E)
= - (D\cdot E)[X].$$
In general, choose a resolution of singularities $X'\to X$.
Since $X'$ has $c$ disjoint components, the degree map on
$X$ factors through the degree map on $X'$ as
$CH_0(X)\to CH_0(\tilde X)\to\Z^c$.
By naturality of $c_2^{an}$, the isomorphism
$H^4_{an}(X,\Z)\cong H^4_{an}(X',\Z)$ allows us to
deduce the result for $X$ from the result for $X'$.
\E{proof}
\B{num}\label{simplicial}
As observed by Beilinson \cite[2.3]{Bei}
({\it cf.\/}\ \cite[\S5]{GDuke}), the formalism of
Deligne cohomology allows us to extend the Chern classes
from $K_0(X)$ to higher $K$-theory as well. The higher
Deligne Chern classes are homomorphisms
$$c_i\colon K_q(X) \to H^{2i-q}_{\cal D}(X,\Z(i)).$$
Composition with $\varepsilon_X$ yields the higher analytic
Chern classes $c^{an}_i$, and reduction mod $n$ yields
the higher \'etale Chern classes $c^{et}_i$.
Moreover, the following holds.
\B{description}
\item{(\thethm.1)}
There is a connected simplicial presheaf
$K\simeq\Omega_0BQP$ and a simplicial sheaf ${\cal D}$
on $X_{zar}$ such that $\pi_q K(U) = K_q(U)$ for $q\ge1$,
and $\pi_q{\cal D}(U) = H^{2i-q}(U_{\mbox{\Large $\cdot $}},\Z(i)_{\cal D})$ for $q\ge0$.
Moreover, there is a map of simplicial presheaves
$C_i^{ss}\colon K \to {\cal D}$
such that $\pi_q C_i^{ss}(X)$ is the
Deligne cohomology Chern class $c_i$ on $K_q(X)$.
({\it cf.\/}\ \cite[5.4]{GDuke}, which differs somewhat from
\cite{Bei} and \cite{GBAMS}.)
Indeed, ${\cal D}$ is the simplicial sheaf of abelian groups associated
by the Dold-Kan theorem (\cite[8.4.1]{W-homo})
to the good truncation $\tau^{\le0}\R\omega_*\Z(i)_{\cal D}[2i]$
of the total derived direct image of $\Z(i)_{\cal D}[2i]$ under
$\omega\colon {X_{\bul}}{}_{,an}\to X_{zar}$.
\item{(\thethm.2)} Let ${\cal E}$ denote the simplicial sheaf associated
by the Dold-Kan theorem to the good truncation
$\tau^{\le0}\R\omega_*\Z/n[2i]$ of the total derived direct
image of $\Z/n[2i]$. Then $\pi_q{\cal E}(U)=H_{an}^{2i-q}(U,\Z/n)
\cong\Het{2i-q}(U,\mu_n^{\otimes i})$. If we define $L$ to be
the homotopy fiber of $K\by{n}K$ then we have
$\pi_q L(U) = K_{q+1}(U;\Z/n)$. This all gives
a homotopy commutative diagram whose rows are
homotopy fibration sequences
\B{equation}\label{fib}\B{array}{ccccccccc}
\Omega K &\to& L &\to& K &\by{n}& K && \\
\kern18pt\downarrow\Omega C^{ss}_2\kern-10pt &&
\kern13pt\downarrow C^{ss}_2 \kern-10pt&&
\kern13pt\downarrow C^{ss}_2 \kern-10pt&&
\kern13pt\downarrow C^{ss}_2 \kern-8pt&& \\
\Omega{\cal D} &\to& \Omega{\cal E} &\by\delta&{\cal D}&\by{n}&{\cal D}&
\by{\bar\varepsilon}&{\cal E}.\\
\E{array}\E{equation}
From Corollary~\ref{c-etale}
and a standard argument with $\Het{*}(X,G,\mu_n^{\otimes i})$
it is easy to see that not only does
$K\to{\cal E}$ induce the higher \'etale Chern class $c^{et}_i$
on $K_*(X)$ but the map $L\to\Omega{\cal E}$ induces
the usual \'etale Chern classes on
$K$-theory with coefficients mod $n$.
$$c^{et}_i\colon K_q(X;\Z/n)\longby{}
\Het{2i-q}(X,\mu_n^{\otimes i})
$$
Applying $\pi_2$ to (\ref{fib}) with $i=2$ and $U=X$ yields
the commutative diagram
\B{equation}\label{et-HD}\B{array}{ccccccc}
K_3(X)&\to&K_3(X;\Z/n) &\longby{}& K_2(X) &\by{n}& K_2(X) \\
&&\downarrow c^{et}_2 && \downarrow c_2 && \downarrow c_2 \\
0&\to& \Het1(X,\mu_n^{\otimes2}) &\longby{\delta}& H_{\cal D}^2(X,\Z(2))
&\by{n}& H_{\cal D}^2(X,\Z(2)).
\E{array}\E{equation}
By (\ref{Kum-etHD}) we see that $c^{et}_2$ vanishes on
$K_3(X)$ and factors through ${}_nK_2(X)$.
Applying $\pi_2$ to (\ref{fib}) with $i=2$ and
sheafifying yields the commutative diagram of sheaves
in which the bottom row is part of (\ref{Rq}):
\B{equation}\label{et-HD-sheaf}\B{array}{ccccccc}
{\cal K}_3&\to&{\cal K}_3(\Z/n) &\longby{}&
\kern-5pt{\cal K}_2 &\by{n}& \kern-5pt{\cal K}_2 \\
&&\downarrow c^{et}_2 && \downarrow c_2 && \downarrow c_2 \\
0&\to& {\cal H}^1(\mu_n^{\otimes2}) &\longby{\delta}& {\cal H}_{\cal D}^2(2)
&\by{n}& {\cal H}_{\cal D}^2(2)
\E{array}\E{equation}
By (\ref{Rq}) we see that $c^{et}_2$ vanishes on
${\cal K}_3$ and factors through the sheaf ${}_n{\cal K}_2$.
\item{(\thethm.3)}
There is a morphism of spectral sequences
between the Brown--Gersten spectral sequence for $K_{-*}(X)$
and the Leray spectral sequence in (\ref{simps}) converging to
$H^{2i+*}_{\cal D}(X,\Z(i))$. At the $E_2^{pq}$-level the
morphisms are the cohomology of $c_i$:
$$H^p_{zar}(X,{\cal K}_{-q}) \by{c_i} H^p_{zar}(X,{\cal H}_{\cal D}^{2i+q}(i))
.$$
Here ${\cal K}_q$ is the sheaf on $X_{zar}$
associated to the presheaf $K_q$ and the sheaves
${\cal H}_{\cal D}^j(i)$ are $R^j\omega_*\Z(i)_{\cal D}$, as in the proof
of Proposition~\ref{sheafHD}.
By \cite{TT}, the first spectral sequence converges to
$K_{-p-q}(X)$ whenever $X$ is quasi-projective.
The second spectral sequence is an obvious
reindexing of (\ref{simps}) and converges to
$H_{\cal D}^{2i+p+q}(X,\Z(i))$.
\E{description}
\E{num}
Here are three applications of the morphism of spectral sequences
in (2.4.3). First, if $X$ is a projective surface
we have a commutative diagram
$$\begin{array}{ccccc}
CH_0(X)&\cong\kern-8pt& H^2(X,{\cal K}_2) &\hookrightarrow& K_0(X)\\
\strut & & c_2\downarrow& & c_2\downarrow \\
& & H^2(X,{\cal H}_{\cal D}^2(2)) &\to & H_{\cal D}^4(X,\Z(2)).\\
\end{array}$$
where the bottom horizontal map is given by
Proposition~\ref{sheafHD}$(vi)$.
Second, suppose that $Y$ is 1-dimensional. \kern-1pt Then we may
identify the group $H^1\kern-1pt(X\kern-1pt,{\cal K}_2\kern-1pt)$
with the subgroup $SK_1(X)$ of $K_1(X)$,
and $c_2\colon SK_1(X)\to H_{\cal D}^3(X,\Z(2))$ is identified with
the composite
$H^1(X,{\cal K}_2)\by{c_2}H^1(X,{\cal H}_{\cal D}^2(2))\to H_{\cal D}^3(X,\Z(2))$.
Third, suppose that $X$ is an irreducible projective surface.
Then $c_2$ vanishes on the image of $H^2(X,{\cal K}_3)$ in $K_1(X)$
because it factors through $H^2(X,{\cal H}_{\cal D}^1(2))=H_{zar}^2(X,\C^*)$,
which is zero because $X$ is irreducible, as we saw in the
proof of Proposition~\ref{sheafHD}.
Since $SK_1(X)$ is an extension of $H^1(X,{\cal K}_2)$ by this image,
we may summarize this as follows.
\B{lemma}\label{c2-SK1}
Let $X$ be an irreducible projective surface over $\C$. Then
the Chern class
$c_2\colon SK_1(X)\to H_{\cal D}^3(X,\Z(2))$ factors as:
$$SK_1(X)\vlongby{\mbox{\rm onto}}H^1(X,{\cal K}_2)
\by{c_2}H^1(X,{\cal H}_{\cal D}^2(2))\hookrightarrow H_{\cal D}^3(X,\Z(2)).
$$
\E{lemma}
\goodbreak
\section{Mayer--Vietoris sequences}
Since we are going to deal with resolutions of singularities or
normalizations we will need some Mayer--Vietoris sequences.
In this section we do this for mixed Hodge structures,
Deligne cohomology and $K$-theory.
Associated to a proper birational morphism
$f\colon X'\to X$ of $\C$-algebraic schemes, and every
closed subscheme $i\colon Y\hookrightarrow X$ we have the commutative square
\B{equation}\label{birsquare}
\B{array}{rcl} Y'& {\stackrel{i'}{\hookrightarrow}}& X'\\
\p{f'}\downarrow & &\downarrow\p{f}\\
Y &{\stackrel{i}{\hookrightarrow}}& X \E{array}
\E{equation}
where $Y' = f^{-1}(Y)\ (\ =Y\times_X X')$.
We shall always assume that $Y$ is chosen so that the restriction
$f\colon X'-Y'\by{\simeq} X-Y$ is an isomorphism.
\goodbreak
\B{prop}\label{M-V-H} {\rm (Mayer--Vietoris for mixed
Hodge structures)\,} Associated with any square
(\ref{birsquare}) we have a long exact sequence of
mixed Hodge structures
$$\cdots\to H^n(X,\Z)\by{u}H^n(X',\Z)\oplus H^n(Y,\Z)\by{v}
H^n(Y',\Z)\by{\partial} H^{n+1}(X,\Z)\to\cdots $$ in which
$$ u=\left(\begin{array}{c}{f^*}\\{i^*}\end{array} \right)
\qquad{\rm and}\qquad v= (i'^*,-f'^*)$$
\E{prop}
\B{proof} We have a map of long exact sequences
$$ \begin{array}{ccccccccc}
\cdots & \to & H^n(X{\rm mod\,} Y,\Z) &
\to & H^n(X,\Z) & \by{i^*} &
H^{n}(Y,\Z) & \to & \cdots \\
& &\p{f^*}\downarrow\cong &
&{\p{f^*}\downarrow} & & {\p{f'^*}\downarrow} & &\\
\cdots & \to & H^n(X'{\rm mod\,} Y',\Z) &
\to & H^n(X',\Z) & \by{i'^*} &
H^{n}(Y',\Z) & \to & \cdots
\end{array} $$
where $H^*(-{\rm mod\,}\dag ,\Z)$ is the relative singular
cohomology functor (defined in \cite[8.3.8]{D}).
By excision $f^*\colon H^*(X{\rm \,mod\,}Y,\Z)\cong
H^*(X'{\rm\,mod\,}Y',\Z)$ ({\it cf.\/}\ \cite[8.3.10]{D}).
By \cite[8.3.9 and 8.2.2]{D} the diagram above is a diagram
in the abelian category of mixed Hodge structures.
The Mayer--Vietoris exact sequence now follows by
a standard diagram chase.
\E{proof}
We then have, as well:
\B{schol} \label{M-V-D}
{\rm (Mayer--Vietoris for Deligne cohomology)\,}
Associated with any square (\ref{birsquare})
we have a long exact sequence in Deligne cohomology
$$\cdots\to H_{{\cal D}}^n(X',\Z(r))\oplus
H_{{\cal D}}^n(Y,\Z(r)) \to H_{{\cal D}}^n(Y',\Z(r))\to
H_{{\cal D}}^{n+1}(X,\Z(r))\to\cdots $$
\E{schol}
\B{proof} The proof of Proposition~\ref{M-V-H} goes through,
once we know that Deligne cohomology satisfies excision.
But since we have excision for the mixed Hodge
structure on relative singular cohomology, one can see it
holds for Deligne cohomology by arguing with the
relative cohomology sequence (\ref{modf-rel}).
\E{proof}
\B{thm}\label{M-V-K}
{\rm (Mayer--Vietoris for $K$-theory)\,}
Let $X$ be a reduced quasiprojective surface over a field
with normalization $\tilde X$.
Then there is a 1-dimensional subscheme $Y$ with
$Y_{red}={\rm Sing}\, X$ such that the
normalization square ({\it cf.\/}\ (\ref{birsquare}))
$$\B{array}{rcl} \tilde Y& \hookrightarrow &
\tilde X\\ \p{\tilde \pi}\downarrow & &\downarrow\p{\pi}\\
Y&\hookrightarrow & X \E{array}$$
induces exact sequences in $K$-theory:
$
K_1(\tilde X)\oplus K_1(Y)\to K_1(\tilde Y)\by{\partial}
K_0(X)\to K_0(\tilde X)\oplus K_0(Y) \to K_0(\tilde Y)
$
$
SK_1(\tilde X)\oplus SK_1(Y)\to SK_1(\tilde Y)\by{\partial}
SK_0(X)\to SK_0(\tilde X)\to 0
$
\E{thm}
\B{proof}
Let $K_*(X,\tilde X)$ and $K_*(Y,\tilde Y)$ be the
relative groups fitting into the long exact sequences in the
commutative diagram
$$\begin{array}{ccccccccccc}
K_1(X) &\to& K_1(\tilde X) &\to& K_0(X,\tilde X) &\to& K_0(X) &\to&
K_0(\tilde X) &\to& K_{-1}(X,\tilde X)\\
\downarrow && \downarrow&&\downarrow &&\downarrow && \downarrow&&\mbox{\large $\parallel$}\\
K_1(Y) &\to& K_1(\tilde Y) &\to& K_0(Y,\tilde Y) &\to& K_0(Y) &\to&
K_0(\tilde Y) &\to& K_{-1}(Y,\tilde Y). \\
\end{array}$$
(The far right terms are isomorphic by \cite[A.6]{PW3}.)
To establish the existence and exactness of the $K_1-K_0$ sequence
we must show that ``excision'' holds for $K_0$, {\it i.e.,\/}\ that
$K_0(X,\tilde X)\cong K_0(Y,\tilde Y)$ for some $Y$
with $Y_{red}={\rm Sing}\, X$ (see \cite[5.1]{GW}).
If $Y$ is a subscheme of $X$ defined by an ${\cal O}_{\tilde X}$-ideal
${\cal I}\subset{\cal O}_X$
then by \cite[A.6]{PW3} there is a natural exact sequence
\B{equation}\label{K0rel}
H^1(Y,{{\cal I}}/{{\cal I}}^2\otimes\Omega_{\tilde X/X})
\vlongby{\eta(Y)}
K_0(X,\tilde X) \to K_0(Y,\tilde Y) \to0.
\E{equation}
We define $Y_1$ using the conductor ideal ${\cal J}$, and $Y$ using
the ideal ${{\cal I}}={{\cal J}}^2$.
Then $Y_{red}={\rm Sing}\, X$, and the map from
${\cal I}/{\cal I}^2$ to ${\cal J}/{\cal J}^2$ is zero.
By naturality in $Y\to Y_1$, the map $\eta(Y)$ in (\ref{K0rel})
is the composite map
$
H^1(Y,{\cal I}/{\cal I}^2\otimes\Omega_{\tilde X/X}) \by0
H^1(Y,{\cal J}/{\cal J}^2\otimes\Omega_{\tilde X/X})
\vlongby{\eta(Y_1)} K_0(X,\tilde X)$$
so $\eta(Y)=0$ in (\ref{K0rel}).
Hence excision holds for $Y$, as claimed.
There is a natural map from the $K_1-K_0$ sequence onto the
``Units-Pic'' sequence, and the kernel is the $SK_1-SK_0$ sequence.
A standard diagram chase, described in \cite[8.6]{PW1},
shows that the latter sequence sequence is also exact.
\E{proof}
We remark that if $Y$ is reduced, or 0-dimensional, or even affine, then
the obstruction $H^1(Y)$ in (\ref{K0rel}) automatically vanishes,
and excision is immediate. Theorem \ref{M-V-K} was proven in these
special cases in \cite[7.5]{PW1} and \cite[A.3]{PW3}.
\B{cor}\label{KtoD}
With the notation of \ref{M-V-K}, the following diagram commutes.
$
\begin{array}{cccccccc}
SK_1(\tilde X)\oplus SK_1(Y) &\kern-8pt\to\kern-9pt&
SK_1(\tilde Y) &\kern-7pt\to\kern-8pt &SK_0(X) &
\kern-7pt\to\kern-8pt & SK_0(\tilde X) &\kern-7pt\to0\\
\downarrow&&\downarrow&&\downarrow&&\downarrow&\\
K_1(\tilde X)\oplus K_1(Y) &\kern-8pt\to\kern-9pt&
K_1(\tilde Y) &\kern-7pt\to\kern-8pt &K_0(X) &
\kern-7pt\to\kern-8pt & K_0(\tilde X) &\kern-7pt\to0\\
c_2\downarrow\ &&c_2\downarrow\ &&c_2\downarrow\ &&c_2\downarrow\ &\\
H_{\cal D}^3(\tilde X,\Z(2))\oplus H_{\cal D}^3(Y,\Z(2)) &
\kern-8pt\to\kern-9pt & H_{\cal D}^3(\tilde Y,\Z(2))&
\kern-7pt\to\kern-8pt &
H_{\cal D}^4(X,\Z(2)) &\kern-7pt\to\kern-8pt &
H_{\cal D}^4(\tilde X,\Z(2)) &\kern-7pt\to0
\end{array}
$
\E{cor}
\B{proof} We use the notation of (2.4.1).
For each open $U$ in $X$,
let $F(U)$ denote the homotopy fiber of
$K(U\times_X Y)\times K(U\times_X\tilde X)
\to K(U\times_X\tilde Y)$. By Proposition \ref{M-V-D} the
corresponding homotopy fiber for Deligne cohomology is
${\cal D}(U)$. In addition, there is a natural map from
$K(U)$ to $F(U)$ which is an isomorphism on $\pi_0$ by
Theorem \ref{M-V-K}. Therefore the natural map $C_2^{ss}$
of (2.4.1)
induces a map
$F(U)\to{\cal D}(U)$ on homotopy fibers, making the diagram
$$\begin{array}{ccccc}
K_1(\tilde Y) &\by{\partial}& \pi_0F(X)
&{\stackrel{\simeq}{\leftarrow}}& K_0(X)\\ \strut
C_2^{ss}\downarrow\ &&\ \downarrow C_2^{ss}&&\ \downarrow C_2^{ss}\\
H_{\cal D}^3(\tilde Y,\Z(2)))&\by{\partial}& \pi_0{\cal D}(X)
&{\stackrel{\simeq}{\leftarrow}}& H_{\cal D}^4(X,\Z(2))
\end{array}$$
commute. But the top composite is the $K$-theory boundary map
in Theorem \ref{M-V-K}.
\E{proof}
Using (2.4) and Lemma~\ref{c2-SK1}, we may refine
Corollary~\ref{KtoD} as follows.
\B{schol}\label{SKtoD}
With the notation of \ref{M-V-K}, the following diagram commutes.
$
\begin{array}{cccccccc}
SK_1(\tilde X)\oplus SK_1(Y) &\kern-8pt\to\kern-9pt&
SK_1(\tilde Y) &\kern-7pt\to\kern-8pt &SK_0(X) &
\kern-7pt\to\kern-8pt & SK_0(\tilde X) &\kern-7pt\to0\\
\mbox{\rm onto}\downarrow\ \ &&\cong\downarrow&&\cong\downarrow&&
\cong\downarrow&\\
H^1(\tilde X,{\cal K}_2)\oplus H^1(Y,{\cal K}_2) &\kern-8pt\to\kern-9pt&
H^{1}(\tilde Y,{\cal K}_2) &\kern-7pt\to\kern-8pt &
H^{2}(X,{\cal K}_2) & \kern-7pt\to\kern-8pt &
H^{2}(\tilde X,{\cal K}_2)& \\
c_2\downarrow\ &&c_2\downarrow\ &&c_2\downarrow\ &&c_2\downarrow\ &\\
H_{\cal D}^3(\tilde X,\Z(2))\oplus H_{\cal D}^3(Y,\Z(2)) &
\kern-8pt\to\kern-9pt & H_{\cal D}^3(\tilde Y,\Z(2))&
\kern-7pt\to\kern-8pt &
H_{\cal D}^4(X,\Z(2)) &\kern-7pt\to\kern-8pt &
H_{\cal D}^4(\tilde X,\Z(2)) &\kern-7pt\to0
\end{array}
$
\E{schol}
\goodbreak
\section{The Albanese 1-motive of a proper surface}
In this section a {\it surface}\, will mean a
\underline{proper}
reduced $2$-dimensional scheme $X$ of finite type
over the complex numbers $\C$.
We will consider the intermediate jacobian $$J^2(X) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\,
\frac{H^{3}(X,\C)}{F^2H^{3}+ H^{3}(X,\Z(2))}$$
This is the mixed Hodge theoretic
generalization of the classical Albanese group variety of a
smooth surface.
We begin with an elementary result
(cf. \cite[Remark 5.5]{GDuke}).
\B{lemma}\label{filt}
Suppose that $X$ is a proper surface. Then
$
F^2H^i(X,\C)\cap H^i(X,\R)=0 \mbox{\ \ for \ } i=2,3.
$
Hence in sequence (\ref{modf}) we have
$$\B{array}{ccc} H^i(X,\Z)_{tors}
\strut&=&\mbox{kernel of } H^i(X,\Z)\to H^i(X,\C)/F^2H^i\\
\strut&=&\mbox{image of } H_{\cal D}^i(X,\Z(2))\by{\varepsilon}H_{an}^i(X,\Z).
\E{array}$$
\E{lemma}
\kern-6pt
\B{proof}
We will show that $H^i(X,\R)$ injects into $H^i(X,\C)/F^2$.
When $X$ is smooth, then $H^i(X)$ has pure weight $i$.
In this case complex conjugation on $H^i(X,\C)$ fixes
$H^i(X,\R)$ but the subspace $F^2H^i(X,\C)$ meets its
conjugate in $0$.
If $X$ is a singular surface, choose a resolution of
singularities $X'\to X$. If $Y$ is a curve containing
the singular locus of $X$, then we are in the situation
of square (\ref{birsquare}). Since
$F^2H^1=F^2H^2=0$ for the curves $Y$ and $Y'$,
the Mayer--Vietoris sequence in Proposition~\ref{M-V-H}
yields $F^2H^i(X,\C)=F^2H^i(X',\C)$ for $i=2,3$.
Comparing the $\R$ and $\C$ structures in the
Mayer--Vietoris long exact sequence of Proposition~\ref{M-V-H}
yields the following diagram has exact rows:
$$\begin{array}{ccccc}
H^1(Y',\R)&\to&H^2(X,\R) & \to & H^2(X',\R)\\
\downarrow & &\downarrow & &\downarrow \\
H^1(Y',\C)&\to& \displaystyle\frac{H^2(X,\C)}{F^2H^2}&
\to &\displaystyle\frac{H^2(X',\C)}{F^2H^2}
\end{array}$$
The right-most vertical arrow in the diagram
is injective because $X'$ is smooth.
A diagram chase shows that the middle vertical
arrow is injective, whence the lemma.
\E{proof}
\subsection*{Normal surfaces}
Consider a surface $X$ with normal singularities;
its singular locus $\Sigma$ is a finite set of closed points.
Choose a desingularization $f\colon X' \to X$ and consider the
exceptional divisor $E=f^{-1}(\Sigma)$; $E$ is the finite disjoint
union of the inverse images $E_{\sigma}\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, f^{-1}(\sigma)$
of the $\sigma\in \Sigma$.
Associated to $f$ is the square (\ref{birsquare}),
with $Y=\Sigma$ and $Y'=E$. Because the fibers $E_\sigma$ of $f$
are connected (by Zariski's Main Theorem), we have
$H^0(\Sigma,\Z)\cong H^0(E,\Z)$.
From Proposition~\ref{M-V-H} we get a
long exact sequence of mixed Hodge structures:
\B{equation}\label{exnorm}
\B{array}{c}
0\to H^1(X,\Z)\by{f^*} H^1(X',\Z)\to
H^1(E,\Z)\to H^2(X,\Z)\by{f^*} \qquad\\ \qquad\qquad
H^2(X',\Z)\to H^2(E,\Z)\to H^3(X,\Z)\by{f^*} H^3(X',\Z) \to 0
\E{array} \E{equation}
\noindent
Recall that if $X$ is proper then
each $H^n(X')$ has pure weight $n$, and
that $W_{n-1}H^n(X,\Q)$ is the kernel of
$f^*\colon H^n(X,\Q)\to H^n(X',\Q)$ by \cite[8.2.5]{D}.
That is, $H^n(X)$ has pure weight $n$ if and only if
$H^n(X,\Q)$ injects into $H^n(X',\Q)$.
There are examples of normal surfaces for which $H^2(X)$
does not have pure weight 2, i.e., with $W_1H^2\neq 0$
({\it cf.\/}\ \cite{BVS1},\cite{BVS2}). The following result
quantifies this impurity.
\B{prop}\label{pure} Let $X$ be a proper normal surface.
If $n\ne2$ then $H^n(X)$ has pure weight $n$. If $n=2$ and
$E$ is the exceptional divisor in a desingularization $X'$, then
$$W_{1}H^2(X,\Q)=\coker H^1(X',\Q)\to H^1(E,\Q).$$
\E{prop}
\B{proof} If $n\ne3$ this follows from the
sequence of mixed Hodge structures (\ref{exnorm}).
For $n=3$ we must show that $H^3(X,\Q)$ embeds in
$H^3(X',\Q)$. Nothing is lost if we replace $X'$
by a quadratic transformation, so
we may assume that all the irreducible components
$E_1,\dots,E_n$ of $E$ are non-singular and that
if $i\neq j$ and $E_i\cap E_j \neq\emptyset$ then
$E_i$ and $E_j$ intersect transversally in exactly one point
not belonging to any other $E_k$.
From (\ref{exnorm}) and the commutative square
$$\B{array}{ccc} \Pic(X') &\by{\cap E} & \Pic(E) \\
\p{c_1}\downarrow& & \downarrow \p{c_1}\\
H^2(X',\Z) &\to & H^2(E,\Z)
\E{array}$$
we see that it suffices to prove that
$\Pic(X')\otimes\Q \to H^2(E,\Q)$ is a surjection.
Now $H^2(E,\Q)=\oplus H^2(E_i,\Q) \cong \Q^n$,
and $\Pic(E)\otimes\Q \by{c_1} H^2(E,\Q)$
is just the degree map.
Moreover, the intersection pairing on the divisors
on $X'$ satisfies $(D\mbox{\Large $\cdot $} E_i) ={\rm deg}(D\cap E_i)$.
Thus if we represent an element of
$\Pic(X')$ by a divisor $D$, its image in
$H^2(E,\Q)\cong\Q^n$ is given by the intersection vector
$(D\mbox{\Large $\cdot $} E_1,\dots,D\mbox{\Large $\cdot $} E_n)$.
Now each $E_i$ represents an element of $\Pic(X')$,
and their intersection vectors form a basis of $H^2(E,\Q)$
because the intersection matrix
$(E_i\mbox{\Large $\cdot $} E_j)$ is negative definite
(see \cite[\S 1]{M} or \cite[14.1]{Lip}).
\E{proof}
\B{cor}\label{abnorm} Let $f\colon X' \to X$ be a
resolution of singularities of a proper normal surface. Then
$J^2(X)$ is an abelian variety, because there is an isomorphism
$$f^*\colon J^2(X)\by{\simeq} J^2(X')$$
\E{cor}
\B{proof} By (\ref{exnorm}) and \ref{pure},
$f^*\colon H^3(X,\Z)\by{} H^3(X',\Z)$
is onto with torsion kernel. \E{proof}
\subsection*{Normalization}
Now let $X$ be a non-normal surface.
The singular locus $\Sigma$ of $X$ is
$1$-dimensional.
Letting $\pi: \tilde X \to X$ denote its normalization,
we have $\pi\colon\tilde X-\tilde\Sigma\by{\simeq}X-\Sigma$,
where $\tilde\Sigma=\pi^{-1}\Sigma$.
By Proposition~\ref{M-V-H},
$\pi$ induces a long exact
sequence of mixed Hodge structures.
\B{equation}\label{exnon-norm}
\B{array}{c}
H^1(X,\Z)\to H^1(\tilde X,\Z)\oplus H^1(\Sigma
,\Z)\to H^1(\tilde\Sigma ,\Z)\to H^2(X,\Z)\to\\
H^2(\tilde X,\Z)\oplus H^2(\Sigma,\Z)\to H^2(\tilde\Sigma ,\Z)
\to H^3(X,\Z) \to H^3(\tilde X,\Z) \to 0 \E{array}
\E{equation}
Since the Hodge structure on $H^2$ of a curve is pure
of type $(1,1)$, the abelian group
$$M = \frac{\coker H^2(\Sigma,\Z)\by{\pi^*}H^2(\tilde\Sigma,\Z)}
{\coker H^2(X,\Z)\by{\pi^*}H^2(\tilde X,\Z)}$$
has a mixed Hodge structure which is pure of type $(1,1)$,
and there is an extension of mixed Hodge structures
\B{equation}\label{exMHS}
0\to M \to H^3(X,\Z) \to H^3(\tilde X,\Z) \to0.
\E{equation}
\B{prop}\label{non-norm} Let $X$ be a proper surface,
with normalization $\pi\colon\tilde X \to X$.
Then we have an extension
$$0\to (\C/\Z)^s \to J^2(X)\by{\pi^*} J^2(\tilde X)\to 0$$
where $s$ is the rank of the abelian group $M$.
\E{prop}
\B{proof} $J^2(X)$ is the cokernel of the natural map
$H^3(X,\Z(2))\to H^3(X,\C)/F^2H^3$. Given this, the result
is a formal consequence of (\ref{exMHS})
and the fact that $F^2M=0$, which implies
that $F^2H^3(X,\C) \cong F^2H^3(\tilde X,\C)$.
We remark that the complex torus $(\C/\Z)^s$ that arises in this
extension is a quotient of the complex torus
$(\C/\Z(2))^s = M\otimes(\C/\Z(2))$ by a finite group.
\E{proof}
\B{cor}\label{1-motive}
Let $f\colon X'\to X$ be a desingularization of a proper
surface $X$, obtained by resolving the singularities of its
normalization $\tilde X$.
Then there is an exact sequence
$$0\to (\C/\Z)^s \to J^2(X)\by{f^*} J^2(X')\to 0$$
where $s$ is the rank of $M$, as in Proposition~\ref{non-norm}.
In particular, if $X'$ has irregularity $q$ then the torsion
subgroup of $J^2(X)$ is isomorphic to $(\Q/\Z)^{2q+s}$.
\E{cor}
Recall from \cite[10.1.2]{D} that a ``1-motive''
$M=(L,A,T,J,u)$ is defined to be an extension $J$
of an abelian variety $A$ by a complex torus $T$, a lattice
$L$ and a homomorphism $L\by{u} J$. Since we may
canonically identify the group of $\C$-points of the
abelian variety $Alb(X')$ with $J^2(X')$,
the conclusion of Corollary~\ref{1-motive}
is just that $J^2(X)$ is part
of a 1-motive $Alb(X)$ in which the lattice $L$ is zero.
\B{defi} Let $X$ be a proper surface over $\C$.
The {\it Albanese 1-motive} of $X$ is the 1-motive $Alb(X)$
given by $$(0,Alb(X'),(\C/\Z(2))^s,J^2(X),{\rm zero}).$$
As the construction in \ref{non-norm} shows,
$Alb$ is a functor from proper surfaces to
1-motives.
\E{defi}
\goodbreak
\subsection*{Torsion in $J^2(X)$}
For simplicity, let us write $\Q/\Z$ for the torsion subgroup
$\Q(2)/\Z(2)$ of $\C/\Z(2)$, so that
$H^i(-,\Q/\Z)\cong H^i(-,\C/\Z(2))_{tors}$.
The maps $H^i(-,\C/\Z(2))\to H_{\cal D}^{i+1}(-,\Z(2))$
of (\ref{star}) induce canonical maps
$$H^i(-,\Q/\Z)\cong
H^i(-,\C/\Z(2))_{tors}\to H_{\cal D}^{i+1}(-,\Z(2))_{tors}.$$
These are the maps in the following Proposition:
\B{prop}\label{HD-tors}
Let $Z$ be a proper scheme over $\C$.
Then
\B{description}
\item[{\it i)}] $H^1(Z,\Q/\Z)\by{\cong} H_{\cal D}^2(Z,\Z(2))_{tors}$
\item[{\it ii)}] If $Z$ is either a curve or a surface then
$$H^2(Z,\Q/\Z)\by{\cong} H_{\cal D}^3(Z,\Z(2))_{tors} $$
\item[{\it iii)}] If $Z$ is a surface then
$$H^3(Z,\Q/\Z)\by{\cong}H_{\cal D}^4(Z,\Z(2))_{tors}\cong
J^2(Z)_{tors}$$
\E{description}
\E{prop}
\B{proof} The first assertion was proven in \ref{sheafHD}.
If $Z$ is a curve then $F^2H^2(Z,\C)=0$, so by
(\ref{star}) we have $H_{\cal D}^3(Z,\Z(2))\cong H^2(Z,\C/\Z(2))$,
and the result is immediate.
We may therefore suppose that $Z$ is a surface, say
with $c$ irreducible components, so that $H^4(Z,\Z)=\Z^c$.
We deduce from (\ref{extJZ}) that
$J^2(Z)_{tors}\cong H_{\cal D}^4(Z,\Z(2))_{tors}$.
Moreover, since $F^2H^4(Z,\C)=H^4(Z,\C)=\C^c$
the sequence (\ref{star}) ends in
\B{equation}\label{star3} H_{\cal D}^3(Z,\Z(2))\to
F^2H^3(Z,\C)\to H^3(Z,\C/\Z(2))\to H_{\cal D}^4(Z,\Z(2))
\by{\varepsilon}\Z^c\to0. \E{equation}
Lemma \ref{filt} states that for $i=2,3$ the image of
$\varepsilon$ in the exact sequence
$$H^{i-1}(Z,\C)/F^2H^{i-1}\to H_{\cal D}^i(Z,\Z(2))\by{\varepsilon}
H^i(Z,\Z)\to H^i(Z,\C)/F^2H^i$$
of (\ref{modf}) is the torsion subgroup $H^i(Z,\Z)_{tors}$.
Combining this with the universal coefficient theorem,
we have a commutative diagram with exact rows:
$$\begin{array}{ccccccc}
0\to\kern-7pt&H^{2}(Z,\C)/F^2H^{2}\oplus H^{2}(Z,\Z)
&\to&H_{\cal D}^3(Z,\Z(2)) &\by{\varepsilon}&H^3(Z,\Z)_{tors}&\to0\\
&\uparrow & &\uparrow & &\mbox{\large $\parallel$} & \\
0\to\kern-7pt&H^{2}(Z,\Z)\otimes\C/\Z(2) &\to &
H^{2}(Z,\C/\Z(2))&\to & H^3(Z,\Z)_{tors}&\to 0.
\end{array}$$
By the five-lemma, sequence (\ref{star}) and (\ref{star3})
we get the extensions
\B{equation}
0\to F^2H^2(Z,\C)\to H^{2}(Z,\C/\Z(2))\to H_{\cal D}^3(Z,\Z(2))\to0,
\E{equation}
\B{equation}
0\to F^2H^3(Z,\C)\to H^3(Z,\C/\Z(2))\to J^2(Z)\to0.
\E{equation}
Since $F^2H^i(Z)$ is uniquely divisible, we may
pass to torsion subgroups.
This proves the remainder of the proposition.
\E{proof}
\medski
\section{Curves}
The singular locus of a reduced surface is usually an
(unreduced) curve. For this reason, we need information
about $K_1$ and $K_2$ of curves in order to study
surfaces. This information is given by theorems
\ref{smooth} and \ref{sing} below.
Part {\it i)\,} of Theorem \ref{smooth}
is of course well-known and
almost classical; a reference is \cite[1.1]{R}.
Since these results are of independent
interest, we have expanded our exposition to include
the case of characteristic $p$.
By a `curve' over a field $k$ we mean a $1$-dimensional
quasiprojective scheme $Y$ over $k$;
a curve is not necessarily reduced.
There is a natural map from $K_1(Y)$ to the
group $H^0(Y,{\cal O}_Y)$ of global units of $Y$; the kernel
of this map is usually written as $SK_1(Y)$.
When $Y$ is a curve there is a natural isomorphism
$SK_1(Y)\cong H^1(Y,{\cal K}_2)$, as well as
a natural short exact sequence
$$0\to H^1(Y,{\cal K}_3)\to K_2(Y)\to H^0(Y,{\cal K}_2)\to 0$$
given by the Brown-Gersten spectral sequence \cite{TT}.
\B{thm}\label{smooth} Let $Y$ be a smooth curve over an
algebraically closed field $k$. Let $r\geq 0$ denote the
number of irreducible components of $Y$ which are proper.
Then \B{description}
\item[{\it i)}] $SK_1(Y)\cong(k^*)^r\oplus V_1$
where $V_1$ is a uniquely divisible group;
\item[{\it ii)}] $K_2(Y)$ and $H^0(Y,{\cal K}_2)$ are both
divisible abelian groups.
\E{description}
\E{thm}
\noindent\B{proof
If $Y$ is a smooth connected curve over an algebraically
closed field $k$ then the localization sequence is
\B{equation}\label{localization}
\coprod_{y\in Y(k)}^{} K_2(k) \to K_2(Y) \to
K_2(k(Y))\by{tame} \coprod_{y\in Y(k)}^{} k^* \to
SK_1(Y)\to 0
\E{equation}
and the image of $K_2(Y)$ in $K_2(k(Y))$ is
$H^0(Y,{\cal K}_2)$. Since $\coprod K_2(k)$
is divisible \cite[1.3]{BT}, the divisibility of $K_2(Y)$
is equivalent to the divisibility of $H^0(Y,{\cal K}_2)$. If
char$(k)=0$, the result now follows from Suslin's exact
sequence \cite[4.4]{S2} for $n$ invertible in $k$:
$$0\to H^0(Y,{\cal K}_2)/n\to \Het{2}(Y,\mu_n^{\otimes2})\to {}_nSK_1(Y) \to0$$
Indeed, if $Y$ is affine then $\Het{2}(Y)=0$,
and if $Y$ is projective then the composite $\mu_n \cong
\Het{2}(Y,\mu_n^{\otimes2})\to SK_1(Y)\to k^*$
is the standard inclusion.
If char$(k)=p>0$, we need only a slight additional argument.
Because $k(Y)$ is the function field of a curve,
we know from \cite[p.391]{BT} that $K_2(k(Y))$ is
$p$-divisible, and from \cite[1.10]{S2}
(which is implicit in \cite[p.397]{BT}) that it
has no $p$-torsion. Hence both $K_2(k(Y))$ and
$\coprod k^*$ are uniquely $p$-divisible groups, {\it i.e.,\/}\
$\Z[\frac{1}{p}]$-modules. It follows that both the
kernel $H^0(Y,{\cal K}_2)$ and cokernel $SK_1(Y)$ of the
`tame symbol' map in (\ref{localization}) must be
uniquely $p$-divisible. This proves Theorem~\ref{smooth}
in characteristic $p$.
\E{proof}
\B{lemma} Let $Y$ be a smooth connected projective curve
over $\C$. Then
$$c_2\colon SK_1(Y) \to H_{\cal D}^3(Y,\Z(2))\cong \C^*$$
is a split surjection.
In particular, it is an isomorphism on torsion subgroups.
\E{lemma}
\B{proof} This is implicit in p.219 of Gillet's paper \cite{GDuke}.
The isomorphism $H_{\cal D}^3(Y,\Z(2))\cong \C/\Z(2)\cong\C^*$
follows from (\ref{star}) or \ref{HD-tors}.
If $y\in Y(\C)$ is considered as an
element of $Pic(Y)$ and $z\in\C^*$ then we can form
$\{ y,z\}\in SK_1(Y)$ and the product formula yields
$c_2(\{ y,z\})=-c_1(y)\cup c_1(z) = z^{-1}\in \C^*$.
\E{proof}
\B{thm}\label{sing} Let $Y$ be any curve over an
algebraically closed field $k$,
and let $r\geq 0$ denote the number of irreducible
components of $Y$ which are proper. Then \B{description}
\item[{\it i)}] If char$(k)=0$, or if $Y$ is reduced, then
$$SK_1(Y)\cong (k^*)^r\oplus V_1,$$
where $V_1$ is a uniquely divisible abelian group;
\item[{\it ii)}] If char$(k)=p>0$ then
$$SK_1(Y)\cong (k^*)^r\oplus V_1\oplus P,$$
where $V_1$ is uniquely divisible and $P$ is a
$p$-group of bounded exponent.
\item[{\it iii)}] If $k=\C$ then the Chern class
$$c_2\colon SK_1(Y) \to H_{\cal D}^3(Y,\Z(2))\cong (\C^*)^r$$
is a split surjection.
In particular, it is an isomorphism on torsion subgroups.
\E{description}
\E{thm}
\noindent\B{proof
We proceed in three steps.\\
{\it Step 1.}
Suppose that $Y$ is any reduced curve over $k$. If we
pick $r$ smooth points $y_i$ on $Y$, one on each
proper component of $Y$, then $Y_0=Y-\{y_1,\ldots ,y_r\}$
is affine. The localization sequence for $Y_0\subset Y$ is
\B{equation}\label{notloc}
\coprod_{i=1}^{r} K_2(k) \to K_2(Y) \to
K_2(Y_0)\to \coprod_{i=1}^{r} k^* \to
SK_1(Y)\to SK_1(Y_0)\to 0
\E{equation}
If $\tilde Y$ is the normalization of $Y$, then we may
indentify the $y_i$ with points on the smooth curve
$\tilde Y$. By the smooth case \ref{smooth}, the composition of
$$\coprod k^* \to SK_1(Y)\to SK_1(\tilde Y)$$
is an injection. Hence $SK_1(Y)$ is the direct sum of
$\coprod k^*$ and $SK_1(Y_0)$ while $K_2(Y)$ is the direct
sum of the image of the divisible group $\coprod K_2(k)$
and the group $K_2(Y_0)$.
Part (iii) now follows from the above Lemma.
This argument also shows that we may replace $Y$ by
$Y_0$ in proving parts (i) and (ii) of Theorem~\ref{sing}
for reduced curves.\\
{\it Step 2.} Now suppose that $Y= \mbox{Spec}(A)$ is any
reduced affine curve over $k$. Let $B$ be the
normalization of $A$, and $I$ the conductor ideal from
$B$ to $A$. By \cite[3.1 and 4.2]{GR}, excision holds for
$K_1$ and there is an exact sequence
$$K_2(B)\oplus K_2(A/I)\to K_2(B/I)\to SK_1(A)\to
SK_1(B)\to 0$$
Since $B$ is a finite product of Dedekind domains, $B/I$
is a finite principal ideal ring. By Corollary~\ref{k2div}
below, $K_2(B/I)$ is uniquely divisible.
By Theorem~\ref{smooth}, $SK_1(B)$ is uniquely divisible
and $K_2(B)$ is divisible. Finally, since $A/I$ is finite
dimensional, we know from Corollary~\ref{k2div} that
$K_2(A/I)$ is uniquely divisible (modulo bounded $p$-torsion if
char$(k)=p\neq 0$). A diagram chase shows that $SK_1(A)$ is
uniquely divisible (modulo bounded $p$-torsion if $p\ne0$).
This proves theorem \ref{sing} for reduced curves.
\B{lemma}\label{nilpo} Let $I$ be a nilpotent ideal in an
algebra $A$ over a field $k$.
\B{description} \item[{\it (a)}] If char$(k)=0$,
$K_n(A,I)$ is a uniquely divisible group for every $n$.
\item[{\it (b)}] If char$(k)=p>0$, $K_2(A,I)$ is a
$p$-group of bounded exponent.
\E{description}
\E{lemma}
\B{proof} Part {\it (a)\,} is proven in
\cite[1.4]{W966}.
If char$(k)=p$, chose $m$ such that
$I^{p^m}=0$; we will show that $p^mK_2(A,I)$. Indeed,
$K_2(A,I)$ is generated by Steinberg symbols $\{a,1+x\}$
with $a\in A$ and $x\in I$, and $p^m\,\{a,1+x\}$ is
$\{a,1+x^{p^m}\}=\{a,1\}=0$.
\E{proof}
\B{cor}\label{k2div} Let $A$ be a finite algebra over
an algebraically closed field $k$.
\B{description} \item[{\it (a)}] If char$(k)=0$ or if $A$
is a principal ideal ring, the group $K_2(A)$ is uniquely divisible.
\item[{\it (b)}] If char$(k)=p$, $K_2(A)$ is the sum of
the uniquely divisible group $K_2(A_{red})$ and a
$p$-group of bounded exponent.
\E{description}
\E{cor}
\B{proof} Let $I$ be the nilradical of $A$, so that
$A_{red}=A/I$ is semisimple and hence $A\to A_{red}$
splits. Then $K_2(A)\cong K_2(A_{red})\oplus K_2(A,I)$,
and $K_2(A_{red})$ is uniquely divisible by
\cite[1.3]{BT}. Finally, if $A$ is a principal ideal
ring then $A$ is a product of truncated polynomial rings
$k[s]/(s^n)$ and a direct calculation (\cite[p.485]{Graham})
shows that $K_2(k[s]/(s^n))\cong K_2(k)$.
\E{proof}
\noindent{\it Step 3.} Finally, suppose that $Y$ is a curve
which is not reduced.
Let ${\cal I}$ denote the nilradical
ideal sheaf of ${\cal O}_Y$, and write ${\cal K}_2{\cal I}$ for the
sheafification of the presheaf
$U\mapsto K_2({\cal O}_Y(U),{\cal I}(U))$.
If ${\cal K}_{2,red}$ denotes the sheafification of
$U\mapsto K_2(U_{red})$, there is an
exact sequence of sheaves on $Y_{Zar}$
\B{equation}\label{Ired}
{\cal K}_2{\cal I}\to {\cal K}_2\to {\cal K}_{2,red}\to 0
\E{equation}
Let $U$ denote the smooth locus of $Y_{red}$. Since
$U_{red}$ is smooth, the ring map ${\cal O}_U\to{\cal O}_{U_{red}}$
splits. Therefore ${\cal K}_2{\cal I}\mid_U$ injects into
${\cal K}_2\mid_U$, {\it i.e.,\/}\ the kernel of ${\cal K}_2{\cal I}\to{\cal K}_2$ is a
skyscraper sheaf supported on $Y-U$. It follows that we
have an exact sequence
$$H^0(Y,{\cal K}_{2,red})\to H^1(Y,{\cal K}_2{\cal I})\to H^1(Y,{\cal K}_2)\to
H^1(Y_{red},{\cal K}_2)\to 0$$
which we may rewrite as follows
\B{equation}\label{reduced}
K_2(Y_{red})\to H^1(Y,{\cal K}_2{\cal I})\to SK_1(Y)\to
SK_1(Y_{red})\to 0
\E{equation}
By Step 2, $SK_1(Y_{red})$ is uniquely divisible. If
char$(k)=p$, we know by Lemma~\ref{nilpo}(b) that
$H^1(Y,{\cal K}_2{\cal I})$ is a $p$-group of bounded exponent,
and this part {\it (ii)\,} of Theorem~\ref{sing} because
a uniquely divisible group has no nontrivial extensions
by a $p$-group. Finally, suppose that char$(k)=0$.
By Lemma~\ref{nilpo}(a), $H^1(Y,{\cal K}_2{\cal I})$ is uniquely
divisible. By Proposition~\ref{n-divisible} below,
$K_2(Y_{red})$ is divisible.
In this case part {\it (i)\,} of Theorem~\ref{sing}
follows from Step 2 and the exact sequence (\ref{reduced}).
\E{proof}
\B{prop}\label{n-divisible} If $\frac{1}{n}\in k$ and
$Y$ is a curve then $K_2(Y)$ is $n$-divisible.
\E{prop}
\B{proof} We consider the $K$-theory of $Y$ with
coefficients $\Z/n$, which is related to the usual
Quillen $K$-theory of $Y$ by exact sequences such
as
$$0\to K_2(Y)\otimes\Z/n\to K_2(Y;\Z/n)\to
K_1(Y)_{n-tors}\to 0$$
We know by \cite[1.4]{W966} that $K_2(Y;\Z/n)\cong
K_2(Y_{red};\Z/n)$, and hence that $K_2(Y)\otimes\Z/n$ is
a subgroup of $K_2(Y_{red})\otimes\Z/n$. Thus we may
assume that $Y$ is reduced. Let $\tilde Y$ be the
normalization of $Y$. The conductor ideal defines a
zero-dimensional subscheme Spec($C$) of $Y$, and also its
preimage Spec($D$) in $\tilde Y$. Because excision holds
(see \cite[1.2]{W966}) we have an exact sequence
$$K_3(D;\Z/n)\to K_2(Y;\Z/n)\to K_2(\tilde Y;\Z/n)\oplus
K_2(C;\Z/n)$$
Now $K_3(D;\Z/n)\cong K_3(D_{red};\Z/n)$, again by
\cite[1.4]{W966}. Since $D_{red}$ is a finite product of
copies of $k$, and $K_3(k;\Z/n)=0$ by Suslin \cite{Suslin},
we
have $K_3(D;\Z/n)=0$. Hence $K_2(Y;\Z/n)$ injects into
$K_2(\tilde Y;\Z/n)\oplus K_2(C;\Z/n)$. By naturality,
the subgroup $K_2(Y)\otimes\Z/n$ of $K_2(Y;\Z/n)$ injects
into the corresponding subgroup $K_2(\tilde Y)\otimes\Z/n
\oplus K_2(C)\otimes\Z/n$ of $K_2(\tilde Y;\Z/n)\oplus
K_2(C;\Z/n)$, but: $K_2(\tilde Y)$ is divisible by
Theorem~\ref{smooth} and $K_2(C)$ is divisible by
Corollary~\ref{k2div}, so this latter subgroup is zero,
hence $K_2(Y)\otimes\Z/n =0$ as claimed.
\E{proof}
\section{$K$-theory results}
In this section we collect some results on the relation
between the Zariski sheaves ${\cal K}_2$ and ${\cal H}^q(\mu_n^{\otimes2})$,
namely \ref{HoobH2}, \ref{nK2} and \ref{square},
which will be used in the proof of the main theorem.
In this section, our field $k$ will always contain $\frac1n$.
The first result, which we cite without proof,
concerns the sheafification of the \'etale Chern class
$c_2^{et}\colon K_2(X)/n\to\Het2(X,\mu_n^{\otimes2})$.
It is an extension by Hoobler of a
well known result for smooth schemes.
\B{prop}\label{HoobH2}
(Hoobler \cite{Hoob}; {\it cf.\/}\ \cite[Thm. 0.2]{PW3})
Let $X$ be a scheme of finite type over a
field containing $\frac1n$.
Then the \'etale Chern class $c_2^{et}$ induces
an isomorphism of Zariski sheaves on $X$: \quad
${\cal K}_2/n \by{\simeq} {\cal H}^2(\mu_n^{\otimes2})$.
\E{prop}
Our other results concern the n-torsion subsheaf
${}_n{\cal K}_2$ of ${\cal K}_2$. We begin with the local version.
\B{lemma}\label{HoobH1} Let $A$ be a semilocal ring
essentially of finite type over a field $k$.
Assume $k$ contains a primitive $n^{th}$ root of unity $\zeta$.
Define a map $$\varphi:
\Het{1}(A,\mu_n^{\otimes2})\cong A^*/A^{*n}\to{}_nK_2(A)$$ by
$\varphi(a)=\{a,\zeta\}$, $a\in A^*$. Then
$\varphi$ is surjective. If $A$ is regular and $k$ contains
an algebraically closed field
then $\varphi$ is an isomorphism.
\E{lemma}
\B{proof} $\varphi$ is well defined because
$\{ a^n,\zeta\}=\{ a,1\}=0$. Suppose first that $A$ is regular.
Then $\varphi$ is onto by the Merkurev-Suslin Theorem.
If in addition its field of fractions $F$ contains an
algebraically closed field, then
$\Het{1}(F,\mu_n^{\otimes2})\cong{}_nK_2(F)$
by \cite[3.7]{S2}. Comparing the Bloch-Ogus resolution of
$\Het{1}(A,\mu_n^{\otimes2})$ to the Gersten--Quillen
resolution of ${}_nK_2(A)$, one gets that
$\Het{1}(A,\mu_n^{\otimes2})\cong {}_nK_2(A)$.
The promotion to any semilocal ring $A$ follows from the same
arguments used by R.Hoobler in \cite{Hoob}. Since $A$ is a
localization of a finitely generated $k$-algebra, there
exists localization $B$ of a polynomial ring over $k$
and an ideal $I$ in $B$ such that $A=B/I$.
Let $(B^h,I^h)$ be the henselization of the pair $(B,I)$.
As $B^h$ is a direct limit of semilocal regular rings
finite over $B$,
the map $\Het{1}(B^h,\mu_n^{\otimes2})\by{\varphi}{}_n{\cal K}_2(B^h)$
is an isomorphism. By a result of O. Gabber \cite[Th.1]{Gabber}
we have $K_3(B^h;\Z/n)\cong K_3(B^h/I^h;\Z/n)\cong K_3(A;\Z/n)$.
By proper base change
$$\Het{1}(B^h,\mu_n^{\otimes2})\cong
\Het{1}(B^h/I^h,\mu_n^{\otimes2})\cong
\Het{1}(A,\mu_n^{\otimes2}).$$
The universal exact sequence for $K$-theory with
coefficients yields a commutative diagram
$$\begin{array}{ccccccc}0\to & K_3(B^h)\otimes\Z/n&\to
&K_3(B^h;\Z/n)&\to &{}_nK_2(B^h)&\to 0\\
&\downarrow & &\mbox{\large $\parallel$} & &\downarrow & \\
0\to&K_3(A)\otimes\Z/n &\to&
K_3(A;\Z/n)&\by{}&{}_nK_2(A)&\to 0
\end{array}$$
Thus the right-most vertical arrow is surjective.
We then conclude from commutativity of the diagram:
$$\begin{array}{ccc}
\Het{1}(B^h,\mu_n^{\otimes2}) &\by{\simeq}&{}_nK_2(B^h)\\
\mbox{\large $\parallel$} & &\quad\downarrow\mbox{\rm onto}\\
\Het{1}(A,\mu_n^{\otimes2}) &\by{\varphi}&{}_nK_2(A)\
\end{array}$$
\kern-14pt \E{proof}
\B{schol}\label{phi-bar} Let $A$ and $k$ be as in
Lemma~\ref{HoobH1}. If $n$ is even, assume that $k$
contains a square root of $-1$.
If $\beta$ is a Bott element in $K_2(k;\Z/n)$ mapping to
$\zeta\in K_1(k)=k^*$, then
multiplication by $\beta$ lifts the map $\varphi$ to a map
$$\bar\varphi\colon\Het{1}(A,\mu_n^{\otimes2})\to K_3(A;\Z/n).$$
This map is a split injection, because the \'etale Chern class
satisfies $c^{et}_2\bar\varphi=-1$.
\E{schol}
\B{proof} The assumption on $k$ implies that $\beta$ exists
and has order $n$, so $\bar\varphi(a)=\{ a,\beta\}$ is
well-defined and lifts $\varphi$.
The product formula (see \cite[Theorem 3.2(ii)]{W-chern}) states
that $c_2(\{a,\beta\})=-[a]\otimes\zeta$
in $A^*/A^{*n}\otimes\mu_n(k) \cong \Het{1}({\rm Spec}\,(A),\mu_n^{\otimes2})$
for every $a\in A^*$.
Thus up to sign $c_2$ is a left inverse of $\bar\varphi$.
\E{proof}
\B{thm}\label{nK2}
Let $X$ be a scheme of finite type over $\C$.
Then the \'etale Chern class defines
an isomorphism of Zariski sheaves:
$$c^{et}_2\colon\; {}_n{\cal K}_2 \by{\simeq} {\cal H}^1(\mu_n^{\otimes2}).$$
\E{thm}
\B{proof} We saw in (2.4.2) that
$c^{et}_2\colon {\cal K}_3(\Z/n) \to {\cal H}^1(\mu_n^{\otimes2})$ vanishes on ${\cal K}_3$.
Hence $c^{et}_2$ is well-defined on ${}_n{\cal K}_2$.
To verify that it is an isomorphism, we check the stalks at a
point $x\in X$. If $A={\cal O}_{X,x}$ we see from \ref{phi-bar}
that the surjection $\varphi\colon \Het1(A,\mu_n^{\otimes2})\to{}_nK_2(A)$
of Lemma~\ref{HoobH1} satisfies $c^{et}_2\ \varphi=-1$.
Elementary algebra now implies that
$c^{et}_2$ is an isomorphism on ${}_nK_2(A)$
and hence on ${}_n{\cal K}_2$.
\E{proof}
\B{cor}\label{square} By (\ref{et-HD-sheaf}),
the following diagram commutes:
$$
\B{array}{ccc} {}_n{\cal K}_2 &\longby\tau& \kern-4pt{\cal K}_2 \\
c^{et}_2\downarrow\cong & &\downarrow{c_2}\\
{\cal H}^1(\mu_n^{\otimes2}) &\longby{\delta}& {\cal H}_{\cal D}^2(2)
\E{array}$$
where $\tau$ is the obvious inclusion
and $\delta$ is defined in (\ref{triangle}) and (\ref{Rq}).
\E{cor}
\B{rmk} This gives the following explicit formula for $\delta$.
Given a unit $a\in A^*$, where $U={\rm Spec}\,(A)$, write $[a]$
for the class of $c_1(a)$ in
$H_{an}^1(U_{\mbox{\Large $\cdot $}},\Z(1)_{\cal D})\cong
H_{an}^0(U_{\mbox{\Large $\cdot $}},{\cal O}_{U_{\mbox{\Large $\cdot $}}}^*)$.
Then the product formula for $c_2$ shows that $\delta$ sends
$a\otimes\zeta\in{\cal O}_X^*(U)\otimes\mu_n\cong{\cal H}^1(\mu_n^{\otimes2})(U)$
to $[\zeta]\cup[a]\in H_{an}^2(U_{\mbox{\Large $\cdot $}},\Z(2)_{\cal D})={\cal H}_{\cal D}^2(U)$.
\E{rmk}
\section{An exact sequence for ${\cal K}_2$-cohomology}
We now give some exact sequences relating $H^1(X,{\cal K}_2)$
and $H^2(X,{\cal K}_2)$. The first is a reinterpretation of
\cite[Theorem D]{PW3} in terms of hypercohomology.
Let ${\cal K}_2^\bullet$ denote the complex ${\cal K}_2\by{n}{\cal K}_2$
concentrated in degrees 0 and 1. The short exact sequence
$0\to{\cal K}_2[-1]\to{\cal K}_2^\bullet\to{\cal K}_2\to0$
gives rise to a long exact sequence, reminiscient of
\cite[(4.4)]{S2}:
\B{equation}\B{array}{c}
0\to H^0(X,{\cal K}_2)/n \by\iota \H^1(X,{\cal K}_2^\bullet)
\by{} H^1(X,{\cal K}_2)\by{n}H^1(X,{\cal K}_2)_{\mathstrut} \by\iota \\
\qquad\H^2(X,{\cal K}_2^\bullet)\to H^2(X,{\cal K}_2) \by{n}
H^2(X,{\cal K}_2) \to\cdots^{\mathstrut}\E{array}
\E{equation}
From this we extract short exact ``Kummer'' sequences, such as
\B{equation}\label{Kummer}
0\to H^1(X,{\cal K}_2)/n \by\iota \H^2(X,{\cal K}_2^\bullet)
\by{\pi} {}_nH^2(X,{\cal K}_2) \to0.
\E{equation}
We also have the exact sequence of low degree terms in
the hypercohomology spectral sequence for ${\cal K}_2^\bullet$,
the relevant part of which is:
\B{equation}\label{lowdeg}
H^0(X,{\cal K}_2/n)\vlongby{d_2} H^2(X,{}_n{\cal K}_2) \to
\H^2(X,{\cal K}_2^\bullet)\by\eta H^1(X,{\cal K}_2/n) \to H^3(X,{}_n{\cal K}_2).
\E{equation}
\B{prop}\label{hyperK2}(\cite[Theorem D]{PW3})
Let $X$ be a quasi-projective over a field $k$ containing $\frac1n$.
Then:
\B{description} \item[{\it (a)}] The $d_2$-differential
$H^0(X,{\cal K}_2/n) \longby{d_2} H^2(X,{}_n{\cal K}_2)$
in the hypercohomology spectral sequence (\ref{lowdeg})
is the composite
$$H^0(X,{\cal K}_2/n) \by{\partial} H^1(X,n\cdot{\cal K}_2)
\by{\partial} H^2(X,{}_n{\cal K}_2)$$
of the boundary maps in the usual interlocking sequences
for ${\cal K}_2$.
\item[{\it (b)}] If $X$ is a surface with isolated singularities,
the map $\pi$ in the Kummer sequence (\ref{Kummer}) for
$\H^2(X,{\cal K}_2^\bullet)$ factors through the surjection $\eta$ in
the hypercohomology spectral sequence (\ref{lowdeg}).
\E{description}
\E{prop}
\B{proof} Part (a) is a special case of a more general result
which we have isolated in Lemma~\ref{hyper} below.
For part (b), it suffices to show that
the following diagram commutes.
$$\begin{array}{ccccccccc}
H^0(X,{\cal K}_2/n)\kern-5pt&
\by{\gamma}\kern-5pt&
H^2(X,{}_n{\cal K}_2)\kern-8pt &\by{\beta}\kern-7pt&
H^1(X,{\cal K}_2)/n\kern-7pt&\to\kern-7pt&H^1(X,{\cal K}_2/n)\kern-5pt &
\to\kern-5pt& {}_nH^2(X,{\cal K}_2) \kern-2pt\to0\\
\mbox{\large $\parallel$} &
&\mbox{\large $\parallel$} &&\iota\downarrow &&\mbox{\large $\parallel$} &&\\
H^0(X,{\cal K}_2/n)\kern-5pt&
\by{d_2}\kern-5pt&
H^2(X,{}_n{\cal K}_2)\kern-8pt &\to\kern-7pt&
\H^2(X,{\cal K}_2^\bullet
\kern-7pt&\to\kern-7pt& H^1(X,{\cal K}_2/n)
\kern-5pt&\to\kern-5pt&\ 0\hfill\\
\end{array} $$
The top row is the exact sequence of
\cite[Theorem D]{PW3}, the bottom row is the exact sequence of
low degree terms (\ref{lowdeg}) and the vertical arrow $\iota$
comes from the Kummer sequence (\ref{Kummer}).
Since ${\cal K}_2\to{\cal K}_2/n$ factors through ${\cal K}^\bullet[1]$,
the right square commutes. The left square commutes by part (a).
The map $\beta$ is constructed as follows.
Let ${\cal L}^\bullet$ denote the subcomplex
${\cal K}_2\to n\cdot{\cal K}_2$ of ${\cal K}_2^\bullet$;
${\cal L}^\bullet$ is quasi-isomorphic to ${}_n{\cal K}_2$.
The inclusion of
$n\cdot{\cal K}_2[-1]$ into ${\cal L}^\bullet$ induces a natural map
$H^1(X,n\cdot{\cal K}_2)\to H^2(X,{}_n{\cal K}_2)$, and
we know that this map is onto by \cite[4.8.1]{PW3}.
We showed in \cite[Proposition 4.9]{PW3} that
$H^1(X,n\cdot{\cal K}_2)\to H^1(X,{\cal K}_2)$ factors through
this surjection, and the induced map is $\beta$.
Thus $\iota\beta$ is induced from the composite map
$n\cdot{\cal K}_2\to{\cal K}_2\to{\cal K}_2^\bullet[1]$
upon taking $H^1$. But this composite map
factors through the subcomplex ${\cal L}^\bullet[1]$
of ${\cal K}^\bullet[1]$, so it follows that the left square commutes.
\E{proof}
Here is the general result about hypercohomology
spectral sequences used to prove part (a) above.
It works for any topos $X$.
\B{lemma}\label{hyper} For any sheaf ${\cal F}$, let ${\cal C}^\bullet$
denote the complex ${\cal F} \by{n} {\cal F}$ concentrated in degrees 0 and 1.
Then up to the sign $(-1)^{p}$, the $d_2$-differentials
$$H^p(X,{\cal F}/n)=H^p(X,H^1{\cal C}) \longrightarrow
H^{p+2}(X,H^0{\cal C})=H^{p+2}(X,{}_n{\cal F})$$
in the hypercohomology spectral sequence of ${\cal C}^\bullet$
are the composites
$$H^p(X,{\cal F}/n)\by{\partial}H^{p+1}(X,n\cdot{\cal F})
\by{\partial}H^{p+2}(X,{}_n{\cal F})$$
of the boundary maps $\partial$ associated
respectively to the exact sequences
$$0\to n\cdot{\cal F}\to{\cal F}\to{\cal F}/n\to0,\qquad
0\to{}_n{\cal F}\to{\cal F}\to n\cdot{\cal F}\to0$$
\E{lemma}
\B{proof}
Given injective resolutions ${}_n{\cal F}\to{\cal I}^\bullet$,
$n\cdot{\cal F}\to{\cal J}^\bullet$ and ${\cal F}/n{\cal F}\to{\cal K}^\bullet$
we can form injective resolutions
${\cal F}\to{\cal E}^{0\bullet}={\cal I}^\bullet\oplus{\cal J}^\bullet$ and
${\cal F}\to{\cal E}^{1\bullet}={\cal J}^\bullet\oplus{\cal K}^\bullet$
using the Horseshoe Lemma.
These form the two columns of a Cartan-Eilenberg resolution
${\cal E}^{\bullet\bullet}$ of the complex ${\cal F}\by{n}{\cal F}$;
by the sign trick, the single horizontal differential
in this complex is $(-1)^{p}$ times the projection/inclusion
$I^p\oplus{\cal J}^p\to {\cal J}^p\to {\cal J}^p\oplus{\cal K}^p$.
Given a class $[s]\in H^p(X,{\cal F}/n)$, represent it by
$s\in H^0(X,K^p)$ with $\partial s=0$ in $H^0(X,K^{p+1})$.
Applying $\partial^v$ to $(0,s)\in H^0(X,J^p\oplus K^p)$
gives an element $(t,0)$ of $H^0(X,J^{p+1}\oplus K^{p+1})$.
Thus $\partial\colon H^p(X,{\cal F}/n)\to H^{p+1}(X,n\cdot{\cal F})$
sends $[s]$ to $[t]$. Applying $\partial^v$
to $(0,t)\in H^0(X,I^p\oplus J^p)$ gives $(u,0)$ for some
$u\in H^0(X,I^{p+1})$. By construction, $u$ is a cycle in
$I^\bullet$ and
$\partial\colon H^{p+1}(X,n\cdot{\cal F})\to H^{p+2}(X,{}_n{\cal F})$
sends $[t]$ to $[u]$.
Now the hypercohomology spectral sequence arises from the row
filtration on the Cartan-Eilenberg resolution ${\cal E}^{\bullet\bullet}$.
Since the pair $((0,(-1)^{p}t),(0,s))\in\mbox{Tot}^p({\cal I})$
has $(((-1)^pu,0),(0,0))$ for its boundary,
the $d_2$-differential in the spectral sequence
takes $[s]$ to $(-1)^{p}[u]$.
\E{proof}
We are now going to connect Proposition~\ref{hyperK2}
with \'etale cohomology using $c^{et}_2$. For this we need to
resort to some standard topological constructions.
Our main result will be Theorem~\ref{NH3} below.
Recall from (2.4.1) that there is a simplicial presheaf $K$
on $X_{zar}$ such that $\pi_qK(U)=K_{q}(U)$.
Let $\tilde K(U)$ be the universal covering space of the
basepoint component of $K(U)$; $\tilde K$ is a simplicial presheaf
by \cite[8.3 or 16.4]{May}. Let $\tilde K^{(2)}(U)$ denote
the second layer of the Postnikov tower of $\tilde K(U)$,
defined in \cite[8.1]{May}; it is an Eilenberg-MacLane complex
of type $(K_2U,2)$ and $\tilde K^{(2)}$ is a simplicial presheaf.
Moreover by \cite[8.2]{May} there are Kan fibrations
$\tilde K^{(2)} \leftarrow \tilde K \to K$.
Now let $\tilde L(U)$ denote the homotopy fiber of the map
$\tilde K(U) \by{n} \tilde K(U)$,
and let $M(U)$ denote the homotopy fiber of the map
$\tilde K^{(2)}(U) \by{n} \tilde K^{(2)}(U)$.
Each $\tilde L(U)$ is a connected space with
$\pi_1\tilde L(U)=K_2(U)/n$
and $\pi_q\tilde L(U)=K_{q+1}(U;\Z/n)$ for $q\ge2$,
while $M(U)$ has only two nontrivial homotopy groups:
$\pi_1M(U)=K_2(U)/n$ and $\pi_2M(U)={}_nK_2(U)$.
In fact, it is not hard to see that $M(U)$ is homotopy
equivalent to the simplicial space obtained by applying the
Dold-Kan theorem to the presheaf of chain complexes
$K_2 \by{n} K_2$ concentrated in degrees 2 and 1.
We can perform the above constructions so that
there is a commutative diagram of simplicial presheaves
(in which the diagram (\ref{fib}) forms the right side):
\B{equation}\label{grid}\B{array}{ccccccc}
M &\leftarrow& \tilde L &\to& L &\vlongby{C^{ss}_2}& \Omega{\cal E} \\
\downarrow && \downarrow && \downarrow &&\delta\downarrow\enskip\\
\tilde K^{(2)}&\leftarrow& \tilde K &\to& K &\longby{C_2^{ss}}&{\cal D}\\
n\downarrow\enskip&&n\downarrow\enskip&&n\downarrow\enskip&&
n\downarrow\enskip\\
\tilde K^{(2)}&\leftarrow& \tilde K &\to& K &\vlongby{C_2^{ss}}&{\cal D}
\E{array}\E{equation}
\B{num}
Given any simplicial presheaf $F$ on $X$, the
{\it generalized sheaf cohomology groups} $\H^q(X,F)$
were defined for $q\le0$ by Brown and Gersten \cite[p.280]{BG}.
(The homotopy categories of simplicial presheaves and
simplicial sheaves are equivalent by \cite[2.8]{J}.
In particular, if $\tilde F$ is the simplicial sheaf
associated to $F$ then $\H^q(X,F)=\H^q(X,\tilde F)$.)
If $F$ is the simplicial sheaf associated by the Dold-Kan theorem
to a cochain complex ${\cal F}$ (concentrated in negative degrees),
then $\H^q(X,F)\cong \H_{zar}^{q}(X,{\cal F})$ for $q\le0$
by \cite[p.281]{BG}. Since the simplicial sheaf associated to
${\cal K}_2^\bullet[2]$ is the sheafification of $M$ we have
$\H^q(X,M)=\H_{zar}^{q+2}(X,{\cal K}_2^\bullet)$.
Similarly, by (2.4.2) we have
$$\H^q(X,\Omega{\cal E})=\H^{q-1}(X,{\cal E})\cong \Het{3-q}(X,\mu_n^{\otimes2}).$$
In particular, diagram (\ref{grid}) induces maps
\B{equation}\label{branch}
\H^2(X,{\cal K}_2^\bullet)=\H^0(X,M) \stackrel\lambda\leftarrow
\H^0(X,\tilde L)\longby{c_2^{et}} \H^0(X,\Omega{\cal E}) = \Het3(X,\mu_n^{\otimes2}).
\E{equation}
If $F$ is a simplicial presheaf on $X$, we write $\tilde\pi_qF$
for the sheaf associated to the presheaf $U\mapsto \pi_qF(U)$.
For example, we have
$$\tilde\pi_qM = \cases{{\cal K}_2/n& if $q=1$; \cr
{}_n{\cal K}_2& if $q=2$; \cr
0 & else.\cr} \qquad
\tilde\pi_q\tilde L=\cases{{\cal K}_2/n& if $q=1$; \cr
{\cal K}_{q+1}(\Z/n)& if $q\ge2$; \cr
0 & else.\cr}
$$
Now recall that $X$ is quasi-projective over $\C$.
By \cite[Theorem 3]{BG} there is a ``Brown-Gersten''
spectral sequence in the fourth quadrant:
$$E_2^{pq}=H_{zar}^p(X,\tilde\pi_{-q}F)
\Rightarrow\H^{p+q}(X,F).$$
In general, this spectral sequence is ``fringed''
\cite[p.285]{BG}, but since all the $F$ we consider here
are infinite loop spaces this fringing does not affect
$\H^q(X,F)$ for $q\le0$.
\E{num}
\B{example}
Here is an example of the fringing phenomenon.
If $F$ is associated to a cochain complex ${\cal F}$,
with ${\cal F}^q=0$ for $q>0$, then
it is well known that the Brown-Gersten spectral sequence
for $F$ is the same as the hypercohomology spectral sequence
for ${\cal F}$. For example, the simplicial sheaf ${\cal E}$ was defined
in (2.4.2) as being associated to $\tau^{\le0}\R\omega_*\Z/n[2i]$.
The hypercohomology spectral sequence of this complex
coincides with the Leray spectral sequence
for $\Het{2i+*}(X,\mu_n^{\otimes i})$ in the region $q\le0$.
Thus it is a fringed spectral sequence converging in the region
$p+q\le0$. The line $p+q=+1$ converges to the kernel of
$\Het{2i+1}(X,\mu_n^{\otimes2})\to H^0(X,{\cal H}^{2i+1}(\mu_n^{\otimes i}))$.
On the other hand, the sheafification $\tilde M$ of $M$ is
associated to the complex of sheaves ${\cal K}_2^\bullet[2]$.
Hence the Brown-Gersten spectral sequence for $M$
is the same as the hypercohomology spectral sequence for
${\cal K}_2^\bullet[2]$, and there is no fringing effect.
\E{example}
\medskip
Any morphism $E\to F$ of simplicial presheaves induces a morphism of
Brown-Gersten spectral sequences.
Thus (\ref{branch}) gives us a commutative diagram:
\def\rod{\stackrel{c^{et}_2\quad\cong}{\kern-20pt
\hbox to115pt{\rightarrowfill}\kern-35pt}}
\B{equation}\label{maze}\B{array}{ccccc}
H^0(X,{\cal K}_2/n) && \rod && H^0(X,{\cal H}^2(\mu_n^{\otimes2})) \\
\mbox{\large $\parallel$} && && \mbox{\large $\parallel$} \\
H^0(X,\tilde\pi_1M) &\stackrel\cong\leftarrow&
H^0(X,\tilde\pi_1\tilde L)
&\vlongby{c^{et}_2\enspace\cong}& H^0(X,\tilde\pi_2{\cal E}) \\
d_2\downarrow\quad && d_2\downarrow\quad && d_2\downarrow\quad \\
H^2(X,\tilde\pi_2M) &\leftarrow& H^2(X,\tilde\pi_2\tilde L)
&\vlongby{c^{et}_2}& H^2(X,\tilde\pi_3{\cal E}) \\
\mbox{\large $\parallel$} && && \mbox{\large $\parallel$} \\
H^2(X,{}_n{\cal K}_2)&& \rod && H^2(X,{\cal H}^1(\mu_n^{\otimes2})) \\
\E{array}\E{equation}
(The bottom square of (\ref{maze}) commutes because,
as noted in (2.4.2), the Chern class map
$c_2^{et}\colon{\cal K}_3(\Z/n)\to{\cal H}^1(\mu_n^{\otimes2})$
factors through ${}_n{\cal K}_2$.)
The following description of the differential in the Leray
spectral sequence was suggested in \cite[(0.4)]{PW3}.
\B{prop}\label{diff}
If we identify ${\cal K}_2/n$ with ${\cal H}^2(\mu_n^{\otimes2})$ by \ref{HoobH2}
and ${}_n{\cal K}_2$ with ${\cal H}^1(\mu_n^{\otimes2})$ by \ref{nK2}, then the
differential $d_2\colon H^0(X,{\cal H}^2(\mu_n^{\otimes2})\to H^2(X,{\cal H}^1(\mu_n^{\otimes2})$
in the Leray spectral sequence for $\Het*(X,\mu_n^{\otimes2})$ becomes
identified with the differential in \ref{hyperK2}(a), {\it i.e.,\/}\
$$H^0(X,{\cal H}^2(\mu_n^{\otimes2}))\cong H^0(X,{\cal K}_2/n) \by{\partial}
H^1(X,n\cdot{\cal K}_2)\by{\partial} H^2(X,{}_n{\cal K}_2)
\cong H^2(X,{\cal H}^1(\mu_n^{\otimes2})).
$$
\E{prop}
\B{proof}
The left vertical map in (\ref{maze}) is the differential in the
hypercohomology spectral sequence for $\H^{*}(X,{\cal K}_2^\bullet)$
by Example 7.4, and was described in Proposition~\ref{hyperK2}(a).
Again by Example~7.4, the right vertical map in (\ref{maze})
is the corresponding differential
in the Leray spectral sequence for $\Het{4+*}(X,\mu_n^{\otimes2})$.
A diagram chase on (\ref{maze}),
starting at $H^0(X,\tilde\pi_1\tilde L)$, yields the result.
\E{proof}
\medskip
\B{defi} Following Suslin \cite{S2},
we define $N\Het3(X)$ to be the kernel
of the natural map
$\Het{3}(X,\mu_n^{\otimes 2})\to H^0(X,{\cal H}^3(\mu_n^{\otimes2}))$.
Here $X$ can be any scheme in which $n$ is invertible.
Of course, when $X$ is a surface over an algebraically closed
field the sheaf ${\cal H}^3(\mu_n^{\otimes2})$ vanishes and we have
$N\Het3(X)=\Het3(X,\mu_n^{\otimes2})$.
\E{defi}
\smallskip
The following result was proven by Suslin \cite[p. 19]{S2}
for smooth varieties. It is a partial answer to
\cite[Question~2]{BV1} and was conjectured in \cite[(0.4)]{PW3}.
\B{thm} \label{NH3}
Let $X$ be a surface with isolated singularities over
a field $k$ containing an algebraically closed field and $\frac1n$.
Then
$$N\Het3(X)\cong\H^2(X,{\cal K}_2\kern-3pt\by{n}\kern-3pt{\cal K}_2).$$
In particular, by (\ref{Kummer})
there is a functorial short exact sequence:
$$0\to H^1(X,{\cal K}_2)/n\to N\Het3(X)\to{}_nCH_0(X)\to 0.$$
\E{thm}
\B{proof}
Since $X$ is a surface, the Brown-Gersten spectral sequences
associated to the simplicial presheaves in (\ref{branch})
have only three nonzero columns. Using the computations
given in (7.3) for $\tilde\pi_qM$ and $\tilde\pi_q\tilde L$,
the resulting exact sequences form the rows of a commutative diagram.
\B{equation}\label{M-L-et}\B{array}{cccccccc}
\kern-8pt H^0(X,{\cal K}_2/n)&\kern-3pt\by{d_2}&
\kern-3pt H^2(X,{}_n{\cal K}_2) \kern-6pt&\to&
\kern-6pt \H^2(X,{\cal K}_2^\bullet)
\kern-2pt&\to&\kern-6pt H^1(X,{\cal K}_2/n) \kern-6pt&\to0\\
\kern-8pt\uparrow\cong&&\uparrow\mbox{\rm onto}&&
\uparrow\lambda&&\uparrow\cong&\\
\kern-8pt H^0(X,{\cal K}_2/n)&\kern-3pt\by{d_2}&
\kern-3pt H^2(X,{\cal K}_3(\Z/n)) \kern-6pt&\to&
\kern-6pt \H^0(X,\tilde L)
\kern-2pt&\to&\kern-6pt H^1(X,{\cal K}_2/n) \kern-6pt&\to0\\
\kern-8pt\downarrow\cong&&
c_2^{et}\downarrow\mbox{\rm onto}\kern1.5em
&&\downarrow\kern1em&&\downarrow\cong&\\
\kern-8pt H^0(X,{\cal H}^2(\mu_n^{\otimes2}))&\kern-3pt\by{d_2}&
\kern-3pt H^2(X,{\cal H}^1(\mu_n^{\otimes2})) \kern-6pt&\to&\kern-6pt N\Het3(X)
\kern-2pt&\to&\kern-6pt H^1(X,{\cal H}^2(\mu_n^{\otimes2})) \kern-6pt&\to0
\E{array}\E{equation}
The outside vertical maps are isomorphisms by \ref{HoobH2}.
The two vertical maps marked `onto' in (\ref{M-L-et}) are
actually split surjections with the same kernel,
and are identified by Lemma~\ref{HoobH1}
since $\varphi\colon{\cal H}^1(\mu_n^{\otimes2})\to{}_n{\cal K}_2$ yields an
isomorphism on $H^2$.
Indeed, by \ref{phi-bar} we know that the map
$c_2^{et}\colon{\cal K}_3(\Z/n)\to{\cal H}^1(\mu_n^{\otimes2})$ is a
surjection, split up to sign by
$\bar\varphi\colon {\cal H}^1(\mu_n^{\otimes2})\to{\cal K}_3(\Z/n)$.
A diagram chase on (\ref{M-L-et}) shows that the two maps
$\H^0(X,\tilde L)\longby{\lambda}\H^2(X,{\cal K}_2^\bullet)$
and $\H^0(X,\tilde L)\longby{c_2^{et}} N\Het3(X)$
are both onto with the same kernel. Thus the quotients
$\H^2(X,{\cal K}_2^\bullet)$ and $N\Het3(X)$ are isomorphic.
\E{proof}
\B{cor}\label{H3} If $k$ is algebraically closed
then the short exact sequence is:
$$0\to H^1(X,{\cal K}_2)/n\to \Het{3}(X,\mu_n^{\otimes 2})
\to{}_nCH_0(X)\to 0.$$
\E{cor}
\B{thm} \label{indivisible} Let $X$ be a normal projective
surface over an algebraically closed field $k$. Let $\ell$
a prime number, $\ell\neq$char$(k)$. Then
$$H^1(X,{\cal K}_2)\otimes \Q_{\ell}/\Z_{\ell} =0\quad\mbox{and}
\quad \Het{3}(X,\Q_\ell/\Z_\ell) \cong CH_0(X)_{\ell-tors}$$
\E{thm}
\B{proof} Choose a resolution of singularities
$\pi\colon X'\to X$.
Passing to the direct limit as $\nu\to\infty$, with
$n=\ell^\nu$, the short exact sequences of
Corollary~\ref{H3} become the rows of the commutative diagram
$$\begin{array}{ccccccc}
0\to & H^1(X,{\cal K}_2)\otimes\Q_\ell/\Z_\ell
&\to&\Het{3}(X,\Q_\ell/\Z_\ell)&\to &CH_0(X)_{\ell-tors}&\to 0\\
&\downarrow & &\downarrow & &\downarrow\cong & \\
0\to & H^1(X',{\cal K}_2)\otimes\Q_\ell/\Z_\ell
&\to&\Het{3}(X',\Q_\ell/\Z_\ell)&\to &CH_0(X')_{\ell-tors}&\to 0.
\end{array}$$
The right-hand vertical map is an isomorphism by the
Collino-Levine Theorem \cite{C2} \cite{L-Alb}.
By \cite{CTR}, we have
$H^1(X',{\cal K}_2)\otimes\Q_{\ell}/\Z_{\ell} =0$.
Therefore it suffices to show that
$$\Het3(X,\Q_{\ell}/\Z_{\ell})\cong
\Het3(X',\Q_{\ell}/\Z_{\ell}).$$
There is a Mayer--Vietoris sequence for $\ell$-adic cohomology
similar to (\ref{exnorm}) for the square (\ref{birsquare}).
This yields an exact sequence
$$0\to T\to\Het{3}(X,\Z_{\ell})\to \Het{3}(X',\Z_{\ell})\to0$$
with $T = \Het{2}(E,\Z_{\ell})/\im(\Het{2}(X',\Z_{\ell}))$.
The proof of Proposition~\ref{pure} goes through in the
$\ell$-adic setting as well to show that $T$
is a torsion group ({\it cf.\/}\ \cite[2.1]{C2}).
Since we also have
$\Het{4}(X,\Z_\ell)\cong \Het{4}(X',\Z)\cong\Z_\ell^c$,
the universal coefficient theorem yields the result:
$$\Het{3}(X,\Q_{\ell}/\Z_{\ell})\cong
\Het{3}(X,\Z_\ell)\otimes\Q_{\ell}/\Z_{\ell}\cong
\Het{3}(X',\Z_\ell)\otimes\Q_{\ell}/\Z_{\ell}\cong
\Het{3}(X',\Q_{\ell}/\Z_{\ell}).$$
\kern-24pt
\E{proof}
\section{Proof of the Main Theorem}
Let $X$ be a complex projective surface.
In Lemma~\ref{degree}
we constructed the Abel--Jacobi map
$\rho\colon A_0(X)\to J^2(X)$.
Our Main Theorem, stated in the Introduction,
states that $\rho$ induces an isomorphism
$A_0(X)_{tors}\cong J^2(X)_{tors}$.
We now proceed to prove the Main Theorem.
If $X$ is a normal surface then the result
$A_0(X)_{tors}\cong J^2(X)_{tors}$ is a paraphrase of
the theorem of Levine and Collino (see \cite{C2}, \cite{L-Alb})
that $A_0(X)_{tors}\cong J^2(\tilde X)_{tors}$ for any
resolution of singularities $\tilde X\to X$, because
$J^2(X)\cong J^2(\tilde X)$ by Corollary~\ref{abnorm}.
Granting the normal case, we shall establish the general case
of our Main Theorem by comparing a singular surface $X$ with its
normalization $\tilde X$. For this, we need the following
crucial Lemma.
Let ${\cal H}_{an}^2(\Z)$ denote the Zariski sheaf on $X$
associated to the presheaf $U\mapsto H_{an}^2(U,\Z)$
\B{lemma}\label{BVS} Let $X$ be an irreducible proper
surface over $\C$. Then the following composition is zero.
$$H_{{\cal D}}^2(X,\Z(2))\by\varepsilon
H_{an}^2(X,\Z)\to H^0(X,{\cal H}_{an}^2(\Z))$$
\E{lemma}
\B{proof} By Lemma~\ref{filt} the image of $\varepsilon$ is
the torsion subgroup of $H_{an}^2(X,\Z)$.
However, the sheaf ${\cal H}_{an}^2(\Z)$ and hence its global sections
are torsion free by \cite[Cor. 3]{BVS1}.
\E{proof}
\B{prop}\label{crux} If $X$ is an irreducible proper surface
over $\C$, the following natural map is zero.
$$H^0(X,{\cal K}_2)\to H^0(X,{\cal K}_2/n)$$
\E{prop}
\B{proof} By Proposition~\ref{sheafHD} the natural map
$H_{\cal D}^2(X,\Z(2))\to H^0_{zar}(X,{\cal H}_{\cal D}^2(2))$ is an isomorphism.
The Proposition follows from Lemma~\ref{BVS} and a chase on
the following diagram, the left part of which commutes
by (2.4.1).
$$\B{array}{ccccccc}
K_2(X) &\by{c_2}& H_{\cal D}^2(X,\Z(2))&\by\varepsilon&
H_{an}^2(X,\Z) &\to& H_{an}^2(X,\Z/n) \\
\downarrow&&\downarrow\cong &&\downarrow&&\downarrow\\
H^0(X,{\cal K}_2) &\by{c_2}& H^0(X,{\cal H}_{\cal D}^2(2))&\to&
H^0(X,{\cal H}_{an}^2(\Z)) &\to& H^0(X,{\cal H}^2(Z/n)).
\E{array}$$
\vskip-17pt
\E{proof}
\B{rmk} When $X$ is a {\it smooth} proper variety over an
algebraically closed field of characteristic zero,
Proposition~\ref{crux} was proven by Colliot-Th\'el\`ene
and Raskind \cite{CTR},
and also by H. Esnault \cite{Esn} over $\C$.
\E{rmk}
\B{prop}\label{SK1}
Let $Z$ be a scheme which is proper over $\C$.
If $Z$ is either a curve or a normal surface then
\B{description}
\item[{\it i)}] $c_2\colon H^1(Z,{\cal K}_2)_{tors}\cong H^2(Z,\Q/\Z)$
\item[{\it ii)}] $H^1(Z,{\cal K}_2)\otimes \Q/\Z = 0$
\E{description}
\E{prop}
\goodbreak
\B{proof} The hypothesis on $Z$ allows us to use
\ref{HD-tors} for the isomorphism
$H^2(Z,\Q/\Z)\cong H_{\cal D}^3(X,\Z(2))_{tors}$.
When $Z$ is a curve both assertions follow from
Theorem~\ref{sing} and this remark. When $Z$ is a normal surface,
part {\it ii)\,} was proven in Theorem~\ref{indivisible}.
In order to prove part {\it i)\,} for a normal surface $Z$,
we apply $H^1$ to Corollary~\ref{square}
and combine with the diagram of Corollary~\ref{morph}
to get a commutative diagram for each $n$:
\B{equation}\label{square2}\B{array}{ccc}
H^1(Z,{}_n{\cal K}_2)&\by{\strut\tau_n} &H^1(Z,{\cal K}_2)_{n-tors}\\
c_2^{et}\downarrow\cong & &\downarrow{c_2}\quad\\
H^1(Z,{\cal H}^1(\mu_n^{\otimes2}))&\by\delta & H^1(Z,{\cal H}_{\cal D}^2(2))_{n-tors}\\
\downarrow && \downarrow\qquad \\
\Het2(Z,\mu_n^{\otimes2}) &\by\delta & H_{\cal D}^3(Z,\Z(2))_{n-tors}.
\E{array}\E{equation}
Taking the direct limit as $n\to\infty$ turns
$\mu_n^{\otimes2}$ into $\Q/\Z$. Since
$H^2(Z,\Q/\Z)$ is the torsion subgroup of
$H_{\cal D}^3(Z,\Z(2))$ by Proposition~\ref{HD-tors},
we have a commutative diagram
$$\B{array}{cccccccc}
&\kern-5pt H^1(Z,{\cal K}_{2,tors})\kern-5pt
&\by{\tau}&\kern-3pt H^1(Z,{\cal K}_2)_{tors}&
\kern-5pt\to & \kern-6pt\coker(\tau)&\to0&\\
&c_2^{et}\downarrow\;\cong&&\downarrow c_2&&\kern-5pt\downarrow&&\\
0\to\kern-3pt&\kern-5pt H^1(Z,{\cal H}^1(\Q/\Z))\kern-5pt&\to&\kern-3pt
\Het2(Z,\Q/\Z)&\kern-5pt\to& \kern-6pt H^0(Z,{\cal H}^2(\Q/\Z))
&\kern-7pt\longby{d_2}&\kern-8pt H^2(Z,{\cal H}^1(\Q/\Z))\\
\E{array}$$
in which the bottom row is exact by Corollary~\ref{morph}.
Therefore in order to prove $(i)$
we are reduced to the claim that
$$\coker\ta
\cong \ker H^0(Z,{\cal H}^2(\Q/\Z))\by{d_2} H^2(Z,{\cal H}^1(\Q/\Z))$$
For each $n$, let $\gamma_n$ denote the composition
$H^0(Z,{\cal K}_2/n)\kern-1.5pt\by\partial\kern-2pt H^1(Z,n\cdot{\cal K}_2)
\kern-1.5pt\by\partial\kern-2pt H^2(Z,{}_n{\cal K}_2)$
in the usual interlocking long exact sequences
\B{equation}\label{interlock}\B{array}{ccccccc}
H^1(Z,{}_n{\cal K}_2)&\by{\mathstrut\tau_n}&H^1(Z,{\cal K}_2)
&\to&H^1(Z,n\cdot{\cal K}_2)&\by\partial&H^2(Z,{}_n{\cal K}_2)\\
&&&&\mbox{\large $\parallel$}&&\\
H^0(Z,{\cal K}_2)&\by0& H^0(Z,{\cal K}_2/n)
&\stackrel\partial\hookrightarrow&
H^1(Z,n\cdot{\cal K}_2) &\to &H^1(Z,{\cal K}_2).
\E{array}\E{equation}
The arrow marked `0' in this diagram is the zero map
by Proposition~\ref{crux}.
The other zig-zag composition in (\ref{interlock}),
from $H^1(Z,{\cal K}_2)$ to
$H^1(Z,{\cal K}_2)$,
is multiplication by $n$.
It follows from (\ref{interlock}) that
$$\ker(\gamma_n) \cong H^0(Z,{\cal K}_2/n)\cap \im(H^1(Z,{\cal K}_2))
\cong \frac{H^1(Z,{\cal K}_2)_{n-tors}}{H^1(Z,{}_n{\cal K}_2)}
= \coker(\tau_n).
$$
By Proposition \ref{hyperK2}(a), $\gamma_n$ is
the differential $d_2$ in the hypercohomology
spectral sequence for ${\cal K}_2\by{n}{\cal K}_2$.
By Proposition~\ref{diff}, we may also identify $\gamma_n$ with the
$d_2$-differential in the Leray spectral sequence for
$\Het{*}(Z,\mu_n^{\otimes2})$. Passing to the direct limit
we obtain the claimed formula:
$\coker\tau = \lim\limits_{n\to\infty}\coker\tau_n\cong
\lim\limits_{n\to\infty} \ker\gamma_n=\ker(d_2)$.
\E{proof}
We are now ready to prove our Main Theorem for an arbitrary
projective surface $X$. Letting $\tilde X$ be its normalization
and $Y$ a subscheme chosen as in Theorem \ref{M-V-K},
we have a Mayer-Vietoris Sequence in $K$-theory,
and also for Deligne cohomology by \ref{M-V-D}.
Taking the torsion subgroups of the diagram in
Corollary~\ref{SKtoD} yields the following
commutative diagram (in which we have abbreviated the
left-hand terms for legibility).
\B{equation}\label{diagram}\begin{array}{cccccccc}
\kern-5pt\biggl\{{SK_1(\tilde X)\oplus\atop SK_1(Y)}\biggr\}_{tors}
\kern-7pt&\to\kern-6pt & SK_1(\tilde Y)_{tors}
\kern-5pt&\to\kern-5pt&
SK_0(X)_{tors} \kern-5pt&\to\kern-7pt&
SK_0(\tilde X)_{tors} & \kern-5pt\to0\\
\kern-5pt\downarrow\mbox{}\kern-5pt
& &\downarrow\cong & &\downarrow\cong&&\downarrow\cong&\\
\kern-5pt\biggl\{{H^1(\tilde X)\oplus\atop H^1(Y,{\cal K}_2)\ }
\biggr\}_{tors} \kern-7pt&\to\kern-6pt &
H^{1}(\tilde Y\kern-2pt,{\cal K}_2)_{tors}
\kern-5pt&\to\kern-5pt&
H^{2}(X,{\cal K}_2)_{tors} \kern-5pt&\to\kern-7pt&
H^{2}(\tilde X,{\cal K}_2)_{tors} & \kern-5pt\to0\\
\kern-5ptc_2\downarrow\cong\ &&c_2\downarrow\cong
&&c_2\downarrow\ \ &&c_2\downarrow\cong \ &\\
\kern-5pt\biggl\{{H_{\cal D}^3(\tilde X)\atop H_{\cal D}^3(Y)}\biggr\}_{tors}
\kern-7pt&\to\kern-6pt & H_{\cal D}^3(\tilde Y\kern-2pt,\Z(2))_{tors}
\kern-5pt&\to\kern-5pt&
H_{\cal D}^4(X,\Z(2))_{tors}\kern-5pt&\to\kern-7pt&
H_{\cal D}^4(\tilde X,\Z(2))_{tors}& \kern-5pt\to0
\end{array}
\E{equation}
Some discussion of diagram (\ref{diagram}) is in order.
The 3 isomorphisms between the terms in the top two rows
come from \ref{SKtoD}.
The 2 vertical maps in the lower left of (\ref{diagram})
are isomorphisms by Proposition \ref{SK1}.
The lower right vertical map
$H^{2}(\tilde X,{\cal K}_2)_{tors}\cong J^2(\tilde X)_{tors}
\cong H_{\cal D}^4(\tilde X,\Z(2))_{tors}$ is an isomorphism
because $\tilde X$ is normal.
The bottom row of (\ref{diagram}) is exact, because
by Proposition \ref{HD-tors} it is isomorphic to
$$H^2(\tilde X,\Q/\Z)\oplus H^2(Y,\Q/\Z)
\to H^2(\tilde Y,\Q/\Z) \to
H^3(X,\Q/\Z) \to H^3(\tilde X,\Q/\Z) \to0.
$$
The top two rows of (\ref{diagram}) are exact except
at $SK_1(\tilde Y)_{tors}$ and $H^1(\tilde Y,{\cal K}_2)_{tors}$
by Proposition~\ref{SK1} and the
elementary lemma~\ref{tors-exact} below,
whose proof is left as an exercise.
The 5-lemma implies that we have an isomorphism
$$c_2\colon H^{2}(X,{\cal K}_2)_{tors}\cong H_{\cal D}^4(X,\Z(2))_{tors}$$
and this finishes the proof of our Main Theorem.
\hfil$\bullet$
\B{rmk}In order for the diagram chase of (\ref{diagram}) to work,
it suffices to know the crude surjectivity of the left vertical
map as $n\to\infty$:
$$H^1(\tilde X,{\cal K}_2)_{tors}\oplus H^{1}(Y,{\cal K}_2)_{tors}
\longby{c_2} H^2(\tilde X,\Q/\Z)\oplus H^2(Y,\Q/\Z).$$
\E{rmk}
\B{lemma}\label{tors-exact}
Let $A\to B\to C\to D$ be an exact sequence of abelian
groups. If $A\otimes\Q/\Z=0$ then the following sequence is exact.
$$B_{tors} \to C_{tors} \to D_{tors}$$
\E{lemma}
Here is a motivic version of our Main Theorem.
For a 1-motive $M=(L,A,T,J,u)$ we let $M_{tors}$ denote the
extension of torsion subgroups.
$$0\to T_{tors}\to J_{tors}\to A_{tors}\to 0$$
Then our Main Theorem says that
$Alb(X)_{tors}$ can be described via {\it algebraic} zero-cycles,
{\it i.e.,\/}\ that $J^2(X)_{tors}$ is isomorphic to $A_0(X)_{tors}$
in a way compatible with normalization and desingularization.
\B{schol} Let $Alb(X)$ the Albanese 1--motive of
a projective surface. We then have the
following identification of $Alb(X)_{tors}$:
$$\begin{array}{ccccccc}
0\to&(\Q/\Z)^s &\to&A_0(X)_{tors} & \to &
A_0(\tilde X)_{tors}&\to 0\\
&\mbox{\large $\parallel$} &&\downarrow\cong&&\downarrow\cong& \\
0\to&(\Q/\Z)^s &\to & J^2(X)_{tors}&
\to & J^2(\tilde X)_{tors} &\to 0. \end{array}$$
\E{schol}
\B{rmk} If $X$ is an {\it affine} surface over $\C$ then
$CH_0(X)=A_0(X)$ is uniquely divisible. Indeed, the fact that
$A_0(X)_{tors} =0$ was proven in \cite[Theorem 2.6]{L2}.
And divisibility of $CH_0(X)=SK_0(X)$ is classical, probably
attributable to Murthy:
Every smooth point $x$ on $X$ is in the image of a map
$j\colon C\to X$ in which $C$ is a smooth affine curve.
The group $\Pic(C)$ is divisible, and the class of $x$
is in the image of the map $j_*\colon \Pic(C)\to SK_0(X)$.
Since $H^3(X,\C)=0$ as well, we also have $J^2(X)=0$.
Thus Roitman's Theorem holds by default in the affine case.
\E{rmk}
|
1995-03-22T06:20:14 | 9503 | alg-geom/9503011 | en | https://arxiv.org/abs/alg-geom/9503011 | [
"alg-geom",
"math.AG"
] | alg-geom/9503011 | Christoph Lossen | Gert-Martin Greuel, Christoph Lossen | Equianalytic and equisingular families of curves on surfaces | LaTeX v 2.09 | null | null | null | null | We consider flat families of reduced curves on a smooth surface S such that
each member C has the same number of singularities of fixed singularity types
and the corresponding (locally closed) subscheme H of the Hilbert scheme of S.
We are mainly concerned with analytic resp. topological singularity types and
give a sufficient condition for the smoothness of H (at C). Our results for
S=P^2 seem to be quite sharp for families of cuves of small degree d.
| [
{
"version": "v1",
"created": "Tue, 21 Mar 1995 14:42:27 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Greuel",
"Gert-Martin",
""
],
[
"Lossen",
"Christoph",
""
]
] | alg-geom | \section*{Introduction}\addcontentsline{toc}{section}{Introduction}
We consider flat families of reduced curves on a smooth surface $S$ such that
for each member $C$ of the family the number of singular points of $C$ and for
each singular point $x \in C$ the ``singularity type'' of $(C,x)$ is fixed.
Fixing these data imposes conditions on the space of all curves and we obtain
in this way a locally closed subscheme of the Hilbert scheme $H_S$ of $S$.
We are mainly concerned with the study of the equianalytic
$(H^{ea}_S)$ respectively the equisingular Hilbert scheme $(H^{es}_S)$, which
are defined
by fixing the analytic respectively the (embedded) topological type of the
singularities. We show that fixing the analytical (respectively topological)
type of $(C,x)$ imposes, at most, $\tau (C,x) =$ Tjurina number of $(C,x)$
(respectively, at most, $ \mu (C,x)-$ mod$(C,x)$, where $\mu$ denote the
Milnor number, mod the
modality in the sense of \cite{AGV}) conditions with equality if
$H^1(C,{\cal N}^{ea}_{C/S})$ (respectively $H^1(C,{\cal N}^{es}_{C/S})$) vanish. Here
${\cal N}^{ea}_{C/S}$ (respectively ${\cal N}^{es}_{C/S}$) denote the equianalytic
(respectively equisingular) normal bundle. The vanishing of $H^1$ implies
the independence of the imposed conditions and the smoothness of $H^{ea}_S$
(respectively $H^{es}_S$) at $C$ (cf.\ \S 3).
In Theorem 3.7 we prove sufficient conditions for the vanishing of
$H^1(C,{\cal N}^{ea}_{C/S})$ (respectively $H^1(C,{\cal N}^{es}_{C/S})$); for the
special case $S = \P^2$ we obtain an additional criterion in Corollary 3.12.
For the proof we use a
vanishing theorem of \cite{GrK} which is an improvement upon the usual
vanishing theorem for sheaves which are not locally free. The local
isomorphism defect isod$_x ({\cal N}^{ea}_{C/S}, {\cal O}_C)$, which is introduced in
3.4, measures how much ${\cal N}^{ea}_{C/S}$ fails to be free at $x$, and similar
for ${\cal N}^{es}_{C/S}$. In many cases of interest, in particular for
${\cal N}^{es}_{C/S}$ and related sheaves, the isomorphism defect is quite big and
gives a considerable improvement of the desired vanishing results.
Therefore, we make some effort to compute respectively estimate it for certain
classes of singularities in \S 4. In \S 5 we give some explicit examples and
applications.
The present work is, in some sense, a continuation of some part of \cite{GrK},
where only equianalytic families were considered.
Our results about the smoothness of $H^{es}_{\P^2}$, which are valid for
arbitrary singularities, contain the previously known
facts for curves with only ordinary multiple points (cf.\ \cite{Gia}) as a
special case; concerning the smoothness of $H^{ea}_{\P^2}$ they are an
improvement of \cite{Sh1}. For concrete applications of the theorems of this
paper it is important to have good estimates for the isomorphism defects.
Most of the formulas concerning these, together with further refinements and
detailed
proofs, appeared in \cite{Lo}. Our results for $\P^2$ seem to be quite sharp
for small $d$ but are asymptotically weaker than those of Shustin \cite{Sh3}
which are quadratic in $d$. However, the methods presented here work for
arbitrary surfaces and may be
combined with Shustin's to provide asymptotically optimal results for curves
on (some classes of) rational surfaces. This will be the subject of a
forthcoming joint paper.
\newpage
\section{Equisingular deformations of plane curve singularities}
In this paragraph we recall some definitions and results due to J.\ Wahl in
the framework of formal deformation theory (cf.\ \cite{Wa2}), transfer them to
the complex analytic category and obtain some additional results which are
used later.
\begin{sub}\label{1.1}{\rm
Let $(C,0) \subset ({\Bbb C}^2,0)$ be a reduced plane curve singularity and $f \in
{\cal O}_{{\Bbb C}^2,0} = {\Bbb C}\{u,v\}$ a convergent power series defining the germ $(C,0)$.
Furthermore, let $m = \mbox{ mult}_0(C)$ denote the multiplicity of $(C,0)$,
that is $f \in \frak{m}^m_{{\Bbb C}^2,0}\backslash\frak{m}^{m+1}_{{\Bbb C}^2,0}$ where
$\frak{m}_{X,x}$ denotes the maximal ideal of a germ $(X,x)$. Consider a
deformation $\varphi : ({\cal C},0) \to (T,0)$ of $(C,0)$ over an arbitrary complex
germ $(T,0)$ together with a section $\sigma : (T,0) \to ({\cal C},0)$. Without
loss
of generality we may assume $\varphi$ to be embedded, that is $\varphi$ is
given by a commutative diagram
\unitlength1cm
\begin{picture}(5,2.5)
\put(3.5,2){$(C,0)$}
\put(4,1.75){\vector(0,-1){1}}
\put(3.9,0.25){0}
\put(4.75,2){$\hookrightarrow$}
\put(4.75,0.25){$\in$}
\put(5.5,2){$({\cal C},0)$}
\put(5.75,0.75){\vector(0,1){1}}
\put(6,1.75){\vector(0,-1){1}}
\put(5.4,1.25){$\sigma$}
\put(6.15,1.25){$\varphi$}
\put(5.5,0.25){$(T,0)$}
\put(6.75,2){$\hookrightarrow$}
\put(7.5,2){$({\Bbb C}^2 \times T,0)$}
\put(7.8,1.75){\vector(-3,-2){1.5}}
\put(7,1){$pr$}
\end{picture}
where $pr$ is the (natural) projection, $\sigma$ maps to the trivial section
and $({\cal C},0)$ is a hypersurface germ of
$({\Bbb C}^2 \times T,0)$ defined by a power series $F \in {\cal O}_{{\Bbb C}^2 \times T,0}$.
Let $I_{\sigma(T)}$ denote the ideal of $\sigma(T,0) \subset ({\Bbb C}^2 \times
T,0)$, then we call the deformation with section $(\varphi,\sigma)$ {\sl
equimultiple}, if $F \in I^m_{\sigma(T)}$ (which is, of course, independent of
the choice of the embedding and the choice of $F$).
}
\end{sub}
\begin{sub}\label{1.2}{\rm
Before defining equisingular deformations, let us recall that the {\sl
equisingularity type}\/ (or {\sl topological type}) of $(C,0)$ may be defined
as
follows: consider an embedded resolution of $(C,0) \subset ({\Bbb C}^2,0)$ given by
a sequence of blowing up points $(N \ge 1)$:
\begin{equation}
M_N \buildrel\pi_N\over \to M_{N-1} \to \cdots \to M_1 \buildrel\pi_1\over\to
M_0 = ({\Bbb C}^2,0).
\end{equation}
Let $C_i \subset M_i$ be the strict and $\hat{C}_i \subset M_i$ the reduced
total transform of $C_0 := (C,0) \subset M_0$ under $\psi_i := \pi_1 \circ
\cdots \circ \pi_i$. Assume that
\begin{itemize}
\item $\pi_1$ blows up $0$;
\item for $i = 2, \ldots, N$, $\pi_i$ blows up singular points of $\hat{C}_i$;
\item $\hat{C}_N$ has only singularities of type $A_1$.
\end{itemize}
Hence, $C_N$ is smooth and $\psi_N$ induces a resolution of $(C,0)$. If we
choose the minimal resolution (that is blowing up only non--nodes of
$\hat{C}_i$) we obtain a well--defined system of multiplicity sequences (cf.\
\cite{BrK}), which defines the equisingularity type of $(C,0)$. It is
well--known (and was proved by Zariski in \cite{Zar}) that this system of
multiplicity sequences determines the embedded topological type of $(C,0)$ and
vice versa (cf.\ \cite{BrK}, 8.4 and \cite{Zar} for further
characterizations).
}
\end{sub}
\begin{sub}\label{1.3}{\rm
Consider a deformation $\varphi : ({\cal C},0) \to (T,0)$ of $(C,0)$ (without
section) and assume it to be embedded
\unitlength1cm
\begin{picture}(4,2.5)
\put(4.5,2){$({\cal C},0)$}
\put(5,1.75){\vector(0,-1){1}}
\put(4.7,1.25){$\varphi$}
\put(4.5,0.25){$(T,0)$}
\put(5.75,2){$\hookrightarrow$}
\put(6.5,2){$({\Bbb C}^2 \times T,0)$}
\put(6.75,1.75){\vector(-3,-2){1.5}}
\put(6.2,1.15){$pr$}
\end{picture}
$\varphi$ is called {\sl equisingular}\/ (cf.\ \cite{Wa2}, \S 3, \S 7) if
there exists a sequence of blowing up subspaces
\begin{equation}
{\cal M}_N \buildrel\tilde{\pi}_N\over\to {\cal M}_{N-1} \to \cdots \to {\cal M}_1
\buildrel\tilde{\pi}_1\over \to {\cal M}_0 = ({\Bbb C}^2 \times T,0)
\end{equation}
such that if ${\cal C}_i \subset {\cal M}_i$ denotes the strict and $\hat{{\cal C}}_i \subset
{\cal M}_i$ the reduced total transform of ${\cal C}_0 = ({\cal C},0)$ under $\tilde{\psi}_i
:= \tilde{\pi}_1 \circ \cdots \circ \tilde{\pi}_i$, the following holds:
\begin{itemize}
\item[(i)] sequence (1) is induced by (2) via the base change $0 \mapsto T$;
\item[(ii)] there is a section $\sigma : (T,0) \to {\cal C}_0$ of $\varphi_0 =
\varphi$ such that $\varphi$ is equimultiple along $\sigma$ and
$\tilde{\pi}_1$ blows up $\sigma(T,0) \subset {\cal M}_0$;
\item[(iii)] for $i = 1, \ldots, N$ there are sections $(T,0) \to {\cal C}_i$ of
$\varphi_i = \varphi \circ \tilde{\psi}_i \mid_{{\cal C}_i} : {\cal C}_i \to (T,0)$
through all singular points of $\hat{C}_i$ (each of those sections being
mapped via $\tilde{\pi}_i$ to such a section of $\varphi_{i-1}$) such that
$\varphi_i$ is equimultiple along them. $\tilde{\pi}_{i+1}: {\cal M}_{i+1}
\to {\cal M}_i$ blows up the sections going through those singular points of
$\hat{C}_i$ which are
blown up by $\pi_{i+1}$ ($i \le N-1$).
\end{itemize}
The sections of (ii), (iii) are called a (compatible) system of {\sl
equimultiple sections} of $\varphi$ through all infinitely near points of
$(C,0)$.
This definition is obviously independent of the embedding of $\varphi$.
Moreover, since an equimultiple deformation of an $A_1$--singularity is
trivial, the definition is also independent of the embedded resolution (1).
The section $\sigma$ of (ii) is called a {\sl singular section}\/ of
$\varphi$. If $(T,0)$ is reduced, then $\varphi : ({\cal C},0) \to (T,0)$ is
equisingular if and only if for a small good representative $\varphi : {\cal C} \to
T$ and for all $t \in T$ there exists an $x \in \varphi^{-1}(t)$ such that the
Milnor number $\mu(\varphi^{-1}(t),x)$ is equal to the Milnor number $\mu
(C,0)$. The existence of a singular section was shown by B.\ Teissier
(\cite{Te}, \S 5).
}
\end{sub}
The following theorem is basically due to J.\ Wahl (\cite{Wa2}, Theorem 7.4):
\begin{theorem} Let $\varphi : ({\cal C},0) \to (T,0)$ be any equisingular
deformation of the reduced plane curve singularity $(C,0) \subset ({\Bbb C}^2,0)$.
\begin{itemize}
\item[(i)] The equimultiple sections through all infinitely near points of
$(C,0)$ which
are required to exist for $\varphi$ are uniquely determined.
\item[(ii)] Let $\phi : {\cal C}_{(C,0)} \to S_{(C,0)}$ be the semiuniversal
deformation of $(C,0)$. Then there exists a smooth subgerm $S^{es}_{(C,0)}
\subset S_{(C,0)}$ such that if $\varphi$ is induced from $\phi$ via the base
change $\psi : (T,0) \to S_{(C,0)}$, then $\psi$ factors through
$S^{es}_{(C,0)}$. In particular, the restriction of $\phi$ to
$S^{es}_{(C,0)}$ is a {\rm semiuniversal equisingular deformation} of
$(C,0)$.
\item[(iii)] Let $T_\varepsilon := \mbox{ Spec}({\Bbb C}[\varepsilon]/\varepsilon^2)$
be the
base space of first order infinitesimal deformations. The set
\begin{center}
$I^{es}(C,0) := \{g \in {\Bbb C}\{u,v\}\; \Big|$ \raisebox{1.5ex}{$F = f +
\varepsilon g$
defines an equisingular}\hspace{-5.25cm}\raisebox{-1.5ex}{deformation of
$(C,0)$
over $T_\varepsilon$}\hspace{1cm}$\Big\}$
\end{center}
is an ideal, the {\rm equisingularity ideal}\/ of $(C,0)$. Especially it
contains the Jacobian ideal
\[
j(C,0) = (f,\frac{\partial f}{\partial u},\; \frac{\partial f}{\partial v})
\cdot {\Bbb C} \{u,v\}
\]
and the vector space $I^{es}(C,0)/j(C,0)$ is isomorphic to the tangent
space of $S^{es}_{(C,0)} \subset S_{(C,0)}$.
\end{itemize}
\end{theorem}
{\bf Proof}: Wahl considers only deformations over Artinian spaces $(T,0)$
but the above facts follow easily from his results:
\begin{itemize}
\item[(i)] Since we require the existence of holomorphic sections over
arbitrary complex germs $(T,0)$, by Wahl these are unique modulo arbitrary
powers of the maximal ideal $\frak{m}_{T,0}$, hence unique.
\item[(ii)] The existence of a smooth formal semiuniversal equisingular
deformation of $(C,0)$ was proved by Wahl. The existence of a convergent
representative can be deduced from his result by applying Artin's and Elkik's
algebraization theorems. A simple direct proof, using the deformation of the
parametrization, is given in \cite{Gr}.
\item[(iii)] follows directly from (\cite{Wa2}, Proposition 6.1).\hfill $\Box$
\end{itemize}
\begin{proposition}\label{1.5} Openness of versality holds for equisingular
deformations, that is if $\varphi : ({\cal C},0) \to (T,0)$ is an equisingular
deformation of $(C,0)$, then for any equisingular representative $\varphi :
{\cal C} \to T$ together with the singular section $\sigma : T \to {\cal C}$ the set of
points $t \in T$ such that $({\cal C}, \sigma(t)) \to (T,t)$ is a versal
deformation of $(\varphi^{-1}(t), \sigma(t))$ is a Zariski--open subspace of
$T$.
\end{proposition}
{\bf Proof}: This follows quite formally from a criterion for openness of
versality due to Flenner (\cite{Fl}, Satz 4.3).\hfill $\Box$
\begin{sub}\label{1.6}{\rm
Let $\mu(c,0) = \dim_{\Bbb C}({\Bbb C}\{u,v\}/(\frac{\partial f}{\partial u},
\frac{\partial f}{\partial v}))$ respectively $\tau(C,0) =
\dim_{\Bbb C}({\Bbb C}\{u,v\}/(f, \frac{\partial f}{\partial u},\; \frac{\partial
f}{\partial v}))$ denote the Milnor respectively Tjurina number of $(C,0)$.
It is well--known that a
deformation of $(C,0)$ over a reduced base $(T,0)$ is equisingular if and only
if the Milnor number is constant along the (unique) singular section. Hence
$S^{es}_{(C,0)}$, being smooth, coincides with the $\mu$--constant stratum of
$S_{(C,0)}$.
The codimension of $S^{es}_{(C,0)}$ in $S_{(C,0)}$ is (by Theorem 1.4
(ii) and (iii)) equal to
\[
\tau^{es}(C,0) = \dim_{\Bbb C}({\Bbb C}\{u,v\}/I^{es}(C,0)).
\]
Together with a result of Gabrielov (\cite{Gab}), which states that the {\sl
modality}\/ mod$(f)$ of the function $f$ with respect to right equivalence
(cf.\ \cite{AGV}) is equal to the dimension of the $\mu$--constant stratum of
$f$ in the ($\mu$--dimensional) semiuniversal unfolding of $f$, we obtain the
following
}
\end{sub}
\begin{lemma}
For any reduced plane curve singularity $(C,0)$ defined by $f
\in {\Bbb C}\{u,v\}$, we have
\[
\tau^{es}(C,0) = \mu(C,0) - \mbox{ mod}(f).
\]
\end{lemma}
\begin{sub}\label{1.9}{\rm
If $\sim$ denotes any equivalence relation of plane curve singularities, a
$\sim$--{\sl singularity type} is by definition a (not ordered) tuple ${\cal S}
=
((C_1,x_1)/\sim, \ldots, (C_m,x_m)/\sim)$ of $\sim$-equivalence classes with
$m$ a non--negative integer. In this paper we are mainly interested in the
following two
cases:
\begin{itemize}
\item $\sim =$ analytic equivalence (isomorphism of complex space germs), in
this
case we call the $\sim$--singularity type {\sl analytic type} and denote it by
${\cal A}$.
\item $\sim =$ topological equivalence (embedded homeomorphism of complex
space germs
(cf.\ 1.3)), the corresponding singularity type is called {\sl
equisingularity type} or {\sl topological type} and denoted by ${\cal T}$.
\end{itemize}
If $(C,x)$ is a reduced plane curve singularity, then ${\cal S}(C,x) =
(C,x)/\sim$
denotes its singularity type and if $C$ is a reduced curve with finitely many
singular points $x_1, \ldots, x_m$ which are all planar, then ${\cal S}(C) =
((C,x_1)/\sim, \ldots, (C,x_m)/\sim)$ is the $\sim$--singularity type of $C$.
For ${\cal S} = {\cal A}$ we obtain ${\cal A}(C)$, the {\sl equianalytic type} of $C$, and
for ${\cal S} = {\cal T}$ we obtain ${\cal T}(C)$, the {\sl equisingular type} of $C$.
As equisingular deformations preserve the topological type, the equianalytic
deformations preserve the analytic type of each fibre, where a deformation
$\varphi : ({\cal C},0) \to (T,0)$ of $(C,0)$ is called {\sl equianalytic} if
$({\cal C},0)$ is analytic isomorphic to $(C \times T,0)$ over $(T,0)$, that is,
$\varphi$ is analytically trivial.}
\end{sub}
\newpage
\section{The equianalytic and equisingular Hilbert scheme}
\begin{sub}\label{2.1}{\rm
Let $S$ be a smooth surface, $T$ a complex space, then by a {\sl family of
embedded (reduced) curves over} $T$ we mean a commutative diagram
\[
\begin{array}{lcl}
{\cal C} & \buildrel j\over\hookrightarrow & \;\;S \times T\\
\varphi\searrow & & \swarrow pr\\
& T &
\end{array}
\]
where $\varphi$ is a proper and flat morphism such that for all
points $t \in T$ the fibre $\varphi^{-1}(t)$ is a {\sl curve}\/ (that is a
reduced pure 1--dimensional complex space), moreover, $j : {\cal C}
\hookrightarrow S \times T$ is a closed embedding and $pr$ denotes the natural
projection. Such a family is called {\sl equianalytic} (respectively {\sl
equisingular}) if for all $t \in T$ the induced (embedded) deformation of each
singular point of $\varphi^{-1}(t)$ over ($T,t$) is equianalytic (respectively
equisingular) --- along the unique singular section $\sigma$ (cf.\ \ref{1.3}).
}
\end{sub}
\begin{sub}\label{2.2}{\rm
The Hilbert functor ${\cal H} ilb_S$ on the category of complex spaces defined by
\[
{\cal H} ilb_S(T) := \{\mbox{ subspaces of } S \times T, \mbox{ proper and flat over
}
T\}
\]
is well--known to be representable by a complex space $H_S$ (cf.\ \cite{Bin}).
This means there is a universal family
\[
\begin{array}{lcl}
{\cal U} & \buildrel j\over\hookrightarrow & \;\; S \times H_S\\
\varphi\searrow & & \swarrow pr\\
& H_S &
\end{array}
\]
such that each element of ${\cal H} ilb_S(T)$, $T$ a complex space, can be induced
from
$\varphi$ via base change by a {\sl unique} map $T \to H_S$.
We define the {\sl equianalytic} (respectively {\sl equisingular}) {\sl
Hilbert functor} ${\cal H} ilb^{ea}_S$ (respectively ${{\cal H}}ilb^{es}_S$) to be the
subfunctor of ${\cal H} ilb_S$ with
\[
\begin{array}{lcl}
{\cal H} ilb^{ea}_S(T) & := & \{\mbox{equianalytic families of embedded curves
over }T\}\\[0.5ex]
{\cal H} ilb^{es}_S(T) & := & \{\mbox{equisingular families of embedded curves
over } T\}
\end{array}
\]
Moreover, fixing the analytic (respectively topological) singularity type
(cf.\ 1.8), we define
\[
\begin{array}{lcl}
{\cal H} ilb^{\cal A}_S(T) & := & \{\mbox{families in } {\cal H} ilb^{ea}_S(T)
\mbox{ whose fibres have (the fixed) analytic singularity type }
{\cal A}\}\\[0.5ex]
{\cal H} ilb^{\cal T}_S(T) & := & \{ \mbox{families in } {\cal H} ilb^{es}_S(T) \mbox{ whose
fibres have (the fixed) topological singularity type } {\cal T}\}
\end{array}
\]
}
\end{sub}
\begin{proposition}
Let ${\cal A}$ be an analytic singularity type corresponding to the topological
type ${\cal T}$, then the functors ${\cal H} ilb^{\cal A}_S$ and ${\cal H} ilb^{\cal T}_S$ are
representable
by locally closed subspaces $H^{\cal A}_S \subset H^{\cal T}_S \subset H_S$.
\end{proposition}
\begin{corollary}
The functor ${\cal H} ilb^{ea}_S$ (respectively ${\cal H} ilb^{es}_S$) is representable
by a
complex space $H^{ea}_S$ (respectively $H^{es}_S$) which is given as the
disjoint union of all $H^{{\cal A}}_S$ (respectively $H^{\cal T}_S$).
\end{corollary}
\begin{remark}{\rm If $S = \P^2$ and if we fix the degree of all fibres to be
$d$, then
$H^{es,d}_{\P^2} \subset \P^N$ (where $N = \frac{d^2 + 3d}{2}$) is given as a
{\sl finite}\/ disjoint union of locally closed subspaces, while in general
$H^{ea,d}_{\P^2}
\subset \P^N$ is an {\sl infinite}\/ union.
}
\end{remark}
\begin{sub}\label{2.3}{\rm
For the proof of Proposition 2.3 we need the following Lemma, which, for
the equianalytic case, is proven in (\cite{GrK}, Lemma 1.4). A proof for
the equisingular case is given in \cite{Gr}.
}
\end{sub}
\begin{lemma}
Let $(C,x)$ be the germ of an isolated plane curve singularity and $\varphi
:({\cal C},x) \to (B,b)$ a deformation of $(C,x)$, then there are unique closed
subgerms $(B^{ea},b) \subset (B^{es},b) \subset (B,b)$ such that for any
morphism $f : (T,t) \to (B,b)$:
\[
\begin{array}{l}
f^\ast\varphi \mbox{ is an equianalytic deformation if and only if } f(T,t)
\subset (B^{ea},b)\\
f^\ast\varphi \mbox{ is an equisingular deformation if and only if } f(T,t)
\subset (B^{es},b)
\end{array}
\]
Moreover, if $\phi :{\cal C}_{(C,x)} \to S_{(C,x)}$ denotes the semiuniversal
deformation of $(C,x)$ and if $\psi : (B,b) \to S_{(C,x)}$ is any morphism
inducing $\varphi$ via pull--back, then $(B^{ea},b) = (\psi^{-1}(0),b)$ and
$(B^{es},b) = (\psi^{-1}(S^{es}_{(C,x)}),b)$.
\end{lemma}
\begin{sub}\label{2.4}{\rm
{\bf Proof of Proposition 2.3}: First we have to remark that the
condition for all fibres to be reduced curves defines an open subspace
$\widetilde{H}_S \subset H_S$. Now, let $b \in \widetilde{H}_S$ be such that
the fibre $\varphi^{-1}(b)$ in the universal family has topological type
${\cal T}$, then by Lemma 2.7 for each $x \in \varphi^{-1}(b)$ there is a
unique closed subspace $(H_x^{es},b) \subset (\widetilde{H}_S,b)$ such that a
morphism $f : (T,t) \to (\widetilde{H}_S,b)$ factors through $(H^{es}_x,b)$ if
and only if $f^\ast\varphi$ is an equisingular deformation.
Let
\[
(H^{es},b) := \bigcap\limits_{x \in \varphi^{-1}(b)} (H_x^{es},b) \subset
(\widetilde{H}_S,b)
\]
and $H^{es}(b) \subset \widetilde{H}_S$ be a small (unique) representative,
then $\cup H^{es}(b)$, where the union is taken over all $b$ whose fibre
$\varphi^{-1}(b)$ has topological type ${\cal T}$, defines a locally closed subspace
of $H_S$ which obviously represents
${\cal H} ilb^{\cal T}_S$. The statement for ${\cal H} ilb^{\cal A}_S$ follows in the
same manner. \hfill$\Box$
}
\end{sub}
\newpage
\section{Completeness of the equianalytic and equisingular characteristic
linear series}
\begin{sub}\label{3.1}{\rm
Let $S$ be a smooth complex surface and $C \subset S$ a reduced compact curve.
Then a {\sl deformation of} $C/S$ over the pointed complex space $T$, $0 \in
T$, is
a triple $({\cal C}, \tilde{i}, j)$ such that we obtain a Cartesian diagram
\unitlength1cm
\begin{center}
\begin{picture}(7,2.5)
\put(2.5,2){$C$}
\put(3,2){$\hookrightarrow$}
\put(3.15,2.2){$j$}
\put(3.75,2){${\cal C}$}
\put(4.3,2.1){\line(1,0){0.5}}
\put(2.3,1.5){$i\;\cap$}
\put(3.75,1.5){$\cap\;\tilde{i}$}
\put(2.5,1){$S$}
\put(3,1){$\hookrightarrow$}
\put(3.75,1){$S \times T$}
\put(5,1){$\Phi$ flat}
\put(2.5,0.5){$\downarrow$}
\put(3.75,0.5){$\downarrow\; \pi$}
\put(2.5,0){$0$}
\put(3.2,0){$\in$}
\put(3.75,0){$T$}
\put(4.8,0.1){\vector(-1,0){0.5}}
\put(4.8,0.1){\line(0,1){2}}
\end{picture}
\end{center}
where $j$ is a closed embedding and the composed morphism $\Phi := \pi \circ
\tilde{i}$ is flat ($S \hookrightarrow S \times T$ denotes the canonical
embedding with image $S \times \{0\}$ and $\pi$ is the projection). Two
deformations $({\cal C}, \tilde{i}, j)$ and $({\cal C}', \tilde{i}', j')$ of $C/S$
over $T$ are {\sl isomorphic} if there exists an isomorphism ${\cal C} \simeq
{\cal C}'$ such that the obvious diagram (with the identity on $S \times T$)
commutes. ${\cal D} e\!f_{C/S}$ denotes the deformation functor from pointed
complex
spaces to sets defined by
\[
{\cal D} e\!f_{C/S} (T) := \{\mbox{isomorphism classes of deformations of } C/S
\mbox{
over } T\}
\]
and we have the natural forgetful morphism ${\cal D} e\!f_{C/S} \to {\cal D} e\!f_C$
given by
$({\cal C}, \tilde{i}, j) \mapsto$ \hbox{$({\cal C}, \Phi = \pi \circ \tilde{i}, j)$},
where
${\cal D} e\!f_C$ denotes the functor of isomorphism classes of deformations of $C$
(forgetting the embedding). Furthermore, for each point $x \in C$, we
consider the morphism ${\cal D} e\!f_C \to {\cal D} e\!f_{C,x}$ where ${\cal D} e\!f_{C,x}$
denotes the
functor of isomorphism classes of deformations of the analytic germ
$(C,x)$.
Let $T_\varepsilon =$ Spec $({\Bbb C}[\varepsilon]/\varepsilon^2)$ be the base space
of first
order infinitesimal deformations. We turn our attention to a subfunctor
${\cal D} e\!f^\prime_{C,x} \subset {\cal D} e\!f_{C,x}$ such that
\[
(T^1)^\prime := {\cal D} e\!f_{C,x}^{\prime}(T_\varepsilon)
\]
is an ideal in ${\cal D} e\!f_{C,x} (T_\varepsilon) \cong {\Bbb C}\{u,v\}/j(C,x)$ and the
corresponding
``global'' subfunctor ${\cal D} e\!f^\prime_{C/S} \subset {\cal D} e\!f_{C/S}$ where ${\cal D}
e\!f^\prime_{C/S}(T)$
consists exactly of all those elements of ${\cal D} e\!f_{C/S}(T)$ which are mapped
to
${\cal D} e\!f^\prime_{C,x}(T)$ for all points $x \in C$.
}
\end{sub}
{\bf Examples}:
\begin{enumerate}
\ite
${\cal D} e\!f^{ea}_{C/S}$ the subfunctor of ${\cal D} e\!f_{C/S}$ consisting of all
isomorphism classes of equianalytic deformations of $C/S$, that is of those
deformations whose induced deformations of the analytic germs $(C,x)$
happen to be equianalytic for all $x \in C$. Here ${\cal D}
e\!f^{ea}_{C,x}(T_\varepsilon) =
0$ in ${\Bbb C}\{u,v\}/j(C,x)$.
\ite
${\cal D} e\!f^{es}_{C/S}$ the subfunctor of ${\cal D} e\!f_{C/S}$ consisting of all
isomorphism classes of equisingular deformations of $C/S$, that is of
those deformations whose induced deformations of $(C,x)$ are
equisingular for all $x \in C$. Here ${\cal D} e\!f^{es}_{C,x}(T_\varepsilon)
= I^{es}
(C,x)/j(C,x)$.
\ite
Further examples are the equimultiple, equigeneric and equiclassical
deformation functors (cf.\ \cite{DH}).
\end{enumerate}
\begin{remark}{\rm ${\cal D} e\!f^{ea}_{C/S}$ coincides with ${\cal D} e\!f^{es}_{C/S}$
if
(and only if) $C$ has only simple (ADE)--singularities.
}
\end{remark}
Now, let $J_C$ be the ideal sheaf of $C$ in ${\cal O}_S$, then we have the natural
exact sequence
\[
0 \to J_C/J^2_C \to \Omega^1_S \otimes_{{\cal O}_S} {\cal O}_C \to \Omega^1_C \to 0
\]
respectively its dual
\[
0 \to \theta_C \to \theta_S \otimes_{{\cal O}_S} {\cal O}_C \buildrel\Psi\over\to
{\cal N}_{C/S} \to {\cal T}^1_C \to 0.
\]
Here ${\cal N}_{C/S} = {\cal O}_S(C) \otimes_{{\cal O}_S} {\cal O}_C$ denotes the normal sheaf of
$C$ in $S$ and ${\cal T}^1_C := \mbox{ Coker } (\Psi)$ is a skyscraper sheaf
concentrated in the singular points of $C$ with $H^0(C,{\cal T}^1_C) \cong
{\cal D} e\!f_C(T_\varepsilon)$ and with stalk in $x \in C$ equal to ${\cal T}^1_{C,x}
\cong
{\cal D} e\!f_{C,x}(T_\varepsilon) = T^1_{(C,x)}$ (for
details cf.\ \cite{Art}). Furthermore, for each subfunctor ${\cal D}
e\!f^\prime_{C/S}$
as above, let $({\cal T}^1_C)'$ denote the subsheaf of ${\cal T}^1_C$ with stalk in $x$
isomorphic to $(T^1)^\prime \subset T^1$ and
\[
{\cal N}'_{C/S} := \mbox{ Ker}({\cal N}_{C/S} \to {\cal T}^1_C/({\cal T}^1_C)').
\]
In particular,
\vspace{-1cm}
\begin{eqnarray*}
{\cal N}^{ea}_{C/S} & = & \mbox{ Ker }({\cal N}_{C/S} \to {\cal T}^1_C),\\
{\cal N}^{es}_{C/S} & = & \mbox{ Ker } ({\cal N}_{C/S} \to {\cal T}^1_C/({\cal T}^1_C)^{es})
\end{eqnarray*}
where ${\cal T}^1_{C,x} \cong {\Bbb C} \{u,v\}/j(C,x),\; ({\cal T}^1_C)^{es}_x \cong
I^{es}(C,x)/j(C,x)$.
\begin{lemma}\label{3.2} There is a canonical isomorphism
\[
\Phi: \quad {\cal D} e\!f^\prime_{C/S}(T_\varepsilon)
\buildrel\cong\over\longrightarrow
H^0(C,{\cal N}'_{C/S}).
\]
\end{lemma}
{\bf Proof}: Each representative of an element in ${\cal D}
e\!f^\prime_{C/S}(T_\varepsilon)$
is given by local equations $(f_i + \varepsilon g_i = 0)_{i\in I}$ (where
$f_i,g_i \in \Gamma(U_i,{\cal O}_S)$ for an open covering $(U_i) _{i\in I}$ of
$S$), which satisfy
\begin{itemize}
\item $(f_i = 0)_{i\in I}$ are local equations for $C \subset S$
\item $f_i + \varepsilon g_i = (a_{ij} + \varepsilon b_{ij}) \cdot (f_j +
\varepsilon g_j)$ on $U_i \cap U_j =: U_{ij}$ with $a_{ij}$ a unit in
$\Gamma(U_{ij}, {\cal O}_S)$ and $b_{ij} \in \Gamma(U_{ij}, {\cal O}_S)$
\item the germ $g_{i,x}$ of $g_i$ projects to an element of
${\cal D} e\!f^\prime_{C,x}(T_\varepsilon) = (T^1)^\prime$ for all $x \in C \cap
U_i$.
\end{itemize}
For the induced sections $\frac{g_i}{f_i} \in \Gamma(U_i, {\cal O}_S(C))$ it
follows immediately
\[
\frac{g_i}{f_i} - \frac{g_j}{f_j} = \frac{a_{ij}g_j + b_{ij}f_j}{a_{ij}f_j} -
\frac{g_j}{f_j} = \frac{b_{ij}}{a_{ij}} \equiv 0 \in \Gamma(U_{ij}, {\cal N}_{C/S})
\]
and $\frac{g_i}{f_i}$ maps to an element of $({\cal T}^1_C)' \subset {\cal T}^1_C$.
Hence,
$(\frac{g_i}{f_i})_{i \in I}$ defines a global section in ${\cal N}'_{C/S}$. It is
easy to check that in this way we get the isomorphism we were looking
for (cf.\ \cite{Mu}, \cite{Lo}).\hfill $\Box$
\begin{sub}\label{3.3}
{\rm
Let $C$ be a compact reduced curve, ${\cal F}$ and ${\cal G}$ coherent torsion--free
sheaves on $C$,
which have rank 1 on each irreducible component $C_i$ of $C$ and $x \in C$.
Then we define the {\sl local isomorphism defect} of ${\cal F}$ in ${\cal G}$ in $x$ as
\[
\mbox{isod}_x ({\cal F}, {\cal G}) := \min(\dim_{{\Bbb C}} \mbox{ Coker} (\varphi :
{\cal F}_x \to {\cal G}_x))
\]
where the minimum is taken over
all (injective) local homomorphisms $\varphi : {\cal F}_x \to {\cal G}_x$. In
particular,
isod$_x({\cal F},{\cal G})$ is a non--negative integer and not zero only in finitely many
points (in \cite{GrK} isod$_x({\cal F},{\cal G})$ was denoted by $-$ind$_x({\cal F},{\cal G})$).
We call
\[
\mbox{isod}({\cal F},{\cal G}) := \sum_{x \in C} \mbox{ isod}_x ({\cal F},{\cal G})
\]
the {\sl total (local) isomorphism defect} of ${\cal F}$ in ${\cal G}$. For an
irreducible component
$C_i$ of $C$ and $x \in C_i$ we set
\[
\begin{array}{l}
{\cal F}_{C_i} := {\cal F} \otimes {\cal O}_{C_i}\mbox{ modulo torsion}\\
\mbox{isod}_{C_i,x}({\cal F},{\cal G}) := \min(\dim_{\Bbb C} \mbox{ Coker} (\varphi_{C_i} :
{\cal F}_{C_i,x} \to {\cal G}_{C_i,x}))
\end{array}
\]
where the minimum is taken over all $\varphi_{C_i}$, which are induced by
local homomorphisms $\varphi : {\cal F}_x \to {\cal G}_x$, and
\[
\mbox{isod}_{C_i}({\cal F}, {\cal G}) := \sum_{x\in C_i} \mbox{ isod}_{C_i,x} ({\cal F}, {\cal G}).
\]
Note that this is again a non--negative integer. In Chapter 4 we present
some explicit calculations.}
\end{sub}
\begin{proposition}\label{3.4}
(\cite{GrK}, Proposition 5.2)
Let $S$ be a smooth surface, $C \subset S$ a compact reduced curve and ${\cal F}$ a
torsion--free coherent ${\cal O}_C$--module which
has rank 1 on each irreducible component $C_i$ of $C$ $(i = 1, \ldots, s)$.
Then $H^1(C,{\cal F}) = 0$ if for $i = 1, \ldots, s$
\[
\chi({\cal F}_{C_i}) > \chi (w_{C,C_i}) - \mbox{ isod}_{C_i} ({\cal F},w_C).
\]
Here $\chi({\cal M}) = \dim H^0((C,{\cal M}) - \dim H^1(C,{\cal M})$ for a
coherent sheaf ${\cal M}$ on $C$ and $w_C$ denotes the dualizing sheaf, $w_{C,C_i}
:= w_C \otimes
{\cal O}_{C_i}$.
\end{proposition}
\begin{remark}{\rm Using Riemann--Roch and the adjunction formula, the
condition above reads
\[
\deg({\cal F}_{C_i}) > (K_S + C) \cdot C_i - \mbox{ isod}_{C_i}({\cal F},{\cal O}_C)
\]
where $K_S$ is the canonical divisor on $S$. Since isod is a local invariant
and since $C$ has planar singularities, we can replace $w_C$ by ${\cal O}_C$.
}
\end{remark}
\begin{theorem}\label{3.5}
Let $S$ be a smooth complex surface and $C \subset S$ a reduced compact curve,
$H^{ea}_S$ respectively $H^{es}_S$ be the representing spaces for the
equianalytic respectively equisingular Hilbert functor, then
\begin{itemize}
\item[(i)] $\dim(H^{ea}_S, C) \ge C^2 + 1 - p_a(C) - \tau(C)$,
with $\tau(C)
= \sum_{x \in Sing(C)} \tau(C,x),$\\[1.0ex]
$\dim(H^{es}_S, C) \ge C^2 + 1 - p_a(C) - \tau^{es}(C)$, with $\tau^{es}(C)
=
\sum_{x \in Sing(C)} \tau^{es}(C,x),$\\[1.0ex]
where $\tau(C,x) = \dim_{\Bbb C}({\cal O}_{C,x}/j(C,x))$ and $\tau^{es}(C,x) =
\dim_{\Bbb C}({\cal O}_{C,x}/I^{es}(C,x))$
\item[(ii)] If $H^1(C, {\cal N}^{ea}_{C/S}) = 0$ (respectively
$H^1(C,{\cal N}^{es}_{C/S}) = 0)$ then $H^{ea}_S$ (respectively $H^{es}_S$) is
smooth at $C$ of dimension
\[
C^2 + 1 - p_a (C) - \tau(C) \quad (\mbox{respectively } C^2 + 1 - p_a(C) -
\tau^{es}(C)).
\]
\item[(iii)] Let $C = C_1 \cup \ldots \cup C_s$ be the decomposition into
irreducible components, then
\begin{itemize}
\item[$\bullet$] $H^1(C, {\cal N}^{ea}_{C/S}) = 0$ if for $i = 1,
\ldots, s$
\[
- K_S \cdot C_i > D \cdot C_i + \tau(C_i) - \mbox{
isod}_{C_i}({\cal N}^{ea}_{C/S}, {\cal O}_C)
\]
\item[$\bullet$] $H^1(C,{\cal N}^{es}_{C/S}) = 0$ if for $i = 1,
\ldots, s$
\[
- K_S \cdot C_i > \sum\limits_{x \in\; Sing (C)} \dim_{\Bbb C}
(({\cal O}_{C,x}/I^{es}(C,x)) \otimes {\cal O}_{C_i,x}) - \mbox{ isod}_{C_i}
({\cal N}^{es}_{C/S}, {\cal O}_C)
\]
\end{itemize}
where $D = \cup_{j\not= i} C_j$ and $K_S$ denotes the
canonical divisor on $S$. Moreover, the isomorphism defects
isod$_{C_i}({\cal N}^{ea}_{C/S}, {\cal O}_C)$ (respectively
isod$_{C_i}({\cal N}^{es}_{C/S},
{\cal O}_C))$ have the lower bound $\#(C_i \cap \mbox{ Sing } C)$.
\end{itemize}
\end{theorem}
\begin{remark}{\rm
\begin{enumerate}
\item
If all singularities of $C$ are quasi--homogeneous or {\sl ordinary $k$--tuple
points} (all branches are smooth with distinct tangents) then we obtain as an
equivalent criterium for the vanishing of $H^1(C,{\cal N}^{es}_{C/S})$
\[
- - - K_S \cdot C_i > D \cdot C_i + \tau^{es}(C_i) - \mbox{ isod}_{C_i}
({\cal N}^{es}_{C/S},{\cal O}_C).
\]
\item
{}From the adjunction formula we obtain
\[
- - - K_S \cdot C_i = C^2_i - 2 p_a (C_i) + 2.
\]
\item
If $C$ is irreducible, we have
\begin{itemize}
\item $H^1(C,{\cal N}^{ea}_{C/S}) = 0$ if $-K_SC > \tau(C) - \mbox{
isod}({\cal N}^{ea}_{C/S}, {\cal O}_C)$
\item $H^1(C,{\cal N}^{es}_{C/S}) = 0$ if $-K_SC > \tau^{es}(C) - \mbox{
isod}({\cal N}^{es}_{C/S}, {\cal O}_C)$.
\end{itemize}
\end{enumerate}}
\end{remark}
\begin{sub}\label{3.6}
{\rm {\bf Proof}: Most parts of the proof are identical for the equianalytic
$(ea)$ and the equisingular $(es)$ case, there we use again the notation $H'$
respectively ${\cal N}'_{C/S}$ as above:
\vspace{-0.5cm}
\begin{itemize}
\item[(ii)]
Let $H^1(C, {\cal N}'_{C/S}) = 0$ and $A \twoheadrightarrow A/(\eta) = \bar{A}$
be a small extension of Artinian ${\Bbb C}$--algebras. For the smoothness of
$(H', C)$, we have to show that each equianalytic (respectively
equisingular) family $\bar{{\cal C}}$ over $\bar{A}$ lifts to an equianalytic
(respectively equisingular) family ${\cal C}$ over $A$.
$\bar{{\cal C}}$ is given by local equations $\bar{F_i} \in \Gamma(U_i, {\cal O}_S
\otimes \bar{A})$, where $(U_i)_{i \in I}$ is an open covering of $S$,
such that
$\bullet$ on $U_{ij} := U_i \cap U_j$, $\bar{F}_i = \bar{G}_{ij}
\cdot \bar{F}_j$ with a unit $\bar{G}_{ij}$
$\bullet$ the image $F^{(0)}_i$ of $\bar{F}_i$ in $\Gamma(U_i, {\cal O}_S
\otimes {\Bbb C})$ is a local equation for $C \subset S$
$\bullet$ the germs $\bar{F}_{i,x}$ describe an equianalytic
(respectively equisingular) deformation of $(C,x)$.
On the other hand, we know that the equianalytic (respectively
equisingular) functor $E'$, which associates to each Artinian local
${\Bbb C}$--algebra the set of all equianalytic (respectively equisingular)
deformations of $(C,x)$ over Spec $A$, is smooth and has a very good
deformation theory (cf.\ \cite{Wa1},
for the equisingular case). Using the results of M.\
Schlessinger (\cite{Schl}, Remark 2.17),
this guarantees in particular
\begin{itemize}
\item[$\bullet$] the existence of an equianalytic (respectively
equisingular) lifting $F_i \in \Gamma(U_i, {\cal O}_S \otimes A)$ of
$\bar{F}_i$
\item[$\bullet$] for any lifting $G_{ij} \in \Gamma(U_{ij}, {\cal O}_S \otimes
A)$
of $\bar{G}_{ij}$ the existence of $h_{ij} \in \Gamma(U_{ij}, {\cal O}_S \otimes
A)$
with $F_i = G_{ij} \cdot F_j + \eta \cdot h_{ij}$ and $(h_{ij})_x \in
j(C,x)$ (respectively $(h_{ij})_x \in I^{es}(C,x)$).
\end{itemize}
To obtain the lifted family we are looking for, we have to modify the
$F_i$ and $G_{ij}$ in a suitable way, such that the $h_{ij}$ become 0.
We know
\begin{eqnarray*}
\eta \cdot h_{ij} + \eta \cdot G_{ij} \cdot h_{jk} & = & F_i - G_{ij}
\cdot F_j + G_{ij} \cdot (F_j - G_{jk} \cdot F_k)\\
& = & \eta \cdot h_{ik} + (G_{ik} - G_{ij} \cdot G_{jk}) \cdot F_k
\end{eqnarray*}
where $(G_{ik} - G_{ij} \cdot G_{jk}) \in \Gamma(U_{ijk}, {\cal O}_S \otimes
(\eta))$ and $(\eta) \cdot \frak m_A = 0$. As sections in ${\cal O}_S
\otimes A/\frak m_A = {\cal O}_S \otimes {\Bbb C}$ we obtain
\[
h_{ij} + G_{ij}^{(0)} \cdot h_{jk} = h_{ik} + \left[\frac{1-G_{ij} \cdot
G_{jk} \cdot G_{ik}^{-1}}{\eta}\right] \cdot G^{(0)}_{ik} \cdot F_k^{(0)}.
\]
Furthermore, $F_i^{(0)} = G_{ij}^{(0)} \cdot F_j^{(0)}$, which implies in
$\Gamma(U_{ijk}, {\cal N}_{C/S})$ the cocycle condition
\[
\frac{h_{ij}}{F_i^{(0)}} + \frac{h_{jk}}{F_j^{(0)}} =
\frac{h_{ik}}{F_i^{(0)}}.
\]
From the definition of the $h_{ij}$ it follows that
$\left(\frac{h_{ij}}{F_i^{(0)}} \mid i, j \in I\right)$ represents an
element in $H^1(C,{\cal N}'_{C/S}) = 0$. Hence, there exist $f_i \in
\Gamma(U_i, {\cal O}_S \otimes {\Bbb C})$ such that
\[
\frac{h_{ij}}{F_i^{(0)}} = \frac{f_j}{F_j^{(0)}} - \frac{f_i}{F_i^{(0)}}
\]
as sections in ${\cal N}'_{C/S}$, especially $h_{ij} + f_i - f_j \cdot
G_{ij}^{(0)} \in \Gamma(U_{ij}, J_C)$ and all germs $(f_i)_x$ lie in the
Jacobian (respectively equisingularity) ideal. Defining $g_{ij} :=
\frac{h_{ij} + f_i - f_j \cdot G_{ij}^{(0)}}{F_j^{(0)}},\; \tilde{F}_i :=
F_i + \eta \cdot f_i$ and $\widetilde{G}_{ij} := G_{ij} + \eta \cdot
g_{ij}$ we obtain the lifted family we were looking for.
\item[(i)] The germ $(H'_S, C)$ is the fibre over the origin of a
(non--linear) obstruction map $H^0(C, {\cal N}'_{C/S}) \to H^1(C, {\cal N}'_{C/S})$
(cf.\ \cite{La}, Theorem 4.2.4). Hence
\vspace{-0.5cm}
\begin{eqnarray*}
\dim\; H^0(C, {\cal N}'_{C/S}) & \ge & \dim(H'_S, C)\\
& \ge & \dim(H^0(C, {\cal N}'_{C/S})) - \dim(H^1(C, {\cal N}'_{C/S}))\\
& = & \chi({\cal N}_{C/S}) - \chi({\cal T}^1_C/({\cal T}^1_C)')\\
& = & \deg({\cal N}_{C/S}) + \chi({\cal O}_C) - \tau'(C)\\
& = & C^2 + 1 -p_a(C) - \tau'(C)
\end{eqnarray*}
where $\tau'(C)$ denotes the total Tjurina number of $C$ (respectively
$\tau^{es}(C))$. Both inequalities become an equality if
$H^1(C,{\cal N}'_{C/S}) =
0$.
\item[(iii)] By Proposition \ref{3.4}, $H^1(C, {\cal N}'_{C/S}) = 0$, if for $i
= 1,
\ldots, s$
\[
\deg(\overline{{\cal N}'_{C/S} \otimes {\cal O}_{C_i}}) > (K_S + C) \cdot C_i -
\mbox{ isod}_{C_i} ({\cal N}'_{C/S}, {\cal O}_C)
\]
where $\overline{\phantom{xxx}}$ denotes reduction modulo torsion. On
the other hand, we have the exact sequence
\[
0 \to \overline{{\cal N}'_{C/S} \otimes {\cal O}_{C_i}} \to {\cal N}_{C/S} \otimes
{\cal O}_{C_i} \to ({\cal T}^1_C/({\cal T}^1_C)^\prime), \otimes {\cal O}_{C_i} \to 0
\]
which implies
\[
\deg(\overline{{\cal N}'_{C/S} \otimes {\cal O}_{C_i}}) = C \cdot C_i - \dim_{\Bbb C}
H^0(C, ({\cal T}^1_C/({\cal T}^1_C)^\prime) \otimes {\cal O}_{C_i}).
\]
Finally, we obtain the above criteria by
\[
\dim_{\Bbb C} H^0(C, {\cal T}^1_C/({\cal T}^1_C)^{es} \otimes {\cal O}_{C_i}) = \sum_{x \in C}
\dim_{\Bbb C}(({\cal O}_{C,x}/I^{es}(C,x)) \otimes {\cal O}_{C_i,x})
\]
respectively using the Leibniz rule (with $g$ as the equation of
$(D,x))$ by
\vspace{-0.5cm}
\begin{eqnarray*}
\dim_{\Bbb C} H^0(C, {\cal T}^1_C \otimes {\cal O}_{C_i}) & = & \sum_{x \in C} \dim_{\Bbb C}
(({\cal O}_{C,x}/j(C,x)) \otimes {\cal O}_{C_i,x})\\
& = & \sum_{x \in C} \dim_{\Bbb C}({\cal O}_{C_i,x}/g \cdot j(C_i,x))\\
& = & \sum_{x \in C} (\dim_{\Bbb C}({\cal O}_{C_i,x}/j(C_i,x)) +
\dim_{\Bbb C}({\cal O}_{C_i,x}/g \cdot {\cal O}_{C_i,x}))\\
& = & \tau(C_i) + C_i \cdot D.
\end{eqnarray*}
\end{itemize}
\begin{flushright} $\Box$\end{flushright}
}
\end{sub}
\begin{sub}\label{3.7}{\rm
{\bf Curves in} $\P^2({\Bbb C})$
Let $C \subset \P^2 := \P^2({\Bbb C})$ be a reduced curve of degree $d$ with
(homogeneous) equation $F(X,Y,Z) = 0$, then we define the {\sl polar} of $C$
relative to
the point ($\alpha : \beta : \gamma) \in \P^2$ to be the curve
$C_{\alpha\beta\gamma}$ with equation $\alpha \cdot F_X(X,Y,Z) + \beta \cdot
F_Y(X,Y,Z) + \gamma \cdot F_Z(X,Y,Z) = 0$.
}\end{sub}
\begin{lemma}
The {\sl generic polar} $C_{\alpha\beta\gamma}$ (with
$(\alpha:\beta:\gamma) \in \P^2$ a generic point) is an irreducible curve of
degree $d-1$, if and only if $C$ is not the union of $d \ge 3$ lines through
the same point.
\end{lemma}
{\bf Proof}: Applying Bertini's theorem, we have irreducibility if there is no
algebraic relation between $F_X, F_Y$ and $F_Z$. Considering such a
(homogeneous) relation of minimal degree
\[
\sum_\alpha a_\alpha F^{\alpha_1}_X \cdot F^{\alpha_2}_Y \cdot F^{\alpha_3}_Z =
0
\]
and differentiating, we obtain a system of equations
\[
A \cdot \;\left(\begin{array}{c}F_{XX}\{\Bbb F}_{XY}\{\Bbb F}_{XZ}\end{array}\right)
\; + B \cdot\;\left(\begin{array}{c}F_{YX}\{\Bbb F}_{YY}\{\Bbb F}_{YZ}\end{array}\right)
\; + \Gamma
\cdot\;\left(\begin{array}{c}F_{ZX}\{\Bbb F}_{ZY}\{\Bbb F}_{ZZ}\end{array}\right)
\; = 0
\]
where $A, B$ and $\Gamma$ generically do not vanish. Now the lemma follows
from the fact that the Hessian covariant vanishes identically only if $C$ is
the union of $d$ lines through one point (cf.\ \cite{He}, Lehrsatz 6).
\hfill $\Box$\\
In the following we choose suitable coordinates such that Sing $C$ lies in the
affine plane $Z \not= 0$ and a
generic polar $C' \subset P^2$ relative to $(\alpha : \beta : 0)$ with equation
$\alpha \cdot F_X + \beta
\cdot F_Y = 0$ is irreducible.
Then we have an obvious morphism ${\cal O}_{C'}(d) \to {\cal T}^1_C$ given by the natural
projections
\[
{\cal O}_{C',x} = {\cal O}_{\P^2,x}/(\alpha f_X + \beta f_Y) \to {\cal O}_{\P^2,x}/j(C,x) =
{\cal T}^1_{C,x}
\]
where $f(X,Y) = F(X,Y, 1)$ is the affine equation of $C$.
We define
\vspace{-0.5cm}
\begin{eqnarray*}
\tilde{{\cal N}}^{ea}_{C'/\P^2} & := & Ker({\cal O}_{C'}(d) \to {\cal T}^1_C)\\
\tilde{{\cal N}}^{es}_{C'/\P^2} & := & Ker({\cal O}_{C'}(d) \to {\cal T}^1_C/({\cal T}_C^1)^{es}).
\end{eqnarray*}
\begin{corollary}\label{3.8}
Let $C \subset \P^2$ be a reduced projective curve of degree $d$,
$C_i(i = 1,\ldots, s)$ its irreducible components and $d_i$
the degree of $C_i$.
\begin{enumerate}
\item[(i)] $H^1(C, {\cal N}^{ea}_{C/\P^2}) = 0$ if and only if the forgetful
morphism ${\cal D} e\!f_{C/\P^2} \to \prod_{x \in Sing\, C}$ ${\cal D} e\!f_{C,x}$ is
surjective.
\item[(ii)] If $H^1(C, {\cal N}^{ea}_{C/\P^2}) = 0$ (respectively $H^1(C,
{\cal N}^{es}_{C/S}) = 0)$ then $(H^{ea}_{\P^2}, C)$ (respectively
$(H^{es}_{\P^2},C)$)
is smooth of dimension $\frac{1}{2} d(d+3) - \tau(C)$ (respectively
$\frac{1}{2} d \cdot (d+3) - \tau^{es}(C))$.
\item[(iii)] $H^1(C, {\cal N}^{ea}_{C/\P^2}) = 0$ if for $i = 1, \ldots, s$
\[
3d_i > d_i \cdot (d - d_i) + \tau(C_i) - \mbox{ isod}_{C_i}
({\cal N}^{ea}_{C/\P^2}, {\cal O}_C),
\]
moreover, isod$_{C_i}({\cal N}^{ea}_{C/\P^2}, {\cal O}_C) \ge \#$ (Sing$(C) \cap C_i)$,\\
$H^1 (C,{\cal N}^{es}_{C/S}) = 0$ if for $i = 1, \ldots, s$
\[
3 d_i > d_i \cdot (d-d_i) + \tau^{es}(C_i) - \mbox{
isod}_{C_i}({\cal N}^{es}_{C/\P^2}, {\cal O}_C).
\]
\item[(iv)] If $C$ is not the union of $d \ge 3$ lines through one point
$H^1(C, {\cal N}^{ea}_{C/\P^2})$ (respectively $H^1(C, {\cal N}^{es}_{C/\P^2})$)
vanishes,
if
\[
4 d > 4 + \tau(C) - \mbox{ isod}(\tilde{{\cal N}}^{ea}_{C'/\P^2}, {\cal O}_{C'})
\]
\[
(\mbox{respectively } 4 d > 4 + \tau^{es}(C) - \mbox{ isod}
(\tilde{{\cal N}}^{es}_{C'/\P^2}, {\cal O}_{C'}))
\]
where $C'$ denotes the generic polar as defined above.
\end{enumerate}
\end{corollary}
{\bf Proof}: (ii) and (iii) follow immediately from Theorem \ref{3.5}. To
prove (iv), we consider the exact sequences
\[
\begin{array}{ccccccccc}
0 & \to & {\cal N}'_{C/\P^2} & \to & {\cal N}_{C/\P^2} & \to & {\cal T}^1_C/({\cal T}^1_C)' & \to &
0\\
0 & \to & \tilde{{\cal N}}'_{C'/\P^2} & \to & {\cal O}_{C'}(d) & \to & {\cal T}^1_C/({\cal T}^1_C)'
&
\to & 0
\end{array}
\]
and the corresponding long exact cohomology sequences where $'$ represents
again both the equianalytic and the equisingular case. We know that
${\cal N}_{C/\P^2} \cong {\cal O}_C(d)$ and (by Proposition \ref{3.4}) $H^1(C,{\cal O}_C(d))
= 0$. Furthermore, if $C$ is not the union of $d \ge 3$ lines through one
point, $C'$ is
irreducible and
\vspace{-0.5cm}
\begin{eqnarray*}
\deg(\tilde{{\cal N}}'_{C'/\P^2}) - (K_{\P^2} + C') \cdot C' & = &
\deg({\cal O}_{C'}(d)) - \tau'(C) - (d-4) \cdot (d-1)\\
& = & 4 \cdot (d-1) - \tau'(C).
\end{eqnarray*}
Hence, applying Proposition \ref{3.4} the conditions in (iv) guarantee the
vanishing of $H^1(C', \tilde{{\cal N}}'_{C'/\P^2})$. Additionally, the exact
sequence
\[
0 \to {\cal O}_{\P_2} \to {\cal O}_{\P_2} (d) \to {\cal O}_C (d) \to 0
\]
respectively an analogous sequence for $C'$ induce surjective mappings
\[
\Phi : H^0(\P^2, {\cal O}_{\P^2}(d)) \twoheadrightarrow H^0(C, {\cal O}_C(d)) \mbox{
respectively }
\Phi' : H^0(\P^2, {\cal O}_{\P^2}(d))\twoheadrightarrow H^0(C', {\cal O}_{C'}(d))
\]
which lead to a commutative diagram with exact horizontal rows
\unitlength1cm
\begin{picture}(10,3)
\put(0.5,1.5){$H^0(\P^2, {\cal O}_{\P^2}(d))$}
\put(3.5,1.8){\vector(2,1){1.2}}
\put(3.8,2.2){$\Phi$}
\put(5.0,2.4){$H^0(C, {\cal O}_C(d)) \to H^0(C, {\cal T}^1_C/({\cal T}^1_C)')
\to H^1(C, {\cal N}'_{C/\P^2}) \to 0$}
\put(9.0,1.5){$\|$}
\put(3.8,0.75){$\Phi'$}
\put(3.5,1.4){\vector(2,-1){1.2}}
\put(5.0,0.5){$H^0(C', {\cal O}_{C'}(d)) \to H^0(C, {\cal T}^1_C/({\cal T}^1_C)') \to 0.$}
\end{picture}
This shows that $H^0(C,{\cal O}_C(d)) \to H^0(C,{\cal T}^1_C/({\cal T}^1_C)')$ is surjective
and, hence, $H^1(C, {\cal N}'_{C/\P^2}) = 0$.
The same argument shows that $H^0(C, {\cal O}_{C}(d)) \to H^0(C,{\cal T}^1_C)$ is
surjective if and only if $H^1(C, {\cal N}^{ea}_{C/\P^2}) = 0$. Since
${\cal D} e\!f_{C/\P^2}(T_\varepsilon) = H^0(C, {\cal O}_C(d))$
and $H^0(C, {\cal T}^1_C) \cong \prod_{x\in Sing\, C}$ ${\cal D}
e\!f_{C,x}(T_\varepsilon)$
this is equivalent to ${\cal D} e\!f_{C/\P^2}(T_\varepsilon) \to \prod_{x
\in Sing(C)}$
${\cal D} e\!f_{C,x}(T_\varepsilon)$ being surjective (and ${\cal D} e\!f_{C/\P^2}$ being
unobstructed). But ${\cal D} e\!f_{C/\P^2}$ and $\prod$ ${\cal D} e\!f_{C,x}$ being
unobstructed, the surjectivity on the tangent level implies the surjectivity
of the functors.\hfill $\Box$
\begin{remark}{\rm
\begin{itemize}
\item[(i)] We call the inequalities in \ref{3.8} (iii) respectively \ref{3.8}
(iv) the $3d$-- respectively $4d$--criteria.
\item[(ii)] The use of a generic polar is due to Shustin \cite{Sh1}, who
obtained (with a different proof) the weaker inequality $4d > 4 + \mu(C)$
instead of \ref{3.8} (iv), where $\mu(C)$ is the total Milnor number of $C$.
\end{itemize}
}
\end{remark}
\begin{sub}\label{3.9}{\rm {\bf A generalization}:
We are also interested in families of curves in $\P^2$ of degree $d$ where for
some singularities the analytic type is fixed, for others only the topological
type is fixed and for the remaining singularities any deformation is allowed.
Let $C \subset \P^2$ be of degree $d$ and Sing$(C) = \{x_1, \ldots, x_k\} \cup
\{y_1, \ldots, y_\ell\} \cup \{z_1, \ldots, z_m\}$. We define the subsheaf
$({\cal T}^1_C)^\prime$ of ${\cal T}^1_C$ by
\[
({\cal T}^1_C)'_x = \left\{
\begin{array}{ll}
0 & \mbox{ if } x \in \{x_1, \ldots, x_k\}\\
I^{es}(C,x)/j(C,x) & \mbox{ if } x \in \{y_1, \ldots, y_\ell\}\\
T^1_{(C,x)} & \mbox{else}
\end{array}\right.
\]
and put
\[
\tau'(C) := \dim_{\Bbb C} H^0(C,{\cal T}^1_C/({\cal T}^1_C)^\prime) = \sum\limits^k_
{i=1} \tau(C,x_i)
+ \sum\limits^\ell_{j=1} \tau^{es}(C,y_j).
\]
Assume there exists a reduced curve $C' \subset \P^2$ of degree $d'$ with the
following
properties:
\begin{itemize}
\item[(a)] $C'$ is irreducible,
\item[(b)] $\{x_1, \ldots, x_k\} \cup \{y_1, \ldots, y_\ell\} \subset C'$,
\item[(c)] if $f_j$ is a local equation of $(C', x_j),\; j = 1, \ldots, k$,
then $f_j \in j(C,x_j)$; if $f_j$ is a local equation of $(C',y_j),\; j = 1,
\ldots, \ell$, then $f_j \in I^{es}(C,y_j)$.
\end{itemize}
Define
\vspace{-0.5cm}
\begin{eqnarray*}
{\cal N}' & := & \mbox{Ker} ({\cal N}_{C/\P^2} = {\cal O}_C(d) \to {\cal T}^1_C/({\cal T}^1_C)
^\prime),\\
\widetilde{{\cal N}}' & := & \mbox{Ker} ({\cal O}_{C'}(d) \to {\cal T}^1_C/({\cal T}^1_C)^\prime).
\end{eqnarray*}
Let ${\cal A}$ be the analytic singularity type defined by $(C,x_1), \ldots,
(C,x_k), {\cal T}$ the topological singularity type defined by $(C,y_1), \ldots,
(C,y_\ell)$ and let ${\cal H} ilb^{{\cal A},{\cal T}}_{\P^2}$ denote the functor
parametrising
proper and flat families of reduced curves in $\P^2$ which have $k$ singular
points of fixed analytic type ${\cal A}$ and $\ell$ singular points of fixed
topological type ${\cal T}$ (see \ref{2.2} for a precise definition). This functor
is represented by a locally closed subspace $H^{{\cal A},{\cal T}}_{\P^2} \subset
H_{\P^2}$ (cf.\ Proposition 2.3).
}
\end{sub}
\begin{proposition}
Let $C \subset \P^2$ be a reduced projective curve of degree $d$, Sing$(C) =
\{x_1, \ldots, x_k\} \cup \{y_1, \ldots, y_\ell\} \cup \{z_1, \ldots,
z_m\}$
and assume that there exists a curve $C' \subset \P^2$ of degree $d'$
satisfying (a) -- (c) above.
\vspace{-0.5cm}
\begin{itemize}
\item[(i)] If $\ell = 0$, then $H^1(C,{\cal N}') = 0$ if and only if ${\cal D}
e\!f_{C/\P^2}
\to \prod\limits^k_{i=1}{\cal D} e\!f_{(C,x_i)}$ is surjective.
\item[(ii)] If $H^1(C,{\cal N}') = 0$ then $H^{{\cal A},{\cal T}}_{\P^2}$ is smooth at $C$ of
dimension $\frac{1}{2} d (d+3) - \tau'(C)$.
\item[(iii)] $H^1(C,{\cal N}') = 0$ if $d'(d-d'+3) > \tau'(C) -$
isod$(\widetilde{{\cal N}}',{\cal O}_{C'})$.
\end{itemize}
\end{proposition}
{\bf Proof}: Use the same argumentation as for Corollary 3.12.\hfill $\Box$
\begin{remark}{\rm
\begin{enumerate}
\item If $\ell = m = 0$, (i) is the same as 3.12 (i). If $\ell = m = 0$
(respectively $k = m = 0)$ (ii) is the same as 3.12\ (ii). If $C$ is
irreducible we may take $C' = C$ and then (iii) is equivalent to 3.12 (iii)
for $\ell = m = 0$ respectively $k = m = 0$. If $C$ is not the union of $d$
lines through one point, we may take $C'$ to be a generic polar and then (iii)
is the same as 3.12 (iv) for $\ell = m = 0$ (respectively $k = m = 0$).
\item We obtain the best possible result for a curve $C'$ of degree $d' =
\frac{d+3}{2}$ satisfying (a) -- (c). In \cite{Sh3}, Shustin has proven the
existence of such an irreducible curve $C'$ of degree
\[
d' \le (2\kappa^2 + \sqrt{\kappa}) \sqrt{\mu(C)} + (1 - \frac{1}{\kappa}) d,
\]
where $\mu(C)$ denotes the total Milnor number and
\[
\begin{array}{ll}
\kappa = &\; \max\;\{\mu(x_i, C) + \mbox{ mult}_{x_i}(C),\; \mu(y_j,C) + \mbox{
mult}_{y_j}(C)\} - 1\\[-0.5ex]
& {1 \le i \le k\atop 1 \le j \le \ell}
\end{array}
\]
\end{enumerate}
}
\end{remark}
\newpage
\section{Local isomorphism defects of plane curve singularities}
Let $u, v$ be local coordinates of the smooth surface $S$ in a singular
point $x$ of the reduced compact curve $C \subset S,\; C = C_1 \cup \ldots
\cup C_s$ the decomposition into irreducible components and $f(u,v) = 0$
(respectively $f_i(u,v) = 0$) be local equations of $(C,x)$ (respectively
($C_i,x$)). In the following, we give estimations for the (local)
isomorphism defects occurring in Chapter 3:
\begin{lemma}\label{4.1}
For a reduced plane curve singularity $(C,x) \subset (S,x)$ we have:
\begin{itemize}
\item [(i)] isod$_x ({\cal N}^{ea}_{C/S}, {\cal O}_C) = 1$ if $(C,x)$ is
quasihomogeneous.\\
isod$_x ({\cal N}^{ea}_{C/S}, {\cal O}_C) > 1$ if $(C,x)$ is
not quasihomogeneous.
\item[(ii)] isod$_{C_i,x} ({\cal N}^{ea}_{C/S}, {\cal O}_C) \ge 1$ for $i = 1, \ldots,
s$.
\end{itemize}
\end{lemma}
{\bf Proof}:
\vspace{-0.5cm}
\begin{itemize}
\item[(i)] By definition isod$_x ({\cal N}^{ea}_{C/S}, {\cal O}_C) = \min\;
\dim_{\Bbb C}(\mbox{
coker }\, \varphi : j(C,x) \to {\cal O}_{C,x})$ where the minimum is taken over all
${\cal O}_{C,x}$--linear maps. Now the Jacobian ideal $j(C,x)$ of an isolated
singularity cannot be generated by a single element and there is an
isomorphism $\varphi : j(C,x) \buildrel \cong\over\to \frak{m}_{C,x}$ exactly
if
$(C,x)$ is quasihomogeneous.
\item[(ii)] We have to look for an ${\cal O}_{C,x}$--linear map $\varphi : j(C,x)
\to {\cal O}_{C,x}$ whose restriction to $(C_i,x)$ has minimal cokernel. The
above statement follows immediately. \hfill $\Box$
\end{itemize}
\begin{sub}\label{4.2}{\rm
Let $(C,x)$ be quasihomogeneous with positive weight vector $w = (w_1, w_2)$
and (weighted) degree $d$, then (as an
${\cal O}_{C,x}$--ideal) $I^{es}(C,x)$ is generated by the Jacobian ideal $j(C,x)$
and all monomials $u^\alpha v^\beta$ with $w_1 \cdot \alpha + w_2 \cdot
\beta \ge d$. Furthermore we have the normalization
\[
n : {\cal O}_{C,x} \hookrightarrow \bar{{\cal O}} := \prod\limits^r_{i=1} {\Bbb C}\{t\}
\]
where $r$ denotes the number of local irreducible components $(C^{(i)},x)$ of
$(C,x)$ (not to be confused with the global components $C_i$).
In the following, we use the notations cond$({\cal O})$, cond$(j)$ respectively
cond$(I^{es})$ for the {\sl conductor ideals} of ${\cal O}_{C,x}$, the Jacobian
respectively the equisingularity ideal in ${\cal O}_{C,x}$, where, for an
${\cal O}_{C,x}$--ideal $I$,
\[
\mbox{ cond} (I) := \{ g \in I \mid g \cdot \bar{{\cal O}} \subset I\}.
\]
Furthermore, for all these ${\cal O}_{C,x}$--ideals, we denote by $\Gamma(I)
\subset {\Bbb N}^r$ the {\sl set of values} of $I$ and by $\underline{c} (I) \in
{\Bbb N}^r$ the
{\sl conductor} of $I$, that is $\Gamma(\mbox{cond}(I)) = \underline{c} (I) +
{\Bbb N}^r$.}
\end{sub}
\begin{lemma}
Let $(C,x)$ be quasihomogeneous of degree $d$,
then
\begin{itemize}
\item[(i)] isod$_x ({\cal N}^{es}_{C/S}, {\cal O}_C) = \delta(C,x) -
\dim_{\Bbb C}(I^{es}(C,x)/$cond$(I^{es}))$\\
{\it especially}: isod$_x({\cal N}^{es}_{C/S}, {\cal O}_C) \ge 1$ with equality if
and only if $j(C,x) = I^{es}(C,x)$.
\item[(ii)] isod$_{C_i,x} ({\cal N}^{es}_{C/S}, {\cal O}_C) =
\dim_{\Bbb C}(({\cal O}_{C,x}/\mbox{cond}({\cal O})) \otimes {\cal O}_{C_i,x}) -
\dim_{\Bbb C}((I^{es}(C,x)/\mbox{cond}(I^{es})) \otimes {\cal O}_{C_i,x})$.
\end{itemize}
\end{lemma}
\newpage
{\bf Proof}:
\begin{itemize}
\item[(i)] To calculate isod$_x({\cal N}^{es}_{C/S}, {\cal O}_C)$ we have to consider an
${\cal O}_{C,x}$--linear mapping
\[
\Psi : I^{es}(C,x) \to {\cal O}_{C,x}
\]
with minimal cokernel. We know that such a $\Psi$ maps cond$(I^{es})$ to
cond$({\cal O})$, hence we obtain the estimate
\vspace{-0.5cm}
\begin{eqnarray*}
\mbox{isod}_x({\cal N}^{es}_{C/S}, {\cal O}_C) & \ge &
\dim_{\Bbb C}({\cal O}_{C,x}/\mbox{cond}({\cal O}))
- - - \dim_C(I^{es}(C,x)/\mbox{cond}(I^{es}))\\
& = & \delta(C,x) - \dim_{\Bbb C}(I^{es}(C,x)/\mbox{cond}(I^{es}))
\end{eqnarray*}
with equality if and only if there exists a $\Psi$ that maps cond$(I^{es})$
onto cond$({\cal O})$, or equivalently (using $\varphi : j(C,x) \buildrel
\cong\over\to \frak m_{C,x})$ if and only if we can find an
${\cal O}_{C,x}$--linear mapping
\[
\Phi : I^{es}(C,x) \to j(C,x)
\]
of weighted degree $\underline{c} (j) - \underline{c} (I^{es})$. Now
$\underline{c}(I^{es})-\underline{1}$ is a maximal (in the sense of
\cite{De}) in the semigroup $\Gamma(I^{es}) \supset \Gamma(j)$.
Hence, $(\underline{c}({\cal O}) - \underline{c}(j)) + \underline{c} (I^{es}) -
\underline{1}$
is a maximal in $\Gamma({\cal O})$ and using the symmetry of $\Gamma({\cal O})$ we see
that
\[
(\underline{c}({\cal O}) - \underline{1}) - (\underline{c}({\cal O}) - \underline{c}(j)
+ \underline{c}
(I^{es}) - \underline{1}) = \underline{c}(j) - \underline{c}(I^{es}) \in
\Gamma({\cal O}).
\]
The additional statement is an immediate consequence from the fact that all
monomials of degree at least $d$ are contained in
cond$(I^{es})$; thus,
\vspace{-0.5cm}
\begin{eqnarray*}
\dim_{\Bbb C}(I^{es}(C,x)/\mbox{cond}(I^{es})) & = &
\dim_{\Bbb C}(j(C,x)/\mbox{cond}(I^{es}) \cap j(C,x))\\
& \le & \dim_{\Bbb C}(j(C,x)/\mbox{cond}(j))\\
& = & \delta(C,x) - 1.
\end{eqnarray*}
\item[(ii)] Follows from the considerations above.\hfill $\Box$
\end{itemize}
\begin{remark}{\rm
If $(C,x)$ is quasihomogeneous, then it is
easy to see that
\vspace{-0.5cm}
\begin{eqnarray*}
\dim_{\Bbb C}(({\cal O}_{C,x}/\mbox{cond}({\cal O})) \otimes {\cal O}_{C_i,x}) & = &
\dim_{\Bbb C}({\cal O}_{C_i,x}/\mbox{cond}({\cal O}_{C_i})) + (D \cdot C_i,x)\\
\dim_{\Bbb C}((I^{es}(C,x)/\mbox{cond}(I^{es})) \otimes {\cal O}_{C_i,x}) & \ge &
\dim_{\Bbb C}(I^{es}(C_i,x)/\mbox{cond}(I^{es}(C_i,x)))
\end{eqnarray*}
(where $C_j$, $j = 1, \ldots, s)$ are the irreducible components of $C$ and
($D = \bigcup\limits_{j\not= i} C_j)$. Thus we obtain as an upper bound
\[
\mbox{isod}_{C_i,x}({\cal N}^{es}_{C/S}, {\cal O}_C) \le \mbox{ isod}_x({\cal N}^{es}_{C_i/S},
{\cal O}_{C_i}) + (D \cdot C_i,x).
\]
Furthermore, for integer weights $w_1 \ge w_2$, gcd$(w_1, w_2) = 1$, the
difference is bounded by $\frac{w_1 - 1}{w_2} + 2$.
}
\end{remark}
\begin{sub}\label{4.3}{\rm
{\bf Examples}:
\begin{itemize}
\item[(i)] If $(C,x)$ is an ADE--singularity, then isod$_x({\cal N}^{es}_{C/S},
{\cal O}_C) = 1$.
\item[(ii)] If $(C,x)$ is homogeneous of degree $r \ge 3$, then
\[
\mbox{isod}_x({\cal N}^{es}_{C/S}, {\cal O}_C) = \frac{r \cdot(r-1)}{2} -2 =
\tau^{es}(C,x) - \mbox{ mult}_x(C)
\]
furthermore, let $C = C_i \cup D$ as above, then we obtain for $(C_i,x)$ a
smooth branch
\[
\mbox{isod}_{C_i,x} ({\cal N}^{es}_{C/S}, {\cal O}_C) = r-2,
\]
while in the singular case
\[
\mbox{isod}_{C_i,x} ({\cal N}^{es}_{C/S}, {\cal O}_C) = (C_i \cdot D, x) + \mbox{ isod}_x
({\cal N}^{es}_{C_i/S}, {\cal O}_{C_i}).
\]
More generally, these statements are valid, if $(C,x)$ is an ordinary
$r$--tuple point ($r$ smooth branches with different tangents, $r \ge 3$);
in this case each equimultiple deformation is equisingular.
\item[(iii)] If $(C,x)$ has the local equation $u^p - v^q = 0$ where $q \ge
p \ge 3$ and $(C_i,x)$ consist of $b \le r = \mbox{ gcd } (p,q)$ irreducible
branches, then
\[
\mbox{isod}_{C_i,x} ({\cal N}^{es}_{C/S}, {\cal O}_C) = \frac{b}{2r} \cdot
(pq(2-\frac{b}{r}) + (r-p-q)) - \Big[\frac{q-2}{p}\Big] - M,
\]
where $M = 2$ unless $b = 1$ and $q = k \cdot p\;\; (k \in {\Bbb N})$, then $M = 1$.
\end{itemize}
}
\end{sub}
\begin{sub}\label{4.4}{\rm
Now let $S = \P^2$ and $C \subset \P^2$ be different from the union of $d \ge
3$ lines through one point and $C' \subset \P^2$ denote the (irreducible)
generic polar of $C$ (cf.\ \ref{3.7}) with affine equation $\alpha \cdot f_X
+ \beta f_Y = 0$.
}
\end{sub}
\begin{lemma}
\begin{itemize}
\item[(i)] isod$_x (\widetilde{{\cal N}}^{ea}_{C'/\P^2}, {\cal O}_{C'}) \ge 0$ with
equality if and only if $(C,x)$ is quasihomogeneous.
\item[(ii)] isod$_x (\widetilde{{\cal N}}^{es}_{C'/\P^2}, {\cal O}_{C'}) \ge \delta(C',x)
- - - \dim_{\Bbb C}(I^{es}(C,x) \otimes {\cal O}_{C',x}/\mbox{cond}(I^{es}(C,x) \otimes
{\cal O}_{C',x}))$.
\end{itemize}
\end{lemma}
{\bf Proof}: By definition, equality in (i) is equivalent to the statement
that the ${\cal O}_{C',x}$--ideal generated by $f, f_X$ and $f_Y$ is generated by
one single element. Obviously, this holds exactly if $(C,x)$ is
quasihomogeneous, (ii) follows from the considerations in the proof of Lemma
4.3.\hfill
$\Box$
\begin{remark}{\rm The generic polar $C'$ depends on the whole curve $C$ and
not only on the germ $(C,x)$, hence, in general it is {\sl not} enough to know
the local
equation of $(C,x)$ to be able to calculate
isod$_x(\widetilde{{\cal N}}^{es}_{C'/\P^2}, {\cal O}_{C'})$. For example, if $(C,x)$
is
homogeneous of degree $d \ge 6$, $(C',x)$ need not be quasihomogeneous. But,
in special cases, we are able to give explicit formulas:
}
\end{remark}
\begin{sub}\label{4.5}{\rm
{\bf Examples}:
\begin{itemize}
\item[(i)] If $(C,x)$ is an ADE--singularity, then
isod$_x(\widetilde{{\cal N}}^{es}_{C'/\P^2}, {\cal O}_{C'}) = 0$.
\item[(ii)] If $(C,x)$ has the (homogeneous) local equation $(u^r - v^r = 0)\;
(r \ge 3)$
then $(C',x)$ has an equation $(\tilde{u}^{r-1} - \tilde{v}^{r-1} = 0)$ and
\[
\mbox{isod}_x (\widetilde{{\cal N}}^{es}_{C'/\P^2}, {\cal O}_{C'}) = \frac{r \cdot
(r-3)}{2}.
\]
Moreover, in
the case of a (not homogeneous) local equation $(u^p - v^q = 0)\; (q > p \ge
3)$, $(C',x)$ has an equation $(\tilde{u}^{p-1} - \tilde{v}^{q-1} = 0)$ and we
obtain the estimate
\[
\mbox{isod}_x(\widetilde{{\cal N}}^{es}_{C'/\P^2}, {\cal O}_{C'}) \ge
\frac{(p-3)(q-1)}{2} + \frac{2gcd(p-1, q-1) - gcd(p,q) -1}{2}-
\Big[\frac{q}{p}\Big] + \varepsilon
\]
where $\varepsilon = 0$ unless $p$ divides $q$, then $\varepsilon = 1$.
\end{itemize}
}
\end{sub}
\begin{sub}{\rm {\bf Problem}: Do the different isomorphism defects
considered above behave (lower)
semicontinuous under equianalytic respectively equisingular deformations of
$C/S$?
}
\end{sub}
\newpage
\section{Applications and Examples}
\begin{corollary}\label{5.1}
\begin{itemize}
\item[(i)]
Let $C \subset \P^2$ be a curve of degree $d$ with {\sl ordinary} $(
k_i${--}) {\sl multiple points} $(i = 1, \ldots, N)$ as the only
singularities, $C = C_1 \cup \ldots \cup C_s$ its decomposition into
irreducible components (deg $C_i = d_i)$ then
\[
(H^{es}_{\P^2}, C) \mbox{ is smooth of dimension }\frac{d
\cdot(d+3)}{2} - \sum\limits^N_{j=1} \bigg(\frac{k_j \cdot(k_j+1)}{2} -
2\bigg), \mbox{
if for } i = 1, \ldots, s
\]
\[
3 \cdot d_i > \sum_{{x \in C_i \cap Sing\, C\atop mult_x(C)>2}}
\mbox{mult}_x(C_i)
\]
where mult$_x(C)$ (respectively mult$_x (C_i)$) denote the multiplicity of $C$
(respectively $C_i$) at $x$.
\item[(ii)] Let $C \subset \P^2$ be a curve of degree $d$ whose singularities
are all of local equations $(u^{p_i} - v^{q_i} = 0)\; (q_i \ge p_i)$ or
ADE--singularities, then
\[
(H^{es}_{\P^2}, C)\mbox{ is smooth of dimension }
\frac{d(d+3)}{2} - \sum\limits^N_{i=1} \bigg(\frac{(p_i + 1) \cdot (q_i + 1) -
gcd(p_i, q_i) - 5}{2} - \Big[\frac{q_i}{p_i}\Big] + \varepsilon_i\bigg)
\]
where
$\varepsilon_i = 0$ unless $p_i$ divides $q_i$, then $\varepsilon_i = 1$, if
\[
4d > 4 + \sum_{\{ADE\}} \mu(C,x) + \sum_{\{not\; ADE\}} (p_i +
2q_i -3 - gcd(p_i -1, q_i - 1)).\]
\end{itemize}
\end{corollary}
{\bf Proof}: The statements follow immediately from Corollary 3.12 and
Example 4.5 (ii), respectively Example 4.9 (ii). \hfill $\Box$
\begin{remark}\label{5.2}{\rm The result in (i) was already obtained by C.\
Giacinti-Diebolt
(\cite{Gia}) using vanishing theorems on the normalization of $C$. It implies
the ancient result of Severi \cite{Sev} that for a curve $C$ with no other
singularities but ordinary double points $(H^{es}_{\P^2}, C)$ is smooth.
Another consequence of the calculations in Chapter 4 is the following: the
contribution of a quasihomogeneous singularity $(C,x)$ with local equation
$(u^p - v^q + uv \cdot \tilde{f}(u,v) = 0)$, $q \ge p \ge 3$, to the
right--hand side in the $3d$--criterion for an irreducible curve $C$ is
\[
\tau^{es}(C,x) - \mbox{ isod}_x({\cal N}^{es}_{C/S}, {\cal O}_C) = p + q - gcd(p,q) -
\varepsilon
\]
where $\varepsilon = 0$, unless $q \equiv 1 \mbox{ mod } p$, then $\varepsilon
=
1$, while the contribution of an $A_k$--singularity is $k-1$. This
corresponds to the result of E.\ Shustin in \cite{Sh2}.
Nevertheless, in some cases the new $4d$--criterion gives more information:
}
\end{remark}
\begin{sub}{\rm
{\bf Example}:
\begin{itemize}
\item[(a)] $(x^4 - x^2 z^2 + y^2 z^2 + y^3 z) \cdot y \cdot (x + 2y + z) \cdot
(x-2y-z) = 0$ defines a {\it reducible}\/ curve $C \subset \P^2$ having exactly
3 ordinary triple points lying on 1 line (hence, Corollary \ref{5.1} does {\it
not}\/ apply) and 7 ordinary double points. But $4d = 28 > 23 = 4 +
\tau(C)$; hence, $(H^{ea}_{\P^2}, C)$, respectively $(H^{es}_{\P^2}, C)$ are
smooth of dimension 16.
\end{itemize}}
\end{sub}
\begin{sub}{\rm
In general, it is a difficult problem to determine for a given $d$ whether
there exists a projective plane curve of degree $d$ having a fixed number of
singularities of given {\sl analytic} type. On the other hand, the local
deformations of a plane curve singularity are well understood. Hence,
knowing about the existence of one low degree curve with ``big'' singularities
our $4d$--criterion allows us to give positive answers to some of the above
existing problems (answers which we did not obtain with the $3d$--criterion
in \cite{GrK}).}
\end{sub}
\begin{remark}{\rm
\begin{enumerate}
\item The surjectivity statement in \ref{3.8} (i) implies: let $C \subset
\P^2$ be of degree $d$ such that $H^1(C,{\cal N}^{ea}_{C/\P^2}) = 0$ and let $\{x_1,
\ldots, x_n\}$ be any subset of Sing$(C)$. If, for $i = 1, \ldots, n$, the
germ $(C,x_i)$ admits a deformation with nearby fibre having singularities
$y^1_i, \ldots, y^{s_i}_i$, then $C$ admits an embedded deformation with
nearby curve $C_t \subset \P^2$ having $y^1_1, \ldots, y^{s_1}_1, \ldots,
y^1_n, \ldots, y^{s_n}_n$ as singularities. Hence, there exists a curve of
degree $d$ with the $y$'s as singularities.
\item The $4d$--criterion in \ref{3.8} has the advantage that $C$ need not be
irreducible. On the other hand, in the $3d$--criterion in \ref{3.5} and
\ref{3.8} we
can completely forget about $A_1$--singularities on $C$. By Lemma 4.1,
for a node we have $\tau(C_i,x) -$ isod$_{C_i,x}({\cal N}^{ea}_{C/\P^2}, {\cal O}_C) \le
0$, which can be neglected in the right--hand side of the $3d$--criterion (we
actually obtain
$-1$ if the node results from the intersection of two global components of
$C$).
\item Since, for a node ${\cal N}^{ea}_{C/S,x} = {\cal N}^{es}_{C/S,x}$, $\tau_{C,x} =
\tau^{es}_{C,x}$ and since the isomorphism defect is a local invariant, we can
neglect nodes also in the $3d$--formulas for $es$. For the $4d$--criterion,
however, nodes have to be counted with 1 (Lemma 4.7).
\end{enumerate}}
\end{remark}
\begin{sub}{\rm
{\bf Examples}:
\begin{itemize}
\item[(b)] the irreducible curve $C \subset \P^2$ with affine equation
\vspace{-0.5cm}
\begin{eqnarray*}
f(x,y)& = & y^2 - 2x^2y + c_1 xy^2 +
c_2y^3+x^4-2c_1x^3y+c_3x^2y^2+c_4xy^3+c_5y^4+c_1x^5\\
& & \phantom{y^2} -(3c_2+2c_3)x^4y
- - -(2c_4+2)x^3y^2-(2c_1+2c_5)x^2y^3+c_6xy^4-(c_4+2)y^5\\
& & \phantom{y^2}+(2c_2+c_3)x^6+(c_4+2)x^5y+ (2c_1+c_5)
x^4y^2-c_6x^3y^3+(c_4+3)x^2y^4\\
& & \phantom{y^2}+ c_1xy^5-(3c_2+c_3+c_6)y^6
\end{eqnarray*}
where $c_1 := 16\alpha\beta^2 - 66\beta^2,\; c_2 := 3 \alpha\beta -
\frac{23}{2}\beta,\; c_3 := -(\beta + 7c_2),\; c_4 := -4\alpha + 13,\; c_5 :=
\beta^2-c_1,\; c_6 := 24\alpha\beta - 92\beta$ and $\alpha,\beta$ (complex)
solutions of $4\alpha^2 - 30\alpha + 55 = 0$ and $\beta^3 = \alpha^2 - 7\alpha
+ 12$ of degree 6 has exactly one singularity, which is of type $A_{19}$.
(The equation of this curve was found by H.\ Yoshihara (cf.\ \cite{Yos})).
Now $4d = 24 > 23 = \tau(C) + 4$ and, hence, each combination of
$A$--singularities given by an adjacent subdiagram of $A_{19}$ occurs on a
curve of degree 6 (this is a very simple proof of a well--known result which
was previously proved by using moduli theory of $K3$--surfaces).
\item[(c)] The curve $C \subset \P^2$ with homogeneous
equation $x^9 + zx^8 + z(xz^3+y^4)^2 = 0$ has exactly one singularity at
$(0:0:1)$ which is of type $A_{31}$. Again we have $4d=36 > 35 = \tau(C) +
4$, hence
$H^{ea}_{\P^2}$ is smooth at $C$. $C$ is obtained by a small deformation of
Luengo's
example of a degree 9 curve ($x^9 + z(xz^3+y^4)^2=0$, having an
$A_{35}$--singularity) with non--smooth
$(H^{ea}_{\P^2}, C)$. Of course, our criterion supports also this
non--smoothness since $4d = 36 < 39 = \tau(C) + 4$.
\item[(d)] $x^7 + y^7 + (x-y)^2 x^2y^2z = 0$ defines an irreducible curve $C
\subset \P^2$ which has 3 transverse cusps (not quasihomogeneous!) at
$(0:0:1)$ and no other singularities. Since $4d = 28 > 24 + 4 - 1 \ge
\tau(C) + 4 -$ isod$_{(0:0:1)} (\widetilde{{\cal N}}^{ea}_{C'/\P^2}, w_{C'})$ we see
that every local deformation of 3 transverse cusps can be realized by curves
of degree 7.
\end{itemize}
}
\end{sub}
\newpage
\addcontentsline{toc}{section}{References}
|
1997-08-22T23:14:19 | 9503 | alg-geom/9503013 | en | https://arxiv.org/abs/alg-geom/9503013 | [
"alg-geom",
"math.AG"
] | alg-geom/9503013 | Claus Hertling | G.-M. Greuel, C. Hertling, and G. Pfister | Moduli spaces of semiquasihomogeneous singularities with fixed principal
part | 31 pages. AMSLaTeX | null | null | null | null | We construct coarse moduli spaces of semiquasihomogeneous hypersurface
singularities with respect to right equivalence and contact equivalence. We
have to fix the principal part of the semiquasihomogeneous singularities. For
the moduli spaces with respect to contact equivalence we also fix the Hilbert
function of the Tjurina algebra induced by the weights.
| [
{
"version": "v1",
"created": "Wed, 22 Mar 1995 12:58:16 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Greuel",
"G. -M.",
""
],
[
"Hertling",
"C.",
""
],
[
"Pfister",
"G.",
""
]
] | alg-geom | \section*{Introduction}\addcontentsline{toc}{section}{Introduction}
One of the important achievements of singularity theory is the explicit
classification of certain ``generic'' classes of isolated hypersurface
singularities via normal forms and the analysis of its properties (cf.\
\cite{AGV}). More complicated singularities deform into a
collection of singularities from these classes and deformation theory is a
powerful tool in studying specific singularities. For a further
classification of more complicated classes of singularities the explicit
determination of normal forms seems to be impossible and not appropriate.
The aim of this article is to start towards a classification of isolated
hypersurface singularities of any dimension via geometric methods, that is by
explicitely constructing a (coarse) moduli space for such singularities with
certain
invariants being fixed. Our method starts from deformation theory and leads
to the
construction of geometric quotients of quasiaffine spaces by certain algebraic
groups whose main part is unipotent. This last part is a major ingredient
and uses the general results of \cite{GP 2}. In
projective algebraic geometry, the theory of moduli spaces is highly developed
but in singularity theory only a few attempts have been made so far, for
example by Ebey, Zariski, Laudal, Pfister, Luengo, Greuel (cf.\ \cite{LP} for
a systematic approach and \cite{GP 1}
for a short survey). In this paper we consider only
semiquasihomogeneous singularities given as a power series $f \in {\Bbb C} \{x_1,
\ldots, x_n\}$ or as a complex space germ $(X,0) = (f^{-1}(0),0) \subset
({\Bbb C}^n,0)$, together with positive weights $w_1, \ldots, w_n$ of the variables
such that the principal part $f_0$ of $f$ (terms of lowest degree) has an
isolated singularity.
For the classification we first fix the Milnor number, probably the most basic
invariant of an isolated hypersurface singularity. Fixing the Milnor number
is known (for $n \not= 3)$ to be equivalent
to fixing, in a family, the embedded topological type of the singularity. If
the Milnor number is fixed, the classification of semiquasihomogeneous
singularities falls naturally into two parts.
Firstly, the classification of the quasihomogeneous principal parts or, which
amounts to the same, the classification of hypersurfaces in a weighted
projective space. Secondly, the classification of semiquasihomogeneous
hypersurface singularities with fixed principal part. These two parts differ
substantially, since the group actions whose orbits describe isomorphism
classes of singularities are of a completely different nature. This article
is devoted to the second task.
The most important equivalence relations for hypersurface singularities are
right equivalence (change of coordinates in the source) and contact
equivalence (change of coordinates and multiplication with a unit or,
equivalently, preserving the isomorphism class of space germs). It turns out
that right equivalence, which is really a classification of functions, is
easier to handle. We prove the existence of a finite group $E_{f_0}$
acting on the affine space $T_-$, the base space of the semiuniversal
$\mu$--constant
deformation of $f_0$ of strictly negative weight, such that $T_-/E_{f_0}$ is
the
desired coarse moduli space. We also show that a fine moduli space almost
never exists. See \S 1 for definitions and precise statements.
Hence, $T_-/E_{f_0}$ classifies, up to right equivalence, semiquasihomogeneous
power series with fixed principal part.
An important step in the construction of moduli spaces with respect to right
equivalence as well as with respect to contact equivalence is to prove that
isomorphisms between two semiquasihomogeneous
functions have necessarily non--negative degree. This is proved in \S 2 and
uses the fact that the filtration on the Brieskorn lattice $H''_0(f)$ induced
by the weights coincides with the $V$--filtration, which is independent of the
coordinates. The proof relies on an analysis of this filtration given in
\cite{He}.
In order to obtain a moduli space with respect to contact equivalence we have
to fix, in addition to the Milnor number, also the Tjurina number. This is
clear because the dimensions of the
orbits of the contact group acting on $T_-$ depend on the
Tjurina number. But fixing the Tjurina number is not sufficient. The orbit
space of the contact group for fixed Tjurina number is, as a topological
space, in general not separated, hence, cannot carry the structure of a
complex space. It turns out, however, that if we fix the whole Hilbert
function of the Tjurina algebra induced by the weights, the orbit space is a
complex space and a coarse moduli space which classifies, up to contact
equivalence, semiquasihomogeneous hypersurface singularities with fixed
principal part and fixed Hilbert function of the Tjurina algebra. For
precise statements see \S 4. These moduli spaces are actually locally closed
algebraic varieties in a weighted projective space.
The orbits of the contact group acting on $T_-$ can also be described as
orbits of an algebraic group $G = U \rtimes (E_{f_0} \cdot {\Bbb C}^\ast)$ where
$E_{f_0}$
is the finite group mentioned above and $U$ is a unipotent algebraic group.
The main ingredient for the proof in the case of contact equivalence is the
theorem on the existence of geometric quotients for unipotent groups in
\cite{GP 2}. But, in order to give the above simple description of the
strata, we have to use, in a non--trivial way, also the symmetry of the Milnor
algebra, a fact which was already noticed in \cite{LP}.
The stratification with respect to the Hilbert function of the Tjurina algebra
and the proof for the existence of a geometric quotient are constructive and
allow the explicit determination of the moduli spaces and families of normal
forms for specific examples.
\newpage
\section{Moduli spaces with respect to right equivalence}
Let ${\Bbb C}\{x_1, \ldots, x_n\} = {\Bbb C} \{x\}$ be the convergent power series
ring. Two power series $f, g \in {\Bbb C}\{x\}$ are called {\bf right equivalent}
$(\buildrel r\over\sim)$ if there exists a $\psi \in$ Aut$({\Bbb C}\{x\})$ such
that $f = \psi(g)$; $f$ and $g$ are called {\bf contact equivalent}
$(\buildrel c\over\sim)$ if there exists a $\psi \in$ Aut$({\Bbb C}\{x\})$ and $u \in
{\Bbb C}\{x\}^\ast$ such that $f = u \psi(g)$. (Equivalently, the local algebras
${\Bbb C}\{x\}/(f)$ and ${\Bbb C}\{x\}/(g)$ are isomorphic
respectively the complex germs $(X,0) \subset ({\Bbb C}^n,0)$ and $(Y,0) \subset
({\Bbb C}^n,0)$ defined by $f$ and $g$ are isomorphic.)
Let $d$ and $w_1, \ldots, w_n$ be any integers. A polynomial $f_0 \in
{\Bbb C}[x_1, \ldots, x_n] = {\Bbb C}[x]$ is {\bf quasihomogeneous} of {\bf type} $(d;
w_1, \ldots, w_n)$ if for any monomial $x^\alpha = x_1^{\alpha_1} \cdot \ldots
\cdot x_n^{\alpha_n}$ occurring in $f_0$,
\[
\deg\, x^\alpha := |\alpha| := w_1 \alpha_1 + \cdots + w_n\alpha_n
\]
is equal to $d$. $w_1, \ldots, w_n$ are called {\bf weights} and $\deg\,
x^\alpha$
is called the (weighted) {\bf degree} of $x^\alpha$.
For an arbitrary power series $f = \sum c_\alpha x^\alpha,\; f \not= 0$, we set
\[
\deg\, f = \inf \{|\alpha|\, \mid\, c_\alpha \not= 0\},
\]
and call it the degree of $f$. For a family of power series $F = \sum
c_{\alpha,
\beta} x^\alpha
s^\beta \in {\Bbb C} \{x,s\}$, parametrized by ${\Bbb C}\{s\}$, we put $\deg_x F= \inf
\{|\alpha| \mid \exists\, \beta$
such that $c_{\alpha,\beta} \not= 0\}$.
$f$ is called quasihomogeneous if it is a
quasihomogeneous polynomial (of some type). $f$ is called {\bf
semiquasihomogeneous} of type $(d; w_1, \ldots, w_n)$, if
\[
f = f_0 + f_1,
\]
where $f_0$ is a quasihomogeneous polynomial of type $(d; w_1, \ldots, w_n)$,
$f_1$ is a power series such that $\deg\, f_1 > \deg\, f_0$ and, moreover,
$f_0$ has an isolated singularity at the origin. $f_0$ is called the {\bf
principal part} of $f$.
Two right equivalent semiquasihomogeneous power series of the same type have
right equivalent principal parts
Recall (\cite{SaK 1}) that a power series $f$ with isolated singularity is
right equivalent to a quasihomogeneous polynomial
with respect to positive weights if and only if
\[
f \in j(f) := (\partial f/\partial x_1, \ldots, \partial f / \partial x_n).
\]
Moreover, in this case the {\bf normalized weights} $\overline{w}_i =
\frac{w_i}{d}
\in {\Bbb Q} \cap (0,\frac{1}{2}]$ are uniquely determined.
We may consider $f \in {\Bbb C}\{x\}, f(0) = 0$ as a map germ $f : ({\Bbb C}^n,0) \to
({\Bbb C},0)$.
An {\bf unfolding} of $f$ over a complex germ or a pointed complex space
$(S,0)$ is by definition a cartesian diagram
\[
\begin{array}{ccc}
({\Bbb C}^n,0) & \hookrightarrow & ({\Bbb C}^n,0) \times (S,0)\\[1.0ex]
f\; \downarrow & & \downarrow \; \phi\\[1.0ex]
({\Bbb C},0) & \hookrightarrow & ({\Bbb C},0) \times (S,0)\\[1.0ex]
\downarrow & & \downarrow \\[1.0ex]
0& \hookrightarrow & (S,0).
\end{array}
\]
Hence, $\phi(x,s) = (F(x,s),s)$ and the unfolding
$\phi$ is determined by $F : ({\Bbb C}^n,0) \times (S,0) \to ({\Bbb C},0),\; F(x,s) = f(x)
+
g(x,s),\; g(x,0) = 0$, and we say that $F$ defines an unfolding of $f$. Two
unfoldings
$\phi$ and $\phi'$ defined by $F$ and $F'$ over $(S,0)$ are called right
equivalent if
there is an isomorphism $\Psi :({\Bbb C}^n,0) \times (S,0) \buildrel \cong\over\to
({\Bbb C}^n,0) \times (S,0),\; \Psi(x,s) = (\psi(x,s),s)$, such that $\phi \circ \Psi
= \phi'$.
For the construction of moduli spaces we have to consider, more generally,
families of unfoldings over arbitrary complex base spaces. Let $S$ denote a
{\bf category of base spaces}, for example the category of complex germs or of
pointed complex spaces or of complex spaces. A {\bf family of
unfoldings} over $S \in {\cal S}$ is a commutative diagram
\[
\begin{array}{rcl}
({\Bbb C}^n,0) \times S & \buildrel\phi\over\longrightarrow & ({\Bbb C},0) \times S\\
\searrow & & \swarrow\\
& S &.
\end{array}
\]
Hence, $\phi (x,s) = (G(x,s),s) = (G_s(x),s)$ and for each $s \in S$, the germ
$\phi : ({\Bbb C}^n,0) \times (S,s) \to ({\Bbb C},0) \times (S,s)$ is an unfolding of $G_s
:
({\Bbb C}^n,0) \to ({\Bbb C},0)$. A morphism of two families of unfoldings $\phi$ and
$\phi' = (G', id_s)$ over $S$ is a morphism $\Psi : ({\Bbb C}^n,0) \times S \to
({\Bbb C}^n,0) \times S,\; \Psi (x,s) = (\psi(x,s),s) = (\psi_s(x), s)$ such that
$\phi \circ \Psi = \phi'$ (equivalently : $G_s(\psi(x,s)) = G'_s(x)$).
$\phi$ and $\phi'$ are called {\bf right equivalent families of unfoldings} if
there is a morphism $\Psi$ of $\phi$ and $\phi'$ such that for each fixed $s
\in S$, $\psi_s \in \mbox{ Aut}({\Bbb C}^n,0)$.\\
{}From now on let $f_0 \in {\Bbb C}[x_1, \ldots, x_n]$ denote a quasihomogeneous
polynomial with isolated singularity of type $(d; w_1, \ldots, w_n)$ with $w_i
> 0$ for $i = 1, \ldots, n$.
Consider a power series $f$ which is right equivalent to a
semiquasihomogeneous power series $f'$ of type $(d; w_1, \ldots, w_n)$.
We say that an unfolding $F$ defines an {\bf unfolding of
$\bf f$ of negative weight} over $(S,0)$ if $F$ is right equivalent to $f'(x)
+ g(x,s)$ with $g(x,0) = 0$ and $\deg_x g > d$. This holds, for
instance, if there exists a ${\Bbb C}^\ast$--action
with (strictly) negative weights on $(S,0)$ such that $\deg\, g = d$, with
respect to the ${\Bbb C}^\ast$--actions on $({\Bbb C}^n,0)$ and on $(S,0)$. By Theorem
2.1 the definition is independent of the choice of $f'$.
We shall now describe the semiuniversal unfolding of $f_0$ of negative weight.
Let $x^\alpha$, $\alpha \in B \subset {\Bbb N}^n$, be a monomial basis of the Milnor
algebra ${\Bbb C}\{x\}/(\partial f_0/\partial x_1, \ldots, \partial f_0/\partial
x_n)$
which is of
${\Bbb C}$--dimension $\mu$ (the Milnor number of $f_0$), and let $\bar{F} (x,t) =
f_0(x) + \sum_{\alpha \in B} x^\alpha s_\alpha,\; s = (s_\alpha)_{\alpha
\in B} \in {\Bbb C}^\mu$ be the semiuniversal unfolding of $f_0$. We are mainly
interested in the sub--unfolding over the affine pointed space $T_- =
({\Bbb C}^k,0)$,
\[
F(x,t) = f_0(x) + \sum^k_{i=1} t_i m_i,\;\;\; t = (t_1, \ldots, t_k) \in T_-,
\]
where the $m_i$ are the ``upper'' monomials, that is
\[
\{m_1, \ldots, m_k\} = \{ x^\alpha \,\mid\, \alpha\in B,\, |\alpha| > d\}.
\]
For fixed $t \in T_-, F_t(x) = F(x,t) \in {\Bbb C}[x]$ is a semiquasihomogeneous
polynomial with principal part $f_0$.
Let $A = {\Bbb C}[(s_\alpha)_{\alpha \in B}]$ and $A_- = {\Bbb C}[t_1, \ldots, t_k]$. If
we give weights to $s_\alpha$ and $t_i$ by $w(s_\alpha) = d- |\alpha|$ and
$w(t_i) = d - \deg(m_i)$, then $\bar{F}$ and $F$ are quasihomogeneous
polynomials in ${\Bbb C}[x,s]$ respectively ${\Bbb C}[x,t]$ and $F$ is the restriction of
$\bar{F}$ to $T_-$, the negative weight part of $T =$ Spec\,$A$, defined by
$\{t_1, \ldots, t_k\} = \{s_\alpha\mid
w(s_\alpha) < 0\}$.
{\bf Example}:
$f_0 = x^3 + y^3 + z^7$ is quasihomogeneous of type $(d; w_1, w_2, w_3) = (21;
7, 7, 3)$ with Milnor number $\mu = 24$. The upper monomials of a monomial
basis of the Milnor algebra ${\Bbb C}\{x,y,z\}/(x^2, y^2, z^6)$ are $m_1 = x z^5,\;
m_2 = yz^5,\; m_3 = xyz^3,\; m_4= xyz^4,\; m_5 = xyz^5$ and, hence, $A_- =
{\Bbb C}[t_1, \ldots, t_5],\; T_- = {\Bbb C}^5$,
\[
F(x,y,z,t) = f_0 +
\sum^5_{i=1}t_i m_i = f_0 + t_1xz^5 + t_2yz^5 + t_3xyz^3 + t_4xyz^4 +
t_5xyz^5,
\]
$w(t_1, \ldots, t_5) = (-1, -1, -2, -5, -8)$.
\begin{remark}\label{1.1}{\rm
Fix any $t \in T_-$. $F$ defines an unfolding of $F_t$ of negative weight
over the pointed space $(T_-, t)$. If we restrict this unfolding to the
germ $(T_-,t)$ this is actually a semiuniversal unfolding of $F_t$ of
negative weight because of the following:
The monomials $m_1, \ldots, m_k$ represent certainly a basis of
${\Bbb C} \{x\} /(\frac{\partial F_t}{\partial x_1}, \ldots, \frac{\partial F_t}
{\partial x_n})$
for $t$ sufficiently close to $0$, since $\mu(F_t) = \mu(f_0)$. But, using
the ${\Bbb C}^\ast$--actions on $T_-$ and on ${\Bbb C}^n$, we see that any $F_t$ is contact
equivalent to
some $F_{t'}$, $t'$ close to $0$.
Hence, ${\cal O}_{{\Bbb C}^n \times T_-,0\times T_-}/(\frac{\partial
F}{\partial x_1}, \ldots, \frac{\partial F}{\partial x_n})$ is actually free
over
$T_-$ with basis $m_1, \ldots, m_k$ and the result follows.
We call the affine family
\[
F : {\Bbb C}^n \times T_- \to {\Bbb C},
\]
$(x,t) \mapsto f_0(x) + \sum^k_{i=1} t_i m_i$ the {\bf semiuniversal family of
unfoldings of negative weight of semiquasihomogeneous
power series with fixed principal part} $\bf f_0$.
}
\end{remark}
\begin{lemma}\label{1.2}
The family of unfoldings $F$ has the following property. If $f$ is any
semiquasihomogeneous power series with principal part $f_0$, then:
\begin{enumerate}
\item[(i)] $T_- = \{0\}$ if and only if $f_0$ is simple or simple elliptic.
\item[(ii)] There exists a $t \in T_-$ such that $f \buildrel r\over\sim F_t$.
\item[(iii)] Let $f \buildrel r\over\sim F_t$ and let $G(x,s) = f(x) + g(x,s)$
be any
unfolding of $f$ of negative weight over the germ $(S,0)$. Then there exists
a morphism, unique on the tangent level, of germs $\varphi : (S,0) \to (T_-,
t)$ such that
$\varphi^\ast F$ is right equivalent to $G$ (that is $T_-$ does not contain
trivial subfamilies of unfoldings).
\item[(iv)] Assume $f_0$ is neither simple nor simple elliptic. There
exist $t, t' \in T_- , t\neq t'$, arbitrarily close to $0$,
such that $F_t \buildrel
r\over\sim F_t'$ (that is $F$ is not universal in any neighbourhood of $0 \in
T_-$).
\end{enumerate}
\end{lemma}
\newpage
{\bf Proof}: \begin{enumerate}
\item[(i)] is due to Saito \cite{SaK 2}.
\item[(ii)] follows from \cite{AGV}, 12.6, Theorem (p.\ 209).
\item[(iii)] If $T_-$ would contain trivial subfamilies of unfoldings there
must be
a $t \in T_-$ with $\mu(F_t) < \mu(f_0)$, which is not the case.
\item[(iv)] The group $\mu_d$ of $d$--th roots of unit acts on
$T_-$, has $0$ as fixed point and a non--trivial orbit for any $t \not= 0$.
Since for $\xi \in \mu_d,\; F_{\xi\circ t}(\xi \circ x) = \xi^d F_t(x) =
F_t(x)$, two different points of an orbit of $\mu_\alpha$ correspond to right
equivalent functions, we obtain (iv).
\end{enumerate}
Let us introduce the notion of a fine and coarse moduli space for unfoldings of
negative weight with principal part $f_0$ (the weights $w_1, \ldots, w_n$ and
$f_0$ are given as above): let ${\cal S}$ be a category of base spaces.
For $S \in {\cal S}$, a {\bf family of unfoldings of negative weight
with principal part} $\bf f_0$ over $S$ is a family of unfoldings
\[
\phi : ({\Bbb C}^n,0) \times S \to ({\Bbb C},0)\times S,\; (x,s) \mapsto (G(x,s),s)
= (G_s(x),s)
\]
such that: for any $s \in S$, $G_s :({\Bbb C}^n,0) \to ({\Bbb C},0)$ is right equivalent
to a semiquasihomogeneous power series with principal part $f_0$ and the germ
of $G$ at $s$, $G :({\Bbb C}^n,0) \times (S,s) \to ({\Bbb C},0)$, is an unfolding of $G_s$
of negative weight. For any
morphism of base spaces $\varphi : T \to S$, the induced map $\varphi^\ast
\phi : ({\Bbb C}^n,0) \times T \to ({\Bbb C},0)\times T,\; (x,t) \mapsto
(G(x,\varphi(t)),t)$, is an
unfolding of negative weight with principal part $f_0$ over $T$. Hence, we
obtain a
functor
\[
\mbox{Unf}^-_{f_0} : {\cal S} \to \mbox{ sets}
\]
which associates to $S \in {\cal S}$ the set of right equivalence classes of
families of
unfoldings
of negative weight with principal part $f_0$ over $S$.
If $pt \in {\cal S}$ denotes the base space consisting of one reduced point, then
\begin{tabular}{lp{12cm}}
Unf$^-_{f_0}(pt) =$ & $\{$ right equivalence classes of power series
$f \in {\Bbb C}\{x_1, \ldots, x_n\}$ which are right equivalent to a
semiquasihomogeneous power series with principal part $f_0\}$.
\end{tabular}
A {\bf fine moduli space} for the functor Unf$^-_{f_0}$ consists of a base
space $T$ and a natural transformation of functors
\[
\psi : \mbox{ Unf}^-_{f_0} \to \mbox{ Hom}(-,T)
\]
such that the pair $(T, \psi)$ represents the functor Unf$^-_{f_0}$.
The pair $(T,\psi)$ is a {\bf coarse moduli space} for Unf$^-_{f_0}$ if
\begin{enumerate}
\item[(i)] if $\psi(pt)$ is bijective, and
\item[(ii)] given the solid arrows (natural transformations) in the following
diagram
\vspace{-0.5cm}
\unitlength1cm
\begin{picture}(8,3)
\put(4,0){Hom$(-,T)$}
\put(6.5,0){\line(1,0){0.15}}
\put(6.7,0){\line(1,0){0.15}}
\put(6.9,0){\line(1,0){0.15}}
\put(7.1,0){\line(1,0){0.15}}
\put(7.3,0){\line(1,0){0.15}}
\put(7.5,0){\line(1,0){0.15}}
\put(7.7,-0.1){$>$}
\put(8.5,0){Hom$(-,T')$,}
\put(6.8,1.5){\vector(-3,-2){1.3}}
\put(7.8,1.5){\vector(3,-2){1.3}}
\put(6.8,2){Unf$^-_{f_0}$}
\end{picture}
there exists a unique dotted arrow (natural transformation) making the diagram
commutative. \end{enumerate}
A fine moduli space is, of course, coarse.
The definitions of fine and coarse moduli spaces still depend on the category
of base spaces ${\cal S}$. If
${\cal S}$ is the category of complex germs and if $(S,0) \in {\cal S}$, then Hom$((S,0),
T)$
denotes the set of morphisms of germs $(S,0) \to (T,t)$ where $t$ may be any
point of $T$. In this case, if $(T, \psi)$ is a fine moduli space, given any
$t \in T$, there exists a unique (up to right equivalence)
universal unfolding of negative weight with principal part $f_0$ over the germ
$(T,t)$ which corresponds to id $\in$ Hom$((T,t), (T,t))$. But we may not
have a universal family over all of $T$. If ${\cal S}$ is the category of all
complex spaces, the existence of a fine moduli space implies the existence of
a global universal family over $T$. But we shall see that even for complex
germs as base spaces a fine moduli space may not exist. A coarse moduli
space, however, does exist even if ${\cal S}$ is the category of all complex spaces.
The reason is that for a coarse moduli space we do not require any kind of
a universal family.
\begin{theorem}\label{1.3}
Let $E_{f_0}$ be the finite group defined in Definition \ref{2.5}, acting on
$T_-$. The geometric quotient $T_-/E_{f_0}$ is a coarse moduli space for the
functor Unf$^-_{f_0} : \mbox{ complex spaces } \to$ sets.
\end{theorem}
{\bf Proof}: Since $E_{f_0}$ is finite, and the action is holomorphic, the
geometric quotient $T_-/E_{f_0}$ exists as a complex space. According to
Theorem 2.1, Proposition \ref{2.3} and Corollary \ref{2.5}, for any
semiquasihomogeneous
power series $f$ with principal part $f_0$ there exists a unique point
$\underline{t} \in T_-/E_{f_0}$ such that if $f_t \buildrel r\over\sim f$, $t
\in
T_-$ maps to $\underline{t}$. In this way we obtain a bijection $\psi(pt)$
from the set of right equivalence classes of semiquasihomogeneous power series
with principal part $f_0$ to $T_-$.
Now let $G : ({\Bbb C}^n,0) \times S \to ({\Bbb C},0)$ define an element of
Unf$^-_{f_0}(S)$ for some complex space $S$. We may cover $S$ by open sets
$U_i$ such that there exist morphisms $\varphi_i : U_i \to T_-$ with
$\varphi^\ast F \buildrel r\over\sim G|_{U_i}$. Even if the $\varphi_i$
are not unique, by the properties of a quotient the compositions $U_i
\buildrel \varphi_i\over \to T_- \to T_-/E_{f_0}$ glue together to give a
morphism
$S \to T_-/E_{f_0}$. This construction is functorial and provides the desired
natural transformation Unf$^-_{f_0} \to \mbox{ Hom}(-,T_-/E_{f_0})$. This
finishes
the proof of Theorem \ref{1.3} (for further details for construction of moduli
spaces via geometric quotients cf.\ \cite{Ne}).
\begin{remark}{\rm
(i)\quad If $f_0$ is simple or simple elliptic, then the coarse moduli space
constructed above consists of one reduced point. Hence, it is even a fine
moduli space.
(ii)\quad If $f_0$ is neither simple nor simple elliptic, Unf$^-_{f_0}$ does
not admit a
fine moduli space, even not if we take complex germs as base spaces. This
can be seen as follows: assume there exists such a fine moduli space then,
since it is also coarse, it must be isomorphic to $T_-/E_{f_0}$. Moreover,
there
exists a universal unfolding over the germ $(T_-/E_{f_0}, 0)$ which can be
induced
from the semiuniversal unfolding $F$ over the germ $(T_-,0)$ and vice versa.
Since $T_-$ does not contain trivial subfamilies, the semiuniversal family $F$
over $(T_-,0)$ would be universal, which contradicts Lemma \ref{1.2} (iv).
}
\end{remark}
{\bf Example}: Let $f_0(x,y) = x^4 + y^5$. We obtain $T_- = {\Bbb C}$ and
$F(x,y,t) = x^4 + y^5 + tx^2y^3,\; (d;w_1,w_2;w(t)) = (20;4,5;-2)$.
In this case $E_{f_0} = \mu_d$ and the ring of invariant functions on $T_-$ is
${\Bbb C}[t^{10}]$, hence $T_-/E_{f_0} \cong {\Bbb C}$. We give a computational argument
that a fine moduli space does not exist:\\
A local universal family over $(T_-/E_{f_0},0)$ would be given by $G :({\Bbb C}^n,0)
\times (T_-/E_{f_0}) \to ({\Bbb C},0),\; (x,y,s) \mapsto G(x,y,s)$. The proof of
Theorem \ref{1.3} shows that then $F$ would be induced from $G$ by the
canonical map $T_- \to T_-/E_{f_0}$, which is not an isomorphism.
Moreover, the fibre $F^{-1}(0)$ would be isomorphic to
$G^{-1}(0)$ under the map $(x,y,t) \mapsto (x,y,s = t^{10})$. The image of
this map can be computed by eliminating $t$ from $F(x,y,t) = 0,\; s - t^{10} =
0$. The result is the hypersurface defined by
$G = (x^4 + y^5)^{10} - sx^{20}y^{30}$.
The special fibre for $s = 0$ has a non--isolated singularity, hence is not
isomorphic to $f_0 = 0$.
\begin{remark}\label{1.5}{\rm
Since the group $E_{f_0}$ acts even algebraically on $T_-$ by Proposition
\ref{2.4}, $T_-/E_{f_0}$ is an algebraic variety. We may take the category of
base spaces ${\cal S}$ to be the category of (separated) algebraic spaces and
define (families of) unfoldings in the same manner as above, replacing the
analytic local ring ${\Bbb C}\{x\}$ by the henselization of ${\Bbb C}[x]$. With the same
proof as above we obtain that $T_-/E_{f_0}$ is a coarse moduli space for the
functor
\[
\mbox{Unf}^-_{f_0} : \mbox{ algebraic spaces } \to \mbox{ sets.}
\]
}
\end{remark}
\newpage
\section{Isomorphism of semiquasihomogeneous singularities}
We fix weights $w_1,...,w_n \in {\Bbb N}$ and a degree $d\in {\Bbb N}$
such that the normalized weights $\overline{w}_i = {w_i \over d}$
fulfill $0 < \overline{w}_i \leq {1\over 2}$.
The weights induce a filtration on ${\Bbb C} \{x\}$.
An automorphism $\varphi \neq id$ of ${\Bbb C} \{x\} $ has degree
$m=\deg \varphi$ if $m$ is the maximal number such that
\[
\deg (\varphi (x_i) - x_i) \geq w_i + m \ \ \forall \ i=1,...,n.
\]
The automorphisms of degree $\geq 0$ form the group
$\mbox{Aut}_{\geq 0}({\Bbb C} \{x\} )$ of all automorphisms of
${\Bbb C} \{x\} $ which respect the filtration.
The automorphisms of degree $>0$ form a normal subgroup
$\mbox{Aut}_{>0}({\Bbb C} \{x\} )$ in Aut$_{\geq 0}({\Bbb C} \{x\} )$.
Automorphisms will be called quasihomogeneous if they map each
quasihomogeneous polynomial to a quasihomogeneous polynomial of the same
degree.
They form a group $G_w \subset \mbox{ Aut}_{\geq 0}({\Bbb C} \{x\} ),$ which is
isomorphic
to the quotient Aut$_{\geq 0}({\Bbb C} \{x\} )/\mbox{Aut}_{>0}({\Bbb C} \{x\} )$.
The image $\varphi (f)$ of a semiquasihomogeneous power series $f$ of degree
$d$ by an automorphism $\varphi$ of ${\Bbb C} \{x\} $ is semiquasihomogeneous
of the same degree if $\deg \varphi \geq 0 $.
The converse is true, too:
\begin{theorem} \label{2.1}
Let $f$ and $g$ be semiquasihomogeneous of degree $d$, and let $\varphi$ be an
automorphism of ${\Bbb C} \{x\} $ such that $\varphi (f) = g$. Then
$\deg \varphi \geq 0$.
\end{theorem}
{\bf Proof:}
The proof uses some facts which come from the Gauss--Manin connection
for isolated hypersurface singularities
(\cite{SS}, \cite{SaM}, \cite{AGVII}, \cite{He}). The main idea is the
following:
in the case of a semiquasihomogeneous singularity the weights $\overline{w}_i$
induce a filtration on ${\Bbb C} \{x\}$
and a filtration on the Brieskorn lattice $H_0''(f)$.
This last filtration coincides with the $V$--filtration and is independent
of the coordinates.
The Brieskorn lattice $H_0''(f)$ is
\[
H_0'' = \Omega^n / df\land d\Omega^{n-1} .
\]
Here $\Omega^k = \Omega^k_{{\Bbb C}^n,0}$ denotes the space of germs of
holomorphic $k$--forms. The class of $\omega \in \Omega^n$ in $H_0''(f)$
is denoted by $s[\omega ]_0 \in H_0''(f)$.
The $V$--filtration on $H_0''(f)$ is determined by the orders
$\alpha_f (\omega) = \alpha_f(s[\omega]_0)$ of $n$--forms
$\omega \in \Omega^n$.
The most explicit description of the order $\alpha_f (\omega)$ might be
the following (\cite{AGVII}, \cite{He}):
\newpage
\begin{eqnarray*}
\alpha_f(\omega) = \min\,\{ \alpha & | & \exists
\mbox{ (manyvalued) continuous family of cycles}\\
&&\delta (t)\in H_{n-1}(X_t,{\Bbb Z}) \mbox{ on the Milnor fibers }X_t \\
&&\mbox{of the singularity }f:({\Bbb C}^n,0) \to ({\Bbb C},0),\\
&&\mbox{such that } a_{\alpha,k} \neq 0 \mbox{ in } \\
&&\int\limits_{\delta (t)} {\omega \over df} =
\sum_{\beta,k} a_{\beta,k}\cdot t^\beta \cdot (\ln t)^k \\
&&\mbox{for a } k \mbox{ with } 0\leq k \leq n-1\ \}.
\end{eqnarray*}
The description shows that we have
\[
\alpha_f (\omega) = \alpha_g (\varphi (\omega))
= \alpha_g (\varphi(h) d\varphi(x))
\]
for $\omega = h(x)dx_1...dx_n = hdx \in \Omega^n$.
Since $f$ is semiquasihomogeneous it is possible to give a simple algebraic
description of the order $\alpha_f (\omega)$.
Indeed, we define mappings
\vspace{-0.5cm}
\begin{eqnarray*}
\nu_C & : & {\Bbb C}\{ x_1,...,x_n\} \to {\Bbb Q}_{\geq 0}\cup \{\infty\},\\
\nu_\Omega & : & \Omega^n \to {\Bbb Q}_{>-1}\cup \{\infty\},\\
\nu_f & : & H_0''(f) \to {\Bbb Q}_{>-1}\cup \{\infty\}
\end{eqnarray*}
by
\[ \nu_C (x^\alpha) = \sum_{i=1}^n \overline{w}_i \alpha_i\ , \
\nu_C(0) = \infty, \
\nu_C (\sum b_\alpha x^\alpha) =
\min \{\nu_C (x^\alpha)\ |\ b_\alpha\neq 0\}
\]
and
\[
\nu_\Omega (hdx) = \nu_C (hx_1...x_n) -1
\]
and
\[
\nu_f (s[\omega]_0) = \nu_f(\omega) =
\max \{\nu_\Omega (\eta)\ |\ s[\eta]_0 = s[\omega]_0\}.
\]
Then, from \cite{He}, Chapter 2.4, it follows that
\[
\nu_f (\omega) = \alpha_f (\omega) = \alpha_g (\varphi(\omega))
= \nu_g (\varphi(\omega)).
\]
For all $\eta\in \Omega^{n-2}$ we have
\[
\nu_f (df\land d\eta)
\geq -1 + \sum_j \overline{w}_j + (1-\max( \overline{w}_i ))
\geq \sum_j \overline{w}_j - \frac{1}{2}.
\]
For $\omega$ with
\[
\min \{\nu_\Omega (\omega ), \nu_f (\omega ),\nu_g (\varphi (\omega )),
\nu_\Omega (\varphi (\omega )) \}
< \sum_j \overline{w}_j - \frac{1}{2}
\]
this implies
\[
\nu_\Omega (\omega ) = \nu_f (\omega ) = \nu_g (\varphi (\omega))
= \nu_\Omega (\varphi (\omega )).
\]
We obtain
\[
\sum_j \overline{w}_j -1 = \nu_\Omega (dx) = \nu_f (dx)
= \nu_g (d\varphi(x)) = \nu_\Omega (d\varphi(x)).
\]
For $i$ with $\overline{w}_i < \frac{1}{2}$ we obtain
\vspace{-0.5cm}
\begin{eqnarray*}
\overline{w}_i + \nu_\Omega (dx) & = & \nu_C(x_i) + \nu_\Omega (dx)
= \nu_\Omega (x_idx) = \nu_f (x_idx) \\
& = & \nu_g (\varphi (x_i) d\varphi(x))
= \nu_\Omega (\varphi (x_i) d\varphi(x))
= \nu_C (\varphi (x_i)) + \nu_\Omega (d\varphi(x)) \\
& = & \nu_C (\varphi (x_i)) + \nu_\Omega (dx)
\end{eqnarray*}
and $ \nu_C (\varphi (x_i)) = \overline{w}_i $.
For $i$ with $\overline{w}_i = \frac{1}{2}$ the equality $\nu_\Omega (x_idx) =
\sum \overline{w}_i - \frac{1}{2} $
implies
\[
\nu_\Omega (\varphi (x_idx)) \geq
\sum \overline{w}_i - \frac{1}{2}
\]
and $ \nu_C (\varphi (x_i)) \geq \frac{1}{2} $.
Therefore, $\nu_C (\varphi (x_i)) \geq \nu_C (x_i) = w_i \ \ \forall
i=1,...,n,$ and thus $\deg \varphi \geq 0$. \hfill
\begin{remark}\label{2.2} \rm
In the following, Theorem \ref{2.1} will be used to describe
a finite group $E_{f_0}\subset \mbox{ Aut}(T_{-})$ which operates transitively
on each set of parameters in $T_{-}$ which belong to one right equivalence
class.
Theorem \ref{2.1} also shows that the Hilbert function
of the Tjurina algebra (cf.\ Chapter 4) is an invariant of the
contact equivalence class.
\end{remark}
Now let $f_0\in {\Bbb C}[x_1,...,x_n]$ be quasihomogeneous of degree $d$
with an isolated singularity in 0. Let $m_1,...,m_k$ denote the monomials
of degree $>d$ in a monomial base of the Milnor algebra of $f_0$.
Consider the semiuniversal unfolding of $f_0$ of negative weight,
\[
F = f_0 + \sum_{i=1}^k m_it_i.
\]
For a fixed value of $t$ we write $F_t = f_0 + \sum m_it_i$.
With $\deg t_i = w(t_i) = d - \deg m_i < 0$ we obtain a filtration on
${\Bbb C}[t_1,...,t_k] = A_{-}$ such that $F \in {\Bbb C}[x,t]$ is
quasihomogeneous of degree $d$ in $x$ and $t$. We write $T_{-}=Spec\ A_{-}$
(cf.\ \S 1).
\begin{proposition} \label{2.3}
For any semiquasihomogeneous power series $f$ with principal part $f_0$ there
exist an automorphism $\varphi \in Aut_{>0}({\Bbb C}\{x\})$ and a
parameter $t\in T_{-}$ such that $\varphi (f) = F_t$.
The $t\in T_{-}$ is uniquely determined.
\end{proposition}
{\bf Proof}:
The existence of $\varphi$ and $t$ is proved in \cite{AGV}, 12.6,
Theorem (p.\ 209).
The following proves the uniqueness of $t$.
Let $t$ and $t'\in T_{-}$ and $\psi \in \mbox{ Aut}_{>0}({\Bbb C}\{x\})$ be given
such
that $\psi (F_t) = F_{t'}$.
With $\psi_s(x_i) = x_i + s(\psi(x_i)-x_i)$ we obtain a family $\psi_s$ of
automorphisms in Aut$_{>0}({\Bbb C}\{x\})$.
The family $\psi_s(F_t)$ of semiquasihomogeneous functions with principal
part $f_0$ connects $\psi_0(F_t) = F_t$ and $\psi_1(F_t) = F_{t'}$.
The family may not be contained in $T_{-},$ but can be induced from $T_{-}$
by a suitable base change:
Following the proof of the theorem in [AGV], 12.6 (p. 209), we can find a
family $\chi_s$ of automorphisms and a holomorphic map $\sigma: {\Bbb C} \to T_-$
such that $\chi_s \circ \psi_s (F_t) = F_{\sigma (s)}$ and
$\chi_s\in Aut_{>0}({\Bbb C} \{x\})$ and even $\chi_0 = id = \chi_1,\
\sigma (0)=t,\, \sigma (1)=t'$.
But since $T_{-}$ is part of the
semiuniversal deformation,
which is miniversal on the $\mu$--constant stratum,
and since $T_{-}$ does not contain trivial subfamilies
with respect to right equivalence, $t=t'$ as desired. \hfill
\begin{proposition} \label{2.4}
\begin{enumerate}
\item For any $\varphi \in G_w^{f_0} = \{\psi\in G_w\ |\ \psi(f_0)=f_0\}$
and any $t\in T_{-}$ there exist $s = \theta(\varphi)(t) \in T_{-}$
and an automorphism $\psi\in Aut_{>0}({\Bbb C}\{x\})$
such that $\psi \circ \varphi (F_t) = F_s$.
\item The function $\theta (\varphi) : T_{-} \to T_{-}$ is uniquely
determined, bijective and fulfills
$\theta(\varphi^{-1}) = \theta^{-1}(\varphi)$ and
$\theta(\varphi)\circ \theta(\psi)
=\theta(\varphi \circ \psi)$ for any $\psi \in G_w^{f_0}$.
\item The components $\theta (\varphi)(t_i)$ are quasihomogeneous
polynomials in $A_{-}$ of degree $\deg (t_i)$.
\end{enumerate}
\end{proposition}
{\bf Proof}:
The statements 1.\ and 2.\ follow from Proposition \ref{2.3}
and from the fact that
Aut$_{>0}({\Bbb C}\{x\})$ is a normal subgroup of
Aut$_{\geq 0}({\Bbb C}\{x\})$. Statement 3.\ follows from the proof of the theorem
in \cite{AGV}, 12.6 (p.\ 209).
Along the lines of this proof one can construct power series
$\psi_1,...,\psi_n \in {\Bbb C}\{x,t\}$
and a family of automorphisms $\psi (t)$ such that
$\psi (t)(x_i) = \psi_i(t)$
with the following properties:
\begin{quote}
$\psi_i$ is quasihomogeneous in $x$ and $t$ of degree $w_i,$ \\
$\psi_i - x_i$ has degree $>w_i$ in $x,$ \\
for any fixed $t$ the automorphism $\psi (t)\in
\mbox{ Aut}_{>0}({\Bbb C}\{x\})$
with $\psi (t)(x_i) = \psi_i(t)$ gives
$\psi (t) \circ \varphi (F_t) = F_{\theta (\varphi)(t)}$.
\end{quote}
The power series $F=f_0+\sum m_it_i,$ and
$\varphi (F)=f_0+\ldots$ and
$\psi (t)\circ \varphi (F) = f_0 + \sum m_i\theta (\varphi)(t_i)$
are all quasihomogeneous of degree $d$
with respect to $x$ and $t$.
This proves 3. \hfill \\
The functions $\theta (\varphi)$ are biholomorphic.
\begin{definition}
The image
$\theta (G_w^{f_0})$ in Aut$(T_{-})$ will be denoted by $E_{f_0}$.
\end{definition}
\begin{corollary}\label{2.5}
The map $\theta : G_w^{f_0} \to E_{f_0} \subset Aut (T_{-})$
is a group homomorphism.
The automorphisms $\theta (\varphi)$ of $T_{-}$ commute with the
${\Bbb C}^{*}$-operation on $T_{-}$.
Each orbit of $E_{f_0}$ consists of all parameters in $T_{-}$ which belong to
one right equivalence class.
\end{corollary}
{\bf Proof}:
The first two statements follow from Proposition \ref{2.4},
the third statement follows from Theorem \ref{2.1}. \hfill
\begin{proposition}\label{2.6}
\begin{enumerate}
\item The group $G_w^{f_0}$ is finite if
$\overline{w}_1,...,\overline{w}_{n-1} < \frac{1}{2}$ and
$\overline{w}_n \leq \frac{1}{2}$.
\item The group $E_{f_0}$ is finite.
\end{enumerate}
\end{proposition}
{\bf Proof}:\\
{\bf 1.}\quad The dimension of the algebraic group $G_w$ is
\[
\dim G_w = \sum_{i=1}^n \# (\mbox{ monomials } x^\alpha
\mbox{ of degree }w_i \ ).
\]
The group $G_w$ operates on
\[
V = \bigoplus_{\deg x^\alpha = d} {\Bbb C}\cdot x^\alpha.
\]
Let $j(f_0)$ denote the Jacobi ideal of $f_0$ and $j_i(f_0)$ the ideal
\[
j_i(f_0) = (\frac{\partial f_0}{\partial x_{1}},\ldots,
\frac{\partial f_0}{\partial x_{i-1}},
\frac{\partial f_0}{\partial x_{i+1}},\ldots,
\frac{\partial f_0}{\partial x_{n}}).
\]
The tangent space $T_{f_0}G_w f_0
\subset T_{f_0}V$ of $G_w f_0$ in $f_0$ is
\[
T_{f_0} G_w f_0 \cong j(f_0)\cap V.
\]
For any relation
\[
0 = \sum_{i=1}^n \sum_{\deg x^\alpha = w_i}
a_{\alpha,i}\cdot x^\alpha \cdot
\frac{\partial f_0}{\partial x_{i}}
= \sum_{i=1}^n b_i \frac{\partial f_0}{\partial x_{i}}
\]
with $a_{\alpha,i}\in {\Bbb C}$ and $b_i = \sum_{\deg x^\alpha = w_i} a_{\alpha,i}
\cdot x^\alpha $ we have $\deg b_i = w_i$ and
$\deg \frac{\partial f_0}{\partial x_{j}} = d - w_j > w_i$ for $j\neq i$.
Therefore, $b_i \not\in j_i(f_0)$ or $b_i = 0$.
But since $f_0$ has an isolated singularity, the sequence
$(\frac{\partial f_0}{\partial x_{1}},\ldots, \frac{\partial f_0}{\partial
x_{n}}) $ is a regular sequence and $\frac{\partial f_0}{\partial x_{i}}$
is not a zero divisor in $j_i(f_0)$. This implies $b_i = 0 $ for any $i,$ and
\[
j(f_0)\cap V = \bigoplus_{i=1}^n\;
\bigoplus_{\deg x^\alpha = w_i} {\Bbb C} \cdot x^\alpha \cdot
{\partial f_0 \over \partial x_{i}},
\]
and
\[
\dim G_w^{f_0} = \dim G_w - \dim j(f_0) \cap V = 0.
\]
{\bf 2.}\quad One can order the weights $w_i$ such that
$\overline{w}_1,\ldots,\overline{w}_r < \frac{1}{2},$
$\overline{w}_{r+1},\ldots,\overline{w}_n = \frac{1}{2}$.
The generalized Morse lemma and Theorem \ref{2.1} imply
the existence of an automorphism $\varphi \in G_w$ and of a
quasihomogeneous polynomial $g_0 \in {\Bbb C}[x_1,\ldots,x_r]$ of degree $d$
such that
$\varphi (f_0) = g_0 + x^2_{r+1} + \ldots + x^2_n $.
Now let $\widetilde{m}_1,\ldots,\widetilde{m}_k$ be the monomials of degree $>
d$
in a
monomial base of the Jacobi algebra of $g_0$.
Analogously to $F$ we obtain families
\vspace{-0.5cm}
\begin{eqnarray*}
\widetilde{G} & = & g_0 + \sum_{i=1}^k
\widetilde{m}_i \widetilde{t}_i \\
\mbox{and } G & = & \widetilde{G} + x^2_{r+1} + \ldots + x^2_n .
\end{eqnarray*}
It is well known that $G_t$ and $G_{t'}$ are right equivalent
if and only if $\widetilde{G}_t$ and $\widetilde{G}_{t'}$ are right equivalent.
Let
$\widetilde{w}$ be the tuple of weights $\widetilde{w} = (w_1,\ldots,w_r)$.
The group $G_{\widetilde{w}}^{g_0}$ is finite by
the first part of this proposition and induces a finite
group $\widetilde{E}_{\widetilde{w}}$ of automorphisms of
$\widetilde{T}_{-} = Spec\ {\Bbb C}[\tilde{t}]$.
In fact this is the largest subgroup of Aut$(\widetilde{T}_{-})$ which respects
the right equivalence classes.
Similarly to Proposition \ref{2.4} one can prove that $\varphi$
induces a biholomorphic mapping from $T_{-}$ to $\widetilde{T}_{-}$
which respects the right equivalence classes.
This gives an injective (in fact bijective) mapping from $E_{f_0}$ to
$\widetilde{E}_{\widetilde{w}}$.
Hence, $E_{f_0}$ is finite. \hfill
\begin{example}\label{2.7} \rm
$f_0 = x^3 + y^3 + z^7,\ (d;w_1,w_2,w_3) = (21;7,7,3),\ T_{-} = {\Bbb C}^5$,
$F = f_0 + \sum_{i=1}^5 t_im_i = f_0 + t_1xz^5 + t_2yz^5 + + t_3xyz^3 +
t_4xyz^4
+ t_5xyz^5$,
the weights of $(t_1,\ldots,t_5)$ are $(-1,-1,-2,-5,-8)$.
$G_w^{f_0}$ contains $6\cdot 3 \cdot 7$ elements:
obviously, $G_w^{f_0} \cong G_{(1,1)}^{g_0} \times {\Bbb Z}_7$
where $g_0 = x^3 + y^3$.
The group $G_{(1,1)}^{g_0}$ is isomorphic to a subgroup of
${\bf Gl}(2,{\Bbb C})$.
The image in ${\bf PGl}(2,{\Bbb C})$ permutes three points in ${\bf P^1C}$
and is isomorphic to $S_3,$ the kernel is isomorphic to
$\{ id , \xi \cdot id , \xi^2 \cdot id\},$ where $\xi = e^{2\pi i/3}$.
Therefore $G_{(1,1)}^{g_0}$ is
\[
G_{(1,1)}^{g_0} = (\langle\alpha\rangle \ltimes \langle\beta\rangle)
\times \langle\gamma\rangle
\times \langle\delta\rangle
\cong S_3 \times {\Bbb Z}_3 \times {\Bbb Z}_7
\]
with
\vspace{-0.5cm}
\begin{eqnarray*}
\alpha & : \ (x,y,z)\ \to & ( y, x,z), \\
\beta & : \ (x,y,z)\ \to & (\xi x,\xi^2 y,z), \\
\gamma & : \ (x,y,z)\ \to & (\xi x,\xi y,z), \\
\delta & : \ (x,y,z)\ \to & ( x, y,e^{2\pi i/7}z).
\end{eqnarray*}
The mapping $\theta: G_w^{f_0} \to E_{f_0}$ is an isomorphism with
\vspace{-0.5cm}
\begin{eqnarray*}
\theta(\alpha) & :\ (t_1,t_2,t_3,t_4,t_5)\ \to &
( t_2, t_1, t_3, t_4 , t_5) ,\\
\theta(\beta ) & :\ (t_1,t_2,t_3,t_4,t_5)\ \to &
(\xi t_1,\xi^2 t_2, t_3, t_4, t_5) ,\\
\theta(\gamma) & :\ (t_1,t_2,t_3,t_4,t_5)\ \to &
(\xi t_1,\xi t_2,\xi^2 t_3,\xi^2 t_4 ,\xi^2 t_5) ,\\
\theta(\delta) & :\ (t_1,t_2,t_3,t_4,t_5)\ \to &
(\zeta^5 t_1,\zeta^5 t_2,\zeta^3 t_3,\zeta^4 t_4 ,\zeta^5 t_5)
\mbox{ with } \zeta = e^{2\pi i/7}.
\end{eqnarray*}
Let ${\Bbb C}^{\ast}$ denote the group of ${\Bbb C}^{\ast}$-operations on $T_{-}$.
Then $E_{f_0} \cap {\Bbb C}^{\ast} = \langle \theta(\gamma),\theta(\delta)\rangle$
and $E_{f_0}\cdot {\Bbb C}^{\ast} \cong \langle\theta(\alpha),\theta(\beta)\rangle
\times {\Bbb C}^{\ast} \cong S_3\times{\Bbb C}^{\ast}$.
\end{example}
\newpage
\section{Kodaira--Spencer map and integral manifolds}
Let $f_0$ be semiquasihomogeneous of type $(d; w_1, \ldots, w_n)$, $w_i > 0$,
and $F : {\Bbb C}^n \times T_- \to {\Bbb C},\; (x,t) \mapsto f_0(x) + \sum\limits^k_{i=1}
t_im_i$,\enspace the semiuniversal family of unfoldings of negative weight as
in \S
1. In order to describe the orbits of the contact group acting on $T_-$ we
study the Kodaira--Spencer map of the induced semiuniversal family of
deformations (of space germs) defined as follows. Let
\[
{\cal X} = \{(x,t) \in {\Bbb C}^n \times T_- \mid F(x,t) = 0\}
\]
and let $({\cal X}, 0 \times T_-)$ denote the germ of ${\cal X}$ along the trivial
section $0 \times T_-$ which is a subgerm of
$({\Bbb C}^n \times T_- , 0 \times T_-) = ({\Bbb C}^n,0) \times T_-$.
The composition with the projection gives a morphism
\[
\phi : ({\cal X}, 0 \times T_-) \hookrightarrow ({\Bbb C}^n,0) \times T_- \to T_-
\]
such that, for any $t \in T_-$, $(\phi^{-1}(t), (0,t)) \cong ({\cal X}_t,0)\subset
({\Bbb C}^n,0)$ is a semiquasihomogeneous hypersurface singularity with principal
part equal to $({\cal X}_0,0) = (f_0^{-1} (0),0) =: (X_0,0)$. We call this family
the {\bf semiuniversal family of deformations of negative weight of
semiquasihomogeneous hypersurface singularities with fixed principal part}
$(\bf X_0,0)$ (see also \S 4).
For the study of the Kodaira--Spencer map of $({\cal X},0 \times T_-) \to T_-$ it
is more convenient to work on the ring level $A_- \to A_-\{x\}/F$.
The Kodaira--Spencer map (cf.\ \cite{LP}) of the family $A_- \to A_-\{x\}/F$,
\[
\rho : \mbox{Der}_{\Bbb C} A_- \to (x)A_-\{x\}/\left(F + (x) (\frac{\partial
F}{\partial x_1}, \ldots, \frac{\partial F}{\partial x_n})\right),
\]
is defined by $\rho(\delta) = \mbox{ class}(\delta F) = \mbox{
class}(\sum\limits^k_{i=1} \delta(t_i)m_i)$.
Let ${\cal L}$ be the kernel of $\rho$. ${\cal L}$ is a Lie--algebra and along the
integral manifolds of ${\cal L}$ the family is analytically trivial (cf.\
\cite{LP}).
In our situation it is possible to give generators of ${\cal L}$ as $A_-$--module:
Let $I = A_-\{x\}/(\frac{\partial F}{\partial x_1}, \ldots, \frac{\partial
F}{\partial x_n})$, then $I$ is a free $A_-$--module and $\{m_i\}_{i=1,
\ldots, k}$ can be extended to a free basis.
Multiplication by $F$ defines an endomorphism of $I$ and $F I \subseteq
\bigoplus\limits^k_{i=1} m_iA_-$.
Define $h_{\alpha,j}$ by
\[
x^\alpha F = \sum h_{\alpha,j} m_j \mbox{ in } I.
\]
Then $h_{\alpha j}$ is homogeneous of degree $|\alpha| + \deg(t_j) = |\alpha|+
d -
\deg(m_j)$. This implies $h_{\alpha j} = 0$ if $|\alpha| + \deg(t_j) \ge 0$,
in
particular $h_{\alpha j} = 0$ if $|\alpha| \ge (n-1)d -2 \sum w_i$. For
$\alpha$
and $|\alpha| < (n-1)d -2 \sum w_i$ let $\delta_\alpha := \sum
h_{\alpha,j} \frac{\partial}{\partial t_j}$.
\begin{proposition}\label{3.1} (cf.\ \cite{LP}, Proposition 4.5):
\begin{enumerate}
\item $\delta_{\alpha}$ is homogeneous of degree $|\alpha|$.
\item ${\cal L} = \sum A_- \delta_\alpha$.
\end{enumerate}
\end{proposition}
Now there is a non--degenerate pairing on $I$ (the residue pairing) which is
defined by $\langle h, k\rangle = \mbox{ hess} (h \cdot k)$. Here
$hess(h)$ is the evaluation of $h$ at the socle (the hessian of $f$).
Using the pairing one can prove the following:
\begin{proposition}\label{3.2}
There are homogeneous elements $n_1, \ldots, n_k \in A_- \{x\}$ with the
following properties:
\begin{enumerate}
\item If $n_i F = \sum^k_{j = 1} h_{ij} m_j$ in $I$
then $h_{ij} = h_{k-j+1, k-i+1}$.
\item If $\delta_i := \sum^k_{j=1} h_{ij}
\frac{\partial}{\partial t_j}$ then
$\delta_i$ is homogeneous of degree $\deg(n_i)$ and
${\cal L} = \sum^k_{i = 1} A_-\delta_i$.
\end{enumerate}
\end{proposition}
In \cite{LP} (Proposition 5.6) this proposition is proved for $n = 2$. The
proof can easily
be extended to arbitrary $n$. The important fact is the symmetry, expressed
in 1.\\
Let $L_+$ be the Lie--algebra of all vector fields of ${\cal L}$ of degree $\ge w =
\min\{w_i\}$. Then
$L$ is finite dimensional and nilpotent. $\delta_2, \ldots, \delta_k \in
L_+$ and $\delta_1 = \sum\limits^k_{i=1} \deg(t_i) t_i
\frac{\partial}{\partial t_i}$ is the Euler vector field (cf. \cite{LP}).
Let $L = L_+ \oplus {\Bbb C} \delta_1$ then $L$ is a finite dimensional and solvable
Lie--algebra and ${\cal L} = \sum A_- L,\; L/L_+ \cong {\Bbb C}\delta_1$.
\begin{corollary}\label{3.3}
The integral manifolds of ${\cal L}$ coincide with the orbits of the
algebraic group $exp(L)$.
\end{corollary}
Now consider the matrix $M(t) := (\delta_i(t_j))_{i, j = 1,
\ldots, k} = (h_{ij})_{i,j=1, \ldots, k}$. Evaluating this matrix
at $t \in T_-$ we have
\begin{eqnarray*}
\mbox{rank } M(t) & = & \mbox{dimension of a maximal integral manifold of }
{\cal L}\\
& & \mbox{(resp.\ of the orbit of exp(L)) at}\; t\\
& = & \mu - \tau(t),
\end{eqnarray*}
where $\tau(t)$ denotes the Tjurina number of the singularity defined by $t$
\enspace i.e.\ of $F(x,t)$.
\begin{example}\label{3.4}{\rm
We continue with Example \ref{2.7}, $f_0 = x^3 + y^3 + z^7$. Let
\vspace{-0.5cm}
\begin{eqnarray*}
n_1 & = & -21\\
n_2 & = & -21 z + \left(\frac{250}{49} t_1^3 t_2 + \frac{55}{7}
t_1^2 t_3 - \frac{250}{49} t_2^4\right) y - \frac{55}{7} t_2^2 t_3
x\\
n_3 & = & -21 z^2 - 30 t_2 y\\
n_4 & = & -21x\\
n_5 & = & -21y
\end{eqnarray*}
then the matrix defined by Proposition \ref{3.2} is
\[
(\delta_i(t_j)) = \left(
\begin{array}{ccccc}
t_1 & t_2 & 2t_3 & 5t_4 & 8t_5\\
0 & 0 & 0 & 2t_3-\frac{10}{7}t_1t_2 & 5t_4\\
0 & 0 & 0 & 0 & 2t_3\\
0 & 0 & 0 & 0 & t_2\\
0 & 0 & 0 & 0 & t_1
\end{array}
\right).
\]
We have $\mu = 24$ and
\begin{tabular}{lp{14cm}}
$\tau = 21$ & if and only if $2t_3 - \frac{10}{7} t_1t_2 \not= 0$,\\
$\tau = 22$ & if and only if $2t_3 - \frac{10}{7} t_1t_2 = 0$ and $t_1
\not= 0$ or $t_2 \not= 0$ or $t_3 \not= 0$ or $t_4 \not= 0$,\\
$\tau = 23$ & if and only if $t_1 = t_2 = t_3 = t_4 = 0$ and $t_5 \not= 0$,\\
$\tau = 24$ & if and only if $t_1 = t_2 = t_3 = t_4 = t_5 = 0$.
\end{tabular}}
\end{example}
\newpage
\section{Moduli spaces with respect to contact equivalence}
In this section we want to construct a coarse moduli space for
semiquasihomogeneous hypersurface singularities with fixed principal part with
respect to contact equivalence, that is isomorphism of space germs.
Such a moduli space does only exist if we
fix further numerical invariants. We shall use the Hilbert function of the
Tjurina algebra induced by the given weights.
Let us first define the functor for which we are going to construct the moduli
space.
A complex germ $(X,0) \subset ({\Bbb C}^n,0)$ is called a {\bf quasihomogeneous}
(respectively {\bf semiquasihomogeneous}) {\bf hypersurface singularity} of
type $(d; w_1, \ldots, w_n)$ if there exists a quasihomogeneous polynomial $f
\in {\Bbb C}[x_1, \ldots, x_n]$ (respectively a semiquasihomogeneous power series $f
\in {\Bbb C}\{x_1, \ldots, x_n\})$ of type $(d; w_1, \ldots, w_n)$ such that $(X,0)
= (f^{-1} (0),0)$. If $f_0$ is the principal part of $f$ then $(X_0,0) =
(f_0^{-1}(0),0)$ is called the {\bf principal part} of $(X,0)$. Multiplying
$f$ with a unit changes $f_0$ by a constant, hence the principal part if
well--defined. Two power series are contact equivalent if and only if the
corresponding space germs are isomorphic.
A {\bf deformation} ({\bf with section}) of $(X,0)$ over a complex germ or a
pointed complex space $(S,0)$ is a cartesian diagram
\[
\begin{array}{ccc}
0 & \hookrightarrow & (S,0)\\
\downarrow & & \downarrow\; \sigma\\
(X,0) & \hookrightarrow & ({\cal X},0)\\
\downarrow & & \downarrow\; \phi\\
0 & \hookrightarrow & (S,0)
\end{array}
\]
such that $\phi$ is flat and $\phi \circ \sigma =$ id. Two deformations
$(\phi, \sigma)$ and $(\phi', \sigma')$ of $(X,0)$ over $(S,0)$ are isomorphic
if there is an isomorphism $({\cal X},0) \buildrel\cong\over\to ({\cal X}',0)$ such that
the obvious diagram commutes. We shall only consider deformations with
section.
If $(X,0) = (f^{-1}(0),0)$ and if $F : ({\Bbb C}^n,0) \times (S,0) \to ({\Bbb C},0)$ is an
unfolding of $f$ then the projection $({\cal X},0) = (F^{-1}(0),0) \to (S,0)$ is a
deformation of $(X,0) \hookrightarrow ({\cal X},0)$ with trivial section $\sigma(s)
= (0,s)$. Conversely, any deformation of $(X,0)$ is isomorphic to a
deformation induced by an unfolding in this way. A deformation
($\phi,\sigma$) of a hypersurface singularity $(X,0)$, which is isomorphic to a
semiquasihomogeneous hypersurface singularity $(X',0) = (f^{-1}(0),0)$ of type
$(d; w_1, \ldots, w_n)$ over $(S,0)$, is called {\bf deformation of negative
weight} if it is isomorphic to a deformation induced by an unfolding of $f$
of negative weight.
We have to show that the definition is independent of the chosen unfolding:
two inducing unfoldings differ by a right equivalence and a multiplication
with a unit. We have shown in \S 1 that the definition depends only on the
right equivalence class. Hence, we have to show the following: if $f(x)$ is
a semiquasihomogeneous power series, $f(x) + g(x,s),\; g(x,0) = 0,\; \deg_x g >
d$, an unfolding of negative weight and $u(x,s) \in {\cal O}^\ast_{{\Bbb C}^n \times
S,0}$ a unit, then $u(f+g) \buildrel r\over\sim f'(x) + g'(x,s)$ with
$f^{-1}(0) = f'^{-1}(0),\; g'(x,0) = 0$ and $\deg_x g'>d$. Replacing
$u(x,s)$ by $(u(x,0))^{-1} u(x,s)$ we may assume that $u(x,s) = u_0(s) +
su_1(x,s),\; u_0(0) = 1,\; u_1(0,s) = 0$.
If $\nu \in {\cal O}_{S,0}$ is a $d$--th root of $u_0$ and if $\psi$ denotes the
automorphism of degree 0, $\psi (x,s) = (\nu(s)^{w_1}x_1, \ldots, \nu(s)^
{w_n}x_n)$, then $u_0(s) f(x)
= f(\psi(x,s)) + s \tilde{f}(x,s),\; \deg_x \tilde{f} > d$. But this implies
$u(f+g) \circ \psi^{-1} = f + g'$ with $g'(x,0) = 0$ and $\deg\, g'_x > d$ as
desired.
Again, we have to consider not only germs but also arbitrary complex spaces as
base spaces. A {\bf family of deformations} of hypersurface singularities
over a base space $S\in {\cal S}$ is a morphism $\phi : {\cal X} \to S$ of complex
spaces together with a section $\sigma : S \to {\cal X}$ such that for each $s \in
S$ the morphism of germs $\phi : ({\cal X}, \sigma(s)) \to (S,s)$ is flat and the
fibre $({\cal X}_s, \sigma(s)) = (\phi^{-1} (s), \sigma(s))$ is a hypersurface
singularity. This is, of course, only a condition on the germ $({\cal X},
\sigma(S))$ of ${\cal X}$ along $\sigma(S)$. A morphism of two families
$(\phi,\sigma)$ and $(\phi',\sigma')$ over $S$ is a morphism $\psi : {\cal X} \to
{\cal X}'$ such that $\phi = \phi' \circ \psi$ and $\sigma' = \psi \circ \sigma$.
$(\phi,\sigma)$ and $(\phi',\sigma')$ are called {\bf contact equivalent} or
{\bf isomorphic families of deformations} if there exists a morphism $\psi$
such that for any $s \in S$, $\psi$ induces an isomorphism of the germs of the
fibres $({\cal X}_s,\sigma(s)) \cong ({\cal X}'_s, \sigma'(s))$.
Let us fix a quasihomogeneous hypersurface singularity $(X_0,0) \subset
({\Bbb C}^n,0)$ of type $(d; w_1, \ldots, w_n)$. For $S \in {\cal S}$, a {\bf family of
deformations of negative weight with
principal part} $(\bf X_0, 0)$ over $S$ is a family of deformations
\[
S \buildrel \sigma\over\to ({\cal X}, \sigma(S)) \buildrel\phi\over\to S
\]
with section such that: for any $s \in S$ the fibre $({\cal X}_s, \sigma(s))$ is
isomorphic to a semiquasihomogeneous hypersurface singularity with principal
part $(X_0,0)$ and the germ $(S,s) \buildrel\sigma\over\to ({\cal X},\sigma(s))
\buildrel \phi\over\to (S,s)$ is a deformation of $({\cal X}_s,\sigma(s))$ of
negative weight.
For any morphism of base spaces $\varphi : T \to S$, the induced deformation
$T \to (\varphi^\ast {\cal X},\; \varphi^\ast \sigma(T)) \to T$ is a family of
deformations with negative weight and principal part $(X_0,0)$. We obtain a
functor
\[
\mbox{Def}^-_{X_0} : {\cal S} \to \mbox{ sets}
\]
which associates to $S \in {\cal S}$ the set of isomorphism classes of families of
deformations of negative weight with principal part $(X_0,0)$ over $S$. The
notations of {\bf fine} and {\bf coarse moduli space} for the functor
Def$^-_{X_0}$ are defined in the same manner as for the functor Unf$^-_{f_0}$
in \S 1. The objects we are going to classify are elements of
\begin{tabular}{lp{10cm}}
Def$^-_{X_0}(pt) =$ & $\{$ isomorphism classes of complex space germs $(X,0)$
which are isomorphic to a semiquasihomogeneous hypersurface singularity with
principal part $X_0\}$.
\end{tabular}
Again, as for Unf$^-_{f_0}$, we cannot expect to obtain fine moduli spaces in
general.
In order to obtain a coarse moduli space, we have to stratify $T_-$ into
$G$--invariant strata on which the geometric quotient with respect to $G$
exists, where $G = \exp\, L_+ \rtimes (E_{f_0} \cdot {\Bbb C}^\ast) \subset \mbox{
Aut}(T_-)$. Once we have this, the proof is the same as for Theorem
\ref{1.3}.
We want to apply Theorem 4.7 from \cite{GP 2} to the action of $L_+$ on
$T_-$.
\begin{theorem}\label{4.1} (\cite{GP 2})\quad Let $A$ be a noetherian
${\Bbb C}$--algebra and $L_+ \subseteq \mbox{ Der}_{\Bbb C}^{nil}A$ a finite dimensional
nilpotent Lie algebra. Suppose $A$ has a filtration
\[
F^\bullet :\; 0 = F^{-1}(A) \subset F^0 (A) \subset F^1 (A) \subset \ldots
\]
by subvector spaces $F^i(A)$ such that\\
$({\bf F})\qquad\qquad\qquad\qquad\qquad \delta F^i(A) \subseteq F^{i-1} (A)\,
\mbox{ for all } i \in {\Bbb Z},\; \delta \in L_+$.\\
Suppose, moreover, $L_+$ has a filtration
\[
Z_\bullet : L_+ = Z_1 (L_+) \supseteq Z_2 (L_+) \supseteq \ldots \supseteq
Z_e(L_+) \supseteq Z_{e+1}(L_+) = 0
\]
by sub Lie algebras $Z_j(L_+)$ such that\\
$({\bf Z})\qquad\qquad\qquad\qquad\qquad\qquad
[L_+, Z_j(L_+)] \subseteq Z_{j+1}(L_+)\, \mbox{for all}\, j \in
{\Bbb Z}$.\\
Let $d : A \to \mbox{ Hom}_{\Bbb C} (L_+, A)$ be the differential defined by $d(a)
(\delta) = \delta (a)$ and let \hbox{Spec $A = \cup U_\alpha$} be the
flattening
stratification of the modules
\[ \mbox{Hom}_{\Bbb C} (L_+, A) / A d (F^i(A))\quad i = 1, 2,\ldots \]
and
\[ \mbox{Hom}_{\Bbb C} (Z_j(L_+), A) / \pi_j(A(dA))\quad j = 1, \ldots, e, \]
where $\pi_j$ denotes the projection Hom$_{\Bbb C}(L_+, A) \to$
Hom$_{\Bbb C}(Z_j(L_+), A)$.\\
Then $U_\alpha$ is invariant under the action of $L_+$ and $U_\alpha
\to U_\alpha / L_+$ is a geometric quotient which is a principal fibre bundle
with fibre $\exp(L_+)$. Furthermore, the closure $\bar{U}_\alpha$ of
$U_\alpha$ is affine, $\bar{U}_\alpha = \mbox{ Spec } A_\alpha$, and the
canonical map $U_\alpha/L_+ \to \mbox{ Spec } A_\alpha^{L_+}$ is an open
embedding.
\end{theorem}
To apply the theorem we have to construct these filtrations and interpret the
corresponding stratification in terms of the Hilbert function of the Tjurina
algebra.\\
There are natural filtrations $H^\bullet ({\Bbb C}\{x\})$
respectively $F^\bullet (A_-)$ on ${\Bbb C}\{x\}$ respectively $A_-$ defined
as follows:\\
Let $F^i(A_-) \subseteq A_-$ be the ${\Bbb C}$--vectorspace
generated by all quasihomogeneous polynomials of degree $> - (i+1)w$ and
$H^i({\Bbb C}\{x\})$ be the ideal generated by all quasihomogeneous polynomials
of degree $\ge i w$, where
\[w := \min\{w_1, \ldots, w_n\}.\]
The filtration $F^\bullet(A_-)$ has the property (${\bf F}$) because every
homogeneous vector field of $L_+$ is of degree $\ge w$.
We also have $A_- dA_- = A_- dF^sA_-$ with
$s = \left[\frac{(n-1)d-2\sum w_i}{w}\right]$, since $nd - 2 \sum w_i$
is the degree of the Hessian of $f$ and $t_k$ is the variable of smallest
degree.
To define $Z_\bullet$ let $Z_i(L_+) :=$ the Lie algebra generated by the
vectorfields $\delta \in L_+,\; \delta$ homogeneous and $\deg(\delta) \ge r_i$,
\[
r_i := \min\{\deg(\delta_j) \mid t_{k+1-j} \in F^{s-i}(A_-)\}.
\]
$Z_\bullet(L_+)$ has the property $({\bf Z})$ because
$\deg([\delta,\delta']) \ge
\deg(\delta) + \deg(\delta')$ for all $\delta, \delta' \in L_+$.
\begin{example}\label{4.2}
{\rm We continue with Example \ref{3.4}, $f_0 = x^3 + y^3 + z^7$.
$w = 3$.\\
$F^\circ(A_-)$ is the ${\Bbb C}$--vector space generated by $t_1, t_2, t_3, t_1^2,
t_1t_2, t_2^2$.\\
$F^1(A_-)$ is the ${\Bbb C}$--vector space generated by $t_4, \{t_1^\nu
t_2^\mu t_3^\lambda\}_{\nu+\mu+2\lambda \le 5}$.\\
$F^2(A_-)$ is the ${\Bbb C}$--vector space generated by $t_5, \{t_1^\nu
t_2^\mu t_3^\lambda t_4\}_{\nu+\mu+2\lambda \le 3}, \{t_1^\nu t_2^\mu
t_3^\lambda\}_{\nu + \mu + 2\lambda \le 8}$.\\
We have $s = 2 = \left[\frac{2 \cdot 21 - 2 \cdot 17}{3}\right]$.\\
$A_-dF^\circ(A_-) = \bigoplus\limits^3_{i=1} A_-dt_i$.\\
$A_-dF^1 (A_-) = \bigoplus\limits^4_{i=1} A_-dt_i$.\\
$A_-dF^2 (A_-) = A_-dA_-$.\\
$r_1 = 3, r_2 = 6$.\\
$L_+ = Z_1(L_+)$.\\
$Z_2(L_+)$ generated by the homogeneous vector fields $\delta \in L_+$ with
$\deg(\delta) \ge 6$.\\
Especially $A_-Z_2(L_+) = \sum\limits^5_{i=3} A_- \delta_i$.\\
$Z_3(L_+) = 0$.
}
\end{example}
We can use Theorem \ref{4.1} to obtain a geometric quotient of the action of
$L_+$ on the flattening stratification defined by the filtrations $F^\bullet$
and $Z_\bullet$. Before doing this we shall prove that this flattening
stratification is also the flattening stratification of the modules defining
the Hilbert function of the Tjurina algebra.\\
For $t \in T_-$ the {\bf Hilbert function of the Tjurina algebra}
\[
{\Bbb C}\{x\}/\left(F(t),\frac{\partial F(t)}{\partial x_1}, \ldots, \frac{\partial
F(t)}{\partial x_n}\right)
\]
corresponding to the singularity defined by $t$ with respect to $H^\bullet$ is
by definition the function
\[
m \mapsto \tau_m(t) := \dim_{\Bbb C} {\Bbb C}\{x\}/\left(F(t), \frac{\partial
F(t)}{\partial x_1}, \ldots, \frac{\partial F(t)}{\partial x_n}, H^m\right).
\]
Notice that $\tau_m(t) = \tau(t)$ if $m$ is large and $\tau_m(t)$ does not
depend on $t$ for small $m$. On the other hand, $\mu_m := \mu_m(t) :=
\dim_{\Bbb C} {\Bbb C}\{x\}/(\frac{\partial F(t)}{\partial x_1}, \ldots, \frac{\partial
F(t)}{\partial x_n}, H^m)$ does not depend on $t \in T_-$ and
\[
\mu_m - \tau_m(t) = \mbox{ rank } (\delta_i(t_j) (t))_{\deg(t_j) > d- mw}.
\]
This is an immediate consequence of the following fact:\\
Let
\[
T^m := A_-\{x\}/\left(F, \frac{\partial F}{\partial x_1}, \ldots,
\frac{\partial F}{\partial x_n}, H^m\right),
\]
then the following sequence is exact and splits: let $\{X^\alpha\}_{\alpha
\in B}$ be a monomial base of $A_-\{x\}/\left(\frac{\partial F}{\partial x_1},
\ldots, \frac{\partial F}{\partial x_n}\right)$.
\[
\begin{array}{rcrllllll}
0 & \to & \bigoplus\limits_{\buildrel |\alpha|\le d\over \alpha \in B}
A_-x^\alpha & \to & T^{\frac{d}{w} + i} & \to
& \mbox{Der}_{\Bbb C} A_-/\left({\cal L} + \sum_{\deg(t_j) \le - iw} A_-
\frac{\partial}{\partial t_j}\right) & \to & 0\\
& & x^\alpha & \mapsto & \mbox{class}(x^\alpha) & & & & \\
& & & &\mbox{class}(m_j) & \mapsto & \mbox{class}(\frac{\partial}{\partial
t_j}), & &
\end{array}
\]
and with the identification $\sum\limits_{\deg(t_j) > -iw} A_-
\frac{\partial}{\partial t_j} \simeq A_-^{N_i}$ we obtain\\
Der$_{\Bbb C} A_-/({\cal L} + \sum\limits_{\deg(t_j) \le -iw} A_- \frac{\partial}{\partial
t_j}) \simeq A_-^{N_i}/M_i$, where $M_i$ is the $A_-$--submodule
generated by the rows of the matrix $(\delta_i(t_j))_{\deg(t_j) >
-iw}$.
We have $F \in H^m$, hence $\mu_m = \tau_m$, if $m \le \frac{d}{w}$
and $H^m \subset
(\frac{\partial F}{\partial x_1}, \ldots, \frac{\partial F}{\partial x_n})$,
hence $\mu_m - \tau_m(t)$ is independent of $m$ and equal to $\mu - \tau(t)$,
if $m \ge \frac{d}{w} + s + 1$ .
Therefore, we have $s + 1$ relevant values for $\tau_i$, and we denote
\vspace{-0.5cm}
\begin{eqnarray*}
\underline{\tau}(t) & := & (\tau_{\frac{d}{w} + 1}(t), \ldots,
\tau_{\frac{d}{w} + s + 1}(t)),\\
\underline{\mu} & := & (\mu_{\frac{d}{w}+1}, \ldots, \mu_{\frac{d}{w}+s+1}).
\end{eqnarray*}
Moreover, let $\Sigma = \{\underline{r} := (r_1, \ldots, r_{s+1}) \mid
\exists\, t
\in T_-$ so that $\underline{\mu} - \underline{\tau}(t) = \underline{r}\}$ and
$T_- = \cup_{\underline{r} \in \Sigma}U_{\underline{r}}$ be the flattening
stratification of the modules
$T^{\frac{d}{w}+1}, \ldots, T^{\frac{d}{w}+s+1}$. That is,
$\{U_{\underline{r}}\}$ is the stratification of $T_-$ defined by fixing the
Hilbert function $\underline{\tau} = \underline{\mu} - \underline{r}$ with the
scheme structure defined by the flattening property.
Let us now consider an arbitrary deformation $\phi : ({\cal X},\{0\} \times S)
\hookrightarrow ({\Bbb C}^n,0) \times S \to S$ of $(X,0) \subset ({\Bbb C}^n,0)$ of
negative weight over a base space $S \in {\cal S}$ where, for each $s \in S$, the
ideal of the germ $({\cal X},(0,s)) \subset ({\Bbb C}^n \times S, (0,s))$ is defined by
$F(x,s) = f(x) + g(x,s),\; g(x,0) = 0$.
Let us denote by ${\cal O}_S \{x\} = {\cal O}_{{\Bbb C}^n \times S, 0 \times S}$ the
topological restriction of ${\cal O}_{{\Bbb C}^n \times S}$ to $0 \times S$,
considered as a sheaf on $S$. Then
$J(I_{{\cal X} , 0 \times S})$, the Jacobian ideal sheaf of $({\cal X}, \{0\} \times S)
\subset ({\Bbb C}^n,0) \times S$, is locally defined by $(F, \frac{\partial
F}{\partial x_1}, \ldots, \frac{\partial F}{\partial x_n}) \subset {\cal O}_S
\{x\}$ and $H^m_S \subset {\cal O}_S\{x\}$ is the ideal sheaf generated by $g \in
{\cal O}_S \{x\}$ such that $\deg_x g \ge mw,\; w = \min\{w_1, \ldots, w_n\}$ as
above. We say that the {\bf family} $\phi$ is {\bf
$\underline{\tau}$--constant} if the coherent ${\cal O}_S$--sheaves
\[
T^m_S := {\cal O}_S \{x\}/J(I_{{\cal X},\{0\}\times S}) + H^m_S
\]
are flat for $\frac{d}{w} + 1\le m \le \frac{d}{w} + s + 1$ (equivalently, for
all $m$). Of course, if $T^m_S$ is flat, then
\[
\tau_m(s) := \dim_{\Bbb C} T^m_{S,s} \otimes {\cal O}_{S,s}/\frak m_{S,s}
\]
is independent of $s \in S$. The converse holds for reduced base spaces:
\begin{lemma}
If $S$ is reduced, then the sheaf $T^m_S$ is flat if and only if $\tau_m(s)$ is
independent of $s \in S$.
\end{lemma}
The proof is standard (cf.\ \cite{GP 3}). Hence, over a reduced base space
$S$, $\underline{\tau}$--constant means just that the Hilbert function
$\underline{\tau}(s) = (\tau_{\frac{d}{w} + 1} (s), \ldots, \tau_{\frac{d}{w}
+ s + 1} (s))$ of the Tjurina algebra is constant. But for arbitrary base
spaces we have to require flatness of the corresponding $T^m_S$.
{\bf Example} ($f_0 = x^3 + y^3 + z^7$, continued)
\begin{eqnarray*}
\underline{\tau}(t) & = & (\tau_8(t), \tau_9(t), \tau_{10}(t))\\
\underline{\mu} & = & ( \mu_8, \mu_9, \mu_{10}) = (22, 23, 24)\\
\Sigma & = & \{(0, 0, 0), (0, 0,1), (0, 1, 2), (1, 1, 2), (1, 2, 3)\}\\
U_{(1,2,3)} & = & D(2t_3 - \frac{10}{7} t_1 t_2) \subseteq T_- = {\Bbb C}^5\\
U_{(1, 1, 2)} & = & V(2t_3 - \frac{10}{7}t_1t_2) \cap D(t_1, t_2) \subseteq
T_-\\
U_{(0, 1, 2)} & = & V(t_1, t_2, t_3) \cap D(t_4) \subseteq T_-\\
U_{(0, 0, 1)} & = & V(t_1, t_2, t_3, t_4) \cap D(t_5) \subseteq T_-\\
U_{(0, 0, 0)} & = & \{(0, 0, 0, 0, 0)\}.
\end{eqnarray*}
\begin{lemma}\label{4.3}
\begin{enumerate}
\item $(0, \ldots, 0,1)$ and $(0, \ldots, 0) \in \Sigma$. $U_{(0, \ldots,0)}
=
\{0\}$ is a smooth point and $U_{(0, \ldots, 1)}$ is defined by $t_1 = \cdots
= t_{k-1} = 0$ and $t_k \not= 0$.
\item Let $\bar{\Sigma} = \Sigma\backslash\{(0, \ldots, 0)\}$ and for
$\underline{r} \in
\bar{\Sigma}$ put
\[
\widetilde{U}_{\underline{r}} = \left\{
\begin{array}{ll}
U_{\underline{r}} & \mbox{ if } \underline{r} \not= (0, \ldots, 0, 1)\\
U_{(0, \ldots, 0, 1)} \cup U_{(0, \ldots, 0)} & \mbox{ if } \underline{r} =
(0, \ldots, 0, 1).
\end{array}\right.
\]
\end{enumerate}
Then $\{\widetilde{U}_{\underline{r}}\}_{\underline{r} \in \bar{\Sigma}}$ is
the
flattening stratification of the modules $\{\mbox{Hom}_{\Bbb C}(L_+, A_-)/A_- dF^i
A_-\}$ and $\{\mbox{Hom}_{\Bbb C}(Z_i(L_+), A_-)/\pi_i(A_-dA_-)\}$.
\end{lemma}
{\bf Proof of Lemma \ref{4.3}}: Because of the exact sequence above the
flattening stratification of the modules $\{T^{\frac{d}{w}+i}\}$ is also the
flattening stratification of $\{\mbox{Der}_{\Bbb C} A_-/({\cal L} + \sum_{\deg(t_j) \le
-iw} A_- \frac{\partial}{\partial t_j})\}$ respectively the flattening
stratification of $\{A_-^{N_i}/M_i\},\; M_i$ the submodule generated by the
rows of the matrix $(\delta_i(t_j))_{\deg(t_j) > -iw}$.
Now we have
$(\ast)$\hspace{4.5cm}$\delta_i(t_j) = \delta_{k-j+1}(t_{k-i+1})$.
By definition of $Z_i(L_+)$ we have
\[
A_-Z_i(L_+) = \sum_{t_{k+1-j} \in F^{s-i}} A_- \delta_j
\]
and with the identification
\[
\sum\limits A_- \frac{\partial}{\partial t_j} = A_-^k,
\]
and $M^i$ the submodule generated by the rows of the matrix
$(\delta_\ell(t_j))_{\ell \ge r_i}$ we obtain
\[
\mbox{Der}_{\Bbb C} A_- / A_- Z_i(L_+) \cong A_-^k/M^i.
\]
(*) implies that the flattening stratification of the modules
$\{T^{\frac{d}{w}+1}, \ldots, T^{\frac{d}{w}+s}\}$, which is $T_- =
\cup_{\underline{r} \in
\bar{\Sigma}} \widetilde{U}_{\underline{r}}$, is the flattening stratification
of the
modules $\{\mbox{Der}_{\Bbb C} A_-/A_- Z_i(L_+)\}_{i=1, \ldots, s}$.
Furthermore the modules $\{\mbox{Hom}_{\Bbb C}(L_+, A_-)/A_-dF^iA_-\}$ and\\
$\{\mbox{Der}_{\Bbb C} A_-/A_- L_+ + \sum_{\deg(t_j)\le -iw}
A_-\frac{\partial}{\partial t_j}\}$ have the same flattening stratification
and they are flat on $U_{\underline{r}}$, because
\[
0 \to A_- \to \mbox{ Der}_{\Bbb C} A_-/A_-L_+ + \sum_{\deg(t_j)\le -iw}
A_-\frac{\partial}{\partial t_j} \to \mbox{ Der}_{\Bbb C} A_- /{\cal L} +
\sum_{\deg(t_j)\le -iw} A_-\frac{\partial}{\partial t_j} \to 0
\]
is exact and splits on $T_-\backslash\{0\}$.
This proves the lemma.
\begin{remark}\label{4.4}{\rm
The main point of the lemma is that the flattening stratification of the
modules $\{\mbox{Hom}_{\Bbb C}(L_+, A_-)/A_- dF^iA_-\}$ is equal to the
flattening stratification of the modules $\{\mbox{Hom}_{\Bbb C}(Z_i(L_+),
A_-)/\pi_i(A_- d A_-)\}$, hence, is defined by the Hilbert function of the
Tjurina algebra alone, without any reference to the action of $L$. This is a
consequence of the symmetry expressed in Proposition \ref{3.2}.}
\end{remark}
As a corollary we obtain the following
\begin{theorem}\label{4.5}
For $\underline{r}
\in \Sigma,\; \widetilde{U}_{\underline{r}}$ is invariant under the action of
$L_+$.
Let Spec $A_{\underline{r}}$ be the closure of $\widetilde{U}_{\underline{r}}$
then
$\widetilde U_{\underline{r}} \to \widetilde{U}_{\underline{r}}/L_+$ is a
geometric
quotient contained in Spec $A^{L_+}_{\underline{r}}$ as an open subscheme
of Spec $A^{L_+}_{\underline{r}}$.
\end{theorem}
{\bf Example} ($f_0 = x^3 + y^3 + z^7$, continued)
\[
\begin{array}{lccc}
1) & \widetilde{U}_{(1, 2, 3)} = D(2t_3 - \frac{10}{7} t_1 t_2) &
\longrightarrow &
\widetilde{U}_{(1, 2, 3)}/L_+ = \mbox{ Spec}\, {\Bbb C}[t_1, t_2,
t_3]_{2t_3-\frac{10}{7} t_1t_2}\\[1.0ex]
& \bigcap\mid & & \bigcap\mid\\[1.0ex]
& \mbox{Spec }{\Bbb C}[t_1, \ldots, t_5] & \longrightarrow & \mbox{Spec }{\Bbb C}[t_1,
t_2, t_3]\\[2.0ex]
2) & \widetilde{U}_{(1, 1, 2)} & \longrightarrow & \widetilde{U}_{(1, 1,
2)}/L_+ =
D(t_1, t_2)\\[1.0ex]
& \bigcap\mid & & \bigcap\mid\\[1.0ex]
& \mbox{Spec }{\Bbb C}[t_1, t_2, t_4, t_5] & \longrightarrow & \mbox{Spec }{\Bbb C}[t_1,
t_2, t_4]\\[1.0ex]
\multicolumn{4}{l}{(\mbox{identifiying } {\Bbb C}[t_1, \ldots, t_5]/2t_3 -
\frac{10}{7} t_1t_2 = {\Bbb C}[t_1, t_2, t_4, t_5].)}\\[2.0ex]
3) & \widetilde{U}_{(0,1,2)} & \longrightarrow & \widetilde{U}_{(0,1,2)}/L_+ =
D(t_4)\\[1.0ex]
& \bigcap\mid & & \bigcap\mid\\[1.0ex]
& \mbox{Spec }{\Bbb C}[t_4, t_5] & \longrightarrow & \mbox{Spec }{\Bbb C}[t_4]\\[2.0ex]
4) & \widetilde{U}_{(0,0,1)} & = & \widetilde{U}_{(0,0,1)}/L_+\\[1.0ex]
& \| & & \|\\[1.0ex]
& \mbox{Spec }{\Bbb C}[t_5] & = & \mbox{Spec }{\Bbb C}[t_5]
\end{array}
\]
\vspace{1cm}
Now $L/L_+ \simeq {\Bbb C} \delta_1$ acts on the geometric quotients
$\widetilde{U}_{\underline{r}}/L_+$ (the ${\Bbb C}^\ast$--action defined by the Euler
vector field $\delta_1$). Also the group $E_{f_0}$ acts and this action
commutes
with the ${\Bbb C}^\ast$--action (cf.\ \ref{2.5}). If we combine this fact with
Theorem 4.6 we obtain the main theorem of this article. In order to
formulate it properly let us denote by
\[
\mbox{Def}^-_{X_0, \underline{\tau}} : {\cal S} \to \mbox{ sets}
\]
the subfunctor of Def$^-_{X_0}$ which associates to a base space $S \in {\cal S}$
the set of isomorphism classes of $\underline{\tau}$--constant families of
deformations of negative weight with principal part $(X_0,0)$ over $S$. For
such a family $\underline{\tau}(s)$ is constant and equal to some tuple
$\underline{\mu} - \underline{r} \in {\Bbb N}^{s+1}$.
\begin{theorem}\label{4.6}
Let $G = \exp L_+ \rtimes (E_{f_0} \cdot {\Bbb C}^\ast) \subseteq \mbox{ Aut }(T_-)$.
\begin{enumerate}
\item The orbits of $G$ are unions of finitely many integral manifolds of
${\cal L}$.
\item Let $T_- = \cup_{\underline{r}\in \Sigma} U_{\underline{r}}$ be the
stratification fixing the
Hilbert function $\underline{\tau}$ of the Tjurina algebra described above.
$U_{\underline{r}}$ is invariant under the action of $G$ and the geometric
quotient $U_{\underline{r}} \to U_{\underline{r}}/G$ exists
and is locally closed in a weighted projective space.
\item $U_{\underline{r}}/G$ is the coarse moduli space for the functor
Def$^-_{X_0, \underline{\tau}} : $ complex spaces $\to$ sets with
$\underline{\tau} = \underline{\mu} - \underline{r}$.
\end{enumerate}
\end{theorem}
\begin{remark}{\rm
As in the case of right equivalence (see Remark 1.5) we may take (separated)
algebraic spaces as category of base spaces. That is, $U_{\underline{r}}/G$
is a coarse moduli space for the functor
\[
\mbox{Def}^-_{X_0, \underline{\tau}} : \mbox{ algebraic spaces }\to \mbox{
sets}.
\]
}
\end{remark}
{\bf Proof} (of Theorem 4.7): We first prove that $U_{\underline{r}}$ is
invariant under the
action of $G$ and that $U_{\underline{r}} \to U_{\underline{r}}/G$ is a
geometric quotient.
To prove that $U_{\underline{r}}$ is invariant under the action of $G$ it is
enough by definition of $U_{\underline{r}}$ that it is invariant under the
action of $E_{f_0}$. The Hilbert function $\underline{\tau}$ of the Tjurina
algebra is invariant under contact equivalence. This is a consequence of
Theorem \ref{2.1} because an automorphism $\varphi$ of ${\Bbb C}\{x\}$ inducing the
isomorphy of two semiquasihomogeneous singularities with principal part $f_0$
has degree $\ge 0$. More precisely, let $f,g$ be semiquasihomogeneous with
principal part $f_0$ and $uf = \varphi(g)$ for a unit $u$ then $\deg(\varphi)
\ge 0$ and consequently $(f, \frac{\partial f}{\partial x_1}, \ldots,
\frac{\partial f}{\partial x_n}, H^m)$ is mapped isomorphically to
$(g, \frac{\partial g} {\partial x_1}, \ldots, \frac{\partial g}{\partial x_n},
H^m)$ for all $m$, in particular $\underline{\tau}(f) = \underline{\tau}(g)$.
Moreover, let $\sigma \in E_{f_0}$, then there is a $\varphi : A_-\{x\} \to
A_-\{x\},\; \deg_x(\varphi) \ge 0$ and $\varphi|_{A_-} = id_{A_-}$
such that
\[\varphi(F(x,t)) \equiv F(x,\sigma (t))\hbox{ mod } A_- H^N
\hbox{ \ for sufficiently large }N\]
(cf. proof of Proposition 2.4).
This implies $\sigma(T^m) = T^m$ for all $m$ and proves that $E_{f_0}$ and,
therefore, $G$ acts on the strata $U_{\underline{r}}$ of the flattening
stratification of the modules $\{T^m\}$.
Now we prove that $U_{\underline{r}} \to U_{\underline{r}}/G$ is a geometric
quotient. First of all it is obvious that the geometric quotients
\[
U_{(0, \ldots, 0,1)} \to U_{(0, \ldots, 0,1)}/G = \{\ast\}
\]
and
\[
U_{(0, \ldots, 0)} = \{\ast\} = U_{(0, \ldots, 0)}/G = \{\ast\}
\]
exist.
Let $\underline{r} \not= (0, \ldots, 0,1),\; (0,\ldots, 0)$ then
$\widetilde{U}_{\underline{r}} = U_{\underline{r}}$. Let
$U_{\le\underline{r}} =
\mbox{ Spec}A_{\underline{r}}$ be the closure of $U_{\underline{r}}$ then we
obtain
\[
\begin{array}{ccc}
\mbox{Spec}A_{\underline{r}} & \buildrel \pi\over\longrightarrow &
\mbox{Spec}A_{\underline{r}}^{L_+}\\[1.0ex]
\cup|\; i & & \cup|\; j\\[1.0ex]
U_{\underline{r}} & \buildrel\pi|_{U_{\underline{r}}}\over\longrightarrow &
U_{\underline{r}}/L_+.
\end{array}
\]
$\pi|_{U_{\underline{r}}}$ defines a geometric quotient and $i,j$ are open
embeddings (Theorem \ref{4.5}). Notice that $\pi$ itself is not necessarily a
geometric quotient.
Now Spec$A_{\underline{r}}^{L_+}$ is affine and $E_{f_0}$ acts on
Spec$A_{\underline{r}}^{L_+}$ and also on $U_{\underline{r}}/L_+$. This
implies (cf.\ \cite{MF}) that
\[
\mbox{Spec}A_{\underline{r}}^{L_+} \buildrel\lambda\over\to \mbox{
Spec}(A_{\underline{r}}^{L_+})^{E_{f_0}}
\]
is a geometric quotient (not necessarily as algebraic schemes since
$A^{L_+}_{\underline{r}}$ need not be of finite type over ${\Bbb C}$) and
consequently
\[
\lambda|_{U_{\underline{r}}/L_+} : U_{\underline{r}}/L_+ \to
(U_{\underline{r}}/L_+)/E_{f_0}
\]
is a geometric quotient which is an algebraic scheme. Especially
$(U_{\underline{r}}/L_+)/E_{f_0} \subseteq
\mbox{ Spec}(A_{\underline{r}}^{L_+})^{E_{f_0}}$ is an open subset.
Finally, ${\Bbb C}^\ast$ acts on Spec$(A_{\underline{r}}^{L_+})^{E_{f_0}}$. It has
one
fixed point $\{\ast\}$ corresponding to $U_{(0, \ldots, 0)} \subseteq
\bar{U}_r =$ Spec$A_{\underline{r}}$. Outside this fixed point the
${\Bbb C}^\ast$--action leads to a geometric quotient:
\[
\begin{array}{ccc}
\mbox{Spec}(A_{\underline{r}}^{L_+})^{E_{f_0}}\backslash\{\ast\} &
\longrightarrow
& \mbox{Proj}(A_{\underline{r}}^{L_+})^{E_{f_0}}\\[1.0ex]
\cup & & \cup\\[1.0ex]
(U_{\underline{r}}/L_+)/E_{f_0} & \longrightarrow &
((U_{\underline{r}}/L_+)/E_{f_0})/{\Bbb C}^\ast\\[1.0ex]
& & \|\\[1.0ex]
& & U_{\underline{r}}/G.
\end{array}
\]
This proves part (1) and (2) of the theorem.
It remains to prove that if $t,t' \in T_-$
define isomorphic singularities then
$t$ and $t'$ are in the same orbit of $G$.
Let $F_t = u\varphi(F_{t'})$ for $t, t' \in T_-, u \in {\Bbb C}\{x\}^\ast$ a unit and
$\varphi$ an automorphism of ${\Bbb C}\{x\}$.
Using the ${\Bbb C}^\ast$--action we find
$t''\in T_-,\ u_1 =\frac{u}{u(0)}\in {\Bbb C}\{x\}^\ast$ and an automorphism
$\varphi_1$ of ${\Bbb C}\{x\}$ such that $F_t = u_1\varphi_1 (F_{t''}),\ u_1(0)=1$
and $t'$ and $t''$ are in one ${\Bbb C}^\ast$--orbit. Then
\[
G(z) := (1+z(u_1-1))\varphi_1(F_{t''})
\]
is an unfolding of $G(0) = F_t$ of negative weight.
This unfolding can be induced
by the semiuniversal unfolding, that is there exists a family of coordinate
transformations $\underline{\psi}(z, -)$ and a path $v$ in $T_-$ such that
\[
G(z) = F(\psi_1(z,x), \ldots, \psi_n(z,x), v(z))
\]
and $v(0) = t$ and $F_{t''} \buildrel r\over\sim F(\psi(1,x), v(1))$.
Now $t=v(0)$ and $v(1)$ are in one orbit of $\exp L$, and $v(1)$ and $t''$
are in one orbit of $E_{f_0}$.
Hence the result.
Now (3) follows in the same manner as the proof of Theorem 1.3.
{\bf Example} ($f_0 = x^3 + y^3 + z^7$, continued)
\begin{enumerate}
\item $U_{(1,2,3)} \longrightarrow U_{(1,2,3)}/G \simeq {\Bbb C}^2,\;
\underline{\tau} = (21, 21, 21),\; \tau = 21$ \\
normal form: $f_0 + t_1 xz^5 + t_2 yz^5 + t_3xyz^3,\\
(t_1:t_2:t_3) \in D_+ (2t_3 - \frac{10}{7} t_1t_2)/S_3 \subset
\P^2_{(1:1:2)}/S_3$\\
$(D_+(2t_3 - \frac{10}{7} t_1t_2)/S_3 \simeq {\Bbb C}^2$, the $S_3$--action being
explained in Example 2.8).
\item $U_{(1,1,2)} \longrightarrow U_{(1,1,2)}/G \simeq
\P^2_{(2,3,5)}\backslash(0:0:1),\; \underline{\tau} = (21, 22, 22),\; \tau =
22$\\
normal form: $f_0 + t_1xz^5+t_2yz^5+\frac{10}{7} t_1t_2xyz^3 +t_4 xyz^4$,\\
$(t_1:t_2:t_4) \in \P^2_{(1:1:5)}/S_3 \;\; (\simeq \P^2_{(2,3,5)})$
\item $U_{(0,1,2)} \longrightarrow U_{(0,1,2,)}/G = \{\ast\},\;
\underline{\tau} = (22, 22, 22),\; \tau = 22$\\
normal form: $f_0 + xyz^4$
\item $U_{(0,0,1)} \longrightarrow U_{(0,0,1,)}/G = \{\ast\},\;
\underline{\tau} = (22, 23, 23),\, \tau = 23$\\
normal form: $f_0 + xyz^5$
\item $U_{(0,0,0)} \longrightarrow U_{(0,0,0,)}/G = \{\ast\},\;
\underline{\tau} = (22, 23, 24),\; \tau = 24$\\
normal form: $f_0$
\end{enumerate}
Hence the moduli space of semiquasihomogeneous hypersurface singularities $X =
\{(x,y,z) \mid f (x,y,z) = 0\}$ with principal part $X_0 = \{(x,y,z) \mid x^3
+ y^3 + z^7 = 0\}$ consists of 5 strata (${\Bbb C}^2,\; \P^2_{(2,3,5)} \backslash (0
: 0 : 1)$, and 3 isolated points) corresponding to 5 possible Hilbert
functions $\underline{\tau}$ of the Tjurina algebra ${\Bbb C}\{x,y,z\}/(f,
\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial
f}{\partial z})$. The generic stratum $U_{(1,2,3)}$ (minimal
$\underline{\tau}$) is an open subset in ${\Bbb C}^5$, the quotient being
2--dimensional, as well as the quotient of the 4--dimensional ``subgeneric''
stratum $U_{(1,1,2)}$.
Note that the families of normal forms are not universal. It just means that
each semiquasihomogeneous singularity with principal part $f_0$ occurs and
that different parameters do not give contact equivalent singularities, except
modulo the ${\Bbb C}^\ast$-- and $S_3$--action.
We see that $U_{(1,1,2)}/G$ can be compactified by $U_{(0,1,2)}/G$, that is
\[
U_{(1,1,2)} \cup U_{(0,1,2)} \to (U_{(1,1,2)} \cup U_{(0,1,2)})/G =
\P^2_{(2,3,5)}
\]
is a geometric quotient. So in this example there exist
geometric quotients of the strata with constant Tjurina number and, hence, a
coarse moduli space for fixed principal part and fixed Tjurina number.
In general this is false (cf.\ \cite{LP}, \S 7).
\begin{remark}{\rm
1.\quad The generic stratum $U_{\underline{\tau}\min}$ corresponding to minimal
Hilbert function $\underline{\tau}$ (with respect to lexicographical ordering)
is an open, quasiaffine subset of $T_-$ and, hence,
$U_{\underline{\tau}\min}/L_+$ is smooth by Theorem 4.1. In particular, the
generic moduli space $U_{\underline{\tau}\min}/G$ has, at most, quotient
singularities (coming from the ${\Bbb C}^\ast$--action and the finite group
$E_{f_0}$).
It is not known whether the bigger stratum $U_{\tau\min}$ corresponding to
minimal Tjurina number $\tau$ admits a geometric quotient, except for $n = 2$
(cf.\ \cite{LP}).
2.\quad We always have two special strata, the most special $U_{(0, \ldots,0)}
=
\{\ast\}$ (corresponding to $f_0$) and the ``subspecial'' $U_{(0, \ldots, 1)}
\cong {\Bbb C}\backslash \{\ast\}$ (corresponding to the singularity $f_0 + m_k,\;\;
m_k$ generating the socle of ${\Bbb C}\{x\}/j(f_0)$, that is the monomial of
maximal degree). The $G$--quotients of these strata give two reduced,
isolated points.
3.\quad As we have seen for $x^3 + y^3 + z^7$, the finite group $E_{f_0}$ need
not
be abelian. If $f_0 = x_1^{a_1} + \cdots + x_n^{a_n}$ is of Brieskorn--Pham
type and gcd$(a_i, a_j) = 1$ for $i \not= j$, then $E_{f_0} \cong \mu_d$, the
group of $d$'th roots of unity, $d = \deg\, f_0$.
4.\quad Note that a coarse moduli space is more than just a bijection between
its points and the corresponding set of isomorphism classes. For instance,
let $U_{\underline{r}}/G$ be affine and let $S \buildrel\sigma\over\to ({\cal X},
\sigma(S)) \buildrel\phi\over\to S$ be a family of deformations from
Def$^-_{X_0} (S)$ with $\underline{\tau}({\cal X}_s, \sigma(s)) = \underline{\mu} -
\underline{r}$. If $S$ is compact then $\phi$ must be locally trivial since
any morphism from $S$ to $U_{\underline{r}}/G$ maps $S$ onto finitely many
points.}
\end{remark}
\newpage
\addcontentsline{toc}{section}{References}
|
1996-02-14T06:20:08 | 9503 | alg-geom/9503010 | en | https://arxiv.org/abs/alg-geom/9503010 | [
"alg-geom",
"hep-th",
"math.AG",
"nlin.SI",
"solv-int"
] | alg-geom/9503010 | Benjamin Enriquez | B. Enriquez and V. Rubtsov | Hitchin systems, higher Gaudin operators and $r$-matrices | null | null | null | null | null | We adapt Hitchin's integrable systems to the case of a punctured curve. In
the case of $\CC P^{1}$ and $SL_{n}$-bundles, they are equivalent to systems
studied by Garnier. The corresponding quantum systems were identified by B.
Feigin, E. Frenkel and N. Reshetikhin with Gaudin systems. We give a formula
for the higher Gaudin operators, using results of R. Goodman and N. Wallach on
the center of the enveloping algebras of affine algebras at the critical level.
Finally we construct a dynamical $r$-matrix for Hitchin systems for a punctured
elliptic curve, and $GL_{n}$-bundles, and (for $n=2$) the corresponding quantum
system.
| [
{
"version": "v1",
"created": "Mon, 20 Mar 1995 20:46:36 GMT"
},
{
"version": "v2",
"created": "Tue, 13 Feb 1996 11:35:19 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Enriquez",
"B.",
""
],
[
"Rubtsov",
"V.",
""
]
] | alg-geom | \section{Introduction.}{}
In [13], N. Hitchin introduced a class of integrable
systems, attached to a complex curve $X$ and a semisimple Lie group $G$.
The problem of quantization of these systems was then connected by
A. Beilinson and V. Drinfel'd to the Langlands program. This quantization
makes use of differential operators on the moduli space of $G$-bundles
on $X$, obtained from the action of the center of the local completion
of the enveloping algebra of a Kac-Moody algebra, at the critical
level.
This program can also be carried out in the case of curves
with marked points. In the particular case of the punctured ${\bf C} P^{1}$,
the action of the center of the enveloping algebra was studied by B.
Feigin, E. Frenkel and N. Reshetikhin in [6]; they obtained an
integrable system whose first operators are identical to Gaudin's
operators ([9]).
In this paper, we consider the question of expressing the action of
higher central elements. We first note, that the Adler-Kostant-Symes (AKS)
scheme, which allows to write families of commuting operators ([2],
[14], [21]), can be applied in the present situation, and then show that
the higher Hamiltonians obtained in [6], coincide with those. So our
problem turns out to be equivalent to expressing higher central elements
in the enveloping algebras at critical level, a problem which was
solved by several authors ([10], [12]). Here we show how the results
of [10] can be used to derive a simple expression of higher Gaudin
Hamiltonians.
We then turn to the case of punctured elliptic curves. We show that
the integrability of Hitchin's system can be deduced
from an $r$-matrix argument. Here $r$-matrix relations contain
additional terms, due to an invariance under the Cartan algebra action.
The $r$-matrix depends on the moduli parameters, so it reminds
dynamical $r$-matrices. In the case of one puncture, our $L$-operator
and $r$-matrix seem connected with those considered respectively
by I. Krichever and A. Gorsky and N. Nekrasov in [15] and [11], and
H. Braden, T. Suzuki and E. Sklyanin [5],
[19]. It is also analogous to the $r$-matrix appearing in
the work of G. Felder and C. Wieczerkowski on the
Knizhnik-Zamolodchikov-Bernard equation on elliptic curves ([7]).
We give the form of the first Hamiltonians in this case; one of them
contains a Weierstass potential, and so is analogous to the
Calogero-Moser Hamiltonian. We compute the corresponding quantum
Hamiltonians, in the case $G=GL_{2}$.
We would like to thank V. Drinfel'd, B. Feigin, G. Felder, E. Frenkel,
A. Gorsky, N. Nekrasov,
A. Reyman, and A. Stoyanovsky for discussions connected with the subject
of this work. We are thankful to A. Beilinson and V. Drinfel'd for
sending us their paper [4]. The work of V.R. was supported by the CNRS,
and partially by grant RFFI 95-01-01101; he is grateful to the Centre de
Math\'ematiques de l'\'Ecole Polytechnique, where this work was done,
for hospitality.
\section{1.}{Hitchin and Beilinson-Drinfeld systems in the case of a
general punctured curve.}
\medskip\noindent
\it 1.1. Hitchin systems. \rm
Let $\overline X$ be a smooth compact complex curve, $x_{i}$
($i=1,\cdots,N$) be distinct points on $\overline X$. Set $X=\overline
X-\{x_{i}\}$.
Let $G$ be a reductive complex Lie group, $B\subset G$ and $T\subset B$
Borel and Cartan subgroups of $G$; let ${\bf g}$, ${\bf b}$ and ${\bf
t}$ be their Lie
algebras. Let ${\cal M}_{G}(X)$ be the moduli space
of principal $G$-bundles on $\overline X$ with choices of a $B$-orbit in each
fibre over $x_{i}$. Let us identify ${\bf g}$ with
its dual, using a non-degenerate invariant form $\langle,
\rangle_{{\bf g}}$. Let $P
\in{\cal M}_{G}(X)$, then $T^{*}_{P}{\cal M}_{G}(X)$ is formed of the
$\xi\in H^{0}(\overline X,\Omega_{\overline X}(\sum_{i=1}^{N}(x_{i}))\otimes
{\bf g}_{P})$, such that
$\xi$ has the expansion near $x_{i}$, $\xi=
{1\over{u_{i}}}\xi_{i}+ {\rm regular}$, and $\xi_{i}\in
({\bf b}_{x_{i}})^{\perp}$; ${\bf b}_{x_{i}}$
is the subspace of the fibre of ${\bf g}_{P}$
at $x_{i}$, corresponding to the $B$-orbit at $P_{x_{i}}$, $u_{i}$ is a
local coordinate at $x_{i}$.
The Hitchin map
$$
H : T^{*}{\cal M}_{G}(X)\to {\cal H}_{X}=
\bigoplus_{i=1}^{r}H^{0}\big(\overline X,\Omega_{\overline X}^{d_{i}}
((d_{i}-1)\sum_{l=1}^{N}(x_{l}))\big),
$$
is then
defined by $(H(P,\xi))_{l}= P_{d_{l}}(\xi)$; $r$ is the rank of $G$,
$d_{l}$ ($1\le l\le r$)
are the characteristic degrees of ${\bf g}$ and $P_{d_{l}}$ corresponding
invariant polynomials.
The generic fiber of the
natural projection ${\cal M}_{G}(X)\to{\cal M}_{G}(\overline X)$
is $(G/B)^{N}$ if genus$(\overline X)>1$, the generic bundle having no
automorphisms; on the other hand, we have for genus$(\overline X)>0$, $\dimm
{\cal H}_{X} =\dimm{\cal H}_{\overline X}+\sum_{l=1}^{r}(d_{l}-1)N
=\dimm{\cal H}_{X}+N(\dimm B-r)$.
If genus$(\overline X)=1$, an open subset of ${\cal M}_{G}(X)$ is identified
with $T/W$ ($W$
is the Weyl group of $G$) if $N=0$, and with
$T\rtimes W\setminus [T\times(G/B)^{N}]$ for $N\ge 1$
(only $W$ acts in the first factor, and $T\rtimes W$ acts diagonally on
$(G/B)^{N}$); on the other
hand, $\dimm{\cal H}_{X}=\sum_{i=1}^{r}(d_{i}-1)N$ if $N\ge 1$, and $r$
if $N=0$.
If genus$(\overline X)=0$ and $N\ge 3$, an open subset of
${\cal M}_{G}(X)$ is identified
with $G\setminus(G/B)^{N}$, whereas $\dimm {\cal H}_{X}=\sum_{l=1}^{r}
[(d_{l}-1)(N-2)-1]$. The cases $N\le 2$ give trivial moduli spaces and
${\cal H}_{X}$. So in all cases
$$
\dimm{\cal M}_{G}(X)= \dimm{\cal H}_{X}.
$$
We can see as in [13] that the functions on $T^{*}{\cal M}_{G}(X)$,
defined by the coordinates of $H$, Poisson commute. Moreover, for
$G=GL_{n}({\bf C})$ we can consider the spectral cover of
$\overline X$, defined as
$\{(x,\lambda)| \lambda^{n}+ \sum_{l\ge 1}H_{i}\lambda^{n-l} = 0\}$,
for $(H_{l})\in {\cal H}$ fixed, in the total space of
$\Omega_{\overline X}(\sum_{i=1}^{N}(x_{i}))$; it has ramification of order
$n$ at the
points $x_{i}$, in the generic situation. It is possible to build a line
bundle over the spectral cover, and to
study the integrability of the system as in [13].
\medskip\noindent
\it 1.2. Beilinson-Drinfeld systems. \rm
To quantize the Hitchin systems, Beilinson and Drinfeld ([4]) define
$\dimm{\cal M}_{G}(X)$ commuting differential operators on ${\cal M}_{G}(X)$, with
symbols the coordinates of the map $H$ (here we assume no marked
points). They are constructed as follows:
a base point $x$ on $X$ being fixed, ${\cal M}_{G}(X)$ is identified with
$G({\cal O}_{x})\setminus G(k_{x})/G(A)$ (Siegel-Weil); ${\cal O}_{x}$ and $k_{x}$
are respectively the local ring and field at $x$, and $A=H^{0}(X-\{x\},
{\cal O}_{X})$. Then the center $Z(U_{-h^{\vee}}({\bf g}(k_{x}))_{loc})$
of $U_{-h^{\vee}}({\bf g}(k_{x}))_{loc}$ (local
completion of the enveloping algebra of the critical level extension of
${\bf g}(k_{x})$) acts by differential operators on the line bundle
$(\det)^{-h^{\vee}}$ over ${\cal M}_{G}(X)$. This procedure can easily be
extended to the punctured case: remark that
${\cal M}_{G}(X)=
G({\cal O}_{x})\setminus G(k_{x})/\Gamma$, where $\Gamma\subset G(A)$ is
formed of the regular maps from $\overline X-\{x\}$ to $G$, taking values in
$B$ at points $x_{i}$. Let $(\lambda_{i})_{1\le i\le N}$ be a system of
dominant weights of $G$. We define a line bundle ${\cal L}_{(\lambda_{i})}$ on
${\cal M}_{G}(X)$ as follows: $(\lambda_{i})$ defines a character of $\Gamma$ (by
the maps $\Gamma\to B^{N}\to T^{N}$) and so a line bundle
${\cal L}'_{(\lambda_{i})}$ on $G(k_{x})/\Gamma$, then ${\cal L}'_{(\lambda_{i})}\otimes
(\det)^{-h^{\vee}}$ has a natural action of $G({\cal O}_{x})$;
${\cal L}_{(\lambda_{i})}$ is then the quotient
bundle. The center of $U_{-h^{\vee}}({\bf g}(k_{x}))_{loc}$ then acts on this
bundle by differential operators as before.
\section{2.}{Hitchin systems in the rational case.}
In this section and the following, we set $\overline X={\bf C} P^{1}$, and
denote by $z_{i}$ the coordinate of the marked point $x_{i}$
($i=1,...,N$); we assume that no $x_{i}$ coincides with $\infty$.
We will express the corresponding Hitchin systems, and recall an $r$-matrix
result of Semenov about them.
An open subset of ${\cal M}_{G}(X)$ is formed by parabolic structures on the
trivial bundle; this subset, that we call ${\cal M}_{G}^{(0)}(X)
$ is isomorphic to $G\setminus (G/B)^{N}$ [the left action of $G$ is
diagonal].
Recall the Springer resolution $T^{*}(G/B)\to{\cal N}$, ${\cal N}$ the nilpotent
cone of ${\bf g}$ ([20]). Then we construct, by reduction, the resolution
$$
T^{*}[G\setminus
(G/B)^{N}]\to\{(\eta^{(i)})\in{\cal N}^{N}|\sum_{i=1}^{N}\eta^{(i)}=0\}/G
$$
(the action of $G$ on the last term is by conjugation).
When the $\eta^{(i)}$ are regular, the parabolic structure
corresponding to $(\eta^{(i)})_{i=1,\cdots,N}$ is
$(g_{i}B)_{i=1,\cdots,N}$, where $g_{i}$ are elements of $G$
conjugating $\eta^{(i)}$ to elements of ${\bf b}\subset {\bf g}$. The
$1$-form $\xi$ is then
$$
\xi=\sum_{i=1}^{N}{\eta^{(i)}\over{z-z_{i}}}dz.\leqno(1)
$$
The Poisson structure on $T^{*}{\cal M}_{G}^{(0)}(X)$
corresponds, in terms of the $(\eta^{(i)})$, to the product of
Kostant-Kirillov structures on each ${\cal N}$. In tensor notation: $\{
\eta^{(i)}\otimes_{,}\eta^{(j)}\}=\delta_{ij}[P,1\otimes\eta^{(j)}]
=-\delta_{ij}[P,\eta^{(i)}\otimes 1]$, $P$ the permutation operator.
We deduce from this:
$$
\{\eta(z)\otimes_{,}\eta(w)\}=[{P\over{z-w}},\eta(z)\otimes
1+1\otimes \eta(w)],\leqno(2)
$$
where $\eta(z)=\sum_{i=1}^{N}{\eta^{(i)}\over{z-z_{i}}}$. So we have:
\proclaim{Proposition 2.1} ([18]) Let us endow ${\cal N}^{N}$ with the
product of Kostant-Kirillov structures on each factor. Then the mapping
${\cal N}^{N}\to {\bf g}[[z^{-1}]]$, $(\eta^{(i)})_{1\le i\le N}
\mapsto \eta(z)=\sum_{i=1}^{N}{\eta^{(i)}\over{z-z_{i}}}$, is
Poisson. \endgroup\par\medbreak
We deduce from this that the coefficients of the forms
$P_{d_{i}}(\eta(z))$ are in involution, because all the central
functions on ${\bf g}[[z^{-1}]]$ are in involution. (This gives
another proof of
involutivity of Hitchin's Hamiltonians.) Let
us give now the expression of the corresponding flows:
\proclaim{Proposition 2.2} Decompose $P_{d_{l}}(\eta(z))$ as
$\sum_{a_{1}+\cdots+a_{N}=d_{l}-1} {H_{d_{l},(a_{i})}\over{
\prod_{i=1}^{N}(z-z_{i})^{a_{i}}
}}(dz)^{d_{l}}$, and denote by $\pr_{d_{l},(a_{i})}$ the flow generated
by $H_{d_{l},(a_{i})}$. Then we have the identity of rational functions
in $z$
$$
\sum_{a_{1}+\cdots+a_{N}=d_{l}-1}
{\pr_{d_{l},(a_{j})}(\eta^{(i)})
\over{\prod_{j=1}^{N}(z-z_{j})^{a_{j}}}}
=
[P'_{d_{l}}(\eta(z)),{\eta^{(i)}\over{z-z_{i}}}].
\leqno(3)
$$
For ${\bf g}=sl_{n}({\bf C})$, the r.h.s. is
$[d_{l}(\sum_{j=1}^{N}{\eta^{(j)}\over{z-z_{j}}})^{d_{l}-1},\eta^{(i)}]$.
For ${\bf g}$ arbitrary, the flows corresponding to $d_{1}=2$ are
$$
\pr_{i}\eta^{(j)}=-{[\eta^{(i)},\eta^{(j)}]\over{z_{i}-z_{j}}}
{\rm \ \ for \ \ }j\ne i, {\rm \ \ and \ \ }
\pr_{i}\eta^{(i)}=\sum_{j\ne i}{[\eta^{(i)},\eta^{(j)}]
\over{z_{i}-z_{j}}}.
\leqno(4)
$$
\endgroup\par\medbreak
We note that in the case $g=sl_{n}({\bf C})$, the flows $\pr_{i}$ already appeared
in [8] (we thank J. Harnad for pointing out this reference to
us). Their integration was studied by many authors (cf. e.g. [1], [3]).
\section{3.}{Gaudin systems.}
\medskip\noindent
\it 1. The moduli stack in the rational case. \rm
Let again ${\bf g}$ be an arbitrary reductive complex Lie algebra, $G$
be the adjoint group. Let $\Delta$ be the set of the roots of ${\bf g}$
w.r.t. ${\bf t}$, ${\bf g}={\bf t}\oplus
\bigoplus_{\alpha\in\Delta}{\bf g}_{\alpha}$ the associated decomposition of
${\bf g}$, $R$ be
the root lattice of ${\bf g}$, $W$ the Weyl group of ${\bf g}$. Classes
of principal $G$-bundles on $\overline X={\bf C} P^{1}$ are parametrized by
$\mathop{\rm Hom}\limits(R,{\bf Z})/W$; to $\varpi\in\mathop{\rm Hom}\limits(R,{\bf Z})$ we associate the Lie algebra
bundle on ${\bf C} P^{1}$
$$
{\bf g}(\varpi)={\bf t}
\oplus\bigoplus_{\alpha\in\Delta}{\bf g}_{\alpha}(\varpi(\alpha)\infty),\leqno(5)
$$
and the associated $G$-bundle $G(\varpi)={\rm Ad}}\def\Diff{{\rm Diff}}\def\Spec{{\rm Spec} {\bf g}(\varpi)$. Its
automorphism group is a subgroup of $G({\bf C}[z])$,
$$
P_{\varpi}=L_{\varpi}U_{\varpi},
\quad U_{\varpi}=\prod_{\alpha\in\Delta}N_{\alpha}(H^{0}(\varpi(\alpha)\infty)),
\leqno(6)
$$
$L_{\varpi}$ the subgroup of ${\bf g}$ with Lie algebra
${\bf l}_{\varpi}={\bf t}\oplus\bigoplus_{\varpi(\alpha)=0}{\bf g}_{\alpha}$.
The moduli space of $G$-bundles on ${\bf C} P^{1}-\{z_{i}\}$ is
$$
{\cal M}_{G}(X)=\prod_{[\varpi]\in\mathop{\rm Hom}\limits(R,{\bf Z})/W}
{\cal M}_{G}^{\varpi}(X),
\leqno(7)
$$
where ${\cal M}_{G}^{\varpi}(X)$ is identified with
$P_{\varpi}\setminus (G/B)^{N}$, where the action of $G({\bf C}[z])$ is the
composition of the morphism $G({\bf C}[z])\to G^{N}$, $g(z)\mapsto
(g(z_{i}))_{i}$, and the left translation. Let $(\lambda_{i})_{i}$ be
integral dominant weights of $G$, ${\cal L}_{\lambda_{i}}$ be the associated line
bundles on $G/B$; $\boxtimes_{i=1}^{N}{\cal L}_{\lambda_{i}}$ is a
$G^{N}$-equivariant bundle on $(G/B)^{N}$, so it is
$P_{\varpi}$-equivariant; let ${\cal L}_{(\lambda_{i})}$ be the quotient bundle
on ${\cal M}_{G}^{\varpi}(X)$.
\medskip\noindent
\it 2. The FFR scheme. \rm
The procedure of sect. 1.2 was applied in [6] to the case of the
punctured ${\bf C} P^{1}$. Let us set some notations.
Let for each $i$, $k_{i}$ and ${\cal O}_{i}$ be the local field and ring at
$z_{i}$; let $\tilde {\bf g}$ the
central extension of $\oplus_{i}{\bf g}(k_{i})$ by the cocycle
$c((a_{i}),(b_{i}))=\sum_{i=1}^{N}\res_{x_{i}}
\langle a_{i}, db_{i}\rangle_{{\bf g}} K$,
with values in the abelian algebra ${\bf C} K$. Let $\tilde {\bf g}_{+}$
be the preimage of ${\bf g}(\oplus_{i}{\cal O}_{i})$ in $\tilde {\bf g}$;
$\tilde {\bf g}_{+}$ is then
isomorphic to ${\bf g}(\oplus_{i}{\cal O}_{i})\oplus
{\bf C} K$. Let for $\lambda$ integral dominant weight of ${\bf g}$,
$V_{\lambda}$ be the corresponding irreducible representation of ${\bf g}$;
and let for $k\in {\bf C}$, and $\lambda_{1},...,\lambda_{N}$ integral dominant
weights of ${\bf g}$, $V_{(\lambda_{i})}^{k}$ be the representation of
$\tilde {\bf g}_{+}$ in $V_{\lambda_{1}}\otimes...\otimes V_{\lambda_{N}}$,
where elements of ${\bf g}(\oplus_{i}{\cal O}_{i})$ act as their
images in ${\bf g}^{\oplus N}$, and $K$ by $k$.
Let $\bar {\bf g}_{(z_{i})}$ be the Lie algebra of
regular maps from $X$ to ${\bf g}$; choose and denote the same way a
lifting of this algebra to $\tilde {\bf g}$.
Let $\hat {\bf g}$ be the universal central extension of ${\bf g}((u))$,
and ${\bf V}_{0}^{-h^{\vee}}$ be the critical level vacuum module over
it. Central
fields $T(\zeta)\in Z(U_{-h^{\vee}}(\hat{\bf g})_{loc})[[\zeta^{\pm1}]]$
are in
correspondance with imaginary weight singular vectors $\sum
I(-l)J(-k)...v_{0}\in{\bf V}_{0}^{-h^{\vee}}$, $I,J,...\in {\bf g}$.
Following [6], the action of $T(\zeta)$ on $H^{0}({\cal M}_{G}(X),
{\cal L}_{(\lambda_{i})})$ can be described as follows. We have an identification
$$
H^{0}({\cal M}_{G}(X), {\cal L}_{(\lambda_{i})})=\bar
H_{(\lambda_{i})}^{-h^{\vee}}=\{\mu\in
({\bf V}_{(\lambda_{i})}^{-h^{\vee}})^{*}|\mu {\rm \ is \
}\bar {\bf g}_{(z_{i})}{\rm -invariant}\},\leqno(8)
$$
where for any $k$, ${\bf V}^{k}_{(\lambda_{i})}$ is the induced
representation $\mathop{\rm Ind}_{\tilde {\bf g}_{+}}^{\tilde {\bf g}}
V^{k}_{(\lambda_{i})}$.
According to [6], 3, lemma 1,
$$
\bar H_{(\lambda_{i})}^{-h^{\vee}}\simeq (V_{\lambda_{1}}\otimes\cdots\otimes
V_{\lambda_{N}})^{*}.
\leqno(9)
$$
The field $T$ corresponds to an imaginary weight singular vector $\sum
I(-l)J(-k)...v_{0}\in{\bf V}_{0}^{-h^{\vee}}$, $I,J,...\in {\bf g}$. Due
to the ``swapping''
identity (3.1) of [6], the action of this singular vector on
$(V_{\lambda_{1}}\otimes\cdots\otimes V_{\lambda_{N}})^{*}$ is
$$
\sum{1\over{(l-1)!}}\pr^{l-1}I(u){1\over{(k-1)!}}\pr^{l-1}J(u)...,
\leqno(10)
$$
where $I(u)=\sum_{i=1}^{N}{{I^{(i)}}\over{u-z_{i}}}$.
For example, the operators corresponding to the degree two Casimir
element are the Gaudin Hamiltonians $H_{2,i}$, such that the combination
$H_{2}(\zeta)=\sum_{i=1}^{N}{H_{2,i}\over{\zeta-z_{i}}}$ satisfies
$$
H_{2}(\zeta)=\sum_{i} e_{i}(z)e_{i}(z),
\leqno(11)
$$
with $(e_{i})$ an orthomormal basis of ${\bf g}$.
\medskip\noindent
\it 3. The AKS scheme. \rm
On the other hand, the expression (1) gives a realization of the
Lie algebra $u^{-1}{\bf g}[[u^{-1}]]$. More precisely, we have a Lie
algebra morphism $\pi:u^{-1}{\bf g}[[u^{-1}]]\to {\bf g}^{\oplus N}$,
defined by
$\pi(Iu^{-k})=\sum_{i=1}^{N}I^{(i)}z_{i}^{k-1}$. Let us show how the AKS
scheme enables us to construct a commuting family in
$U(u^{-1}{\bf g}[[u^{-1}]])$. Let us decompose the central extension
${\bf C} K\to
\hat {\bf g}\to {\bf g}((u))$ as $\hat {\bf g}={\bf a}\oplus {\bf b}$,
${\bf a}=\sigma(u^{-1}{\bf g}[[u^{-1}]])$
and ${\bf b}=\alpha^{-1}({\bf g}[u])$, $\alpha$ being the projection and
$\sigma$ being a
section of $u^{-1}{\bf g}[[u^{-1}]]$ to $\hat {\bf g}$. Then, $U\hat
{\bf g}=U{\bf a}\oplus (U\hat {\bf g}){\bf b}$. We have then an algebra
morphism $Z(U\hat {\bf g})\to U{\bf a}$,
given by the projection to the first factor, whose image is a commuting
family in $U{\bf a}$. Let us specialize this construction to the critial
level. We have then a sequence of morphisms
$$
Z(U_{-h^{\vee}}\hat {\bf g})\to U(u^{-1}{\bf g}[[u^{-1}]])\to
(U{\bf g})^{\otimes N},
$$
the last one being given by $\pi$. This gives a family of commuting
differential operators on $(G/B)^{N}$.
\noindent
\it Remark. \rm According to [17], Gaudin systems can be obtained from
quantum tops systems by a reduction procedure, which explains that the
AKS scheme can be applied to them.
\medskip\noindent
\it 4. Coincidence of the AKS and FFR systems. \rm
To see that these operators are the same as those obtained by the
previous construction, let us work out the AKS scheme more
explicitly. The central field $T(u)$ associated to $\sum I(-l)J(-k)...$,
is the normally ordered product
$$
\sum{1\over{(l-1)!}}{1\over{(k-1)!}}...(\partial^{l-1}\bar
I(u)(\partial^{k-1}\bar J(u)...()))\leqno(12)
$$
(where the parenthesis stand for the normal ordering operation); here
$$
\bar I(u)=\sum_{n\in{\bf Z}}I(-n-1)u^{n}=I_{+}(u)+I_{-}(u),\leqno(13)
$$
$I_{+}(u)=\sum_{n\ge 0}I(-n-1)u^{n}$. The transform of this expression
by the AKS procedure will be, due to the conventions
$(AB)(u)=(A_{+}B+BA_{-})(u)$,
$$
\sum{1\over{(l-1)!}}{1\over{(k-1)!}}...(\partial^{l-1}
I_{+}(u)(\partial^{k-1}J_{+}(u)...())).
\leqno(14)
$$
But the image by $\pi$ of $I_{+}(u)$ is
$I(u)=\sum_{i=1}^{N}{{I^{(i)}}\over{u-z_{i}}}$; this shows
\proclaim{Proposition 3.1}
The expressions (10) and (14) for AKS and FFR Hamiltonians, coincide.
\endgroup\par\medbreak
\medskip\noindent
\it 5. Application: expression of the higher Gaudin operators in the
$sl_{n}$ case. \rm
The following can then be deduced from [10], using the Newton
identities.
\proclaim{Proposition 3.2} Let $(s_{p})_{p\ge 0}$ be the sequence of
polynomials in $n$, defined by $s_{1}=0$, $s_{2}=n/2$, $s_{3}=-2n/3$,
and for $p\ge 2$,
$$
(n-p)s_{p}-2(p+1)s_{p+1}-(p+2)s_{p+2}=0;\leqno(15)
$$
let $\lambda_{1},\cdots,\lambda_{n}$ be the solutions of the equation
$$
\lambda_{n}-s_{1}(n)\lambda^{n-1}+s_{2}(n)\lambda^{n-2}-s_{3}(n)\lambda^{n-3}\cdots=0,
\leqno(16)
$$
and let $H=\diag(\lambda_{1},\cdots,\lambda_{n})$. Let us set,
${}^{k}H={\rm Ad}}\def\Diff{{\rm Diff}}\def\Spec{{\rm Spec}(k)H$, for $k\in K=SU(n,{\bf C})$; and let $dk$ be a Haar
measure on $K$. Then the higher Gaudin
Hamiltonians are the operators $H_{l,a_{i}}$, ($\sum_{i=1}^{N}a_{i}=l-1$),
defined for each $l=2,...,N$ by
$$
\sum_{a_{1}+...+a_{N}=l-1}
{H_{l,a_{i}}\over{\prod_{i=1}^{N}(\zeta-z_{i})^{a_{i}}}}=\int_{K}\bigg(
{{({}^{k}H)^{(i)}}\over{\zeta-z_{i}}}\bigg)^{l}dk.\leqno(17)
$$
\endgroup\par\medbreak
\section{4.}{An $r$-matrix for the case of punctured elliptic curves.}
Let us turn now to the case where $\overline X$ is an elliptic curve
${\bf C}^{\times}/q^{{\bf Z}}$ ($q<1$). We denote by $z_{i}$ ($i=1,\cdots,N$)
the coordinates of the marked points. We fix from now on,
$G=GL_{n}({\bf C})$.
Consider the open subset ${\cal M}^{(0)}_{G}(\overline X)$ of
${\cal M}_{G}(\overline X)$, formed of the space of bundles on $\overline X$,
direct
sums of line bundles of degree $0$. These bundles are parametrized by
the symmetric product $\overline X^{(n)}$; to
$(t_{1},\cdots,t_{n})\in ({\bf C}^{\times})^{n}$, we associate the bundle
${\cal E}_{(t_{\alpha})}={\bf C}^{\times}\times{\bf C}^{n}/\{(z,\xi)\sim
(qz,\diag(t_{\alpha})\xi)\}$ over $X$; changing $(t_{\alpha})$ into
$(q^{a_{\alpha}}t_{\alpha})$ (with the $a_{\alpha}$ integers)
gives an isomorphic bundle, the isomorphism being $(z,\xi)\mapsto
(z,\diag(z^{a_{\alpha}})\xi)$.
Now, the preimage in ${\cal M}_{G}(X)$ of this open subset can be
identified with
$$
T\rtimes S_{n}\setminus ({\bf C}^{\times})^{n}\times
(G/B)^{N}/[(t_{\alpha},g_{i}B)
\sim (q^{a_{\alpha}}t_{\alpha},\diag(z_{i}^{a_{\alpha}})g_{i}B)],
$$
$T\rtimes S_{n}$ acting diagonally on $(G/B)^{N}$, and by permutations on
$({\bf C}^{\times})^{n}$. We denote it by ${\cal M}^{(0)}_{G}(X)$.
The cotangent to ${\cal M}^{(0)}_{G}(X)$ is now the quotient by
$S_{n}$ of the
reduction by $T$ of $T^{*}(({\bf C}^{\times})^{n}\times (G/B)^{N})$.
The Springer resolution gives now a mapping from
$T^{*}{\cal M}^{(0)}_{G}(X)$ to
$$
\eqalign{
T\rtimes
S_{n}\setminus\{(p_{\alpha},t_{\alpha},\eta_{i})
\in{\bf C}^{n}\times({\bf C}^{\times})^{n}
& \times{\cal N}^{N} |
(\sum_{i=1}^{N}\eta_{i})_{t}=0\}/
\parskip0pt\par\noindent}\noindent#1}}} &
/ \{(p_{\alpha},t_{\alpha},\eta_{i})\sim(p_{\alpha},
q^{a_{\alpha}}t_{\alpha}, {\rm Ad}}\def\Diff{{\rm Diff}}\def\Spec{{\rm Spec}(\diag(z_{i}^{a_{\alpha}}))\eta_{i})\},
\parskip0pt\par\noindent}\noindent#1}}}}$$
bijective over the open subset, defined by the condition that each
$\eta_{i}$ be regular. This map is
compatible with the Poisson bracket given by
the product of $\{p_{\alpha},t_{\beta}\}=\delta_{\alpha\beta}t_{\beta}$, and Kostant-Kirillov
on each copy of ${\cal N}$.
The corresponding $1$-form $\xi\in
H^{0}(X,gl({\cal E}_{(t_{\alpha})})(-\sum_{i=1}^{N}(z_{i})))$ can be seen as a
$1$-form $\tilde{\xi}$ on ${\bf C}^{\times}$ with values in $gl_{n}({\bf C})$,
with simple poles at $z_{i}q^{{\bf Z}}$, and such that
$\tilde{\xi}(qz)={\rm Ad}}\def\Diff{{\rm Diff}}\def\Spec{{\rm Spec}(t_{\alpha})\tilde{\xi}(z)$; it is given by
$\tilde{\xi}(z)=\bar{\xi}(z){{dz}\over z}$, with
$$
\bar{\xi}(z)_{\alpha\beta}=\sum_{i=1}^{N}\eta_{\alpha\beta}^{(i)}
{{\theta}\def\vare{\varepsilon(t_{\alpha}^{-1}t_{\beta}z z_{i}^{-1})}\over {\theta}\def\vare{\varepsilon(t_{\alpha}^{-1}t_{\beta})\theta}\def\vare{\varepsilon(z
z_{i}^{-1})}} {\rm \ if } \alpha\ne \beta,
\bar{\xi}(z)_{i}={1\over\theta}\def\vare{\varepsilon'(1)}p_{\alpha}+\sum_{i=1}^{N}{1\over\theta}\def\vare{\varepsilon'(1)}
{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(zz_{i}^{-1})\eta_{i}^{i}.
\leqno(18)
$$
Here $\theta}\def\vare{\varepsilon(z)=\prod_{i\ge 0}(1-q^{i}z)\prod_{i\ge 1}(1-q^{i}z^{-1})$;
$\theta}\def\vare{\varepsilon$ has the properties $\theta}\def\vare{\varepsilon(qz)=-z^{-1}\theta}\def\vare{\varepsilon(z)$,
$\theta}\def\vare{\varepsilon(z^{-1})=-z^{-1}\theta}\def\vare{\varepsilon(qz)$; we denote $\dot\theta}\def\vare{\varepsilon(z)=z{d\theta}\def\vare{\varepsilon\over dz}(z)$.
We will show that the commutativity of the coordinates of the
$\mathop{\rm tr}\limits\tilde{\xi}(z)^{k}$ in a basis of the space of $k$-forms
on $X-\{z_{i}\}$ can be deduced, as in prop. 2.1, from an $r$-matrix
argument:
\proclaim{Proposition 4.1}
Let $r(z,w,t_{\alpha})$ and $\rho(z,w,t_{\alpha})$ be the matrices acting on
${\bf C}^{n}\otimes{\bf C}^{n}$, with elements
$$
r(z,w,t_{\alpha})_{\alpha\beta}^{\gamma\delta}=\bigg(
-{\theta}\def\vare{\varepsilon(t_{\alpha}^{-1}t_{\beta}zw^{-1})\over {\theta}\def\vare{\varepsilon(t_{\alpha}^{-1}t_{\beta})\theta}\def\vare{\varepsilon(z
w^{-1})}}\delta_{\alpha\beta}^{\delta\gamma}
+{1\over\theta}\def\vare{\varepsilon'(1)}{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(zw^{-1})\delta_{\alpha\beta}^{\gamma\delta}
\bigg)(1-\delta_{\alpha\beta})\leqno(19)
$$
and
$$
\rho(z,w,t_{\alpha})_{\alpha\beta}^{\gamma\delta}={1\over\theta}\def\vare{\varepsilon'(1)}
{\theta}\def\vare{\varepsilon(t_{\alpha}^{-1}t_{\beta}zw^{-1})\over {\theta}\def\vare{\varepsilon(t_{\alpha}^{-1}t_{\beta})\theta}\def\vare{\varepsilon(z
w^{-1})}}
\bigg[
{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(t_{\alpha}t_{\beta}^{-1})+{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(t_{\alpha}^{-1}t_{\beta}
zw^{-1}) \bigg]
\delta_{\alpha\beta}^{\delta\gamma}(1-\delta_{\alpha\beta});\leqno(20)
$$
let $\bar{\xi}(z)$ be given by formula (18); let us endow the system of
variables $(p_{\alpha},t_{\alpha},\eta_{i})$ with the Poisson brackets, product
of $\{p_{\alpha},t_{\beta}\}=\delta_{\alpha\beta}t_{\beta}$ and Kostant-Kirillov on each copy of
${\cal N}$; then we have
$$
\eqalign{
\{\bar\xi(z,t_{\alpha})\otimes_{,}\bar\xi(w,t_{\alpha})\}=
[r(z,w,t_{\alpha}), & \bar\xi(z,t_{\alpha})\otimes 1+1\otimes \bar\xi (w,t_{\alpha})]
\parskip0pt\par\noindent}\noindent#1}}} & +\rho(z,w,t_{\alpha})[(\sum_{i=1}^{N}\eta_{i})_{t}\otimes 1
-1\otimes(\sum_{i=1}^{N}\eta_{i})_{t}].}\leqno(21)
$$
\endgroup\par\medbreak
\noindent
{\bf Proof.} In the case of the brackets $\{\bar\xi_{\alpha\beta},
\bar\xi_{\beta\gamma}\}$,
$\alpha,\beta,\gamma$ all different, it is a consequence of the formula
$$
{\theta}\def\vare{\varepsilon(tzw^{-1})\over{\theta}\def\vare{\varepsilon(t)\theta}\def\vare{\varepsilon(zw^{-1})}}
{\theta}\def\vare{\varepsilon(tt'w)\over{\theta}\def\vare{\varepsilon(tt')\theta}\def\vare{\varepsilon(w)}}
-{\theta}\def\vare{\varepsilon(t^{\prime -1}zw^{-1})\over{\theta}\def\vare{\varepsilon(t^{\prime -1})\theta}\def\vare{\varepsilon(zw^{-1})}}
{\theta}\def\vare{\varepsilon(tt'z)\over{\theta}\def\vare{\varepsilon(tt')\theta}\def\vare{\varepsilon(z)}} =
{\theta}\def\vare{\varepsilon(tz)\over{\theta}\def\vare{\varepsilon(t)\theta}\def\vare{\varepsilon(z)}}
{\theta}\def\vare{\varepsilon(t'w)\over{\theta}\def\vare{\varepsilon(t')\theta}\def\vare{\varepsilon(w)}};
$$
to show it, let $F(z,w,t,t')$ be the difference of both sides. We have
$F(qz,w,t,t')=t^{-1}F(z,w,t,t')$; moreover $F$ has no poles for $z\to 0$
or $z\to w$; since $t\notin q^{{\bf Z}}$, this shows $F=0$.
In the case of the brackets $\{\bar\xi_{\alpha\a},\bar\xi_{\alpha\beta}\}$, $\alpha\ne
\beta$,
it follows from
$$
\eqalign{
{1\over\theta}\def\vare{\varepsilon'(1)}t{d\over dt}[{\theta}\def\vare{\varepsilon(tw)\over{\theta}\def\vare{\varepsilon(t)\theta}\def\vare{\varepsilon(w)}}]
+{1\over\theta}\def\vare{\varepsilon'(1)}{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z){\theta}\def\vare{\varepsilon(tw)\over{\theta}\def\vare{\varepsilon(t)\theta}\def\vare{\varepsilon(w)}}
= &
-{\theta}\def\vare{\varepsilon(tzw^{-1})\over{\theta}\def\vare{\varepsilon(t)\theta}\def\vare{\varepsilon(zw^{-1})}}
{\theta}\def\vare{\varepsilon(tz)\over{\theta}\def\vare{\varepsilon(t)\theta}\def\vare{\varepsilon(z)}} \parskip0pt\par\noindent}\noindent#1}}}
&
+{1\over\theta}\def\vare{\varepsilon'(1)}{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(zw^{-1}){\theta}\def\vare{\varepsilon(tw)\over{\theta}\def\vare{\varepsilon(t)\theta}\def\vare{\varepsilon(w)}};
}$$
this equality is proven as follows; let $F(z,w)$ be the difference of
the two sides, then $F(qz,w)=F(z,w)$ and $F(z,w)$ has no poles for
$z\to w $ or $z\to 0$, which shows that $F(z,w)$ does not depend on $z$;
pose $F(z,w)=\varphi(w)$, then $\varphi(qw)=t\varphi(w)$ (this follows
from
${\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(qz)=-1+{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z)$, obtained by derivation
of the functional equation in $\theta}\def\vare{\varepsilon$); and $\varphi(z)$ has no poles
either, so $F(z,w)=0$.
In the case of the brackets $\{\bar\xi_{\alpha\beta},\bar\xi_{\beta\alpha}\}$, $\alpha\ne
\beta$, it follows from the fact, that if we pose
$$
F(z,w)= {\theta}\def\vare{\varepsilon(t^{-1}z)\over{\theta}\def\vare{\varepsilon(t^{-1})\theta}\def\vare{\varepsilon(z)}}{\theta}\def\vare{\varepsilon(tw)\over{\theta}\def\vare{\varepsilon(t)\theta}\def\vare{\varepsilon(w)}}
+{\theta}\def\vare{\varepsilon(t^{-1}zw^{-1})\over{\theta}\def\vare{\varepsilon(t^{-1})\theta}\def\vare{\varepsilon(zw^{-1})}}{1\over\theta}\def\vare{\varepsilon'(1)}
[{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z)-{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(w)],
$$
we have $F(z,w)=F(z\zeta,w\zeta)$ for any $\zeta\in{\bf C}^{\times}$.
Indeed, $F(qz,w)=tF(z,w)-{1\over\theta}\def\vare{\varepsilon'(1)}t
{\theta}\def\vare{\varepsilon(t^{-1}zw^{-1})\over{\theta}\def\vare{\varepsilon(t^{-1})\theta}\def\vare{\varepsilon(zw^{-1})}}$,
so with $\varphi(z,w)=F(z,w)-F(z\zeta,w\zeta)$, we have
$\varphi(qz,w)=t\varphi(z,w)$; as $F(z,w)$ has no poles in $z$ (or in
$w$), $\varphi$ has no poles either, and so it vanishes. So $F$ is only
a function of $zw^{-1}$, that we can evaluate when $w\to 1$; this
evaluation gives the matrix elements of $\rho$.
The brackets $\{\bar\xi_{\alpha\a},\bar\xi_{\beta\b}\}$ are all zero, and the
$[r,\bar\xi\otimes 1 + 1\otimes\bar\xi]_{\alpha\a}^{\beta\b}$ also; finally, the
brackets $\{\bar\xi_{\alpha\beta},\bar\xi_{\gamma\delta}\}$ ($\alpha,\beta,\gamma,\delta$ all
different) are
all zero, as well as the matrix elements
$[r,\bar\xi\otimes 1 + 1\otimes\bar\xi]_{\alpha\gamma}^{\beta\delta}$. \hfill $~\vrule height .9ex width .8ex depth -.1ex$
Now, after the reduction by $T$, the $\mathop{\rm tr}\limits\bar\xi(z)^{s}$ will be
in involution.
Let us give now the explicit form of the Hamiltonians generated by
$\mathop{\rm tr}\limits\bar\xi(z)^{2}$. We have
$$\eqalign{
\mathop{\rm tr}\limits\bar\xi(z)^{2} & =\sum_{\alpha=1}^{n}(p_{\alpha}+\sum_{i=1}^{N}\eta_{\alpha\a}^{(i)}
{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(zz_{i}^{-1}))^{2}
+2\sum_{1\le \alpha<\beta\le n}
\bigg(
\sum_{i=1}^{N}\eta_{\alpha\beta}^{(i)}{\theta}\def\vare{\varepsilon(t_{\alpha}t_{\beta}^{-1}zz_{i}^{-1})\over
\theta}\def\vare{\varepsilon(t_{\alpha}t_{\beta}^{-1})\theta}\def\vare{\varepsilon(zz_{i}^{-1})}
\bigg)
\cdot \parskip0pt\par\noindent}\noindent#1}}}
&\cdot
\bigg(
\sum_{i=1}^{N}\eta_{\beta\alpha}^{(i)}{\theta}\def\vare{\varepsilon(t_{\beta}t_{\alpha}^{-1}zz_{i}^{-1})\over
\theta}\def\vare{\varepsilon(t_{\beta}t_{\alpha}^{-1})\theta}\def\vare{\varepsilon(zz_{i}^{-1})}
\bigg);\parskip0pt\par\noindent}\noindent#1}}}}
$$
since
$$
\bar\xi(qz)={\rm Ad}}\def\Diff{{\rm Diff}}\def\Spec{{\rm Spec}(t_{\alpha})\bar\xi(z)-(\sum_{i=1}^{N}\eta^{(i)})_{t},
$$
we have
$$
(\mathop{\rm tr}\limits\bar\xi^{2})(qz)=(\mathop{\rm tr}\limits\bar\xi^{2})(z)+\mathop{\rm tr}\limits(\sum_{i=1}^{N}
\eta^{(i)})_{t}^{2}-2\sum_{i=1}^{N}{{\dot\theta}\over{\theta}}(zz_{i}^{-1})
\{
\sum_{\alpha=1}^{n}\eta_{\alpha\a}(\sum_{i}\eta_{\alpha\a}^{(i)})\},
$$
so that
$$
\mathop{\rm tr}\limits\bar\xi(z)^{2}=H_{0}+\sum_{i=1}^{N}H_{i}
{{\dot\theta}\over{\theta}}(zz_{i}^{-1})+\sum_{i=1}^{N}\bigg(
{{\dot\theta}\over{\theta}}(zz_{i}^{-1})\bigg)^{2}
\{\sum_{\alpha=1}^{n}\eta_{\alpha\a}(\sum_{i}\eta_{\alpha\a}^{(i)})\};
$$
using
$$
\eqalign{
({\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(zz_{i}^{-1})-{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(zz_{j}^{-1}))^{2}
& =\wp(\ln (zz_{i}^{-1}))
+\wp(\ln (zz_{j}^{-1})) \parskip0pt\par\noindent}\noindent#1}}}
&
-2{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z_{i}z_{j}^{-1})
[{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(zz_{i}^{-1})-{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(zz_{j}^{-1})]
+[{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z_{i}z_{j}^{-1})]^{2},
}
$$
$$\eqalign{
{\theta}\def\vare{\varepsilon(tzz_{i}^{-1})\over{\theta}\def\vare{\varepsilon(t)\theta}\def\vare{\varepsilon(zz_{i}^{-1})}}
{\theta}\def\vare{\varepsilon(t^{-1}zz_{j}^{-1})\over{\theta}\def\vare{\varepsilon(t^{-1})\theta}\def\vare{\varepsilon(zz_{j}^{-1})}}
& =[{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(zz_{i}^{-1})-{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(zz_{j}^{-1})]
{\theta}\def\vare{\varepsilon(t^{-1}z_{i}z_{j}^{-1})\over{\theta}\def\vare{\varepsilon(t^{-1})\theta}\def\vare{\varepsilon(z_{i}z_{j}^{-1})}} \parskip0pt\par\noindent}\noindent#1}}}
& -[{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(t^{-1})-{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(t^{-1}z_{i}z_{j}^{-1})]
{\theta}\def\vare{\varepsilon(t^{-1}z_{i}z_{j}^{-1})\over{\theta}\def\vare{\varepsilon(t^{-1})\theta}\def\vare{\varepsilon(z_{i}z_{j}^{-1})}}
{\rm\ if\ } j\ne i, \parskip0pt\par\noindent}\noindent#1}}}
& =\wp(\ln (zz_{i}^{-1}))-\wp(\ln (tz_{i})) {\rm\ \ else }
\parskip0pt\par\noindent}\noindent#1}}}}$$
[we set $z=e^{\tau}$, so near $\tau=0$,
${\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z)\sim{1\over\tau}$,
$\wp(\tau)\sim{1\over{\tau^{2}}}$], we find
$$\eqalign{
H_{i} & =2\sum_{\alpha=1}^{n}p_{\alpha}\eta_{\alpha\a}^{(i)}+2\sum_{\alpha=1}^{n}
\sum_{j\ne i}\eta_{\alpha\a}^{(i)}
\eta_{\alpha\a}^{(j)}
[{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z_{i}z_{j}^{-1})-{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z_{j}z_{i}^{-1})] \parskip0pt\par\noindent}\noindent#1}}}
& +2\sum_{\alpha\ne \beta}\sum_{j\ne i}\eta_{\alpha\beta}^{(i)}\eta_{\beta\alpha}^{(j)}
{\theta}\def\vare{\varepsilon(t_{\beta}t_{\alpha}^{-1}z_{i}z_{j}^{-1})\over{\theta}\def\vare{\varepsilon(t_{\beta}t_{\alpha}^{-1})}\theta}\def\vare{\varepsilon(z_{i}
z_{j}^{-1})} \parskip0pt\par\noindent}\noindent#1}}}
}\leqno(22)
$$
(a less symmetric form could be obtained using
${\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z)+{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z^{-1})=1$, and the irrelevance
of combinations of the
$\sum_{i=1}^{N}\eta_{\alpha\a}^{(i)}$), and
$$
\eqalign{
H_{0} & =\sum_{\alpha=1}^{n}p_{\alpha}^{2}+\sum_{j<i} \eta_{\alpha\a}^{(i)}
\eta_{\alpha\a}^{(j)}[{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z_{i}z_{j}^{-1})]^{2}
-2\sum_{\alpha<\beta}\sum_{i=1}^{N} \eta_{\alpha\beta}^{(i)}
\eta_{\beta\alpha}^{(i)}\wp(\ln(t_{\alpha}t_{\beta}^{-1})) \parskip0pt\par\noindent}\noindent#1}}}
&
-2\sum_{\alpha<\beta}\sum_{i\ne j}\eta_{\alpha\beta}^{(i)}\eta_{\beta\alpha}^{(j)}
[{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(t_{\beta}t_{\alpha}^{-1})-{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}
(t_{\beta}t_{\alpha}^{-1}z_{i}z_{j}^{-1})]
{\theta}\def\vare{\varepsilon(t_{\beta}t_{\alpha}^{-1}z_{i}z_{j}^{-1})\over{\theta}\def\vare{\varepsilon(t_{\beta}t_{\alpha}^{-1})
\theta}\def\vare{\varepsilon(z_{i}z_{j}^{-1})}}.
\parskip0pt\par\noindent}\noindent#1}}}}\leqno(23)
$$
\section{Remark.}{} It is interesting to compare these results with
those of [15], [5], [19], [11]. The system considered in these papers is
connected with the case $N=1$. Also there should be some connection
between the $r$-matrix (19) and the ones from [5] and [19].
\section{5.}{Gaudin-Calogero system in the $sl_{2}$ case.}
Let us see now how to construct a quantization of the system of the last
section when $G=GL_{2}$. We will construct differential operators on
$$
{\cal M}^{(0)}_{G}(X)=
T\rtimes S_{n}\setminus ({\bf C}^{\times})^{n}\times
(G/B)^{N}/[(t_{\alpha},g_{i}B)
\sim (q^{a_{\alpha}}t_{\alpha},\diag(z_{i}^{a_{\alpha}})g_{i}B)],
$$
whose symbols will be the Hitchin's Hamiltonians, $\mathop{\rm tr}\limits\bar\xi(z)^{s}$,
for $n=2$.
For this, we consider an integer $k$ and a system of dominant weights
$(\lambda_{i})_{i=1,\cdots,N}$, and the algebra
${\cal A}=\Diff(({\bf C}^{\times})^{n},{\cal L}_{k}^{\boxtimes n})\otimes
\otimes_{i=1}^{N}\Diff(G/B,{\cal L}_{\lambda_{i}})$ [here
${\cal L}_{k}=\pi^{*}{\cal O}(k(1))$, $\pi$ the projection ${\bf C}^{\times}\to X$,
$\Diff(X,{\cal L})=H^{0}(X,{\cal L}\otimes{\cal D}_{X}\otimes{\cal L}^{-1})$, for $X$ an
analytic variety and ${\cal L}$ a line bundle on $X$]. Let $(t_{\alpha})_{1\le
\alpha\le n}$ be the coordinates on $({\bf C}^{\times})^{n}$, and
$\hat{p}_{\alpha}=t_{\alpha}{\pr\over{\pr t_{\alpha}}}+k{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(t_{\alpha})$;
let again, $e_{\alpha\beta}^{(i)}$
denote the action of $e_{\alpha\beta}\in gl_{n}({\bf C})$ on the $i$-th factor of
the second part of ${\cal A}$. Consider now the matrix $L(z)\in
gl_{n}({\bf C})\otimes {\cal A}$, defined by
$$
\eqalign{
L(z)_{\alpha\beta}=\sum_{i=1}^{N} e_{\alpha\beta}^{(i)}
{{\theta}\def\vare{\varepsilon(t_{\alpha}^{-1}t_{\beta}z z_{i}^{-1})}\over {\theta}\def\vare{\varepsilon(t_{\alpha}^{-1}t_{\beta})\theta}\def\vare{\varepsilon(z
z_{i}^{-1})}} & {\rm \ if\ } \alpha\ne \beta, \parskip0pt\par\noindent}\noindent#1}}} &
L(z)_{\alpha\a}={1\over\theta}\def\vare{\varepsilon'(1)}\hat{p}_{\alpha}
+\sum_{i=1}^{N}{1\over\theta}\def\vare{\varepsilon'(1)}{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(zz_{i}^{-1})e_{\alpha\a}^{i}.
}\leqno(24)
$$
Let us perform now the reduction of ${\cal A}$ w.r.t. $T$. It can be done as
follows: let ${\cal A}[0]$ be the weight zero subalgebra of ${\cal A}$, ${\cal A}[0]=\{x\in
{\cal A}| [h_{\alpha\a},x]=0, 1\le \alpha \le n\}$ and ${\cal A}^{red}={\cal A}[0]/(h_{\alpha\a})_{1\le \alpha
\le n}$ (where $(h_{\alpha\a})_{1\le \alpha\le n}$ is the left, or right ideal
generated by the $h_{\alpha\a}$ in ${\cal A}[0]$). Then ${\cal A}^{red}$ is the algebra of
globally defined differential operators on $({\bf C}^{\times})^{n}\times
[T\setminus(G/B)^{N}]$, twisted by the quotient of ${\cal L}_{k}^{\boxtimes
n}\boxtimes\boxtimes_{i=1}^{N}
{\cal L}_{\lambda_{i}}$.
{}From $\mathop{\rm tr}\limits L(qz)^{2}=\mathop{\rm tr}\limits L(z)^{2}+\mathop{\rm tr}\limits(L(z)h+hL(z))+\mathop{\rm tr}\limits(h^{2})$, we see
that $\mathop{\rm tr}\limits L(z)^{s}$, $s=1,2$ define elements of $[{\cal A}/\sum_{\alpha=1}^{2}
{\cal A} h_{\alpha\a}]\otimes H^{0}(X,{\cal O}(s\sum_{i=1}^{N}(z_{i})))$, which also
belong to
${\cal A}^{red}\otimes H^{0}(X,{\cal O}(s\sum_{i=1}^{N}(z_{i})))$. Then
\proclaim{Proposition 5.1} The expansions of $\mathop{\rm tr}\limits L(z)^{s}$, $s=1,2$,
along bases of $ H^{0}(X,{\cal O}(s\break\sum_{i=1}^{N}(z_{i})))$, form a
commutative family in ${\cal A}^{red}$. These operators are $S_{2}$-invariant
and invariant under the action of ${\bf Z}^{2}$ defined by
$(a_{\alpha})\cdot
(t_{\alpha},g_{i}B)=(q^{a_{\alpha}}t_{\alpha},\diag(z_{i}^{a_{\alpha}})g_{i}B)$, and hence
define operators on
${\cal M}^{(0)}_{GL_{2}}(X)$, twisted by the line bundle associated
with $(k,\lambda_{i})$. Their symbols coincide with Hitchin's Hamiltonians.
\endgroup\par\medbreak
\noindent
{\bf Proof.} If $w\in S_{2}$, then $w^{*}L(z)=L(z)$; let
$\vare_{\alpha}$ be the $\alpha$-th basis vector of ${\bf Z}^{2}$, then
$\vare_{\alpha}^{*}L(z)={\rm Ad}}\def\Diff{{\rm Diff}}\def\Spec{{\rm Spec}(\diag(1,\cdots,z,\cdots,1))L(z)$ ($z$ in $\alpha$-th
position). The last statement follows from the fact that the symbol of
$\hat{p}_{\alpha}$ is $p_{\alpha}$, and the symbol of $e_{\alpha\beta}^{(i)}$ is
$\eta_{\alpha\beta}^{(i)}$. The first statement follows from a direct
computation, using the explicit form of the Hamiltonians:
$$
\eqalign{
\hat{H}_{i}
& = \hat{p}h^{(i)}
+2\sum_{j\ne i}h^{(i)}h^{(j)}
[{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z_{i}z_{j}^{-1})-{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z_{j}z_{i}^{-1})]
\parskip0pt\par\noindent}\noindent#1}}}
& +2\sum_{j\ne i} \big(e^{(i)}f^{(j)}
{\theta}\def\vare{\varepsilon(t^{2}z_{i}z_{j}^{-1})\over{\theta}\def\vare{\varepsilon(t^{2})}\theta}\def\vare{\varepsilon(z_{i}
z_{j}^{-1})}
+e^{(j)}f^{(i)}
{\theta}\def\vare{\varepsilon(t^{-2}z_{i}z_{j}^{-1})\over{\theta}\def\vare{\varepsilon(t^{-2})}\theta}\def\vare{\varepsilon(z_{i}
z_{j}^{-1})}\big)
}\leqno(25)
$$
and
$$
\eqalign{
\hat{H}_{0} & =\hat{p}^{2}+\sum_{j<i} h^{(i)}
h^{(j)}[{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(z_{i}z_{j}^{-1})]^{2}
-2\sum_{i=1}^{N}e^{(i)}f^{(i)}\wp(\ln(t^{2})) \parskip0pt\par\noindent}\noindent#1}}}
&
-2\sum_{i\ne j}e^{(i)}f^{(j)}
[{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(t^{2})-{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}
(t^{2}z_{i}z_{j}^{-1})]
{\theta}\def\vare{\varepsilon(t^{2}z_{i}z_{j}^{-1})\over{\theta}\def\vare{\varepsilon(t^{2})
\theta}\def\vare{\varepsilon(z_{i}z_{j}^{-1})}}.
\parskip0pt\par\noindent}\noindent#1}}}}\leqno(26)
$$
\hfill $~\vrule height .9ex width .8ex depth -.1ex$
Equations (25) and (26) define differential operators acting on
${\bf C}^{\times}\times
[T\setminus ({\bf C} P^{1})^{N}]$; $(t,t_{i})$
being the product coordinates on
${\bf C}^{\times}\times ({\bf C} P^{1})^{N}$, we have
$\hat{p}=2t{\pr\over{\pr t}}+2k{\dot\theta}\def\vare{\varepsilon\over\theta}\def\vare{\varepsilon}(t^{2})$,
$h^{(i)}=2(t_{i}{\pr\over{\pr t_{i}}}+\lambda_{i})$,
$e^{(i)}=t_{i}^{2}{\pr\over{\pr t_{i}}}+2\lambda_{i}t_{i}$,
$f^{(i)}=-{\pr\over{\pr t_{i}}}$.
For $N=1$, this system is reduced to
$$
\hat{H}_{0}=\hat{p}^{2}-2e^{(1)}f^{(1)}\wp(\ln t^{2}) ,
\quad\hat{H}_{1}=e^{(1)}f^{(1)}.
$$
\section{Remarks.}{}
1. A natural module for ${\cal A}^{red}$ is ${\rm Fun}({\bf C}^{\times})\otimes
(V_{\lambda_{1}}\otimes\cdots\otimes V_{\lambda_{N}})[0]$.
More precisely, we can pose the eigenvalue problem
$\hat{H}_{i}\psi=\mu_{i}\psi$, $\hat{H}_{0}\psi=\mu_{0}\psi$, $\psi$ a
function of ${\bf C}^{\times}$, with values in
$\otimes_{i=1}^{N}V_{\lambda_{i}}$, whose component in
$(\otimes_{i=1}^{N}V_{\lambda_{i}})[\bar\lambda_{i}]$, $\psi_{\bar\lambda_{i}}(t)$,
satisfies $\psi_{\bar\lambda_{i}}(qt)=z_{1}^{\bar\lambda_{1}}\cdots
z_{N}^{\bar\lambda_{N}}z^{\ell}
\psi_{\bar\lambda_{i}}(t)$, for each system of weights $(\bar\lambda_{i})$,
$\ell$ being a fixed integer.
The space of such functions, with only poles at
$q^{{\bf Z}}$, is stable under the actions of $\hat{H}_{0}$ and the
$\hat{H}_{i}$.
2. Prop. 5.1 suggests that the operators constructed here coincide with
the result of the action of the center of the enveloping algebra at the
critical level, when $k=2$. Indeed in this case, after [22], the
quotient of ${\cal L}_{k}^{\boxtimes 2}$ by $S_{2}$ coincides with
$(\det_{|{\cal M}^{(0)}_{GL_{2}}(X)})^{-2}$, on which this center should act.
After obtaining the main results of this paper, we learnt about the
paper of N. Nekrasov [16], where Hitchin systems for degenerate curves
are described as many-body problems.
\vskip 1truecm
\noindent
{\bf References}
\bigskip
\item{[1]} M.R. Adams, J. Harnad, E. Previato, {\sl Isospectral
Hamiltonian flows in finite and infinite dimensions II. Integration of
flows,} Commun. Math. Phys. 134 (1990), 555-85.
\medskip
\item{[2]} M. Adler, {\sl On a trace functional for formal
pseudo-differential operators and the symplectic structure for the KdV
type equations,} Invent. Math. 50 (1979), 219-48.
\medskip
\item{[3]} A. Beauville, {\sl Jacobiennes des courbes spectrales et
syst\`emes compl\`etement int\'e-grables,} Acta Math., 169 (1990),
211-35.
\medskip
\item{[4]} A.A. Beilinson, V.G. Drinfeld, {\sl Quantization of
Hitchin's fibration and Langlands program,} preprint.
\medskip
\item{[5]} H.W. Braden, T. Suzuki, {\sl $R$-matrices for Elliptic
Calogero-Moser Models}, Lett. Math. Phys. 30, 147-59 (1994).
\medskip
\item{[6]} B.L. Feigin, E.V. Frenkel, N. Reshetikhin, {\sl Gaudin
model, Bethe ansatz and correlation functions at the critical level,}
Commun. Math. Phys. 166 (1), 27-62 (1995).
\medskip
\item{[7]} G. Felder, C. Wieczerkowski, {\sl Conformal field theory on
elliptic curves and Knizhnik-Zamolodchikov-Bernard equations,}
hep-th/9411004.
\medskip
\item{[8]} R. Garnier, Rend. Circ. Mat. Palermo 43, 155-91 (1919).
\medskip
\item{[9]} M. Gaudin, Jour. Physique 37 (1976), 1087-1098.
\medskip
\item{[10]} R. Goodman, N.R. Wallach, {\sl Higher-order Sugawara operators
for affine Lie algebras,} Trans. AMS, 315:1 (1989), 1-55.
\medskip
\item{[11]} A.S. Gorsky, N.A. Nekrasov,{\sl Elliptic Calogero-Moser
system from two-dimensio-nal current algebra,} hep-th/9401021.
\medskip
\item{[12]} N. Hayashi, {\sl Sugawara operators and Kac-Kazhdan
conjecture,} Invent. Math. 54 (1988), 13-52.
\medskip
\item{[13]} N. Hitchin, {\sl Stable bundles and integrable systems,} Duke
Math. Jour., 54 (1), 91-114 (1987).
\medskip
\item{[14]} B. Kostant, {\sl The solution to a generalized Toda lattice
and representation theory,} Adv. Math. 34 (1980), 13-53.
\medskip
\item{[15]} I.M. Krichever, {\sl Elliptic solutions of the
Kadomtsev-Petviashvili equation and integrable systems of particles,}
Funct. An. Appl., 14 (1), 282-90 (1990).
\medskip
\item{[16]} N. Nekrasov, {\sl Holomorphic bundles and many-body systems,}
PUPT-1534, ITEP-N95/1, hep-th/9503157.
\medskip
\item{[17]} A.G. Reyman, {\sl Quantum tops,} Int. J. Mod. Phys. B,
7:20-21 (1993), 3707-13.
\medskip
\item{[18]} M.A. Semenov-Tian-Shansky, {\sl D. Sci. thesis,} LOMI,
Leningrad (1985).
\medskip
\item{[19]} E.K. Sklyanin, {\sl Dynamical $r$-matrices for the elliptic
Calogero-Moser system,} LPTHE 93-42, hep-th/9308060.
\medskip
\item{[20]} T.A. Springer, {\sl Trigonometric sums, Green functions of
finite groups and representations of Weyl groups,}
Inv. Math. 36, 173-207 (1976).
\medskip
\item{[21]} W. Symes, {\sl Systems of Toda type, inverse spectral
problems and representation theory,} Invent. Math. 59 (1990),
195-338.
\medskip
\item{[22]} L.W. Tu, {\sl Semistable bundles over an Elliptic Curve,}
Adv. Math. 98, 1-26 (1993).
\medskip
\medskip\medskip
\section{}{}
B.E., V.R.: Centre de Math\'{e}matiques, URA 169
du CNRS, Ecole Polytechnique, 91128 Palaiseau, France
V.R.: ITEP, Bol. Cheremushkinskaya, 25, 117259,
Moscow, Russia.
\bye
|
1995-08-08T07:51:28 | 9503 | alg-geom/9503021 | en | https://arxiv.org/abs/alg-geom/9503021 | [
"alg-geom",
"math.AG"
] | alg-geom/9503021 | Nitin Nitsure | Nitin Nitsure and Claude Sabbah | Moduli of pre-$\cal D$-modules, perverse sheaves and the Riemann-Hilbert
morphism -I | LaTeX, 28 pages | null | null | null | null | We construct a moduli scheme for semistable pre-$\D$-modules with prescribed
singularities and numerical data on a smooth projective variety. These
pre-$\D$-modules are to be viewed as regular holonomic $\D$-modules with `level
structure'. We also construct a moduli scheme for perverse sheaves on the
variety with prescribed singularities and other numerical data, and represent
the de Rham functor (which gives the Riemann-Hilbert correspondence) by an
analytic morphism between the two moduli schemes.
| [
{
"version": "v1",
"created": "Tue, 28 Mar 1995 11:35:00 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Nitsure",
"Nitin",
""
],
[
"Sabbah",
"Claude",
""
]
] | alg-geom | \subsection*{\hbox{}\hfill{\normalsize\sl #1}\hfill\hbox{}}}
\textheight 23truecm \textwidth 15truecm
\addtolength{\oddsidemargin}{-1.05truecm}
\addtolength{\topmargin}{-1.5truecm}
\makeatletter \def\l@section{\@dottedtocline{1}{0em}{1.2em}} \makeatother
\begin{document}
\title{Moduli of pre-${\cal D}$-modules, perverse sheaves\\ and the
Riemann-Hilbert morphism -I}
\author{Nitin Nitsure\thanks{Tata Institute of Fundamental
Research, Bombay} \and Claude Sabbah\thanks{CNRS, URA D0169,
Ecole Polytechnique, Palaiseau}} \date{March 28, 1995}
\maketitle
\begin{abstract} We construct a moduli scheme for semistable
pre-${\cal D}$-modules with prescribed singularities and numerical
data on a smooth projective variety. These pre-${\cal D}$-modules are
to be viewed as regular holonomic ${\cal D}$-modules with `level
structure'. We also construct a moduli scheme for perverse
sheaves on the variety with prescribed singularities and other
numerical data, and represent the de Rham functor (which gives
the Riemann-Hilbert correspondence) by an analytic morphism
between the two moduli schemes.
\end{abstract}
\vfill
\tableofcontents
\vfill
\newpage
\section{Introduction} This paper is devoted to the moduli
problem for regular holonomic ${\cal D}$-modules and perverse sheaves
on a complex projective variety $X$. It treats the case where
the singular locus of the ${\cal D}$-module is a smooth divisor $S$
and the characteristic variety is contained in the union of the
zero section $T^*_XX$ of the cotangent bundle of $X$ and the
conormal bundle $N^*_{S,X}$ of $S$ in $X$ (also denoted
$T_S^*X$). The sequel (part II) will treat the general case of
arbitrary singularities.
A moduli space for ${\cal O}$-coherent ${\cal D}$-modules on a smooth
projective variety was constructed by Simpson [S]. These are
vector bundles with integrable connections, and they are the
simplest case of ${\cal D}$-modules. In this moduli construction, the
requirement of semistability is automatically fulfilled by all
the objects.
Next in order of complexity are the so called `regular
meromorphic connections'. These ${\cal D}$-modules can be generated by
vector bundles with connections which have logarithmic
singularities on divisors with normal crossing. These
${\cal D}$-modules are not ${\cal O}$-coherent, but are torsion free as
${\cal O}$-modules. A moduli scheme does not exist for these
${\cal D}$-modules themselves (see section 1 of [N]), but it is
possible to define a notion of stability and construct a moduli
for vector bundles with logarithmic connections. This was done
in [N]. Though many logarithmic connections give rise to the
same meromorphic connection, the choice of a logarithmic
connection is infinitesimally rigid if its residual eigenvalues
do not differ by nonzero integers (see section 5 of [N]).
In the present paper and its sequel, we deal with general
regular holonomic ${\cal D}$-modules. Such modules are in general
neither ${\cal O}$-coherent, nor ${\cal O}$-torsion free or pure
dimensional. We define objects called pre-${\cal D}$-modules, which
play the same role for regular holonomic ${\cal D}$-modules that
logarithmic connections played for regular meromorphic
connections. We define a notion of (semi-)stability, and
construct a moduli scheme for (semi-) stable pre-${\cal D}$-modules
with prescribed singularity stratification and other numerical
data. We also construct a moduli scheme for perverse sheaves
with prescribed singularity stratification and other numerical
data on a nonsingular variety, and show that the Riemann-Hilbert
correspondence defines an analytic morphism between (an open set
of) the moduli of pre-${\cal D}$-modules and the moduli of perverse
sheaves.
The contents of this paper are as follows. Let $X$ be a smooth
projective variety, and let $S$ be a smooth hypersurface on $X$.
In section 2, we define pre-${\cal D}$-modules on $(X,S)$ which may be
regarded as ${\cal O}_X$-coherent descriptions of those regular
holonomic ${\cal D}_X$-modules whose characteristic variety is
contained in $T^*_XX\cup T^*_SX$. The pre-${\cal D}$-modules form an
algebraic stack in the sense of Artin, which is a property that
does not hold for the corresponding ${\cal D}$-modules.
In section 3, we define a functor from the pre-${\cal D}$-modules to
${\cal D}$-modules (in fact we mainly use the presentation of
holonomic ${\cal D}$-modules given by Malgrange [Mal], that we call
Malgrange objects). This is a surjective functor, and though not
injective, it has an infinitesimal rigidity property (see
proposition \ref{prop4}) which generalizes the corresponding
result for meromorphic connections.
In section 4, we introduce a notion of (semi-)stability for
pre-${\cal D}$-modules, and show that semistable pre-${\cal D}$-modules with
fixed numerical data form a moduli scheme.
Next, we consider perverse sheaves on $X$ which are
constructible with respect to the stratification $(X-S)\cup S$.
These have finite descriptions through the work Verdier,
recalled in section 5.
We observe that these finite descriptions are objects which
naturally form an Artin algebraic stack. Moreover, we show in
section 6 that S-equivalence classes (Jordan-H\"older classes)
of finite descriptions with given numerical data form a coarse
moduli space which is an affine scheme. Here, no hypothesis
about stability is necessary.
In section 7, we consider the Riemann-Hilbert correspondence.
When a pre-${\cal D}$-module has an underlying logarithmic connection
for which eigenvalues do not differ by nonzero integers,
we functorially associate to it a finite description, which is
the finite description of the perverse sheaf associated to the
corresponding ${\cal D}$-module by the Riemann-Hilbert correspondence
from regular holonomic ${\cal D}$-modules to perverse sheaves. We show
that this gives an analytic morphism of stacks from the analytic
open subset of the stack (or moduli) of pre-${\cal D}$-modules on
$(X,S)$ where the `residual eigenvalues are good', to the stack
(or moduli) of finite descriptions on $(X,S)$.
In section 8, we show that the above morphism of analytic stacks
is in fact a spread (surjective local isomorphism) in the
analytic category. We also show that it has removable
singularities in codimension 1, that is, is can be defined
outside codimension two on any parameter space which is smooth
in codimension 1.
\paragraph{Acknowledgement} The authors thank the exchange
programme in mathematics of the Indo-French Center for the
Promotion of Advanced Research, New Delhi, the Ecole
Polytechnique, Paris, and the Tata Institute of Fundamental
Research, Bombay, for supporting their collaboration. The first
author also thanks ICTP Trieste and the University of
Kaiserslautern for their hospitality while part of this work was
done.
\section{Pre-${\cal D}$-modules}
Let $X$ be a nonsingular variety and let $S\subset X$ be a
smooth divisor (reduced). Let ${\cal I}_S\subset {\cal O}_X$ be the ideal
sheaf of $S$, and let $T_X[\log S]\subset T_X$ be the sheaf of
all tangent vector fields on $X$ which preserve ${\cal I}_S$. Let
${\cal D}_X[\log S]\subset {\cal D}_X$ be the algebra of all partial
differential operators which preserve $I_S$; it is generated as
an ${\cal O}_X$ algebra by $T_X[\log S]$.
The ${\cal I}_S$-adic filtration on ${\cal O}_X$ gives rise to a
(decreasing) filtration of ${\cal D}_X$ as follows: for $k\inZ\!\!\!Z$
define $V^k{\cal D}_X$ as the subsheaf of ${\cal D}_X$ whose local sections
consist of operators $P$ which satisfy $P\cdot {\cal I}_S^j\subset
{\cal I}_S^{k+j}$ for all $j$. By construction, one has ${\cal D}_X[\log
S]=V^0{\cal D}_X$ and every $V^k({\cal D}_X)$ is a coherent ${\cal D}_X[\log
S]$-module.
Let $p:N_{S,X}\to S$ denote the normal bundle of $S$ in $X$. The
graded ring $\mathop{\rm gr}\nolimits_V{\cal D}_X$ is naturally identified with
$p_*{\cal D}_{N_{S,X}}$. Its $V$-filtration (corresponding to the
inclusion of $S$ in $N_{S,X}$ as the zero section) is then
split.
There exists a canonical section $\theta$ of the quotient ring
${\cal D}_X[\log S]/{\cal I}_S{\cal D}_X[\log S]=\mathop{\rm gr}\nolimits^0_V{\cal D}_X$, which is locally
induced by $x\partial_x$, where $x$ is a local equation for $S$.
It is a central element in $\mathop{\rm gr}\nolimits^0_V{\cal D}_X$. This ring contains
${\cal O}_S$ as a subring and ${\cal D}_S$ as a quotient (one has
${\cal D}_S=\mathop{\rm gr}\nolimits^0_V{\cal D}_X/\theta\mathop{\rm gr}\nolimits^0_V{\cal D}_X$). One can identify locally
on $S$ the ring $\mathop{\rm gr}\nolimits^0_V{\cal D}_X$ with ${\cal D}_S[\theta ]$.
A coherent $\mathop{\rm gr}\nolimits^0_V{\cal D}_X$-module on which $\theta$ acts by $0$ is
a coherent ${\cal D}_S$-module. The locally free rank one
${\cal O}_S$-module ${\cal N}_{S,X}={\cal O}_X(S)/{\cal O}_X$ is a $\mathop{\rm gr}\nolimits^0_V{\cal D}_X$-module
on which $\theta$ acts by $-1$.
\begin{definition}\rm
A {\sl logarithmic module} on $(X,S)$ will mean a sheaf of
${\cal D}_X[\log S]$-modules, which is coherent as an ${\cal O}_X$-module. A
{\sl logarithmic connection} on $(X,S)$ will mean a logarithmic
module which is coherent and torsion-free as an ${\cal O}_X$-module.
\end{definition}
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
It is known that when $S$ is nonsingular, any logarithmic
connection on $(X,S)$ is locally free as an ${\cal O}_X$-module.
\begin{definition}[Family of logarithmic modules]\rm
Let $f:Z\to T$ be a smooth morphism of schemes. Let $Y\subset Z$
be a divisor such that $Y\to T$ is smooth. Let $T_{Z/T}[\log
Y]\subset T_{X/Y}$ be the sheaf of germs of vertical vector
fields which preserve the ideal sheaf of $Y$ in ${\cal O}_Z$. This
generates the algebra ${\cal D}_{Z/T}[\log Y]$. A family of
logarithmic modules on $Z/T$ is a ${\cal D}_{Z/T}[\log Y]$-module
which is coherent as an ${\cal O}_Z$-module, and is flat over ${\cal O}_T$.
When $f:Z\to T$ is the projection $X\times T\to T$, and
$Y=S\times T$, we get a {\sl family of logarithmic modules on
$(X,S)$ parametrized by $T$}.
\end{definition}
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
The restriction to $S$ of a logarithmic module is acted on by
$\theta$: for a logarithmic connection, this is the action of
the residue of the connection, which is an ${\cal O}_S$-linear
morphism.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
There is an equivalence (restriction to $S$) between logarithmic
modules supported on the reduced scheme $S$ and
$\mathop{\rm gr}\nolimits^0_V{\cal D}_X$-modules which are ${\cal O}_S$-coherent, (hence locally
free ${\cal O}_S$-modules, since they are locally ${\cal D}_S$-modules). In
the following, we shall not make any difference between the
corresponding objects.
\bigskip We give two definitions of pre-${\cal D}$-modules. The two
definitions are `equivalent' in the sense that they give not
only equivalent objects, but also equivalent families, or more
precisely, the two definitions give rise to isomorphic algebraic
stacks. To give a familier example of such an equivalence, this
is the way how vector bundles and locally free sheaves are
`equivalent'. Note also that mere equivalence of objects is not
enough to give equivalence of families --- for example, the
category of flat vector bundles is equivalent to the category of
$\pi_1$ representations, but an algebraic family of flat bundles
gives in general only a holomorphic (not algebraic) family of
$\pi_1$ representations.
In their first version, pre-${\cal D}$-modules are objects that live
on $X$, and the functor from pre-${\cal D}$-modules to ${\cal D}$-modules
has a direct description in their terms. The second version of
pre-${\cal D}$-modules is more closely related to the Malgrange
description of ${\cal D}$-modules and the Verdier description of
perverse sheaves, and the Riemann-Hilbert morphism to the stack
of perverse sheaves has direct description in its terms.
\begin{definition}[Pre-${\cal D}$-module of first kind on $(X,S)$]\rm
Let $X$ be a nonsingular variety, and $S\subset X$ a smooth
divisor. A pre-${\cal D}$-module ${\bf E} = (E,F,t,s)$ on $(X,S)$ consists
of the following data
(1) $E$ is a logarithmic connection on $(X,S)$.
(2) $F$ is a logarithmic module on $(X,S)$ supported on the
reduced scheme $S$ (hence a flat connection on $S$).
(3) $t:(E\vert S) \to F$ and $s:F \to (E\vert S)$ are ${\cal D}_X[\log
S]$ linear maps,
which satisfies the following conditions:
(4) On $E\vert S$, we have $st = R$ where $R\in End(E\vert S)$
is the residue of $E$.
(5) On $F$, we have $ts = \theta_F$ where $\theta_F:F\to F$ is
the ${\cal D}_X[\log S]$ linear endomorphism induced by any Eulerian
vector field $x\partial /\partial x$.
\end{definition}
If $(E,F,t,s)$ and $(E',F',t',s')$ are two pre-${\cal D}$-modules, a
morphism between them consists of ${\cal D}_X[\log S]$ linear
morphisms $f_0:E\to E'$ and $f_1:F\to F'$ which commute with
$t,t'$ and with $s,s'$.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
It follows from the definition of a pre-${\cal D}$-module $(E,F,t,s)$
that $E$ and $F$ are locally free on $X$ and $S$ respectively,
and the vector bundle morphisms $R$, $s$ and $t$ all have
constant ranks on irreducible components of $S$.
\paragraph{Example} Let $E$ be a logarithmic connection on
$(X,S)$. We can associate functorially to $E$ the following
three pre-${\cal D}$-modules. Take $F_1$ to be the restriction of $E$
to $S$ as an ${\cal O}$-module. Let $t_1 = R$ (the residue) and $s_1 =
1_F$. Then ${\bf E}_1=(E,F_1,t_1,s_1)$ is a pre-${\cal D}$-module, which
under the functor from pre-${\cal D}$-modules to ${\cal D}$-modules defined
later will give rise to the meromorphic connection corresponding
to $E$. For another choice, take $F_2 = E\vert S$, $t_2=1_F$ and
$s_2=R$. This gives a pre-${\cal D}$-module ${\bf E}_2 = (E,F_2,t_2,s_2)$
which will give rise to a ${\cal D}$-module which has nonzero torsion
part when $R$ is not invertible. For the third choice (which is
in some precise sense the minimal choice), take $F_3$ to be the
image vector bundle of $R$. Let $t_3 =R:(E\vert S)\to F_3$, and
let $s_3:F_3\hookrightarrow (E\vert S)$. This gives a
pre-${\cal D}$-module ${\bf E}_3 = (E,F_3,t_3,s_3)$. We have functorial
morphisms ${\bf E}_3\to {\bf E}_2 \to {\bf E}_1$ of pre-${\cal D}$-modules.
\begin{definition}[Families of pre-${\cal D}$-modules]\rm
Let $T$ be a complex scheme. A family ${\bf E}_T$ of
pre-${\cal D}$-modules on $(X,S)$ parametrized by the scheme $T$, a
morphism between two such families, and pullback of a family
under a base change $T'\to T$ have obvious definitions (starting
from definition of families of ${\cal D}_X[\log S]$-modules), which we
leave to the reader. This gives us a fibered category $PD$ of
pre-${\cal D}$-modules over the base category of $C\!\!\!\!I$ schemes. Let
$\cal PD$ be the largest (nonfull) subcategory of $PD$ in which
all morphisms are isomorphisms. This is a groupoid over $C\!\!\!\!I$
schemes.
\end{definition}
\begin{proposition}
The groupoid $\cal PD$ is an algebraic stack in the sense of
Artin.
\end{proposition}
\paragraph{Proof} It can be directly checked that $\cal PD$ is a sheaf,
that is, descent and effective descent are valid for faithfully
flat morphisms of parameter schemes of families of
pre-${\cal D}$-modules. Let $Bun_X$ be the stack of vector bundles on
$X$, and let $Bun_S$ be the stack of vector bundles on $S$. Then
$\cal PD$ has a forgetful morphism to the product stack
$Bun_X\times_{C\!\!\!\!I} Bun_S$. The later stack is algebraic and the
forgetful morphism is representable, hence the desired
conclusion follows.
\bigskip
Before giving the definition of pre-${\cal D}$-modules of the second
kind, we observe the following.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{rem1} Let $N$ be any line bundle on a smooth variety
$S$, and let $\ov{N} = P(N\oplus {\cal O}_S)$ be its projective
completion, with projection $\pi : \ov{N} \to S$. Let
$S^{\infty} = P(N)$ be the divisor at infinity. For any
logarithmic connection $E$ on $(\ov{N} ,S\cup S^{\infty})$, the
restriction $E\vert S$ is of course a ${\cal D}_{\ov{N}}[\log S\cup
S^{\infty}]$-module. But conversely, for any ${\cal O} $-coherent
${\cal D}_{\ov{N}}[\log S\cup S^{\infty}]$-module $F$ scheme
theoretically supported on $S$, there is a natural structure of
a logarithmic connection on $(\ov{N} ,S\cup S^{\infty})$ on its
pullup $\pi ^*(F)$ to $\ov{N}$. The above correspondence is well
behaved in families, giving an isomorphism between the algebraic
stack of ${\cal D}_{\ov{N}}[\log S\cup S^{\infty}]$-modules $F$
supported on $S$ and the algebraic stack of logarithmic
connections $E$ on $(\ov{N} ,S\cup S^{\infty})$ such that the
vector bundle $E$ is trivial on the fibers of $\pi :\ov{N} \to
S$. The functors $\pi ^*(-)$ and $(-)\vert S$ are fully
faithful.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{rem2} Let $S\subset X$ be a smooth divisor, and let
$N=N_{S,X}$ be its normal bundle. Then the following are
equivalent in the sense that we have fully faithful functors
between the corresponding categories, which give naturally
isomorphic stacks.
(1) ${\cal D}_X[\log S]$-modules which are scheme theoretically
supported on $S$.
(2) ${\cal D}_N[\log S]$-modules which are scheme theoretically
supported on $S$.
(3) ${\cal D}_{\ov{N}}[\log S \cup S^{\infty}]$-modules which are
scheme theoretically supported on $S$.
The equivalence between (2) and (3) is obvious, while the
equivalence between (1) and (2) is obtained as follows. The
Poincar\'e residue map $res:\Omega ^1_X[\log S] \to {\cal O}_S$ gives
the following short exact sequence of ${\cal O}_S$-modules. $$0\to
\Omega ^1_S \to \Omega ^1_X[\log S]|S \to {\cal O}_S\to 0$$ By taking
duals, this gives $$0 \to {\cal O}_S \to T_X[\log S]|S \to T_S\to 0.$$
It can be shown that there exists a unique isomorphism $T_X[\log
S]\vert S \to T_N[\log S]\vert S$ which makes the following
diagram commute, where the rows are exact.
$$\matrix{ 0 & \to & {\cal O}_S & \to & T_N[\log S]|S & \to & T_S &
\to & 0 \cr & & \Vert & & \downarrow & & \Vert & & \cr 0 & \to
& {\cal O}_S & \to & T_X[\log S]|S & \to & T_S & \to & 0 \cr }$$
\refstepcounter{theorem}\paragraph{Remarks \thetheorem}
(1) Observe that the element $\theta$ is just the image of $1$
under the map ${\cal O}_S \to T_X[\log S]\vert S$.
(2) Using the notations of the beginning of this section, one
can identify the ring $\pi_*{\cal D}_{\ov{N}}[\log S \cup S^{\infty}]$
with $\mathop{\rm gr}\nolimits^0_V{\cal D}_X$. Hence $\theta$ is a global section of
${\cal D}_{\ov{N}}[\log S \cup S^{\infty}]$.
\bigskip
We now make the following important definition.
\begin{definition}[Specialization of a logarithmic module]\rm
Let $E$ be a logarithmic module on $(X,S)$. Then the
specialization $\mathop{\rm sp}\nolimits_SE$ will mean the logarithmic connection
$\pi ^*(E\vert S)$ on $(\ov{N_{S,X}} , S\cup S^{\infty})$.
\end{definition}
Now we are ready to define the second version of
pre-${\cal D}$-modules.
\begin{definition}[Pre-${\cal D}$-modules of the second kind on
$(X,S)$]\label{def1}\rm A pre-${\cal D}$-mo\-dule (of the second kind)
${\bf E} = (E_0,E_1,c,v)$ on $(X,S)$ consists of the following data
(1) $E_0$ is a logarithmic connection on $(X,S)$,
(2) $E_1$ is a logarithmic connection on $(\ov{N_{S,X}},S\cup
S^\infty)$,
(3) $c:\mathop{\rm sp}\nolimits_SE_0 \to E_1$ and $v:E_1 \to \mathop{\rm sp}\nolimits_SE_0$ are
${\cal D}_{\ov{N_{S,X}}}[\log S\cup S^\infty]$-linear maps,
which satisfies the following conditions:
(4) on $\mathop{\rm sp}\nolimits_SE_0$, we have $cv = \theta_{\mathop{\rm sp}\nolimits_SE_0}$,
(5) on $E_1$, we have $vc = \theta_{E_1}$,
(6) the vector bundle underlying $E_1$ is {\sl trivial} in the
fibers of $\pi:\ov{N_{S,X}}\to S$ (that is, $E_1$ is locally
over $S$ isomorphic to $\pi^*(E_1|S)$).
\end{definition}
If $(E_0,E_1,c,v)$ and $(E'_0,E'_1,c',v')$ are two
pre-${\cal D}$-modules, a morphism between them consists of ${\cal D}_X[\log
S]$ linear morphisms $f_0:E_0\to E'_0$ and $f_1:E_1\to E'_1$
such that $\mathop{\rm sp}\nolimits_Sf_0$ and $f_1$ commute with $v,v'$ and with
$c,c'$.
\begin{definition}[Families of pre-${\cal D}$-modules of the second kind]\rm
Let $T$ be a complex scheme. A family ${\bf E}_T$ of
pre-${\cal D}$-modules on $(X,S)$ parametrized by the scheme $T$, a
morphism between two such families, and pullback of a family
under a base change $T'\to T$ have obvious definitions which we
leave to the reader. This gives us a fibered category $PM$ of
pre-${\cal D}$-modules of second kind over the base category of $C\!\!\!\!I$
schemes.
\end{definition}
\begin{proposition}
The functor which associates to each family of pre-${\cal D}$-module
$(E_0,E_1,c,v)$ of the second kind parametrized by $T$ the
family of pre-${\cal D}$-module of the first kind $(E_0,E_1|S, c|S,
v|S)$ is an equivalence of fibered categories.
\end{proposition}
\paragraph{Proof} This follows from remarks \ref{rem1} and \ref{rem2} above.
\section{From pre-${\cal D}$-modules to ${\cal D}$-modules}
In this section we first recall the description of regular
holonomic ${\cal D}$-modules due to Malgrange [Mal] and we associate a
`Malgrange object' to a pre-${\cal D}$-module of the second kind
(Proposition \ref{prop2}), which has good residual eigenvalues
(definition \ref{goodres}), each component of $S$ do not differ
by positive integers. Having such a direct description of the
Malgrange object enables us to prove that every regular
holonomic ${\cal D}$-module with characteristic variety contained in
$T^*_XX\cup T^*_SX$ arises from a pre-${\cal D}$-module (Corollary
\ref{cor3}), and also helps us to prove an infinitesimal
rigidity property for the pre-${\cal D}$-modules over a given
${\cal D}$-module (Proposition
\ref{prop4}).
\inter{Malgrange objects}
Regular holonomic ${\cal D}$-modules on $X$ whose characteristic
variety is contained in $T^*_XX\cup T^*_SX$ have an equivalent
presentation due to Malgrange and Verdier, which we now
describe.
Let us recall the definition of the {\sl specialization}
$\mathop{\rm sp}\nolimits_S(M)$ of a regular holonomic ${\cal D}_X$-module $M$: fix a
section $\sigma$ of the projection $C\!\!\!\!I\toC\!\!\!\!I/Z\!\!\!Z$ and denote $A$
its image; every such module admits a unique (decreasing)
filtration $V^kM$ ($k\inZ\!\!\!Z$) by ${\cal D}_X[\log S]$-submodules which
is good with respect to $V{\cal D}_X$ and satisfies the following
property: on $\mathop{\rm gr}\nolimits^k_VM$, the action of $\theta$ admits a minimal
polynomial all of whose roots are in $A+k$. Then by definition
one puts $\mathop{\rm sp}\nolimits_SM=\oplus_{k\inZ\!\!\!Z}\mathop{\rm gr}\nolimits^k_VM$. One has
$(\mathop{\rm sp}\nolimits_SM)[*S]=\mathop{\rm sp}\nolimits_S(M[*S])=(\mathop{\rm gr}\nolimits_{V}^{\geq k}M)[*S]$ for all
$k\geq 1$, if we put $\mathop{\rm gr}\nolimits_{V}^{\geq k}M=\oplus_{\ell\geq
k}\mathop{\rm gr}\nolimits^\ell_VM$. The $p_*{\cal D}_{N_SX}$-module $\mathop{\rm sp}\nolimits_SM$ does not
depend on the choice of $\sigma$ (if one forgets its gradation).
If $\theta$ acts in a locally finite way on a $\mathop{\rm gr}\nolimits^0_V{\cal D}_X$ or a
$p_*{\cal D}_{N_{S,X}}$-module, we denote $\Theta$ the action of
$\exp(-2i\pi\theta)$.
Given a regular holonomic ${\cal D}_X$-module, we can functorially
associate to it the following modules:
(1) $M[*S]={\cal O}_X[*S]\otimes_{{\cal O}_X}M$ is the $S$-localized
${\cal D}_X$-module; it is also regular holonomic;
(2) $\mathop{\rm sp}\nolimits_S M$ is the specialized module; this is a regular
holonomic $p_*{\cal D}_{N_SX}$-module, which is also {\sl monodromic},
i.e. the action of $\theta$ on each local section is locally (on
S) finite.
The particular case that we shall use of the result proved in
[Mal] is then the following:
\begin{theorem}
There is an equivalence between the category of regular
holonomic ${\cal D}_X$-modules and the category which objects are
triples $({\cal M},\overline M,\alpha)$, where ${\cal M}$ is a $S$-localized
regular holonomic ${\cal D}_X$-module, $\overline M$ is a monodromic
regular holonomic $p_*{\cal D}_{N_SX}$-module and $\alpha$ is an
isomorphism (of localized $p_*{\cal D}_{N_SX}$-modules) between
$\mathop{\rm sp}\nolimits_S{\cal M}[*S]$ and $\overline M[*S]$.
\end{theorem}
In fact, the result of [Mal] does mention neither holonomicity
nor regularity. Nevertheless, using standard facts of the
theory, one obtains the previous proposition. Regularity
includes here regularity at infinity, i.e. along $S^\infty$.
This statement can be simplified when restricted to the category
of regular holonomic ${\cal D}$-modules which characteristic variety
is contained in the union $T^*_XX\cup T^*_SX$.
\begin{definition}\rm
A {\sl Malgrange object} on $(X,S)$ is a tuple $(M_0,M_1,C,V)$ where
(1) $M_0$ is an $S$-localized regular holonomic ${\cal D}_X$-module
which is a regular meromorphic connection on $X$ with poles on
$S$,
(2) $M_1$ is a $S$-localized monodromic regular holonomic
$p_*{\cal D}_{N_SX}$-module which is a regular meromorphic connection
on $N_{S,X}$ (or $\ov{N_{S,X}}$) with poles on $S$ (or on $S\cup
S^\infty$),
(3) $C,V$ are morphisms (of $p_*{\cal D}_{N_{S,X}}$-modules) between
$\mathop{\rm sp}\nolimits_SM_0$ and $M_1$ satisfying $VC=\Theta-\mathop{\rm id}\nolimits$ on
$\mathop{\rm sp}\nolimits_SM_0$ and $CV=\Theta-\mathop{\rm id}\nolimits$ on $M_1$.
\end{definition}
The morphisms between two Malgrange objects are defined in an
obvious way, making them an abelian category.
The previous result can be translated in the following way, using [Ve]:
\begin{corollary} There is an equivalence between the category
of regular holonomic ${\cal D}$-modules which characteristic variety
is contained in $T^*_XX\cup T^*_SX$ and the category of
Malgrange objects on $(X,S)$.
\end{corollary}
\inter{From pre-${\cal D}$-modules to Malgrange objects}
\begin{definition}\label{goodres}\rm
(1) We say that a logarithmic connection $F$ on $(X,S)$ has
{\sl good residual eigenvalues} if for each connected component $S_a$
of the divisor $S$, the residual eigenvalues $(\lambda _{a,k})$
of $F$ along $S_a$ do not include a pair $\lambda _{a,i},\lambda
_{a,j}$ such that $\lambda _{a,i}-\lambda _{a,j}$ is a nonzero
integer.
(2) We say that a pre-${\cal D}$-module ${\bf E} =(E_0,E_1,s,t)$ has
{\sl good residual eigenvalues} if the logarithmic connection
$E_0$ has good residual eigenvalues as defined above.
\end{definition}
We now functorially associate a Malgrange object ${\bf M}=\eta
({\bf E})=(M_0,M_1,C,V)$ to each pre-${\cal D}$-module ${\bf E} =
(E_0,E_1,c,v)$ on $(X,S)$ with $E_0$ having good residual
eigenvalues.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{rem3} By definition of a pre-${\cal D}$-module it follows
that the nonzero eigenvalues of $\theta_a$ on $E_0|S_a$ (the
residue along $S_a$) are the same as the nonzero eigenvalues of
$\theta_a$ on $E_{1,a}$.
\begin{proposition}{\bf(The Malgrange object associated to a
pre-${\cal D}$-module with good residual
eigenvalues)}\quad\label{prop2} Let ${\bf E}=(E_0,E_1,c,v)$ be a
pre-${\cal D}$-module on $(X,S)$ of the second kind (definition
\ref{def1}), such that $E_0$ has good residual eigenvalues. Let
$\eta ({\bf E} ) = (M_0,M_1,C,V)$ where
(1) $M_0=E_0[*S]$,
(2) $M_1=E_1[*S]$,
(3) $C = c\circ
\displaystyle{e_{}^{-2i\pi\theta_{E_0}}- 1\over\theta_{E_0}}$.
(4) $V=v$
Then $\eta ({\bf E})$ is a Malgrange object, and $\eta$ is
functorial in an obvious way.
\end{proposition}
\paragraph{Proof} Because $E_0$ has good residual eigenvalues, one can use
the filtration $V^kE_0[*S]$ $=I_{S}^{k}E_0\subset E_0[*S]$ in
order to compute $\mathop{\rm sp}\nolimits_SE_0[*S]$. It follows that the
specialization of $E_0[*S]$ when restricted to $N_{S,X}-S$ is
canonically isomorphic to the restriction of $\mathop{\rm sp}\nolimits_SE_0=\pi
^*(E_0\vert S)$ to $N_{X,S}-S$.
\inter{Essential surjectivity}
\begin{proposition}\label{prop3}
Every Malgrange object $(M_0,M_1,C,V)$ on $(X,S)$ can be
obtained in this way from a pre-${\cal D}$-module.
\end{proposition}
\paragraph{Proof} This follows from [Ve]: one chooses Deligne lattices in
$M_0$ and $M_1$. One uses the fact that every ${\cal D}$-linear map
between holonomic ${\cal D}$-modules is compatible with the
$V$-filtration, so sends the specialized Deligne lattice of
$M_0$ to the one of $M_1$. Moreover, the map $v$ can be obtained
from $V$ because the only integral eigenvalue of $\theta$ on the
Deligne lattice is $0$, so
$\displaystyle{e_{}^{-2i\pi\theta}- 1\over\theta}$ is
invertible on it.
The previous two propositions give the following.
\begin{corollary}\label{cor3} The functor from pre-${\cal D}$-modules
on $(X,S)$ to regular holonomic ${\cal D}$-modules on $X$ with
characteristic variety contained in $T^*_XX\cup T^*_SX$ is
essentially surjective.
\end{corollary}
\inter{Infinitesimal rigidity}
For a regular holonomic ${\cal D}$-module ${\bf M}$ with characteristic
variety $T^*_XX\cup T^*_SX$, there exist several nonisomorphic
pre-${\cal D}$-modules ${\bf E}$ which give rise to the Malgrange object
associated to ${\bf M}$. However, we have the following
infinitesimal rigidity result, which generalizes the
corresponding results in [N].
\begin{proposition}[Infinitesimal rigidity]\label{prop4} Let
$T=\mathop{\rm Spec}\nolimits\displaystyle{C\!\!\!\!I [\epsilon ]\over (\epsilon ^2)}$. Let ${\bf E}_T$
be a family of pre-${\cal D}$-modules on $(X,S)$ parametrized by $T$.
Let the associated family ${\bf M}_T$ of ${\cal D}$-modules on $X$ be
constant (pulled back from $X$). Let ${\bf E}$ (which is the
specialization at $\epsilon =0$) be of the form ${\bf E} =
(E,F,s,t)$ where along any component of $S$, no two distinct
eigenvalues of the residue of the logarithmic connection $E$
differ by an integer. Then the family ${\bf E}_T$ is also constant.
\end{proposition}
\paragraph{Proof} By [N], the family $E_{T}$ is constant, as well as the
specialization $\mathop{\rm sp}\nolimits_SE_{T}$. As a consequence, the residue
$\theta_{E_T}$ is constant. Let us now prove that the family
$F_T$ is constant.
Let $S_a$ be a component of $S$ along which the only possible
integral eigenvalue of $\theta_E$ is $0$. Then it is also the
only possible integral eigenvalue of $\theta_F$ along $S_a$
because the generalized eigenspaces of $\theta_E$ and $\theta_F$
corresponding to a nonzero eigenvalue are isomorphic by $s$ and
$t$ (see remark \ref{rem3}). We also deduce from [N] that $F_T$
is constant as a logarithmic module along this component.
Assume now that $0$ is not an eigenvalue of $\theta_E$ along
$S_a$ but is an eigenvalue of $\theta_F$ along this component.
Then $\theta_F$ may have two distinct integral eigenvalues, one
of which is $0$. Note that, in this case, $\theta_E$ is an
isomorphism (along $S_a$), as well as $\theta_{E_T}$ which is
obtained by pullback from $\theta_E$. It follows that on $S_a$
we have an isomorphism $F_T\simeq E_T|S_a\oplus
\mathop{\rm Ker}\nolimits\theta_{F_T}$. Consequently $\mathop{\rm Ker}\nolimits\theta_{F_T}$ is itself a
family. It is enough to show that this family is constant. But
the corresponding meromorphic connection on $N_{S,X}^{}-S$ is
constant, being the cokernel of the constant map $C_T:M_{0T}\to
M_{1T}$. We can then apply the result of [N] because the only
eigenvalue on $\mathop{\rm Ker}\nolimits\theta_F$ is $0$.
The maps $s_T$ and $t_T$ are constant if and only if for each
component $S_a$ of $S$ and for some point $x_a\in S_a$ their
restriction to $F_T|{x_a}\times T$ and $E_T|{x_a}\times T$ are
constant. This fact is a consequence of the following lemma.
\begin{lemma} Let $E$ and $F$ be finite dimensional complex
vector spaces, and let $\theta_E\in End (E)$ and $\theta_F\in
End (F)$ be given. Let $V\subset W=Hom(F,E) \times Hom(E,F)$ be
the closed subscheme consisting of $(s,t)$ with $st=\theta_E$
and $ts=\theta_F$. Let $\phi :W\to W$ be the holomorphic map
defined by $$\phi (s,t) = (s, t {e^{st} -1 \over st}).$$ Then the
differential $d\phi$ is injective on the Zariski tangent space
to $V$ at any closed point $(s,t)$.
\end{lemma}
\paragraph{Proof} Let $(a, b)$ be a tangent vector to $V$ at $(s,t)$. Then
by definition of $V$, we must have $at+sb=0$ and $ta+bs=0$.
Using $at+sb=0$, we can see that $d\phi (a, b) = (a, bf(st))$
where $f$ is the entire function on $End(E_0)$ defined by the
power series $(e^x-1)/x$. Suppose $(a,bf(st))=0$. Then $a=0$ and
so the condition $ta+bs=0$ implies $bs=0$. As the constant term
of the power series $f$ is $1$ and as $bs=0$, we have
$bf(st)=b$. Hence $b=0$, and so $d\phi$ is injective.
\section{Semistability and moduli for pre-${\cal D}$-modules.}
In order to construct a moduli scheme for pre-${\cal D}$-modules, one needs a
notion of semistability. This can be defined in more than one way.
What we have chosen below is a particularly simple and canonical
definition of semistability. (In an earlier version of this
paper, we had employed a definition of semistability in terms of
parabolic structures, in which
we had to fix the ranks of $s:E_1\to E_0|S$ and $t:E_0|S \to
E_1$ and a set of parabolic weights.)
Let $S_a$ be the irreducible
components of the smooth divisor $S\subset X$. For a pre-${\cal D}$-module
${\bf E} =(E_0,E_1,s,t)$, we denote by $E_a$ the restriction of
$E_1$ to $S_a$, and we denote by $s_a$ and $t_a$ the
restrictions of $s$ and $t$.
\inter{Definition of semistability}
We fix an ample line bundle on $X$, and denote the resulting
Hilbert polynomial of a coherent sheaf $F$ by $p(F,n)$.
For constructing a moduli, we fix the Hilbert
polynomials of $E_0$ and $E_a$, which we denote by $p_0(n)$ and
$p_a(n)$.
Recall (see [S]) that an ${\cal O} _X$-coherent ${\cal D} _X[\log S]$-module $F$
is by definition {\sl semistable} if it is pure dimensional, and
for each ${\cal O} _X$ coherent
${\cal D} _X[\log S]$ submodule $F'$, we have the inequality
$p(F',n)/rank (F') \le p(F,n)/rank (F) $
for large enough $n$. We call $p(F,n)/rank (F)$ the {\sl
normalized Hilbert polynomial} of $F$.
\begin{definition}\rm
We say that the pre-${\cal D}$-module ${\bf E}$ is {\sl semistable} if
the ${\cal D}_X[\log S]$-modules $E_0$ and $E_a$ are semistable.
\end{definition}
\refstepcounter{theorem}\paragraph{Remarks \thetheorem}
(1) It is easy to prove that the semistability of the ${\cal D}
_X[\log S]$-module $E_a$ is equivalent to the semistability of
the logarithmic connection $\pi ^*_a(E_a)$ on $P(N_{S_a,X}\oplus
1)$ with respect to a natural choice of polarization.
(2) When $X$ is a curve, a pre-${\cal D}$-module ${\bf E}$ is semistable if
and only if the logarithmic connection $E_0$ on $(X,S)$ is
semistable, for then $E_1$ is always semistable.
(3) Let the dimension of $X$ be more than one.
Then even when a pre-${\cal D}$-module ${\bf E}$ is a pre meromorphic connection
(equivalently, when $s:E_1 \to E_0\vert S$ is an isomorphism),
the definition of semistability of ${\bf E}$ does not
reduce to the semistability of the underlying logarithmic
connection $E_0$ on $(X,S)$. This is to be expected because we
do not fix the rank of $s$ (or $t$) when we consider families of
pre-${\cal D}$-modules. Also note that even in dimension one,
meromorphic connections are not a good subcategory of the
abelian category of all regular holonomic ${\cal D}$-modules with
characteristic variety contained in $T^*_XX\cup T^*_SX$, in the
sense that a submodule or a quotient module of a meromorphic
connection is not necessarily a meromorphic connection.
\inter{Boundedness and local universal family}
We let the index $i$ vary over $0$ and over the $a$.
\begin{proposition}[Boundedness] Semistable pre-${\cal D}$-modules
with given Hilbert po\-lynomials $p_i$ form a bounded set, that
is, there exists a family of pre-${\cal D}$-modules parametrized by a
noetherian scheme of finite type over $C\!\!\!\!I$ in which each
semistable pre-${\cal D}$-module with given Hilbert polynomials occurs.
\end{proposition}
\paragraph{Proof} This is obvious as each $E_i$ (where $i=0,a$) being
semistable with fixed Hilbert polynomial, is bounded.
Next, we construct a local universal family. By boundedness,
there exists a positive integer $N$ such that for all $n\ge N$,
the sheaves $E_0(N)$ and $E_1(N)$ are generated by global
sections and have vanishing higher cohomology. Let $\Lambda =
D_X[\log S]$. Let ${\cal O} _X =\Lambda_0 \subset \Lambda_1 \subset
\cdots \subset \Lambda$ be the increasing filtration of
$\Lambda$ by the order of the differential operators. Note that
each $\Lambda_k$ is an ${\cal O}_X$ bimodule, coherent on each side.
Let $r$ be a positive integer larger than the ranks of the
$E_i$. Let $Q_i$ be the quot scheme of quotients
$q_i:\Lambda_r\otimes {\cal O}_X (-N)^{p_i(N)}\to\!\!\!\!\to E_i$
where the right ${\cal O}_X$-module structure on $\Lambda_r$ is used
for making the tensor product. Note that $G_i=PGL(p_i(N))$ has a
natural action on $Q_i$. Simpson defines a locally closed
subscheme $C_i\subset Q_i$ which is invariant under $G_i$, and
a local universal family $E$ of $\Lambda$-modules parametrized
by $C_i$ with the property that for two morphisms $T\to C_i$,
the pull back families are isomorphic over an open cover
$T'\to T$ if and only if the two morphisms define $T'$ valued
points of $C_i$ which are in a common orbit of $G_i(T')$.
On the product $C_0\times C_a$, consider the linear schemes
$A_a$ and $B_a$ which respectively correspond to
$Hom_{\Lambda}(E_1,E_0)$ and $Hom_{\Lambda}(E_0,E_1)$ (see Lemma
2.7 in [N] for the existence and universal property of such linear
schemes). Let $F_a$ be the fibered product of $A_a$ and $B_a$
over $C_0\times C_a$. Let $H_a$ be the closed subscheme of $F_a$
where the tuples $(q_0,q_1,t,s)$ satisfy $st=\theta$ and
$ts=\theta$. Finally let $H$ be the fibered product of the
pullbacks of the $H_a$ to $C= C_0 \times \prod_a C_a$. Note that
$H$ parametrizes a tautological family of pre-${\cal D}$-modules on
$(X,S)$ in which every semistable pre-${\cal D}$-module with given
Hilbert polynomials occurs.
The group $${\cal G} = G_0 \times \prod_a (G_a \times GL(1))$$ has a
natural action on $H$, with
$$(q_0,q_a,t_a,s_a)\cdot (g_0,g_a,\lambda_a) =
(q_0g_0,q_ag_a,(1/\lambda_a)t_a,\lambda_a s_a)$$
It is clear from the definitions of $H$ and this action that two
points of $H$ parametrise isomorphic pre-${\cal D}$-modules if and only
if they lie in the same $G$ orbit.
The morphism $H\to C\times \prod _aC_a$ is an affine morphism
which is ${\cal G}$-equivariant, and by Simpson's construction of
moduli for $\Lambda$-modules, the action of ${\cal G}$ on $C\times
\prod _aC_a$ admits a good quotient in the sense of geometric
invariant theory. Hence a good quotient $H//{\cal G}$ exists by
Ramanathan's lemma (see Proposition 3.12 in [Ne]), which by
construction and universal properties of good quotients
is the coarse moduli scheme of semistable pre-${\cal D}$-modules with
given Hilbert polynomials.
Note that under a good quotient in the sense of geometric
invariant theory, two different orbits can in some cases get mapped
to the same point (get identified in the quotient).
In the rest of this section, we determine what are the closed
points of the quotient $H//{\cal G}$.
\refstepcounter{theorem}\paragraph{Remark \thetheorem} For simplicity of notation, we assume in the rest of this
section that $S$ has only one connected component. It will be
clear to the reader how to generalize the discussion when $S$
has more components.
\inter{Reduced modules}
Assuming for simplicity that $S$ has only one connected
component, so that ${\cal G} = {\cal H} \times GL(1)$ where
$H=G_0 \times G_1$, we can construct the quotient
$H//{\cal G}$ in two steps: first we go modulo
the factor $GL(1)$, and then take the quotient of $R=H//GL(1)$
by the remaining factor ${\cal H}$. The following lemma is obvious.
\begin{lemma}\label{lem4.5}
Let $T$ be a scheme of finite type over $k$, and let $V\to T$
and $W\to T$ be linear schemes over $T$. Let $V\times W$ be
their fibered product (direct sum) over $T$, and let $V\otimes
W$ be their tensor product. Let $\phi :V\times W\to V\otimes W$
be the tensor product morphism. Then its schematic image
$D\subset V\otimes W$ is a closed subscheme which (i) parametrizes all
decomposable tensors, and (ii) base changes correctly.
Let $GL(1,k)$ act on $V\times W$ by the formula
$\lambda \cdot (v,w) = (\lambda v, (1/\lambda )w)$. Then $\phi
:V\times W\to D$ is a good quotient for this action.
\end{lemma}
\paragraph{Proof}
The statement is local on the base, so we can assume
that (i) the base $T$ is an affine scheme, and (ii) both the linear
schemes are closed linear subschemes of trivial vector bundles
on the base, that is, $V\subset A^m_T$ and $W\subset A^n_T$ are
subschemes defined respectively by homogeneous linear equations
$f_p(x_i)=0$ and $g_q(y_j)=0$ in the coordinates $x_i$ on
$A^m_T$ and $y_j$ on $A^n_T$.
Let $z_{i,j}$ be the coordinates on $A^{mn}_T$, so that
the map $\otimes :A^m_T\times _T A^n_T \to A^{mn}$ sends
$(x_i,y_j) \mapsto (z_{i,j})$ where $z_{i,j}=x_iy_j$.
Its schematic image is the subscheme $C$ of $A^{mn}_T$
defined by the equations $z_{a,b}z_{c,d} - z_{a,d}z_{b,c} = 0$,
that is, the matrix $(z_{i,j})$ should have rank $ < 2$.
Take $D$ to be the subscheme of $C$ defined by the equations
$f_p(z_{1,j},\ldots ,z_{m,j}) = 0$ and
$g_q(z_{i,1},\ldots ,z_{i,n}) = 0$. Now the lemma \ref{lem4.5} follows
trivially from this local coordinate description.
\paragraph{}
The above lemma implies the following.
To get the quotient $H//GL(1)$, we just have to
replace the fibered product $A\times B$ over
$C_0\times C_1$ by the closed subscheme $Z\subset D\subset
A\otimes B$, where $D$ is the closed subscheme consisting
of decomposable tensors $u$, and $Z$ is the closed subscheme of
$D$ defined as follows. Let $\mu _0$ and $\mu _1$ be the
canonical morphisms from $A\otimes B$ to the linear schemes
representing $End_{\Lambda} (E_0|S)$ and $End_{\Lambda} (E_1)$
respectively. Then
$Z$ is defined to consist of all $u$ such that $\mu
_0(u)=\theta \in End _{\Lambda}(E_0|S) $ and $\mu _1(u) = \theta
\in End_{\Lambda}(E_1)$. There is a canonical
$GL(1)$ quotient morphism $A\times B \to D$ over
$C_0\times C_1$, which sends $(s,t)\mapsto u=s\otimes t$.
These give the $GL(1)$ quotient map $H\to Z$.
Note that the map $H\to C_0\times C_1$ is ${\cal G}$ equivariant, and the action
of $GL(1)$ on $C_0\times C_1$ is trivial, so we get a ${\cal H}$-equivariant map
$Z\to C_0\times C_1$.
In order to describe the identifications brought about by the
above quotient, we make the following definition.
\begin{definition}\rm
A {\sl reduced module} is a tuple $(E_0,E_1,u)$ where $E_0$ and
$E_1$ are as in a pre-${\cal D}$-module, and $u\in
Hom_{\Lambda}(E_1,E_0|S)\otimes Hom_{\Lambda}(E_0,E_1)$ is a
decomposable tensor,
such that the canonical maps $\mu _0:Hom_{\Lambda}(E_1,E_0|S)\otimes
Hom_{\Lambda}(E_0,E_1) \to End_{\Lambda}(E_0|S)$ and
$\mu _1: Hom_{\Lambda}(E_1,E_0|S)\otimes Hom_{\Lambda}(E_0,E_1)
\to End_{\Lambda}(E_1)$,
both map $u$ to the endomorphism $\theta$ of the appropriate
module. In other words,
there exist $s$ and $t$ such that $(E_0,E_1,s,t)$ is a
pre-${\cal D}$-module, and $u=s\otimes t$.
We say that the reduced module $(E_0,E_1,s\otimes t)$ is the associated
reduced module of the pre-${\cal D}$-module $(E_0,E_1,s,t)$.
Moreover, we say that a reduced module is semistable if it is
associated to a semistable pre-${\cal D}$-module.
\end{definition}
\begin{lemma} Let $V$ and $W$ are two vector spaces, $v,v'\in V$ and
$w,w'\in W$, then
(1) If $v\otimes w=0$ then $v=0$ or $w=0$.
(2) If $v\otimes w=v'\otimes w'\ne 0$, then there exists a
scalar $\lambda \ne 0$ such that $v=\lambda v'$ and $w =
(1/\lambda ) w'$.
\end{lemma}
\refstepcounter{theorem}\paragraph{Remark \thetheorem} The above lemma shows that if ${\bf E}$ and ${\bf E} '$ are two
non-isomorphic pre-${\cal D}$-modules whose associated reduced modules
are isomorphic, then
we must have $s\otimes t =s'\otimes t'=0$. In particular,
$\theta$ will act by zero on $E_0|S$ and also on $E_1$ for such
pre-${\cal D}$-modules as $st=0$ and $ts=0$.
\inter{S-equivalence and stability}
\begin{definition}\rm
By a {\sl filtration} of a logarithmic connection $E$ we shall
mean an increasing filtration $E_p$ indexed by $Z\!\!\!Z$ by subvector
bundles which are logarithmic connections.
Similarly, a filtration on a ${\cal D} _X[\log S]$-module $F$
supported on $S$ will mean a filtration of the vector bundle
$F\vert S$ by subbundles $F_p$ which are ${\cal D} _X[\log
S]$-submodules. A filtration of a
pre-${\cal D}$-module $(E_0,E_1,s,t)$ is an increasing filtration
$(E_i)_p$ of the logarithmic connection $E_i$ ($i=0,1$) such
that $s$ and $t$ are filtered morphisms with respect to the
specialized filtration of $E_0$ and the filtration of $E_1$.
A filtration of a reduced module
$(E_0,E_1,u)$, with $u=s\otimes t$ where we take $s=0$ and $t=0$
if $u=0$, is a filtration of the pre-${\cal D}$-module
$(E_0,E_1,s,t)$. We shall always assume that this filtration is
exhaustive, that is, $(E_i)_p=0$ for $p\ll0$ and $(E_i)_p=E_i$
for $p\gg0$. A filtration is {\sl nontrivial} if some $(E_i)_p$
is a proper subbundle of $E_i$ for $i=0$ or $1$.
\end{definition}
For a filtered pre-${\cal D}$-module (or reduced module), each step of
the filtration as well as the graded object are pre-${\cal D}$-modules
(or reduced modules).
\refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{deform}
There is a natural family $({\bf E}_\tau)_{\tau\in A^1}^{}$ of
pre-${\cal D}$-modules or reduced modules parametrized by the affine
line $A^1=\specC\!\!\!\!I[\tau]$, which fibre at $\tau=0$ is the graded
object ${\bf E}'$ and the fibre at $\tau_0\neq0$ is isomorphic to
the original pre-${\cal D}$-module or reduced module ${\bf E}$: put (for
$i=0,1$) ${\cal E}_i=\oplus_{p\inZ\!\!\!Z}^{}(E_i)_p\tau^p\subset E_i\otimes
C\!\!\!\!I[\tau,\tau_{}^{-1}]$ and the relative ${\cal D}\log$-structure is
the natural one.
\begin{definition}\rm
A {\sl special filtration of a coherent ${\cal O} _X$-module} $E$ is a
filtration for which each $E_p$ has the same normalized Hilbert
polynomial as $E$. A {\sl special filtration of a reduced
module} $(E_0,E_1,u)$ is a filtration of this reduced module
which is special on $E_0$ and on $E_1$.
\end{definition}
The graded reduced module ${\bf E}'$ associated with a special
filtration of a semistable reduced module ${\bf E}$ is again
semistable.
\begin{definition}\rm
The equivalence relation on the set of isomorphism classes
of all semistable reduced modules generated by this relation
(by which ${\bf E} '$ is related to ${\bf E}$) will be called S-equivalence.
\end{definition}
\begin{definition}\label{defstable}\rm
We say that a semistable reduced module is {\sl stable} if it
does not admit any nontrivial special filtration.
\end{definition}
\refstepcounter{theorem}\paragraph{Remarks \thetheorem}
(1) Note in particular that if each $E_0$, $E_a$ is stable
as a $\Lambda$-module, then the
reduced module ${\bf E} $ is
stable. Consequently we have the following. Though the
definition of stability depends on the ample line bundle $L$ on
$X$, irrespective of the choice of the ample bundle, for any
pre-${\cal D}$-module such that the monodromy representation of
$E_0\vert (X-S)$ is irreducible, and the monodromy
representation of $\pi _a ^*E_a \vert (N_{S_a,X}-S_a)$ is
irreducible for each component $S_a$, the corresponding reduced
module is stable. The converse is not true
-- a pre-${\cal D}$-module, whose reduced module is stable, need not
have irreducible monodromies. The example 2.4.1 in [N] gives a
logarithmic connection, whose associated pre-${\cal D}$-module in
which $s$ is identity and $t$ is the residue, gives a stable
reduced module, but the monodromies are not irreducible.
(2) If $u=0$, the reduced module is stable if and only $E_0$ and
each $E_a$ is stable.
(3) When $X$ is a curve, a reduced module with $u\ne 0$ is stable if
and only if the logarithmic connection $E_0$ is stable. If
$u=0$, each $E_a$ must moreover have length at most one as an
${\cal O}_X$-module. Hence over curves, there is a plentiful supply of
stable reduced modules.
\begin{lemma}\label{uisflat}
Let $(E_0,E_1,u)$ be a reduced module and let $(E_i)_p$ be
filtrations of $E_i$ ($i=0,1$). Then $s$ and $t$ are filtered
morphisms with respect to the specialized filtrations if and only
if there exists some point $P\in S$ such that the restrictions
of $s$ and $t$ to the fibre $E_{i,P}$ at $P$ are filtered with
respect to the restricted filtrations.
\end{lemma}
\paragraph{Proof}
This follows from the fact that if a section $\sigma$ of a
vector bundle with a flat connection has a value $\sigma(P)$ in
the fibre at $P$ of a sub flat connection, then it is a section
of this subbundle: we apply this to $s$ (resp. $t$) as a section
of $Hom((E_0)_{p|S},(E_1)_{|S})$ (resp.
$Hom((E_1)_{p|S},(E_0)_{|S})$).
\inter{A criterion for stability}
Let ${\bf E}=(E_0,E_1,u=s\otimes t)$ be a reduced module. Assume
that we are given filtrations $0=F_0(E_i)\subset
F_1(E_i)\subset\cdots\subset F_{\ell_i}(E_i)=E_i$ of $E_i$
($i=0,1$) by vector subbundles which are ${\cal D}_X[\log
S]$-submodules.
For $j=0,\ldots,\ell_i$ let $k(j)$ be the smallest $k$ such
that $s(\mathop{\rm sp}\nolimits_SF_j(E_0))\subset F_k(E_1)$ and let $J(s)$ be the
graph of the map $j\to k(j)$. A {\sl jump point} is a point
$(j,k(j))$ on this graph such that $k(j-1)<k(j)$. Consider also
the set $G_s$ made by points under the graph: $G_s=\{ (j,k)\mid
k\leq k(j)\}$. For $t$ there is an equivalent construction: we
have a map $k\to j(k)$ and a set $G_t$ on the left of the graph
$I(t)$: $G_t=\{ (j,k)\mid j\leq j(k)\}$.
\begin{definition}\rm
$u=s\otimes t$ is {\sl compatible} with the filtrations if the
two sets $G_s$ and $G_t$ intersect at most at (common) jump
points (where if $u=0$, take $s=0$ and $t=0$).
\end{definition}
\begin{proposition}\label{nonstable}
Let ${\bf E}=(E_0,E_1,u)$ be a semistable reduced module. The
following conditions are equivalent:
(1) ${\bf E}$ is not stable,
(2) there exists a nontrivial special filtration $F_j(E_i)$
($j=0,\ldots\ell_i$) of each $E_i$ where all inclusions are
proper and $u$ is compatible with these filtrations.
\end{proposition}
\paragraph{Proof}
$(1)\Rightarrow(2)$: If ${\bf E}$ is not stable, we can find two
nontrivial special filtrations $(E_0)_p$ and $(E_1)_q$ such that
$s$ and $t$ are filtered morphisms. Let $p_j$ ($j=1,\ldots
,\ell_0$) be the set of jumping indices for $(E_0)_p$ and $q_k$
($k=1,\ldots ,\ell_1$) for $(E_1)_q$. For each $j_0$ and $k_0$
we have $j(k(j_0))\leq j_0$ and $k(j(k_0))\leq k_0$. We define
$F_j(E_0)=(E_0)_{p_j}$ and $F_k(E_1)=(E_1)_{q_k}$. We get
nontrivial filtrations of $E_0$ and $E_1$ where all inclusions
are proper. Moreover there cannot exist two distinct points
$(j_0,k(j_0))$ and $(j(k_0),k_0)$ with $j_0\leq j(k_0)$ and
$k_0\leq k(j_0)$ otherwise we would have $j_0\leq j(k_0)\leq
j(k(j_0))\leq j_0$ and the same for $k_0$ so the two points
would be the equal. Consequently $u$ is compatible with these
filtrations.
$(2)\Rightarrow(1)$: We shall construct a special filtration
$((E_0)_p,(E_1)_q)$ of
the reduced module from the filtrations $F_j(E_i)$ of each $E_i$.
Choose a polygonal line with only positive slopes, going through
each jump point of $G_s$ and for which each jump point of $G_t$
is on or above this line (see figure \ref{fig1}).
\setlength{\unitlength}{.5truecm}
\begin{figure}[htb]
\begin{center}
\begin{picture}(10,8)(0,0)
\put(0,0){\line(1,0){10}}
\put(0,0){\line(0,1){8}}
\put(9.5,-.7){$j$}
\put(-.5,7.5){$k$}
\put(3,4){\circle*{.2}}
\put(6,6){\circle*{.2}}
\put(8,8){\circle*{.2}}
\put(3,4){\line(1,0){3}}
\put(6,6){\line(1,0){2}}
\put(8,8){\line(1,0){2}}
\put(3,0){\line(0,1){4}}
\put(6,4){\line(0,1){2}}
\put(8,6){\line(0,1){2}}
\put(1,1){\circle{.2}}
\put(2,3){\circle{.2}}
\put(3,4){\circle{.2}}
\put(5,5){\circle{.2}}
\put(7,7){\circle{.2}}
\put(0,1){\line(1,0){1}}
\put(1,3){\line(1,0){1}}
\put(2,4){\line(1,0){1}}
\put(3,5){\line(1,0){2}}
\put(5,7){\line(1,0){2}}
\put(1,1){\line(0,1){2}}
\put(2,3){\line(0,1){1}}
\put(3,4){\line(0,1){1}}
\put(5,5){\line(0,1){2}}
\put(7,7){\line(0,1){1}}
\put(6,3){$G_s$}
\put(2,6){$G_t$}
\multiput(.2,.2)(.2,.2){4}{\circle*{.1}}
\multiput(1,1)(.2,.3){10}{\circle*{.1}}
\multiput(3,4)(.4,.2){5}{\circle*{.1}}
\multiput(5,5)(.2,.2){15}{\circle*{.1}}
\end{picture}
\caption{\label{fig1}$\bullet=$ jump points of $s$, $\circ=$
jump points of $t$}
\end{center}
\end{figure}
Choose increasing functions $p(j)$ and $q(k)$ such that
$p(j)-q(k)$ is identically $0$ on this polygonal line, is $<0$
above it and $>0$ below it (for instance, on each segment
$[(j_0,k_0),(j_1,k_1)]$ of this polygonal line, parametrised by
$j=j_0+m\varepsilon_1$, $k=k_0+m\varepsilon_2$, put
$p(j)=p(j_0)+\varepsilon_2(j-j_0)$ and
$q(k)=q(k_0)+\varepsilon_1(k-k_0)$, and $p(0)=q(0)=0$). For
$p(j)\leq p<p(j+1)$ put $(E_0)_p=F_j(E_0)$ and for $q(k)\leq
q<q(k+1)$ put $(E_1)_q=F_k(E_1)$. The filtration
$((E_0)_p,(E_1)_q,u)$ is then a nontrivial special filtration of
the reduced module ${\bf E}$.
\begin{proposition}\label{open}
Semistability and stability are Zariski open conditions on the
parameter scheme of any family of reduced modules.
\end{proposition}
\paragraph{Proof} As semistability is an open condition on ${\cal D} _X[\log
S]$-modules, it follows it is an open condition on reduced
modules. Now, for any family of semistable reduced modules
parametrised by a scheme $T$, all possible special filtrations
of the form given by \ref{nonstable}
on the specializations of the family are parametrised by
a scheme $U$ which is projective over $T$. The image of $U$ in
$T$ is the set of non stable points in $T$, hence its complement
is open.
\inter{Points of the moduli}
We are now ready to prove the following theorem.
\begin{theorem} Let $X$ be a projective variety together with an
ample line bundle, and let $S\subset X$ be a smooth divisor.
(1) There exists a coarse moduli scheme ${\cal P}$ for
semistable pre-${\cal D}$-modules on $(X,S)$ with given Hilbert
polynomials $p_i$. The scheme ${\cal P}$ is quasiprojective, in
particular, separated and of finite type over $C\!\!\!\!I$.
(2) The points of ${\cal P}$
correspond to S-equivalence classes of semistable
pre-${\cal D}$-modules.
(3) The S-equivalence class of a semistable reduced module ${\bf E} $
equals its isomorphism class if and only if ${\bf E} $ is stable.
(4) ${\cal P}$ has an open subscheme ${\cal P} ^s$ whose points are the
isomorphism classes of all stable reduced modules. This is a
coarse moduli for (isomorphism classes of) stable reduced
modules.
\end{theorem}
\paragraph{Proof} Let ${\cal P} = H//{\cal G}$. Then (1) follows by the construction of
${\cal P}$. To prove (2), first note that by the existence of the
deformation ${\bf E} _t$ (see \ref{deform}) of any reduced module
${\bf E}$ corresponding to a
weighted special filtration, and by the separatedness of ${\cal P}$,
the reduced module ${\bf E} $ and its limit ${\bf E} '$ go to the same point
of ${\cal P}$. Hence an S-equivalence class goes to a common point of
${\cal P}$. For the converse, first recall that ${\cal G} = {\cal H} \times
GL(1)$, and the quotient ${\cal P}$ can
be constructed in two steps: ${\cal P} = R//{\cal H}$ where $R=H/{\cal G}$.
The scheme $R$ parametrizes a canonical family of reduced modules.
Let the ${\cal H}$ orbit of a point $x$ of $R$ corresponding the reduced
module ${\bf E} $ not be closed in $R$. Let $x_0$ be any of its limit points.
Then there exists a 1-parameter subgroup $\lambda$ of ${\cal H}$ such
that $x_0 = \lim _{t\to 0} \lambda (t) x$. This defines a map
from the affine line $A^1$ to $R$, which sends $t\mapsto \lambda
(t)x$. Let ${\bf E} _t$ be the pull back of the tautological family
of reduced modules parametrized by $R$.
Then from the description of the limits of the actions of
1-parameter subgroups on a quot scheme given in section 1 of
Simpson [S], it follows that ${\bf E}$ has a special filtration such
that the family ${\bf E} _t$ is isomorphic to a deformation of the type
constructed in \ref{deform} above. Hence the reduced modules
parametrized by $x$ and $x_0$ are S-equivalent. This proves (2).
If the orbit of $x$ is not closed, then it has a limit $x_0$
outside it under a 1-parameter subgroup, which by above represents a
reduced module ${\bf E} '$ which is the limit of ${\bf E} $ under a special
filtration. As by assumption ${\bf E} '$ is not isomorphic to ${\bf E} $, the
special filtration must be nontrivial. Hence ${\bf E} $ is not stable.
Hence stable points have closed orbits in $R$. If $x$ represents
a stable reduced module, then $x$ cannot be the limit point of
any other orbit. For, if $x$ is a limit point of the orbit of
$y$, then by openness of stability (see \ref{open}), $y$ should
again represent a stable reduced module. But then by above, the
orbit of $y$ is closed. This proves (3).
Let $H^s\subset H$ be the open subscheme where the corresponding
pre-${\cal D}$-module is stable. By (2) and (3) above, $H^s$ is
saturated under
the quotient map $H \to {\cal P}$, hence by properties of a good
quotient, its image ${\cal P} ^s$ is open in ${\cal P}$. Moreover by (2) and
(3) above, $H^s$ is the inverse image of ${\cal P}^ s$. Hence $H^s \to
{\cal P} ^s$ is a good quotient, which again by (2) and (3) is an
orbit space. Hence points of ${\cal P} ^s$ are exactly the
isomorphism classes of stable reduced modules, which proves (4).
\section{Perverse sheaves, Verdier objects and finite descriptions}
Let $X$ be a nonsingular projective variety and let $S$ be a
smooth divisor. The abelian category of perverse sheaves
constructible with respect to the stratification $(X-S,S)$ of
$X$ is equivalent to the category of `Verdier objects' on
$(X,S)$. Before defining this category, let us recall the notion
of specialization along $S$.
Let ${\cal E}$ be a local system (of finite dimensional vector spaces)
on $X-S$. The {\sl specialization} $\mathop{\rm sp}\nolimits_S{\cal E}$ is a local system
(of the same rank) on $N_{S,X}^{}-S$ equipped with an
endomorphism $\tau_{\cal E}$. It is constructed using the nearby cycle
functor $\psi$ defined by Deligne applied to the morphism which
describes the canonical deformation from $X$ to the normal
bundle $N_{S,X}^{}$.
A local system ${\cal F}$ on $N_{S,X}^{}-S$ equipped with an
endomorphism $\tau_{\cal F}$ is said to be {\sl monodromic} if
$\tau_{\cal F}$ is equal to the monodromy of ${\cal F}$ around $S$. Then
$\mathop{\rm sp}\nolimits_S{\cal E}$ is monodromic.
\begin{definition}\rm
A {\sl Verdier object} on $(X,S)$ is a tuple ${\bf V}=({\cal E},{\cal F},C,V)$
where
(1) ${\cal E}$ is a local system on $X-S$,
(2) ${\cal F}$ is a monodromic local system on $N_{S,X}^{}-S$,
(3) $C:\mathop{\rm sp}\nolimits_S{\cal E}\to{\cal F}$ and $V:{\cal F}\to\mathop{\rm sp}\nolimits_S{\cal E}$ are morphisms of
(monodromic) local systems on $N_{S,X}^{}-S$ satisfying
(4) $CV=\tau_{\cal F}-\mathop{\rm id}\nolimits$ and $VC=\tau_{\cal E}-\mathop{\rm id}\nolimits$.
\end{definition}
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
The morphisms between Verdier objects on $(X,S)$ are defined in
an obvious way, and the category of Verdier objects is an
abelian category in which each object has finite length.
Hence the following definition makes sense.
\begin{definition}\rm
We say that two Verdier objects are {\sl S-equivalent} if they
admit Jordan-H\"older filtrations such that the corresponding
graded objects are isomorphic.
\end{definition}
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
Let $B$ be a tubular neighbourhood of $S$ in $X$, diffeomorphic
to a tubular neighbourhood of $S$ in $N_{S,X}^{}$. Put
$B^*=B-S$. The specialized local system $\mathop{\rm sp}\nolimits_S{\cal E}$ can be
realized as the restriction of ${\cal E}$ to $B^*$, its monodromy
$\tau_{\cal E}$ at some point $x\in B^*$ being the monodromy along the
circle normal to $S$ going through $x$. Hence a Verdier object
can also be described as a tuple ${\bf V}$ where ${\cal F}$ is a local
system on $B^*$ and $C$, $V$ are morphisms between ${\cal E}|B^*$ and
${\cal F}$ subject to the same condition (4).
\bigskip
The notion of a family of perverse sheaves is not
straightforward. We can however define the notion of a family of
Verdier objects. Let us define first a family of local systems
on $X-S$ (or on $N_{S,X}^{}-S$) parametrized by a scheme $T$.
This is a locally free $p^{-1}{\cal O}_T$-module of finite rank, where
$p$ denotes the projection $X-S\times T\to T$. Morphisms between
such objects are $p^{-1}{\cal O}_T$-linear. The notion of a family of
Verdier objects is then straightforward.
In order make a moduli space for Verdier objects, we shall
introduce the category of `finite descriptions' on $(X,S)$. Let
us fix the following data (D):
(D1) finitely generated groups $G$ and $G_a$ for each component
$S_a$ of $S$,
(D2) for each $a$ an element $\tau_a$ which lies in the center
of $G_a$ and a group homomorphism $\phi_a:G_a\to G$.
\begin{definition}\label{def2}\rm
A finite description ${\bf D}$ (with respect to the data (D)) is a
tuple $(E,\rho,F_a,\rho_a,C_a,V_a)$ where
(1) $\rho:G\to GL(E)$ is a finite dimensional complex
representation of the group $G$; for each $a$ we will regard
$E$ as a representation of $G_a$ via the homomorphism
$\phi_a:G_a\to G$;
(2) for each $a$, $\rho_a:G_a\to GL(F_a)$ is a finite
dimensional complex representation of the group $G$;
(3) for each $a$, $C_a:E\to F_a$ and $V_a: F_a\to E$ are
$G_a$-equivariant linear maps such that
$V_aC_a=\rho(\tau_a)-\mathop{\rm id}\nolimits$ in $GL(E)$ and
$C_aV_a=\rho_a(\tau_a)-\mathop{\rm id}\nolimits$ in $GL(F_a)$.
\end{definition}
A morphism between two finite descriptions has an obvious
definition.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{rem4}
Let $P_0\in X-S$ and let $P_a$ be a point in the component
$B*_a$ of $B^*$. Choose paths $\sigma_a:[0,1]\to X-S$ with
$\sigma_a(0)=P_0$ and $\sigma_a(1)=P_a$. Let $G$ be the
fundamental group $\pi_1(X-S,P_0)$, and let $G_a = \pi_1
(B^*_a,P_a )$. Let $\tau_a \in G_a $ be the positive loop based
at $P_a $ in the fiber of $B^*_a\to S_a $. Finally, let
$\phi_a:G_a\to G$ be induced by the inclusion
$B^*_a\hookrightarrow X-S$ by using the path $\sigma_a$ to
change base points. Then, under the equivalence between
representations of fundamental group and local system, the
category of finite description with respect to the previous data
is equivalent to the category of Verdier objects on $(X,S)$.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
The category of finite descriptions is an abelian category in
which each object has finite length. Therefore the notion of
S-equivalence as in definition 5.3 above makes sense for finite
descriptions.
\begin{definition}\rm
A family of finite descriptions parametrized by a scheme $T$ is
a tuple $(E_T, \rho_T,F_{T,a}, \rho_{T,a}, C_{T,a}, V_{T,a})$
where $E_T$ and the $F_{T,a}$ are locally free sheaves on $T$,
$\rho$ and $\rho_{T,a}$ are families of representations into
these, and the $C_{T,a}$ and $V_{T,a}$ are ${\cal O}_T$-homomorphisms
of sheaves satisfying the analogues of condition \ref{def2}.3
over $T$. The pullback of a family under a morphism $T'\to T$ is
defined in an obvious way, giving a fibered category. Let $PS$
denote the corresponding groupoid.
\end{definition}
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
It can be checked (we omit the details) that the groupoid $PS$
is an Artin algebraic stack.
\section{Moduli for perverse sheaves}
Let us fix data (D) as above.
\begin{theorem}
There exists an affine scheme of finite type over $C\!\!\!\!I$, which is
a coarse moduli scheme for finite descriptions
${\bf D}=(E,\rho,F_a,\rho_a,C_a,V_a)$ relative to {\rm (D)} with
fixed numerical data $n=\dim E$ and $n_a=\dim F_a$. The closed
points of this moduli scheme are the S-equivalence classes of
finite descriptions with given numerical data $(n,n_a)$.
\end{theorem}
Using remark \ref{rem4} we get
\begin{corollary}
There exists an affine scheme of finite type over $C\!\!\!\!I$, which is
a coarse moduli scheme for Verdier objects ${\bf V}=({\cal E},{\cal F},C,V)$ (or
perverse sheaves on $(X,S)$) with fixed numerical data $n={\rm
rank} {\cal E}$ and $n_a={\rm rank}{\cal F}|B^*_a$. The closed points of
this moduli scheme are the S-equivalence classes of Verdier
objects with given numerical data $(n,n_a)$.
\end{corollary}
The above corollary and its proof does not need $X$ to be a
complex projective variety, and the algebraic structure of $X$
does not matter. All that is needed is that the fundamental
group of $X-S$ and that of each $S_a$ is finitely generated.
The rest of this section contains the proof of the above
theorem.
\begin{proposition}\label{prop5}
(1) Let ${\bf D}$ be a finite description, and let $\mathop{\rm gr}\nolimits({\bf D})$ be its
semisimplification. Then there exists a family ${\bf D}_T$ of finite
descriptions parametrized by the affine line $T=A^1$ such that
the specialization ${\bf D}_0$ at the origin $0\in T$ is isomorphic
to $\mathop{\rm gr}\nolimits({\bf D})$, while ${\bf D}_t$ is isomorphic to ${\bf D}$ at any $t\ne
0$.
(2) In any family of finite descriptions parametrized by a
scheme $T$, each S-equiva\-len\-ce class (Jordan-H\"older class)
is Zariski closed in $T$.
\end{proposition}
\paragraph{Proof}
The statement (1) has a proof by standard arguments which we
omit. To prove (2), first note that if ${\bf D}_T$ is any family and
${\bf D}'$ a simple finite description, then the condition that
${\bf D}' \times \{ t \}$ is a quotient of ${\bf D}_t$ defines a closed
subscheme of $T$. From this, (2) follows easily.
\paragraph{Construction of Moduli} Let $E$ and $F_a$ be vector
spaces with $\dim(E)=n$ and $\dim(F_a) =n_a$. Let $\cal R$ be
the affine scheme of all representations $\rho$ of $G$ in $E$,
made as follows. Let $h_1,\ldots , h_r$ be generators of $G$.
Then $\cal R$ is the closed subscheme of the product $GL(E)^r$
defined by the relations between the generators. Similarly,
choose generators for each $G_a$, and let ${\cal R}_i$ be the
corresponding affine scheme of all representations $\rho_a$ of
$G_a$ in $F_a$.
Let $$A \subset {\cal R} \times \prod_a ({\cal R}_a \times
Hom(E,F_a) \times Hom(F_a,E))$$ be the closed subscheme defined
by condition \ref{def2}.3 above. Its closed points are tuples
$(\rho,\rho_a, C_a, V_a)$ where the linear maps $C_a:E\to F_a$
and $V_a:F_a\to E$ are $G_a$-equivariant under the
representations $\rho \phi_a: G_a\to GL(E)$ and $\rho_a: G_a\to
GL(F_a)$, and satisfy $V_aC_a = \rho (\tau_a )-1$ in $GL(E)$,
and $C_aV_a = \rho_a(\tau_a) -1$ in $GL(F_a)$ for each $a$.
The product group ${\cal G} =GL(E) \times (\prod_a GL(F_a))$ acts on
the affine scheme $A$ by the formula
$$(\rho ,\rho_a, C_a,V_a)\cdot (g,g_a) =
(g^{-1}\rho g, g_a^{-1}\rho_a g_a,g_a^{-1}C_ag, g^{-1}V_ag_a).$$
The orbits under this action are exactly the isomorphism classes
of finite descriptions. The moduli of finite descriptions is
the good quotient ${\cal F} =A//{\cal G}$, which exists as $A$ is affine and
${\cal G}$ is reductive. It is an affine scheme of finite type over
$C\!\!\!\!I$. It follows from \ref{prop5}.1 and \ref{prop5}.2 and
properties of a good quotient that the Zariski closures of two
orbits intersect if and only if the two finite descriptions are
S-equivalent. Hence closed points of ${\cal F}$ are S-equivalence
classes (Jordan-H\"older classes) of finite descriptions.
\section{Riemann-Hilbert morphism}
To any Malgrange object ${\bf M}$, there is an obvious associated
Verdier object ${\bf V}({\bf M})$ obtained by applying the de~Rham
functor to each component of ${\bf M}$. This defines a functor,
which is in fact an equivalence of
categories from Malgrange objects to Verdier objects. We have
already defined a functor $\eta$ from pre-${\cal D}$-modules with good
residual eigenvalues to Malgrange objects. Composing, we get
an exact functor from pre-${\cal D}$-modules with good residual eigenvalues
to Verdier objects. Choosing base points in $X$ and paths as in
remark \ref{rem4} we get an exact functor ${\cal R\!\!H}$ from pre-${\cal D}$-modules
to finite descriptions. This construction works equally well
for families of pre-${\cal D}$-modules, giving us a holomorphic family
${\cal R\!\!H} ({\bf E}_T)$ of Verdier objects (or finite descriptions)
starting from a holomorphic family ${\bf E}_T$ of pre-${\cal D}$-modules
with good residual eigenvalues.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
Even if ${\bf E}_T$ is an algebraic family of pre-${\cal D}$-modules with
good residual eigenvalues, the
associated family ${\cal R\!\!H} ({\bf E}_T)$ of Verdier objects may not be
algebraic.
\refstepcounter{theorem}\paragraph{Remark \thetheorem} If a semistable pre-${\cal D}$-module has good residual
eigenvalues, then any other semistable pre-${\cal D}$-module in its
S-equivalence class has (the same) good residual eigenvalues. Hence the
analytic open subset $T_g$ of the parameter space $T$ of any
analytic family of semistable pre-${\cal D}$-modules defined by the
condition that residual eigenvalues are good is saturated under
S-equivalence.
\begin{lemma} If two semistable pre-${\cal D}$-modules with good
residual eigenvalues are S-equivalent (in the sense of
definition \ref{defstable} above), then the associated finite
descriptions are S-equivalent (that is,
Jordan-H\"older equivalent).
\end{lemma}
\paragraph{Proof} Let ${\bf E} =(E_0,E_1,s,t)$ be a pre-${\cal D}$-module with good
residual eigenvalues (that is, the logarithmic connection $E_0$
has good residual eigenvalues on each component of $S$) such that
$s\otimes t=0$. Then one can easily construct a family of
pre-${\cal D}$-modules parametrized by the affine line $A^1$ which is the
constant family ${\bf E} $ outside some point $P\in A^1$, and
specializes at $P$ to ${\bf E} '=(E_0,E_1,0,0)$. Let $\phi :A^1 \to F$
be the resulting morphism to the moduli ${\cal F}$ of finite
descriptions. By construction, $\phi$ is constant on $A^1 -P$,
and so as ${\cal F}$ is separated, $\phi$ is constant. As the points
of ${\cal F}$ are the S-equivalence classes of finite descriptions, it
follows that the finite descriptions corresponding to ${\bf E} $ and
${\bf E} '$ are S-equivalent. Hence the S-equivalence class of the
finite description associated to a pre-${\cal D}$-module depends only
on the reduced module made from the pre-${\cal D}$-module. Now we must
show that any two S-equivalent (in the sense of \ref{defstable})
reduced semistable modules have associated finite descriptions
which are again S-equivalent (Jordan-H\"older equivalent). This
follows from the deformation given in \ref{deform} by using the
separatedness of ${\cal F}$ as above.
\bigskip
Now consider the moduli ${\cal P} = H//{\cal G}$ of semistable
pre-${\cal D}$-modules. Let $H_g$ be the analytic open subspace of $H$
where the family parametrized by $H$ has good residual
eigenvalues. By the above remark, $H_g$ is saturated under $H\to
{\cal P}$. Hence its image ${\cal P} _g\subset {\cal P}$ is analytic open. Let
$\phi :H_g \to {\cal F}$ be the classifying map to the moduli ${\cal F}$ of
finite descriptions for the tautological family of
pre-${\cal D}$-modules parametrized by $H$, which is defined because
of the the above lemma. By the analytic universal property of
GIT quotients (see Proposition 5.5 of Simpson [S] and the remark
below), $\phi$ factors through an analytic map ${\cal R\!\!H} :P_g \to
{\cal F}$, which we call as the {\sl Riemann-Hilbert morphism}.
\refstepcounter{theorem}\paragraph{Remark \thetheorem} In order to apply Proposition 5.5 of [S], note that
a ${\cal G}$-linear ample line bundle can be given on $H$ such that
all points of $H$ are semistable. Moreover, though the
proposition 5.5 in [S] is stated for semisimple groups, its proof
works for reductive groups.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
The Riemann-Hilbert morphism can also be thought of as a
morphism from the analytic stack of pre-${\cal D}$-modules with good
residual eigenvalues to the
analytic stack of perverse sheaves.
\section{Some properties of the Riemann-Hilbert morphism}
In this section we prove some basic properties of the morphism
${\cal R\!\!H}$, which can be interpreted either at stack or at moduli
level.
\begin{lemma}[Relative Deligne construction]\label{lemdel}
(1) Let $T$ be the spectrum of an Artin local algebra of finite
type over $C\!\!\!\!I$, and let $\rho_T$ be a family of representations
of $G$ (the fundamental group of $X-S$ at base point $P_0$)
parametrized by $T$. Let $E$ be a logarithmic connection with
eigenvalue not differing by nonzero integers, such that the
monodromy of $E$ equals $\rho$, the specialization of $\rho_T$.
Then there exists a family $E_T$ of logarithmic connections
parametrized by $T$ such that $E_0=E$ and $E_T$ has monodromy
$\rho_T$.
(2) A similar statement is true for analytic germs of
$G$-representations.
\end{lemma}
\paragraph{Proof} For each $a$, choose a fundamental domain $\Omega_a$ for
the exponential map ($z\mapsto \exp (2\pi \sqrt{-1} z)$) such
that the eigenvalues of the residue $R_a(E)$ of $E$ along $S_a$
are in the interior of the set $\Omega_a$. As the differential
of the exponential map $M(n,C\!\!\!\!I )\to GL(n,C\!\!\!\!I)$ is an isomorphism
at all those points of $M(n,C\!\!\!\!I )$ where the eigenvalues do not
differ by nonzero integers, using the fundamental domains
$\Omega_a$ we can carry out the Deligne construction locally to
define a family $E_T$ of logarithmic connections on $(X,S)$ with
$E_0=E$, which has the given family of monodromies.
Note that for the above to work, we needed the inverse function
theorem, which is valid for Artin local algebras.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
If in the above, the family $\rho_T$ of monodromies is a
constant family (that is, pulled back under $T\to \mathop{\rm Spec}\nolimits (C\!\!\!\!I )$),
then $E_T$ is also a constant family as follows from Proposition
5.3 of [N].
\begin{proposition}[`Injectivity' of ${\cal R\!\!H} $]\label{propinj}
Let ${\bf E}=(E,F,t,s )$ and ${\bf E}'=(E',F',t',s')$ be
pre-${\cal D}$-modules having good residual eigenvalues, such that for
each $a$, the eigenvalues of the residues of $E$ and $E'$ over
$S_a$ belong a common fundamental domain $\Omega_a$ for the
exponential map $exp :C\!\!\!\!I \to C\!\!\!\!I ^*:z\mapsto \exp (2\pi
\sqrt{-1}z)$. Then ${\bf E}$ and ${\bf E}'$ are isomorphic if and only
if the finite descriptions ${\cal R\!\!H}({\bf E})$ and ${\cal R\!\!H}({\bf E}')$ are
isomorphic.
\end{proposition}
\paragraph{Proof}
It is enough to prove that if the Malgrange objects ${\bf M}$ and
${\bf M}'$ are isomorphic, then so are the pre-${\cal D}$-modules ${\bf E}$
and ${\bf E}'$. First use the fact that, in a given meromorphic
connection $M$ on $X-S$ (or on $N_{S,X}^{}-S$), there exists one
and only one logarithmic connection having its residue along
$S_a$ in $\Omega_a$ for each $a$, to conclude that $E$ and $E'$
(resp. $F$ and $F'$) are isomorphic logarithmic modules. To
obtain the identification between $s$ and $s'$ (resp. $t$ and
$t'$), use the fact that these maps are determined by their
value at a point in each connected component $N_{S_a,X}^{}-S_a$
of $N_{S,X}^{}-S$ and this value is determined by the
corresponding $C_a$ or $C'_a$ (resp. $V_a$ or $V'_a$).
\begin{proposition}[Surjectivity of ${\cal R\!\!H}$]\label{propsurj}
Let ${\bf D}$ be a finite description, and let $\sigma_a:C\!\!\!\!I ^*\to C\!\!\!\!I
$ be set theoretic sections of $z\mapsto \exp (2\pi
\sqrt{-1}z)$. Then there exists a pre-${\cal D}$-module ${\bf E}$ whose
eigenvalues of residue over $S_a$ are in image$(\sigma_a)$, for
which ${\cal R\!\!H}({\bf E})$ is isomorphic to ${\bf D}$.
\end{proposition}
\paragraph{Proof} This follows from proposition \ref{prop3}.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
The propositions \ref{propinj} and \ref{propsurj} together say
that the set theoretic fiber of ${\cal R\!\!H}$ over a given finite
description is in bijection with the choices of `good'
logarithms for the local monodromies of the finite description
(here `good' means eigenvalues do not differ by nonzero
integers).
\begin{proposition}[Tangent level injectivity for ${\cal R\!\!H}$]\label{propinfinj}
Let $(E,F,t,s)_T$ be a family of pre-${\cal D}$-modules having good
residual eigenvalues parametrized by the spectrum $T$ of an
Artinian local algebra. Let the family ${\cal R\!\!H}(E,F,t,s)_T$ of
finite descriptions parametrized by $T$ be constant (pulled back
under $T\to \specC\!\!\!\!I$). Then the family $(E,F,t,s)_T$ is
also constant.
\end{proposition}
\paragraph{Proof} This is just the rigidity result of proposition \ref{prop4}.
\begin{proposition}[Infinitesimal surjectivity for ${\cal R\!\!H}$]\label{propinfsurj}
Let $T$ be the spectrum of an Artin local algebra of finite type
over $C\!\!\!\!I$, and let ${\bf D}$ be a family of finite descriptions
parametrized by $T$. Let ${\bf E}$ be a pre-${\cal D}$-module having good
residual eigenvalues such that ${\cal R\!\!H}({\bf E})={\bf D}_{\xi}$, the
restriction of ${\bf D}$ over the closed point $\xi$ of $T$. Then
there exists a family ${\bf E}'_T$ of pre-${\cal D}$-modules having good
residual eigenvalues with ${\bf E}'_{\xi}={\bf E}$ and
${\cal R\!\!H}({\bf E}_T)={\bf D}_T$.
\end{proposition}
\paragraph{Proof} This follows from lemma \ref{lemdel} and the proof of
proposition \ref{prop3} which works for families over Artin
local algebras.
\begin{theorem}
The analytic open substack of the stack (or analytic open subset
of the moduli) of pre-${\cal D}$-modules on $(X,S)$, where ${\bf E} $ has
good residual eigenvalues, is an analytic spread over the stack
(or moduli) of perverse sheaves on $(X,S)$ under the
Riemann-Hilbert morphism.
\end{theorem}
\paragraph{Proof} This follows from propositions \ref{propsurj},
\ref{propinfinj} and \ref{propinfsurj} above.
Note that we have not defined ${\cal R\!\!H}$ on the closed analytic
subset $T_o$ of the parameter space of a family where ${\bf E} $ does
not have good residual eigenvalues. Note that $T_o$ is defined
by a `codimension one' analytic condition, that is, if $T$ is
nonsingular, and if $T_o$ is a nonempty and proper subset of
$T$, then $T_o$ has codimension 1 in $T$. However, it follows
from Proposition \ref{propremov} below that the morphism ${\cal R\!\!H}$
on $T-T_o$ can be extended to an open subset of $T$ of
complementary codimension at least two. However, on the extra
points to which it gets extended, it may not represent the de
Rham functor.
\begin{proposition}[Removable singularities for ${\cal R\!\!H}$]\label{propremov}
Let $T$ be an open disk in $C\!\!\!\!I$ centered at $0$. Let
${\bf E}_T=(E,F,t,s)_T$ be a holomorphic family of pre-${\cal D}$-modules
parametrized by $T$. Let the restriction $E_z$ have good
residual eigenvalues for all $z\in T-\{0\}$. Then there exists a
holomorphic family ${\bf D}_U$ of finite descriptions parametrized
by a neighbourhood $U$ of $0\in T$ such that on $U-\{0\}$, the
families ${\cal R\!\!H}({\bf E}_U\vert U-\{0\}) $ and ${\bf D}_{U- \{0\}}$ are
isomorphic.
\end{proposition}
If at $z=0$ the logarihmic connection $E$ does not have good
residual eigenvalues, it is possible to change it to obtain a
new logarithmic connection having good residual eigenvalues.
This is done by the classical `shearing transformation' that we
adapt below ({\sl inferior and superior modifications} for
pre-${\cal D}$-modules). This can be done in family and has no effect
on the Malgrange object at least locally.
\begin{definition}\rm
If $E$ is a vector bundle on $X$, and $V$ a subbundle of the
restriction $E\vert S$, then the inferior modification ${_VE}$
is the sheaf of all sections of $E$ which lie in $V$ at points
of $S$. This is a locally free subsheaf of $E$ (but not
generally a subbundle). The superior modification $^VE$ is the
vector bundle ${\cal O}_X(S)\otimes {_VE}$.
\end{definition}
\refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{rem6}
If $E\vert S =V\oplus V'$, then we have a canonical isomorphism
$${_VE}\vert S \to V \oplus ({\cal N}^*_{S,X}\otimes V')$$ and hence
also a canonical isomorphism $$^VE|S\to ({\cal N}_{S,X}\otimes
V)\oplus V'$$
\refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{rem7}
If $(E,\nabla)$ is a logarithmic connection on $(X,S)$ and $V$
is invariant under the residue, then it can be seen that $_VE$
is invariant under $\nabla$, so is again a logarithmic
connection. We call it the inferior modification of the
logarithmic connection $E$ along the residue invariant subbundle
$V\subset E\vert S$. It has the effect that the residual
eigenvalues along $V$ get increased by $1$ when going from $E$
to $_VE$. As ${\cal O}_X(S)$ is canonically a logarithmic connection,
the superior modification $^VE$ is also a logarithmic
connection, with the residual eigenvalues along $V$ getting
decreased by $1$.
\bigskip
Let $(E,F,t,s)$ be pre-${\cal D}$-module on $(X,S)$ such that $E$ has
good residual eigenvalues. Let us for simplicity of writing
assume that $S$ is connected. Let $E|S = \oplus_\alpha E^\alpha$
and $F=\oplus_\alpha F^\alpha$ be the respective direct sum
decompositions into generalized eigen subbundles for the action
of $\theta$. Then (see also remark \ref{rem3}) as $\theta$
commutes with $s$ and $t$, it follows that $t(E^\alpha) \subset
F^\alpha$ and $s(F_\alpha)\subset E^\alpha$. Moreover, when
$\alpha\ne 0$, the maps $s$ and $t$ are isomorphisms between
$E^\alpha$ and $F^\alpha$.
Now let $\alpha\ne 0$. Let $V=E^\alpha$ and $V'=\oplus_{\beta\ne
\alpha}E^\beta$. Let $F'' = \oplus_{\beta\ne \alpha}F^\beta$.
Let $F' = F^\alpha \oplus {\cal N}^*_{S,X}\otimes F''$. Let
$E'={_VE}$. Then using \ref{rem6} and the above, we get maps
$t':E'|S \to F'$ and $s':F'\to E'|S$ such that $(E',F',s',t')$
is a pre-${\cal D}$-module.
\begin{definition}\label{definfmod}\rm
We call the pre-${\cal D}$-module $(E',F',s',t')$ constructed above as
the inferior modification of $(E,F,s,t)$ along the generalized
eigenvalue $\alpha\ne 0$.
\end{definition}
Similarly, we can define the superior modification along a
generalized eigenvalue $\alpha\ne 0$ by tensoring with
${\cal O}_X(S)$.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
The construction of inferior or superior modification of
pre-${\cal D}$-modules can be carried out over a parameter space $T$
(that is, for families) provided the subbundles $V$ and $V'$
form vector subbundles over the parameter space $T$ (their ranks
are constant).
\paragraph{Proof of \protect\ref{propremov}}
If the restriction $E= E_{T\vert z=0}$ has good residual
eigenvalues, then ${\cal R\!\!H} {\bf E}_T$ has the desired property. So
suppose $E$ does not have good residual eigenvalues.
We first assume for simplicity of writing that $E$ fails to have
good residual eigenvalues because its residue $R_a$ on $S_a$ has
exactly one pair $(\alpha,\alpha-1)$ of distinct eigenvalues
which differ by a positive integer, with $\alpha-1\ne 0$. Let
$f_T$ be the characteristic polynomial of $R_{a,T}$. Then $f_0$
has a factorization $f_0 = gh$ such that the polynomials $g$ and
$h$ are coprime, $g(\alpha)=0$ and $h(\alpha-1)=0$. On a
neighbourhood $U$ of $0$ in $T$ we get a unique factorization
$f_T\vert U = g_Uh_U$ where $g_U$ specializes to $g$ and $h_U$
specializes to $h$ at $0$. By taking $U$ small enough, we may
assume that $g_U$ and $h_U$ have coprime specializations at all
points of $U$. Let $V_U$ be the kernel of the endomorphism
$g_U(R_{a,U})$ of the bundle $E_{a,U}$. If $U$ is small enough
then $F_U$ is a subbundle. Now take the inferior modification
${\bf E}'= ({_V}E_U, F'_U,t'_U,s'_U)$ of the family $(E,F,t,s)_U$ as
given by construction \ref{definfmod}. Then ${_VE}_U$ is a
family of logarithmic connections having good residual
eigenvalues, so by definition ${\bf E}'$ has good residual
eigenvalues.
If $(0,1)$ are the eigenvalues, then use superior modification
along the eigenvalue $1$.
If $R_a$ has eigenvalues $(\alpha,\alpha-k)$ for some integer
$k\ge 1$, then repeat the above inferior (or superior)
modification $k$ times (whether to choose an inferior or
superior modification is governed by the following restriction :
the multiplicity of the generalized eigenvalue $0$ should not
decrease at any step). By construction, we arrive at the desired
family $(E',F',s',t')$.
\section*{References} \addcontentsline{toc}{section}{References}
[L] Laumon, G. : Champs alg\'ebriques. Preprint no. 88-33,
Universit\'e Paris Sud, 1988.
[Mal] Malgrange, B. : Extension of holonomic ${\cal D}$-modules, in
Algebraic Analysis (dedicated to M. Sato), M. Kashiwara and T.
Kawai eds., Academic Press, 1988.
[Ne] Newstead, P.E. : {\sl Introduction to moduli problems and
orbit spaces}, TIFR lecture notes, Bombay (1978).
[N] Nitsure, N. : Moduli of semistable logarithmic connections.
J. Amer. Math. Soc. 6 (1993) 597-609.
[S] Simpson, C. : Moduli of representations of the fundamental
group of a smooth projective variety - I, Publ. Math. I.H.E.S.
79 (1994) 47-129.
[Ve] Verdier, J.-L. : Prolongements de faisceaux pervers
monodromiques, Ast\'erisque 130 (1985) 218-236.
\bigskip
Addresses:
School of Mathematics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Bombay 400 005, India. e-mail:
[email protected]
Centre de Mathematiques, CNRS ura169, Ecole Polytechnique,
Palaiseau cedex, France. e-mail: [email protected]
\end{document}
|
1995-08-28T06:20:09 | 9503 | alg-geom/9503004 | en | https://arxiv.org/abs/alg-geom/9503004 | [
"alg-geom",
"dg-ga",
"math.AG",
"math.DG"
] | alg-geom/9503004 | Rogier Brussee | Rogier Brussee | The canonical class and the $C^\infty$ properties of K\"ahler surfaces | 38 pages. Hardcopy available upon request. Latex 2e with amsart v.
1.2 or AMSlaTeX version 1.1. reason for resubmission: Changed title,
corrected serious error in the argument for $p_g = 0$, major technical
improvements in the handling of the localised Euler class of infinite
dimensional bundles, referred some analysis to the literature and made some
general improvements in the exposition | null | null | Bielefeld Preprint 95-037 | null | We give a self contained proof using Seiberg Witten invariants that for
K\"ahler surfaces with non negative Kodaira dimension (including those with
$p_g = 0$) the canonical class of the minimal model and the $(-1)$-curves, are
oriented diffeomorphism invariants up to sign. This implies that the Kodaira
dimension is determined by the underlying differentiable manifold (Van de Ven
Conjecture). We use a set up that replaces generic metrics by the construction
of a localised Euler class of an infinite dimensional bundle with a Fredholm
section. This allows us to compute the Seiberg Witten invariants of all
elliptic surfaces with excess intersection theory. We then reprove that the
multiplicities of the elliptic fibration are determined by the underlying
oriented manifold, and that the plurigenera of a surface are oriented
diffeomorphism invariants.
| [
{
"version": "v1",
"created": "Fri, 10 Mar 1995 21:48:55 GMT"
},
{
"version": "v2",
"created": "Fri, 25 Aug 1995 16:23:58 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Brussee",
"Rogier",
""
]
] | alg-geom | \section{Preparation}
We first prove the corollaries from the main theorems~\ref{main}
and~\ref{ellmult}.
\begin{pf}
Corollary \ref{spheres}. Let $S$ be a positive sphere on a surface with
$\kappa \ge 0$. Blow up $n =S^2 + 1$ times. Now $e = S + E_1 + \cdots
+E_n$ is a $(-1)$-sphere. Hence there is a $(-1)$-curve $E_0$ such that
$e = \pm E_0 \in H_2(X,{\Bbb Z})$. Then $S = \pm E_0$ , or $e = E_0 = E_1$
say. The first possibility leads to the contradiction $E_0^2 \ge 0$,
the second to $S = 0 \in H_2(X,{\Bbb Z})$. (Reducing non negative spheres to
$(-1)$-spheres is a well known trick, but I forgot where I read it
precisely.)
Corollary~\ref{-infty} follows from corollary~\ref{spheres}.
Corollary~\ref{Kodaira}.By the above,
a K\"ahler surface is of Kodaira dimension $-\infty$ if it
contains a non trivial $(0)$-sphere. Clearly all ruled
surfaces contain one. To deal with ${\Bbb P}^2$, note that there is no surface
with $b_+ =b_1=0$. Thus diffeomorphisms between surfaces with $b_2=1$,
$b_1=0$ are automatically orientation preserving. Then a surface
diffeomorphic to ${\Bbb P}^2$ must contain a $(+1)$-sphere, and is therefore of
Kodaira dimension $-\infty$. Since $b_2 = 1$ it must in fact be equal to
${\Bbb P}^2$ (alternatively use Yau's result that
${\Bbb P}^2$ is the only surface with the homotopy type of ${\Bbb P}^2$
\cite[Theorem 1.1]{BPV}, but this is a
deep theorem).
We conclude that
Kodaira dimension $-\infty$ can be characterised by just diffeomorphism
type. Without loss of generality we can therefore assume that
$\kappa \ge 0$.
If $\Kmin^2 > 0$, then $X$ is of general type. If $\Kmin^2 = 0 $
and $\Kmin$ is not torsion, then $\kappa(X) = 1$, finally if $\Kmin$ is
torsion, $\kappa(X) = 0$. This proves that Kodaira dimension is determined
by the oriented diffeomorphism type. If $X$ and $Y$ are orientation
reversing diffeomorphic, both are minimal, otherwise one of them would
contain a positive sphere. Then necessarily either $K_X^2 = K_Y^2 = 0$,
or both have $K_X^2 ,K_Y^2 >0$, i.e. $X$ and $Y$ are of general type.
Now copy the argument of \cite[lemma S.4]{FM}: for minimal surfaces
with $\kappa = 0,1$, the signature $\sign = \numfrac13(K^2 - 2e)\le 0$.
Thus $\sigma(X) = -\sigma(Y) = 0$, and $e(X) = e(Y) = 0$. In Kodaira
dimension $0$, this leaves only tori and hyperelliptic surfaces, which
can fortunately be recognised by homotopy type \cite[lemma 2.7]{FM}.
Corollary~\ref{plurigenera}. Since $P_1=p_g$ is an oriented topological
invariant we will whence assume that $n\ge 2$.
We have to distinguish between the
different Kodaira dimensions.
For surfaces of general type (i.e $\kappa = 2$) we argue as follows.
The plurigenera $P_n$ and $\chi(O_X)$ are birational invariants.
Then by Ramanujan vanishing and Riemann Roch (cf. \cite[corollary
VII.5.6]{BPV}) we have
\begin{equation}\label{gtpluri}
P_n(X) = P_n(\Xmin) = {\numfrac12} n(n-1) \Kmin^2 + \chi({\mathcal O}_X)
\end{equation}
Since $\chi({\mathcal O}_X)$ is an oriented topological
invariant the $P_n$ are oriented diffeomorphism
invariants in this case.
For surfaces with Kodaira dimension $0$ or
$1$ with a fundamental group that is not finite cyclic, we simply quote
\cite[S.7]{FM}. For surfaces with finite cyclic fundamental group, it
follows from the invariance of the multiplicities and the canonical
bundle formula which gives an explicit formula for $P_n(X)$ in terms of
the multiplicities and $\chi(O_X)$. (see \cite[lemma I.3.18, prop.
I.3.22]{FM}).
\end{pf}
Here is an other easy corollary
\begin{Corollary}
Every $(-2)$-sphere $\tau$ is orthogonal to $\Kmin$ . If there is a
$(-1)$-curve $E_1$ such that $\tau\cdot E_1 \ne 0$, then there is a
$(-1)$-curve $E_2$ such that $\tau = \pm E_1 \pm E_2 \in H_2(X,{\Bbb Z})$.
\end{Corollary}
\begin{pf}
Let $R_\tau $ be the reflection in $\tau$. It is represented by a
diffeomorphism with support in a neighborhood of $\tau$. By the
invariance of $\Kmin$ up to sign, $R_\tau \Kmin = \Kmin + (\tau \cdot
\Kmin) \tau = \pm \Kmin$. But if $\Kmin \ne 0 \in H^2(X,{\Bbb Q})$, then
$\tau$ and $\Kmin$ are indepent, since $\tau^2 = -2 $ and $\Kmin^2 \ge
0$. Thus in either case $(\tau, \Kmin) = 0$. Moreover if $E_1$ is a
$(-1)$-curve then either $R_\tau E_1 = E_1$, $R_\tau E_1 = -E_1$, or
there is a different $(-1)$-curve $E_2$ such that $R_\tau(E_1) = \pm
E_2$. The first possibility gives $\tau\cdot E_1 = 0$, the second
$(\tau \cdot E_1)^2 = 2$ i.e. is impossible, and the third $(\tau\cdot
E_1) = \pm 1$. The statement follows.
\end{pf}
It will be convenient to first prove the main theorem~\ref{main} with
(co)homology groups with ${\Bbb Q}$ coefficients, and later mop up to prove
the theorem over ${\Bbb Z}$. Theorem~\ref{main} mod torsion is a formal
consequence of the existence of a set of basic classes
$$
\K(X) = \{K_1, K_2 \ldots \} \subset H^2(X,{\Bbb Z})
$$
functorial under oriented diffeomorphism between $4$-manifolds with
$b_+ \ge 1$, and having the following properties:
\begin{Properties} \label{*}
For every K\"ahler surface $X$ of non negative Kodaira dimension
\begin{enumerate}
\item \label{i} the $K_i$ are of type $(1,1)$ i.e. represented by divisors,
\item \label{ii} if $X$ is minimal, then for every K\"ahler form $\Phi$,
$\deg_\Phi(K_X )\ge |\deg_\Phi(K_i)|$,
\item \label{iii} if $\~X \m@p--\rightarrow{\sigma} X$ is the blow-up of a point $x
\in X$,
then $\sigma_*\K(\~X) = \K(X)$.
\item \label{iv} every $K_i$ is characteristic i.e.
$K_i \equiv w_2(X) \pmod 2$,
\item \label{v} $K_X \in \K$.
\end{enumerate}
\end{Properties}
In the case that $X$ is an algebraic surface we could replace
item~\ref{ii} by weaker and
more geometric requirement that
$2g(H) - 2 \ge H^2 + |K_i\cdot H|$ for every
very ample divisor $H$ without changing the results.
We will see later that
Seiberg Witten theory will give such an inequality for all
surfaces minimal or not.
This should not be confused with a Thom conjecture type of
statement, since our methods do not give information about the minimal
genus for arbitrary smooth real surfaces in a homology class.
It is also clearly impossible to have a degree inequality like
property \ref{ii}
for all K\"ahler forms if $X$ is rational or ruled.
Recall that for algebraic surfaces, the Mori cone $\NEbar(X) \subset
H_2(X,{\Bbb R})$ is the closure of the cone generated by effective curves. It
is dual to the nef (or K\"ahler) cone. In other words, the numerical
equivalence class of a curve $D$ lies in $\NEbar(X)$ if and only if
$H\cdot D \ge 0$ for all $H$ ample. For a K\"ahler surface $(X,\Phi)$,
it will be convenient to define the
nef cone as closure of the positive cone in $H^{1,1}(X)
\subset H^2(X,{\Bbb R})$ spanned by all K\"ahler forms, and containing
$\Phi$. The Mori cone $\NEbar$ is then just the dual cone in
$H_2(X,{\Bbb R})\cap {H^{1,1}}^{\scriptscriptstyle\vee}$ i.e.
$$
\NEbar=
\{C \in {H^{1,1}}^{\scriptscriptstyle\vee} \subset H_2(X,{\Bbb R}) \mid \int_C \omega \ge
0, \txt{for all K\"ahler forms $\omega$}\}.
$$
(With this definition, a line bundle is nef iff for all $\epsilon > 0$,
it admits a metric such that the curvature form $F$ has
$\numfrac{\sqrt{-1}}{2\pi} F \ge -\epsilon \Phi$. A class $\omega \in
\NEbar$ if there exists a sequence of closed positive currents of type
$(1,1)$ converging to the dual of $\omega$, i.e is $\NEbar$ dual to
$N_{\text{psef}}$ in \cite[proposition 6.6]{Demailly}. I am grateful to
Demailly for explaining this to me). We will freely identify homology
and cohomology by Poincar\'e duality.
\begin{Lemma} \label{decomp}
If a class $L \in H^{1,1}(X)$ satisfies $\deg_\Phi(K_X)\ge
|\deg_\Phi(L)|$ for all K\"ahler forms $\Phi$, then
there is a unique decomposition of the canonical divisor
$K_X = D_+ + D_- $ with $D_+$, $D_- \in \NEbar(X)$ such that
$L= D_+ - D_-$.
\end{Lemma}
\begin{pf}
Define $D_\pm = {\numfrac12}(K_X \pm L)$. Then $K_X = D_+ + D_-$,
$L= D_+ - D_- $, and $D_\pm \in \NEbar$.
\end{pf}
The following simple lemma is a minor generalisation of the
fact that the canonical divisor of a surface of general type is
numerically connected \cite[VII.6.1]{BPV}.
\begin{Lemma} \label{connectedness}
Let $X$ be a minimal K\"ahler surface of non negative Kodaira
dimension. Suppose there is a decomposition $ K_X = D_+ + D_-$ with
$D_+$, $D_- \in \NEbar(X)\subset H^{11}(X)$. Then $D_+ \cdot D_- \ge 0
$, with equality if and only if say $K_X \cdot D_+ = D_+^2 = 0$. Thus
if $X$ is of general type then $D_+ = 0$, if $\kappa(X) = 1$, then $D_+
= \lambda K_X$ with $0 \le \lambda \le 1$, and finally $D_+ = D_- = 0$
if $\kappa(X) = 0$.
\end{Lemma}
\begin{pf}
First assume that $D_+^2 \le 0$. Since $K_X$ is nef, $D_+ \cdot D_- =
(K_X-D_+)\cdot D_+ \ge -D_+^2 \ge 0$, with equality iff $K_X \cdot D_+ =
D_+^2 = 0$. If $D_+^2 >0$ and $D_-^2 >0 $, then using the K\"ahler
form $\Phi$, we can write $D_+ = \alpha \Phi + C_+$ and $D_- = \beta
\Phi + C_-$ with $\alpha$, $\beta >0$ and $C_\pm \in \Phi^\perp$. By
the Hodge index theorem,
$$
D_+ \cdot D_- = \alpha \beta \Phi^2 + C_+\cdot C_-
\ge \alpha\beta \Phi^2 - \sqrt{-C_+^2}\sqrt{-C_-^2} >0.
$$
The statement for surfaces of general type follows directly
from Hodge index and the fact that $K_X^2 > 0$.
If $\kappa(X) = 1$, then $K_X$ is
a generator of the unique isotropic subspace of $K_X^\perp$,
so $D_+ = \lambda K_X$, and $D_- = (1-\lambda) K_X$.
Since $K_X$, $D_+$ and $D_- \in \NEbar(X)$, $\lambda$ is bounded by
$0 \le \lambda \le 1$.
Finally if $\kappa(X) = 0$, $K_X$ is numerically trivial and,
$D_+$ and $D_-$ must be zero as well.
\end{pf}
\begin{Lemma} \label{inequality}
Let $X$ be a surface of non negative Kodaira dimension with
$(-1)$-curves $E_1, \ldots E_m$. Assume that $\K$ has
properties~\ref{*}. Then $K_i^2 \le K_X^2$ for all $K_i \in \K$, with
equality if and only if
$$
K_i = \lambda \Kmin + \sum \pm E_i \in H^2(X,{\Bbb Q})
$$
where $\lambda = \pm 1$ if $X$ is of general type, $\lambda$ is a
rational number with $|\lambda | \le 1 $ if $\kappa(X) = 1$, and
where $\lambda = 0$ if $\kappa(X) = 0$.
\end{Lemma}
\begin{pf}
By property \eqref{iii}, and \eqref{iv},
$K_i = K_{i,\min} + \sum_j (2a_{ij} + 1)E_j$.
Thus
$$
K_i^2 \le K_{i,\min}^2 - \#(-1)\hbox{-curves},
$$
with equality if and only if $a_{ij} =0$, or $-1$ for all $i,j$. Since
$K_X^2 = \Kmin^2 - \#(-1)$-curves, we can assume that $X$ is minimal.
Using property~\eqref{i} and \eqref{ii} and lemma~\ref{decomp}, write
$K_X = D_+ + D_-$ and $K_i = D_+ - D_-$, with $D_\pm \in \NEbar(X)$.
Then by lemma~\ref{connectedness} $K_i^2 = K_X^2 - 4 D_+\cdot D_- \le
K_X^2 $ with equality under the stated condition. Note that this lemma
does not use diffeomorphism invariance, nor that $K_X \in \K$.
\end{pf}
We are now in a position to formulate and prove half of the main theorem
\begin{Proposition} \label{Kcharacterisation}
Assume that for all 4-manifolds $X$ with $b_+ \ge 1$ there is a set of
basic classes $\K(X) = \{K_1, K_2, \ldots\} \subset H^2(X,{\Bbb Z})$
functorial under oriented diffeomorphism having properties~\ref{*}.
Then $\Kmin$ is an oriented $C^\infty$ invariant up to sign and
torsion, and every $(-1)$-sphere is represented by a $(-1)$-curve up
to sign and torsion.
\end{Proposition}
\begin{pf}
Using lemma~\ref{inequality} we can easily reduce the invariance of
$\Kmin$ up to sign and torsion to
showing that $(-1)$-spheres are represented by $(-1)$-curves up to sign
and torsion.
Since $K_X \in \K$, there is a nonempty subset
$\K_0 = \{ K_j\} \subset \K$ with
$K_j^2 = K_X^2 = 2e(X) + 3\sign(X)$.
Consider the projection $K_{j,\min}$ of $K_j$ to the minimal model i.e.
the projection to the orthogonal complement of the $(-1)$-spheres.
If $K_{j,\min}^2 >0$, then by lemma~\ref{inequality}, $X$ is of general
type, and $K_{j,\min} = \pm \Kmin$ up to torsion.
If $K_{j,\min}^2 = 0$, there are two possibilities. If $K_{j,\min}$ is
torsion for all $j$, then again by lemma~\ref{inequality},
$X$ is of Kodaira dimension $0$ i.e. $\Kmin$ is also
torsion.
Otherwise we choose $j$ such that $K_{j,\min} \ne 0$ has maximal
divisibility. Since $K_X \in \K_0$ our little lemma shows that,
the Kodaira dimension is $1$ and $K_{j,\min} = \pm \Kmin$.
Now let $e$ be the class of a $(-1)$-sphere in $H^2(X,{\Bbb Q})$. Without loss of
generality, we can assume that $K_X\cdot e < 0$.
Consider $R_e$ the reflection generated by a $(-1)$-sphere $e$. It is
represented by an orientation preserving diffeomorphism.
Since $\K$ is invariant under oriented diffeomorphisms, the
characterisation of basic classes with square $K_X^2$ tells us that
\begin{align}
R_e K_X &= \Kmin + \sum E_i + 2 (K_X\cdot e)e
\label{line1}
\\
&= \lambda \Kmin + \sum \pm E_i
\label{line2}
\end{align}
with $|\lambda| \le 1$.
Taking intersection with $E_i$ we find that $(E_i\cdot e)( e\cdot K_X)
=0$~or~$1$. Since $ K_X\cdot e \equiv e^2$ is odd, $e$ is either
orthogonal to all $(-1)$ curves (i.e. $ e\in H^2(\Xmin,{\Bbb Q})$) or there is
a $(-1)$-curve, say $E_1$, such that
$K_X\cdot e =E_1 \cdot e= -1$. However, $e \in H^2(\Xmin)$ implies
that $e = {\lambda -1 \over 2K_X \cdot e}\Kmin$,
which is impossible because $\Kmin^2 \ge 0$.
Thus, after renumbering the $(-1)$-curves, \eqref{line1} and
\eqref{line2} can be rewritten to
\begin{equation}\label{(-1)-sphere}
e = {\numfrac12}(1-\lambda) \Kmin + \sum _{i=1}^N E_i
\end{equation}
with $N = {\numfrac14}(1-\lambda)^2 \Kmin^2 + 1$.
Now reflect $e$ in $E_1^\perp$. $R_{E_1} e$ is yet another
$(-1)$-sphere,
so it has a representation as in equation~\eqref{(-1)-sphere}, except
possibly for an overall sign
\begin{align*}
R_{E_1}e &= {\numfrac12}(1-\lambda)\Kmin - E_1 + \sum _{i=2}^N E_i
\\
&= \pm \big({\numfrac12}(1-\mu) \Kmin + \sum_{j=1}^M E_{i_j}\big).
\end{align*}
Upon comparison, we see that the sign is minus, that $N=M =1$, and that
$0 \le 1-\lambda = \mu - 1 \le 0$ unless
$\Kmin = 0$. In other words $ e = E_1 \in H^2(X,{\Bbb Q})$.
\end{pf}
\section{The localised Euler class of a Banach bundle.}\label{top}
We will use a construction pioneered by Pidstrigatch and Pidstrigatch
Tjurin \cite{Pidstrigatch:instanton}, \cite[\S 2]{PT}, which is a
convenient and general way to define fundamental cycles for moduli
spaces arising from elliptic equations. Unfortunately their
construction is not quite in the generality we will need it, and we will
therefore set it up in fairly large generality here. The cycle is the
localised homological Euler class of an infinite dimensional bundle. It
can be used to give definitions that avoid transversality arguments
needing small deformations, generic metrics etcetera, although
transversality will be extremely useful for computations and proofs. The
construction is modeled on Fultons intersection theory and in the
complex case it makes the whole machinery of excess intersection theory
available. However, although the construction is very simple in principle,
the whole thing has turned a bit technical. On
first reading it is best to ignore the difference between \v Cech and
singular homology, and continue to proposition \ref{locEuler}, the
construction of the Euler class in the proof of this proposition
and corollary~\ref{locChern}. Some readers might even
want to continue to the next section, since we will use rather little of
the general machinery for the proofs of the theorems and corollaries in
the introduction.
\smallskip\noindent
We first make some algebraic topological preparations.
For any pair of topological spaces $A \subset X$,
homology with closed support and
with local coefficients $\locsys$ is defined as
$$
H_i^{\cl}(X,A;\locsys) = {\displaystyle \lim_{\leftarrow}}_K H_i(X,A \cup (X-K);\locsys)
$$
where we take the limit over all compacta $K \subset X - \o A$.
$H^{\cl}_*$ is functorial under proper maps. Unfortunately this
``homology theory'' suffers the same tautness problems that singular
homology has. To be able to work with well behaved cap products we will
have to complete it. The following works well enough for our purposes but
is a bit clumsy.
Suppose that $X$ is {\sl locally modelable} i.e.
is locally compact Hausdorff and has local models
which are each subsets of some ${\Bbb R}^n$. Obviously locally compact
subsets of locally modelable spaces are locally
modelable, in particular a closed subset of a local modelable space is
locally modelable.
Then for every compact subset $K \subset X - \o A$ there is a
neighborhood $U_K \supset K$ in $X$ which embeds in ${\Bbb R}^N$.
We now define
$$
\cH_i^{\cl}(X,A,\locsys) = {\displaystyle \lim_{\leftarrow}}_K \cH_i(U_K,A\cap U_K \cup
(U_K -K); \locsys)
$$
where for every pair $(Y,B)$ in a manifold $M$, \v Cech homology is defined
as
$$
\cH_i(Y,B) = {\displaystyle \lim_{\leftarrow}} \{H_i(V,W),\ (V,W) \txt{neighborhoods of}
(Y,B) \txt{in} M\}
$$
This definition depends neither on the choice of $U_K$,
nor on the embedding $U_K \lhook\nobreak\joinrel\nobreak\to {\Bbb R}^N$, since two embeddings are
dominated by the diagonal embedding, and $\cH_*(Y,B)$ does not depend on
$M$ but only on $(Y,B)$ (c.f \cite[VIII.13.16]{Dold}).
Fortunately we do not usually have to bother with \v Cech homology.
Suppose in addition that $X$ is locally contractible e.g. locally a
sub analytic set (c.f. \cite[\S I.1.7]{GoreskyMcPherson}, and the fact
that Whitney stratified spaces admit a triangulation). Then $X$ is
locally an Euclidean neighborhood retract (ENR) by \cite[IV
8.12]{Dold} and since in a Hausdorff space a finite union of ENR's is
an ENR by \cite[IV 8.10]{Dold} we can assume that $U_K$ is an ENR. Now
assume that $A$ is open. Then by \cite[prop. VIII 13.17]{Dold}
$$
\cH_* (U_K,U_K\cap A \cup (U_K-K)) \iso
H_*(U_K,U_K \cap A \cup (U_K - K)) \iso H_i(X,A \cup X-K).
$$
Thus in this case $\cH_*^{\cl}(X,A) = H_*^{\cl}(X,A)$. If $A$ is closed
and locally contractible then one should be able to organise things such
that $U_K \cap A$ is an ENR and the same conclusion would hold.
\begin{Lemma}\label{capproduct}
Let $X$ be a locally modelable space, and $Z$ a locally compact (e.g.
closed) subspace, then there are cap products
$$
\cH^i(X,X - Z,\locsys) \tensor \cH^{\cl}_j(X,\locsys') \m@p--\rightarrow{\cap}
\cH^{\cl}_{j-i}(Z, \locsys\tensor\locsys')
$$
with the following properties.
\begin{enumerate}
\item
If $Y$ is locally embeddable, $f: Y\to X$ is proper, and
$\sigma' \in \cH^{\cl}_j(Y, Y-f^{-1}(Z))$, then the push-pull formula
holds:
$$
f_*(f^* c \cap \sigma') = c \cap f_* \sigma'.
$$
\item
If $Z \lhook\nobreak\joinrel\nobreak\m@p--\rightarrow{i} W$ is proper and $W$ is locally compact,
we can increase supports i.e.
$$
c|_{(X,X-W)} \cap \sigma = i_*(c \cap \sigma).
$$
\end{enumerate}
\end{Lemma}
\begin{pf}
For every $c \in \cH^i(X,X-Z)$ and $\sigma \in \cH_j^{\cl}(X)$,
we have to construct a class $c \cap \sigma \in \cH_{i-j}(Z, Z-K)$
for a cofinal family of compacta $\{K\}$. Since $Z$ is locally compact,
every compactum $K$ is contained in a compactum $L
\subset Z$ with $L \supset \o L \supset K$. Likewise there exists a
compactum $L' \supset\!\supset L$.
By excision it suffices to construct a class in $\cH_{i-j}(L,L-K)$. Let
$U_{L'}$ be a neighborhood of $L'$ in $X$ which embeds in ${\Bbb R}^N$.
Let $V_L$, $W_{L-K} \subset V_L$, and $V_K \subset V_K$ be neighborhoods
of respectively $L$, $L_K$ and $K$ in ${\Bbb R}^N$. Define $U_L = V_L \cap
U_{L'}$. We can assume that $U_L \cap Z = U_L \cap L'$, $V_K \cap Z =
V_K \cap L$, and after replacing $V_{L-K}$ by $(V_{L-K} - (L' \cap
W_{L-K}^c) \cup V_K$, that $V_L \cap (L'-K) = W_{L-K} \cap (L' - K)$.
Then our task is to construct a class $c_L\cap \sigma_L \in
H_{i-j}(V_L,W_{L-K})$ possibly after shrinking $V_L$ and $W_{L-K}$.
We have a restriction map $\cH^i(X,X-Z) \to \cH^i(U_L,U_L-L')$. After
shrinking $V_L$ if necessary, $c|_{(U_L,U_L-L')}$ comes from a
class $c_L \in H^i(V_L,V_L -L')$.
By definition there is map
$$
\cH^{\cl}_j(X) \to \cH_j(U_L,U_L-K) \to H_j(V_L,V_L-K).
$$
Let $\sigma_L \in H_j(V_L,V_L-K)$ be the image of $\sigma$.
Now write $V_L- K = (V_L - L') \cup (W_{L-K} - K)$. Then the standard
cap product \cite[VII Def. 12.1]{Dold} gives a map
$$
H^i(V_L,V_L-L') \tensor H_j(V_L,V_L-K) \m@p--\rightarrow{\cap}
H_{j-i}(V_L,W_{L-K}-K)
$$
so we get a class $c_L \cap \sigma_L \in H_{j-i}(V_L, W_{L-K})$ as
required. Since if $K' \supset K$, choices for $K'$ will work a
fortiori for $K$, we can pass to the limit.
To prove the first property, note that since $f$ is proper, $f^{-1} Z$
is locally compact. Choose compacta $K \subset\!\subset L \subset\!\subset L'
\subset Z$ giving compacta $f^{-1}K \subset\!\subset f^{-1}L \subset\!\subset
f^{-1}L'$. Note that compacta of the form $f^{-1}K$ are a cofinal
family of compacta in $f^{-1}(Z)$. Embed neighborhoods $U_{L'} \subset
V_{L'} \subset {\Bbb R}^N$ and $U_{f^{-1}L'} \subset {\Bbb R}^M$. Now we
carry out the construction above with the diagonal embedding of
$U_{f^{-1}L'}$ in $ {\Bbb R}^{N + M}$. Let $V_{f^{-1}L'}$ be a neighborhood
of $U_{f^{-1}L'} \in {\Bbb R}^{N+M}$. We can assume that $V_{f^{-1}L'} \to
V_{L'}$ under the projection $\pi$ to ${\Bbb R}^N$. We can also assume that
$c|_{(U_L,U_L-L')}$ comes from a class $c_L \in H^i(V_L,V_L - L')$.
Finally let $\sigma_{f^{-1}L'}$ be an image of $\sigma$ in
$H_j(V_{f^{-1}L'}, \pi^{-1}W_{K-L})$. Then the first property follows
from the identity
$$
\pi_*(\pi^* c_L \cap \sigma'_{f^{-1}L'}) = c_L \cap \pi_*
\sigma'_{f^{-1}L'}
$$
in $H_j(V_l, W_{K-L})$. The second property is left to reader.
\end{pf}
A smooth manifold $X$ of dimension $n$, has an orientation system
$\orr(X)$, the sheafification of the presheaf $U \to H^n(X,X-U)$.
Equivalently, we can define $\orr(X)$ as the sheaf
$R^d\pi_*(X\times X, X\times X
-\Delta, {\Bbb Z})$, where $\Delta$ is the diagonal of $X\times X$, $\pi$
the projection on the first coordinate, and $R^d\pi_*$ the
parametrised version of the $d$ th cohomology.
Likewise
for a real vector bundle $E$ of rank $r$ there is an orientation system
$\orr(E)$, the sheafification of $H_q(E|_U,E|_U - U)$.
We have $\orr(X) = \orr (TX)^{\scriptscriptstyle\vee}$,
as can be seen immediately from the alternative description of $\orr(X)$
and excision.
A manifold $X$ has a unique fundamental class
$[X] \in H^{cl}_n(X,\orr(X))$ in singular or \v Cech homology such that
for small $U$
$$
[X]|_{X-U}\in H_d(X,X-U,H^d(X,X-U))= \rmmath{Hom}(H^d(X,X-U),H^d(X,X-U))
$$
is identified with the identity (cf \cite[p. 357]{Spanier}).
Similarly, a bundle $E$ has a
Thom class $\Phi_E \in \cH^r(E,E-X, \orr(E))$ \cite[p. 283]{Spanier}.
In turn for every section $s$ in $E$ with zero set $Z(s)$,
the Thom class defines a localised cohomological
Euler class $e(E,s) = s^*\Phi_E \in \cH^r(X, X - Z(s), \orr(E))$.
\smallskip\noindent
Let $M$ be a Banach manifold, $E$ a real Banach vector bundle on $M$
and $s$ a section of $E$ with zero set $Z(s)$.
The section induces an exact sequence
\begin{equation}\label{tosplit}
0 \m@p--\rightarrow{} E \m@p--\rightarrow{} s^* TE \m@p--\rightarrow{\pi} TM \to 0,
\end{equation}
which expresses that the vertical tangent bundle of the total space of
$E$ is canonically isomorphic to the bundle $E$. On $Z(s)$ we have a
canonical splitting of this sequence,
given by the sequence
$$
0 \m@p--\rightarrow{} TM \m@p--\rightarrow{Ts_0} s_0^*TE \m@p--\rightarrow{} E \m@p--\rightarrow{} 0
$$
defined by the zero section $s_0$, and the
identification $s^*TE|_{Z(s)} = s_0^*TE|_{Z(s)}$ over $Z(s)$.
This gives a canonical map
$$
Ds: TM|_{Z(s)} \m@p--\rightarrow{Ts} s^*TE = s_0^*TE \m@p--\rightarrow{} E|_{Z(s)}.
$$
If $D$ is a connection on $E$ then
$D(s)$ is a splitting that extends the canonical splitting over $Z(S)$
(hence the notation) but in general connections need not exist on Banach
manifolds. We will avoid choosing non canonical splittings.
\begin{Proposition}\label{locEuler}
Let $M$ be a smooth Banach manifold, $E$ a banach bundle over $M$ and $s$
a section in $E$. Assume that
\begin{enumerate}
\item
{\sloppy
The map $Ds$ is a section in the bundle $\Fred^d(TM|_{Z(s)},E|_{Z(s)})$
of Fredholm maps of index $d$. We say that
$Z(s)$ has virtual dimension $d$, and that $Ds$ is Fredholm of index $d$.
}
\item
The real line bundle
$\det(\Ind(Ds))$ is trivialised over $Z(s)$.
\end{enumerate}
Then these data define a \v Cech homology class with closed support
$$
{\Bbb Z}(E,s)= {\Bbb Z}(s) \in \cH^{\cl}_d(Z(s),{\Bbb Z})
$$
with the following properties.
\begin{enumerate}
\item\label{smoothcase}
The class ${\Bbb Z}(s) = [Z(s)]$ if $Z(s)$ is smooth of dimension $d$ and
carries the natural orientation defined by the trivialisation of
$\det(\Ind Ds)$,
\item\label{homotopy}
if $\{C\}$ is a family of closed subsets of $M$ such that $C \cap Z(s)$ is
compact for all $C$, then there is a natural map $\cH_j(Z(s)) \to
{\displaystyle \lim_{\leftarrow}}_C H_j(M,M-C,{\Bbb Z})$, and if $s_t$ is a one parameter family of
sections with this property then
${\Bbb Z}(s_0) = {\Bbb Z}(s_1)) \in {\displaystyle \lim_{\leftarrow}}_C H_d(M,M-C,{\Bbb Z})$.
\end{enumerate}
For every exact sequence
$$
0 \to E' \to E \to E'' \to 0,
$$
defined over a neighborhood of $Z(s)$,
let $s''$ be the induced section in $E''$, and $s'$ the induced
section of $E'|_{Z(s'')}$ with zero set $Z(s)$. Then
\begin{enumerate}
\setcounter{enumi}{2}
\item \label{fdeuler}
if $E'$ has finite rank
$$
{\Bbb Z}(s) = e(E'|_{Z(s'')},s')\cap {\Bbb Z}(s''),
$$
\item \label{stability}
if $Ds''|_{Z(s)}$ is surjective,
then $Z(s'')$ is smooth in a neighborhood of $Z(s)$,
$Ds':TZ(s'')|_{Z(s)} \to E'|_{Z(s)}$
is Fredholm, with $\Ind Ds' \iso \Ind Ds$ and
$$
{\Bbb Z}(E,s) = {\Bbb Z}(E'|_{Z(s'')},s').
$$
\end{enumerate}
\end{Proposition}
For property \ref{homotopy} there are two typical situations we have in
mind. One is that we have a natural connected family of sections $s_t$
such that $Z(s_t)$ is compact for all $t\in T$. In this situation we get
a homology class ${\Bbb Z}(s_{t_0}) \in H_d(M)$ independent of the choice of
$t_0$ (take $\{C\} = \{M\}$). Such will be the case in Seiberg Witten
theory. In the other case we again have a family of sections $s_t$ but
there is ``bubbling'' which invariably means we lack some a priori
estimate. For example in Donaldson theory, the moduli space of ASD
connections with curvature bounded in the $L^4$ norm is compact.
Therefore it is natural to define a family of subsets $\{\B^{\le
C}\}_{C \in {\Bbb R}^+}$ in the space $\B^*$ of all irreducible $L^2_2$
connections mod gauge, where $\B^{\le C}$ the subset of connections
with $L^4$ norm of the curvature bounded by $C$.
\begin{pf}
If $M$ (hence $E$) is a finite dimensional manifold of dimension $N+d$
then $E$ is a real vector bundle of rank $N$ with an isomorphism
$\det(E) = \det(TM)$ over $Z(s)$.
Let $[M] \in H^{\cl}_N(M,\orr(M))$ be the fundamental class,
and $\Phi_E$ the twisted Thom class of $E$ in
$H^{N-d}(E,E-M,\orr(E))$.
Define
$$
{\Bbb Z}(s) = e(E,s) \cap [M] \in \cH^{\cl}_d(Z, \orr(E)\tensor \orr(M))
= \cH^{\cl}_d(Z(s),{\Bbb Z})
$$
i.e. ${\Bbb Z}(s)$ is the Poincar\'e dual of the localised cohomological
Euler class. In the last step we used the chosen trivialisation of
$\orr(E) \tensor \orr(M) = \orr(\det TM^{\scriptscriptstyle\vee}\tensor\det E) =\orr
(\det(\Ind(Ds)))$ given by the trivialisation of the index.
In the infinite dimensional case we proceed similarly but we have to go
through a limiting process and use that we know what to do when the
section is regular. For each
compactum $K \subset Z$ we have to construct a class ${\Bbb Z}_K \in
\cH_d(Z,Z-K)$ such that for $K' \supset K$ the class ${\Bbb Z}_{K'}|_{Z-K} =
{\Bbb Z}_K$ under the restriction map $H_d(Z,Z-K') \to
H_d(Z,Z-K)$.
Over a neighborhood $U$ of $K$ in $M$ we can find a
finite rank $N$ subbundle $F$ of $E$ such that $\rmmath{Im}(Ds)|_K + F|_K =
E|_K$. Such a bundle certainly exists: we can choose a finite number of
sections $s_1, \ldots s_N$ such that the $s_i$ span
$\rmmath{Coker}(Ds_x)$ for
every $x \in K$, and possibly after perturbing we can assume that the
$s_i$ are linearly independent in a neighborhood.
Let $\~E$ be the quotient bundle $E/F$ defined over $U$, and $\~s$ the
induced section with zero set $M_f = Z(\~s)$ ($f$ is for finite, $M$ is
for, well, manifold).
Clearly the map $TM|_{Z(s)} \m@p--\rightarrow{Ds} E|_{Z(s)} \m@p--\rightarrow{} \~E $ is
surjective. Since the canonical map $D\~s$ on $M_f$ restricts to this
composition on $Z(s)$, $D\~s$ is surjective on $M_f$ possibly after
shrinking $U$. Hence $M_f$ is a smooth manifold. Let $T =
\ker(TM|_{M_f} \to \~E)$. There is a canonical identification $T \iso
TM_f$. Now $T$ is a bundle of rank $N+d$ since
\begin{equation}\label{theindex}
\Ind(Ds)|_K = T - F.
\end{equation}
Thus $M_f$ has dimension $N+d$.
On $M_f$, the section $s$ in $E$ lifts to a section $s_f$ of the
subbundle $F$. Clearly $Z(s_f) = Z(s) \cap U$.
Define
$$
{\Bbb Z}_K = e(F|_{M_f},s_f) \cap [M_f] \in \cH_d(Z(s), Z(s) - K;{\Bbb Z})).
$$
Here we have used the restriction map
$$
\cH^{\cl}_d(Z(s)\cap U; \orr(F)\tensor \orr(M_f))
\to\cH_d(Z(s),Z(s)-K; \orr(F) \tensor \orr(M_f)),
$$
the identification
$\orr(\det(\Ind(Ds))) = \orr(F) \tensor \orr(M_f)$
and the chosen trivialisation of $\det(\Ind(Ds))$
as in the finite dimensional case.
This construction does not depend on the choices. If $F$ and $F'$ are
two choices of subbundles of $E$ then there is third bundle
$F''$ containing $F + F'$. We can therefore assume that $F$ is a
subbundle of $F'$. Then using primes to denote objects we get out of the
construction above using $F'$ instead of $F$, we have a section
$s'_f$ in $F'$, a section $s''_f$ in $F'/F$ cutting out $M_f$ in $M'_f$
and the identity
\begin{align*}
{\Bbb Z}_K' &= e(F'|_{M'_f} s'_f)\cap [M'_f]
\\
&= e(F|_{M_f},s_f)\cap e(F'/F|_{M'_f}, s''_f) \cap [M'_f]
\\
&= e(F|_{M_f},s_f) \cap [M_f] = {\Bbb Z}_K.
\end{align*}
Note that in the third step we have used the identification
$\orr(M_f) = \orr(M'_f) \tensor \orr(F'/F))|_{M_f}$.
In particular, if $K' \supset K$ all choices on $K'$ work a fortiori for
$K$, so we can pass to the limit.
The relation ${\Bbb Z}(s) = [Z(s)]$ for regular sections
(property~\ref{smoothcase}), and the
compatibility with Euler classes of finite rank bundles
(property~\ref{fdeuler})
are now clear from the construction.
The stability property~\ref{stability} also follows from
the construction. For every compactum $K$,
we can choose the finite rank subbundle $F$
as a subbundle of $E'$. Then $\~E \surj\to E''$. Now one checks
that by a diagram chase that
$$
Z(\~E,\~s) = Z(E'/F|_{Z(E'',s'')}, s'\bmod F)
$$
and that
{
\let\to=\rightarrow
\begin{align*}
TZ(\~E,\~s) &= \rmmath{Ker}(TM \to \~E)
\\
&= \rmmath{Ker}(\rmmath{Ker}(TM \to E'') \to E'/F)
\\
&= \rmmath{Ker}(TZ(s'') \to E'/F) = TZ(E'/F).
\end{align*}
}
In particular, the orientations agree. Thus we see that
\begin{align*}
{\Bbb Z}_K(E,s) &= e(F,s_f)\cap [Z(\~E,\~s)]
\\
&= e(F,s_f)\cap [Z(E'/F|_{Z(E'',s'')}, s'\bmod F)]
= {\Bbb Z}_K(E'|_{Z(s'')},s').
\end{align*}
It only remains to pass to the limit over $K$.
To see that $\cH^{\cl}_j(Z(s))$ maps to ${\displaystyle \lim_{\leftarrow}}_C H_j(M,M-C)$ note that
for every compact subset $K = C \cap Z(s)$, we constructed a finite
dimensional manifold $M_f \supset K$.
Then we have maps
\begin{align*}
\cH_j^{\cl}(Z(s)) \to &\cH_j(Z(s),Z(s)-K)
= \cH_j(Z(s) \cap M_f, Z(s) \cap M_f -K)
\\
&\to H_j(M_f, M_f - C) \to H_j(M,M-C).
\end{align*}
Again this map is independent of choices, and we can pass to the limit.
The homotopy property of ${\Bbb Z}$ is a formal consequence of the
compatibility with finite dimensional Euler classes. Consider the
trivial bundle ${\Bbb R}$ over the interval $[-1,2]$ with the one parameter
family of sections $\theta - \tau$ where $\theta :[-1,2] \to {\Bbb R}$ is the
inclusion and $0 \le \tau \le 1$. Then clearly $e({\Bbb R},\theta) =
e({\Bbb R},\theta-1) \in H^1([-1,2],\{-1,2\})$ is the canonical generator.
Consider $M\times [-1,2]$. Let $\pi:M \times [-1,2] \to M$
be the projection and $S: M\times [-1,2] \to \pi^*E$
an extension of our one parameter family of sections e.g. $S_t = s_0$
for $t \le 0$ and $S_t = s_1$ for $ t\ge 1$.
The bundle $\pi^*E \mathop\oplus {\Bbb R}$ has a one parameter family of
sections $(S,\theta- \tau)$. Now
\begin{align*}
{\Bbb Z}(s_0) &\buildrel \ref{stability}\over=
\pi_* {\Bbb Z}(\pi^*E \mathop\oplus {\Bbb R}; (S,\theta))
\\
&\buildrel \ref{fdeuler}\over=
\pi_* e({\Bbb R},\theta) \cap {\Bbb Z}(\pi^*E;S)
\\
&= \pi_* e({\Bbb R},\theta - 1)\cap {\Bbb Z}(\pi^*E;S)
\\
&= \pi_* {\Bbb Z}(\pi^*E \mathop\oplus {\Bbb R};(S,\theta-1))
= {\Bbb Z}(s_1)
\end{align*}
\end{pf}
\begin{Corollary}\label{locChern} (compare \cite[prop. III.2.4]{PT})
Let $M$ be a complex Banach manifold, $E$ a holomorphic vector bundle
and $s$ a holomorphic section with zero set $Z(s)$
Assume that $Ds$ is a section of $\Fred_{\Bbb C}^d(TM|_{Z(s)},E|_{Z(s)})$.
We say that $Z(s)$ has complex virtual dimension $d$, and that $Ds$ is
Fredholm of complex index $d$.
Then the localised Euler class ${\Bbb Z}(s)= [Z(s)] \in H^{\cl}_{2d}(Z(s),{\Bbb Z})$,
if $Z(s)$ is a local complete intersection
of dimension $d$, and more generally
\begin{equation}\label{magic}
{\Bbb Z}(s) = [c(\Ind(Ds))^{-1} c_*(Z(s))]_{2d}
\end{equation}
where $c_*(Z(s))$ is the total homological chern class of $Z(s)$ defined
analogous to \cite[example 4.2.6]{Fulton} by equation \eqref{homchernclass}
and coincides with the Poincar\'e dual of the
cohomological chern classes of the tangent bundle if $Z(s)$ is
smooth.
\end{Corollary}
\begin{Remark}
If $Z(s)$ is smooth we can
even get away with an almost complex manifold $M$ and the assumption
that $Ds$ is complex linear.
\end{Remark}
\begin{Remark}
I have tacitly removed $M$ and $E$ from the notation of the
homological Chern class $c_*(Z(s))$. I strongly believe that
$c_*(Z(s))$ is independent of the embedding but I did not prove this.
There is one case where independence of $c_*(Z(s))$ on the embedding can
be proved completely analogous to \cite[Example 4.2.6]{Fulton} by
simply replacing algebraic arguments by complex analytic ones: if for
every $K \subset Z(s)$ compact, there exists a {\sl holomorphic}
finite rank sub bundle $F \lhook\nobreak\joinrel\nobreak\to E$ defined over a neighborhood of
$K$ such that $F|_K + \rmmath{Im}(Ds)|_K = E|_K$. Then a neighborhood $U_K$ of
$K$ in $Z(s)$ sits in a complex rather then almost complex finite
dimensional manifold
$M_f$. Such a bundle should typically exist if $Z(s)$ has the
structure of a quasi projective variety, and $\rmmath{Coker} DS$ has the
interpretation of a coherent sheaf as in \cite[\S 5,
\S6]{Pidstrigatch:instanton}.
\end{Remark}
\begin{pf}
We will use Mac Phersons graph construction, that is we consider the
limit $\lambda \to \infty$ of the map $(\lambda s: 1)$ in ${\Bbb P}(E \oplus
{\mathcal O})$ or finite dimensional approximations thereof. We use the notations
of the proof of proposition~\ref{locEuler}.
For a compactum $K \subset Z(s)$ we choose the finite rank bundle $F$ as
follows. It is a complex bundle, and in every point of $Z(s)$ there are
sections of $F$ which restricted to a neighborhood are holomorphic
sections of $E$ and which span locally a subbundle $F^{\hol} \lhook\nobreak\joinrel\nobreak\to
F$, such that $Ds: TE|_{Z(s)} \surj\to E/F^{\hol}|_{Z(s)}$ is a
surjection. We do not assume that $F$ is a holomorphic subbundle,
because I do not see a reason why such a bundle should exist. However
since $F$ is a complex bundle, the quotient $\~E = E/F$ and
$$
TM_f|_{Z(s)} = T|_{Z(s)} =
\rmmath{Ker} (TM|_{Z(s)} \m@p--\rightarrow{Ds} E|_{Z(s)} \to \~E|_{Z(s)})
$$
are complex bundles.
We extend this complex structure on $TM_f$
over all of $M_f$, possibly after shrinking $M_f$,
making it into an almost complex
manifold of complex dimension $d+ N$.
\def\overline{(0,1)}{\overline{(0,1)}}
\def\overline{(\lambda s_f,1)}{\overline{(\lambda s_f,1)}}
\def\overline{(s_f,0)}{\overline{(s_f,0)}}
\def\overline{(s_f,1/\lambda)}{\overline{(s_f,1/\lambda)}}
Consider the space ${\Bbb P}(F \oplus {\mathcal O}) \m@p--\rightarrow{\pi} M_f$. Then the total space
of $F$ embeds in ${\Bbb P}(F \oplus {\mathcal O})$. The image of the zero section will
also be called the zero section, and the complement of $F$ the divisor
at infinity.
Let $Q$ be the universal quotient bundle.
The bundle $Q$ has sections $\overline{(0,1)}$ , and $\overline{(\lambda s_f,1)}$
cutting out the zero section and the graph of $\lambda s_f$
respectively. Equivalently we can cut out the graph of $\lambda s_f$ by
$\overline{(s_f,1/\lambda)}$. Then clearly as $\lambda \to \infty$ the graph degenerates to
a set contained in the zero set of $\overline{(s_f,0)}$.
Now $Z(\overline{(s_f,0)})$ has two ``irreducible components''. One
component $\~M_f \lhook\nobreak\joinrel\nobreak\to {\Bbb P} F$ is the closure of the image of
$(s_f:0): M_f -Z(s) \to {\Bbb P} F|_{M_f - Z(s)} \subset {\Bbb P}(F\oplus {\mathcal O})$
It will be called the strict transform.
The other component is just ${\Bbb P}(F \oplus {\mathcal O})|_{Z(s)}$. Let ${\mathcal E}_f =
\~M_f \cap {\Bbb P}(F \oplus {\mathcal O})|_{Z(s)}$. It will be called the exceptional
divisor.
I claim that
\begin{equation}\label{dimclaim}
\cH^{\cl}_{2d+2N-1 + i}({\mathcal E}_f) =0 \txt{for} i \ge 0.
\end{equation}
Accepting this claim we see that
$\~M_f$ carries a unique fundamental class $[\~M_f]$
restricting to $[\~M_f - {\mathcal E}_f]$ by the exact sequence
$$
\cH_{2d + 2N}^{\cl}({\mathcal E}_f) \to
\cH_{2d+2N}^{\cl}(\~M_f) \to
H^{\cl}_{2d+2N}(\~M_f - {\mathcal E}_f) \to
\cH^{\cl}_{2d+2N - 1}({\mathcal E}_f)
$$
Consider $ C' = {\Bbb Z}(\overline{(s_f,0)}) - [\~M_f] \in \cH^{\cl}_{2d +
2N}(Z(\overline{(s_f,0)}))$. Then $C'$ comes from a unique class $C \in
\cH^{\cl}_{2d+2N}({\Bbb P}(F \oplus {\mathcal O})|_{Z(s)})$ because of the sequence.
$$
0 \to \cH^{\cl}_{2d+2N}({\Bbb P}(F \oplus {\mathcal O})|_{Z(s)}) \to
\cH^{\cl}_{2d+2N}(Z(\overline{(s_f,0)})) \to H^\cl_{2d+2N}(\~M_f - {\mathcal E}_f).
$$
Now note that $Q$ restricted
to the zero section is canonically isomorphic to $F$.
We therefore have the following chain of equivalences
\begin{align*}
{\Bbb Z}(s)_K &= e(F,s_f) \cap [M_f]
\\
&=\pi_* e(\pi^*F,\lambda s_f) \cap e(Q,\overline{(0,1)}) \cap [{\Bbb P}(F\oplus{\mathcal O})]
\\
&=\pi_* \(e(Q,\overline{(\lambda s_f,1)})\cup e(Q,\overline{(0,1)})\) \cap [{\Bbb P}(F \oplus {\mathcal O})]
\\
&=\pi_* e(Q,\overline{(0,1)}) \cap e(Q,\overline{(s_f,1/\lambda)}) \cap [{\Bbb P}(F \oplus {\mathcal O})]
\\
&=\pi_* e(Q,\overline{(0,1)}) \cap \(e(Q,\overline{(s_f,0)})\cap [{\Bbb P}(F \oplus{\mathcal O})]\).
\\
&=\pi_* e(Q,\overline{(0,1)}) \cap {\Bbb Z}(\overline{(s_f,0)})
\end{align*}
If we accept the claim~\eqref{dimclaim}
for a moment and we note that the support of
$\~M_f$ and $e(Q,\overline{(0,1)})$
are disjoint we see further that
$$
{\Bbb Z}(s)_K = \pi_*e(Q,\overline{(0,1)}) \cap C' = \pi_* e(Q) \cap C
$$
If we use that $e(Q)= c_\Top(Q)$
this can be rewritten further to
\begin{align*}
{\Bbb Z}(s)_K &= [\pi_* c(Q)\cap C)]_{2d}
\\
&= [c(F) \pi_*\((1-h)^{-1}\cap C \)]_{2d}
\\
&= [c(F-T) \(c(T) s_*(Z(s),M_f)\)]_{2d}
\end{align*}
where we used the notation $h = c_1({\mathcal O}_{{\Bbb P}(F \oplus {\mathcal O})}(-1))$ and
\begin{equation}\label{Segreclass}
s_*(Z(s),M_f) \buildrel\mathrm{def}\over= \pi_* (1-h)^{-1} C
\end{equation}
for the total homological Segre class of the normal cone (this
terminology will be justified in a minute).
But $c(F-T)= c(\Ind Ds)^{-1}$
and since $T = TM_f$,
\begin{equation}\label{homchernclass}
c_*(Z(s))\buildrel\mathrm{def}\over= c(T) s_*(Z(s),M_f)
\end{equation}
is exactly the analogue of the
homological chern classes of \cite[example 4.2.6]{Fulton}.
We show that $c_*(Z(s))$ does not depend on the choice of $F$. Again it
suffices to treat the case that $F'\subset F$. We use primes whenever
an object is associated to $F'$. The independence follows directly
from a formula for the Segre classes which expresses how it behaves under
the extension $M'_f \subset M_f$ in terms of the normal bundle
$F/F'$ of $M'_f \subset M_f$.
\begin{equation}\label{coneext}
s_*(Z(s),M_f) = c(F/F')^{-1} s_*(Z(s),M'_f).
\end{equation}
Assuming \eqref{coneext}, we see that
\begin{align*}
c_*(Z(s)) &= c(T)s_*(Z(s),M_f)
\\
&= c(T) c(F/F')^{-1}s_*(Z(s),M'_f) =
c(T')s_*(Z(s),M'_f).
\end{align*}
In particular we can take the limit over $K$.
Formula \eqref{coneext} is
well known for integrable complex manifolds \cite[example
4.1.5]{Fulton}, and we will follow the proof closely.
There are two terms in the class $C$ occurring in the definition
\eqref{Segreclass} of the Segre class, which we treat separately.
\nc\sfprimenul{\overline{(s'_f,0)}}
Note that there is a regular section $\sigma$
of $F/F'(1)$ on ${\Bbb P}(F\oplus{\mathcal O})|_{M_f}$ cutting out
${\Bbb P}(F' \oplus {\mathcal O})|_{M_f}$. Therefore
\begin{align*}
[{\Bbb P}(F'\oplus {\mathcal O})|_{M'_f}]
&= e(F/F', s_f \bmod F') \cap [{\Bbb P}(F'\oplus {\mathcal O})|_{M_f}]
\\
&= e(F/F',s_f \bmod F') \cap e(F/F'(1),\sigma)
\cap [{\Bbb P}(F\oplus {\mathcal O})|_{M_f}].
\end{align*}
Since on ${\Bbb P}(F'\oplus {\mathcal O})|_{M_f}$ there is an exact sequence
$$
0 \to Q'\to Q \to F/ F'\to 0,
$$
we have $e(Q',\sfprimenul)\cup e(F/F', s_f \bmod F') = e(Q,\overline{(s_f,0)})$. Then
the above implies that
\begin{align*}
{\Bbb Z}(Q',\sfprimenul)
&= e(Q',\sfprimenul) \cap [{\Bbb P}(F'\oplus {\mathcal O})|_{M'_f}]
\\
&=e(Q,\overline{(s_f,0)} )\cap e(F/F'(1),\sigma) \cap [{\Bbb P}(F\oplus {\mathcal O})|_{M_f}]
\\
&= e(F/F'(1),\sigma)\cap {\Bbb Z}(Q,\overline{(s_f,0)})
\end{align*}
As for the other term,
on $\~M_f$ there is a smooth section in ${\mathcal O}(-1)$ given
by $(s_f,0)$ which is an isomorphism ${\mathcal O} \iso {\mathcal O}(-1)$ on
$\~M_f -{\mathcal E}$. It follows that
$$
[\~M'_f - {\mathcal E}] = e(F/F',s_f \bmod F') \cap [\~M_f - {\mathcal E}]
= e(F/F'(1),\sigma) \cap [\~M_f - {\mathcal E}].
$$
Then we have the equality
$$
[\~M'_f] = e(F/F'(1),\sigma) \cap [\~M_f].
$$
because both left and right hand side are cycles supported on
$\~M'_f -{\mathcal E} \cup {\Bbb P}(F'\oplus {\mathcal O})|_{Z(s)}$ restricting to $[{\mathcal M}'_f
-{\mathcal E}]$.
For the computation of the Segre class we can forget
about the support given by $\sigma$ and use
$$
e(F/F'(1)) = c_\Top(F/F'(1))
= \sum c_{\Top-j}(F/F')h^j.
$$
Thus we finally get the expression
\begin{align*}
s_*(Z(s), M'_f) &=
\pi_*\(\sum h^{i+j} c_{\Top-j}(F'/F)
\cap ({\Bbb Z}(Q,\overline{(s_f,0)}) - [\~M_f])\)
\\
&= c(F'/F) s_*(Z(s),M_f)
\end{align*}
which we set out to prove.
\bgroup
\def\eN{\eN}
It remains to prove the claim~\eqref{dimclaim}.
We first turn to the case that $Z(s)$ is smooth
but possibly of the wrong
dimension. This condition implies that $\rmmath{Im} Ds|_T \subset F$
has constant rank over $Z(s)$ because $\ker Ds|_{T}= \ker Ds = TZ(s)$.
Then $\rmmath{Im} Ds|_T$ is just the normal bundle $\eN$ of $Z(s)$ in $M_f$.
Now let us identify the limit set $(s_f: 1/\lambda)(M_f)$ when $\lambda \to
\infty$. If we have a smooth path $\gamma$ with $\gamma(0) = x_0 \in
Z(s)$, then
we see that
$
\lim_{t\to 0} (s_f:0)(\gamma(t)) = (Ds_f(\ddt|_0\gamma):0).
$
Therefore $\~M_f$ is just the blowup $\^M_f$ of $Z(s)$ in $M_f$.
This makes sense even though $M_f$ is only an almost complex manifold
since the normal bundle $\eN$ has a complex structure.
The blow up is obtained
abstractly by identifying a tubular neighborhood $N_\epsilon$
of $Z(s)$ with the normal bundle, and replacing $N_\epsilon$ with
$I = \{(l, x) \in {\Bbb P}\eN \times N_\epsilon \mid l \ni x\}$. It is an
almost complex manifold, so certainly carries a fundamental class
$[\~M_f]$. It is also clear that ${\mathcal E}_f = {\Bbb P}\eN$ is a submanifold
of real codimension $2$, and certainly satisfies the
claim~\eqref{dimclaim}.
Let ${\mathcal O}({\mathcal E}_f)$ be the smooth complex line bundle on the blow-up
$\^M_f$ defined by
the exceptional divisor ${\mathcal E}_f$, and let
$z \in A^0({\mathcal O}(E))$ be a section cutting out ${\mathcal E}_f= {\Bbb P}\eN$ with the
proper orientation i.e. ${\Bbb Z}({\mathcal O}({\mathcal E}_f),z) = [{\mathcal E}_f]$.
On $\^M_f$ the pulled back section is of the form $s_f = z \^s_f$ with
$\^s$ nowhere vanishing. Therefore the limit set of
$(s_f:1/\lambda)(\^M_f)$ in ${\Bbb P}(F \oplus {\mathcal O})|_{\^M_f}$ as
$\lambda \to \infty$ is just $(\^s:0)(\^M_f) \cup D$ where
$D \subset {\Bbb P}(F \oplus{\mathcal O})|_{{\mathcal E}_f}$ is the ${\Bbb P}^1$ bundle
joining the zero section
$(0:1)|_{{\mathcal E}_f}$ and the section $(\^s_f:0)$.
Then down on $M_f$ the limit set of $(s_f: 1/\lambda)(M_f)$ is just
$\~M_f \cup C{\mathcal E}_f$, where $C{\mathcal E}_f$ is cone bundle over $Z(s)$ joining ${\mathcal E}_f
\subset \~M_f$ and the zero section.
Now $C{\mathcal E}_f$ represents the homology class $C$. Thus
$$
s_*(Z(s),M_f) = \pi_* (1-h)^{-1}C{\mathcal E}_f = \pi_* (1-h)^{-1} {\mathcal E}_f
= \pi_* (1-h)^{-1} {\Bbb P}\eN = s(\eN) \cap [Z(s)]
$$
Therefore if $Z(s)$ is smooth we find the expected formula
$$
c_*(Z(s)) = c(TM_f)s(\eN) \cap [Z(s)] = c(TZ(s)) \cap [Z(s)].
$$
Note that in deriving this formula we have not really used the
holomorphicity of $s$. It was sufficient that $M$ has an almost complex
structure and that $Ds$ is complex linear. Replacing manifolds by
stratified spaces the proof carries over essentially verbatim if $Z(s)$
is a local complete intersection since this condition implies that
$Ds|_{T}$ has constant rank, and that we have a well defined normal
bundle.
\egroup
In proving the claim \eqref{dimclaim} in the general case we use
holomorphicity more strongly. We first blow up $Z(s)^{\mathrm{red}}$
in $M$ to get a new infinite dimensional analytic space $\^M$. That
this is possible follows from the local analysis of the normal cone
in~\cite[\S III.1]{PT}.
Locally on $M$, the exceptional divisor ${\mathcal E} \subset \^M$ can be
described as follows. Locally on $M$ we have an exact sequence of
holomorphic bundles
$$
0 \to F^{\hol} \to E \to \~E^{\hol} \to 0,
$$
such that $TM|_{Z(s)} \surj\to \~E|_{Z(s)}$ is surjective, i.e. locally
$F^{\hol}$ can take the role of $F$. Further, locally we can split the
sequence since $F^{\hol}$ has finite rank. Let the holomorphic subbundle
$\~{\~E} \subset E$ be a lift of $\~E^{\hol}$. We write $s =
s_f^{\hol} \oplus \~{\~s}$ corresponding to the decomposition $E =
F^{\hol} \oplus \~{\~ E}$. Then locally ${\mathcal E} \iso {\mathcal E}^{\hol}_f
\times_{Z(s)} {\Bbb P}\~{\~E}$, where ${\mathcal E}^{\hol}$ is the exceptional divisor
of the blow up of $Z(s)$ in $M_f^{\hol}$, and where $M_f^{\hol}$ is the
integrable finite dimensional complex manifold $Z(\~s^{\hol})$.
Moreover ${\mathcal E}_f^{\hol}$ is naturally embedded in ${\Bbb P}(F^{\hol} \oplus
{\mathcal O})|_{Z(s)} \subset {\Bbb P}(E \oplus {\mathcal O})|_{Z(s)}$. If we are a little more
careful and choose $\~{\~E}$ such that ${\Bbb P}\~{\~E}|_{Z(s)} \subset {\mathcal E}$
then ${\mathcal E} = {\mathrm{Join}}({\mathcal E}_f^{\hol}, {\Bbb P}\~{\~E}|_{Z(s)}) \subset {\Bbb P}
E|_{Z(s)}$.
Let $z \in H^0({\mathcal O}({\mathcal E}))$ be a section vanishing exactly along ${\mathcal E}$.
On $\^M$ we can decompose the section as $s = z^n \^s$.
Therefore, just as
in the previous finite dimensional case,
$(s: 1/\lambda)(\^M) \to {\Bbb P}(E \oplus {\mathcal O})$ degenerates to $(\^s:
0)(\^M)\cup n D$ where $D$ is the ${\Bbb P}^1$ bundle over ${\mathcal E}$ joining the
zero section $(0:1)|_{{\mathcal E}}$ and $(\^s_f:0)|_{{\mathcal E}}$.
Down on $M$, this means that $(s: 1/\lambda)(M) \subset {\Bbb P}(E \oplus {\mathcal O})$
degenerates to $\~M \cup C{\mathcal E}$ where $\~M \subset {\Bbb P} E$
is isomorphic to $\^M$ with $\~M \cap {\Bbb P}(E \oplus {\mathcal O})|_{Z(s)} \iso {\mathcal E}$,
and $C{\mathcal E}$ is the cone bundle over $Z(s)$ joining the zero section and
${\mathcal E}$.
Now we finally come to our claim~\eqref{dimclaim}. The set
${\mathcal E}_f = {\Bbb P}(F \oplus {\mathcal O}) \cap {\mathcal E}$.
At the very beginning we chose $F$
such that $F \supset F^{\hol}$. Locally we define $\~{\~F} = F \cap
\~{\~E}$, then locally $F = F^{\hol} \oplus \~{\~F}$ and locally
${\mathcal E}_f = {\mathrm {Join}}({\mathcal E}^{\hol}_f , {\Bbb P}\~{\~F}|_{Z(s)})$. Thus ${\mathcal E}_f$ is
a stratified space of real dimension $2d +2N-2$, and we are done.
\end{pf}
\begin{Remark}\label{Zhat}
In the complex case we have obviously defined a class containing more
information about the section. Let
$$
\widehat{\Bbb Z}(s) = c(\Ind(Ds))^{-1}c_*(Z(s)).
$$
\end{Remark}
\section{Seiberg Witten classes}
We will collect a few facts about Seiberg Witten basic classes in a
formulation suitable for arbitrary K\"ahler surfaces.
In the usual formulation, these classes
are the support of a certain function on the set of $\Spin^c$-structures.
However in the presence of 2-torsion, $\Spin^c$-structures cause
endless confusion
which is why I have chosen to base my exposition on
SC-structures \cite{Karrer}. This notion catches the essence of
$\Spin^c$-structures, the existence of spinors.
It is well suited to the K\"ahler case
and is equivalent to that of a $\Spin^c$-structure in dimension 4.
For more details see \cite{Karrer}.
Let $X$ be a closed oriented manifold of dimension $2n$.
Choose a Riemannian metric $g$
with Levi-Civita connection $\nabla^g$,
and Clifford algebra bundle
$C(X,g) = C(T^{\scriptscriptstyle\vee} X,g)$.
There is a natural isomorphism of bundles
$c:\wedge ^* T^{\scriptscriptstyle\vee} X \to C(X,g)$ given
by anti-symmetrisation. It induces a connection
and metric on $C(X,g)$ also denoted $\nabla^g$ and $g$.
An {\sl SC-structure} is a smooth complex vector bundle $W$ of rank
$2^n$ together with an algebra bundle isomorphism $\rho: C(X,g) \to
\mathop{{\mathcal E}\mkern-3mu{\mathit nd}}\nolimits(W)$. In other words an SC structure is a bundle with the
irreducible Clifford algebra representation $\Delta$ in every fibre. A
section $\phi \in A^0(W)$ is called a (smooth) spinor.
An SC-structure exists if and only if $w_2(X)$ can be
lifted to the integers \cite[\S 3.4]{Karrer}.
Existence will be clear in the case of K\"ahler surfaces.
SC-structures admit an invariant hermitian metric i.e. one such that
Clifford multiplication by 1-forms is skew hermitian (sh). The chirality
operator $\Gamma = (\sqrt{-1})^n c(\Vol_g)$ has square $1$, and is
hermitian. Thus $\Gamma$ has an orthogonal eigenbundle decomposition
$W = W^+ \oplus W^-$ with eigenvalue $\pm 1$, the positive and negative
spinors of the SC-structure. A one form $\omega \in A^1(X)$ defines an
skew hermitian map $c(\omega): W^\pm \to W^\mp$ which is an isomorphism
away from the zero set of $\omega$.
In this paragraph we assume $\dim(X)= 4$.
Then $T^{\scriptscriptstyle\vee}_X \iso \mathop{{\mathcal H}\mkern-3mu{\mathit om}}\nolimits(W^+,W^-)^{\mathrm{sh}}$.
Let $L_W = \det W^+$. Then $L_W \iso \det W^-$, by the isomorphism
induced from
Clifford multiplication by a generic $1$-form,
which is an isomorphism outside codimension 4.
Thus $W$ is a $\Spin^c(4)$-bundle if we identify
$$
\Spin^c(4) =
\{(U_1,U_2) \in U(2) \times U(2) \mid \det(U_1) = \det(U_2)\}.
$$
We recover the usual definition $\Spin^c(4) = \Spin(4)\times_{{\Bbb Z}/2/Z}
U(1)$ from the isomorphism $\Spin(4) = SU(2) \times SU(2)$.
In any case by chasing around the cohomology
sequences of the diagram
$$
\arrowlen3em
\begin{matrix}
0 &\m@p--\rightarrow{}&{\Bbb Z}/2{\Bbb Z}&\m@p--\rightarrow{}&\Spin^c(4)&\m@p--\rightarrow{}&\SO(4)\times U(1)&\m@p--\rightarrow{}1
\\
& &\vm@p\Vert{}& &\vm@p\uparrow{} & &\vm@p\uparrow{}
\\
0 &\m@p--\rightarrow{}&{\Bbb Z}/2{\Bbb Z}&\m@p--\rightarrow{}&\Spin(4) &\m@p--\rightarrow{}&\SO(4)
&\m@p--\rightarrow{}1
\end{matrix}
$$
we see that $L_W + w_2(X) \equiv 0 \pmod 2$, and that this is the only
obstruction to lifting the $SO(4)\times U(1)$ bundle to $\Spin^c(4)$.
If $H^2(X,{\Bbb Z})$
has no 2-torsion, the line bundle $L\equiv w_2(X)$ determines such a
lift completely, and it is common to speak of the $\Spin^c$-structure $L$.
An {\sl SC-Clifford module} $(S,\<,>,\nabla)$,
is an SC-structure with a non-degenerate invariant
hermitian metric $\<,>$ and
a unitary Clifford connection $\nabla$ i.e. a unitary connection such
that for all vector fields $X$, spinors $\phi \in A^0(S)$, and $1$-forms
$\omega$ we have
$$
\nabla_X (\omega \cdot \phi) = (\nabla^g_X \omega)\cdot \phi +
\omega \cdot \nabla_X \phi.
$$
The {\sl Dirac operator} $\delbar$ of a Clifford module is the
composition
$$
A^0(W) \m@p--\rightarrow{\nabla} A^1(W) \m@p--\rightarrow{\cdot} A^0(W).
$$
It is an elliptic self adjoint first order differential operator, and
it maps positive spinors to negative ones and vice versa (i.e.
$\delbar:A^0(W^\pm) \to A^0(W^\mp)$).
Since $\rho$ is parallel, $\nabla$ respects the decomposition
$W = W^+ \oplus W^-$. Thus $\nabla$
induces a connection on $L_W$ with curvature $F$.
Much of the usefulness of SC-structures is a consequence of
the following easy lemma.
\begin{Lemma}
The set of isomorphism classes $\SC$ of SC-structures is an $H^2(X,{\Bbb Z})$
torsor i.e. if $\SC \ne \emptyset$ and we fix an SC-structure $W_0$,
then for every SC-structures $W_1$, there exits a unique line bundle ${\mathcal L}$
such that $W_1 = W_0\tensor{\mathcal L}$. Every SC-structure $S$ admits a
Clifford module structure $(W, \<,>, \nabla)$. If we fix one
SC-Clifford module $(W_0, \<,>_0, \nabla_0)$, there is a unique triple
$({\mathcal L},h,d)$ of a smooth line bundle ${\mathcal L}$, with hermitian metric $h$ and
unitary connection $d$, such that
\begin{equation}\label{SCrepr}
(W,\<,>,\nabla) \iso (W_0, \<,>_0, \nabla_0) \tensor ({\mathcal L}, h,d).
\end{equation}
\end{Lemma}
\begin{pf}
Clearly if $W_0$ is an SC structure, so is $W_0\tensor {\mathcal L}$ for every line
bundle ${\mathcal L}$. Conversely,
the bundle of Clifford linear homomorphisms
${\mathcal L}(W_0,W) = \mathop{{\mathcal H}\mkern-3mu{\mathit om}}\nolimits_C(W_0,W)$ has rank~1, and the natural map $W_0
\tensor {\mathcal L}(W_0, W) \to W$ is an isomorphism.
For existence of a Clifford module structure see
\cite[prop. 4.2.1, 4.5.1]{Karrer}.
It will be clear for K\"ahler surfaces. It
follows directly from the definition of a Clifford module that the
natural connection and metric on $\mathop{{\mathcal H}\mkern-3mu{\mathit om}}\nolimits(W_0,W)$ leaves ${\mathcal L}(W_0,W)$
invariant. Hence there is an induced metric and connection $(h,d)$ on
${\mathcal L}(W_0,W)$, which has property \eqref{SCrepr}. Conversely
if $(W,\<,>,\nabla)$ is defined by equation \eqref{SCrepr}, then
$$
({\mathcal L},h,d) =
\mathop{{\mathcal H}\mkern-3mu{\mathit om}}\nolimits_C\((W_0,\<,>_0,\nabla_0)\,,\,
(W_0,\<,>_0,\nabla_0)\tensor({\mathcal L},h,d)\)
$$
which proves uniqueness.
\end{pf}
If a base SC-structure is chosen, the line bundle ${\mathcal L}$ will be called
the twisting line bundle.
There is a natural gauge group $\G^{\Bbb C}$ acting on a Clifford module, the
group of all smooth invertible Clifford linear endomorphisms. $\G^{\Bbb C}$
can be canonically identified with $C^\infty(X, {\Bbb C}^*)$. In the
representation~\eqref{SCrepr}, $\G^{\Bbb C} = C^\infty(X, {\Bbb C}^*)$ acts in the
usual way on the set of metrics and unitary connections on the twisting
line bundle ${\mathcal L}$. Since every hermitian metric on a line bundle is gauge
equivalent, so is every Clifford invariant metric on a Clifford module.
Thus, up to gauge we can fix an invariant metric and we are left with a
residual gauge group $\G = C^\infty(X, \mathrm{U}(1))$.
The set of Clifford connections $\A$ on a fixed hermitian SC structure
$(W,\<,>)$ (i.e. Clifford module structures) is an affine space
modeled on $\sqrt{-1} A^1_{\Bbb R}(X)$.
Using the representation~\eqref{SCrepr} and harmonic representatives,
one shows that the set of connections mod gauge is
$$
\B = \A / \G \iso \sqrt{-1} A^1_{\Bbb R}(X) / d\log C^\infty(X,\mathrm{U}(1))
\iso H^1_{DR}(X)/H^1(X,{\Bbb Z}) \oplus \ker d^*
$$
We set $\Pee^* = \A \times A^0(W^+)^* / \G$. It is a
${\Bbb C}{\Bbb P}^\infty \times {\Bbb R}^+$ bundle over $\B$. Thus $\Pee^*$ has the
homotopy type of $(S^1)^{b_1(X)} \times {\Bbb C}{\Bbb P}^\infty$.
There is an alternative description of $\B$ and $\Pee^*$ that will be
useful. Let $\A^{\Bbb C}$ be the set of all Clifford connections, and $\Herm$
the set of all hermitian metrics on ${\mathcal L}$. Let
$$
\Amod = \{(\nabla,<,>),\ \nabla
\txt{is} <,>\hbox{-unitary}\} \subset \A\times\Herm
$$
be the set of Clifford module structures.
Fix a metric $<,>_0$ and a $<,>_0$-unitary connection $\nabla_0$. The
representation $\nabla = \nabla_0 + a$, models $\A^{\Bbb C}$ on $A^1_{\Bbb C}(X)$,
and the representation $<,> = e^f < , >_0$ models $\Herm$ on $A^0_{\Bbb R}(X)$.
A pair $(\nabla,<,> ) \in \Amod$ if and only if $a+ \bar a = df$.
In particular $a$ is determined by $f$ and its imaginary part,
so $\Amod$ is modeled on $A^0_{\Bbb R}(X)\times A^1_{\Bbb R}(X)$.
Now the diagonal action of $\G^{\Bbb C}$ on $\A^{\Bbb C} \times \Herm$
leaves $\Amod$ invariant. Our alternative description of $\B$ and
$\Pee^*$ is
\begin{equation}\label{alternative}
\Pee^* = \Amod \times A^0(W^+)^*/ \G^{\Bbb C} \to \B = \Amod/ \G^{\Bbb C}
\end{equation}
Finally, to do decent gauge theory we have to complete to Banach
spaces and -manifolds. Seiberg Witten theory works fine with an $L^p_1$
completion of $\A$, $\A^{\Bbb C}$, and $A^0(W^+)$ and an $L^p_2$ completion
of $\G$, $\G^{\Bbb C}$ and $\Herm$ if $p > \dim X$. In this range $L^p_1
\lhook\nobreak\joinrel\nobreak\to C^0$, and therefore the two possible $L_p$ descriptions of
$\Pee^*$ and $\B$ coincide. On the other hand, the Sobolev range does
not seem optimal: with more care and work one can probably use all
$p$-completions with $2 - \dim(X) / p > 0$. We will suppress completions
from the notation, explicitly mentioning completions if necessary.
{}From now on we assume $\dim X = 4$.
Fix an SC structure $W$ and choose an invariant hermitian metric $\<,>$.
Choose a Riemannian metric $g$ and a real 2
form $\epsilon$, which are {\sl admissible} in the following sense:
$L_W$ admits no connection with $F^+ = -2\pi \sqrt{-1}\epsilon^+$,
where as usual $+$ means taking the self dual part.
Admissible metrics and forms exist if $b_+ \ge 1$, since the condition is
certainly satisfied if
$c_1(L_W) \not\in \epsilon^{\mathrm{harm}} + H^-_g$ where $H^-_g$
is the space of $g$-anti-self-dual closed forms, and ``harm'' means
projection to the harmonic part.
Note that no use of Sard-Smale is made to define admissibility.
Actually for most of our purposes it would be enough to
let $\epsilon$ be a closed (hence harmonic) self-dual form.
By a transversality argument \cite{Donaldson:intersectionform},
the admissible (metrics,forms) form a connected set if $b_+ \ge 2$.
We say that a metric $g$ is admissible if
$(g,0)$ is.
Even if $b_+ = 1$, all metrics are admissible when $L_W^2 \ge 0$,
and $L_W$ is not torsion.
In dimension $4$, the anti-symmetrisation map gives an isomorphism
$c:\Lambda ^+ \iso \rmmath{End}_0^{sh}(W^+)$
between the real self-dual
forms and the traceless skew hermitian
endomorphisms of $W^+$. This special phenomenon allows us (or rather
Seiberg and Witten) to write down the monopole equations \cite{Witten}
\begin{align}
\label{SW1}
\delbar \phi &= 0 \qquad \phi \in A^0(W^+)
\\
\label{SW2}
c(F^+) &= 2\pi \phi\<\phi,-> - \pi |\phi|^2
-2\pi\sqrt{-1} c(\eps^+).
\end{align}
Let ${\mathcal M} = {\mathcal M}(W,g,\epsilon) \subset \Pee^*$
be the space of solutions modulo gauge.
As a technical remark, note that we use the conventions of \cite{BGV},
and that in their conventions the Weitzenb\"ock (Lichnerowitz)
formula restricted to $W^+$ reads
$$
\delbar^2 = \nabla^* \nabla + r /4 + c(F^+/2)
$$
(\cite[th. 3.52]{BGV} and the observation that the
twisting curvature of an SC
structure is $1/\rmmath{rank}(W^+)$ times the curvature on $\det(W^+)$.)
The sign difference in the $c(F^+)$ term in \cite[lemma 2]{KM:Thom}
explains the relative change of
sign with respect to \cite[formula $(*)$]{KM:Thom}
in the Seiberg Witten equations. It is
chosen in such a way that the Weitzenb\"ock formula gives $C^0$ control
on the harmonic positive spinor $\phi$.
A basic property of the monopole equation noted by Witten,
which follows from the Weitzenb\"ock formula \cite[lemma 2]{KM:Thom} or a
variational description \cite[Section 3]{Witten}, is the following
\begin{Proposition}\label{Bochner}
The monopole equations have no solution with $\phi \ne 0$ if the
metric has positive scalar curvature.
\end{Proposition}
Alternatively we can define ${\mathcal M}$ as the zero of a Fredholm section in an
infinite dimensional vector bundle. Let
$\W^\pm = (\A\times A^0(W^+)^* \times_\G A^0(W^\pm) \to \Pee^*$.
Then ${\mathcal M}$ is the zero of the section in $\W^- \oplus A^+(X)$
given by the monopole equations~\eqref{SW1}, and \eqref{SW2}.
To see that it is actually a Fredholm section
we linearise the equations, assuming
that $(\nabla,\phi)$ is a solution, and $(\nabla + \eps a, \phi + \eps
\psi)$ with $a \in \sqrt{-1}A_{\Bbb R}^1(X)$ and $\psi \in A^0(W^+)$ is
a solution up to order 1 in $\eps$. We get (c.f \cite[eq.2.4]{Witten})
\begin{gather*}
\delbar \psi + a\cdot \phi = 0
\\
c^{-1}(2\pi(\phi\<\psi,-> + \psi\<\phi,-> - \rmmath{Re}\<\phi,\psi>)
-d^+a = 0.
\end{gather*}
The tangent space of the $\G$-orbit of $(\nabla,\phi)$ is
$\{(a,\psi) = (- d u, u \phi), \ u \in \sqrt{-1}A^0_{\Bbb R}(X)\}$.
Thus the Zariski tangent space of ${\mathcal M}$ in $(\nabla, \phi)$ is the first
cohomology of the Fredholm complex
$$
\sqrt{-1} A_{\Bbb R}^0(X) \to \sqrt{-1}A_{\Bbb R}^1(X) \oplus A^0(W^+) \to
\sqrt{-1} A^+_{\Bbb R}(X)
\oplus
A^0(W^-),
$$
where the maps are given by the left hand side of the linearised equations.
The virtual
dimension is given by Atiyah Singer index formula and is
\begin{equation}\label{vdim}
d(W) = \vdim_{\Bbb R}({\mathcal M}) = {\numfrac14}(L_W^2 - (2 e(X) + 3 \sign(X))),
\end{equation}
where $e(X)$ is the topological Euler characteristic, and $\sign(X)$ the
signature \cite[eq. 2.5]{Witten}.
The crucial property that makes Seiberg Witten theory so much easier
than Donaldson theory is
\begin{Proposition}\cite[Corollary 3]{KM:Thom},\cite[\S 3]{Witten}
The moduli space ${\mathcal M}$ is compact.
For fixed $c >0$ there are only finitely many SC-structure $W$ with
$d({\mathcal M}(W)) \ge -c$ and ${\mathcal M}(W,g,\epsilon) \ne \emptyset$.
\end{Proposition}
Note that for generic pairs $(g,\epsilon)$, moduli spaces of negative
virtual dimension are empty, but I do not see an a priori reason why
moduli spaces of arbitrary negative virtual dimension should not exist
for special pairs. Likewise for generic pairs the moduli space is
smooth of dimension $d(W)$ \cite{KM:Thom}. However we have no need
for this fact.
The index bundle $\Ind(Ds)$ of the deformation complex can be deformed
by compact operators (over a compact space !) into the sum of the index
of the signature complex and the index of the complex dirac operator.
Thus the determinant line bundle $\det(\Ind(Ds)$ of the index is
naturally oriented by choosing an orientation for $\det H^1(X,{\Bbb R})^{\scriptscriptstyle\vee}
\tensor H^+(X,{\Bbb R})$. We will in fact assume that an orientation for both
$H^+$ and $H^1$ is chosen. Suppose further that the pair $(g,\epsilon)$
is admissable (i.e. ${\mathcal M}((W,g,\epsilon) \subset \Pee^*$), then
proposition \ref{locEuler} in the previous section gives us a
homology class $\MM \in H_{d(W)}(\Pee^*)$,
i.e. a homology class
of the proper virtual dimension even if
${\mathcal M}$ is not smooth, not reduced and not of the proper dimension (note that
in our case the moduli space ${\mathcal M} = Z(s)$ is compact, and homology with
closed support is just ordinary homology).
In case ${\mathcal M}$ is smooth and has the proper dimension it is just the
fundamental class. The class $\MM$ depends only on the
connected component of $(g,\epsilon)$ in the space of admissable pairs,
by the homotopy property of the localised Euler class proposition
\ref{locEuler}.\ref{homotopy}.
In particular ${\mathcal M}$ is independent of the admissable pair if $b_+ \ge 2$.
If $b_+ = 1$ the choice of an orientation of $H^+$ is the choice of a
connected component in
$\{\omega^2 >0\} \subset H^2(X,{\Bbb R})$. It will be called the forward timelike
cone. For every metric $g$ let $\omega_g$ be the unique self dual form
in the forward timelike cone with $\int \omega^2 = 1$. For a pair
$(g,\epsilon)$ and an SC-structure $W$ define the {\sl discriminant}
\begin{equation}\label{discriminant}
\Delta_W(g,\epsilon) = \int (c_1(L_W) - \epsilon) \omega_g
\end{equation}
A pair $(g,\epsilon)$ is admissable if the
discriminant $\Delta_W(g,\epsilon) \ne 0$, because it means precisely
that $c_1(L_W) \notin \epsilon^{\harm} + H^-$. Clearly the discriminant
depends only on the period $(\omega_g, {\epsilon^+}^{\mathrm harm})$.
\begin{Lemma}
If $b_+ = 1$ a pair $(g,\epsilon)$ is admissable if and only if
the discriminant $\Delta_W(g,\epsilon) \ne 0$.
There are exactly two connected components of admissable
pairs labeled by the sign of the discriminant.
\end{Lemma}
\begin{pf}
Suppose two pairs $(g_i,\epsilon_i)$, $i=0,1$,
have discriminants $\Delta_i$ of equal sign.
Connect them by a path $(g_t, \epsilon_t)$ in the space
of all pairs. Let $(\omega_t, \epsilon^{+,\harm}_t)$ be the corresponding
path of periods.
Then the discriminant
$$
\Delta_t = \int (c_1(L_W) - \epsilon^{+,\harm}_t)\omega_t
$$
is continuous in $t$ but may change sign.
However if we modify the path by setting
$$
\epsilon'_t = \epsilon_t +
(\Delta_t - (1-t)\Delta_0- t\Delta_1)\omega_t
$$
then using
$\Delta_W(g,\epsilon + \delta) = \Delta_W(g,\epsilon) - \int
\delta\wedge\omega_g$ and $\int \omega^2 = 1$ we see that
$$
\Delta'_t= \Delta_W(g_t,\epsilon'_t) = (1-t)\Delta_0 + t\Delta_1.
$$
In particular $\Delta'_t$ does not change sign, so that
$(g_t,\epsilon'_t)$ is a path of admissable pairs.
Conversely if $c_1(L_W) \in \epsilon^\harm + H^-$, then any connection
$\nabla$ with induced Chern form $\epsilon^\harm$ determines a
``reducible'' solution $(\nabla,0) \in \Pee- \Pee^*$ of the monopole
equations.
\end{pf}
\begin{Definition}
If $b_+ \ge 2$, the {\sl SW-multiplicity} is the map
\begin{align*}
n:\SC &\to \Lambda^*H^1(X,{\Bbb Z})[t] \iso
H_*(\Pee^*,{\Bbb Z})
\\
W &\mapstochar\nobreak\to \MM(W,g,\epsilon)
\end{align*}
where $(g,\epsilon)$ is any $W$-admissable pair.
If $b_+ = 1$ the {\sl SW-multiplicities} $n_+$ and $n_-$ are defined
similarly but with pairs $(g_\pm,\epsilon_\pm)$ having positive
respectively negative discriminant.
\end{Definition}
It should be remarked that the SW-multiplicity (ies) depend(s)
implicitly on the orientation of $H^+$ and $H^1$. For $b_+ > 1$ this is
only a matter of sign, but for $b_+ = 1$ the orientation of $H^+$
determines in addition which invariant is $n_+$ and which is $n_-$.
All known examples with $b_+ \ge 2$ have
non trivial multiplicities only when the virtual dimension
$d(W) = 0$. However
for surfaces with $p_g =0$ it is easy to give examples
with one of $n_\pm$ is non trivial for $d(W) >0$ we will in fact use such
an invariant. If $b_1 \ne 0$,
the $H^1$ part of the multiplicity becomes essential.
\begin{Remark}
Since $H_i(\Pee^*) = 0$ for $i <0$,
a moduli space of negative virtual dimension
never defines a nontrivial class.
Thus if for a class $L \in H^2(X,{\Bbb Z})$ there exists an SC-structure $W$
with $L = c_1(L_W)$ and the multiplicity $n(W) \ne 0$ (respectively one
of $n_{\pm}(W) \ne 0$ then
$L^2 \ge 3 e(X) + 2\sign(X)$ (c.f. equation~\eqref{vdim}).
\end{Remark}
\begin{Remark}\label{specialchamber}
In the case $b_+ = 1$ we can alternatively consider the multiplicity as
depending in addition on a chamber structure in
$$
\Gamma = \{(\omega,\epsilon) \in H^2(X,{\Bbb R})^2
\mid \omega^2 = 1,\ \omega_0 > 0\}
$$
where a chamber is defined by walls which are in turn defined by
all classes $L \equiv w_2(X)$ through equation~\eqref{discriminant}. This
is particularly useful when we consider structures with $L_W^2 \ge
0$, $L_W$ is not torsion. Then all pairs $(g,0)$ are admissable and
have discriminant of equal sign, because the forward timelike cone is
strictly on one side of the hyperplane $L_W^\perp \subset H^2(X,{\Bbb R})$.
Thus for this subset we have a preferred chamber.
\end{Remark}
We will say that $L\in H^2(X,{\Bbb Z})$ with $L \equiv w_2(X)$
has non trivial multiplicity if there is an SC-structure $W$ such that $L
= c_1(L_W)$ and $W$ has non trivial multiplicity. If $b_+ =1$ we will
further qualify which multiplicity is non trivial (i.e. $n_+$ or $n_-$)
or which chamber is chosen. We will simply write $n(L) \ne 0$ or
$n_+(L) \ne 0$ etc.
A final and important piece of general theory is the following blow-up
formula \cite{Stern:talk},\cite[\S 8]{FS:rational}.
We will give a proof valid for K\"ahler surfaces in
section~\ref{computations}.
\begin{Theorem}\label{blowup4}
Let $X$ be a closed oriented 4-manifold with $b_+ \ge 1$. An
SC-structure $\~W$ on $X\#\Pbar^2$ can be decomposed as $\~W = W \#
W_k^{\Pbar^2}$, with determinant lines $L_{\~W} = L_W + (2k+1)E$. If
the multiplicity $n_{(\pm)}(\~W) \ne 0$ then $d(\~W) = d(W) - k(k+1) \ge
0$, and the multiplicity $n_{(\pm)}(W) \ne 0$. Moreover if
$L_{W_{\Pbar^2}} = \pm E$ (i.e. $E \cdot L_{\~W} = \pm 1$) then
$n_{(\pm)} (\~W) = n_{(\pm)}(W)$ under the identification $H^1(X,{\Bbb Z})
\iso H^1(\~X,{\Bbb Z})$.
\end{Theorem}
Here, $ n_{(\pm)} = n$ if $b_+ > 1$, and
if $b_+ = 1$, it is understood that we compare say $n_+(W \#
W_k^{\Pbar^2})$ with $n_+(W)$.
\section{Seiberg Witten classes of K\"ahler surfaces}
{}From now on, $(X, \Phi)$ denotes a K\"ahler surface.
Then $X$ has a natural base SC-structure
$$
W_0 = \Lambda^{0,*} X
$$
with Clifford multiplication given by
$$
c(\omega^{10} + \omega^{01}) =
\sqrt2\(- i(\omega^{10}) +\eps(\omega^{01}) \),
$$
where $i$ is contraction and $\eps$ is exterior
multiplication. The metric and connection induced by the K\"ahler
structure on $\Lambda^{0*} X$ define a Clifford module structure on
$W_0$.
For an arbitrary SC structure $W$ = $W({\mathcal L})$ the spinor bundles
are of form
$$
W^+ = (\Lambda^{00} \oplus \Lambda^{02})\tensor {\mathcal L},
\qquad W^- = \Lambda^{01}({\mathcal L}).
$$
and $L_W = \det(W^+) = -K \tensor {\mathcal L}^2$ (c.f. lemma~\ref{SCrepr}).
We call ${\mathcal L}$ the twisting line bundle.
We now turn to the monopole equations (see also \cite[Section
4]{Witten}). In the decomposition of $W^+$, a positive spinor will be
written $\phi = (\alpha,\beta)$. The Dirac equation is then
\cite[Propos. 3.67]{BGV}.
$$
\delbar \phi = \sqrt2 (\dbar\alpha + \dbar^* \beta)= 0.
$$
\nc\dzdz{dz_1\wedge dz_2}
\nc\dzbardzbar{d\bar z_1 \wedge d \bar z_2}
\def\dzdzbar#1{dz_#1\wedge d\bar z_#1}
Since $X$ is K\"ahler, we can locally choose holomorphic geodesic
coordinates $(z_1,z_2)$. A basis of the self dual forms is then the
K\"ahler form
$\Phi = \numfrac {\sqrt{-1}}2(\dzdzbar1 + \dzdzbar2)$,
$\dzdz$ and $\dzbardzbar$. Let $h$ be an hermitian metric on ${\mathcal L}$.
Choose a unit generator $e$ for ${\mathcal L}$,
then an orthonormal basis for $W^+ $ is $e$ and
${\numfrac12} e\dzbardzbar$.
Using the definition of Clifford Multiplication we compute:
\begin{align*}
c(\Phi)e &= \numfrac{\sqrt{-1}}2
(-i(dz_1) \eps(d \bar z_1) + \eps(d \bar z_1)i(d z_1)
-i(dz_2) \eps(d \bar z_2) + \eps(d \bar z_2)i(d z_2))e
\\
&= - 2\sqrt{-1} e.
\end{align*}
In exactly the same way we compute $c(\Phi)$, ${\numfrac12} e \dzbardzbar$,
and the action of $c(\dzdz)$ and $c(\dzbardzbar)$ on $e$ and
${\numfrac12} e \dzbardzbar$. The result in matrix form is given by
$$
c(\Phi) = \begin{pmatrix}
- 2\sqrt{-1} & 0
\\
0 & 2\sqrt{-1}
\end{pmatrix}
\quad
c(\dzdz) = \begin{pmatrix}
0 & -4
\\
0 & 0
\end{pmatrix}
\quad
c(\dzbardzbar) = \begin{pmatrix}
0 & 0
\\
4 & 0
\end{pmatrix}.
\hskip 0pt minus 1 fil
$$
\def\bet#1#2{\beta_{\dot #1\dot#2}}
On the other hand, writing $\alpha = \alpha_e e$, and $\beta = {\numfrac12}
\bet12 e \dzbardzbar$,
$$
(\alpha + \beta) \< \alpha +\beta,-> =
\begin{pmatrix}
|\alpha_e|^2 & \alpha_e\bar\bet12
\\
\bar\alpha_e\bet12 & |\bet12|^2
\end{pmatrix}.
$$
Thus if we define $\alpha^* = h(\alpha,-)$, $\beta^*= h(\beta,-)$ and
take the trace free part,
we get the healthy global expression
$$
(2\pi(\alpha + \beta)\< \alpha + \beta, ->)_0=
-2\pi\sqrt{-1} c\({\numfrac12} (|\beta|_h^2 - |\alpha|_h^2) \Phi
+ \sqrt{-1}(- \alpha \beta^* + \beta \alpha^*)) \)
$$
Plug all this in the monopole equations~\eqref{SW1},\eqref{SW2}.
Writing $c_1(F) =\numfrac{-1}{2 \pi i} F$, and using that $\Lambda \Phi
=2$ the monopole equation for a K\"ahler metric and perturbation
$\epsilon = \lambda \Phi$ can be rewritten to
\begin{align}
&\dbar \alpha + \dbar^* \beta = 0 \label{cSW1}
\\
& F^{02} = 2\pi \beta \alpha^* \label{cSW2}
\\
&F^{20} = -2\pi \alpha \beta^* \label{cSW3}
\\
&\Lambda c_1(F)^{11} = (|\beta|^2 - |\alpha|^2) + 2\lambda.
\label{cSW4}
\end{align}
Note that $F$ is the curvature on $L_W$, but that these are equations
for a unitary connection $d =\dee + \dbar$ on ${\mathcal L}$ and sections $\alpha
\in A^{00}({\mathcal L})$, and $\beta \in A^{02}({\mathcal L})$ through the identity $F =
-F(K) + 2 F({\mathcal L},d)$. Here $F(K)$ is the curvature of the canonical line
bundle i.e. minus the Ricci form.
In terms of the twisting bundle the virtual (real) dimension of the
moduli space reads
\begin{equation}\label{cvdim}
d({\mathcal L}) = d(\Lambda^{0*}({\mathcal L})) = {\numfrac14}(L^2 - K^2)
= {\mathcal L}\cdot ({\mathcal L}- K).
\end{equation}
A more precise description is given by
\begin{Proposition} \label{Kahlermonopoles}
{\sloppy
A necessary condition for the existence of solutions
to the mono\-pole equations~\eqref{cSW1} to \eqref{cSW4}, is that
$({\mathcal L},\dbar)$ is a holomorphic line bundle, and that
}
\begin{align}
-\deg_\Phi(K) &\le \deg_\Phi(L) < \int(\lambda \Phi^2),
\txt{or} \label{case0}
\\
\int \lambda \Phi^2 &< \deg_\Phi(L) \le \deg_\Phi(K),
\txt{or} \label{case2}
\\
\int \lambda \Phi^2 &= \deg_\Phi(L)
\label{singcase}
\end{align}
In particular $L_W = -K \tensor {\mathcal L}^2$ has a natural holomorphic structure.
In case \eqref{case0} the moduli space ${\mathcal M} = {\mathcal M}({\mathcal L}, \Phi, \lambda)$ of
solutions can be
identified as a real analytic space with the moduli space of pairs of a
holomorphic structures $\dbar$ on ${\mathcal L}$, and a divisor $\alpha \in
|({\mathcal L},\dbar)|$, in particular the Zariski tangent space in
$(\dbar,\alpha)$ is canonically identified with
$H^0({\mathcal L}|_{Z(\alpha)})$.
In case \eqref{case2} the moduli space ${\mathcal M}$ of solutions
can be identified with the moduli
space of pairs of a holomorphic structure $\dbar$ on ${\mathcal L}$, and an
element $\beta \in {\Bbb P} H^2({\mathcal L}) = |K\tensor {\mathcal L}^{\scriptscriptstyle\vee}|^{\scriptscriptstyle\vee}$, in
particular the Zariski tangent space at $(\dbar,\beta)$ is isomorphic to
$\overline{H^0(K\tensor {\mathcal L}|_{Z(\bar\beta)})}$.
In case \eqref{singcase} the ``moduli space'' ${\mathcal M} \subset \Pee - \Pee^*$
(i.e. $\alpha = \beta = 0$)
can be identified with the space of holomorphic structures $\dbar$ on ${\mathcal L}$.
\end{Proposition}
\begin{pf}
Combining~\eqref{cSW1} and ~\eqref{cSW2} yields
\begin{equation}\label{positivity}
\dbar\dbar^* \beta = -\dbar^2 \alpha
= - F^{02}\alpha
= -2\pi |\alpha|^2 \beta.
\end{equation}
Integrating both sides against $\<\beta,->$, immediately gives that
$\alpha\beta= 0$ and $\dbar\beta = \dbar\alpha = 0$. Thus $F^{02} =
F^{20} =0$, Since $F^{02} = 2 F^{02}({\mathcal L},d)$, $\dbar$ is a
holomorphic structure on ${\mathcal L}$, and either $0 \ne \alpha \in H^0({\mathcal L})$ and
$\beta= 0$ or $0 \ne \beta \in H^2({\mathcal L})$ and $\alpha = 0$, or $\alpha =
\beta = 0$. Note
that if for example $\alpha \ne 0$, then $\beta = 0$ is cut out
transversely by equation~\eqref{positivity}.
The last monopole equation~\eqref{cSW4} gives the condition
$$
\deg(L) = -\deg(K) + 2\deg({\mathcal L}) = {\numfrac12}\int\Lambda c_1(F)\Phi^2
= {\numfrac12}\int ( |\beta|^2 - |\alpha|^2 + 2\lambda) \Phi^2
$$
which fixes the global $L_2$ norm of $\alpha$ and $\beta$, and determines
whether $\alpha \ne 0$ or $\beta \ne 0$ or $\alpha = \beta = 0$.
Finally we deal with equation~\eqref{cSW4}. If $\alpha \ne 0$ and
$\beta =0$ then we are dealing essentially with the abelian vortex
equation studied by Steve Bradlow \cite[\S 4]{Bradlow}
Oscar Garcia-Prada
and earlier in a different guise by Kazdan Warner \cite{KazdanWarner}.
See also \cite{Bradlow:nonabelian} and \cite{OkonekTeleman:coupledSW}.
I thank Steve Bradlow for pointing out that almost all of the work had
already been done by him and Oscar Garcia-Prada.
To identify the moduli space as a real analytic space we just
jazz up Bradlow's results a bit. This is necessary because we have to
to understand how
the moduli space is cut out in order to apply the localised Euler
class machinery in the next section.
It is slightly more convenient to use our alternative
description~\eqref{alternative} of $\Pee^*$, and solve for a pair
$(d_L, h)$ where $h = e^f h_0$ is a hermitian metric on $L$ and $d_L=
\dee + \dbar = d_0 + a$ is $h$ unitary, and mod out the full gauge
group $\G^{\Bbb C}$ of all complex nowhere vanishing functions. To be
precise we take $d_L$ in $L^p_1$, and $\G^{\Bbb C}$ and $f$ in $L^p_2$ with
$p > 4$. The sections $\alpha$ and $\beta$, being disguised spinors, are
as before in $L^p_1$.
For an $h$-unitary connection we have , $\dee h(s,t) = h(\dee s, t) +
h(s, \dbar t)$ for all sections $s,t \in A^0({\mathcal L})$. Thus $d_L$ is
determined by $\dbar$ and $h$, or equivalently, $a^{01}$ and $f$.
Expressed in $a^{01}$ and $f$, equation~\eqref{cSW4} becomes
\begin{equation}\label{masterf}
\laplace f = 2\pi (|\beta|^2_{h_0} - |\alpha|_{h_0}^2)e ^f -
2\sqrt{-1}\Lambda(\dee_0 a^{01} - \dbar_0\bar{a^{01}}) + \mu
\end{equation}
where
$\mu = 2\pi(2\lambda + (\Lambda c_1(F(K)) - 2 \Lambda c_1({\mathcal L},\nabla_0)$
(compare \cite[lemma 4.1]{Bradlow}).
If $\beta$ is small in $L^p_1$ hence in $C^0$,
we can solve for $f$ in equation~\eqref{masterf} with
the solution depending real analytically on $(a^{01},\alpha)$ by
the analytical lemma~\ref{fsoln}.
Moreover, variation of ~\eqref{masterf} with respect
to $f$ when $\beta = 0$ gives
\begin{equation}\label{deltamasterf}
\delta\hbox{``eqn \eqref{masterf}''} =
(\laplace + 2\pi|\alpha|^2e^f)\delta f.
\end{equation}
Thus, equation~\eqref{masterf} cuts out this solution
transversely. More invariantly, if $\beta$ is small, there is a unique
metric $h(\dbar,\alpha,\beta) = h_0e^{f(\dbar-\dbar_0,\alpha,\beta)}$
solving the last monopole equation~\eqref{cSW4}.
\begin{Lemma} \label{fsoln}
Let $X$ be a compact Riemannian manifold,
and $\dim(X) < p < \infty$ a Sobolev weight.
Then for every real non negative function
$0 \le w_0 \in L^p$, with $\int w_0 > 0$
and real function $\mu_0 \in L^p$, with $\int \mu_0 > 0$, there exists
a neighborhood $U_{(w_0,\mu_0)} \subset L^p \times L^p$ such that
for all $(w,\mu)\in U_{(w_0,\mu_0)}$ the equation
\begin{equation}\label{rawfeq}
\laplace f = - w e^f + \mu
\end{equation}
has a unique $L^p_2$ solution depending analytically on $w$ and $\mu$.
The solution is smooth if $w$ and $\mu$ are smooth.
\end{Lemma}
\begin{pf}
As in \cite[lemma 4]{Bradlow} make the substitution
$f = \~f - g$ where $g$ is the unique solution of $\laplace g =
\int \mu - \mu $ to reduce to the case where $\mu$ is constant.
Then apply \cite[theorem 10.5(a)]{KazdanWarner}
to solve the equation for $w_0,\mu_0$ (note that Kazdan Warners Laplacian
is negative definite and that the proof
works fine with $w \in L^p$ instead of
$C^\infty$).
Since at a solution $f_0$ for $(w_0,\mu_0)$ we have
$$
\delta \hbox{``eqn \eqref{rawfeq}''} =
(\laplace + w_0 e^{f_0})\delta f
$$
and $(\laplace + w_0 e^{f_0})$ is invertible, we conclude with the
implicit function theorem that there continues to exist a solution for
$(w,\mu)$ in a small neighborhood of $(w_0,\mu_0)$, and that this
solution depends real analytically on $(w,\mu)$. Regularity follows from
standard bootstrapping techniques. Uniqueness follows from the weak
maximum principle (\cite[theorem 8.1]{GilbargTrudinger}, c.f.
\cite[remark 10.12]{KazdanWarner}).
\end{pf}
In geometric terms, this has the following consequence. let $\A^{01}$ be
(the $L^p_1$-completion) of the space of $\dbar$-operators on ${\mathcal L}$
modeled on $A^{01}(X)$ through $\dbar = \dbar_0 + a^{01}$. The complex
gauge group $\G^{\Bbb C}$ acts naturally by conjugation. Let
$$
\Pee^{01*} = \A^{01} \times (A^{00}({\mathcal L})\oplus A^{02}({\mathcal L}))^* /\G^{\Bbb C}
$$
Clearly there is a projection
$\Pee^* \to \Pee^{01*}$ forgetting $h$. What we have done is
showing that there is section
\begin{align*}
\Pee^{01*} & \to \Pee^*
\\
(\dbar,\alpha,\beta) &\to (\dbar,\alpha,\beta,h(\dbar,\alpha,\beta))
\end{align*}
in a neighborhood of $\beta = 0$,
whose image is cut out as a real analytic space by the last monopole
equation~\eqref{cSW4}.
So far we have not used the other equations. Suppose we are in
case~\eqref{case0}, i.e. where a solution corresponds to sections.
Then ${\mathcal M}$ is cut out by $\dbar^2 =0$, $\dbar \alpha
= 0$, $\beta = 0$ and, by the preceding argument, $h =
h(\dbar,\alpha,\beta)$. Thus projection identifies ${\mathcal M}$
with
$$
\MBN = \{ (\dbar,\alpha,\beta) \in \Pee^{01},
\ \dbar^2 = 0,\ \dbar\alpha =0, \beta=0\}
$$
For the Zariski tangent space it gives
\begin{align*}
T_{(\nabla,\alpha,0,h)}{\mathcal M} &= T_{(\dbar,\alpha,0)}\MBN
\\
&= \rmmath{Ker}\left.
\begin{pmatrix}
\dbar & \alpha \\
& \dbar
\end{pmatrix}
\right/ \rmmath{Im}
\begin{pmatrix}
\alpha \\
-\dbar
\end{pmatrix}
\\
&= {\Bbb H}^1( 0 \m@p--\rightarrow{} {\mathcal O}
\m@p--\rightarrow{\alpha} {\mathcal L} \m@p--\rightarrow{} 0)
\\
&= H^0({\mathcal L}|_{Z(\alpha)}).
\end{align*}
It is easy to check that the linearised versions of equations~\eqref{cSW1},
\eqref{cSW2}, \eqref{cSW3},
and~\eqref{masterf} give the same result (as it should).
\ifcomment\bgroup\par\medskip\noindent\small{
Substituting $(\nabla + \eps a,\alpha + \eps \xi,\eps \eta, h e^{\eps
f})$ in \eqref{complexSW}, when $(\nabla,\alpha,0,h) \in {\mathcal M}$ yields
\begin{align*}
&\dbar^*\eta + \dbar \xi + a^{01} \alpha = 0
\\
&\dbar a^{01} = \bar\alpha \eta
\\
&\dee a^{10} = - \alpha \bar \eta
\\
&\numfrac{\sqrt{-1}}{2\pi}\Lambda (\dbar a^{10} + \dee a^{01}) +
\numfrac1{4\pi} \laplace f = -|\alpha|^2_h f - 2 \rmmath{Re}
h(\alpha,\xi).
\end{align*}
As above, the equations give $\eta = 0$ and we can just solve for $f$.
We mod out the infinitesimal gauge transformations sending $u \in
A^0_{\Bbb C}(X)$ to
$(a,\xi,\eta,f) = (-du,u\alpha,0,2\rmmath{Re} u)$
The conclusion is the same.
}
Case \eqref{case2} is reduced to the previous case by Serre duality.
In case \eqref{singcase} the metric $h$ we look for is an
(almost) Hermite-Einstein metric.
\end{pf}
\begin{Corollary} \label{genus}
Let $X$ be K\"ahler surface.
and $L \equiv w_2(X)$ be a class in $H^2(X,{\Bbb Z})$ with $n(L) \ne 0$.
Then $L$ is of type $(1,1)$.
Moreover if $p_g >0$, then for all K\"ahler forms $\Phi$ on
$X$, the class $L$ satisfies
$$
\deg_\Phi(K_X) \ge \deg_\Phi(L) \ge -\deg_\Phi K_X
$$
If $p_g = 0$, and $n_-(L) \ne 0$ (resp. $n_+(L) \ne 0$), then
$$
\deg_\Phi(L) \ge -\deg_\Phi(K_X)\
\txt{(resp.} \deg_\Phi(L)\le \deg_\Phi(K_X))
$$
\end{Corollary}
\begin{pf}
First we consider the case $p_g >0$. Under the conditions of the
corollary, there is an SC-structure $W$ with $L_W = L$ which admits at
least one solution to the monopole equation for {\sl every} admissable
pair $(g,\epsilon)$. In particular $W$ admits a solution for every
K\"ahler metric and $\epsilon = \lambda \Phi$. Thus $L = L_W$ is of type
$(1,1)$. Moreover the necessary condition for the existence of a
solution of section {\sl or} cosection type (i.e. equation \ref{case0}
{\sl or} \ref{case2} in proposition~\ref{Kahlermonopoles}) gives
precisely the required inequality in the limit $\lambda \to 0$.
If $p_g =0$, then $L$ is automatically of type $(1,1)$ and
say the condition $n_-(L) \ne 0$ means that there is an SC structure $W$
with $L_W = L$ such that for any K\"ahler metric,
$W$ admits solutions of section type (i.e. equation
\ref{case0}) if $\lambda$ is sufficiently large.
This gives a lower bound but no upper bound on $\deg_\Phi(L)$.
\end{pf}
\begin{Remark}\label{easyp_g=0}
If $p_g =0$ and we restrict to perturbation $\epsilon =0$ (or small),
then the same argument as in the $p_g >0$ case gives the stronger
degree inequality if $L^2 \ge 0$, $L$ is not torsion, since in this
case all metrics are admissable and have discriminant of equal sign. In
particular on a Del Pezzo surface such classes do not exist.
\end{Remark}
\begin{Corollary} \label{Kisthere}
Let $X$ be a K\"ahler surface with base SC structure $W_0 =
\Lambda^{0*} X$. Then $n(W_0) = 1$ if $p_g >0 $ and $n_-(W_0) = 1$ if
$p_g =0$, in particular $n(-K_X) \ne 0$ resp. $n_-(-K_X) \ne 0$.
Likewise, $n(W_0(K_X) = \pm 1$ if $p_g>0$ and $n_+(W_0(K_X) = \pm 1$,
in particular $n(K_X)\ne 0$ resp. $n_+(K_X) \ne 0$. Moreover $W_0$ is
the only SC-structure $W$ with $L_W = -K_X$ mod torsion and non
trivial multiplicity $n$ respectively $n_-$. In particular if $L \in
H^2(X,{\Bbb Z})$, such that $L = -K \in H^2(X,{\Bbb Q})$ and $n(L) \ne 0$ resp.
$n_-(L) \ne 0$ then $L = -K \in H^2(X,{\Bbb Z})$.
\end{Corollary}
\begin{pf}
We will prove the statement for $-K_X$. Then we have to consider
SC-structures $W = \Lambda^{0*}({\mathcal L})$ with $c_1({\mathcal L})$ torsion. Choose a
K\"ahler metric and $\lambda \gg 0$. Then ${\mathcal M}(W) \iso \MBN({\mathcal L})$ the
moduli space of line bundles with a section. But ${\mathcal M}^{BN}({\mathcal L})$ is
just a reduced point if ${\mathcal L}$ is trivial, and empty if $c_1({\mathcal L})$ is non
trivial torsion. Thus $W_0 = \Lambda^{0*} X$ is unique among the
SC-structures $W$ with $L_W = -K_X$ mod torsion with $n(W) \ne 0$
(resp. $n_-(W) \ne 0$). In fact its multiplicity is $1$.
The case $+K_X$ can be dealt similarly with Serre duality.
Its multiplicity is $\pm 1$ because of the unpleasant orientation
switches.
\end{pf}
\begin{Corollary}\label{slickdivisor}
Let $D$ be an effective divisor with $D\cdot(D-K) = 0$, $h^0({\mathcal O}(D))
=1$, $h^0({\mathcal O}_D(D)) =0$, and $h^0({\mathcal L}(D)) = 0$ for every line bundle ${\mathcal L}
\in \rmmath{Pic}^0(X)$. Then $n(-K_X + 2D) \ne 0$ if $p_g >0$ and $n_-(-K_X +
2D) \ne 0$ if $p_g =0$. Likewise, $n(K_X -2D) \ne 0$ if $p_g >0$ and
$n_+(K_X - 2D) \ne 0$ if $p_g = 0$.
\end{Corollary}
\begin{pf}
This corollary is proved just as the previous one, and reduces to it if
$D =0$. The conditions of the corollary ensure precisely that
$\MBN({\mathcal O}(D))$ consists of one smooth point and that
$\vdim(\Lambda^{0*}(D)) = 0$.
\end{pf}
We are finally in the position to prove the main theorem~\ref{main}
and corollary~\ref{poscurv}.
Our first task is to define a set $\K$ of basic classes.
\begin{Definition}
If $b_+ \ge 2$ then the basic classes are defined by
$$
\K = \{ K \in H^2(X,{\Bbb Z}) \mid n(K) \ne 0\}
$$
If $b_+ = 1$ then $\K = \K_- \cup \K_+$ where
\begin{align*}
\K_- =\{ K &\in H^2(X,{\Bbb Z}) \mid n_-(K) \ne 0, \txt{and}
\exists L \txt{with} n_-(L) \ne 0
\\
&\txt{such that} n_-(L-m(K+L))\ne 0 \txt{for some $m \ge 1$} \}.
\end{align*}
The set $\K_+$ is defined similarly in terms of $n_+$. Here we are allowed
to take $m\ge 1$ rational as long as $m(K+L)$ is two divisible.
\end{Definition}
These basic classes
are rightfully {\sl the} Seiberg-Witten basic classes when $b_+
\ge 2$, but for $b_+ =1$ the definition is geared towards the specific
application we have in mind.
We will show that $\K$ has all properties~\ref{*}.
It is clear that $\K$ is an oriented diffeomorphism invariant, and that
the basic classes are characteristic. The pushforward property
\ref{*}.\ref{iii} follows immediately from the blow up formula
theorem~\ref{blowup4} or~\ref{blowup}. For K\"ahler surfaces
the classes are of type $(1,1)$ by
corollary~\ref{genus}.
The degree property~(\ref{*}.\ref{ii})
(for all surfaces minimal or not) follows also from
corollary~\ref{genus}. This is immediate for $p_g >0$.
If $p_g = 0$ assume that $K \in \K_+$ say, the case $K \in \K_-$ being
essentially the same.
Then the corollary gives the three inequalities
\begin{align}
\deg K &\le \deg K_X,
\\
\deg L &\le \deg K_X,
\\
-m\deg K &\le \deg K_X + (m-1) \deg L \le m\deg K_X.
\end{align}
If $p_g >0$ then $K_X \in \K$ by corollary~\ref{Kisthere}.
Thus it remains to check that $K_X \in \K$ if $p_g =0$.
In fact we will check that $-K_X \in \K$.
We have already seen in corollary~\ref{Kisthere} that $n_-(-K_X) \ne 0$.
Either directly from corollary~\ref{slickdivisor},
or using the invariance under the reflection in the
exceptional curves $E_1, \ldots, E_n$ we see that
$n_-(-K_X + 2\sum E_i) \ne 0$. Then
denoting
$$
{\mathcal L}_m = m\Kmin + \sum E_i,
$$
we have to check that $n_-(-K_X + 2{\mathcal L}_m) \ne 0$.
We will distinguish four cases.
If $\kappa(X) = 0$, then $\Kmin$ is torsion and
we can take $m=\rmmath{ord}(\Kmin)$, since $n_-(-K_X + 2\sum E_i) \ne 0$.
If $\kappa(X) = 1$, then $\Xmin$ has a unique elliptic fibration $\Xmin
\m@p--\rightarrow{\pi} C$. By the canonical bundle formula, $\Kmin =
\pi^*{\mathcal L}_C(\pi^*K_C + \sum (p_i - 1)F_i)$, where ${\mathcal L}_C$ is a line bundle
on $C$ of degree $\chi$. Since $p_g = 0$ and $\chi \ge 0$, we have $0
\le g\le q\le 1$, and we distinguish further between $g=0$ and $g =1$.
If $g= 0$, then $c_1(\pi^*{\mathcal L}_C(K_C)) =(\chi-2) F$, where $F$ is a
general fibre, and there are at least $3-\chi$ multiple fibers because
$\Kmin >0$. Now the class $\Kmin + \sum_{i=1}^{2-\chi} F_i =
\sum_{j=3-\chi}^n (p_j -1) F_j$ is of the form $m\Kmin$ with rational
$m >1$. Again by corollary~\ref{slickdivisor}, we have
$$
n_-(-K_X + 2{\mathcal L}_m) = n_-(-K_X + 2(\sum_{j=3-\chi}^n (p_j-1)F_j +
\sum E_i)) \ne 0
$$
If $g=1$, then $\chi = 0$, and $K_C =0$.
In this case we can take $m = 1$ since $c_1({\mathcal L}_C) =0\in H^2(X,{\Bbb Z})$
and by corollary~\ref{slickdivisor}
$$
n_-(-K_X +2{\mathcal L}_1) = n_-(-K_X + 2(\sum (p_i-1)F_i + \sum E_i)) \ne 0.
$$
The most instructive case is when $X$ is of general type. Then the
irregularity $q=0$ since $p_g = 0$ and $\chi({\mathcal O}_X) >0$.
Take $m=2$,
then $\MBN({\mathcal L}_2) = |2\Kmin + \sum E_i|$. By
formula~\eqref{gtpluri} (or directly by Ramanujan vanishing)
$$
\dim_{\Bbb C}\MBN({\mathcal L}_2) = P_2 - 1 = \Kmin^2 = {\numfrac12} \vdim_{\Bbb R}(W_2).
$$
Thus the moduli space is again smooth of the proper dimension and we
conclude that $n_-(-K_X + 2{\mathcal L}_2) \ne 0$. In fact
$n_-(\Lambda^{0*}({\mathcal L}_2)) = t^{\Kmin^2}$ since the ${\mathcal O}(1)$ on $\Pee^*$
corresponds to the ${\mathcal O}(1)$ on $\MBN$. This is because both measure the
weight of the action of the constant gauge transformations on the
spinors respectively sections.
It now follows from lemma~\ref{inequality} that if $\kappa(X) \ge 0$,
all SW-structure have a moduli space of
virtual dimension $d=0$, and up to torsion, the basic classes are of
type
\begin{equation}
K =\lambda \Kmin + \sum \pm E_i \bmod
\hbox{Torsion}, \qquad |\lambda| \le 1.
\end{equation}
Moreover by proposition~\ref{Kcharacterisation}, $\Kmin$ is invariant
up to sign and torsion and every $(-1)$-sphere is represented by a
$(-1)$-curve up to sign and torsion.
We first get rid of torsion in the $(-1)$-curve conjecture i.e. theorem
{}~\ref{main} part~\ref{mainb}.
Let $e$ be a $(-1)$-sphere, giving a connected sum decomposition $X =
X'\# \Pbar^2$.
As we have used before, there is a
diffeomorphism $R_e =\rmmath{id} \# {\Bbb C}$-conjugation representing the reflection
in $e$.
I claim that for any SC-structure $W$ on a 4-manifold
$$
R_e^*(W) = W \tensor{\mathcal O}((c_1(L_W),e)e),
$$
where ${\mathcal O}(e)$ is the
line bundle corresponding to the Poincar\'e dual of $e$.
In fact if we write $R_e^* W = W \tensor {\mathcal L}$, then
${\mathcal L} = \mathop{{\mathcal H}\mkern-3mu{\mathit om}}\nolimits_C(W, R_e^*W)$ (c.f. the proof of \ref{SCrepr}).
Now we can just identify $W$ and $R_e^* W$ on $X'$, i.e. ${\mathcal L}$ is
trivialised on $X'$. Thus
$$
c_1({\mathcal L}) \in \rmmath{Im} H^2(X, X- X', {\Bbb Z}) \iso
H^2(\Pbar^2) \subset H^2(X,{\Bbb Z}).
$$
Write ${\mathcal L} = {\mathcal O}(a e)$ for some integer $a$. Since
$$
L_W + 2 a e = L_{R_e^* W} = R_e^*L_W = L + 2(e,L_W) e
$$
the claim is proved.
Going back to the K\"ahler case, we can assume that $e$ is homologous
to a $(-1)$-curve $E$ up to torsion. Consider $W =
R_e^*R_E^*(\Lambda^{0*} X) = \Lambda^{0*}(E-e)$. By oriented
diffeomorphism invariance $n_{(-)}(W) \ne 0$ (in case $p_g =0$ we have
tacitly used the fact that $R_e^*R_E^*$ induces the identity on rational
cohomology so in particular does not change the orientation of $H^+$).
Moreover $c_1(L_W) = -K_X$ up to torsion. By corollary~\ref{Kisthere},
we conclude that $W = \Lambda^{0*} X$, so $e = E \in H^2(X,{\Bbb Z})$.
Finally for the invariance $\pm \Kmin$, consider any basic class of the
form $L = \pm \Kmin + \sum \pm E_i$ up to torsion. After reflections in
the $(-1)$-curves, we get a class $L'$ equal to $\pm K_X$ up to
torsion. By corollary~\ref{Kisthere} $L' = \pm K_X \in H^2(X,{\Bbb Z})$. Now
for any basis $E'_1, \ldots E'_n$ of the lattice in $H^2(X,{\Bbb Z})$ spanned
by the $(-1)$-spheres (e.g. the $(-1)$-curves) we have the identity
$$
\pm \Kmin = L' + \sum (E'_i,L') E'_i
= L + \sum ( E'_i, L) E'_i \in H^2(X,{\Bbb Z}).
$$
This finally proves theorem~\ref{main}.
\begin{Remark}
It is easy to give a
definition of basic classes for $b_+ = 1$
that satisfies all properties ~\ref{*} except the
invariance under blow down (i.e. property~\ref{*}.\ref{iii}).
A class $K$ is then basic if there exists a metric $g$ such
for all $\delta >0$
there exists an admissable pair $(g,\epsilon)$ with
$\|\epsilon^{+,\harm}\| < \delta$ such that
$n(g,\epsilon,K) \ne 0$. The degree inequality for minimal surfaces
then follows from remark \ref{specialchamber}. But alas, if $K^2 <0$ one
can not avoid the possibility that a chamber on the blow up
realisable with small $\epsilon$
can only be realised for large $\epsilon$ on the blow down.
In my original treatment I used this definition. I am grateful to
Robert Friedman whose
insistent questions about my definitions made me realise this mistake.
\end{Remark}
\begin{Remark}\label{whycastelnuovo}
An easy application of the
techniques of the next section gives the following.
If ${\mathcal L}$ is a holomorphic line bundle on a surface with $p_g = q = 0$ with
$h^0({\mathcal L}) \ge \chi({\mathcal L})\ge 1$,
then $n_-(\Lambda^{0*}({\mathcal L})) = t^{{\mathcal L}({\mathcal L}-K_X) \over 2}$.
If $p_g= q = 0$ and $\kappa(X) \ge 0$
we can apply this to ${\mathcal L}_2= 2\Kmin + \sum E_i$. Then by the Castelnuovo
criterion and the above we conclude
$n_-(-K_X + 2L_2) \ne 0$. This gives an alternative way to prove
that $-K_X \in \K$ in this case.
Conversely the degree inequality~\ref{*}.\ref{ii} cannot hold true for
rational and ruled surfaces for K\"ahler forms $\Phi$ such that
$\deg_\Phi(K_X) < 0$. Since in deriving the degree inequality we
did not use that $\kappa(X) \ge 0$, we conclude that for $\kappa(X) =
-\infty$ the set of the above defined basic classes $\K= \emptyset$.
In particular we see that the following proposition is a rather
direct analog of to the classical Castelnuovo criterion.
\end{Remark}
\begin{Proposition}\label{Castelnuovo}
A K\"ahler surface
is rational if and only if $b_1= 0$, and $\K = \emptyset$.
\end{Proposition}
\begin{Remark}
After reading \cite{FM:SW} I realised the following. The blow up
formula~\ref{blowup4} can be generalised to connected sum decompositions
$X = X'\# N$ with $N$ negative definite and $H_1(N,{\Bbb Z}) = 0$. The latter
condition is automatic for K\"ahler surfaces of non negative Kodaira
dimension by a beautiful observation of Kotschick
(an unramified covering $\~N \to N$ of
degree $d$ gives an unramified covering $\~X = d X' \# \~N \to X'\# N$
which is an algebraic surface of non negative Kodaira dimension with a
connected sum decomposition with a factor with $b_+ >0$). Such smooth
negative definite manifolds $N$ have $H_2(N) = \mathop\oplus_{i=1}^n {\Bbb Z}
n_i$. SC structures $W_N$ on $N$ are determined by $L_N = \sum (2a_i +
1) n_i$. Thus the reflections $R_{n_i}$ in $n_i^{\perp}$, act on the SC
structures on $N$. SC -structures on $X'\# N$ are of the form $W= W_{X'}
\# W_N$. Now the blow up formula is as if $N = n \Pbar^2$: $W = W_{X'}
\# W_N$ is an SW-structure on $X'\# N$ if and only if $W_{X'}$ is a
SW-structure on $X'$ and $d(W) \ge 0$. In particular the Seiberg Witten
structures are invariant under the operation $R_{n_i}: W_{X'}\# W_N \to
W_{X'} \# R_{n_i} W_N$, and $\mathop{{\mathcal H}\mkern-3mu{\mathit om}}\nolimits_C(W, R_{n_i}W)$ has a
trivialisation over $X'$. With these remarks the arguments for
$(-1)$-spheres carry over directly to prove that for K\"ahler surfaces
$X$ with $\kappa(X)\ge 0$, with a connected sum decomposition $X = X'\#
N$, $H_2(N) \subset H_2(X)$ is spanned by $(-1)$-curves.
\end{Remark}
Stefan Bauer showed me how to use the Seiberg Witten multiplicities and
the basic classes to determine the multiplicities of the elliptic
surface. If the surface does not have finite cyclic fundamental group,
the multiplicities can be read off from the topology. Thus we consider
a minimal elliptic surface $X_{pq}$ fibred over ${\Bbb P}^1$ with 2 multiple
fibers of multiplicity $p$ and $q$ We will assume that $p\le q$.
\begin{Corollary} (Bauer)
The multiplicities $p$ and $q$ are determined by the underlying oriented
differentiable manifold, unless $p_g = 0$, $p=1$ and $q$ arbitrary. The
surfaces $X_{1q}$ are all rational and diffeomorphic.
\end{Corollary}
\begin{pf}
If the canonical class $K_X$ is not torsion , we can
write $K_X$ in terms of the primitive vector $\kappa$ in
the ray spanned by $K_X$, normalised so that $\kappa \Phi >0$
$$
K_X = (p_g - 1)F + (p-1)F_p + (q-1) F_q
= {(p_g + 1)pq - p -q \over \gcd(p,q)} \kappa
\in H^2(X,{\Bbb Z})/\mathord{\hbox{Torsion}}.
$$
Let $d(p,q) = \((p_g+1)pq-p-q\)/\gcd(p,q)$ be the oriented
divisibility of $K_X$. If $K_X$ is torsion we simply set $d(p,q) = 0$.
The divisibility $d(p,q)<0$ if and only if $p_g=0$, $p=1$ and $q$ is
arbitrary. But this implies that $K_X$ is rational. We have already seen
that we can recognise rationality as Kodaira dimension $-\infty$ and
$b_1 =0$ (corollary~\ref{Kodaira} or proposition~\ref{Castelnuovo}).
Thus we can assume that $X_{pq}$ has non negative Kodaira dimension.
Then $\pm K_X \in \K$ are the basic classes with the highest
divisibility (or torsion) and the oriented divisibility $d(p,q) \ge 0$
is just the unoriented divisibility of $\pm K_X$. The number
$\gcd(p,q)$ is also determined by the oriented manifold, being the order
of the fundamental group. Choose one of these classes, say $-K_X$.
First consider the case $p_g >0$. Suppose that $K= -K_X + 2 F_q \le
0$, (i.e. on the same side of $0$ as $-K_X$), then it is the basic
class with second largest divisibility since $F_q$ is the smallest
effective vertical divisor, and $n(-K_X + 2F_q)) \ne 0$ by
lemma~\ref{slickdivisor} above. Thus if there exist basic classes other
then $\pm K_X$, we can reconstruct $p$ from $(2p/ \gcd(p,q))\kappa = K -
(-K_X)$. Since $d(p,q)$, $p_g$ and $\gcd(p,q)$ are known, this
determines $q$ as well. Obviously if we have chosen $+K_X$ the same
arguments works with $K = K_X - 2F_q$, there is nothing that prefers
$K_X$ over $-K_X$.
In the case $p_g =0$ we make a small modification. We choose an
orientation of $H^+$, which for a moment we assume is the standard one.
Consider the classes $K \in H^2(X,{\Bbb Z})$ mod torsion in the half ray
spanned by $0$ and $ -K_X$ with unoriented divisibility at most
$d(p,q)$ (i.e. in between $0$ and $-K_X$) such that $n_-(K) \ne 0$.
Note that $-K_X$ is just the basic class with largest divisibility in
$\K_-$. Then if $K = -K_X + 2 F_q \le 0$ we can use exactly the same
argument as in the case $p_g >0$.
If we choose a different orientation of $H^+$, we replace
$-K_X$ by $+K_X$ but just as above the conclusion is the same.
If $\K = \pm K_X$ or
for $p_g =0$ if $\{K \in [-K_X, 0] \mid n_-(K) \ne 0\} = -K_X $
then $d(p,q)\gcd(p,q)< 2p$.
The few possibilities are listed in the following table
$$
\begin{array}{|l|c|c|c|l|}
\hline
\strut&(p,q) &\gcd(p,q) &d(p,q) &\text{Type} \\
\hline
p_g=0 &(2,2) &2 &0 &\text{Enriques} \\
&(2,3) &1 &1 & \\
&(2,4) &2 &1 & \\
&(2,5) &1 &3 & \\
&(3,3) &3 &1 & \\
&(3,4) &1 &5 & \\
\hline
p_g=1 &(1,1) &1 &0 &\text{K3} \\
&(1,2) &1 &1 & \\
\hline
\end{array}
$$
Clearly, in this case the pair $(p,q)$ is determined by the
oriented differentiable manifold as well.
\end{pf}
To prove that no surface with $\kappa \ge 0$ admits a metric with positive
scalar curvature (corollary~\ref{poscurv}), first consider the case $p_g
>0$. Then the statement is clear, and one of Witten's basic
observations. By proposition~\ref{Bochner}, for 4-manifolds with positive
scalar curvature $n(K) = 0$ for all $K \in H^2(X,{\Bbb Z})$, since
for our metric with positive scalar curvature $g$ and small
perturbations $\epsilon$, we have ${\mathcal M}(W,g,\epsilon) = \emptyset$ for all
SC-structures $W$. On the other hand we just showed that $n(-K_X) \ne 0$
using a K\"ahler metric.
The same argument works if $p_g = 0$ and $K_X^2 \ge 0$:
$n(-K_X,g,\epsilon)$ is independent of the metric $g$ and of $\epsilon$ as
long as $\epsilon$ is small, with the exception of the case $-K_X$
torsion in which case we have to choose $\epsilon$ in the forward light
cone. But we can do better.
For the general case $p_g =0$, we choose a perturbation $\epsilon =
\lambda \Phi$ with $0<\lambda \ll 1$ say. Now suppose that the metric
with positive scalar curvature $g$ has period $\omega_g = \omega_{\min}
+ \sum \eta_i E_i$ where $\omega_{\min} $ is the projection to the
cohomology of minimal model. Then since $\omega_g$ is in the interior
of the forward light cone, and $\Kmin$ is in the closure of the forward
light cone, $\omega\cdot \Kmin = \omega_{\min}\cdot \Kmin\ge 0$ with
equality iff $\Kmin$ is torsion. Then for {\sl some} choice of signs in
$-\Kmin - \sum \pm E_i$ we have
$$
\omega_g\cdot (-\Kmin - \sum \pm E_i)\le 0 < \lambda \int\omega_g \Phi
$$
Thus for {\sl some} choice of signs we compute $n_-$ (rather than $n_+$)
with our metric of positive scalar curvature and small perturbation.
Hence $n_-(-\Kmin-\sum \pm E_i) = 0$. On the other hand $n_-(-\Kmin
-\sum \pm E_i) = n_-(-K_X) \ne 0$, a contradiction just like before.
\section{Some computations of Seiberg-Witten multiplicities}%
\label{computations}
In this section we will go beyond determining potential basic classes and
compute the Seiberg Witten multiplicity of elliptic surfaces. We also
prove an algebraic version of the blow up formula.
It is here that our excess intersection formulas pay off.
We first show how to go over to a fully complex point of view.
Then we use the special geometry of elliptic surfaces to compute the
multiplicities and finally we prove a blow up formula.
{}From now on we identify an SC-structure with the corresponding
twisting line bundle ${\mathcal L}$.
We will consider the solutions of the monopole equations of section
type, i.e. corresponding to equation~\eqref{case0}.
We have already seen that the variation of the last monopole
equation~\eqref{cSW4} with respect to the hermitian metric is $h$ is
given by $(\laplace + |\alpha|_h^2)h^{-1}\delta h$ (c.f.
equation~\ref{masterf}). Therefore the solutions to the fourth
monopole equation ~\eqref{cSW4} is a smooth submanifold of $\Pee^*$ in a
neighborhood of the moduli space ${\mathcal M}({\mathcal L})$. In the proof of
proposition~\ref{Kahlermonopoles} we have seen that we can identify this
submanifold with the ``vortex locus'' $\{h = h(\dbar,\alpha,\beta)\}$
i.e. the image of the section $\Pee^{01*} \to \Pee^*$. The vortex locus
is well defined in a neighborhood of the moduli space ${\mathcal M}({\mathcal L})$ only, but
this will not affect our arguments, as the construction of the localised
Euler class in section \ref{top} depends only on a neighborhood of
${\mathcal M}({\mathcal L})$. Since the vortex locus is given by a function, we can identify
it with its domain $\Pee^{01*}$ which carries a natural complex
structure.
By property \ref{stability} of proposition~\ref{locEuler} we are
allowed to compute the localised Euler class $\MM({\mathcal L})$ of the moduli
space by considering ${\mathcal M}({\mathcal L})$ as a zero set of a section $S$ over the
vortex locus cut out by the remaining equations, which define the same
ideal as
$$
\dbar^2 = 0, \qquad \dbar \alpha =0, \quad \beta = 0
$$
i.e. complex equations ! Moreover the deformation complex of these
equations on
$\Pee^{01*}$ in a point $(\dbar,\alpha,0)$ is
$$
A^{00}(X) \m@p--\rightarrow{}
A^{01}(X) \oplus A^{00}({\mathcal L}) \oplus A^{02}({\mathcal L}) \m@p--\rightarrow{}
A^{02}(X) \oplus A^{01}({\mathcal L})
$$
where the map is complex linear. We trivialise the determinant of the
index using the complex structure. This has brought us safely in
complex waters, and allows us to use proposition ~\ref{locChern} and in
particular formula~\ref{magic}.
{}From now on we identify ${\mathcal M}({\mathcal L})$ with $\MBN({\mathcal L})$.
Define the vector bundles
$$
\sfA^{pq}({\mathcal L}) =\(\A^{01} \times
(A^{00}({\mathcal L} \oplus A^{02}({\mathcal L}))^*\)\times_{\G^{\Bbb C}} A^{pq}({\mathcal L})
$$
over $\Pee^{01*}$. Then $\MBN$ is given by a section $s$ in $E = A^{02}(X)
\oplus \sfA^{01}({\mathcal L})$, and the tangent space is given by
$$
T\Pee^{01*} \iso \(A^{01}(X) \oplus \sfA^{00}({\mathcal L})
\oplus \sfA^{02}({\mathcal L})\)/ A^{00}(X).
$$
The deformation complex can be considered as a map
$T\Pee^* \to E$ and is exactly what we called $Ds$ in section~\ref{top}.
To identify the index $\Ind Ds$ we first make a compact
perturbation, keeping only the differential operator part of the
deformation complex. Then it splits naturally in the $\dbar$ complex on
$X$ with index ${\Bbb C}^{\chi({\mathcal O}_X)}$ and the index of complex
$$
0 \m@p--\rightarrow{} \sfA^{00}({\mathcal L}) \m@p--\rightarrow{\dbar} \sfA^{01}({\mathcal L})
\m@p--\rightarrow{\dbar} \sfA^{02}({\mathcal L}) \to 0
$$
where $\dbar$ is the universal $\dbar$ operator descended to $\Pee^{01*}$.
To rewrite this index
in holomorphic terms, consider the universal divisor
$$
\Delta = \{ (\dbar,\alpha,x) \mid \alpha(x) = 0\}
$$
on the pull back of $X \times \MBN$. Now if $\Omega^{pq}$ is the sheaf
of $C^{\infty}$ $(p,q)$-forms on $X$ considered as an ${\mathcal O}(X)$ module,
then I claim that on the pull back of $X \times \MBN$ to $\A^{01} \times
(A^{00}({\mathcal L} \oplus A^{02}({\mathcal L}))^*$, there is a $\G^{\Bbb C}$ equivariant exact
sequence
$$
0 \m@p--\rightarrow{} {\mathcal O}(\Delta)
\m@p--\rightarrow{\dbar} p_1^*\Omega^{00}({\mathcal L})
\m@p--\rightarrow{\dbar} p_1^*\Omega^{01}({\mathcal L})
\m@p--\rightarrow{\dbar} p_1^*\Omega^{02}({\mathcal L}) \to 0.
$$
In fact this only says that $(\dbar,\alpha, x) \to \alpha(x)$ is a
$\G^{\Bbb C}$ invariant section vanishing along $\Delta$ with multiplicity $1$
lying in the kernel of $\dbar$, which is obvious. Now descend this
whole complex to $X \times \MBN$ and take push forward to $\MBN$. Then
we get an exact sequence of complexes
$$
0 \m@p--\rightarrow{} Rp_*{\mathcal O}(\Delta) \m@p--\rightarrow{\dbar} \sfA^{00}({\mathcal L})
\m@p--\rightarrow{\dbar} A^{01}({\mathcal L}) \m@p--\rightarrow{\dbar} \sfA^{02}({\mathcal L}) \m@p--\rightarrow{} 0
$$
where we are considering $Rp_*{\mathcal O}(\Delta)$) as a complex with zero
boundary operator and $\sfA^{pq}({\mathcal L})$ as a complex concentrated in
degree $0$. Thus for the index we find
\begin{equation}\label{index}
\Ind(Ds) = \Ind\(Rp_* {\mathcal O}(\Delta)\) + {\Bbb C}^{\chi}
\end{equation}
A more precise description of $\MBN$ depends on the surface.
Here we will do the case of elliptic surfaces. The author has succeeded
in treating ruled surfaces in a similar way.
\begin{Proposition}\label{multiplic}
Let $X \m@p--\rightarrow{\pi} C$ be a K\"ahlerian elliptic surface
over a curve $C$ of genus $g$, with
multiple fibers $F_1, \ldots F_r$ of multiplicity $p_1, \ldots p_r$
of holomorphic Euler characteristic $\chi$.
Consider the line bundle ${\mathcal L} = {\mathcal O}(\pi^*D + \sum a_i F_i)$ where
$D$ is a divisor on $C$ of degree $d$, and $0 \le a_i < p_i$.
Then the Seiberg Witten multiplicity is zero if $d < 0$, and if $d\ge 0$
it is given by
$$
n_{(-)}(\Lambda^{0*}({\mathcal L})) = \begin{cases}
(-1)^d{\chi + 2g-2\choose d} & \txt{if} \chi + g -2 \ge 0
\\
\sum (-1)^j { 1-g-\chi + d-j \choose d-j}{g \choose j}
& \txt{if} \chi + g - 2 < 0
\end{cases}
$$
\end{Proposition}
Note that if the topological Euler characteristic $e >0$ (or equivalently
$\chi >0$) then $g =q = {\numfrac12} b_1(X)$ \cite[corollary II.2.4]{FM}, so in
this case $\chi + g-2 = p_g - 1$. Note that the second formula is just
1 if $p_g =q =0$ (i.e. $e>0$). This illustrates
remark~\ref{whycastelnuovo}.
If $p_g >0$ and $q = g = 0$, so in particular $e = 12 \chi > 0$, Witten
proves this formula by choosing a general $\omega \in H^0(K_X)$ and
using the perturbation $\epsilon = \omega + \bar\omega$. He then argues
that the multiplicity $n({\mathcal L})$ is the number of ways we can decompose a
fixed canonical divisor $K_0$ as $K_0 = D_+ + D_-$ with $D_+\in
|({\mathcal L},\dbar_0)|$, and $D_- \in |K\tensor({\mathcal L},\dbar_0)^{\scriptscriptstyle\vee}|$, where
$\dbar_0$ is the unique holomorphic structure that ${\mathcal L}$ admits \cite[eq.
(4.23) e.v.]{Witten}.
To be honest, this is
what I read out of it. Note for example that his sheaf $R$ is just
${\mathcal L}|_Z(\alpha)$, and that
$$
h^0(R) = h^0({\mathcal L}|_{Z(\alpha))} = \dim T_{(\dbar,\alpha)} = d
$$
(the last equality we will see in a minute).
Actually I think that the
computations below are the mathematical version of (I paraphrase)
``integrating over
the bosonic and fermionic collective coordinates in the path integral''
and ``computing the Euler class of the bundle of the cokernel of the
operator describing the linearised monopole equations over the moduli
space (the bundle of
antighost zero modes)'' \cite[above (4.11)]{Witten}. In fact with
hindsight, the latter seems a dual description
of the localised Euler class in the case that the
cokernel has constant rank.
\begin{pf}
We choose a K\"ahler metric and $\lambda$ such that
$\deg_\Phi({\mathcal L}^{\tensor 2}(-K)) < \lambda \Vol(X)$. This means that if
${\mathcal L}$ has non zero multiplicity, it must carry a holomorphic structure
with a section. In case $p_g =0$ it also means we are looking at $n_-$.
But $({\mathcal L},\dbar)$ has a section if and only if $D$ is an effective
divisor on $C$. In fact a family of vertical line bundles with a section
gives a family of effective divisors on $C$ by pushforward of the line
bundle, and conversely a family of effective divisors on $C$ gives a
family of vertical line bundles with a section by pull back and
multiplication with a fixed section in ${\mathcal O}(B) = {\mathcal O}(\sum a_i F_i)$ ($B$
for base locus). Thus there is a natural isomorphism
$$
\MBN \iso \MBN_C = C^d
$$
where $C^d$ is the $d^{\text{th}}$ symmetric power of $C$. The functorial
isomorphism comes with an isomorphism ${\mathcal O}(\Delta_X) = {\mathcal O}(\pi^*\Delta_C +
B)$.
Next we use Grothendieck Riemann Roch (an alias of the family index
theorem). Let $q: C \times C^d \to C^d$ be the projection map. Then the
projection $p: X \times \MBN \to \MBN$ can be factored as $p = q\circ
\pi\times \rmmath{id}$. Thus writing $\pi\times \rmmath{id}$ as $\pi$,
\begin{align*}
\rmmath{ch}(Rp_*{\mathcal O}(\Delta))
&= \rmmath{ch}\(Rq_* \({\mathcal O}(\Delta_C) \tensor R\pi_*{\mathcal O}(B)\)\)
\\
&= q_*\(ch({\mathcal O}(\Delta_C) \rmmath{ch} R\pi_*{\mathcal O}(B) \rmmath{td}(C)\)
\\
&= q_*\(ch({\mathcal O}(\Delta_C))
\pi_*\(e^B(1 - K/2 + \chi({\mathcal O}_X)(pt\times C^d)\)\)
\\
&= \chi({\mathcal O}_X) q_*\(ch({\mathcal O}(\Delta_C))(pt\times C^d)\)
\\
&= \rmmath{ch}({\mathcal O}(1)^\chi),
\end{align*}
where we have abbreviated the holomorphic Euler characteristic by
$\chi$. If we denote by $x$ the chern class of ${\mathcal O}(1)$, then our
computation shows that
$$
c_t(\Ind Ds) = (1 + tx)^\chi,
$$
at least over the rationals.
The chern classes of the tangent bundle of $C^d$ are computed in
\cite[eq. VII.5.4]{ACGH}. Denoting the pullback of the $\theta$ divisor
on $\rmmath{Pic}^d$ to
$C^d$ by $\theta$ the result is
$$
c_t(T_{C^d}) = (1+ tx)^{d+1-g} e^{-t\theta / (1+ tx)}
$$
Combining these two expressions, our multiplicity drops out
\begin{align*}
n(\Lambda^{0*}({\mathcal L})) &= c(\Ind Ds)^{-1} c(T_C^d) \cap [C^d]
\\
&= [(1 + tx) ^{d +1-g - \chi} e^{-t\theta /1+tx}]_{t^d}
\end{align*}
With the following identity of formal power series
\cite[eq. VIII.3.1]{ACGH}
$$
[(1 + xt)^a f( - t/(1+ xt))]_{t^b} = [ (1 - xt)^{b - a - 1}
f(-t)]_{t^b}.
$$
the expression becomes
$$
n(\Lambda^{0*}({\mathcal L})) = [(1-tx)^{\chi + g - 2} e^{-t\theta}]_{t^d}
=
\begin{cases}
(-1)^d\sum_{j=0}^d {\chi + g - 2 \choose d-j} {\theta^j x^{d-j} \over j!}
& \txt{if} \chi + g- 2 \ge 0
\\
\sum_{j=0}^d (-1)^j {1-g-\chi + d-j \over d-j}{\theta^j x^{d-j} \over j!}
& \txt{if} \chi + g -2 < 0
\end{cases}
\hskip 0pt minus 1fil
$$
Now $\theta^jx^{d-j}\cap [C^d] = j! {g \choose j}$ \cite[below eq.
VIII.3.3]{ACGH}. The elementary identity $\sum_j {a \choose j}{b \choose
c-j} = {a + b \choose c}$ then gives the answer as stated.
\end{pf}
As a second application of the methods developed we give
a complex analytic version of the blow up formula.
\begin{Proposition}\label{blowup}
Let $(X,\Phi)$ be a K\"ahler surface, and ${\mathcal L}$ a line bundle on $X$.
Suppose that $\deg_\Phi({\mathcal L}^{\tensor 2}(-K) < \lambda \Vol(X)$. Let
$\sigma:\~X \to X$ be the blow up of $X$ in a point,
with K\"ahler form $\~\Phi$, and let
$\~{\mathcal L} = {\mathcal L}(a E)$ be a line bundle on $\~X$ with $a\ge 0$. Suppose
that the cohomology class of $\~\Phi$ is close to $\Phi$.
Then there is a natural identification ${\mathcal M}(\~{\mathcal L}) =
{\mathcal M}({\mathcal L})$, and
$$
\MM(\~{\mathcal L}) = [(1+x)^{a(a-1)/2}\widehat \MM({\mathcal L})]_{\dim_{\Bbb R}=
{\mathcal L}\cdot({\mathcal L}-K)- a(a-1)}.
$$
Here $\widehat\MM$ is the class defined in remark~\ref{Zhat}, and $x$
the class of the natural bundle ${\mathcal O}(1)$ over ${\mathcal M}$.
In particular if $a=0,1$ then $n(\~{\mathcal L}) = n({\mathcal L})$.
\end{Proposition}
Of course this proposition determines the multiplicity
$$
n_{(-)}(\Lambda^{0*}({\mathcal L}(aE))) = n_{(-)}(\Lambda^{0*}({\mathcal L}(-aE)))).
$$
Since quite in general $n_+(\Lambda^{0*}({\mathcal L})) = \pm n_-(\Lambda^{0*}(K
\tensor {\mathcal L}^{{\scriptscriptstyle\vee}})$ it determines the corresponding relation for
$n_+$ up to sign which is really all we need here.
\begin{pf}
The conditions on the degree imply that a solution of the monopole
equations correspond to a holomorphic structure on ${\mathcal L}$ with a section.
Since $\~\Phi$ is close to $\Phi$ we have (by definition of close)
$\deg_{\~\Phi}(\~L) < \lambda\Vol(\~X)$, hence solutions on the blowup
also correspond to holomorphic structures on $\~{\mathcal L}$ with a section.
Now $aE$ is contained in the base locus of the sections. Thus similar to
what we did for elliptic surfaces, we get an identification of ${\mathcal M}({\mathcal L})$
with ${\mathcal M}(\~{\mathcal L})$ by multiplication of the
section with a section in ${\mathcal O}(aE)$, and the universal divisor
on $\~X\times {\mathcal M}(\~{\mathcal L})$ is $\~\Delta = \Delta + aE$.
Again, identify the chern class of the index of the
deformation complex with formula~\eqref{index}.
Let $\~p$ be the projection $\~X \times {\mathcal M}({\mathcal L}) \to {\mathcal M}({\mathcal L})$, and
$p$ the projection $X\times {\mathcal M}({\mathcal L}) \to {\mathcal M}({\mathcal L})$.
Then the total chern class of the index is
$$
c(R\~p_*(\~\Delta)) = c\(Rp_*\({\mathcal O}(\Delta)\tensor R\sigma_*{\mathcal O}(aE)\) \).
$$
By induction on $a$, one shows that
$$
R\sigma_* {\mathcal O}(aE) = {\mathcal O} -{\mathcal O}_{pt}^{a(a-1)/2}.
$$
Since ${\mathcal O}(\Delta|_{pt\times {\mathcal M}({\mathcal L})}) = {\mathcal O}(1)$ it gives
$$
c(R\~p_*(\~\Delta)) = c(Rp_*{\mathcal O}(\Delta))/c({\mathcal O}(1))^{a(a-1)/2}.
$$
Formula~\eqref{magic} gives us
$$
\MM(\~L) =
[(1+x)^{a(a-1)/2} \(c(Rp_*(\Delta))^{-1}c_*({\mathcal M}({\mathcal L})\)]_{d(\~L)}
$$
Since the real virtual dimension of ${\mathcal M}(\~{\mathcal L})$ is
$d(\~{\mathcal L}) = {\mathcal L}\cdot ({\mathcal L}-K) -a(a-1)$ and the term in brackets is exactly
$\widehat\MM({\mathcal L})$, we have proved the formula.
\end{pf}
|
1995-03-28T07:20:32 | 9503 | alg-geom/9503018 | en | https://arxiv.org/abs/alg-geom/9503018 | [
"alg-geom",
"math.AG"
] | alg-geom/9503018 | Bruce Hunt | Bruce Hunt | A gem of the modular universe | LaTeX 2.09 | null | null | null | null | We introduce one of the most beautiful algebraic varieties known, a quintic
hypersurface in projective five-space, which is invariant under the action of
the Weyl group of $E_6$. This variety is intricately related with many other
moduli problems, some of which are: marked hyperelliptic curves of genus two,
Picard curves of genus four with a $\sqrt{-3}$-level structure, six points on
the projective line, abelian surfaces with (1,3) polarisations, quartic
surfaces invariant under the action of the Heisenberg group in projective
three-space, K3-surfaces which are double covers of the projective plane
branched along six lines, and last but not least, cubic surfaces in projective
three-space. These relationships are developed in some detail, with particular
care on the birational aspects.
| [
{
"version": "v1",
"created": "Mon, 27 Mar 1995 06:45:27 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Hunt",
"Bruce",
""
]
] | alg-geom | \part{Projective embeddings of modular varieties}\label{chapter10}
\section{The tetrahedron in ${\Bbb P}^3$}
\subsection{Arrangements defined by Weyl groups}
Let $\Phi(G,T)\subset} \def\nni{\supset} \def\und{\underline \tt^*$ be a root system of a simple group $G$ (over
$\komp$). Using notations as in Bourbaki we have the roots (for those systems
which will be of interest to us in the sequel)
\begin{equation}\label{e108.1}\begin{minipage}{14cm}\begin{itemize}
\item[$\bf A_n$] $\pm (\ge_i-\ge_j),\ 1\leq i< j \leq n+1$;
\item[$\bf B_n$] $\pm (\ge_i\pm \ge_j),\ \pm\ge_i, 1\leq i < j \leq n$;
\item[$\bf C_n$] $\pm (\ge_i\pm \ge_j),\ \pm2\ge_i, 1\leq i < j \leq
n$;
\item[$\bf D_n$] $\pm (\ge_i\pm\ge_j),\ 1\leq i < j \leq n$;
\item[$\bf F_4$] $\pm (\ge_i\pm \ge_j),\ \pm\ge_k,\ \pm{1\over
2}(\ge_1\pm\ge_2\pm\ge_3\pm\ge_4), 1\leq i < j \leq 4,\
k=1,\ldots,4$;
\item[$\bf E_6$] $\pm (\ge_i \pm \ge_j), 1\leq i<j \leq 5,\ \pm{1\over
2}(\ge_1\pm\ge_2\pm\ge_3\pm\ge_4\pm\ge_5-\ge_6-\ge_7+\ge_8)$, with
an even number of ``$-$'' signs in the parenthesis;
\end{itemize}
\end{minipage}
\end{equation}
Each root $\ga$ determines an orthogonal plane $\ga^{\perp}$, and for any
arrangement $\bf X_n$,
\begin{equation}\label{e108.10} \ifmmode {\cal A} \else$\cA$\fi({\bf X_n}):=\{\ga^{\perp} \Big| \ga
\hbox{ a root}\}
\end{equation}
is a central arrangement in $\komp^n$, i.e., each of the planes passes
through the origin. This induces a projective arrangement in ${\Bbb P}^{n-1}$,
as follows. Blow up the origin of $\komp^n$; the exceptional divisor is a
${\Bbb P}^{n-1}$. The {\em projective arrangement} is the union of the
intersections $[H]\cap {\Bbb P}^{n-1}$ in the exceptional divisor, where $[H]$
is the proper transform of the hyperplane $H=\ga^{\perp}$ under the blow
up. The projective arrangements for $\bf B_n$ and $\bf C_n$ coincide, and
these arrangements are given in ${\Bbb P}^{n-1}$ with projective coordinates
$(x_1:\ldots :x_n)$ as follows:
\begin{equation}\label{e108.2} \begin{minipage}{14cm}\begin{tabbing}
$\ifmmode {\cal A} \else$\cA$\fi({\bf A_n})$: \quad \= $\{ x_i=0,\ i=1,\ldots, n;\ x_i=x_j,\
1\leq i<j\leq n\}$;\\ $\ifmmode {\cal A} \else$\cA$\fi({\bf B_n})$: \> $\{ x_i=0,\ i=1,\ldots,
n;\ x_i=\pm x_j,\ 1\leq i<j\leq n\}$;\\ $\ifmmode {\cal A} \else$\cA$\fi({\bf D_n})$: \> $\{
x_i=\pm x_j,\ 1\leq i<j\leq n\}$;\\ $\ifmmode {\cal A} \else$\cA$\fi({\bf F_4})$: \> $\{ x_i=0,\
i=1,\ldots, n;\ x_i=\pm x_j,\ 1\leq i<j\leq 4,\ {1\over 2}(x_1\pm
x_2\pm x_3\pm x_4)\}$;\\ $\ifmmode {\cal A} \else$\cA$\fi({\bf E_6})$: \> $\{ x_i=\pm x_j,\ 1\leq
i<j \leq 5,\ {1\over 2}(x_1\pm x_2 \pm x_3\pm x_4 \pm x_5 + x_6)\}$.
\end{tabbing}
\end{minipage}
\end{equation}
For the arrangement of type $\bf A_n$ we have made the coordinate
transformation $x_1=\ge_1-\ge_{n+1}, \ldots, x_n=\ge_n-\ge_{n+1}$, so
$x_i-x_j=\ge_i-\ge_j$ for $1\leq i< j\leq n$, and for $\bf E_6$ we have
taken $x_6$ to replace $x_8-x_7-x_6$.
The arrangements above are the arrangements defined by the projective
reflection groups $PW({\bf X_n})$. Each hyperplane is the reflection plane
for the reflection on the corresponding root. From this point of view these
arrangements are studied in \cite{OS2}.
\subsection{Rank 4 arrangements}
As described above, the groups $W({\bf A_4})$, $W({\bf B_4})$, $W({\bf
D_4})$ and $W({\bf F_4})$ give rise to projective arrangements in
${\Bbb P}^3$. They consists of ten, 16, 12 and 24 planes, respectively. They
may also be described as follows (see \cite{GS})):
\begin{tabbing}
$\ifmmode {\cal A} \else$\cA$\fi({\bf A_4})$: \quad \= four faces of a tetrahedron plus the six
symmetry planes; \\ $\ifmmode {\cal A} \else$\cA$\fi({\bf B_4})$: \> \parbox{12cm}{six faces of a cube
plus the nine symmetry planes plus the plane at infinity;}\\ $\ifmmode {\cal A} \else$\cA$\fi({\bf
D_4})$: \> \parbox[t]{12cm}{six faces of a cube plus the six symmetry
planes through two edges each, OR: eight faces of an octahedron plus
three symmetry planes containing four vertices each plus the plane at
infinity;}\\ $\ifmmode {\cal A} \else$\cA$\fi({\bf F_4})$: \> \parbox[t]{12cm}{ ``desmic figure'':
six faces of the cube, eight faces of an inscribed octahedron, nine
symmetry planes and the plane at infinity; this is also determined by
the regular 24-cell;}
\end{tabbing}
The combinatorial description of these arrangements can be encoded in
numbers:
\begin{equation}\label{e109.1} t_q(j):=\#\{{\Bbb P}^j \hbox{'s of the arrangement
through which $q$ of the reflection planes pass}\}.
\end{equation}
In the case of the above arrangements we have the following data
($t_q:=t_q(0)$, the number of points):
\begin{equation}\label{e109.2}\begin{minipage}{14cm}
\begin{tabbing}
$\ifmmode {\cal A} \else$\cA$\fi({\bf A_4})$: \quad \= $t_6=5,\ t_4=10;\ t_3(1)=10,\ t_2(1)=15.$\\
$\ifmmode {\cal A} \else$\cA$\fi({\bf B_4})$: \> $t_9=4, t_6=8,\ t_5=12,\ t_4=16;\ t_4(1)=6,\
t_3(1)=16,\ t_2(1)=36$. \\ $\ifmmode {\cal A} \else$\cA$\fi({\bf D_4})$: \> $t_6=12,\ t_3=12;\
t_3(1)=16,\ t_2(1)=18.$ \\ $\ifmmode {\cal A} \else$\cA$\fi({\bf F_4})$: \> $t_9=24,\ t_4=96;\
t_4(1)=18,\ t_3(1)=32,\ t_2(1)=72.$
\end{tabbing}
\end{minipage}
\end{equation}
\begin{definition}\label{d109.1} An arrangement $\ifmmode {\cal A} \else$\cA$\fi\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^n$ is said to be
in (combinatorial) {\em general position}, if $t_q(j)=0$ for all $q>n-j$.
All ${\Bbb P}^j$'s $\subset} \def\nni{\supset} \def\und{\underline \ifmmode {\cal A} \else$\cA$\fi$ for which $j>n-q$ holds are the {\em
singularities} of the arrangement. The singularities are {\it genuine}
if they are not the intersection of higher-dimensional singular loci with
one of the planes of the arrangement. The union of all genuine
singularities is the {\em singular locus}.
\end{definition}
In the above arrangements we have the following singular loci:
\begin{equation}\label{e109.3}\begin{minipage}{14cm}
\begin{tabbing}
$\ifmmode {\cal A} \else$\cA$\fi({\bf A_4})$: \quad \= five singular points, ten singular lines;\\
$\ifmmode {\cal A} \else$\cA$\fi({\bf B_4})$: \> 12=4+8 (genuine) singular points, 22=6+16 singular
lines; \\ $\ifmmode {\cal A} \else$\cA$\fi({\bf D_4})$: \> 12 singular points, 16 singular lines; \\
$\ifmmode {\cal A} \else$\cA$\fi({\bf F_4})$: \> 24 (genuine) singular points, 50=18+32 singular
lines.
\end{tabbing}
\end{minipage}
\end{equation}
\subsection{The tetrahedron}
Consider now the arrangement $\ifmmode {\cal A} \else$\cA$\fi({\bf A_4})$ in (\ref{e109.2}). By
(\ref{e109.3}) the singular locus consists of five points and ten lines. We
introduce the following notation: $P_1=(1,0,0,0),\ P_2=(0,1,0,0),\
P_3=(0,0,1,0),\ P_4=(0,0,0,1),\ P_5=(1,1,1,1)$, and $l_{ij}$ will denote
the line joining $P_i$ and $P_j$. Each line contains two of the five
points, and at each of the points four of the ten lines meet. The
arrangement is {\em resolved} by performing the following birational
modification of ${\Bbb P}^3$:
\begin{equation}\label{e109.4} \begin{minipage}{14cm}\begin{itemize}\item[a)]
Blow up the five points, $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1:\widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}^3\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^3$;
\item[b)] Blow up the proper transforms of the ten lines,
$\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2:\tilde{{\Bbb P}}^3\longrightarrow} \def\sura{\twoheadrightarrow \widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}^3,\ \varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta:\tilde{{\Bbb P}}^3\longrightarrow} \def\sura{\twoheadrightarrow
{\Bbb P}^3$.
\end{itemize}
\end{minipage}
\end{equation}
In the resolution 15 exceptional divisors $E_1,\ldots,E_5$ and
$L_{12},\ldots, L_{45}$ are introduced. The $E_i$ are the proper transforms
of the exceptional divisors introduced under $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1$, and are isomorphic
to ${\Bbb P}^2$ blown up in the four points $(1:0:0),\ (0:1:0),\ (0:0:1),\
(1:1:1)$, as are the proper transforms $H_i$ of the ten planes of the
arrangement. The ten exceptional divisors $L_{ij}$ are each isomorphic to
${\Bbb P}^1\times {\Bbb P}^1$. The symmetry group of $\tilde{{\Bbb P}}^3$ consists of
projective linear transformations of ${\Bbb P}^3$ which preserve the arrangement
$\ifmmode {\cal A} \else$\cA$\fi({\bf A_4})$, together with certain {\em birational} transformations of
${\Bbb P}^3$ which are {\em regular} on $\tilde{{\Bbb P}}^3$, i.e., which contain the
singular locus (\ref{e109.3}) with simple multiplicity in their
ramification locus. Hence the Weyl group itself, $W({\bf A_4})=\gS_5$
(symmetric group on five letters) is contained in the symmetry group. But
in fact, $\gS_6$ is the symmetry group, and the extra generator is a
permutation of one of the $E_i$ and $H_j$, which clearly can be done {\em
on } $\tilde{{\Bbb P}}^3$.
\subsection{A birational transformation}
Note that since each of the ten lines in (\ref{e109.3}) contains two of the
five points which are blown up under $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1$, the normal bundle of the
proper transform of each line on $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}^3$ is $\ifmmode {\cal O} \else$\cO$\fi(1-2)\oplus\ifmmode {\cal O} \else$\cO$\fi(1-2)=
\ifmmode {\cal O} \else$\cO$\fi(-1)\oplus \ifmmode {\cal O} \else$\cO$\fi(-1)$. By general results of threefold birational
geometry, it follows that
\begin{equation}\label{e110.1} \begin{minipage}{14cm}\begin{itemize}\item[a)]
The divisors $L_{ij}$ on $\tilde{{\Bbb P}}^3$ may be blown down to an
ordinary threefold rational point (node), i.e., a singularity given
by the equation $x^2+y^2+z^2+t^2=0$, OR:
\item[b)] The ten lines on $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}^3$ may be blown down to the nodes
mentioned in a).
\end{itemize}\end{minipage}\end{equation}
In other words, there is a threefold which we denote by $T$, which contains
ten threefold nodes, with a birational triangle:
\begin{equation}\label{e110.2}
\unitlength1cm
\begin{picture}(3,2)(0,-.5)
\put(-.5,1){$\tilde{{\Bbb P}}^3$}\put(0,1.1){\vector(1,0){1.5}} \put(.7,1.3){$
\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2$} \put(1.8,1){$\widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}^3 \stackrel{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1}{\longrightarrow} \def\sura{\twoheadrightarrow} {\Bbb P}^3$}
\put(-.4,0){$\Pi_2$} \put(-.3,.9){\vector(1,-1){1}} \put(1.8,0){$\Pi_1$}
\put(1.9,.9){\vector(-1,-1){1}} \put(.7,-.5){$T$}
\end{picture}
\end{equation}
The map $\Pi_2$ blows down the union of ten disjoint ``quadric surfaces''
(i.e., divisors isomorphic to ${\Bbb P}^1\times {\Bbb P}^1$) to ordinary nodes, while
$\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2$ blows these quadric surfaces down to ten disjoint lines, which
$\Pi_1$ then blows down to the same ten isolated nodes. The 5+10 divisors
$E_i$ and $H_j$ on $\tilde{{\Bbb P}}^3$ have the following properties:
\begin{equation}\label{e111.1}\begin{minipage}{14cm} \begin{itemize}\item[a)]
Each is isomorphic to ${\Bbb P}^2$ blown up in four points;
\item[b)] Each contains ten lines of intersection with the other 15,
forming an arrangement in the blown up ${\Bbb P}^2$ of ten lines meeting
in 15 points.
\item[c)] Under the birational map $\Pi_2$ each of the divisors $E_i$
and $H_j$ are blown down to a ${\Bbb P}^2$; the image of the ten lines of
b) lie four at a time in each of these ${\Bbb P}^2$'s, as the four
$t_3$-points of the following arrangement, which is the union of the
intersections of the given ${\Bbb P}^2$ with the others:
$$\unitlength1cm
\begin{picture}(8,3.5)(1,0.8)
\thinlines \put(2,1.5){\line(1,0){6.5}} \put(2,1.5){\line(4,1){5}}
\put(2,1.5){\line(6,5){3.5}}
\put(2,1.5){\line(-1,0){0.5}} \put(2,1.5){\line(-4,-1){0.5}}
\put(2,1.5){\line(-6,-5){0.5}}
\put(5,.75){\line(0,1){4}} \put(5,4){\line(6,-5){3.5}}
\put(5,4){\line(-6,5){0.5}} \put(8,1.5){\line(-4,1){5}}
\put(8,1.5){\line(4,-1){0.5}}
\put(2,1.5){\circle*{.2}} \put(5,2.25){\circle*{.2}}
\put(5,4){\circle*{.2}} \put(8,1.5){\circle*{.2}}
\end{picture}$$
\item[d)] The composition $\Pi_2\circ \varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta^{-1}$ restricted to each of the
planes $H_j$ is a usual Cremona transformation, blowing up three non-colinear
points and blowing down the three joining lines. {\it Proof:} Take a face
$H_j$ of the tetrahedron; $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1$ blows up the three vertices it
contains, so $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1^{-1}(H_j)$ (the proper transform of $H_j$) is
${\Bbb P}^2$ blown up in three points. Under $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2$, a fourth point of
$\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1^{-1}(H_j)$ is blown up, but it is blown down again under $\Pi_2$,
as are the proper transforms of the three lines (in the plane $H_j$)
joining the three vertices. By symmetry the same holds for all the $H_j$.
\end{itemize}
\end{minipage}
\end{equation}
It follows that on $T$, the images $\tilde{H}_j=\Pi_2(H_j)$ and
$\tilde{E}_i=\Pi_2(E_i)$ are copies of ${\Bbb P}^2$, each containing four of the
ten nodes of $T$. Furthermore, in each of $\tilde{H}_j$ and $\tilde{E}_j$
we have the four $t_3$-points of the arrangement (\ref{e111.1}), which are
these four nodes of $T$. Finally, since there are 15 ${\Bbb P}^2$'s, ten nodes
and four of them in each of the 15 ${\Bbb P}^2$'s, there are five of these
divisors passing through a given node. Explicitly, take the node $n_{ij}$
corresponding to the line $l_{ij}$ in (\ref{e109.3}). Then it meets the
exceptional divisors $\~E_i,\ \~E_j$, as well as the three of the
$\~H_{\nu}$ for which $H_{\nu}$ contains the line $l_{ij}$.
\subsection{Fermat covers associated with arrangements}
Let $\ifmmode {\cal A} \else$\cA$\fi\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^n$ be an arrangement of hyperplanes, i.e., a union
$\ifmmode {\cal A} \else$\cA$\fi=\cup _{i=1}^k H_i$ of $k$ hyperplanes, and let $d\geq2 $ be an
integer. To the pair $(\ifmmode {\cal A} \else$\cA$\fi,d)$ there is an associated function field
$\ifmmode {\cal L} \else$\cL$\fi(\ifmmode {\cal A} \else$\cA$\fi,d)$, an algebraic extension of the rational function field
$\ifmmode {\cal M} \else$\cM$\fi({\Bbb P}^n)$. It defines, in a unique way, a branched cover $Y(\ifmmode {\cal A} \else$\cA$\fi,d)\longrightarrow} \def\sura{\twoheadrightarrow
{\Bbb P}^n$, and a unique desingularisation $\~Y(\ifmmode {\cal A} \else$\cA$\fi,d)$. The function field is
defined by:
\begin{equation}\label{e111a.1} \ifmmode {\cal L} \else$\cL$\fi(\ifmmode {\cal A} \else$\cA$\fi,d)=\komp\left({x_1 \over x_0},\ldots
,{x_n\over x_0}\right)\left[
\sqrt{H_2/H_1}\hspace{-1.5cm}\raisebox{.2cm}{$\scriptstyle d$}
\hspace{1.5cm},\ldots,
\sqrt{H_k/H_1}\hspace{-1.5cm}\raisebox{.2cm}{$\scriptstyle d$}
\hspace{1.5cm}\right],
\end{equation}
and the cover $Y(\ifmmode {\cal A} \else$\cA$\fi,d)$ is the so-called Fox closure of the \'etale cover
over ${\Bbb P}^n-\ifmmode {\cal A} \else$\cA$\fi$ which is defined by (\ref{e111a.1}). $Y(\ifmmode {\cal A} \else$\cA$\fi,d)$ is smooth
outside of the {\em singular locus} of $\ifmmode {\cal A} \else$\cA$\fi$ (Definition \ref{d109.1}), and
the singularities of $Y(\ifmmode {\cal A} \else$\cA$\fi,d)$ are resolved by resolving the singularities
of $\ifmmode {\cal A} \else$\cA$\fi$. This is done by first blowing up all (genuine, i.e., not near
pencil) singular points, then all singular lines, and so forth. The
resolution (\ref{e109.4}) is a typical example. This is described in more
detail in the author's thesis; the desingularisation $\~Y(\ifmmode {\cal A} \else$\cA$\fi,d)$ is the
fibre product in the following diagram:
\begin{equation}\label{e111a.2}
\begin{array}{ccc}\~Y(\ifmmode {\cal A} \else$\cA$\fi,d) & \longrightarrow} \def\sura{\twoheadrightarrow & Y(\ifmmode {\cal A} \else$\cA$\fi,d) \\ \~{\pi}\downarrow &
& \downarrow\pi \\ \~{{\Bbb P}}^n & \stackrel{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}{\longrightarrow} \def\sura{\twoheadrightarrow} & {\Bbb P}^n
\end{array}
\end{equation}
where the horizontal arrows are modifications and the vertical arrows are
Galois covers with Galois group $(\integer/d\integer)^{k-1}$, which is the Galois
group of the field extension of (\ref{e111a.1}). $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta$ is the modification
of ${\Bbb P}^n$ which resolves the singularities of $\ifmmode {\cal A} \else$\cA$\fi$. For example, each
singular point $P$ on $Y(\ifmmode {\cal A} \else$\cA$\fi,d)$ is resolved by an {\em irreducible}
divisor $D_P$, which itself is a Fermat cover $Y(\ifmmode {\cal A} \else$\cA$\fi',d)$, where $\ifmmode {\cal A} \else$\cA$\fi'\subset} \def\nni{\supset} \def\und{\underline
{\Bbb P}^{n-1}$ is the arrangement induced in the exceptional ${\Bbb P}^{n-1}$ which
resolves the point $P'=\pi(P)$. It consists of $k'$ planes, where $k'$ is
the number of the $k$ hyperplanes which meet at the point $P'$, and in the
process of resolving $Y(\ifmmode {\cal A} \else$\cA$\fi,d)$, the cover $Y(\ifmmode {\cal A} \else$\cA$\fi',d)$ is resolved also.
Hence on $\~Y(\ifmmode {\cal A} \else$\cA$\fi,d)$ there is a {\em smooth} divisor $D_P$ which resolves
the singular point $P$ of $Y(\ifmmode {\cal A} \else$\cA$\fi,d)$.
The singular covers $Y(\ifmmode {\cal A} \else$\cA$\fi,d)$ can also be realised as complete
intersections, namely the intersections of $N=k-n$ Fermat hypersurfaces in
${\Bbb P}^{k-1}$:
\begin{eqnarray}\label{e111a.3}
F_1 & = & a_{11}x_1^d+\cdots + a_{1k}x_k^d \nonumber\\ \vdots & & \vdots
\\ F_N & = & a_{N1}x_1^d+\cdots +a_{Nk}x_k^d \nonumber
\end{eqnarray}
where $a_{11}H_1+\cdots +a_{1k}H_k, \ldots , a_{N1}H_1+\cdots + a_{Nk}H_k$
are the $(k-n)$ linear relations among the $k$ hyperplanes $H_i$. The map
$Y(\ifmmode {\cal A} \else$\cA$\fi,d)\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^n$ is realised explicitly by the map
$(x_1,\ldots,x_k)\mapsto (x_1^d,\ldots,x_k^d)$.
\subsection{The hypergeometric differential equation}
The Fermat covers for the arrangement $\ifmmode {\cal A} \else$\cA$\fi({\bf A_4})$ are closely related
to solutions of the hypergeometric differential equation on ${\Bbb P}^3$, which
is an algebraic differential equation with regular singular points, whose
singular locus {\em coincides} with the arrangement $\ifmmode {\cal A} \else$\cA$\fi({\bf A_4})$,
meaning that solutions are locally branched along the planes of the
arrangement.
First we introduce a new notation for the 15 surfaces $E_i,\ H_j$. These
can be numbered by pairs $(i,j),\ i<j\in \{0,\ldots,5\}$, with $E_i=H_{0i}$
and
\begin{equation}\label{e111b.1} H_{ij}\cap H_{kl}\neq \emptyset \iff
i\neq j\neq k\neq l.
\end{equation}
We denote by $0i$ the point $P_i$ in ${\Bbb P}^3$, and by $0ij$ the singular
line joining $0i$ and $0j$ in ${\Bbb P}^3$. We then let $L_{0ij}$ denote the
exceptional divisor on $\~{{\Bbb P}}^3$. We have (in ${\Bbb P}^3$)
\begin{equation}\label{e111b.2} H_{ij}\cap H_{kl}=0mn \iff \{i,j,k,l\}\cap
\{0,m,n\}=\emptyset.
\end{equation}
We want to consider branched covers $Y\longrightarrow} \def\sura{\twoheadrightarrow \~{{\Bbb P}}^3$ (with $\~{{\Bbb P}}^3$ as
in (\ref{e109.4})), which are branched along the $H_{ij}$ and the
$L_{0ij}$. Hence we let
\begin{equation}\label{e111b.3} n_{ij}:=\hbox{ branching degree along
$H_{ij}$};\quad n_{0ij}:=\hbox{ branching degree along $L_{0ij}$},
\end{equation}
and of course $n_{ij},\ n_{0ij}\in \integer\cup \infty$. (It makes sense to
allow negative branching degrees, as we will see below.)
To define the hypergeometric differential equation we may just as well work
on ${\Bbb P}^n$ with homogenous coordinates $(x_0:\ldots:x_n)$, and consider the
arrangement $\ifmmode {\cal A} \else$\cA$\fi({\bf A_n})$ of (\ref{e108.2}). Let $\gl_i\in \rat,\ \
i=0,\ldots,n+1, \infty,\ \sum_i\gl_i=n+1$. The hypergeometric differential
equation is:
\begin{equation}\label{e111b.4} \left\{
\begin{minipage}{14cm}$\ds(x_i-x_j)\del_i\del_jF + (\gl_i-1)(\del_iF-\del_jF) =
0,\ 1\leq i<j \leq n$ \\ $x_i(x_i-1)\del_i^2F+P_i(x,\gl)\del_iF
+(\gl_i-1)\sum{x_{\ga}(x_{\ga}-1) \over (x_i-x_{\ga})}\del_{\ga}F +
\gl_{\infty}(1-\gl_i)F=0,\ 1\leq i \leq n$.
\end{minipage} \right.
\end{equation}
where
$$P_i(x,\gl)=x_i(x_i-1)\sum{1-\gl_{\ga} \over x_i-x_{\ga}} + \gl_0+\gl_i
-3-(2\gl_i+\gl_0+\gl_{n+1})x_i.$$ A solution of (\ref{e111b.4}) turns out
to be a period of an algebraic curve (the periods are many valued, as are
the solutions of (\ref{e111b.4})). The curve is
\begin{equation}\label{e111b.5}
y^{\nu}=x^{\mu_0}(x-1)^{\mu_{n+1}}(x-t_1)^{\mu_1}\cdots (x-t_n)^{\mu_n},
\end{equation}
where the $\mu_i,\ \nu$ are related to the $\gl_i$ by the relation
\begin{equation}\label{e111b.6} {\mu_i \over \nu} = 1-\gl_i.
\end{equation}
The equation (\ref{e111b.4}) has an $(n+1)$-dimensional solution space,
spanned by $(n+1)$ periods of differentials of the curve (\ref{e111b.5}):
\begin{equation}\label{e111c.1} \go_i=\int_{\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_i}{dx \over y},\quad
<\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_0,\ldots,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_n>=H^1(C,\integer).\end{equation} Taking these gives a
homogenous many valued map
\begin{equation}\label{e111c.2} (\go_0,\ldots, \go_n):D\subset} \def\nni{\supset} \def\und{\underline \~{{\Bbb P}}^n
\stackrel{\phi}{\longrightarrow} \def\sura{\twoheadrightarrow} {\Bbb P}^n,
\end{equation}
where $D$ is some Zariski open set (see (\ref{e111d.2}) below). The map is
well-defined, since not all $\go_i$ vanish simultaneously. For very special
values of the parameters $\gl_i$, the image of $\phi$ is the complex ball
$\ball_n\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^n$ (this is just the Borel embedding of $\ball_n$ in its
compact dual). In fact, one has the following theorem:
\begin{theorem}[\cite{DM}, \cite{T}]\label{t111c.1} If the following
conditions are satisfied, then $\phi(D)=\ball_n$:
$$\sum\mu_i=2,\quad \forall_{i,j}:\ (1-\mu_i-\mu_j)^{-1}\in \integer\cup
\infty.$$ In this case there exists a finite cover
$$Y\longrightarrow} \def\sura{\twoheadrightarrow D$$ branched along the total transform of $\ifmmode {\cal A} \else$\cA$\fi({\bf A_n})$, which
is a quotient $\gG\backslash \ball_n$ with $\gG$ torsion free.
\end{theorem}
The integers $n_{ij}:=(1-\mu_i-\mu_j)^{-1}$ are then just the branching
degrees of $Y\longrightarrow} \def\sura{\twoheadrightarrow D$ along the divisor $H_{ij}$. In fact the numbering
introduced in (\ref{e111b.1}) can be done analogously for any $n$.
In the special case of $\ifmmode {\cal A} \else$\cA$\fi({\bf A_4})$ on ${\Bbb P}^3$, the integers $n_{0ij}$
of (\ref{e111b.3}) are determined by the relation
$$n_{0ij}=2\left({1\over n_{kl}}+{1\over n_{lm}}+{1\over
n_{km}}\right)^{-1},
$$ where the line $0ij$ is the intersection of $H_{kl},\ H_{lm},\ H_{km}$,
and these together with the $n_{ij}$ describe the branching degrees along
the entire branch locus. The solutions of the equations in Theorem
\ref{t111c.1} are as follows: \renewcommand{\arraystretch}{1.5}
\begin{equation}\label{e111d.1}\begin{array}{cl} 1) &
{1\over 3}, {1\over 3}, {1\over 3}, {1\over 3}, {1\over 3}, {1\over 3},
\\ 2) & {1 \over 2}, {1\over 2}, {1\over 4}, {1\over 4}, {1\over 4},
{1\over 4} \\ 3) & {3 \over 4}, {1\over 4}, {1\over 4}, {1\over 4},
{1\over 4}, {1\over 4} \\ 4) & {1\over 2}, {1\over 3}, {1\over 3},
{1\over 3}, {1\over 3}, {1\over 6}\\ 5) & {3\over 8}, {3\over 8},
{3\over 8}, {3\over 8}, {3\over 8}, {1\over 8} \\ 6) & {5\over 12},
{5\over 12}, {5\over 12}, {1\over 4}, {1\over 4}, {1\over 4} \\ 7) & {7
\over 12}, {5 \over 12}, {1\over 4}, {1\over 4}, {1\over 4}, {1\over
4}
\end{array}
\end{equation}
\renewcommand{\arraystretch}{1.2} The set $D$ of (\ref{e111c.2}) is
determined as the complement of
\begin{equation}\label{e111d.2} H_{\infty}=\{H_{ij}\Big| n_{ij}=\infty; \
L_{0ij}\Big| n_{0ij}=\infty\}\subset} \def\nni{\supset} \def\und{\underline \~{{\Bbb P}}^3.
\end{equation}
This is the locus which the uniformizing map (\ref{e111c.2}) maps onto the
{\em boundary} of $\ball_3\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^3$, i.e., $\phi(D)=\ball_n,\
\phi(H_{\infty})\subset} \def\nni{\supset} \def\und{\underline \del\ball_n$. This requires of course that the
corresponding covers of the divisors on the cover $\~Y$ be abelian
varieties (as these are the compactification divisors on ball quotients).
This can happen as follows
\begin{equation}\label{e111d.3}\begin{minipage}{12cm}
\begin{itemize}\item[(i)] On one of the
$H_{ij}$, this can occur if the branching degrees are: 2 for the six
lines of (\ref{e111.1}), and $-4$ for the four exceptional curves.
\item[(ii)] On $L_{0ij}$, this can happen if $\mu_k+\mu_l+\mu_m=1,\
\mu_0+\mu_i+\mu_j=1$.
\end{itemize}
\end{minipage}
\end{equation}
In the second case, the surface $S_{0ij}$ covering $L_{0ij}$ is of the form
$C_1\times C_2$, where $C_1\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^1$ (respectively $C_2\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^1$) is a
cover, with branching determined by $(\mu_k,\mu_l,\mu_m)$ (respectively
determined by $(\mu_0,\mu_i,\mu_j)$). It is an abelian variety $\iff$ both
curves $C_i$ are elliptic. Note that $Y\longrightarrow} \def\sura{\twoheadrightarrow \~{{\Bbb P}}^3$ will be a Fermat
cover $\iff$ all $n_{ij}$ coincide $\iff$ all $\mu_i$ conicide. In
particular,
\begin{proposition}\label{p111d.1} The only ball quotient in the list
(\ref{e111d.1}) which is a Fermat cover which is a ball quotient is the
solution 1), namely $Y(\ifmmode {\cal A} \else$\cA$\fi({\bf A_4}),3)$ is a ball quotient.
\end{proposition}
\begin{remark} We will see later (see I3 following Lemma \ref{lq4.1} below)
that the solution 4) gives rise also to a Fermat cover which is a ball
quotient, namely $Y(\ifmmode {\cal A} \else$\cA$\fi({\bf D_4}),3)$.
\end{remark}
\section{The Segre cubic ${\cal S}_3$}
In this section we will show that the variety $T$ of (\ref{e110.2}) has a
projective embedding as a cubic hypersurface known as the Segre cubic,
which we denote by $\ifmmode {\cal S} \else$\cS$\fi_3$.
\subsection{Segre's cubic primal}
In ${\Bbb P}^5$ with homogenous coordinates $(x_0:\ldots:x_5)$ consider the
locus
\begin{equation}\label{e111.3} \ifmmode {\cal S} \else$\cS$\fi_3:=\{\sum_{i=0}^5x_i=0;\quad
\sum_{i=0}^5x_i^3=0\}.
\end{equation}
As the first equation is linear, this shows that $\ifmmode {\cal S} \else$\cS$\fi_3$ is a hypersurface,
i.e., $\ifmmode {\cal S} \else$\cS$\fi_3\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^4=\{x\in {\Bbb P}^5\big|\sum x_i=0\}$. Using
$(x_0:\ldots:x_5)$ as projective coordinates, the relation $x_5=-x_0-\cdots
-x_4$ gives the equation of $\ifmmode {\cal S} \else$\cS$\fi_3$ as a hypersurface; however, the
equation in ${\Bbb P}^5$ shows that $\ifmmode {\cal S} \else$\cS$\fi_3$ is invariant under the symmetry
group $\gS_6$, acting on ${\Bbb P}^5$ by permuting coordinates, which is not so
immediate from the hypersurface equation.
It is known that for any degree $d$ there is an upper bound on the number
of ordinary double points which a hypersurface of degree $d$ can have, the
so-called Varchenko bound. For cubic threefolds this number is ten, and it
has been known since the last century that $\ifmmode {\cal S} \else$\cS$\fi_3$ is the {\em unique} (up
to isomorphism) cubic with ten nodes. The nodes on $\ifmmode {\cal S} \else$\cS$\fi_3$ are given by the
points of ${\Bbb P}^5$ for which three of the coordinates are 1 and the other
three are $-1$. This is just the $\gS_6$-orbit of
\begin{equation}\label{e112.0} (1,1,1,-1,-1,-1).
\end{equation}
There is another interesting locus on $\ifmmode {\cal S} \else$\cS$\fi_3$. Consider, in ${\Bbb P}^5$, the
planes $P_{\gs}$ given by
\begin{equation}\label{e112.1} P_{\gs}=\{x_{\gs(0)}+x_{\gs(3)}=x_{\gs(1)} +
x_{\gs(4)}=x_{\gs(2)}+x_{\gs(5)}=0\},
\end{equation}
where $\gs\in \gS_6$. There are 15 such $P_{\gs}$'s, the $\gS_6$-orbit of
\begin{equation} P_{id}=\{x_0+x_3=x_1+x_4=x_3+x_5=0\}.
\end{equation}
One checks easily that each $P_{\gs}$ contains four of the double points;
for example $P_{id}$ contains the following:
$$(1,1,-1,1,-1,-1),\ (1,-1,1,1,-1,-1),\ (1,-1,-1,1,1,-1),\
(1,-1,-1,1,-1,1).$$
Furthermore, the intersection of $P_{id}$ with the other $P_{\gs}$ is the
line arrangement (\ref{e111.1}). It is easily checked that each $P_{\gs}$
is contained entirely in $\ifmmode {\cal S} \else$\cS$\fi_3$. One can also argue as follows. Any line
in ${\Bbb P}^5$ which contains two of the nodes of $\ifmmode {\cal S} \else$\cS$\fi_3$ meets $\ifmmode {\cal S} \else$\cS$\fi_3$ with
multiplicity 4, hence is contained in $\ifmmode {\cal S} \else$\cS$\fi_3$. Similarly, each $P_{\gs}$
meets $\ifmmode {\cal S} \else$\cS$\fi_3$ in the six lines of the arrangement (\ref{e111.1}), hence is
contained in $\ifmmode {\cal S} \else$\cS$\fi_3$.
We just remark that the hyperplane sections $\{x_i=0\}$ of $\ifmmode {\cal S} \else$\cS$\fi_3$ are
cubic surfaces with equation
\begin{equation}\label{e112.3}
S_3 = \{ \sum_{i=0}^4x_i=\sum_{i=0}^4x_i^3=0\}.
\end{equation}
This cubic surface is known as the Clebsch diagonal surface and is a
remarkably beautiful object. It is the unique cubic surface having $\gS_5$
as symmetry group. The relation between $S_3$ and the icosahedral group was
studied by Hirzebruch. It turns out that $S_3$ is $A_5$-equivariantly
birational to the Hilbert modular surface for $\ifmmode {\cal O} \else$\cO$\fi_{\rat(\sqrt{5})}$, of
level $\sqrt{5}$.
Other interesting hyperplane sections are given by the hyperplanes
$\ifmmode {\cal T} \else$\cT$\fi_{ij}=\{x_i-x_j=0\}$; indeed, $\ifmmode {\cal T} \else$\cT$\fi_{ij}$ also contains four of the ten
nodes, hence $\ifmmode {\cal T} \else$\cT$\fi_{ij}\cap \ifmmode {\cal S} \else$\cS$\fi_3$ is a four-nodal cubic surface. This
four-nodal cubic surface is projectively unique, and is called the Cayley
cubic.
\subsection{A birational transformation}
\begin{theorem}\label{t113.1} The variety $T$ of equation (\ref{e110.2}) is
biregular to $\ifmmode {\cal S} \else$\cS$\fi_3$; the isomorphism $\psi:T\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal S} \else$\cS$\fi_3$ defined below is
$\gS_6$-equivariant.
\end{theorem}
{\bf Proof:} Following Baker \cite{Baker}, IV, p.~152, we define a
birational map
$$\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda:{\Bbb P}^3- - \ra \ifmmode {\cal S} \else$\cS$\fi_3.$$ Consider all quadric surfaces in ${\Bbb P}^3$
passing through the points $P_i$ of (\ref{e109.3}). A base of this linear
system is given by the following degenerate quadrics. Let
$(z_0:\ldots:z_3)$ be homogenous coordinates on ${\Bbb P}^3$, and set
\begin{equation}\label{e113.1}
\begin{array}{lll} \xi=z_0(z_3-z_1), & \eta=z_1(z_3-z_2), &
\gz=z_2(z_3-z_0); \\ \xi'=z_1(z_3-z_0), & \eta'=z_2(z_3-z_1), &
\gz'=z_0(z_3-z_2).
\end{array}
\end{equation}
These quadrics satisfy the relations $\xi+\eta+\gz=\xi'+\eta'+\gz'$ and
$\xi\eta\gz=\xi'\eta'\gz'$. Now change coordinates by setting
\begin{equation}\label{e113.a}
\begin{array}{lll} \xi=X+Y, & \eta=Y+Z, & \gz=X+Z; \\
\xi'=-(X'+Y'), & \eta'=-(Y'+Z'), & \gz'=-(X'+Z').
\end{array}\end{equation}
Then the relations $\xi+\eta+\gz=\xi'+\eta'+\gz'$ and
$\xi\eta\gz=\xi'\eta'\gz'$ become
\begin{eqnarray}\label{e113.2} X+Y+Z+X'+Y'+Z' & = & 0 \\
X^3+Y^3+Z^3+(X')^3+(Y')^3+(Z')^3 & = & 0. \nonumber
\end{eqnarray}
One sees this is just equation (\ref{e111.3}) of the Segre cubic. This
yields a rational map $\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda:{\Bbb P}^3- - \ra \ifmmode {\cal S} \else$\cS$\fi_3$, $\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda(z_0:z_1:z_2:z_3)=
(X,Y,Z,X',Y',Z')$. The base locus of the linear system of quadrics defining
$\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda$ (\ref{e113.1}) is the five points of (\ref{e109.3}), as the quadrics
all contain these points. It follows that $\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda$ blows up all five points,
the exceptional divisors $E_1,\ldots ,E_5$ being projective planes. Now
consider one of the ten lines of (\ref{e109.3}); for example, the one given
by $z_2=z_3=0$. Then $\eta=\gz=\eta'=\gz'=0$ and $\xi=\xi'=-z_0z_1$. In
other words, $\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda$ maps that line to the point $(1,0,0,1,0,0)$ in the
$(\xi,\eta,\gz,\xi',\eta',\gz')$ space, which is the point
$(1,1,-1,-1,-1,1)$ in the $(X,Y,Z,X',Y',Z')$ space. But that is just one of
the ten nodes of $\ifmmode {\cal S} \else$\cS$\fi_3$. From $\gS_6$-symmetry we conclude that
$\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda:{\Bbb P}^3- - \ra \ifmmode {\cal S} \else$\cS$\fi_3$ coincides with the map $\Pi=\Pi_1\circ
\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1^{-1}$, with $\Pi_1$ as in (\ref{e110.2}) and $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1$ as in
(\ref{e109.4}). In other words, $\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda=\Pi$ is the composition of morphisms
\begin{equation}\label{e113.3} \unitlength1cm \begin{picture}(3,2)
\put(1.5,1.66){$\widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}^3$} \put(1.5,1.5){\vector(-1,-1){.9}}
\put(1.7,1.5){\vector(1,-1){.9}} \put(.2,.33){${\Bbb P}^3$}
\put(2.76,.33){$T,$} \put(.76,.3){$- - - - \ra$} \put(1.5,.66){$\Pi$}
\put(.66,1.33){$\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1$} \put(2.33,1.33){$\Pi_2$}
\end{picture}
\end{equation}
and since (\ref{e113.2}) states that $\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda=\Pi$ maps onto $\ifmmode {\cal S} \else$\cS$\fi_3$, this
gives an isomorphism $T\ifmmode\ \cong\ \else$\isom$\fi \ifmmode {\cal S} \else$\cS$\fi_3$. Explicitly, $t\in T,\ t\mapsto
(\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1\circ\Pi^{-1}_2)(t)\mapsto
\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda((\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1\circ\Pi^{-1}_2)(t))=\psi(t)\in \ifmmode {\cal S} \else$\cS$\fi_3$ is the desired map. The
$\gS_6$-equivariance follows from the fact that the whole diagram
(\ref{e113.3}) is $\gS_6$-equivariant. \hfill $\Box$ \vskip0.25cm
Now consider the Picard group of $\ifmmode {\cal S} \else$\cS$\fi_3$. From the explicit form of
birational map as given by Theorem \ref{t113.1} and (\ref{e110.2}), we see
that $\hbox{Pic}} \def\Jac{\hbox{Jac}(\ifmmode {\cal S} \else$\cS$\fi_3)$ is generated by the image of the hyperplane class, call
it $H$, and the five exceptional classes $E_i$. It follows that
$\hbox{Pic}} \def\Jac{\hbox{Jac}(\ifmmode {\cal S} \else$\cS$\fi_3)$ has rank 6, and the primitive part $\hbox{Pic}} \def\Jac{\hbox{Jac}^0(\ifmmode {\cal S} \else$\cS$\fi_3)$, i.e., the
complement of the hyperplane class, has rank 5. The 15 classes $H_{ij}$
introduced in (\ref{e111b.1}) (these are the 15 linear ${\Bbb P}^2$'s on $\ifmmode {\cal S} \else$\cS$\fi_3$
noted in (\ref{e112.1})) give classes in $\hbox{Pic}} \def\Jac{\hbox{Jac}(\ifmmode {\cal S} \else$\cS$\fi_3)$ and in
$\hbox{Pic}} \def\Jac{\hbox{Jac}^0(\ifmmode {\cal S} \else$\cS$\fi_3)$. The 15 hyperplanes \begin{equation}
\label{e112b.3} \ifmmode {\cal H} \else$\cH$\fi_{ij}=\{x_i+x_j=0\},
\end{equation}
each of which meets $\ifmmode {\cal S} \else$\cS$\fi_3$ in three of the 15 ${\Bbb P}^2$'s, give 15 {\em
relations} in $\hbox{Pic}} \def\Jac{\hbox{Jac}^0(\ifmmode {\cal S} \else$\cS$\fi_3)$: since $\ifmmode {\cal H} \else$\cH$\fi_{ij}\cap \ifmmode {\cal S} \else$\cS$\fi_3$ is a hyperplane
section, the sum of the three ${\Bbb P}^2$'s cut out by $\ifmmode {\cal H} \else$\cH$\fi_{ij}$, i.e.,
$H_{i_1,j_1}+H_{i_2,j_2}+H_{i_3.j_3}=\ifmmode {\cal H} \else$\cH$\fi_{ij}\cap \ifmmode {\cal S} \else$\cS$\fi_3$, is linearly
equivalent to the hyperplane class. This yields the following exact
sequence of $\integer$-modules:
\begin{equation}\label{e112b.1} \begin{array}{ccccccccccc} 1 & \longrightarrow} \def\sura{\twoheadrightarrow & K &
\longrightarrow} \def\sura{\twoheadrightarrow & \integer\{\ifmmode {\cal H} \else$\cH$\fi_{ij}\} & \longrightarrow} \def\sura{\twoheadrightarrow & \integer\{H_{ij}\} & \longrightarrow} \def\sura{\twoheadrightarrow & \hbox{Pic}} \def\Jac{\hbox{Jac}^0(\ifmmode {\cal S} \else$\cS$\fi_3) &
\longrightarrow} \def\sura{\twoheadrightarrow & 1 \\ & & \Big\|\wr & & \Big\|\wr & & \Big\|\wr & & \Big\|\wr & &
\\ 1 & \longrightarrow} \def\sura{\twoheadrightarrow & \integer^5 & \longrightarrow} \def\sura{\twoheadrightarrow & \integer^{15} & \longrightarrow} \def\sura{\twoheadrightarrow & \integer^{15} & \longrightarrow} \def\sura{\twoheadrightarrow & \integer^5
& \longrightarrow} \def\sura{\twoheadrightarrow & 1. \end{array}\end{equation}
\begin{lemma}\label{l112b.1} In the sequence (\ref{e112b.1}), all
$\integer$-modules are $\gS_6$-modules, i.e., the exact sequence is one of
$\gS_6$-modules.
\end{lemma}
{\bf Proof:} This is visible for the right three entries of the first
sequence in (\ref{e112b.1}), and it then follows for $K$. \hfill $\Box$ \vskip0.25cm Now
consider a generic hyperplane section of $\ifmmode {\cal S} \else$\cS$\fi_3$; this is a smooth cubic
surface. Let $\nu:S=\ifmmode {\cal S} \else$\cS$\fi_3\cap H\hookrightarrow} \def\hla{\hookleftarrow \ifmmode {\cal S} \else$\cS$\fi_3$ denote the inclusion of the
section, and let $\nu*:H^2(\ifmmode {\cal S} \else$\cS$\fi_3,\integer)\longrightarrow} \def\sura{\twoheadrightarrow H^2(S,\integer)$ be the induced map on
cohomology. Then by the Lefschetz hyperplane theorem, this map is {\em
injective}, and since both $S$ and $\ifmmode {\cal S} \else$\cS$\fi_3$ are regular (i.e., not
irregular, that is, have no holomorphic one forms), we may view this as an
injective map of the Picard groups: $\hbox{Pic}} \def\Jac{\hbox{Jac}(\ifmmode {\cal S} \else$\cS$\fi_3)\hookrightarrow} \def\hla{\hookleftarrow \hbox{Pic}} \def\Jac{\hbox{Jac}(S)$, and a
corresponding inclusion on the primitive part. Recall also that we have on
the cubic surface 27 generators (the 27 lines), 45 relations among these
(the 45 tritangents), and an exact sequence on $\hbox{Pic}} \def\Jac{\hbox{Jac}^0(S)$ as in
(\ref{eB3.2}). All in all we get the following map of sequences as in
(\ref{e112b.1}):
\begin{equation}\label{e112b.2}\begin{array}{ccccccccccc}
1 & \longrightarrow} \def\sura{\twoheadrightarrow & \integer^5 & \longrightarrow} \def\sura{\twoheadrightarrow & \integer^{15} & \longrightarrow} \def\sura{\twoheadrightarrow & \integer^{15} & \longrightarrow} \def\sura{\twoheadrightarrow & \integer^5 &
\longrightarrow} \def\sura{\twoheadrightarrow & 1 \\ & & \downarrow & & \downarrow & & \downarrow & & \downarrow
& & \\ 1 & \longrightarrow} \def\sura{\twoheadrightarrow & \integer^{24} & \longrightarrow} \def\sura{\twoheadrightarrow & \integer^{45} & \longrightarrow} \def\sura{\twoheadrightarrow & \integer^{27} & \longrightarrow} \def\sura{\twoheadrightarrow &
\integer^6 & \longrightarrow} \def\sura{\twoheadrightarrow & 1. \end{array}\end{equation} where the right hand groups
are $\hbox{Pic}} \def\Jac{\hbox{Jac}^0(\ifmmode {\cal S} \else$\cS$\fi_3)$ and $\hbox{Pic}} \def\Jac{\hbox{Jac}^0(S)$, respectively, and the down arrows are
inclusions (by Lefschetz). Note that this corresponds to a symmetry
breaking. Indeed, on the first sequence there is a symmetry group $\gS_6$
acting, as already noted, while on the group $\hbox{Pic}} \def\Jac{\hbox{Jac}^0(S)$, in fact on the
whole second sequence, the group $W(E_6)$ acts naturally, as is well-known.
\begin{proposition}\label{p112b.1} The ideal $\ifmmode \hbox{{\script I}} \else$\scI$\fi(10)$ of the ten nodes is
generated by the five quadrics $\ifmmode {\cal R} \else$\cR$\fi_{\gl}$ of the Jacobian ideal of
$\ifmmode {\cal S} \else$\cS$\fi_3$.
\end{proposition}
{\bf Proof:} The inclusion $Jac(\ifmmode {\cal S} \else$\cS$\fi_3)\subset} \def\nni{\supset} \def\und{\underline \ifmmode \hbox{{\script I}} \else$\scI$\fi(10)$ is obvious, and the
five elements of $Jac(\ifmmode {\cal S} \else$\cS$\fi_3)$ are clearly independent. The fact that
$\ifmmode \hbox{{\script I}} \else$\scI$\fi(10)$ has rank 5 has been verified by standard basis computations
(with the algebra program Macaulay). \hfill $\Box$ \vskip0.25cm
\begin{corollary}\label{c112b.1} The ideal of the ten nodes of $\ifmmode {\cal S} \else$\cS$\fi_3$,
$\ifmmode \hbox{{\script I}} \else$\scI$\fi(10)$, is the Jacobian ideal of $\ifmmode {\cal S} \else$\cS$\fi_3$. \hfill $\Box$ \vskip0.25cm
\end{corollary}
\subsection{Uniformisation}
In this section we will show that the Segre cubic $\ifmmode {\cal S} \else$\cS$\fi_3$ is actually the
Satake compactification of a Picard modular variety. Let $K=\rat(\sqrt{-3})$
be the field of Eisenstein numbers, and consider the $\rat$-group
$G=U(3,1;K)$, the unitary group of a hermitian form on a four-dimensional
$K$-vector space with signature (3,1). Consider the arithmetic group
$\gG:=G_{\integer}=U(3,1;\ifmmode {\cal O} \else$\cO$\fi_K)\subset} \def\nni{\supset} \def\und{\underline G(K)$, where $\ifmmode {\cal O} \else$\cO$\fi_K$ denotes the ring of
integers in $K$. It acts on the three-ball with non-compact quotient $\ifmmode {X_{\gG}} \else$\xg$\fi$.
Consider the principal congruence subgroups $\gG(\sqrt{-3})$ and $\gG(3)$,
as defined in \cite{J}. These determine a corresponding level structure in
the sense of Definition 2.5 of \cite{J}. Now note the following well-known
isomorphisms:
\begin{equation}\label{e114.1} \gG/\gG(\sqrt{-3})=\gS_6, \quad
\gG(3)/\gG(\sqrt{-3})=(\integer/3\integer)^9.
\end{equation}
It follows from this that the corresponding quotients $X(a):=X_{\gG(a)}$,
$a=1,\sqrt{-3},3$, yield Galois covers
\begin{equation}\label{e114.2} X(3)\stackrel{(\integer/3\integer)^9}{\longrightarrow} \def\sura{\twoheadrightarrow} X(\sqrt{-3})
\stackrel{\gS_6}{\longrightarrow} \def\sura{\twoheadrightarrow} X(1),
\end{equation}
which explicitly describe the level structures involved. As usual let
$X(a)^*$ denote the Satake compactification.
\begin{theorem}\label{t114.1} There is a commutative diagram
\begin{equation}\label{e115.0}
\unitlength.4cm
\begin{picture}(18,18)
\put(8,1){$\ifmmode {\cal S} \else$\cS$\fi_3$} \put(3,5.7){$\tilde{{\Bbb P}}^3$}
\put(14,2.5){$X(\sqrt{-3})^*$} \put(8.8,7.7){$\-X(\sqrt{-3})$}
\put(7.6,12){$Y^{\wedge}$} \put(3,17){$\tilde{Y}$}
\put(14.5,14){$X(3)^*$} \put(9,19){$\-X(3)$}
\put(3.5,16.5){\vector(1,-1){4}} \put(8.5,18.5){\vector(-3,-1){4.3}}
\put(10,18.5){\vector(1,-1){4}} \put(14,14){\vector(-3,-1){4.3}}
\put(9,18.5){\vector(0,-1){9.5}} \put(3,16.5){\vector(0,-1){9.5}}
\put(8,11.5){\vector(0,-1){9.5}} \put(15,13.5){\vector(0,-1){9.5}}
\put(8.5,7.5){\vector(-3,-1){4}} \put(9.5,7.5){\vector(1,-1){4}}
\put(13.5, 2.5){\vector(-3,-1){4}} \put(3.5,5.5){\vector(1,-1){4}}
\end{picture}
\end{equation}
where the horizontal maps from right to left are isomorphisms, those from
left to right are birational, and the vertical maps are $(\integer/3\integer)^9$
covers.
\end{theorem}
{\bf Proof:} First we have, over ${\Bbb P}^3$, a singular cover $T_{DM}$ defined
by the solution 1) of (\ref{e111d.1}). This is desingularised by blowing up
the ${\Bbb P}^3$ along the singular locus of the arrangement, $\tilde{{\Bbb P}}^3
\longleftarrow} \def\rar{\rightarrow \tilde{T}_{DM}$. From the fact that all $\mu_i=1/3$, we see that all
$n_{ij}$ and $n_{0ij}$ are equal to three, that is $T_{DM}$ is the Fermat
cover $Y(\ifmmode {\cal A} \else$\cA$\fi({\bf A_4}),3)$, and $\tilde{T}_{DM}$ is its desingularisation
$\tilde{Y}:=\tilde{Y}(\ifmmode {\cal A} \else$\cA$\fi({\bf A_4}),3)$ as in (\ref{e111a.2}); see also
Proposition \ref{p111d.1}. By Theorems \ref{t111c.1} and
\ref{t113.1}, $\tilde{Y}$ is the
desingularisation of the ball quotient $\gG'\backslash \ball_3$, for some torsion
free group $\gG'$. Blowing $\tilde{Y}$ down from $\tilde{{\Bbb P}}^3$ to $\ifmmode {\cal S} \else$\cS$\fi_3$
gives the singular variety $Y^{\wedge}$, which we will see in a minite is
the Satake compactification of the ball quotient. Hence we only need to
identify the groups and check the compactifications coincide. As to the
first, we start with
\begin{Lemma}\label{l115.1} Let $\GQ$ be an isotropic
$\rat$-form of $U(3,1)$, $G_{\rat}\sim U(3,1;L)$, $L$ imaginary quadratic
over $\rat$, and let $\gG\subset} \def\nni{\supset} \def\und{\underline G_{\rat}$ a torsion free arithmetic subgroup
with arithmetic quotient $\ifmmode {X_{\gG}} \else$\xg$\fi$, Baily-Borel compactification $\ifmmode {X_{\gG}^*} \else$\xgs$\fi$ and
toroidal compactification $\ifmmode {\overline{X}_{\gG}} \else$\xgc$\fi$. Then the isomorphism class of a single
compactification divisor determines the field $L$, and hence $G_{\rat}$ up
to isogeny.
\end{Lemma}
{\bf Proof:} First note that for $U(3,1)$ the parabolic (there is only one
conjugacy class of parabolics, as the $\fR$-rank is one) takes on the
particularly simple form
\begin{equation}\label{e115.1} \begin{minipage}{14cm} \hspace*{\fill} $
P \ifmmode\ \cong\ \else$\isom$\fi (\ifmmode {\cal R} \else$\cR$\fi\ifmmode {\cal K} \else$\cK$\fi)\sdprod \ifmmode {\cal Z} \else$\cZ$\fi V, \quad \ifmmode {\cal R} \else$\cR$\fi\ifmmode\ \cong\ \else$\isom$\fi \fR^{\times}, \quad
\ifmmode {\cal K} \else$\cK$\fi=SU(2)\times U(1)$ \hspace*{\fill}
\hspace*{\fill} $\ifmmode {\cal Z} \else$\cZ$\fi= \fR, \quad V=\komp^2 $\hspace*{\fill}
\end{minipage}
\end{equation}
For the $\rat$-form of $P$, it follows that $V_{\rat}\ifmmode\ \cong\ \else$\isom$\fi L^2$ for some
imaginary quadratic field $L$, and for the arithmetic parabolic $\gG_P\subset} \def\nni{\supset} \def\und{\underline
P,\ \gG_P\cap V_{\rat}\subset} \def\nni{\supset} \def\und{\underline (\ifmmode {\cal O} \else$\cO$\fi_L)^2$ is some lattice. Furthermore, the
theory of toroidal embeddings shows that a compactification divisor of
$\ifmmode {\overline{X}_{\gG}} \else$\xgc$\fi$ is of the form $\komp^2/(\gG_P\cap V_{\rat})$, which has complex
multiplication by $L$, so its isomorphism class determines $L$, which was
to be shown. \hfill $\Box$ \vskip0.25cm Now an easy calculation shows what the compactification
divisors on $\tilde{Y}$ are. Namely, these are the irreducible components
of the inverse image in $\tilde{Y}$ of the exceptional divisors
$L_{0ij}\subset} \def\nni{\supset} \def\und{\underline \tilde{{\Bbb P}}^3,\ L_{0ij}\ifmmode\ \cong\ \else$\isom$\fi {\Bbb P}^1\times {\Bbb P}^1$. The local
geometry of the arrangement shows the branch locus in $L_{0ij}$ is of the
form $p_1^*(\ifmmode {\cal O} \else$\cO$\fi(3)) \otimes p_2^*(\ifmmode {\cal O} \else$\cO$\fi(3))$, i.e., of the form $\{0\}\times
{\Bbb P}^1,\ \{1\}\times {\Bbb P}^1,\ \{\infty\}\times {\Bbb P}^1$ and ${\Bbb P}^1\times
\{0\},\ {\Bbb P}^1\times \{1\},\ {\Bbb P}^1\times \{\infty\}$. It is well-known that
the elliptic curve $E\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^1$, branched at $(0,1,\infty)$ to degree 3,
with Galois group $\integer/3\integer$, is the elliptic curve $E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}=\komp/\integer\oplus
\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta\integer,\ \varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta=e^{2\pi i / 3}$. From this it follows
\begin{Lemma}\label{l115.2} The compactification divisors $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_i$ of
$\tilde{Y}$ are products
$$ \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_i\ifmmode\ \cong\ \else$\isom$\fi E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}\times E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta},$$ where $E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}$ is the unique
elliptic curve with $\integer/6\integer$ as automorphism group, i.e.,
$E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}=\komp/\integer\oplus \varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta\integer=\{x^3+y^3+z^3=0\}$ and
$\hbox{Aut}} \def\Im{\hbox{Im}(E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta})=<\pm 1,\pm\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta, \pm\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta^2>$.\hfill $\Box$ \vskip0.25cm
\end{Lemma}
Note that the morphism $\tilde{Y}\longrightarrow} \def\sura{\twoheadrightarrow Y^{\wedge}$ blows down the $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_i$ to
singular points (just as $\tilde{{\Bbb P}}^3\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal S} \else$\cS$\fi_3$ blows down the $L_{0ij}$
to the nodes of $\ifmmode {\cal S} \else$\cS$\fi_3$) which lie over the nodes of $\ifmmode {\cal S} \else$\cS$\fi_3$. From this
and the well-known fact that the Satake compactification of a ball quotient
has only isolated, zero-dimensional singularities, which are resolved in a
torus embedding by means of complex tori or quotients thereof, we get the
following
\begin{Corollary}\label{c115.1} The variety $Y^{\wedge}$ is the Satake
compactification of the quotient $\gG'\backslash \ball_3$, with $\gG'\subset} \def\nni{\supset} \def\und{\underline G_{\rat}$
and $G_{\rat}$ isogenous to $U(3,1;K)$, $K$ the field of Eisenstein
numbers as above. \hfill $\Box$ \vskip0.25cm
\end{Corollary}
Now that it is established that $\gG'$ is (isogenous to) an arithmetic
subgroup of $U(3,1;K)$, group-theoretic methods can be applied to determine
the arithmetic subgroup. This is done in detail in \cite{J}, Lemma 2.9 and
Theorem 2.11. The result is: $\gG'=P\gG(3)$, and the group $\gG_{\ifmmode {\cal S} \else$\cS$\fi_3}$
giving rise to the Segre cubic is $\gG_{\ifmmode {\cal S} \else$\cS$\fi_3}=P\gG(\sqrt{-3})$. This
yields the statements of the theorem on the arithmetic groups. The
compactification divisors of $\tilde{Y}$ coincide by Lemma \ref{l115.2}
with those of $\-X_{\gG'}$, and these are blown down under $\tilde{Y}\longrightarrow} \def\sura{\twoheadrightarrow
Y^{\wedge}$ to the singularities on the Satake compactification which is
$Y^{\wedge}$, $Y^{\wedge}\ifmmode\ \cong\ \else$\isom$\fi X_{\gG'}^*$. The cover $\tilde{Y}\longrightarrow} \def\sura{\twoheadrightarrow
\tilde{{\Bbb P}}^3$ (respectively $Y^{\wedge}\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal S} \else$\cS$\fi_3$) is now readily
identified with $\-X(3)\longrightarrow} \def\sura{\twoheadrightarrow \-X(\sqrt{-3})$ (respectively
with $X(3)^*\longrightarrow} \def\sura{\twoheadrightarrow X(\sqrt{-3})^*$) of (\ref{e114.2}), from the fact
that the branching loci,
degrees and group actions coincide. Details can be found in \cite{J}.
\hfill $\Box$ \vskip0.25cm
\subsection{Moduli interpretation}
Now applying Shimura's theory we get the following moduli description of
$\ifmmode {\cal S} \else$\cS$\fi_3$ (see \cite{J}, \S2 for details).
\begin{theorem}\label{t116.1} Any point $x\in \ifmmode {\cal S} \else$\cS$\fi_3-\{\hbox{ten nodes}\}$
determines a unique isomorphism class of principally polarised abelian
fourfolds with complex multiplication by $K=\rat(\sqrt{-3})$ and a level
$\sqrt{-3}$ structure. The signature of the complex multiplication is
(3,1). Any point $x\in Y^{\wedge}-\{\hbox{inverse image under $\phi$ of
(\ref{e115.0}) of the ten nodes}\}$ determines a unique isomorphy class
of abelian fourfolds as above with a level 3 structure.
\end{theorem}
Moreover, the moduli interpretation of the 15 ${\Bbb P}^2$'s on $\ifmmode {\cal S} \else$\cS$\fi_3$ is given
in \cite{J}.
\begin{proposition}\label{p116.1} The 15 ${\Bbb P}^2$'s on $\ifmmode {\cal S} \else$\cS$\fi_3$ are
compactifications of two-dimensional ball quotients which are moduli
spaces of those abelian fourfolds $A_x^4$ as above which split:
$$A_x^4\ifmmode\ \cong\ \else$\isom$\fi A_x^3\times E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}.$$ The intersections of the 15 planes
determine moduli points of $A_x^4$ which further decompose, i.e., $A_x^3$
splits.
\end{proposition}
\begin{remark} It is natural to ask whether, given a point $x\in \ifmmode {\cal S} \else$\cS$\fi_3$, one
can give the equations defining the abelian variety $A_x$ occuring in
Theorem \ref{t116.1}. In some sense one can.
First it turns out the $A_x$ is the
Jacobian of an algebraic curve, as described by the hypergeometric
equation as in equation (\ref{e111b.5}). Since the parameters are by
Proposition \ref{p111d.1} the set 1) in (\ref{e111d.1}), these curves
have the form:
\begin{equation}\label{e116.1} C_{\tau}=\{y^3=\prod_{i=1}^6(x-t_i(\tau))\};
\end{equation}
$C_{\tau}$ obviously has an automorphism of order 3, given by $y\mapsto
\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta y$ with the third root of unity $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta$. This yields an automorphism of
the Jacobian of $C_{\tau}$. Without much difficulty one finds
\begin{itemize}\item[(i)] $\Jac(C_{\tau})=A_{\tau}$ has complex
multiplication by $\ifmmode {\cal O} \else$\cO$\fi_K$, the signature is (3,1).
\item[(ii)] The automorphism group is $\ifmmode {\cal O} \else$\cO$\fi_K^*$, and is given by
multiplication by $\pm\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta$ in $\ifmmode {\cal O} \else$\cO$\fi_K$.
\end{itemize}
The most direct way to see this is to write down the Jacobian of the curve
(\ref{e116.1}) and show that its periods have the complex multiplication. A
basis of the (1,0) differentials on $C_{\tau}$ written in the normal form
\begin{equation}\label{e116a.1} y^3=x(x-1)(x-t_1)(x-t_2)(x-t_3)
\end{equation}
is given by
\begin{equation}\label{e116a.3} \int{dx \over
\sqrt[3]{x(x-1)(x-t_1)(x-t_2)(x-t_3)}};
\end{equation}
choosing a base of $H_1(C_{\tau},\integer)$ and taking the integrals over the
elements of that base gives the Jacobian; the multiplication by $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta$ is
then evident. Hence one may invoke Shimura's theory to conclude:
\begin{lemma}\label{l116a.1} The isomorphism classes of the Jacobians of the
curves (\ref{e116a.1}) are given as the points of the arithmetic quotient
$PU(3,1;\ifmmode {\cal O} \else$\cO$\fi_K)\backslash \ball_3$. Putting a $\sqrt{-3}$ level structure on the
Jacobians yields the moduli space $\gG(\sqrt{-3})\backslash \ball_3$.
\end{lemma}
The latter space has already been identified with the open subset of smooth
points on $\ifmmode {\cal S} \else$\cS$\fi_3$.
The precise relation between the moduli {\em point} $\tau\in \ball_3$ and the
{\em values} of the $t_i$ has been derived for surfaces, i.e., for $\tau$
in one of the subballs covering one of the 15 ${\Bbb P}^2$'s on $\ifmmode {\cal S} \else$\cS$\fi_3$, by
Holzapfel. The result is: there are automorphic forms $G_2,G_3$ and $G_4$
of indicated weights on $\ball_2$ such that
\begin{equation}\label{e116a.2}
C_{\tau}=\{y^3=x^4-G_2(\tau)x^2-G_3(\tau)x-G_4(\tau)\},
\end{equation}
much akin to the Weierstra\ss\ equation for an elliptic curve. (The
variable $x$ in (\ref{e116a.2}) is of course different than that in
(\ref{e116a.1})). There is no doubt a similar expression for $\tau\in
\ball_3$.
\end{remark}
\section{The Igusa quartic ${\cal I}_4$}
This variety has been known since the last century, and it is related to
the configuration in ${\Bbb P}^4$ which is dual to the 15 hyperplanes of
(\ref{e112b.3}) and the 15 planes of (\ref{e112.1}) which they cut out on
$\ifmmode {\cal S} \else$\cS$\fi_3$, and in fact $\ifmmode {\cal I} \else$\cI$\fi_4$ is just the dual variety of $\ifmmode {\cal S} \else$\cS$\fi_3$. It was
also known in the last century that the tangent hyperplane sections of
$\ifmmode {\cal I} \else$\cI$\fi_4$ are Kummer surfaces, giving $\ifmmode {\cal I} \else$\cI$\fi_4$ a moduli interpretation. Igusa,
in the 1960's, made this rigorous and showed that $\ifmmode {\cal I} \else$\cI$\fi_4$ is the Satake
compactification of $\gG(2)\backslash \sieg_2$, the Siegel modular threefold of
level 2. We begin by discussing the projective variety, then turn to
Igusa's results.
\subsection{The quartic locus associated to a configuration of 15 lines}
Let $l_{\gs}$ be the line dual in ${\Bbb P}^4$ to the ${\Bbb P}^2$ of (\ref{e112.1}),
and let $h_{ij}$ denote the point dual to $\ifmmode {\cal H} \else$\cH$\fi_{ij}$ of (\ref{e112b.3}).
Then these 15 lines meet at the 15 points $h_{ij}$, and three of the 15
lines meet at each, corresponding to the three ${\Bbb P}^2$'s which are
contained in each $\ifmmode {\cal H} \else$\cH$\fi_{ij}$. Furthermore, each of the 15 lines contains
three of the 15 points, as each ${\Bbb P}^2$ is contained in three of the
$\ifmmode {\cal H} \else$\cH$\fi_{ij}$. It is useful to introduce the following notation: each line is
given a notation $(ij)$, and two such lines $(ij),\ (kl)$ meet if and only
if the sets $(ij),\ (kl)$ are disjoint. Hence the 15 points are numbered by
{\em synthemes} $(ij,kl,mn)$ and the three lines meeting each point are the
indicated {\it duads} (pairs) $(ij),\ (kl),\ (mn)$. Then there are ten
sets such as 23, 31, 12 and 56, 64, 45 with the property that the first and
last three do not meet, but each of the first meets each of the last.
Therefore the six lines are generators of a quadric surface
\begin{equation}\label{e117a.1}
Q_{ijk}=\parbox{12cm}{quadric with $(ij), (jk), (ik)$ in one ruling and
$(lm), (mn), (ln)$ in the other ruling}
\end{equation}
Then $Q_{ijk}$ lies in a ${\Bbb P}^3$, and there are ten such, corresponding to
the ways of dividing the six numbers into two {\it triads} (triples). Let
us denote the corresponding ${\Bbb P}^3$ by $K_{ijk}$, so
\begin{equation}\label{e117a.1a} Q_{ijk}\subset} \def\nni{\supset} \def\und{\underline K_{ijk}.
\end{equation}
Then each of the 15 lines is contained in four of the $K_{ijk}$, and six of
the $K_{ijk}$ meet at each of the 15 points.
Consider now a set of four mutually skew of the 15 lines, for example 12,
23, 24, 25. Then there will be a two-dimensional space of ${\Bbb P}^2$'s which
meet all four lines (as we are in ${\Bbb P}^4$, generically a plane and a line
will not intersect). Of all of these planes, there are exactly two passing
through a given point of space $x\in {\Bbb P}^5$. The locus we are interested in
is:
\begin{equation}\label{e117a.2} \ifmmode {\cal Q} \else$\cQ$\fi:=\left\{x\in {\Bbb P}^5 \left|
\parbox{6cm}{ the two planes meeting four skew lines of the 15
$(ij)$ and passing through $x$ {\em coincide}}\right.\right\}.
\end{equation}
If, as in (\ref{e113.1}), we take coordinates
$\xi,\eta,\gz,\xi',\eta',\gz'$ satisfying $\xi+\eta+\gz=\xi'+\eta'+\gz'$ as
coordinates on ${\Bbb P}^4$, then the condition (\ref{e117a.2}) yields a locus
with equation (\cite{Baker}, p.~125):
\begin{equation}\label{e117a.3} \sqrt{(\eta-\gz')(\eta'-\gz)} +
\sqrt{(\gz-\xi')(\gz'-\xi)} + \sqrt{(\xi-\eta')(\xi'-\eta)}=0.
\end{equation}
To find the dual variety of the locus $\ifmmode {\cal Q} \else$\cQ$\fi$, Baker does the following.
Letting $a,b,c$ be variables, $a'=(1-a),\ b'=(1-b),\ c'=(1-c)$, consider
the six points which are the vertices of a coordinate simplex in ${\Bbb P}^5$, and
call them $A, B, C, A', B', C'$. Then any point of our ${\Bbb P}^4$ can be
written as $x=A/bc'+B/ca'+ C/ab'+ A'/b'c+ B'/c'a+ C'/a'b$. Calculating the
tangent plane of $\ifmmode {\cal Q} \else$\cQ$\fi$ at a point $x\in \ifmmode {\cal Q} \else$\cQ$\fi$ which satisfies
(\ref{e117a.3}), in terms of the coordinates used in (\ref{e117a.3}), one
gets:
\begin{equation}\label{e117a.4}
bc'\xi+ca'\eta+ab'\gz-b'c\xi'-c'a\eta'-a'b\gz'=0.
\end{equation}
Now putting $u=bc', v=ca', w=ab', u'=-b'c, v'=-c'a, w'=-a'b$, the equation
becomes
\begin{equation}\label{e117a.5} u\xi+v\eta+w\gz+u'\xi'+v'\eta'+w'\gz'=0,
\end{equation}
with the two identities
\begin{equation}\label{e117a.6} u+v+w+u'+v'+w'=0,\quad uvw+u'v'w'=0.
\end{equation}
Since the identities (\ref{e117a.6}) do not depend on the point, it follows
that these equations define the dual variety. Now comparing with
(\ref{e113.1}), we have
\begin{proposition}\label{p117a.1} The dual variety of the quartic locus
$\ifmmode {\cal Q} \else$\cQ$\fi$ is the Segre cubic $\ifmmode {\cal S} \else$\cS$\fi_3$.
\end{proposition}
It is easy to see that $\ifmmode {\cal Q} \else$\cQ$\fi$ is singular along the 15 lines. It was also
noted classically that a tangent hyperplane section of $\ifmmode {\cal Q} \else$\cQ$\fi$ is a Kummer
quartic surface, with 16 nodes, 15 from the intersections with the 15
singular lines, and one from the point of tangency.
\subsection{Igusa's results}
The relation to the Kummer quartic surfaces is correctly understood by
studying theta constants for the theta functions with 1/2-characteristics.
This was done by Igusa in \cite{igusa}, and we now recall some of his
results.
\subsubsection{Theta functions}
Let $\tau\in \sieg_g=\{M\in M_g(\komp)\big| \tau={^t(\tau)},\, \Im(\tau) \hbox{
positive definite}\}$, $z\in \komp^g$, and $m=(m',m'')\in \rat^{2g}$. Note
that $\sieg_g$ is a hermitian symmetric space of type $\bf III_g$.
\begin{definition}\label{d117.1} The {\em theta function of degree $g$ and
characteristic $m$} is defined by the power series
$$\gt_m(\tau,z)=\sum_{n\in \integer^g}\exp\left({1\over 2}{^t(n+m')}\tau(n+m')
+ {^t(n+m')}(z+m'')\right).$$
\end{definition}
As a function of $\tau$ the series $\gt_m$ converges precisely for $\tau\in
\sieg_g$, while as functions of $z$ by fixed $\tau$ these are theta functions
on $A_{\tau}=\komp^g/(\integer^g+\tau\integer^g)$. As such the zeros on $A_{\tau}$ are
determined by the characteristic $m$. The corresponding {\em theta
constant} is \begin{equation}\label{e117.2} \gt_m(\tau):=\gt_m(\tau,0).
\end{equation}
Igusa has studied in \cite{igusa} these theta constants, in particular the
theta functions with characteristics $m\in {1\over 2^n}\integer$. Some of his
results are the following.
\begin{lemma}\label{l3.1.2} $\gt_m(\tau)\equiv 0 \iff m\hbox{\em mod}(1)$
satisfies $\exp(4\pi i{(^tm')}m'')=-1.$
\end{lemma}
The Siegel modular group $\gG_g(1)=Sp(2g,\integer)$ acts on the arguments
$(\tau, z)$ as follows:
\begin{equation}\label{e117.3} M=\left(\begin{array}{cc}A & B \\ C & D
\end{array}\right),\ M(\tau,z)=\left((A\tau+B)(C\tau+D)^{-1}, (C\tau +
D)^{-1}z\right),
\end{equation}
and on the characteristic itself by \begin{equation}\label{e117.4}
M(m)=\left(\begin{array}{cc}D & -C \\ -B & A \end{array}\right)\cdot m
+{1 \over 2} \left(\begin{array}{c} \diag(C^tD) \\ \diag(A^tB)\end{array}
\right).
\end{equation}
The behavior of the theta functions under $M$ is given by
\begin{lemma}\label{l117.1} Let $M\in \gG_g(1)$ act on $(\tau,z)$ as in
(\ref{e117.3}) and on the characteristic $m$ as in (\ref{e117.4}). Then
the theta functions transform according to the rule:
\begin{equation} \gt_{M(m)}(M(\tau,z)) = \gk(M)\exp(2\pi i
\phi_m(M))\hbox{det}} \def\Ker{\hbox{Ker}(C\tau+D)^{1/2} \times \exp(\pi i {^tz}(C\tau
+D)^{-1}Cz)\gt_m(\tau,z),
\end{equation}
where $\gk(M)$ is some eighth root of unity and $\phi_m(M)$ is defined by
the formula
$$\phi_m(M)=-{1\over2}{^tm'}BDm' + {^tm''}{^tA}Cm'' - 2{^tm'}{^tB}Cm''
-{^t\diag}(A^tB)(Dm'-Cm'').$$
\end{lemma}
In particular for the theta constants the formula becomes
\begin{equation}\label{e117.5} \gt_{M(m)}(M\tau)=\gk(M)\exp(2\pi i
\phi_m(M))\hbox{det}} \def\Ker{\hbox{Ker}(C\tau+D)^{1/2}\gt_m(\tau).
\end{equation}
What the equation (\ref{e117.5}) says for $g=2$ is that up to an eighth
root of unity, non-vanishing theta constants with 1/2-characteristics are
automorphic forms of weight 1/2 for the main congruence subgroup of level 2
in $Sp(4,\integer)$. Indeed, for $M\in \gG(2)$, it holds that $e^{2\pi
i\phi_m(M)}=1$, as Igusa shows. There are 16 characterstics $m$; six are
{\em odd} (i.e., $\gt_m(\tau,z)=-\gt_m(\tau,-z)$) so give rise to vanishing
theta constants, while ten are even. The fourth powers $\gt_m^4$ are
genuine automorphic forms for $\gG(2)$, and determine a morphism
\begin{equation}\label{e118.2} f:\gG(2)\backslash \sieg_2 \longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^9=(\gt_{m_1}^4:
\cdots :\gt_{m_{10}}^4),
\end{equation}
where $m_1,\ldots,m_{10}$ are the ten even characteristics.
\subsubsection{The ring of automorphic forms}
Among the ten coordinate theta functions there are five linear relations,
the Riemann relations. This implies that the map $f$ in (\ref{e118.2}) maps
into a ${\Bbb P}^4$, displaying the quotient $X_{\gG(2)}$ as a hypersurface. In
fact, since this is an embedding by means of automorphic functions whose
closure $X_{\gG(2)}^*\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^4$ is normal (see below), it follows that $f$
gives a Baily-Borel embedding of the arithmetic quotient. The proof that
$f$ is an embedding given by Igusa is quite deep, involving showing that
the ring of modular forms of $\gG(2)$ is the integral closure of the ring
generated by the said theta functions. More precisely, his result is
\begin{theorem}[\cite{igusa},p.~397]\label{t118.1}
Take as coordinates in ${\Bbb P}^4$ the following theta constants:
$$y_0=\gt^4_{(0110)}(\tau),\ y_1=\gt^4_{(0100)}(\tau),\
y_2=\gt^4_{(0000)}(\tau),$$
$$y_3=\gt^4_{(1000)}(\tau)-\gt^4_{(0000)}(\tau),\
y_4=-\gt^4_{(1100)}(\tau)-\gt^4_{(0000)}(\tau),$$ where we let
$(ijkl)$ denote the characteristic $({i\over 2}{j\over 2}{k\over
2}{l\over 2})$. Set also $$\chi_{10}=\prod_{\hbox{even
$m$}}\gt_m^2.$$ Then the ring of modular forms of $\gG(2)$ is given
by:
$$R(\gG(2))=\komp[y_0,\ldots,y_4,\chi_{10}]/\ifmmode {\cal E} \else$\cE$\fi,$$ where $\ifmmode {\cal E} \else$\cE$\fi$ is the ideal
generated by the following two relations:
\begin{minipage}{14cm}$\ifmmode {\cal E} \else$\cE$\fi=\left\{\begin{array}{rcl}
R_1 & = & (y_0y_1+y_0y_2+y_1y_2-y_3y_4)^2-4y_0y_1y_2(\sum y_i) \\ R_2
& = & \chi_{10}^2-{1\over 4}s(y_0,\ldots,y_4),\ s \hbox{ homogenous
of degree 5}
\end{array}\right.$
\end{minipage}
\end{theorem}
However, the formula $R_1$ relating the theta functions was known long
before Igusa. Since the five linear relations determining the image ${\Bbb P}^4$
of $f$ are known, it is sufficient to give a single relation of minimal
degree among the $\gt^4_m$ to determine the image. This relation can be
found as early as in the 1887 paper of Maschke \cite{maschke},
p.~505\footnote{the equation is somewhat hidden: ``...da\ss\ dagegen die
symmetrische Function vierter Dimension sich bis auf einen Zahlenfactor
als das Quadrat der zweiten Dimension erweist.''}. In terms of the theta
constants above, this equation is
\begin{equation}\label{e118.1}
\left(\sum\gt^8_m\right)^2-4\left(\sum \gt_m^{16}\right)=0,
\end{equation}
which, as can be checked, is the same quartic as that given by $R_1$ in
\ref{t118.1}, as well as that given by (\ref{e117a.3}).
\begin{definition}\label{d118.1} The {\em Igusa quartic} $\ifmmode {\cal I} \else$\cI$\fi_4$ is the
quartic threefold defined in ${\Bbb P}^4$ by the relation $R_1$ of Theorem
\ref{t118.1} or the equation (\ref{e118.1}).
\end{definition}
As a corollary we have
\begin{corollary}\label{c118.1} The Igusa quartic
$\ifmmode {\cal I} \else$\cI$\fi_4$ and the quartic locus $\ifmmode {\cal Q} \else$\cQ$\fi$ of (\ref{e117a.3}) coincide, and this
quartic is the Satake compactification of $X_{\gG(2)}$.
\end{corollary}
Hence we have described $X_{\gG(2)}^*$ as a singular quartic hypersurface
in ${\Bbb P}^4$. There are the two interesting loci:
\begin{itemize}\item[(i)] the singular locus, which is the boundary of the
Baily-Borel embedding of $X_{\gG(2)}$;
\item[(ii)] the intersection of $\ifmmode {\cal I} \else$\cI$\fi_4$ with the coordinate hyperplanes in
${\Bbb P}^9$, which are the modular subvarieties $\-Y_m(2)$ of \cite{J}, Thm.
3.19; these are quotients of symmetric subdomains isomorphic to a product
of discs.
\end{itemize}
As already mentioned, the singular locus of $\ifmmode {\cal I} \else$\cI$\fi_4$ consists of 15 lines;
this can be directly calculated from the equation. Alternatively, applying
general formula for the number of cusps (see for example \cite{Yam}) we see
that $X_{\gG(2)}$ has 15 one-dimensional boundary components and 15
zero-dimensional boundary components; by \ref{c118.1} this is then the
singular locus of $\ifmmode {\cal I} \else$\cI$\fi_4$. (That these boundary components are rational
curves is obvious ($\gG(2)\backslash \sieg_1$ is rational); that they are actually
{\em lines} is not so obvious, but an easy calculation). This line of
reasoning also requires the result, also due to Igusa, that, although
$\gG(2)$ is not torsion-free, there are nonetheless no singularities on
$X_{\gG(2)}$.
\subsection{Moduli interpretation}
The embedding (\ref{e118.2}) of $X_{\gG(2)}^*$ as the quartic $\ifmmode {\cal I} \else$\cI$\fi_4$ shows
that $\ifmmode {\cal I} \else$\cI$\fi_4$ has a moduli interpretation. In fact, $X_{\gG(2)}$ is a rough
moduli space of principally polarised abelian surfaces with a level 2
structure. However, $\gG(2)$ contains torsion, namely the element $-1$, so
$X_{\gG(2)}$ is {\em not} a fine moduli variety. This corresponds to the
fact that the automorphism $z\mapsto -z$ of $A_{\tau}$ {\em preserves} the
level 2 structure, hence the actual {\em object} which is parameterised by
$X_{\gG(2)}$ is the {\em quotient} $A_{\tau}/(z\mapsto -z)$. This is just
the Kummer quartic surface which already occured above. The precise
relation is given by
\begin{theorem}\label{t119.1} For a point $x\in \ifmmode {\cal I} \else$\cI$\fi_4-\{\hbox{intersections
of $\ifmmode {\cal I} \else$\cI$\fi_4$ with the ten coordinate planes in (\ref{e118.2})}\}$, the
corresponding Kummer quartic surface $K_x=A_{\tau}/\{\pm1\}$, where
$x=p(\tau)$ for the natural projection $p:\sieg_2\longrightarrow} \def\sura{\twoheadrightarrow X_{\gG(2)}$, is the
intersection of $\ifmmode {\cal I} \else$\cI$\fi_4$ with the tangent hyperplane at $x$, $T_x\ifmmode {\cal I} \else$\cI$\fi_4$:
$$K_x=\ifmmode {\cal I} \else$\cI$\fi_4\cap T_x\ifmmode {\cal I} \else$\cI$\fi_4.$$
\end{theorem}
This statement can be found for example in \cite{Baker}. It amounts to the
fact, true in any dimension, that for $n\geq3$ the theta functions with
characteristics $\in \integer/n\integer$ on a fixed $A_{\tau}$ give an embedding of
$A_{\tau}$, while for $n=2$ they map onto the Kummer variety.
The reason one must exclude the ten hyperplane sections in Theorem
\ref{t119.1} is the following result.
\begin{proposition}\label{p120.1} The ten hyperplane sections $\{\gt_m^4=0\}
\cap \ifmmode {\cal I} \else$\cI$\fi_4$ are tangent hyperplane sections, i.e., the intersection is of
degree 2 and multiplicity 2.
\end{proposition}
A proof, based only on the equation of $\ifmmode {\cal I} \else$\cI$\fi_4$, can be found in
\cite{Baker}. To understand the meaning of this, note that a general
hyperplane section meets $\ifmmode {\cal I} \else$\cI$\fi_4$ in a quartic surface, while the
intersections here are quadric surfaces, hence to preserve degree must be
counted twice (i.e., multiplicity 2). Consider the symmetric subdomain
$\sieg_1\times \sieg_1\subset} \def\nni{\supset} \def\und{\underline \sieg_2$, which in this case is the set of reducible
matrices:
$$\sieg_1\times \sieg_1 = \left\{\left(\begin{array}{cc} \tau_1 & 0 \\ 0 &
\tau_2\end{array} \right)\right\}\subset} \def\nni{\supset} \def\und{\underline \left\{\left(\begin{array}{cc}
\tau_1 & \tau_{12} \\ \tau_{12} & \tau_2 \end{array}\right)\right\}
=\sieg_2.$$ Then an easy calculation shows that the theta function of
Definition \ref{d117.1} is a {\em product} of two theta functions of a
single variable (i.e., $z\in \komp$). This is equivalent to the fact that
for reducible $\tau \in \sieg_2$, the abelian surface $A_{\tau}$ is a product
of two elliptic curves, $A_{\tau}=E_1\times E_2$. In this case, the map
given onto the ``product Kummer'' variety is a map $s:E_1\times E_2\longrightarrow} \def\sura{\twoheadrightarrow
E_1/\{\pm1\}\times E_2/\{\pm1\}={\Bbb P}^1\times {\Bbb P}^1$, and this ${\Bbb P}^1\times
{\Bbb P}^1$ is the quadric surface occuring in \ref{p120.1}. Since ${\Bbb P}^1\times
{\Bbb P}^1$ has no moduli, we see that {\em formally} the statement of Theorem
\ref{t119.1} remains true for all $x\in \ifmmode {\cal I} \else$\cI$\fi_4-\{\hbox{15 singular
lines}\}$, if we consider product Kummer varieties instead of the usual
ones, and the hyperplane section is the quadric surface of Proposition
\ref{p120.1}. Note however, that this quadric surface, being a modular
subvariety, can also be described as:
\begin{equation}\label{e120.1} E_1/\{\pm1\}\times E_2/\{\pm1\}\ifmmode\ \cong\ \else$\isom$\fi
(\gG_1(2)\backslash \sieg_1)^*\times (\gG_1(2)\backslash \sieg_1)^*,
\end{equation}
describing the product Kummer surface of a reducible abelian surface as a
compactification of an arithmetic quotient, that is, as a Janus-like
variety. We then get the following moduli interpretation of the quadric
surfaces.
\begin{proposition}\label{p120.2} The ten quadric surfaces of Proposition
\ref{p120.1} are modular subvarieties which correspond to abelian
surfaces which split. More precisely, for any $x$ on one of the quadric
surfaces, but not on any of the singular lines (there are six such
singular lines on each quadric surface, see (\ref{e117a.1})), determines
a smooth abelian surface which splits, with a level 2 structure.
\end{proposition}
Finally we note that this geometry can be described, as discussed already
in \cite{J} and many other places, in terms of the finite geometry of
\begin{equation}\label{e121.1} V=(\integer/2\integer)^4.
\end{equation}
Let $<\ ,\ >$ denote the induced symplectic form on $V$; every vector $v\in
V$ is isotropic with respect to $<\ ,\ >$. Since there are 15 non-zero
vectors, there are 15 one-dimensional boundary components. Similarly, there
are 15 isotropic planes in $V$, giving 15 zero-dimensional boundary
components. The modular subvarieties of Proposition \ref{p120.2}
correspond in this setting to non-singular pairs $\{\gd,\gd^{\perp}\}$,
where $\gd$ is a two-dimensional subspace of $V$ on which $<\ ,\ >$ is non
degenerate, and $\gd^{\perp}$ denotes the orthocomplement with respect to
$<\ ,\ >$. Of these there are exactly ten, as is easily checked. We leave
further details to the reader.
\subsection{Birational transformations}\label{section3.4}
We have seen above in Proposition \ref{p117a.1} that $\ifmmode {\cal S} \else$\cS$\fi_3$ and $\ifmmode {\cal I} \else$\cI$\fi_4$
are dual varieties. It follows from general theory that they are then in
fact {\em birational}. In this section we describe the ensuing birational
map explicitly. We consider the following modifications of ${\Bbb P}^4$.
\begin{itemize}\item[a)] Blow up the ten nodes (\ref{e112.0}) of $\ifmmode {\cal S} \else$\cS$\fi_3$;
denote this by $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1:\widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}^4\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4$. There are ten exceptional
${\Bbb P}^3$'s, each with normal bundle $\ifmmode {\cal O} \else$\cO$\fi_{{\Bbb P}^3}(-1)$. Consider one of the
15 hyperplanes $\ifmmode {\cal H} \else$\cH$\fi_{ij}$ of (\ref{e112b.3}). Since each hyperplane
contains $4+2+1=7$ nodes, its proper transform on $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}^4$ is a
${\Bbb P}^3$ blown up in those seven points; each of the 15 ${\Bbb P}^2$'s of
(\ref{e112.1}) lying on $\ifmmode {\cal S} \else$\cS$\fi_3$ contains four of the nodes, so their
proper transforms are copies of ${\Bbb P}^2$ blown up in four points. Finally,
let $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal S} \else$\cS$\fi}_3\subset} \def\nni{\supset} \def\und{\underline \widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}^4$ denote the proper transform of $\ifmmode {\cal S} \else$\cS$\fi_3$
in $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}^4$; $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal S} \else$\cS$\fi}_3$ is smooth, and
${\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1}_{|\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal S} \else$\cS$\fi}_3}:\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal S} \else$\cS$\fi}_3\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal S} \else$\cS$\fi_3$ is a desingularisation of
$\ifmmode {\cal S} \else$\cS$\fi_3$, replacing each node with a quadric surface $\ifmmode\ \cong\ \else$\isom$\fi {\Bbb P}^1\times
{\Bbb P}^1$.
\item[b)] Let $\ifmmode \hbox{{\script I}} \else$\scI$\fi(15)$ denote the ideal of the 15 singular lines of
$\ifmmode {\cal I} \else$\cI$\fi_4$; blow up $\ifmmode \hbox{{\script I}} \else$\scI$\fi(15)$, and let $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2:\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}}^4\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4$
denote this modification. Under $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2$, each of the lines is replaced
by a ${\Bbb P}^2$-bundle over that line, and each point is replaced by a union
of ${\Bbb P}^1$'s, one each for each {\em pair} $(l_1,l_2)$ of {\em lines}
meeting at the point; this mentioned ${\Bbb P}^1$ is then the intersection of
the fibre ${\Bbb P}^2$ of $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2$ at that point with the (two) exceptional
${\Bbb P}^2$-bundles over the lines $l_1$ and $l_2$. Note that the proper
transforms of the ten quadrics of Proposition \ref{p120.1} on $\ifmmode {\cal I} \else$\cI$\fi_4$ are
still biregular to ${\Bbb P}^1\times {\Bbb P}^1$, while the proper transforms of
each of the lines turns out to be a {\em Kummer modular surface}, that
is, ${\Bbb P}^2$ blown up in four points. Let $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_4$ be the proper
transform of $\ifmmode {\cal I} \else$\cI$\fi_4$ in $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}}^4$; then
${\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2}_{|\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_4}:\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_4\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal I} \else$\cI$\fi_4$ is a desingularisation of
$\ifmmode {\cal I} \else$\cI$\fi_4$.
\end{itemize}
\begin{theorem}\label{t123.1} The varieties $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal S} \else$\cS$\fi}_3$ and $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_4$
are biregular, and the explicit birational map $\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha:\ifmmode {\cal S} \else$\cS$\fi_3- - \ra \ifmmode {\cal I} \else$\cI$\fi_4$
is the birational morphism completing the following diagram:
$$\begin{array}{ccccc} & \widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal S} \else$\cS$\fi}_3 & \stackrel{\tilde{\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha}}{\longrightarrow} \def\sura{\twoheadrightarrow} &
\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_4 & \\ \varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1 & \downarrow & & \downarrow & \varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2 \\ & \ifmmode {\cal S} \else$\cS$\fi_3
& \stackrel{\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha}{- - \ra} & \ifmmode {\cal I} \else$\cI$\fi_4. & \end{array}$$ Moreover, $\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha$ is
$\gS_6$-equivariant.
\end{theorem}
{\bf Proof:} As $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1$ and $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2$ are $\gS_6$-equivariant, the second
statement follows from the first. Let $D\subset} \def\nni{\supset} \def\und{\underline \ifmmode {\cal S} \else$\cS$\fi_3$ be the open set:
\begin{equation}\label{e123.1} D=\ifmmode {\cal S} \else$\cS$\fi_3-\{\hbox{15 hyperplanes $P_{\gs}$ of
(\ref{e112.1})}\};
\end{equation}
here we may take the regular map of $D$ onto the set of tangent hyperplanes
(now viewing $\ifmmode {\cal I} \else$\cI$\fi_4$ as the projective dual of $\ifmmode {\cal S} \else$\cS$\fi_3$), and set
\begin{eqnarray}\label{e123.2} \varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha_{|D}:D & \longrightarrow} \def\sura{\twoheadrightarrow & D'\subset} \def\nni{\supset} \def\und{\underline \ifmmode {\cal I} \else$\cI$\fi_4 \\
x & \mapsto & ({\Bbb P}^3)_x=\hbox{tangent hyperplane to $\ifmmode {\cal S} \else$\cS$\fi_3$ at $x$}
\nonumber
\end{eqnarray}
\begin{Lemma} The subset $D'\subset} \def\nni{\supset} \def\und{\underline \ifmmode {\cal I} \else$\cI$\fi_4$ is: $D'=\ifmmode {\cal I} \else$\cI$\fi_4-\{\hbox{10 quadric
surfaces of Proposition \ref{p120.2}}\}$.
\end{Lemma}
{\bf Proof:} Suppose $x\in D$; then $({\Bbb P}^3)_x$ meets $D$ in an irreducible
cubic (the union of the $P_{\gs}$ are {\em all} the linear subspaces
contained in $\ifmmode {\cal S} \else$\cS$\fi_3$, so outside of this locus $({\Bbb P}^3)_x\cap \ifmmode {\cal S} \else$\cS$\fi_3$ cannot
have a linear factor, so, being cubic, must be irreducible), while the ten
quadric surfaces are the locus of the tangent hyperplanes meeting $\ifmmode {\cal S} \else$\cS$\fi_3$
in one of the nodes, all of which are excluded in $D$. \hfill $\Box$ \vskip0.25cm Now we glue
$D$ onto the rest of $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal S} \else$\cS$\fi}_3$, and $D'$ onto the rest of
$\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_4$. The locus $\gL_1=\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal S} \else$\cS$\fi}_3-D$ coincides with
$\gL_2=\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_4-D'$, as follows from the descriptions of the rational
maps $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1$ and $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2$ above. Both the $\gL_i$ consist of ten
${\Bbb P}^1\times {\Bbb P}^1$'s and 15 rational surfaces, each isomorphic to ${\Bbb P}^2$
blown up in four points. Hence we can complete $\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha_{|D}$ to a biregular
isomorphism $\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha:\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal S} \else$\cS$\fi}_3\longrightarrow} \def\sura{\twoheadrightarrow \widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_4$, by fixing an isomorphism
$\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha_{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi}:\gL_1\longrightarrow} \def\sura{\twoheadrightarrow \gL_2$, and setting
$$\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha(x)=\left\{\parbox{6cm}{$\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha_{|D}(x),$ if $x\in D$ \\
$\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha_{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi}(x)$, if $x\in \gL_1$}\right.$$ completing the proof of
Theorem \ref{t123.1}. \hfill $\Box$ \vskip0.25cm The following description is more concrete. If
$x$ is one of the nodes of $\ifmmode {\cal S} \else$\cS$\fi_3$, there is a quadric cone of tangent (to
$\ifmmode {\cal S} \else$\cS$\fi_3$) hyperplanes at $x$; so closing up $\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha$ maps $x$ to the quadric
surface over which the above is a cone, i.e., $x$ is blown up. If $x$ is
{\em not} a node, then there is a unique tangent hyperplane $T_x\ifmmode {\cal S} \else$\cS$\fi_3$,
determining a point of $\ifmmode {\cal I} \else$\cI$\fi_4$. Furthermore, $T_x\ifmmode {\cal S} \else$\cS$\fi_3$ and $T_y\ifmmode {\cal S} \else$\cS$\fi_3$
{\em coincide} for $x\neq y$, if and only if $x$ and $y$ are contained in a
common Segre plane (\ref{e112.1}), and the line joining $x$ and $y$ in that
Segre plane passes through one of the four nodes, say $N$, in that Segre
plane. This is because $T_x\ifmmode {\cal S} \else$\cS$\fi_3\cap \ifmmode {\cal S} \else$\cS$\fi_3=\ifmmode {\cal H} \else$\cH$\fi\cup Q_x$, where $Q_x$ is a
residual quadric cone, and the quadric cone is the intersection of
$T_x\ifmmode {\cal S} \else$\cS$\fi_3$ with the cone $C_N$ which is the tangent cone of the node $N$ in
the Segre plane. So if $x$ and $y$ lie on a line through $N$, $Q_x$ and
$Q_y$ coincide, so $T_x\ifmmode {\cal S} \else$\cS$\fi_3$ and $T_y\ifmmode {\cal S} \else$\cS$\fi_3$ coincide also.
\begin{theorem}\label{t122a.1} The duality map $d:\ifmmode {\cal S} \else$\cS$\fi_3- - \ra \ifmmode {\cal I} \else$\cI$\fi_4$ is
given by the linear system of quadrics \ref{p112b.1}, i.e., by the
elements of the ideal $\ifmmode \hbox{{\script I}} \else$\scI$\fi(10)$ of the ten nodes: $d=\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha$.
\end{theorem}
{\bf Proof:} It suffices to check that $d$, viewed as a modification of
$\ifmmode {\cal S} \else$\cS$\fi_3$, coincides with the birational map $\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha$ of Theorem \ref{t123.1}.
But this is easy. As the base locus is the set of nodes, these are blown
up. As just explained, $x$ and $y$ in one of the Segre planes map to the
same point on the image line precisely when the line joining them passes
through one of the nodes in the Segre plane. As these lines are precisely
what the map $\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha$ blows down, $d$ certainly coincides with $\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha$. \hfill $\Box$ \vskip0.25cm
We also have the following analogue of Corollary \ref{c112b.1}.
\begin{lemma}\label{l122a.1} The ideal $\ifmmode \hbox{{\script I}} \else$\scI$\fi(15)$ of the 15 singular lines of
the Igusa quartic coincides with the Jacobian ideal of $\ifmmode {\cal I} \else$\cI$\fi_4$.
\end{lemma}
{\bf Proof:} Once again the inclusion $\ifmmode \hbox{{\script J}} \else$\scJ$\fi ac(\ifmmode {\cal I} \else$\cI$\fi_4)\subset} \def\nni{\supset} \def\und{\underline \ifmmode \hbox{{\script I}} \else$\scI$\fi(15)$ is
obvious, and the inverse inclusion can be verified by means of standard
basis computations, namely that $\ifmmode \hbox{{\script I}} \else$\scI$\fi(15)$ is generated by five cubics.
\hfill $\Box$ \vskip0.25cm Along the same lines as Theorem \ref{t122a.1} we then get
\begin{theorem} \label{t122b.1} The duality map $d:\ifmmode {\cal I} \else$\cI$\fi_4- - \ra\ifmmode {\cal S} \else$\cS$\fi_3$ is given
by the system of cubics containing the 15 lines, i.e., by the Jacobian
ideal of $\ifmmode {\cal I} \else$\cI$\fi_4$: $d=\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha^{-1}$.
\end{theorem}
{\bf Proof:} As above, it suffices to show that $d$, viewed as a
modification of $\ifmmode {\cal I} \else$\cI$\fi_4$, coincides with the map $\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha^{-1}$ of Theorem
\ref{t123.1}. This is readily verified, as the base locus, the 15 lines,
are blown up, while the tangent planes for any two points $x$ and $y$ in a
common quadric of $\ifmmode {\cal I} \else$\cI$\fi_4$ (of Proposition \ref{p120.1}) coincide, blowing
down the quadric surface to a node. \hfill $\Box$ \vskip0.25cm
\subsection{The Siegel modular threefold of level 4}
{}From the general theory of congruence subgroups, $X_{\gG(4)}\longrightarrow} \def\sura{\twoheadrightarrow
X_{\gG(2)}$ is a Galois cover, with Galois group $\gG(2)/\gG(4)\ifmmode\ \cong\ \else$\isom$\fi
(\integer/2\integer)^9$. Indentifying $X_{\gG(2)}^*$ with $\ifmmode {\cal I} \else$\cI$\fi_4$ and identifying
$\ifmmode {\cal I} \else$\cI$\fi_4$ birationally with $\ifmmode {\cal S} \else$\cS$\fi_3$, we can consider Fermat covers over
$X_{\gG(2)}^*$, i.e., given by a diagram
\begin{equation}\label{e122b.1} \begin{array}{cccccc} Z({\bf A_4},n) &
\stackrel{\-{\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha}^{-1}}{- - \ra} & Y^{\wedge}({\bf A_4},n) & \longleftarrow} \def\rar{\rightarrow &
\tilde{Y}({\bf A_4},n) & \\ \downarrow & & \downarrow & & \downarrow &
(\integer/2\integer)^9 \\ \ifmmode {\cal I} \else$\cI$\fi_4 & \stackrel{\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha^{-1}}{- - \ra} & \ifmmode {\cal S} \else$\cS$\fi_3 & \longleftarrow} \def\rar{\rightarrow &
\tilde{\ifmmode {\cal S} \else$\cS$\fi}_3 &
\end{array}
\end{equation}
where $\-{\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha}^{-1}$ is {\em induced} by $\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha^{-1}$, that is,
(\ref{e122b.1}) is a fibre square (cf. (\ref{e115.0}), where
$\tilde{\ifmmode {\cal S} \else$\cS$\fi}_3$ is denoted $\tilde{{\Bbb P}}^3$).
\begin{theorem}\label{t122b.1} The Fermat cover $Z({\bf A_4},2)$ is the
Satake compactification of the Siegel modular threefold of level 4.
\end{theorem}
{\bf Proof:} It suffices to show that $\~Y({\bf A_4},2)$ is the induced
cover over $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_4$, where $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2:\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_4\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal I} \else$\cI$\fi_4$ is the
desingularisation of $\ifmmode {\cal I} \else$\cI$\fi_4$ of Theorem \ref{t123.1}. Now the
identification can be reduced to identifying what is in the branch locus of
$\~Y({\bf A_4},2)\longrightarrow} \def\sura{\twoheadrightarrow \~{\ifmmode {\cal S} \else$\cS$\fi}_3$. There are two kinds of components:
\begin{itemize}\item[a)] covers $\~Y({\bf A_3},2)$ of blown up ${\Bbb P}^2$'s, the
$H_{ij}$ of (\ref{e111b.1});
\item[b)] covers $\~Y({\bf A_2},2)\times \~Y({\bf A_2},2)$ of ${\Bbb P}^1\times
{\Bbb P}^1$'s, the $L_{0ij}$.
\end{itemize}
\begin{Lemma}\label{l122b.1} $\~Y({\bf A_3},2)\ifmmode\ \cong\ \else$\isom$\fi S(4)$, Shioda's elliptic
modular surface of level 4.
\end{Lemma}
{\bf Proof:} This is well-known. $\~Y({\bf A_3},2)$ is K3 since it is a
Fermat cover branched over six lines. One constructs structures of fibre
space $\~Y({\bf A_3},2)\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^1$ with elliptic curves as fibres by taking
the cover of the pencil of lines through a node (each such line meets four
of the six lines outside the node, so the cover is branched at four points,
i.e., is elliptic). The six fibres of type $I_4$ are readily identified, as
are the 16 sections. \hfill $\Box$ \vskip0.25cm
\begin{Lemma}\label{l122c.1} The cover $\~Y({\bf A_2},2)\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^1$ coincides
with the cover $(\gG_1(4)\backslash \sieg_1)^*\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^1$, by which we mean the
Galois actions coincide.
\end{Lemma}
{\bf Proof:} This is even more well-known. \hfill $\Box$ \vskip0.25cm The theorem now follows,
provided we accept that $\~Y({\bf A_4},2)$ is a quotient of $\sieg_2$ at all,
i.e., that the cover $\sieg_2\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal I} \else$\cI$\fi_4^0$ factorises (here
$\ifmmode {\cal I} \else$\cI$\fi_4^0=\ifmmode {\cal I} \else$\cI$\fi_4-\{\hbox{15 lines}\}$), $\tilde{Y}({\bf A_4},2)^0:= \~Y({\bf
A_4},2)-q^{-1}(\hbox{15 lines})$:
\begin{equation}\label{e122c.1}\unitlength1.6cm \begin{picture}(1.6,1.4)
\put(.2,1.33){$\sieg_2$} \put(.6,1.4){\vector(1,0){1}}
\put(1.8,1.33){$\ifmmode {\cal I} \else$\cI$\fi_4^0$} \put(.4,1.25){\vector(1,-1){.6}}
\put(.7,.45){$\~Y({\bf A_4},2)^0.$} \put(1.2,.66){\vector(1,1){.6}}
\put(1.7,.8){$q$}
\end{picture}
\end{equation}
But there is an easy way to see that this is the case: we can, for any
given $x\in \~Y({\bf A_4},2)-\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$ and $y=q(x)$, put a level 4 structure on
$A_y$, such that the Galois group just permutes the level 4 over level 2
structures, that is, we make the identification $\gG(2)\backslash\gG(4)\ifmmode\ \cong\ \else$\isom$\fi
(\integer/2\integer)^9\ifmmode\ \cong\ \else$\isom$\fi$ the Galois group of the cover.
So $\~Y({\bf A_4},2)$, being a moduli space as in Shimura's
theory, is a quotient of $\sieg_2$. \hfill $\Box$ \vskip0.25cm
One could also imagine arguing with uniqueness of Galois covers, since we
know the branch locus, branch degrees and Galois group. However there is in
general no such uniqueness of covers, so we have to be careful. In our
situation, there are two possible approaches to show uniqueness:
\begin{itemize}\item[1)] Since the modular subvarieties determine, on the
group-theoretic side, generators of the corresponding arithmetic group,
we could conclude, from the isomorphisms \ref{l122b.1} and \ref{l122c.1},
the desired result.
\item[2)] Since the branch divisors are totally geodesic with respect to
the Bergmann metric, on the cover the metric retains its symmetry
property.
\end{itemize}
Method 1) has been applied in \cite{J}, and 2) can be carried out for ball
quotients.
\section{The Hessian varieties of ${\cal S}_3$ and ${\cal I}_4$}
\subsection{The Nieto quintic}\label{section4.1}
Let $(x_0:\ldots:x_5)$ be the projective coordinates on ${\Bbb P}^5$ used to
define $\ifmmode {\cal S} \else$\cS$\fi_3$ in (\ref{e111.3}), and let $\gs_i=\gs_i(x_0,\ldots,x_5)$ be
the $i$-th elementary symmetric function $\gs_{\gl}=\sum_{i_1<\ldots <
i_{\gl}}x_{i_1}\cdots x_{i_{\gl}}$ in $(x_0:\ldots:x_5)$. Define the {\em
Nieto quintic} $\ifmmode {\cal N} \else$\cN$\fi_5$ by the equations\begin{equation}\label{e124.1}
\ifmmode {\cal N} \else$\cN$\fi_5=\left\{\begin{array}{l}\gs_1 = 0 \\ \gs_5=0\end{array} \right. \subset} \def\nni{\supset} \def\und{\underline
{\Bbb P}^4=\{\gs_1=0\}\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^5.
\end{equation}
The symmetry of $\ifmmode {\cal N} \else$\cN$\fi_5$ under the symmetric group $\gS_6$ is evident from
the equation. This quintic was discovered in the thesis \cite{Nie} and
further studied in \cite{BN}, which will be our general reference for this
section. We just briefly describe the geometry of $\ifmmode {\cal N} \else$\cN$\fi_5$ without
discussing details.
The singular locus is relatively easy to determine, just by calculating the
Jacobian of (\ref{e124.1}). The result is
\begin{proposition}[\cite{BN}, 3.1]\label{p124.1} $\ifmmode {\cal N} \else$\cN$\fi_5$ has the following
singular locus:
\begin{itemize}\item[(i)] 20 lines $L_{ijk}=\{x_i=x_j=x_k=0=\sum x_i\}$;
\item[(ii)] ten isolated points, the $\gS_6$-orbit of $(1,1,1,-1,-1,-1)$,
which are the points $P_{ij}=(1,\pm1,\ldots \pm 1)$, with $+1$ in the
$i$-th and $j$-th positions.
\end{itemize}
\end{proposition}
We will give a different proof of this below, see the discussion following
Proposition \ref{piq8.1}. Note that the ten points occuring in (ii) are
just the ten nodes of $\ifmmode {\cal S} \else$\cS$\fi_3$ (see (\ref{e112.0}), cf.~also
Remark \ref{r126.2} below). Furthermore, a local
calculation shows that the singularities of $\ifmmode {\cal N} \else$\cN$\fi_5$ along the lines of (i)
are of the type $\{\hbox{disc}\}\times A_1$, and at the points of (ii) are
ordinary double points. Hence the former are resolved by a ${\Bbb P}^1$-bundle
over the line $L_{ijk}$, while the points are resolved, as with the case of
$\ifmmode {\cal S} \else$\cS$\fi_3$, by quadric surfaces. The 20 lines $L_{ijk}$ of \ref{p124.1} meet
at the following 15 points:
\begin{equation}\label{e124.2} Q_{ij}=(0,\ldots,1,\ldots,-1,\ldots)
=\{\gS_6-\hbox{orbit of } Q_{56}=(0:0:0:0:1:-1)\}.
\end{equation}
\begin{lemma}\label{l124.1}
The 20 lines $L_{ijk}$ of Proposition \ref{p124.1} meet four at a time at
the 15 points $Q_{ij}$; each line $L_{ijk}$ contains three of the points,
namely we have $Q_{ij}\in L_{klm}\iff \{i,j\}\cap \{k,l,m\}=\emptyset$.
\end{lemma}
{\bf Proof:} The line $L_{123}$ contains the three points $Q_{46},\ Q_{45}$
and $Q_{56}$, so by $\gS_6$-invariance each line contains three of the
$Q_{ij}$. The point $Q_{56}$ is contained in the four lines $L_{123},\
L_{124},\ L_{134}$ and $L_{234}$, so by $\gS_6$-invariance, each point is
contained in four lines. \hfill $\Box$ \vskip0.25cm Also, $\ifmmode {\cal N} \else$\cN$\fi_5$ contains a finite number of
linear planes.
\begin{lemma}\label{l125.1} $\ifmmode {\cal N} \else$\cN$\fi_5$ contains the following 30 ${\Bbb P}^2$'s:
\begin{itemize}\item[(i)] 15 planes $N_{ijkl}=\{x_i+x_j=x_k+x_l=x_m+x_n=0\}$;
\item[(ii)] 15 planes $N_{ij}=\{x_i=x_j=0=\sum_{k\neq i,j}x_k\}$;
\end{itemize}
\end{lemma}
{\bf Proof:} It is immediately verified that these planes satisfy the
equation (\ref{e124.1}).\hfill $\Box$ \vskip0.25cm Presumably these are in fact {\em all} the
linear planes contained in $\ifmmode {\cal N} \else$\cN$\fi_5$. Note that the $N_{ijkl}$ are just the
15 planes (\ref{e112.1}) lying on the Segre cubic $\ifmmode {\cal S} \else$\cS$\fi_3$.
Among the 15 planes $N_{ijkl}$ the common intersections were described in
the discussion of the planes $P_{\gs}$ on the Segre cubic (see
(\ref{e111.1})).
\begin{lemma}\label{l125.2} Each plane $N_{ijkl}$ contains the following four
of
the ten points of \ref{p124.1}, (ii): $$P_{km},\ P_{kn},\ P_{lm} \hbox{
and } P_{ln};$$ it also contains the following three of the 15 points
$Q_{ij}$ of (\ref{e124.2}): $Q_{ij},\ Q_{kl}$ and $Q_{mn}$.
\end{lemma}
{\bf Proof:} Consider $N_{0123}$; it contains the four nodes
$(1:-1:1:-1:1:-1),\ (1:-1:-1:1:1:-1),\ (1:-1:1:-1:-1:1)$ and
$(1:-1:-1:1:-1:1)$ which are the points $P_{24},\ P_{25},\ P_{34}$ and
$P_{35}$, which gives the first statement by $\gS_6$-symmetry (there is an
asymmetry in the notation, since we may take $i<j,k<l$ in the notation for
$N_{ijkl}$, and since the first coordinate of $P_{ij}$ may be assumed to be
$+1$). Similarly, $N_{0123}$ contains the three points $Q_{01},\ Q_{23}$
and $Q_{45}$, giving the second statement by $\gS_6$-symmetry.\hfill $\Box$ \vskip0.25cm We now
note that these seven points lie in the plane $N_{0123}$ as in Figure
\ref{Figure1}.
\begin{figure}[thb]
$$\fbox{\unitlength1.5cm \begin{picture}(6,4) \put(.5,.5){\line(1,1){3}}
\put(1,1){\line(3,1){4.5}}\put(1,1){\line(-3,-1){.5}}
\put(1,1){\circle*{.2}}\put(1,.7){$P_{35}$}
\put(.5,2.5){\line(3,-1){5}} \put(.7,3.1){$N_{01}$}
\put(2,2){\circle{.2}}\put(1.9,2.2){$Q_{23}$}
\put(3,1){\circle{.2}}\put(2.4,.9){$Q_{45}$}
\put(2.5,3.5){\line(1,-1){3}}\put(3,3.5){\line(0,-1){3}}
\put(3,1.75){\circle*{.2}} \put(2.8,1,4){$P_{24}$}
\put(3,3){\circle*{.2}}\put(3.2,3){$P_{25}$}
\put(4,2){\circle{.2}}\put(3.8,2.2){$Q_{01}$}
\put(5,1){\circle*{.2}}\put(4.6,.7){$P_{34}$}
\put(5.2,3.1){$N_{23}$}\put(5.6,1.8){$N_{45}$} \put(5,3.5){in
$N_{0123}$} \put(.5,1){\line(1,0){5}} \thicklines
\put(1,3){\line(1,-1){2.5}} \put(2.5,.5){\line(1,1){2.5}}
\put(.5,2){\line(1,0){5}}
\end{picture}}$$
\caption[15 planes of type 1 on ${\cal N}_5$]{\label{Figure1}\small
The plane $N_{0123}$}
\end{figure}
This is in fact easily checked. Note that the lines in $N_{0123}$, i.e.,
the intersections with the other $N_{ijkl}$ are {\em not} the lines of
Proposition \ref{p124.1}; those lines have equations such as
$x_0=x_1+x_2=x_1+x_3=x_4+x_5$. However, in the 15 planes $N_{ij}$ of
\ref{l125.1}, several of the 20 singular lines $L_{ijk}$ {\em do} lie. In
fact, we have
\begin{lemma}\label{l125.3} Each $N_{ij}$ contains the four lines $L_{ijk}.\
L_{ijl},\ L_{ijm}$ and $L_{ijn}$. There are three planes passing through
$L_{ijk}$, namely $N_{ij},\ N_{ik}$ and $N_{jk}$. $N_{ij}$ contains none
of the nodes of \ref{p124.1}, (i), but contains six of the points
$Q_{ij}$ of (\ref{e124.2}), namely $Q_{kl},\ Q_{km},\ Q_{kn},\ Q_{lm},\
Q_{ln}$ and $Q_{mn}$. These six points lie three at a time on the
$L_{ijk}$ and form in each $N_{ij}$ a configuration as shown in Figure
\ref{Figure2}.
\begin{figure}[tbh]
$$\fbox{\unitlength1.5cm \begin{picture}(6,4)\put(.5,1){\line(1,0){5}}
\put(.5,2){\line(1,0){5}} \put(3,3.5){\line(0,-1){3}}
\put(3,3){\circle{.2}}\put(1,1){\circle{.2}}\put(2,2){\circle{.2}}
\put(3,1.66){\circle{.2}}\put(4,2){\circle{.2}}\put(5,1){\circle{.2}}
\thicklines \put(.5,.5){\line(1,1){3}}\put(5.5,2.5){\line(-3,-1){5}}
\put(.5,2.5){\line(3,-1){5}}\put(2.5,3.5){\line(1,-1){3}}
\put(.5,2.7){$L_{015}$}\put(1.6,3.6){$L_{012}$}\put(3.6,3.6){$L_{013}$}
\put(5,2.7){$L_{014}$}\put(5.5,1.2){$N_{0125}$}
\put(5.5,1.8){$N_{0124}$}\put(2.6,.2){$N_{0123}$}
\put(1.1,.7){$Q_{25}$}\put(4.6,.7){$Q_{34}$}\put(2.6,1.3){$Q_{23}$}
\put(1.8,2.2){$Q_{24}$}\put(3.8,2.2){$Q_{35}$}\put(3.2,2.8){$Q_{45}$}
\put(5,3.5){in $N_{01}$}
\end{picture}}$$
\caption[15 planes of type 2 on ${\cal N}_5$]{\label{Figure2}\small
The plane $N_{01}$}
\end{figure}
The three light lines are intersection of $N_{01}$ with $N_{ijkl}$ as
indicated.
\end{lemma}
{\bf Proof:} This is once again easily verified. \hfill $\Box$ \vskip0.25cm Finally we note that
there are hyperplanes in ${\Bbb P}^4$ cutting out these ${\Bbb P}^2$'s on $\ifmmode {\cal N} \else$\cN$\fi_5$.
\begin{lemma}\label{l126.1} The six hyperplanes $\ifmmode {\cal H} \else$\cH$\fi_{ij}=\{x_i+x_j=0\}$ meet
$\ifmmode {\cal N} \else$\cN$\fi_5$ each in the union of the three planes $N_{ijkl},\ N_{ijkm}$ and
$N_{ijkn}$, and a residual quadric; the six hyperplanes $x_i=0$ meet
$\ifmmode {\cal N} \else$\cN$\fi_5$ each in the union of five planes $N_{ij},\ N_{ik},\ N_{il},\
N_{im}$ and $N_{in}$.
\end{lemma}
{\bf Proof:} This is once again just a computation. \hfill $\Box$ \vskip0.25cm Now let us
consider the intersection of $\ifmmode {\cal S} \else$\cS$\fi_3$ and $\ifmmode {\cal N} \else$\cN$\fi_5$. As is obvious from the
above description, they both contain the 15 planes $N_{ijkl}$, and, the
intersection being of degree 15, this is the entire intersection. From
general arguments on projective varieties, from the fact that the dual of
$\ifmmode {\cal S} \else$\cS$\fi_3$, namely the Igusa quartic $\ifmmode {\cal I} \else$\cI$\fi_4$, is {\em normal}, it follows that
the parabolic divisor on $\ifmmode {\cal S} \else$\cS$\fi_3$, which is the intersection of $\ifmmode {\cal S} \else$\cS$\fi_3$ with
the {\em Hessian variety}, must get blown down under the duality map, i.e.,
the intersection $Hess(\ifmmode {\cal S} \else$\cS$\fi_3)\cap \ifmmode {\cal S} \else$\cS$\fi_3$ consists of the 15 planes on
$\ifmmode {\cal S} \else$\cS$\fi_3$! Since the Hessian has degree 5, this is the entire intersection,
and it is natural to ask whether $\ifmmode {\cal N} \else$\cN$\fi_5$ and $Hess(\ifmmode {\cal S} \else$\cS$\fi_3)$ are related. In
fact, we have
\begin{lemma}\label{l126.2} The Nieto quintic is the Hessian of $\ifmmode {\cal S} \else$\cS$\fi_3$,
i.e., $\ifmmode {\cal N} \else$\cN$\fi_5= Hess(\ifmmode {\cal S} \else$\cS$\fi_3)$, with equality, not just isomorphism.
\end{lemma}
{\bf Proof:} This is an easy computation (at least for a computer).\hfill $\Box$ \vskip0.25cm
\begin{remark}\label{r126.2}
Since the Hessian variety $\hbox{Hess}} \def\rank{\hbox{rank}(V)$ aquires nodes where $V$
has nodes, this ``explains'' the ten isolated singularities on $\ifmmode {\cal N} \else$\cN$\fi_5$.
\end{remark}
\subsection{Two birational transformations}
We consider in this section two particularly interesting birational maps
from $\ifmmode {\cal N} \else$\cN$\fi_5$.
\subsubsection{ \ }
The first arises through the duality map. Consider the birational map of
${\Bbb P}^4$ given in the following diagram:\begin{equation}\label{e126.1}
\begin{array}{ccccccc} {\Bbb P}^4 & \stackrel{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1}{\longleftarrow} \def\rar{\rightarrow}
& \widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}^4 & \stackrel{\~{\ga}}{- - - \ra} & \widehat} \def\tilde{\widetilde} \def\nin{\not\in{\widehat} \def\tilde{\widetilde} \def\nin{\not\in{{\Bbb P}}}^4 &
\stackrel{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2}{\longrightarrow} \def\sura{\twoheadrightarrow} & {\Bbb P}^4 \\ \cup & & \cup & & \cup & & \cup \\
{\cal S}_3 & \longleftarrow} \def\rar{\rightarrow & \widehat} \def\tilde{\widetilde} \def\nin{\not\in{\cal S}_3 & \stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow} &
\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\cal I}_4 & \longrightarrow} \def\sura{\twoheadrightarrow & {\cal I}_4
\end{array}\quad;
\end{equation}
$\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1$ and $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2$ were described in section \ref{section3.4}, and this
diagram extends the one of Theorem \ref{t123.1} to the ambient rational
fourfolds. It is easily seen that the ensuing rational map of ${\Bbb P}^4$,
$\ga:=\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2\circ \~{\ga}\circ \varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1^{-1}:{\Bbb P}^4\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4$, is the
map given by the Jacobian ideal of $\ifmmode {\cal S} \else$\cS$\fi_3$, that is, by the
linear system of quadrics on the ten nodes of $\ifmmode {\cal S} \else$\cS$\fi_3$ (see Corollary
\ref{c112b.1}). This ``defines'' the map $\tilde{\ga}$; although we could
in principle take any extension of $\tilde{\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha}:\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal S} \else$\cS$\fi_3}\longrightarrow} \def\sura{\twoheadrightarrow
\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi_4}$, for our purposes it is convenient to use $\ifmmode \hbox{{\script I}} \else$\scI$\fi(10)$. Then set
\begin{equation}\label{e127.1} \ifmmode {\cal W} \else$\cW$\fi_{10}:=\ga(\ifmmode {\cal N} \else$\cN$\fi_5)
\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^4.
\end{equation}
We now describe $\ifmmode {\cal W} \else$\cW$\fi_{10}$ and show it is a hypersurface of degree 10,
explaining the notation. First we have the
\begin{lemma}\label{l25aux} The map $\ga:{\Bbb P}^4\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4$ blows up the ten
nodes $P_{ij}$ of $\ifmmode {\cal S} \else$\cS$\fi_3$, with exceptional divisors $\ifmmode {\cal E} \else$\cE$\fi_{ij}$. Let
$\ifmmode {\cal C} \else$\cC$\fi_{ij}$ denote the tangent cone at the point $P_{ij}$, a quadric cone
fibred in lines passing through $P_{ij}$. Then each line of the cone gets
blown down to the corresponding point in $\ifmmode {\cal E} \else$\cE$\fi_{ij}$.
\end{lemma}
{\bf Proof:} Since all the quadrics of $\ifmmode \hbox{{\script I}} \else$\scI$\fi(10)$
vanish at the $P_{ij}$, these points
are blown up. To say the lines of $\ifmmode {\cal C} \else$\cC$\fi_{ij}$ get blown down is to say the
ratios of the quadrics are constant along the line. This follows from the
fact that the quadrics are the partial derivatives of $f$ (the defining
polynomial of $\ifmmode {\cal S} \else$\cS$\fi_3$), and the line is tangent to the zero locus of $f$.
\hfill $\Box$ \vskip0.25cm From this we get
\begin{lemma} $\ga$, restricted to $\ifmmode {\cal N} \else$\cN$\fi_5$,
is an isomorphism on the complement in $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1^{-1}(\ifmmode {\cal N} \else$\cN$\fi_5)$ of the
intersection of $\ifmmode {\cal N} \else$\cN$\fi_5$ with the tangent cones at the ten isolated
singularities $P_{ij}$.
\end{lemma}
{\bf Proof:} This is clear from construction, taking into account the
following fact, proved in \cite{BN}: the intersection of $\ifmmode {\cal N} \else$\cN$\fi_5$ with the
tangent cone of one of the nodes $P_{ij}$ consists of the six Segre planes
through the node, and an irreducible quartic {\em ruled} surface. It is
then clear that these ruled surfaces get blown down, and that outside the
Segre cubic and the ruled quartics, the birational map is a morphism. \hfill $\Box$ \vskip0.25cm
{}From this it follows:
\begin{lemma}\label{l127.1} The birational map $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2\circ \~{\ga} \circ
\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_1^{-1}:\ifmmode {\cal N} \else$\cN$\fi_5- - \ra \ifmmode {\cal W} \else$\cW$\fi_{10}$ has image $\ifmmode {\cal W} \else$\cW$\fi_{10}$, whose singular
locus contains the following.
\begin{itemize}\item[(i)] ten singular quadric surfaces (the tangent
hyperplane intersections of $\ifmmode {\cal I} \else$\cI$\fi_4$);
\item[(ii)] 20 singular lines, coming from the singular locus of $\ifmmode {\cal N} \else$\cN$\fi_5$.
\end{itemize}
\end{lemma}
{\bf Proof:} This follows from the description above. \hfill $\Box$ \vskip0.25cm Now note that
both $Hess(\ifmmode {\cal I} \else$\cI$\fi_4)$ and $\ifmmode {\cal W} \else$\cW$\fi_{10}$ are symmetric under $\gS_6$, and both are
singular along the ten quadrics of the Igusa $\ifmmode {\cal I} \else$\cI$\fi_4$. It may very well be
that the two coincide, but we have not checked this. At any rate,
$\ifmmode {\cal W} \else$\cW$\fi_{10}$ meets $\ifmmode {\cal I} \else$\cI$\fi_4$ in the union of quadric surfaces, each with
multiplicity 2. Hence
\begin{theorem}\label{t127.1} $\ifmmode {\cal I} \else$\cI$\fi_4\cap\ifmmode {\cal W} \else$\cW$\fi_{10}$ consists of the ten quadric
surfaces (\ref{e117a.1}), each with multiplicity 2. Consequently, the
degree of $\ifmmode {\cal W} \else$\cW$\fi_{10}$ is 10, justifying the notation.
\end{theorem}
{\bf Proof:} The intersection has reduced degree 20, and each surface
component is counted twice, hence the degree of the intersection is 40, so
the degree of $\ifmmode {\cal W} \else$\cW$\fi_{10}$ is 10. \hfill $\Box$ \vskip0.25cm
\begin{problem}\label{p131.1}
Is $\ifmmode {\cal W} \else$\cW$\fi_{10}$ also a compactification of an arithmetic quotient?
\end{problem}
\subsubsection{ \ }
The other birational transformation is the following.
\begin{equation}\label{e127.2}\begin{minipage}{14cm}\begin{itemize}
\item[a)] Blow up the 15 points $Q_{ij}$ of (\ref{e124.2}); let
$p_1:\~{\ifmmode {\cal N} \else$\cN$\fi}_5\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal N} \else$\cN$\fi_5$ denote this blow up.
\item[b)] As each of the lines $L_{ijk}$ contains three points (see
Lemma \ref{l124.1}), each $L_{ijk}$ can be blown down to an isolated
singular point (the normal bundle is $\ifmmode {\cal O} \else$\cO$\fi(-2)\oplus\ifmmode {\cal O} \else$\cO$\fi(-2)$,
cf.~(\ref{e110.1})). Let $p_2:\~{\ifmmode {\cal N} \else$\cN$\fi}_5\longrightarrow} \def\sura{\twoheadrightarrow
\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal N} \else$\cN$\fi}_5$ denote this blow down.
\end{itemize}
\end{minipage}
\end{equation}
The following is easy to see (see Figures \ref{Figure1} and \ref{Figure2}).
\begin{lemma}\label{l127.3} The singular locus of $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal N} \else$\cN$\fi}_5$ consists of
the 20 isolated cusps from (\ref{e127.2}) b), and the ten cusps, the images
of the singular points $P_{ij}$ of Proposition \ref{p124.1}, (ii). The
proper transforms of the $N_{ijkl}$ of Lemma \ref{l125.1}, (i) on $\ifmmode {\cal N} \else$\cN$\fi_5$
are ${\Bbb P}^2$'s blown up in three points, a del Pezzo surface; the proper
transforms of the $N_{ij}$ of Lemma \ref{l125.1}, (ii) are ${\Bbb P}^2$'s
blown up in six points, then the $L_{ijk}$ are blown down to four nodes,
so this is the singular cubic surface with four nodes, the Cayley cubic.
\end{lemma}
\subsection{Moduli interpretation}
The Nieto quintic was discovered as the solution of a certain moduli
problem, and we briefly state the results of \cite{BN} describing this.
The point of departure is the action of the Heisenberg group $H_{2,2}$ on
${\Bbb P}^3$, and the study of quartics which are invariant under the action.
$H_{2,2}$ is a group of order 32 generated by the following linear
transformations of ${\Bbb P}^3$ with coordinates $(z_0:z_1:z_2:z_3)$:
\begin{equation}\label{e128.1}\begin{array}{rclcl} \gs_1 & : &
(z_0:z_1:z_2:z_3) & \mapsto & (z_2:z_3:z_0:z_1) \\ \gs_2 & : &
(z_0:z_1:z_2:z_3) & \mapsto & (z_1:z_0:z_3:z_2) \\ \tau_1 & : &
(z_0:z_1:z_2:z_3) & \mapsto & (z_0:z_1:-z_2:-z_3) \\ \tau_2 & : &
(z_0:z_1:z_2:z_3) & \mapsto & (z_0:-z_1:z_2:-z_3) \end{array}
\end{equation}
The center of the group is $\pm1$ and $PH_{2,2}=H_{2,2}/\pm1$ has a nice
interpretation:
\begin{equation}\label{e128.2} PH_{2,2}\ifmmode\ \cong\ \else$\isom$\fi (\integer/2\integer)^4,
\end{equation}
which carries, as in (\ref{e121.1}), an induced symplectic form. This
means that one can speak of isotropic elements of the {\em group}
$PH_{2,2}$. The normaliser of $H_{2,2}$ in $SL(4,\komp)$ maps surjectively to
$\gS_6\ifmmode\ \cong\ \else$\isom$\fi Sp(4,\integer/2\integer)$, which acts transitively on diverse geometric
loci of the sympectic form {\em inside} the group $PH_{2,2}$. These loci
are:
\begin{equation}\label{e128.3}\begin{minipage}{14cm}\begin{itemize}
\item[a)] 15 pairs of skew lines
\item[b)] 15 invariant tetrahedra
\item[c)] ten fundamental quadrics.
\end{itemize}\end{minipage}\end{equation}
The moduli problem considered is a special set of quartics which are
invariant under (\ref{e128.2}). The set of {\em all} invariant quartics is
just a ${\Bbb P}^4$, spanned for example by the five quartics:
$$\begin{array}{ccc} & g_0:=z_0^4+z_1^4+z_2^4+z_3^4 & \\
g_1:=2(z_0^2z_1^2+z_2^2z_3^2) & g_2=2(z_0^2z_2^2+z_1^2z_3^2) &
g_3:=2(z_0^2z_3^2+z_1^2z_2^2) \\ & g_4:=4z_0z_1z_2z_3. & \end{array}$$
Let $(A,B,C,D,E)$ denote the coordinates of a particular quartic
$Q_{(A,B,C,D,E)}=\{Ag_0+Bg_1+Cg_2+Dg_3+Eg_4=0\}$. The generic quartic
$Q_{(A,B,C,D,E)}$ is smooth, and the locus of singular quartics can be
determined as an equation in $(A,\ldots,E)$. Note that the $(A,\ldots,E)$
are functions of $(z_0:\cdots:z_3)$, so the answer as to whether
$Q_{(A,B,C,D,E)}$ is singular depends on the point $z\in {\Bbb P}^3$. This is
discussed in detail in \cite{BN}. The result is given in Table
\ref{table19}.
\begin{table}
\begin{center}
\caption{\label{table19}Singular Heisenberg invariant quartics }
\begin{tabular}{|l|l|l|l|}\hline $z\in {\Bbb P}^3$ & dim$Q^{sing}$ &
$Q_{(A,B,C,D,E)}$ & $S_{(A,B,C,D,E)}$ \\ \hline\hline $\notin$ fix line &
0 & Kummer surface & Segre cubic \\ \hline $\in$ one fix line & 2 &
singular in four coordinate vertices & $A=0$ \\ \hline $\in$ the
intersection of two fixed lines & 3 & singular along two fixed lines &
$A=B=0$ \\ \hline
\end{tabular}
\end{center}
{\small Notations: $Q^{sing}$ denotes the space of quartics singular at
$z$, $S_{(A,B,C,D,E)}$ denotes the equation of the locus $Q^{sing}$ in
the coordinates $(A,B,C,D,E)$.}
\end{table}
As one sees, the first row of the table is equivalent to Theorem \ref{t119.1}
above! The special class of quartics to be considered here is, however, a
quite different set, consisting of generically smooth quartics. This is the
set of Kummer surfaces of (1,3)-polarised abelian surfaces, which, as it
turns out, can be smoothly embedded in ${\Bbb P}^3$. This was discovered
independently by Naruki and Nieto (see \cite{na} and \cite{Nie}). The 16
exceptional ${\Bbb P}^1$'s resolving the 16 double points of the Kummer surface
are 16 disjoint {\em lines} on the quartics. Also, by a result of Nikulin
\cite{Ni}, the converse is true, i.e., any quartic containing 16 lines is a
Kummer surface. Furthermore, the quartic being invariant under $PH_{2,2}$,
if it contains one line, it contains all 16 transforms, so the moduli
involved is the condition:
\begin{equation}\label{e129.1} L\ \parbox[t]{12cm}{ is a line in ${\Bbb P}^3$
lying on a smooth Heisenberg invariant quartic surface}
\end{equation}
The equation describing this in the Grassmannian $\fG(2,2)=\{x_0^2+\cdots +
x_5^2=0\}$ is calculated in \cite{Nie}. It is
\begin{equation}\label{e130.1}
\ifmmode {\cal M} \else$\cM$\fi_{20}=\{\gs_5(x_0^2,\ldots,x_5^2)=0=\gs_1(x_0^2,\ldots,x_5^2)\}.
\end{equation}
Now one considers the natural 2-power map
\begin{eqnarray}\label{e130.2} m_2:{\Bbb P}^5 & \longrightarrow} \def\sura{\twoheadrightarrow & {\Bbb P}^5 \\
(x_0,\ldots,x_5) & \mapsto & (x_0^2,\ldots,x_5^2)=(u_0,\ldots,u_5)
\nonumber
\end{eqnarray}
and the image of $\ifmmode {\cal M} \else$\cM$\fi_{20}$ in ${\Bbb P}^5$. Comparing the equations
(\ref{e124.1}) and (\ref{e130.1}) we have
\begin{lemma}\label{l130.1} $m_2(\ifmmode {\cal M} \else$\cM$\fi_{20})=\ifmmode {\cal N} \else$\cN$\fi_5$.
\end{lemma}
The main results of \cite{BN} can be described as follows. First we define
a Zariski open subset $M^s\subset} \def\nni{\supset} \def\und{\underline \ifmmode {\cal M} \else$\cM$\fi_{20}$. The following 15 quadric surfaces
$q_{ij}\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^5$ actually lie on $\ifmmode {\cal M} \else$\cM$\fi_{20}$, as is easily verified:
\begin{equation}\label{e130a.1} q_{ij}=\{x_i=x_j=0=\sum_{m\neq i,j}x_m^2\}.
\end{equation}
Under the squaring map $m_2$ (\ref{e130.2}) the quadric $q_{ij}$ maps to
the plane
\begin{equation} N_{ij}=\{u_i=u_j=0=\sum_{m\neq i,j}u_m\};
\end{equation}
so the image of ${\bf Q}:=\cup_{i,j}q_{ij}$ is ${\bf N}:=\cup_{i,j}N_{ij}$,
and the planes $N_{ij}$ are the 15 planes of Lemma \ref{l125.1} (ii).
Furthermore the $N_{ij}$ are contained in the branch locus of
${m_2}_{|\ifmmode {\cal M} \else$\cM$\fi_{20}}:\ifmmode {\cal M} \else$\cM$\fi_{20}\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal N} \else$\cN$\fi_5$; this locus is {\em singular} on
$\ifmmode {\cal M} \else$\cM$\fi_{20}$ because $\ifmmode {\cal N} \else$\cN$\fi_5$ is tangent to $u_i=0$ and $u_j=0$ in all of
$N_{ij}$.
Next consider the inverse image under $m_2$ of the ten nodes; since the
nodes lie on {\em none} of the branch planes $u_i=0$, each node has
$\deg(m_2)=32$ inverse images, so $\ifmmode {\cal M} \else$\cM$\fi_{20}$ has 320 singular points
(clearly also nodes), which are the $\gS_6$-orbit, call it ${\bf P}$, of
the points
\begin{equation}\label{e130a.2} (\pm1:\pm1:\pm1:\pm i:\pm i: \pm i).
\end{equation}
Finally consider the inverse images of the 15 Segre planes of Lemma
\ref{l125.1} (i). This locus is given by the 15 equations which are the
$\gS_6$-orbit of
\begin{equation}\label{e130a.3} x_0^2+x_1^2=x_2^2+x_3^2=x_4^2+x_5^2=0.
\end{equation}
Inspection shows that this degree 8 surface on $\ifmmode {\cal M} \else$\cM$\fi_{20}$ splits into eight
planes, giving altogether 120=15.8 planes on $\ifmmode {\cal M} \else$\cM$\fi_{20}$; let ${\bf R}$
denote their union. Now define:
\begin{equation}\label{e130a.5} M^s:=\ifmmode {\cal M} \else$\cM$\fi_{20}-{\bf Q}-{\bf P}-{\bf N},\
\ifmmode {\cal N} \else$\cN$\fi_5^s:=m_2(M^s).
\end{equation}
Then the statement proved in \cite{BN} is
\begin{theorem}\label{t130.1} \begin{itemize}\item[a)] $M^s$ is isomorphic to
a Zariski open subset of the moduli space $\ifmmode {\cal A} \else$\cA$\fi_{(1,3)}(2)$ of abelian
surfaces with a $(1,3)$ polarisation and a level $2$ structure;
\item[b)] There is a double cover $p:\~{\ifmmode {\cal N} \else$\cN$\fi}\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal N} \else$\cN$\fi_5$ for which
$p^{-1}(\ifmmode {\cal N} \else$\cN$\fi_5^s)$ is isomorphic to a Zariski open set of the moduli
space $\ifmmode {\cal A} \else$\cA$\fi_{(2,6)}(2)$;
\item[c)] $\ifmmode {\cal N} \else$\cN$\fi_5^s$ is the moduli space of $PH_{2,2}$-invariant smooth
quartic surfaces containing $16$ skew lines.
\end{itemize}
\end{theorem}
Since the varieties $\ifmmode {\cal M} \else$\cM$\fi_{20}$, $\~{\ifmmode {\cal N} \else$\cN$\fi}$ and $\ifmmode {\cal N} \else$\cN$\fi_5$ are compactifications
of the Zariski open sets of (\ref{e130a.5}), we have the following:
\begin{corollary} \label{c130.1} There are birational equivalences:
$$\ifmmode {\cal M} \else$\cM$\fi_{20}- - \ra (\gG_{(1,3)}(2)\backslash \sieg_2)^*,\quad \~{\ifmmode {\cal N} \else$\cN$\fi}- - \ra
(\gG_{(2,6)}(2)\backslash \sieg_2)^*,\quad \ifmmode {\cal N} \else$\cN$\fi_5- - \ra (\gG\backslash \sieg_2)^*,$$ where
$ \gG_{(1,3)} \subset} \def\nni{\supset} \def\und{\underline \gG_{(2,6)} \subset} \def\nni{\supset} \def\und{\underline \gG,\
[\gG:\gG_{(2,6)}(2)]=2.$
\end{corollary}
As is shown in \cite{BN}, the map $\~{\ifmmode {\cal N} \else$\cN$\fi}\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal N} \else$\cN$\fi_5$ is given in the
following way. It just happens to turn out the {\em any} of the
$PH_{2,2}$-invariant quartics with 16 skew lines actually contains 32
lines, the first skew set of 16 and a second set of 16 skew lines. The
second set of sixteen is found as the image of the first set under the
involution
\begin{equation}\label{e130.3} (x_0:\ldots:x_5)\mapsto \left({-1 \over x_0}:
{1\over x_1}: \cdots :{1\over x_5}\right),
\end{equation}
which can be adjoined to the group $PH_{2,2}$ to form a group of order 32.
Altogether the 32 lines have the following properties.
\begin{equation}\label{e130.4} \begin{minipage}{14cm}\begin{itemize}
\item[a)] The 32 lines intersect in 32 points;
\item[b)] Each line contains ten of the 32 intersection points;
\item[c)] Each intersection point is contained in ten of the 32 lines.
\end{itemize}\end{minipage}\end{equation}
A configuration with the properties (\ref{e130.4}) is called a
$(32_{10})$-configuration.
{}From Nikulin's results just mentioned, it follows that the second set of 16
lines are also the images of blown-up torsion points on another abelian
surface, so there are {\em two} abelian surfaces with $(2,6)$ polarisation
and level $2$ structure giving rise to the {\em same} resolved Kummer
surface, i.e., the map is given by
\begin{eqnarray}\label{e130.5} \~{\ifmmode {\cal N} \else$\cN$\fi} & \longrightarrow} \def\sura{\twoheadrightarrow & {\cal N}_5 \\
(A_{\tau_1},A_{\tau_2}) & \mapsto & \-{(A_{\tau_1}/\{\pm 1\})}\ifmmode\ \cong\ \else$\isom$\fi
\-{(A_{\tau_2}/\{\pm 1\})},\nonumber
\end{eqnarray}
where the isomorphism permutes the two sets of 16 skew lines.
The next step is to identify the modular subvarieties on the arithmetic
quotients of Corollary \ref{c130.1}. From the structure of the periods we
know that in terms of abelian surfaces, these modular subvarieties
parameterise the abelian surfaces which split. These loci are described to
some extent in \cite{BN}.
\begin{theorem}\label{t131.1}\begin{itemize}\item[a)] Points on $\ifmmode {\cal N} \else$\cN$\fi_5$
parameterise smooth quartic surfaces unless they lie on one of the 30
planes of Lemma \ref{l125.1};
\item[b)] points on $\ifmmode {\cal N} \else$\cN$\fi_5$ parameterise quartic surfaces containing more
that 32 lines if an only if the corresponding abelian surfaces are
products. Furthermore, a line on a surface of this set of quartic
surfaces has coordinates in ${\Bbb P}^5$ which is in the $\gS_6$-orbit of
$$x_0^4(x_1^2+x_2^2)+x_1^4(x_2^2+x_0^2)+x_2^4(x_0^2+x_1^2)-6x_0^2x_1^2x_2^2
=0.$$
\end{itemize}
\end{theorem}
Unfortunately, these result do not allow us to explicitly describe the
relation between the compactification $\ifmmode {\cal M} \else$\cM$\fi_{20}$ and compactifications of
$\ifmmode {\cal A} \else$\cA$\fi_{(1,3)}(2)$, in particular the Baily-Borel embedding. This must be
considered an interesting open problem.
\subsection{A conjecture}
To end this section we make a conjecture on one of the birational models of
the variety $\ifmmode {\cal N} \else$\cN$\fi_5$. Consider the birational map $\ifmmode {\cal N} \else$\cN$\fi_5- - \ra \widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal N} \else$\cN$\fi}_5$
of (\ref{e127.2}). Recalling now the Janus-like isomorphism between the
Picard modular variety $\-{X}_{\gG_{\sqrt{-3}}(\sqrt{-3})}$ and the Siegel
modular variety $\-{X}_{\gG(2)}$ (see \cite{J}), it is natural to ask
about an analogue
here, since the involved Siegel modular varieties of Corollary \ref{c130.1}
all are related to level
2, albeit with different polarisations. So consider abelian fourfolds with
complex multiplication by $\rat(\sqrt{-3})$ of signature (3,1), with a level
$\sqrt{-3}$ structure, but with (1,1,1,3) polarisations.
\begin{problem}\label{p132.1} Is $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal N} \else$\cN$\fi}_5$ the Satake compactification of
$X_{(1,1,1,3)}(\sqrt{-3}):=\gG_{(1,1,1,3)}(\sqrt{-3})\backslash \ball_3$, where
$\gG_{(1,1,1,3)}(\sqrt{-3})$ denotes the arithmetic group giving
equivalence of complex multiplication by $\rat(\sqrt{-3})$, signature
$(3,1)$, with a level
$\sqrt{-3}$ structure and a $(1,1,1,3)$-polarisation?
\end{problem}
I conjecture that for {\em some} subgroup of $\gG_{(1,1,1,3)}(\sqrt{-3})$,
this does in fact hold.
Evidence for the conjecture:
\begin{itemize}\item[i)] The proper transforms of the 15 Segre planes are
by Proposition \ref{p116.1} the moduli space of principally polarised
abelian threefolds with complex multiplication by $\rat(\sqrt{-3})$,
signature (2,1), with a level
$\sqrt{-3}$ structure, (although these moduli spaces are blown up in three
points on $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal N} \else$\cN$\fi}_5$). These could parameterize
abelian fourfolds with said CM, signature (3,1) with a level
$\sqrt{-3}$ structure and polarisation $(1,1,1,3)$ which split:
$$A_4\ifmmode\ \cong\ \else$\isom$\fi A_3\times A_1,$$ where $A_3$ has CM, signature (2,1),
polarisation (1,1,1), and $A_1$ has CM, but a polarisation 3.
\item[ii)] The proper transforms of the 15 planes $N_{ij}$ of Lemma
\ref{l125.1}, (ii), are four nodal cubic surfaces (Lemma \ref{l127.3}).
These surfaces occur also on the ball quotient $\ifmmode {\cal S} \else$\cS$\fi_3$ above: pick any
four of the nodes which are not coplanar; they determine a unique ${\Bbb P}^3$
in ${\Bbb P}^4$, and its intersection with $\ifmmode {\cal S} \else$\cS$\fi_3$, a cubic surface, has four
nodes in the four nodes of $\ifmmode {\cal S} \else$\cS$\fi_3$ in that ${\Bbb P}^3$. (Note that there is a
unique four-nodal cubic surface, as it is ${\Bbb P}^2$ blown up in the six
intersection points of four (general) lines, a complete quadrilateral in
${\Bbb P}^2$, and any two such quadrilaterals are projectively equivalent.
This cubic surface is usually called the Cayley cubic, mentioned above.)
\item[iii)] The singular locus consists of isolated singular points,
resolved by quadric surfaces, so these singularities are rational. Recall
that at each $P_{ij}$, six of the 15 Segre planes meet. At each $Q_{ij}$
(the 15 points (\ref{e124.2})), three of the Segre planes and six of the
$N_{ij}$ of Lemma \ref{l125.1}, (ii) meet. In both cases, the quadric
${\Bbb P}^1\times {\Bbb P}^1$ can be covered equivariantly by a product
$E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}\times E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}$
of the elliptic curve $E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}$ with branching only at the intersection
with the proper transforms of the 30 planes above, as follows:
\begin{itemize}\item $P_{ij}$: $E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}\times E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^1\times
{\Bbb P}^1$ a Galois $\integer/3\integer$-quotient;
\item $Q_{ij}$: $E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}\times E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^1\times{\Bbb P}^1$ is the
product of two double covers branched at $0,1,\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta, \varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta^2$.
\end{itemize}
This supports by Lemma \ref{l115.1} the idea that this could be the
compactification locus of $X_{(1,1,1,3)}(\sqrt{-3})$.
\end{itemize}
\section{The Coble variety $\cal Y$}
\subsection{Arithmetic quotients of domains of type $\bf IV_n$}
Let $V$ be a $k$ vector space of dimension $n$, $k$ a totally real field,
and $b$ a bilinear symmetric form on $V$. Let $G(V,b)$ be the symmetry
group, and $G_{\rat}=Res_{k|\rat}G(V,b)$ the $\rat$-group it defines. We
assume that $G_{\rat}$ is of hermitian type, so that for every infinite prime
$\nu$ of $k$ the signature of
$b_{\nu}$ is $(n-2,2)$ or $b_{\nu}$ is definite.
$G_{\rat}$ is (absolutely) simple (defines an irreducible
domain) only if $k=\rat$ (or if $b_{\nu}$ is definite for all but a single
$\nu$, in which case the $\rat$-group is anisotropic, but we will not
consider this situation), and the corresponding real group gives rise to a
bounded symmetric domain only if $b$ has Witt index 2. This is the case we
consider here. The classification of such forms is well-known; since we
require the Witt index to be 2, two such forms $b$ and $b'$ are equivalent
over $\rat$ $\iff$ $det(b)=det(b')$, where $det(b)$ is to be viewed as an
element of $\rat^{\times}/(\rat^{\times})^2$.
Now let $\ifmmode {\cal L} \else$\cL$\fi\subset} \def\nni{\supset} \def\und{\underline V$ be a (maximal) lattice, and let $G_{\ifmmode {\cal L} \else$\cL$\fi}$ be the
arithmetic group it defines, $\gG\subset} \def\nni{\supset} \def\und{\underline G_{\ifmmode {\cal L} \else$\cL$\fi}$ a subgroup of finite index.
We first remark on the moduli interpretation of the arithmetic quotient
$\ifmmode {X_{\gG}} \else$\xg$\fi$.
\begin{proposition}\label{p106.1} $\ifmmode {X_{\gG}} \else$\xg$\fi$ is a moduli space of (pure) Hodge
structures of weight 2 on $V$ with $h^{2,0}=1$ (and $h^{1,1}=dim(\ifmmode {X_{\gG}} \else$\xg$\fi)$)
with respect to the lattice $\ifmmode {\cal L} \else$\cL$\fi\subset} \def\nni{\supset} \def\und{\underline V$.
\end{proposition}
{\bf Proof:} The symmetry group of such a Hodge structure is of real type
$SO(n-2,2)$, and $G(V,b)$ is a $\rat$-form in which $G_{\ifmmode {\cal L} \else$\cL$\fi}$ is an
arithmetic subgroup. Since the corresponding ``period'' (i.e., position of
the varying complex subspace $H^{1,1}$ in $H^2_{\komp}$) is clearly the same
exactly when the two periods differ by an element of $G_{\ifmmode {\cal L} \else$\cL$\fi}$, while $\gG$
defines a level structure of some kind, the result follows. \hfill $\Box$ \vskip0.25cm This
proposition is often used in the study of polarised K3-surfaces, which have
a pure Hodge structure of type (1,19,1). In fact, for each polarisation
degree (i.e., the number $C^2$ for the ample divisor $C$ on the K3-surface
which gives the projective embedding) $2e,\ (e\geq1)$ one has an arithmetic
group $\gG_e$ such that the arithmetic quotient $X_{\gG_e}$ is the moduli
space of K3 surfaces with the given polarisation. Recall the {\em Picard
number} $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta$ is the rank of the group of algebraic cycles, i.e.,
$\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta=rk_{\integer}H^2(S,\integer)\cap H^{1,1}$. Then one has the following.
\begin{proposition}\label{p106.2} Let $S$ be a K3 surface with $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta$= the
Picard number of $S$. Then the dimension of the moduli space of K3's
which are in the family preserving the lattice of algebraic cycles
$H^2(S,\integer)\cap H^{1,1}$ is 20-$\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta$.
\end{proposition}
{\bf Proof:} Recall that for a K3 surface $H^1(S,\gT)\ifmmode\ \cong\ \else$\isom$\fi H^1(S,\Omega} \def\go{\omega} \def\gm{\mu} \def\gn{\nu} \def\gr{\rho^1)$,
so $H^{1,1}(S)$ may be viewed as the tangent space of the local deformation
space, which should be thought of as a varying complex subspace of
$H^2(S,\komp)_{prim}$, while $H^2(S,\integer)$ is fixed. Let $\ifmmode \hbox{{\script A}} \else$\scA$\fi=H^2(S,\integer)\cap
H^{1,1}$ be the lattice of algebraic cycles, $\ifmmode \hbox{{\script T}} \else$\scT$\fi=H^2(S,\integer)\cap
(H^{2,0}(S)\oplus H^{0,2}(S))$ the lattice of transcendental cycles. We
have $rk_{\integer}\ifmmode \hbox{{\script A}} \else$\scA$\fi=rk_{\integer}(H^2(S,\integer))-rk_{\integer}\ifmmode \hbox{{\script T}} \else$\scT$\fi$ in general and
$rk_{\integer}\ifmmode \hbox{{\script A}} \else$\scA$\fi=\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta$ by assumption, so $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta=22-rk_{\integer}\ifmmode \hbox{{\script T}} \else$\scT$\fi$, while the
moduli space is defined by the group $G(V',b')$, where
$V'=\ifmmode \hbox{{\script A}} \else$\scA$\fi^{\perp}\otimes \rat$, since we are requiring $\ifmmode \hbox{{\script A}} \else$\scA$\fi$ to be
preserved. (Recall that for an algebraic cycle $\ifmmode {\cal C} \else$\cC$\fi$ the integral
$\int_{\ifmmode {\cal C} \else$\cC$\fi}\go$ over the holomorphic two-form $\go$ vanishes, hence the
algebraic cycles contribute nothing to the periods). Thus
$G(\fR)=SO(20-\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta,2)$, giving rise to a domain of type $\bf IV_{20-\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}$.
\hfill $\Box$ \vskip0.25cm Of course in this particular case, the lattice $\ifmmode {\cal L} \else$\cL$\fi\subset} \def\nni{\supset} \def\und{\underline V$ is very
special; the ``intersection form'' $b$ restricted to $\ifmmode {\cal L} \else$\cL$\fi$ is even and
unimodular, and as is well-known, decomposes as
\begin{equation}\label{e106.1} \ifmmode {\cal L} \else$\cL$\fi\ifmmode\ \cong\ \else$\isom$\fi <-2e>\oplus {\bf H}^2 \oplus {\bf
E_8}^2,
\end{equation}
where ${\bf H}$ is the two-dimensional hyperbolic lattice, and ${\bf E_8}$
is the root lattice of type ${\bf E_8}$. Let us remark that the
compactification of these arithmetic quotients has been carried out in the
thesis \cite{scat}, but we will not need this. We will very quickly
describe a particulary interesting family of K3 surfaces which has been
thoroughly studied by Yoshida and his collaborators, see \cite{MSY} for
details on all matters here.
\subsection{A four-dimensional family of K3's}\label{s106a.1}
The family of K3 surfaces to be described here is the set of surfaces which
are double covers of ${\Bbb P}^2$ branched along the union of six disjoint
lines. Recall that there is a 19-dimensional family of K3 surfaces which
are double covers of the plane branched along a sextic curve; they are
smooth as long as the sextic is smooth, and generically have Picard
number $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta=1$. An arrangement of six lines in ${\Bbb P}^2$ is a maximally singular
sextic; there are 15 intersection points of the six lines (if they are in
general position), and each such gives rise to an $A_1$-singularity on the
double cover. Resolving the 15 double points introduces 15 exceptional
curves with self intersection number $-2$, so together with the pullback of
the generic line, this gives 16 independent cycles on the surface:
$\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta=16$. Hence the transcendental lattice $\ifmmode \hbox{{\script T}} \else$\scT$\fi$ has rank 4, so by
Proposition \ref{p106.2}, the moduli space is four-dimensional. Let
\begin{equation}\label{e106a.1} \gG=\{g\in G(\ifmmode \hbox{{\script T}} \else$\scT$\fi_{\fR},Q)(\ifmmode\ \cong\ \else$\isom$\fi
SO(4,2))\Big| g(\ifmmode \hbox{{\script T}} \else$\scT$\fi)\subset} \def\nni{\supset} \def\und{\underline \ifmmode \hbox{{\script T}} \else$\scT$\fi\},
\end{equation}
where $Q$ is the intersection form on $H^2(S,\integer)$, extended to $\fR$, then
restricted to $\ifmmode \hbox{{\script T}} \else$\scT$\fi_{\fR}$. This is clearly an arithmetic subgroup, and by
Proposition \ref{p106.1}, the arithmetic quotient $\ifmmode {\xg = \gG\bs\cD} \else$\xgeq$\fi$ is the
four-dimensional moduli space. We list some of the interesting loci for
this family. Let $L$ be the given arrangement, $L=l_1\cup \ldots \cup l_6$,
and let $t_p:=$ the number of $p$-fold points of the arrangement, i.e., the
number of points at which $p$ of the lines meet (see (\ref{e109.1})), and
let $\pi:S\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^2$ denote the (singular) double cover.
\subsubsection{Three-dimensional loci}
\begin{itemize}
\item[1)] Suppose there is a conic which is {\em tangent} to all six lines.
Then the inverse image of this quadric is a ${\Bbb P}^1$, which, as is easily
checked, has self-intersection number $4-6=-2$, so the double cover has
16 exceptional cycles, hence $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta=17$. It is in fact easy to see that
the surface $S$ is in this case a classical Kummer surface, i.e., a
quartic surface in ${\Bbb P}^3$ with 16 nodes which is the Kummer variety of a
principally polarised abelian surface $A_S$. The projection from a node
gives the double cover $\pi:S\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^2$, and the tangent conic is the
image of the (blown up) node used to project. The abelian surface is the
Jacobian of a genus 2 curve, and this curve is the double cover of the
conic, {\em branched at the six points of tangency}. This is well-known.
\item[2)] If $t_3=1,\ t_2=12$, then the threefold point induces an
$A_2$-singularity on the double cover which is resolved by two ${\Bbb P}^1$'s,
so there are now 2+12 exceptional ${\Bbb P}^1$'s and the hyperplane section.
We have the following picture. \\ \fbox{
\begin{minipage}{5cm}
\unitlength1cm
\begin{picture}(5,4)(0.5,0.3)
\put(1,3){\line(2,-1){4}} \put(2,2.5){\circle*{.2}}
\put(1,2.5){\line(1,0){4}} \put(1,2){\line(2,1){4}}
\put(1,4){\line(1,-1){3.5}} \put(2.5,4){\line(1,-3){1.2}}
\put(1.7,.5){\line(1,2){1.75}}
\end{picture}
\end{minipage}}
\begin{minipage}{10cm} There are in fact three more exceptional ${\Bbb P}^1$'s,
which are the inverse image on the double cover of the three lines which
pass through the triple point and one of the three double points not
lying on a line through the triple point. It is easy to see that these
three double points are independent parameters of such arrangements, so
this defines a three-dimensional family, so by Proposition \ref{p106.2},
we have $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta=17$ for the generic member of this family.
\end{minipage}
\end{itemize}
\subsubsection{Two-dimensional loci}
\begin{itemize}
\item[3)]
\fbox{
\begin{minipage}{5cm}
\unitlength1cm
\begin{picture}(5,4)(0.5,0.3)
\put(1,2.5){\line(2,-1){4}} \put(2,2){\circle*{.2}}
\put(1,2){\line(1,0){4}} \put(1,1.5){\line(2,1){4}}
\put(2.5,.5){\line(1,2){1.75}} \put(3.35,2.3){\circle*{.2}}
\put(1,3.5){\line(2,-1){4}} \put(2.55,4){\line(1,-2){1.75}}
\end{picture}
\end{minipage}}
\begin{minipage}{10cm}If $t_3=2, t_2=9$, there are two possibilities.
Suppose first that the two threefold points do {\em not} lie on one of
the six lines. Then we have the picture to the left. This gives rise to
two isolated $A_2-$singularities. The inverse image of the line joining
the two threefold points is also an exceptional ${\Bbb P}^1$. In this case the
generic double cover has $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta=18$, and as parameters one can take two
double ratios: consider two of the lines $l_1, l_2$, both passing through one
of the threefold points $p$; the three intersection points with the other
lines, together with $p$, give four points on each line -- hence two
double ratios.
\end{minipage}
\item[4)] It may also occur that both threefold points lie on a line, but
in this case we also have $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta=18$, i.e., a two-dimensional family.
\end{itemize}
\subsubsection{One-dimensional loci}
\begin{itemize}
\item[5)] If $t_4=1$, then the double cover has an {\em elliptic}
singularity over the point, so is not K3. Hence this is a genuine {\em
degeneration} of the K3, i.e., belongs to the boundary of the
compactification. It turns out that then a line must also be double, so
that the double cover has two components.
\item[6)] As a further specialisation of 4) it may happen that there are
three triple points. Since four of the lines may be choosen fixed (for
example $x_0=x_1=x_2=0,\ x_0-x_1=0$), there is only one modulus, given
for example by the intersection point of the two variable lines. Here we
have $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta=19$.
\end{itemize}
\subsubsection{Zero-dimensional loci}
\begin{itemize}
\item[7)] If three lines are taken, each {\em double}, then the double
cover splits into two copies of ${\Bbb P}^2$. This is in the closure of the set
of degenerations of type 5).
\item[8)] The arrangement is the complete quadrilateral. The picture is: \\
\fbox{
\begin{minipage}{6.5cm}
\unitlength.8cm
\begin{picture}(8,5)(1.5,0)
\thinlines \put(2,1.5){\line(1,0){6.5}} \put(2,1.5){\line(4,1){5}}
\put(2,1.5){\line(6,5){3.5}}
\put(2,1.5){\line(-1,0){0.5}} \put(2,1.5){\line(-4,-1){0.5}}
\put(2,1.5){\line(-6,-5){0.5}}
\put(5,.75){\line(0,1){4}} \put(5,4){\line(6,-5){3.5}}
\put(5,4){\line(-6,5){0.5}} \put(8,1.5){\line(-4,1){5}}
\put(8,1.5){\line(4,-1){0.5}}
\end{picture}
\end{minipage}}
\begin{minipage}{9cm}It is known that the {\em Fermat} cover (not the double
cover) of this arrangement is Shioda's elliptic modular surface of level
4, $S(4)$, so it follows that the double cover is isogenous to $S(4)$,
i.e., a quotient of $S(4)$ by a group isomorphic to $(\integer/2\integer)^4$. This
is the most special K3 surface in the family and has $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta=20$.
\end{minipage}
\end{itemize}
\subsubsection{Level 2 structure}
Now consider, in addition to the above data, a level 2 structure.
Geometrically this amounts to fixing an {\em order} of the six lines. In
terms of the lattice $\ifmmode \hbox{{\script T}} \else$\scT$\fi$ it is not so easy to see what it means. In
\cite{MSY} it is shown by explicit computation that the subgroup $\gG(2)$
is the group generated by reflections on the ``roots'' of $\ifmmode \hbox{{\script T}} \else$\scT$\fi$, that is
the integral elements of norm $-2$. Furthermore it is shown there that
$\gG$ is generated by the reflections on the elements of norm $-2$ or $-4$,
and that $\gG/\gG(2)\ifmmode\ \cong\ \else$\isom$\fi \gS_6\times \integer/(2)$. Hence by the results 2.7.1,
2.7.7, 2.8.2 of \cite{MSY} we have
\begin{proposition}\label{p106b.1} The arithmetic quotient $\gG(2)\backslash \ifmmode {\cal D} \else$\cD$\fi$ is
the moduli space for K3 surfaces which are double covers of ${\Bbb P}^2$,
branched over an {\em ordered} set of six lines.
\end{proposition}
\begin{table}
\caption{\label{table18} Loci of a four-dimensional family of K3 surfaces }
\vspace*{.5cm}
\begin{minipage}{16.5cm}
\hspace*{2.5cm}\fbox{\begin{minipage}{5cm} Locus 1)
Igusa quartic \unitlength1cm
\begin{picture}(2,2)(0,.2)
\put(0,1){\circle{.2}} \put(1,0.5){\circle{.2}} \put(2,1){\circle{.2}}
\put(0,2){\circle{.2}} \put(1,2.5){\circle{.2}} \put(2,2){\circle{.2}}
\bezier{100}(0,1)(-.2,1.5)(0,2) \bezier{100}(0,2)(0.5,2.5)(1,2.5)
\bezier{100}(1,2.5)(1.5,2.5)(2,2) \bezier{100}(2,2)(2.3,1.5)(2,1)
\bezier{100}(2,1)(1.5,0.45)(1,.5) \bezier{100}(1,.5)(0.5,0.45)(0,1)
\end{picture}
{\hspace*{\fill} \fbox{1} \hfill}
\end{minipage}}
\hspace*{\fill} \fbox{\begin{minipage}{5cm} Locus 2)
$X^{\{ijk\}}$ \unitlength1cm
\begin{picture}(2,2)(0,.2)
\put(1,1){\circle{.2}} \put(2,1.5){\circle{.2}} \put(1,2){\circle{.2}}
\put(3,1.8){\circle{.2}} \put(1,1){\circle{.2}}
\put(1.1,1){\line(1,0){.9}} \put(2,1){\circle{.2}}
\put(2.1,1){\line(1,0){.9}} \put(3,1){\circle{.2}}
\end{picture}
{\hspace*{\fill} \fbox{20} \hfill}
\end{minipage}} \hspace{\fill}
\begin{minipage}{15cm}\unitlength1cm \begin{picture}(15,2)
\put(5,.7){\vector(-1,1){.8}} \put(8,.7){\vector(1,1){.8}}
\put(13,.7){\vector(-1,1){.8}}
\end{picture}
\end{minipage}
\hspace*{3cm} \fbox{\begin{minipage}{5cm} Locus 3)
$X^{\{ijk;lmn\}}$ \unitlength1cm
\begin{picture}(2,2)(.2,.3)
\put(0,1){\put(1,1){\circle{.2}} \put(1.1,1){\line(1,0){.9}}
\put(2,1){\circle{.2}} \put(2.1,1){\line(1,0){.9}}
\put(3,1){\circle{.2}} } \put(1,1){\circle{.2}}
\put(1.1,1){\line(1,0){.9}} \put(2,1){\circle{.2}}
\put(2.1,1){\line(1,0){.9}} \put(3,1){\circle{.2}}
\end{picture}
{\hspace*{\fill} \fbox{10} \hfill}
\end{minipage}}
\hspace{\fill} \fbox{\begin{minipage}{5cm} Locus 4)
$X^{\{ijk;imn\}}$ \unitlength1cm
\begin{picture}(2,2)
\put(.5,1){\circle{.2}} \put(.6,1.1){\line(2,1){.9}}
\put(.6,1.1){\line(2,-1){.9}} \put(1.5,.5){\circle{.2}}
\put(1.5,1.5){\circle{.2}} \put(1.6,.6){\line(2,-1){.9}}
\put(1.6,1.6){\line(2,1){.9}} \put(2.5,1.4){\circle{.2}}
\put(2.5,0){\circle{.2}} \put(2.5,2){\circle{.2}}
\end{picture}
{\hspace*{\fill} \fbox{90} \hfill}
\end{minipage}}
\begin{minipage}{15cm} \unitlength1cm \begin{picture}(15,2)
\put(5,.7){\vector(0,1){.8}} \put(9,.7){\vector(1,1){.8}}
\put(13,.7){\vector(0,1){.8}}
\end{picture}
\end{minipage}
\hspace*{3cm} \fbox{\begin{minipage}{5cm} Locus 5)
$X^{\{ij\}}$ \unitlength1cm
\begin{picture}(2,2)
\put(.5,1){\circle{.2}}\put(.6,1){\line(1,0){.9}}
\put(1.5,1){\circle{.2}}\put(1.6,1){\line(1,0){.9}}
\put(2.5,1){\circle{.2}}\put(2.6,1){\line(1,0){.9}}
\put(2,2){\circle{.3}}\put(2,2){\circle*{.2}}
\end{picture}
{\hspace*{\fill} \fbox{15} \hfill}
\end{minipage}}
\hspace{\fill} \fbox{\begin{minipage}{5cm} Locus 6)
$X^{\{ijk;klm;mni\}}$ \unitlength1cm
\begin{picture}(2,2)(0,.2)
\put(.5,.5){\circle{.2}}\put(.6,.5){\line(1,0){.9}}
\put(.5,.5){\line(1,2){.4}}
\put(1.5,.5){\circle{.2}}\put(1.6,.5){\line(1,0){.9}}
\put(1,1.5){\circle{.2}}\put(1,1.5){\line(1,2){.4}}
\put(1.5,2.5){\circle{.2}}\put(1.5,2.5){\line(1,-2){.4}}
\put(2,1.5){\circle{.2}} \put(2,1.5){\line(1,-2){.4}}
\put(2.5,.5){\circle{.2}}
\end{picture}
{\hspace*{\fill} \fbox{120} \hfill}
\end{minipage}}
\begin{minipage}{15cm} \unitlength1cm \begin{picture}(15,2)
\put(5,.7){\vector(0,1){.8}} \put(9,.7){\vector(1,1){.8}}
\put(13,.7){\vector(0,1){.8}}
\end{picture}
\end{minipage}
\hspace*{3cm} \fbox{\begin{minipage}{5cm} Locus 7)
$X^{\{ijk;kl;mn\}}$ \unitlength1cm
\begin{picture}(2,2)
\put(1,1){\circle{.3}}\put(1,1){\circle*{.2}}
\put(2,1){\circle{.3}}\put(2,1){\circle*{.2}}
\put(1.5,2){\circle{.3}}\put(1.5,2){\circle*{.2}}
\end{picture}
{\hspace*{\fill} \fbox{15} \hfill}
\end{minipage}}
\hspace{\fill} \fbox{\begin{minipage}{5cm} Locus 8)
$X^{\{ijk;klm;mni;jln\}}$ \unitlength1cm
\begin{picture}(2,2)(.9,.3)
\put(.5,1.5){\circle{.2}}\put(1.6,.9){\circle{.2}}\put(1.6,2.1){\circle{.2}}
\put(2,1.5){\circle{.2}}\put(2.7,.3){\circle{.2}}\put(2.7,2.7){\circle{.2}}
\put(0,1.25){\line(2,1){3}}\put(0,1.75){\line(2,-1){3}}
\put(1.33,.5){\line(2,3){1.5}}\put(1.33,2.5){\line(2,-3){1.5}}
\end{picture}
{\hspace*{\fill} \fbox{30} \hfill}
\end{minipage}}
\end{minipage}
\vspace*{.5cm} {\small The notations $X^{\{ijk;klm;mni;jln\}}$, etc, are
taken from \cite{MSY}; the arrows indicate inclusions among the various
loci. \hfill\newline
\vspace*{-1.2cm}The symbol \unitlength1cm
\begin{picture}(.15,.1)\put(0.2,0.1){\circle{.3}}\put(0.2,0.1){\circle*{.2}}
\end{picture} \quad means a double line. The number of each kind of loci is
indicated by \fbox{x}; the dimensions are three in the top row down to zero
in the last row. Locus 1) is where the six points lie on a conic, while the
20 $X^{\{ijk\}}$ are the loci where there are three of the six points on a
line. The 15 $X^{\{ijk;kl;mn\}}$ lie on the boundary of the moduli space,
while the 30 $X^{\{ijk;klm;mni;jln\}}$ lie in ``the farthest interior'' of
the domain. A more complete description is given in Corollary
\ref{c133.1}.}
\end{table}
We refer the reader to \cite{MSY} for a detailed description of the loci
described above, of the periods and of the corresponding Picard-Fuchs
equations (and much more). We give in Table \ref{table18} a description of
the loci, giving the dual graph of the six lines (i.e., a vertex for each
line, two vertices lying on a line $\iff$ the corresponding lines meet), as
well as the number of loci, and the names given to them in \cite{MSY}.
We now give an explicit projective description of the
Baily-Borel compactification of the arithemetic quotient $\gG(2)\backslash \ifmmode {\cal D} \else$\cD$\fi$ of
Proposition \ref{p106b.1}. All the facts presented here were proved
originally by Coble \cite{C} or by Yoshida and his collaborators in
\cite{MSY}. We have the four-dimensional family of K3 surfaces just
discussed, defined in terms of a set of (ordered) six lines in
the plane. Dual to the six lines are six points, and so the relation with
the moduli space of cubic surfaces is evident. Let two ordered sets of six
lines, $(l_1,\ldots,l_6),\ (l_1',\ldots,l_6')$ be given.
\begin{definition}\label{d133.0} The two sets of lines $(l_1,\ldots,l_6),\
(l_1',\ldots,l_6')$ are said to be {\em associated}, if the following
relation holds. Since the set $(l_1,\ldots,l_6)$ is ordered, we can form
two triangles, $$\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi(l_1,l_2,l_3),\quad \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi(l_4,l_5,l_6);$$ these two
triangles have together six vertices, which come equipped with a
numbering, say $(p_1,\ldots,p_6)$, and these correspond dually to another
ordered set of six lines, $(l_{p_1},\ldots,l_{p_6})$. Then
$(l_1,\ldots,l_6)$ and $(l_1',\ldots,l_6')$ are associated, if:
$(l_{p_1},\ldots,l_{p_6})=(l_1',\ldots,l_6')$, as a set of six ordered
lines.
\end{definition}
Of course, starting with two sets of ordered six points, one can define in
the same way the notion of association. Since, as abstract moduli spaces,
the space of ordered sets of six lines is the ``same'' (by duality) as the
set of ordered sets of six points, we see that we are dealing here with the
space of sets of six ordered points in ${\Bbb P}^2$. This problem was dealt with
in the papers of Coble \cite{C}, and has been given a modern treatment in
\cite{DO}. It can be described as follows. The relevant moduli space is
easy to describe: let $(p_1,...,p_n)$ be a set of $n$ points in ${\Bbb P}^k$;
this is represented by $M$, the $n\times (k+1)$ matrix whose $i^{th}$
column gives the coordinates of the point $p_i$. The moduli space is then
the GIT quotient
\begin{equation}\label{e133.2}
\hbox{{\helv P}$^k_n$\ } \def\helva{\hbox{\helv A}} = GL(k+1) \backslash M(n,k+1)/(\komp^*)^n.
\end{equation}
By taking the set of semistable points in $M(n,k+1)$ the above quotient is
compact, although singular. It is classical that \ifmmode \hbox{\helv P}^1_6 \else $\pos$\fi\ is a threefold whose
compactification can be identified with a cubic threefold in ${\Bbb P}^4$ with
ten ordinary double points, which is just the Segre cubic $\ifmmode {\cal S} \else$\cS$\fi_3$. Note
that the similar moduli problem, namely six points on a conic in ${\Bbb P}^2$,
is realised by the Igusa quartic $\ifmmode {\cal I} \else$\cI$\fi_4$, so these are very closely
related, but not identical moduli problems.
Our interest here is in \pts, a fourfold. In this case we may represent
elements by matrices
\begin{equation}\label{e133.3}
\hbox{{\helv P}$^k_n$\ } \def\helva{\hbox{\helv A}} \ni M = \left[ \begin{array}{c c c c c c} 1 & 0 & 0 & 1 & x &
w \\ 0 & 1 & 0 & 1 & y & z \\ 0 & 0 & 1 & 1 & u & u \\
\end{array} \right] ,
\end{equation}
and as Coble shows, the map \pts $\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4$, $M \mapsto [x:y:w:z:u]$ is a
birational map (it is clear that $\pts$ is rational, this map simply gives
an explicit birationalisation). The GIT theory here consists of finding
$G$-invariant functions on \pts, and these turn out to be generated by
$3\times 3$ minors of $M$.
In terms of the matrix $M$ the process of association can be described as
follows. Each such matrix $M$ determines a second one: since the six points
are ordered, one can define six lines by $l_{12}=\overline{p_1p_2}$,
$l_{13}=\overline{p_1p_3}$, $l_{23}=\overline{p_2p_3}$, $l_{45}=\overline{p_4p_5}$,
$l_{46}=\overline{p_4p_6}$, $l_{56}=\overline{p_5p_6}$; these six lines determine
dually six points, whose coordinates are then brought into the normal form
given above. It turns out that the entries of the second matrix are
determined by the fact that the maximal minors are proportional to the
maximal minors of the first. More precisely, if we let $(ijk)$ denote the
$3\times 3$ minor of $M$ which is given by the columns $i,j,k$, and if we
let $M'$ be the associated matrix, $(ijk)'$ the corresponding minor, then
the minors of $M$ and $M'$ are related by:
\begin{equation}\label{e133.1} (123)(145)(246)(356)=(124)'(135)'(236)'(456)'.
\end{equation}
Now association is an involution on $\pts$, and one can take the {\em
quotient} by this involution.
\begin{definition}\label{d133.1} Let $\ifmmode {\cal Y} \else$\cY$\fi$ be the double cover of ${\Bbb P}^4$
branched along the Igusa quartic $\ifmmode {\cal I} \else$\cI$\fi_4$, $\pi:\ifmmode {\cal Y} \else$\cY$\fi\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4$.
\end{definition}
Clearly $\ifmmode {\cal Y} \else$\cY$\fi$ will be singular precisely along the singular locus of
$\ifmmode {\cal I} \else$\cI$\fi_4$, i.e.,
\begin{lemma}\label{l133.1} The singular locus of $\ifmmode {\cal Y} \else$\cY$\fi$ consists of 15 lines,
the inverse images of the 15 singular lines of $\ifmmode {\cal I} \else$\cI$\fi_4$.
\end{lemma}
\begin{theorem}[\cite{DO},Example 4, p.~37]\label{t133.0}
The moduli space of six ordered points in ${\Bbb P}^2$ is equal to the double
cover $\ifmmode {\cal Y} \else$\cY$\fi$, and the double cover involution on $\ifmmode {\cal Y} \else$\cY$\fi$ coincides with the
association involution on $\pts$.
\end{theorem}
In other words, a set $(p_1,\ldots,p_6)$ is {\em associated to itself}, if
and only if the six points lie on a conic in ${\Bbb P}^2$.
Consider one of the hyperplanes $H$ in ${\Bbb P}^4$, $H=\{\gt_m^4=0\}$ of
Proposition \ref{p120.1}. Since $H$ is {\em tangent} to $\ifmmode {\cal I} \else$\cI$\fi_4$, the
inverse image $\pi^{-1}(H)$ in $\ifmmode {\cal Y} \else$\cY$\fi$ will {\em split into two copies of
${\Bbb P}^3$}. In this way, we get a union of 20 ${\Bbb P}^3$'s on $\ifmmode {\cal Y} \else$\cY$\fi$,
\begin{lemma}\label{l133.2} The inverse images $\pi^{-1}(H)$ of the tangent
hyperplanes $H=\{\gt_m^4=0\}$ consist of two copies each of ${\Bbb P}^3$, and
these two ${\Bbb P}^3$'s on $\ifmmode {\cal Y} \else$\cY$\fi$ meet in the quadric surface which is the
inverse image under $\pi$ of the quadric on $\ifmmode {\cal I} \else$\cI$\fi_4$ to which $H$ is
tangent. This gives a total of 20 such ${\Bbb P}^3$'s on $\ifmmode {\cal Y} \else$\cY$\fi$.
\end{lemma}
A resolution of singularities of $\pts$ is affected by resolving the Igusa
quartic by blowing up the ideal of the 15 lines; this is the map $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_2$
of Theorem \ref{t123.1}. Let $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\hbox{\pts}}$ denote this
desingularisation $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\hbox{\pts}} \longrightarrow} \def\sura{\twoheadrightarrow \hbox{\pts}$. On
$\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\hbox{\pts}}$ we have a set of 36 divisors, the {\em discriminant
locus}, the proper transforms of the Igusa quartic, the 20 ${\Bbb P}^3$'s and
the 15 exceptional divisors of the blow up.
It is clear how this variety is the moduli space of cubic surfaces: blow up
${\Bbb P}^2$ in the six points, and embed by the linear system of cubic curves
through the six points. The ordering of the six points of course determines
a marking of the 27 lines in the well-known manner. The symmetry group of
$\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\hbox{\pts}}$ is $\gS_6\times \integer/2\integer$; although the Weyl group
$W(E_6)$ acts birationally on it, the action is not regular. For that it is
neccessary to modify $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\hbox{\pts}}$ even more. Dolgachev mentions in
\cite{DO} that he suspects it is sufficient to blow up $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\hbox{\pts}}$
in the intersection of the 36 divisors.
One of the many things proved in \cite{MSY} is the following.
\begin{theorem}\label{t133.1} The variety $\ifmmode {\cal Y} \else$\cY$\fi$ is the Baily-Borel
compactification of the arithmetic quotient $\gG(2)\backslash \ifmmode {\cal D} \else$\cD$\fi$ of
Proposition \ref{p106.1}.
\end{theorem}
The proof given in \cite{MSY} of this fact simply (!) calculates the image
of the period map, and in determining when the periods lie on the boundary
of the period domain $\ifmmode {\cal D} \else$\cD$\fi$, the authors find that this locus
coincides with the set of
K3 surfaces whose set of six lines correspond to those singularities of
Lemma \ref{l133.1} of $\ifmmode {\cal Y} \else$\cY$\fi$.
\begin{corollary}\label{c133.1} The Loci 5) and 7) of Table \ref{table18}
are the inverse images on $\ifmmode {\cal Y} \else$\cY$\fi$ of the 15 singular lines and 15 singular
points, respectively, of the branch locus $\ifmmode {\cal I} \else$\cI$\fi_4$. The Loci 3) of Table
18 are the inverse images of the ten special hyperplane sections of Lemma
\ref{l133.2}, i.e., the quadrics. The loci 2) of Table 18 are the 20
${\Bbb P}^3$'s of Lemma \ref{l133.2}, and Locus 1) is just the branch locus of
the double cover.
\end{corollary}
\part{A Gem of the modular universe}\label{chapter13}
\renewcommand{\arraystretch}{1}
\section{The Weyl group $W(E_6)$}
\subsection{Notations}
We use the same notation as above for the 27 lines on a cubic surface in
${\Bbb P}^3$: $a_1,...a_6,\ b_1,...,b_6,\ c_{12},...,c_{56}$. The 36 double
sixes are:
$$N= \left[ \matrix{ a_1 & a_2 & a_3 & a_4 & a_5 & a_6 \cr b_1 & b_2 & b_3
& b_4 & b_5 & b_6 \cr } \right], \hspace{1cm} (1) $$
$$N_{ij}=\left[ \matrix{ a_i & b_i & c_{jk} & c_{jl} & c_{jm} & c_{jn} \cr
a_j & b_j & c_{ik} & c_{il} & c_{im} & c_{in} \cr } \right],
\hspace{1cm} (15)$$
$$N_{lmn}\footnote{here we switch notations from $N_{ijk}$ in equation
(\ref{eB2.2}) to $N_{lmn}$ for convenience} =\left[ \matrix{ a_i & a_j &
a_k & c_{mn} & c_{ln} & c_{lm} \cr c_{jk} & c_{ik} & c_{ij} & b_l & b_m
& b_n \cr} \right] \hspace{1cm} (20). $$ The 45 tritangents
are:\begin{equation}\label{eQ1.1}\begin{minipage}{6cm}
\begin{center}
$(ij)=<a_i\ b_j\ c_{ij}>,\ \ i\neq j \ \ (30)$
$(ij.kl.mn)=<c_{ij}\ c_{kl}\ c_{mn}>\ \ (15).$
\end{center}
\end{minipage}
\end{equation}
Two double sixes are {\em syzygetic} it they contain four lines in common,
for example: $$N\ \ \hbox{and} \ \ N_{12}\ \ \hbox{have } a_1,\ a_2,\ b_1,\
b_2\ \hbox{in common},$$ and {\em azygetic} if they have six lines in
common, for example: $$N\ \ \hbox{and} \ \ N_{456}\ \ \hbox{have } a_1,\
a_2,\ a_3,\ \ b_4,\ b_5,\ b_6\ \hbox{in common}.$$ Two azygetic double
sixes have six lines in common and contain 12 other lines; these 12 lines
form another double six, azygetic with respect to both, for example $N,\
N_{123},\ N_{456}$. Such triples are refered to as triples of azygetic
double sixes or, because of the interpretation in terms of tritangents, a
trihedral pair. Each double six is syzygetic to 15 others, forming 270 such
pairs, and azygetic to 20 others, forming 120 triples. Our notation for
the 120 triples are:
\begin{equation}\label{eQ1.2}
\begin{array}{rclr}
\{ijk\} & = & <N,N_{ijk},N_{lmn}>, & (10) \\ \{ij.jk\} & = &
<N_{ij},N_{ik},N_{jk}>, & (20) \\ \{ij.kl\} & = &
<N_{ij},N_{ikl},N_{jkl}> & (90).\\
\end{array}
\end{equation}
We recognize these as the trihedral pairs of (\ref{eB2.1}) under the
correspondence
$$\left[\begin{array}{ccc} a_i & b_j & c_{ij} \\ b_k & c_{jk} & a_j \\
c_{ik} & a_k & b_i \end{array}\right] \longleftrightarrow} \def\ra{\rightarrow} \def\Ra{\Rightarrow
<N_{ij},N_{ik},N_{jk}>,\quad \left[\begin{array}{ccc} c_{il} & c_{jm} &
c_{kn} \\ c_{mn} & c_{ik} & c_{jl} \\ c_{jk} & c_{ln} & c_{im}
\end{array}\right] \longleftrightarrow} \def\ra{\rightarrow} \def\Ra{\Rightarrow <N,N_{ijk},N_{lmn}>, $$
$$\left[\begin{array}{ccc} a_i & b_j & c_{ij} \\ b_l & a_k & c_{kl} \\
c_{il} & c_{jk} & c_{mn} \end{array}\right] \longleftrightarrow} \def\ra{\rightarrow} \def\Ra{\Rightarrow
<N_{ij},N_{ikl},N_{jkl}>.$$ Hence the triads of trihedral pairs discussed
there are expressed in condensed form as follows:
\begin{equation}\label{eQ1.3}\begin{minipage}{14cm}
$$ [ijk.lmn]=\left[ \matrix{ N_{ij} & N_{jk} & N_{ik} \cr N_{lm} &
N_{mn} & N_{ln} \cr N & N_{ijk} & N_{lmn} \cr }
\right],\hspace{1cm} (10)$$
$$ [ij.kl.mn]=\left[ \matrix{ N_{ij} & N_{ikl} & N_{jkl} \cr N_{kl} &
N_{kmn} & N_{lmn} \cr N_{mn} & N_{nij} & N_{mij} \cr } \right],
\hspace{1cm} (30).$$
\end{minipage}
\end{equation}
The group of incidence preserving permutations of the 27 lines, a group of
order 51840, can be generated by the following six operations:
$$(i,i+1),\ i=1,...,5:\hbox{transposition of the indices},$$ and
$$ (123) :\hbox{map } N \mapsto N_{123},$$ and the graph of this
presentation is shown in Figure \ref{Figure4}.
\begin{figure}[hbt]
$$ \unitlength0.8cm
\begin{picture}(10.5,3.5)
\put(0.25,3.0){\circle*{0.5}} \put(2.75,3.0){\circle*{0.5}}
\put(5.25,3.0){\circle*{0.5}} \put(7.75,3.0){\circle*{0.5}}
\put(10.25,3.0){\circle*{0.5}} \put(5.25,0.5){\circle*{0.5}}
\linethickness{0.4mm}
\put(0.25,3.0){\line(1,0){2.5}} \put(2.75,3.0){\line(1,0){2.5}}
\put(5.25,3.0){\line(1,0){2.5}} \put(7.75,3.0){\line(1,0){2.5}}
\put(5.25,0.5){\line(0,1){2.5}}
\put(0.25,3.6){\makebox(0,0){(12)}} \put(2.75,3.6){\makebox(0,0){(23)}}
\put(5.25,3.6){\makebox(0,0){(34)}} \put(7.75,3.6){\makebox(0,0){(45)}}
\put(10.25,3.6){\makebox(0,0){(56)}}
\put(5.25,-0.1){\makebox(0,0){(123)}}
\end{picture}
$$
\caption[The graph of the group of the 27 lines]{\label{Figure4}\small
The graph of the group of the permutations of the 27 lines}
\end{figure}
This is the graph whose vertices correspond to generators, two vertices A,
B being connected if ABA=BAB and not connected if AB=BA.
\subsection{Roots}
Let $\tt$ be a maximal abelian subalgebra of the compact Lie algebra
$\ee_{6,u}$ over $\fR$, i.e., $\tt \ifmmode\ \cong\ \else$\isom$\fi \fR^6$. Let $x_1,...,x_6$ be
coordinates such that the root forms of $E_6$ are:
\begin{eqnarray*}
(40) & & \pm(x_i\pm x_j), \hspace{1cm} 1\leq i <j\leq 5 \\ (32) & & \pm{1
\over 2}(\pm x_1\pm x_2\pm x_3\pm x_4 \pm x_5 + x_6), \ \ \hbox{even
number of ``$-$'' signs inside the parenthesis.}
\end{eqnarray*}
(Note that in Bourbaki notation, our variables $x_i=\ge_i,\ i=1,..,5$,
while our coordinate $x_6$ is denoted $\ge_8-\ge_7-\ge_6$ there). The 36
positive root forms are given by $\pm x_i+x_j$ and ${1\over 2}(\pm x_1\pm
x_2\pm x_3\pm x_4 \pm x_5 + x_6)$, and they correspond to the 36 double
sixes of the 27 lines on a cubic surface. We use the following notations
for these forms
\begin{equation}\label{eQ2.1}
\begin{array}{lcl} h & = & {1 \over 2}(x_1+...+x_6), \\
h_{1j} & = & x_{j-1}-{1\over 2}(x_1+...+x_5-x_6),\ \ j=2,...,6 \\ h_{jk}
& = & -x_{j-1}+x_{k-1},\ \ 1\neq j<k \\ h_{1jk} & = & x_{j-1}+x_{k-1}, \
\ j,k=2,...,6 \\ h_{jkl} & = &+x_{j-1}+x_{k-1}+x_{l-1}-{1 \over
2}(x_1+...+x_5-x_6),\ \ j,k,l\neq 1. \\
\end{array}
\end{equation}
The Weyl group of $E_6$ is generated by the reflections on these 36
hyperplanes; we denote these reflections by $s,\ s_{ij},$ and $s_{ijk}$.
As a system of simple roots we take :
\begin{equation}\label{eQ2.q}
\begin{array}{cclcl}
\ga_1 & = & -{1 \over 2}(-x_1+...+x_5-x_6) & = & h_{12} \\ \ga_2 & = &
x_1+x_2 & = & h_{123} \\ \ga_3 & = & -x_1+x_2 & = & h_{23} \\ \ga_4 & = &
-x_2+x_3 & = & h_{34} \\ \ga_5 & = & -x_3+x_4 & = & h_{45} \\ \ga_6 & = &
-x_4+x_5 & = & h_{56}. \\
\end{array}
\end{equation}
Then the Dynkin diagram is as shown in Figure \ref{Figure5}; we recover
Figure \ref{Figure4} by replacing $\ga_i$ by the corresponding {\it
reflection} $s, s_{ij}, s_{ijk}$ on the hyperplanes where $h, h_{ij},
h_{ijk}$, respectively, vanish. This shows clearly the isomorphism of
$W(E_6)$ and the group of the permutations of the 27 lines, $$Aut(\ifmmode {\cal L} \else$\cL$\fi)\ifmmode\ \cong\ \else$\isom$\fi
W(E_6).$$
\begin{figure}[hbt]
$$ \unitlength0.8cm
\begin{picture}(10.5,3.5)
\put(0.25,3.0){\circle*{0.5}} \put(2.75,3.0){\circle*{0.5}}
\put(5.25,3.0){\circle*{0.5}} \put(7.75,3.0){\circle*{0.5}}
\put(10.25,3.0){\circle*{0.5}} \put(5.25,0.5){\circle*{0.5}}
\linethickness{0.4mm}
\put(0.25,3.0){\line(1,0){2.5}} \put(2.75,3.0){\line(1,0){2.5}}
\put(5.25,3.0){\line(1,0){2.5}} \put(7.75,3.0){\line(1,0){2.5}}
\put(5.25,0.5){\line(0,1){2.5}}
\put(0.25,3.6){\makebox(0,0){$\ga_1$}}
\put(2.75,3.6){\makebox(0,0){$\ga_3$}}
\put(5.25,3.6){\makebox(0,0){$\ga_4$}}
\put(7.75,3.6){\makebox(0,0){$\ga_5$}}
\put(10.25,3.6){\makebox(0,0){$\ga_6$}}
\put(5.25,-0.1){\makebox(0,0){$\ga_2$}}
\end{picture}
$$
\caption[The Dynkin diagram of $E_6$]{\label{Figure5}\small
The Dynkin diagram of the Weyl group of $E_6$}
\end{figure}
The action of the reflections on the root forms can be described as
follows:\par \vspace{.5cm} \renewcommand{\arraystretch}{1.4}
$\begin{array}{ccc|ccc|ccc|ccc} s(h_{ij}) & = & h_{ij} & s(h_{ijk}) & = &
h_{lmn} & s_{ijk}(h) & = & h_{lmn} & s_{ij}(h) & = & h \\ \hline
s_{ijk}(h_{lmn}) & = & h & s_{ijk}(h_{kmn}) & = & h_{kmn} &
s_{ijk}(h_{jkn}) & = & h_{in} & s_{ijk}(h_{ln}) & = & h_{ln} \\ \hline
s_{ijk}(h_{kn}) & = & h_{ijn} & s_{ijk}(h_{jk}) & = & h_{jk} &
s_{ij}(h_{klm}) & = & h_{klm} & s_{ij}(h_{jlm}) & = & h_{ilm} \\ \hline
s_{ij}(h_{ijm}) & = & h_{ijm} & s_{ij}(h_{ij}) & = & h_{ij} &
s_{ij}(h_{jk}) & = & h_{ik} & s_{ij}(h_{lk}) & = & h_{lk} \\ \hline
\end{array}$\renewcommand{\arraystretch}{1}
\subsection{Vectors}
The Killing form of $E_6$, a quadratic invariant, can be calculated as the
sum of the squares of all roots, and evaluates to (a constant times):
$$I_2=x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+{1 \over 3}x_6^2.$$ With respect to the
Killing form we have the vectors dual to the root forms:
\begin{equation}\label{eQ3.1}
\begin{array}{lcll}
H & = & {1 \over 2}(1,1,1,1,1,3); & \\ H_{1j} & = & -{1 \over
2}(1,...,-1,...,-3), & 1 \hbox{ in the $(j-1)^{st}$ spot},\ j=2,\ldots,6;
\\ H_{jk} & = & -{1 \over 2}(0,..1,..,-1,..,0), & \pm 1 \hbox{ in the
$(j-1)^{st}, (k-1)^{st}$ spot}, 1<j<k\leq 6; \\ H_{1jk} & = & {1 \over
2}(0,..1,..,1,..,0), & 1 \hbox{ in the $(j-1)^{st}, (k-1)^{st}$ spot},\
1<j<k\leq 6; \\ H_{jkl} & = & -{1 \over 2}(1,-1..,-1,..,-1,..,-3), & 1's
\hbox{ in the $(j-1),\ (k-1),\ (l-1)$ spots}, \\ & & & \ 1<j<k<l\leq 6; \\
\end{array}
\end{equation}
which may be thought of as the root vectors (of the positive roots; the
negative roots have a ``$-$'' sign in front).
\begin{table}
\caption{\label{table23}
The arrangement in ${\Bbb P}^5$} \renewcommand{\tabcolsep}{4pt}
\begin{tabular}{|l|r|rr|rrr|rrrrr|rrrr|} \hline $N(\ifmmode {\cal O} \else$\cO$\fi)$ & $A_1$ & $A_1^2$
& $A_2$ & $A_1^3$ &$A_{1,2}$ & $A_3$ & $A_{1^2,2}$ & $A_2^2$ & $A_{1,3}$
& $A_4$ & $D_4$ & $A_{1,2^2}$ & $A_{1,4}$ & $A_5$ & $D_5$ \\ \hline
$\ifmmode {\cal O} \else$\cO$\fi$ & k & $t_2(3)$ & $t_3(3)$ & $t_3(2)$ &$t_4(2)$ & $t_6(2)$ &
$t_5(1)$ & $t_6(1)$ & $t_7(1)$ & $t_{10}(1)$ & $t_{12}(1)$ & $t_7$ &
$t_{11}$ & $t_{15}$ & $t_{20}$ \\ \hline $\#$ & 36 & 270 & 120 & 540 &
720 & 270 & 1080 & 120 & 540 & 216 & 45 & 360 & 216 & 36 & 27 \\ \hline
$t(4)$ & 1 & 15 & 10 & 45 & 80 & 45 & 150 & 20 & 105 & 60 & 15 & 70 & 66
& 15 &15 \\ \hline $t_2(3)$ & & 1 & 0 & 6 & 8 & 3 & 28 & 4 & 18 & 12 & 3
& 20 & 20 & 6 & 7 \\ $t_3(3)$ & & & 1 & 0 & 6 & 9 & 9 & 2 & 18 & 18 & 6 &
6 & 18 & 6 & 9 \\ \hline $t_3(2)$ & & & & 1 & 0 & 0 & 6 & 0 & 3 & 0 & 1 &
6 & 6 & 1 & 3 \\ $t_4(2)$ & & & & & 1 & 0 & 3 & 1 & 3 & 3 & 0 & 4 & 6 & 3
& 3 \\ $t_6(2)$ & & & & & & 1 & 0 & 0 & 2 & 4 & 2 & 0 & 4 & 2 & 5 \\
\hline $t_5(1)$ & & & & & & & 1 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 1 \\
$t_6(1)$ & & & & & & & & 1 & 0 & 0 & 0 & 3 & 0 & 3 & 0 \\ $t_7(1)$ & & &
& & & & & & 1 & 0 & 0 & 0 & 2 & 1 & 1 \\ $t_{10}(1)$ & & & & & & & & & &
1 & 0 & 0 & 1 & 1 & 2 \\ $t_{12}(1)$ & & & & & & & & & & & 1 & 0 & 0 & 0
& 3 \\ \hline
\end{tabular}
\end{table}
As is well-known, there is also a set of 27 fundamental weights which form
an orbit of $W(E_6)$, namely:
\begin{equation}\label{eQ3.3}
\begin{array}{cclccl}
a_1 & = & -{2 \over 3}x_6; & a_j & = & x_{j-1}-{1 \over 2}(x_1+...+x_5+{1
\over 3}x_6); \\ b_1 & = & {1 \over 2}(x_1+...+x_5-{1 \over 3}x_6); &
b_j & = & x_{j-1}+{1 \over 3}x_6; \\ c_{1j} & = & -x_{j-1}+{1 \over 3}x_6;
& c_{ij} & = & -x_{j-1}-x_{i-1}+{1 \over 2}(x_1+...+x_5-{1 \over 3}x_6)
.\\
\end{array}
\end{equation}
These form the $W(E_6)$ orbit of the fundamental weights denoted
$\overline{\go}_1$ and $\overline{\go}_6$ in Bourbaki, which are just our $-a_1$ and
$b_6$, respectively. Note that the following relation holds:
\begin{equation}\label{eQ3.4} \sum_{i=1}^6 a_i = -3h =-3(\sum_{i=1}^6 x_i)
=-\sum_{i=1}^6 b_i.
\end{equation}
Also note that the $a_i$ and $b_i$ are related by
\begin{equation}\label{eQ3.5} b_i=a_i-{1\over 3}(a_1+\cdots +a_6).
\end{equation}
The corresponding vectors which are dual with respect to the Killing form
are:
\begin{equation}\label{eQ3.2}
\begin{array}{cclcclr}
A_1 & = & (0,...,0,-2); & A_j & = & {1 \over 2}(-1,...,+1,..,-1) & +1\
\hbox{in the $j-1$ spot;} \\ B_1 & = & {1 \over 2}(1,...,1,-1); & B_j & =
& (0,...,1,..,1) & +1\ \hbox{in the $j-1$ spot;} \\ C_{1j} & = &
(0,..,-1,...,1); & C_{ij} & = & {1 \over 2}(1,..,-1,..,-1,..,-1) & -1\
\hbox{in the $j-1$, $i-1$ spots.} \\
\end{array}
\end{equation}
\subsection{The arrangement defined by $W(E_6)$}
The 36 hyperplanes in ${\Bbb P}^5$ defined by the vanishing of the 36 root forms
form the arrangement $\ifmmode {\cal A} \else$\cA$\fi({\bf E_6})$ of (\ref{e108.2}).
For later reference we give the combinatorial data of
the arrangement here. We denote as in (\ref{e109.1}) a ${\Bbb P}^m$ through
which $k$ of the hyperplanes pass by $t_k(m)$. For the normalisers we use
the notation $A_{i^k,j^l}$ for $A_i^k\times A_j^l$. The data of the
arrangement is given in Table \ref{table23}.
\subsection{Special Loci}
In Table \ref{table24} we give a list of special loci which will be
particularly important in what follows, so we give a brief description of
each.
\begin{table}[htb]
\caption{\label{table24} Special loci in ${\Bbb P}^5$ \hfill}
$$\begin{array}{|r|l| l| c| l|}\hline \# & \hbox{space} & \hbox{Symmetry} &
N(\ifmmode {\cal O} \else$\cO$\fi) & \hbox{notation in Table \ref{table23}} \\
\hline \hline 36 & {\Bbb P}^4 & A_5 &
A_1 & - \\ \hline 120 & {\Bbb P}^3 & D_4 & A_2 & t_3(3) \\ \hline 120 &
{\Bbb P}^1 & A_2 & A_2\times A_2 & t_6(1) \\ \hline 216 & {\Bbb P}^1 & A_2 & A_4 &
t_{10}(1) \\ \hline 45 & {\Bbb P}^1 & A_1 & D_4 & t_{12}(1) \\ \hline 36 &
\hbox{point} & - & A_5 & t_{15} \\ \hline 27 & \hbox{point} & - & D_5 &
t_{20} \\ \hline \end{array}$$
\end{table}
\subsubsection{36 ${\Bbb P}^4$'s}\label{i1}
In each of the 36 hyperplanes given by the vanishing of one of the 36 forms
(\ref{eQ2.1}), $h$ say, the induced group is $\gS_6$, and as a reflection
group on ${\Bbb P}^4$ it defines a projective arrangement of 15 planes; since
each double six is syzygetic to 15 and azygetic to 20 others, there are 15
hyperplanes through which one of the other 35 intersect $h$, and ten planes
through which two others of the 35 meet $h$. We immediately recognize this
geometry as that in ${\Bbb P}^4$ discussed in the first part of the paper. The
15 hyperplanes are the 15 $\ifmmode {\cal H} \else$\cH$\fi_{ij}$ of (\ref{e112b.3}), each of which cuts
out three planes on $\ifmmode {\cal S} \else$\cS$\fi_3$, and the ten are the hyperplanes mentioned in
(\ref{e117a.1a}) and Proposition \ref{p120.1}. These in turn are the dual
hyperplanes to the ten nodes on $\ifmmode {\cal S} \else$\cS$\fi_3$.
\subsubsection{120 ${\Bbb P}^3$'s}\label{i2}
These ${\Bbb P}^3$'s correspond to the 120 triples
of azygetic double sixes,
i.e., each is cut out by three of the 36 hyperplanes of \ref{i1}. In each
such hyperplane, these ${\Bbb P}^3$'s correspond to the ten hyperplanes in $h$
just mentioned, given by the $K_{ijk}$ of (\ref{e117a.1a}). Each of these
contains 15 planes, and one can check that these are just the faces and
symmetry planes of a cube. The six lines in $K_{ijk}$ which are the
singular locus $\ifmmode {\cal I} \else$\cI$\fi_4\cap K_{ijk}$, are easily identified with the six
12-fold lines $t_{12}(1)$ which are contained in $K_{ijk}$\footnote{The
arrangement is $\ifmmode {\cal A} \else$\cA$\fi({\bf D_4})$, minus the plane at infinity. Of the 16
$t_3(1)$ of $\ifmmode {\cal A} \else$\cA$\fi({\bf D_4})$ (see (\ref{e109.2})), four lie in the plane
at infinity.}, and the nine points $t_{20}$ contained in $K_{ijk}$ are
the intersection points of those six lines\footnote{Likewise, nine of the
12 $t_6$ of $\ifmmode {\cal A} \else$\cA$\fi({\bf D_4})$ lie in the plane at infinity}. Equations of
the 120 ${\Bbb P}^3$'s are given by a triple of azygetic double sixes, e.g., by
$<h,h_{ijk},h_{lmn}>$.
\subsubsection{120 ${\Bbb P}^1$'s} \label{i3} The
120 lines correspond exactly to $A_2$ subroot systems, each containing
three (positive) roots, so that each line contains three of the 36 points.
The 120 lines are determined as follows. Consider a triad of triples of
azygetic double sixes and the corresponding matrix of linear forms (see
(\ref{eQ1.3})), say
$$[ijk.lmn]=\left[ \matrix{ h_{ij} & h_{jk} & h_{ik} \cr h_{lm} & h_{mn} &
h_{ln} \cr h & h_{ijk} & h_{lmn} \cr } \right].$$ Taking the ideal
defined by the vanishing of two rows defines the corresponding line, i.e.,
\begin{equation}\label{eQ5.1}
\begin{array}{lcl}
L_{\{ij.jk\}} & = & <h_{lm}, h_{mn}, h_{ln}, h, h_{ijk}, h_{lmn}> \\
L_{\{lm.mn\}} & = & <h_{ij}, h_{jk}, h_{ik}, h, h_{ijk}, h_{lmn}> \\
L_{\{ijk\}} & = & <h_{ij}, h_{jk}, h_{ik},h_{lm}, h_{mn}, h_{ln}>.\\
\end{array}
\end{equation}
Each of the 120 lines contains three of the nodes, so for example,
\begin{equation}\label{eQ5.2}
H_{ij},\ H_{jk},\ H_{ik} \in L_{\{ij.jk\}}.
\end{equation}
There are 40 such triples of the 120 lines, which have the characterising
property that they span ${\Bbb P}^5$. These correspond to subroot systems of the
type $A_2 \times A_2 \times A_2$, where all three copies are orthogonal to
one another. Note that given an $A_2$ subroot system, there is a unique
$A_2 \times A_2$ subroot system orthogonal to it. Thus the $A_2$ subroot
system is defined by the vanishing of the six root forms of the
complementary $A_2 \times A_2$. There are 120 of each of both types of
subroot systems. Summing up, there are six of the 36 hyperplanes passing
through each of these 120 lines while each such line contains three of the
36 nodes.
The {\it induced arrangement} is as follows.
Blowing up along the line introduces
an exceptional ${\Bbb P}^3$ over each point of the line; the intersection of it
with the proper transforms of the six planes passing through it is the
induced arrangement. It is of type $A_2\times A_2$, i.e., is given by two
skew lines in ${\Bbb P}^3$ and two sets of three hyperplanes through each line.
\subsubsection{216 ${\Bbb P}^1$'s}\label{i4}
Consider a pair of {\em skew} lines, say $a_1, a_2.$ There is a unique
double six containing the given pair as a column, e.g.,
$$N_{12}=\left[ \matrix{ a_1 & b_1 & c_{23} & c_{24} & c_{25} & c_{26} \cr
a_2 & b_2 & c_{13} & c_{14} & c_{15} & c_{16} \cr } \right]. $$ There
are 216 lines in ${\Bbb P}^5$ which join points such as $A_1, A_2, H_{12}$ (see
(\ref{eQ3.1}) and (\ref{eQ3.2})). The
ideal of these 216 lines is generated by 24 sextics, forming the
irreducible representation denoted 24$_p$ in \cite{BL}. We can exhibit
these sextics explicitly, as follows. The 216 lines are given by the
equations:
\begin{equation}\label{eQ7.1}
\begin{array}{lclc}
<A_i,A_j,H_{ij}> & = & <h_{kl}|k,l\neq i,j; h_{ijk}> & (15) \\
<B_i,B_j,H_{ij}> & = & <h_{kl}|k,l\neq i,j; h_{klm}|k,l,m\neq i,j> & (15)
\\ <A_i,B_i,H> & = & <h_{kl}|k,l\neq i> & (6) \\ <A_i,C_{jk},H_{lmn}> & =
& <h_{jk}, h_{\gl \gm}|\gl,\gm\neq i,j,k; h_{ij\gl}, h_{ik\gl}|\gl\neq
i,j,k> & (60) \\ <B_i,C_{jk},H_{ijk}> & = & <h_{jk}, h_{\gl
\gm}|\gl,\gm\neq i,j,k; h_{\gl\gm\gn}|\gl\neq i,j,k,\gm\neq i,k,
\gn\neq i> & (60) \\ <C_{ik},C_{jk},H_{ij}> & = &
<h_{ijm},h_{mn},h,h_{kmn}|m,n\neq i,j,k> & (60), \\
\end{array}
\end{equation}
i.e., each is defined by the vanishing of ten of the $h$'s; these lines are
the $t_{10}(1)$ listed in the table of the arrangement. We claim the
sextics are the products of the six root forms of an $A_2\times A_2$
subroot system. To see this, pick one, say $\Phi=h_{12}\cdot h_{13}\cdot
h_{23}\cdot h_{45}\cdot h_{46}\cdot h_{56}$. It will suffice to check that
for any of the 216 lines listed in (\ref{eQ7.1}), at least one of the
hyperplanes on the right hand side is among the set $h_{12}, h_{13},
h_{23},h_{45}, h_{46}, h_{56}$. This is at most a tedious, but
straightforward task.
The dual ${\Bbb P}^3$'s, which are defined by the vanishing of the forms which
are dual to the points of
the left-hand sides, each {\em contain} the ten points which are
dual to the forms on
the right, for example
$$P_{<A_i,A_j>}=\{a_i=a_j=h_{ij}=0\}\ni (H_{kl},H_{ijk}).$$
The induced arrangement over each line is the arrangement $\ifmmode {\cal A} \else$\cA$\fi(W({\bf
A_4}))$ of (\ref{e109.2}). Among the ten hyperplanes defining the line,
say $<A_1,B_1,H>$, there are ten triples of azygetic double sixes,
$\{ij.jk\}$ in (\ref{eQ1.2}), with $i\neq 1$: $\{23.34\},\ \{23.35\},\
\{23.36\},\ \{24.45\},\ \{24.46\},\ \{25.56\},\ \{34.45\},\ \{34.46\},\
\{35.56\}$, and $\{45.56\}$, and these determine the ten $t_3(1)$ of
(\ref{e109.2}).
\subsubsection{45 ${\Bbb P}^1$'s}\label{i5}
The 45 lines are the lines joining the 27 points of (\ref{eQ3.2}) in
threes, for example, $$L_{(ij)}=<A_i,B_j,C_{ij}>.$$ These lines are defined
by the vanishing of 12 of the $h$'s, so for example
\begin{equation}\label{eQ9.1}
L_{(12)}=<h_{34},h_{35},h_{36},h_{45},h_{46},h_{56},h_{234},h_{235},h_{236},
h_{245},h_{246},h_{256}>;
\end{equation}
these are the hyperplanes corresponding to the 12 double sixes {\em not}
containing any of $a_1, b_2, c_{12}$.
The induced arrangement is $\ifmmode {\cal A} \else$\cA$\fi(W({\bf D_4}))$, with 12 planes
corresponding to the 12 hyperplanes through the line. Again there will be
$t_2(1)$'s and $t_3(1)$'s, corresponding to triples of azygetic double
sixes (respectively to syzygetic double sixes).
The ideal of the 45 lines is generated by 15 quartics which form the
irreducible representation denoted 15$_q$ in \cite{BL}. It is easy to see
that this space of quartics is generated by a product of four pairwise
azygetic $h$'s, for example by $h_{24}\cdot h_{124}\cdot h_{35}\cdot
h_{135}$. In fact, each hyperplane of type $h_{ij}$ contains the 15 lines
numbered (like the tritangents) by: \par
$$\begin{array}{lccl} (kl) & \hbox{for} & k,l\neq i,j & \hbox{(12 of
these)}\\ (ij.kl.mn) & \hbox{for} & k,l,m,n \neq i,j\ & \hbox{(3 of
these)}
\end{array}$$ \\
while the hyperplanes of type $h_{ijk}$ contain the 15 lines numbered by:
$$\begin{array}{lccl} (mn) & \hbox{for} & n=i\ or \ j, m\neq i,j & \hbox{(9
of these)}\\ (il.jm.kn) & & & \hbox{(6 of these).}
\end{array}$$\\
It is now easy to check that every line is contained in at least one of the
four hyperplanes. Alternatively we can argue as follows: each $h$ contains
15 of the lines; there are six ${\Bbb P}^3$'s which are the intersection of two
of the four, three of which are contained in each $h$. These three meet in
a common line in the $h$, so the number of lines contained in the union is:
$4\cdot(15-7)+2\cdot 6+1=45$, where the 7=number of lines in the union of
the three ${\Bbb P}^2$'s in each $h$, 2=3-1 is the number of lines in each such
${\Bbb P}^2$, not in the others, and one is the common line. Note that this is
the Macdonald representation corresponding to the four roots of an
$A_1\times A_1\times A_1\times A_1\subset} \def\nni{\supset} \def\und{\underline D_4$ subroot system. Five of these
lines meet at each of the 27 points, corresponding to the five tritangents
through each of the 27 lines.
\subsubsection{36 points}\label{i6}
These are the 36 points (\ref{eQ3.1}) dual to the 36 hyperplanes of
\ref{i1}. The induced arrangement is of course just the arrangement in
${\Bbb P}^4$ above. There are 15 hyperplanes passing through each of the 36
points, and these are just the hyperplanes which are coded by the double
sixes which are syzygetic to the one with the notation of the point as in
(\ref{eQ3.1}). So, for example, the 15 ${\Bbb P}^4$'s through the point $H$ are
the 15 $h_{ij}$.
These points correspond to ($\pm$) the roots of $E_6$. The orthogonal
complement in $\fR^6$ of the root $\ga$ is projectively equivalent to the
{\em dual} hyperplane to the point. For example, $H$ is dual to $h$, and
one of the hyperplanes $P$ will contain $H$ $\iff$ the dual point $p$ is
contained in $h$. The ideal of the 36 points is generated by 20 cubics,
forming the irreducible representation of $W(E_6)$ denoted 20$_p$ in
\cite{BL}. We can find these cubics explicitly as follows. Consider the
hyperplanes $a_1, b_2, c_{12}$ corresponding to a tritangent. From Table
\ref{table23} above we see that each of these hyperplanes contains 20 of
the 36 points (actually, the table contains the dual information: there are
20 of the $h_{ij}$, etc, passing through each of the 27 points), and the
${\Bbb P}^3$ which is the common intersection of these three contains 12 of the
36 (the dual information is contained in the table: the 45 lines are
12-fold lines). Hence the product $a_1\cdot b_2\cdot c_{12}$ contains
3.(20-12)+12 = 36 of the 36 points.
Through each of the 36 points, also 15 of the 27 hyperplanes of
(\ref{eQ3.3}) pass, corresponding to the 15 lines {\em not} contained in
the double six whose notation the point has. For example, the point
$H={1\over 2}(1,\ldots,1,3)$ is contained in all the $c_{ij}$. In the
exceptional ${\Bbb P}^4$ at the point, both sets of 15 hyperplanes (coming from
the 36, respectively 27 hyperplanes) {\it coincide}.
\subsubsection{27 points}\label{i7}
These are the points $A_i,B_i,C_{ij}$ of (\ref{eQ3.2}). There are 20 of the
36 hyperplanes meeting at each, so the induced arrangement is one of 20
${\Bbb P}^3$'s in ${\Bbb P}^4$, and one sees easily that it is $\ifmmode {\cal A} \else$\cA$\fi(W({\bf D_5}))$.
This arrangement is also induced in any of the hyperplanes $a_i,
b_i,c_{ij}$ of (\ref{eQ3.3}); we note that there are two kinds of ${\Bbb P}^2$,
namely $t_2(2)$'s, corresponding to pairs of skew lines, and $t_3(2)$'s,
corresponding to tritangents. Since each line is contained in five
tritangents, there are five of the latter and 15 of the former (in each
$a_i$, etc.). These 15 form an arrangement of type $\ifmmode {\cal A} \else$\cA$\fi(W({\bf A_5}))$ as
discussed above. The ideal of these 27 points is generated by 30 cubics,
forming the irreducible representation denoted 30$_p$ in \cite{BL}. It is
easy to see that this space of cubics is generated by a product of three
members of a syzgetic triple as $h_{12}\cdot h_{13}\cdot h_{23}$, for
example: Each of the hyperplanes contains 15 of the 27, the ${\Bbb P}^3$ which
is thier common intersection contains nine, so the union contains
3.(15-9)+9=27, or all of the points. Note that this is just the Macdonald
representation corresponding to the (3) roots of an $A_2$ subroot system.
We need, in addition to the above, certain information on the dual spaces.
\subsubsection{45 ${\Bbb P}^3$'s} \label{i8}
Consider one of the 45 ${\Bbb P}^3$'s which is dual to one of the 45 lines of
\ref{i5}; it is cut out by three of the forms (\ref{eQ3.3}), and can be
denoted as one of the 45 tritangents, for example, if $L_{(ij)}$ denotes
the line $<A_i,B_j,C_{ij}>$ as in (\ref{eQ9.1}), the dual ${\Bbb P}^3$ may be
denoted by $l_{(ij)}$, and \begin{eqnarray}\label{eq3.1} l_{(ij)} & = &
a_i\cap b_j \cap c_{ij} \\ l_{(ij:kl:mn)} & = & c_{ij}\cap c_{kl}\cap
c_{mn}.\nonumber
\end{eqnarray}
Consider the ${\Bbb P}^3$ $l_{(12)}$, given by $a_1=b_2=c_{12}=0$, or
$x_1=x_6=0$. Then one checks easily that the hyperplanes (\ref{eQ2.1})
reduce in $l_{(12)}$ to the arrangement $\ifmmode {\cal A} \else$\cA$\fi({\bf F_4})$ of (\ref{e108.2}).
Considering how the 27 hyperplanes (\ref{eQ3.3}) intersect $l_{(12)}$, we
find that these restrict to the set of short roots, that is, give a
subarrangement of type $\ifmmode {\cal A} \else$\cA$\fi({\bf D_4})$. See also Proposition \ref{pQ7.1}
below.
\subsection{Invariants}
Since the 27 forms (\ref{eQ3.3}) are (as a set) invariant under the Weyl
group the expression
\begin{equation}\label{invariantsQ3}
I_k:=\sum_{i,j} \left\{ a_i^k + b_i^k + c_{ij}^k \right\},
\end{equation}
if non-vanishing, is an invariant of degree $k$. The ring of invariants of
$W(E_6)$ is generated by elements in degrees 2, 5, 6, 8, 9 and 12, which
can be taken to be $I_2,\ldots, I_{12}$. We note that while $I_2$ and
$I_5$ are {\em unique}, the other invariants are only defined up to
addition of terms coming from lower degrees.
\section{The invariant quintic}
\subsection{Equation}
There is a unique (up to scalars) $W(E_6)$-invariant polynomial of degree
5. Written with integer coefficients in the variables $x_i$ it is
\begin{equation}\label{eiq1.1} f(x_1,\ldots,x_6)=x_6^5-6x_6^3\gs_1(x)
-27x_6\left(\gs_1^2(x)-4\gs_2(x)\right) -648\sqrt{\gs_5(x)},
\end{equation}
where $\gs_i(x)$ is the $i$th elementary symmetric polynomial of the
$x_1^2,\ldots,x_5^2$, so in particular $\sqrt{\gs_5(x)}=x_1x_2x_3x_4x_5$.
The polynomial $f(x)$ displays manifestly the $W(D_5)$-invariance of the
quintic. Under the change of variables from the $x_i$ to the $a_i$ of
(\ref{eQ3.3}), the equation $g(a)$ can be derived as follows. By
(\ref{eQ3.5}), we have $b_i=a_i-{1\over3}(a_1+\cdots+a_6)$, which by
equation (\ref{eQ3.4}) can be written $b_i=a_i-h$. The following trick was
shown to me by I. Naruki. Consider $\prod_{i=1}^6 a_i -\prod_{i=1}^6 b_i$.
This sextic divides the root $h$, and the quotient is $W(E_6)$-invariant.
To see this, calculate
\begin{eqnarray} a_1\cdot \cdots \cdot a_6 -(b_1\cdot \cdots \cdot b_6) & = &
\prod a_i -\prod (a_i-h) \label{eiq1.2} \\ & = & \gs_6(a)-\left[
\gs_6(a)-h\gs_5(a) +h^2\gs_4(a) -h^3 \gs_3(a) +h^4\gs_2(a)
-h^5\gs_1(a)\right], \nonumber \end{eqnarray} where here $\gs_i(a)$ are
the elementary symmetric functions of the $a_i$. Consequently,
\begin{eqnarray*} a_1\cdot \cdots \cdot a_6 -(b_1\cdot \cdots \cdot b_6) & = &
h\left(\gs_5(a) -h\gs_4(a) +h^2\gs_3(a)-h^3\gs_2(a) +h^4\gs_1(a)\right),
\end{eqnarray*}
and since by (\ref{eQ3.4}) $h=-{1\over 3}\gs_1(a)$, this yields
\begin{eqnarray}
g(a) & = & 81\gs_5(a)+27\gs_4(a)\gs_1(a) +9\gs_3(a)\gs_1^2(a)
+3\gs_2(a)\gs_1^3(a) +\gs_1^5(a),\label{eiq1.3}
\end{eqnarray}
giving the
expression of the invariant quintic expressing manifestly the
$W(A_5)=\gS_6$-invariance.
\begin{definition} \label{diq1.1} The {\em invariant quintic} $\ifmmode {\cal I} \else$\cI$\fi_5$ is the
hypersurface of degree 5
$$\ifmmode {\cal I} \else$\cI$\fi_5:=\{x\in {\Bbb P}^5\Big| f(x)=0\}\ifmmode\ \cong\ \else$\isom$\fi \{a\in {\Bbb P}^5\Big| g(a)=0\},$$
where the isomorphism is given by the change of coordinates from the
$x_i$ to the $a_i$.
\end{definition}
\subsection{Singular locus}
Because of the equivalence of the coordinates $x_i,\ i=1,\ldots,5$, there
are essentially two different partial derivatives of $f$, namely
\begin{eqnarray}\label{eiq2.1} j_1: & = & {\del f \over \del x_1}\ifmmode\ \cong\ \else$\isom$\fi
\cdots \ifmmode\ \cong\ \else$\isom$\fi {\del f \over \del x_5} \\ j_2: & = & {\del f \over \del
x_6}. \nonumber \end{eqnarray} Calculating these forms gives
\begin{eqnarray}\label{eiq2.2} -{\del f \over \del x_i} & = & 12x_6^3x_i
+54x_6x_i(x_j^2+x_k^2+x_l^2+x_m^2)+648 x_jx_kx_lx_m, \\ {\del f\over \del
x_6} & = & 5x_6^4-18x_6^2\gs_1(x) +27(\gs_1^2(x) -4\gs_4(x)). \nonumber
\end{eqnarray} These are quartics with manifest $W(D_4)$ and $W(D_5)$
symmetry, respectively.
\begin{theorem}\label{tiq4.1} The singular locus of $\ifmmode {\cal I} \else$\cI$\fi_5$ consists of the 120
lines of \ref{i3}, which meet ten at a time in the 36 points of \ref{i6}.
\end{theorem}
{\bf Proof:} Consider first the hyperplane section $x_6=0$. Then the
equations to be solved are
\begin{eqnarray}\label{eiq4.1} x_ix_jx_kx_l & = & 0,\quad (i,j,k,l<6); \\
\label{eiq4.2} \gs_1^2(x)-4\gs_2(x) & = & 0. \end{eqnarray}
{}From (\ref{eiq4.1}) we get: two of the $x_i$ must vanish, say $x_4,\ x_5$,
and then (\ref{eiq4.2}) takes the form \begin{eqnarray}
\left(x_1^2+x_2^2+x_3^2\right)^2
-4\left(x_1^2x_2^2+x_1^2x_3^2+x_2^2x_3^2\right) & = & 0, \nonumber \\
(x_1+x_2+x_3)(x_1-x_2-x_3)(x_1+x_2-x_3)(x_1-x_2+x_3) & = & 0
\label{eiq4.3}
\end{eqnarray}
which splits into a product of four lines. Since the product $x_1\cdot
\cdots \cdot x_5=0$ is a coordinate simplex in ${\Bbb P}^4=\{x_6=0\}$, it
follows that the 2-simplices of this simplex correspond to planes where two
of the coordinates vanish, hence there are ${ 5 \choose 2}=10 $ such
2-simplices; in each we have the four lines given by (\ref{eiq4.3}). This
gives the 40 of the 120 lines contained in $x_6=0$. This implies that, in
the union of the 27 hyperplanes (\ref{eQ3.3}), the singular locus of
$\ifmmode {\cal I} \else$\cI$\fi_5$ consists of 120 lines.
Suppose that $x_6\neq 0$. Then the simultaneous vanishing of the partials
${\del f \over \del x_i},\ i=1,\ldots,5$ imply that four of the $x_i$ must
vanish, say $x_2=x_3=x_4=x_5=0$. But the intersection of $\ifmmode {\cal I} \else$\cI$\fi_5$ with the
line $\{x_2=x_3=x_4=x_5=0\}$ is given by
\begin{equation}\label{eiq4.4}
x_6^5-6x_6^3x_1^2-27x_6x_1^4=x_6(x_6+i\sqrt{3}x_1)(x_6-i\sqrt{3}x_1)
(x_6+3x_1) (x_6-3x_1),
\end{equation}
and the last two terms are the equations of $b_2$ and $c_{12}$, two other
of the 27 hyperplanes of (\ref{eQ3.3}). From this we conclude that any
singular point is contained in one of the 27 hyperplanes, and by the above,
that the singular locus of $\ifmmode {\cal I} \else$\cI$\fi_5$ consists of the 120 lines, as stated.
\hfill $\Box$ \vskip0.25cm
The types of singularities are given by the following.
\begin{proposition}\label{piq5.1} The singularities of $\ifmmode {\cal I} \else$\cI$\fi_5$ can be
described as follows.
\begin{itemize}\item[i)] At a generic point $x\in$ one of the 120 lines, a
transversal hyperplane section has an ordinary $A_1$-singularity, so the
singularity is of type disc$\times A_1$.
\item[ii)] At one of the 36 intersection points $p$, the singularity has
multiplicity 3, and the tangent cone is of the form
$$s_5+s_4t+s_3t^2,$$ where $s_3$ is the cone over the Segre cubic,
$s_4=s_3\cdot h_p$, where $h_p$ is the hyperplane of \ref{i1} dual to
$p$, and $s_5$ is the cone over the intersection $\ifmmode {\cal I} \else$\cI$\fi_5\cap h_p$.
\end{itemize}
\end{proposition}
{\bf Proof:} i) follows from consideration of generic hyperplane sections
of $\ifmmode {\cal I} \else$\cI$\fi_5$, which are quintic threefolds with 120 isolated singularities,
so singularities worse than $disc\times A_1$ are impossible. ii) is just a
computation, done as follows. Suppose the point is
$p=H_{23}=(1,-1,0,0,0,0)$. Then inhomogenizing by setting
$t_i=x_i/x_1-tp_i$ (where $p_i$ denotes the $i$th coordinate of $p$),
inserting into the equation of $\ifmmode {\cal I} \else$\cI$\fi_5$ gives the stated result. The fact
that $s_3$ is the cone over $\ifmmode {\cal S} \else$\cS$\fi_3$ can be seen as follows. We can write
\begin{equation}\label{eiq6.1} f=s_5+s_3(th_p +t^2)
\end{equation}
and it follows that blowing up $\ifmmode {\cal I} \else$\cI$\fi_5$ at $p$ is given by setting
$t=\infty$ and that the proper transform of $\ifmmode {\cal I} \else$\cI$\fi_5$ in the exceptional
${\Bbb P}^4$ (of the blow up of ${\Bbb P}^5$ at the point $p$) is a cubic $S_3=0$,
where $s_3=0$ is the cone over $S_3=0$. Since there are ten of the 120
lines meeting at $p$, the resolving divisor of the blow up, which is a
cubic threefold, will have ten isolated singularities. As mentioned already
above, this implies the cubic threefold is isomorphic to $\ifmmode {\cal S} \else$\cS$\fi_3$. One can
also see the 15 special hyperplane sections of $\ifmmode {\cal S} \else$\cS$\fi_3$: these are the
proper transforms, under the blowing up of $p$, of the 15 of the 36
hyperplanes \ref{i1} passing through the point. The rest is calculation.
\hfill $\Box$ \vskip0.25cm
We have (using Macaulay) calculated the ideal $\ifmmode \hbox{{\script I}} \else$\scI$\fi(120)$ of the 120 lines,
and it turns out to be just the Jacobian ideal of $\ifmmode {\cal I} \else$\cI$\fi_5$. I know of no
simple proof of this fact.
\subsection{Resolution of singularities} It turns out to be very easy to
desingularize $\ifmmode {\cal I} \else$\cI$\fi_5$. By the proof of Proposition \ref{piq5.1}, we know
the 36 triple points can be resolved by blowing up each such point $p$. Let
$\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta^{(1)}:\ifmmode {\cal I} \else$\cI$\fi_5^{(1)}\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal I} \else$\cI$\fi_5$ denote this blow up of $\ifmmode {\cal I} \else$\cI$\fi_5$. This has
the effect of seperating all 120 lines of $\ifmmode {\cal I} \else$\cI$\fi_5$, and the singularities
along the lines are just $A_1$, by Proposition \ref{piq5.1}, i). Hence
a desingularisation is achieved by resolving each of the 120 lines. There
are two possible ways to do this. First, one can blow up the lines in
${{\Bbb P}^5}^{(1)}$=${\Bbb P}^5$ blown up in the 36 points, and take the proper
transform of $\ifmmode {\cal I} \else$\cI$\fi_5^{(1)}$; this has the effect of replacing each singular
line by a quadric surface bundle, a ${\Bbb P}^1\times {\Bbb P}^1$-bundle, over the
line. Hence there are 120 exceptional divisors, each isomorphic to
${\Bbb P}^1\times {\Bbb P}^1\times {\Bbb P}^1$. We call this resolution of $\ifmmode {\cal I} \else$\cI$\fi_5$ the
{\em big resolution} and denote it by $\~{\ifmmode {\cal I} \else$\cI$\fi}_5$. Secondly, we can take a
small resolution by blowing down one of the fiberings in the exceptional
${\Bbb P}^1\times {\Bbb P}^1$ over a point of the line. In this way, each of the
singular lines is replaced by a ${\Bbb P}^1$-bundle over the line, in other
words by a ${\Bbb P}^1\times {\Bbb P}^1$. We call this the {\em small resolution} and
denote it by $\ifmmode {\cal I} \else$\cI$\fi_5^{(s)}$. Here no further (beyond the 36 on
$\ifmmode {\cal I} \else$\cI$\fi_5^{(1)}$) exceptional divisors are introduced.
\begin{lemma}\label{liq7.1} The quintic $\ifmmode {\cal I} \else$\cI$\fi_5$ has two resolutions, which we
denote by $\~{\ifmmode {\cal I} \else$\cI$\fi}_5$ and $\ifmmode {\cal I} \else$\cI$\fi_5^{(s)}$. On $\~{\ifmmode {\cal I} \else$\cI$\fi}_5$ there are 36+120
exceptional divisors, 36 copies of the resolution of the Segre cubic, and
120 copies of ${\Bbb P}^1\times {\Bbb P}^1\times {\Bbb P}^1$. On $\ifmmode {\cal I} \else$\cI$\fi_5^{(s)}$ there are
only 36 exceptional divisors, each a small resolution of the Segre cubic.
\end{lemma}
\subsection{${\cal I}_5$ is rational}
Quite generally, in ${\Bbb P}^5$, taking four ${\Bbb P}^3$'s which meet only in
lines, there is a unique line which meets each of them and passes through a
given point $P\in {\Bbb P}^5$, namely the line $<\ga,P>\cap<\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda,P>\cap
<\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta,P>\cap <\gd,P>$, if $\ga,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta,\gd$ denote the ${\Bbb P}^3$'s and
$<\ga,P>$ is the hyperplane spanned by $\ga$ and $P$. Now let $P\in
\ifmmode {\cal I} \else$\cI$\fi_5$, and choose four of the 15 of the 45 ${\Bbb P}^3$'s through one of the
triple points $p$, such that the four ${\Bbb P}^2$'s on $(\ifmmode {\cal S} \else$\cS$\fi_3)_p$ meet each
other only in points; then the four ${\Bbb P}^3$'s meet only in lines, and we
may apply this reasoning to conclude:
\begin{center}
\parbox{14cm}{for each $P\in \ifmmode {\cal I} \else$\cI$\fi_5-\{4\ {\Bbb P}^3$'s \}, there is a unique
line $L_p$, which joins $P$ and $\ga,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta,\gd$.}
\end{center}
Then, fixing a generic hyperplane $F\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^5$. the line $L_p$ intersects
$F$ in a single point; we get a rational map:
$$\ifmmode {\cal I} \else$\cI$\fi_5 -\ -\ -\ \rightarrow F$$
$$P\mapsto L_p\cap F.$$
We now carry out this argument to derive an explicit rationalisation. I am
indebted to B. v. Geemen for help in performing this. We {\em choose} four
convienient ${\Bbb P}^3$'s which only meet in lines (although these do not all
pass through a point). The four ${\Bbb P}^3$'s will be defined as follows:
\begin{equation}\label{eQR.1} \begin{array}{clccc}
P_1 & = & \{a_1=b_4=0\} & = & \{l_1=m_1=0\} \\ P_2 & = & \{a_4=b_5=0\}
& = & \{l_2=m_2=0\} \\ P_3 & = & \{a_5=b_6=0\} & = & \{l_3=m_3=0\} \\
P_4 & = & \{c_{35}=c_{12}=0\} & = & \{l_4=m_4=0\}
\end{array}
\end{equation}
Letting $F$ be an auxilliary ${\Bbb P}^4$ with homogenous coordinates
$(y_0:\ldots:y_4)$, the intersection of the line $<P_1,\ga>\cap \cdots \cap
<P_4,\ga>$ with $F$ is given by
$$y_0l_i-y_im_i=0,$$ which leads to \begin{equation}\label{eQR.2}
\begin{array}{ccl} y_0 & = & m_1\cdots m_4 \\ y_i & = & l_i\cdot
m_1\cdots \widehat} \def\tilde{\widetilde} \def\nin{\not\in{m}_i \cdots m_4,
\end{array}\end{equation}
a system of quartics in ${\Bbb P}^5$, which, when restricted to $\ifmmode {\cal I} \else$\cI$\fi_5$, give
the rational map $\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha:\ifmmode {\cal I} \else$\cI$\fi_5 - - \ra {\Bbb P}^4(=F)$. Inverting the equations
for $x_1,\ldots,x_6$ we get
\begin{equation}\label{eQR.3} \begin{array}{ccl}
x_1 & = & y_0^6y_1y_3+y_0^5y_1^2y_3-2y_0^6y_2y_3-y_0^5y_1y_2y_3
+y_0^4y_1^2y_2y_3+2y_0^4y_1y_2y_3^2+2y_0^3y_1^2y_2y_3^2 \\ & & \
-y_0^6y_1y_4-y_0^5y_1^2y_4-y_0^5y_1y_2y_4-2y_0^4y_1^2y_2y_4
-y_0^3y_1^2y_2^2y_4-y_0^5y_1y_3y_4-y_0^4y_1^2y_3y_4 \\ & & \ \
-2y_0^4y_1y_2y_3y_4-4y_0^3y_1^2y_2y_3y_4-2y_0^3y_1y_2^2y_3y_4
-3y_0^2y_1^2y_2^2y_3y_4-2y_0^3y_1y_2y_3^2y_4 \\ & & \ \ \
-3y_0^2y_1^2y_2y_3^2y_4-2y_0^2y_1y_2^2y_3^2y_4
-2y_0y_1^2y_2^2y_3^2y_4+y_0^2y_1^2y_2y_3y_4^2 \\ & & \ \ \ \
+y_0y_1^2y_2^2y_3y_4^2+y_0y_1^2y_2y_3^2y_4^2+y_1^2y_2^2y_3^2y_4^2 \\
x_2 & = & y_0^6y_1y_3+y_0^5y_1^2y_3+y_0^5y_1y_2y_3+y_0^4y_1^2y_2y_3
+2y_0^5y_1y_3^2+2y_0^4y_1^2y_3^2+2y_0^4y_1y_2y_3^2 \\ & & \
+2y_0^3y_1^2y_2y_3^2-y_0^6y_1y_4-y_0^5y_1^2y_4-y_0^5y_1y_2y_4
-2y_0^4y_1^2y_2y_4-y_0^3y_1^2y_2^2y_4-5y_0^5y_1y_3y_4 \\ & & \ \
-5y_0^4y_1^2y_3y_4-6y_0^4y_1y_2y_3y_4-8y_0^3y_1^2y_2y_3y_4
-2y_0^3y_1y_2^2y_3y_4-3y_0^2y_1^2y_2^2y_3y_4-4y_0^4y_1y_3^2y_4 \\ & & \
\ \ -4y_0^3y_1^2y_3^2y_4-6y_0^3y_1y_2y_3^2y_4
-7y_0^2y_1^2y_2y_3^2y_4-2y_0^2y_1y_2^2y_3^2y_4
-2y_0y_1^2y_2^2y_3^2y_4+2y_0^5y_1y_4^2 \\ & & \ \ \ \
+2y_0^4y_1^2y_4^2+2y_0^4y_1y_2y_4^2+4y_0^3y_1^2y_2y_4^2+2y_0^2y_1^2y_2^2y_4^2
+4y_0^4y_1y_3y_4^2+4y_0^3y_1^2y_3y_4^2 \\ & & \ \ \ \ \
+6y_0^3y_1y_2y_3y_4^2+9y_0^2y_1^2y_2y_3y_4^2+2y_0^2y_1y_2^2y_3y_4^2
+5y_0y_1^2y_2^2y_3y_4^2+2y_0^3y_1y_3^2y_4^2 +2y_0^2y_1^2y_3^2y_4^2 \\ &
& \ \ \ \ \ \
+4y_0^2y_1y_2y_3^2y_4^2+5y_0y_1^2y_2y_3^2y_4^2+2y_0y_1y_2^2y_3^2y_4^2
+3y_1^2y_2^2y_3^2y_4^2 \\
x_3 & = & 2y_0^7y_3+3y_0^6y_1y_3+y_0^5y_1^2y_3+2y_0^6y_2y_3
+3y_0^5y_1y_2y_3+y_0^4y_1^2y_2y_3-2y_0^7y_4-3y_0^6y_1y_4 \\ & & \
-y_0^5y_1^2y_4-2y_0^6y_2y_4-5y_0^5y_1y_2y_4-2y_0^4y_1^2y_2y_4
-2y_0^4y_1y_2^2y_4-y_0^3y_1^2y_2^2y_4-2y_0^6y_3y_4 \\ & & \ \
-3y_0^5y_1y_3y_4-y_0^4y_1^2y_3y_4-4y_0^5y_2y_3y_4
-6y_0^4y_1y_2y_3y_4-2y_0^3y_1^2y_2y_3y_4-2y_0^3y_1y_2^2y_3y_4 \\ & & \
\ \ -y_0^2y_1^2y_2^2y_3y_4+2y_0^3y_1y_2y_3^2y_4
+y_0^2y_1^2y_2y_3^2y_4-2y_0^3y_1y_2y_3y_4^2 \\ & & \ \ \ \
-y_0^2y_1^2y_2y_3y_4^2-2y_0^2y_1y_2^2y_3y_4^2
-y_0y_1^2y_2^2y_3y_4^2-2y_0^2y_1y_2y_3^2y_4^2-y_0y_1^2y_2y_3^2y_4^2 \\
& & \ \ \ \ \ -2y_0y_1y_2^2y_3^2y_4^2-y_1^2y_2^2y_3^2y_4^2 \\
x_4 & = & 2y_0^7y_3+3y_0^6y_1y_3+y_0^5y_1^2y_3+y_0^5y_1y_2y_3
+y_0^4y_1^2y_2y_3-2y_0^5y_1y_3^2-2y_0^4y_1^2y_3^2-2y_0^7y_4 \\ & & \
-3y_0^6y_1y_4-y_0^5y_1^2y_4-3y_0^5y_1y_2y_4-2y_0^4y_1^2y_2y_4
-y_0^3y_1^2y_2^2y_4-2y_0^6y_3y_4-y_0^5y_1y_3y_4 \\ & & \ \
+y_0^4y_1^2y_3y_4-2y_0^4y_1y_2y_3y_4-y_0^2y_1^2y_2^2y_3y_4+4y_0^4y_1y_3^2y_4
+4y_0^3y_1^2y_3^2y_4+2y_0^3y_1y_2y_3^2y_4 \\ & & \ \ \
+3y_0^2y_1^2y_2y_3^2y_4-2y_0^4y_1y_3y_4^2-2y_0^3y_1^2y_3y_4^2
-2y_0^3y_1y_2y_3y_4^2-3y_0^2y_1^2y_2y_3y_4^2 \\ & & \ \ \ \
-y_0y_1^2y_2^2y_3y_4^2-2y_0^3y_1y_3^2y_4^2-2y_0^2y_1^2y_3^2y_4^2
-2y_0^2y_1y_2y_3^2y_4^2-3y_0y_1^2y_2y_3^2y_4^2-y_1^2y_2^2y_3^2y_4^2 \\
x_5 & = & -y_0^6y_1y_3-y_0^5y_1^2y_3-y_0^5y_1y_2y_3-y_0^4y_1^2y_2y_3
+y_0^6y_1y_4+y_0^5y_1^2y_4+2y_0^6y_2y_4+3y_0^5y_1y_2y_4 \\ & & \
+2y_0^4y_1^2y_2y_4+2y_0^4y_1y_2^2y_4+y_0^3y_1^2y_2^2y_4
+3y_0^5y_1y_3y_4+3y_0^4y_1^2y_3y_4+2y_0^4y_1y_2y_3y_4 \\ & & \ \
+4y_0^3y_1^2y_2y_3y_4+2y_0^3y_1y_2^2y_3y_4+y_0^2y_1^2y_2^2y_3y_4
+y_0^2y_1^2y_2y_3^2y_4-2y_0^5y_1y_4^2-2y_0^4y_1^2y_4^2 \\ & & \ \ \
-2y_0^4y_1y_2y_4^2-4y_0^3y_1^2y_2y_4^2-2y_0^2y_1^2y_2^2y_4^2
-2y_0^4y_1y_3y_4^2-2y_0^3y_1^2y_3y_4^2-2y_0^3y_1y_2y_3y_4^2 \\ & & \ \
\ \
-5y_0^2y_1^2y_2y_3y_4^2-3y_0y_1^2y_2^2y_3y_4^2-y_0y_1^2y_2y_3^2y_4^2
-y_1^2y_2^2y_3^2y_4^2 \\ x_6 & = &
-3y_0^6y_1y_3-3y_0^5y_1^2y_3-3y_0^5y_1y_2y_3-3y_0^4y_1^2y_2y_3
+3y_0^6y_1y_4+3y_0^5y_1^2y_4+3y_0^5y_1y_2y_4 \\ & & \
+6y_0^4y_1^2y_2y_4+3y_0^3y_1^2y_2^2y_4+3y_0^5y_1y_3y_4+3y_0^4y_1^2y_3y_4
+6y_0^4y_1y_2y_3y_4+6y_0^3y_1^2y_2y_3y_4 \\ & & \ \
+3y_0^2y_1^2y_2^2y_3y_4-3y_0^2y_1^2y_2y_3^2y_4+3y_0^2y_1^2y_2y_3y_4^2
+3y_0y_1^2y_2^2y_3y_4^2+3y_0y_1^2y_2y_3^2y_4^2 \\ & & \ \ \
+3y_1^2y_2^2y_3^2y_4^2
\end{array}
\end{equation}
a system of octics in ${\Bbb P}^4$, yielding the rational map $$ \psi:{\Bbb P}^4\longrightarrow} \def\sura{\twoheadrightarrow
\ifmmode {\cal I} \else$\cI$\fi_5.$$ These rational maps are morphisms outside of the base locus.
\begin{lemma}\label{lQR.1} The base locus of $\varphi} \def\gt{\theta} \def\gT{\Theta} \def\ga{\alpha$ consists of the four
${\Bbb P}^3$'s $P_1,\ P_2,\ P_3,\ P_4$. The base locus of $\psi$ is a surface
of degree 32.\hfill $\Box$ \vskip0.25cm
\end{lemma}
The first statement is clear from construction, while the second is a
computation. We performed this with the help of Macaulay to calculate a
standard basis of the ideal; the base locus is the intersection of the six
octics.
\section{Hyperplane sections}
\subsection{Reducible hyperplane sections}
Consider the hyperplane section $H_5:=\ifmmode {\cal I} \else$\cI$\fi_5\cap \{x_6=0\}$; it is the union
of five ${\Bbb P}^3$'s which form a coordinate simplex $x_1\cdot x_2\cdot
x_3\cdot x_4\cdot x_5$ in the ${\Bbb P}^4$ given by $\{x_6=0\}$. Now
$a_1=-2/3x_6$ and invariance implies that the 27 hyperplane sections
$a_i=0,\ b_i=0,\ c_{ij}=0$ all have the same property. Each such hyperplane
contains 40 of the 120 ${\Bbb P}^1$'s, which meet six at a time in 20 of the 36
points. Consider three lines in a tritangent, say $(a_1,\ b_2,\ c_{12})$.
These three hyperplanes pass through a common ${\Bbb P}^3$, namely
$$l_{(12)}:=\{x_6=0,\ x_1=0\}.$$ Such ${\Bbb P}^3$'s therefore correspond to the
tritangents and there are 45 such on $\ifmmode {\cal I} \else$\cI$\fi_5$; these are just the 45
${\Bbb P}^3$'s of \ref{i8}. Hence we have
\begin{proposition}\label{pQ7.1} The quintic $\ifmmode {\cal I} \else$\cI$\fi_5$ contains 45 ${\Bbb P}^3$'s,
which are cut out by the 27 hyperplane sections (\ref{eQ3.3}), and each
such hyperplane section meets $\ifmmode {\cal I} \else$\cI$\fi_5$ in the union of five of the 45
${\Bbb P}^3$'s. These can be numbered in terms of the tritangents of a cubic
surface, i.e., for any 3 lines in a tritagent plane of a cubic surface,
the corresponding hyperplanes of (\ref{eQ3.3}) intersect in a common
${\Bbb P}^3$, and this ${\Bbb P}^3$ lies on $\ifmmode {\cal I} \else$\cI$\fi_5$.
\end{proposition}
Also the intersections of the 45 ${\Bbb P}^3$'s can be described. Each such
${\Bbb P}^3$ contains 16 of the 120 lines which meet in 12 of the 36 points;
these 12 points are the vertices of a triad of desmic tetrahedra. Consider
the ${\Bbb P}^3$ $l_{(12)}$; the corresponding tritangent meets 12 others,
namely (13), (14), (15), (16), (32), (42), (52), (62), (12.34.56),
(12.35.46), (12.36.45) and (21), and the 12 ${\Bbb P}^3$'s corresponding to them
meet $l_{(12)}$ in a ${\Bbb P}^2$ (the generic intersection has dimension 1).
These 12 planes in $l_{(12)}$ form the arrangement $\ifmmode {\cal A} \else$\cA$\fi({\bf D_4})$ in
${\Bbb P}^3$.
\subsection{Special hyperplane sections}
We now consider the intersections of $\ifmmode {\cal I} \else$\cI$\fi_5$ with the 36 reflection
hyperplanes \ref{i1}. Take for example the reflection hyperplane $\{h=0\}$;
since $h$ is just a multiple of $\gs_1(a)$ (see (\ref{eQ3.4})), it follows
from the equation (\ref{eiq1.3}) that the intersection $\{h=0\}\cap \ifmmode {\cal I} \else$\cI$\fi_5$
is a quintic hypersurface in ${\Bbb P}^4$ with the equation:
\begin{equation}\label{eiq8.1} Q_1:=\{h=0\}\cap \ifmmode {\cal I} \else$\cI$\fi_5 = \left\{
\begin{array}{c} \gs_1(a) = 0 \\ \gs_5(a) = 0 \end{array} \right.
\end{equation}
Comparing with the equation (\ref{e124.1}), we see that this is a copy of
the Nieto quintic! By symmetry, each of the 36 hyperplane sections is
isomorphic to this one, and we denote them by
\begin{equation}\label{eiq8.2} T=\{h=0\}\cap \ifmmode {\cal I} \else$\cI$\fi_5,\
T_{ij}=\{h_{ij}=0\}\cap \ifmmode {\cal I} \else$\cI$\fi_5,\ T_{ijk}=\{ h_{ijk}=0\}\cap \ifmmode {\cal I} \else$\cI$\fi_5.
\end{equation}
So we have:
\begin{proposition}\label{piq8.1} There are 36 copies of the Nieto quintic
(\ref{e124.1}) on $\ifmmode {\cal I} \else$\cI$\fi_5$.
\end{proposition}
We can determine the singular locus of these hyperplane sections,
independently of the discussion given in section \ref{section4.1}. The
reflection hyperplane contains 20 of the 120 lines, which meet in 15 of the
36 points (corresponding to the 15 roots of an ${\bf A_5}$ subsystem), so
the quintic threefold has 20 singular lines, with 15 singular points of
multiplicity 3. In fact, the resolving divisor of each of these 15 points
is a four-nodal cubic surface, which is a hyperplane section of the Segre
cubic $\ifmmode {\cal S} \else$\cS$\fi_3$ (see the discussion following Problem \ref{p132.1}, ii)).
Furthermore, recalling that there are ten of the 120 lines which pass
through the triple point which is {\em dual} to the given reflection
hyperplane, each such intersects the reflection hyperplane transversally,
giving the ten isolated ordinary double points on that quintic (see
Proposition \ref{p124.1}), and in some sense ``explains'' these isolated
singularities.
\subsection{Generic hyperplane sections}
A generic hyperplane section is a quintic threefold in ${\Bbb P}^4$ with 120
nodes. This is a fascinating family of Calabi-Yau threefolds, which has a
beautiful geometric configuration associated with it, in some sense
``dual'' to the configuration of the 27 lines on a cubic surface.
\begin{proposition} Let $H\in {\Bbb P}^5$ be a generic hyperplane and let
$Q_H=\ifmmode {\cal I} \else$\cI$\fi_5\cap H$ be the hyperplane section. Then we have
\begin{itemize}\item[1)] There are 45 ${\Bbb P}^2$'s on $Q_H$, which are cut out
by 27 hyperplanes; these could appropriately be called {\em quintangent
planes}.
\item[2)] The group of incidence preserving permutations of the 45
${\Bbb P}^2$'s is $W(E_6)$; this is also the group of incidence preserving
permutations of the 27 hyperplanes.
\item[3)] There are 36 hyperplane sections of $Q_H$, each of which is a
20-nodal quintic surface.
\item[4)] The 120 nodes of $Q_H$ form an orbit under $W(E_6)$.
\end{itemize}\end{proposition}
{\bf Proof:} For any of the 45 ${\Bbb P}^3$'s in $\ifmmode {\cal I} \else$\cI$\fi_5$ and hyperplane section
$H$, it holds that ${\Bbb P}^3\cap H={\Bbb P}^2\subset} \def\nni{\supset} \def\und{\underline H\cap \ifmmode {\cal I} \else$\cI$\fi_5=Q_H$, showing 1). The
second point is evident, and in a sense ``dual'' to the situation with
cubic surfaces. We have seen that a special hyperplane section as in
(\ref{eiq8.2}) is isomorphic to the Nieto quintic and has 20 singular lines
in its singular locus; therefore any generic hyperplane section has exactly
20 nodes. 4) follows since the $W(E_6)$ orbit consisting of the 120 lines,
restricted to the hyperplane section is still an orbit. \hfill $\Box$ \vskip0.25cm We now
consider some of the invariants of the nodal quintic threefolds. Let $V$
denote a nodal quintic, $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{V}\longrightarrow} \def\sura{\twoheadrightarrow V$ a small resolution and
$\tilde{V}\longrightarrow} \def\sura{\twoheadrightarrow V$ a big resolution. Letting $s$ denote the number of nodes,
the betti numbers are
\begin{equation}\label{e8.3.1}\begin{minipage}{15cm}$$\begin{array}{ll}
b_1(V)=1=b_1(\widehat} \def\tilde{\widetilde} \def\nin{\not\in{V}), & b_2(\tilde{V})=1+d+s; \\
b_2(V)=1+d=b_2(\widehat} \def\tilde{\widetilde} \def\nin{\not\in{V}), & b_4(\tilde{V})=1+d+s; \\
b_3(V)=b_3(V_t)-s+d, & b_3(\widehat} \def\tilde{\widetilde} \def\nin{\not\in{V})=b_3(V)-s+d=b_3(\tilde{V}),
\end{array}$$
\end{minipage}\end{equation}
where $V_t$ is a smooth hypersurface of same degree as $V$ and $d$ is the
{\it defect}. The defect may be calculated by the following result.
\begin{theorem}[\cite{W}, p.~27]\label{twerner}
Let $V\in {\Bbb P}^4$ be a nodal hypersurface of degree $n\geq 3$. Then
$$\hbox{dim}} \def\deg{\hbox{deg}(\ifmmode \hbox{{\script P}} \else$\scP$\fi_{2n-5}(V))=\hbox{dim}} \def\deg{\hbox{deg}\left\{ \parbox{6cm}{homogenous polynomials of
degree $2n-5$ in ${\Bbb P}^4$, containing all nodes of $V$}\right\} =
{2n-1 \choose 4} -s +d.$$
\end{theorem}
Applied to the case at hand, we need the dimension of the space of {\it
quintics} vanishing at all the nodes. Clearly this is the degree five
component in the ideal of the 120 points. As we mentioned above, we {\it
know} the ideal of the 120 lines (it is the Jacobian ideal of $\ifmmode {\cal I} \else$\cI$\fi_5$
$\ifmmode \hbox{{\script J}} \else$\scJ$\fi ac(\ifmmode {\cal I} \else$\cI$\fi_5)$), so we know also the ideal of the 120 points; it is the
restriction of $\ifmmode \hbox{{\script J}} \else$\scJ$\fi ac(\ifmmode {\cal I} \else$\cI$\fi_5)$ to the hyperplane, generated by six
quartics.
\begin{proposition} The dimension of the space $\ifmmode \hbox{{\script P}} \else$\scP$\fi_5(Q_H)$ is 30.
\end{proposition}
{\bf Proof:} Each of the quartics (which are clearly independent for a
generic hyperplane $H$) of $\ifmmode \hbox{{\script J}} \else$\scJ$\fi ac(\ifmmode {\cal I} \else$\cI$\fi_5)$ can be multiplied by any
hyperplane, giving a quintic which contains the 120 nodes. The set of
hyperplanes is $({\Bbb P}^5)^{\vee}$, so the dimension of $\ifmmode \hbox{{\script P}} \else$\scP$\fi_5(Q_H)$ is
$6\cdot 5=30$. \hfill $\Box$ \vskip0.25cm We can now apply Theorem \ref{twerner} to calculate
the defect $d$ for $Q_H$. The formula is $126-120+d=30$, from which is
follows that $d=24$.
\begin{corollary} The small resolutions $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{Q}_H$ of the quintic
threefolds $Q_H$ have the following betti and Hodge numbers:
$$\begin{array}{lll} b_2(\widehat} \def\tilde{\widetilde} \def\nin{\not\in{Q}_H) = 25, & h^{1,1}=25, & \\
b_3(\widehat} \def\tilde{\widetilde} \def\nin{\not\in{Q}_H)=12=2+2h^{2,1}, & h^{2,1} =5, & e=2h^{1,1}-2h^{2,1}=40.
\end{array}$$
\end{corollary}
{\bf Proof:} Insertion of $d=24$ in (\ref{e8.3.1}).\hfill $\Box$ \vskip0.25cm In the well known
manner for Calabi-Yau threefolds the isomorphism $H^2(V,\Omega} \def\go{\omega} \def\gm{\mu} \def\gn{\nu} \def\gr{\rho^1)\ifmmode\ \cong\ \else$\isom$\fi
H^1(V,\Theta)$ identifies the Hodge space $H^{2,1}$ with the space of
infinitesimal deformations of $V$, $H^1(V,\Theta)$. This is by the above
five-dimensional, hence the moduli space of these 120 nodal quintics (a
Zariski open subset of $({\Bbb P}^5)^{\vee}$) is also a global space of
complex deformations of the small resolution. We can describe the space
$H^{2,1}$ more concretely as follows. Consider the space $\ifmmode \hbox{{\script P}} \else$\scP$\fi_5(Q_H)$; let
$\ifmmode \hbox{{\script J}} \else$\scJ$\fi\subset} \def\nni{\supset} \def\und{\underline \ifmmode \hbox{{\script P}} \else$\scP$\fi_5(Q_H)$ be the subspace generated by the Jacobi ideal of
$Q_H$; since $Q_H$ has five partial derivatives, $\ifmmode \hbox{{\script J}} \else$\scJ$\fi$ is $5\cdot 5=25$
dimensional, and $\ifmmode \hbox{{\script J}} \else$\scJ$\fi$ cannot contribute to infinitesimal deformations, so
we have
$$H^{2,1}(\widehat} \def\tilde{\widetilde} \def\nin{\not\in{Q}_H)\ifmmode\ \cong\ \else$\isom$\fi \ifmmode \hbox{{\script P}} \else$\scP$\fi_5(Q_H)/\ifmmode \hbox{{\script J}} \else$\scJ$\fi.$$
As a final remark consider the Picard group $\hbox{Pic}} \def\Jac{\hbox{Jac}(\widehat} \def\tilde{\widetilde} \def\nin{\not\in{Q}_H)$ and the
orthocomplement of the hyperplane section $\hbox{Pic}} \def\Jac{\hbox{Jac}^0(\widehat} \def\tilde{\widetilde} \def\nin{\not\in{Q}_H)$. Then
$rk_{\integer}\hbox{Pic}} \def\Jac{\hbox{Jac}^0(\widehat} \def\tilde{\widetilde} \def\nin{\not\in{Q}_H)=24$, and the 45 ${\Bbb P}^2$'s give us privledged
representatives in $\hbox{Pic}} \def\Jac{\hbox{Jac}^0$; the 27 hyperplanes represent relations, so we
have an exact sequence
$$\integer^{27}\longrightarrow} \def\sura{\twoheadrightarrow \integer^{45}\longrightarrow} \def\sura{\twoheadrightarrow \hbox{Pic}} \def\Jac{\hbox{Jac}^0(\widehat} \def\tilde{\widetilde} \def\nin{\not\in{Q}_H) \longrightarrow} \def\sura{\twoheadrightarrow 1,$$ and the kernel is
six-dimensional. The sum sequence is then
\begin{equation}\label{e48a.1} 1 \longrightarrow} \def\sura{\twoheadrightarrow \integer^6 \longrightarrow} \def\sura{\twoheadrightarrow \integer^{27}\longrightarrow} \def\sura{\twoheadrightarrow \integer^{45} \longrightarrow} \def\sura{\twoheadrightarrow
\integer^{24}\longrightarrow} \def\sura{\twoheadrightarrow 1,
\end{equation}
and this is really dual to the sequence (\ref{eB3.2}) for cubic surfaces.
\begin{remark}
The period map for this five-dimensional family of Calabi-Yau threefolds
maps to the domain $\ifmmode {\cal D} \else$\cD$\fi = Sp(6,\fR)/U(1)\times U(5)$. Note that any
hyperplane passing through one of the 45\ ${\Bbb P}^3$'s will intersect
$\ifmmode {\cal I} \else$\cI$\fi_5$ in the union of that ${\Bbb P}^3$ and a residual quartic; clearly
these constitute the set of cusps for the period map, i.e., on the 45
lines in $({\Bbb P}^5)^{\vee}$ (the dual ${\Bbb P}^5$) which parameterise the set of
hyperplanes passing through one of the 45 ${\Bbb P}^3$'s, the period map maps
to the boundary of the domain $\ifmmode {\cal D} \else$\cD$\fi$ above. These 45 one-dimensional
cusps meet in 27 points, i.e., zero-dimensional cusps, which correspond
to the 27 hyperplane sections which split into the union of five
${\Bbb P}^3$'s. But we can say more. Noting that, excepting the hyperplanes
above, all hyperplane sections of $\ifmmode {\cal I} \else$\cI$\fi_5$ are irreducible quintics, the
worst that can happen is that the hyperplane passes through one of the 36
triple points of $\ifmmode {\cal I} \else$\cI$\fi_5$. We will see below that these are still
Calabi-Yau (Proposition \ref{p158.1}), hence {\it not} contained in the
boundary.
\end{remark}
\subsection{Tangent hyperplane sections}
We now consider the case of a hyperplane tangent to $\ifmmode {\cal I} \else$\cI$\fi_5$ at a point
$p\in \ifmmode {\cal I} \else$\cI$\fi_5$. In this case the section $Q_p$ aquires an additional node.
Note that the 121 nodes fall into two ``orbits'', one set of 120 on which
$W(E_6)$ acts as a permutation group, and the additional point $p$. For a
121-nodal quintic the same calculation as above gives $e(Q_p)=42$,
$h^{2,1}=4,\ h^{1,1}=25$. It follows that the $H_4(Q_p,\rat)$ is the same
as for $Q_x,\ x\in {\Bbb P}^5$ generic. The difference to the generic case is
in $H_3$, more precisely in $H^{2,1}$. Indeed, we now require
$\ifmmode \hbox{{\script P}} \else$\scP$\fi_5(Q_p)$, that is, quintics through all 121 nodes, so as opposed to
the general case, we now only have, for each of the five quartics in the
Jacobi ideal of $Q_p$, since each contains $p$, a five-dimensional family
of quintics, as above. But for the quartics through the 120 nodes which are
{\it not} in the Jacobi ideal, we must take a hyperplane {\it through the
point} $p$, so
\begin{proposition} For a 121-nodal quintic $Q_p$, $p\in \ifmmode {\cal I} \else$\cI$\fi_5$, we have
$\hbox{dim}} \def\deg{\hbox{deg}\ifmmode \hbox{{\script P}} \else$\scP$\fi_5(Q_p)=5\cdot 5 + 1\cdot 4 =29$. \hfill $\Box$ \vskip0.25cm
\end{proposition}
We can now apply Theorem \ref{twerner} to calculate the defect:
$$d=29-126+121 = 24.$$
\begin{corollary} The betti numbers for the small resolution $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{Q}_p$
are
$$\begin{array}{ll}b_2(\widehat} \def\tilde{\widetilde} \def\nin{\not\in{Q}_p) =25, & h^{1,1} = 25, \\ b_3(\widehat} \def\tilde{\widetilde} \def\nin{\not\in{Q}_p) =
10, & h^{2,1}=4,\ \ \ \ e=42. \\
\end{array}$$ \end{corollary}
We remark that since $h^{1,1}$ is still 25, the sequence (\ref{e48a.1})
still holds for $Q_p$.
\section{Birational maps and the projection from a triple point}
\subsection{The cuspidal model}
First we recall the notations $\ifmmode {\cal I} \else$\cI$\fi_5^{(1)}$ for the blow up of $\ifmmode {\cal I} \else$\cI$\fi_5$ at
the 36 triple points, $\~{\ifmmode {\cal I} \else$\cI$\fi}_5$ for the big resolution of $\ifmmode {\cal I} \else$\cI$\fi_5$, and
$\ifmmode {\cal I} \else$\cI$\fi_5^{(s)}$ for the small resolution. Note that on $\ifmmode {\cal I} \else$\cI$\fi_5^{(1)}$, each of
the 120 lines has normal bundle $\ifmmode {\cal O} \else$\cO$\fi(-2)^{\oplus 3}$, hence each line can
be blown down to an isolated singular point.
\begin{definition}\label{dq4.1} Consider the following birational
transformation of $\ifmmode {\cal I} \else$\cI$\fi_5$:
\begin{itemize}\item[i)] Blow up the 36 triple points,
$\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta^{(1)}:\ifmmode {\cal I} \else$\cI$\fi_5^{(1)} \longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal I} \else$\cI$\fi_5$;
\item[ii)] Blow down the proper transforms of the 120 lines to 120 isolated
singularities, $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta^{(2)}:\ifmmode {\cal I} \else$\cI$\fi_5^{(1)}\longrightarrow} \def\sura{\twoheadrightarrow \widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_5$.
\end{itemize}
Step ii) defines the {\em cuspidal model} $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_5$.
\end{definition}
This is a four-dimensional analogue of $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal N} \else$\cN$\fi}_5$ of (\ref{e127.2}).
Indeed, for each of the 36 hyperplane sections of Proposition \ref{piq8.1},
the proper transform on $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_5$ is isomorphic to $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal N} \else$\cN$\fi}_5$:
\begin{lemma}\label{lq4.1} Let $T\ifmmode\ \cong\ \else$\isom$\fi \ifmmode {\cal N} \else$\cN$\fi_5$ be one of the 36 special
hyperplane sections of (\ref{eiq8.2}), and let $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{T}$ denote its
proper transform on $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_5$. Then $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{T}\ifmmode\ \cong\ \else$\isom$\fi \widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal N} \else$\cN$\fi}_5$.
\end{lemma}
{\bf Proof:} Just check that the steps i) and ii) of Definition
\ref{dq4.1}, when restricted to $T$, coincide with those of
(\ref{e127.2}).\hfill $\Box$ \vskip0.25cm Let us mention that $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_5$ ``looks like'' a
ball quotient too, at least assuming a positive answer to Problem
\ref{p132.1}. We explain what ``looks like'' means in the following items.
\begin{itemize}
\item[I1] Each isolated singularity is resolved by a ${\Bbb P}^1\times
{\Bbb P}^1\times {\Bbb P}^1$; the arrangement induced in each by the proper
transforms of the 36 hyperplanes and 36 exceptional divisors is a {\em
product}, consisting of three fibres in each fibering (i.e., $\{\hbox{3
points}\}\times {\Bbb P}^1\times {\Bbb P}^1,\ {\Bbb P}^1\times \{\hbox{3
points}\}\times {\Bbb P}^1,\ {\Bbb P}^1\times {\Bbb P}^1\times \{\hbox{3 points}\}$,
see \ref{i3}). Hence this can be covered in an equivariant way by
$E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}\times E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}\times E_{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}$ (see Lemma \ref{l115.2}).
\item[I2] The proper transforms of the 36 hyperplane sections of
Proposition \ref{piq8.1} are by Lemma \ref{lq4.1} isomorphic to
$\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal N} \else$\cN$\fi}_5$, so, if the Problem \ref{p132.1} has an affirmative
solution, these are ball quotients, with cusps being those isolated
singularities of $\widehat} \def\tilde{\widetilde} \def\nin{\not\in{\ifmmode {\cal I} \else$\cI$\fi}_5$ which are contained in the given
$\widehat} \def\tilde{\widetilde} \def\nin{\not\in{T}$.
\item[I3] Consider the 45 ${\Bbb P}^3$'s of Proposition \ref{pQ7.1}. These are
(the proper transforms of) the 45 ${\Bbb P}^3$'s of \ref{i8}. These ${\Bbb P}^3$'s
are also ball quotients, in fact in two different ways.
\begin{itemize}\item[1)] There is a cover $Y\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^3$, branched over the
arrangement $\ifmmode {\cal A} \else$\cA$\fi(W({\bf D_4}))$ in ${\Bbb P}^3$, which is a ball quotient.
This example can be derived from the solution 4) of (\ref{e111d.1}) by
means of the natural squaring map $m_2:{\Bbb P}^3\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^3,\ (x_0:\ldots
:x_3)\mapsto (x_0^2:\ldots : x_3^2)$. Then the arrangement $\ifmmode {\cal A} \else$\cA$\fi(W({\bf
D_4}))$ is the pullback under $m_2$ of the six symmetry planes of the
tetrahedron in the arrangement $\ifmmode {\cal A} \else$\cA$\fi(W({\bf A_4}))$, and pulling back the
solution 4), we get the cover $Y\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^3$, branched along $\ifmmode {\cal A} \else$\cA$\fi(W({\bf
D_4}))$ (with branching degree 3 at each hyperplane), which is a ball
quotient by a fix point free group.
\item[2)] There is a cover $Z\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^3$, branched along the arrangement
$\ifmmode {\cal A} \else$\cA$\fi(W({\bf F_4}))$ in ${\Bbb P}^3$ (but not a Fermat cover), which is a ball
quotient; this example is explained in \cite{hunt}, Thm.~7.6.5, and is
the {\it only} known ball quotient related to a plane arrangement in
${\Bbb P}^4$ which does {\it not} derive from those given by solutions of the
hypergeometric differential equation.
\end{itemize}
Both of the arrangements mentioned, $\ifmmode {\cal A} \else$\cA$\fi(W({\bf D_4}))$ and $\ifmmode {\cal A} \else$\cA$\fi(W({\bf
F_4}))$, arise naturally on the 45 ${\Bbb P}^3$'s: the first is the
intersection with the 27 hyperplanes, the second is the intersection with
the 36 hyperplanes.
\end{itemize}
\subsection{Projection from a triple point}
Let $p\in \ifmmode {\cal I} \else$\cI$\fi_5$ be one of the 36 triple points, and let $h_p$ be the dual
hyperplane (one of the 36 ${\Bbb P}^4$'s of \ref{i1}). The projection of
${\Bbb P}^5$ from $p$ is defined as follows. Consider the ${\Bbb P}^4$ of all lines
through $p$; this is just the dual $h_p$, and each line $l_p$ through $p$
corresponds to a unique point of $h_p$ (its intersection with $h_p$). Since
any point $x$ of ${\Bbb P}^5$ is on a unique line $(l_x)_p$ through $p$, the map
\begin{eqnarray}\label{e154.1} \pi_p:{\Bbb P}^5 & \longrightarrow} \def\sura{\twoheadrightarrow & h_p \\
x & \mapsto & (l_x)_p\cap h_p \nonumber
\end{eqnarray} gives the {\em projection from $p$}. Restricting to $\ifmmode {\cal I} \else$\cI$\fi_5$
this gives a generically finite (rational) map, which we also denote by
$\pi_p$, $\pi_p:\ifmmode {\cal I} \else$\cI$\fi_5- - \ra h_p$.
\begin{lemma}\label{l154.1} $\pi_p:\ifmmode {\cal I} \else$\cI$\fi_5- - \ra h_p$ is generically
a double cover.
\end{lemma}
{\bf Proof:} Since the triple point has multiplicity 3, a generic line will
meet $\ifmmode {\cal I} \else$\cI$\fi_5$ in $(5-3)=2$ further points. \hfill $\Box$ \vskip0.25cm
\begin{lemma}\label{l154.2} $\pi_p:\ifmmode {\cal I} \else$\cI$\fi_5- - \ra h_p$ is a quotient map by the
group $G_p\ifmmode\ \cong\ \else$\isom$\fi \integer/2\integer$ generated by the reflection $\gs_p$ on the root
$p$.
\end{lemma}
{\bf Proof:} The reflection $\gs_p$ fixes $h_p$; it is the inversion
($(z_0:z_1) \mapsto (z_1:z_0)$) on any line $l_p$ through $p$, where the
homogenous coordinates are choosen such that $l_p\cap h_p=(1:1)$. Since
$\ifmmode {\cal I} \else$\cI$\fi_5$ is mapped by $\gs_p$ onto itself, it follows that two points of
$\ifmmode {\cal I} \else$\cI$\fi_5\cap l_p$ are related by inversion on $l_p$. So the group action is
manifest. \hfill $\Box$ \vskip0.25cm We now describe how to make $\pi_p$ into a {\em morphism}.
First of all, one must blow up $p$; let $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta_p:\ifmmode {\cal I} \else$\cI$\fi_{5,p}\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal I} \else$\cI$\fi_5$ denote
this blow up. Let $(\ifmmode {\cal S} \else$\cS$\fi_3)_p$ be the copy of $\ifmmode {\cal S} \else$\cS$\fi_3$ which is the
exceptional divisor at $p$. For any $x\in (\ifmmode {\cal S} \else$\cS$\fi_3)_p$, the line $(l_x)_p$
through $p$ and intersecting $h_p$ in the Segre cubic there, is tangent to
$\ifmmode {\cal I} \else$\cI$\fi_5$ {\em at the triple point} $p$. Secondly, certain subvarieties get
{\em blown down}. Indeed, suppose $(l_x)_p$ is {\em contained in} $\ifmmode {\cal I} \else$\cI$\fi_5$
for some $x\in \ifmmode {\cal I} \else$\cI$\fi_5$. Then, clearly, $(l_x)_p\mapsto (l_x)_p\cap h_p$, the
whole line maps to a point, or in other words, gets blown down.
\begin{lemma}\label{l155.1} The projection $\pi_p:\ifmmode {\cal I} \else$\cI$\fi_5- - \ra h_p$, which is
well-defined on $\ifmmode {\cal I} \else$\cI$\fi_{5,p}$, blows down all linear subspaces on $\ifmmode {\cal I} \else$\cI$\fi_5$
which pass through $p$, and is a double cover outside the union $\ifmmode \hbox{{\script L}} \else$\scL$\fi_p$
of all such linear subspaces on $\ifmmode {\cal I} \else$\cI$\fi_5$ passing through $p$.\hfill $\Box$ \vskip0.25cm
\end{lemma}
We now describe $\ifmmode \hbox{{\script L}} \else$\scL$\fi_p$. Recall that the linear subspaces on $\ifmmode {\cal I} \else$\cI$\fi_5$ are
the 45 ${\Bbb P}^3$'s and their intersections. Hence $\ifmmode \hbox{{\script L}} \else$\scL$\fi_p$ consists of all
the ${\Bbb P}^3$'s and their intersections, which pass through $p$. Recall from
\ref{i6} that this is the set of 15 of the 45 ${\Bbb P}^3$'s of Proposition
\ref{pQ7.1}. Therefore, we get
\begin{lemma}\label{l155.2} The projection $\pi_p,p:\ifmmode {\cal I} \else$\cI$\fi_{5,p}\longrightarrow} \def\sura{\twoheadrightarrow h_p$ blows
down the union of 15 ${\Bbb P}^3$'s to the 15 planes in $h_p$ which are the
intersection of $\ifmmode {\cal S} \else$\cS$\fi_3$ and $\ifmmode {\cal N} \else$\cN$\fi_5$.
\end{lemma}
Now let $X=\ifmmode {\cal I} \else$\cI$\fi_{5,p}^{\%}$, the double cover of $h_p$ branched along the
union of $\ifmmode {\cal S} \else$\cS$\fi_3$ and $\ifmmode {\cal N} \else$\cN$\fi_5$ (which is of degree 8, so a double cover
exists). $X$ is clearly {\em singular along} the 15 planes. Indeed:
\begin{lemma}\label{l155.3} $\pi_p,p:\ifmmode {\cal I} \else$\cI$\fi_{5,p}\longrightarrow} \def\sura{\twoheadrightarrow h_p$ factors over
$\ifmmode {\cal I} \else$\cI$\fi_{5,p}^{\%}$, and $\Pi:X\longrightarrow} \def\sura{\twoheadrightarrow h_p$ is the double cover of ${\Bbb P}^4$
branched along the union $\ifmmode {\cal S} \else$\cS$\fi_3\cup \ifmmode {\cal N} \else$\cN$\fi_5$.
\end{lemma}
{\bf Proof:} This follows from the discussion above; the branch locus $\ifmmode {\cal R} \else$\cR$\fi$
is the set:
$$\ifmmode {\cal R} \else$\cR$\fi=\{x\in \ifmmode {\cal I} \else$\cI$\fi_{5,p}\Big| (l_x)_p \hbox{ is tangent to $\ifmmode {\cal I} \else$\cI$\fi_5$ at
$x$}\}.$$ This happens if either
\begin{itemize}\item[i)] $x\in h_p$, since then $x$ is fixed by $\gs_p$;
\item[ii)] $x\in (\ifmmode {\cal S} \else$\cS$\fi_3)_p$, the exceptional divisor over $p$.
\end{itemize}
Therefore $\ifmmode {\cal R} \else$\cR$\fi=\ifmmode {\cal S} \else$\cS$\fi_3\cup \ifmmode {\cal N} \else$\cN$\fi_5$. By Lemmas \ref{l155.1} and \ref{l155.2},
15 ${\Bbb P}^3$'s are blown down to ${\Bbb P}^2$'s, and outside of this locus, $\Pi$
is 2:1. \hfill $\Box$ \vskip0.25cm
\subsection{Double octics and quintic hypersurfaces}
With the result of Lemma \ref{l155.3} at hand, we can get a new slant on
the quintic threefolds which are hyperplane sections of $\ifmmode {\cal I} \else$\cI$\fi_5$. For this,
consider a hyperplane section of the cover $\Pi:X\longrightarrow} \def\sura{\twoheadrightarrow h_p$, that is, let
$H\subset} \def\nni{\supset} \def\und{\underline h_p$ be a hyperplane, and let $X_H$ be its inverse image in $X$:
$$\Pi_H:X_H\longrightarrow} \def\sura{\twoheadrightarrow H,$$ a double cover of ${\Bbb P}^3$. The branch locus is $H\cap
(\ifmmode {\cal S} \else$\cS$\fi_3\cup \ifmmode {\cal N} \else$\cN$\fi_5)$, which is the union of a cubic and a quintic surface in
${\Bbb P}^3$. Note the $H\cap \{\hbox{ one of the 15 ${\Bbb P}^2$'s $\subset} \def\nni{\supset} \def\und{\underline \ifmmode {\cal S} \else$\cS$\fi_3\cap
\ifmmode {\cal N} \else$\cN$\fi_5$}\}$ is a {\em line}, contained in both $H\cap \ifmmode {\cal S} \else$\cS$\fi_3$ and in $H\cap
\ifmmode {\cal N} \else$\cN$\fi_5$. In other words, $H\cap (\ifmmode {\cal S} \else$\cS$\fi_3\cup \ifmmode {\cal N} \else$\cN$\fi_5)=S_H\cup Q_H$, where $S_H$
is the cubic surface, $Q_H$ is the quintic surface, and $$S_H\cap
Q_H=\{\hbox{15 lines}\}.$$
\begin{proposition}\label{p158.1} Let $X_H=\Pi^{-1}(H)$, the double cover of
${\Bbb P}^3$ branched along $S_H\cup Q_H$. Then there is a canonical model
$\-X_H$ of $X_H$ which is Calabi-Yau.
\end{proposition}
{\bf Proof:} We know the resolution of $X$; it is given by ``inverting''
the projection from the node, by blowing up along the 15 planes $\ifmmode {\cal S} \else$\cS$\fi_3\cap
\ifmmode {\cal N} \else$\cN$\fi_5$, yielding $\ifmmode {\cal I} \else$\cI$\fi_{5,p}$. Let $\-X_H$ be the proper transform of $X_H$
in $\ifmmode {\cal I} \else$\cI$\fi_{5,p}$. Assuming $H$ to be sufficiently general, $\-X_H$ clearly
has canonical singularities (as $\ifmmode {\cal I} \else$\cI$\fi_{5,p}$ does), so we must only show
that it is Calabi-Yau. We note, however, that $\-X_H$ is (the proper
transform on $\ifmmode {\cal I} \else$\cI$\fi_{5,p}$ of) a hyperplane section of $\ifmmode {\cal I} \else$\cI$\fi_5$! This is
because the degree is invariant under projection, hence under $\Pi$. But
this is a hyperplane section of $\ifmmode {\cal I} \else$\cI$\fi_5$ through the triple point $p$. Hence
the proper transform on $\-X_H$ of the exceptional divisor $(\ifmmode {\cal S} \else$\cS$\fi_3)_p$ is a
hyperplane section of $\ifmmode {\cal S} \else$\cS$\fi_3$, i.e., a (generically smooth) cubic surface.
This singularity is known to be canonical, and $\-X_H$ is, just as a nodal
quintic, canonically Calabi-Yau. \hfill $\Box$ \vskip0.25cm
\begin{table}
\caption{\label{table25}
Degenerations of double octics and quintic hypersurfaces}
\begin{minipage}{16.5cm}
\begin{center}
\fbox{\begin{minipage}{12cm}\begin{center} Space of all quintic
hypersurfaces
101-dimensional
\end{center}
\end{minipage}}
$$\cup$$
\fbox{\begin{minipage}{6cm}\begin{center} 120-nodal quintics
5-dimensional
\end{center}
\end{minipage}}
$$\cup$$
\fbox{\begin{minipage}{5cm}\begin{center} quintic hypersurfaces with 111
nodes and one multiplicity 3 singular point
4-dimensional
\end{center}\end{minipage}}
$$\|$$
\fbox{\begin{minipage}{5cm}\begin{center} double cover $Y\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^3$,
branched over $S\cup Q$
$S\cap Q=\{\hbox{ 15 lines}\}$
\end{center}
\end{minipage}}
$$\cap$$
\fbox{\parbox{6cm}{double cover branched over cubic and quintic, such that
$S\cup Q$ is stable}}
$$\cap$$ \fbox{\begin{minipage}{12cm}\begin{center} Space of all double
octics
149-dimensional
\end{center}\end{minipage}}
\end{center}
\end{minipage}
\end{table}
\begin{corollary}\label{c158.1} The family of hyperplane sections of $\ifmmode {\cal I} \else$\cI$\fi_5$
passing through one of the 36 triple points $p$ is, via projection, a
family of Calabi-Yau threefolds which are degenerations of double octics.
\end{corollary}
It is natural to ask the meaning of this in terms of variations of Hodge
structures. Recalling that a Type III degeneration of a K3 surface,
corresponding to a zero-dimensional boundary component of the period
domain, is one like a quartic degenerating into four planes, it is natural
to ask
\begin{question} Is a double cover of ${\Bbb P}^3$ branched over the union of a
cubic and a quintic a semistable degeneration of a double octic?
\end{question}
\begin{remark} There is a notion of ``connecting''
moduli spaces of CY threefolds by degenerations, and the Corollary
\ref{c158.1} shows that the moduli space of quintic hypersurfaces in
${\Bbb P}^4$ and the moduli space of double octics are connected; the
birational transformations which are required for such ``connections''
are given here by projection in projective space, very geometric.
\end{remark}
In Table \ref{table25} we give a rough description of these relations.
\subsection{The dual picture}
Now consider $\ga(\ifmmode {\cal S} \else$\cS$\fi_3\cup \ifmmode {\cal N} \else$\cN$\fi_5)$, with $\ga$ the map (\ref{e126.1})
given by the quadrics on the ten nodes of $\ifmmode {\cal S} \else$\cS$\fi_3$. By Theorem \ref{t122a.1}
and by definition of $\ifmmode {\cal W} \else$\cW$\fi_{10}$ (\ref{e127.1}), we have
\begin{equation}\label{e156.1} \ga(\ifmmode {\cal S} \else$\cS$\fi_3\cup \ifmmode {\cal N} \else$\cN$\fi_5) = \ifmmode {\cal I} \else$\cI$\fi_4\cup \ifmmode {\cal W} \else$\cW$\fi_{10},
\end{equation}
and by Theorem \ref{t127.1}, the intersection $\ifmmode {\cal I} \else$\cI$\fi_4\cap \ifmmode {\cal W} \else$\cW$\fi_{10}$ consist
of 10 quadric surfaces. Define $\ifmmode {\cal W} \else$\cW$\fi$ to be the double cover of ${\Bbb P}^4$
branched along $\ifmmode {\cal W} \else$\cW$\fi_{10}$:
\begin{equation}\label{e156.2} \tau:\ifmmode {\cal W} \else$\cW$\fi\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4=(h_p)^{\vee}.
\end{equation}
We may consider the fibre square:
\begin{equation}\label{e156.3} \begin{array}{rcl} {\cal Z} & \longrightarrow} \def\sura{\twoheadrightarrow & {\cal Y} \\
\downarrow & & \downarrow \pi \\ \tau:{\cal W} & \longrightarrow} \def\sura{\twoheadrightarrow & {\Bbb P}^4
\end{array}
\end{equation}
where $\pi:\ifmmode {\cal Y} \else$\cY$\fi\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4$ is defined in Definition \ref{d133.1}. Then
$\pi_{\ifmmode {\cal Z} \else$\cZ$\fi}:\ifmmode {\cal Z} \else$\cZ$\fi\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4$ is a Galois cover with Galois group $G_{\ifmmode {\cal Z} \else$\cZ$\fi}\ifmmode\ \cong\ \else$\isom$\fi
\integer/2\integer\times \integer/2\integer$. Let $H\ifmmode\ \cong\ \else$\isom$\fi \integer/2\integer\subset} \def\nni{\supset} \def\und{\underline G_{\ifmmode {\cal Z} \else$\cZ$\fi}$ be the diagonal
subgroup; it is a normal subgroup, and we may form the quotient
$$\eta:\ifmmode {\cal Z} \else$\cZ$\fi\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal Z} \else$\cZ$\fi',\quad \ifmmode {\cal Z} \else$\cZ$\fi'=\ifmmode {\cal Z} \else$\cZ$\fi/H.$$
\begin{lemma}\label{l156.1} $\pi_{\ifmmode {\cal Z} \else$\cZ$\fi}:\ifmmode {\cal Z} \else$\cZ$\fi\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4$ factors over $\eta$,
and $\eta':\ifmmode {\cal Z} \else$\cZ$\fi'\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4$ is a double cover, hence Galois.
\end{lemma}
{\bf Proof:} This is a general fact about fibre squares of double covers
like (\ref{e156.3}). \hfill $\Box$ \vskip0.25cm
\begin{theorem}\label{t156.1} The rational map $\ga$ induces a rational map
of the double covers $\Pi:X\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4$ of Lemma \ref{l155.3} and
$\eta':\ifmmode {\cal Z} \else$\cZ$\fi'\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb P}^4$ of Lemma \ref{l156.1}. Furthermore, the rational
map
$$\Xi:X- - \ra \ifmmode {\cal Z} \else$\cZ$\fi'$$ is $\gS_6$-equivariant.
\end{theorem}
{\bf Proof:} Recall from Lemma \ref{l25aux} that $\ga$ blows up the ten
nodes and blows down the tangent cones of the nodes to the quadric surfaces
(on $\ifmmode {\cal I} \else$\cI$\fi_4$). $\Pi:X\longrightarrow} \def\sura{\twoheadrightarrow h_p$ is a double cover branched along $\ifmmode {\cal S} \else$\cS$\fi_3\cup
\ifmmode {\cal N} \else$\cN$\fi_5$, and we can calculate the image of the branch locus under $\ga$.
The ten nodes get blown up, the ten quadric cones (in ${\Bbb P}^4$) get blown
down to quadric surfaces (in the exceptional ${\Bbb P}^3$'s). Let $\~C\subset} \def\nni{\supset} \def\und{\underline X$ be
the inverse image in $X$ of the union of the ten quadric cones; then on
$X\backslash \~C$, $\ga$ is {\it biregular}. On the other hand, $\ga(\~C)$ is just
the union of the ten quadric surfaces of the intersection $\ifmmode {\cal I} \else$\cI$\fi_4\cap
\ifmmode {\cal W} \else$\cW$\fi_{10}$. Consequently
$$\ga_{|X\backslash \~C}:X\backslash\~C \longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal Z} \else$\cZ$\fi'\backslash(\eta')^{-1}(\ifmmode {\cal I} \else$\cI$\fi_4\cap \ifmmode {\cal W} \else$\cW$\fi_{10})$$ is
a reguar morphism of double covers, and letting $C\subset} \def\nni{\supset} \def\und{\underline h_p$ denote the ten
quadric cones, $X\backslash \~C\longrightarrow} \def\sura{\twoheadrightarrow h_p\backslash C$ is a double cover, as is also
$$\ifmmode {\cal Z} \else$\cZ$\fi'\backslash(\eta')^{-1}(\ifmmode {\cal I} \else$\cI$\fi_4\cap \ifmmode {\cal W} \else$\cW$\fi_{10}) \longrightarrow} \def\sura{\twoheadrightarrow (h_p)^{\vee}\backslash \ifmmode {\cal I} \else$\cI$\fi_4\cap
\ifmmode {\cal W} \else$\cW$\fi_{10}, $$ while $\ga(C)=\ifmmode {\cal I} \else$\cI$\fi_4\cap \ifmmode {\cal W} \else$\cW$\fi_{10}$. Hence in the diagram
$$\begin{array}{rcl} X & \stackrel{\Xi}{\longrightarrow} \def\sura{\twoheadrightarrow} & \ifmmode {\cal Z} \else$\cZ$\fi' \\ \downarrow & &
\downarrow \\ h_p & \stackrel{\ga}{\longrightarrow} \def\sura{\twoheadrightarrow} & (h_p)^{\vee} \end{array}$$
$\Xi$ is regular outside of $\~C$ and maps $\~C$ to $(\eta')^{-1}(\ifmmode {\cal I} \else$\cI$\fi_4\cap
\ifmmode {\cal W} \else$\cW$\fi_{10})$. Furthermore, everything is defined $\gS_6$-equivariantly. This
proves the Theorem. \hfill $\Box$ \vskip0.25cm
\begin{corollary}\label{c157.1} $\ifmmode {\cal I} \else$\cI$\fi_5$ sits $\gS_6$-equivariantly
birationally in the center of the diagram \unitlength1cm
$$\begin{picture}(2,2)\put(0,0){${\cal W}$}
\put(.5,0.15){\vector(1,0){1.3}} \put(1.9,0){${\Bbb P}^4$.}
\put(0,1.6){${\cal Z}$} \put(.5,1.75){\vector(1,0){1.3}}
\put(1.9,1.6){${\cal Y}$}
\put(.25,1.5){\vector(0,-1){1}}\put(1.95,1.5){\vector(0,-1){1}}
\put(.4,1.6){\vector(1,-1){.5}} \put(.9,.9){${\cal Z}'$}
\put(1.2,.8){\vector(1,-1){.5}}
\end{picture}$$
This shows the relation between the quintic $\ifmmode {\cal I} \else$\cI$\fi_5$ and the Coble variety
$\ifmmode {\cal Y} \else$\cY$\fi$. \end{corollary} {\bf Proof:} We have the series of modifications
$$\begin{array}{ccccccc} \ifmmode {\cal I} \else$\cI$\fi_5 & \longrightarrow} \def\sura{\twoheadrightarrow & \ifmmode {\cal I} \else$\cI$\fi_{5,p} & \stackrel{\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda}{\longrightarrow} \def\sura{\twoheadrightarrow} &
X & \stackrel{\Xi}{\longrightarrow} \def\sura{\twoheadrightarrow} & \ifmmode {\cal Z} \else$\cZ$\fi' \\ & & & & \downarrow & & \downarrow \\ &
& & & {\Bbb P}^4 & \stackrel{\ga}{\longrightarrow} \def\sura{\twoheadrightarrow} & {\Bbb P}^4, \end{array}$$ where
$\ifmmode {\cal I} \else$\cI$\fi_5\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {\cal I} \else$\cI$\fi_{5,p}$ blows up the node $p$, $\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda$ blows down the 15
${\Bbb P}^3$'s through the node to the 15 ${\Bbb P}^2$'s of the intersection
$\ifmmode {\cal S} \else$\cS$\fi_3\cap \ifmmode {\cal N} \else$\cN$\fi_5$, $\ga$ and $\Xi$ are as described above. Since $\ifmmode {\cal Z} \else$\cZ$\fi'$
clearly sits in the center of the diagram and all modifications are
$\gS_6$-equivariant, the Corollary follows. \hfill $\Box$ \vskip0.25cm
\section{${\cal I}_5$ and cubic surfaces}
\subsection{The Picard group}
Let $A_1(\ifmmode {\cal I} \else$\cI$\fi_5)$ be the Chow group of Weil divisors modulo algebraic
equivalence. Clearly a generic hyperplane section yields an element in
$A_1(\ifmmode {\cal I} \else$\cI$\fi_5)$, which we denote by $n$. Recall the reducible hyperplane
sections of Proposition \ref{pQ7.1} which split each into the union of five
copies of ${\Bbb P}^3$. These subvarieties are divisors on $\ifmmode {\cal I} \else$\cI$\fi_5$, hence also
yield classes in the Chow group. These 45 divisors are related by 27
relations, the sum of the five classes in the Chow group being equivalent
to $n$. Since $\ifmmode {\cal I} \else$\cI$\fi_5$ is normal, we have an injection $\hbox{Pic}} \def\Jac{\hbox{Jac}(\ifmmode {\cal I} \else$\cI$\fi_5)\hookrightarrow} \def\hla{\hookleftarrow
A_1(\ifmmode {\cal I} \else$\cI$\fi_5)$. Let $\hbox{Pic}} \def\Jac{\hbox{Jac}^0(\ifmmode {\cal I} \else$\cI$\fi_5)$ denote the orthogonal complement of the
class $n$ in $A_1(\ifmmode {\cal I} \else$\cI$\fi_5)$ with respect to this injection. Then we have
\begin{lemma}\label{liq6.1} We have an exact sequence of $\integer$-modules,
$$0\longrightarrow} \def\sura{\twoheadrightarrow \integer^6\longrightarrow} \def\sura{\twoheadrightarrow \integer^{27}\longrightarrow} \def\sura{\twoheadrightarrow \integer^{45} \longrightarrow} \def\sura{\twoheadrightarrow {\em\hbox{Pic}} \def\Jac{\hbox{Jac}}^0(\ifmmode {\cal I} \else$\cI$\fi_5) \longrightarrow} \def\sura{\twoheadrightarrow 0.$$
\end{lemma}
{\bf Proof:} The 45 ${\Bbb P}^3$'s are classes in $A_1(\ifmmode {\cal I} \else$\cI$\fi_5)$ which generate
$\hbox{Pic}} \def\Jac{\hbox{Jac}^0(\ifmmode {\cal I} \else$\cI$\fi_5)$ (as they contain all singularities), and the 27 relations
are those just mentioned, given by the 27 hyperplane sections. So the
sequence is clear as soon as we have shown that the rank of $\hbox{Pic}} \def\Jac{\hbox{Jac}^0(\ifmmode {\cal I} \else$\cI$\fi_5)$
is 24 (see the sequence (\ref{e48a.1})). This now follows from the
Lefschetz hyperplane theorem, as the dimension of $\ifmmode {\cal I} \else$\cI$\fi_5$ is four, so there
is an isomorphism between the $H^2$'s of $\ifmmode {\cal I} \else$\cI$\fi_5$ and a hyperplane section.
We may apply the Lefschetz theorem because the singularities of $\ifmmode {\cal I} \else$\cI$\fi_5$ and
of a hyperplane section are local complete intersections (see the book by
Goresky \& MacPherson for details). \hfill $\Box$ \vskip0.25cm Note that this sequence displays
$\hbox{Pic}} \def\Jac{\hbox{Jac}^0(\ifmmode {\cal I} \else$\cI$\fi_5)$ as an {\em irreducible} $W(E_6)$-module. Furthermore, we
see that just as in (\ref{e48a.1}), this sequence is dual to the
corresponding sequence for cubic surfaces.
\subsection{${\cal I}_5$ and cubic surfaces: combinatorics}
We collect the facts relating the combinatorics of the 27 lines with those of
$\ifmmode {\cal I} \else$\cI$\fi_5$ in Table \ref{table26}.
\begin{table}[htb]
\caption{\label{table26} Combinatorics of ${\cal I}_5$ and the 27 lines}
\vspace*{.5cm}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|l|l|}\hline
Locus on a cubic surface (see Table \ref{table20}) &
Locus on ${\cal I}_5$ \\ \hline \hline
27 lines $a_i, b_i, c_{ij}$ & 27 hyperplane sections $\{a_i=0\}\cap {\cal
I}_5$, etc. \\ \hline
2 lines are skew & \parbox{7cm}{the hyperplanes intersect in one of 216
${\Bbb P}^3$'s dual to the lines of \ref{i4}; this ${\Bbb P}^3$ intersects ${\cal I}_5$
in the union of three planes and a quadric (see Lemma \ref{l126.1})} \\ \hline
\parbox{7cm}{two lines are in a tritangent} & \parbox{7cm}{the
hyperplanes intersect in one of the 45 ${\Bbb P}^3$'s of \ref{i8} } \\ \hline
45 tritangents & the 45 ${\Bbb P}^3$'s of \ref{i8} \\ \hline
\parbox{7cm}{Two tritangents meet in a line of the cubic surface} &
\parbox{7cm}{two of the 45 ${\Bbb P}^3$'s meet in a ${\Bbb P}^2$; this is one of the
planes in the ${\Bbb P}^3$ defining the arrangement ${\cal A}(W({\bf D_4}))$ as
discussed in \ref{i8}} \\ \hline
\parbox{7cm}{Two tritangents meet in a line outside of the cubic surface} &
\parbox{7cm}{two of the 45 ${\Bbb P}^3$'s are {\em skew}, i.e., meet only in a
line; this line is part of the singular locus of the arrangement ${\cal
A}(W({\bf D_4}))$ just mentioned} \\ \hline
36 double sixes & \parbox{7cm}{36 triple points of ${\cal I}_5$ AND 36 copies
of the Nieto quintic ${\cal N}_5$} \\ \hline
\parbox{7cm}{Two double sixes are azygetic} & \parbox{7cm}{two of the triple
points lie on one of the 120 lines of the singular locus of ${\cal I}_5$} \\
\hline
\parbox{7cm}{Two double sixes are syzygetic} & \parbox{7cm}{two of the triple
points do not lie on one of the 120 lines} \\ \hline
\parbox{7cm}{A line is {\em not} contained in a double six} &
\parbox{7cm}{the hyperplane dual to the line contains the triple point which
corresponds to the double six} \\ \hline \end{tabular}
\end{table}
\section{The dual variety}
Let $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$ be the projective dual variety to $\ifmmode {\cal I} \else$\cI$\fi_5$; since
$\ifmmode {\cal I} \else$\cI$\fi_5$ is invariant under $W(E_6)$, so is $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$. Although we
do not have explicit equations for $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$, we can say quite a
bit about its geometry, just from the fact that it is dual to $\ifmmode {\cal I} \else$\cI$\fi_5$.
\subsection{Degree}
First we show that {\em degree of\ $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$=10m+4k}. Quite
generally, one can say the following. Suppose we are given a variety
$X\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^n$ which has singular locus consisting of a set of {\em lines},
meeting each other in a set of {\em points}, and let us further assume
that the situation is symmetric, i.e., each line contains the same
number of points, each point being hit by the same number of lines;
let us denote these numbers by $N=\#$ lines, $M=\#$ points, $\gn=\#$
points in each line and $\gm=\#$ lines meeting at each point. Consider
the dual variety $X^{\vee}$. We claim:
\begin{itemize}
\item[-] There are $N$ ${\Bbb P}^{n-2}$'s $\subset} \def\nni{\supset} \def\und{\underline X^{\vee}$.
\item[-] Each ${\Bbb P}^{n-2}$ is cut out by $\gn$ hyperplanes.
\item[-] There are $M$ such special hyperplane sections of $X^{\vee}$.
\item[-] Each of the $M$ hyperplanes meets $X^{\vee}$ in $\gm$ of the
$N\ {\Bbb P}^{n-2}$'s.
\item[-] Hence, deg($X^{\vee}$)=$m\gm+rest$,
\end{itemize}
where the $rest$ is given in terms of the local geometry around the given
point.
The proofs of these are immediate: each of the points corresponds to a
hyperplane (=set of all hyperplanes through the point), each line defines
dually a ${\Bbb P}^{n-2}$, and since $X$ is singular along the line, each hyperplane
through the line is {\em tangent} to $X$ there $\Rightarrow$ the dual
${\Bbb P}^{n-2}\subset} \def\nni{\supset} \def\und{\underline X^{\vee}$. The other statements are then clear.
To determine $rest$, consider the following. The set theoretic
image of the given point in the dual variety is the {\em total} transform
({\em not} the proper transform) of the given point. This is set
theoretically easy to compute, but there may be a multiplicity coming in.
We apply these considerations to $\ifmmode {\cal I} \else$\cI$\fi_5$ and $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$: on $\ifmmode {\cal I} \else$\cI$\fi_5$
we have singular lines, $N$=120, $M$=36, $\gn$=3, $\gm$=10, and hence
$deg(\ifmmode {\cal I} \else$\cI$\fi_5^{\vee})=m10+rest.$ In our case $rest$ is easy to figure out:
recall that we resolved the singularities of \ifmmode {\cal I} \else$\cI$\fi$_5$ by blowing up the 36
points, then the 120 lines; the resolving divisors over the points were copies
of the Segre cubic. The variety dual to the Segre cubic is the Igusa quartic,
and the image of the ten nodes on the Segre cubic are ten quadric
surfaces (\ref{e117a.1}) which are {\em tangent hyperplane sections},
i.e., the hyperplanes which meet the Igusa quartic in one such quadric
and are tangent to it there. These ten hyperplanes are of course just the
10 ${\Bbb P}^3$'s on the dual variety being cut out by the chosen hyperplane
section (see Proposition \ref{p160.1} below).
This hyperplane section of $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$ may be {\em tangent}
to $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$ along the Igusa quartic, hence
$$deg(\ifmmode {\cal I} \else$\cI$\fi_5^{\vee})=10m+k\cdot4.$$
\subsection{Singular locus}
Consider the 45 ${\Bbb P}^3$'s on \cIf; since there is a pencil of hyperplanes
through each, the dual variety $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$\ will have 45 singular
lines, which meet in 27 points (which are dual to the 27 hyperplanes
cutting out the 45 ${\Bbb P}^3$'s).
These 27 points are of course $A_i, B_i, C_{ij}$.
Applying our reasoning from above to this we see that
deg(${\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}}^{\vee}$)=5+ rest. We conclude rest=0, or in other words,
{\em a resolution of singularities of ${\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}}$ is affected by blowing
up the 45 lines simultaneously; there is no exceptional divisor over the 27
points.}
However, since we are dealing with fourfolds, $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$ could even be
normal and still have a singular locus of dimension two. For example, it is
reasonable to believe that the ten quadrics on each copy of the Igusa quartic
$\ifmmode {\cal I} \else$\cI$\fi_4$ on the reducible hyperplane sections discussed below might be {\em
singular} on $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$, but that is of course just a guess.
Furthermore, there is no reason whatsoever why the dual variety should be
normal. In fact, it is a case of great exception when the dual variety is
normal, the general case being that there is a singular parabolic divisor
(coming from the intersection $\hbox{Hess}} \def\rank{\hbox{rank}(X)\cap X$), as well as a double point
locus, also (in general) a divisor, coming from the set of bitangents.
In our case, however, since $\hbox{Hess}} \def\rank{\hbox{rank}(\ifmmode {\cal I} \else$\cI$\fi_5)\cap \ifmmode {\cal I} \else$\cI$\fi_5$ consists of the union of
the 45 ${\Bbb P}^3$'s, all of which get blown down, there is no parabolic {\em
divisor}. But there is no easy way to exclude a double point divisor.
\subsection{Reducible hyperplane sections}
As already mentioned, $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$\ contains 120 ${\Bbb P}^3$'s, each being cut
out by three of the $h$'s, (in fact by a triple of azygetic double sixes), and
each
such intersection $h\cap \ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$ consists of ten such ${\Bbb P}^3$'s, plus a
copy of the Igusa quartic. There are 36 such hyperplane sections which
decompose
into ten ${\Bbb P}^3$'s and a copy of the Igusa quartic:
\begin{proposition}\label{p160.1} The 36 hyperplane sections ${\bf h}\cap
\ifmmode {\cal I} \else$\cI$\fi_5$, ${\bf h}=h,\ h_{ij},\ h_{ijk}$, are reducible, consisting of ten
${\Bbb P}^3$'s and a copy of the Igusa quartic $\ifmmode {\cal I} \else$\cI$\fi_4$. The ten ${\Bbb P}^3$'s are just
the $K_{ijk}$ of (\ref{e117a.1a}), each a bitangent plane to $\ifmmode {\cal I} \else$\cI$\fi_4$.
\end{proposition}
{\bf Proof:} These are the 36 hyperplanes dual to the 36 triple points of
$\ifmmode {\cal I} \else$\cI$\fi_5$; at each such $p$ ten of the 120 lines meet, and the triple point
itself yields the copy of $\ifmmode {\cal I} \else$\cI$\fi_4$ (it is blown up with exceptional divisor
$(\ifmmode {\cal S} \else$\cS$\fi_3)_p$, which is dual to $(\ifmmode {\cal I} \else$\cI$\fi_4)_p$, a copy of $\ifmmode {\cal I} \else$\cI$\fi_4$). \hfill $\Box$ \vskip0.25cm
So restricted to the triple point, the duality $\ifmmode {\cal I} \else$\cI$\fi_5- - \ra \ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$
yields precisely the dual map $\ga$ of (\ref{e126.1})!
The 120 ${\Bbb P}^3$'s meet two at a
time in 270 ${\Bbb P}^2$'s, each of which is cut out by six of the $h$'s (2 triples
of
azygetic double sixes, two rows in a triple). Note that these 270 ${\Bbb P}^2$'s
are
the $t_6(2)$ of Table \ref{table23}.
Each ${\Bbb P}^2$ contains two nodes
and five of the 27 points, as well as two of the 45 lines. Through each
such line two of these ${\Bbb P}^2$'s pass (as each line is cut out by 12 of the
$h$'s). Therefore in each $h$ we have ten ${\Bbb P}^3$'s meeting in ${10 \choose
2}=45\ {\Bbb P}^2$'s which meet in 15 of the 45 ${\Bbb P}^1$'s, and contain 15 of the 27
points. The 15 lines and 15 points are just the singular locus of the Igusa
quartic, and the ten ${\Bbb P}^3$'s are tangent to the Igusa quartic along quadrics,
as mentioned earlier.
\subsection{Special hyperplane sections}
Inspection of the 27 forms and 27 points in ${\Bbb P}^5$ shows that each of the 27
hyperplanes contains {\em none} of the 27 points and {\em none} of the 45
lines;
it follows that hyperplane sections such as $\ifmmode {\cal K} \else$\cK$\fi:=\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}\cap \{a_1=0\}$
are irreducible hypersurfaces in ${\Bbb P}^4$ with 45 isolated singularities,
coming from the intersections with the singular lines of $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$.
As mentioned above, there may also be a singular locus coming from other
singularities on $\ifmmode {\cal I} \else$\cI$\fi_5^{\vee}$.
Furthermore, there are 40 ${\Bbb P}^2$'s lying on this threefold, and 16 hyperplanes
in
$a_i$ which cut out ten of these on $\ifmmode {\cal K} \else$\cK$\fi$. The 16 hyperplanes are those 16 of
the 216 ${\Bbb P}^3$'s which lie in $a_i$, corresponding to the 16 lines which $a_i$
is skew to. The symmetry group of this threefold is $W(D_5)$. This is a {\em
degeneration} of a {\em generic} hyperplane section, which will contain 120
${\Bbb P}^2$'s.
\bigskip
|
1995-03-30T07:20:48 | 9503 | alg-geom/9503024 | en | https://arxiv.org/abs/alg-geom/9503024 | [
"alg-geom",
"math.AG"
] | alg-geom/9503024 | Heath Martin | Heath M. Martin and Juan C. Migliore | Degrees of generators of ideals defining subschemes of projective space | 27 pages, LaTeX v 2.09 | null | null | null | null | For an arithmetically Cohen--Macaulay subscheme of projective space, there is
a well-known bound for the highest degree of a minimal generator for the
defining ideal of the subscheme, in terms of the Hilbert function. We prove a
natural extension of this bound for arbitrary locally Cohen--Macaulay
subschemes. We then specialize to curves in $\pthree$, and show that the curves
whose defining ideals have generators of maximal degree satisfy an interesting
cohomological property. The even liaison classes possessing such curves are
characterized, and we show that within an even liaison class, all curves with
the property satisfy a strong Lazarsfeld--Rao structure theorem. This allows us
to give relatively complete conditions for when a liaison class contains curves
whose ideals have maximal degree generators, and where within the liaison class
they occur.
| [
{
"version": "v1",
"created": "Wed, 29 Mar 1995 18:57:19 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Martin",
"Heath M.",
""
],
[
"Migliore",
"Juan C.",
""
]
] | alg-geom | \section{Degrees of Generators}
Throughout this paper, we will work over an algebraically closed field $k$,
of arbitrary characteristic. Let $S = k[x_0, \dots, x_n]$
be the polynomial ring over $k$, and ${\Bbb P}^n$ the $n$-dimensional
projective space over $k$. We will furthermore consider only
subschemes of ${\Bbb P}^n$ which are locally Cohen--Macaulay and equidimensional.
It is well-known that this is equivalent to requiring that all
the intermediate cohomology modules $H^i_*({\Bbb P}^n, V)$, $1 \leq i \leq \dim V$,
have finite length.
Given a subscheme $V$ of ${\Bbb P}^n$, with $\dim V = d$, let $I_V$ denote
its homogeneous, saturated defining ideal in $S$. Thus $S/I_V$ is
a standard graded $k$-algebra, and so we can define the Hilbert
function of $S/I_V$ by
$$
H(S/I_V, t) = \dim_k [S/I_V]_t.
$$
Alternatively, we sometimes write $H(V, t)$ for $H(S/I_V, t)$. It
is a standard fact that there is a polynomial $P(S/I_V, t)$, having
degree $d$, such that $H(S/I_V, t) = P(S/I_V, t)$ for
all $t \gg 0$. We furthermore define the $n$th difference
of $H(S/I_V, t)$ inductively as follows:
\begin{eqnarray*}
\Delta^1 H(S/I_V,t) &=& H(S/I_V, t) - H(S/I_V, t-1) \\
\Delta^n H(S/I_V,t) &=& \Delta^1 (\Delta^{n-1} H(S/I_V, t)).
\end{eqnarray*}
Now, since $H(S/I_V, t)$ is eventually a polynomial of degree $d$,
the function $\Delta^{d+1} H(S/I_V, t)$ is eventually zero, and we
define
$$
\sigma(S/I_V) = \min \{\, k : \Delta^{d+1} H(S/I_V, t) = 0 \mbox{ for all
$t \geq k$} \, \}.
$$
Again, we will sometimes write $\sigma(V)$ for $\sigma(S/I_V)$.
It is worth noting that if $\dim V = d$, then
the Hilbert function of $V$ and the Hilbert polynomial
of $V$ are equal in all degrees $\geq t + d + 1$ if
and only if $\sigma(V) = t$.
The degree at which the Hilbert function and
the Hilbert polynomial agree from then on is sometimes called,
at least in the context of local algebra, the postulation number,
and has played an important role in
questions about Cohen--Macaulayness and related invariants in local rings.
Given an ideal $I$, we will write $\alpha(I)$ for the minimal degree
of a minimal generator, and $\omega(I)$ for the maximal
degree of a minimal generator of $I$.
Next, let $H^i_*({\Bbb P}^n, {\cal I}_V) = \oplus_{s\in {\Bbb Z}} H^i({\Bbb P}^n, {\cal
I}_V(s))$
be the cohomology modules of $V$. We will put
$h^i({\Bbb P}^n, {\cal I}_V(t)) = \dim_k H^i({\Bbb P}^n, {\cal I}_V(t))$.
We let $e(V) = \max\{\, t : h^{d+1}({\Bbb P}^n, {\cal I}_V(t)) \not=0\,\}$
denote the index of speciality of $V$. Also, note
that by our assumption that subschemes be locally Cohen--Macaulay
and equidimensional, $h^i({\Bbb P}^n, {\cal I}_V(t))$ is non-zero
for only finitely many $t$, when $1 \leq i \leq d$.
Hence for a non-arithmetically Cohen--Macaulay curve $C$ in ${\Bbb P}^n$,
with notation following Martin-Deschamps and Perrin, we can define
$$
r_a(C) = \min \{\, n \in {\Bbb Z} : h^1({\Bbb P}^n, {\cal I}_C(n)) \not = 0 \,\} \quad\quad
r_o(C) = \max \{\, n \in {\Bbb Z} : h^1({\Bbb P}^n, {\cal I}_C(n)) \not = 0 \,\},
$$
and $\mathop{\rm diam\,} H^1_*({\Bbb P}^n, C) = r_o(C) - r_a(C) + 1$, the number of
components between the first and the last non-zero components, inclusive.
Note that some of the intermediate components may have dimension zero,
but we also count them. If $C$ is arithmetically Cohen--Macaulay,
then $H^1_*({\Bbb P}^3, C) = 0$, and we will put $\mathop{\rm diam\,} H^1_*({\Bbb P}^3, C) = 0$.
In this section, we prove a statement about the maximal degree
of a generator for the defining ideal of a curve,
in terms of the Hilbert function and the cohomology
of the curve. The relationship between these two objects is well-known,
and we spell it out explicitly in the first lemma.
\begin{lemma} \label{hilbfcn_eqn}
Let $C$ be a curve in ${\Bbb P}^n$. Then
\begin{eqnarray*}
\Delta^2 H(C,t) &=& h^2({\Bbb P}^n, {\cal I}_C(t)) - 2h^2({\Bbb P}^n, {\cal I}_C(t-1)) + h^2({\Bbb P}^n, {\cal I}_C(t-2))
\\
&& \quad\quad \mbox{} - h^1({\Bbb P}^n, {\cal I}_C(t)) + 2h^1({\Bbb P}^n, {\cal I}_C(t-1)) - h^1({\Bbb P}^n,
{\cal I}_C(t-2))
\end{eqnarray*}
\end{lemma}
\begin{proof} First, recall that the Hilbert polynomial is given
by $P(C,t) = h^0({\Bbb P}^n, {\cal O}_C(t)) - h^1({\Bbb P}^n, {\cal O}_C(t))$
(see \cite[Exercise III.5.2]{hart}). Thus,
from the short exact sequence
$$
0 \rightarrow I_t \rightarrow S_t \rightarrow H^0({\Bbb P}^n, {\cal O}_C(t))
\rightarrow H^1({\Bbb P}^n, {\cal I}_C(t)) \rightarrow 0,
$$
we obtain
\begin{eqnarray*}
H(C,t) &=& h^0({\Bbb P}^n, {\cal O}_C(t)) - h^1({\Bbb P}^n, {\cal I}_C(t)) \\
&=& P(C,t) + h^2({\Bbb P}^n, {\cal I}_C(t)) - h^1({\Bbb P}^n, {\cal I}_C(t)).
\end{eqnarray*}
Now, since $P(C, t)$ is a polynomial of degree $1$, on taking second
differences we obtain
\begin{eqnarray*}
\Delta^2 H(C,t) &=& h^2({\Bbb P}^n, {\cal I}_C(t)) - 2h^2({\Bbb P}^n, {\cal I}_C(t-1)) + h^2({\Bbb P}^n, {\cal I}_C(t-2))
\\
&& \quad\quad \mbox{} - h^1({\Bbb P}^n, {\cal I}_C(t)) + 2h^1({\Bbb P}^n, {\cal I}_C(t-1)) - h^1({\Bbb P}^n,
{\cal I}_C(t-2))
\end{eqnarray*}
which is what we wanted to show.
\end{proof}
The following corollary is an immediate consequence.
\begin{cor} \label{Delta-coh}
$\Delta^2 H(C, t) = 0$ for all $t \geq k+2$ if and only
if $h^2({\Bbb P}^n, {\cal I}_C(t)) = h^1({\Bbb P}^n, {\cal I}_C(t))$ for all $t \geq k$. \mbox{\hskip 1cm $\rlap{$\sqcap$}\sqcup$}
\end{cor}
\begin{prop} \label{equal-coh}
Suppose $C$ is a curve in ${\Bbb P}^n$ defined by an ideal $I = I_C$.
If $\omega(I) = \sigma(C) + k$ for some $k \geq 1$, then
$e(C) = r_o(C)$ and $h^2({\Bbb P}^n, {\cal I}_C(t)) = h^1({\Bbb P}^n, {\cal I}_C(t))$
for $t \geq e(C) -k+1$.
\end{prop}
\begin{proof} We first show that $e(C) = r_o(C)$. If not,
let $m = \max \{\, e(C), r_o(C) \,\}$. Then by Castelnuovo--Mumford
regularity \cite{mumford}, we have
$$
\sigma(C) < \sigma(C) + k = \omega(I) \leq \mathop{\rm reg\,}(I) \leq m+3.
$$
But $\Delta^2 H(C, m+2) \not=0$, because of Lemma~\ref{hilbfcn_eqn}
and so $\sigma(C) = m+3$, which is a contradiction.
Thus, we must have $e(C) = r_o(C) = m$. In particular,
$\mathop{\rm reg\,}(I) = m+3$, and so again by Castelnuovo--Mumford
regularity, we have $\sigma(C) = \omega(I) - k \leq \mathop{\rm reg\,}(I) - k = m + 3 - k$.
Thus, we have $\Delta^2 H(C, t) = 0$ for $t \geq m+3-k$, and so by
Lemma~\ref{hilbfcn_eqn}, $h^1({\Bbb P}^n, {\cal I}_C(t)) = h^2({\Bbb P}^n, {\cal I}_C(t))$
for $t \geq m-k+1$.
\end{proof}
\begin{prop} \label{maxdegree}
Suppose $I = I_C$ defines a curve $C$ in ${\Bbb P}^n$.
Then $\omega(I) \leq \sigma(S/I) + \mathop{\rm diam\,} H^1_*({\Bbb P}^n, {\cal I}_C)$.
\end{prop}
\begin{proof}
This follows immediately from the previous proposition, since
$h^1({\Bbb P}^n, {\cal I}_C(t))$ and $h^2({\Bbb P}^n, {\cal I}_C(t))$ can only possibly be
non-zero and
equal for $k=\mathop{\rm diam\,} H^1_*({\Bbb P}^n, {\cal I}_C)$ degrees.
\end{proof}
\begin{remark} Also note that Proposition~\ref{maxdegree} includes
the case that $C$ is arithmetically Cohen--Macaulay, and says
that $\omega(I_C) \leq \sigma(S/I_C)$. This is a result in
\cite{DGM}; see also \cite[Proposition 1.2]{CGO}.
\end{remark}
\begin{remark} Some comments about the proof of this result are in order.
First of all, if either $e(C) \not= r_o(C)$ or if $e(C) = r_o(C)$ and
$h^1({\Bbb P}^n, {\cal I}_C(r_o)) \not= h^2({\Bbb P}^n, {\cal I}_C(r_o))$, we get
$\omega(I) \leq \sigma(S/I)$. This is the same bound as when $C$
is assumed to be arithmetically Cohen--Macaulay. Thus,
the cases of most interest occur when $H^1_*({\Bbb P}^n, {\cal I}_C)$ and
$H^2_*({\Bbb P}^n, {\cal I}_C)$ both become zero at the same degree, and
moreover have equal dimensions for some number of preceding degrees.
Essentially, Proposition~\ref{equal-coh} says that
having a generator of high degree forces $h^1({\Bbb P}^n, {\cal I}_C(t))$
and $h^2({\Bbb P}^n, {\cal I}_C(t))$ to be equal in a large number of degrees.
\end{remark}
Curves which are not arithmetically Cohen--Macaulay and have a generator
of maximum degree in the sense of the above proposition, must be
``almost Buchsbaum.'' This means that a general linear form $L$
induces a multiplication on $H^1_*({\Bbb P}^n, {\cal I}_C)$ which has non-trivial
kernel in each degree. As notation, if $L$ is a linear form,
let $K_L$ be the kernel of the multiplication on $H^1_*({\Bbb P}^n, {\cal I}_C)$
induced by $L$.
\begin{prop} \label{almostBuchsbaum}
Let $C$ be a non-arithmetically Cohen--Macaulay curve
in ${\Bbb P}^3$, and suppose that $h^1({\Bbb P}^n, {\cal I}_C(t)) = h^2({\Bbb P}^n, {\cal I}_C(t))$
in the last $r$ degrees. Let $L$ be a general linear form and
let $K = K_L$. Then $\dim_k K_{t} > \dim_k K_{t+1} > 0$ for
all $t = r_o(C) - r + 1, \dots, r_o(C) - 1$.
\end{prop}
\begin{proof} Let $L$ be a general general linear form defining
a hyperplane $H$, and let $Z = C \cap H$ be the hyperplane section
of $C$ considered as a subscheme of ${\Bbb P}^{n-1}$. Then it is easy to
see that
$$
\Delta^2 H(C, t) = \Delta^1 H(Z, t) + \Delta^1 \dim_k K_{t-1}.
$$
Let $p = r_o(C)$, and note that by the condition on cohomology
and by Lemma~\ref{hilbfcn_eqn}, we have
$$
0 = \Delta^2 H(C, p+2) = \Delta^1 H(Z, p+2) + \Delta^1 K_{p+1}.
$$
But $\dim_k K_{p+1} = 0$ and $\dim_k K_p > 0$, so
$\Delta^1 K_{p+1} < 0$. This implies that $\Delta^1 H(Z, p+2) > 0$,
and since $\Delta^1 H(Z, t)$ is non-increasing in the range
in which we are interested (see \cite{DGM}),
then $\Delta^1 H(Z,t) > 0$ for all
$t = r_o(C) - r + 3, \dots, r_o(C) + 2$. But the assumptions
on the cohomology of $C$ then imply
$$
0 = \Delta^2 H(C, t) = \Delta^1 H(Z,t) + \Delta^1 \dim _k K_{t-1}
$$
for $t = r_o(C) - r + 3, \dots, r_o(C) + 2$, and so
$\Delta^1 \dim_k K_{t-1} < 0$. That is, $\dim_k K_{t} > \dim_k K_{t+1} > 0$
for $t = r_o(C) - r+1, \dots, r_o(C) - 1$.
\end{proof}
As an immediate corollary, we obtain the following statement:
\begin{cor} \label{almostBuchs-deg}
Suppose $C$ is a curve in ${\Bbb P}^n$ having a generator
of degree $\sigma(C) + \mathop{\rm diam\,} H^1_*({\Bbb P}^n, {\cal I}_C)$. Then
for each general linear form $L$, $\dim K_t > \dim K_{t+1} > 0$,
for $t = r_a(C), \dots, r_o(C) - 1$. \mbox{\hskip 1cm $\rlap{$\sqcap$}\sqcup$}
\end{cor}
We can also use Proposition~\ref{almostBuchsbaum} to refine
the bound on the maximum degree of a generator.
\begin{cor} If $C$ is a curve in ${\Bbb P}^n$ defined by an ideal $I = I_C$,
then $\omega(I) \leq \sigma(C) + \mathop{\rm diam\,} K$.
\end{cor}
\begin{proof} Again, this follows from Proposition~\ref{equal-coh} and
Proposition~\ref{almostBuchsbaum}.
\end{proof}
More generally, we have the following result for subschemes of dimension
$d$ in ${\Bbb P}^n$:
\begin{prop}\label{general_maxdegree}
Let $V$ be a subscheme of ${\Bbb P}^n$ having dimension $d$,
defined by an ideal $I = I_V$. Then
$\omega(I) \leq \sigma(V) + \max \{\, \mathop{\rm diam\,} H^i_*({\Bbb P}^n, {\cal I}_V) :
i = 1, \dots, d\,\}$.
\end{prop}
\begin{proof} Since the proof of this result is quite similar to
the case of curves, we will only give an outline.
The Hilbert polynomial of $V$ is given by
$$
P(V, t) = \sum_{i=0}^d (-1)^i h^i({\Bbb P}^n, {\cal O}_V(t)),
$$
and so from the exact sequence
$$
0 \rightarrow I_t \rightarrow S_t \rightarrow H^0({\Bbb P}^n, {\cal O}_V(t))
\rightarrow H^1({\Bbb P}^n, {\cal I}_V(t)) \rightarrow 0,
$$
we see that the Hilbert function of $V$ is
$$
H(V, t) = P(V, t) + \sum_{i=1}^d (-1)^i h^i({\Bbb P}^n, {\cal I}_V(t)).
$$
Since $P(V,t)$ is a polynomial of degree $d$, when we take
$(d+1)$th differences, we get
$$
\Delta^{d+1} H(V, t) = \sum_{i=1}^d (-1)^i
\sum_{j=0}^{d+1} (-1)^j{{d+1} \choose j} h^i({\Bbb P}^n, {\cal I}_V(t-j)).
$$
Now we argue by cases. If none of the cohomology modules
end in the same place, it is easy to see by Castelnuovo--Mumford
regularity that $\omega(I) \leq \sigma(V)$.
Suppose, on the other hand, that some of the cohomology modules
end in the same degree $t$, say, and the others end
in degrees $< t$, and let $m$ be the
maximum of the diameters of the intermediate cohomologies.
Then $\mathop{\rm reg\,}(V) = t + d + 1$, and by the formula above,
$\sigma(V) \geq t - m +d + 1$, and since $\omega(I) \leq \mathop{\rm reg\,}(I)$,
the required inequality follows.
\end{proof}
We can be a bit more precise in a few cases. For instance,
suppose $V$ is a surface in ${\Bbb P}^n$. Then there are only
three non-zero cohomology modules, and as in the proof above
the Hilbert function of $V$ is given by
$$
H(V, t) = P(V, t) - h^3({\Bbb P}^n, {\cal I}_V(t)) + h^2({\Bbb P}^n, {\cal I}_V(t))
- h^1({\Bbb P}^n, {\cal I}_V(t)).
$$
Because the $h^3$ term and the $h^1$ term have the same sign,
the only cancellation that can occur comes from the $h^2$ term, and using
the same argument as above, we get the inequality
$\omega(I) \leq \sigma(I) + \mathop{\rm diam\,} H^2_*({\Bbb P}^n, {\cal I}_V)$.
In particular, if $V$ were a non-arithmetically Cohen--Macaulay
surface, with $H^2_*({\Bbb P}^n, {\cal I}_V) = 0$, then
$\omega(I) \leq \sigma(V)$, which is the same bound
as in the arithmetically Cohen--Macaulay case.
\section{Curves with equal cohomology}
The previous section showed that the property of having a generator
of high degree is very closely related to having equal (non-zero) cohomology
dimensionally in a large number of degrees. This section is devoted
to characterizing when an even liaison class of curves in ${\Bbb P}^3$
has this property. By a slight abuse of terminology, we make the
following definition:
\begin{defn} A curve $C$ in ${\Bbb P}^3$ is said to have {\em equal
cohomology} if $e(C) = r_o(C)$ and $h^1({\Bbb P}^3, {\cal I}_C(t)) = h^2({\Bbb P}^3,
{\cal I}_C(t))$
for $t = r_a(C), \dots, r_o(C)$.
\end{defn}
We first recall the structure theory for curves in ${\Bbb P}^3$
(which holds more generally for codimension $2$ subschemes of ${\Bbb P}^n$) initiated
by Lazarsfeld and Rao, and developed
in a series of papers, of which \cite{BBM} contains
the most general statement and proof.
See also the book \cite{juan-book} for a comprehensive overview
of liaison theory and the Lazarsfeld--Rao structure theory. First,
let $C$ be a curve, and choose a form $F \in I_C$ of degree $f$ and
a form $G \in S$ of degree $g$ so that $F$ and $G$ have no common
components. Then $I_Z = G \cdot I_C + (F)$ defines a curve $Z$ in ${\Bbb
P}^3$,
called a basic double link of $C$, and denoted
$$
C:(g,f) \rightarrow Z.
$$
It is easy to see that $Z$ is evenly linked to $C$, and that there is
a short exact sequence
$$
0 \rightarrow S(-g-f) \stackrel{[F\;G]}{\longrightarrow}
I_C(-g) \oplus S(-f) \stackrel{\phi}{\longrightarrow}
I_Z \rightarrow 0,
$$
where $\phi(r,s) = rG+sF$.
The Lazarsfeld--Rao property says essentially that
even liaison classes of curves are built up by this process
of basic double linkage. More precisely, let ${\cal L}$
be an even liaison class of curves in ${\Bbb P}^3$. Then the cohomology
module $M = H^1_*(C)$ is invariant up to shifts in grading as $C$ varies
in ${\cal L}$, and there is a leftmost shift of $M$ which
actually occurs as the deficiency module of a curve in ${\cal L}$, and
every rightward shift is realized. Thus ${\cal L}$ is parameterized
by shifts of $M$, and a curve $C_0$ which has $H^1_*(C_0)$
in the leftmost shift is called a {\em minimal curve}. Every other
curve $C$ in ${\cal L}$ is obtained from $C_0$ as follows: there is
a curve $C_m$ which is a deformation of $C$ through curves having
constant cohomology, and a series of basic double links
\begin{equation}\label{bdl-chain}
C_0 : (1, d_0) \rightarrow C_1 : (1, d_1) \rightarrow \cdots
\rightarrow C_{m-1} : (1, d_{m-1}) \rightarrow C_m.
\end{equation}
We can moreover choose the degrees to satisfy
$d_0 = \cdots = d_s < d_{s+1} < \cdots < d_{m-1}$. Note furthermore
that $C_i$ is in the $i$th shift of ${\cal L}$; that is, $H^1_*(C_i)$
is a rightward shift by $i$ degrees of $H^1_*(C_0)$. Also,
for curves in ${\Bbb P}^3$, the book \cite{mdp1} gives much more
information about the behavior of invariants along liaison classes,
and moreover gives an algorithm for computing the minimal curve
in the liaison class from the deficiency module
associated to the class.
We will need to have some information about how Hilbert functions
change as we move along an even liaison class by basic double linkage.
The following result is quite elementary.
\begin{lemma} \label{hilbfcn-bdl}
Suppose $I = I_C$ defines a curve $C$ in ${\Bbb P}^3$. Let $L$ be a
general hyperplane and let $F$ define a surface of degree $d$ containing
$C$. Form the basic double link $Z$ of $C$ by $L$ and $F$. Then
$$
\Delta^2 H(Z, t) = \left\{ \begin{array}{ll}
\Delta^2 H(C, t-1) + 1 & \mbox{\rm if $1 \leq t \leq d-1$} \\
\Delta^2 H(C, t-1) & \mbox{\rm if $t \geq d$.}
\end{array}\right.
$$
\end{lemma}
\begin{proof} Since $Z$ is a basic double link of $C$, we have a short
exact sequence
$$
0 \rightarrow S(-d-1) \rightarrow I_C(-1) \oplus S(-d) \rightarrow I_Z
\rightarrow 0.
$$
Using the additive properties of Hilbert functions, it is easy to see that
$\Delta^2 H(Z, t) = \Delta^2 H(C, t-1) + \Delta^3 H(F, t)$. Then
the statement follows, since $\Delta^3 H(F, t) = 1$ for
$0 \leq t \leq d-1$, and is zero otherwise.
Note that this is also in \cite[Corollary 2.3.5]{nollet}, in terms
of postulation characters.
\end{proof}
\begin{definition} \label{deltadef}
Suppose there is a chain
$$
{\cal C} : C_0 : (1, d_0) \rightarrow C_1 : (1, d_1) \rightarrow \cdots
\rightarrow C_{m-1} : (1, d_{m-1}) \rightarrow C_m
$$
of basic double links by surfaces $F_i$ having degrees $d_i$. Then
define $\delta({\cal C}, t, s)$ to be the number of $F_i$ such that
$d_i \geq t-s+i+1$, for $i = 1, \dots, s-1$.
\end{definition}
It is easy to see that
$\delta({\cal C}, t, s) \leq \delta({\cal C}, t-1, s)$.
The following result is a straightforward calculation using
Lemma~\ref{hilbfcn-bdl}.
\begin{cor} \label{hilbfcn-chain}
Let ${\cal C}$ as above be a chain of basic double links. Then for
each $s = 0, \dots, m$,
$$
\Delta^2 H(C_s, t) = \Delta^2 H(C_0, t-s) + \delta({\cal C},t,s).\mbox{\hskip 1cm $\rlap{$\sqcap$}\sqcup$}
$$
\end{cor}
The main result of this section concerns when an even liaison class has a curve
with $h^1$ and $h^2$ equal for some number of places, and
gives a characterization
in terms of the Hilbert function of the minimal curve in the liaison
class. We first state a more general version, from which we
can trivially make a statement about minimal curves.
\begin{thm} \label{coh-prop} Let ${\cal L}$ be an even liaison
and let $C_0$ be a curve in ${\cal L}$.
Then a sequence
$$
C_0 \rightarrow \dots \rightarrow C_m = C
$$
of basic double links can be constructed
with $C$ having $h^1({\Bbb P}^3, {\cal I}_C(t)) = h^2({\Bbb P}^3, {\cal I}_C(t))$ in the last
$r$ places if and only if $C_0$ has
$e(C_0) \leq r_o(C_0)$ and
has Hilbert function satisfying
$$
\Delta^2 H(C_0, r_o(C_0) - r + 3) \leq \Delta^2 H(C_0, r_o(C_0) - r + 4) \leq
\cdots \leq \Delta^2 H(C_0, r_o(C_0) + 2) \leq 0.
$$
\end{thm}
\begin{proof}
Suppose that $C_0$ has $e(C_0) \leq r_o(C_0)$ and Hilbert function
$$
\Delta^2 H(C_0, t) = \quad \cdots \quad t_1 \quad t_2 \quad \cdots \quad
t_r \quad 0 \quad \cdots
$$
where $t_1 \leq t_2 \leq \cdots \leq t_r \leq 0$, and the term $t_r$ occurs
in degree $r_o(C_0) + 2$.
If not all the $t_i$ are equal, let $s$, $1 \leq s \leq r$, be
the first integer for which $t_{s-1} < t_s$. Otherwise, let $s=r+1$.
Note that $I_{C_0}$ must have elements in degree $\leq r_o(C_0)-r+3$, so we can
form the basic double link
$C_0:(1, r_o(C_0)-r+s+1) \rightarrow C_1$.
Then by Lemma~\ref{hilbfcn-bdl}, $C_1$ has Hilbert function
$$
\Delta^2 H(C_1, t) = \quad \cdots \quad t_1 + 1 \quad \cdots \quad t_{s-1}+1
\quad t_s \quad \cdots \quad t_r \quad 0 \quad \cdots,
$$
where now the term $t_r$ occurs in degree $r_o(C_0)+3$.
Continuing by induction,
we construct a sequence of basic double links to a curve $C_m$ having
Hilbert function
$$
\Delta^2 H(C_m, t) = \quad \cdots \quad u_1 \quad \cdots \quad u_r \quad 0
\quad \cdots,
$$
where $u_1 = \cdots = u_r = 0$, and where the term $u_r$
occurs in degree $r_o(C_0) + 2 +m$.
We claim that $C_m$ has the cohomology property. To see this, note that
$r_o(C_m) = r_o(C_0) + m$, and that
$\sigma(C_m) \leq r_o(C_0)+m-r+3$, so that $\Delta^2 H(C_m, t) = 0$
for all $t \geq r_o(C_0)+m-r+3$. But by Corollary~\ref{Delta-coh}, we see that
$h^2({\Bbb P}^3, {\cal I}_{C_m}(t)) = h^1({\Bbb P}^3, {\cal I}_{C_m}(t))$
for all $t \geq r_0(C_0)+m-r-1 = r_o(C_m) - r + 1$. That is,
$h^2({\Bbb P}^3, {\cal I}_{C_m}(t)) = h^1({\Bbb P}^3, {\cal I}_{C_m}(t))$
in the last $r$ places.
For the converse, suppose $C \in {\cal L}$ has the cohomology property
and that there
is a sequence of basic double links
$$
{\cal C} : C_0:(1, d_0) \rightarrow C_1:(1, d_1) \rightarrow
\dots \rightarrow C_m = C.
$$
First,
note that by \cite[Lemma 1.14]{BM1}, $e(Z)$ increases by at least one
each time we move up in the liaison class. But $r_o(Z)$
increases by exactly one each time. Since the cohomology
property for $C_m$ in particular means that $e(C_m) = r_o(C_m)$,
then we must have $e(C_0) \leq r_o(C_0)$.
Next we show that the Hilbert function of $C_0$ has the
given form. First,
$$
\Delta^2 H(C_0, r_o(C_0)+2)= \Delta^2 H(C_m, r_o(C_0) + 2 +m) -
\delta({\cal C}, r_o(C_0)+m, m),
$$
and since $r_o(C_0) + 2 + m = r_o(C_m)+2$, we have
$\Delta^2 H(C_m, r_o(C_0)+2+m) = 0$, because of Corollary~\ref{Delta-coh}
and the
fact that $h^1({\Bbb P}^3, {\cal I}_C(t)) = h^2({\Bbb P}^3, {\cal I}_C(t))$ for $t\geq r_0(C_m)$.
Thus, $\Delta^2 H(C_0, r_o(C_0)+2) \leq 0$. Moreover, for each
$i = 1, \dots, r-1$, we have
\begin{eqnarray*}
\Delta^2 H(C_0, r_o(C_0)+2-i) &=& \Delta^2 H(C_m, r_o(C_0)+2-i+m) -
\delta({\cal C}, r_o(C_0)+2-i+m, m) \\
&=& -\delta({\cal C}, r_o(C_0)+2-i+m, m).
\end{eqnarray*}
As above, the second equality follows from the fact that
$r_o(C_0) + 2 -i + m = r_o(C_m) + 2 - i$ and because the
cohomology property for $C_m$ implies $\Delta^2 H(C_m, r_o(C_m) + 2 - i) = 0$
via Corollary~\ref{Delta-coh}. But by our observation following
Definition~\ref{deltadef}, we have
\begin{eqnarray*}
\Delta^2 H(C_0, r_o(C_0)+2-i) &=& - \delta({\cal C}, r_o(C_0)+2-i+m, m) \\
&\leq& -\delta({\cal C}, r_o(C_0)+2-(i-1)+m, m) \\
&=& \Delta^2 H(C_0, r_o(C_0)+2-(i-1)),
\end{eqnarray*}
and so the proof is finished.
\end{proof}
Since for any curve $C$ in an even liaison class ${\cal L}$ there
is a sequence of basic double links from a minimal curve
in ${\cal L}$ to a curve $C_m$, followed by a deformation
with constant cohomology to $C$, the Theorem has the immediate
consequence:
\begin{cor} \label{mincoh-prop}
An even liaison class of curves in ${\Bbb P}^3$ contains a curve $C$
having $h^1({\Bbb P}^3, {\cal I}_C(t)) = h^2({\Bbb P}^3, {\cal I}_C(t))$ in
the last $r$ places if and only if the minimal curve $C_0$
in ${\cal L}$ has $e(C_0) \leq r_o(C_0)$ and
has Hilbert function satisfying
$$
\Delta^2 H(C_0, r_o(C_0) - r + 3) \leq \Delta^2 H(C_0, r_o(C_0) - r + 4)
\leq \dots \leq \Delta^2 H(C_0, r_o(C_0) + 2) \leq 0.
$$
\end{cor}
\section{Liaison Classes of Curves with Equal Cohomology}
As we saw in the previous section,
the presence or non-presence of a curve in an even liaison
class ${\cal L}$ having equal cohomology can be detected by looking at
the Hilbert function of the minimal curve in ${\cal L}$. This raises some
immediate questions: if an even liaison class ${\cal L}$ contains
curves with equal cohomology, in what shifts of ${\cal L}$ do they occur,
and ``how many'' such curves are there? To answer these questions,
we first make some remarks concerning how the equal cohomology property
behaves with respect to basic double linkage.
\begin{remark}\label{eqcoh&bdls}
\begin{enumerate}
\item[(a.)] \label{bdl-rem1}
Let ${\cal L}$ be an even liaison class with associated
deficiency module of diameter $r$.
Suppose ${\cal L}$ contains curves with equal cohomology.
Then by the previous corollary, the minimal curve in the liaison class
has Hilbert function ending in a non-decreasing
sequence of $r$ non-positive terms,
beginning in degree $r_a(C_0) + 2$.
Say $\Delta^2H(C_0, t) = \cdots\quad t_1 \quad \cdots\quad t_r$ is this
sequence. Let
$$
C_0:(1,b_0) \rightarrow C_1:(1, b_1) \rightarrow C_m:(1,b_{m-1})
\rightarrow C_m
$$
be a sequence of basic double links. By Lemma~\ref{hilbfcn-bdl},
if $\Delta^2 H(C_i, t)$ ends in negative terms, and if the
basic double linkage $C_i:(1,b_i) \rightarrow C_{i+1}$
changes one of these negatives, then it also changes every term
preceding. In particular, it must change the left-most negative term.
Moreover, each basic double link which changes negatives increases
the left-most negative term by exactly one. Note that a
link $C_i:(1, b_i) \rightarrow C_{i+1}$ changes negative terms
if and only if $b_i \geq r_a(C_i) + 2 = r_a(C_0) + i + 2$.
\item[(b.)] \label{bdl-rem2}
Related to this is the observation that if
there are more than $-t_1$ basic double links which change negative
terms, then $C_m$ cannot have equal cohomology. More precisely,
if we have $b_i \geq r_a(C_0) + i + 2$ for more than
$-t_1$ indices $i$, then the $t_1$ term eventually becomes positive,
and this forces $h^1({\Bbb P}^3, {\cal I}_C(t))$ and $h^2({\Bbb P}^3, {\cal I}_C(t))$
to be non-equal in at least the leftmost degree.
\item[(c.)] \label{bdl-rem3}
Continuing with that theme, it follows rather trivially
that if we have more than $-t_1$ links which change
negative terms, then no further basic double link can possibly
produce a curve with equal cohomology.
\item[(d.)] \label{bdl-rem4}
As a final remark, note that if $C :(1, d) \rightarrow D$
is a basic double link, and if $C$ has equal cohomology, then
$D$ has equal cohomology if and only if $d \leq r_a(C) + 3$. This follows
directly from Corollary~\ref{Delta-coh} and Lemma~\ref{hilbfcn-bdl}.
\end{enumerate}
\end{remark}
Now, we begin our description of which curves in the liaison
class have equal cohomology by showing that there is a
unique minimal such curve.
\begin{prop} Suppose ${\cal L}$ is an even liaison class which
contains curves having equal cohomology. Then up to deformation
through curves with constant cohomology, there is a unique
minimal curve with equal cohomology.
\end{prop}
\begin{proof} This is quite easy, given our remarks above.
If $\Delta^2 H(C_0, t) = \cdots\quad t_1\quad\cdots\quad t_r$ are
the final non-decreasing non-positive terms in the Hilbert function of $C_0$,
as guaranteed by Corollary~\ref{mincoh-prop},
then it requires at least $-t_1$ basic double links to reach
a curve $D$ for which $\Delta^2 H(D, t) = 0$ for $t \geq r_a(D) + 2$,
and by Corollary~\ref{Delta-coh}, $D$ then has equal cohomology.
On the other hand, the construction in Theorem~\ref{coh-prop}
produces a curve with equal cohomology in exactly $-t_1$ steps.
Also, if we reach a curve in equal cohomology in $-t_1$ steps,
then every basic double link of degree $b_i$ satisfies
$b_i \geq r_a(C_0) + i + 2$. Thus, by Lemma~\ref{hilbfcn-bdl},
no matter what sequence of basic double links we take, the Hilbert
function of the resulting curve is invariant. Thus
any two curves in this shift with equal cohomology are deformations
through curves with constant cohomology; see \cite[Proposition 3.1]{BM2}.
\end{proof}
Now, we need to recall a result from \cite{BM2} on ``equivalence''
of basic double links. In the proof of \cite[Lemma 5.2]{BM2},
they show the following: suppose
$$
C_1:(1, b_1) \rightarrow C_2:(1, b_2) \rightarrow C_3
$$
are basic double links with $b_1 < b_2$.
Then the sequence $b_1, b_2$ is equivalent
to the sequence $b_2 - 1, b_1 + 1$ in the sense that if we make
the basic double links
$$
C_1:(1, b_2 - 1) \rightarrow C_2':(1, b_1+1) \rightarrow C_3',
$$
then $C_3$ and $C_3'$ are deformations of each other, through
curves with constant cohomology. Note that implicit in this
is the fact that the basic double linkage of degree $b_2 - 1$
can actually be made; this is also noted in their proof.
We will use this idea of ``flipping'' adjacent degrees in the
next result.
\begin{prop} Suppose ${\cal L}$ is an even liaison
class containing curves with equal cohomology, and that
$C$ is a curve in ${\cal L}$ having equal cohomology.
Then $C$ is obtained from the minimal curve having
equal cohomology by a sequence of basic double linkages
followed by a deformation through curves with constant cohomology,
if necessary. Each curve in the sequence also has equal cohomology.
\end{prop}
\begin{proof} Assume that the deficiency module associated
to ${\cal L}$ has diameter $r$. First, since $C$ is in ${\cal L}$, then
by the Lazarsfeld--Rao property, after deforming $C$ through curves
with constant cohomology if necessary, we may assume that
there is a sequence of basic double links
\begin{equation}\label{bdl1}
C_0:(1,b_0) \rightarrow C_1:(1,b_1) \rightarrow\cdots \rightarrow C_m = C
\end{equation}
where $C_0$ is the (absolute) minimal curve in ${\cal L}$,
and we can assume that $b_0 \leq \dots \leq b_{m-1}$.
Since ${\cal L}$ possesses curves with equal cohomology,
$C_0$ has Hilbert function given by
$$
\Delta^2 H(C_0, t) = \quad\cdots\quad t_1 \quad\cdots\quad t_r
$$
where $t_1\leq\dots\leq t_r \leq 0$, and $t_1$ is in
degree $r_a(C_0) + 2$.
Now, since $C$ has equal cohomology, then exactly $-t_1$
of the basic double links in the sequence~(\ref{bdl1})
change negatives. That is, exactly $-t_1$ of the $b_i$
satisfy $b_i \geq r_a(C_0) + i + 2$. This follows from
Remark~\ref{bdl-rem2}(a).
Choose the first index $s \geq 0$ for which $b_i \geq r_a(C_0) + i + 2$,
and using the equivalence outlined above, flip this degree down
to the first position. Note that
this is possible since our original sequence of $b_i$'s is non-decreasing
and $b_{s-j} < r_a(C_0) + s + 2 - j \leq b_s - j$
for $0 \leq j \leq s$.
This creates an equivalent sequence
of basic double links of degrees $b_0', \dots, b_{m-1}'$,
where $b_0' = b_s - s$, $b_i' = b_{i-1} + 1$ for $0 < i \leq s$,
and $b_i' = b_i$ for $i \geq s+1$. In particular, exactly
$-t_1$ of the $b_i'$ satisfy $b_i' \geq r_a(C_0) + i + 2$,
and moreover $b_0' \geq r_a(C_0) + 2$. Continue in the same manner:
find the second time that a $b_i'$ changes negatives, and flip
it down to the second position, and so forth. Then we end up
with a sequence $c_0, \dots, c_{m-1}$ of basic double links
which is equivalent to the one we started with, and which
moreover has $c_i \geq r_a(C_0) + i + 2$ for $i = 0, \dots, -t_1-1$.
Hence, by Remark~\ref{eqcoh&bdls}, since we change exactly
$-t_1$ negative terms, all in the first $-t_1$ links, and since
we eventually end up with a curve having equal cohomology,
then the curve $C_{-t_1}$ and each curve from $C_{-t_1}$ on,
must also have equal cohomology. This follows from Remark~\ref{bdl-rem1}.
\end{proof}
We recapitulate what we have proven in the next statement:
\begin{thm} \label{LR-prop}
Suppose ${\cal L}$ is an even liaison class
containing curves with equal cohomology. Then there is a minimal
shift ${\cal L}^t$ which contains a curve with equal cohomology;
the curves with equal cohomology in the minimal shift are
unique up to deformation
through curves with constant cohomology; every curve in ${\cal L}$
with equal cohomology is obtained from the minimal one
by basic double linkage and deformation through curves with
constant cohomology; and finally
every rightward shift of ${\cal L}^t$ contains, up
to deformation, a finite, non-zero number of curves
with equal cohomology.
\end{thm}
\begin{proof} The only part which remains to be proven is
the final statement. If $C$ is a curve with equal cohomology
in some shift ${\cal L}^s$ of the liaison class, then
$\Delta^2 H(C, t) = 0$ for all $t \geq r_a(C) + 2$.
This implies in particular that $I_C$ contains non-zero
elements of degree $\geq r_a(C) + 2$. Thus, we can make a
basic double link $C:(1, r_a(C) + 2) \rightarrow D$,
and $D \in {\cal L}^{s+1}$ also has equal cohomology.
On the other hand, if we make a basic double link
of degree $> r_a(C) + 3$, then the resulting curve does not
have equal cohomology, so there is only a finite number
of allowable degrees.
\end{proof}
Theorem~\ref{LR-prop} shows that the curves with equal
cohomology have a strong Lazarsfeld-Rao property,
in the sense that there are unique minimal curves,
every other curve is obtained by basic double linkage, and
in each allowable shift, there are only a finite number
of curves, up to deformation.
In the case of Buchsbaum liaison classes, we can actually count
the number of curves in each shift which have equal cohomology.
\begin{prop} \label{number-Buchsbaum} Suppose ${\cal L}$ is a
Buchsbaum even liaison class having curves with equal cohomology,
and let ${\cal L}^s$ be the minimal shift in which such
a curve occurs. Then for each $t \geq s$, there are
exactly $2^{t-s}$ curves, up to flat deformations, having equal cohomology.
\end{prop}
\begin{proof} This follows from the fact that if $D \in {\cal L}^h$,
then $\alpha(I_D) = 2N + h$, where $N = \sum \dim H^1_*({\Bbb P}^3, {\cal
I}_D(i))$,
and the description of the Hilbert function of minimal Buchsbaum
curves in \cite[Proposition 2.1]{BM1}.
In particular, the minimal curve $C$ having equal cohomology has
Hilbert function
$$
\Delta^2 H(C, t) = 1 \quad 2 \quad \cdots \quad 2\alpha + s.
$$
In order to move from ${\cal L}^s$ to ${\cal L}^{s+1}$ and preserve
the cohomology property, we can only make basic double
links of degree $2\alpha+s$ or $2\alpha+s+1$.
Similarly, we can only move from ${\cal L}^{s+1}$ to ${\cal L}^{s+2}$
by basic double links of degree $2\alpha+s+1$ or $2\alpha+s+2$.
Continuing inductively, the statement is proven.
\end{proof}
We can also give a proof more along the lines of the original
proof that the liaison classes of curves in ${\Bbb P}^3$ have
the Lazarsfeld--Rao property. It is much less constructive
in nature, but, in some sense, points out the naturality
of our cohomological criterion of equal cohomology.
Since it is so different in spirit from our previous argument, we
felt it necessary to include it here.
\begin{lemma} Let $\cal F$ be a rank $(r+1)$ vector bundle on ${\Bbb P}^3$ with
$H^2_* ({\Bbb P}^3, {\cal F}) = 0$. Let
\[
\phi_1 : \bigoplus_{i=1}^{r+1} {\cal O}_{{\Bbb P}^3} (-a_i ) \rightarrow
{\cal F},
\hskip .5in a_1 \leq \dots \leq a_r
\]
\[
\phi_1 : \bigoplus_{i=1}^{r+1} {\cal O}_{{\Bbb P}^3} (-b_i ) \rightarrow
{\cal F},
\hskip .5in b_1 \leq \dots \leq b_r
\]
be morphisms whose degeneracy loci are curves $C_1$ and $C_2$ with equal
cohomology. Then there exists a morphism
\[
\phi : \bigoplus_{i=1}^{r+1} {\cal O}_{{\Bbb P}^3} (-c_i ) \rightarrow {\cal F},
\hskip .5in c_i = \min \{ a_i ,b_i \}
\]
whose degeneracy locus is also a curve with equal cohomology.
\end{lemma}
\begin{proof}
Notice that $C_1$ and $C_2$ are evenly linked, in the even liaison class
determined by the stable equivalence class of $\cal F$, according to Rao's
classification \cite{rao}. By \cite[Lemma 2.1]{BBM}, there exists such a
$\phi$ whose degeneracy locus is a curve
$C$. We just have to prove that $C$ has equal cohomology.
Twisting and relabeling if necessary, we may assume that we have locally free
resolutions
\[
0 \rightarrow \bigoplus_{i=1}^{r+1} {\cal O}_{{\Bbb P}^3} (-a_i ) \rightarrow
{\cal
F} \rightarrow {\cal I}_{C_1} \rightarrow 0
\]
\[
0 \rightarrow \bigoplus_{i=1}^{r+1} {\cal O}_{{\Bbb P}^3} (-b_i ) \rightarrow
{\cal
F} \rightarrow {\cal I}_{C_2} (h) \rightarrow 0.
\]
Notice that the deficiency module of $C_2$ is shifted $h$ places to the right
of that of $C_1$. We then get
\[
0 \rightarrow H^2 ({\Bbb P}^3, {\cal I}_{C_1} (t)) \rightarrow
H^3 ({\Bbb P}^3, \bigoplus {\cal O}_{{\Bbb P}^3} (t-a_i )) \rightarrow
H^3 ({\Bbb P}^3, {\cal F}(t)) \rightarrow 0
\]
\[
0 \rightarrow H^2 ({\Bbb P}^3, {\cal I}_{C_2} (t+h)) \rightarrow
H^3 ({\Bbb P}^3, \bigoplus {\cal O}_{{\Bbb P}^3} (t-b_i )) \rightarrow
H^3 ({\Bbb P}^3, {\cal F}(t)) \rightarrow 0.
\]
(The first 0 comes from the assumption on the vanishing of the cohomology of
$\cal F$ and the second from the fact that $h \geq 0$.) By the assumption of
equal cohomology, for $t \geq r_a(C_1)$
the first term in the first sequence has the
same dimension as the first term in the second sequence.
Hence for $t \geq r_a(C_1)$, also the second terms are equal. Therefore
\[
\left \{ a_i \ | \ r_a(C_1) - a_i \leq -4 \right \} =
\left \{ b_i \ | \ r_a(C_1) - b_i \leq -4 \right \}
\]
(since these are the terms which contribute to the middle cohomology space in
the degrees $t \geq r_a$). That is,
\[
\left \{ a_i \ | \ a_i \geq r_a + 4 \right \} =
\left \{ b_i \ | \ b_i \geq r_a + 4 \right \}
\]
Call this set $A$.
Now, for any curve $Y$ with locally free resolution
\[
0 \rightarrow \bigoplus_{i=1}^{r+1} {\cal O}_{{\Bbb P}^3} (-d_i ) \rightarrow
{\cal F} \rightarrow {\cal I}_{Y} (\delta ) \rightarrow 0,
\]
$Y$ has equal cohomology if and only if
$\{ d_i \ | \ d_i \geq r_a(Y) + 4 \} = A$
(since this set determines $h^3({\Bbb P}^3, \bigoplus {\cal O}(-d_i + t))$
and hence $h^2 ({\cal I}_Y (t+\delta )$ in the desired range).
The proof of the lemma follows immediately from this fact.
\end{proof}
\begin{cor} \label{LRprop}
The set of curves in a given even liaison class which have equal cohomology
satisfy the Lazarsfeld--Rao property.
\end{cor}
\begin{proof}
The proof is identical to that in \cite{BBM}. The lemma above replaces
\cite[Lemma 2.1]{BBM}. Then \cite[Proposition 2.2]{BBM}
goes through to prove
the uniqueness of the minimal element. Similarly, \cite[Proposition 2.3]{BBM}
goes through to show the relation between the minimal element and any other
curve in the even liaison class with equal cohomology. Finally,
\cite[Theorem 2.4]{BBM}
still works to show how to produce a curve with equal cohomology as
a sequence of basic double links followed by a deformation, starting with a
minimal curve with equal cohomology. The proof in \cite{BBM} shows that such a
sequence exists. The fact that we start with equal cohomology and end
with equal cohomology shows that every step in between has equal cohomology
too.
\end{proof}
\begin{remark}\label{mincurves} We do not yet know of any examples
of even liaison classes for which the absolute minimal curve
$C_0$ is also the minimal curve with equal cohomology.
\end{remark}
\section{Integral Curves with Equal Cohomology} \label{integral}
There has been much recent progress on further clarifying the
structure of even liaison classes by giving conditions for the
presence within the liaison classes of nice curves. In particular,
there is information on where in a given class one
can find integral curves \cite{nollet},
or smooth and connected curves in Buchsbaum classes \cite{MDP2}, and
on how these curves are related to each other and to the minimal
curve in the class.
The paper \cite{PR} shows that at least in Buchsbaum
classes, smooth and connected curves share the same
Lazarsfeld--Rao properties as irreducible curves, and
imply that one can obtain the integral curves within
a given shift of a liaison class by deforming irreducible curves.
Their calculations are based also on the work of Nollet, as well
as on \cite{MDP2}.
In this section, we are interested in using some of the results
of \cite{nollet} to obtain some information on
when an even liaison class contains curves which are integral
and have equal cohomology. We first recall the relevant
definitions and some results from \cite{nollet}.
Let $C$ be a curve in ${\Bbb P}^3$, defined by an ideal $I=I_C$.
The postulation character of $C$ is given by $\gamma_C(n) = -\Delta^3 H(C, n)$.
There are three natural invariants to attach to $C$:
\begin{eqnarray*}
s(C) &=& \min\{\,n : \gamma_C(n) \geq 0 \,\} \\
t(C) &=& \min\{\,n : \gamma_C(n) > 0\,\} \\
t_1(C) &=& \mbox{smallest degree of a surface containing $C$ which meets}\\
&&\quad\quad\mbox{a surface of degree $s(C)$ containing $C$ properly}.
\end{eqnarray*}
We note for clarity that $s(C) = \alpha(C)$, the minimal degree of
a generator of $I_C$, and $t_1(C) = \beta(C)$, the minimal degree
for which $I_{\leq t} = \oplus_{i \le t} [I]_i$ generates
an ideal of codimension $2$.
Next, say that $C$ dominates a curve $D$ at height $h$ if
$C$ can be obtained from $D$ by a sequence of $h$ basic double
links, followed by a deformation. The central definition for this
section is the following:
suppose $C$ dominates the minimal curve $C_0$ in ${\cal L}$ at height $h$.
Then
$$
\theta_C(n) = \left\{ \begin{array}{ll}
\gamma_C(n), & \mbox{if $s(C) \le n < s(C_0) + h$} \\
\gamma_C(n) - \gamma_{C_0}(n-h), &
\mbox{if $n \geq s(C_0) + h$} \\
0, & \mbox{otherwise.}
\end{array} \right.
$$
(This definition appears in \cite{PR} and is clearly equivalent
to the one in \cite{nollet}.)
We say $\theta_C$ is connected in degrees $\ge a$ if
$\theta_C(b) > 0$ for $b \geq a$ implies $\theta_C(n) > 0$
for all $a \leq n \leq b$, and similarly $\theta_C$ is connected
in degrees $\le b$ if $\theta_C(a) > 0 $ for some $a \leq b$
implies $\theta_C(n) > 0$ for all $a \leq n \leq b$. Finally,
$\theta_C$ is connected about an interval $[a,b]$ if it
is connected in degrees $\geq a$ and in degrees $\leq b$,
and if $\theta_C(n) > 0$ for all $a \leq n \leq b$.
Now, Nollet proves the following theorem in \cite{nollet}:
\begin{thm} Let ${\cal L}$ be an even liaison class of curves
in ${\Bbb P}^3$ with minimal curve $C_0$.
\begin{enumerate}
\item[{\rm (}a.{\rm )}] {\rm (\cite[Theorem 5.2.1]{nollet})}
If $C \in {\cal L}$ is an integral curve of height $h$,
then $\theta_C$ is connected about $[t(C_0)+h, t_1(C_0) + h - 1]$.
\item[{\rm (}b.{\rm )}] {\rm (\cite[Theorem 5.2.5]{nollet})}
Conversely, suppose
$C$ dominates at height $h$ an integral curve $D$ in ${\cal L}$,
which is generically Cartier on a surface of minimal degree and
has either $\theta_D \not=0$ or $t(D) \leq e(D) + 4$. If
$\theta_C$ is connected about $[t(C_0) + h, t_1(C_0) + h - 1]$, then
$C$ can be deformed to an integral curve.
\end{enumerate}
\end{thm}
Thus, having $\theta_C$ connected about the interval
$[t(C_0) + h, t_1(C_0) + h - 1]$ is very close to having $C$ integral.
As it turns out, this condition is relatively easy to check
for curves with equal cohomology. We begin with some
elementary calculations. Throughout, let ${\cal L}$ be an
even liaison class of curves, which contains curves having equal
cohomology, and let $C_0$ be the (absolute) minimal curve in ${\cal L}$
and $C$ the minimal curve in ${\cal L}$ with equal cohomology.
\begin{lemma}\label{s_t_and_theta}
Suppose the minimal curve $C$ with equal cohomology has height $h$
over $C_0$. Then:
\begin{eqnarray*}
s(C) &=& s(C_0) + h \\
t(C) &=& t(C_0) + h \\
\theta_C(n) &=& \left\{ \begin{array}{ll}
-\gamma_{C_0}(n-h) & \mbox{\rm if $r_a(C_0) + h + 2 < n \leq \sigma(C_0) +
h$}\\
0 & \mbox{\rm otherwise }.\end{array}\right.
\end{eqnarray*}
\end{lemma}
\begin{proof} Note that in order to move up the liaison class from the
minimal curve $C_0$ to the curve $C$, in order to get $C$ with equal
cohomology, we must take basic double links $C_i \rightarrow C_{i+1}$
of degree large enough to change the negative signs in $\Delta^2 H(C_i, t)$.
Clearly, this degree $d$, say, is strictly larger than $t(C_i)$.
By \cite[Corollary 2.3.5]{nollet}, then,
$\gamma_{C_{i+1}}(n) = \gamma_{C_i}(n-1)$ for $n \leq d$. In particular,
$s(C_{i+1}) = s(C_i) + 1$ and $t(C_{i+1}) = t(C_i) + 1$, and so the
first two statements are done by induction. The assertion about
$\theta_C$ then follows from the definition of $\theta_C$ and
the fact that $\gamma_C(n) = 0$ for $n \geq r_a(C_0) + h + 2$,
since $C$ has equal cohomology, and using that $s(C_0) \leq r_a(C_0) + 2$
and $s(C) = s(C_0) + h$.
\end{proof}
Now we are ready to determine which curves have both equal cohomology
and connected $\theta$. Our first result takes care of a rather
trivial case.
\begin{prop} Suppose the minimal curve $C$ with equal cohomology has
height $h$ over $C_0$ and has $\theta_C$ connected about the interval
$[t(C_0) + h, t_1(C_0) + h - 1]$. Then $C=C_0$, up to deformation,
and $t(C_0) = t_1(C_0)$.
Conversely, if $C_0$ has equal cohomology and $t(C_0) = t_1(C_0)$,
then $\theta_{C_0}$ is connected about the interval $[t(C_0), t_1(C_0) - 1]$.
\end{prop}
\begin{proof} First note that since $C$ has equal cohomology,
we must have $t(C) \leq r_a(C) + 2$, and this then implies by
Lemma~\ref{s_t_and_theta} that $t(C_0) \leq r_a(C_0) + 2$.
But again by Lemma~\ref{s_t_and_theta}, this means that
$\theta_C(t(C_0)+h) = 0$, so $\theta_C > 0$ on the
interval $[t(C_0)+h, t_1(C_0) + h -1]$ if and only if this
interval is empty. Clearly, this is equivalent to $t(C_0) = t_1(C_0)$.
Now, $\theta_C$ is connected in degrees $\geq t(C_0) + h$ if and
only $\theta_C = 0$ in degrees $\geq t(C_0) + h$, and again
by Lemma~\ref{s_t_and_theta}, this occurs if and only if
$\sigma(C_0) = r_a(C_0) + 2$. By Corollary~\ref{Delta-coh},
this implies that $C_0$ has equal cohomology, and so $C = C_0$,
up to deformation.
The other direction is quite trivial, since $\theta_{C_0} = 0$ and
$[t(C_0), t_1(C_0) - 1]$ is empty.
\end{proof}
Our next proposition is the main result of this section, and tells
us when an even liaison class contains curves with
equal cohomology and connected $\theta$. Note that it is identical
in spirit to Theorem~\ref{mincoh-prop}.
\begin{prop} \label{theta-coh}
Suppose ${\cal L}$ is an even liaison class of curves with minimal
curves satisfying $t(C_0) < t_1(C_0)$. Then ${\cal L}$ contains
a curve $D$ of height $\overline{h}$, say, having equal cohomology
and $\theta_D$ connected about $[t(C_0) + \overline{h}, t_1(C_0) + \overline{h} - 1]$
if and only if $t_1(C_0) \leq \sigma(C_0) + 1$ and the Hilbert function
of $C_0$ satisfies
$$
\Delta^2H(C_0, r_a(C_0) + 2) < \Delta^2H(C_0, r_a(C_0) + 3) < \cdots
<\Delta^2H(C_0, \sigma(C_0) - 1) < 0.
$$
\end{prop}
\begin{proof} First, suppose $C_0$ has $t_1(C_0) \leq \sigma(C_0) + 1$
and satisfies the condition on the Hilbert function. Note
that the condition on the Hilbert function implies
that $t(C_0) \leq r_a(C_0) + 2$. Then by
Lemma~\ref{s_t_and_theta}, the minimal curve $C$ with equal
cohomology has $\theta_C(t) = 0$ for $t(C_0) + h \leq t \leq r_a(C_0) + h + 2$
and $\theta_C(t) > 0$ for $r_a(C_0)+h+2 < t \leq \sigma(C_0) + h$,
where $C$ has height $h$ over $C_0$.
Now perform a sequence of $r_a(C_0) + 2 - t(C_0)$ basic double links
all of degree $r_a(C_0) + h + 3$ to reach a curve $D$ of
height $\overline{h} = h + r_a(C_0) + 2 - t(C_0)$. Then by a repeated
application of \cite[Corollary 2.3.5]{nollet}, it is easy to
check that
$$
\theta_D(t) = \left\{
\begin{array}{ll}
1 & \mbox{ for $t(C_0) + \overline{h} \leq t \leq r_a(C_0) + \overline{h} + 2$} \\
\theta_C(t+h-\overline{h}) & \mbox{ for $r_a(C_0) + \overline{h} + 2 < t \leq \sigma(C_0) +
\overline{h}$} \\
0 & \mbox{ otherwise. }
\end{array} \right.
$$
In particular, $\theta_D$ is connected about
$[t(C_0) + \overline{h}, t_1(C_0) + \overline{h} - 1]$. Also, since $D$ was
obtained from $C$ by basic double links of low degree, $D$ still
has equal cohomology; see Remark~\ref{bdl-rem4}(d.).
Conversely, suppose $D$ is a height $\overline{h}$ curve with equal cohomology,
and with $\theta_D$ connected about the interval
$[t(C_0) + \overline{h}, t_1(C_0) + \overline{h} - 1]$. Then there is a sequence
of basic double links
$$
C:(1,b_0) \rightarrow C_1:(1, b_1) \rightarrow \cdots
\rightarrow D
$$
where $C$ is the minimal curve in ${\cal L}$ with equal cohomology
(of height $h$, say) and where each $b_i$ satisfies
$b_i \le r_a(C_0) + h + i + 3$ (see Remark~\ref{eqcoh&bdls}(d.) or
the proof of Corollary~\ref{LRprop}). If $C=C_0$, then
$r_a(C_0) + 2 = \sigma(C_0)$, so there is nothing
to show. Hence we may assume that $C_0$ does not have
equal cohomology, and this means that $\Delta^2 H(C_0, r_a(C_0) + 2) < 0$,
which in turn implies $\gamma_{C_0}(r_a(C_0) + 3) < 0$
and $t(C_0) \leq r_a(C_0) + 2$.
By a repeated use of \cite[Corollary 2.3.5]{nollet},
$\gamma_D(n) = \gamma_C(n-\overline{h}+h)$ for $r_a(C_0) + \overline{h}+3 \leq n$.
But by Lemma~\ref{s_t_and_theta},
$\gamma_C(n-\overline{h}+h) = -\gamma_{C_0}(n-\overline{h})$ for
$r_a(C_0) + \overline{h} + 3 \leq n \leq \sigma(C_0) + \overline{h}$.
Now, $\theta_D$ is connected in degree $\geq t(C_0) + \overline{h}$, and
our assumption that $t(C_0) < t_1(C_0)$ implies in particular that
$\theta_D(t(C_0) + \overline{h}) > 0$.
Also, $\theta_D(\sigma(C_0) + \overline{h}) = -\gamma_{C_0}(\sigma(C_0)) > 0$,
so $\theta_D > 0$ on the interval $[t(C_0) + \overline{h}, \sigma(C_0) + \overline{h}]$.
Next, note that $\theta_D$ is positive on the
interval $[r_a(C_0) + \overline{h} + 3, \sigma(C_0) + \overline{h}]$, since this
interval is contained in the interval $[t(C_0) + \overline{h}, \sigma(C_0) +
\overline{h}]$.
Thus, we have $0 < \theta_D(t) = -\gamma_{C_0}(t-\overline{h})$
for $r_a(C_0) + \overline{h} + 3 \leq t \leq \sigma(C_0) + \overline{h}$.
This clearly implies that $\gamma_{C_0}(t) < 0$ for
$r_a(C_0) + 3 \leq t\leq \sigma(C_0)$, and this means
that $\Delta^2 H(C_0, t)$ is strictly increasing in the given range.
Finally, to see that $t_1(C_0) \leq \sigma(C_0) + 1$, note
that the connectedness property of $\theta_D$ implies that
$\theta_D > 0$ on $[r_a(C_0) + \overline{h} + 3, t_1(C_0) + \overline{h}]$,
but clearly $\theta_D(t) = 0$ for $t >\sigma(C_0) + \overline{h}$,
by the argument above.
\end{proof}
\section{Degrees of Generators and Liaison Classes of Curves}
In this section, we go back to studying degrees of generators of
the ideals defining space curves, by using the results on cohomology
given in the previous sections. We are able to give some nice conditions
on the degrees of the components of the deficiency module associated
to a liaison class in order for the class to contain a curve whose
ideal has a generator of maximal degree. Since knowledge about curves
with equal cohomology in a given liaison class depends so crucially
on knowing the Hilbert function of the minimal curve in the liaison
class, our characterizations for when a liaison class contains
curves with generators of high degree work best when we already
know the Hilbert function of the minimal curve. To do this, we
have to make some extra assumptions on the liaison class.
We have concentrated on cohomological criteria, and two
very clean statements are given in Proposition~\ref{degree-maxcorank}
and in Proposition~\ref{degree-Buchsbaum}. Similarly, using
our results on integral curves, we give some results on
existence within a liaison class of integral curves with
generators of maximal degree.
As we showed in the previous sections, if $C$ is defined by
an ideal $I = I_C$, then the property of $I_C$ having
a generator of high degree is very closely related to
having $h^1({\Bbb P}^3, {\cal I}_C(t)) = h^2({\Bbb P}^3, {\cal I}_C(t))$ in
a large number of places, and furthermore, curves with
this cohomology property are easily constructed by basic double linkage,
as long as we know the minimal curve in a liaison class. However,
we should remark that in order to construct a curve whose
ideal has a high degree generator, we need to choose the degrees
of the basic double links carefully, since it is possible to
make $h^1 = h^2$ in the maximum number of places, without
introducing a high degree generator. The following example
should clarify this somewhat.
\begin{example} Start with the Buchsbaum liaison class
${\cal L} = {\cal L}_{4,1}$, whose deficiency module has two
consecutive components of dimensions $4$ and $1$, respectively.
Then the minimal curve $C_0$ in ${\cal L}$ has
Hilbert function (see \cite[Corollary 2.18]{BM1})
$$
\Delta^2 H(C_0, t) = 1 \quad 2 \quad \cdots \quad 10 \quad -2 \quad -1.
$$
Taking the two basic double links
$$
C_0:(1, 13) \rightarrow C_1:(1, 13) \rightarrow C_2
$$
produces the curve $C_2$, for which $h^1 = h^2$ in the last two places,
and which has $\sigma(C_2) = 12$, but
$\omega(I_{C_2}) = 13$. Thus the bound in Proposition~\ref{maxdegree}
is not obtained, even though $C_2$ does have the cohomology property.
On the other hand, note that if we take the basic double links
$$
C_0:(1, 12) \rightarrow C'_1:(1,14) \rightarrow C'_2,
$$
then $C'_2$ has $h^1 = h^2$ in the last two places, and
also has $\sigma(C'_2) = 12 = \omega(I_{C'_2}) - 2$. So the
bound is achieved in this case.
\end{example}
Generally speaking, this second sequence of basic double links
produces curves whose defining ideals have a high degree generator.
Note that it is essentially the procedure given by Theorem~\ref{coh-prop}.
However, we need to require that there are no ``trailing zeroes''
on the end of the second difference of the Hilbert function. We
formalize this in the next statement.
\begin{prop}\label{degree} Suppose ${\cal L}$ is an even liaison
class of curves in ${\Bbb P}^3$, with minimal curve $C_0$, and
let $r = \mathop{\rm diam\,} H^1_*({\Bbb P}^3, {\cal I}_C)$. If the Hilbert function
of $C_0$ satisfies
$$
\Delta^2 H(C_0, r_a(C_0)+2) \leq \dots \leq \Delta^2 H(C_0, r_o(C_0)+2) < 0,
$$
then there exists a curve $C$ in ${\cal L}$ whose
defining ideal $I_C$ satisfies $\omega(I_C) = \sigma(S/I_C) + r$.
\end{prop}
\begin{proof}
We follow the construction of basic double links in the
first part of Theorem~\ref{coh-prop}. Perform the given
sequence of basic double links, up to the next to the last stage. Since
$\Delta^2 H(C_0, r_o(C_0) + 2) < 0$, then at this stage
we see that
$$
\Delta^2 H(C_{m-1}, t) = \cdots \quad -1 \quad -1 \quad \cdots \quad -1 \quad 0
\quad \cdots,
$$
where there are $r$ terms of $-1$, and the rightmost term
occurs in degree $r_o(C_0)+2+m$. Thus our final basic double link
$C_{m-1}:(1, r_o(C_0)+m+4) \rightarrow C_m$ produces
the curve $C_m$, whose ideal $I_{C_m}$ has a generator
of degree $r_o(C_0) + m + 4$, and such that
$\sigma(C_m) = r_o(C_0) + m - r + 4$. Hence
$\omega(I_{C_m}) = \sigma(C_m) + r$, which is what we wanted
to show.
\end{proof}
In fact, using the structure theory for curves with equal cohomology
developed in the previous section, we can say more about the
curves in an even liaison class having maximal degree generators.
Namely, the minimal such curve occurs in the shift
${\cal L}^s$ of ${\cal L}$, where $s=-\Delta^2 H(C_0, r_a+2)$, and
every other such curve is obtained by a sequence of basic double
links from this minimal one, followed if necessary by a deformation.
Moreover, up to a flat deformation, there are only a finite number
of curves with a maximal degree generator in each allowable shift.
\begin{remark} It is interesting to note that curves in a
non-arithmetically Cohen--Macaulay liaison class which
have a generator of maximal degree in fact have all
of their generators of relatively high degree. Indeed,
since the minimal such curve $C$ lies in the shift ${\cal L}^s$
as above, $\alpha(I_C) = \alpha(I_{C_0}) + s$.
So, at least if the minimal curve does not already have
a maximal degree generator (see Remark~\ref{mincurves}),
we are forced to increase all the degrees of the generators.
This is in some contrast with the arithmetically Cohen--Macaulay
case, where there are curves $D$ with quadric generators
and for which $\omega(D) = \sigma(D)$, the maximum possible.
For example, let $D$ be the union of a plane curve of
degree $m$
and a line, which meet at one point. Then it is easy to see
that $I_D$ has quadric generators and $\omega(D) = \sigma(D) = m$.
On the other hand, even within a non-arithmetically Cohen--Macaulay
liaison class, we can make the difference $\omega(I_C) -\alpha(I_C)$
arbitrarily large when $\omega(I_C)$ is maximal. For this,
simply take the minimal curve $C$ with a maximal degree
generator, and form $m$ basic double links all of
degree $\alpha(I_C)$. Then the resulting curve $C_m$
still has $\omega(I_{C_m})$ maximal (see Remark~\ref{eqcoh&bdls}(d.)),
and has
$\omega(I_{C_m}) - \alpha(I_{C_m}) = \omega(I_C) - \alpha(I_{C}) + m$.
\end{remark}
Next, we want to interpret the conditions on Hilbert functions
only in terms of cohomology. As we noted above, to do this
we need to make some extra assumptions to allow us to calculate
the Hilbert function of the minimal curve. Our first result
is for maximal corank curves, and the second for Buchsbaum curves.
\begin{cor} \label{degree-maxcorank}
Suppose ${\cal L}$ is an even liaison
class of curves in ${\Bbb P}^3$, with minimal curve $C_0$. Assume that
$e(C_0) < r_a(C_0)$ {\rm (}i.e., $C_0$ has maximal corank{\rm )}.
Then ${\cal L}$ contains a curve $C$ such that
$\omega(I_C) = \sigma(C) + \mathop{\rm diam\,} H^1_*({\Bbb P}^3, {\cal I}_C)$ if and only if
$$
h^1({\Bbb P}^3, {\cal I}_{C_0}(t)) \geq 3 h^1({\Bbb P}^3, {\cal I}_{C_0}(t+1))
- 3 h^1({\Bbb P}^3, {\cal I}_{C_0}(t+2))
+ h^1({\Bbb P}^3, {\cal I}_{C_0}(t+3)),
$$
for $t=r_a(C_0), \dots, r_o(C_0)$.
\end{cor}
\begin{proof} The conditions on the cohomology modules
guarantee via Lemma~\ref{hilbfcn_eqn} that
$$
\Delta^2 H(C_0, r_o - r + 2) \leq \dots \leq \Delta^2(r_o + 2, C_0)
< 0,
$$
and so sufficiency follows from Proposition~\ref{degree}.
To see necessity, note that if ${\cal L}$ contains a curve
with a generator of maximal degree, then by Proposition~\ref{equal-coh},
that curve has equal cohomology, and so by Corollary~\ref{mincoh-prop},
the minimal curve in ${\cal L}$ has Hilbert function whose
second difference ends in a sequence of non-decreasing negative
terms, and then Lemma~\ref{hilbfcn_eqn} translates this
back to the required statement about cohomology.
\end{proof}
Recall that a Buchsbaum liaison class is completely determined
by the dimensions of the graded components of the associated deficiency module.
We will write ${\cal L}_{n_1\dots n_r}$ for the Buchsbaum class
associated to the graded module $M = \oplus [M]_i$, where
$\dim_k [M]_i = n_i$ and is zero otherwise, and where we assume that
$n_1, n_r > 0$ and $n_i \ge 0$ for $1 < i < r$.
\begin{prop} \label{degree-Buchsbaum}
Suppose ${\cal L} = {\cal L}_{n_1 \dots n_r}$ is a Buchsbaum even liaison
class. Then ${\cal L}$ contains a curve $C$ such that
$\omega(I_C) = \sigma(I_C) + r$ if and only if
$n_i \geq 3 n_{i+1}$ for $i = 1, \dots r-1$.
\end{prop}
\begin{proof} It follows from \cite[Corollary 2.18]{BM1} that
the conditions on the deficiency module guarantee that the
Hilbert function of $C_0$ satisfies the conditions given in
Proposition~\ref{degree}, and we can therefore use that result to
prove sufficiency.
On the other hand, if $C$ is a curve in ${\cal L}$ whose ideal has a generator
of maximal degree, then by Proposition~\ref{equal-coh}, $C$ has
equal cohomology.
Thus by Corollary~\ref{mincoh-prop}, the
minimal curve $C_0 \in {\cal L}$ has Hilbert function
satisfying
$$
\Delta^2 H(C_0, r_o(C_0) - r + 2) \leq \dots \leq \Delta^2 H(C_0, r_o(C_0) + 2)
\leq 0.
$$
But now it follows from the description of the Hilbert function for minimal
Buchsbaum curves given in \cite[Corollary 2.18]{BM1} that
$n_i \geq 3 n_{i+1}$, for each $i = 1, \dots, r-1$.
\end{proof}
\begin{remark}
Proposition \ref{degree} is, in a sense, a complete answer to the problem of
determining which even liaison classes $\cal L$ contain a curve $C$ whose
defining ideal $I_C$ satisfies
$\omega (I_C ) = \sigma (S/I_C ) + \hbox{diam } H^1_* ({\Bbb P}^3, {\cal I}_C )$.
However, it is generally not easy to tell, for a given even liaison class,
what the Hilbert function of the corresponding minimal
curve is. So it is worth noting that there are also times when one can tell
directly from the associated deficiency module $M$ (defined up to shift)
that such a curve does not exist.
If $M$ is annihilated by the maximal ideal (i.e. if $C$ is Buchsbaum),
then the necessary and sufficient condition for the existence of such a curve
is given in terms of the dimensions of the components of $M$
(Proposition~\ref{degree-Buchsbaum}).
Similarly, if $\cal L$ contains any curve
with maximal corank then the minimal curve has
maximal corank as well, since any
basic double link increases $r_a $ by exactly 1 and $e$ by at least 1
(\cite[Lemma~1.14]{BM1}). Hence again it reduces to a question of
the dimensions of the module components.
It is also true that $\cal L$ contains curves of maximal rank
(i.e.\ $r_0 < \alpha$) if and only if the minimal curve in
$\cal L$ has maximal rank, since a
basic double link increases $r_o$ by exactly 1 and increases $\alpha$ by at
most 1. (This was first observed in \cite[Theorem~2.1]{BM3}.) Then it follows
from Theorem~\ref{coh-prop} that if the deficiency module $M$ associated to
$\cal L$ has diameter 3 or more, and if $\cal L$ contains
any curve of maximal rank,
then $\cal L$ does not contain any curve achieving our bound on $\omega$.
Finally, we observe that it follows from Proposition~\ref{equal-coh} and
Corollary~\ref{almostBuchs-deg}
that if $\mathop{\rm diam\,} K < \mathop{\rm diam\,} M$ then $\cal L$ contains no curve
achieving our bound. This immediately rules out a huge number of even liaison
classes, since ``most'' classes will have a module containing at least one pair
of consecutive components, for which multiplication by a general linear form is
injective.
\end{remark}
By using the results in Section~\ref{integral}, we can prove
similar results for the existence of integral curves with generators
of maximal degree. However, because the theorems in that
section only go one direction, and because deformations do not in general
preserve integrality or degrees of generators, we can only
get necessity.
\begin{thm} Suppose ${\cal L}$ is a liaison class whose minimal
curve $C_0$ satisfies $t(C_0) < t_1(C_0)$. If ${\cal L}$ contains
an integral curve with a generator of maximal degree, then the
minimal curve $C_0$ has Hilbert function which satisfies
$$
\Delta^2 H(C_0, r_a(C_0) + 2) < \cdots < \Delta^2 H(C_0, \sigma(C_0) - 1) < 0.
$$
\end{thm}
\begin{proof} This follows immediately from Proposition~\ref{theta-coh}.
\end{proof}
As before, with extra assumptions on the liaison class, we
can give a cohomological criterion.
\begin{prop} Suppose ${\cal L}$ is a liaison class whose minimal
curve $C_0$ satisfies $t(C_0) < t_1(C_0)$.
\begin{enumerate}
\item[{\rm (}a.{\rm )}] If $C_0$ has maximal corank, and if ${\cal L}$
contains integral curves with generators of maximal degree,
then
$$
h^1({\Bbb P}^3, {\cal I}_{C_0}(t)) > 3 h^1({\Bbb P}^3, {\cal I}_{C_0}(t+1))
- 3 h^1({\Bbb P}^3, {\cal I}_{C_0}(t+2))
+ h^1({\Bbb P}^3, {\cal I}_{C_0}(t+3)),
$$
for $t = r_a(C_0), \dots, r_o(C_0)$.
\item[{\rm (}b.{\rm )}] If ${\cal L}_{n_1\dots n_r}$ is a Buchsbaum
liaison class containing integral curves with generators of maximal
degree, then $n_i > 3n_{i+1}$ for $i = 1, \dots, n_r$.
\end{enumerate}
\end{prop}
\begin{proof} This follows exactly as before, using
Proposition~\ref{theta-coh}.
\end{proof}
|
1995-03-28T07:20:38 | 9503 | alg-geom/9503019 | en | https://arxiv.org/abs/alg-geom/9503019 | [
"alg-geom",
"math.AG"
] | alg-geom/9503019 | Martin Pikaart | Martin Pikaart | An orbifold partition of ${\overline{M}_g^n}$ | 16 pages, Latex Version 2.09, will appear in The Moduli space of
Curves (eds. Dijkgraaf, Faber, van der Geer), Progress in Math., Birkh"auser | null | null | Utrecht preprint Nov. 1994, nr. 882 | null | We define a partition of ${\overline{M}_g^n}$ and show that the cohomology of
${\overline{M}_g^n}$ in a given degree admits a filtration whose respective
quotients are isomorphic to the shifted cohomology groups of the parts if $g$
is sufficiently large. This implies that the map $H^k({\overline{M}_g^n}) \ra
H^k(M_g^n)$ is onto and that the Hodge structure of $H^k(M_g^n)$ is pure of
weight $k$ if $g \geq 2k+1$. Our main ingredient is the stability theorem of
Harer and Ivanov.
| [
{
"version": "v1",
"created": "Mon, 27 Mar 1995 08:47:43 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Pikaart",
"Martin",
""
]
] | alg-geom | \section{Introduction}
The moduli space of smooth $n$-pointed complex curves of genus
$g$ is a quasi-projective orbifold $M_g^n$ of dimension $3g-3+n$
(where as usual, we assume that $2g-2+n>0$).
It is compactified by the moduli space of stable pointed curves, $\Mgnbar$,
which is a projective orbifold. We will write $M_g$ and $\Mgbar$ if $n=0$.
We need some terminology in order to state our main result.
Let $X$ be an irreducible orbifold. An {\it orbifold partition} of $X$ is a
finite filtration by closed subvarieties
$$X_\bullet:= (X =X_0 \supset X_1 \supset \cdots \supset X_{m+1} =\emptyset),$$
such that $ Y_\alpha:=X_\alpha \setminus X_{\alpha+1} $ is an orbifold.
If $\dim Y_\alpha >\dim Y_{\alpha +1}$ this is called a
{\it stratification}. A connected component of a $Y_\alpha $ is called a {\it
part} (respectively a {\it stratum}). For example,
a stratification of $\Mgnbar$ is defined by the subvarieties
$X_i:=\{ \mbox{curves with at least $i$ singularities}\}$ for $i=0,1,\dots$;
the strata are just the loci that parametrize stable pointed curves
of a fixed topological type. We will therefore refer to this as the {\it
stratification of $\Mgnbar$ by topological type}. A simple example of an
orbifold partition (of $ {\bf C}^2$) that is not a stratification is given by
${\bf C}^2 \supset L_1 \cup L_2 \supset L_1$, where $L_i$
is the ith coordinate axis; the parts are ${\bf C}^2 \setminus (L_1\cup L_2)$,
$L_2 \setminus \{ 0 \}$ and $L_1$.
Let us say that an orbifold partition $X_\bullet $ of $X$ {\it filters
cohomology up to degree $k$} if the Gysin map
$$H^{j-2 {\it codim} Y_\alpha}(Y_\alpha)(-{\it codim} Y_\alpha) \rightarrow
H^j(X \setminus X_{\alpha+1})$$
is injective for all $j \leq k$ and all $\alpha$. (Here and throughout this
paper cohomology is taken with rational coefficients.) For
$2{\it codim} Y_\alpha > k$ this condition is empty. Thus, if one
is only interested in a partition filtering cohomology up to degree $k$,
it suffices to consider $X_0 \supset \cdots \supset X_{\alpha+1}$ such that
$X_{\alpha+1}$ has codimension at least $\frac{1}{2}k$. It is easily seen that
if a partition filters cohomology up to degree $k$ there
exists a filtration on $H^j(X)$ for $j \leq k$, such that there is a canonical
morphism of mixed Hodge structures which maps the respective
quotients isomorphically onto the groups
$ H^{j -2 {\it codim} Y_\alpha}(Y_\alpha)(-{\it codim} Y_\alpha)
\mbox{ for } j<k \mbox{ and all } \alpha$.
We can now formulate our main result.
\medbreak\noindent{\bf Corollary \ref{coho Mgnbar}}
{\it Given $k \geq 0$, then for $g$ large enough, $\Mgnbar$ admits an orbifold
partition which has $M_g^n$ as its open part, is coarser than the
stratification by topological type and filters cohomology up to degree $k$.}
\medbreak
We shall see that the stratification by topological type does not
have this property. We deduce from the main result the following corollary.
\medbreak\noindent{\bf Corollary \ref{pure Hodge}}
{\it If $g \geq 2k+1$, then the restriction map $H^k(\Mgnbar) \ra H^k(M_g^n)$
is surjective; consequently the mixed Hodge structure on $H^k(M_g)$ is pure of
weight $k$.}
\medbreak
Corollary \ref{coho Mgnbar} enables us to define the ``stable cohomology''
of $\Mgnbar$ as $g$ goes to infinity and $n$ is fixed. This stable cohomology
is not finitely generated. For example, if $n=0$,
the generators in degree 2 are the tautological class and the boundary classes,
naturally indexed by $0,1,2, \dots$.
Using additional properties of the partition that we construct we prove that
for any
$n \geq 0$ the cohomology of $\Mgnbar$ is not of Tate type if $g$ is
sufficiently large.
Let us sketch the proof of corollary \ref{coho Mgnbar}
and give the motivation for the partition we define. (See
Sections \ref{graphs and partition} and \ref{results} for details.)
We take $n=0$ for simplicity. Let $D_0$ be the locus in $\Mgbar$
parametrizing irreducible curves with one node and $D_1$ the
locus in $\Mgbar$ parametrizing curves with one smooth
component of genus $g-1$ and one of genus 1, joined in
one point. Consider the maps $f_i:H^{l-2}(D_i) \ra H^l(D_i)$
for $i=0$ or 1, given
by taking the cup product with the first Chern class of the normal bundle
of $D_i$. The normal direction of $D_i$ corresponds to smoothening
the unique singular point. Notice that $f_i$ can be obtained as the composite
of the Gysin map $H^{l-2}(D_i)(-1) \ra H^l(M_g \cup D_i)$ and
the obvious restriction map.
The unique singular point lies in both cases on at least one local component
of high genus.
As we shall see in Section \ref{results}, this will imply that
the $f_i$ are injective in low degree and
a fortiori the Gysin maps are injective.
However, this does not carry over to codimension 2. Let $D_{01}$
be the locus in $\Mgbar$ parametrizing curves with two nodes
and two components,
one smooth component of genus $g-1$ and one singular component of geometric
genus 0. Clearly one of the nodes lies on two local components of genus 0.
As we shall see, this implies that $H^{l-4}(D_{01})(-2) \ra H^l(D_{01})$ is
not injective (in fact zero).
A way to restore injectivity is by coarsening the
partition. If we let $D$ be the union of $ D_1$ and $D_{01}$, then the normal
direction of $D$ corresponds to smoothening the singular point on
the component of genus $g-1$ and we get an injective map as we wanted.
Smoothening the singularity on the genus 0 component has become a direction
along the part.
It will turn out that we can define the partition completely in
terms of graphs and certain subgraphs. Our parts will be (irreducible) unions
of strata of the stratification by topological type; locally they are like
the parts in the given example of a non-stratification of ${\bf C}^2$.
The unique open dense stratum of the stratification
by topological type in a given part will be refered to
as its {\it generic} stratum. For every part, its generic stratum
will have a graph with the
property that every edge corresponds to a singularity lying on at
least one local component of high genus.
Unfortunately, I was unable to define such a subgraph canonically; Definition
\ref{specifiek} involves the choice of the number $\alpha$.
The point is that smoothening many singularities lying on low genus components
may yield a component of high genus. See Remark \ref{infinite genus}
for further comments.
In Section \ref{graphs and partition} we prove the existence of an
orbifold partition using only some formal combinatorial properties of graphs.
In Section \ref{stable and weak} we define weak subgraphs and prove the
combinatorial properties needed in the previous section. Section \ref{results}
contains the results.
\smallskip
I thank Eduard Looijenga for suggesting the problem to me and
for many helpful conversations. I am grateful to the referee for useful
comments.
\section{Stable graphs and the orbifold partition}\label{graphs and partition}
All graphs considered in this paper are weighted, that is,
each vertex has a non-negative integer associated to it.
For a stable $n$-pointed curve $C$, we define its
{\it stable graph} as follows. Each irreducible component of $C$ defines a
vertex, the weight of the
vertex being the genus of the normalisaton of the component.
Omitting from the normalisation of
such an irreducible component the inverse images of the
singular and marked points on $C$ yields a smooth projective curve
minus a finite number of points.
For each missing point we draw a half edge emanating from the vertex defined by
that connected component and for every singular point on $C$ we join the two
half edges associated to it to obtain a whole edge. We omit the word whole if
it is clear that we consider a whole edge.
The remaining half edges, which correspond to the $n$ marked points
on the curve, will be called {\it loose half edges}.
Thus, unless we specify, a half edge can be either
half of a whole edge or a loose half edge.
We define a {\it stable graph}
($n$-pointed of genus $g$) as the stable
graph of some stable $n$-pointed curve of genus $g$.
An automorphism of a stable graph is an automorphism of the underlying
unweighted graph preserving the weights.
Given a stable graph one can recover a topological model for its
stable $n$-pointed curve by taking one curve for every vertex with genus equal
to the weight of the vertex, omitting small open discs for every half edge
emanating from that vertex, glueing the appropriate
boundaries and contracting them. Notice that if we had incorporated the
ordering of the $n$ points in the definition of a stable graph, then our graphs
would correspond bijectively to strata of the stratification by topological
type.
A {\it subgraph} is a subset of a stable graph with the property that if it
contains a half edge it contains its unique vertex.
By a {\it full subgraph} we mean a
subgraph which contains every half edge emanating from one of its vertices.
A connected component of a full subgraph is a stable graph.
In the same way as above, one can
associate to a connected component of a full subgraph a topological
model of a curve. We define the {\it genus} of a connected component of a full
subgraph as the arithmetic genus of the curve associated to that connected
component.
The {\it genus} of a full subgraph is defined as the sum of the
genera of its connected components. In other words, the genus of a full
subgraph equals the first Betti number of that full subgrabh plus the sum
of the weights taken over the vertices in the full subgraph.
We denote the greatest integer less than or equal to a real number $a$ by
$[a]$.
Fix $k$ and let ${\cal G}_{k}(g,n)$ be the set of
(isomorphism classes of)
stable graphs, $n$-pointed of genus $g$ and at most $[k/2+1]$ whole edges.
If there is no chance of confusion, we write ${\cal G}$ for
${\cal G}_{k}(g,n)$.
(For the reason we only consider graphs with this many edges,
see the remark made in the introduction
following the definition of a orbifold partition which filters cohomology.)
The maximal number of vertices for a graph
in ${\cal G}$ is $[k/2+1]+1$. From now on we will only consider graphs in
${\cal G}$.
We may regard a stable graph as a one dimensional topological space.
If $e$ is a whole edge, we denote the contraction of $e$ by $\pi_e: \Gamma
\rightarrow \Gamma /e$ and the image of its end vertices by $\pi_e(e)$.
We make $\Gamma/e$ into a weighted graph as follows.
If $p$ is a vertex of $\Gamma$ not incident with $e$, then the weight of
$\pi_e(p)$ is that of $p$.
The weight of $\pi_e(e)$ is defined as the sum of the weights of the two
end vertices if $e$ is not a loop and the weight of its unique end vertex plus
one otherwise. Clearly this makes $\Gamma/e$ into an element of
${\cal G}_k(g,n)$, corresponding to the stable curve obtained by smoothening
the singularity corresponding to the edge $e$.
If $\Delta$ is a full subgraph of $\Gamma$, we denote the quotient of
$\Gamma$ obtained by
contracting all whole edges of $\Delta$ by
$\Gamma/ \Delta$.
The image of a full subgraph under $\pi_e$ is full,
provided that
the contracted edge has either both or neither of its end vertices in that
full subgraph.
The permutation group on $n$ elements, $S_n$, acts on $\Mgnbar$
fixing the locus of points parametrizing curves with at least a fixed number
of singularities. Such loci define the stratification by topological type,
see introduction. Thus $S_n$ acts on the stratification by topological type.
The orbits of the $S_n$-action on the stratification by topological type
of $\Mgnbar $ minus loci of codimension at least $[k/2+1]+1$ are in one to
one correspondence with the elements of ${\cal G}_k(g,n)$.
Giving a partition of the first set is therefore equivalent to giving a
partition of ${\cal G}_k(g,n)$.
This, in turn, corresponds to a function
$$\phi : {\cal G}_k(g,n) \ra {\cal G}_k(g,n),$$
as follows. Recall that we want the parts we seek to be irreducible unions of
strata of
the stratification by topological type. A graph will be mapped to
the graph of the generic stratum of the part it will belong to. (Compare the
example in the introduction: the graph of $\Delta_{01}$ is mapped to
the graph of $\Delta_1$.) This implies that $\phi$ contracts
a certain subgraph.
So we have to assign to every graph in
${\cal G}_k(g,n)$ a subgraph in such a way that the function $\phi$,
defined as contraction of that subgraph, corresponds to an orbifold
partition.
One important ingredient we need to obtain an orbifold partition is a partial
order on the image of $\phi$,
such that the union of all parts greater than or equal to a given part will be
a Zariski open neighbourhood of that part.
Now we will state the formal properties of the graphs needed to
obtain an orbifold partition- which will turn out to filter cohomology.
\begin{definition}
Suppose we are given, for every $\Gamma$ in ${\cal G}_k(g,n)$, a subgraph
$\Gamma_W$. Define $\phi:{\cal G}_k(g,n) \ra {\cal G}_k(g,n)$ by $\phi(\Gamma)
=\Gamma/\Gamma_W$. Denote the image of $\phi$ by $I$.
For $ \Gamma$ in $I$ define $S_\Gamma^0$ to be the locus in $\Mgnbar$ whose
points correspond to stable $n$-pointed curves with
graph $\Gamma$.
Define $S_\Gamma$ (respectively, $S$) to be the locus in $\Mgnbar$ whose
points correspond to stable $n$-pointed curves with graph
in $\phi^{-1}(\Gamma)$ (respectively, ${\cal G}$).
\end{definition}
If $C$ is a curve with graph $\Gamma$, then we will call the irreducible
components of $C$ corresponding to vertices in $\Gamma_W$ {\it weak
components}. Analogously we define {\it strong components}.
\begin{proposition} \label{formal}
Notations as in the previous definition.
Suppose a partial order is given on $I$, such that the following properties
hold:
\begin{enumerate}
\item[a.] $\Gamma_W$ is a proper full subgraph of $\Gamma$ which is invariant
under the automorphism group of the stable graph $\Gamma$;
\item[b.] if $e$ is a whole edge of $\Gamma$ not contained in $\Gamma_W$,
then $\Gamma/e > \Gamma$;
\item[c.] if $e$ is a whole edge contained in $\Gamma_W$ , then the image of
$\Gamma_W$ under contraction of $e$ is $(\Gamma/e)_W$;
\item[d.] a whole edge $e$ is contained in $\Gamma_W$ if and only if
$\pi_e(e)$ is contained in $(\Gamma/e)_W$;
\item[e.] if $k=1$ (respectively $k \geq 2$) and $p$ a vertex of $\Gamma$ such
that the weight of $p$ is 0 (respectively the weight of $p$ is at most
$k+2$), then $\Gamma_W$ contains the vertex $p$.
\end{enumerate}
Then the following hold :
\begin{enumerate}
\item $S_{{\it max }(I)} = M_g^n$;
\item for all $\Gamma \in I$: (the connected components of) $S_\Gamma$ and
$S_\Gamma^0$ are
orbifolds; the generic curve $C_\Gamma$ has graph $\Gamma$;
\item for all $\Gamma \in I$: $\cup_{\Delta \geq \Gamma} S_\Delta$ is a Zariski
open neighbourhood of $S_\Gamma$;
\item the (connected components of the) $S_\Gamma$ are parts of an orbifold
partition;
\item for all $\Gamma \in I$:
the strong (respectively, weak) components of $C_\Gamma$ do not (respectively,
do) degenerate in $S_\Gamma$;
\item for all $\Gamma \in I$:
every singular point of $C_\Gamma$ lies on at least one strong component
of $C_\Gamma$;
\item for all $\Gamma \in I$:
if $C'$ is a strong component of $C_\Gamma$ and $k=1$ (respectively $k \geq 2$)
then the genus of $C'$ is at least 1 (respectively at least $k+3$).
\end{enumerate}
\end{proposition}
\begin{proof}
{\it 1}. Notice that by property {\it a}, $\Gamma_W$ is a proper full subgraph,
which means that $\Gamma/\Gamma_W$ has no whole edges if and only if $\Gamma$
has no whole edges.
By property {\it b}, the stable graph of a smooth $n$-pointed curve
is larger than every other graph; consequently, $S_{max(I)}=M_g^n$.
{\it 2}. We already remarked that (a connected component of) $S_\Gamma^0$ is an
orbifold, namely it is a stratum
of the stratification by topological type. By the definition of $\phi$
we have that $S_\Gamma$ is contained in the closure of $S_\Gamma^0$.
So it suffices to prove that it is locally closed and without self
intersection. It follows from property {\it c}
that $S_\Gamma$ is locally closed.
Suppose $S_\Gamma$ has self intersection; this is necessarily a normal
crossing.
Let $\Delta$ be the graph corresponding to the generic locus of self
intersection of $S_\Gamma$ and suppose for simplicity that there are two local
branches of $S_\Gamma$. Thus, locally in a neighbourhood of $S_\Delta$,
there is an involution permuting the two branches of $S_\Gamma$.
This involution induces an involution $i$ on $\Delta$ which keeps $\Delta_W$
fixed but not pointwise fixed. If $e$ is an edge of $\Delta_W$ such that
$\Delta/e$ is the graph of one of the local branches, then $\Delta/i(e)$
is the (isomorphic) graph of the other local branch. But by property {\it a},
$\Delta_W$ contains both $e$ and $i(e)$, so $S_\Gamma$ contains the plane
spanned by the two local branches, contradiction.
{\it 3}. This follows directly from statement {\it 2} and property {\it b}.
{\it 4}. Choose an order reversing injective map $m: I \ra {\bf N}$ and
define $X_i:= \cup_{m(\Gamma) \leq i} S_\Gamma$.
The $X_i$ form a filtration which defines an orbifold partition by statements
{\it 2} and {\it 3} whose parts are the $S_\Gamma$.
{\it 5}. Follows from property {\it d}.
{\it 6}. By property {\it c}, we have that the image of $\phi$ equals the fix
point set of $\phi$, which means that for $\Gamma$ in the image of $\phi$,
every whole
edge of $\Gamma$ has at least one end vertex not in $\Gamma_W$.
{\it 7}. This is clear from property {\it e}.
\unskip\nolinebreak\hfill\hbox{\quad $\Box$} \end{proof}
Now we have reduced the problem of finding a partition to a purely
combinatorial one, which will be dealt with in the next section.
\section{Stable graphs, weak subgraphs}\label{stable and weak}
We will now start out to define a full subgraph $\Gamma_W$ for every
$\Gamma$ in ${\cal G}_k(g,n)$ and prove the properties a,b,c,d and e of
Proposition \ref{formal}.
For any full subgraph $\Delta$ of a stable graph $\Gamma$,
we denote its number of
vertices respectively its genus by $v(\Delta)$ resp. $g(\Delta)$. We denote
the weight of a vertex $P$ by $w(P)$.
Let $k \in {\bf Z}_{\geq 0}$ be given.
Let $\Phi_k :{\bf Z}_{\geq 0} \ra {\bf R}_{\geq 0} $ be a function
satisfying
$$2 \Phi_k(n) +[\frac{1}{2}k+1] \leq \Phi_k(n-1), \mbox{ if } n > 0.$$
In particular, it is a decreasing function.
\begin{proposition} \label{union} Let $k \geq 0$ be given.
Let $\Delta_1$ and $\Delta_2$ be full subgraphs of a stable graph $\Gamma$.
Let $\Delta$ be the full subgraph on the vertices of $\Delta_1$ and
$\Delta_2$.
If $g(\Delta_i) \leq \Phi_k(v(\Gamma)-v(\Delta_i)) $ for $i=1$ and $i=2$,
then also $g(\Delta) \leq \Phi_k(v(\Gamma)-v(\Delta)) $.
\end{proposition}
\begin{proof} We may suppose $ v(\Delta_1) \geq 1$ and $ v(\Delta_1) \geq
v(\Delta_2)$.
The statement is trivial if $\Delta_2$ is contained in $\Delta_1$, so we may
assume that this is not the case. Thus $v(\Delta) > v(\Delta_1)$.
We have:
$$ g(\Delta) \leq g(\Delta_1) + g(\Delta_2) +[\frac{1}{2}k+1] \leq
2 \Phi_k(v(\Gamma)-v(\Delta_1)) +[\frac{1}{2}k+1] $$
$$ \leq \Phi_k(v(\Gamma)-v(\Delta)). \eqno{\Box}$$
\end{proof}
Here and in the next proposition the term $[\frac{1}{2}k+1]$ comes from the
fact that we take a {\it full} subgraph; compare the formula for the genus
of a full subgraph given in Section \ref{graphs and partition}.
\begin{definition} \label{weak}
A full subgraph $\Delta$ of a stable graph $\Gamma$ is called
$\Phi_k$-weak if $g(\Delta) \leq \Phi_k(v(\Gamma)-v(\Delta))$.
\end{definition}
Notice that this definition only depends on $k$ and not on $g$.
By the proposition above the maximal $\Phi_k$-weak subgraph is well defined.
\begin{definition} \label{maximal weak}
Denote the maximal $\Phi_k$-weak subgraph of $\Gamma$ by $\Gamma_W$.
A vertex is called
strong if it is contained in the complement of $\Gamma_W$. Denote by $\Gamma_S$
maximal full subgraph on the strong vertices.
\end{definition}
Not every full subgraph of the maximal $\Phi_k$-weak subgraph is $\Phi_k$-weak,
see the example following Proposition \ref{abcde hold}. Therefore we don't use
the term $\Phi_k$-weak vertex.
\begin{proposition}\label{full proper invariant}
If $g > \Phi_k(0)$, then for every $\Gamma$ in ${\cal G}_k(g,n)$ we have that
$\Gamma_W$ is a proper full subgraph which is invariant under
the automorphisms of the stable graph $\Gamma$.
\end{proposition}
\begin{proof}
By definition, $\Gamma_W$ is full. It is a proper subgraph because
$g(\Gamma)>\Phi_k(v(\Gamma)-v(\Gamma))$ and thus $\Gamma$ is not
$\Phi_k$-weak. Invariance under automorphisms follows because both
the genus and the number of vertices are kept fixed by automorphisms
of the stable graph.
\unskip\nolinebreak\hfill\hbox{\quad $\Box$} \end{proof}
Recall that if $e$ is a whole edge of $\Gamma$, we write $\pi_e$ for the
contraction of $e$ and $\pi_e(e)$ for the image of its end vertices.
The inverse image of a full subgraph under $\pi_e$ is full.
For $e$ a whole edge of $\Gamma$, we put $\delta(e):=0$ if $e$ is a
loop, $\delta(e):=1$ otherwise.
\begin{proposition} \label{sub}
If $e$ is a whole edge contained in $\Gamma_W$ or in its complement,
then we have $\pi_e(\Gamma_W) \subset (\Gamma /e)_W$.
\end{proposition}
\begin{proof} We have:
$$g(\pi_e(\Gamma_W))=g (\Gamma_W) \leq \Phi_k(v(\Gamma)-v(\Gamma_W)) =$$
\[ \left\{ \begin{array}{ll}
\Phi_k((v(\Gamma/e) +\delta(e)-v(\pi_e(\Gamma_W))) &
\mbox{ $e$ outside $\Gamma_W$}\\
\Phi_k(v(\Gamma/e) +\delta(e)-[v(\pi_e(\Gamma_W))+\delta(e)]) &
\mbox{ $e$ contained in $\Gamma_W$}
\end{array} \right. \]
$$ \leq \Phi_k(v(\Gamma/e)-v(\pi_e(\Gamma_W))). \eqno{\Box}$$
\end{proof}
\begin{proposition} \label{gammaw maps to gamma/ew}
If $e$ is a whole edge contained in $\Gamma_W$, then the image of
$\Gamma_W$ under contraction of $e$ is $ (\Gamma /e)_W$.
\end{proposition}
\begin{proof}
Proposition \ref{sub} yields us one inclusion.
Put $\Delta :=\pi_e^{-1}((\Gamma/e)_W)$, then we have:
$$g(\Delta) =g ((\Gamma/e)_W) \leq \Phi_k(v(\Gamma/e)-v((\Gamma/e)_W)) $$
$$=\Phi_k(v(\Gamma)-\delta(e)-[v(\Delta)-\delta(e)])
=\Phi_k(v(\Gamma)-v(\Delta)). \eqno{\Box}$$\end{proof}
\begin{proposition}\label{not weak kept fixed under degeneration}
A whole edge $e$ is contained in $\Gamma_W$ if and only if
$\pi_e(e)$ is contained in $(\Gamma/e)_W$.
\end{proposition}
\begin{proof}
The implication "$\Rightarrow$" follows from Proposition \ref{gammaw maps to
gamma/ew}. For
the other implication, suppose $(\Gamma/e)_W$ contains $\pi_e(e)$.
Put $\Delta :=\pi_e^{-1}((\Gamma/e)_W)$, then the same computation as in the
proof of Proposition \ref{gammaw maps to gamma/ew} implies that $\Delta$ is
$\Phi_k$-weak and therefore
contained in $\Gamma_W$. Consequently, $\Gamma_W$ contains $e$.
\unskip\nolinebreak\hfill\hbox{\quad $\Box$} \end{proof}
\begin{definition} \label{phi}
Define a function $\phi_k(g,n): {\cal G}_k(g,n) \rightarrow {\cal G}_k(g,n)$
by $\phi_k(\Gamma) =\Gamma /\Gamma_W$. When there is no chance of
confusion, we will write $\phi$ instead of $\phi_k$.
\end{definition}
We call $\phi$ contraction of the maximal $\Phi_k$-weak subgraph.
It is clear that we have: $\Gamma$ is contained in $Im(\phi)$
if and only if $ \Gamma_W$
does not contain any whole edges. It follows that the image of $\phi$ equals
the fix point set of $\phi$.
Put $I_k(g,n):=Im(\phi_k(g,n))$; we will write $I$ instead of $I_k(g,n)$
if there is no chance of confusion. The set $I$ will be the index set for our
orbifold partition.
Before we can define a partial order on $I$, we need two more propositions.
\begin{proposition} \label{low inside}
If $P$ is a vertex of $\Gamma$ and $w(P) \leq {\mbox max}
\{ w(Q) \}$, where $Q$ runs over the vertices of $\Gamma_W$, then $\Gamma_W$
contains $P$.
\end{proposition}
\begin{proof}
Suppose not, let $\Delta$ be the full subgraph on $P$ and $\Gamma_W$.
We have :
$$ g(\Delta) \leq g(\Gamma_W)+g(P) +[\frac{1}{2}k+1] \leq 2g(\Gamma_W) +
[\frac{1}{2}k+1] $$
$$ \leq \Phi_k(v(\Gamma)-(v(\Gamma_W) +1))
=\Phi_k(v(\Gamma)-v(\Delta)). \eqno{\Box}$$
\end{proof}
\begin{proposition}\label{joins}
Let $e$ be an edge which joins $\Gamma_S$ and $\Gamma_W$.
Let $\Lambda $ be the full subgraph of $\Gamma/e$ on the vertices of
$\pi_e(\Gamma_W) -\pi_e(e)$. Then $\Lambda$ is $\Phi_k$-weak.
\end{proposition}
\begin{proof} We have:
$$ g(\Lambda) \leq g(\Gamma_W) \leq \Phi_k(v(\Gamma)-v(\Gamma_W))
=\Phi_k(v(\Gamma)-[v(\Lambda)+1])$$
$$=\Phi_k(v(\Gamma/e)-v(\Lambda)). \eqno{\Box}$$
\end{proof}
\begin{definition} Let $\Gamma$ be in ${\cal G}$.
Let $s(\Gamma)$ be the number of strong vertices.
Define $\{ w_i(\Gamma) \}_{i=1}^{s(\Gamma)}$ to be the set of weights of
strong vertices, ordered such that $w_i(\Gamma) \geq w_{i+1}(\Gamma)$.
For $i \in \{ s(\Gamma)+1,...,[k/2+1]+1 \}$, define $w_i(\Gamma):=g$.
Let $n(\Gamma)$ be the number of half edges parting from strong vertices.
Finally, define {\rm index}$(\Gamma)$ of a graph $\Gamma$ to be the vector
$(w_1(\Gamma),....,w_{[k/2+1]+1}(\Gamma),n(\Gamma)).$
\end{definition}
Notation: ${\rm index}(\Gamma)> {\rm index}(\Delta)$ refers to lexicographical
ordering.
\begin{proposition} \label{indeks}
a) If $e$ is a whole edge of $\Gamma$ not contained in $\Gamma_W$, then
${\rm index}(\Gamma/e)>{\rm index}(\Gamma)$. \hfill \break
b) If $e$ is a whole edge contained in $\Gamma_W$, then ${\rm index}(\Gamma/e)
={\rm index}(\Gamma)$.
\end{proposition}
\begin{proof}
a) Suppose first that neither of the end vertices of $e$ is in $\Gamma_W$
and suppose that ${\rm index}(\Gamma) $ $= (\dots,w_i , \dots , w_j,\dots)$.
If $e$ is a loop and its end vertex has weight $w_i$,
then by Propositions \ref{low inside} and \ref{not weak kept fixed under
degeneration} we have
${\rm index}(\Gamma/e)=(\dots,w_i+1,\dots) > {\rm index}(\Gamma)$.
If $e$ is not a loop and its end vertices have weights $w_i$ and $w_j$,
then ${\rm index}(\Gamma/e)=(\dots,w_i+w_j,\dots) > {\rm index}(\Gamma)$.
Secondly suppose $e$ has end vertices $P \in \Gamma_S$ and $Q \in \Gamma_W$.
By Proposition \ref{joins} we have $s(\Gamma) \geq s(\Gamma/e)
$. If $s(\Gamma) > s(\Gamma/e)$, then we are done.
If $s(\Gamma) = s(\Gamma/e)$ and $w(Q) >0$ the argument above applies.
If $s(\Gamma) = s(\Gamma/e)$ and $w(Q)=0$, then the first $[k/2+1]+1$
coefficients of both indices are equal.
We define the following numbers:
\[ \begin{array}{l}
a:=\# \{ \mbox{edges joining $P$ and $Q$} \},\\
b:=\# \{ \mbox{loops at $Q$} \},\\
c:=\# \{ \mbox{edges joining $Q$ and another vertex in $\Gamma_W$} \},\\
d:=\# \{ \mbox{edges joining $Q$ and a vertex in $\Gamma_S$ not equal to
$P$}\}, \\
f:=\# \{ \mbox{loose half edges at $Q$}\}.
\end{array} \]
One sees easily that $n(\Gamma/e)=n(\Gamma) +a+2b+c+d+f-2$.
Stability of the vertex $Q$ implies $a+2b+c+d+f\geq 3$, so we have
${\rm index}(\Gamma/e)> {\rm index}(\Gamma)$. \hfill \break
b) Follows immediately from Proposition \ref{gammaw maps to gamma/ew}.
\unskip\nolinebreak\hfill\hbox{\quad $\Box$} \end{proof}
\begin{corollary}
The index is preserved under contraction of the maximal $\Phi_k$-weak
subgraph.
\end{corollary}
\begin{definition}\label{partial order}
We define a partial order on $I$ as follows:
$\Gamma \geq \Delta$ if and only if ${\rm index}(\Gamma)$ $ >
{\rm index}(\Delta)$ or $\Gamma =\Delta$.
\end{definition}
\begin{corollary} \label{order OK}
If $e$ is a whole edge of $\Gamma$ not contained in $\Gamma_W$, then $\Gamma/e
> \Gamma$.
\end{corollary}
Now we are almost ready to prove the remaining property e of Proposition
\ref{formal}. We do this by making a suitable choice for $\Phi_k$.
For a given $k$, define $ \alpha := 1/ [k/2+1]+2$ and
$\beta:=[k/2+1]^2 \alpha^{[k/2+1]+1}$. We fix $\alpha$ and $\beta$
for the rest of this paper.
\begin{definition} \label{specifiek}
Define $\Phi_k :{\bf Z}_{\geq 0} \ra {\bf R}_{\geq 0} $ by $\Phi_k(n):=
\alpha^{-n}\beta$.
\end{definition}
An easy calculation yiels that
$$2 \Phi_k(n) +[\frac{1}{2}k+1] \leq \Phi_k(n-1), \mbox{ if } n > 0.$$
\begin{proposition}\label{gammaW bevat laag geslacht}
If $k=1$ (respectively $k \geq 2$) and $P$ a vertex of $\Gamma$ such
that the weight of $P$ is 0 (respectively the weight of $P$ is at most
$k+2$), then $\Gamma_W$ contains the vertex $P$.
\end{proposition}
\begin{proof}
Let $l$ be the number of loops at the vertex $P$. One has to check that
$0+l \leq \Phi_k([k/2+1]-l) $ (respectively $k+2+l \leq \phi_k([k/2+1]-l)$),
which is an easy computation.
\unskip\nolinebreak\hfill\hbox{\quad $\Box$} \end{proof}
\begin{proposition} \label{abcde hold}
Notations as above.
Let $k$ be given, and let $g$ be at least $\beta$. Define $\Phi_k$ as in
Definition \ref{specifiek}. Define $\phi$ and $I$ as in Definition
\ref{phi}. Let a partial order on $I$ be defined by Definition \ref{partial
order}. Then the properties a, b, c, d and e of Proposition \ref{formal}
hold.
\end{proposition}
\begin{proof}
The properties a, b, c, d and e are precisely the Propositions
\ref{full proper invariant}, \ref{order OK}, \ref{gammaw maps to gamma/ew},
\ref{not weak kept fixed under degeneration} and \ref{gammaW bevat laag
geslacht}. \unskip\nolinebreak\hfill\hbox{\quad $\Box$}
\end{proof}
Not every full subgraph of the maximal $\Phi_k$-weak subgraph is
$\Phi_k$-weak.
Consider the
following example: $k=9$, so $[k/2+1]=5$ and $\alpha=11/5$.
We have $\beta=11^6/5^4 \approx 2834$ and thus we have to take $g > 2834$.
Let $\Gamma$ be the graph with vertices $p_1, \dots , p_6$ and and
edges $e_1, \dots , e_5$ such that $e_i$ joins $p_i$ and $p_{i+1}$.
Suppose $w(p_1)=w(p_2)=11,~w(p_3)=22,~w(p_4)=44,~w(p_5)=88$ and
$w(p_6)=g-176$. One can check that the maximal $\Phi_k$-weak subgraph is the
full subgraph on the vertices $p_1, \dots,p_5$, but the full subgraph on
the vertex $p_5$ is not $\Phi_k$-weak.
\begin{remark} \label{infinite genus} {\rm
If one is willing to consider graphs of infinite genus, i.e.\ several
vertices can have infinite genus, then a canonical
definition of weak subgraph is readily available: just take the full subgraph
on the vertices of finite genus. (The definition of index
needs to be adapted.) Properties analogous to those of the previous
propositions, and in some cases even stronger results, can be proved.}
\end{remark}
\section{The results}\label{results}
We keep the notations and assumptions of Proposition \ref{abcde hold}, unless
the contrary is explicitely stated.
As we have seen in Section \ref{graphs and partition}, a connected component of
the $S_\Gamma^0$ is a stratum of the
stratification by topological type; it is an orbifold of
codimension equal to the number of singular points of its topological model
which we will denote by $C_\Gamma$.
In fact, a stratum is isomorphic to a product of lower dimensional moduli
spaces $M_h^m$ modulo a finite group.
Recall that the symmetric group on $n$ elements acts transitively on the
connected components of the $S_\Gamma^0$.
Furthermore, the orbifold
normal bundle of $S_\Gamma^0$ has fibre over $(C,x_1, \dots, x_n)$
isomorphic to $\oplus(T_xC' \otimes T_xC'')$, where the sums runs over
the singular points of $C$ and $C',~C''$ are the local components of $C$ in a
suitable neighbourhood of $x$. This isomorphism globalizes to an
isomorphism of bundles on $S_\Gamma^0$.
It follows from Proposition \ref{formal} that the orbifold
normal bundle of $S_\Gamma$ has fibre over $(C,x_1, \dots, x_n)$ isomorphic to
$\oplus(T_xC' \otimes T_xC'')$. Here the sums runs over the singular points
of $C$ which are specializations of singular points on the topological
model of $S_\Gamma^0$ and $C',~C''$ are the local components of $C$ in
a suitable neighbourhood of $x$. Thus, the sums runs over the edges of
$\Gamma$.
Notice that $S=\cup_{\Gamma \in I} S_\Gamma$ and that the real codimension
of the complement of $S$ in $\Mgnbar$ is larger than $k$.
Before going on we explain the Stability Theorem of Harer and Ivanov in an
algebro-geometric way. These theorems
are essential in the proof of Theorem \ref{filters coarse}.
Let $S$ be the locus in $\overline{M_{g+1}}$ of irreducible stable curves
of genus $g$ with one singularity; $S$ is a stratum of the stratification by
topological type, it is isomorphic to $M_g^2$ modulo the involution permuting
the two points. There is a map $p:S \rightarrow M_g $
which forgets the two points.
Let $i:U_S \hookrightarrow \overline{M_{g+1}}$ be the inclusion of
a suitable $C^\infty$ tubular neighbourhood of $S$ and let $\pi :U_S
\rightarrow S$ be
the natural retraction. Consider the diagram
$$M_{g+1} \stackrel{i}{\longleftarrow} U_S \setminus S \stackrel{p \circ
\pi}{\longrightarrow} M_g.$$
The Stability Theorem of Harer and Ivanov now says that if $g \geq 2k+1$, then
both maps induce isomorphisms on cohomology in degree up to $k$ (see
\cite{Harer1},\cite{Ivanov}). Furthermore $H^0(M_g) \cong {\bf Q}$ for $g \geq
0$ and $H^1(M_g) \cong 0$ for $g \geq 1$, see \cite[Ch. 7]{Harer2}.
These facts account for the conditions of property e of Proposition
\ref{formal}.
Hence we can define the kth stable cohomology group $H^k(M_\infty)$ of the
moduli space by $H^k(M_\infty):=H^k(M_g)$ when $g \geq 2k+1$.
If $g \geq 2k+1$ (respectively $g \geq 0$ if $k=0$, respectively $g>0$ if
$k=1$), we say $g$ is in the stable range with respect to $k$.
Moreover $i^*$ and $(p \circ \pi)^*$ are morphisms of mixed Hodge structures,
so $H^k(M_\infty)$ carries a well-defined mixed Hodge structure.
\begin{remark} {\rm
In \cite{Mumford} classes $\kappa_i \in H^{i,i}(M_\infty)$ are constructed
and in \cite{Miller} it is proved that the symmetric algebra on these classes
injects into $H^\bullet(M_\infty)$. Mumford conjectures that this is
actually an isomorphism in low degree, see \cite[Introduction]{Mumford}.
The conjecture would imply our corollary \ref{pure Hodge}. } \end{remark}
We need a corollary of Harer's results.
Assigning to a pointed curve the tangent space at its ith point defines
a line bundle ${\cal L}_i$ on $\Mgnbar$.
Consider the natural forgetful map
$M_g^n \rightarrow M_g$. Define $H^\bullet (M_g)[u_1, \dots, u_n]
\rightarrow H^\bullet(M_g^n)$, where the $u_i$ have degree 2,
by sending $u_i$ to the first Chern class
$c_1({\cal L}_i)$. Then we have (see \cite{Looijenga}): this is an isomorphism
up to degree $k$ if $g \geq 2k+1$.
\begin{theorem} \label{filters coarse}
Let $k$ be given. If $g > \beta ,$ then the partition $S=\cup_{\Gamma \in I}
S_\Gamma$ is coarser than the stratification by topological type,
has $S_{{\it max }(I)} =M_g^n$ and filters cohomology up to degree $k$.
\end{theorem}
\begin{proof}
We have already seen in Proposition \ref{formal}, properties 1 and 4,
that the $S_\Gamma$ form a partition which has $M_g^n$ as open
part and is coarser than the stratification by topological type.
So it remains to show that for all $\Gamma \in I$ and all $l<k$, the Gysin maps
on cohomology $H^{l-2codim S_\Gamma}(S_\Gamma)(-codim S_\Gamma)
\ra H^l(\cup_{\Delta \geq \Gamma} S_\Delta)$
induced by the inclusions $S_\Gamma \rightarrow
\cup_{\Delta \geq \Gamma} S_\Gamma$ are injections.
When we write property x we mean property x of Proposition \ref{formal}.
The orbifold $S_\Gamma^0$ is the quotient of
$$ \prod M_{g_s}^{n_s} \times \prod M_{g_w}^{n_w}$$
by a finite group. Here the first product runs over the strong vertices
of $\Gamma$ and the second over those which are not strong.
Because of properties 2 and 5 we have that $S_\Gamma$ is contained in the
quotient of
$$\prod M_{g_s}^{n_s} \times \prod \overline{M_{g_w}^{n_w}}$$
by a finite group, where again the products
run over the vertices which are strong respectively not strong.
Property 7 implies that the $g_a$ are in the stable range w.r.t. $[k/2+1]$.
By the result of Looijenga mentioned above we get :
$$H^i(M_{g_s}^{n_s}) \cong \mbox{degree $i$ part of }H^i(M_\infty)[u_1, \dots,
u_{n_s}], \eqno{(1)}$$
where the $u_i$ have degree 2.
We have seen that the normal bundle of $S_\Gamma$ splits as a direct
sum of line bundles, and thus its top Chern class becomes the product of the
first Chern classes of these line bundles.
We claim that these first Chern classes are all of the form $u_i+u_j$ or
$u_i+a$, where the $u_i$ are as in $(1)$ and $a$ is
an element of $H^*(\overline{M_{g_w}^{n_w}})$. We postpone the proof of the
claim for a moment.
By properties 2 and 3, $S_\Gamma$ is a closed suborbifold of
$\cup_{\Delta \geq \Gamma}S_\Delta$.
Consider the Gysin sequence for the inclusion $S_\Gamma$ in $\cup_{\Delta \geq
\Gamma}S_\Delta$:
$$
\dots \ra H^{l-2 {\it codim} S_\Gamma}(S_\Gamma)(-codim S_\Gamma)
\ra H^l(\cup_{\Delta \geq \Gamma} S_\Delta) \ra
H^l(\cup_{\Delta > \Gamma} S_\Gamma) \ra \dots
$$
Composing $H^{l-2codimS_\Gamma}(S_\Gamma)(-codim S_\Gamma) \ra H^l(\cup_{\Delta
\geq \Gamma} S_\Delta)$
with the restriction morphism to $ H^l(S_\Gamma)$
we get a morphism $H^{l-2codimS_\Gamma}(S_\Gamma)(-codim S_\Gamma)
\ra H^l(S_\Gamma)$
which is given by taking the cup product with the top Chern class of
the normal bundle of $S_\Gamma$. From what we have said above it follows that
cupping with this Chern class is injective up to degree $k$. A fortiori
the Gysin maps are injective.
It remains to prove the claim.
As explained, the line bundles of which we are taking the first Chern classes
correspond to whole edges of the graph $\Gamma$. Let $f$ be a whole
edge and let $P$ and $Q$ be its end vertices (which possibly coincide).
The line bundle under consideration is the tensor product of the two
line bundles corresponding to $P$ and half of $f$ respectively to $Q$ and
the other half of $f$.
By property 6 we have that either $P$ and $Q$ are both strong
vertices or precisely one of them is a strong vertex. In the first case the
first Chern classes of both line bundles are of the form $u_i$ and so
we get $u_i+u_j$ as first Chern class of the tensor product. In the second case
we get $u_i+a$ for some element $a$ of $H^*(\overline{M_{g_w}^{n_w}})$.
This proves the claim.
\unskip\nolinebreak\hfill\hbox{\quad $\Box$} \end{proof}
\begin{remark} {\rm
The parts $S_\Gamma$ depend upon
the definition of $\Phi_k$-weak subgraphs, which depends upon $\alpha$, which
in turn depends upon $k$.
This implies that the filtration in Theorem \ref{filters coarse}
depends upon $k$. In this remark we will write $\alpha_k$ and $\beta_k$ to
stress dependence.
There is a natural inclusion $i: {\cal G}_{k}(g,n) \hookrightarrow
{\cal G}_{k+1}(g,n)$ which is a bijection if $k$ is even.
We claim that $i(\Gamma_W) \subset (i(\Gamma))_W$.
This is clear if $k$ is even because then
$\alpha_k^{v(\Delta)-v(\Gamma)}\beta_k
=\alpha_{k+1}^{v(i(\Delta))-v(i(\Gamma))}\beta_{k+1}$ and
we even have equality. If $k$ is odd
one has to check that $\alpha_k^{v(\Delta)-v(\Gamma)}\beta_k
\leq \alpha_{k+1}^{v(i(\Delta))-v(i(\Gamma))}\beta_{k+1}.$
If we put $l:=[k/2+1]$
and $n=v(\Delta)-v(\Gamma)$,
then this amounts to checking the inequality
$(\frac{2l+3}{l+1})^{l+1-n}(l+1)^2 \geq (\frac{2l+1}{l})^{l-n}l^2$,
which is tedious but elementary.
It follows that the parts $S'_\Delta$ for $k+1$ contain unions of parts
$S_\Gamma$ for $k$:
$S'_\Delta \supset \cup_{\Gamma \in J} S_\Gamma$. We can now apply Theorem
\ref{filters coarse} to this union to get a filtration on $H^l(S_\Delta')$ some
of whose
subquotients are isomorphic to $H^{l+2 codimS_\Delta'-2
codimS_\Gamma}(S_\Gamma)(-codim S_\Delta' + codim S_\Gamma)$
for $l<k$. We conclude that enlarging $k$ amounts to taking
the union of some parts.
} \end{remark}
\begin{remark} {\rm
Theorem \ref{filters coarse} does not hold if we replace the partition by the
stratification by topological type. To see this,
let $k$ be given and take $g>5\alpha^{[k/2+1]+1}$. Take $n=0$ for simplicity.
Choose natural numbers $d>c>b>a>\alpha^{[k/2+1]+1}$ such that
$a+b+c+d=g$. Consider the graph $\Gamma$
which has five vertices, of weights $0,~a~,b,~c,~d$ and four edges,
joining the weight $0$ component to the other four. The automorphism group
of the stable graph $\Gamma$ clearly is trivial. One checks easily that
$\Gamma_W$ is
the full subgraph on the weight $0$ vertex, which implies that $\Gamma$ defines
the open stratum $S_\Gamma^0$ of a part $S_\Gamma$. The part $S_\Gamma$ is
obtained by letting
the genus $0$ curve degenerate in all possible ways. There are three
possible degenerations, corresponding to the three possibilities of
partioning the four other vertices into
two sets of two. So we have: $S_\Gamma^0 \cong N \times M_0^4$ and $S_\Gamma
\cong N \times \overline{M_0^4}$, where $N:=M_a^1 \times M_b^1 \times
M_c^1 \times M_d^1$.
$M_0^4$ is a ${\bf P}^1$ minus 3 points and $\overline{M_0^4}$ is ${\bf P}^1$.
Thus $S_\Gamma=S_\Gamma^0 \cup \cup_1^3 N \times {\it point}$.
We have $H^2(N \times \overline{M_0^4})=H^2(N) \otimes H^0(\overline{M_0^4})
\oplus H^0(N) \otimes H^2(\overline{M_0^4})=H^2(N) \oplus
{\bf Q}$. Suppose Theorem \ref{filters coarse} would hold with the
partition replaced by the
stratification by topological type. Then we would have, using the above:
$H^2(N \times \overline{M_0^4})=
H^2(N \times {M_0^4}) \oplus \oplus_{i=1}^3 H^0(N \times {\it point})=
H^2(N) \oplus {\bf Q}^3$, which leads to a
contradiction. }
\end{remark}
\begin{corollary} \label{coho Mgnbar}
Given $k \geq 0$, then for $g>\beta$, $\Mgnbar$ admits an orbifold partition
which has $M_g^n$ as its open part, is coarser than the stratification by
topological type and filters cohomology up to degree $k$.
\end{corollary}
\begin{proof}
Using Theorem \ref{filters coarse}
we get the statement with $\Mgnbar$ replaced by $S$.
The real codimension of the complement of $S$ in $\Mgnbar$ is
$2[k/2+1]+2 >k$, thus
$H^l(\Mgnbar) \cong H^l(S)$, for all $l<k$. (Compare the remark following
the definition of a partition which filters cohomology in the introduction.)
Combining these gives the desired result.
\unskip\nolinebreak\hfill\hbox{\quad $\Box$} \end{proof}
\begin{corollary} \label{pure Hodge}
If $g \geq 2k+1$, then $H^k(\Mgnbar) \ra H^k(M_g^n)$ is onto; consequently,
the mixed Hodge structure on $H^k(M_g^n)$ is pure of weight $k$.
\end{corollary}
\begin{proof}
By corollary \ref{coho Mgnbar}, we have that $H^k(M_g^n)$ is a quotient of
$H^k(\Mgnbar)$, so the last statement holds for $g > \beta$.
Now use that the image of $H^k(\Mgnbar)$
in $H^k(M_g^n)$ is $W^kH^k(M_g^n)$, which equals $H^k(M_g^n)$ if $g \geq
2k+1$.
\unskip\nolinebreak\hfill\hbox{\quad $\Box$} \end{proof}
\begin{corollary}\label{not Tate}
If $g >>0$, then the cohomology of $\Mgnbar$ is not of Tate type.
\end{corollary}
\begin{proof}
We use the fact that the modular form $\Delta$ can be seen as a non vanishing
holomorphic 11-form on $\overline{M_1^{11}}$ (see \cite{Deligne}). Thus,
$H^{11,0}(\overline{M_1^{11}})$ is not zero.
Take $g$ large enough and $n$ arbitrary. One of the strata in
corollary \ref{coho Mgnbar}
is the following: the generic graph consists of two vertices, one of weight
$g-11$ and one of weight 1. There are 11 edges joining them and $n$ loose half
edges at the vertex of weight $g-11$. Clearly, the maximal $\Phi_k$-weak
subgraph of
this graph consists of the vertex of weight 1 with its 11 half edges.
This graph is therefore in the image of
the map $\phi$ and defines a generic point of a part of codimension 11.
The open dense topological stratum in it is $M_{g-11}^{n+11}
\times {M_1^{11}}$.
The corresponding part is obtained by
letting the genus 1 curve degenerate, thus it is $M_{g-11}^{n+11} \times
\overline{M_1^{11}}$.
Consider the cohomology group $H^{22,11}(\Mgnbar)
\subset H^{33}(\Mgnbar)$. By corollary \ref{coho Mgnbar}
it has a subquotient $H^{11,0}(M_{g-11}^{n+11} \times
\overline{M_1^{11}})$, which has as direct summand
$H^0(M_{g-11}^{n+11}) \bigotimes H^{11,0}(\overline{M_1^{11}}) \neq 0 $.
By considering different strata we find that other cohomology groups don't
vanish either. For example: suppose $n \geq 10$ and take a graph consisting of
two vertices, one
of weight 1 and one of weight $g-1$, joined by one edge. Furthermore the weight
one vertex has 10 loose half edges and the other vertex $n-10$. This
defines the generic graph of a part of codimension 1.
By the same argument as above, we see that $H^{12,1}(\Mgnbar)$ is not zero.
\unskip\nolinebreak\hfill\hbox{\quad $\Box$} \end{proof}
\begin{proposition}\label{stabiele cohomologie voor Moneindignstreep}
There exists a ``stable cohomology" of $\Mgnbar$ for $n$ fixed and $g \ra
\infty$.
\end{proposition}
\begin{proof}
Notation as before.
Let $k$ be given and suppose $g > \beta$. Let $\Gamma$ be in the image of
$\phi_k(g,n)$, that is, $\Gamma$ defines a generic stratum of a part $S$.
Choose a strong vertex $P$ of $\Gamma$, let $h$ be the weight of $P$.
Consider the graph we get by changing $h$ to $h+1$. Call this graph
$\Delta$. Because the definition of $\Phi_k$-weak subgraph does not depend on
$g$,
we see that $\Gamma_W$ and $\Delta_W$ can be identified. This means that
$\Delta$ is in the image of $\phi_k(g+1,n)$ and therefore defines a part $T$ in
$\overline{M_{g+1}^n}$. Furthermore we get that $H^l(S) \cong H^l(T)$ for all
$l <k$, because $h$ is in the stable range with respect to $k$.
For every $x \in {\bf N}$
we define $\psi_x: {\cal G}_k(g+x,n) \ra
{\cal G}_k(g+x+([k/2+1]+1)!,n)$ as follows:
for any vertex $P$ in $\Gamma_S$ add
$([k/2+1]+1)!/ \# \Gamma_S$ to its weight.
$\psi$ clearly respects the parts and
thus defines a map $\psi_x :I_k(g+x,n) \ra I_k(g+x+([k/2+1]+1)!,n)$. Because
the maximal $\Phi_k$-weak subgraphs don't change, this map is an injection.
Furthermore it induces isomorphisms on the cohomology of the parts, as
explained above. So we get an injection induced by $ \psi_x$:
$$H^l(M_{g+x}^n) \hookrightarrow H^l(M_{g+x+([\frac{1}{2}k+1]+1)!}^n),$$
for all $l<k$, which maps the subquotients isomorphically onto the
corresponding subquotients. We claim that the inductive limit
over these maps is independent of $x$. We will compare the inductive limits
for 0 and $x>0$. For any $\Gamma$ in $Im(\phi_k(g,n))$ we choose
one strong vertex $P_\Gamma$.
Define $\chi_x(\Gamma) \in {\cal G}_k(g+x,n)$
by adding $x$ to the weight of $P$. $\chi_{x}(\Gamma)$
is in the image of $\phi_k(g+x,n)$. Define $\overline{\chi_x}:
Im(\phi_k(g+([k/2+1]+1)!,n)) \ra
Im(\phi_k(g+x+([k/2+1]+1)!,n))$ by adding $x$ to the
weight of the strong vertex $\psi_x (P_\Gamma)$. We clearly have
$\overline{\chi_x} \circ \psi_0=\psi_x \circ \chi_x$.
Now we replace 0 by $x$ and $x$ by
$([k/2+1]+1)!$.
Playing the same trick and using that the inductive limits are
canonically isomorphic if the indices differ by a multiple of
$([k/2+1]+1)!$, we prove our claim. We define $H^l(\overline{M^n_\infty})$ as
this inductive limit.
\unskip\nolinebreak\hfill\hbox{\quad $\Box$} \end{proof}
One may regard this as the ``stable cohomology" of $\Mgnbar$ for $g \ra
\infty$. We note however that this stable cohomology is not finitely
generated.
For example, if $n=0$, the stable generators in degree $2$ are the tautological
class, and the boundary classes naturally indexed by $0,1,2,\dots$.
We note also that the part defined in Corollary \ref{not Tate} defines
a part for every $g$ sufficiently large; consequently the proof
of Theorem \ref{stabiele cohomologie voor Moneindignstreep} shows that
even the stable
cohomology of $\Mgnbar$ for $g \ra \infty$ is not of Tate type.
|
1996-12-06T08:03:17 | 9503 | alg-geom/9503001 | en | https://arxiv.org/abs/alg-geom/9503001 | [
"alg-geom",
"math.AG"
] | alg-geom/9503001 | Nitin Nitsure | Nitin Nitsure | Quasi-parabolic Siegel Formula | LaTeX, 6 pages. Reason for re-submission : A factor that was missing
in the first version is now included in the formula | null | null | null | null | The result of Siegel that the Tamagawa number of $SL_r$ over a function field
is 1 has an expression purely in terms of vector bundles on a curve, which is
known as the Siegel formula. We prove an analogous formula for vector bundles
with quasi-parabolic structures. This formula can be used to calculate the
Betti numbers of the moduli of parabolic vector bundles using the Weil
conjucture.
| [
{
"version": "v1",
"created": "Thu, 2 Mar 1995 06:49:00 GMT"
},
{
"version": "v2",
"created": "Fri, 6 Dec 1996 07:04:00 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Nitsure",
"Nitin",
""
]
] | alg-geom | \subsection*{\hbox{}\hfill{\normalsize\sl #1}\hfill\hbox{}}}
\textheight 23truecm \textwidth 15truecm
\addtolength{\oddsidemargin}{-1.05truecm}
\addtolength{\topmargin}{-1.5truecm}
\makeatletter \def\l@section{\@dottedtocline{1}{0em}{1.2em}} \makeatother
\title{Quasi-parabolic Siegel Formula}
\author{Nitin Nitsure}
\begin{document}
\date{Corrected version, 6 December 1996}
\maketitle
School of Mathematics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400 005, India. e-mail:
[email protected]
\begin{abstract} The result of Siegel that the Tamagawa number
of $SL_r$ over a function field is $1$ has an expression purely
in terms of vector bundles on a curve, which is known as the
Siegel formula. We prove an analogous formula for vector bundles
with quasi-parabolic structures. This formula can be used to
calculate the betti numbers of the moduli of parabolic vector
bundles using the Weil conjucture.
\end{abstract}
\section{Introduction}
The Betti numbers of the moduli of stable vector bundles on a
complex curve, in all the
cases where the rank and degree are coprime,
were first determined by Harder and Narasimhan [H-N] as an
application of the Weil conjuctures. For this, they made use of
the result of Siegel that the Tamagawa number of the special
linear group over a function field is 1. In their refinement of
the same Betti number calculation in [D-R], Desale
and Ramanan expressed the result of Siegel in purely vector
bundle terms. This result about the Tamagawa number, called the
Siegel formula, was later given a simple proof in the language
of vector bundles by Ghione and Letizia [G-L], by introducing a
notion of effective divisors of higher rank on a curve, and
counting the number of effective divisors which correspond to a
given vector bundle. This purpose of this note is to introduce
the notion of a quasi-parabolic divisor of higher rank on a curve
(Definition 3.1 below), and to prove a quasi-parabolic analogue
(Theorem 3.4 below) of the Siegel formula, which is done here by
suitable generalizing the method of [G-L]. In a note to follow,
this formula is used to calculate the Zeta function and thereby
the Betti numbers of the moduli of parabolic bundles in the case
`stable = semistable' (these Betti numbers have already been
calculated by a guage theoretic method for genus $\ge 2$ in [N]
and for genus $0$ and $1$ by Furuta and Steer in [F-S]).
{\bf Acknowledgement} I thank M. S. Narasimhan for suggesting
the problem of extending [H-N] to parabolic bundles.
\section{Divisors supported on $X-S$}
Let $X$ be an absolutely irreducible, smooth projective curve
over the finite field $k={\bf F}_q$, and let $S$ be any closed
subset of $X$ whose points are $k$-rational. Let $K$ denote the
function field of $X$, and let $K_X$ denote the constant sheaf $K$ on
$X$. Let $g$ denote the genus of $X$. Let $r$ be a positive
integer.
Recall that (see [G-L]) a coherent subsheaf $D\subset K _X^r$ of
generic rank $r$ is called an $r$-divisor, and the $r$-divisor
is called effective (or positive) if ${\cal O} _X^r\subset D$. The
support of the divisor is by definition
the support of the quotient $D/{\cal O} _X^r$, which is a torsion sheaf.
The lenght $n$ of $D/{\cal O} _X^r$ is called the degree of the divisor.
Note that $D$ is a locally free sheaf of rank $r$ and degree
$n$.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
Let $Z_X(t)$ be the zeta function of $X$. Then as $S$ consists
of $k$-rational points, it can be seen that the zeta function
$Z_{X-S}$ of $X-S$ is given by the formula
$$Z_{X-S}(t) = (1-t)^sZ_X(t)\eqno(1)$$
where $s$ is the cardinality of $S$.
Note that an effective $r$-divisor on $X-S$ is the same as an
effective $r$-divisor on $X$ whose support is disjoint from $S$.
The part (1) of the proposition 1 of [G-L] gives the following,
with $X-S$ in place of $X$.
\begin{proposition}
Let $b_n^{(r)}$ be the number of effective $r$-divisors of degree
$n$ on $X$ whose support is disjoint from $S$. Let
$Z_{X-S}^{(r)}(t) = \sum _{n\ge 0} b_n^{(r)}t^n$.
Then we have
$$Z_{X-S}^{(r)}(t) = \prod _{1\le j\le r} Z_{X-S}(q^{j-1}t)\eqno(2)$$
\end{proposition}
In order to have the analogue of the part (2) of the proposition
1 of [G-L], we need the following lemmas.
\begin{lemma}
Let $V$ be a finite dimensional vector space over $k={\bf F}_q$,
and $s$ a positive integer. For any $1\le i\le s$,
let $\pi _i:k^s\to k$ be the linear
projection. For any surjective linear map $\phi :V\to k^s$, let
$V_i$ be the kernel of $\pi _i\phi :V\to k$, which is a
hyperplane in $V$ as $\phi$ is surjective.
Let $P=P(V)$, and $P_i=P(V_i)$ denote the corresponding
projective spaces. Let $N(\phi )$ denote the number of
$k$-rational points of $P - \cup _{1\le i\le s} P_i$. Then for
any other surjective $\psi : V\to k^s$,
we have $N(\phi )=N(\psi )$. In other words, given $s$, this
number depends only on $dim(V)$.
\end{lemma}
\paragraph{Proof} Given any two surjective maps $\phi ,\psi :V\to k^s$, there
exists an $\eta \in GL(V)$ such that $\phi \eta = \psi$. From
this, the result follows.
\begin{lemma}
Let $n$ be a positive integer, such that $n>2g-2+s$
where $g$ is the genus of $X$ and $s$ is the cardinality of $s$.
Let $b_n$ is the total number
of effective $1$-divisors of degree $n$ supported on $X-S$.
Then for any line bundle $L$ on $X$ of degree $n$, the number of effective
$1$-divisors supported on $X-S$ which define $L$ is $b_n/P_X(1)$, where
$P_X(1)$ is the number of isomorphism classes of line bundles of
any fixed degree on $X$.
\end{lemma}
(Here, $P_X(t)$ is the polynomial $(1-t)(1-qt)Z_X(t)$.)
\paragraph{Proof} Let $L$ be any line bundle on $X$ of degree $n$, where
$n>2g-2+s$. Then $H^1(X,L(-S))=0$, so the natural map
$\phi :H^0(X,L)\to H^0(X,L|S)$ is surjective. Let $V=H^0(X,L)$.
Then $dim(V)=n+1-g$. Choose a basis for each fiber $L_P$ for
$P\in S$. This gives an identification of $H^0(X,L|S)$ with
$k^s$. Now it follows that the number $N(\phi )$ defined in the
preceeding lemma depends only on $n$, and is independent of the
choice of $L$ as long as it has degree $n$. But $N(\phi )$ is
precisely the number of effective $1$-divisors supported on $X-S$,
which define the line bundle $L$ on $X$.
Using the above lemma, the following proposition follows, by an
argument similar to the proof of part (2) of proposition 1 in
[G-L]. The proof in [G-L] expresses the number
of $r$-divisors in terms of the number of $1$-divisors, and
the above lemma tells us the number of $1$-divisors with support
in $X-S$ corresponding to a given line bundle on $X$.
\begin{proposition}
For $L$ a line bundle of degree $n$, let $b_n^{(r,L)}$ be the
number of effective $r$-divisors on $X$ supported on $X-S$,
having determinant isomorphic to $L$. Then provided that
$n>2g-2+s$, we have
$$b_n^{(r,L)} = b_n^{(r)}/P_X(1)\eqno(3)$$
\end{proposition}
\begin{proposition}
$$\lim _{n\to\infty} {b_n^{(r)}\over{q^{rn}}} =
P_X(1){(q-1)^{s-1}\over{q^{g-1+s}}}
Z_{X-S}(q^{-2})\cdots Z_{X-S}(q^{-r})\eqno(4)$$
\end{proposition}
\paragraph{Proof} The above statement is the analogue of proposition 2 of
[G-L], with the following changes. Instead of all $r$-divisors
on $X$ in [G-L], we consider only those which are supported over
$X-S$, and instead of $Z_X(t)$, we use $Z_{X-S}(t)$. As
$Z_{X-S}(t) = (1-t)^sZ_X(t)$, the property of $Z_X(t)$ that it
has a simple pole at $t=q^{-1}$ and is regular at $1/q^j$ for
$j\ge 2$ is shared by $Z_{X-S}(t)$.
Hence the proof in [G-L] works also in our case, proving the
proposition.
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
There is a minor misprint in the equation labeled (1) in [G-L]
(page 149); the factor $q^{g-1}$ should be read as $q^{1-g}$.
Let $L$ be any given line bundle on $X$.
Choose any closed point $P\in X-S$, and let $l$ denote its
degree. For any ${\cal O} _X$ module $E$, set $E(m)=E\otimes
{\cal O}_X(mP)$. If a vector bundle $E$ of rank $r$ degree $n$ has
determinant $L$, then $E(m)$ has determinant $L(rm)$, degree
$n+rml$ and Euler characteristic $\chi(m)=n+rml+r(1-g)$.
The equations (3) and (4) above imply the following.
$$\lim _{m\to\infty}
{b_{n+rml}^{(r,L(rm))}\over{q^{r\chi(m)}}} =
(q-1)^{s-1}
q^{(r^2-1)(g-1)-s}
Z_{X-S}(q^{-2})\cdots Z_{X-S}(q^{-r})\eqno(5)$$
\section{Quasi-parabolic divisors}
For basic facts about parabolic bundles, see [S] and [M-S]. We
now introduce the notion of a quasi-parabolic effective divisor
of rank $r$. Let $S\subset X$ be a finite subset consisting of
$k$-rational points. For each $P_i\in S$, let there be given
positive integers $p_i$ and $r_{i,1},\ldots ,r_{i,p_i}$ with
$r_{i,1}+\ldots +r_{i,p_i} =r$. This will be called, as usual,
the quasi-parabolic data. Recall that a quasi-parabolic
structure on a vector bundle $E$ of rank $r$ on $X$ by
definition consists of flags
$E_{P_i}=F_{i,1}\supset\ldots\supset F_{i,p_i}\supset F_{i,p_i+1}=0$
of vector subspaces in the fibers over the points of $S$ such that
$dim(F_{i,j}/F_{i,j+1})=r_{i,j}$ for each $j$ from $1$ to $p_i$.
\begin{definition}\rm
Let $X$, $S$, and the numerical data $(r_{i,j})$ be as above.
A positive quasi-parabolic divisor $(F,D)$ on $X$ consists of
(i) a quasi-parabolic structure $F$ on the trivial bundle ${\cal O}
_X^r$, consisting of flags $F_i$ in $k^r$ at points $P_i\in S$
of the given numerical type $(r_{i,j})$,
together with
(ii) an effective $r$-divisor $D$ on $X$, supported on $X-S$.
\end{definition}
Note that if $(F,D)$ is a quasi-parabolic $r$-divisor, then the
rank $r$ vector bundle $D$ has a parabolic structure given by
$F$. We denote by $P^{(r)}_E$ the set of all effective parabolic
$r$-divisors whose associated parabolic bundle is isomorphic to
a given parabolic bundle $E$.
For any vector bundle $E$ of rank $r$, let
$Hom^S_{inj}({\cal O} _X^r,E)$ denote the set of all injective
sheaf homomorphisms ${\cal O} _X^r \to E$ which are injective when
restricted to $S$. For any quasi-parabolic bundle $E$, the group
of all quasi-parabolic automorphisms of $E$ will be denoted by
$ParAut(E)$. Then $ParAut(E)$ acts on $Hom_{inj}^S({\cal O} _X^r,E)$
by composition. This action is free, and $P^{(r)}_E$ has a
canonical bijection with the quotient set $Hom_{inj}^S({\cal O}
_X^r,E)/ParAut(E)$. Hence the cardinality of $P^{(r)}_E$ is
given by
$$|P^{(r)}_E|=
{{|Hom_{inj}^S({\cal O} _X^r,E)|}\over |{ParAut(E)|}}\eqno(6)$$
For $1\le i\le s$, let ${\rm Flag}_i$ be the variety of flags in
$k^r$ of the numerical type $(r_{i,1},\ldots ,r_{i,p_i})$.
Let ${\rm Flag}_S = \prod _{1\le i\le s}{\rm Flag}_i$. Let
$f(q,r_{i,j})$ denote the number of $k$-rational points of ${\rm
Flag}_S$. If $a_n^{(r,L)}$ denotes the number
of quasi-parabolic divisors of flag data $(r_{i,j})$ with
degree $n$, rank $r$ and determinant $L$, then we have
$$a_n^{(r,L)} = f(q,r_{i,j})b_n^{(r,L)}\eqno(7)$$
Now let $J(r,L)$ denote the set of all isomorphism classes of
quasi-parabolic vector bundles of rank $r$, degree $n$, determinant
$L$ having the given quasi-parabolic data $(r_{i,j})$ over $S$. Hence
the equation (6) above implies the following.
$$a^{(r,L)}_n = \sum _{E\in J(r,L)}
{{|Hom_{inj}^S({\cal O} _X^r,E)|}\over |{ParAut(E)|}}\eqno(8)$$
For any integer $m$, the map from $J(r,L) \to J(r, L(rm)$ which sends
$E$ to $E(m) =E\otimes O_X(mP)$ is a bijection which preserves
$|ParAut|$. Hence for each $m$, we have
$$a^{(r,L(rm))}_{n+rml} = \sum _{E\in J(r,L)}
{{|Hom_{inj}^S({\cal O} _X^r,E(m))|}\over |{ParAut(E)|}}\eqno(9) $$
\begin{lemma}
With the above notations,
$$\lim_{m\to \infty}{{|Hom_{inj}^S({\cal O} _X^r,E(m))|}\over {q^{r\chi (E(m))}}}
= {(q^r-1)^s(q^r-q)^s\cdots(q^r-q^{r-1})^s\over q^{r^2s}}\eqno(10)$$
If $S$ is non-empty, the limit is already attained for all
large enough $m$ (where `large enough' depends on $E$).
\end{lemma}
\paragraph{Proof} If $S$ is empty, the above lemma reduces to
lemma 3 in [G-L]. If $S$ is nonempty, then any morphism of locally free sheaves
on $X$ which is injective when restricted to $S$ is injective.
Let $m$ be large enough, so that $E(m)$ is generated by
global sections, $H^1(X,E(m))=0$, and $h^0(X,E(m))=\chi(E(m)) \ge rs$.
Then $H^0(X,E(m))$ has a basis consisting of
sections $\sigma_{i,P_j}$, $\tau_{\ell}$ for $i=1,\ldots, r$, $j=
1,\ldots,s$, and $\ell=1,\ldots, \chi(E(m))-rs$, such that
(1) the sections $\tau_{\ell}$ are zero on $S$,
(2) the sections $\sigma_{i,P_j}$ are zero at all other points of $S$
except $P_j$ (and hence $\sigma_{i,P_j}$ restrict at $P_j$ to a basis
of the fiber of $E(m)$ at $P_j$.
Any element of $Hom_{{\cal O}_X}({\cal O}_X^r,E(m))
= Hom_{{\bf F}_q}({\bf F_q}^r, H^0(X,E(m)))$ is given in terms of this
basis by a $r\times q^{\chi(E(m))}$ matrix $A$. The condition that
this lies in
$$Hom_{inj}^S({\cal O} _X^r,E(m)) \subset Hom({\cal O} _X^r, E(m))$$
is the condition that each of the $s$ disjoint $r\times r$-minors,
corresponding to the part $\sigma_{1,P_j},\ldots, \sigma_{r,P_j}$ of
the basis, has nonzero determinant. This contributes the factor
$${|GL_r({\bf F}_q)|\over |M_r({\bf F}_q)|}=
{(q^r-1)(q^r-q)\cdots(q^r-q^{r-1})\over q^{r^2}}
$$
for each $P_j$, which
proves the lemma.
\begin{lemma}
The following sum and limit can be interchanged to give
$$\sum _{E\in J(r,L)} \lim _{m\to\infty}
{{|Hom_{inj}^S({\cal O} _X^r,E(m))|}\over {q^{r\chi (E(m))}|ParAut(E)|}}
= \lim _{m\to\infty} \sum _{E\in J(r,L)}
{{|Hom_{inj}^S({\cal O} _X^r,E(m))|}\over {q^{r\chi (E(m))}
|ParAut(E)|}}
$$
\end{lemma}
This lemma has a proof entirely analogous to the corresponding
statement in [G-L], so we omit the details.
By equation (10), the left hand side in the above
lemma equals
$${(q^r-1)^s(q^r-q)^s\cdots(q^r-q^{r-1})^s\over q^{r^2s}}
\sum _{E\in J(r,L)} {1\over{|ParAut(E)|}}
$$
On the other hand, by (9), the right hand side is
$\lim _{m\to\infty} a^{(r,L(rm))}_{n+rml}/q^{r\chi (m)}$. By
equations (5) and (7), this limit has the following value.
$$f(q,r_{i,j})(q-1)^{s-1} q^{(r^2-1)(g-1)-s}
Z_{X-S}(q^{-2})\cdots Z_{X-S}(q^{-r})$$
By putting $Z_{X-S}(t) = (1-t)^sZ_X(t)$ in the above, and cancelling
common factors from both sides, we get the following.
\begin{theorem
{\rm (Quasi-parabolic Siegel formula)}
$$\sum _{E\in J(r,L)} {1\over{|ParAut(E)|}}=
f(q,r_{i,j}) {q^{(r^2-1)(g-1)}\over q-1 }
Z_X(q^{-2})\cdots Z_X(q^{-r}) $$
\end{theorem}
\refstepcounter{theorem}\paragraph{Remark \thetheorem}
If $S$ is empty or more generally if the quasi-parpbolic structure at
each point of $S$ is trivial (that
is, each flag consists only of the zero subspace and the whole space),
then on one hand $ParAut(E)=Aut(E)$, and on the other hand each flag
variety is a point, and so $f(q,r_{i,j})=1$. Hence in this situation
the above formula reduces to the original Siegel formula
$$\sum _{E\in J(r,L)} {1\over{|Aut(E)|}}=
{q^{(r^2-1)(g-1)}\over q-1 } Z_X(q^{-2})\cdots Z_X(q^{-r}) $$
\section*{References}
[D-R] Desale, U. V. and Ramanan, S. : Poincar\'e Polynomials of
the Variety of Stable Bundles, {\sl Math. Annln.} {\bf 216}
(1975), 233-244.
[F-S] Furuta, M. and Steer, B. : Siefert-fibered homology
3-spheres and Yang-Mills equations on Riemann surfaces with
marked points, {\sl Adv. Math.} {\bf 96} (1992) 38-102.
[G-L] Ghione, F. and Letizia, M. : Effective divisors of higher
rank on a curve and the Siegel formula, {\sl Composito Math.}
{\bf 83} (1992), 147-159.
[H-N] Harder, G. and Narasimhan, M. S. : On the Cohomology
Groups of Moduli Spaces of Vector Bundles over Curves, {\sl
Math. Annln.} {\bf 212} (1975), 215-248.
[M-S] Mehta, V. B. and Seshadri, C. S. : Moduli of vector bundles
on curves with parabolic structures, {\sl Math. Annln.} {\bf 248}
(1980) 205-239.
[N] Nitsure, N. : Cohomology of the moduli of parabolic vector
bundles, {\sl Proc. Indian Acad. Sci. (Math. Sci.)} {\bf 95}
(1986) 61-77.
[S] Seshadri, C. S. : Fibres vectoriels sur les courbes
algebriques, {\sl Asterisque} {\bf 96} (1982).
\end{document}
|
1993-09-30T22:08:20 | 9309 | alg-geom/9309007 | en | https://arxiv.org/abs/alg-geom/9309007 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9309007 | David R. Morrison | Paul S. Aspinwall, Brian R. Greene and David R. Morrison | The Monomial-Divisor Mirror Map | 22 pages, LaTeX | Internat. Math. Res. Notices (1993), 319-337 | null | IASSNS-HEP-93/43, CLNS 93/1237 | null | For each family of Calabi-Yau hypersurfaces in toric varieties, Batyrev has
proposed a possible mirror partner (which is also a family of Calabi-Yau
hypersurfaces). We explain a natural construction of the isomorphism between
certain Hodge groups of these hypersurfaces, as predicted by mirror symmetry,
which we call the monomial-divisor mirror map. We indicate how this map can be
interpreted as the differential of the expected mirror isomorphism between the
moduli spaces of the two Calabi-Yau manifolds. We formulate a very precise
conjecture about the form of that mirror isomorphism, which when combined with
some earlier conjectures of the third author would completely specify it. We
then conclude that the moduli spaces of the nonlinear sigma models whose
targets are the different birational models of a Calabi-Yau space should be
connected by analytic continuation, and that further analytic continuation
should lead to moduli spaces of other kinds of conformal field theories. (This
last conclusion was first drawn by Witten.)
| [
{
"version": "v1",
"created": "Thu, 30 Sep 1993 21:08:10 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Aspinwall",
"Paul S.",
""
],
[
"Greene",
"Brian R.",
""
],
[
"Morrison",
"David R.",
""
]
] | alg-geom | \section*{References\@mkboth
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\begin{document}
\@dblarg\@xtitle{The Monomial-Divisor Mirror Map}
\@dblarg\@xauthor{Paul S. Aspinwall, Brian R. Greene and David R. Morrison}
\address{Aspinwall:\ School of Natural Sciences, Institute for Advanced
Study, Princeton, NJ 08540\\
Greene:\ F.R. Newman Laboratory of Nuclear Studies, Cornell University,
Ithaca, NY 14853\\
Morrison:\ School of Mathematics, Institute for Advanced Study,
Princeton, NJ 08540}
\date{}
\thanks{Research partially supported by
DOE grant
DE-FG02-90ER40542, a National Young Investigator award,
NSF grant DMS-9103827, and
an American Mathematical Society Centennial Fellowship.
}
\renewcommand{\LARGE}{\Large\bf}
\maketitle
\renewcommand{\Large}{\large}
\begin{abstract}
For each family of Calabi-Yau hypersurfaces in toric varieties,
Batyrev has
proposed a possible mirror partner (which is also a family of
Calabi-Yau hypersurfaces).
We explain a natural construction
of the isomorphism
between certain Hodge groups of these hypersurfaces, as predicted
by mirror symmetry, which we call the {\em monomial-divisor mirror map}.
We indicate how this map can be
interpreted as the differential of the expected mirror isomorphism between
the moduli spaces of the two Calabi-Yau manifolds.
We formulate a very precise conjecture about the form of that mirror
isomorphism,
which when combined with some earlier conjectures of the third author
would completely specify it.
We then
conclude that the moduli spaces of the nonlinear
sigma models whose targets are the different birational models of a
Calabi-Yau space should be connected by analytic continuation,
and that further analytic continuation should lead to moduli spaces
of other kinds
of conformal field theories.
(This last conclusion was first drawn by Witten.)
\end{abstract}
\section{Reflexive polyhedra}
Mirror symmetry, which proposes that Calabi-Yau manifolds should come
in pairs with certain remarkable
properties, is a phenomenon that was first observed
in the physics literature \cite{dixon,LVW,CLS,GP}.%
\footnote{For general mathematical background on
mirror symmetry and mirror pairs, we refer the reader to
\cite{mirrorbook} and \cite{guide}.}
The most concrete realization of this phenomenon---actually the only
one in which there is a physical argument linking the conformal
field theories associated to the pair of Calabi-Yau manifolds---is given
by the Greene-Plesser
orbifolding construction \cite{GP} for
Fermat hypersurfaces in weighted projective spaces
and certain quotients
of them by finite groups. Roan \cite{roan-mirror,roan-topological}
has given a natural description of this construction in terms of
toric geometry, and he showed that the mirror phenomenon in that case
can be interpreted as a kind of duality between toric hypersurfaces.
This enabled him to give rigorous mathematical proofs of certain
formulas discovered by physicists.
Batyrev \cite{batyrev1}
has recently found an elegant characterization of Calabi-Yau
hypersurfaces which are ample Cartier divisors in (mildly singular)
toric varieties.
The characterization is stated in terms of the
{\em Newton polyhedron} of the hypersurface, which is the convex hull
of the monomials appearing in its equation.
This is always an integral polyhedron, that is,
a compact convex polyhedron $P$ whose vertices are elements
of a lattice $M$ in a real affine space $M_{\Bbb R}:=M\otimes\Bbb R$.
Batyrev's characterization states that
the general hypersurface with Newton polyhedron $P$ is Calabi-Yau
(that is, has trivial canonical bundle and at worst Gorenstein
canonical singularities),
provided
that $0$ is in the interior of $P$, and that each
affine hyperplane $H\subset M_{\Bbb R}$
which meets $P$ in a face of codimension one
has the form
\[H:=\{y\in M_{\Bbb R}\ |\ \langle \ell,y\rangle=-1\}\]
for some $\ell=\ell(H)$ in the dual lattice $N:=\operatorname{Hom}(M,\Bbb Z)$.
An integral polyhedron with this property is called {\em reflexive}.
The normals $\ell(H)$ of supporting hyperplanes $H$ for codimension-one
faces of a reflexive polyhedron $P$ have as their convex
hull the {\em polar polyhedron} $\polyhedron^\circ$, which is defined to be
\[\polyhedron^\circ:=\{x\in N_{\Bbb R}\ |\
\langle x,y\rangle\ge-1 \text{ for all } y\inP\}.\]
Batyrev showed that the polar polyhedron $\polyhedron^\circ$ of a reflexive integral
polyhedron $P$ is itself a reflexive integral polyhedron
(with respect to the dual lattice $N$).
This led him to propose that hypersurfaces $X$ and $Y$ with Newton
polyhedra $P$ and $\polyhedron^\circ$, respectively, should form a
mirror pair.
The evidence for Batyrev's proposal is of several kinds. First and
foremost is the fact that this polar polyhedron construction specializes
to Roan's interpretation of the Greene-Plesser
orbifolding construction in
the case of quotients of Fermat hypersurfaces in weighted projective spaces.
This is encouraging, since as noted above the
Greene-Plesser construction provides the {\em only} complete example of mirror
symmetry for hypersurfaces---the only example for which there is a physical
argument for the existence of a mirror isomorphism of the corresponding
conformal field theories. A second piece of evidence (which we discuss
more fully below) is an isomorphism between certain Hodge groups associated
to $X$
and $Y$ (extending the work of Roan),
as would be predicted by mirror symmetry. And finally,
Batyrev shows that his construction is compatible with the existence
of certain ``quantum symmetries'' as expected based on physical reasoning.
This quantum
symmetry behavior looks somewhat unnatural mathematically, so verifying
it is an important check.
This evidence falls short of fully establishing a mirror symmetry
relationship between $X$ and $Y$, since it does not link the
corresponding conformal field theories. However, it does provide
strong
grounds for suspecting the existence of a mirror isomorphism. And the
naturality of Batyrev's polar polyhedron construction
is extremely compelling (at least to
mathematicians).
If mirror symmetry does hold between $X$ and $Y$, there will be an isomorphism
between Hodge groups $H^{1,1}(\widehat{X})$ and $H^{d-1,1}(\widehat{Y})$,
where $\widehat{X}\to X$ and $\widehat{Y}\to Y$ are appropriate (partial)
resolutions of singularities, and $d$ is the common
dimension of $X$ and $Y$.
The existence of such an isomorphism had been
shown quite explicitly by Roan \cite{roan-mirror,roan-topological}
in the weighted Fermat hypersurface case---%
the general case is addressed by Batyrev.
In the earlier preprint versions of \cite{batyrev1}, Batyrev
found an equality between the dimensions of certain subspaces
$H^{1,1}_{\text{toric}}(\widehat{X})$ and
$H^{d-1,1}_{\text{poly}}(\widehat{Y})$
of the Hodge groups,
mistakenly believed to have been the entire spaces. In the final version
of \cite{batyrev1}, he shows that the full Hodge groups are
isomorphic, following suggestions made by the present authors. The
error in the earlier version of the paper was fortuitous, however,
as it revealed that the mirror isomorphism might be expected to preserve
those subspaces.
In this note, we explain a very natural construction of the isomorphism
between $H^{1,1}_{\text{toric}}(\widehat{X})$ and
$H^{d-1,1}_{\text{poly}}(\widehat{Y})$ ,
and indicate how it can be
interpreted as the differential of the expected mirror map between
the moduli spaces (when restricted to appropriate subspaces of
those moduli spaces).
The space $H^{d-1,1}_{\text{poly}}(\widehat{Y})$
is isomorphic to the space
of first-order polynomial deformations of
$Y$,
and can be generated by {\em monomials}; the space
$H^{1,1}_{\text{toric}}(\widehat{X})$ consists of that part of the second
cohomology of $\widehat{X}$
coming from the ambient toric variety, and can be generated by
toric {\em divisors}. Our map comes from a natural one-to-one
correspondence between monomials and toric divisors
whose definition is inspired
by the constructions of Roan and Batyrev;
we have named it the {\em monomial-divisor
mirror map}.
\section{Divisors} \label{divisorsection}
Our first task is to describe the partial resolutions of singularities
we will use, and the divisors on them.
Let $\Delta$ be a fan determining a toric variety
(see \cite{Oda} or \cite{Fulton} for the definitions,
and for proofs of the facts we review below).
The support $|\Delta|$ of $\Delta$ is a subset of a real vector space
$N_{\Bbb R}$, and the convex cones $\sigma$ in the fan $\Delta$ are rational
polyhedral cones with respect to a lattice $N$ in $N_{\Bbb R}$;
the algebraic torus which acts on the toric variety is
$T:=N\otimes\Bbb{C}^*$.
We let $\Delta(1)$ denote the set of one-dimensional cones in $\Delta$.
There is a natural {\em generator} map $\operatorname{gen}:\Delta(1)\to N$ which assigns to
each
one-dimensional cone $\rho$ the unique generator $\operatorname{gen}(\rho)$
of the semigroup
$\rho\cap N$.
Each such $\rho$ also has an associated $T$-invariant Weil divisor
$D_\rho$ in the toric variety,
which is the closure of the $T$-orbit corresponding to the cone $\rho$.
One can always describe a {\em projective} toric variety by beginning with
a compact convex polyhedron $P$ in
a real affine space $M_{\Bbb R}$,
integral with respect to a lattice $M$ in $M_{\Bbb R}$. The
projective toric variety is then determined by the
{\em normal
fan} of the polyhedron $P$; this
is the fan ${\cal N}(P)$ consisting of all cones
${\cal N}(P,p)$ to $P$ at $p\inP$, where
\[{\cal N}(P,p):=\{x\in N_{\Bbb R}\ |\
\langle x,p\rangle\le\langle x,y\rangle \text{ for all } y\inP\},\]
and where $N:=\operatorname{Hom}(M,\Bbb Z)$ is the dual lattice, and $N_{\Bbb R}:=N\otimes\Bbb R$.
Each proper face of the polar polyhedron $\polyhedron^\circ$ of $P$
is contained in a unique cone in ${\cal N}(P)$, which is
the cone over that face.
The toric variety $V$ determined by the fan ${\cal N}(P)$
is the natural one in which the hypersurfaces $X$ with Newton polyhedron
$P$ are ample divisors. In the case of a reflexive
integral polyhedron $P$, the general such $X$
is an anti-canonical divisor in $V$ and will be a Calabi-Yau
variety with canonical singularities, as proved by Batyrev \cite{batyrev1}.
We need to partially resolve
the singularities of $V$ while retaining the triviality of the canonical
bundle of the hypersurface,
getting as close as possible to a complete resolution.
To do this, construct a blowup $\widehat{V}\to V$, determined by a fan
$\Delta$ which is a subdivision of the fan ${\cal N}(P)$. There will
be an induced blowup $\widehat{X}\to X$ of hypersurfaces, where $\widehat{X}$
is the proper transform of $X$ on $\widehat{V}$. In order to maintain
the triviality of the canonical bundle of $\widehat{X}$,
restrict the set $\Delta(1)$
of one-dimensional cones as follows: the image
${\Xi}:=\operatorname{gen}(\Delta(1))$ of $\Delta(1)$ in $N$
should lie in the set
$\polyhedron^\circ\cap N$, where $\polyhedron^\circ$
is the polar polyhedron of $P$.
(We will sometimes restrict ${\Xi}$ to lie in the subset
$(\polyhedron^\circ\cap N)_0\subset{\polyhedron^\circ\cap N}$
consisting of those
lattice points in $\polyhedron^\circ$
which do not lie in the interior of a codimension-one face of $\polyhedron^\circ$.)
Oda and Park \cite{OP} (cf.\ also \cite{Stanley}) have
shown the existence of a simplicial subdivision $\Delta$
of the fan ${\cal N}(P)$ such that
$\operatorname{gen}(\Delta(1))=(\polyhedron^\circ\cap N){-}\{0\}$
(or any subset thereof),
and such that the corresponding $\widehat{V}$ is projective.
In general, there will be many such fans $\Delta$.
Since the fan $\Delta$ is simplicial, the toric variety $\widehat{V}$
has the structure of an {\em orbifold} (formerly called
{\em $V$-manifold} \cite{satake3}):
it can be covered by open sets of the form
$U/G_U$ where $G_U$ is a finite group acting on a manifold $U$ such that
the fixed locus of any $1\ne g\in G_U$ has real codimension at least $2$.
The open sets $U/G_U$ are used to define the notion of
{\em orbifold-smooth
differential forms}, pieced together from $G_U$-invariant smooth forms
on the open sets $U$. Many of the theorems about the differential
geometry of smooth algebraic varieties have natural orbifold versions.
In particular, there are orbifold
de~Rham cohomology groups $H^k_{\operatorname{DR}}(\widehat{V},\Bbb R)$ isomorphic to
the real \v Cech cohomology \cite{satake3}, and
orbifold Hodge groups $H^{p,q}(\widehat{V})$ which satisfy a version
of the Dolbeault
theorem \cite{baily1}.
The general hypersurface $\widehat{X}\subset\widehat{V}$ is also an
orbifold, and has orbifold de Rham and Hodge groups of its own.
We can describe the group $\operatorname{WDiv}_T(\widehat{V})$ of
toric Weil divisors on $\widehat{V}$
and their images in the Chow group $A_{n-1}(\widehat{V})$
(where $n=d+1$ is the dimension of $\widehat{V}$),
as follows (cf.\ Cox \cite{cox}).
There is a natural isomorphism\footnote{We use the notation
$\Bbb Z\langle S\rangle$ for the free abelian group on the set $S$,
and $\Bbb Z^S$ for the $\Bbb Z$-module of maps from $S$ to $\Bbb Z$,
which is naturally isomorphic to the dual lattice
$\operatorname{Hom}(\Bbb Z\langle S\rangle,\Bbb Z)$ of $\Bbb Z\langle S\rangle$.
The map determined by $\varphi\in\Bbb Z^S$ is denoted by $s\mapsto\varphi_s$.
}
$\alpha:\Bbb Z^{\Xi}\to\operatorname{WDiv}_T(\widehat{V})$
which sends the function $\varphi\in\Bbb Z^{\Xi}$ to the divisor
\[\sum\varphi_aD_{\operatorname{gen}^{-1}(a)}.\]
Under this isomorphism, if we define an embedding
$\operatorname{ad}_{\Xi}:M\to\Bbb Z^{\Xi}$ by sending
$m\in M$ to the function $\operatorname{ad}_{\Xi}(m)$ defined by
$\operatorname{ad}_{\Xi}(m):a\mapsto\langle a,m\rangle$,
then
\[\div(\chi^m)=-\alpha(\operatorname{ad}_{\Xi}(m)),\]
where $\chi^m:T\to\Bbb{C}^*$ is the character of $T$ associated to $m$, regarded
as a meromorphic function on $\widehat{V}$. Thus, $M$ gives rise to
linear equivalences among toric divisors. In fact,
there is an exact sequence
\begin{equation} \label{exactseq}
0 \longrightarrow M \stackrel{\operatorname{ad}_{\Xi}}{\longrightarrow}
\Bbb Z^{{\Xi}} \stackrel{\bar{\alpha}}{\longrightarrow}
A_{n-1}(\widehat{V}) \longrightarrow 0 ,
\end{equation}
where $\bar{\alpha}$ denotes the composite of $\alpha$ with the projection
to the Chow group.
This is nothing other than the usual description of $A_{n-1}$ as
``divisors modulo linear equivalence'', since $\Bbb Z^{\Xi}$ represents
toric divisors and $M$ represents the linear equivalences among them.
\medskip
To compute the group of toric divisors on the hypersurface
$\widehat{X}$, we use the natural restriction maps from
divisors on $\widehat{V}$ (which exists since each toric divisor
on $\widehat{V}$ meets $\widehat{X}$ in a subvariety of codimension $1$):
\[\begin{array}{ccccccccc}
0&\longrightarrow&M&\longrightarrow&\operatorname{WDiv}_T(\widehat{V})&\longrightarrow&
A_{n-1}(\widehat{V})&\longrightarrow&0\\
&&{\scriptstyle||}&&\downarrow&&\downarrow&&\\
0&\longrightarrow&M&\longrightarrow&\operatorname{WDiv}_T(\widehat{X})&\longrightarrow&
A_{d-1}(\widehat{X})&&
\end{array}.\]
This time, the toric divisors need not generate the entire Chow group;
we denote the image of $\operatorname{WDiv}_T(\widehat{X})$ in $A_{d-1}(\widehat{X})$
by $A_{d-1}(\widehat{X})_{\text{toric}}$. Its complexification we call
the {\em toric part of $H^{1,1}$}, denoted by
$H^{1,1}_{\text{toric}}(\widehat{X}):=A_{d-1}(\widehat{X})_{\text{toric}}\otimes\Bbb{C}$.
The kernel of the
restriction map $\operatorname{WDiv}_T(\widehat{V})\to\operatorname{WDiv}_T(\widehat{X})$ is easy
to describe. A divisor with trivial restriction must be supported on
divisors which are disjoint from the general hypersurface
$\widehat{X}\subset\widehat{V}$. Since
the line bundle $\O_V(X)$ is generated by its global sections, the
general hypersurface $X\subset V$
will not meet the zero-dimensional strata of $V$ (in the
stratification by $T$-orbits). So any divisor on $\widehat{V}$
which maps to such a stratum will be disjoint from $\widehat{X}$,
the proper transform of $X$.
Such divisors are characterized by the property that the
corresponding point in ${\Xi}$ lies in the interior of some
codimension-one face of $\polyhedron^\circ$.
Other toric divisors on $\widehat{V}$ cannot be disjoint from
$\widehat{X}$, since they map to larger strata of $V$ which are not
disjoint from $X$.
Thus, if we let ${\Xi}_0={\Xi}\cap(\polyhedron^\circ\cap N)_0$
be the subset of ${\Xi}$ consisting
of all points which do {\em not} lie in interiors of
codimension-one faces of
$\polyhedron^\circ$, we find that $\operatorname{WDiv}_T(\widehat{X})\cong\Bbb Z^{{\Xi}_0}$
and that
\begin{equation} \label{divisorsA}
A_{d-1}(\widehat{X})_{\text{toric}}\cong\operatorname{Coker}(\operatorname{ad}_{{\Xi}_0})\cong
\Bbb Z^{{\Xi}_0}/M.
\end{equation}
In particular, if ${\Xi}\supset(\polyhedron^\circ\cap N)_0-\{0\}$
then
$A_{d-1}(\widehat{X})_{\text{toric}}\cong\Bbb Z^{(\polyhedron^\circ\cap N)_0-\{0\}}/M$,
and hence
\begin{equation} \label{divisors}
H^{1,1}_{\text{toric}}(\widehat{X})\cong(\Bbb Z^{(\polyhedron^\circ\cap N)_0-\{0\}}/M)\otimes\Bbb{C}.
\end{equation}
\section{Monomials}
Our task in this section is to describe moduli spaces for hypersurfaces
in $\widehat{V}$.
We retain the notation of the previous section: $\widehat{V}$ is the
toric variety associated to a subdivision $\Delta$ of the normal fan
${\cal N}(P)$ of a reflexive polyhedron $P$.
We assume that $\Delta$ is simplicial, so that $\widehat{V}$ is
$\Bbb Q$-factorial; we also assume that $\widehat{V}$ is projective.
Given a hypersurface $\widehat{X}\subset\widehat{V}$, the space of
first order
deformations of complex structure of $\widehat{X}$ is isomorphic
to $H^1(\Theta_{\widehat{X}})$. The simplest way to deform the
complex structure on $\widehat{X}$ is to perturb the equation of
the hypersurface; this leads to a subspace
$H^1(\Theta_{\widehat{X}})_{\text{poly}}\subset H^1(\Theta_{\widehat{X}})$
of {\em polynomial} first-order deformations.
(It is quite possible for this to be a proper subspace \cite{pdm}.)
In the case that $\widehat{X}\subset\widehat{V}$ is a Calabi-Yau
hypersurface, we can use the isomorphism $H^1(\Theta_{\widehat{X}})
\cong H^{d-1,1}(\widehat{X})$ to also specify a ``polynomial'' subspace
$H^{d-1,1}_{\text{poly}}(\widehat{X})\subset H^{d-1,1}(\widehat{X})$
of the corresponding Hodge group.
In principle, the moduli spaces of the hypersurfaces $\{\widehat{X}\}$
should be fairly easy to describe. Global sections of
$\O_{\widehat{V}}(\widehat{X})$ provide equations for the hypersurfaces,
and the entire family can be described as
$\P H^0(\O_{\widehat{V}}(\widehat{X}))$. But we need to mod out by
automorphisms of $\widehat{V}$, and this is where technical complications
arise.
Let $D$ be a toric divisor on $\widehat{V}$, and write
$D=\sum_{a\in{\Xi}} d_a D_{\operatorname{gen}^{-1}(a)}$. There is a natural
isomorphism between $H^0(\O(D))$ and the space $\Bbb{C}^{P_D\cap M}$,
where $P_D$ is the polytope
\[P_D:=\{y\in M_{\Bbb R} \ |\ \langle a,y\rangle\ge-d_a \text{ for all }
a\in{\Xi}\} \]
(cf.~\cite{Fulton}).
In fact, if we identify $H^0(\O(D))$ with the space of meromorphic
functions on $\widehat{V}$ which have (at worst) poles along $D$,
then to each $m\in P_D\cap M$ we can associate the meromorphic
function $\chi^m$: it has at worst poles along $D$ thanks to the
definition of $P_D$.
In the special case $D=\sum_{\rho\in\Delta(1)}D_\rho\in|-K_{\widehat{V}}|$,
the polytope
$P_{\Sigma\, D_\rho}$ coincides with the original polyhedron
$P\subset M_{\Bbb R}$ used to describe $V$.
The automorphism group $\operatorname{Aut}(\widehat{V})$ of a $\Bbb Q$-factorial toric
variety $\widehat{V}$ has been described recently by Cox \cite{cox},
generalizing some results from the smooth case due to
Demazure \cite{demazure}. Cox's description is in terms of a central
extension of $\operatorname{Aut}(\widehat{V})$ by a torus $G$, which
fits in an exact sequence
\begin{equation}\label{auttilde}
1\longrightarrow G\longrightarrow \mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})
\longrightarrow \operatorname{Aut}(\widehat{V})\longrightarrow1,
\end{equation}
where $G:=\operatorname{Hom}(A_{n-1}(\widehat{V}),\Bbb{C}^*)$. The advantage of working
with $\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$ is that it acts naturally on all cohomology
groups $H^0(\O(D))$
at once. The clearest way to see these actions is to follow Cox again
and introduce the {\em homogeneous coordinate ring}\/
$S:=\Bbb{C}[x_a]_{(a\in{\Xi})}$ of $\widehat{V}$. This ring can be
(multi) graded by defining the {\em degree}\/ of the monomial
$\prod x_a^{\varphi_a}$ to be the divisor class
$[\sum \varphi_a D_{gen^{-1}(a)}]$
in $A_{n-1}(\widehat{V})$. For a fixed divisor $D$, the set of
elements of degree $[D]$ in the homogeneous coordinate ring can be
identified with $H^0(\O(D))$: the meromorphic function $\chi^m$ with
$m\in P_D\cap M$ corresponds to the homogeneous monomial
$x^{\div(\chi^m)+D}:=\prod x_a^{\langle a,m\rangle + d_a}$.
The torus $T:=\operatorname{Hom}(M,\Bbb{C}^*)$ which acts on $\widehat{V}$ is naturally
a subgroup of $\operatorname{Aut}(\widehat{V})$; the induced extension $\widetilde{T}$
of $T$ by $G$ has the form $\widetilde{T}:=\operatorname{Hom}(\Bbb Z^{\Xi},\Bbb{C}^*)$.
In fact, if we
apply the functor $\operatorname{Hom}(\mbox{---},\Bbb{C}^*)$ to the natural exact sequence
\eqref{exactseq},
we get a sequence for $\widetilde{T}$ which fits as the first row in
the commutative diagram
\[\begin{array}{ccccccccc}
1&\longrightarrow&G&\longrightarrow&\widetilde{T}&\longrightarrow&
T&\longrightarrow&1\\
&&{\scriptstyle||}&&\cap&&\cap&&\\
1&\longrightarrow&G&\longrightarrow&\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})&\longrightarrow&
\operatorname{Aut}(\widehat{V})&\longrightarrow&1
\end{array}.\]
The grading of the homogeneous coordinate ring $S$ can also be described
in terms of the action of $G$ on $S$. The torus $\widetilde{T}$ acts
on $S$ in a transparent way: each monomial in $S$ can be written in the
form $x^\varphi=\prod x_a^{\varphi_a}$ for some $\varphi\in\Bbb Z^{{\Xi}}$,
and the action of $t\in\operatorname{Hom}(\Bbb Z^{{\Xi}},\Bbb{C}^*)$ sends $x^\varphi$ to
$t(\varphi)\cdot x^\varphi$. When we restrict this action to the subgroup
$G=\operatorname{Hom}(A_{n-1}(\widehat{V},\Bbb{C}^*))$, then for each divisor class $[D]$,
the subspace of $S$ on which
$G$ acts via the character $\gamma\mapsto\gamma([D])$ is precisely
$H^0(\O(D))\cong\bigoplus\Bbb{C}\cdot x^{\div(\chi^m)+D}$.
The induced action of $t\in\widetilde{T}$ on $H^0(\O(D))$ then sends
$\chi^m$ to $t(\alpha^{-1}(\div(\chi^m)+D))\cdot\chi^m$, for every
$m\in P_D\cap M$. This action can be described in terms of the map
$\Bbb Z\langle P_D\cap M\rangle\to\Bbb Z^{\Xi}$ defined by
\begin{equation}\label{action}
m\mapsto(a\mapsto\langle a,m\rangle+d_a),
\end{equation}
which induces a homomorphism of tori
$\widetilde{T}\to(\Bbb{C}^*)^{P_D\cap M}$
that determines the
action of $\widetilde{T}$ on $\Bbb{C}^{P_D \cap M}$.
Notice that the map \eqref{action} factors
as a composite of two maps
\begin{equation} \label{twomaps}
\Bbb Z\langle P_D\cap M\rangle\to M\oplus\Bbb Z\to\Bbb Z^{\Xi}
\end{equation}
with the first map given by
$m\mapsto(m,1)$ and the second map given by
$(m,k)\mapsto \operatorname{ad}_{\Xi}(m)+k\cdot\alpha^{-1}(D)$.
The corresponding homomorphism of tori factors as
\begin{equation} \label{torusfactor}
\widetilde{T}\to T\times\Bbb{C}^*\to(\Bbb{C}^*)^{P_D\cap M}.
\end{equation}
In the special case $D=\sum D_\rho$, the induced map
$\Bbb{C}^*\to(\Bbb{C}^*)^{P\cap M}$ is simply the diagonal embedding.
The groups $\operatorname{Aut}(\widehat{V})$ and $\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$ are
not in general reductive. Thus, to construct moduli spaces for
hypersurfaces\footnote{Batyrev \cite{batyrev2} has constructed moduli
spaces for {\em affine}\/ hypersurfaces, obtaining a somewhat
different space than ours if
$(P\cap M)_0\ne(P\cap M)$ (it even has a different
dimension).
Batyrev and Cox \cite{BC} have recently considered a construction
similar to the one described here.}
on $\widehat{V}$, we should use Fauntleroy's extension \cite{fauntleroy1}
of Mumford's Geometric Invariant Theory (GIT) \cite{GIT}, and attempt to
construct a quotient for the
action of $\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$ on $H^0(\O(D))$. It would be
interesting to know if
this construction of moduli spaces
can be carried out in general---Fauntleroy has carried it out in
some special cases \cite{fauntleroy2}.
We can at least obtain a birational model of the desired moduli
space by using a fairly standard result (cf.\ \cite{Rosenlicht,CDT})
which guarantees the existence of an $\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$-stable
Zariski-open set $U\subset H^0(\O(D))$ such that the geometric quotient
$U/\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$ exists.
We
indicate the birational class of such quotients
with the notation
$H^0(\O(D))\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu} \mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$.
In the case of interest in this paper, $D=-K_{\widehat{V}}$. To
study this particular moduli space,
we take a simpler course of action, and restrict our attention
to a subspace of $H^0(\O(-K_{\widehat{V}}))$
on which $\widetilde{T}$ acts in such
a way that the quotient exists and
has the ``expected'' dimension for the entire
moduli space.
In a wide class of examples, $\widetilde{T}$ is in fact the connected
component of the identity in $\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$, and the only
differences between our moduli space and the ``true'' moduli space
for hypersurfaces are
a remaining quotient by a finite group, and a possible ambiguity in the
choice of Zariski-open set
used in constructing the quotient.
In particular, the map from our space to the true moduli space is a
dominant map between spaces of the same dimension.
We hope that this latter property is true
in general, but postpone that question to a future investigation.
Our construction of a simplified model for the
hypersurface moduli
space relies on another result of Demazure and Cox about
$\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$. They show that
\[\dim\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})=\dim\widetilde{T}+\#(R(N,\Delta)),\]
where
\begin{eqnarray*}
R(N,\Delta):=\{m\in M&\ |\ &\langle \operatorname{gen}(\rho),m\rangle\le1
\text{ for all }\rho\in\Delta(1),\\
&&
\text{with equality for a unique } \rho=\rho_m\in\Delta(1)\}
\end{eqnarray*}
is the set of {\em roots} of the toric variety $\widehat{V}$ associated
to the fan $\Delta$.
Note that for each root
$m\in R(N,\Delta)$, we have $-m\inP\cap M$. In fact,
the set $-R(N,\Delta)$ can be characterized
as the subset of $P\cap M$
consisting of lattice points which lie in the interiors of codimension-one
faces of $P$. We can thus decompose
\[P\cap M=-R(N,\Delta)\cup(P\cap M)_0,\]
and write
\[\Bbb{C}^{P\cap M}=\Bbb{C}^{-R(N,\Delta)}\oplus\Bbb{C}^{(P\cap M)_0}.\]
The subgroup $\widetilde{T}\subset\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$ preserves
this direct sum decomposition, so we can let $\widetilde{T}$ act on the
second factor $\Bbb{C}^{(P\cap M)_0}$ alone.
Our ``simplified hypersurface
moduli space'' will be
the
GIT\ quotient
$\Bbb{C}^{(P\cap M)_0}_{\text{ss}}/\widetilde{T}$.
(We regard the action of $\widetilde{T}$ on $\Bbb{C}^{(P\cap M)_0}$
as specifying a linearization of the action on
$\P(\Bbb{C}^{(P\cap M)_0})$, so there is no ambiguity in
the choice of GIT\ quotient.)
There is then a natural rational map
\begin{equation}\label{rationalmap}
\Bbb{C}^{(P\cap M)_0}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\widetilde{T}\mathrel{\dabar@\dabar@\mathchar"0\msafam@4B}
\Bbb{C}^{P\cap M}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})
\end{equation}
which could be refined to a regular map
\begin{equation}\label{map}
\Bbb{C}^{(P\cap M)_0}_{\text{ss}}/\widetilde{T}\to
U/\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})
\end{equation}
if an appropriate set of
``semistable'' points $U\subset\Bbb{C}^{P\cap M}$
were available from (generalized) GIT.
Note that $\operatorname{Ker}(\xi_{[-K]})$, which is a subgroup of both $\widetilde{T}$
and $\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$, acts trivially on both spaces.
By equation \eqref{torusfactor},
$\widetilde{T}/\operatorname{Ker}(\xi_{[-K]})\cong T\times\Bbb{C}^*$.
Note also that the two
quotient spaces can be expected to have the same dimension.
\begin{definition}
We say that the family
$\{\widehat{X}\}$ has the {\em dominance property} if the
natural rational map
$\Bbb{C}^{(P\cap M)_0}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\widetilde{T}\mathrel{\dabar@\dabar@\mathchar"0\msafam@4B}
\Bbb{C}^{P\cap M}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$
is a dominant map between two varieties
of the same dimension.
(Note that this property is independent of the choice of quotients.)
\end{definition}
This dominance property clearly holds if $R(N,\Delta)=\emptyset$;
we expect that it should hold
in general, but have not checked this.
The ``simplified hypersurface
moduli space'' parameterizes hypersurfaces with
equations of the form
\[\sum_{m\in(P\cap M)_0}c_m\chi^m=0\]
modulo the equivalences given by the action of
$\widetilde{T}/\operatorname{Ker}(\xi_{[-K]})\cong T\times\Bbb{C}^*$.
The $\Bbb{C}^*$ factor is diagonally embedded in $(\Bbb{C}^*)^{(P\cap M)_0}$,
and so gives an overall scaling of the equation.
We can describe a Zariski-open
subset\footnote{The apparent lack of naturality in this step of
our construction---why restrict to a subset?---will be redressed
later in the paper.}
of our moduli space by restricting to equations with $c_0\ne0$, and
using the overall scaling of the equation
to set that coefficient $c_0$ equal to $1$.
Thus, the open subset
can be described as a quotient
$\Bbb{C}^{(P\cap M)_0{-}\{0\}}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu} T$
with a point $c\in\Bbb{C}^{(P\cap M)_0{-}\{0\}}$
corresponding to the hypersurface with equation
\[\chi^0+\sum_{m\in(P\cap M)_0{-}\{0\}}c_m\chi^m=0.\]
Let ${\Upsilon}_0=(P\cap M)_0-\{0\}$ to simplify notation.
Here is the crucial observation for the construction of the
monomial-divisor mirror map: the action of $T=N\otimes\Bbb{C}^*$ on $\Bbb{C}^{{\Upsilon}_0}$
is induced by tensoring the homomorphism $\operatorname{ad}_{{\Upsilon}_0}:N\to \Bbb Z^{{\Upsilon}_0}$
with $\Bbb{C}^*$. (The explicit identification
of $\operatorname{ad}_{{\Upsilon}_0}$ as the homomorphism
needed to specify the $T$-action
follows immediately from
the definition of the maps in equation \eqref{twomaps},
since $\operatorname{ad}_{{\Upsilon}_0}$ is dual to the natural map $\Bbb Z\langle{{\Upsilon}_0}\rangle\to M$
induced by the inclusion ${{\Upsilon}_0}\subset M$.) In particular,
the tangent space to the simplified moduli space
$\Bbb{C}^{(P\cap M)_0{-}\{0\}}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu} T$
has the form
$(\Bbb Z^{(P\cap M)_0{-}\{0\}}/N)\otimes\Bbb{C}$.
When the family $\{\widehat{X}\}$ has the dominance property
(e.g., when
$R(N,\Delta)=\emptyset$), the induced rational map
$\Bbb{C}^{(P\cap M)_0{-}\{0\}}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu} T\mathrel{\dabar@\dabar@\mathchar"0\msafam@4B}
\Bbb{C}^{P\cap M}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$
is also dominant.
In this case,
we can describe the tangent space to the
space of polynomial deformations of a general Calabi-Yau hypersurface
$\widehat{X}\subset\widehat{V}$
as
\begin{equation} \label{monomials}
T_{[\widehat{X}],\,\,\Bbb{C}^{P\cap M}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})}=
H^{d-1,1}_{\text{poly}}(\widehat{X})\cong
(\Bbb Z^{(P\cap M)_0{-}\{0\}}/N)\otimes\Bbb{C},
\end{equation}
where $[\widehat{X}]$
represents the class of $\widehat{X}$ in
$\Bbb{C}^{P\cap M}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V})$.
We can apply these same considerations to the family of hypersurfaces
determined by the polar polyhedron $\polyhedron^\circ$, which Batyrev has proposed
as a mirror partner for the family $\{\widehat{X}\}$. To do this,
we need to choose a simplicial subdivision $\fan^\circ$ of
${\cal N}(\polyhedron^\circ)$ which determines a projective toric variety $\widehat{V}^\circ$
and a family of hypersurfaces $\widehat{Y}\subset\widehat{V}^\circ$.
Replacing
$\widehat{X}$, $P$, $M$, $N$ by
$\widehat{Y}$, $\polyhedron^\circ$, $N$, $M$, respectively, in equation
\eqref{monomials}, we find that
\begin{equation} \label{Ymonomials}
T_{[\widehat{Y}],\,\,\Bbb{C}^{\polyhedron^\circ\cap N}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V}^\circ)}=
H^{d-1,1}_{\text{poly}}(\widehat{Y})\cong
(\Bbb Z^{(\polyhedron^\circ\cap N)_0{-}\{0\}}/M)\otimes\Bbb{C},
\end{equation}
whenever $\{\widehat{Y}\}$ has the dominance property.
The monomial-divisor mirror map is now evident, when one compares
equations \eqref{divisors} and \eqref{Ymonomials}.
(Note that the same map
$\operatorname{ad}_{(\polyhedron^\circ\cap N)_0-\{0\}}$
is used to define the embedding $M\to\Bbb Z^{(\polyhedron^\circ\cap N)_0-\{0\}}$
in both cases.)
\begin{theorem} Let $P$ be a reflexive polyhedron,
integral with respect
to the lattice $M$, and let $\polyhedron^\circ$ be its polar polyhedron,
which is integral with respect to the dual lattice $N$. Let $\Delta$
and $\fan^\circ$ be simplicial subdivisions of ${\cal N}(P)$
and ${\cal N}(\polyhedron^\circ)$, respectively, and let $\{\widehat{X}\}$
and $\{\widehat{Y}\}$ be the corresponding families of Calabi-Yau
hypersurfaces.
Assume that
$(\polyhedron^\circ\cap N)_0{-}\{0\}\subset \operatorname{gen}(\Delta(1))
\subset (\polyhedron^\circ\cap N){-}\{0\}$.
Assume also that $R(M,\fan^\circ)=\emptyset$, or more
generally, simply assume that $\{\widehat{Y}\}$ has the dominance
property. Then there is a natural isomorphism
\begin{equation}\label{mdmm}
H^{d-1,1}_{\text{poly}}(\widehat{Y})\stackrel{\cong}{\to} H^{1,1}_{\text{toric}}(\widehat{X})
\end{equation}
induced by equations \eqref{divisors} and \eqref{Ymonomials},
since both spaces are naturally isomorphic to
$\operatorname{Coker}(\operatorname{ad}_{(\polyhedron^\circ\cap N)_0-\{0\}})\otimes\Bbb{C}$.
We call the isomorphism \eqref{mdmm} the {\em monomial-divisor mirror map}.
\end{theorem}
\section{K\"ahler cones}
Let $\widehat{V}$ be a $\Bbb Q$-factorial toric variety,
determined by a simplicial fan $\Delta$, and let ${\Xi}=\operatorname{gen}(\Delta(1))$.
We can describe the group $\operatorname{Div}_T(\widehat{V})$ of toric
Cartier divisors on $\widehat{V}$ as follows.
In order for $D=\sum d_aD_{\operatorname{gen}^{-1}(a)}$ to be Cartier,
there must be a continuous piecewise linear (PL) function
$\psi_D:|\Delta|\to\Bbb R$
(called the {\em support function determined by $D$})
which is linear on each cone $\sigma\in\Delta$,
which takes integer values on $|\Delta|\cap N$, and which satisfies
\begin{equation}\label{PL}
\psi_D(a)=- d_a\qquad\text{ for all }a\in{\Xi}.
\end{equation}
Since the fan $\Delta$ is simplicial, the PL\ function $\psi_D$
is completely determined by the values specified in equation \eqref{PL}
and the fan $\Delta$:
one just extends by linearity to each (simplicial) cone in $\Delta$.
The integrality
condition $\psi_D(|\Delta|\cap N)\subset\Bbb Z$ remains nontrivial, however.
(If we require instead that $\psi_D(|\Delta|\cap N)\subset\Bbb Q$, we get
the group of $\Bbb Q$-Cartier divisors.) Since $\Delta$ is simplicial,
every Weil divisor is $\Bbb Q$-Cartier, that is, $\widehat{V}$ is
$\Bbb Q$-factorial. Put another way, there is a natural isomorphism
between $\operatorname{Pic}(\widehat{V})\otimes\Bbb Q$ and $A_{n-1}(\widehat{V})\otimes\Bbb Q$.
The ample Cartier divisors (or ample $\Bbb Q$-Cartier divisors) are characterized
by {\em strict convexity} of $-\psi_D$, where $\psi_D$ is the
support function determined by $D$ (cf.\ \cite{Fulton}).\footnote{Our
signs are chosen to conform to the literature as closely as possible,
while giving the term ``convexity'' its conventional meaning.}
This means the following: given a PL\ function $\eta$
which is linear on each cone $\sigma\in\Delta$, let $u_\sigma\in M=\operatorname{Hom}(N,\Bbb Z)$ be
the
linear function such that
\[\langle x,u_\sigma\rangle =\eta(x)\quad\text{ for all } x\in\sigma.\]
The convexity condition is:
\[\eta(x)\ge\langle x,u_\sigma\rangle\quad\text{ for all } x\in|\Delta|;\]
the convexity if {\em strict} if the inequality is strict for all
$x\not\in\sigma$.
(In fact, it suffices to check this at the points of ${\Xi}$:
for convexity, we must have
\[\eta(a)\ge\langle a,u_\sigma\rangle\quad\text{ for all } a\in{\Xi}\]
with equality whenever $a\in\sigma$;
strict convexity requires a strict inequality whenever
$a\not\in\sigma$.)
The cone of real convex PL\ functions is denoted by $\operatorname{CPL}(\Delta)$,
following the notation of Oda and Park \cite{OP}.
If there exists a strictly convex function in $\operatorname{CPL}(\Delta)$, the fan
$\Delta$ is called {\em regular}.
There is
a related cone (introduced by
Gel'fand, Zelevinski\v\i, and Kapranov \cite{GKZ}):
\[\operatorname{CPL}^\sim(\Delta)=\{\varphi\in\Bbb R^{\Xi}\ |\ \exists\
\eta\in\operatorname{CPL}(\Delta)
\text{ with } \varphi_a=\eta(a) \text{ for all } a\in{\Xi}\}.\]
The definitions are constructed so that if $D$ is an ample $\Bbb Q$-Cartier
divisor, then $\alpha^{-1}(D)\in\operatorname{CPL}^\sim(\Delta)$. (One can choose
$\eta=-\psi_D$ in the definition.)
Note that $M_{\Bbb R}$ is contained in $\operatorname{CPL}^\sim(\Delta)$, and the corresponding
$\eta$'s are precisely the
{\em smooth} PL\ functions.
The image of $\operatorname{CPL}^\sim(\Delta)$ in $\Bbb R^{\Xi}/M_{\Bbb R}$,
which we may think of as the set of convex PL\ functions modulo
smooth PL\ functions,
is denoted
by $\operatorname{cpl}(\Delta)$. Under the isomorphism
$\Bbb R^{\Xi}/M_{\Bbb R}\cong A_{n-1}(\widehat{V})\otimes\Bbb R
(\cong\operatorname{Pic}(\widehat{V})\otimes\Bbb R)$,
this cone
$\operatorname{cpl}(\Delta)$ maps to the closed real cone generated by the ample divisor
classes on $\widehat{V}$.
An effective method of calculating all possible cones $\operatorname{cpl}(\Delta)$
in terms of the ``linear Gale transform'' is described in \cite{OP}.
The exponential sheaf sequence gives rise to an isomorphism
$\operatorname{Pic}(\widehat{V})\cong H^2(\widehat{V},\Bbb R)\cong H^{1,1}(\widehat{V},\Bbb R)$,
since $h^i(\O_{\widehat{V}})=0$
for $i>0$.
Now there is a natural notion of {\em positivity} for orbifold-smooth
$(1,1)$-forms: one requires that the $G_U$-invariant $(1,1)$-forms
on the local uniformizing sets $U$ be positive. The K\"ahler form of
every orbifold-K\"ahler metric is easily seen to be
a positive, orbifold-smooth
$(1,1)$-form. The set
${\cal K}(\widehat{V})\subset H^{1,1}(\widehat{V},\Bbb R)$
consisting of orbifold de~Rham
classes which have such a positive representative is called the
{\em K\"ahler cone} of $\widehat{V}$.
We could not find the following lemma in the literature
(although it should be known).
\begin{lemma}
Under the natural map $\operatorname{Pic}(\widehat{V})\to H^2_{\operatorname{DR}}(\widehat{V},\Bbb R)$
which assigns to a line bundle the corresponding orbifold de Rham class,
the ample line bundles map to positive de Rham classes.
\end{lemma}
Note that the lemma is not as obvious as is the analogous lemma
in the smooth case, since
even if we are given a projective embedding $\widehat{V}\to\P^N$,
the pullback of the Fubini-Study form (which establishes the positivity
of the
de Rham class of $\O_{\P^N}(1)$) is {\em not}
necessarily positive
as an orbifold $2$-form.
\medskip
\begin{pf*}{Sketch of proof}
We modify an argument of Guillemin and Sternberg
\cite{GS1}.
By a result of Delzant \cite{delzant} and Audin \cite{audin},
the toric variety $\widehat{V}$ can be described
as a symplectic reduction of the action of $G=\operatorname{Hom}(A_{n-1}(\widehat{V}),\Bbb{C}^*)$
on $\Bbb{C}^{{\Xi}}$; the specific symplectic reduction which produces
$\widehat{V}$ is $\Phi^{-1}(\alpha)/G_{\Bbb R}$, where $\Phi$ is the moment
map for the action, $G_{\Bbb R}$ is the maximal compact subgroup of $G$,
and $\alpha\in\operatorname{cpl}(\Delta)$. An ample line bundle $L$ on $\widehat{V}$
corresponds to a character $\chi^L$ of $G$, since $A_{n-1}(\widehat{V})$
is the character lattice of $G$; moreover, the corresponding point in
$A_{n-1}(\widehat{V})\otimes\Bbb R$ lies in $\operatorname{cpl}(\Delta)$. If we apply
the constructions described on pp.~520--521 of \cite{GS1}, we produce
a line bundle on $\widehat{V}$ with a specific (orbifold-)connection,
whose curvature is the symplectic form obtained by symplectic reduction
from the standard form on $\Bbb{C}^{{\Xi}}$. That curvature form
provides the desired positive orbifold $2$-form.
\end{pf*}
The converse statement---that if the class of a line bundle is represented
by a positive, orbifold-smooth $2$-form then the line bundle is ample---%
is a theorem of Baily \cite{baily2}. Putting the two together, we conclude
that the image of the cone $\operatorname{cpl}(\Delta)$ in $H^2_{\operatorname{DR}}(\widehat{V},\Bbb R)$
is precisely the closure of the K\"ahler cone $\overline{{\cal K}(\widehat{V})}$.
\medskip
As remarked earlier, the hypersurface $\widehat{X}$ is itself an
orbifold, so it has an orbifold K\"ahler cone
${\cal K}(\widehat{X})\subset H^{1,1}(\widehat{X},\Bbb R)$.
The positive orbifold-smooth K\"ahler forms
on $\widehat{V}$ corresponding to classes in the interior of
$\operatorname{cpl}(\Delta)$ will
restrict to positive orbifold-smooth forms on $\widehat{X}$, since
$\widehat{X}$ meets all singular strata of $\widehat{V}$ transversally.
We call the resulting cone
${\cal K}_{\text{toric}}(\widehat{X})
\subset{\cal K}(\widehat{X})\cap H^{1,1}_{\text{toric}}(\widehat{X})$
the {\em cone of toric K\"ahler classes} on $\widehat{X}$.
\section{The K\"ahler moduli space}
Mirror symmetry predicts a close relationship between the moduli space
of complex structures on one Calabi-Yau manifold $\widehat{Y}$
and the so-called ``K\"ahler moduli space'' of its mirror partner
$\widehat{X}$. This K\"ahler moduli space, which arises in the study of
nonlinear sigma models with
target $\widehat{X}$,\footnote{The physics of these models
is believed to be as well-behaved on orbifolds as on manifolds
\cite{DHVW,DFMS}.} is
an open subset of ${\cal D} /\Gamma$, where
\[{\cal D} :=
\{B+i\,J\in H^2(\widehat{X},\Bbb{C})\ |\ J\in{\cal K}(\widehat{X})\},\]
and where
\[\Gamma:= H^2(\widehat{X},\Bbb Z)\rtimes \operatorname{Aut}(\widehat{X})\]
(cf.~\cite{compact}.) The precise open subset of ${\cal D} /\Gamma$
which constitutes the moduli space is difficult to determine,
since general convergence criteria for the sigma model are unknown at
present.\footnote{But as we will observe below, the region of convergence
in this particular case can be inferred from mirror symmetry.}
However, the open set is expected
to include points sufficiently
far out along any path in ${\cal D} $ whose imaginary part is moving
towards infinity while staying away from the boundary of the K\"ahler cone.
Such paths should approach a common
point, called the {\em large radius limit},
in an appropriate partial compactification of ${\cal D} /\Gamma$.
A general discussion of conditions under which such a limit point exists
can be found in \cite{compact}.
We will be interested in
a ``toric subspace'' of the K\"ahler moduli space,
defined by intersecting the moduli space with
${\cal D}_{\text{toric}}(\widehat{X})/\Gamma_{\text{toric}}$, where
\[{\cal D}_{\text{toric}}(\widehat{X}):= \{B+i\,J\in H^{1,1}_{\text{toric}}
(\widehat{X})\ |\ J\in{\cal K}_{\text{toric}}(\widehat{X})\},\]
and $\Gamma_{\text{toric}}:=A_{d-1}(\widehat{X})_{\text{toric}}
\rtimes \operatorname{Aut}(\widehat{X})_{\text{toric}}$.
(The toric automorphisms $\operatorname{Aut}(\widehat{X})_{\text{toric}}$
are those automorphisms of $\widehat{X}$ induced by an automorphism
of the ambient toric variety $\widehat{V}$.)
If the GIT\
of the family of hypersurfaces is well-behaved (which will be the
case for the families of primary interest to us), then for the general
hypersurface $\widehat{X}$ the group $\operatorname{Aut}(\widehat{X})_{\text{toric}}$
will be finite.
Let $L=A_{d-1}(\widehat{X})_{\text{toric}}$ and consider the torus
$L\otimes\Bbb{C}^*$ which contains ${\cal D}/L$ as an open subset.
The cone ${\cal K}_{\text{toric}}\subset L\otimes\Bbb R$, which is a rational polyhedral
cone, determines an affine torus embedding ${\cal M}$ with a unique
$0$-dimensional orbit $p\in{\cal M}$ under the action of the torus $L\otimes\Bbb{C}^*$.
We let $\overline{{\cal D}/L}$ be the closure of ${\cal D}/L$ in ${\cal M}$, and
let $({\cal D}/L)^-$ be the interior of $\overline{{\cal D}/L}$.
(This contains the point $p$.) If $\operatorname{Aut}(\widehat{X})_{\text{toric}}$ is finite, then
since everything in sight is $\operatorname{Aut}(\widehat{X})_{\text{toric}}$-equivariant,
we may take the quotient and get a partial compactification $({\cal D}/\Gamma)^-$
of ${\cal D}/\Gamma$ with a distinguished boundary point (again denoted by
$p$), the {\em large radius limit} point.\footnote{This construction
is identical to the one described in \cite{compact}, since Looijenga's
semi-toric compactification \cite{Looijenga} coincides with the
``toroidal embeddings'' of \cite{AMRT} when ${\cal K}$ is rational polyhedral
and $\Gamma/L$ is finite.}
This point is the common limit of the paths described earlier.
The torus $L\otimes\Bbb{C}^*=A_{d-1}(\widehat{X})_{\text{toric}}\otimes\Bbb{C}^*$ which
is being compactified can be described in the form
\begin{eqnarray*}
A_{d-1}(\widehat{X})_{\text{toric}}\otimes\Bbb{C}^*&=&
(\Bbb Z^{{\Xi}_0}/M)\otimes\Bbb{C}^*\\&=&
(\Bbb{C}^*)^{{\Xi}_0}/(M\otimes\Bbb{C}^*)
\end{eqnarray*}
using equation \eqref{divisorsA}.
Now the orbits of $M\otimes\Bbb{C}^*$ on $(\Bbb{C}^*)^{{\Xi}_0}$ are all good
orbits of the same dimension.
The action of $M\otimes\Bbb{C}^*$ on the larger space $\Bbb{C}^{{\Xi}_0}$
may be less well-behaved, but in any case we can regard
$(\Bbb{C}^*)^{{\Xi}_0}/(M\otimes\Bbb{C}^*)$ as a representative of the
birational class of quotients
$\Bbb{C}^{{\Xi}_0}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}(M\otimes\Bbb{C}^*)$.
Suppose that the family $\{\widehat{Y}\subset\widehat{V}^\circ\}$
associated to the polar polyhedron
of $\{\widehat{X}\}$
has the dominance property (introduced in section 3), and that $\Delta$ is chosen
so that
$(\polyhedron^\circ\cap N)_0{-}\{0\}\subset \operatorname{gen}(\Delta(1))
\subset (\polyhedron^\circ\cap N){-}\{0\}$.
Then we deduce from the monomial-divisor mirror map a diagram
in which the vertical maps are dominant:
\[\begin{array}{ccccc}
{\cal D}_{\text{toric}}(\widehat{X})/A_{d-1}(\widehat{X})_{\text{toric}}&\subset&
(\Bbb{C}^*)^{(\polyhedron^\circ\cap N)_0{-}\{0\}}/(M\otimes\Bbb{C}^*)&=&
\Bbb{C}^{(\polyhedron^\circ\cap N)_0{-}\{0\}}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}(M\otimes\Bbb{C}^*)\\
\raise6pt\hbox{$\Big\downarrow$}&&
\raise6pt\hbox{$\Big\downarrow$}&&
\stackrel{\stackrel{\scriptscriptstyle|}{\scriptscriptstyle|}}
{\scriptscriptstyle\downarrow}\\
{\cal D}_{\text{toric}}(\widehat{X})/\Gamma_{\text{toric}}&\subset&
(A_{d-1}(\widehat{X})\otimes\Bbb{C}^*)/\operatorname{Aut}(\widehat{X})_{\text{toric}}&&
\Bbb{C}^{\polyhedron^\circ\cap N}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V}^\circ).
\end{array}\]
The left and center vertical maps are simply the quotient maps
by $\operatorname{Aut}(\widehat{X})_{\text{toric}}$, and the right map is the dominant rational
map from the simplified moduli space to the actual moduli space
of the family $\{\widehat{Y}\}$.
We can now formulate a ``mirror symmetry'' conjecture for these families,
which generalizes some (less precise) earlier conjectures of
Aspinwall and L\"utken \cite{AL} and Batyrev \cite{batyrev2}.
\begin{conjecture}
Suppose that $\operatorname{Aut}(\widehat{X})_{\text{toric}}$ is finite, and that
the dominance property holds for $\{\widehat{Y}\}$. Then
\begin{enumerate}
\item
there is an open set $U\subset({\cal D}_{\text{toric}}(\widehat{X})/\Gamma_{\text{toric}})^-$
containing the large radius limit point $p$ such that
$U\cap({\cal D}_{\text{toric}}(\widehat{X})/\Gamma_{\text{toric}})$ is the toric part of the K\"ahler
moduli space,
\item
there is an appropriate quotient
$\Bbb{C}^{\polyhedron^\circ\cap N}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V}^\circ)$ and a
{\em ``mirror map''}\footnote{We denote this map by $\mu^{-1}$
in order to match the conventions established in \cite{compact}.}
\[\mu^{-1}:U\to
\Bbb{C}^{\polyhedron^\circ\cap N}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V}^\circ),\]
which is an isomorphism onto its image, and which,
when restricted to $U\cap({\cal D}_{\text{toric}}(\widehat{X})/\Gamma_{\text{toric}})$,
serves to identify points whose conformal field theories are
mirror-isomorphic, such that
\item
the differential of the inverse map $\mu$ at the ``large complex structure
limit point'' $\mu^{-1}(p)$
\[d\mu:
T_{\mu^{-1}(p),\,\,\Bbb{C}^{\polyhedron^\circ\cap N}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V}^\circ)}
\to
T_{p,\,\,U}
\]
coincides with the monomial-divisor mirror map
\[
H^{d-1,1}_{\text{poly}}(\widehat{Y})
\to
H^{1,1}_{\text{toric}}(\widehat{X})
\]
up to signs,
once we have made the canonical identifications
\begin{eqnarray*}
T_{\mu^{-1}(p),\,\,\Bbb{C}^{\polyhedron^\circ\cap N}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V}^\circ)}&=&
H^{d-1,1}_{\text{poly}}(\widehat{Y})\\
T_{p,\,\,U}&=&H^{1,1}_{\text{toric}}(\widehat{X}).
\end{eqnarray*}
That is, there is some element
\[ \theta_{\Delta} \in A_{d-1}(\widehat{X})_{\text{toric}} \otimes \Bbb{C}^*
\subset \operatorname{Aut} ( H^{1,1}_{\text{toric}} (\widehat{X}) ) \]
of order $2$, which when composed with the monomial-divisor mirror map
yields $d\mu$. (When $d \ge 3$, the automorphism $\theta_{\Delta}$ which
specifies the signs is unique.)
\end{enumerate}
In particular, the
location of the large complex structure limit point $\mu^{-1}(p)$ can be
calculated using the monomial-divisor mirror map and the knowledge
of the cone ${\cal K}_{\text{toric}}(\widehat{X})$.
\end{conjecture}
In \cite{compact}, a very general
conjecture was formulated which specifies the
``canonical coordinates'' to be used in the mirror map, up to some
constants of integration. Those constants can be determined if one knows
the differential of the mirror map at the large radius limit point---%
for these toric hypersurfaces, that differential is supplied by the
conjecture above.
So the two conjectures together completely specify the canonical
coordinates.
A similar conjecture about canonical coordinates
for toric hypersurfaces has been independently made by Batyrev and
van Straten \cite{BvS}.
If the conjecture stated above is true, then among other things the so-called
``$3$-point functions'' (part of the conformal field theories)
must coincide under this mapping. The $3$-point
function on the moduli space $\Bbb{C}^{\polyhedron^\circ\cap N}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V}^\circ)$
can be calculated in terms of the variation of Hodge structure
of the family $\{\widehat{Y}\}$ (cf.~\cite{guide}),
and this gives a method to identify precisely which subset of
${\cal D}_{\text{toric}}(\widehat{X})/\Gamma_{\text{toric}}$ constitutes the toric part of the
K\"ahler
moduli space. For that subset can be characterized as the domain of
convergence of a power series expansion of the $3$-point functions,
when calculated in the canonical coordinates on
${\cal D}_{\text{toric}}(\widehat{X})/\Gamma_{\text{toric}}$.
Using the mirror map, this power series calculation can actually be made
in $\Bbb{C}^{\polyhedron^\circ\cap N}\mathrel{\mskip-4.5mu/\!/\mskip-4.5mu}\mathop{\widetilde{\rm Aut}}\nolimits(\widehat{V}^\circ)$, by calculating
the variation of
Hodge structure of the family $\{\widehat{Y}\}$.
The leading term in the power series expansion of the $3$-point function
on ${\cal D}_{\text{toric}}(\widehat{X})/\Gamma_{\text{toric}}$ is the cubic form
\[H^{1,1}_{\text{toric}}(\widehat{X})\times H^{1,1}_{\text{toric}}(\widehat{X})\times
H^{1,1}_{\text{toric}}(\widehat{X})\to\Bbb{C}\]
given by the cup product; higher terms are given by
``quantum corrections'' that depend
on the point in ${\cal D}_{\text{toric}}(\widehat{X})/\Gamma_{\text{toric}}$, and that all vanish
at the large radius limit point $p\in({\cal D}_{\text{toric}}(\widehat{X})/\Gamma_{\text{toric}})^-$.
One consequence of our conjecture would therefore be an agreement between
the leading term in the variation of Hodge structure calculation for
the family $\{\widehat{Y}\}$ near the large complex structure limit point,
and the cup product cubic form on $\widehat{X}$.
This consequence was
first checked in an example by Aspinwall, L\"utken, and Ross \cite{ALR}
several years ago.\footnote{It is possible to verify that the large
complex structure limit point used in \cite{ALR} agrees with the
one predicted by the monomial-divisor mirror map.}
More recently, Batyrev \cite{batyrev2} checked his version of this
statement in the case that $X$ itself is smooth,
and the authors \cite{AGM} checked it
in an example in which
there are five different birational choices for $\widehat{X}$ (with the same
$X\subset V$). After learning of the results of \cite{AGM} and of the
present paper, Batyrev \cite{batyrev3} checked this consequence in general.
The $3$-point function coming from variation of Hodge structure can be
used to determine the choice of signs $\theta_{\Delta}$:
the poles in that function should occur at {\em positive}\/ real values
of the canonical coordinates (cf.~\cite{AGMiii}).
The automorphism $\theta_{\Delta}$ with this property can be
calculated explicitly using methods of Gel'fand, Zelevinski\v\i, and
Kapranov \cite{GKZ}; we will discuss this in detail elsewhere.
Another consequence of our conjecture is that the K\"ahler moduli
spaces for different birational models $\widehat{X}$ of the function
field of $X$ can naturally be regarded as analytic continuations of
one another. (For after applying mirror symmetry, they are seen to
occupy different regions in the same moduli space.) This was the
principal conclusion of our earlier paper \cite{AGM};
a similar idea is
due independently to Manin \cite{manin}.
\section{Phases and the secondary fan}
In the course of defining the
monomial-divisor mirror map,
we made a somewhat unnatural restriction to a Zariski-open subset
of the simplified hypersurface moduli space. We now return to
the study of the full moduli space.
The ``simplified hypersurface
moduli space'' will be birational to the quotient
$(\Bbb{C}^*){}^{(P\cap M)_0}/\widetilde{T}$.
In fact, the moduli space of primary interest is the space parameterizing
those hypersurfaces whose singularities are no worse than generic.
This is the complement of the ``principal discriminant'' of
Gel'fand, Zelevinski\v\i, and Kapranov \cite{GKZ}.
One natural compactification
of the moduli space would be the one in which this ``principal discriminant''
is an ample divisor.
Whatever compactification we use,
the compactified moduli space is itself a toric variety (since it contains
the torus $(\Bbb{C}^*){}^{(P\cap M)_0}/\widetilde{T}$
as a dense open subset).
If we compactify so that the principal discriminant is ample,
then the toric variety is determined by
the Newton
polyhedron for the principal discriminant which,
as Gel'fand, Zelevinski\v\i, and Kapranov show, has a convenient
combinatorial description as a so-called ``secondary polytope''.
To explain the combinatorics, we note that the
action of $\widetilde{T}$
on $(\Bbb{C}^*){}^{(P\cap M)_0}$
is induced by a homomorphism
$\operatorname{ad}^+_{(P\cap M)_0}:N^+\to\Bbb Z^{(P\cap M)_0}$,
dual to the map $\Bbb Z\langle{(P\cap M)_0}\rangle\to M^+$
given by equation \eqref{twomaps}, where $M^+=M\oplus\Bbb Z$.
We should imagine embedding the set ${(P\cap M)_0}$
into $M^+$ via the map
$b\mapsto (b,1)$; the image is a finite set of points in the affine
hyperplane $\{(m,1)\}\subset M^+$. The convex cone spanned by
these points we denote by $P^+$; it is the cone over the
image of the original polyhedron $P$ (generated by the
points ${(P\cap M)_0}$)
in the affine hyperplane $\{(m,1)\}\subset M^+$.
The dual cone to $P^+$ has the form $(\polyhedron^\circ)^+$,
where $\polyhedron^\circ$ is the polar polyhedron of $P$.
The {\em secondary fan} (which
is the normal fan of the secondary polytope) is the fan consisting of
all cones $\operatorname{cpl}(\Sigma)$, where $\Sigma$ is a regular
refinement of the fan ${\cal N}(\,(\polyhedron^\circ)^+)$
(cf.~\cite{BFS}).
Among the possible regular refinements $\Sigma$ we find fans of the form
\[(\fan^\circ)^+:=
\{\text{cone over }(\sigma\capP)\ |\ \sigma\in\fan^\circ\},\]
for regular fans $\fan^\circ$ which refine ${\cal N}(\polyhedron^\circ)$. (But there
are others, which do not have this form.) For such fans, it
is easy to see that $\operatorname{cpl}(\,(\fan^\circ)^+)=\operatorname{cpl}(\fan^\circ)$, regarding both
as cones in the same space
$\Bbb Z^{{(P\cap M)_0}}/N^+\cong\Bbb Z^{{(P\cap M)_0}-\{0\}}/N$.
So our chosen
compactification of the moduli space is the toric variety which is
specified by the secondary fan. It has the pleasant property that
it includes all of the partial compactifications that were needed to
describe the ``large complex structure limits'' coming from mirror
symmetry of sigma models (since those were given by the cones
$\operatorname{cpl}(\fan^\circ)$). It has another nice property as well, first
observed by Kapranov et al.\ \cite{KSZ} and Batyrev \cite{batyrev2}:
the compactification constructed in this way dominates all possible
GIT\ compactifications, coming from different choices of linearization.
What does this structure correspond to under mirror symmetry? The
embedding $(\polyhedron^\circ\cap N)_0\subset N$, together with a (regular)
refinement $\Delta$ of
the fan ${\cal N}(P)$, was used to determine the projective
toric variety
$\widehat{V}$.
The
new extended embedding $(\polyhedron^\circ\cap N)_0\subset N^+$ (whose image lies in
the affine hyperplane $\{(n,1)\}\subset N^+$), together with a regular
refinement
$\Sigma$ of the fan ${\cal N}(P^+)$, can also be used to
determine a toric variety, of dimension one larger than the previous variety.
Among these toric varieties we find the total spaces of canonical
bundles over the various choices of $\widehat{V}$ (when we take $\Sigma$
of the form $\Delta^+$).
This is precisely the structure that Witten has found to be relevant
in his study of Landau-Ginzburg theories and their deformations
\cite{phases}. Each choice of fan $\Sigma$ determines a different
``phase'' of the physical theory. When the fan $\Sigma$ is of the form
$\Delta^+$, the physical theory is related to the nonlinear sigma model
with target $\widehat{X}$ (where $\widehat{X}\subset\widehat{V}$
is generic). But for other fans $\Sigma$, the physical theory is quite
different. (We refer the reader to \cite{phases} for more details.)
Thus, not only can the different sigma models with birationally equivalent
targets be viewed as analytic continuations of one another, there are
further analytic continuations (to regions in the moduli space corresponding
to $\operatorname{cpl}(\Sigma)$ for $\Sigma\ne\Delta^+$) to other kinds of physical
theory.
\ifx\undefined\leavevmode\hbox to3em{\hrulefill}\,
\newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
|
1993-09-02T13:47:17 | 9309 | alg-geom/9309002 | en | https://arxiv.org/abs/alg-geom/9309002 | [
"alg-geom",
"math.AG"
] | alg-geom/9309002 | Kirti | Kirti Joshi (School of Mathematics, Tata Institute of Fundamental
Research, Bombay) | A family of \'etale coverings of the affine line | The file is in standard LateX, eight pages | null | null | null | null | In the note we construct a family of \'etale coverings of the affine line.
More specifically, let $F$ be a finite field of characteristic $p$ and suppose
that the cardinality of $F$ is at least 4. Let $A = F[T]$ be the polynomial
ring in one variable $T$, $K=F(T)$. Let $K_\infty$ be the completion of $K$
along the valuation given by $1/T$, and let $C$ be the completion of the
algebraic closure of $K_\infty$. We prove in this note that there is a
continous surjection
$$\pi_1^{alg}(\A^1_C) \to \lim_{\leftarrow \atop I} SL_2( A/I )/{\pm 1},$$
where $\pi_1^{alg}(\A^1_C)$ is the algebraic fundamental group of the affine
line over $C$, and the inverse limit on the right (above) is taken over all
nonzero proper ideals in $A=F[T]$.
We use the theory of Drinfel'd modular curves to obtain these coverings.
| [
{
"version": "v1",
"created": "Thu, 2 Sep 1993 15:17:00 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Joshi",
"Kirti",
"",
"School of Mathematics, Tata Institute of Fundamental\n Research, Bombay"
]
] | alg-geom | \section{Introduction}
This note was inspired by a colloquium talk given by
S.~S.~Abhyankar at the Tata Institute\footnote{During his visit in the
month of December, 1992}, on the work of Abhyankar, Popp and Seiler
(see \cite{Popp}). It was pointed out in this talk that classical
modular curves can be used to construct (by specialization) coverings
of the affine line in positive characteristic. In this ``modular''
optic it seemed natural to consider Drinfel'd modular curves for
constructing coverings of the affine line. This note is a direct
outgrowth of this idea.
While it is trivial to see that affine line in characteristic
zero has no non-trivial \'etale coverings, in \cite{Abhyankar} it was shown
that the situation in positive characteristic is radically different
and far more interesting. Let us, for the sake of definiteness, work
over a field $k$ of characteristic $p>0$. In \cite{Abhyankar},
Abhyankar conjectured that any finite group whose order is divisble
by $p$ and which is generated by its $p$-Sylow subgroups (such a
finite group is sometimes called a ``quasi-$p$-group''), occurs as
quotient of the algebraic fundamental group of the affine line. It is
customary to write $\pi_1^{\rm alg}({\bf A}^1_k)$ to denote the algebraic
fundamental group of the affine line. While Abhyankar's conjecture
indicates that the algebraic fundamental group of the affine line is
quite complicated, our result perhaps illustrates its cyclopean
proportions. The result we prove (see Theorem~\ref{main theorem}
below) is the analogue of the following well-known classical result,
which falls out of the theory of elliptic modular curves over the
field of complex numbers: there is a continous quotient
$\pi_1^{\rm alg}({\bf P}^1_{{\bf C}}-\{0,1728,\infty\}) \surjects \prod_{p}
SL_2({\bf Z}_p)/\{\pm 1\}$, where ${\bf Z}_p$ denotes the ring of $p$-adic
integers.
We would like to thank N. Mohan Kumar for numerous suggestions
and conversations; he also explained to us a variant of Abhyankar's
Lemma, which is crucial to our argument. We would also like to thank
E.-U.~Gekeler, Dipendra Prasad for electronic correspondence while
this note was being written -- their remarks and suggestions have been
extremely useful; in particular we would like to point out that the
conjecture stated at the end of this note was formulated with the help
of Dipendra Prasad. We would also like to thank M.~V.~Nori for useful
comments and Dinesh Thakur for encouragement.
\section{Resum\'e of Drinfel'd modules and their moduli}
In this section we recall a few of the standard facts about
Drinfel'd modules. Since the basic theory of Drinfel'd modules and
their moduli is well documented we will be brief; all the facts
which we will need can be found in the following standard references:
\cite{Drinfel'd}, \cite{Deligne-Husemoller}, \cite{Gekeler2}.
Since we do not need the full strength of Drinfel'd's work, we will
work with a very special situation which is required for our purpose.
In this section, we will outline this special situation.
Let us fix some notations. Let ${\bf F}_q$ denote a finite field
with $q=p^m$ elements and of characteristic $p$. We will write $A =
{\bf F}_q[ T ]$, $K = {\bf F}_q(T)$. Further denote by $K_\infty$, the
completion of $K$ along the valuation corresponding to $1/T$. Denote
by $C$ the completion of the algebraic closure of $K_\infty$. The
field $C$ is thus a ``universal domain'' of characteristic $p$. There
is a natural inclusion $K \into C$.
Let ${\cal G}_a$ denote the additive group scheme. Then it is easy
to see that the ring of endomorphisms of ${\cal G}_a$ defined over $C$,
denoted by $\mathop{\rm End}\nolimits_C({\cal G}_a)$, is a noncommutative ring generated by the
Frobenius endomorphism. More precisely, for an indeterminate $\tau$,
consider the noncommutative ring $C\{\tau_p\}$ of all polynomials in
$\tau$ with coefficients in $C$, and where the multiplication rule is
given by $\tau_p a = a^p \tau_p$ for all $a\in C$. Then one checks that
$\mathop{\rm End}\nolimits_C({\cal G}_a) \isom C\{\tau_p\}$. Observe that this isomorphism gives
rise to a ring homomorphism $\partial:\mathop{\rm End}\nolimits_C({\cal G}_a) \to C$ which sends
an endomorphism of ${\cal G}_a$ to the constant term of the polynomial in
$\tau$ associated to it.
A {\em Drinfel'd module} $\phi$ over $C$ is a ring
homomorphism $\phi:A \to \mathop{\rm End}\nolimits_C({\cal G}_a)$ such that composite map
$\partial \phi $ is the natural inclusion of $A\to C$. It is easy to
check that any such map factors through the subring
$C\{\tau_p^m\}\subset C\{\tau_p\}$. For simplicity of notation, we
will write $\tau=\tau_p^m$. Thus for any $a\in A$ we have an
endomorphism $\phi_a$ of ${\cal G}_a$ defined over $C$. Moreover, note that,
any such $\phi$ is ${\bf F}_q$ linear. So as $A$ is generated as an
${\bf F}_q$-algebra by $T$, giving a $\phi$ thus amounts to specifying a
single endomorphism of ${\cal G}_a$ corresponding to $T\in A$, $\phi_T =
\sum_{i=0}^r a_i \tau^i$, with $a_0 =T$. Then this can be extended to
all of $A$. The $\tau$-degree of $\phi_T$ is called the {\em rank} of
the Drinfel'd module $\phi$.
If $\phi,\phi'$ are two Drinfel'd modules then a {\em morphism
of Drinfel'd modules} is an endomorphism $u\in\mathop{\rm End}\nolimits_C({\cal G}_a)$ such that
for all $a\in A$ we have $\phi_a u = u \phi_a' $. It is easy to see
that any such $u$ is in fact contained in $C\{\tau\}$.
For any $a\in A$ a Drinfeld module $\phi$ specifies a closed
subgroup-scheme of ${\cal G}_a$: the kernel of the endomorphism $\phi_a$,
$\ker(\phi_a)$. For example, if $a=T$, then the kernel of $\phi_T$ is
simply the roots of the polynomial $T+\sum_{i=1}^r a_i X^{q^i-1} =0$
together with $0$. Moreover, for any ideal $I\subset A$, we can
define a subgroup scheme of ${\cal G}_a$ using the ring structure. One checks
that $\ker(\phi_I)(C)$ is a free $(A/I)$-module of rank $r$, where $r$
is the rank of $\phi$. This lets us define a notion of an $I$-level
structure on $\phi$. Drinfel'd has shown (see \cite{Drinfel'd})
that there is a moduli of Drinfel'd modules with level structure (in
general we have only a ``coarse moduli scheme''). Our main interest is
the case of rank two Drinfel'd modules. And henceforth, we shall work
with Drinfel'd modules of rank two.
The theory of Drinfel'd modules of rank two behaves like the
theory of elliptic curves. The fact that the moduli of Drinfel'd
modules of rank two over $C$, is a smooth affine curve over $C$ is a
very special case of a fundamental result of Drinfel'd (see
\cite{Drinfel'd}). We need several facts about these Drinfel'd
modular curves. The first fact we need is the following:
\begin{thm}
The ``coarse moduli'' of rank two Drinfel'd modules over $C$
is the affine line over $C$.
\end{thm}
\begin{proof}
For a proof see \cite{Goss} paragraph 1.32, or
\cite{Gekeler1}.
\end{proof}
Thus the above situation is analogous to the classical
situation for elliptic curves. In fact the identification with the
line is given by a ``j-invariant''. If $\phi_T = T+ a\tau + b \tau^2$,
is a rank 2 module then $b\neq 0$ and then its $j$-invariant is
$a^{q+1}/b$.
Before we need some notations. For any ${\bf F}_q$ algebra $R$, let
$$G_1(R) = \left\{g\in GL_2(R)\big| \det(g)\in {\bf F}_q^* \right\},$$
$$ G(R) = G_1(R)/Z({\bf F}_q),$$
where $Z({\bf F}_q)$ is the group of ${\bf F}_q$-valued scalar matrices with
nonzero determinant.
The following result is really the heart of our construction.
\begin{thm}\label{Drinfel'd-Gekeler}
For every non-zero ideal $I\subset A$, there is a ``coarse
moduli'' of Drinfel'd modules of rank two with full $I$-level structure
exists and is an affine curve over $C$. There is a ``forget the level
structure'' morphism to ${\bf A}^1_C$. This map is branched over $0\in
{\bf A}^1_C$. The covering is Galois and the Galois group is $G(A/I)$. The
ramification index of any point lying over $0$ is $q+1$ and is
independent of $I$. In particular, the ramification is tame.
\end{thm}
\begin{proof}
As mentioned earlier, the existence of the moduli is due to
Drinfel'd (see \cite{Drinfel'd}). These curves have been studied in
great detail by Goss and Gekeler. In the our case the Galois group
can easily be calculated, a convenient reference for it is
\cite{Gekeler2}, for instance see Lemma 1.4 on page 79, also see the
first section of \cite{Gekeler1}. The ramification information is
computed in \cite{Goss}, Lemma 4.2. One also finds it computed in
\cite{Gekeler2}, on page 87.
As in the classical situation, the ramification takes place
over the Drinfel'd module with extra automorphisms. From the
definitions, it is clear that an automorphism of a Drinfel'd module of
rank two over $C$ is firstly an automorphism $u$ of ${\cal G}_a$, defined
over $C$. Clearly any such automorphism must be an invertible element
of $C$. Then a simple calculation shows that if a nonzero element of
$C$ is an automorphism of a Drinfel'd module then it must be a root of
unity. One checks that with the exception of the module with
$j$-invariant equal to zero, the automorphism group of the Drinfel'd
module is ${\bf F}_q^*$. The module with $j$-invariant equal to zero has
automorphism group $F_{q^2}^*$. Thus in particular, the ramification
is tame. These facts are easily proved by explicit calculations.
\end{proof}
\section{The main theorem}
We are now ready to state and prove our main theorem. One
should note that most of the work has already gone in the
construction and analysis the of the moduli of Drinfel'd modules.
\begin{thm}\label{main theorem}
If $p=2,3$ then assume that $q=p^m, m\geq 2$. There is a
family of \'etale coverings, $Y_I$ of ${\bf A}^1_C$, indexed by the nonzero
proper ideals $I\subset A$. The curves $Y_I$ are affine curves over
$C$ and the covering $Y_I \to {\bf A}^1_C$ is Galois with Galois group
$SL_2(A/I)/\{\pm1\}$. Moreover, these coverings form an inverse system
indexed by $I$. Thus in particular we have a continous quotient
$$\pi_1^{\rm alg}( {\bf A}^1_C) \to \lim_{\longleftarrow \atop I}
\left( SL_2(
A/I )/\{\pm 1\} \right).$$
\end{thm}
\begin{proof}
By Theorem~\ref{Drinfel'd-Gekeler}, we have a tamely ramified
covering of the affine line which is branched over one point. Also
note that the ramification index of any point over $0\in {\bf A}^1_C$ is
$q+1$, independent of the ideal $I\subset A$.
Now we apply a suitable variant of Abhyankar's Lemma to
remove the tame ramification. The crucial thing is to ensure, if
possible, that the ``pull back'' coverings remain irreducible. The
following variant of Abhyankar's Lemma (see Lemma~\ref{Mohan's
lemma}) which was pointed out to me by N. Mohan Kumar, gives an
explicit criterion to check irreducibility, then we apply this
criterion to the case at hand. This is an easy exercise in elementary
group theory. We have stated all the necessary results as a sequence
lemmas, and since the proofs of all the individual statements are
easy, we leave the details to the reader.
\end{proof}
\begin{lemma}\label{Mohan's lemma}
Let $k$ be an algebraically closed field of characteristic
$p$. Let $X \to {\bf A}^1_k$ be a finite Galois cover defined over $k$,
with Galois group $G$. Further assume that the cover is branched over
$0\in {\bf A}^1_k$, and any point lying over it is tamely ramified with
ramification index $n$. Let ${\bf A}^1_k \to {\bf A}^1_k$ be a ${\bf Z}/n$ covering
ramified completely at $0$, and unramified elsewhere. Let $X'$ be the
normalization of the fibre product $X \times_{{\bf A}^1_k} {\bf A}^1_k$. Suppose
that there are no nontrivial homomorphisms $G \to {\bf Z}/n$. Then $X'$ is
irreducible, and the Galois group of the covering $X' \to {\bf A}^1$ is
$G$.
\end{lemma}
\begin{proof}
Clearly one is reduced to proving the following field theory
statement:
Let $L/k(t)$ be a finite Galois extension with Galois group
$G$. Let $E=k(t^{1/n})$. Then $L/k(t)$ and $E/k(t)$ are linearly
disjoint over $k(t)$ if and only if there are no nontrivial
homomorphisms $G \to {\bf Z}/n$.
And the above statement is immediate from the fact that
$E/k(t)$ is a Kummer extension. This finishes the proof.
\end{proof}
We now need some elementary group theoretic lemmas to apply
the above criterion. Since the proofs are easy we will state the
lemmas without proofs.
\begin{lemma}\label{lemma1}
Let $\wp \neq 0$ is a prime ideal of $A$. Then for all $k
\geq 1$ and for all $n\geq 2$, the natural morphism
$$GL_k(A/\wp^n) \to GL_k( A/\wp^{n-1})$$
is surjective.
\end{lemma}
Recall that for any ${\bf F}_q$ algebra $R$, we had defined the groups
$$G_1(R) = \left\{g\in GL_2(R)\big| \det(g)\in {\bf F}_q^* \right\},$$
$$ G(R) = G_1(R)/Z({\bf F}_q),$$
where $Z({\bf F}_q)$ is the group of ${\bf F}_q$-valued scalar matrices with
nonzero determinant. Further, for any prime ideal $\wp\neq 0$ of $A$,
write $G_1^n = G_1(A/\wp^n)$, and $G^n = G(A/\wp^n)$, for all $n\geq
1$. The results which follow are valid for any non-zero prime ideal
$\wp$, so we have supressed $\wp$ in our notations.
\begin{lemma}\label{lemma2}
The natural map $G_1^n \to G_1^{n-1}$ is surjective for all
$n\geq 2$.
\end{lemma}
\begin{lemma}\label{lemma3}
The natural map $G^n \to G^{n-1}$ is surjective for all
$n\geq 2$.
\end{lemma}
Denote by $G^{(n,n-1)}$ the kernel of the map $G^n \to
G^{n-1}$, similarly write $G_1^{(n,n-1)}$, for the kernel of the
corresponding map for $G_1$.
\begin{lemma}\label{lemma4}
Let $q=2^m, m\geq 2$. Let $F/{\bf F}_q$ be any finite
extension. Then there are no nontrivial morphisms $G(F) \to
{\bf Z}/(q+1)$.
\end{lemma}
\begin{lemma} \label{lemma5}
Let $q=2^m,m\geq 2$. Then for any $n\geq 1$ there are no
nontrivial maps $G^n \to {\bf Z}/(q+1)$.
\end{lemma}
\begin{proof}
The proof is by induction on $n$. For $n=1$, we are done by the
previous lemma. Now show that the kernel of any map $G^n \to {\bf Z}/(q+1)$
contains $G^{(n,n-1)}$. Then we are done by induction.
\end{proof}
\begin{lemma}\label{lemma6}
Let $q = p^m, p\neq 2$. If $p=3$ then $m\geq 2$.
Then there is a canonical morphism $G^n \to {\bf F}_q^*/{\bf F}_q^{*2}$. In
particular we have a natural map $G^n \to {\bf Z}/2$, obtained by
identifying ${\bf F}_q^*/{\bf F}_q^{*2}$ with ${\bf Z}/2$.
\end{lemma}
\begin{lemma}\label{lemma7}
Let $q=p^m, p\neq 2$. If $p=3$ then $m\geq2$. Let $F/{\bf F}_q$ be
any finite extension. Then any morphism $G(F)\to {\bf Z}/(q+1)$ factors
through the canonical morphism given by the above lemma, followed by
the inclusion of ${\bf Z}/2\to {\bf Z}/(q+1)$.
\end{lemma}
\begin{lemma}\label{lemma8}
Let $q = p^m, p\neq 2$. If $p=3, m\geq 2$. Then any
nontrivial morphism $G^n \to {\bf Z}/(q+1)$ factors through the canonical
morphism.
\end{lemma}
\begin{proof}
This is again proved by induction on $n$. As before, one
checks that any such map is trivial on $G^{(n,n-1)}$.
\end{proof}
Now we can identify our Galois groups.
\begin{lemma}\label{lemma9}
Let $q=2^m, m \geq 2$. For any non zero prime ideal
$\wp\subset A$, we have:
$$G^n = SL_2(A/\wp^n).$$
\end{lemma}
\begin{lemma}\label{lemma10}
Let $q=p^m, p\neq 2$, if $p=3$ then $m\geq 2$. let
$\wp$ be any non zero prime ideal in $A$. For any $n\geq 1$, let
$$\tilde{G^n} = ker(G^n \to {\bf Z}/2).$$
Then $\tilde{G^n} = SL_2(A/\wp^n)/\{\pm 1\}$.
\end{lemma}
Thus we can now prove our main theorem. If $p=2$, then the
pull back coverings remain irreducible and the Galois group is $G^n$.
If $p$ is odd, then there is a quadratic subfield in common. And the
Galois group is $\tilde{G^n}$. Then we are done by the above lemmas.
After the result of Madhav Nori (see \cite{Madhav}) and the
one proved above, we would like to advance the following conjecture:
\begin{conj}
Let $G/K$ be any isotropic, semisimple, simply-connected
algebraic group over $K={\bf F}_q(T)$, with center $Z$. Let ${\bf A}^{{\rm
fin}}_K$ denote the finite adeles of $K$. Then any maximal compact
subgroup of $G({\bf A}^{{\rm fin}}_K)/Z({\bf F}_q)$ occurs as a continous
quotient of the fundamental group of the affine line over $C$.
\end{conj}
|
1997-07-01T06:56:43 | 9707 | alg-geom/9707002 | en | https://arxiv.org/abs/alg-geom/9707002 | [
"alg-geom",
"math.AG"
] | alg-geom/9707002 | Aaron Bertram | Aaron Bertram (University of Utah) | Stable pairs and log flips | 17 pages, LaTeX2e. Will appear in the Proceedings of the AMS Santa
Cruz conference (1995) | null | null | null | null | This paper has two parts. In the first part, we review stable pairs and
triples on curves, leading up to Thaddeus' diagram of flips and contractions
starting from the blow-up of projective space along a curve embedded by a
complete linear series of the form K + ample. In the second part, we identify
log canonical divisors which exhibit Thaddeus' flips and contractions as "log"
flips and contractions in the sense of the log-minimal-model program.
| [
{
"version": "v1",
"created": "Tue, 1 Jul 1997 04:56:34 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Bertram",
"Aaron",
"",
"University of Utah"
]
] | alg-geom | \section{Stable Bradlow Pairs.}
Let ${\mathcal A}$ be a category with a zero object in which kernels and cokernels
exist, as well as direct sums.
Let $S \subset Ob({\mathcal A})$ be a subset which is closed under direct sums.
\begin{definition}
A function $\mu: S \rightarrow {{\bf R}}$
is called a {\it slope function} if
for all short exact sequences $0 \rightarrow B \rightarrow A \rightarrow
C \rightarrow 0$ of elements of $S$,
$$\mu(B) < \mu(A) \Leftrightarrow \mu(A) < \mu(C)\ \mbox{and}\
\mu(B) = \mu(A) \Leftrightarrow \mu(A) = \mu(C)$$
\end{definition}
Given a slope function $\mu:S \rightarrow {{\bf R}}$:
\begin{definition}
An object
$A\in S$ is called {\it stable} if $\mu(B) < \mu(A)$
whenever $B\in S$ and there exists an injection $B \hookrightarrow A$
other than the identity.
$A$ is called semistable if $\mu(B) \le \mu(A)$ above,
strictly semistable if it is semistable but not stable, and
unstable if it is not semistable.
\end{definition}
\begin{definition}
(a) If $A\in S$ is unstable, then a filtration:
$$0 \hookrightarrow A_1 \hookrightarrow A_2 \hookrightarrow ...
\hookrightarrow A_n = A$$
by elements of $S$ is called a Harder-Narasimhan filtration if
the $A_i/A_{i-1}$ are all in $S$ and semistable and
$\mu(A_1) > \mu(A_2/A_1) > ... > \mu(A_n/A_{n-1})$.
\medskip
(b) If $A\in S$ is semistable, then a filtration
$$0 \hookrightarrow A_1 \hookrightarrow A_2 \hookrightarrow ...
\hookrightarrow A_n = A$$
is called a Jordan-H\"older filtration if the
$A_i/A_{i-1}$ are all in $S$ and stable.
Given a Jordan-H\"older filtration of $A$, the object
$gr(A) := \oplus_{i=1}^nA_i/A_{i-1} \in S$ is called the associated graded of
the filtration. Two Jordan-H\"older filtrations are called
{\it s-equivalent} if
their associated graded objects are isomorphic.
\end{definition}
\begin{example}{(Vector Bundles on $C$):}
$\bullet$ ${\mathcal A}$ is
the category of
isomorphism classes of vector bundles on $C$.
$\bullet$ $S = Ob({\mathcal A}) - \{0\}$.
$\bullet$ $\mu:S \rightarrow {{\bf Q}}$ is the usual
slope function $\mu(E) = \ \mbox{deg}(E)/\mbox{rk}(E)$. Then:
(a) Harder-Narasimhan filtrations always exist (and up to
isomorphism only depend upon
the isomorphism class of $E$).
(b) Jordan-H\"older filtrations always exist, producing
associated gradeds which only depend upon the isomorphism
class of $E$. In particular, $s$-equivalence
becomes an equivalence relation on isomorphism classes of
semistable bundles.
(c) For fixed invariants $r$ (the rank) and either $d$ (the
degree) or ${\mathcal O}_C(D)$ (the isomorphism class of the determinant),
there are projective coarse moduli spaces
$M_C(r,d)$ (respectively, $M_C(r,D)$) for the functors ``families of semistable
vector bundles
modulo s-equivalence with the given invariants''. (See \cite{S} for details.)
\end{example}
The next example is due to Bradlow and Garcia-Prada
(\cite{BG}).
\begin{example}{(Triples on $C$):}
$\bullet$ ${\mathcal A}$ is the category
of isomorphism classes of triples $(E,f,F)$, where $E,F$ are vector
bundles on $C$ and $f:E \rightarrow F$ is a homomorphism. A triple is called
{\it nontrivial} if $F\ne 0$ and {\it nondegenerate} if
$f$ has maximal rank at some point. A morphism in this category is a pair
$(\alpha,\beta)$ consisting of morphisms
$\alpha:E\rightarrow E'$ and $\beta:F \rightarrow F'$ in the category of
vector bundles,
such that the following diagram commutes:
$$\begin{array}{ccc}E & \stackrel\alpha\rightarrow & E' \\
f\downarrow & & f'\downarrow \\ F & \stackrel \beta\rightarrow & F' \end{array}$$
Both $\alpha$ and $\beta$ need to be injective to make
$(\alpha,\beta): (E,f,F) \rightarrow (E',f',F')$
injective as a morphism of triples. Direct sums obviously
exist as the ``free'' sum of triples:
$(E,f,F) \oplus (E',f',F') = (E\oplus E',f\oplus f',F\oplus F')$.
$\bullet$ $S$ is the set of nontrivial triples.
$\bullet$ For each $\sigma \in {{\bf R}}$ and $(E,f,F) \in S$, let:
$$\mu_\sigma(E,f,F) = \frac{\mbox{deg}(E) + \ \mbox{deg}(F) +
\sigma (\mbox{rk}(E) + \ \mbox{rk}(F))}{\mbox{rk}(F)}.$$
Note the asymmetry in the slope function!
One says $(E,f,F)$ is $\sigma$-stable if it is stable with respect to
the slope function $\mu_\sigma$.
\end{example}
\begin{theorem}{(\cite{BG} Theorem 6.1)} For fixed invariants:
$$r_1 = \ \mbox{rk}(E), r_2 = \ \mbox{rk}(F), d_1 = \ \mbox{deg}(E),
d_2 = \ \mbox{deg}(F) \ \mbox{and}\ \sigma \in {{\bf R}}$$
a coarse moduli space exists for the functor
``families of non-degenerate $\sigma$-stable triples with the given invariants'',
which is moreover projective if $r_1 + r_2$ is relatively prime
to $d_1 + d_2$ and $\sigma$ is ``generic'' (see \cite{BG}).
\end{theorem}
\noindent {\bf Explanation of the Parameter:} The idea is to relate
stable triples $(E,f,F)$ on $C$ to (equivariantly) stable equivariant
bundles $G$ on $C\times {\bf P}^1$. (The action is the
automorphism group of ${\bf P}^1$ acting on the second factor.)
This is a consequence of K\"unneth, which gives an isomorphism
$$\mbox{Hom}_{{\mathcal O}_C}(E,F) \cong \
\mbox{Ext}^1_{{\mathcal O}_{C\times {\bf P}^1}}
(p^*E\otimes q^*{\mathcal O}_{{\bf P}^1}(2),p^*F),$$
telling us to look for $G$ in the corresponding extension:
$$0 \rightarrow p^*F \rightarrow G \rightarrow p^*E \otimes q^*{\mathcal O}_{{\bf P}^1}(2)
\rightarrow 0.$$
(this technique is often called ``dimensional reduction''.)
The main point is now that ample line bundles on $C\times {\bf P}^1$
are of the form $p^*L\otimes q^*M$,
and (an equivariant version of) Gieseker stability
for bundles on $C\times {\bf P}^1$ depends upon a parameter,
namely the {\bf ratio} deg$(L)$/deg$(M)$. Stability of $G$ with respect to
a given ratio translates into $\sigma$-stability for
triples for a fixed value of $\sigma$.
\bigskip
We are most concerned with the following,
first considered by Bradlow in \cite{Br}.
\begin{example}{(Pairs on $C$):}
$\bullet$ Restrict the category of
triples to the objects: $({\mathcal O}_C,f,E)$ and
$(0,0,E)$. These objects we will call pairs, following the literature.
This full subcategory is closed under kernels and cokernels. It is not
closed under arbitrary direct sums, but if a set of
pairs is given, with the property that at most one of
them is of the form
$({\mathcal O}_C,f,E)$, then their direct sum does lie in the
subcategory.
(This will be enough to construct associated gradeds for
Jordan-H\"older filtrations!)
$\bullet$ $S$ is the set of nontrivial pairs (i.e. $E \ne 0$) and a pair
of the form $({\mathcal O}_C,f,E)$ is nondegenerate if
and only if $f \ne 0$.
$\bullet$ The slope functions $\mu_\sigma$ are the same as for triples.
\end{example}
With respect to this slope function, observe that
a pair $({\mathcal O}_C,f,E)$ is
$\sigma$-stable if and only if:
\medskip
(i)\ $\mu(F) < \mu(E) - \sigma(\frac 1{\mbox{\small rk}(F)} -
\frac 1{\mbox{\small rk}(E)})$
for each $({\mathcal O}_C,g,F) \hookrightarrow ({\mathcal O}_C,f,E)$, and
\medskip
(ii)\ $\mu(F) < \mu(E) + \sigma(\frac 1{\mbox{\small rk}(E)})$
for each $(0,0,F) \hookrightarrow ({\mathcal O}_C,f,E)$.
\begin{theorem}{(\cite{T1} (1.1)-(1.19))} Fix invariants:
$$\mbox{rk}(E) = 2\ \mbox{and}\
\mbox{det}(E) \cong {\mathcal O}_C(D),\ \mbox{with} \
\mbox{deg}(D) = d.\ \mbox{Then:}$$
(a) Harder-Narasimhan and Jordan-H\"older filtrations exist if $\sigma > 0$ and
yield a well-defined $s$-equivalence for $\sigma$-semi-stable pairs.
(b) For each $\sigma > 0$, a projective variety $M_C(2,D,\sigma)$
(abbreviated $M_\sigma$)
coarsely represents the functor:
``families of nondegenerate $\sigma$-semistable
pairs $({\mathcal O}_C,f,E)$ modulo $s$-equivalence''.
There is a universal family over the open locus parametrizing
stable pairs, which is smooth and irreducible.
\end{theorem}
Full proofs of the properties listed below can be
found in \cite{T1}.
(Please note that our $\sigma$ differs from the $\sigma$
in \cite{T1} by a factor of $2$.)
\bigskip
\noindent {\bf Properties of Stable Pairs:}
{\it Fix rk$(E) = 2$ and det$(E) \cong {\mathcal O}_C(D)$.
Also assume that $g \ge 2$ (but see the note at the end of this section). Then:
\medskip
(a) There are no $\sigma$-semi-stable pairs if
$\sigma < 0$ or $\sigma > d$.
\medskip
(b) There are always $0$-semi-stable pairs, though no $0$-stable pairs.
\medskip
(c) $M_{d}$ is a point.}
\bigskip
{\bf Proof:} If $\sigma < 0$, then by (ii) above,
$(0,0,E) \hookrightarrow ({\mathcal O}_C,f,E)$
destabilizes any pair. If $\sigma > d$ and $({\mathcal O}_C,f,E)$
is given, let $L$ be the line-bundle image of ${\mathcal O}_C$ in $E$
with induced map $s:{\mathcal O}_C \rightarrow L$.
Then using (i) above, the pair is destabilized by the natural inclusion $({\mathcal O}_C,s,L) \hookrightarrow
({\mathcal O}_C,f,E)$.
\medskip
If $\sigma = 0$, then conditions (i) and (ii) coincide, telling us that
$({\mathcal O}_C,f,E)$ is semistable if
and only if $E$ is semistable, and that no pair is $0$-stable. Note that
there is no Jordan-H\"older filtration.
\medskip
If $\sigma = d$, then by the analysis in the proof of (a),
$({\mathcal O}_C,f,E)$ is $\sigma$-semi-stable
if and only if $f:{\mathcal O}_C \rightarrow E$ has no zeroes,
and all such $({\mathcal O}_C,f,E)$ are $s$-equivalent, with associated graded
$({\mathcal O}_C,\mbox{id},{\mathcal O}_C) \oplus (0,0,{\mathcal O}_C(D))$. So
there are no $\sigma$-stable pairs, and the
moduli space is a point.
\medskip
In contrast to the
boundary cases presented here, the stable locus in $M_\sigma$ will be
nonempty if $0 < \sigma < d$.
\bigskip
\noindent {\bf Critical Points and Local Triviality:}
\begin{center}
{\it Let $\Gamma = \{0 < c < d \ | \
c \equiv d \ (\mbox{mod}\ 2)\}$.}
\end{center}
{\it \noindent For each $\sigma$, let $Z_\sigma \subset M_\sigma$ be the locus of
$\sigma$-strictly-semistable pairs.
\medskip
(d) If $\sigma \not \in \Gamma$, then $Z_\sigma = \emptyset$ (i.e.
$\sigma$-semistable $\Rightarrow$
$\sigma$-stable).
\medskip
(e) If $c = d - 2n \in \Gamma$, then $Z_c \cong C_n$, the $n$-th symmetric product of $C$.
\medskip
(f) If $ I \subset (0,d) - \Gamma$ is an interval and $\sigma,\sigma' \in I$,
then $M_\sigma \cong M_{\sigma'}$.
\medskip
(g) Suppose that $c\in \Gamma$ and $c^- < c < c^+$ are real numbers
in the neighboring intervals of $(0,d) - \Gamma$. Then there are surjective
morphisms:
$$\begin{array}{ccccc}M_{c^-} &&&& M_{c^+} \\
& \stackrel{f^-}\searrow & & \stackrel{f^+}\swarrow \\ && M_c \end{array}$$
\noindent {\bf Key Point:} $f^-$ and $f^+$ are isomorphisms away from
$Z_c \subset M_c$ and projective bundles over $Z_c$. (The
projective bundles are identified in the proof).}
\bigskip
{\bf Proof:} If $\sigma > 0$ and if $F$ is the bundle in a destabilizing
subpair of $({\mathcal O}_C,f,E)$, then it is easy to see that $F$ is a line
bundle. But if $\sigma \not \in \Gamma$, then the right side of
(i) and (ii) are not integers, whereas $\mu(F)$ is an integer. So
we cannot have equality. This proves (d).
\medskip
Suppose $c = d - 2n \in \Gamma$. Then a $c$-strictly semistable
pair $({\mathcal O}_C,f,E)$ has a subpair which is either isomorphic
to $({\mathcal O}_C,s,{\mathcal O}_C(A))$ (where $s$ is the tautological
section, deg$(A) = n$, and ${\mathcal O}_C(A)$ is the image of ${\mathcal O}_C$ in $E$) or
else it is isomorphic to $(0,0,L)$, where deg$(L) = d - n$. But
{\bf either} possibility forces the associated graded
for the pair $({\mathcal O}_C,f,E)$ to be
of the form $({\mathcal O}_C,s,{\mathcal O}_C(A)) \oplus (0,0,{\mathcal O}_C(D-A))$,
and these are parametrized by $C_n$.
\medskip
The stability conditions do not
change when $\sigma$ moves within an interval $I \subset (0,d) - \Gamma$
(again because $\mu(F) \in {\bf Z}$) so the moduli spaces are isomorphic
by the universal property of a coarse moduli space.
\medskip
If $c\in \Gamma$, then apart from $Z_c$, the stability conditions do not change when
$c$ is replaced by $c^-$ or $c^+$, so the first part of the key point follows
as in the previous paragraph.
\medskip
Let $c = d - 2n$, and consider
$({\mathcal O}_C,s,{\mathcal O}_C(A)) \oplus (0,0,{\mathcal O}_C(D-A)) \in Z_c$. Then
it follows that among all
pairs with this associated graded,
exactly those pairs of the form:
$$\begin{array}{ccccccccc}&&{\mathcal O}_C \\ && \downarrow & \searrow\\
0 & \rightarrow & {\mathcal O}_C(A) & \rightarrow & E & \rightarrow
& {\mathcal O}_C(D-A) \rightarrow 0\end{array}$$
are $c^-$-stable, and these are parametrized by $|K_C + D - 2A|$, which has
dimension $d-2n+g-2$ (independent of $A$) since $d-2n > 0$.
\medskip
On the other hand, among all pairs with this associated graded,
exactly those pairs of the form:
$$\begin{array}{ccccccccc}&&&&&&{\mathcal O}_C \\ &&&&&\swarrow & \downarrow \\
0 & \rightarrow & {\mathcal O}_C(D-A) & \rightarrow & E & \rightarrow
& {\mathcal O}_C(A) &\rightarrow & 0\end{array}$$
are $c^+$-stable. Such pairs are parametrized by ${\bf P}(V)$, where
$V$ sits in the long exact sequence:
$$H^0(C,{\mathcal O}_C(D-A)) \rightarrow V \rightarrow H^1(C,O_C(D-2A)) \rightarrow H^1(C,{\mathcal O}_C(D-A)).$$
(in fact, ${\bf P}(V)$ is naturally isomorphic to ${\bf P}(H^0(C,{\mathcal O}_C(D-A)
\otimes {\mathcal O}_A)^*))$.
\medskip
Thus the dimension of ${\bf P}(V)$ is $n-1$, independent of $A$.
\bigskip
So there are a {\bf finite} number of moduli spaces $M_\sigma$, linked by
morphisms as in the following diagram:
$$\begin{array}{cccccccccccc} &X_2&&&& X_1 & & & & X_0\\
\cdots & & \searrow & & \swarrow & & \searrow & & \swarrow & & \searrow\\
& & & M_{d-4} & & & & M_{d-2} & & & & M_d \\
\end{array}$$
where each $X_n \cong M_{(d-2n)^-} \cong M_{(d-2n-2)^+}$.
\medskip
Theorems ~\ref{old1} and ~\ref{old2} are embedded in this diagram because of:
\medskip
\noindent {\bf Large Values of $\sigma$:}
\medskip
{\it (h) If $d > 0$, then $X_0 \cong |K_C+D|^*$.
\medskip
\hskip .22in If $d > 2$, then $M_{d-2} \cong |K_C+D|^*$ and $X_1 \cong X =
\ \mbox{bl}(|K_C+D|^*,C)$.
\medskip
\hskip .22in Moreover, the morphism $f^-: X_1 \rightarrow M_{d-2}$ is the blow-down.
\medskip
\hskip .22in If $d > 4$, then $f^+:X_1 \rightarrow M_{d-4}$
is the contraction $\gamma:X \rightarrow Y$.}
\medskip
{\bf Proof:} A special case of the proof of (g) shows that $X_0 \cong |K_C+D|^*$.
\medskip
Another special case of the proof of (g) shows that
$f^+:X_0 \rightarrow M_{d-2}$ is an isomorphism, because the ``exceptional''
part of the map is a ${\bf P}^0$-bundle(!)
\medskip
When (g) is applied to the map $f^-:X_1 \rightarrow M_{d-2} = |K_C + D|^*$,
one discovers that the exceptional locus is a divisor, which is
a projective bundle over $C$, hence $f^-$ is the blow-down.
\medskip
Finally, when (g) is applied to the map $f^+:X_1 \rightarrow M_{d-4}$,
the exceptional set consists of lines spanned by two points of $C$
(i.e. the secant lines) which are contracted to points. This means that
the linear series which realizes $f^+$ must be a multiple of
$|2H - E|$, so the fact that $f^+$ has connected fibers
implies $f^+$ is equal to $\gamma$.
\bigskip
So we've got Theorem ~\ref{old1} and (a very precise) Theorem ~\ref{old2} when $d > 4$.
To see what happens for $d = 4$ from this point of view (for example,
in the case $g = 2$ and $d = 4$ considered in the introduction), we need
to analyze:
\medskip
\noindent {\bf All Values of $\sigma$:}
\medskip
{\it (i) If $n > 1$, then $X_n$ (if defined) is isomorphic to $X_1$ off codimension $2$.
\medskip
\hskip .2in The maps $f^+:X_n \rightarrow M_{d-2n-2}$ are multiples of
$|(n+1)H - nE|$.
\medskip
\hskip .2in The maps $f^-:X_n \rightarrow M_{d-2n}$ are
multiples of $|nH - (n-1)E|$.
\medskip
(j) If $\sigma$ is in the first interval of $(0,d) - \Gamma$, then
there is a morphism:
$$f:M_\sigma = X_{[\frac {d-1}2]} \rightarrow M_C(2,D)$$
which is the contraction determined by high multiples of
$|dH - (d-2)E|$.}
\medskip
{\bf Proof:} The first part of (i) is a dimension count using
(g), which allows us to transfer linear series from $X_1 = X$ over
to each $X_n$. The reader is referred to \cite{T1}, where the
ample cone is constructed for each $X_n$, the boundary of which gives
properties (i) and (j). Notice in particular, that each $f^+$ and $f^-$
is a very simple contraction by property (g), but that the final
map $f$ in (j) can have rather more complicated behavior, as in the
example of the introduction.
\bigskip
Thus the mirror image of the diagram following property (g) gives us:
$$\begin{array}{cccccccccccc}X = X_1&&&&&&X_2 && & & X_{[\frac {d-1}2]}\\
\downarrow &&\searrow &&& \swarrow && \searrow &\cdots & \swarrow & \downarrow \\
|K_C+D|^*&&&&Y = M_{d-4}&&&&& &M_C(2,D)
\end{array}$$
which is the advertised generalization of Theorems ~\ref{old1} and ~\ref{old2}.
\bigskip
\noindent Note: Most of this analysis also applies to curves of genus
$0$ and $1$.
\medskip
{\bf genus 1:} All properties (a)-(j) apply. The only difference between
this and the general case occurs when $d$ is even, in which case
$M_C(2,D)$ is isomorphic to ${\bf P}^1$, rather than a point, as one
would expect by a dimension count. For example, if $d = 4$, then
$\gamma: X = \mbox{bl}({\bf P}^3,C) \rightarrow Y = {\bf P}^1$ is the
contraction determined by the pencil of quadrics vanishing along $C$.
\medskip
{\bf genus 0:} Properties (a)-(j) apply if $d$ is even and
$M_C(2,D)$ is a point, corresponding to the vector bundle
${\mathcal O}_{{\bf P}^1}(\frac d2) \oplus {\mathcal O}_{{\bf P}^1}(\frac d2)$.
On the other hand, if $d = 2n + 1$, then $M_\sigma = \emptyset$ if
$\sigma < 1$, because all bundles are unstable. Other than
this, which forces obvious changes to properties (b),(c),(g) and (j),
everything is as in the general case. Notice that in this case,
$M_1$ is isomorphic to ${\bf P}^n = ({\bf P}^1)_n$, by property (e).
For example, when $d = 5$, then $\gamma: X = \mbox{bl}({\bf P}^3,C)
\rightarrow Y = M_1 = {\bf P}^2$ is the contraction determined by
the web of quadrics vanishing along the twisted cubic. This contraction
is a ${\bf P}^1$ bundle, a special case of the key point of property (g).
\section{Log Flips.}
The goal of this section is to interpret the
birational maps:
$$X = X_1 --\!\!\!> X_2 --\!\!\!> ... --\!\!\!> X_{[\frac {d-1}2]}$$
as flips, in the sense of the minimal model program.
In fact, they are not flips, but rather their {\bf inverses} are flips
(at least initially),
in the traditional sense. While this is an interesting observation,
it is not the one I want to pursue, because the inverses point in the
wrong direction, from the point of view of
Theorems ~\ref{old1} and ~\ref{old2}. For example, with this interpretation, the first contraction
$\gamma:X\rightarrow Y$ is not a flipping contraction, but rather
contracts curves whose intersection with $K_X$ are positive. Such
contractions are hard to understand, in general. Fortunately, the
theory of log minimal models provides a means for turning the flips
around, provided we can find suitable divisors on the $X_k$.
We begin with a quick tour of the parts of the log minimal model program
relevant to our discussion.
\bigskip
Let $X$ be a smooth projective variety.
\begin{definition} A ${{\bf Q}}$-divisor on $X$ is a finite
sum of distinct prime divisors with rational coefficients.
It is effective if all the
coefficients are non-negative. Intersections with curves, self intersections
and numerical equivalence are all defined as with ordinary divisors.
Let $F$ be an effective ${{\bf Q}}$-divisor on $X$. If $F = \sum \alpha_iF_i$,
then the support of $F$, denoted Supp$(F)$, is the union of the prime
divisors $F_i$ which appear in $F$ with positive coefficients.
\end{definition}
\begin{definition} If $F$ is an effective divisor
on $X$, then a {\bf log resolution} of $(X,F)$ is a morphism
$f:\widetilde X \rightarrow X$ with the property that $\widetilde X$ is
smooth, and
$\sum f\mbox{-exceptional divisors}\ + f_*^{-1}(\mbox{Supp}(F))$ is a
normal crossings divisor.
\end{definition}
\noindent Note: If $f:Y \rightarrow X$ is any birational morphism
of smooth varieties, and
if $D$ is a ${{\bf Q}}$-divisor on $X$,
then $E_f$ will denote the sum of the $f$-exceptional divisors, and $f^*(D)$ and
$f_*^{-1}(D)$ will denote the total transform and the strict transform
of $D$ on $Y$, respectively. (They are well-defined by linearity.)
\medskip
Let $F = \sum \alpha_i F_i$ be an effective ${{\bf Q}}$-divisor.
\medskip
\begin{definition} $F$ is {\bf log canonical} if
each coefficient $\alpha_i \le 1$, and there is a log resolution of
$(X,F)$ with the property that all coefficients of the components of $E_f$
are at least $-1$ in the
${{\bf Q}}$-divisor:
$$(K_{\widetilde X} -f^*K_X) + (f_*^{-1}(F) - f^*(F))$$
\end{definition}
\noindent Note: This property is independent of the log resolution.
\begin{definition} Suppose that $B \subset X$ is a curve
(which we also identify with its image in $\mbox{H}_2(X,{\bf R})$). Then
$B$ spans an {\bf extremal ray} of the cone of
effective curves on $X$ if there is an element
$\lambda \in \ \mbox{H}^2(X,{\bf R})$
such that:
(i) $\lambda(B) = 0$ and
(ii) if $\beta \in \ \mbox{H}_2(X,{\bf R})$ is
a limit of sums $\sum c_iB_i$ of curves with positive (real) coefficients,
then $\lambda(\beta) \ge 0$ with equality if and only if $\beta$ is
a multiple of $B$.
\end{definition}
\begin{definition} Suppose that $B$ spans an extremal ray and
$f:X \rightarrow Y$ is a morphism satisfying $f_*({\mathcal O}_X) = {\mathcal O}_Y$.
If $B$ is
contained in a fiber of $f$, and if moreover every curve
contained in every fiber of $f$ is homologous to a (rational) multiple
of $B$, then $f$ (which is uniquely determined if it exists) is called
the {\bf extremal contraction} associated to $B$.
\end{definition}
Suppose that $F$ is a log-canonical divisor on $X$. A basic result
of the log minimal model program is the following (see \cite{CKM} and
\cite{Ketal}):
\bigskip
\noindent {\bf Contraction Theorem:} {\it If $B\subset X$
spans an extremal ray
and $B.(K_X + F) < 0$, then there is an extremal contraction
$\gamma:X \rightarrow Y$ associated to $B$.}
\bigskip
A central question of the minimal model program is:
\bigskip
\noindent {\bf Do Log Flips Exist?:} Suppose the contraction $f:
X \rightarrow Y$ of the theorem is an isomorphism off
codimension $2$ in $X$. Then
does there exist a morphism $f^+:X^+ \rightarrow Y$ with the
following properties:
\medskip
(a) $f^+$ is
an isomorphism off codimension 2 in $X^+$. Let $(K_X + F)^+$ be the
strict transform of $K_X+F$ in $X^+$.
\medskip
(b) If $B^+ \subset X^+$ is a curve lying in a fiber of $f^+$,
then $B^+.(K_X+F)^+ > 0$.
\medskip
(c) The singularities of $X^+$ (or rather, of $(X^+,F^+)$) are
not too bad (for example, so that we can even define the intersections
$B^+.(K_X+F)^+$ in (b)).
\bigskip
When $F = \emptyset$ and the dimension of $X$ is $3$, then
the affirmative answer to this question is a deep theorem of
Mori (together with a definition of ``not too bad'', of course).
The answer is also known to be yes for arbitrary $F$ and dimension
$3$. The interested reader is urged to consult \cite{CKM}
and \cite{Ketal}, as well as Koll\'ar's notes in this Proceedings
for an introduction to the minimal
and log minimal model programs and other applications.
\bigskip
Next, we construct a
morphism which will eventually be
a log resolution.
\bigskip
Let $M$ be a line bundle
on $C$, let $C_k$ be the $k$-th symmetric product of $C$,
and let $V = H^0(C,M)$. If $M$ has the following property:
$$(*)_k: \ \mbox{For all} \ D \in C_k, \ \mbox{dim}(H^0(C,M(-D)) =
\mbox{dim}(V) - k$$
then each such divisor $D$ determines a ${\bf P}^{k-1} \subset {\bf P}(V)$,
which is called the span of $D$. Given
that property $(*)_k$ holds, the $k$-th secant variety is:
$$\Sigma_k(C) = \bigcup _{D\in C_k} \ \mbox{span}(D) \subset {\bf P}(V).$$
If $M$ satisfies $(*)_2$ (i.e. $M$ is very ample),
let $X = \ \mbox{bl}({\bf P}(V),C)$ (as in \S 1).
\bigskip
\noindent Observation: ${\mathcal O}_C(K_C + D)$ satisfies $(*)_{d-1}$.
(Riemann-Roch!)
\medskip
The following construction
blows up the secant varieties of $C$.
\begin{theorem}\label{logres} (\cite{B1} Theorem 1)
(a) Suppose $n \ge 1$ and $M$ is a line bundle
with property $(*)_{2n}$.
Then there is a birational morphism
$f:\widetilde X \rightarrow X$
which is a composition of the following blow-ups:
\medskip
$f^{(2)}: X^{(2)} \rightarrow X^{(1)} = X \ \mbox{blows up the strict transform
of $\Sigma_2(C)$}$,
\medskip
$f^{(3)}:X^{(3)} \rightarrow X^{(2)} \ \mbox{blows up the strict transform
of $\Sigma_3(C)$}$
\medskip
\hskip .5in $\vdots$
$f^{(n)}:\widetilde X = X^{(n)} \rightarrow X^{(n-1)}\ \mbox{blows up
the strict transform
of $\Sigma_n(C)$}$
\medskip
Moreover, the strict transform of each $\Sigma_k(C)$ in $X^{(k-1)}$ is smooth
and irreducible of dimension
$2k-1$, transverse to all exceptional divisors,
so in particular $\widetilde X$ is smooth.
\medskip
For consistency, let $f^{(1)}: X \rightarrow {\bf P}(V)$ be the blow-down.
Let $E^{(k)}$ be the strict transform in $\widetilde X$ of
each $f^{(k)}$-exceptional divisor.
Then $E^{(1)} + ... + E^{(n)}$ is
a normal crossings divisor on $\widetilde X$ with $n$ smooth components.
\medskip
If $M$ is a line bundle
that does not satisfy $(*)_2$,
let $\widetilde X = {\bf P}(V)$. Then:
\medskip
(b) (Terracini recursiveness)
Suppose $k \le n$ and
$x\in \Sigma_k(C) - \Sigma_{k-1}(C)$. Then
the fiber
$$(f^{(k)})^{-1}(x) \subset X^{(k)}$$
is naturally
isomorphic to ${\bf P}(H^0(C,M(-2A)))$, where $A$ is the unique divisor of degree $k$
whose span contains $x$.
Moreover, the fiber
$$f^{-1}(x) \subset E^{(k)} \subset \widetilde X$$
is isomorphic to $\widetilde X_A$,
the variety obtained by applying (a) of the Theorem to
the line bundle $M(-2A)$.
\medskip
(c) If $g \ge 2$ and if $M ={\mathcal O}_C(K_C+D)$, then there is a
{\bf morphism}
$$\widetilde \psi_{|K_C+D|}: \widetilde X \rightarrow M_C(2,D)$$
which extends $\psi_{|K_C+D|}$.
When restricted to a fiber
$f^{-1}(x)$ of part (b), $\widetilde \psi_{|K_C+D|}$ agrees with
$\widetilde \psi_{|K_C+D-2A|}$ (and this property determines
$\widetilde \psi_{|K_C+D|}$ uniquely!)
\end{theorem}
For the proof, see \cite{B1}. Notice that
parts (a) and (b) make
no reference to moduli, hence generalize to, for example, canonical
embeddings, where condition $(*)_d$ is equivalent to the nonexistence
of $g^1_d$'s. As for part (c), the idea is to construct a vector bundle
on $C \times \widetilde X$ by a sequence of elementary modifications
of the bundle (constructed from the universal
extension) on $C \times |K_C+D|^*$
along the exceptional divisors for each $f^{(k)}$, and to use
this bundle to get the map to moduli.
\medskip
In fact, though, the proof really constructs families of nondegenerate pairs
$({\mathcal O}_C,f,E)$ parametrized by the $\widetilde X^{(k)}$ (in all genera)
with the following property. For every $y\in \widetilde X$ and every
$\sigma \in [0,d)$
(or $[1,d)$ if $g = 0$ and $d$ is odd) there is an $X^{(k)}$ such that
the image of $y$ in $X^{(k)}$ parametrizes a $\sigma$-semistable pair.
Thus, for each $\sigma$, there is a natural morphism:
$$\psi_\sigma:\widetilde X \rightarrow M_\sigma.$$
\bigskip
Now we construct log-canonical divisors on $X = \ \mbox{bl}(|K_C+D|^*,C)$.
\medskip
\noindent {\bf Linear Algebra Construction:} Given any vector bundle $F$ on $C$,
the cup product gives rise to a linear map:
$$c: \ \mbox{Ext}^1(F(D),F) \rightarrow
\ \mbox{Hom}(H^0(C,F(D)),H^1(C,F))$$
Also, the summand ${\mathcal O}_C \hookrightarrow F \otimes F^*$
produces an inclusion of vector spaces:
$$\iota: \ \mbox{Ext}^1({\mathcal O}_C(D),{\mathcal O}_C) \hookrightarrow
\ \mbox{Ext}^1(F(D),F)$$
One can think of the composition $c\circ \iota$
pointwise as follows. Given
$$\epsilon:\ 0 \rightarrow {\mathcal O}_C \rightarrow E \rightarrow {\mathcal O}_C(D)
\rightarrow 0$$
one tensors each term by $F$, and $c(\iota(\epsilon))$ is the
connecting homomorphism:
$$c(\iota(\epsilon)) = \delta: H^0(C,F(D)) \rightarrow H^1(C,F).$$
When we lift $c\circ \iota$ to a map of trivial bundles
on $|K_C + D|^*$,
it determines a matrix $M(F)$ of linear forms on $|K_C+D|^*$ via:
$${\mathcal O}_{|K_C+D|^*}(-1) \rightarrow \ \mbox{Hom}(H^0(C,F(D)),H^1(C,F))
\otimes {\mathcal O}_{|K_C+D|^*}.$$
\begin{proposition}\label{linalg} (a) For each $0 < k \le \frac d2$, there is a
nonempty open subset $U \subset \ \mbox{Pic}^{k-(g-1)}(C)$ such that
$$(*)\ \ L \in U \Rightarrow \ h^0(C,L^{-1}(D)) = d-k \ \mbox{and}\
h^1(C,L^{-1}) = k.$$
If $L \in U$, choose a basis for $H^0(C,L^{-1}(D))$, and let
$I = (i_1,...,i_k)$ be a mutiindex with $1 \le i_1 < ... < i_k \le d-k$.
Then $I$ determines a $k\times k$ minor $M_I(L^{-1})$
(choosing columns $i_1,...,i_k$ from the matrix $M(L^{-1})$) yielding a
divisor on $\widetilde X$:
$$D_{L,I} \in
|kH - (k-1)E^{(1)} - (k-2)E^{(2)} - ... - E^{(k-1)}|.$$
(i.e. the generic multiplicity of $\mbox{det}(M_I(L^{-1}))$
along $\Sigma_i(C)$ is at least $k-i$).
\medskip
Finally, if we let $V_k$ be the sub-linear-series spanned by the $D_{L,I}$,
then $V_k$ is base-point-free (and independent of choices of basis).
\medskip
(b) Suppose $g > 0$. Then for each $0 < l \le d$, there is a nonempty
open subset $U \subset M_C(2,l-(2g-2))$ such that
$$(*)\ \ F \in U \Rightarrow \ h^0(C,F^{-1}(D)) = 2d - l\ \mbox{and}\
h^1(C,F^{-1}) = l.$$
If $F \in U$, choose $J = (j_1,...,j_l)$ such that
$1 \le j_1 < ... < j_l \le d-l$ and the minor $M_J(F^{-1})$ as in (a). Then
det$(M_J(F^{-1}))$ determines a divisor:
$$D_{F,J} \in |lH - (l-2)E^{(1)} - (l-4)E^{(2)} - ... |,$$
and the sub-linear-series $W_l$
spanned by the $D_{F,J}$ is base-point-free.
\end{proposition}
{\bf Proof:} The values for
$h^0$ and $h^1$ in $(*)$ are generic in Pic and $M_C(2,*)$
respectively. $U$ is
an intersection of
two nonempty open subsets.
For the next part, the following observation is crucial. Given
an effective divisor $A$ on $C$, an extension
$\epsilon \in \ \mbox{Ext}^1({\mathcal O}_C(D),{\mathcal O}_C)$ determines
a point $\overline \epsilon \in \ \mbox{span}(A) \subset |K_C+D|^*$
if and only if the extension splits when pushed forward:
$$\begin{array}{ccccccccc}0 & \rightarrow & {\mathcal O}_C & \rightarrow
& E & \rightarrow & {\mathcal O}_C(D) & \rightarrow & 0 \\
&&\downarrow & \swarrow \\ &&{\mathcal O}_C(A) \end{array}$$
Now suppose $\overline \epsilon \in \Sigma_i(C)$, so is in the
span of some divisor $A$ of degree $i < k$. It follows
(tensoring the inclusion ${\mathcal O}_C(D-A) \hookrightarrow E$
by $L^{-1}$) that
$H^0(C,L^{-1}(D-A)) \subset \ \mbox{ker}(c(\iota(\epsilon)))$, which
by Riemann-Roch has dimension at least $(d-k)-i$. Thus the rank of each
$M_I(L^{-1})$ is at most $i$, and therefore its determinant has
multiplicity at least $k-i$ at $\overline \epsilon$, from which the
linear series computation in (a) follows. The linear series in (b) is
computed similarly.
\medskip
We prove base-point-freeness first when $g=0$ and $d = 2n+1$.
Given $k$, if
$y \in \widetilde X -
(E^{(1)} \cup ...\cup E^{(k-1)})$, then
the bundle $E$ associated to $\overline \epsilon = y$
is isomorphic
to ${\mathcal O}_{{\bf P}^1}(m) \oplus {\mathcal O}_{{\bf P}^1}(d-m)$ where
$d-m > m \ge k$. This
is because of the crucial observation.
It follows that $h^0(C,E(-k-1)) = d-2k$, so some
$M_I({\mathcal O}_{{\bf P}^1}(-k-1))$ has full rank at $\overline \epsilon$,
and thus $y$ is not a base point. If
$y\in E^{(i)}$ for some (minimal) $i < k$, then Theorem ~\ref{logres} (b) allows us to
place $y$ in a fiber over $\Sigma_i(C)$ isomorphic to
$\widetilde X_A$ for some
divisor $A$ satisfying deg$(A) = i$. The restriction
to this fiber of $V_k$ is identified with the linear series $V_{k-i}$
under the isomorphism with $\widetilde X_A$, and so we can conclude
base-point-freeness by induction.
\medskip
The proof of base-point-freeness is similar in general. Suppose
$d = 2n+1$ or $2n+2$, so ${\mathcal O}_C(K_C+D)$ satisfies $(*)_{2n}$.
Given $k$ (or $l = 2k$ or $2k-1$), first consider the points
$y \in \widetilde X - (E^{(1)} \cup ... \cup E^{(k-1)})$.
If $y \in E^{(k)} \cup ... \cup E^{(n)}$, then
we can
find an $L \in U$ (or $F\in U$) such that
$h^0(C,E\otimes L^{-1}) = d - 2k$ (or $h^0(C,E\otimes F^{-1}) = 2d - 2l$)
because the bundle $E$ associated to $\overline \epsilon = f(y)$
fits in an exact sequence:
$0 \rightarrow {\mathcal O}_C(D-A) \rightarrow E
\rightarrow {\mathcal O}_C(A) \rightarrow 0$
($k \le \ \mbox{deg}(A) \le n$).
(This is a consequence of the crucial observation.)
If $y$ does not
lie in an exceptional divisor, then the bundle $E$ associated to the extension
$\overline \epsilon = y$ is semistable
(Theorem ~\ref{logres} (c) or the crucial observation) in which case
the same fact about $h^0(C,E\otimes L^{-1})$ (or $h^0(C,E\otimes F^{-1})$)
is a standard result,
for example, see \cite {B3}, Lemma 3.6. Once this is
achieved,
one has base-point-freeness off the exceptional
divisors $E^{(1)} \cup ... \cup E^{(k-1)}$ and
the points of these exceptional divisors
are treated by induction using Theorem ~\ref{logres} (b) and the same identification
of $V_k$ with $V_{k-i}$ (or $W_l$ with $W_{l-2i}$) as above.
\bigskip
There are two exceptional cases (given in detail below) where the linear series
($V_k$ in case (a) and $W_l$ in case (b)) are trivial, which occur when
$\Sigma_n(C)$ is a divisor and $k$ (or $l$) is maximal. In all other cases,
we can use Bertini to find smooth members of the linear series which meet
all the exceptional divisors $E^{(1)},...,E^{(n)}$ transversally.
\bigskip
\noindent {\bf Exceptional Cases:} (a) Suppose $g = 0$ and $d = 2n+2 = 2k$. Then
there is only one $k\times k$ matrix $M_I({\mathcal O}_{{\bf P}^1}(-k-1))$,
and $V_k$ has only one element. (So since it is base-point-free, it has
to be trivial!)
With a suitable choice of basis,
$M_I({\mathcal O}_{{\bf P}^1}(-k-1))$) is the
standard square matrix:
\medskip
$$\left(\begin{array}{ccccc}z_1 & z_2 & z_3 & \cdots & z_k \\
z_2 & z_3 & z_4 & \cdots & z_{k+1} \\
\vdots & \vdots & \vdots & & \vdots \\
z_k & z_{k+1} & z_{k+2} & \cdots & z_{d-1}\end{array}\right)$$
\medskip
\noindent whose $2\times 2$ minors cut out the rational normal curve
$C \subset {\bf P}^{d-2} = |K_C+D|^*$ (see Proposition 9.7 in \cite{H}).
Its determinant cuts out $\Sigma_n(C) \subset {\bf P}^{d-2}$.
The linear series $V_k$ is trivial because here
$E^{(n)} \equiv nH - (n-1)E^{(1)} - ... - 2E^{(n-1)}$ on $\widetilde X$.
Notice that although we cannot use Bertini to find smooth divisors
in $V_k$, we can use Theorem ~\ref{logres} (a) to conclude that $E^{(n)}$ itself
is smooth and meets the other exceptional divisors transversally.
\medskip
(b) Suppose $g = 1$, and $d = l = 2n+1$. Then there is
one $l\times l$ matrix $M_J(F^{-1})$ for each stable bundle $F$
of rank $2$ and degree $d = d-(2g-2)$. There is no reason a priori
why this determines a trivial linear series,
however, as in Exception (a), one computes that
$\Sigma_n(C) \subset |K_C+D|^*$ is a divisor, of degree $d$,
which must therefore be the zero locus of each determinant,
and $E^{(n)} \equiv dH - (d-2)E^{(1)} - ... - 3E^{(n-1)}$.
(The degree can be computed using Lemma 2.5 (Chapter VIII)
from \cite{ACGH}). Again, we will use the fact that $E^{(n)}$
is smooth, intersecting the other
exceptional divisors transversally.
\bigskip
\noindent {\bf Remarks:} If $l$ is even, the conditions
$h^0(C,F^{-1}(D)) = 2d-l$ and $h^0(C,F^{-1}) = l$ may not be
independent of the choice of a representative $F$
for a semistable point in $M_C(2,l-(2g-2))$. However, if these properties are
true for the associated graded, then they hold for all representatives,
as is easily checked.
Moreover, if we let
$l = 2k$, then the split bundles determine an inclusion of
linear series: $V_k \cdot V_k \subseteq W_l$. (In genus $0$ and $1$, this is
an equality!)
\bigskip
Next, we use the linear series to find:
\bigskip
\noindent {\bf Some Log Canonical Divisors on $X$:} Let $F = f_*A$, where:
\medskip
\noindent {\bf Genus 0:} (a) If $d = 2n+1$, then $A \in W_{2n}$
is a general member.
\medskip
(b) If $d = 2n+2$, then $A = E^{(n)} + A'$, where
$A' \in V_{n}$ is a general member.
\medskip
\noindent {\bf Genus 1:} (a) If $d = 2n+1$, then $A = E^{(n)}$.
\medskip
(b) If $d = 2n+2$, then $A \in W_d$ is a general member.
\medskip
\noindent {\bf Genus $\ge$ 2:} $A \in W_d$ is general.
\bigskip
Alternatively, one can think of $F$ in each case
as the strict transform in $X$ of a
hypersurface in $|K_C+D|^*$ (highly singular along the secant varieties).
However, when we think of $F$ as the push-forward of a divisor $A$
on $\widetilde X$, then
the following becomes almost immediate.
\bigskip
\noindent {\bf Claim:} In all the cases above, $f:\widetilde X \rightarrow X$ is
a log resolution of $(X,F)$ and $F$ is a log canonical divisor.
\medskip
{\bf Proof:} By Theorem ~\ref{logres} (a), all the $f$-exceptional divisors
are smooth with normal crossings. In each case, the strict transform of
the support of $F$
is the support of $A$, which is a sum of smooth divisors which
intersect all others with normal
crossings, either by Bertini or Theorem ~\ref{logres}(a) again. So $f$ is a
log resolution of $(X,F)$.
\medskip
Since each blow-up $f^{(k)}$ was along a smooth center trasnverse to all
exceptional divisors, it follows that the coefficient of $E^{(k)}$ in
$K_{\widetilde X} - f^*K_X$ is the codimension of
$\Sigma_k(C)$ in $|K_C+D|^*$ minus $1$, a consequence
of Riemann-Hurwitz. It also follows that since we constructed
$F$ as $f_*A$, the coefficient
of $E^{(k)}$ in $f_*^{-1}F - f^*F = A-f^*F$ is the (negative of the)
generic multiplicity
of $F$ along the strict transform of $\Sigma_k(C)$ in $X$, which
is computed directly from the linear series in which $A$
lies.
This is the information we need to check that $F$ is log canonical.
The computations in genus $0$ are left to the reader. Here
is the data for genus $\ge 1$:
\medskip
$\bullet$ \ codimension of $\Sigma_k(C)$ in $|K_C+D|^*$:\ \ $d + g - 2k -1$.
\medskip
$\bullet$ \ multiplicity of $F$ along $E^{(k)}$:\ \ $d - 2k$.
\medskip
Since it follows that the coefficient of each $E^{(k)}$ in
$(K_{\widetilde X} - f^*K_X)
+ (A - f^*F)$ is $g-2$, we see that $F$ is log canonical.
\medskip
We can (and need to!) do a little better when $g \ge 2$
if we use ${{\bf Q}}$-divisors. If $p,q$ are positive integers, let
$(W_d)^{p}$ be the linear series spanned by products of $p$ elements of $W_d$,
and given a smooth element $G \in (W_d)^p$ (this linear series is
base-point free), consider $F' = \frac 1qf_*(G)$. This is
not only numerically equivalent to $\frac pq F$ (as is easy to see), but
all coefficients of the $E^{(k)}$ in the expression
$f_*^{-1} F' - f^*F'$
are $\frac pq$ times the corresponding coefficients for $F$.
We will abuse notation and say that this divisor is a member of $\frac pq F$, keeping in
mind that if $p > q$, then the literal ${{\bf Q}}$-divisor $\frac pq F$
cannot be log canonical, by definition, while a member
constructed in this way might be log canonical. In fact,
if $d > 4$, then
$$\left(\frac{d+g-5}{d-4}\right)F \ \mbox{has a log canonical member}$$
by the data above (keep in mind that $E^{(1)}$ is not
$f$-exceptional).
\bigskip
Now we will relate these log canonical divisors to
the diagram at the end of \S 2 constructed by stable pairs. Namely, recall that
whenever $d > 2k$, there was a diagram:
$$\begin{array}{ccccc} X_{k-1} & & & & X_k \\
&\stackrel{f^+}\searrow \ & & \stackrel{f^-}\swarrow \\ & &
M_{d-2k}\ \ \ \ \end{array}$$
Moreover, $f^-$ and $f^+$ are obviously extremal ray contractions
since each contracts a projective bundle over $C_k$ and
dim$(H_2(X,{\bf R})) = 2$. (Take any
curve in a projective-space fiber to span the extremal ray.)
Finally, each contraction is an isomorphism off of codimension $2$.
\bigskip
\begin{proposition}\label{logflip} If $k = 2$ or $d > 2g -2$ and $k$ is arbitrary,
then the diagram above
is a log flip for (the strict transform on $X_{k-1}$ of)
$K_X + \left(\frac {d+g-5}{d-4}\right)F$.
\end{proposition}
{\bf Proof:} We need to show: (a) the member of
$\left(\frac {d+g-5}{d-4}\right)F$
constructed as above is log canonical
on each $X_{k}$ (this will certainly suffice for the condition
``not too bad'' in the ``definition'' of log flips), and
(b) If $B \subset X_{k-1}$ and
$B^+ \subset X_k$ are curves spanning extremal rays, then
$B.(K_X+\left(\frac {d+g-5}{d-4}\right)F) < 0$
and $B^+.(K_X+\left(\frac {d+g-5}{d-4}\right)F) > 0$.
\bigskip
We prove (b) first.
Recall (property (i) from \S 2) that the map $f^+$ is a multiple of
$|kH - (k-1)E|$. Thus, if $B \subset X_{k-1}$ is an
extremal ray, then $B.(H - \frac{k-1}kE) = 0$. Moreover,
$|(k-1)H - (k-2)E|$ is nef on $X_{k-1}$, and $B$ is
{\bf not} contracted in this linear series, so it follows that
$B.(H-\frac{k-2}{k-1}E) > 0$, and $B.E > 0$.
\medskip
\noindent From the data:
\medskip
$K_X \equiv -(d+g-1)H + (d+g-4)E$, and
\medskip
$F \equiv dH - (d-2)E$,
\medskip
\noindent we get $K_X+\left(\frac{d+g-5}{d-4}\right)F \equiv \frac {4g-4}{d-4}(H - \frac{d+2g-6}{4g-4}E)$,
from which it follows that its intersection with $B$ is negative
when $k = 2$ or
$d > 2g-2$. Moreover, the mirror
image of this argument shows that if $B^+ \subset X_k$ is
an extremal ray for $f^-$, then its intersection
with $K_X+\left(\frac{d+g-5}{d-4}\right)F$ is positive in the same cases.
\medskip
So the only thing left to see is the fact that
the member of $\left(\frac {d+g-5}{d-4}\right)F$ we constructed is
log canonical
on all $X_k$, not just $X = X_1$, as was shown earlier. In fact, I claim
a stronger result, which will explain all the maps as log flips:
\begin{lemma} If $d > 2k$, then a general member of
$\left(\frac{d+g-2k-1}{d-2k}\right)F$ is
log canonical on $X_{k-1}$.
\end{lemma}
{\bf Proof:} The construction is as before, pushing down
a general element of $(W_d)^{d+g-2k-1}$ and dividing by
$d-2k$.
After $k-1$ elementary modifications, the
proof of Theorem ~\ref{logres}(c) (see \cite{B1}) produces a family
of $\sigma = n-2k+1$-stable pairs on $C$ parametrized by $X^{(k-1)}$, hence
a morphism $\psi^{(k-1)}:X^{(k-1)} \rightarrow X_{k-1}$ since
$X_{k-1} = M_\sigma$. Moreover,
the morphism $\psi_\sigma : \widetilde X \rightarrow X_{k-1}$ factors
through $\psi^{(k-1)}$ via the composition of blow-downs
$f_{k-1}:\widetilde X \rightarrow X^{(k-1)}$ from Theorem ~\ref{logres}(a).
\medskip
One checks that if
$G \in (W_d)^p$ (for any $p > 0$), then
$(f_{k-1})_*G$ descends to a divisor on $X_{k-1}$.
Thus when we log resolve $(X_{k-1},\frac pq F)$ by the map
$\psi_\sigma$, then only the
exceptional divisors $E^{(k)}$ and above appear with a nonzero
coefficient in $\psi_\sigma^*\frac pq F - (\psi_\sigma)_*^{-1}(\frac pq F)$,
and those, it is easy to see, appear with the same coefficients
as in the earlier computation. The lemma immediately follows.
\medskip
\begin{corollary}\label{logcan} Each rational map $X_{k-1} --\!\!\!> X_{k}$ is
a log flip.
\end{corollary}
{\bf Proof:} Using the log canonical divisor on $X_{k-1}$ from the Lemma:
$$K_X + \left(\frac{d+g-2k-1}{d-2k}\right)F \equiv
\frac {k(2g-2)}{d-2k}\left(H - \frac{(2k-1)(2g-4) + d-2}{k(2g-2)}E\right)$$
has negative intersection with $B$ and positive intersection
with $B^+$ (as in the proof of Proposition ~\ref{logflip}, keeping in
mind the fact that $d > 2k$).
\bigskip
\noindent {\bf Final Remark:} I have split off Proposition ~\ref{logflip}
from Corollary ~\ref{logcan} (which is in a sense more powerful!) to point
out a curious fact. Namely, if $d > 2g-2$, which is precisely when
$\psi_{|K_C+D|}:|K_C+D|^* --\!\!\!>
M_C(2,D)$ is dominant, then we can construct a single ${{\bf Q}}$-divisor on $X$
for which all the maps $X_{k-1} --\!\!\!> X_k$ simultaneously become log flips.
When $d \le 2g-2$, however, one needs to tailor the divisor to
the variety $X_{k-1}$ and in fact it seems that no single ${{\bf Q}}$-divisor
on $X$ will be log canonical and have the desired intersection properties with
all the extremal rays. (At least the linear algebra construction does not
produce such a divisor.)
\medskip
\noindent {\bf Acknowledgements:} I would like to thank Michael Thaddeus
and the referee for their careful reading and useful comments on an
earlier version of this paper.
|
1998-03-25T20:05:02 | 9707 | alg-geom/9707010 | en | https://arxiv.org/abs/alg-geom/9707010 | [
"alg-geom",
"math.AG"
] | alg-geom/9707010 | Michael Finkelberg | Michael Finkelberg, Ivan Mirkovi\'c (Independent University of Moscow
and University of Massachusetts at Amherst) | Semiinfinite Flags. I. Case of global curve $P^1$ | References updated | null | null | null | null | The Semiinfinite Flag Space appeared in the works of B.Feigin and E.Frenkel,
and under different disguises was found by V.Drinfeld and G.Lusztig in the
early 80-s. Another recent discovery (Beilinson-Drinfeld Grassmannian) turned
out to conceal a new incarnation of Semiinfinite Flags. We write down these and
other results scattered in folklore. We define the local semiinfinite flag
space attached to a semisimple group $G$ as the quotient $G((z))/HN((z))$ (an
ind-scheme), where $H$ and $N$ are a Cartan subgroup and the unipotent radical
of a Borel subgroup of $G$. The global semiinfinite flag space attached to a
smooth complete curve $C$ is a union of Quasimaps from $C$ to the flag variety
of $G$. In the present work we use $C=P^1$ to construct the category $PS$ of
certain collections of perverse sheaves on Quasimaps spaces, with factorization
isomorphisms. We construct an exact convolution functor from the category of
perverse sheaves on affine Grassmannian, constant along Iwahori orbits, to the
category $PS$. Conjecturally, this functor should correspond to the restriction
functor from modules over quantum group with divided powers to modules over the
small quantum group.
| [
{
"version": "v1",
"created": "Wed, 9 Jul 1997 17:31:57 GMT"
},
{
"version": "v2",
"created": "Wed, 25 Mar 1998 19:05:02 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Finkelberg",
"Michael",
"",
"Independent University of Moscow\n and University of Massachusetts at Amherst"
],
[
"Mirković",
"Ivan",
"",
"Independent University of Moscow\n and University of Massachusetts at Amherst"
]
] | alg-geom | \section{Introduction}
\subsection{} We learnt of the {\em Semiinfinite Flag Space} from B.Feigin
and E.Frenkel in the late 80-s. Since then we tried to understand this
remarkable object. It appears that it was essentially constructed, but
under different disguises, by V.Drinfeld and G.Lusztig in the early 80-s.
Another recent discovery ({\em Beilinson-Drinfeld Grassmannian}) turned out
to conceal a new incarnation of Semiinfinite Flags.
We write down these and other results scattered in the folklore.
\subsection{}
Let $\bG$ be an almost simple simply-connected group with
a Cartan datum $(I,\cdot)$ and a simply-connected simple
root datum $(Y,X,\ldots)$ of finite type as in ~\cite{l}, ~2.2.
We fix a Borel subgroup $\bB\subset\bG$, with a Cartan subgroup
$\bH\subset\bB$, and the unipotent radical $\bN$.
B.Feigin and E.Frenkel define the Semiinfinite Flag Space
$\CZ$ as the quotient of
$\bG((z))$ modulo the connected component of $\bB((z))$
(see ~\cite{ff}).
Then they study the category $\PS$ of perverse sheaves on $\CZ$ equivariant
with respect to the Iwahori subgroup $\bI\subset \bG[[z]]$.
In the first two chapters we are trying to make sense of this definition.
We encounter a number of versions of this space.
In order to give it a structure of an ind-scheme, we define the (local)
semiinfinite flag space as
$
\widetilde{\bf Q}=
\bG((z))/\bH\bN((z))$ (see section 4).
The (global) semiinfinite space
attached to a smooth complete curve $C$ is the system of
varieties $\CQ^\al$ of ``quasimaps'' from
$C$ to the flag variety of $\bG$
--- the Drinfeld compactifications of the degree $\al$ maps.
In the present work we restrict ourselves to the case $C=\BP^1$.
The main incarnation of the semiinfinite flag space in this paper
is a collection $\CZ$ (for {\em zastava})
of (affine irreducible finite dimensional) algebraic varieties
$\CZ^\alpha_\chi\sub \CQ^\al$, together with certain closed embeddings and
{\em factorizations}.
Our definition of $\CZ$ follows the scheme suggested by G.Lusztig in
~\cite{l2}, \S11: we approximate the ``closures'' of Iwahori orbits by their
intersections with the transversal orbits of the opposite Iwahori subgroup.
However, since the set-theoretic intersections of the above
``closures'' with the opposite Iwahori orbits can not be equipped with the
structure of algebraic varieties, we postulate $\CZ^\alpha_\chi$
for the ``correct'' substitutes of such intersections.
Having got the collection of $\CZ^\alpha_\chi$ with factorizations, we
imitate the construction of ~\cite{fs}
to define the category $\PS$ (for
{\em polubeskrajni snopovi})
of certain collections of perverse sheaves with $\BC$-coefficients
on $\CZ^\alpha_\chi$ with {\em factorization isomorphisms}. It is defined
in chapter 2; this category is the main character of the present work.
\subsection{}
\label{quantum}
If $\bG$ is of type $A,D,E$ we set $d=1$; if $\bG$ is of type
$B,C,F$ we set $d=2$; if $\bG$ is of type $G_2$ we set $d=3$. Let $q$
be a root of unity of sufficiently large degree $\ell$ divisible by $2d$.
Let $\fu$ be the small (finite-dimensional) quantum group associated to $q$
and the root datum $(Y,X,\ldots)$ as in ~\cite{l}.
Let $\CC$ be the category of $X$-graded $\fu$-modules
as defined in ~\cite{ajs}. Let $\CC^0$ be the block of $\CC$ containing the
trivial $\fu$-module. B.Feigin and G.Lusztig conjectured (independently)
that the category $\CC^0$ is equivalent to $\PS$.
Let $\fU\supset\fu$ be the quantum group with divided powers associted to $q$
and the root datum $(Y,X,\ldots)$ as in ~\cite{l}. Let $\fC$ be the category
of $X$-graded finite dimensional $\fU$-modules, and let $\fC^0$ be the block
of $\fC$ containing the trivial $\fU$-module.
The works ~\cite{kl}, ~\cite{l4} and
~\cite{kt} establish an equivalence of $\fC^0$ and the category $\CP(\CG,\bI)$.
Here $\CG$ denotes the affine Grassmannian $\bG((z))/\bG[[z]]$, and
$\CP(\CG,\bI)$ stands for the category of perverse sheaves
on $\CG$ with finite-dimensional support constant along the orbits of $\bI$.
\subsection{}
\label{quantum res}
The chapter 3 is devoted to the construction of the {\em convolution} functor
$\bc_\CZ:\ \CP(\CG,\bI)\lra\PS$
which is the geometric counterpart
of the restriction functor from $\fC^0$ to $\CC^0$, as suggested by V.Ginzburg
(cf. ~\cite{gk} ~\S4).
One of the main results of this chapter is the Theorem ~\ref{Satake} which is
the sheaf-theoretic version of the classical Satake isomorphism. Recall that
one has a {\em Frobenius homomorphism} $\fU\lra U(\fg^L)$ (see ~\cite{l})
where $U(\fg^L)$ stands for the universal enveloping algebra of the Langlands
dual Lie algebra $\fg^L$. Thus the category of finite dimensional
$\bG^L$-modules is naturally embedded into $\fC$ (and in fact, into $\fC^0$).
On the geometric level this corresponds to the embedding $\CP(\CG,\bG[[z]])
\subset\CP(\CG,\bI)$. The Theorem ~\ref{Satake} gives a natural interpretation
(suggested by V.Ginzburg)
of the weight spaces of $\bG^L$-modules in terms of the composition
$$\bG^L-mod\simeq\CP(\CG,\bG[[z]])\subset\CP(\CG,\bI)\stackrel{\bc_\CZ}{\lra}
\PS.$$
\subsection{}
Let us also mention here the following conjecture which might be known to
specialists (characteristic $p$ analogue of conjecture in ~\ref{quantum}).
Let $\bG^L$ stand for the Langlands dual Lie group. Let
$p$ be a prime number bigger than the Coxeter number of $\fg^L$, and let
$\overline\BF_p$ be
the algebraic closure of finite field $\BF_p$.
Let $\fC_p$ be the category of algebraic $\bG^L(\overline\BF_p)$-modules,
and let $\fC^0_p$
be the block of $\fC_p$ containing the trivial module.
Let $\CC_p$ be the category of graded modules over the Frobenius kernel of
$\bG^L(\overline\BF_p)$, and let $\CC^0_p$
be the block of $\CC_p$ containing the trivial module (see ~\cite{ajs}).
Finally, let $\PS_p$ be the category of snops {\em with
coefficients in} $\overline\BF_p$, and let $\CP(\CG,\bI)_p$ be the category
of perverse sheaves on $\CG$ constant along $\bI$-orbits {\em with coefficients
in} $\overline\BF_p$. Then the categories $\CC^0_p$ and $\PS_p$
are equivalent, the categories $\fC^0_p$ and $\CP(\CG,\bI)_p$ are equivalent,
and under these equivalences the restriction functor $\fC^0_p\lra\CC^0_p$
corresponds to the convolution functor $\CP(\CG,\bI)_p\lra\PS_p$ (cf.
~\ref{quantum res}).
The equivalence $\CP(\CG,\bI)_p\iso\fC^0_p$ should be an extension of
the equivalence between $\CP(\CG,\bG[[z]])_p\subset\CP(\CG,\bI)_p$ and
the subcategory of $\fC^0_p$ formed by the
$\bG^L(\overline\BF_p)$-modules which factor through the Frobenius homomorphism
$Fr:\ \bG^L(\overline\BF_p)\lra\bG^L(\overline\BF_p)$. The latter equivalence
is the subject of forthcoming paper of K.Vilonen and the second author.
\subsection{}
The Zastava space $\CZ$ organizing all the ``transversal slices"
$\CZ^\alpha_\chi$ may seem cumbersome.
At any rate the existence of various models of the slices
$\CZ^\alpha_\chi$
(chapter 1), is undoubtedly beautiful by itself. Some of the
wonderful properties of $\CQ^\al$ and $\CZ^\alpha_\chi$
are demonstrated in ~\cite{ku}, ~\cite{fk}, ~\cite{fkm} in the case
$\bG=SL_n$. We expect all these properties to hold for the general $\bG$.
\subsection{}
To guide the patient reader through the
notation, let us list
the key points of this paper. The Theorem ~\ref{Z} identifies the
different models of $\CZ^\alpha_\chi$ (all essentially due to V.Drinfeld)
and states the factorization property. The exactness of the convolution
functor $\bc_\CZ:\ \CP(\CG,\bI)\lra\PS$ is proved in the
Theorem ~\ref{tough} and Corollary ~\ref{bunk}. The Theorem ~\ref{Satake}
computes the value of the convolution functor on $\bG[[z]]$-equivariant sheaves
modulo the parity vanishing conjecture ~\ref{parity}.
\subsection{}
In the next parts we plan to study
$D$-modules on the local variety $\widetilde{\bf Q}$
(local construction of the category $\PS$, global sections as modules over
affine Lie algebra $\hat\fg$,
action of the affine Weyl group by Fourier transforms),
the relation of the local and global
varieties (local and global Whittaker sheaves, a version of the
convolution functor twisted by a character of $N((z))$),
and the sheaves on Drinfeld compactifications of maps
into partial flag varieties.
\subsection{} The present work owes its very existence to V.Drinfeld. It
could not have appeared without the generous help of many people who shared
their ideas with the authors.
Thus, the idea of {\em factorization} (section 9)
is due to V.Schechtman.
A.Beilinson and V.Drinfeld taught us the
{\em Pl\"ucker} picture of the (Beilinson-Drinfeld)
affine Grassmannian
(sections 6 and 10).
G.Lusztig has computed the local singularities of the
Schubert strata closures in the spaces $\CZ^\alpha_\chi$ (unpublished,
cf ~\cite{l1}).
B.Feigin and V.Ginzburg taught us their understanding of
the Semiinfinite Flags for many years (in fact, we learnt of Drinfeld's
Quasimaps' spaces from V.Ginzburg in the Summer 1995).
A.Kuznetsov was always
ready to help us whenever we were stuck in the geometric problems (in fact,
for historical reasons, the section 3 has a lot in common with ~\cite{ku} \S1).
We have also benefited from the discussions with R.Bezrukavnikov and
M.Kapranov.
Parts of this work were done while the authors were enjoying
the hospitality and support of the University of Massachusetts at Amherst,
the Independent Moscow University and the Sveu\v{c}ili\v{s}te u Zagrebu.
It is a great pleasure to thank these institutions.
\section{Notations}
\subsection{}
\label{group}{\bf Group $\bG$ and its Weyl group $\CW_f$.}
We fix a Cartan datum $(I,\cdot)$ and a simply-connected simple
root datum $(Y,X,\ldots)$ of finite type as in ~\cite{l}, ~2.2.
Let $\bG$ be the corresponding simply-connected almost
simple Lie group with the Cartan subgroup $\bH$ and the Borel subgroup
$\bB\supset \bH$ corresponding to the set of simple roots $I\subset X$.
We will denote by $\CR^+\subset X$ the set of positive roots.
We will denote by $2\rho\in X$ the sum of all positive roots.
Let $\bB_+=\bB$ and let $\bB_-\supset \bH$ be the opposite Borel subgroup.
Let $\bN$ (resp. $\bN_-$)
be the radical of $\bB$ (resp. $\bB_-$).
Let $\bbH=\bB/\bN=\bB_-/\bN_-$ be the abstract
Cartan group. The corresponding Lie algebras are denoted, respectively,
by $\fb,\fb_-,\fn,\fn_-,\fh$.
Let $\bX$ be the flag manifold $\bG/\bB$, and let $\bA=\bG/\bN$ be the
principal affine space. We have canonically
$H_2(\bX,\BZ)=Y;\ H^2(\bX,\BZ)=X$.
For $\nu\in X$ let $\bL_\nu$ denote the corresponding $\bG$-equivariant
line bundle on $\bX$.
Let $\CW_f$ be the Weyl group of $\bG$.
We have a canonical bijection $\bX^\bH=\CW_f$
such that the neutral element $e\in \CW_f=\bX^\bH\subset\bX$ forms a single
$\bB$-orbit.
We have a Schubert stratification of $\bX$ by $\bN$- (resp. $\bN_-$-)orbits:
$\bX=\sqcup_{w\in \CW_f}\bX_w$ (resp. $\bX=\sqcup_{w\in \CW_f}\bX^w)$ such that
for $w\in \CW_f=\bX^\bH\subset\bX$ we have $\bX^w\cap\bX_w=\{w\}$.
We denote by $\ol\bX_w$ (resp. $\ol\bX^w$) the Schubert variety --- the
closure of $\bX_w$ (resp. $\bX^w$). Note that $\ol\bX_w=\sqcup_{y\leq w}
\bX_y$ while $\ol\bX^w=\sqcup_{z\geq w}\bX^z$ where $\leq$ denotes the
standard Bruhat order on $\CW_f$.
Let $e\in \CW_f$ be the shortest element (neutral element),
let $w_0\in \CW_f$ be
the longest element, and let $s_i,\ i\in I$,
be the simple reflections in $\CW_f$.
\subsection{}
\label{reps} {\bf Irreducible representations of $\bG$.}
We denote by $X^+$ the cone of positive weights (highest weights of finite
dimensional $\bG$-modules). The fundamental weights $\omega_i:\
\langle i,\omega_j\rangle=\delta_{ij}$ form the basis of $X^+$.
For $\lambda\in X^+$ we denote by $V_\lambda$ the finite dimensional
irreducible representation of $\bG$ with highest weight $\lambda$.
We denote by $V_\lambda^\vee$ the representation dual to $V_\lambda$;
the pairing: $V_\lambda^\vee\times V_\lambda\lra\BC$ is denoted by
$\langle,\rangle$.
For each $\lambda\in X^+$ we choose
a nonzero vector $y_\lambda\in V_\lambda^{\bN_-}$.
We also choose a nonzero vector $x_\lambda\in (V_\lambda^\vee)^\bN$ such that
$\langle x_\lambda,y_\lambda\rangle=1$.
\subsection{}
\label{config} {\bf Configurations of $I$-colored divisors.}
Let us fix $\alpha\in\BN[I]\subset Y,\ \alpha=\sum_{i\in I}a_ii$.
Given a curve $C$ we consider the configuration space $C^\alpha
\df\prod_{i\in I} C^{(a_i)}$
of colored
effective divisors of multidegree $\alpha$ (the set of colors is $I$).
The dimension of $C^\alpha$ is equal to the length
$|\alpha|=\sum_{i\in I}a_i$.
Multisubsets of a set $S$ are defined as elements of some symmetric power
$S^{(k)}$ and we denote the image of $(s_1,...,s_k)\in S^k$ in $S^{(k)}$
by $\{\{s_1,...,s_k\}\}$.
We denote by $\fP(\alpha)$
the set of all partitions of $\alpha$, i.e multisubsets
$\Ga=
\{\{\ga_1,...,\ga_k\}\}$ of $\BN[I]$ with $\gamma_r\not=0$ and
$\sum_{r=1}^k \ga_i=\al$.
For $\Gamma\in\fP(\alpha)$ the corresponding stratum $C^\alpha_\Gamma$
is defined as follows. It is formed by configurations which can be
subdivided into $m$ groups of points, the $r$-th group containing $\gamma_r$
points; all the points in one group equal to each other, the different
groups being disjoint. For example, the main diagonal in $C^\alpha$
is the closed stratum given by partition $\alpha=\alpha$, while the complement
to all diagonals in $C^\alpha$ is the open stratum given by partition
$$
\alpha=\sum_{i\in I}(\underbrace{i_k+i_k+\ldots+i_k}_{a_k\operatorname{ times}})
$$
Evidently, $C^\alpha=\bigsqcup\limits_{\Gamma\in\fP(\alpha)}C^\alpha_\Gamma$.
\bigskip
\centerline{\bf CHAPTER 1. The spaces $Q$ and $Z$}
\section{Quasimaps from a curve to a flag manifold}
\subsection{}
We fix a smooth projective curve $C$ and $\alpha\in\BN[I]$.
\subsubsection{Definition} An algebraic map $f:\ C\lra\bX$ has degree $\alpha$
if the following equivalent conditions hold:
a) For the fundamental class $[C]\in H_2(C,\BZ)$ we have $f_*[C]=\alpha\in
Y=H_2(\bX,\BZ)$;
b) For any $\nu\in X$ the line bundle $f^*\bL_\nu$ on $C$ has degree
$\langle\alpha,\nu\rangle$.
\subsection{}
\label{maps}
The Pl\"ucker embedding of the flag manifold $\bX$ gives rise to the
following interpretation of algebraic maps of degree $\al$.
For any irreducible $V_\lambda$ we consider the trivial vector bundle
$\CV_\lambda=V_\lambda\otimes\CO$ over $C$.
For any $\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$ we denote
by the same letter the induced morphism $\phi:\ \CV_\lambda\otimes \CV_\mu
\lra \CV_\nu$.
Then
a map of degree $\al$ is a collection of {\em line subbundles}
$\fL_\lambda\subset\CV_\lambda,\ \lambda\in X^+$ such that:
a) $\deg\fL_\lambda=-\langle\alpha,\lambda\rangle$;
b) For any surjective
$\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$
such that $\nu=\lambda+\mu$ we have $\phi(\fL_\lambda\otimes\fL_\mu)=
\fL_\nu$;
c) For any $\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$
such that $\nu<\lambda+\mu$ we have $\phi(\fL_\lambda\otimes\fL_\mu)=0$.
Since the surjections
$V_\la\ten V_\mu\ra V_{\la+\mu}$ form
one $\cs$-orbit, systems $\LL_\la$ satisfying (b) are
determined by a choice of $\fL_{\om_i}$ for the fundamental weights
$\om_i,\ i\in I$.
If we replace the curve $C$ by a point, we get the Pl\"ucker description of
the flag variety $\bX$ as the set of collections of lines $L_\la\sub V_\la$
satisfying conditions of type (b) and (c).
Here, a Borel subgroup $B$ in $\bX$ corresponds to
a system of lines $(L_\la,\ \la\in X^+)$ if the lines are the fixed
points of the unipotent radical $N$ of $B$, $L_\la=(V_\la)^N$,
or equivalently, if $N$ is the common stabilizer for all lines
$N=\bb{\la\in X^+}\cap G_{L_\la}$.
The space of degree $\alpha$ quasimaps from $C$ to $\bX$ will be denoted
by $\qc^\alpha$.
\subsection{Definition}
\label{quasimaps}
(V.Drinfeld)
The space
$\CQ^\al=\CQ^\al_C$ of {\em quasimaps} of degree $\alpha$
from $C$ to $\bX$ is the space of
collections of {\em invertible subsheaves}
$\fL_\lambda\subset\CV_\lambda,\ \lambda\in X^+$ such that:
a) $\deg\fL_\lambda=-\langle\alpha,\lambda\rangle$;
b) For any surjective
$\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$
such that $\nu=\lambda+\mu$ we have $\phi(\fL_\lambda\otimes\fL_\mu)=
\fL_\nu$;
c) For any $\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$
such that $\nu<\lambda+\mu$ we have $\phi(\fL_\lambda\otimes\fL_\mu)=0$.
\subsubsection{Lemma}
a) The evident inclusion $\qc^\alpha\subset\CQ^\alpha$ is an open
embedding;
b) $\CQ^\alpha$ is a projective variety.
{\em Proof.} Obvious. $\Box$
\subsubsection{}
\label{praf}
Here is another version of the Definition, also due to V.Drinfeld.
The principal affine space $\bA=\bG/\bN$ is an $\bH_a$-torsor over $\bX$.
We consider its affine closure $\bbA$,
that is, the spectrum of the ring of functions on $\bA$.
Recall that $\bbA$ is the space of collections of vectors $v_\lambda\in
V_\lambda,\ \lambda\in X^+$, satisfying the following Pl\"ucker relations:
a) For any surjective
$\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$
such that $\nu=\lambda+\mu$, and $\phi(y_\lambda\otimes y_\mu)=y_\nu$,
we have $\phi(v_\lambda\otimes v_\mu)=v_\nu$;
b) For any $\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$
such that $\nu<\lambda+\mu$ we have $\phi(v_\lambda\otimes v_\mu)=0$.
The action of $\bH_a$ extends to $\bbA$ but it is not free anymore.
Consider the
quotient stack $\hat\bX=\bbA/\bH_a$. The flag variety $\bX$ is an
open substack in $\hat\bX$. A map
$\hat{\phi}:\ C\to\hat\bX$ is nothing else than an $\bH_a$-torsor
$\Phi$ over $C$ along
with an $\bH_a$-equivariant morphism $f:\ \Phi\to\bbA$. The degree of this map
is defined as follows.
Let $\lambda:\ \bH_a\to\BC^*$ be the character of $\bH_a$ corresponding
to a weight $\lambda\in X$. Let $\bH_\lambda\subset \bH_a$ be the kernel
of the morphism $\lambda$. Consider the induced $\BC^*$-torsor
$\Phi_\lambda=\Phi/\bH_\lambda$ over $C$. The map $\hat\phi$ has
degree $\alpha\in\BN[I]$ if
$$
\text{for any }\lambda\in X\quad\text{we have}\quad
\deg(\Phi_\lambda)=\langle\lambda,\alpha\rangle.
$$
{\bf Definition.}
The space $\CQ^\alpha$ is the space of maps $\hat{\phi}:\ C\to\hat\bX$
of degree $\alpha$ such that the generic point of $C$ maps into
$\bX\subset\hat\bX$.
The equivalence of the two versions of Definition follows by comparing their
Pl\"ucker descriptions.
\subsection{}
In this subsection we describe a stratification of $\CQ^\alpha$
according to the singularities of quasimaps.
\subsubsection{}
\label{sigma}
Given $\beta,\gamma\in\BN[I]$ such that $\beta+\gamma=\alpha$, we
define the proper map
$
\sigma_{\beta,\gamma}:\
\CQ^\beta \times C^\gamma \lra \CQ^\alpha$.
Namely, let $f=(\fL_\lambda)_{\lambda\in X^+}\in\ \CQ^\beta$
be a quasimap of degree $\beta$;
and let
$
D=\sum_{i\in I}D_i
\cdd i
$
be an effective colored divisor of multidegree
$\gamma=\sum_{i\in I}d_ii$, that is, $\deg(D_i)=d_i$.
We define $\sigma_{\beta,\gamma}(f,D)\df f(-D)
\df (
\fL_\lambda(-\langle D,\lambda\rangle )
)_{\lambda\in X^+}
\in\CQ^\alpha
,$
where we use the pairing $Div^I(C)\bb{\Z}\ten X\ra Div(C)$ given by
$\langle D,\lambda \rangle =
\sum_{i\in I}\langle i,\lambda\rangle \cdd D_i$.
\subsubsection{}
\label{strat M}
{\bf Theorem.} ${\displaystyle \ \CQ^\alpha=\bigsqcup_{0\leq\beta\leq\alpha}
\sigma_{\beta,\alpha-\beta}(\qc^\beta\times C^{\alpha-\beta})}$
{\em Proof.} Any invertible subsheaf $\fL_\la\sub \VV_\la$ lies in a unique
line subbundle $\ti\fL_\la\sub \VV_\la$ called the {\em normalization} of $\fL$.
So any quasimap $\fL$ defines a map $\ti\fL$ (called the {\em normalization}
of $\fL$) of degree $\be\le\al$
and an $I$-colored effective divisor $D$ (called the {\em defect} of $\fL$)
corresponding to the torsion sheaf $\ti\fL/\fL$, such that $\fL=\ti\fL(-D)$.
$\Box$
\subsubsection{Definition}
\label{domain}
Given a quasimap $f=(\fL_\lambda)_{\lambda\in X^+}
\in\ \CQ^\alpha$, its {\em domain of definition} $U(f)$
is the maximal Zariski open $U(f)\subset C$ such that for any $\lambda$
the invertible subsheaf $\fL_\lambda\subset\CV_\lambda$
restricted to $U(f)$ is actually a line subbundle.
\subsubsection{Corollary}
\label{big domain}
For a quasimap $f=(\fL_\lambda)_{\lambda\in X^+}\in\ \CQ^\alpha$ of degree
$\alpha$ the complement $C-U(f)$ of its domain of definition consists of
at most $|\alpha|$ points. $\Box$
\subsection{}
\label{C=line}
From now on, unless explicitly stated otherwise, $C=\BP^1$.
{\bf Proposition.} (V.Drinfeld) $\qc^\alpha$ is a smooth manifold of dimension
$2|\alpha|+\dim(\bX)$.
{\em Proof.} We have to check that at a map $f\in\qc^\alpha$ the first
cohomology $H^1(\BP^1,f^*\CT\bX)$ vanishes (where $\CT\bX$ stands for the
tangent bundle of $\bX$), and then the tangent space $\Theta_f\qc^\alpha$
equals $H^0(\BP^1,f^*\CT\bX)$.
As $\CT\bX$ is generated by the global sections, $f^*\CT\bX$ is
generated by global sections as well, hence
$H^1(\BP^1,f^*\CT\bX)=0$. To compute the dimension of $\Theta_f\qc^\alpha=
H^0(\BP^1,f^*\CT\bX)$ it remains to compute the Euler characteristic
$\chi(\BP^1,f^*\CT\bX)$. To this end we may replace $\CT\bX$ with its
associated graded bundle $\oplus_{\theta\in\CR^+}\bL_\theta$. Then
$$
\chi(\BP^1,f^*(\bigoplus_{\theta\in\CR^+}\bL_\theta))=\sum_{\theta\in\CR^+}
(\langle\alpha,\theta\rangle+1)=\langle\alpha,2\rho\rangle+\sharp\CR^+=
2|\alpha|+\dim\bX$$
$\Box$
\subsection{}
\label{CZ}
Now we are able to introduce our main character.
First we consider the open subspace $U^\alpha\subset\ \CQ^\alpha$ formed by
the quasimaps containing $\infty\in\BP^1$ in their domain of definition
(see ~\ref{domain}). Next we define the closed subspace $\CZ^\alpha\subset
U^\alpha$ formed by quasimaps with value at $\infty$ equal to $\bB_-\in\bX$:
$$\CZ^\alpha\df\ \{f\in U^\alpha | f(\infty)=\bB_-\}$$
We will see below that $\CZ^\alpha$ is an affine algebraic variety.
\subsubsection{}
\label{dimension}
It follows from Proposition ~\ref{C=line} that $\dim\CZ^\alpha=2|\alpha|$.
\section{Local Flag space}
In this section we define a version of $\CQ^\alpha$ where one replaces
the global curve $C$ by the formal neighbourhood of a point.
\subsection{}
\label{SS}
We set $\CO=\BC[[z]]\stackrel{p_n}{\lra}\CO_n=\BC[[z]]/z^n,\CK=\BC((z))$.
We define the scheme $\bbA(\CO)$ (of infinite type): its points are the
collections of vectors $v_\lambda\in V_\lambda\otimes\CO,\ \lambda\in X^+$,
satisfying the Pl\"ucker equations like in ~\ref{praf}.
It is a closed subscheme of $\prod_{i\in I}V_{\omega_i}\otimes\CO$.
We define the open subscheme $\bbA(\CO)_n\subset\bbA(\CO)$: it is formed
by the collections $(v_\lambda)_{\lambda\in X^+}$ such that $p_n(v_{\omega_i})
\not=0$ for all $i\in I$. Evidently, for $0\leq n\leq m$, one has
$\bbA(\CO)_n\subset\bbA(\CO)_m$.
We define the open subscheme $\CS\subset\bbA(\CO)$ as the union
$\bigcup_{n\geq0}\bbA(\CO)_n$. One has $\CS=\bA(\CO)$.
The scheme $\CS$ is equipped with the free action of $\bH_a:\
h(v_\lambda)_{\lambda\in X^+}=(\lambda(h)v_\lambda)_{\lambda\in X^+}$.
The quotient scheme $\bQ=\CS/\bH_a$ is a closed subscheme in
$\prod_{i\in I}\BP(V_{\omega_i}\otimes\CO)$. It is formed by the collections
of lines satisfying the Pl\"ucker equations. We denote the natural projection
$\CS\lra\bQ$ by $pr$.
\subsection{}
\label{SSeta}
For $\eta\in\BN[I]$ we define the closed subscheme $\CS^{-\eta}\subset\CS$
formed by the collections $(v_\lambda)_{\lambda\in X^+}$ such that
$v_\lambda=0\ \modul\ z^{\langle\eta,\lambda\rangle}$. We have the natural
isomorphism $\CS\iso\CS^{-\eta},\ (v_\lambda)_{\lambda\in X^+}\mapsto
(z^{\langle\eta,\lambda\rangle}v_\lambda)_{\lambda\in X^+}$.
Now we can extend the definition of $\CS^\chi$ to arbitrary $\chi\in Y$.
Namely, we define $\CS^\chi$ to be formed by the collections
$(v_\lambda\in V_\lambda\otimes\CK)_{\lambda\in X^+}$ such that
$(z^{\langle\chi,\lambda\rangle}v_\lambda)_{\lambda\in X^+}\in\CS$.
Evidently, $\CS^\chi\subset\CS^\eta$ iff $\chi\leq\eta$, and then the inclusion
is the closed embedding. The open subscheme $\CS^\eta-
\bigcup_{\chi<\eta}\CS^\chi\subset\CS^\eta$
will be denoted by $\dCS^\eta\subset\CS^\eta$.
The ind-scheme $\bigcup_{\eta\in Y}\CS^\eta$ will be denoted by $\tCS$.
The ind-scheme $\tCS$ is equipped with the natural action of the proalgebraic
group $\bG(\CO)$ (coming from the action on $\prod_{i\in I}V_{\omega_i}\otimes
\CK$), and the orbits are exactly $\dCS^\eta,\ \eta\in Y$.
\subsection{}
\label{QQ}
All the above (ind-)schemes are equipped with the free action
of $\bH_a$, and taking quotients we obtain the schemes
$\bQ^\eta=\CS^\eta/\bH_a,\ \eta\in Y$. They are all closed subschemes of
the ind-scheme $\prod_{i\in I}\BP(V_{\omega_i}\otimes\CK)$.
We have $\bQ^\chi\subset\bQ^\eta$ iff $\chi\leq\eta$, and then the inclusion
is the closed embedding. The ind-scheme $\tbQ=\tCS/\bH_a$ is the union
$\tbQ=\bigcup_{\eta\in Y}\bQ^\eta$.
The ind-scheme $\tbQ$ is equipped with the natural action of the proalgebraic
group $\bG(\CO)$ (coming from the action on
$\prod_{i\in I}\BP(V_{\omega_i}\otimes\CK)$),
and the orbits are exactly $\dbQ^\eta=\dCS^\eta/\bH_a,\ \eta\in Y$.
\subsection{}
We consider $C=\BP^1$ with two marked points $0,\infty\in C$. We choose a
coordinate $z$ on $C$ such that $z(0)=0,z(\infty)=\infty$.
\subsubsection{}
For $\alpha\in\BN[I]$ we define the space $\hCQ^\alpha\stackrel{pr}{\lra}
\CQ^\alpha$ formed by the collections $(v_\lambda\in\fL_\lambda\subset
\CV_\lambda)_{\lambda\in X^+}$ such that
a) $(\fL_\lambda\subset\CV_\lambda)_{\lambda\in X^+}\in\CQ^\alpha$;
b) $v_\lambda$ is a regular nonvanishing section of $\fL_\lambda$ on
$\BA^1=\BP^1-\infty$;
c) $(v_\lambda)_{\lambda\in X^+}$ satisfy the Pl\"ucker equations like in
~\ref{praf}.
It is easy to see that $\hCQ^\alpha\stackrel{pr}{\lra}\CQ^\alpha$ is a
$\bH_a$-torsor: $h(v_\lambda,\fL_\lambda)=(\lambda(h)v_\lambda,\fL_\lambda)$.
\subsubsection{}
\label{m}
Taking a formal expansion at $0\in C$ we obtain the closed embedding
$\fs_\alpha:\ \hCQ^\alpha\hookrightarrow\CS$. Evidently, $\fs_\alpha$ is
compatible with the $\bH_a$-action, so it descends to the same named
closed embedding $\fs_\alpha:\ \CQ^\alpha\hookrightarrow\bQ$.
\subsubsection{Lemma}
\label{codime}
Let $\beta\in\BN[I]$. Then $\codim_\bQ\bQ^{-\beta}\geq2|\beta|$.
{\em Proof.} Choose $\alpha\geq\beta$, and consider
the closed embedding $\fs_\alpha:\ \CQ^\alpha\hookrightarrow\bQ$.
Then $\fs_\alpha^{-1}(\bQ^{-\beta})=\CQ^{\alpha-\beta}$ embedded into
$\CQ^\alpha$ as follows: $(\fL_\lambda\subset\CV_\lambda)_{\lambda\in X^+}
\mapsto(\fL_\lambda(-\langle\beta,\lambda\rangle0)
\subset\CV_\lambda)_{\lambda\in X^+}$.
Now $\codim_\bQ\bQ^{-\beta}\geq\codim_{\CQ^\alpha}\CQ^{\alpha-\beta}=2|\beta|$.
$\Box$
\section{Pl\"ucker sections}
In this section we describe another model of the space $\CZ^\alpha$ introduced
in ~\ref{CZ}.
\subsection{}
\label{polynom}
We fix a coordinate $z$ on the affine line $\BA^1=\BP^1-\infty$.
We will also view the configuration space
$\BA^\alpha\df(\BA^1)^\alpha$ (see ~\ref{config})
as the space of collections of unitary
polynomials $(Q_\lambda)_{\lambda\in X^+}$ in $z$, such that
(a) $\deg(Q_\lambda)=\langle\alpha,\lambda\rangle$, and
(b) $Q_{\lambda+\mu}=Q_\lambda Q_\mu$.
\subsection{}
\label{fZ}
Recall the notations of ~\ref{reps}.
For each $\lambda\in X^+$ we will use the decomposition
$V_\lambda=\BC y_\la\oplus(\Ker x_\la)=
(V_\lambda)^\bN\oplus\fn_- V_\lambda$,
compatible with the action of $\fh=\fb_-\cap\fb$, i.e., with the weight
decomposition.
For a section $v_\lambda\in\Ga(\BA^1,\CV_\lambda)=
V_\la\ten\BC[z]\df V_\la[z]$, we will use a polynomial
$Q_\la\df\ \langle x_\lambda,v_\lambda\rangle\in\BC[z]$,
to write down the decomposition
$v_\la=Q_\la\cdd y_\la\pl {v''}_\la\in
\BC[z]\cdd y_\la\pl(\Ker x_\la)[z] = V_\la[z]$.
{\bf Definition.} (V.Drinfeld) The space $\sZ^\alpha$ of
{\em Pl\"ucker sections} of degree $\alpha$ is the space of collections
of sections $v_\lambda\in\Ga(\BA^1,\CV_\lambda)=
V_\la\ten\BC[z]\df V_\la[z]
,\ \lambda\in X^+$;
such that for $v_\la=Q_\la\cdd y_\la\pl {v''}_\la\in
\BC[z]\cdd y_\la\pl(\Ker x_\la)[z]$, one has
a) Polynomial $Q_\lambda$ is unitary
of degree $\langle\alpha,\lambda\rangle$;
b) Component ${v''}_\la$ of $v_\la$ in $(\Ker x_\la)[z]$ has degree strictly
less than $\langle\alpha,\lambda\rangle$;
c) For any $\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$ such that
$\nu=\lambda+\mu$ and $\phi^\vee(x_\nu)=x_\lambda\otimes x_\mu$ we have
$\phi(v_\lambda\otimes v_\mu)=v_\nu$;
d) For any $\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$ such that
$\nu<\lambda+\mu$ we have $\phi(v_\lambda\otimes v_\mu)=0$.
\subsubsection{}
\label{affine}
Collections
$(v_\lambda)_{\lambda\in X^+}$ that satisfy (c),
are determined by a choice of $v_{\omega_i},\ i\in I$.
Hence $\sZ^\alpha$ is an affine algebraic variety.
\subsubsection{}
\label{pi}
Due to the properties a),c) above, the collection of polynomials $Q_\lambda$
defined in a) satisfies the conditions of ~\ref{polynom}. Hence we
have the map $$\pi_\alpha:\ \sZ^\alpha\lra\BA^\alpha$$
\section{Beilinson-Drinfeld Grassmannian}
In this section we describe yet another model of the space $\CZ^\alpha$
introduced in ~\ref{CZ}.
\subsection{}
Let $C$ be an arbitrary smooth projective curve;
let $\CT$ be a left
$\bG$-torsor
over $C$, and let $\tau$ be a section of $\CT$ defined over a Zariski open
subset $U\subset C$, i.e., a trivialization of $\CT$ over $U$.
We will define a $\bB$- (resp. $\bB_-$-) type
$d(\tau)$ (resp. $d_-(\tau)$): a measure of singularity of $\tau$ at $C-U$.
\subsubsection{}
\label{type}
Section $\tau$ defines a $\bB$-subtorsor $\bB\cdd\tau\sub\CT$. This
reduction of $\CT$ to $\bB$ over $U$ is the same as a section of
$\bB\bss\CT$ over $U$. Since $\bG/\bB$ is
proper, this reduction (i.e. section),
extends uniquely to the whole $C$. Thus we obtain
a $\bB$-subtorsor
$\barr{\bB\cdd\tau}\sub\CT$ (the closure of $\bB\cdd\tau\sub\CT|U$ in $\CT$),
equipped with a section
$\tau$ defined over $U$.
Using the projection $\bB\lra\bbH$ we can induce
$\barr{\bB\cdd\tau}$ to
a torsor over $C$ for the abstract Cartan group $\bbH\cong \bB/\bN$ of $\bG$;
namely, $\CT_{\tau,\bB}\df \bN\bss \barr{\bB\cdd\tau}$, equipped with a section
$\tau_\bB$ defined over $U$.
The choice of simple coroots (cocharacters of $\bbH$) $I\subset Y$ identifies
$\bbH$ with $(\BC^*)^I$. Thus the section $\tau_\bB$
of $\CT_{\tau,\bB}$ produces
an $I$-colored divisor $d(\tau)$
supported at $C-U$. We will call $d(\tau)$
the $\bB$-$type$ of $\tau$.
Replacing $\bB$ by $\bB_-$ in the above construction we define the
$\bB_-$-$type$
$d_-(\tau)$.
\subsection{}
\label{bZ}
Recall that A.Beilinson and V.Drinfeld have introduced the {\em relative
Grassmannian} $\CG_C^{(n)}$ over $C^n$ for any $n\in\BN$ (see
~\cite{todisappear}): its fiber $p_n^{-1}(x_1,\ldots,x_n)$
over an $n$-tuple $(x_1,\ldots,x_n)\in C^n$
is the space of isomorphism classes of $\bG$-torsors $\CT$ equipped with a
section $\tau$ defined over $C-\{x_1,\ldots,x_n\}$.
We will consider a certain finite-dimensional subspace of
a partialy symmetrized version of the
relative Grassmannian.
{\bf Definition.} (A.Beilinson and V.Drinfeld)
$\bZ^\alpha$ is the space of isomorphism classes of the following data:
a) an $I$-colored effective divisor $D\in\BA^\alpha$;
b) $\bG$-torsor $\CT$ over $\BP^1$ equipped with a section $\tau$ defined over
$\BP^1-supp(D)$ such that:
i) $\bB$-type $d(\tau)=0$;
ii) $\bB_-$-type $d_-(\tau)$ is a negative divisor (opposite to effective)
such that $d_-(\tau)+D$ is effective.
\subsubsection{}
\label{bZU}
By the definition, the space $\bZ^\alpha$ is equipped with a projection
$p_\alpha$ to $\BA^\alpha:\ (D,\CT,\tau)\mapsto D$. For a subset $U\subset\BA^1$
we will denote by $\bZ_U^\alpha$ the preimage $p_\alpha^{-1}(U)$.
\subsubsection{}
The reader may find another realization of $\bZ^\alpha$ in ~\ref{PBD} below.
\subsection{}
In this subsection we will formulate the crucial {\em factorization} property
of $\bZ^\alpha$.
\subsubsection{}
Recall the following property of the Beilinson-Drinfeld relative Grassmannian
$\CG_C^{(n)}\overset{p_n}{\lra}C^n$ (see ~\cite{todisappear}).
Suppose an $n$-tuple $(x_1,\ldots,x_n)\in C^n$ is represented as a union
of an $m$-tuple $(y_1,\ldots,y_m)\in C^m$ and
a $k$-tuple $(z_1,\ldots,z_k)\in C^k,\ k+m=n$, such that all the points of the
$m$-tuple are disjoint from all the points of the $k$-tuple.
Then $p_n^{-1}(x_1,\ldots,x_n)$ is canonically isomorphic to the product
$p_m^{-1}(y_1,\ldots,y_m)\times p_k^{-1}(z_1,\ldots,z_k)$
\subsubsection{}
\label{factorization}
Suppose we are given a decomposition $\alpha=\beta+\gamma,\ \beta,\gamma
\in\BN[I]$ and two disjoint subsets
$U,\Upsilon\subset\BA^1$.
Then $U^\beta\times\Upsilon^\gamma$ lies in $\BA^\alpha$, and we will denote
the preimage $p_\alpha^{-1}(U^\beta\times\Upsilon^\gamma)$ in $\bZ^\alpha$ by
$\bZ^{\beta,\gamma}_{U,\Upsilon}=
\bZ^\al|_{(U^\beta\times\Upsilon^\gamma)}$ (cf. ~\ref{bZU}).
The above property of relative Grassmannian immediately implies the
following
{\bf Factorization property.} There is a canonical factorization isomorphism
$\bZ^{\beta,\gamma}_{U,\Upsilon}\cong\bZ^\beta_U
\times\bZ^\gamma_\Upsilon$, i.e.,
$$
\bZ^\al |_{(U^\beta\times\Upsilon^\gamma)}\cong
\bZ^\be|_{U^\beta}
\times
\bZ^\ga|_{\Upsilon^\gamma}
.$$
\subsection{Remark} Let us describe the fibers of $p_\alpha$
in terms of the normal slices to the semiinfinite Schubert cells
in the loop Grassmannian.
\subsubsection{}
\label{Iwasawa}
Let $\CG$ be the usual affine Grassmannian $\bG((z))/\bG[[z]]$. It is
naturally identified with the fiber of $\CG^{(1)}_{\BP^1}$ over the point
$0\in\BP^1$. Due to the
Iwasawa decomposition in p-adic groups,
there is a natural bijection
between $Y$ and the set of orbits of the group
$\bN((z))$ (resp. $\bN_-((z))$) in $\CG$; for $\gamma\in Y$ we will denote the
corresponding orbit by $S_\gamma$ (resp. $T_\gamma$). We will denote by
$\ol{T}_\gamma$ the ``closure'' of $T_\gamma$, that is, the union
$\cup_{\beta\geq\gamma}T_\gamma$.
It is proved in ~\cite{mv} that the intersection
$\ol{T}_\gamma\cap S_\beta$ is not empty iff $\gamma\leq\beta$.
Then it is
an affine algebraic variety, a kind of a normal slice to
$T_\be$ in $\barr T_\ga$.
Let us call it $TS_{\gamma,\beta}
\df \ol{T}_\gamma\cap S_\beta$ for short.
If rank$(\bG)>1$ then $TS_{\gamma,\beta}=
\ol{T}_\gamma\cap S_\beta$ is not necessarily irreducible.
But it is always equidimensional of dimension $|\beta-\gamma|$. There is
a natural bijection between the set of irreducible components of
$TS_{\gamma,\beta}=\ol{T}_\gamma\cap S_\beta$
and the canonical basis of $U^+_{\beta-\gamma}$ (the
weight $\beta-\gamma$ component of the quantum universal enveloping algebra
of $\fn$) (see ~\cite{l} for the definition of canonical basis of $U^+$).
\subsubsection{}
Recall the diagonal stratification of $\BA^\alpha$ defined in ~\ref{config}
and the map $p_\al:\bZ^\al\ra\BA^\alpha$.
We consider a partition $\Gamma:\ \alpha=\sum_{k=1}^m\gamma_k$ and a divisor
$D$ in the stratum $\BA_\Gamma^\alpha$. The interested reader will check
readily the following
{\em Claim.} $p_\alpha^{-1}(D)$ is isomorphic to the product
$\prod_{k=1}^m TS_{-\gamma_k,0}
=\prod_{k=1}^m \ol{T}_{-\ga_k}\cap S_0\cong
\prod_{k=1}^m \ol{T}_0\cap S_{\ga_k}$.
In particular, the fiber over a point in the closed stratum is isomorphic to
$TS_{-\alpha,0}= \ol{T}_{-\al}\cap S_0 \cong \ol{T}_0\cap S_\al $,
while the fiber over a generic point is isomorphic to
the product of affine lines
$TS_{-i,0}\cong \ol{T}_0\cap S_{-i}\cong\BA^1$, that is,
the affine space $\BA^{|\alpha|}$.
\subsubsection{Corollary}
\label{irred}
$\bZ^\alpha$ is irreducible.
\section{Equivalence of the three constructions}
\subsection{}
In this subsection we construct an isomorphism
$\varpi:\ \CZ^\alpha\iso\sZ^\alpha$, i.e., from the subsheaves
$\fL_\la\sub\CV_\la$ we construct the sections $v_\la\in\Ga(\BA^1,\CV_\la)$.
\subsubsection{}
\label{tuda}
Let $f\in\CZ^\alpha$ be a quasimap given by a collection $(\fL_\lambda
\subset\CV_\lambda=V_\lambda\otimes\CO_{\BP^1})_{\lambda\in X^+}$.
Since $\fL_\lambda|_{\BA^1}$ is trivial, it has a unique up to proportionality
section $v_\lambda$ generating it over $\BA^1$.
We claim that the pairing
$\langle x_\lambda,v_\lambda\rangle$ does not vanish identically. In effect,
since $\deg(f)=\alpha$, the meromorphic section
$\frac{v_\lambda}{z^{\langle\alpha,\lambda\rangle}}$
of $\CV_\lambda$ is regular nonvanishing
at $\infty\in\BP^1$. Moreover, since $f(\infty)=\bB_-$, we have
$\frac{v_\lambda}{z^{\langle\alpha,\lambda\rangle}}(\infty)\in
V_\lambda^{\bN_-}$.
Thus, $\langle x_\lambda,\frac{v_\lambda}{z^{\langle\alpha,\lambda\rangle}}
\rangle(\infty)\not=0$.
Now we can normalize $v_\lambda$ (so far defined up to a multiplication by
a constant) by the condition that $\langle x_\lambda,v_\lambda\rangle$ is a
unitary polynomial. Let us denote this polynomial by $Q_\lambda$. It has degree
$d_\lambda\leq\langle\alpha,\lambda\rangle$ since $\deg(f)=\alpha$.
Since
$\frac{v_\lambda}{z^{\langle\alpha,\lambda\rangle}}(\infty)\in
V_\lambda^{\bN_-}$,
we see that $\deg\langle e,v_\lambda\rangle<d_\lambda$ for any $e\perp y_\la$.
Moreover, since $\deg(f)=\alpha$ we must then have
$d_\lambda=\langle\alpha,\lambda\rangle$.
Thus we have checked that the collection $(v_\lambda)_{\lambda\in X^+}$
satisfies the conditions a),b) of the Definition ~\ref{fZ}. The conditions
c),d) of {\em loc. cit.} follow from the conditions b),c) of the Definition
~\ref{quasimaps}. In other words, we have defined the Pl\"ucker section
$$\varpi(f)\df\ (v_\lambda)_{\lambda\in X^+}\in\sZ^\alpha$$
\subsubsection{}
\label{obratno}
Here is the inverse construction. Given a Pl\"ucker section
$(v_\lambda)_{\lambda\in X^+}\in\sZ^\alpha$ we define the corresponding
quasimap $f=(\fL_\lambda)_{\lambda\in X^+}\in\CZ^\alpha$ as follows.
We can view $v_\lambda$ as a regular section of
$\CV_\lambda(\langle\alpha,\lambda\rangle\infty)$ over the whole $\BP^1$.
It generates an invertible subsheaf $\fL_\lambda'\subset
\CV_\lambda(\langle\alpha,\lambda\rangle\infty)$. We define
$$\fL_\lambda\df\ \fL_\lambda'(-\langle\alpha,\lambda\rangle\infty)
\subset\CV_\lambda$$
\subsubsection{}
It is immediate to see that the above constructions are inverse to each other,
so that $\varpi:\ \CZ^\alpha\lra\sZ^\alpha$ is an isomorphism.
\subsubsection{Remark} Note that the definition of the space $\CZ^\alpha$
depends only on the choice of Borel subgroup $\bB_-\subset \bG$, while the
definition of $\sZ^\alpha$ depends also on the choice of the opposite Borel
subgroup $\bB\subset \bG$ or, equivalently, on the choice of the Cartan subgroup
$\bH\subset \bB_-$.
We want to stress that the projection $\pi_\alpha:\ \CZ^\alpha=\sZ^\alpha\lra
\BA^\alpha$ {\em does depend} on the choice of $\bB$. Let us describe
$\pi_\alpha\varpi(f)$ for a genuine map (as opposed to quasimap)
$f\in\CZ^\alpha$. To this end recall (see ~\ref{group}) that the
$\bB$-invariant
Schubert
varieties $\ol\bX_{s_iw_0},\ i\in I$, are divisors in $\bX$. Their formal
sum may be viewed as an $I$-colored divisor $\fD$ in $\bX$. Then $f^*\fD$ is
a well defined $I$-colored divisor on $\BP^1$ since $f(\BP^1)\not\subset\fD$
since $f(\infty)=\bB_-\in\bX_{w_0}$. For the same reason the point $\infty$
does not lie in $f^*\fD$, so $f^*\fD$ is really a divisor in $\BA^1$.
It is easy to see that $f^*\fD\in\BA^\alpha$ and $f^*\fD=\pi_\alpha\varpi(f)$.
\subsection{}
In this subsection we construct an isomorphism
$\xi:\ \sZ^\alpha\iso\bZ^\alpha$, so from a system of sections
$v_\la$ we construct a $\bG$-torsor $\CT$ with a section $\tau$
and an $I$-colored divisor $D$.
\subsubsection{Lemma}({\em The Pl\"ucker picture of $\bG$.})
\label{drinf}
The map
$\psi:\ g\mapsto (gx_\lambda,gy_\lambda)_{\lambda\in X^+}$ is a bijection
between $\bG$ and the space of collections $\{(u_\lambda\in V_\lambda^\vee,
\upsilon_\lambda\in V_\lambda)_{\lambda\in X^+})\}$ satisfying the following
conditions:
a) For any $\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$ such that
$\nu=\lambda+\mu$ and $\phi^\vee(x_\nu)=x_\lambda\otimes x_\mu$ we have
$\phi(\upsilon_\lambda\otimes \upsilon_\mu)=\upsilon_\nu$;
b) For any $\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$ such that
$\nu<\lambda+\mu$ we have $\phi(\upsilon_\lambda\otimes\upsilon_\mu)=0$;
c) For any $\bG$-morphism
$\varphi:\ V_\lambda^\vee\otimes V_\mu^\vee\lra V_\nu^\vee$ such that
$\nu=\lambda+\mu$ and $\varphi(x_\lambda\otimes x_\mu)=x_\nu$ we have
$\varphi(u_\lambda\otimes u_\mu)=u_\nu$;
d) For any $\bG$-morphism
$\varphi:\ V_\lambda^\vee\otimes V_\mu^\vee\lra V_\nu^\vee$ such that
$\nu<\lambda+\mu$ we have $\varphi(u_\lambda\otimes u_\mu)=0$;
e) $\langle u_\lambda,\upsilon_\lambda\rangle=1$.
{\em Proof.} We are considering the
systems $(v,u)=\left(v_\la\in V_\lambda,\ u_\la\in V_\lambda^\vee,
\ {\la\in X^+}\right)$ such that both $v$ and $u$ are Pl\"ucker sections
and $\langle v,u\rangle =1$, i.e., $\langle v_\la,u_\la\rangle =1$
for each $\la$.
These form a $\bG$-torsor and we have fixed its element $(y,x)$,
which we will use to think of this torsor as a Pl\"ucker picture
of $\bG$.
The stabilizers $\bG_v$ and $\bG_u$ are the
unipotent radicals of the opposite Borel subgroups, for instance
$\bG_x=\bN$ and $\bG_y=\bN_-$.
So this torsor canonicaly maps into the
open $\bG$-orbit in $\bX\tim\bX$ and the fiber at $(\bB',{\bB}'')$ is
a torsor for a Cartan subgroup $\bB'\cap\bB''$.
$\Box$
\subsubsection{}
\label{suda}
Given a Pl\"ucker section $(v_\lambda)_{\lambda\in X^+}$, the collection
of meromorphic sections
$(x_\lambda\in\CV_\lambda^\vee,\frac{v_\lambda}{Q_\lambda}\in\CV_\lambda)$
evidently satisfies the conditions
a)--e) of the above Lemma, and hence defines a meromorphic function
$g:\ \BA^1\lra \bG$. Let us list the properties of this function.
a) By the definition ~\ref{drinf} of the isomorphism $\psi$,
since $g$ fixes the Pl\"ucker section $x$ the function
$g$ actually takes values in $\bN\subset \bG$;
b) The argument similar to that used in ~\ref{tuda} shows that $g$ can be
extended to $\BP^1$, is regular at $\infty$, and $g(\infty)=1\in \bN$
(since $\frac{v_\lambda}{Q_\lambda}(\infty)=y_\la$,
$g(\infty)$ stabilizes $x_\la$ and $y_\la$ so it lies
in $\bN\cap\bN_-$);
c) Let $D=\pi_\alpha(v_\lambda)$ (see ~\ref{pi}) be the $I$-colored divisor
supported at the roots of $Q_\lambda$. Then $g$ is regular on $\BP^1-D$.
\subsubsection{}
\label{sjuda}
We define $\xi(v_\lambda)=(D,\CT,\tau)\in\bZ^\alpha$ as follows:
$D=\pi_\alpha(v_\lambda);\ \CT$ is the trivial $\bG$-torsor; the section $\tau$
is given by the function $g$.
Let us describe the corresponding $\bbH$-torsor
$\CT_{\tau,\bB_-}$
with meromorphic
section
$\tau_{\bB_-}$.
To describe an $\bbH$-torsor $\fL$ with a section $s$
it suffices to describe the induced $\BC^*$-torsors $\fL_\lambda$
with sections $s_\lambda$ for all characters $\lambda:\ \bbH\lra\BC^*$.
In fact, it suffices to consider only
$\lambda\in X^+$.
Then $\fL_\lambda$ is given by the construction of
~\ref{obratno}, and $s_\lambda=\frac{v_\lambda}{Q_\lambda}$.
Thus, the conditions i),ii) of the Definition ~\ref{bZ} are evidently satisfied.
\subsubsection{}
\label{trivial}
To proceed with the inverse construction, we will need the following
{\em Lemma.} Suppose $(D,\CT,\tau)\in\bZ^\alpha$. Then $\CT$ is trivial
and has a canonical section $\varsigma$.
{\em Proof.} By the construction ~\ref{type}, $\CT$ is induced from the
$\bB$-torsor
$\barr{\bB\cdd \tau}$.
By the Definition ~\ref{bZ}, the induced $\bbH$-torsor
$\CT_{\tau,\bB}$ is trivial, that is,
$\barr{\bB\cdd \tau}$
can be further reduced to an
$\bN$-torsor. But any $\bN$-torsor over $\BP^1$ is trivial since
$H^1(\BP^1,\bV)=0$ for any unipotent group $\bV$
(induction in the lower central series). $\Box$
\subsubsection{}
According to the above Lemma, we can find a unique section $\varsigma$ of
$\CT$ defined over the whole $\BP^1$ and such that $\varsigma(\infty)=
\tau(\infty)$. Hence a triple $(D,\CT,\tau)\in\bZ^\alpha$ canonically
defines a meromorphic function
$$
g\df\ \tau\varsigma^{-1}:\ \BP^1\lra \bG
,$$
i.e., $g(x)\cdd\varsigma(x)=\tau(x),\ x\in\BP^1$.
One sees immediately that $g$ enjoys the properties ~\ref{suda}a)--c).
Now
we can apply the Lemma ~\ref{drinf} in the opposite direction and obtain
from $g$ a collection
$\psi^{-1}(g)=(x_\lambda,\ti\upsilon_\lambda)_{\lambda\in X^+}$
with $\ti\upsilon_\la$ a certain meromorphic sections of $\CV_\la$.
According to ~\ref{polynom}, the divisor $D$ defines a collection
of unitary polynomials $(Q_\lambda)_{\lambda\in X^+}$, and we can define
$v_\lambda\df\ Q_\lambda\ti\upsilon_\lambda$.
One checks easily that $(v_\lambda)\in\sZ^\alpha$, and
$(D,\CT,\tau)=\xi(v_\lambda)$.
In particular, $\xi:\ \sZ^\alpha\lra\bZ^\alpha$ is an isomorphism.
\subsection{}
\label{Z}
We conclude that $\CZ^\alpha,\sZ^\alpha,\bZ^\alpha$ are all the same
and all maps to $\BA^\al$ coincide.
We preserve the notation $\CZ^\alpha$ for this {\em Zastava}
space, and $\pi_\alpha$ for its
projection onto $\BA^\alpha$. We combine the properties ~\ref{dimension},
~\ref{affine},~\ref{factorization},~\ref{irred} into the following
{\bf Theorem.} a) $\CZ^\alpha$ is an irreducible affine algebraic variety
of dimension $2|\alpha|$;
b) For any decomposition $\alpha=\beta+\gamma,\ \beta,\gamma\in\BN[I]$, and
a pair of disjoint subsets $U,\Upsilon\subset\BA^1$, we have the
{\em factorization property}
(notations of ~\ref{bZU} and ~\ref{factorization}):
$$\CZ^{\beta,\gamma}_{U,\Upsilon}=\CZ^\beta_U\times\CZ^\gamma_\Upsilon$$
\bigskip
\centerline{\bf CHAPTER 2. The category $\PS$}
\section{Schubert stratification}
\subsection{}
\label{sig}
We will stratify $\ZZ^\al$ in stages.
We denote by $\CQ^\al\suppp\qp^\al\suppp \qc^\al $,
respectively the variety
of all quasimaps of degree $\al$ and the subvarieties of
the quasimaps defined at $0$ and of genuine maps.
In the same way we denote the varieties of based quasimaps
$\ZZ^\al\suppp\zp^\al\df\ZZ^\al\cap\qp^\al
\suppp
\zc^\al\df\ZZ^\al\cap \qc^\al=$ based maps of degree $\al$.
Recall (see ~\ref{sigma}) the map
$\sigma_{\beta,\ga}: \CQ^\be\tim C\gaa\ra \CQ^{\be+\ga},
\ \si_{\be,\ga}(f,D)=f(-D)$.
For $\be\le \al(=\be+\ga)$, it restricts
to the embedding
$ \CQ^\be\inj \CQ^{\al},
\ f\mm f(-(\al-\be)\cdd 0)=f(\ (\be-\al)\cdd 0)$, and in particular
$\CZ^\beta\hra\CZ^\alpha$.
\subsection{}
\label{coarse}
In the first step we stratify $\ZZ^\al$ according to the singularity at
$0$.
It follows immediately
from the Theorem ~\ref{strat M} that
$$
\CZ^\alpha\cong
\bigsqcup_{0\leq\beta\leq\alpha}\dZ^\beta
.$$
The closed embedding of a stratum $\dZ^\beta$ into $\CZ^\alpha$ will be
denoted by $\sigma_{\beta,\alpha-\beta}$.
\subsection{}
Next, we stratify the quasimaps $\dZ^\al$ defined at $0$, according to the
singularity on $\ccs$.
Again, it follows immediately
from the Theorem ~\ref{strat M} that
$$
\zp^\alpha\cong
\bigsqcup_{0\leq\beta\leq\alpha}\zc^\beta\tim (\ccs)^{\al-\be}
.$$
\subsection{}
\label{Schubert}
One more refinement comes from the decomposition of the
flag variety $\bX$ into the $\bB$-invariant Schubert cells.
Given an element $w$ in the Weyl group $\CW_f$,
we define the locally closed subvarieties
({\em Schubert strata}) $\dZ^{\alpha}_w\subset\dZ^\alpha$
and $\zc^{\alpha}_w\subset\zc^\alpha$, as the sets of
quasimaps $f$ such that $f(0)\in\bX_w$.
The closure of $\dZ^{\alpha}_w$ in $\CZ^\alpha$ will be denoted by
$\oCZ^{\alpha}_w$.
Evidently,
$$\dZ^\alpha=\bigsqcup_{w\in \CW_f}\dZ^{\alpha}_w
\aand \zc^\alpha=\bigsqcup_{w\in \CW_f}\zc^{\alpha}_w.$$
Beware that $\dZ^{\alpha}_w$ may happen to be empty: e.g. for $\alpha=0,
w\not=w_0$.
\subsubsection{}
\label{fineS Q}
Finally, the last refinement comes from the
diagonal stratification of the configuration space $(\ccs)^\delta=
\bigsqcup_{ \Ga\in\fP(\delta)}(\ccs)_\Ga^\delta$.
Altogether,
we obtain the following stratifications of $\ZZ^\al$:
$$
\CZ^\alpha\cong
\bigsqcup_{\alpha\ge \beta}\dZ^\beta
\ \ (\text{{\em coarse stratification}})
$$
$$
\cong
\bigsqcup^{\alpha\ge \beta\ge\ga}_{\Gamma\in\fP(\beta-\gamma)}
\zc^\ga\tim (\ccs)^{\be-\ga}_\Gamma \ \
(\text{{\em fine stratification}})
$$
$$
\cong\bigsqcup^{\alpha\ge \beta\ge\ga}_{ w\in \CW_f,\ \Ga\in\fP(\be-\ga)}
\zc^\ga_w \tim (\ccs)_\Ga^{\beta-\gamma}
\ \ (\text{{\em fine Schubert stratification}})
.$$
Similarly, we have the {\em fine stratification}
(resp. {\em fine Schubert stratification}) of $\CQ^\alpha$:
$$\CQ^\alpha=
\bigsqcup^{\alpha\ge \beta\ge\ga}_{\Gamma\in\fP(\beta-\gamma)}
\qc^\ga\tim (\BP^1-0)^{\be-\ga}_\Gamma=
\bigsqcup^{\alpha\ge \beta\ge\ga}_{ w\in \CW_f,\ \Ga\in\fP(\be-\ga)}
\qc^\ga_w \tim (\BP^1-0)_\Ga^{\beta-\gamma}$$
Here $\qc^\gamma_w\subset\qc^\gamma$ denotes the locally closed subspace of
maps $\BP^1\to\bX$ taking value in $\bX_w\subset\bX$ at $0\in\BP^1$.
The strata $\qc^\gamma_w\times(\BP^1-0)_\Gamma^{\beta-\gamma}$ are evidently
smooth.
Note that the strata $\zp^\al_w$
are not necessarily smooth in general, e.g. for
$\bG=SL_3,\ \alpha$ the sum of simple coroots, $w=w_0$.
To understand the ``fine Schubert strata''
$\zc^\ga_w \tim (\ccs)_\Ga^\beta$ we need to understand the varieties
$\zc^\ga_w$.
\subsection{Conjecture}
\label{conj}
For $\gamma\in\BN[I],w\in\CW_f$ the variety
$\zc^\gamma_w$ is smooth. Hence the ``fine Schubert stratification''
is really a stratification.
\subsubsection{Lemma}
\label{cheap}
For $\gamma$ sufficiently dominant (i.e. $\langle\gamma,i'\rangle>10$) and
arbitrary $w\in\CW_f$ the variety $\zc^\gamma_w$ is smooth.
{\em Proof.} Let us consider the map
$\varrho_\gamma:\ \qc^\gamma\lra\bX\times\bX,\ f\mapsto(f(0),f(\infty))$.
We have $\oZ^\gamma=\varrho_\gamma^{-1}(\bX_w,\bB_-)$.
It suffices to prove that $\varrho_\gamma$ is smooth and surjective.
Recall that the tangent
space $\Theta_f$ to $\qc^\gamma$ at $f\in\qc^\gamma$ is canonically
isomorphic to $H^0(\BP^1,f^*\CT\bX)$. Let us interpret $\bX$ as a variety of
Borel subalgebras of $\fg$. We denote $f(0)$ by $\fb_0$, and $f(\infty)$ by
$\fb_\infty$. So we have to prove that the canonical map
$H^0(\BP^1,f^*\CT\bX)\lra\CT_{\fb_0}\bX\oplus\CT_{\fb_\infty}\bX$ is surjective.
To this end it is enough to have $H^1(\BP^1,f^*\CT\bX(-0-\infty))=0$.
This in turn holds whenever $\gamma$ is sufficiently dominant.
$\Box$
\subsubsection{Lemma}
\label{dim Schubert}
For $\gamma$ sufficiently dominant we have
$\dim\oZ^\gamma_w=2|\gamma|-\dim\bX+\dim\bX_w$.
{\em Proof.} The same as the proof of ~\ref{cheap}. $\Box$
\subsubsection{Remark} Unfortunately, one cannot prove the conjecture
~\ref{conj} for arbitrary $\gamma$ the same way
as the Lemma ~\ref{cheap}: for arbitrary
$\gamma$ the map $\varrho_\gamma$ is not smooth. The simplest example
occurs for $\bG=SL_4$ when $\gamma$ is twice the sum of simple coroots.
This example was found by A.Kuznetsov.
\section{Factorization}
This section follows closely \S4 of ~\cite{fs}.
\subsection{}
\label{fake}
Now we replace the maps into the flag variety $\bX$ with the maps
from $\BP^1$ to the product $\bX\tim Y=\sqcup_{\chi\in Y}\ \bX_\chi$.
So for arbitrary $\chi\in Y$ and $\alpha\in\BN[I]$
we obtain the spaces $\CZ^\alpha_\chi$ of based maps into $\bX_\chi$
and it makes sense now
to add the subscript $\chi$ to all the
strata (coarse, Schubert, fine) defined in the previous section.
We will consider a system $\ZZ$ of varieties
$\CZ^\alpha_\chi,\ \al,\ga\in Y$, together with two kinds of maps
defined for any $\beta,\ga\in\BN[I]$:
a) closed embeddings,
$$\sigma^{\beta,\gamma}_\chi:\
\CZ^\beta_\chi\hra\CZ^{\beta+\gamma}_{\chi+\gamma}
,$$
b) factorization identifications
$$\CZ^{\beta,\gamma}_{\chi,\Ue,\Upe}=
\CZ^\beta_{\chi,\Ue}\times\CZ^\gamma_{\chi-\beta,\Upe}$$
defined for $\varepsilon>0$ and
$U_\varepsilon\df\
\{z\in\BC,\ |z|<\varepsilon\}$, and
$\Upsilon_\varepsilon\df\
\{z\in\BC,\ |z|>\varepsilon\}$.
Of course, without the subscript these are the
factorizations from ~\ref{factorization}
and the embeddings
from ~\ref{sig}.
\subsection{}
\label{snop}
We will denote by $\IC^\alpha_\chi$ the perverse $IC$-extension of the
constant sheaf at the generic point of $\CZ^\alpha_\chi$.
The following definition makes sense only modulo the validity of conjecture
~\ref{conj}.
{\bf Definition.} A {\em snop} $\CK$ is the following collection of data:
a) $\chi=\chi(\CK)\in Y$, called the {\em support estimate} of $\CK$;
b) For any $\alpha\in\BN[I]$, a perverse sheaf $\CK^\alpha_\chi$ on
$\CZ^\alpha_\chi$ smooth along the fine Schubert stratification;
c) For any $\beta,\gamma\in\BN[I],\ \varepsilon>0$,
a {\em factorization isomorphism}
$$
\CK^{\beta+\gamma}_\chi|_{\CZ^{\beta,\gamma}_{\chi,\Ue,\Upe}}
\iso
\CK^\beta_\chi|_{\CZ^\beta_{\chi,\Ue}}
\boxtimes
\IC^\gamma_{\chi-\beta}|_{\CZ^\gamma_{\chi-\beta,\Upe}}$$
satisfying the {\em associativity constraints} as in ~\cite{fs}, \S\S 3,4.
We spare the reader the explicit formulation of these constraints.
\subsection{}
\label{awful}
Since at the moment the conjecture ~\ref{conj} is unavailable we will provide
an ugly provisional substitute of the Definition ~\ref{snop}. Namely, recall
that $\CZ^\alpha=
\sqcup_{\alpha\geq\beta\geq\gamma}\oZ^\gamma\times(\BC^*)^{\beta-\gamma}$.
We introduce an open subvariety
$$\ddZ^\alpha=\bigsqcup_{\alpha\geq\beta\geq\gamma\gg0}\oZ^\gamma\times
(\BC^*)^{\beta-\gamma}$$
The union is taken over sufficiently dominant $\gamma$, i.e. such that
$\langle\gamma,i'\rangle>10$ for any $i\in I$. Certainly, if $\alpha$ itself
is not sufficiently dominant, $\ddZ^\alpha$ may happen to be empty.
We have the fine Schubert stratification
$$\ddZ^\alpha=\bigsqcup^{\alpha\geq\beta\geq
\gamma\gg0}_{w\in\CW_f,\ \Gamma\in\fP(\beta-\gamma)}\oZ^\gamma_w\times
(\BC^*)^{\beta-\gamma}_\Gamma$$ with smooth strata (see the Lemma ~\ref{cheap}).
Now we can repeat the Definition ~\ref{snop} replacing $\CZ^\alpha_\chi$
by $\ddZ^\alpha_\chi$. Thus in ~\ref{snop} b) we have to restrict ourselves
to sufficiently dominant $\alpha$, and in ~\ref{snop} c) $\beta$ has to be
sufficiently dominant as well.
\subsubsection{}
In what follows we use the Definition ~\ref{snop}. The reader unwilling to
believe in the Conjecture ~\ref{conj} will readily substitute the conjectural
Definition ~\ref{snop} with the provisional working Definition ~\ref{awful}.
\subsection{Examples}
We define the {\em irreducible} and {\em standard} snops.
\subsubsection{}
\label{CL}
Let us describe a snop $\CL(w,\chi)$ for $\chi\in Y,\ w\in \CW_f$.
a) The support of $\CL(w,\chi)$ is $\chi$.
b) $\CL(w,\chi)^\alpha_\chi$ is the irreducible $IC$-extension
$\IC(\oCZ^{\alpha}_{w,\chi})=j_{!*}\IC(\dZ^{\alpha}_{w,\chi})$
of the perverse $IC$-sheaf on the Schubert stratum $\dZ^{\alpha}_{w,\chi}\subset
\dZ^\alpha_\chi$. Here $j$ stands for the affine open embedding
$\dZ^{\alpha}_{w,\chi}\hra\oCZ^{\alpha}_w$.
In particular, $\IC(\oCZ^\alpha_{w_0,\chi})=\IC^\alpha_\chi$.
c) Evidently, $\oCZ^{\beta}_{w,\chi,\Ue}$
(resp. $\oCZ^{\beta,\gamma}_{w,\chi,\Ue,\Upe}$) is open in
$\oCZ^{\beta}_{w,\chi}$ (resp. $\oCZ^{\alpha}_{w,\chi}$) for any
$\beta+\gamma=\alpha$. Moreover, $\oCZ^{\beta,\gamma}_{w,\chi,\Ue,\Upe}
=\oCZ^{\beta}_{w,\chi,\Ue}\times\CZ^\gamma_{\chi-\beta,\Upe}$.
This induces the desired factorization isomorphism.
\subsubsection{}
\label{CM}
If we replace in ~\ref{CL}b) above $j_{!*}\IC(\dZ^{\alpha}_{w,\chi})$ by
$j_!\IC(\dZ^{\alpha}_{w,\chi})=:\CM(w,\chi)^\alpha_\chi$
(resp. $j_*\IC(\dZ^{\alpha}_{w,\chi})=:\CalD\CM(w,\chi)^\alpha_\chi$)
we obtain the snop $\CM(w,\chi)$ (resp. $\CalD\CM(w,\chi)$).
\subsection{}
Given a snop $\CK$ with support $\chi$, and $\eta\geq\chi,\
\alpha\in\BN[I]$, we define a sheaf $'\CK^\alpha_\eta$ on $\CZ^\alpha_\chi$
as follows. We set $\gamma\df\ \eta-\chi$. If $\alpha\geq\gamma$ we set
$$'\CK^\alpha_\eta\df\ (\sigma^{\alpha-\gamma,\gamma}_\chi)_*
\CK^{\alpha-\gamma}_\chi$$
(for the definition of $\sigma$ see ~\ref{fake}).
Otherwise we set $'\CK^\alpha_\eta\df\ 0$.
It is easy to see that the factorization isomorphisms for $\CK$ induce
similar isomorphisms for $'\CK$, and thus we obtain a snop $'\CK$ with support
$\eta\geq\chi$.
\subsection{}
We define the category $\widetilde\PS$ of snops.
\subsubsection{}
\label{morphism}
Given two snops $\CF,\CK$ we will define the morphisms $\Hom(\CF,\CK)$
as follows. Let $\eta\in Y$ be such that $\eta\geq\chi(\CF),\chi(\CK)$.
For $\alpha=\beta+\gamma\in\BN[I]$ we consider the following composition:
$$\vartheta^{\beta,\gamma}_\eta:\ \Hom_{\CZ^\alpha_\eta}('\CF^\alpha_\eta,
'\CK^\alpha_\eta)\lra\Hom_{\CZ^{\beta,\gamma}_{\Ue,\Upe}}
('\CF^\alpha_\eta|_{\CZ^{\beta,\gamma}_{\Ue,\Upe}},
'\CK^\alpha_\eta|_{\CZ^{\beta,\gamma}_{\Ue,\Upe}})\iso$$
$$\Hom_{\CZ^\beta_{\eta,\Ue}\times\CZ^\gamma_{\eta-\beta,\Upe}}
('\CF^\beta_\eta|_{\CZ^\beta_{\eta,\Ue}}\boxtimes\IC^\gamma_{\eta-\beta}
|_{\CZ^\gamma_{\eta-\beta,\Upe}},
'\CK^\beta_\eta|_{\CZ^\beta_{\eta,\Ue}}\boxtimes\IC^\gamma_{\eta-\beta}
|_{\CZ^\gamma_{\eta-\beta,\Upe}})=
\Hom_{\CZ^\beta_{\eta,\Ue}}('\CF^\beta_\eta|_{\CZ^\beta_{\eta,\Ue}},
'\CK^\beta_\eta|_{\CZ^\beta_{\eta,\Ue}})$$
(the second isomorphism is induced by the factorization isomorphisms for
$'\CF$ and $'\CK$, and the third equality is just K\"unneth formula).
Now we define
$$\Hom(\CF,\CK)\df\ \dirlim_\eta\invlim_\alpha
\Hom_{\CZ^\alpha_\eta}('\CF^\alpha_\eta,'\CK^\alpha_\eta)$$
Here the inverse limit is taken over $\alpha\in\BN[I]$, the transition
maps being $\vartheta^{\beta,\alpha-\beta}_\eta$, and the direct limit is
taken over $\eta\in Y$ such that $\eta\geq\chi(\CF),\chi(\CK)$, the
transition maps being induced by the obvious isomorphisms
$\Hom_{\CZ^\alpha_\eta}('\CF^\alpha_\eta,'\CK^\alpha_\eta)=
\Hom_{\CZ^{\alpha+\gamma}_{\eta+\gamma}}('\CF^{\alpha+\gamma}_{\eta+\gamma},
'\CK^{\alpha+\gamma}_{\eta+\gamma})$.
\subsubsection{}
With the above definition of morphisms and obvious composition, the snops
form a category which we will denote by $\widetilde\PS$.
\subsection{}
\label{PS}
Evidently, the snops $\CL(w,\chi)$ are irreducible objects of $\widetilde\PS$.
It is easy to see that any irreducible object of $\widetilde\PS$ is isomorphic
to some $\CL(w,\chi)$.
We define the category $\PS$ of {\em finite snops}
as the full subcategory of $\widetilde\PS$ formed
by the snops of finite length. It is an abelian category. We will see later
that $\CM(w,\chi)$ and $\CalD\CM(w,\chi)$ (see ~\ref{CM}) lie in $\PS$ for any
$w,\chi$.
One can prove the following very useful technical lemma exactly as in
~\cite{fs}, 4.7.
\subsubsection{Lemma}
\label{stabilize}
Let $\CF,\CK$ be two finite snops. Let $\eta\geq\chi(\CF),\chi(\CK)$.
There exists $\beta\in\BN[I]$ such that for any $\alpha\geq\beta$
the canonical maps $\Hom(\CF,\CK)\lra
\Hom_{\CZ^\alpha_\eta}('\CF^\alpha_\eta,'\CK^\alpha_\eta)$
are all isomorphisms. $\Box$
\bigskip
\centerline{\bf CHAPTER 3. Convolution with affine Grassmannian}
\section{Pl\"ucker model of affine Grassmannian}
\subsection{}
Let $\CG$ be the usual affine Grassmannian $\bG((z))/\bG[[z]]$. It is the
ind-scheme representing the functor of isomorphism classes of pairs
$(\CT,\tau)$ where $\CT$ is a $\bG$-torsor on $\BP^1$, and $\tau$ is its
section (trivialization) defined off $0$ (see e.g. ~\cite{mv}). It is equipped
with a natural action of proalgebraic group $\bG[[z]]$, and we are going
to describe the orbits of this action. It is known (see e.g. {\em loc. cit.})
that these orbits are numbered by dominant cocharacters $\eta\in Y^+\subset Y$.
Here $Y^+\subset Y$ stands for the set of cocharacters $\eta$ such that
$\langle\eta,i'\rangle\geq0$ for any $i\in I$. For $\eta\in Y^+$ we denote
the corresponding $\bG[[z]]$-orbit in $\CG$ by $\CG_\eta$, and we denote
its closure by $\oCG_\eta$.
Recall that for a dominant character $\lambda\in X^+$ we denote by $V_\lambda$
the corresponding irreducible $\bG$-module, and we denote by $\CV_\lambda$
the trivial vector bundle $V_\lambda\otimes\CO_{\BP^1}$ on $\BP^1$.
\subsection{Proposition}
\label{closure}
The orbit closure $\oCG_\eta\subset\CG$ is the space of collections
$(\CU_\lambda)_{\lambda\in X^+}$ of vector bundles on $\BP^1$ such that
a) $\CV_\lambda(-\langle\eta,\lambda\rangle0)\subset\CU_\lambda\subset
\CV_\lambda(\langle\eta,\lambda\rangle0)$, or equivalently,
$\CU_\lambda(-\langle\eta,\lambda\rangle0)\subset\CV_\lambda\subset
\CU_\lambda(\langle\eta,\lambda\rangle0)$;
b) $\deg\CU_\lambda=\deg\CV_\lambda=0$, or in other words,
$\dim\CV_\lambda(\langle\eta,\lambda\rangle0)/\CU_\lambda=
\langle\eta,\lambda\rangle\dim V_\lambda$;
c) For any surjective $G$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$
and the corresponding morphism $\phi:\ \CV_\lambda\otimes\CV_\mu\lra\CV_\nu$
(hence $\phi:\ \CV_\lambda(\langle\eta,\lambda\rangle0)\otimes
\CV_\mu(\langle\eta,\mu\rangle0)\lra
\CV_\nu(\langle\eta,\lambda+\mu\rangle0)$) we have
$\phi(\CU_\lambda\otimes\CU_\mu)=\CU_\nu$.
{\em Proof.} $\bG$-torsor on a curve $C$ is the same as a tensor functor from
the category of $\bG$-modules to the category of vector bundles on $C$. $\Box$
\subsection{}
\label{loc}
Let us give a local version of the above Proposition. Recall that
$\CO=\BC[[z]]\subset\CK=\BC((z))$. For a finite-dimensional vector space $V$,
a {\em lattice} $\fV$ in $V\otimes\CK$ is an $\CO$-submodule of $V\otimes\CK$
{\em commeasurable} with $V\otimes\CO$, that is, such that
$(V\otimes\CO)\cap \fV$
is of finite codimension in both $V\otimes\CO$ and $\fV$.
{\bf Proposition.}
The orbit closure $\oCG_\eta\subset\CG$ is the space of collections
$(\fV_\lambda)_{\lambda\in X^+}$ of lattices in $V_\lambda\otimes\CK$ such that
a) $z^{\langle\eta,\lambda\rangle}(V_\lambda\otimes\CO)\subset
\fV_\lambda\subset
z^{-\langle\eta,\lambda\rangle}(V_\lambda\otimes\CO)$, or equivalently,
$z^{\langle\eta,\lambda\rangle}\fV_\lambda\subset V_\lambda\otimes\CO\subset
z^{-\langle\eta,\lambda\rangle}\fV_\lambda$;
b) $\dim(z^{-\langle\eta,\lambda\rangle}(V_\lambda\otimes\CO)/\fV_\lambda)=
\langle\eta,\lambda\rangle\dim V_\lambda$;
c) For any surjective $G$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$
and the corresponding morphism
$\phi:\ (V_\lambda\otimes\CO)\otimes(V_\mu\otimes\CO)\lra(V_\nu\otimes\CO)$
(hence $\phi:\ z^{-\langle\eta,\lambda\rangle}(V_\lambda\otimes\CO)\otimes
z^{-\langle\eta,\mu\rangle}(V_\mu\otimes\CO)\lra
z^{-\langle\eta,\lambda+\mu\rangle}(V_\nu\otimes\CO)$), we have
$\phi(\fV_\lambda\otimes \fV_\mu)=\fV_\nu$.
$\Box$
\subsection{}
\label{Iwahori}
Let $\bI\subset\bG[[z]]$ be the Iwahori subgroup; it is formed
by all $g(z)\in\bG[[z]]$ such that $g(0)\in\bB\subset\bG$. We will denote by
$\CP(\CG,\bI)$ the category of perverse sheaves on $\CG$
with finite-dimensional support, constant along $\bI$-orbits.
The stratification of $\CG$ by $\bI$-orbits is a certain refinement of
the stratification $\CG=\sqcup_{\eta\in Y^+}\CG_\eta$. Namely, each $\CG_\eta$
decomposes into $\bI$-orbits numbered by $\CW_f/\CW_\eta$ where $\CW_\eta$
stands for the stabilizer of $\eta$ in $\CW_f$. For $w\in\CW_f/\CW_\eta$
we will denote the corresponding $\bI$-orbit by $\CG_{w,\eta}$. Let us
introduce a Pl\"ucker model of $\CG_{w,\eta}$.
\subsubsection{} For $\eta\in Y^+$ let $I_\eta\subset I$ be the set of all $i$
such that $\langle\eta,i'\rangle=0$ (thus for $i\not\in I_\eta$ we have
$\langle\eta,i'\rangle>0$). Then $\CW_\eta$ is generated by the simple
reflections $\{s_i, i\in I_\eta\}$. Let $\bP(I_\eta)$ be the corresponding
parabolic subgroup (e.g. for $I_\eta=\emptyset$ we have $\bP(I_\eta)=\bB$,
while for $I_\eta=I$ we have $\bP(I_\eta)=\bG$). Let $\bX(I_\eta)=
\bG/\bP(I_\eta)$ be the corresponding partial flag variety. The $\bB$-orbits
on $\bX(I_\eta)$ are naturally numbered by $\CW_f/\CW_\eta:\ \bX(I_\eta)=
\sqcup_{w\in\CW_f/\CW_\eta}\bX(I_\eta)_w$. The Pl\"ucker embedding realizes
$\bX$ as a closed subvariety in $\prod_{i\in I}\BP(V_{\omega_i})$. Its image
under the projection
$\prod_{i\in I}\BP(V_{\omega_i})\lra\prod_{i\not\in I_\eta}
\BP(V_{\omega_i})$ exactly coincides with $\bX(I_\eta)$.
\subsubsection{Lemma-Definition}
a) For $\eta=\sum_{i\in I}n_ii$,
and $(\CU_\lambda)_{\lambda\in X^+}\in\CG_\eta$ we have\\
$\dim(\CU_{\omega_i}+
\CV_{\omega_i}((n_i-1)\cdot0)/\CV_{\omega_i}((n_i-1)\cdot0))=
\dim V_{\omega_i}^{{\bf U}(I_\eta)}$ where ${\bf U}(I_\eta)$ is the unipotent
radical of $\bP(I_\eta)$.
b) Thus $\CU_{\omega_i}, i\in I,$ defines a subspace $K_i$ in
$\CV_{\omega_i}(n_i\cdot0)/\CV_{\omega_i}((n_i-1)\cdot0)=
V_{\omega_i}$. This collection
of subspaces $(K_i)_{i\in I}\in\prod_{i\in I}\operatorname{Gr}(V_{\omega_i})$
satisfies the Pl\"ucker relations and thus gives a point in $\bX(I_\eta)$;
c) We will denote by $\br$ the map $\CG_\eta\lra\bX(I_\eta)$ defined in b);
d) For $w\in\CW_f/\CW_\eta$ we have $\CG_{w,\eta}=\br^{-1}(\bX(I_\eta)_w)$.
$\Box$
We are obliged to D.Gaitsgory who pointed out a mistake in the earlier
version of the above Lemma.
\subsubsection{}
For $\theta\in Y$ we consider the corresponding homomorphism $\theta:\
\BC^*\lra\bH\subset\bG$ as a formal loop $\theta(z)\in\bG((z))$.
It projects to the same named point $\theta(z)\in\bG((z))/\bG[[z]]=\CG$.
There is a natural bijection between the set of $\theta(z),\ \theta\in Y$,
and the set of Iwahori orbits: each Iwahori orbit $\CG_{w,\eta}$ contains
exactly one of the above points, namely, the point $w\eta(z)$.
\subsection{}
\label{PBD}
Recall the Beilinson-Drinfeld avatar $\bZ^\alpha$ of the Zastava space
$\CZ^\alpha$ (see ~\ref{bZ}). In this subsection we will give a Pl\"ucker model
of $\bZ^\alpha$.
{\bf Proposition.} $\bZ^\alpha$ is the the space of pairs
$(D,(\fU_\lambda)_{\lambda\in X^+})$ where $D\in\BA^\alpha$ is an
$I$-colored effective divisor, and
$(\fU_\lambda)_{\lambda\in X^+}$ is a collection of vector bundles on $\BP^1$
such that
a) $\CV_\lambda(-\infty D)\subset\fU_\lambda\subset\CV_\lambda(+\infty D)$;
b) $\CV_\lambda^\bN\subset\fU_\lambda\supset
\CV_\lambda^{\bN_-}(-\langle D,\lambda\rangle)$ (notations of ~\ref{sigma}),
the first inclusion being a {\em line subbundle} (and the second an invertible
subsheaf);
c) $\deg\fU_\lambda=0$;
d) For any surjective $G$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$
and the corresponding morphism $\phi:\ \CV_\lambda\otimes\CV_\mu\lra\CV_\nu$
(hence $\phi:\ \CV_\lambda(+\infty D)\otimes
\CV_\mu(+\infty D)\lra
\CV_\nu(+\infty D)$) we have
$\phi(\fU_\lambda\otimes\fU_\mu)=\fU_\nu$.
{\em Proof.} Obvious. $\Box$
\subsubsection{Remark}
\label{?}
Recall the isomorphism $\varpi^{-1}\xi:\ \bZ^\alpha\iso\CZ^\alpha$ constructed
in section 7. Let us describe it in terms of ~\ref{PBD}. The Lemma
~\ref{trivial} says that there is a unique system of isomorphisms
$\iota_\lambda:\ \fU_\lambda\iso\CV_\lambda,\ \lambda\in X^+$, identical at
$\infty\in\BP^1$ and compatible with tensor multiplication.
Then $\varpi^{-1}\xi(D,(\fU_\lambda)_{\lambda\in X^+})=(\fL_\lambda\subset
\CV_\lambda)_{\lambda\in X^+}$ where $\fL_\lambda=
\iota_\lambda(\CV_\lambda^{\bN_-}(-\langle D,\lambda\rangle))$.
\subsection{}
\label{stack}
Let $\fM$ be the scheme representing the
functor of isomorphism classes of $\bG$-torsors on $\BP^1$ equipped with
trivialization in the formal neighbourhood of $\infty\in\BP^1$
(see ~\cite{ka} and ~\cite{kt1}).
\subsubsection{}
\label{strat stack}
The scheme $\fM$ is stratified by the locally closed subschemes $\fM_\eta:\
\fM=\sqcup_{\eta\in Y^+}\fM_\eta$ according to the isomorphism types of
$\bG$-torsors. Namely, due to Riemann's classification, for a $\bG$-torsor
$\CT$ and any
$\lambda\in X^+$ the associated vector bundle $\CV_\lambda^\CT$ decomposes
as a direct sum of line bundles $\CO(r_k^\lambda)$ of well-defined degrees
$r_1^\lambda\geq\ldots\geq r^\lambda_{\dim V_\lambda}$. Then $\CT$ lies in the
stratum $\fM_\eta$ iff $r_1^\lambda=\langle\eta,\lambda\rangle$.
For any $\eta\in Y^+$ the union of strata $\fM^\eta:=
\sqcup_{Y^+\ni\chi\leq\eta}\fM_\chi$ forms an open subscheme of $\fM$.
This subscheme is a projective limit of schemes of finite type, all the maps
in projective system being fibrations with affine fibers. Moreover,
$\fM^\eta$ is equipped with a free action of a prounipotent group $\bG^\eta$
(a congruence subgroup in $\bG[[z^{-1}]]$) such that the quotient $\ufM^\eta$
is a smooth scheme of finite type. The theory of perverse sheaves on $\fM$
smooth along the stratification by $\fM_\eta$ is developed in ~\cite{kt1}.
We will refer the reader to this work, and will freely use such perverse
sheaves, e.g. $\IC(\fM_\eta)$.
\subsubsection{}
\label{thin}
Restricting a trivialization of a $\bG$-torsor from $\BP^1-0$ to
the formal neighbourhood of $\infty\in\BP^1$ we obtain the closed
embedding $\bi:\ \CG\hra\fM$. The intersection of $\fM_\eta$ and $\CG_\chi$
is nonempty iff $\eta\leq\chi$, and then it is transversal. Thus, $\oCG_\eta
\subset\fM^\eta$. According to ~\cite{kt}, the composition $\oCG_\eta
\hookrightarrow\fM^\eta\lra\ufM^\eta$ is a closed embedding.
\subsubsection{}
\label{mapstack}
For a $\bG$-torsor $\CT$ and an irreducible $\bG$-module $V_\lambda$ we
denote by $\CV_\lambda^\CT$ the associated vector bundle. Following
~\ref{maps} and ~\ref{quasimaps} we define
for {\em arbitrary} $\alpha\in Y$ the scheme $\ofQ^\alpha$
(resp. $\fQ^\alpha$) representing the functor of isomorphism classes of pairs
$(\CT,(\fL_\lambda)_{\lambda\in X^+})$
where $\CT$ is a $\bG$-torsor trivialized in the formal neighbourhood of
$\infty\in\BP^1$, and $\fL_\lambda\subset\CV_\lambda^\CT,\
\lambda\in X^+$, is a collection of line subbundles (resp. invertible
subsheaves) of degree $\langle-\alpha,\lambda\rangle$ satisfying the Pl\"ucker
conditions (cf. {\em loc. cit.}). The evident projection $\ofQ^\alpha\lra\fM$
(resp. $\fQ^\alpha\lra\fM$) will be denoted by $\obp$ (resp. $\bp$).
The open embedding $\ofQ^\alpha\hra\fQ^\alpha$ will be denoted by $\bj$.
Clearly, $\bp$ is projective, and $\obp=\bp\circ\bj$.
The free action of prounipotent group $\bG^\eta$ on $\fM^\eta$ lifts to
the free action of $\bG^\eta$ on the open subscheme
$\bp^{-1}(\fM^\eta)\subset\fQ^\alpha$.
The quotient is a scheme of finite type $\ufQ^{\alpha,\eta}$ equipped with
the projective morphism $\bp$ to $\ufM^\eta$. There exists a
$\bG[[z^{-1}]]$-invariant stratification $\fS$ of $\fQ^\alpha$ such that
$\bp$ is stratified with respect to $\fS$ and the stratification
$\fM=\sqcup_{\eta\in Y^+}\fM_\eta$. One can define perverse sheaves on
$\fQ^\alpha$ smooth along $\fS$ following the lines of ~\cite{kt1}.
In particular, we have the irreducible Goresky-Macpherson sheaf
$\IC(\fQ^\alpha)$.
Following ~\ref{strat M} we introduce a decomposition of $\fQ^\alpha$
into a disjoint union of
locally closed subschemes according to the isomorphism types of $\bG$-torsors
and defects of invertible subsheaves:
$$\fQ^\alpha=\bigsqcup^{\eta\in Y^+}_{\beta\leq\alpha}\ofQ^\beta_\eta
\times C^{\alpha-\beta}$$ where $C=\BP^1$ and $\ofQ^\beta_\eta=
\obp^{-1}(\fM_\eta)\subset\ofQ^\beta$.
\subsection{IC sheaves}
\subsubsection{}
\label{GMZ}
The Goresky-MacPherson sheaf $\IC^\alpha$ on $\CZ^\alpha$
is smooth along stratification
$$\CZ^\alpha=
\bigsqcup^{\alpha\geq\beta\geq\gamma\geq0}_{\Gamma\in\fP(\beta-\gamma)}
\zc^\gamma\times(\BC^*)^{\beta-\gamma}_\Gamma$$
(cf. ~\ref{Schubert}). It is evidently constant along strata, so its stalk
at a point in $\zc^\gamma\times(\BC^*)^{\beta-\gamma}_\Gamma$
depends on the stratum only. Moreover, due to factorization property, it
depends not on $\alpha\geq\beta$ but only on their difference $\alpha-\beta\in
\BN[I]$. We will denote it by $\IC^{\alpha-\beta}_\Gamma$.
In case $\bG=SL_n$ these stalks were computed in ~\cite{ku}.
\subsubsection{}
\label{GMQ}
Recall (see ~\ref{strat M}) that $\CQ^\beta,\ \beta\in\BN[I]$, is stratified
by the type of defect:
$$\CQ^\beta=\bigsqcup^{\beta\geq\gamma\geq0}_{\Gamma\in\fP(\beta-\gamma)}
\oQ^\gamma\times C^{\beta-\gamma}_\Gamma$$
The Goresky-Macpherson sheaf $\IC(\CQ^\beta)$ on $\CQ^\beta$ is constant
along the strata. It is immediate to see that its stalk at any point in the
stratum $\oQ^\gamma\times C^{\beta-\gamma}_\Gamma$ is isomorphic, up to a
shift, to $\IC^0_\Gamma$. In particular, it depends on the defect only.
\subsubsection{}
\label{GM}
{\bf Proposition.}
a) The Goresky-Macpherson sheaf $\IC(\fQ^\beta)$ on $\fQ^\beta,\
\beta\in Y$, is constant along the locally closed subschemes
$$\fQ^\beta=\bigsqcup^{\beta\geq\gamma}_{\Gamma\in\fP(\beta-\gamma)}
\ofQ^\gamma\times C^{\beta-\gamma}_\Gamma$$
b) The stalk of $\IC(\fQ^\beta)$ at any point in the
$\ofQ^\gamma\times C^{\beta-\gamma}_\Gamma$ is isomorphic,
up to a shift, to $\IC^0_\Gamma$.
{\em Proof.} Will be given in ~\ref{later}. $\Box$
\subsubsection{}
\label{parity}
Let $\phi\in\fQ^\beta$. The stalk $\IC(\fQ^\beta)_\phi$ is a graded vector
space.
{\bf Conjecture.} {\em (Parity vanishing)} Nonzero graded parts of
$\IC(\fQ^\beta)_\phi$ appear in cohomological degrees of the same parity.
\subsubsection{Remark} In case $\bG=SL_n$ the conjecture follows from
the Proposition ~\ref{GM} and ~\cite{ku} 2.5.2. In general case the conjecture
follows from the unpublished results of G.Lusztig. We plan to prove it in
the next part.
\section{Convolution diagram}
\subsection{Definition} For $\alpha\in Y,\eta\in Y^+$ we define the
{\em convolution diagram} $\CG\CQ_\eta^\alpha$ as the space of collections
$(\CU_\lambda,\fL_\lambda)_{\lambda\in X^+}$ of vector bundles with invertible
subsheaves such that
a) $(\CU_\lambda)_{\lambda\in X^+}\in\oCG_\eta$, or in other words,
$(\CU_\lambda)_{\lambda\in X^+}$ satisfies the conditions ~\ref{closure} a)-c);
b) $\fL_\lambda\subset\CU_\lambda$ has degree $-\langle\alpha,\lambda\rangle$;
c) For any surjective $\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$
such that $\nu=\lambda+\mu$ we have (cf. ~\ref{closure} c)
$\phi(\CL_\lambda\otimes\CL_\mu)=\CL_\nu$;
d) For any $\bG$-morphism $\phi:\ V_\lambda\otimes V_\mu\lra V_\nu$ such that
$\nu<\lambda+\mu$ we have $\phi(\CL_\lambda\otimes\CL_\mu)=0$.
\subsubsection{} Let us denote by $\oGQ_\eta^\alpha$ the open subvariety in
$\CG\CQ_\eta^\alpha$ formed by all the collections $(\CU_\lambda,\fL_\lambda)$
such that $\fL_\lambda$ is a {\em line subbundle} in $\CU_\lambda$ for
any $\lambda\in X^+$. The open embedding
$\oGQ_\eta^\alpha\hra\CG\CQ_\eta^\alpha$ will be denoted by ~$\bj$.
\subsection{Definition} a) We define the projection $\bp:\ \CG\CQ_\eta^\alpha
\lra\oCG_\eta$ as $\bp(\CU_\lambda,\fL_\lambda)=(\CU_\lambda)$;
b) We define the map $\bq:\ \CG\CQ_\eta^\alpha\lra\CQ^{\alpha+\eta}$ as follows:
$$\bq(\CU_\lambda,\fL_\lambda)=(\fL_\lambda(-\langle\eta,\lambda\rangle0)
\subset\CU_\lambda(-\langle\eta,\lambda\rangle0)\subset\CV_\lambda)$$
(cf. ~\ref{closure} a) and the Definition ~\ref{quasimaps}).
\subsubsection{}
\label{some}
We will denote by $\obp$ the restriction of $\bp$ to the open subvariety
$\oGQ_\eta^\alpha\stackrel{\bj}{\hookrightarrow}\CG\CQ_\eta^\alpha$.
\subsubsection{Remark}
Note that $\CG\CQ_0^\alpha=\CQ^\alpha,\ \oGQ_0^\alpha=\qc^\alpha,\ \bq=\id$,
and $\bp$ is the projection to the point $\oCG_0$.
Note also that while in the Definition ~\ref{quasimaps} we imposed the
positivity condition $\alpha\in\BN[I]$ (otherwise $\CQ^\alpha$ would be empty)
here we allow arbitrary $\alpha\in Y$. It is easy to see that
$\CG\CQ_\eta^\alpha$ (as well as $\oGQ_\eta^\alpha$)
is nonempty iff $\eta+\alpha\in\BN[I]$.
\subsection{}
Let us give a local version of the convolution diagram. Recall the notations
of ~\ref{loc} and ~\ref{SSeta}.
{\bf Definition.} For $\eta\in Y^+$ we define the {\em extended local
convolution
diagram} $\CG\CS_\eta$ as the ind-scheme formed by the collections
$(\fV_\lambda,v_\lambda)_{\lambda\in X^+}$ of lattices in $V_\lambda\otimes\CK$
and vectors in $V_\lambda\otimes\CK$ such that
a) $(\fV_\lambda)_{\lambda\in X^+}\in\oCG_\eta$, or in other words,
$(\fV_\lambda)_{\lambda\in X^+}$ satisfies the condition ~\ref{loc}a)-c);
b) $v_\lambda\in\fV_\lambda$;
c) $(v_\lambda)_{\lambda\in X^+}\in\tCS$ (see ~\ref{SSeta}).
\subsection{Definition}
a) We define the projection $\bp:\ \CG\CS_\eta\lra\oCG_\eta$ as
$\bp(\fV_\lambda,v_\lambda)=(\fV_\lambda)$;
b) We define the map $\bq:\ \CG\CS_\eta\lra\CS$ as follows (cf. ~\ref{loc}a):
$$\bq(\fV_\lambda,v_\lambda)_{\lambda\in X^+}=
(z^{\langle\eta,\lambda\rangle}v_\lambda\in
z^{\langle\eta,\lambda\rangle}\fV_\lambda
\subset V_\lambda\otimes\CO)_{\lambda\in X^+}$$
\subsection{}
\label{lcd}
The torus $\bH_a$ acts in a natural way on $\CG\CS_\eta:\
h(\fV_\lambda,v_\lambda)=(\fV_\lambda,\lambda(h)v_\lambda)$. The action is
evidently free, and we denote the quotient by $\CG\bQ_\eta$,
the {\em local convolution diagram}. The map $\bp:\ \CG\CS_\eta\lra\oCG_\eta$
commutes with the action of $\bH_a$ (trivial on $\oCG_\eta$), so it descends
to the same named map $\bp:\ \CG\bQ_\eta\lra\oCG_\eta$.
The map $\bq:\ \CG\CS_\eta\lra\CS$ commutes with the action of $\bH_a$
(for the action on $\CS$ see ~\ref{SS}), so it descends to the same named
map $\bq:\ \CG\bQ_\eta\lra\bQ$.
The proalgebraic group $\bG(\CO)$ acts on $\CG\bQ_\eta$ and on $\bQ$, and
the map $\bq$ is equivariant with respect to this action.
\subsection{}
\label{cartes}
Let us compare the local convolution diagram with the global one.
For $\alpha\in Y$, taking formal expansion at $0\in C$ as in ~\ref{m},
we obtain the closed embedding $\fs:\ \CG\CQ^\alpha_\eta\hookrightarrow
\CG\bQ_\eta$. It is easy to see that the following diagram is cartesian:
$$
\begin{CD}
\CG\CQ^\alpha_\eta @>\fs>> \CG\bQ_\eta \\
@V{\bq}VV @V{\bq}VV \\
\CQ^{\eta+\alpha} @>{\fs}>> \bQ
\end{CD}
$$
\subsection{}
Recall the locally closed embedding $\CZ^\alpha\subset\CQ^\alpha$.
{\bf Definition.} We define the {\em restricted convolution diagram}
$\CG\CZ_\eta^\alpha\subset\CG\CQ_\eta^\alpha$ as the preimage
$\bq^{-1}(\CZ^{\eta+\alpha})$. The open subvariety $\CG\CZ_\eta^\alpha\cap
\oGQ_\eta^\alpha$
will be denoted by $\oGZ_\eta^\alpha$.
We will preserve
the notations $\bp,\bq$ (resp. $\obp,\bj$) for the restrictions of
these morphisms to $\CG\CZ_\eta^\alpha$ (resp. $\oGZ_\eta^\alpha$).
\subsection{}
\label{idiot}
We construct the {\em Beilinson-Drinfeld} avatar
$\CG\bZ^\alpha_\eta$ of the restricted convolution diagram $\CG\CZ_\eta^\alpha$.
{\bf Definition.} $\CG\bZ^\alpha_\eta$ is the space of triples
$(D,(\CU_\lambda)_{\lambda\in X^+},(\fU_\lambda)_{\lambda\in X^+})$
where $D\in\BA^{\eta+\alpha}$ is an effective $I$-colored divisor, and
$(\CU_\lambda)_{\lambda\in X^+},(\fU_\lambda)_{\lambda\in X^+}$ are the
collections of vector bundles on $\BP^1$ such that
a) $(\CU_\lambda)_{\lambda\in X^+}\in\oCG_\eta$, or in other words,
$(\CU_\lambda)_{\lambda\in X^+}$ satisfies the conditions ~\ref{closure} a)-c);
b) $(D,(\fU_\lambda)_{\lambda\in X^+})\in\bZ^{\eta+\alpha}$, or in other words,
$(D,(\fU_\lambda)_{\lambda\in X^+})$ satisfies the conditions ~\ref{PBD} a)-d);
c) $\iota_\lambda(\CV_\lambda^{\bN_-}(-\langle D,\lambda\rangle))\subset
\CU_\lambda(-\langle\eta,\lambda\rangle0)$
(notations of ~\ref{?}).
\subsubsection{}
The identification $\CG\CZ_\eta^\alpha=\CG\bZ_\eta^\alpha$ easily follows from
~\ref{PBD}. Under this identification, for
$(D,(\CU_\lambda)_{\lambda\in X^+},(\fU_\lambda)_{\lambda\in X^+})
\in\CG\bZ_\eta^\alpha$, we have
$$\bp(D,(\CU_\lambda)_{\lambda\in X^+},(\fU_\lambda)_{\lambda\in X^+})=
(\CU_\lambda)_{\lambda\in X^+}$$ and
$$\bq(D,(\CU_\lambda)_{\lambda\in X^+},(\fU_\lambda)_{\lambda\in X^+})=
(D,(\fU_\lambda)_{\lambda\in X^+})$$
\subsection{} We will introduce the {\em fine} stratifications of
$\CG\CQ_\eta^\alpha$ and $\CG\CZ_\eta^\alpha$ following the section 7.
\subsubsection{Fine stratification of $\CG\CQ_\eta^\alpha$}
\label{fine GQ}
We have $$\CG\CQ_\eta^\alpha=
\bigsqcup\ooGQ_\chi^\gamma\times(\BP^1-0)^{\beta-\gamma}_\Gamma$$
Here the union is taken over dominant $\chi\leq\eta$ in $Y^+$, arbitrary
$\gamma\leq\beta\leq\alpha\in Y$, and partitions $\Gamma\in\fP(\beta-\gamma)$.
Furthermore, $\ooGQ_\chi^\gamma\subset\oGQ_\chi^\gamma$ is an open subvariety
formed by all the collections $(\CU_\lambda,\fL_\lambda)_{\lambda\in X^+}$ such
that $(\CU_\lambda)\in\CG_\chi\subset\oCG_\chi$, and $\fL_\lambda$ is a
{\em line subbundle} in $\CU_\lambda$.
The stratum $\ooGQ_\chi^\gamma\times(\BP^1-0)^{\beta-\gamma}_\Gamma$ is formed
by the collections $(\CU_\lambda,\fL_\lambda)_{\lambda\in X^+}$ such that
$\bp(\CU_\lambda,\fL_\lambda)\in\CG_\chi\subset\oCG_\eta$; the normalization
(see ~\ref{strat M}) of $\fL$ in $\CU$ has degree $\gamma$; and the defect $D$
(see {\em loc. cit.}) of $\fL$ in $\CU$ equals $(\alpha-\beta)0+D'$ where
$D'\in(\BP^1-0)^{\beta-\gamma}_\Gamma$.
{\bf Proposition.} $\ooGQ^\alpha_\eta$ is smooth for arbitrary
$\alpha,\eta$, i.e. the fine strata are smooth.
{\em Proof} will be given in ~\ref{sometimes}.
\subsubsection{Fine Schubert stratification of $\CG\CQ_\eta^\alpha$}
\label{fineS GQ}
We have $$\CG\CQ_\eta^\alpha=
\bigsqcup\ooGQ_{w,\chi}^\gamma\times(\BP^1-0)^{\beta-\gamma}_\Gamma$$
Here the union is taken over dominant $\chi\leq\eta$ in $Y^+$,
representatives $w\in\CW_f/\CW_\chi$, arbitrary
$\gamma\leq\beta\leq\alpha\in Y$, and partitions $\Gamma\in\fP(\beta-\gamma)$.
Furthermore, $\ooGQ_{w,\chi}^\gamma\subset\oGQ_\chi^\gamma$
is an open subvariety
formed by all the collections $(\CU_\lambda,\fL_\lambda)_{\lambda\in X^+}$ such
that $(\CU_\lambda)\in\CG_{w,\chi}\subset\oCG_\chi$, and $\fL_\lambda$ is a
{\em line subbundle} in $\CU_\lambda$.
The stratum $\ooGQ_\chi^\gamma\times(\BP^1-0)^{\beta-\gamma}_\Gamma$ is formed
by the collections $(\CU_\lambda,\fL_\lambda)_{\lambda\in X^+}$ such that
$\bp(\CU_\lambda,\fL_\lambda)\in\CG_{w,\chi}\subset\oCG_\eta$;
the normalization of $\fL$ in $\CU$ has degree $\gamma$; and the defect $D$
(see {\em loc. cit.}) of $\fL$ in $\CU$ equals $(\alpha-\beta)0+D'$ where
$D'\in(\BP^1-0)^{\beta-\gamma}_\Gamma$.
{\bf Proposition.} $\ooGQ^\alpha_{w,\eta}$ is smooth for arbitrary
$\alpha,\eta,w$, i.e. the fine Schubert strata are smooth.
{\em Proof} will be given in ~\ref{sometimes}.
\subsubsection{Fine Schubert stratification of $\CG\CZ_\eta^\alpha$}
\label{fine GZ}
Similarly, we have $$\CG\CZ_\eta^\alpha=
\bigsqcup\oGZ_{w,\chi}^\gamma\times(\BC^*)^{\beta-\gamma}_\Gamma$$
Here the union is taken over dominant $\chi\leq\eta$ in $Y^+$,
representatives $w\in\CW_f/\CW_\chi$, arbitrary
$\gamma\leq\beta\leq\alpha\in Y$, and partitions $\Gamma\in\fP(\beta-\gamma)$.
Furthermore, $\oGZ_{w,\chi}^\gamma\subset\oGZ_\chi^\gamma$ is an open subvariety
formed by all the collections $(\CU_\lambda,\fL_\lambda)_{\lambda\in X^+}$ such
that $(\CU_\lambda)\in\CG_{w,\chi}\subset\oCG_\chi$,
and $\fL_\lambda$ is a {\em line subbundle} in $\CU_\lambda$.
The stratum $\oGZ_{w,\chi}^\gamma\times(\BC^*)^{\beta-\gamma}_\Gamma$ is formed
by the collections $(\CU_\lambda,\fL_\lambda)_{\lambda\in X^+}$ such that
$\bp(\CU_\lambda,\fL_\lambda)\in\CG_{w,\chi}\subset\oCG_\eta$; the normalization
of $\fL$ in $\CU$ has degree $\gamma$; and the defect $D$
of $\fL$ in $\CU$ equals $(\alpha-\beta)0+D'$ where
$D'\in(\BC^*)^{\beta-\gamma}_\Gamma$.
\subsubsection{}
\label{unwilling}
The reader unwilling to believe that $\oGZ^\gamma_{w,\chi}$
is smooth for arbitrary $\gamma\in Y$ may repeat the trick of
~\ref{awful}. Namely, one can replace $\CG\CZ^\alpha_\eta$ with an open
subvariety ${\ddot{\CG\CZ}}^\alpha_\eta$ formed by the union of the above
strata for sufficiently dominant $\gamma$; they are easily seen to be smooth.
Moreover, $\bq({\ddot{\CG\CZ}}^\alpha_\eta)\supset\ddZ^{\eta+\alpha}$, and
${\ddot{\CG\CZ}}^\alpha_\eta\supset\bq^{-1}(\ddZ^{\eta+\alpha})$.
\subsection{}
\label{fine local}
Let us introduce the {\em fine Schubert stratification} of the local
convolution diagram (see ~\ref{lcd}). Iwahori subgroup
$\bI\subset\bG(\CO)$ acts on $\tbQ$, and defines the {\em fine Schubert
stratification} of $\tbQ$ by Iwahori orbits $\dbQ_w^\alpha\subset\dbQ^\alpha,\
\alpha\in Y,w\in\CW_f$. Furthermore, we have
$$\CG\bQ_\eta=\bigsqcup\CG\dbQ_{w,\chi}^{-\alpha}$$
Here the union is taken over dominant $\chi\leq\eta$ in $Y^+$, representatives
$w\in\CW_f/\CW_\chi$, and $\alpha\in\BN[I]$. The stratum
$\CG\dbQ_{w,\chi}^{-\alpha}$ consists of collections
$(\fV_\lambda,v_\lambda)_{\lambda\in X^+}$ (vectors $v_\lambda\in\fV_\lambda$
are defined up to multiplication by $\BC^*$) such that
a) $(\fV_\lambda)_{\lambda\in X^+}\in\CG_{w,\chi}\subset\oCG_\eta$;
b) $z^{-\langle\alpha,\lambda\rangle}v_\lambda\in\fV_\lambda$ for all $\lambda$,
but $z^{-\langle\alpha,\lambda\rangle-1}v_\lambda\not\in\fV_\lambda$ for some
$\lambda$.
\section{Convolution}
\subsection{}
\label{hell}
Let $\fA,\fB$ be smooth varieties, and let $\fp:\ \fA\lra\fB$ be a map.
Suppose $\fA$ (resp. $\fB$) is equipped with a stratification $\fS$ (resp.
$\fT$), and $\fp$ is stratified with respect to the stratifications.
Let $\fR$ be another stratification of $\fB$, transversal to $\fT$.
Let $\CB$ (resp. $\CA$) be a perverse sheaf on $\fB$ (resp. $\fA$) smooth
along $\fR$ (resp. $\fS$). Let $b=\dim\fB$.
{\em Lemma.} a) $\CA\otimes\fp^*\CB[-b]$ is perverse;
b) $\CA\otimes\fp^*\CB[-b]=\CA\stackrel{!}{\otimes}\fp^!\CB[b]$;
c) Let $\ol{R}$ (resp. $\ol{S}$) be the closure of a stratum $R$ in $\fR$
(resp. $S$ in $\fS$). Then $\IC(\ol{S})\otimes\fp^*\IC(\ol{R})[-b]=
\IC(\ol{S}\cap\fp^{-1}\ol{R})$;
d) Let $R$ be a stratum of stratification $\fR$. Then $\fp^{-1}R$ is smooth.
{\em Proof.} a,b)
Let $g:\ \fG\lra\fA\times\fB$ denote the closed embedding of
the graph of $\fp$. The perverse sheaf $\CA\boxtimes\CB$ on $\fA\times\fB$
is smooth along the product stratification $\fS\times\fR$. The transversality
of $\fR$ and $\fT$ implies that the embedding $g$ is noncharacteristic with
respect to $\CA\boxtimes\CB$ (see ~\cite{ks}, Definition 5.4.12).
Furthermore, by definition, $\CA\otimes\fp^*\CB=g^*(\CA\boxtimes\CB)$, and
$\CA\stackrel{!}{\otimes}\fp^!\CB=g^!(\CA\boxtimes\CB)$. Now a) is nothing
else than ~\cite{ks}, Corollary 10.3.16(iii), while b) is nothing else than
~\cite{ks}, Proposition 5.4.13(ii). $\Box$
c) We consider $\ol{S}\times\ol{R}$ as a subvariety of $\fA\times\fB$.
Since $g:\ \fG\lra\fA\times\fB$ is noncharacteristic with respect to
$\IC(\ol{S})\boxtimes\IC(\ol{R})=\IC(\ol{S}\times\ol{R})$, we conclude that
$g^*\IC(\ol{S}\times\ol{R})[-b]=\IC(\fG\cap(\ol{S}\times\ol{R}))$.
It remains to note that $\fG\cap(\ol{S}\times\ol{R})=\ol{S}\cap\fp^{-1}\ol{R}$,
and $g^*\IC(\ol{S}\times\ol{R})=\IC(\ol{S})\otimes\fp^*\IC(\ol{R})$. $\Box$
d) We will view $\fp^{-1}R$ as a subscheme of $\fA$ (scheme-theoretic fiber
over $R$). Let $a\in\fp^{-1}R\subset\fA$.
The Zariski tangent space $T_a(\fp^{-1}R)$ equals $d\fp_a^{-1}(T_bR)$
where $b=\fp(a)$, and $d\fp_a:\ T_a\fA\lra T_b\fB$
stands for the differential of $\fp$ at $a$.
Let $\CT$ be a stratum of $\fT$ containing $b=\fp(a)$. Then $T_b\CT$ is
contained in $d\fp_a(T_a\fA)$ since $\fp$ is stratified with respect to $\fT$.
Furthermore, $T_bR+T_b\CT=T_b\fB$ due to the transversality of $\CT$ and $R$.
Hence $T_bR+d\fp_a(T_a\fA)=T_b\fB$. Hence $\dim(d\fp_a^{-1}(T_bR))=
\dim\fA-\dim\fB+\dim R$. We conclude that the dimension of the Zariski tangent
space $T_a(\fp^{-1}R)$ is independent of $a\in\fp^{-1}R$, and thus $\fp^{-1}R$
is smooth. $\Box$
\subsection{}
\label{j}
Consider the following cartesian diagram:
$$
\begin{CD}
\CG\CQ^\alpha_\eta @>\bi>> \fQ^\alpha \\
@V{\bp}VV @V{\bp}VV \\
\oCG_\eta @>{\bi}>> \fM
\end{CD}
$$
{\bf Proposition.}
a) For a $\bG[[z]]$-equivariant perverse sheaf $\CF$ on $\oCG_\eta$ the sheaf
$\IC(\fQ^\alpha)\otimes\bp^*\bi_*\CF[-\dim\ufM^\eta]$ is supported on
$\CG\CQ^\alpha_\eta$ and is perverse;
b) $\IC(\CG\CQ_\eta^\alpha)=\IC(\fQ^\alpha)\otimes\bp^*\bi_*\IC(\oCG_\eta)
[-\dim\ufM^\eta]$.
{\em Proof.} a) Let us restrict the right column of the above diagram to
the open subscheme $\fM^\eta\subset\fM$, and take its quotient by $\bG^\eta$.
We obtain the cartesian diagram
$$
\begin{CD}
\CG\CQ^\alpha_\eta @>\bi>> \ufQ^{\alpha,\eta} \\
@V{\bp}VV @V{\bp}VV \\
\oCG_\eta @>{\bi}>> \ufM^\eta
\end{CD}
$$
of schemes of finite type. Here the rows are closed embeddings, and
$\ufM^\eta$ is smooth. The stratification $\fS$ of $\fQ^\alpha$ is invariant
under the action of $\bG^\eta$, and descents to the same named quotient
stratification of $\ufQ^{\alpha,\eta}$. Similarly, the stratification of $\fM$
by the isomorphism type of $\bG$-torsors descents to the stratification
$\fT$ of $\ufM^\eta$. The sheaf $\bi_*\CF$ on $\ufM^\eta$ is smooth along
the stratification $\fR$ transversal to $\fT$. We have to prove that
$\IC(\ufQ^{\alpha,\eta})\otimes\bp^*\bi_*\CF[-\dim\ufM^\eta]$ is perverse.
In order to apply the Lemma ~\ref{hell}a) we only have to find an embedding
$\ufQ^{\alpha,\eta}\stackrel{u}{\hookrightarrow}\fA$ into a smooth scheme
such that the map $\bp$ and stratification $\fS$ extend to $\fA$. Then we
apply the Lemma ~\ref{hell}a) to the sheaf $\CA=u_*\IC(\ufQ^{\alpha,\eta})$
on $\fA$.
We will construct $\fA$ as a projective bundle over $\ufM^\eta$. The points of
$\ufM^\eta$ are the $\bG$-torsors $\CT$ over $C$ trivialized in some
infinitesimal neighbourhood of $\infty\in C$.
The points of $\ufQ^{\alpha,\eta}$ are the
$\bG$-torsors $\CT$ over $C$ trivialized
in some infinitesimal neighbourhood of $\infty\in C$ along with collections
of invertible subsheaves $\fL_\lambda\subset\CV^\CT_\lambda, \lambda\in X^+$,
satisfying Pl\"ucker relations.
Now, if $\alpha=\sum_{i\in I}a_ii$ is dominant enough,
the fiber of $\fA$ over $\CT\in\ufM^\eta$
is $\prod_{i\in I}\BP(\Gamma(C,\CV^\CT_{\omega_i}(a_i)))$. The map $u$
sends $(\CT,(\fL_\lambda\subset\CV^\CT_\lambda)_{\lambda\in X^+})$ to
$(\fL_{\omega_i}(a_i))\in
\prod_{i\in I}\BP(\Gamma(C,\CV^\CT_{\omega_i}(a_i)))$.
If $\alpha$ is not dominant enough, we first embed $\ufQ^{\alpha,\eta}$
into $\ufQ^{\beta+\alpha,\eta}$ for dominant enough $\beta$ as follows:
$(\CT,(\fL_\lambda\subset\CV^\CT_\lambda)_{\lambda\in X^+})\mapsto
(\CT,(\fL_\lambda(-\langle\beta,\lambda\rangle0)\subset
\CV^\CT_\lambda)_{\lambda\in X^+})$. Then we compose with the above projective
embedding of $\ufQ^{\beta+\alpha,\eta}$.
This completes the proof of a). $\Box$
b) We apply the Lemma ~\ref{hell}c) to $\ol{R}=\oCG_\eta,\
\ol{S}=\ufQ^{\alpha,\eta}$. $\Box$
\subsection{Proposition}
\label{+}
a) For a perverse sheaf $\CF$ in $\CP(\oCG_\eta,\bI)$ the sheaf
$\IC(\fQ^\alpha)\otimes\bp^*\bi_*\CF[-\dim\ufM^\eta]$ is supported on
$\CG\CQ^\alpha_\eta$ and is perverse;
b) $\IC(\CG\CQ_{w,\eta}^\alpha)=\IC(\fQ^\alpha)\otimes\bp^*\bi_*
\IC(\oCG_{w,\eta})[-\dim\ufM^\eta]$.
{\em Proof.} The same as the proof of the Proposition ~\ref{j}; we need only
to find a refinement $\fW$ of the stratification
$\fM=\bigsqcup_{\eta\in Y^+}\fM_\eta$ which would be transversal to the
Iwahori orbits $\CG_{w,\chi}$ in $\CG$.
Now $\fM=\bigsqcup_{\eta\in Y^+}\fM_\eta$ is the stratification by the
orbits of proalgebraic group $\bG[[z^{-1}]]$ acting naturally on $\fM$.
The desired refinement is the stratification by the orbits of subgroup
$\bI_-\subset\bG[[z^{-1}]]$ formed by the formal loops $g(z)\in\bG[[z^{-1}]]$
such that $g(\infty)\in\bB_-$. $\Box$
\subsubsection{}
Recall the notations of ~\ref{some}.
{\bf Conjecture.} a) The map $\obp:\ \oGQ^\alpha_\eta\lra\oCG_\eta$ is smooth
onto its image;
b) Up to a shift,
$\IC(\fQ^\alpha)\otimes\bp^*\bi_*\CF[-\dim\ufM^\eta]=\bj_{!*}\obp^*\CF$
for any perverse sheaf $\CF$ in $\CP(\oCG_\eta,\bI)$.
\subsection{Proof of the Propositions ~\ref{fine GQ} and ~\ref{fineS GQ}}
\label{sometimes}
We apply the Lemma ~\ref{hell}d) to the following situation:
$\fA=\overset{\circ}\ufQ{}^{\alpha,\eta}\subset\ufQ^{\alpha,\eta},\
\fB=\ufM^\eta,\ R=\CG_{w,\eta},\ \fp=\obp$. The stratification $\fT$ is defined
as follows. Recall the stratification
$\fW$ of $\fM$ by $\bI_-$-orbits introduced
in the proof of ~\ref{+}. It is invariant under the action of $\bG^\eta$
and descends to the desired stratification $\fT$ of $\ufM^\eta$ transversal
to $R$.
It remains to note that
$\fA=\overset{\circ}\ufQ{}^{\alpha,\eta}=\obp^{-1}(\fM^\eta)/\bG^\eta$
is smooth being a quotient by the free group action of an open subscheme
$\obp^{-1}(\fM^\eta)$ of the smooth scheme $\ofQ^\alpha$. Thus the
assumptions of ~\ref{hell}d) are in force, and we conclude that
$\ooGQ_{w,\eta}^\alpha=\obp^{-1}(\CG_{w,\eta})$ is smooth.
The proof of smoothness of $\ooGQ_\eta^\alpha$ is absolutely similar. $\Box$
\subsection{}
\label{z}
We define the following locally closed subscheme
$\fZ^\alpha\subset\fQ^\alpha$. Its points are
the $\bG$-torsors $\CT$ over $C$ trivialized
in the formal neighbourhood of $\infty\in C$ along with collections
of invertible subsheaves $\fL_\lambda\subset\CV^\CT_\lambda, \lambda\in X^+$,
satisfying Pl\"ucker relations plus two more conditions:
a) in some neighbourhood of $\infty\in C$ the invertible subsheaves
$\fL_\lambda\subset\CV^\CT_\lambda$ are line subbundles. Thus they may be
viewed as a reduction of $\CT$ to $\bB\subset\bG$ in this neighbourhood.
Since $\CT$ is trivialized in the formal neighbourhood of $\infty\in C$,
we obtain a map from this neighbourhood to the flag manifold $\bX$.
b) The value of the above map at $\infty\in C$ equals $\bB_-\in\bX$.
We have the following cartesian diagram:
$$
\begin{CD}
\CG\CZ^\alpha_\eta @>\bi>> \fZ^\alpha \\
@V{\bp}VV @V{\bp}VV \\
\oCG_\eta @>{\bi}>> \fM
\end{CD}
$$
{\bf Proposition.}
For a perverse sheaf $\CF\in\CP(\oCG_\eta,\bI)$ the sheaf
$\IC(\fZ^\alpha)\otimes\bp^*\bi_*\CF[-\dim\ufM^\eta]$ is supported on
$\CG\CZ^\alpha_\eta$ and is perverse.
{\em Proof.} Similar to the proof of the Proposition ~\ref{j}. $\Box$
\subsubsection{Remark}
\label{none}
Let us denote the embedding of $\fZ^\alpha$ into $\fQ^\alpha$ by $s$.
One can easily check that $\IC(\fZ^\alpha)=s^*\IC(\fQ^\alpha)[-\dim\bX]$.
\subsection{}
\label{isomor}
Recall the notations of ~\ref{sig}.
{\bf Proposition.} a) $\bq:\ \CG\CQ_\eta^\alpha\lra\CQ^{\eta+\alpha}$
(resp. $\bq:\ \CG\CZ_\eta^\alpha\lra\CZ^{\eta+\alpha}$) is proper;
b) Restriction of $\bq:\ \CG\CQ_\eta^\alpha\lra\CQ^{\eta+\alpha}$
(resp. $\bq:\ \CG\CZ_\eta^\alpha\lra\CZ^{\eta+\alpha}$) to $\qp^{\eta+\alpha}
\subset\CQ^{\eta+\alpha}$ (resp. to $\zp^{\eta+\alpha}\subset\CZ^{\eta+\alpha}$)
is an isomorphism.
{\em Proof.} a) is evident.
b) It suffices to consider the case of
$\bq:\ \CG\CQ^\alpha_\eta\lra\CQ^{\eta+\alpha}$.
Let $(\fL_\lambda)_{\lambda\in X^+}\in\CQ^{\eta+\alpha}$.
Then $(\CU_\lambda,\fL_\lambda')_{\lambda\in X^+}\in\bq^{-1}
(\fL_\lambda)_{\lambda\in X^+}$ iff for any $\lambda\in X^+$ we have
$\fL_\lambda'=\fL_\lambda(\langle\eta,\lambda\rangle0)$, and
$\CU_\lambda\supset\CV_\lambda(-\langle\eta,\lambda\rangle0)+\fL_\lambda'$.
Consider $\fL_\lambda=\CV_\lambda^{\bN_-}(-\langle\eta+\alpha,
\lambda\rangle\infty)$,
so that $\varphi=(\fL_\lambda)_{\lambda\in X^+}\in\qp^{\eta+\alpha}$.
Then $(\CU_\lambda,\fL_\lambda')_{\lambda\in X^+}\in\bq^{-1}
(\fL_\lambda)_{\lambda\in X^+}$ iff for any $\lambda\in X^+$ we have
$\fL_\lambda'=\fL_\lambda(\langle\eta,\lambda\rangle0)=
\CV_\lambda^{\bN_-}(\langle\eta,\lambda\rangle0-\langle\eta+
\alpha,\lambda\rangle\infty)$, and
$\CU_\lambda\supset\CV_\lambda(-\langle\eta,\lambda\rangle0)+\fL_\lambda'=
\CV_\lambda(-\langle\eta,\lambda\rangle0)+
\CV_\lambda^{\bN_-}(\langle\eta,\lambda\rangle0)$.
In other words, $(\CU_\lambda)_{\lambda\in X^+}$ lies in the intersection of
$\oCG_\eta$ with the semiinfinite orbit $T_\eta$ (see ~\ref{Iwasawa}). This
intersection consists exactly of one point (see ~\cite{mv}). Thus
$\bq^{-1}(\varphi)$ consists of one point.
Recall the cartesian diagram ~\ref{cartes}.
We have $\fs(\qp^{\eta+\alpha})\subset\dbQ^0$ (notations of ~\ref{QQ}),
in particular, $\fs(\varphi)\in\dbQ^0$. Since the map $\bq:\ \CG\bQ_\eta\lra
\bQ$ is $\bG(\CO)$-equivariant, and its fiber over $\fs(\varphi)$ consists
of one point, we conclude that $\bq$ is isomorphism over the $\bG(\CO)$-orbit
$\dbQ^0$. Since $\qp^{\eta+\alpha}=\fs^{-1}(\dbQ^0)$, applying the cartesian
diagram ~\ref{cartes}, we deduce that $\bq$ is isomorphism over
$\qp^{\eta+\alpha}.\ \Box$
\subsection{Proof of the Proposition ~\ref{GM}}
\label{later}
We are interested in the stalk of $\IC(\fQ^\alpha)$ at a point
$(\CT,(\fL_\lambda)_{\lambda\in X^+})\in\ofQ^\gamma\times
C^{\beta-\gamma}_\Gamma\subset\fQ^\alpha$. Suppose that the isomorphism class
of $\bG$-torsor $\CT$ equals $\eta\in Y^+$, i.e. $\CT\in\fM_\eta$.
The stalk in question evidently does not depend on a choice of $\CT\in\fM_\eta$
and the defect $D\in C^{\beta-\gamma}_\Gamma$. In particular, we may (and will)
suppose that $\CT\in\bi(\CG_\eta)$, and $D\in (C-0)^{\beta-\gamma}_\Gamma$.
Then one can see easily that the stalk in question is isomorphic, up to a shift,
to the stalk of Goresky-MacPherson sheaf $\IC(\CG\CQ^\alpha_\eta)$ at the
point $(\CT,(\fL_\lambda)_{\lambda\in X^+})\in\CG\CQ^\alpha_\eta$.
On the other hand, according to the Proposition ~\ref{isomor} b), the latter
stalk is isomorphic to the stalk of $\IC(\CQ^{\eta+\alpha})$ at the point
$\bq(\CT,(\fL_\lambda)_{\lambda\in X^+})$. This point has the same defect $D$.
Applying ~\ref{GMQ} we complete the proof of the Proposition ~\ref{GM}. $\Box$
\subsection{}
\label{semismall}
Recall that a map $\pi:\ \CX\lra\CY$ is called {\em dimensionally
semismall} if the following condition holds: let $\CY_m$ be the set
of all points $y\in\CY$ such that $\dim(\pi^{-1}y)\geq m$, then for $m>0$
we have $\codim_\CY\CY_m\geq2m$.
Let us define $\CX_m=\pi^{-1}\CY_m$. Then we can formulate an equivalent
condition of semismallness as follows: for any $m\geq0$ we have
$\codim_\CX\CX_m\geq m$.
Suppose $\CX$ (resp. $\CY$) is equipped with a stratification $\fS$
(resp. $\fT$), and $\pi$ is stratified with respect to $\fS$ and $\fT$.
Then $\pi$ is called {\em stratified semismall} (see ~\cite{mv}) if
$\pi$ is proper, and the restriction $\pi|_S$ to any stratum in $\fS$ is
dimensionally semismall. In this case
$\pi_*=\pi_!$ takes perverse sheaves on $\CX$ smooth along $\fS$ to perverse
sheaves on $\CY$ smooth along $\fT$ (see {\em loc. cit.}).
\subsection{}
\label{q}
Recall the fine Schubert stratifications of $\CQ^{\eta+\alpha}$
(resp. $\CG\CQ_\eta^\alpha$) introduced in ~\ref{fineS Q}
(resp. ~\ref{fineS GQ}). The map
$\bq:\ \CG\CQ_\eta^\alpha\lra\CQ^{\eta+\alpha}$ is stratified with respect
to these stratifications.
{\bf Proposition.}
$\bq:\ \CG\CQ_\eta^\alpha\lra\CQ^{\eta+\alpha}$ is stratified semismall.
{\em Proof} will use a few Lemmas.
\subsubsection{Lemma}
\label{lemma1}
$\bq:\ \CG\CQ_\eta^\alpha\lra\CQ^{\eta+\alpha}$ is dimensionally semismall.
{\em Proof.}
Recall that we have $\CQ^{\eta+\alpha}=\sqcup_{\beta\leq
\eta+\alpha}\qp^\beta$. It is enough to prove that for
$(\fL_\lambda)_{\lambda\in X^+}\in\qp^\beta$ we have
$\dim\bq^{-1}(\fL_\lambda)_{\lambda\in X^+}\leq |\eta+\alpha-\beta|$.
Let us start with the case $\fL_\lambda=
\CV_\lambda^{\bN_-}(\langle\beta-\eta-\alpha,\lambda\rangle0-
\langle\beta,\lambda\rangle\infty)$. Then, like in the Proposition
~\ref{isomor}, we have
$\bq^{-1}(\fL_\lambda)_{\lambda\in X^+}=\oCG_\eta\cap
\overline{T}_{\beta-\alpha}$, and
according to ~\cite{mv}, we have $\dim(\oCG_\eta\cap
\overline{T}_{\beta-\alpha})\leq
|\eta+\alpha-\beta|$.
Now $\qp^\beta$ is stratified by the defect: $\qp^\beta=
\sqcup^{\beta\geq\gamma\geq0}_{\Gamma\in\fP(\beta-\gamma)}
\qc^\gamma\times (C-0)^{\beta-\gamma}_\Gamma$, and $\bq$ is evidently
stratified with respect to this stratification. The point
$(\CV_\lambda^{\bN_-}(\langle\beta-\eta-\alpha,\lambda\rangle0-\langle\beta,
\lambda\rangle\infty))_{\lambda\in X^+}$ lies in the smallest (closed) stratum
$\gamma=0,\Gamma=\{\{\beta\}\}$. Since the dimension of preimage is a
lower semicontinuous function on $\qp^\beta$, we conclude that for
any point $(\fL_\lambda)_{\lambda\in X^+}\in\qp^\beta$ we have
$\dim\bq^{-1}(\fL_\lambda)_{\lambda\in X^+}\leq|\eta+\alpha-\beta|$.
$\Box$
\subsubsection{}
\label{lemma2}
Recall the fine stratification of $\CG\CQ^\alpha_\eta$
(resp. of $\CQ^{\eta+\alpha}$) introduced in ~\ref{fine GQ}
(resp. in ~\ref{fineS Q}).
{\em Lemma.}
$\bq:\ \CG\CQ_\eta^\alpha\lra\CQ^{\eta+\alpha}$ is stratified semismall
with respect to fine stratifications.
{\em Proof.} We consider a fine stratum
$\ooGQ_\chi^\gamma\times(\BP^1-0)^{\beta-\gamma}_\Gamma\subset
\CG\CQ^\alpha_\eta$ (see ~\ref{fine GQ}).
Temporarily we will write $\bq^\alpha_\eta$ for $\bq$ to stress its dependence
on $\eta$ and $\alpha$.
The restriction of $\bq^\alpha_\eta$ to the stratum
$\ooGQ_\chi^\gamma\times(\BP^1-0)^{\beta-\gamma}_\Gamma$ decomposes into
the following composition of morphisms:
$$\ooGQ_\chi^\gamma\times(\BP^1-0)^{\beta-\gamma}_\Gamma
\stackrel{a\times\id}{\hookrightarrow}
\CG\CQ_\chi^\gamma\times(\BP^1-0)^{\beta-\gamma}_\Gamma
\stackrel{\bq^\gamma_\chi\times\id}{\lra}
\CQ^{\chi+\gamma}\times(\BP^1-0)^{\beta-\gamma}_\Gamma
\stackrel{b}{\hookrightarrow}
\CQ^{\eta+\alpha}$$
Here $a$ is the open inclusion; and $b(\fL,D')=
\fL((\beta-\alpha+\chi-\eta)0-D')$.
Now $\bq^\gamma_\chi\times\id$ is semismall according to the Lemma
~\ref{lemma1}. This completes the proof of the Lemma. $\Box$
\subsubsection{Lemma}
\label{lemma3}
The restriction of $\bq$ to the fine Schubert stratum
$\ooGQ^\alpha_{w,\eta}\subset\CG\CQ^\alpha_\eta$ is dimensionally
semismall for any $w\in\CW_f/\CW_\eta$.
{\em Proof.} Let $\bK\subset\bI\subset\bG(\CO)$
denote the first congruence subgroup
formed by the loops $g(z)\in\bG(\CO)$ such that $g(0)=1$.
The point $\eta(z)\in\CG_{e,\eta}\subset\CG_\eta$ was introduced in
~\ref{Iwahori}. For a positive integer $m$ the subset $(\CG\CQ^\alpha_\eta)_m$
(with respect to $\bq:\ \CG\CQ^\alpha_\eta\lra\CQ^{\eta+\alpha}$)
was introduced in ~\ref{semismall}.
{\em Claim 1.} $\codim_{\ooGQ_\eta^\alpha}(\ooGQ_\eta^\alpha)_m=
\codim_{\obp^{-1}(g\cdot\bK\cdot\eta(z))}[\obp^{-1}(g\cdot\bK\cdot\eta(z))
\cap(\ooGQ^\alpha_\eta)_m]$ for any $m\geq0$ and $g\in\bG$.
In effect, due to the $\bG$-equivariance of $\obp$ and $\bq$, the RHS does
not depend on a choice of $g\in\bG$, so it suffices to consider $g=e$.
The stabilizer of $\eta(z)$ in $\bG$ is nothing else than the parabolic
subgroup $\bP(I_\eta)$ introduced in ~\ref{Iwahori}.
We have
$$\CG_\eta=\bG\times_{\bP(I_\eta)}[\bK\cdot\eta(z)];\
\ooGQ_\eta^\alpha=\bG\times_{\bP(I_\eta)}[\obp^{-1}(\bK\cdot\eta(z))];\
(\ooGQ_\eta^\alpha)_m=\bG\times_{\bP(I_\eta)}
[\obp^{-1}(\bK\cdot\eta(z))\cap(\ooGQ^\alpha_\eta)_m]$$
The Claim follows.
{\em Claim 2.}
$\codim_{\obp^{-1}(g\cdot\bK\cdot\eta(z))}
[\obp^{-1}(g\cdot\bK\cdot\eta(z))\cap(\ooGQ^\alpha_\eta)_m]=
\codim_{\ooGQ^\alpha_{w,\eta}}
[\ooGQ^\alpha_{w,\eta}\cap(\ooGQ^\alpha_\eta)_m]$
for any $m\geq0,\ g\in\bG$, and $w\in\CW_f/\CW_\eta$.
In effect, let us choose $g$ in the normalizer of $\bH$ representing $w$.
Let us denote by $\bP_w$ the intersection $\bP(I_\eta)\cap g\bP(I_\eta)g^{-1}$.
Then we have
$$\ooGQ^\alpha_{w,\eta}=\bP(I_\eta)\times_{\bP_w}
[\obp^{-1}(g\cdot\bK\cdot\eta(z))];\
\ooGQ^\alpha_{w,\eta}\cap(\ooGQ^\alpha_\eta)_m=\bP(I_\eta)\times_{\bP_w}
[\obp^{-1}(g\cdot\bK\cdot\eta(z))\cap(\ooGQ^\alpha_\eta)_m]$$
The Claim follows.
Comparing the two Claims we obtain
$$\codim_{\ooGQ^\alpha_{w,\eta}}(\ooGQ^\alpha_{w,\eta})_m\geq
\codim_{\ooGQ^\alpha_{w,\eta}}[\ooGQ^\alpha_{w,\eta}\cap(\ooGQ^\alpha_\eta)_m]=
\codim_{\ooGQ^\alpha_\eta}(\ooGQ^\alpha_\eta)_m\geq m$$
The last inequality holds by the virtue of the Lemma ~\ref{lemma1}.
This completes the proof of the Lemma. $\Box$
\subsubsection{} Now we are ready to finish the proof of the Proposition.
It remains to show that the restriction of $\bq$ to any fine Schubert stratum
is dimensionally semismall. It follows from the Lemma ~\ref{lemma3}
in the same way as the Lemma ~\ref{lemma2} followed from the
Lemma ~\ref{lemma1} (``twisting by defect'').
This completes the proof of the Proposition. $\Box$
\subsection{Corollary}
\label{exact}
The functor $\bq_*=\bq_!$ takes perverse sheaves on $\CG\CQ_\eta^\alpha$
smooth along the fine Schubert stratification to perverse sheaves on
$\CQ^{\eta+\alpha}$ smooth along the fine Schubert stratification. $\Box$
\subsection{} A few remarks are in order.
\subsubsection{Remark}
Recall the fine Schubert stratification of the local convolution diagram
introduced in ~\ref{lcd} and ~\ref{fine local}.
The arguments used in the proof of the Proposition ~\ref{q} along with the
Lemma ~\ref{codime} show that the map $\bq:\ \CG\bQ_\eta\lra\bQ$ is stratified
semismall with respect to the fine Schubert stratification.
\subsubsection{Remark}
The same arguments as in the proof of Lemma ~\ref{lemma3} show that the
convolution $\CA*\CB$ of perverse sheaves on $\CG$ (see
~\cite{lus}, or ~\cite{g}, ~\cite{mv})
is perverse if $\CB$ is $\bG(\CO)$-equivariant.
In the particular case $\CA\in\CP(\CG,\bI)$ this was also proved by
G.Lusztig in ~\cite{l5} using calculations in the affine Hecke algebra.
\subsection{Theorem}
\label{tough}
Let $\eta\in Y^+,\alpha\in\BN[I]$.
Consider the following diagram:
$$
\begin{CD}
\oCG_\eta @<{\bp}<< \CG\CQ^\alpha_\eta @>{\bq}>> \CQ^{\eta+\alpha} \\
@V{\bi}V{\subset}V @V{\bi}V{\subset}V @. \\
\fM @<\bp<< \fQ^\alpha @. {}
\end{CD}
$$
For a perverse sheaf $\CF$ in $\CP(\oCG_\eta,\bI)$, the sheaf
$\bc^\alpha_\CQ(\CF):=\bq_*(\IC(\fQ^\alpha)\otimes\bp^*\CF)[-\dim\ufM^\eta]$ on
$\CQ^{\eta+\alpha}$ is perverse and smooth along
the fine Schubert stratification.
{\em Proof.} By the virtue of ~\ref{+} we know that
$\IC(\fQ^\alpha)\otimes\bp^*\CF[-\dim\ufM^\eta]$ is a perverse sheaf on
$\CG\CQ^\alpha_\eta$. In order to apply the Corollary ~\ref{exact} we have
to check that
$\IC(\fQ^\alpha)\otimes\bp^*\CF[-\dim\ufM^\eta]$ is smooth along the fine
Schubert stratification. The sheaf $\bp^*\CF$ is evidently smooth along
the fine Schubert stratification. The sheaf $\IC(\fQ^\alpha)$ is constant
along the stratification by defect (see ~\ref{GM}), hence
$\bi^*\IC(\fQ^\alpha)$ is smooth along the fine Schubert stratification.
This completes the proof of the Theorem. $\Box$
\subsection{Conjecture}
Let $\eta\in Y^+,\alpha\in Y$ be such that $\eta+\alpha\in\BN[I]$.
Consider the following diagram:
$$
\begin{CD}
\oCG_\eta @<{\bp}<< \CG\CQ^\alpha_\eta @>{\bq}>> \CQ^{\eta+\alpha} \\
@V{\bi}V{\subset}V @V{\bi}V{\subset}V @. \\
\fM @<\bp<< \fQ^\alpha @. {}
\end{CD}
$$
For a perverse sheaf $\CF$ in $\CP(\oCG_\eta,\bI)$, the sheaf
$\bq_*(\IC(\fQ^\alpha)\otimes\bp^*\CF)[-\dim\ufM^\eta]$ on
$\CQ^{\eta+\alpha}$ is perverse and smooth along
the fine Schubert stratification.
\subsection{Corollary}
\label{bunk}
Let $\eta\in Y^+,\alpha\gg0$.
Consider the following diagram:
$$
\begin{CD}
\oCG_\eta @<{\bp}<< {\ddot{\CG\CZ}}^\alpha_\eta @>\bq>> \ddZ^{\eta+\alpha} \\
@V{\bi}V{\subset}V @V{\bi}V{\subset}V @. \\
\fM @<\bp<< \fZ^\alpha @. {}
\end{CD}
$$
(notations of ~\ref{unwilling} and ~\ref{z}).
For a perverse sheaf $\CF\in\CP(\oCG_\eta,\bI)$, the sheaf
$\bc^\alpha_\CZ(\CF):=\bq_*(\IC(\fZ^\alpha)\otimes\bp^*\CF)[-\dim\ufM^\eta]$
on $\ddZ^{\eta+\alpha}$ is perverse and smooth along
the fine Schubert stratification.
{\em Proof.} Let us denote by $s$ the locally closed embedding
$\ddZ^{\eta+\alpha}\stackrel{s}{\hookrightarrow}\CQ^{\eta+\alpha}$.
Also, temporarily, let us denote the maps $\bp$ and $\bq$ from the diagram
~\ref{tough} (resp. ~\ref{bunk}) by $\bp_\CQ$ and $\bq^\CQ$ (resp. $\bp_\CZ$
and $\bq^\CZ$) to stress their difference.
Then we have
$\bq^\CZ_*(\IC(\fZ^\alpha)\otimes\bp_\CZ^*\CF)[-\dim\ufM^\eta]=
s^*\bq^\CQ_*(\IC(\fQ^\alpha)\otimes\bp_\CQ^*\CF)[-\dim\ufM^\eta-\dim\bX]$
(cf. ~\ref{none}).
We also have $\codim_{\CQ^{\eta+\alpha}}\ddZ^{\eta+\alpha}=\dim\bX$, and
the fine Schubert strata in $\ddZ^{\eta+\alpha}$ are intersections of
$\ddZ^{\eta+\alpha}$ with the fine Schubert strata in $\CQ^{\eta+\alpha}$.
One can check readily that the functor $s^*[-\dim\bX]$ takes perverse
sheaves on $\CQ^{\eta+\alpha}$ smooth along the fine Schubert stratification
to perverse sheaves on $\ddZ^{\eta+\alpha}$ smooth along the fine Schubert
stratification. The application of ~\ref{tough} completes the proof of the
Corollary. $\Box$
\subsection{Conjecture}
Let $\eta\in Y^+,\alpha\in Y$ be such that $\eta+\alpha\in\BN[I]$.
Consider the following diagram:
$$
\begin{CD}
\oCG_\eta @<{\bp}<< \CG\CZ^\alpha_\eta @>{\bq}>> \CZ^{\eta+\alpha} \\
@V{\bi}V{\subset}V @V{\bi}V{\subset}V @. \\
\fM @<\bp<< \fZ^\alpha @. {}
\end{CD}
$$
For a perverse sheaf $\CF$ in $\CP(\oCG_\eta,\bI)$, the sheaf
$\bq_*(\IC(\fZ^\alpha)\otimes\bp^*\CF)[-\dim\ufM^\eta]$ on
$\CZ^{\eta+\alpha}$ is perverse and smooth along
the fine Schubert stratification.
\subsection{}
Now we will compare $\bc_\CZ^\alpha(\CF)$ for a fixed
$\CF\in\CP(\oCG_\eta,\bI)$
and various $\alpha$. Recall the notations of ~\ref{snop} and ~\ref{awful}.
{\bf Proposition.} For any $\beta,\gamma\in\BN[I], \varepsilon>0,$ there is a
factorization isomorphism
$$\bc_\CZ^{\beta+\gamma-\eta}\CF|_{\ddZ^{\beta,\gamma}_{\Ue,\Upe}}\iso
\bc_\CZ^{\beta-\eta}\CF|_{\ddZ^\beta_\Ue}\boxtimes
\IC^\gamma|_{\ddZ^\gamma_\Upe}$$
{\em Proof.} Follows easily from ~\ref{idiot}. $\Box$
\subsection{} The above Proposition shows that we can organize the collection
$(\bc_\CZ^{\alpha-\eta}\CF)$ for $\alpha\in\BN[I]$ into a snop $\bc_\CZ\CF$.
Namely, we set the support estimate $\chi(\bc_\CZ\CF)=\eta,\
(\bc_\CZ\CF)^\alpha_\eta=\bc_\CZ^{\alpha-\eta}\CF$. This way we obtain an
exact functor $\bc_\CZ:\ \CP(\CG,\bI)\lra\PS$.
\section{Examples of convolution}
\subsection{}
\label{stalks}
Let $\CF$ be a perverse sheaf in $\CP(\oCG_\eta,\bI)$.
For $\chi\leq\eta, w\in\CW_f/\CW_\chi$ we have $\CG_{w,\chi}\subset\oCG_\eta$.
The sheaf $\CF$ is constant along $\CG_{w,\chi}$, and we denote by
$\CF_{w,\chi}$ its stalk at any point in $\CG_{w,\chi}$.
{\em Lemma.} The stalk of $\IC(\fQ^\alpha)\otimes\bp^*\CF$ at any
point in a fine Schubert stratum $\ooGQ^\gamma_{w,\chi}\times
(\BC^*)^{\beta-\gamma}_\Gamma\subset\CG\CQ_\eta^\alpha$ (see ~\ref{fineS GQ})
equals $\CF_{w,\chi}\otimes\IC^{\alpha-\beta}_\Gamma$ (see ~\ref{GMZ}).
{\em Proof.} Follows immediately from the Proposition ~\ref{GM}.
$\Box$
\subsection{}
\label{Satake}
Let $\bG^L$ be the Langlands dual group. Its character lattice
coincides with $Y$, and the dominant characters are exactly $Y^+$. For
$\eta\in Y^+$ we denote by $W_\eta$ the irreducible $\bG^L$-module with the
highest weight $\eta$. For $\chi\in Y$ we denote by $_{(\chi)}W_\eta$ the
weight $\chi$-subspace of $W_\eta$.
Let $\IC(\oCG_\eta)$ denote the Goresky-MacPherson sheaf of $\oCG_\eta$.
A natural isomorphism $H^\bullet(\oCG_\eta,\IC(\oCG_\eta))\iso W_\eta$ is
constructed in ~\cite{mv}.
Recall that for $\chi\in Y,\ w\in\CW_f,$ the irreducible snop $\CL(w,\chi)$
was introduced in ~\ref{CL}. The following result was suggested by V.Ginzburg.
{\bf Theorem.} There is a natural isomorphism of snops:
$$\bc_\CZ\IC(\oCG_\eta)\iso\bigoplus_{\chi\in Y}\ _{(\chi)}W_\eta\otimes
\CL(w_0,\chi)$$
{\em Proof.} It is a reformulation of the main result of ~\cite{mv}.
In effect, by the Proposition ~\ref{j}b) we know that
$\IC(\fQ^\alpha)\otimes\bp^*\IC(\oCG_\eta)[-\dim\ufM^\eta]=
\IC(\CG\CQ_\eta^\alpha)$. So
we have to prove that $\bq_*\IC(\CG\CQ_\eta^\alpha)=
\bigoplus_{0\leq\beta\leq\eta+\alpha}\
_{(\beta-\alpha)}W_\eta\otimes\IC(\CQ^\beta)$. Here we make use of the
filtration $\CQ^{\eta+\alpha}=\bigcup_{0\leq\beta\leq\eta+\alpha}\CQ^\beta$
subject to the stratification $\CQ^{\eta+\alpha}=
\bigsqcup_{0\leq\beta\leq\eta+\alpha}\qp^\beta$ by the defect at $0\in C$.
We know that $\bq$ is proper, semismall, and stratified with respect to the
above stratification. By the Decomposition Theorem (see ~\cite{bbd}), we
have {\em a priori}
$\bq_*\IC(\CG\CQ_\eta^\alpha)=\bigoplus_{0\leq\beta\leq\eta+\alpha}
L_\beta\otimes\IC(\CQ^\beta)$ for some vector spaces $L_\beta$. To identify
$L_\beta$ with $_{(\beta-\alpha)}W_\eta$ it suffices to compute the stalks at
$\phi=(\fL_\lambda)_{\lambda\in X^+}\in\qp^\beta$ where $\fL_\lambda=
\CV_\lambda^{\bN_-}(\langle\beta-\eta-\alpha,\lambda\rangle\cdot0-
\langle D,\lambda\rangle)$ for some $D\in(\BP^1-0)^\beta_\Gamma$.
As in the proof of ~\ref{lemma1} we have
$\bq^{-1}(\phi)=\oCG_\eta\cap\overline{T}_{\beta-\alpha}=
\bigsqcup_{\gamma\geq0}
(\oCG_\eta\cap T_{\beta-\alpha+\gamma})$.
According to the Lemma ~\ref{stalks}, we have
$\IC(\CG\CQ_\eta^\alpha)|_{\oCG_\eta\cap T_{\beta-\alpha+\gamma}}=
\IC(\oCG_\eta)|_{\oCG_\eta\cap T_{\beta-\alpha+\gamma}}\otimes
\IC^\gamma_\Gamma$. According to ~\cite{mv}, we have
$H^\bullet_c(\oCG_\eta\cap T_{\beta-\alpha+\gamma},\IC(\oCG_\eta))=\
_{(\beta-\alpha+\gamma)}W_\eta$. Due to the parity vanishing (see
{\em loc. cit.} and ~\ref{parity}), the spectral sequence computing
$H^\bullet(\oCG_\eta\cap\overline{T}_{\beta-\alpha},\IC(\CG\CQ_\eta^\alpha))$
collapses and gives
$H^\bullet(\oCG_\eta\cap\overline{T}_{\beta-\alpha},\IC(\CG\CQ_\eta^\alpha))=
\bigoplus_{0\leq\gamma\leq\eta+\alpha-\beta}\
_{(\beta-\alpha+\gamma)}W_\eta\otimes\IC^0_{\{\{\gamma\}\}}=
\bigoplus_{0\leq\gamma\leq\eta+\alpha-\beta}\
_{(\beta-\alpha+\gamma)}W_\eta\otimes\IC(\CQ^{\beta+\gamma})_\phi$.
This completes the proof of the Theorem. $\Box$
|
1997-08-11T19:46:39 | 9707 | alg-geom/9707018 | en | https://arxiv.org/abs/alg-geom/9707018 | [
"alg-geom",
"math.AC",
"math.AG",
"q-alg"
] | alg-geom/9707018 | James M. Turner | James M. Turner | On Simplicial Commutative Rings with Vanishing Andr\'e-Quillen Homology | 14 pages, LaTeX2e | null | null | null | null | We propose a generalization of a conjecture of D. Quillen, on the vanishing
of Andr\'e-Quillen homology, to simplicial commutative rings. This conjecture
characterizes a notion of local complete intersection, extended to the
simplicial setting, under a suitable hypothesis on the local characteristic.
Further, under the condition of finite-type homology, we then prove the
conjecture in the case of a simplicial commutative algebra augmented over a
field of non-zero characteristic. As a consequence, we obtain a proof of
Quillen's conjecture for a Noetherian commutative algebra - again augmented
over a field of non-zero characteristic.
| [
{
"version": "v1",
"created": "Thu, 31 Jul 1997 17:38:39 GMT"
},
{
"version": "v2",
"created": "Mon, 11 Aug 1997 17:46:38 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Turner",
"James M.",
""
]
] | alg-geom | \section{Introduction}
In \cite{And} and \cite{Qui2}, M. Andr\'e and D. Quillen constructed the
notion of a homology\\ $D_*(A|R;M)$ for a commutative algebra $A$,
over a ring $R$, and an $A$-module $M$. It was then conjectured
(see Section 5 of \cite{Qui2}) that, under suitable conditions on
$R$ and $A$, the vanishing of the homology in sufficiently
high degrees determines $A$ as a local complete intersection. In
particular, for local rings, the conjecture takes the following form.
\begin{conjecture}\label{con1.1}
Let $R$ be a (Noetherian, commutative) local ring with residue
field $\Bbb{F}$, and let $D_s(\Bbb{F}|R) = D_s(\Bbb{F}|R;\Bbb{F}), \ s
\geq 0$. Then the following are equivalent:
\begin{itemize}
\item[(1)]
$D_s(\Bbb{F}|R) = 0, \quad s >> 0$;
\item[(2)] $D_s(\Bbb{F}|R) = 0, \quad s \geq 3$;
\item[(3)] $R$ is a complete intersection.
\end{itemize}
\end{conjecture}
In this form, \ref{con1.1} was proven by L. Avramov, in outline form,
in \cite{Avr}, and, in much greater generality, in \cite{Avr2}.
Recall that a local ring $R$ is a complete intersection if its $I$-adic
completion $\hat{R}$ is a quotient of a complete regular ring by an
ideal generated by a regular sequence. From this description and the
properties of Andr\'e-Quillen homology, the implications (3)
$\Rightarrow$ (2) $\Rightarrow$ (1) in \ref{con1.1} are immediate.
The objective of this paper is to extend a version of \ref{con1.1} for
simplicial local rings in an effort to bring the full power of
simplicial homotopy theory to bear on this type of problem and thereby
obtain a different proof of \ref{con1.1} closer in spirit to the
topological results of J.-P. Serre in \cite{Serre} and Y. Umeda in
\cite{Ume}.
In \cite{Qui3}, D. Quillen gave a construction of Andr\'e-Quillen
homology $D_*(A|B;M)$ where $B$ is a simplicial commutative ring, $A$ a
simplicial commutative $B$-algebra, and $M$ a simplicial $A$-module.
Let ${\mathcal R}_{\Bbb{F}}$ be the category of (commutative) local rings, with
residue field $\Bbb{F}$, and $s {\mathcal R}_{\Bbb{F}}$ the category of
simplicial objects over ${\mathcal R}_{\Bbb{F}}$. It follows from \cite{Qui1}
that $s {\mathcal R}_{\Bbb{F}}$ has three classes of maps, called weak
equivalences, fibrations, and cofibrations, giving it the structure of a
closed simplicial model category. Using this structure, we say that a
simplicial local ring $R$ is an {\em $n$-extension} if there
is a cofibration sequence
$$
\Sigma^{n-1} S_0 \rightarrow R \rightarrow S_{1},
$$
in the homotopy category $Ho(s {\mathcal R}_{\Bbb{F}})$, such that $S_{0}$
is polynomial in ${\mathcal R}_{\Bbb{F}}$ and $\hat{S_{1}} \cong
\Sigma^{n}\bar{S}_{1}$ in $Ho(s {\mathcal R}_{\Bbb{F}})$. Here $\Sigma$ denotes
the suspension in $Ho(s {\mathcal R}_{\Bbb{F}})$. (See $\S$I.2 and $\S$I.3 of
\cite{Qui1} for the theory of suspension and cofibration sequences in
homotopical algebra.)
\begin{definition}\label{def1.2}
Let $R$ be an object of $s {\mathcal R}_{\Bbb{F}}$. Then:
\begin{itemize}
\item[(1)] $R$ is {\bf regular} if $R$ is a 1-extension
with $\bar{S}_{1}$ smooth in ${\mathcal R}_{\Bbb{F}}$.
\item[(2)] $R$ is a {\bf complete intersection} if it is a
1-extension with $\bar{S}_1$ regular in $s {\mathcal R}_{\Bbb{F}}$.
\item[(3)] $R$ is {\bf $Q$-bounded} if $(Q \pi_* R)_s = 0$ for $s >> 0$,
and {\em bounded} if $\pi_s R = 0$ for $s \gg 0$.
\item[(4)] The {\bf simplicial dimension} of $R$ is the integer
$$
s \cdot \dim R = \max\{s| \, D_s(\Bbb{F}|R) \neq 0\}.
$$
We then say that $R$ has {\bf finite simplicial dimension} if $s \cdot
\dim R < \infty$.
\item[(5)] $R$ is said to have {\bf finite-type homology} provided each
$D_q(\Bbb{F}|R)$ is a finite dimensional $\Bbb{F}$-vector space.
\item[(6)] If $R$ has both finite-type homology and finite simplicial
dimension, we call $R$ {\bf finite}.
\end{itemize}
Given a simplicial commutative ring $R$, then $R$ is said to be {\em
locally} of any one of (1) -- (6) provided $R_{\wp}$ is such, for each
simplicial prime ideal $\wp$ in $R$. Given a simplicial prime ideal $\wp$
in $R$, we denote by $\Bbb{F}(\wp)$ the residue field of $R_{\wp}$ and we say
that $R$ is {\em locally of non-zero characteristic} provided
char$\Bbb{F}(\wp)\neq 0$ for all such $\wp$.
\end{definition}
We can now state our proposed simplicial generalization of \con{con1.1}.
\begin{vanishingconjecture}\label{vancon1.3}
Let $R$ be a locally finite simplicial commutative ring which is locally
of non-zero characteristic. Then $R$ is a locally complete intersection
if and only if $R$ is locally $Q$-bounded.
\end{vanishingconjecture}
In the rational case, while a complete intersection may be both
$Q$-bounded and of finite simplicial dimension, the converse is not true.
See the note following \prop{prop3.7} regarding counter-examples.
To demonstrate the validity of \ref{vancon1.3}, we consider the
subcategory, ${\mathcal A}_{\Bbb{F}}$, of ${\mathcal R}_{\Bbb{F}}$, consisting of
augmented $\Bbb{F}$-algebras, i.e., unitary $\Bbb{F}$-algebras $A$
together with a fixed $\Bbb{F}$-algebra map $A \rightarrow \Bbb{F}$, called
the augmentation of $A$.
In this paper, we give evidence for \vancon{vancon1.3} by proving:
\begin{theorem}\label{thm1.5}
Suppose $A$ is a finite simplicial augmented commutative
$\Bbb{F}$-algebra with char$\Bbb{F} > 0$. Then $A$ is bounded if and
only if $A$ is a complete intersection.
\end{theorem}
\begin{corollary}\label{cor1.5}
Let $A$ be an augmented Noetherian commutative $\Bbb{F}$-algebra,
$char\Bbb{F} > 0$. Then $A$ is a complete intersection, as a local algebra,
if and only if $A$ has finite simplicial dimension.
\end{corollary}
{\em Proof.} By IV.55 of \cite{And}, $H^{Q}_{*}(A)$ is of finite-type.
Thus, by \thm{thm1.5}, $A$ has finite simplicial dimension if and
only if $A$ is a complete intersection, as a simplicial algebra, if
and only if $A$ is a complete intersection, as a local algebra, by
the classical implication of (2) $\Rightarrow$ (3) in \con{con1.1}
(see Proposition 26 of \cite{And}). \hfill $\Box$
\bigskip
\subsection*{Organization of this paper}
In this section, we review the closed simplicial model category
structure for $s {\mathcal A}_{\Bbb{F}}$ and the construction and properties of
homotopy and Andr\'e-Quillen homology. In Section 3, we describe
the notion Postnikov envelopes for objects
of $s {\mathcal A}_{\Bbb{F}}$ and explore its properties. In Section 4, we
study the homotopy of n-extensions. Finally, in Section 5, we
introduce and study the notion of a Poincar\'e series for a simplicial
algebra, obtaining just enough information to prove \thm{thm1.5}.
\subsection*{Acknowledgements}
The author would like to thank Haynes Miller, for suggesting this
project along with the direction it should take, and Jean Lannes for
many useful directions as well as for making his stay in France
worthwhile. Most of the work on this project was done while the
author was visiting the Institut des Hautes \'Etudes Scientifique and
the Ecole Polytechnique. He would like to thank them for their
hospitality and use of their facilities during his stay. Finally, the
author would like to thank Julie Riddleburger for putting this paper
into LaTeX form.
\section{Homotopy Theory of Simplicial Augmented\\ Commutative Algebras}
\setcounter{equation}{0}
We now review the closed simplicial model category structure for $s
{\mathcal A}_{\Bbb{F}}$. We will assume the reader is familiar with the
general theory of homotopical algebra given in \cite{Qui1}.
We call a map $f: \ A \rightarrow B$ in $s {\mathcal A}_{\Bbb{F}}$ a
\begin{itemize}
\item[(i)] weak equivalence ($\stackrel{\sim}{\rightarrow}$)
$\Leftarrow\!\!\Rightarrow \pi_* f$ is an isomorphism;
\item[(ii)] fibration ($\rightarrow\!\!\!\!\rightarrow$)
$\Leftarrow\!\!\Rightarrow f$
surjects in positive degrees;
\item[(iii)] cofibration($\hookrightarrow$) $\Leftarrow \!\!\Rightarrow f$
is a retract of
an almost free map.
\end{itemize}
Here a map $f: \ A \rightarrow B$ in $s {\mathcal A}_{\Bbb{F}}$ is {\em almost
free} if there is an almost simplicial $\Bbb{F}$-vector space (no $d_0$)
$V$ (see \cite{Goe1}) together with a map of almost simplicial
$\Bbb{F}$-vector spaces $V \rightarrow IB$ such that the induced map $A
\otimes S(V) \stackrel{\cong}{\longrightarrow} B$ is an isomorphism of
almost simplicial algebras. Here $S$ is the symmetric algebra functor.
Now given a finite simplicial set $K$ and a simplicial algebra $A$,
define $A \wedge K$ and $A^K$ by
$$
(A \wedge K)_n = \bigotimes_{K_{n}} A_n
$$
and
$$
(A^K)_n = \prod_{K_{n}} A_n.
$$
Here the tensor product $\otimes $ is the coproduct in $s
{\mathcal A}_{\Bbb{F}}$. The product in $s {\mathcal A}_{\Bbb{F}}$ is defined as
$\Lambda \times_{\Bbb{F}} \Gamma$, for $\Lambda, \Gamma$ in $s
{\mathcal A}_{\Bbb{F}}$, so that the diagram
$$
\begin{array}{ccc}
\Lambda \times_{\Bbb{F}}\Gamma & \longrightarrow & \Gamma \\[2mm]
\downarrow & & \hspace*{10pt} \downarrow \epsilon \\[2mm]
\Lambda & \longrightarrow & \Bbb{F} \\[-3mm]
& \epsilon &
\end{array}
$$
is a pullback of simplicial vector spaces.
\begin{theorem}\label{thm2.1} (\cite{Qui1}, \cite{Mil}, and \cite{Goe1})
With these definitions, $s {\mathcal A}_{\Bbb{F}}$ is a closed simplicial
model category.
\end{theorem}
Given a simplicial vector space $V$, define its normalized chain complex
$NV$ by
\begin{equation}
N_nV = V_n/(\mbox{Im} s_0 + \cdots + \mbox{Im} s_n)
\end{equation}
and $\partial: \ N_n V \rightarrow N_{n-1}V$ is $\partial = \sum^n_{i=0}
(-1)^id_i$. The homotopy groups $\pi_* V$ of $V$ is defined as
$$
\pi_n V = H_n(NV), \quad n \geq 0.
$$
Thus for $A$ in $s {\mathcal A}_{\Bbb{F}}$ we define $\pi_* A$ as above.
The Eilenberg-Zilber theorem (see \cite{Mac}) shows that the algebra
structure on $A$ induces an algebra structure on $\pi_*A$.
If we let ${\mathcal V}$ be the category of $\Bbb{F}$-vector spaces, then there
is an adjoint pair
$$
S: \ {\mathcal V} \Leftarrow\!\!\Rightarrow {\mathcal A}_{\Bbb{F}}: \, I,
$$
where $I$ is the augmentation ideal function and $S$ is the symmetric
algebra functor. For an object $V$ in ${\mathcal V}$ and $n \geq 0$, let
$K(V,n)$ be the associated Eilenberg-MacLane object in $s {\mathcal V}$ so that
$$
\pi_s K(V,n) = \left\{\begin{array}{ll}
V & s = n; \\[2mm]
0 & s \neq n.
\end{array}\right.
$$
Let $S(V,n) = S(K(V,n))$, which is an object of $s {\mathcal A}_{\Bbb{F}}$.
For $A$ in ${\mathcal A}_{\Bbb{F}}$, the indecomposable functor $QA =
I(A)/I^2(A)$ which is an object of ${\mathcal V}$. Furthermore, we have an
adjoint pair
$$
Q: \ {\mathcal A}_{\Bbb{F}} \Leftarrow \!\!\Rightarrow {\mathcal V}: \ (-)_+
$$
where $V_+$, for $V$ in ${\mathcal V}$, is the simplicial algebra $V \oplus
\Bbb{F}$ where
$$
(v,r) \cdot (w,s) = (sv+rw,rs)
$$
for $(v,r), (w,s) \in V \oplus \Bbb{F}$. $(-)_+, Q$ provides an
equivalence between ${\mathcal V}$ and the category of abelian group objects
in ${\mathcal A}_{\Bbb{F}}$.
For $A$ in $s {\mathcal A}_{\Bbb{F}}$, we define its Andr\'e-Quillen homology,
as per \cite{Goe1} and \cite{Goe2}, by
$$
H^Q_s(A) = \pi_s QX, \quad s \geq 0,
$$
where we choose a factorization
$$
\Bbb{F} \hookrightarrow X \stackrel{\sim\hspace*{5pt}}{\rightarrow \!\!\!\!\!
\rightarrow} A
$$
of the unit $\Bbb{F} \rightarrow A$ as a cofibration and a trivial
fibration. This definition is independent of the choice of
factorization as any two are homotopic over $A$ (note that every object
of $s {\mathcal A}_{\Bbb{F}}$ is fibrant). It is known (see, for example,
\cite{Mil}) that
$$
H^Q_s(A) = D_s(A|\Bbb{F};\Bbb{F}).
$$
From the transitivity sequence, one can easily check that
$D_0(\Bbb{F}|A) = 0$, and $D_{s+1}(\Bbb{F}|A)\\ \cong H^Q_s(A)$ for all
$s \geq 0$.
Now, as shown in \cite{Goe2},
$$
\pi_n A = [S(n),A],
$$
where $S(n) = S(\Bbb{F},n)$ and $[\quad,\quad]$ denotes the morphisms in
$Ho(s {\mathcal A}_{\Bbb{F}})$. Thus the primary operational structure for
the homotopy groups in $s {\mathcal A}_{\Bbb{F}}$ is determined by $\pi_*
S(V_0)$ for any $V_0$ in $s {\mathcal V}$. By Dold's theorem \cite{Dold} there
is a triple ${\mathcal S}$ on graded vector spaces so that
\begin{equation}
\pi_* S(V) \cong {\mathcal S}(\pi_* V)
\end{equation}
encoding this structure. If char$\Bbb{F} = 0$, ${\mathcal S}$ is the free skew
symmetric functor and, if char$\Bbb{F} > 0$, ${\mathcal S}$ is the free divided
power algebra on the underlying vector space of a certain free algebra
constructed from the input (see, for example, \cite{Bou} and \cite{Goe1}).
Now recall that maps $A \stackrel{f}{\rightarrow} B
\stackrel{g}{\rightarrow} C$ is a {\em cofibration sequence} in $Ho(s
{\mathcal A}_{\Bbb{F}})$ if $f$ is isomorphic to a cofibration $X
\stackrel{u}{\rightarrow} Y$, of cofibrant objects, with cofibre $Y
\stackrel{v}{\rightarrow} Z$ isomorphic to $g$. Thus given any map $f: \
A \rightarrow B$ in $s {\mathcal A}_{\Bbb{F}}$ there is a cofibration sequence $A
\stackrel{f}{\rightarrow} B \rightarrow M(f)$ in $Ho(s {\mathcal A}_{\Bbb{F}})$ formed
by factoring $\Bbb{F} \rightarrow A$ into $\Bbb{F} \hookrightarrow \bar{A}
\stackrel{\sim}{\rightarrow\!\!\!\!\rightarrow} A$, form the diagram
$$
\begin{array}{ccc}
\bar{A} & \hookrightarrow & X \\[2mm]
s
\begin{picture}(1,1)\put(1,1) {$\downarrow$} \put(1,5)
{$\downarrow$}\end{picture}
\hspace*{10pt} &&
\begin{picture}(1,1)\put(1,1) {$\downarrow$} \put(1,5)
{$\downarrow$}\end{picture}
\hspace*{10pt} \wr \\[2mm]
A & \stackrel{f}{\rightarrow} & B
\end{array}
$$
and then let $M(f) = X \otimes_{\bar{A}}\Bbb{F}$, which is cofibrant. As
an example, the {\em suspension} $\Sigma A$ of an object
$A$ in $s {\mathcal A}_{\Bbb{F}}$ by $M(\epsilon)$, where $\epsilon: \ A \rightarrow
\Bbb{F}$ is the augmentation.
Finally, recall that the {\it completion} $\hat{A}$ of a simplicial augmented
algebra $A$ is defined as
$$
\hat{A} = lim_{t} A/I^{t}.
$$
If $f: A \to B$ is a map of simplicial algebras, we denote by
$\hat{f}: \hat{A} \to \hat{B}$ the induced map of completions.
We can now summarize methods for computing homotopy and Andr\'e-Quillen
homology that we will need for this paper.
\begin{proposition}\label{prop2.4}
\begin{itemize}
\item[(1)] If $f: \ A \stackrel{\sim}{\rightarrow} B$ is a weak equivalence
in $s {\mathcal A}_{\Bbb{F}}$, then $H^Q_*(f): \ H^Q_*(A)
\stackrel{\cong}{\rightarrow} H^Q_*(B)$ is an isomorphism.
\item[(2)] for any $A$ in $s {\mathcal A}_{\Bbb{F}}$ there is a spectral sequence
$$
E^1_{s,t} = {\mathcal S}_{s} (H^Q_*(A)) \Rightarrow \pi_t \hat{A},
$$
called Quillen's Fundamental spectral sequence (see \cite{Qui2} and
\cite{Qui3}), which converges when $H^{Q}_{0}(A)=0$.
\item[(3)] There is a Hurewicz homomorphism $h: \ I\pi_* A \rightarrow
H^Q_*(A)$ such that if $A$ is connected and $H^Q_s (A) = 0$ for $s<n$
then $A$ is $n$-connected and $h: \ \pi_n A \stackrel{\cong}{\rightarrow}
H^Q_n(A)$ is an isomorphism.
\end{itemize}
\end{proposition}
{\em Proof.}
(1) is a standard result. See, for example, \cite{Qui2} or \cite{Goe2}.
For (2), see chapter IV of \cite{Goe1} and \cite{Tur}. Finally, (3) is
in \cite{Goe1}. \hfill $\Box$
\bigskip
The following is a selection of results from \cite{Tur}.
\begin{proposition}\label{compprop}
Let $A$ and $B$ be in $s {\mathcal A}_{\Bbb{F}}$. Then
\begin{itemize}
\item[(1)] if $f:A \to B$ is an $H^{Q}_{*}$-isomorphism then
$\hat{f}: \hat{A} \to \hat{B}$ is a weak equivalence,
\item[(2)] if $A$ is connected then $\pi_{*}\hat{A} \cong \pi_{*}A$,
\item[(3)] $H^{Q}_{*}(\hat{A}) \cong H^{Q}_{*}(A)$, and
\item[(4)] $Q\pi_{*}\hat{A} \cong Q\pi_{*}A$.
\end{itemize}
\end{proposition}
{\em Remark.} If $H^{Q}_{0}(A) = 0$ then \prop{compprop} (4) follows
from a Quillen fundamental spectral sequence argument. This is
due to the fact that while this spectral sequence doesn't directly
converge to $\pi_{*}A$ it does allow, under the above condition on $H^{Q}_{0}$,
sufficient information to be extracted about the indecomposables (see \cite{Tur}
for further details). This case is sufficient for our needs.
\bigskip
\begin{proposition}\label{prop2.5}
Let $A \stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} C$ be a cofibration
sequence in $Ho(s {\mathcal A}_{\Bbb{F}})$. Then:
\begin{itemize}
\item[(1)]
There is a long exact sequence
$$
\begin{array}{l}
\cdots \rightarrow H^Q_{s+1} (C) \stackrel{\partial}{\rightarrow} H^Q_s(A)
\stackrel{H^Q_*(f)}{\rightarrow} H^Q_s(B) \\[3mm]
\hspace*{20pt} \stackrel{H^Q_*(g)}{\rightarrow} H^Q_s(C)
\stackrel{\partial}{\rightarrow} H^Q_{s-1}(C) \rightarrow \cdots
\end{array}
$$
\item[(2)] There is a first quadrant spectral sequence of algebras
$$
E^2_{s,t} = Tor^{\pi_*A}_{s} (\pi_* B, \Bbb{F})_t
\Rightarrow \pi_{s+t} C,
$$
which we refer to as the Eilenberg-Moore spectral sequence.
\item[(3)] If $A$ is connected, there is a first quadrant spectral
sequence of algebras
$$
E^2_{s,t} = \pi_s (C \otimes \pi_t A) \Rightarrow \pi_{s+t} B,
$$
which we refer to as the Serre spectral sequence.
\item[(4)] If $A$ is connected and $C$ is $n$-connected, then there is a
homomorphism $\tau: \ \pi_{n+1} C \rightarrow \pi_n A$, called the {\em
transgression}, such that the diagram
$$
\begin{array}{ccccccc}
\pi_{n+1}B & \stackrel{\pi_* f}{\rightarrow} & \pi_{n+1}C & \stackrel{\tau}{\rightarrow}
& \pi_nA & \stackrel{\pi_* f}{\rightarrow} & \pi_n B \\[1mm]
h \downarrow \hspace*{8pt}
&& h \downarrow \hspace*{8pt}
&& \hspace*{8pt} \downarrow h && \hspace*{8pt} \downarrow h \\[1mm]
H^Q_{n+1}B & \stackrel{H^Q_*f}{\rightarrow} & H^Q_{n+1}(C) &
\stackrel{\partial}{\rightarrow} & H^Q_n(A) & \stackrel{H^Q_*f}{\rightarrow} &
H^Q_n B
\end{array}
$$
commutes and the top sequence is exact.
\end{itemize}
\end{proposition}
{\em Proof.}
(1) is just the transitivity sequence for $H^Q_*$. See \cite{Goe1}.
(2) is the spectral sequence of Theorem 6(b) in $\S$II.6 of \cite{Qui1}.
See also \cite{Goe1}. By Theorem 6(d) in $\S$II.6 of \cite{Qui1}, there
is a $1^{st}$-quadrant spectral sequence
$$
E^2_{*,*} = \pi_* (B \otimes_A \pi_* A) \Rightarrow \pi_* B,
$$
where $\pi_*A$ is an $A$-module via the augmentation $A \rightarrow \pi_0
A$. Here we can assume our cofibration sequence is a cofibration with
cofibre $C$. Since $A$ is connected, then $B \otimes_A \pi_* A \cong C
\otimes \pi_* A$. The algebra structure follows from the construction
of the spectral sequence and the fact that $A \stackrel{f}{\rightarrow} B$
is a map of simplicial algebras. This gives us (3).
For (4), since $A$ is connected and $C$ is $n$-connected, then in the
Serre spectral sequence
$$
d^{n+1}: \ \pi_{n+1} C \cong E^{n+1}_{n+1,0} \rightarrow E^{n+1}_{0,n} \cong
\pi_n A,
$$
which we propose is our desired map $\tau$. From this same spectral
sequence, we have $\pi_s A \cong \pi_s B$, $s < n$, and, using methods
modified from the next section, we can assume that $N_s IC = 0$ for $s
\leq n$ and $N_{n+1}B \rightarrow\!\!\!\!\rightarrow N_{n+1}C$ is surjective.
Since $E^{1}_{n+1,0} = N_{n+1}C$ and $E^2_{n+1,0} \cong E^{n+1}_{n+1,0}
\cong \pi_{n+1}C$, then $d^{n+1}$ is constructable in precisely the same
way as the boundary map in homological algebra. Since we can assume
cofibrancy of our objects under consideration, then the diagram
$$
\begin{array}{ccccccc}
H_{n+1}(NB) & \rightarrow & H_{n+1}(NC) & \stackrel{d^{n+1}}{\rightarrow}
& H_n(NA) & \rightarrow & H_n(NB) \\
\downarrow && \downarrow && \downarrow && \downarrow \\
H_{n+1}(NQB) & \rightarrow & H_{n+1}(NQC) & \stackrel{\partial}{\rightarrow} &
H_n(NQA) & \rightarrow & H_n(NQB)
\end{array}
$$
commutes by naturality. The result follows. \hfill $\Box$
\section{Postnikov Envelopes}
\setcounter{equation}{0}
In this section, we construct and determine some
properties of a useful tool for studying simplicial algebras.
First, we recall the following standard result which will be useful for
us (see section II.4 of \cite{Qui1}).
\begin{lemma}\label{lma3.4}
Let $V$ and $W$ be simplicial vector spaces. Then the map
$$
[V,W] \rightarrow \mbox{Hom}_{{\mathcal V}_{*}}(\pi_* V, \pi_* W)
$$
is an isomorphism.
\end{lemma}
\begin{proposition}\label{prop3.5}
Let $A$ in $s {\mathcal A}_{\Bbb{F}}$. Then
\begin{itemize}
\item[(1)] There is a
map of simplicial algebras
$$
f_{0}:S(H^{Q}_{0}(A),0)\to A
$$
which induces an isomorphism on $H^{Q}_{0}$.
\item[(2)] Suppose $A$ is $(n-1)$-connected for $n \geq 1$.
Then there exists a map in $s {\mathcal A}_{\Bbb{F}}$,
$$
f_n: \ S(H^Q_nA,n) \rightarrow A,
$$
which is an isomorphism on $\pi_n$ and $H^Q_n$.
\end{itemize}
\end{proposition}
{\em Proof.}
(1) Let $\iota: H^{Q}_{0}(A)\to I\pi_{0}A$ be a choice of splitting for
the surjection $I\pi_{0}A \to H^{Q}_{0}(A)$. By \lma{lma3.4}, $f$ can be
chosen to be the adjoint of
the map of simplicial vector spaces $K(H^{Q}_{0}(A),0)\to IA$ induced
by $\iota$. By the transitivity sequence, $H^{Q}_{0}(A(1))=0$ so the
fundamental spectral sequence for $A(1)$ converges, by \prop{prop2.4}
(2), so $A(1)$ is connected.
(2) By the Hurewicz theorem, \prop{prop2.4} (3), the map
$h: \ \pi_nA \rightarrow H^Q_n A$ is an isomorphism. Now the adjoint functors
$$
S: \ s {\mathcal V} \Leftarrow\!\!\Rightarrow s {\mathcal A}_{\Bbb{F}}: \ I
$$
induce an adjoint pair
$$
S: \ Ho(s {\mathcal V}) \Leftarrow\!\!\Rightarrow Ho(s {\mathcal A}_{\Bbb{F}}): \ I.
$$
Thus we have isomorphisms
\begin{eqnarray*}
[S(H^Q_nA,n),A]
& \cong & [K(H^Q_nA,n),IA] \\[2mm]
& \cong & \mbox{Hom}_{{\mathcal V}}(H^Q_n A, \pi_n IA),
\end{eqnarray*}
using \lma{lma3.4}. Choosing $f_n$ to correspond to the inverse of $h$
gives the result. \hfill $\Box$
\bigskip
Now given $A$ we form the {\em Postnikov envelopes} as the
sequence of cofibrations
$$
A(1) \stackrel{j_{2}}{\hookrightarrow} A(2)
\stackrel{j_{3}}{\hookrightarrow} \cdots
\stackrel{j_{n}}{\hookrightarrow}
A(n) \stackrel{j_{n+1}}{\hookrightarrow} \cdots
$$
with the following properties:
\begin{itemize}
\item[(1)] $A(1) = \widehat{M(f_{0})}$,
\item[(2)] for each $n\geq 1$, $A(n)$ is a $(n-1)$-connected and for $s \geq n$,
$$
H^Q_s A(n) \cong H^Q_sA.
$$
\item[(3)] There is a cofibration sequence
$$
S(H^Q_{n} A, n) \rightarrow A(n) \stackrel{j_{n+1}}{\rightarrow} A(n+1).
$$
\end{itemize}
The existence of a Postnikov envelopes follows easily from
\prop{compprop}, \prop{prop3.5}, and
\begin{lemma}\label{lma3.6}
If $A$ is $(n-1)$-connected, for $n\geq 1$, then the cofibre $M(f_n)$ of $f_n: \
S(H^Q_nA,n) \rightarrow A$ is $n$-connected and satisfies $H^Q_sM(f_n) \cong
H^Q_sA$ for $s > n$.
\end{lemma}
{\em Proof.}
This follows from \ref{prop3.5} and the transitivity sequence
$$
H^Q_{s+1}M(f_n) \rightarrow H^Q_s S(H^Q_n A,n) \rightarrow H^Q_s A \rightarrow
H^Q_s M(f_n).
$$
\hfill $\Box$
\bigskip
{\bf Note:} We have been implicitly using the computation
$$
H^Q_s S(V,n) = \pi_s QS(V,n) = \pi_s K(V,n) = V
$$
for $s = n$ and 0 otherwise. The converse holds as well.
\begin{proposition}\label{prop3.7}
Let $A$ be connected in $s {\mathcal A}_{\Bbb{F}}$ and suppose $H^Q_s A = 0, \
s \neq n > 0$. Then $A \cong S(H^Q_n A,n)$ in $Ho(s {\mathcal A}_{\Bbb{F}})$.
\end{proposition}
{\em Proof.} Since $A$ is connected, then $A$ is $(n-1)$-connected by
the Hurewicz theorem. By \ref{prop3.5}, $f_n: \ S(H^Q_nA,n) \rightarrow A$ is
an $H^Q_n$-isomorphism and hence a weak equivalence by
\ref{prop2.4}(2). \hfill $\Box$
\bigskip
{\bf Note.} From this proposition, if char$\Bbb{F}$=0 then $S(V,n)$
has simplicial dimension $n$ and $\pi_{*}S(V,n)$ is free
skew-commutative on a basis of $V$ concentrated in degree $n$. Thus
$S(V,n)$ is $Q$-bounded, for any $n$, showing that \vancon{vancon1.3} fails
in the zero characteristic case.
\bigskip
\section{The Homotopy and Homology of $n$-Extensions}
\setcounter{equation}{0}
Call an object $A$ in $s {\mathcal A}_{\Bbb{F}}$ a {\em simple
$n$-extension} if $A$ is an $n$-extension in $s
{\mathcal A}_{\Bbb{F}}$ with $\bar{S}_1 = S(V_1,0)$, $V_1$ in ${\mathcal V}$.
Also, for this section and the next, we define the {\em simplicial
dimension} of $A$ to be
$$
s \cdot \dim A = \max\{s| \, H^{Q}_{s}(A) \neq 0\}
$$
We now proceed to prove:
\begin{theorem}\label{thm1.4}
Let $A$ be in $s {\mathcal A}_{\Bbb{F}}$. Then:
\begin{itemize}
\item[(1)] If $A$ is a connected simple $n$-extension for $n \geq 2$,
then, in $Ho(s {\mathcal A}_{\Bbb{F}})$, we have
$$
A \cong S(H^Q_{n-1}(A),n-1) \otimes S(H^Q_n(A),n).
$$
\item[(2)] $A$ is a complete intersection if and only if $A$ is a simple
1-extension.
\item[(3)] If $A$ is a complete intersection then $H^Q_s(A) = 0$
for $s \geq 2$ and if $H^{Q}_{*}(A)$ is of finite-type then $A$ is
bounded.
\item[(4)] If $H^{Q}_{0}(A)=0$ and $H^{Q}_{*}(A)$ is of finite-type
then $\pi_{*}(\hat{A})$ is of finite-type.
\item[(5)] The Postnikov envelope $A(1)$ has the following properties:
\begin{itemize}
\item[(a)] If $A$ has finite simplicial dimension, then so does $A(1)$;
\item[(b)] If $H^{Q}_{0}(A)$ is finite and $\pi_{*}A$ is bounded
then $\pi_{*}A(1)$ is $Q$-bounded.
\item[(c)] If $H^{Q}_{*}(A)$ is of finite-type then $H^{Q}_{*}(A(1))$
is also of finite-type.
\end{itemize}
\end{itemize}
\end{theorem}
We begin with
\begin{lemma}\label{lma4.1}
Let $A$ in $s {\mathcal A}_{\Bbb{F}}$ be a connected simple $n$-extension for $n
\geq 2$. Then $A$ is an $n$-extension
of the form
$$
S(H^Q_{n-1} A,n-1) \rightarrow A \rightarrow S(H^Q_n A,n).
$$
\end{lemma}
{\em Proof.} Let $V_0, V_1$ be vector spaces so that there is a
cofibration sequence
$$
S(V_0, n-1) \rightarrow A \rightarrow S(V,n).
$$
Then the transitivity sequence tells us that $H^Q_s A = 0$, $s \neq n, \
n-1$ and there is an exact sequence
$$
0 \rightarrow H^Q_nA \rightarrow V_1 \rightarrow V_0 \rightarrow H^Q_{n-1}A \rightarrow 0.
$$
Thus $A$ is $n-2$ connected and Postnikov tower gives us a cofibration sequence
$$
S(H^Q_{n-1}A, n-1) \rightarrow A \rightarrow A(n-1) = S(H^Q_n A,n).
\eqno\Box
$$
\bigskip
{\em Proof of \thm{thm1.4} (1).} By \lma{lma4.1}, there is a cofibration
sequence
$$
S(H^Q_{n-1},A,n-1) \stackrel{i}{\rightarrow} A \stackrel{j}{\rightarrow} S(H^Q_nA,n),
$$
where we can assume $A$ is cofibrant, $i$ is a cofibration, and $j$ is
the cofibre. Consider the commuting diagram
$$
\begin{array}{ccc}
[S(H^Q_nA,n),A] & \stackrel{j_*}{\longrightarrow} &
[S(H^Q_nA,n),S(H^Q_nA,n)] \\[1mm]
\cong \downarrow \hspace*{10pt}
&&
\hspace*{10pt} \downarrow \cong \\[1mm]
[K(H^Q_n A,n),IA] && [K(H^Q_n A,n), IS(H^Q_n A,n)] \\[1mm]
\cong \downarrow \hspace*{10pt}
&&
\hspace*{10pt} \downarrow \cong \\[1mm]
\mbox{Hom}(H^Q_nA,I \pi_n A) & \stackrel{h_*}{\longrightarrow}
& \mbox{Hom}(H^Q_n A, H^Q_n A)
\end{array}
$$
Then $j$ will split, up to homotopy, if we can show that $h: \ \pi_n A
\rightarrow H^Q_n A$ is onto.
By \prop{prop2.5}(4), there is a commutative diagram
$$
\begin{array}{ccccccc}
\pi_n A & \stackrel{\pi_* j}{\rightarrow} & \pi_n S(H^Q_n A,n)
& \stackrel{\tau}{\rightarrow} & \pi_{n-1} S(H^Q_{n-1},A, n-1) &
\stackrel{\pi_* i}{\rightarrow} & \pi_{n-1} A \\[2mm]
h \downarrow \hspace*{10pt} && \cong \downarrow \hspace*{10pt} &&
\hspace*{10pt} \downarrow \cong && \hspace*{10pt} \downarrow \cong \\[2mm]
H^Q_n A & \stackrel{\cong}{\rightarrow} & H^Q_n A & \stackrel{\partial = 0}{\rightarrow}
& H^Q_{n-1}A & \stackrel{\cong}{\rightarrow} & H^Q_{n-1}A
\end{array}
$$
with the rows exact. Thus $\pi_n j$ is onto and, hence, $h: \ \pi_n A
\rightarrow H^Q_n A$ is onto. \hfill $\Box$
\bigskip
\begin{lemma}\label{lma4.3}
Suppose $A$ in $s {\mathcal A}_{\Bbb{F}}$ is regular. Then
$S(H^{Q}_{0}(A),0) \cong \hat{A}$ in $Ho(s {\mathcal A}_{\Bbb{F}})$.
\end{lemma}
{\em Proof.} By the standard transitivity sequence for $D_*$ applied to
$\Bbb{F} \rightarrow A \rightarrow \Bbb{F}$, $D_0(\Bbb{F}|A) = 0$ and
$D_{s+1}(\Bbb{F}|A) \cong H^Q_s(A)$, so since $A$ is regular, then
$H^Q_s(A) = 0$, $s > 0$. Thus $f_{0}$ is an $H^{Q}_{*}$-isomorphism
and so $\hat{f_{0}}$ is a weak equivalence by \prop{compprop}. \hfill $\Box$
\bigskip
{\em Proof of \thm{thm1.4} (2).} If $A$ is a complete intersection
then it is a 1-extension of the form
$$
S_{0}\rightarrow A \rightarrow S_{1}
$$
with $S_{0}$ polynomial and $\bar{S}_{1}$ regular as simplicial augmented
algebra. By \lma{lma4.3}, A is thus a simple 1-extension. The converse is
clear. \hfill $\Box$
\bigskip
{\em Proof of \thm{thm1.4} (3).} If $A$ is a complete intersection,
then $H^Q_s(A) = 0, \ s \geq 2$ follows (2) and the transitivity
sequence. Consider now the Eilenberg-Moore spectral sequence
$$
E^{2}_{s,t}=Tor^{S(H^{Q}_{0}(A))}_{s}(\pi_{*}A,\Bbb{F})_{t}\Longrightarrow
\pi_{s+t}M(f_{0})
$$
which is a first quadrant homology-type spectral sequence of algebras.
Since $H^{Q}_{0}(A)$ is finite, $S(H^{Q}_{0}(A))$ has finite flat
dimension and since, by \prop{compprop},
$Q\pi_{*}M(f_{0}) = Q\pi_{*}A(1) = Q\pi_{*}S(H^{Q}_{1}(A),1)$ is
finite concentrated in degree 1 then
$\pi_{*}M(f_{0})$ is bounded and we can conclude, by an induction on
$dim_{\Bbb{F}}H^{Q}_{0}(A)$, that $\pi_{*}A$ is bounded.
\hfill $\Box$
\bigskip
{\em Example.} Suppose an augmented commutative $\Bbb{F}$-algebra $B$ is a
complete intersection. Then there is a complete regular algebra $\Gamma$ and
an ideal
$I$, generated by a regular sequence, so that $\Gamma/I \cong \hat{B}$. As we
saw, $\Gamma \cong S(V_0)$, so the condition of regularity on $I$ is
equivalent to there being
a {\em projective extension}, that is, (see \cite{Goe1}) an extension
$$
\Bbb{F} \rightarrow S(V_1) \stackrel{i}{\rightarrow} S(V_0) \rightarrow \hat{B} \rightarrow
\Bbb{F},
$$
so that $i$ makes $S(V_0)$ into a projective $S(V_1)$-module. In
$Ho(s {\mathcal A}_{\Bbb{F}})$, $\hat{B}$ is equivalent $M(i)$ and so there is
a cofibration sequence of the form
$$
S(V_0,0) \rightarrow \hat{B} \rightarrow S(V_1,1).
$$
Thus, $\hat{B}$, and hence $B$, is a complete intersection as a simplicial
algebra.
\hfill $\Box$
\bigskip
{\em Proof of \thm{thm1.4} (4).} Since $H^{Q}_{0}(A)=0$, the
fundamental spectral sequence
$$
E^1_{s,t} = {\mathcal S}_{s} (H^Q_*(A))_{t} \Rightarrow \pi_t \hat{A},
$$
converges. From the known structure of ${\mathcal S}$ (see e.g. \cite{Bou}),
if $V$ is a finite-dimensional vector space then each
${\mathcal S}_{s}(V)_{t}$ is finite and ${\mathcal S}_{s}(V)_{t}=0, s \gg 0$ for
each fixed t. The result follows. \hfill $\Box$
\bigskip
{\em Proof of \thm{thm1.4} (5).} First, (a) is immediate from the
transitivity sequence. For (b), $V = H^{Q}_{0}(A)$ is finite and the
Eilenberg-Moore spectral sequence has the form
$$
E^{2}_{s,t}=Tor^{S(V)}_{s}(\pi_{*}A,\Bbb{F})_{t}\Longrightarrow
\pi_{s+t}M(f_{0})
$$
Since $S(V)$ has finite flat dimension and $\pi_{*}A$ is a graded
$S(V)$-module then
$$
Tor^{S(V)}_{s}(\pi_{*}A,\Bbb{F})_{t} = Tor^{S(V)}_{s}(\pi_{t}A,\Bbb{F})
$$
vanishes for $s \gg 0$ and vanishes for $t \gg 0$ if $\pi_{*}A$ is
bounded. We conclude $\pi_{*}M(f_{0})$ is bounded and hence
$\pi_{*}A(1)$ is $Q$-bounded, by \prop{compprop}. Finally, for (c),
\lma{lma3.6} tells us that $H^{Q}_{s}(A)=H^{Q}_{s}(A(1))$ for
$s \geq 1$. Thus if $H^{Q}_{*}(A)$ is of finite-type
then $H^{Q}_{*}(A(1))$ is of finite-type. \hfill $\Box$
\bigskip
\section{The Poincar\'e Series of a Simplicial Algebra}
\setcounter{equation}{0}
For this section, we assume char$\Bbb{F} = p > 0$.
Let $A$ be a connected simplicial augmented commutative
$\Bbb{F}$-algebra such that $\pi_{*}A$ is of finite-type. We define its
{\em Poincar\'e series} by
$$
\vartheta(A,t) = \sum_{n\geq 0}(dim_{\Bbb{F}}\pi_{n}A)t^{n}.
$$
If $V$ is a finite-dimensional vector space and $n>0$ we write
$$
\vartheta(V,n,t) = \vartheta(S(V,n),t).
$$
Given power series $f(t) = \sum a_{i}t^{i}$ and $g(t) = \sum b_{i}t^{i}$
we define the relation $f(t) \leq g(t)$ provided $a_{i}\leq
b_{i}$ for each $i\geq 0$.
\begin{lemma}\label{poiprop}
Given a cofibration sequence
$$
A \rightarrow B \rightarrow C
$$
of connected objects in ${\mathcal A}_{\Bbb{F}}$ with finite-type homotopy
groups, then
$$
\vartheta(B,t) \leq \vartheta(A,t)\vartheta(C,t)
$$
which is an equality if the sequence is split.
\end{lemma}
{\em Proof.} From the Serre spectral sequence
$$
E^{2}_{s,t}=\pi_{s}(C\otimes\pi_{t}A) \Longrightarrow \pi_{s+t}B
$$
we have
$$
\vartheta(A,t)\vartheta(C,t) =
\sum_{n}(\sum_{s+t=n}dim_{\Bbb{F}}E^{2}_{s,t})t^{n} \geq
\vartheta(B,t).
$$
If the cofibration sequence is split then the spectral sequence
collapses, giving an equality. \hfill $\Box$
\bigskip
If $\Pi$ is a finitely-generated abelian group and $n>0$ let
$$
\vartheta(\Pi,n,t) = \sum_{s}(dim_{\Bbb{F}}H_{s}(K(\Pi,n);\Bbb{F}))t^{s}.
$$
\begin{lemma}\label{lmapoi}
Let $V$ be a finite-dimensional vector space and $\Pi$ a free abelian
group of the same dimension. Then for any $n>0$
$$
\vartheta(V,n,t) = \vartheta(\Pi,n,t).
$$
\end{lemma}
{\em Proof.} As shown in \cite{Car}, there is a weak equivalence of
simplicial vector spaces
$$
S(V,n) \rightarrow \Bbb{F}[K(\Pi,n)]
$$
which gives us the desired result. \hfill $\Box$
\bigskip
\begin{proposition}\label{phiprop}
Given a finite-dimensional vector space $V$ and any $n>0$ the
Poin-\\car\'e series $\vartheta(V,n,t)$ converges in the open unit disc.
\end{proposition}
{\em Proof.} This follows from \lma{lmapoi} and the results of J.P.
Serre in \cite{Serre} and Y. Umeda in \cite{Ume}. \hfill $\Box$
\bigskip
Now given two power series $f(t)$ and $g(t)$ we say $f(t) \sim g(t)$
provided $lim_{t\to \infty }f(t)/g(t)\\ = 1$. Given a Poincar\'e series
$\vartheta(V,n,t)$, for a finite-dimensional $\Bbb{F}$-vector space $V$
and $n>0$, let
$$
\varphi(V,n,t) = log_{p}\vartheta(V,n,1-p^{-t}).
$$
\begin{proposition}\label{poieq}
For $V$ an $\Bbb{F}$-vector space of finite dimension
$q$ and $n>0$ then $\varphi(V,n,t)$ converges on the real line and
$$
\varphi(V,n,t)\sim qt^{n-1}/(n-1)!.
$$
\end{proposition}
{\em Proof.} This follows from \lma{lmapoi} and Th\'eor\`eme 9b in
\cite{Serre}, for char$\Bbb{F}=2$, and its generalization in \cite{Ume}.
\hfill $\Box$
\bigskip
A major step in proving \thm{thm1.5} will be accomplished with
\begin{theorem}\label{connprop}
Let $A$ be a connected finite simplicial augmented commutative
$\Bbb{F}$-algebra. Then if $A$ is $Q$-bounded we have $A \cong
S(H^{Q}_{1}(A),1)$ in $Ho(s {\mathcal A}_{\Bbb{F}})$.
\end{theorem}
{\em Proof.} By \thm{thm1.4} (4) and \prop{compprop}, $\pi_{*}A$ is of
finite-type and
hence, as it is also $Q$-bounded, bounded as well. Let $n = s \cdot \dim
A$. We must show that $n = 1$.
Consider the Postnikov envelope
$$
S(H^{Q}_{s-1}(A),s-1) \rightarrow A(s-1) \rightarrow
A(s)
$$
for each s. From the theory of cofibration sequences (see section I.3 of
\cite{Qui1}) the above sequence extends to a cofibration sequence
$$
A(s-1) \rightarrow A(s) \rightarrow
S(H^{Q}_{s-1}(A),s).
$$
Thus, by \lma{poiprop}, we have
$$
\vartheta(A(s),t) \leq
\vartheta(A(s-1),t)\vartheta(H^{Q}_{s-1}(A),s,t).
$$
Starting at $s=n-1$ and iterating this relation, we arrive at the
inequality
$$
\vartheta(A(n-1),t) \leq \vartheta(A,t)\prod_{s=1}^{n-2}
\vartheta(H^{Q}_{s}(A),s+1).
$$
Now, $A(n) \cong S(H^{Q}_{n}(A),n)$ by \prop{prop3.7},
but, by \thm{thm1.4} (1) and \lma{poiprop}, we have
$$
\vartheta(A(n-1),t) =
\vartheta(H^{Q}_{n-1}(A),n-1,t)\vartheta(H^{Q}_{n}(A),n,t).
$$
Since $\pi_{*}(A)$ is of finite-type and
bounded then there exists a $D>p$ such that
$\vartheta(A,t) \leq D$, in the open unit disc. Combining, we have
$$
\vartheta(H^{Q}_{n-1}(A),n-1,t)\vartheta(H^{Q}_{n}(A),n,t)
\leq D\prod_{s=1}^{n-2}\vartheta(H^{Q}_{s}(A),s+1).
$$
Applying a change of variables and $log_{p}$ to the above inequality, we get
$$
\varphi(H^{Q}_{n-1}(A),n-1,t)+\varphi(H^{Q}_{n}(A),n,t)
\leq d+\sum_{s=1}^{n-2}\varphi(H^{Q}_{s}(A),s+1).
$$
By \prop{poieq}, there is a polynomial $f(t)$ of degree $n-2$,
non-negative integer $a$, and positive integers $b$ and $d$ such that
$$
at^{n-2}+bt^{n-1} \leq d+f(t)
$$
which is clearly false for $n>1$. Thus $n=1$. The rest of the proof
follows from \prop{prop3.7}. \hfill $\Box$
\bigskip
{\em Proof of \thm{thm1.5}.} The ``if'' part is \thm{thm1.4} (3). We
thus concentrate on the ``only if'' part. We are given a finite simplicial
augmented commutative $\Bbb{F}$-algebra $A$ such that $\pi_{*}A$ is
bounded. By \thm{thm1.4} (4) and (5), $A(1)$ is connected, $Q$-bounded,
finite, and $\pi_{*}A(1)$ is of finite-type. Thus $A(1)$ is bounded as
well and we conclude $A(1) \cong S(H^{Q}_{1}(A), 1)$ by \thm{connprop}.
\hfill $\Box$
\bigskip
|
1997-07-24T10:02:49 | 9707 | alg-geom/9707015 | en | https://arxiv.org/abs/alg-geom/9707015 | [
"alg-geom",
"math.AG"
] | alg-geom/9707015 | Arnaud Beauville | Arnaud Beauville | Fano contact manifolds and nilpotent orbits | Plain TeX. Postscript file available at
http://www..dmi.ens.fr/users/beauvill/ | null | null | null | null | A contact structure on a complex manifold M is a corank 1 subbundle F of T(M)
such that the bilinear form on F with values in the quotient line bundle
L=T(M)/F deduced from the Lie bracket of vector fields is everywhere
non-degenerate. In this paper we consider the case where M is a Fano manifold;
this implies that L is ample. If g is a simple Lie algebra, the unique closed
orbit in P(g) (for the adjoint action) is a Fano contact manifold; it is
conjectured that every Fano contact manifold is obtained in this way. A
positive answer would imply an analogous result for compact quaternion-Kahler
manifolds with positive scalar curvature, a longstanding question in Riemannian
geometry. In this paper we solve the conjecture under the additional
assumptions that the group of contact automorphisms of M is reductive, and that
the image of the rational map M - - -> P(H0(M,L)*) associated to L has maximum
dimension. The proof relies on the properties of the nilpotent orbits in a
semi-simple Lie algebra, in particular on the work of R. Brylinski and B.
Kostant.
| [
{
"version": "v1",
"created": "Thu, 24 Jul 1997 08:02:40 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Beauville",
"Arnaud",
""
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\newlabel{main}{0.1
\centerline{\bf Fano contact manifolds and nilpotent orbits}
\smallskip
\smallskip \centerline{Arnaud {\pc BEAUVILLE\note{1}{Partially supported by the
European HCM project ``Algebraic Geometry in Europe" (AGE).}}}
\vskip1truecm
{\bf Introduction}
\smallskip
\par\hskip 1truecm\relax A {\it contact structure} on a complex manifold $M$ is a corank
$1$ subbundle $F\i T_M$ such that the bilinear form on $F$ with values in the
quotient line bundle $L=T_M/F$ deduced from the Lie bracket on $T_M$ is
everywhere non-degenerate. This implies that the dimension of $M$ is odd, say
$\mathop{\rm dim}\nolimits M=2n+1$, and that the canonical bundle $K_M$ is isomorphic to $L^{-n-1}$.
In this paper we will
consider the case where $M$ is compact and $L$ is
{\it ample}, that is, $M$ is a {\it Fano manifold}.
\par\hskip 1truecm\relax This turns out to be a strong restriction on the manifold $M$; the only
examples known so far are obtained as follows (see Prop.\ \ref{Omin} and
\ref{nilorb} below). Let ${\goth g}$ be a simple complex Lie algebra; the
adjoint group acting on ${\bf P}({\goth g})$ has exactly one closed orbit
${\bf P}{\cal O}_{\rm min}$, which is the projectivization of the {\it minimal
nilpotent orbit} ${\cal O}_{\rm min}\i {\goth g}$. The Kostant-Kirillov
symplectic structure on ${\cal O}_{\rm min}$ defines a contact structure on
${\bf P}{\cal O}_{\rm min}$.
\par\hskip 1truecm\relax It is generally conjectured that
{\it every Fano contact manifold is obtained in this way}. This problem is
motivated by Riemannian geometry, more precisely by the study of
compact {\it quaternion-K\"ahler} manifolds. I~will say only a few words here,
referring for instance to [L-S], [L] and the bibliography therein for a more
complete treatment. A quaternion-K\"ahler manifold $\Q$ is a Riemannian
manifold with holonomy ${\it Sp}(n){\it Sp}(1)$. It carries a natural $S^2$\kern-1.5pt -
bundle $M\rightarrow \Q$, the {\it twistor space}, which turns out to be a
complex contact manifold; moreover if $\Q$ is compact and its scalar curvature
is positive, $M$ is a Fano contact manifold. The only known examples of
positive quaternion-K\"ahler manifolds are certain symmetric spaces
associated to each compact simple Lie group, the so-called ``Wolf spaces";
thanks to the work of LeBrun and Salamon, a positive answer to the above
conjecture would imply that every compact quaternion-K\"ahler manifold with
positive scalar curvature is isometric to a Wolf space.
\par\hskip 1truecm\relax Our result is the following:
\th Theorem
\enonce Let $M$ be a Fano contact manifold, satisfying the following
conditions:
\par\hskip 0.5truecm\relax{\rm (H1)} The rational map $\varphi^{}_L:M\dasharrow {\bf
P}(\H^0(M,L)^*)$ associated to the line bundle $L$ is generically finite {\rm
(}that is, $\mathop{\rm dim}\nolimits
\varphi^{} _L(M)=\mathop{\rm dim}\nolimits M${\rm );}
\par\hskip 0.5truecm\relax{\rm (H2)} The group $G$ of contact automorphisms of $M$ is reductive.
\par\hskip 1truecm\relax Then the Lie algebra ${\goth g}$ of $G$ is simple, and $M$ is isomorphic
to the minimal orbit
${\bf P}{\cal O}_{\rm min}\i{\bf P}({\goth g})$.
\endth\label{main}
\par\hskip 1truecm\relax While hypothesis (H1) is rather strong,
(H2) is harmless from the point of view of Riemannian geometry: by the
results of [L], it always holds for the twistor spaces of positive
quaternion-K\"ahler manifolds.
\par\hskip 1truecm\relax We will get an apparently stronger result, namely that $M$ and
${\bf P}{\cal O}_{\rm min}$ are isomorphic as {\it contact} complex manifolds.
It is however a general fact that whenever two
compact simply-connected contact manifolds are isomorphic, the isomorphism can
be chosen compatible with the contact structures ([L], Prop.\ 2.3).
\par\hskip 1truecm\relax The strategy of the proof is as follows. Using some elementary symplectic
geometry, the map $\varphi^{}_L$ can be viewed as a ``contact moment map"
$M\rightarrow {\bf P}({\goth g})$. Then (H1) implies that
$G$ has an open orbit in $M$, whose image by $\varphi^{}_L$ is a nilpotent
orbit ${\bf P}{\cal O}\i {\bf P}({\goth g})$. We are thus led to classify
finite $G$\kern-1.5pt - equivariant coverings $M\rightarrow \overline{{\bf
P}{\cal O}}$, where $M$ is smooth. Examples of such coverings appear in
[B-K], with $M$ being the minimal orbit in ${\bf P}({\goth g}')$ for some
simple Lie algebra ${\goth g}'$ containing ${\goth g}$; our key result is
that all possible examples arise essentially in this way. Theorem \ref{main}
follows then easily.
\vskip1truecm
\section {Contact geometry}
\par\hskip 1truecm\relax Let $M$ be a complex contact projective manifold. Recall that the
contact structure is given by an exact sequence $$0\rightarrow
F\longrightarrow T_M\qfl{\theta }L\rightarrow 0\ ,$$such that the
(${\cal O}_M$\kern-1.5pt - bilinear) alternate form $(X,Y)\mapsto \theta ([X,Y])$ on $F$
is everywhere non-degenerate. Alternatively the contact structure can be
described by the twisted 1-form $\theta\in \H^0(M,\Omega^1_M\otimes L)$,
the {\it contact form}. \par\hskip 1truecm\relax We denote by $G$ the neutral component of the
group of automorphisms of
$M$ preserving $F$. This is an algebraic group, whose Lie
algebra
${\goth g}$ consists of the vector
fields $X\in \H^0(M, T_M)$ such that $[X,F]\i F$. The following result is
well-known (see e.g.\ [L]):
\th Proposition
\enonce The map
$\H^0(\theta ):\H^0(M,T_M)\rightarrow
\H^0(M, L)$ maps ${\goth g}$ isomorphically onto $\H^0(M,L)$.
\endth\label{split}
{\it Proof}: Let us first prove the decomposition $\H^0(M,T_M)=\H^0(M,F)\oplus
{\goth g}$. Let $X\in
\H^0(M,T_M)$. The map $U\mapsto \theta([X,U])$ from $F$ to $L$ is ${\cal
O}_M$\kern-1.5pt - linear, hence there exists a unique vector field $X'$ in $F$ such
that $\theta([X,U])=\theta([X',U])$ for all $U$ in $F$. This means that
$[X-X',U]$ belongs to $F$, that is that $X-X'$ belongs to ${\goth g}$. Writing
$X=X'+(X-X')$ provides the required direct sum decomposition.
\par\hskip 1truecm\relax Let ${\cal L}\i T_M$ be the subsheaf of infinitesimal contact
transformations. Applying the above result to each open subset of $M$ we get
$T_M=F\oplus {\cal L}$, so that $\theta$ induces a (${\bf C}$\kern-1.5pt - linear)
isomorphism of ${\cal L}$ onto
$L$. Our statement follows by taking global sections.\cqfd
\smallskip
\global\def\currenvir{subsection\label{equi} For each $g\in G$ the automorphism $T(g)$ of $T_M$
induces an automorphism of $L$ above $g$; in other words, the
line bundle $L$ has a canonical $G$\kern-1.5pt - linearization. In particular the group
$G$ acts on $\H^0(M,L)$; the isomorphism $\theta :{\goth g}\rightarrow
\H^0(M,L)$ is $G$\kern-1.5pt - equivariant with respect to this action and the adjoint
action on ${\goth g}$. Also the rational map $\varphi^{} _L:M\dasharrow {\bf
P}(\H^0(M,L)^*)$ associated to the line bundle $L$ is $G$\kern-1.5pt - equivariant.
\global\def\currenvir{subsection Let
$L^{\scriptscriptstyle\times }$ be the principal ${\bf C}^*$\kern-1.5pt - bundle associated to the {\it dual}
line bundle $L^*$ -- that is the complement of the zero section in $L^*$, on
which ${\bf C}^*$ acts by homotheties. We will say that a $p$\kern-1.5pt - form
$\omega $ on $L^{\scriptscriptstyle\times }$ is ${\bf C^*}$\kern-1.5pt - {\it equivariant} if
$\lambda ^*\omega =\lambda \omega $ for every $\lambda \in{\bf C}^*$.
\par\hskip 1truecm\relax We have a canonical linear form $\tau :p^*L\rightarrow
{\cal O}_{L^*}$, which is bijective on $L^{\scriptscriptstyle\times }$: if $s$ is a local
section of $L$ on $M$, the function $\tau (p^*s)$ maps a point $(m,\xi )$ of
$L^*$ $(\xi \in L(m)^*)$ to $\langle s(m),\xi \rangle$. We use $\tau $ to
trivialize $p^*L$ on $L^{\scriptscriptstyle\times }$. We can therefore consider
$p^*\theta $ as a $1$\kern-1.5pt - form on $L^{\scriptscriptstyle\times }$; it is ${\bf C}^*$\kern-1.5pt -
equivariant. The following lemma is classical (see for instance [A], App.\ 4 E,
or [L], p. 425):
\th Lemma
\enonce The $2$\kern-1.5pt - form $d(p^*\theta ) $
is a symplectic structure on $L^{\scriptscriptstyle\times }$. Conversely, any ${\bf C}^*$\kern-1.5pt -
equivariant symplectic $2$\kern-1.5pt - form on $L^{\scriptscriptstyle\times }$ is of the form $d(p^*\theta)$,
where
$\theta$ is a contact form on $M$, which is uniquely determined.\cqfd
\endth\label{symplectization}
\smallskip
\global\def\currenvir{subsection\label{diagramme} To each point $(m,\xi )$ of $L^*$ $(m\in M$, $\xi
\in L(m)^*)$, we associate the linear form $\mu_L(m,\xi ) $ on $\H^0(M,L)$
defined by $\langle\mu_L(m,\xi ),s\rangle=\langle s(m),\xi \rangle$ for each
$s\in \H^0(M,L)$. This gives a morphism $\mu _L:L^*\rightarrow \H^0(M,L)^*$
which is ${\bf C}^*$\kern-1.5pt - equi\-va\-riant and induces on the projectivizations
the
rational map
$\varphi^{} _L:M \dasharrow {\bf P}(\H^0(M,L)^*)$. Using the isomorphism
$\theta:{\goth g}\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} \H^0(M,L)$ (Prop.\ \ref{split}), we get a
commutative $G$\kern-1.5pt - equi\-variant diagram
\def\dia#1{\def\baselineskip=0truept{\baselineskip=0truept}
\offinterlineskip \matrix{#1}}
\def\vrule height 12pt depth 5pt width 0pt\vrule{\vrule height 2pt depth 0pt width 0.4pt}
\def\vrule height 12pt depth 5pt width 0pt{\vrule height 2pt depth 1pt width 0pt}
$$\dia{
{L^{\scriptscriptstyle\times }} &\kern-5pt \hfl{\mu }{}\kern-5pt & {\goth g}^*&\cr
\vrule height 2pt depth 1pt width 0pt\cr
\vrule height 12pt depth 5pt width 0pt\vrule & & \vrule height 12pt depth 5pt width 0pt\vrule&\cr
\vrule height 12pt depth 5pt width 0pt\vrule & &\cr
\vrule height 12pt depth 5pt width 0pt\vrule & & \vrule height 12pt depth 5pt width 0pt\vrule\cr\vrule height 12pt depth 5pt width 0pt\vrule & &\cr
\llap{$\scriptstyle p\ $} \vrule height 12pt depth 5pt width 0pt\vrule & & \vrule height 12pt depth 5pt width 0pt\vrule&\cr
\vrule height 12pt depth 5pt width 0pt\vrule & &\cr
\downarrow & &\downarrow\cr
M & \kern-8pt\longdash{\varphi }{} \kern-8pt& {\bf P}({\goth g}^*)&\kern-5pt .
}$$
\par\hskip 1truecm\relax As we have seen in (\ref{equi}), the action of $G$ on $M$ lifts to an
action on $L^{\scriptscriptstyle\times }$, which is linear on the fibres; similarly any field
$X\in{\goth
g}$ lifts to a vector field $\widetilde{X}$ on
$L^{\scriptscriptstyle\times }$ which projects to $X$ on $M$.
\th Proposition
\enonce $\mu $
is a moment map for the action
of $G$ on the symplectic manifold $L^{\scriptscriptstyle\times }$.
\endth\label{momentmap}
{\it Proof}: This means by definition
that for each $X\in{\goth g}$, the vector field $\widetilde{X}$ is the
Hamiltonian vector field associated to the function $\langle\mu ,X\rangle$ on
$L^{\scriptscriptstyle\times }$. To prove this, we first observe that since the $1$\kern-1.5pt - form
$\eta =p^*\theta $ is preserved by $G$, its Lie derivative
$L_{\widetilde{X}}\eta $ vanishes for each $X\in{\goth g}$. By the Cartan
homotopy
formula, this implies $i(\widetilde{X})\,d\eta = -d\langle\eta
,\widetilde{X}\rangle$. But we have $\langle\eta
,\widetilde{X}\rangle = \tau (p^*\theta (X) )=\langle\mu
,X\rangle$, thus
$i(\widetilde{X})\,d\eta = -d\langle\mu ,X\rangle$, which proves our
claim.\cqfd
\smallskip
\par\hskip 1truecm\relax The classical computation of the differential of the moment map
gives: \th Proposition
\enonce Let $m\in M$, and $\xi$ a point of $L^{\scriptscriptstyle\times }$ above $m$. The following
conditions are equivalent:
\par\hskip 0.5truecm\relax{\rm (i)} $\varphi$ is defined at $m$ and its differential $T_m(\varphi)$
is injective;
\par\hskip 0.5truecm\relax{\rm (ii)} the $G$\kern-1.5pt - orbit of $\xi$ is open in $L^{\scriptscriptstyle\times }$;
\par\hskip 0.5truecm\relax{\rm (iii)} the $G$\kern-1.5pt - orbit of $m$ is open in $M$ and $\xi$ is
conjugate
under $G$ to $\ell \xi$ for every $\ell \in{\bf C}^*$.
\endth\label{openorbit}
{\it Proof}: Since $\mu$ is ${\bf C}^*$\kern-1.5pt - equivariant, condition (i) is
equivalent to:
\par\hskip 1truecm\relax ${\rm (i')}$ $\mu(\xi)\not=0$ {\it and} $ T_\xi(\mu)$ {\it is injective}.
Let $\omega$ be the symplectic 2-form on
$L^{\scriptscriptstyle\times }$; for $v\in T_\xi(L^{\scriptscriptstyle\times })$ and $X\in{\goth g}$, the formula
$i(\widetilde{X})\,\omega = -d\langle\mu ,X\rangle$ (\ref{momentmap}) gives
$$\langle T_\xi(\mu)\cdot v\,,\,X\rangle = -\langle
i(\widetilde{X})\omega_\xi\,,\,v \rangle =
\omega_\xi(v,\widetilde{X}(\xi))\ ,
$$
so that the kernel of $T_\xi(\mu)$ is the orthogonal of $T_\xi(G\cdot \xi)$ in
$T_\xi(L^{\scriptscriptstyle\times })$ (with respect to $\omega_\xi$). This gives the equivalence of
${\rm (i')}$ and (ii); since the action of
$G$ commutes with the homotheties, (ii) is equivalent to (iii).\cqfd
\smallskip
\th Corollary
\enonce {\rm a)} If $L$ is very ample, $M$ is homogeneous.
\par\hskip 1truecm\relax {\rm b)} If $\varphi$ is generically finite, $M$ contains an open $G$\kern-1.5pt -
orbit. \endth \label{ample}
{\it Proof}: Under the hypothesis of a), each point of $M$ has an open
orbit, thus necessarily equal to $M$. The hypothesis of b) implies that
$\varphi$ is an immersion at a general point of $M$.\cqfd
\smallskip
\par\hskip 1truecm\relax Cor.\ \ref{ample} a) has also been obtained by J. Wisniewski (private
communication).
\vskip1truecm
\section {Coadjoint orbits}
\global\def\currenvir{subsection\label{koki} Let ${\goth g}$ be a Lie algebra; the adjoint group
$G$ acts on the dual ${\goth g}^*$ of ${\goth g}$ through the coadjoint
representation. Recall that each coadjoint orbit ${\cal O}$ carries a
canonical $G$\kern-1.5pt - invariant symplectic structure $\Omega $, the {\it
Kostant-Kirillov} structure: for $\xi \in{\cal O}$, the tangent space
$T_\xi ({\cal O})$ is canonically isomorphic to ${\goth g}/{\goth z}^{}_\xi $,
where ${\goth z}^{}_\xi=\mathop{\rm Ker}\nolimits(\xi\kern 1pt{\scriptstyle\circ}\kern 1pt\mathop{\rm ad}\nolimits)$ is the annihilator of $\xi $ in
${\goth g}$; the 2-form $\Omega _\xi $ is induced by the alternate form
$(X,Y)\mapsto
\xi ([X,Y])$ on ${\goth g}$. The following result shows that whenever ${\cal
O}$ is invariant under homotheties, its image ${\bf P}{\cal O}$ in ${\bf
P}({\goth g}^*)$ carries a natural contact structure:
\th Proposition \enonce Let ${\goth g}$ be a Lie algebra, $G$ its adjoint
group, $\xi $ a nonzero linear form on ${\goth g}$, ${\cal O}$ its coadjoint
orbit in ${\goth g}^*$, ${\bf P}{\cal O}$ the image of ${\cal O}$ in ${\bf
P}({\goth g}^*)$. The following conditions are equivalent:
\par\hskip 0.5truecm\relax{\rm (i)} ${\bf P}{\cal O}$ is odd-dimensional;
\par\hskip 0.5truecm\relax{\rm (ii)} the orbit ${\cal O}\i{\goth g}^*$ is invariant by homotheties;
\par\hskip 0.5truecm\relax{\rm (iii)} for each $\ell \in{\bf C}^*$, $\ell \, \xi $ is $G$\kern-1.5pt -
conjugate to $\xi$;
\par\hskip 0.5truecm\relax{\rm (iv)} there exists $H\in {\goth g}$ such that $\xi \kern 1pt{\scriptstyle\circ}\kern 1pt
\mathop{\rm ad}\nolimits(H)=\xi$;
\par\hskip 0.5truecm\relax{\rm (v)} the annihilator ${\goth z}^{}_\xi $ of $\xi $ in
${\goth g}$ is contained in $\mathop{\rm Ker}\nolimits\xi $.
\par\hskip 1truecm\relax When these conditions are satisfied, the Kostant-Kirillov symplectic
structure on ${\cal O}$ comes from a $G$\kern-1.5pt - invariant contact structure on
${\bf P}{\cal O}$. \endth\label{nilorb}
\par\hskip 0.5truecm\relax ${\rm (i)\Leftrightarrow (iii)}$: Let $Z_\xi$ be the stabilizer of $\xi
$ in $G$, and $Z_{[\xi ]}$ the stabilizer of the image $[\xi ]$ of $\xi$ in
${\bf P}({\goth g}^*)$. The action of $Z_{[\xi]} $ on the line $[\xi]$
defines a homomorphism $\ell :Z_{[\xi ]}\rightarrow {\bf C}^*$, and we have
an exact sequence $$0\rightarrow Z_\xi \longrightarrow Z_{[\xi ]}\qfl{\ell
}{\bf C}^*\ . $$Since the orbit ${\cal O}$ is
even-dimensional, (i) is equivalent to $\mathop{\rm dim}\nolimits Z_{[\xi ]}=\mathop{\rm dim}\nolimits Z_\xi +1$, that
is to the surjectivity of $\ell $, which is nothing but condition (iii).
\par\hskip 0.5truecm\relax ${\rm (ii)\Leftrightarrow (iii)}$: Clear.
\par\hskip 0.5truecm\relax ${\rm (iii)\Leftrightarrow (iv)}$: The Lie algebra ${\goth z}^{}_{[\xi]}
$ of $Z_{[\xi]} $ consists of the elements $H$ of
${\goth g}$ such that $\xi \kern 1pt{\scriptstyle\circ}\kern 1pt \mathop{\rm ad}\nolimits(H)=\lambda \xi $ for some
$\lambda=\lambda (H) \in{\bf C}$. The homomorphism $\lambda :{\goth
z}^{}_{[\xi]}\rightarrow {\bf C}$ thus defined is the Lie derivative of $\ell
$, so the surjectivity of $\ell $ is equi\-valent to the surjectivity of
$\lambda $, that is to (iv). \par\hskip 0.5truecm\relax ${\rm (iv)\Leftrightarrow (v)}$: The linear
map $u:H\mapsto \xi \kern 1pt{\scriptstyle\circ}\kern 1pt \mathop{\rm ad}\nolimits(H)$ of ${\goth g}$ into ${\goth g}^*$ is
antisymmetric, hence $\mathop{\rm Im}\nolimits u = (\mathop{\rm Ker}\nolimits u)^\perp$. But (iv) is equivalent to
$\xi \in\mathop{\rm Im}\nolimits u$ and (v) to $\xi \in (\mathop{\rm Ker}\nolimits
u)^\perp$.
\smallskip
\par\hskip 1truecm\relax Finally when ${\cal O}$ is invariant by homotheties, the Kostant-Kirillov
2-form on ${\cal O}$ is ${\bf C}^*$\kern-1.5pt - equivariant, and therefore comes
from a $G$\kern-1.5pt - invariant contact structure on ${\bf P}{\cal
O}$ (lemma \ref{symplectization}).\cqfd
\smallskip
\rem{Remark} Assume that the equivalent conditions of Prop.\ \ref{nilorb}
hold;
the contact structure on ${\bf P}{\cal O}$ can be described explicitely as
follows.
Let $\psi\in{\cal O}$; the tangent space
$T_{[\psi]}({\bf P}{\cal O})$ is canonically isomorphic to ${\goth g}/{\goth
z}_{[\psi]}$. Observe that
${\goth z}_{[\psi]}$ {\it is contained in} $\mathop{\rm Ker}\nolimits \psi$:
each element $Z$ of ${\goth z}_{[\psi]}$
satisfies $\psi\kern 1pt{\scriptstyle\circ}\kern 1pt \mathop{\rm ad}\nolimits(Z)=\lambda \psi$ for some $\lambda\in{\bf C}$;
if $\lambda=0$ we have $\psi(Z)=0$ by (v) above, while if
$\lambda\not=0$ we have $\psi(Z)=\lambda^{-1} \psi([Z,Z])=0$. Then the
contact structure $F\i T_{{\bf P}{\cal O}}$ is defined by
$F_{[\psi]}= (\mathop{\rm Ker}\nolimits \psi)/{\goth z}_{[\psi]} $.
\medskip
\global\def\currenvir{subsection\label{semi-simple} Suppose that the Lie algebra ${\goth g}$ is {\it
semi-simple}. Using the Killing form we
identify the $G$\kern-1.5pt - module ${\goth g}^*$ to ${\goth g}$ endowed with the
adjoint
action. The element $\xi $ corresponds to a nonzero element $N$ of ${\goth g}$.
Conditions (iii) to (v) read:
\par\hskip 0.5truecm\relax${\rm (iii')}$ for each $\ell \in{\bf C}^*$, $\ell N $ is
$G$\kern-1.5pt - conjugate to $N$;
\par\hskip 0.5truecm\relax${\rm (iv')}$ there exists $H\in {\goth g}$ such that $[H,N]=N
$;
\par\hskip 0.5truecm\relax${\rm (v')}$ the centralizer ${\goth z}^{}_N $ of $N $ in
${\goth g}$ is orthogonal to $N$.
\par\hskip 1truecm\relax They are equivalent to $N$ being {\it nilpotent}: ${\rm (iii')}$ implies
$\mathop{\rm Tr}\nolimits \rho (N)^p=0$ for any representation $\rho $ of ${\goth g}$ and any $p\ge
1$; conversely, if $N$ is nilpotent, ${\rm (iv')}$ holds by the
Jacobson-Morozov
theorem.
\global\def\currenvir{subsection\label{simple} Let ${\goth h}$ be a Cartan subalgebra of ${\goth
g}$,
$R=R({\goth g},{\goth h})$ the root system of ${\goth g}$ relative to ${\goth
h}$.
We have a direct sum decomposition $\displaystyle {\goth g}={\goth h}\,\oplus
\gdir_{\alpha\in R}^{}{\goth g}^\alpha$. A nonzero vector $X_\alpha\in {\goth
g}^\alpha$ is called a {\it root vector} (relative to $\alpha $).
\par\hskip 1truecm\relax If ${\goth g}$ is simple, the Weyl group acts transitively
on the set of roots with a given length, and the corresponding root vectors are
conjugate. This defines the (nilpotent) orbits ${\cal O}_{min}$ of a long root
vector and ${\cal O}_{short}$ of a short root vector; these orbits coincide if
and
only if
all roots have the same length (types
$A_l,D_l,E_l$).
\th Proposition
\enonce Let ${\goth g}$ be a simple complex Lie algebra. There exists exactly
one
closed orbit in ${\bf P}({\goth g})$ {\rm (}for the adjoint action{\rm ),}
namely
the orbit
${\bf P}{\cal O}_{min}$ of a long root vector. Every orbit contains
${\bf P}{\cal O}_{min}$ in its closure.
\endth\label{Omin}
{\it Proof}: Let $N$ be a nonzero element of ${\goth g}$. The orbit of $[N]$
in
${\bf P}({\goth g})$ is closed if and only if ${\goth z}^{}_{[N]}$
contains a Borel subalgebra
${\goth b}$, so that there exists a linear form $\lambda$ on ${\goth b}$ such
that
$\mathop{\rm ad}\nolimits(X)\cdot N=\lambda(X)N$ for all $X\in{\goth b}$. This means that $N$ is a
highest weight vector for the adjoint representation; since ${\goth g}$ is
simple, the adjoint representation is irreducible, and its highest weight
vector is
$X_\theta $, where $\theta $ is the highest root with respect to the basis of
$R({\goth g},{\goth h})$ such that ${\goth b}={\goth
h}\,\oplus\gdir_{\alpha\ge 0}^{}{\goth g}^\alpha$. We conclude that the orbit
${\bf P}{\cal O}_{min}$ of $X_\theta$ is the unique closed orbit in ${\bf
P}({\goth g})$.\cqfd
\medskip
\rem{Examples} For the classical case, we get the following Fano contact
manifolds:
\par\hskip 0.5truecm\relax$A_l$: the projectivized cotangent bundle ${\bf P}T^*({\bf P}^l)$;
\par\hskip 0.5truecm\relax$B_l,D_l$: the Grassmannian ${\bf G}_{iso}(2,V)$ of isotropic 2-planes
in a quadratic vector space $V$, of dimension $2l+1$ and $2l$ respectively;
\par\hskip 0.5truecm\relax$C_l$: the projective space ${\bf P}^{2l-1}$.
\par\hskip 1truecm\relax For the type $G_2$ we get a Fano 5-fold of index 3 which appears in the
work of Mukai [Mu]. The other exceptional Lie algebras give rise to Fano
contact manifolds of dimension 15, 21, 33 and 57.
\smallskip
\rem{Remark}\label{unique} It follows from [L], Cor.\ 3.2, or from a direct
computation, that if
${\goth g}$ is not of type $C_l$ the manifold ${\bf P}{\cal
O}_{min}$ admits a {\it unique} contact structure; in all cases, the
contact structure we have defined is the unique $G$\kern-1.5pt - invariant contact
structure.
\vskip1truecm
\section{First consequences of (H1) and (H2)}
\global\def\currenvir{subsection\label{center} From now on we assume that $\varphi$ {\it is
generically finite} (or equivalently, $\mathop{\rm dim}\nolimits \varphi (M)=\mathop{\rm dim}\nolimits M$). By Cor.\
\ref{ample}, this implies that $G$ has an open orbit $M^{\rm o}$ in $M$. Since
$\varphi $ is $G$\kern-1.5pt - equivariant, it is everywhere defined on $M^{\rm o}$; the
image $\varphi (M^{\rm o})$ is an orbit ${\bf P}{\cal O}$ of $G$ in ${\bf P}
({\goth g}^*)$, and the induced map $\varphi ^{\rm o}:M^{\rm o}\rightarrow {\bf
P}{\cal O}$ is a finite \'etale covering.
\par\hskip 1truecm\relax Let us mention at once an immediate consequence: if a
connected normal subgroup of $G$ fixes a point $[\xi ]\in{\bf P}{\cal O}$, it
acts
trivially on
$M^{\rm o}$, hence on $M$; it follows that {\it the stabilizer ${\goth z}^{
}_{[\xi ]}$ of $[\xi ]$ in ${\goth g}$ contains no nonzero ideal of} ${\goth
g}$. In particular, {\it the center of ${\goth g}$ is trivial}.
\th Lemma
\enonce Assume that the character group of $G$ is trivial, and $\mathop{\rm Pic}\nolimits(M)={\bf
Z}$.
Then $M\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}} M^{\rm o}$ has codimension $\ge 2$ in $M$.
\endth\label{codim}
{\it Proof}: Let $m$ be a point of $M^{\rm o}$, $[\xi ]$ its image in ${\bf
P}({\goth g}^*)$. The stabilizer $Z_m$ of $m$ in $G$ is a subgroup of finite
index in the stabilizer $Z_{[\xi ]}$ of $[\xi ]$. Since $M^{\rm o}$ and
therefore ${\bf P}{\cal O}$ are odd-dimensional, the equivalent conditions of
Prop.\
\ref{nilorb} are satisfied; hence the homomorphism $\ell :Z_{[\xi
]}\rightarrow
{\bf C}^*$ deduced from the action of $Z_{[\xi ]}$ on the line $[\xi ]$ is
surjective, and so is
its restriction to $Z_m$.
\par\hskip 1truecm\relax Recall that the group $\mathop{\rm Pic}\nolimits^G(M^{\rm o})$ of $G$\kern-1.5pt - linearized line
bundles
on $M^{\rm o}\cong G/Z_m$ is canonically isomorphic to the character group
$\hbox{\san \char88}(Z_m)$. On the other hand, the hypothesis on $G$ ensures that the forgetful
map $\mathop{\rm Pic}\nolimits^G(M^{\rm o})\rightarrow \mathop{\rm Pic}\nolimits(M^{\rm o})$ is injective ([M], Ch.\ 1,
Prop.\ 1.4). Since we have found a surjective character of $Z_m$, it follows
that
$\mathop{\rm Pic}\nolimits(M^{\rm o})$ contains an infinite cyclic group.
\par\hskip 1truecm\relax Let $(D_i)_{i\in I}$ be the family of one-codimensional components of
$M\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}} M^{\rm o}$. We have an exact sequence
$${\bf Z}^I\,\hfl{(D_i)}{}\, \mathop{\rm Pic}\nolimits(M)\longrightarrow \mathop{\rm Pic}\nolimits(M^{\rm o})\rightarrow
0\
.$$ Since $\mathop{\rm Pic}\nolimits(M)={\bf Z}$ and each $D_i$ has a nonzero class in $\mathop{\rm Pic}\nolimits(M)$,
the
only possibility is $I=\varnothing$.\cqfd
\th Lemma
\enonce Let $M$ be a normal projective variety, $L$ an ample line bundle on
$M$, $\varphi :M\dasharrow {\bf P}^r$ the associated rational map, $\N\i{\bf
P}^r$
its image.
Assume
that there are open subsets $M^{\rm o}\i M$ and $\N^{\rm o}\i \N$, whose
complements
have codimension $\ge 2$, such that $\varphi $ is defined
everywhere on $M^{\rm o}$, $\varphi (M^{\rm o})=\N^{\rm o}$ and the
induced morphism $\varphi^{\rm o}:M^{\rm o}\rightarrow \N^{\rm o}$ is finite.
Then
$\varphi $ is everywhere defined and finite. \endth\label{finite}
{\it Proof}: Replacing $\N$ by its normalization we may assume that $\N$
is normal; then the restriction maps $\H^0(\N,{\cal O}_{\N}(n))\rightarrow
\H^0(\N^{\rm o},{\cal O}_{\N}(n))$ and $\H^0(M,L^n)\rightarrow \H^0(M^{\rm
o},L^n)$ are bijective.
Let $CM=\mathop{\rm Spec}\nolimits \sdir_{n\ge 0}^{}\H^0(M,L^n)$ and
$C\N=\mathop{\rm Spec}\nolimits \sdir_{n\ge 0}^{}\H^0(\N,{\cal O}_{\N}(n))$ be the cones
over $M$ and $\N$ respectively associated to the line bundles
$L$ and ${\cal O}_{\N}(1)$. The homomorphism $(\varphi^{\rm o})^*$ induces a
finite morphism $C\varphi :CM\rightarrow C\N$, which is ${\bf C}^*$\kern-1.5pt -
equivariant. The inverse image of the vertex of $C\N$ under $C\varphi $
is finite and stable under ${\bf C}^*$, hence reduced to the vertex of $CM$.
Therefore $C\varphi $ induces a finite morphism $M\rightarrow \N$ which extends
$\varphi ^{\rm o}$.\cqfd
\smallskip
\global\def\currenvir{subsection\label{H3} Let us now assume that ${\goth g}$ is reductive (this is
our hypothesis (H2)). By (\ref{center}) this actually implies that ${\goth
g}$
is {\it semi-simple}. We will always identify ${\goth g}^*$ with ${\goth g}$
using the Killing form. We also make a third hypothesis:
\par\hskip 1truecm\relax (H3) $\mathop{\rm Pic}\nolimits(M)={\bf Z}$.
\par
This is innocuous because Theorem \ref{main} is known to be true when $b_2\ge
2$,
as a consequence of a theorem of Wisniewski (see [L-S], cor.\ 4.2).
\th Proposition
\enonce Under the hypotheses {\rm (H1)} to {\rm (H3)}, the map
$\varphi:M\rightarrow {\bf P}({\goth g})$ is a finite morphism onto the closure
of a nilpotent orbit ${\bf P}{\cal O}$. $M$ has only finitely many orbits; each
orbit is a finite \'etale covering of a nilpotent orbit in ${\bf P}({\goth
g})$.
\endth\label{summary}
{\it Proof}: Since $G$ is semi-simple, the hypotheses of lemma \ref{codim}
hold. We
have already seen that the orbit ${\cal O}$ is ${\bf C}^*$\kern-1.5pt - invariant, hence
nilpotent (\ref{semi-simple}). Therefore $\overline{{\bf P}{\cal O}}$ is a
finite
union of nilpotent orbits in ${\bf P}({\goth g})$. Since such an orbit is
odd-dimensional, the codimension of ${\bf P}{\cal O}$ in $\overline{{\bf P}
{\cal
O}}$ is $\ge 2$, so we can apply lemma \ref{finite}; the Proposition
follows.\cqfd
\smallskip
\rem{Remark}\label{converse} Conversely, suppose given a compact manifold $M$
with an action of $G$ and a finite surjective $G$\kern-1.5pt - equivariant morphism
$\varphi :M\rightarrow \overline{{\bf P}{\cal O}}$ onto the closure of a
nilpotent orbit in ${\bf P}({\goth g})$. Then $M$ {\it is a Fano contact
manifold}. Indeed, let $M^{\rm o}=\varphi^{-1} ({\bf P}{\cal O})$, and
$L=\varphi^*{\cal O}(1)$. The contact structure of ${\bf
P}{\cal O}$ pulls back to a contact structure $\theta^{\rm o}\in \H^0(M^{\rm
o},\Omega^1_{M^{\rm o}}\otimes L)$, which extends to a contact structure
$\theta\in\H^0(M,\Omega^1_{M}\otimes L)$ because $M\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}} M^{\rm o}$ has
codimension $\ge 2$. Since $L$ is ample, $M$ is a Fano contact manifold. \par\hskip 1truecm\relax
We have thus reduced our problem to a question about nilpotent orbits of
semi-simple Lie algebras, which we will study in the next sections.
\vskip1truecm\section{Nilpotent orbits} \global\def\currenvir{subsection\label{dyn}
At this point we need to recall Dynkin's classification of nilpotent
orbits in a semi-simple Lie algebra ${\goth g}$ (a general reference for the
material in this section is [C-M]). We fix a nilpotent element $N_0$ of
${\goth
g}$, and denote by ${\cal O}$ its orbit in ${\goth g}$ (under the adjoint
action).
\par\hskip 1truecm\relax By the Jacobson-Morozov theorem, there exist elements $H$
and $N_1$ in ${\goth g}$ satisfying
$$[H,N_0]=2N_0\qquad [H,N_1]=-2N_1 \qquad [N_1,N_0]=H\ ,$$
so that the subspace of ${\goth g}$ spanned by $N_0,N_1,H$ is a Lie
subalgebra isomorphic to ${\goth s}{\goth l}_2$. As a
${\goth s}{\goth l}_2$\kern-1.5pt - module, ${\goth g}$ is then isomorphic to a direct
sum of simple modules $\hbox{\san \char83}^kV$, where $V$ is the standard 2-dimensional
representation. It follows easily that:
\par\hskip 1truecm\relax (\ref{dyn}.{\it a}) there is a direct sum decomposition ${\goth
g}=\sdir_{i\in{\bf Z}}^{}{\goth g}(i)$, where ${\goth
g}(i)$ is the subspace of elements $ X\in {\goth g}$ with $[H,X]=iX$.
\par\hskip 1truecm\relax (\ref{dyn}.{\it b}) Put ${\goth p}=\sdir_{i\ge 0}^{}{\goth g}(i)$,
${\goth n}=\sdir_{i\ge 2}^{}{\goth g}(i)$. Then ${\goth p}$ is a
parabolic subalgebra of ${\goth g}$; ${\goth n}$ is a unipotent ideal in
${\goth p}$. The map $\mathop{\rm ad}\nolimits(N_0):{\goth p}\rightarrow {\goth n}$ is surjective.
\par\hskip 1truecm\relax (\ref{dyn}.{\it c}) Let ${\goth h}$ be a Cartan subalgebra of ${\goth g}$
containing $H$. There exists a basis $B$ of the root system $R({\goth g},{\goth
h})$ such that $\alpha (H)\in\{0,1,2\}$ for each $\alpha \in B$. The {\it
weighted
Dynkin diagram} of $N_0$ is obtained by labelling each node $\alpha \in B$ of
the
Dynkin diagram of ${\goth g}$ with the number $\alpha (H)\in\{0,1,2\}$. It
depends
only on the orbit ${\cal O}$ of $N_0$; two different nilpotent orbits give
rise
to different weighted diagrams.
\medskip
\defG\times ^P{\kern-1.5pt\goth n}{G\times ^P{\kern-1.5pt\goth n}}
\def{\goth g}\times ^{\goth p}{\kern-1.5pt\goth n}{{\goth g}\times ^{\goth p}{\kern-1.5pt\goth n}}
\def{\goth z}^{}_N{{\goth z}^{}_N}
\global\def\currenvir{subsection\label{gpn} Let $P$ be the parabolic subgroup of $G$ with Lie
algebra ${\goth p}$.
We denote by $G\times ^P{\kern-1.5pt\goth n}$ the quotient of $G\times {\goth n}$ by $P$
acting by $p\cdot (g,N)=(gp^{-1},\mathop{\rm Ad}\nolimits(p)N)$; in other words, $G\times ^P{\kern-1.5pt\goth n}$ is the
$G$\kern-1.5pt -
homogeneous vector bundle on $G/P$ associated to the adjoint action of $P$ on
${\goth n}$. For $g\in G$, $N\in {\goth n}$, we denote by $(g,N)\dot{}$ the
image
of $(g,N)$ in $G\times ^P{\kern-1.5pt\goth n}$; the tangent space to
$G\times ^P{\kern-1.5pt\goth n}$ at $(g,N)\dot{}$ is canonically isomorphic to the quotient of ${\goth
g}\times {\goth n}$ by the subspace of elements $({\it P},[N,{\it P}])$ with
${\it P}\in{\goth p}$.
\par\hskip 1truecm\relax The orbit $G\cdot(1,N_0)\dot{}$ is open in
$G\times ^P{\kern-1.5pt\goth n}$. Since the stabilizer in $G$ of
$(1,N_0)\dot{}$ is $Z_{N_0}$, there is a unique $G$\kern-1.5pt - equivariant
isomorphism
${\cal O}\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} G\cdot(1,N_0)\dot{}$ mapping $N_0$ onto $(1,N_0)\dot{}$. We will
identify
${\cal O}$ to the open orbit of $G\times ^P{\kern-1.5pt\goth n}$ through this isomorphism.
\th Lemma
\enonce The Kostant-Kirillov symplectic $2$\kern-1.5pt - form on
${\cal O}$ extends to a $G$\kern-1.5pt - invariant $2$\kern-1.5pt - form $\omega$ on $G\times ^P{\kern-1.5pt\goth n}$. Let
$(g,N)\dot{}\inG\times ^P{\kern-1.5pt\goth n}$; the kernel of $\omega_{(g,N)\dot{}} $ consists of the
images
of the elements $(X,[N,X])$, with $X\in{\goth n}^\perp=\sdir_{i\ge -1}^{}{\goth
g}(i)$ and $[N,X]\in{\goth n}$.
\endth\label{noyau}
{\it Proof}: Consider the alternate bilinear form on
${\goth g}\times {\goth n}$ defined by
$$((X,Q),(X',Q')) \mapsto
(N\,|\,[X,X'])+(X\,|\,Q')-(X'\,|\,Q)\ .$$
Its kernel consists of pairs $(X,Q)$ with $X\in {\goth n}^\perp$ and $Q=[N,X]$;
in
particular, it contains the elements $({\it P},[N,{\it P}])$ for ${\it P}\in
{\goth p}$, so that our form factors through $T_{(g,N)\dot{}}(G\times ^P{\kern-1.5pt\goth n})$ and
defines a
$G$\kern-1.5pt - invariant 2-form $\omega $ on $G\times ^P{\kern-1.5pt\goth n}$.
\par\hskip 1truecm\relax The isomorphism ${\cal O}\rightarrow G\cdot(1,N_0)\dot{}$ induces on the
tangent spaces the isomorphism ${\goth g}/{\goth z}^{}_{N_0}\rightarrow
T_{(1,N_0)\dot{}}(G\times ^P{\kern-1.5pt\goth n})$ which maps the class of $X\in{\goth g}$ to the class
of
$(X,0)$. Through this isomorphism,
$\omega_{(1,N_0)\dot{}}$ corresponds to the alternate form $(X,X')\mapsto
(N_0\,|\,[X,X'])$, that is to the Kostant-Kirillov $2$\kern-1.5pt - form $\omega _0$ at
$N_0$. Since $\omega$ and $\omega_0$ are $G$\kern-1.5pt - invariant, the restriction of
$\omega$ to ${\cal O}$ is equal to $\omega_0$.\cqfd
\smallskip
\par\hskip 1truecm\relax The following lemma will be the key technical ingredient for our proof
of
the main theorem. We put ${\goth g}^{\scriptscriptstyle\times }={\goth g}\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}\{0\}$, ${\goth
n}^{\scriptscriptstyle\times }={\goth
n}\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}\{0\}$.
\th Lemma
\enonce Let $N\in{\goth n}$. Let $\overline{\cal O}$ be the closure of ${\cal
O}$
in ${\goth g}^{\scriptscriptstyle\times }$.
Assume that the normalization $\widetilde{{\cal O}}$ of $\overline{\cal O}$
is smooth above $N$. Then the centralizer ${\goth z}^{}_N$ is contained in
${\goth n}^\perp$.
\endth\label{key}
{\it Proof}: Consider the morphism $G\times ^P{\kern-1.5pt\goth n}^{\scriptscriptstyle\times } \rightarrow {\goth g}^{\scriptscriptstyle\times }$ which
maps $(g,N)\dot{}$ to $\mathop{\rm Ad}\nolimits(g)N$. Its image is the closure $\overline{\cal O}$
of ${\cal O}$ in ${\goth g}^{\scriptscriptstyle\times }$; since $G\times ^P{\kern-1.5pt\goth n}$ is
smooth, it factors through $\widetilde{\cal O}$. The
induced morphism $\pi :G\times ^P{\kern-1.5pt\goth n}^{\scriptscriptstyle\times } \rightarrow \widetilde{\cal O}$ is proper and
birational: it induces the identity on the open orbit ${\cal O}\iG\times ^P{\kern-1.5pt\goth n}^{\scriptscriptstyle\times }$.
\par\hskip 1truecm\relax Since the complement of ${\cal O}$ in
$\widetilde{{\cal O}}$ has codimension $\ge 2$, the symplectic 2-form on
${\cal O}$ extends to a 2-form $\varpi$ on the smooth part $\widetilde{{\cal
O}}_{sm}$ of $\widetilde{{\cal O}}$; the pull-back of $\varpi$ to $\pi
^{-1}(\widetilde{{\cal O}}_{sm})\iG\times ^P{\kern-1.5pt\goth n} $ coincides with the restriction of
$\omega$.
It follows that every tangent
vector at the point
$x=(1,N)\dot{}$ of $G\times ^P{\kern-1.5pt\goth n}^{\scriptscriptstyle\times }$ killed by
$T_x(\pi)$ belongs to the kernel of $\omega
_x $.
Since the orbit of $x$ under $Z^{\rm o}_N$ maps to a point in
$\widetilde{{\cal
O}}$, the vectors $({\it Z},0)$ with ${\it Z}\in{\goth z}^{}_N$
must belong to the kernel of $\omega_x $; in view of Lemma \ref{noyau}, this
means that ${\goth z}^{}_N$ is contained in ${\goth n}^\perp$.\cqfd
\vskip1truecm
\section{The birational case}
\par\hskip 1truecm\relax In this section we will prove Theorem \ref{main} in the simpler case when
the map $\varphi^{}_L$ is assumed to be birational.
We start with a technical lemma about Lie algebras; we keep the notation of
(\ref{dyn}).
\th Lemma
\enonce Assume that $N_0$ is not contained in a proper ideal of ${\goth g}$,
and
that for every nonzero elements $N\in{\goth g}(2)$ and $Q\in{\goth g}(-2)$ the
bracket
$[N,Q]$ is nonzero.
Then ${\goth g}$ is simple, and either ${\cal O}$ is the minimal orbit, or
${\goth
g}$ is of type $G_2$ and ${\cal O}$ is the orbit of a short root vector.
\endth\label{classif}
{\it Proof}: Assume first that ${\goth g}$ is a product of two
nonzero semi-simple Lie algebras ${\goth g}'$ and ${\goth g}''$.
Write $N_0=(N'_0,N''_0)$, $H=(H',H'')$, $N_1=(N'_1,N''_1)$; the hypothesis on
$N_0$ ensures that $N'_0$ and $N''_0$ (and therefore also
$H',H'',N'_1,N''_1$) are nonzero. We have $N'_1\in {\goth g}(-2)$,
$N''_0\in{\goth g}(2)$ and $[N'_1,N''_0]=0$, contrary to the hypothesis. Thus
${\goth g}$ is simple.
\par\hskip 1truecm\relax For any
nonzero $N\in{\goth g}(2)$, we have ${\goth z}^{}_N\cap{\goth g}(-2)=(0)$; by
[C-M], 3.4.17, this implies that $N$ is conjugate to $N_0$. There exists a root
$\alpha $ with $\alpha (H)=2$ (the corresponding root vectors span ${\goth
g}(2)$); therefore $N_0$ is conjugate to $X_\alpha $. \par\hskip 1truecm\relax Assume that ${\goth
g}$ is of type $B_l,C_l$ or $F_4$, and that
$\alpha $ is a short root. According to [C-M] the weighted Dynkin diagram of
$X_\alpha $ is one of the following:
\def\kern-12pt\raise2pt\vbox{\hrule width .94truecm{\kern-12pt\raise2pt\vbox{\hrule width .94truecm}
\kern-11.5pt}
\def\kern-12pt\Longrightarrow\!\!=\!=\kern-12pt{\kern-12pt\Longrightarrow\!\!=\!=\kern-12pt}
\def\kern-12pt =\!=\!\!\Longleftarrow\kern-12pt{\kern-12pt =\!=\!\!\Longleftarrow\kern-12pt}
\def\diaram#1{\def\baselineskip=0truept{\baselineskip=0truept
\lineskip=4truept\lineskiplimit=1truept} \matrix{#1}}
\def\scriptstyle 2{\scriptstyle 2}
\def\scriptstyle 1{\scriptstyle 1}
\def\scriptstyle 0{\scriptstyle 0}
\vskip-10pt$$\diaram{\baselineskip4pt
\scriptstyle 2 && \scriptstyle 0 & & \scriptstyle 0 & & \scriptstyle 0 && \scriptstyle 0\cr
\circ & \kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ &\kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ &
\kern-2pt\cdots\cdots \kern-2pt& \circ &
\kern-12pt\Longrightarrow\!\!=\!=\kern-12pt &\circ}$$\vskip-20pt
$$\diaram{\baselineskip4pt
\scriptstyle 0 && \scriptstyle 1 & & \scriptstyle 0 & & \scriptstyle 0 && \scriptstyle 0\cr
\circ & \kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ &\kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ &
\kern-2pt\cdots\cdots \kern-2pt& \circ &
\kern-12pt =\!=\!\!\Longleftarrow\kern-12pt &\circ}$$\vskip-20pt
$$\diaram{\baselineskip4pt
\scriptstyle 0 && \scriptstyle 0 & & \scriptstyle 0 & & \scriptstyle 1 \cr
\circ & \kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ &\kern-12pt\Longrightarrow\!\!=\!=\kern-12pt &\circ
&\kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ}$
In each case the highest root $\theta $ satisfies $\theta (H)=2$, hence
$X_\theta $ should be conjugate to $X_\alpha $ -- a contradiction.
Therefore either $\alpha $ is a long root, or ${\goth g}$ is of type
$G_2$.\cqfd
\medskip
\th Proposition
\enonce Let ${\cal O}$ be a nilpotent orbit in ${\goth g}$ and $\overline{\cal
O}$ its closure in ${\goth g}^{\scriptscriptstyle\times }$.
Assume that ${\cal O}$ is not contained in a proper ideal of ${\goth g}$, and
that the normalization of $\overline{\cal O}$
is smooth. Then ${\goth g}$ is simple, and either ${\cal O}$ is
the minimal nilpotent orbit, or ${\goth g}$ is of type $G_2$ and ${\cal
O}$ is the orbit of a short root vector.
\endth\label{closmooth}
\par\hskip 1truecm\relax In the first case $\overline{{\cal O}}$ is equal to ${\cal O}$, hence
smooth.
In the second case $\overline{\cal O}$ is not normal, and its normalization
is isomorphic to the minimal nilpotent orbit in ${\goth
s}{\goth o}(7)$ [L-Sm].
\smallskip
{\it Proof}: By Lemma \ref{key}, we have ${\goth z}^{}_N\i{\goth n}^\perp$ for each
nonzero
element $N$ of ${\goth n}$.
Taking $N$ in ${\goth g}(2)$, we see that the hypotheses of Lemma
\ref{classif}
are satisfied, hence the result.\cqfd
\smallskip
\th Corollary
\enonce Let $M$ be a Fano contact manifold, such that
\par\hskip 1truecm\relax {\rm (i)} the rational map $\varphi:M\dasharrow {\bf P}({\goth g})$ is
generically injective; \par\hskip 1truecm\relax {\rm (ii)} the group $G$ of contact automorphisms
of
$M$ is reductive.\par Then $\varphi $ induces an isomorphism of $M$ onto the
minimal nilpotent orbit in ${\bf P}({\goth g})$. \endth
{\it Proof}: Consider the commutative diagram (\ref{diagramme})
$$\diagram{
L^{\scriptscriptstyle\times } & \hfl{\mu }{} & {\goth g}^{\scriptscriptstyle\times }&\cr
\vfl{}{} & & \vfl{}{}& \cr
M & \hfl{\varphi }{} & {\bf P}({\goth g})&\kern-10pt.
}$$
By Prop.\ \ref{summary} $\varphi $ is a finite birational morphism onto the
closure of a nilpotent orbit ${\bf P}{\cal
O}$ in ${\bf P}({\goth g})$; since the diagram is cartesian, $\mu $
is finite and birational onto $\overline{\cal O}$, hence realizes
$L^{\scriptscriptstyle\times }$ as the normalization of $\overline{{\cal O}}$. Since the image
$\overline{{\bf P}{\cal O}}$ of $\varphi $ spans ${\bf P}({\goth g})$, ${\cal
O}$ cannot be contained in any proper subspace of ${\goth g}$. By Prop.\
\ref{closmooth}, this implies either that ${\cal O}$ is a minimal orbit, or
that
${\goth g}$ is of type $G_2$ and ${\cal O}$ is the orbit of a short root
vector;
in that case $M$ is isomorphic to ${\bf P}{\cal O}'$, where ${\cal O}'$ is the
minimal orbit in ${\goth s}{\goth o}(7)$, and this isomorphism preserves the
contact structures (remark \ref{unique}). But then ${\goth g}$ contains
${\goth
s}{\goth o}(7)$, a contradiction.\cqfd
\vskip1truecm
\section{The general case}
\global\def\currenvir{subsection As explained in Remark \ref{converse}, we want to classify finite
$G$\kern-1.5pt - equivariant surjective morphisms
$\varphi:M\rightarrow \overline{{\bf P}{\cal O}}$, where $M$ is smooth and
${\cal
O}\i{\goth g}$ is a nilpotent orbit; such a morphism will be called for short
a
$G$\kern-1.5pt - {\it covering} of
$\overline{{\bf P}{\cal O}}$. Examples of $G$\kern-1.5pt - coverings appear in the
classification of ``shared orbit pairs" [B-K], associated to certain pairs
${\goth g}\i {\goth g}'$ of simple Lie algebras: the manifold $M$ is the
minimal
orbit ${\bf P}{\cal O}'_{min}$ for ${\goth g}'$, while the orbit ${\cal
O}\i{\goth
g}$ is given in the list below. Brylinski and Kostant find the following cases:
\def\vrule height 12pt depth 5pt width 0pt{\vrule height 12pt depth 5pt width 0pt}
\def\vrule height 15pt depth 15pt width 0pt{\vrule height 15pt depth 15pt width 0pt}
\def\vrule height 12pt depth 5pt width 0pt\vrule{\vrule height 12pt depth 5pt width 0pt\vrule}
\def\noalign{\hrule}{\noalign{\hrule}}
\def\hfill}\def\hq{\hfill\quad{\hfill}\def\hq{\hfill\quad}
$$\vcenter{\offinterlineskip
\halign{\vrule height 12pt depth 5pt width 0pt\vrule\hq#\hq& \vrule height 12pt depth 5pt width 0pt\vrule\hq#\hq& \vrule height 12pt depth 5pt width 0pt\vrule\hq#\hq& \vrule height 12pt depth 5pt width 0pt\vrule\hq#\hq\vrule height 12pt depth 5pt width 0pt\vrule\cr\noalign{\hrule}
\vrule height 15pt depth 15pt width 0pt ${\goth g}$ & ${\goth g}'$ & ${\cal O}$ & $\deg\varphi$ \cr\noalign{\hrule}
$A_2$ & $G_2$ & ${\cal O}_{(3)}$ & 3 \cr\noalign{\hrule}
$B_l$ & $D_{l+1}$ & ${\cal O}_{(3,1,\ldots,1)}$ & 2 \cr\noalign{\hrule}
$B_4$ & $F_4$ & ${\cal O}_{(2,2,2,2,1)}$ & 2 \cr\noalign{\hrule}
$C_l$ & $A_{2l-1}$ & ${\cal O}_{(2,2,1,\ldots,1)}$ & 2 \cr\noalign{\hrule}
$D_l$ & $B_{l}$ & ${\cal O}_{(3,1,\ldots,1)}$ & 2 \cr\noalign{\hrule}
$D_4$ & $F_4$ & ${\cal O}_{(3,2,2,1)}$ & 4 \cr\noalign{\hrule}
$F_4$ & $E_6$ & ${\cal O}_{short}$ & 2 \cr\noalign{\hrule}
$G_2$ & $B_3$ & ${\cal O}_{short}$ & 1 \cr\noalign{\hrule}
$G_2$ & $D_4$ & ${\cal O}_{sub}$ & 6 \cr\noalign{\hrule}
}}\leqno{\global\def\currenvir{subsection}$$
\label{list}
\par\hskip 1truecm\relax The notation for the orbit ${\cal O}$ requires some explanation: in the
classical cases, ${\goth g}$ is viewed as an algebra of matrices via the
standard
representation; then ${\cal O}_{(d_1,\ldots,d_k)}$ denote the conjugacy class
of
a matrix in ${\goth g}$ with Jordan type $(d_1,\ldots,d_k)$. As in
(\ref{simple}),
${\cal O}_{short}$ is the orbit of a short root vector. Finally ${\cal
O}_{sub}$
is the so-called subregular orbit, that is the unique codimension 2 orbit in
the
nilpotent cone.
\th Proposition
\enonce Let $G$ be a simple complex Lie group acting on a manifold $M$, ${\goth
g}$ the Lie algebra of $G$,
${\cal O}\i{\goth g}$ a nilpotent orbit,
$\varphi:M\rightarrow \overline{{\bf P}{\cal O}}$ a finite
$G$\kern-1.5pt - equivariant surjective morphism. Then either ${\cal O}={\cal
O}_{min}$ and $\varphi$ is an isomorphism, or $\varphi $ is {\rm (}up to
isomorphism{\rm )} one of the $G$\kern-1.5pt - coverings appearing in the list
$(\ref{list})$. \endth\label{coverings}
{\it Proof}:\global\def\currenvir{subsection\label{Galois} Let $M^{\rm o}$ be the open $G$\kern-1.5pt -
orbit in $M$; let
$m$ be a point of $M^{\rm o}$, $\H^{\rm o}$
its stabilizer in $G$ and $\H$ the stabilizer of $\varphi(m)$. Since $M$ is
Fano,
$M$ and therefore $M^{\rm o}$ are simply connected;
this implies that $\H^{\rm o}$ is the neutral
component of $\H$. So
the covering $M^{\rm o}\rightarrow {\bf P}{\cal O}$ is a Galois covering, with
Galois group $\Gamma := \H/\H^{\rm o}$. Since $M={\rm Proj}\sdir_{n\ge
0}^{}\H^0(M^{\rm o},L^n)$, the action of $\Gamma $ on $M^{\rm o}$ extends to an
action on $M$, which commutes with the $G$\kern-1.5pt - action.
\par\hskip 1truecm\relax Observe that the $G$\kern-1.5pt - covering $M\rightarrow \overline{{\bf
P}{\cal O}}$ is uniquely determined by ${\cal O}$: the open $G$\kern-1.5pt - orbit
$M^{\rm o}\i M$
is the simply-connected covering of ${\bf P}{\cal O}$, and $M$ is the
integral closure of $\overline{{\bf P}{\cal O}}$ in $M^{\rm o}$.
Thus our task is to prove that only the
orbits listed in (\ref{list}) can occur.
\global\def\currenvir{subsection\label{arg} We will prove this by induction on the dimension of
${\cal O}$, the case ${\cal O}={\cal O}_{min}$ being clear in view of
(\ref{Galois}). By Prop.\ \ref{closmooth} we can assume $\deg(\varphi)>1$. Let
$\gamma\in\Gamma
$, and let
$F$ be a component of the fixed locus of
$\gamma$. Then $F$ is a closed submanifold of $M$, stable under $G$; the map
$\varphi$
induces a $G$\kern-1.5pt - covering $F\rightarrow \overline{{\bf P}{\cal O}}_F$ for
some
orbit ${\cal O}_F\i\overline{\cal O}$. By the induction hypothesis, $F$ is
isomorphic to the minimal orbit ${\bf P}{\cal O}'_{min}$ for some simple Lie
algebra ${\goth g}'$ containing ${\goth g}$; either ${\goth g}'={\goth g}$,
or the pair
$({\goth g},{\goth g}')$ is one of the pairs appearing in the list
(\ref{list}).
\par\hskip 1truecm\relax Let us
say for short that an orbit ${\cal O}'\i\overline{\cal O}$ is {\it ramified} if
$\varphi^{-1} ({\bf P}{\cal O}')$ is contained in the fixed locus of some
nontrivial
element of $\Gamma $. Let ${\cal O}'\i\overline{\cal O}$ an orbit which is not
ramified; since $\varphi$ induces an isomorphism of $M/\Gamma $ onto the
normalization $\widetilde{{\bf P}{\cal O}}$ of $\overline{{\bf P}{\cal O}}$, we
have:
\par\hskip 0.5truecm\relax (\ref{arg}.{\it a}) $\widetilde{{\bf P}{\cal
O}}$ is smooth along ${\bf P}{\cal O}'$; in particular, the centralizer of any
element of ${\cal O}'\cap{\goth n}$ is contained in ${\goth n}^\perp$ (lemma
\ref{key}).
\par\hskip 0.5truecm\relax (\ref{arg}.{\it b}) Any
nonzero element $N\in {\cal O}'\cap{\goth g}(2)$ satisfies ${\goth z}^{}_N \cap{\goth
g}(-2)=(0)$, hence is conjugate to $N_0$ by [C-M], 3.4.17; therefore if
${\cal O}'\cap{\goth g}(2)\not=(0)$, then ${\cal O}'={\cal O}$.
\par\hskip 0.5truecm\relax (\ref{arg}.{\it c}) Assume that $\overline{\cal O}$ is normal
along ${\cal O}'$. Then $\varphi$ is \'etale above ${\bf P}{\cal O}'$, so that
$T_m(\varphi)$ is injective at each point $m$ of $\varphi^{-1} ({\bf P}{\cal
O}')$. But this implies that $m$ belongs to the open orbit $M^{\rm o}$ (Prop.\
\ref{openorbit}), hence ${\cal O}'={\cal O}$ again.
\par\hskip 0.5truecm\relax (\ref{arg}.{\it d}) Assume that the Galois group $\Gamma $ is cyclic of
prime order, and that $\overline{\cal O}$ is normal. Let $M^{\Gamma}$ be the
fixed
locus of $\Gamma $ in $M$. Then $\varphi $ induces an isomorphism of
$M^{\Gamma }$ onto its image; in particular, $\varphi (M^\Gamma )$ is smooth.
By
Prop.\ \ref{closmooth}, this implies that the only ramified orbit is ${\cal
O}_{min}$, so by (\ref{arg}.{\it c}) we have $\overline{\cal O}={\cal O}\cup
{\cal
O}_{min}$.
\smallskip
\global\def\currenvir{subsection Now we examine which orbits ${\cal O}\i{\goth g}$ may occur. We
order
the nilpotent orbits by the relation ``${\cal O}'\le {\cal O}$ iff ${\cal
O}'\i\overline{{\cal O}}$". Given
the Lie algebra
${\goth g}$, the possible ramified orbits are those contained in the closure
of
the orbit ${\cal O}$ in (\ref{list}).
Using the above arguments we will show that only one more orbit is allowed:
its
boundary must contain only ramified orbits.
This gives us for each Lie algebra ${\goth g}$ a small list of orbits, among
which we may eliminate those which are simply connected; we will show that
the remaining ones are those which appear in the list (\ref{list}).
\medskip
{\it Type $A_l\ (l\ge 4)$}\vglue0pt
\par\hskip 1truecm\relax All orbit closures in case $A_l$ are normal [K-P1], so by
(\ref{arg}.{\it c}) there is only one orbit which is not ramified. There is
no
shared orbit pair, so the only ramified orbit is the minimal one. The next
orbit
in the partial ordering is
${\cal O}_{(2,2,1,\ldots)}$, which is simply-connected [C-M, p.\ 92].\smallskip
{\it Type $A_3$}
\par\hskip 1truecm\relax The possible ramified orbits are ${\cal O}_{min}$ and ${\cal O}_{(2,2)}$;
the
next orbit in the partial ordering is ${\cal O}_{(3,1)}$. The orbit ${\cal
O}_{(2,2)}$ gives rise to case $(D_3,B_3)$ of (\ref{list}); ${\cal O}_{(3,1)}$
is simply-connected.
\smallskip
{\it Type $A_2$}
\par\hskip 1truecm\relax There are only two orbits, ${\cal O}_{min}$ and the
principal orbit ${\cal O}_{(3)}$, which gives rise to case $(A_2,G_2)$ of
(\ref{list}). \medskip
\par\hskip 1truecm\relax For the types $B_l,C_l \hbox{ or }D_l$, most orbit closures are normal,
with
the following exceptions [K-P2]:
\par\hskip 0.5truecm\relax a) There may exist an orbit
${\cal O}$
whose closure is non-normal along a codimension 2 orbit ${\cal O}'$, but
whose
normalization is singular along ${\cal O}'$. In this case by (\ref{arg}.{\it
a})
${\cal O}'$ is ramified;
\par\hskip 0.5truecm\relax b) When ${\goth g}$ is of type $D_l$, there are orbits (corresponding to
the
so-called ``very even" classes) whose closure is not known to be normal.
However
these orbit closures have a boundary component of codimension 2 along which
they
are normal, so that (\ref{arg}.{\it c}) still applies.\smallskip
{\it Type $B_l$ and $D_l$, $l\ge 5$}
\par\hskip 1truecm\relax The Lie algebra ${\goth g}$ is ${\goth s}{\goth o}(n)$ $(n\ge 10)$. The
possible ramified orbits are ${\cal O}_{min}$ and ${\cal O}_{(3,1,\ldots)}$;
the
only possible next orbit is ${\cal O}_{(2,2,2,2,1,\ldots)}$ (${\cal
O}_{(3,2,2,1,\ldots)}$ is excluded because its closure contains ${\cal
O}_{(2,2,2,2,1,\ldots)}$ which is not ramified). The orbit ${\cal
O}_{(3,1,\ldots)}$ gives rise to cases $(B_l,D_{l+1} )$ and $(D_l,B_l)$; ${\cal
O}_{(2,2,2,2,1,\ldots)}$ is simply-connected (\hbox{[C-M]}, p.\ 92).
\smallskip
{\it Type $B_4$}
\par\hskip 1truecm\relax The configuration of orbits is the same as above, but here the orbit
${\cal
O}_{(2,2,2,2,1)}$ can be ramified. Therefore the next orbit ${\cal
O}_{(3,2,2,1,1)}$
might occur. However its fundamental group is ${\bf
Z}/2$, and its closure is normal \hbox{[K-P2]}, so we deduce from
(\ref{arg}.{\it d}) that this orbit does not occur.
\par\hskip 1truecm\relax The orbit ${\cal
O}_{(2,2,2,2,1)}$ is no longer simply-connected; it gives rise to case
$(B_4,F_4)$ in
(\ref{list}).
\smallskip
{\it Type $B_3$}\vglue0pt
\par\hskip 1truecm\relax Again the orbit ${\cal O}_{(3,2,2)}$ can occur a priori; the same argument
as for $B_4$ applies.
\smallskip
{\it Type $D_4$}
\par\hskip 1truecm\relax The possible ramified orbits are ${\cal O}_{min}$, the three orbits next
to
${\cal O}_{min}$ in the partial ordering (namely ${\cal O}_{(3,1,\ldots)}$ and
the
two orbits ${\cal O}_{(2,2,2,2)}$), and ${\cal O}_{(3,2,2,1)}$; the next orbit
is
${\cal O}_{(3,3,1,1)}$.
\par\hskip 1truecm\relax The three orbits next to ${\cal
O}_{min}$ have the same weighted Dynkin diagram up to automorphisms, and are
therefore isomorphic; they give the case $(D_4,B_4)$. The orbit ${\cal
O}_{(3,2,2,1)}$ gives the case $(D_4,F_4)$. Finally ${\cal O}_{(3,3,1,1)}$ has
fundamental group ${\bf Z}/2$ and normal closure [K-P2], so is excluded by
(\ref{arg}.{\it d}).
\medskip
{\it Type} $C_l$ $(l\ge 2)$
\par\hskip 1truecm\relax The possible ramified orbits are ${\cal O}_{min}$ and ${\cal
O}_{(2,2,1,\ldots)}$; the next orbit is ${\cal
O}_{(2,2,2,1,\ldots)}$ if $l\ge 3$, and ${\cal O}_{(4)}$ if $l=2$. This orbit
has
fundamental group ${\bf Z}/2$ and is normal [K-P2], so it is excluded again by
(\ref{arg}.{\it d}).
The orbit
${\cal O}_{(2,2,1,\ldots)}$ gives the case $(C_l,A_{2l-1})$.
\medskip
{\it Type} $E_l$
\par\hskip 1truecm\relax The only possible ramified orbit is the minimal one. If ${\cal
O}$ is not reduced to ${\cal O}_{min}$ it contains the next orbit ${\cal O}_1$
in the partial ordering, which is the orbit of $X_\lambda +X_\mu $, where
$\lambda $ and $\mu $ are two orthogonal roots. By (\ref{arg}.{\it a}) the
centralizer of an element of ${\cal O}_1\cap{\goth n}$ is contained in ${\goth
n}^\perp$. \par\hskip 1truecm\relax Let $\sigma $ be the sum of the simple roots, and $\alpha
,\beta ,\gamma $ the simple roots corresponding to the three ends of the
Dynkin graph. Then $\sigma ,\,\sigma -\alpha ,\,\sigma -\beta ,\,\sigma
-\gamma $ are roots ([B], \S 1, n\up{o}\kern 2pt 6, cor.\ 3 of prop.\ 19) and $\sigma
-\alpha $ and $\sigma -\beta $ are orthogonal; the element $N=X_{\sigma
-\alpha }+X_{\sigma -\beta }$ satisfies $[N\,,\,X_{\gamma -\sigma }]=0$. Let
$s=\sigma (H)$ and $m=\max \{\alpha (H),\beta (H),\gamma (H)\}$. If $s-m\ge 2$
we have $N\in {\goth n}$ and $X_{\gamma -\sigma }\notin {\goth n}^\perp$, a
contradiction.
\par\hskip 1truecm\relax Suppose $s=2$ and $\alpha (H)=\beta (H)=0$. Then $N$ belongs to ${\goth
g}(2)$, which by (\ref{arg}.{\it b}) implies ${\cal O}={\cal O}_1$; this is
excluded because ${\cal O}_1$ is simply-connected ([C-M], pp.\ 129, 130, 132).
\par\hskip 1truecm\relax Looking at the list of possible weighted Dynkin diagrams in {\it loc.\
cit.}\ and eli\-mi\-nating the simply-connected orbits, the above constraints
leave
us with only one possible case, the weighted Dynkin diagram
\def\scriptstyle 2{\scriptstyle 2}
\def\scriptstyle 1{\scriptstyle 1}
\def\scriptstyle 0{\scriptstyle 0}
\def\vrule height 3pt depth 0pt width 0pt{\vrule height 3pt depth 0pt width 0pt}
$$\dia{
\scriptstyle 1 && \scriptstyle 0 & & \scriptstyle 0 & & \scriptstyle 0 && \scriptstyle 0 && \scriptstyle 0 && \scriptstyle 1 \cr
\vrule height 3pt depth 0pt width 0pt \cr
\circ & \kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ &\kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ & \kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ &\kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ &
\kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ &\kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ \cr
&& && \vrule height10pt depth4pt width 0.4pt &&&&&&&&\cr
&& && \circ &&&&&&&&\cr\vrule height 3pt depth 0pt width 0pt\cr
&& && \scriptstyle 0 &&&&&&&&\cr}$$
for $E_8$. In that case one finds easily two orthogonal roots $\lambda $ and
$\mu $ with $\lambda (H)=\mu (H)=2$, for instance (with the notation of [B],
planche VII) $\lambda ={1\over 2}\sum_i\varepsilon _i$ and $\mu
=\varepsilon_8-\varepsilon _7$; we conclude again by (\ref{arg}.{\it b})
that ${\cal O}={\cal O}_1$.
\medskip
{\it Type $F_4$}\vglue0pt
\par\hskip 1truecm\relax The orbits which can be ramified are ${\cal O}_{min}$ and ${\cal
O}_{short}$. If ${\cal O}$ is bigger than ${\cal O}_{short}$, it contains
the orbit ${\cal O}_1$ next to ${\cal O}_{short}$; this is the orbit of
$X_\alpha +X_\beta $, where $\alpha $ and $\beta $ are two orthogonal roots of
distinct lengths. Let $$\diaram{
{\scriptstyle l_1} && {\scriptstyle l_2} & & {\scriptstyle l_3} & &
{\scriptstyle
l_4} \cr \circ & \kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ
&\kern-12pt\Longrightarrow\!\!=\!=\kern-12pt &\circ &\kern-12pt\raise2pt\vbox{\hrule width .94truecm &\circ}$$
be the weighted Dynkin diagram of ${\cal O}$. Assume first $l_1+l_2+l_3\ge
2$. Using the notation of [B], planche VIII, let
$$\nospacedmath\displaylines{
\alpha =\varepsilon _2=\alpha _1+\alpha _2+\alpha _3\ ,\quad \beta
=\varepsilon _1-\varepsilon _4=\alpha _1+2\alpha _2+2\alpha _3+2\alpha _4\
,\cr
\gamma =\varepsilon _1+\varepsilon _4=\alpha _1+2\alpha _2+4\alpha _3+2\alpha
_4\ .}$$
We have $[X_\alpha +X_\beta \,,\,X_{-\gamma }]=0$, $X_\alpha
+X_\beta\in{\goth n}$ and $X_{-\gamma }\notin {\goth n}^\perp$, contradicting
(\ref{arg}.{\it a}).
\par\hskip 1truecm\relax A glance at the tables ([C-M], p.\ 128) shows that the nilpotent orbits
with $l_1+l_2+l_3\le 1$ are simply-connected, with the exception of ${\cal
O}_{short}$; the latter gives the case $(F_4,E_6)$.
\medskip
{\it Type} $G_2$
\par\hskip 1truecm\relax The only orbit which is not simply-connected is the subregular orbit
([C-M], p.~128), which gives rise to case $(G_2,D_4)$.\cqfd
\medskip
\rem{Example}\label{sp} Let us give an example of a $G$\kern-1.5pt - covering when
${\goth g}$ is not simple. Let ${\bf n}=(n_1,\ldots,n_k)$ be a sequence of
positive integers; for each $i$, let ${\goth g}_i$ be the Lie algebra ${\goth
s}{\goth p}(2n_i)$, and
$V_i\ (\cong {\bf C}^{2n_i})$ its standard representation. Then
${\goth g}_i$ can be identified
with $\hbox{\san \char83}^2V_i$; the minimal nilpotent orbit ${\cal O}_i\i{\goth g}_i$ is then
identified with the cone of rank one tensors, so that we have a 2-to-1 map
$\mu _i:V_i\rightarrow \overline{\cal O}_i={\cal O}_i\cup\{0\}$ mapping a
vector $v$ to $v^2$. We put ${\goth g}=\pprod_{i}^{}{\goth g}_i$, ${\cal
O}=\pprod_i^{}{\cal O}_i$, $M={\bf P}(V)$ with $V=\ \sdir_i^{}V_i$. The maps
$\mu _i$ define a $G$\kern-1.5pt - covering $\varphi_{\bf n} :{\bf P}(V)\rightarrow
\overline{{\bf P}{\cal O}}$, of degree $2^{k-1}$. Note that $M$ is a
minimal orbit in ${\bf P}({\goth g}')$, with ${\goth g}'={\goth s}{\goth
p}(V)$.
\th Proposition
\enonce Assume that ${\goth g}$ is a product of simple Lie algebras
${\goth g}_1,\ldots,{\goth g}_k$ $(k>1)$. Let $\varphi:M\rightarrow
\overline{{\bf P}{\cal O}}$ be a $G$\kern-1.5pt - covering. Then there exists
a sequence ${\bf n}=(n_1,\ldots,n_k)$ of
positive integers such that $\varphi $ is isomorphic to the $G$\kern-1.5pt - covering
$\varphi _{\bf n}$ of example $\ref{sp}$. In particular,
${\goth g}_i$ is isomorphic to ${\goth s}{\goth p}(2n_i)$ for each $i$, the
orbit ${\cal O}$ is the product of the minimal orbits ${\cal O}_i\i{\goth
g}_i$, and $M$ is isomorphic to ${\bf P}^{2n-1}$ with $n=\sum n_i$.
\endth\label{nonsimple}
{\it Proof}: The orbit ${\cal O}$ is a product of nontrivial orbits ${\cal
O}_i\i{\goth g}_i$. Let ${\cal O}_i^{sc}$ be the
simply-connected covering of ${\cal O}_i$, and $\overline{{\cal O}_i^{sc}}$
the
integral closure of $\overline{\cal O}_i$ in ${\cal O}_i^{sc}$ (contrary to an
earlier notation, we denote by $\overline{\cal O}_i$ the closure of ${\cal
O}_i$ {\it in} ${\goth g}$). The action of $G\times {\bf C}^*$ on
${\cal O}_i$ extends to an action on ${\cal O}_i^{sc}$ and $\overline{{\cal
O}_i^{sc}}$. There is only one point
$o^{}_i$ of $\overline{{\cal O}_i^{sc}}$ above $0\in{\goth g}$; the open subset
$\overline{{\cal O}_i^{sc}}\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}\{o_i\}$ is a principal ${\bf C}^*$\kern-1.5pt - bundle
over a variety
$M_i$ which admits a finite $G$\kern-1.5pt - equivariant morphism onto $\overline{{\bf
P}{\cal O}}_i$.
\par\hskip 1truecm\relax Let
$M'=(\pprod_i^{} \overline{{\cal O}_i^{sc}})^{\scriptscriptstyle\times }/{\bf C}^*$, where the
superscript $^{\scriptscriptstyle\times }$ means that we take out the point $(o^{}_1,\ldots,o^{}_k)$.
This is a normal variety, with a finite morphism onto $\overline{{\bf P}{\cal
O}}$; the open subset $(\pprod_i^{}{\cal O}_i^{sc})/{\bf C}^*$ is
simply-connected and its complement has codimension $\ge 2$. This implies that
$M'$ is isomorphic to $M$.
\par\hskip 1truecm\relax Since $M$ is smooth, it follows that each
$\overline{{\cal O}_i^{sc}}$ must be smooth. This implies first of all that $
{\cal O}_i^{sc}$ is smooth, hence by Prop.\ \ref{closmooth} and
\ref{coverings} isomorphic to the minimal orbit ${\bf P}{\cal O}'_i$
for some simple Lie algebra ${\goth g}'_i$ containing ${\goth g}_i$. Then
${\cal O}_i^{sc}$ is the simply-connected cover of ${\cal O}'_i$, and
$\overline{{\cal O}_i^{sc}}$ is its integral closure in $\overline{\cal O}_i$.
Since $\overline{{\cal O}_i^{sc}}$ is smooth, this happens if and only if
${\goth g}_i={\goth g}'_i\cong {\goth s}{\goth p}(2n_i)$ for some integer
$n_i\ge 1$ ([B-K], thm.\ 4.6); then ${\cal O}_i={\cal O}'_i$ by Prop.\
\ref{coverings}, so we are in the situation of example
\ref{sp}.\cqfd\medskip
\par\hskip 1truecm\relax The above results imply directly Theorem \ref{main}, in a slightly more
precise form:
\th Theorem
\enonce Let $M$ be a Fano contact manifold, satifying the conditions $(\H1)$
and $(\H2)$ of Theorem $\ref{main}$. Then the Lie algebra ${\goth g}$ of
infinitesimal contact transformations of $M$ is simple, and the canonical
map $\varphi:M\rightarrow {\bf P}({\goth g})$ induces an isomorphism of $M$
onto the minimal orbit ${\bf P}{\cal O}_{min}\i {\bf P}({\goth g})$.
\endth
{\it Proof}: By (\ref{H3}), we can assume that $M$ satisfies also (H3); then
$\varphi$ induces a $G$\kern-1.5pt - covering $M\rightarrow \overline{{\bf P}{\cal O}}$
onto the closure of some nilpotent orbit in ${\bf P}({\goth g})$ (Prop.\
\ref{summary}). By Prop.\ \ref{coverings} and
\ref{nonsimple}, $M$ is isomorphic to the minimal orbit in ${\bf P}({\goth
g}')$ for some simple Lie algebra ${\goth g}'$ containing ${\goth g}$;
moreover if $\varphi$ is not an embedding, ${\goth g}'$ contains strictly
${\goth g}$, which is impossible since ${\goth g}'$ is an algebra of
infinitesimal contact transformations of $M$ (see remark \ref{unique}).
Therefore
$\varphi$ is an embedding and ${\goth g}'={\goth g}$.\cqfd
\vskip2cm
\centerline{ REFERENCES} \vglue15pt\baselineskip13.4pt
\def\num#1{\smallskip \item{\hbox to\parindent{\enskip [#1]\hfill}}}
\parindent=1.3cm
\num{A} V.\ {\pc ARNOLD}: {\sl Mathematical methods of classical mechanics}.
Graduate Texts in
Math.\ {\bf 60}; Springer-Verlag, New York-Heidelberg (1978).
\num{B} N.\ {\pc BOURBAKI}: {\sl Groupes et alg\`ebres de Lie}, Chap.\ VI.
Hermann, Paris (1968).
\num{B-K} R.\ {\pc BRYLINSKI}, B.\ {\pc KOSTANT}: {\sl
Nilpotent orbits, normality and Hamiltonian group actions}. J.\ of the A.M.S.\
{\bf 7}, 269-298 (1994).
\num{C-M} D.\ {\pc COLLINGWOOD}, W.\ {\pc MC}{\pc GOVERN}: {\sl Nilpotent
orbits in
semi-simple Lie algebras}. Van Nostrand Reinhold Co., New York (1993).
\num{K-P1} H.\ {\pc KRAFT}, C.\ {\pc PROCESI}: {\sl Closures of conjugacy
classes of matrices are normal}. Invent.\ math.\ {\bf 53}, 227-247 (1979).
\num{K-P2} H.\ {\pc KRAFT}, C.\ {\pc PROCESI}: {\sl On the geometry of the
conjugacy classes in classical groups}. Comment.\ Math.\ Helvetici {\bf 57},
539-602 (1982).
\num{L} C.\ {\pc LE}{\pc BRUN}: {\sl Fano manifolds, contact
structures, and quaternionic geometry}. Int.\ J.\ of Math. {\bf 6}, 419-437
(1995). \num{L-S} C.\ {\pc LE}{\pc BRUN}, S.\ {\pc SALAMON}: {\sl Strong
rigidity of quaternion-K\"ahler manifolds}. Invent.\ math.\ {\bf 118}, 109-132
(1994). \num{L-Sm} T.\ {\pc LEVASSEUR}, S.\ {\pc SMITH}: {\sl Primitive ideals
and nilpotent orbits in type $G_2$}. J.\ of Algebra {\bf 114}, 81-105 (1988).
\num{M} D.\ {\pc MUMFORD}, J.\ {\pc FOGARTY}: {\sl Geometric invariant
theory}. 2\up{nd} edition. Springer-Verlag, New York-Heidelberg (1982).
\smallskip
\num{Mu} S.\ {\pc MUKAI}: {\sl Biregular classification of Fano $3$\kern-1.5pt - folds
and Fano manifolds of coindex $3$}. Proc.\ Nat.\ Acad.\ Sci.\ USA {\bf 86},
3000-3002 (1989).
\vskip1cm
\def\pc#1{\eightrm#1\sixrm}
\hfill\vtop{\eightrm\hbox to 5cm{\hfill Arnaud {\pc BEAUVILLE}\hfill}
\hbox to 5cm{\hfill DMI -- \'Ecole Normale
Sup\'erieure\hfill} \hbox to 5cm{\hfill (URA 762 du CNRS)\hfill}
\hbox to 5cm{\hfill 45 rue d'Ulm\hfill}
\hbox to 5cm{\hfill F-75230 {\pc PARIS} Cedex 05\hfill}}
\end
|
1998-02-23T16:31:47 | 9707 | alg-geom/9707004 | en | https://arxiv.org/abs/alg-geom/9707004 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9707004 | Robert Friedman | Robert Friedman, John W. Morgan, Edward Witten | Principal G-bundles over elliptic curves | AMS-TeX, 20 pages, amsppt style; minor errors corrected | Math.Res.Lett.5:97-118,1998 | null | null | null | Let $G$ be a simple and simply connected complex Lie group. We discuss the
moduli space of holomorphic semistable principal $G$ bundles over an elliptic
curve $E$. In particular we give a new proof of a theorem of Looijenga and
Bernshtein-Shvartsman, that the moduli space is a weighted projective space.
The method of proof is to study the deformations of certain unstable bundles
coming from special maximal parabolic subgroups of $G$. We also discuss the
associated automorphism sheaves and universal bundles, as well as the relation
between various universal bundles and spectral covers.
| [
{
"version": "v1",
"created": "Mon, 7 Jul 1997 17:43:09 GMT"
},
{
"version": "v2",
"created": "Mon, 23 Feb 1998 15:31:46 GMT"
}
] | 2010-04-07T00:00:00 | [
[
"Friedman",
"Robert",
""
],
[
"Morgan",
"John W.",
""
],
[
"Witten",
"Edward",
""
]
] | alg-geom | \section{1. Introduction.}
Let $E$ be an elliptic curve with origin $p_0$, and let $G$ be
a complex simple algebraic group. For simplicity, we shall
only consider the case where $G$ is simply connected, although
all of the methods discussed below can be extended to the case
of a general group $G$. The goal of this note is to announce
some results concerning the moduli of principal holomorphic
$G$-bundles over $E$. Detailed proofs, as well as a more
thorough discussion of the case where $E$ is allowed to be
singular or to vary in families and of the connections with del
Pezzo surfaces, elliptic
$K3$ surfaces, and Calabi-Yau manifolds which are elliptic or
$K3$ fibrations, will appear elsewhere.
Grothendieck \cite{21} considered principal holomorphic
$G$-bundles over $\Pee ^1$, and showed that it was always
possible to reduce the structure group to a Cartan subgroup,
i\.e\. to a maximal (algebraic) torus in $G$. Atiyah \cite{1}
classified all holomorphic vector bundles over an elliptic
curve (in other words, the cases
$G=SL(n,
\Cee)$ or $G =PGL(n,
\Cee)$), without however considering the problem of trying to
construct a moduli space or find a universal bundle. In
\cite{16}, \cite{17}, and \cite{18}, this problem is studied
in the rank two case with a view toward constructing relative
moduli spaces in families. This approach has been generalized
to arbitrary rank in \cite{20}. A great deal of work has
been done on the moduli spaces and stacks of
$G$-bundles over a curve of genus at least two, partly
motivated by the study of conformal blocks and the Verlinde
formulas, by very many authors, e\.g\.
\cite{5}, \cite{15}. A basic method here is to
relate the moduli stack to an appropriate loop group. Related
constructions in the case of genus one have been carried out
by Baranovsky-Ginsburg \cite{4}, based on unpublished work of
Looijenga (see for example \cite{13}). They relate semistable
$G$-bundles to conjugacy classes in a corresponding affine
Kac-Moody group. Recently Br\"uchert
\cite{9} has constructed a Steinberg-type cross-section for
the adjoint quotient of the affine Kac-Moody group whose image
lies in the set of regular elements, and this construction
leads to a moduli space for semistable
$G$-bundles which is equivalent to the one we construct in
Section 4 below. (We are indebted to Slodowy for calling our
attention to the work of Br\"uchert and sketching an argument
for the equivalence of the approach described above with the
one we give in this paper.) Finally, many of the results in
this note, along with applications to physics, are discussed in
\cite{19}.
The contents of this note are as follows. We will be concerned
with the classification of semistable $G$-bundles. As is
typical in invariant theory or moduli problems, the
classification will be up to a coarser equivalence than
isomorphism, which is usually called S-equivalence and will be
defined more precisely in Section 2. In Section 2, we describe
the moduli space of semistable
$G$-bundles over $E$ via flat connections for the maximal
compact subgroup $K$ of $G$, or equivalently via conjugacy
classes of representations $\rho\: \pi _1(E) \to K$. Such
bundles, which for a simply connected group
$G$ are exactly the bundles whose structure group reduces to a
Cartan subgroup, have an automorphism group which is as large
as possible in a certain sense within a fixed S-equivalence
class. The main result here is a theorem due to Looijenga and
Bernshtein-Shvartsman which describes this moduli space as a
weighted projective space. At the end of the section, we
connect this description, in the case where
$G=E_6, E_7, E_8$, with the moduli space of del Pezzo surfaces
of degree $3,2,1$ respectively and with the deformation theory
of simple elliptic singularities. In Section 3, we describe
regular $G$-bundles, which by contrast with flat bundles have
automorphism groups whose dimensions are as small as
possible within a fixed S-equivalence
class. The generic $G$-bundle is both flat and regular.
However at special points of the moduli space we can choose
either a unique flat representative or a unique regular
representative, and it is the regular representatives which
fit together to give holomorphic families. In Section 4, we
show how special unstable bundles over certain maximal
parabolic subgroups can be used to give another description of
the moduli space in terms of regular bundles and obtain a new
proof of the theorem of Looijenga and Bernshtein-Shvartsman.
Finally, in the last section we discuss the existence of
universal bundles and give a brief description of how our
construction can be twisted with the help of a certain
spectral cover.
\section{2. Split semistable bundles.}
We fix notation for the rest of this paper. As before, $E$
denotes an elliptic curve with origin $p_0$. Let $G$ be a
simple and simply connected complex Lie group of rank $r$, and
let $\xi \to E$ be a holomorphic principal $G$-bundle over
$E$. The following definition differs from that given in
Ramanathan
\cite{32}, but is equivalent to it.
\definition{Definition 2.1} The principal bundle $\xi \to E$
is {\sl semistable\/} if the associated vector bundle $\ad
\xi$ is a semistable vector bundle. The principal bundle $\xi
\to E$ is {\sl unstable\/} if it is not semistable.
\enddefinition
Note that, if $\xi$ is stable in the sense of \cite{32}, it is
still possible for the vector bundle
$\ad
\xi$ to be strictly semistable. However, in our case ($G$
simply connected), there are essentially no properly stable
bundles over $E$, and so the above definition will suffice for
our purposes.
If $\xi$ is an unstable bundle, the structure group of $\xi$
reduces canonically to a parabolic subgroup $P$ of $G$, the
{\sl Harder-Narasimhan parabolic\/} associated to $\xi$ (see
for example \cite{31} or \cite{2}, pp\. 589--590). The
canonical reduction holds over a general base curve. In the
case of a base curve $E$ of genus one, it is easy to see that
the structure group further reduces to a Levi factor of $P$.
Recall the following standard terminology: a {\sl family\/} of
principal
$G$-bundles over $E$ parametrized by a complex space (or
scheme) $S$ is a principal $G$-bundle
$\Xi$ over $E\times S$. The family $\Xi$ is a family of {\sl
semistable\/} principal $G$-bundles over $E$ if $\Xi|E\times
\{s\}=\Xi _s$ is semistable for all $s\in S$. Finally, let
$\xi$ and
$\xi'$ be two semistable bundles over
$E$. We say that
$\xi$ and
$\xi'$ are {\sl S-equivalent\/} if there exists a family of
semistable bundles
$\Xi$ parametrized by an irreducible $S$ and a point $s\in S$
such that, for
$t\neq s$, $\Xi |E \times \{t\} \cong \xi$ and $\Xi |E \times
\{s\} \cong
\xi'$. More generally, we let S-equivalence be the equivalence
relation generated by the above relation.
The following holds only under our assumption that $G$ is
simply connected.
\proposition{2.2} Let $\xi$ be a semistable principal
$G$-bundle, and suppose that the rank of $G$ is $r$. Then
$h^0(E; \ad \xi) \geq r$. Equivalently, $\dim \Aut _G\xi \geq
r$, where $\dim \Aut _G\xi$ denotes the group of global
automorphisms of $\xi$ \rom(as a
$G$-bundle\rom).
\endstatement
\definition{Definition 2.3} Let $\xi$ be a semistable
principal $G$-bundle. We call $\xi$ {\sl regular\/} if $h^0(E;
\ad \xi) = r$, or equivalently if $\dim
\Aut _G(\xi) = r$. We call $\xi$ {\sl split\/} if its
structure group reduces to a Cartan subgroup of
$G$, i\.e\. a maximal (algebraic) torus.
\enddefinition
It is easy to check that split bundles have the following
closure property: if there exists a family of semistable
bundles
$\Xi$ parametrized by an irreducible $S$ and a point $s\in S$
such that, for
$t\neq s$, the bundles $\Xi |E \times \{t\}$ are split and all
isomorphic to each other, then
$\Xi |E \times \{s\}$ is isomorphic to $\Xi |E \times \{t\}$,
$t\neq s$, and thus it is split as well. In general, however,
the condition of being split is neither open nor closed. On the
other hand, by the upper semicontinuity theorem, regularity is
an open condition: if
$\Xi$ is a family of semistable bundles parametrized by $S$
and $\Xi |E \times
\{s\}$ is regular, then $\Xi |E \times \{t\}$ is regular for
all $t$ in an open neighborhood of $s$.
To describe the set of split bundles, we introduce flat
bundles on the compact group. Let $K$ be a maximal compact
subgroup of $G$. Then $K$ is a compact, simple and simply
connected Lie group. If $\frak k$ is the Lie algebra of $K$ and
$\frak g$ is the Lie algebra of $G$, then $\frak g$ is the
complexification of
$\frak k$. Given a representation $\rho\: \pi _1(E) \cong \Zee
\oplus \Zee \to K$, we can form the associated principal
$K$-bundle $(\tilde E \times K)/\pi _1(E) \to E$, where $\pi
_1(E)$ acts on $\tilde E$, the universal cover of $E$, in the
usual way, and on $K$ via $\rho$. We shall call such a
$K$-bundle a {\sl flat\/} $K$-bundle. Using the inclusion
$K\subset G$, we can also view a flat $K$-bundle as a
$G$-bundle, and we shall also incorrectly refer to the induced
$G$-bundle as a flat $K$-bundle. We will need the following
version of the theorem of Narasimhan-Seshadri
\cite{29} and Ramanathan
\cite{32} (see also Atiyah-Bott \cite{2} and Donaldson
\cite{12}):
\theorem{2.4} Let $\xi\to E$ be a semistable principal
$G$-bundle. Then there is a flat $K$-bundle $S$-equivalent to
$\xi$, and it is unique up to
isomorphism of flat $K$-bundles. More precisely, there is a
family of semistable principal $G$-bundles $\Xi$ over
$E\times \Cee$, such that, for
$t\neq 0$, $\xi_t=\Xi|E\times \{t\}\cong \xi$, and such that
$\xi _0= \Xi|E\times \{0\}$ is the $G$-bundle associated to a
flat $K$-bundle via the inclusion $K\subset G$. Finally, two
flat
$K$-bundles are isomorphic as
$G$-bundles if and only if they are isomorphic as $K$-bundles.
\endstatement
We note that Theorem 2.4 also holds for a non-simply connected
group. The special feature of simply connected groups which we
need to describe the moduli space of flat
$K$-bundles is contained in the following result of Borel
\cite{7} (see also
\cite{22} for the analogous algebraic result, due to Springer
and Steinberg):
\theorem{2.5} Let $K$ be a compact, simple, and simply
connected Lie group, and let $r_1$ and $r_2$ be two commuting
elements of $K$. Then there exists a maximal torus $T$ in $K$
with $r_1, r_2\in T$.
\endstatement
Since $\pi _1(E) \cong \Zee\oplus \Zee$, to give a
representation $\rho\: \pi _1(E) \to K$ is to give two
commuting elements $r_1, r_2\in K$. Thus a flat $K$-bundle
reduces to a $T$-bundle. In particular, we see that for a
simply connected group
$G$, every $G$-bundle associated to a flat bundle is split, and
conversely. On the other hand, if $G$ is not simply connected,
every split bundle lifts to the universal cover $\tilde G$ of
$G$, so that a $G$-bundle which does not lift to $\tilde G$
cannot be split. Thus the correct notion for unliftable
bundles is that of a flat bundle.
Returning to the case of a simply connected group $G$, let $T$
be a maximal torus in the compact group $K$. One checks that
two homomorphisms from $\pi _1(E)$ to $T$ are conjugate by an
element of $K$ if and only if they are conjugate by an element
of the normalizer of $T$ in $K$. Thus we have:
\theorem{2.6} There is a natural bijection from the set of
flat $K$-bundles up to isomorphism, or equivalently the set of
semistable $G$-bundles up to S-equivalence, to the set $\Hom
(\pi _1(E), T)/W$, where $W$ is the Weyl group of $K$, acting
in the usual way on the maximal torus $T$.
\endstatement
Fix a maximal torus $T$ in
$K$. If $\Lambda = \pi _1(T)$, then $T\cong U(1) \otimes
_{\Zee}\Lambda$. Moreover, since $K$ is simply connected, if
$\frak t_\Ar$ denotes the real Lie algebra of $T$, then
$\Lambda\subset
\frak t_\Ar$ is the lattice generated by the coroots $\alpha
\spcheck$, where
$\alpha \in \frak t_\Ar^*$ is a root. Now given a
homomorphism
$\rho\: \pi _1(E) \cong \Zee\oplus \Zee\to K$, the image of
$\rho$ is generated by two commuting elements of $K$ and so,
after conjugation, lies in $T$. The set of flat $T$-bundles is
naturally
$$\Hom (\pi _1(E), T) = \Hom (\pi _1(E),U(1) \otimes
_{\Zee}\Lambda ) \cong
\Hom (\pi _1(E), U(1))\otimes _\Zee \Lambda.$$ Now $\Hom (\pi
_1(E), U(1))$ is the set of flat line bundles on $E$, and is
naturally identified with $\Pic ^0E$. Since we have fixed a
base point $p_0\in E$, we can further identify $\Pic ^0E$ with
$E$. Thus the space of flat
$T$-bundles is naturally $E\otimes _\Zee \Lambda$. On the
other hand, as we are classifying not flat $T$-bundles but
flat $K$-bundles, we must take the quotient of $E\otimes _\Zee
\Lambda$ by the action of the Weyl group $W$ of $G$ acting on
$E\otimes _\Zee \Lambda$ via the natural action of $W$ on
$\Lambda$. We have thus described the coarse moduli space
of semistable $G$-bundles over $E$ as $(E\otimes _\Zee
\Lambda)/W$. A different proof of this result has been
given by Laszlo \cite{25}.
The varieties $(E\otimes _\Zee
\Lambda)/W$ have been studied by
Looijenga
\cite{27} and Bernshtein-Shvartsman
\cite{6}, who proved the following theorem:
\theorem{2.7} Let $E$ be an elliptic curve and let $\Lambda$
be the coroot lattice of a simple root system $R$ with Weyl
group $W$. Then $(E\otimes _\Zee
\Lambda)/W$ is a weighted projective space $WP(g_0, \dots,
g_r)$, where the weights $g_i$ are given as follows: $g_0 =
1$, and the remaining roots $g_i$ are found by choosing a set
of simple roots $\alpha _1, \dots , \alpha _r$, and then
writing the coroot $\tilde \alpha
\spcheck$ dual to the highest root
$\tilde \alpha$ as a linear combination $\sum _ig_i\alpha
_i\spcheck$ of the coroots dual to the simple roots. In case
$R$ is simply laced, we can identify the dual coroot $\alpha
\spcheck \in R\spcheck$ to $\alpha$ with
$\alpha$, and consequently the
$g_i$ are the coefficients of $\tilde \alpha$ in terms of the
basis
$\alpha _1, \dots , \alpha _r$.
\endstatement
The proof of \cite{27} and \cite{6} makes use of formal
theta functions for a complexified affine Weyl group. We shall
outline a different proof of (2.7) below.
Since it will be important to motivate the construction of
Section 4, let us give Looijenga's reason for studying the
space $(E\otimes _\Zee\Lambda)/W$. Let $(X, x_0)$ be the germ
of a simple elliptic singularity whose minimal resolution has
a single exceptional component which is a smooth elliptic
curve $E$ with self-intersection $-3, -2$, or $-1$. These are
exactly the simple elliptic singularities which can be
realized as hypersurface singularities in $(\Cee ^3, 0)$, and
we shall refer to them as being of type
$\tilde E_6, \tilde E_7, \tilde E_8$ respectively. These
singularities are weighted cones over
$E$ corresponding to a line bundle $L$ on $E$ of degree
$3$, $2$, or $1$, and thus have a
$\Cee ^*$-action. Moreover $\Cee ^*$ also acts on the tangent
space to the deformations of $(X, x_0)$. The zero weight
directions (in other words those directions fixed by the
$\Cee^*$-action) correspond to deforming
$(X, x_0)$ in an equisingular family by deforming $E$. The
remaining weights are negative, and deformations in the
neagative weight space correspond to deforming $(X, x_0)$ to
a rational double point (RDP) singularity or smoothing it. The
local action of
$\Cee ^*$ on the negative weight deformations may be
globalized, and the quotient corresponding to the singularity
$\tilde E_r$ is a weighted projective space
$WP(g_0, \dots, g_r)$, where the weights $g_i$ are those
defined above for the root system
$E_r$. On the other hand, by the general theory of negative
weight deformations of singularities with $\Cee ^*$-actions,
and in particular by work of Pinkham \cite{30}, Looijenga
\cite{26}, and later M\'erindol \cite{28}, the points of
this weighted projective space parametrize triples $(\bar S, D,
\varphi)$, where $\bar S$ is a generalized del Pezzo surface
of degree $9-r$ (i\.e\., $\bar S$ has at worst rational double
point singularities and the inverse of the dualizing sheaf
$K_{\bar S}$ is ample on $\bar S$, with $K_{\bar S}^2 = 9-r$),
$D \in |-K_{\bar S}|$ is a smooth divisor, not passing through
the singularities of
$\bar S$, and $\varphi$ is an isomorphism from $D$ to the
fixed elliptic curve
$E$ such that $\varphi ^*L = N_{D/\bar S}$. The moduli of such
triples
$(\bar S, D, \varphi)$ can be described directly in terms of
the defining equations for $\bar S$ and can also be checked
directly to be a weighted projective space with the correct
weights. (Similar but slightly more involved arguments also
handle the case of degree $4$ and $5$, in which case the
singularity is a codimension two complete intersection, in the
case of degree $4$, and the corresponding root system is
$D_5$, or a Pfaffian singularity in case the degree is $5$,
and the root system is $A_4$.)
Now an elementary Torelli-type theorem shows that the
pair
$(\bar S, D)$ (ignoring the extra structure of $\varphi$) is
determined by the homomorphism
$\psi_0\: H^2_0(S; \Zee) \to D$, where $S$ is the minimal
resolution of $\bar S$ and $H^2_0(S;
\Zee)$ is the orthogonal complement of
$[K_S]$ in $H^2(S; \Zee)$, given as follows: represent a class
$\lambda\in H^2_0(S; \Zee)$ by a holomorphic line bundle $L$
on $S$ such that $\deg ( L|D) = 0$, and define $\psi_0
(\lambda)$ to be the element $L|D\in \Pic ^0D \cong D$. But
$H^2_0(S; \Zee)$ is isomorphic to the root lattice for the
corresponding root system $E_r$, and this isomorphism is
well-defined up to the action of the Weyl group. The choice of
the isomorphism $\varphi$ enables one to extend the map
$\psi_0$ to a map $\psi\: H^2(S;\Zee)/\Zee[D] \to E$,
essentially because on the fixed curve $E$ we can choose a
$(9-r)^{\text{th}}$ root of the line bundle $L$, and
conversely the choice of such a root fixes an isomorphism from
$D$ to $E$ which lines up $L$ with $N_{D/\bar S}$. Now
$H^2(S;\Zee)/\Zee[D]$ is dual to the coroot lattice $\Lambda$
of the root system $E_r$, and
$\psi$ defines an element of $E\otimes _\Zee\Lambda$,
well-defined modulo the action of $W$. In this way, we have
identified $WP(g_0, \dots, g_r)$ with $(E\otimes
_\Zee\Lambda)/W$. Let $\bar S$ be the result of contracting
all of the curves on $S$ not meeting $D$. Thus $\bar S$ has
certain rational double point (RDP) singularities. Under the
identification of the moduli space of pairs
$(S,D)$ with the set of $\psi\:
\Lambda \spcheck
\to E$, it is not difficult to show that the RDP singularities
on $S$ correspond to homomorphisms $\psi$ such that there is a
sub-root lattice
$\Lambda ' \subseteq \Ker \psi$. In fact, the maximal such
lattice $\Lambda '$ describes the type of the RDP singularities
on $\bar S$. Here the main point is to show, by a Riemann-Roch
argument, that if
$\gamma \in \Ker \psi$ with $\gamma ^2 = -2$, then $\pm
\gamma$ is represented by an effective curve on $S$ disjoint
from $D$, and thus gives a singular point on the surface
obtained by contracting all such curves. In this way, there is
a link between subgroups of $E_r$, $r = 6,7,8$, and
singularities of the corresponding del Pezzo surfaces.
\section{3. Regular bundles.}
Recall that, for a simply connected group $G$, the bundle $\xi$
is {\sl regular\/} if
$h^0(E; \ad \xi)$ is equal to the rank of $G$. We begin by
giving a detailed description of the set of regular bundles in
case
$G$ is one of the classical groups. At the end of the section
we shall outline the general structure of regular bundles. Let
us give a preliminary definition:
\definition{Definition 3.1} Let $I_n$ be the vector bundle of
rank $n$ and trivial determinant on $E$ defined inductively as
follows: $I_1=\scrO_E$, and
$I_n$ is the unique nonsplit extension of $I_{n-1}$ by
$\scrO_E$. More generally, if $\lambda$ is a line bundle on
$E$ of degree zero, we define
$I_n(\lambda) = I_n\otimes \lambda$.
\enddefinition
An easy argument shows that the algebra $\Hom (I_n, I_n)$ is
isomorphic to
$\Cee[t]/(t^n)$, and in particular it is a commutative
unipotent
$\Cee$-algebra of dimension $n$.
If $V$ is an arbitrary semistable vector bundle of degree zero
over $E$ and
$\lambda$ is a line bundle of degree zero over $E$, let
$V_\lambda \subseteq V$ be the sum of all of the subbundles of
$V$ which are filtered by a sequence of subbundles whose
successive quotients are isomorphic to $\lambda$. An easy
argument shows that $V_\lambda$ itself is the maximal such
subbundle with this property and that $V = \bigoplus _\lambda
V_\lambda$. A straightforward induction classifies the
possible $V_\lambda$ as a direct sum
$\bigoplus _jI_{k_j}(\lambda)$. From this, it is easy to check:
\proposition{3.2} Let $V$ be a semistable vector bundle over
$E$ with trivial determinant, i\.e\. $V$ is a principal
$SL(n)$-bundle over $E$. If
$V\cong \bigoplus _{i=1}^rI_{d_i}(\lambda _i)$, where the
$\lambda _i$ are line bundles on $E$ of degree zero, such that
$\lambda _1^{d_1} \otimes \cdots
\otimes \lambda _r^{d_r} =\scrO_E$ and $\sum _id_i = n$, then
$V$ is regular if and only if $\lambda _i\neq \lambda _j$ for
all $i\neq j$.
\endstatement
To deal with the case of the symplectic or orthogonal group,
the main point is to decide when a bundle $V$ carries a
nondegenerate alternating or symmetric form. The crucial case
is that of $I_n$. In this case, we have the following:
\proposition{3.3} There exists a nondegenerate alternating
pairing on $I_n$ if and only if $n$ is even. There exists a
nondegenerate symmetric pairing on
$I_n$ if and only if $n$ is odd. In both cases, every two such
nondegenerate pairings on $I_n$ are conjugate under the action
of $\Aut I_n$.
\endstatement
With this said, we can describe the regular symplectic
bundles. It is simplest to describe them via the standard
representation:
\proposition{3.4} Let $V$ be a vector bundle of rank $2n$ over
$E$ with a nondegenerate alternating form, and suppose that
the dimension of the group of symplectic automorphisms of $V$
is $n$. Then there exist positive integers
$d_i$ and nonnegative integers $a_j$, $0\leq j\leq 3$, with
$\sum _id_i +
\sum _ja_j = n$, such that $V$ is isomorphic to
$$\bigoplus _i\left(I_{d_i}(\lambda _i) \oplus I_{d_i}(\lambda
_i^{-1})\right)\oplus I_{2a_0} \oplus I_{2a_1}(\eta _1)
\oplus I_{2a_2}(\eta _2)
\oplus I_{2a_3}(\eta _3),$$ where the $\lambda _i$ are line
bundles of degree zero, not of order two, such that, for all
$i\neq j$, $\lambda _i\neq \lambda _j^{\pm1}$, and $\eta_1,
\eta _2, \eta _3$ are the three distinct line bundles of order
two on $E$. Conversely, suppose that $V$ is a vector bundle as
given above. Then $V$ has a nondegenerate alternating form,
all such forms have a group of symplectic automorphisms of
dimension exactly $n$, and every two nondegenerate
alternating forms on
$V$ are equivalent under the action of
$\Aut V$.
\endstatement
In particular, we see that a regular symplectic bundle is
always a regular bundle in the sense of $SL(2n)$-bundles.
For $SO(2n)$ and $SO(2n+1)$, the situation is a little more
complicated for two reasons. First, we shall only consider
those bundles which can be lifted to $Spin (2n)$ or $Spin
(2n+1)$, but shall not describe here the actual choice of a
lifting. Secondly, because of (3.3), it turns out that a
regular $SO(n)$-bundle does not always give a regular
$SL(n)$-bundle.
\proposition{3.5} Let $V$ be a vector bundle of rank $2n$ over
$E$ with a nondegenerate symmetric form, and suppose that the
dimension of the group of orthogonal automorphisms of $V$ is
$n$. Finally suppose that $V$ can be lifted to a principal
$Spin (2n)$-bundle. Then
$V$ is isomorphic to
$$\bigoplus _i\left(I_{d_i}(\lambda _i) \oplus I_{d_i}(\lambda
_i^{-1})\right)\oplus \bigoplus _j\left(I_{2a_j+1}(\eta
_j)\oplus\eta _j\right)$$ where the $\lambda _i$ are line
bundles of degree zero, not of order two, such that, for all
$i\neq j$, $\lambda _i\neq \lambda _j^{\pm1}$, $\eta _0 =
\scrO_E, \eta_1,
\eta _2, \eta _3$ are the four distinct line bundles of order
two on $E$, and the second sum is over some subset
\rom(possibly empty\rom) of $\{0,1,2,3\}$. Conversely, every
such vector bundle
$V$ has a nondegenerate symmetric form, all such forms have a
group of orthogonal automorphisms of dimension exactly $n$,
and every two nondegenerate symmetric forms on $V$ are
equivalent under the action of $\Aut V$.
\endstatement
Here the symmetric form on $I_{2a_0+1}\oplus\scrO_E$ consists
of the orthogonal direct sum of the nondegenerate form on the
factor $I_{2a_0+1}$ given by (3.3), together with the obvious
form on $\scrO_E$, and similarly for the summands
$I_{2a_i+1}(\eta _i)\oplus \eta _i$. Moreover, not all of the
summands $I_{2a_j+1}(\eta _j)\oplus\eta _j$ need be present in
$V$. We remark that, if a vector bundle
$\bigoplus _jI_{d_j}(\lambda _j)$ is isomorphic to its dual,
and the sum of all the factors where $\lambda _j =
\eta _i$ for some $i$ has odd rank, then the same must be true
for all of the
$\eta _i$. Thus, if the automorphism group of $V$ is to be as
small as possible, then either $V$ is as described in (3.5) or
$V$ is of the form
$$\bigoplus _i\left(I_{d_i}(\lambda _i) \oplus I_{d_i}(\lambda
_i^{-1})\right)\oplus I_{2a_0+1}\oplus I_{2a_1+1}(\eta
_1)\oplus I_{2a_2+1}(\eta_2)\oplus I_{2a_3+1}(\eta _3).$$
But in this last case $V$ does not lift to a $Spin
(2n)$-bundle.
The case of $SO(2n+1)$, which we shall not state explicitly,
is completely analogous, except that the summand
$I_{2a_0+1}\oplus\scrO_E$ is replaced by the odd rank summand
$I_{2a_0+1}$, which must always be present.
We return now to the study of regular bundles over a general
group $G$.
\proposition{3.6} Let $\xi$ be a semistable principal
$G$-bundle over $E$. Then the structure group of $\xi$ reduces
to an abelian subgroup of $G$. If furthermore $\xi$
is regular, the structure group of $\xi$ reduces to an abelian
subgroup of
$\Aut _G\xi$, which naturally sits inside $G$
up to conjugation.
\endstatement
In fact, one can take the structure group of $\xi$ to be of
the following form. Let $\xi _0$ be the split bundle
S-equivalent to $\xi$, corresponding to the
representation $\rho\: \pi _1(E) \to T\subset K$. Let $T_0$
be the image of $\rho$. Then there exists a
subgroup $U$ of $G$ commuting with $T_0$, which
is either trivial or a $1$-parameter commutative unipotent
subgroup, such that the structure group of $\xi$ reduces to
$T_0U$.
We now describe
the set of bundles which are simultaneously regular and
split. If $\xi$ is split, then
$\xi$ corresponds to a point of $(E\otimes_{\Zee} \Lambda)/W$.
After lifting this point to an element $\mu$ of
$E\otimes_{\Zee} \Lambda$, we see that we can describe
$\ad \xi$ as follows. A root $\alpha$ defines a homomorphism
$\Lambda \to
\Zee$, and thus a homomorphism $E\otimes _{\Zee}\Lambda \to E
\cong \Pic ^0E$. Denote the image of $\mu$ in $E$ by
$\alpha(\mu)$ and the corresponding line bundle by
$\lambda_{\alpha(\mu)}$. Then, as vector bundles,
$$\ad \xi \cong \scrO_E^r \oplus \bigoplus
_\alpha\lambda_{\alpha(\mu)}.$$ Hence $\xi$ is regular if and
only if, for every root $\alpha$, $\alpha (\mu)
\neq 0$.
In particular, there is a nonempty Zariski open subset of
$(E\otimes_{\Zee}
\Lambda)/W$ such that all of the corresponding split bundles
are regular. In fact, on this open subset, S-equivalence is
the same as isomorphism.
At the other extreme, we can consider bundles which are
S-equivalent to the trivial bundle. The split representative
for the S-equivalence class corresponds to the image of $0\in
E\otimes_{\Zee} \Lambda$ in $(E\otimes_{\Zee}
\Lambda)/W$, which has the unique preimage $0\in
E\otimes_{\Zee} \Lambda$. To describe the regular
representative, or more precisely its adjoint bundle, we first
recall the definition of the {\sl Casimir weights\/}
$d_1, \dots, d_r$ of a root system
$R$. These can be defined to be the numbers $m_i +1$, where
the $m_i$ are the exponents of $R$ (cf\.
\cite{8}, V (6.2)), and they are also the degrees of a set of
homogeneous generators for the invariants of the symmetric
algebra of the vector space corresponding to the root system
$R$ under the action of the Weyl group. To describe a regular
bundle S-equivalent to the trivial bundle, we shall describe
its adjoint bundle. (Here an $(\ad G)$-bundle has in general
finitely many liftings to a $G$-bundle, but exactly one of
these will turn out to be S-equivalent to the trivial bundle.)
\proposition{3.7} There is a unique regular $G$-bundle $\xi$
S-equivalent to the trivial bundle. As vector bundles over $E$,
$$\ad \xi \cong \bigoplus _iI_{2d_i-1},$$
where the $d_i$ are the Casimir weights of the root system of
$G$.
\endstatement
The bundle $\ad \xi$ can be seen to be an ($\ad G$)-bundle as
follows: start with the bundle $I_3 = \Sym ^2I_2$. It is an
$SL(2)$ bundle which descends to an
$SO(3)$-bundle. Now there is a ``maximal" embedding of $SO(3)$
in $G$, unique up to conjugation. Thus there is a
representation $\rho$ of $SO(3)$ on the Lie algebra $\frak g$.
Under this representation $\frak g$ decomposes as a direct sum
$$\frak g = \bigoplus _i\Sym ^{2d_i-2}(\Cee ^2),$$ where we
view $\Cee ^2$ as the standard representation of $SL(2)$, and
thus its odd symmetric powers give representations of $SO(3)$.
In particular, the
$G$-bundle induced by $\rho$ gives rise to the ($\ad
G$)-bundle described above.
We can generalize the above picture for the trivial bundle to
an arbitrary bundle. Let $\xi$ be an arbitrary semistable
$G$-bundle and let $\xi _0$ be the unique split bundle
S-equivalent to $\xi$. Then $\Aut \xi _0$ is up to isogeny a
product of $N$ factors $G_i$, where each factor $G_i$ is
either simple or isomorphic to $\Cee^*$. Let $\mu \in
E\otimes_{\Zee} \Lambda$ be a representative for the class of
$\xi _0$. The Lie algebra
$H^0(E;\ad \xi _0)$ of $\Aut \xi _0$ is identified with
$$\frak h \oplus \bigoplus _{ \alpha(\mu) = 0}\frak g^\alpha,$$
where $\frak g^\alpha$ is the root space corresponding to the
root $\alpha$ (and thus in particular the rank of this
reductive Lie algebra is $r$). We then have:
\proposition{3.8} With notation as above, let
$\xi_{\text{reg}}$ be a regular semistable bundle S-equivalent
to $\xi_0$. Let $r_i$ be the rank of $G_i$, where by
definition $r_i =1$ if $G_i\cong \Cee^*$, and let $d_{ij},
1\leq j\leq r_i$ be the Casimir weights of $G_i$, where we set
$d_{i1} = 1$ if $G_i\cong
\Cee ^*$. Then the maximal subbundle of $\ad \xi_{\text{reg}}$
which is filtered by subbundles whose successive quotients are
$\scrO_E$ is
$$\left(\ad \xi_{\text{reg}}\right)_{\scrO_E} = \bigoplus
_{i=0}^N\bigoplus _{j=1}^{r_i}I_{2d_{ij}-1}.$$
\endstatement
From this, it is possible in principle to give a complete
description of $\ad
\xi_{\text{reg}}$.
As a consequence of Proposition 3.6, one can show:
\proposition{3.9} Let $\xi$ be a semistable principal
$G$-bundle. Then $\xi$ is S-equivalent to a unique regular
semistable bundle and to a unique split bundle.
\endstatement
There are thus two canonical representatives for every
S-equivalence class, depending on whether we choose the
regular or the split representative. For an open dense subset
of bundles, these two representatives will in fact coincide.
As should be clear from Section 2, the split representatives
arise most naturally from the point of view of flat
connections. However, if we try to find a universal
holomorphic $G$-bundle, then we must work instead with regular
bundles. In fact, even working locally, it is not possible to
fit the split bundles together into a universal bundle, even
for $SL(n)$.
Finally, we make some comments about the automorphism group of
a regular bundle.
\proposition{3.10} Let $\xi$ be a regular semistable
$G$-bundle. If $\Aut _G(\xi)$ is the automorphism group of
$\xi$ and $(\Aut_G(\xi))^0$ is the component of $\Aut
_G(\xi)$ containing the identity, then $(\Aut _G(\xi))^0$
is abelian. Moreover, $\Aut _G(\xi)$ is itself abelian if and
only if $\xi$ corresponds to a smooth point of the moduli
space of S-equivalence classes of semistable $G$-bundles.
\endstatement
In fact, a careful analysis of the root systems involved shows
that the singular locus of the moduli space corresponding to
$\Zee/d\Zee$-isotropy is smooth and irreducible, of dimension
equal to the number of $i$ such that $d|g_i$, in the notation
of Theorem 2.7. Of course, this statement also follows
directly from Theorem 2.7.
\section{4. The parabolic construction.}
In this section, we describe a method of constructing families
of regular semi\-stable $G$-bundles. The motivation is as
follows: we seek to find an analogue for bundles of the
singularities picture outlined above in Section 2. That is, we
seek to find a mildly ``singular" (in other words, unstable)
$G$-bundle
$\xi _0$ together with a $\Cee ^*$-action on its deformation
space, such that the weighted projective space corresponding
to the quotient of the negative weight deformations of $\xi
_0$ by $\Cee^*$ is both the weighted projective space
$WP(g_0,
\dots, g_r)$ and is the coarse moduli space of semistable
$G$-bundles modulo S-equivalence. (Actually, with our
conventions the action of $\Cee^*$ will be by positive
weights.) It will also turn out that the points of the weighted
projective space parametrize regular
$G$-bundles, as opposed to split bundles, and will thus enable
us to find locally a universal
$G$-bundle away from the orbits where $\Cee ^*$ does not act
freely. In fact, in many cases we can use this construction to
produce a global universal $G$-bundle.
To pursue this idea further, we have seen that unstable
$G$-bundles over $E$ reduce to a parabolic subgroup of $E$,
and further to a Levi factor $L$. Conversely, fix a maximal
parabolic subgroup $P$ of $G$ and a Levi decomposition $P =
LU$, where $U$ is the unipotent radical of $P$ and $L$ is the
reductive or Levi factor. Then $U$ is normal, all Levi factors
are conjugate in $P$, and the quotient homomorphism
$P \to L$ is well-defined. The group $L$ is never semisimple;
in fact, since
$P$ is a maximal parabolic, the connected component of the
center of
$L$ is $\Cee ^*$. The maximal parabolic subgroup $P$ has a
canonical character $\chi\: P \to \Cee ^*$ (the unique
primitive dominant character), which is induced from a
character $L\to \Cee$. Using this character, we can define the
determinant line bundle of a principal
$L$-bundle over $E$. Fix an
$L$-bundle
$\eta$, such that $\det \eta$ has negative degree. The induced
$G$-bundle $\xi _0$ is unstable, because $\xi_0$ also reduces
to the opposite parabolic to $P$, and the determinant line
bundle for the primitive dominant character of the opposite
parabolic has positive degree. Consider the set of all
$P$-bundles
$\xi$ such that the associated
$L$-bundle (via the homomorphism $P\to L$) is $\eta$. It is
straightforward to classify all such bundles: the group $L$
acts by conjugation on $U$, and the
$L$-bundle
$\eta$ and the action of $L$ on $U$ define a sheaf of
unipotent groups
$\underline{U}(\eta)$ on $E$, which is in general nonabelian.
The set of all isomorphism classes of $P$-bundles
$\xi$ which reduce to $\eta$ may then be identified with the
cohomology set $H^1(E;
\underline{U}(\eta))$. The $\Cee^*$ in the center of $L$ then
acts on $H^1(E; \underline{U}(\eta))$. Cohomology
sets similar to
$H^1(E; \underline{U}(\eta))$, arising from the $H^1$ of a
sheaf of unipotent groups over a base curve, have been studied
in a different context by Babbitt and Varadarajan
\cite{3}, following ideas of Deligne, as well as by Faltings
\cite{14}. Using similar ideas, one can show that $H^1(E;
\underline{U}(\eta))$ has a (non-canonical) linear structure
and that $\Cee ^*$ acts linearly in this structure with
positive weights (following certain standard conventions), so
that the quotient is isomorphic to a weighted projective
space.
In the case of $SL(n)$, it is easy to make these ideas
explicit. The maximal parabolic subgroups of $SL(n)$ correspond
to filtrations $\{0\} \subset \Cee ^d \subset
\Cee ^n$, where $0< d < n$. For each such $d$, there is a
unique stable bundle
$W_d$ over $E$ of rank $d$ such that $\det W_d =
\scrO_E(p_0)$. The unstable bundle which we consider is then
$W_d^* \oplus W_{n-d}$, and it has a nontrivial
$\Cee^*$-action which acts trivially on $\det (W_d^* \oplus
W_{n-d})$. In this case, a straightforward argument shows:
\theorem{4.1} Let $V$ be a regular semistable vector bundle of
rank $n$. Then there is an exact sequence
$$0 \to W_d^* \to V \to W_{n-d} \to 0. $$ Moreover, the
automorphism group of $V$ acts transitively on the set of
subbundles of $V$ isomorphic to
$W_d^*$ whose quotients are isomorphic to $W_{n-d}$. Finally,
if $V$ is a nonsplit extension of $W_{n-d}$ by $W_d^*$, then
$V$ is in fact a regular semistable vector bundle.
\endstatement
We note that in this case the parabolic subgroup in question is
$$P = \left\{\,\pmatrix A&B\\O&D\endpmatrix: A\in GL(d), D\in
GL(n-d), \det A\cdot
\det D = 1\,\right\},$$ the Levi factor of $P$ is given by
$$L= \left\{\,\pmatrix A&O\\O&D\endpmatrix: A\in GL(d), D\in
GL(n-d), \det A\cdot
\det D = 1\,\right\},$$ and the unipotent radical $U$ of $P$,
which in this case is abelian, is given by
$$U= \left\{\,\pmatrix I&B\\O&I\endpmatrix: B \text{ is a
$d\times (n-d)$ matrix}\,\right\}.$$ It is easy then to
identify $H^1(E; \underline{U}(\eta))$ with the usual sheaf
cohomology group $H^1(E; W_{n-d}^*\otimes W_d^*)$ and the
$\Cee ^*$-action with the usual one, up to a factor. In this
way, the moduli space of regular semistable vector bundles
over $E$ of rank $n$ and trivial determinant is identified
with $\Pee ^{n-1}$, a fact which could also be established by
spectral cover methods \cite{20}. The full
tangent space to the deformations of the unstable bundle
$W_d^* \oplus W_{n-d}$ keeping the determinant trivial is
$H^1(E; \ad(W_d^* \oplus W_{n-d}))$. This group contains the
subgroup $H^1(E; W_{n-d}^*\otimes W_d^*)$ which is tangent to
the set of extensions described above. The one remaining
direction has $\Cee^*$-weight zero, which correponds to moving
the point $p_0$ on $E$ and which should be viewed as a one
parameter family of locally trivial deformations.
In the case of $SL(n)$, or equivalently the root system
$A_{n-1}$, every maximal parabolic subgroup has an abelian
unipotent radical and there is an appropriate construction
from any such subgroup giving the moduli space of regular
semistable
$G$-bundles. In all other cases, we have the following:
\theorem{4.2} Let $G$ be a complex, simple, and simply
connected group, not of type $A_n$. Then there exists a unique
maximal parabolic subgroup $P$ of $G$, up to conjugation, such
that, if $L$ is the Levi factor of $P$, then there exists an
$L$-bundle $\eta$ with the following properties:
\roster
\item"{(i)}" The connected component of the automorphism group
of $\eta$ as an
$L$-bundle is $\Cee ^*$.
\item"{(ii)}" The line bundle $\det \eta$ has negative
degree, and so the
$G$-bundle
$\xi _0$ induced by
$\eta$ is unstable.
\item"{(iii)}" If $U$ is the unipotent radical of $P$, then
the nonabelian cohomology set $H^1(E; \underline{U}(\eta))$
has the structure of affine $(r+1)$-dimensional space.
\item"{(iv)}" There exists a linear structure on $H^1(E;
\underline{U}(\eta))$ for which the natural copy of $\Cee
^*\subseteq \Aut_G\xi _0$ acts linearly, fixing the trivial
element, and with negative weights. The
stabilizer of every nontrivial element of $H^1(E;
\underline{U}(\eta))$ is finite, and the quotient $(H^1(E;
\underline{U}(\eta)) -
\{0\})/\Cee^*$ is a weighted projective space $WP(g_0, \dots,
g_r)$.
\item"{(v)}" If $\xi$ is a $P$-bundle over $E$ corresponding
to an element of
$H^1(E; \underline{U}(\eta)) - 0$, then $\xi$ is a regular
semistable bundle.
\endroster In all cases, the bundle $\eta$ with the above
properties is uniquely specified by requiring that $\det \eta
= \scrO_E(-p_0)$.
\endstatement
In fact, (iv) and (v) are a consequence of the other
properties. If we do not specify that $\det \eta =
\scrO_E(-p_0)$, then it is still the case that $\det\eta$ must
have degree $-1$, and so
$\eta$ is specified up to translation on $E$.
We note that all of the weights are equal, in other words the
weighted projective space is an ordinary projective space,
exactly in the cases $A_n$ and $C_n$, in other words for the
groups $SL(n+1)$ and $Sp(2n)$. In all other cases, for a simply
connected group $G$, the weighted projective space will in fact
have singularities.
To describe the maximal parabolic subgroups which arise in
Theorem 4.2, note first that maximal parabolic subgroups of
$G$, up to conjugation, are in one-to-one correspondence with
the vertices of the Dynkin diagram of the corresponding root
system. In case $G$ is
$D_n$ or
$E_6, E_7, E_8$, the maximal parabolic subgroup in Theorem 4.2
corresponds to the unique trivalent vertex of the Dynkin
diagram. In the remaining cases, the vertex in question is the
unique vertex meeting the multiple edge which is the long
root. (Such vertices will be trivalent in an appropriate sense
except for the case $C_n$.)
Let us describe the construction explicitly for the remaining
classical groups. The simplest case after $A_n$ is the case of
$Sp(2n)$, in other words $C_n$. In this case the parabolic in
question corresponds to those elements of $Sp(2n)$ which
preserve a totally isotropic subspace of dimension $n$. Thus
$$P = \left\{\,\pmatrix T&B\\O&{}^tT^{-1}\endpmatrix: T\in
GL(n), T^{-1}B = {}^t(T^{-1}B)\,\right\},$$ the Levi factor of
$P$ is given by
$$L= \left\{\,\pmatrix T&O\\O&{}^tT^{-1}\endpmatrix: T\in
GL(n)
\,\right\},$$ and the unipotent radical $U$ of $P$, which in
this case is also abelian, is given by
$$U= \left\{\,\pmatrix I&B\\O&I\endpmatrix: {}^tB =
B\,\right\}.$$ The unstable symplectic bundle corresponding to
$\eta$ is the bundle
$W_n^*\oplus W_n$, with the first factor embedded as a totally
isotropic subbundle and the second as its dual. It is easy
then to identify
$H^1(E; \underline{U}(\eta))$ with the usual sheaf cohomology
$H^1(E; \Sym ^2W_n^*)$. Here $\Cee^*$ acts with constant
weight, so that the quotient is an ordinary (smooth)
$\Pee^{n-1}$.
Next we consider $Spin(2n)$, although here it will be more
convenient to work in $SO(2n)$. The natural analogue of the
construction for the symplectic group would lead to the
unstable bundle $W_n^*\oplus W_n$, together with the symmetric
nondegenerate form for which $W_n^*$ is isotropic and which
identifies the dual of $W_n^*$ with the complementary $W_n$.
Such orthogonal bundles do not lift to $Spin(2n)$, although
this construction does identify all of the regular semistable
$SO(2n)$-bundles with $w_2\neq 0$ with the projective space on
$H^1(E; \bigwedge ^2W_n^*)$, which is a $\Pee ^{n-2}$. For
liftable $SO(2n)$-bundles, we use the parabolic subgroup
corresponding to the trivalent vertex, which is the subgroup of
$g\in SO(2n)$ preserving an isotropic subspace of rank $n-2$.
In this case the unipotent radical is nonabelian. The bundle
$\eta$, viewed as an unstable
$SO(2n)$-bundle $\xi _0$, is the bundle
$$\xi _0 = W_{n-2}^* \oplus Q_4 \oplus W_{n-2},$$ where
$W_{n-2}^*$ is an isotropic subspace, $Q_4$ is the
$SO(4)$-bundle
$\scrO_E\oplus \eta _1\oplus \eta _2 \oplus \eta _3$, in the
notation of Section 3, with a diagonal nondegenerate symmetric
pairing, and
$Q_4$ is orthogonal to the direct sum $W_{n-2}^* \oplus
W_{n-2}$. More invariantly $Q_4 = Hom (W_2, W_2)$ with the
quadratic form given by the trace. Note that neither $Q_4$
nor $W_{n-2}^* \oplus W_{n-2}$ lifts to a
$Spin$-bundle, and hence the direct sum is liftable. In this
case, the nonabelian cohomology set
$H^1(E; \underline{U}(\eta))$ (for $Spin(2n)$) has a weight
$1$ piece given by
$H^1(Q_4\otimes W_{n-2}^*)$, of rank $4$, and a weight $2$
piece given by
$H^1(E; \bigwedge ^2W_{n-2}^*)$, which has rank $n-3$. Similar
results hold for
$Spin (2n+1)$, by replacing $Q_4$ by
$$Q_3 = \eta _1\oplus \eta _2 \oplus \eta _3 = \ad W_2.$$
Returning to the general case, let us show that the weighted
projective space
$WP(g_0, \dots, g_r)$ arising from the parabolic construction
can be naturally identified with $(E\otimes _\Zee\Lambda)/W$,
thus giving a new proof of Looijenga's theorem. One first
shows that there exists a universal $G$-bundle over
$E\times H^1(E; \underline{U}(\eta))$ in the appropriate
sense. By general properties, there is a $\Cee^*$-equivariant
map from $H^1(E; \underline{U}(\eta))-0$ to the moduli space
of semistable $G$-bundles, in other words to $(E\otimes
_\Zee\Lambda)/W$.
\theorem{4.3} The induced map $WP(g_0, \dots, g_r) \to
(E\otimes _\Zee\Lambda)/W$ is an isomorphism.
\endstatement
The essential point of the proof is to compare
the determinant line bundles on the two sides, and then to use
the elementary fact that a degree one morphism from a weighted
projective space to a normal variety is an isomorphism. On the
weighted projective side, the determinant line bundle is
always Cartier, and in fact it is the line bundle
$K_{WP^r}^{2}$. On the other side, it is easy to calculate
the preimage of the determinant line bundle in $E\otimes
_\Zee\Lambda$. At least in the case of a simply laced root
system $R$, the fact that the degree of the morphism in
question is one then follows from the fact that the order of
the Weyl group is $r!(g_1\cdots g_r)\det R$ \cite{8}.
The parabolic construction also leads to a proof of the
existence of universal bundles in certain cases. For a fixed
$G$, we denote by $\Cal M_E=\Cal M_E(G)$ the moduli space of
regular semistable $G$-bundles over $E$ and by $\Cal M_E^0$
the smooth locus of $\Cal M_E$.
\theorem{4.4} If $G= SL(n)$, let $P_d$ be the maximal
parabolic subgroup of $SL(n)$ stabilizing the flag $\{0\}
\subset \Cee^d\subset \Cee^n$, and if $G\neq SL(n)$, let
$P$ be the maximal parabolic subgroup of $G$ described in
Theorem \rom{4.2}. Let
$n_P$ be the positive integer defined as follows\rom:
\roster
\item"{(i)}" If $G=SL(n)$ and $P=P_d$, then $n_{P_d} =
n/\gcd(d,n)$.
\item"{(ii)}" If $G$ is of type $C_n$, $B_n$ with $n$ even, or
$D_n$ with $n$ odd, then $n_P = 2$.
\item"{(iii)}" In all other cases, $n_P =1$.
\endroster
Let $\bar G$ be the quotient of $G$ by the unique subgroup of
the center of $G$ of order $n_P$. Then the universal
$G$-bundle over $E\times H^1(E; \underline{U}(\eta))$ descends
to a universal
$\bar G$-bundle $\overline{\Xi}$ on $E\times \Cal M_E^0$.
\endstatement
Let us mention the analogous results for families of elliptic
curves over a base $B$. Let $\pi \: Z \to B$ be a flat family,
all of whose fibers are smooth elliptic curves or more
generally irreducible curves of arithmetic genus one (i\.e\.
smooth, nodal, or cuspidal curves). Let $\sigma$ be a section
of $\pi$ meeting each fiber in a smooth point. Associated to
$Z$ is the line bundle $L$ on $B$ defined by
$L^{-1} = R^1\pi _*\scrO_Z$, which can be identified with
$\scrO_Z(\sigma)|\sigma$ under the isomorphism $\sigma \to B$
induced by $\pi$. We want to describe the parabolic
construction along the family
$Z$. To do so, recall that we have the weights $g_i$ of (2.8),
which we assume ordered so that
$g_i \leq g_{i+1}$. Recall also that we have defined the
Casimir weights
$d_1, \dots, d_r$ of a root system $R$ in Section 3. We order
the $d_i$ by increasing size, except in the case of $D_n$,
where we order the $d_i$ by: $2, 4,n,6, 8, \dots, 2n-2$.
Our result in families can then be somewhat loosely stated as
follows:
\theorem{4.5} Suppose that $G\neq E_8$. The parabolic
construction then globalizes over $Z$ to give a bundle of
nonabelian cohomology groups over
$B$. This bundle is a bundle of affine spaces with a
$\Cee^*$-action which is isomorphic to the vector bundle
$$\scrO_B \oplus L^{-d_1} \oplus \cdots \oplus L^{-d_r}.$$ Via
this isomorphism $\Cee ^*$ acts diagonally on the line bundles
in the direct sum, by the weight
$g_i$ on the factor $ L^{-d_i}$ \rom(and with weight $g_0 =
1$ on the factor
$\scrO_B$\rom). The associated bundle of weighted projective
spaces is then a universal relative moduli space for
$G$-bundles which are regular and semistable on every fiber.
\endstatement
A result closely related to Theorem 4.5 was established by
Wirthm\"uller \cite{33}, who also noted the exceptional
status of
$E_8$. We note that, from our point of view, in the case of
$E_8$ there is a family of weighted projective spaces over the
open subset $B'$ of $B$ over which the fibers of $\pi$ are
either smooth or nodal. However, this family is not the
quotient of a vector bundle minus its zero section by $\Cee^*$
acting diagonally. Furthermore, the construction degenerates
in an essential way at the cuspidal curves. A similar
phenomenon appears if we try to classify generalized del Pezzo
surfaces of degree one with an appropriate hyperplane section.
\section{5. Automorphism sheaves and spectral covers.}
In this section, we fix $G$ and denote by $\Cal M_E(G)=\Cal
M_E$ the moduli space of regular semistable $G$-bundles over
$E$. Likewise, given an elliptic fibration with a section $\pi
\: Z \to B$ whose fibers are smooth elliptic curves or nodal or
cuspidal cubics (except in the case $G=E_8$ where we will not
allow cuspidal fibers), we have a relative
moduli space $\Cal M_{Z/B}=
\Cal M_{Z/B}(G)$. Thus in all cases $\Cal M_{Z/B}$ is a
bundle of weighted projective spaces.
Because of fixed points for the
$\Cee ^*$ action, the universal $G$-bundle over $E\times
H^1(E;
\underline{U}(\eta))$ does not descend to a universal
$G$-bundle over $E\times \Cal M_E$, even locally, near the
singular points of
$\Cal M_E$, and a similar statement holds in families.
However, let
$\Cal M_E^0$ denote the smooth locus of $\Cal M_E$, and
similarly for $\Cal M_{Z/B}^0$. Then locally in either the
classical or
\'etale topology there exists a universal bundle $\Xi$ over
$E\times \Cal M_E^0$, and similarly for $Z\times _B\Cal
M_{Z/B}^0$. As we have seen in Theorem 4.4, there also exists
a $\bar G$-bundle $\overline{\Xi}$ over $E\times \Cal M_E^0$,
where $\bar G$ is a quotient of $G$ by a subgroup of the
center of order at most two. In particular, a universal
adjoint bundle always exists. In this section, we describe the
issues of the existence and uniqueness of a global universal
bundle over $E\times \Cal M_E^0$ or $Z\times _B\Cal
M_{Z/B}^0$.
There are other questions closely related to these. Given a
family $\pi \: Z \to B$ as above, suppose that
$\Xi$ is a $G$-bundle over $Z$ such that $\Xi |\pi ^{-1}(b)$
is a semistable bundle for all $b$ for which $\pi ^{-1}(b)$ is
smooth. Then $\Xi$ defines a section of $\Cal M_{Z/B}$ over the
open subset of $B$ consisting of such $b$. At the singular
points of $\Cal M_{Z/B}$, the section is locally liftable to
the affine bundle of cohomology groups over $B$. Conversely,
a locally liftable section defines local
$G$-bundles over $\pi ^{-1}(U)$ for all sufficiently small
open sets $U$ of $B$ (in the classical or \'etale topology).
Note that the parabolic construction extends over the singular
fibers of $\pi$ (except for cuspidal fibers in case $G=E_8$),
dictating the correct definition of regular semistable
$G$-bundles for a singular fiber. When does a locally liftable
section of $\Cal M_{Z/B}$ actually determine a
$G$-bundle over
$Z$? More generally, how can we describe the set (possibly
empty) of all bundles corresponding to a given section? For
simplicity, we shall assume that the section does not pass
through the singular points of
$\Cal M_{Z/B}$. Thus, if there existed a relative universal
bundle over $Z\times _B\Cal M_{Z/B}^0$, we could simply pull
this bundle back by the section to obtain a bundle over $B$.
While a relative universal bundle does not usually exist,
there are many cases where a section does indeed determine a
$G$-bundle. However, our answers are complete only in the
cases $G = SL(n), Sp(2n)$.
Working for the moment with a single curve $E$, over an open
subset of $\Cal M_E^0$ where there exists a local universal
bundle $\Xi$, there is an associated group scheme
$\underline{\Aut}(\Xi)$. Because the associated automorphism
groups are abelian on $\Cal M_E^0$, as follows from (3.10),
these local group schemes piece together to give an abelian
group scheme over
$\Cal M_E^0$, whose associated sheaf of sections will be
denoted
$\Cal A$. In the usual way, the obstruction to finding a
global universal principal $G$-bundle over $E\times
\Cal M_E^0$ lies in $H^2(\Cal M_E^0; \Cal A)$, and if this
obstruction is zero, then the set of all such principal
bundles is a principal homogeneous space over $H^1(\Cal
M_E^0; \Cal A)$. More generally, given an elliptic fibration
$\pi\: Z \to B$ as above, we can fit together the automorphism
group schemes of local universal bundles to find an abelian
group scheme over $\Cal M_{Z/B}^0$ whose fiber over every
point $b\in B$ is the group scheme constructed above. Let $\Cal
A_B$ denote the sheaf of sections of this group scheme. Given a
section $s$ of $\Cal M_{Z/B}^0
\to B$, we can pull back the the above group scheme to obtain
a group scheme over $B$, whose sheaf of sections we denote by
$\Cal A_B(s)$. Just as in the
case of a single smooth elliptic curve, the obstruction to
finding a $G$-bundle over $Z$ corresponding to the section $s$
lies in
$H^2(B; \Cal A_B(s))$, and if this obstruction is zero, then
the set of all such bundles is a principal homogeneous space
over $H^1(B; \Cal A_B(s))$.
Let us describe the sheaf $\Cal A$ in the case of $SL(n)$ and
a fixed elliptic curve $E$ in more detail. For each integer
$d$, $1\leq d
\leq n-1$, one can construct a universal extension $\Cal E_d$
over $E\times \Pee ^{n-1}$, viewing $\Pee^{n-1}$ as $\Ext
^1(W_{n-d}, W_d^*)$, which fits into an exact sequence
$$0 \to \pi _1^*W_d^* \otimes \pi _2^*\scrO_{\Pee ^{n-1}}(1)
\to \Cal E_d \to
\pi _1^*W_{n-d} \to 0.$$ Clearly, $\det \Cal E_d$ has trivial
restriction to each slice $E\times \{s\}$ but is not in fact
trivial. On the other hand, since the restriction of $\Cal
E_d$ to every fiber is regular and semistable,
$\pi _2{}_*Hom (\Cal E_d, \Cal E_d)$ is a sheaf of locally
free commutative
$\Cee$-algebras over $\Pee^{n-1}$ of rank $n$, and thus
corresponds to a finite morphism $\nu \: T \to \Pee ^{n-1}$ of
degree $n$, which we shall call the {\sl spectral cover\/} of
$\Pee ^{n-1}$. It is straightforward to identify the base
$\Pee ^{n-1}$ with the complete linear system $|np_0|$ and the
cover $T$ with the incidence correspondence in $\Pee
^{n-1}\times E$, in other words
$$T = \left\{\, (\sum _{i=1}^ne_i, e): \sum _{i=1}^ne_i \in
|np_0|, e = e_i {\text{ for some $i$}}\,\right\}.$$ Thus $T$
is smooth, and it has the structure of a $\Pee ^{n-2}$-bundle
over $E$ such that the $\Pee ^{n-2}$ fibers are mapped to
hyperplanes in $\Pee^{n-1}$ under $\nu$. Another way to
describe
$T$ is as follows: let
$\Lambda\cong \Zee ^{n-1}$ as the sublattice of $\Zee ^n$ of
vectors whose sum is zero, acted on by the Weyl group $\frak
S_n$, so that $\Pee ^{n-1} = |np_0| = (E\otimes \Lambda)/\frak
S_n$. Let
$W_0 =\frak S_{n-1}\subset \frak S_n$ be the stabilizer of the
vector
$e_n \in \Zee ^n$. Then $T = (E\otimes \Lambda)/W_0$.
A standard argument shows that, if
$\Cal V$ is a vector bundle over $E\times \Pee^{n-1}$ whose
restriction to every slice is isomorphic to the corresponding
restriction of $\Cal E_d$, then
$\pi _2{}_*Hom (\Cal V, \Cal E_d)$ is locally free of rank one
over $\pi _2{}_*Hom (\Cal E_d,
\Cal E_d) = \nu _*\scrO_T$, and thus corresponds to a line
bundle on $T$, and conversely every line bundle on $T$ defines
a vector bundle $\Cal V$ with the above property. It is
helpful to compare this situation with the one usually
encountered in algebraic geometry, where we try to make a
moduli space of simple vector bundles and then the only choice
is to twist by the pullback of a line bundle from the moduli
space factor.
From this it follows that, in the case of $SL(n)$, the
automorphism sheaf $\Cal A$ is given by the kernel of the norm
homomorphism $\nu_*\scrO_T^* \to \scrO_{\Pee ^{n-1}}^*$. Hence
there is an exact sequence
$$0 \to H^1(\Pee^{n-1}; \Cal A) \to \Pic T \to \Pic \Pee
^{n-1} \to H^2(\Pee^{n-1}; \Cal A) \to H^2(\scrO_T^*)\to 0.$$
Thus,
$H^1(\Pee^{n-1}; \Cal A) \cong \Zee\times E$ for $n>2$ and
$H^1(\Pee^1; \Cal A) \cong E$, and
$H^2(\Pee^{n-1}; \Cal A)
\cong H^3(T; \Zee)$. There is also an analogue of the above
exact sequence where we take \'etale cohomology. In this case,
$H^1(\Pee^{n-1}; \Cal A)$ is unchanged and $H^2(\Pee^{n-1};
\Cal A) \cong H_{\text{\'et}}^2(T; \Bbb G_m)$, which is a
torsion group. The obstruction to gluing together local
families (in either the classical or \'etale topology) of
$SL(n)$-bundles to make a global
$SL(n)$-bundle over
$E\times \Pee^{n-1}$ lives in $H^2(\Pee^{n-1}; \Cal A)$, and in
case the obstruction is zero the set of all such bundles in
then a principal homogeneous space over
$H^1(\Pee^{n-1}; \Cal A)$. In our case, a direct construction
using the pushforward of appropriate line bundles on $E\times
T$ shows that the obstruction in $H^2(\Pee^{n-1};
\Cal A)$ vanishes. Thus the family of universal
$SL(n)$-bundles
$\Cal V$ over $E\times \Pee^{n-1}$ is parametrized by
$\Zee\times E$ for $n>2$ and by $E$ if $n=2$. In case we
consider the corresponding situation in families $Z\to B$,
then there exist mod
$2$ obstructions to finding a principal $SL(n)$-bundle over the
entire family, and these obstructions are not in general zero.
On the other hand, there always exists a universal
$GL(n)$-bundle
$V$ such that
$V|\pi ^{-1}(b)$ has trivial determinant for all $b$, so that
$\det V$ is pulled back from $B$. See
\cite{20} for more detail in the case of vector bundles.
Similar explicit constructions can be carried out for the
symplectic group. Let $\Lambda = \Zee ^n$, and let the Weyl
group $W = \frak S_n \ltimes (\Zee/2\Zee)^n$ act on $\Lambda$,
where the symmetric group acts by permuting the basis elements
and $(\Zee/2\Zee)^n$ acts by sign changes. Then $(E\otimes
\Lambda)/W = \Sym ^n\Pee ^1 = \Pee ^n$. Let $W_0 = \frak
S_{n-1}
\ltimes (\Zee/2\Zee)^{n-1}$ be the subgroup of $W$ fixing the
last basis vector, and set $T^{\text{sp}} = (E\otimes
\Lambda)/W_0 = \Pee ^{n-1} \times T$. The group $W_0$ is a
subgroup of index two in the larger group $W_1 = \frak S_{n-1}
\ltimes (\Zee/2\Zee)^n$, and there is an induced involution
$\iota$ on $T^{\text{sp}}$, with quotient $T^{\text{sp}}/\iota
= S = \Pee ^{n-1} \times \Pee ^1$. We then have:
\proposition{5.1} For the symplectic group $Sp(2n)$, the
automorphism sheaf $\Cal A^{\text{sp}}$ over $\Cal M_E(Sp(2n))
\cong \Pee^n$ is given by
$$\Cal A^{\text{sp}} = \{\, f\in \nu_*\scrO_{T^{\text{sp}}}^*:
\iota ^*f= f^{-1}\,\}.$$
\endproclaim
Using (5.1) one can show that there is a universal bundle over
$E\times
\Cal M_E(Sp(2n))$ as well, and that the set of all universal
bundles is parametrized by $E$. Thus we have constructed
universal bundles over $E\times \Cal M_E$ in the two cases
where the moduli space is smooth. It then follows from Theorem
4.4 that a universal bundle exists over
$\Cal M_E^0(G)$ in all cases, with the possible exception of
$G= Spin (4n+1)$ and $G = Spin (4n+2)$.
We return to the case of a general $G$ and analyze the
structure of the sheaf $\Cal A$ over $\Cal M_E^0$. Since
$\Cal A$ is the sheaf of sections of an abelian algebraic
group scheme, there is the exponential map $\exp$ from the
corresponding sheaf of Lie algebras
$\operatorname{Lie}\Cal A$ to $\Cal A$. The kernel of $\exp$
is a constructible sheaf, which we denote by
$\underline{\Lambda}$, and the image of $\exp$ is the sheaf
$\Cal A^0$ which, locally, consists of all sections of
$\underline{\Aut}(\Xi)$ passing through the identity component
of every fiber. First we note that $\Cal A = \Cal A^0$ on the
Zariski open subset $U$ of $\Cal M^0$ consisting of split
bundles, where the fiber over $x\in U$ of the group scheme
corresponding to $\Cal A$ is $(\Cee ^*)^r$ and is connected. If
the root system for
$G$ is simply laced, we can say more:
\proposition{5.2} Suppose that $G$ is simply laced. If $G\neq
SL(2)$, then the set
$$\{\, \xi \in \Cal M^0: \Cal A_\xi \neq \Cal A^0_\xi\,\}$$
has codimension at least two in $\Cal M^0$.
\endproclaim
As a consequence, in the relative setting, for $G$ simply
laced, if
$\dim B = 1$ and $G \neq SL(2)$, then for a generic section
$s$ of
$\Cal M_{Z/B}^0$, we can always assume that $\Cal A_B(s) = \Cal
A_B^0(s)$. The above
proposition does not hold if $G$ is not simply laced; for
example, it fails for
$Sp(2n)$.
Next we turn to $\Cal A^0 =\operatorname{Lie}\Cal A
/\underline{\Lambda}$. Note that, in case there is a universal
bundle $\Xi$ over $E\times \Cal M^0$, then
$\operatorname{Lie}\Cal A = R^0p_2{}_*(\ad \Xi)$ is dual to
$R^1p_2{}_*(\ad \Xi)$, which is the tangent bundle to $\Cal
M^0$. Thus
$\operatorname{Lie}\Cal A =
\Omega ^1_{\Cal M^0}$ is the cotangent bundle. In fact, this
statement always holds, since a universal bundle exists
locally and the automorphism sheaf is abelian. Another way to
describe the cotangent bundle is as follows: let $\frak h$ be
the Lie algebra of a Cartan subgroup of $G$. Then the Weyl
group acts on $E\otimes
\Lambda$ and on the trivial vector bundle $\scrO_{E\otimes
\Lambda}\otimes _\Cee\frak h$, and the sheaf of $W$-invariant
sections is a coherent sheaf over $(E\otimes \Lambda)/W= \Cal
M$ whose restriction to $\Cal M^0$ is locally free, and in
fact is
$\Omega ^1_{\Cal M^0}$. The constructible sheaf
$\underline{\Lambda}$ can be described as follows. Let $U$ be
the open subset of $\Cal M^0$ over which the map $E\otimes
\Lambda \to
\Cal M$ is unramified, and let $i\: U \to \Cal M^0$ be the
inclusion. Then the action of $W$ on $\Lambda$ gives a locally
constant sheaf $\underline{\Lambda} _0$ on $U$, and
$\underline{\Lambda} = i_*\underline{\Lambda}_0$. The map
$\Lambda \to \frak h$ induces an inclusion
$\underline{\Lambda} \to \left(\scrO_{E\otimes
\Lambda}\otimes _\Cee\frak h\right)^W$, and this is the same
as the inclusion $\underline{\Lambda} \to
\operatorname{Lie}\Cal A$.
This picture is related to the general theory of spectral
covers of \cite{24} and \cite{10} (as has also been
noted by Donagi in
\cite{11}). Suppose that
$\varpi$ is an element of $\frak h$ such that $W\cdot\varpi$
spans
$\frak h$ over
$\Cee$. In the typical application, $\varpi$ is (the dual of) a
minuscule weight, if such exist. Let $W_0$ be the stabilizer of
$\varpi$. If we set $T = (E\otimes \Lambda)/W_0$, then there
is a surjection $\nu \: T \to \Cal M$. By pure algebra,
$$\nu _*\scrO_T = \left(\scrO_{E\otimes \Lambda}\otimes_\Cee
\Cee[W/W_0]\right)^W.$$ On the other hand, there is a
surjection $\Cee[W/W_0] \to \frak h$ whose kernel consists of
the relations in the orbit $W\cdot
\varpi$. Correspondingly, there is a surjection
$$\left(\scrO_{E\otimes \Lambda}\otimes_\Cee
\Cee[W/W_0]\right)^W \to \left(\scrO_{E\otimes
\Lambda}\otimes_\Cee
\frak h\right)^W.$$ In particular $H^1(\Cal
M;\operatorname{Lie}
\Cal A)$ is a quotient of
$H^1(\Cal M; \nu_*\scrO_T)$.
Now suppose that we are in the relative case of an elliptic
fibration $\pi \: Z\to B$. There is then a relative universal
moduli space $\Cal M_{Z/B}$ (with the usual care in the case of
$E_8$). The covers $T \to \Cal M$ defined over every smooth
fiber extend to a finite morphism $\Cal T_{Z/B} \to \Cal
M_{Z/B}$. A section $s$ of the map $\Cal M^0_{Z/B} \to B$
defines a finite cover
$C_s$ of $B$, which we will call the {\sl spectral cover\/} in
this case. Of course, $C_s$ need not be smooth or even
reduced. In case
$\dim B = 1$ and $s$ is generic, the above discussion
identifies the connected components of
$H^1(B;\Cal A_B(s))$ with an abelian variety which is a
quotient of the Jacobian $J(C_s)$, and which is called the {\sl
Prym-Tyurin variety\/} of the spectral cover.
A straightforward dimension argument shows:
\proposition{5.3} Suppose that $\dim B = 1$ and that $\Cal
A_B(s)_b = \Cal A_B^0(s)_b$ for at least one point $b\in B$.
Then $H^2(B; \Cal A_B(s)) = 0$. In other words, there exists a
universal $G$-bundle over $B$ corresponding to the section $s$.
\endstatement
If however $\Cal A_B(s)_b \neq \Cal A_B^0(s)_b$ for all $b\in
B$, then it is possible for there not to exist a universal
$G$-bundle over $B$ corresponding to $s$, even when $G=
SL(2)$. For $\dim B$ arbitrary, the possible obstructions in
the case of $SL(n)$ are analyzed in detail in \cite{20}.
Let us work out the twisting group $H^1(B;\Cal A_B(s))$
explicitly in the simplest cases $G= SL(n), Sp(2n)$, with
$\dim B$ arbitrary:
\proposition{5.4} Suppose that $G = SL(n)$. Let $C_s \to B$ be
the spectral cover defined above. Then
$$H^1(B;\Cal A_B(s)) = \Ker \{\, \operatorname{Norm}\: \Pic
(C_s) \to
\Pic B\,\}.$$ If $G = Sp(2n)$, let $C_s$ be the corresponding
degree $2n$ cover of $B$, let $\iota\: C_s\to C_s$ be the
induced involution, and let
$f\: C_s\to D_s$ be the degree two quotient of $C_s$ by
$\iota$. Then
$$H^1(B;\Cal A_B(s)) = \Ker \{\, \operatorname{Norm}\: \Pic
(C_s) \to
\Pic D_s\,\}.$$
\endstatement
Thus, in case $B$ is a curve, $H^1(B;\Cal A_B(s))$ is the
generalized Prym variety of the cover $C_s\to D_s$. Similar
results hold for the remaining classical groups $Spin (2n)$ and
$Spin (2n+1)$.
On the other hand, suppose that $G= E_6, E_7, E_8$, that $\dim
B = 1$, and that the section $s$ is generic. In this case,
there is an associated fibration of del Pezzo surfaces $p\: Y
\to B$, where $Y$ is a smooth threefold. Moreover $Z$ is
included as a smooth divisor on $Y$ so that $p|Z = \pi \: Z
\to B$. Let
$J^3(Y)$ denote the intermediate Jacobian of $Y$. There is an
induced morphism
$J^3(Y) \to J(B)$, where $J(B)$ is the ordinary Jacobian of
$B$ coming from the homomorphism $H^*(Y) \to H^*(Z) \to
H^{*-2}(B)$. Denote the kernel of the morphism $J^3(Y) \to
J(B)$ by
$J^3(Y/B)$. Finally set
$$H^{2,2}_0(Y; \Zee) = \left.\Ker \{\, H^4(Y; \Zee) \to H^2(B;
\Zee)\,\}\right/\Zee\cdot [Y_t],$$ where $Y_t$ is a general
fiber of $p$. In general $H^{2,2}_0(Y;
\Zee)$ is a finite group. We then obtain the following
theorem, first proved by Kanev
\cite{24} in the case
$B =\Pee^1$ via the Abel-Jacobi homomorphism:
\proposition{5.5} In the above situation, there is an exact
sequence
$$0\to J^3(Y/B) \to H^1(\Cal A_B(s)) \to H^{2,2}_0(Y; \Zee)
\to 0.$$
\endstatement
We note that we can interpret $H^1(\Cal A_B(s))$ as a relative
Deligne cohomology group.
\Refs
\ref \no 1\by M. Atiyah \paper Vector bundles over an
elliptic curve
\jour Proc. London Math. Soc. \vol 7\yr 1957 \pages
414--452\endref
\ref \no 2\by M. Atiyah and R. Bott \paper The Yang-Mills
equations over Riemann surfaces \jour Phil. Trans. Roy. Soc.
London A\vol 308\yr 1982\pages 523--615\endref
\ref \no 3\by D.G. Babbitt and V.S. Varadarajan \paper Local
moduli for meromorphic differential equations \jour
Ast\'erisque \vol 169--170 \yr 1989 \pages 1--217
\endref
\ref \no 4\by V. Baranovsky and V. Ginzburg \paper Conjugacy
classes in loop groups and $G$-bundles on elliptic curves
\jour Internat. Math. Res. Notices \vol 15 \yr 1996 \pages
733--751
\endref
\ref \no 5\by A. Beauville and Y. Laszlo \paper Conformal
blocks and generalized theta functions \jour Comm. Math. Phys.
\vol 164 \yr 1994 \pages 385--419
\endref
\ref \no 6 \by I.N. Bernshtein and O.V. Shvartsman \paper
Chevalley's theorem for complex crystallographic Coxeter
groups \jour Funct. Anal. Appl. \vol 12 \yr 1978 \pages
308--310 \endref
\ref \no 7 \by A. Borel \paper Sous-groupes commutatifs et
torsion des groupes de Lie compactes \jour T\^ohoku Math.
Jour. \vol 13 \yr 1961 \pages 216--240 \endref
\ref \no 8\by N. Bourbaki \book Groupes et Alg\`ebres de Lie
\bookinfo Chap. 4, 5, et 6 \publ Masson \publaddr Paris \yr
1981 \endref
\ref \no 9\by G. Br\"uchert \paper Trace class elements and
cross-sections in Kac-Moody groups\paperinfo PhD. thesis,
Hamburg \yr 1995
\endref
\ref \no 10\by R. Donagi \paper Spectral covers
\inbook Current topics in complex algebraic geometry (Berkeley
1992/93) \bookinfo MSRI Publications \vol 28 \publ Cambridge
University Press \publaddr Cambridge \yr 1995
\endref
\ref \no 11\bysame \paper Principal bundles on
elliptic fibrations
\jour Asian J. Math. \vol 1 \yr 1997 \pages 214--223\endref
\ref \no 12 \by S.K. Donaldson \paper A new proof of a theorem
of Narasimhan and Seshadri \jour J. Differential Geometry \vol
18 \yr 1983 \pages 269--277
\endref
\ref \no 13\by P. Etingof and I. Frenkel
\paper Central extensions of current algebras in two dimensions
\jour Comm. Math. Phys.
\vol 165 \yr 1994\pages 429--444\endref
\ref \no 14 \by G. Faltings \paper Stable $G$-bundles and
projective connections \jour J. Algebraic Geometry \vol 2\yr
1993 \pages 507--568
\endref
\ref \no 15 \by G. Faltings \paper A proof for the
Verlinde formula \jour J. Algebraic Geometry \vol 3\yr
1994 \pages 347--374
\endref
\ref \no 16\by R. Friedman \paper Rank two vector bundles over
regular elliptic surfaces \jour Inventiones Math. \vol 96 \yr
1989 \pages 283--332
\endref
\ref \no 17 \bysame \paper Vector bundles and
$SO(3)$-invariants for elliptic surfaces \jour J. Amer. Math.
Soc. \vol 8 \yr 1995
\pages 29--139 \endref
\ref \no 18\by R. Friedman and J.W. Morgan \book Smooth
Four-Manifolds and Complex Surfaces \bookinfo Ergebnisse der
Mathematik und ihrer Grenzgebiete 3. Folge
\vol 27 \publ Springer-Verlag \publaddr Berlin Heidelberg New
York
\yr 1994
\endref
\ref \no 19 \by R. Friedman, J.W. Morgan and E. Witten \paper
Vector bundles and
$F$ theory \jour Commun. Math. Phys. \vol 187 \yr 1997 \pages
679--743
\endref
\ref \no 20 \bysame \paper
Vector bundles over elliptic fibrations \paperinfo
alg-geom/9709029 \endref
\ref \no 21 \by A. Grothendieck \paper Sur la classification
des fibr\'es holomorphes sur la sph\`ere de Riemann \jour
Amer. J. Math \vol 79 \yr 1956 \pages 121--138 \endref
\ref \no 22 \by J. Humphreys \book Conjugacy Classes in
Semisimple Algebraic Groups
\bookinfo Mathematical Surveys and Monographs \vol 43\publ
Amer. Math. Soc. \publaddr Providence \yr 1995 \endref
\ref \no 23\by V. Kanev \paper Intermediate Jacobians and
Chow groups of threefolds with a pencil of del Pezzo
surfaces \jour Annali Mat. pura appl. \vol 154
\yr 1989 \pages 13--48
\endref
\ref \no 24\bysame \paper Spectral covers and Prym-Tjurin
varieties I\inbook Abelian Varieties \bookinfo Proceedings of
the International Conference (Egloffstein 1993) \eds W.
Barth, K. Hulek, and H. Lange
\publ de Gruy\-ter \publaddr Berlin-New York \yr 1995 \endref
\ref \no 25\by Y. Laszlo \paperinfo preprint \endref
\ref \no 26\by E. Looijenga \paper On the semi-universal
deformation of a simple elliptic hypersurface singularity
II: the discriminant\jour Topology
\vol 17 \yr 1975 \pages 23--40
\endref
\ref \no 27\bysame \paper Root systems and elliptic curves
\jour Invent. Math. \vol 38 \yr 1976
\pages 17--32
\endref
\ref \no 28\by J.-Y. M\'erindol \paper Les singularit\'es
simples elliptiques, leurs d\'eformations, les surfaces de
del Pezzo et les transformations quadratiques \jour Ann.
Scient. E.N.S. \vol 15 \yr 1982 \pages 17--44
\endref
\ref \no 29\by M. S. Narasimhan and C. S. Seshadri \paper
Stable and unitary bundles on a compact Riemann surface \jour
Annals of Math. \vol 82
\yr 1965 \pages 540--567 \endref
\ref \no 30\by H. Pinkham \paper Simple elliptic
singularities, del Pezzo surfaces, and Cremona
transformations \inbook Several Complex Variables \bookinfo
Proc. Symp. Pure Math. \vol 30 \publ Amer. Math. Soc.
\publaddr Providence
\yr 1977 \pages 69--71
\endref
\ref \no 31\by S. Ramanan and A. Ramanathan\paper Some
remarks on the instability flag \jour T\^ohoku Math. Jour.
\vol 36 \yr 1984 \pages 269--291
\endref
\ref \no 32\by A. Ramanathan\paper Stable principal
bundles on a compact Riemann surface \jour Math. Ann.
\vol 213 \yr 1975 \pages 129--152
\endref
\ref \no 33\by K. Wirthm\"uller \paper Root systems and
Jacobi forms\jour Compositio Math.
\vol 82 \yr 1992 \pages 293--354
\endref
\endRefs
\enddocument
\end
|
1997-10-27T07:41:28 | 9707 | alg-geom/9707003 | en | https://arxiv.org/abs/alg-geom/9707003 | [
"alg-geom",
"math.AG"
] | alg-geom/9707003 | Hosono Shinobu | S.Hosono | GKZ Systems, Gr\"obner Fans and Moduli Spaces of Calabi-Yau
Hypersurfaces | 28 pages, LaTeX. (The statements in Prop.4.7 and Claim 5.8 are
clarified.) | null | null | null | null | We present a detailed analysis of the GKZ(Gel'fand, Kapranov and Zelevinski)
hypergeometric systems in the context of mirror symmetry of Calabi-Yau
hypersurfaces in toric varieties. As an application we will derive a concise
formula for the prepotential about large complex structure limits. (Talk given
at the Taniguchi Symposium ``Topological Field Theory, Primitive Forms and
Related Topics'' December, 1996)
| [
{
"version": "v1",
"created": "Thu, 3 Jul 1997 09:57:30 GMT"
},
{
"version": "v2",
"created": "Mon, 27 Oct 1997 06:41:28 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Hosono",
"S.",
""
]
] | alg-geom | \section{1. Introduction}
Mirror symmetry of Calabi-Yau manifolds
has been playing a central role in revealing non-perturbative
aspects of the type II string vacua, i.e., the moduli spaces for
a family of Calabi-Yau manifolds. Since the success in determining the
quantum geometry on the IIA moduli space made by Candelas et al\cite{CdGP}
in 1991, there have been many progresses and a lot of
communications between physics and mathematics on this topics\cite{GY}.
In this article, we will be concerned with the mirror
symmetry of Calabi-Yau hypersurfaces in toric varieties.
In this case, the mirror symmetry may be traced to a rather
combinatorial properties of the reflexive polytopes which determines
the ambient toric varieties due to ref.\cite{Bat1}.
Furthermore since the period integrals of Calabi-Yau hypersurfaces turn out
to satisfy the hypergeometric differential equation, ${\cal A}$-hypergeometric
system, introduced by Gel'fand, Kapranov and Zelevinski (GKZ),
we can study in great detail the moduli spaces of Calabi-Yau hypersurfaces.
Based on the analysis of GKZ-hypergeometric system in our context, we
will derive a closed formula for the prepotential, which defines the special
K\"ahler geometry on the moduli spaces.
In section 2, we will review the mirror symmetry of Calabi-Yau
hypersurfaces in toric varieties. This is meant to fix our notations as
well as to introduce the mirror symmetry due to Batyrev. In section 3,
we will introduce GKZ-hypergeometric system (${\Delta^*}$-hypergeometric system)
as an infinite set of differential equations satisfied by period integrals
and summarize basic results following \cite{GKZ1}. We also define the extended
${\Delta^*}$-hypergeometric system incorporating the automorphisms of the toric
varieties. We will remark that the ${\Delta^*}$-hypergeometric system in our
context is resonant in general. In section 4, we will review basic
properties of the toric ideal and the Gr\"obner fan as an equivalence
classes of the term orders in the toric ideal. We will use the Gr\"obner
fan to compactify the space of the variables in the ${\Delta^*}$-hypergeometric
system, and propose it as a natural compactification of the corresponding
family of Calabi-Yau hypersurfaces. In section 5, we will prove general
existence of the so-called large complex structure limits, at which
the monodromy becomes maximally unipotent\cite{Mor}. We will also present
a general formula for the local solutions about these points. In the
final section, we will derive a closed formula for the prepotential,
which is valid about a large complex structure limit for arbitrary
Calabi-Yau hypersurfaces in toric varieties. Our formula determines
the special K\"ahler geometry about a large complex structure limit
as well as the quantum corrected Yukawa coupling. Claim 5.8, Claim 5.11,
and Claim 6.8 in the last two sections are meant to state those results
that are verified in explicit calculations by many examples without
general proofs.
All the results except Prop.6.7 for the prepotential in the
final section have already reported in refs.\cite{HKTY1}\cite{HKTY2}
\cite{HLY1}\cite{HLY2}.
\vspace{0.3cm}\noindent
{\bf Acknowledgments.}
This article is based on the joint works
with A. Klemm, B.H. Lian, S. Theisen and S.-T. Yau.
The author would like to thank them for their enjoyable collaboration.
He is also grateful to express his thanks to the organizers of the
Taniguchi symposium, as well as the Taniguchi Foundation,
where he had valuable discussions with many participants.
He also express his thanks to L. Borisov and the referee for their
valuable comments on the first version of this article.
\section{2. Mirror Symmetry of Calabi-Yau Hypersurfaces}
In this section, we will summarize mirror symmetry of
Calabi-Yau hypersurfaces in toric varieties due to
Batyrev. We refer the paper\cite{Bat1} for details.
Let $M\cong {\bf Z}^d$ be a lattice of rank $d$ and $N$ be its
dual. We denote the dual pairing $M\times N \rightarrow {\bf Z}$ by
$ \langle , \rangle $. A (convex) polytope ${\Delta}$ is a convex hull of a finite set
of points in $M_{{\Bbb R}}:=M\otimes{\Bbb R}$. In the following, we assume ${\Delta}$
contains the origin in its interior.
The polar dual ${\Delta^*}\subset N_{{\Bbb R}}$ is defined by
\begin{equation}
{\Delta^*}=\{ x \in N_{\Bbb R} \; \vert\; \langle x,y \rangle \geq -1 ,\; y\in {\Delta} \;\} \;\;.
\label{eqn: Ds}
\end{equation}
\vspace{0.3cm}\noindent
{\bf Definition 2.1.}
A polytope ${\Delta}$ is called reflexive if it is a convex hull of
a finite set of integral points in $M_{\Bbb R}$ and contains only the
origin in its interior.
\vspace{0.3cm}\noindent
{\bf Proposition 2.2.}
When a polytope ${\Delta}$ is reflexive, its dual ${\Delta^*}$ is also reflexive.
\vspace{0.2cm}
Since $({\Delta^*})^*={\Delta}$, reflexive polytopes come with a pair $({\Delta},{\Delta^*})$.
The following descriptions about ${\Delta^*}$ with $N$ equally apply to
${\Delta}$ with $M$ by symmetry.
\vspace{0.3cm}\noindent
{\bf Definition 2.3.}
A maximal triangulation $T_o$ of ${\Delta^*}$ is a simplicial decomposition
of ${\Delta^*}$ with properties; 1) every $d$-simplex contains the origin
as its vertex, 2) 0-simplices consist of all integral points of ${\Delta^*}$.
\vspace{0.2cm}
For a maximal triangulation $T_o$ of ${\Delta^*}$, we consider a complete fan
$\Sigma({\Delta^*},T_o)$ over the triangulation $T_o$ in $N_{\Bbb R}$.
Associated to the data $(\Sigma({\Delta^*},T_o),$ $N)$
we consider a toric variety ${\Bbb P}_{\Sigma({\Delta^*},T_o)}$ \cite{Oda}\cite{Ful}.
Due to the property that ${\Delta^*}$ is reflexive, we have
\vspace{0.3cm}\noindent
{\bf Proposition 2.4.} (Prop.2.2.19 in \cite{Bat1}) {\it
${\Bbb P}_{\Sigma({\Delta^*},T_o)}$ is a projective variety for at least
one maximal triangulation with its anti-canonical class
$-K=\displaystyle{ \sum_{\rho \in N\cap {\Delta^*} \setminus \{ 0\} } } D_\rho$
ample. }
\vspace{0.3cm} \noindent
{\bf Note.} In \cite{Bat1}, the maximal triangulations with the property
in this proposition are called {\it projective}.
In case of $d\leq 4$, we can observe widely that
every maximal triangulation is projective. More generally we observe
that every triangulation of a reflexive polytope
is {\it regular} which generalize projective(, see right after
eq.(\ref{eqn:convex}) for the definition).
For a restricted class of reflexive polytopes
(the type I or II in the following classification),
it has been proved (Th.4.10 in \cite{HLY2}) that
every nonsingular maximal triangulation is projective,
see also Remark after Th.2.5.
In the following, we will write a projective toric variety by
${\Bbb P}_{\Sigma({\Delta^*},T_o)}$ choosing a projective maximal triangulation
$T_o$ of ${\Delta^*}$.
\vspace{0.2cm}
Let us fix a basis $\{ n_1,\cdots,n_d\}$ of $N$ and denote its
dual basis by $\{m_1,\cdots,m_d\}$.
With respect to this basis, we denote the coordinate ring of
the torus $T_N:={\rm Hom}\,_{\bf Z}(M,{\Bbb C}^*) \subset {\Bbb P}_{\Sigma({\Delta^*},T_o)}$
by ${\Bbb C}[Y_1^{\pm1},\cdots,Y_d^{\pm1}]$ with the generators
$Y_k={\rm{\bf e}}(m_k):T_N\rightarrow {\Bbb C}^*$ defined by
{\bf e}$(m_k)(t)=t(m_k)$. Consider a Laurent
polynomial $f_{\Delta} = \sum_{\nu\in {\Delta}\cap M} c_{\nu} Y^\nu$ with
complex coefficients $c_\nu$. We denote by $X_{\Delta}$ the Zariski closure of
the zero locus $(f_{\Delta}=0)$ in ${\Bbb P}_{\Sigma({\Delta^*},T_o)}$ for generic $c_\nu$'s.
Similarly, we consider a projective toric variety ${\Bbb P}_{\Sigma({\Delta},T_o)}$
associated to a projective maximal triangulation $T_o$ of ${\Delta}$, and
denote the coordinate ring of $T_M:={\rm Hom}\,_{\bf Z}(N,{\Bbb C}^*)
\subset {\Bbb P}_{\Sigma({\Delta},T_o)}$ by ${\Bbb C}[X_1^{\pm1},\cdots,X_d^{\pm1}]$ with
$X_k={\rm{\bf e}}(n_k)$.
\vspace{0.3cm}\noindent
{\bf Theorem 2.5.} (Th.4.2.2, Corollary 4.2.3, Th.4.4.3 in \cite{Bat1})
{\it Let $({\Delta},{\Delta^*})$ be a pair of reflexive polytopes in
dimensions $d\leq4$ (, in $M_{\Bbb R}$ and $N_{\Bbb R}$, respectively). Then;
1) generic hypersurfaces $X_{\Delta}\subset {\Bbb P}_{\Sigma({\Delta^*},T_o)}$ and
$X_{\Delta^*} \subset {\Bbb P}_{\Sigma({\Delta},T_o)}$
define smooth Calabi-Yau manifolds,
2) these two hypersurfaces are mirror symmetric in their
Hodge numbers, i.e., $h^{1,1}(X_{\Delta})=h^{d-2,1}(X_{\Delta^*}) ,
\;$ $ h^{d-2,1}(X_{\Delta})= h^{1,1}(X_{\Delta^*})$. }
\vspace{0.3cm}
\noindent
{\bf Remark.}
Depending on the toric data of the reflexive polytopes,
the ambient spaces have (Gorenstein) singularities (Prop.2.2.2 in \cite{Bat1})
in general. We call a maximal triangulation is {\it nonsingular}
if its simplices of maximal dimensions consists of unit simplices, i.e.,
simplices with unit volume. It is easy to deduce that the toric variety
is nonsingular if the maximal triangulation is so.
Now we classify the reflexive polytopes into the following
three types:
\par\noindent
$\bullet$ type I;
the polytope
has no integral point in the interior of all codimension one faces,
and has a nonsingular maximal triangulation,
\par\noindent
$\bullet$ type II;
the polytope has at least one integral point in the interior of some
codimension one face, and has a nonsingular maximal triangulation,
\par\noindent
$\bullet$ type III;
the polytope does not have a nonsingular maximal triangulation.
In the following, we always consider a nonsingular maximal triangulation
$T_o$ for the polytopes of type I and II. Then the toric varieties
${\Bbb P}_{\Sigma({\Delta^*},T_o)}$ are projective and nonsingular for both
the polytopes ${\Delta^*}$ of type I and II (Th.4.10 in \cite{HLY2}),
however we distinguish these two
because of the difference in the {\it root system} for their
dual polytopes ${\Delta}$;
\begin{equation}
\begin{array}{rcl}
R({\Delta},M)=\{ \alpha \in {\Delta^*}\cap N \;\vert\;
&& \makebox[-1em]{}
\exists m_\alpha \in {\Delta}\cap M \; s.t.
\langle m_\alpha, \alpha \rangle =-1 \; \\
&&\makebox[-1em]{} {\rm and}\;
\langle m , \alpha \rangle \geq 0 \;(m\not= m_\alpha \in {\Delta}\cap M) \;\; \} \\
\end{array}
\label{eqn: rootD}
\end{equation}
The root system determines
the automorphisms of the toric variety ${\Bbb P}_{\Sigma({\Delta},T_o)}$ infinitesimally
due to the following result, which we will utilize in the next section;
\vspace{0.3cm}\noindent
{\bf Proposition 2.6.} (Prop.3.13 in \cite{Oda}) {\it
For a nonsingular toric variety ${\Bbb P}_{\Sigma({\Delta},T_o)}$, we have a direct sum
decomposition via the root system $R({\Delta},M)$;
\begin{equation}
\begin{array}{rcl}
{\rm Lie}({\rm Aut}\,({\Bbb P}_{\Sigma({\Delta},T_o)})) &=&
H^0({\Bbb P}_{\Sigma({\Delta},T_o)},\Theta_{{\Bbb P}_{\Sigma({\Delta},T_o)}}) \\
&=& {\rm Lie}(T_M) \oplus \(
\oplus_{\alpha\in R({\Delta},M)}
{\Bbb C} {\rm{\bf e}}(\alpha)\delta_{m_{\alpha}}\) \\
\end{array}
\label{eqn: lie}
\end{equation}
where $\delta_m \; (m \in M)$ is the derivation on $T_M$ defined by
$\delta_m {\rm{\bf e}}(n):= \langle m,n \rangle {\rm{\bf e}}(n)$. }
\newpage
\section{3. Resonance in GKZ Hypergeometric System}
We consider a family of Calabi-Yau hypersurfaces $X_{{\Delta^*}}(a)
\subset {\Bbb P}_{\Sigma({\Delta},T_o)}$ varying the coefficients $a_{\ns{{}}}$ in the
defining equation $f_{\Delta^*}(a)=\sum_{\ns{{}}\in {\Delta^*}\cap N} a_{\ns{}} X^{\ns{}}$.
By this polynomial deformation, we describe the complex structure
deformation of $X_{\Delta^*}$. This deformation space is mapped to that
of (complexified) K\"ahler class of $X_{\Delta}\subset {\Bbb P}_{\Sigma({\Delta^*},T_o)}$
under the mirror symmetry. According to the local Torelli theorem \cite{BG},
we can introduce a local coordinate on the moduli space in terms of
period integrals. In case of hypersurfaces in toric varieties, we
have one canonical period integral \cite{Bat2}\cite{BC}
\begin{equation}
\Pi(a)={1\over (2\pi i)^d} \int_{C_0} {1\over f_{\Delta^*}(X,a)}
\prod_{i=1}^{d} {d X_i \over X_i} \;\;,
\label{eqn: Pi}
\end{equation}
with the cycle $C_0=\{ \vert X_1\vert=\cdots=\vert X_d\vert=1\}$ in $T_M$.
Here we study the differential equation satisfied by (\ref{eqn: Pi}).
\newsubsec{(3-1) Extended GKZ hypergeometric system }
Let ${\cal A}=\{ \bar\chi_0,\cdots,\bar\chi_p\}$ be a finite
set of integral points in $\{1\}\times{\Bbb R}^n\subset{\Bbb R}^{n+1}$. We assume
the vectors $\bar\chi_0,\cdots,\bar\chi_p$ span ${\Bbb R}^{n+1}$.
\vspace{0.3cm}\noindent
{\bf Definition 3.1.}
Consider the lattice of relations among the set ${\cal A}$,
\begin{equation}
L=\{ \; (l_0,\cdots,l_p)\in {\bf Z}^{p+1}\; \vert\;
\sum_{i=0}^p l_i\bar\chi_{i,j}=0 \;, (j=1,\cdots,n+1) \;\} \;,
\label{eqn: LA}
\end{equation}
where $\bar\chi_{i,j}$ represents the $j$-th component of
the vector $\bar\chi_i$. {\it ${\cal A}$-hypergeometric
system with exponent $\beta \in {\Bbb C}^{n+1}$ } is a system of
differential equations for a complex function $\Psi(a)$
on ${\Bbb C}^{\cal A}$;
\begin{eqnarray}
{\cal D}_l \Psi(a)&=&\{
\prod_{l_i>0}\({\partial \; \over \partial a_i}\)^{l_i} -
\prod_{l_i<0}\({\partial \; \over \partial a_i}\)^{-l_i} \} \Psi(a)=0
\;\; (l \in L ) \\
{\cal Z}\Psi(a)&=&\{ \sum_{i=0}^p \bar\chi_i a_i {\partial \; \over \partial a_i}
-\beta \} \Psi(a) =0 \;\; .
\label{eqn: gkz}
\end{eqnarray}
\vspace{0.3cm}\noindent
{\bf Proposition 3.2.} (\cite{Bat2}) {\it
The period integral (\ref{eqn: Pi}) satisfies
the ${\cal A}$-hyper-\break geometric system
with ${\cal A}=\{1\}\times ({\Delta^*}\cap N)$ and
$\beta=(-1)\times\vec 0$. We call this hypergeometric system as
${\Delta^*}$-hypergeometric system. }
\vspace{0.2cm}
By direct evaluation of the action of ${\cal D}_l$ and ${\cal Z}$
on the period integral (\ref{eqn: Pi}), we obtain this proposition.
Here we consider the meaning of the linear operator ${\cal Z}$.
The first component of this vector operator is exactly the
Euler operator, and just says that the period integral has homogeneous
degree $-1$ as a function of $a_i$'s. For the other components, it is
easy to deduce that these come from the invariance of the period
integral under the torus actions, which act infinitesimally on
the coordinate $X_k={\rm{\bf e}}(n_k)$ by
$\delta_m X_k = \langle m, n_k \rangle X_k$. It is now clear that these actions
should be considered for all elements in
${\rm Lie}({\rm Aut}\, ({\Bbb P}_{\Sigma({\Delta},T_o)}))$.
Then we may write the invariance of the period integral under the
infinitesimal action of $\xi \in {\rm Lie}({\rm Aut}\, ({\Bbb P}_{\Sigma({\Delta},T_o)}))$
by a linear differential operator ${\cal Z}_\xi$ acting on $\Pi(a)$ through
\begin{equation}
{\cal Z}_{\xi} \Pi(a)=
\int_{C_0} \xi
\( {1\over f_{\Delta^*}(X,a)} \)
\prod_{i=1}^{d} {d X_i \over X_i} = 0 \;\;.
\label{eqn: Zxi}
\end{equation}
For explicit forms of the operators ${\cal Z}_\xi$, we refer to the examples
given in p.541 of \cite{HLY1}.
\vspace{0.3cm}\noindent
{\bf Proposition 3.3.} ((2.13) in \cite{HLY1})
{\it The period integral $\Pi(a)$ satisfies
\begin{equation}
\begin{array}{crl}
&& {\cal D}_l \Pi(a) =0 \; (l\in L) \;,\; \\
&& \\
&& {\cal Z}_E \Pi(a)=0 \;,\;
{\cal Z}_\xi \Pi(a)= 0 \;\;
(\xi \in {\rm Lie}({\rm Aut}\, ({\Bbb P}_{\Sigma({\Delta},T_o)}))), \\
\end{array}
\label{eqn: extGKZ}
\end{equation}
where we denote the Euler operator by
${\cal Z}_E=\sum_{i=0}^p a_i{\partial\; \over \partial a_i} +1$. }
\vspace{0.2cm}
We call this system as {\it extended GKZ-hypergeometric system} or
{\it extended ${\Delta^*}$-hypergeometric system}. By Prop.2.6, it is
clear that this extended system reduces to the GKZ system if the
polytope ${\Delta^*}$ is of type I. In the following, we take an approach to
study mainly the ${\Delta^*}$-hypergeometric system because the set of the solutions
of the extended ${\Delta^*}$-hypergeometric system can be found in that of the
${\Delta^*}$-hypergeometric system.
\newsubsec{(3-2) Convergent series solutions }
Here we summarize general results in \cite{GKZ1} about the
convergent series
solution of the ${\cal A}$-hypergeometric system with exponent $\beta$.
This is to introduce the notion of the secondary fan as well as to
fix our conventions and notations. Since our interest is in the period
integrals, we assume ${\cal A}=\{1\}\times ({\Delta^*}\cap N)$ and
$\beta=(-1)\times \vec 0$. Hereafter we write the integral points
explicitly by ${\Delta^*}\cap N=\{\ns{0},\cdots,\ns{p}\} \;(\ns{0}\equiv\vec
0)$ and $\bn{i}:=1\times\ns{i} \; (i=0,\cdots,p)$.
We start with a formal solution of the ${\cal A}$-hypergeometric system
with exponent $\beta$ given by
\begin{equation}
\Pi(a,\gamma)=\sum_{l\in L} {1 \over
\prod_{0\leq i\leq p} \Gamma(l_i+\gamma_i+1)}a^{l+\gamma}
\;\;,
\label{eqn: fsol}
\end{equation}
where $\beta=\sum_i \gamma_i \bn{i}$. Now define an affine space
$\Phi(\beta):=\{ \gamma\in {\Bbb R}^{p+1}\vert \beta=\sum \gamma_i\bn{i} \}$.
A subset $I\subset \{ 0,\cdots,p\}$ is called a base if $\bn{I}:=\{\bn{i}\vert
i\in I\}$ form a basis of ${\Bbb R}^{d+1}$. Given a base $I$, we may solve
$\sum_{j\in I}\gamma_j \bn{j} = \beta -\sum_{j\not\in I}\gamma_j\bn{j}$
for $\gamma_j \;(j\in I)$ and define $\Phi_{{\bf Z}}(\beta,I)=\{
\gamma\in \Phi(\beta) \;\vert\; \gamma_j \in {\bf Z} \;(j\not\in I) \}$.
We choose an integral basis $A=\{ \l1, \cdots, \l{p-d} \}$ of the lattice
$L$, and define $\Phi^A_{{\bf Z}}(\beta,I)=\{ \gamma\in \Phi_{{\bf Z}}(\beta,I)
\;\vert\; \gamma_j=\sum_{k=1}^{p-d} \lambda_k \l{k}_j \; (0\leq \lambda_k<1,
\; j\not\in I) \}$. Then $\Phi^A_{{\bf Z}}(\beta,I)$ provides
a set of representatives of the quotient $\Phi_{\bf Z}(\beta,I)/L$
and kills the invariance $\gamma \rightarrow \gamma+v \; (v\in L)$ in the
formal solution (\ref{eqn: fsol}).
\vspace{0.3cm}\noindent
{\bf Definition 3.4.} For a base $I$, define a cone in $L_{\Bbb R}=L\otimes{\Bbb R}$
by ${\cal K}({\cal A},I)=\{ l \in L_{\Bbb R} \;\vert\; l_i\geq0\;(i\not\in I)\}$.
A ${\bf Z}$-basis $A\subset L$ is said {\it compatible}
with a base $I$ if it generates a cone that contains ${\cal K}({\cal
A},I)$.
\vspace{0.3cm}\noindent
{\bf Theorem 3.5.} (Prop.1 in \cite{GKZ1}) {\it
Fix a base $I$ and choose a ${\bf Z}$-basis $A=\{\l1,\cdots,\l{p-d}\}$
compatible with
it. Then the formal solution (\ref{eqn: fsol})
takes the form $\Pi(a,\gamma)=a^\gamma
\sum_{m\in {\bf Z}^{p-d}_{\geq0}}c_m(\gamma) x^m$ for each $\gamma\in
\Phi_{\bf Z}^A(\beta,I)$ with $x_k=a^{\l{k}}$. This powerseries converges
for sufficiently small $\vert x_k \vert$. }
\vspace{0.3cm}\noindent
{\bf Remark.}
The coefficient $c_m(\gamma)$ is given explicitly by
$$
c_m(\gamma)={1\over \prod_{i=0}^p \Gamma(\sum_k m_k\l{k}_i + \gamma_i+1)} \;.
$$
For some index $i$ of the base $I$ in the above theorem it can happen that
$\sum m_k\l{k}_i+\gamma_i+1$ is non-positive for all $m \in
{\bf Z}_{\geq0}^{p-d}$ , which means we have the trivial solution
$\Pi(a,\gamma)\equiv0$. In this case, we multiply an infinite number
$\Gamma(\gamma_i+1)$ to obtain
nonzero powerseries, i.e.,
$
{\Gamma(\gamma_i+1) \over \Gamma(\sum m_k\l{k}_i+\gamma_i+1)} :=
\lim_{\varepsilon\rightarrow0}
{\Gamma(\gamma_i+1+\varepsilon) \over
\Gamma(\sum m_k\l{k}_i+\gamma_i+1+\varepsilon)} .
$
\vspace{0.3cm}\noindent
{\bf Definition 3.6.}
Consider $P={\rm Conv.}\( \{ 0, \bn{0},\cdots,\bn{p} \}\)$ in
${\Bbb R}^{d+1}$. A collection of bases $T=\{I\}$ is called a triangulation
of $P$ if $\cup_{I\in T} \langle \bn{I} \rangle =P$ for simplices
$ \langle \bn{I} \rangle ={\rm Conv.}\(\{0\}\cup\bn{I}\)$ $(I\in T)$,
and $ \langle \bn{I_1} \rangle \cap \langle \bn{I_2} \rangle \;(I_1,I_2\in T)$ is
a lower dimensional common face.
\vspace{0.3cm}\noindent
{\bf Note.}
Since $(d+1)$-simplices in $P$ are in one-to-one correspondence to
$d$-simplices in ${\Delta^*}$, we identify a triangulation of $T$ of $P$
with its corresponding triangulation of ${\Delta^*}$. We call
a triangulation $T$ of $P$ is maximal if it corresponds to a maximal
triangulation of ${\Delta^*}$(Def.2.3).
\vspace{0.2cm}
For a base $I$ and a point $\eta\in {\Bbb R}^{p+1}$, we consider a linear
function $h_{I,\eta}$ on ${\Bbb R}^{p+1}$ by $h_{I,\eta}(\bn{i})=\eta_i \;(i\in I)$.
We define ${\cal C}({\cal A},I)=\{ \eta\in {\Bbb R}^{p+1} \;\vert\;
h_{I,\eta}\;(\bn{i}) \leq \eta_i \;(i\not\in I)\}$ and ${\cal C}({\cal
A},T):=\cap_{I\in T}{\cal C}({\cal A},I)$ for a triangulation $T$.
Then it is easy to see that ${\cal C}({\cal A},T)$ consists of
$\eta \in {\Bbb R}^{p+1}$ for which we have a convex piecewise
linear function $h_{T,\eta}$ on $T$ determined by
$h_{T,\eta}\vert_{ \langle \bn{I} \rangle }= h_{I,\eta} \; (I\in T)$ and satisfies
$h_{T,\eta}(\bn{i})\leq \eta_i$ for $\bn{i}$
not a vertex of $T$, i.e.,
\begin{equation}
\begin{array}{crl}
{\cal C}({\cal A},T)=\{ \eta \in {\Bbb R}^{p+1}\; \vert \;\;
&& \makebox[-2em]{}
h_{I_1,\eta}(v)\leq h_{I_2,\eta}(v) \;\;
(v\in \langle \bn{I_2} \rangle , \; I_1,I_2\in T), \\
&& \makebox[-2em]{}
h_{T,\eta}(\bn{i})\leq \eta_i \;\;
(\bn{i} \;\,{\rm is} \;\, {\rm not}\;\, {\rm a} \;\,
{\rm vertex}\;\,{\rm of}\; T)\; \}\\
\end{array}
\label{eqn:convex}
\end{equation}
A triangulation is called {\it regular} if ${\cal C}({\cal A},T)$ has
interior points.
We say a ${\bf Z}$-basis $A\subset L$ is compatible with a triangulation
$T$ if it is compatible with all bases $I$ in $T$.
\vspace{0.3cm}\noindent
{\bf Proposition 3.7.} (Prop.5 in \cite{GKZ1}) {\it
For every regular triangulation $T$, there exists (infinitely many)
${\bf Z}$-basis of $L$ compatible with $T$. }
\vspace{0.2cm}
The exponent $\beta$ is called {\it $T$-nonresonant} if the set
$\Phi^A_{\bf Z}(\beta,I)\; (I\in T)$ are pairwise disjoint.
We set $\Phi^A_{\bf Z}(\beta,T):=\cup_{I\in T}\Phi^A_{\bf Z}(\beta,I)$.
We normalize the volume of the standard $(d+1)$-simplex to $1$.
\vspace{0.3cm}\noindent
{\bf Theorem 3.8.} (Th.3 in \cite{GKZ1}) {\it
For a regular triangulation $T$ of the polytope $P$, and a ${\bf Z}$-basis
$A=\{\l{1},\cdots,\l{p-d}\}$ of $L$ compatible with $T$, there are integral
powerseries in the variables $x_k=a^{\l{k}}$ for
$a^{-\gamma}\Pi(a,\gamma) \;(\gamma\in \Phi^A_{\bf Z}(\beta,T))$, which
converge for sufficiently small $\vert x_k\vert$. If the exponents
$\beta$ is $T$-nonresonant, these series constitute $vol(P)$ linearly
independent solutions. }
\vspace{0.3cm}\noindent
{\bf Remark.}
In our case of ${\Delta^*}$-hypergeometric system with $\beta=(-1)\times\vec0$,
we have one special element $\gamma=(-1,0,\cdots,0)$ in the set
$\Phi_{\bf Z}(\beta,I)$ for any base $I$. If the polytope ${\Delta^*}$ is of
type I or II in our classification, a nonsingular maximal
triangulation $T$ of $P$ consists of those bases $I$ for which
$\vert \det (\bar\nu_{j,i}^*)_{1\leq i\leq d+1,\; j\in I}\vert =1$.
Because of this unimodularity, we have
\begin{equation}
\Phi_{\bf Z}(\beta,I)=(-1,0,\cdots,0)+L \;\; ,
\label{eqn: phiL}
\end{equation}
and $\Phi_{\bf Z}^A(\beta,I)=\{ (-1,0,\cdots,0)\}$ for every base
$I$ of the maximal triangulation and any ${\bf Z}$-basis $A$ compatible with it.
Thus we encounter a ``maximally $T$-resonant'' situation.
\vspace{0.3cm}\noindent
{\bf Definition 3.9.}
For a regular triangulation $T$ and a ${\bf Z}$-basis
$A=\{ \l{1},\cdots,$ $\l{p-d}\}$
compatible with $T$, we define a
power series $w_0(x,\rho; A)=a_0\Pi(a,\gamma)$
(with $\gamma=\sum_{k=1}^{p-d}\rho_k\l{k}+(-1,0,\cdots,0)$) by
\begin{equation}
w_0(x,\rho;A)=\sum_{m\in {\bf Z}_{\geq0}^{p-d} }
{ \Gamma(-\sum_k (m_k+\rho_k)\l{k}_0+1) \over
\prod_{1\leq i\leq p}
\Gamma(\sum_k(m_k+\rho_k)\l{k}_i+1) } x^{m+\rho} \;\;,
\label{eqn: wnot}
\end{equation}
where
$x_k=(-1)^{\l{k}_0}a^{\l{k}}$.
\vspace{0.3cm}\noindent
{\bf Remark.}
Here we have applied our recepie of multiplying the constant
$\Gamma(\gamma_0+1)$ to the formal solution $\Pi(a,\gamma)$.
We adopt this definition because for a maximall triangulation $T_o$,
we encounter the situation $\Pi(a,\gamma)\equiv 0$, namely,
$\sum m_k \l{k}_0 +\gamma_0+1 \in {\bf Z}_{\leq0} \;
(m\in {\bf Z}^{p-n}_{\geq0})$ for a ${\bf Z}$-basis
$A=\{\l{1},\cdots,\l{p-n}\}$ compatible with $T_o$.
(See Prop.4.8, Prop.4.9 and Th.4.10 in \cite{HLY2}).
In general a basis $A$ compatible with a regular triangulation $T$
contains both the bases vectors $\l{k}$ with positive 0-th component
and nonpositive 0-th component. Taking a value
$\rho_0 \in {\bf Z}^{p-n}_{\geq0}$ under this situation cause infinity
for some $m$ in the numerator of the coefficients of (\ref{eqn: wnot}).
In this case we understand in our definition (\ref{eqn: wnot})
to take a limit $\rho \rightarrow \rho_0$ in a generic way.
(If we encounter infinities under this limit in some coefficient,
we go back to the original definition (\ref{eqn: fsol}).
For a maximall triangulation $T_o$, we observe that this limit exists
for all coefficients in (\ref{eqn: wnot}), see Claim5.8.)
\newsubsec{(3-3) Secondary fan}
It is known that the set ${\cal C}({\cal A},T)$ is a closed
polyhedral cone and that these cones cover ${\Bbb R}^{p+1}$ when we vary the
triangulations. Thus the set of these cones and their lower dimensional faces
all together define a complete, polyhedral fan ${\cal F}({\cal A})$
called the {\it secondary fan}\cite{BFS}\cite{OP}.
Let $\overline M = {\bf Z} \times M$ and
$\overline
N = {\bf Z} \times N$. We extend naturally the pairing $ \langle , \rangle $ to that
of $\overline M$ and $\overline N$. Consider the lattice
$\overline {\cal M}:= \oplus_{\ns{} \in {\Delta^*}\cap N}
{\bf Z} \en{}\;(=\oplus_{i=0}^p {\bf Z}\en{i})$ and its dual
$\overline {\cal N}={\rm Hom}\,_{\bf Z}(\overline {\cal M},{\bf Z})$. Then we have the
following exact sequences
\begin{equation}
\begin{array}{crl}
&& \makebox[2.5cm]{}
0
\smash{\mathop{\longrightarrow}\limits^{}} \; \overline M \;
\smash{\mathop{\longrightarrow}\limits^{\quad {\bf A}\quad}}
\; \overline{\cal M} \;
\smash{\mathop{\longrightarrow}\limits^{\quad {\bf B}\quad }}
\; \Xi(\overline M) \;
\smash{\mathop{\longrightarrow}\limits^{}} 0 \;\;, \\
&&
0 \smash{\mathop{\longleftarrow}\limits^{}} \; {\rm Coker}{\bf A}^* \;
\smash{\mathop{\longleftarrow}\limits^{}} \; \overline N \;
\smash{\mathop{\longleftarrow}\limits^{\quad {\bf A}^* \quad}}
\; \overline{\cal N} \;
\smash{\mathop{\longleftarrow}\limits^{\quad {\bf B}^* \quad}}
\; \Xi(\overline N) \;
\smash{\mathop{\longleftarrow}\limits^{}} 0 \;\;, \\
\end{array}
\label{eqn: exacts}
\end{equation}
where ${\bf A}(\bar m)=\sum_{i=0}^p \langle \bar m,\bn{i} \rangle \en{i} \;\;
(\bar m\in \overline M)$ and ${\bf B}$ is the quotient. The dual is given by
${\bf A}^*(\mu)=\sum_{i=0}^p \mu(\en{i})\bn{i} \;\; (\mu\in \overline N)$.
The pair $\{ {\cal B}, \Xi(\overline M) \}$ with
${\cal B}:=\{ {\bf B}(\en{0}), \cdots, {\bf B}(\en{p}) \}$ is
called {\it Gale transform} of a pair
$\{ {\cal A}, $ $\overline N \}$. Under this general
setting, let us consider a polyhedral cone in $\Xi(\overline M)_{\Bbb R}$
\begin{equation}
{\cal C}'({\cal A},T)=\cap_{I\in T} \( \sum_{i\not\in I}
{\Bbb R}_{\geq0}{\bf B}(\en{i}) \)\;\;.
\end{equation}
\vspace{0.3cm}\noindent
{\bf Proposition 3.10.}(Lemma 4.2 in \cite{BFS}) {\it
The map ${\bf B}$ induces the following decomposition
\begin{equation}
{\cal C}({\cal A},T)={\rm Ker}({\bf B})\oplus {\cal C}'({\cal A},T) \;\;.
\label{eqn:decomp}
\end{equation} }
\vspace{0.3cm}
By definition, the cone ${\cal C}'({\cal A},T)$ is strongly convex.
Using the above decomposition, we redefine the secondary fan to be
\begin{equation}
{\cal F}({\cal A})=\{ {\cal C}'({\cal A},T) \vert \; T:
{\rm regular} \;\; {\rm triangulation} \} .
\end{equation}
Now the secondary fan consists of strongly convex, polyhedral cones.
If the polytope ${\Delta^*}$ is of type I or II, then the quotient $\Xi(\overline M)$
is torsion free and thus
$({\cal F}({\cal A}),\Xi(\overline M))$ defines a toric
variety. Even in the case of type III, we may consider the corresponding
toric variety by simply projecting out the torsion part of $\Xi(\overline M)$.
We will use this toric variety for the compactification of the moduli
space in the next section.
Now let us note that $\Xi(\overline N)\cong {\rm Ker}({\bf A}^*)$ is
identified with the lattice $L$, and thus ${\cal K}({\cal A},T) \subset
\Xi(\overline N)_{\Bbb R}$. By definition of ${\cal C}'({\cal A},T)$, we may
deduce
\begin{equation}
{\cal K}({\cal A},T)={\cal C}'({\cal A},T)^\vee \;\;.
\label{eqn: Cdual}
\end{equation}
Since for a regular triangulation $T$, ${\cal C}'({\cal A},T)$ is a strongly
convex polyhedral cone with interior points, the dual cone
${\cal K}({\cal A},T)$ is also strongly convex polyhedral cone.
Therefore we see that there are infinitely many ${\bf Z}$-basis of the
lattice $L$ compatible with $T$ (Prop.3.7).
\section{4. Toric Ideal and Gr\"obner Fan}
In this section we will reduce the infinite set of
operators ${\cal D}_l\; (l\in L)$ in our ${\Delta^*}$-hypergeometric system
to a finite set. This will be related to the compactification
problem of the moduli spaces.
\newsubsec{(4-1) Toric ideal and Gr\"obner fan}
We write the operators ${\cal D}_l$ in (6) simply by
${\cal D}_l=({\partial\;\over\partial a})^{l_+} - ({\partial\;\over\partial a})^{l_-}$ with
$l=l_+-l_-$. Keeping this form in mind we define
{\it toric ideal} in a polynomial ring:
\vspace{0.3cm}\noindent
{\bf Definition 4.1.} Consider a polynomial ring
${\Bbb C}[y]:={\Bbb C}[y_0,\cdots,y_p]$.
Toric ideal ${\cal I}_{\cal A}$ is defined to be an ideal generated by {\it binomials}
$y^{l_+}-y^{l_-} \; (l\in L)$,
\begin{equation}
{\cal I}_{\cal A}= \langle \; y^{l_+}-y^{l_-} \;\vert\; l\in L \; \rangle \;.
\end{equation}
\vspace{0.3cm}
Toric ideal has been extensively studied in ref.\cite{Stu1}\cite{GKZ2}.
Here we summarize relevant results for our purpose.
A {\it term order} (monomial ordering) on ${\Bbb C}[y]$ is a total order $\prec$
on the set of monomials $\{ y^\alpha \;\vert\; \alpha\in {\bf Z}_{\geq0}^{p+1}\}$
satisfying, 1) $y^\alpha \prec y^\beta$ implies $y^{\alpha+\gamma} \prec
y^{\beta+\gamma}$ and 2) $1$ is the unique minimal element.
When we have an ideal ${\cal I} \subset {\Bbb C}[y]$ and fix a term order,
we can speak of the leading term $LT_\prec(f)$ for
every non-zero polynomial in ${\cal I}$.
Then we define {\it initial ideal} of ${\cal I}$ by
\begin{equation}
\langle LT_\prec ({\cal I}) \rangle = \langle LT_\prec (f) \;\vert\; f\in {\cal I}, f\not=0 \rangle \;.
\end{equation}
A finite set ${\cal G}\subset {\cal I}$ is called {\it Gr\"obner basis} with
respect to a term order $\prec$ if it generates the initial ideal;
\begin{equation}
\langle LT_\prec ({\cal I}) \rangle = \langle LT_\prec(g) \;\vert\; g \in {\cal G} \rangle \;.
\label{eqn: groebner}
\end{equation}
\vspace{0.3cm}\noindent
{\bf Theorem 4.2.} (Th.1.2 in \cite{Stu2}) {\it
For every ideal ${\cal I}\subset {\Bbb C}[y]$, there are only finitely many
distinct initial ideals. }
\vspace{0.3cm}
We consider representing the term orders by weight vectors
${\omega} = (w_0,\cdots,$ ${\omega}_p)\in {\Bbb R}^{p+1}$. For a polynomial $f=\sum_\alpha
c_\alpha y^\alpha$, we define its {\it leading terms} $LT_{\omega}(f)$ to
be a sum of terms $c_\alpha y^\alpha$ whose weight $t_{\omega}(y^\alpha)
:={\omega}_0\alpha_0+\cdots+{\omega}_p\alpha_p$ is maximal. It is obvious that if
the components of ${\omega} \in {\Bbb R}_{\geq0}^{p+1}$ are rationally independent,
the weight determines a term order on ${\Bbb C}[y]$. When we fix an ideal
${\cal I}\subset {\Bbb C}[y]$, we may relax the condition for the weight ${\omega}$
to be a term order;
we say a weight ${\omega}\in {\Bbb R}^{p+1}$ defines a term order
of ${\cal I}$ if $ \langle LT_{\omega}({\cal I}) \rangle = \langle LT_\prec({\cal I}) \rangle $ for
some term order $\prec$. The following proposition provides a 'converse'
statement,
\vspace{0.3cm}\noindent
{\bf Proposition 4.3.} (Prop.1.11 in \cite{Stu2}) {\it
For any term order $\prec$, there exists a weight
${\omega}\in {\Bbb R}_{\geq0}^{p+1}$ such that $ \langle LT_w({\cal I}) \rangle
= \langle LT_\prec({\cal I}) \rangle $. }
\vspace{0.3cm}
Now {\it Gr\"obner region} is defined to be a set
\begin{equation}
GR({\cal I})=\{ {\omega}\in {\Bbb R}^{p+1} \vert
\langle LT_{\omega}({\cal I}) \rangle = \langle LT_{{\omega}'}({\cal I}) \rangle \;
{\rm for} \;{\rm some} \; {\omega}'\in {\Bbb R}_{\geq0}^{p+1} \} \; .
\end{equation}
\vspace{0.3cm}\noindent
{\bf Proposition 4.4.} (Prop.1.12 in \cite{Stu2}) {\it
If an ideal ${\cal I}\in {\Bbb C}[y]$ is a homogeneous ideal with some grading
$deg(y_i)=d_i >0$, then $GR({\cal I})={\Bbb R}^{p+1}$. }
\vspace{0.3cm}
Since the toric ideal ${\cal I}_{\cal A}$ is homogeneous ideal with
$deg(y_0)=\cdots=$ $deg(y_p)$ $=1$, we see $GR({\cal I}_{\cal A})={\Bbb R}^{p+1}$.
For a term order ${\omega}$ of ${\cal I}_{\cal A}$, we define
\begin{equation}
{\cal C}({\cal I}_{\cal A},{\omega})=\{ {\omega}' \in {\Bbb R}^{p+1} \;\vert\;
\langle LT_{\omega}({\cal I}_{\cal A}) \rangle = \langle LT_{{\omega}'}({\cal I}_{\cal A}) \rangle \;\} \;.
\label{eq:Cw}
\end{equation}
It is known that this set constitutes an open, convex, polyhedral cone
in ${\Bbb R}^{p+1}$ (Prop.2.1 in \cite{Stu1}). In the following, we mean by
${\cal C}({\cal I}_{\cal A}, {\omega})$ the closure of the set (\ref{eq:Cw}).
Then due to Th.4.2 and Prop.4.4, the collection $\{ {\cal C}({\cal I}_{\cal A}, {\omega}) \}$ is
finite and defines a complete polyhedral fan ${\cal F}({\cal I}_{\cal A})$ in ${\Bbb R}^{p+1}$,
called the {\it Gr\"obner fan}.
\newsubsec{(4-2) Indicial ideal and compactification of ${\rm Hom}\,_{\bf Z}(L,{\Bbb C}^*)$ }
In the previous section, we called a triangulation $T$ of the
polytope $P$ regular if the cone ${\cal C}({\cal A},T)$ has interior
points. Here we characterize the regular triangulation in a geometrical
way. To this aim let us first consider a polytope $P_{\omega}:={\rm Conv.}\(\{
{\omega}_0\times\ns{0},\cdots, {\omega}_p\times\ns{p}\} \)$ in ${\Bbb R}^{d+1}$ for
a weight ${\omega} \in {\Bbb R}^{p+1}$. If we
project a polytope $P_{\omega}$ to $1\times {\Bbb R}^d$, then we have the polytope
$1\times {\Delta^*}$. Thus we may regard the weight ${\omega}$ giving a hight to each
vertex of $1\times{\Delta^*}$. For generic weight ${\omega}$, the 'lower' faces of
the polytope $P_{\omega}$ consist of simplices and define, under the
projection, a simplicial decomposition of ${\Delta^*}$ and thus induce a
triangulation $T_{\omega}$. The regular triangulation of the polytope $P$ is a
triangulation $T_{\omega}$ obtained for some weight ${\omega}$ in this way
(see Def.5.3 of \cite{Zie} for more details).
It is not difficult to see the relation of the polytope $P_{\omega}$ to
the piecewise linear function $h_{T,\eta}$ in (\ref{eqn:convex})
with $\eta={\omega}$.
Given a (regular) triangulation $T$ of the polytope $P$, the
{\it Stanley-Reisner ideal} $SR_T$ in ${\Bbb C}[y]$ is defined to be the ideal
generated by all monomials $y_{i_1}\cdots y_{i_k}$ for which the
vertices $\bn{i_1},\cdots,\bn{i_k}$ do not make a simplex in $T$.
The following theorem is due to Sturmfels:
\vspace{0.3cm}\noindent
{\bf Theorem 4.5.} (Thm. 3.1 in \cite{Stu1}) {\it
If a weight ${\omega}$ defines a term order of the toric ideal ${\cal I}_{\cal A}$, then it
induces a regular triangulation $T_{\omega}$. Moreover the Stanley-Reisner ideal
$SR_{T_{\omega}}$ is equal to the radical of the initial ideal
$ \langle LT_{\omega}({\cal I}_{\cal A}) \rangle $. }
\vspace{0.3cm}
As an immediate corollary to this theorem, we see that the Gr\"obner fan
is a refinement of the fan $\{ {\cal C}({\cal A},T_{\omega}) \}$. Since the cone
${\cal C}({\cal A},T_{\omega})$ decomposes according to (\ref{eqn:decomp}),
we have similar
decomposition of ${\cal C}({\cal I}_{\cal A},{\omega})$ to ${\cal C}'({\cal I}_{\cal A},{\omega})$. In the
following we call the collection $\{{\cal C}'({\cal I}_{\cal A},{\omega})\}$ as the
Gr\"obner fan ${\cal F}({\cal I}_{\cal A})$.
Now we determine a finite set of operators ${\cal D}_l$ which characterize the
power series $w_0(x,\rho;A)$ for each regular triangulation and a
${\bf Z}$-basis $A$ compatible with it.
This provides us a way to analyze our resonant GKZ hypergeometric
system.
Let us consider a term order ${\omega}$ of ${\cal I}_{\cal A}$. According to
Th.4.5, the term oder ${\omega}$ determines a regular triangulation
$T_{\omega}$ and also a cone ${\cal C}'({\cal I}_{\cal A},{\omega}) \subset {\cal C}'({\cal
A},T_{\omega})$.
If the cone ${\cal C}'({\cal I}_{\cal A},{\omega})$ is simplicial and regular,
i.e., the integral generators of its one-dimensional boundary cones
generate the lattice points ${\cal C}'({\cal I}_{\cal A},{\omega}) \cap \Xi(\overline M)$,
we simply make its dual cone ${\cal C}'({\cal I}_{\cal A},{\omega})^\vee$ and take the
integral generators of this cone as a canonical ${\bf Z}$-basis $A$ of $L$
which is compatible with $T_{\omega}$. If not, we
subdivide the cone ${\cal C}'({\cal I}_{\cal A},{\omega})$ into simplicial, regular cones
and reduce the problem to the former case.
More generally, we may take a ${\bf Z}$-basis $A_\tau=\{
\l{1}_\tau,\cdots,\l{p-d}_\tau \}$ of $L$ compatible with $T_{\omega}$
considering any simplicial, regular cone $\tau$ contained in ${\cal
C}'({\cal I}_{\cal A},{\omega})$ and making its dual $\tau^\vee$.
Associated to ${\omega}$, we have a Gr\"obner basis ${\cal B}_{\omega} \subset {\cal I}_{\cal A}$.
By B\"uchberger's algorithms to construct the (reduced) Gr\"obner basis, we
see that every generator $g \in {\cal B}_{\omega}$ is a binomial of the form
$y^{l_+}-y^{l_-}$ with some $l \in L$. In the following, we assume
${\cal B}_{\omega}$ to be the reduced Gr\"obner basis which is determined
uniquely for a term order ${\omega}$ (, see Chapter 2 of \cite{CLO} for the
properties of the reduced Gr\"obner basis). Translating this to differential
operator, we write the Gr\"obner basis ${\cal B}_{\omega}=\{ {\cal D}_{l_1}, \cdots,
{\cal D}_{l_s} \} \; (1\leq s < \infty)$. Now, for each generator, we define
\begin{equation}
J_l(\rho;A_\tau):= a_0 x_\tau^{-\rho} a^{l_\pm}\(\da{}\)^{l_\pm}
a_0^{-1}x_\tau^{\rho}
\;\;,
\label{eqn: Jl}
\end{equation}
where the choice in $l_\pm$ is made respectively by
${\omega}\cdot l_+ - {\omega}\cdot l_- >0 \;\; (<0) $.
(The factor $a_0$ originate from the definition
$w_0(x,\rho;A):= a_0 \Pi(a,\gamma)$ in Def.3.9.)
\vspace{0.3cm}\noindent
{\bf Definition 4.6.}
For a term order ${\omega}$ of ${\cal I}_{\cal A}$ and an arbitrary regular cone $\tau$ contained
in ${\cal C}'({\cal I}_{\cal A},{\omega})$, we define,
through the Gr\"obner basis ${\cal B}_{\omega}=\{{\cal D}_{l_1},\cdots, {\cal D}_{l_s}\}$,
an {\it indicial ideal} in ${\Bbb C}[\rho_1,\cdots,\rho_{p-d}]$;
\begin{equation}
Ind_{\omega}(\tau)= \langle J_{l_1}(\rho,A_\tau),\cdots,J_{l_s}(\rho,A_\tau) \rangle \;\;.
\end{equation}
Similarly to the indicial equations of the differential equations of
Fuchs type, we also consider the {\it indicial equations} for our
${\Delta^*}$-hypergeometric system as algebraic equations for $\rho$
coming from the leading terms of the operators $D_l \;(l\in L)$.
(Note that the leading term of an operator $D_l$ varies in general
when a term order ${\omega}$ varies. Here we consider for the indicial
equations all possible leading terms when ${\omega}$ varies inside $\tau$.)
\vspace{0.3cm}\noindent
{\bf Proposition 4.7.} {\it
In the notation above, the indicial ideal $Ind_{\omega}(\tau)$
coincides with the ideal generated by the indicial equations
for the indices $\rho$ of the powerseries $w_0(x_\tau,\rho;A_\tau)$. }
\noindent
{\bf (Proof)}
Consider an operator ${\cal D}_l \in {\cal B}_{\omega}$. If
${\omega}\cdot l_+ - {\omega}\cdot l_- >0$, we multiply $a^{l_+}$ to obtain
\begin{equation}
a^{l_+}{\cal D}_l =a^{l_+}\(\da{}\)^{l_+}
-a^{l_+-l_-}a^{l_-}\(\da{}\)^{l_-} \; .
\end{equation}
Since the initial ideal $ \langle LT_{\omega}({\cal I}_{\cal A}) \rangle $ and thus
the reduced Gr\"obner basis ${\cal B}_{\omega}$ does not change for
${\omega}$ in the interior of $\tau$, $Int(\tau)$, we have ${\omega}\cdot(l_+-l_-)>0$
for all ${\omega}\in Int(\tau)$, i.e., $l_+-l_-\in \tau^\vee\cap L$.
Since we have chosen the ${\bf Z}$-basis
$A_\tau=\{ \l{1}_\tau,\cdots,\l{p-d}_\tau\}$ so that it generates
all integral points in $\tau^\vee \cap L$,
$a^{l_+-l_-}$ is a monomial of $x_\tau$, which
vanish in the limit $x_\tau \rightarrow 0$. The same argument applies
to the case ${\omega}\cdot l_+ - {\omega}\cdot l_- <0$. Therefore the indicial equations
arising from the operators $D_l \in {\cal B}_{\omega}$ exactly coincide
with the generators of the indicial ideal (\ref{eqn: Jl}).
For general operators $D_l \; (l\in L)$ , depending on
the weight ${\omega} \in Int(\tau)$, we have two possible leading terms.
However for both of them,
owing to the defining property of the Gr\"obner basis,
we have $LT_{\omega}(D_l)=\(\da{}\)^\mu LT_{\omega}(D_{l_k})$
for some $k$ and $\mu$. Multiplying a monomial $a^{\mu+l_{k\pm}}$, we
obtain
\begin{equation}
a_0x_\tau^{-\rho} a^{\mu+l_{k\pm}} LT_{\omega}(D_l) a_0^{-1} x_\tau^{\rho}
=F(\rho) \, J_{l_k}(\rho;A_\tau) \;\;,
\end{equation}
with some polynomial $F(\rho)$.
Thus we see all polynomial relations of $\rho$ comming from the
leading terms are in $Ind_{\omega}(\tau)$.
\par
Conversely, since all generators of the ideal $Ind_{\omega}(\tau)$
give the indicial equations related to ${\cal B}_{\omega}$,
the ideal $Ind_{\omega}(\tau)$ is contained in the other.
Therefore the two ideals are the same. \hfill $\Box$
\vspace{0.3cm}
Now based on Prop.4.7, we may claim the following;
\vspace{0.3cm}\noindent
{\bf Proposition 4.8.} {\it
Consider a compact toric variety ${\Bbb P}_{{\cal F}({\cal I}_{\cal A})}$ associated to
the Gr\"obner fan $({\cal F}({\cal I}_{\cal A}), \Xi(\overline M))$. Then
for any resolution ${\Bbb P}_{\tilde{\cal F}({\cal I}_{\cal A})} \rightarrow {\Bbb P}_{{\cal
F}({\cal I}_{\cal A})}$ associated to a refinement $(\tilde{\cal F}({\cal I}_{\cal A}), \Xi(\overline M))
\rightarrow ({\cal F}({\cal I}_{\cal A}), \Xi(\overline M))$, we have integral powerseries
of the form $w_0(x_\tau,\rho;A_\tau)$ $(\rho\in V(Ind_{\omega}(\tau)))$
at each boundary point given by the normal
crossing toric divisors, namely at the origin of
$Hom_{s.g.}(\tau^\vee\cap L,{\Bbb C})$. We will call this compactification
Gr\"obner compactification. }
\vspace{0.3cm}\noindent
{\bf Remark.}
Since Prop.4.7 provides us only a necessary condition for the indices
$\rho$ to give a powerseries solution $w_0(x,\rho;A_\tau)$, we do not claim
by Prop.4.8, although we expect, that all $\rho\in V(Ind_{\omega}(\tau))$
form the powerseries solutions of our ${\Delta^*}$-hypergeometric system.
\newsubsec{(4-3) Resonance of ${\Delta^*}$-hypergeometric system}
When the polytope ${\Delta^*}$ is of type I or II, we have seen in
the Remark right after Th.3.8 that the
${\Delta^*}$-hypergeometric system becomes ``maximally resonant'' for a maximal
triangulation $T_o$. Here we study this resonance in detail restricting
our attention to the polytopes of type I or II. We also comment about
the case of type III.
We call a collection of vertices ${\cal P}=\{\bn{i_1},\cdots,\bn{i_a}\}$
{\it primitive} if ${\cal P}$ does not form a simplex in $T_o$ but
${\cal P}\setminus \{ \bn{i_s}\}$ does for any $\bn{i_s}\in {\cal P}$.
By definition of the Stanley-Reisner ideal, it is easy to deduce that
the monomials that corresponds to primitive collections generate the
ideal $SR_{T_o}$.
Let us denote by $\Sigma(1\times{\Delta^*},T_o)$ the fan in $\overline N_{\Bbb R}$ that
is naturally associated to the triangulation $T_o$ of $P$. Since the
volumes of all $d+1$ simplices in $T_o$ are unimodular for the polytope
${\Delta^*}$ of type I or II, the fan $\Sigma(1\times{\Delta^*},T_o)$ consists of regular
cones. Therefore if we have a primitive collection
${\cal P}=\{\bn{i_1},\cdots,\bn{i_a}\}$, we obtain
\begin{equation}
\bn{i_1}+\cdots+\bn{i_a}=\sum_k c_k \bn{j_k} \;\; (c_k \in {\bf Z}_{\geq0})
\label{eqn: prim}
\end{equation}
where $\{\bn{j_k} \vert c_k\not= 0\}$ generates a cone that contains
the vector in the left hand side.
Writing (\ref{eqn: prim}) as
$\bn{i_1}+\cdots+\bn{i_a}-\sum c_k \bn{j_k}=0$, we read
the corresponding {\it primitive relation} $l({\cal P}) \in L$.
\vspace{0.3cm}\noindent
{\bf Lemma 4.9.} {\it
Every primitive collection of a maximal triangulation $T_o$ does not
contain the point $\bn{0}=1\times \vec 0$. }
\par
\noindent
{\bf (Proof)}
Suppose a primitive collection is given by ${\cal P}=\{ \bn{0}, \bn{i_1},
\cdots,\bn{i_a} \} $ $(1\leq i_1,\cdots,i_a \leq p)$. Since it is primitive,
the simplex $ \langle \bn{i_1},\cdots,\bn{i_a} \rangle $ must be a simplex in
the triangulation $T_o$, which means that this simplex is a face of
some maximal dimensional simplex in $T_o$. Since $T_o$ is a maximal
triangulation in which every maximal dimensional simplex contains
the vertex $\bn{0}$, we see the simplex
$ \langle \bn{0},\bn{i_1},\cdots,\bn{i_a} \rangle $ must be a simplex
in $T_o$, which is a contradiction. \hfill $\Box$
\vspace{0.3cm}\noindent
{\bf Proposition 4.10.} {\it
For a term order
${\omega}$ such that $T_{\omega}$ is a maximal triangulation,
the initial ideal $ \langle LT_{\omega}({\cal I}_{\cal A}) \rangle $ is radical and
$ \langle LT_{\omega}({\cal I}_{\cal A}) \rangle =SR_{T_{\omega}}$. }
\par\noindent
{\bf (Proof)}
Consider the primitive collections for the triangulation $T_{\omega}$,
which generate the Stanley-Reisner ideal $SR_{T_{\omega}}$. Write a primitive
collection ${\cal P}=\{ \bn{i_1},\cdots,\bn{i_a}\}$ and the corresponding
primitive relation as $l({\cal P})$ considering the relation
(\ref{eqn: prim}).
For a term order ${\omega}$, the regular triangulation $T_{\omega}$
is induced from the lower faces of the polytope
$P_{\omega}={\rm Conv.}(\,\{\tilde\ns{0}, \cdots ,\tilde\ns{p}
\;\vert\; \tilde\ns{k}=$ ${\omega}_k\times\ns{k}\;(k=0,\cdots,p)\}\,)$.
Then the convex hull
${\rm Conv.}(\{\tilde\ns{i}\;\vert\; \bn{i}\in{\cal P}\})$ is not a simplex
that corresponds to a lower face of $P_{\omega}$. Therefore
we have a ``height'' inequality
$( \tilde\ns{i_1}+\cdots+\tilde\ns{i_a} )_1 > (\sum c_k \tilde\ns{j_k})_1$,
namely,
\begin{equation}
{\omega}_{i_1}+\cdots+{\omega}_{i_a} > \sum_k c_k {\omega}_{j_k} \;\;.
\label{eqn: height}
\end{equation}
This means that $LT_{\omega}(y^{l({\cal P})_+}-y^{l({\cal P})_-})=
y_{i_1}\cdots y_{i_a}$, which is one of the generators of the ideal
$SR_{T_{\omega}}$. Since this argument applies to all primitive collections,
we conclude $SR_{T_{\omega}}\subset \langle LT_{\omega}({\cal I}_{\cal A}) \rangle $. Since the opposite
inclusion follows from $SR_{T_{\omega}}=\sqrt{ \langle LT_w({\cal I}_{\cal A}) \rangle }$ (Th.4.5),
we conclude $SR_{T_{\omega}}= \langle LT_w({\cal I}_{\cal A}) \rangle $, which proves the initial
ideal is radical. \hfill $\Box$
\vspace{0.3cm}\noindent
{\bf Corollary 4.11.} {\it
Under the hypothesis in the previous proposition,
the set of all possible primitive collections
$\{ {\cal P}_1,\cdots,{\cal P}_s \}$ of $T_{\omega}$ determines the
Gr\"obner basis by ${\cal B}_{\omega}=\{ {\cal D}_{l({\cal P}_1)}, \cdots,
{\cal D}_{l({\cal P}_s)} \}$.
And the indicial ideal $Ind_{\omega}(\tau)$
is homogeneous for an arbitrary regular cone $\tau$ contained
in ${\cal C}'({\cal I}_{\cal A},{\omega})$. }
\par\noindent
{\bf (Proof)}
By definition, $SR_{T_{\omega}}$ is generated by the monomials
corresponding to primitive collections. From the argument in the
proof of Prop.4.10, we know
$SR_{T_{\omega}}= \langle LT_{\omega}( {\cal D}_{l({\cal P}_1)}),$
$\cdots, LT_{\omega}( {\cal D}_{l({\cal P}_s)}) \rangle $.
This combined with $ \langle LT_{\omega}({\cal I}_{\cal A}) \rangle =SR_{T_{\omega}}$ establishes
that ${\cal B}_{\omega}$ is the Gr\"obner basis.
{}For the rest, we note that any primitive collection ${\cal P}=
\{ \bn{i_1},\cdots,\bn{i_a}\}$ does not contain $\bn0$ according to
Lemma 4.9. Now we have
\begin{equation}
J_{l({\cal P})}(\rho;A_\tau)=a_0x_\tau^{-\rho}
a^{l({\cal P})_+} \(\da{}\)^{l({\cal P})_+} a_0^{-1}x_\tau^\rho
= x_\tau^{-\rho} \ta{{i_1}}\cdots\ta{{i_a}} x_\tau^\rho ,
\end{equation}
where $ \ta{{}} =a \da{} $, which proves that the generator
$J_{l({\cal P})}(\rho;A_\tau)$ is
homogeneous in $\rho$. \hfill $\Box$
\vspace{0.3cm}\noindent
{\bf Remark.}
If we combine a general result that the GKZ system is holonomic
\cite{GKZ1}, i.e., its solution space is finite dimensional,
with our Corol. 4.11, we may
conclude that the zero is the only solution for the indices $\rho$.
This is the maximal $T$-resonance in our approach. We will give
following \cite{HLY2} an independent proof about this in the next section.
As we remarked before, our ${\Delta^*}$-hypergeometric
system for the polytope ${\Delta^*}$ of
type III does not share this property. Here we can explain the difference.
We first note that the primitive collections
generate the Stanley-Reisner ideal and has the property in Lemma 4.9
irrespective to the type of polytopes.
The only change in the above arguments
is in the definition of the primitive relation. Namely, since all
cones are not regular in type III case, for some primitive collection
the equation (\ref{eqn: prim}) should be replaced by
\begin{equation}
\lambda_{i_1}\bn{i_1}+ \cdots + \lambda_{i_a} \bn{i_a}
= \sum_k c_k \bn{j_k} \;\;,
\label{eqn: sing}
\end{equation}
with some positive integers $\lambda_{i_1},\cdots,\lambda_{i_a}$ not all
equal to one.
Accordingly the leading term $LT_{\omega}(y^{l({\cal P})_+}-y^{l({\cal P})_-})$
will be replaced by $(y_{i_1})^{\lambda_{i_1}}\cdots
(y_{i_a})^{\lambda_{i_a}}$. This indicates that the initial ideal
$LT_{\omega}({\cal I}_{\cal A})$ is no longer radical, and therefore the generators
$J_l(\rho;A_\tau)$ become inhomogeneous. When translating the monomial
$y^{l({\cal P})_+}$ to the differential operator
$a^{l({\cal P})_+} \( \da{} \)^{l({\cal P})_+}$, each $\lambda_i$-fold
degeneration to zero 'splits' to simple zeros.
Thus every index does not degenerate to zero,
although we still have a simple zero.
\section{5. Large compex Structure Limit}
Here we will study in detail the maximal resonance of the
${\Delta^*}$-hypergeo-metric system. We will identify this resonance with the
large complex structure limit (LCSL), i.e., a celebrated boundary point
in the moduli space of Calabi-Yau manifolds\cite{Mor}.
\newsubsec{(5-1) Maximal degeneration} In this subsection, we will restrict
our arguments to the polytopes of type I or II. In these two cases,
we have a nonsingular projective toric variety
${\Bbb P}_{\Sigma( {\Delta^*},T_o)}$ for a maximal triangulation. We focus on
the Chow ring of this toric variety. The Chow ring of a variety is a
free abelian group generated by irreducible closed subvarieties, modulo
rational equivalence, which is endowed with the ring structure via the
intersection products. In case of non-singular compact toric varieties
${\Bbb P}_{\Sigma({\Delta^*},T_o)}$, it has a simple description in terms of the
(toric) divisors;
\vspace{0.3cm}\noindent
{\bf Proposition 5.1.} (sect.3.3 of \cite{Oda}, sect.5.2 of \cite{Ful})
{\it
The Chow ring $A^*({\Bbb P}_{\Sigma({\Delta^*},T_o)})$ is isomorphic to the
cohomology ring $H^{2*}({\Bbb P}_{\Sigma({\Delta^*},T_o)},{\bf Z})$ and is given by
\begin{equation}
A^*({\Bbb P}_{\Sigma({\Delta^*},T_o)})={\bf Z}[D_0,\cdots,D_p]/(SR_{T_o}+\bar R),
\end{equation}
where $D_k \; (k>0)$ represents the toric-divisor determined by
the integral point $\ns{k}$. $SR_{T_o}$ is the Stanley-Reisner
ideal and $\bar R$ is the ideal generated by linear relations
$\sum_{k=0}^p \langle 1\times u, \bn{k} \rangle D_{k} =0 \;\;(u\in M)\;. $ }
\vspace{0.3cm}\noindent
{\bf Note.}
Owing to lemma 4.9, we can take the generators of $SR_{T_o}$ that
do not contain $D_0$. Therefore the generator $D_0$ plays only a dummy
role, although it makes sense as a divisor if we consider a toric
variety defined by the (non-complete) fan
$\Sigma(1\times{\Delta^*},T_o) \subset \overline N_{\Bbb R}$.
\vspace{0.3cm}
Now consider a term order ${\omega}$ of the toric ideal ${\cal I}_{\cal A}$ and denote the
Gr\"obner basis by ${\cal B}_{\omega}=\{ {\cal D}_{l_1},\cdots,{\cal D}_{l_s}\}$. We define
\begin{equation}
I_l(\ta{{}}):=a_0 a^{l_\pm}\(\da{{}} \)^{l_\pm} a_0^{-1}
\end{equation}
for each $LT_{\omega}({\cal D}_l)=(\da{{}})^{l_\pm}$ in a similar
way to $J_l(\rho;A_\tau)$. Obviously these two are related by
$J_l(\rho;A_\tau)=x_\tau^{-\rho}I_l(\ta{{}} )x_\tau^{\rho}$.
We consider the following ideals in ${\Bbb C}[\ta0,\cdots,\ta{p}]$,
\begin{equation}
I_{\omega}:= \langle I_{l_1}(\ta{{}} ),\cdots, I_{l_s}(\ta{{}} ) \rangle \;,\;
\bar R_a:= \langle \sum_{i=0}^p \langle 1\times u, \bn{i} \rangle \ta{i}
\;\vert\; u\in M \rangle \;\;.
\end{equation}
\vspace{0.3cm}\noindent
{\bf Proposition 5.2.} {\it
For a term order ${\omega}$ of ${\cal I}_{\cal A}$ and an arbitrary regular cone $\tau$
contained in ${\cal C}'({\cal I}_{\cal A},{\omega})$, we have
\begin{equation}
{\Bbb C}[\rho]/Ind_{\omega}(\tau) \cong {\Bbb C}[\ta{{}}]/(I_{\omega}+\bar R_a) \;\;.
\end{equation} }
\par
\noindent
{\bf (Proof)}
When we take the ${\bf Z}$-basis $A_\tau=\{ \l1_\tau, \cdots,\l{p-d}_\tau
\}$, we have $\ta{i}=\sum_{k=1}^{p-d} (\l{k}_\tau)_i
\theta_{x^{(k)}_\tau}$. Then the homomorphism $\phi : {\Bbb C}[\ta{{}}]\rightarrow
{\Bbb C}[\theta_{x_\tau}]\cong{\Bbb C}[\rho]$ induced by this relation is surjective,
since rank$(L)=p-d$, and satisfies
${\rm Ker}\, \phi = \bar R_a $ and $\phi(I_{\omega})= Ind_{\omega} (\tau)$.
This proves the assertion.
\hfill $\Box$
\vspace{0.3cm}\noindent
{\bf Proposition 5.3.} {\it
Consider a term order ${\omega}$ of ${\cal I}_{\cal A}$ with $T_{\omega}$ a maximal
triangulation $T_o$. Then for any regular cone $\tau$ contained in
${\cal C}'({\cal I}_{\cal A},{\omega})$, the variety associated to the indicial ideal
$Ind_{\omega}(\tau)$ consists only one point, i.e.,
\begin{equation}
V(Ind_{\omega}(\tau))=\{0\}
\;\;.
\end{equation}
}
\par
\noindent
{\bf (Proof)}
By Corol. 4.11, we know the indicial ideal is homogeneous
for a term order ${\omega}$ of the given property. Moreover the
ideal $I_{\omega}$ coincides with the Stanley-Reisner ideal $SR_{T_o}$.
Therefore we have
\begin{equation}
{\Bbb C}[\rho]/Ind_{\omega}(\tau) \cong {\Bbb C}[\ta{{}}]/(I_{\omega}+\bar R_a)
\cong A^*({\Bbb P}_{\Sigma({\Delta^*},T_o)})\otimes{\Bbb C} \;\;,
\end{equation}
which is finite dimensional. Since the ideal is homogeneous,
the claim follows. \hfill $\Box$
We write our series (\ref{eqn: wnot}) for generic $\rho$ by $w_0(x,\rho;A)=
\sum_{m\in {\bf Z}^{p-n}_{\geq 0}} c(m+\rho)x^{m+\rho}$. As remarked after
Def.3.9, the value $\rho=0$ might cause the infinity in the numerator for
some coefficient $c(m+\rho)$. One way to treat this infinity problem is
to take the limit $\rho\rightarrow 0$ as discussed there and
we will come back to this recepie in Claim 5.8. Here following
\cite{HLY2}, we introduce the series
\begin{equation}
w_0(x_\tau,0;A_\tau)_{\geq0}:=
\sum_{m\in{\bf Z}^{p-n}_{\geq0},\; -\sum_k m_k\l{k}_\tau\geq0} c(m) x^m \;\;.
\label{eqn: wnotpositive}
\end{equation}
\vspace{0.3cm}\noindent
{\bf Theorem 5.4.} (Th.5.2 in \cite{HLY2}) {\it
For a term order ${\omega}$ with $T_{\omega}$ a maximal triangulation
and any regular cone $\tau$ contained in ${\cal C}'({\cal I}_{\cal A},{\omega})$,
the series $w_0(x_\tau,0; A_\tau)_{\geq0}$ is
the only powerseries solution of the ${\Delta^*}$-hypergeometric system
about the origin of $U_\tau=Hom_{s.g.}(\tau^\vee\cap L,{\Bbb C})$. }
\vspace{0.3cm}
To prove this theorem, we prepare the following lemma;
\vspace{0.3cm}\noindent
{\bf Lemma 5.5.} {\it
Consider a subset $S\not=\{\phi\}$ that is contained in $\tau^\vee\cap L$.
There exist an element $\delta\in S$ and a simplicial, regular cone
$C_\delta^\vee \subset L_{\Bbb R}$ such that $C_\delta^\vee$ contains
both the subset $S-\delta$ and the cone $\tau^\vee$. }
\par\noindent
{\bf (Proof)}
Consider a hyperplane $H(v;z_0)$ with a normal vector $v\in \tau$ and
passing through a point $z_0$ in $\tau^\vee$. When we consider a parallel
transport $H(v,t z_0)$ $(t\geq0)$ of the hyperplane, we may find
the minimal $t_0$ such that $H(v, t_0 z_0)\cap S \not=\{\phi\}$ while
$H(v,t z_0)\cap S =\{\phi\} \;\; (t <t_0)$. Changing the normal vector
$v$ slightly, if necessary, we may assume the intersection
$H(v, t_0 z_0)\cap S$ occurs at a point $\delta$. Now for this $\delta$,
we see that the union $U:=(\tau^\vee\cap L)
\cup (S-\delta) \setminus\{0\}$ is contained in the half space $H_>(v,0)$.
Therefore the normal cone to the set $U$ at the origin is
strongly convex, polyhedral cone. Since a strongly convex, polyhedral cone
can be inside a simplicial, regular cone, the assertion follows. \hfill $\Box$
\noindent
{\bf (Proof of Th.5.4.)}
To prove the theorem, we write the series $w_0(x_\tau,0;$ $A_\tau)$ in
terms of $a_0,\cdots,a_p$ by
\begin{equation}
w_0(a,0,\tau)=\sum_{l \in \tau^\vee\cap L} c_l a^l \;\; ,
\end{equation}
with $c_0=1$. Now suppose we have two different series of this form.
Then the difference of the two may be written by
$r(a,0,S)=\sum_{l \in S} d_l a^l $
with a subset $S\subset \tau^\vee \cap L\setminus \{0\}$. Using the
result in the lemma 5.5, we may write this series via nonzero $\delta$ as
\begin{equation}
r(a,0,C_\delta)=a^\delta \sum_{l \in C_\delta^\vee \cap L}
d_{l+\delta}a^{l} \;\;,
\label{eqn: delta}
\end{equation}
or $r(x_\tau,0,A_{C_\delta})=x_\tau^{\rho(\delta)} \sum_{n \in
{\bf Z}^{p-d}_{\geq 0}} d(n) x_\tau^n $ with $\rho(\delta)\not=0 ,d(0)\not=0$ and
$C_\delta \subset \tau$. This is a contradiction to Prop.5.3. \hfill $\Box$
\vspace{0.3cm}\noindent
{\bf Remark.}
By direct evaluation of the period integral (\ref{eqn: Pi})\cite{Bat2},
we can verify that $a_0\Pi(a)$ exactly coincides with the powerseries in
Th.5.4 when expressed in terms of the ${\bf Z}$-basis $A_\tau$ (Prop.5.15
\cite{HLY2}).
\newsubsec{(5-2) All solutions about maximal degeneration points}
Here we determine other solutions about maximal
degeneration points, all of which contains logarithmic singularities.
As in the previous subsection, our arguments are restricted to the
polytopes of type I or II.
Let us note that the first degree elements of the Chow ring,
$A^1({\Bbb P}_{\Sigma({\Delta^*},T_o)})$, describe the Picard group of the toric
variety ${\Bbb P}_{\Sigma({\Delta^*},T_o)}$ and may be expressed by
\begin{equation}
A^1({\Bbb P}_{\Sigma({\Delta^*},T_o)})={\bf Z} D_0\oplus\cdots\oplus{\bf Z} D_p / \bar R
\;\cong\; \Xi(\overline M) \;\;.
\end{equation}
From this we see a dual pairing between the Picard group and the lattice
$L\cong \Xi(\overline N)$;
\begin{equation}
A^1({\Bbb P}_{\Sigma({\Delta^*},T_o)})\times L \rightarrow {\bf Z} \;\;.
\label{eqn: dualpL}
\end{equation}
\vspace{0.3cm}\noindent
{\bf Definition 5.6.}
For a ${\bf Z}$-basis $A_\tau=\{ \l{1}_\tau, \cdots, \l{p-d}_\tau \}$ of $L$
determined from a term order ${\omega}$ with $T_{\omega}$ equal to a maximal
triangulation $T_o$, we denote its dual by
$A_\tau^\vee=\{ J_{\tau,1},\cdots,J_{\tau,p-d} \}$
or simply by $\{ J_1\cdots,J_{p-d}\}$ when its dependence on $\tau$ is
obvious.
\vspace{0.3cm}\noindent
{\bf Note.}
By construction, the basis $A_\tau^\vee$ consists of
the integral generators of the simplicial,
regular cone $\tau$ contained in ${\cal C}'({\cal I}_{\cal A},{\omega})$ ($=
{\cal C}'({\cal A},T_o)$ by Prop.4.10). ${\cal C}'({\cal A},T_o)$ consists
of convex functions on $T_o$ which may be identified with the convex
functions on the fan $\Sigma({\Delta^*},T_o)$. Since the set of all convex functions
on the fan $\Sigma({\Delta^*},T_o)$ determines the closure of the K\"ahler cone
of ${\Bbb P}_{\Sigma({\Delta^*},T_o)}$ (see Corol.2.15 in \cite{Oda}),
the bases $J_{\tau,1},\cdots, J_{\tau,p-d}$ generate a simplicial,
regular cone contained in this closure of the K\"ahler cone.
\vspace{0.3cm}{\noindent}
{\bf Definition 5.7.}
For the powerseries $w_0(x_\tau,\rho;A_\tau)=
\sum_{n\in{\bf Z}_{\geq0}^{p-d}} c(n+\rho)
x_\tau^{n+\rho}$ in (\ref{eqn: wnot}), we
define
\begin{equation}
w_0(x_\tau,J;A_\tau):= \sum_{n\in{\bf Z}_{\geq0}^{p-d}}
c\(n+{J \over 2\pi i}\) x_\tau^{n+{J\over 2\pi i}} \;\;,
\label{eqn: wJ}
\end{equation}
as the Taylor series expansion of $w_0(x_\tau,\rho;A_\tau)$ about $\rho=0$
followed by the substitution $\rho={J\over 2\pi i}$, where $J$'s
are defined in Def.5.6.
\vspace{0.3cm}
In refs.\cite{HKTY1}\cite{HKTY2}\cite{HLY1}\cite{HLY2},
it is widely verified
\vspace{0.3cm}\noindent
{\bf Claim 5.8.} {\it
The expansion (\ref{eqn: wJ}) exists as an element in
$A^*({\Bbb P}_{\Sigma({\Delta^*},T_o)})\otimes {\Bbb C}\{x_\tau\}[\log x_\tau]$,
and the coefficient series constitute a complete set of
the local solutions about the maximal degeneration points.
Especially the limit $w_0(x_\tau,\rho;A_\tau)\vert_{\rho\rightarrow0}$
coincides with $w_0(x_\tau,0; A_\tau)_{\geq0}$ in Th.5.4. }
\vspace{0.3cm}\noindent
{\bf Remark.}
We comment about the case of the polytopes of type III.
In this case, since the toric variety ${\Bbb P}_{\Sigma({\Delta^*},T_o)}$ is singular,
the Chow ring should be considered over ${\Bbb Q}$. Under this modification the
expansion (\ref{eqn: wJ}) makes sense in
$A^*({\Bbb P}_{\Sigma({\Delta^*},T_o)})_{\Bbb Q} \otimes {\Bbb C}\{x_\tau\}[\log x_\tau]$.
Then the coefficient series should be in a subspace of
the whole solution space of the ${\Delta^*}$-hypergeometric system. More
precisely, as we see in Remark after Corol. 4.11, the initial ideal
$LT_{\omega}({\cal I}_{\cal A})$ is no longer radical but we have a strict inclusion
$LT_{\omega}({\cal I}_{\cal A}) \subset \sqrt{LT_{\omega}({\cal I}_{\cal A})}$.
As discussed there, we have
$LT_{\omega}({\cal D}_l)=(\da{i_1})^{\lambda_{i_1}}
\cdots (\da{i_a})^{\lambda_{i_a}}$
for some element of the Gr\"obner basis ${\cal B}_{\omega}=\{ {\cal D}_{l_1},
\cdots, {\cal D}_{l_s} \}$. If we define ${\rm rad}(LT_{\omega}({\cal D}_l))
:=\da{i_1}\cdots\da{i_a}$,
then the radical may be expressed by
$\sqrt{LT_{\omega}({\cal I}_{\cal A})}= \langle {\rm rad}(LT_w({\cal D}_{l_1})),
\cdots, {\rm rad}(LT_w({\cal D}_{l_s})) \rangle $. Correspondingly, if we define
$\tilde I_{\omega}(\ta{{}}):=
a_0 a_{i_1}\cdots a_{i_a} {\rm rad}(LT_{\omega}({\cal D}_l)) a_0^{-1}$,
we naturally come to the ``radical'' of the indicial ideal
$\tilde{Ind}_{\omega}(\tau):= \langle \tilde J_{l_1}(\rho;A_\tau), \cdots, \tilde
J_{l_s}(\rho;A_\tau) \rangle $ with $\tilde J_l(\rho;A_\tau):=x_\tau^{-\rho}\tilde
I_{\omega}(\ta{{}}) x_\tau^{\rho}$. By definition, we have strict inclusions
$Ind_{\omega}(\tau)$ $\subset \tilde{Ind}_{\omega}(\tau)$ and $V(Ind_{\omega}(\tau)) \supset
V(\tilde{Ind}_{\omega}(\tau))$. As is clear now, our Prop.5.2 and Prop.5.3
apply to the ``radical'' $\tilde{Ind}_{\omega}(\tau)$ under the replacements
$I_{\omega}$ by $\tilde I_{\omega}$ and the Chow ring by that over ${\Bbb Q}$.
The expansion (\ref{eqn: wJ}) gives all logarithmic solutions
which arise from the degeneration $V(\tilde{Ind}_{\omega}(\tau))$ $=\{ 0\}$.
\newsubsec{(5-3) LCSL of Calabi-Yau hypersurfaces}
So far we have been concerned with the ${\Delta^*}$-hypergeometric system.
Since the period integral
(\ref{eqn: Pi}) of Calabi-Yau hypersurface $X_{\Delta^*}$ satisfies the
(extended) ${\Delta^*}$-hypergeometric system, a complete set of
the period integrals of $X_{\Delta^*}$
should be found in the set of solutions of the ${\Delta^*}$-hypergeometric system.
We will find that the expansion (\ref{eqn: wJ}) contains the period integrals
in a natural way from the mirror symmetry.
Before going into this topic, we need to discuss about the compactification
of the moduli space ${\cal M}(X_{\Delta^*}(a))$ of the polynomial deformation
of the Calabi-Yau hypersurface $X_{\Delta^*}$. Through a detailed analysis of
the local solutions of the ${\Delta^*}$-hypergeometric system, we have arrived
at a natural compactification, the Gr\"obner compactification
${\Bbb P}_{{\cal F}({\cal I}_{\cal A})}$ in Prop.4.8. Now it is natural to adopt this
compactification as that of the moduli space ${\cal M}(X_{\Delta^*}(a))$.
However, one problem arises when the hypersurface (, precisely its ambient
space,) has non-trivial automorphisms. We need to mod out the space
${\Bbb P}_{{\cal F}({\cal I}_{\cal A})}$ by the induced actions from the automorphisms,
whose infinitesimal forms are described in (\ref{eqn: extGKZ}).
Here to avoid getting involved in the problems related to the
actions of the automorphisms, we take a ``gauge choice'' that sets to
zero all polynomial deformations corresponding to integral points
on codimension-one faces of ${\Delta^*}$. Note that, in view of Prop.2.6,
the degree of the freedom associated to the non-trivial automorphisms
would be fixed by this gauge choice.
In the following, we use the subscript $s$
(, $s$ of simply-minded!,) to indicate this
naive choice of the ``gauge''; for example $\Delta_s^*$-hypergeometric system,
the toric ideal ${\cal I}_{{\cal A}_s}$ etc. Note that all the polytopes of type II
will be treated as the polytopes of type III under this prescription.
\vspace{0.3cm}\noindent
{\bf Definition 5.9.}
As an compactification of ${\cal M}(X_{\Delta^*}(a))$, we define
\begin{equation}
\overline{\cal M}(X_{{\Delta^*}}(a))={\Bbb P}_{{\cal F}({\cal I}_{{\cal A}_s})} \;\;.
\end{equation}
\vspace{0.3cm}
Now we consider the toric part of the Chow ring of the Calabi-Yau
hypersurface $X_{{\Delta}_s}$, which comes from the ambient space by
restriction. Since we have $[X_{{\Delta}_s}]=D_1+\cdots+D_p$ for the divisor of
the Calabi-Yau hypersurface, the restriction may be attained by the
quotient as follow;
\vspace{0.3cm}\noindent
{\bf Definition 5.10.}
\begin{equation}
A^*(X_{{\Delta}_s})_{toric}=
A^*({\Bbb P}_{\Sigma(\Delta_s^*,T_o)})_{\Bbb Q}/ Ann(D_1+\cdots+D_p) \;\;,
\label{eqn: chowX}
\end{equation}
where $Ann(x)$ is defined by
$Ann(x)=\{ \; y\in {\cal R} \;\vert\; x\,y=0 \;\}$ for a ring ${\cal R}$.
\vspace{0.3cm}\noindent
{\bf Claim 5.11.} {\it
Period integrals about a LCSL of Calabi-Yau hypersurface $X_{\Delta_s^*}$ are
extracted from the series $w_0(x_\tau,J;A_\tau)$ (\ref{eqn: wJ})
expanded in $A^*(X_{{\Delta}_s})_{toric}\otimes {\Bbb C}\{x_\tau\}[\log x_\tau]$. }
\vspace{0.3cm}\noindent
{\bf Remark.}
In general, the period integrals of Calabi-Yau hypersurfaces satisfy the
differential equations of Fuchs type, so-called the Picard-Fuchs equations
\cite{PF}.
Picard-Fuchs equations determines the period integrals as its solutions.
Our Claim 5.11 says that our ${\Delta^*}$-hypergeometric system is reducible in
general and contains the Picard-Fuchs equation as a component of it.
In refs.\cite{HKTY1}\cite{HLY1}, it is verified in several examples
of all types of the polytopes that Picard-Fuchs
equations are derived from the (extended) ${\Delta^*}$-hypergeometric system
after a factorization of the operator $\ta{1}+\cdots+\ta{p}$ from the
left, which we may identify with the quotient by $Ann(D_1+\cdots+D_p)$ in
(\ref{eqn: chowX}).
\section{6. Prepotential}
In this section, we study so-called the
prepotential \cite{Saito} near a LCSL in detail.
Under the mirror map, a LCSL is mapped to a large
radius limit in which the instanton corrections to the prepotential
are suppressed exponentially, and has important applications to
the enumerative geometry. Also the prepotential determines the
{\it special K\"ahler geometry} on the moduli space
$\overline {\cal M}(X_{\Delta_s^*})$ and
that on the complexified K\"ahler moduli space of the
mirror $X_{{\Delta}_s}$.
In this section, we fix a term order ${\omega}$ for which $T_{\omega}$ is a maximal
triangulation of $\Delta_s^*$ and take the ${\bf Z}$-basis $A_\tau$ choosing a
regular cone $\tau$ in ${\cal C}'({\cal I}_{{\cal A}_s},{\omega})$.
Based on Claim 5.11,
we expand the series $w_0(x_\tau,J;A_\tau)$ defined in (\ref{eqn: wJ})
(see also \cite{Sti}) as;
\begin{equation}
w_0(x_\tau,J;A_\tau)=
w^{(0)}(x_\tau,J)+w^{(1)}(x_\tau,J)+{1\over2!}w^{(2)}(x_\tau,J)
+{1\over3!}w^{(3)}(x_\tau,J) \;,
\label{eqn: wJexp}
\end{equation}
where the superscripts indicate the degree in the Chow ring
$A^*(X_{{\Delta}_s})_{toric}\otimes {\Bbb C}\{x_\tau\}[\log x_\tau]$.
\vspace{0.3cm}\noindent
{\bf Definition 6.1.}
The {\it special coordinate} $(t_1,\cdots, t_{p-d})$ of
the special K\"ahler geometry is defined
by the ratios of the period integrals;
\begin{equation}
t{\hskip-0.02cm\cdot\hskip-0.02cm} J = {w^{(1)}(x_\tau,J) \over w^{(0)}(x_\tau,J) } \;\;,
\label{eqn: special}
\end{equation}
where we abuse the letters $J_1, \cdots, J_{p-d}$ to represent the images
of the $J$'s under the quotient (\ref{eqn: chowX}).
The inverse series of this relation will be called the {\it mirror map}.
\vspace{0.3cm}\noindent
{\bf Note.} Since $w^{(1)}(x_\tau,J)$ is linear in log$x_\tau$, the mirror
map takes the form $x_\tau(q):=x_\tau(q_1,\cdots,q_{p-d})$ with
$q_k:={\rm e}^{2\pi i t_k}$. It is easy to see that $x_\tau^{(k)}(q)
=q_k(1+{\cal O}(q))$.
\vspace{0.3cm}\noindent
{\bf Definition 6.2.}
We define the prepotential in the special coordinate by
\begin{equation}
F(t):=\int_{X_{{\Delta}_s}} {\cal F}(x_\tau(q), J) \;\;,
\label{eqn: Ft}
\end{equation}
with the {\it invariant} density
\begin{equation}
{\cal F}(x_\tau(q), J)={1\over2} \({1\over w^{(0)}}\)^2
\{w^{(0)}(-{1\over3!}w^{(3)}-{c_2(X_{{\Delta}_s})\over 12}w^{(1)})+
w^{(1)}({1\over2!}w^{(2)}) \} \;.
\label{eqn: Fd}
\end{equation}
The integration symbol
$\int_{X_{{\Delta}_s}} :=\int_{{\Bbb P}_{\Sigma({\Delta_s^*},T_o)}}
[X_{{\Delta}_s}]$ is meant to take the coefficient of the 'volume form' in the
Chow ring $A^*({\Bbb P}_{\Sigma(\Delta_s^*,T_o)})_{\Bbb Q}$ normalized by
$\int_{{\Bbb P}_{\Sigma({\Delta_s^*},T_o)}} [X_{{\Delta}_s}]c({\Bbb P}_{\Sigma(\Delta_s^*,T_o)})/(1+[X_{{\Delta}_s}])=\chi(X_{{\Delta}_s})$. (For the normalization when
$\chi(X_{{\Delta}_s})=0$, see ref.\cite{HLY1}.)
\vspace{0.3cm}\noindent
{\bf Note.}
It would be instructive to summarize the general description\cite{Str} of
the special K\"ahler geometry on the complex structure moduli space of
Calabi-Yau threefolds. Let us denote the holomorphic 3-form of
a family of Calabi-Yau threefolds $W_\psi$ by $\Omega(\psi)$.
We take a symplectic basis $\{A_a,B_b\}$ $(a,b=0,\cdots,h^{2,1}(W))$
of $H_3(W,{\bf Z})$ and construct the period integrals $z_a(\psi)=\int_{A_a}
\Omega(\psi)$ and ${\cal G}_b(\psi)=\int_{B_b}\Omega(\psi)$.
Then the holomorphic 3-form may be written by $\Omega(\psi)=\sum_a z_a(\psi)
\alpha_a + \sum_b {\cal G}_b(\psi)\beta_b$ in terms of the dual
bases $\alpha_a$ and $\beta_b$ in $H^3(W,{\bf Z})$. Locally we can introduce
on the moduli space a K\"ahler metric,
so-called the Weil-Peterson metric \cite{Tian},
through the K\"ahler potential $K(\psi,\bar\psi)=-\log i\int_M \Omega(\psi)
\wedge \bar\Omega(\psi)$ $= -\log i\sum_a (z_a(\psi)
\overline{ {\cal G}_a(\psi)}$
$- {\cal G}_a(\psi) \overline{ z_a(\psi)} )$.
It is shown in ref.\cite{Str} that the prepotential
$F(\psi)=$
${1\over 2}\sum_a z_a(\psi){\cal G}_a(\psi)$ describes the potential
$K(\psi,\bar\psi)$ by
\begin{equation}
K(\psi,\bar\psi)=
-{\rm log}i\sum_a \( z_a(\psi)\overline{{\partial F(\psi)\over \partial z_a}}
-\overline{z_a(\psi)} {\partial F(\psi) \over \partial z_a} \) \;,
\end{equation}
and defines the {\it special K\"ahler geometry} on the moduli space.
Our definition (\ref{eqn: Fd}) of the prepotential, up to the prefactor
$(w^{(0)})^{-2}$ which makes the prepotential invariant under
$\Omega(\psi)\mapsto f(\psi)\Omega(\psi)$, implicitly contains a claim that
$(w^{(0)},w^{(1)},{1\over2!}w^{(2)},-{1\over3!}w^{(3)}-
{c_2(X_{{\Delta}_s}) \over 12} w^{(1)})$ form the period integrals for a symplectic
basis of $H_3(X_{\Delta_s^*},{\bf Z})$. Several evidences for this claim are
reported in \cite{HLY3}.
In the following, we will restrict our attention to the form of
the prepotential near a LCSL assuming its application to the enumerative
geometry (the instanton counting).
\vspace{0.3cm}
Now consider the following expansion in the Chow ring associated to
the series
$w_0(x_\tau,0,A_\tau)=\sum_{n\in {\bf Z}_{\geq0}^{p-d}}c(n)x_\tau^n$;
\begin{equation}
\sum_{n\in {\bf Z}^{p-d}_{\geq0}}c\(n+{J\over 2\pi i}\)x_\tau^n=
\tilde w^{(0)}(x_\tau,J)+\tilde w^{(1)}(x_\tau,J)+
{1\over2!}\tilde w^{(2)}(x_\tau,J)
+{1\over3!}\tilde w^{(3)}(x_\tau,J) \;.
\label{eqn: wJexpp}
\end{equation}
\vspace{0.3cm}\noindent
{\bf Lemma 6.3.} {\it
The two definitions of the series (\ref{eqn: wJexp}) and (\ref{eqn: wJexpp})
are related by
\begin{equation}
\begin{array}{crl}
w^{(0)}&=&\tilde w^{(0)} \;, \\
w^{(1)}&=&(\log x{\hskip-0.02cm\cdot\hskip-0.02cm} \hat J) w^{(0)} + \tilde w^{(1)} \;,\\
w^{(2)}&=&(\log x{\hskip-0.02cm\cdot\hskip-0.02cm} \hat J)^2 w^{(0)} +
2(\log x{\hskip-0.02cm\cdot\hskip-0.02cm} \hat J)\tilde w^{(1)} + \tilde w^{(2)}\;,\\
w^{(3)}&=&(\log x{\hskip-0.02cm\cdot\hskip-0.02cm} \hat J)^3 w^{(0)} +
3(\log x{\hskip-0.02cm\cdot\hskip-0.02cm} \hat J)^2\tilde w^{(1)} +
3(\log x{\hskip-0.02cm\cdot\hskip-0.02cm} \hat J) \tilde w^{(2)} +
\tilde w^{(3)}\;,\\
\end{array}
\end{equation}
where we have introduced an abbreviation $\log x{\hskip-0.02cm\cdot\hskip-0.02cm}\hat J=
\sum_{a=1}^{p-d} (\log x_a){1\over 2\pi i}J_a$. }
\vspace{0.3cm}\noindent
{\bf Lemma 6.4.} {\it
The series $\tilde w^{(d)}(x_\tau,J) \;\;(d=1,2,3)$ in (\ref{eqn: wJexpp})
have the form
\begin{equation}
\begin{array}{crl}
\tilde w^{(1)}(x_\tau,J)&=&\sum_n c(n) \Psi^{(1)}(n) x_\tau^n \;\;, \\
\tilde w^{(2)}(x_\tau,J)&=&\sum_n c(n)
\{(\Psi^{(1)}(n))^2+\Psi^{(2)}(n)\}x_\tau^n \;\;, \\
\tilde w^{(3)}(x_\tau,J)&=&\sum_n c(n) \{(\Psi^{(1)}(n))^3+
3\Psi^{(1)}(n)\Psi^{(2)}(n)+\Psi^{(3)}(n)\} x_\tau^n \;\;, \\
\end{array}
\label{eqn: wPsi}
\end{equation}
where $\Psi^{(k)}(n)$'s are elements in the Chow ring of degree $k$
defined by
\begin{equation}
\begin{array}{crl}
&&\Psi^{(1)}(n)
=-(\hat J \cdt l_0)\psi(1-n \cdt l_0)-\sum_{i=1}^{p} (\hat J \cdt l_i) \psi(1+n \cdt l_i) \;,\\
&&\Psi^{(2)}(n)=(\hat J \cdt l_0)^2\psi'(1-n \cdt l_0)-
\sum_{i=1}^{p}(\hat J \cdt l_i)^2\psi'(1+n \cdt l_i) \;, \\
&&\Psi^{(3)}(n)=-(\hat J \cdt l_0)^3\psi''(1-n \cdt l_0)-
\sum_{i=1}^{p}(\hat J \cdt l_i)^3\psi''(1+n \cdt l_i) \;, \\
\end{array}
\end{equation}
with $\hat J{\hskip-0.02cm\cdot\hskip-0.02cm} l_k =
\sum_{a=1}^{p-d} {J_a \over 2\pi i} l^{(a)}_k \; (k=0,\cdots, p)$,
$\psi(z)={d \; \over dz}\log\Gamma(z)$ and the derivatives of $\psi(z)$. }
\vspace{0.3cm}\noindent
{\bf Lemma 6.5.} {\it
\begin{equation}
\Psi^{(1)}(0)=0 \;\;,\;\;
\Psi^{(2)}(0)=-{c_2(X_{{\Delta}_s})\over 12} \;\;,\;\;
\Psi^{(3)}(0)=-{6\zeta(3) \over (2\pi i)^3} c_3(X_{{\Delta}_s}) \;\;.
\end{equation} }
\par
\noindent
{\bf (Proof)}
These constant terms originate from those of the $\psi$-functions;
$\psi(1)=-\gamma \;,\; \psi'(1)={\pi^2 \over 6} \;,\;
\psi''(1)=-2 \zeta(3)$. These values of the $\psi$-functions combined
with the adjunction formula for the total Chern class, with
$D_i=J{\hskip-0.02cm\cdot\hskip-0.02cm} l_i$ under the rational equivalence in the Chow ring,
\begin{equation}
c(X_{{\Delta}_s})={\prod_{i=1}^p(1+D_i) \over 1+[X_{{\Delta}_s}] }
={\prod_{i=1}^p(1+J{\hskip-0.02cm\cdot\hskip-0.02cm} l_i) \over 1-J{\hskip-0.02cm\cdot\hskip-0.02cm} l_0} \;,
\end{equation}
result in our claim for the leading terms.
(Note that $c_1(X_{{\Delta}_s})=0$ for $\Psi^{(1)}(0)$.) \hfill $\Box$
\vspace{0.3cm}\noindent
{\bf Remark.}
We can subtract these constant terms $\Psi^{(k)}(0)$ in a systematic way
modifying the expansion (\ref{eqn: wJexpp}) slightly as follows;
\begin{equation}
\sum_{n\in {\bf Z}^{p-d}_{\geq 0}}
{c(n+{J\over2\pi i}) \over c({J\over2\pi i})} x_\tau^n
= w^{(0)}(x_\tau)+\tilde w_r^{(1)}(x_\tau,J)+
{1\over 2}\tilde w_r^{(2)}(x_\tau,J)
+ {1\over 3!} \tilde w_r^{(3)}(x_\tau,J) \;.
\end{equation}
This is because this change of normalization in the series $w_0$
simply results in the replacement $\Psi^{(k)}(n)$ with
$\Psi^{(k)}_r(n):= \Psi^{(k)}(n)-\Psi^{(k)}(0)$ in (\ref{eqn: wPsi}).
\vspace{0.3cm}
Now it is immediate from Lemmas 6.4 and 6.5 to obtain
\vspace{0.3cm}\noindent
{\bf Lemma 6.6.}
\begin{eqnarray*}
\tilde w^{(1)}&=&\tilde w^{(1)}_r \;\;,\;\;
\tilde w^{(2)}=-{c_2(X_{{\Delta}_s}) \over 12}w^{(0)} + \tilde w_r^{(2)} \;,\\
\tilde w^{(3)}&=&-{6\zeta(3) \over (2\pi i)^3} c_3(X_{{\Delta}_s})w^{(0)}
-{c_2(X_{{\Delta}_s}) \over 4} \tilde w_r^{(1)} + \tilde w_r^{(3)} . \\
\end{eqnarray*}
\vspace{-2.5cm}
\begin{equation}
\label{eqn: wr} \end{equation}
\par\vspace{0.8cm}\noindent
\vspace{0.3cm}
Now using the results in Lemmas 6.3-6.6, we may arrive at our final
form of the prepotential, see also \cite{Sti}, modulo the kernel of
the integration $\int_{X_{{\Delta}_s}}$ in Def.6.2;
\vspace{0.3cm}\noindent
{\bf Proposition 6.7.} {\it
The invariant form of the prepotential ${\cal F}(x,J)$
may be expressed by
\begin{eqnarray}
{\cal F}(t,J)
&=&{1\over6}(t{\hskip-0.02cm\cdot\hskip-0.02cm} J)^3-{c_2(X_{{\Delta}_s}) \over 24}(t{\hskip-0.02cm\cdot\hskip-0.02cm} J)+
{\zeta(3) \over 2(2\pi i)^3}c_3(X_{{\Delta}_s}) \nonumber \\
&& \quad\quad -{1\over 2}\log \left( \sum_{n\in {\bf Z}_{\geq0}^{p-d}}
{ c(n+{J\over 2\pi i}) \over c({J\over 2\pi i}) } x_\tau^n \right) \;\;,
\nonumber \\
\end{eqnarray}
with the mirror map $x_\tau=x_\tau(q)$. }
\vspace{0.3cm}\noindent
{\bf Claim 6.8.} {\it
Three times derivatives of the prepotential give the
instanton corrected Yukawa couplings;
\begin{eqnarray}
&&K_{t_a t_b t_c}(t)={\partial^3 \;\; \over \partial t_a \partial t_b \partial t_c} F(t)
\nonumber \\
&&= \int_{X_{{\Delta}_s}} J_aJ_bJ_c +
\sum_{ \Gamma\in H_2(X_{{\Delta}_s},{\bf Z}) \atop
\Gamma\not=0 } (\Gamma{\hskip-0.02cm\cdot\hskip-0.02cm} J_a)(\Gamma{\hskip-0.02cm\cdot\hskip-0.02cm} J_b)
(\Gamma{\hskip-0.02cm\cdot\hskip-0.02cm} J_c) N(\Gamma)
{{\rm e}^{2\pi i \Gamma{\hskip-0.02cm\cdot\hskip-0.02cm} (t{\hskip-0.02cm\cdot\hskip-0.02cm} J)} \over 1-
{\rm e}^{2\pi i \Gamma{\hskip-0.02cm\cdot\hskip-0.02cm} (t{\hskip-0.02cm\cdot\hskip-0.02cm} J)}} \;, \nonumber \\
\end{eqnarray}
where $\Gamma{\hskip-0.02cm\cdot\hskip-0.02cm} J:= \int_\Gamma J$ and $N(\Gamma)$ counts the number of
the rational curves of class $\Gamma$ on the Calabi-Yau manifolds $X_{{\Delta}_s}$.}
\vspace{0.3cm}\noindent
{\bf Note.}
Since the mirror map has the $q$-expansion
$x_\tau^{(k)}(q)=q_k(1+{\cal O}(q))$,
it is immediate to deduce that the number of lines $N(\Gamma)$ in
$X_{{\Delta}_s}$ is counted by
\begin{equation}
N(\Gamma)=\int_{X_{{\Delta}_s}} -{1\over2}{c((\Gamma{\hskip-0.02cm\cdot\hskip-0.02cm} J)+J) \over c(J)} \;\;.
\end{equation}
We see that the famous number $2785$ for the quintic in ${\Bbb P}^4$ \cite{CdGP}
is counted by this formula as
\begin{equation}
N(1)=-{1\over2}\int_{{\Bbb P}^4} 5J {(5+5J)(4+5J)(3+5J)(2+5J)(1+5J)
\over (1+5J)^5}\;\;.
\end{equation}
The invariant form of the prepotential ${\cal F}(x,J)$ may have significant
applications to extracting the predicted numbers of the rational curves
$N(\Gamma)$. In a recent work \cite{HSS}, this form has been utilized
efficiently to verify that the numbers $N(\Gamma)$ of a certain Calabi-Yau
manifold (Schoen's Calabi-Yau manifold) are related to the modular forms,
the theta function of the $E_8$ lattice and Dedekind's eta function.
\def\thebibliography#1{\vskip 1.2pc{\centerline {\bf References}}\vskip 4pt
\list
{[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
\advance\leftmargin\labelsep
\usecounter{enumi}}
\def\hskip .11em plus .33em minus .07em{\hskip .11em plus .33em minus .07em}
\sloppy\clubpenalty4000\widowpenalty4000
\sfcode`\.=1000\relax}
\let\endthebibliography=\endlist
|
1997-07-31T12:26:47 | 9707 | alg-geom/9707017 | en | https://arxiv.org/abs/alg-geom/9707017 | [
"alg-geom",
"math.AG"
] | alg-geom/9707017 | Andre. Hirschowitz | A. Hirschowitz and S. Ramanan | New evidence for Green's conjecture on syzygies of canonical curves | Tex-type: LaTeX | null | null | null | null | We prove that two weakened forms of Green's conjectures for canonical curves
are equivalent when the genus $g$ is odd.
| [
{
"version": "v1",
"created": "Thu, 31 Jul 1997 11:27:37 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Hirschowitz",
"A.",
""
],
[
"Ramanan",
"S.",
""
]
] | alg-geom | \section{Introduction}
Some twelve years ago, Mark Green [G] made a few conjectures
regarding the behaviour of syzygies of a curve $C$ imbedded in
$\mbox{$I\!\!P$}^n$ by a complete linear system. The so-called {\it generic} Green
conjecture on canonical curves pertains to this question when
the linear system is the canonical one and the curve is generic
in the moduli, and predicts what are the numbers of syzygies in that
case. Green and Lazarsfeld [GL] have observed that curves with
nonmaximal Clifford index have extra syzygies and
we will call
{\it specific} Green
conjecture on canonical curves the stronger prediction that the
curves which have the numbers of
syzygies expected for generic curves
are precisely those with maximal Clifford index ($[(g-1)/2]$).
(As a matter of fact, the full Green
conjecture on canonical curves relates more closely the Clifford
number
with the existence of extra syzygies.)
Many attempts have been made to settle
this question, and some nice results have been obtained ([Sch][V]).
In this note, we work over an algebraically closed field of
arbitrary
characteristic and prove that, as stated above, the generic and specific
Green conjectures for canonical curves are equivalent at least when
the genus $g$ is odd.
Let $C$ be a curve canonically imbedded in $\mbox{$I\!\!P$}^{g-1}$
with ideal sheaf
${\cal
I}_C$. We denote by $Q$ the universal quotient on
$\mbox{$I\!\!P$}^{g-1}$, so that $Q(1)$ is the tangent bundle, and by $Q_C$ its
restriction to $C$.
It is generally known (see [P-R] for example)
that extra syzygies appear when,
for some $i\leq [(g-1)/2]$, the natural map
$\Lambda ^i (\Gamma (C,Q_C)) \rightarrow \Gamma (C, \Lambda
^iQ_C)$ is not surjective. It is easy to see that the relevant quotient
of $\Gamma (C, \Lambda
^iQ_C)$ by $\Lambda ^i (\Gamma (C,Q_C))$ is isomorphic to
$\Gamma (\Lambda^{i+1}Q \otimes {\cal I}_C(1))$ (cf 2.1).
\begin{thm} Let $g = 2k-1 \geq 5$ be an odd integer. If the generic curve $C$
of
genus $g$ has the expected number of syzygies (i.e.
$\Gamma (\Lambda ^{k} Q \otimes {\cal I}_C(1))=0$),
then so does any curve of genus $g$ with maximal Clifford index, namely
$k-1.$
\end{thm}
To prove this,
we compute
a virtual (divisor) class $v$ for the
locus (in the moduli) of curves $C$ for which
the cohomology group
$\Gamma (\Lambda^{k} Q \otimes {\cal I}_C(1))$ does not vanish.
Once $v$ is computed, we compare it with
the class $c$ of
the locus of $k$-gonal curves (these are
curves with non-maximal Clifford index in our
case), which,
thanks to Harris and Mumford [HM], is already known,
and we find that $v=(k-1)c$. We conclude by proving that the
generic $k$-gonal curve has at least $k-1$ extra syzygies, which
implies that the $k$-gonal locus
occurs with multiplicity
$k-1$ in $v$,
leaving
no room for another component. Our proof gives another
consequence
of the generic Green's conjecture, namely that the number of extra
syzygies (more precisely
$h^0(\Lambda^{k} Q \otimes {\cal I}_C(1))$)
is exactly $k-1$ for any $k$-gonal curve $C$ in the smooth part of the
$k$-gonal locus.
Finally, we observe
that our argument fails completely in the case of even genus, where the
expected codimension of our jump locus is no more one.
\section { Preliminaries on syzygies} \label {prel}
We collect here a few useful remarks on syzygies.
\begin{prop} Let $S$ be a linearly normal
subscheme of $\mbox{$I\!\!P$}^n $
(i.e. $\Gamma (\mbox{$I\!\!P$}^n,{\cal O}(1)) \to
\Gamma (S, {\cal O}(1))$ is an isomorphism) with ideal sheaf ${\cal
I}_S$.
Then the cokernel of
$$
\Lambda^ i\Gamma (\mbox{$I\!\!P$}^n,Q) = \Gamma (\mbox{$I\!\!P$}^n, \Lambda ^i Q)\rightarrow
\Gamma (S,\Lambda^i Q_S)
$$
is canonically isomorphic to
$$\Gamma (\mbox{$I\!\!P$}^n,\Lambda^{i+1} Q\otimes {\cal I}_S(1)).$$
\end{prop}
\noindent{\bf Proof.}
Consider the following exact sequence of sheaves on $P(V) =
\mbox{$I\!\!P$}^n$:
$$
0\rightarrow {\cal O}(-1) \rightarrow V_P \rightarrow Q
\rightarrow 0
$$
By taking the exterior $(i+1)$-th power and tensoring with ${\cal
O}(1)$, we get the exact sequence:
$$0\rightarrow \Lambda^{i}Q \rightarrow \Lambda^{i+1} V_P (1)\rightarrow
\Lambda^ {i+1}Q (1) \rightarrow 0.
$$
This exact sequence of vector bundles remains
exact on tensorisation by ${\cal I}_S$ as well as ${\cal O}_S$.
Thus we get the commutative diagram
$$\begin{array}{ccccccccc}
& & 0 & & 0 & & 0 & &\\
& & \downarrow & & \downarrow & & \downarrow & &\\
0 & \rightarrow & \Lambda^i Q \otimes {\cal I}_S & \rightarrow &
\Lambda^{i+1} V_P \otimes {\cal I}_S(1) & \rightarrow & \Lambda^{i+1} Q
\otimes {\cal I}_S(1) & \rightarrow & 0 \\
& & \downarrow & & \downarrow & & \downarrow & &\\
0 & \rightarrow & \Lambda^i Q & \rightarrow & \Lambda^{i+1} V_P (1) &
\rightarrow & \Lambda^{i+1} Q(1) & \rightarrow & 0\\
& & \downarrow & & \downarrow & & \downarrow & &\\
0 & \rightarrow & \Lambda^i Q_S & \rightarrow & \Lambda^{i+1} V_S(1)
& \rightarrow & \Lambda^{i+1} Q_S(1) & \rightarrow & 0 \\
& & \downarrow & & \downarrow & & \downarrow & &\\
& & 0 & & 0 & & 0 & &
\end{array}
$$
Now apply the section functor $\Gamma $ : the middle row remains
exact. Thus we may apply the snake lemma to the two lower rows. This
yields the desired isomorphism because under our
assumption, $\Gamma (\Lambda ^{i+1}V_P(1)) \to \Gamma (\Lambda
^{i+1}V_S(1))$ is an isomorphism as well. $\hfill\square$
\begin{rem}
Thus we will think of $\Gamma (\Lambda ^j Q \otimes {\cal I}_C(1))$
as the space of extra syzygies. From this point of view,
extra syzygies behave in a monotonic way with respect to the
degree $j$ and the subvariety $C$:
a) If $C \subset S$ are two subvarieties of $\mbox{$I\!\!P$}^{g-1}$,
then $h^0 (\Lambda ^j Q \otimes {\cal I}_C(1))\geq
h^0 (\Lambda ^j Q \otimes {\cal I}_S(1))$. We will estimate
syzygies of our canonical curves by using a scroll $S$ containing
them.
b) If $i < j$, then $h^0 (\Lambda ^j Q \otimes {\cal I}_C(1))\geq
h^0 (\Lambda ^i Q \otimes {\cal I}_C(1)).$
The above proposition is applicable with our canonical curve: $S = C$.
Also, the Clifford index of the generic curve of genus $g$, is well-known
to be $[(g-1)/2]$. Finally, if $ i < j$ then
$\Gamma (\Lambda ^j Q \otimes {\cal I}_C(1))=0$ implies
$\Gamma (\Lambda ^i Q \otimes {\cal I}_C(1))=0$
so that we have an equivalent formulation of the
specific conjecture of Green:
\noindent{\bf Specific Green's conjecture.} {\it Let $C$ be a
canonically imbedded curve with maximum Clifford index
$[(g - 1)/2]$. Then
$\Gamma (\Lambda ^j Q \otimes {\cal I}_C(1))$ is zero for
$j = [(g+1)/2]$.}
\end {rem}
We will use in Section \ref {scroll} the following semi-continuity statement:
\begin{prop} \label {scs} Let $p: W \rightarrow T$
be a smooth family of projective
varieties parametrized by the spectrum $T$ of a discrete valuation
ring. We suppose that $W$ is
endowed with a line bundle ${\cal L}$,
that $h^0(W_t, {\cal L}_t)$ is constant and that
for each point $t \in T$, ${\cal L}_t$ is generated by global
sections. This
yields a $T$-morphism $m$ from $W$ to $\mbox{$I\!\!P$}(p_*{\cal L})$. We denote
by $I_t$
the ideal sheaf of $m(W_t)$ and by $Q_t$ the tautological quotient
bundle
on the fibre $\mbox{$I\!\!P$}_t=\mbox{$I\!\!P$}(H^0(W_t, {\cal
L}_t))$.
Then for any $i$, the dimension
$h^0 (\mbox{$I\!\!P$}_t, \Lambda^iQ_t \otimes I_t(1))$ is upper-semi-continuous.
\end{prop}
\noindent {Proof.}
By properness of the Hilbert scheme, there exists
a $T$-flat subscheme $\bar W$ of $\mbox{$I\!\!P$}(p_*{\cal L})$ with the property
that
its fibre over the general point $t_1$ of $T$ is $m(W_{t_1})$. By
continuity,
its special fibre $\bar W_{t_0}$ contains $m(W_{t_0})$ (indeed, they
are equal, but we don't need this).
Thus, by inclusion, we have
$$h^0 (\mbox{$I\!\!P$}_{t_0}, \Lambda^iQ_{t_0} \otimes I_{t_0}(1)) \geq
h^0 (\mbox{$I\!\!P$}_{t_0}, \Lambda^iQ_{t_0} \otimes I_{\bar W_{t_0}}(1)),$$
and by semi-continuity,
$$h^0 (\mbox{$I\!\!P$}_{t_0}, \Lambda^iQ_{t_0} \otimes I_{\bar W_{t_0}}(1)) \geq
h^0 (\mbox{$I\!\!P$}_{t_1}, \Lambda^iQ_{t_1} \otimes I_{\bar W_{t_1}}(1)),$$
which altogether prove our claim.$\hfill \square$
\section{The syzygy locus in the case of odd genus} \label {S3}
In this section, we write ${\cal M}={\cal M}^{o}_g$ for
the open subvariety of ${\cal M}_g$ consisting
of points that represent isomorphism classes of smooth curves with
trivial automorphism group. What we need to know of ${\cal M}$ is
that an effective divisor on it which is rationally equivalent to zero
is indeed zero: this is for instance because ${\cal M}$ has a
projective compactification with two-codimensional boundary (cf e.g. [A]).
Let $x$ be a point in ${\cal M}$ and $C$ the corresponding curve.
We consider the canonical imbedding of $C$ in $\mbox{$I\!\!P$}^{g-1}$
($C$ is not hyperelliptic);
we denote by ${\cal I}_C$ the ideal sheaf of $C$
and by $Q$ the tautological quotient bundle of rank $g-1$ on $\mbox{$I\!\!P$}^{g-1}$.
Finally, we denote by $S_{g}$
the locus in ${\cal M}$ of (points corresponding to) curves
$C$ satisfying $\Gamma (\Lambda^{k} Q \otimes {\cal I}_C(1)) \neq 0.$
As a jump locus, $S_g$ has
a natural Cartier divisor structure (see e.g. the proof of the next proposition)
and
we
compare its class in the Picard group of ${\cal
M}$ with
the class $c$ of the $k$-gonal locus (cf [HM]).
\begin{prop} Let $g = 2k-1\geq 5$ be an odd
integer
such that the generic curve $C$ of genus $g$ satisfies
$\Gamma (\Lambda^{k} Q \otimes {\cal I}_C(1))=0.$ Then, in the Picard group of
${\cal M}$,
the rational class $v$ of $S_{g}$ is $(k-1)c$.
\end{prop}
\noindent{\bf Proof.}
There exists a universal curve ${\cal C}$ over ${\cal M}$,
that is to say a smooth variety $\cal C$ and a smooth
projective morphism $\pi :\cal C\to \cal M$ such that for any
$x\in \cal M$ the fibre of $\pi $ over $x$ is the curve of
genus $g$ whose isomorphism class is given by the point $x$. Let
$\omega = \omega_{\pi }$ be the cotangent bundle along the
fibres and $E$ its direct image on ${\cal M}$ by $\pi $. Then $\pi$
factors
through
the natural canonical imbedding of ${\cal C}$ in the projective
bundle $p:\mbox{$I\!\!P$} = \mbox{$I\!\!P$} (E) \to {\cal M}$. Let ${\cal I}$ be the ideal
sheaf
of ${\cal C}$ in $\mbox{$I\!\!P$}$.
The relatively ample (hyperplane) line bundle along the fibres
of $\mbox{$I\!\!P$} $ will be denoted as usual by $ {\cal O}_p(1)$.
Finally we write again $Q$ for the vector bundle on $\mbox{$I\!\!P$} $ given by the
exact sequence
$$
0\rightarrow {\cal O}_p(-1) \rightarrow p ^*(E)^*
\rightarrow Q \rightarrow 0.
$$
Observe that $p _*(\Lambda ^l Q(1))$ is a vector bundle of rank
${g\choose l}g - {g\choose l - 1}$.
Similarly, on each fibre ${\cal C}_x$, $\Lambda ^l Q_{{\cal C}_x}\otimes
\omega_{{\cal C}_x}$
is semi-stable (cf [PR]) of slope $2l + 2g -
2$, thus non-special, and
$p _*(\Lambda ^l Q(1)\otimes {\cal O}_{\cal C})$ is also a vector
bundle, of rank
${g-1\choose l}(2l + g -
1)$, for each $l>0.$
Substituting $k$ for $l$, the
restriction from $\mbox{$I\!\!P$}$ to the universal curve yields a morphism
$r$ from $p _*(\Lambda ^k Q(1))$ to
$p _*(\Lambda ^k Q(1)\otimes {\cal O}_{\cal C})$, and our assumption
means that this morphism is injective (at the generic point).
We observe
that the
two vector bundles have the same rank, namely ${2k - 2\choose k
}(4k - 2) = {2k - 1\choose k - 1}(2k - 2)$. Thus the above
map defines a (degeneracy) divisor in ${\cal M}$ and this is $S_g$,
by definition. Its (virtual) rational class is
$v=c_1(p _*(\Lambda ^k Q(1)\otimes {\cal O}_{\cal C}))-c_1(p _*(\Lambda ^k Q(1))).$
We will
compute this class in $Pic({\cal M})$ as a
multiple of $\lambda= c_1(E)$. We will start with the following
computation in the appropriate Grothendieck group $K$. Let $t$
be an indeterminate and for any vector bundle $V$, let $\lambda
_t(V)$ denote the element $\sum t^i \Lambda ^i(V)$ in $K[[t]]$.
This extends to a homomorphism of $K$ into the multiplicative
group consisting of power series with constant term 1 in
$K[[t]]$, and this map is still denoted by $\lambda _t$.
Consider now $x = p _!(\lambda _t(Q).{\cal I}(1)) = p
_!(\lambda _t(Q).({\cal O}_p(1) - {\cal O}_{\cal C}(1)))$. Substitute $Q =
p ^*(E^*) - {\cal O}_p(-1)$. Then we obtain
\begin{eqnarray*}
x & = & p _!( \frac {\lambda _t(p ^*(E^*))}{\lambda _t({\cal
O}(-1))} ({\cal O}_p(1) - {\cal O}_{\cal C}(1))\cr
& = & \lambda _t(E^*) p _!
\left ( \frac {{\cal O}_p(1) - {\cal O}_{\cal C}(1)}{1
+ t{\cal O}_p(-1)}\right )\cr
& = & {\lambda }_t(E^*) \sum _{j = 0} ^{j = \infty } (-1)^j t^j
(p _!({\cal O}_p(1-j)) - p _!({\cal O}_{\cal C}(1-j))).
\end{eqnarray*}
Our class $v$ is the coefficient of $t^k$ in the first
Chern class of $-x$. Observe that $p _! ({\cal O}_p(1 - j)) = 0$,
whenever $2 \leq j \leq g$. Also we have $p _!({\cal O}) = 1$
and $p _!({\cal O}_p(1)) = E. $ On the other hand, $p _!({\cal
O}_{\cal C}(1 - j))$ can be seen to be $ E^* - 1$ for $j = 0$ and to
be $1 - E$ for $j = 1$. The first Chern class of the direct
images for $j\geq 2$ can be computed by the Grothendieck-Riemann-Roch
theorem to be $(1 - 6(1 - j) + (1-j)^2) \lambda $ (see
[M]). The first Chern class of $\lambda _t(E^*)$ is clearly
equal to $-t(1 + t)^{g - 1}\lambda $. Also the rank of $\lambda
_t(E)$ is $(1 + t)^g$, while the rank of $p _!({\cal O}_{\cal C}(n))$
is $(g - 1)(2n - 1)$. Thus $v$
is equal to $N\lambda $ where $-N$
is the coefficient of $t^k$ in
$$
(1 + t)^g\{ 1 - \sum ^{i =
\infty}_{i = 0} (-1)^i (1 + 6i + 6i^2)t^i \} - t(1 + t)^{g -
1}\{ g - t - (g - 1)\sum _{ i = 0}^{i = \infty } (- 1)^i t^i (1 - 2i)\}.
$$
On the one hand, we have
$(1 + t)^g (1 - \sum _{i = 0} ^{i = \infty } (-1)^i(1 - 6i + 6i^2)t^i)$
$=
(1 + t)^g ( 1 - \sum _{i = 0} ^{i = \infty }(-1)^i(6(i + 1)(i + 2)
- 24(i + 1) + 13)t^i )$
$=(1 + t)^g ( 1 - {12\over (1 + t)^3} + {24\over (1 + t)^2} -
{13\over 1 + t})$
$=(1 + t)^{g - 3}( (1+t)^3 - {13}(1 + t)^2 + {24} (1 + t) - 12)$
$=t(1 + t)^{g - 3} (t^2 - 10t +1),$
and on the other,
$t(1 + t)^{g - 1} ( g - t - (g-1)\sum _{i = 0} ^{i = \infty } (-1)^i(
3 - 2(i + 1))t^i)$\\
$=t(1 + t)^{g - 1}(g - t - (g - 1)({3\over 1 +
t} - {2\over (1 + t)^2}))$\\
$=t(1 + t)^{g - 3}((1 + t)^2(g - t) -
(g - 1)3(1 + t) - 2(g - 1))$\\
$=t(1 + t)^{g - 3}(-t^3 + (g-2)t^2
+ (-g + 2)t + 1).$
This leads to the determination of $N$ to be the coefficient of
$t^k$ in
$$t^2(1 +t)^{2k - 4}(-t^2 +(2k - 4)t - (2k - 13)),$$
namely
$$- {2k - 4\choose k - 4} + (2k - 4){2k - 4\choose k - 3} -
(2k - 13){2k - 4\choose k - 2}$$
and this simplifies to
$$ 6(k +1)(k - 1){(2k - 4)!\over (k-2)!k!}.$$
Now Harris and Mumford [HM] have studied the locus of $k$-gonal
curves in ${\cal M}$ and have shown that this variety
is a divisor whose class is $6(k+1){(2k - 4)!\over (k-2)!k!}\lambda $,
which proves our claim. $\hfill \square$
\section{Syzygies of scrolls} \label {scroll}
Extra syzygies of $k$-gonal curves arise because they lie on scrolls.
So we start with estimating some syzygies of scrolls.
\begin{prop} Let $W$ be a vector bundle on
$\mbox{$I\!\!P$} ^1$ of rank $k-1$ and degree $k$. We suppose $W$
to be globally
generated. We denote by $I_W$ the ideal of the image of the natural morphism
from $\mbox{$I\!\!P$} (W)$ into $\mbox{$I\!\!P$} \Gamma (W)$ and by $Q$ the tautological
quotient
bundle on this projective space. Then
the dimension $h^0 (\mbox{$I\!\!P$} \Gamma (W), \Lambda ^{k
} Q\otimes I_W(1))$ is at least $k - 1$.
\end{prop}
\noindent {\bf Proof.} By \ref {scs}, we may suppose that $W$ is generic
namely
$W= {\cal O}(1)^{\oplus {k-2}} \oplus {\cal
O}(2)$. In this case,
the natural morphism $\mbox{$I\!\!P$}(W) \to \mbox{$I\!\!P$}
\Gamma (W)$ is an imbedding. We will use freely the identification (2.1).
Consider $X = \mbox{$I\!\!P$} ^1 \times \mbox{$I\!\!P$} ^1$ and the variety $Y = \mbox{$I\!\!P$}^1
\times \mbox{$I\!\!P$} (W)$. Let us denote as usual by $p_1$ and $p_2$ the
two projections (in both cases) and by $\pi $ the fibration
$\mbox{$I\!\!P$} (W)\to \mbox{$I\!\!P$} ^1$, as well as the morphism $Y\to X$ given by $I
\times \pi $. Let $\Delta $ be the diagonal divisor in $X$ and
$D$ its inverse image in $Y$. Let $Q$ be the universal
quotient bundle on $\mbox{$I\!\!P$} \Gamma (W)$ and its restriction to $\mbox{$I\!\!P$}
(W)$. Now consider on $Y$ the bundle homomorphism
$p_1^*(W)^* \to p_2^*(Q)$ obtained as the composition of the pull
back by $p_1$ of the natural inclusion $W^* \to \Gamma
(W)^*\otimes {\cal O}$ and the pull-back by $p_2$ of the
tautological map $\Gamma (W)^* \otimes {\cal O} \to Q$. This
homomorphism is injective as a sheaf morphism but has one-dimensional
kernel on
the fibres over points of $D$. Thus we obtain an inclusion of
${\cal L}:= p_1^*(\Lambda ^{k - 1}W^*)\otimes {\cal O}(D)$ into $p_2^*(\Lambda
^{k - 1}Q)$. Note that ${\cal O}(D)$ is isomorphic to $ p_1^*{\cal O}(1)
\otimes p_2^* \pi ^*{\cal O}(1)$ so that ${\cal L}$
is isomorphic to $p_1^*({\cal O}(-k + 1))\otimes
p_2^*({\pi ^*(\cal O}(1)))$. Taking direct image by $p_1$ we get
a homomorphism of ${\cal O}(-k + 1) \otimes \Gamma (\mbox{$I\!\!P$} ^1, {\cal
O}(1))$ into $\Gamma (\Lambda ^{k - 1}Q)$. This fits in the
following commutative diagram
$$
\begin{array}{ccc}
{\cal O}(-k)& \rightarrow & \Lambda ^{k - 1}\Gamma (Q)\otimes {\cal O}\\
\downarrow & & \downarrow \\
{\cal O}(-k + 1)\otimes \Gamma (\mbox{$I\!\!P$} ^1, {\cal O}(1)) & \rightarrow &
\Gamma (\Lambda ^{k - 1}Q) \otimes {\cal O} \\
\downarrow & & \downarrow \\
{\cal O}(-k + 2)&\rightarrow & coker\Lambda ^{k - 1}\Gamma
(Q)\otimes {\cal O}\to \Gamma (\Lambda ^{k - 1}Q)\otimes {\cal O}.
\end{array}
$$
We wish to make two remarks here. Firstly the lower horizontal
arrow is nonzero. In fact, for any point $x$ of $\mbox{$I\!\!P$} ^1$, the
middle horizontal arrow gives a two-dimensional space of
sections of $\Lambda ^{k - 1}Q$. This is obtained as follows.
Consider the sheaf inclusion of the trivial subbundle $W^*_x$ in
$\Gamma (W)^*$ on $\mbox{$I\!\!P$}(W)$ and compose it with the natural
homomorphism of the trivial bundle with fibre $\Gamma (W)^*$
into $Q$. Take the $(k - 1)$-th exterior power of this map. This
becomes an inclusion of ${\cal O}(x) = {\cal O}(1)$ in $\Lambda ^{k -
1}Q$. Thus at the $\Gamma $-level this gives the
two-dimensional space of sections required. Clearly the sections
of the trivial bundle $\Lambda ^{k -1}(W_x^*)$ give a
one-dimensional subspace of this. This is the top horizontal
arrow in our diagram. Conversely let $s$ be an element
of $\Lambda ^{k - 1}\Gamma (Q) = \Lambda ^{k - 1}(\Gamma W)^*$,
then its exterior product with any element of $W_x^*$ gives an
element of $\Lambda ^k\Gamma (Q) = \Lambda ^k (\Gamma W)^*$.
If $s$ is actually a section
of the sub-bundle generated by $W_x$, then this exterior product
should be zero at the generic point and hence 0. This implies
that $s$ belongs to $\Lambda ^{k - 1}(W_x)^* = {\cal O}(-k)$.
Secondly, since all our constructions are canonical and $W$ is a
homogeneous bundle, it follows that the lower horizontal arrow
is $SL(2)$-equivariant. Now the proposition is a consequence of
the following claim: If ${\cal O}(-n)$ admits a non-zero map
into a trivial bundle, which is equivariant for the natural
$SL(2)$-actions, then the rank of the trivial bundle is at least
$n + 1$. To prove it, use the dual map of the trivial bundle into
${\cal O}(n)$ and use the fact that $\Gamma ({\cal O}(n))$ is an
irreducible $SL(2)$-module. This implies that the induced map at
the $\Gamma $-level, which is nonzero by assumption, is actually
injective. $\hfill \square$
\section { Extra syzygies of gonal curves} \label {S5}
We say that a curve of genus $g = 2k - 1$ is $k$-gonal if it carries
a line
bundle $L$ of degree $k$ whose linear system has no base points and thus
yields a $k$-sheeted morphism $\pi $ onto $\mbox{$I\!\!P$}^1$. In this paragraph,
we prove that $k$-gonal curves of genus $2k-1$
have at least $k-1$ extra syzygies.
\begin{prop} Let $C$ be a nonhyperelliptic $k$-gonal curve of
genus $2k - 1$, with $L$ the special line bundle of degree $k$ and
$Q_C$
the restriction of the tautological quotient bundle on $\mbox{$I\!\!P$}^{g
-1}$ to the canonically imbedded curve $C$. Then the dimension
of $H^0(C, \Lambda ^{k - 1}Q_C)$ is at least ${g\choose k - 1} + k
- 1$.
\end{prop}
\noindent {\bf Proof.}
Consider the direct
image $V$ of the canonical line bundle $K$ of $C$ by $\pi $. The
so-called trace map gives a homomorphism of $V$ onto $K_{\mbox{$I\!\!P$}
^1} = {\cal O}(-2)$. Let $W$ be its kernel. Thus we have an
exact sequence
$$
0 \rightarrow W \rightarrow V \rightarrow {\cal
O}(-2)\rightarrow 0
$$
Since ${\cal O}(-2)$ has no nonzero sections it follows that
$\Gamma (W) = \Gamma (V) = \Gamma (C,K)$. Moreover, the kernel
of the evaluation map $\Gamma (C,K) \to W_p$ at any point $p\in
\mbox{$I\!\!P$} ^1$ is simply the set of sections vanishing on $\pi
^{-1}(p)$, that is to say $s\Gamma (K\otimes L^{-1})$ where $s$
is a nonzero section of $L$ vanishing on this fibre. On
computing the dimension of this space to be $k$ by Riemann-Roch,
we find that the evaluation map from $\Gamma (C,K)_{\mbox{$I\!\!P$} ^1}$ to
$V$ is actually onto $W$. Thus, $W$ is generated by global sections
and we get a morphism of $\mbox{$I\!\!P$} (W)$ into $\mbox{$I\!\!P$} \Gamma (C,K)$.
Finally the pull-back of $V$ to $C$, namely $\pi ^*\pi _*(K)$
comes with a natural homomorphism onto $K$. Indeed the natural
surjection of $\Gamma (K)$ onto $K$ factors through this map,
which can be thought of as `evaluation along fibres'. Thus we
have a morphism from $C$ to $\mbox{$I\!\!P$} (W)$ the composition of which with
the above mentioned morphism from $\mbox{$I\!\!P$} (W) \to \mbox{$I\!\!P$} \Gamma(K)$
to $C$ is the canonical imbedding. Thus our claim follows from Section
\ref {scroll}.
$\hfill \square$
\section{Proof of the theorem}
In this section, we give the proof of our theorem.
We start with the
\begin{lem} Let $S$ be a smooth variety and $E$ and
$F$ two vector bundles of the same rank $n$. Let $f:E\to F$ be a
homomorphism which is generically an isomorphism, and $D$ a
subvariety of codimension 1 in $S$ on which $f$ has kernel of
rank $\geq r$, then the degeneracy divisor of $f$ contains $D$
as a component of multiplicity at least $r$.
\end{lem}
\noindent {\bf Proof.}
Note that the question is local and
localising at the generic point of $D$, we may assume that $S$
is a discrete valuation ring with maximal ideal $\gotm M$
and that $f$ is a square
matrix of nonzero determinant. Then by a proper choice of basis
we may assume $f$ to be diagonal of the form $\delta_{i,j}t^{m_i}, 0
\leq
i,j \leq n$,
where $t$ is a generating parameter. Our assumption ensures
$m_i > 0$ for at least $r$ indices. Then clearly $det(f)$
is in $\gotm M^r$.
Since the degeneracy locus is defined by $det (f)$, this
proves our assertion. $\hfill \square$
Before turning to the proof, we state again our
\begin{thm} Let $g = 2k-1 \geq 5$ be an odd integer. If the generic curve $C$
of
genus $g$ has no extra syzygies (i.e.
$\Gamma (\Lambda ^{k} Q \otimes {\cal I}_C(1))=0$),
then so does any curve of genus $g$ with maximal Clifford index, namely
$k-1.$
\end{thm}
\noindent{\bf Proof.}
We have shown (see Section 3) that the syzygy divisor $S_g$ is the
degeneracy locus of a homomorphism of a vector bundle into
another of same rank, and (see Section 5)
that at the generic $k$-gonal curve this
homomorphism has kernel of dimension at least $k - 1$. Thanks to the
previous
lemma, this implies that the locus of $k$-gonal curves is contained in
$S_g$ with
multiplicity at least $k - 1$.
By our computation in Section 3, the residual divisor has rational class
zero,
thus is the zero divisor (this is what we need
to know about ${\cal M}$). Thus in ${\cal M}$, curves with extra syzygies
are in the $k$-gonal divisor. Now even around curves with
automorphisms,
we can see by going to a covering where a universal curve exists, that
the locus of curves with extra syzygies is a divisor. Since the locus
of curves with automorphisms is of codimension at least two in ${\cal
M}$, we get
that
even in ${\cal M}_g$, curves with extra syzygies
are in the $k$-gonal divisor, thus have nonmaximal
Clifford index. This proves our theorem.$\hfill \square$
\begin {rem}
We may even conclude that curves
with nonmaximal Clifford index (which have extra syzygies by [GL])
are all in the $k$-gonal divisor. Note that this result is true
(without our assumption on the generic curve), cf [ELMS].
\end {rem}
|
1998-07-20T11:38:20 | 9707 | alg-geom/9707008 | en | https://arxiv.org/abs/alg-geom/9707008 | [
"alg-geom",
"math.AG"
] | alg-geom/9707008 | Pelham Wilson | P. M. H. Wilson | Flops, Type III contractions and Gromov-Witten invariants on Calabi-Yau
threefolds | 20 pages, latex2e, minor changes to previous version | null | null | null | null | We investigate Gromov-Witten invariants associated to exceptional classes for
primitive birational contractions on a Calabi-Yau threefold X. It was observed
in a previous paper that these invariants are locally defined, in that they can
be calculated from knowledge of an open neighbourhood of the exceptional locus
of the contraction; in this paper, we make this explicit.
For Type I contractions (i.e. only finitely many exceptional curves), a
method is given for calculating the Gromov-Witten invariants, and these in turn
yield explicit expressions for the changes in the cubic form $D^3$ and the
linear form $D.c_2$ under the corresponding flop. For Type III contractions
(when a divisor E is contracted to a smooth curve C of singularities), there
are only two relevant Gromov-Witten numbers n(1) and n(2). Here n(2) is the
number (suitably defined) of simple pseudo-holomorphic rational curves
representing the class of a fibre of E over C, and n(1) the number of simple
curves representing half this class. Explicit formulae for n(1) and n(2) are
given (n(1) in terms of the singular fibres of E over C and n(2)=2g(C)-2). An
easy proof of these formulae is provided when g(C)>0. The main part of the
paper then gives a proof valid in general (including the case g(C)=0).
| [
{
"version": "v1",
"created": "Wed, 9 Jul 1997 16:46:00 GMT"
},
{
"version": "v2",
"created": "Tue, 14 Oct 1997 16:58:47 GMT"
},
{
"version": "v3",
"created": "Mon, 16 Mar 1998 17:44:10 GMT"
},
{
"version": "v4",
"created": "Mon, 20 Jul 1998 09:38:18 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Wilson",
"P. M. H.",
""
]
] | alg-geom | \section{Introduction}
In this paper, we investigate Gromov--Witten invariants associated to
exceptional classes for primitive birational contractions on a Calabi--Yau
threefold $X$. As already remarked in \cite{18}, these invariants are
locally defined, in that they can be calculated from knowledge of an open
neighbourhood of the exceptional locus of the contraction; intuitively, they are
the numbers of rational curves in such a neighbourhood. In \S\ref{sec1},
we make this explicit in the case of Type~I contractions, where the
exceptional locus is by definition a finite set of rational curves.
Associated to the contraction, we have a flop; we deduce furthermore in
Proposition~\ref{prop_1.1} that the changes to the basic invariants (the
cubic form on $H^2(X,\mathbb Z)$ given by cup product, and the linear form given by
cup product with the second Chern class $c_2$) under the flop are explicitly
determined by the Gromov--Witten invariants associated to the exceptional
classes.
The main results of this paper concern the Gromov--Witten invariants
associated to classes of curves contracted under a Type~III primitive
contraction. Recall \cite{17} that a primitive contraction $\varphi\colon X\to\Xbar$ is of
Type~III if it contracts down an irreducible divisor $E$ to a curve of
singularities $C$. For $X$ a smooth Calabi--Yau threefold, such contractions
were studied in \cite{18}; in particular, it was shown there that the curve
$C$ is smooth and that $E$ is a conic bundle over $C$. We denote by
$2\eta\in H_2(X,\mathbb Z)/\operatorname{Tors}$ the numerical class of a fibre of $E$ over $C$. In
the case when $E$ is a $\mathbb P ^1$-bundle over $C$, this may in fact be a
primitive class, and so the notation is at slight variance with that adopted
in \S\ref{sec1}, where $\eta$ is assumed to be the primitive class. In the
case when the class of a fibre is not primitive (for instance, when $E$ is not
a $\mathbb P ^1$-bundle over $C$), the primitive class contracted by $\varphi$ will be
$\eta$. We denote the Gromov--Witten numbers associated to $\eta$ and
$2\eta$ by $n_1$ and $n_2$, with the convention that $n_1=0$ if $2\eta$ is the
primitive class. The above conventions have been adopted so as to achieve
consistency of notation for all Type~III contractions.
If the genus $g$ of the curve $C$ is strictly positive, under a
general holomorphic deformation of the complex structure on $X$, the
divisor $E$ disappears leaving only finitely many of its fibres, and (except
in the case of elliptic quasiruled surfaces, where all the Gromov--Witten
invariants vanish) we have a Type~I contraction. The results of \S\ref{sec1}
may then be applied to deduce the Gromov--Witten invariants associated to
the classes $m\eta$ for $m>0$. These are all determined by the Gromov--Witten
numbers $n_1$ and $n_2$, and explicit
formulas for $n_1$ and $n_2$ are given in Proposition~\ref{prop_2.3}; in
particular $n_2=2g-2$.
The formulas for $n_1$ and $n_2$ remain valid also for $g=0$, although the
slick proof given in Proposition~\ref{prop_2.3} for the case $g>0$ no longer
works. The formula for $n_1$ is proved for all values of $g(C)$ by local
deformation arguments in Theorem~\ref{thm_2.5}. Verifying that $n_2=-2$ in
the case when $g(C)=0$ is rather more difficult, and involves the technical
machinery of moduli spaces of stable pseudo\-holomorphic maps and the virtual
neighbourhood method, as used in \cite{2,9} in order to construct
Gromov--Witten invariants for general symplectic manifolds. In particular,
we shall need a cobordism result from \cite{13}, which we show in
Theorem~\ref{thm_3.1} applies directly in the case where no singular fibre
of $E$ is a double line. The general case may be reduced to this one by
making a suitable almost complex small deformation of complex structure. In
\S\ref{sec4}, we give an application of our calculations. In \cite{18}, it
was shown that if $X_1$, $X_2$ are Calabi--Yau threefolds which are
symplectic deformations of each other (and general in their complex moduli),
then their K\"ahler cones are the same. Now we can deduce
(Corollary~\ref{cor4.1}) that corresponding codimension one faces of these
cones have the same contraction type.
The author thanks Yongbin Ruan for the benefit of conversations concerning
material in \S\ref{sec3} and his preprint \cite{13}.
\section{Flops and Gromov--Witten invariants}\label{sec1}
If $X$ is a smooth Calabi--Yau threefold with K\"ahler cone $\mathcal K$, then the
nef cone $\overline{\mathcal K}$ is locally rational polyhedral away from the cubic cone
\[
W^*=\bigl\{ D\in \HR 2 X \ ; \ D^3=0\bigr\};
\]
moreover, the codimension one faces of $\overline{\mathcal K}$ (not contained in $W^*$)
correspond to primitive birational contractions $\varphi\colon X\to\Xbar$ of one of three
different types \cite{17}.
In the numbering of \cite{17}, Type~I contractions are those where only a
finite number of curves (in fact $\mathbb P^1$s) are contracted. The singular
threefold $\overline X$ then has a finite number of cDV singularities. Whenever one
has such a small contraction on $X$, there is a flop of $X$ to a different
birational model $X'$, also admitting a birational contraction to $\overline X$;
moreover, identifying $\HR 2 {X'}$ with $\HR 2 {X}$, the nef cone of $X'$
intersects the nef cone of $X$ along the codimension one face which defines
the contraction to $\overline X$ \cite{6, 7}. It is well known \cite{7} that
$X'$ is smooth, projective and has the same Hodge numbers as $X$, but that
the finer invariants, such as the cubic form on $\HZ 2 X$ given by cup
product, and the linear form on $\HZ 2 X$ given by cup product with
$c_2(X)=p_1(X)$, will in general change. Recall that, when $X$ is simply
connected, these two forms along with $\HZ 3 X$ determine the diffeomorphism
class of $X$ up to finitely many possibilities \cite{14}, and that if
furthermore $H_2(X,\mathbb Z)$ is torsion free, this information determines the
diffeomorphism class precisely \cite{16}.
When the contraction $\varphi\colon X\to\Xbar$, corresponding to such a {\em flopping face}
of $\overline{\mathcal K}$, contracts only isolated $\mathbb P^1$s with normal bundle $(-1,-1)$
(that is, $\overline X$ has only simple nodes as singularities), then it is a
standard calculation to see how the above cubic and linear forms (namely the
cup product $\mu\colon \HZ 2 X\to\mathbb Z$, and the form $c_2\colon \HZ 2 X\to\mathbb Z$)
change on passing to $X'$ under the flop. Since any flop is an isomorphism
in codimension one, we have natural identifications
\[
\HR 2 {X'}\cong \operatorname{Pic}_{\mathbb R}(X')\cong \operatorname{Pic}_{\mathbb R}(X)\cong \HR 2 X.
\]
If
we are in the case where the exceptional curves $C_1,\dots,C_N$ are
isolated $\mathbb P^1$s with normal bundle $(-1,-1)$, and if we denote by $D'$ the
divisor on $X'$ corresponding to $D$ on $X$, then
\[
(D')^3=D^3-\sum(D\cdot C_i)^3
\quad \text{and}\quad c_2(X')\cdot D'=c_2(X)\cdot D+2\sum D\cdot C_i \ .
\]
This is an easy verification -- see for instance \cite{1}.
\begin{prop}\label{prop_1.1} Suppose that $X$ is a smooth Calabi--Yau
threefold, and $\varphi\colon X\to\Xbar$ is any Type~I contraction, with $X'$ denoting the
flopped Calabi--Yau threefold. The cubic and linear forms $(D')^3$ and
$D'\cdot c_2(X')$ on $X'$ are then explicitly determined by the cubic and
linear forms $D^3$ and $D\cdot c_2(X)$ on $X$, and the $3$-point
Gromov--Witten invariants $\Phi_A $ on $X$, for $A\in H_2(X,\mathbb Z)$ ranging
over classes which vanish on the flopping face.
\end{prop}
\begin{rem} This is essentially the statement from physics that the A-model
3-point correlation function on $\mathcal K(X)$ may be analytically continued to
give the A-model 3-point correlation function on $\mathcal K(X')$.
\end{rem}
\begin{pf} We use the ideas from \cite{18}; in particular, we know that on
a suitable open neighbourhood of the exceptional locus of $\varphi$, there
exists a small holomorphic deformation of the complex structure for which
the exceptional locus splits up into disjoint $(-1,-1)$-curves (\cite{18},
Proposition~1.1).
Let $A\in H_2(X,\mathbb Z)$ be a class with $\varphi_* A=0$. The argument from
\cite{18}, Section~1 then shows how the Gromov--Witten invariants
$\Phi_A(D,D,D)$ can be calculated from local information. Having fixed a
K\"ahler form $\omega$ on $X$, a small deformation of the holomorphic
structure on a neighbourhood of the exceptional locus may be patched
together in a
$C^{\infty}$ way with the original complex structure to yield an almost
complex structure tamed by $\omega$, and the Gromov--Witten invariants can
then be calculated in this almost complex structure. The Gromov Compactness
Theorem is used in this argument to justify the fact that all of the
pseudo\-holomorphic rational curves representing the class $A$ have images
which are $(-1,-1)$-curves in the deformed local holomorphic structure.
Here we also implicitly use the Aspinwall--Morrison formula for the
contribution to Gromov--Witten invariants from multiple covers of
infinitesimally rigid $\mathbb P^1$s, now proved mathematically by Voisin
\cite{15}. So if $n(B)$ denotes the number of $(-1,-1)$-curves representing
a class given $B$, then
\[
\Phi_A(D,D,D)=(D\cdot A)^3 \sum_{kB=A} n(B)/k^3,
\]
where the sum is taken over all integers $k>0$ and classes $B\in H_2(X,\mathbb Z)$
such that $kB=A$. So if $H_2(X,\mathbb Z)$ is torsion free and $A$ is the primitive
class vanishing on the flopping face, this says that
\[
\Phi_{mA}(D,D,D)=(D\cdot A)^3 \sum_{d|m} n(dA)d^3.
\]
Recall that the Gromov--Witten invariants used here are the ones (denoted
$\widetilde\Phi$ in \cite{12}) which count marked parametrized curves
satisfying a perturbed pseudo\-holomorphicity condition. Knowledge of the
numbers $n(A)$ for the classes A with $\varphi_* A=0$ determines the
Gromov--Witten invariants $\Phi_{A}$ for classes A with $\varphi_*A=0$, and
vice-versa.
If we can now show that the local contributions to $(D')^3$ and $D'\cdot
c_2(X')$ are well-defined and invariant under the holomorphic deformations
of complex structure we have made locally, then the obvious formulas for
them will hold. Let $\eta\in H_2(X,\mathbb Z)/\operatorname{Tors}$ be the primitive class with
$\varphi_* \eta=0$ and $n_d$ denote the total number of $(-1,-1)$-curves on the
deformation which have numerical class $d\eta$; the $n_d$ are therefore
nonnegative integers (cf.\ \cite{10}, Remark~7.3.6). Then
\begin{align}
(D')^3&=D^3-(D\cdot\eta)^3\sum_{d>0}n_dd^3, \tag{2.1.1}\\
D'\cdot c_2(X')&=D\cdot c_2(X)+2(D\cdot\eta)\sum_{d>0}n_dd. \tag{2.1.2}
\end{align}
To justify the premise in the first sentence of the paragraph, the basic result
needed is that of local conservation of number, as stated in \cite{3},
Theorem~10.2.
For calculating the change in $D^3$ for instance, let $X$ now denote the
neighbourhood of the exceptional locus of $\varphi$ and $\pi\colon \mathcal X\to B$ the
small deformation under which the exceptional locus splits up into
$(-1,-1)$-curves. So we have a regular embedding (of codimension six)
\[
\renewcommand{\arraystretch}{1.3}
\begin{matrix}
\mathcal X &\hookrightarrow & \mathcal X \times \mathcal X \times \mathcal X &=\mathcal Y \\
\downarrow && \downarrow \\
B & \kern1.2em=\kern-1.2em & B
\end{matrix}
\]
In order to calculate the triple products $D_1'\cdot D_2'\cdot D_3'$ from
$D_1\cdot D_2\cdot D_3$ and the numbers $n_d$, we may assume {\em wlog}
that the $D_i$ are very ample, and so in particular we get effective
divisors $\mathcal D_1$, $\mathcal D_2$ and $\mathcal D_3$ on $\mathcal X /B$. Applying \cite{8},
Theorem~11.10, we can flop in the family $\mathcal X\to B$, hence obtaining a
deformation $\mathcal X'\to B$ of the flopped neighbourhood $X'$. We wish to
calculate the local contribution to $D_1'\cdot D_2'\cdot D_3'$; with the
notation as in \cite{3}, Theorem~10.2, we have a fibre square
\[
\renewcommand{\arraystretch}{1.3}
\begin{matrix}
\mathcal W & \longrightarrow & \mathcal D_1' \times \mathcal D_2'\times \mathcal D_3' \\
\downarrow && \downarrow \\
\mathcal X' &\longrightarrow & \mathcal X' \times \mathcal X'\times \mathcal X'
\end{matrix}
\]
with $\operatorname{Supp}(\mathcal W)=\bigcap \operatorname{Supp}(\mathcal D_i')$. Furthermore, we may assume that the
divisors $\mathcal D_i $ were chosen so that $\mathcal D_1\cap\mathcal D_2\cap\mathcal D_3$ has no
points in $\mathcal X$, and so in particular $\mathcal W$ is proper over $B$. Letting
$D_i'(t)$ denote the restriction of $\mathcal D_i'$ to the fibre $X_t'$, we
therefore have a well-defined local contribution to $D_1'(t)\cdot
D_2'(t)\cdot D_3'(t)$ (concentrated on the flopping locus of $X_t'$), which
is moreover independent of $t\in B$. Thus by making the local calculation as
in (7.4) of \cite{1}, we deduce that
\[
D_1\cdot D_2\cdot D_3-D_1'\cdot D_2'\cdot D_3'
=(D_1\cdot \eta)(D_2\cdot \eta)(D_3\cdot \eta)
\sum_{d>0} n_d d^3
\]
as required.
The proof for $c_2\cdot D$ is similar. Here we consider the graph
$\widetilde X \subset X \times X'$ of the flop, with $\pi_1\colon \widetilde X\to X
$ and $\pi_2\colon \widetilde X\to X' $ denoting the two projections, and
$E \subset \widetilde X$ the exceptional divisor for both $\pi_1$ and $\pi_2$.
Then $\pi_2^*(T_{X'})\rest{\widetilde X \setminus E}=\pi_1^*(T_X)\rest{\widetilde X
\setminus E} $, and so in particular $c_2(\pi_2^* T_{X'})-c_2(\pi_1^*T_X)$
is represented by a 1-cycle $Z$ on $E$. Suppose {\em wlog} that $D$ is very
ample, and that $D'$ denotes the corresponding divisor on $X'$. Set $\pi_1^*
D=\widetilde D$ and $ \pi_2^* D'=\widetilde D +F$, with $F$ supported on $E$. Then
$c_2(X')\cdot D'=c_2(\pi_2^* T_{X'})\cdot(\widetilde D+F)$. Hence
\[
c_2(X')\cdot D'-c_2(X)\cdot D=c_2(\pi_2^* T_{X'})\cdot F+Z\cdot
\widetilde D=c_2((\pi_2^* T_{X'})\rest F)+(Z\cdot \widetilde D)_E
\]
where the right-hand side is purely local. Note the slight abuse of notation
here that $F$ denotes also the fixed {\em scheme} for the linear system $
|\pi_2^* D' |$.
Now taking $X$ to be a local neighbourhood of the flopping locus, and taking
a small deformation $\mathcal X\to B$ as before, we obtain families $\mathcal X'$,
$\widetilde \mathcal X $, $\mathcal D$, $\mathcal E$, $\mathcal F$ and $\mathcal Z$ over B (corresponding to
$X'$, $\widetilde X$, $D$, $E$, $F$ and $Z$). For ease of notation, we shall use
$\pi_1$ and $\pi_2$ also for the morphisms of families $\widetilde \mathcal X\to
\mathcal X$, respectively $\widetilde \mathcal X\to \mathcal X'$. Applying \cite{3},
Theorem~10.2 to the family of vector bundles $(\pi_2^* T_{\mathcal X'/B})\rest{\mathcal F}$
on the scheme $\mathcal F$ over $B$ yields that $c_2((\pi_2^*
T_{\mathcal X'/B})\rest{F_t})$ is independent of $t\in B$. Noting that $\widetilde\mathcal D
\hookrightarrow \widetilde\mathcal X$ is a regular embedding, we apply the same
theorem to the fibre square
\[
\renewcommand{\arraystretch}{1.3}
\begin{matrix}
\widetilde\mathcal D \times_{\widetilde \mathcal X} \mathcal E & \longrightarrow & \mathcal E \\
\downarrow && \downarrow \\
\widetilde\mathcal D &\longrightarrow & \widetilde\mathcal X
\end{matrix}
\]
and the cycle $\mathcal Z$ on $\mathcal E$. This yields that $(Z_t\cdot\widetilde D_t)_{E_t}$
on $E_t$ is independent of $t\in B$, where by definition
\[
Z_t=c_2(\pi_2^* T_{\mathcal X'/B})\rest{X_t}-c_2(\pi_1^* T_{\mathcal X /B})
\rest{X_t}.
\]
Thus the local contribution to $D'(t)\cdot c_2(X_t')$ is
well-defined and independent of $t$, and so we need only make the local
calculation for generic $t$ (where the exceptional locus of the flop consists of
disjoint $(-1,-1)$-curves). This calculation may be found in \cite{1}, (7.4).
\end{pf}
\begin{spec} There are reasons for believing that only the numbers $n_1$
and $n_2$ are nonzero, and hence that the Gromov--Witten invariants
associated to classes $m\eta$ for $m>2$ all arise from multiple covers. If
this speculation is true, then the changes under flopping to the cubic form
and the linear form would be determined by these two integers, and
conversely.
\end{spec}
\section{Type~III contractions and Gromov--Witten invariants}\label{sec2}
The main results of this paper concern the Gromov--Witten invariants
associated to classes of curves contracted under a Type~III primitive
contraction. Recall \cite{17} that a primitive contraction $\varphi\colon X\to\Xbar$ is of
Type~III if it contracts down an irreducible divisor $E$ to a curve of
singularities $C$. For $X$ a smooth Calabi--Yau threefold, such contractions
were studied in \cite{18}; in particular, it was shown there that the curve
$C$ is smooth and that $E$ is a conic bundle over $C$. We denote by
$2\eta\in H_2(X,\mathbb Z)/\operatorname{Tors}$ the numerical class of a fibre of $E$ over $C$. As
explained in the Introduction, we denote by $n_1$ and $n_2$ the Gromov--Witten
numbers associated to the classes $\eta$ and $2\eta$, where
$n_1=0$ if $E$ is a $\mathbb P^1$-bundle over $C$. If the generic fibre of $E$
over $C$ is reducible (consisting of two lines, each with class $\eta$), then,
except in two cases, it follows from the arguments of \cite{18}, \S4 that, by
making a global holomorphic deformation of the complex structure, we may
reduce down to the case where the generic fibre of $E$ over $C$ is
irreducible. The two exceptional cases are:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $g(C)=1$ and $E$ has no double fibres.
\item $g(C)=0$ and $E$ has two double fibres.
\end{enumerate}
However, Case~(a) is an {\em elliptic quasi-ruled} surface in the terminology
of \cite{18}, and hence disappears completely under a generic global
holomorphic deformation. In particular, we know that all the Gromov--Witten
invariants $\Phi_A$ are zero, for $A\in H_2(X,\mathbb Z)$ having numerical class
$m\eta$ for any $m>0$.
In Case~(b), $E$ is a nonnormal generalized
del Pezzo surface $\overline\F_{3;2}$ of
degree 7 (see \cite{18}). As argued there however, we may make a holomorphic
deformation in a neighbourhood of $E$ so that $E$ deforms to a {\em smooth}
del Pezzo surface of degree 7, and where the class $\eta$ is then represented
by either of two `lines' on the del Pezzo surface (which are $(-1,-1)$-curves
on the threefold); hence $n_1=2$. In fact, the smooth del Pezzo surface is
fibred over $\mathbb P^1$ with one singular (line pair) fibre. The arguments we
give below may be applied locally (more precisely with the global almost
complex stucture obtained by suitably patching the local small holomorphic
deformation on an open neighbourhood of $E$ with the original complex
structure), and the Gromov--Witten invariants may be calculated as if the
original contraction $\varphi$ had contracted such a smooth del Pezzo surface
of degree 7. In particular, $n_1=2$ comes from the two components of the
singular fibre (Theorem~\ref{thm_2.5}), and $n_2=-2$ is proved in
\S\ref{sec3} (see also Remark~\ref{rem_2.4}).
Let us therefore assume that the generic fibre of $E$ over $C$ is
irreducible, and so in particular $E\to C$ is obtained from a $\mathbb P^1$-bundle
over $C$ by means of blowups and blowdowns. Moreover $E$ itself is a conic
bundle over $C$, and so its singular fibres are either line pairs or double
lines.
\begin{lem} In the above notation, $E$ has only singularities on the
singular fibres of the map $E\to C$. When the singular fibre is a line pair,
we have an $\rA{n}$ singularity at the point where the two components
meet (we include here the possibility $n=0$ when the point is a smooth point
of $E$). When the singular fibre is a double line, we have a
$\rD{n}$ singularity on the fibre (here we need to include the case $n=2$,
where we in fact have two $\rA1$ singularities, and $n=3$, where we
have an $\rA3$ singularity).
\end{lem}
\begin{pf} The proof is obvious, once the correct statement has been found.
The statement of this result in \cite{17} omits (for fibre a double line)
the cases $\rD{n}$ for $n>2$.
\end{pf}
\begin{lem}\label{lem_2.2} Suppose that $E\to C$ as above has $a_r$ fibres
which are line pairs with an $\rA{r}$ singularity and $b_s$ fibres which are
double lines with a $\rD{s}$ singularity (for $r\ge0$ and $s\ge2$), then
\[
K_E^2=8(1-g)-\sum_{r\ge 0} a_r (r+1)-\sum_{s\ge 2} b_s s,
\]
where $g$ denotes the genus of $C$.
\end{lem}
This enables us to give a slick calculation of the Gromov--Witten invariants
when the base curve has genus $g>0$. In this case, it was shown in
\cite{17} that for a generic deformation of $X$, only finitely many fibres
from $E$ deform, and hence the Type~III contraction deforms to a Type~I
contraction. Thus Gromov--Witten numbers $n_1$ and $n_2$ may be defined as
in Section 1, and are nonnegative integers.
\begin{prop}\label{prop_2.3} When $g>0$, we have
\[
n_1=2\sum_{r\ge 0} a_r (r+1)+2\sum_{s\ge 2} b_s s \quad
\text{and} \quad n_2=2g-2.
\]
\end{prop}
\begin{pf} We take a generic 1-parameter
deformation of $X$, for which the
Type~III contraction deforms to a Type~I contraction. We therefore have a
diagram
\[
\renewcommand{\arraystretch}{1.3}
\begin{matrix}
\mathcal X &\longrightarrow &\overline\mathcal X \\
\downarrow && \downarrow \\
\Delta &=& \Delta
\end{matrix}
\]
where $\Delta\subset\mathbb C$ denotes a small disc. Since the singular locus of
$\overline\mathcal X $ consists only of curves of cDV singularities, we may again
apply \cite{8}, Theorem~11.10 to deduce the existence of a (smooth)
flopped fourfold $\mathcal X'\to \overline\mathcal X$. The induced family $\mathcal X'\to\Delta$ is
given generically by flopping the fibres, and at $t=0$ it is easily checked
that $X_0'\cong X_0$; this operation is often called an {\em elementary
transformation} on the family. Identifying the groups $H^2 (X_t,\mathbb Z)\cong
H^2 (X_t',\mathbb Z)$ as before, this has the effect (at $t=0$) of sending $E$ to
$-E$ (cf.\ the discussion in \cite{5}, \S3.3). So if $E'$ denotes the class
in $H^2(X_t',\mathbb Z)$ corresponding to the class $E$ in
$H^2(X_t,\mathbb Z)$, we have $(E')^3=-E^3$. For $t\ne 0$, we just have a flop, and
so $(E')^3$ can be calculated from equation (2.1.1), namely
$(E')^3=E^3+n_1+8n_2$. Therefore, using Lemma~\ref{lem_2.2}
\[
n_1+8n_2=-2E^3=16(g-1)+2\sum_{r\ge0}a_r(r+1)+2\sum_{s\ge2}b_ss.
\]
Similarly, we have $c_2(X')\cdot E'=- c_2(X)\cdot E$, and so from
equation (2.1.2) it follows that $2n_1+4n_2=2 c_2\cdot E$. An
easy calculation of the right-hand side then provides the second equation
\[
2n_1+4n_2=8(g-1)+4\sum_{r\ge0}a_r(r+1)+4\sum_{s\ge2}b_ss.
\]
Solving for $n_1$ and $n_2$ from these two equations gives the desired
result. \end{pf}
\begin{rem}\label{rem_2.4} This result remains true even when
$g=0$, although the slick proof given above is no longer valid. The formula
for $n_1$ is checked in Theorem~\ref{thm_2.5} by local deformation arguments
(for which the genus $g$ is irrelevant), showing that the contribution to
$n_1$ from a line pair fibre with $\rA{r}$ singularity is $2(r+1)$, and
from a double line fibre with $\rD{s}$ singularity is $2s$. Let $A\in
H_2(X,\mathbb Z)$ denote the class of a fibre of $E\to C$. Observe that any
pseudo\-holomorphic curve representing the numerical class $\eta$ will be a
component of a singular fibre of $E\to C$. Moreover, the components $l$ of a
singular fibre represent the same class in $H_2(X,\mathbb Z)$, and so in
particular twice this class is $A$. Thus the Aspinwall--Morrison formula (as
proved in \cite{15}) yields the contribution to the Gromov--Witten
invariants $\Phi_A (D,D,D)$ from double covers, purely in terms of $n_1$ and
$D\cdot A$. The difference may be regarded as the contribution to $\Phi_A
(D,D,D)$ from simple maps, and taking this to be $n_2 (D\cdot A)^3$
determines the number $n_2$ (in \S\ref{sec3}, we shall see how $n_2$ may be
determined directly from the moduli space of simple stable holomorphic
maps). If $g>0$, the above argument shows that this is in agreement with our
previous definition, and yields moreover the equality $n_2=2g-2$. The fact
that $n_2=-2$ when $g=0$ requires a rather more subtle argument involving
technical machinery -- see Theorem~\ref{thm_3.1}. I remark that the value
$n_2=-2$ is needed in physics, and that there is also a physics argument
justifying it (see \cite{4}, \S5.2 and \cite{5}, \S3.3) -- essentially, it
comes down to a statement about the A-model 3-point correlation functions.
In \S\ref{sec3} below, we give a rigorous mathematical proof of the
assertion.
\end{rem}
\begin{thm}\label{thm_2.5} The formula for $n_1$ in
Proposition~\ref{prop_2.3} is valid irrespective of the value of the genus
$g=g(C)$.
\end{thm}
\begin{pf} By making a holomorphic deformation of the complex structure on
an open neighbourhood $U$ in $X$ of the singular fibre $Z$ of $E\to C$, we
may calculate the contribution to $n_1$ from that singular fibre -- see
\cite{18}, (4.1). The deformation of complex structure is obtained as in
\cite{18} by considering the one dimensional family of Du Val singularities
in $\overline X$, and deforming this family locally in a suitable neighbourhood
$\overline U$ of the dissident point. Our assumption is that the family
$\overline U\to\Delta$ has just an $\rA1$ singularity on $\overline U_t$ for $t\ne 0$,
and we may assume also that $\overline U\to \Delta$ is a good representative (in the
sense explained in \cite{18}). The open neighbourhood $U$ is then the blowup
of $\overline U$ in the smooth curve of Du Val singularities (\cite{18}, p.\ 569).
The contribution to $n_1$ may be calculated locally, and will not change
when we make small holomorphic deformations of the complex structure on $U$,
which in turn corresponds to making small deformations to the family
$\overline U\to\Delta$.
First we consider the case where the singular fibre $Z$ is a line pair --
from this, it will follow that the dissident singularity on $\overline U$ is a
$\rcA{n}$ singularity with $n>1$, and that $\overline U$ has a local analytic
equation of the form
\[
x^2+y^2+z^{n+1}+tg(x,y,z,t)=0
\]
in $\mathbb C^3\times\Delta$ (here $t$ is a local coordinate on $\Delta$, and $x=y=z=0$
the curve $C$ of singularities). For $t\ne 0$, we have an $\rA1$
surface singularity, which implies that $g$ must contain a term of the form
$t^r z^2$ for some $r\ge 0$. By an appropriate analytic change of
coordinates, we may then assume that $\overline U$ has a local analytic equation of
the form
\[
x^2+y^2+z^{n+1}+t^{r+1}z^2+t h(x,y,z,t)=0,
\]
where $h$ consists of terms which are at least cubic in $x,y,z$. By making a
small deformation of the family $\overline U\to \Delta$, we may reduce to the case
$n=2$, that is, $\overline U$ having local equation $x^2+y^2+z^3+t^{r+1}z^2+th=0$.
At this stage, we could in fact also drop the term $th$ (an easy check using
the versal deformation family of an $\rA2$ singularity), but this will
not be needed.
We now make a further small deformation to get $\overline U_{\varepsilon} \subset
\mathbb C^3 \times \Delta $ given by a polynomial
\[
x^2+y^2+z^3+t^{r+1} z^2+\varepsilon z^2+th=x^2+y^2+z^2 (z+t^{r+1}+\varepsilon)+th \ .
\]
This then has $r+1$ values of $t$ for which the singularity is an $\rA2$
singularity -- for other values of $t$, it is an $\rA1$ singularity. If we
blow up the singular locus of $\overline U_{\varepsilon}$, we therefore obtain a smooth
exceptional divisor for which $r+1$ of the fibres over $\Delta$ are line pairs.
By the argument of \cite{18}, (4.1), this splitting of the singular fibre
into $r+1$ line pair singular fibres of the simplest type can be achieved by
a local holomorphic deformation on a suitable open neighbourhood of the
fibre in the original threefold $X$.
It is however clear that a line pair coming from a dissident $\rcA2$
singularity of the above type contributes precisely two to the
Gromov--Witten number $n_1$ -- one for each line in the fibre. In terms of
equations, we have a local equation for $\overline X$ of the form $x^2+y^2+z^3+w
z^2=0$; deforming this to say $x^2+y^2+z^3+w z^2+\varepsilon w=0$, we get two simple
nodes, and hence two disjoint $(-1,-1)$-curves on the resolution.
The argument of \cite{18}, (4.1) shows that the Gromov--Witten number $n_1$
may be calculated purely from these local contributions, and so the total
contribution to $n_1$ from the line pair singular fibre of $E$ with
$\rA{r}$ singularity is indeed $2(r+1)$, as claimed.
For the case of the singular fibre $Z$ of $E$ being a double line, the
dissident singularity must be $\rcE6$, $\rcE7$, $\rcE8$, or $\rcD{n}$ for
$n\ge4$. Thus $\overline U$ has a local analytic equation of the form
$f(x,y,z)+tg(x,y,z,t)$ in $\mathbb C^3\times\Delta$ for $f$ a polynomial of the
appropriate type ($t$ a local coordinate on $\Delta$, and $x=y=z=0$ the curve
of singularities). To simplify matters, we may deform $f$ to a polynomial
defining a $\rD4$ singularity, and hence make a small deformation of the
family to one in which the dissident singularity is of type $\rcD4$. We then
have a local analytic equation of the form
\[
x^2+y^2 z+z^3+t g(x,y,z,t)=0.
\]
For $t\ne 0$, we have an $\rA1$ singularity, and so the terms of $g$ must be
at least quadratic in $x,y,z$. Moreover, by changing the $x$-coordinate, we may
take the equation to be of the form
\[
x^2+y^2 z+z^3+t^a y^2+t^b yz+t^c z^2+t h(x,y,z,t)=0,
\]
with $a,b,c$ positive, and where the terms of $h$ are at least cubic in $x,y,z$.
The fact that the blowup $U$ of $\overline U$ in $C$ is smooth is easily checked to
imply that $a=1$. Since
\[
ty^2+2 t^b yz=t(y+t^{b-1}z)^2-t^{2b-1}z^2,
\]
we have an obvious change of $y$-coordinate which brings the
equation into the form
\[
x^2+y^2 z+z^3 +t y^2+t^r z^2+t h_1
(x,y,z,t)=0,
\]
where $r=\operatorname{min} \{ c, 2b-1 \}$ and $h_1$ has the same
property as $h$.
When we blow up $\overline U$ along the curve $x=y=z=0$, we obtain an exceptional locus
$E$ with a double fibre over $t=0$, on which we have a $\rD{r+1}$ singularity (including the case $r=1$ of two $\rA1$ singularities,
and $r=2$ of an $\rA3$ singularity). Moreover, this was also true of our
original family, since the small deformation of $f$ we made did not affect the
local equation of the exceptional locus.
Moreover, by adding a term $\varepsilon_1 y^2+\varepsilon_2 z^2$, we may
deform our previous equation to one of the form
\[
x^2+y^2(z+t+\varepsilon_1)+z^2(z+t^r+\varepsilon_2)+th_1(x,y,z,t)=0.
\]
When $ t+\varepsilon_1=0$, we have an $\rA3$ singularity, and when $t^r+\varepsilon_2=0$,
an $\rA2$ singularity. Moreover, when we blow up the singular locus of this
deformed family, the resulting exceptional divisor is smooth and has line
pair fibres for these $r+1$ values of $t$. Thus, as seen above, the
contribution to $n_1$ from the original singular fibre (a double line with a
$\rD{r+1}$ singularity) is $2(r+1)$ as claimed.
\end{pf}
\section{Calculation of $n_2$ for Type~III contractions}\label{sec3}
Let $\varphi\colon X\to\Xbar $ be a Type~III contraction on a Calabi--Yau threefold $X$,
which contracts a divisor $E$ to a (smooth) curve $C$ of genus $g$. When
$g>0$, it was proved in Proposition~\ref{prop_2.3} that the Gromov--Witten
number $n_2$ (defined for arbitrary genus via Remark~\ref{rem_2.4}) is
$2g-2$. The purpose of this Section is to extend this result to include the
case $g=0$ ($C$ is isomorphic to $\mathbb P^1$), and to prove $n_2=2g-2$ in
general.
Arguing as in \cite{18}, it is clear that the desired result is a local one,
depending only on a neighbourhood of the exceptional divisor $E$. As remarked
in \S\ref{sec2}, we may then always reduce down to the case that the generic
fibre of $E\to C$ is irreducible. If all the fibres of $E\to C$ are smooth (so
$E$ is a $\mathbb P^1$-bundle over $C$), the fact that $n_2=2g-2$ was proved in
Proposition 5.7 of \cite{11}, using a cobordism argument. This latter result
was extended by Ruan in \cite{13}, Proposition~2.10, using the theory of
moduli spaces of stable maps and the virtual neighbourhood technique
(cf.~\cite{2,9}). If the singular fibres of $E\to C$ are line pairs, Ruan's
result applies directly. We prove below that the linearized Cauchy--Riemann
operator has constant corank for the stable (unmarked) rational curves given
by the fibres of $E$ over $C$, and hence by Ruan's result that there is an
obstruction bundle $\mathcal H$ on $C$, with $n_2$ determined by the Euler class of
$\mathcal H$. By Dolbeault cohomology, there is a natural identification of
$\mathcal H$ with the cotangent bundle $T_C^*$ on $C$, and hence the formula for $n_2$
follows. We note however that for Ruan's result to hold, we do not need an
integrable almost complex structure on $X$. Provided we have a natural
identification between the cokernel of the linearized Cauchy--Riemann operator
and the cotangent space at the corresponding point of $C$, we can still deduce
that $n_2=2g-2$. In the general case of a Type~III contraction which has
double fibres, we show below that we can make a small local deformation of the
almost complex structure on $X$ so that $E$ deforms to a family of
pseudo\-holomorphic rational curves over $C$ with at worst line pair singular
fibres, and for which Ruan's method applies.
\begin{thm}\label{thm_3.1} For any Type~III contraction $\varphi\colon X\to\Xbar $,
the Gromov--Witten number $n_2=2g-2$.
\end{thm}
\begin{pf} We saw above that we may assume that the generic fibre of $E\to
C$ is irreducible. Furthermore, we initially assume also that the singular
fibres are all line pairs, and later reduce the general case to this one.
We let $J$ denote the almost complex structure on $X$, which we know is
integrable (at least in a neighbourhood of $E$), and tamed by a symplectic
form $\omega$. Let $A\in H_2(X,\mathbb Z)$ be the class of a fibre of $E\to C$.
Adopting the notation from \cite{13}, we consider the moduli space
$\overline{\mathcal M}_A(X,J)=\overline{\mathcal M}_A(X,0,0,J)$ of stable unmarked rational
holomorphic maps, a compactification of the space of (rigidified)
pseudo\-holomorphic maps $\mathbb C\mathbb P^1\to X$, representing the class
$A$. The theory of stable maps, as explained in Section 3 of \cite{13}, goes
through for unmarked stable maps, by taking each component of the domain as
a bubble component, and adding marked points (in addition to the double
points) as in \cite{13} in order to stabilize the components (thus taking a
local slice of the automorphism group).
In the case that all the singular fibres of $E\to C$ are line pairs,
$\overline{\mathcal M}_A (X,J)$ has two components, one corresponding to simple maps and
the other to double covers. It is now a simple application of Gromov
compactness to see that these two components are disjoint, since a sequence
of double cover maps cannot converge to a simple map. A similar argument
will show that for all almost complex structures $J_t$ in some neighbourhood
of $J=J_0$, the moduli space $\overline{\mathcal M}_A (X,J_t)$ will consist of two
disjoint components, one corresponding to the simple maps and the other to
the double covers.
Since any stable unmarked rational holomorphic map must be an embedding, it
is clear that the component $\overline{\mathcal M}'_A (X,J)$ corresponding to the simple
maps can be identified naturally with the smooth base curve $C$, and that
for all almost complex structures in some neighbourhood of $J=J_0$, the
moduli space $\overline{\mathcal M}'_A (X,J_t)$ of simple unmarked stable holomorphic maps
is compact. The Gromov--Witten invariant $n_2$ that we seek can then be
defined via Ruan's virtual neighbourhood invariant $\mu_{\mathcal S}$, and may be
evaluated on $(X,J)$ by using \cite{13}, Proposition~2.10.
Let us now go into more details of this. We consider $C^{\infty}$ stable
maps $f\in\overline B_A (X)=\overline B_A (X,0,0)$ in the sense of \cite{13}, Definition
3.1, where Ruan shows later in the same Section that the naturally
stratified space $\overline B_A (X)$ satisfies a property which he calls {\em
virtual neighbourhood technique admissable} or {\em VNA}, and as he says, for
the purposes of the virtual neighbourhood construction, behaves as if it were a
Banach $V$-manifold. Since any simple marked holomorphic stable map $f$ in
$\overline{\mathcal M}'_A (X,J)$ is forced to be an embedding, we may restrict our
attention to $C^{\infty}$ stable maps whose domain $\Sigma$ comprises at most
two $\mathbb P^1$s. We stratify $\overline B_A (X)$ according to the combinatorial type
$D$ of the domain $\Sigma$. Thus any $f\in \overline{\mathcal M}'_A (X,J)$ belongs to one of
two strata of $\overline B_A (X)$.
In general, for $k$-pointed $C^{\infty}$ stable maps of genus $g$, Ruan shows
that for any given combinatorial type $D$, the substratum $ B_D (X,g,k)$ is
a Hausdorff Frechet V-manifold (\cite{13}, Proposition 3.6). As mentioned
above, he needs to add extra marked points in order to stabilize the
nonstable components of the domain $\Sigma$, thus taking a local slice of the
action of the automorphism group on the unstable marked components of $\Sigma$.
Moreover, the tangent space $T_f B_D (X,g,k)$ is identified with $\Omega^0 (f^*
T_X)$, as defined in his equation \cite{13}, (3.29). The tangent space $T_f
\overline B_A (X,g,k)$ can then be defined as $T_f B_D (X,g,k) \times \mathbb C_f $,
where $\mathbb C_f$ is the space of gluing parameters (see \cite{13}, equation
before (3.67)).
In our case, however, things are a bit simpler. Given $f\in \overline{\mathcal M}'_A
(X,J)$ with domain $\Sigma$ consisting of two $\mathbb P^1$s, the tangent space $T_f
\overline B_A (X)$ is of the form $\Omega^0 (f^* T_X) \times \mathbb C$, and we have a
neighbourhood $\widetilde U_f$ of $f$ in $\overline B_A (X)$ defined by
\cite{13}, (3.43), consisting of stable maps $\overline f^{v,w}$ parametrized
locally by
\[
\bigl\{w\in\Omega^0(f^*T_X)\ ;\ \|w\|_{C^1}<\varepsilon'\bigr\}
\]
(corresponding to deformations within the stratum $B_D(X)$), and by
$v\in\mathbb C_f^\varepsilon $ (an $\varepsilon$-ball in $\mathbb C_f=\mathbb C$ giving the gluing parameter at
the double point). This then corresponds to the above decomposition of
$T_f\overline B_A (X)$ into two factors. On the first factor, the linearization
$D_f\overline\partial_J $ of the Cauchy--Riemann operator restricts to
\[
\overline\partial_{J,f}\colon\Omega^0(f^*T_X)\to\Omega^{0,1}(f^*T_X)
\]
in the notation of \cite{13}. The index of this operator may be calculated
using Riemann--Roch on each component of $\Sigma$ (cf.~the proof of Lemma
3.16 in \cite{13}, suitably modified to take account of the extra marked
points), and is seen to be $-2$.
Let us now consider the stable maps $f^v=f^{v,0}$ for
$v\in\mathbb C_f^\varepsilon\setminus\{0\}$. These are stable maps $\mathbb C\mathbb P^1\to X$ which
differ from $f$ only in small discs around the double point, and in this
sense are approximately holomorphic. Set $v=r e^{i\theta}$; then the
gluing to get $f^v\colon \Sigma^v\to X$ is only performed in discs around the
double point of radius $2r^2 /\rho$ in the two components ($\rho$ a suitable
constant). It can then be checked for any $2<p<4$ that $\| \overline\partial_J (f^v)\|_{L^p_1} \le Cr^{4/p}$ (see \cite{13} Lemma 3.23, and
\cite{10} Lemma A.4.3), from which it follows that the linearization
\[
L_A=D_f \overline\partial_J
\]
of the Cauchy--Riemann operator should be taken as zero on the factor $\mathbb C_f$
in $T_f \overline B_A (X)$. Thus we deduce that the index of $L_A$ is zero, and
that $\operatorname{coker} L_A$ is same as the cokernel of $\overline\partial_{J,f}\colon \Omega^0 (f^*
T_X)\to \Omega^{0,1} (f^* T_X)$, which by Dolbeault cohomology may be
identified as
\[
H^1(f^*T_X)=H^1(Z,T_X{}\rest Z),
\]
where $Z$ is the fibre of $E\to C$ (over a point $x\in C$) corresponding to
the image of $f$.
We note that these are exactly the same results as are obtained in the smooth
case, when $\Sigma$ consists of a single $\mathbb P^1$. Here, we need to add three
marked points to stabilize $\Sigma$, and Riemann--Roch then gives immediately
that the index of $L_A$ is zero.
Observe that $Z$ is a complete intersection in $X$, and so for our purposes
is as good as a smooth curve. Via the obvious exact sequence,
$H^1(T_X{}\rest Z)$ may be naturally identified with $H^1(N_{Z/X})$, which
in turn may be naturally identified with $H^0 (N_{Z/X})^*$ (since
$K_Z=\bigwedge^2 N_{Z/X}$, we have a perfect pairing $H^0 (N_{Z/X}) \times
H^1(N_{Z/X})\to H^1(K_Z)\cong \mathbb C$). Observing that $H^0 (N_{Z/X})=H^0
(\mathcal O_Z\oplus\mathcal O_Z (E))\cong \mathbb C$, we know that $\operatorname{coker} L_A $ has complex
dimension one and is naturally identified with $T^*_{C,x}$, the dual of the
tangent space at $x$ to the Hilbert scheme component $C$. This we have seen
is true for all $f\in\overline{\mathcal M}'_A(X,J)$.
We now apply \cite{13}, Proposition~2.10, (2) to our set-up, where
$C=\overline{\mathcal M}'_A (X,J)=\mathcal M_{\mathcal S}=\mathcal S^{-1}(0)$ for $\mathcal S$ the Cauchy--Riemann
section of $\overline \mathcal F_A(X)$ (as constructed in \cite{13}, \S3) over a
suitable neighbourhood of $\mathcal M_{\mathcal S}$ in $\overline B_A(X)$. The above calculations
verify that the conditions of Proposition~2.10, (2) are satisfied, with
$\ind(L_A)=0$, $\operatorname{dim}(\operatorname{coker} L_A)=2$ and $\operatorname{dim} (\mathcal M_{\mathcal S})=2$. Moreover, we
deduce that the obstruction bundle $\mathcal H$ on $\mathcal M_{\mathcal S}$ is just the cotangent
bundle $T_C^*$ on $C$.
The Gromov--Witten number $n_2$ may then be defined to be $\mu_{\mathcal S}(1)$.
It follows from the basic Theorem~4.2 from \cite{13} that this is
independent of any choice of tamed almost complex structure and is a
symplectic deformation invariant. Thus by considering a small deformation of
the almost complex structure and using \cite{13}, Proposition~2.10, (1), it
is the invariant $n_2$ that we seek. Applying Ruan's crucial
Proposition~2.10, (2), the invariant can be expressed as
\[
\mu_{\mathcal S}(1)=\int_{\mathcal M'_A (X,J)} e(T_C^*),
\]
from which it follows that $n_2=2g-2$ as claimed.
The general case (where $E\to C$ also has double fibres) can now be reduced
to the case considered above. Suppose we have a point $Q\in C$ for which the
corresponding fibre is a double line. We choose an open disc $\Delta \subset C$
with centre $Q$, and a neighbourhood $U$ of $Z$ in $X$, with $U$ fibred over
$\Delta$, the fibre $U_0$ over $Q$ containing the fibre $Z$. Letting $\overline U\to
\Delta$ denote the image of $U$ under $\varphi$, a family of surface Du Val
singularities, we make a small deformation $\overline\mathcal U\to \Delta'$ of $\overline U$,
as in the proof of Theorem~\ref{thm_2.5} of this paper, and in this way
obtain a holomorphic deformation $\mathcal U\to \Delta'$ of $U$ under which
$E_0=E\rest{\Delta}$ deforms to a family of surfaces $E_t$ ($t\in \Delta'$), all
fibred over $\Delta$, and with at worst line pair singular fibres for $t\ne 0$.
Considering $\overline\mathcal U\to \Delta \times \Delta'$ as a two parameter deformation
of the surface singularity $ \overline U_0$, we may take a good representative and
apply Ehresmann's fibration theorem (with boundary) to the corresponding
resolution $\mathcal U\to\Delta\times\Delta'$ (cf.\ \cite{18}, proof of Lemma 4.1).
In this way, we may assume that $\mathcal U\to \Delta \times \Delta'$ is differentiably
trivial over the base. In particular, the family $\mathcal U\to\Delta'$ is also
differentiably trivial, and hence determines a holomorphic deformation of
the complex structure on a fixed neighbourhood $U$ of $Z$, where $U\to \Delta$
is also differentiably trivial.
We perform this procedure for each singular fibre $Z_1,\dots,Z_N $ of
$E\to C$, obtaining, for each $i$, an open neighbourhood $U_i$ of $Z_i$
fibred over $\Delta_i \subset C$, and a holomorphic complex structure $J_i$ on
$U_i$ with the properties explained above (of course, if $Z_i$ is a line
pair, we may take $J_i$ to be the original complex structure $J$). Let
$\frac{1}{2}\overline\Delta_i$ denote the closed subdisc of $\Delta_i$ with half the
radius, $C^*=C\setminus\bigcup_{i=1}^N\frac{1}{2}\overline\Delta_i$, and
$E^*=E\rest{C^*}\to C^*$ the corresponding open subset of $E$. We then take
a tubular neighbourhood $U^*\to C^*$ of $E^*\to C^*$, equipped with the
original complex structure $J$. By taking deformations to be sufficiently
small and shrinking radii of tubular neighbourhoods if necessary, all these
different complex structures may be patched together in a $C^{\infty}$ way
(tamed by the symplectic form) over the overlaps in $C$. In this manner, we
obtain an open neighbourhood $W$ of $E$ in $X$, and a tamed almost complex
structure $J'$ on $W$, which is a small deformation of the original complex
structure $J$ and which satisfies the following properties:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item Each singular fibre $Z_i$ of $E\to C$ has an open neighbourhood $U_i
\subset W$ fibred over $\Delta_i \subset C$ with $J'$ inducing an integrable
complex structure on each fibre (thus $U_i\to \Delta_i$ is a $C^{\infty}$
family of holomorphic surface neighbourhoods).
\item The almost complex structure $J'$ is integrable in a smaller
neighbourhood $U'_i \subset U_i$ of each singular fibre, with the
corresponding family $U'_i\to \Delta'_i$ being holomorphic.
\item On the complement of $\bigcup U_i$ in $W$, the almost complex
structure $J'$ coincides with the original complex structure $J$.
\item $E$ deforms to a $C^{\infty}$ family of pseudo\-holomorphic rational
curves $E'\to C$ in $(W,J')$, with generic fibre $\mathbb C\mathbb P^1$ and the only
singular fibres being line pairs. Moreover, we may assume that any such
singular fibre is contained in one of the above open sets $U'_i$.
\end{enumerate}
Of course, we may now patch $J'$ on $W$ with the original complex structure $J$
on $X$ to get a global tamed almost complex structure on $X$, which we shall also
denote by $J'$. Provided we have taken our deformations sufficiently small, the
standard argument via Gromov compactness ensures that any pseudo\-holomorphic
stable map representing the class $A$ has image contained in a fibre of $E'\to
C$.
The theory of \cite{13} applies equally well to almost complex structures, and hence
to our almost complex manifold $X'$ with complex structure $J'$. Clearly, all
the calculations remain unchanged for stable maps whose image (a fibre of
$E'\to C$) has a neighbourhood on which $J'$ is integrable, and in particular
this includes all the singular fibres. Suppose therefore that $f\colon \mathbb C\mathbb P^1\to
X'$ is a pseudo\-holomorphic rational curve whose image $Z$ is contained in an
overlap $U_i \setminus U'_i$ (where $J'$ may be nonintegrable). The linearized
Cauchy--Riemann operator $L_A$ still has index zero, since by the argument
of \cite{10}, p.~24, the calculation via Riemann--Roch continues to give the
correct value. We therefore need to show that $\operatorname{coker} L_A$ is still
identified naturally as $T^*_{C,x}$, and hence that the obstruction bundle
is $\mathcal H=T^*_C$ as before.
Setting $U=U_i$ and $\Delta=\Delta_i$, we know that $U\to \Delta$ is locally
(around the image $Z$ of $f$) a $C^{\infty}$ family of holomorphic surface
neighbourhoods. Moreover, the linearized
Cauchy--Riemann operator $L_A = D_f\colon C^{\infty} (f^*T_U)\to \Omega^{0,1}
(f^*T_U)$ fits into the
following commutative diagram (with exact rows)
\[
\renewcommand{\arraystretch}{1.3}
\begin{matrix}
0&\to&C^{\infty}(f^*T_{U/\Delta})&\to
&C^{\infty}(f^*T_U)&\to&C^{\infty}(g^*T_{\Delta})&\to&0 \\
&&\Bigm\downarrow\overline\partial_{f}&&\Bigm\downarrow D_f&&\Bigm\downarrow&& \\
0&\to&\Omega^{0,1}(f^*T_{U/\Delta})&\to
&\Omega^{0,1}(f^*T_U)&\to&\Omega^{0,1}(g^*T_{\Delta})&\to&0
\end{matrix}
\]
where $g$ is the constant map on $\mathbb C\mathbb P^1$ with image the point $x\in \Delta$,
and where the fibre of $E'$ over $x$ is $Z$. Let us denote by $U_x$ the
corresponding holomorphic surface neighbourhood, the fibre of $U$ over $x$. The
cokernel of
\[
\overline\partial_{f}\colon C^{\infty}(f^*T_{U/\Delta})\to\Omega^{0,1}(f^*T_{U/\Delta})
\]
is then naturally identified via Dolbeault cohomology with
$H^1(T_{U_x}{}\rest Z)\cong H^1(N_{Z/U_x})$. This latter space is in turn
naturally identified with $H^1(N_f)\cong H^0(N_f)^*\cong T^*_{C,x}$.
I claim now that $J'$ may be found as above for which $\operatorname{coker} L_A$ has the
correct dimension (namely real dimension two) for all fibres of $E'\to C$.
Since $L_A$ has index zero and $\ker L_A$ has dimension at least two, we
need to show that that the dimension of $\operatorname{coker} L_A$ is not more than two.
This follows by a Gromov compactness argument. Suppose that the dimension is
too big for some fibre of $E'\to C$, however close we take $J'$ to $J$. We
can then find sequences of almost complex structures $J'_\nu$ (with the
properties (a)--(d) described above) converging to $J=J_0$, and
pseudo\-holomorphic rational curves $f_\nu\colon\mathbb C\mathbb P^1\to (X, J'_\nu)$ at
which $\operatorname{coker} L_A$ has real dimension $>2$. By construction, the image of
such a map is not contained in any $U'_i$ (since $J'_\nu$ would then be
integrable on some neighbourhood of the image, and then we know that $\operatorname{coker}
L_A$ has the correct dimension). Thus the image of $f_\nu$ has nontrivial
intersection with the compact set $X \setminus \bigcup U'_i$. By Gromov
compactness, the $f_\nu$ may be assumed to converge to a pseudo\-holomorphic
rational curve on $(X,J)$ whose image is not contained in any $U'_i$. This is
therefore just an embedding $f\colon\mathbb C\mathbb P^1\to (X, J)$ of some smooth fibre
of $E\to C$, at which we know that $\operatorname{coker} L_A$ has real dimension
precisely two; this then is a contradiction. A similar argument, via Gromov
compactness, then yields the fact that $J'$ may be found as above such that
the linear map $\operatorname{coker} (\overline\partial_{f})\to \operatorname{coker} (D_f)$ is an isomorphism
for all smooth fibres of $E'\to C$, since this is
true for all the smooth fibres of $E\to C$ on $(X,J)$.
For such a $J'$, we deduce that $\operatorname{coker} L_A$ is naturally identified with
$T^*_{C,x}$ for all fibres, and hence the obstruction bundle identified as
$T^*_C$. The previous argument may then be applied directly with the almost
complex structure $J'$, showing that the symplectic invariant $n_2$ is
$2g-2$ in general. The proof of Theorem~\ref{thm_3.1} is now complete.
\end{pf}
\section{An application to symplectic deformations of Calabi--Yaus}
\label{sec4}
If $X$ is a Calabi--Yau threefold which is general in moduli, we know that
any codimension one face of its nef cone $\overline{\mathcal K} (X)$ (not contained in the
cubic cone $W^*$) corresponds to a primitive birational contraction
$\varphi\colon X\to\Xbar $ of Type~I, II or $\text{III}_0$, where Type~$\text{III}_0$
denotes a Type~III contraction for which the genus of the curve $C$ of
singularities on $\overline X$ is zero.
In \cite{18}, we studied Calabi--Yau threefolds which are symplectic
deformations of each other. One of the results proved there (Theorem~2) said
that if $X_1$ and $X_2$ are Calabi--Yau threefolds, general in their complex
moduli, which are symplectic deformations of each other, then their K\"ahler
cones are the same. The proof of this essentially came down to showing that
certain Gromov--Witten invariants associated to exceptional classes were
nonzero. Using the much more precise information obtained in this paper, we
are able to make a stronger statement.
\begin{cor}\label{cor4.1} With the notation as above, any codimension one
face (not contained in $W^*$) of $\overline{\mathcal K} (X_1)=\overline{\mathcal K} (X_2)$ has the same
contraction type (Type~I, II or $\text{\sl III}_0$) on $X_1$ as on
$X_2$.
\end{cor}
\begin{pf} The fact that Type~II faces correspond is easy, since for $D$ in
the interior of such a face, the quadratic form $q(L)=D\cdot L^2$ is
degenerate, which is not the case for $D$ in the interior of a Type~I or
Type~$\text{III}_0$ face. Stating it another way, if we consider the Hessian
form associated to the topological cubic form $\mu$, then $h$ is a form of
degree $\rho=b_2$ which has a linear factor corresponding to each Type~II
face. Thus the condition that a face is of Type~II is topologically
determined.
The result will therefore follow if we can show that a face of the nef cone
which is Type~I for one of the Calabi--Yau threefolds is not of
Type~$\text{III}_0$ for the other. However, for a Type~I face, we saw in
\S\ref{sec1} that $n_d$ is always nonnegative; for a Type~$\text{III}_0$
face, we proved in Theorem~\ref{thm_3.1} that $n_2=-2$. Since Gromov--Witten
invariants are invariant under symplectic deformations, the result is proved.
\end{pf}
\begin{rem} It is still an open question whether there exist examples of
Calabi--Yau threefolds $X_1$ and $X_2$ which are symplectic deformations of
each other but not in the same algebraic family.
\end{rem}
|
1997-07-09T19:51:21 | 9707 | alg-geom/9707009 | en | https://arxiv.org/abs/alg-geom/9707009 | [
"alg-geom",
"hep-th",
"math.AG",
"math.QA",
"q-alg"
] | alg-geom/9707009 | Martin Markl | Martin Markl | Simplex, associahedron, and cyclohedron | 42 pages, LaTeX, article 12pt + leqno style | null | null | null | null | The aim of the paper is to give an `elementary' introduction to the theory of
modules over operads and discuss three prominent examples of these objects -
simplex, associahedron (= the Stasheff polyhedron) and cyclohedron (= the
compactification of the space of configurations of points on the circle).
Keywords: (right) module over an operad, module associated to a cyclic
operad, Koszul module over an operad.
| [
{
"version": "v1",
"created": "Wed, 9 Jul 1997 17:55:00 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Markl",
"Martin",
""
]
] | alg-geom | \section{\@startsection {section}{1}{\z@ }%
{-3.5ex plus -1ex minus -.2ex}{2.3ex plus .2ex}{\bf }}
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\begin{document}
\pagestyle{myheadings}
\bibliographystyle{plain}
\baselineskip20pt plus 2pt minus 1pt
\parskip3pt plus 1pt minus .5pt
\begin{center}
{\Large \bf
Simplex, associahedron, and cyclohedron}
\end{center}
\begin{center}
{\large Martin Markl}
\end{center}
\footnote{\noindent{\bf Mathematics Subject Classification:}
57P99}
\footnote{Supported by a Fulbright grant, by the grant
AV \v CR \#1019507 and by the grant GA \v CR \#201/96/0310}
\section*{Introduction.}
\vskip2mm
\odstintro
The paper deals with three types of convex polyhedra. The most
classical is the
$n$-dimensional {\em simplex\/}
$\Delta^n$~\cite[\S10.1]{switzer:75}, the basic ingredient of
simplicial topology and perhaps one of the most important
mathematical objects at all~\cite{may:67}.
Another polyhedron is the Stasheff polytope $K_n$, also
called the
{\em associahedron\/}, the basic tool for the study of
homotopy associative Hopf
spaces~\cite[page~277]{stasheff:TAMS63}.
The last type is the polyhedron $W_n$, defined as the
Axelrod-Singer
compactification of the configuration space of $n$ points on the
circle,
and introduced by R.~Bott and
C.~Taubes~\cite[page~5249]{bott-taubes:JMP94}
in connection with the study of nonperturbative link invariants,
recently dubbed by J.~Stasheff the
{\em cyclohedron\/}~\cite{stasheff:from-ops}.
\odstintro
The crucial property of
the collection $K =\{K_n\}_{n\geq 1}$ is that it forms a
cellular {\em operad\/}~\cite[page~278]{stasheff:TAMS63}.
J.~Stasheff observed in~\cite{stasheff:from-ops}
that the collection $W = \{W_n\}_{n\geq 1}$ is a
right {\em module\/}, in the sense
of~\cite[page~1476]{markl:zebrulka}, over the
operad $K$. In Theorem~\ref{vicko} we prove
that also the collection $\Delta = \{\Delta^n\}_{n\geq 0}$ is a
natural right module over the operad ${\it Ass}$ for associative
algebras.
\odstintro
Operads were introduced
to encode varieties of algebras. We show that, in the same
spirit,
also modules over operads describe varieties of some objects. We
call
these objects
{\em traces\/} (Definition~\ref{el}),
since they naturally generalize traces on
associative algebras.
For a so-called {\em cyclic\/} operad
${\cal P}$~\cite[Definition~2.1]{getzler-kapranov:cyclic} we construct
a natural ${\cal P}$-module $M_{{\cal P}}$, the module {\em associated\/} to
the operad ${\cal P}$ (Definition~\ref{Turmo}).
We show that $M_{\cal P}$-traces are exactly
{\em invariant bilinear forms\/} in the sense of E.~Getzler and
M.M.~Kapranov~\cite[Definition~4.1]{getzler-kapranov:cyclic}.
\odstintro
Algebras over the cellular chain operad $CC_*(K)$ of the
associahedron are {A($\infty$)}-algebras introduced by J.~Stasheff
in~\cite[page~294]{stasheff:TAMS63}.
They can be understood as algebras with
the usual associativity condition
\[
(ab)c = a(bc)
\]
satisfied only up to a system of coherent homotopies.
In Proposition~\ref{nuzky}
we show that
the traces over the cellular chain complex $CC_*(W)$ of the
cyclohedron are {\em homotopy traces\/} on {A($\infty$)}-algebras,
for which the usual
condition
\[
T(ab) = T(ba)
\]
is satisfied only up to a system of coherent homotopies.
The traces over the cellular chain complex $CC_*(\Delta)$ of the
simplex are described in Theorem~\ref{vicko}.
\odstintro
The cellular chain complex $CC_*(K)$ of the associahedron has
a very
effective description -- it is the {\em operadic bar
construction\/}
on the operad ${\it Ass}$ for associative
algebras~\cite[Example~4.1]{markl:zebrulka}.
We introduce the bar construction on a {\em module\/} over an
operad (this definition was independently made by V.~Ginzburg
and
A.A.~Voronov in~\cite{ginzburg-voronov}) and show that the
cellular
chain complex $CC_*(W)$ of the cyclohedron is the bar
construction
on the ${\it Ass}$-module ${\it Cycl}$, which describes traces (ordinary,
not
homotopy) on associative algebras (Theorem~\ref{ucpavka}).
A fully algebraic
description of $CC_*(\Delta)$ is given in Theorem~\ref{resiz}.
\odstintro
V.Ginzburg and
M.M.~Kapranov~\cite[Definition~4.1.3]{ginzburg-kapranov:DMJ94}
introduced so-called {\em Koszul operads\/}, with all
expected nice properties, and the related notion of the
{\em Koszul
dual\/} of a
{\em quadratic
operad\/}~\cite[\S2.1.9]{ginzburg-kapranov:DMJ94}.
We introduce analogous notions for {\em modules\/} over operads,
i.e.~we introduce {\em quadratic modules\/}, their
{\em Koszul (quadratic) duals\/} and the
property of {\em Koszulness\/} for these modules. These
definitions
were
again independently made by V.~Ginzburg and
A.A.~Voronov in~\cite{ginzburg-voronov}.
\odstintro
The operad ${\it Ass}$ for associative algebras is
Koszul~\cite[Corollary~4.2.7]{ginzburg-kapranov:DMJ94}.
Since the operadic bar construction on ${\it Ass}$
is the cellular chain complex of the associahedron, the
Koszulness of
${\it Ass}$ follows from the acyclicity of $K$, which in turn
follows from
the fact that it is a convex polyhedron.
In a similar manner, we show in Theorem~\ref{Amphora1}
that the module ${\it Cycl}$ describing
traces on associative algebras is Koszul, as a consequence
of the
convexity of the cyclohedron $W$. A more general
argument is to observe that ${\it Cycl}$ is the module associated
to the
cyclic operad ${\it Ass}$ (Example~\ref{kacirek})
and then apply Theorem~\ref{Katalogizacni} saying that a
module associated to a Koszul operad is Koszul.
\odstintro
We show in Lemma~\ref{whoop} that,
for each module over an operad, there exists a
{\em spectral sequence\/}, converging to the homology of the bar
construction. We also prove in Proposition~\ref{myska}
that for modules over Koszul
operads this spectral sequence collapses.
Our spectral sequence, applied to an ${\it Ass}$-module ${\it Cycl}$,
carries
a strong geometrical message -- the initial term is the cellular
chain complex of the cyclohedron, while the next term is
the cellular
chain complex of the simplex. If we interpret the cyclohedron
as the compactification of the simplex constructed by a
sequence of
blow-ups~\cite[page~5249]{bott-taubes:JMP94},
then the spectral sequence
describes the inverse process -- `deblowing-up' the cyclohedron
back
to the simplex, see Section~\ref{22}.
\odstintro
{\em Some further suggestions.\/}
Consider the following `standard situation' closely related to
the topological quantum field theory. Let $C(S^m)$ be
the Axelrod-Singer
compactification~\cite[Section~5]{axelrod-singer:preprint}
of the configuration
space of distinct points in the sphere $S^m$, and ${\sf F}_m$
the compactification
of the moduli space of configurations of distinct
points in ${\bf R}^m$~\cite[\S3.2]{getzler-jones:preprint}.
It is known that ${\sf F}_n$ is a topological
operad~\cite[\S3.2]{getzler-jones:preprint} and that this
operad acts on
the
right module $C(S^m)$~\cite[Theorem~5.2]{markl:cf}
(to be precise, if $m \not= 1,3,7$,
the sphere $S^m$ is not parallelizable and we
need a suitable framed versions of the objects above).
The homology operad $H_*({\sf F}_m)$ describes a form of graded
Poisson algebras~\cite[Theorem~3.1]{getzler-jones:preprint}
(or Batalin-Vilkovisky algebras, in the framed
case)~\cite[Section~4]{G2}, and it is not difficult to see that
$H_*(C(S^n))$
is the module associated to the cyclic operad $H_*({\sf F}_m)$
in the
sense of our Definition~\ref{Turmo}.
Our paper deals with the above situation for $n=1$, while
all the
machinery cries for an application to a general situation.
Another suggestion for further research is the following.
E.~Getzler and M.M.~Kapranov introduced
in~\cite[Definition~5.2]{getzler-kapranov:cyclic} the cyclic
homology
of an algebra over a cyclic operad as the (left, nonabelian)
derived
functor of the universal invariant bilinear form functor
$\lambda({\cal P},-)$. We propose to study, for a (noncyclic)
operad ${\cal P}$
and a ${\cal P}$-module $M$, the derived functor of the universal
$M$-{\em trace\/} as a natural generalization of the
cyclic homology. The cyclic homology
will be then a special case for ${\cal P}$ cyclic and $M = M_{\cal P}$.
There are two ways to read the paper -- either as an exposition
of the properties of the associahedron, cyclohedron and simplex,
with some generalizations, or as a paper on general theory
of modules
over operads, with a special attention paid to the three
examples above.
\noindent
{\em Acknowledgment:\/}
I would like to express my thanks to Jim Stasheff for numerous
discussions
and hospitality during my stay at the University of North
Carolina.
Also the communication with Sasha Voronov, who was working
independently on~\cite{ginzburg-voronov}, was very useful. I
am also
very grateful to Steve Shnider and
the referee for careful reading the manuscript and
many useful remarks.
\section*{Plan of the paper:}
\noindent
\hangindent=5mm
\hangafter=1
{\em Section~\ref{1968}:
Associahedron and the cyclohedron as a truncation of the
simplex.\/}
We recall the convex realization of the associahedron as a
truncation of the simplex, due to S.~Shnider and S.~Sternberg,
and
construct a similar realization of the cyclohedron.
\noindent
\hangindent=5mm
\hangafter=1
{\em Section~\ref{bolehlav}: Cyclohedron as a module over the
associahedron.\/}
We recall (right) modules over operads and introduce traces as
algebraic objects described by these modules. We introduce
the module
${\it Cycl}$ for traces on associative algebras. We prove that the
cyclohedron is a module over the associahedron
and describe the corresponding traces.
\noindent
\hangindent=5mm
\hangafter=1
{\em Section~\ref{hrnicek1}: Simplex as a module over the
operad for
associative algebras.\/}
We show that the simplex is a module over the
operad ${\it Ass}$ for associative algebras. We prove that the
associated
cellular chain complex is free and describe the corresponding
traces.
\noindent
\hangindent=5mm
\hangafter=1
{\em Section~\ref{cervena-tuzka}:
Quadratic operads and modules;
modules associated to cyclic operads.\/}
We present a class of
operads and modules having a particularly easy description. We
recall
cyclic operads and introduce the module associated to a cyclic
operad.
\noindent
\hangindent=5mm
\hangafter=1
{\em Section~\ref{penezenka}: Cyclohedron as the cobar
construction.\/} We introduce the cobar construction on a
module over
an operad. We define quadratic Koszul modules. We show that the
cellular chain complex of the cyclohedron is the cobar
construction
on the module ${\it Cycl}$ and deduce from this fact that ${\it Cycl}$ is
Koszul.
\noindent
\hangindent=5mm
\hangafter=1
{\em Section~\ref{22}: Cyclohedron as a compactification of the
simplex.\/}
We view the cyclohedron as a compactification of the simplex,
constructed as a sequence of blow-ups. We show that the spectral
sequence related to the cobar construction `deflates' the
cyclohedron
back to the simplex.
\noindent
\hangindent=5mm
\hangafter=1
{\em Appendix: Traces versus invariant bilinear forms.\/}
We show that traces over the module associated to a cyclic
operad are
exactly invariant bilinear forms of E.~Getzler and
M.M.~Kapranov.
\section{Associahedron and the cyclohedron as a truncation
of the
simplex}
\label{1968}
Let ${\cal B}(n)$ denote the set of all meaningful bracketings of $n$
independent variables $\rada1n$.
The {\em associahedron\/} $K_n$ is a convex $(n-2)$-dimensional
polyhedron whose faces are indexed by elements of ${\cal B}(n)$. To
be more precise, ${\cal B}(n)$ is a poset
(= partially-ordered set) ordered by saying that $b' \prec
b''$ if
$b''$ is
obtained from $b'$ by removing one or more pair of
brackets. Then
$K_n$ is a convex polyhedron whose poset of faces is (isomorphic
to) ${\cal B}(n)$. See Figure~\ref{k3andk4} for $K_3$ and $K_4$. A
nice
picture of $K_5$ can be found
in~\cite[page~151]{markl-stasheff:JofAlg94}.
\begin{figure}[hb]
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\caption{$K_3$ (left) and $K_4$ (right).\label{k3andk4}}
\end{figure}
We recall a very cute `linear convex realization' of $K_n$
as a truncation of the $(n-2)$-dimensional simplex, due to
S.~Shnider
and S.~Sternberg~\cite{shnider-sternberg:book}.
Our exposition follows the corrected version given
in~\cite[Appendix~B]{stasheff:from-ops}.
We need an alternative description of the poset ${\cal B}(n)$. Let
$P(n)$ denote the set of all proper subintervals of the interval
$[1,n-1] = \{\rada1{n-1}\}$.
Two intervals $I,J \in P(n)$ are
called {\em compatible\/}, if $I\cup J$ is not an interval
properly containing both $I$ and $J$, i.e.~if either $J\subset
I$, or
$I\subset J$, or $I\cup J$ is not an interval. Let ${\cal I}(n)$
be the set of all subsets $\iota$ of $P(n)$ such that $I$
and $J$ are
compatible for any $I,J \in \iota$.
The poset structure on ${\cal I}(n)$ is given by the set
inclusion: $\iota \preceq \kappa$ if $\kappa\subset \iota$.
\begin{lemma}
\label{prim}
(Shnider-Sternberg)
The posets ${\cal B}(n)$ and ${\cal I}(n)$ are isomorphic.
\end{lemma}
\noindent
{\bf Proof.}
For $I = [i,j]\in P(n)$, let $b(I)$ be the bracketing $1\cdots(i
\cdots j+1)\cdots n$. This correspondence is easily seen
to induce
a poset isomorphism ${\cal I}(n) \cong {\cal B}(n)$.\hspace*{\fill
\noindent
Define the function $c: P(n)\to {\bf R}_{>0}$ by $c(I):= 3^{\#I}$,
for $I
\in P(n)$. Let $K_n \subset {\bf R}^{n-1}$
be the convex polytope defined by
\[
K_n = \left\{(t_1,\ldots,t_{n-1})\in {\bf R}^{n-1};\
\sum_{k=1}^{n-1}t_k = c([1,n-1]),\
\sum_{k\in I}t_k \geq c(I),\
I\in P(n)\right\}.
\]
Denote also, for $I\in P(n)$, by $P_I$ the hyperplane
\[
P_I := \left\{(t_1,\ldots,t_{n-1})\in {\bf R}^{n-1};\
\sum_{k\in I}t_k =c(I)\right\}.
\]
The proof of the following proposition is given
in~\cite[Appendix~B]{stasheff:from-ops}.
\begin{proposition}(Shnider-Sternberg)
\label{Skoda}
The polytope $K_n$ has nonempty interior in the
$(n-2)$-dimensional
hyperplane $\{(t_1,\ldots,t_{n-1})\in {\bf R}^{n-1};\
\sum_{k=1}^{n-1}t_k = 3^{n-1}\}$. The intersection
\[
K_n \cap \bigcap\{P_{I},\ I \in \iota\}
\]
defines a nonempty $(n-\#I-2)$-dimensional face of $K_n$ for any
$\iota \in {\cal I}(n)$. All faces of $K_n$ are
obtained in this way.
\end{proposition}
If we denote, for $\iota \in {\cal I}(n)$, by
$P_{\iota}$ the intersection $\bigcap\{P_{I},\ I \in \iota\}$,
then
the above proposition immediately implies that
the correspondence $\iota \mapsto K_n \cap P_\iota$ defines an
isomorphism of the poset ${\cal I}(n)$ and the poset of faces
of $K_n$.
This is the promised
convex realization of $K_n$. The case $n=4$ is illustrated on
Figure~\ref{realK4}.
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\put(54.50,23.16){\makebox(0,0)[cc]{$\cdot$}}
\put(22.00,57.33){\makebox(0,0)[cc]{$\cdot$}}
\put(30.83,49.00){\makebox(0,0)[cc]{$\cdot$}}
\put(39.67,40.67){\makebox(0,0)[cc]{$\cdot$}}
\end{picture}
\end{center}
\caption{$(t_1,t_3)$-projection of the
convex realization of $K_4$.\label{realK4}}
\end{figure}
As observed in~\cite[Appendix~B]{stasheff:from-ops},
the above construction works also
for other choices of the function $c : P(n)\to {\bf R}_{>0}$
provided it
is
admissible in the sense that
\[
c(I)+c(J)< c(I\cup J), \mbox{ if $I\cup J$ properly contains
both $I$
and $J$.}
\]
Let us proceed to the definition of the cyclohedron $W_n$.
As the associahedron, it will be a convex polyhedron
characterized by the poset ${{\cal B}C}(n)$ indexing its faces. Consider
again
$n$
independent formal variables, labeled by natural
numbers $1,\ldots,n$. The elements of ${{\cal B}C}(n)$ will be
equivalence
classes represented by
bracketing of the chain $\rada{\sigma(1)}{\sigma(n)}$, where
$\sigma \in \Sigma_n$ is a {\em cyclic\/} permutation. In
contrast
to the case of ${\cal B}(n)$, we allow also the bracketing which
embraces all elements. Thus, for instance, $(3(12))$
represents an
element of ${{\cal B}C}(n)$.
The equivalence relation is given as follows. Let $\sigma\in
\Sigma_n$
be a cyclic permutation, let $b'$ be a bracketing of
$\sigma(1)\cdots\sigma(s)$ and $b''$ a bracketing of
$\sigma(s+1)\cdots\sigma(n)$, for some $1\leq s\leq n$. Then
we identify $b'b''$ to $b''b'$. Thus, for example,
$3(12)= (12)3$ in ${{\cal B}C}(3)$
(but $(3(12))\not= ((12)3)$). The partial
order on ${{\cal B}C}(n)$ is defined, as for ${\cal B}(n)$, by deleting
pairs of
brackets.
Each element $b$ of ${{\cal B}C}(n)$ can be uniquely represented by
a symbol,
obtained from a representative of $b$ by forcing the
indeterminates
into the natural order. We call such symbols {\em cyclic
bracketings\/}. The formal definition will be obvious from the
following example.
\begin{example}{\rm\
The poset ${{\cal B}C}(2)$ contains three elements, $(12)$, $(21)$ and
$12$, where $(21)$ is represented by the cyclic bracketing
$1)(2$.
The poset structure is depicted by the interval, see
Figure~\ref{W2}.
\begin{figure}[hbtp]
\begin{center}
\unitlength=1mm
\begin{picture}(53.00,15.00)(10,10)
\thicklines
\put(20.00,10.00){\line(1,0){30.00}}
\put(20.00,10.00){\makebox(0,0)[cc]{$\bullet$}}
\put(50.00,10.00){\makebox(0,0)[cc]{$\bullet$}}
\put(35.00,15.00){\makebox(0,0)[cc]{$12$}}
\put(17.00,15.00){\makebox(0,0)[cc]{$(12)$}}
\put(53.00,15.00){\makebox(0,0)[cc]{$1)(2$}}
\end{picture}
\end{center}
\caption{$W_2$.\label{W2}}
\end{figure}
\noindent
Below are listed elements of the poset ${{\cal B}C}(3)$:
\[
\begin{array}[b]{|c|c||c|c|}
\hline
\mbox{elements of ${{\cal B}C}(3)$}&\mbox{cyclic
bracketings}&\mbox{cont.}&\mbox{\hskip11mm cont.\hskip11mm}
\\
\hline \hline
(1(23))&(1(23))&(231)&1)(23
\\
((12)3)&((12)3)&2(31)=(31)2&1)2(3
\\
((23)1)&1)((23)&(312)&12)(3
\\
(2(31))&1))(2(3&(12)3=3(12)&(12)3
\\
((31)2)&1)2)((3&(123)&(123)
\\
(3(12))&(12))(3&123=231=312&123
\\
1(23)=(23)1&1(23)&&\\
\hline
\end{array}
\]
The poset structure of ${{\cal B}C}(3)$ is depicted on Figure~\ref{W3}.
}\end{example}
\begin{figure}[hbtp]
\begin{center}
\unitlength2mm
\begin{picture}(62.00,35.00)(10,3)
\thicklines
\put(20.00,20.00){\line(1,1){10.00}}
\put(30.00,30.00){\line(1,0){20.00}}
\put(50.00,30.00){\line(1,-1){10.00}}
\put(60.00,20.00){\line(-1,-1){10.00}}
\put(50.00,10.00){\line(-1,0){20.00}}
\put(30.00,10.00){\line(-1,1){10.00}}
\put(20.00,20.00){\makebox(0,0)[cc]{$\bullet$}}
\put(30.00,30.00){\makebox(0,0)[cc]{$\bullet$}}
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\put(30.00,10.00){\makebox(0,0)[cc]{$\bullet$}}
\put(28.00,35.00){\makebox(0,0)[cc]{$((12)3)$}}
\put(52.00,35.00){\makebox(0,0)[cc]{$(1(23))$}}
\put(62.00,20.00){\makebox(0,0)[lc]{$1)((23)$}}
\put(52.00,5.00){\makebox(0,0)[cc]{$1))(2(3$}}
\put(28.00,5.00){\makebox(0,0)[cc]{$1)2)((3$}}
\put(18.00,20.00){\makebox(0,0)[rc]{$(12))(3$}}
\put(40.00,35.00){\makebox(0,0)[cc]{$(123)$}}
\put(40.00,5.00){\makebox(0,0)[cc]{$1)2(3$}}
\put(27.00,25.00){\makebox(0,0)[lc]{$(12)3$}}
\put(27.00,15.00){\makebox(0,0)[lc]{$12)(3$}}
\put(53.00,25.00){\makebox(0,0)[rc]{$1(23)$}}
\put(53.00,15.00){\makebox(0,0)[rc]{$1)(23$}}
\put(40.00,20.00){\makebox(0,0)[cc]{$123$}}
\end{picture}
\end{center}
\caption{$W_3$.\label{W3}}
\end{figure}
The structure of ${{\cal B}C}(4)$ is indicated on Figure~\ref{W4}. The
picture
is already rather complicated, so we labeled only the vertices
(= the
minimal elements of ${{\cal B}C}(4)$). The label of an arbitrary face
can be easily found -- it is the least upper bound
of all vertices of the face. For example, the pentagon on the
top of $W_4$ is labeled by $123)(4$, the front hexagon is
labeled by
$(12)34$, etc.
\begin{figure}[hbtp]
\begin{center}
\unitlength=0.67mm
\begin{picture}(200.11,131.11)
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\put(20.00,10.00){\line(1,0){160.05}}
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$1)2)\!)\!(3(\!(4$}}
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$(12)\!)\!)\!(3(4$}}
\put(18.00,7.00){\makebox(0,0)[ct]{\scriptsize
$(12)\!)\!(\!(34)$}}
\put(32.00,23.00){\makebox(0,0)[lc]{\scriptsize
$1)2)\!(\!(\!(34)$}}
\put(28.00,34.00){\makebox(0,0)[lc]{\scriptsize
$1)\!)2)\!(\!(3(4$}}
\put(88.00,38.00){\makebox(0,0)[cc]{\scriptsize
$1)\!)\!(2(\!(34)$}}
\put(112.00,38.00){\makebox(0,0)[cc]{\scriptsize
$1)\!(\!(\!(2(34)\!)$}}
\put(83.00,58.00){\makebox(0,0)[rc]{\scriptsize
$1)\!)\!)\!(2(3(4$}}
\put(117.00,58.00){\makebox(0,0)[lc]{\scriptsize
$1)\!(\!(\!(23)4)$}}
\put(103.00,67.00){\makebox(0,0)[lb]{\scriptsize
$1)\!)\!(\!(23)\!(4$}}
\put(78.00,126.00){\makebox(0,0)[rb]{\scriptsize
$1)2)3)\!(\!(\!(4$}}
\put(122.00,126.00){\makebox(0,0)[lb]{\scriptsize
$(1(23)\!)\!(4$}}
\put(100.00,135.00){\makebox(0,0)[cb]{\scriptsize
$1)\!(23)\!)\!(\!(4$}}
\put(112.00,110.00){\makebox(0,0)[lc]{\scriptsize
$(\!(\!12\!)3)\!)\!(\!4$}}
\put(88.00,110.00){\makebox(0,0)[rc]{\scriptsize
$(\!12\!)\!)3)\!(\!(\!4$}}
\put(180.00,5.00){\makebox(0,0)[lc]{\scriptsize
$(\!(12)\!(34)\!)$}}
\put(200.00,15.00){\makebox(0,0)[lc]{\scriptsize
$(\!(\!(12)3)4)$}}
\put(200.00,47.00){\makebox(0,0)[lc]{\scriptsize
$(\!(1(23)\!)4)$}}
\put(166.00,22.00){\makebox(0,0)[rc]{\scriptsize
$(1(2(34)\!)\!)$}}
\put(172.00,34.00){\makebox(0,0)[rc]{\scriptsize
$1(\!(23)4)\!)$}}
\end{picture}
\end{center}
\caption{$W_4$.\label{W4}}
\end{figure}
We construct, mimicking the approach of Shnider and Sternberg, a
convex realization of the poset ${{\cal B}C}(n)$. First some
terminology. By a {\em cyclic subinterval\/} of $[1,n]$ we mean
either a `normal' subinterval $[i,j]$, $1\leq i\leq j \leq n$,
representing the subset $\{\rada ij\}$ of
$\{\rada 1n\}$, or the symbol $i][j$,
$1\leq i <j \leq n$, representing $\{\rada 1i\}\cup \{\rada
jn\}$. We
will always suppose that the corresponding sets are proper
subsets of
$\{\rada 1n\}$, i.e.~we exclude the intervals $[1,n]$ and
$i][i+1$,
for $1\leq i <n$. Let us denote by $PC(n)$ the set of all cyclic
subintervals in the above sense. We denote by ${{\cal I}C}(n)$ the
set of
all subsets of $PC(n)$ consisting of {\em nested\/}
subintervals,
meaning that, for $I,J \in \iota \in {{\cal I}C}(n)$, either $I\subset
J$ or
$J \subset I$.
Again, ${{\cal I}C}(n)$ is a poset, the order being induced by the
inclusion.
We have the following analog of Lemma~\ref{prim}.
\begin{lemma}
The posets ${{\cal B}C}(n)$ and ${{\cal I}C}(n)$ are isomorphic.
\end{lemma}
\noindent
{\bf Proof.}
Define, for $I \in PC(n)$, the cyclic bracketing $b(I)\in
{{\cal B}C}(n)$ by
\[
b(I)=\left\{
\begin{array}{ll}
1\cdots(i\cdots j+1)\cdots n, &\mbox{ for $I = [i,j],\ j< n$,}
\\
1)\cdots(i\cdots n, &\mbox{ for $I = [i,n]$, and}
\\
1\cdots i+1)\cdots(j\cdots n, &\mbox{ for $I = i][j$.}
\end{array}
\right.
\]
This correspondence induces the desired poset
isomorphism.\hspace*{\fill
Our convex realization of ${{\cal B}C}(n)$, whose possibility
was predicted in~\cite[Appendix~B]{stasheff:from-ops}),
is defined as follows. Let
$W_n \subset {\bf R}^n$ be the convex polyhedron
\[
W_n = \left\{(t_1,\ldots,t_n)\in {\bf R}^n;\
\sum_{k=1}^nt_k = c([1,n]),\
\sum_{k\in I}t_k \geq c(I),\
I\in PC(n)\right\}.
\]
For $I\in P(n)$, let $P_I$ be the hyperplane
\[
P_I := \left\{(t_1,\ldots,t_n)\in {\bf R}^n;\
\sum_{k\in I}t_k =c(I)\right\}.
\]
The proof of the following proposition is a straightforward
modification of the proof of Proposition~\ref{Skoda} as given
in~\cite[Appendix~B]{stasheff:from-ops}).
\begin{proposition}
The polytope $W_n$ has nonempty interior in the
$(n-1)$-dimensional
hyperplane $\{(t_1,\ldots,t_n)\in {\bf R}^n;\
\sum_{k=1}^n t_k = c([1,n])\}$. The intersection
\[
W_n \cap \bigcap\{P_{I},\ I \in \iota\}
\]
defines a nonempty $(n-\#I-1)$-dimensional face of $W_n$ for any
$\iota\in {{\cal I}C}(n)$ and all faces of $W_n$ are
obtained in this way.
\end{proposition}
For $\iota \in {{\cal I}C}(n)$, let
$P_{\iota}$ be the intersection $\bigcap\{P_{I},\ I \in
\iota\}$.
Then the correspondence $\iota \mapsto W_n \cap P_\iota$
defines an
isomorphism between the poset ${{\cal I}C}(n)$ and the poset of faces of
the polytope $W_n$. This is our
convex realization of $W_n$. The convex realization of $W_3$
is shown
on Figure~\ref{realW3}.
\begin{figure}[hbtp]
\begin{center}
\unitlength 0.95mm
\linethickness{0.4pt}
\begin{picture}(100.00,100.00)(0,-7)
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\put(26.17,45.17){\makebox(0,0)[cc]{.}}
\put(45.17,22.67){\makebox(0,0)[cc]{.}}
\put(32.00,42.17){\makebox(0,0)[cc]{.}}
\put(51.00,19.67){\makebox(0,0)[cc]{.}}
\put(27.33,39.83){\makebox(0,0)[cc]{.}}
\put(46.33,17.33){\makebox(0,0)[cc]{.}}
\put(34.83,37.00){\makebox(0,0)[cc]{.}}
\put(53.83,14.50){\makebox(0,0)[cc]{.}}
\put(35.83,32.67){\makebox(0,0)[cc]{.}}
\put(41.67,29.67){\makebox(0,0)[cc]{.}}
\put(37.00,27.33){\makebox(0,0)[cc]{.}}
\put(44.50,24.50){\makebox(0,0)[cc]{.}}
\put(40.00,27.00){\makebox(0,0)[cc]{.}}
\put(41.00,22.67){\makebox(0,0)[cc]{.}}
\put(42.17,17.33){\makebox(0,0)[cc]{.}}
\put(18.17,43.50){\makebox(0,0)[cc]{.}}
\put(13.50,41.17){\makebox(0,0)[cc]{.}}
\put(21.00,38.33){\makebox(0,0)[cc]{.}}
\end{picture}
\end{center}
\caption{$(t_1,t_3)$-projection of the convex realization of
$W_3$.\label{realW3}}
\end{figure}
\begin{observation}{\rm\
The cyclohedron $W_n$ has $n(n-1)$ codimension-one faces,
represented
by the bracketings
\begin{equation}
\label{csa}
b_{k,n} :=
(\rada{\sigma(1)}{\sigma(k)})\rada{\sigma(k+1)}{\sigma(n)},\ 1 <
k\leq n,
\end{equation}
where $\sigma \in \Sigma_n$ is a cyclic permutation. The face
represented by the bracketing $b_{k,n}$ is isomorphic to the
product $W_{n-k+1}\times K_k$. For example, $W_4$ depicted on
Figure~\ref{W4}, has
\begin{itemize}
\item[-]
$4$ hexagonal faces,
corresponding to $(12)34$, $(23)41$, $(34)12$ and $(41)23$,
isomorphic to $W_3 \times K_2 = W_3 \times \mbox{point}$,
\item[-]
4 square faces, corresponding to $(123)4$, $(234)1$, $(341)2$
and
$(412)3$, isomorphic to $W_2 \times K_3$, and
\item[-]
4 pentagonal faces, corresponding to $(1234)$, $(2341)$,
$(3412)$ and
$(4123)$, isomorphic to $W_1 \times K_4 = \mbox{point}
\times K_4$.
\end{itemize}
This was observed by J.~Stasheff who realized that this is
a strong
motivation for the existence of a module structure which we will
discuss in the following Section~\ref{bolehlav}.
}\end{observation}
\begin{observation}{\rm\
It is clear from our constructions that the cyclohedron $W_n$
is a
truncation of the associahedron $K_{n+1}$, for $n\geq 1$.
This is, of course, a trivial statement -- any convex
polyhedron is
a truncation of an arbitrary other convex polyhedron of the same
dimension, so we must be more precise.
For any $n\geq 1$ there is an obvious map
$P(n+1)\hookrightarrow PC(n)$ which decomposes $PC(n)$ as
\[
PC(n)= P(n+1) \sqcup E(n),
\]
where $E(n)$ is the subset of `exotic' cyclic intervals of
the form
$i][j$, $1\leq i<j\leq n$. The polyhedron $W_n$ is then the
truncation of $K_{n+1}$ by hyperplanes $P_I$ indexed by the
`exotic'
intervals $I\in E(n)$. Compare Figures~\ref{realK4}
and~\ref{realW3}
for $n=3$. We do not
know whether this observation has any deeper meaning.
}\end{observation}
\begin{observation}{\rm
\label{sdff}
Choose, for each $t=\rada 1n$, a point $P_t$
in the interior of the codimension
one face of $W_n$ corresponding to $(\rada tn,\rada 1{t-1})$.
The convex hull of the set
$\set{\rada{P_1}{P_n}}$ is a simplex, closely related to the
`deblowing up' of $W_n$ described in Section~\ref{22}. We
will use
this simplex to introduce an orientation of $W_n$.
}\end{observation}
\section{Cyclohedron as a module over the associahedron}
\label{bolehlav}
We believe that there is no need to give a detailed definition
of an
operad. Recall only that an {\em operad\/} (in a symmetric
monoidal
category ${\cal C}= ({\cal C},\times)$) is a
sequence ${\cal P} = \{{\cal P}(n); n\geq 1\}$ of objects of ${\cal C}$
together with morphisms
\[
\gamma=\gamma_{m_1,\ldots,m_l}:{\cal P}(l)\times
{\cal P}(m_1)\times\cdots\times{\cal P}(m_l)
\longrightarrow
{\cal P}(m_1+\cdots+m_l),
\]
given for any $l,m_1,\ldots,m_l \geq 1$,
satisfying the usual axioms~\cite[Definition~3.12]{may:1972}.
If not stated otherwise, we assume our operads to be
{\em symmetric\/}, i.e.~we assume
that each ${\cal P}(n)$ has a right action of the symmetric group
$\Sigma_n$, $n\geq 2$,
which has again to satisfy some
axioms~\cite[Definition~1.1]{may:1972}. We frequently write
$p(\rada{p_1}{p_l})$ instead of $\gamma(p,\rada{p_1}{p_l})$.
One comment concerning the action of the symmetric group is
in order
here. Our convention is determined by the conventional choice
of the
multiplication in the symmetric group. We accepted the standard
one with
$\sigma \cdot \tau$ meaning $\sigma(\tau)$, i.e.~the permutation
(= a
map) $\tau$ followed by $\sigma$. Then ${\cal P}(n)$ must
be a
{\em right\/} $\Sigma_n$-module, which is the convention used
in the
original definition of P.~May quoted above.
Recall that, for any object $V\in {\cal C}$, there
exist the so-called {\em endomorphism operad\/} ${\cal E}_V =
\coll{{\cal E}_V}$
with ${\cal E}_V(n):= \Hom {V^{\times n}}V$. If ${\cal P}$ is an operad
in ${\cal C}$,
then a {\em${\cal P}$-algebra
structure on $V$\/} is an operad map $a: {\cal P} \to {\cal E}_V$.
\begin{example}{\rm\
\label{hrnicek}
The collection ${\cal B} = \coll {{\cal B}}$ introduced in
Section~\ref{1968}
has a structure of a (nonsymmetric) operad in the
category of posets. The composition $\gamma(b;\rada{b_1}{b_l})$
is,
for $b\in {\cal B}(l)$ and $b_i \in {\cal B}(m_i)$, defined as the
bracketing
$b(\rada{b_1}{b_l})$ obtained by inserting $b_i$ at the $i$-th
position in $b$, $1\leq i\leq l$. We believe that it is clear
what we
mean by this. For example
\begin{eqnarray*}
\gamma_{{\cal B}}(12;1(23),12)& =& 1(23)45,
\\
\gamma_{{\cal B}}((12)3;(12)3,12,(12)(24))
& =& ((12)345)(67)(89),\ \mbox{etc.}
\end{eqnarray*}
A classical result of
J.~Stasheff~\cite[page~278]{stasheff:TAMS63} says
that
the collection of the associahedra $K = \{K_n\}_{n\geq 1}$ has a
cellular
(nonsymmetric) operad structure which induces on the collection
${\cal B} =
\coll {{\cal B}}$ of
its faces the operad structure of Example~\ref{hrnicek}.
As a consequence, cellular chains on $K$ form an
operad ${\cal A} = \coll {\cal A}$, ${\cal A}(n)= CC_*(K_n)$, in the category of
differential graded
vector spaces. Algebras over the operad ${\cal A}$ are
{A($\infty$)}-algebras~\cite[page~294]{stasheff:TAMS63}.
}\end{example}
Probably the most effective way to describe the operad ${\cal A}$
is to say
that ${\cal A} = {\Omega}(\dualI{{\bf s} {\it Ass}})$, the operadic cobar
construction on the
dual cooperad $\dualI {{\bf s} {\it Ass}}$, where ${\bf s} {\it Ass}$ is the
suspension of the
operad for associative
algebras. This is the same, since the operad ${\it Ass}$ is Koszul,
as to say that ${\cal A}$ is the {\em minimal model\/} of the
associative
operad ${\it Ass}$. All this is explained
in~\cite{markl:zebrulka}. We are
going
to
make
a similar analysis for the cyclohedron.
\begin{definition}
\label{Amphora}
A (right) module over an operad ${\cal P}$ is a collection
$M = \coll M$ such that each $M(n)$ is, for $n\geq 1$, a
$\Sigma_n$-module,
together with morphisms
\begin{equation}
\label{dymka}
\nu= \nu_{\rada{m_1}{m_l}}: M(l) \times {\cal P}(m_1)\times \cdots
\times
{\cal P}(m_l)
\longrightarrow M(m_1+\cdots+ m_l),
\end{equation}
given for any $l, m_1,\ldots,m_l \geq 1$.
The structure maps $\nu_{\rada{m_1}{m_l}}$ must satisfy
the axioms
which are obtained by replacing, in the May's definition of an
operad~\cite[Definition~3.12]{may:1972}, the first occurrence
of ${\cal P}$
by $M$, see~\cite[Definition~1.3]{markl:zebrulka} for details.
\end{definition}
\begin{remark}{\rm\
Observe the resemblance of the above definition to the
definition of
an operad. This is due to the fact that right modules over
an operad
are
special cases of general modules, which are abelian groups
object in
a certain comma category of operads, see the discussion
in~\cite[page~1476]{markl:zebrulka}.
}\end{remark}
\begin{remark}{\rm\
\label{mince}
The structure map $\nu$ is, as in the
case of operads, determined by the system of `comp' maps
\begin{equation}
\label{comp}
\circ_i :M(m) \otimes {\cal P}(n) \to M(m+n-1),\ m,n \geq 1,\
1\leq i\leq m,
\end{equation}
defined by $\circ_i (x,p) := \nu(x;1,\ldots,1,p,1,\ldots,1)$
($p$ at
the $i$-th position) which have to satisfy certain
axioms~\cite[Formula~(1)]{markl:zebrulka}.
}\end{remark}
\begin{example}{\rm\
An operad ${\cal P}$ is a
module over itself. Very important nontrivial examples are
provided by the Axelrod-Singer compactification of configuration
spaces of points in a manifold. The result is a module over
the operad
of `local' configurations, see~\cite{markl:cf}.
There is the following analog of the endomorphism operad. Let
$A,W$ be objects of the category ${\cal C}$. Then the
collection ${\cal E}_{A,W}= \coll{{\cal E}_{A,W}}$ with ${\cal E}_{A,W}(n):=
\Hom{A^{\times n}}W$ is a module over the endomorphism operad
${\cal E}_A$,
the module structure being given by the obvious composition
of maps,
as in the case of the endomorphism operad. A ${\cal P}$-algebra
structure $a: {\cal P} \to {\cal E}_A$ on $A$ induces a ${\cal P}$-module
structure on ${\cal E}_{A,W}$.
}\end{example}
We are going now to introduce
objects described, in the similar sense as algebras are
described by
operads, by {\em modules\/} over
operads. We will call them, from the reasons which will be
explained
in Example~\ref{why-traces},
{\em traces\/} over algebras.
\begin{definition}
\label{el}
Let $M$ be a ${\cal P}$-module and let $A$ be a ${\cal P}$-algebra. An
$M$-trace
over
$A$ is a map $t : M \to {\cal E}_{A,W}$ of ${\cal P}$-modules, where
${\cal E}_{A,W}$
has the ${\cal P}$-module structure induced from the ${\cal P}$-algebra
structure
on $A$.
\end{definition}
\begin{example}{\rm\
Rather dull examples of traces are given by taking $M= {\cal P}$. For
example, an ${\it Ass}$-trace over an associative algebra is (given
by) a
bilinear map $B : A \times A \to W$ such that $B(ab,c)
= B(a,bc)$,
$a,b,c\in A$,
i.e.~by an (not necessary symmetric) invariant bilinear form.
}\end{example}
We will need the following notation.
Let, for permutations $\sigma \in \Sigma_l$ and
$\sigma_i\in \Sigma_{m_i}$, $1\leq i\leq l$,
\begin{equation}
\label{myska1}
\sigma(\rada{\sigma_1}{\sigma_l})\in \Sigma_{m_1+\cdots+ m_l}
\end{equation}
denote the permutation $\sigma(\rada{m_1}{m_l})\cdot (\sigma_1
\oplus \cdots \oplus \sigma_l)$, where the meaning of $\sigma_1
\oplus \cdots \oplus \sigma_l$ is clear and
$\sigma(\rada{m_1}{m_l})$
permutes the blocks of $\rada{m_1}{m_l}$-elements via $\sigma$.
This defines a map $\Sigma_l
\times \Sigma_{m_1}\squeeze \Sigma_{m_l}
\to \Sigma_{m_1+\cdots +m_l}$.
\begin{example}{\rm\
\label{tuzka}
More interesting example of a trace can be constructed as
follows.
Take again the (symmetric) operad {\it Ass}\ for associative
algebras.
Recall that ${\it Ass}(n) = {\bf k}[\Sigma_n]$, the group ring of the
symmetric group
over the ground field ${\bf k}$. The operad structure map $\gamma =
\gamma_{{\it Ass}}$ is defined by
$\gamma(\sigma;\rada{\sigma_1}{\sigma_l}) =
\sigma(\rada{\sigma_1}{\sigma_l})$, where
$\sigma(\rada{\sigma_1}{\sigma_l})$ has the same meaning as
in~(\ref{myska1}).
The group of cyclic permutations ${\bf Z}_n = {\bf Z}/n{\bf Z}$
acts from the left on
$\Sigma_n$.
The group $\Sigma_{n-1}$ is imbedded in $\Sigma_n$ as
permutations
leaving $1$ fixed. This embedding is a cross-section to
the ${\bf
Z}_n$-action, thus we can identify $\Sigma_{n-1}$ as a {\em
set\/} to
the coset space ${\bf Z}_n \backslash \Sigma_n$.
The projection $\pi_n : \Sigma_n \to
{\bf Z}_n \backslash \Sigma_n \cong \Sigma_{n-1}$ then induce on
${\bf k}[\Sigma_{n-1}]$ a structure of a right $\Sigma_n$-module.
Define the collection ${\it Cycl} = \coll{{\it Cycl}}$ by
${\it Cycl}(n):= {\bf k}[\Sigma_{n-1}]$, $n\geq 1$.
The system of maps $\{\pi_n: \Sigma_n \to
\Sigma_{n-1}\}_{n\geq 1}$ is the projection $\pi :
{\it Ass} \to {\it Cycl}$ of collections.
\begin{lemma}
The projection $\pi : {\it Ass} \to {\it Cycl}$ induces on ${\it Cycl}$
the structure
of a module over the operad ${\it Ass}$.
\end{lemma}
\noindent
{\bf Proof.}
The structure maps $\nu = \nu_{{\it Cycl}}$ are determined, for
$\sigma \in \Sigma_l$ and $\sigma_i \in \Sigma_{m_i}$, $1\leq i
\leq l$, by
\[
\nu(\pi(\sigma);\rada{\sigma_1}{\sigma_l}) :=
\gamma_{{\it Ass}}(\sigma;\rada{\sigma_1}{\sigma_l}),
\]
where $\gamma_{{\it Ass}}(\sigma;\rada{\sigma_1}{\sigma_l})=
\sigma(\rada{\sigma_1}{\sigma_l})$.
The proof is then finished by an easy verification
that, if $\sigma' \equiv \sigma''$ mod ${\bf Z}_l$, then
\[
\sigma'(\rada{\sigma_1}{\sigma_l}) \equiv
\sigma''(\rada{\sigma_1}{\sigma_l}) \mbox{ mod }
{\bf Z}_{m_1+\cdots +m_l},
\]
which we leave to the reader.\hspace*{\fill
}\end{example}
\begin{example}{\rm\
\label{why-traces}
A ${\it Cycl}$-trace over an
associative algebra $A$ is (characterized by)
a map $T: A \to W$ such that
\begin{equation}
\label{nabla}
T(ab) = \znamenko{|a|\cdot |b|}T(ba),\ a,b\in A,
\end{equation}
i.e.~$T$ is a trace in the usual sense.
We postpone the verification of this statement to
Example~\ref{nabijecka}.
}\end{example}
In the rest of this section we show that the collection $W :=
\colla W$ of the cyclohedra is a natural cellular (right)
module over
the cellular operad $K = \colla K$ and describe $W$-traces on an
A($\infty$)- (= $K$)-algebra.
It is convenient to consider harmless symmetrizations. Recall
that
${\cal B}(n)$ was the poset of all bracketings of $\rada 1n$. Take
instead
be the poset ${\overline {\cal B}}(n)$ of all bracketings of
$\rada{\sigma(1)}{\sigma(n)}$, $\sigma \in \Sigma_n$. Obviously
${\overline {\cal B}}(n) = \Sigma_n \times {\cal B}(n)$ and ${\overline {\cal B}}(n)$ is the
poset of
faces of the $n!$-connected polyhedron ${\overline K}_n :=
\Sigma_n\times K_n$. The collection ${\overline K} := \colla {\overline K}$ is a
(symmetric) cellular operad and the corresponding operad
of cellular
chains ${\overline {\cal A}} := CC_*({\overline K})$ is the (symmetric) operad for
{A($\infty$)}-algebras.
There is a similar symmetrization of the cyclohedron.
We introduced ${{\cal B}C}(n)$ as the poset of equivalence classes of
bracketings of
$\rada {\sigma(1)}{\sigma(n)}$ with a {\em cyclic\/} permutation
$\sigma \in \Sigma_n$. If we admit all permutations, we
obtain the
poset ${\overline {{\cal B}C}}(n)= \Sigma_{n-1}\times {{\cal B}C}(n)$ whose realization
is the $(n-1)!$-connected polyhedron ${\overline W}_n := \Sigma_{n-1}
\times W_n$.
\begin{lemma}
\label{paska}
The collection ${\overline {{\cal B}C}} := \coll{{\overline {{\cal B}C}}}$ is a natural module
over
the operad ${\overline {\cal B}}:= \coll{{\overline {\cal B}}}$ in the symmetric monoidal
category
of posets.
\end{lemma}
\noindent
{\bf Proof.}
The easiest way to define the module structure is the following.
Let $b$ be a bracketing of $\rada{1}{l}$ representing
an element $[b]\in {{\cal B}C}(l)\subset {\overline {{\cal B}C}}(l)$. Let $b_i \in
{\cal B}(m_i)\subset {\overline {\cal B}}(m_i)$
be, for $1\leq i \leq l$ ,
a bracketing of $\rada{1}{m_i}$. Then we define
$\nu_{{\overline {{\cal B}C}}}(b;\rada{b_1}{b_l}) \in {\overline {{\cal B}C}}(m)$, $m =
m_1+\cdots m_l$, to be
the element represented by the composite (in the same sense
as in
Example~\ref{hrnicek}) bracketing $b(\rada{b_1}{b_l})$
of $m$.
The set ${\overline {{\cal B}C}}(l)$ is $\Sigma_l$-generated by elements of the
same form as $b$, i.e.~by elements represented by a bracketing
of
the `unpermuted' string $\rada1l$,
and the same is true also for ${\overline {\cal B}}(m_i)$, $1\leq i\leq l$.
Thus the formula for the composition of arbitrary elements is
dictated by the equivariance of the module composition map. We
leave
the verification that this definition is correct to the reader.
\hspace*{\fill
\begin{theorem}
\label{sirky}
The collection ${\overline W} := \colla{{\overline W}}$ carries a structure of
a module over the operad ${\overline K} := \colla{{\overline K}}$ in the
category of
cellular complexes which induces, on the level of the poset of
faces, the structure of Lemma~\ref{paska}.
The homology collection $H_*({\overline W})= \{H_*({\overline W}_n)\}_{n\geq 1}$
coincides, as an ${\it Ass} = H_*({\overline K})$-module, to the module
${\it Cycl}$
introduced in Example~\ref{why-traces}.
\end{theorem}
\noindent
{\bf Proof.}
The proof is a modification of the proof of the existence of an
operad structure on the associahedron, given by J.~Stasheff
in~\cite[page~278]{stasheff:TAMS63}.
By Remark~\ref{mince}, the ${\overline K}$-module
structure on ${\overline W}$ is given by the `comp' maps
\[
\circ_i :{\overline W}_m \times {\overline K}_n \to {\overline W}_{m+n-1},\
m,n\geq 1,\ 1\leq i\leq m.
\]
As a matter of fact, in our case it is enough to specify
\begin{equation}
\label{vrsek}
\circ_1 :W_m \times K_n \to W_{m+n-1},
\end{equation}
the remaining `comp' maps are determined by the equivariance. We
define $\circ_1$ of~(\ref{vrsek}) to be the identification
of the
product $W_m \times K_n$ to the face of $W_{m+n-1}$ indexed by
the bracketing $b_{n,n+m-1}$ of~(\ref{csa}). The second part is
immediate.\hspace*{\fill
\begin{observation}{\rm\
The ${\overline K}$-module structure on the `symmetrized'
cyclohedron ${\overline W}$ restricts to the right action of the
nonsymmetric operad $K$ on the `nonsymmetric' cyclohedron $W$.
This is the structure observed by J.~Stasheff
in~\cite[Section~4]{stasheff:from-ops}.
Another argument for the existence of the module
structure of Theorem~\ref{sirky} is the interpretation of the
cyclohedron to the compactification of a configuration space,
see
Section~\ref{22}.
}\end{observation}
Let us consider the ${\overline {\cal A}}$-module ${\cal M}:= CC_*({\overline W})$ of
cellular chains on the cyclohedron.
To describe traces over ${\cal M}$,
it is convenient to accept the following
notation. For graded indeterminates $\rada{a_1}{a_n}$
and a permutation $\sigma\in
\Sigma_n$, the {\em Koszul sign\/}
$\epsilon(\sigma)=\epsilon(\sigma;\rada{a_1}{a_n})$
is defined by
\[
a_1\land\dots\land a_n = \epsilon(\sigma;a_1,\dots,a_n)
\cdot a_{\sigma(1)}\land \dots \land a_{\sigma(n)},
\]
which has to be satisfied in the free graded commutative algebra
$\mbox{\large$\land$}(\rada{a_1}{a_n})$. Denote also
\[
\chi(\sigma)=\chi(\sigma;\rada{a_1}{a_n}) :=
{\rm sgn}(\sigma)\cdot \epsilon(\sigma;\rada{a_1}{a_n}).
\]
For an expression
$X(\rada {a_1}{a_n})$ in indeterminates
$\rada {a_1}{a_n}$, let the {\em cyclic sum\/}
\begin{equation}
\label{cyclsum}
\cyclsum X(\rada{a_1}{a_n}) :=
\sum_\sigma \chi(\sigma) X(\rada {a_{\sigma(1)}}{a_{\sigma(n)}})
\end{equation}
be the summation over all cyclic permutations. A convincing
example
of the use of this convention is the
(graded) Jacobi identity written as
\[
\cyclsum [a_1,[a_2,a_3]] = 0.
\]
The following proposition, whose proof we postpone after
Theorem~\ref{ucpavka},
describes ${\cal M}$-traces over {A($\infty$)}-algebras.
\begin{proposition}
\label{nuzky}
Let $A = (A; m_1=\partial,m_2,m_3,\ldots)$ be an
{A($\infty$)}-algebra~(\cite[page~294]{stasheff:TAMS63}, but we use
the sign
convention of~\cite[\S1.4]{markl:JPAA92}). Then
an
${\cal M}$-trace is given by a differential graded vector space $W =
(W,\delta)$, $\deg{\delta} = -1$,
and a system $T_n :A^{\otimes n}\to W$
of degree-$(n-1)$ linear
maps, $n\geq 1$, such that, for all $\rada{a_1}{a_n} \in A$,
\begin{itemize}
\item[(i)]
$T_n(\rada{a_1}{a_n})= \chi(\sigma)\cdot
T_n(\rada{a_{\sigma(1)}}{a_{\sigma(n)}})$
for all
cyclic permutations $\sigma \in \Sigma_n$, and
\item[(ii)]
for all $n\geq 1$,
\begin{equation}
\label{Ax}
\delta T_n(\rada{a_1}{a_n}) =
\cyclsum \sum_{1\leq k\leq n}
\znamenko{k+n} T_{n-k+1}(m_k(\rada{a_1}{a_{k}}),
\rada{a_{k+1}}{a_n}).
\end{equation}
\end{itemize}
We call such objects homotopy traces over an {A($\infty$)}-algebra
$A$.
\end{proposition}
\noindent
Let us write down the axiom~(\ref{Ax}) explicitly for small
$n$. For
$n=1$
it gives
\[
\partial T_1(a) = T_1 (\delta(a)),\ a\in A,
\]
which means that $T_1$ is a homomorphism of differential graded
spaces $(A,\partial)$ and $(W,\delta)$. For $n=2$ it becomes
\[
\delta T_2(a,b) +T_2(\partial a,b) -\znamenko{|a|\cdot |b|}
T_2(\partial b,a) =
T_1(m_2(a,b)) -\znamenko{|a|\cdot |b|}
T_1(m_2(b,a)),\ a,b\in A,
\]
i.e.~$T_1$ is a trace for the `multiplication' $m_2$ up to
a homotopy $T_2$.
For higher $n$'s, the axiom~(\ref{Ax})
represents `coherence conditions' for
the homotopy $T_2$. An important special case is when $A$ is an
ordinary associative algebra, that is the only nontrivial
structure map is $m_2$, which is an associative multiplication
$\cdot$. The axioms for the corresponding trace are obtained by
putting, in
Proposition~\ref{nuzky},
$m_k=0$ for $k\geq 3$. We also substitute
$(-1)^\frac{n(n-1)}{2}T_n$ for
$T_n$, to get rid of the overall sign $(-1)^n$.
A homotopy trace is then a system $\{T_n:A^{\otimes n}\to W
\}_{n\geq 1}$ of degree-$(n-1)$ linear maps,
satisfying~\ref{nuzky}(i) and
\begin{eqnarray}
\delta T_1(a)\!\! &=&\!\! 0\nonumber
\\
\delta T_2 (a,b)\!\! &=& \!\!
T_1(a\cdot b)-\znamenko{|a|\cdot |b|}
T_1(b\cdot a)\nonumber
\\
\delta T_3 (a,b,c)\!\! &=&\!\!T_2(a\cdot b,c)
+\znamenko{|a|\cdot(|b|+|c|)} T_2(b\cdot c,a)
+\znamenko{|c|\cdot(|a|+|b|)} T_2(c\cdot a,b)\nonumber
\\
&\vdots&\nonumber
\\
\label{Blatter}
\hskip5mm \delta T_n(\rada {a_1}{a_n})\!\!&=& \!\!
\cyclsum T_{n-1}(a_1\cdot a_2,\rada{a_3}{a_n}),\ n\geq 4.
\end{eqnarray}
Equation~(\ref{Blatter}) describes a trace over a certain
${\it Ass}$-module ${\overline {\cal D}}$, closely related to the simplex. The
following
Section~\ref{hrnicek1} is devoted to the study of this module.
\section{Simplex as a module over the operad for associative
algebras}
\label{hrnicek1}
Let $\Delta_n$ be the standard $(n-1)$-dimensional simplex
(!observe
that the conventional notation for
our $\Delta_n$ is $\Delta^{n-1}$!). An
explicit description of $\Delta_n$ is the following. Denote, for
$1\leq i\leq n$, by $e_i$ the point
$(\rada 00,1,\rada 00)\in {\bf R}^n$ ($1$ at
the $i$-th position). Then $\Delta_n \subset {\bf R}^n$ is
the convex
hull of the set $\{\rada{e_1}{e_n}\}$.
Figure~\ref{simplex} shows $\Delta_n$ for $n=3$.
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\put(9.33,29.66){\makebox(0,0)[cc]{$\{3\}$}}
\put(1.66,17.66){\makebox(0,0)[cc]{$\{31\}$}}
\put(22.66,18.16){\makebox(0,0)[cc]{$\{23\}$}}
\put(10.83,-1.28){\makebox(0,0)[cc]{$\{12\}$}}
\put(10.50,6.33){\makebox(0,0)[cc]{$\{123\}$}}
\put(12.33,27.33){\line(-1,0){1.50}}
\put(10.83,27.33){\line(1,0){3.00}}
\put(16.17,13.16){\makebox(0,0)[cc]{$\cdot$}}
\put(3.83,11.00){\makebox(0,0)[cc]{$\cdot$}}
\put(13.00,8.83){\makebox(0,0)[cc]{$\cdot$}}
\put(7.33,11.50){\makebox(0,0)[cc]{$\cdot$}}
\put(11.33,16.66){\makebox(0,0)[cc]{$\cdot$}}
\put(8.17,12.33){\makebox(0,0)[cc]{$\cdot$}}
\put(6.67,15.83){\makebox(0,0)[cc]{$\cdot$}}
\put(20.50,9.33){\makebox(0,0)[cc]{$\cdot$}}
\put(21.50,5.83){\makebox(0,0)[cc]{$\cdot$}}
\put(17.33,5.00){\makebox(0,0)[cc]{$\cdot$}}
\put(10.50,18.00){\makebox(0,0)[cc]{$\cdot$}}
\put(14.00,18.50){\makebox(0,0)[cc]{$\cdot$}}
\put(0.83,6.16){\makebox(0,0)[cc]{$\cdot$}}
\put(-5.22,3.50){\line(1,-1){2.44}}
\put(27.56,0.94){\line(1,1){3.00}}
\put(2.56,3.83){\makebox(0,0)[cc]{$\cdot$}}
\put(17.67,16.17){\makebox(0,0)[cc]{$\cdot$}}
\put(11.00,22.94){\makebox(0,0)[cc]{$\cdot$}}
\thicklines
\put(12.33,27.33){\line(-2,-3){16.67}}
\put(-4.34,2.33){\line(1,0){33.33}}
\put(29.00,2.33){\line(-2,3){16.78}}
\end{picture}
\end{center}
\caption{$\Delta_3$.\label{simplex}}
\end{figure}
There is a classical
correspondence between the poset of subsets of $\{\rada1n\}$
and the
poset of faces of $\Delta_n$ given by
\[
\mbox{subset $S$ of $\{\rada1n\}$}
\longleftrightarrow
\mbox{convex hull of the set $\{e_i\}_{i\in S}\subset {\bf
R}^n$.}
\]
See~\cite[\S10.1]{switzer:75} for details.
Let $\Delta := \{\Delta_n\}_{\geq 1}$. In fact, it is more
convenient
to consider the symmetrized version ${\overline \Delta} :=
\{{\overline \Delta}_n\}_{\geq 1}$, where ${\overline \Delta}_n$ is the disjoint
union
of $(n-1)!$ copies of $\Delta_n$, indexed by cyclic orders
of its
vertices. This means that the poset of faces of ${\overline \Delta}_n$
consists of elements of the form
\begin{equation}
\label{bookshop}
\set{\rada{i_1}{i_l}}\times [\sigma],
\end{equation}
where $\set{\rada{i_1}{i_l}}$ is a subset of $\set{\rada
1n}$, $1\leq
l\leq n$, and $[\sigma]$ is an equivalence class from the
left coset
${\bf Z}_n \backslash \Sigma_n$. We define the
right action of $\Sigma_n$ by
\[
(\set{\rada{i_1}{i_l}}\times [\sigma])\cdot \rho :=
\set{\rada{\rho^{-1}(i_1)}{\rho^{-1}(i_l)}}\times [\sigma\rho].
\]
Let $e = \set{\rada{i_1}{i_l}}\times [\sigma]$ be a face (=
cell) of
${\overline \Delta}(n)$ as in~(\ref{bookshop}). An orientation of $e$
is given
by choosing an order of elements of $\set{\rada{i_1}{i_l}}$. Two
such
orders induce the same orientation if and only if they differ
by a
permutation of signature $+1$. Thus the cellular cell complex
${\overline {\cal D}}(n) := CC_*({\overline \Delta}(n))$ is a vector space with
the basis
\begin{equation}
\label{Y}
\langle\rada{i_1}{i_l}\rangle\times [\sigma]
\end{equation}
where $\langle\rada{i_1}{i_l}\rangle$ denotes
the cell $\set{\rada{i_1}{i_l}}\times [\sigma]$ with the
orientation
induced by the order $i_1<
\cdots < i_l$. The right action of $\Sigma_n$ is given by
\[
(\langle\rada{i_1}{i_l}\rangle\times [\sigma])\cdot \rho :=
(\langle\rada{\rho^{-1}(i_1)}{\rho^{-1}(i_l)}\rangle)\times
[\sigma\rho].
\]
\begin{theorem}
\label{vicko}
The collection ${\overline \Delta} :=
\{{\overline \Delta}_n\}_{\geq 1}$ of cell complexes
has a natural structure of a (right)
module over the operad ${\it Ass}$ for associative algebras. The
traces
over the cellular chain complex ${\overline {\cal D}} := CC_*({\overline \Delta})$
are the
objects described by~(\ref{Blatter}).
\end{theorem}
\noindent
{\bf Proof.}
We observed in Remark~\ref{mince}
that the action is determined by a system of `comp'
maps
\[
\circ_i : {\overline \Delta}_n
\otimes {\it Ass}(m) \to {\overline \Delta}_{m+n-1},\ n,m\geq 1.
\]
Since ${\overline \Delta}_n$ is $\Sigma_n$-generated by
$\Delta_n =
\Delta_n \times [\mbox{$1\!\!1$}_n] \subset {\overline \Delta}_n$
(the copy corresponding to the
`normal' cyclic order $(\rada 1n)$) and ${\it Ass}(m) =
{\bf k}[\Sigma_n]$ is
$\Sigma_m$-generated by the identity permutation $\mbox{$1\!\!1$}_m
\in \Sigma_m$, it is enough to specify $\circ_i(t,x)$ for
$t\in \Delta_n$ and $x = \mbox{$1\!\!1$}_m$. We define $\circ_i(-,\mbox{$1\!\!1$}_m):
\Delta_n
\to \Delta_{m+n-1}$ to be the unique simplicial map such that
\[
\circ_i(\set j \times [\mbox{$1\!\!1$}_n], [\mbox{$1\!\!1$}_m]) :=
\left\{
\begin{array}{ll}
\set j,& \mbox{ for $1 \leq j \leq i$, and}
\\
\set{j+m-1},& \mbox{ for $i < j \leq n$.}
\end{array}
\right.
\]
In other words, $\circ_i(-,\mbox{$1\!\!1$}_m)$ is the canonical inclusion
$\Delta_n
\hookrightarrow \Delta_{m+n-1}$, identifying $\Delta_n$ to the
$(n-1)$-dimensional face of $\Delta_m$ corresponding to
the subset
$\{\rada 1i,\rada{i+m}{n+m-1}\}$.
The induced map of the cellular chain complex satisfies
\begin{equation}
\label{napoleon}
\circ_i(\langle \rada 1n \rangle \times [\mbox{$1\!\!1$}_n], [\mbox{$1\!\!1$}_m])=
\langle \rada 1i,\rada{i+m}{n+m-1}\rangle \times [\mbox{$1\!\!1$}_{m+n-1}].
\end{equation}
It is a straightforward verification to prove that this
really defines
an ${\it Ass}$-action. The second part will be proved after we
formulate
Theorem~\ref{resiz}.\hspace*{\fill
We are going to give an algebraic characterization of
the ${\it Ass}$-module ${\overline {\cal D}}$. To do this, we need some more or less
standard
notions, which we will also find useful later. From now on,
if not stated
otherwise, the underlying symmetric monoidal category
will be the category of (differential) graded
vector spaces.
For any collection $E= \coll E$ there exists
the {\em free operad\/} ${\cal F}(E)$ on
$E$~\cite[page~226]{ginzburg-kapranov:DMJ94}. The operad
${\cal F}(E)$
has the following very explicit description in terms of trees.
Denote by ${\cal T}$ the set of (labeled rooted)
trees and by ${\cal T}_n$ the subset of ${\cal T}$ consisting of
trees having $n$ input edges. Let $E({\tt T})$ denote, for ${\tt T}
\in {\cal T}$,
the set of `multilinear' colorings of the vertices of ${\tt T}$
by the elements of
$E$ such that a vertex with $k$ input edges is colored by
an element
of $E(k)$. The free operad ${\cal F}(E)$ on $E$ may be then
described as
\begin{equation}
\label{fax}
{\cal F}(E)(n):=
\bigoplus_{{\tt T}\in{\cal T}_n}E({\tt T})
\end{equation}
with the operad structure
on ${\cal F}(E)$ given by the operation of `grafting'
trees. We will in fact always assume that $E(1)= 0$, thus
we consider
in~(\ref{fax}) only trees whose all vertices are at
least binary, i.e.\
they have at least two incoming edges.
The details may be found
in~\cite{ginzburg-kapranov:DMJ94,getzler-jones:preprint}.
\begin{example}
\label{zajicek_usacek}{\rm\
The set ${\cal T}_2$ has only one element (Figure~\ref{t2andt3}) and
${\cal F}(E)(2)= E(2)$. The set ${\cal T}_3$ has four elements (see again
Figure~\ref{t2andt3})
\begin{figure}[hbtp]
\begin{center}
\unitlength 0.80mm
\thicklines
\begin{picture}(157.62,22.95)
\put(7.62,22.95){\line(0,-1){10.00}}
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\put(7.62,12.95){\line(1,-1){10.00}}
\put(57.62,22.95){\line(0,-1){10.00}}
\put(57.62,12.95){\line(-1,-1){10.00}}
\put(57.62,12.95){\line(1,-1){10.00}}
\put(52.62,7.95){\line(1,-1){5.00}}
\put(7.62,12.95){\makebox(0,0)[cc]{$\bullet$}}
\put(52.62,7.95){\makebox(0,0)[cc]{$\bullet$}}
\put(57.62,12.95){\makebox(0,0)[cc]{$\bullet$}}
\put(-2.38,-2.05){\makebox(0,0)[cc]{$1$}}
\put(17.62,-2.05){\makebox(0,0)[cc]{$2$}}
\put(47.62,-2.05){\makebox(0,0)[cc]{$1$}}
\put(57.62,-2.05){\makebox(0,0)[cc]{$2$}}
\put(67.62,-2.05){\makebox(0,0)[cc]{$3$}}
\put(87.62,22.95){\line(0,-1){10.00}}
\put(87.62,12.95){\line(-1,-1){10.00}}
\put(87.62,12.95){\line(1,-1){10.00}}
\put(82.62,7.95){\line(1,-1){5.00}}
\put(82.62,7.95){\makebox(0,0)[cc]{$\bullet$}}
\put(87.62,12.95){\makebox(0,0)[cc]{$\bullet$}}
\put(77.62,-2.05){\makebox(0,0)[cc]{$2$}}
\put(87.62,-2.05){\makebox(0,0)[cc]{$3$}}
\put(97.62,-2.05){\makebox(0,0)[cc]{$1$}}
\put(117.62,22.95){\line(0,-1){10.00}}
\put(117.62,12.95){\line(-1,-1){10.00}}
\put(117.62,12.95){\line(1,-1){10.00}}
\put(112.62,7.95){\line(1,-1){5.00}}
\put(112.62,7.95){\makebox(0,0)[cc]{$\bullet$}}
\put(117.62,12.95){\makebox(0,0)[cc]{$\bullet$}}
\put(107.62,-2.05){\makebox(0,0)[cc]{$3$}}
\put(117.62,-2.05){\makebox(0,0)[cc]{$1$}}
\put(127.62,-2.05){\makebox(0,0)[cc]{$2$}}
\put(147.62,22.95){\line(0,-1){10.00}}
\put(147.62,12.95){\line(-1,-1){10.00}}
\put(147.62,12.95){\line(1,-1){10.00}}
\put(147.62,12.95){\line(0,-1){10.00}}
\put(147.62,12.95){\makebox(0,0)[cc]{$\bullet$}}
\put(137.62,-2.05){\makebox(0,0)[cc]{$1$}}
\put(147.62,-2.05){\makebox(0,0)[cc]{$2$}}
\put(157.62,-2.05){\makebox(0,0)[cc]{$3$}}
\end{picture}
\end{center}
\caption{The sets ${\cal T}_2$ (left) and ${\cal T}_3$
(right).\label{t2andt3}}
\end{figure}
and ${\cal F}(E)(3)$ consists of three copies of $E(2)\otimes E(2)$
which corresponds to the three binary trees in ${\cal T}_3$ and
one copy of
$E(3)$ corresponding to the corolla (= the tree with one
vertex).
Compare also~\cite[Figure~7]{ginzburg-kapranov:DMJ94}.
}\end{example}
In the same manner, for each operad ${\cal P}$ and for each collection
$X =
\coll X$ there exists the {\em free (right) ${\cal P}$-module\/}
generated
by the collection $X$, which we denote $X\circ {\cal P}$.
An explicit description is~\cite[page~312]{markl:dl}
\begin{equation}
\label{cajicek}
(X\circ {\cal P})(m) =
\def.7{.7}
\bigoplus
\left(
\mbox{Ind}^{\Sigma_m}_{\Sigma_{m_1}
\times \cdots \times \Sigma_{m_l}}
(X(l)\otimes {\cal P}(m_1)\otimes \cdots \otimes {\cal P}(m_l))
\right)_{\Sigma_l},
\end{equation}
where the summation is taken over all $m_1+\cdots +m_l =n$,
$l\geq 1$.
On the right-hand side,
$\mbox{Ind}^{\Sigma_m}_{\Sigma_{k_1}
\times \cdots \times \Sigma_{k_l}}(-)$ denotes the induced
representation and $(-)_{\Sigma_l}$ the quotient under the
obvious
action of $\Sigma_l$.
The term $X(l)\otimes {\cal P}(m_1)\otimes \cdots \otimes {\cal P}(m_l)$
on the right-hand side of~(\ref{cajicek})
can be interpreted as colorings of the tree
${\tt T}_{m_1,\ldots,m_l}$
from Figure~\ref{thetree} such that the output vertex is colored
by an element of $X(l)$ and
the remaining vertices by elements of ${\cal P}$.
\begin{figure}[hbtp]
\begin{center}
\unitlength 1.00mm
\thicklines
\begin{picture}(59.69,22.89)
\put(6.69,5.39){\line(-6,-5){6.83}}
\put(6.69,5.56){\line(-1,-2){2.75}}
\put(6.69,5.23){\line(1,-1){5.33}}
\put(17.02,5.73){\line(-2,-5){2.20}}
\put(17.02,5.73){\line(1,-1){5.83}}
\put(53.02,6.39){\line(-6,-5){7.50}}
\put(52.85,6.56){\line(-2,-5){2.73}}
\put(52.69,6.73){\line(1,-1){7.00}}
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\put(6.52,5.39){\makebox(0,0)[cc]{$\bullet$}}
\put(17.02,5.73){\makebox(0,0)[cc]{$\bullet$}}
\put(52.85,6.56){\makebox(0,0)[cc]{$\bullet$}}
\put(34.69,12.73){\makebox(0,0)[cc]{$\cdots$}}
\put(7.35,-0.44){\makebox(0,0)[cc]{$\cdots$}}
\put(19.52,-0.61){\makebox(0,0)[cc]{$\cdots$}}
\put(54.19,-0.61){\makebox(0,0)[cc]{$\cdots$}}
\put(32.71,22.89){\makebox(0,0)[lb]{$l$ inputs}}
\put(4.85,7.23){\makebox(0,0)[rb]{$m_1$ inputs}}
\put(19.85,6.06){\makebox(0,0)[lb]{$m_2$ inputs}}
\put(55.52,8.56){\makebox(0,0)[lb]{$m_l$ inputs}}
\put(6.59,5.41){\line(0,1){15.19}}
\put(52.90,6.41){\line(0,1){14.18}}
\put(52.90,20.59){\line(-1,0){46.32}}
\put(27.50,20.62){\line(0,-1){9.30}}
\put(17.19,5.53){\line(-1,-6){0.92}}
\put(17.02,20.56){\line(0,-1){15.17}}
\end{picture}
\end{center}
\caption{The tree ${\tt T}_{m_1,\ldots,m_l}$. The output vertex is
symbolized as a `rake' with no output edge,
to underline its distinguished character.
\label{thetree}}
\end{figure}
\begin{example}
\label{haficek}
{\rm\
We have $(X\circ {\cal P})(1) = X(1)$, corresponding to the tree
${\tt T}_1$ on
Figure~\ref{odrazka}.
\begin{figure}[hbtp]
\begin{center}
\unitlength 1.00mm
\thicklines
\begin{picture}(142.67,43.00)
\put(131.67,15.00){\line(-1,-1){10.00}}
\put(131.67,15.33){\line(1,-1){10.33}}
\put(12.34,35.00){\makebox(0,0)[cc]{$\bullet$}}
\put(72.01,34.38){\makebox(0,0)[cc]{$\bullet$}}
\put(131.67,34.33){\makebox(0,0)[cc]{$\bullet$}}
\put(12.34,15.00){\makebox(0,0)[cc]{$\bullet$}}
\put(59.29,15.33){\makebox(0,0)[cc]{$\bullet$}}
\put(83.34,15.67){\makebox(0,0)[cc]{$\bullet$}}
\put(131.67,15.00){\makebox(0,0)[cc]{$\bullet$}}
\put(-0.00,42.33){\makebox(0,0)[cc]{${\tt T}_1:$}}
\put(57.67,42.66){\makebox(0,0)[cc]{${\tt T}_{1,1}:$}}
\put(118.67,43.00){\makebox(0,0)[cc]{${\tt T}_2:$}}
\put(16.34,36.33){\makebox(0,0)[lc]{$X(1)$}}
\put(16.00,17.33){\makebox(0,0)[lc]{${\cal P}(1)={\bf k}$}}
\put(74.36,38.21){\makebox(0,0)[lc]{$X(2)$}}
\put(62.67,12.33){\makebox(0,0)[lc]{${\cal P}(1)={\bf k}$}}
\put(87.01,19.33){\makebox(0,0)[lc]{${\cal P}(1)={\bf k}$}}
\put(135.33,37.00){\makebox(0,0)[lc]{$X(1)$}}
\put(135.33,18.00){\makebox(0,0)[lc]{${\cal P}(2)$}}
\put(12.34,0.00){\makebox(0,0)[cc]{$1$}}
\put(59.34,0.00){\makebox(0,0)[cc]{$1$}}
\put(83.34,0.33){\makebox(0,0)[cc]{$2$}}
\put(122.00,0.00){\makebox(0,0)[cc]{$1$}}
\put(142.67,0.00){\makebox(0,0)[cc]{$2$}}
\put(12.34,35.00){\line(0,-1){29.00}}
\put(59.34,34.33){\line(1,0){24.00}}
\put(83.34,34.33){\line(0,-1){28.00}}
\put(59.34,34.33){\line(0,-1){27.67}}
\put(131.67,34.33){\line(0,-1){19.33}}
\end{picture}
\end{center}
\caption{$(X\circ {\cal P})(1)$ (left) and $(X\circ {\cal P})(2)$
(right).\label{odrazka}}
\end{figure}
The vector space $(X\circ {\cal P})(2)$ consists of a
copy of $X(2)$ corresponding to ${\tt T}_{1,1}$ and a copy of
$X(1)\otimes {\cal P}(2)$ corresponding to ${\tt T}_2$, see again
Figure~\ref{odrazka}. Note that we still assume that ${\cal P}(1)=
{\bf k}$.
}\end{example}
For a graded vector space $V=\bigoplus_p V_p$ let $\uparrow\! V$
(resp.
$\downarrow\! V$) be the {\em suspension\/} (resp. the {\em
desuspension\/}) of $V$, i.e.
the graded
vector space defined by $(\uparrow\! V)_p := V_{p-1}$
(resp. $(\downarrow\! V)_p
:= V_{p+1}$).
We have the obvious natural maps $\uparrow : V \to \uparrow\! V$ and
$\downarrow: V\to \downarrow\! V$.
For a collection $E$, the {\em suspension\/} ${\bf s\hskip0mm} E$
is the collection with
\begin{equation}
\label{sign-factor}
({\bf s\hskip0mm} E)(n):=
{\rm sgn}\ \otimes \uparrow^{n-1}E(n),
\end{equation}
$n\geq 1$, where
$\uparrow^{n-1}$ is the $(n-1)$-fold
suspension introduced above and ${\rm sgn}$ is the signum
representation of
$\Sigma_n$ on ${\bf k}$. The reason why we need the signum
factor is that we intend to apply the suspension to operads and
modules over operads.
Without this factor, the composition induced on
the suspension will
not be equivariant,
compare also~\cite[page~8]{getzler-jones:preprint}. There is an
obvious similar notion of the {\em desuspension\/} ${\bf s\hskip0mm}^{-1}
E$ of the
collection $E$.
Let us come back to the promised algebraic characterization
of the
${\it Ass}$-module ${\overline {\cal D}}$.
Consider the free ${\it Ass}$-module ${\bf s\hskip0mm}{\it Cycl} \circ {\it Ass}$
generated by
the suspension of the collection ${\it Cycl}$ introduced in
Example~\ref{tuzka}.
It will be useful
to have an explicit description of
the elements of ${\bf s\hskip0mm} {\it Cycl} \circ
{\it Ass}$. Consider the free graded right $\Sigma_n$-module $H(n)$
generated by the trees ${\tt T}_{m_1,\ldots,m_l}$ introduced in
Figure~\ref{thetree}, with $m_1+\cdots+m_l=n$.
The grading is given by $\deg({\tt T}_{m_1,\ldots,m_l}):= l-1$
Thus $H(n)$ is, by
definition, the graded vector space with the basis
\[
\left\{
{\tt T}_{m_1,\ldots,m_l}\times \sigma,\ \sigma \in \Sigma_n,\
1\leq l\leq n,\ m_1+\cdots+m_l=n
\right\},\ \deg({\tt T}_{m_1,\ldots,m_l}\times \sigma)= l-1.
\]
A neat graphical presentation of the symbol
${\tt T}_{m_1,\ldots,m_l}\times
\sigma$ is the tree ${\tt T}_{m_1,\ldots,m_l}$ with the inputs
labeled by
$\rada{\sigma^{-1}(1)}{\sigma^{-1}(n)}$. For example,
${{\tt T}_{2,1}}\times (123)$ can be depicted as
\begin{center}
\unitlength 1.20mm
\thicklines
\begin{picture}(10,13)(53,23)
\put(52.63,33.53){\line(1,0){8.67}}
\put(61.30,33.53){\line(0,-1){4.33}}
\put(61.30,29.19){\line(-1,-1){4.00}}
\put(61.30,29.19){\line(1,-1){4.00}}
\put(52.63,33.53){\line(0,-1){8.00}}
\put(52.66,22.95){\makebox(0,0)[cc]{2}}
\put(57.19,22.95){\makebox(0,0)[cc]{3}}
\put(57.19,33.53){\makebox(0,0)[cc]{$\bullet$}}
\put(61.35,28.95){\makebox(0,0)[cc]{$\bullet$}}
\put(65.21,22.95){\makebox(0,0)[cc]{1}}
\end{picture}
\end{center}
Define now the left action of the group ${\bf Z}_l$, considered
as the group
of cyclic permutations of order $l$, on $H(n)$ as
follows. Recall that, for $\zeta \in {\bf Z}_l \subset \Sigma_l$,
we denoted by $\zeta(\rada{m_1}{m_l}) \in
\Sigma_n$ the permutation which permutes the blocks of
$\rada{m_1}{m_l}$-elements via $\zeta$. Then we put
\begin{equation}
\label{pejska_Mikinka}
\zeta
\left(
\tttr{m_1}{m_l} \times \sigma
\right)
:= {\rm sgn}(\zeta) \cdot
\tttr{m_{\zeta^{-1}(1)}}{m_{\zeta^{-1}(l)}} \times
\zeta(\rada{m_1}{m_l})\cdot\sigma.
\end{equation}
The following lemma is an easy consequence of definitions and
formula~(\ref{cajicek}).
\begin{lemma}
\label{MGD}
The graded vector space $({\bf s\hskip0mm}{\it Cycl}
\circ {\it Ass})(n)$ can be identified with the graded vector space
with the
basis given by equivalence classes of symbols
\begin{equation}
\label{3M}
\tttr{m_1}{m_l} \times \sigma \in H(n),\
\deg({\tt T}_{m_1,\ldots,m_l})= l-1,
\end{equation}
modulo the left action of the group ${\bf Z}_l$ defined
in~(\ref{pejska_Mikinka}). Under this identification,
the right action of $\Sigma_n$ on the
equivalence class $[\tttr{m_1}{m_l} \times \sigma]$ is
described as
\[
[\tttr{m_1}{m_l} \times \sigma]\cdot \rho :=
{\rm sgn}(\sigma)\cdot [\tttr{m_1}{m_l} \times \sigma\rho].
\]
\end{lemma}
Let $\xi_n\in {\it Cycl}(n)$ be the generator
represented by the identical permutation $\mbox{$1\!\!1$}_n\in \Sigma_n$
and let
$\alpha_2 = \mbox{$1\!\!1$}_2 \in {\it Ass}(2) = {\bf k}[\Sigma_2]$. Define the
differential $\partial$ on ${\bf s\hskip0mm}{\it Cycl} \circ {\it Ass}$ by
\begin{eqnarray}
\label{guma}
\partial (\uparrow\!^{n-1} \xi_n) &:=& -\sum_{\sigma}
{\rm sgn}(\sigma)\cdot \nu(\uparrow\!^{n-2}
\xi_{n-1};\alpha_2,\rada 11)\cdot \sigma
\\
\nonumber
&=&-\cyclsum
\nu(\uparrow\!^{n-2}\xi_{n-1};\alpha_2,\rada 11).\
\mbox{ /the cyclic sum notation of~(\ref{cyclsum})/}
\end{eqnarray}
Using the identification of
Lemma~\ref{MGD}, this could be also written as
\[
\partial ({{\tt T}}_{\underbrace%
{\mbox{\scriptsize $1,\ldots,1$}}_{n\times}}
\times [\mbox{$1\!\!1$}_n])= -\cyclsum
({{\tt T}}_{2,\underbrace%
{\mbox{\scriptsize $1,\ldots,1$}}_{n-1\times}}\times[\mbox{$1\!\!1$}_n] ),
\]
or, in a diagrammatic shorthand,
\vskip2mm
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Since the elements $\{ \uparrow\!^{n-1} \xi_n\}_{n\geq 1}$ generate
${\bf s\hskip0mm}{\it Cycl} \circ {\it Ass}$, formula~(\ref{guma}) is enough to
determine
the differential $\partial$.
We leave to the reader to verify that the definition is
correct and
that $\partial^2=0$.
\begin{theorem}
\label{resiz}
The ${\it Ass}$-module ${\overline {\cal D}} = CC_*({\overline \Delta})$
is isomorphic to the free differential
${\it Ass}$-module $({\bf s\hskip0mm}{\it Cycl} \circ {\it Ass},\partial)$ constructed
above.
\end{theorem}
\noindent
{\bf Proof.}
Any differential ${\it Ass}$-module map $\omega:
({\bf s\hskip0mm}{\it Cycl} \circ {\it Ass},\partial) \to ({\overline {\cal D}},\partial_{{\overline {\cal D}}})$ is
determined by the values $\omega(\uparrow\!^{n-1}\xi_n)$, $n\geq 1$.
We define $\omega(\uparrow\!^{n-1}\xi_n):= e_n$, where $e_n \in
{\overline {\cal D}}(n)$
is the top $(n-1)$-dimensional oriented cell
$\langle \rada 1n\rangle$.
We shall verify that the above defined map $\omega$ commutes
with the
differentials,
\[
\omega(\partial \uparrow\!^{n-1}\xi_n) = -\omega(\cyclsum
\nu(\uparrow\!^{n-2}\xi_{n-1},\alpha_2,\rada11)) =
\partial e_n.
\]
Because $\omega$ is a module homomorphism,
the above equation can be rewritten as
\begin{equation}
\label{hvezdicka}
\cyclsum\nu_{{\overline {\cal D}}}(e_{n-1};\alpha_2,\rada11) = -\partial e_n.
\end{equation}
The standard formula for the boundary of
$\langle \rada 1n \rangle$~\cite[\S10.1]{switzer:75}
says that
\begin{equation}
\label{ja}
\partial e_n = \partial \langle\rada 1n\rangle
= \sum_{1\leq i \leq n}
\znamenko{i+1}
\langle \rada 1{i-1},\rada {i+1}n\rangle
\end{equation}
while the defining formula~(\ref{napoleon})
for the ${\it Ass}$-module action on ${\overline \Delta}$ gives
\[
\nu_{{\overline {\cal D}}}(e_{n-1};\alpha_2,\rada11) =
\langle1,\rada3n\rangle.
\]
Now it is enough to observe that
\[
\cyclsum\langle1,\rada3n\rangle= - \sum_{1\leq i
\leq n}\znamenko{i+1}\langle\rada1{i-1},\rada{i+1}n\rangle
\]
which, together with~(\ref{ja}), gives~(\ref{hvezdicka}).
It remains to prove that $\omega$ is an isomorphism.
To this end, we give an explicit formula for the map
$\omega$. Let
$\tttr{m_1}{m_l} \times \sigma \in H(n)$ be as
in Lemma~\ref{MGD}. The
numbers $\rada{m_1}{m_l}$ determine a sequence $\rada{i_1}{i_l}$
by
$i_s:= m_1+\cdots m_{s-1}+1$, $1\leq s\leq
l$. Consider a map $\varphi: H(n)\to {\overline {\cal D}}(n)$ defined by
\[
\varphi(\tttr{m_1}{m_l} \times \sigma)
:= \langle\rada{\sigma^{-1}(i_1)}{\sigma^{-1}(i_l)}\rangle
\times [\sigma],
\]
where we denoted elements of ${\overline {\cal D}}(n)$ (= cells of
${\overline \Delta}(n)$) as
in~(\ref{Y}).
It is immediate to see that $\varphi$ is an
$\Sigma_n$-equivariant
epimorphism. For the left action of $\zeta \in {\bf Z}_l$ we have
\begin{eqnarray*}
\varphi(\zeta(\tttr{m_1}{m_l} \times \sigma)
&=&
{\rm sgn}(\zeta)\!\cdot\!
\varphi(
\tttr{m_{\zeta^{-1}(1)}}%
{m_{\zeta^{-1}(l)}} \times \zeta(\rada{m_1}{m_l})\sigma)
\\
&=& {\rm sgn}(\zeta)\!\cdot\! \langle
\rada{\sigma^{-1}(i_{\zeta(1)})}{\sigma^{-1}(i_{\zeta(l)})}
\rangle \times
[\sigma]
= \langle\rada{\sigma^{-1}(i_1)}{\sigma^{-1}(i_l)}
\rangle\times
[\sigma],
\end{eqnarray*}
which shows that $\varphi(x) = \varphi(\zeta y)$. On the
other hand,
a moment's
reflection show that $\varphi(x) = \varphi(y)$, for $x,y\in
H(n)$,
implies the existence of some $\zeta \in {\bf Z}_l$ such that
$x = \zeta y$.
Thus the map $\varphi$ induces an equivariant
isomorphism $({\bf s\hskip0mm}{\it Cycl} \circ {\it Ass})(n) = {\bf Z}_l \backslash H(n)
\cong {\overline {\cal D}}(n)$, which is exactly our map $\omega$. The nature
of the
map $\omega$ is illustrated on Figure~\ref{peniz}.\hspace*{\fill
\begin{figure}[hbtp]
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\end{picture}
\end{center}
\caption{
A representation of oriented faces of $\Delta_1$, $\Delta_2$
and $\Delta_3$ by
equivalence classes of labeled planar trees, representing
elements of $H(1)$, $H(2)$ and $H(3)$.
\label{peniz}}
\end{figure}
A ${\overline {\cal D}}$-trace $t:{\overline {\cal D}} \to {\cal E}_{A,V}$ is, under the
identification
of Theorem~\ref{resiz}, given by the values $T_n :=
t(\uparrow\!^{n-1}\xi_n)$, $n\geq 1$. The axiom~(\ref{Blatter}) then
reflects~(\ref{hvezdicka}).
The relation between the axiom~(\ref{Blatter}) and the
geometry of
the simplex is visualized on Figure~\ref{delta}.
\begin{figure}[hbtp]
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(74.16,52.17)
\put(-40.33,49.67){\makebox(0,0)[lc]{$\delta T_1 (a) = 0$:}}
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\put(-10.50,52.17){\makebox(0,0)[cc]{$a$}}
\put(-8.00,49.67){\makebox(0,0)[lc]{$)=0$}}
\put(-40.33,39.67){\makebox(0,0)[lc]
{$\delta T_2 (a,b) = T_1(a\cdot b)-\znamenko{|a|\cdot
|b|}T_1(b\cdot a):$}}
\put(37.34,39.67){\makebox(0,0)[lc]{$\delta($}}
\put(43.00,39.67){\makebox(0,0)[cc]{$\bullet$}}
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\put(55.50,39.67){\makebox(0,0)[lc]{$)=$}}
\put(64.66,39.67){\makebox(0,0)[cc]{$\bullet$}}
\put(64.66,43.17){\makebox(0,0)[cc]{$ab$}}
\put(69.16,39.84){\makebox(0,0)[cc]{$-$}}
\put(74.16,39.67){\makebox(0,0)[cc]{$\bullet$}}
\put(74.16,43.17){\makebox(0,0)[cc]{$ba$}}
\put(43.00,39.67){\vector(1,0){10.00}}
\put(-40.50,26.34){\makebox(0,0)[lc]
{$\delta T_3 (a,b,c) = T_2(a\cdot
b,c)+\znamenko{|a|\cdot(|b|+|c|)}
T_2(b\cdot c,a)+\znamenko{|c|\cdot(|a|+|b|)} T_2(c\cdot a,b)$:}}
\put(-40.50,8.67){\makebox(0,0)[lc]{$\delta\left(\rule{0mm}{10mm}\right.$}}
\put(-22.65,14.50){\vector(-3,-4){8.13}}
\put(-14.53,3.67){\vector(-3,4){8.13}}
\put(-30.80,3.67){\vector(1,0){16.25}}
\put(-7.17,8.84){\makebox(0,0)[cc]{$\left.\rule{0mm}{10mm}\right)=$}}
\put(4.49,8.84){\vector(1,0){14.00}}
\put(21.99,8.84){\makebox(0,0)[cc]{$+$}}
\put(42.49,8.84){\makebox(0,0)[cc]{$+$}}
\put(24.99,8.84){\vector(1,0){14.00}}
\put(45.49,8.84){\vector(1,0){14.00}}
\put(-26.45,10.67){\makebox(0,0)[rb]{$(ca)b$}}
\put(-18.61,10.34){\makebox(0,0)[lb]{$(bc)a$}}
\put(-22.78,2.84){\makebox(0,0)[ct]{$(ab)c$}}
\put(11.50,11.50){\makebox(0,0)[cb]{$(ab)c$}}
\put(32.00,11.67){\makebox(0,0)[cb]{$(bc)a$}}
\put(52.00,11.67){\makebox(0,0)[cb]{$(ca)b$}}
\put(42.99,43.17){\makebox(0,0)[cc]{$ab$}}
\put(53.16,43.00){\makebox(0,0)[cc]{$ba$}}
\end{picture}
\end{center}
\caption{
Relation of the axiom~(9)
to the geometry of the
simplex.
\label{delta}
}
\end{figure}
\section{Quadratic operads and modules; modules associated
to cyclic
operads}
\label{cervena-tuzka}
Each operad ${\cal P}$ can be presented as a quotient ${\cal F}(E)/I$,
for a
collection $E$ and an `operadic' ideal $I$. The operad ${\cal P}$
is {\em quadratic\/}~\cite[page~228]{ginzburg-kapranov:DMJ94}
if it has a presentation
such that the
collection $E$ is concentrated in degree $2$, $E = E(2)$,
and the
ideal $I$ is generated by a subspace $R \subset {\cal F}(E)(3)$. In
this
case we write ${\cal P} = \prez ER$.
\begin{example}{\rm\
\label{genius}
Quadratic operads are omnipresent. Just
recall that ${\it Ass} = \prez ER$ for $E = E(2) = {\bf k}[\Sigma_2]$,
the
regular representation of $\Sigma_2$. Choosing a generator
$\mu \in E$, we can write $E = {\rm Span}(\mu,\mu S_{21})$, where
$S_{21}\in \Sigma_2$
is the transposition. Then $R\subset {\cal F}(E)(3)$ is the
$\Sigma_3$-subspace generated by $\mu(1,\mu)- \mu(\mu,1)$. If we
think of $\mu$ as corresponding to a multiplication, then the
generator of $R$ expresses the associativity. Sometimes
we simplify
the notation and write ${\it Ass}= \prez \mu{\mu(1,\mu)-
\mu(\mu,1)}$.
}\end{example}
Similarly, for each ${\cal P}$-module $M$ there exists a collection
$X$ such
that $M = (X\circ {\cal P})/J$ for some right submodule
$J\subset X\circ {\cal P}$.
The following definition was introduced independently also
in~\cite{ginzburg-voronov}.
\begin{definition}
\label{celenka}
The module $M$ is called quadratic, if, in the above
presentation,
the collection $X$ is
concentrated in degree $1$, $X= X(1)$, and the right submodule
$J$
is generated by a subspace $G \subset (X \circ {\cal P})(2)$.
In this case we write $M = \prezmod X{{\cal P}}G$.
\end{definition}
\begin{example}{\rm\
\label{nabijecka}
Let $X=X(1)$ be generated by one element $g$; $X(1)=
{\rm Span}(g)$. Let
$G \subset (X\circ {\it Ass})(2)$ be defined as $G = {\rm Span}(g(\mu)(1-
S_{21}))$, where $\mu$ and $S_{21}\in \Sigma_2$
has the same meaning as in Example~\ref{genius}.
Then it is not
difficult to see that
${\it Cycl} = \prezmod X{{\it Ass}}G$ or, in a more explicit notation,
\begin{equation}
\label{Spitfire}
{\it Cycl} = \prezmod g{{\it Ass}}{g(\mu)- g(\mu)S_{21}}.
\end{equation}
Now we can give the characterization of ${\it Cycl}$ traces
promised in Example~\ref{why-traces}.
Since the ${\it Ass}$-module ${\it Cycl}$ is generated by
$g\in {\it Cycl}(1)$, any ${\it Cycl}$-trace $t: {\it Cycl} \to {\cal E}_{A,W}$ is
determined by the image $T := t(g)\in {\cal E}_{A,V}(1)= \Hom AW$. The
symmetry~(\ref{nabla}) then follows from the condition
$t(g(\mu)(1-
S_{21})) = 0$.
}\end{example}
In fact, we show that ${\it Cycl}$ is a very special case of a
module over
the {\em cyclic\/} operad ${\it Ass}$. Cyclic operads were
introduced by
E.~Getzler and
M.M.~Kapranov~\cite[Section~2]{getzler-kapranov:cyclic}.
The definition we are going to recall is based on the following
convention.
We interpret the symmetric group $\Sigma_{n+1}$ as the group of
permutations of the set $\{\rada 0n\}$ and $\Sigma_n$
as the subgroup of $\Sigma_{n+1}$ consisting of permutations
$\sigma \in \Sigma_{n+1}$ with $\sigma(0)= 0$. If $\tau_n
\in \Sigma_{n+1}$ is the cycle $(\rada 0n)$, then $\tau_n$
and $\Sigma_n$ generate $\Sigma_{n+1}$.
\begin{definition}
\label{tabal}
Cyclic operad is an ordinary operad ${\cal P}$ such that the right
$\Sigma_n$-action on ${\cal P}(n)$ extends, for $n\geq 1$, to
an action
of $\Sigma_{n+1}$ such that
\begin{itemize}
\item[(i)]
$\tau_1(1)= 1$, where $1\in {\cal P}(1)$ is the unit and
\item[(ii)]
for $p\in {\cal P}(m)$ and $q\in {\cal P}(n)$,
\[
\gamma(p;q,\rada 11)\cdot \tau_{m+n-1}= \znamenko{|p|\cdot|q|}
\gamma(q\cdot\tau_n;\rada11,p\cdot\tau_m).
\]
\end{itemize}
\end{definition}
An intuitive feeling for the action of $\tau_n$ is suggested by
Figure~\ref{feel}.
\begin{figure}[hbtp]
\begin{center}
\unitlength 1.00mm
\thinlines
\begin{picture}(60.84,24.84)
\put(49.67,16.00){\line(-1,-1){10.17}}
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\put(41.33,5.83){\line(0,-1){4.83}}
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\put(48.00,5.83){\line(0,-1){4.83}}
\put(57.33,5.83){\line(0,-1){4.83}}
\put(53.00,3.33){\makebox(0,0)[cc]{$\cdots$}}
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\put(13.17,3.33){\makebox(0,0)[cc]{$\cdots$}}
\put(39.08,0.91){\oval(4.50,3.17)[b]}
\put(36.83,1.00){\line(0,1){22.33}}
\put(55.17,23.09){\oval(11.33,3.50)[t]}
\put(60.83,23.67){\line(0,-1){23.00}}
\put(9.67,10.33){\makebox(0,0)[cc]{$p$}}
\put(49.50,10.33){\makebox(0,0)[cc]{$p\tau_n$}}
\put(27.00,10.33){\makebox(0,0)[cc]{$\longmapsto$}}
\put(1.50,-2.67){\makebox(0,0)[cc]{$1$}}
\put(5.00,-2.67){\makebox(0,0)[cc]{$2$}}
\put(8.17,-2.67){\makebox(0,0)[cc]{$3$}}
\put(17.50,-2.67){\makebox(0,0)[cc]{$n$}}
\put(44.83,-2.67){\makebox(0,0)[cc]{$1$}}
\put(48.00,-2.67){\makebox(0,0)[cc]{$2$}}
\put(60.83,-2.67){\makebox(0,0)[cc]{$n$}}
\end{picture}
\end{center}
\caption{A `visualization' of the action of $\tau_n$. The
element
$\tau_n$ turns
$p\in {\cal P}(n)$, represented as a `thing' with $n$ inputs and one
output, a bit so that the first
input becomes the output and the output
becomes the last input of $p\cdot \tau_n$.\label{feel}}
\end{figure}
Let ${\cal P} = \prez ER$ be a quadratic operad. Thus $E(2)$ is a
$\Sigma_2= {\bf Z}_2$ space and the homomorphism ${\rm sgn}:
\Sigma_3\to {\bf Z}_2$ equips $E(2)$ with a $\Sigma_3$-action which
induces on ${\cal F}(E)$ a cyclic operad structure. We say, according
to~\cite[\S3.2]{getzler-kapranov:cyclic},
that ${\cal P}$ is {\em cyclic quadratic\/},
if the subspace $R
\subset {\cal F}(E)(3)$ is $\Sigma_4$-invariant.
In this case the operad ${\cal P}$ carries a natural cyclic structure
induced from the cyclic structure of ${\cal F}(E)$.
An example of a cyclic quadratic operad is the operad ${\it Ass}$,
see~\cite[Proposition~2.4]{getzler-kapranov:cyclic}
for a very explicit description of the cyclic
structure, and also for other examples of cyclic operads.
Let ${\cal P} = ({\cal P},\gamma)$ be a cyclic operad in
the sense of Definition~\ref{tabal}. Define the ${\cal P}$-module
$M_{{\cal P}}$ as follows. As a collection, $M_{{\cal P}}(n+1)= {\cal P}(n)$,
for $n\geq 0$. The structure maps are given by
\begin{eqnarray}
\label{eqv}
\nu(x;p_0,\rada 11)&:=&\znamenko{|p_0|\cdot |x|}
\gamma(p_0\cdot\tau_{m_0};\rada 11,x), \mbox{ and}
\\
\nonumber
\nu(x;1,\rada{p_1}{p_n}) &:=& \gamma(x;\rada{p_1}{p_n}),
\end{eqnarray}
for $x\in M_{{\cal P}}(n+1)={\cal P}(n)$, $p_i\in {\cal P}(m_i)$, $0\leq i\leq n$.
These conditions, along with the module axioms
(Definition~\ref{Amphora}), imply that
\[
\nu(x;\rada{p_0}{p_n})= \znamenko{|p_0|\cdot |x|}
\gamma(p_0\cdot\tau_{m_0};\rada11, \nu(x;1,\rada{p_1}{p_n})).
\]
The verification of module axioms of $M_{{\cal P}}$
is routine.
\begin{definition}
\label{Turmo}
We call
the module $M_{\cal P}$ the module associated to the cyclic
operad ${\cal P}$.
\end{definition}
We will also consider {\em unital\/} operads
which describe algebras {\em with unit\/}.
Unital operad is an
operad such that ${\cal P}(0)$ is nonempty, generated by an element
$\vartheta$, encoding the unit ${\bf k} \to A$ in the corresponding
algebra
$A$. The element $\vartheta$ determines,
for $n\geq 1$ and $1\leq i\leq n$, the
`degeneracy' maps $s_i : {\cal P}(n)\to {\cal P}(n-1)$ by $s_i(p):=
\gamma(p;\rada 11,\vartheta,\rada 11)$ ($\vartheta$ at the
$i$-th
position). These maps satisfy certain commutation
relations~\cite[page~278, Proposition~3]{stasheff:TAMS63}
which follow from the axioms of an operad. For us, the most
important
is the relation $s_1(p)= s_2(p S_{21})$ for $p\in {\cal P}(2)$, which
follows from the equivariance of structure maps. This,
together with
a natural requirement that $s_1 = s_2$ on ${\cal P}(2)$, gives
that the
maps $s_1=s_2$ are $\Sigma_2$-equivariant on ${\cal P}(2)$. We
denote both
maps $s_1$ and $s_2$ (which are the same) by $s$.
\begin{example}
\label{vypocetni}
{\rm\
There is the operad ${\it UAss}$ for associative algebras with unit,
which is the same as the operad ${\it Ass}$ except that ${\it UAss}(0)=
{\rm Span}(\vartheta)$. The map $s :{\it Ass}(2) = {\bf k}[\Sigma_2]\to {\bf k}$
is the
standard augmentation of the group ring ${\bf k}[\Sigma_2]$.
Another example is the operad ${\it UComm}$ for unital commutative
algebras. It has, for $n\geq 1$, ${\it UComm}(n)= {\it Comm}(n)= \mbox{$1\!\!1$}$ (the
trivial
one-dimensional representation), and
${\it UComm}(0)= {\rm Span}(\vartheta)$. The map $s : {\it UComm}(2)= {\bf k}
\to {\bf k}$
is the identity.
A more complicated example is the operad ${\it UPoiss}$ for
Poisson algebras with unit. Here by a Poisson algebra with
unit we
mean an ordinary Poisson
algebra~\cite[Example~3.3]{markl:dl} $P = (P, \cdot, [-,-])$
with a distinguished element $1\in P$ which is a two-sided
unit for
the commutative multiplication $\cdot$, while $[x,1]= [1,x]=
0$, for
all $x\in P$. The operad ${\it UPoiss}$ coincides with the operad
${\it Poiss}$
for Poisson algebras (which is very explicitly described
in~\cite{fox-markl:ContM97}),
except that ${\it UPoiss}(0) ={\rm Span}(\vartheta)$. The
component ${\it UPoiss}(2)$ is the direct sum $\mbox{$1\!\!1$} \oplus {\rm sgn}$,
with the trivial one-dimensional representation $\mbox{$1\!\!1$}$
concentrated in degree zero, and the signum representation
${\rm sgn}$ in degree 1.
The map $s : {\it UPoiss}(2)\to {\bf k} = {\it UPoiss}(1)$ is the
projection on the zero-dimensional component.
}\end{example}
We saw that a natural way to construct cyclic operads was to
take a
quadratic operad ${\cal P} = \prez ER$ for which the relations
$R$ were
invariant under the natural $\Sigma_4$-action; the operad
${\cal P}$ had
then a natural cyclic structure. There is a similar approach to
unital operads.
So, let ${\cal P} = \prez ER$ be a quadratic operad and suppose we are
given an epimorphism $s :E \to {\bf k}$. This will be a model for
the two
degeneracy maps $s_1 = s_2 : {\cal P}(2)= E \to {\cal P}(1)= {\bf k}$.
These two maps induce degeneracy maps on the free operad
${\cal F}(E)$ satisfying the correct commutation relations. The
following
definition expresses the conditions assuring that this structure
preserves the relations $R$.
\begin{definition}
\label{virgo}
Let ${\cal P} = \prez ER$ be a quadratic (resp.~cyclic quadratic)
operad.
Suppose that we are given an epimorphism $s : E \to {\bf k}$
such that
\begin{itemize}
\item[(i)]
$s$ is invariant under the $\Sigma_2$ (resp.~$\Sigma_3$, in the
cyclic case) action, and
\item[(ii)]
the induced maps $s_1,s_2,s_3: {\cal F}(E)(3)\to {\cal F}(E)(2)$ send the
subspace $R\subset {\cal F}(E)(3)$ to zero.
\end{itemize}
Then the collection ${\it U}{\cal P}$, defined by ${\it U}{\cal P}(n):= {\cal P}(n)$ for
$n\geq 1$
and ${\it U}{\cal P}(0)= {\bf k}$, has a natural structure of a unital
operad. We
call operads of this form quadratic unital (resp.~cyclic
quadratic
unital)
operads.
\end{definition}
All the three operads ${\it UAss}$, ${\it UComm}$ and ${\it UPoiss}$ from
Example~\ref{vypocetni} are cyclic quadratic unital operads
in the sense of
Definition~\ref{virgo}.
\begin{proposition}
\label{zabacek}
Let ${\cal P} = \prez ER$ be a cyclic unital quadratic operad in
the sense
of Definition~\ref{virgo}. Then the associated module
$M_{{\it U}{\cal P}}$ is
quadratic,
\[
M_{{\it U}{\cal P}} =
\prezmod{{\rm Span}(\vartheta)}{{\cal P}}{{\rm Ker}(s):E \to {\bf k}}.
\]
\end{proposition}
\noindent
{\bf Proof.}
Let $X = X(1):= {\rm Span}(g)$ and consider the map $\psi :
X\circ {\cal P} \to M_{{\it U}{\cal P}}$ defined by $p(g) :=
\vartheta \in M_{{\it U}{\cal P}}(1)= {\it U}{\cal P}(0)$.
Because clearly $X\circ {\cal P} = {\cal P}$, $\psi$ is, by~(\ref{eqv}),
given as
\[
(X\circ {\cal P})(n)= {\cal P}(n)\ni p \longmapsto \nu(\vartheta;p) =
\gamma(p\cdot\tau_n; \rada 11,\vartheta)\in M_{{\it U}{\cal P}}(n)
= {\it U}{\cal P}(n-1).
\]
Thus $\psi$ is a sequence
$\psi(n):{\cal P}(n)\mapsto {\it U}{\cal P}(n-1)$ given by
$\psi(n)(p):= \gamma(p\cdot\tau_n;
\rada 11,\vartheta)$. These maps are epimorphisms, because $E$
generates ${\cal P}$ and $s:E \to {\bf k}$ (= the composition with
$\vartheta$)
is epi, by assumption.
On the other hand, $\psi(2):{\cal P}(2)=E \to {\it U}{\cal P}(1)={\bf k}$ is exactly
the map
$s$, thus the submodule of $X\circ {\cal P}$ generated by ${\rm Ker}(s)$ is
certainly contained in the kernel of $\psi$. A moment's
reflections
shows that the whole kernel of $\psi$ is generated by
${\rm Ker}(s)$.\hspace*{\fill
\begin{example}{\rm\
\label{kacirek}
If ${\cal P} = {\it UAss}$ is the operad for unital associative algebras
from
Example~\ref{vypocetni},
then ${\rm Ker}(s:E= {\bf k}[\Sigma_2] \to {\bf k})=
{\rm sgn}$. So, by Proposition~\ref{zabacek}, $M_{{\it UAss}}=
\prezmod{\vartheta}{{\it Ass}}{{\rm sgn}}$, thus $M_{{\it UAss}} = {\it Cycl}$,
by~(\ref{Spitfire}).
For the operad ${\it UComm}$ the kernel of $s$ is trivial,
hence
\[
M_{{\it UComm}} = {\rm Span}(\vartheta)\circ {\it Comm} \cong {\it Comm}.
\]
In other words, $M_{{\it UComm}}$ is the operad
${\it Comm}$ considered as a right
module over itself. We recommend to the reader to make the
similar
discussion for the operad ${\it UPoiss}$.
}\end{example}
\section{Cyclohedron as the cobar construction}
\label{penezenka}
For a graded vector space $V$, let
$\dual V$ denote the graded dual of $V$, i.e.\ the graded
vector space $\bigoplus_p(\dual V)_p$ with $(\dual V)_p
={\rm Hom}^{-p}(V,{{\bf k}})={\rm Hom}(V_{p},{{\bf k}})$, the space
of linear maps from $V_{p}$ to ${\bf k}$. If $V$ is a right
$\Sigma_n$-module, then we equip $\dual V$ with the transposed
$\Sigma_n$-action.
We believe that there is no real risk of confusion of the ${}^*$
indicating the dual with the star indicating the degree.
Recall
also~\cite[\S3.5.1]{ginzburg-kapranov:DMJ94}
that a {\em cooperad\/} is a collection ${\cal Q}=\coll {{\cal Q}}$
together with a system of maps
\[
\omega = \omega_{\rada{m_1}{m_l}}: {\cal Q}(m_1+\cdots+m_l)
\to {\cal Q}(l)\otimes {\cal Q}(m_1)\otimes \cdots \otimes {\cal Q}(m_l),
\]
which satisfy the axioms which are exactly the duals of
the axioms
for an operad. A typical example of a cooperad is the dual
$\dual{\cal P}$ of
an operad ${\cal P}$, i.e.~the collection $\dual{\cal P} =
\{\dualI{{\cal P}(n)}\}_{n\geq 1}$
with the cooperad structure defined by the dualization of the
structure maps of ${\cal P}$; here an obvious finite type assumption
is
necessary, but it will be always satisfied in the paper and
we will
make no explicit comments about it.
Observe that if ${\cal P}$ is an operad, then both the suspension
${\bf s\hskip0mm} {\cal P}$ and the desuspension ${\bf s\hskip0mm}^{-1} {\cal P}$ introduced
in Section~\ref{hrnicek1}, with the sign convention
of~(\ref{sign-factor}),
have a natural operad
structure induced from the operad structure on ${\cal P}$.
The structure maps of a cooperad ${\cal Q}$ determine (and are
determined
by) a map ${\overline \nu}:
{\cal Q}\to {\cal F}({\cal Q})$ of collections. Composing this
map with the (de)suspensions gives a degree -1
map $\downarrow\! {\cal Q} \to {\cal F}(\downarrow\! {\cal Q})$
which uniquely (because of the freeness of ${\cal F}(\downarrow\! {\cal Q})$)
extends
to
a degree -1 derivation ${\partial_{\Cob}}$ of the operad ${\cal F}(\downarrow\!
{\cal Q})$ which
satisfies, as a consequence of the axioms of a cooperad,
${\partial_{\Cob}} \circ{\partial_{\Cob}} =0$.
The differential graded operad
\[
{\Omega}({\cal Q}) := ({\cal F}(\downarrow\! {\cal Q}),{\partial_{\Cob}})
\]
is called the {\em cobar construction\/} on the cooperad
${\cal Q}$~\cite[\S3.2.12]{ginzburg-kapranov:DMJ94}.
Let ${\cal P} = \prez ER$ be a quadratic operad. Take the dual $\dual
E$ of
$E$ and let $R^{\perp}$ be the annihilator of the space
$R\subset {\cal F}(E)(3)$ in
${\cal F}(\dual E \otimes {\rm sgn})(3)$. The quadratic operad ${\cal P}^! :=
\prez {\dual E \otimes {\rm sgn}}{R^{\perp}}$ is,
according to~\cite[\S2.1.9]{ginzburg-kapranov:DMJ94}, called the
{\em Koszul\/} (or {\em quadratic\/}) {\em dual\/}
of the operad ${\cal P}$. We always have a map
$\downarrow\! \dualI{{\bf s\hskip0mm} {\cal P}} \to {\cal P}^!$ of collections defined as
the composition
\[
\downarrow\! \dualI{{\bf s\hskip0mm} {\cal P}}\stackrel{\rm proj.}{\longrightarrow}
(\downarrow\! \dualI{{\bf s\hskip0mm} {\cal P})(2)} = \dual E\otimes {\rm sgn} =
{\cal P}^!(2) \hookrightarrow {\cal P}^!
\]
which extends, by the freeness of
${\cal F}(\downarrow\! \dualI{{\bf s\hskip0mm} {\cal P}})$, to a differential graded
operad map
\begin{equation}
\label{budicek}
\pi : {\Omega}(\dualI{{\bf s\hskip0mm} {\cal P}}) = ({\cal F}(\downarrow\! \dualI{{\bf s\hskip0mm} {\cal P}}),{\partial_{\Cob}})
\to ({\cal P}^!,\partial=0).
\end{equation}
The quadratic operad ${\cal P}$ is called
{\em Koszul\/}~\cite[Definition~4.1.3]{ginzburg-kapranov:DMJ94}
if the map in~(\ref{budicek}) is a homology isomorphism.
In the rest of the paper, the space of generators $E=E(2)$ of a
quadratic
operad will always be ungraded, concentrated in degree
zero. Then the
components of
both the operad ${\cal P}$ and its dual ${\cal P}^!$ are concentrated
in degree
zero as well, and the Koszulness
implies that the complex $({\Omega}(\dualI{{\bf s\hskip0mm} {\cal P}})(n),
{\partial_{\Cob}})$ is acyclic in positive dimensions, for all $n$. On
the other
hand, as shown
in~\cite[Theorem~4.1.13]{ginzburg-kapranov:DMJ94},
this acyclicity condition implies the
Koszulness of ${\cal P}$.
\begin{example}{\rm\
\label{yhr}
The operad ${\it Ass}$ is well-known to be Koszul self-dual, ${\it Ass} =
{\it Ass}^!$~\cite[Theorem~2.1.11]{ginzburg-kapranov:DMJ94}, and
Koszul~\cite[Corrolary~4.2.7]{ginzburg-kapranov:DMJ94}.
We have already remarked that the
operad ${\overline {\cal A}}$ of cellular chains on the (symmetrized)
associahedron
${\overline K}$ coincides, as a differential graded operad, with
the cobar
construction ${\Omega}(\dualI{{\bf s\hskip0mm}{\it Ass}})$. Let us give an explicit
illustration of this statement.
The $n$-th piece $(\downarrow\! \dualI{{\bf s\hskip0mm} {\it Ass}})(n)$ of the
collection
$(\downarrow\! \dualI{{\bf s\hskip0mm} {\it Ass}})$ is isomorphic to
one copy of the regular
representation ${\bf k}[\Sigma_n]$ concentrated in degree $(n-2)$.
The isomorphism is not unique, but we may choose, for example,
$\lambda : {\bf k}[\Sigma_n] \to \downarrow\! ({\bf s\hskip0mm} {\it Ass})^*(n)$ given by
$\lambda(\sigma)(\uparrow\!^{n-2}\rho) := {\rm sgn}(\sigma)\cdot
\delta_{\sigma,\rho}$, where $\sigma, \rho \in \Sigma_n$ and the
meaning of the `Kronecker delta' $\delta_{\sigma,\rho}$
is clear.
Formula~(\ref{fax})
then describes ${\cal F}(\downarrow\! \dualI{{\bf s\hskip0mm} {\it Ass}})(n)$ as the
vector space
spanned by the set of all {\em planar\/} (rooted, labeled)
$n$-trees,
having at least binary vertices. The identification of these
trees
with the bracketings of $\rada{\sigma(1)}{\sigma(n)}$,
$\sigma \in \Sigma_n$, i.e.~with the elements of the set
${\overline {\cal B}}(n)$,
is classical -- see~\cite[\S1.4]{boardman-vogt:73};
two examples are
shown on Figure~\ref{hreben}.
\begin{figure}[hbtp]
\begin{center}
\unitlength 0.70mm
\thicklines
\begin{picture}(174.84,50.34)
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\put(130.84,16.67){\line(6,-5){7.50}}
\put(138.84,40.00){\line(2,-3){15.56}}
\put(154.00,16.67){\line(-1,-1){6.67}}
\put(154.17,17.17){\line(6,-5){7.83}}
\put(138.84,40.00){\line(6,-5){36.00}}
\put(138.84,39.83){\makebox(0,0)[cc]{$\bullet$}}
\put(130.84,16.50){\makebox(0,0)[cc]{$\bullet$}}
\put(154.17,17.00){\makebox(0,0)[cc]{$\bullet$}}
\put(108.50,5.00){\makebox(0,0)[cc]{$1$}}
\put(123.50,5.00){\makebox(0,0)[cc]{$6$}}
\put(138.17,5.00){\makebox(0,0)[cc]{$2$}}
\put(147.50,4.83){\makebox(0,0)[cc]{$3$}}
\put(161.84,4.83){\makebox(0,0)[cc]{$4$}}
\put(174.67,4.83){\makebox(0,0)[cc]{$5$}}
\put(108.33,31.34){\makebox(0,0)[rc]{$1(62)(34)5:$}}
\put(43.24,16.56){\line(-1,-1){6.97}}
\put(16.74,16.74){\line(1,-1){6.97}}
\end{picture}
\end{center}
\caption{Two examples of an identification of planar trees with
elements of ${\cal B}$.\label{hreben}}
\end{figure}
The fact that the cellular differential coincides with
${\partial_{\Cob}}$ is a routine combinatorics.
See~\cite[Example~4.1]{markl:zebrulka} for details.
The identification above shows that the operad ${\it Ass}$
is Koszul. More precisely, we know that $H_*({\overline K})= H_0({\overline K})=
{\it Ass}$, because ${\overline K}(n)$ is the union of convex polyhedra
indexed by
the elements of the symmetric group. On the other hand, due
to the
identification above, $H_*({\Omega}(\dualI{{\bf s\hskip0mm} {\cal P}}))= H_*({\overline K})$,
thus the map
$\pi$ of~(\ref{budicek}) is a homology isomorphism.
}\end{example}
A {\em comodule\/} over a cooperad ${\cal Q}$
is a collection $N = \coll N$ together with structure maps
\[
\kappa = \kappa_{\rada{m_1}{m_l}}: N(m_1+\cdots+m_l)
\to N(l)\otimes {\cal Q}(m_1)\otimes \cdots \otimes {\cal Q}(m_l),
\]
satisfying axioms dual to the axioms of a module over an operad.
An example is the dual $\dual {\cal M}$ of a ${\cal P}$-module ${\cal M}$,
which is a
natural comodule over the cooperad $\dual{\cal P}$.
Let $N$ be a ${\cal Q}$-comodule. As in the case of
cooperads, the structure maps induce a degree -1 differential
${\partial_{\Cob}}$
on the free module $N\circ {\cal F}(\downarrow\! {\cal Q})$. The right
differential graded ${\Omega}({\cal Q})$-module
\[
{\Omega}(N;{\cal Q}) :=(N\circ {\cal F}(\downarrow\! {\cal Q}),{\partial_{\Cob}})
\]
is called the (right) {\em cobar construction\/} on the
${\cal Q}$-comodule
$N$.
Suppose that $M = \prezmod X{{\cal P}}G$ is a
quadratic ${\cal P}$-module, in the
sense of Definition~\ref{celenka}, over a quadratic operad ${\cal P} =
\prez ER$. Take
the dual $\dual X$ and let $G^{\perp}$ be the annihilator of
$G$ in $(\dual
X\circ (\dual E \otimes {\rm sgn}))(2)$. Then the quadratic
${\cal P}^!$-module
$M^! := \prezmod {\dual X}{{\cal P}^!}{G^{\perp}}$
is called the {\em Koszul\/} (or {\em quadratic\/})
{\em dual\/} of $M$. The above definitions were
independently made in~\cite{ginzburg-voronov}.
\begin{proposition}
The ${\it Ass}$-module ${\it Cycl}$ is Koszul self-dual, ${\it Cycl}^! =
{\it Cycl}$.
\end{proposition}
\noindent
{\bf Proof.}
Under the notation of Example~\ref{nabijecka}, the
$\Sigma_2$-space
$(X\circ {\it Ass})(2)$ is the direct sum $\mbox{$1\!\!1$} \oplus {\rm sgn}$ of
the trivial
and signum representations, while clearly $R= {\rm sgn}$,
generated by
$g(\mu) - g(\mu)S_{21}$. Then $R^\perp$ is easily seen to
be ${\rm sgn}$
and the proposition follows.\hspace*{\fill
Observe that, for a ${\cal P}$-module $M = \coll M$, the collection
${\bf s\hskip0mm} M$ has an induced
${\bf s\hskip0mm} {\cal P}$-module structure.
For a quadratic ${\cal P}$-module $\prezmod X{{\cal P}}G$ over a quadratic
operad
${\cal P} =
\prez ER$ we have, as in~(\ref{budicek}), the map
\begin{equation}
\label{Opusem}
\pi :{\Omega}(\dualI {{\bf s\hskip0mm} M},\dualI {{\bf s\hskip0mm} {\cal P}}) \to (M^!,\partial
= 0),
\end{equation}
induced by the composition
\[
\dualI {{\bf s\hskip0mm} M} \stackrel{\rm proj.}{\longrightarrow}
\dualI{{\bf s\hskip0mm} M}(1) = \dual X = M^!(1) \hookrightarrow M^!.
\]
The following definition was independently made
in~\cite{ginzburg-voronov}.
\begin{definition}
\label{telefon}
A quadratic module $M$ over a quadratic operad ${\cal P}$
is called Koszul if the
map $\pi$ in~(\ref{Opusem}) is a homology isomorphism.
\end{definition}
\begin{theorem}
\label{Amphora1}
The module ${\it Cycl}$ is Koszul.
\end{theorem}
The theorem will follow from Theorem~\ref{ucpavka} which
explicitly identifies the cobar
construction ${\Omega}(\dualI{{\bf s\hskip0mm}{\it Cycl}}, \dualI{{\bf s\hskip0mm}{\it Ass}})$
to the cellular chain
complex
${\cal M}$ of the symmetrized cyclohedron ${\overline W}$. Thus the
Koszulness of
${\it Cycl}$ is, as in the case of the operad ${\it Ass}$,
a consequence of the fact that the cyclohedron is a convex
polyhedron. At the end of the
section we formulate a more general statement which also implies
Theorem~\ref{Amphora1}.
\begin{theorem}
\label{ucpavka}
The cobar construction ${\Omega}(\dualI{{\bf s\hskip0mm}{\it Cycl}}, \dualI{{\bf s\hskip0mm}
{\it Ass}})$ is
isomorphic
to the cellular chain complex ${\cal M} = CC_*({\overline W})$
of the cyclohedron ${\overline W}$.
\end{theorem}
\noindent
{\bf Proof.}
As we observed in Example~\ref{yhr}, the collection
$\downarrow\!\dualI{{\bf s\hskip0mm} {\it Ass}}(n)$ consists of one copy of the regular
representation ${\bf k}[\Sigma_n]$ concentrated in degree
$(n-2)$. Similarly,
the collection $\dualI{{\bf s\hskip0mm} {\it Cycl}}$ is
isomorphic to the collection ${\bf s\hskip0mm} {\it Cycl}$.
We are going to give an explicit description of the space
$(\dualI{{\bf s\hskip0mm}{\it Cycl}}\
\circ \downarrow\!\dualI{{\bf s\hskip0mm} {\it Ass}})(n)$, similar to that of
Lemma~\ref{MGD}.
Suppose that $T_i$ is,
for each $1\leq i \leq l$, a planar $m_i$-tree,
whose all vertices are at least binary. Let $R(\rada{T_1}{T_l})$
be
the tree obtained by grafting the trees $\rada{T_1}{T_l}$ at the
inputs of the `$l$-rake,' or, pictorially:
\begin{center}
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Let $J(n)$ be the free graded right $\Sigma_n$-module generated
by the
symbols $R(\rada{T_1}{T_l})$, $m_1+\cdots+ m_l = n$, with
the degree
defined by
\[
\deg(R(\rada{T_1}{T_l}))= n-1-\sum_{i=1}^l \#{\rm vert}(T_i),
\]
where $ \#{\rm vert}(T_i)$ is the number of vertices of
the tree $T_i$. For $\zeta \in
{\bf Z}_l$ we put (see~(\ref{pejska_Mikinka}) for the notation)
\begin{equation}
\label{monce1}
\zeta(R(\rada{T_1}{T_l}) \times \sigma) :={\rm sgn}(\zeta) \cdot
R(\rada{T_{\eta^{-1}(1)}}{T_{\eta^{-1}(l)}}) \times
\zeta(\rada{m_1}{m_l}) \sigma.
\end{equation}
Then the graded vector space $(\dualI{{\bf s\hskip0mm}{\it Cycl}}\
\circ \downarrow\!\dualI{{\bf s\hskip0mm} {\it Ass}})(n)$ is spanned by equivalence
classes of
elements
\[
R(\rada{T_1}{T_l}) \times \sigma \in J(n),
\]
modulo the left action of the group ${\bf Z}_l$
introduced in~(\ref{monce1}). The right
action of the group $\Sigma_n$ is given by
\[
[R(\rada{T_1}{T_l}) \times \sigma]\cdot \rho :=
[R(\rada{T_1}{T_l}) \times \sigma\rho].
\]
We may symbolize the element $R(\rada{T_1}{T_l}) \times
\sigma \in
J(n)$ as the tree $R(\rada{T_1}{T_l})$ with the inputs
labeled by $\rada{\sigma^{-1}(1)}{\sigma^{-1}(l)}$.
There is an almost obvious one-to-one correspondence between
these
labeled planar $n$-trees and cyclic bracketings of $n$
indeterminates from
${\overline {{\cal B}C}}(n)$. This
becomes absolutely clear after
looking at Figure~\ref{master}.
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\put(32.79,8.63){\makebox(0,0)[cc]{$1$}}
\put(47.78,8.81){\makebox(0,0)[cc]{$3$}}
\put(55.66,8.81){\makebox(0,0)[cc]{$1$}}
\put(60.51,8.81){\makebox(0,0)[cc]{$2$}}
\put(6.50,167.33){\makebox(0,0)[cc]{$= (1)$}}
\end{picture}
\end{center}
\caption{A representation of elements of ${{\cal B}C}(1)$, ${{\cal B}C}(2)$ and
${{\cal B}C}(3)$ by equivalence classes of planar trees.\label{master}}
\end{figure}
To be more formal, the isomorphism ${\varphi} :
(\dualI{{\bf s\hskip0mm}{\it Cycl}}\
\circ \downarrow\!\dualI{{\bf s\hskip0mm} {\it Ass}}) \to {\cal M}$ is defined by
${\varphi}(\uparrow\!^{n-1}
\xi_n) := f_n$, where $f_n = (\rada 1n)$ is the top
$n$-dimensional
cell of the cyclohedron $W_n$ and $\xi_n$ the generator of
${\it Cycl}(n)$
represented by $\mbox{$1\!\!1$}_n \in \Sigma_n$. We must specify also the
orientation of $f_n$. In Observation~\ref{sdff} we
constructed points $\rada {P_1}{P_n}$ spanning a simplex
$\Delta_{f_n}\in f_n$. We orient $f_n$ coherently with the
orientation
of $\Delta_{f_n}\in f_n$ induced by the order $P_1 < \cdots <
P_n$ of
its vertices.
The cobar
differential $\partial_{{\Omega}}$ is given by
\begin{equation}
\label{tuti}
\partial_{{\Omega}}(\uparrow\!^{n-1} \xi_n) =
\cyclsum \sum_{1\leq k\leq n}\znamenko{n+k}
\nu(\uparrow\!^{n-k}\xi_{n-k+1};\alpha_k,\rada11),
\end{equation}
where $\alpha_k= \mbox{$1\!\!1$}_k\in {\bf k}[\Sigma_k]= {\it Ass}$ is the generator,
$k\geq 1$. We shall compare now~(\ref{tuti}) to the geometric
boundary of the top-dimensional cell $f_n$ of the cyclohedron
$W_n$.
This can be done exactly as in the proof of Theorem~\ref{resiz}
and we
leave it to the reader.\hspace*{\fill
The {\em proof of Theorem~\ref{nuzky}\/} is now immediate. An
${\cal M}$-trace $T: {\cal M}\to {\cal E}_{A,W}$ is determined by a system
$\{T_n :=
t(\uparrow\!^{n-1}\xi_n): A^{\otimes n}\to W\}_{n \geq 1}$. The
axiom~(\ref{Ax}) then reflects~(\ref{tuti}).
We finish this section by the following theorem whose proof,
based on
a straightforward but involved spectral sequence argument,
we omit.
\begin{theorem}
\label{Katalogizacni}
Let $M_{{\it U}{\cal P}}$ be a module associated to a cyclic unital
quadratic
operad ${\cal P}$. Then $M_{U{\cal P}}$ is Koszul if and only if ${\cal P}$ is.
\end{theorem}
Because ${\it Cycl} = M_{{\it UAss}}$
(Example~\ref{kacirek}) and the operad ${\it Ass}$ is
Koszul~\cite[Corollary~4.2.7]{ginzburg-kapranov:DMJ94},
Theorem~\ref{Katalogizacni} gives an alternative proof of
Theorem~\ref{Amphora1}.
\section{Cyclohedron as a compactification of the simplex}
\label{22}
For a compact Riemannian manifold $V$, let $C^0_n(V)
=\{(\rada{v_1}{v_n});\ v_i\not= v_j\}$ be the configuration
space
of $n$ distinct points in $V$. Axelrod and Singer constructed
in~\cite{axelrod-singer:preprint}
a compactification $C_n(V)$ of this space,
by adding to $C^0_n(V)$
the blow-ups along the diagonals. The space $C_n(V)$ is
a manifold
with corners, whose open part (= top-dimensional stratum) is
$C^0_n(V)$.
There exists a similar compactification of the {\em moduli
space\/}
${\buildrel \circ \over {{\sf F}}}_m(n)$ of configurations of $n$
distinct points in the $m$-dimensional
Euclidean
plane ${\bf R}^m$ modulo the action of the affine group,
described by Getzler and
Jones in~\cite[\S3.2]{getzler-jones:preprint}
and denoted by ${\sf F}_m(n)$. The authors
of~\cite{getzler-jones:preprint} also
observed that the collection
${\sf F}_m := \{{\sf F}_m(n)\}_{n\geq 1}$ has a
natural structure of a topological operad. In~\cite{markl:cf} we
proved
that
\begin{theorem}
\label{1725}
If $V$ is an $m$-dimensional parallelizable Riemannian
manifold, then
the collection $C(V) := \{C_n(V)\}_{n\geq 1}$ forms a right
module
over the operad ${\sf F}_m$ in the category of manifolds with
corners.
\end{theorem}
There is also a `framed' version of the above
theorem for manifolds which are not parallelizable, but we
will not
need it.
Take $V=S^1$. Then it is immediately seen that the space
$C_n^0(V)$
has $(n-1)!$ components indexed by cyclic orders of $n$
points on the circle. Each of these components is isomorphic to
$\oDelta^n \times S^1$,
where $\oDelta^n$
is the open $n$-dimensional simplex. It is `well-known' (see
Remark~\ref{beuo}
below) that the compactification
$C_n(S^1)$ is isomorphic to ${\overline W}_n \times S^1$, the product
of the
symmetrized cyclohedron with the
circle~\cite[page~5249]{bott-taubes:JMP94}.
Similarly, ${\buildrel \circ \over {{\sf F}}}_1(n)$ is easily seen to have $n!$ components
indexed
by orders of the set of $n$ points on the line, each component
being
isomorphic to $\oDelta^{n-2}$. Again, it is `well-known'
that the compactification ${\sf F}_1(n)$ is the (symmetrized)
associahedron ${\overline K}_n$~\cite[3.2(1)]{getzler-jones:preprint}.
This assumed,
our statement (Theorem~\ref{sirky}) about the existence of a
${\overline K}$-module structure on the cyclohedron follows from
Theorem~\ref{1725} applied on $C(S^1)= {\overline W} \times S^1$
(the extra factor $S^1$ plays no r\^ole).
\begin{remark}{\rm
\label{beuo}
The Axelrod-Singer compactification $C_n(S^1)$ is a manifold
with
corners, constructed by a very explicit sequence of blow-ups. We
do
not know any `universal' characterization of this space. Thus
to prove that $C_n(S^1) \cong {\overline W}_n \times S^1$ would require
an explicit construction of an isomorphism of two manifolds with
corners,
which is certainly not a tempting challenge. But a reflection
on the
structure of these two object `proves' the isomorphism
`beyond any
doubts', which is the opinion shared by many authors. The
same remark applies also to the isomorphism ${\sf F}_1(n)\cong
{\overline K}_n$.
}\end{remark}
\begin{remark}{\rm
It follows from general properties of manifold-with-corners
that both
$K_n$ and $W_n$ are truncations of a
simplex~\cite[Proposition~6.1]{markl:cf}, but this existence
statement
says
nothing about an explicit linear convex realization of
Section~\ref{1968}.
}\end{remark}
As we observed above,
the cyclohedron $W_n$ can be viewed as the simplex $\Delta^n$,
some faces of whose were blown-up. In the rest of this
section we
show that a very natural spectral sequence related to the cobar
construction can be interpreted as an inverse process --
`deblowing-up' of the cyclohedron back to the closed simplex.
For a collection $X$ and $p\geq 1$, let $\skel Xp\subset X$
be the subcollection
defined by $\skel Xp(n) = X(n)$ for $n\leq p$ and $\skel Xp(n)
= 0$
otherwise.
Let us consider, for a comodule $N$ over an cooperad ${\cal Q}$
and for a natural $n$, the subspace
\[
F_p(n) := (\skel N{p+1} \circ {\cal F}(\downarrow\! {\cal Q}))(n) \subset
(N \circ {\cal F}(\downarrow\! {\cal Q}))(n).
\]
It is easily seen that $F_p(n)$ is ${\partial_{\Cob}}$-invariant, thus
$\{F_p(n)\}_{p\geq 0}$ is an increasing filtration
of the $n$-th piece ${\Omega}(N,{\cal Q})(n)$ of the
cobar construction ${\Omega}(N,{\cal Q}) = (N \circ {\cal F}(\downarrow\! {\cal Q}),
{\partial_{\Cob}})$.
Let ${\bf E}(n) = (E^r_{pq}(n), d^r)$ be the corresponding spectral
sequence.
The following lemma is an easy exercise.
\begin{lemma}
\label{whoop}
The spectral sequence ${\bf E}(n) = (E^r_{pq}(n), d^r)$
constructed above converges to $H_*({\Omega}(N,{\cal Q})(n))$.
The first term $E^1$ is described as
\[
E^1_{pq}(n)= (N(p+1)\circ H_*({\Omega}({\cal Q})))(n)_{p+q},
\]
the space of elements of degree $p+q$ in the $n$-th piece of
the free
$H_*({\Omega}({\cal Q}))$-module on the $(p+1)$-th piece of the
collection $N$.
\end{lemma}
If the cooperad ${\cal Q}$ and the comodule $N$ are Koszul, the
spectral
sequence above collapses at the 1st term, which has a very
explicit
description. Since we did not formulate the Koszulness for
cooperads
and comodules (though the definition is an exact dual),
we suppose
from now on that
$N = \dualI{{\bf s\hskip0mm} M}$ and ${\cal Q} = \dualI{{\bf s\hskip0mm} {\cal P}}$, for a module $M$
over an
operad ${\cal P}$. We also suppose that $M$ and ${\cal P}$ are not graded,
i.e.~
that both $M(n)$ and ${\cal P}(n)$ are concentrated in degree $0$,
$n\geq 1$.
\begin{proposition}
\label{myska}
If the operad ${\cal P}$ is Koszul, then $E^1_{pq}(n)= 0$ for
$q\geq 1$,
while
\[
E^1_{p0}(n)= (\dualI{{\bf s\hskip0mm} M}(p+1) \circ {\cal P}^!)(n),
\]
and the spectral sequence collapses at $E^1$.
The module $M$ is Koszul if
and only if the complex
\[
0 \stackrel{d^1}{\longleftarrow} E^1_{00}(n)
\stackrel{d^1}{\longleftarrow} E^1_{10}(n)
\stackrel{d^1}{\longleftarrow} E^1_{20}(n) \longleftarrow \cdots
\]
is acyclic in positive dimensions, for all $n\geq 1$.
\end{proposition}
\noindent
{\bf Proof.}
If ${\cal P}$ is Koszul, then $H_*({\Omega}({\cal P}))=
H_0({\Omega}({\cal P}))={\cal P}^!$, by definition. Thus, by Lemma~\ref{whoop},
$E^1_{pq}(n)= (\dualI{ {\bf s\hskip0mm} M}(p+1)\circ {\cal P}^!)(n)_{p+q}$ which
may be
nonzero only for $q=0$, because $\dualI{ {\bf s\hskip0mm} M}(p+1)$ is
concentrated
in degree $p$.
The collapsing is obvious from degree
reasons.
The second part of the statement follows immediately
from the definition of the
Koszulness of a module (Definition~\ref{telefon}).\hspace*{\fill
Our spectral sequence has,
for ${\cal P} = {\it Ass}$ and $M ={\it Cycl}$, a beautiful
geometric meaning.
The initial term ${\bf E}^0 = (E^0_{pq},d^0)$ is the cobar
construction ${\Omega}(\dualI{{\bf s\hskip0mm}{\it Cycl}}, \dualI{{\bf s\hskip0mm}{\it Ass}})$
which is isomorphic, by
Theorem~\ref{ucpavka}, to the
cellular chain complex of the cyclohedron ${\overline W}_n$, while
${\bf E}^1 =
(E^1_{pq},d^1) = ({\bf s\hskip0mm}{\it Cycl} \circ {\it Ass},\partial)$ is isomorphic,
by Theorem~\ref{resiz}, to the cellular chain
complex of the simplex ${\overline \Delta}_n$. The passage from
${\bf E}^0$ to
${\bf E}^1$ can be interpreted as the `deblowing-up' of
the cyclohedron back
to the simplex. This process is visualized on
Figure~\ref{deblow}.
\begin{figure}[hbtp]
\begin{center}
\unitlength 1.50mm
\thicklines
\begin{picture}(91.66,52.08)
\put(0.50,37.00){\line(1,1){10.00}}
\put(10.72,47.41){\line(1,0){20.00}}
\put(30.72,47.08){\line(1,-1){10.00}}
\put(40.39,37.41){\line(-1,-1){10.00}}
\put(30.72,27.08){\line(-1,0){20.00}}
\put(10.86,27.21){\line(-1,1){10.00}}
\put(0.72,37.08){\makebox(0,0)[cc]{$\bullet$}}
\put(10.72,47.08){\makebox(0,0)[cc]{$\bullet$}}
\put(30.72,47.08){\makebox(0,0)[cc]{$\bullet$}}
\put(40.72,37.08){\makebox(0,0)[cc]{$\bullet$}}
\put(30.72,27.08){\makebox(0,0)[cc]{$\bullet$}}
\put(10.72,27.08){\makebox(0,0)[cc]{$\bullet$}}
\put(8.72,52.08){\makebox(0,0)[cc]{$((12)3)$}}
\put(32.72,52.08){\makebox(0,0)[cc]{$(1(23))$}}
\put(42.72,37.08){\makebox(0,0)[lc]{$1)((23)$}}
\put(32.72,22.08){\makebox(0,0)[cc]{$1))(2(3$}}
\put(8.72,22.08){\makebox(0,0)[cc]{$1)2)((3$}}
\put(-1.28,37.08){\makebox(0,0)[rc]{$(12))(3$}}
\put(20.72,52.08){\makebox(0,0)[cc]{$(123)$}}
\put(20.72,22.08){\makebox(0,0)[cc]{$1)2(3$}}
\put(7.72,42.08){\makebox(0,0)[lc]{$(12)3$}}
\put(7.72,32.08){\makebox(0,0)[lc]{$12)(3$}}
\put(33.72,42.08){\makebox(0,0)[rc]{$1(23)$}}
\put(33.72,32.08){\makebox(0,0)[rc]{$1)(23$}}
\put(20.72,37.08){\makebox(0,0)[cc]{$123$}}
\put(55.83,4.33){\line(5,6){16.67}}
\put(72.50,24.33){\line(5,-6){16.81}}
\put(89.30,4.16){\line(-1,0){33.64}}
\put(10.72,46.75){\line(1,0){20.00}}
\put(40.86,36.94){\line(-1,-1){10.00}}
\put(10.39,26.75){\line(-1,1){10.00}}
\put(72.33,24.11){\makebox(0,0)[cc]{$\bullet$}}
\put(55.66,4.11){\makebox(0,0)[cc]{$\bullet$}}
\put(89.44,4.11){\makebox(0,0)[cc]{$\bullet$}}
\put(53.00,1.89){\makebox(0,0)[cc]{$\{3\}$}}
\put(91.66,2.11){\makebox(0,0)[cc]{$\{2\}$}}
\put(72.55,27.44){\makebox(0,0)[cc]{$\{1\}$}}
\put(63.25,16.77){\makebox(0,0)[rb]{$\{13\}$}}
\put(81.44,17.00){\makebox(0,0)[lb]{$\{12\}$}}
\put(72.33,1.44){\makebox(0,0)[ct]{$\{23\}$}}
\put(72.33,12.33){\makebox(0,0)[cc]{$\{123\}$}}
\put(44.84,29.43){\line(6,-5){9.02}}
\put(44.20,28.74){\line(6,-5){9.01}}
\put(53.93,21.00){\line(-1,0){2.61}}
\put(53.93,21.00){\line(0,1){2.16}}
\end{picture}
\end{center}
\caption{$\Delta_3$ as deblowing-up of $W_3$. The faces of
$W_3$ which
are contracted by $d^0$ to a vertex
are indicated by double lines, $(123)$ is contracted to
$\set 1$,
$1)(23$ to $\set 2$ and $12)(3$ to $\set 3$.\label{deblow}}
\end{figure}
\section*{Appendix: Traces versus invariant bilinear forms.}
Let us recall the following notion
of~\cite[Definition~4.1]{getzler-kapranov:cyclic}.
If ${\cal P}$ is a cyclic operad and $A$ a ${\cal P}$-algebra, then
a bilinear
form $B: A\otimes A\to W$ with values in a vector space $W$
is called
{\em invariant\/} if, for all $n\geq 0$, the map
$B_n:{\cal P}(n)\otimes A^{\otimes (n+1)} \to W$ defined by the
formula
\begin{equation}
\label{plus}
B_n(p{\otimes} x_0 {\otimes} x_1 {\otimes} \cdots {\otimes} x_n):=
\znamenko {|x_0|\cdot |p|}B(x_0,p(\rada {x_1}{x_n})),
\end{equation}
is invariant under the action of the symmetric group
$\Sigma_{n+1}$
on ${\cal P}(n){\otimes} A^{\otimes (n+1)}$.
\vskip2mm
\noindent
{\bf Proposition A.1.}
{\it Let ${\cal P}$ be a cyclic operad and let $M_{{\cal P}}$ be the
associated
module introduced in Definition~\ref{Turmo}.
Let $A$ be a ${\cal P}$-algebra. Then
there exists a 1-1 correspondence between $M_{{\cal P}}$-traces on the
${\cal P}$-algebra $A$ in the sense of Definition~\ref{el}, and
invariant bilinear forms on $A$.}
\vskip2mm
\noindent
{\bf Proof.}
Let $t : M_{\cal P} \to {\cal E}_{A,W}$ be an $M_{\cal P}$-trace. Because
$M_{\cal P}(n+1)= {\cal P}(n)$, the trace is represented by a system $\{t_n
:{\cal P}(n-1)\to \Hom{A^{\otimes n}}W\}_{n\geq 2}$ of linear maps. We
claim that $B := t_2(1): A\otimes A \to W$, where $1\in {\cal P}(1)$
is the
unit, is an invariant bilinear form.
To see it, observe that~(\ref{plus}) can be rewritten as
\[
B_n(p\otimes x_0 \otimes \cdots \otimes x_n) =
\nu_{{\cal E}_{A,W}}(t_2(1);1,p)(\rada{x_0}{x_n}),
\]
while
\[
\nu_{{\cal E}_{A,W}}(t_2(1);1,p) =
\nu_{{\cal E}_{A,W}}(\nu_{M_{\cal P}}(1;1,p)) = t_{n+1}(p \cdot \tau_n),
\]
thus
\begin{equation}
\label{xplus}
B_n(p\otimes x_0 \otimes \cdots \otimes x_n) =
t_{n+1}(p \cdot \tau_n)(\rada{x_0}{x_n})
\end{equation}
and the equivariance of $B_n$ follows from the equivariance of
$t_{n+1}$. On the other hand, if $B$ is an invariant bilinear
form,
then~(\ref{xplus}) defines a trace.\hspace*{\fill
|
1997-07-13T19:55:18 | 9707 | alg-geom/9707012 | en | https://arxiv.org/abs/alg-geom/9707012 | [
"alg-geom",
"math.AG"
] | alg-geom/9707012 | Kalle Karu | Dan Abramovich and Kalle Karu | Weak semistable reduction in characteristic 0 | AMS-LaTeX, 22 pages | null | null | null | null | Let X->B be a morphism of varieties in characteristic zero. Semistable
reduction has been proved for dim(B)=1 (Kempf, Knudsen, Mumford, Saint-Donat),
dim(X)=dim(B)-1 (de Jong) and dim(X)=dim(B)+2 (Alexeev, Kollar,
Shepherd-Barron). In this paper we consider the general case. First we define
what we mean by a semistable morphism in terms of toroidal embeddings. Then we
reduce the varieties to toroidal embeddings and solve a slightly weaker version
of semistable reduction. We also state the full semistable reduction problem in
terms of combinatorics of the associated polyhedral complexes.
| [
{
"version": "v1",
"created": "Sun, 13 Jul 1997 17:55:10 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Abramovich",
"Dan",
""
],
[
"Karu",
"Kalle",
""
]
] | alg-geom | \section{INTRODUCTION}
Regretfully, we work over an algebraically closed field $k$ of
characteristic 0.
\subsection{The problem} Roughly speaking, the semistable reduction problem we
address here asks for the following:
\begin{quote}
Let $X\rightarrow} \newcommand{\dar}{\downarrow B$ be a surjective morphism of complex projective
varieties with geometrically integral generic fiber. Find a generically finite
proper surjective morphism (that is, an alteration) $B_1\rightarrow} \newcommand{\dar}{\downarrow B$, and a proper
birational morphism (that is, a modification) $Y\rightarrow} \newcommand{\dar}{\downarrow X\times_B B_1$, such that
the morphism $Y\rightarrow} \newcommand{\dar}{\downarrow B_1$ is nice.
\end{quote}
Of course, one needs to decide what a ``nice morphism'' means.
The question was posed, among other places, in the introduction of
\cite{te}, p. vii. It can be viewed as a natural extension of
Hironaka's theorem on resolution of singularities, which is in a sense
``the general fiber'' of semistable reduction.
\subsection{Brief history} The case { $\dim B=1, \dim
X=2$ is very old, see \cite{aw}. When $\dim B=1,$ semistable reduction
was obtained in \cite{te}, in the best possible sense: $Y$ is
nonsingular, and all the fibers are reduced, strict divisors of normal
crossings.
Using a result of Kawamata on ramified covers (see \cite{kawamata},
theorem 17), one can obtain semistable reduction ``in codimension 1''
over a base of arbitrary dimension. Below, we will refer to the
result of Kawamata as {\bf ``Kawamata's trick''.} We will discuss it
in detail in section \ref{reduced-fibers}.
The case where $\dim X = \dim B + 1$ has recently been proven by de
Jong \cite{dj}. Here one shows that any family of curves can be made
into a family of nodal curves, which are indeed as ``nice'' as one may
expect.
Using recent difficult results of Alexeev, Koll\'ar and
Shepherd-Barron (see \cite{alex}, \cite{alex2}), one obtains a version
of the case $\dim X = \dim B + 2$. Here each fiber is a
semi-log-canonical surface.
Up until recently, not much has been known about the case $\dim X
>\dim B+2$. Often one finds remarks of the following flavor: ``since
we do not have a semistable reduction result over a base of higher
dimension, we will work around it in the following technical
manner...''.
\subsection{Definition of semistable families} We give here a
description of the best possible kind of morphisms we have in mind.
Let $f:X\to B$ be a flat morphism of nonsingular projective varieties
with connected fibers. Somewhat informally, we say that $f$ is {\bf
semistable} if for each point $x\in X$ with $f(x) = b$ there is a
choice of formal coordinates $\hat{B}_b = {\operatorname{Spec\ }}\ k[[t_i]]$ and $\hat{X}_x =
{\operatorname{Spec\ }}\
k[[x_j]]$, such that $f$ is given by: $$t_i =
\prod_{j=l_{i-1}+1}^{l_i} x_j. $$ Here $0 = l_0 < l_1 \cdots < l_m
\leq n$, where $n=\dim X$ and $m = \dim B$. To be more precise, we
give things a more global structure using the notion of a toroidal
morphism. At the same time we describe a slightly weaker condition
which will appear below:
\begin{dfn}
The morphism $f:X\to B$ above is called {\bf weakly semistable} if
\begin{enumerate}
\item the varieties $X$ and $B$ admit toroidal structures $U_X\subset
X$ and $U_B\subset B$, with $U_X=f^{-1}U_B$;
\item with this structure, the morphism $f$ is toroidal;
\item the morphism $f$ is equidimensional;
\item all the fibers of the morphism $f$ are reduced;
and
\item $B$ is nonsingular.
\end{enumerate}
If also $X$ is nonsingular, we say that the morphism $f:X\to B$ is
{\bf semistable}.
\end{dfn}
\subsection{The ultimate goal} The result one would really like to have is:
\begin{conj}\label{conj-semistable}
Let $X\rightarrow} \newcommand{\dar}{\downarrow B$ be a surjective morphism of complex projective
varieties with geometrically integral generic fiber. There is a projective
alteration $B_1\rightarrow} \newcommand{\dar}{\downarrow B$, and a projective
modification $Y\rightarrow} \newcommand{\dar}{\downarrow X\times_B B_1$, such that $Y\rightarrow} \newcommand{\dar}{\downarrow B_1$ is semistable.
\end{conj}
Na\"{\i}vely one might hope to have each fiber isomorphic to a divisor
of normal crossings. But already in the case of a 2-parameter family
of surfaces $t_1 = x_1x_2;\, t_2 = x_3 x_4$, this is impossible. It
seems that the definition above is the best one can hope for.
\subsection{A. J. de Jong's results} In \cite{dj}, Johan de Jong shows,
among many other results, that if one allows $Y\rightarrow} \newcommand{\dar}{\downarrow X\times_B B_1$ to
be an alteration instead of a modification, one can make $Y\rightarrow} \newcommand{\dar}{\downarrow B_1$
very nice indeed: $Y$ is nonsingular, $Y\rightarrow} \newcommand{\dar}{\downarrow B_1$ is semistable as in
the definition above, and moreover it can be written as a composition
of nodal curve fibrations $Y=Y_0\to Y_1\to \cdots \to Y_k=B_1$.
De Jong's methods and ideas will serve as a starting point for
investigating the semistable reduction conjecture.
\subsection{Our main result} The main result of this paper is the
following:
\begin{th}[Weak semistable reduction]\label{th-weak-semistable-reduction}
Let $X\rightarrow} \newcommand{\dar}{\downarrow B$ be a surjective morphism of complex projective
varieties with geometrically integral generic fiber. There exist an
alteration $B_1\rightarrow} \newcommand{\dar}{\downarrow B$ and a modification
$Y\rightarrow} \newcommand{\dar}{\downarrow X\times_B B_1$, such that $Y\rightarrow} \newcommand{\dar}{\downarrow B_1$ is weakly semistable.
\end{th}
With a little more work we will get $X$ to have only quotient
singularities. There are many cases (such as when $f$ is a family of
surfaces) where we can actually prove the semistable reduction
conjecture. These will be pursued elsewhere. Hopefully, by the time
this paper achieves its final form the conjecture will be fully
proven.
\subsection{Mild morphisms} A few words are in order about the significance of
our result. Note that the property of a morphism being semistable is
far from being stable under base changes. One may ask, what remains
from semistability after at least {\em dominant} base changes? Here is
a suggestion:
\begin{dfn}
We define a morphism $X\rightarrow} \newcommand{\dar}{\downarrow B$ as above to be {\bf mild}, if for any
dominant $B_1\rightarrow} \newcommand{\dar}{\downarrow B$ where $B_1$ has at most rational Gorenstein
singularities, we have that $X\times_B B_1$ has at most rational
Gorenstein singularities as well.
\end{dfn}
Mild morphisms arise naturally in moduli theory. Indeed, mild families
of curves are precisely nodal families; families of Gorenstein
semi-log-canonical surfaces mentioned above are mild. For a discussion
of why mild morphisms are useful, see \cite{fibered}. In fact, the
paper \cite{fibered} would have been much simplified, had mild
reduction been available.
Already in the case $\dim B=1$, mild reduction is a much easier task
than semistable reduction. Indeed, lemma 2 on page 103 of \cite{te},
and the discussion there, already give mild reduction in this
case. The delicate combinatorics of chapter III of \cite{te} is not
used for this purpose.
It will be shown (see section \ref{mildness}) that weakly semistable morphisms
are indeed mild.
\subsection{Structure of the proof}
After the introduction, section \ref{toroidal-morphisms} will be
devoted to a general discussion of toroidal morphisms. The proof
itself will begin with section \ref{toroidal-reduction}.
Semistable reduction has at least two flavors: first, the fibers of
the morphism $Y\rightarrow} \newcommand{\dar}{\downarrow B_1$ should have nice local defining
equations. Second, the family should have nice algebraic
properties. We will perform a number of reduction steps, incrementally
improving one or the other of these flavors.
\subsubsection{Toroidal reduction} In the first step,
carried out in section \ref{toroidal-reduction}, we will show that any
morphism can be modified to a toroidal morphism. The construction is
inspired by the inductive procedure of \cite{dj}, and follows closely
the proofs in \cite{aj}.
Just as in \cite{aj}, the construction we give is very
non-canonical. Even when the generic fiber of $X\to B$ is smooth, it
will be blown up during the construction. One hopes that methods such
as those of \cite{bm} or \cite{villa} could be adapted to this
situation and give a more canonical procedure.
It is tempting to state the following conjecture.
\begin{conj} Let $X\to B$ be a morphism as in the theorem. Let $U\subset B$ be
an open set over which $X$ is toroidal, and let $\Sigma = B \setmin U$. There
exists modifications $X'\to X$ and $B'\to B$, each of which is the composition
of a sequence of blowings up with smooth centers lying over $\Sigma$, and a
lifting $X'\to B'$ which is toroidal.
\end{conj}
It should be noted, that in view of recent results of Morelli \cite{m} and
W{\l}odarczyk \cite{w}, this conjecture implies the strong blow-up - blow-down
conjecture.
\subsubsection{Improving the toroidal morphism} In sections
\ref{remove-horizontal} and \ref{equidimensional} we perform a couple
of simple reduction steps to improve our situation. {Let $f:(U_X
\subset X) \rightarrow} \newcommand{\dar}{\downarrow(U_B\subset B)$ be any toroidal morphism, with $B$
nonsingular. By the results of \cite{te}, we can find a toroidal
resolution of singularities $X'\rightarrow} \newcommand{\dar}{\downarrow X$. Let $f':X'\rightarrow} \newcommand{\dar}{\downarrow B$ be the
resulting projection. We first show that now $f^{-1} U_B\subset X'$ is
also a toroidal embedding, which is easier to handle: there are no
horizontal divisors. For convenience, we replace $X\rightarrow} \newcommand{\dar}{\downarrow B$ by the new
morphism. We remark that one can proceed a fair distance without
removing these horizontal divisors, and, we believe, the results one
can obtain are of interest (e.g., the inductive structure of de Jong
can be preserved), but this would make the present paper much more
cumbersome, so we delay that investigation to a future occasion.
Now, our morphism $X\rightarrow} \newcommand{\dar}{\downarrow B$ is not necessarily equidimensional. We
repair this by an appropriate decomposition of the associated conical
polyhedral complexes $\Delta_X$ and $\Delta_B$. We make sure that,
after the modification, the base remains nonsingular, and then the morphism
is automatically flat.
\subsubsection{Kawamata's trick and reduced fibers.} We start section
\ref{reduced-fibers} with a discussion of Kawamata's trick and its
relation with toroidal morphisms in some detail. Then we use
Kawamata's trick to find a finite base change, after which all the
fibers are reduced. This finishes the proof of the main theorem, since
the resulting morphism is weakly semistable. A variant of Kawamata's trick for
global ``index 1 covers'' is discussed in section \ref{sec-cartier}.
\subsubsection{Mild reduction.}
We begin section \ref{mildness} by checking that the resulting fibers
are Gorenstein.
Using a base change and descent argument, and the fact that toroidal
singularities are always rational, we then prove that the resulting
family is mild.
\subsubsection{Combinatorial restatement} In section \ref{combinatorial} the
semistable reduction conjecture is restated purely in combinatorial terms. We
end the paper with a discussion of the problems one encounters when
trying to go from weak semistable reduction to semistable reduction.
\subsection{Acknowledgments} We would like to thank O. Gabber, A.J. de Jong,
H. King and K. Matsuki for helpful and inspiring discussions.
\subsection{Terminology}
A {\bf modification} is a proper birational morphism of irreducible
varieties.
An {\bf alteration} $a:B_1\rightarrow} \newcommand{\dar}{\downarrow B$ is a proper, surjective, generically
finite morphism of irreducible varieties, see \cite[2.20]{dj}. The
alteration $a$ is a {\bf Galois alteration} if there is a finite group
$G\subset {\operatorname{Aut}}_B (B_1)$ such that the associated morphism $B_1/G\rightarrow} \newcommand{\dar}{\downarrow
B$ is birational, compare \cite[5.3]{dj2}.
\section{Toroidal morphisms}\label{toroidal-morphisms}
We have collected in this section some notations and preliminaries
about toric varieties, toroidal embeddings, and their morphisms (see
\cite{te} for details).
\begin{rem}
Our approach here is based on the formalism of \cite{te}. A different approach,
using {\em logarithmic structures}, was developed by K. Kato, see \cite{kato},
\cite{kato1}. It is our belief, that the approach via logarithmic structures
should eventually prevail - it provides us with a flexible category, in which
toroidal embeddings ($=$ logarithmically regular schemes) and toroidal
morphisms ($=$ logarithmically smooth(!) morphisms) play a special role.
Some of our statements below are rendered almost trivial with Kato's formalism,
e.g. Lemmas \ref{lem-tor-composition} and \ref{lem-tor-prod}.
The reason we decided to stick with the formalism of \cite{te} is, that the
theory of logarithmic structures is not yet in stable form (see the many
flavors of such structures introduced in Kato's papers), and, more importantly,
it has not yet gained widespread acceptance as a basic formalism. It might have
turned away some readers (especially those combinatorially inclined) had we
used the theory of logarithmic structures throughout.
It is also worth noting, that Kato's notion of a fan, although it has a nice
structural morphism, is much less amenable to combinatorial manipulation than
the polyhedral complexes of \cite{te}.
\end{rem}
\subsection{Toric varieties} Given a lattice $N\cong{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}^n$, its dual
$M={\operatorname{Hom }}(N,{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})$,
a strictly convex rational polyhedral cone
$\sigma\subset N_{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}} = N\otimes{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$ with its dual $\sigma^\vee =
\{m\in M_{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}} | m(u)\geq 0$ for all $u\in \sigma\}$, we define the {\bf affine
toric variety} $X_\sigma = {\operatorname{Spec\ }} S[\sigma]$ where
$S[\sigma]$ is the semigroup algebra of $\sigma^\vee\cap M$ over the
ground field. If more than one toric variety is considered, we use a
subscript: $N_\sigma$, $M_\sigma$.
We denote by $\sigma^{(1)}$ the 1-dimensional edges of $\sigma$.
The indivisible points $v$ in $\sigma^{(1)}\cap N$ are called the {\bf
primitive
points} of $\sigma$. The variety $X_\sigma$ is nonsingular if and only if the
primitive points of $\sigma$ form a part of a basis of $N$. In that
case we say that $\sigma$ is nonsingular.
The toric variety $X_\sigma$ contains an n-dimensional algebraic torus
$T=\bfg^n_m$ as an open dense subset, and the action of $T$ on itself
extends to an action on $X_\sigma$. Thus, $X_\sigma$ is a disjoint
union of orbits of this action. There is a one-to-one correspondence
between the orbits and the faces of $\sigma$. In particular,
1-dimensional faces ${\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}_{+}v_i$ correspond to codimension~1 orbits
$\bfo_{v_i}$.
A {\bf toric morphism} $f:X_\sigma\rightarrow} \newcommand{\dar}{\downarrow X_\tau$ is a dominant equivariant
morphism of toric varieties defined by a linear map $f_\Delta:
(N_\sigma,\sigma) \rightarrow} \newcommand{\dar}{\downarrow (N_\tau,\tau)$. We use the same notation for
the scalar extension $f_\Delta: N_\sigma\otimes{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}} \rightarrow} \newcommand{\dar}{\downarrow
N_\tau\otimes{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$.
\subsection{Toroidal embeddings}
Given a normal variety $X$ and an open subset $U_X\subset X$,
the embedding $U_X\subset X$ is called {\bf toroidal} if for every
closed point $x\in X$ there exist a toric variety $X_\sigma$, a point
$s\in X_\sigma$, and an isomorphism of complete local algebras
\[ \hat{{\cal{O}}}_{X,x} \cong \hat{{\cal{O}}}_{X_{\sigma},s} \]
so that the ideal of $X\setmin U_X$ corresponds to the ideal of
$X_{\sigma}\setmin T$. Such a pair $(X_\sigma, s)$ is called a local
model at $x\in X$. By restricting $X_{\sigma}$ if necessary, we can
assume that the orbit of $s$ is the unique closed orbit in
$X_{\sigma}$.
\begin{dfn}\label{def-toroidal-map}
A dominant morphism ${f}:(U_X\subset X)\rightarrow} \newcommand{\dar}{\downarrow(U_B\subset B)$ of toroidal
embeddings is called {\bf toroidal} if for every closed point $x\in X$
there exist local models $(X_{\sigma},s)$ at $x$, $(X_{\tau},t)$ at
${f}(x)$ and a toric morphism $g: X_{\sigma}\rightarrow} \newcommand{\dar}{\downarrow X_{\tau}$ so that the
following diagram commutes
\[
\begin{CD}
\hat{{\cal{O}}}_{X,x} @<{\cong}<< \hat{{\cal{O}}}_{X_{\sigma},s}\\
@A{\hat{f}^{*}}AA @AA{\hat{g}^{*}}A\\ \hat{{\cal{O}}}_{B,f(x)} @<{\cong}<<
\hat{{\cal{O}}}_{X_{\tau},t}
\end{CD}
\]
where $\hat{f}^{*}$ and $\hat{g}^{*}$ are the ring homomorphisms
induced by $f$ and $g$.
\end{dfn}
\subsection{Cones and polyhedral complexes} Let $X\setmin U_X = \cup_{i\in
I}E_i$ where $E_i$ are irreducible
and have codimension~1. We will assume that all the $E_i$ are normal,
that is, $U_X\subset X$ is a {\bf toroidal embedding without
self-intersection} (also known as a {\bf strict} toroidal embedding). In that
case, we can use the irreducible components of $\cap_{i\in J}
E_i$ for all $J\subset I$ to define a stratification of $X$ (these components
are the closures of strata). Closures of strata
formally corresponds to closures of orbits in local models. Since a toric
morphism maps orbits to orbits, a toroidal morphism maps strata to
strata.
Let Y be a stratum in $X$, which is by definition an open set in an irreducible
component of $\cap_{i\in J} E_i$ for some $J\subset I$. The star of
$Y$ is the union of strata in whose closure $Y$ lies (each of these corresponds
to some $K\subset J\subset I$). To
the stratum $Y$ we associate
\begin{enumerate}
\item $M^Y$: -- the group of Cartier divisors in ${\operatorname{Star}}(Y)$ supported
in ${\operatorname{Star}}(Y)\setmin U_X$
\item $N^Y$: -- ${\operatorname{Hom }}(M^Y,{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})$
\item $M_{+}^Y \subset M^Y$: -- effective Cartier divisors
\item $\sigma^Y\subset N^Y_{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$: -- the dual of $M_{+}^Y$
\end{enumerate}
It is shown in \cite{te} (Corollary 1, page 61) that if
$(X_{\sigma},s)$ is a local model at $x\in X$ in the stratum $Y$, then
\begin{eqnarray*}
M^Y &\cong& M_\sigma/\sigma^{\bot} \\ \sigma^Y &\cong& \sigma
\end{eqnarray*}
The cones $\sigma^Y$ glue together to form a polyhedral complex
$\Delta_X = (|\Delta_X|,\{\sigma^Y\},\{M^Y\})$, where the lattices $M^Y$ form
an
integral
structure on $\Delta_X$. Equivalently, instead of $M^Y$ we may give
the lattices $N^Y$ and embeddings $\sigma^Y\hookrightarrow
N^Y_{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$. We also denote $M_{\sigma^Y}=M^Y$,
$N_{\sigma^Y}=N^Y$. Then, comparing to the lattices of local models,
\begin{eqnarray*}
M_{\sigma^Y} &\cong& M_\sigma/\sigma^{\bot} \\ N_{\sigma^Y} &\cong&
N_\sigma\cap \mbox{Span}(\sigma) \\
\end{eqnarray*}
If $N_{\sigma^Y}\neq N_\sigma$ we can choose a splitting of $N_\sigma$
so that at a point $x\in Y$ the local model is in the form
$(X_{\sigma'}\times\bfg_m^l,(s',1))$ where $(N_{\sigma'},\sigma')\cong
(N_{\sigma^Y},\sigma^Y)$.
\begin{lem}
A toroidal morphism ${f}:X\rightarrow} \newcommand{\dar}{\downarrow B$ induces a morphism
$f_\Delta:\Delta_X \rightarrow} \newcommand{\dar}{\downarrow
\Delta_B$, which for each cone $\sigma^Y$ is the
restriction of $g_\Delta: (\sigma,N_\sigma)\rightarrow} \newcommand{\dar}{\downarrow(\tau,N_\tau)$ where
$(X_{\sigma},s)$, $(X_{\tau},t)$ are local models at $x\in Y\subset
X$, $f(x)\in Z\subset B$, and $g_\Delta$ is the linear map determined
by the toric morphism $g: X_{\sigma}\rightarrow} \newcommand{\dar}{\downarrow X_{\tau}$ in
Definition~\ref{def-toroidal-map}.
\end{lem}
{\bf Proof.} To see that the maps $f_\Delta$ defined for different
cones $\sigma^Y$ agree on the overlaps it suffices to notice that the
dual morphism $f_\Delta^\vee:M_{\tau^Z}\rightarrow} \newcommand{\dar}{\downarrow M_{\sigma^Y}$ is defined
by pulling back a Cartier divisor and restricting it to
${\operatorname{Star}}(Y)\setmin U_X$. Since the pullback is defined independently of
the stratum, we see that $f_\Delta$ is well defined. \qed
\begin{rem}
Note that the polyhedral morphism $f_\Delta:\Delta_X \rightarrow} \newcommand{\dar}{\downarrow
\Delta_B$ is well defined even if $f$ is not toroidal, as long as
$f(U_X)\subset U_B$.
\end{rem}
\begin{lem}\label{lem-tor-composition}
If $e:X\rightarrow} \newcommand{\dar}{\downarrow Y$ and $f:Y\rightarrow} \newcommand{\dar}{\downarrow Z$ are toroidal morphisms, then $f\circ
e:X\rightarrow} \newcommand{\dar}{\downarrow Z$ is also toroidal.
\end{lem}
\begin{rem} This lemma is a triviality if one uses logarithmic structures.
\end{rem}
{\bf Proof.} Let $x\in X$, $y=e(x)\in Y$, $z=f(y)\in Z$, and choose
local models at $x$, $y$ and $z$ as in
Definition~\ref{def-toroidal-map}. Consider the tower
\[
\begin{CD}
\hat{{\cal{O}}}_{X,x} @<{\cong}<< \hat{{\cal{O}}}_{X_{\sigma},s}\\
@A{\hat{e}^{*}}AA @AA{\hat{g}^{*}}A\\ \hat{{\cal{O}}}_{Y,y} @<{\cong}<<
\hat{{\cal{O}}}_{X_{\tau_1},t_1}\\ @A{id}AA @AA{\alpha}A\\ \hat{{\cal{O}}}_{Y,y}
@<{\cong}<< \hat{{\cal{O}}}_{X_{\tau_2},t_2}\\ @A{\hat{f}^{*}}AA
@AA{\hat{h}^{*}}A\\ \hat{{\cal{O}}}_{Z,z} @<{\cong}<< \hat{{\cal{O}}}_{X_{\rho},r}
\end{CD}
\]
where the upper and lower squares commute by the definition of
toroidal morphism, and where $\alpha$ is defined by tracing the other
three sides of the middle square. Then the middle square also
commutes.
Since $\tau_1$ and $\tau_2$ are isomorphic, we can take
$\tau_1=\tau_2=\tau$ and $X_{\tau_1}= X_{\tau_2}= X_{\tau}$. The map
$\alpha$, of course, need not be the identity. Let the coordinate
rings of the tori in $X_{{\sigma}}$, $X_{{\tau}}$ and $X_{{\rho}}$ be
$k[x_1,x_1^{-1},\ldots,x_n,x_n^{-1}]$,
$k[y_1,y_1^{-1},\ldots,y_m,y_m^{-1}]$ and
$k[z_1,z_1^{-1},\ldots,z_l,z_l^{-1}]$, respectively, so that the toric
morphisms $g$ and $h$ are defined by
\[ y_i \mapsto \prod_{j=1}^{n} x_j^{a_{ij}}, \hspace{.5in}
z_i \mapsto \prod_{j=1}^{m} y_j^{b_{ij}} \] for some $a_{ij},
b_{ij} \in {\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}$.
The maps in the third line of the tower identify the group of
T-invariant Cartier divisors in $X_{{\tau_2}}$ with the group of
Cartier divisors in $Y$ supported in $Y\setmin U_Y$ and passing
through $y$. The maps in the second line of the tower identify the
latter group with the group of T-invariant Cartier divisors in
$X_{{\tau_1}}$. Hence $\alpha$ induces a group homomorphism between
T-invariant Cartier divisors of $X_{{\tau_2}}$ and
$X_{{\tau_1}}$. Since T-invariant Cartier divisors in $X_{{\tau}}$ are
given by products of $y_i$, then (using the same letter $y_i$ for the
image of $y_i$ in the completed local ring) $\alpha$ maps
\[ y_i \mapsto u_i \prod_{j=1}^{m} y_j^{c_{ij}} \]
where $u_i$ are units in $\hat{{\cal{O}}}_{X_{{\tau}},t_1}$, and $c_{ij}\in
{\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}$.
The composition $\hat{g}^{*}\circ\alpha\circ\hat{h}^{*}$ is then
defined by
\[ z_i \mapsto v_i \prod_{j=1}^{m} x_i^{d_{ij}} \]
where the matrix with entries $d_{ij}$ is the product of the matrices
with entries $a_{ij}$, $c_{ij}$ and $b_{ij}$, and where
$v_i=\hat{g}^{*}(\prod_j u^{c_{ij}})$ are units in
$\hat{{\cal{O}}}_{X_{{\sigma}},s}$. The matrix $(d_{ij})$ is equivalent to a
matrix in the form
\[
\begin{pmatrix} D \\ 0\end{pmatrix}
\]
where D is an $l\times l$ diagonal matrix with diagonal entries
$d_1,\ldots,d_l \in {\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}$. Hence we can change the coordinate
functions $x_i$ and $z_i$ of the respective tori so that
$\hat{g}^{*}\circ\alpha\circ\hat{h}^{*}$ maps
\[ z_i \mapsto v_i x_i^{d_i}\]
If we now set
\[ \tilde{x_i} = v_i^{\frac{1}{d_i}} x_i \]
then in these new coordinates the composition
$\hat{g}^{*}\circ\alpha\circ\hat{h}^{*}$ is induced by a toric
morphism defined by
\[ z_i \mapsto x_i^{d_i} \]
and thus $f\circ e:X\rightarrow} \newcommand{\dar}{\downarrow Z$ is toroidal. \qed
Given a toroidal embedding $U_X\subset X$ with polyhedral complex
$\Delta_X$, and a subdivision $\Delta'_X$ of $\Delta_X$, one
constructs (see \cite{te}) a new toroidal embedding $U_{X'}\subset X'$ with
polyhedral complex $\Delta'_X$, and a birational toroidal morphism $f': X'\rightarrow} \newcommand{\dar}{\downarrow
X$ such that the induced map of the polyhedral complexes $ \Delta'_X \rightarrow} \newcommand{\dar}{\downarrow
\Delta_X$ is the given subdivision. If $Y$ is a stratum in $X$ corresponding
to the cone $\sigma^Y \in \Delta_X$, and if $\sigma'\subset \sigma$ is
a cone in the subdivision, define
\[ V_{\sigma'} = {\operatorname{Spec\ }}_{{\operatorname{Star}}(Y)} \sum_{D\in {\sigma'}^{\vee} \cap M^Y}
{\cal O}} \def\nor{{\rm nor}} \def\question{{\bf (?)}_{{\operatorname{Star}}(Y)}(-D), \]
where the sum is taken inside the field of rational functions of
${\operatorname{Star}}(Y)$.
Then $X'$ is formed by gluing together the open sets $V_{\sigma'}$.
A subdivision $\Delta_X'$ of $\Delta_X$ is called {\bf projective} if
there exists a continuous function $\psi:|\Delta_X|\rightarrow} \newcommand{\dar}{\downarrow {\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$, taking
rational values on $\sigma\cap N_\sigma$, which is
convex and piecewise linear on each cone $\sigma\in\Delta_X$, and the
largest pieces in $\sigma$ where $\psi$ is linear are the cones of the
subdivision. Such $\psi$ is called a {\bf good function} (or {\em lifting
function}, or {\em order function}), for the
subdivision $\Delta_X'$ of $\Delta_X$. A projective subdivision
corresponds to a projective modification $X'\rightarrow} \newcommand{\dar}{\downarrow X$.
\begin{lem}\label{lem-lifting} Let $f:X\to B$ be a toroidal morphism,
$f_\Delta:\Delta_X\to
\Delta_B$ the associated morphism of polyhedral complexes. Let $X'\to
X$ and $B'\to B$ be toroidal modifications, with associated
subdivisions $\Delta_{X'}$ and $\Delta_{B'}$. Then $f$ lifts to a
morphism $f': X' \to B'$ if and only if for each cone $\sigma' \in
\Delta_{X'}$, there exists a cone $\tau' \in \Delta_{B'}$ such that
$f_\Delta(\sigma') \subset \tau'$.
\end{lem}
{\bf Proof.} Let $\sigma'\subset \sigma^Y \in \Delta_X$ and
$\tau'\subset\tau^Z\in\Delta_B$ be cones in the subdivisions such that
$f_\Delta(\sigma') \subset \tau'$. The homomorphism
${\cal{O}}_{{\operatorname{Star}}(Z)} \rightarrow} \newcommand{\dar}{\downarrow {\cal{O}}_{{\operatorname{Star}}(Y)}$ extends to
\[ \sum_{D\in {\tau'}^{\vee} \cap M^Z} {\cal O}} \def\nor{{\rm nor}} \def\question{{\bf (?)}_{{\operatorname{Star}}(Z)}(-D) \rightarrow} \newcommand{\dar}{\downarrow
\sum_{E\in {\sigma'}^{\vee} \cap M^Y} {\cal O}} \def\nor{{\rm nor}} \def\question{{\bf (?)}_{{\operatorname{Star}}(Y)}(-E) \]
because $f^{\vee}_{\Delta} ({\tau'}^\vee) \subset
{\sigma'}^{\vee}$. This shows that the rational map $f'$ is a morphism
on each $V_{\sigma'}$. Since $V_{\sigma'}$ cover $X'$, $f'$ is a
morphism.
Conversely, suppose $f'$ is a morphism. Let $x'\in X', f'(x')=b'\in
B'$. Since the three morphisms $X'\rightarrow} \newcommand{\dar}{\downarrow X$, $X\rightarrow} \newcommand{\dar}{\downarrow B$ and $B'\rightarrow} \newcommand{\dar}{\downarrow B$
are toroidal, we have a toric morphism $g: X_{\sigma'}\rightarrow} \newcommand{\dar}{\downarrow X_{\tau'}$
where $(X_{\sigma'},s')$ and $(X_{\tau'},t')$ are local models at $x'$
and $b'$. Thus $g_{\Delta}(\sigma') \subset \tau'$. But
$g_{\Delta}|_{\sigma'} = f_{\Delta}|_{\sigma'}$, hence
$f_{\Delta}(\sigma') \subset \tau'$. Since this is true for $x'$ in
any stratum of $X$, we get that for any cone $\sigma' \in
\Delta_{X'}$, $f_\Delta(\sigma') \subset \tau'$ for some $\tau' \in
\Delta_{B'}$. \qed
\section{Toroidal reduction}\label{toroidal-reduction}
\subsection{Statement of result}
The purpose of this section is to modify any family of varieties into
a toroidal morphism.
\begin{th}\label{th-toroidal-reduction}
Let $f:X\to B$ be a projective, surjective morphism with geometrically
integral generic fiber, and assume $B$ integral. Let $Z\subset X$ be a proper
closed subscheme. There exist a diagram as follows:
$$\begin{array}{lclcl} U_X & \subset & X' &\stackrel{m_X}{\to} &X \\
\dar & & \dar f' & & \dar f\\ U_B & \subset & B'
& \stackrel{m_B}{\to} &B \end{array}
$$
such that $m_B$ and $m_X$ are modifications, $X'$ and $B'$ are
nonsingular, the inclusions on the left are toroidal embeddings, and
such that
\begin{enumerate}
\item $f'$ is toroidal.
\item Let $Z' = m_X^{-1}Z$. Then $Z'$ is a strict normal crossings
divisor, and $Z'\subset X'\setmin U_{X'}$.
\end{enumerate}
\end{th}
\subsection{To begin the proof,} we proceed by induction on the relative
dimension of $f$.
If the relative dimension of $f$ is 0, let $m_X:X'\to X$ be a
resolution of singularities such that $Z'=m_X^{-1}Z$ is a strict
normal crossings divisor, let $B'=X'$ and $m_B = f\circ m_X$, and
$f'=id$ the identity.
Assume we have proven the result for morphisms of relative dimension
$n-1$, and consider the case $\operatorname{rel.dim } f = n$.
\subsection{Preliminary reduction steps}
First, we may replace $X$ by its normalization, therefore we may
assume $X$ normal, and by blowing up $Z$ in $X$ we may assume $Z$ a
Cartier divisor. Let $\eta\in B$ be the generic point of $B$. By the
projectivity assumption we have $X\subset {\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^N_B$ for some
$N$. Choosing a generic projection ${\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^N_\eta \das {\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^{n-1}_\eta$
we get a rational map $X_\eta \das {\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^{n-1}_\eta$. Replacing $X$ by
the closure of the graph of this map, we may assume that we have a
morphism $g:X \to {\Bbb{P}}} \newcommand{\bfc}{{\Bbb{C}}^{n-1}_B = P$.
\subsection{Semistable reduction of a family of curves}
By \cite{dj2}, Theorem 2.4, we have a diagram as follows:
$$\begin{array}{lcl} X_1 & \stackrel{\alpha}{\rightarrow} \newcommand{\dar}{\downarrow} & X \\ \dar g_1 & &
\dar g \\ P_1 & \stackrel{a}{\rightarrow} \newcommand{\dar}{\downarrow} & P \\ & &
\dar \\ & & B \end{array}
$$
and a finite group $G\subset {\operatorname{Aut}}_PP_1$, with the following
properties:
\begin{enumerate}
\item The morphism $a:P_1\to P$ is a Galois alteration with Galois
group $G$.
\item The action of $G$ lifts to ${\operatorname{Aut}}_XX_1$, and $\alpha:X_1\to X$ is
a Galois alteration with Galois group $G$.
\item There are $n$ disjoint sections $\sigma_i:P_1\rightarrow} \newcommand{\dar}{\downarrow X_1$ such that
the strict altered transform $Z_1\subset X_1$ of $Z$ is the union of
their images, and $G$ permutes the sections $\sigma_i$.
\item The morphism $g_1: X_1 \to P_1$ is a nodal family of curves, and
$\sigma_i(P_1)$ is disjoint from $\operatorname{Sing}g_1$.
\end{enumerate}
We may replace $X$, $P$ and $Z$ by $X_1/G$ and $P_1/G$, and
$\alpha^{-1}Z/G$. Note that $\alpha^{-1}Z/G$ is not necessarily equal
to the union of the images of $\sigma_i$, but the complement lies over
a proper closed subset in $P_1$.
\subsection{Using the inductive hypothesis}
Let $\Delta\subset P$ be the union of the loci over which $Z,P_1$ or
$X_1$ are not smooth. We apply the inductive assumption to
$\Delta\subset P \to B$, and obtain a diagram as follows:
$$\begin{array}{lclcl} U_P & \hookrightarrow & P' &\stackrel{m}{\to} &P \\
\dar & & \dar & & \dar\\
U_B & \hookrightarrow & B' &\to &B
\end{array}
$$
Such that $P', B'$ are nonsingular, $P' \to P$ and $B' \to B$ are
modifications, the left square is a toroidal morphism, and
$m^{-1}\Delta$ is a divisor of strict normal crossings contained in
$P' \setmin U_P$.
We may again replace $P, B$ by $P', B'$, and further we may replace
$X,X_1, P_1,Z$ and $ \sigma_i$ by their pullback to $P'$. In
particular $P\to B$ has a toroidal structure, and $P_1\to P$ is
unramified over $U_P$. By Abhyankar's lemma, since $P_1$ is normal, it inherits
a toroidal structure given by $U_{P_1} = m^{-1}U_P$ as well, so that $P_1\to P$
is a toroidal finite morphism.
\subsection{Conclusion of proof} Now $X_1\to P_1$ is a nodal family which is
smooth over $U_P$, therefore it as well inherits a toroidal structure
$U_{X_1}\subset U_X$, where $U_{X_1} = ({g_1}^{-1} U_{P_1}) \setmin
(\cup\sigma_i(P_1))$; e.g. local equations around a node are of the form
$ uv = f(t)$, where $f(t)$ is a monomial on $P_1$. Notice that
$\alpha^{-1} Z $ is a divisor contained in $U_{X_1}$ (see
\cite{aj}, 1.3).
In this situation we can apply the procedure of \cite{aj}, section 1.4
to make the group $G$ act toroidally on $X_1$: first we blow up the
scheme $\operatorname{Sing} g_1$ to separate the branches of the
nodes. Then we are in the situation of Proposition 1.8 of \cite{aj},
namely there is a canonical $G$-equivariant blowup $d:\tilde{X_1} \to
X_1$ such that $G$ acts strictly toroidally on
$b^{-1}U_{X_1}\subset\tilde{X_1}$. Let ${X'} =\tilde{X_1}/G$, then $
{X'}\to B $ inherits a toroidal structure and ${X'}\to X$ is
birational; moreover, ${Z'}\subset {X'}$ is a divisor contained in
${X'}\setmin U_{{X'}}$. Applying toroidal resolution of singularities,
the induction step is proven. \qed
\section{Removing horizontal divisors}\label{remove-horizontal}
We may now replace $X\to B$ by $X'\to B'$, and thus we may assume that the
morphism $f$ is toroidal.
Our goal in this section is to arrive at a situation where $f^{-1} U_B = U_X$.
The rough idea is, that a morphism between nonsingular toroidal
embeddings $f:X\to B$ is locally given by monomials $t_i = x_1^{k_1}
\cdots x_r^{k_r}$, in which the variables defining horizontal divisors
cannot appear, so these divisors are unnecessary in the toroidal
description. We make this precise by a simple translation argument.
\begin{prp}
Let $U_X\subset X$ and $U_B \subset B$ be nonsingular toroidal
embeddings and $f: X\rightarrow} \newcommand{\dar}{\downarrow B$ a surjective toroidal morphism. Then,
denoting $U_X'= f^{-1}(U_B) \supset U_X$, we have that $U_X'\subset X$ is a
toroidal embedding, and $f: (U_X'\subset X)\rightarrow} \newcommand{\dar}{\downarrow (U_B\subset B)$ is a
toroidal morphism.
\end{prp}
{\bf Proof.} Since $f$ maps $U_X$ into $U_B$, $f^{-1}(B\setmin
U_B)$ as a set is a union of divisors supported in $X\setmin
U_X$. In local models these divisors are all T-invariant.
Consider local models $(X_{\sigma},s)$ at $x$, $(X_{\tau},t)$ at
$f(x)$, and the toric morphism $g:X_\sigma\rightarrow} \newcommand{\dar}{\downarrow X_\tau$. We may assume
that $v_1,\ldots,v_n$ is a basis of $N_\sigma$ and $\sigma$ is
generated by $v_1,\ldots,v_k$. Then $X_\sigma \cong
{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^k\times\bfg_m^{n-k}$, and we may take $s=(0,1)$. Let the closures
of the orbits corresponding to $v_1,\ldots,v_j$ be the horizontal
divisors. That means, $g_\Delta(v_i)=0$ for $i=1,\ldots,j$, and $g$
factors through the projection:
\[ g: X_\sigma \cong {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^j\times{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^{k-j}\times\bfg^{n-k}_m \rightarrow} \newcommand{\dar}{\downarrow
{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^{k-j}\times\bfg^{n-k}_m \to X_\tau \]
Now take $s'=(1,0,1)\subset
{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^j\times{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^{k-j}\times{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^{n-k}$. From the factorization of $g$
we see that $g(s')=g(s)$. Translation by $(1,0,0)$ gives an
isomorphism of the local rings at $s$ and $s'$ so that the ideals of
the T-invariant divisors passing trough $s'$ corresponds to the ideal
of the vertical T-invariant divisors passing through $s$. Thus,
$(X_\sigma,s')$ is a local model for the embedding $U_X'\subset X$ at
$x\in X$ and $g:(X_\sigma,s')\rightarrow} \newcommand{\dar}{\downarrow(X_\tau,t)$ is the toric morphism of
the local models representing $f$. \qed
\section{Making the morphism equidimensional}\label{equidimensional} The goal
of this section is to perform modifications on $B$ and $X$, after which
the morphism becomes equidimensional. First a lemma which
characterizes equidimensional toroidal morphisms:
\begin{lem} Let $f:X\to B$ be a surjective toroidal morphism, $f_\Delta:
\Delta_X\to\Delta_B$ the associated morphism of polyhedral complexes.
Then $f$ is equidimensional if and only if for any cone $\sigma \in
\Delta_X$, we have $f_\Delta(\sigma) \in \Delta_B$. That is, the image
of a cone of $\Delta_X$ is a cone of $\Delta_B$.
\end{lem}
{\bf Proof.} Computing the dimension of a local ring commutes with
taking the completion. Thus, it suffices to consider local models. The
generic fiber of a toric morphism $f: X_\sigma \rightarrow} \newcommand{\dar}{\downarrow X_\tau$ has
dimension ${\operatorname{Rank}}(N_\sigma) - {\operatorname{Rank}}(N_\tau)$. Now $f$ maps a
$k$-dimensional orbit corresponding to a
$({\operatorname{Rank}}(N_\sigma)-k)$-dimensional face $\sigma'$ of $\sigma$ onto some
$l$-dimensional orbit corresponding to a $({\operatorname{Rank}}(N_\tau)-l)$-dimensional
face $\tau'$ of $\tau$. Thus $f$ is equidimensional if and only if
\[ ({\operatorname{Rank}}(N_\sigma)-k) - ({\operatorname{Rank}}(N_\tau)-l) \leq {\operatorname{Rank}}(N_\sigma) - {\operatorname{Rank}}(N_\tau), \]
that means $l \leq k$. Every $k$-dimensional cone maps to an
$l$-dimensional cone for $l\leq k$ if and only if the image of every
cone is a cone. \qed
\begin{rem} In case $\tau$ is simplicial the condition of the theorem
is equivalent to the statement that all 1-dimensional faces of
$\sigma$ map to 0 or 1-dimensional faces of $\tau$.
\end{rem}
The following lemma is a slight generalization of the toric Chow's
lemma (\cite{danilov} 6.9.2 page 119).
\begin{lem} Given a polyhedral complex $\Delta$ and a subdivision
$\Delta'$ of $\Delta$, there exists a projective subdivision
$\Delta''$ of $\Delta$ which refines $\Delta'$.
\end{lem}
{\bf Proof.} First, we show that it suffices to find a ``good''
function $\psi$ (see Section~\ref{toroidal-morphisms}) on each cone
$\sigma\in\Delta$. Indeed, if for each $\sigma\in\Delta$ we have found
a good function $\psi_\sigma: \sigma\rightarrow} \newcommand{\dar}{\downarrow{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$, then a good function on
$|\Delta|$ is the sum
\[ \psi = \sum_{\sigma\in\Delta} \bar{\psi}_\sigma \]
where $\bar{\psi}_\sigma: |\Delta|\rightarrow} \newcommand{\dar}{\downarrow{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$ is a good extension of
$\psi_\sigma$ constructed as follows. To extend $\psi_\sigma$ to a
cone $\tau\in\Delta$ we proceed by induction on the dimension of
$\tau$ ({\em cf.} \cite{te} Lemma~1, page~33). If $\dim \tau =
1$ and $\tau$ is not a face of $\sigma$, define
$\bar{\psi}_\sigma|_\tau\equiv 0$. Now assume that $\dim {\tau} >
1$ and $\bar{\psi}_\sigma$ is defined on $\partial\tau$. Choose a
point $x$ in the relative interior of $\tau$ and define
\[ \bar{\psi}_\sigma (\lambda x + (1-\lambda) y) = \lambda C +
(1-\lambda) \bar{\psi}_\sigma(y), \hspace{0.3in} y\in\partial\tau,
0\leq\lambda\leq 1.\]
For big $C\in{\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}$ the extension $\bar{\psi}_\sigma|_\tau$ is convex.
Now let $\sigma\in\Delta$ with $\dim {\sigma}=n$. For every $n-1$
dimensional cone $\tau\in\Delta'$ in the subdivision of $\sigma$ choose
a linear rational function $l_\tau: \sigma\rightarrow} \newcommand{\dar}{\downarrow{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$ such that $\tau$
is the subset of $\sigma$ defined by $l_\tau=0$. Then the sum
\[ \psi_\sigma(x) = -\sum_{\tau}|l_\tau(x)| \]
is a good function on $\sigma$. \qed
\begin{prp}
Let $U_X\subset X$ and $U_B \subset B$ be toroidal embeddings with
polyhedral complexes $\Delta_X$ and $\Delta_B$ respectively, and
assume that $B$ is nonsingular. Let $ f: X\rightarrow} \newcommand{\dar}{\downarrow B$ be a surjective
toroidal morphism. Then there exist projective subdivisions
$\Delta_X'$ of $\Delta_X$ and $\Delta_B'$ of $\Delta_B$ with
$\Delta_B'$ nonsingular, such that the induced map $f':X'\rightarrow} \newcommand{\dar}{\downarrow B'$ is an
equidimensional toroidal morphism.
If $f^{-1}(U_B)=U_X$ then $(f')^{-1}(U_{B'})=U_{X'}$.
\end{prp}
{\bf Proof.} There exists a subdivision of $\Delta_B$ ``induced'' by
$\Delta_X$. For $x\in\tau\in\Delta_B$ let $S_x$ be the set of cones
$\sigma\in\Delta_X$ such that $\sigma\cap
f_\Delta^{-1}(x)\neq\{0\}$. Since $f_\Delta$ is surjective, $S_x\neq
\emptyset$. The relation $x\sim y \Leftrightarrow S_x = S_y$ for
$x,y\in\tau$ is clearly an equivalence, hence it defines a partition
of $\Delta_B$. The equivalence class of $x$ is
\[ \bigcap_{\sigma\in S_x} f_\Delta(\sigma) \]
which is a convex rational polyhedral subcone of $\tau$. Thus the
partition defines a subdivision $\Delta_B^0$ of $\Delta_B$ such that
$f_\Delta(\sigma)$ for any cone $\sigma\in\Delta_X$ is a union of
cones in $\Delta_B^0$. By the previous lemma there exists a refinement
$\Delta_B^1$ of $\Delta_B^0$ which is a projective subdivision of
$\Delta_B$. Finally, we let $\Delta_B'$ be a nonsingular projective
subdivision of $\Delta_B^1$.
For $f_\Delta$ to map cones of $\Delta_{X}$ to cones of $\Delta_B'$,
the complex $\Delta_{X}$ has to be subdivided. Since the subdivision
$\Delta_B'$ of $\Delta_{B}$ is projective, there exists a good
function $\psi$ on $|\Delta_B'|=|\Delta_B|$. The piecewise linear
function $\psi\circ f_\Delta$ defines a projective subdivision
$\Delta_X'$ of $\Delta_{X}$ whose cones map into cones of
$\Delta_B'$. If $\sigma \in \Delta_X'$ then $f_\Delta(\sigma)$ is a
union of faces of some cone $\tau\in\Delta_B$. Since $f_\Delta$ is
linear on $\sigma$, $f_\Delta(\sigma)$ is convex in $\tau$, hence
$f_\Delta(\sigma)$ is a face of $\tau$.
If we assume from the beginning that $f^{-1}(U_B)=U_X$, that means
$f_\Delta^{-1}(0)\cap |\Delta_X| = 0$, then clearly the same is true
for any subdivision $|\Delta_X'| = |\Delta_X|$, hence
$(f')^{-1}(U_{B'})=U_{X'}$. \qed
\section{Kawamata's trick and reduced fibers}\label{reduced-fibers}
\subsection{Statement of result} The goal in this section is to find a
finite base change, after which
all the fibers in the resulting morphism are reduced. The base change
we perform will not necessarily be toroidal, but the morphism
after base change will still be toroidal.
\begin{prp}\label{prop-red}
Let $U_X\subset X$ and $U_B \subset B$ be projective toroidal
embeddings, and assume that $B$ is nonsingular. Let $f: X\rightarrow} \newcommand{\dar}{\downarrow B$ be a
surjective equidimensional toroidal morphism with
$f^{-1}(U_B)=U_X$. Then there exists a finite surjective morphism $p:
B'\rightarrow} \newcommand{\dar}{\downarrow B$ so that, denoting by $X'$ the normalization of $X\times_B
B'$, we have that $B'$ and $X'$ are toroidal embeddings, the projection $f':
X'\rightarrow} \newcommand{\dar}{\downarrow
B'$ is an equidimensional toroidal morphism with reduced fibers, and
$(f')^{-1}(U_{B'})=U_{X'}$.
\end{prp}
The construction of $X'$ and $B'$ is more explicitly given in
Proposition~\ref{prop-compl}, where the polyhedral complexes
of $X'$ and $B'$ are also described.
\subsection{The toric pictures} To start, we characterize equidimensional
toroidal morphisms with reduced fibers in terms of polyhedral
complexes.
\begin{lem} Let $f:X\to B$ be an equidimensional toroidal morphism,
$f_\Delta:\Delta_X\to \Delta_B$ the associated morphism of polyhedral
complexes. Then $f$ has reduced fibers if and only if for any cone
$\sigma \in \Delta_X$, with image $\tau\in \Delta_B$, we have
$f_\Delta(N_\sigma\cap\sigma) = N_\tau\cap\tau$. That is, the image of
the lattice in any cone of $X$ is the lattice in the image cone.
\end{lem}
{\bf Proof.} It suffices to consider the toric morphism of local
models $f: X_\sigma \rightarrow} \newcommand{\dar}{\downarrow X_\tau$ and the fiber over a point $t\in
X_\tau$ lying in the closed orbit of $X_\tau$. If the orbit of $t$ is
not $\{t\}$ then $X_\tau$ is a product $X_\tau = X_{\tau'}\times
\bfg_m^q$ for some $q>0$. Without loss of generality we may then
replace $X_\tau$ by $X_{\tau'}$, and replace $f$ by $p\circ f$ where $p:
X_\tau\rightarrow} \newcommand{\dar}{\downarrow X_{\tau'}$ is the projection. Indeed, the fiber $(p\circ
f)^{-1}(p(t))$ is isomorphic to $f^{-1}(t)\times \bfg_m^q$, and
$p_\Delta$ gives an isomorphism $p_\Delta: N_\tau\cap\tau \cong
N_{\tau'}\cap\tau'$. Thus we may assume that $\{t\}$ is the unique
closed orbit of $X_\tau$. The ideal of $f^{-1}(t)$ is generated by
$k[f_\Delta^{\vee}(\tau^\vee\cap(M_\tau\setmin \{0\}))] \subset
k[\sigma^\vee\cap M_\sigma]$. The fiber is reduced if and only if the
image $f_\Delta^{\vee}(\tau^\vee\cap(M_\tau\setmin \{0\}))$ is
saturated in $\sigma^\vee\cap M_\sigma$. This happens if and only if
$f_\Delta(N_\sigma\cap\sigma) = N_\tau\cap\tau$. \qed
When $X_\tau$ is nonsingular, the condition of the lemma is that
primitive points of $\sigma$ map to primitive points of $\tau$.
\begin{lem}\label{lem-tor-prod} Let $X_\sigma\rightarrow} \newcommand{\dar}{\downarrow X_\tau$ be a toric
morphism with $X_\tau$ nonsingular. Let $X_{\tau'}$ be a toric variety
given by $\tau'=\tau$ and $N_{\tau'}\subset N_\tau$ a sublattice of
finite index. Then every irreducible component of the normalization
of $X_\sigma \times_{X_\tau} X_{\tau'}$ is a toric variety
$X_{\sigma'}$ given by the cone $\sigma'=\sigma$ and integral lattice
$N_{\sigma'} = N_\sigma \cap f_\Delta^{-1}(N_{\tau'})$.
\end{lem}
{\bf Proof.} The ring of regular functions of $X_{\tau'}
\times_{X_\tau} X_\sigma$ is
\[ {\cal{O}}_{X_{\tau'}}\otimes_{{\cal{O}}_{X_\tau}} {\cal{O}}_{X_\sigma} =
k[(\tau')^{\vee}\cap M_{\tau'}]
\otimes_{k[\tau^{\vee}\cap M_\tau]} k[\sigma^{\vee}\cap M_\sigma] =
k[\pi] \]
where $\pi$ is the pushout of $j: \tau^{\vee}\cap M_\tau \to (\tau')^{\vee}\cap
M_{\tau'} $ and $f^\vee_\Delta: \tau^{\vee}\cap M_\tau \to \sigma^{\vee}\cap
M_\sigma$:
\[ \pi = ((\tau')^{\vee}\cap M_{\tau'}) \times (\sigma^{\vee}\cap
M_\sigma) /\sim.\]
Here $\sim$ is the equivalence relation generated by:
$(v_1,w_1)\sim
(v_2,w_2)$
whenever there exists $u\in \tau^{\vee}\cap M_\tau$ such that
$ (v_1,w_1) = (v_2,w_2) \pm (u,-f^\vee_\Delta(u)). $
Let $M$ be the abelian group
\[ M = M_{\tau'}\times M_\sigma /((u,-f^\vee_\Delta(u))) |
u\in \tau^{\vee}\cap M_\tau). \]
We will show that the semigroup homomorphism $\iota: \pi \rightarrow} \newcommand{\dar}{\downarrow M$ is injective.
Suppose that $\iota(v_1,w_1) = \iota(v_2,w_2)$, where $v_1,v_2 \in
(\tau')^{\vee}\cap M_{\tau'} $ and $w_1,w_2 \in \sigma^{\vee}\cap
M_\sigma$. Say $ (v_1,w_1) = (v_2,w_2) + (u,-f^\vee_\Delta(u))$ for some $u\in
M_\tau$. Let $t_1,\ldots,t_m \in
\tau^\vee$ be a basis of $M_\tau$, so that in this basis $\tau^\vee$ is given
by the inequalities $t_i\geq 0$, for $i=1,\ldots,k$. Write $u=\sum_i
\alpha_it_1$ with $\alpha_i\in {\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}$. Collecting positive and negative terms,
we get $u = u_1-u_2$ with $u_1,u_2\in \tau^{\vee}\cap M_\tau.$ Writing
$v_1,v_2$ in terms of $t_i$ and expanding the equation $v_1=v_2+u_1-u_2$, we
see that $v_2-u_2 \in \tau^{\vee}\cap M_{\tau'}$. Let $(v_3,w_3) =
(v_2-u_2,w_2+f^\vee_\Delta(u_2))$. Clearly
$$(v_1,w_1)\sim (v_3,w_3)\sim (v_2,w_2)$$
and
$\iota$ is injective.
The integral closure of $k[\pi]$ in $k[M]$ is the semigroup algebra
$k[\tilde{\pi}]$ where $\tilde{\pi}$ is the saturation of $\pi$ in
$M$. Write $M=F\oplus T$ where $F$ is the free part and $T$ is
torsion. Then $T\subset\tilde{\pi}$ because $m T=0$ for some
$m>0$. For any $f+t \in \tilde{\pi}$ where $f\in F$ and $t\in T$, we
have $-t\in T\subset \tilde{\pi}$, hence $f\in \tilde{\pi}$. Thus
\[ \tilde{\pi} \cong (\tilde{\pi}\cap F) \oplus T.\]
If $t_i$ are generators of $T$ of order $m_i$, and if $x_i$ is the
image of $t_i$ in $k[\tilde{\pi}]$, then
\[ k[\tilde{\pi}] \cong k[\tilde{\pi}\cap F][\ldots,x_i,\ldots] / (x_i
^{m_i}=1) \] Thus the normalization of ${\operatorname{Spec\ }} {k[\tilde{\pi}]}$ has
$|T|$ components with $x_i = \zeta_i$ where $\zeta_i$ are $m_i$'th
roots of unity, each component isomorphic to ${\operatorname{Spec\ }} {k[\tilde{\pi}\cap
F]}$.
Next we show that $F$ can be embedded in $M_\sigma\otimes{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$ so that
$M_\sigma\subset F$. The image of the homomorphism $\phi: M\rightarrow} \newcommand{\dar}{\downarrow
M_\sigma\otimes{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$ defined by $\phi(v,w)=f_\Delta^\vee(v)+w$ for
$v\in M_{\tau'}, w\in M_\sigma$ contains $M_\sigma$. Since $M$ and
$M_\sigma$ have the same rank, $\phi$ embeds $F$ in
$M_\sigma\otimes{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$ as a lattice of full rank containing
$M_\sigma$. As $f_\Delta^\vee$ takes $\tau^\vee$ into $\sigma^\vee$,
we see that $\phi$ maps $\tilde{\pi}\cap F$ into $\sigma^\vee$ so that
the image contains $\sigma^{\vee}\cap M_\sigma$. Therefore, ${\operatorname{Spec\ }}
{k[\tilde{\pi}\cap F]}$ is a toric variety defined by the cone
$\sigma'=\sigma$ and integral lattice $N_{\sigma'} = F^\vee$. To
determine $F^{\vee}$, first, we have $M_\sigma\subset F$, hence
$F^{\vee}\subset N_\sigma$; second, $f_\Delta^{\vee}(M_{\tau'})\subset
F$ implies that $f_\Delta(F^{\vee})\subset N_{\tau'}$. Conversely,
$N_\sigma\cap f_\Delta^{-1}(N_{\tau'}) \subset F^{\vee}$, so the two
are equal. \qed
To get a toric morphism with reduced fibers, one can take a base
change $X_{\tau'} \rightarrow} \newcommand{\dar}{\downarrow X_\tau$, where $\tau'=\tau$ and
$N_{\tau'}\subset N_{\tau}$ is a sublattice of finite index. By
Lemma~\ref{lem-tor-prod} every component $X_{\sigma'}$ of the
normalization of $X_\sigma \times_{X_\tau} X_{\tau'}$ is a toric
variety defined by the cone $\sigma' = \sigma$ and integral lattice
$N_{\sigma'} = N_\sigma \cap f_\Delta^{-1}(N_{\tau'})$. By a judicious
choice of $N_{\tau'}$ the fibers of $X_{\sigma'}\rightarrow} \newcommand{\dar}{\downarrow X_{\tau'}$ are
reduced.
\subsection{Kawamata's covering}
To perform a similar base change in the toroidal case we need a
toroidal morphism $B'\rightarrow} \newcommand{\dar}{\downarrow B$ which ramifies over a divisor $D$ with a
certain index.
\begin{dfn} \label{def-cyclic-cover} Let $L$ be a Cartier divisor on $B$, and
let $D\in|m L|$. Choose a
rational section $s_L$ of $ {\cal{O}}_B(L)$ defining $L$ and a regular section
$s_D$ of ${\cal{O}}_B(mL)$ defining $D$. Consider the rational function $\phi =
s_D/s_L^m$ on $B$. Then the field $K(B)(\sqrt[m]{\phi})$
depends only on $D$ and ${\cal{O}}(L)$. The normalization of $B$ in
$K(B)(\sqrt[m]{\phi})$ is called {\bf the cyclic cover ramified along $D$
with index $m$. }
\end{dfn}
\begin{rem} Another way to define the cyclic cover, is as the normalization of
$${\cal S}pec_B{\operatorname{Sym}}^\bullet\bigl({\cal{O}}_B(L)^\vee \bigr)/(\{f- s(f)\}_f)
$$
where we view $s$ as a morphism $s:{\cal{O}}_B(kL)\to{\cal{O}}_B((k+m)L)$.
\end{rem}
When $B$ is nonsingular and $D$ a divisor of normal crossings so that
$s_D=x_1\cdots x_l$ for some local parameters $x_1,\ldots,x_l$, then
the cyclic cover has a local equation
\[ z^m = x_1\cdots x_l. \]
It is nonsingular if and only if $l=1$.
Let $U_B\subset B$ be a nonsingular projective (strict) toroidal
embedding. Then $B\setmin U_B = \sum D_i$ is a strict divisor of
normal crossings. Consider the data $(D_i, m_i)$ where $D_i$ are the
irreducible components of $B\setmin U_B$ and $m_i$ are positive
integers, $i=1,\ldots,m$.
A {\it Kawamata covering
package} consists of $(D_i, m_i, H_{ij})$ with the following properties:
\begin{enumerate}
\item $H_{ij}$ are effective reduced nonsingular divisors on $B$, for
$i=1,\ldots,m$, $j=1,\ldots,\dim B$.
\item $\sum_i D_i + \sum_{i,j} H_{ij}$ is a reduced divisor of normal
crossings (in particular, $H_{ij}$ are distinct).
\item $D_i+ H_{ij} \in |m_i L_i|$ for some Cartier divisor $L_i$ for all
$i,j$.
\end{enumerate}
To find $H_{ij}$, let $M$ be an ample divisor. Take a multiple of $M$
if necessary so that $m_i M-D_i$ is very ample for all $i$, and choose
$H_{ij}$ general members in $|m_i M-D_i|$.
Now let $B_{ij}$ be the cyclic cover ramified along $D_i+H_{ij}$ with
index $m_i$; let $B'$ be the normalization of
\[ B_{1,1} \times_B \ldots \times_B B_{m,{\operatorname{dim}}{B}}\]
and let $p:B'\rightarrow} \newcommand{\dar}{\downarrow B$ be the projection.
\begin{lem} [Kawamata] The variety $B'$ is nonsingular, ramified with
index $m_i$ along $D_i+H_{ij}$. The reduction of the inverse image
$p^*(\sum_i D_i + \sum_{i,j} H_{ij})_{\red}$ is a divisor of normal crossings.
\end{lem}
{\bf Proof.} Let $x_i$ be local equations of $D_i$, and let $y_{ij}$
be local equations of $H_{ij}$ at $b\in B$. Then $B_{ij}$ is locally
given by the equation
\[ z_{ij}^{m_i} = x_i y_{ij} \]
It suffices to prove that the normalization of $\times_j B_{ij}$ is
nonsingular for all $i$. If $b\in D_i$
then since $\bigcup_j H_{ij} \cap D_i = \emptyset$, say $b\notin
H_{i,0}$, and $y_{i,0}$ is a unit in ${\cal{O}}_{B,b}$. The normalization of
the product $\times_j B_{ij}$ is locally defined by the equations
\begin{eqnarray*}
z_{i,1}^{m_i} &=& x_i \\ (\frac{z_{i,j}}{z_{i,1}})^{m_i} &=& y_{i,j}
\hspace{.5in} j=2,\ldots \dim B.
\end{eqnarray*}
Since $x_i, y_{i,2},\ldots,y_{i,\dim B}$ are either units or local
parameters in ${\cal{O}}_{B,b}$, it follows that the normalization of the
product $\times_j B_{ij}$ is nonsingular at $b$. A similar situation happens
when $b\notin D_i$, since then $x_i$ is a unit in
${\cal{O}}_{B,b}$. Thus, $B'$ is
nonsingular and ramified with index $m_i$ along $D_i$ and
$H_{ij}$. Replacing $B$ by $D_i$ or $H_{ij}$, we get that $p^*(\sum_i
D_i + \sum_{i,j} H_{ij})_{\red}$ is a divisor of normal
crossings. \qed
Let $\tilde{U}_B = B\setmin (\bigcup_{i,j} H_{ij}\cup D_i)$, and
$\tilde{U}_{B'} = p^{-1} (\tilde{U}_B)$. Both $\tilde{U}_B \subset B$
and $ \tilde{U}_{B'} \subset B'$ are toroidal embeddings because $B,
B'$ are nonsingular and the divisors $B\setmin\tilde{U}_B$,
$B'\setmin\tilde{U}_{B'}$ cross normally. From the construction and local
equations we
also see that $p$ is a toroidal morphism with respect to this structure. If
$b'\in B'$, $b=p(b')\in
B$, then there exist local parameters $\{x_l\}$ at $b$ and $\{z_l\}$
at $b'$ such that $p$ is defined by $x_l = z_l^{a_l}$ where $a_l =
m_i$ if the divisor defined by $x_l$ is $D_i$ or $H_{ij}$, otherwise
$a_l =1$. We denote the polyhedral complexes of $\tilde{U}_B \subset
B$ and $ \tilde{U}_{B'} \subset B'$ by $\tilde{\Delta}_B$ and
$\tilde{\Delta}_{B'}$, respectively. So, every cone $\tau' \in
\tilde{\Delta}_{B'}$ is mapped homeomorphically to a cone $\tau \in
\tilde{\Delta}_{B}$ by $p_\Delta$. If $N_\tau$ has basis
$u_1,\ldots,u_m$ then $N_{\tau'}$ has basis $a_1 u_1, \ldots, a_m u_m$
where $a_l$ are as above.
\subsection{The toroidal picture} We need to see how adding the divisors
$H_{ij}$ to the toroidal structure of $B$ affects the toroidal structure of
$X$.
\begin{lem} Let $f: X\rightarrow} \newcommand{\dar}{\downarrow B$ be toroidal, $B$ nonsingular, H a generic
hyperplane section of $B$. Let $x\in X$, $b=f(x)\in H$. Then
\begin{itemize}
\item[(i)] there exist local models $(X_\sigma = X_\sigma'\times
\bfg_m, z\times 1)$ at $x$ and $(X_\tau = X_\tau'\times \bfg_m,
y\times 1)$ at $b$ such that $H$ corresponds to the divisor
$X_\tau'\times \{1\}$ and the morphism $f$ is a product of toroidal
morphisms
\[ f=g \times id: X_\sigma'\times \bfg_m \rightarrow} \newcommand{\dar}{\downarrow X_\tau'\times \bfg_m \]
\item[(ii)] $U_B\setmin H\subset B$, $U_X\setmin f^{-1}(H)\subset
X$ are toroidal embeddings and $f$ is a toroidal morphism of these
embeddings.
\end{itemize}
\end{lem}
{\bf Proof.} Let $(X_{\sigma},s)$ and $(X_{\tau},t)$ be local models
at $x\in X$ and $b=f(x)\in B$, respectively, and let $f$ also denote
the toric morphism of the local models defined by
\[ f_\Delta: (N_\sigma,\sigma)\rightarrow} \newcommand{\dar}{\downarrow(N_\tau,\tau) \]
Clearly $U_B\setmin H\subset B$ is toroidal and we may assume that
it has a local model $(X_{<\tau,v>},t')$, where $<\tau,v>$ is the cone
spanned by $\tau$ and some indivisible $v\in N_{\tau}$, and where $t'$
lies in the unique closed orbit of $X_{<\tau,v>}$. Write $N_{\tau,2}$
for the saturated sublattice of $N_{\tau}$ generated by $v$, and
choose a splitting
\begin{eqnarray*}
N_{\tau} &\cong& N_{\tau,1} \oplus N_{\tau,2} \\ X_{\tau} &\cong&
X_{\tau}' \times \bfg_m \\ X_{<\tau,v>} &\cong& X_{\tau}' \times
{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1
\end{eqnarray*}
Since $f$ is dominant, $\tilde{N}_{\tau,2} = f_\Delta(f_\Delta^{-1}(
N_{\tau,2}))$ is a sublattice of finite index of $N_{\tau,2}$. The
inclusion $\tilde{N}_{\tau,2} \subset N_{\tau,2}$ corresponds to an
\'etale cover of $\bfg_m$. Since the completed local rings are
isomorphic, we may replace the local model by a local model in the
\'etale cover and assume that $f_\Delta^{-1}(N_{\tau,2})$ surjects
onto $N_{\tau,2}$.
Let $N_{\sigma,1}=f_\Delta^{-1}(N_{\tau,1}) \subset N_{\sigma}$ and
let $N_{\sigma,2} \subset N_{\sigma}$ be generated by some $u\in
N_{\sigma}$ such that $f_\Delta(u)=v$. For any $w\in N_{\sigma}$,
$f_\Delta(w) = v_1+v_2 \in N_{\tau,1} \oplus N_{\tau,2}$, there exists
$m\in {\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}$ so that $f_\Delta(m u)=v_2$. Hence $w-m u \in
N_{\sigma,1}$, and $N_{\sigma}=N_{\sigma,1}+N_{\sigma,2}$. Since
$N_{\sigma,1}\cap N_{\sigma,2} = \{0\}$ the sum is direct,
\begin{eqnarray*}
N_{\sigma} &\cong& N_{\sigma,1} \oplus N_{\sigma,2} \\ X_{\sigma}
&\cong& X_{\sigma}' \times \bfg_m
\end{eqnarray*}
and $f_\Delta$ maps $N_{\sigma,1}$ to $N_{\tau,1}$ and $N_{\sigma,2}$
to $N_{\tau,2}$. Thus the toric morphism $f$ is a product $f=g\times
h$ where $g:X_{\sigma}'\rightarrow} \newcommand{\dar}{\downarrow X_{\tau}'$ and $h=id:\bfg_m \rightarrow} \newcommand{\dar}{\downarrow \bfg_m$
are the toric morphisms induced by the restriction of $f_\Delta$ to
$N_{\sigma,1}$ and $N_{\sigma,2}$, respectively. Since $f_\Delta$ maps
$<\sigma,u>$ to $<\tau,v>$, $f$ extends to a toric morphism
\[ f: X_{<\sigma,v>} \cong X_{\sigma}' \times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1 @>{g\times id}>>
X_{\tau}' \times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1 \cong X_{<\tau,v>}\]
For $t=(y,1)\in X_{\tau}' \times \bfg_m \subset X_{\tau}' \times
{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1$ the complete local rings $\hat{{\cal{O}}}_{X_{\tau}' \times
\bfg_m,(y,1)}$ and $\hat{{\cal{O}}}_{X_{\tau}' \times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1,(y,0)}$ are
isomorphic via translation. The closures of codimension 1 orbits
through $(y,0)$, are those through $(y,1)$ plus $X_{\tau}' \times
\{0\}$ which formally corresponds to $H$. Similarly, for $s=(z,1)\in
X_{\sigma}' \times \bfg_m \subset X_{\sigma}' \times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1$ the ring
$\hat{{\cal{O}}}_{X_{\sigma}' \times \bfg_m,(z,1)}$ is isomorphic to
$\hat{{\cal{O}}}_{X_{\sigma}' \times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1,(z,0)}$ via translation, and the
closures of codimension 1 orbits through $(z,0)$, are those through
$(z,1)$ plus $X_{\sigma}' \times \{0\} = f^{*}(X_{\tau}' \times
\{0\})$ which formally corresponds to $f^{*}(H)$.
Thus $(X_{<\sigma,u>},(z,0))$ and $(X_{<\tau,v>},(y,0))$ are local
models at $x\in X$ and $b\in B$, respectively, and the morphism
\[ f: (U_X\setmin f^{-1}(H)\subset X)\rightarrow} \newcommand{\dar}{\downarrow(U_B\setmin H\subset B)\]
is toroidal. \qed
It follows from the lemma that $f:(\tilde{U}_B\subset B) \rightarrow} \newcommand{\dar}{\downarrow
(\tilde{U}_X\subset X)$ is toroidal, where $\tilde{U}_X =
f^{-1}(\tilde{U}_B)$. By Lemma~\ref{lem-tor-prod}, the normalization
$X'$ of $X\times_B B'$ is toroidal. Let $f': X'\rightarrow} \newcommand{\dar}{\downarrow B'$, $p':X'\rightarrow} \newcommand{\dar}{\downarrow X$
be the (toroidal) projections, and let $\tilde{\Delta}_X$,
$\tilde{\Delta}_{X'}$ be the polyhedral complexes of $X, X'$.
\[
\begin{CD}
X' @>{p'}>> X\\ @V{f'}VV @VV{f}V\\ B' @>{p}>> B
\end{CD}
\]
Then $p_\Delta'$ maps a cone $\sigma' \in \tilde{\Delta}_{X'}$
homeomorphically to a cone $\sigma \in \tilde{\Delta}_{X}$; the
integral lattice of $\sigma'$ can then be identified with a sublattice
$N_{\sigma'} \subset N_\sigma$.
The following lemma shows that the added divisors $p^{-1}(H_{ij})$ and
$(p\circ f')^{-1}(H_{ij})$ can be removed from the toroidal structures
of $B'$ and $X'$ so that $f'$ (but not $p$ or $p'$) remains toroidal.
\begin{lem}
$p^{-1}(U_B) \subset B'$ and $(p\circ f')^{-1}(U_B) \subset X'$ are
toroidal embeddings and $f'$ is a toroidal morphism of these
embeddings.
\end{lem}
We show how to remove one irreducible divisor $H=H_{ij}$ for some
$i,j$. Since the question is local, choose local models $X_{\sigma}$,
$X_{\tau}$ and $X_{\tau'}$ of $X$, $B$, and $B'$ so that both $f$ and
$p$ are products
\[ f: X_{\sigma} \cong X_{\sigma}' \times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1 @>{g\times id}>>
X_{\tau}' \times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1 \cong X_{\tau}\]
\[ p: X_{\tau'} \cong X_{\rho}' \times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1 @>{q\times r}>>
X_{\tau}'\times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1 \cong X_{\tau} \]
where all morphisms are toric and $H$ corresponds to $X_{\tau}' \times
\{0\}$. Since $p$ ramifies along $H$, the morphism
$r:{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1\rightarrow} \newcommand{\dar}{\downarrow{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1$ has degree $m\geq 2$.
The fiber product is then
\begin{eqnarray*} X'' & = & (X_{\sigma}' \times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1)\times_{X_{\tau}'
\times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1} (X_{\tau'}' \times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1) \\
& \cong & (X_{\sigma}' \times_{X_{\tau}'}
X_{\tau'}')\times ({\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1 \times_{{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1} {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1) \\
& \cong & (X_{\sigma}'
\times_{X_{\tau}'} X_{\tau'}')\times {\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1
\end{eqnarray*}
and the projection
$X''\rightarrow} \newcommand{\dar}{\downarrow X_{\tau'}$ is the product of the projection $X_{\sigma}'
\times_{X_{\tau}'} X_{\tau'}' \rightarrow} \newcommand{\dar}{\downarrow X_{\tau'}'$ and $id:
{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1\rightarrow} \newcommand{\dar}{\downarrow{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1$. The divisor $(p\circ f')^{*}(H)$ in $X''$ is
$X_{\sigma}' \times_{X_{\tau}'} X_{\tau'}' \times \{0\}$ and by the
same translation argument as above we can remove $p^{*}(H)$ and
$(p\circ f')^{*}(H)$ from the toroidal structures of $B'$ and $X''$,
respectively. \qed
Let $\Delta_{X'}, \Delta_{B'}$ be the polyhedral complexes of the
embeddings $p^{-1}(U_B) \subset B'$ and $(p\circ f')^{-1}(U_B) \subset
X'$. Removing the divisors $p^{-1}(H_{ij})$ and $(p\circ
f')^{-1}(H_{ij})$ means removing the corresponding edges (and
everything attached to them) from the polyhedral complexes
$\tilde{\Delta}_{B'}$ and $\tilde{\Delta}_{X'}$. As $p_\Delta$ and
$p_\Delta'$ map cones of $\tilde{\Delta}_{B'}$ and
$\tilde{\Delta}_{X'}$ homeomorphically to cones of
$\tilde{\Delta}_{B}$ and $\tilde{\Delta}_{X}$, restrictions of
$p_\Delta$ and $p_\Delta'$ map cones of ${\Delta}_{B'}$ and
${\Delta}_{X'}$ homeomorphically to cones of ${\Delta}_{B}$ and
${\Delta}_{X}$. We summarize the previous constructions in the
following proposition.
\begin{prp}\label{prop-compl}
With the assumptions of Proposition~\ref{prop-red}, let $u_i$ be the
primitive points of $\Delta_B$, and let $m_i>0$ for
$i=1,\ldots,l$. There exists a finite covering $p:B'\rightarrow} \newcommand{\dar}{\downarrow B$ so that, if
$X'$ is the normalization of $X\times_B B'$ and $f':X'\rightarrow} \newcommand{\dar}{\downarrow B'$,
$p':X'\rightarrow} \newcommand{\dar}{\downarrow X$ the two projections, we have
\begin{itemize}
\item[(i)] $U_{X'}\subset X'$ and $U_{B'}\subset B'$ are toroidal
embeddings with polyhedral complexes $\Delta_{X'}$ and $\Delta_{B'}$;
moreover, $f'$ is a toroidal morphism of these embeddings.
\item[(ii)] There exist morphisms
$$p_\Delta:\Delta_{B'}\rightarrow} \newcommand{\dar}{\downarrow \Delta_{B}$$
and $$p_\Delta':
\Delta_{X'}\rightarrow} \newcommand{\dar}{\downarrow \Delta_{X},$$
such that
\begin{list}{$\circ$}{}
\item for any cone $\tau'\in \Delta_{B'}$, the morphism $p_\Delta$ maps
$\tau'$ isomorphically to $\tau$, and identifies $N_{\tau'}$ with the
sublattice of $N_{\tau}$ generated by $m_i u_i$;
\item for any cone $\sigma'\in \Delta_{X'}$, $p_\Delta'$ maps
$\sigma'$ isomorphically to $\sigma$, and identifies
$N_{\sigma'}$ with the sublattice $N_\sigma \cap f_\Delta^{-1}
(N_{\tau'})$ of $N_{\sigma}$
\item The following diagrams commute:
\begin{minipage}{2.5in}
\[
\begin{CD}
\sigma' @>{\cong}>p_\Delta'> \sigma\\ @V{f_\Delta'}VV @VV{f_\Delta}V\\
\tau' @>{\cong}>p_\Delta> \tau
\end{CD}
\]
\end{minipage}
\begin{minipage}{2.5in}
\[
\begin{CD}
N_{\sigma'} @>{\subset}>p_\Delta'> N_\sigma\\ @V{f_\Delta'}VV
@VV{f_\Delta}V\\ N_{\tau'} @>{\subset}>p_\Delta> N_{\tau}
\end{CD}
\]
\end{minipage} \\
\end{list}
\end{itemize}
\end{prp} \qed
{\bf Proof of \ref{prop-red}} Let $\Delta_X$, $\Delta_B$ be the
polyhedral complexes of $X$, $B$, and $f_\Delta:
\Delta_X\rightarrow} \newcommand{\dar}{\downarrow\Delta_B$. By equidimensionality, $f_\Delta$ maps
$\sigma^{(1)}$ to $\tau^{(1)}$. If $u_i$ are the primitive
points of $\Delta_B$, let $v_{ij}$ be the
primitive points of $\Delta_X$ such that $f_\Delta(v_{ij}) = m_{ij}
u_i$ for some $m_{ij} >0$. Set $m_i = {\operatorname{lcm}}_j \{m_{ij}\}$. Then for all
$i,j$, some multiple of $v_{ij}$ maps to $m_i u_i$. Now we use the
covering data $(D_i, m_i)$ to define the toroidal morphism $f':X'\rightarrow} \newcommand{\dar}{\downarrow
B'$ as in the previous proposition. Let $\sigma' \in \Delta_{X'}$,
$f_\Delta'(\sigma') = \tau' \in \Delta_{B'}$. There exist
$\sigma\in\Delta_X$, $\tau\in\Delta_B$ such that $\sigma'\cong\sigma$,
$\tau'\cong\tau$. The integral lattice $N_{\tau'}$ is generated by
$m_i u_i$ with $u_i$ lying on the edges of $\tau$, and
$N_{\sigma'}=N_\sigma \cap f_\Delta^{-1} (N_{\tau'})$. For any edge of
$\sigma'$ spanned by $v_{ij}$, some multiple $a_{ij} v_{ij}$ maps to
$m_i u_i$. Hence $a_{ij} v_{ij} \in N_{\sigma'}$ is primitive and maps
to the primitive point $m_i u_i$. This proves that $f'$ has reduced
fibers. \qed
\section{Mildness of the morphism}\label{mildness}
Recall that a normal variety $X$ is said to have Gorenstein
singularities if it is Cohen-Macaulay and has an invertible dualizing
sheaf $\omega_X$. It is said to have rational singularities, if for a
resolution of singularities $r:X'\to X$ we have $r_*\omega_{X'} =
\omega_X$. Every toric variety is Cohen-Macaulay with rational
singularities (\cite{te} Theorem~14, p. 52). The dualizing sheaf of an affine
toric variety $X_\tau$ is the coherent sheaf $\omega_{X_\tau} =
{\cal{O}}(-\sum D_i)$ where $D_i$ are closures of the codimension 1
orbits. This sheaf is invertible if and only if there exists an element of $M$
(namely a linear function on $\tau$ taking integer values on $N$) such that
$\psi(v) = -1$ for all $v\in \tau^{(1)}$.
\begin{lem}
Let $X_\sigma$ and $X_\tau$ be affine toric varieties with $X_\tau$
nonsingular. Let $f: X_\sigma \rightarrow} \newcommand{\dar}{\downarrow X_\tau$ be an equidimensional toric
morphism without horizontal divisors, having only reduced fibers. Then
$X_\sigma$ has rational Gorenstein singularities.
\end{lem}
{\bf Proof.} Since $X_\tau$ is nonsingular, it has rational
Gorenstein singularities. Let $\psi:\tau\rightarrow} \newcommand{\dar}{\downarrow{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$ be the linear
interpolation of $\psi(u_i)=-1$ for every primitive point $u_i$ of
$\tau$. Then $\psi\circ f_\Delta:\sigma\rightarrow} \newcommand{\dar}{\downarrow{\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}$ is the required
function. It is linear because $\psi$ and $f_\Delta$ are; since $f$ is
equidimensional and has reduced fibers, primitive points in $\sigma$
map to primitive points in $\tau$, hence $\psi\circ f_\Delta$ takes
the value $-1$ on every primitive point of $\sigma$. \qed
\begin{lem}\label{lem-rational-Gorenstein}
Let $U_X\subset X$ and $U_B \subset B$ be toroidal embeddings, and
assume that B is nonsingular. Let $f: X\rightarrow} \newcommand{\dar}{\downarrow B$ be an equidimensional
toroidal morphism, without horizontal divisors, and with reduced fibers. Then
$X$ has rational Gorenstein singularities.
\end{lem}
{\bf Proof.} Having rational Gorenstein singularities is a local
analytic property. Since all local models have rational Gorenstein
singularities, so does $X$. \qed
To show that the morphism is mild, we need to look at the situation
after dominant base changes. We first look at the cases where the base
change is relatively nice:
\begin{lem}\label{lem-log-smoothness}
Let $f:X\to B$ be as above. Let $g:B'\to B$ be a dominant morphism,
where $B'$ a nonsingular variety, and assume that $ g^{-1} (B\setmin U_B)$ is
a normal
crossings divisor. Let $X'$ be the pullback of $X$ to $B'$. Then
$X'\to B'$ admits a toroidal structure relative to $ g^{-1}U_B \subset B'$.
\end{lem}
{\bf Proof.} We use the formalism of logarithmic structures. By \cite{kato1},
\S 8.1,
the morphism $X\to B$ is logarithmically smooth. Also $ g^{-1}U_B \subset B'$
endows $B'$ with a logarithmically regular structure. Moreover $g:B'\to B$ is a
morphism of logarithmic schemes. The variety $X'$ thus inherits a logarithmic
structure. The morphism $X'\to B'$ clearly satisfies the formal lifting
property for logarithmic smoothness (\cite{kato1},
\S 8.1,(i)). It is left to show that $X'$ satisfies condition (S)
(\cite{kato1} \S 1.5). Indeed, since $f$ is equidimensional, the fibers of $f$
are
reduced and $B'$ is normal, it follows that $X'$ is regular in codimension
1. Since $f$ is a Gorenstein morphism and $B$ is Gorenstein, we have that $X'$
is
Gorenstein, in particular it is Cohen Macaulay. It then follows that $X'$ is
normal. Combining this with the assumption that $f^*M_B$ is saturated in $M_X$,
we have that the monoids giving the logarithmic charts
on $X'$ are integral and saturated. Altogether, we have that $X'$ satisfies
condition (S). Therefore $f':X'\to B'$ is logarithmically smooth, and thus
toroidal.
\begin{lem}\label{lem-descent} Let $f:X\to B$ be a flat Gorenstein morphism and
assume that $B$
has
rational Gorenstein singularities. Assume there is a modification $r:B'\to B$
with rational Gorenstein singularities, such that $X'=X\times_B B'$ has
rational singularities. Then $X$ has rational singularities as well.
\end{lem}
{\bf Proof.} Consider the diagram
$$\begin{array}{rcl} X' &\stackrel{h}{\to} & X \\
f' \downarrow & & \downarrow f\\
B' &\stackrel{r}{\to} & B.
\end{array}$$
By base change we have $h^*\omega_f = \omega_{f'}$. By assumption we have
$\omega_B = r_*\omega_{B'}$. The flatness of $f$ implies $f^*r_*\omega_{B'} =
h_*{f'}^*\omega_{B'}$. Therefore we have
\begin{eqnarray*}
h_*\omega_{X'} & = & h_*(\omega_{f'} \otimes {f'}^*\omega_{B'} )\\
& = & h_*(h^*\omega_f \otimes {f'}^*\omega_{B'}) \\
& = & \omega_f \otimes h_*{f'}^*\omega_{B'} \\
& = & \omega_f \otimes f^*r_*\omega_{B'} \\
& = & \omega_f \otimes f^*\omega_B \\
& = & \omega_X.
\end{eqnarray*}
Thus $X$ has rational singularities. \qed
\begin{prp}
Let $U_X\subset X$ and $U_B \subset B$ be toroidal embeddings and
assume that B is nonsingular. Let $f: X\rightarrow} \newcommand{\dar}{\downarrow B$ be an equidimensional
morphism with reduced fibers, which is toroidal such that $U_X=f^{-1}U_B$.
Then $f$ is mild.
\end{prp}
{\bf Proof.} Let $B_1\to B$ be a dominant morphism such that $B_1$ has rational
Gorenstein singularities. We need to show that $X_1 = X\times_BB_1$ has
rational Gorenstein singularities. By lemma \ref{lem-rational-Gorenstein} it
has Gorenstein singularities.
Pick a resolution of singularities $B'\to B_1$ such that the inverse image of
$B\setmin U_B$ in $B'$ is a divisor with normal crossings. By lemma
\ref{lem-log-smoothness} we have that $X' = X\times_BB'$ is toroidal, therefore
$X'$ has rational singularities. By lemma \ref{lem-descent} we have that $X_1$
has rational singularities as well, which is what we needed. \qed
\section{The Cartier covering}\label{sec-cartier} In section
\ref{combinatorial} below we will
translate the semistable reduction conjecture in purely combinatorial terms. In
order to maximize the flexibility of the combinatorial operations, we need to
generalize Kawamata's trick slightly to accommodate cases where $B$ has
quotient singularities. In such a situation we need a finite cover of $B$ which
is nonsingular, and such that the resulting polyhedral complex is easily
described. The following statement will suffice for this purpose.
\begin{prp}\label{prop-cartier} Let $U_X\subset X$ and $U_B \subset B$ be
projective toroidal
embeddings, and assume that $B$ has only quotient singularities.
Let $f: X\rightarrow} \newcommand{\dar}{\downarrow B$ be a surjective mild toroidal morphism,
satisfying $f^{-1}(U_B)=U_X$. Then there exists a finite surjective morphism
$p: B'\rightarrow} \newcommand{\dar}{\downarrow B$ so that, denoting $X'=X\times_B B'$, $U_{B'}=p^{-1}U_B$ and
$U_{X'}= X\times_B U_{B'}$ we have
\begin{enumerate}
\item $B'$ and $X'$ are toroidal embeddings with polyhedral complexes
$\Delta_{X'}$ and $\Delta_{B'}$, with $B'$ nonsingular;
\item the projection $f': X'\rightarrow} \newcommand{\dar}{\downarrow B'$ is a mild toroidal morphism;
\item There exist morphisms
$$p_\Delta:\Delta_{B'}\rightarrow} \newcommand{\dar}{\downarrow \Delta_{B}$$
and $$p_\Delta':
\Delta_{X'}\rightarrow} \newcommand{\dar}{\downarrow \Delta_{X},$$
such that
\begin{list}{$\circ$}{}
\item for any cone $\tau'\in \Delta_{B'}$, the morphism $p_\Delta$ maps $\tau'$
isomorphically to $\tau$, and identifies
$N_{\tau'}$ with the sublattice of
$N_{\tau}$ generated by the primitive vectors of $\tau$;
\item for any cone $\sigma'\in \Delta_{X'}$, $p_\Delta'$ identifies
$N_{\sigma'}$ with the sublattice
$N_\sigma \cap f_\Delta^{-1} (N_{\tau'})$ of $N_{\sigma}$, and maps
$\sigma'$ isomorphically to $\sigma$.
\item The following diagrams commute:
\begin{minipage}{2.5in}
\[
\begin{CD}
\sigma' @>{\cong}>p_\Delta'> \sigma\\ @V{f_\Delta'}VV @VV{f_\Delta}V\\
\tau' @>{\cong}>p_\Delta> \tau
\end{CD}
\]
\end{minipage}
\begin{minipage}{2.5in}
\[
\begin{CD}
N_{\sigma'} @>{\subset}>p_\Delta'> N_\sigma\\ @V{f_\Delta'}VV
@VV{f_\Delta}V\\ N_{\tau'} @>{\subset}>p_\Delta> N_{\tau}
\end{CD}
\]
\end{minipage} \\
\end{list}
\end{enumerate}
\end{prp}
{\bf Proof.} We use a construction analogous to Kawamata's covering. First, let
$B$ be a variety, $E$ an effective Weil divisor on $B$. We make the following
assumptions:
\begin{enumerate}
\item as a scheme, $E$ is integral and normal;
\item the divisor $ mE = D$ is a Cartier;
\item There is a Cartier divisor $L$ on $B$ such that ${\cal{O}}(D) = {\cal{O}}(mL)$.
\end{enumerate}
As in Definition \ref{def-cyclic-cover}, we choose a function $\phi$ with
$\operatorname{div}(\phi) = D-mL$, and define the cyclic cover $p:B'\to B$ by
taking $m$-th root of $\phi$. The point is, that the ${\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}$-Cartier divisor
$p^*E$ is in fact Cartier.
Let us see what happens in the toric case.
\begin{lem} Let $X_\tau$ be an affine toric variety with quotient
singularities and $E$ a toric divisor corresponding to a primitive vector $v$.
Write $\tau={\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}_+v\times \tau_1$. Choose $m\in \bfn$ such that $D = mE$ is
Cartier. Then ${\cal{O}}(mD)$ is trivial, and the corresponding cyclic cover is
$X_{\tau'}$, where $|\tau'| = \tau$ and $N_{\tau'} = {\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}} v\times
(N_\tau\cap\tau_1)$.
\end{lem}
This lemma is easy and left to the reader. If we iterate the lemma with all the
toric divisors, we obtain a nonsingular covering, whose lattice is generated by
the primitive vectors of $\tau$.
Assume that in addition we have a mild morphism $X_\sigma\to X_\tau$.
Lemma \ref{lem-tor-prod} works word-for-word in this case. The only thing which
needs to be changed in the proof is, that since $f_\Delta(N_\sigma) = N_\tau$,
the semigroup homomorphism $i:\pi\to M$ is still injective. (This is not a
serious business - we could replace the product by its reduction anyway.)
In order to define a Kawamata covering package, we need the following Bertini
type lemma (this is a special case of the stratified Bertini Theorem) .
\begin{lem}
Let $U_B\subset B$ be a toroidal embedding. Let $H_0$ be a very ample divisor
on $B$ and let $H$ be a general element of $|H_0|$. Let $U_B' = U_B\setmin
H$. We have
\begin{enumerate}
\item $U_B'\subset B$ is a toroidal embedding;
\item let $b\in H$ and assume that $b$ lies on a stratum $Y\subset B$ of the
toroidal embedding $U_B\subset B$. Then $H\cup Y$ is nonsingular at $b$;
\item there is a local model $X_\rho \times \bfg_m^k$ for $U_B\subset B$ at
$b$, where
$Y$ corresponds to the factor $\bfg_m^k$, and $H$ corresponds to $X_\rho
\times \bfg_m^{k-1}\times\{1\}$.
\item there is a local model $X_\rho \times \bfg_m^{k-1}\times{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1$ for
$U_B'\subset B$ at
$b$, where $Y$ corresponds to $\bfg_m^k\times{\Bbb{A}}} \newcommand{\bfn}{{\Bbb{N}}^1$, and $H$ corresponds to
$X_\rho
\times \bfg_m^{k-1}\times\{0\}$.
\end{enumerate}
\end{lem}
{\bf Proof.} Since $B$ is a stratified space, it has a local product structure
$X_\rho\times Y$ at $b$. We need to adapt this
structure to the divisor $H$ as in the lemma. Applying Bertini's theorem to
the closure of the stratum $Y$, we have that $H\cup Y$ is nonsingular. Pick a
regular function $u$ near $b$ defining $H$ on a neighborhood of $b$. Then the
restriction $u_Y$ of $u$ to $Y$ is a regular parameter. We can find a regular
system of parameters on $Y$ at $b$ including $u_Y$ and lift them to $B$. This
lifting gives the desired adaptation of the product structure. \qed
Using this lemma, the proof of Proposition \ref{prop-compl} works word-for-word
here, and yields Proposition \ref{prop-cartier}. \qed
\section{Towards semistable reduction}\label{combinatorial}
\subsection{Combinatorial semistable reduction}
We are going to restate the semistable reduction conjecture (Conjecture
\ref{conj-semistable}) in purely combinatorial terms. First let us define
semistability combinatorially.
\begin{dfn}\label{dfn-comb-semi} Let $\Delta_X$ and $\Delta_B$
be rational conical polyhedral complexes, and let $f_\Delta:\Delta_X\to
\Delta_B$ be a surjective polyhedral map.
We say that $f_\Delta$ is {\bf weakly semistable} if the following conditions
hold.
\begin{enumerate}
\item We have $f_\Delta^{-1}(0) = \{0\}$.
\item For any cone $\tau\in \Delta_X$, we have $f_\Delta(\tau)\in \Delta_B$.
\item\label{item-reduced} For any cone $\tau\in \Delta_X$, we have
$f_\Delta(N_\tau) =
N_{f_\Delta(\tau)}$.
\item $\Delta_B$ is simplicial with index 1.
\end{enumerate}
If also $\Delta_X$ is simplicial with index 1, we say that $f_\Delta$ is {\bf
semistable}.
\end{dfn}
We now define the operations we are allowed to perform.
\begin{dfn} Let $\Delta$ be a rational conical polyhedral complex. A {\bf
lattice alteration} $\Delta_1\to \Delta$ consists of an integral structure
induced by a consistent choice of sublattices $N^1_\sigma\subset N_\sigma$ for
each cone $\sigma\subset \Delta$. An {\bf alteration} $\Delta_1\to \Delta$ is a
composition $\Delta_1\to \Delta'\to \Delta$ of a lattice alteration
$\Delta_1\to \Delta'$ with a subdivision $\Delta'\to \Delta$. The alteration
is {\bf projective} if the corresponding subdivision $\Delta'\to \Delta$ is
projective.
\end{dfn}
Note that the composition of alterations is an alteration. Be warned that the
factorization of a polyhedral alteration above is not analogous to the Stein
factorization in algebraic geometry - it has the opposite order.
\begin{dfn} \begin{enumerate}
\item Let $\Delta_X\to \Delta_B$ be a polyhedral map of rational conical
polyhedral complexes. Let $\Delta_B'\to \Delta_B$ be a subdivision. The induced
subdivision $\Delta_X'\to \Delta_X$ is the minimal subdivision admitting a
polyhedral map to $\Delta_B'$. The cones of $\Delta_X'$ are of the form
$\sigma\cap f_\Delta^{-1}(\tau')$ for $\sigma\in \Delta_X$ and $\tau'\in
\Delta_B'$.
\item Let $\Delta_X\to \Delta_B$ be a polyhedral map of rational conical
polyhedral complexes. Let $\Delta_B^1\to \Delta_B$ be a lattice alteration. The
induced lattice alteration $\Delta_X^1\to \Delta_X$ is the minimal lattice
alteration mapping to $\Delta_B^1$. The lattices of $\Delta_X^1$ are of the
form $N_\sigma\cap f_\Delta^{-1}(N_{\tau^1})$ for $\sigma\in \Delta_X$ and
$\tau^1\in \Delta_B^1$.
\item Let $\Delta_X\to \Delta_B$ be a polyhedral map of rational conical
polyhedral complexes. Let $\Delta_B^1\to \Delta_B'\to \Delta_B$ be an
alteration, factored as a lattice alteration followed by a subdivision. The
induced alteration $\Delta_X^1\to \Delta_X$ is the induced lattice alteration
$\Delta_X^1\to \Delta_X'$ of the induced subdivision $\Delta_X'\to \Delta_X$.
\end{enumerate}\end{dfn}
Note that an alteration induced by a projective alteration is projective.
We are now ready to state our conjecture.
\begin{conj}\label{conj-comb-semistable}
Let $f_\Delta:\Delta_X\to \Delta_B$ be a polyhedral map of rational conical
polyhedral complexes, and assume for simplicity that $f_\Delta^{-1}(0) =
\{0\}$. Then there exists a projective alteration $\Delta_B^1\to \Delta_B$,
with induced
alteration $\Delta_X^1\to \Delta_X$, and a projective subdivision $\Delta_Y\to
\Delta_X^1$, such that $\Delta_Y \to \Delta_B^1$ is semistable.
\end{conj}
The conjectures are tied together by the following proposition:
\begin{prp}
Conjecture \ref{conj-comb-semistable} implies Conjecture \ref{conj-semistable}.
\end{prp}
{\bf Proof.} Let $X\to B$ be a surjective morphism of complex projective
varieties with geometrically integral generic fiber. By Theorem
\ref{th-toroidal-reduction} we may assume the morphism toroidal. Let
$f_\Delta:\Delta_X \to \Delta_B$ be the associated polyhedral map. By
theorem \ref{th-weak-semistable-reduction} we may assume $f_\Delta$ is weakly
semistable. We now invoque Conjecture \ref{conj-comb-semistable}. Let
$\Delta_B^1\to \Delta_B'\to\Delta_B$ be a projective alteration, $\Delta_X^1\to
\Delta_X$ the induced alteration, and $\Delta_Y\to
\Delta_X^1$ a projective subdivision such that $\Delta_Y \to \Delta_B^1$ is
semistable. The subdivision $\Delta_B'\to\Delta_B$ and the induced subdivision
$\Delta_X'\to\Delta_X$ give rise to modifications
$B'\to B$ and $X'\to X$ with a lifting $X'\to B'$ (see lemma
\ref{lem-lifting}). Let $\Delta_B^0\to \Delta_B'$ be the lattice alteration
given by the sublattice generated by the primitive vectors, and $\Delta_X^0\to
\Delta_X'$ the induced lattice alteration. Proposition
\ref{prop-cartier} gives rise to an alteration $\widetilde{B_0} \to B'$, with
pullback $\widetilde{X_0} \to X'$, and a
polyhedral map $\Delta_{\widetilde{B_0}}\to \Delta_B^0$ which is an isomorphism
on each cone. The same holds for $\Delta_{\widetilde{X_0}}\to \Delta_X^0$. The
lattice alteration $\Delta_B^1\to \Delta_B'$, being of index 1, factors through
$\Delta_B^0$. Proposition \ref{prop-compl} gives an alteration
$\widetilde{B_1}\to\widetilde{B_0}$ with pullback
$\widetilde{X_1}\to\widetilde{X_0}$ and a polyhedral map
$\Delta_{\widetilde{B_1}}\to \Delta_B^1$ which is an
isomorphism
on each cone, and the same holds for $\Delta_{\widetilde{X_1}}\to
\Delta_X^1$. The subdivision $\Delta_Y\to \Delta_X^1$ induces a subdivision
$\widetilde{\Delta_{Y}}\to\Delta_{\widetilde{X_1}}$, giving rise to a toroidal
modification $\widetilde{Y}\to \widetilde{X_1}$. Since $\Delta_Y\to
\Delta_B^1$ is semistable, we have that
$\widetilde{\Delta_{Y}}\to\Delta_{\widetilde{B_1}}$ is semistable as well, and
therefore $\widetilde{Y}\to\widetilde{B_1}$ is semistable, as required. \qed
\subsection{How far can we push the results?} In \cite{ar}, it is shown that if
$\Delta_0\subset \Delta$ is a subcomplex, and $\Delta_0'\to \Delta_0$ is a {\em
projective} triangulation, then there is a projective triangulation $\Delta'\to
\Delta$
extending $\Delta_0'$, without new edges: ${\Delta'}^{(1)} = {\Delta}^{(1)}
\cup {\Delta_0'}^{(1)}$. Applying the main theorem of \cite{te}, Chapter III,
it is shown in \cite{ar} that given a weakly semistable $X\to B$, one can
locally find a finite
map $B_1\to B$ and a modification $X_2\to X\times_BB_1$ such that $X_2\to B_1$
is semistable in codimension 1. If we apply Kawamata's trick, then clearly this
can be done globally. Moreover, $X_2$ has only quotient singularities.
A weakly
semistable morphism is said to be {\bf almost semistable} if it is semistable
in codimension 1 and moreover $X$ has quotient singularities. Thus, in theorem
\ref{th-weak-semistable-reduction} we may replace ``weakly semistable'' by
``almost semistable''. An analogous definition can be made on the polyhedral
side.
It is important to note that an almost semistable morphism is not necessarily
semistable. It is easy to give a polyhedral example: let $\tau = ({\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}_+)^2$ be
endowed with the standard lattice ${\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}^2$, and let $\sigma = ({\Bbb{R}}} \newcommand{\bfo}{{\Bbb{O}}_+)^4$
be given the lattice generated by ${\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}^4$ and the vector
$w=(1/2,1/2,1/2,1/2)$.
We have a polyhedral map $\sigma\to \tau$ given by $(a,b,c,d)\mapsto (a+b,
c+d)$. It is easy to see that this is almost semistable, but not semistable,
since $\sigma$ has index 2. Needless to say, a corresponding toroidal example
can be easily constructed as well.
The example we just gave is easy to amend. Indeed, if we subdivide $\tau$ at
its barycenter $(1,1)$, take the induced subdivision of $\sigma$, then its
star subdivision centered at $w$, and extend this to a triangulation using
\cite{ar}, we obtain a semistable map. This can be extended to families of
surfaces in general - the main observation (see \cite{wang} for the case
$\dim(B)=1$) is that one can use Pick's theorem and subdivide, with no need
for additional lattice alteration. We plan to pursue this elsewhere.
One last remark: the second author has shown, that in order to prove semistable
reduction, it is sufficient to produce $B_1$ and $Y$ such that $Y\to B_1$
satisfies all but condition \ref{item-reduced} of the requirements for
semistability in Definition \ref{dfn-comb-semi}. Again, this will be pursued
elsewhere.
|
1997-07-10T11:37:16 | 9707 | alg-geom/9707011 | en | https://arxiv.org/abs/alg-geom/9707011 | [
"alg-geom",
"math.AG"
] | alg-geom/9707011 | Dionisi Carla | Carla Dionisi | The tangent space at a special symplectic instanton bundle on P^{2n+1} | Latex, 11 pages, to appear in Annali di Matematica | null | null | null | null | Let $MI_{Simp,P^{2n+1}}(k)$ be the moduli space of stable symplectic
instanton bundles on $P^{2n+1}$ with second Chern class $c_2=k$ (it is a closed
subscheme of the moduli space $MI_{P^{2n+1}}(k)$), We prove that the dimension
of its Zariski tangent space at a special (symplectic) instanton bundle is
$2k(5n-1)+4n^2-10n+3, k\geq 2$. It follows that special symplectic instanton
bundles are smooth points for $ k \leq 3 $
| [
{
"version": "v1",
"created": "Thu, 10 Jul 1997 09:43:34 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Dionisi",
"Carla",
""
]
] | alg-geom | \section*{Introduction}
Symplectic instanton bundles on $\mbox{{I \hspace{-,18cm}P}}^{2n+1}$ are holomorphic bundles of rank 2n
(see \cite{A} ,\cite{MS} and \cite{OS}) that correspond to the self-dual solutions of
Yang-Mills equations on $\mbox{{I \hspace{-,18cm}P}}^n(\mbox{{I \hspace{-.18cm}H}})$. \
They are given by some monads (see section 2
for precise definitions) and their only topological invariant is $c_2=k$.\\
At present the dimension of their moduli space $MI_{\mbox{Simp},\mbox{{I \hspace{-,18cm}P}}^{2n+1}}(k)$
is not known except in the cases n=1, where the dimension is $8k-3$
(see \cite{HN}), and in
few other cases corresponding to small values of k.\\
$MI_{\mbox{Simp},\mbox{{I \hspace{-,18cm}P}}^{2n+1}}(k)$ is a closed subscheme of
$MI_{\mbox{{I \hspace{-,18cm}P}}^{2n+1}}(k)$ and this last scheme parametrizes stable instanton
bundles with structural group $GL(2n)$.\\
The class of special instanton bundles was introduced in \cite{ST}. \\
Let $E \in MI_{\mbox{{I \hspace{-,18cm}P}}^{2n+1}}(k)$ be a special symplectic instanton bundle. The tangent
dimension $h^1(End(E))$ was computed in \cite{OT} and it is equal to
$4(3n-1)k+(2n-5)(2n-1)$ .\\
The Zariski tangent space of $MI_{\mbox{Simp},\mbox{{I \hspace{-,18cm}P}}^{2n+1}}(k)$
at $E$ is isomorphic to $H^1(S^2E)$
and in this paper we prove that
\begin{equation}
\label{zero}
h^2(S^2E)=\left(\begin{array}{c} k-2 \\ 2 \end{array} \right) \cdot
\left(\begin{array}{c} 2n-1 \\ 2 \end{array} \right) \qquad \forall \ k \geq 2
\end{equation}
By the Hirzebruch-Riemann-Roch formula , since $h^0(S^2E)=0$ and $h^i(S^2E)=0 \
\forall\ i\geq3$, it follows that:
\begin{center}
$\chi(S^2E)=h^2(S^2E)-h^1(S^2E)=2n^2+n+\frac{1}{2}\left[ k^2\left( \begin{array}{c} 2n-1 \\ 2 \end{array} \right) -k(10n^2-5n-1) \right] $
\end{center}
and
\begin{theorem} \label{acca1}
Let $E$ be a special symplectic instanton bundle.Then
$$h^1(S^2E)=2k(5n-1)+4n^2-10n+3 \qquad ,k\geq 2 $$
\end{theorem}
(for $n=1$ it is well known that \ $ h^1(S^2E)=8k-3$ ).\\
Now, since by the Kuranishi map $H^2(S^2E)$ is the space
of obstructions to the smoothness at $E$ of $MI_{\mbox{Simp},\mbox{{I \hspace{-,18cm}P}}^{2n+1}}(k)$,
we obtain
\begin{cor}
$\forall k\geq 2$
the dimension of any irreducible component of
$MI_{\mbox{Simp},\mbox{{I \hspace{-,18cm}P}}^{2n+1}}(k)$,
containing a special symplectic instanton bundle is bounded by the value \\
$2k(5n-1)+4n^2-10n+3 \quad $ ( linear in k )
\end{cor}
\begin{cor}
$\forall n$ \ $MI_{\mbox{Simp},\mbox{{I \hspace{-,18cm}P}}^{2n+1}}(3)$ is smooth at a special instanton
bundle $E$,and the dimension of any irreducible component containing $E$ is
$ 4n^2+20n-3 $.
\end{cor}
The main remark of this paper is that it is easier to compute $H^2(S^2E)$
and $H^2(\stackrel{2}{\wedge} E)$ together as SL(2)-modules (although this second cohomology
space has a geometrical meaning only for orthogonal bundles) than to compute
$H^2(S^2E)$ alone.
\section{Preliminaries}
Throughout this paper $\mbox{{\itshape I \hspace{-.38cm} K}}$ denotes an algebraically closed field of characteristic
zero.
$U$ denotes a 2-dimensional $\mbox{{\itshape I \hspace{-.38cm} K}}$ vector space \((U=<s,t>)\), \ $S_n=S^nU$
its n-th symmetric power $\ms{0.2}(S_n=<s^n, s^{n-1}t,\ldots, t^n>$) ,
$V_n=U\otimes S_n $ $\ms{0.2}(V_n=<s\otimes s^n, s\otimes s^{n-1}t,\ldots s\otimes t, \ldots t\otimes t^n>)$ \
and $\mbox{{I \hspace{-,18cm}P}}^{2n+1}=\mbox{{I \hspace{-,18cm}P}}(V_n)$.
\begin{defin}
\label{def 1.1}
A vector bundle $E$ on $\mbox{{I \hspace{-,18cm}P}}^{2n+1}$ of rank $2n$ is called an {\bf instanton bundle of
quantum number $k$} if:
\begin{itemize}
\pagebreak
\item $E$ has Chern polinomial $c_t(E) \ = \ (1-t^2)^{-k}$;
\item $E(q)$ has natural cohomology in the range $-(2n+1) \ \leq \ q \ \leq 0$,
that is $H^i(E(q)) \ \neq \ 0$ for at most one $i = i(q)$.
\end{itemize}
\end{defin}
By \cite{OS},\cite{AO1},the Definition ~\ref{def 1.1} is
equivalent to : \\
i)$E$ is the cohomology bundle of a monad:
\[
0 \rightarrow O(-1)^k \rightarrow \Omega^1(1)^k \rightarrow O^{2n(k-1)} \rightarrow 0
\]
or ii) $E$ is the cohomology bundle of a monad:
\[
0 \rightarrow O(-1)^k \fr{A} O^{2n+2k} \fr{B^t} O(1)^k \rightarrow 0
\]
(where, after we have fixed a coordinate system, A and B can be identified
with matrices in the space
$Mat(k,2n+2k,S_1)$)
\begin{defin}
An instanton bundle $E$
is called {\bf symplectic} if there is
an isomorphism $\varphi:E \rightarrow E^ {\vee}$ satisfying
$\varphi = -\varphi^{\vee}$.
\end{defin}
\begin{defin}
An instanton bundle is called {\bf special} if it arises from a monad
where the morfism $B^t$
is defined in some system of homogeneous coordinates $x_0, \cdots x_n, y_0 \cdots y_n$
on $\mbox{{I \hspace{-,18cm}P}}^{2n+1}$ by the trasposed of the matrix:
\[
B = \left(
\begin{array}{cccccccccccc}
x_0 & \cdots & x_n & 0 & \cdots & 0 & y_0 & \cdots & y_n & 0 & \cdots & 0 \\
0 & x_0 & \cdots & x_n & 0 & \cdots & 0 & y_0 & \cdots & y_n & 0 & \cdots \\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\
\cdots & 0 & x_0 & \cdots & x_n & 0 & \cdots & 0 & y_0 & \cdots & y_n & 0 \\
0 & \cdots & 0 & x_0 & \cdots & x_n & 0 & \cdots & 0 & y_0 & \cdots & y_n
\end{array}
\right)
\] \\
\end{defin}
The following lemma is well known (and easy to prove)
\pagebreak
\begin{lemma}
$$H^{0} (O(1)) \cong V^{\vee}$$
$$H^{0}(\Omega^1(2)) \cong \stackrel{2}{\wedge} V^{\vee}$$
$$H^i (\mbox{{I \hspace{-,18cm}P}}^n, S^2 \Omega^1(1)) = \left \{ \begin{array}{ll}
0 & \mbox{se } i\neq 1 \\
\stackrel{2}{\wedge} V^{\vee} & \mbox{se } i= 1
\end{array} \right.$$
\end{lemma}
\section{Existence of a special symplectic instanton bundle}
There is a natural exact sequence of GL(U)-equivariant maps for any $k , n\geq 1$
(Clebsch-Gordan sequence):
\begin{equation}
\label{uno_3}
0 \rightarrow \stackrel{2}{\wedge} U \otimes S_{k-1} \otimes V_{n-1} \fr{\beta} S_k \otimes V_n \fr{\mu} V_{k+n} \rightarrow 0
\end{equation}
where $\mu$ is the multiplication map and $\beta$ is defined by
$(s \wedge t) \otimes f \otimes g \rightarrow (sf \otimes tg - tf \otimes sg)$
We can define (see \cite{OT}) the morphism \\
\centerline{
$\ac{b}: S_{k-1}^{\vee} \otimes \Omega^1(1) \rightarrow \stackrel{2}{\wedge} U^{\vee} \otimes S_{k-2}^{\vee} \otimes V_{n-1}^{\vee} \otimes O$
}
and it is induced the complex
\begin{equation}
\label{cinque_3}
A \otimes O(-1) \fr{\ac{a}} S^{\vee}_{k-1} \otimes \Omega^1(1) \fr{\ac{b}} \stackrel{2}{\wedge} U^{\vee} \otimes S^{\vee}_{k-2} \otimes V_{n-1}^{\vee} \otimes O
\end{equation}
where A is a k-dimensional subspace of $S^{\vee}_{2n+k-1} \otimes \stackrel{2}{\wedge} U^{\vee}$
such that ~\refeq{cinque_3} is a monad and the cohomology bundle $E$ is a special symplectic
instanton bundle.
It was proved in \cite{OT} that
$$ H^2(EndE)\cong Ker(\Phi^{\vee})^{\vee}$$
where
\centerline{
$\Phi^{\vee}: S_{k-2}^{\otimes 2} \otimes V_{n-1}^{\otimes 2} \rightarrow S_{k-1}^{\otimes 2} \otimes \stackrel{2}{\wedge} V_{n}$
}
and there is an isomorphism of
SL(2)-representations
$$
\varepsilon : S^{\vee}_{k-3} \otimes S^{\vee}_{k-3} \otimes S^2V^{\vee}_{n-2} \rightarrow Ker(\Phi^{\vee})$$
\section{How to identify $H^2(S^2E)$ and $H^2(\stackrel{2}{\wedge} E)$}
\begin{prop}
Let $E$ be special symplectic instanton bundle ,cohomology of monad
~\refeq{cinque_3} and $N=Ker \ac{b}$.Then
\begin{description}
\item[(i)] $H^2 (S^2E) \cong H^2(S^2N) $
\item[(ii)] $H^2 (\stackrel{2}{\wedge} E) \cong H^2(\stackrel{2}{\wedge} N) $
\end{description}
\end{prop}
\begin{proof} \\
We denote $B:= S_{k-1}^{\vee}$ \qquad and \qquad $C:=\stackrel{2}{\wedge} U^{\vee} \otimes S_{k-2}^{\vee} \otimes V_{n-1}^{\vee} $ \\
The result follows from the two exact sequences given by
monad ~\refeq{cinque_3} :
\begin{equation}
\label{cinque_4}
0 \rightarrow N \rightarrow B \otimes \Omega^1 (1) \rightarrow C \otimes O \rightarrow 0
\end{equation}
\begin{equation}
0 \rightarrow A \otimes O(-1) \rightarrow N \rightarrow E \rightarrow 0
\end{equation}
In fact, by performing the second symetric and alternating power
of sequence ~\refeq{cinque_4}, we have
\centerline{
\begin{minipage}{2in}
\begin{tabbing}
$ 0 \rightarrow S^2 N \rightarrow $\=$\tilde{A}$\=$\rightarrow B \otimes C \otimes \Omega^1(1) \rightarrow \stackrel{2}{\wedge} C \otimes O \rightarrow 0$\\
\> \>$\searrow$\=\hspace{0.5 cm}$\nearrow$\=\\
\> \> \> $ M^1$\\
\> \>$\nearrow$\hspace{0.5 cm}$\searrow$\\
\> 0 \> \> \> 0
\end{tabbing}
\end{minipage}
}
\begin{equation}
\label{DUE}
\end{equation}
where $\tilde{A}:= S^2(B \otimes \Omega^1(1)) = (S^2 B \otimes S^2(\Omega^1(1))) \oplus ( \stackrel{2}{\wedge} B \otimes \Omega^2(2))$ \\
and \\
\centerline{
\begin{minipage}{2in}
\begin{tabbing}
$ 0 \rightarrow \stackrel{2}{\wedge} N \rightarrow $\=$\overline{A} $\=$\ \rightarrow B \otimes C \otimes \Omega^1(1) \rightarrow O \otimes S^2 C \rightarrow 0$\\
\> \>$\searrow$\=\hspace{0.5 cm}$\nearrow$\=\\
\> \> \> $ M$\\
\> \>$\nearrow$\hspace{0.5 cm}$\searrow$\\
\> 0 \> \> \> 0
\end{tabbing}
\end{minipage}
}
\begin{equation}
\label{TRE}
\end{equation}
where $\overline{A}:= \stackrel{2}{\wedge}(B \otimes \Omega^1(1)) = (\stackrel{2}{\wedge} B \otimes S^2(\Omega^1(1))) \oplus ( S^2 B \otimes \Omega^2(2))$ \\
\end{proof}
\subsection{Identifying $ H^2(S^2 N)$ and $H^2 ( \stackrel{2}{\wedge} N) $}
i)
\label{par43}
Diagram ~\refeq{DUE} gives the following two exact sequences:\\
\begin{minipage}{5 in}
\begin{equation}\label{quattro4}
O \rightarrow H^0(M^1) \rightarrow H^1(S^2N) \rightarrow H^1(\ac{A}) \rightarrow H^1(M^1) \rightarrow H^2(S^2(N)) \rightarrow H^2(\ac{A}) \rightarrow \cdots
\end{equation}
\begin{tabbing}
$O \rightarrow H^0(M^1) \rightarrow B \otimes C \otimes H^0$\=$(\Omega^1(1))\rightarrow \stackrel{2}{\wedge} C\rightarrow H^1(M^1) \rightarrow B \otimes C \otimes H^1$\=$(\Omega^1(1))\rightarrow \cdots$\\
\>$\parallel$\> $\parallel$\\
\>$0 $ \> $0$
\end{tabbing}
\begin{equation}\label{cinque4}
\end{equation}
\end{minipage}
Sequence \refeq{cinque4} implies:
$H^0(M^1) = 0$ \quad and \quad $H^1(M^1) \cong \stackrel{2}{\wedge} C$
Then, by using the two formulas:
$H^1(\tilde{A}) = (S^2 B \otimes H^1(S^2 \Omega^1(1))) \oplus ( \stackrel{2}{\wedge} B \otimes H^1(\Omega^2(2))
=S^2 B \otimes \stackrel{2}{\wedge} V^{\vee}$
and:\
$H^2(\tilde{A})= (S^2 B \otimes H^2(S^2 \Omega^1(1))) \oplus ( \stackrel{2}{\wedge} B \otimes H^2(\Omega^2(2)) =0$
sequence ~\refeq{quattro4} becomes:
\[0 \rightarrow H^1(S^2N) \rightarrow H^1(\ac{A}) \rightarrow H^1(M^1) \rightarrow H^2(S^2(N)) \rightarrow 0\]
i.e.\[0 \rightarrow H^1(S^2 N) \rightarrow S^2 B \otimes \stackrel{2}{\wedge} V^{\vee} \fr{\ac{\Phi}} \stackrel{2}{\wedge} C \rightarrow H^2(S^2N) \rightarrow 0 \]
\[ \Longrightarrow \qquad H^2 ( S^2 N ) \cong \mbox{Coker}(\ac{\Phi}) = (\mbox{Ker}(\ac{\Phi}^{\vee}))^{\vee} \]
Then:
\[H^2 ( S^2 N) ^{\vee} = \mbox{Ker} \left[ \stackrel{2}{\wedge} (S_{k-2} \otimes V_{n-1} ) \fr{\ac{\Phi}^{\vee}} S^2(S_{k-1}) \otimes \stackrel{2}{\wedge} V_n\right]\]
ii)
Diagram ~\refeq{TRE} gives the following two exact sequences:
\begin{minipage}{5in}
\begin{equation}\label{sei4}
O \rightarrow H^0(M) \rightarrow H^1(\stackrel{2}{\wedge} N) \rightarrow H^1(\overline{A}) \rightarrow H^1(M) \rightarrow H^2(\stackrel{2}{\wedge} N) \rightarrow H^2(\overline{A}) \rightarrow \cdots
\end{equation}
\begin{tabbing}
$O \rightarrow H^0(M) \rightarrow B \otimes C \otimes H^0$\=$(\Omega^1(1))\rightarrow S^2 C $\=$\otimes H^0(O) \rightarrow H^1(M) \rightarrow 0 \rightarrow \cdots$\\
\>$\parallel$\> $\parallel$\\
\>$0 $ \> $S^2 C$
\end{tabbing}
\begin{equation}\label{sette4}
\end{equation}
\end{minipage}\\
and, from sequence ~\refeq{sette4}, we get \\
$H^0(M) = 0$ \quad and \quad $H^1(M) \simeq S^2 C$ \\
Then, since :\\
$H^1(\overline{A})=(H^1(S^2(\Omega^1(1))\otimes \stackrel{2}{\wedge} B) \oplus (S^2 B \otimes H^1(\Omega^2(2)))=
\stackrel{2}{\wedge} B \otimes \stackrel{2}{\wedge} V^{\vee}$ \\
and \qquad $H^2(\overline A) = 0$\\
sequence ~\refeq{sei4} becomes :
\begin{minipage}{5 in}
\begin{tabbing}
$O \rightarrow H^0$\=$(M) \rightarrow H^1(\stackrel{2}{\wedge} N)$\=$ \rightarrow H^1(\overline{A}) \rightarrow H^1(M) \rightarrow H^2(\stackrel{2}{\wedge} N) \rightarrow 0$\\
\>$\parallel$\\
\>0
\end{tabbing}
\end{minipage}
i.e. \qquad
$0 \rightarrow H^1(\stackrel{2}{\wedge} N) \rightarrow \stackrel{2}{\wedge} B \otimes \stackrel{2}{\wedge} V^{\vee} \fr{\overline{\Phi}} S^2 C \rightarrow H^2(\stackrel{2}{\wedge} N ) \rightarrow 0 $ \\
\[
\Longrightarrow \qquad H^2 ( \stackrel{2}{\wedge} N ) \cong \mbox{Coker}(\overline{\Phi}) = (\mbox{Ker}(\overline{\Phi}^{\vee}))^{\vee} \]
Then we obtain :
\[(H^2(\stackrel{2}{\wedge} N))^{\vee} = \mbox{Ker} \left[ S^2(S_{k-2}\otimes V_{n-1})\fr{\overline{\Phi}^{\vee}}\stackrel{2}{\wedge} S_{k-1}\otimes \stackrel{2}{\wedge} V_n \right]\]
\subsection{\bf Identifying $H^2(S^2 E)$}
We have \qquad \qquad $H^2(S^2E)^{\vee} \cong \mbox{Ker} \ \ac{\Phi}^{\vee}$ \\
where \qquad
$\ac{\Phi}^{\vee} : \stackrel{2}{\wedge}(S_{k-2}\otimes V_{n-1}) \rightarrow S^2 S_{k-1} \otimes \stackrel{2}{\wedge} V_n$ \\
is explicitly given by
\begin{tabbing}
$\ac{\Phi}^{\vee}((g \otimes v)\wedge (g^1 \otimes v^1)) =$\=$sg \cdot sg^1 \otimes (tv \wedge tv^1)-sg\cdot tg^1 \otimes (tv \wedge sv^1)+$\\
\>$-tg\cdot sg^1 \otimes (sv \wedge tv^1)+tg\cdot tg^1 \otimes (sv \wedge sv^1) $ \\
\end{tabbing}
i.e. \qquad $ \ac{\Phi}^{\vee} = \ac{p} \circ (\stackrel{2}{\wedge}\beta)$ \\
where \qquad
$\beta : \ \ \stackrel{2}{\wedge} U \otimes S_{k-2} \otimes V_{n-1} \rightarrow S_{k-1} \otimes V_n \; \; \; $
is such that
\[
(s \wedge t) \otimes (g \otimes v) \mapsto (sg \otimes tv) - (tg \otimes sv)
\]
and \begin{tabbing}
$\ac{p} \ : \ $\=$\stackrel{2}{\wedge}($\=$S_{k-1} \otimes V_n) \rightarrow S^2 S_{k-1} \otimes \stackrel{2}{\wedge} V_n$ \\
\>\>$\|$ \\
\>$(\stackrel{2}{\wedge} S_{k-1} \otimes S^2V_n) \oplus (S^2S_{k-1} \otimes \stackrel{2}{\wedge} V_n)$\\
\end{tabbing}
is such that
$$
(f \otimes u) \wedge (f' \otimes u^1) \mapsto f\cdot f' \otimes u \wedge u^1 \mbox{.}
$$
Now, we consider the $SL(2)$-equivariant morphism:\\
\[
\ac{\varepsilon}^1 \ : \ \stackrel{2}{\wedge} (S_{k-3} \otimes V_{n-2}) \rightarrow \stackrel{2}{\wedge}(S_{k-2} \otimes V_{n-1})
\]
where, up to the order of factors, the map
$\ac{\varepsilon}^1 := \beta^1 \wedge \beta^1$ and $\beta^1:S_{k-3} \otimes V_{n-2} \rightarrow S_{k-2} \otimes V_{n-1}$
is defined as $\beta$. Hence, $\ac{\varepsilon}^1 $ is injective.
Finally, we define
\[
\ac{\varepsilon} \ : \ \stackrel{2}{\wedge} S_{k-3} \otimes S^2V_{n-2} \rightarrow \stackrel{2}{\wedge}(S_{k-2} \otimes V_{n-1})
\]
as $\ac{\varepsilon} = \ac{\varepsilon}^1 \circ \ac{i}$, \ where
\begin{tabbing}
$\ac{i} \ :\ $\=$ \stackrel{2}{\wedge} S_{k-3} \otimes S^2V_{n-2} \rightarrow \stackrel{2}{\wedge}(S_{k-3} \otimes V_{n-2}) \ \ $ such that \\
\> $f \wedge f' \otimes u\cdot u^1 \longmapsto (f \otimes u) \wedge (f' \otimes u^1) + (f \otimes u^1) \wedge (f' \otimes u)$\\
\end{tabbing}
is an injective map. Then,
also $\ac{\varepsilon}$ is injective.
\begin{lem}
\label{lemma442}
Im $ \ac{\varepsilon} \subset \mbox{Ker} \ \ac{\Phi}^{\vee}$
\end{lem}
\begin{proof}
Straightforward computation.
\end{proof}
\subsection {\bf Identifying $H^2(\stackrel{2}{\wedge} E)$}
We have $$H^2(\stackrel{2}{\wedge} E)^{\vee} \cong \mbox{Ker} \ \ol{\Phi}^{\vee}$$
where \qquad
$
\ol{\Phi}^{\vee} \ : \ S^2(S_{k-2}\otimes V_{n-1}) \rightarrow \stackrel{2}{\wedge} S_{k-1} \otimes \stackrel{2}{\wedge} V_n
$ \qquad
is explicity given by
\begin{tabbing}
$\ol{\Phi}^{\vee}((g \otimes v)\cdot (g^1 \otimes v^1)) =$\=$sg \wedge sg^1 \otimes (tv \wedge tv^1) - sg \wedge tg^1 \otimes(tv \wedge sv^1)$\\
\>$-tg \wedge sg^1 \otimes (sv \wedge tv^1)+(tg \wedge tg^1) \otimes (sv \wedge sv^1)$
\end{tabbing}
i.e. \qquad
$\ol{\Phi}^{\vee} = \ol{p} \circ(S^2 \beta)$ \qquad where
\pagebreak
\begin{tabbing}
$\ol{p} \ : \ $\=$S^2($\=$S_{k-1}\otimes V_n) \rightarrow \stackrel{2}{\wedge} S_{k-1} \otimes \stackrel{2}{\wedge} V_n$\\
\>\>$\|$\\
\>$(\stackrel{2}{\wedge} S_{k-1}\otimes \stackrel{2}{\wedge} V_n) \oplus (S^2 S_{k-1} \otimes S^2 V_n)$
\end{tabbing}
is such that
\[
\ol{p}((f \otimes u) \cdot (f' \otimes u^1)) = f \wedge f' \otimes u \wedge u^1
\]
We consider the $SL(2)$-equivariant morphism:
\[
\ol{\varepsilon}^1 \ :\ S^2(S_{k-3} \otimes V_{n-2}) \rightarrow S^2 (S_{k-2} \otimes V_{n-1})
\]
such that:
\begin{tabbing}
$\ol{\varepsilon}^1((f \otimes u) \cdot (f' \otimes u^1)) =$\=$(sf \otimes tu) \cdot(sf' \otimes tu^1)-(sf \otimes su) \cdot (tf' \otimes tu^1)+$\\
\>$-(tf \otimes tu) \cdot (sf' \otimes su^1) + (sf \otimes tu) \cdot (sf' \otimes tu^1)$
\end{tabbing}
($\ol{\varepsilon}^1 = S^2 \beta^1 $ hence $ \ol{\varepsilon}^1$ is injective).
Finally, we define
\[
\ol{\varepsilon} \ : \ S^2 S_{k-3} \otimes S^2 V_{n-2} \rightarrow S^2 (S_{k-2} \otimes V_{n-1})
\]
as $\ol{\varepsilon} = \ol{\varepsilon}^1 \circ \ol{i}$ \quad where
\begin{tabbing}
$\ol{i}\ : \ $\=$ S^2 S_{k-3} \otimes S^2V_{n-2} \rightarrow S^2 (S_{k-3} \otimes V_{n-2})$ \ such that \\
\>$f \cdot f' \otimes uu^1 \mapsto (f \otimes u)(f' \otimes u^1) + (f \otimes u^1)(f' \otimes u)$
\end{tabbing}
is an injective map. Then, also $\ol{\varepsilon}$ is injective
\begin{lem}
\label{lemma443}
Im $ \ol{\varepsilon} \subset \mbox{Ker} \ \ol{\Phi}^{\vee}$
\end{lem}
\begin{proof}
Straightforward computation.
\end{proof}
\begin{theorem}
For any special symplectic instanton bundle $E$
\[
H^2(S^2E)\ \simeq\ \stackrel{2}{\wedge} (S_{k-3})^{\vee} \otimes S^2(V_{n-2})^{\vee}\]
\end{theorem}
\begin{proof}
By lemma ~\ref{lemma442} and ~\ref{lemma443} we have the following diagram
with exact rows and columns:
\begin{minipage}{4in}
\begin{tabbing}
$0 \rightarrow H^2(\stackrel{2}{\wedge}$\=$ N)^{\vee} \rightarrow S^2(S_{k-2} \otimes$\=$V_{n-1}) \fr{\ol{\Phi}^{\vee}} \stackrel{2}{\wedge} S_{k-1} \otimes$\=$\stackrel{2}{\wedge} V_n \rightarrow H^1(\stackrel{2}{\wedge}$\=$N)^{\vee}\rightarrow 0$ \kill
\> $0$ \> $0$ \> $0$ \> $0$\\
\>$\downarrow$ \> $\downarrow$ \> $\downarrow$\> $\downarrow$\\
$0 \rightarrow H^2(\stackrel{2}{\wedge} N)^{\vee} \rightarrow S^2(S_{k-2} \otimes V_{n-1}) \fr{\ol{\Phi}^{\vee}} \stackrel{2}{\wedge} S_{k-1} \otimes \stackrel{2}{\wedge} V_n \rightarrow H^1(\stackrel{2}{\wedge} N)^{\vee}\rightarrow 0$ \\
\>$\downarrow$ \> $\downarrow$ \> $\downarrow$\> $\downarrow$\\
$0 \rightarrow H^2(N \otimes N)^{\vee} \rightarrow S_{k-2} ^{\otimes 2} \otimes V_{n-1}^{\otimes 2} \fr{\Phi^{\vee}} S_{k-1}^{\otimes 2} \otimes \stackrel{2}{\wedge} V_n \rightarrow H^1(N \otimes N)^{\vee}\rightarrow 0$ \\
\>$\downarrow$ \> $\downarrow$ \> $\downarrow$\> $\downarrow$\\
$0 \rightarrow H^2(S^2 N)^{\vee} \rightarrow \stackrel{2}{\wedge} (S_{k-2} \otimes V_{n-1}) \fr{\ac{\Phi}^{\vee}} S^2 S_{k-1} \otimes \stackrel{2}{\wedge} V_n \rightarrow H^1(S^2 N)^{\vee}\rightarrow 0$ \\
\>$\downarrow$ \> $\downarrow$ \> $\downarrow$\> $\downarrow$\\
\> $0$ \> $0$ \> $0$ \> $0$\\
\end{tabbing}
\end{minipage} \\
It was shown in \cite{OT} that:
$H^2(EndE) \simeq{Ker}\ \Phi^{\vee} = H^2(N \otimes N)^{\vee} \simeq S_{k-3}^{\otimes 2} \otimes S^2 V_{n-2}$
We have proved that there are two injective maps:
\[
\ac{\varepsilon}\ : \ \stackrel{2}{\wedge} (S_{k-3}) \otimes S^2V_{n-2} \rightarrow \mbox{Ker} \ \ac{\Phi}^{\vee} \simeq H^2(S^2N)^{\vee} \simeq H^2(S^2E)^{\vee}
\]
\[
\ol{\varepsilon}\ : \ S^2(S_{k-3}) \otimes S^2 V_{n-2} \rightarrow \mbox{Ker} \ \ol{\Phi}^{\vee} \simeq H^2(\stackrel{2}{\wedge} N)^{\vee} \simeq H^2(\stackrel{2}{\wedge} E)^{\vee}
\]
Then, we can consider the following diagram:
\begin{tabbing}
$0 \rightarrow S^2 S_{k-3} $\=$ \otimes S^2 V_{n-2} \rightarrow S_{k-3}^{\otimes 2} \otimes $\=$ S^2V_{n-2} \rightarrow \stackrel{2}{\wedge} S_{k-3}$\=$\otimes V_{n-2}$\kill
\> $0$ \> $0$ \> $0$ \\
\> $\downarrow$ \> $\downarrow$ \> $\downarrow$ \\
$0 \rightarrow S^2 S_{k-3} \otimes S^2 V_{n-2} \rightarrow S_{k-3}^{\otimes 2} \otimes S^2V_{n-2} \rightarrow \stackrel{2}{\wedge} S_{k-3} \otimes S^2V_{n-2} \rightarrow 0 $\\
\> $\downarrow \ol{\varepsilon}$ \> $\downarrow \varepsilon$ \> $\downarrow \ac{\varepsilon}$ \\
\ \ \ \ \ \ \ \ \ $0\rightarrow $ \> $H^2(\stackrel{2}{\wedge} E)^{\vee} \rightarrow H^2(EndE)^{\vee} \rightarrow H^2(S^2E)^{\vee} \rightarrow 0 $ \\
\>\> $\downarrow$\\
\>\> $0$\\
\end{tabbing}
and by the {\bf Snake-Lemma} there is the exact sequence :
\begin{tabbing}
$0 \rightarrow$ Ker \ \= $\ol{\varepsilon} \rightarrow $ Ker\ \= $\varepsilon \rightarrow $ Ker \= $\ac{\varepsilon} \rightarrow \mbox{Coker}\ \ol{\varepsilon} \rightarrow \mbox{Coker}\ $\=$
\varepsilon \rightarrow \mbox{Coker}\ \ac{\varepsilon} \rightarrow 0$\\
\> $\|$ \> $\|$ \> $\|$ \> $\|$ \\
\> $0$ \> $0$ \> $0$ \> $0$\\
\end{tabbing}
$
\Rightarrow \mbox{Coker} \ \ol{\varepsilon} = 0 \Rightarrow \ol{\varepsilon}\ \mbox{is an
isomorphism} \ \Rightarrow \ac{\varepsilon}\ \mbox{is an isomorphism}.
$ \\
Thus:
\[H^2(S^2E)^{\vee}\ \cong\ \stackrel{2}{\wedge} (S_{k-3}) \otimes S^2(V_{n-2})\]
i.e. \qquad \
$H^2(S^2E)\ \simeq\ \stackrel{2}{\wedge} (S_{k-3})^{\vee} \otimes S^2(V_{n-2})^{\vee}$ \quad as we wanted.\\
\end{proof}
\begin{oss} By
this theorem formula ~\ref{zero} and theorem ~\ref{acca1} are easily
proved.
\end{oss}
|
1997-07-09T16:28:38 | 9707 | alg-geom/9707005 | en | https://arxiv.org/abs/alg-geom/9707005 | [
"alg-geom",
"math.AG"
] | alg-geom/9707005 | Jesper Funch Thomsen | Jesper Funch Thomsen | Irreducibility of \bar{M}_{0,n}(G/P,\beta) | 8 pages, AmsLaTeX | null | null | null | null | Let G be a linear algebraic group, P be a parabolic subgroup of G and \beta
be a cycle of dimension 1 in the Chow group of the quotient G/P. Using
geometric arguments and Borel's fixed point theorem, we prove that the moduli
space \bar{M}_{0,n}(G/P, \beta) of n-pointed genus 0 stable maps representing
\beta is irreducible.
| [
{
"version": "v1",
"created": "Wed, 9 Jul 1997 14:28:33 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Thomsen",
"Jesper Funch",
""
]
] | alg-geom | \section{Introduction}
Let $G$ be a complex connected linear algebraic group,
$P$ be a parabolic subgroup of $G$ and $\beta \in A_1(G/P)$
be a 1-cycle class in the Chow group of G/P.
An $n$-pointed genus $0$ stable map into $G/P$ representing
the class $\beta$, consists of data $( \mu : C \rightarrow X ;
p_1, \dots , p_n )$, where $C$ is a connected, at most nodal,
complex projective curve of arithmetic genus $0$, and $\mu$
is a complex morphism such that $\mu_*[C] = \beta$ in $A_1(G/P)$.
In addition $p_i$, $i=1, \dots, n$ denote $n$ nonsingular
marked points on $C$ such that every component of $C$, which by
$\mu$ maps to a point, has at least 3 points which is either nodal
or among the marked points (this we will refer to as every component
of $C$ being stable). The set of $n$-pointed genus $0$ stable maps into
$G/P$ representing the class $\beta$, is parameterized by a coarse
moduli space $\overline{M}_{0,n}(G/P,\beta)$.
In general it is known that $\overline{M}_{0,n}(G/P,\beta)$ is a
normal complex projective scheme with finite quotient singularities.
In this paper we will prove that $\overline{M}_{0,n}(G/P,\beta)$ is
irreducible. It should also be noted that we in addition
will prove that the boundary divisors
in $\overline{M}_{0,n}(G/P,\beta)$
, usually denoted by
$D(A,B,\beta_1, \beta_2)$ ($\beta = \beta_1 + \beta_2$, $A \cup B$
a partition of $\{ 1, \dots , n \}$)
, are irreducible.
After this work was carried out we learned that B. Kim and
R. Pandharipande \cite{KimPan} had proven the same results, and even
proved connectedness of the corresponding moduli spaces
in higher genus.
Our methods however differs in many ways.
For example in this paper we consider the action of a Borel
subgroup of $G$ on $\overline{M}_{0,n}(G/P,\beta)$, while
Kim and Pandharipande mainly concentrate on maximal torus action.
Another important difference is that we in this presentation
proceed by induction on $\beta$. This means that
the question of $\overline{M}_{0,n}(G/P,\beta)$ being irreducible,
can be reduced to simple cases.
This work was carried out while I took part in the program
``Enumerative geometry and its interaction with theoretical
physics'' at the Mittag-Leffler Institute. I would like to
use this opportunity to thank the Mittag-Leffler Institute
for creating a stimulating atmosphere.
Thanks are also due to Niels Lauritzen and S{\o}ren Have
Hansen for useful discussions concerning the generalization
from full flag varieties to patial flag varieties.
\section{Summary on $\overline{M}_{0,n}(G/P,\beta)$}
\label{summary}
In this section we will summarize the properties
of the coarse moduli space $\overline{M}_{0,n}(G/P,\beta)$
which we will make use of. The notes on quantum cohomology
by W. Fulton and R. Pandharipande \cite{FulPan} will serve as our main
reference.
As mentioned in the introduction the moduli
space $\overline{M}_{0,n}(G/P,\beta)$ parameterizes $n$-pointed
genus $0$ stable maps into $G/P$ representing the class $\beta$.
By definition $\beta$ is effective if it is represented
by some $n$-pointed genus $0$ stable map.
In the following we will only consider values of
$n$ and $\beta$ where $\overline{M}_{0,n}(G/P,\beta)$ is non-empty.
This means $\beta$ must be effective and $n \geq 0$, and if
$\beta = 0$ we must have $n \geq 3$.
The moduli space $\overline{M}_{0,n}(G/P,\beta)$ is known to be a
normal projective scheme (see \cite{FulPan}). This implies that
$\overline{M}_{0,n}(G/P,\beta)$ splits up into a finite
disjoint union of its components. This we will use several times.
\subsection{Contraction morphism}
On $\overline{M}_{0,n+1}(G/P,\beta)$ we have a contraction morphism
$$ \overline{M}_{0,n+1}(G/P,\beta) \rightarrow \overline{M}_{0,n}(G/P,\beta)$$
which ``forget'' the $(n+1)$'th marked point. The contraction morphisms
value on a closed point in $\overline{M}_{0,n+1}(G/P,\beta)$, represented
by $(\mu : C \rightarrow G/P; p_1, \dots , p_{n+1})$, is the point in
$\overline{M}_{0,n}(G/P,\beta)$ represented by
$( \mu^{\circ} : C^{\circ} \rightarrow G/P; p_1 , \dots , p_n)$,
where $C^{\circ}$ denote $C$
with the unstable components collapsed, and $\mu^{\circ}$ is the
map induced from $\mu$.
From the construction of $\overline{M}_{0,n}(G/P,\beta)$
it follows, that the contraction map is a surjective map with connected
fibres.
\subsection{Evaluation map}
For each element $a \in \{1 , \dots n \} $ we have an evaluation map
$$ \delta_a :\overline{M}_{0,n}(G/P,\beta) \rightarrow G/P.$$
Its value on a closed point in $\overline{M}_{0,n}(G/P,\beta)$
corresponding to $(\mu :C \rightarrow G/P ; p_1, \dots , p_n)$ is
defined to be $\mu(p_a)$.
\subsection{Boundary}
By a boundary point in $\overline{M}_{0,n}(G/P,\beta)$ we will mean
a point which correspond to a reducible curve. Let
$ A \cup B =\{1, \dots n \}$ be a partition of $\{ 1, \dots n \} $
in disjoint sets,
and let $\beta_1, \beta_2 \in A_1(X)$ be effective classes such that
$\beta = \beta_1 + \beta_2$.
We will only consider the cases when $\beta_1 \neq 0$ (resp.
$\beta_2 \neq 0$) or $|A| \geq 2$ (resp. $|B| \geq 2$).
With these conditions on $\beta_1, \beta_2, A$ and $B$
we let $ D(A,B,\beta_1, \beta_2)$
denote the set of elements in $\overline{M}_{0,n}(G/P,\beta)$ where the
corresponding curve $C$ is of the following form :
\begin{itemize}
\item $C$ is the union of (at most nodal) curves $C_A$ and $C_B$ meeting in a
point.
\item The markings of $A$ and $B$ lie on $C_A$ and $C_B$
respectively.
\item $C_A$ and $C_B$ represent the classes $\beta_1$ and $\beta_2$
respectively.
\end{itemize}
Notice here that our restrictions on $A$,$B$,$\beta_1$ and $\beta_2$
is the stability conditions on $C_A$ and $C_B$.
\noindent It is clear that every boundary element lies in at least one of
these $D(A,B,\beta_1,\beta_2)$.
The sets $D(A,B,\beta_1, \beta_2)$ are in fact closed,
and we will regard them as subschemes of
$\overline{M}_{0,n}(G/P,\beta)$ by giving them the reduced scheme structure.
Closely related to $D(A,B,\beta_1, \beta_2)$ is the
scheme $M(A,B, \beta_1, \beta_2)$ defined by the fibre
square
$$
\begin{CD}
M(A,B,\beta_1, \beta_2) @>p_2>>
\overline{M}_{0,A \cup \{ \centerdot \}
}(G/P,\beta_1) \times \overline{M}_{0,B \cup \{ \centerdot \}
}(G/P,\beta_2) \\
@Vp_1VV @VV\delta_{\centerdot}^A \times \delta_{\centerdot}^BV \\
G/P @>\Delta >> G/P \times G/P
\end {CD}
$$
Here $\Delta$ is the diagonal embedding and $\delta_{\centerdot}^A$
and $\delta_{\centerdot}^B$ denotes the evaluation maps with respect to
the point $ \{ \centerdot \}$. In \cite{FulPan} it is proved that
$M(A,B,\beta_1,\beta_2)$ is a normal projective variety
and that we have a canonical map
$$ M(A,B,\beta_1,\beta_2) \longrightarrow D(A,B,\beta_1,
\beta_2). $$
This map is clearly surjective.
As
$M(A,B,\beta_1,\beta_2)$ is a closed subscheme of $\overline{M}_{0,A \cup
\{ \centerdot \} }(G/P,\beta_1) \times \overline{M}_{0,B \cup
\{ \centerdot \} }(G/P,\beta_2)$ we can regard the closed points
of $M(A,B,\beta_1, \beta_2)$ as elements of the form $(z_1,z_2)$,
where $z_1 \in \overline{M}_{0,A \cup \{ \centerdot \} }(G/P,\beta_1)$ and
$z_2 \in \overline{M}_{0,B \cup \{ \centerdot \} }(G/P,\beta_2)$. The image
of $(z_1,z_2)$ in $D(A,B,\beta_1,\beta_2)$ will then be denoted
by $z_1 \sqcup z_2$. Given $z_1 \in \overline{M}_{0,A \cup \{ \centerdot
\} }(G/P,\beta_1)$, $z_2 \in \overline{M}_{0,B \cup \{ \centerdot \} \cup
\{ * \}}(G/P,\beta_2)$ and $z_3 \in \overline{M}_{0,C \cup \{ * \}
}(G/P,\beta_3)$
, with $\delta_{\centerdot}(z_1) = \delta_{\centerdot}(z_2)$ and
$\delta_*(z_2) = \delta_*(z_3)$,
we then have the identity $(z_1 \sqcup z_2)
\sqcup z_3 = z_1 \sqcup (z_2 \sqcup z_3)$ inside $ \overline{M}_{0,A
\cup B \cup C}(G/P,\beta_1+ \beta_2 +\beta_3)$.
\subsection{G-action}
As mentioned in the introduction we have a $G$-action
$$G \times \overline{M}_{0,n}(G/P,\beta) \rightarrow
\overline{M}_{0,n}(G/P,\beta).$$
On closed points we can describe the action in the following
way. Let $x \in \overline{M}_{0,n}(G/P,\beta)$ be a closed point
corresponding to the data $(\mu : C \rightarrow G/P ; p_1 , \dots , p_n )$,
and let $g$ be a closed point in $G$. Then $g \cdot x$ is the point
in $\overline{M}_{0,n}(G/P,\beta)$ corresponding to
$(\mu_g : C \rightarrow G/P ; p_1 , \dots , p_n )$, where $\mu_g =
(g \cdot) \circ \mu$. Here $g \cdot$ denotes multiplication with
$g$ on $G/P$.
\subsection{Special cases}
The following special cases of our main result follows from the
construction and formal properties of our moduli spaces.
\noindent $\beta = 0$ : Here the moduli space
$\overline{M}_{0,n}(G/P,\beta)$ is canonical isomorphic to
$\overline{M}_{0,n} \times G/P$, where $\overline{M}_{0,n}$
denote the moduli space of stable $n$-pointed curves of genus $0$.
As $\overline{M}_{0,n}$ is known to be irreducible \cite{Knudsen}
we get that
$\overline{M}_{0,n}(G/P,0)$ is irreducible.
\noindent $G/P = \P^1$ : The irreducibility of $\overline{M}_{0,n}(\P^1,d)$
follows from the construction of the moduli space in \cite{FulPan}.
First of all $\overline{M}_{0,0}(\P^1, 1) \cong \operatorname{Spec} ({\Bbb C})$
so we may assume that $(n,d) \neq (0,1)$. With this assumption
$\overline{M}_{0,n}(\P^1, d)$ is the quotient of a variety $M$
by a finite group. Now $M$ is glued together by the moduli spaces
$\overline{M}_{0,n}(\P^1,d,\overline{t})$ of $\overline{t}$-maps
spaces (here $\overline{t}=(t_0,t_1)$ is a basis of $\O_{\P^1}(1)$).
See section 3 in \cite{FulPan} for a definition of
$\overline{M}_{0,n}(\P^1, d, \overline{t})$. The moduli spaces
$\overline{M}_{0,n}(\P^1, d, \overline{t})$ are irreducible
(in fact they are ${\Bbb C}^*$-bundles over an open subscheme of
$\overline{M}_{0,m}$ for a suitable $m$). This follows from
the proof of Proposition 3.3 in \cite{FulPan}.
It is furthermore
clear that $\overline{M}_{0,n}(\P^1, d, \overline{t})$,
and $\overline{M}_{0,n}(\P^1, d, \overline{t}')$ intersect
non-trivially for different choices of bases $\overline{t}$ and
$\overline{t}'$. This imply that $\overline{M}_{0,n}(\P^1,d)$
is connected, and as it is locally normal
it must be irreducible.
\section{Flag varieties}
\label{flag}
In this section we will give a short review on flag varieties.
Main references will be \cite{Springer}, \cite{Demazure} and
\cite{Kock}. In \cite{Springer} one can find the general
theory on the structure of linear algebraic groups. The Chow
group of $G/B$, where $B$ is a Borel subgroup, can be found in
\cite{Demazure}. From this one easily recovers the Chow group
for a general flag variety $G/P$ (e.g. \cite{Kock} Section 1).
\subsection{Schubert varieties.}
Let $G$ be a complex connected linear algebraic group and
$P$ be a parabolic subgroup of $G$. As we will only be interested
in the quotient $G/P$, we may assume that $G$ is semisimple.
Fix a maximal torus $T$ and a Borel subgroup $B$ such that
$$ T \subseteq B \subseteq P \subseteq G. $$
Let $W$ (resp. $R$) denote the Weyl group (resp. roots) associated to
$T$ and let $R^+$ denote the positive roots with respect to $B$.
Let further $D \subseteq R^+$ denote the simple
roots.
Given $\alpha \in R$ we let $s_{\alpha} \in W$ denote the corresponding
reflection.
From general theory on algebraic groups we know that
$P$ is associated to a unique subset $I \subseteq D$, such that
$P = B W_I B$, where $W_I$ is the subgroup of $W$ generated by
the reflections $s_{\alpha}$ with $\alpha \in I$.
The flag variety $G/P$ is then the disjoint union of a finite number
of $B$-invariant subsets $C(w) = B w P / P$ with $w \in W^I$, where
$$W^I = \{ w \in W | w \alpha \in R^+ \text{ for all } \alpha \in I \}.$$
Each $C(w)$, $w \in W^I$ is isomorphic to ${\Bbb A}^{l(w)}$. Here $l(w)$
denotes the length of a shortest expression of $w$ as a product
of simple reflections $s_{\alpha}$, $\alpha \in D$. The closures
of $C(w)$, $w \in W$, inside
$G/P$ is called the generalized Schubert varieties.
We will denote them by $X_w$, $w \in W$, respectively.
In case $l(w)=1$ we have $X_w \cong \P^{\,1}$.
\subsection{Chow group.}
The Chow group $A_*(G/P)$ is freely generated. As a basis we can pick
$[X_w]$, $w \in W^I$. In \cite{Kock} it is proved that this basis
is orthogonal. Using that positive classes intersect in positive
classes on G/P (Cor. 12.2 in \cite{FulInt}), we conclude that
a class in
$A_*(G/P)$ is positive (or zero) if and only if it is of the form
$$ \sum _{w \in W^I} a_w [X_w] \text{ with } a_w \geq 0.$$
\subsection{Effective classes.}
Let $\beta \in A_1(G/P)$.
From above it is clear that $\beta$
can only be effective (in the
sense of Section \ref{summary}), if $\beta$ is a positive linear
combination of $[X_{s_{\alpha}}]$ with $\alpha \in D \cap W^I
= D \setminus I$. Noticing that $X_{s_{\alpha}} \cong \P^{\,1}$,
$\alpha \in D \setminus I$, implies
the inverse, that is, a positive linear combination of $[
{X_{s_{\alpha}}}]$, $\alpha \in D \setminus I$ is effective.
Using the above we can introduce a partial ordering on the set of effective
classes in $A_1(G/P)$.
\begin{defn}
Let $\beta_1$ and $\beta_2$ be effective classes.
If there exist an effective class $\beta_3$ such that
$\beta_2 = \beta_1 + \beta_3$ we write $\beta_1 \prec
\beta_2$. If $\beta$ is an effective class with the property
$$\beta' \prec \beta \Rightarrow \beta' = 0 \text{ or } \beta'
= \beta$$
we say that $\beta$ is irreducible. An effective class $\beta$
is reducible if it is not irreducible.
\end{defn}
Notice that a non-zero effective class $\beta$ is irreducible if and
only if $\beta = [X_{s_{\alpha}}]$ for some $\alpha \in
D \setminus I$.
In the proof of the irreducibility of $\overline{M}_{0,n}(G/P,\beta)$
we will use induction on $\beta$ with respect to this ordering.
This is possible because given an effective class $\beta \in
A_1(G/P)$, there is only finitely many other effective classes $\beta'$
with $\beta' \prec \beta$.
\subsection{Summary.}
We are ready to summarize what will be important for us
\begin{itemize}
\item The set of effective classes in $A_1(G/P)$ has
a ${\Bbb Z}_{\geq 0}$-basis represented by $B$-invariant closed subvarieties
$X_{s_{\alpha}}$, $\alpha \in D \setminus I$, of
$G/P$.
\item The subsets $X_{s_{\alpha}}$, $\alpha \in D
\setminus I$ are the only $B$-invariant irreducible
1-dimensional closed subsets of $G/P$.
\item $X_{s_{\alpha}} \cong \P^{\,1}$, $\alpha \in D \setminus I$.
\end{itemize}
\section{Boundary of $\overline{M}_{0,n}(X,\beta)$}
\label{boundary}
In this section we begin the proof of our main result.
Remember that our convention is that whenever we write
$\overline{M}_{0,n}(G/P,\beta)$, $D(A,B,\beta_1,\beta_2)$
or $M(A,B,\beta_1,\beta_2)$, we assume that these are well defined
and non-empty.
From now on we will assume that $G$, a semisimple linear algebraic
group, and a parabolic subgroup $P$ have been fixed. We let $X$ denote
$G/P$.
We will need to know when $D(A,B,\beta_1,\beta_2)$
is irreducible and for this purpose we have the following proposition.
\begin{prop}
\label{divisor}
Suppose that $\overline{M}_{0,A
\cup \{ \centerdot \} }(X,\beta_1)$ and $\overline{M}_{0,B
\cup \{ \centerdot \} }(X,\beta_2)$ are irreducible. Then
the scheme $M(A,B,\beta_1,\beta_2)$ is also irreducible. In particular
$D(A,B,\beta_1,\beta_2)$ will be irreducible in this case.
\begin{pf}
As $M(A,B,\beta_1,\beta_2)$ is a normal scheme it splits
up into a disjoint union of irreducible components $C_1,C_2, \dots,
C_l$. Our task is to show that $l=1$. Consider the natural map
$\pi : G \rightarrow G/P$. Locally (in the Zariski topology)
this map has a section (\cite{Jantzen} p.183) , i.e.
there exists an open cover $ \{U_i \}_{i \in I}$ of $X$
(we assume $U_i \neq \emptyset$) and
morphisms $s_i : U_i \rightarrow G$ such that $\pi \circ s_i$
is the identity map.
By pulling back the covering $ \{U_i \}_{i \in I}$ of $X$,
by the evaluation maps $\delta_{\centerdot}^A$ and
$\delta_{\centerdot}^B$, we get open coverings
$ \{ V_i^A \}_{i \in I} $
and $ \{ V_i^B \}_{i \in I} $ of $\overline{M}_{0,A
\cup \{ \centerdot \} }(X,\beta_1)$ and $\overline{M}_{0,B
\cup \{ \centerdot \} }(X,\beta_2)$ respectively.
Finally an open cover $ \{W_i \}_{i \in I}$ of
$M(A,B,\beta_1,\beta_2)$ is obtained
by setting $W_i = p_1^{-1}(U_i)= p_2^{-1}(V_i^A \times V_i^B)$.
We claim
$$\forall i,j \in I : W_i \cap W_j
\neq \emptyset.$$
To see this consider $U_i,U_j \subseteq
X$. As $X$ is irreducible there exists a closed point
$x \in U_i \cap U_j$. Using that $G$ acts
transitively on $X$ we can choose elements $z_1 \in
\overline{M}_{0,A \cup \{ \centerdot \} }(X,\beta_1)$ and $z_2 \in
\overline{M}_{0,B \cup \{ \centerdot \} }(X,\beta_2)$, with
$\delta_{\centerdot}^A(z_1) = \delta_{\centerdot}^B(z_2)=x$.
With these choices it is clear that $(z_1,z_2)$ correspond to a point in
$W_i \cap W_j$.
Next we want to show that $W_i$ is
irreducible. For this consider the map
$$
\begin{array}{cccc}
\psi_i : & V_i^A \times V_i^B & \rightarrow &
\overline{M}_{0,A
\cup \{ \centerdot \} }(X,\beta_1) \times \overline{M}_{0,B \cup
\{ \centerdot \} }(X,\beta_2)
\vspace{2ex}\\
& (z_1,z_2) & \mapsto & (z_1, ((s_i \circ \delta_{\centerdot}^A)(z_1))
((s_i \circ \delta_{\centerdot}^B)(z_2))^{-1} z_2)
\end{array}
$$
where we use the group action of $G$ on $\overline{M}_{0,B
\cup \{ \centerdot \} }(X,\beta_2)$. By definition $\psi_i$
factors through $W_i$. We therefore have an induced map
$$\psi_i^{'} : V_i^A \times V_i^B \rightarrow W_i. $$
Clearly $\psi_i^{'} \circ p_2$ is the identity map. This
implies that $\psi_i^{'}$ is surjective, and as $V_i^A
\times V_i^B$ is irreducible, we get that $W_i$ is irreducible.
At last we notice
that as $W_i$ is irreducible it must be contained in one of the
components $C_1,C_2, \dots ,C_l$ of $M(A,B,\beta_1, \beta_2)$.
On the other hand the $W_i$'s intersect non-trivially so all of
them must be contained in the same component. But
$ \{ W_i \} _{i \in I }$ was an
open cover of $M(A,B,\beta_1,\beta_2)$. We conclude that $l = 1$,
as desired. Being a surjective image of $M(A,B,\beta_1,\beta_2)$
this implies that $D(A,B,\beta_1,\beta_2)$ is also irreducible.
\end{pf}
\end{prop}
\section{ Properties of the components of $\overline{M}_{0,n}
(X,\beta)$}
In this section we study the behaviour of the components of
$\overline{M}_{0,n}(X,\beta)$.
Let $K_1, K_2, \dots,
K_l$ denote the components of $\overline{M}_{0,n}(X,\beta)$. As
$ \overline{M}_{0,n}(X,\beta)$ is normal,
the $K_i$'s are disjoint. Remember that we had a group action
of $G$ on $\overline{M}_{0,n}(X,\beta)$ which was introduced in Section
\ref{summary}. We claim
\begin{lem}
\label{inv}
Let $K$ be a component of $\overline{M}_{0,n}(X,\beta)$. Then $K$
is invariant under the group action of $G$ on $\overline{M}_{0,n}
(X,\beta)$.
\begin{pf}
Let $\eta : G \times \overline{M}_{0,n}(X,\beta) \rightarrow
\overline{M}_{0,n}(X,\beta)$ denote the group action, and consider
the image $\eta(G \times K)$ of $G \times K$. As $G \times K$ is
irreducible $\eta(G \times K)$ will also be irreducible. This
means that $\eta(G \times K)$ is contained in a component,
say $K_1$, of $\overline{M}_{0,n}(X,\beta)$. On the other hand
$\eta( \{ e \} \times K) \subseteq K$ (here $e$ denotes the identity
element in $G$) so we conclude that $K=K_1$.
\end {pf}
\end{lem}
The next lemma concerns the boundary of components of $\overline{M}_{0,0}
(X,\beta)$ when $\beta$ is reducible.
\begin{lem}
\label{bound}
Let $\beta$ be a reducible effective class and
$K$ be a component of $\overline{M}_{0,0}(X,\beta)$.
Then there exist boundary elements in $K$.
\begin{pf}
Assume $K$ do not have boundary elements. Then by definition
of boundary points, each element in $K$ would correspond to an
irreducible curve. Using Lemma \ref{inv} we have an induced
$B$-action on $K$. As $K$ is projective, and $B$ is a connected
solvable linear algebraic group, we can use Borel's fixed point
Theorem (see \cite{Springer} p.159) to conclude that this action
has a fixed point. This means that there exist $z \in K$ such that
$b z = z$, for all $b \in B$. Let $\P^{\,1} \stackrel{\mu}{\rightarrow}
X$ be the stable curve with its morphism to $X$ which
correspond to $z$. By definition of the group action of $B$ on
$z$ we conclude that $\mu(\P^{\,1})$ must be a $B$-invariant subset of $X$.
On the other hand $\mu (\P^{\,1})$ is closed, irreducible and of
dimension 1. By the properties stated in Section \ref{flag}
$\mu(\P^{\,1})$ must be equal to a 1-dimensional
Schubert variety $X_{s_{\alpha}}$ of $X$.
From this we conclude that
$\mu_*[\P^{\,1}] = m [X_{s_{\alpha}}]$, where $m$
is a positive integer.
As $\beta$ is reducible $m \geqq 2$.
The closed embedding $i : X_{s_{\alpha}}
\rightarrow X$ induces a map
$i_* : \overline{M}_{0,0}(X_{s_{\alpha}}, m
[X_{s_{\alpha}}])
\rightarrow \overline{M}_{0,0}(X,\beta)$, where an element $(C \stackrel{f}
{\rightarrow} X_{s_{\alpha}}) \in \overline{M}_{0,0}
(X_{s_{\alpha}},
m [X_{s_{\alpha}}])$ goes to $i_*(C \stackrel{f}
{\rightarrow} X_{s_{\alpha}}) =
(C\stackrel{i \circ f}{\rightarrow}
X)$. As $X_{s_{\alpha}}$ is isomorphic to $\P^{\,1}$ we know that
$ \overline{M}_{0,0}(X_{s_{\alpha}}, m
[{X}_{s_{\alpha}}])$ is irreducible.
On the other hand $z$ is in the
image of $i_*$ so
we conclude that $i_*( \overline{M}_{0,0}(X_{s_{\alpha}}, m
[X_{s_{\alpha}}])) \subseteq K$.
But a boundary element in
$\overline{M}_{0,0}(X_{s_{\alpha}}, m [
X_{s_{\alpha}}])$ is easy to construct by hand
(as $m \geq 2$),
which gives us the desired contradiction.
\end{pf}
\end{lem}
The following will also be useful.
\begin{lem}
\label{hit}
Let $\beta \in A_1(X)$ be a reducible effective
class and suppose that $\overline{M}_{0,0}(X,\beta')$ is
irreducible for $\beta' \prec \beta$.
Furthermore let $K$ be a component of $\overline{M}_{0,0}(X,\beta)$.
Then there exists a
non-zero irreducible class $\beta'$, with $\beta -\beta'$ effective,
such that $D(\emptyset, \emptyset,
\beta', \beta - \beta') \cap K \neq \emptyset$.
\begin{pf}
By Lemma \ref{bound} we can choose a boundary point $z \in K$.
There exists effective classes $\beta_1$ and $\beta_2$ such
that $z \in D(\emptyset, \emptyset, \beta_1, \beta_2)$.
We may assume that $\beta_1$ is reducible. Choose an effective
non-zero irreducible class $\beta'$ and an effective class $\beta''$
such that $\beta_1 = \beta' + \beta''$. Choose also
$z_1 \in \overline{M}_{0, \{ Q_1 \} }(X,\beta')$,
$z_2 \in \overline{M}_{0, \{ Q_1 \} \cup \{ Q_2 \} }(X,\beta'')$ and
$z_3 \in \overline{M}_{0, \{ Q_2 \} }(X,\beta_2)$,
such that $\delta_{Q_1}(z_1) = \delta_{Q_1}(z_2)$ and
$\delta_{Q_2}(z_2) = \delta_{Q_2}(z_3)$. Then
$z_1 \sqcup z_2 \in \overline{M}_{0, \{ Q_2 \} }(X,\beta_1)$
from which we conclude
$(z_1 \sqcup z_2) \sqcup z_3 \in D(\emptyset, \emptyset,
\beta_1, \beta_2)$. On the other hand
$z_2 \sqcup z_3 \in \overline{M}_{0, \{ Q_1 \} }(X,\beta - \beta')$
by which we conclude
$z_1 \sqcup (z_2 \sqcup z_3) \in D(\emptyset, \emptyset,
\beta', \beta - \beta')$.
Using Proposition \ref{divisor} we know that $ D(\emptyset, \emptyset,
\beta_1, \beta_2)$ is irreducible and as
$z \in D(\emptyset, \emptyset,\beta_1, \beta_2) \cap
K$, we must have $ D(\emptyset, \emptyset,\beta_1, \beta_2)
\subseteq K$, in particular $(z_1 \sqcup z_2) \sqcup z_3
\in K$. On the other hand
$$ (z_1 \sqcup z_2) \sqcup z_3 = z_1 \sqcup ( z_2 \sqcup z_3)
\in D(\emptyset, \emptyset,
\beta', \beta - \beta').$$
This proves the lemma.
\end{pf}
\end{lem}
\section{Irreducibility of $\overline{M}_{0,n}(X,\beta)$}
In this section we will prove that the moduli spaces
$\overline{M}_{0,n}(X,\beta)$ are irreducible. First we notice
that for $\beta \neq 0$ we can restrict our attention to a fixed
$n$.
\begin{lem}
\label{n-inv}
Let $n_1,n_2 \geqq 0$ be integers and $\beta \in A_1(X) \setminus
\{ 0 \}$ be an effective class. Then $\overline{M}_{0,n_1}(X,\beta)$
is irreducible if and only if $\overline{M}_{0,n_2}(X,\beta)$ is
irreducible.
\begin{pf}
It is enough to consider the case $n_2=n+1$ and $n_1=n$ for
a positive integer $n$. The contraction
morphism $f : \overline{M}_{0,n+1}(X,\beta) \rightarrow
\overline{M}_{0,n}(X,\beta)$ which forgets the $(n+1)$'th point
is a surjective map with connected fibres. Let $K_1,K_2, \dots,
K_s$ (resp. $C_1,C_2, \dots, C_t$) be the components of
$\overline{M}_{0,n+1}(X,\beta)$ (resp. $\overline{M}_{0,n}(X,\beta)$).
As the components are mutually disjoint and $f$ is surjective
we must have $s \geqq t$. Let us now restrict our attention to one
of the components of $\overline{M}_{0,n}(X,\beta)$, say $C_1$. Assume
that $K_1,K_2, \dots, K_r$ ($r \leqq s$) are the components
which by $f$ maps to $C_1$. It will be enough to show that $r=1$.
Assume $r \geqq 2$. As
$$ C_1 = \bigcup_{i=1}^{r} f(K_i) $$
and as $C_1$ is irreducible, at least one of the components $K_1
,K_2, \dots, K_r$ maps surjectively onto $C_1$.
So there must exist a point $x$ in $C_1$ which is in
the image of at least 2 of the components in $\overline{M}_{0,n+1}(X,\beta)$.
But then the fibre of $f$ over $x$ is not connected, which
is a contradiction.
\end{pf}
\end{lem}
The idea in proving the irreducibility of $\overline{M}_{0,n}(X,\beta)$
is to use induction on the class $\beta \in A_1(X)$. By
this we mean that we will prove that $\overline{M}_{0,n}(X,\beta)$
is irreducible assuming the same condition is true for $\beta' \prec
\beta$.
The first step in the induction procedure will be to show
that $\overline{M}_{0,0}(X,\beta)$ is irreducible, when $\beta$ is
a non-zero irreducible class.
\begin{lem}
\label{nonred}
Let $\beta$ be a non-zero irreducible class.
Then $\overline{M}_{0,0}(X,\beta)$ is irreducible.
\begin{pf}
As $\beta$ is a non-zero irreducible class, $\beta$ must be the class of a
1-dimensional Schubert variety $X_{s_{\alpha}}$.
Let $K$ be a component
of $\overline{M}_{0,0}(X,\beta)$. As in the proof of Lemma \ref{bound}
we have
a $B$-action on $K$, which by Borel's fixed point theorem
is forced to have a fixed point. Let $x \in K$ be a fixed point.
As $\beta$ is irreducible $x$ must correspond to an irreducible
curve, i.e. $x$ correspond to a map of the form $ \P^{\,1}
\stackrel{\mu}{\rightarrow} X$. The image $\mu(\P^{\,1})$ is
a closed 1-dimensional $B$-invariant irreducible
subset of $X$. As by assumption $\mu_*[\P^{\,1}] = [\overline
{X}_{s_{\alpha}}]$, we conclude that
$\mu(\P^{\,1}) = X_{s_{\alpha}}$.
Now $X_{s_{\alpha}}$ is isomorphic to $\P^{\,1}$,
so $\mu$ must be an isomorphism
onto its image. But clearly every map $ \P^{\,1}
\stackrel{f}{\rightarrow} X$ with $f(\P^{\,1})= X_{s_{\alpha}}$,
which is an
isomorphism onto its image, represent the same point in
$\overline{M}_{0,0}(X,\beta)$. Above we have shown that this point
belongs to every component of $\overline{M}_{0,0}(X,\beta)$.
Using that the components of $\overline{M}_{0,0}(X,\beta)$ are disjoint
the lemma follows.
\end{pf}
\end{lem}
Now we are ready for the general case.
\begin{thm}
\label{thm}
Let $\beta \in A_1(X)$ be an effective class and $X = G/P$ be a
flag variety. Then $\overline{M}_{0,n}(X,\beta)$ is irreducible
for every positive integer n.
\begin{pf}
The case $\beta = 0 $ is trivial as noted in Section \ref{summary}.
By Lemma \ref{n-inv} we may therefore assume that $n = 0$. As
remarked above we will proceed by induction. Assume
that the theorem has been proven for $\beta '$ with
$\beta ' \prec \beta$. Referring to Lemma \ref{nonred} we may assume
that $\beta$ is reducible. Write $\beta = \sum_{i=1}^{m}
\beta_i$ as a sum of non-zero irreducible effective classes
$\beta_i$.
Then $m \geq 2$.
We divide into 2 cases.
Assume first that
$m=2$. So $\beta = \beta' + \beta''$,
where $\beta'$ and $\beta''$ are effective irreducible
classes. In this case every boundary element lie in
$D(\emptyset , \emptyset, \beta', \beta'')$, which
we by induction know is irreducible (Proposition \ref{divisor}).
On the other hand do every component of $\overline{M}_{0,0}
(X,\beta)$ contain a boundary point (by Lemma \ref{bound}).
Using that the components of $\overline{M}_{0,0}(X,\beta)$
are disjoint, the theorem follows in this case.
Assume therefore that $m \geq 3$.
For each $i=1, \dots,m$ choose $z_i \in
\overline{M}_{0,\{ Q_i \} }(X,\beta_i)$, a point such that $\delta_{Q_i}
(z_i)= eP$, where $\delta_{Q_i}$ is the evaluation map onto
$X$.
Let $Q = \{ Q_1, Q_2, \dots, Q_m\}$, and choose a
point $z_0 \in \overline{M}_{0,Q}(X,0)$ corresponding to
a curve $C \cong \P^{\,1}$ and a map $\mu : C \rightarrow X$ such that
$\mu(C) = eP$. Define
$$ z = z_0 \sqcup (\sqcup_{i=1}^{m}
z_i) \in \overline{M}_{0,0}(X,\beta).$$
Then clearly $z \in D(\emptyset , \emptyset, \beta_i, \beta
-\beta_i)$ for all $i$. Let $K$
be the component of $\overline{M}_{0,0}(X,\beta)$ which
contains $z$. By the induction hypothesis and Proposition 1,
$D(\emptyset , \emptyset, \beta_i, \beta - \beta_i)$ is
irreducible for all $i$, which implies $D(\emptyset , \emptyset,
\beta_i, \beta -\beta_i) \subseteq K$ for all $i$.
On the other hand, by Lemma \ref{hit}, every component of
$\overline{M}_{0,0}(X,\beta)$ will intersect at least
one of the sets $D(\emptyset , \emptyset, \beta_i, \beta
- \beta_i)$. Using, and now for the last time, that the
components of $\overline{M}_{0,0}(X,\beta)$ are
disjoint, the theorem follows.
\end{pf}
\end{thm}
\begin{cor}
Let $X=G/P$ be a flag variety. Then the boundary divisors
$D(A,B,\beta_1,\beta_2)$ of $\overline{M}_{0,n}(X,\beta)$
are irreducible.
\begin{pf}
Use Proposition \ref{divisor} and Theorem \ref{thm}.
\end{pf}
\end{cor}
\bibliographystyle{amsplain}
\ifx\undefined\leavevmode\hbox to3em{\hrulefill}\,
\newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
|
1997-12-19T22:43:55 | 9712 | alg-geom/9712022 | en | https://arxiv.org/abs/alg-geom/9712022 | [
"alg-geom",
"math.AG"
] | alg-geom/9712022 | Alexander Polishchuk | Alexander Polishchuk | Poisson structures and birational morphisms associated with bundles on
elliptic curves | 20 pages, AMSLatex | null | null | null | null | In this paper we define a Poisson structure on some moduli spaces related to
principal G-bundles on elliptic curves, the simplest example being the moduli
space of stable pairs: a vector bundle and its global section. We also study
birational morphisms between projective spaces appearing as such moduli spaces.
| [
{
"version": "v1",
"created": "Fri, 19 Dec 1997 21:43:54 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Polishchuk",
"Alexander",
""
]
] | alg-geom | \section{Stable triples}
Let us recall the definition of stable triples
from \cite{BG}. Let $T=(E_1,E_2,\Phi)$ be a triple consisting
of two vector bundles $E_1$ and $E_2$ on $X$ and a homomorphism
$\Phi:E_2\rightarrow E_1$. For a real parameter $\sigma$ the $\sigma$-degree
of $T$ is defined as follows:
$$\deg_{\sigma}(T)=\deg(E_1)+\deg(E_2)+\sigma\cdot\operatorname{rk}(E_2).$$
Now the $\sigma$-slope of $T$ is defined by the formula
$$\mu_{\sigma}(T)=\frac{\deg_{\sigma}(T)}{\operatorname{rk}(E_1)+\operatorname{rk}(E_2)}.$$
Note that if $L$ is a line bundle then we can define a tensor
of a triple $T$ with $L$ naturally, so that one has
$\mu_{\sigma}(T\otimes L)=\mu_{\sigma}(T)+\deg L$.
The triple $T$ is called $\sigma$-stable if for every
non-zero proper subtriple $T'\subset T$ one has
$\mu_{\sigma}(T')<\mu_{\sigma}(T)$.
Sometimes it is convinient to introduce another stability parameter
$\tau=\mu_{\sigma}(T)$.
The category of triples $T=(E_1,E_2,\Phi)$ is equivalent to the
category of extensions
\begin{equation}\label{equivext}
0\rightarrow p^*E_1\rightarrow F\rightarrow p^*E_2(2)\ra0
\end{equation}
on $X\times\P^1$ where $p:X\times\P^1\rightarrow X$ is the projection.
Indeed, the space of such extensions is
$\operatorname{Ext}^1_{X\times\P^1}(p^*E_2(2),p^*E_1)\simeq\operatorname{Hom}_X(E_2,E_1)$.
This extension has a unique $\operatorname{SL}_2$-equivariant structure
and as shown in \cite{BG} the $\sigma$-stability condition on $T$
is equivalent to the $\operatorname{SL}_2$-equivariant stability
of $F$ with respect to some polarization on $X\times\P^1$
depending on $\sigma$.
Let us denote by ${\cal M}_{\sigma}={\cal M}_{\sigma}(d_1,d_2,r_1,r_2)$ the moduli space of
$\sigma$-stable triples $T=(E_1,E_2,\Phi)$ on $X$ with $\deg(E_i)=d_i$,
$\operatorname{rk} E_i=r_i$.
When using another stability parameter $\tau$ we will denote
the same moduli space by ${\cal M}_{\tau}$.
This moduli space can be constructed using
geometric invariant theory as in \cite{Be}.
We claim that in the case of elliptic curve all these moduli
spaces are smooth.
\begin{lem} The moduli space ${\cal M}_{\sigma}$ is smooth.
\end{lem}
\noindent {\it Proof}. According to \cite{BG} we have to show that
$H^2(X\times\P^1,\underline{\operatorname{End}} F)^{\operatorname{SL}_2}=0$ for the
$\operatorname{SL}_2$-equivariant vector bundle $F$ associated with a
$\sigma$-stable triple. Consider the exact sequence
$$0\rightarrow K\rightarrow \underline{\operatorname{End}} F\rightarrow\underline{\operatorname{Hom}}(p^*E_1,p^*E_2(2))\rightarrow 0.$$
Since the direct image $Rp_*$ of the last term will have
no $\operatorname{SL}_2$-invariant part we have
$H^*(X\times\P^1,\underline{\operatorname{End}} F)^{\operatorname{SL}_2}\simeq H^*(X\times\P^1,K)$.
Now $K$ sits in the exact triangle
$$\underline{\operatorname{Hom}}(p^*E_2(2),p^*E_1))\rightarrow K\rightarrow\underline{\operatorname{End}} p^*E_1\oplus
\underline{\operatorname{End}} p^*E_2\rightarrow\underline{\operatorname{Hom}}(p^*E_2(2),p^*E_1)[1]\rightarrow\ldots$$
It follows that equivariant direct image of $K$ with respect
to the projection $p$ is quasi-isomorphic to the complex
\begin{equation}\label{tangcomp}
C^{\cdot}:\underline{\operatorname{End}} E_1\oplus\underline{\operatorname{End}} E_2\stackrel{d}{\rightarrow}
\underline{\operatorname{Hom}}(E_2,E_1)
\end{equation}
concentrated in degrees $0$ and $1$, where $d(A,B)=A\Phi-\Phi B$.
We have the exact sequence of cohomologies
$$H^1(X,\underline{\operatorname{End}}E_1\oplus\underline{\operatorname{End}} E_2)\rightarrow
H^1(X,\underline{\operatorname{Hom}}(E_2,E_1))\rightarrow H^2(X, C^{\cdot})\rightarrow 0.$$
Now by Serre duality we have
$H^1(X,\underline{\operatorname{Hom}}(E_2,E_1))^*\simeq H^0(X,\underline{\operatorname{Hom}}(E_1,E_2))$.
According to Lemma 4.4 of \cite{BG} this space is zero
unless $\Phi$ is an isomorphism. In the latter case the
first arrow in the above exact sequence is surjective, so in
either case we get $H^2(X\times\P^1,K)=H^2(X, C^{\cdot})=0$.
\qed\vspace{3mm}
The proof of this lemma also shows that the tangent space to
${\cal M}_{\sigma}$ at a triple $T$ is identified with the
hypercohomology space $H^1(X,C^{\cdot})$ where $C^{\cdot}$
is the complex (\ref{tangcomp}). This can also be
shown directly considering infinitesemal deformations
of the first order for triples.
\section{Poisson structure}
\label{mainsec}
Let us fix a trivialization $\omega_X\simeq\O_X$ of the
canonical bundle of $X$. Then we can
define a Poisson structure on the moduli space of
triples ${\cal M}_{\sigma}$. As we have seen above the tangent space
to ${\cal M}_{\sigma}$ at a triple $T$ is identified with
$H^1(X,C)$ where $C$ is the complex
(\ref{tangcomp}). By Serre duality the cotangent space
is isomorphic to $H^0(X,C^*)=H^1(X,C^*[-1])$,
where the complex $C^*[-1]=((C^1)^*\stackrel{-d^*}{\rightarrow}(C^0)^*)$ is
concentrated in degrees $0$ and $1$. Using the natural
autoduality of $\operatorname{End} E_i$ the complex $C^*[-1]$ can be identified with
$$\underline{\operatorname{Hom}}(E_1,E_2)\stackrel{-d^*}\rightarrow
\underline{\operatorname{End}}E_1\oplus\underline{\operatorname{End}}E_2$$
where $-d^*(\Psi)=(-\Phi\Psi,\Psi\Phi)$.
Now let us consider the
morphism of complexes $\phi:C^*[-1]\rightarrow C$ with components
$\phi_1=0$ and
\begin{equation}\label{phi0}
\phi_0:\underline{\operatorname{Hom}}(E_1,E_2)\rightarrow
\underline{\operatorname{End}}E_1\oplus\underline{\operatorname{End}}E_2:
\Psi\mapsto (\Phi\Psi,\Psi\Phi).
\end{equation}
Since $d\circ\phi_0=0$, we have indeed the morphism of
complexes. Therefore, we can take the induced map on
hypercohomologies
$$H_T=\phi_*:H^1(X,C^*[-1])\rightarrow H^1(X,C).$$
Note that we get a map from the cotangent space
to the tangent space of ${\cal M}_{\sigma}$ at $T$.
This construction easily globalizes to give a morphism
$H$ from the cotangent bundle to the tangent bundle of
${\cal M}_{\sigma}$.
\begin{thm} $H$ defines a Poisson structure on ${\cal M}_{\sigma}$.
\end{thm}
\noindent {\it Proof} . Let us check that $H^*=-H$. First of all we claim that
$\phi^*[-1]=\phi$ in the homotopy category of complexes.
Indeed, by definition $\phi^*[-1]$ has components
$(\phi^*[-1])_0=0$ and
$$(\phi^*[-1])_1:\underline{\operatorname{End}}E_1\oplus\underline{\operatorname{End}}E_2\rightarrow
\underline{\operatorname{Hom}}(E_2,E_1):(A,B)\mapsto A\Phi+\Phi B.$$
Now let us consider the map
$$h:\underline{\operatorname{End}}E_1\oplus\underline{\operatorname{End}}E_2\rightarrow\underline{\operatorname{End}}E_1\oplus
\underline{\operatorname{End}}E_2:(A,B)\mapsto (-A,B).$$
Immediate check shows that $h$ provides a homotopy from
$\phi^*[-1]$ to $\phi$. Now the skew-commutativity of $H$
follows immediately from the skew-commutativity of the
natural pairing
$H^1(C)\otimes H^1(C')\rightarrow H^2(C\otimes C')$
that comes from the minus sign in the commutativity
constraint for the tensor product of complexes.
The Jacoby identity will be proven in section \ref{gener}
in more general situation.
\qed\vspace{3mm}
We can interpret the Poisson bivector $H$ in terms of
$\operatorname{SL}_2$-equivariant bundles on $X\times\P^1$ as follows.
Let $F$ be the extension (\ref{equivext}) associated with a triple
$T$. Then the tangent space to ${\cal M}_{\sigma}$ at $T$ is
identified with $H^1(X\times\P^1, \underline{\operatorname{End}} F)^{\operatorname{SL}_2}$,
hence by Serre duality the cotangent space is identified with
$H^1(X\times\P^1, \underline{\operatorname{End}}(F)(-2))$. Now we claim that $H_T$
is induced by some canonical morphism
$$\a:\underline{\operatorname{End}}F(-2)\rightarrow\underline{\operatorname{End}}F$$
on $X\times\P^1$.
Namely, let $\underline{\operatorname{End}}(F,p^*E_1)$ be the kernel of the
natural projection
$\underline{\operatorname{End}}F\rightarrow\underline{\operatorname{Hom}}(p^*E_1,p^*E_2(2))$.
The dual morphism to the embedding gives a morphism
$\underline{\operatorname{End}}F\rightarrow\underline{\operatorname{End}}(F,p^*E_1)^*$.
Thus, to construct $\a$ it is sufficient to construct a morphism
$$\widetilde{\a}:\underline{\operatorname{End}}(F,p^*E_1)^*(-2)\rightarrow\underline{\operatorname{End}}(F,p^*E_1).$$
Now the bundle $\underline{\operatorname{End}}(F,p^*E_1)$ sits in the following
exact triple
$$0\rightarrow\underline{\operatorname{Hom}}(p^*E_2(2),p^*E_1)\rightarrow \underline{\operatorname{End}}(F,p^*E_1)
\rightarrow\underline{\operatorname{End}}(p^*E_1)\oplus\underline{\operatorname{End}}(p^*E_2)\rightarrow 0.$$
We have the morphism to the last term of this triple
$$p^*\phi_0:\underline{\operatorname{Hom}}(p^*E_1,p^*E_2)\rightarrow\underline{\operatorname{End}}(p^*E_1)\oplus
\underline{\operatorname{End}}(p^*E_2)$$
where $\phi_0$ is defined in (\ref{phi0}). It is easy to check
that $p^*\phi_0$ lifts uniquely to a morphism
$$\underline{\operatorname{Hom}}(p^*E_1,p^*E_2)\rightarrow \underline{\operatorname{End}}(F,p^*E_1).$$
Now we define $\widetilde{\a}$ to be the composition of the latter
morphism with the natural projection
$\underline{\operatorname{End}}(F,p^*E_1)^*(-2)\rightarrow\underline{\operatorname{Hom}}(p^*E_1,p^*E_2)$.
One has the natural morphism $\det:{\cal M}_{\sigma}\rightarrow\operatorname{Pic}(X)^2$
associating to a triple $(E_1,E_2,\Phi)$ the pair of line
bundles $(\det E_1,\det E_2)$. We claim that $\det$ is
a Casimir morphism, i.e. preimage of any local function
downstairs is a Casimir function upstairs (that is a function
having zero Poisson bracket with any other function).
Indeed, the cotangent map to $\det$ is just the natural
embedding
$$i:H^0(X,\O_X)^2\rightarrow H^1(X, C^{\cdot})$$
which factors through $H^0(X,C^1)=H^0(X,\underline{\operatorname{End}} E_1)\oplus
H^0(X,\underline{\operatorname{End}} E_2)$. On the other hand, $H$ factors through
the map $H^1(X, C^{\cdot})\rightarrow H^1(X,C^0)$, hence the image
of $i$ is killed by $H$.
In particular, the Poisson bracket on ${\cal M}_{\sigma}$
induces Poisson brackets on the fibers of the morphism $\det$.
These fibers can be identified with moduli spaces
${\cal M}_{\sigma}(L_1,L_2,r_1,r_2)$ of triples with
fixed determinants $\det E_i\simeq L_i$.
Tensoring with a fixed line bundle $L$ gives a Poisson isomorphism
${\cal M}_{\sigma}(L_1,L_2,r_1,r_2)\simeq
{\cal M}_{\sigma}(L_1\otimes L^{\otimes r_1}, L_2\otimes L^{\otimes r_2},r_1,r_2)$.
An automorphism $\phi:X\rightarrow X$ induces an isomorphism of moduli
spaces ${\cal M}_{\sigma}(L_1,L_2,r_1,r_2)\rightarrow{\cal M}_{\sigma}(\phi^*L_1,
\phi^*L_2,r_1,r_2)$ compatible with Poisson structures.
\section{Special cases}
For any bundle $E_1$ and a subbundle $E_2\subset E_1$ let us
denote by $\underline{\operatorname{End}}(E_1,E_2)$ the sheaf of local
homomorphisms of $E_1$ preserving $E_2$. In other words,
this is the kernel of the natural projection
$\underline{\operatorname{End}}(E_1)\rightarrow\underline{\operatorname{Hom}}(E_2,E_1/E_2)$.
\begin{lem}\label{subbun}
Let $T=(E_1,E_2,\Phi)$
be a $\sigma$-stable triple such that $\Phi:E_2\rightarrow E_1$ is an embedding of
$E_2$ as a subbundle. Then
the tangent space to ${\cal M}_{\sigma}$ at $T$ can be identified
with $H^1(X,\underline{\operatorname{End}}(E_1,E_2))$.
\end{lem}
\noindent {\it Proof}. The natural embedding
$\underline{\operatorname{End}}(E_1,E_2)\rightarrow\underline{\operatorname{End}}(E_1)\oplus\underline{\operatorname{End}}(E_2)$
induces the quasi-isomorphism $\underline{\operatorname{End}}(E_1,E_2)\rightarrow
C^{\cdot}$. Hence the assertion.
\qed\vspace{3mm}
Under the identification of this lemma our Poisson structure
at the triple $T$ for which $\Phi$ is an embedding of a
subbundle can be described as follows. From the exact triple
$$0\rightarrow\underline{\operatorname{Hom}}(E_1/E_2,E_2)\rightarrow\underline{\operatorname{End}}E_1
\rightarrow\underline{\operatorname{End}}(E_1,E_2)^*\rightarrow 0$$
we get a boundary homomorphism
$$H^0(X,\underline{\operatorname{End}}(E_1,E_2)^*)\rightarrow
H^1(X,\underline{\operatorname{Hom}}(E_1/E_2,E_2)).$$
Composing it with the natural morphism
$$H^1(X,\underline{\operatorname{Hom}}(E_1/E_2,E_2))\rightarrow
H^1(X,\underline{\operatorname{End}}(E_1,E_2))$$
we get a morphism
$$H^1(X,\underline{\operatorname{End}}(E_1,E_2))^*\simeq
H^0(X,\underline{\operatorname{End}}(E_1,E_2)^*)\rightarrow H^1(X,\underline{\operatorname{End}}(E_1,E_2)),$$
which coincides with $H_T$ under identification of Lemma
\ref{subbun}. Equivalently, we may start with
the natural morphism
$$H^0(X,\underline{\operatorname{End}}(E_1,E_2)^*)\rightarrow
H^0(X,\underline{\operatorname{Hom}}(E_2,E_1/E_2))$$ and compose it with the
boundary homomorphism
$$H^0(X,\underline{\operatorname{Hom}}(E_2,E_1/E_2))\rightarrow
H^1(X,\underline{\operatorname{End}}(E_1,E_2))$$
coming from the exact triple
\begin{equation}\label{imptriple}
0\rightarrow\underline{\operatorname{End}}(E_1,E_2)\rightarrow\underline{\operatorname{End}} E_1\rightarrow
\underline{\operatorname{Hom}}(E_2, E_1/E_2)\rightarrow 0.
\end{equation}
The equivalence of this description with the previous one
follows immediately from the commutative diagram
\begin{equation}
\begin{array}{ccccc}
\underline{\operatorname{Hom}}(E_1/E_2,E_2) &\lrar{} &\underline{\operatorname{End}}(E_1,E_2)
&\lrar{} &\underline{\operatorname{End}} E_2\oplus\underline{\operatorname{End}} E_1/E_2 \\
\ldar{\operatorname{id}} & & \ldar{} & & \ldar{} \\
\underline{\operatorname{Hom}}(E_1/E_2,E_2) &\lrar{} &\underline{\operatorname{End}} E_1
&\lrar{} &\underline{\operatorname{End}}(E_1,E_2)^* \\
&&\ldar{} & & \ldar{}\\
&&\underline{\operatorname{Hom}}(E_2,E_1/E_2) &\lrar{\operatorname{id}}&
\underline{\operatorname{Hom}}(E_2,E_1/E_2)
\end{array}
\end{equation}
It is easy to check using the above descriptions
that the restriction of our Poisson bracket to the space of triples
for which $\Phi$ is an embedding of a subbundle is a
particular case of the Poisson bracket on moduli of
principal bundles over
parabolic subgroups defined by Feigin and Odesskii in
\cite{FO}.
\begin{lem}\label{simple}
Keep the assumptions of Lemma \ref{subbun}.
Then we have an exact triple
$$0\rightarrow \operatorname{End} E_2\oplus\operatorname{End} (E_1/E_2)\rightarrow\ker H_T\rightarrow
\operatorname{End} E_1/\operatorname{End}(E_1,E_2)\rightarrow 0.$$
If $\Phi(E_2)$ is a direct summand of $E_1$ then $H$ vanishes
at $T$. Otherwise, the dimension of $\ker H_T$ is minimal
(and equals to $2$) if and only if the bundles $E_1$, $E_2$
and $E_1/E_2$ are simple.
\end{lem}
\noindent {\it Proof} . Considering the second of the above descriptions of
$H_T$ we see immediately that there is an exact sequence
\begin{equation}\label{triplepf}
0\rightarrow \operatorname{End} E_2\oplus \operatorname{End} (E_1/E_2)\rightarrow\ker H_T\rightarrow
\ker (H^0(X,\underline{\operatorname{Hom}}(E_2,E_1/E_2))\rightarrow
H^1(\underline{\operatorname{End}}(E_1,E_2)).
\end{equation}
Using (\ref{imptriple}) the last term can be identified with
$\operatorname{End} E_1/ \operatorname{End}(E_1,E_2)$. Moreover, the last arrow in
(\ref{triplepf}) is
surjective since we have a natural map
$\operatorname{End} E_1\rightarrow\ker H_T$ coming from the morphism
$\underline{\operatorname{End}} E_1\rightarrow\underline{\operatorname{End}}(E_1,E_2)^*$, and the composition
of this map with the last arrow of (\ref{triplepf})
is just the canonical projection to
$\operatorname{End} E_1/\operatorname{End}(E_1,E_2)$.
If $\Phi(E_2)$ is a direct summand in $E_1$ then the
boundary homomorphism used in the definition of $H_T$ is zero,
hence $H_T=0$. Otherwise, $\dim \ker H_T=2$ if and only if
$E_2$ and $E_1/E_2$ are simple and all endomorphisms of $E_1$
preserve $\Phi(E_2)$. We claim this can happen only when
$E_1$ is also simple. Indeed, let $A:E_1\rightarrow E_1$ be any
endomorphism. By assumption $A$ preserves $\Phi(E_2)$.
Adding a constant to $A$ we may assume that
$A|_{\Phi(E_2)}=0$. Then it induces a map $E_1/E_2\rightarrow E_2$.
However, $\sigma$-stability of our triple implies by Lemma 4.4
of \cite{BG} that $\operatorname{Hom}(E_1,E_2)=0$ since $\Phi$ is not an
isomorphism in our situation. It follows that
$\operatorname{Hom}(E_1/E_2,E_2)=0$, hence, $A=0$. Thus, $\operatorname{End} E_1={\Bbb C}$.
\qed\vspace{3mm}
We are mainly interested in the case when $E_2=\O_X$, $\det E_1$
is fixed. In terms of parameter $\tau$ the stability condition
on $\Phi:\O_X\rightarrow E_1$
is that $\mu(E'_1)<\tau$ for every proper non-zero subbundle
$E'_1\subset E_1$ and $\mu(E_1/E'_1)>\tau$ for every proper subbundle
$E'_1\subset E_1$ such that $\Phi\in H^0(X,E'_1)$.
Now let $E$ be a stable bundle on $X$ of degree $d$ and rank
$r$ (in particular, $gcd(d,r)=1$).
Set $\tau=\mu(E)=\frac{d}{r}$ and consider the moduli space
${\cal M}_{\tau}(\det E,\O_X,r+1,1)$. It is easy to see that the
stability condition on such a triple $\Phi:E_2=\O_X\rightarrow E_1$ is
equivalent to the condition that $\Phi$ is nowhere vanishing
section and the quotient $E_1/\Phi(\O_X)$ is a stable bundle.
Moreover, since there exists a unique stable bundle of rank
$r$ and determinant $\det E$ it follows that
$E_1/\Phi(\O_X)\simeq E$. Thus, we can identify the moduli
space of such triples with the projective space
$\P\operatorname{Ext}^1(E,\O_X)$. If $gcd(d,r+1)=1$ then generic extension of $E$ by
$\O_X$ is stable. Hence, according to Lemma \ref{simple} in this case the
Poisson bracket on $\P\operatorname{Ext}^1(E,\O_X)$ is symplectic at general point.
Let $t_x:X\rightarrow X$ be the translation by
$x\in X$. Then by functoriality we have a natural Poisson
isomorphism
$$\P\operatorname{Ext}^1(E,\O_X)\widetilde{\rightarrow} \P\operatorname{Ext}^1(t_x^*E,\O_X).$$
Note that as $x$ varies $t_x^*E$ runs through all stable
bundles of given degree $d$ and rank $r$. Let $K\subset X$ be
the finite subgroup of order $d^2$ consisting of $x$ such that
$t_x^*\det E\simeq\det E$.
Then for $x\in K$ one has $t_x^*E\simeq E$, therefore, $K$
acts on $\P\operatorname{Ext}^1(E,\O_X)$ by linear transformations
preserving the Poisson structure.
Another special moduli space associated with a fixed stable
bundle $E$ is ${\cal M}_{\tau}(\det E,\O_X,\operatorname{rk} E, 1)$ where
$\tau=\mu(E)$. Then the condition on a triple just means that
$E_1$ is stable, hence isomorphic to $E$, and $\Phi$ is
an arbitrary non-zero section. Therefore, this moduli space
can be identified with $\P H^0(X,E)$.
The Poisson bracket in this case can be described as follows.
The tangent space $T_{[s]}$ to the line generated by $s\in H^0(X,E)$ is
identified with $\operatorname{coker}(H^0(X,\O_X)\stackrel{s}{\rightarrow}
H^0(X,E))$. Thus the cotangent space is
$T^*_{[s]}=\ker((H^1(X,E^*)\rightarrow H^1(X,\O_X))$. Let $D\subset X$ be the
divisor of zeroes of $s$, so that $s:\O_X\rightarrow E$ factors
as $\O_X\rightarrow\O_X(D)\rightarrow E$ where $\O_X(D)$ is embedded as a
subbundle into $E$. Then we have the natural map
\begin{equation}\label{map}
H^1(X,E^*)\rightarrow H^1(X,E^*(D))\rightarrow
H^1(X,\underline{\operatorname{End}}(E,\O_X(D))).
\end{equation}
The latter space sits in the exact sequence
$$0\rightarrow H^0(X,E(-D)/\O_X)\rightarrow
H^1(X,\underline{\operatorname{End}}(E,\O_X(D)))\rightarrow H^1(X,\underline{\operatorname{End}} E)
\simeq H^1(X,\O_X).$$
It follows that (\ref{map}) induces a map
$T^*_{[s]}\rightarrow H^0(X,E(-D)/\O_X)$. Furthermore, it is easy to
check that its composition with the boundary homomorphism
$H^0(X,E(-D)/\O_X)\rightarrow H^1(X,\O_X)$ is zero, hence, we get
a map
$$T^*_{[s]}\rightarrow H^0(X,E(-D))/H^0(X,\O_X).$$
Now the latter space is naturally embedded into $T_{[s]}$
and the composition with this embedding gives our Poisson
structure at $[s]\in\P H^0(X,E)$.
\begin{lem} Let $H_{[s]}:T_{[s]}^*\rightarrow T_{[s]}$ be the above
Poisson structure on $\P H^0(X,E)$. If $\operatorname{rk} E=1$ then
$H_{[s]}=0$. Otherwise, one has an exact sequence
$$0\rightarrow H^1(X,\operatorname{ad}(E/\O_X(D)))\rightarrow\operatorname{coker} H_{[s]}\rightarrow
H^0(D,E|_D)\rightarrow 0$$
where $D$ is the zero divisor of $s$.
\end{lem}
\noindent {\it Proof} . Let us denote by $V\subset T_{[s]}$ the subspace
$H^0(X,E(-D))/H^0(X,\O_X)$. Since the image of $H_{[s]}$
is contained in $V$ we have the exact sequence
$$0\rightarrow\operatorname{coker}(T^*_{[s]}\rightarrow V)\rightarrow\operatorname{coker} H_{[s]}\rightarrow
T_{[s]}/V\rightarrow 0.$$
Since $E(-D)$ is stable of positive slope, it follows that
$H^1(X, E(-D))=0$, hence we have an isomorphism
$$T_{[s]}/V\simeq H^0(X,E)/H^0(X,E(-D))\simeq H^0(D,E|_D).$$
Now the assertion follows easily from the exact sequence
$$T_{[s]}^*\rightarrow H^1(X,\underline{\operatorname{End}}(E,\O(D)))\rightarrow
H^1(X,\underline{\operatorname{End}}(E/\O_X))\oplus H^1(X,\O_X)\rightarrow 0.$$
\qed\vspace{3mm}
More generally, we can consider the moduli space ${\cal M}_{\tau}(L,\O_X,r,1)$
where $L$ is a fixed line bundle of degree $d$,
$\tau=\frac{d}{r}+\epsilon$ where $\epsilon>0$ is sufficiently small.
Then we get the moduli space of pairs $s:\O_X\rightarrow E$
where $E$ is a semistable bundle with determinant $L$, $\operatorname{rk} E=r$, $s$ is
a section which doesn't belong to any destabilizing subbundle of $E$. We have
a Casimir morphism from this moduli space to the moduli stack of semistable
bundles, so the fibers inherit the Poisson structure. In particular,
if we take the semistable bundle $E=(E_0)^{\oplus k}$ where $E_0$ is a
stable bundle, then the corresponding fiber is identified with the
Grassmannian $G(k,H^0(E_0))$ of $k$-dimensional subspaces in $H^0(E_0)$,
so we get some family of Poisson structures on the Grassmannians.
\section{Fourier transforms}
Let $m:X\times X\rightarrow X$ be the group law on $X$, $x_0\in X$ be
the neutral element.
Let
$${\cal P}=m^*\O_X(x_0)\otimes p_1^*\O_X(-x_0)\otimes
p_2^*\O_X(-x_0)$$
be the Poincar\'e line bundle on $X\times X$
inducing an isomorphism of $X$ with the dual elliptic
curve. We denote by ${\cal F}$ the corresponding Fourier-Mukai
transform which is an autoequivalence of the
the derived category ${\cal D}^b(X)$ of coherent sheaves on $X$ given
by
$${\cal F}(E)=Rp_{2*}(Lp_1^*E\sideset{^L}{}{\otimes}{\cal P}).$$
One has ${\cal F}\circ{\cal F}\simeq (-\operatorname{id}_X)^*[-1]$ (see \cite{Mukai}).
It is easy to see that for every $E\in{\cal D}^b(X)$ one has
$\operatorname{rk}{\cal F}(E)=\deg E$, $\deg{\cal F}(E)=-\operatorname{rk} E$.
It follows that if $T:{\cal D}^b(X)\rightarrow{\cal D}^b(X)$ is a composition of
some sequence of Fourier transforms and
tensorings with line bundles, then the vector
$v(T(E))=(\operatorname{rk} T(E), \deg T(E))$ is obtained from the vector
$v(E)=(\operatorname{rk} E, \deg E)$ by applying some matrix $A\in\operatorname{SL}_2({\Bbb Z})$.
Furthermore, one can lift the natural action of $\operatorname{SL}_2({\Bbb Z})$
on vectors $(\operatorname{rk}, \deg)$ to the action of a central extension
of $\operatorname{SL}_2({\Bbb Z})$ by ${\Bbb Z}$ on ${\cal D}^b(X)$.
More precisely, we can consider the
standard presentation of $\operatorname{SL}_2({\Bbb Z})$ by generators
$S=\left( \matrix 0 & 1\\ -1 & 0 \endmatrix \right)$ and
$R=\left( \matrix 1 & 0\\ 1 & 1 \endmatrix \right)$
subject to relations
$$S^2=(RS)^3,\ S^4=1.$$
Then the central extension in question is the group $\widetilde{\operatorname{SL}}_2({\Bbb Z})$
generated by $S$ and $R$ with the only relation $S^2=(RS)^3$.
The action of this group on ${\cal D}^b(X)$ is the following:
$S$ acts as the Fourier-Mukai transform while
$R$ acts as tensoring with $\O_X(x_0)$ (see \cite{Mukai}).
We will consider the action of $\widetilde{\operatorname{SL}}_2({\Bbb Z})$
on morphisms of stable bundles.
For this it will be useful to know the orbits of $\operatorname{SL}_2({\Bbb Z})$
on pairs of primitive vectors in ${\Bbb Z}^2$. First of all,
for a pair of vectors
$v_1=(r_1,d_1)$, $v_2=(r_2,d_2)$ such that $gcd(r_i,d_i)=1$ for $i=1,2$
the determinant $\det(v_1,v_2)\in{\Bbb Z}$ is invariant of $\operatorname{SL}_2({\Bbb Z})$.
We consider only pairs $v_1,v_2$ with $\det(v_1,v_2)\neq 0$.
For such pairs there is a second $\operatorname{SL}_2({\Bbb Z})$-invariant
$\a(v_1,v_2)\in({\Bbb Z}/\det(v_1,v_2))^*$ defined from the condition
$$v_1\equiv\a(v_1,v_2)v_2\operatorname {mod}\det(v_1,v_2){\Bbb Z}^2$$
It is easy to see that the $\operatorname{SL}_2({\Bbb Z})$-orbit of such $(v_1,v_2)$
consists of all pairs with the same $\det$ and $\a$.
Henceforward, we restrict ourself to considering stable bundles
on $X$ with
determinant isomorphic to $\O_X(nx_0)$ for some $n$. The
reason is that the group $\widetilde{\operatorname{SL}}_2({\Bbb Z})$
preserves the set $S_{x_0}$ of objects of the form
$E[k]$ where $k\in{\Bbb Z}$, $E$ is either a stable bundle
with determinant $\O_X(nx_0)$ for some $n$,
or $\O_{x_0}$. An element of $S_{x_0}$ is determined by
its degree and rank uniquely up to a shift. It follows that
the group $\widetilde{\operatorname{SL}}_2({\Bbb Z})$ acts transitively
on $S_{x_0}$. Furthermore, an element $T\in\widetilde{\operatorname{SL}}_2({\Bbb Z})$
is completely determined by its action on a pair of
elements of $S_{x_0}$ which are not isomorphic up to shift.
The first immediate consequence of the action of
$\widetilde{\operatorname{SL}}_2({\Bbb Z})$ is that for stable bundles $E_1$ and $E_2$
such that $\det E_1\simeq\det E_2\simeq\O_X(dx_0)$ and
$\operatorname{rk} E_1\equiv \operatorname{rk} E_2\operatorname {mod}(d)$ there is a canonical
isomorphism
$$\P\operatorname{Ext}^1(E_1,\O_X)\simeq\P\operatorname{Ext}^1(E_2,\O_X).$$
Indeed, under these conditions there is a unique element
$T\in\widetilde{\operatorname{SL}}_2({\Bbb Z})$ such that $T(\O_X)\simeq\O_X$ and
$T(E_1)\simeq E_2$. Considering the action of $T$
on morphisms from $E_1$ to $\O_X[1]$ we get the above
isomorphism.
Now for every stable bundle $E$ with $\det
E\simeq\O_X(dx_0)$ where $d>1$ we can find an element
$T\in\widetilde{\operatorname{SL}}_2({\Bbb Z})$ such that $T(E)\simeq \O_X$.
Then $T(\O_X)=E'[n]$ for
some stable bundle $E'$ and some $n\in{\Bbb Z}$. Since $\operatorname{Hom}(\O_X,E)\neq 0$
we should have $\operatorname{Hom}(T(\O_X), T(E))\neq 0$, hence $n=0$ or $-1$.
Consider first the case $n=-1$.
Then one has $\det E'\simeq \O_X(dx_0)$ and
$$r\cdot r'\equiv -1\operatorname {mod}(d)$$
where $r=\operatorname{rk} E$, $r'=\operatorname{rk} E'$
(this is deduced comparing invariants $\a$ for the pair of vectors
$(v(\O_X),v(E))$ and its image under $T$).
Conversely, for every $E'$ satisfying these conditions there
exists a unique element $T\in\widetilde{\operatorname{SL}}_2({\Bbb Z})$ sending $E$ to
$\O_X$ and $\O_X$ to $E'[-1]$.
The transformation $T$ induces an isomorphism
\begin{equation}\label{linisom}
T_*:\P H^0(E)\widetilde{\rightarrow}\P\operatorname{Ext}^1(E',\O_X).
\end{equation}
(an isomorphism of this kind with $r=1$, $r'=d-1$
was constructed in \cite{FMW} by a different method.)
Note that in the previous section we identified both sides of the
isomorphism (\ref{linisom})
with some special moduli spaces of pairs, in particular,
they carry natural Poisson structures.
\begin{prop} The isomorphism $T_*$ is compatible with Poisson structures.
\end{prop}
\noindent {\it Proof} . Let $s:\O_X\rightarrow E$ be a non-zero section,
$\O_X\rightarrow\widetilde{E}\rightarrow E'$ be the corresponding extension of $E'$
by $\O_X$ with class $T_*(s)\in\operatorname{Ext}^1(E',\O_X)$.
It suffices to prove that $T_*$ preserves Poisson structure
over a non-empty open subset, hence we can assume that
$E/\O_X$ has no torsion.
By Serre duality the cotangent space
$T^*_{[s]}\P H^0(E)$ can be identified with
$$\ker(\operatorname{Ext}^1(E,\O_X)\rightarrow H^1(\O_X))\simeq
\operatorname{Ext}^1(E/\O_X,\O_X)/{\Bbb C}\cdot e$$
where $e\in\operatorname{Ext}^1(E/\O_X, \O_X)$ is the class of the extension
$\O_X\rightarrow E\rightarrow E/\O_X$. Under this identification the Poisson bracket
on $\P H^0(E)$ at the point $[s]$ is induced by the natural morphism
$$H_{[s]}:\operatorname{Ext}^1(E/\O_X,\O_X)/{\Bbb C}\cdot e\rightarrow T_{[s]}\P H^0(E)$$
which comes from the identification of $T_{[s]}\P H^0(E)$
with $\ker(H^1(\underline{\operatorname{End}}(E,\O_X))\rightarrow H^1(\O)^2)$ and
from the natural morphism
$$\operatorname{Ext}^1(E/\O_X,\O_X)=H^1(\underline{\operatorname{Hom}}(E/\O_X,\O_X))\rightarrow
H^1(\underline{\operatorname{End}}(E,\O_X)).$$
In other words, we have a morphism from a neighborhood of $[e]$
in the space of extensions
$\P\operatorname{Ext}^1(E/\O_X,\O_X)$ to $\P H^0(E)$ (since in the
neighborhood of $[e]$ such an extension is necessarily
isomorphic to $E$), and $H_{[s]}$ is just the
tangent map to this morphism at the point $[e]$.
Similarly, the Poisson bracket on $\P\operatorname{Ext}^1(E',\O_X)$
at the point $[T_*s]$ can be identified with
the tangent map
$$\operatorname{Hom}(\widetilde{E},E')/{\Bbb C}\cdot f\rightarrow
T_{[T_*(s)]}\P\operatorname{Ext}^1(E',\O_X)$$
to the local morphism from
$\P\operatorname{Hom}(\widetilde{E},E')$ to $\P\operatorname{Ext}^1(E',\O_X)$ at the point
$[f]$ where $f:\widetilde{E}\rightarrow E'$ is the canonical morphism.
Here we use the natural identification of
$\operatorname{Hom}(\widetilde{E},E')/{\Bbb C}\simeq\ker(H^0(E')\rightarrow H^1(\O_X))$
with the cotangent space to $\P\operatorname{Ext}^1(E',\O_X)$ at
$[T_*(s)]$.
Now we have the following commutative square of local
morphisms in the neighborhood of points induced by $s$:
\begin{equation}
\begin{array}{ccc}
\P\operatorname{Ext}^1(E/\O_X,\O_X) & \lrar{} & \P H^0(E)\\
\ldar{T_*} && \ldar{T_*}\\
\P\operatorname{Hom}(\widetilde{E},E') &\lrar{} & \P\operatorname{Ext}^1(E',\O_X)
\end{array}
\end{equation}
Considering the corresponding commutative square of
tangent maps we get the compatibility of $T_*$ with
Poisson brackets.
\qed\vspace{3mm}
For some other choice of autoequivalence $T:{\cal D}^b(X)\rightarrow{\cal D}^b(X)$
sending $E$ to $\O_X$ one has $T(\O_X)\simeq (E'')^*$ for a
stable bundle $E''$ of degree $\deg E''=\deg E=d$ and rank
$r''$ satisfying the congruence
$$r''\cdot r\equiv 1\operatorname {mod}(d)$$
where $r=\operatorname{rk} E$.
In this case we get an isomorphism
$$\P H^0(E)\simeq \P H^0(E'').$$
We claim that it is also compatible with the natural Poisson
structures on both sides. Indeed, this is proven exactly
as in the above proposition using the following commutative
diagram of local morphisms:
\begin{equation}
\begin{array}{ccc}
\P\operatorname{Ext}^1(E/\O_X,\O_X) & \lrar{} & \P H^0(E)\\
\ldar{T_*} && \ldar{T_*}\\
\P\operatorname{Hom}(E'',E''/\O_X) &\lrar{} & \P H^0(E'')
\end{array}
\end{equation}
Combining these isomorphisms we also get Poisson isomorphisms
between $\P\operatorname{Ext}^1(E_1,\O_X)$ and $\P\operatorname{Ext}^1(E_2,\O_X)$ for
stable bundles $E_1$ and $E_2$ of the same degree $d$ and
ranks $r_1$ and $r_2$ satisfying $r_1 r_2\equiv 1\operatorname {mod} (d)$.
We denote by ${\cal M}(d,r)$ the projective space
$\P\operatorname{Ext}^1(E,\O_X)$ where $E$ is a stable bundle with
determinant $\O_X(dx_0)$ and
rank $r$ (in particular, $gcd(d,r)=1$), considered as a Poisson
variety. Then the above results show
that ${\cal M}(d,r)$ depends only on $d$ and on the residue
$r\in({\Bbb Z}/d{\Bbb Z})^*$, furthermore, we have
\begin{equation}\label{fourisom}
{\cal M}(d,r)\simeq{\cal M}(d,r^{-1}).
\end{equation}
Also for every stable bundle $E$ of degree $d>0$ and rank $r$
we have an isomorphism of Poisson varieties
$$\P H^0(E)\simeq{\cal M}(d,-r^{-1}).$$
The Poisson isomorphism (\ref{fourisom}) is the classical
limit of the following isomorphism of Sklyanin algebras
$$Q_{d,r}(x)\simeq Q_{d,r'}(x)$$
for every $\tau\in X$, where $rr'\equiv 1\operatorname {mod}(d)$
(cf. \cite{FOf}).
To see this isomorphism let us make the substitutions
$i=r'j'$, $j=r'i'$, $n=r'(n'+i'-j')$
in the
defining relation (\ref{relation}) of $Q_{d,r}(x)$.
Then using the relation $\theta_{-i}(-x)=a\cdot b^i\cdot\theta_i(x)$
(where $a$ and $b$ are some non-zero constants, $b^d=1$)
we can rewrite the quadratic relations in the form
\begin{equation}\label{relation2}
\sum_{n'\in{\Bbb Z}/d{\Bbb Z}}\frac{\theta_{j'-i'-(r'-1)n'}(0)}
{\theta_{r'n'}(x)\theta_{j'-i'-n'}(-x)}
t_{j'-n'}t_{i'+n'}=0.
\end{equation}
Now it is obvious
that the map $t_i\mapsto t_{r'i}$ defines an isomorphism
from $Q_{d,r}(x)$ to $Q_{d,r'}(x)$ as required.
\section{Birational transformations}
If one changes the stability parameter $\tau$ the moduli spaces
${\cal M}_{\tau}(L_1,L_2,r_1,r_2)$ undergo some birational transformations,
see \cite{T}, \cite{BDW}. Clearly,
these birational transformations are compatible with the Poisson
structures on their domain of definition.
In particular, considering moduli of pairs $\O_X\rightarrow E$ where
$\deg E=d$, $\operatorname{rk} E=r$ are such that $gcd (r,d)=1$ and $gdc(r+1,d)=1$
we get a Poisson birational transformation from ${\cal M}(d,r)$ to
${\cal M}(d,-(r+1)^{-1})\simeq\P H^0(E)$ where $E$ is a stable bundle
of degree $d$ and rank $r+1$.
Let us denote by $R_d\subset{\Bbb Z}/d{\Bbb Z}$ the set of residues $r$ such that
$r\in({\Bbb Z}/d{\Bbb Z})^*$ and $r+1\in({\Bbb Z}/d{\Bbb Z})^*$.
The map $\phi:r\mapsto -(r+1)^{-1}$ preserves $R_d$
and satisfies $\phi^3=\operatorname{id}$.
On the other hand, the involution
$\b:R_d\rightarrow R_d: r\mapsto r^{-1}$ also preserves $R_d$ and
we have $\phi\b=\b\phi^{-1}$. It follows that $\b$ and $\phi$
generate the action of the symmetric group $S_3$ on $R_d$.
Recall that in the previous section we defined an isomorphism
${\cal M}(d,r)\rightarrow{\cal M}(d,r^{-1})={\cal M}(d,\b(r))$.
\begin{thm}\label{s3} The birational morphisms
${\cal M}(d,r)\rightarrow{\cal M}(d,\phi(r))$ and ${\cal M}(d,r)\rightarrow{\cal M}(d,\b(r))$
defined above extend to an action of $S_3$ on
$\sqcup_{r\in R_d}{\cal M}(d,r)$.
\end{thm}
\noindent {\it Proof} . For every residue $r\in R_d$ let us denote by $E_r$ a
stable bundle with determinant $\O_X(dx_0)$ and rank
$\operatorname{rk} E\equiv r\operatorname {mod}(d)$ such that $0<\operatorname{rk} E<d$.
Let us check the relation $\phi^3=\operatorname{id}$. For this we have
to show that the corresponding composition of
birational transformations
$${\cal M}(d,r)\rightarrow{\cal M}(d,\phi(r))\rightarrow{\cal M}(d,\phi^2(r))\rightarrow{\cal M}(d,r)$$
is the identity. By definition the first arrow
is the composition of the map that associates to a
generic morphism $f:E_r\rightarrow\O_X[1]$ the corresponding morphism
$\operatorname{Cone}(f)[-1]:\O_X\rightarrow E_{r+1}$ (considered up to constant) with the
autoequivalence $T_r\in\widetilde{\operatorname{SL}}_2({\Bbb Z})$ which sends
$E_{r+1}$ to $\O_X[1]$ and $\O_X$ to $E_{\phi(r)}$.
It follows that the above triple composition amounts to
applying the construction $\operatorname{Cone}(\cdot)[-1]$ thrice (note
that in our situation this construction is functorial) and
applying the functor $T_{\phi^2(r)}T_{\phi(r)}T_r$.
The triple composition of $\operatorname{Cone}(\cdot)[-1]$ is isomoprhic to
the shift $\operatorname{id}[-2]$, while
$T_{\phi^2(r)}T_{\phi(r)}T_r=\operatorname{id}[2]$, hence the assertion.
It remains to check the relation between our birational
transformations corresponding to the relation
$\phi\b=\b\phi^{-1}$. This amount to checking the following
relation between contravariant functors from $D^b(X)$ to
itself:
$$DT_{\b\phi(r)}DU_{\phi(r)}T_r=U_r$$
where $D:{\cal D}^b(X)^{op}\rightarrow{\cal D}^b(X)$ is the duality functor
$E\mapsto R\underline{\operatorname{Hom}}(E,\O_X)$, $U_r\in\widetilde{\operatorname{SL}}_2({\Bbb Z})$
is the unique element sending $E_r$ to $\O_X$ and $\O_X$
to $E_{\b(r)}^*$. Note that for every $T\in\widetilde{\operatorname{SL}}_2({\Bbb Z})$
we have $DTD\in\widetilde{\operatorname{SL}}_2({\Bbb Z})$ (this follows from the
compatibility between the Fourier-Mukai transform and duality,
cf. \cite{Mukai}), moreover the corresponding involution on
$\operatorname{SL}_2({\Bbb Z})$ is just the conjugation by the matrix
$\left(\matrix 1 & 0 \\ 0 & -1 \endmatrix \right)$. Using this one can check
the above identity up to shift. Finally, since both parts send
$E_r$ to $\O_X$ the identity follows.
\qed\vspace{3mm}
It follows from the above theorem that
for every $\sigma\in S_3$ and $r\in R_d$ such that $\sigma(r)=r$
we get a birational automorphism $f_{\sigma}$ of ${\cal M}(d,r)$.
Since $\b$
acts as an isomorphism on our moduli spaces
there are essentially two different
cases to consider: the residue $r\in R_d$ is fixed $\phi$
or by $\phi\b$. The fixed points of $\phi$ are residues $r$ satisfying
$$r^2+r+1\equiv 0\operatorname {mod}(d).$$
In this case we get a Poisson birational automorphism
$f_{\phi}$ of ${\cal M}(d,r)$ such that $f_{\phi}^3=\operatorname{id}$.
The map $\phi\b$ has the only fixed point $r=-2\operatorname {mod}(d)$
(provided that $d$ is odd), so we get a Poisson birational
involution $f_{\phi\b}$ of ${\cal M}(d,d-2)$.
Let us describe these birational automorphisms more
explicitly.
Consider first the case when $r^2+r+1\equiv 0\operatorname {mod}(d)$, i.e.
$\phi(r)=r$.
Using the notation of the proof of the above theorem
we have $T_r(E_{r+1})=\O_X[1]$, $T_r(\O_X)=E_r$, and
$T_r(E_r)= E_{r+1}[1]$.
Now let us consider the closed subvariety of
$$Z\subset \P\operatorname{Hom}(E_{r+1},E_r)\times\P\operatorname{Ext}^1(E_r,\O_X)\times
\P\operatorname{Hom}(\O_X,E_{r+1})$$
consisting of triples of lines $([v_1],[v_2],[v_3])$
such that all three pairwise compositions $v_2\circ v_1$,
$v_3\circ v_2$ and $v_1\circ v_3$ are zeroes.
It is easy to see that any of three projections of $Z$ to
the projective spaces are birational. In fact, general point
of $Z$ corresponds to the exact triangle
$$\O_X\stackrel{v_3}{\rightarrow} E_{r+1}\stackrel{v_1}{\rightarrow} E_r
\stackrel{v_2}{\rightarrow}\O_X[1].$$
In particular, $Z$ is birational to
${\cal M}(d,r)=\P\operatorname{Ext}^1(E_r,\O_X)$. On the other hand,
the functor $T_r$ gives rise to the automorphism of $Z$ given by
$$\Phi:(v_1,v_2,v_3)\mapsto (T_r(v_2)[-1],T_r(v_3),T_r(v_1)[-1]).$$
Now $\Phi$ induces our birational automorphism of ${\cal M}(d,r)$ with cube
equal to the identity.
In the simplest non-trivial case $d=7$, $r=2$ the functor
we use has form $T_r:A\mapsto {\cal F}(A(-2x_0))(3x_0)[1]$.
In this case one can describe $Z$ as the double blow-up of
${\cal M}(7,2)$. For this it is more convenient to use the
isomorphism of ${\cal M}(7,2)$ with $\P H^0(E_3)$.
Then we have a natural embedding
$S^2 C\rightarrow\P H^0(E_3): D\mapsto H^0(E_3(-D))$ where $D$ is an effective divisor
of degree 2 on $X$. We also define the 4-dimensional
variety $V\subset \P H^0(E_3)$ containing $S^2 C$ as
the union of chords of $S^2 C\subset \P H^0(E_3)$ connecting
$D_1$ and $D_2$ in $S^2 C$ such that $D_1\cap
D_2\neq\emptyset$.
Then our variety $Z$ is obtained by first blowing-up of $\P
H^0(E_3)$ along
$S^2 C$, and then blowing up the proper transform of $V$.
The automorphism $\Phi$ cyclically permutes the following three
divisors on $Z$: two exceptional divisors and the proper
transform of the chord
variety of $S^2 C$ (which is a hypersurface in $\P H^0(E_3)$).
In the case $r\equiv -2\operatorname {mod}(d)$ (where $d$ is odd) the
birational autoequivalence of ${\cal M}(d,d-2)$ is described as
follows. Again using the notation from the proof of
Theorem \ref{s3} we have
$U_{d-1}(E_{d-1})=\O_X$, $U_{d-1}(\O_X)=E_{d-1}^*$ and
$U_{d-1}(E_{d-2})=E_{d-2}^*[1]$.
Now let us consider the subvariety
$$Y\subset \P\operatorname{Hom}(E_{d-1},E_{d-2})\times\P\operatorname{Ext}^1(E_{d-2},\O_X)$$
consisting of pairs $([v_1],[v_2])$ such that
the composition $v_2v_1\in\operatorname{Ext}^1(E_{d-1},\O_X)$ is zero.
Then both projections of $Y$ to projective spaces are
birational. On the other hand, the functor $U_{d-1}$ induces an
involution of $Y$ sending $(v_1,v_2)$ to
$(U_{d-1}(v_2)^*[1], U_{d-1}(v_1)^*[1])$,
hence our birational involution of
${\cal M}(d,d-2)=\P\operatorname{Ext}^1(E_{d-2},\O_X)$.
\section{Generalization}\label{gener}
In this section we consider a generalization of the main construction
of section \ref{mainsec} to the case of principal bundles with
other structural groups than $\operatorname{GL}$. For this note that the
datum of a triple $(E_1,E_2,\Phi)$ with $\operatorname{rk} E_i=r_i$ for
$i=1,2$ is equivalent to that of a pair $(P,s)$
where $P$ is a principal bundle with structure group
$\operatorname{GL}_{r_1}\times\operatorname{GL}_{r_2}$ and a section $s\in V(P)$
of the vector bundle $V(P)$ associated with the natural
representation of $\operatorname{GL}_{r_1}\times\operatorname{GL}_{r_2}$
on the space of $r_1\times r_2$-matrices.
To generalize this
let us consider a general reductive group $G$ and
its representation $V$. Then one can consider the moduli stack
${\cal M}_{G,V}$ of pairs $(P,s)$ where $P$ is a principal $G$-bundle,
$s\in H^0(X,V(P))$ is a global section of the corresponding
vector bundle associated to $V$ and $P$.
Let $\gg$ be the Lie algebra of $G$.
Assume that we are given a symmetric invariant tensor
$t\in S^2(\gg)^{\gg}$. Then $t$ induces a morphism of
$\gg$-modules $t_*:S^2(V)\rightarrow S^2(V)$ as follows.
Let $t=\sum_i x_i\otimes y_i$, then
$$t_*(v\otimes v)=\sum (x_i\cdot v)\otimes (y_i\cdot v)$$
where $\cdot$ denotes the action of $\gg$ on $V$.
Assume that $t_*=0$. Then fixing a trivialization
$\omega_X\simeq\O_X$ we can construct a Poisson bracket
on the smooth locus of ${\cal M}$ as follows.
The tangent space to ${\cal M}$ at a point $(P,s)$ can be identified with
the hypercohomology space $H^1(X,C^\cdot)$
where $C^{\cdot}$ is the complex
$\gg(P)\stackrel{d}{\rightarrow} V(P)$ concentrated in degrees $0$ and $1$,
where $\gg(P)$ is the vector bundle associated with $P$ and the
adjoint representation, the map $d$ is induced by the
Lie action of $\gg(P)$ on $V(P)$: $d(A)=A\cdot s$.
Hence, the contangent space can be identified with
$H^1(C^*[-1])$. Now we can construct the morphism
of complexes $\phi:C^*[-1]\rightarrow C$ as before
setting $\phi_1=0$ and
$\phi_0:V^*(P)\rightarrow\gg(P)$ to be the composition of
$d^*$ with the map $\gg^*(P)\rightarrow\gg(P)$ induced by $t$.
We claim that our condition on $t$ and $V$
implies that $d\circ\phi_0=0$ and that the obtained morphism
$H$ from the cotangent space of ${\cal M}$ to the tangent space is
skew-symmetric. Indeed, essentially we have to check
that for every $v\in V$ the following composition is zero:
$$V^*\stackrel{d_v^*}{\rightarrow}\gg^*\stackrel{t}{\rightarrow}\gg
\stackrel{d_v}{\rightarrow} V$$
where $d_v(A)=A\cdot v$. This is equivalent to the condition
$t_*(v\otimes v)=0$. Now the skew-symmetry follows as before:
the homotopy beteen $\phi$ and $\phi^*[-1]$ is constructed
using the map $\gg^*(P)\rightarrow\gg(P)$ induced by $t$.
\begin{thm} The above construction defines
a Poisson bracket on the smooth locus of ${\cal M}$.
\end{thm}
\noindent {\it Proof} . We have to check the Jacoby identity for our bracket.
We will use the approach similar to that of \cite{Bo},
\cite{Bo2}. The Jacoby identity can be rewritten in terms of the morphism
$H:T_{{\cal M}}^*\rightarrow T_{{\cal M}}$ as follows:
\begin{equation}\label{Jacobi}
H(\omega_1)\cdot\langle H(\omega_2),\omega_3\rangle-\langle [H(\omega_1),H(\omega_2)],
\omega_3\rangle+ cp(1,2,3)=0
\end{equation}
where $\omega_i\in T_{{\cal M}}^*$ are local 1-forms on ${\cal M}$,
$[\cdot,\cdot]$ is the commutator of vector fields,
$cp(1,2,3)$ indicates terms obtained by cyclic permutation
of 1,2 and 3 from the first two terms.
Working over an affine \'etale open $U\rightarrow{\cal M}$ we can represent
every 1-form $\omega\in T_{{\cal M}}^*(U)$ by a Cech cocycle
$(\phi_{ij},\psi_i)$ for some open covering $\{ U_i\}$
of $U\times X$, where $\phi_{ij}\in \Gamma(U_i\cap U_j,V^*(P))$,
$\psi_i\in\Gamma(U_i,\gg^*(P))$ are such that
$-d^*\phi_{ij}=\psi_j-\psi_i$ over $U_i\cap U_j$,
$P$ is the universal $G$-bundle on ${\cal M}$.
Similarly, every vector field $v\in T_{{\cal M}}(U)$
can be represented by a Cech cocycle
$(\a_{ij},\nu_i)$, where $\a_{ij}\in\Gamma(U_{ij},\gg(P))$,
$\nu_i\in\Gamma(U_i, V(P))$ are such that $d\a_{ij}=\nu_j-\nu_i$.
In terms of these representatives the pairing between $T^*_{{\cal M}}$
and $T_{{\cal M}}$ takes form
\begin{equation}
\langle (\a_{ij},\nu_i),(\phi_{ij},\psi_i)\rangle=
\operatorname{Tr}(\langle\a_{ij},\psi_j\rangle+\langle\phi_{ij},\nu_i\rangle),
\end{equation}
where $\operatorname{Tr}:H^1(U\times X,\O_{U\times X})\rightarrow H^0(U,\O_U)$ is the morphism
induced by the trivialization of $\omega_X$.
The map $H$ sends a 1-form $(\phi_{ij},\psi_i)$ to
the vector field represented by the cocycle $(t\circ d^*\phi_{ij},0)$
where $t$ is considered as a map $\gg^*\rightarrow\gg$.
Since $d^*\phi_{ij}=\psi_i-\psi_j$ we have
$$H(\phi_{ij},\psi_i)=(0,d\circ t(\psi_i))\operatorname {mod}(\operatorname{im}(\delta))$$
where $\delta$ is the differential in the Cech complex of
$C^{\cdot}$. Note that since $d\circ t\circ d^*=0$, we
have $d\circ t(\psi_i)=d\circ t(\psi_j)$ over $U_i\cap U_j$,
hence we obtain the global section $d\circ t(\psi_{\cdot})\in
\Gamma(U\times X, V(P))$.
It follows that
\begin{equation}
\langle H(\phi_{ij},\psi_i),(\phi'_{ij},\psi'_i)\rangle=
\operatorname{Tr}(\langle\phi'_{ij},d\circ t(\psi_i)\rangle)=
\operatorname{Tr}(\langle\psi'_i-\psi'_j, t(\psi_i)\rangle.
\end{equation}
Let us consider the relative Atiyah extension for the universal bundle $P$:
$$0\rightarrow\gg(P)\rightarrow\AA(P)\rightarrow p^*T_{{\cal M}}\rightarrow 0.$$
where $p:{\cal M}\times X\rightarrow{\cal M}$ is the projection, $\AA(P)$ is the
bundle of relative infinitesemal symmetries of $P$.
Then for sufficiently fine covering $\{ U_i\}$ a Cech cocycle
$(\a_{ij},\nu_i)$ representing a local vector field on
${\cal M}$ can be written as follows: $\a_{ij}=D_j-D_i$,
$\nu_i=D_i(s)$ where $D_i\in\Gamma(U_i,\AA(P))$, $s\in V(P)$
is the universal section, the symbol of $D_i$ is equal to the
restriction of a given vector field to $U_i$.
In particular, for a vector field
represented by a cocycle $(0,\nu)$ where
$\nu\in\Gamma(U\times X, V(P))$ we have $\nu=D(s)$ for some
$D\in\Gamma(U\times X,\AA(P))$.
After these remarks we can start proving (\ref{Jacobi}).
Let us denote by $(\phi^h_{ij},\psi^h_i)$
Cech cocycles representing 1-forms $\omega_h$ for $h=1,2,3$.
Let $D^h\in\Gamma(U\times X,\AA(P))$ be infinitesemal
symmetries corresponding to $H(\omega_h)$ so that
$D^h(s)=d\circ t(\psi_i)=t(\psi_i)(s)$ over $U_i$.
Then we have
$$H(\omega_1)\cdot\langle H(\omega_2),\omega_3\rangle=
D^1\cdot\operatorname{Tr}(\langle\psi^3_i-\psi^3_j, t(\psi^2_i)\rangle)=
\operatorname{Tr}(\langle D^1(\psi^3_i-\psi^3_j), t(\psi^2_i)\rangle+
\langle \psi^3_i-\psi^3_j, D^1(t(\psi^2_i))\rangle).$$
On the other hand, it is easy to compute the commutator:
$$[H(\omega_1),H(\omega_2)]=(0,D^1D^2(s)-D^2D^1(s))=
(0,D^1(t(\psi^2_i)(s))-D^2(t(\psi^1_i)(s))).$$
Hence, we have
\begin{align*}
&\langle [H(\omega_1),H(\omega_2)], \omega_3\rangle=
\operatorname{Tr}\langle\phi^3_{ij},
D^1(t(\psi^2_i)(s))-D^2(t(\psi^1_i)(s))\rangle=\\
&\operatorname{Tr}\langle\phi^3_{ij},D^1(t(\psi^2_i))(s)-D^2(t(\psi^1_i))(s)
+t(\psi^2_i)(D^1(s))-t(\psi^1_i)(D^2(s))\rangle=
\\
&\operatorname{Tr}\langle\psi^3_i-\psi^3_j,D^1(t(\psi^2_i))-D^2(t(\psi^1_i))-
[t(\psi^1_i),t(\psi^2_i)]\rangle.
\end{align*}
It follows that
\begin{align*}
&H(\omega_1)\cdot\langle H(\omega_2),\omega_3\rangle-
\langle [H(\omega_1),H(\omega_2)], \omega_3\rangle=\\
&\operatorname{Tr}(\langle D^1(\psi^3_i-\psi^3_j), t(\psi^2_i)\rangle+
\langle\psi^3_i-\psi^3_j,t(D^2(\psi^1_i))+
[t(\psi^1_i),t(\psi^2_i)]\rangle=\\
&
\operatorname{Tr}(\langle D^1(\psi^3_i-\psi^3_j), t(\psi^2_i)\rangle+
\langle D^2(\psi^1_i), t(\psi^3_i-\psi^3_j)\rangle+
\langle\psi^3_i-\psi^3_j,[t(\psi^1_i),t(\psi^2_i)]\rangle)
\end{align*}
Since this is equal to
$D^1\cdot\operatorname{Tr}\langle\phi^3_{ij},dt(\psi^2_{\cdot})\rangle-
\operatorname{Tr}\langle\phi^3_{ij},[D^1,D^2](s)\rangle$
which is skew-symmetric in $i,j$,
we can also skew-symmetrize in $i,j$ the expression obtained above.
Then after adding terms obtained
by cyclic permutation of $1$, $2$ and $3$ we obtain the
trace of the following expression
$$\langle\psi^3_i-\psi^3_j,
[t(\psi^1_i),t(\psi^2_i)]+[t(\psi^1_j),t(\psi^2_j)]\rangle+
cp(1,2,3).$$
Up to a coboundary this is equal to
$$\langle\psi^3_i,\psi^1_j,\psi^2_j\rangle_t-
\langle\psi^3_j,\psi^1_i,\psi^2_i\rangle_t+ cp(1,2,3),$$
where we denote $\langle x,y,z\rangle_t=\langle x, [t(x),t(y)]\rangle$.
From the fact that $t\in (S^2\gg)^*$ one can deduce
easily that $\langle \cdot,\cdot,\cdot\rangle_t$ is $\gg$-invariant
and skew-symmetric. This implies the following identity:
$$\langle\psi^3_i,\psi^1_j,\psi^2_j\rangle_t-
\langle\psi^3_j,\psi^1_i,\psi^2_i\rangle_t+ cp(1,2,3)
=\langle\psi^1_i-\psi^1_j,\psi^2_i-\psi^2_j,\psi^3_i-
\psi^3_j\rangle_t=
-\langle d^*\phi^1_{ij},d^*\phi^2_{ij},d^*\phi^3_{ij}\rangle_t.$$
It remains to notice that $\langle d^*\phi^1, d^*\phi^2,
d^*\phi^3\rangle_t=0$
for any $\phi^1,\phi^2,\phi^3\in V^*(P)$.
Indeed, we have to show that for any triple of elements
$\varphi_1,\varphi_2,\varphi_3\in V^*$ and any $v\in V$ one has
$$\langle d^*_v\varphi_1, d^*_v\varphi_2, d^*_v\varphi_3\rangle_t=0$$
where $d^*_v\varphi_h\in\gg^*$ is defined by
$d^*_v\varphi_h(x)=\varphi_h(x\cdot v)$, $h=1,2,3$.
Let $t=\sum_i x_i\otimes y_i$. Then
\begin{align*}
& \langle d^*_v\varphi_1,d^*_v\varphi_2,d^*_v\varphi_3\rangle_t=
\langle d^*\varphi_1,
\sum_{i,j}[d^*\varphi_2(y_i)x_i,d^*\varphi_3(y_j)x_j]\rangle=\\
&\sum_{i,j}\varphi_1([x_i,x_j]\cdot v)\varphi_2(y_i\cdot v)
\varphi_3(y_j\cdot v).
\end{align*}
Now
$$\sum_{i,j} [x_i,x_j]\cdot v\otimes y_i\cdot v\otimes
y_j\cdot v=
\sum_{i,j}x_ix_jv\otimes y_iv\otimes y_jv-
\sum_{i,j}x_jx_iv\otimes y_iv\otimes y_jv=0$$
since $\sum_i x_iv\otimes y_iv=0$.
\qed\vspace{3mm}
Notice that the condition $t_*=0$ is usually not satisfied
when $\gg$ is simple. However, for example if $S^2(V)$
is irreducible and $\gg$ is simple then we necessarily
have $t_*=\lambda\cdot\operatorname{id}$ for some scalar $\lambda$. It follows
that we can replace $G$ by its product $G\times{\Bbb G}_m$ with one-dimensional
torus, $t$ by its sum with the appropriate multiple of the
square of the generator of $\operatorname{Lie}({\Bbb G}_m)$, so that for the new
tensor $t'$ and the same representation $V$ (on which ${\Bbb G}_m$ acts
via identity character) the condition $t'_*=0$ will be satisfied.
In the case $G=\operatorname{GL}_{r_1}\times \operatorname{GL}_{r_2}$ and $V=\operatorname{Mat}(r_1,r_2)$
the tensor $t$ is equal to $(t_1, -t_2)$ where
$t_1=\sum E_{ij}\otimes E_{ji}$ is the standard symmetric
invariant tensor for ${\frak gl}_{r_1}$, $t_2$ is the similar
tensor for ${\frak gl}_{r_2}$.
An interesting case is $G=\operatorname{GSp}_{2r}$, the group of invertible
matrices preserving the symplectic from up to a scalar.
In this case we can take $V$ to be the standard
representation of $G$ of rank $2r$, then $S^2(V)$ is
isomorphic to the adjoint representation of $\Sp_{2r}$
on which $\operatorname{GSp}_{2r}$ acts naturally, and one can easily
find a non-zero invariant tensor $t\in (S^2\gg)^{\gg}$ with $t_*=0$
(in fact, such $t$ is unique up to a constant).
The corresponding moduli stack is the stack of the following
data: a vector bundle $E$ together with a symplectic form
$$E\otimes E\rightarrow L$$
inducing an isomorphism $E\simeq E^*\otimes L$, and a section
$s:\O_X\rightarrow E$. In particular, taking $E$ to be a fixed bundle
with a symplectic form as above, we can consider sometimes
the appropriate quotient space of $\P H^0(E)$ by the group
of $\operatorname{GSp}$-automorphisms of $E$
as a Poisson substack in the above stack.
More precisely, we can define the stability condition for
such pairs $(E, s)$ depending on a parameter $\tau$:
the only difference with the case of $\operatorname{GL}$ is that one
should consider totally isotropic subbundles of $E$.
Then for $\tau=\mu(E)+\epsilon$ we have the Casimir morphism
from such moduli space to the stack of semistable
$\operatorname{GSp}$-bundles, hence its fibers inherit the Poisson structure.
For example, if $E_0$ is a stable bundle of degree 2
then there is a natural $\operatorname{GSp}_4$-structure on $E=E_0\oplus E_0$
such that both summands are totally isotropic. Then
the $\tau$-stability condition (with $\tau=\mu(E_0)+\epsilon)$
allows only sections $s\in H^0(E)=
H^0(E_0)\oplus H^0(E_0)$ with non-zero projections to both summands.
Hence, the space of such sections up to the action of symplectic
automorphisms of $E$ is $S^2\P H^0(E)$, so we get a Poisson
structure on the latter variety.
|
1999-02-08T22:55:15 | 9712 | alg-geom/9712028 | en | https://arxiv.org/abs/alg-geom/9712028 | [
"alg-geom",
"math.AG"
] | alg-geom/9712028 | Victor Vinnikov | Joseph A. Ball (Virginia Tech) and Victor Vinnikov (Weizmann
Institute) | Zero-pole interpolation for matrix meromorphic functions on a compact
Riemann surface, and a matrix Fay trisecant identity | AMS-LaTeX, 37 pages; final version (several references added and some
misprints corrected), to appear in Amer. J. of Math | null | null | null | null | This paper presents a new approach to constructing a meromorphic bundle map
between flat vector bundles over a compact Riemann surface having a prescribed
Weil divisor (i.e., having prescribed zeros and poles with directional as well
as multiplicity information included in the vector case). This new formalism
unifies the earlier approach of Ball-Clancey (in the setting of trivial bundles
over an abstract Riemann surface) with an earlier approach of the authors
(where the Riemann surface was assumed to be the normalizing Riemann surface
for an algebraic curve embedded in ${\bold C}^2$ with determinantal
representation, and the vector bundles were assumed to be presented as the
kernels of linear matrix pencils). The main tool is a version of the Cauchy
kernel appropriate for flat vector bundles over the Riemann surface. Our
formula for the interpolating bundle map (in the special case of a single zero
and a single pole) can be viewed as a generalization of the Fay trisecant
identity from the usual line bundle case to the vector bundle case in terms of
Cauchy kernels. In particular we obtain a new proof of the Fay trisecant
identity.
| [
{
"version": "v1",
"created": "Tue, 23 Dec 1997 23:04:31 GMT"
},
{
"version": "v2",
"created": "Mon, 8 Feb 1999 21:55:15 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Ball",
"Joseph A.",
"",
"Virginia Tech"
],
[
"Vinnikov",
"Victor",
"",
"Weizmann\n Institute"
]
] | alg-geom | \section{Introduction} \label{S:intro}
The following zero-pole interpolation problem is one of the main objects
of study in the recent monograph \cite{bgr}. We state here the simplest
case where all zeros and poles are assumed to be simple and disjoint.
{\sl Given a finite collection $\lambda^1, \dots,
\lambda^{n_0}, \mu^1, \dots, \mu^{n_\infty}$ of distinct points in the
complex plane ${\bold C}$, nonzero column vectors $u_1, \dots,
u_{n_\infty} \in {\bold C}^{r \times 1}$ and nonzero row vectors $x_1,
\dots, x_{n_0} \in {\bold C}^{1 \times r}$, find (if possible) an $r
\times r$ matrix function $T(z)$ having value equal to the identity matrix
$I$ at infinity such that
(1) $T(z)$ is analytic on $({\bold C} \cup \{\infty\})
\backslash \{\mu^1, \dots,
\mu^{n_\infty}\}$ and $T(z)^{-1}$ has analytic continuation to $({\bold
C} \cup \{\infty \}) \backslash \{\lambda^1, \dots, \lambda^{n_0} \}$,
(2) for $i=1,
\dots, n_0$, $T(\lambda^i)$ has rank $r-1$ and $x_i T(\lambda^i) = 0$,
and (3) for $j=1, \dots, n_\infty$, $T(\mu_j)^{-1}$ (i.e., the analytic
continuation of $T(z)^{-1}$ to $z=\mu^j$) has rank $r-1$ and $T(\mu^j)^{-1}
u_j = 0$.} A complete solution, along with numerous applications to
problems in factorization, matrix interpolation and $H_\infty$-control,
is given in \cite{bgr}. The solution (for this simple case) is
as follows. {\sl A solution exists if and only if the $n_0 \times
n_\infty$ matrix
\begin{equation}
\Gamma = [\Gamma_{ij}]\text{ with } \Gamma_{ij} = \dfrac{x_i u_j}{\mu^j -
\lambda^i}
\label{Gamma}
\end{equation}
is square and invertible. In this
case the unique solution is given by
\begin{equation}
T(z) = I + \sum_{j=1}^{n_\infty} u_j (z-\mu^j)^{-1} c_j
\label{genus0solution}
\end{equation}
where $c= \begin{bmatrix} c_1 & \dots & c_{n_\infty} \end{bmatrix} ^T$ is
given by}
\begin{equation}
c= \Gamma^{-1} \begin{bmatrix} x_1^T & \dots & x_{n_0}^T \end{bmatrix}^T.
\label{scalarcoefficients}
\end{equation}
The solution in \cite{bgr} uses system theory ideas, especially the state
space similarity theorem specifying the level of uniqueness for two
realizations of the same rational matrix function as the transfer function
of a linear system. Later work (see \cite{bgrak}) handles the nonregular
case (where $\det T(z)$ vanishes identically and the nature of the zero
structure must be enlarged) by elementary linear algebra, without
recourse to the state space similarity theorem.
There have now appeared two seemingly distinct generalizations of this
result to higher genus. In \cite{bc}, the problem is posed to construct
a (global, single-valued) meromorphic matrix function on the compact
Riemann surface $X$ satisfying conditions as in (1), (2) and (3) above.
A matrix analogous to $\Gamma$ appears, but the solution criterion is not
as simple; nevertheless, an explicit formula analogous to
\eqref{genus0solution} and \eqref{scalarcoefficients} was found for the
solution when it exists. The approach in \cite{bc} can be seen as an analogue
of that in \cite{bgrak} (i.e., system theory ideas are avoided
and a simple ansatz is used to reduce the problem to an analysis of a
linear system of equations). The
paper \cite{hip}, on the other hand, while formulating a more general
problem (involving bundle maps between certain types of flat vector
bundles rather than global meromorphic matrix functions) in an abstract
setting, works primarily in a more concrete setting, where the Riemann
surface $X$ is taken to be the normalizing Riemann surface for an
algebraic curve $C$ having a determinantal representation
\[
C = \{ (\lambda_1, \lambda_2) \in {\bold C}^2 \colon \det (\lambda_1
\sigma_2 - \lambda_2 \sigma_1 + \gamma) = 0 \}
\]
(where $\sigma_1, \sigma_2, \gamma$ are $M \times M$ matrices and
$\lambda = (\lambda_1, \lambda_2)$ are affine coordinates),
and the input and output
bundles $E$ and $\widetilde{E}$ are assumed to have kernel
representations, e.g.,
\[
E(\lambda) = \{ v \in {\bold C}^M \colon (\lambda_1 \sigma_2 - \lambda_2
\sigma_1 + \gamma) v = 0\}.
\]
In this
setting, a non-metric version of the several-variable system theory
connected with the model theory for commuting operators due to Livsic
(i.e., a version with all Hilbert space inner products dropped) applies,
any meromorphic bundle map satisfying appropriate conditions at the
points at infinity can be realized as the transfer function, or the
joint characteristic function, of a Livsic-Kravitsky 2D system, and the
zero-pole bundle-map interpolation problem can be solved using the
state-space similarity theorem for this setting in a manner
completely parallel to that of
\cite{bgr} for the genus 0 case. (For a recent systematic treatment of
the Livsic theory, we refer to \cite{lkmv} and \cite{vinsurvey}).
In this solution there is a
matrix $\Gamma$ analogous to the $\Gamma$ in \eqref{Gamma} along with an
explicit formula for the solution (when it exists) as in
\eqref{genus0solution}. In this setting, one specifies an output bundle
$\widetilde{E}$ having a kernel bundle representation as well as the zero-pole
interpolation data. The invertibility of $\Gamma$ is then equivalent to
the existence of an input bundle $E$ also having a kernel representation
together with a bundle map $T \colon E \to \widetilde{E}$ meeting the
zero-pole interpolation conditions.
The purpose of this paper is to synthesize these two approaches. We
obtain a generalization of the approach of \cite{bc} which handles the
vector bundle problem, and clarify the solution criterion as well as the
role of the invertibility of $\Gamma$ in this abstract setting. To
obtain an analogue of the basic ansatz in \cite{bc} used for the form of
the solution for the general bundle case, we need a version of the
Cauchy kernel $(z,w) \to \frac{1}{z-w}$ for sections of a flat vector
bundle $\chi$ satisfying $h^0(\chi \otimes \Delta) = 0$ where $\Delta$
is a line bundle of differentials of order 1/2 (a theta characteristic or
a spin structure).
This object was introduced in \cite{hip} (see also \cite{AV2}, \cite{vin1}
for the line bundle case)
but a proof of its existence for the vector bundle case relied on the
theory of determinantal representations of algebraic curves and of kernel
representations of bundles over such curves. Here we give a simple,
direct existence proof using only some cohomology theory of vector
bundles and the Riemann-Roch theorem. A similar proof of the same result
for the line bundle case is given in \cite{raina1} and \cite{raina2}; the
general case is also handled in \cite{newfay} but by using completely
different techniques (involving the theory of the Green's function for
the heat equation over $X$). Various other forms of the Cauchy kernel
for a Riemann surface have appeared earlier in the literature, in
particular in connection with the Riemann-Hilbert problem (see
\cite{Rodin}, \cite{Zverovich}). However, these are developed within the
framework of meromorphic differentials whereas our Cauchy kernel is
defined as a multiplicative meromorphic differential of order $1/2$. The
use of half-order differentials has the advantage that no extraneous
poles are introduced in the Cauchy kernel. We mention that the paper
\cite{AV} applies our Cauchy kernel to the study of indefinite Hardy
spaces on a finite bordered Riemann surface.
Taking the constant term in the Laurent expansion of the Cauchy kernel
around the diagonal allows us to define a certain flat connection on the
flat bundle $\chi$. In the concrete setting of the determinantal
representations, this connection was already introduced in \cite{hip}.
This flat connection is determined canonically up to a choice of a bundle
$\Delta$ of
half-order differentials.
We also make explicit the mappings between the concrete and abstract
settings; in this way we are able to see explicitly the equivalence
between the solution in \cite{hip} and the solution in \cite{bc}.
The main ingredient is the explicit formula for the determinantal
representation of an algebraic curve
with a given kernel bundle in terms of the Cauchy kernel
of the bundle.
We also specialize the results to the line bundle case. For this case,
both the solution to the zero-pole interpolation problem and the Cauchy
kernel can be expressed explicitly in terms of theta functions (see
\cite{oldfay}, \cite{mumford} and \cite{farkaskra} for background
material on theta functions). When this is done, the equality between
these two forms of the solution of the interpolation problem leads to a
new proof of the trisecant identity due to Fay (see \cite{oldfay}
Corollary 2.19 or \cite{mumford} Volume II page 3.214).
In the general vector bundle case, the formula for the solution of the
interpolation problem in terms of the Cauchy kernels (in the case of a
single zero and a single pole---see \eqref{3.12a}) can be viewed as a
matrix version of the Fay trisecant identity.
We mention that
in the genus 1 case one can obtain an explicit formula for the Cauchy
kernel for the general case of flat vector bundles (see \cite{bcv}).
We close the introduction by mentioning three other possible further
applications of our Cauchy kernel.
First of all, as will be shown in Section \ref{S:detrep}, the vector
bundle $\chi$ is completely determined by the values $K(\chi; x^i,x^j)$
of the Cauchy kernel at a certain finite collection of points $x^1,
\dots, x^m$ (forming a line section in a birational planar embedding of
$X$). This suggests that these values can be used as affine coordinates
for the bundle $\chi$ in the corresponding
moduli space of semistable bundles on the complement of the generalized
theta divisor. A very
similar construction for line bundles on hyperelliptic curves is due to
Jacobi (see \cite{Jacobi}) and has been given a modern treatment by
Mumford (see Volume II of \cite{mumford}).
Secondly, in the line bundle case consideration of the Fay trisecant
identity when some of the points come together leads to very interesting
identities, showing in particular that the theta function satisfies the
KP equations (see \cite{oldfay}, Volume II of \cite{mumford} and
\cite{Shiota}). An interesting line of research is to consider similar
limiting versions of our matrix Fay trisecant identity \eqref{3.12a}.
A related problem is to find the relation between the Cauchy kernel
and the matrix Baker-Akhiezer function of Krichever and Novikov \cite{KrNo}
whose definition involves the so called Tjurin parameters \cite{Tju1,Tju2}
of the vector bundle.
Thirdly, the absence of explicit formulas for the Cauchy kernel makes it
interesting to try to find formulas for the Cauchy kernel of one vector
bundle in terms of the Cauchy kernel for another. In particular it would
be interesting to find how the Cauchy kernel behaves under pullback and
direct image.
The paper is organized as follows. Section \ref{S:intro} is this
introduction. Section \ref{S:cauchyker} develops the Cauchy kernel for a
flat line bundle. Section \ref{S:absint} then formulates and solves the
zero-pole interpolation problem in the abstract setting. Section
\ref{S:linebundle} obtains explicit formulas for all the results in the
line bundle case and obtains the new proof of the Fay trisecant
identity. Section \ref{S:detrep} explains how to use Cauchy kernels to
obtain a canonical map from the abstract to the concrete setting.
Finally, Section \ref{S:conint} explains the connections with the
concrete interpolation problem solved in \cite{hip}.
\section{The Cauchy kernel for a flat vector bundle}
\label{S:cauchyker}
We assume that we are given a compact Riemann surface $X$.
Let $\Delta$ be a line bundle of differentials of order
$\frac{1}{2}$ on $X$, i.e., a line bundle satisfying $\Delta \otimes
\Delta \cong K$, where $K$ is the canonical line bundle (i.e., the
line bundle with local holomorphic sections equal to local holomorphic
differentials on $X$). Note that since $\deg(K) = 2g-2$, $\deg(\Delta) =
g-1$ where $g$ is the genus of $X$.
In addition we assume that we are given a holomorphic
complex vector bundle $\chi$ of degree $0$ (and of rank $r$, say)
over $X$ such that
\[
h^0(\chi \otimes \Delta) = 0,
\]
i.e., $\chi \otimes \Delta$ has no nonzero global holomorphic sections.
The condition implies that $\chi$ is necessarily semistable, and means
that the (equivalence class of) $\chi$ lies on the complement of the
generalized theta divisor in the moduli space of semistable vector
bundles of rank $r$ and degree $0$ on $X$ (see \cite{newfay},
\cite{Seshadri} and \cite{Drezet}).
It also follows immediately from Weil's criterion for flatness \cite{Gunning}
that $\chi$ is actually a flat vector bundle
(see \cite{hip} page 275 for details). By definition,
since $\chi$ is flat, sections $h$ of $\chi$ have the property that they
lift to ${\bold C}^r$-vector functions $\widetilde{h}$
defined on the universal cover
$\widetilde X$ of $X$ such that
\[
h(R \widetilde{p}) = \chi(R) h(\widetilde{p})
\]
for all $\widetilde{p} \in \widetilde{X}$ where $R$ is any element of the
group of deck transformations Deck($\widetilde{X}/X)$ $ \cong \pi_1(X)$
and where $R \to \chi(R) \in GL(r)$ is a (constant)
factor of automorphy associated with the bundle $\chi$.
(We somewhat abuse the notation denoting a constant factor of automorphy
and the corresponding flat vector bundle by the same letter.)
The main object of this section is to define an object $K(\chi; \cdot,
\cdot)$ associated with any such bundle $\chi$ which we shall
call the {\it Cauchy kernel} for the bundle $\chi$. Let $M$ denote the
Cartesian product $M=X \times X$ and let $\pi_1 \colon M \to X$ be the
projection map onto the first coordinate and $\pi_2 \colon M \to X$ the
projection onto the second coordinate. The defining property of $K(\chi;
\cdot, \cdot)$ is that $K(\chi; \cdot, \cdot)$ be a meromorphic
mapping of the vector bundles $\pi_2^* \chi$ and $\pi_1^* \chi \otimes
\pi_1^*\Delta \otimes \pi_2^* \Delta$ on $M$ which is holomorphic outside
of the diagonal ${\cal D} = \{(p,p) \in M \colon p \in X\}$, where it has a
simple pole with residue $I_r$. More precisely, the latter condition
means the following: for any $\widetilde{p}_0 \in \widetilde{X}$ and any local
parameter $t$ on $X$, if we let $\sqrt{dt}$ be the corresponding local
holomorphic frame for $\Delta$ lifted to a neighborhood of
$\widetilde{p}_0$ on $\widetilde{X}$, then near $(\widetilde{p}_0,
\widetilde{p}_0) \in \widetilde{X} \times \widetilde{X}$ the lift of
$K(\chi; \cdot, \cdot)$ to $\widetilde{X} \times \widetilde{X}$ has the form
\[
\frac{ K(\chi; \widetilde{p},\widetilde{q}) }{ \sqrt{dt}(\widetilde{p})
\sqrt{dt}(\widetilde{q}) } =
\frac{1}{ t(\widetilde{p}) - t(\widetilde{q}) }
\left[ I_r + O\left(\sqrt{ |t(\widetilde{p})|^2 + |t(\widetilde{q})|^2 }
\right) \right].
\]
Thus, if $e$ is in the fiber of $\chi$ at a point $q$ on $X$, and $t$ is
a local parameter of $X$ centered at $q$, then $K(\chi; \cdot, q)
\frac{e}{\sqrt{dt}(q)}$ is a meromorphic section of $\chi \otimes \Delta$
that has a single simple pole at $q$, with a residue (in terms of the
local parameter $t$) equal to $e \sqrt{dt}(q)$. Note that since $\chi
\otimes \Delta$ has no nontrivial global holomorphic sections, such a
meromorphic section is unique whenever it exists.
Note that when $X$ is the Riemann sphere and $\chi$ is (necessarily)
trivial, then
\[K(\chi;p,q) = \frac{I_r}{t(p) - t(q)} \sqrt{dt}(p) \sqrt{dt}(q)
\]
where $t$ is the standard coordinate on the complex plane, i.e., we get
the usual Cauchy kernel.
Existence was shown in \cite{hip} by exhibiting an explicit formula for
$K(\chi; \cdot, \cdot)$; the construction involved
using a representation of $X$ as the normalized Riemann surface for an
algebraic curve $C$ embedded in ${\bold P}^2$ and representing the bundle $E
= \chi \otimes \Delta \otimes {\cal O}(1)$ as the kernel bundle
associated with a determinantal representation of the curve $C$
\begin{gather}
C = \{[\mu_0,\mu_1,\mu_2] \in {\bold P}^2 \colon \det (\mu_1\sigma_2 - \mu_2
\sigma_1 + \mu_0 \gamma) = 0\} \notag \\
E(\mu) = \ker (\mu_1\sigma_2 - \mu_2
\sigma_1 + \mu_0 \gamma).
\notag
\end{gather}
When the rank $r$ of the vector bundle $\chi$ is 1, one can get an
explicit formula (in terms of the Abel-Jacobi map and classical theta
functions on the Jacobian variety of $X$) for the Cauchy kernel (see
\cite{hip}). Details of this formula will be reviewed in Section
\ref{S:linebundle} of this paper, where other special aspects of the line bundle
case will also be discussed.
Our purpose in this section is to give an alternative proof of the existence
of such a Cauchy kernel $K(\chi; \cdot, \cdot)$ for
a flat vector bundle $\chi$ by a direct, simple, more abstract argument
(without relying on representing $X$ as the normalizing Riemann surface for a
curve $C$ having a determinantal representation as in \cite{hip}).
This is the content of the following theorem.
\begin{theorem} \label{T:cauchyker}
Let $\chi$ be a flat vector bundle over the Riemann surface $X$ with
$h^0(\chi \otimes \Delta) = 0$ as above. Then the Cauchy kernel $K(\chi;
\cdot, \cdot)$ exists, i.e., there is a unique meromorphic
mapping of the vector bundles $\pi_2^* \chi$ and $\pi_1^* \chi \otimes
\pi_1^*\Delta \otimes \pi_2^* \Delta$ on $M=X \times X$
which is holomorphic outside
of the diagonal ${\cal D} = \{(p,p) \in M \colon p \in X\}$, where it has a
simple pole with residue $I_r$.
\end{theorem}
\begin{pf}
Note that we have the following exact sequence of vector bundles over
$M=X \times X$:
\begin{equation} \label{exseq1}
0 \to {\cal O}(-{\cal D}) \to {\cal O} \to {\cal O}|_{\cal D} \to 0.
\end{equation}
For ease of notation, define vector bundles $F$ and $W$ over $M$ and $V$
and $K$ over $X$ by
\begin{gather}
F= \pi_1^* \chi \otimes \pi_1^* \Delta \otimes \pi_2^* \chi^\vee \otimes
\pi_2^* \Delta \notag \\
W = F \otimes {\cal O}({\cal D}) \notag \\
V = \chi \otimes \Delta \notag \\
K = \text{ the canonical line bundle on } X \notag
\end{gather}
where $\chi^\vee$ is the dual bundle of $\chi$.
Tensoring the exact sequence \eqref{exseq1} with $W$ gives us
\begin{equation} \label{exseq2}
0 \to F \to W \to W \otimes {\cal O}|_{\cal D} \to 0.
\end{equation}
The map of taking the residue along the diagonal defines a linear mapping
$$
{\cal R} \colon H^0(M,W) \rightarrow H^0(X, End\ V).
$$
Our goal is to
show that there exists a unique element $K(\chi; \cdot, \cdot)$ of
$H^0(M,W)$ so that ${\cal R}(K(\chi; \cdot, \cdot)) = I_V$.
Note that the bundle $W \otimes {\cal O}|_{\cal D}$ can be identified
with the bundle $End\ V$ of endomorphisms of $V$. Moreover the residue
mapping ${\cal R}$ is exactly the mapping from $H^0(M,W)$ into $H^0(X,
End\ V)$ induced by the mapping $W \rightarrow \left. W \otimes
{\cal O}\right|_{\cal D}$ in \eqref{exseq2}.
The vector bundle exact sequence \eqref{exseq2}
induces (see the Basic Fact
on page 40 of \cite{gh}) the exact cohomology sequence
\begin{align} \notag
0 \to H^0(M,F) & \to H^0(M,W) \to H^0(X, End\ V) \to \\
\to H^1(M, F) & \to \cdots.
\label{exseq3}
\end{align}
Next we argue that (i) $h^0(F) = 0$ and (ii) $h^1(F) = 0$. The
statement (i) follows easily from our assumption that $h^0(V) = 0$.
As for statement (ii) it follows from the Kunneth
formulas (see page 58 of \cite{gh}) that
\begin{align}
H^1(M,F) = & H^1(M, \pi_1^*(V) \otimes \pi_2^*( V^\vee
\otimes K))
\notag \\
\cong & \left(H^0(M, \pi_1^*(V)) \otimes H^1(M, \pi_2^*(V^\vee
\otimes K)) \right) \\
& \oplus \left( H^1(M, \pi_1^*(V)) \otimes
H^0(M, \pi_2^*(V^\vee \otimes K) ) \right). \label{kunneth}
\end{align}
By our assumption that $h^0(V)=0$ it follows that the
first term on the right hand side of \eqref{kunneth} is 0. Since
$h^0(V) = 0 $ we also have $h^0(V^\vee \otimes
K) = 0$ as well, by the Riemann-Roch Theorem for vector bundles on an
algebraic curve (see \cite{Gunning}) and the assumption that deg $V=r(g-1)$.
Hence the second term in \eqref{kunneth} is zero as
well. This verifies the desired fact (ii).
Hence the exact sequence \eqref{exseq3} collapses to
\begin{equation} \label{exseq4}
0 \to H^0(M,W) \overset{{\cal R}}{\to} H^0(X,End\ V) \to 0.
\end{equation}
It follows that ${\cal R} \colon H^0(M,W) \to H^0(X,End\ V)$
is an isomorphism and the Theorem follows.
\end{pf}
Let $\widetilde{p}_0$ be an arbitrary point of $\widetilde{X}$, and let
$t$ be a local coordinate for $\widetilde{X}$ near $\widetilde{p}_0$.
Then by definition the Cauchy kernel $K(\chi; \cdot, \cdot)$ is such that
\[
\left(t(\widetilde p) - t(\widetilde q) \right)
\frac{K(\chi; \widetilde p, \widetilde q)}{\sqrt{dt}(\widetilde p)
\sqrt{dt}(\widetilde q)}
\]
is analytic in $(\widetilde p, \widetilde q)$ near $(\widetilde p_0,
\widetilde p_0)$ with value at $(\widetilde p_0, \widetilde p_0)$ equal
to $I_r$; hence, for $(\widetilde p, \widetilde q)$ close to $(\widetilde
p_0, \widetilde p_0)$, $K(\chi; \cdot, \cdot)$ has a representation of
the form
\begin{multline} \label{cauchyexp}
\frac{K(\chi; \widetilde p, \widetilde q)}{\sqrt{dt}(\widetilde p)
\sqrt{dt}(\widetilde q)} \\
= \frac{1}{t(\widetilde p) - t(\widetilde q)}
\left[ I_r + \frac{A_\ell}{dt}(\widetilde p_0) t(\widetilde p) +
\frac{A}{dt}(\widetilde p_0) t(\widetilde q) + O\left( |t(\widetilde
p)|^2 + |t(\widetilde q)|^2 \right) \right]
\end{multline}
for appropriate holomorphic matrix $K$-valued coefficients $A(\widetilde
p_0)$ and $A_\ell(\widetilde p_0)$. An important point for us is the
following result.
\begin{lemma} \label{L:cauchyexp}
Let $A_\ell$ and $A$ be the linear coefficients appearing
in the Laurent expansion of the Cauchy kernel $K(\chi; \cdot, \cdot)$ as
in \eqref{cauchyexp}. Then
\[A(\widetilde p_0) + A_\ell(\widetilde p_0) = 0
\]
for all $\widetilde p_0 \in \widetilde X$.
\end{lemma}
\begin{pf}
Let $\widetilde{p}_0$ be an arbitrary point of $\widetilde X$ and let $t$
be a local coordinate for $\widetilde X$ near $\widetilde p_0$. For
$\widetilde p \in X$ near $\widetilde p_0$, define
$$
f(\widetilde p) = (t(\widetilde p) - t(\widetilde q)) \cdot
\left. \frac{K(\chi; \widetilde p, \widetilde q)}{\sqrt{dt}(\widetilde p)
\sqrt{dt}(\widetilde q)} \right|_{\widetilde p = \widetilde q}.
$$
From \eqref{cauchyexp} we see that
$$
f(\widetilde p) = I_r + \left[ \frac{A_\ell}{dt}(\widetilde p_0) +
\frac{A}{dt}(\widetilde p_0)\right] t(\widetilde p) + O(|t(\widetilde p)|^2).
$$
In particular,
\begin{equation} \label{der1}
\left. \frac{df}{dt}(\widetilde p) \right|_{\widetilde p = \widetilde p_0} =
\frac{A_\ell}{dt}(\widetilde p_0) + \frac{A}{dt}(\widetilde p_0).
\end{equation}
On the other hand, we can use $t'(\widetilde p') = t(\widetilde p') -
t(\widetilde p)$ as a local coordinate for the variable $\widetilde p'$
near the point $\widetilde p \in X$. From \eqref{cauchyexp} again we have
\begin{multline}
[(t(\widetilde p') - t(\widetilde p)) - (t(\widetilde q') - t(\widetilde p))]
\frac{K(\chi; \widetilde p', \widetilde q')}{\sqrt{dt}(\widetilde p')
\sqrt{dt}(\widetilde q')} \\
\notag = I_r + \frac{A_\ell}{dt}(\widetilde p) (t(\widetilde p') -
t(\widetilde p)) + \frac{A}{dt}(\widetilde p) (t(\widetilde q') -
t(\widetilde p)) + O(|t(\widetilde p') - t(\widetilde p)|^2 +
|t(\widetilde q') - t(\widetilde p)|^2).
\end{multline}
Evaluation of both sides of this equation at $\widetilde p' = \widetilde
q' = \widetilde p$ yields $f(\widetilde p) = I_r$ from which we get
\begin{equation} \label{der2}
\frac{df}{dt}(\widetilde p) = 0
\end{equation}
for all $\widetilde p \in X$. Comparison of \eqref{der1} and
\eqref{der2} now gives $A(\widetilde p_0) + A_\ell(\widetilde p_0) = 0$
as asserted.
\end{pf}
We can use these coefficients $A(\widetilde p)$ and $A_\ell(\widetilde q)$
defined by \eqref{cauchyexp} to define connections $\nabla_\chi$ on
$\chi$ and $\nabla^*_\chi$ on $\chi^\vee$ according to the formulas
\begin{align}
\nabla_\chi y &= A y + dy \notag \\
\nabla^*_\chi x &= A_\ell^T x + dx. \label{connection}
\end{align}
for local holomorphic sections $y$ of $\chi$ and $x$ of $\chi^\vee$.
The result $A_\ell + A = 0$ from Lemma \ref{L:cauchyexp} is equivalent to the
fact that $\nabla_\chi$ and $\nabla_\chi^*$ are {\it dual to each other}, i.e.
\[
d(x^Ty) = x^T (\nabla_\chi y) + (\nabla_\chi^* x)^T y
\]
for local holomorphic sections $y$ of $\chi$ and $x$ of $\chi^\vee$,
where $(y, x) \to x^Ty$ is the pairing between $\chi$ and $\chi^\vee$.
Moreover, from the formula for $\nabla_\chi$ we see that the connection
matrix associated with $\nabla_\chi$ is a matrix of holomorphic
$(1,0)$-forms.
Hence, $\nabla_\chi$ is {\it compatible with the complex structure of}
$X$ and moreover, since we are in complex dimension 1, the connection
$\nabla_\chi$ is {\it flat}, i.e., $\nabla_\chi$ has {\it zero curvature}
(see Section~5 of Chapter~0 of \cite{gh} for all relevant definitions).
The existence of such a flat connection on $\chi$ in turn implies that
$\chi$ itself is a flat vector bundle (see \cite{mst} pages 294--295).
In general there are many choices of distinct flat connections on a flat
vector bundle; our construction via the Cauchy kernel provides a canonical
choice of such a flat connection (up to a choice of a bundle $\Delta$
of half-order
differentials). An explicit formula for $\nabla$ in the line
bundle case is given in Section \ref{S:linebundle}.
{\bf Remark:} The proof of Theorem \ref{T:cauchyker}
used only the fact that $\deg\chi = 0$ and did not use the flatness of $\chi$.
Since the existence of a flat connection (compatible with the
complex structure) implies that the
bundle is flat, our construction
gives a direct proof of
the flatness of $\chi$ independent of Weil's theorem.
\section{The abstract interpolation problem} \label{S:absint}
In this section we consider as given two flat vector bundles $\chi$ and
$\widetilde \chi$ over the Riemann surface $X$ for which both $h^0(\chi
\otimes \Delta)=0$ and $h^0(\widetilde \chi \otimes \Delta)=0$, where
again, $\Delta$ is a line bundle of half-order differentials over $X$.
We are interested in studying pole-zero interpolation conditions imposed
on a bundle map of $\chi$ to $\widetilde \chi$. The data for the
interpolation problem is as follows. We assume that we are given
$n_\infty$ distinct points $\mu^1, \dots, \mu^{n_\infty}$ (the
prescribed poles) together with $n_0$ distinct points $\lambda^1, \dots,
\lambda^{n_0}$ (the prescribed zeros). For each fixed index $j$ ($j=1,
\dots, n_\infty$) we specify a linearly independent set $\{u_{j1}, \dots,
u_{j,s_j}\}$ of $s_j$ vectors in the fiber $\widetilde \chi(\mu^j)$ of
$\widetilde \chi$ over $\mu^j$ (the prescribed pole vectors) and for each
fixed index $i$ ($i=1, \dots, n_0$) we specify a linearly independent set
$\{x_{i1}, \dots, x_{i,t_i}\}$ of $t_i$ vectors in the fiber $\widetilde
\chi^\vee(\lambda^i)$ of the dual bundle $\widetilde \chi^\vee$ of
$\widetilde \chi$ (the prescribed null vectors). Also, for each pair of
indices ($i,j$) for which $\lambda^i = \mu^j=:\xi^{ij}$, we specify a
collection $\{\rho_{ij,\alpha\beta} \colon 1 \le \alpha \le t_i, 1 \le
\beta \le s_j\}$ of numbers that depend on the choice of the local
parameter at the point $\xi^{ij}$. The {\it Abstract Interpolation
Problem} (ABSINT) which we study in this section is the following:
{\it determine if
there exists a bundle map $T \colon \chi \to \widetilde \chi$ with
transpose $T^\vee \colon \widetilde \chi^\vee \to \chi^\vee$ such that:
\begin{enumerate}
\item[(i)] $T$ has poles only at the points $\{\mu^1, \dots, \mu^{n_\infty}\}$;
for each $j=1, \dots, n_\infty$, the pole of $T$ at $\mu^j$ is simple,
and the residue $R_j = \text{ Res }_{p=\mu^j} T \colon \chi(\mu^j) \to
\widetilde \chi(\mu^j)$ of $T$ at $\mu^j$ is such that $\{u_{j1}, \dots,
u_{j,s_j}\}$ spans the image space $\text{im } R_j$ of $R_j$.
\item[(ii)] The bundle map $(T^\vee)^{-1} \colon \chi^\vee \to \widetilde
\chi^\vee$ has poles only at $\{\lambda^1, \dots, \lambda^{n_0}\}$; for
each $i=1, \dots, n_0$, the pole of $(T^\vee)^{-1}$ at $\lambda^i$ is
simple and the residue $\widehat R_i = \text{
Res}_{p=\lambda^i}(T^\vee)^{-1} \colon \chi^\vee(\lambda^i) \to
\widetilde \chi^\vee (\lambda^i)$ of $(T^\vee)^{-1}$ at $\lambda^i$ is
such that $\{x_{i1}, \dots, x_{i,t_i}\}$ spans the image space $\text{ im
} \widehat R_i$ of $\widehat R_i$.
\item[(iii)] For each pair of indices ($i,j$) for which $\lambda^i = \mu^j =:
\xi^{ij}$, and for $\alpha = 1, \dots, t_i$, let $x_{i \alpha}(p)$ be a
local holomorphic section of $\widetilde \chi^\vee$ with $x_{i
\alpha}(\xi^{ij}) = x_{i \alpha}$ such that $T^\vee(p)x_{i \alpha}(p)$
has analytic continuation to $p = \xi^{ij}$ with value at $p = \xi^{ij}$
equal to 0. Then
\[
(\nabla^*_{\widetilde \chi} x_{i \alpha}(\xi^{ij}))^T u_{j \beta} = \rho_{ij,
\alpha \beta}
\]
for $\beta = 1, \dots , s_j$.
\end{enumerate}
When such a bundle map $T$ exists, give an
explicit formula for the construction of $T$.}
In order for solutions to exist, the compatibility condition
\begin{equation} \label{comp}
x_{i \alpha}u_{j \beta} = 0
\text{ whenever } \lambda^i = \mu^j.
\end{equation}
must hold. This follows from the requirement that the meromorphic
section
$$
x_{i \alpha}(p)T(p)
$$
be analytic at the point $p=\xi^{ij}:=
\lambda^i = \mu^j$. Hence we shall always assume that our data
collection
\begin{equation} \label{dataset}
{\boldsymbol \omega} = \{(x_{i\alpha},\lambda^i), (u_{j\beta}, \mu^j),
\rho_{ij, \alpha \beta} \}
\end{equation}
also satisfies this compatibility condition.
It will be convenient to work with an alternate form of the interpolation
condition (iii) in (ABSINT). Suppose that $u(p) = T(p) \varphi(p)$,
where $\varphi$ is a local holomorphic section of $\chi$ near $\xi^{ij}$
chosen so that $\text{Res}_{p=\xi^{ij}} u(p) = u_{j \beta}$ with respect
to the local coordinate $t^{ij}$ centered at $\xi^{ij}$. Then
\begin{align}
\left(x_{i \alpha}(p) \right)^T\left( t^{ij}(p) u(p) \right)
& = t^{ij}(p) \cdot \left(x_{i \alpha}(p)\right)^T T(p) \varphi(p) \\
& = \left( t^{ij}(p) T^\vee(p) x_{i \alpha}(p)\right)^T \varphi(p)
\end{align}
has a double order zero at $p=\xi^{ij}$, and hence
\[
d\left( x_{i \alpha}(p)^T (t^{ij}(p) u(p) ) \right)|_{p=\xi^{ij}} = 0.
\]
Since $\nabla_{\widetilde{\chi}}^*$ and $\nabla_{\widetilde{\chi}}$ are
dual connections as a consequence of Lemma \ref{L:cauchyexp}, we
therefore have
\[
\left( \nabla_{\widetilde{\chi}}^* x_{i \alpha}(\xi^{ij})\right)^T u_{j
\beta} + x_{i \alpha}^T \nabla_{\widetilde{\chi}} (t^{ij}(p) u(p))|_{ p=
\xi^{ij}} = 0.
\]
Thus the interpolation condition in part (iii) of (ABSINT) can be
expressed alternatively as
\begin{equation} \label{coupledint}
x_{i \alpha}^T \nabla_{\widetilde{\chi}}(t^{ij}(p) u(p))|_{p = \xi^{ij}} =
- \rho_{ij, \alpha \beta}
\end{equation}
where $u(p) = T(p) \varphi(p)$ for a local holomorphic section
$\varphi$ of $\chi$
near $\xi^{ij}$ such that $\text{Res}_{p=\xi^{ij}}u(p) = u_{j \beta}$.
In the scalar case ($r=1$), the compatibility condition \eqref{comp} can
never be satisfied in a nontrivial way and hence the third set of
interpolation conditions is absent under our assumptions; this
corresponds to the fact that a scalar meromorphic function cannot have a
zero and a pole at the same point $\xi^{ij}$. Moreover, for the case
$r=1$, necessarily $t_i=1$ for all $i$ and $s_j=1$ for all $j$. In the
case where both $\chi$ and $\widetilde{\chi}$ are trivial (or more
generally if we use coordinates with respect to a local holomorphic frame
for $\widetilde{\chi}$ near $\mu^j$ or $\widetilde{\chi}^\vee$ near
$\lambda^i$), there is no loss of generality in taking $x_i:=x_{i1}=1$
and $u_j:=u_{j1}=1$ for all $i$ and $j$. Thus the only remaining
relevant data are the zeros $\lambda^1, \dots \lambda^{n_0}$
and the poles $\mu^1, \dots, \mu^{n_\infty}$ (all assumed here to
be distinct). As is standard in algebraic geometry, the formal sum
\[
{\boldsymbol \lambda} - {\boldsymbol \mu}:= \lambda^1 + \dots +
\lambda^{n_0} - \mu^1
- \dots - \mu^{n_\infty}
\]
is said to be a {\it divisor} on $X$. If $f$ is a meromorphic function,
the associated principal divisor $(f)$ is defined to be the formal sum
$ p^1 + \dots +
p^{n_0} - q^1 - \dots - q^{n_\infty}$ where the $p^i$'s are the zeros of
$f$ and the $q^j$'s are the poles of $f$ (with repetitions according to
respective multiplicities). Associated with any divisor ${\boldsymbol
\lambda} - {\boldsymbol \mu}$ as above is the vector bundle
${\cal O}({\boldsymbol \mu} -
{\boldsymbol \lambda})$
whose holomorphic sections can be identified
with global
meromorphic functions $h$ such that
\[
(h) \ge {\boldsymbol \lambda} - {\boldsymbol \mu},
\]
i.e., such that the zeros of $h$ include the points $\lambda^1, \dots,
\lambda^{n_0}$ (all with multiplicity at least 1) and the poles of $h$
are a subset of $\mu^1, \dots, \mu^{n_\infty}$ (all with multiplicity at
most 1).
It is convenient for us to introduce matrix analogues of these ideas.
Let $\boldsymbol \omega$ be an interpolation data set as in \eqref{dataset}.
Let us introduce the notation
\begin{equation} \label{poledata}
({\boldsymbol \mu}, {\bold u}) = \{ (\mu^j, u_{j \beta}) \colon 1 \le j \le
n_\infty, 1 \le \beta \le s_j \}
\end{equation}
for the pole part of $\boldsymbol \omega$.
We let ${\cal M}(\widetilde{\chi}\otimes \Delta)$ be the sheaf of
meromorphic sections of $\widetilde{\chi} \otimes \Delta$.
For $U$ an open subset of $X$ we
define
\begin{align} \notag
{\cal O}(\widetilde{\chi} \otimes \Delta)({\boldsymbol \mu}, {\bold u}) (U) = &
\{u \in {\cal M}(\widetilde{\chi} \otimes \Delta ) (U) \colon u
\text{ has poles only at } \mu^j, \\
& u_{-1}:=\text{Res}_{p=\mu^j} \in \text{span }\{u_{j \beta} \colon 1 \le
\beta \le s_j\} \}
\notag
\end{align}
and
\begin{align} \notag
{\cal O}(\widetilde{\chi}\otimes \Delta)({\boldsymbol \omega})(U) =
& \{u \in {\cal O}(\widetilde{\chi} \otimes \Delta) ({\boldsymbol \mu}, {\bold
u})(U) \colon \text{ if } \lambda^i \ne \mu^j, \text{ then } x_{i
\alpha}u(\lambda^i) = 0; \\
\notag
& \text{if } \lambda^i = \mu^j =:\xi^{ij}, \text{ there is a local
holomorphic section } x_{i \alpha}(p) \text{ of } \widetilde{\chi}^\vee \\
\notag
& \text{such that (i) } x_{i \alpha}(\xi^{ij}) = x_{i \alpha}, \text{ (ii) }
x_{i \alpha}(p)^T \frac{u}{ \sqrt{ dt^{ij} } }(p) \text{ has analytic} \\
\notag
& \text{continuation to } p = \xi^{ij}
\text{with value } 0 \text{ there, and} \\
\notag
& \text{ (iii) }
x_{i \alpha}^T \nabla_{\chi}\left(t^{ij} \frac{u}{ \sqrt{dt^{ij}} }\right)
= - \sum_{\beta = 1}^{s_j}\rho_{ij,\alpha \beta}
c_\beta \frac{1}{ \sqrt{dt^{ij}} (\xi^{ij})} \\
\notag
& \text{if Res}_{p=\mu^j} u = \sum_{\beta=1}^{s_j} u_{j \beta} c_\beta.\}
\end{align}
It is obvious that ${\cal O}(\widetilde{\chi} \otimes
\Delta)({\boldsymbol \mu}, {\bold u})$ and ${\cal O}(\widetilde{\chi}
\otimes \Delta)({\boldsymbol \omega})$ are locally free sheaves of rank
$r$, and we denote the corresponding rank $r$ vector bundles by
$(\widetilde{\chi} \otimes \Delta)({\boldsymbol \mu}, {\bold u})$ and
$(\widetilde{\chi} \otimes \Delta)({\boldsymbol \omega})$. It is also
obvious that $T$ is a solution of (ABSINT) if and only if $T$ is an
isomorphism from $\chi \otimes \Delta$ to $(\widetilde{\chi} \otimes
\Delta)({\boldsymbol \omega})$.
The solution of the zero-pole interpolation problem introduced at the
beginning of this section is as follows.
\begin{theorem} \label{T:absint}
Define a $n_0 \times n_\infty$ block matrix $\Gamma = [\Gamma_{ij}]$
($1 \le i \le n_0$, $1 \le j \le n_\infty$)
where the block entry $\Gamma_{ij}$ in turn is a $t_i \times s_j$ matrix
$\Gamma_{ij} = [\Gamma_{ij, \alpha \beta}]$ ($1 \le \alpha \le t_i$, $1
\le \beta \le s_j$) with matrix entries $\Gamma_{ij, \alpha \beta}$ given by
\begin{equation} \label{defGamma}
\Gamma_{ij, \alpha \beta} =
\begin{cases} - x_{i \alpha}^T K(\widetilde{\chi};\lambda^i, \mu^j) u_{j
\beta}, &\text{if } \lambda^i \ne \mu^j; \\
-\rho_{ij, \alpha \beta} & \text{if } \lambda^i = \mu^j.
\end{cases}
\end{equation}
In addition we introduce the block matrices
\begin{gather} \notag
{\bold u}_i = \begin{bmatrix} u_{i1} & \dots & u_{i s_i} \end{bmatrix},
\quad
{\bold x}_j = \begin{bmatrix} x_{j1} & \dots & x_{j t_i} \end{bmatrix}, \\
K_{{\boldsymbol \mu}, {\bold u}}(p) =
\begin{bmatrix} K(\widetilde{\chi}; p, \mu^1) {\bold u}_1 & \dots &
K(\widetilde{\chi}; p, \mu^{n_\infty}) {\bold u}_{n_\infty} \end{bmatrix}, \\
K^{{\bold x},{\boldsymbol \lambda}}(q) = \begin{bmatrix} {\bold x}^T_1
K(\widetilde{\chi}; \lambda^1, q) \\
\vdots \\ {\bold x}^T_{n_0} K(\widetilde{\chi}; \lambda^{n_0}, q)\end{bmatrix}.
\end{gather}
Let $q$ be a point of $X$ disjoint from all the interpolation nodes
$\lambda^1, \dots, \lambda^{n_0}$, $\mu^1, \dots, \mu^{n_\infty}$
and let $Q$ be an invertible linear map of the fiber
space $\chi(q)$ to the fiber
space $\widetilde{\chi}(q)$.
Then the abstract interpolation problem (ABSINT) has a solution $T$ with
value $Q$ at the point $q$ if and only if the matrix $\Gamma$ is square
and invertible and
\begin{equation} \label{residues}
[K(\widetilde{\chi}; p^i,q) + K_{{\boldsymbol\mu}, {\bold u}}(p^i) \Gamma^{-1}
K^{{\bold x},{\boldsymbol \lambda}}(q)]
Q (\text{Res}_{p^i}\ K(\chi; \cdot , q)^{-1}) = 0
\end{equation}
at each pole $p^i$ of $K(\chi; \cdot, q)^{-1}$.
In this case the unique solution $T$ of the interpolation problem (ABSINT)
with value
$Q$ at $q$ is given by
\begin{equation} \label{solution}
T(p) = [K(\widetilde{\chi}; p,q) + K_{{\boldsymbol \mu}, {\bold u}}(p)
\Gamma^{-1} K^{{\bold x},{\boldsymbol \lambda}}(q)] Q K(\chi; p,q)^{-1}
\end{equation}
with inverse given by
\begin{equation} \label{inversesolution}
T^{-1}(p) = K(\chi;p,q)^{-1} T^{-1}(q)[K(\widetilde{\chi};q,p) +
K_{{\boldsymbol \mu},{\bold u}}(q) \Gamma^{-1} K^{ {\bold x},
{\boldsymbol \lambda} }(p)].
\end{equation}
\end{theorem}
Two special cases of formula \eqref{solution} deserve to be mentioned.
The first is the case where $n_0=n_\infty = 1$, $t_1=s_1=1$ and
$\lambda^1 \ne \mu^1$. If we set $x =x_1$, $\lambda = \lambda^1$, $\mu
= \mu^1$ and $u = u_1$, then $\Gamma = -x K(\widetilde{\chi}; \lambda,
\mu) u$ is just a number and the formula \eqref{solution} becomes
\begin{equation} \label{3.12a}
T(p) K(\chi; p,q) T(q)^{-1} =
\dfrac{ K(\widetilde{\chi}; p,q) - K(\widetilde{\chi}; p,\mu) u x^T
K(\widetilde{\chi}; \lambda,q)}{x^T K(\widetilde{\chi}; \lambda, \mu) u}.
\end{equation}
In the line bundle case, the identity \eqref{3.12a} reduces to the Fay trisecant
identity and will be discussed in Section \ref{S:linebundle}. The second
special case of interest
is the case where the given zero and pole vectors at each
interpolation node span the whole fiber space. In this case there is an
explicit multiplicative formula for the interpolant $T$
in terms of the prime form
$E(p,q)$; this will be discussed in detail at the end of Section
\ref{S:linebundle}.
A second version of the abstract interpolation problem (ABSINT) has the
same form (i), (ii) and (ii) as (ABSINT), but with the input bundle
$\chi$ left also as an unknown to be found, subject to the proviso that
it also be flat and have $h^0(\chi \otimes \Delta)=0$. This version of
the problem was studied in \cite{hip} in a more concrete setting where
$X$ is the normalizing Riemann surface for an algebraic curve $C$
embedded in ${\bold P}^2$ having a maximal rank $r$
determinantal representation; we will discuss the connections of this
setup with ours in Sections 5 and 6. At this time we also state the
solution to the modified (ABSINT).
\begin{theorem} \label{T:bvabsint}
Let $\boldsymbol \omega$ be a data set for (ABSINT)
as above and form the matrix
$\Gamma$ as in \eqref{defGamma}. Then there exists a flat bundle
$\chi$ with $h^0(\chi \otimes \Delta) = 0$ and a meromorphic bundle map
$T \colon \chi \to \widetilde{\chi}$ satisfying the interpolation
conditions (i), (ii) and (iii) of (ABSINT) if and only if $\Gamma$ is
square and invertible.
\end{theorem}
To prove Theorem \ref{T:absint} we need some preliminary lemmas.
\begin{lemma} \label{L1:absint}
A global meromorphic section of $\widetilde{\chi}\otimes \Delta$ is in
${\cal O}(\widetilde{\chi} \otimes \Delta) ({\boldsymbol \mu},{\bold u})(X)$
if and only if $h$ has the form
\[
h= \sum_{j=1}^{n_\infty} \sum_{\beta = 1}^{s_j} K(\widetilde{\chi}; p,
\mu^j) u_{j \beta} c_{j \beta}
\]
for some scalars $c_{j \beta}$.
\end{lemma}
\begin{pf} Suppose $h \in {\cal O}(\widetilde{\chi} \otimes \Delta)
( {\boldsymbol \mu}, {\bold u})(X)$. Choose scalars $c_{j \beta}$ so that
\[
\text{Res}_{p = \mu^j} h(p) = \sum_{\beta=1}^{s_j} u_{j \beta} c_{j \beta}
\]
and set
\[
\widehat{h}(p) = \sum_{j=1}^{n_\infty} \sum_{\beta=1}^{s_j}
K(\widetilde{\chi}; p, \mu^j) u_{j \beta} c_{j \beta}.
\]
Then $\widehat{h} \in {\cal M}(\widetilde{\chi} \otimes \Delta)(X)$ and $h
- \widehat{h} \in {\cal O}(\widetilde{\chi} \otimes \Delta)(X)$. Thus
$h = \widehat{h}$ since ${\cal O}(\widetilde{\chi} \otimes \Delta)(X)
= H^0(X, \widetilde{\chi} \otimes \Delta) = 0$ by our standing
assumptions on $\widetilde{\chi}$. The converse direction follows easily
from the defining properties of the Cauchy kernel $K(\widetilde{\chi};
\cdot, \cdot)$.
\end{pf}
\begin{lemma} \label{L2:absint}
The map
\[
[[c_{j \beta}]_{1 \le \beta \le s_j}]_{1 \le j \le n_\infty} \to
h(p)=\sum_{j=1}^{n_\infty} \sum_{\beta = 1}^{t_j} K(\widetilde{\chi}; p,
\mu^j) u_{j \beta} c_{j \beta}
\]
establishes a one-to-one correspondence between $\ker \Gamma$ and
${\cal O}(\widetilde{\chi} \otimes \Delta)( {\boldsymbol \omega})(X)$.
In particular
\[
\dim \ker \Gamma = h^0( (\widetilde{\chi} \otimes \Delta)
({\boldsymbol \omega}) ).
\]
\end{lemma}
\begin{pf}
Suppose first that $h \in {\cal O}(\widetilde{\chi} \otimes \Delta)(
{\boldsymbol \omega})(X)$. In particular
$h \in {\cal O}(\widetilde{\chi} \otimes
\Delta) ({\boldsymbol \mu}, {\bold u})(X)$, so by Lemma \ref{L1:absint} there
exists a collection of complex numbers $\{c_{j \beta}\}_{1\le j \le
n_\infty, 1 \le \beta \le s_j}$ so that
\[
h(p) = \sum_{j=1}^{n_\infty} \sum_{\beta=1}^{t_j} K(\widetilde{\chi}; p,
\mu^j) u_{j \beta} c_{j \beta}.
\]
We next see what conditions the other requirements on $h$ for admission
to the class ${\cal O}(\widetilde{\chi} \otimes \Delta) ({\boldsymbol
\omega})(X)$ impose on the scalars $c_{j \beta}$.
If $i$ is any index for which $\lambda^i \ne \mu^j$ for all $j$, we must have
\begin{align} \notag
0 = x_{i \alpha}h(\lambda^i) & = \sum_{j=1}^{n_\infty} \sum_{\beta = 1}
^{t_j} x_{i \alpha} K(\widetilde{\chi}; \lambda^i, \mu^j) u_{j \beta}
c_{j \beta} \\
\label{kerGamma1}
&= - \sum_{j=1}^{n_\infty} \sum_{\beta = 1}^{t_j} \Gamma_{ij, \alpha,
\beta} c_{j \beta}.
\end{align}
If, on the other hand, $i$ is an index such that $\lambda^i = \mu^j =:
\xi^{ij}$ for some index $j$, then we write the Laurent series expansion
for $\frac{h}{\sqrt{dt^{ij}}}$ near $\xi^{ij}$ (where $t^{ij}(p)$ is a
local coordinate for $X$ centered at $\xi^{ij}$)
\[
\frac{h}{ \sqrt{dt^{ij}} }= \left[ \frac{h}{\sqrt{dt^{ij}}}\right]_{-1}
\frac{1}{t^{ij}} + \left[\frac{h}{\sqrt{dt^{ij}}}\right]_0 + O(|t^{ij}(p)|).
\]
We compute
\begin{equation} \label{res}
\left[\frac{h}{\sqrt{dt^{ij}}}\right]_{-1} = \text{Res}_{p=\xi^{ij}}\
\frac{h}{\sqrt{t^{ij}}}(p)
=\sum_{\beta=1}^{s_j} u_{j \beta} \sqrt{dt^{ij}}(\xi^{ij}) c_{j \beta}
\end{equation}
and hence
\begin{equation} \label{rescoeff}
\text{Res}_{p=\xi^{ij}} \frac{h}{\sqrt{dt^{ij}}}(p) = \sum_{\beta =
1}^{s_j} u_{j \beta} {\bold c}_{j \beta}
\end{equation}
where ${\bold c}_{j \beta} = \sqrt{dt^{ij}}(\xi^{ij}) c_{j \beta}$.
Moreover,, we have
\begin{align} \notag
\left[ \frac{h}{\sqrt{dt^{ij}}}\right]_0 = & \sum_{k \ne j} \sum_{\beta =
1} ^{s_k} \frac{K(\widetilde{\chi}; \xi^{ij},
\mu^k)}{\sqrt{dt^{ij}}(\xi^{ij})} u_{k \beta}c_{k \beta} \\
\label{0coeff}
&+ \sum_{\beta=1}^{s_j} \frac{A_{\ell}(\xi^{ij})}{dt(\xi^{ij})}
\sqrt{dt^{ij}}(\xi^{ij}) u_{j \beta} c_{j \beta}.
\end{align}
We next compute
\begin{gather} \notag
x_{i \alpha}^T (
A(\xi^{ij})\left[\frac{h}{\sqrt{dt^{ij}}}\right]_{-1} +
\left[ \frac{h}{ \sqrt{dt^{ij}} } \right]_0 dt^{ij}(\xi^{ij}) ) =
\sum_{\beta=1}^{s_j} x^T_{i \alpha} A(\xi^{ij}) u_{j \beta}
\sqrt{dt^{ij}}(\xi^{ij}) c_{j \beta} \\
\notag
+\sum_{k \ne j} \sum_{\beta=1}^{s_j} x^T_{i \alpha}
\frac{ K(\widetilde{\chi}; \xi^{ij}, \mu^k) }{ \sqrt{dt^{ij}}(\xi^{ij}) }
u_{j \beta} c_{j \beta} dt^{ij}(\xi^{ij})
+ \sum_{\beta=1}^{s_j} x_{i \alpha}
\frac{ A_\ell(\xi^{ij}) }{ \sqrt{dt^{ij}}(\xi^{ij}) } u_{j \beta} c_{j \beta}
dt^{ij}(\xi^{ij}) \\
\notag
= \left\{ \sum_{\beta=1}^{s_j} x_{i \alpha}^T (A(\xi^{ij}) +
A_\ell(\xi^{ij})) u_{j \beta} c_{j \beta} + \sum_{k \ne j}
\sum_{\beta=1}^{s_k} x_{i \alpha}^T K(\widetilde{\chi}; \xi^{ij}, \mu^k)
u_{j \beta} c_{j \beta} \right\} \sqrt{dt^{ij}}(\xi^{ij}).
\end{gather}
By Lemma \ref{L:cauchyexp} the first term in the braces vanishes.
Combining this fact with the formula \eqref{rescoeff} for the
coefficients ${\bold c}_{j \beta}$, we see that the interpolation condition
\[
x_{i \alpha}^T \nabla_{\widetilde{\chi}}\left( t^{ij}
\frac{h}{\sqrt{dt^{ij}}}\right) = - \sum_{\beta=1}^{s_j} \rho_{ij, \alpha
\beta} {\bold c}_{j \beta}
\]
becomes
\[
\left\{ \sum_{k \ne j} \sum_{\beta = 1}^{s_k} x_{i \alpha}^T
K(\widetilde{\chi}; \xi^{ij}, \mu^k) u_{j \beta} c_{j \beta} \right\}
\sqrt{dt^{ij}}(\xi^{ij}) =
- \left( \sum_{\beta = 1} ^{s_j} \rho_{ij, \alpha \beta} c_{j \beta} \right)
\sqrt{dt^{ij}}(\xi^{ij}).
\]
After canceling off $\sqrt{dt^{ij}}(\xi^{ij})$ and recalling that
$\xi^{ij} = \lambda^i$ we see that this can be rewritten as
\begin{equation} \label{kerGamma2}
\sum_{j=1}^{n_\infty} \sum_{\beta=1}^{s_j} \Gamma_{ij, \alpha, \beta} c_{j
\beta} = 0
\end{equation}
and this equation holds for all pairs of indices $(i, \alpha)$ such that
$\lambda^i = \mu^j$ for some $j$. Combining \eqref{kerGamma2} and
\eqref{kerGamma1} we see that the column vector $[[c_{j \beta}]_{1 \le
\beta \le s_j}]_{1\le j \le n_\infty}]$ is in $\ker \Gamma$ as claimed.
Conversely, if $[[c_{j \beta}]_{1 \le \beta \le s_j}]_{1 \le j \le n_\infty}$
is in $\ker \Gamma$ and we set
\[
h(p) = \sum_{j=1}^{n_\infty} \sum_{\beta =1}^{s_j} K(\widetilde{\chi}; p,
\mu^j) c_{j \beta},
\]
then one can verify that $h \in {\cal O}(\widetilde{\chi} \otimes \Delta)
({\boldsymbol \omega})(X)$ by reversing the steps of the above argument.
\end{pf}
The analogue of Lemma \ref{L1:absint} at the level of bundle
endomorphisms is the following.
\begin{lemma} \label{L1':absint}
Suppose that $T$ is a holomorphic bundle map from $\chi \otimes \Delta$
to $ (\widetilde{\chi} \otimes \Delta) ( {\boldsymbol \mu}, {\bold u})$
such that $T(q) = Q \colon
\chi(q) \to \widetilde{\chi}(q)$. Then there exists a unique choice of
operators $\widehat{x}_{j \beta} \colon \widetilde{\chi}(q) \to {\bold C}$
such that
\[
T(p) = \left[ K(\widetilde{\chi}; p, q) +
\sum_{j=1}^{n_\infty} \sum_{\beta = 1} ^{s_j} K(\widetilde{\chi}; p,
\mu^j) u_{j \beta} \widehat{x}_{j \beta} \right] Q K(\chi;p,q)^{-1}.
\]
\end{lemma}
\begin{pf}
Choose operators $\widehat{x}_{j \beta} \colon \widetilde{\chi}(q) \to
{\bold C}$ so that
\[
\text{Res}_{\mu^j} T(\cdot) K(\chi; \mu^j,q)Q^{-1} = \sum_{ \beta}^{s_j}
u_{j \beta} \widehat{x}_{j \beta}.
\]
Set
\[
\widehat{T}(p) = \left[ K(\widetilde{\chi}; p,q) + \sum_{j=1}^{n_\infty}
\sum_{\beta = 1}^{s_j} K(\widetilde{\chi}; p, \mu^j) u_{j \beta}
\widehat{x}_{j \beta} \right] Q(K(\chi, p,q)^{-1}.
\]
Then, for any vector $v \in \chi(q)$ we have
\begin{gather} \notag
T(\cdot) K(\chi; \cdot, q)v - \widehat{T}(\cdot)
K(\chi; \cdot, q) v \\ \notag
=T(\cdot) K(\chi; \cdot, q) v - \left[ K(\widetilde{\chi}; \cdot, q)Q
+\sum_{j=1}^{n_\infty} \sum_{\beta=1}^{s_j} K(\widetilde{\chi}; \cdot,
\mu^j) u_{j \beta} \widehat{x}_{j \beta}Q \right] v \\
\end{gather}
is an element of ${\cal O}(\widetilde{\chi} \otimes \Delta)(X)$. By our
standing assumption that $h^0(\widetilde{\chi} \otimes \Delta) = 0$, we
conclude that $(T-\widehat{T})(\cdot) K(\chi; \cdot, q)v=0$ for all $v
\in \chi(q)$. This is enough to force $T=\widehat{T}$, and the lemma follows.
\end{pf}
\begin{pf*}{Proof of Theorem \ref{T:absint}}
We first argue that necessarily $\Gamma$ is square and invertible if
a solution $T$ to the interpolation problem (ABSINT) exists.
To do this, we show first that $\ker \Gamma =
\{0\}$ and secondly, that $\Gamma$ is square.
To see that $\ker \Gamma =\{0\}$, we proceed as follows.
If $T$ is a solution of (ABSINT), then multiplication by $T$ induces a
biholomorphic bundle map between $\chi \otimes \Delta$ and
$(\widetilde{\chi} \otimes \Delta)({\boldsymbol \omega})$.
By assumption, $h^0(\chi \otimes \Delta) = 0$. Hence we also have
$h^0( (\widetilde{\chi} \otimes \Delta) ({\boldsymbol \omega})) = 0$. Now it
follows from Lemma \ref{L2:absint} that $\ker \Gamma = \{0\}$.
Next we argue that $\Gamma$ is square. Since $\chi$ and
$\widetilde{\chi}$ by assumption are both flat, both $\chi$ and
$\widetilde{\chi}$ have degree 0, as do $\det \chi$ and $\det
\widetilde{\chi}$. Then
\[
\text{deg}(\det T) =
\text{deg}(\det \widetilde{\chi}) - \text{ deg} (\det \chi) = 0.
\]
If $T$ is a solution of (ABSINT), then the total number of
zeros $n_0(T)$ of $T$ (counted with multiplicities as appropriate
for meromorphic matrix functions---see Chapter 3 of \cite{bgr})
is equal to
$\sum_{i=1}^{n_0} t_i =:N_0$ which is the number of rows of
$\Gamma$, while the total number of poles $n_\infty(T)$ (again counted with
multiplicities) is equal to $ \sum_{j=1}^{n_\infty} s_j =:N_\infty$ which
is equal to
the number of columns of $\Gamma$. In general we have $\text{deg}(\det T)
= n_0(T) - n_\infty(T)$. Hence the equality $\text{deg}(\det T) = 0$ for
$T$ a solution
of (ABSINT) implies that $\Gamma$ is square. Combining this with the
result of the previous paragraph, we see that $\Gamma$ is invertible as well.
If $T$ is a solution of (ABSINT), then in particular $T$
satisfies the hypotheses of Lemma \ref{L1':absint} and hence $T$ has the
form
\begin{equation} \label{ansatz}
T(p) = \left[ K(\widetilde{\chi};p,q) + \sum_{j=1}^{n_\infty} \sum_{\beta
= 1}^{s_j} K(\widetilde{\chi}; p, \mu^j) u_{j \beta} \widehat{x}_{j
\beta} \right] Q K(\chi, p,q)^{-1}
\end{equation}
for appropriate operators $\widehat{x}_{j \beta} \colon
\widetilde{\chi}(q) \to {\bold C}$. We now find what additional
restrictions on $\widehat{x}_{j \beta}$ are forced by the zero and
coupled zero-pole interpolation conditions (ii) and (iii) in (ABSINT).
Suppose that $i$ is an index for which $\lambda^i \ne \mu^j$ for
any $j$. Then the zero interpolation condition $x_{i \alpha}^T T(\lambda^i)
=0$ forces, for all $\alpha$ between $1$ and $t_i$,
\[
x_{i \alpha}^T K(\widetilde{\chi}; \lambda^i,q) Q K(\chi; \lambda^i, q)^{-1}
+ \sum_{j=1}^{n_\infty} \sum_{\beta=1}^{s_i}x_{i \alpha}^T
K(\widetilde{\chi}; \lambda^i, \mu^j) u_{j \beta}
\widehat{x}_{j \beta} Q K(\chi;
\lambda^i,q)^{-1} = 0.
\]
Recalling the definition of $\Gamma_{ij, \alpha \beta}$, we can rewrite
this as
\begin{equation} \label{Gamma1}
\sum_{j=1}^{n_\infty} \sum_{\beta=1}^{s_j} \Gamma_{ij, \alpha \beta} \widehat{x}_{j \beta} = x_{i \alpha}
K(\widetilde{\chi}, \lambda^i, q)
\end{equation}
for all index pairs $(i,\alpha)$ such that $\lambda^i \ne \mu^j$ for any $j$.
We next consider an index $i$ for which $\lambda^i =
\mu^j:= \xi^{ij}$ for some $j$. Let $x_{i \alpha}(p)$ be a local
holomorphic section of $\widetilde{\chi}^\vee$ as in the third set of
interpolation conditions. Let $\varphi(p)$ be the meromorphic local
section of $\chi$ given by
\[
\varphi(p) = \frac{K(\chi; p,q)}{\sqrt{dt^{ij}}(p)} e
\]
for a vector $e \in \chi(q)$ where $t^{ij}(p)$ is a local coordinate on
$X$ centered at $\xi^{ij}$, and let $u(p)$ be the local meromorphic
section of $\widetilde{\chi}$ given by $u(p) = T(p) \varphi(p)$. From
\eqref{ansatz} we have then
\[
u(p) = \frac{K(\widetilde{\chi};p,q)}{\sqrt{dt^{ij}}(p)} Q e
+ \sum_{j=1}^{n_\infty} \sum_{\beta = 1}^{s_j}
\frac{K(\widetilde{\chi};p,\mu^j)}{\sqrt{dt^{ij}}(p)} u_{j \beta}
\widehat{x}_{j \beta} Q e.
\]
Then the coefficients $[u]_{-1}$ and $[u]_0$ in the Laurent expansion of
$u(p)$ centered at $\xi^{ij}$ with respect to local coordinate $t^{ij}$
are given by
\begin{align} \label{-1Laurent}
[u]_{-1} &= \sum_{\beta = 1}^{s_j} u_{j \beta} \sqrt{dt^{ij}}(\xi^{ij})
\widehat{x}_{j \beta} Q e, \\
\label{0Laurent}
[u]_0 &= \frac{K(\widetilde{\chi}; \xi^{ij},q)}{\sqrt{dt^{ij}}(\xi^{ij})}
Q e + \sum_{k \ne j} \sum_{\beta = 1}^{s_k} \frac{K(\widetilde{\chi};
\xi^{ij}, \mu^k)}{\sqrt{dt^{ij}}(\xi^{ij})} u_{k \beta}
\widehat{x}_{k \beta} Q e \\
\notag
& + \sum_{\beta=1}^{t_j} \frac{A_\ell(\xi^{ij})}{dt(\xi^{ij})}
\sqrt{dt}(\xi^{ij}) u_{j \beta} \widehat{x}_{j \beta} Q e
\end{align}
so the alternate coupled interpolation condition (iii) given by
\eqref{coupledint} implies that
\[
x_{i \alpha}^T \left( A(\xi^{ij}) [u]_{-1} + [u]_0 dt^{ij}(\xi^{ij}) \right)
= - \sum_{\beta = 1}^{s_j} \rho_{ij, \alpha \beta} \widehat{x}_{j \beta} Q e
\cdot \sqrt{dt^{ij}}(\xi^{ij}).
\]
Substitution of the expressions \eqref{-1Laurent} and \eqref{0Laurent}
for $[u]_{-1}$ and $[u]_0$ gives
\begin{align} \notag
\sum_{\beta=1}^{s_j} x_{i \alpha}^T A(\xi^{ij}) u_{j \beta}
\widehat{x}_{j \beta} Q e \sqrt{dt^{ij}}(\xi^{ij})
& + x_{i \alpha}^T
\frac{K(\widetilde{\chi}; \xi^{ij}, q)}{\sqrt{dt^{ij}}(\xi^{ij})} Q e
dt^{ij}(\xi^{ij}) \\
\notag
+ \sum_{k \ne j} \sum_{\beta = 1}^{s_k} x_{i \alpha}^T
\frac{K(\widetilde{\chi}; \xi^{ij}, \mu^k)}{\sqrt{dt^{ij}}(\xi^{ij})}
u_{k \beta} \widehat{x}_{k \beta} Q e \cdot dt^{ij}(\xi^{ij})
& +
\sum_{\beta = 1}^{t_j} x_{i \alpha}^T
\frac{A_\ell(\xi^{ij})}{\sqrt{dt^{ij}}(\xi^{ij})} u_{j \beta}
\widehat{x}_{j \beta} Q e \cdot dt^{ij}(\xi^{ij}) \\
\notag
= &- \sum_{\beta = 1}^{s_j} \rho_{ij, \alpha \beta} \widehat{x}_{j \beta}
Q e \cdot \sqrt{dt^{ij}}(\xi^{ij}).
\end{align}
By using the result of Lemma \ref{L:cauchyexp} and recalling the definition
\eqref{defGamma} of $\Gamma_{ij, \alpha \beta}$ we see that this
expression collapses to
\[
\sum_{k=1}^{n_\infty} \sum_{\beta = 1}^{s_k} \Gamma_{ik, \alpha \beta}
\widehat{x}_{k \beta} Q e = x_{i \alpha}^T
K(\widetilde{\chi}; \xi^{ij},q) Qe.
\]
Since this must hold for all $e \in \chi(q)$, we arrive at the operator
equation
\begin{equation} \label{Gamma2}
\sum_{k=1}^{n_\infty} \sum_{\beta=1}^{s_k} \Gamma_{ik, \alpha \beta}
\widehat{x}_{k \beta} = x_{i \alpha}^T K(\widetilde{\chi}; \xi^{ij},q)
\end{equation}
which must hold for all index pairs $(i, \alpha)$ for which $\lambda^i =
\mu^j$ for some $j$. Combining \eqref{Gamma1} and \eqref{Gamma2} gives us
\[
\Gamma \widehat{{\bold x}} = K^{{\boldsymbol \lambda}, {\bold x}}(q)
\]
where we have set $\widehat{{\bold x}}$ equal to the column vector
$[[ \widehat{x}_{j \beta}]_{ 1 \le \beta \le s_j}]_{1 \le j \le n_\infty}$.
Plugging this
value into \eqref{ansatz} leaves us with the formula \eqref{solution} for
the solution $T$. This also establishes the uniqueness of the solution
of (ABSINT) whenever it exists.
Since $T$ is analytic at the points $p^1, \dots, p^n$ where $K(\chi;
\cdot, q)^{-1}$ has poles, necessarily the residue conditions
\eqref{residues} must hold as well. The necessity and uniqueness parts
of the theorem are now established.
Conversely, assume that $\Gamma$ is invertible and that the residue
conditions \eqref{residues} hold. We define $T(p)$ by the formula
\eqref{solution}. Then $T$ is a meromorphic bundle map of $\chi$ and
$\widetilde{\chi}$ with only simple poles which occur at most at the
points $\mu^1, \dots, \mu^{n_\infty}$ with
\[
\text{im Res}_{\mu^i} T(\cdot) \subset \text{span }\{u_{j \alpha} \colon
1 \le \alpha \le s_j\}
\]
for $j=1, \dots, n_\infty$. Since $T$ has the form \eqref{ansatz} with
operators $\widehat{x}_{j \beta}$ ($1 \le j \le n_\infty$, $1 \le \beta
\le s_j$) satisfying \eqref{Gamma1} and \eqref{Gamma2}, we see that also
$T$ satisfies the interpolation conditions (ii) and (iii) in (ABSINT)
as well. Hence, the number of poles $n_\infty(T)$ of $T$ (counting
multiplicities for a meromorphic matrix function as in Chapter 3 of
\cite{bgr}) is at most
$\sum_{j=1}^{n_\infty} s_j =: N_\infty$ and the number of zeros $n_0(T)$
of $T$ (counting multiplicities) is at least
$\sum_{i=1}^{n_\infty} t_i =: N_0$. As $T$ is a bundle map of the flat
bundles $\chi$ and $\widetilde{\chi}$, we know that $n_0(T) =
n_\infty(T)$. On the other hand, since $\Gamma$ is square we have
$N_0=N_\infty$. From the chain of inequalities
\[
N_0 \le n_0(T) = n_\infty(T) \le N_\infty
\]
combined with the equality $N_0 = N_\infty$, we get that $n_0(T) = N_0$
and $n_\infty(T) = N_\infty$. This implies that necessarily
\[
\text{im Res}_{p = \mu^j} T(p) = \text{span}\{u_{j1}, \dots, u_{j s_j} \}
\]
and
\[
{\text{im Res}}_{p=\lambda^i}(T^{\vee})^{-1}(p) = \text{span}\{x_{i1}, \dots,
x_{i t_i}\}
\]
and that $T(p)$ is analytic and invertible at every point $p \in X$
outside of $\mu^1, \dots, \mu^{n_\infty}$, $\lambda^1, \dots, \lambda^{n_0}$.
This verifies that $T$ is a bona fide solution of the interpolation
problem (ABSINT).
It remains only to verify the formula \eqref{inversesolution} for the
inverse of $T$. To see this, we note that $(T^{-1})^T$ is also the
solution of an interpolation problem of the type (ABSINT), namely the one
with data set $[{\boldsymbol \omega}]^\vee$ given by
\begin{enumerate}
\item $( {\boldsymbol \lambda},{\bold x})$ in place of
$( {\boldsymbol \mu}, {\bold u})$,
\item $({\boldsymbol \mu}, {\bold u})$ in place of $({\boldsymbol \lambda},
{\bold x})$, and
\item $-\rho_{ji, \beta \alpha}$ in place of $\rho_{ij, \alpha \beta}$.
\end{enumerate}
The matrix $\Gamma$ associated with this interpolation problem turns out
to be exactly $-\Gamma^T$ where $\Gamma$ is as in \eqref{defGamma}.
Hence by Theorem \ref{T:absint} the bundle map $(T^{-1})^T$ must be given by
\begin{equation} \label{Tinvtr}
(T^{-1})^T = [K(\widetilde{\chi}^\vee;p,q) - K_{{\boldsymbol \lambda},
{\bold x}}(p) (\Gamma^{-1})^T K^{{\bold u},
{\boldsymbol \mu}}(q)] (T(q)^{-1})^T K(\chi^\vee;
p,q)^{-1}.
\end{equation}
By the uniqueness property of Cauchy kernels, it is easy to see that
\[
K(\chi^\vee; p,q)^T = - K(\chi; q,p).
\]
Hence, taking transpose on both sides of \eqref{Tinvtr} gives
\begin{align} \notag
T^{-1}(p) = & -K(\chi;q,p)^{-1} T(q)^{-1}[-K(\widetilde{\chi}; q,p)
- K_{ {\boldsymbol \mu}, {\bold u}}(q) \Gamma^{-1} K^{{\bold x},
{\boldsymbol \lambda} }(p) ] \\
\notag = & K(\chi;q,p)^{-1} T(q)^{-1}[-K(\widetilde{\chi}; q,p)
+ K_{{\boldsymbol \mu}, {\bold u}}(q) \Gamma^{-1} K^{{\bold x},{\boldsymbol
\lambda}}(p) ]
\end{align}
and the formula \eqref{inversesolution} follows.
\end{pf*}
{\bf Remark:} In case $\widetilde{\chi} = \chi =:\chi_0$ are both taken
to be the trivial bundle of rank $r$, the Cauchy kernel $K(\chi_0; \cdot,
\cdot)$ has the scalar form $k_0(\cdot, \cdot) I_r$ where $k_0(\cdot,
\cdot)$ is the Cauchy kernel for the trivial line bundle over $X$. In
this case the ansatz \eqref{ansatz} simplifies to
\[
T(p) = Q + \sum_{j=1}^{n_\infty} \sum_{\beta=1}^{s_j} f_{\mu^j}(p) u_{j
\beta} \widehat{x}^\prime_{j \beta}
\]
where we have set
\[f_{\mu}(p) = \frac{k_0(p,\mu)}{k_0(p,q)}
\]
and the row vectors $\widehat{x}^\prime_{j \beta} = \widehat{x}_{j \beta}
Q$ are now taken to be the unknowns. Note that $k_0(\cdot, q)$ is a
half-order differential with divisor of degree $g-1$ and a pole at $q$;
if we assume that the zeros are distinct, this divisor has the form $p^1 +
\dots + p^g - q$ for distinct points $p^1, \dots, p^g,q \in X$. If the
image of the divisor $p^1 + \dots + p^g$ under the Abel-Jacobi map is not
on the classical theta divisor in the Jacobian (i.e. if $(p^1 + \dots +
p^g$ is a {\it non-special} divisor), then there are no nonzero constant
meromorphic functions with only poles equal to at most simple poles at
the points $p^1, \dots, p^g$; this
corresponds to our assumption that $h^0(\chi_0 \otimes \Delta)=h^0(\Delta)=0$.
Furthermore, in this case, the global scalar meromorphic function
$f_\mu(\cdot)$ on $X$ (for $\mu$ a point of $X$ disjoint from $p^1,
\dots, p^g,q$) is uniquely determined (up to a nonzero scalar multiple)
by the condition that it have a pole at $\mu$ and that its divisor
$(f_\mu)$ satisfy
\[
(f_\mu) \ge q - \mu - p^1 - \dots - p^g.
\]
In this way our results and analysis on the (ABSINT) problem reduce to
the work in \cite{bc} for the trivial bundle case.
Notice that $\chi_0$ can be replaced here by $\xi\otimes\chi_0$
for any line bundle $\xi$ of degree $0$ satisfying $h^0(\xi \otimes \Delta)=0$,
replacing the Cauchy kernel $k_0(\cdot,\cdot)$ for the trivial line bundle
by the Cauchy kernel for $\xi$;
this corresponds to letting $p^1 + \dots + p^g$ be any non-special
effective divisor of degree $g$.
\vspace{.1in}
One remaining piece of business in this section is the proof of Theorem
\ref{T:bvabsint}. The problem of identifying the unknown input bundle in
a more explicit fashion will be addressed in Section \ref{S:conint}.
\begin{pf*}{Proof of Theorem \ref{T:bvabsint}}
If there exists such an input bundle $\chi$ and meromorphic bundle map
$T$, then $T$ implements a biholomorphic bundle map between $\chi \otimes
\Delta$ and $(\widetilde{\chi} \otimes \Delta)({\boldsymbol \omega})$.
Since $h^0(\chi
\otimes \Delta) = 0$, it then follows
from Lemma \ref{L2:absint} that $\Gamma$ is injective. Since deg$(\chi
\otimes \Delta) = r(g-1)$, it must be the case that deg$(\widetilde{\chi}
\otimes \Delta)({\boldsymbol \omega})=r(g-1)$ as well. This means that
$\Gamma$ is square,
and hence invertible.
Conversely, suppose that $\Gamma$ is square and invertible. Define a
bundle $\chi$ so that $\chi \otimes \Delta \cong (\widetilde{\chi}
\otimes \Delta)({\boldsymbol \omega})$. Since $\Gamma$ is square,
it follows that
\[
\text{deg}\left( (\widetilde{\chi}\otimes \Delta)({\boldsymbol \omega})\right) =
\text{deg}(\widetilde{\chi} \otimes \Delta) = r(g-1),
\]
and hence $\text{deg}(\chi \otimes \Delta) = r(g-1)$. Since $\ker \Gamma
= \{0\}$, we know by Lemma \ref{L2:absint} that $h^0( (\widetilde{\chi}
\otimes \Delta)({\boldsymbol \omega}))=0$; thus $h^0(\chi \otimes \Delta) = 0$. It
follows from these two facts as in the proof of Theorem 3.1 in \cite{hip}
that $\chi$ is flat.
Let now $S \colon \chi \otimes \Delta \to (\widetilde{\chi} \otimes
\Delta)({\boldsymbol \omega})$ be an implementation of the holomorphic bundle
isomorphism between $\chi \otimes \Delta$ and $(\widetilde{\chi} \otimes
\Delta)({\boldsymbol \omega})$.
Define $T \colon \chi \to \widetilde{\chi}$ so that $S
= T \otimes I_{{\cal O}(\Delta)}$. Then $T$ is a meromorphic bundle map
from $\chi$ to $\widetilde{\chi}$ which solves the interpolation problem
(ABSINT).
\end{pf*}
\section{The line bundle case and Fay's identity}
\label{S:linebundle}
In this section we specialize the work of the preceding sections to the
line bundle case.
We shall need here some basic facts concerning the Jacobian variety,
the Abel-Jacobi map and associated theta functions (theta function, theta
functions with characteristics and prime form)
for the Riemann surface $X$. The review here is quite
sketchy; for complete details the reader should consult \cite{oldfay},
\cite{farkaskra} or \cite{mumford}.
When the rank $r$ of the vector bundle $\chi$ is 1, one can get an
explicit formula for $K(\chi; \cdot, \cdot)$ in terms of the Abel-Jacobi
map for the surface $X$ and various variants of the classical theta
function associated with the Jacobian variety of $X$ (see \cite{hip}).
Specifically, in this case we may assume that $\chi$ is a flat unitary line
bundle with factor of automorphy (also called $\chi$) given by $\chi(A_j)
= \exp (- 2 \pi i a_j)$, $\chi(B_j) = \exp (2 \pi i b_j)$, $j=1, \dots, g$,
where $A_1, \dots, A_g, B_1, \dots, B_g$ form a canonical integral
homology basis on $X$. Let $\Omega$ be the corresponding period matrix,
let $J(X)= {\bold C}^g / {\bold Z}^g + \Omega {\bold Z}^g$ be the
Jacobian variety of $X$ and let $\phi \colon X \to J(X)$ be the
Abel-Jacobi map. As is standard, we extend $\phi$ by linearity to any
divisor on $X$, and, using the correspondence between linear equivalence
classes of divisors and isomorphism classes of line bundles, we consider
$\phi$ to be defined on any line bundle on $X$ as well. One can verify
that then $\phi(\chi) = z$ where $z = \Omega a + b$ and $a,b \in {\bold
R}^g$ have respective coordinates $a_j,b_j$.
Then the explicit formula
for the Cauchy kernel (as given in \cite{hip}) is the following. The
verification is straightforward, once one has in hand the properties and
factors of automorphy for the various objects involved.
\begin{theorem} \label{T:scalarCauchykernel}
For the case where $\chi$ is a flat unitary line bundle as above, the Cauchy
kernel as defined in Section \ref{S:cauchyker} is given explicitly by
\begin{equation}
K(\chi;p,q) = \frac{ \theta \begin{bmatrix} a \\ b \end{bmatrix} (\phi(q)
- \phi(p)) } {\theta \begin{bmatrix} a \\ b \end{bmatrix} (0) E(q,p)}.
\label{lineCauchyker}
\end{equation}
\end{theorem}
In the statement of Theorem \ref{T:scalarCauchykernel}
$\theta \begin{bmatrix} a \\ b \end{bmatrix} ( \cdot)$ is the
associated theta function with characteristics $\begin{bmatrix} a \\ b
\end{bmatrix}$, $E(\cdot, \cdot)$ is the prime form on $X \times X$, and
we assume the line bundle $\Delta$ of differentials of order
$\frac{1}{2}$ has been chosen so that $\phi(\Delta) = - \kappa$, where
$\kappa \in J(X)$ is Riemann's constant (see \cite{oldfay} and
\cite{mumford}). Note that a consequence of Riemann's theorem is that
$\theta(z) \ne 0$ if and only if $h^0(\chi \otimes \Delta) = 0$, and
hence $\theta \begin{bmatrix} a \\ b \end{bmatrix} (0) \ne 0$ in
\eqref{lineCauchyker} and the formula makes sense. No such explicit
formula is known at present for the higher rank case except in genus 1
(see \cite{bcv}).
In the line bundle case one can also give an explicit formula for the
canonical connections $\nabla_\chi$, $\nabla_\chi^*$ associated with the
flat unitary line bundle $\chi$. This is the content of the following
Proposition.
\begin{proposition} \label{P:connection}
For the case where $\chi$ is a flat unitary line bundle with
normalizations as above, then the canonical connections $\nabla_\chi$
and $\nabla_\chi^*$ are given by
\begin{align}
\nabla_\chi y = & \left[ \sum_{j=1}^g \frac{\partial}{\partial z_j} \log
\theta \begin{bmatrix} a \\ b \end{bmatrix} (0) \ \omega_j(p) \right] y +
dy \\
= & \left[ \sum_{j=1}^g [2 \pi i a_j + \frac{\partial}{\partial z_j} \log
\theta(z) ] \omega_j(p) \right] y + dy,\\
\nabla_\chi x = & - \left[ \sum_{j=1}^g \frac{\partial}{\partial z_j} \log
\theta \begin{bmatrix} a \\ b \end{bmatrix} (0) \ \omega_j(p) \right] x +
dx \\
= & - \left[ \sum_{j=1}^g [2 \pi i a_j + \frac{\partial}{\partial z_j} \log
\theta(z) ] \omega_j(p) \right] x + dx.
\end{align}
\end{proposition}
\begin{pf}
This follows directly by comparing the general expansion for the Cauchy kernel
$$
\frac{K(\chi; p,p_0)}{\sqrt{dt}(p) \sqrt{dt}(p_0)} =
\frac{1}{t(p) - t(p_0)} \left[ I_r + \frac{A_\ell}{dt}(p_0) t(p) + O(|t(p)|^2)
\right]
$$
on the one hand and substituting the expansion of the theta function
$$
\theta \begin{bmatrix} a \\ b \end{bmatrix} \left(\phi(p_0) - \phi(p)\right)
= \theta \begin{bmatrix} a \\ b \end{bmatrix} (0) - \sum_{j=1}^g
\frac{\partial \theta \begin{bmatrix} a \\ b \end{bmatrix}} {\partial z_j}
( 0 ) \frac{\omega_j}{dt} \left( p_0 \right) t(p) + O(|t(p)|^2)
$$
and the expansion of the prime form (see Corollary 2.5 in \cite{oldfay})
$$
E(p_0,p) = t(p) + O(|t(p)|^3)
$$
into \eqref{lineCauchyker}.
\end{pf}
{\bf Remark.} From the formula for $\nabla_\chi$ and $\nabla_\chi^*$ it
follows that the coefficients $A(p)$ and $A_\ell(p)$ are independent of
the choice of homology bases (i.e., marking) on the Riemann surface $X$
as long as the bundle $\Delta$ of half-order differentials defined by
$\phi(\Delta) = \kappa$ remains the same,
since the unitary flat representative for a flat line bundle is unique.
It is an amusing exercise to verify this independence directly by using
the transformation law for theta functions (see \cite{mumford} and
\cite{Igusa}).
We next specialize the work of Section \ref{S:absint} to the scalar (or
line bundle) case, where $\chi$ and $\widetilde{\chi}$ are flat unitary
line bundles. As explained in Section \ref{S:absint}, necessarily the
multiplicities $s_j$ and $t_i$ are all 1 and without loss of generality
we may take $u_{j1}=1$, $x_{i1}=1$ for all $i$ and $j$.
Then the
compatibility condition \eqref{comp} forces the third interpolation
condition to be absent. The data of the problem consists simply of the
set of $n_\infty + n_0$ distinct points $\mu^1, \dots, \mu^{n_\infty},
\lambda^1, \dots, \lambda^{n_0}$ together with the flat unitary line bundles
$\chi$ and $\widetilde{\chi}$.
The problem then is to produce a bundle
map $T \colon \chi \to \widetilde{\chi}$ with divisor equal to
${\boldsymbol{\lambda}} -
{\boldsymbol{\mu}}$ (where we have set ${\boldsymbol{\lambda}} = \lambda^1
+ \dots + \lambda^{n_0}$ and ${\boldsymbol \mu} =
\mu^1 + \dots + \mu^{n_\infty}$).
If we view the bundles in terms of factors of automorphy, we can view $T$
simply as a multivalued function on $X$ having divisor equal to
${\boldsymbol \lambda} - {\boldsymbol \mu}$ and factor of automorphy
$\chi_T$ given by
\[
\chi_T(A_j) = e^{-2 \pi i a_j}, \ \chi_T(B_j) = e^{2 \pi i b_j}
\text{ for } j=1, \dots, g
\]
where $\phi(\widetilde{\chi}) - \phi(\chi) = \Omega a + b$ (where we have
set $a= \begin{bmatrix} a_1 & \dots & a_g \end{bmatrix}^T$ and $b=
\begin{bmatrix} b_1 & \dots & b_g \end{bmatrix}^T$).
In the genus zero case where $X= \bold{C} \cup \{\infty\}$ is the Riemann
sphere, any flat unitary line bundle is trivial and the problem is to
produce a global meromorphic function with divisor equal to ${\boldsymbol
\lambda}
- {\boldsymbol \mu}$.
Trivially a solution exists if and only if $n_0=n_\infty$ and then the
unique solution with value 1 at infinity is given in the multiplicative
form
\begin{equation}
T(z) = \dfrac{\prod_{i=1}^{n_0} (z-\lambda^i)}{\prod_{j=1}^{n_\infty}
(z-\mu^j)}.
\label{g0prod}
\end{equation}
or in the partial fraction form
\begin{equation}
T(z) = 1 + \sum_{j=1}^{n_\infty} c_j (z-\mu^j)^{-1}
\label{g0partialfrac}
\end{equation}
where $c^T= \begin{bmatrix} c_1 & \dots & c_{n_\infty} \end{bmatrix}$ is
the unique solution of the linear system of equations $S c
= \begin{bmatrix} 1 &
\dots & 1 \end{bmatrix}^T$ with $S$ equal to the Sylvester matrix
\[
S = [S_{ij}] \text{ with } S_{ij} = \frac{1}{\mu^j - \lambda^i},
\]
or, in other words,
\begin{equation}
c = S^{-1} \begin{bmatrix} 1 & \dots & 1 \end{bmatrix}^T.
\label{g0partialfraccoef}
\end{equation}
It is possible to evaluate the vector $c$ explicitly from
\eqref{g0partialfraccoef} once one knows the entries of $S^{-1}$
explicitly. This in turn can be done once one knows an explicit formula
for the determinant of a Sylvester matrix $S$, since the cofactor
matrices are again of the same form. In this way one can verify directly
the equivalence of the two formulas \eqref{g0prod} and
\eqref{g0partialfrac}. For details on the algebra of
this computation, we refer to
Theorem 4.3.2 of \cite{bgr}.
We shall see that an analogous pair of formulas holds for the solution of
the abstract interpolation problem
(ABSINT) for the higher genus case (for the line bundle setting). This is
the content of the next Theorem.
\begin{theorem} \label{T:lineabsint}
Consider the problem (ABSINT) for the case where $\chi$ and
$\widetilde{\chi}$ are flat unitary line bundles and given data set equal to
$\boldsymbol{\lambda} - \boldsymbol{\mu} = \lambda^1 + \dots + \lambda^{n_0} -
\mu^1 - \dots -
\mu^{n_\infty}$ as above. Then a solution exists if and only if
\begin{equation}
n_0 = n_\infty \text{ and }
\phi(\widetilde{\chi}) - \phi(\chi) = \phi({\boldsymbol \lambda}) -
\phi(\boldsymbol \mu).
\label{necessity}
\end{equation}
In this case, if $q$ is a point of $X$ disjoint from the set
$\{\lambda^1, \dots, \lambda^{n_0}, \mu^1, \dots, \mu^{n_\infty}\}$ of
prescribed zeros and poles and $Q$ is any invertible fiber map from
$\chi(q)$ onto $\widetilde{\chi}(q)$, then a solution of (ABSINT) having
value $Q$ at $q$ is given in multiplicative form as
\begin{equation} T(p) =
\dfrac{ \prod_{i=1}^{n_0}
E(p,\lambda^i)/E(q,\lambda^i)}{\prod_{j=1}^{n_\infty} E(p,\mu^j)/E(q,\mu^j)}
\exp(-2 \pi i a^T (\phi(p) - \phi(q)) Q
\label{prod}
\end{equation}
where $a^T = \begin{bmatrix} a_1 & \dots & a_g \end{bmatrix}$ and
$\phi({\boldsymbol{\lambda}}) - \phi({\boldsymbol{\mu}}) =
\Omega a + b$ with $a,b \in {\bold R}^g$.
If $\chi \otimes \Delta$ and $\widetilde{\chi} \otimes \Delta$ have no
nontrivial holomorphic sections, then the solution
$T$ with $T(q)=Q$ is unique and
alternatively is given by the partial fraction formula
\begin{align}
T(p)=& \left\{ \dfrac{\theta[\widetilde{z}](\phi(q) -
\phi(p))}{\theta[\widetilde{z}](0) E(q,p)}
+ \sum_{j=1}^{n_\infty} \sum_{k=1}^{n_0}
\dfrac{\theta[\widetilde{z}](\phi(\mu^j) -
\phi(p))}{\theta[\widetilde{z}](0)E(\mu^j,p)} \cdot [\Gamma^{-1}]_{jk}
\cdot \dfrac{\theta[\widetilde{z}](\phi(q)-
\phi(\lambda^k))}{\theta[\widetilde{z}](0) E(q,\lambda^k) } \right\} \notag \\
& \times Q \cdot \dfrac{\theta[z](0) E(q,p)}{\theta[z](\phi(q) - \phi(p))}
\label{partialfrac}
\end{align}
where the $n_0 \times n_\infty$ matrix $\Gamma$ is given by
\[
\Gamma = [\Gamma_{ij}] \text{ with }
\Gamma_{ij} =- \dfrac{\theta[\widetilde{z}](\phi(\mu^j) -
\phi(\lambda^i))}{\theta[\widetilde{z}](0) E(\mu^j, \lambda^i)}.
\]
and where we have set
\[
z = \phi(\chi), \ \widetilde{z} = \phi(\widetilde{\chi}).
\]
Here we write $\theta[z](\lambda)$ rather than $\theta\begin{bmatrix} a
\\ b \end{bmatrix}(\lambda)$ if $z = \Omega a + b \in \bold{C}^g$ with
$a, b \in {\bold R}^g$.
\end{theorem}
\begin{pf} If such a bundle map exists, then the bundle $\chi \otimes
{\cal O}(\boldsymbol{\lambda} - \boldsymbol{\mu})$ and
$\widetilde{\chi}$ are holomorphically equivalent. In particular, the
divisor $\boldsymbol{\lambda} - \boldsymbol{\mu}$
must have degree 0 since both $\chi$ and
$\widetilde{\chi}$ are flat bundles. The equality $\phi(\widetilde{\chi}) =
\phi(\chi) + \phi(\boldsymbol{\lambda}) - \phi(\boldsymbol{\mu})$
then follows from the
correspondence between flat bundles and linear equivalence classes of
divisors mentioned above. This verifies the necessity condition
\eqref{necessity}.
Conversely, assume that \eqref{necessity} holds and define $T$ by the
right-hand side of \eqref{prod}.
That the zero-pole divisor of $T$
is $\boldsymbol{\lambda} - \boldsymbol{\mu}$ follows directly from the fact
that the divisor of the
prime form $(p,q) \to E(p,q)$ is the diagonal $\{(p,p) \colon p \in X\}
\subset X \times X$. One can next check from the known period relations
of $E$ that the right-hand side of \eqref{prod} has the factor of
automorphy $\chi_T$
\[
\chi_T(A_j) = \exp (-2 \pi i a_j), \ \chi_T(B_j) = \exp (2 \pi i b_j)
\text{ for } j=1, \dots, g
\]
where $a_1, \dots, a_g, b_1, \dots, b_g$ are respective components of
$a,b \in {\bold R}^g$ chosen so that $\Omega a + b =
\phi(\boldsymbol{\lambda}) -
\phi(\boldsymbol{\mu})$. The second condition in \eqref{necessity} now
guarantees that $T(\cdot)$ so defined is a
bundle map from $\chi$ into $\widetilde{\chi}$. The uniqueness assertion
is a consequence of Lemma \ref{L1':absint}.
The alternative formula \eqref{partialfrac} is
simply a rewriting of the formula \eqref{solution} from Theorem
\ref{T:absint} specialized to the line bundle case,
where we have substituted the explicit formula
\eqref{lineCauchyker} for the Cauchy kernel from
Theorem \ref{T:scalarCauchykernel}.
\end{pf}
Note that part of the content of Theorem \ref{T:lineabsint} is that the
matrix $\Gamma$ is invertible whenever $\chi$ and $\widetilde{\chi}$ have
no nontrivial holomorphic sections and (ABSINT) has a solution.
It is of interest to specialize Theorem \ref{T:lineabsint} to the case
of one prescribed zero and pole $\lambda - \mu = \lambda^1 - \mu^1$.
When this is done we obtain the following result.
\begin{theorem} \label{T:redlineabsint}
If $\chi$ and $\widetilde{\chi}$ are two flat unitary line bundles
such that neither $\chi \otimes \Delta$ nor $\widetilde{\chi}
\otimes \Delta$ have
nontrivial holomorphic sections and $\lambda$ and $\mu$ are two distinct
points of $X$, then the unique meromorphic bundle map from $\chi$ to
$\widetilde{\chi}$ with zero-pole divisor equal to $\lambda - \mu$
and value $Q \neq 0$ at the point $q \in X$ is given by either
\begin{equation} \label{specialprod}
T(p) = \dfrac{E(p,\lambda)}{E(p, \mu)} \dfrac{E(q,\mu)}{E(q,\lambda)}
\exp(-2 \pi i a^T(\phi(p) - \phi(q)) Q
\end{equation}
or
\begin{align}
T(p) =& \exp(-2 \pi i a^T(\phi(p) - \phi(q)))
\left\{ \dfrac{\theta(z + \phi(\lambda) - \phi(\mu) + \phi(q) -
\phi(p)) \theta(z) }{\theta(z + \phi(\lambda) - \phi(\mu))
\theta(z + \phi(q) - \phi(p))} \right. \notag \\
& \left. - \dfrac{\theta(z + \phi(\lambda) - \phi(p)) \theta(z+ \phi(q) -
\phi(\mu)) E(\mu, \lambda) E(q,p)}
{\theta(z+\phi(\lambda) - \phi(\mu)) \theta(z + \phi(q) - \phi(p))
E(\mu,p) E(q,\lambda)} \right\} Q.
\label{specialpartialfrac}
\end{align}
\end{theorem}
\begin{pf} The starting point of course is formula \eqref{prod} and
\eqref{partialfrac} specialized to the case $\boldsymbol{\lambda} -
\boldsymbol{\mu}
= \lambda - \mu$. The formula \eqref{specialprod} is an immediate
consequence of \eqref{prod}. Derivation of \eqref{specialpartialfrac}
requires a little bit of algebra. We use the definition
\[
\theta[z](\lambda) = \exp( \pi i a^T \Omega a + 2 \pi i a (\lambda + b))
\theta(\lambda + z)
\]
(where $z = \Omega a + b$ with $a, b \in {\bold R}^g$) to express all
theta functions with characteristic $\theta[z](\cdot)$ in terms of the
theta function itself $\theta(\cdot)$. When this is plugged into
\eqref{partialfrac} and little bit of algebra is used to collect the
exponential factor (noting that $a = a_{\widetilde{z}} - a_z$
if $\widetilde{z} - z (=\phi(\lambda) - \phi(\mu)) = \Omega a + b$ and
$\widetilde{z} = \Omega a_{\widetilde{z}} + b_{\widetilde{z}}$,
$z = \Omega a_z + b_z$ with $a$, $b$, $a_{\widetilde{z}}$,
$b_{\widetilde{z}}$, $a_z$, $b_z$ in ${\bold R}^g$), we get
\begin{align}
T(p) = & \exp(- 2 \pi i a^T(\phi(p) - \phi(q) ) ) \left\{
\dfrac{ \theta(z + \phi(\lambda) - \phi(\mu) + \phi(q) -
\phi(p))}{\theta(z + \phi(\lambda) - \phi(\mu)) E(q,p)} \right. \notag \\
& - \dfrac{\theta(z + \phi(\lambda) -
\phi(p))}{\theta(z+\phi(\lambda) - \phi(\mu)) E(\mu,p)} \cdot
\dfrac{ \theta(z + \phi(\lambda) - \phi(\mu) E(\mu, \lambda)}{\theta(z)}
\notag \\
& \left. \cdot \dfrac{\theta(z+\phi(q) - \phi(\mu))}
{\theta(z + \phi(\lambda) - \phi(\mu)) E(q, \lambda)} \right\}
\cdot Q \dfrac{\theta(z) E(q,p)}{\theta(z+ \phi(q) - \phi(p))}.
\notag
\end{align}
The formula \eqref{specialpartialfrac} now follows by simple algebraic
manipulation.
\end{pf}
As a Corollary we obtain a version of Fay's Trisecant Identity (see
\cite{oldfay} formula (45) page 34 or \cite{mumford} Volume II page 3.214).
\begin{corollary} \label{C:faytrisecant}
For $X$ a compact Riemann surface, $\phi$ its Abel-Jacobi map,
$\theta(\lambda)$ and $E(p,q)$ its associated respective
theta function and prime
form, $p,q,\lambda, \mu$ points of $X$ and $z \in {\bold C}^g$, the
following identity holds:
\begin{gather}
\theta(z+\phi(\lambda) - \phi(\mu)) \theta(z+\phi(q) - \phi(p))
E(p,\lambda) E(q,\mu) \notag \\
+ \theta(z+\phi(\lambda) - \phi(p)) \theta(z +
\phi(q) - \phi(\mu)) E(\lambda, \mu) E(q,p) \notag \\
= \theta(z + \phi(\lambda) - \phi(\mu) + \phi(q) - \phi(p)) \theta(z)
E(p,\mu) E(q,\lambda).
\label{faytrisecant}
\end{gather}
\end{corollary}
\begin{pf} From the identity of the two expressions \eqref{specialprod}
and \eqref{specialpartialfrac} for $T(p)$ in Theorem
\ref{T:redlineabsint}, we have equality of the following two expressions
for $\exp(2 \pi i a^T(\phi(p) - \phi(q))) T(p) Q^{-1}$:
\begin{align}
\dfrac{E(p,\lambda)}{E(p,\mu)} \dfrac{E(q,\mu)}{E(q,\lambda)} = &
\dfrac{\theta(z + \phi(\lambda) - \phi(\mu) + \phi(q) -
\phi(p)) \theta(z) }{\theta(z + \phi(\lambda) - \phi(\mu))
\theta(z + \phi(q) - \phi(p))} \notag \\
& - \dfrac{\theta(z + \phi(\lambda) - \phi(p)) \theta(z+ \phi(q) -
\phi(\mu)) E(\mu, \lambda) E(q,p)}
{\theta(z+\phi(\lambda) - \phi(\mu)) \theta(z + \phi(q) - \phi(p))
E(\mu,p) E(q,\lambda)}
\notag
\end{align}
Multiplication of both sides by $\theta(z+\phi(\lambda)-\phi(\mu))
\theta(z + \phi(q) - \phi(p)) E(\mu,p) E(q, \lambda)$ along with
a liberal use of
the general identity $E(p,q) = -E(q,p)$ along with some algebra now
leads to Fay's identity \eqref{faytrisecant} as desired.
\end{pf}
The identity \eqref{faytrisecant} is actually a special case of a more
general identity (see Corollary 2.19 in \cite{oldfay}) which gives an
explicit expression for the determinant of a matrix $M$ of the form
\[
M = [M_{ij}] \text{ where } M_{ij} = \dfrac{\theta(z + \phi(\mu^j)
- \phi(\lambda^i))}{E(\mu^j, \lambda^i)}.
\]
Since the cofactor matrices of such a matrix are of the same form, one
can then compute explicitly (in terms of theta functions and prime
forms) the entries of the inverse of the matrix $\Gamma$ appearing in
Theorem \ref{T:lineabsint}. In this way one can verify by direct
computation the identity of the two expressions \eqref{prod} and
\eqref{partialfrac} for $T(p)$ in Theorem \ref{T:lineabsint}. This then
is a canonical higher genus generalization of Theorem 4.3.2 in \cite{bgr}.
Note that this proof of Fay's identity arises from equating a
multiplicative formula for the solution of a zero-pole interpolation
problem to a partial-fraction expression for the same solution. Formula
\eqref{solution} in Theorem
\ref{T:absint} gives an analogue of the partial fraction expression for
the solution of a zero-pole interpolation problem for a vector bundle
endomorphism. Formula \eqref{solution}, giving a connection between the Cauchy
kernel $K(\chi; p,q)$ and $K(\widetilde \chi; p,q)$, can be viewed as a
matrix-valued version of the Fay trisecant identity.
There is one case in higher rank when a multiplicative representation does
exist, namely the case of full rank zero-pole interpolation (where the
given pole vectors $\{u_{j \beta} \colon 1 \le \beta \le s_j = r\}$ span
the fiber space $\widetilde{\chi}(\mu^j)$ and
the given null vectors $\{ x_{i \alpha} \colon 1 \le \alpha \le t_i =r\}$
span the fiber space $\widetilde{\chi}^\vee(\lambda^i)$) for each $i$ and
$j$. Then $T(p)$ is again given by \eqref{prod} where $Q$ now is a
product of a scalar from the fiber of ${\cal O}({\boldsymbol\lambda} -
{\boldsymbol \mu})(q)$ and a value at $q$ of an automorphism of $\widetilde
\chi$. Without loss of generality we may assume that $\{x_{i \alpha}
\colon 1 \le \alpha \le r\}$ and $\{u_{j \beta} \colon 1 \le \beta \le
r\}$ consist of the standard basis vectors for each $i$ and $j$. Then in
the partial fraction expansion \eqref{solution} of $T(p)$ we have that
$\Gamma$ has the block matrix form
\begin{equation} \label{Gammafull}
\Gamma = -[K(\widetilde{\chi}; \lambda^i, \mu^j)]_{i=1, \dots, n_0; j=1,
\dots, n_\infty},
\end{equation}
and that $K_{ {\boldsymbol \mu},{\bold u}}(p)$ and $K^{\bold x,
{\boldsymbol \lambda}}(q)$
are block row and column matrices respectively
\begin{gather}
K_{{\boldsymbol \mu}, {\bold u}}(p) =
\begin{bmatrix} K(\widetilde{\chi}; p, \mu^1) & \dots &
K(\widetilde{\chi}; p, \mu^{n_\infty}) \end{bmatrix}, \\
K^{{\bold x},{\boldsymbol \lambda}}(q) = \begin{bmatrix}
K(\widetilde{\chi}; \lambda^1, q) \\
\vdots \\ K(\widetilde{\chi}; \lambda^{n_0}, q)\end{bmatrix}.
\end{gather}
Equating this multiplicative formula to the partial fraction expansion
leads to the same result as in formula (2.16) in \cite{newfay}.
\section{Determinantal representations of algebraic curves and kernel
bundles via Cauchy kernels} \label{S:detrep}
In \cite{hip} zero-pole interpolation problems of the sort discussed here
were studied in a more concrete setting of vector bundles over an
algebraic curve embedded in projective space with fiber space given as
the kernel of a two-variable matrix pencil. In this section we make the
connections between that setting and the abstract compact
Riemann surface setting
of Section~\ref{S:absint} of this paper explicit. As we shall see, the
link between
the two settings is provided by the Cauchy kernels introduced in
Section~\ref{S:cauchyker}.
We first review the setting from \cite{hip}. Suppose that we are given
three $M \times M$ matrices $\sigma_1, \sigma_2, \gamma$ and let
$U_0(z) = U_0(z_1, z_2)$ be the two-variable linear
matrix pencil
\[
U_0(z) = z_1 \sigma_2 - z_2 \sigma_1 + \gamma, \quad
z=(z_1, z_2).
\]
We will also often consider the homogenization $U(\mu)$ (where $\mu =
[\mu_0,\mu_1,\mu_2]$ are projective coordinates in ${\bold P}^2$) given by
\[
U(\mu) = \mu_0 U_0(\frac{\mu_1}{\mu_0}, \frac{\mu_2}{\mu_0})
=\mu_1 \sigma_2 - \mu_2 \sigma_1 + \mu_0 \gamma.
\]
Although $\det U(\mu)$ is not well-defined as a function of the
projective variable $\mu=[\mu_0,\mu_1,\mu_2]$, nevertheless its zero set
is well-defined and defines a curve $C \subset {\bold P}^2$ by
\[
C=\{\mu=[\mu_0,\mu_1,\mu_2] \in {\bold P}^2\colon \det U(\mu)=0\}.
\]
We shall assume that $U(\mu)$ defines a maximal irreducible determinantal
representation of rank $r$; this means that $\det U(\mu) = F(\mu)^r$
where $F$ is an irreducible homogeneous polynomial of degree $m$ (so
$M=rm$), and that $\ker U(\mu) = r$ for all smooth points $\mu$ of $C$,
i.e., points $\mu^0$ where at least one of $\frac{\partial F}{\partial
\mu_0}(\mu^0)$, $\frac{\partial F}{\partial \mu_1}(\mu^0)$,
and $\frac{\partial
F}{\partial \mu_2}(\mu^0)$ is not zero. In case of a singular point
$\mu^0$, we assume that $\dim \ker U(\mu^0)$ is as large as possible,
namely $sr$ where $s$ is the multiplicity of $\mu^0$. Under these
conditions $E(\mu) = \ker U(\mu)$ lifts to a vector bundle $E$ of rank
$r$ over the normalizing Riemann surface $X$ of $C$; note that the bundle
$E$ is realized concretely as a rank $r$ subbundle of the trivial bundle
of rank $M$ over $X$. The normalizing Riemann surface $X$ is a
Riemann surface
such that there is a holomorphic mapping $\pi \colon X \to {\bold P}^2$
whose image equals $C$ such that $\pi$ is a one-to-one immersion on the
inverse image of smooth points of $C$; we call $\pi \colon X \rightarrow C$
a birational embedding of $X$ in ${\bold P}^2$.
For more details, see \cite{hip}.
As in \cite{hip}, we shall assume for simplicity that all the singular
points of $C$ are nodes (i.e., $\pi^{-1}(q) = \{p^1, p^2 \}$ where $p^1$
and $p^2$ are distinct points on $X$ with neighborhoods $U_1$ and $U_2$
such that $\pi$ is an immersion at both $p^1$ and $p^2$ and the analytic
arcs $\pi(U_1) $ and $\pi(U_2)$ intersect transversally at $q$).
We also assume that the line at infinity $\{\mu_0 = 0\}$ is nowhere
tangent to $C$.
The holomorphic vector bundle $E_\ell$ which is dual to $E$ can be
realized concretely as a subbundle of the trivial rank $M$ bundle over $X$
(with fibers now written as row vectors) via
\[
E_\ell(\mu) = \ker_\ell U(\mu)
=\{x \in {\bold C}^{1 \times M} \colon x U(\mu) = 0\}.
\]
A concrete pairing between $E_\ell \otimes {\cal O}(1) \otimes \Delta$ and
$E \otimes {\cal O}(1) \otimes \Delta$ is given by
\begin{equation} \label{pairing}
\{u_\ell, u\} = \frac{u_\ell}{\mu_0} \frac{\xi_1\sigma_1 + \xi_2
\sigma_2}{\xi_1\ d\lambda_1 + \xi_2\ d\lambda_2} \frac{u}{\mu_0}.
\end{equation}
Here $u_\ell$ and $u$ are local holomorphic sections of $E_\ell \otimes
{\cal O}(1) \otimes \Delta$ and $E\otimes {\cal O}(1) \otimes \Delta$
respectively, $\lambda_1$ and $\lambda_2$ are meromorphic functions on
$X$ given by $\lambda_1 = z_1 \circ \pi$ and $\lambda_2 = z_2 \circ \pi$,
and $\xi_1,\xi_2$ are arbitrary (not both zero)
complex parameters.
If $E$ and $E_\ell$ are right and left kernel bundles determined by a
rank $r$ maximal determinantal representation $U(\mu)$ of a curve $C$ as
above, then it can be shown that necessarily $E \otimes {\cal O}(1)
\otimes \Delta$ is isomorphic to a flat bundle $\chi$ over $X$ with the
property that $h^0(\chi \otimes \Delta) = 0$, and that $E_\ell \otimes
{\cal O}(1) \otimes \Delta$ is isomorphic to the dual $\chi^\vee$ of $\chi$.
This isomorphism of $E \otimes {\cal O}(1) \otimes \Delta$ with $\chi$
is implemented
explicitly by a {\it matrix of normalized sections} $u^\times(p)$.
Explicitly, $u^\times$ is an $M \times r$ matrix whose columns are
meromorphic sections of the pullback of $E \otimes \Delta$ to the
universal cover $\widetilde{X}$ of $X$ such that:
\begin{enumerate}
\item[1.] $\dfrac{1}{\sqrt{dt}(R \widetilde{p})} u^\times(R \widetilde{p}) =
\frac{1}{\sqrt{dt}(\widetilde{p})}u^\times(\widetilde{p})
\chi^{-1}(R)$ for all $\widetilde{p}
\in \widetilde{X}$ and all $R \in \text{Deck}(\widetilde{X}/X)$ $ \cong
\pi_1(X)$, where $t$ is a local parameter on $X$ and $\sqrt{dt}$ is the
corresponding local holomorphic frame for $\Delta$ lifted to the
neighborhoods of $\widetilde{p}$ and $R\widetilde{p}$ on $\widetilde{X}$.
\item[2.] Each column of $u^\times$ has first order poles at (the points
of $\widetilde{X}$ over) the points of $C$ at infinity, and is
holomorphic everywhere else.
\item[3.] For each $p \in X$, the columns of $u^\times(\widetilde{p})$
form a basis for the fiber $(E \otimes \Delta)(p)$, where $\widetilde{p}
\in \widetilde{X}$ is over $p$ (if $p$ is a point of $C$ at infinity we
have first to multiply $u^\times$ by a local parameter centered at $p$).
\end{enumerate}
Simply speaking, $u^\times$ consists of a multiplicative $\Delta$-valued
meromorphic frame for $E$, normalized to have poles exactly at the points
of $C$ at infinity. An isomorphism $\chi \to E \otimes {\cal O}(1)
\otimes \Delta$ is now given explicitly by $y \to \mu_0 u^\times y$
where $y$ is a
local holomorphic section of $\chi$. An $r \times M$ matrix of normalized
sections $u_\ell^\times$ of $E_\ell$, whose rows are meromorphic sections
of the pullback of $E_\ell \otimes \Delta$ to the universal covering
$\widetilde{X}$ of $X$, is defined similarly, with item (1) replaced by
\begin{enumerate}
\item[$1_\ell .$] $\dfrac{1}{\sqrt{dt}}
(R \widetilde{p}) u_\ell^\times(R \widetilde p) =
\frac{1}{\sqrt{dt}(\widetilde{p})} \chi(R) u_\ell^\times(\widetilde{p})$
for all $\widetilde{p} \in \widetilde{X}$ and all $R \in
\text{Deck}(\widetilde{X}\backslash X) \cong \pi_1(X)$, where $t$ and
$\sqrt{dt}$ are as before.
\end{enumerate}
An isomorphism $\chi^\vee \to E_\ell \otimes
{\cal O}(1) \otimes \Delta$ is given explicitly by $x \to \mu_0 x^T
u_\ell^\times$, where $x$ is a local holomorphic section of $\chi^\vee$.
Given $u^\times$, the {\it dual }
matrix of normalized section $u_\ell^\times$ is determined uniquely by
\[
u_\ell^\times \frac{\xi_1 \sigma_1 + \xi_2 \sigma_2}{\xi_1\ d\lambda_1 +
\xi_2\ d\lambda_2} u^\times = I_r
\]
(where $I_r$ is the $r \times r$ identity matrix), so that under the
isomorphisms $\chi \cong E \otimes {\cal O}(1) \otimes \Delta$ and
$\chi^\vee \cong E_\ell \otimes {\cal O}(1) \otimes \Delta$ the natural
duality pairing between $\chi^\vee$ and $\chi$ equals the pairing
\eqref{pairing}.
If we now define $K(\chi; \widetilde{p}, \widetilde{q})$ by
\begin{equation} \label{cauchypairing}
K(\chi; \widetilde{p}, \widetilde{q}) = u_\ell^\times(\widetilde{p})
\frac{\xi_1 \sigma_1 + \xi_2 \sigma_2}{\xi_1(\lambda_1(\widetilde{p}) -
\lambda_1(\widetilde{q})) + \xi_2 (\lambda_2(\widetilde{p}) -
\lambda_2(\widetilde{q}))} u^\times(\widetilde{q}),
\end{equation}
then $K$ has all the properties of the Cauchy kernel as defined in
Section \ref{S:cauchyker}. This method of constructing the Cauchy
kernel, via a dual pair of normalized sections of the kernel bundles
associated with a maximal determinantal representation of an algebraic
curve $C$ embedded in ${\bold P}^2$ which has the Riemann surface $X$ as
its normalizing surface, was presented in \cite{hip}.
Here we wish to make explicit the reverse path. We start with a
compact Riemann
surface $X$ and a flat holomorphic vector bundle $\chi$ over $X$ for which
$h^0(\chi \otimes \Delta) = 0$. We assume as given the associated Cauchy
kernel as developed in Section \ref{S:cauchyker}. We then construct a
birational embedding of $X$ into ${\bold P}^2$ with image equal to the
curve $C$ together with a rank $r$ maximal determinantal representation
of $C$ in such a way that we recover the Cauchy kernel $K(\chi; \cdot,
\cdot)$ from a dual pair of normalized sections for the associated left
and right kernel bundles associated with this determinantal
representation of $C$, as in \eqref{cauchypairing}.
We first need some preparations. Let $\chi$ be a flat vector bundle over
the Riemann surface $X$ such that $h^0(\chi \otimes \Delta)=0$.
In addition choose two scalar meromorphic functions $\lambda_1$,
$\lambda_2$ on $X$ such that ${\cal M}(X) = {\bold C}(\lambda_1,
\lambda_2)$, i.e., rational functions in $\lambda_1,\lambda_2$ generate
the whole field of (scalar) meromorphic functions on $X$. Assume that
all poles of $\lambda_1$ and $\lambda_2$ are simple, and denote the set
of poles by $x^1, \dots, x^m \in X$. Define complex numbers $c_{ik}$ ($1
\le i \le m$, $k=1,2$) by
\[
c_{ik} = -\text{Res}_{p=x^i}\lambda_k(p)
\]
where the residue is with respect to some fixed local coordinate $t^i =
t^i(p)$ centered at $p=x^i$. On occasion we shall also need the next
coefficient $-d_{ik}$ in the Laurent expansion of $\lambda_k$ at $x^\i$:
\[
\lambda_k(p) = -\frac{c_{ik}}{t^i} - d_{ik} + O(|t^i|).
\]
Define $M \times M$ matrices (where $M = mr$) $\sigma_1, \sigma_2,\gamma$ by
\begin{equation} \label{pencilcoef}
\sigma_1 = \underset{1\le i \le m}{\text{diag.}} (c_{i1} I_r), \quad
\sigma_2 = \underset{1\le i \le m}{\text{diag.}}(c_{i2}I_r) \quad
\gamma = [\gamma_{ij}]_{i,j=1,\dots,m}
\end{equation}
where
\[
\gamma_{ij} = \begin{cases}
d_{i1}c_{i2} -d_{i2} c_{i1}, & i=j \\
(c_{i1} c_{j2}- c_{j1}c_{i2}) \dfrac{K(\chi; x^i, x^j)}{dt^j(x^j)}, & i \ne j.
\end{cases}
\]
Also define
\begin{equation} \label{normsec}
u^\times(p) = \begin{bmatrix} K(\chi; x^1, p) \\ \vdots \\ K(\chi; x^m,p)
\end{bmatrix}, \quad
u_\ell^\times(p) = -\begin{bmatrix} K(\chi; p, x^1) & \dots & K(\chi; p, x^m)
\end{bmatrix}
\end{equation}
Then we have the following result.
\begin{theorem} \label{T:detrep} Let $\chi$ be a flat vector bundle over
the Riemann surface $X$ such that $h^0(\chi \times \Delta) = 0$ with
associated Cauchy kernel $K(\chi; \cdot, \cdot)$ and use a pair of
meromorphic functions $\lambda_1(p),\lambda_2(p)$ on $X$ which generate
the field ${\cal M}(X)$ of meromorphic functions on $X$ to define matrices
$\sigma_1$, $\sigma_2$ and $\gamma$ as in \eqref{pencilcoef}. Then:
(i) The map $\pi_0 \colon X \to {\bold C}^2$ given by
\[
\pi_0(p) = (\lambda_1(p), \lambda_2(p))
\]
maps $X \backslash \{x^1, \dots, x^m\}$ onto the affine part $C_0$ of
an algebraic curve $C \subset {\bold P}^2$ and extends to a
birational embedding $\pi \colon X \rightarrow C$ of $X$ in ${\bold P}^2$.
The defining irreducible
homogeneous polynomial $F(\mu_0, \mu_1, \mu_2)$ of $C$ is such that
$ \det (\mu_1 \sigma_2 -
\mu_2 \sigma_1 + \mu_0 \gamma) = F(\mu_0, \mu_1, \mu_2)^r$.
(ii) Denote by $E$ the kernel bundle over $C$ given in affine
coordinates by
\[
E(\lambda) = \ker (\lambda_1 \sigma_2 - \lambda_2 \sigma_1 + \gamma),\quad
\lambda= (\lambda_1, \lambda_2).
\]
Then $\chi \cong E \otimes {\cal O}(1) \otimes \Delta$ with $u^\times$ and
$u_\ell^\times$ given by \eqref{normsec} equal to the dual matrices of
normalized sections of $E$ and $E_\ell$.
\end{theorem}
We shall prove Theorem \ref{T:detrep} under the assumption that all the
singular points of $C$ are nodes.
\begin{pf}
We define the curve $C$ as the compactification in projective space of
the image
of the map $\pi$ in the statement of the theorem
\[
C_0 = \{(\lambda_1(p), \lambda_2(p))\colon p \in X\}.
\]
Then $X$ is the normalizing Riemann surface for the curve $C$ and the
degree of $C$ is equal to the number of intersections with the line at
infinity, namely, deg $C=m$. Let $f(z_1,z_2)=0$ be an irreducible
polynomial of degree $m$ such that $f(z_1,z_2)=0$ is the defining equation
for $C$ (in affine coordinates). Thus $f(\lambda_1(p),\lambda_2(p))=0$
for all $p \in X$. We must show that $z_1\sigma_2 - z_2 \sigma_1 +
\gamma$ is a maximal determinantal representation of $f(z_1,z_2)^r=0$,
that $E \otimes {\cal O}(1) \otimes \Delta \cong \chi$ and that
$u^\times$ and $u_\ell^\times$ are dual matrices of normalized sections
of $E$ and $E_\ell$.
The first step is to prove the identities
\begin{align}
(\lambda_1(p) \sigma_2 -\lambda_2(p)\sigma_1+\gamma)u^\times(p) & =0
\label{identity1} \\
u_\ell^\times(p) (\lambda_1(p) \sigma_2 - \lambda_2(p) \sigma_1 + \gamma)
& = 0
\label{identity2} \\
\frac{u_\ell^\times(p) (\xi_1 \sigma_1 + \xi_2 \sigma_2 ) u^\times(p)}
{\xi_1 \ d\lambda_1(p) + \xi_2\ d\lambda_2(p)} & = 1.
\label{identity3}
\end{align}
To prove \eqref{identity1}, set
\[
h(p) = (\lambda_1(p) \sigma_2 - \lambda_2(p) \sigma_1 + \gamma)
u^\times(p).
\]
Note that $h(p)$ is a meromorphic section of $\chi \otimes \Delta$. To
check that $h=0$ it suffices to check that $h$ has no poles (since
$h^0(\chi \otimes \Delta)=0$). The only possible poles in the formula
for $h$ occur at the points $x^1, \dots, x^m$ and these are at most
double poles. For $\alpha = 1, \dots, m$, let us write down the Laurent
expansion for $h$ near $x^\alpha$ as
\[
h(p) = [h]^{\alpha,-2}(t^\alpha)^{-2} + [h]^{\alpha, -1} (t^\alpha)^{-1}
+ [\text{analytic at $x^\alpha$]}.
\]
We must show that $[h]^{\alpha, -2}=0$ and $[h]^{\alpha, -1} = 0$
for $1 \le
\alpha \le m$.
Each of $[h]^{\alpha, -2}$ and $[h]^{\alpha, -1}$ in turn is a block $m
\times 1$ column matrix: $[h]^{\alpha, -2} = \left[ [h]^{\alpha, -2}_i\right]$
and
$[h]^{\alpha, -1} = \left[ [h]^{\alpha, -1}_i \right]$ with $i = 1,
\dots m$. We compute
\begin{align} \notag
[h]^{\alpha, -2}_i =& \sum_{j=1}^m (-c_{\alpha 1} c_{i2} \delta_{ij} +
c_{\alpha 2} c_{i 1} \delta_{ij}) \delta_{j \alpha}\cdot
(- dt^\alpha(x^\alpha))\\
\notag
=& ( c_{\alpha 1} c_{\alpha 2} - c_{\alpha 2} c_{\alpha 1})
dt^\alpha(x^\alpha) = 0
\end{align}
where $\delta_{ij}$ is the Kronecker delta.
Similarly,
\begin{align} \notag
[h]^{\alpha, -1}_i =& \sum_{j,\ j\ne \alpha} (-c_{\alpha 1} c_{i 2}
\delta_{ij} +c_{\alpha 2} c_{i 1} \delta_{ij}) K(\chi, x^j, x^\alpha) \\
\label{hresidue}
& + \sum_j^m \{-d_{\alpha 1} c_{i 2} \delta_{ij} + d_{\alpha 2} c_{i 1}
\delta_{ij} + \gamma_{ij} \} \delta_{j \alpha}(- dt^\alpha(x^\alpha)).
\end{align}
For $i = \alpha$ \eqref{hresidue} becomes
\[
(-d_{\alpha 1} c_{\alpha 2} + d_{\alpha 2} c_{\alpha 1} + d_{\alpha 1}
c_{\alpha 2} - c_{\alpha 1} d_{\alpha 2})(- dt^\alpha(x^\alpha))= 0
\]
while, for $i \ne \alpha$, \eqref{hresidue} becomes
\[
(-c_{\alpha 1} c_{i2} + c_{\alpha 2} c_{i 1}) K(\chi; x^i, x^\alpha) +
(c_{i1} c_{\alpha 2} - c_{\alpha 1} c_{i 2}) \frac{K(\chi; x^i,
x^\alpha)}{dt^\alpha(x^\alpha)}(- dt^\alpha(x^\alpha)) = 0.
\]
Thus $h=0$ and \eqref{identity1} follows.
By a similar calculation of Laurent series coefficients one can verify
\eqref{identity2}.
To verify \eqref{identity3},
by a standard lemma (see Proposition 2.3 in \cite{hip}), it suffices to
show that
\[
\frac{u^\times_\ell(p) \sigma_k u^\times(p)}{d\lambda_k(p)} = 1 \text{
for } k=1,2.
\]
The numerator of the expression on the left is given by
\begin{gather} \notag
\begin{bmatrix} K(\chi; p,x^1) & \dots & K(\chi; p,x^m) \end{bmatrix}
\begin{bmatrix} -c_{ik} & & \\ & \ddots & \\ & & - c_{mk} \end{bmatrix}
\begin{bmatrix} K(\chi; x^1,p) \\ \vdots \\ K(\chi; x^m, p)
\end{bmatrix} \\
\notag
= -\sum_{i=1}^m K(\chi; p, x^i) [\text{Res}_{p=x^i} \lambda_k(p)] K(\chi;
x^i,p)
\end{gather}
and hence
\[
\frac{u^\times_\ell(p) \sigma_k u^\times(p)}{d\lambda_k(p)} =
- \frac{\sum_{i=1}^m K(\chi; p, x^i) [\text{Res}_{p=x^i}\lambda_k(p)]
K(\chi; x^i,p)}
{d\lambda_k(p)}.
\]
The double pole at each $x^i$ in the numerator is cancelled by a double
pole at each $x^i$ in the denominator. Note also that the product of two
half-order differentials in the numerator is cancelled by the
differential in the denominator. The resulting quotient is a
well-defined holomorphic section of $Hom(\chi,\chi)$ which has the value
$I$ at $x^1, \dots, x^m$, and hence must equal $I$ at all $p \in X$.
Equation \eqref{identity3} now follows.
Denote by $d(z_1,z_2)$ the polynomial $d(z_1,z_2)=\det
(z_1\sigma_2-z_2\sigma_1 + \gamma)$. By \eqref{identity1} or
\eqref{identity2} we know that $d(\lambda_1(p),\lambda_2(p))=0$
identically in $p \in X$. Since $f$ is by assumption the irreducible
defining polynomial for $X$, it follows that $f(z_1,z_2) | d(z_1,z_2)$,
and hence,
\begin{equation} \label{fact}
d(z_1,z_2) = f(z_1,z_2)^s g(z_1,z_2)
\end{equation}
for some positive integer $s$ and some polynomial $g$ relatively prime
with respect to $f$. By inspection we see that $d$ has
degree equal to $M=mr$,
while, as already mentioned, $f$ has degree equal to $m$. From the
factorization we see that deg $d \ge s(\text{deg } f)$, or
$r \ge s$. On the other hand, at a smooth point $(z_1,z_2) \in C$, we
know that $\dim E(z) \ge r$ since the $r$ linearly independent columns of
$u^\times(p)$ are in $\ker (\lambda_1(p) \sigma_2 - \lambda_2(p) \sigma_1
+ \gamma)$. But also, from the factorization \eqref{fact} and an
inductive argument working with the minors the matrix pencil
$z_1 \sigma_2 - z_2 \sigma_1 + \gamma$ (see the proof of
Theorem 3.2 in \cite{hip}) one can show that $\dim E(z) \le s$.
From $r \ge s$ and $r \le \dim E(z) \le s$ we conclude that $r=s$.
From \eqref{fact} and degree counting we conclude that the polynomial $g$
is a constant, and without loss of generality, $d=f^r$. Thus
$z_1\sigma_2 - z_2 \sigma_1 + \gamma$ is a maximal determinantal
representation of $f(z_1,z_2)^r=0$ as asserted, except that we still must
check that for any node $q$ on $C$ the columns of $u^\times(p^1)$ and
$u^\times(p^2)$ are linearly independent, where $\pi^{-1}(q) = \{p^1, p^2\}$.
It then follows from \eqref{identity1}, \eqref{identity2} and
\eqref{identity3} that $u^\times(p)$ and $u^\times_\ell(p)$ form the
associated dual pair of normalized cross-sections for $E$ and $E_\ell$
respectively.
We first claim if $L$ is a straight line nowhere tangent to $C$ and
$y^1, \dots, y^m$ are the preimages on $X$ of the points of intersection
of $C$ with $L$, then the block matrix $[K(\chi; x^i, y^j]_{i,j = 1,
\dots, m}$ is invertible. This follows immediately from the discussion
of the full rank zero-pole interpolation problem at the end of Section 4,
as we now show.
Since the divisor $x^1 + \dots + x^m - y^1 - \dots - y^m$ is equivalent
to $0$, the input bundle in the full rank zero-pole
interpolation problem with zeros $x^1, \dots, x^m$ and poles $y^1, \dots, y^m$
and output bundle $\chi$ is again (isomorphic to) $\chi$. Since $h^0(\chi
\otimes \Delta) = 0$, the matrix $\Gamma$ \eqref{Gammafull} is invertible.
Next, by taking any line $L$ through the node $q$ which is nowhere tangent to
$C$, we see that the columns of $u^\times(p^1)$ and $u^\times(p^2)$ are
columns in a $M \times M$ invertible matrix (namely, the associated
matrix $\Gamma$), and hence are linearly
independent.
\end{pf}
We remark that the same proof works when the singularities of $C$ are
any ordinary singular points, or more generally, are such that a singular
point of multiplicity $s$ has $s$ distint preimages on $X$.
\section{The concrete interpolation problem for meromorphic bundle maps
between kernel bundles of determinantal representations
of an algebraic curve} \label{S:conint}
In the paper \cite{hip} the following problem was considered. We are
given an irreducible algebraic curve $C$ in ${\bold P}^2$ together with
its normalizing compact Riemann surface $X$ and the normalization map
$\pi \colon X \to C$. We assume that the defining polynomial for $C$ is
an irreducible polynomial $f$ of degree $m$ (in affine coordinates). For
simplicity we assume again that the only singularities of $C$ are nodes and
that $C$ intersects the line at infinity in $m$ distinct smooth points.
We suppose in addition that $f^r$ has a maximal determinantal representation
\[
f^r(z_1,z_2) = \det (z_1 \sigma_2 - z_2 \sigma_1 + \widetilde{\gamma})
\]
where $\sigma_1$, $\sigma_2$ and $\widetilde{\gamma}$ are $M \times M$
matrices ($M=mr$), with which is associated the kernel bundle
$\widetilde{E}$ of rank $r$ over $C \backslash C_{sing}$ ($C_{sing}$ is
the set of the singular points of $C$) with fibers (over affine points)
given by
\begin{equation} \label{outputbundle}
\widetilde{E}(z) = \ker (z_1 \sigma_2 - z_2 \sigma_1 +
\widetilde{\gamma}).
\end{equation}
As explained in Section \ref{S:detrep}, we may consider the pullback of
$\widetilde{E}$ to $X \backslash \pi^{-1}(C_{sing})$ as extended to a
rank $r$ vector bundle over all of $X$. We also have the left kernel
bundle $\widetilde{E}_\ell$ where
\begin{equation} \label{dualoutputbundle}
\widetilde{E}_\ell(z) = \ker_\ell (z_1 \sigma_2 - z_2 \sigma_1 +
\widetilde{\gamma})
\end{equation}
with pullback under $\pi$ also extendable to a rank $r$ vector bundle
defined over all of $X$. These bundles, or more precisely their twists
$\widetilde{E} \otimes {\cal O}(1) \otimes \Delta$ and
$\widetilde{E}_\ell \otimes {\cal O}(1) \otimes \Delta$, have the canonical
pairing \eqref{pairing} with each other, as explained in Section
\ref{S:detrep}.
The data for the concrete interpolation (CONINT)
problem to be considered in this
section consists of:
\begin{enumerate}
\item[(D1)] $n_\infty$ distinct smooth, finite points $\mu^1 = (\mu^1_1,
\mu^1_2), \dots, \mu^{n_\infty} = (\mu^{n_\infty}_1, \mu^{n_\infty}_2)$
of $C$ (the preassigned poles),
\item[(D2)] for each $j=1, \dots, n_\infty$, a linearly independent set
$\{\varphi_{j1}, \dots, \varphi_{j,s_j} \}$ of $s_j$ vectors in the fiber
$\widetilde{E}(\mu^j)$ (the preassigned pole vectors),
\item[(D3)] $n_0$ distinct smooth, finite points $\lambda^1 = (\lambda^1_1,
\lambda^1_2), \dots, \lambda^{n_0}=(\lambda^{n_0}_1, \lambda^{n_0}_2)$ of
$C$ (the preassigned zeros),
\item[(D4)] for each $i=1, \dots, n_0$, a linearly independent set
$\{\psi_{i1}, \dots, \psi_{i,t_i}\}$ of $t_i$ vectors in the fiber
$\widetilde{E}_\ell(\lambda^i)$ (the preassigned null vectors), and
\item[(D5)] for each pair of indices $(i,j)$ for which $\lambda^i = \mu^j =:
\xi^{ij}$, a choice of a local coordinate $t^{ij}$ on $X$ centered at
$\xi^{ij}$ and a collection of numbers $\{\rho_{ij, \alpha \beta} \colon
1 \le \alpha \le t_i, 1 \le \beta \le s_j\}$ (the preassigned coupling
numbers with respect to the chosen local coordinate).
\end{enumerate}
The interpolation problem then is to find an $M \times M$ matrix $\gamma$
defining a maximal determinantal representation of $f^r$
\begin{equation} \label{detrepforfr}
f^r(z_1,z_2) = \det (z_1 \sigma_2 - z_2 \sigma_1 + \gamma)
\end{equation}
giving the kernel bundle $E$ over $X$ with fiber over a smooth finite
point $z \in C$ given by
\begin{equation} \label{inputbundle}
E(z) = \ker (z_1 \sigma_2 - z_2 \sigma_1 + \gamma)
\end{equation}
and the left kernel bundle $E_\ell$ given by
\begin{equation} \label{dualinputbundle}
E_\ell(z) = \ker_\ell (z_1 \sigma_2 - z_2 \sigma_1 + \gamma)
\end{equation}
together with meromorphic bundle maps
\[
S \colon E \to \widetilde{E},\quad S_\ell \colon \widetilde{E}_\ell \to
E_\ell
\]
(where we write bundle maps on left kernel bundles as acting from the
right), where $S \otimes I_{{\cal O}(1) \otimes \Delta}$ and $S_\ell
\otimes I_{{\cal O}(1) \otimes \Delta}$ are transposes of each other with
respect to the pairing \eqref{pairing}, so that $S$ (and $S_\ell$) act as
the identity operator $I$ on the corresponding fibers at the points at
infinity, and the following set of interpolation conditions is satisfied:
\begin{enumerate}
\item[(I1)] $S$ has poles only at $\mu^1, \dots, \mu^{n_\infty}$; for each
$j=1, \dots, n_\infty$, the pole of $S$ at $\mu^j$ is simple, and the
vectors $\{\varphi_{j1}, \dots, \varphi_{j,n_\infty}\}$ span the image
space of the residue $R_j \colon E(\mu^j) \to \widetilde{E}(\mu^j)$ of
$S$ at $\mu^j$.
\item[(I2)] The bundle map $S_\ell^{-1} \colon
E_\ell \to \widetilde{E}_\ell$ has
poles only at the points $\{\lambda^1, \dots, \lambda^{n_0}\}$; for each
$i=1, \dots, n_0$, the pole of $S_\ell^{-1}$ at $\lambda^i$ is simple and
the vectors $\{\psi_{i1} \dots, \psi_{i t_i}\}$ span the image space of
the residue $\widehat{R}_i \colon E_\ell(\lambda^i) \to
\widetilde{E}_\ell(\lambda^i)$ of $S_\ell^{-1}$ at $\lambda^i$.
\item[(I3)] For each pair of indices $(i,j)$ where $\lambda^i = \mu^j =:
\xi^{ij}$, and for $\alpha = 1, \dots, t_i$, let $\psi_{i \alpha}(p)$ be a
local holomorphic section of $\widetilde{E}_\ell$ near $\xi^{ij}$ with
\[
\psi_{i \alpha}(t^{ij}) = \psi_{i \alpha} + \psi_{i \alpha 1} t^{ij} +
o(t^{ij})
\]
such that $\psi_{i \alpha} S_\ell(p)$ has analytic continuation to $p =
\xi^{ij}$ with value there equal to $0$. Then, for any choice of complex
parameters $\xi_1$ and $\xi_2$
\[
\psi_{i \alpha 1}
\frac{\xi_1 \sigma_1 + \xi_2 \sigma_2}
{\xi_1 \lambda_1^{\prime}(\xi^{ij}) + \xi_2
\lambda_2^{\prime}(\xi^{ij})}\varphi_{j \beta}
- \psi_{i \alpha} (\xi_1 \sigma_1 + \xi_2 \sigma_2) \varphi_{j \beta}
\frac{\xi_1 \lambda_1^{\prime \prime}(\xi^{ij}) + \xi_2 \lambda_2^{\prime
\prime}(\xi^{ij})}
{2(\xi_1 \lambda_1^{\prime}(\xi^{ij}) + \xi_2
\lambda_2^{\prime}(\xi^{ij}))^2} = \rho_{ij, \alpha \beta}.
\]
Here ${}^{\prime} = \dfrac{d}{dt^{ij}}$.
\end{enumerate}
It can be shown that a necessary consistency condition
on the data set for the problem to have a
solution is that
\begin{equation} \label{ZP}
\psi_{i \alpha} (\xi_1 \sigma_1 + \xi_2 \sigma_2) \varphi_{j \beta} = 0.
\end{equation}
Given the data set (D1)--(D5) we form the $n_0 \times n_\infty$
block matrix $\Gamma^0 =
[\Gamma^0_{ij}]$ ($1 \le i \le n_0, 1 \le j \le n_\infty$) where
$\Gamma^0_{ij}=[\Gamma^0_{ij,\alpha \beta}]$ ($1 \le \alpha \le t_i$, $1
\le \beta \le s_j$) in turn is the $t_i \times s_j$ matrix with entries
given by
\begin{equation} \label{Gamma0}
\Gamma^0_{ij, \alpha \beta} = \begin{cases}
\psi_{i \alpha} \frac{\xi_1 \sigma_1 + \xi_2 \sigma_2}
{\xi_1(\mu^j_1 - \lambda^i_1) + \xi_2(\mu^j_2 - \lambda^i_2)}
\varphi_{j \beta} & \text{if } \lambda^i \ne \mu^j \\
-\rho_{ij, \alpha \beta} & \text{if } \lambda^i = \mu^j.
\end{cases}
\end{equation}
Additional matrices which we shall need are
\begin{align}
A_1 = \begin{bmatrix} \mu^1_1I_{s_1} & & \\
& \ddots & \\
& & \mu^{n_\infty}_1 I_{s_{n_\infty}} \end{bmatrix},
& \quad
A_2 = \begin{bmatrix} \mu^1_2I_{s_1} & & \\
& \ddots & \\
& & \mu^{n_\infty}_2 I_{s_{n_\infty}} \end{bmatrix}, \notag
\\
Z_1=\begin{bmatrix}\lambda^1_1 I_{t_1} & & \\
& \ddots & \\
& & \lambda^{n_0}_1 I_{t_{n_0}}
\end{bmatrix},
& \quad
Z_2=\begin{bmatrix}\lambda^1_2 I_{t_1} & & \\
& \ddots & \\
& & \lambda^{n_0}_2 I_{t_{n_0}}
\end{bmatrix}, \notag
\\
\varphi_j = \begin{bmatrix} \varphi_{j1} & \dots & \varphi_{j s_j}
\end{bmatrix},
& \quad
\varphi = \begin{bmatrix} \varphi_1 & \dots & \varphi_{n_\infty}
\end{bmatrix}, \notag
\\
\psi_i = \begin{bmatrix} \psi_{i1} \\ \vdots \\ \psi_{i t_i}
\end{bmatrix},
& \quad
\psi = \begin{bmatrix} \psi_1 \\ \vdots \\ \psi_{n_0} \end{bmatrix}.
\label{matrices}
\end{align}
The solution of the concrete interpolation problem (CONINT) obtained in
\cite{hip} is as follows.
\begin{theorem} \label{T:conint} (See Theorem 4.1 of \cite{hip}.)
Assume that we are given a curve $C$
with defining irreducible polynomial $f$, a maximal determinantal
representation for $f^r$ as in \eqref{detrepforfr} together with
associated kernel bundle $\widetilde{E}$ \eqref{outputbundle}
and left kernel bundle
$\widetilde{E}_\ell$ \eqref{dualoutputbundle}
and a data set (D1)--(D5) for the interpolation
problem (I1)--(I3). Then the interpolation problem has a solution if and
only if the interpolation data satisfy the compatibility conditions
\eqref{ZP}
at the overlapping zeros and poles, and the matrix $\Gamma^0$ given by
\eqref{Gamma0} is square and invertible.
In this case the unique solution of the
interpolation problem (I1)--(I3) is given by
\begin{equation} \label{gamma}
\gamma = \widetilde{\gamma} - \sigma_1 \varphi (\Gamma^0)^{-1} \psi \sigma_2
+ \sigma_2 \varphi (\Gamma^0)^{-1} \psi \sigma_1
\end{equation}
with associated kernel bundle $E$ \eqref{inputbundle} and left kernel
bundle $E_\ell$ \eqref{dualinputbundle},
with $S(z)$ given by
\begin{equation} \label{formulaforS}
S(z) = [I + \varphi (\xi_1(z_1I - A_1) + \xi_2(z_2 I -A_2))^{-1}
(\Gamma^0)^{-1} \psi (\xi_1 \sigma_1 + \xi_2 \sigma_2)]|_{E(z)}
\end{equation}
and with $S_\ell^{-1}(z)$ given (as a right multiplication operator) by
\begin{equation} \label{formulaforSell}
S_\ell^{-1}(z) = [I-(\xi_1 \sigma_1 + \xi_2 \sigma_2) \varphi
(\Gamma^0)^{-1} \left(\xi_1 (z_1 I - Z_1) + \xi_2( z_2 I - Z_2) \right)^{-1}
\psi] |_{E_\ell(z)}.
\end{equation}
Here the matrices $A_1,A_2,Z_1,Z_2,\psi,\varphi$ are as in
\eqref{matrices}.
\end{theorem}
The main goal of this section is to use the machinery developed in
Section \ref{S:detrep} to make explicit the connections between Theorem
\ref{T:conint} and Theorem \ref{T:absint} of Section \ref{S:absint}.
Suppose therefore that $\chi$ and $\widetilde{\chi}$ are two flat bundles
over the Riemann surface $X$ with $h^0(\chi \otimes
\Delta)=0=h^0(\widetilde{\chi} \otimes \Delta)$. We use a fixed pair
$(\lambda_1(p), \lambda_2(p))$ of meromorphic functions on $X$ generating
${\cal M}(X)$ to produce a map $\pi \colon X \to C$. The respective
Cauchy kernels $K(\chi; \cdot, \cdot)$ and $K(\widetilde{\chi}; \cdot,
\cdot)$ generate corresponding maximal determinantal representations
$z_1 \widetilde{\sigma}_2 - z_2 \widetilde{\sigma}_1 +
\widetilde{\gamma}$ and $z_1 \sigma_2^{\prime} - z_2 \sigma_1^{\prime} +
\gamma^{\prime}$ for $f(z_1,z_2)^r=0$ as in Theorem \ref{T:detrep}. Note
however that the formulas for $\sigma_1$ and $\sigma_2$ in
\eqref{pencilcoef} depend only on the choice of embedding functions
$\lambda_1(p)$ and $\lambda_2(p)$, and not on the particular flat bundle
$\chi$; hence we may and shall write simply $\sigma_i$ in place of
$\widetilde{\sigma}_i$ and $\sigma_i^{\prime}$ for $i=1,2$. We also have
associated dual pairs of matrices of normalized sections:
$\widetilde{u}^\times, \widetilde{u}_\ell^\times$ for $\widetilde{E}=
\ker(z_1 \sigma_2 - z_2 \sigma_1 + \widetilde{\gamma})$ and
$\widetilde{E}_\ell = \ker_\ell (z_1 \sigma_2 - z_2 \sigma_1 +
\widetilde{\gamma})$ respectively, and $u^{\times \prime}, u^{\times
\prime}_\ell$ for $E= \ker (z_1 \sigma_2 - z_2 \sigma_1 +
\gamma^{\prime})$ and $E_\ell = \ker_\ell (z_1 \sigma_2 - z_2 \sigma_1 +
\gamma^{\prime})$. Since $\widetilde{u}^\times$ implements an
isomorphism between $\widetilde{\chi}$ and $\widetilde{E} \otimes {\cal
O}(1) \otimes \Delta$ and $u^{\times \prime}$ implements an isomorphism
between $\chi$ and $E^\prime \otimes {\cal O}(1) \otimes \Delta$, any
meromorphic bundle map $T \colon \chi \to \widetilde{\chi}$ induces a
meromorphic bundle map $S \colon E^\prime \to \widetilde{E}$ determined by
\[
S(p) u^{\times \prime}(p) = \widetilde{u}^\times(p) T(p).
\]
However, in the solution of (CONINT) from \cite{hip} stated in Theorem
\ref{T:conint}, the solution $S$ is normalized to act as the identity
operator over the points of $C$ at infinity. In order for the map $S$
constructed as above from the abstract bundle map $T \colon \chi \to
\widetilde{\chi}$ to achieve this normalization, we must make an adjustment
\begin{equation} \label{adjustment}
\alpha (z_1 \sigma_2 - z_2 \sigma_1 + \gamma^\prime) \beta = z_1 \sigma_2
- z_2 \sigma_1 + \gamma
\end{equation}
on the input determinantal representation, where $\alpha, \beta \in
GL(M^{mr}, {\bold C})$. If $u^\times, u^\times_\ell$ is the dual pair of
normalized sections for $E=\ker (z_1 \sigma_2 - z_2 \sigma_1 + \gamma)$
and $E_\ell = \ker_\ell (z_1 \sigma_2 - z_2 \sigma_1 + \gamma)$, then
\[
u^\times(p) = \beta^{-1} u^{\times \prime}(p), \quad u^\times_\ell(p) =
u^{\times \prime}_\ell(p) \alpha^{-1}
\]
and we seek to solve instead
the equation
\[
S(p) u^\times(p) = \widetilde{u}^\times(p) T(p),
\]
or equivalently
\begin{equation} \label{intertwining}
S(p) \beta^{-1} u^{\times \prime}(p) = \widetilde{u}^\times(p) T(p)
\end{equation}
for $S,\alpha,\beta$ subject to the proviso that $S(x^i)=
I_{E(x^i)=\widetilde{E}(x^i)}$ for $i=1, \dots, m$. Since the columns of
$u^{\times \prime}(p)$ and $\widetilde{u}^\times(p)$ (after multiplication
by a local parameter on $X$ at $p$) simply form a
standard basis in ${\bold C}^M$ when evaluated at $x^1, \dots, x^m$, we
see that we should take
\[
\beta = \begin{bmatrix} T(x^1) & & \\ & \ddots & \\
& & T(x^m) \end{bmatrix}^{-1}.
\]
In order to guarantee $\alpha \sigma_k \beta = \sigma_k$ for $k=1,2$ as
required in \eqref{adjustment} we then take
\[
\alpha = \begin{bmatrix} T(x^1) & & \\ & \ddots & \\ & & T(x^m)
\end{bmatrix}.
\]
Thus $\gamma = \alpha \gamma^{\prime} \beta$ is given by $\gamma =
[\gamma_{ij}]_{i,j=1,\dots, m}$ with
\begin{equation} \label{inputgamma}
\gamma_{ij} = \begin{cases} d_{i1}c_{i2} - c_{i1} d_{i2} & \text{if } i=j \\
(c_{i1}c_{j2} - c_{j1}c_{i2})T(x^i)K(\chi;x^i,x^j)T(x^j)^{-1} & \text{if
} i \ne j.
\end{cases}
\end{equation}
We remark that the ``adjustment'' of $\gamma$ by the values of a bundle
map at the points of $X$ over the points of $C$ at infinity plays a
central role in the construction of triangular models for commuting
nonselfadjoint operators; see \cite{vin1} and \cite{lkmv}, Chapter 12.
We now suppose that we are given an abstract interpolation data set
$\omega$ as in \eqref{dataset} for an Abstract Interpolation Problem
(ABSINT) as in Section \ref{S:absint} with output bundle
$\widetilde{\chi}$, and let $y \to \widetilde{u}^\times y$ and $x \to
x^T \widetilde{u}^\times_\ell$ be the associated bundle isomorphisms
from $\widetilde{\chi}$ to $\widetilde{E} \otimes {\cal O}(1) \otimes
\Delta$ and from $\widetilde{\chi}^\vee$ to $\widetilde{E}_\ell \otimes
{\cal O}(1) \otimes \Delta$, where $\widetilde{E}$ and
$\widetilde{E}_\ell$ are the right and left kernel bundles respectively
associated with the maximal determinantal representation $z_1 \sigma_2 -
z_2 \sigma_1 + \widetilde{\gamma}$ (with
$\sigma_1,\sigma_2,\widetilde{\gamma}$ given by \eqref{pencilcoef},
and with
$\widetilde{u}^\times$ and $\widetilde{u}^\times_\ell$ given by
\eqref{normsec}, all with $\widetilde{\chi}$ in place of $\chi$). We
assume that the pair of meromorphic functions $\lambda_1(p)$ and
$\lambda_2(p)$ is chosen in such a way that the set of poles $x^1,
\dots, x^m$ is disjoint from the preassigned poles $\mu^1, \dots,
\mu^{n_\infty}$ and the set of preassigned zeros $\lambda^1, \dots,
\lambda^{n_0}$. Define a data set $\omega_0$ for a (CONINT) problem as
follows:
\begin{enumerate}
\item The preassigned poles consist of the points
$\pi(\mu^1) = (\mu^1_1, \mu^1_2),
\dots, \pi(\mu^{n_\infty}) = (\mu^{n_\infty}_1, \mu^{n_\infty}_2)$ with
associated pole vectors $\varphi_{j \beta} \in
\widetilde{E}(\pi(\mu^j))$ given by $\varphi_{j \beta} =
\widetilde{u}^\times(\mu^j) u_{j \beta}$ for $j=1, \dots, n_\infty$ and
$\beta = 1, \dots, s_j$.
\item The preassigned zeros consist of the points $\pi(\lambda^1) =
(\lambda^1_1, \lambda^1_2), \dots, \pi(\lambda^{n_0}) = ( \lambda^{n_0}_1,
\lambda^{n_0}_2)$ with associated sets of null vectors $\psi_{i \alpha}
\in \widetilde{E}_\ell(\pi(\lambda))$
given by $\psi_{i \alpha} = x^T_{i \alpha}
\widetilde{u}^\times_\ell(\lambda^i)$ for $i=1, \dots, n_0$ and
$\alpha=1, \dots, t_i$.
\item For those pairs of indices $(i,j)$ for which $z^i=w^j=:\xi^{ij}$
we take the associated coupling numbers $\rho_{ij, \alpha \beta}$ to be
the same as those specified for the (ABSINT) problem.
\end{enumerate}
With this choice of data set, the reader can check that the matrices
$\varphi$ and $\psi$ as defined in \eqref{matrices} reduce to
\[
\varphi = \begin{bmatrix} K_{\boldsymbol \mu, {\bold u}}(x^1) \\ \vdots \\
K_{\boldsymbol \mu,{\bold u}}(x^m) \end{bmatrix}, \quad
\psi = -\begin{bmatrix} K^{{\bold x}, \boldsymbol \lambda}(x^1) &
\dots & K^{{\bold
x}, \boldsymbol \lambda}(x^m) \end{bmatrix}
\]
where the notation $K_{\boldsymbol \mu,{\bold u}}(p)$ and
$K^{{\bold x}, \boldsymbol \lambda}(p)$
is as in the statement of Theorem \ref{T:absint}.
It turns out that the (ABSINT) problem with data set $\omega$ is
equivalent to the (CONINT) problem with data set $\omega_0$ under the
identifications $\pi \colon X \to C$ and $\widetilde{u}^\times:
\widetilde{\chi} \to \widetilde{E} \otimes {\cal O}(1) \otimes \Delta$
and $\widetilde{u}^\times_\ell: \widetilde{\chi}^\vee \to
\widetilde{E}_\ell \otimes {\cal O}(1) \otimes \Delta$ sketched above,
in that a solution $T$ of (ABSINT) corresponds to a solution $S$ of
(CONINT) under the correspondence (including the normalization
at the points over infinity) between
abstract bundle maps $T$ and concrete bundle maps $S$ discussed above.
(For the first two interpolation conditions, this observation is rather
transparent. For the third interpolation condition (I3), this requires
the relation between the interpolation condition (I3) for the (CONINT)
problem with a flat connection on the bundle $\widetilde{E}_\ell \otimes
{\cal O}(1) \otimes \Delta$ and the correspondence of this
connection with the coefficient $A_\ell(p)$ in the Laurent expansion of
the Cauchy kernel $K(\widetilde{\chi}; \cdot, \cdot)$ along the
diagonal; this is explained in Section 3.2 of \cite{hip}.)
Hence the formula for the solution $T$ of (ABSINT) in \eqref{solution} must
correspond to the formula for the solution $S$ of (CONINT) in
\eqref{formulaforS} under the correspondence \eqref{intertwining}. The
point of the next result is to verify this directly; in addition we see
that the matrix $\Gamma$ appearing in Theorem \ref{T:absint} is
identical to the matrix $\Gamma^0$ appearing in Theorem \ref{T:conint}.
\begin{theorem} \label{T:comparison}
Let $\omega$ be the data set for an (ABSINT) problem with
$\omega_0$ the corresponding data set for a (CONINT) problem. Then
$\Gamma = \Gamma_0$. Furthermore, if $\Gamma$ is invertible and $\chi$
is the input bundle for which (ABSINT) has a solution, then a solution
$T$ of (ABSINT) is related to the unique solution $S$ of (CONINT) having
value identity on the fibers over the points at infinity according to
the intertwining condition \eqref{intertwining}.
\end{theorem}
\begin{pf} The fact that $\Gamma = \Gamma_0$ is a simple consequence of
the definitions and of the formula \eqref{cauchypairing} expressing the
Cauchy kernel $K(\widetilde{\chi}; \cdot, \cdot)$ in terms of a dual
pair $\widetilde{u}^\times, \widetilde{u}^\times_\ell$ of normalized sections
of $\widetilde{E}$ and $\widetilde{E}_\ell$.
It remains to verify the intertwining relation \eqref{intertwining}
\[
S(p) \beta^{-1} u^{\times \prime}(p) = \widetilde{u}^\times(p) T(p)
\]
where $S$ is given by \eqref{formulaforS}, $T$ by \eqref{solution} and
$\beta^{-1} = \underset{1\le i\le m}{\text{diag.}} \{T(x^i)\}$.
We compute
\begin{align} \notag
[S(p) \beta^{-1} u^{\times \prime}(p)]_i = &
\sum_{j=1}^m S_{ij}(p) T(x^j) K(\chi; x^j,p) \\
\notag
= T(x^i) K(\chi; x^i,p) & -
K_{\boldsymbol \mu,{\bold u}}(x^i) \cdot
\underset{i'}{\text{diag.}}
\{( \xi_1 \lambda_1(p)+\xi_2 \lambda_2(p) - \xi_1 \mu^{i^\prime}_1
-\mu^{i^\prime}_2)^{-1} I_{s_{i^\prime}} \} \cdot
\\
& \cdot
\Gamma^{-1} \cdot \sum_{j=1}^m (\xi_1 c_{j1} + \xi_2 c_{j2})
K^{{\bold x}, \boldsymbol \lambda}( \lambda^j)
T(x^j)
K(\chi; x^j,p).
\label{Sside1}
\end{align}
From \eqref{solution} with $x^i$ in place of $p$ and $p$ in place of $q$
(and hence $T(p)$ in place of $Q$) we see that
\[
T(x^i) K(\chi; x^i,p)=[K(\widetilde{\chi};x^i,p)+
K_{\boldsymbol \mu,{\bold u}}(x^i) \Gamma^{-1} K^{{\bold x},
\boldsymbol \lambda}(p)] T(p).
\]
We use this identity both in the form indicated and with $x^j$ in place
of $x^i$ to convert \eqref{Sside1} to
\begin{align} \notag
[S(p) \beta^{-1} u^{\times \prime}(p)]_i = &
[K(\widetilde{\chi};x^i,p) +K_{\boldsymbol \mu,{\bold u}}(x^i) \Gamma^{-1}
K^{{\bold x},\boldsymbol \lambda}(p)] T(p) - \\
\notag
- K_{\boldsymbol \mu,{\bold u}}(x^i) \cdot & \underset{i^\prime}{\text{diag.}}
\{(\xi_1 \lambda_1(p) + \xi_2 \lambda_2(p) - \xi_1 \mu^{i^\prime}_1 -
\xi_2 \mu^{i^\prime}_2)^{-1}I_{s_{i^\prime}}\} \cdot \\
\label{Sside2}
\cdot \Gamma^{-1} \cdot \sum_{j=1}^m (\xi_1 c_{j1} + \xi_2 c_{j2}) &
K^{{\bold x},\boldsymbol \lambda}(x^j) [K(\widetilde{\chi};x^j,p)
+K_{\boldsymbol \mu,{\bold u}}(x^j) \Gamma^{-1}
K^{{\bold x},\boldsymbol \lambda}(p)] T(p).
\end{align}
Next we use the general identity (see also \cite{AV})
\begin{gather}
\notag
\sum_{j=1}^m (\xi_1 c_{j1} + \xi_2 c_{j2}) K(\widetilde{\chi};p,x^j)
K(\widetilde{\chi};x^j,q) = \\
\label{collection}
=(\xi_1 \lambda_1(q) + \xi_2 \lambda_2(q) - \xi_1 \lambda_1(p) - \xi_2
\lambda_2(p)) K(\widetilde{\chi}; p,q)
\end{gather}
which is valid for all distinct points $p,q$ in $X$ which are disjoint
from $x^1, \dots, x^m$.
To prove this ``collection formula'' \eqref{collection}, consider each
side as a function of $p$ with $q$ fixed. Since $h^0(\widetilde{\chi}
\otimes \Delta) = 0$, it suffices to show that the local principal part
in the Laurent series expansion at each pole of each side matches with
the local principal part of the other side. One can check that the
only possible poles are all simple and occur at $x^1, \dots, x^m$ with
residue of each side at $x^i$ equal to the common value $-(\xi_1c_{i1} +
\xi_2 c_{i2}) dt^i(x^i) K(\widetilde{\chi}; x^i,q)$.
Immediate consequences of the identity \eqref{collection} which are
important for our context here are:
\begin{align} \notag
\sum_{j=1}^m (\xi_1 c_{j1} + \xi_2 c_{j2}) & K^{{\bold x},
\boldsymbol \lambda}(\lambda^i) K(\widetilde{\chi};x^j,p) = \\
\label{consequence1}
= & \underset{i^\prime}{\text{diag.}} \{ (\xi_1 \lambda_1(p) + \xi_2
\lambda_2(p) - \xi_1 \lambda^{i^\prime}_1 - \xi_2
\lambda^{i^\prime}_2)I_{t_{i^\prime}}\} \cdot
K^{{\bold x},\boldsymbol\lambda}(p),
\end{align}
and, if $(i^\prime, j^\prime)$ is a pair of indices for which
$\lambda^{i^\prime} \ne \mu^{j^\prime}$ then the $(i^\prime,
j^\prime)$-matrix entry of
$\sum_{j=1}^m (\xi_1 c_{j1} + \xi_2 c_{j2})
K^{{\bold x}, \boldsymbol \lambda}(x^j) K_{\boldsymbol \mu,{\bold u}}(x^j)$
is given by
\begin{gather} \notag
[ \sum_{j=1}^m (\xi_1 c_{j1} + \xi_2 c_{j2})
K^{{\bold x}, \boldsymbol\lambda}(x^j)
K_{\boldsymbol \mu,{\bold u}}(x^j)]_{i^\prime, j^\prime}
= \\ \notag
= (\xi_1 \mu^{j^\prime}_1 + \xi_2 \mu^{j^\prime}_2 - \xi_1
\lambda^{i^\prime}_1 - \xi_2 \lambda^{i^\prime}_2) {\bold x}_{i^\prime}
K(\widetilde{\chi}; \lambda^{i^\prime}, \mu^{j^\prime}) {\bold u}_{j^\prime}
\\ \notag
= - (\xi_1 \mu^{j^\prime}_1 + \xi_2 \mu^{j^\prime}_2 - \xi_1
\lambda^{i^\prime}_1 - \xi_2 \lambda^{i^\prime}_2) \Gamma_{i^\prime,
j^\prime}.
\end{gather}
Hence in matrix form we have
\begin{gather} \notag
\sum_{j=1}^m (\xi_1 c_{j1} + \xi_2 c_{j2})
K^{{\bold x}, \boldsymbol \lambda}(x^j) K_{\boldsymbol\mu,{\bold u}}(x^j) =
\\ \label{consequence2}
= \underset{i^\prime}{\text{diag.}} \{(\xi_1 \lambda^{i^\prime}_1
+ \xi_2 \lambda^{i^\prime}_2)I_{t_{i^\prime}} \} \cdot \Gamma - \Gamma
\cdot \underset{j^\prime}{\text{diag.}}\{(\xi_1 \mu^{j^\prime}_1 + \xi_2
\mu^{j^\prime}_2) I_{s_{j^\prime}} \}.
\end{gather}
In the case where $p=q$, the collection
formula \eqref{collection} takes the limiting form
\begin{gather} \notag
\sum_{j=1}^m (\xi_1 c_{j1} + \xi_2 c_{j2}) K(\widetilde{\chi};p,x^j)
K(\widetilde{\chi};x^j,p) = \\
\label{degencollection}
-(\xi_1 \lambda^\prime_1(p) + \xi_2 \lambda^\prime_2(p)) \ dt(p)
\end{gather}
where ${}^\prime = \dfrac{d}{dt}$ where $t$ is a local coordinate
centered at $p$. An application of this degenerate collection formula
\eqref{degencollection}
gives, for $\lambda^{i^\prime} =
\mu^{j^\prime} =: \xi^{i^\prime j^\prime}$,
\begin{gather} \notag
[ \sum_{j=1}^m (\xi_1 c_{j1} + \xi_2 c_{j2})
K^{{\bold x}, \boldsymbol \lambda}(x^j)
K_{\boldsymbol \mu,{\bold u}}(x^j)]_{i^\prime, j^\prime}
= \\ \notag
= -(\xi_1 \lambda^\prime_1(\xi^{i^\prime j^\prime}) + \xi_2
\lambda^\prime_2(\xi^{i^\prime j^\prime}) )\ dt(\xi^{i^\prime j^\prime})
\ {\bold x}^T_{i^\prime} {\bold u}_{j^\prime} = 0
\end{gather}
where we used the compatibility condition \eqref{comp} for the last
step. We conclude that \eqref{consequence2} continues to be valid even
in the case where $\lambda^{i^\prime} = \mu^{j^\prime}$.
Making the substitutions \eqref{consequence1} and \eqref{consequence2}
in \eqref{Sside2}, we obtain
\begin{gather} \notag
[S(p) \beta^{-1} u^{\times \prime}(p)]_i =
[K(\widetilde{\chi};x^i,p)+
K_{\boldsymbol \mu,{\bold u}}(x^i) \Gamma^{-1}
K^{{\bold x},\boldsymbol \lambda}(p)] T(p) -
\\ \notag
- K_{\boldsymbol\mu, {\bold u}}(x^i) \cdot
\underset{i'}{\text{diag.}}
\{( \xi_1 \lambda_1(p)+\xi_2 \lambda_2(p) - \xi_1 \mu^{i^\prime}_1
-\xi_2 \mu^{i^\prime}_2)^{-1} I_{s_{i^\prime}} \} \cdot
\\ \notag
\cdot
\Gamma^{-1} \cdot \left( \underset{i^\prime}{\text{diag.}}\{(\xi_1
\lambda_1(p) + \xi_2\lambda_2(p) - \xi_1 \lambda^{i^\prime}_1 -\xi_2
\lambda^{i^\prime}_2) I_{t_{i^\prime}}\}
K^{{\bold x}, \boldsymbol \lambda}(p) +
\right.
\\ \notag
+ \left[ \underset{i^\prime}{\text{diag.}}\{ (\xi_1
\lambda^{i^\prime}_1 + \xi_2 \lambda^{i^\prime}_2) I_{t_{i^\prime}} \}
\Gamma \right. - \Gamma \underset{j^\prime}{\text{diag.}} \{ ( \xi_1
\mu^{j^\prime}_1 + \xi_2 \mu^{j^\prime}_2) I_{s_{j^\prime}}\}
\left. \right]
\Gamma^{-1} K^{{\bold x}, \boldsymbol\lambda}(p) \left. \right) =
\\ \notag
= K(\widetilde{\chi};x^i,p) T(p) + K_{\boldsymbol\mu, {\bold u}}(x^i)
\Gamma^{-1} K^{{\bold x}, \boldsymbol\lambda}(p) T(p) -
\\ \notag
- K_{\boldsymbol \mu, {\bold u}}(x^i) \cdot
\underset{i^\prime}{\text{diag.}}
\{ (\xi_1 \lambda_1(p) + \xi_2 \lambda_2(p) - \xi_1 \mu^{i^\prime}_1 -
\xi_2 \mu^{i^\prime}_2)^{-1} I_{s_{i^\prime}} \} \cdot
\\ \notag
\cdot \left( (\xi_1 \lambda_1(p) + \xi_2 \lambda_2(p)) \Gamma^{-1}
K^{{\bold x}, \boldsymbol \lambda}(p) \right.
- \Gamma^{-1} \underset{i^\prime}{\text{diag.}}
\{( \xi_1 \lambda^{i^\prime}_1 + \xi_2 \lambda^{i^\prime}_2)
I_{t_{i^\prime}} \} K^{{\bold x}, \boldsymbol \lambda}(p) +
\\ \notag
+ \Gamma^{-1} \underset{i^\prime}{\text{diag.}}
\{ (\xi_1 \lambda^{i^\prime}_1 + \xi_2 \lambda^{i^\prime}_2)
I_{t_{i^\prime}} \}
K^{{\bold x}, \boldsymbol \lambda}(p)
\left.
- \underset{j^\prime}{\text{diag.}}
\{(\xi_1 \mu^{j^\prime}_1 + \xi_2 \mu^{j^\prime}_2) I_{s_{j^\prime}} \}
\Gamma^{-1} K^{{\bold x}, \boldsymbol\lambda}(p)
\right)
\\ \notag
= K(\widetilde{\chi}; x^i,p) T(p)
\end{gather}
and the intertwining relation \eqref{intertwining} follows.
\end{pf}
{\bf REMARK:} Note that the results of this section in principle give a
means of computing explicitly the unknown input bundle $\chi$ appearing
in Theorem \ref{T:bvabsint} in Section \ref{S:absint}. Indeed, given a
data set $\omega$ for an (ABSINT) problem, we can convert it to a data
set $\omega_0$ for a (CONINT) problem, as explained in the discussion
preceding the statement of Theorem \ref{T:comparison}. In the context of
the (CONINT) problem, Theorem \ref{T:conint} solves the problem of
identifying the input bundle. Namely, the input bundle $E$ is defined to
be $E=\ker (z_1\sigma_2 - z_2 \sigma_1 + \gamma)$ where $\sigma_1,
\sigma_2 $ and $\widetilde{\gamma}$ are given by \eqref{pencilcoef} and
$\gamma$ is given by \eqref{gamma}. The vector bundle $\chi$ is then
determined up to biholomorphic equivalence by the condition
\[
\chi \cong E \otimes {\cal O}(1) \otimes \Delta.
\]
Moreover, as explained in \cite{hip}, it is possible to construct a
matrix of normalized sections $u^\times$ for $E$ from working with the
minors of the matrix pencil $z_1 \sigma_2 - z_2 \sigma_1 + \gamma$. Such
a matrix of normalized sections $u^\times$ in turn implements concretely
the biholomorphic equivalence between $\chi$ and $E \otimes {\cal O}(1)
\otimes \Delta$.
|
1998-12-07T01:29:44 | 9712 | alg-geom/9712031 | en | https://arxiv.org/abs/alg-geom/9712031 | [
"alg-geom",
"math.AG"
] | alg-geom/9712031 | James S. Milne | J.S. Milne | Descent for Shimura Varieties | 6 pages | Michigan Math. J. 46 (1999), no. 1, 203--208 | null | null | null | We verify that the descent maps provided by Langlands's Conjugacy Conjecture
do satisfy the continuity condition necessary for them to be effective. Thus
Langlands's conjecture does imply the existence of canonical models.
This replaces an earlier version of the paper --- the proof in this version
is simpler, and the exposition more detailed.
| [
{
"version": "v1",
"created": "Tue, 30 Dec 1997 00:59:16 GMT"
},
{
"version": "v2",
"created": "Mon, 7 Dec 1998 00:29:43 GMT"
}
] | 2021-01-19T00:00:00 | [
[
"Milne",
"J. S.",
""
]
] | alg-geom | \subsubsection{Notations and Conventions}
A variety over a field $k$ is a geometrically reduced scheme of finite type
over $\Spec k$ (\emph{not} necessarily irreducible). For a variety $V$ over
a field $k$ and a homomorphism $\sigma \colon k\rightarrow k^{\prime }$, $%
\sigma V$ is the variety over $k^{\prime }$ obtained by base change. The
ring of finite ad\`{e}les for $\mathbb{Q}$ is denoted by $\mathbb{A}_{f}{}$.
\section{Descent of Varieties.}
In this section, $\Omega$ is an algebraically closed field of characteristic
zero. For a field $L\subset\Omega$, $A(\Omega/L)$ denotes the group of
automorphisms of $\Omega$ fixing the elements of $L$.
Let $V$ be a variety over $\Omega$, and let $k$ be a subfield of $\Omega$. A
family $(f_{\sigma})_{\sigma\in A(\Omega/k)}$ of isomorphisms $f_{\sigma
}\colon\sigma V\rightarrow V$ will be called a \emph{descent system\/} if $%
f_{\sigma\tau}=f_{\sigma}\circ\sigma f_{\tau}$ for all $\sigma,\tau\in
A(\Omega/k)$. We say that a model $(V_{0},f\colon V_{0,\Omega}\rightarrow V)$
of $V$ over $k$ \emph{splits} $(f_{\sigma})$ if $f_{\sigma}=f\circ(\sigma
f)^{-1}$ for all $\sigma\in A(\Omega/k)$, and that a descent system is \emph{%
effective\/} if it is split by some model over $k$. The next theorem
restates results of Weil 1956.
\begin{theorem}
Assume that $\Omega $ has infinite transcendence degree over $k$. A descent
system $(f_{\sigma })$$_{\sigma \in A(\Omega /k)}$ on a quasiprojective
variety $V$ over $\Omega $ is effective if, for some subfield $L$ of $\Omega
$ finitely generated over $k$, the descent system $(f_{\sigma })_{\sigma \in
A(\Omega /L)}$ is effective.
\end{theorem}
\begin{proof}
Let $k^{\prime}$ be the algebraic closure of $k$ in $L$ --- then $k^{\prime}$
is a finite extension of $k$ and $L$ is a regular extension of $k^{\prime}$.
Let $(V_{t^{\prime}},f^{\prime}\colon V_{t^{\prime},\Omega}\rightarrow V)$
be the model of $V$ over $L$ splitting $(f_{\sigma})_{\sigma\in A(\Omega/L)}$%
. Let $t\colon L\rightarrow k_{t}$ be a $k^{\prime}$-isomorphism from $L$
onto a subfield $k_{t}$ of $\Omega$ linearly disjoint from $L$ over $%
k^{\prime}$, and let $V_{t}=V_{t^{\prime}}\otimes_{L,t}k_{t}$. Zorn's Lemma
allows us to extend $t$ to an automorphism $\tau$ of $\Omega$ over $%
k^{\prime}$. The isomorphism
\begin{equation*}
f_{t,t^{\prime}}\colon V_{t^{\prime},\Omega}\overset{f^{\prime}}{\rightarrow
}V\overset{f_{\tau}^{-1}}{\rightarrow}\tau V\overset{(\tau f^{\prime})^{-1}}{%
\rightarrow}V_{t,\Omega}
\end{equation*}
is independent of the choice of $\tau$, is defined over $L\cdot k_{t}$, and
satisfies the hypothesis of Weil 1956, Theorem 6, which gives a model $(W,f)$
of $V$ over $k^{\prime}$ splitting $(f_{\sigma})_{\sigma\in A(\Omega
/k^{\prime})}$.
For $\sigma \in A(\Omega /k)$, $g_{\sigma }\overset{\text{df}}{=}f_{\sigma
}\circ \sigma f\colon \sigma W_{\Omega }\rightarrow V$ depends only on $%
\sigma |k^{\prime }$. For $k$-homomorphisms $\sigma ,\tau \colon k^{\prime
}\rightarrow \Omega $, define $f_{\tau ,\sigma }=g_{\tau }^{-1}\circ
g_{\sigma }\colon \sigma W\rightarrow \tau W$. Then $f_{\tau ,\sigma }$ is
defined over the Galois closure of $k^{\prime }$ in $\Omega $ and the family
$(f_{\tau ,\sigma })$ satisfies the hypotheses of Weil 1956, Theorem 3,
which gives a model of $V$ over $k$ splitting $(f_{\sigma })_{\sigma \in
A(\Omega /k)}.$
\end{proof}
\begin{corollary}
Let $\Omega$, $k$, and $V$ be as in the theorem, and let $%
(f_{\sigma})_{\sigma\in A(\Omega/k)}$ be a descent system on $V$. If there
is a finite set $\Sigma$ of points in $V(\Omega)$ such that
\begin{enumerate}
\item any automorphism of $V$ fixing all $P\in \Sigma $ is the identity
map, and
\item there exists a subfield $L$ of $\Omega$ finitely generated over $k$
such that $f_{\sigma}(\sigma P)=P$ for all $P\in\Sigma$ and all $\sigma\in
A(\Omega/L)$,
\end{enumerate}
then $(f_{\sigma})_{\sigma\in A(\Omega/k)}$ is effective.
\end{corollary}
\begin{proof}
After possibly replacing the $L$ in (b) with a larger finitely generated
extension of $k$, we may suppose that $V$ has a model $(W,f)$ over $L$ for
which the points of $\Sigma$ are rational, i.e., such that for each $%
P\in\Sigma$, $P=f(P^{\prime})$ for some $P^{\prime}\in W(L)$. Now, for each $%
\sigma\in A(\Omega/L)$, $f_{\sigma}$ and $f\circ\sigma f^{-1}$ are both
isomorphisms $\sigma V\rightarrow V$ sending $\sigma P$ to $P$, and so
hypothesis (a) implies they are equal. Hence $(f_{\sigma})_{\sigma\in
A(\Omega/L)}$ is effective, and the theorem applies.
\end{proof}
\begin{remark}
\begin{enumerate}
\item It is easy to construct noneffective descent systems. For example,
take $\Omega$ to be the algebraic closure of $k$, and let $V$ be a variety $%
k $. A one-cocycle $h\colon A(\Omega/k)\rightarrow\Aut(V_{\Omega})$ can be
regarded as a descent system --- identify $h_{\sigma}$ with a map $\sigma
V_{\Omega }=V_{\Omega}\rightarrow V_{\Omega}$. If $h$ is not continuous, for
example, if it is a homomorphism into $\Aut(V)$ whose kernel is not open,
then the descent system will not be effective.
\item An example (Dieudonn\'{e} 1964, p 131) shows that the hypothesis that
$V$ be quasiprojective in (1.1) is necessary unless the model $V_{0}$ is
allowed to be an algebraic space in the sense of M. Artin.
\item Theorem 1.1 and its corollary replace Lemma 3.23 of Milne 1994, which
omits the continuity conditions.
\end{enumerate}
\end{remark}
\subsection{Application to moduli problems.}
Suppose we have a contravariant functor $\mathcal{M}$ from the category of
algebraic varieties over $\Omega $ to the category of sets, and equivalence
relations $\sim $ on each of the sets $\mathcal{M}(T)$ compatible with
morphisms. The pair $(\mathcal{M},\sim )$ is then called a \emph{moduli
problem} over $\Omega $. A $t\in T(\Omega )$ defines a map
\begin{equation*}
m\mapsto m_{t}\overset{\text{df}}{=}t^{\ast }m\colon \mathcal{M}%
(T)\rightarrow \mathcal{M}(\Omega ).
\end{equation*}
\noindent A \emph{solution to the moduli problem }is a variety $V$ over $%
\Omega$ together with an isomorphism $\alpha\colon\mathcal{M}(\Omega
)/\!\!\sim\rightarrow V(\Omega)$ such that:
\begin{enumerate}
\item for all varieties $T$ over $\Omega$ and all $m\in\mathcal{M}(T)$, the
map $t\mapsto\alpha(m_{t})\colon T(\Omega)\rightarrow V(\Omega)$ is regular
(i.e., defined by a morphism $T\rightarrow V$ of $\Omega$-varieties);
\item for any variety $W$ over $\Omega$ and map $\beta\colon\mathcal{M}%
(\Omega)/\!\!\sim\rightarrow W(\Omega)$ satisfying the condition (a), the
map $P\mapsto\beta(\alpha^{-1}(P))\colon V(\Omega)\rightarrow W(\Omega)$ is
regular.
\end{enumerate}
\noindent Clearly, a solution to a moduli problem is unique up to a unique
isomorphism when it exists.
Let $(\mathcal{M},\sim )$ be a moduli problem over $\Omega $, and let $k$ be
a subfield $\Omega $. For $\sigma \in A(\Omega /k)$, define $^{\sigma }%
\mathcal{M}$ to be the functor sending an $\Omega $-variety $T$ to $\mathcal{%
M}(\sigma ^{-1}T)$. We say that $(\mathcal{M},\sim )$ is \emph{rational over}
$k$ if there is given a family $(g_{\sigma })_{\sigma \in A(\Omega /k)}$ of
isomorphisms $g_{\sigma }\colon ^{\sigma }\mathcal{M}\rightarrow \mathcal{M}$, compatible with $\sim$,
such that $g_{\sigma \tau }=g_{\sigma }\circ \sigma g_{\tau }$ for all $%
\sigma $, $\tau \in A(\Omega /k)$ --- the last equation means that $%
g_{\sigma \tau }(T)=g_{\sigma }(T)\circ g_{\tau }(\sigma ^{-1}T)$ for all
varieties $T$. Note that $^{\sigma }\mathcal{M}(\Omega )=\mathcal{M}(\Omega )
$, and that the rule $\sigma m=g_{\sigma }(m)$ defines an action of $%
A(\Omega /k)$ on $\mathcal{M}(\Omega )$. A \emph{solution to a moduli problem%
} $(\mathcal{M},\sim ,(g_{\sigma }))$ \emph{rational over} $k$ is a variety $%
V_{0}$ over $k$ together with an isomorphism $\alpha \colon \mathcal{M}%
(\Omega )/\!\!\sim \rightarrow V_{0}(\Omega )$ such that
\begin{enumerate}
\item $(V_{0,\Omega},\alpha)$ is a solution to the moduli problem $(%
\mathcal{M},\sim)$ over $\Omega$, and
\item $\alpha $ commutes with the actions of $A(\Omega /k)$ on $\mathcal{M}%
(\Omega )$ and $V_{0}(\Omega )$.
\end{enumerate}
\noindent Again, $(V_{0},\alpha)$ is uniquely determined up to a unique
isomorphism (over $k$) when it exists.
\begin{theorem}
Assume that $\Omega $ has infinite transcendence degree over $k$. Let $(%
\mathcal{M},\sim ,(g_{\sigma }))$ be a moduli problem rational over $k$ for
which $(\mathcal{M},\sim )$ has a solution $(V,\alpha )$ over $\Omega $.
Then $(\mathcal{M},\sim ,(g_{\sigma }))$ has a solution over $k$ if there
exists a finite subset $\Sigma \subset \mathcal{M}(\Omega )$ such that
\begin{enumerate}
\item any automorphism of $V$ fixing $\alpha(P)$ for all $P\in\Sigma$ is
the identity map, and
\item there exists a subfield $L$ of $\Omega $ finitely generated over $k$
such that $g_{\sigma }(P)\sim P$ for all $P\in \Sigma $ and all $\sigma \in
A(\Omega /L)$.
\end{enumerate}
\end{theorem}
\begin{proof}
The family $(g_{\sigma})$ defines a descent system on $V$, which Corollary
1.2 shows to be effective.
\end{proof}
\section{Descent of Shimura Varieties.}
In this section, all fields will be subfields of $\mathbb{C}$. For a
subfield $E$ of $\mathbb{C}$, $E^{\text{ab}}$ denotes the composite of all
the finite abelian extensions of $E$ in $\mathbb{C}$.
Let $(G,X)$ be a pair satisfying the axioms (2.1.1.1--2.1.1.3) of Deligne
1979 to define a Shimura variety, and let $\Sh(G,X)$ be the corresponding
Shimura variety over $\mathbb{C}$. We regard $\Sh(G,X)$ as a pro-variety
endowed with a continuous action of $G(\mathbb{A}_{f})$ --- in particular
(ibid. 2.7.1) this means that $\Sh(G,X)$ is a projective system of varieties
$(\Sh_{K}(G,X))$ indexed by the compact open subgroups $K$ of $G(\mathbb{A}%
_{f})$. Let $[x,a]$ $=([x,a]_{K})_{K}$ denote the point in $\Sh(G,X)(\mathbb{%
C})$ defined by a pair $(x,a)\in X\times G(\mathbb{A}_{f})$, and let $E(G,X)$
be the reflex field of $(G,X)$. For a special point $x\in X$, let $%
E(x)\supset E(G,X)$ be the reflex field for $x$ and let
\begin{equation*}
r_{x}\colon \Gal(E(x)^{{ \text{ab} }}/E(x))\rightarrow T(\mathbb{A}_{f})/T(\mathbb{Q}%
)^{-}
\end{equation*}
be the reciprocity map defined in Milne 1992, p164 (inverse to that in
Deligne 1979, 2.2.3). Here $T$ is a subtorus of $G$ such that $\text{Im}%
(h_{x})\subset T_{\mathbb{R}}$ and $T(\mathbb{Q})^{-}$ is the closure of $T(%
\mathbb{Q})$ in $T(\mathbb{A}{}_{f})$. A\emph{\ model\/} of $\Sh(G,X)$ over
a field $k$ is a pro-variety $S$ over $k$ endowed with an action of $G(%
\mathbb{A}_{f})$ and a $G(\mathbb{A}_{f})$-equivariant isomorphism $f\colon
S_{\mathbb{C}}\rightarrow \Sh(G,X)$. A model of $\Sh(G,X)$ over $E(G,X)$ is
\emph{canonical } if, for each special point $x\in X$ and $a\in G(\mathbb{A}%
_{f})$, $[x,a]$ is rational over $E(x)^{\text{ab}}$ and $\sigma \in \Gal%
(E(x)^{\text{ab}}/E(x))$ acts on $[x,a]$ according\footnote{%
More precisely, the condition for $(S,f)$ to be canonical is the following:
if $P\in S(\mathbb{C})$ corresponds under $f$ to $[x,a]$, then $\sigma P$
corresponds under $f$ to $[x,r_{x}(\sigma )\cdot a].$} to the rule:
\begin{equation*}
\sigma \lbrack x,a]=[x,r_{x}(\sigma )\cdot a].
\end{equation*}
Let $k$ be a field containing $E(G,X)$. A \emph{descent system} for $\Sh(G,X)
$ over $k$ is a family of isomorphisms
\begin{equation*}
(f_{\sigma }\colon \sigma \Sh(G,X)\rightarrow \Sh(G,X))_{\sigma \in A(%
\mathbb{C}/k)}
\end{equation*}
such that,
\begin{enumerate}
\item for all $\sigma,\tau\in A(\mathbb{C}/k)$, $f_{\sigma\tau}=f_{\sigma
}\circ\sigma f_{\tau}$, and
\item for all $\sigma\in A(\mathbb{C}/k) $, $f_{\sigma}$ is equivariant for
the actions of $G(\mathbb{A}_{f})$ on $\Sh(G,X)$ and $\sigma\Sh(G,X)$.
\end{enumerate}
\noindent We say that a model $(S,f)$ of $\Sh(G,X)$ over $k$ \emph{splits }$%
(f_{\sigma})$ if $f_{\sigma}=f\circ\sigma f^{-1}$ for all $\sigma\in
A(\Omega/k)$, and that a descent system if \emph{effective} if it is split
by some model over $k$. A descent system $(f_{\sigma})$ for $\Sh(G,X)$ over $%
E(G,X)$ is \emph{canonical} if
\begin{equation*}
f_{\sigma}(\sigma\lbrack x,a])=[x,r_{x}(\sigma|E(x)^{{ \text{ab} }})\cdot a]
\end{equation*}
\noindent whenever $x$ is a special point of $X$, $\sigma\in A(\mathbb{C}%
/E(x))$, and $a\in G(\mathbb{A}_{f})$.
\begin{remark}
\begin{enumerate}
\item For a Shimura variety $\Sh(G,X)$, there exists at most one canonical
descent system for $\Sh(G,X)$ over $E(G,X)$. (Apply Deligne 1971, 5.1, 5.2.)
\item Let $(S,f)$ be a model of $\Sh(G,X)$ over $E(G,X)$, and let $%
f_{\sigma }=f\circ(\sigma f)^{-1}$. Then $(f_{\sigma})_{\sigma\in A(\mathbb{C%
}/k)}$ is a descent system for $\Sh(G,X)$, and $(f_{\sigma})$ is canonical
if and only if $(S,f)$ is canonical.
\item Suppose $\Sh(G,X)$ has a canonical descent system $%
(f_{\sigma})_{\sigma\in A(\mathbb{C}/E(G,X))}$; then $\Sh(G,X)$ has a
canonical model if and only if $(f_{\sigma})$ is effective. (Follows from
(a) and (b).)
\item A descent system $(f_{\sigma})_{\sigma\in A(\mathbb{C}/k)}$ on $\Sh%
(G,X)$ defines for each compact open subgroup $K$ of $G(\mathbb{A}_{f})$ a
descent system $(f_{\sigma,K})_{\sigma\in A(\mathbb{C}/k)}$ on the variety $%
\Sh_{K}(G,X)$ (in the sense of \S1). If $(f_{\sigma})$ is effective, then so
also is $(f_{\sigma,K})$ for all $K$; conversely, if $(f_{\sigma,K})_{\sigma%
\in A(\mathbb{C}/k)}$ is effective (in the sense of \S1) for all
sufficiently small $K$, then $(f_{\sigma})_{\sigma\in A(\mathbb{C}/k)}$ is
effective (in the sense of this section).
\end{enumerate}
\end{remark}
\begin{lemma}
The automorphism group of the quotient of a bounded symmetric domain by a
neat arithmetic group is finite.
\end{lemma}
\begin{proof}
According to Mumford 1977, Proposition 4.2, such a quotient is an algebraic
variety of logarithmic general type, which implies that its automorphism
group is finite (Iitaka 1982, 11.12).
Alternatively, one sees easily that the automorphism group of the quotient
of a bounded symmetric domain $D$ by a neat arithmetic subgroup $\Gamma$ is $%
N/\Gamma$ where $N$ is the normalizer of $\Gamma$ in $\Aut(D)$. Now $N$ is
countable and closed (because $\Gamma$ is closed), and hence is discrete
(Baire category theorem). Because the quotient of $\Aut(D)$ by $\Gamma$ has
finite measure, this implies that $\Gamma$ has finite index in $N$. Cf.
Margulis 1991, II 6.3.
\end{proof}
\begin{theorem}
Every canonical descent system on a Shimura variety is effective.
\end{theorem}
\begin{proof}
Let $(f_{\sigma})_{\sigma\in A(\mathbb{C}/E(G,X))}$ be a canonical descent
system for the Shimura variety $\Sh(G,X)$. Let $K$ be a compact open
subgroup of $G(\mathbb{A}_{f})$, chosen so small that the connected
components of $\Sh_{K}(G,X)$ are quotients of bounded symmetric domains by
\emph{neat} arithmetic groups. Let $x$ be a special point of $X$. According
to Deligne 1971, 5.2, the set $\Sigma=\{[x,a]_{K}\mid a\in G(\mathbb{A}%
_{f})\}$ is Zariski dense in $\Sh_{K}(G,X)$. Because the automorphism group
of $\Sh
_{K}(G,X)$ is finite, there is a finite subset $\Sigma_{f}$ of $\Sigma$ such
that any automorphism $\alpha$ of $\Sh_{K}(G,X)$ fixing each $P\in\Sigma_{f}$
is the identity map.
The rule
\begin{equation*}
\sigma \ast \lbrack x,a]_{K}=[x,r_{x}(\sigma )\cdot a]_{K}
\end{equation*}
defines an action of $\Gal(E(x)^{\text{ab}}/E(x))$ on $\Sigma $ for which
the stabilizer of each point of $\Sigma $ is open. Therefore, there exists a
finite abelian extension $L$ of $E(x)$ such that $\sigma \ast P=P$ for all $%
P\in \Sigma _{f}$ and all $\sigma \in \Gal(E(x)^{\text{ab}}/L)$.
Now, because $(f_{\sigma})_{\sigma\in A(\Omega/E(G,X))}$ is canonical, $%
f_{\sigma,K}(\sigma P)=P$ for all $P\in\Sigma_{f}$ and all $\sigma\in A(%
\mathbb{C}{}/L)$, and we may apply Corollary 1.2 to conclude that $%
(f_{\sigma,K})_{\sigma\in A(\mathbb{C}/E(G,X))}$ is effective. As this holds
for all sufficiently small $K$, $(f_{\sigma})_{\sigma\in A(\mathbb{C}%
/E(G,X))}$ is effective.
\end{proof}
\begin{remark}
\begin{enumerate}
\item If Langlands's Conjugacy Conjecture (Langlands 1979, p232, 233) is
true for a Shimura variety $\Sh(G,X)$, then $\Sh(G,X)$ has a canonical
descent system (ibid. \S 6; also Milne and Shih 1982, \S 7).
\item Langlands's Conjugacy Conjecture is true for all Shimura varieties
(Milne 1983). Hence canonical models exist for all Shimura varieties.
\end{enumerate}
\end{remark}
Another proof, based on different ideas, that the descent maps given by Langlands's conjecture are effective can be found in Moonen 1998. (I thank the referee for this reference.)
\section*{References}
Deligne, P., Travaux de Shimura, in S\'{e}minaire Bourbaki, 23\`{e}me
ann\'{e}e (1970/71), Exp. No. 389, 123--165. Lecture Notes in Math., 244,
Springer, Berlin, 1971.
Deligne, P., Vari\'{e}t\'{e}s de Shimura: Interpr\'{e}tation modulaire, et
techniques de construction de mod\`{e}les canoniques, Proc. Symp. Pure Math.
33 Part 2, pp. 247--290, 1979.
Dieudonn\'{e}, J., Fondements de la G\'{e}om\'{e}trie Alg\'{e}brique
Moderne, Presse de l'Universit\'{e} de Montr\'{e}al, 1964.
Iitaka, S., Algebraic Geometry, Springer, Heidelberg, 1982.
Langlands, R., Automorphic representations, Shimura varieties, and motives,
Ein M\"{a}rchen, Proc. Symp. Pure Math. 33 Part 2, pp. 205--246, 1979.
Margulis, G.A., Discrete subgroups of semisimple Lie groups, Springer,
Heidelberg, 1991.
Milne, J.S., The action of an automorphism of $\mathbb{C}$ on a Shimura
variety and its special points, Prog. in Math., vol. 35, Birkh\"{a}user,
Boston, pp. 239--265, 1983.
Milne, J.S., The points on a Shimura variety modulo a prime of good
reduction, in The Zeta Function of Picard Modular Surfaces (Langlands and
Ramakrishnan, eds), Les Publications CRM, Montr\'{e}al, pp. 153--255, 1992.
Milne, J. S., Shimura varieties and motives, in Motives (Seattle,
WA, 1991), 447--523, Proc. Sympos. Pure Math., Part 2, Amer. Math. Soc.,
Providence, RI, 1994.
Milne, J.S. and Shih, K-y., Conjugates of Shimura varieties, in Hodge
Cycles, Motives, and Shimura Varieties, Lecture Notes in Math., vol. 900,
Springer, Heidelberg, pp. 280--356, 1982.
Moonen, B., Models of Shimura varieties in mixed characteristics, in Galois Representations in Arithmetic Geometry (A.J. Scholl and R.L Taylor, editors), Cambridge University Press, pp. 267--350, 1998.
Mumford, D., Hirzebruch's proportionality theorem in the non-compact case,
Invent. Math. 42, 239--272, 1977.
Weil, A., The field of definition of a variety, Amer. J. Math 78, pp.
509--524, 1956.
\end{document}
|
1997-12-16T01:39:01 | 9712 | alg-geom/9712009 | en | https://arxiv.org/abs/alg-geom/9712009 | [
"alg-geom",
"hep-th",
"math.AG",
"math.DG",
"q-alg"
] | alg-geom/9712009 | Andrei Okounkov | Spencer Bloch and Andrei Okounkov | The Character of the Infinite Wedge Representation | Latex, 57 pages; typos in bibliography corrected | null | null | null | null | We study the character of the infinite wedge projective representation of the
algebra of differential operators on the circle. We prove quasi-modularity of
this character and also compute certain generating functions for traces of
differential operators which we call correlation functions. These correlation
functions are sums of determinants built from genus 1 theta functions and their
derivatives.
| [
{
"version": "v1",
"created": "Tue, 9 Dec 1997 20:55:53 GMT"
},
{
"version": "v2",
"created": "Tue, 16 Dec 1997 00:39:01 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Bloch",
"Spencer",
""
],
[
"Okounkov",
"Andrei",
""
]
] | alg-geom | \section{Introduction}\label{sec:intro} The purpose of this paper is to study the character
associated to a certain basic projective representation, the infinite wedge representation
\cite{KP}, of the Lie algebra ${\mathcal D}$ of differential operators on the circle. Concretely, we write
\begin{equation}\label{01} D_n := \left(t\frac{d}{dt}\right)^n,\quad n\ge 0
\end{equation}
acting as differential operators on the ring $R={\Bbb C}[t,t^{-1}]$ of Laurent polynomials. (Here $D_0 =
1$ is the identity operator. We occasionally write $D=D_1$.) ${\mathcal D}$ has a vector space basis
$t^mD_n,\ m\in {\Bbb Z},\ n\ge 0$.
The Witt algebra of derivations of $R,\ {\mathcal W}=\text{\rm Der}(R)\subset {\mathcal D}$
is spanned by $t^mD,\ m\in{\Bbb Z}$. These algebras are graded, with $t^mD_n$ having weight $m$. One has
a central extension \cite{KR} $0\to {\Bbb C}\cdot c \to {\widehat{\mathcal D}} \to {\mathcal D} \to 0$ inducing the Virasoro central
extension $\widehat{{\mathcal W}}$ of ${\mathcal W}$, and these central extensions are also graded, with $c$ having
degree $0$. Certain graded highest weight representations of $\widehat{{\mathcal W}}$ arise in connection
with conformal field theory and have been of considerable interest to physicists. These
representations have the property that for a suitable choice $\widetilde{D}\in \widehat{{\mathcal W}}$
lifting $D$, the character
$$\text{\rm Trace}(q^{\widetilde{D}})
$$
is well-defined (i.e. the action of $\widetilde{D}$ is semi-simple with finite eigenspaces) and is
the $q$-expansion of a modular form \cite{ZHU}. Notice that a different choice of lifting of $D$
to $\widehat{{\mathcal W}}$ multiplies the character by $q^a$ for some constant $a$. In particular, there
is at most one lifting for which the character is modular.
For representations of ${\widehat{\mathcal D}}$ one may consider the Abelian subalgebra
\begin{equation}\label{02} {\mathcal H} := {\Bbb C} D_0\oplus{\Bbb C} D_1\oplus{\Bbb C} D_2\oplus\ldots\subset {\mathcal D}.
\end{equation}
Suppose the action of ${\mathcal H}$ is semi-simple with finite dimensional simultaneous eigenspaces. If we
choose liftings $\widetilde{D}_n \in {\widehat{\mathcal D}}$ of $D_n$ we may define the character
\begin{equation}\label{03}
\Omega(q_0,q_1,\ldots)
:=
\text{\rm Trace}\left(q_0^{\widetilde{D}_0}q_1^{\widetilde{D}_1}q_2^{\widetilde{D}_2}\cdots
\right)\,.
\end{equation}
It is also frequently convenient to write $q_r = e^{2\pi i\tau_r}$ so the character becomes (with
abuse of notation)
\begin{equation}\Omega(\tau_0,\tau_1,\ldots)
:= \text{\rm Trace}\left( \exp\left(2\pi i\sum_{r\ge 0}\widetilde{D}_r\tau_r\right)\right)\,.
\end{equation}
Assuming the eigenspaces of $D_1$ are themselves finite dimensional, we may specialize the $\tau_r
\mapsto 0$ for $r\ne 1$ and develop $\Omega(\tau_0,\tau_1,\ldots)$ in a formal Taylor series
expansion in
$\tau_0,\tau_2,\ldots$ with coefficients functions in $\tau_1$:
\begin{equation}\label{05} \Omega(\tau_0,\tau_1,\ldots) = \sum_A\omega_A(\tau_1)\tau^A/A!\,.
\end{equation}
Here $A=(a_0,a_2,a_3,\ldots)$ with almost all $a_j=0$, and $\tau^A/A!$ is multi-index notation.
Finally, $\widetilde{D}_0$ is central in ${\widehat{\mathcal D}}$ so its eigenspaces are stable under the ${\widehat{\mathcal D}}$ action.
We write
\begin{equation}\label{06}V(q_1,q_2,\ldots) = \text{\rm Coeff. of $q_0^0$ in }\Omega(q_0,q_1,\ldots)
\end{equation}
for the character of the $\widetilde{D}_0=0$ eigenspace. Again, by abuse of notation, we also
write $V(\tau_1,\tau_2,\ldots)$ and we expand in a series
\begin{equation}\label{07} V = \sum_{B=(b_2,b_3,\ldots)} v_B(\tau_1)\tau^B/B!\,,
\end{equation}
where, by construction,
\begin{equation}\label{07a}
\frac{v_B(\tau_1)}{(2\pi i)^{b_2+b_3+\dots}}=
\text{\rm Trace}\big|_{\widetilde{D}_0=0} \left(
q_1^{\widetilde{D}_1} \prod_{k=2}^\infty
\left(\widetilde{D}_k\right)^{b_k}
\right)\,.
\end{equation}
Here the trace is taken in the $\widetilde{D}_0=0$ subspace.
In the case of the infinite wedge representation, these characters have the following shape
\begin{multline}\label{08} \Omega(q_0,q_1,q_2,\ldots) = \\
q_1^{-\xi(-1)}q_3^{-\xi(-3)}\cdots
\prod_{r\ge 0}
(1+q_0q_1^{r+{\frac{1}{2}}}q_2^{(r+{\frac{1}{2}})^2}\cdots)(1+q_0^{-1}q_1^{r+{\frac{1}{2}}}q_2^{-(r+{\frac{1}{2}})^2}\cdots)
\end{multline}
\begin{multline}\label{09}V(q_1,q_2,\ldots) =
q_1^{-\xi(-1)}q_3^{-\xi(-3)}\cdots \\
\times\sum_{(m|n)}q_1^{\sum_j (m_j+{\frac{1}{2}})+(n_j+{\frac{1}{2}})}q_2^{\sum_j (m_j+{\frac{1}{2}})^2-(n_j+{\frac{1}{2}})^2}
q_3^{\sum_j (m_j+{\frac{1}{2}})^3+(n_j+{\frac{1}{2}})^3}\cdots
\end{multline}
Here
\begin{equation}\label{010}\xi(s) := \sum_{n\ge 1}\left(n-{\textstyle \frac{1}{2}}\right)^{-s} = (2^s-1)\zeta(s).
\end{equation}
Also, the sums are over all
\begin{gather}\label{011}(m|n) = (m_1,m_2,\ldots,m_a|n_1,\ldots,n_a) \\
m_a>m_{a-1}>\cdots >m_1\ge 0,\quad n_a>n_{a-1}>\cdots>n_1\ge 0\notag \,.
\end{gather}
Such data are {\it Frobenius coordinates} for partitions $\lambda$ (cf. \cite{MacD}, p. 3.) In
particular, the exponents in \eqref{09} are functions on the set of partitions. We define
\begin{equation}\label{012}p_r(\lambda) := \sum_j
\left(m_j+{\textstyle \frac{1}{2}}\right)^r+
(-1)^{r+1}\left(n_j+{\textstyle \frac{1}{2}}\right)^r \,.
\end{equation}
Thus $p_0(\lambda)=0$ and $p_1(\lambda) = |\lambda|$ is the number being partitioned.
The formulas \eqref{08} and \eqref{09} appear (without the anomaly terms) in \cite{FKRW} and
\cite{AFMO}. The anomaly terms depend on a choice of liftings of the $D_n$ to ${\widehat{\mathcal D}}$. For
representations of the Virasoro algebra, this lifting is determined by modularity. The correct
analog, quasimodularity, for representations of ${\widehat{\mathcal D}}$ was suggested by Dijkgraaf \cite{D} based
on an interpretation of $V(\tau_1,\tau_2,0,0,\ldots)$ as a generating function counting covers of a
fixed elliptic curve E with given degree and genus. From this point of view, a mirror symmetry
argument suggested an interpretation in terms of (nonholomorphic) modular forms on the dual moduli
space. More precisely, note that in \eqref{05} and \eqref{07} the variable $\tau_1$ plays a
distinguished role. The analogue of modularity involves the behavior of $\Omega$ or $V$
under the transformations
\begin{equation}\label{12} \tau_1 \mapsto
\frac{a\tau_1+b}{c\tau_1+d},\qquad \tau_j \mapsto \frac{\tau_j}{(c\tau_j+d)^{j+1}},\ j\ne 1\,,
\end{equation}
for $\begin{pmatrix}a & b \\ c & d \end{pmatrix}\in \Gamma$, where $\Gamma\subset \text{\rm SL}_2({\Bbb Z})$
is some subgroup of finite index. Given $C=(c_0,c_2,c_3,\ldots)$ (resp. $C=(c_2,c_3,\ldots)$) with
almost all $c_i=0$ define the weight of $C$
\begin{equation}\label{0013} \text{\rm wt}(C) := \sum (i+1)c_i \,.
\end{equation}
To say that the $\omega_A(\tau_1)$ (resp. the $v_B(\tau_1)$) were modular of weight
$\text{\rm wt}(A)+w$ (resp. $\text{\rm wt}(B)+w$) for some $w$ would imply that $\Omega$ (resp. $V$) was
modular of weight $w$ under the transformation \eqref{12}. This doesn't happen in the examples we
know. Instead, the $\omega_A$ and $v_B$ are {\it quasimodular} of these weights. The notion of
quasimodular form is developed in \cite{KZ}, where a rigorous proof of quasimodularity for the
$\omega_A$ (resp. $v_B$) is given for $A=(a_0,a_2,0,0,\ldots)$ (resp. $B=(b_2,0,0,\ldots)$). By
definition, the ring of quasimodular forms is the graded algebra generated over the ring of
modular forms by the Eisenstein series of weight $2$
$$ G_2(q) := -\frac{B_2}{4}+\sum_{n=1}^\infty\sum_{d|n}dq^n.
$$
Unlike the ring of modular forms, this ring is closed under the derivation $q\frac{d}{dq}$. One
can canonically associate to a quasimodular form a certain nonholomorphic ({\it almost
holomorphic} in the terminology of op.\ cit.) modular form.
\begin{defn}\label{defn01} A series $F = \sum_A f_A(\tau_1)\tau^A/A!$ for $A=(a_0,a_2,\ldots)$ is
quasimodular of weight $w$ if each $f_A(\tau_1)$ is quasimodular of
weight $w+\text{\rm \rm wt}(A)$.
\end{defn}
\begin{thm}\label{th02}$\Omega(\tau_0,\tau_1,\ldots)$ (resp. $V(\tau_1,\tau_2,\ldots)$) is
quasimodular of weight $0$ (resp. weight $-{\frac{1}{2}}$).
\end{thm}
Although the anomaly factors $q_{2r+1}^{-\xi(-2r-1)}$ are uniquely determined by quasimodularity,
they can also be understood solely in representation-theoretic terms. Roughly speaking, if one
tries to construct the infinite wedge representation as a representation of ${\mathcal D}$ (rather than
${\widehat{\mathcal D}}$) one is lead to meaningless infinite constants of the form $\sum_{n\ge 1} \left(n-{\frac{1}{2}}\right)^r,\
r\ge 1$. Regularizing these sums via analytic continuation of $\xi(s)$ and using the fact that
$\xi(-2r)=0$ for $r\ge 0$ leads to the stated values.
The analogy between \eqref{08} and the triple product formulas for the
genus 1 theta functions continues with the following elliptic
transformation formula for $\Omega(\tau_0,\tau_1,\ldots)$.
\begin{thm}\label{th03} Define the transformation $T$ by
$$
T(\tau_j) = \tau_j -\binom{j+1}{1}\tau_{j+1}+\binom{j+2}{2}\tau_{j+2}
-\ldots\,,
$$
and set $q'_i=\exp(2\pi i T(\tau_i))$. Then we have
$$
\Omega(q_0',q_1',q_2',\ldots) =
q_0q_1^{-{\frac{1}{2}}}q_2^{+\frac{1}{3}}q_3^{-\frac{1}{4}}\cdots\Omega(q_0,q_1,\ldots)
$$
\end{thm}
The above results constitute a pleasant but perhaps not terribly surprising generalization of the
work in \cite{D} and \cite{KZ}. The character $V$ possesses some other
hidden structure
and relation to the genus 1 theta functions,
which seems to us quite surprising and new.
The idea is to replace
the bulky generating function \eqref{09} for the quantities
\eqref{07a} by other generating functions, where the
auxiliary variables are attached not
to the exponents $b_i$ in \eqref{07a}
but to the indices $k$ of $\widetilde{D}_k$. These
new generating functions admit a neat evaluation in terms of
genus 1 theta functions.
We call these generating functions $n$-point \emph{correlation
functions} because both by their definition (as averages of product of
$n$ generating series) and by their analytic structure (determinants built from
theta functions and their derivatives) these functions closely
resemble correlation functions in QFT. However, we were not able
to find any precise connection and in the present text we work with
these functions using solely the classical methods of analysis.
Concretely, the definition of these $n$-point functions is the
following. Let $f(\lambda)$ be a function on partitions, and define
\begin{equation}\label{015} \langle f\rangle_q \ :=
\sum_\lambda f(\lambda)q^{|\lambda|}\Big/ \sum_\lambda
q^{|\lambda|}\,.
\end{equation}
That is, $\langle f\rangle_q$ is the expectation of the
function $f$ provided the the probability of each partition $\lambda$
is proportional to $q^{|\lambda|}$.
Then, the equations \eqref{07a}, \eqref{09} and \eqref{011} can be
restated as follows:
\begin{equation}\label{015a}
\text{\rm Trace}\big|_{\widetilde{D}_0=0} \left(
q_1^{\widetilde{D}_1} \prod_{k=2}^\infty
\left(\widetilde{D}_k\right)^{b_k}
\right)=\eta(q_1) \left\langle \prod_{k=2}^\infty
\left(p_k(\lambda)-\xi(-k)\right)^{b_k} \right\rangle_{q_1}\,.
\end{equation}
One checks (see section \ref{sec:prepart}) that
\begin{equation}\label{016a}
p_k(\lambda)-\xi(-k)=\left.\left(t\frac{d}{dt}\right)^k
\left(\sum_{i=1}^\infty t^{\lambda_i-i+1/2} - \frac1{\log t}\right)
\right|_{t=1}\,.
\end{equation}
By definition, set for $n=1,2,3,\dots$
\begin{equation}\label{016}
F(t_1,\dots,t_n;q)= \left\langle \prod_{k=1}^n \left(\sum_{i=1}^\infty
t_k^{\lambda_i-i+{\frac{1}{2}}}\right) \right\rangle_{q} \,,
\end{equation}
that is, $F(t_1,\dots,t_n;q)$
is the expectation of the product of $n$ generating
series for the quantities \eqref{016a}.
By linearity, all quantities
\eqref{015a} satisfying $b_2+b_3+\dots\le m$ for some $m$
are encoded in the functions \eqref{016} with $n\le m$.
\begin{defn}\label{defn02} The functions \eqref{016} are called
$n$-point correlation functions.
\end{defn}
The equation \eqref{016a} can be restated as follows.
Write $t_k = e^{u_k}$ and
define a differential operator
\begin{equation}\label{017a}
\delta(u) := \frac1{u} +(2\pi i)^{-1}\sum_{r=1}^\infty
\frac{\partial}{\partial\tau_r}u^r/r!\,.
\end{equation}
Then
\begin{equation}\label{017}
\Big<\prod_{k=1}^n \Big(\sum_{i=1}^\infty
t_k^{\lambda_i-i+{\frac{1}{2}}}\Big)\Big>_{q_1}
= \eta(q_1)\delta(u_1)\circ\ldots\circ\delta(u_n)V|_{\tau_2=\tau_3=\dotsc = 0}.
\end{equation}
This is also proved in section \ref{sec:prepart}.
The form of the singular term $1/u$ and in \eqref{017a} and
the corresponding term in \eqref{016a} very much depends on our choice of
anomaly factors in the character.
Write $q=q_1$, write $F(t_1,\dotsc,t_n)$ for the $n$-point
function, and write $\Theta(t)=\Theta(t;q)$ for the
following theta function:
$$ \Theta(t) :=
\eta(q)^{-3}\sum_{n\in {\Bbb Z}}
(-1)^nq^{\frac{(n+{\frac{1}{2}})^2}{2}}x^{n+{\frac{1}{2}}}= (q)^{-2}_\infty(x^{\frac{1}{2}} -
x^{-{\frac{1}{2}}})(qx)_\infty(q/x)_\infty\,.
$$
Let $\Theta^{(p)}(t) = (t\frac{d}{dt})^p\Theta(t)$. Then our main result on the $n$-point function
is following:
\begin{thm}\label{thm04}
\begin{equation}\label{018}
\vspace{-2 \jot}
F(t_1,\dots,t_n)=
\sum_{\sigma\in\mathfrak{S}(n)}\,
\frac
{\displaystyle \det\left( \frac{\displaystyle \Theta^{(j-i+1)}(t_{\sigma(1)}\cdots
t_{\sigma(n-j)})}{\displaystyle (j-i+1)!}
\right)}
{\displaystyle \Theta(t_{\sigma(1)})\,\Theta(t_{\sigma(1)} t_{\sigma(2)}) \dots
\Theta(t_{\sigma(1)}\cdots t_{\sigma(n)})} \,\,.
\end{equation}
Here $\sigma$ runs through all permutations $\mathfrak{S}(n)$
of $\{1,\dotsc,n\}$, the
matrices in the numerator have size $n\times n$, and we define
$1/(-n)!=0$ if $n\ge 1$.
\end{thm}
For the $1$-point function, this is simply
$$\Big<\sum_{i=1}^\infty
t^{\lambda_i-i+{\frac{1}{2}}}\Big>_q = \frac{1}{\Theta(t)}\,,
$$
because
$$
\Theta'(1)=1\,.
$$
For $n=3$ the equation \eqref{018} becomes:
\begin{multline}
F(t_1,t_2,t_3)=\\
\frac1{\displaystyle\Theta(t_1 t_2 t_3)}
\sum_{\sigma\in\mathfrak{S}(3)}
\det
\left(
\begin{array}{rrc}
\frac{\displaystyle\Theta'\left(t_{\sigma(1)} t_{\sigma(2)}\right)}
{\displaystyle \Theta\phantom{{}'}\left(t_{\sigma(1)} t_{\sigma(2)}\right)}&
\frac{\displaystyle 1}{\displaystyle 2} \,
\frac{\displaystyle\Theta''\left(t_{\sigma(1)}\right)}
{\displaystyle\Theta\phantom{{}''}\left(t_{\sigma(1)}\right)}&
\frac{\displaystyle\Theta'''(1)}{\displaystyle 3!}\\
{\displaystyle 1}&
\frac{\displaystyle\Theta'\phantom{{}'}\left(t_{\sigma(1)}\right)}
{\displaystyle\Theta\phantom{{}''}\left(t_{\sigma(1)}\right)}&
0\\
&
{\displaystyle 1}&
1
\end{array}
\right)
\end{multline}
The traces \eqref{015a} can be computed from \eqref{018}
using repeated differentiation, L'Hospital's rule and
formulas for the derivatives $\Theta^{(p)}(1;q)$ in terms of
the Eisenstein series:
\begin{align*}
\Theta^{(3)}(1;q)&= - 6 G_2(q)\,,\\
\Theta^{(5)}(1;q)&= - 10 G_4(q)+ 60 G_2(q)^2\,, \\
\Theta^{(7)}(1;q)&= - 14 G_6(q)+420 G_4(q) G_2(q)- 840 G_2(q)^3\,,
\end{align*}
and so on, see \eqref{th'}. Note that all even order derivatives
of the odd function $\Theta(x)$ vanish at $x=1$.
Sections \ref{sec:results}-\ref{sec:pf} are devoted
to the proof of \eqref{018}.
An essential ingredient of this proof are the following $q$-difference
equations for the functions $F(t_1,\dots,t_n)$
which is established in section 8:
\begin{thm}\label{thm05} For $n=1,2,\dots$ we have
\begin{multline*}
F(qt_1,t_2,\dots,t_n)=-q^{1/2} t_1 \dots t_n \times \\ \sum_{s=0}^{n-1}
(-1)^s
\sum_{1<i_1<\dots < i_s \le n} F(t_1 t_{i_1} t_{i_2} \cdots t_{i_s},
\dots,\widehat{\,t_{i_1}}, \dots,\widehat{\,t_{i_s}}, \dots) \,.
\end{multline*}
\end{thm}
These equations are analogs of elliptic transformations for the
$n$-point functions $F(t_1,\dots,t_n)$\,.
Here is another point of view about these things. Let ${\mathcal A}$ be an associative algebra with $1$, and let
$\rho : {\mathcal A} \to \text{\rm End}(F)$ be a representation of the associated Lie algebra (with commutator
bracket). We do {\it not} assume $\rho$ compatible with the associative algebra structures. Let
$D\in {\mathcal A}$ be given and assume $\rho(D)$ is semisimple with finite dimensional eigenspaces. Let
$F_a \subset F$ be the eigenspace of $\rho(D)$ with eigenvalue $a$. Since $[D, D^n]=0$, $\rho(D^n)$
stabilizes $F_a$, and it makes sense to consider for a variable $u$ the expression
\begin{equation}\label{019}f_\rho(u,z):= \chi_\rho(z)^{-1}\sum_a\Big(\text{\rm Tr}_{F_a}\sum_{n\ge 0}
\rho(D^n)u^n/n!\Big)e^{az}\,.
\end{equation}
where $\chi_\rho(x) := \text{\rm Tr}e^{x\rho(D)}$ is the trace of $\rho$. If in fact
$\rho(D^n)=\rho(D)^n$ for all $n$, we find
\begin{equation}f_\rho(u,z) = \chi_\rho(z)^{-1}\sum_a
\text{\rm Tr}_{F_a}e^{at}e^{az}=\chi_\rho(u+z)/\chi_\rho(z) \,.
\end{equation}
More generally, letting
$n=(n_1,\ldots,n_r)$ run through $r$-tuples of non-negative integers and replacing $\rho(D^n)$ with
$\prod_i \rho(D^{n_i})$ and $u^n/n!$ with $\prod u_i^{n_i}/n_i!$ we may define
$f_\rho(u_1,\ldots,u_r;z)$. When $\rho$ is compatible with associative multiplication, we have
$f_\rho = \chi_\rho(u_1+\ldots+u_r+z)/\chi_\rho(z)$.
When $\rho$ is a projective representation, $\rho(D^n)$ is well defined only up to adding
$\alpha_n\cdot \text{\rm Id}$. Such a modification replaces $f_\rho(u,z)$ with
$f_\rho(u,z)+(\sum_n\alpha_nu^n/n!)$. If ${\mathcal H}$ is spanned by the powers of $D$ as in
\eqref{02} we find writing $\chi_\rho(\tau_0,\tau_1,\tau_2,\ldots)$ for the full character that
\begin{multline}\label{020} f_\rho(u_1,\ldots,u_r;\tau_1) = \\
\chi_\rho(\tau_1)^{-1}\sum_{n_1,\ldots,n_r\ge
0}\left(\prod_{i=1}^r\frac{\partial}{\partial\tau_{n_i}}\right)
\chi_\rho(\tau_0,\tau_1,\tau_2,\ldots)
|_{\tau_0=\tau_2=\ldots=0}\prod_i u_i^{n_i}/n_i! \,.
\end{multline}
For the representation \eqref{06}, this is \eqref{017}.
Finally, section \ref{sec:example} proves quasimodularity and computes the $n$-point function for
a representation of a subalgebra of ${\mathcal D}$ studied in \cite{B}.
The first author would like to
acknowledge considerable inspiration from conversations and correspondence with V. Kac and D.
Zagier. It was Zagier who suggested that these characters might be quasimodular.
\section{The Infinite Wedge Representation}\label{sec:rep}
Material in this section is quite well known. References are \cite{K}, \cite{K2}, \cite{PS},
\cite{FKRW}. We work throughout with vector spaces, Lie algebras, and representations of Lie
algebras over the field
${\Bbb C}$. Recall that a (projective) representation of a Lie algebra ${\mathcal L}$ is a Lie algebra
homomorphism
${\mathcal L} \to \text{\rm End}(F)$; (resp. ${\mathcal L} \to \text{\rm End}(F)/{\Bbb C}\cdot\text{\rm Id}$) for a vector space $F$.
A projective representation gives rise to a central extension of Lie algebras via pullback:
\begin{equation}\begin{CD}0 @>>> {\Bbb C}\cdot c @>>> \widetilde{{\mathcal L}} @>>> {\mathcal L} @>>> 0 \\
@. @VVV @VVV @VVV @. \\
0 @>>>{\Bbb C}\cdot\text{\rm Id} @>>> \text{\rm End}(F) @>>> \text{\rm End}(F)/{\Bbb C}\cdot\text{\rm Id} @>>> 0\,.
\end{CD}
\end{equation}
Let $V$
be a vector space with basis
$v_n,\ n\in
{\Bbb Z}$. Define the finite matrices
\begin{equation}{\mathcal A}_0 := \bigoplus_{i,j\in {\Bbb Z}} {\Bbb C} E_{ij}\subset \text{\rm End}(V)\,.
\end{equation}
where $E_{ij}(v_i)=v_j$ and $E_{ij}(v_k)=0;\ k\ne i$. Let ${\mathcal A}\supset{\mathcal A}_0$ be the larger space of
all matrices supported in a bounded strip. An infinite matrix
$M=\sum_{\text{\rm infinite}}a_{ij}E_{ij}$ lies in $A$ if and only if there exists a constant $c$ such
that $a_{ij}=0$ if $|i-j|>c$. For example, ${\rm Id}\in{\mathcal A}-{\mathcal A}_0$. Suppose given a representation
$\alpha_0$ of ${\mathcal A}_0$ which extends to a projective representation $\alpha$ of ${\mathcal A}$:
\begin{equation}\label{13} \begin{array}{ccc} {\mathcal A}_0 & \stackrel{\iota}{\hookrightarrow} &
\makebox[.8in][l]{${\mathcal A}$}
\\
\rule[-4mm]{0cm}{9mm}\makebox[.1in][l]{$\downarrow\! \alpha_0$} &
\makebox[.1in]{$\swarrow\!\sigma$} &
\makebox[.7in][l]{$\downarrow\!
\alpha$}
\\
\text{\rm End}(F) & \twoheadrightarrow & \text{\rm End}(F)/{\Bbb C}\cdot{\rm Id}\,.
\end{array}
\end{equation}
Let $\sigma$ be a lifting of $\alpha$ which is a map of vector spaces, not necessarily compatible
with Lie algebra structures and not necessarily extending $\alpha_0$. Assume, however, that
$\sigma$ is continuous in the sense that for any $f\in F$ and any
$x=\sum_{\text{\rm infinite}}x_{ij}\iota(E_{ij})\in {\mathcal A}$ there exists $N=N(x,f)$ such that
$x_{ij}\sigma\iota(E_{ij})(f)=0$ for $|i|+|j|>N$, and
$$\sigma(x)(f) = \sum_{|i|+|j|\le N}x_{ij}\sigma\iota(E_{ij})(f)\,.
$$
Notice that such a $\sigma$ is unique up to a finite modification. More precisely, if $\sigma'$ is
another map with the same property, $\sigma\iota(E_{ij})=\sigma'\iota(E_{ij})$ for almost all
$i,j$. We obtain in this way a map
\begin{equation}\label{14} \epsilon := \alpha_0-\sigma\circ\iota : {\mathcal A}_0 \to {\Bbb C}\cdot{\rm Id}\,.
\end{equation}
The functional $\epsilon$ is uniquely determined (up to a finite modification) by $\alpha_0,\alpha$.
\begin{defn}\label{reg} Let ${\mathcal S}$ be a collection of sequences of complex numbers
$c=\{c_{ij}\}_{i,j\in
{\Bbb Z}}$. A regularization scheme for ${\mathcal S}$ is a map $r:{\mathcal S} \to {\Bbb C}$ such that if $\{c_{ij}\},
\{c_{ij}'\}\in{\mathcal S}$ differ for only finitely many pairs $i,j$, then
$$r(\{c_{ij}\}) = r(\{c_{ij}'\})+\sum_{i,j}(c_{ij}-c_{ij}').
$$
(We will usually assume ${\mathcal S}$ is saturated in the sense that if $c\in {\mathcal S}$ and $c'$ differs from $c$
for only finitely many $i,j$ then $c'\in {\mathcal S}$.)
\end{defn}
Assume now we have $\alpha, \alpha_0$, and that there exists a $\sigma$ as in \eqref{13} above.
Let $\widehat{{\mathcal A}}$ be the corresponding central extension of ${\mathcal A}$. Let ${\mathcal S},r$ be as in
definition (\ref{reg}). Suppose given $x=\sum x_{ij}\iota(E_{ij})\in {\mathcal A}$, such that
$\{x_{ij}\epsilon(E_{ij})\}\in {\mathcal S}$, where $\epsilon$ is defined in \eqref{14}. Then
$$\sigma(x) + r(\{x_{ij}\epsilon(E_{ij})\})\cdot{\rm Id} \in \text{\rm End}(F)
$$
is well-defined independent of the choice of $\sigma$. Thus we have a lifting $\hat x\in
\widehat{{\mathcal A}}$ depending on the regularization scheme $r$ but not on the choice of lifting
$\sigma$.
\begin{exam}\label{ex12} For the infinite wedge representation,
\begin{equation}\epsilon(E_{ij}) = \begin{cases} 1 & i=j\le 0\,, \\ 0
& \text{\rm else\,.}
\end{cases}
\end{equation}
The elements $x$ we want to lift have the form $x=\sum_{m\in{\Bbb Z}}\left(m-{\textstyle \frac{1}{2}}\right)^kE_{mm}$ for
$k=0,1,2,\ldots$, so the sequences we need to regularize are
$$\left\{\left(m-{\textstyle \frac{1}{2}}\right)^k\right\}_{m\le 0}\,.
$$
The natural way to do this is by analytic continuation of the function
\begin{equation}\label{16}\xi(s) = \sum_{n\ge 1} \left(n-{\textstyle \frac{1}{2}}\right)^{-s}
\end{equation}
We define $r(\{(m-{\frac{1}{2}})^k\}_{m\le 0})=(-1)^k\xi(-k)$\,.
\end{exam}
The infinite wedge representation $F:=\Lambda^\infty V$ is the vector space with basis
\begin{equation}v_I = v_{i_m}\wedge v_{i_{m-1}}\wedge v_{i_{m-2}}\wedge\ldots\,,
\end{equation}
where $i_{k-1}<i_k$ and $i_k = k$ for $k\ll 0$. By definition $m$ is the {\it charge} of $v_I$ and
$$\text{\rm energy}(v_I) := \sum_{k=m}^{k=-\infty}(i_k-k)\,.
$$
For example, the elements
\begin{equation} |m\rangle := v_m\wedge v_{m-1}\wedge v_{m-2}\wedge\ldots
\end{equation}
have charge $m$ and energy $0$. The wedge here is formal, but we can use the expected alternating
linearity to make sense for $A\in {\mathcal A}_0$ of expressions like
\begin{equation}\label{19} Av_I := \sum_{k=m}^{k=-\infty}v_{i_m}\wedge\ldots\wedge Av_{i_k}\wedge
v_{i_{k-1}}\wedge \ldots \quad .
\end{equation}
We obtain a representation $\alpha_0 : {\mathcal A}_0 \to \text{\rm End}(F)$. To extend $\alpha_0$ to ${\mathcal A}$, we
would have to define expressions like
\begin{equation}\label{110} \sum_{p\in {\Bbb Z}}a_pE_{p,p+r}(v_I)\,.
\end{equation}
When $r\ne 0$ it is easy to see that $E_{p,p+r}v_I = 0$ for $|p|>>0$, so \eqref{110} makes sense.
Similarly $E_{p,p}v_I=0$ for $p>>0$. Define $\sigma : {\mathcal A}_0 \to \text{\rm End}(F)$ by
\begin{equation} \sigma(E_{ij}) = \begin{cases}\alpha_0(E_{ij}) & i\ne j \text{\rm or
}i=j>0 \\
\alpha_0(E_{ii})-{\rm Id} & i\le 0\end{cases}
\end{equation}
It is straightforward to check that for any $r$ and any $I$ there exists an $N$ such
that $\sigma(E_{p,p+r})(v_I)=0$ for $|p|\ge N$, so $\sigma$ extends to a map ${\mathcal A}\to\text{\rm End}(F)$
as in \eqref{13}. Define $\alpha$ to be the composition
$${\mathcal A}\stackrel{\sigma}{\to}\text{\rm End}(F)\to
\text{\rm End}(F)/{\Bbb C}\cdot{\rm Id}\,.
$$
\begin{lem} The map $\alpha$ is a projective representation of ${\mathcal A}$.
\end{lem}
\begin{proof} The assertion is there exists a bilinear map $a: {\mathcal A}\times {\mathcal A} \to {\Bbb C}$ with
$$[\sigma(x),\sigma(y)] = \sigma([x,y]) + a(x,y){\rm Id}.
$$
One defines
$$a(E_{ij},E_{k\ell}) = \begin{cases}1 & i=\ell\le 0 \text{\rm\ and }j=k\ge 1\,, \\
-1 & i=\ell\ge 1 \text{\rm\ and } j=k\le 0 \,,\\
0 & \text{\rm else}\,.\end{cases}
$$
Because of the conditions on the signs of the indices, $a$ extends to ${\mathcal A}\times{\mathcal A}$ as desired.
\end{proof}
Given $u\in {\Bbb C}$, we have an action as differential operators of ${\mathcal D}$ on ${\Bbb C}[t,t^{-1}]t^u$.
Identifying this space with $V$ via $t^{n+u} \mapsto v_n$, we get a mapping of associative
algebras (and hence a fortiori of Lie algebras)
\begin{gather*} \delta_u : {\mathcal D} \to {\mathcal A}\subset \text{\rm End}(V)\,, \\
\delta_u(t^pD^r) = \sum_{n\in {\Bbb Z}}(n+u)^rE_{n,n+p} \notag\,.
\end{gather*}
We define a projective representation
\begin{gather*}\rho_u := \alpha\circ\delta_u : {\mathcal D} \to \text{\rm End}(F)/{\Bbb C}\cdot{\rm Id}\,, \\
\rho := \rho_{-{\frac{1}{2}}}\notag\,.
\end{gather*}
We will customarily view $\rho$ as a representation on a central extension, $\rho : {\widehat{\mathcal D}} \to
\text{\rm End}(F)$. Our objective now is to compute the character of the representation $\rho_u$.
Define operators $\psi^*_{-r},\ r\ge 0$ and $\psi_{-r},\ r>0$ on $F$ by
$$\psi_{-r}(v_I) := v_r\wedge v_I;\quad \psi^*_{-r}(\ldots\wedge v_{-r}\wedge\ldots) =
\ldots\wedge\widehat{\rule{0cm}{3mm}v_{-r}}\wedge\ldots\quad.
$$
A basis for $F$ can be written
\begin{multline*} \psi_{-i_1}\cdots\psi_{-i_a}\psi^*_{-j_1}\cdots\psi^*_{-j_b}|0\rangle\,,\\
0<i_1<i_2<\ldots<i_a,\ 0\le j_1<j_2<\ldots<j_b\,.
\end{multline*}
We have
\begin{multline*}\sigma\circ\delta_u(D_r)\Big(\psi_{-i_1}\cdots\psi_{-i_a}\psi^*_{-j_1}
\cdots\psi^*_{-j_b}|0\rangle\Big) = \\
\sigma\begin{pmatrix}\ddots &&\\ &(n+u)^r &\\ &&\ddots\end{pmatrix}\Big(\psi_{-i_1}\cdots
\psi_{-i_a}\psi^*_{-j_1}
\cdots\psi^*_{-j_b}|0\rangle\Big) = \\
\Big(\sum_{\ell=1}^a (i_\ell+u)^r - \sum_{m=1}^b (-j_m+u)^r\Big)\Big(\psi_{-i_1}\cdots
\psi_{-i_a}\psi^*_{-j_1}
\cdots\psi^*_{-j_b}|0\rangle\Big).
\end{multline*}
For example, writing $q_r = \exp(2\pi i\tau_r)$ we get
\begin{gather*}\exp\Big(2\pi i\sum_{r\ge 0}\tau_r\cdot\sigma\circ\delta_u(D_r)\Big)(\psi_{-n}|0\rangle)
= q_0q_1^{n+u}q_2^{(n+u)^2}\cdots \,,\\
\exp\Big(2\pi i\sum_{r\ge 0}\tau_r\cdot\sigma\circ\delta_u(D_r)\Big)(\psi^*_{-n}|0\rangle)
= q_0^{-1}q_1^{n+u}q_2^{-(n+u)^2}\cdots \,.\notag
\end{gather*}
On all of $F$ we find
\begin{multline*}\text{\rm Tr}\exp\Big(2\pi i\sum_{r\ge 0}\tau_r\cdot\sigma\circ\delta_u(D_r)\Big) = \\
\prod_{n\ge
0}\big(1+q_0q_1^{n+u+1}q_2^{(n+u+1)^2}\cdots\big)\big(1+q_0^{-1}q_1^{n-u}q_2^{-(n-u)^2}\cdots\big).
\end{multline*}
We now specialize to the case $u=-{\frac{1}{2}}$. The reason why this particular value of $u$ yields a
quasimodular character is not clear, but it may have to do with the fact that for $u=p+{\frac{1}{2}}\in
{\Bbb Z}+{\frac{1}{2}}$ the space $V={\Bbb C}[t,t^{-1}]t^u$ has a nondegenerate, symmetric bilinear form
$$(t^{n+u},t^{m+u}) := \text{\rm res}_{t=0} t^{n+m+2u-1}dt.
$$
Further, if we consider the involution
$$\sigma : {\mathcal D} \to {\mathcal D};\quad \sigma(t^aD^b) := -t^a(-D-a)^b,
$$
we find
$$\sigma^2 = I;\quad \sigma[x,y] = [\sigma(x),\sigma(y)];\ (x(a),b)+(a,\sigma(x)(b)) = 0.
$$
With reference
to example (\ref{ex12}), we note that
$\xi(s) = (2^s-1)\zeta(s)$ vanishes at $s=0,-2,-4,\ldots$ We therefore define the character of the
infinite wedge representation of
${\mathcal D}$ to be
\begin{multline}\label{1_18}\Omega(q_0,q_1,\ldots) = \Omega(\tau_0,\tau_1,\ldots) = \\
q_1^{-\xi(-1)}q_3^{-\xi(-3)}\cdots\prod_{n\ge
0}\big(1+q_0q_1^{n+{\frac{1}{2}}}q_2^{(n+{\frac{1}{2}})^2}\cdots\big)\big(1+q_0^{-1}q_1^{n+{\frac{1}{2}}}
q_2^{-(n+{\frac{1}{2}})^2}\cdots\big)\,.
\end{multline}
Note that $\delta_u(D_0)=\sum_{i\in {\Bbb Z}} E_{ii}$ is the charge operator
\begin{multline*}\delta_u(D_0)\psi_{-i_1}\cdots
\psi_{-i_a}\psi^*_{-j_1}
\cdots\psi^*_{-j_b}|0\rangle = \\
(a-b)\psi_{-i_1}\cdots\psi_{-i_a}\psi^*_{-j_1}\cdots\psi^*_{-j_b}|0\rangle.
\end{multline*}
Operators in ${\mathcal D}$ preserve the charge, so the charge eigenspaces are stable under $\rho_u$. For
$u=-{\frac{1}{2}}$, the characters of the subrepresentation of charge $n$ is the coefficient of $q_0^n$
in \eqref{1_18}. Of particular interest is the charge $0$ part. Using \eqref{09} and \eqref{012},
this can be written
\begin{multline}\label{120} V(q_1,q_2,\ldots) = V(\tau_1,\tau_2,\ldots) := \\
q_1^{-\xi(-1)}q_3^{-\xi(-3)}\cdots\{\text{\rm Coeff. of $q_0^0$ in $\Omega(q_0,q_1,\ldots)$}\}= \\
q_1^{-\xi(-1)}q_3^{-\xi(-3)}\cdots\sum_\lambda q_1^{p_1(\lambda)}q_2^{p_2(\lambda)}\cdots\,.
\end{multline}
Here the sum is over all partitions $\lambda$.
\section{Elliptic transformation of the character $\Omega$}\label{sec:transf}
In this section we want to expose some interesting analogies between $\Omega(q_0,q_1,q_2,\ldots)$
\eqref{1_18} and the classical genus $1$ theta function, which we will write
\begin{equation}\theta(q_0,q_1) := \sum_{n\in{\Bbb Z}}q_0^nq_1^{n^2/2}.
\end{equation}
The triple product formula (cf. \cite{MUM}, p. 70) implies
\begin{multline}\label{22}\theta(q_0,q_1) = \prod_{m\ge 1}(1-q_1^m)\prod_{n\ge
0}(1+q_0q_1^{n+{\frac{1}{2}}}) (1+q_0^{-1}q_1^{n+{\frac{1}{2}}}) = \\
\eta(q_1)\Omega(q_0,q_1,1,1,\ldots).
\end{multline}
Here $\eta(q_1)=q_1^{\frac{1}{24}}\prod_{m\ge 1}(1-q_1^m)$ is the classical eta function, and we
have used the fact that the first anomaly factor in $\Omega$ is given by
$$q_1^{-\xi(-1)}=q_1^{-(2^{-1}-1)\zeta(-1)} = q_1^{-\frac{1}{24}}.
$$
We want to generalize to $\Omega$ the elliptic transformation law
\begin{equation}\theta(q_0q_1^{-1},q_1) = q_0q_1^{-{\frac{1}{2}}}\theta(q_0,q_1).
\end{equation}
For this, we define infinite matrices
\begin{equation} U:= \begin{pmatrix}0 & -1 & 0 & 0 & \hdots \\ 0 & 0 & -2 & 0 & \hdots \\
0 & 0 & 0 & -3 & \hdots \\
\vdots & \vdots & \vdots & \vdots
\end{pmatrix}
\end{equation}
and
\begin{equation} T:= \exp(U) = \begin{pmatrix}1 & -1 & 1 & -1 & 1 &\hdots \\\vspace{4pt}
0 & 1 & -\binom{2}{1} & \binom{3}{2} & -\binom{4}{3} & \hdots \\
0 & 0 & 1 & -\binom{3}{1} & \binom{4}{2} & \hdots \\
\vdots & \vdots & \vdots & \vdots & \vdots
\end{pmatrix} \quad.
\end{equation}
Define for $j\ge 0$
\begin{gather*}\tau_j' := T(\tau_j) = \tau_j -\binom{j+1}{1}\tau_{j+1}+\binom{j+2}{2}\tau_{j+2}
-\ldots\,, \\
q_j' := T(q_j) = q_jq_{j+1}^{-\binom{j+1}{1}}q_{j+2}^{\binom{j+2}{2}}\cdots\,. \notag
\end{gather*}
\begin{thm}\label{thm21} $\Omega(q_0',q_1',q_2',\ldots) =
q_0q_1^{-{\frac{1}{2}}}q_2^{+\frac{1}{3}}q_3^{-\frac{1}{4}}\cdots\Omega(q_0,q_1,\ldots)$.
\end{thm}
\begin{lem}\label{lem22} For $n\ge 2$ we have
$$\sum_{i=1}^{n-1}(-1)^i\binom{n}{i}\xi(i-n) = \frac{(-1)^{n+1}}{n+1}+\frac{(-1)^n}{2^n}\,.
$$
\end{lem}
\begin{proof}[Proof of lemma] Recall one has Bernoulli numbers $B_n$, $n\ge 0$ satisfying $B_0=1,\
B_1=-{\frac{1}{2}},\ \zeta(1-n)=-\frac{B_n}{n}$ for $n\ge 2$. Substituting and using the identity
$\frac{n+1}{j+1}\binom{n}{j}= \binom{n+1}{j+1}$, the desired formula becomes
\begin{equation}\label{27}\sum_{k=1}^n(-1)^{k-1}\binom{n+1}{k}(1-2^{1-k})B_k \stackrel{?}{=}
\frac{n+1}{2^n} - 1 \,.
\end{equation}
Consider the Bernoulli polynomials
$$B_N(x) := \sum_{k=0}^N\binom{N}{k}B_kx^{N-k} \,,
$$
which may be defined by the generating function
$$\sum_{N=0}^\infty B_N(x)t^N/N! = \frac{te^{xt}}{e^t-1}\,.
$$
As a consequence we get
\begin{equation}\label{28}B_N(x+1)-B_N(x) = Nx^{N-1} \,.
\end{equation}
Summing for $-1\ge x\ge -p$ we get for $N=n+1$ and $p\ge 1$ the equivalent identities
\begin{gather}(n+1)\sum_{\ell=-1}^{-p}\ell^n+B_{n+1}(-p)-B_{n+1}(0)=0 \,,\notag \\
(n+1)\sum_{\ell=-1}^{-p}\ell^n+(-p)^{n+1}+\sum_{k=1}^n B_k
\binom{n+1}{k}(-p)^{n+1-k} = 0\,.
\label{29}
\end{gather}
(We are grateful to V. Kac for suggesting \eqref{29}.)
It is straightforward to deduce \eqref{27}
from \eqref{29} taking $p=1,2$\,.
\end{proof}
\begin{proof}[Proof of theorem] The transformation $T$ satisfies
\begin{gather*}T(q_0)T(q_1)^sT(q_2)^{s^2}\cdots = q_0q_1^{s-1}q_2^{(s-1)^2}\cdots\,, \\
T(q_0)^{-1}T(q_1)^sT(q_2)^{-s^2}\cdots = q_0^{-1}q_1^{s+1}q_2^{-(s+1)^2}\cdots\,. \notag
\end{gather*}
Note the second identity follows from the first, replacing $s$ by $-s$ and inverting. The first
identity is left for the reader. It follows that
\begin{multline}\label{211}\frac{\prod_{n\ge
0}\big(1+T(q_0)T(q_1)^{n+{\frac{1}{2}}}\cdots\big)\big(1+T(q_0)^{-1}T(q_1)^{(n+{\frac{1}{2}})}
T(q_2)^{-(n+{\frac{1}{2}})^2}\cdots\big)}{\prod_{n\ge
0}\big(1+q_0q_1^{n+{\frac{1}{2}}}\cdots\big)\big(1+q_0^{-1}q_1^{(n+{\frac{1}{2}})}
q_2^{-(n+{\frac{1}{2}})^2}\cdots\big)} \\
=
\frac{1+q_0q_1^{-{\frac{1}{2}}}q_2^{\frac{1}{4}}q_3^{-\frac{1}{8}}
\cdots}{1+q_0^{-1}q_1^{{\frac{1}{2}}}q_2^{-\frac{1}{4}} \cdots} = q_0q_1^{-{\frac{1}{2}}}q_2^{\frac{1}{4}}\cdots\,.
\end{multline}
As a consequence of the lemma we have
\begin{equation}\label{212}
\frac{T(q_1)^{-\xi(-1)}T(q_3)^{-\xi(-3)}\cdots}{q_1^{-\xi(-1)}q_3^{-\xi(-3)}\cdots} =
q_2^{\frac{1}{3}-\frac{1}{4}}q_3^{-\frac{1}{4}+\frac{1}{8}}q_4^{\frac{1}{5}-\frac{1}{16}}
\cdots \,.
\end{equation}
The proof follows by combining \eqref{211} and \eqref{212}.
\end{proof}
Recall we have defined $V(q_1,q_2,\ldots)$ to be the coefficient of $q_0^0$ in
$\Omega(q_0,q_1,\ldots)$.
\begin{thm}\label{thm23} We have the following series expansion for $\Omega$:
$$\Omega(q_0,q_1,q_2,\ldots) = \sum_{n=-\infty}^{n=\infty}V(T^{-n}(q_1),T^{-n}(q_2),\ldots)
q_0^nq_1^{n^2/2}q_2^{n^3/3}\cdots .
$$
\end{thm}
\begin{proof} First note
\begin{equation}T(q_0)^nT(q_1)^{n^2/2}T(q_2)^{n^3/3}\cdots =
q_0q_1^{-{\frac{1}{2}}}q_2^{\frac{1}{3}}\cdots q_0^{n-1}q_1^{(n-1)^2/2}\cdots.
\end{equation}
Write $\Omega = \sum_{-\infty}^\infty V_n(q_1,\ldots)q_0^nq_1^{n^2/2}\cdots$. Then
\begin{multline*}q_0q_1^{-{\frac{1}{2}}}q_2^{\frac{1}{3}}\cdots\sum_nV_{n-1}(q)q_0^{n-1}q_1^{(n-1)^2/2}\cdots
= \\
q_0q_1^{-{\frac{1}{2}}}q_2^{\frac{1}{3}}\cdots\Omega(q) = \Omega(T(q)) = \\
\sum V_n(T(q))T(q_0)^nT(q_1)^{n^2/2}\cdots = \\
q_0q_1^{-{\frac{1}{2}}}q_2^{\frac{1}{3}}\cdots\sum_nV_{n}(T(q))q_0^{n-1}q_1^{(n-1)^2/2}\cdots.
\end{multline*}
Taking coefficients of $q_0^{n-1}$ we get $V_n(T(q))=V_{n-1}(q)$. The formula follows since
$V_0=V$.
\end{proof}
\begin{exam} Consider the formula in the theorem with $q_j\mapsto 1,\ j\ge 2$. We have
$$T(q)|_{q_j\mapsto 1,j\ge 2} = q_1
$$
and by \eqref{012}
$$V(q_1,1,1,\ldots) = q_1^{-\frac{1}{24}}\sum_\lambda q_1^{p_1(\lambda)} =
q_1^{-\frac{1}{24}}\prod_{m\ge 1}(1-q_1^m)^{-1}=\eta(q_1)^{-1}.
$$
The assertion of the theorem is then
\begin{equation}\eta(q_1)^{-1}\sum_{n\in {\Bbb Z}} q_0^nq_1^{n^2/2} = q_1^{\frac{-1}{24}}\prod_{m\ge
0}(1+q_0q_1^{m+{\frac{1}{2}}})(1+q_0^{-1}q_1^{m+{\frac{1}{2}}}).
\end{equation}
Multiplying through by $\eta(q_1)$ yields the triple product formula
\eqref{22}.
\end{exam}
\section{Quasimodular forms}\label{sec:quasi}
In this section, we recall the theory of quasimodular forms as developed in \cite{KZ}.
All results are due to Kaneko and Zagier and are recalled here solely for the convenience of the
reader.
We fix a subgroup of finite index $\Gamma\subset\Gamma_1 := \text{\rm SL}(2,{\Bbb Z})$. A holomorphic modular
form of weight $k$ (for $\Gamma$) is a holomorphic function $f(\tau)$ on the upper half-plane
${\mathcal H}=\{\tau=x+iy\ |\ y>0\}$ satisfying
\begin{equation}\label{31}f\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^kf(\tau),\quad \begin{pmatrix}a & b
\\ c & d\end{pmatrix}\in\Gamma\,.
\end{equation}
Note for some $\ell\ge 1$
we have $(\begin{smallmatrix}1 & \ell \\ 0 & 1\end{smallmatrix})\in\Gamma$, so $f$ may be expanded
in a Fourier series
\begin{equation}\label{32}f(\tau) = \sum a_n\exp(2\pi in\tau/\ell) = \sum a_n q^{n/\ell}\,.
\end{equation}
We assume that $f$ is holomorphic at $i\infty$, i.e. $a_n=0$ for $n<0$ and the Fourier series
converges for $|q|<1$. The holomorphic modular forms constitute a graded ring
$$M_*(\Gamma) := \oplus M_k(\Gamma)
$$
graded by the weight $k$.
An {\it almost holomorphic} function on ${\mathcal H}$ will be a function
\begin{equation} F(\tau) = \sum_{m=0}^N f_m(\tau)y^{-m}
\end{equation}
on ${\mathcal H}$ where $y=\text{\rm Im}(\tau)$ and each $f_m$ has a Fourier expansion as in \eqref{32}. An
almost holomorphic modular form of weight $k$ is an almost holomorphic function $f(\tau)$
satisfying the weight $k$ modularity property \eqref{31}.
\begin{exam} The classical Eisenstein series ($B_k$ Bernoulli number as in Lemma (\ref{lem22}))
\begin{equation}\label{34}G_k(q):=\frac{-B_k}{2k}+\sum_{n=1}^\infty\Big(\sum_{d|n}
d^{k-1}\Big)q^n,\quad k=2,4,6,\ldots
\end{equation}
is modular of weight $k$ for $\Gamma=\Gamma_1$ and $k\ge 4$. On the other hand, $G_2$ satisfies
the transformation
\begin{equation}\label{35} G_2\left(\frac{a\tau+b}{c\tau+d}\right) =
(c\tau+d)^2G_2(\tau)-\frac{c(c\tau+d)}{4\pi i} \,.
\end{equation}
Notice, however, that
\begin{equation}\label{36} \left[\text{\rm Im}\left(\frac{a\tau+b}{c\tau+d}\right)\right]^{-1} -(c\tau+d)^2y^{-1} =
-2ic(c\tau+d) \,.
\end{equation}
It follows that the function
\begin{equation}\label{37} G_2^*(\tau) := G_2(\tau)+\frac{y^{-1}}{8\pi}
\end{equation}
is an almost holomorphic modular form of weight $2$ for $\Gamma_1$.
\end{exam}
\begin{defn}\label{defn32} A quasimodular form of weight $k$ is a holomorphic function $f(\tau)$ on
${\mathcal H}$ such that there exists an almost holomorphic modular form $F = \sum_{m=0}^N f_my^{-m}$ of
weight $k$ with $f_0=f$.
\end{defn}
\begin{exam} $G_2$ is quasimodular of weight $2$. Indeed, one can take $F=G_2^*$.
\end{exam}
Let us write $AHM_*(\Gamma)$ (resp. $QM_*(\Gamma)$) for the graded ring of almost holomorphic
(resp. quasimodular) forms. (Kaneko and Zagier write $\widehat{M}$ and $\widetilde{M}$, but this
makes it difficult to remember which is which.)
\begin{prop}\label{prop34} The assignment $F=\sum_{j=0}^My^{-j}f_j\mapsto f_0$ defines an
isomorphism of graded rings, $AHM_*(\Gamma)\cong QM_*(\Gamma)$.
\end{prop}
\begin{proof} Note this is well defined, i.e. $\sum_{j=0}^My^{-j}f_j(\tau) \equiv 0$ for
holomorphic $f_j$ if and only if all the $f_j$ are zero. To see this one can e.g.\ apply the
differential operator $\frac{iM}{2}y^{-1}+\frac{d}{d\bar\tau}$ and argue by induction on $M$. The
map in question is surjective by definition, so it suffices to show injectivity. Suppose for some
$r\ge 1$ and $f_r \ne 0$ that
$F=y^{-r}f_r+\ldots+y^{-M}f_M$ is almost holomorphic modular of weight $k$. Let
$A=(\begin{smallmatrix}a & b \\ c & d\end{smallmatrix}) \in\Gamma$ and write $j=c\tau+d$. Using
\eqref{36} we get
\begin{multline*} (j^2y^{-1}-2icj)^rf_r(A\tau)+\ldots +(j^2y^{-1}-2icj)^Mf_M(A\tau) = \\
j^ky^{-r}f_r(\tau)+\ldots+j^ky^{-M}f_M(\tau).
\end{multline*}
Now identify coefficients of powers of $y^{-1}$
\begin{align}\label{39} f_M(A\tau) &= j^{k-2M}f_M(\tau) \,,\\
f_{M-1}(A\tau)-2icj\binom{M}{1}f_M(A\tau) &= j^{k-2M+2}f_{M-1}(\tau) \notag \,,\\
f_{M-2}(A\tau)-2icj\binom{M-1}{1}f_{M-1}(A\tau) & +(2icj)^2\binom{M}{2}f_M(A\tau) \notag\\
&= j^{k-2M+4}f_{M-2}(\tau)\notag\,, \\
\makebox[5cm][c]{$\vdots$} & \makebox[3cm][c]{$\vdots$} \notag
\displaybreak[0]\\
f_r(A\tau)-2icj\binom{r+1}{1}f_{r+1}(A\tau) \qquad\quad& \notag \\
+(2icj)^2\binom{r+2}{2}f_{r+2}(A\tau) -\ldots &=
j^{k-2r}f_r(\tau)\notag \,.
\end{align}
The terms not involving $y$ give
\begin{equation}(-2icj)^rf_r(A\tau)+\ldots+(-2cij)^Mf_M(A\tau) = 0. \label{310}
\end{equation}
Solve \eqref{39} recursively:
\begin{align*}f_M(A\tau) &= j^{k-2M}f_M(\tau) \,,\\
f_{M-1}(A\tau) &= 2ic\binom{M}{1}j^{k-2M+1}f_M(\tau)+j^{k-2M+2}f_{M-1}(\tau)\notag \,,\\
f_{M-2}(A\tau) &= \ldots + j^{k-2M+4}f_{M-2}(\tau) \notag \,, \\
\makebox[2cm][c]{$\vdots$} & \makebox[5cm][c]{$\vdots$} \notag \\
f_r(A\tau) & = \ldots + j^{k-r}f_r(\tau)\,. \notag
\end{align*}
Finally, substituting in \eqref{310} yields
\begin{multline}\label{312} (c\tau+d)^{k-r}c^rf_r(\tau)
+\alpha_{r+1}(c\tau+d)^{k-r-1}c^{r+1}f_r(\tau) +\ldots \\
+ \alpha_M (c\tau+d)^{k-M}c^Mf_M(\tau) = 0.
\end{multline}
Here the $\alpha_j$ are constants independent of $c$ and $d$. Varying $A\in \Gamma$ yields a
contradiction. Indeed, the map $\Gamma\to{\Bbb C}^2,\ A\mapsto (c,d)$ has Zariski dense image, so in the
above identity, $c$ and $d$ can be taken to be independent variables. The coefficient of
$c^rd^{k-r}$ is nontrivial.
\end{proof}
\begin{prop}\begin{enumerate}\item[(i)] $M_*(\Gamma)\subset QM_*(\Gamma)$.
\item[(ii)] $QM_*(\Gamma) = M_*(\Gamma)\otimes{\Bbb C}[G_2]$.
\item[(iii)] $QM_*(\Gamma)$ is stable under the operator (of degree $2$) $D:= \frac{d}{d\tau}$. We
have
$$QM_k(\Gamma) = \oplus_{0\le i\le k/2}D^iM_{k-2i}(\Gamma)\oplus{\Bbb C}\cdot D^{k/2-1}G_2.
$$
\end{enumerate}
\end{prop}
\begin{proof} (i) is clear. To prove (ii), we claim first that the map
\begin{equation}\label{313}M_*(\Gamma)\otimes{\Bbb C}[G_2^*] \to AHM_*(\Gamma)
\end{equation}
is an isomorphism, where $G_2^*$ is as in \eqref{37}. Indeed, for $F = f_0+\ldots+y^{-M}f_M$ almost
holomorphic modular of weight $k$, it follows from the first line of \eqref{39} that $f_M$ is
holomorphic modular of weight $k-2M$. We have
$$F - f_M\cdot (8\pi G_2^*)^M = g_0+\ldots+y^{-M+1}g_{M-1}
$$
is almost holomorphic modular of weight $k$. Surjectivity of \eqref{313} follows by induction on
$M$. Injectivity is straightforward, keeping track of powers of $y^{-1}$. Assertion (ii) now
follows from proposition (\ref{prop34}). Finally, (iii) is left for the reader.
\end{proof}
\section{Quasimodularity for characters $\Omega$ and $V$}\label{sec:qmov}
The purpose of this section is to prove
\begin{thm} The series $\Omega(\tau_0,\tau_1,\ldots)$ and $V(\tau_1,\tau_2,\ldots)$ are
quasimodular of weights $0$ and $-{\frac{1}{2}}$ respectively.
\end{thm}
We focus first on $\Omega$. Recall (definition (\ref{defn32})) a series
$F(\tau_0,\tau_1,\tau_2,\ldots)$ is said to be quasimodular of weight
$k$ if it can be expanded in a formal Taylor series
$$\sum_{J=(j_0,j_2,j_3,\ldots)} B_J(\tau_1)\tau^J/J!
$$
with $B_J(\tau_1)$ quasimodular of weight
\begin{equation}\label{4wt} k+{\rm wt}(J):= k+j_0+3j_2+4j_3+\ldots.
\end{equation}
Since we have not
specified a group $\Gamma$, there will be no harm e.g. in replacing $\tau_1$ by $2\tau_1$.
Define
$$\Phi(q)=\Omega(-q_0,q_1,q_2,\ldots).$$
One has $\Omega(q) =
\Phi(q_0^2,q_1^2,q_2^2,\ldots)/\Phi(q)$, so, since the space of quasimodular forms is a ring, it
will suffice to prove
$\Phi(q)$ is quasimodular. We have
$$\Phi(1,x_1,1,1,\ldots) = x_1^{-\frac{1}{24}}\prod_{r\ge
0}(1-x_1^{r+\frac{1}{2}})^2 = \big(\eta(x^{1/2})/\eta(x)\big)^2,
$$
which is modular of weight $0$. It therefore suffices to show
$$G :=\log\big(\Phi(x)/\Phi(1,x_1,1,1,\ldots)\big)
$$
is quasimodular. We have (with $\xi(s)$ as in \eqref{16})
\begin{multline}\label{41}
G = -2\pi i\big(\sum_{n\ge 1}\xi(-2n-1)\tau_{2n+1}\big)+ \\
\sum_{r\ge
0}\big[\log(1-x_0x_1^{n+\frac{1}{2}}x_2^{(n+\frac{1}{2})^2}\cdots)+
\log(1-x_0^{-1}x_1^{n+\frac{1}{2}}x_2^{-(n+\frac{1}{2})^2}\cdots)- \\
2\log(1-x_1^{n+\frac{1}{2}})\big]\,.
\end{multline}
We expand the final sum in \eqref{41}
\begin{multline*}B:= \\ -\sum_{n\ge 0\,,\ l\ge 1}\frac{x_1^{(n+{\frac{1}{2}})l}}{l}
\left(\exp\left(2\pi
il\left(\tau_0+\left(n+{\textstyle \frac{1}{2}}\right)^2\tau_2+\left(n+{\textstyle \frac{1}{2}}\right)^3\tau_3+\ldots\right)\right)\right. \\
+\left.
\exp\left(2\pi
il\left(-\tau_0-\left(n+{\textstyle \frac{1}{2}}\right)^2\tau_2+\left(n+{\textstyle \frac{1}{2}}\right)^3\tau_3-\ldots\right)\right) - 2\right).
\end{multline*}
Define coefficients $\alpha_m(J)$, where $J=(j_0,j_2,j_3,\ldots)$ and $\ j_k \ge 0$, by
\begin{equation}\frac{\partial^J}{\partial \tau^J}B|_{\tau_0=\tau_2=\ldots=0} =
\sum_{m=1}^\infty\alpha_m(J)x_1^{m/2}\,.
\end{equation}
Note $\alpha_m(0)=0$. For $J\ne 0$, write $|J|=j_0+j_2+\ldots$. We get
\begin{equation}\alpha_m(J) = -(2\pi i)^{|J|}\sum_{n\ge 0\,,\ l\ge
1}l^{|J|-1}(1+(-1)^{{\rm wt}(J)})\left(n+{\textstyle \frac{1}{2}}\right)^{{\rm wt}(J)-|J|}x_1^{(n+{\frac{1}{2}})l}.
\end{equation}
Thus, $\alpha_m(J)=0$ for ${\rm wt}(J)$ odd. For even weight, we find
\begin{multline}\label{45}
\frac{\partial^J}{\partial \tau^J}B|_{\tau_0=\tau_2=\ldots=0} = -2(2\pi
i)^{|J|}2^{|J|-{\rm wt}(J)}\sum_{m\ge 1}m^{|J|-1}x_1^{m/2}\sum_{d|m;\ d \text{\rm odd}}d^{{\rm wt}(J)-2|J|+1}
\\ = -2(2\pi i)^{|J|}2^{|J|-{\rm wt}(J)}\sum_{m\ge 1}m^{{\rm wt}(J)-|J|}x_1^{m/2}\sum_{l|m;\ \frac{m}{l}
\text{\rm odd}}l^{2|J|-{\rm wt}(J)-1} \,.
\end{multline}
The Eisenstein series of weight $k$ and level $1$ was given in \eqref{34}.
As in \cite{KZ}, we work with the level two Eisenstein series
\begin{gather*}F_k^{(1)}(q) := G_k(q^{{\frac{1}{2}}}) - G_k(q) =
\sum_{n=1}^\infty\Big(\sum_{d|n,2\not\,\mid d}(n/d)^{k-1}\Big)q^{n/2}\,, \\
F_k^{(2)} := G_k(q^{{\frac{1}{2}}}) - 2^{k-1}G_k(q) = (1-2^{k-1})\zeta(1-k)/2+
\sum_{n=1}^\infty\Big(\sum_{d|n,2\not\,\mid d}d^{k-1}\Big)q^{n/2}.
\end{gather*}
Suppose first that the inequality $k(J) := {\rm wt}(J)-2|J|+2>1$ holds. It follows from the first
equality in \eqref{45} that
\begin{equation}\label{48}
\frac{\partial^J}{\partial \tau^J}B|_{\tau_0=\tau_2=\ldots=0} =
-4\pi
i(2^{-k(J)+1})(\partial/\partial\tau_1)^{|J|-1}(F_{k(J)}^{(2)}(x_1)-F_{k(J)}^{(2)}(0))\,.
\end{equation}
Note $k(J)$ is even, so the remaining possibility is $k(J) \le 0$. In this case, the second
equality in \eqref{45} implies
\begin{equation}\frac{\partial^J}{\partial \tau^J}B|_{\tau_0=\tau_2=\ldots=0} = -2(2\pi
i)^{2-k(J)}(\partial/\partial\tau_1)^{{\rm wt}(J)-|J|}(F_{2-k(J)}^{(1)}(x_1))\,.
\end{equation}
Since the $F^{(i)}_k$ are quasimodular, it follows that the $\frac{\partial^J}{\partial
\tau^J}B|_{\tau_0=\tau_2=\ldots=0}$ are quasimodular except possibly (because of the constant
term on the right) in in the case $k(J)\ge 2;\ |J|=1$ where the constant term is not correct.
This only happens if $J=(0,\ldots,0,1,0,\ldots)$ where $j=1$ for some $j\ge 3$ odd and all
the other entries are zero. In this case $k(J)=j+1$ and it follows from \eqref{48} and
the identity $\xi(s)=(2^s-1)\zeta(s)$ that
\begin{multline*}\frac{\partial}{\partial\tau_j} B|_{\tau=0} = (-4\pi
i)2^{-j}\big(F^{(2)}_{j+1}(x_1)+(2^j-1)\zeta(-j)/2\big)= \\
(-4\pi i)2^{-j}F^{(2)}_{j+1}(x_1)+2\pi i\xi(-j)\,.
\end{multline*}
The last constant term exactly cancels the $-2\pi i\xi(-j)$ which appears in the first sum on
the right in \eqref{41}. This completes the proof of quasimodularity for $\Omega$.
It remains to prove quasi-modularity for $V(x_1,\ldots)$. The proof parallels that
in \cite{KZ}. We begin by expanding
\begin{gather} V(\tau_1,\tau_2,\ldots) = \sum_{K =
(k_2,k_3,\ldots)}A_K(\tau_1)\tau^K/K!\,, \\
V(T^{-n}(\tau)) = \sum_{m=0}^\infty\sum_{K}(\partial/\partial\tau_1)^mA_K(\tau_1)
(T^{-n}(\tau_1)-\tau_1)^m(m!)^{-1}(T^{-n}(\tau))^K/K!\,, \\
\label{413}
\Omega(\tau) =
\sum_{n=-\infty}^\infty\sum_{J=(j_0,j_2,j_3,\ldots)}\sum_{m=0}^\infty\sum_{K}
(\partial/\partial\tau_1)^mA_K(\tau_1)(T^{-n}(\tau_1)-\tau_1)^m(m!)^{-1}\times \\
\times (T^{-n}(\tau))^K(K!)^{-1}a_Jn^{\text{\rm wt}(J)}\tau^J(J!)^{-1}x_1^{n^2/2}\,. \notag
\end{gather}
Here $\sum_Ja_Jn^{\text{\rm wt}(J)}\tau^J(J!)^{-1}$ is the power series expansion of
$x_0^nx_2^{n^3/3}x_3^{n^4/4}\cdots$, with $a_J$ independent of $n$.
One verifies
the expansion
\begin{equation}\label{414}
T^{-n}(\tau_j) =
\tau_j+\binom{j+1}{1}n\tau_{j+1}+\binom{j+2}{2}n^2\tau_{j+2}+\ldots\quad
.
\end{equation}
Fix $m\ge 0$ and $K=(k_2,k_3,\ldots)$. Let $P(\tau)=\tau_2^{p_2}\tau_3^{p_3}\ldots$
and $Q_i(\tau)=\tau_i^{q_{ii}}\tau_{i+1}^{q_{i,i+1}}\ldots;\ i\ge 2$ be monomials
with
$\deg(P)=\sum p_j = m$ and $\deg(Q_i)=k_i$. Define the weights by ${\rm wt}(P) = \sum
(j+1)p_j$ and ${\rm wt}(Q_i) = \sum_{j\ge i} (j+1)q_{ij}$. As a consequence of
\eqref{414}, $P$ appears in $(T^{-n}(\tau_1)-\tau_1)^m(m!)^{-1}$ with a
coefficient $c_{P,m}n^{{\rm wt}(P)-2m}$, where $c_{P,m}$ is constant independent
of $n$. Similarly, $Q:=\prod Q_i$ appears in $(T^{-n}(\tau))^K(K!)^{-1}$ with
coefficient $c_{\{Q_i\},K}n^{\sum_i {\rm wt}(Q_i)-(i+1)k_i}$. Now fix
$J=(j_0,j_2,\ldots)$. We obtain a contribution to the coefficient of the monomial
$P(\tau)Q(\tau)\tau^J$ in \eqref{413} of the form
\begin{equation}\label{415}
c_{J,K,m,\{Q_i\},P}\frac{\partial^m}{\partial\tau_1^m}A_K(\tau_1)
\sum_{n=-\infty}^\infty n^{{\rm wt}(P)-2m+\sum_i ({\rm wt}(Q_i)-(i+1)k_i)+{\rm wt}(J)}x_1^{n^2/2}.
\end{equation}
If the exponent of $n$ is odd, this cancels. Assume this exponent equals $2r$
for $r\ge 0$ an integer. Then \eqref{415} can be rewritten
\begin{equation}\label{416}
b_{J,K,m,\{Q_i\},P}\frac{\partial^m}{\partial\tau_1^m}A_K(\tau_1)
\frac{\partial^r}{\partial\tau_1^r}\theta_{00}(\tau_1)\,,
\end{equation}
where $\theta_{00}(\tau_1)=\sum \exp(\pi in^2\tau_1)$ is a modular form of weight
$1/2$. Since differentiation preserves quasi-modularity and increases weight by
$2$, it follows that, assuming $A_K(\tau_1)$ is quasimodular of weight
${\rm wt}(K)-{\frac{1}{2}}$, the expression in \eqref{416} is quasi-modular of weight
${\rm wt}(P)+\sum{\rm wt}(Q_i)+{\rm wt}(J)$.
We will prove $A_K(\tau_1)$ is quasimodular of weight ${\rm wt}(K)-1/2$ by induction on
${\rm wt}(K)$. Write
\begin{multline*}\theta_{00}(x) = \sum_{-\infty}^\infty x^{n^2/2}= \prod_{m\ge
1}(1-x^m)\prod_{m\ge 0}(1+x^{m+{\frac{1}{2}}})^2 = \\
\eta(x)\Omega(1,x,1,1\ldots)\,.
\end{multline*}
Substituting $x_j=1;\ j\ne 1$ in Theorem (\ref{thm23}) yields
\begin{gather*}\Omega(1,x_1,1,\ldots) = V(x_1,1,\ldots)\sum x_1^{n^2/2} =
V(x_1,1,\ldots)\theta_{00}(x_1) \,,\notag \\
V(x_1,1,\ldots) = \eta(x_1)^{-1} \,.
\end{gather*}
This is modular of weight $-1/2$ as desired. Now fix an index set
$M=(m_2,m_3,\ldots)$ with ${\rm wt}(M)>0$ and assume
$A_K(\tau_1)$ is quasimodular of weight ${\rm wt}(K)-1/2$ for all $K$ with
${\rm wt}(K)<{\rm wt}(M)$. Consider the coefficient $B_M(\tau_1)$ of $\tau^M$ in the
expansion of $\Omega(\tau)$ \eqref{413}. We know that $B_M$ is quasimodular of
weight ${\rm wt}(M)$. It is a sum of terms \eqref{416}, of which all but one involve
$A_K$ with ${\rm wt}(K)<{\rm wt}(M)$ and are quasimodular of weight ${\rm wt}(M)$ by our inductive
hypothesis. The one remaining term, which appears with coefficient $(M!)^{-1}$, is
$A_M(\tau_1)\theta_{00}(\tau_1)$ (take $J=0=m,\ P=1,\ Q_i = \tau_i^{m_i}$.)
It follows that $A_M(\tau_1)$ is quasimodular of weight ${\rm wt}(M)-1/2$ as desired.
\section{Preliminaries on partitions}\label{sec:prepart}
Let $\lambda = \lambda_1\ge \lambda_2\ge\ldots\ge\lambda_\ell >0=\lambda_{\ell+1}=\ldots$ be a
partition. $\ell=\ell(\lambda)$ is the length of the partition, and $|\lambda| = \sum\lambda_i$
is the number being partitioned.
Let $f(\lambda)$ be a function on the set of partitions. Assuming $f$ does not grow too
rapidly with $|\lambda|$ we may consider for $|q|\ll 1$ the ratio
\begin{equation}\label{51} \langle f\rangle_q\ := \frac{\sum_\lambda f(\lambda)q^{|\lambda|}}{\sum_\lambda
q^{|\lambda|}} = (q)_\infty \sum_\lambda f(\lambda)q^{|\lambda|}.
\end{equation}
Here, we write for $q$ given and $n\le \infty$,
\begin{equation}\label{52}(a)_n = (1-a)(1-aq)(1-aq^2)\cdots (1-aq^n).
\end{equation}
Fix an integer $n\ge 1$ and variables $t_1,\ldots,t_n$. We will be interested in the following
``$n$-point correlation function''
\begin{equation}\label{53} F(t_1,\ldots,t_n) := \Big<\prod_{k=1}^n\Big(\sum_{i=1}^\infty
t_k^{\lambda_i-i+{\frac{1}{2}}}\Big)\Big>_q.
\end{equation}
To understand the relation between $F$ and the character $V(q)$, we take a small detour, beginning
with some basic ideas from the theory of partitions. The {\it diagram} of a partition
$\lambda$ is simply the array of dots (or squares) with
$\lambda_1$ dots in the first row, $\lambda_2$ dots in the second row, etc. Here is $4,4,3,2,1$.
$$\begin{array}{cccc} \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet \\
\bullet & \bullet \\
\bullet
\end{array}
$$
The transposed partition $\lambda'$ is obtained by flipping along the diagonal
$$\begin{array}{ccccc} \bullet & \bullet & \bullet & \bullet & \bullet\\
\bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet \\
\bullet & \bullet
\end{array}
$$
For $\lambda_i\ge i$ let $m_i=\lambda_i-i$ be the number of dots to the right of the $i$-th point
on the diagonal, and let $n_i$ be the number of points below it. (so
$n_i(\lambda)=m_i(\lambda')$.) Note $m_1>m_2>\ldots \ge 0$ and similarly for the $n_i$. The $m_i$
and $n_i$ are called Frobenius coordinates of $\lambda$ and written
$(m_i,\ldots,m_p|n_1,\ldots,n_p)$. For example, Frobenius coordinates for $4,4,3,2,1$ are
$(3,2,0|4,2,0)$. Frobenius coordinates are related to the $\lambda_i$
via the following generating
function identity.
\begin{lem}\label{lem51} Let $\lambda$ be a partition of length $\ell$ with Frobenius
coordinates\newline
$(m_i,\ldots,m_p|n_1,\ldots,n_p)$. Then
$$\sum_i (t^{\lambda_i-i+{\frac{1}{2}}}-t^{-i+{\frac{1}{2}}}) = \sum_k (t^{m_k+{\frac{1}{2}}}-t^{-(n_k+{\frac{1}{2}})}).
$$
\end{lem}
\begin{proof} It is clear that the $m_k$ correspond to the nonnegative $\lambda_i-i$. Thus it will
suffice to identify the terms with negative exponents on the two sides. This follows from the
identity of sets of numbers
$$\{\lambda_{p+1}-(p+1),\dots,\lambda_{\ell}-\ell,-n_1-1,\dots,-n_p-1\} = \{-\ell,\dots,-2,-1\},,
$$
which, in turn, follows by comparing the negative elements from the identity
$$
\{\lambda_1-1,\dots,\lambda_{\ell}-\ell,-\lambda'_1,\dots,-\lambda'_k +
k-1\}=\{-\ell,\dots, k-1\}\,, \forall k \ge \lambda_1\,,
$$
established in \cite{MacD}, Chapter I, (1.7).
\end{proof}
Substituting $t=\exp(u)$ and expanding in the previous lemma yields identities
\begin{multline}\label{54} \sum_i \left(\lambda_i-i+{\textstyle \frac{1}{2}}\right)^r
+(-1)^{r+1} \left(i-{\textstyle \frac{1}{2}}\right)^r =
\\
\sum_j \left(m_j+{\textstyle \frac{1}{2}}\right)^r + (-1)^{r+1}\left(n_j+{\textstyle \frac{1}{2}}\right)^r =: p_r(\lambda).
\end{multline}
For example, $|\lambda| = p_1(\lambda)$. Recall we have already encountered the $p_r(\lambda)$ in
the expansion for the character $V(q)$:
\begin{equation} V(q_1,q_2,q_3,\ldots) = q_1^{-\xi(-1)}q_3^{-\xi(-3)}\cdots \sum_\lambda
\prod_{r\ge 1}q_r^{p_r(\lambda)}.
\end{equation}
To see the connection more specifically, write $q_n = \exp(2\pi i \tau_n)$ and consider the
differential operator
\begin{equation}\label{56} \delta = \delta(u) := u^{-1}+(2\pi i)^{-1}\sum_{n=1}^\infty
\frac{u^n}{n!}\frac{\partial}{\partial \tau_n}.
\end{equation}
\begin{prop} We have with $F$ as in \eqref{53} and $t=\exp(u)$
$$F(t) = \eta(q_1)\delta V|_{\tau_2=\tau_3=\ldots =0}.
$$
\end{prop}
\begin{proof} Consider the Bernoulli polynomials, defined by
\begin{equation}\label{57} \frac{u\exp(xu)}{\exp(u)-1} = \sum_{k=0}^\infty B_k(x)u^k/k!;\quad
B_0(x) = 1,\ B_1(x) = x-{\frac{1}{2}}
\end{equation}
One has (\cite{hida}, theorem 1, p. 43)
\begin{equation}\label{58} \xi(1-m) = \frac{-B_m({\frac{1}{2}})}{m}.
\end{equation}
Combining these identities yields
\begin{equation}\label{59} -\sum_{n=1}^\infty
\xi(-n)u^n/n! =
\frac{1}{t^{\frac{1}{2}} - t^{-{\frac{1}{2}}}}-u^{-1} = \sum_{i=1}^\infty t^{-i+{\frac{1}{2}}} -u^{-1}.
\end{equation}
On the other hand, using lemma \ref{lem51} and the identity
\begin{equation}\label{510} \eta(q_1)^{-1} = q_1^{-\xi(-1)}\sum_\lambda q_1^{|\lambda|},
\end{equation}
one computes
\begin{multline*}\eta(q_1)\delta V|_{\tau_2=\ldots = 0} = u^{-1} -\sum \xi(-n)u^n/n! + \\
+ \Big(\sum_\lambda q_1^{|\lambda|}\Big)^{-1}\sum_\lambda q_1^{|\lambda|}\sum_{i=1}^\infty
(t^{\lambda_i - i+{\frac{1}{2}}} - t^{-i+{\frac{1}{2}}} ) = \\
= u^{-1}+\Big(\sum_\lambda q_1^{|\lambda|}\Big)^{-1}\sum_\lambda q_1^{|\lambda|}\sum_{i=1}^\infty
t^{\lambda_i - i+{\frac{1}{2}}} - u^{-1} = \\
=\Big<\sum_{i=1}^\infty t^{\lambda_i - i+{\frac{1}{2}}}\Big>_{q_1} = F(t).
\end{multline*}
\end{proof}
More generally, one finds
\begin{equation}\label{512} \eta(q_1)\delta(u_1)\circ\ldots\circ\delta(u_n)V|_{\tau_2=\ldots=0}
= \Big<\prod_{k=1}^n\Big(\sum_{i=1}^\infty
t_k^{\lambda_i-i+{\frac{1}{2}}}\Big)\Big>_{q_1} = F(t_1,\dotsc,t_n).
\end{equation}
This is also equivalent to the formula \eqref{016a} from the introduction.
\section{The formula for correlation functions}\label{sec:results}
In this section we give a detailed statement of our results on the generating function
$F(t_1\dotsc,t_n)$ defined in \eqref{53}. To simplify notation, we
write $\Theta(x)=\Theta(x;q)$ for the
following theta function
\begin{align}\label{61}
\Theta(x) = \T_{11}(x;q)&=\eta^{-3}(q)\sum_{n\in{\Bbb Z}} (-1)^n q^{\frac{(n+1/2)^2}2} x^{n+1/2} \\
&=(q)_\infty^{-2} (x^{1/2}-x^{-1/2}) (q x)_\infty (q/x)_\infty \,.
\end{align}
The function $\Theta(x)$ is odd,
\begin{equation*}
\Theta(x^{-1})=-\Theta(x),
\end{equation*}
and satisfies the difference equation
\begin{equation}\label{62}
\Theta(q^m x)=(-1)^m q^{-m^2/2} x^{-m} \Theta(x)\,,
\quad m\in{\Bbb Z} \,.
\end{equation}
Define
\begin{equation}
\Theta^{(k)}(x):= \left(x\d x\right)^k \T_{11}(x;q)\, ,\ \Theta' = \Theta^{(1)},\ \text{\rm etc.}
\end{equation}
where $x\d x$ is the natural invariant vector
field on the group ${\Bbb C}^* = {\Bbb C}\setminus 0$.
We shall prove the following
\begin{thm}\label{thm61}
\begin{equation}\label{64}
\vspace{-3 \jot}
F(t_1,\dots,t_n)=
\sum_{\sigma\in\mathfrak{S}(n)}\, \frac
{\displaystyle \det\left( \frac{\displaystyle \Theta^{(j-i+1)}(t_{\sigma(1)}\cdots
t_{\sigma(n-j)})}{\displaystyle (j-i+1)!}
\right)_{i,j=1}^n}
{\displaystyle \Theta(t_{\sigma(1)})\, \Theta(t_{\sigma(1)} t_{\sigma(2)}) \dots
\Theta(t_{\sigma(1)}\cdots t_{\sigma(n)})}
\end{equation}
Here $\sigma$ runs through all permutations $\mathfrak{S}(n)$ of $\{1,\dotsc,n\}$,the
matrices in the numerator have size $n\times n$, and we define
$1/(-n)!=0$ if $n\ge 1$.
\end{thm}
In particular, since $\Theta'(1)=1$ and $\Theta''(1)=0$, we have
\begin{align*}
F(t_1) &=\frac{1}{\Theta(t_1)}\,,\notag\\
F(t_1,t_2) &=\frac{1}{\Theta(t_1 t_2)}
\left(
\frac{\Theta'(t_1)}{\Theta(t_1)} + \frac{\Theta'(t_2)}{\Theta(t_2)} \right)\,. \notag
\end{align*}
For $n=3$ the formula \eqref{64} looks as follows:
\begin{multline}\label{66}
F(t_1,t_2,t_3)=\\
\frac1{\displaystyle\Theta(t_1 t_2 t_3)}
\sum_{\sigma\in\mathfrak{S}(3)}
\det
\left(
\begin{array}{rrc}
\frac{\displaystyle\Theta'\left(t_{\sigma(1)} t_{\sigma(2)}\right)}
{\displaystyle \Theta\phantom{{}'}\left(t_{\sigma(1)} t_{\sigma(2)}\right)}&
\frac{\displaystyle 1}{\displaystyle 2} \,
\frac{\displaystyle\Theta''\left(t_{\sigma(1)}\right)}
{\displaystyle\Theta\phantom{{}''}\left(t_{\sigma(1)}\right)}&
\frac{\displaystyle\Theta'''(1)}{\displaystyle 3!}\\
{\displaystyle 1}&
\frac{\displaystyle\Theta'\phantom{{}'}\left(t_{\sigma(1)}\right)}
{\displaystyle\Theta\phantom{{}''}\left(t_{\sigma(1)}\right)}&
0\\
&
{\displaystyle 1}&
\Theta'(1)
\end{array}
\right)\\
= \frac{1}{\Theta(t_1 t_2 t_3)}
\left( \sum_{1\le i \ne j \le 3}
\frac{\Theta'(t_i)}{\Theta(t_i)}
\frac{\Theta'(t_i t_j)}{\Theta(t_i t_j)} -
\sum_{i=1}^3 \frac{\Theta''(t_i)}{\Theta(t_i)} + \Theta'''(1)
\right) \,.
\end{multline}
The determinants in \eqref{64}, having only one non-zero diagonal
below the main diagonal,
have a nice combinatorial expansion, see
the formula \eqref{78} below.
One has the following simple
\begin{lem}\label{lem61a}
\begin{equation}\label{th'}
\Theta^{(2m+1)}(1;q)=(2m+1)!
\sum_{k_1+2k_2+\dots=m} \frac{(-2)^{k_1+k_2+\dots}}
{k_1!\, k_2!\, \cdots}
\prod_i \left(\frac{G_{2i}(q)}{(2i)!}\right)^{k_i} \,.
\end{equation}
\end{lem}
Observe that the sum in \eqref{th'} is over all
partitions $1^{k_1} 2^{k_2} 3^{k_3} \dots$ of
the number $m$ and $k_1+k_2+\dots$ is the length
of such a partition. In particular,
\begin{align*}
\Theta^{(3)}(1;q)&= - 6 G_2(q)\,,\\
\Theta^{(5)}(1;q)&= - 10 G_4(q)+ 60 G_2(q)^2\,, \\
\Theta^{(7)}(1;q)&= - 14 G_6(q)+420 G_4(q) G_2(q)- 840 G_2(q)^3\,.
\end{align*}
This lemma will be proved below.
\begin{remark} The equality in Theorem \ref{thm61} may look like an
equality of multivalued functions but in fact
the only ambiguity is the factor
$$
\sqrt{\,t_1\cdots t_n}
$$
which appears on both sides of of the theorem and,
therefore, can be ignored.
\end{remark}
\begin{remark} Alternatively, one may consider the $(n+1)\times (n+1)$ matrix
\begin{equation}\label{67}\left(\begin{array}{ccccc} \Theta(t_1\cdots t_n) & \Theta'(t_1\cdots t_{n-1}) &\hdots &
\frac{1}{n!}\Theta^{(n)}(1) \\
0 & \Theta(t_1\cdots t_{n-1}) & \hdots & \frac{1}{(n-1)!}\Theta^{(n-1)}(1) \\
\vdots & \vdots & \hdots & \vdots \\
0 & 0 & \hdots & \Theta'(1) \\
0 & 0 & \hdots & 0
\end{array}\right)
\end{equation}
(Note the bottom right hand entry is $\Theta(1)=0$.) This matrix has rank $n$, so there will be a
unique column vector
$$\begin{pmatrix}v_n(\td{n}) \\
v_{n-1}(\td{n-1}) \\
\vdots \\
v_1(t_1) \\
1\end{pmatrix}
$$
which is killed by multiplication on the left by the matrix \eqref{67}. Then
$$F(\td{r}) = \sum_{\sigma} v_r(t_{\sigma(1)},\dotsc,t_{\sigma(r)}).
$$
\end{remark}
\begin{proof}[Proof of Lemma \ref{lem61a}]
Let $f(a)$ be some function of a variable $a$ and let $b$ be
another variable. We have:
\begin{align*}
e^{f(a+b)-f(a)}&=\exp\left(\sum_{m=1}^\infty \frac{f^{(m)}(a)}{m!}
b^m\right)\\
&=\prod_{m=1}^\infty \exp\left(\frac{f^{(m)}(a)}{m!}
b^m\right)\\
&=\prod_{m=1}^\infty \sum_{k=0}^\infty \left(\frac{f^{(m)}(a)}{m!}
\right)^k \frac{b^{mk}}{k!}\\
&=\sum_{s=0}^\infty
\sum_{k_1+2k_2+3k_3+\dots=s}
\frac{b^s}{k_1!\, k_2!\, k_3!\cdots}
\prod_{i=1}^\infty\left(\frac{f^{(i)}(a)}{i!}\right)^{k_i}\,.
\end{align*}
In other words, we have
\begin{multline}\label{67a}
e^{-f(a)} \frac{d^s}{d a^s} e^{f(a)} = \\
s! \sum_{k_1+2k_2+3k_3+\dots=s} \frac1{k_1!\, k_2!\, k_3!\cdots}
\left(\frac{f'}{1!}\right)^{k_1}
\left(\frac{f''}{2!}\right)^{k_2}
\left(\frac{f'''}{3!}\right)^{k_3} \cdots \,.
\end{multline}
We wish to apply the above formula to the
triple product formula for
the theta function and use the fact that
$$
\left.\left(x\frac{d}{dx}\right)^{m}
\log\left((qx)_\infty (q/x)_\infty\right)
\right|_{x=1}=
\begin{cases}
-2 G_{m}(q) - B_{m}/m\,, &\text{$m$ is even}\\
0\,, &\text{$m$ is odd} \,.
\end{cases}
$$
The derivative $\Theta^{(2m+1)}(1;q)$ is quasi-modular of weight $2m$.
It is, therefore, sufficient, to compute the term
$$
\left. (2m+1) \left(x\frac{d}{dx} (x^{1/2}-x^{-1/2}) \right)
\left(x\frac{d}{dx}\right)^{2m} (qx)_\infty (q/x)_\infty
\right|_{x=1}\,,
$$
which gives the contribution of the maximal weight. Using
\eqref{67a} we obtain that the above term equals
the RHS of \eqref{th'} modulo terms of lower weight.
This establishes \eqref{th'}.
\end{proof}
Before diving into the proof of Theorem \ref{thm61}
in the next section, we give as a warmup the proof for the $1$-point
function.
\begin{thm}\label{thm64}
$$\Big<\sum_i t^{\lambda_i -i+{\frac{1}{2}}}\Big>_q = \frac{1}{\Theta(t)}.
$$
\end{thm}
\begin{lem}\label{lem65}
$$\Big<t^{\lambda_k}\Big>_q = \frac{(q)_{\infty}}{(1-q)\cdots (1-q^{k-1})(1-q^kt)(1-q^{k+1}t)\cdots }
$$
\end{lem}
\begin{proof} The notation is as in \eqref{52}. Recall in section (\ref{sec:prepart}) we defined
the transpose $\lambda'$ of a partition $\lambda$. The lemma is straightforward once one observes that
$$\Big<t^{\lambda_k}\Big>_q = \Big<t^{\lambda_k'}\Big>_q = \Big<t^{\#\{i|\lambda_i\ge k\}}\Big>_q.
$$
\end{proof}
Recall the following Heine's $q$-analog of the Gauss
${}_2F_1$-summation. Given any $a$, $b$, $c$ satisfying
$|c|<|ab|$ we have
\begin{equation}\label{69}
\sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n (q)_n}
\left(\frac{c}{ab}\right)^n = \frac{(c/a)_\infty
(c/b)_\infty}{(c)_\infty (c/ab)_\infty}\,,
\end{equation}
see for example \cite{GR}, Section 1.5. Recall that the symbol
$(a)_n$ was defined in \eqref{52}.
\begin{proof}[Proof of theorem \ref{thm64}]
\begin{multline*} \Big<\sum_i t^{\lambda_i -i+{\frac{1}{2}}}\Big>_q = \sum_{k=1}^\infty t^{{\frac{1}{2}}
-k}\frac{(q^k)_\infty}{(q^kt)_\infty} = \\
= t^{-{\frac{1}{2}}}\frac{(q)_\infty}{(qt)_\infty}\sum_{r=0}^\infty
t^{-r}\frac{(qt)_i}{(q)_i} = \frac{(q)_\infty^2}{(t^{\frac{1}{2}} -
t^{-{\frac{1}{2}}})(qt)_\infty(q/t)_\infty}= \frac{1}{\Theta(t)}.
\end{multline*}
\end{proof}
\section{Beginning of the proof of Theorem \ref{thm61}}\label{sec:not}
Our strategy for proving theorem \ref{thm61} will be based on the
following simple implication
$$
\Bigg(
\begin{matrix}
\text{\rm $f(x)$ is holomorphic on ${\Bbb C}\setminus 0$ and }\\
f(qx)=f(x)
\end{matrix}
\Bigg)
\Rightarrow
\Bigg(
\text{\rm $f(x)$ is a constant}
\Bigg)\quad .
$$
We shall show that both sides of theorem \ref{thm61} satisfy
the same difference equation and have the
same singularities.
We now introduce some useful notation.
Given indices
$
1< i_1<\dots<i_n \,.
$
set
\begin{align}\label{71}
\ex{i_1,\dots,i_r}{t_1,\dots,t_r}
:&= \left\langle \prod_{k=1}^n t_k^{\lambda_{i_k}-i_k+1/2}
\right\rangle_q \\
&= \prod_{k=1}^n t_k^{1/2-i_k}
\left\langle \prod_{k=1}^n t_k^{\lambda'_{i_k}}
\right\rangle_q \notag \\
&= (q)_\infty
\frac
{t_1^{1/2-i_1} \dots t_r^{1/2-i_r}}
{(q)_{i_1-1} (q^{i_1} t_1)_{i_2-i_1} (q^{i_2} t_1 t_2)_{i_3-i_2}
\dots
(q^{i_r} t_1 \cdots t_r)_\infty}
\,,\notag
\end{align}
The notation $(a)_n$ is as in \eqref{52}, and the last equality in \eqref{71} is a
straightforward generalization of lemma (\ref{lem65}). For brevity, write
$$
\ex{i}{t} :=\ex{i_1,\dots,i_r}{t_1,\dots,t_r} \,.
$$
Set
\begin{align}
H(t_1,\dots,t_n)&:=\sum_{i_1<\dots<i_n} \ex{i}{t} \,, \label{72} \\
G(t_1,\dots,t_n)&:=\sum_{\sigma\in\mathfrak{S}(n)} H(t_{\sigma(1)},\dotsc,t_{\sigma(n)}) \,. \label{73}
\end{align}
By $\P n$ denote the set of all partitions
of the
set $\{1,\dots,n\}$. An element $\pi$ of $\P n$
$$
\pi=\{\pi_1,\dots,\pi_\ell\} \in \P n\,, \quad \pi_i\subset\{1,\dots,n\}\,,
$$
is by definition an unordered collection of subsets
of $\{1,\dots,n\}$ such that
\begin{align*}
&\pi_i \cap \pi_j = \emptyset\,, \quad i\ne j\,, \\
&\pi_1 \cup \dots \cup \pi_\ell = \{1,\dots,n\} \,.
\end{align*}
Note the difference between partitions of the
\emph{set} $\{1,\dots,n\}$ and partitions of the \emph{number} $n$.
The number $\ell$ is called the \emph{length} of $\pi$ and is denoted
by $\ell(\pi)$. The subsets $\pi_i$ are called the \emph{blocks} of $\pi$.
Given a partition $\pi\in\P n$ set
$$
G^\pi(t_1,\dots,t_n):=G\left(\prod_{k\in\pi_1} t_k\,, \dots ,
\prod_{k\in\pi_{\ell(\pi)}} t_k \right) \,.
$$
For example, if $n=3$ and $\pi=\{\{1,2\},\{3\}\}$ then
$$
G^\pi(t_1,t_2,t_3)=G(t_1 t_2, t_3) \,.
$$
With this notation we have
\begin{equation}\label{76}
F(t_1,\dots,t_n)=\sum_{\pi\in\P n} G^\pi(t) \,.
\end{equation}
Set also
\begin{equation}\label{77}
F^\pi(t_1,\dots,t_n):= F\left(\prod_{k\in\pi_1} t_k\,, \dots ,
\prod_{k\in\pi_{\ell(\pi)}} t_k \right)\,, \quad \pi\in\P n \,.
\end{equation}
We shall also need the set ${\Bbb G} n$ of all {\it
compositions} of the set $\{1,\dots,n\}$. By definition,
the set ${\Bbb G} n$ consists of all {\it ordered}
collections
$$
\gamma=(\gamma_1,\dots,\gamma_\ell)\,, \quad \gamma_i\subset\{1,\dots,n\}\,,
$$
such that
\begin{align*}
&\gamma_i \cap \gamma_j = \emptyset\,, \quad i\ne j\,, \\
&\gamma_1 \cup \dots \cup \gamma_\ell = \{1,\dots,n\} \,.
\end{align*}
The number $\ell$ is called again the {\it length} of $\gamma$ and is denoted
by $\ell(\g)$. The set ${\Bbb G} n$ is in a natural one-to-one
correspondence with the set ${\Upsilon}_n$ of all partial flags
$$
\emptyset={\upsilon}_1\varsubsetneq{\upsilon}_2\varsubsetneq\dots\varsubsetneq{\upsilon}_\ell
\varsubsetneq{\upsilon}_{\ell+1}=\{1,\dots,n\}
$$
in the set $\{1,\dots,n\}$, namely, we set
\begin{equation}\label{77a}
{\upsilon}(\gamma)_k=\gamma_1 \cup \dots \cup \gamma_{k-1}\,.
\end{equation}
For any partition $\pi$ or composition $\gamma$ of an $n$-element set of
length $\ell$ define
its sign as
$$
(-1)^\pi = (-1)^\gamma = (-1)^{n+\ell}\,.
$$
Then, in particular, the sign of a permutation equals the sign of
the corresponding partition into disjoint cycles.
Denote by
$
U(t_1,\dots,t_n)
$
the RHS of \eqref{64}, and set
\begin{multline*}
T(t_1,\dots,t_n):= \Theta(t_1 \dotsc t_n) U(t_1,\dots,t_n) \\
=\sum_{\sigma\in\mathfrak{S}(n)}
\frac 1{\Theta(t_{\sigma(1)})\Theta(t_{\sigma(1)} t_{\sigma(2)}) \dotsc \Theta(t_{\sigma(1)}\cdots
t_{\sigma(n-1)})}\times \notag \\
\times \det\left( \frac{\Theta^{(j-i+1)}(t_{\sigma(1)}\cdots t_{\sigma(n-j)})}{(j-i+1)!}
\right)\,. \notag
\end{multline*}
Expanding the determinants one obtains
\begin{equation}\label{78}
T(t_1,\dots,t_n)=\sum_{\gamma\in{\Bbb G} n} (-1)^{\gamma}
\Theta^{(\#\gamma_1)}(1) \,
\prod_{k=2}^{\ell(\g)}
\frac{\Theta^{(\#\gamma_k)}\left(\prod_{i\in{\upsilon}(\gamma)_{k}} t_i\right)}
{\Theta\left(\prod_{i\in{\upsilon}(\gamma)_{k}} t_i\right)}\,,
\end{equation}
where we used the notation \eqref{77a} for the partial sums of a
composition $\gamma$.
Define the functions $T^\pi(t)$, $\pi\in\P n$, as in \eqref{77}.
We will use two subsets of $\P n$. By
$\Po n$ denote the set of all partitions $\pi$ such
that $\pi$ has at most one block of cardinality $>1$ which,
in addition, contains the number 1.
By $\Pt n$ denote the set of all partitions $\pi$
that have $\{1\}$ as a block. We have, for example,
\begin{gather*}
\{\{1,2\},\{3\}\}\in\Po 3\,,\\
\{\{1\},\{2,3\}\}\in\Pt 3\,.
\end{gather*}
Denote by
$$
\text{Atom}({\{1,\dots,n\}})=\{\{1\},\dotsc,\{n\}\} \in \P n
$$
the partition into 1-element subsets. It is clear
that
$$
\Po n \cap \Pt n = \text{Atom}({\{1,\dots,n\}})\,.
$$
Define the subsets $\Go n,\Gt n \subset {\Bbb G} n$ in the same way as for
partitions of $\{1,\dots,n\}$.
We shall need the following identities
\begin{lem}
\begin{gather}\label{710}
\sum_{\pi\in\P n} (-1)^{\pi} \ell(\pi) ! =
\sum_{\gamma\in{\Bbb G} n} (-1)^{\gamma} = 1 \\
\label{711} \sum_{\pi\in\P n} (-1)^{\ell(\pi)} (\ell(\pi)-1) ! = 0 \,.
\end{gather}
\end{lem}
\begin{proof} We shall prove \eqref{710}. The proof of \eqref{711} is similar. Define $f: \prod_n
\to \prod_{n-1}$ in the obvious way, by simply omitting $n$ from a partition. We argue by
induction on $n$. Note
$$\# f^{-1}(\pi) = \ell(\pi)+1.
$$
More precisely, $f^{-1}(\pi)$ contains $\ell(\pi)$ partitions of length $\ell(\pi)$ and one
partition of length $\ell(\pi)+1$. We compute
\begin{multline*} \sum_{\pi\in\prod_n} (-1)^{n+\ell(\pi)}\ell(\pi)! = \\
\sum_{\pi\in\prod_{n-1}}
(-1)^n\Big((-1)^{\ell(\pi)} \ell(\pi) \ell(\pi)!+(-1)^{\ell(\pi)+1}(\ell(\pi)+1)!\Big) = \\
\sum_{\pi\in\prod_{n-1}}
(-1)^{n-1+\ell(\pi)}\ell(\pi)! = 1.
\end{multline*}
\end{proof}
\section{Difference equations for the correlation functions}\label{sec:difF}
The aim of this section is the following
\begin{thm}\label{thm81}
$$
F(q t_1, t_2,\dots,t_n)= -q^{1/2} t_1\cdots t_n \left(
\sum_{\pi\in\Po n} (-1)^{\pi} F^\pi (t_1,\dots,t_n) \right) \,.
$$
\end{thm}
Recall that, by definition,
\begin{multline*}
\sum_{\pi\in\Po n} (-1)^{\pi} F^\pi (t_1,\dots,t_n)=\\
\sum_{s=0}^{n-1}
(-1)^s
\sum_{1<i_1<\dots < i_s \le n} F(t_1 t_{i_1} t_{i_2} \cdots t_{i_s},
\dots,\widehat{\,t_{i_1}}, \dots,\widehat{\,t_{i_s}}, \dots) \,.\notag
\end{multline*}
We begin with some auxiliary propositions.
\begin{prop}\label{prop82} For any $k=1,\dots,n$
\begin{align*}
H(t_1,\dots,qt_k,\dots,t_n)=
&-q^{1/2} t_1 \cdots t_n H(t_1,\dots,t_n) \notag\\
&+ \frac{q t_k}{1-q t_k} H(t_1,\dots,t_{k-1},
qt_k t_{k+1}, t_{k+2},\dots,t_n) \\
&- \frac{1}{1-q t_k} H(t_1,\dots,t_{k-2},
qt_{k-1} t_{k}, t_{k+1},\dots,t_n) \,.\notag
\end{align*}
Here if $k=1$ (or $k=n$) the third (second) summand
should be omitted.
\end{prop}
\begin{cor}\label{cor83}
\begin{align}
G(q t_1, t_2,\dots,t_n)=& -q^{1/2} t_1\cdots t_n G(t_1,\dots,t_n) \label{82} \\
&-\sum_{k=2}^n G(q t_1 t_k,t_2,\dots,t_{k-1},t_{k+1},\dots, t_n)\notag \\
=& -q^{1/2} t_1\cdots t_n \left(
\sum_{\pi\in\Po n} (-1)^{\pi} (n-\ell(\pi))! \, \, G^\pi (t) \right) \,. \label{83}
\end{align}
\end{cor}
\begin{lem}\label{lem84} Let $u$ and $v$ be two variables.
For any integers $a < b$
\begin{equation}
\sum_{a < i < b} \frac{(q v)^{1/2-i}}
{(q^a u)_{i-a} (q^i u v)_{b-i}} =
\frac{1}{1-q v}
\left(
\frac{(q v)^{3/2-b}}{(q^a u)_{b-a-1}} -
\frac{(q v)^{1/2-a}}{(q^{a+1} u v )_{b-a-1}}
\right)
\,.
\end{equation}
In particular, for $a=1$ and $u=1$ we obtain
\begin{equation}
\sum_{1 \le i < b} \frac{(q v)^{1/2-i}}
{(q)_{i-1} (q^i v)_{b-i}}
=
\frac{1}{1-q v}
\frac{(q v)^{3/2-b}}{(q)_{b-2}} \,.
\end{equation}
\end{lem}
\begin{proof} Follows from the following particular
case of the $q$-Gauss summation formula \eqref{69}: for any $c$ and $d$ one
has
\begin{equation}
\sum_{i=c}^{\infty} (q v)^{1/2 - i}
\frac{(q^d u v)_{i-c}}{(q^d u)_{i-c}}=
-(q v)^{3/2-c} \frac{1-q^{d-1} u}{1- q v} \,.
\end{equation}
\end{proof}
In the rest of the section we shall write $a\ll b$ if $a+1<b$.
\begin{lem}\label{lem85} We have
\begin{align}
&H(t_1,\dots,qt_k,\dots,t_n)= \notag \\
&\quad=
\sum_{i_1<\dots < i_{k} \ll i_{k+1} < \dots < i_n}
q^{1/2-i_k} (1-q^{i_k} t_1 \cdots t_{k}) t_{k+1} \cdots t_n
\ex{i_1,\dots,i_n}{t_1,\dots,t_n}\label{88} \\
&\quad=
\sum_{i_1<\dots < i_{k-1} \ll i_{k} < \dots < i_n}
q^{3/2-i_k} (1-q^{i_k-1} t_1 \cdots t_{k-1}) t_k \cdots t_n
\ex{i_1,\dots,i_n}{t_1,\dots,t_n} \label{89}
\end{align}
\end{lem}
\begin{proof}
Recall that by definition
$$
H(t_1,\dots,qt_k,\dots,t_n) = \sum_{i_1<\dots<i_n}
\ex{i_1,\dots,i_n}{t_1,\dots,qt_k,\dots,t_n} \,.
$$
We have the two following identities:
\begin{align}
&\ex{i_1,\dots,i_n}{t_1,\dots,qt_k,\dots,t_n} = \notag \\
&\qquad=
q^{1/2-i_k} (1-q^{i_k} t_1 \cdots t_{k}) t_{k+1} \cdots t_n
\ex{i_1,\dots,i_k,i_{k+1}+1,\dots,i_n+1}{t_1,\dots,t_n} \label{810} \\
&\qquad=
q^{1/2-i_k} (1-q^{i_k} t_1 \cdots t_{k-1}) t_k \cdots t_n
\ex{i_1,\dots,i_{k-1},i_{k}+1,\dots,i_n+1}{t_1,\dots,t_n} \label{811}
\end{align}
Replacing in \eqref{810} the indices $i_{k+1}+1,\dots,i_n+1$
by $i_{k+1},\dots,i_n$ we obtain \eqref{88}.
Replacing in \eqref{811} the indices $i_{k}+1,\dots,i_n+1$
by $i_{k},\dots,i_n$ we obtain \eqref{89}.
\end{proof}
\begin{proof}[Proof of proposition \eqref{prop82}]
Consider the general case $1 < k < n$. The two other cases $k=1$ and $k=n$ are
similar. Rewrite the sum \eqref{88} as follows
\begin{multline}\label{812}
\sum_{i_1<\dots < i_{k} \ll i_{k+1} < \dots < i_n}
q^{1/2-i_k} (1-q^{i_k} t_1 \cdots t_{k}) t_{k+1} \cdots t_n
\ex{i_1,\dots,i_n}{t_1,\dots,t_n} \\
=
\sum_{i_1< \dots < i_n} (\dots) -
\sum_{\stackrel{i_1<\dots < i_{k} \ll i_{k+2} < \dots < i_n}{
i_{k+1}=i_k+1}} (\dots)\,,
\end{multline}
and denote the two summands in the RHS of \eqref{812} by $\s1$ and $\s2$. The
formula \eqref{88} together with the following identity
\begin{multline*}
q^{1/2-i_k} (1-q^{i_k} t_1 \cdots t_{k}) t_{k+1} \cdots t_n
\ex{i_1,\dots,i_k,i_k+1,i_{k+2},\dots,i_n}{t_1,\dots,t_n} = \\
q^{1/2-i_k} (1-q^{i_k} t_1 \cdots t_{k+1}) t_{k+2} \cdots t_n
\ex{i_1,\dots,i_{k-1},i_k,&i_{k+2},\dots,i_n}{t_1,\dots,t_{k-1},t_k t_{k+1},&t_{k+2},
\dots,t_n}
\end{multline*}
implies that
$$
\s2=H(t_1,\dots,q t_k t_{k+1}, \dots, t_n) \,.
$$
Set
\begin{align}
\s3:&=\s1 + q^{1/2} t_1\cdots t_n H(t_1,\dots,t_n) \notag \\
&=t_{k+1}\cdots t_n \sum_{i_1<\dots<i_n} q^{1/2-i_k} {\ex{i}{t} }\ \,. \label{814}
\end{align}
Using Lemma \ref{lem84} we can evaluate one of the
nested sums in \eqref{814} as follows:
\begin{multline*}
\sum_{i_{k-1} < i_k < i_{k+1}}
\frac {(qt_k)^{1/2-i_k}}
{(q^{i_{k-1}} t_1 \cdots t_{k-1})_{i_k-i_{k-1}}
(q^{i_{k}} t_1 \cdots t_{k})_{i_{k+1}-i_{k}}} \\=
\frac1{1-q t_k}
\left(
\frac{(qt_k)^{3/2-i_{k+1}}}
{(q^{i_{k-1}} t_1 \cdots t_{k-1})_{i_{k+1}-i_{k-1}-1}} -
\frac{(qt_k)^{1/2-i_{k-1}}}
{(q^{i_{k-1}+1} t_1 \cdots t_{k})_{i_{k+1}-i_{k-1}-1}}
\right) \,.
\end{multline*}
Then sum $\s3$ becomes the following difference
\begin{align*}
&\frac{t_k \cdots t_n}{1-q t_k}
\sum_{\dots<i_{k-1} \ll i_{k+1} < \dots}
q^{3/2-i_{k+1}} (1- q^{i_{k+1}-1}t_1\cdots t_{k-1})
\ex{\dots,i_{k-1},i_{k+1},\dots}{
\dots,t_{k-1}, t_k t_{k+1},\dots}
\\
&-\frac{t_{k+1} \cdots t_n}{1-q t_k}
\sum_{\dots<i_{k-1} \ll i_{k+1} < \dots}
q^{1/2-i_{k-1}} (1- q^{i_{k-1}}t_1\cdots t_{k})
\ex{\dots,i_{k-1},i_{k+1},\dots}{
\dots,t_{k-1}t_k, t_{k+1}, \dots} \,.
\end{align*}
By Lemma \ref{lem85} this yields
\begin{multline*}
\s3= \frac1{1-qt_k} H(t_1,\dots,t_{k-1},q t_k t_{k+1}, \dots, t_n)
\\
- \frac1{1-qt_k} H(t_1,\dots,q t_{k-1}t_k, t_{k+1}, \dots, t_n) \,.
\end{multline*}
Thus,
\begin{align*}
H(t_1,\dots,qt_k,\dots,t_n)=& \s3 -
q^{1/2} t_1 \cdots t_n H(t_1,\dots,t_n) -\s2 \\
=&-q^{1/2} t_1 \cdots t_n H(t_1,\dots,t_n)\\
&+ \frac{q t_k}{1-q t_k} H(t_1,\dots,t_{k-1},
qt_k t_{k+1}, t_{k+2},\dots,t_n) \\
&- \frac{1}{1-q t_k} H(t_1,\dots,t_{k-2},
qt_{k-1} t_{k}, t_{k+1},\dots,t_n) \,.
\end{align*}
This concludes the proof.
\end{proof}
\begin{proof}[Proof of corollary \ref{cor83}]
We will prove \eqref{82}; equation \eqref{83} follows easily from \eqref{82}.
Substitute the formula from proposition \ref{prop82} into \eqref{73} and consider the
coefficient of a summand
$$
H(\dots, q t_1 t_j, \dots)\,, \quad j=2,\dots,n \,.
$$
Such a summand arises from the expansion of
$$
H(\dots, q t_1, t_j, \dots)\quad\text{\rm\ and}
\quad H(\dots, t_j, q t_1, \dots) \,.
$$
The sum of the coefficients equals
$$
\frac{q t_1}{1- q t_1} -
\frac{1}{1- q t_1} = -1 \,.
$$
This proves \eqref{82}.
\end{proof}
\begin{prop}\label{prop86}
\begin{equation}\label{815}
F(q t_1, t_2,\dots,t_n)= -q^{1/2} t_1\cdots t_n \left(
\sum_{\pi\in\Pt n} G^\pi(t) \right) \,.
\end{equation}
\end{prop}
\begin{proof}
Substitute \eqref{83} into \eqref{76}. Let $c_\pi$ denote the
coefficient of the summand $G^\pi(t)$ in this
expansion. It is clear that
$$
c_\pi = -q^{1/2} t_1\cdots t_n \,, \pi \in \Pt n \,.
$$
Suppose that $\pi \notin \Pt n$ and show that $c_\pi=0$.
Since $F(q t_1, t_2,\dots,t_n)$
is symmetric in variables $t_2,\dots,t_n$
we can assume that
$$
\pi=\{\{1,2,\dots,m\},\pi_2,\dots,\pi_l\}\,, \quad 1<m\le n\,,
$$
where $\{\pi_2,\dots,\pi_l\}$ is a partition of the set
$\{m+1,\dots,n\}$.
Let $\sigma$ be a partition of $\{1,\dots,m\}$
$$
\sigma=\{\sigma_1,\dots,\sigma_s\} \in \P m \,.
$$
Set
$$
\pi\land\sigma=\{\sigma_1,\dots,\sigma_s,\pi_2,\dots,\pi_l\} \in \P n \,.
$$
It is easy to see that the term $G^\pi(t)$ arises precisely
from the expansion of the terms of the form
$$
G^{\pi\land\sigma} (qt_1,t_2,\dots,t_n)\,, \quad \sigma\in\P m \,.
$$
By \eqref{83} it arises with the coefficient
$$
q^{1/2} t_1\cdots t_n (-1)^{\ell(\sigma)} (\ell(\sigma)-1)! \,.
$$
By \eqref{711}, the sum of these coefficients over all
$\sigma\in\P m$ equals zero.
This concludes the proof.
\end{proof}
\begin{proof}[Proof of theorem \ref{thm81}]
We will show that the RHS in theorem \ref{thm81} equals the RHS of \eqref{815}. Substitute
\eqref{76} into RHS of theorem \ref{thm81} and let $c_\pi$ denote the
coefficient of the summand $G^\pi(t)$ in this
expansion. Again, it is clear that
$$
c_\pi = -q^{1/2} t_1\cdots t_n \,, \pi \in \Pt n \,.
$$
and we have to show that
$$
c_\pi = 0 \,, \pi \notin \Pt n \,.
$$
Again, we can suppose that
$$
\pi=\{\{1,2,\dots,m\},\pi_2,\dots,\pi_l\}\,, \quad 1<m\le n\,,
$$
where $\{\pi_2,\dots,\pi_l\}$ is a partition of the set
$\{m+1,\dots,n\}$.
For any set $S$ denote by $\text{Atom}(S)$ the partition of $S$ into
one-element blocks. The summand $G^\pi(t)$ arises in the RHS of
from the expansion of terms
$$
F^{\left\{\{S\},\text{Atom}(\{1,\dots,n\}\setminus S)\right\}}(t)\,,
\quad 1\in S \subset \{1,\dots,m\}\,,
$$
and therefore the coefficient $c_\pi$ equals
\begin{align*}
c_\pi&=-q^{1/2} t_1\cdots t_n \sum_{1\in S \subset \{1,\dots,m\}} (-1)^{|S|+1} \\
&=-q^{1/2} t_1\cdots t_n \sum_{k=0}^{m-1} (-1)^k \binom{m-1}k \\
&=0 \,.
\end{align*}
This concludes the proof.
\end{proof}
\section{Singularities of correlation functions}\label{sec:sing}
In this section we consider the singularities of the
function $F(t_1,\dots,t_n)$. The series \eqref{76} converges
uniformly on compact subsets of the set
$$
\left\{|t_i|>1\,, i=1,\dots,n\right\}\setminus
\bigcup_{S\subset\{1,\dots,n\}} \bigcup_{m=1}^\infty
\left\{q^m \prod_{k\in S} t_k =1 \right\}
$$
and has simple poles on the divisors
$$
\left\{q^m \prod_{k\in S} t_k =1 \right\}\,,
\quad S\subset\{1,\dots,n\} \,.
$$
The difference equation from theorem \ref{thm81} gives a meromorphic
continuation of the function $F(t)$ onto
(a double cover of) the domain $({\Bbb C}\setminus 0)^n$.
By symmetry, it suffices to consider the divisors
\begin{equation}\label{91}
q^m t_1 t_2 \dots t_k = 1\,, \quad k\le n,\quad m=1,2,\dots \,.
\end{equation}
We shall prove the following
\begin{thm}\label{thm91} We have
\begin{equation}\label{92}
F(t_1,\dots,t_n)=(-1)^{m}
\frac{q^{m^2/2} m^{k-1}}{q^m t_1\dots t_k-1}
\frac{F(t_{k+1},\dots,t_n)}{(t_{k+1}\dots t_n)^m}+ \dots \,,
\end{equation}
where dots stand for terms regular at the divisor \eqref{91} and we
assume that
$$
F(t_{k+1},\dots,t_n)=1\,,\quad k=n\,.
$$
\end{thm}
A curious property of \eqref{92} is that the residue
\begin{equation}\label{93}
\operatorname{Res}_{\{q^m t_1 t_2 \dots t_k = 1\}} F(t_1,\dots,t_n)
\end{equation}
does not depend on the variables
$t_1,\dots,t_k$.
Introduce the following notation. Given a function
$f(i_1,\dots,i_n)$ set
$$
\widetilde{\sum_{1\le i_1 \le \dots \le i_k \le m}} f(i_1,\dots,i_n) =
\sum_{1\le i_1 \le \dots \le i_n \le m}
\frac{f(i_1,\dots,i_n)}
{\# \operatorname{Stab}_{S(n)} \{i_1,\dots,i_n\}} \,.
$$
If the function $f$ is symmetric, we have
\begin{equation}\label{94}
\widetilde{\sum_{1\le i_1 \le \dots \le i_k \le m}} f(i_1,\dots,i_n) = \frac 1{n!} \sum_{i_1,\dots,i_n=1}^m
f(i_1,\dots,i_n) \,.
\end{equation}
With this notation we have
$$
F(t)=
\sum_{\stackrel{\text{\rm over all permutations}}{
\text{\rm of $t_1,\dots,t_n$}}}
\,\,
\widetilde{\sum_{1\le i_1 \le \dots \le i_n}}
\ex{i}{t} \,.
$$
It is clear that the evaluation of the residue \eqref{93} boils down to
the evaluation of the following sum
\begin{multline}\label{95}
\sum_{\stackrel{\text{\rm permutations}}{\text{\rm of $t_1,\dots,t_k$}}} \\
\widetilde{\sum_{1\le i_1 \le \dots \le i_k \le m}} \frac{t_1^{1/2-i_1} \dots t_k^{1/2-i_k}}
{(q)_{i_1-1} (q^{i_1} t_1)_{i_2-i_1} (q^{i_2} t_1 t_2)_{i_3-i_2} \dots
(q^{i_k} t_1 \dots t_k)_{m-i_k}} \,,
\end{multline}
where the variables $t_1,\dots,t_k$ are subject to
constraint
$$
q^m t_1 \cdots t_k =1 \,.
$$
This evaluation follows from a curious rational function identity
which shall now be established. Set
$$
\cbm {i_1,\dots,i_k}{t_1,\dots,t_k}:=
\frac1
{(q)_{i_1-1} (q^{i_1} t_1)_{i_2-i_1} (q^{i_2} t_1 t_2)_{i_3-i_2} \dots
(q^{i_k} t_1 \dots t_k)_{m-i_k}} \,.
$$
Then we have the following
\begin{thm}\label{thm92} Let $m,k\in{\Bbb N}$ and suppose that
$$
q^m t_1 t_2 \dots t_k = 1\,.
$$
Then
\begin{equation}\label{96}
\scy\quad \widetilde{\sum_{1\le i_1 \le \dots \le i_k \le m}} \cbm {i_1,\dots,i_k}{t_1,\dots,t_k} =
\frac{m^{k-1}}{(k-1)!} \,.
\end{equation}
In particular, this sum is independent of variables $q,t_1,\dots,t_k$.
\end{thm}
\begin{proof}[Proof of Theorem \ref{thm92}]
Let us use $q,t_1,\dots,t_{k-1}$ as independent variables.
First, show that LHS of \eqref{96} does not depend on $t_1,\dots,t_{k-1}$. Since
it is a rational function bounded on infinity it suffices
to show that it does not have any poles. The only poles it can
possibly have are simple poles on divisors
$$
t_s t_{s+1} \dots t_{r-1} t_r = q^{-l}\,, \quad s\le r\le k-1,\,
1\le l \le m \,.
$$
By symmetry, it suffices to consider the pole on
\begin{equation}\label{97}
t_1 \dots t_r = q^{-l}\,, \quad \quad r\le k-1,\,
1\le l \le m \,.
\end{equation}
This pole arises from two type of summands in \eqref{96}. First, it
arises from the summands
\begin{equation}\label{98}
\cbm {i_1,\dots,i_k}{t_1,\dots,t_k}\,, \quad i_r \le l < i_{r+1}\,,
\end{equation}
and it also arises from summands
\begin{equation}\label{99}
\cbm {i_1,\dots,i_k}{t_{r+1},\dots,t_k,t_1,\dots,t_r}\,,
\quad i_{k-r} \le m-l < i_{k-r+1}\,.
\end{equation}
We shall match each summand of the form \eqref{98} to a
summand of the form \eqref{99} in such a way that the poles
in each pair will cancel out. Namely, it is easy to see
that the sum
\begin{equation}\label{910}
\cbm {i_1,\dots,i_k}{t_1,\dots,t_k} \,+\,
\cbm {i_{r+1}-l,\dots,i_k-l,i_1+m-l,\dots,i_r+m-l}
{t_{r+1},\dots,t_k,t_1,\dots,t_r}\,,
\end{equation}
where
$$
1\le i_1\le \dots \le i_r \le l < i_{r+1}\le \dots \le i_k \le m\,,
$$
is regular on the divisor \eqref{97}; notice that the
inequalities
$$
1\le i_1 \le \dots \le i_r < i_{r+1} \le \dots \le i_k \le m
$$
imply that
$$
\#\operatorname{Stab}\{i_1,\dots,i_k\}=
\#\operatorname{Stab}\{i_{r+1}-l,\dots,i_k-l,i_1+m-l,\dots,i_r+m-l\}\,.
$$
This proves that the LHS of \eqref{96} is regular on \eqref{97}.
Therefore, the LHS of \eqref{96} does not depend on $t_1,\dots,t_k$
and we can substitute
$$
t_1=t_2=\dots=t_{k-1}=1\,,\quad t_k=q^{-m} \,.
$$
Then the LHS of \eqref{96} becomes
\begin{equation}\label{911}
\widetilde{\sum_{1\le i_1 \le \dots \le i_k \le m}} \,\,
\sum_{r=1}^k \frac1{(q)_{i_r} (q^{i_r-m})_{m-i_r}} \,.
\end{equation}
Since the inner sum in \eqref{911} is symmetric in $i_1,\dots,i_k$
we can use \eqref{94} to obtain
\begin{align*}
&\frac1{k!} \sum_{i_1,\dots,i_k=1}^m \sum_{r=1}^k
\frac1{(q)_{i_r} (q^{i_r-m})_{m-i_r}} \notag\\
&= \frac{m^{k-1}}{(k-1)!} \sum_{i=1}^m
\frac1{(q)_{i} (q^{i-m})_{m-i}} \notag\\
&= \frac{m^{k-1}}{(k-1)!} \frac1{(q^{1-m})_{m-1}}
\sum_{i=1}^m
\frac{(q^{1-m})_{i-1}}{(q)_{i}} \notag\\
&=\frac{m^{k-1}}{(k-1)!} \,,\notag
\end{align*}
where the last equality follows from the q-binomial theorem.
This concludes the proof.
\end{proof}
Replacing in \eqref{96} the variables $q,t_1,\dots,t_k$ by their
reciprocals one obtains the following
\begin{cor}\label{cor93} Suppose that $q^m t_1 t_2 \dots t_k = 1$. Then
the sum
\begin{equation}\label{912}
\scy
\widetilde{\sum_{1\le i_1 \le \dots \le i_k \le m}} \frac{t_1^{1/2-i_1} \dots t_k^{1/2-i_k}}
{(q)_{i_1-1} (q^{i_1} t_1)_{i_2-i_1} (q^{i_2} t_1 t_2)_{i_3-i_2} \dots
(q^{i_k} t_1 \dots t_k)_{m-i_k}}
\end{equation}
equals
$$
(-1)^{m-1} \frac{q^{m^2/2} m^{k-1}}{(k-1)!}\,,
$$
for all $m$ and $k$.
\end{cor}
This corollary gives the evaluation of the sum \eqref{95} and
this proves \eqref{92}
\section{Difference equations for the RHS in \eqref{64}}\label{sec:dif}
In this section we show that the functions $U(t_1,\dots,t_n)$ satisfy the
very same difference equations as the functions $F(t)$ do.
Since we shall use only the difference equation \eqref{62} for
the theta function $\Theta(x)$, let us consider following
general situation.
Let $t_0$ be an auxiliary variable. We shall eventually
let
$$
t_0\to 1 \,.
$$
Modify temporarily the definition \eqref{77a} as
follows
\begin{equation}\label{101}
{\upsilon}(\gamma)_k=\{0\}\cup \gamma_1 \cup \dots \cup \gamma_{k-1}\,.
\end{equation}
Suppose that a function
$$
r(x;m)\,, \quad m=0,1,2,\dots\,,
$$
satisfies the two following properties:
\begin{equation}\label{102}
r(x;0)=1
\end{equation}
identically and
\begin{equation}\label{103}
r(qx;m)=\sum_{i=0}^m (-1)^i \binom mi\, r(x;m-i) \,.
\end{equation}
Consider the function
\begin{equation}\label{104}
R(t_1,\dots,t_n\,|\, t_0)=\sum_{\gamma\in{\Bbb G} n} (-1)^{\gamma} \prod_{k=1}^{\ell(\g)}
r\left({{\textstyle \prod}}_{i\in{\upsilon}(\gamma)_{k}} t_i \,; \#\gamma_k
\right) \,,
\end{equation}
where we use the modified definition \eqref{101}. We want to show that
\begin{thm}\label{thm101} Suppose a function $r(x;m)$ satisfies \eqref{102}
and \eqref{103}. Then
\begin{equation}\label{105}
R(qt_1,t_2,\dots,t_n\,|\, t_0)= \sum_{\pi\in\Po n}
(-1)^{\pi} R^\pi(t_1,\dots,t_n\,|\, t_0) \,,
\end{equation}
where the function $R(t_1,\dots,t_n\,|\, t_0)$ was defined in \eqref{104}
\end{thm}
Here, as usual,
\begin{equation}\label{106}
R^\pi(t_1,\dots,t_n\,|\, t_0):= R\left(\left.\prod_{k\in\pi_1} t_k\,, \dots ,
\prod_{k\in\pi_{\ell(\pi)}} t_k \,\right|\, t_0 \right)\,, \quad \pi\in\P n \,.
\end{equation}
\begin{proof} Given a composition
$$
\gamma=(\gamma_1,\dots,\gamma_l)\,,
$$
let $m=m(\gamma)$ denote the number of the block that contains 1.
Substitute \eqref{103} into \eqref{104}.
We obtain
\begin{equation}\label{107}
R(qt_1,t_2,\dots,t_n\,|\, t_0)=
\sum_{\gamma\in{\Bbb G} n}
\sum_{s_{m+1}=0}^{\#\gamma_{m+1}}
\dots
\sum_{s_l=0}^{\#\gamma_{l}}
\sqb
{\gamma_{m+1},\dots,\gamma_l}
{s_{m+1},\dots,s_l}\,,
\end{equation}
where
\begin{equation}\label{108}
\sqb
{\gamma_{m+1},\dots,\gamma_l}
{s_{m+1},\dots,s_l}
\end{equation}
stands for the following product
\begin{multline*}
(-1)^{n+\ell(\g)+s_{m+1}+\dots+s_{l}}
\prod_{k=1}^{m}
r \left({{\textstyle \prod}}_{i\in{\upsilon}(\gamma)_{k}} t_i\,; \#\gamma_k
\right)\times \\
\prod_{k=m+1}^{l} \binom {\#\gamma_i}{s_i}
r \left({{\textstyle \prod}}_{i\in{\upsilon}(\gamma)_{k}} t_i\,; \#\gamma_k - s_i
\right)
\,.
\end{multline*}
Let us divide all summands \eqref{108} into 3 following types
according to
occurrence of certain patters in the sequence $s_{m+1},\dots,s_l$.
The summands of the form
\begin{equation}\label{109}
\sqb
{\gamma_{m+1},&\dots,&\gamma_k,&\dots}
{\#\gamma_{m+1},&\dots,&\#\gamma_k,&0,\dots,0}\,, \quad m+1\le k \le l \,,
\end{equation}
will be called type I summands.
If \eqref{108} is not of type I then let $k$, $m+1\le k \le l$,
be the minimal number such that
$$
\big(0 < s_k < \#\gamma_k\big)\quad \text{\rm\ or}\quad
\big(s_k=0\text{\rm\ and }s_{k+1}=\#\gamma_{k+1}\big) \,.
$$
We shall say that \eqref{108} is of type II (type III) if the
first (second) parenthesis contains a true statement.
We shall show that the the type II summands cancel with
the type III summand while the type I summands produce
the RHS of \eqref{105}.
The cancelation of the type II and type III summands
follows from the following identity. Suppose
that
$$
0 < s_k < \#\gamma_k\,.
$$
Then
\begin{multline}\label{1010}
\sqb{\dots,\gamma_k,\dots}
{\dots,s_k,\dots} + \\
\sum_{\stackrel{\delta \subset \gamma_k}{
\#\delta=s_k}}
\left[\begin{matrix} \gamma_1,\dots,\gamma_m\\{}\end{matrix}
\left|\begin{matrix}
\dots,&\gamma_k\setminus\delta,&\delta,&\dots \\
\dots,&0,&\#\delta,&\dots
\end{matrix} \right.
\right]
= 0 \,.
\end{multline}
To see \eqref{1010} notice that
all $\binom {\gamma_k}{s_k}$ summands in the sum over subsets
$\delta\subset \gamma_k$ are equal and proportional
to the first summand in \eqref{1010}.
Now consider a type I summand \eqref{109}. Set
\begin{equation}\label{1011}
\delta=\gamma_{m+1}\cup \dots \cup \gamma_k \,.
\end{equation}
We want to fix a subset
$$
\delta\subset \{2,\dots,n\}
$$
and compute the sum of
all type I summands \eqref{109} satisfying \eqref{1011}.
Let us consider the nontrivial case
$$
\delta\ne\emptyset \,.
$$
First sum over all
$$
(\gamma_{m+1}, \dots, \gamma_k)\in\Gamma(\delta)\,,
$$
where $\Gamma(\delta)$ stands for the set of all compositions of
the set $\delta$.
We have
\begin{multline}\label{1012}
\sqb
{\gamma_{m+1},&\dots,&\gamma_k,&\gamma_{k+1},\dots}
{\#\gamma_{m+1},&\dots,&\#\gamma_k,&0,\dots,0}=\\
(-1)^{k-m+1} \sqb
{\delta &\gamma_{k+1},\dots}
{\#\delta,&0,\dots,0} \,.
\end{multline}
Since \eqref{1012} depends only on the parity of number of parts
in the composition \eqref{1011} we can use \eqref{710}
to obtain
\begin{multline*}
\sum_{
(\gamma_{m+1},\dots,\gamma_k)\in\Gamma(\delta)}
\sqb
{\gamma_{m+1},&\dots,&\gamma_k,&\gamma_{k+1},\dots}
{\#\gamma_{m+1},&\dots,&\#\gamma_k,&0,\dots,0} = \\
(-1)^{\#\delta+1}
\sqb
{\delta, &\gamma_{k+1},\dots}
{\#\delta,&0,\dots,0} \,.
\end{multline*}
Now sum over the remaining blocks
$$
(\gamma_1,\dots,\gamma_{m},\gamma_{k+1},\dots,\gamma_l)\,.
$$
It follows from the definition \eqref{104}, \eqref{106} that
\begin{multline*}
(-1)^{\#\delta+1}
\sum_{
(\gamma_1,\dots,\gamma_{m},\gamma_{k+1},\dots,\gamma_l)}
\sqb
{\delta, &\gamma_{k+1},\dots}
{\#\delta,&0,\dots,0}
\\=
(-1)^{n+\ell(\sigma)} R^{\sigma} (t\,|\, t_0) \,,
\end{multline*}
where $\sigma=\sigma(\delta)$ is the following element of $\Po n$
$$
\sigma=\{{1\cup \delta}, \text{Atom}(\{2,\dots,n\}\setminus \delta)\} \,.
$$
Recall that $\text{Atom}(\{2,\dots,n\}\setminus \delta)$ denotes the
partition of the set $\{2,\dots,n\}\setminus \delta$ into
1-element blocks.
Thus, the sum of all type I summands in \eqref{107}
equals the RHS of \eqref{105}. This concludes the proof of
the theorem.
\end{proof}
\begin{cor}\label{cor102}
\begin{equation}\label{1014}
T(qt_1,t_2,\dots,t_n)= \sum_{\pi\in\Po n}
(-1)^{n+\ell(\pi)} T^\pi(t_1,\dots,t_n) \,.
\end{equation}
\end{cor}
\begin{proof}
Take
$$
r(x;m)=\frac{\Theta^{(m)}(x)}{\Theta(x)} \,.
$$
By virtue of \eqref{62} this function satisfies \eqref{103} and
it obviously satisfies \eqref{102}. We have
$$
T(t_1,\dots,t_n)=\operatorname{Res}_{\,t_0=1} R(t_1,\dots,t_n\,|\, t_0) \,.
$$
Taking the residue in \eqref{105} we obtain \eqref{1014}.
\end{proof}
\begin{cor}\label{cor103}
\begin{equation}\label{1015}
{U(qt_1,t_2,\dots,t_n)} = - q^{1/2}
t_1 \cdots t_n \left( \sum_{\pi\in\Po n}
(-1)^{n+\ell(\pi)} {U^\pi(t_1,\dots,t_n)}
\right) \,.
\end{equation}
\end{cor}
\section{Singularities of the RHS in \eqref{64}}\label{sec:Using}
In this section we shall prove that $U(t_1,\dots,t_n)$
has exactly the same singularities as $F(t_1,\dots,t_n)$.
\begin{thm}\label{thm111} For $k=1,\dots,n$ we have
\begin{equation}\label{111}
U(t_1,\dots,t_n)=(-1)^{m}
\frac{q^{m^2/2} m^{k-1}}{q^m t_1\cdots t_k-1}
\frac{U(t_{k+1},\dots,t_n)}{(t_{k+1}\cdots t_n)^m}+ \dots \,,
\end{equation}
where dots stand for terms regular at the divisor
$$
q^m t_1\cdots t_k=1
$$
and we
assume that
$$
U(t_{k+1},\dots,t_n)=1\,,\quad k=n\,.
$$
\end{thm}
Let us again point out that \eqref{111} implies that the
residue
$$
\operatorname{Res}_{q^m t_1\dots t_k=1} U(t_1,\dots,t_n)
$$
is independent of $t_1,\cdots,t_k$.
In the proof we shall use a curious identity good for
any odd smooth function which we shall state as a
separate Theorem \ref{thm113}.
It is clear that since the function $U(t)$ satisfies the
very same difference equation as $F(t)$ does it
suffices to consider the case
$$
m=0 \,.
$$
That is, we have to show that
\begin{equation}\label{112}
U(t_1,\dots,t_n)=\frac1{t_1-1} U(t_2,\dots,t_n)+ \dots\,,
\quad t_1\to 1 \,,
\end{equation}
and that
\begin{equation}\label{113}
\text{\rm
$U(t)$ is regular on $\{t_1\cdots t_k=1\}$ for
$1<k\le n$
}
\end{equation}
It follows from the very definition of the
function $U(t)$ (look, for example, at the
formula \eqref{66}) that
\begin{align*}
U(t_1,\dots,t_n)&=U(t_1,\dots,t_k) U(t_{k+1}\dots t_n) + \dots \\
&=\frac{T(t_1,\dots,t_k)}{t_1 \cdots t_k -1}
U(t_{k+1}\dots t_n) + \dots\,,
\end{align*}
where dots stand for terms regular on the divisor
$$
t_1\cdots t_k=1 \,.
$$
Since
$$
T(1)=1 \,.
$$
the Theorem 4.1 will follow from the following
\begin{thm}\label{thm112} We have
\begin{equation}\label{114}
T(t_1,\dots,t_n)\Big|_{t_1\cdots t_n=1}=0
\end{equation}
provided $n>1$.
\end{thm}
\begin{proof}
Till the end of the proof the variables $t_1,\dots,t_n$ will be
always subject to constraint
\begin{equation}\label{115}
t_1\cdots t_n=1\,.
\end{equation}
Induct on $n$. Since
$$
T(t_1,t_2)=
\frac{\Theta'(t_1)}{\Theta(t_1)} + \frac{\Theta'(t_2)}{\Theta(t_2)}
$$
the case $n=2$ is clear.
First, show that
\begin{equation}\label{116}
\text{\rm $T(t_1,\dots,t_n)$ is a constant.}
\end{equation}
By the difference equation \eqref{1014} and the induction hypothesis
$$
T(q t_1, t_2,\dots,t_n)=T(t_1,\dots,t_n)+(-1)^{n+1} \,.
$$
Similarly
$$
T(q t_1, t_2,\dots,t_n)=T(q t_1, q^{-1}t_2,\dots,t_n)+(-1)^{n+1} \,.
$$
It follows that
$$
T(q t_1, q^{-1}t_2,\dots,t_n)=T(t_1,\dots,t_n) \,.
$$
Hence, by virtue of the strategy enunciated at the beginning of section \ref{sec:not}, the claim
\eqref{116} will follow from the following claim
\begin{equation}\label{117}
\text{\rm $T(t_1,\dots,t_n)$ is regular}\,,
\end{equation}
which will now be established.
By symmetry it suffices to show that $T(t_1,\dots,t_n)$ is
regular on the divisor
\begin{equation}\label{118}
t_1 \dots t_k = 1\,, \quad 1\le k < n-2 \,.
\end{equation}
Again, from the definition \eqref{78} it is clear that
\begin{multline}\label{119}
T(t_1,\dots,t_n)=\\
\left(\frac1{\Theta(t_1\dots t_k)} +
\frac1{\Theta(t_{k+1}\dots t_n)}
\right)T(t_1\dots t_k)T(t_{k+1}\dots t_n) + \dots
\end{multline}
where dots stand for a function regular on \eqref{118}.
By \eqref{115} the sum in parentheses in \eqref{119} vanishes
and this proves \eqref{116} and \eqref{117}.
Thus, what is left is to show that LHS of \eqref{114}
vanishes at some point. This follows from the
general identity established below in Theorem \ref{thm113} where one
has to
substitute
$$
f(x)=\Theta(x) \,.
$$
This concludes the proof of the theorem.
\end{proof}
Let $f(x)$ be an odd function
$$
f(x^{-1})=-f(x) \,.
$$
Set
$$
\p{k}(x):= \left(x\d x\right)^k f(x)\,, k\in{\Bbb N} \,.
$$
Consider the following function
\begin{equation}\label{1110}
\Phi(t_1,\dots,t_n):=\sum_{\gamma\in{\Bbb G} n} \Psi_\gamma (t) \,,
\end{equation}
where
$$
\Psi_\gamma(t)=(-1)^{\ell(\g)} \p{\#\gamma_1}(1)\, \prod_{i=2}^{\ell(\g)}
\frac{
\p{\#\gamma_i}\left({\tprod}_{j\in\ups(\g)_i}t_j\right)}
{f\left({\tprod}_{j\in\ups(\g)_i}t_j\right)} \,.
$$
Note that $\Psi_\gamma(t)=0$ if $\#\gamma_1$ is even. We want to prove
the following
\begin{thm}\label{thm113} Suppose that $n>1$ and the variables $t_1,\dots,t_n$ are subject
to constraint
$$
t_1 \cdots t_n = 1 \,.
$$
Then for any odd function $f(x)$ with a simple zero at $x=1$
$$
f(1)=0\,, \quad f'(1)\ne 0
$$
we have
\begin{equation}\label{1111}
\Phi(t_1,\dots,t_n)\to 0\,, \quad t_1 \to 1 \,,
\end{equation}
where the function $\Phi(t)$ was defined in \eqref{1110}.
\end{thm}
\begin{proof}
Let us divide all summands in \eqref{1110} into 3 following types:
\begin{enumerate}
\item $\gamma_{\ell(\g)}=\{1\}$\,,
\item $\gamma_1=\{1\}$\,,
\item others.
\end{enumerate}
Note that all type 3 summands are regular at $t_1=1$.
Let $\gamma=(\gamma_1,\dots,\gamma_{\ell-1},\{1\})$ be a type 1 composition.
Then the composition
$$
\gamma'=(\{1\},\gamma_1,\dots,\gamma_{\ell-1})
$$
is of type 2. Consider the sum
\begin{equation}\label{1112}
\Psi_\gamma (t) + \Psi_{\gamma'} (t) \,.
\end{equation}
There are two possible cases: $\#\gamma_1$ is even and $\#\gamma_1$ is odd.
If
$$
\text{\rm $\#\gamma_1$ is even}
$$
then the first summand in \eqref{1112} is zero
and the second one is regular at $t_1=1$. In this case
we obtain
\begin{equation}\label{1114}
\lim_{t_1\to 1} \left(\Psi_\gamma (t) + \Psi_{\gamma'} (t)
\right) =
(-1)^{\ell} \p{\#\gamma_1+1}(1)\,\, \prod_{i=2}^{\ell-1}
\frac{
\p{\#\gamma_i}\left({\tprod}_{j\in\ups(\g)_i}t_j\right)}
{f\left({\tprod}_{j\in\ups(\g)_i}t_j\right)} \,.
\end{equation}
It is easy to see that \eqref{1114} will exactly cancel with
the type 3 summand corresponding to the
composition
\begin{equation}\label{1115}
(\{1\}\cup\gamma_1,\dots,\gamma_{\ell-1}) \,.
\end{equation}
Note that if $\#\gamma_1$ is odd then the contribution of
the composition \eqref{1115} is zero.
Now consider the sum \eqref{1112} in the case
$$
\text{\rm $\#\gamma_1$ is odd}\,.
$$
We have
\begin{alignat*}{2}
\Psi_\gamma(t)&=
(-1)^{\ell+1} \p{\#\gamma_1}(1)\,\frac{f'(t_1)}{f(t_1)}
&&\prod_{i=2}^{\ell-1}
\frac{
\p{\#\gamma_i}\left({\tprod}_{j\in\ups(\g)_i}t_j\right)}
{f\left({\tprod}_{j\in\ups(\g)_i}t_j\right)}
\\
\Psi_{\gamma'}(t)&=
(-1)^{\ell} f'(1) \,\frac{\p{\#\gamma_1}(t_1)}{f(t_1)}
&&\prod_{i=2}^{\ell-1}
\frac{
\p{\#\gamma_i}\left(t_1\cdot {\tprod}_{j\in\ups(\g)_i}t_j\right)}
{f\left(t_1 \cdot{\tprod}_{j\in\ups(\g)_i}t_j\right)}
\end{alignat*}
Observe that
\begin{equation}\label{1117}
f'(1) \,\, \frac{\p{\#\gamma_1}(t_1)}{f(t_1)} -
\p{\#\gamma_1}(1)\,\, \frac{f'(t_1)}{f(t_1)} \to 0\,,
\quad t_1 \to 1
\end{equation}
because \eqref{1117} is regular and odd. Therefore
\begin{multline*}
\lim_{t_1\to 1} \left(\Psi_\gamma (t) + \Psi_{\gamma'} (t)
\right) = \\
(-1)^\ell
\p{\#\gamma_1}(1)\,\,
\lim_{t_1\to 1} \frac{f'(t_1)}{f(t_1)}
\left(
\prod_{i=2}^{\ell-1}
\frac{
\p{\#\gamma_i}\left(t_1\cdot {\tprod}_{j\in\ups(\g)_i}t_j\right)}
{f\left(t_1 \cdot{\tprod}_{j\in\ups(\g)_i}t_j\right)} - \right.\\
\left.
\prod_{i=2}^{\ell-1}
\frac{
\p{\#\gamma_i}\left({\tprod}_{j\in\ups(\g)_i}t_j\right)}
{f\left({\tprod}_{j\in\ups(\g)_i}t_j\right)}\right) \,.
\end{multline*}
By L'Hospital rule this limit equals
\begin{multline*}
(-1)^\ell \p{\#\gamma_1}(1)
\left(
\sum_{k=2}^{\ell-1}
\left( \frac
{\p{\#\gamma_k+1}\left({{\textstyle \prod}}_{j\in{\upsilon}(\gamma)_k} t_j\right)}
{f\left({{\textstyle \prod}}_{j\in{\upsilon}(\gamma)_k} t_j\right)} -
\right.
\right. \\
-
\left.
\left.
\frac
{\p{\#\gamma_k}\left({{\textstyle \prod}}_{j\in{\upsilon}(\gamma)_k} t_j\right)f'\left({{\textstyle \prod}}_{j\in{\upsilon}(\gamma)_k} t_j\right)}
{f\left({{\textstyle \prod}}_{j\in{\upsilon}(\gamma)_k} t_j\right)^2}
\right)
\prod_{\stackrel{i=2}{i\ne k}}^{\ell-1}
\frac{
\p{\#\gamma_i}\left({\tprod}_{j\in\ups(\g)_i}t_j\right)}
{f\left({\tprod}_{j\in\ups(\g)_i}t_j\right)}
\right) \,.
\end{multline*}
It is easy to see that the summand
$$
(-1)^\ell \p{\#\gamma_1}(1) \,\,
\frac
{\p{\#\gamma_k+1}\left({{\textstyle \prod}}_{j\in{\upsilon}(\gamma)_k} t_j\right)}
{f\left({{\textstyle \prod}}_{j\in{\upsilon}(\gamma)_k} t_j\right)}
\prod_{\stackrel{i=2}{i\ne k}}^{\ell-1}
\frac{
\p{\#\gamma_i}\left({\tprod}_{j\in\ups(\g)_i}t_j\right)}
{f\left({\tprod}_{j\in\ups(\g)_i}t_j\right)}
$$
cancels with the contribution of the type 3
composition
\begin{equation}\label{1118}
(\gamma_1,\dots, \{1\}\cup \gamma_k ,\dots, \gamma_{\ell-1})
\,, \quad 2\le k \le l-1\,.
\end{equation}
Similarly the summand
\begin{multline*}
(-1)^{\ell+1} \p{\#\gamma_1}(1) \,\,
\frac
{\p{\#\gamma_k}\left({{\textstyle \prod}}_{j\in{\upsilon}(\gamma)_k} t_j\right)f'\left({{\textstyle \prod}}_{j\in{\upsilon}(\gamma)_k} t_j\right)}
{f\left({{\textstyle \prod}}_{j\in{\upsilon}(\gamma)_k} t_j\right)^2} \times \\
\prod_{\stackrel{i=2}{i\ne k}}^{\ell-1}
\frac{
\p{\#\gamma_i}\left({\tprod}_{j\in\ups(\g)_i}t_j\right)}
{f\left({\tprod}_{j\in\ups(\g)_i}t_j\right)}
\,.
\end{multline*}
cancels with the contribution of the type 3
composition
\begin{equation}\label{1119}
(\gamma_1,\dots,\{1\},\gamma_k,\dots,\gamma_{\ell-1})
\,, \quad 2\le k \le l-1 \,.
\end{equation}
It is clear that the compositions of the form
\eqref{1115}, \eqref{1118}, and \eqref{1119} exhaust the set of
type 3 compositions. This concludes the proof.
\end{proof}
\section{Conclusion of the proof of Theorem \ref{thm61}}\label{sec:pf}
Induct on $n$. Suppose that $n=1$. By Theorems \ref{thm81} and \ref{thm91}
the function
$$
\Theta(t_1) F(t_1)
$$
is holomorphic on ${\Bbb C}\setminus 0$, invariant
under the transformation
$$
t_1 \mapsto q t_1
$$
and equal to $1$ for $t_1=1$. It follows that
$$
F(t_1)=\frac{1}{\Theta(t_1)} \,.
$$
Suppose that $n>1$ and consider the function
\begin{equation}\label{121}
\Theta(t_1\cdots t_n) F(t_1,\dots,t_n) - T(t_1,\dots,t_n) \,.
\end{equation}
By induction hypothesis, we have
$$
\Theta(t_1\cdots t_n) F^\pi (t_1,\dots,t_n) - T^\pi(t_1,\dots,t_n)\,,
$$
provided
$$
\ell(\pi) < n \,.
$$
Therefore by Theorem 1.1 and Corollary 3.2 the function
is invariant
under the transformation
$$
t_1 \mapsto q t_1\,.
$$
By symmetry it is invariant under all
transformations
$$
t_i \mapsto q t_i\,, \quad i=1,\dots,n \,.
$$
By Theorems \ref{thm91} and \ref{thm111} the function \eqref{121} is holomorphic and vanishes
if
$$
t_1 \cdots t_n = 0 \,.
$$
Therefore \eqref{121} equals zero. This concludes the proof.
\setcounter{section}{12}
\section{Another Example}\label{sec:example}
Another representation of a subalgebra of the algebra $
{\mathcal D}$ of differential operators on
${\Bbb C}[t,t^{-1}]$ consisting of differential operators which are skew-adjoint in a suitable sense,
was studied in \cite{B}. This algebra contains Virasoro and also the odd powers
$D,D^3,D^5,\ldots$ of $D = t\frac{d}{dt}$. The $D^{2n+1}$ act semisimple with finite eigenspaces,
and the resulting character is
\begin{equation}\label{131}\Psi(\tau_1,\tau_3,\ldots) = q_1^{\zeta(-1)/2}q_3^{\zeta(-3)/2}\cdots
\prod_{n=1}^\infty (1-q_1^nq_3^{n^3}\cdots)^{-1}
\end{equation}
In this section, we will show that the Taylor expansion for $\Psi$ is quasimodular of weight -1/2
(cf. \eqref{4wt}), and we will calculate the $n$-point function
\begin{multline}\label{132}{\mathcal F}_n(\tau_1,z_1,\dotsc,z_n) := \\
\sum_{k_1,\dotsc,k_n\ge 1}
\frac{\partial^n}{\partial\tau_{2k_1-1}\cdots\partial\tau_{2k_n-1}}(\Psi)|_{\tau_3 = \cdots = 0}
\frac{z_1^{2k_1-1}\cdots z_n^{2k_n-1}}{(2k_1-1)!\cdots (2k_n-1)!}.
\end{multline}
As before, we write
$$q_r = \exp(2\pi i \tau_r);\quad r\ge 1.
$$
We compute
\begin{gather*} \frac{1}{2\pi i}\frac{\partial}{\partial \tau_{2j-1}} \log\Psi = \zeta(1-2j)/2 +
\sum_{m,n=1}^\infty m^{2j-1}q_1^{nm}q_3^{nm^3}\cdots \\
\frac{1}{(2\pi i)^r}\frac{\partial^r}{\partial \tau_{2j_1-1}\cdots\partial\tau_{2j_r-1}} \log\Psi
=
\sum_{m,n=1}^\infty m^{2(\sum_k j_k-r)-1}(nm)^{r-1}q_1^{nm}q_3^{nm^3}\cdots \notag
\end{gather*}
Note these expressions depend only on $r$ and $\sum_{k=1}^r j_k$. Define
\begin{gather}\label{134} h_{r, 2(j_1+\ldots+j_r)}(\tau_1,\tau_3,\ldots) := \frac{1}{(2\pi
i)^r}\frac{\partial^r}{\partial \tau_{2j_1-1}\cdots\partial\tau_{2j_r-1}} \log\Psi,\\
g_{r, 2(j_1+\ldots+j_r)}(\tau_1) := h_{r, 2(j_1+\ldots+j_r)}(\tau_1,0,0,\ldots) =
\frac{\partial^{r-1}}{\partial\tau_1^{r-1}}G_{2(j_1+\ldots+j_r-r+1)}(\tau_1) \label{135}
\end{gather}
where $G_{2p}(\tau)$ is the Eisenstein series of weight $2p$ as in \eqref{34}. Note $g_{r,s}$ is
quasimodular of weight $2a$. For $A = (a_3,a_5,\ldots)$ write ${\rm wt}(A) = 4a_3+6a_5+\ldots$
(compare \eqref{4wt}). The coefficient of $\tau^A/A!$ in the Taylor expansion for $\log\Psi$ is
$g_{r,{\rm wt} A}(\tau_1)$. Also
\begin{equation}\label{136} \Psi|_{\tau_3=\cdots = 0} = \eta^{-1}(\tau_1)
\end{equation}
Thus, by \eqref{4wt} we conclude that $\log(\Psi\eta)$ is quasimodular of
weight $0$. Exponentiating yields
\begin{prop}\label{prop131} $\Psi$ is quasimodular of weight $-1/2$.
\end{prop}
We now consider the $n$-point function \eqref{132}
\begin{lem} Let ${\mathcal S} = \{j_1,\dotsc,j_n\}$ be a set of positive integers. Let $\Pi({\mathcal S})$ denote
the set of all partitions $\mu = \{\mu_1,\dotsc,\mu_\ell\}$ of ${\mathcal S}$. For any finite set $\phi$
let $\#\phi$ denote the number of elements in $\phi$. For a subset $\phi\subset {\mathcal S}$ let $|\phi|$
be the sum over the elements. Then
$$\frac{1}{(2\pi
i)^n}\frac{\partial^n}{\partial \tau_{2j_1-1}\cdots\partial\tau_{2j_n-1}}\Psi = \Psi\sum_{\mu\in
\Pi({\mathcal S})} \prod_{k=1}^{\#\mu}h_{\#\mu_k,2|\mu_k|}
$$
\end{lem}
\begin{proof}By induction on $n$. For $n=1$ this is just the logarithmic derivative. Suppose now
$n\ge 2$ and the assertion holds for $n-1$. Note
$$\frac{1}{2\pi i}\frac{1}{\partial\tau_{2j_n-1}}h_{n-1,2(j_1+\ldots+j_{n-1})} =
h_{n,2(j_1+\ldots+j_{n})}
$$
Let ${\mathcal T} = \{j_1,\dotsc,j_{n-1}\}$. We have inductively
\begin{multline*}\frac{1}{(2\pi
i)^n}\frac{\partial^n}{\partial \tau_{2j_1-1}\cdots\partial\tau_{2j_n-1}}\Psi =
\frac{1}{2\pi i}\frac{\partial}{\partial\tau_{2j_n-1}}\Big(\Psi\sum_{\mu\in
\Pi({\mathcal T})} \prod_{k=1}^{\#\mu}h_{\#\mu_k,2|\mu_k|}\Big) = \\
= \Psi h_{1,2j_n}\sum_{\mu\in
\Pi({\mathcal T})} \prod_{k=1}^{\#\mu}h_{\#\mu_k,2|\mu_k|} + \Psi\sum_{\mu\in
\Pi({\mathcal T})}\sum_{p=1}^{\#\mu}h_{\#\mu_p+1,2(|\mu_p|+j_n)}\prod_{k\neq p}h_{\#\mu_k,2|\mu_k|} \\
= \Psi\sum_{\mu\in\Pi({\mathcal S})}\prod_{k=1}^{\#\mu}h_{\#\mu_k,2|\mu_k|}.
\end{multline*}
\end{proof}
Define
\begin{equation}{\mathcal G}_n(\tau,z) := \sum_{r=1}^\infty \frac{\partial^{n-1}}{\partial\tau^{n-1}}
G_{2r}(\tau)z^{2r+n-2}/(2r+n-2)!
\end{equation}
One has, with $\Theta$ as in \eqref{61}
\begin{equation}2{\mathcal G}_1(\tau,2\pi iz) := \frac{1}{2\pi
i}(-\frac{d}{dz}\log\Theta(z)+\frac{1}{z})
\end{equation}
To see this, let $\sigma(z,\tau)$ be the elliptic sigma function as defined e.g. in \cite{L}, p.
247. One has
\begin{multline*}\frac{-1}{2\pi i}\frac{d}{dz}\log\Theta = \frac{-1}{2\pi i} \frac{d}{dz}\log\sigma
+2G_2(\tau)(2\pi iz) = \\
\frac{-1}{2\pi iz} +2G_2(\tau)(2\pi iz)+2G_4(\tau)(2\pi iz)^3/3!+2G_6(\tau)(2\pi iz)^5/5!+\ldots
\end{multline*}
Combining \eqref{132}, \eqref{135}, and \eqref{136} we get
\begin{multline}\label{1310} {\mathcal F}(\tau,z_1,\dotsc,z_n) = \eta(\tau)^{-1}\sum_{k_1,\ldots,k_n\ge 1}
\frac{z_1^{2k_1-1}\cdots z_n^{2k_n-1}}{(2k_1-1)!\cdots (2k_n-1)!}\times \\
\sum_{\mu\in\Pi\{k_1,\dotsc,k_n\}} \prod_{p=1}^{\#\mu} \frac{\partial^{\#\mu_p
-1}}{\partial\tau^{\#\mu_p -1}}G_{2(|\mu_p|-\#\mu_p +1)}(\tau).
\end{multline}
For a function $f(z_1,\dotsc,z_n)$ let $\epsilon f$ denote the ``oddification'' of $f$, e.g.
$$\epsilon f(z_1,z_2) = \frac{1}{4}(f(z_1,z_2)-f(-z_1,z_2)-f(z_1,-z_2)+f(-z_1,-z_2))
$$
Note
$$\epsilon((z_1+\ldots+z_n)^r/r!) = \sum_{\ell_1+\ldots+\ell_n=(r+n)/2}
\frac{z_1^{2\ell_1-1}\cdots z_n^{2\ell_n-1}}{(2\ell_1-1)!\cdots (2\ell_n-1)!}
$$
It follows that
\begin{multline*}\epsilon{\mathcal G}_n(\tau,z_1+\ldots+z_n) = \sum_{r=1}^\infty \frac{\partial^{n-1}}
{\partial\tau^{n-1}} G_{2r}(\tau)\times \\
\sum_{k_1+\ldots+k_n=r+n-1}\frac{z_1^{2k_1-1}\cdots
z_n^{2k_n-1}}{(2k_1-1)!\cdots (2k_n-1)!}
\end{multline*}
We conclude
\begin{prop} The $n$-point function is given by
$${\mathcal F}(\tau,z_1,\dotsc,z_n) =
\eta(\tau)^{-1}\sum_{\mu\in\Pi_n}\prod_{k=1}^{k=\#\mu}\epsilon{\mathcal G}_{\#\mu_k}(\sum_{i\in \mu_k} z_i)
$$
\end{prop}
\begin{proof} We have an obvious identification
$$\Pi_n = \Pi\{1,\dotsc,n\} = \Pi\{k_1,\dotsc,k_n\}
$$
(Note $\{k_1,\dotsc,k_n\}$ is treated formally, i.e. we do not identify the $k_i$ if they are
equal.) Write $\kappa=\{k_1,\dotsc,k_n\}$ and let $\mu_{p,\kappa}\subset \kappa$ be the
corresponding subset. Given $\mu\in\Pi_n$ define
\begin{multline*}{\mathcal F}_\mu(\tau,z_1,\dotsc,z_n) = \eta(\tau)^{-1}\sum_{k_1,\ldots,k_n\ge 1}
\frac{z_1^{2k_1-1}\cdots z_n^{2k_n-1}}{(2k_1-1)!\cdots (2k_n-1)!}\times \\
\prod_{p=1}^{\#\mu} \frac{\partial^{\#\mu_p
-1}}{\partial\tau^{\#\mu_p -1}}G_{2(|\mu_{p,\kappa}|-\#\mu_p +1)}(\tau).
\end{multline*}
so ${\mathcal F} = \sum_{\Pi_n} {\mathcal F}_\mu$. It will suffice to show
\begin{equation}\label{1312}{\mathcal F}_\mu =
\eta(\tau)^{-1}\prod_{p=1}^{p=\#\mu}\epsilon{\mathcal G}_{\#\mu_p}(\sum_{i\in \mu_p} z_i).
\end{equation}
But given $\kappa$, the term $z_1^{2k_1-1}\cdots z_n^{2k_n-1}$ occurs exactly once in
\eqref{1312} and has the correct coefficient in $\tau$.
\end{proof}
\bibliographystyle{plain}
\renewcommand\refname{References}
|
1997-12-03T00:49:42 | 9712 | alg-geom/9712004 | en | https://arxiv.org/abs/alg-geom/9712004 | [
"alg-geom",
"math.AG"
] | alg-geom/9712004 | Janos Kollar | J\'anos Koll\'ar | Real Algebraic Threefolds I: Terminal Singularities | LATEX2e, 24 pages | null | null | null | null | This is the first of a series of papers studying real algebraic threefolds
using the minimal model program. The main results are outlined in Part II. The
present part I. contains the necessary preliminary work concerning terminal
singularities. First I give standard forms for 3-dimensional terminal
singularities over arbitrary fields. This is then used to develop a fairly
complete topological classification over the reals.
| [
{
"version": "v1",
"created": "Tue, 2 Dec 1997 23:49:41 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Kollár",
"János",
""
]
] | alg-geom | \section{Introduction}
In real algebraic geometry, considerable attention has
been paid to the study of real algebraic curves (in
connection with Hilbert's 16th problem) and also to real
algebraic surfaces. See \cite{Viro90}, \cite{Riesler93} and the
references there.
In higher dimensions one of the main avenues of
investigation was initiated by \cite{Nash52}, and later
developed by many others (see
\cite{AK92} for some recent directions). One of these results says
that every compact differentiable manifold can be realized
as the set of real points of an algebraic variety.
\cite{Nash52} posed the problem of obtaining similar results
using a restricted class of varieties, for instance rational
varieties.
The aim of this series of papers is to develop the theory of
minimal models for
real algebraic threefolds. This approach gives very strong
information about the topology of real algebraic threefolds, and
it also answers the above mentioned question of
\cite{Nash52}.
For algebraic threefolds over $\c$, the minimal model
program provides a very powerful tool. The method of the
program is the following. (See \cite{Koll87} or \cite{CKM88}
for introductions)
Starting with a smooth projective variety $X$, we perform a
series of ``elementary" birational transformations
$$ X=X_0\map X_1\map \cdots \map X_n
$$ until we reach a variety $X_n$ whose global structure is
``simple". In essence the minimal model
program allows us to investigate many questions in two
steps: first study the effect of the ``elementary"
transformations and then consider the ``simple" global
situation.
In parctice both of these steps are frequently rather
difficult. For instance, we still do not have a complete
list of all possible ``elementary" steps, despite repeated
attempts to obtain it.
A somewhat unpleasant feature of the theory is
that the varieties $X_i$ are not smooth, but have so called
terminal singularities. In developing the theory of minimal models
for real algebraic threefolds, we again have to understand the
occurring terminal singularities.
The aim of this paper is to give a classification of
terminal 3-fold singularities over
$\r$. Minimal models serve only as a background, the proofs depend
entirely on well established methods of singularity theory.
I do not even use the definition of terminal singularities!
Terminal 3-fold singularities over $\c$ are completely
classified.
\cite{Reid85} is a very readable introduction and survey.
I will take the result of this classification as my definition,
since the theory over $\r$ can be most naturally developed in this
setting.
The classification is, in some sense, not complete.
In a few cases I obtain unique normal forms
(\ref{ca1-top.thm}), but in most cases this seems nearly
impossible (see
\cite{Markushevich85} for a
special case over $\c$). My aim is to write the
singularities in a form that allows one to determine their
topology over $\r$. The resulting lists and algorithms are
given in sections 4--5.
It turns out that the normal forms of 3-fold terminal
singularities are essentially the same over any field of
characteristic zero. Thus in sections 2--3 I work with any
subfield of $\c$.
As a consequence of the classification over $\c$, we know
that 3-fold terminal singularities come in two types. Some
are hypersurface singularities, and the others are quotients
of these hypersurface singularities by a finite cyclic
group. Accordingly, the classification over any field is
done in two steps. Section 2 deals with terminal
hypersurface singularities. These results are mostly routine
generalizations of the theory over $\c$.
Quotient singularities frequently have ``twisted" forms over
a subfield of $\c$. ``Twisted" forms do not
appear for 3-fold terminal singularities, and so the
classification ends up very similar to the one over $\c$.
\begin{ack} I thank G. Mikhalkin for
answering my numerous questions about real algebraic geometry.
Partial financial support was provided by the NSF under grant
number DMS-9622394. Most of this paper was written while I visited
RIMS, Kyoto Univ.
\end{ack}
\section{Terminal hypersurface singularities}
\begin{notation} For a field $K$ let $K[[x_1,\dots,x_n]]$
denote the ring of formal power series in $n$ variables over
$K$. For $K=\r$ or
$K=\c$, let $K\{x_1,\dots,x_n\}$ denote the ring of those
formal power series which converge in some neighborhood of
the origin.
For any $F\in K\{x_1,\dots,x_n\}$ the set $(F=0)$ is a germ
of a real or complex analytic set. I will refer to it as a
singularity. If $F\in K[[x_1,\dots,x_n]]$ then by the
singularity $(F=0)$ I mean the scheme
$\spec_KK[[x_1,\dots,x_n]]/(F)$.
For a power series $F$, $F_d$ denotes the degree $d$
homogeneous part. The multiplicity, denoted by $\mult_0F$,
is the smallest $d$ such that
$F_d\neq 0$. If we write a power series as $F_{\geq d}$ then
it is assumed that its multiplicity is at least $d$.
Two power series $F,G\in K[[x_1,\dots,x_n]]$
are called equivalent over $K$ if there is an automorphism of
$K[[x_1,\dots,x_n]]$ given by
$x_i\mapsto \phi_i(x_1,\dots,x_n)\in K[[x_1,\dots,x_n]]$
and an invertible $u(x_1,\dots,x_n)\in K[[x_1,\dots,x_n]]$
such that
$$
u(x_1,\dots,x_n)G(x_1,\dots,x_n)=F(\phi_1,\dots,\phi_n).
$$ Thus $F$ and $G$ are equivalent iff the corresponding
singularities $(F=0)$ and $(G=0)$ are isomorphic (over
$K$).
We have to pay special attention to cases when $F$
and $G$ are not equivalent over $K$ but are equivalent
over some larger field. For instance, $F=x_1^2+x_2^2$ and
$G=x_1^2-x_2^2$ are not equivalent over $\r$ but are
equivalent over $\c$.
If $K=\r,\c$ and $F,G\in K\{x_1,\dots,x_n\}$ then I am
mainly interested in equivalences where $u,\phi_i\in
K\{x_1,\dots,x_n\}$.
If $F,G\in K\{x_1,\dots,x_n\}$ have isolated critical
points at the origin, then
$F$ and $G$ are equivalent in $K\{x_1,\dots,x_n\}$ iff
they are equivalent in $K[[x_1,\dots,x_n]]$ (cf.
\cite[p.121]{AGV85}), thus we do not have to be careful
about this distinction.
\end{notation}
\begin{defn} Let $K$ be a field of characteristic zero with
algebraic closure $\bar K$. $(F(x,y,z)=0)$ is called a {\it Du
Val} singularity (or a rational double point) iff over $\bar
K$ it is equivalent to one of the standard forms
\begin{enumerate}
\item[$A_n$] \quad $x^2+y^2+z^{n+1}=0$;
\item[$D_n$] \quad $x^2+y^2z+z^{n-1}=0$;
\item[$E_6$] \quad $x^2+y^3+z^4=0$;
\item[$E_7$] \quad $x^2+y^3+yz^3=0$;
\item[$E_8$] \quad $x^2+y^3+z^5=0$.
\end{enumerate} Du Val singularities have many interesting
intrinsic characterizations, (cf. \cite{Durfee79, Reid85})
but I will not use this.
\end{defn}
The following definition introduces our basic objects of study.
\begin{defn}\label{cdv.def} Let $K$ be a field of
characteristic zero with algebraic closure $\bar K$.
$(F(x,y,z,t)=0)$ is called a {\it compound Du Val} singularity (or
{\it cDV} for short) iff over $\bar K$ it is equivalent to
$$ h(x,y,z)+tf(x,y,z,t)=0
$$ where $(h=0)$ is a Du Val singularity.
$(F(x,y,z,t)=0)$ is called a
$cA_n$ (resp. $cD_n$ or $cE_n$) singularity if its equation
can be written as above with $h$ having type
$A_n$ (resp. $D_n$ or $E_n$), but it does not admit such
representation with a smaller value of $n$. It is called a
$cA$ (resp. $cD$ or $cE$) singularity if the value of $n$ is
not specified.
\end{defn}
The reason we are interested in cDV singularities is the
following:
\begin{thm}\cite{Reid80} A 3-dimensional hypersurface
singularity over
$\c$ is terminal iff it is an isolated cDV singularity.\qed
\end{thm}
The aim of this section is to develop ``normal forms" for cDV
singularities over any field $K$. This will then give
``normal forms" for 3-dimensional terminal hypersurface
singularities over $K$.
The proof is a rather standard application of the
methods of
\cite{AGV85}.
\begin{say}\label{nf.meth} We repeatedly use 3 methods:
\begin{enumerate}
\item The Weierstrass preparation theorem. This is
frequently stated only over
$\c$, but it works over any field since the Weierstrass
normal form is unique.
\item The elimination of the $y^{n-1}$-term from the
polynomial
$a_ny^n+a_{n-1}y^{n-1}+\dots$ by a coordinate change
$y\mapsto y-a_{n-1}/na_n$ when $a_n$ is invertible.
\item Let $M_1,\dots,M_k$ be
monomials in the variables
$x_1,\dots,x_m$. Assume that
$x_0M_1,\dots,x_0M_k$ are multiplicatively independent.
Then any power series of the form
$\sum M_i\cdot u_i(x_1,\dots,x_m)$ where $u_i(0)\neq 0$
for all $i$ is equivalent to
$\sum M_i\cdot u_i(0)$ by a suitable coordinate change
$x_i\mapsto x_i\cdot(\mbox{unit})$.
\end{enumerate}
\end{say}
These elementary operations are sufficient to deal with the
$cA$ and
$cE$ cases. In the $cD$ case the following generalization
of (\ref{nf.meth}.2) is needed.
\begin{const}\label{nf.meth.weighted} In
$K[[x_1,\dots,x_m]]$, assign positive integral weights to
the variables $w(x_i)=w_i$. For a monomial set $w(\prod
x_i^{c_i})=\sum c_iw_i$. Write a power series in terms of
its weighted homogeneous pieces
$F=F_d+F_{d+1}+\dots$. Choose $g_i\in K[[x_1,\dots,x_m]]$
such that
$w(g_i)=w(x_i)+e$ for some $e>0$. Then
$$ F(x_i+g_i)=F(x_i)+\sum g_i\frac{\partial F_d}{\partial
x_i}+ R_{>(d+e)}(x_i).
$$ Repeatedly using this for higher and higher degrees, we
see that, for every $N>0$,
$F$ is equivalent to a power series
$F^N+R_{>N}$ where $F^N$ is a polynomial of degree $N$ and
no linear combination of the monomials in $F^N$ can be
written in the form
$\sum g_i(\partial F_d/\partial x_i)$ as above.
In the ring of formal power series this can be continued
indefinitely, thus at the end we can kill all the degree
$>d$ elements of the Jacobian ideal
$$
\Delta(F_d):=\left(\frac{\partial F_d}{\partial x_1},
\dots , \frac{\partial F_d}{\partial x_m}\right).
$$
If $F\in K\{x_1,\dots,x_m\}$ defines an isolated
singularity, then by Tougeron's lemma (cf.
\cite[p.121]{AGV85}), $F^N+R_{>N}$ is equivalent to
$F^N$ by an analytic coordinate change for $N\gg 1$. Thus
the final conclusion is the same.
\end{const}
\begin{prop}\label{morse.lem} Any power series $F_{\geq
2}(x_1,\dots,x_n)$ is equivalent to a power series
$$ a_1x_1^2+\dots+a_kx_k^2+G_{\geq 3}(x_{k+1},\dots,x_n).
$$
\end{prop}
Proof. By a linear change of coordinates we can diagonalize
$F_2$, thus we can assume that
$F_2=a_1x_1^2+\dots+a_kx_k^2$. Repeatedly applying
(\ref{nf.meth}.1) to the variables
$x_1,\dots,x_k$ we reach a situation when $F$ is a quadratic
polynomial in
the variables $x_1,\dots,x_k$. (\ref{nf.meth}.2) can then
be used to eliminate the linear terms in $x_1,\dots,x_k$.\qed
\begin{thm}\label{ca.thm} Assume that $F_{\geq
1}(x,y,z,t)\in K[[x,y,z,t]]$ defines a terminal singularity
of type $cA$. Then $F$ is equivalent to one of the following:
\begin{enumerate}
\item[$cA_0$] \quad $x=0$.
\item[$cA_1$] \quad $ax^2+by^2+cz^2+dt^m=0$, where
$abcd\neq 0$.
\item[$cA_{>1}$] \quad $ax^2+by^2+f_{\geq 3}(z,t)=0$, where
$ab\neq 0$. This has type $cA_n$ for $n=\mult_0f-1$.
\end{enumerate}
\end{thm}
Proof. If $F_1\neq 0$ then (\ref{nf.meth}.1) gives $cA_0$.
Thus assume that $F_1=0$. $F$ has type $cA$, hence $F_2$ is
a quadric of rank at least 2. If the rank is 2 then
(\ref{morse.lem}) gives the
$cA_{>1}$ cases.
Assume finally that $F_2$ has rank 3 or 4.
By (\ref{morse.lem}) we can write $F$ as
$ax^2+by^2+cz^2+g(t)=0$. Using (\ref{nf.meth}.3) we obtain
$ax^2+by^2+cz^2+dt^m=0$.
In all these cases we can multiply through by $a^{-1}$ to get a
somewhat simpler form when the coefficient of $x^2$ is 1.
\qed
\begin{thm}\label{cd.thm} Assume that $F_{\geq
2}(x,y,z,t)\in K[[x,y,z,t]]$ defines a terminal singularity
of type $cD$. Then $F$ is equivalent to one of the following:
\begin{enumerate}
\item[$cD_4$] \quad $x^2+f_{\geq 3}(y,z,t)$, where $f_3$ is
not divisible by the square of a linear form.
\item[$cD_{>4}$] \quad $x^2+y^2z+ayt^r+h_{\geq s}(z,t)$,
where $a\in K$,
$r\geq 3$, $s\geq 4$ and $h_s\neq 0$. This has type $cD_n$
where
$n=\min\{2r, s+1\}$ if $a\neq 0$ and $n=s+1$ if $a=0$.
\end{enumerate}
\end{thm}
Proof. $F_2$ is a rank one quadric, thus in suitable
coordinates the equation becomes $ax^2+f_{\geq 3}(y,z,t)$.
Here $f_3\neq 0$ is not the cube of a linear form since
otherwise we would have a type $cE$ singularity. If $f_3$
is not divisible by the square of a linear form then we have
case
$cD_4$.
If $f_3$ is divisible by the square of a linear form, then
$f_3=l_1^2l_2$ for two linear forms $l_i$, and both of them
are defined over $K$. We can change coordinates $l_1\mapsto
y$ and $l_2\mapsto z$.
At this point our power series is $x^2+y^2z+(\mbox{higher
order terms})$. Assign weights $w(x)=3, w(y)=w(z)=2,
w(t)=6$. The leading term is
$x^2+y^2z$. Using (\ref{nf.meth.weighted}) we can eliminate
all monomials which contain $y^2$ or $yz$.
To see the last part, take the hyperplane section
$t=\lambda z$. The term
$ay\lambda^rz^r$ can be eliminated by a substitution
$y\mapsto y+(a/2)\lambda^rz^{r-1}$. This creates a term
$-(a/2)^2\lambda^{2r} z^{2r-1}$. The only problem could be
that
$h(z,\lambda z)$ has multiplicity $2r-1$ and there is
cancellation. However, $h_{2r-1}(z,\lambda z)=z^{2r-1}
h_{2r-1}(1,\lambda)$ is a polynomial of degree $2r-1$ in
$\lambda$, thus it does not equal
$-(a/2)^2\lambda^{2r} z^{2r-1}$.
\qed
\begin{thm}\label{ce.thm} Assume that $F_{\geq
2}(x,y,z,t)\in K[[x,y,z,t]]$ defines a terminal singularity
of type $cE$. Then $F$ is equivalent to one of the following:
\begin{enumerate}
\item[$cE_6$] \quad $x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq
4}(z,t)$, where
$h_4\neq 0$.
\item[$cE_7$] \quad $x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq
5}(z,t)$, where
$g_3\neq 0$.
\item[$cE_8$] \quad $x^2+y^3+yg_{\geq 4}(z,t)+h_{\geq
5}(z,t)$, where
$h_5\neq 0$.
\end{enumerate}
\end{thm}
Proof. $F_2$ is a rank one quadric by (\ref{cdv.def}), thus
in suitable coordinates the equation becomes $ax^2+f_{\geq
3}(y,z,t)$. Here
$f_3\neq 0$ and it is the cube of a linear form since
otherwise we would have a type $cD$ singularity.
(\ref{nf.meth}.1--2) gives an equation
$$ ax^2+by^3+yg_{\geq 3}(z,t)+h_{\geq 4}(z,t).
$$ Multiply the equation by $a^3b^2$ and then make the
substitutions
$x\mapsto xa^{-2}b^{-1}$ and $y\mapsto ya^{-1}b^{-1}$ to
get the required normal forms.\qed
\section{Higher index terminal singularities}
The classification of non-hypersurface terminal 3-fold
singularities over $\c$ relies on the following construction:
Let $\z_n$ denote the cyclic group of order $n$ and
$\epsilon$ a primitive
$n^{th}$ root of unity. Assume that $\z_n$ acts on $\c^4$ by
$$
\sigma: (x,y,z,t)\mapsto
(\epsilon^{a_x}x,\epsilon^{a_y}y,\epsilon^{a_z}z,\epsilon^{a_t}t).
$$ I will use the shorter notation
$\frac{1}{n}(a_x,a_y,a_z,a_t)$ to denote such an action.
If $F(x,y,z,t)$ is equivariant with respect to this action,
then
$\z_n$ acts on the hypersurface $(F=0)$ and we can take the
quotient,
denoted by $(F=0)/\frac{1}{n}(a_x,a_y,a_z,a_t)$.
By \cite{Reid80}, every terminal 3-fold
singularity $X$ over
$\c$ is of the form
$(F=0)/\frac{1}{n}(a_x,a_y,a_z,a_t)$, where $F$ defines a
terminal hypersurface singularity. The value of $n$ is
uniquely determined by
$X$, it is called the {\it index} of $X$.
It is not easy to come up with a complete list of terminal
3-fold singularities, but by now the list is well
understood; see
\cite{Reid85} for a good survey. It turns out that most
actions do not produce terminal quotients and we have only a
few cases:
\begin{thm}\label{hind.overC.thm}\cite{Mori85}
Let $0\in X$ be a 3-fold terminal
nonhypersurface singularity over
$\c$. Then $0\in X$ is isomorphic to a singularity
described by the following list:
$$
\begin{tabular}{|c|l|c|l|l|}
\hline name &\qquad equation & index & \quad action &
condition\\
\hline cA/n& $xy+f(z,t)$ & $n$ & $(r,-r,1,0)$ & $(n,r)=1$\\
\hline cAx/2& $x^2+ y^2 +f_{\geq 4}(z,t)$ & $2$ & $(0,1,1,1)$&\\
\hline cAx/4 & $x^2+y^2 +f_{\geq 2}(z,t)$ & $4$ & $(1,3,1,2)$&
$f_2(0,1)=0$\\
\hline cD/2&$x^2+f_{\geq 3}(y,z,t)$ & $2$ &$(1,0,1,1)$&\\
\hline cD/3&$x^2+f_{\geq 3}(y,z,t)$ &$3$ &$(0,2,1,1)$&
$f_3(1,0,0)\neq 0$\\
\hline cE/2&$x^2+y^3+f_{\geq 4}(y,z,t)$ &$2$
&$(1,0,1,1)$&\\
\hline
\end{tabular}
$$
\end{thm}
The equations have to satisfy 2 obvious conditions:
\begin{enumerate}
\item The equations define a terminal hypersurface
singularity.
\item The equations are $\z_n$-equivariant. (In fact
$\z_n$-invariant, except for $cAx/4$.)
\end{enumerate}
If we work over a field $K$ which does not contain
the
$n^{th}$ roots of unity, then the action
$\frac{1}{n}(a_1,\dots,a_m)$ is not defined over $K$. There
is, however, another way of loking at the quotient which
does make sense over any field.
Any action of the cyclic group $\z_n$ on $\c^m$ defines a
$\z_n$-grading
$w$ of $\c[[x_1,\dots,x_m]]$ by
$$ w(\prod x_i^{c_i})=a \qtq{iff}
\sigma (\prod x_i^{c_i})=\epsilon^a \cdot \prod x_i^{c_i}.
$$ If $F$ is $\z_n$-equivariant then
$(F)\subset \c[[x_1,\dots,x_m]]$ is a homogeneous ideal,
hence the grading descends to a grading of
$\c[[x_1,\dots,x_m]]/(F)$. The ring of functions on the
quotient
$(F=0)/\frac{1}{n}(a_1,\dots,a_m)$ can be identified with
the ring of grade zero elements of $\c[[x_1,\dots,x_m]]/(F)$.
If $K$ is any field, $n\in \n$ and $a_i\in \z$, then we
obtain a
$\z_n$-grading $w=w(a_1,\dots,a_m)$ of $K[[x_1,\dots,x_m]]$
(or of
$\r\{x_1,\dots,x_m\}$) by
$$ w(\prod x_i^{c_i})=\sum c_ia_i \in \z_n.
$$ Let $R\subset K[[x_1,\dots,x_m]]$ denote the subring of
grade zero elements. Then $\spec_KR$ gives a singularity
over $K$ which is denoted by
$$ {\Bbb A}^m/{\textstyle \frac{1}{n}}(a_1,\dots,a_m).
$$ (Especially when $K=\r$, one might be tempted to write
${\Bbb R}^m/{\textstyle \frac{1}{n}}(a_1,\dots,a_m)$
instead. However,
the set of real points of ${\Bbb A}^m/{\textstyle
\frac{1}{n}}(a_1,\dots,a_m)$ is not in any sense a
quotient of the set $\r^n$ (cf. (\ref{2ind.quot})), so this may
lead to confusion.)
If $F\in K[[x_1,\dots,x_m]]$ is graded homogeneous, then
$w$ gives a grading of $K[[x_1,\dots,x_m]]/(F)$. Let
$R/(R\cap(F))\subset K[[x_1,\dots,x_m]]/(F)$ be the subring
of grade zero elements.
$\spec_K R/(R\cap(F))$ defines a singularity over $K$. By
construction,
$$
\spec_K R/(R\cap(F))\times_{\spec K}\spec \bar K\cong
(F=0)/{\textstyle
\frac{1}{n}}(a_1,\dots,a_m).
$$ Thus $\spec_K R$ is a terminal singularity over $K$ iff
$(F=0)/{\textstyle \frac{1}{n}}(a_1,\dots,a_m)$ is a
terminal singularity over $\bar K$.
Under certain conditions, every $K$-form of a quotient
is obtained this way:
\begin{thm}\label{hind.gen.thm}
$K$ be a field of characteristic zero with algebraic
closure $\bar K$.
Let $\z_n$ denote the cyclic group of order $n$ and
$\epsilon$ a primitive
$n^{th}$ root of unity. Assume that $\z_n$ acts on $\bar K^m$
by
$\sigma: (x_i)\mapsto
(\epsilon^{w_i}x_i)$.
Let $F\in \bar K[[x_1,\dots,x_m]]$ be equivariant with respect to this
action, and assume that the fixed point set of $\sigma$ has
codimension at least 2 in $(F=0)$.
Assume in addition that
$$
w(F)-\sum w_i\qtq{is relatively prime to} n.
$$
Let $0\in X$ be a singularity over $K$
such that
$$
X\times_{\spec K}\spec \bar K\cong (F=0)/
\textstyle{\frac{1}{n}(w_1,\dots,w_m)}.
$$
Then there is an $F^K\in K[[x_1,\dots,x_m]]$ such that
$F$ and $F^K$ are equivalent over $\bar K$ and
$$
X\cong (F^K=0)/
\textstyle{\frac{1}{n}(w_1,\dots,w_m)}.
$$
\end{thm}
It is worthwhile to note that the condition about
$n$ and $w(F)-\sum w_i $ being relatively prime is essential:
\begin{exmp} Consider the quotient singularity
$\c[u,v]/\frac{1}{n}(1,-1)$. It is isomorphic to
$(xy-z^n=0)$ via the
substitutions $x=u^n,y=v^n,z=uv$. Over $\c$ we have a
Du Val singularity
$A_{n-1}=(x^2+y^2+z^n=0)$.
Over $\r$ we see that
$(x^2-y^2-z^n=0)\cong {\Bbb A}^2/\frac{1}{n}(1,-1)$. Another
$\r$-form of $A_{n-1}$ is $x^2+y^2-z^n$. This can also be
obtained as a quotient, but this time we act on ${\Bbb A}^2$
by rotation with angle
$2\pi/n$.
Finally, if $n$ is even, then there is another $\r$-form of
$A_{n-1}$ given by
$(x^2+y^2+z^n=0)$. The only $\r$-point is the origin, so we
do not even have a nonzero map
$\r^2\to (x^2+y^2+z^n=0)$.
As another example, take the 4-dimensional terminal
singularity
$\c^4/\frac{1}{n}(a,-a,b,-b)$ for any $(ab,n)=1$. It has
another
$\r$-form given as ${\Bbb A}^4/\z_n$ where we act on the
first two coordinates by rotation with angle $2a\pi/n$ and
on the last two coordinates by rotation with angle
$2b\pi/n$.
In some special cases there are further $\r$-forms. Take for
instance
$\c^4/\frac{1}{2}(1,1,1,1)$. This can be realized as the
cone over
$\c\p^3$ embedded by the quadrics to $\c\p^9$.
Let $C\subset \r\p^2$ be a smooth conic. Taking symmetric powers
we have $S^3C\subset S^3\r\p^2$ and $S^3H^0(\r\p^2,\o(1))$ embeds
it to $\r\p^9$. If $C$ has a real point, then
$S^3C\cong \r\p^3$ and we get the Veronese embedding. If $C$ has no
real points then the image is a variety over $\r$ without real
points. The cone over it is a real form of
${\Bbb A}^4/\frac12(1,1,1,1)$ with an isolated real point at the
origin.
\end{exmp}
Proof of (\ref{hind.gen.thm}).
Set $S=\bar K[[x_1,\dots,x_m]]/(F)$,
the ring of functions on $\tilde X_{\bar K}:=(F=0)$.
The $\z_n$-action defines a
$\z_n$-grading $S=\sum_{i=0}^{n-1}S_i$. $S_0$, the ring of grade $0$
elements, is exactly the ring of functions on $X_{\bar K}$. Our aim is
to find an algebraic way of reconstructing $S$ from $S_0$, which then
hopefully generalizes to nonclosed fields.
There is another summand which can be easily seen algebraically.
Set $d=w(F)-\sum w_i$. Note that
$$
\frac{1}{\partial F/\partial x_m}dx_1\wedge\dots\wedge dx_{m-1}
$$
is a local generator of $\omega_S$ and it has weight $-d$. Thus
$$
\omega_{S_0}\cong S_d\frac{1}{\partial F/\partial
x_m}dx_1\wedge\dots\wedge dx_{m-1}.
$$
Once $S_d$ is determined, we obtain $S_{jd}$
as follows. The multiplication map
$$
S_a\otimes_{S_0} S_b\to S_{a+b}\qtq{(subscripts modulo $n$)}
$$ are
isomorhisms over the open set where the $\z_n$-action is free.
We assumed that the complement has codimension at least 2, thus
$S_{jd}\cong S_d^{[j]}$, where $ S_d^{[j]}$ denotes the double dual
of
$S_d^{\otimes j}$. If $d$ and $n$ are
relatively prime, then we obtain every summand $S_i$ this way. In
particular,
$$
S=\sum_{i=0}^{n-1}S_i\cong \sum_{j=0}^{n-1}\omega_{S_0}^{[j]}.
$$
Over an arbitrary field, we can thus proceed as follows.
Let $\omega_X$ be the dualizing sheaf of $X$. This is also
the reflexive sheaf $\o_X(K_X)$ where $K_X$ is the canonical
class.
Then $\omega_X^{[n]}$ is isomorphic to $\o_X$, where $n$ is
the index. (We know this over $\bar K$. Isomorphism of two
sheaves
$F,G$ is a question about $\Hom(F,G)$ and this commutes with
base field extensions.) Fix such an isomorphism
$s:\omega_X^{[n]}\to \o_X$.
Consider the $\o_X$-algebra
$$
R(X,s):=\sum_{j=0}^{n-1}\omega_X^{[j]},
$$ where multiplication for $j+k\geq n$ is given by
$$
\omega_X^{[j]}\otimes \omega_X^{[k]}\mapsto
\omega_X^{[j+k]}\cong \omega_X^{[n]}\otimes
\omega_X^{[j+k-n]}\stackrel{s\otimes 1}{\longrightarrow}
\omega_X^{[j+k-n]}.
$$ This has a $\z_n$ grading by declaring $\omega_X^{[j]}$
to have grade
$j$.
(Note. Two isomorphisms $s_1,s_2:\omega_X^{[r]}\to \o_X$
differ by an invertible function $h\in \o_X^*$. If $h$ is an
$n^{th}$-power, then the resulting algebras $R(X,s_i)$
are isomorphic, but they need not be isomorphic otherwise.
This is connected with the topological aspects observed in
(\ref{2ind.quot}).)
Over $\bar K$, $R(X,s)$ is isomorphic to $\o_{\tilde X}$. Thus
$R(X,s)$ is a $K$-form of
$\o_{\tilde X}$. In particular, $R(X,s)$ is an algebra
of the form
$K[[x_1,\dots,x_m]]/(F^K)$, where $F$ and $F^K$ are equivalent over
$\bar K$.
The grading lifts to a grading of
$K[[x_1,\dots,x_m]]$ such that $F^K$ is graded homogeneous. We can
choose
$x_i$ to be homogeneous.\qed
As a corollary, we obtain the following classification of
terminal 3-fold nonhypersurface singularities over nonclosed
fields:
\begin{thm}\label{hind.term.thm} Let $K$ be a field of
characteristic zero and $0\in X$ a 3-fold terminal
nonhypersurface singularity over
$K$. Then $0\in X$ is isomorphic over $K$ to a singularity
described by the following list:
$$
\begin{tabular}{|c|l|c|l|l|}
\hline name &\qquad equation & index & \quad weights &
condition\\
\hline cA/2& $ax^2+by^2+f(z,t)$ & $2$ & $(1,1,1,0)$&\\
\hline cA/n& $xy+f(z,t)$ & $n\geq 3$ & $(r,-r,1,0)$&
$(n,r)=1$\\
\hline cAx/2& $ax^2+by^2 +f_{\geq 4}(z,t)$ & $2$ & $(0,1,1,1)$&\\
\hline cAx/4 & $ax^2+by^2 +f_{\geq 2}(z,t)$ & $4$ & $(1,3,1,2)$&
$f_2(0,1)=0$\\
\hline cD/2&$x^2+f_{\geq 3}(y,z,t)$ & $2$ &$(1,0,1,1)$&\\
\hline cD/3&$x^2+f_{\geq 3}(y,z,t)$ &$3$ &$(0,2,1,1)$&
$f_3(1,0,0)\neq 0$\\
\hline cE/2&$x^2+y^3+f_{\geq 4}(y,z,t)$ &$2$
&$(1,0,1,1)$&\\
\hline
\end{tabular}
$$
\end{thm}
\begin{complement} The corresponding quotient singularity
is terminal iff the equations satisfy 2 obvious
conditions:
\begin{enumerate}
\item The equations define a terminal hypersurface
singularity.
\item The equations are graded homogeneous.
\end{enumerate}
With these assumptions, a terminal singularity corresponds to exactly
one case on the above list.
\end{complement}
Proof. By looking at the list of (\ref{hind.overC.thm}), we see
that the assumptions of (\ref{hind.gen.thm}) are satisfied.
Hence we know that $X$ is of the form
$(F^K=0)/\frac1{n}(a_x,a_y,a_z,a_t)$ where
$\frac1{n}(a_x,a_y,a_z,a_t)$ is on the list of
(\ref{hind.overC.thm}).
Once we know a $\z_n$-grading on $K[[x,y,z,t]]$ and a
graded homogeneous power series $F^K$, we can try to bring it
to some normal form using the methods (\ref{nf.meth}) and
(\ref{nf.meth.weighted}). They are set up in such a way
that if $F^K$ is homogeneous in a
$\z_n$-grading the all coordinate changes respect the
grading.
The proofs of (\ref{ca.thm}, \ref{cd.thm}, \ref{ce.thm})
remain unchanged. The only difference is in
(\ref{morse.lem}). It is not true that a quadratic form can
be diagonalized using a linear transformation which respects
the
$\z_n$-grading. The best one can achieve is a sum of forms in
disjoint sets of variables $\sum q_i$ where each $q_i$ is
either $au_i^2$ or $u_iv_i$. The latter case is necessary
iff the two variables have different
$\z_n$-grading.
In the $cD$ and $cE$ cases the quadric has rank 1, so it can
be diagonalized.
In the $cA/2$ and $cAx/2$ cases every grade 0 quadric is
diagonalizable.
In the $cAx/4$ case $x^2,xz,y^2, z^2$ are the only grade 2
quadratic monomials. A quadratic form like this can again be
diagonalized.
Finally let us look at the $cA/n$-case for $n\geq 3$. The
only grade 0 degree 2 monomials are $xy, t^2$ and $xz$ if
$r=-1$ or $yz$ if $r=1$. We need to get a rank $\geq 2$
quadric, so $xy $ (or $xz$ if $r=-1$,
$yz$ if
$r=1$) must appear. In the $r=\pm 1$ case we may need to
perform a linear change of variables to get the normal form
$xy+f(z,t)$.
\qed
\section{The topology of terminal hypersurface singularities}
Let $0\in X$ be a real singularity. It's real points
$X(\r)$ form a topological space, which can be triangulated
(cf. \cite[9.2]{BCR87}). We may assume that $0$ is a vertex of the
triangulation. Then locally near
$0$, $X(\r)$ is PL-homeomorphic to the cone over a
simplicial complex
$L=L(X(\r))$, which is called the {\it link} of
$0$ in $X(\r)$. The local topology of $X(\r)$ at $0$ is
thus determined by $L$.
In general one needs to contemplate the dependence of $L$ on
various choices made. I am mainly interested in the case
when $X$ is a 3-dimensional isolated singularity. In this
case $L$ is a compact surface (without boundary) and so $L$
and $X(\r)$ determine each other up to homeomorphism.
The aim of this section is to classify terminal
singularities over $\r$ according to their local topology.
To be precise, we give a classification in the $cA$ cases
and provide a procedure in the $cD$ and $cE$ cases which
reduces the 3-dimensional problem to some questions about
plane curve singularities.
\begin{notation} $M\sim N$ denotes that $M$ and $N$ are
homeomorphic.
$\uplus$ denotes disjoint union.
$M\uplus rN$ denotes the disjoint union of $M$ and of $r$ copies
of $N$.
$M_g$ denotes the unique compact, closed and orientable surface of
genus
$g$.
\end{notation}
We start with a general lemma.
\begin{lem}\label{orient} Let $X$ be a smooth real
hypersurface. Then
$X(\r)$ is orientable.
\end{lem}
Proof. Let $X=(f=0)$ be a real equation where $f\in
\r[x_1,\dots,x_n]$ or $f\in \r\{x_1,\dots,x_n\}$. At each
point $p\in X$, $X$ divides a neighborhood of $p$ into two
halves. $f$ is positive on one half and negative on the
other half. Choosing a sign thus determines an orientation.
\qed
\begin{thm}\label{ca1-top.thm} The following table gives a
complete list of 3-dimensional terminal singularities of
type $cA_1$ over
$\r$.
In the table $n\geq 1$. Case 4, $n=1$ and case 5, $n=1$ are
isomorphic. Aside from this, two singularities are
isomorphic iff they correspond to the same case and the same
value of $n$.
$$
\begin{tabular}{|c|l|c|}
\hline case & \qquad equation & $L$ \\
\hline $cA_1(1)$ & $x^2+y^2+ z^2\pm t^{2n+1}$ & $S^2$ \\
\hline $cA_1(2)$ & $x^2+y^2- z^2\pm t^{2n+1}$ & $S^2$ \\
\hline $cA_1(3)$& $x^2+y^2+z^2+ t^{2n}$ & $\emptyset$ \\
\hline $cA_1(4)$& $x^2+y^2+z^2-t^{2n}$ & $S^2\uplus S^2$ \\
\hline $cA_1(5)$& $x^2+y^2-z^2+ t^{2n}$ & $S^2\uplus S^2$ \\
\hline $cA_1(6)$& $x^2+y^2-z^2- t^{2n}$ & $S^1\times S^1$ \\
\hline
\end{tabular}
$$
\end{thm}
Proof. The equations follow from (\ref{ca.thm}), once we
note that
after multiplying by $\pm 1$ we may assume that the
quadratic part has at least 2 positive eigenvalues.
The topology is easy to figure out. Since all the claims are
special cases of the next result, I discuss them in more
detail there.\qed
\begin{thm}\label{can-top.thm} A 3-dimensional terminal
singularity of type $cA_{>1}$ over $\r$ is equivalent to
a form
$$ x^2\pm y^2\pm h(z,t)\prod_{i=1}^m f_i(z,t)=0,
$$
where the $f_i$ are irreducible power series (over $\r$)
such that
$(f_i(z,t)=0)$ changes sign on $\r^2\setminus\{0\}$ and
$h(z,t)$ is positive on $\r^2\setminus\{0\}$. The
following table gives a complete list of the possibilites
for the topology of $X(\r)$.
$$
\begin{tabular}{|l|l|c|}
\hline \quad case & \qquad equation & $L(X(\r))$ \\
\hline $cA_{>1}^+(0,+)$& $x^2+y^2+h$ & $\emptyset$ \\
\hline $cA_{>1}^+(0,-)$& $x^2+y^2-h$ & $S^1\times S^1$ \\
\hline $cA_{>1}^+(m)$& $x^2+y^2\pm hf_1\cdots f_m$ &
$\uplus m S^2$\\
\hline $cA_{>1}^-(0)$& $x^2-y^2\pm h$ & $S^2\uplus S^2$ \\
\hline $cA_{>1}^-(m)$& $x^2-y^2\pm hf_1\cdots f_m$ &$M_{m-1}$\\
\hline
\end{tabular}
$$
\end{thm}
Proof. We already have the form $x^2\pm y^2+f(z,t)$ by
(\ref{ca.thm}). Write
$f$ as a product of irreducible power series over $\r$.
Those factors which do not vanish on $\r^2\setminus\{0\}$
are multiplied together to get $h$. By writing
$\pm h$ we may assume that $h$ is positive on
$\r^2\setminus\{0\}$. (Since the signs of the other factors
are not fixed, the sign of $h$ matters only if there are no
other factors.) Let
$f_i$ be the remaining factors of $f$.
Assume now that we are in the $cA^+$-case: $x^2+ y^2\pm
h\prod f_i$. Projection to the $(z,t)$-plane is a proper map
whose fibers are as follows:
\begin{enumerate}
\item $S^1$ if $\pm h(z,t)\prod f_i(z,t)<0$,
\item a point if $\pm h(z,t)\prod f_i(z,t)=0$,
\item empty if $\pm h(z,t)\prod f_i(z,t)>0$.
\end{enumerate}
If $m=0$ then $X(\r)\setminus\{0\}$ is a circle bundle
over either
$\r^2\setminus\{0\}$ or over the empty set. The first case
gives $L\sim S^1\times S^1$ by (\ref{orient}).
If $m>0$, we have to describe the semi-analytic set
$U:=(\prod f_i(z,t)\leq 0)\subset
\r^2$. Semi-analytic sets can be triangulated
(cf. \cite[9.2]{BCR87}), thus in a neighborhood of the origin, $U$
is the cone over
$U\cap (z^2+t^2=\epsilon)$.
Each $(f_i=0)$ is an irreducible curve germ over $\r$, thus
homeomorphic to $\r^1$. So each $f_i$ has 2 roots on the
circle
$(z^2+t^2=\epsilon)$. Hence $U\cap (z^2+t^2=\epsilon)$ is
the disjoint union of $m$ closed arcs. Therefore $L$ has $m$
connected components, each homeomorphic to $S^2$.
The second possibility is the $cA^-$-case: $x^2- y^2- h\prod
f_i$. (The two choices of $\pm h$ are equivalent by
interchanging
$x$ and $y$.) Here we project to the $(y,z,t)$-hyperplane.
The fiber over a point $(y,z,t)$ is
\begin{enumerate}
\item 2 points if $y^2+ h(z,t)\prod f_i(z,t)>0$,
\item 1 point if $y^2+ h(z,t)\prod f_i(z,t)=0$,
\item empty if $y^2+ h(z,t)\prod f_i(z,t)<0$.
\end{enumerate} Thus we have to determine the region
$$ U:=\{y^2+ h(z,t)\prod f_i(z,t)\geq 0\}\subset
(y^2+z^2+t^2=\epsilon)\sim S^2,
$$ and then take its double cover to get $L$.
If $m=0$ then $U=S^2$ and so $L=S^2\uplus S^2$. If $m>0$
then $h\prod f_i$ is negative on $m$ disjoint arcs in the
circle
$(z^2+t^2=\epsilon)$, and $y^2+ h(z,t)\prod f_i(z,t)$ is
negative in contractible neighborhoods of these intervals.
Thus
$U=S^2\setminus(\mbox{$m$-discs})$ and so $L$ is a surface
of genus
$m-1$, orientable by (\ref{orient}).
\qed
\begin{exmp} It is quite instructive to consider the following
incorrect approach to the topology of $cD$ and $cE$-type
singularities. I illustrate it in the $cE_6$-case.
Over $\r$, a surface singularity of type $E_6$ is $x^2+y^2\pm z^4$.
In both cases, projection to the $(x,z)$-plane is a homeomorphism.
Consider a $cE_6$-type point $X$. If $L(X(\r))$ has several
connected components, then a suitable hyperplane intersects at
least two of them. By a small perturbation we obtain an
$E_6$-singularity as the intersection, thus we conclude that
$L(X(\r))$ is connected. This is especially suggestive if we note
that instead of a plane we could use a small perturbation of any
smooth hypersurface.
Unfortunately the conclusion is false, as we se in
(\ref{ce6.exmp}). $L(X(\r))$ can have several components,
and some of them are not seen by general hypersurface sections.
These look like very ``thin" cones, as opposed to the main component
which is ``thick". It would be interesting to give precise
meaning to this observation and to see its significance in the
study of singularities.
\end{exmp}
The following approach to the topology of $cD$ and $cE$-type
singularities is taken from
\cite[Sec.12]{AGV85}.
\begin{say}[Deformation to the weighted tangent cone]{\ }
Let $X:=(f(x_1,\dots,x_n)=0)$ be a hypersurface singularity.
For simplicity of notation I assume that $f$ converges for
$|x_i|< 1+\delta$. Assign integral weights to the variables
$w(x_i)=w_i$ and write $f$ as the sum of weighted
homogeneous pieces
$$ f=f_d+f_{d+1}+f_{d+2}+\dots,
$$ where $f_s$ is weighted homogeneous of degree $s$. For a
parameter
$\lambda\neq 0$ set
\begin{eqnarray*} f^{\lambda}(x_1,\dots,x_n):=
\lambda^{-d}f(\lambda^{w_1}x_1,\dots,\lambda^{w_n}x_n)\\ =
f_d+\lambda f_{d+1}+\lambda^2f_{d+2}+\dots
\end{eqnarray*} This suggests that if we define $f^0:=f_d$
then
$$ X^{\lambda}:=(f^{\lambda}=0)\qtq{for} \lambda\in
\r
$$
is a ``nice" family of hypersurface singularities. For
$\lambda\neq 0$ they are all isomorphic to $(f=0)$ and for
$\lambda=0$ we obtain the weighted tangent cone
$(f_d=0)$.
This can be used to determine the topology of $X(\r)$ in 2
steps. First describe $X^0$ and then try to relate
$X^{\lambda}$ and $X^0$ for small values of $\lambda$.
Let $w$ be a common multiple of the $w_i$ and set
$u_i=w/w_i$.
\end{say}
\begin{prop}\label{def.nc.prop}
Notation as above. Assume that $X=(f=0)$ is an isolated
hypersurface singularity.
Then there is a $0<\lambda_0$ such that for every
$0<\lambda\leq \lambda_0$
\begin{enumerate}
\item $L^{\lambda}:=X^{\lambda}\cap (\sum x_i^{2u_i}=1)$ is
smooth, and
\item $X^{\lambda}\cap (\sum x_i^{2u_i}\leq 1)$ is
homeomorphic to the cone over
$L^{\lambda}$.
\end{enumerate}
\end{prop}
Proof. The map $\r^m\to \r^+$ given by
$(x_1,\dots,x_n)\mapsto\sum x_i^{2u_i}$ is proper. Thus its
restriction to
$X^{\lambda}$ is also proper. The proposition follows once
we establish that the resulting map
$$ t:X^{\lambda}\to \r
$$ has no critical points with critical value in $(0,1]$ for
$0<\lambda\leq \lambda_0$.
The critical values of a real algebraic morphism form a
semi-algebraic set (cf. \cite[9.5]{BCR87}), thus there is a
$0<\mu_0$ such that
$t:X^1\to \r$ has no critical values in $(0,\mu_0]$. The
following diagram is commutative
$$
\begin{array}{ccc} X^{\lambda} & \stackrel{x_i\mapsto
\lambda^{-w_i}x_i}{\longrightarrow} & X^1\\ t\downarrow\ &
& \
\downarrow t\\
\r & \stackrel{s\mapsto \lambda^{-w}s}{\longrightarrow}& \r
\end{array}
$$ which shows that (\ref{def.nc.prop}) holds with
$\lambda_0=\mu_0^{1/w}$.\qed
So far we have not done much, but the advantage of this
approach is that we can view $L^{\lambda}$ as a deformation
of the compact real algebraic variety $L^0$. If $L^0$ is
smooth then this deformation is locally trivial
differentiably. Thus we obtain:
\begin{cor} Assume that $(f_d=0)$ defines an isolated
singularity. Then $L^{\lambda}$ is diffeomorphic to
$L^0$.\qed
\end{cor}
This is sufficient to describe the the topology of
``general"
members of several families of terminal
singularities:
\begin{cor}\label{ce.top.gen} Let
$X$ be a terminal
singularity given by one of the of the following equations:
\begin{enumerate}
\item[$cD_4$] \quad $x^2+f_{\geq 3}(y,z,t)$, where $f_3=0$
has no real singular point.
\item[$cE_6$] \quad $x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq
4}(z,t)$, where
$h_4$ has no multiple real linear factor.
\item[$cE_8$] \quad $x^2+y^3+yg_{\geq 4}(z,t)+h_{\geq
5}(z,t)$, where
$h_5$ has no multiple real linear factor.
\end{enumerate}
Then:
\begin{enumerate}
\item[$cD_4$] \quad $L(X(\r))\sim S^2$ if $(f_3=0)\subset \r\p^2$
has one connected component and $L(X(\r))\sim S^2\uplus (S^1\times
S^1)$ if
$(f_3=0)$ has two connected components.
\item[$cE_6$] \quad $L(X(\r))\sim S^2$.
\item[$cE_8$] \quad $L(X(\r))\sim S^2$.
\end{enumerate}
\end{cor}
Proof. We use deformation to the weighted tangent cone
with (suitable integral multiples of the) weights
$(1/2,1/3,1/3,1/3)$ in the $cD_4$-case,
$(1/2,1/3,1/4,1/4)$ in the $cE_6$-case, and
$(1/2,1/3,1/5,1/5)$ in the $cE_8$-case. The equations for
$X^0$ are
$x^2+f_3(y,z,t)=0$,
$x^2+y^3+h_4(z,t)=0$ and $x^2+y^3+h_5(z,t)=0$. Our
conditions guarantee that $X^0(\r)$ has isolated
singularities, thus it is sufficient to determine
$L(X^0(\r))$.
In the $cE$-cases, projection to the $(x,z,t)$ hyperplane is a
homeomorphism from $X^0(\r)$ to $\r^3$, thus $L(X^0(\r))\sim
S^2$
In the $cD_4$-cases we project to the $(y,z,t)$-hyperplane.
As in the proof of (\ref{can-top.thm}), we can get
$L(X^0(\r))$ once we know the set
$U\subset (y^2+z^2+t^2=1)$ where $f_3$ is nonnegative. The
boundary
$\partial U$ doubly covers the projective curve
$(f_3=0)\subset \r\p^3$. If
$(f_3=0)\subset \r\p^2$ has one connected component then
it is a pseudo-line and $\partial U$ is a connected double
cover, hence $U$ is a disc. If $(f_3=0)$ has two connected
components, then one is a pseudo-line the other an oval.
$\partial U$ has 3 connected components, and $U$ is a disc
plus an annulus. Thus $L(X^0(\r))\sim S^2\uplus (S^1\times
S^1)$.
\qed
\begin{rem} In the $cE$ cases of the above example, projection to
the
$(x,z,t)$ plane is a homeomorphism from $X^0(\r)$ to $\r^3$
even if $h_4$ or $h_5$ have multiple factors. In these
cases, however, we can not conclude that
$X(\r)$ is also homeomorphic to $\r^3$. In fact we see in
(\ref{ce6.exmp}) that this is not always true.
\end{rem}
Similar arguments work in some of the $cD_{>4}$-cases:
\begin{cor}\label{cd.top.gen} Let $X$ be a terminal
singularity given by equation
$x^2+y^2z+h_{\geq s}(z,t)$, where $z\not\vert h_{s}$ and
$h_{s}$ has no multiple real linear factors. Let $s$ be the
number of real linear factors of $h_{s}$. There are three
cases:
\begin{enumerate}
\item $s=2r+1$ and
$L(X(\r))\sim M_r\uplus rS^2$;
\item $s=2r,\ h(0,1)<0$ and
$L(X(\r))\sim M_r\uplus (r-1)S^2$;
\item $s=2r,\ h(0,1)>0$ and
$L(X(\r))\sim M_{r-1}\uplus rS^2$.
\end{enumerate}
\end{cor}
Proof. We use deformation to the weighted tangent cone with
weights
$(1/2,(s-1)/2s,1/s,1/s)$. Thus we need to figure out the
topology of
$x^2+y^2z+h_{(n-1)}(z,t)=0$. As before, this reduces to
understanding the set where $y^2z+h_{s}(z,t)\leq 0$. This
can be done by projecting to the $(z,t)$ plane. Details are
left to the reader.
\qed
\begin{say}\label{choose.weights} For many terminal
singularities one can not choose weights so that the
weighted tangent cone has an isolated singularity at the origin, but
in all cases it is possible to choose weights so that the
weighted tangent cone has 1-dimensional singular locus:
$$
\begin{tabular}{|c|l|ccc|}
\hline name &\quad equation is \quad $x^2+$ &$w(y)$ &$w(z)$&$w(t)$\\
\hline $cD_4$& $f_{\geq 3}(y,z,t)$ &
$\frac13$&$\frac13$&$\frac13$\\
\hline $cD_{>4}(1)$& $y^2z+ayt^r+h_{\geq s}(z,t)$ &
$\frac{s-1}{2s}$&$\frac1{s}$&$\frac1{s}$\\
\hline $cD_{>4}(2)$& $y^2z\pm yt^r+h_{\geq s}(z,t)$ &
$\frac{r-1}{2r-1}-$\mbox{\scriptsize $\epsilon$}&
$\frac1{2r-1}+$\mbox{\scriptsize $2\epsilon$}
&$\frac1{2r-1}+\frac{\epsilon}{r}$\\
\hline $cE_6$ &$y^3+yg_{\geq 3}(z,t)+h_{\geq 4}(z,t)$ &
$\frac13$&$\frac14$&$\frac14$\\
\hline $cE_7$ &$y^3+yg_{\geq 3}(z,t)+h_{\geq 5}(z,t)$ &
$\frac13$&$\frac29$&$\frac29$\\
\hline $cE_8$ &$y^3+yg_{\geq 4}(z,t)+h_{\geq 5}(z,t)$ &
$\frac13$&$\frac15$&$\frac15$\\
\hline
\end{tabular}
$$
In the $cD_{>4}$ case we asume that $h_s\neq 0$ and use
the first weight sequence if $a=0$ or $2r>s+1$ and the
second weight sequence if
$2r\leq s+1$, where $\epsilon$ is a small positive number.
(We could use $\epsilon=0$ except when $2r=s-1$.) Let
$(w_1,w_2,w_3,w_4)$ be integral multiples of these weights.
The weighted tangent cone, and its singularities are the
following:
$$
\begin{tabular}{|c|l|l|}
\hline name &weighted tangent cone& \qquad singularities \\
\hline $cD_4$& $x^2+f_3(y,z,t)$ &at singular pts of
$(f_3=0)$\\
\hline $cD_{>4}(1)$& $x^2+y^2z+h_s(z,t)$&at multiple real
factors of
$zh_s$\\
\hline $cD_{>4}(2)$& $x^2+y^2z\pm yt^r$ & at the $z$-axis\\
\hline $cE_6$ &$x^2+y^3+h_4(z,t)$& at multiple real factors
of $h_4$\\
\hline $cE_7$ &$x^2+y^3+yg_3(z,t)$& at real factors of
$g_3$\\
\hline $cE_8$ &$x^2+y^3+h_5(z,t)$& at multiple real factors
of $h_5$\\
\hline
\end{tabular}
$$ These equations have the form $x^2+F(y,z,t)$ and the
deformation to the weighted tangent cone leaves this form
invariant:
$$
x^2+F^{\lambda}(y,z,t)=x^2+\lambda^{-d}F(\lambda^{w_2}y,\lambda^{w_3}z,\lambda^{w_4}t).
$$
Set
$$ U^{\lambda}:=\{(y,z,t)\vert F^{\lambda}(y,z,t)\leq
0\subset (y^{2u_2}+z^{2u_3}+t^{2u_4}=1)\}.
$$
$U^{\lambda}$ is a semi-algebraic set and its boundary is
the real algebraic curve
$$ C^{\lambda}:=\{(y,z,t)\vert F^{\lambda}(y,z,t)=0\subset
(y^{2u_2}+z^{2u_3}+t^{2u_4}=1)\}.
$$ We have established the following:
\end{say}
\begin{prop} $C^{\lambda}$ is a deformation of the real
algebraic curve
$C^0$ inside the smooth real algebraic surface
$(y^{2u_2}+z^{2u_3}+t^{2u_4}=1)$.\qed
\end{prop}
\begin{say} The deformations of real algebraic curves can be
understood in two steps (cf. \cite{Viro90}). Put small discs around
the singularities. Outside the discs all small deformations are
topologically trivial and inside the discs we have a local
problem involving real curve singularities. Here we have the
advantage that
$(y^{2u_2}+z^{2u_3}+t^{2u_4}=1)$ is an affine algebraic
surface, thus we can choose the local deformations
independently and they can always be patched together.
Thus we can describe the possible cases for $C^{\lambda}$,
and thereby the topological types of the corresponding
3-dimensional terminal singularities, if we can describe the
deformations of the occurring real plane curve
singularities. By looking at the equations we see that the
only singularities that we have to deal with are the
2-variable versions of the Du Val singularities:
\begin{enumerate}
\item[$A_n$] \quad $y^2\pm z^{n+1}=0$;
\item[$D_n$] \quad $y^2z\pm z^{n-1}=0$;
\item[$E_6$] \quad $y^3+z^4=0$;
\item[$E_7$] \quad $y^3\pm yz^3=0$;
\item[$E_8$] \quad $y^3+z^5=0$.
\end{enumerate}
For all of these cases, a complete list of the topological types of
real deformations is known \cite{Chislenko88}. The list can also
be found in \cite[Figs. 16--28]{Viro90}, which contains many
further examples.
\end{say}
\begin{exmp} Consider for instance the $cD_4$ cases. The
various possibilities for $f_3$ are easy to enumerate. The
most interesting is $f_3=yzt$. Here $C^0$ is the union of
the 3 coordinate hyperplanes intersecting
$(y^{6}+z^{6}+t^{6}=1)$. We have 6 singular points of type
$u^2-v^2=0$. At each of them we can choose a deformation
$u^2-v^2\pm \epsilon=0$. This gives $2^6$ possibilities. The
symetries of the octahedron act on the configurations so it is easy
to get a complete list.
At the end we get $7$ possible topological
types for $L(X(\r)$ where
$X=(x^2+yzt+f_{\geq 4}(y,z,t)=0)$:
$$
M_2, M_1\uplus S^2, M_1, S^2, 2S^2, 3S^2, 4S^2.
$$
It truns out that these exhaust all the cases given by
$cD_4$.
\end{exmp}
\begin{exmp}\label{ce6.exmp} Consider the $cE_6$-type points
$$
x^2+y^3+yg_{\geq 3}(z,t)\pm z^2t^2+h_{\geq 5}(z,t).
$$
Using the methods of (\ref{nf.meth.weighted}) these can be brought
to the form
$$
x^2+y^3\pm z^2t^2+ya(z)+yb(t)+c(z)+d(t).
$$
The weighted tangent cone, $(x^2+y^3\pm z^2t^2=0)$
is singular along the $z$ and $t$-axes. In order to understand the
sigularity type of $C^{\lambda}$, say along the positive
$z$-halfaxis, set
$t=\epsilon$. We get an equation
$$
x^2+y^3\pm z^2\epsilon^2+ya(z)+yb(\epsilon)+c(z)+d(\epsilon).
$$
$\mult_0a\geq 3$ and $\mult_0c\geq 5$, thus all the terms
involving $z$ can be absorbed into $z^2$, and we obtain
the equivalent form
$$
x^2\pm z^2+y^3+yb(\epsilon)+d(\epsilon).
$$
The cubic $y^3+yb(\epsilon)+d(\epsilon)$ has 3 real roots if
$4b(\epsilon)^3+27d(\epsilon)^2<0$ and
1 real root if
$4b(\epsilon)^3+27d(\epsilon)^2>0$.
$C^0$ is homeomorphic to $S^1$ and it has 4 singular points
(along the $z$ and $t$ halfaxes). $C^{\lambda}$ is a smooth curve
which has an oval near a singular point of $C^0$ if the
corresponding cubic has 3 real roots and no ovals if only 1
real root. Thus $C^{\lambda}$ has at most 5 connected components.
We can also determine the location of the ovals relative to the
``main component" of $C^{\lambda}$. In deforming $z^2-y^3=0$,
the oval can appear only toward the negative $y$-direction.
Putting all this together, we get the following possibilities
for $L(X(\r))$:
$$
\begin{array}{ll}
rS^2,\ 1\leq r\leq 5 &\mbox{in the $+z^2t^2$-case, and} \\
M_r,\ 0\leq r\leq 4 &\mbox{in the $-z^2t^2$-case.}
\end{array}
$$
\end{exmp}
\section{The topology of terminal quotient singularities}
Let $0\in X$ be 3-fold terminal singularity and $\pi:\tilde
X\to X$ its index one cover. As we proved,
$X=\tilde X/\frac{1}{n}(a_1,\dots,a_m)$ where $n$ is the
index of $X$ and the $a_i$ are integers. We use this
representation to determine the topology of $X$ in terms of
the already known topology of $\tilde X$.
The main question is to determine the real points of
${\Bbb A}^m/\frac{1}{n}(a_1,\dots,a_m)$. Let
$\sigma:\c^m\to \c^m$ be the corresponding action of $1\in
\z_n$.
The answer depends on the parity of $n$. First we discuss
the odd index cases which are easier.
\begin{prop}\label{odd.quot}
Assume that $n$ is odd and set
$Y={\Bbb A}^m/\frac{1}{n}(a_1,\dots,a_m)$. Then the induced
map
$\r^n\to Y(\r)$ is a homeomorphism.
\end{prop}
Proof. Let $R\subset \r[[x_1,\dots,x_m]]$ denote the ring of
invariant functions. A point $P\in {\Bbb A}^m$ maps to a
real point of $X$ iff
$f(P)\in \r$ for every $f\in R$. Let $\epsilon$ be a
primitive $n^{th}$ root of unity. If $P=(p_1,\dots,p_m)$ is
real then
$$
\sigma^b(p_1,\dots,p_m)=(\epsilon^{ba_1}p_1,\dots,\epsilon^{ba_m}p_m)
$$
is also real iff $\sigma^b(P)=P$. This shows that the
quotient map
$\r^n\to X(\r)$ is an injection.
Let $Q\in X(\r)$ be a point. Then $\pi^{-1}(Q)\subset {\Bbb
A}^m$ has an odd number of closed points over $\c$ (usually
$n$ of them) and as a scheme it is defined over $\r$. Thus
it has a real point, hence
$\r^n\to X(\r)$ is also surjective.\qed
\begin{cor}\label{oddind.termquot} Let $0\in X$ be a 3-fold
terminal singularity of odd index and $\pi:\tilde X\to X$
its index one cover. Then $\pi: \tilde X(\r)\to X(\r)$ is a
homeomorphism.\qed
\end{cor}
The even index case is more subtle. For purposes of
induction we allow the case when $n$ is odd. Consider the
action
$\frac{1}{n}(a_1,\dots,a_m)$ on ${\Bbb A}^m$. Write
$n=2^sn'$ where
$n'$ is odd. Let $\eta$ be a primitive $2^{s+1}$-st root of
unity and
$j:\r^m\to \c^m$ the map
$$ j(x_1,\dots,x_m)=(\eta^{a_1}x_1,\dots,\eta^{a_m}x_m).
$$ (If $n$ is odd then $\eta=-1$, hence $j(\r^m)=\r^m$.)
Write
$a_i=2^ca'_i$ such that $a'_i$ is odd for some $i$. If
$s>c$, let
$\tau:\c^m\to \c^m$ be the $\z_2$-action
$$
\tau(x_1,\dots,x_m)=((-1)^{a'_1}x_1,\dots,(-1)^{a'_m}x_m).
$$ For $s=c$ let $\tau$ be the identity. Note that both
$\r^n$ and
$j(\r^n)$ are $\tau$-invariant.
\begin{prop}\label{2ind.quot} Set
$Y={\Bbb A}^m/\frac{1}{n}(a_1,\dots,a_m)$. Define $j$ and
$\tau$ as above. Then $Y(\r)$ is the quotient of $\r^m\cup
j(\r^m)$ by $\tau$.
\end{prop}
Proof. The proof is by induction on $m$ and $n$. We can
assume that the action is faithful, that is
$\sum b_ia_i=1$ is solvable in integers. Indeed, for non
faithful actions we get the same quotient from a smaller
group action. The definitions of $j$ and $\tau$ are set
up such that they do not change if we change the group this
way.
Set $F:=\prod_i x_i^{b_i}$. By induction on $m$ we know that
(\ref{2ind.quot}) holds on each coordinate hyperplane. Thus
we have to deal with points $P=(p_1,\dots,p_m)$ such that
each
$p_i\neq 0$.
Assume that $\pi(P)$ is real. Let $\epsilon$ be a primitive
$(2n)^{th}$ root of unity.
$p_i^n$ is real, hence
$p_i=\epsilon^{c_i}\cdot(\mbox{real number})$ for some
$c_i\in \z$. Thus $F(P)=\epsilon^{c}\cdot(\mbox{real
number})$.
$F(\sigma(P))=\epsilon^2F(P)$, hence by replacing
$P$ by $\sigma^r(P)$ for some $r$ we may assume that
$F(P)\in\r$ or $F(P)\in\eta\cdot\r$.
Assume first that $F(P)$ is real. For each $i$ the function
$F^{n-a_i}x_i$ is invariant, hence has a real value at $P$.
Thus $P\in
\r^n$. If $F(P)\in\eta\cdot\r$ then the same agrument shows
that
$p_i\in \eta^{a_i}\cdot \r$, thus $P\in j(\r^m)$.
This shows that $\r^m\cup j(\r^m)\to Y(\r)$ is surjective.
It is also
$\tau$-invariant. Finally, if $P=(p_1,\dots,p_m)\in
\r^m\cup j(\r^m)$ then
$$
\sigma^s(P)=(\epsilon^{ba_1}p_1,\dots,\epsilon^{ba_m}p_m)
\in \r^m\cup j(\r^m)
$$ iff
$\sigma^s(P)=P$ or $\sigma^s(P)=\tau(P)$. Thus
$\r^m\cup j(\r^m)\to Y(\r)$ is $2:1$ for $s>c$ and $1:1$ for
$s=c$.
\qed
\begin{defn}
Let $F\in \r[[x_1,\dots,x_m]$ be a power series, homogeneous
of grade $d$ under the grading $\frac{1}{n}(a_1,\dots,a_m)$.
Let $\eta$ be as above. Define the {\it companion} $F^c$ of $F$
with respect to the action $\frac{1}{n}(a_1,\dots,a_m)$
by
$$
F^c(x_1,\dots,x_m):=\eta^{-d}F(\eta^{a_1}x_1,\dots,\eta^{a_m}x_m).
$$ Note that $F^c\in \r[[x_1,\dots,x_m]]$.
\end{defn}
\begin{cor}\label{2ind.termquot} Let $0\in X$ be a 3-fold
terminal singularity of even index and $\pi:\tilde X\to X$
its index one cover. Then
$$
L(X(\r)) \sim
L(\tilde X(\r))/(\tau)\uplus
L(\tilde X^c(\r))/(\tau).
$$
\end{cor}
Proof. We use the notation of (\ref{2ind.quot}). Let $F=0$
be the equation of $\tilde X$ and let
$W:=(F=0)\cap(\r^m\cup j(\r^m))$. Then $X(\r)$ is the
quotient of $W$ by $\tau$.
$(F=0)\cap\r^m=\tilde X(\r)$. $(F=0)\cap j(\r^m)$ can be
identified with the set of real zeros of
$F(\eta^{a_1}x_1,\dots,\eta^{a_m}x_m)=0$. (The normalizing
factor
$\eta^{-d}$ does not change the set of zeros.)
In the terminal case the group action is fixed point free on
$(F=0)\setminus\{0\}$, thus $(F=0)\cap\r^m$ and $(F=0)\cap
j(\r^m)$ intersect only at the origin.\qed
As a byproduct we obtain the following:
\begin{cor}\label{h.ind.not.isol} Let $0\in X$ be a 3-fold
terminal singularity of index$>1$. Then
$0\in X(\r)$ is not an isolated point.
\end{cor}
Proof. Let $\tilde X$ be the index 1 cover. We are done,
unless
$0\in \tilde X(\r)$ is an isolated point. This happens
only in cases
$cA/2, cAx/2$ (and maybe for $cAx/4$) where the equation of
$\tilde X$ is
$F=x^2+y^2+f(z,t)$ and $f(z,t)$ is positive on
$\r^2\setminus\{0\}$.
Let us compute $F^c$. In the $cA/2$ ase we get
$-x^2-y^2+f(iz,t)$ and this has nontrivial solutions in the
$(x,y,t)$-hyperplane. In the $cAx/2$ case we get
$x^2-y^2+f(iz,it)$ and this has nontrivial solutions in the
$(x,y)$-plane.
In the $cAx/4$ case already $\tilde X$ has nontrivial
$\r$-points. Indeed, here $f$ has grade 2, thus every
$t$-power in it has an odd exponent. Thus $f(z,t)$ is not
positive on the $t$-axis.
\qed
\begin{say}[Orientability of index 2 quotients]
We have seen in (\ref{orient}) that every real algebraic hypersurface is
orientable, and so are their quotients by odd order groups
(\ref{odd.quot}). With index 2 quotients, the question of
orientablilty is interesting.
Consider a quotient $(F=0)/\frac12(w_1,\dots,w_m)$.
Let $\sigma$ be the corresponding $\z_2$ action.
We can orient
$X:=(F=0)$ by choosing an orientation of $\r^m$ and at each smooth
point of $X$ we choose the normal vector pointing in the
direction where $F$ is positive. $\sigma$ preserves the orientation of
$\r^m$ iff $\sum w_i$ is even. The parity of $w(F)$ determines the sign
in
$\sigma(F)=\pm F$. Thus $\sigma$ preserves the induced orientation of
$X$ iff $w(F)+\sum w_i$ is even. If $w(F)+\sum w_i$ is odd, the induced
orientation is not preserved. If $\sigma$ fixes a connected component
of (the nonsingular part of) $X(\r)$, then the corresponding quotient
is not orientable. If, however, $\sigma$ only permutes the
connected components of $X(\r)$ then the quotient is still orientable.
Thus we obtain:
\begin{lem}\label{ind2.orient}
Let $0\in X:=(F=0)/\frac12(w_1,\dots,w_m)$ be an isolated singular
point,
with index one cover $\tilde X$ and companion $\tilde X^c$. Then
$L(X(\r))$ is nonorientable iff $w(F)+\sum w_i$ is odd
and $\sigma$ fixes at least one of the connected components
of $L(\tilde X(\r))$ or $L(\tilde X^c(\r))$.\qed
\end{lem}
\end{say}
\begin{exmp}[The topology of $cA/2$ points] {\ }
The simplest case is ${\Bbb A}^3/{\textstyle
\frac1{2}(1,1,1)}$.
The link of ${\Bbb A}^3$ is the sphere
$(x^2+y^2+z^2=\epsilon^2)$. We act by the antipodal map,
and the quotient is $\r\p^2$. The quotient of the purely
imaginary subspace also gives real points, thus
$L(X(\r))\sim 2\r\p^2$.
In the $cA_{>0}/2$ cases write
$$
X:=(x^2\pm y^2+f(z,t)=0)/\frac12(1,1,1,0)
$$
with cover $\tilde X:=(x^2\pm y^2+f(z,t)=0)$.
Only even powers of $z$ occur in $f(z,t)$, thus we can write
$f(z,t)=G(z^2,t)$. As in (\ref{can-top.thm}) we factor it as
$G(z^2,t)=\pm h(z,t)\prod_{i=1}^r f_i(z,t)$.
The companion cover is $\tilde X^c=(x^2\pm y^2-G(-z^2,t)=0)$. We have to
be careful since the product decomposition of $G$ is not preserved. A
factor of $h$ may become indefinite and it can also happen that
two factors become conjugate over
$\r$. Thus we write $-G(-z^2,t)=\pm h'(z,t)\prod_{j=1}^{r'} f'_j(z,t)$.
By (\ref{2ind.termquot}), $L(X(\r))=L(\tilde X(r))/\tau\uplus
L(\tilde X^c(r))/\tau$. In all these cases (\ref{ind2.orient}) shows
that if
$\tau$ fixes a connected component, the quotient is not orientable.
Thus if
$L(\tilde X(r))$ or $L(\tilde X^c(r))$ is connected, the quotient is
not orientable. This holds in all the $cA^-_{>0}(r>0)$ cases.
In the $cA^-_{>0}(0)$ case the equation is $(x^2=y^2+h)$, and each
halfspace of $(x\neq 0)$ contains a unique connected component.
Thus $\tau$ interchanges the two connected components and the quotient
is orentable.
In the $cA^+_{>0}$ case
$(\pm h(z,t)\prod_{i=1}^r f_i(z,t)=0)$ is negative on
$r$ connected regions $P_1,\dots,P_r\subset \r^2$, and
$L(\tilde X(\r))$ consist of $r$ copies of $S^2$, one for
each
$P_j$. The involution $\tau$ fixes the
$t$-axis pointwise, thus if one of the half $t$-axes is
contained in some
$P_j$, then $\tau$ fixes the $S^2$ over that region. The
other copies of
$S^2$ are interchanged. The same holds for
$\tilde X^c$.
Along the $t$-axis $G(z^2,t)$ and $-G(-z^2,t)$ have
opposite signs. Thus among the 4 pairs
$$
\begin{array}{cc} (\mbox{positive half $t$-axis},
G(z^2,t)) & (\mbox{positive half $t$-axis}, -G(-z^2,t))\\
(\mbox{negative half $t$-axis}, G(z^2,t))& (\mbox{negative
half $t$-axis}, -G(-z^2,t))
\end{array}
$$ there are two where the function is negative along the
half axis.
We obtain the following list of possibilities. (We use the notation
$K_r:=S^2\# r\r\p^2$, thus $K_2$ is the Klein bottle.)
$$
\begin{tabular}{|l|l|c|}
\hline \qquad $\tilde X$ & \qquad $\tilde X^c$ & $L(X(\r))$ \\
\hline $cA_0$& $cA_0$ & $\r\p^2\uplus \r\p^2$ \\
\hline $cA_{>0}^-(r>0)$& $cA_{>0}^-(r'>0)$ & $K_r\uplus K_{r'}$ \\
\hline $cA_{>0}^-(r>0)$& $cA_{>0}^-(0)$ & $K_r\uplus S^2$ \\
\hline $cA_{>0}^-(0)$& $cA_{>0}^-(0)$ & $S^2\uplus S^2$ \\
\hline $cA_{>0}^+(r>0)$& $cA_{>0}^+(r'>0)$ &
$2\r\p^2\uplus {\textstyle\frac{r+r'-2}{2}}S^2$ \\
\hline $cA_{>0}^+(r>0)$& $cA_{>0}^+(0,+)$ &
$2\r\p^2\uplus {\textstyle\frac{r-2}{2}}S^2$ \\
\hline $cA_{>0}^+(r>0)$& $cA_{>0}^+(0,-)$ &
$K_2\uplus {\textstyle\frac{r}{2}}S^2$ \\
\hline $cA_{>0}^+(0,-)$& $cA_{>0}^+(0,+)$ &
$K_2$ \\
\hline
\end{tabular}
$$
Thus $X(\r)$ is not orientable, except in the fourth case. This
indeed occurs:
Let $\tilde X:=(x^2-y^2+z^{4m}+t^{2n}=0)$
and
$X:=\tilde X/{\textstyle
\frac12(1,1,1,0)}$ with companion
$\tilde X^c=(-x^2+y^2+z^{4m}+t^{2n}=0)$.
$\tilde X^c\cong \tilde X$ and $L(\tilde X(\r))\sim
S^2\uplus S^2$. $\tau$ interchanges the two
copies of $S^2$. Thus $X(\r)$ is orientable and
$ L(X(\r))\sim S^2\uplus S^2$.
\end{exmp}
|
1997-12-15T18:23:18 | 9712 | alg-geom/9712015 | en | https://arxiv.org/abs/alg-geom/9712015 | [
"alg-geom",
"math.AG"
] | alg-geom/9712015 | H. Esnault | H\'el\`ene Esnault, V. Srinivas, Eckart Viehweg | The universal regular quotient of the Chow group of 0-cycles on a
singular projective variety | Latex 2e | null | null | null | null | We show the existence of a regular universal quotient as a smooth commutative
algebraic group of the Chow group of 0-cycles on a projective reduced variety,
and give over the field of complex numbers an analytic description of it. This
generalizes the classical theory of the Albanese. The Chow group of 0-cycles is
then isomorphic to this smooth algebraic group if and only if it is finite
dimensional in the sense of Mumford. This generalizes the classical theorem of
Roitman.
| [
{
"version": "v1",
"created": "Mon, 15 Dec 1997 17:23:17 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Esnault",
"Hélène",
""
],
[
"Srinivas",
"V.",
""
],
[
"Viehweg",
"Eckart",
""
]
] | alg-geom | \section{Chow groups and regular homomorphisms}
We begin by recalling the definition of the Chow group of
0-cycles $CH^n(X)$, as given in \cite{LW} (see also \cite{BiS}). As in
\cite{BiS}, we adopt the convention that a point lying on a lower
dimensional component of $X$ is deemed to be singular. Let $X_{{\rm sing}}$
denote the (closed) subset of singular points, and $X_{{\rm reg}}=X-X_{{\rm sing}}$ the
complementary open set. The closure of $X_{{\rm reg}}$ is the union of the
$n$-dimensional components of $X$.
The group $Z^n(X)$ of 0-cycles is defined to be the free abelian
group on the closed points of $X_{{\rm reg}}$. The subgroup $R^n(X)$ of cycles
rationally equivalent to 0 is defined using the notion of a
Cartier curve.
\begin{defn}\label{ccurve}
A {\em Cartier curve} is a subscheme $C\subset X$, defined over
$k$, such that
\begin{points}
\item $C$ is pure of dimension 1
\item no component of $C$ is contained in $X_{{\rm sing}}$
\item if $x\in C\cap X_{{\rm sing}}$, then the ideal of $C$ in ${\mathcal O}_{x,X}$ is
generated by a regular sequence (consisting of $n-1$ elements).
\end{points}
\end{defn}
If $C$ is a Cartier curve on $X$, with generic points
$\eta_1,\ldots,\eta_s$, and ${\mathcal O}_{S,C}$ is the
semilocal ring on $C$ of the points of $S=(C\cap
X_{{\rm sing}})\cup\{\eta_1,\ldots,\eta_s\}$, there is a natural map on
unit groups
\[\theta_{C,X}:{\mathcal O}_{S,C}^*\>>> \bigoplus_{i=0}^s{\mathcal O}_{\eta_i,C}^*.\]
Define $R(C,X)={\rm image}\,\theta_{C,X}$. For $f\in R(C,X)$,
define the divisor of $(f)_C$ as
follows: let $C_i$ denote the maximal Cohen-Macaulay subscheme
of $C$ supported on the component with generic point $\eta_i$.
Then for any $x\in C_i$ the map
\[{\mathcal O}_{x,C_i}\>>>{\mathcal O}_{\eta_i,C_i}={\mathcal O}_{\eta_i,C}\]
is the injection of a Cohen-Macaulay local ring of dimension 1
into its total quotient ring. If $f_i$ is the component of $f$ in
${\mathcal O}_{\eta_i,C}$, then $f_i=a_x/b_x$ for some non zero-divisors
$a_x,b_x\in{\mathcal O}_{x,C_i}$. Define
\[(f)_C=\sum_{i=1}^s(f_i)_{C_i}=\sum_{i=1}^s\sum_{x\in
C_i}(\ell({\mathcal O}_{x,C_i}/a_x{\mathcal O}_{x,C_i})-\ell({\mathcal O}_{x,C_i}/b_x{\mathcal O}_{x,C_i}))
\cdot [x].\]
Standard arguments imply that this is well-defined ({\it i.e.\/},\ the
coefficient of $[x]$ is independent of the choice of the
representation $f_i=a_x/b_x$, and vanishes for all but a finite
number of $x$).
Suppose $C$ is reduced. Then in the above considerations, ${\mathcal O}_{x,C_i}$ is
an integral domain with quotient field ${\mathcal O}_{\eta_i,C}$. If $v_1,
\ldots,v_m$ are the discrete valuations of ${\mathcal O}_{\eta_i,C}$ centered at
$x$, then the multiplicity of $x$ in $(f)_{C_i}$ is
\begin{equation}\label{val}
\ell({\mathcal O}_{x,C_i}/a_x{\mathcal O}_{x,C_i})-\ell({\mathcal O}_{x,C_i}/b_x{\mathcal O}_{x,C_i})=
\sum_{j=1}^m v_j(f_i)
\end{equation}
(compare \cite{Fulton}, Example~A.3.1.).
In fact, let $R$ be the integral closure of $O={\mathcal O}_{x,C_i}$ in
${\mathcal O}_{\eta_i,C}$. The Chinese remainder theorem implies that
\[\ell(R/a_xR) = \sum_{j=1}^m v_j(a_x),\]
and similarly for $b_x$. Multiplying $a_x$ and $b_x$ by the same
element of $O$ we may assume that both $a_xR$ and $b_xR$ are contained
in $O$, and $$\ell(O/a_xO)+\ell(R/O) = \ell(R/a_xR) + \ell(a_xR/a_xO).$$
Since $a_x \neq 0$ the second terms on both sides are equal.
\begin{defn}\label{rat-equ}
Let $U \subset X_{\rm reg}$ be an open dense subscheme.
$R^n(X,U)$ is defined to be the subgroup of $Z^n(U)$
generated by elements $(f)_C$ as $C$ ranges over all Cartier
curves with $C\cap U$ dense in $C$, and $f\in R(C,X)$
with $(f)_C \in Z^n(U)$.
For $U=X_{{\rm reg}}$ we write $R^n(X)$ instead of $R^n(X,X_{{\rm reg}})$
and define
$$CH^n(X) = Z^n(X)/R^n(X).$$
Mapping a point $x \in X_{{\rm reg}}$ to its rational equivalence
class defines a map
$$\gamma :X_{{\rm reg}} \>>> CH^n(X).$$
If $U_1, \ldots , U_r$ denote the irreducible components of $X_{\rm reg}$
then $Z^n(X)_{\deg 0}$ and $CH^n(X)_{\deg 0}$
denote the subgroup of $Z^n(X)$ and $CH^n(X)$, respectively, of
cycles $\delta$ with $\deg(\delta|_{U_i})=0$
for $i = 1, \ldots r$.
\end{defn}
As noted in \cite{BiS}, lemma~1.3 of \cite{LW} allows one to
restrict to considering only curves $C$ such that $C\cap
X_{{\rm reg}}$ has no embedded points, and any irreducible component
$C'$ of $C$ which lies entirely in $X_{{\rm reg}}$ occurs in $C$ with
multiplicity 1. The moving lemmas 2.2.2 and 2.2.3 of \cite{BiS}
allow stronger restrictions on $C$:
\begin{lemma}\label{moving}
Let $A \subset X_{{\rm sing}}$ be a closed subset of dimension $\leq n-2$,
and let $D \subset X$ be a closed subset of dimension $\leq n-1$. Then any
element $\delta\in R^n(X)$ can be written in the form
$\delta=(f)_C$
for a single (possibly reducible) Cartier curve $C$, such that
\begin{enumerate}
\item[(a)] $C$ is reduced
\item[(b)] $C\cap A=\emptyset$
\item[(c)] $C\cap D$ is empty or consists of finitely many points.
\end{enumerate}
\end{lemma}
\begin{cor}\label{equidim}
If $U \subset X_{\rm reg}$ is an open and dense subscheme, then
$$CH^n(X)=Z^n(U)/R^n(X,U) \mbox{ \ \ and \ \ }CH^n(X)_{\deg
0}=Z^n(U)_{\deg 0}/R^n(X,U).$$
\end{cor}
\begin{proof}
First note that the zero cycles supported on $U$ generate
$CH^n(X)$ since the corresponding assertion holds true for
curves. The moving lemma \ref{moving} for $D=X-U$
implies that $R^n(X)\cap Z^n(U)=R^n(X,U)$.
\end{proof}
\begin{rmk}
Let $X^{(n)}$ denote the union of the $n$-dimensional irreducible
components of $X$, and let $X^{<n}$ be the union of the lower dimensional
components. Applying the corollary to $X^{(n)}$ and the open subset
$U=X^{(n)}-X_{{\rm sing}}=X-X_{sing}$, we see that the natural map from
$CH^n(X)$ to $CH^n(X^{(n)})$ is surjective. It seems plausible that a
stronger form of lemma~\ref{moving} holds, where $A$ is allowed to be any
closed subset of $X$ of codimension $\geq 2$ which is disjoint from
${\rm supp}\,(\delta)$. If this is true, then applying it to $X^{(n)}$ with
$A=X^{(n)}\cap X^{<n}$, one sees that for any $\delta\in R^n(X^{(n)})\cap
Z^n(X)$ there exists a reduced Cartier curve $C$ in $X^{(n)}$, disjoint from
$A$, and $f \in R(C,X^{(n)})$ with $\delta=(f)_C$. Then $C$ is also a
Cartier curve on $X$, and $\delta\in R^n(X)$. We deduce that $CH^n(X)\to
CH^n(X^{(n)})$ is an isomorphism. We have as yet been unable to prove this.
\end{rmk}
\begin{rmk}\label{rmk-equi} Keeping the notation from the previous remark,
we note further that for $k={\mathbb C}$, the natural maps
\[H^{2n}(X,{\mathbb Z}(n))\>>> H^{2n}(X^{(n)},{\mathbb Z}(n)),\;\; D^n(X)\>>>
D^n(X^{(n)}),\;\; A^n(X)\>>> A^n(X^{(n)})\]
are isomorphisms, since $X^{<n}$ has constructible cohomological
dimension $\leq 2(n-1)$ and coherent cohomological dimension $\leq n-1$.
\end{rmk}
As reflected by the notation, $R(C,X)$ depends on the pair
$(C,X)$, and is not necessarily intrinsic to $C$. In fact,
since we have not imposed any unit condition at singular points
of $C$ which lie in $X_{{\rm reg}}$, the functions $f \in R(C,X)$ are
defined on some curve $C'$, birational to $C$.
\begin{defn}\label{admissible}
Let $C'$ be a reduced projective curve and $\iota:C' \to X$ be a
morphism. Then $(C',\iota)$ will be called {\it admissible}
if $\iota: C' \to C=\iota(C')$ is birational, if $C$ is a
reduced Cartier curve and if for some open neighbourhood $W$ of
$X_{{\rm sing}}$ the restriction of $\iota$ to $\iota^{-1}(W)$ is a
closed embedding.
\end{defn}
If $(C',\iota)$ is admissible one has an inclusion $R(C',C')
\subset R(\iota(C),X)$ which is an equality if
$\iota^{-1}(X_{{\rm reg}})$ is non-singular.
\begin{lemma}\label{gysin}
Let $(C',\iota)$ be admissible. Then there exists a homomorphism
(of abstract groups)
$$ \eta: {\rm Pic}^0(C') \cong CH^1(C')_{\deg 0} \>>> CH^n(X)_{\deg 0}$$
which maps the isomorphism class of ${\mathcal O}_{C'}(p-p')$ to $\gamma
(\iota(p))-\gamma(\iota(p'))$.
\end{lemma}
\begin{proof}
By definition ${\rm Pic}(C')= Z^1(C'_{\rm reg})/R(C',C')$ and one has
a map
$$\gamma \circ \iota : Z^1(C'_{\rm reg}) \>>> CH^n(X).
$$
The equality (\ref{val}) shows that for $f\in R(C',C')$
the image of $(f)_{C'}$ in $CH^n(X)$ is zero.
\end{proof}
\begin{notations}\label{difference}
Let $Y$ be a non-singular scheme with irreducible components $Y_1,
\ldots ,Y_s$, let $G$ be an abstract or an algebraic group, and let
$\pi: Y \to G$ a map or morphism.
\begin{points}
\item After choosing base points $p_i \in Y_i$ a map
$$
\pi_m : S^m(Y):= S^m( \bigcup_{i=1}^s Y_i) \>>> G,
$$
is defined by $\pi_m(y_1, \ldots, y_m)=\sum_{j=1}^m
(\pi(y_j) - \pi(p_{\rho(j)}))$, where $\rho(j) = i$ if $y_j \in Y_i$.
\item To avoid the reference to base points, we will frequently use
different maps:
$$
\pi^{(-)} : \Pi_Y = \bigcup_{i=1}^s Y_i\times Y_i \>>> G
$$
is defined by $\pi^{(-)}(y,y') = \pi(y)-\pi(y')$, and
$\pi^{(-)}_m: S^m(\Pi_Y) \to G$ is the composite
$S^m(\Pi_Y) \to S^m(G) \> {\rm sum} >> G$.
\end{points}
\end{notations}
If $G$ is an algebraic group, then the images of
$\pi^{(-)}_m$ lie in the connected component of $0$.
In particular for $U$ open and dense
in $X_{\rm reg}$ we will frequently consider
\begin{align*}
&\gamma^{(-)} = \gamma_U^{(-)}: \Pi_U \>>> CH^n(X)_{\deg 0}\\
\mbox{and \ \ \ \ }
&\gamma_m = \gamma_{U,m} : S^m(U) \>>> CH^n(X)_{\deg 0}.
\end{align*}
\begin{lemma} \label{generators}
Let $G$ be a $d$-dimensional
smooth connected commutative algebraic group
and let $\Gamma \subset G$ be a constructible subset which
generates $G$ as an abstract group.
Then
\begin{points}
\item the image of the
composite map \ $S^d(\Gamma) \>>> S^d(G) \>{\rm sum} >> G $ \
is dense
\item
$S^{2d}(\Gamma) \>>> S^{2d}(G) \>{\rm sum} >> G$
\
is surjective
\item if $B$ is a non-singular scheme with connected components
$B_1, \ldots ,B_s$ and if $\vartheta: B \to G$ is a morphism
with image $\Gamma$ then the morphism
$$
\vartheta_{d}^{(-)}: S^d(\Pi_B)=S^d \bigl( \bigcup_{i=1}^s
B_i \times B_i \bigr) \>>> G
$$
with $\vartheta_{d}^{(-)}((b_1,b_1'),\ldots (b_d,b_d')) = \sum_{i=1}^d
(\vartheta(b_i)-\vartheta(b_i'))$ is surjective.
\end{points}
\end{lemma}
\begin{proof}
Let $\bar{\Gamma}_1, \ldots, \bar{\Gamma}_s$ be the irreducible
components of the closure $\bar{\Gamma}$ of $\Gamma$, and let
$\Gamma_i= \bar{\Gamma_i}\cap \Gamma$.
It is sufficient to find non-negative integers
$d_1, \ldots ,d_s$ with $\sum_{i=1}^s d_i \leq d$ such that
the image of
$S^{d_1}(\Gamma_1)\times \cdots \times S^{d_s}(\Gamma_s)$
is dense in $G$. To this end, we may assume that the identity
of $G$ lies on each $\Gamma_i$.
Let $\bar{\Gamma}_1^\nu$ be the closure of the image of
$S^\nu(\Gamma_1)$ in $G$. Since
$\bar{\Gamma}_1^\nu \subset \bar{\Gamma}_1^{\nu+1}$
there exists some $d_1 \leq d$ with
$\bar{\Gamma}_1^{d_1} = \bar{\Gamma}_1^{d_1+1}$,
and $d_1$ is minimal with this property.
Hence $\bar{\Gamma}_1^{d_1} = \bar{\Gamma}_1^{2 d_1}$
and $\bar{\Gamma}_1^{d_1}$ is a subgroup of $G$
of dimension larger than or equal to $d_1$.
If $s=1$, i.e. if $\bar{\Gamma}$ is irreducible, then
$\bar{\Gamma}_1^{d_1}=G$.
In general, replacing $G$ by $G/\bar{\Gamma}_1^{d_1}$ one
obtains \ref{generators} (i) by induction on $s$.
The second part is an easy consequence of (i). Let $U$
be an open dense subset of $G$, contained in
$S^{d_1}(\Gamma_1)\times \cdots \times S^{d_s}(\Gamma_s)$.
Given $p \in G$ the intersection of the two open sets $U$ and
$p-U$ is non-empty and hence there are points $a,b \in U$ with
$p-b=a$.
For (iii) we may assume that for some point $b_i' \in B_i$, the image of
$B_i\times\{b'_i\}$ in $\Gamma_i$ is dense, for each $i$.
By (i) one finds
$d_1, \ldots ,d_s$ with $\sum_{i=1}^s d_i = d$ such that
the image
$$
\vartheta^{(-)}_d(S^{d_1}(B_1\times \{b_1'\})\times \cdots \times
S^{d_s}(B_s\times \{b_s'\}))
$$
contains a subset $U$ which is open in $G$. Given $p \in G$
the intersection of $U$ and of $p+U$ is non empty and hence
$p=a-b$ for two points $a$ and $b$ in $U$. Obviously $a-b$ lies
in the image of $\vartheta^{(-)}_d$.
\end{proof}
\begin{cor}\label{generators2} Let $C'$ be a reduced curve,
let $B_1, \ldots B_s$ be the connected components of $B=C'_{\rm reg}$,
let $b'_j \in B_j$ be base points and let $\vartheta: B \to
{\rm Pic}^0(C')$ be the morphism with $\vartheta|_{B_j}(b)={\mathcal O}_{C'}(b-b_j')$.
Then there exists some open connected subscheme $W$ of $S^g(B)$,
for $g=\dim_k({\rm Pic}^0(C'))$, such that
$\vartheta_W := \vartheta_g|_W$
is an open embedding.
\end{cor}
\begin{proof}
By \ref{generators} we find some $W$ with $\vartheta_W(W)$ open
and $\vartheta_W$ finite over its image. On the other hand, any
fibre of $\vartheta_g$ is an open subset of $\P(H^0(C',{\mathcal O}_{C'}(D)))$
for some divisor $D$ on $C'$; hence the projective spaces corresponding to
points of $\vartheta_W(W)$ must be 0-dimensional.
\end{proof}
\begin{lemma}\label{pic}
Let $G$ be a smooth commutative algebraic group, $U \subset X_{\rm reg}$ an
open and dense subset, and $\pi: U \to G$ a morphism.
Then the following two conditions are equivalent.
\begin{enumerate}
\item[(a)] There exists a homomorphism (of abstract groups)
$\phi:CH^n(X)_{\deg 0} \to G$ such that $\pi^{(-)} = \phi \circ
\gamma^{(-)}$ (as maps on the closed points).
\item[(b)] For all admissible pairs $(C',\iota)$ with
$B=(\iota^{-1}(U))_{\rm reg}$ dense in $C'$ there exists a
homomorphism of algebraic groups $\psi:{\rm Pic}^0(C') \to G$ such that
the diagram
$$
\begin{CD}
\Pi_B \>\vartheta^{(-)} >> {\rm Pic}^0(C')\\
{\mathbb V} \iota V V {\mathbb V} V \psi V \\
\Pi_U \> \pi^{(-)} >> G
\end{CD}
$$
commutes. Here $\vartheta: B \to {\rm Pic}(C')$ denotes the natural
morphism, mapping a point $p$ to the isomorphism class of the
invertible sheaf ${\mathcal O}_{C'}(p)$.
\end{enumerate}
Moreover, if the equivalent conditions (a) and (b) are true, the
morphism $\psi$ in (b) factors as
$$ \begin{TriCDV}
{{\rm Pic}^0(C')}{\> \eta >>}{CH^n(X)_{\deg 0}}
{\SE \psi E E }{\SW W \phi W}{G}
\end{TriCDV}
$$
and the image of $\phi:CH^n(X)_{\deg 0}\to G$ is contained in the
connected component of the identity of $G$.
\end{lemma}
\begin{proof}
Assume (a) and let $(C',\iota)$ be admissible and $g=\dim({\rm Pic}^0(C'))$.
Choosing base points $b'_j \in B_j$, one finds by \ref{generators2} an
open subscheme $W$ of $S^{g}(B)$ such that the morphism $\vartheta_W:W \to
{\rm Pic}^0(C')$ is an open embedding. By \ref{gysin} one obtains a
homomorphism $$
\psi:{\rm Pic}^0(C')\>\eta>>CH^n(X)_{\deg 0} \> \phi >> G
$$
of abstract groups. By assumption $\pi^{(-)}$ is a morphism of
schemes and the same holds true for $\pi^{(-)}\circ\iota: \Pi_B \to G.$
Thereby the restriction of $\psi$ to the open subscheme
$W \subset {\rm Pic}^0(C')$ is a morphism of schemes, and
being a homomorphism of abstract groups $\psi$ is a morphism of
algebraic groups.
Since each point of $X_{\rm reg}$ lies on some Cartier curve,
the images of the connected algebraic groups ${\rm Pic}^0(C')$
generate $CH^n(X)_{\deg 0}$ and the image $\phi(CH^n(X)_{\deg
0})$ lies in the connected component of $G$, which contains the
identity.
The morphism $\pi^{(-)}$ induces a map
$\tilde{\phi}:Z^n(U)_{\deg 0} \to G$ and it remains to verify
that (b) implies that $\tilde{\phi}(R^n(X,U))=0$. By
\ref{moving} each $\delta\in R^n(X,U)$ is of the form $(f)_C$
for a reduced Cartier curve $C$. There exists an admissible pair
$(C',\iota)$ with $\iota(C')=C$ and with $\iota^{-1}(X_{\rm reg})$
non-singular.
$(f)_C$ is the image of $(f)_{C'}$ in $Z^n(U)$ and
by assumption $\iota\circ\pi^{(-)}$ factors through ${\rm Pic}^0(C')$.
\end{proof}
\begin{cor}\label{equ-reg}
Let $\phi: CH^n(X)_{\deg 0} \to G$ be a homomorphism to
a smooth commutative algebraic group $G$. Then the following
conditions are equivalent:
\begin{points}
\item $\phi\circ\gamma^{(-)}: \Pi_{X_{\rm reg}} \>>> G$
is a morphism of schemes.
\item There exists an open dense subscheme $U$ of $X_{\rm reg}$
such that $\phi\circ\gamma^{(-)}|_{\Pi_U}$ is a morphism of
schemes.
\item Given a base point $p_i$ on each irreducible component
$U'_i$ of some open dense subscheme $U$ of $X_{\rm reg}$, the map
$\pi: U \to G$ with $\pi|_{U'_i}(x) = \phi(x-p_i)$ is a morphism
of schemes.
\item Given any $m>0$ and base points $p_i$ on each irreducible
component $U'_i$ of some open dense subscheme $U$ of $X_{\rm reg}$,
$\phi\circ\gamma_m : S^m(U) \to G$ is a morphism of schemes.
\end{points}
\end{cor}
Of course, ``$\pi$ is a morphism of schemes'' stands for ``there
exists a morphism of schemes whose restriction to closed points
coincides with $\pi$'', an abuse of terminology which we will repeat
throughout this article.
\begin{proof}
Obviously (i) implies (ii). For $U \subset X_{\rm reg}$ given, the
equivalence of (ii), (iii), and (iv) is an easy exercise. In
fact, the morphism $\pi$ in \ref{equ-reg} (iii) is just
$\phi\circ\gamma_1$.
Assume that (iii) holds true for some $U$.
We will show that the corresponding property holds true for
$X_{\rm reg}$ itself. To this aim consider the map
$\bar{\pi}: X_{\rm reg} \to G$ with $\bar{\pi}|_{U'_i}(x) =
\phi(x-p_i)$ and the graph $\Gamma_{\bar{\pi}}$ of $\bar{\pi}$
in $X_{\rm reg}\times G$. By definition, $\Gamma_{\bar{\pi}} \cap
U\times G$ is the graph $\Gamma_{\pi}$. Let $Z$ be the closure
of $\Gamma_\pi$ in $X_{\rm reg}\times G$.
$\Gamma_{\bar{\pi}}$ is contained in $Z$.
In fact, given a point $x \in X_{\rm reg}$ one can find a Cartier
curve $C$ through $x$ with $U\cap C$ dense in $C$ and with
$B=C \cap X_{\rm reg}$ non-singular. By lemma \ref{pic} the
morphism $(\pi|_{C\cap U})^{(-)} : \Pi_{C\cap U} \to G$
factors through a morphism ${\rm Pic}^0(C)\to G$ of algebraic groups
and, in particular, it extends to a morphism
$\Pi_{B} \to G$.
Again this implies that the restriction of $\bar{\pi}$ to
$B$ is a morphism, hence $\Gamma_{\bar{\pi}}\cap B\times G$
is closed and therefore contained in $Z$.
By construction the morphism $p_1: Z \to X_{\rm reg}$ induced by the
projection is birational and surjective. Let $V \subset X_{\rm reg}$
be the largest open subscheme with $p_1|_{p_1^{-1}(V)}$ an
isomorphism. Then $\bar{\pi}|_V$ is a morphism of schemes and
${\rm codim}_{X_{\rm reg}}(X_{\rm reg}-V) \geq 2$. By theorem 1 in
\cite{BLR}, 4.4, $\bar{\pi}|_V$ extends to a morphism $X_{\rm reg}
\to G$. The graph of this morphism is contained in $Z$, hence it
is equal to $Z$ and $\bar{\pi}$ is a morphism.
\end{proof}
We end this section by giving the definition of a
regular homomorphism, used already in the formulation of the
main theorems in the introduction.
\begin{defn}\label{def-reg}
Let $G$ be a smooth commutative algebraic group.
A homomorphism $\phi:CH^n(X)_{\deg 0} \to G$ (of abstract groups) is
called {\it a regular homomorphism}, if one of the
equivalent conditions in \ref{equ-reg} holds true.
\end{defn}
\begin{lemma}\label{reg_sur}
The image of a regular homomorphism $\phi:CH^n(X)_{\deg 0} \to
G$ is a connected algebraic subgroup of $G$.
\end{lemma}
\begin{proof}
Let $G'$ denote the Zariski closure of $\phi (CH^n(X)_{\deg
0})$. By \ref{pic}, $G'$ is connected and it is generated by the image of
$\Pi_{X_{\rm reg}} \to G'$. Hence \ref{reg_sur} follows from the third
part of Lemma \ref{generators}.
\end{proof}
\section{The cycle class map}
Throughout the next three sections we will assume that the
ground field $k$ is the field of complex numbers.
${\mathcal O}_X$ and $\Omega^m_{X/{\mathbb C}}$ will respectively denote the sheaves of
holomorphic functions and (analytic K\"ahler) differential $m$-forms.
As in the introduction consider the Deligne complex
\[{\mathcal D}(n)_X=\left(0\to
{\mathbb Z}_X(n)\to{\mathcal O}_X\to\Omega^{1}_{X/{\mathbb C}}\to\cdots\to
\Omega^{n-1}_{X/{\mathbb C}}\to 0\right),\]
and associated cohomology group $D^{n}(X)={\mathbb H}^{2n}(X,{\mathcal D}(n)_X).$
In this section we construct the cycle class homomorphism
$CH^n(X) \to D^n(X)$, using Cartier curves $C$ in $X$.
By the moving lemma \ref{moving} it will be sufficient to
consider reduced Cartier curves $C$ in $X$.
Note, however, that we do not have that $C$ is a local complete
intersection in $X$, in general; this is only given to hold at points of
$C\cap X_{{\rm sing}}$. This leads to a slight technical difficulty. We will
need to define `Gysin' maps for Cartier curves $C$ in
$X$. These are directly defined in case $C$ is a local complete
intersection, and in general one has first to make a sequence of point
blow ups centered in $X_{{\rm reg}}$ to reduce to this special case. Indeed,
even to show that the cycle homomorphism $Z^n(X)\to D^n(X)$ respects
rational equivalence, a similar procedure needs to be followed.
Note that the exterior derivative yields a map of complexes
${\mathcal D}(n)_X\to \Omega^n_{X/{\mathbb C}}[-n],$
and there is an obvious map ${\mathcal D}(n)_X\to{\mathbb Z}(n)_X$.
\begin{lemma}\label{local}
For $x\in X_{{\rm reg}}$, there is a unique element
$[x]\in H^{2n}_{\{x\}}(X,{\mathcal D}(n)_X)$
which maps to the topological cycle class of $x$ in
$H^{2n}_{\{x\}}(X,{\mathbb Z}(n))$ as well as to the ``Hodge cycle class''
of $x$ in $H^n_{\{x\}}(X,\Omega^n_{X/{\mathbb C}})$.
This gives rise to a well-defined cycle class homomorphism
$Z^n(X)\to D^n(X)$, whose composition with $D^n(X)\to
H^{2n}(X,{\mathbb Z}(n))$ is the topological cycle class homomorphism.
\end{lemma}
\begin{proof} The element $[x]$ exists because the topological and Hodge
cycle classes both map to the de Rham cycle class of $x$ in
$H^{2n}_{\{x\}}(X,{\mathbb C})={\mathbb H}^{2n}_{\{x\}}(X,\Omega^{\d}_{X/{\mathbb C}})$, by a standard
local computation. See \cite{EV}, \S7, for example (though $X$ is singular, the
terms in the above computation depend only on a neighbourhood of $x$ in
$X$, and we have $x\in X_{{\rm reg}}$; hence \cite{EV}, \S7 is applicable).
\end{proof}
\begin{lemma}\label{curves1}
In the above situation, if $\dim X=1$, then there is a natural
quasi-isomorphism ${\mathcal D}(1)_X\cong{\mathcal O}_X^*[-1]$, yielding an
identification ${\rm Pic}(X)\cong D^1(X)$ (and hence also ${\rm Pic}^0(X)\cong
A^1(X)$). Under the identification, the class of a smooth point $[x]\in
D^1(X)$ corresponds to the class of the invertible sheaf ${\mathcal O}_X(x)$.
\end{lemma}
\begin{proof} The natural quasi-isomorphism is equivalent to the exactness
of the exponential sequence. The description of the class of a
point $x$ as the class of the invertible sheaf ${\mathcal O}_X(x)$ is
also a standard local computation.
\end{proof}
Now we argue as in \cite{BiS}, in order to show that the map
$Z^n(X)\to D^n(X)$ factors through $CH^n(X)$. We follow the convention
that the truncated de Rham complex of K\"ahler differentials
$$
\Omega_{X/{\mathbb C}}^{< n}=(0\to{\mathcal O}_X\to\cdots\to\Omega^{n-1}_{X/{\mathbb C}}\to 0)
$$
has ${\mathcal O}_X$ placed in degree
0; thus we have an exact sequence of complexes
\[0\>>> \Omega_{X/{\mathbb C}}^{< n}[-1]\>>> {\mathcal D}(n)_X\>>> {\mathbb Z}(n)_X \>>> 0.\]
\begin{lemma}\label{curves2} Let $X$ be a projective variety of
dimension $n$ over ${\mathbb C}$, and $C\subset X$ be a reduced Cartier
curve which is a local complete intersection in $X$. Then there
is a commutative diagram
\[
\begin{CD}
Z^1(C) \>>> Z^n(X)\\
{\mathbb V} V V {\mathbb V} V V\\
D^1(C) \> {\rm Gysin}>> D^n(X)\\
{\mathbb V} V V {\mathbb V} V V \\
H^2(C,{\mathbb Z}(1)) \>{\rm Gysin}>> H^{2n}(X,{\mathbb Z}(n))
\end{CD}
\]
\end{lemma}
\begin{proof}
Consider the local (hyper) cohomology sheaves ${\mathcal H}^j_C({\mathcal D}(n)_X)$ of the
complex ${\mathcal D}(n)_X$ with support in $C$. We claim that for any point $x\in
C$, the stalks ${\mathcal H}^j_C({\mathcal D}(n)_X)_x$ vanish for $j\neq 2n-1$, unless $x$
is a singular point of $C$. Indeed, if $x\in C$ is a non-singular point
(so that $x\in X_{{\rm reg}}$ as well), then there is a long exact sequence of
stalks
\begin{multline*}
\cdots\>>>{\mathcal H}^{j-1}_C({\mathbb Z}(n)_X)_x\oplus
{\mathcal H}^{j-1-n}_C(\Omega^n_{X/{\mathbb C}})_x\>>>
{\mathcal H}^{j-1}_C({\mathbb C}_X)_x\>>> {\mathcal H}^j_C({\mathcal D}(n)_X)_x\\
\>>> {\mathcal H}^j_C({\mathbb Z}(n)_X)_x\oplus{\mathcal H}^{j-n}_C(\Omega^n_{X/{\mathbb C}})_x\>>>
{\mathcal H}^j_C({\mathbb C}_X)_x\>>>\cdots
\end{multline*}
However ${\mathcal H}^i_C({\mathbb Z}(n)_X)_x={\mathcal H}^i_C({\mathbb C}_X)_x=0$ for $i \neq2n-2$,
${\mathcal H}^{2n-2}_C({\mathbb Z}(n)_X)$ injects into ${\mathcal H}^{2n-2}_C({\mathbb C}_X)$, and
${\mathcal H}^i_C(\Omega^n_{X/{\mathbb C}})_x=0$ unless $i=n-1$, for a non-singular point
$x\in C$ as above. This implies that ${\mathcal H}^j_C({\mathcal D}(n)_X)_x=0$ for $j\neq
2n-1$, for such $x$. Also ${\mathcal H}^{2n-1}_C({\mathcal D}(n)_X)_x$ fits into an exact
sequence
\[0\>>>{\mathcal H}^{2n-2}_C({\mathbb C}_X/{\mathbb Z}(n)_X)_x\>>>{\mathcal H}^{2n-1}_C({\mathcal D}(n)_X)_x\>>>
{\mathcal H}^{n-1}_C(\Omega^n_{X/{\mathbb C}})_x\>>>
0,\]
with ${\mathcal H}^{2n-2}_C({\mathbb C}_X/{\mathbb Z}(n)_X)_x\cong{\mathbb C}/{\mathbb Z}(1)={\mathbb C}^*$.
Thus, ${\mathcal H}^j_C({\mathcal D}(n)_X)$ is supported at a finite set of points, if
$j\neq 2n-1$. Hence in the local-to-global spectral sequence
\[E_2^{p,q}=H^p(C,{\mathcal H}^q_C({\mathcal D}(n)_X))\Longrightarrow
{\mathbb H}^{p+q}_C(X,{\mathcal D}(n)_X)\]
we have $E_2^{p,q}=0$ for $p>0$, $q\neq
2n-1$. In particular, there is a well-defined injective map
\[\alpha:H^1(C,{\mathcal H}^{2n-1}_C({\mathcal D}(n)_X))\>>> {\mathbb H}^{2n}_C(X,{\mathcal D}(n)_X).\]
We will next construct a natural map of sheaves on $C$
\[{\mathcal O}_C^*\>>>{\mathcal H}^{2n-1}_C({\mathcal D}(n)_X).\]
The desired Gysin map $D^1(C)\to D^n(X)$ is then defined to be the
composition
\begin{multline*}
H^1(C,{\mathcal O}_C^*)\>>> H^1(C,{\mathcal H}^{2n-1}_C({\mathcal D}(n)_X))\>{\alpha}>>
{\mathbb H}^{2n}_C(X,{\mathcal D}(n)_X)\>>>\\
{\mathbb H}^{2n}(X,{\mathcal D}(n)_X)=D^n(X)
\end{multline*}
To construct the map on sheaves ${\mathcal O}^*_C\to
{\mathcal H}^{2n-1}_C({\mathcal D}(n)_X)$, we argue locally, as follows. Let $U$ be
an affine neighbourhood in $X$ of a point $x\in C$, on which
the ideal of $C$ is generated by a regular sequence of functions
$f_1,\ldots,f_{n-1}$, determining a morphism $f:U\to {\mathbb A}^{n-1}_{{\mathbb C}}$ such
that $f^{-1}(0)=C\cap U$. Note that there are well-defined sections (of
the skyscraper sheaves)
\[\alpha\in
\Gamma({\mathcal H}^{2n-2}_{\{0\}}({\mathbb Z}(n)_{{\mathbb A}^{n-1}_{{\mathbb C}}}))={\mathbb Z}(1),\;\;\;\;\beta\in
\Gamma({\mathcal H}^{n-1}_{\{0\}}(\Omega^{n-1}_{{\mathbb A}^{n-1}_{{\mathbb C}}/{\mathbb C}})),\]
which have the same image
$\gamma\in \Gamma({\mathcal H}^{2n-2}_{\{0\}}(\Omega_{{\mathbb A}^{n-1}_{\mathbb C}/{\mathbb C}}^{\bullet})),$
under the obvious maps, and such that $\beta$ is annihilated by the ideal of
$0$ in $\Gamma({\mathcal O}_{{\mathbb A}^{n-1}_{{\mathbb C}}})$, for the natural module structure
on $\Gamma({\mathcal H}^{n-1}_{\{0\}}(\Omega^{n-1}_{{\mathbb A}^{n-1}_{{\mathbb C}}/{\mathbb C}}))$. In fact,
these conditions uniquely determine such a pair of sections
$(\alpha,\beta)$ up to sign, and there is a standard choice, with
$\beta$ determined by
$\dlog(z_1)\wedge\cdots\wedge\dlog(z_{n-1}),$
where $z_j$ are the coordinate functions, so that $\beta$ is the cup
product of the local divisor classes
\begin{equation}\label{ext}
\dlog(z_j)\in
\Gamma({\mathbb A}^{n-1}_{{\mathbb C}},\sext^1_{{\mathbb A}^{n-1}_{\mathbb C}}({\mathcal O}_{\{z_j=0\}},
\Omega^1_{{\mathbb A}^{n-1}_{{\mathbb C}}/{\mathbb C}}))
\subset \Gamma({\mathbb A}^{n-1}_{{\mathbb C}},
{\mathcal H}^1_{\{z_j=0\}}(\Omega^1_{{\mathbb A}^{n-1}_{{\mathbb C}}/{\mathbb C}})).
\end{equation}
Hence $\gamma$ is also determined. Now consider
\begin{gather*}
f^*\alpha\in\Gamma(U,{\mathcal H}^{2n-2}_C({\mathbb Z}(n)_X)), \;\;\;\;
f^*\beta\in \Gamma(U,{\mathcal H}^{n-1}_C(\Omega^{n-1}_{X/{\mathbb C}})),\\
\mbox{and} \;\;\;\; f^*\gamma\in\Gamma(U,{\mathcal H}^{2n-1}_C(\Omega^{<n}_{X/{\mathbb C}})),
\end{gather*}
where $f^*\alpha$ and $f^*\beta$ both map to $f^*\gamma$, and $f^*\beta$
is annihilated by any section of the ideal sheaf of $C\cap U$ in $U$. Thus
$f^*\alpha$ and $f^*\beta$ yield maps of sheaves
\[{\mathbb Z}(1)_C\mid_U\>>>
{\mathcal H}^{2n-2}_C({\mathbb Z}(n)_X)\mid_U,\;\;\;\;
{\mathcal O}_C\mid_U\>>> {\mathcal H}^{n-1}_C(\Omega^{n-1}_{X/{\mathbb C}})\mid_U,\]
giving rise to a commutative diagram of sheaves
$$
\begin{CD}
{\mathbb Z}(1)_C\mid_U \>>> {\mathcal H}^{2n-2}_C({\mathbb Z}(n)_X)\mid_U\\
{\mathbb V} V V {\mathbb V} V V\\
{\mathcal O}_C\mid_U \>>> {\mathcal H}^{2n-2}_C(\Omega^{<n}_{X/{\mathbb C}})\mid_U.
\end{CD}
$$
There is a long exact sequence of sheaves on $C$
$$
\cdots\>>>{\mathcal H}^j_C({\mathbb Z}(n)_X)\>>>{\mathcal H}^j_C(\Omega^{<n}_{X/{\mathbb C}})\>>>
{\mathcal H}^{j+1}_C({\mathcal D}(n)_X)\>>>{\mathcal H}^{j+1}_C({\mathbb Z}(n)_X)\>>>\cdots
$$
Hence from the exponential sequence
\[0\>>> {\mathbb Z}(1)_C\>>> {\mathcal O}_C\>{\rm exp}>>{\mathcal O}_C^*\>>> 0,\]
and the above commutative diagram, we deduce that there is a well-defined
map of sheaves
\[{\mathcal O}_C^*\mid_U\>>> {\mathcal H}^{2n-1}_C({\mathcal D}(n)_X)\mid_U.\]
We will now show that these locally defined maps patch together to give
well-defined sheaf maps
\begin{equation}\label{sheafmap}
{\mathcal O}_C\>>> {\mathcal H}^{2n-2}_C(\Omega^{n-1}_{X/{\mathbb C}}) \mbox{ \ \ \ and \ \
\ }{\mathcal O}_C^*\>>> {\mathcal H}^{2n-1}_C({\mathcal D}(n)_X).
\end{equation}
To do this, it suffices to show that the classes $f^*\alpha$,
$f^*\beta$ and $f^*\gamma$ defined above are in fact independent
of the map $f$, {\it i.e.\/},\ of the choice of generators for the ideal of
$C$ in $U$. This too can be seen ``universally''. Since the
ideal sheaf of $C$ in $X$ is locally generated by a regular
sequence, any two such sets of local generators for ${\mathcal I}_C$
on the affine open set $U$ differ by the operation of an element of
$\GL_{n-1}({\mathcal O}_X(V))$, for some neighbourhood $V$ of $C\cap U$ in $U$.
Hence it suffices to show that if
$p:\GL_{n-1}({\mathbb C})\times{\mathbb C}^{n-1}\to{\mathbb C}^{n-1}$ is the projection, and
$m:\GL_{n-1}({\mathbb C})\times{\mathbb C}^{n-1}\to{\mathbb C}^{n-1}$ the map given by the operation
of $\GL_{n-1}({\mathbb C})$ on ${\mathbb C}^{n-1}$ by invertible linear transformations,
then $p^*\alpha=m^*\alpha$, $p^*\beta=m^*\beta$,
and hence also $p^*\gamma=m^*\gamma$. We leave the verification of this to
the reader, as a simple application of the K\"unneth formula.
Finally, note that for $U=X_{{\rm reg}}$, we have a commutative diagram with
exact rows
\[
\minCDarrowwidth=.8cm
\begin{CD}
0\>>> {\mathbb Z}(1)_C\mid_U \>>> {\mathcal O}_C\mid_U\>>>{\mathcal O}_C^*\mid_U\>>> 0\\
\noarr {\mathbb V}\cong V V {\mathbb V} V V {\mathbb V} V V\\
0\>>>
{\mathcal H}^{2n-2}_C({\mathbb Z}(n)_X)\mid_U\>>>{\mathcal H}^{2n-2}_C(\Omega^{<n}_{X/{\mathbb C}})\mid_U
\>>> {\mathcal H}^{2n-1}_C({\mathcal D}(n)_X)\mid_U\>>> 0
\end{CD}\]
where the left vertical arrow is an isomorphism.
For a smooth point $x\in C$, apply the functors
${\mathcal H}^j_{\{x\}}$ to the rows of the above diagram, and note that
${\mathcal H}^j_{\{x\}}({\mathbb Z}(1)_C)=0$ for $j\neq 2$, and
${\mathcal H}^j_{\{x\}}({\mathcal O}_C)=0$ for $j\neq 1$.
We then obtain another diagram with exact rows
\[
\minCDarrowwidth=.6cm
\hspace*{-.4cm}
\begin{CD}
0\>>> {\mathcal H}^1_{\{x\}}({\mathcal O}_C) \>>> {\mathcal H}^1_{\{x\}}({\mathcal O}_C^*) \>>>
{\mathcal H}^2_{\{x\}}({\mathbb Z}(1)_C)\>>> 0\\
\noarr {\mathbb V} V V {\mathbb V} V V {\mathbb V} V \cong V\\
0\>>> {\mathcal H}^1_{\{x\}}({\mathcal H}^{2n-2}_C(\Omega^{<n}_{X/{\mathbb C}}))
\>>> {\mathcal H}^1_{\{x\}}({\mathcal H}^{2n-1}_C({\mathcal D}(n)_X)) \>>>
{\mathcal H}^2_{\{x\}}({\mathcal H}^{2n-2}_C({\mathbb Z}(n)_X))\>>> 0
\end{CD}\]
The bottom row may be identified (see \cite{Ha2}, III, Ex.~8.7, pg.~161) with
the exact sequence
\[0\>>> {\mathcal H}^{2n-1}_{\{x\}}(\Omega^{<n}_{X/{\mathbb C}})\>>>
{\mathcal H}^{2n}_{\{x\}}({\mathcal D}(n)_X)\>>> {\mathcal H}^{2n}_{\{x\}}({\mathbb Z}(n)_X)\>>> 0.\]
We claim that, under the above identification, the local cycle class of
$x$ in ${\mathcal H}^2_{\{x\}}({\mathcal D}(1)_C)={\mathcal H}^1_{\{x\}}({\mathcal O}_C^*)$ maps to the
corresponding local cycle class of $x$ in ${\mathcal H}^{2n}_{\{x\}}({\mathcal D}(n)_X)$.
Choosing a suitable regular system of parameters on $X$ at $x$, we reduce
to checking this in the special case when $x\in X$ is the origin $0\in
{\mathbb C}^n$, and the curve $C$ is the $z_n$-axis, given by the vanishing of the
first $n-1$ coordinates. We again leave this verification to the reader.
This means that, in the commutative diagram
\[
\begin{CD}
H^1_{\{x\}}(C,{\mathcal O}_C^*) \>>> H^1(C,{\mathcal O}_C^*)=D^1(C)\\
{\mathbb V} V V {\mathbb V} V V\\
{\mathbb H}^{2n}_{\{x\}}(X,{\mathcal D}(n)_X) \>>> D^n(X)
\end{CD}\]
the cycle class of $x$ in $D^1(C)$ maps to that of $x$ in $D^n(X)$. Hence
we have shown that there is a commutative diagram
\[\begin{CD}
Z^1(C) \>>> Z^n(X)\\
{\mathbb V} V V {\mathbb V} V V\\
D^1(C) \>{\rm Gysin}>> D^n(X)
\end{CD}
\]
It remains to show that the Gysin map ${\rm Pic}(C)=D^1(C)\to D^n(X)$ is
compatible with the topological Gysin map $H^2(C,{\mathbb Z}(1))\to
H^{2n}(X,{\mathbb Z}(n))$. Since $Z^1(C)\to D^1(C)$ is surjective, the
compatibility of the two Gysin maps is clear from the fact that each one
maps the class of $x$ on $C$ to the corresponding class on $X$.
\end{proof}
\begin{rmk}\label{gys_rem}
Assume that the local complete intersection curve $C$ lies in the
Cohen-Macaulay locus $X_{\rm CM}$ of $X$. Then the
first sheaf map in (\ref{sheafmap}) factors as
\begin{equation}\label{gys_fac}
{\mathcal O}_C\>>> \sext^{n-1}_X( {\mathcal O}_C,\Omega^{n-1}_{X/{\mathbb C}}) \>>>
{\mathcal H}^{2n-2}_C(\Omega^{n-1}_{X/{\mathbb C}}).
\end{equation}
To see this, note that
$$
\beta \in
\Gamma({\mathbb A}^{n-1}_{{\mathbb C}},\sext^{n-1}_{{\mathbb A}^{n-1}_{\mathbb C}}({\mathcal O}_{\{0\}},
\Omega^{n-1}_{{\mathbb A}^{n-1}_{{\mathbb C}}/{\mathbb C}}))
\subset \Gamma({\mathbb A}^{n-1}_{{\mathbb C}},
{\mathcal H}^{n-1}_{\{0\}}(\Omega^{n-1}_{{\mathbb A}^{n-1}_{{\mathbb C}}/{\mathbb C}}))
$$
as it is the product of the classes $\dlog(z_j)$ in (\ref{ext}).
Further the map $U \to {\mathbb A}^{n-1}_{\mathbb C}$ is flat in a neigbourhood
of $C\cap U$, as it is equidimensional. Thus $f^*\beta$ defines
a class in
$\Gamma(U,\sext^{n-1}_X({\mathcal O}_{C},\Omega^{n-1}_{X/{\mathbb C}}))$
mapping to $\Gamma(U,{\mathcal H}^{n-1}_{C}(\Omega^{n-1}_{X/{\mathbb C}}))$.
As
\begin{multline*}
{\rm Ext}^{n-1}_{\GL_{n-1}({\mathbb C})\times
{\mathbb C}^{n-1}}({\mathcal O}_{\GL_{n-1}({\mathbb C})\times \{0\}},
\Omega^{n-1}_{\GL_{n-1}({\mathbb C})\times {\mathbb C}^{n-1}/{\mathbb C}})\\
\subset H^{n-1}_{\GL_{n-1}({\mathbb C})\times
\{0\}}(\GL_{n-1}({\mathbb C})\times {\mathbb C}^{n-1}, \Omega^{n-1}_{\GL_{n-1}({\mathbb C})\times
{\mathbb C}^{n-1}/{\mathbb C}})),
\end{multline*}
the class $f^*\beta$ defines the factorization (\ref{gys_fac}).
\end{rmk}
\begin{lemma}\label{blowup} Let $X$ be projective of dimension
$n$ over ${\mathbb C}$, $f:Y\to X$ the blow up of a smooth point $x\in
X$. Then the natural maps $f_*:CH^n(Y)\to CH^n(X)$ and
$f^*:D^n(X)\to D^n(Y)$ are isomorphisms, and there is a
commutative diagram
\[\begin{CD}
Z^n(Y) \>>> D^n(Y)\\
{\mathbb V} f_* V V {\mathbb A} \cong A f^* A\\
Z^n(X) \>>> D^n(X)
\end{CD}
\]
\end{lemma}
\begin{proof} The isomorphism on Chow groups is easy to prove, using the
fact that the exceptional divisor $E$ is a projective space (the
details are in \cite{BiS}). That $f^*:D^n(X)\to D^n(Y)$ is an
isomorphism is also easy to see, for the same reason, using also
the exact sequence
\[0\>>> f^*\Omega^1_{X/{\mathbb C}}\>>> \Omega^1_{Y/{\mathbb C}}\>>>
\Omega^1_{E/{\mathbb C}}\>>> 0.\]
So we need to prove that if $y\in Y$ is any smooth point, then
its class in $D^n(Y)$ is the inverse image of that of $f(y)$ in
$D^n(X)$. This is clear if $f(y)\neq x$. If $f(y)=x$, we may
argue as follows. There is a commutative diagram with exact
rows
\[
\minCDarrowwidth=.6cm
\hspace*{-.5cm}
\begin{CD}
0 \>>> {\mathbb H}^{2n}_{\{y\}}(Y,{\mathcal D}(n)_Y)\>>>
H^{2n}_{\{y\}}(Y,{\mathbb Z}(n))\oplus H^n_{\{y\}}(Y,\Omega^n_{Y/{\mathbb C}})
\>>> H^{2n}_{\{y\}}(Y,{\mathbb C}(n))\>>> 0\\
\noarr {\mathbb V} V V {\mathbb V} V V {\mathbb V} V V\\
0 \>>> {\mathbb H}^{2n}_{E}(Y,{\mathcal D}(n)_Y)\>>>
H^{2n}_{E}(Y,{\mathbb Z}(n))\oplus H^n_{E}(Y,\Omega^n_{Y/{\mathbb C}})
\>>> H^{2n}_{E}(Y,{\mathbb C}(n))\>>> 0\\
\noarr {\mathbb A} f^* A A {\mathbb A} f^* A A {\mathbb A} f^* A A\\
0 \>>> {\mathbb H}^{2n}_{\{x\}}(X,{\mathcal D}(n)_X)\>>>
H^{2n}_{\{x\}}(X,{\mathbb Z}(n))\oplus H^n_{\{x\}}(X,\Omega^n_{X/{\mathbb C}})
\>>> H^{2n}_{\{y\}}(X,{\mathbb C}(n))\>>> 0
\end{CD}\]
Here the downward vertical arrows are the natural maps
(``increase support''). It is standard that the topological
local cycle classes of $x$ and $y$ have the same images in
$H^{2n}_E(Y,{\mathbb Z}(n))$. Similarly, the images in
$H^n_E(Y,\Omega^n_{Y/{\mathbb C}})$ of the local cycle classes of $x$
and $y$ in Hodge cohomology are also known to be equal; for
example, this follows from the existence of a Gysin map
$f_*:H^n_{\{y\}}(Y,\Omega^n_{Y/{\mathbb C}})\to
H^n_{\{x\}}(X,\Omega^n_{X/{\mathbb C}})$, which maps the local class of
$y$ to that of $x$, and which factors through
$$H^n_E(Y,\Omega^n_{Y/{\mathbb C}})\>{(f^*)^{-1}}>>H^n_{\{x\}}
(X,\Omega^n_{X/{\mathbb C}}).$$
Thus $f^*[x]=[y]\in
{\mathbb H}^{2n}_E(Y,{\mathcal D}(n)_Y)$, and hence a similar equality is valid
in $D^n(Y)$ as claimed.
\end{proof}
\begin{lemma}\label{equivalence} The map $Z^n(X)\to D^n(X)$
factors through $CH^n(X)$, and hence determines a homomorphism
$\varphi:CH^n(X)_{\deg 0}\to A^n(X)$.
\end{lemma}
\begin{proof} This is similar to the corresponding proof in
\cite{BiS}. Let $C\subset X$ be a reduced Cartier curve, and $f\in
R(C,X)$. Let $\pi:Y\to X$ be a composition of blow ups at
smooth points so that the strict transform $\tilde{C}$ of $C$ in
$Y$ satisfies $\tilde{C}_{{\rm sing}}=\tilde{C}\cap Y_{{\rm sing}}\cong
C\cap X_{{\rm sing}}$. Then
\[R(C,X)=R(\tilde{C},Y)=R(\tilde{C},\tilde{C}),\]
and $\pi_*(f)_{\tilde{C}}=(f)_C\in Z^n(X).$
Now from lemma~\ref{curves1}, $(f)_{\tilde{C}}\mapsto 0\in D^n(Y),$
and so from lemma~\ref{blowup},
$(f)_C=\pi_*(f)_{\tilde{C}}=0\in D^n(X).$
\end{proof}
\begin{cor}\label{corblowup}
If $f:Y\to X$ is a composition of blow ups at smooth points,
then we have a diagram
\[\begin{CD}
CH^n(Y) \>>> D^n(Y)\\
{\mathbb V} f_* V \cong V {\mathbb A} \cong A f^* A\\
CH^n(X) \>>> D^n(X)
\end{CD}
\]
\end{cor}
\begin{cor}\label{curves3}
For any reduced Cartier curve $C\subset X$, there are commutative
diagrams
\[\begin{CD}
Z^1(C) \>>> Z^n(X)\\
{\mathbb V} V V {\mathbb V} V V\\
{\rm Pic}(C) \>{\rm Gysin}>>CH^n(X)\\
{\mathbb V} \cong V V {\mathbb V} V V\\
D^1(C) \>{\rm Gysin}>> D^n(X)\\
{\mathbb V} V V {\mathbb V} V V \\
H^2(C,{\mathbb Z}(1)) \>{\rm Gysin}>> H^{2n}(X,{\mathbb Z}(n))
\end{CD}
\mbox{ \ \ \ \ and \ \ \ \ \ }
\begin{CD}
{\rm Pic}^0(C) \>>> CH^n(X)_{\deg 0}\\
{\mathbb V} \cong V V {\mathbb V} V V \\
A^1(C) \>>> A^n(X)
\end{CD}\]
\end{cor}
\begin{proof} As in the proof of lemma~\ref{equivalence}, by a compositon
of blow-ups at smooth points, we reduce to the case when $C$ is a local
complete intersection in $X$. Then lemma~\ref{curves2} implies the
corollary.
\end{proof}
Considering embedded resolution of singularities one obtains from
\ref{curves3} and \ref{corblowup} a second construction of the
Gysin map in \ref{gysin} over ${\mathbb C}$. At the same time, it gives the
compatibility of this map with the Gysin map for the Deligne cohomology,
constructed in \ref{curves2}.
\section{Some general properties of $A^n(X)$ over ${\mathbb C}$}
It is shown in \cite{BiS} that if $X$ is projective over ${\mathbb C}$ of
dimension $n$, then there is a natural surjection (which is
referred to in \cite{BiS} as the {\em Abel-Jacobi map})
\[AJ^n_X:CH^n(X)_{\deg 0}\>>>
J^n(X):=\frac{H^{2n-1}(X,{\mathbb C}(n))}{F^0H^{2n-1}(X,{\mathbb C}(n))+{\rm
image}\,H^{2n-1}(X,{\mathbb Z}(n))},\] where by results of Deligne,
$J^n(X)$ is a semi-abelian variety (since the non-zero Hodge numbers of
$H^{2n-1}(X,{\mathbb Z}(n))$ lie in the set
$\{(-1,0),(0,-1),(-1,-1)\}$).
\begin{lemma}\label{algebraicity}
Let $X$ be projective of dimension $n$ over ${\mathbb C}$. Then there is
a natural surjection $\psi:A^n(X)\to J^n(X),$
whose kernel is a ${\mathbb C}$-vector space. $A^n(X)$ has a unique structure as an
algebraic group such that $\psi$ is a morphism of algebraic groups, with
additive kernel ({\it i.e.\/},\ with kernel isomorphic to a direct sum of copies of
${\mathbb G}_a$).
\end{lemma}
\begin{proof}
By a result of Bloom and Herrera \cite{BH}, the natural map
\[H^{2n-1}(X,{\mathbb C}(n))\>>>
H^{2n-1}_{DR}(X/{\mathbb C})={\mathbb H}^{2n-1}(X,\Omega^{\d}_{X/{\mathbb C}})\]
is split injective. As explained in \cite{D}~(9.3.2), if $X_{\d}\to X$ is a
suitable hypercovering by a smooth proper simplicial scheme, the splitting
may be given by the composition
$$H^{2n-1}_{DR}(X/{\mathbb C})\to
H^{2n-1}_{DR}(X_{\d}/{\mathbb C})\cong H^{2n-1}(X_{\d},{\mathbb C}(n))\cong
H^{2n-1}(X,{\mathbb C}(n)).
$$
{From} this description, the splitting is a map of
filtered vector spaces, where $H^{2n-1}(X,{\mathbb C}(n))$ has the
Hodge filtration for the mixed Hodge structure while
$H^{2n-1}_{DR}(X/{\mathbb C})$ has the truncation filtration ({\it i.e.\/},\ the
filtration b\^ete).
Hence we obtain a commutative diagram
$$
\begin{TriCDV}
{H^{2n-1}(X,{\mathbb C}(n))}{\>>>}{H^{2n-1}(X,{\mathbb C})/F^0H^{2n-1}(X,{\mathbb C}(n))}
{\SE E E}{\NE E {\vartheta}E}
{{\mathbb H}^{2n-1}(X,\Omega^{< n}_{X/{\mathbb C}})}
\end{TriCDV}
$$
The map $\vartheta$ induces the map $\psi$ taking quotients modulo
$H^{2n-1}(X,{\mathbb Z}(n))$. Note that by weight considerations, the natural map
\[
H^{2n-1}(X,{\mathbb Z}(n))\>>> H^{2n-1}(X,{\mathbb C}(n))/F^0H^{2n-1}(X,{\mathbb C}(n))
\]
has a torsion kernel. Hence the kernels of $\psi$ and $\vartheta$ are the
same (and the latter is a ${\mathbb C}$-vector space). This represents $A^n(X)$ as
an analytic group extension of the semi-abelian variety $J^n(X)$ by an
additive group ${\mathbb G}_a^r$, for some $r$, and hence as an analytic group
extension of an abelian variety by a group ${\mathbb G}_a^r\times {\mathbb G}_m^s$. As noted
in \cite{D}, (10.1.3.3), for any abelian variety $A$ over ${\mathbb C}$, the isomorphism
classes of analytic and algebraic groups extensions of $A$ by either
${\mathbb G}_a$ or by ${\mathbb G}_m$ coincide (as a consequence of GAGA); hence a similar
property is valid for extensions by ${\mathbb G}_a^r\times {\mathbb G}_m^s$. This implies
that $A^n(X)$ has a unique algebraic structure such that $\psi$ is a
homomorphism of algebraic groups over ${\mathbb C}$, as claimed.
\end{proof}
For $X$ a curve, as in (i) of the next corollary, note that
${\rm Pic}^0(X)$ has the natural algebraic structure obtained by
representing a suitable Picard functor. In
particular, given an algebraic family of divisors (of degree 0)
on $X$ parametrized by a variety (or scheme) $T$, the induced
map $T\to{\rm Pic}^0(X)$ is automatically a morphism. On the other
hand, $A^1(X)$ has the algebraic structure given by
lemma~\ref{algebraicity}. Hence, a priori, the induced
map $T\to A^1(X)$ obtained from such a family is only analytic,
since it is essentially given by integration. From (i) of the
corollary, it will follow that it is in fact algebraic. The content
of (ii) of the corollary is similar.
\begin{cor}\label{algcurve}\ \
\begin{points}
\item If $X$ is a curve, then the natural isomorphism ${\rm Pic}^0(X)\cong
A^1(X)$ is an isomorphism of algebraic groups.
\item In general, if $C\subset X$ is a reduced Cartier curve,
then the induced homomorphism $A^1(C)\to A^n(X)$ of
corollary~\ref{curves3} is algebraic.
\end{points}
\end{cor}
\begin{proof}
(i) The identification is certainly analytic, and in both cases,
when one represents the algebraic group as an extension of an
abelian variety by a commutative affine group, the abelian
variety in question is just
${\rm Pic}^0(\tilde{X})=J(\tilde{X})=D^1(\tilde{X})$, the Jacobian of
the normalized curve $\tilde{X}$ (by which we mean the product of the
Jacobians of the connected components of $\tilde{X}$). Now one argues that
the identification must be algebraic as well, since one has the
one-one correspondence between analytic and algebraic extensions
of an abelian variety by ${\mathbb G}_a^r\times{\mathbb G}_m^s$.
(ii) Let $\tilde{X} \to X^{(n)}$ be a desingularization of $X^{(n)}$ such
that the proper transform $\tilde{C}$ of $C$ is the normalization of
$C$. First note that one has a factorization
$$
\begin{CD}
A^1(C) \>>> A^n(X)\\
{\mathbb V} V V {\mathbb V} V V\\
J^1(C) \>>> J^n(X)\\
{\mathbb V} V V {\mathbb V} V V\\
A^1(\tilde{C}) \>>> A^n(\tilde{X})
\end{CD}
$$
where all maps are analytic group homomorphisms, and
the vertical ones are algebraic (lemma \ref{algebraicity}).
Indeed the map $C \to X$ induces a morphism of mixed Hodge
structures $H^1(C) \to H^{2n-1}(X)$, and therefore an analytic
group homomorphim $J^1(C) \to J^n(X)$, which has to be algebraic
as it is compatible with its abelian part
$J^1(\tilde{C}) \to A^n(\tilde{X})$ and all analytic group
homomorphisms ${\mathbb G}_m^s \to {\mathbb G}_m^{s'}$ are algebraic. Similarly,
all group homomorphisms ${\mathbb G}_a^r \to {\mathbb G}_a^{r'}$ are algebraic,
and therefore $A^1(C) \to A^n(X)$ is algebraic as well.
\end{proof}
\begin{defn}\label{deflie}
For any commutative algebraic group $A$ over ${\mathbb C}$, let
$\Omega(A)$ denote the dual vector space to the Lie algebra
$\Lie(A)$. We may then identify $\Omega(A)$ with the vector
space of (closed) translation invariant regular 1-forms on $A$.
\end{defn}
Our next goal is to give a description of $\Omega(A^n(X))$, generalizing
the fact that for a non-singular projective variety $X$, $\Omega(A^n(X))$
is the space of holomorphic 1-forms on $X$ (since in that case, $A^n(X)$
is the Albanese variety of $X$).
\begin{lemma}\label{purity}
Let $X$ be projective of dimension $n$ over ${\mathbb C}$, and let
$\omega_X$ denote the dualizing module of $X$ (in the sense of \cite{Ha},
Ch.~III, \S7). Let $X^{(n)}$ be the union of the $n$-dimensional
components of $X$, and let $\omega_{X^{(n)}}$ denote its dualizing module.
\begin{points}
\item $\omega_X$ is annihilated by the ideal sheaf of $X^{(n)}$ in $X$.
With its natural induced structure as an ${\mathcal O}_{X^{(n)}}$-module,
$\omega_X\cong\omega_{X^{(n)}}$, and is a torsion-free
${\mathcal O}_{X^{(n)}}$-module. Hence for any coherent ${\mathcal O}_X$-module ${\mathcal F}$, the
sheaf $\shom_{{\mathcal O}_X}({\mathcal F}, \omega_X)$ is also naturally an
${\mathcal O}_{X^{(n)}}$-module, which is ${\mathcal O}_{X^{(n)}}$-torsion free, and
for any dense open set $U\subset X^{(n)}$, the restriction map
\[{\rm Hom}_X({\mathcal F},\omega_X)\>>> {\rm Hom}_U({\mathcal F}\mid_U,\omega_X\mid_U)\]
is injective. In particular, taking $U=X_{{\rm reg}}$, so that
$\omega_X\mid_U=\Omega^n_{U/{\mathbb C}}$, and taking ${\mathcal F}=\Omega^{n-1}_{X/{\mathbb C}}$,
we have that
${\rm Hom}_X(\Omega^{n-1}_{X/{\mathbb C}},\omega_X)$
may be identified with a ${\mathbb C}$-subspace of the vector space of holomorphic
1-forms on $X_{{\rm reg}}$ which are meromorphic on $X^{(n)}$.
\item $\Omega(A^{n}(X))$ is naturally identified with the subspace of
${\rm Hom}_X(\Omega^{n-1}_{X/{\mathbb C}},\omega_X)$ consisting of {\em closed} 1-forms.
\item When $n=1$,
\[\Omega(A^{1}(X))=\Omega({\rm Pic}^{0}(X)))=H^{0}(X,\omega_X).\]
\item Let $j:X_{\rm CM}\to X$ be the inclusion of the open subset of
Cohen-Macaulay points. The natural map
\[\Omega(A^{n}(X))\>>>
\left(\mbox{closed 1-forms in
${\rm Hom}_X(\Omega^{n-1}_{X/{\mathbb C}},j^{m}_*j^{*}\omega_X)$}\right)\]
is an isomorphism, where $j^{m}_*$ denotes the meromorphic
direct image.
\end{points}
\end{lemma}
\begin{proof} (i) We note first that $\omega_X\cong\omega_{X^{(n)}}$, and
the latter is a torsion-free ${\mathcal O}_{X^{(n)}}$-module. Indeed, if we fix a
projective embedding $X\hookrightarrow\P^N_{{\mathbb C}}$, then
\[\omega_X=\sext_{\P^N}^{N-n}({\mathcal O}_X,\omega_{\P^N_{{\mathbb C}}}),\]
and there is an analogous formula for $\omega_{X^{(n)}}$. As in \cite{Ha},
we see by Serre duality on $\P^N_{{\mathbb C}}$ that
$\sext^i({\mathcal F},\omega_{\P^N_{{\mathbb C}}})=0$ for all
$i\leq N-n$ for any coherent sheaf ${\mathcal F}$ supported in dimension $<n$. This
gives the desired isomorphism, and implies that any local section of
${\mathcal O}_{X^{(n)}}$, which is a non zero-divisor, is also a non zero-divisor
on $\sext^{N-n}_{\P^N_{{\mathbb C}}}({\mathcal O}_X,\omega_{\P^N_{{\mathbb C}}})$. This means exactly
that $\omega_{X^{(n)}}$ is torsion-free.
We conclude that for any coherent ${\mathcal O}_X$-module ${\mathcal F}$, the
sheaf $\shom_X({\mathcal F},\omega_X)$ is a torsion-free ${\mathcal O}_{X^{(n)}}$-module as
well. Applying this to ${\mathcal F}=\Omega^{n-1}_{X/{\mathbb C}}$ gives (i).
(iii) is a special case of (ii). To prove (ii), first
note that from the definition of $A^{n}(X)$, we have
\begin{equation}\label{lie_descr}
\Lie(A^{n}(X))=\coker(d:H^{n}(X,\Omega^{n-2}_{X/{\mathbb C}})\to
H^{n}(X,\Omega^{n-1}_{X/{\mathbb C}})).
\end{equation}
{From} Serre duality for $H^{n}$ and ${\rm Hom}$, as in the definition of the
dualizing sheaf in \cite{Ha}, we have an identification of the dual vector
space
\[H^{n}(X,\Omega^i_{X/{\mathbb C}})^{*}=
{\rm Hom}_{{\mathcal O}_X}(\Omega^i_{X/{\mathbb C}},\omega_X),\]
for any $i$. Thus $\Omega(A^n(X))$ is identified with the subspace of
${\rm Hom}_{{\mathcal O}_X}(\Omega^{n-1}_{X/{\mathbb C}},\omega_X)$ of elements $\varphi$ such that
the composition
\[\ell: H^n(X,\Omega^{n-2}_{X/{\mathbb C}})\>{d}>> H^n(X,\Omega^{n-1}_{X/{\mathbb C}})
\>{\varphi}>> H^n(X,\omega_X)\cong {\mathbb C}\]
is 0. It remains to show that, identifying elements $\varphi\in
{\rm Hom}_{{\mathcal O}_X}(\Omega^{n-1}_{X/{\mathbb C}},\omega_X)$ with certain holomorphic
1-forms on $X_{{\rm reg}}$, $\Omega(A^n(X))$ is just the subspace of closed
1-forms.
To see this, since we may consider $\varphi$ as a meromorphic 1-form on
$X$ which is holomorphic on $X_{{\rm reg}}$, we can find a coherent sheaf of
ideals ${\mathcal J}$, defining the Zariski closed subset $X_{{\rm sing}}\subset X$ ({\it i.e.\/},\
the subscheme determined by ${\mathcal J}$ has $X_{{\rm sing}}$ as its underlying reduced
scheme), such that
\begin{points}
\item $\eta\mapsto \eta\wedge\varphi$ defines an element of
${\rm Hom}_{{\mathcal O}_X}({\mathcal J}\Omega^{n-2}_{X/{\mathbb C}},\Omega^{n-1}_{X/{\mathbb C}})$
\item $\eta\mapsto \eta\wedge d\varphi$ defines an element of
${\rm Hom}_{{\mathcal O}_X}({\mathcal J}\Omega^{n-2}_{X/{\mathbb C}},\omega_X)$, where we view
$\omega_X$ as a certain coherent extension of $\Omega^n_{X_{{\rm reg}}/{\mathbb C}}$ to
$X$.
\end{points}
(Here ${\mathcal J}{\mathcal F}$ denotes ${\rm image}\,({\mathcal J}\otimes{\mathcal F}\to{\mathcal F})$, for any ideal
sheaf ${\mathcal J}$ and coherent sheaf ${\mathcal F}$). Since ${\mathcal J}$ defines $X_{{\rm sing}}$
within $X$, the natural map
\[H^n(X,{\mathcal J}\Omega^{n-2}_{X/{\mathbb C}})\>>> H^n(X,\Omega^{n-2}_{X/{\mathbb C}})\]
is surjective, and for any
$\varphi\in{\rm Hom}_{{\mathcal O}_X}(\Omega^{n-1}_{X/{\mathbb C}},\omega_X)$, the composition
\[\ell_1:H^n(X,{\mathcal J}\Omega^{n-2}_{X/{\mathbb C}})\>{d}>>
H^n(X,\Omega^{n-1}_{X/{\mathbb C}})\>{\varphi}>>
H^n(X,\omega_X)\cong {\mathbb C}\]
factors through $\ell$. Thus
\[\varphi\in \Omega(A^n(X))\;\;\;\iff\;\;\; \ell_1=0.\]
We have 2 other related linear functionals
\[\ell_2:H^n(X,{\mathcal J}\Omega^{n-2}_{X/{\mathbb C}})\>>>{\mathbb C},\;\;\;\;
\ell_3:H^n(X,{\mathcal J}\Omega^{n-2}_{X/{\mathbb C}})\>>>{\mathbb C},\]
defined by
\begin{gather*}
\ell_2:H^n(X,{\mathcal J}\Omega^{n-2}_{X{\mathbb C}})\>{\wedge d\varphi}>>
H^n(X,\omega_X)\cong {\mathbb C},\\
\ell_3: H^n(X,{\mathcal J}\Omega^{n-2}_{X/{\mathbb C}})\>{\wedge \varphi}>>
H^n(X,\Omega^{n-1}_{X/{\mathbb C}})\>{d}>> H^n(X,\omega_X)\cong{\mathbb C},
\end{gather*}
where in the definition of $\ell_3$, we have let $d$ also denote the
composite of the exterior derivative
$\Omega^{n-1}_{X/{\mathbb C}}\to\Omega^n_{X/{\mathbb C}}$ with the natural map
$\Omega^n_{X/{\mathbb C}}\to\omega_X$. The formula
\[d(\eta\wedge\varphi)=d\eta\wedge\varphi+(-1)^{n-2}\eta\wedge d\varphi,\]
for any $n-2$ form $\eta$, implies that
$\ell_3=\ell_1+(-1)^{n-2}\ell_2.$
Now by Serre duality and the ${\mathcal O}_{X^{(n)}}$-torsion freeness of
$\shom_{{\mathcal O}_X}({\mathcal J}\Omega^{n-2}_{X/{\mathbb C}}, \omega_X)$ (see (i)),
$\ell_2$ vanishes precisely when $d\varphi=0$ as a 2-form on $X_{reg}$. On
the other hand, we claim that for any $\varphi\in
{\rm Hom}_{{\mathcal O}_X}(\Omega^{n-1}_{X/{\mathbb C}},\omega_X)$, the map $\ell_3$ constructed
as above is always 0. This will imply that $\ell_1=0$ \iff $\varphi$ is a
closed meromorphic 1-form.
To prove that $\ell_3$ vanishes, it suffices to prove that the
map
\[H^n(X,\Omega^{n-1}_{X/{\mathbb C}})\>{d}>> H^n(X,\omega_X)\]
vanishes. One way to understand this is to note that if $\pi:Y\to X$ is a
resolution of singularities, then there is a commutative diagram
\[\begin{CD}
H^n(X,\Omega^{n-1}_{X/{\mathbb C}}) \>{d}>> H^n(X,\omega_X)\\
{\mathbb V} \pi^* VV {\mathbb A} A \pi_* A\\
H^n(Y,\Omega^{n-1}_{Y/{\mathbb C}}) \>{d}>> H^n(Y,\omega_Y)
\end{CD}
\]
which reduces us to proving that $H^n(Y,\Omega^{n-1}_{Y/{\mathbb C}})\>{d}>>
H^n(Y,\omega_Y)$ vanishes. This follows from Hodge theory, or alternately
may be proved as in \cite{Ha2}, III, lemma~8.4.
Proof of (iv):\quad We begin by recalling that since $X$ is
reduced, it is Cohen-Macaulay in codimension 1, so that
$Z=X-j(X_{\rm CM})$ has codimension $\geq 2$ in $X$. Let ${\mathcal I}$
denote the ideal sheaf of $Z$ in $X$. Let ${\mathcal D}_m$ be the complex
of sheaves
\[{\mathcal D}_m=(0\to j_!{\mathbb Z}_{X_{\rm CM}}(n)\to
{\mathcal I}^{m+n-1}\by{d}{\mathcal I}^{m+n-2}\Omega^{1}_{X/{\mathbb C}}\by{d} \cdots
\by{d} {\mathcal I}^{m}\Omega^{n-1}_{X/{\mathbb C}}).\]
Then ${\mathcal D}_m$ is a subcomplex of ${\mathcal D}(n)_X$, whose cokernel
complex consists of sheaves supported on $Z$; the 0-th term of
the cokernel is ${\mathbb Z}(n)_Z$, while the other terms are coherent
sheaves supported on $Z$. Since $\dim Z\leq n-2$, we see that
${\mathbb H}^{i}$ of this cokernel complex vanishes for $i\geq 2n-1$.
Hence ${\mathbb H}^{2n}(X,{\mathcal D}_m)\to
{\mathbb H}^{2n}(X,{\mathcal D}(n)_X)$ is an isomorphism, for all $m$. Now as
in the proof of (i), one uses duality, to conclude that for all
$m$, there are isomorphisms
\begin{align*}
{\rm Hom}(\Omega^{n-1}_{X/{\mathbb C}},\omega_X)&\>>>
{\rm Hom}({\mathcal I}^m\Omega^{n-1}_{X/{\mathbb C}},\omega_X),\\
{\rm Hom}(\Omega^{n-2}_{X/{\mathbb C}},\omega_X)&\>>>
{\rm Hom}({\mathcal I}^m\Omega^{n-2}_{X/{\mathbb C}},\omega_X),
\end{align*}
and taking the direct limit over all $m$, we obtain (iv).
\end{proof}
Our next goal is the proof of proposition~\ref{basic}, which gives us
another useful way to recognize elements of the vector space
$\Omega(A^n(X))$. We make use of two lemmas.
\begin{lemma}\label{basic1}
Let $X\subset\P^N_{{\mathbb C}}$ be a reduced projective variety of dimension $n$.
Then we can find a finite number of linear projections $\pi_i:X\to
\P^n_{{\mathbb C}}$, each of which is a finite morphism, such that the induced
sheaf map
\[\bigoplus_i\pi_i^*\Omega^{n-1}_{\P^n/{\mathbb C}}\>>> \Omega^{n-1}_{X/{\mathbb C}}\]
is surjective.
\end{lemma}
\begin{proof} For any linear projection $\pi:X\to\P^n_{{\mathbb C}}$, there is a
factorization
\[\pi^*\Omega^{n-1}_{\P^n_{{\mathbb C}}/{\mathbb C}}\>>>
\Omega^{n-1}_{\P^N_{{\mathbb C}}/{\mathbb C}}\otimes{\mathcal O}_X\>{\psi}>>\Omega^{n-1}_{X/{\mathbb C}},\]
where the natural map $\psi$ is surjective.
So it suffices to prove the stronger assertion that there are projections
$\pi_i$ as above such that the induced sheaf map
\[\bigoplus_i\pi_i^*\Omega^{n-1}_{\P^n/{\mathbb C}}\>>>
\Omega^{n-1}_{\P^N_{{\mathbb C}}/{\mathbb C}}\otimes{\mathcal O}_X\]
is surjective.
We claim that for any $x\in X$, we can find a finite set of such
projections $\pi_i:X\to\P^n_{{\mathbb C}}$ such that the map of ${\mathbb C}$-vector spaces
\[\bigoplus_i\pi_i^*\Omega^{n-1}_{\P^n/{\mathbb C}}\otimes{\mathbb C}(\pi_i(x))\>>>
\Omega^{n-1}_{\P^N_{{\mathbb C}}/{\mathbb C}}\otimes{\mathbb C}(x)\]
is surjective. Indeed, the Grassmannian ${\mathbb G}_{{\mathbb C}}(n+1,N+1)$ (of $n+1$
dimensional subspaces of ${\mathbb C}^{N+1}$) parametrizes linear projections from
$\P^N_{{\mathbb C}}$ to $\P^n_{{\mathbb C}}$, and it contains a dense Zariski open subset
corresponding to projections which are finite morphisms on $X$. Hence the
$n$-dimensional vector subspaces
\[\pi^*\Omega^1_{\P^n_{{\mathbb C}}/{\mathbb C}}\otimes{\mathbb C}(\pi(x))\subset
\Omega^1_{\P^N_{{\mathbb C}}/{\mathbb C}}\otimes{\mathbb C}(x)\]
also range over a Zariski open subset of the Grassmannian of
$n$-dimensional subspaces of the cotangent space of $\P^N_{{\mathbb C}}$ at $x$.
In particular, we can find a finite number of them whose $(n-1)$-th exterior
powers span the $(n-1)$-th exterior power of this cotangent space, namely
$\Omega^{n-1}_{\P^N_{{\mathbb C}}/{\mathbb C}}\otimes{\mathbb C}(x)$.
Now suppose $\pi_1,\ldots,\pi_r$ are chosen finite linear projections
$X\to\P^n_{{\mathbb C}}$, and that
\[\bigoplus_{i=1}^r\pi_i^*\Omega^{n-1}_{\P^n/{\mathbb C}}\>>>
\Omega^{n-1}_{\P^N_{{\mathbb C}}/{\mathbb C}}\otimes{\mathcal O}_X\]
is not surjective. We can then find a point $x\in X$ at which the cokernel
is non-zero. By the above claim, we can augment the set of projections to
$\pi_1,\ldots,\pi_r,\pi_{r+1},\ldots,\pi_s$ so that the cokernel of the
new map
\[\bigoplus_{i=1}^{r+s}\pi_i^*\Omega^{n-1}_{\P^n/{\mathbb C}}\>>>
\Omega^{n-1}_{\P^N_{{\mathbb C}}/{\mathbb C}}\otimes{\mathcal O}_X\]
does not have $x$ in its support. Thus the support of the cokernel has
strictly decreased. Now the lemma follows by Noetherian induction.
\end{proof}
\begin{lemma}\label{basic2}
Let ${\mathcal F}$ be a reflexive coherent sheaf on $\P^n_{{\mathbb C}}$, and $\omega$ a
meromorphic section of ${\mathcal F}\otimes\Omega^1_{\P^n_{{\mathbb C}}/{\mathbb C}}$, which is
regular on some given (nonempty) Zariski open subset $W\subset\P^n_{{\mathbb C}}$.
Suppose there is a non-empty open set $V$ in ${\mathbb G}_{{\mathbb C}}(2,n+1)$, the
Grassmannian of lines in $\P^n_{{\mathbb C}}$, such that
\begin{points}
\item each line $L\in V$ meets $W$, and is disjoint from the non
locally-free locus of ${\mathcal F}$
\item for each $L\in V$, the image of $\omega$ in
$({\mathcal F}\otimes\Omega^1_{L/{\mathbb C}})\mid_{L\cap W}$ extends to a regular section
of ${\mathcal F}\otimes\Omega^1_{L/{\mathbb C}}$ on $L$.
\end{points}
Then $\omega$ extends (uniquely) to a regular section on $\P^n_{{\mathbb C}}$ of
${\mathcal F}\otimes\Omega^1_{\P^n_{{\mathbb C}}/{\mathbb C}}$.
\end{lemma}
\begin{proof}
Since ${\mathcal F}$ is reflexive, it is locally free outside a Zariski closed
set $A$ (of codimension $\geq 3$), and any section of
${\mathcal F}\otimes\Omega^1_{\P^n_{{\mathbb C}}/{\mathbb C}}$ defined in the complement of $A$
extends uniquely to a section on all of $\P^n_{{\mathbb C}}$. Since $\omega$ is a
meromorphic section, it determines a (unique) regular section of some twist
${\mathcal F}\otimes\Omega^1_{\P^n_{{\mathbb C}}/{\mathbb C}}(D)$, for an effective divisor $D$;
there is a unique such twist $D$ which is minimal with respect to the
partial order on effective divisors (determined by inclusion of
subschemes). Our goal is to show that $D=0$.
If $F$ is an irreducible component of ${\rm supp}\, D$ which appears
in $D$ with multiplicity $r>0$, then we can find a point $x\in F$ such that
\begin{points}
\item $x$ is a non-singular point of $F$, and does not lie on any other
component of $D$; further, ${\mathcal F}$ is locally free near $x$
\item $V$ contains a line through $x$
\item there is a regular parameter $t\in{\mathcal O}_{x,\P^n_{{\mathbb C}}}$ ({\it i.e.\/},\ $t$ is
part of a regular system of parameters) such that $t$ defines the
ideal of $F$ at $x$, and such that $t^r\omega$ determines a regular,
non-vanishing section of ${\mathcal F}\otimes\Omega^1_{\P^n_{{\mathbb C}}/{\mathbb C}}$ in a
neighbourhood of $x$.
\end{points}
It then follows that for a non-empty Zariski open set of lines $L$ through
$x$, we have $L\in V$, and $t^r \omega$ maps to a regular, non-vanishing
section of ${\mathcal F}\otimes\Omega^1_{L/{\mathbb C}}$ near $x$, while $\omega$ itself
maps to a regular section of ${\mathcal F}\otimes\Omega^1_{L/{\mathbb C}}$. However, $t$
vanishes at $x$. This is a contradiction.
\end{proof}
If $C\subset X$ is a reduced, local complete intersection Cartier curve,
then in fact $C\subset X_{\rm CM}\cap X^{(n)}$ (recall that $X_{\rm CM}$ denotes
the (dense) Zariski open subset of Cohen-Macaulay points of $X$). The
sheaf map ${\mathcal O}_C\to {\mathcal H}^{2n-1}_C(\Omega^{n-1}_{X/{\mathbb C}})$ in (\ref{sheafmap})
induces a composite map
$$\alpha_C:H^1(C,{\mathcal O}_C)\>>> H^1(C,{\mathcal H}^{2n-1}_C(\Omega^{n-1}_{X/{\mathbb C}}))\>>>
H^n_C(X,\Omega^{n-1}_{X/{\mathbb C}})\mbox{$\,\>>>\hspace{-.5cm}\to\hspace{.15cm}$} {\mathbb H}^{2n-1}(X,\Omega^{<n}_{X/{\mathbb C}}).$$
This is just the map $\Lie({\rm Pic}^0(C))\to \Lie(A^n(X))$ on Lie algebras
induced by the composition of the group homomorphisms ${\rm Pic}^0(C)\to
A^1(C)$ and the Gysin map $A^1(C)\to A^n(X)$.
\begin{propose}\label{basic} \ \
\begin{enumerate}
\item[(a)] Let $C\subset X$ be a reduced, local complete intersection
Cartier curve, and let $U\subset X_{reg}$ be a dense open subset such that
$U\cap C$ is dense in $C_{reg}$. Then the dual $\alpha_C^{\vee}$ of
$\alpha_C:H^1(C,{\mathcal O}_C)\to {\mathbb H}^{2n-1}(X,\Omega^{<n}_{X/{\mathbb C}})$
({\it i.e.\/},\ of $\Lie({\rm Pic}^0(C))\to \Lie(A^n(X))$)
fits into a commutative diagram
$$
\begin{CD}
\Omega(A^n(X)) \> \subset >> H^0(X_{\rm reg}, \Omega^1_{X_{\rm reg}/{\mathbb C}})\\
{\mathbb V} V {\alpha_C^{\vee}}V {\mathbb V} V {\rm restriction}V\\
H^0(C,\omega_C) \> \subset >>
H^0(C\cap U,\Omega^1_{C\cap U/{\mathbb C}}).
\end{CD}
$$
(Here the right hand vertical arrow is given by restriction of 1-forms.)
\item[(b)]
Let $U\subset X_{{\rm reg}}$ be a dense Zariski open set, and let $\omega\in
\Gamma(U,\Omega^1_{U/{\mathbb C}})$ be closed. Then $\omega\in
\Omega(A^n(X))$ if and only if
\begin{enumerate}
\item[(i)] $\omega$ yields a meromorphic section on $X$ of $\Omega^1_{X/{\mathbb C}}$
\item[(ii)] for any reduced, local complete intersection Cartier curve
$C\subset X$ such that $C\cap U$ is dense in $C$, the
restriction of $\omega$ to $B=C_{reg}\cap U$ is in the image of the
natural injective map
\[H^0(C,\omega_C) \>>> H^0(B,\Omega^1_{B/{\mathbb C}}).\]
\end{enumerate}
\end{enumerate}
\end{propose}
\begin{proof}
First we prove (a). From lemma~\ref{purity}, it suffices to prove that if
$\beta_C:H^1(C,{\mathcal O}_C)\to H^n(X,\Omega^{n-1}_{X/{\mathbb C}})$ is the obvious map
through which $\alpha_C$ factors, then the dual map $\beta_C^{\vee}$ fits into
a commutative diagram
$$
\begin{CD}
H^0(X,\shom_X(\Omega^{n-1}_{X/{\mathbb C}},\omega_X)) \> \subset >> H^0(X_{\rm reg},
\Omega^1_{X_{\rm reg}/{\mathbb C}})\\
{\mathbb V} V {\beta_C^{\vee}}V {\mathbb V} V {\rm restriction}V\\
H^0(C,\omega_C) \> \subset >>
H^0(C\cap X_{\rm reg},\omega_{C\cap X_{{\rm reg}}}).
\end{CD}
$$
Here we have used Serre duality on $X$ and $C$ to make the identifications
$$H^n(X,\Omega^{n-1}_{X/{\mathbb C}})^{\vee}=
H^0(X,\shom_X(\Omega^{n-1}_{X/{\mathbb C}},\omega_X)),$$
$$H^1(C,{\mathcal O}_C)^{\vee}=H^0(C,\omega_C).$$
Since $C$ is a reduced, local complete intersection Cartier curve in $X$
(so that $C\subset X_{\rm CM}\cap X^{(n)}$), we have the adjunction formula
\[\omega_C=\shom_C(\bigwedge^{n-1}{\mathcal I}_C/{\mathcal I}_C^2,\omega_X\otimes{\mathcal O}_C).\]
Hence there is a natural sheaf map
\[\psi_C:\shom_X(\Omega^{n-1}_{X/{\mathbb C}},\omega_X)\>>>
\shom_C(\bigwedge^{n-1}{\mathcal I}_C/{\mathcal I}_C^2,\omega_X\otimes{\mathcal O}_C)=\omega_C\]
induced by restriction to $C$, and composition with the natural map
\begin{align*}
\bigwedge^{n-1}{\mathcal I}_C/{\mathcal I}_C^2&\>>>\Omega^{n-1}_{X/{\mathbb C}}\otimes{\mathcal O}_C,\\
f_1\wedge\cdots\wedge f_{n-1}&\longmapsto df_1\wedge\cdots\cdots
df_{n-1}.
\end{align*}
On any open set $U\subset X_{{\rm reg}}$ with $U\cap C\subset C_{{\rm reg}}$, one
verifies at once, from the explicit description, that the map
$\psi_C\mid_U$ is just the restriction map on 1-forms
$\Omega^1_{U/{\mathbb C}}\to\Omega^1_{C\cap U/{\mathbb C}}$.
Hence the desired commutativity (which implies (a)) follows from:
\begin{claim}\label{duality}
$\beta^{\vee}$ is the map induced by $\psi_C$ on global sections.
\end{claim}
To prove the claim, first note that for the local complete intersection
curve $C$ in $X_{\rm CM}$, one also has
$$
\sext^{n-a}_X({\mathcal O}_C,\omega_X)=
\left\{ \begin{array}{ll}
\omega_C & \mbox{ \ for \ } a=1\\
0 & \mbox{ \ for \ } a\neq 1,
\end{array} \right.
$$
Hence there is a Gysin map given as the composite
\begin{multline*}
H^1(C,\omega_C)=H^1(X,\sext^{n-1}_X({\mathcal O}_C,\omega_X)) \> \epsilon
>> {\rm Ext}^n_X({\mathcal O}_C,\omega_X)\\
\>>> H^n_C(X,\omega_X) \>>> H^n(X,\omega_X)
\end{multline*}
where $\epsilon$ is the isomorphism resulting from the (degenerate)
spectral sequence
$$
E_2^{a,b-a}=H^a(X,\sext^{b-a}_X({\mathcal O}_C,\omega_X)) \Longrightarrow
{\rm Ext}^b_X({\mathcal O}_C,\omega_{X}).
$$
The trace map ${\rm Tr}_C:H^1(C,\omega_C) \to {\mathbb C}$ (of Serre duality on
$C$) factors as
$$
{\rm Tr}_C:H^1(C,\omega_C) \>{\rm Gysin}>> H^n(X,\omega_X) \> {\rm Tr}_X >>
{\mathbb C}
$$
(one way to verify this is to show that the composite ${\rm Tr}_X\circ{\rm
Gysin}$ has the universal property of ${\rm Tr}_C$).
Now the claim~\ref{duality} amounts to the assertion that the following
diagram commutes:
\[\begin{CD}
H^1(C,{\mathcal O}_C) \>{\rm Gysin}>> H^n(X,\Omega^{n-1}_{X/{\mathbb C}})\\
{\mathbb V}{\psi_C(\varphi)} V V {\mathbb V}{\varphi} V V\\
H^1(C,\omega_C) \>{\rm Gysin}>> H^n(X,\omega_X)
\end{CD}\]
{From} Remark~\ref{gys_rem}, this will follow if we prove the commutativity
of the diagram of ${\mathcal O}_X$-linear maps
$$
\begin{CD}
{\mathcal O}_C \>>> \sext^{n-1}_X({\mathcal O}_C,\Omega^{n-1}_{X/{\mathbb C}})\\
{\mathbb V} \psi_C(\varphi) V V {\mathbb V} V \varphi V\\
\omega_C \> \cong >> \sext^{n-1}_X({\mathcal O}_C,\omega_X).
\end{CD}
$$
As $\omega_C$ is torsion-free, it is enough to check this commutativity on
a suitable open subset of the regular locus of $C$, where it is easily
verified.
We now show the ``if'' part of (b) (note that the other direction
follows directly from (a)). By lemma~\ref{basic1}, it suffices to prove
that for each finite, linear projection\\
$\pi:X\to\P^n_{{\mathbb C}}$, the meromorphic 1-form $\omega$ determines a
section of
$$\shom_X(\pi^*\Omega^{n-1}_{\P^n_{{\mathbb C}}/{\mathbb C}},\omega_X).$$
Since $\pi$ is a finite morphism,
\[\pi_*\omega_X=\shom_{\P^n_{{\mathbb C}}}(\pi_*{\mathcal O}_X,\omega_{\P^n_{{\mathbb C}}}),\]
and we have a sequence of natural identifications of sheaves
\begin{align*}
&\pi_*\shom_X(\pi^*\Omega^{n-1}_{\P^n_{{\mathbb C}}/{\mathbb C}},\omega_X)\cong
\shom_{\P^n_{{\mathbb C}}}(\Omega^{n-1}_{\P^n_{{\mathbb C}}/{\mathbb C}}\otimes\pi_*{\mathcal O}_X,
\omega_{\P^n_{{\mathbb C}}})\\
&\cong \shom_{\P^n_{{\mathbb C}}}(\pi_*{\mathcal O}_X,
\shom_{\P^n_{{\mathbb C}}}(\Omega^{n-1}_{\P^n_{{\mathbb C}}/{\mathbb C}},\omega_{\P^n_{{\mathbb C}}}))
\cong {\mathcal F}\otimes\Omega^1_{\P^n_{{\mathbb C}}/{\mathbb C}},
\end{align*}
where ${\mathcal F}=\shom_{\P^n_{{\mathbb C}}}(\pi_*{\mathcal O}_X,{\mathcal O}_{\P^n_{{\mathbb C}}})$
is a (non-zero) coherent reflexive sheaf on $\P^n_{{\mathbb C}}$.
Let $W\subset\P^n_{{\mathbb C}}$ be a dense open subset such that
$\pi^{-1}(W)\subset U$. Then $\omega$ determines a section of
${\mathcal F}\otimes\Omega^1_{\P^n_{{\mathbb C}}/{\mathbb C}}$ on $W$, and we want to show it
extends to a global section of this sheaf. We do this by verifying that
the hypotheses of lemma~\ref{basic2} are satisfied.
Let $L$ be a line in $\P^n_{{\mathbb C}}$, disjoint from the
non-flat locus of $\pi:X^{(n)}\to\P^n_{{\mathbb C}}$ (which is a subset of $\P^n_{{\mathbb C}}$
of codimension $\geq 2$, since $X^{(n)}$ is reduced and purely of dimension
$n$). Then the scheme-theoretic inverse image of $L$ in $X^{(n)}$ is a closed,
local complete intersection subscheme of $X^{(n)}$, purely of dimension 1, and
which is contained in the Cohen-Macaulay locus of $X^{(n)}$ (since
$X^{(n)}$ is Cohen-Macaulay precisely at all points $x\in X$ where $\pi$
is flat). If further $L$ is not contained in the branch locus of $\pi$ on
$X^{(n)}$ ({\it i.e.\/},\ $\pi$ is \'etale over all but finitely many points of $L$),
then $\pi^{-1}(L)=D$ is non-singular outside a finite set. Thus $D$ is a
reduced, complete intersection curve in $X^{(n)}$. Further, if $D\cap
X^{<n}=\emptyset$, then $D$ is a reduced local complete intersection curve
in $X$, whose non-singular locus is contained in $X_{{\rm reg}}$. In particular
$D$ is a reduced Cartier curve in $X$. Finally, if $L$ is not contained
in the image of $X-U$, then $D$ has finite intersection with $X-U$, and
hence $D\cap U$ is dense in $D$. Clearly the set of all such lines $L$
contains a non-empty open subset of the Grassmannian of lines.
For a line $L$ as above, we have
\begin{align*}
\pi_*\omega_D&\cong \shom_{L}(\pi_*{\mathcal O}_D,\omega_L)\cong
\shom_L(\pi_*{\mathcal O}_X\otimes{\mathcal O}_L,\omega_L)\\
&\cong \shom_L(\pi_*{\mathcal O}_X\otimes{\mathcal O}_L,\Omega^1_{L/{\mathbb C}})
\cong {\mathcal F}\otimes\Omega^1_{L/{\mathbb C}},
\end{align*}
since ${\mathcal F}\otimes{\mathcal O}_L\cong \shom_L(\pi_*{\mathcal O}_D,{\mathcal O}_L)$ (as $\pi$ is flat
over $L$). Since we are given that the image of $\omega$ in
$\Omega^1_{D\cap U/{\mathbb C}}$ extends to a global section of $\omega_D$,
it follows that the corresponding section of
${\mathcal F}\otimes\Omega^1_{L/{\mathbb C}}\mid_{L\cap W}$ extends to a global section of
${\mathcal F}\otimes\Omega^1_{L/{\mathbb C}}$. Thus we have verified the hypotheses of
lemma~\ref{basic2}.
\end{proof}
\begin{rmk}\label{example}
Two properties of $A^n(Y)$, which are true for smooth projective varieties
$Y$, do not carry over to the general case: the compatibility with
products, and its dimension being constant in a flat family. We give
examples to illustrate these pathologies.
Let $X$ and $Y$ be projective varieties of dimension $n$ and
$m$, respectively, and let $r(X)$ and $r(Y)$ denote the number
of irreducible components of dimensions $n$ and $m$ respectively. By
\cite{D} the K\"unneth decomposition $$
H^{2(n+m)-1}(X\times Y, {\mathbb Z})/_{({\rm torsion})} =
\left[H^{2n-1}(X,{\mathbb Z})^{r(Y)} \oplus H^{2m-1}(Y,{\mathbb Z})^{r(X)}\right]
/_{({\rm torsion})}
$$
is compatible with the Hodge structure. Thus
\begin{equation}\label{product1}
J^{n+m}(X\times Y) = J^n(X)^{r(Y)} \times J^m(Y)^{
r(X)} .
\end{equation}
For $A^{n+m}(X \times Y)$ the picture is wilder.
By (\ref{lie_descr}) in the proof of \ref{purity}, we have
\begin{gather}\label{product2}
\Lie(A^{n+m}(X\times Y))=\\
\frac{H^n(\Omega^{n-1}_{X/{\mathbb C}})\otimes
H^m(\Omega^{m}_{Y/{\mathbb C}})\oplus H^n(\Omega^{n}_{X/{\mathbb C}})\otimes
H^m(\Omega^{m-1}_{Y/{\mathbb C}})}{H^n(\Omega^{n-2}_{X/{\mathbb C}})\otimes
H^m(\Omega^{m}_{Y/{\mathbb C}})\oplus H^n(\Omega^{n-1}_{X/{\mathbb C}})\otimes
H^m(\Omega^{m-1}_{Y/{\mathbb C}})\oplus H^n(\Omega^{n}_{X/{\mathbb C}})\otimes
H^m(\Omega^{m-2}_{Y/{\mathbb C}})} \notag
\end{gather}
where the maps from the denominator are
$$
d_X\otimes {\rm id}_Y, \ \ d_X\otimes {\rm id}_Y + (-1)^{n-1}{\rm id}_X
\otimes d_Y\mbox{ \ \ and \ \ }{\rm id}_X \otimes d_Y.
$$
Consider an elliptic curve $E$, the rational curve
$\Gamma=(x^3-y^2z) \subset \P^2_{\mathbb C}$, with a cusp, and the union
of three rational curves $ C = (xyz) \subset \P^2_{\mathbb C}$. They all
are fibres of the family ${\mathcal C} \to \P=\P(H^0(\P^2,{\mathcal O}_{\P^2}(3)))$
of curves of degree three in $\P^2$.
Hodge theory implies that
$$
d:H^1(E,{\mathcal O}_E) \>>> H^1(E,\Omega_E^1)\mbox{ \ \ and \ \ }
d:H^1(C,{\mathcal O}_C) \>>> H^1(C,\Omega_C^1)\cong {\mathbb C}^3
$$
are both zero. Using (\ref{product2}) this shows
\begin{align*}
&A^2(C \times E) = J^2(C \times E) =
A^1(C)\times A^1(E)^{3}= {\mathbb G}_m
\times E^{3}\\
&A^2(C \times C) = J^2(C \times C) =
A^1(C)^{3}\times A^1(C)^{3}= {\mathbb G}_m^6.
\end{align*}
On the other hand, $\Gamma - (0:1:0)= {\rm Spec \,}({\mathbb C}[t^2,t^3])$ and,
if $\pi: \tilde{\Gamma} \to \Gamma$ denotes the normalization, one has
exact sequences
\begin{gather*}
0\>>> {\mathcal O}_{\Gamma} \>>> \pi_*({\mathcal O}_{\tilde{\Gamma}}) \>>> {\mathbb C} t \>>> 0
\mbox{ \ \ and}\\
0\>>> \Omega^1_{\Gamma/{\mathbb C}} \>>> \pi_*(\Omega^1_{\tilde{\Gamma}/{\mathbb C}}) \>>>
{\mathbb C} dt \>>> 0. \hspace*{1cm}
\end{gather*}
Thus ${\mathbb C} t=H^1(\Gamma,{\mathcal O}_\Gamma) \> d >\cong > {\mathbb C} dt \> > \subset >
H^1(\Gamma,\Omega^1_{\Gamma/{\mathbb C}}) \cong {\mathbb C}^2$
and one obtains by \ref{product2}
\begin{align*}
&A^2(\Gamma \times E) = {\mathbb G}_a \times E = A^1(\Gamma)\times A^1(E)\\
&A^2(\Gamma \times \Gamma) = \frac{{\mathbb C}^2 \times {\mathbb C}^2}{{\mathbb C}} =
{\mathbb G}_a^3 \mbox{ \ \ whereas \ \ } A^1(\Gamma)\times A^1(\Gamma) =
{\mathbb G}_a \times {\mathbb G}_a.
\end{align*}
In particular, a product formula as (\ref{product1}) fails for $A^n$
instead of $J^n$, and the dimension of $J^n$ and $A^n$ are not
constant for the fibres ${\mathcal C} \times {\mathcal C} \to \P \times \P$.
It is amusing to write down the cycle map for the last example.
Writing
$$
\Gamma_{\rm reg}\times \Gamma_{\rm reg} = (\Gamma -(0:0:1))\times
(\Gamma-(0:0:1)) = {\rm Spec \,}({\mathbb C}[u]\otimes_{\mathbb C} {\mathbb C}[v]),
$$
$\Omega(\Gamma\times\Gamma)={\rm
Hom}_{\Gamma\times\Gamma}(\Omega^1_{\Gamma\times\Gamma/{\mathbb C}},
\omega_{\Gamma\times\Gamma})_{\rm cl}$ decomposes as
\begin{align*}
&(H^0(\Gamma,\shom(\Omega^1_{\Gamma/|C},\omega_\Gamma)) \otimes
H^0(\Gamma,\omega_\Gamma) \oplus H^0(\Gamma,\omega_\Gamma)
\otimes H^0(\Gamma,\shom(\Omega^1_{\Gamma/{\mathbb C}},\omega_\Gamma)))_{\rm
cl}\\
&= ({\mathbb C} dv \oplus {\mathbb C} udv \oplus {\mathbb C} du \oplus {\mathbb C} vdu)_{\rm cl} =
{\mathbb C} dv \oplus {\mathbb C} du \oplus {\mathbb C} (udv+vdu).
\end{align*}
The cycle map is
\begin{align*}
\Pi_{\Gamma_{\rm reg} \times \Gamma_{\rm reg}}={\mathbb A}_{\mathbb C}^2 \times {\mathbb A}_{\mathbb C}^2
& \>>> {\mathbb G}_a^3\\
((x_1,x_2),(y_1,y_2)) & \longmapsto \left\{
\begin{array}{rll}
du &\mapsto &y_1-x_1\\
dv &\mapsto &y_2-x_2\\
udv+vdu & \mapsto & y_1y_2-x_1x_2.
\end{array}\right.
\end{align*}
\end{rmk}
\section{The universal property over ${\mathbb C}$}
Let $U_1,\ldots,U_r$ be the connected components of $X_{{\rm reg}}$,
and for each $i$, let $p_i\in U_i$ be a base point.
Let $G$ be a commutative algebraic group.
By \ref{def-reg} and \ref{equ-reg} a homomorphism (of abstract groups)
$\phi:CH^n(X)_{\deg 0}\to G$ is regular, if and only if
$\phi\circ\gamma_m:S^m(X_{\rm reg})\to G$ is a morphism of varieties, for
some $m>0$.
\begin{thm}\label{regular} \ \
\begin{points}
\item The homomorphism $\varphi:CH^n(X)_{\deg 0}\to A^n(X)$ constructed in
lemma~\ref{equivalence} is regular and surjective.
\item The cokernel of the map $H_1(X_{\rm reg},{\mathbb Z}) \to \Lie(A^n(X))$, defined by
integration of 1-forms over homology classes, is naturally
isomorphic to $A^n(X)$ and the composite
$(\varphi\circ\gamma)^{(-)}: \Pi_{X_{\rm reg}} \to CH^n(X)_{\deg
0}\to A^n(X)$ is given by
$$
(x,y) \longmapsto \bigr\{ \omega \mapsto \int^y_x \omega \bigr\}
$$
\item {\em (Universality)} $\varphi$ satisfies the following universal
property: for any regular homomorphism $\phi:CH^n(X)_{\deg 0}\to
G$ to a commutative algebraic group there exists a unique
homomorphism $h:A^n(X)\to G$ of algebraic groups with
$\phi=h\circ\varphi$.
\end{points}
\end{thm}
\begin{proof}[Proof of (i)] It suffices to prove that
$\varphi\circ\gamma_1:U=X_{\rm reg}\to A^n(X)$ is a morphism. Note
that, from the definition, it is clearly analytic. Further, we
have the following.
\begin{enumerate}
\item[(a)] the composition $U\to A^n(X)\to {\rm Alb}\,(\tilde{X})$ is a
morphism, where $\tilde{X}$ is a resolution of singularities of $X^{(n)}$,
since we may then regard $U$ as an open subset of $\tilde{X}$, and the map
$U\to {\rm Alb}\,(\tilde{X})$ is the restriction of the Albanese mapping for
$\tilde{X}$, with appropriate base-points. Here ${\rm Alb}\,(\tilde{X})$ is the
product of the Albanese varities of the connected components of
$\tilde{X}$, and $A^n(X)$ is an extension of ${\rm Alb}\,(\tilde{X})$ by a group
${\mathbb G}_a^r\times{\mathbb G}_m^s$, so that in particular $A^n(X)\to{\rm Alb}\,(\tilde{X})$ is
a Zariski locally trivial fibre bundle.
\item[(b)] For each reduced Cartier curve $C\subset X$, the composite
$$C_{{\rm reg}}\>>> U\>>> A^n(X)$$
is a morphism. Indeed, for each component $B_0$ of
$C_{\rm reg}$, the composition
\[B_0\>>> U\>{\gamma_1}>>CH^n(X)_{\deg 0}\>{\varphi}>> A^n(X)\]
agrees with
\[B_0\>>> {\rm Pic}^0(C)\>>> CH^n(X)_{\deg 0}\>{\varphi}>>A^n(X)\]
up to a translation, and by corollary~\ref{algcurve}, the latter is
algebraic.
\end{enumerate}
Now we may argue as in \cite{BiS}: we are reduced to proving that if $V$ is a
non-singular affine variety, a holomorphic function on $V$ which is
algebraic when restricted to ``almost all'' algebraic curves in $V$, is in
fact an algebraic regular function. This may be proved using Noether
normalization and power series expansions for holomorphic functions on
${\mathbb C}^n$, or deduced from \cite{Si}, (1.1).
Since $\Omega(A^n(X))$ is a finite dimensional subspace of
1-forms on $U$, there exist reduced local complete intersection Cartier
curves $C_i \subset X$, for $i=1,\ldots,s$, such that
$$
\Omega(A^n(X)) \>>> \bigoplus_{i=1}^s H^0(C_i,\omega_{C_i})
$$
is injective. Hence
$$
\bigoplus_{i=1}^s {\rm Pic}^0(C_i) \> \oplus \psi_i >> A^n(X)
$$
is surjective.
\noindent
{\it Proof of (ii) and (iii):}\quad Let $\phi:CH^n(X)_{\deg
0}\to G$ be a regular homomorphism to a commutative algebraic group $G$.
By lemma \ref{pic} the image of $\phi$ is contained in the connected
component of the identity of $G$. Hence we may assume without loss of
generality that $G$ is connected.
Now $\Omega(G)$ consists of closed, translation-invariant 1-forms. Thus
if
$$h=\phi\circ\gamma_1:U\>>> G,$$
then the image of $h^*:\Omega(G)\to
\Gamma(U,\Omega^1_{X/{\mathbb C}})$ is contained in the subspace of closed
1-forms. We claim that in fact
$h^*(\Omega(G))\subset \Omega(A^n(X)).$
This is deduced from the criterion of proposition~\ref{basic},
(b), since we know that for any reduced Cartier curve $C$ in $X$, the
composition
$${\rm Pic}^0(C)\>>> CH^n(X)_{\deg 0}\>{\phi}>> G$$
is a homomorphism of algebraic groups. Now we observe that if
$B_0$ is any component of $C_{{\rm reg}}$, then
$$
B_0 \>>> {\rm Pic}^0(C)=\Lie(A^1(C))/{\rm image}\,H_1(C_{\rm reg},{\mathbb Z})
$$
is given by integration of 1-forms in $H^0(C,\omega_C)$.
Moreover the composite
$$
B_0\>>> U\>{h}>> G
$$
agrees with
$$
B_0\>>>{\rm Pic}^0(C)\>>>G,
$$
up to a translation by an element of $G$ (and elements of
$\Omega(G)$ are translation invariant).
Dualizing the above inclusion on 1-forms, we thus obtain a map
on Lie algebras
$\Lie(A^n(X))\to\Lie(G).$
This fits into a commutative diagram
\[\begin{CD}
H_1(U,{\mathbb Z})\>>> \Lie (A^n(X))\\
{\mathbb V} V V {\mathbb V} V V \\
H_1(G,{\mathbb Z})\>>> \Lie(G)
\end{CD}\]
where the horizontal arrows are given by integration of 1-forms over
homology classes. Further, there is a commutative diagram
\begin{equation}\label{diag}
\begin{CD}
U \>>> \Lie(A^n(X))/{\rm image}\,H_1(U,{\mathbb Z})\\
{\mathbb V} \gamma_1 V V {\mathbb V} V \tilde{\phi}V \\
CH^n(X)_{\deg 0}\> {\phi}>> G=\Lie(G)/{\rm image}\,H_1(G,{\mathbb Z})
\end{CD}
\end{equation}
where $\tilde{\phi}$ is a homomorphism of analytic groups, and
where the upper horizontal arrow is given by integration of
1-forms in $\Omega(A^n(X))$.
We claim that the map $H_1(U,{\mathbb Z})\to \Lie(A^n(X))=\Omega(A^n(X))^*$
factors through the (surjective) composition
\[H_1(U,{\mathbb Z})\>{\cong}>> H^{2n-1}_c(U,{\mathbb Z}(n))\mbox{$\,\>>>\hspace{-.5cm}\to\hspace{.15cm}$} H^{2n-1}(X,{\mathbb Z}(n)),\]
where $H^*_c$ denotes compactly supported cohomology, and the isomorphism
is by Poincar\'e duality. Indeed, let $C\subset X^{(n)}$ be a sufficiently
general reduced complete intersection curve in $X^{(n)}$. Then
\[C\cap X^{<n}=\emptyset,\;\; C_{\rm sing}=C\cap X^{(n)}_{sing}=C\cap
X_{{\rm sing}},\]
and one has a Gysin homomorphism
\[H^1(C,{\mathbb Z}(1))\to H^{2n-1}(X,{\mathbb Z}(n))\cong H^{2n-1}(X^{(n)},{\mathbb Z}(n))\]
which fits into a commutative diagram with exact rows
\[\begin{CD}
H^0(C_{{\rm sing}},{\mathbb Z}(1)) \>>> H^1_c(C\cap U,{\mathbb Z}(1)) \>>>
\hspace*{-.82cm}\to \hspace{.42cm} H^1(C,{\mathbb Z}(1))\hspace{.3cm}\\
{\mathbb V} V V {\mathbb V} V V {\mathbb V} V{\rm Gysin}V\\
H^{2n-2}(X^{(n)}_{{\rm sing}},{\mathbb Z}(n))\>>> H^{2n-1}_c(U,{\mathbb Z}(n))\>>>
\hspace*{-.7cm} \to \hspace{.2cm} H^{2n-1}(X^{(n)},{\mathbb Z}(n))
\end{CD}\]
The left hand vertical arrow is in fact surjective, since
$H^{2n-2}(X^{(n)}_{{\rm sing}},{\mathbb Z}(n-1))$ is the free abelian group on the
$(n-1)$-dimensional components of $X^{(n)}_{{\rm sing}}$, and (since $C$ is a
general complete intersection) $C_{{\rm sing}}$ has non-empty intersection
(which is supported at smooth points, and is transverse) with each such
component of $X^{(n)}_{{\rm sing}}$. Now we note that the composite $H_1(C\cap
U,{\mathbb Z})\to H_1(U,{\mathbb Z})\to \Lie(A^n(X))$ factors through the surjective
composite $H_1(U,{\mathbb Z})\cong H^1_c(U,{\mathbb Z}(1))\to H^1(C,{\mathbb Z}(1))$, since $C\cap
U\to U\to A^n(X)$ is compatible with a homomorphism ${\rm Pic}^0(C)\to A^n(X)$
(here ``compatible'' means that for any component $B_0$ of $C\cap U$, the
composites $B_0\to U\to A^n(X)$ and $B_0\to{\rm Pic}^0(C)\to A^n(X)$ agree up
to translation by an element of $A^n(X)$). Now a diagram chase implies the
claim made at the beginning of the paragraph.
Thus in the diagram (\ref{diag}) we see that $\Lie(A^n(X))/{\rm
image}\,H_1(U,{\mathbb Z})$ is identified with
\[{\Lie(A^n(X))}/{{\rm image}\,H^{2n-1}(X,{\mathbb Z}(n))}\;=A^n(X).\]
Hence there is a homomorphism $\tilde{\phi}: A^n(X)\to G$,
such that $\gamma_1 \circ \phi:U\to G$ factors through $A^n(X)$.
Since $\gamma_1^*:\Omega(A^n(X))\to
\Gamma(U,\Omega^1_{U/{\mathbb C}})$ is injective, the induced map $A^n(X)\to G$
with this property is unique, since the corresponding map on Lie algebras
is uniquely determined. Since ${\rm image}\,\gamma_1$ generates $CH^n(X)_{\deg
0}$, the two homomorphisms
\[\phi:CH^n(X)_{\deg 0}\>>> G,\;\; CH^n(X)_{\deg 0}\>{\varphi}>>
A^n(X)\>{\tilde{\phi}}>> G\]
must coincide. This proves the universal property of $\varphi$, except
that we need to note that $\tilde{\phi}$ is a morphism.
By lemma \ref{equ-reg}, $\phi$ induces an algebraic group
homomorphism ${\rm Pic}^0(C) \to G$ for all admissible pairs $(C',
\iota)$, with $C = \iota(C')$.
As above, we can choose reduced complete intersection curves $C_i$, $i=1,
\ldots, s$, such that
$$
\bigoplus_{i=1}^s {\rm Pic}^0(C_i) \> \oplus \psi_i >> A^n(X)
$$
is surjective. As $\tilde{\phi} \circ (\oplus \psi_i)$ is an
algebraic group homomorphism, $\tilde{\phi}$ is an algebraic group
homomorphism as well.
\end{proof}
\begin{rmk} Lemma~\ref{algebraicity}, combined with the Roitman Theorem
proved in [BiS], imply that $\varphi:CH^n(X)_{\deg 0}\to A^n(X)$ is an
isomorphism on torsion subgroups. In other words, the Roitman Theorem is
valid for $\varphi:CH^n(X)_{\deg 0}\to A^n(X)$, over ${\mathbb C}$. This is another
similarity with the Albanese mapping for a non-singular projective variety.
\end{rmk}
\begin{rmk}
The proof of theorem \ref{regular} is close in spirit to the
construction of a ``generalized Albanese variety'' in
\cite{FW}. There Faltings and W\"ustholz consider a finite
dimensional subspace $V \subset H^0(X_{\rm reg},\Omega^1_{X_{\rm reg}})$,
containing the 1-forms with logarithmic poles on some
desingularization of $X^{(n)}$, and they construct a commutative algebraic
group $G_V$ together with a morphism $X_{\rm reg} \to G_V$, which is
universal among the morphisms $\tau: X_{\rm reg} \to H$ to
commutative algebraic groups $H$, with $\tau^*(\Omega(G))
\subset V$.
\end{rmk}
\section{Picard groups of Cartier curves}
In the next section, we give an algebraic construction of $A^n(X)$
for a reduced projective $n$-dimensional variety $X$, defined over an
algebraically closed field $k$. As in the analytic case, we will use the
Picard scheme for Cartier curves in $X$ and for families of such curves.
In this section, we discuss some properties of such families of curves, and
the corresponding Picard schemes. In particular, we establish the
technical results \ref{mu2} and \ref{mu3}, which are important steps in
the algebraic construction of $A^n(X)$.
Let $S$ be a non-singular variety, and let $f:{\mathcal C} \to S$ be a flat
family of projective curves with reduced geometric fibres
$C_s=f^{-1}(s)$. Then
$$
g(C_s):= \dim_{k(s)} H^1(C_s,{\mathcal O}_{C_s}) \mbox{ \ \ \ and \ \ \ }
\# C_s:= \dim_{k(s)} H^0(C_s,{\mathcal O}_{C_s})
$$
are both constant on $S$. In fact, let
${\mathcal C} \> \tilde{f} >> \tilde{S} \>\varkappa >> S$
be the Stein factorization of $f$. Since the fibres of $f$ are reduced,
$\varkappa:\tilde{S}\to S$ is a finite \'etale morphism and
$\# C_s=\#\varkappa^{-1}(s)$ is constant, as well as
$g(C_s)= \chi(C_s,{\mathcal O}_{C_s}) - \# C_s$.
Let $S' \to S$ be a finite (possibly branched) covering such that
$f':{\mathcal C}' = {\mathcal C} \times_SS'\to S'$ is the disjoint union of families
of curves ${\mathcal C}'_i \to S'$, for $i=1,\ldots ,s$ with connected
fibres. By \cite{BLR}, 8.3, theorem 1, the relative Picard
functors ${\rm Pic}_{{\mathcal C}'_i/S'}$ are represented by an algebraic
space ${\rm Pic}({\mathcal C}'_i/S')$, and we define
$${\rm Pic}({\mathcal C}'/S')={\rm Pic}({\mathcal C}'_i/S')\times_{S'}\cdots\times_{S"}{\rm Pic}({\mathcal C}'/S').
$$
For the smooth locus ${\mathcal C}_{\rm sm}$ of $f$ consider the $g$-th
symmetric product
$$
{f'}^g:S^g({\mathcal C}'_{\rm sm} /S')\>>> S'
$$
over $S'$. For any open subscheme $W'\subset S^g({\mathcal C}'_{\rm sm} /S')$
there is a natural map $\vartheta_{W'} : W' \to {\rm Pic}({\mathcal C}'/S')$.
By \cite{BLR}, 9.3, lemmas 5 and 6, one has the following
generalization of \ref{generators2}:
\begin{lemma}\label{relpic2}
After replacing $S'$ by an \'etale covering, there exists
an open subscheme $W'\subset S^g({\mathcal C}'_{\rm sm} /S')$ with
geometrically connected fibres over $S'$,
such that $\vartheta_{W'} : W' \to {\rm Pic}({\mathcal C}'/S')$ is an open
embedding.
\end{lemma}
Recall that $X^{(n)}$ denotes the union of the $n$-dimensional irreducible
components of $X$, and $X^{<n}$ is the union of the smaller dimensional
components.
\begin{notations}\label{linearsystem}
For a very ample invertible sheaf ${\mathcal L}$ on $X^{(n)}$ we write
$$
|{\mathcal L}|^{n-1} = \P(H^0(X^{(n)},{\mathcal L}))\times \cdots \times \P(H^0(X^{(n)},{\mathcal L}))
\ \ \ \ (n-1)\mbox{-times}
$$
and $|{\mathcal L}|^{n-1}_0$ for the open subscheme defined by $n-1$-tuples
$D_1, \ldots , D_{n-1}$ of divisors such that
\begin{enumerate}
\item[(i)] $C = D_1 \cap \cdots\cap D_{n-1}$ is a reduced complete
intersection curve in $X^{(n)}$,
\item[(ii)] $C\cap X^{<n}=\emptyset$, and
\item[(iii)] $X_{\rm reg} \cap C$ is non-singular and dense in $C$.
\end{enumerate}
Note that by (ii), $C$ is a reduced Cartier curve in $X$ which is
a local complete intersection. By abuse of notation we
will sometimes
write $C \in |{\mathcal L}|^{n-1}$ instead of $(D_1, \ldots , D_{n-1}) \in
|{\mathcal L}|^{n-1}$.
\end{notations}
The normalization $\pi :\tilde{C} \to C$ induces a surjection
$\pi^*:{\rm Pic}^0(C) \to {\rm Pic}^0(\tilde{C})$. By \cite{BLR}, 9.2, the
kernel of $\pi^*$ is the largest linear subgroup $H(C)$ of
${\rm Pic}(C)$.
One has
\begin{align}\label{chieq}
\dim(H(C)) & = \dim({\rm Pic}^0(C)) - \dim({\rm Pic}^0(\tilde{C}))\notag\\
& = \dim_k(H^1(C,{\mathcal O}_C)) - \dim_k(H^1(\tilde{C},{\mathcal O}_{\tilde{C}}))\\
& = \chi(\tilde{C},{\mathcal O}_{\tilde{C}}) - \chi({C},{\mathcal O}_{{C}})
- ( \# \tilde{C} - \# C ),\notag
\end{align}
where again $\# C$ and $\# \tilde{C}$ denote the number of connected
components of $C$ and $\tilde{C}$, respectively.
Given a flat family of projective curves $f:{\mathcal C} \to S$ over an irreducible
variety $S$ with reduced geometric fibres $C_s$, there exists a finite
(possibly branched) covering $S'\to S$ and an open dense subscheme
$S'_0\subset S'$ such that the normalization of ${\mathcal C}\times_S S'_0$ is
smooth over $S'_0$. Hence $\# \tilde{C}_s$, and the dimension of the
linear part $H(C_s)$ of ${\rm Pic}^0(C_s)$, are both constant on the image of
$S'_0$.
\begin{defn}\label{mu}
For a reduced projective curve $C$ we define $r(C)$ to be the
number of irreducible components of $C$, and $\mu(C)$ to be the
dimension of the largest linear subgroup of ${\rm Pic}^0(C)$.
By \cite{F}, Satz 5.2, for a very ample invertible sheaf ${\mathcal L}$
the open subscheme $|{\mathcal L}|^{n-1}_0$ is not empty. Then $r({\mathcal L})$ and
$\mu({\mathcal L})$ denote the values of $r(C)$ and of $\mu(C)$ for
$C \in |{\mathcal L}|^{n-1}_0$ in general position.
\end{defn}
By the equality (\ref{chieq}) one has:
\begin{equation}\label{chieq2}
\chi(\tilde{C},{\mathcal O}_{\tilde{C}}) - \chi({C},{\mathcal O}_{{C}}) \geq
\mu(C) \geq \chi(\tilde{C},{\mathcal O}_{\tilde{C}}) -
\chi({C},{\mathcal O}_{{C}}) - r(C) + 1.
\end{equation}
\begin{lemma}\label{muinequality}
For a very ample invertible sheaf ${\mathcal L}$ and for a positive integer $N$,
$$\mu({\mathcal L}^N) \leq N^{n-1}\cdot (\mu({\mathcal L}) + r({\mathcal L}) -1).$$
\end{lemma}
\begin{proof}
Given $D^{(i)}_j \in |{\mathcal L}|$, for $i=1, \ldots , N$ and $j=1,
\ldots , n-1$, we write
\begin{align*}
I &= \{1, \ldots ,N\}^{n-1}\\
C^{(\underline{i})} &= D^{(i_1)}_1 \cap \cdots \cap
D^{(i_{n-1})}_{n-1}\mbox{ \ \ for \ \ } \underline{i} = (i_1, \ldots ,
i_{n-1}) \in I\\
\mbox{and \ \ \ \ } C &= \bigcup_{\underline{i} \in I}
C^{(\underline{i})} = \bigcap_{j=1}^{n-1} (D_j^{(1)} \cup \cdots
\cup D_j^{(N)}).
\end{align*}
\begin{claim}\label{general}
There exists a choice of the divisors $D^{(i)}_j \in |{\mathcal L}|$ such
that
\begin{enumerate}
\item[(a)] $C^{(\underline{i})} \in |{\mathcal L}|^{n-1}_0$, \
$\mu(C^{(\underline{i})})= \mu({\mathcal L})$ \ and \
$r(C^{(\underline{i})})= r({\mathcal L})$ \vspace{.05cm}
\item[(b)] $C^{(\underline{i})} \cap C^{(\underline{i}')} \cap
X_{\rm sing} = \emptyset$ for $\underline{i} \neq \underline{i}'$\vspace{.05cm}
\item[(c)] each point $x \in C_{\rm sing} \cap X_{\rm reg}$ lies on
exactly two components $C^{(\underline{i})}$ and
$C^{(\underline{i}')}$. In this case, there exists one $\nu$ with
$i_j = i'_j$ for all $j\neq \nu$. Locally in $x$ the
surface
$$Y = D^{(i_1)}_1 \cap \cdots \cap \widehat{D^{(i_{\nu})}_\nu} \cap
\cdots \cap D^{(i_{n-1})}_{n-1}\cap X_{\rm reg}
$$
is nonsingular and contains $C^{(\underline{i})}$ and
$C^{(\underline{i}')}$ as two smooth divisors intersecting
transversally. \vspace{.05cm}
\item[(d)] $C$ is a reduced complete intersection curve in $|{\mathcal L}^N|^{n-1}$.
\end{enumerate}
\end{claim}
\begin{proof} (d) follows from (a), (b) and (c).
Since $|{\mathcal L}|^{n-1}_0$ is open and dense in $|{\mathcal L}|^{n-1}$
(a) holds true for sufficiently general divisors. Counting
dimensions one finds that for $\underline{i}\neq \underline{i}'$
the intersection $C^{(\underline{i})} \cap C^{(\underline{i}')}$
is either empty or consists of finitely many points. The latter
can only happen, if all but one entry in $\underline{i}$ and
$\underline{i}'$ are the same, and obviously one may assume that
the intersection points all avoid $X_{\rm sing}$. Moreover
$$
C^{(\underline{i})} \cap C^{(\underline{i}')} \cap
C^{(\underline{i}'')}= \emptyset
$$
for pairwise different $\underline{i},\ \underline{i}'\
\underline{i}''\in I$. Now (c) follows from the Bertini theorem
\cite{F}, Satz 5.2, saying that for sufficiently general divisors
$D_j^{(i)}$
\begin{gather*}
Y = D^{(i_1)}_1 \cap \cdots \cap \widehat{D^{(i_{\nu})}_\nu} \cap
\cdots \cap D^{(i_{n-1})}_{n-1}\cap X_{\rm reg}\\
C^{(\underline{i})}=Y \cap D_\nu^{(i_\nu)}
\mbox{ \ \ \ and \ \ \ }C^{(\underline{i}')}=Y \cap D_\nu^{(i'_\nu)}
\end{gather*}
are non-singular and that $C^{(\underline{i})}$ and
$C^{(\underline{i}')}$ meet tranversally on $Y$.
\end{proof}
Let ${\mathbb A}^{M+1} \subset |{\mathcal L}^N|^{n-1}$ be an affine open subspace
containing the point $s_0$ which corresponds to the tuple
$\{D_j^{(1)}\cup \cdots \cup D_j^{(N)}\}_{j=1,\ldots,n-1}$,
and let $\P^M$ be the projective space parametrizing lines in
${\mathbb A}^{M+1}$, passing through $s_0$. There is a line $S \in \P^M$ such that
\begin{points}
\item
the total space ${\mathcal C}$ of the restriction
$$
\begin{CD}
{\mathcal C} \>\tau>> X\\
{\mathbb V} f V V\\
S \> \subset >> |{\mathcal L}^N|^{n-1}.
\end{CD}
$$
of the universal family to $S$ is non-singular in a
neighbourhood of each point $x\in C_{\rm sing} \cap X_{\rm reg}$
\item
the intersection of $X_{\rm reg}$ with the
general fibre of $f:{\mathcal C} \to S$ is non-singular.
\end{points}
In fact, using the notation from \ref{general} (c), we can
choose for a point $x\in C_{\rm sing} \cap X_{\rm reg}$ a line $S$
connecting $s_0$ with a point $(D'_1,
\ldots ,D'_{n-1})\in {\mathbb A}^{M+1}$, where
$$
D'_j = D_j^{(1)}\cup \cdots \cup D_j^{(N)} \mbox{ \ \ for \ \ }
j \neq \nu,
$$
where $D'_\nu \cap Y_{\rm reg}$ is non singular, and where $x \not\in
D'_\nu$. By this choice, in a neigbourhood of $x$ the
restriction of the universal family ${\mathcal C}$ to $S$ is just a
fibering of $Y$ over $S$. Hence the condition (i) is valid
for the chosen point $x$.
However, for each point $x \in C_{\rm sing} \cap
X_{\rm reg}$, the condition (i) is an open condition in $\P^M$, and hence for a
general line $S$, (i) hold true for all points in $C_{\rm sing} \cap X_{\rm reg}$;
clearly the second condition (ii) holds as well.
The family $f:{\mathcal C} \to S$ has only finitely many non-reduced
fibres and outside of them $U=\tau^{-1}(X_{\rm reg})$ contains
only finitely many points, which are singularities of the fibres.
Replacing $S$ by an open neighbourhood of $s_0$, we may assume
thereby, that for $s\neq s_0$ the fibre $C_s=f^{-1}(s)$ is
reduced, that $C_s\cap X_{\rm reg}$ is non-singular and dense in $C_s$ and that
$\mu(C_s)=\mu({\mathcal L}^N)$. In particular $U$ is non-singular outside
of the points $C_{\rm sing}\cap X_{\rm reg}$, and by condition (i) $U$ is
non singular. Moreover, $f|_U : U \to
S$ is semi-stable; hence $f|_U$ is a local complete intersection morphism,
smooth outside a finite subset of $U$. Let $L$ be a finite extension of the
function field $k(S)$ such that the normalization of
${\mathcal C}\times_S {\rm Spec \,}(L)$ is smooth over $L$, and let $S'$ be the
normalization of $S$ in $L$. Consider
$$
\begin{CDS}
{\mathcal C}' \> \sigma >> {\mathcal C}\times_S S' \> \eta' >> {\mathcal C}\\
& \SE E f' E {\mathbb V} pr_2 V V & {\mathbb V} V f V \\
&& S' \> \eta >> S
\end{CDS}
$$
where $\sigma$ denotes the normalization. Since
$U\times_S S' \to S'$ is a local complete intersection morphism, smooth
outside a finite subset of the domain, $U\times_S S'$ is normal and
$\sigma$ restricted to $U'=\sigma^{-1}(U\times_S S')$ is an
isomorphism. By construction the general fibre of $f'$ is smooth and
${\mathcal C}'$ is normal. Since for all $s' \in S'$ the fibres
$C'_{s'}={f'}^{-1}(s')$ of $f'$ are reduced on the open dense
subvariety $U'$, they are reduced everywhere. Note also that ${\mathcal C}-U\to S$
is finite, and hence so is ${\mathcal C}'-U'\to S'$.
Let $s', s'_0 \in S'$ be points, with $s_0=\eta(s'_0)$, and with
$s=\eta(s')$ in general position. The inequality (\ref{chieq2})
implies that
\begin{align*}
\mu({\mathcal L}^N) = \mu(C_{s}) &\leq \chi(C'_{s'},{\mathcal O}_{C'_{s'}})-
\chi(C_s,{\mathcal O}_{C_s})
\\
&= \chi(C'_{s'_0},{\mathcal O}_{C'_{s'_0}})-\chi(C_{s_0},{\mathcal O}_{C_{s_0}}).
\end{align*}
Since $C'_{s'_0}\cap U'$ is isomorphic to
$C_{s_0}\cap U$ the curve $C'_{s'_o}$ is
finite over and birational to $C=C_{s_0}$.
Moreover, the fibres $C'_{s'_0}\cap U'$ and $C\cap U$ have the same number
$\delta$ of double points.
Writing ${C'}^{(\underline{i})}$ for the preimage of
${C}^{(\underline{i})}$ in $C'_{s'_0}$ one obtains
\begin{align*}
& \chi(C,{\mathcal O}_{C}) + \delta = \sum_{\underline{i}\in I}
\chi({C}^{(\underline{i})},{\mathcal O}_{{C}^{(\underline{i})}}),\\
& \chi(C'_{s'_0},{\mathcal O}_{C'_{s'_0}})+ \delta = \sum_{\underline{i}\in I}
\chi({C'}^{(\underline{i})},{\mathcal O}_{{C'}^{(\underline{i})}})\\
\mbox{and \ \ \ }
&\mu({\mathcal L}^N)\leq \sum_{\underline{i}\in
I}(\chi({C'}^{(\underline{i})},{\mathcal O}_{{C'}^{(\underline{i})}})-
\chi({C}^{(\underline{i})},{\mathcal O}_{{C}^{(\underline{i})}})
).
\end{align*}
Finally, ${C'}^{(\underline{i})}$ is finite over and birational
to ${C}^{(\underline{i})}$, thus it is dominated by the
normalization of ${C}^{(\underline{i})}$, and (\ref{chieq2})
implies
\begin{align*}
\sum_{\underline{i}\in
I}(\chi({C'}^{(\underline{i})},{\mathcal O}_{{C'}^{(\underline{i})}})-
\chi({C}^{(\underline{i})},{\mathcal O}_{{C}^{(\underline{i})}}))
&\leq \sum_{\underline{i}\in I}(\mu({C}^{(\underline{i})})
+ r({C}^{(\underline{i})}) -1)\\
&=N^{n-1}\cdot (\mu({\mathcal L}) +r({\mathcal L}) -1).
\end{align*}
\end{proof}
Replacing ${\mathcal L}$ by its $N$-th power one obtains by lemma
\ref{muinequality} ample invertible sheaves on $X^{(n)}$ with many more
linearly independent sections than $\mu({\mathcal L})$. For example, if
$X_1, \ldots ,X_r$ are the irreducible components of $X^{(n)}$, then
$$
{\rm image}\,( H^0(X^{(n)},{\mathcal L}^N) \to H^0(X_i,{\mathcal L}^N|_{X_i}))=
H^0(X_i,{\mathcal L}^N|_{X_i}),
$$
for sufficiently large $N$, and its dimension i bounded below by a
non-zero multiple of $N^n$, whereas by \ref{muinequality}, $\mu({\mathcal L}^N)$
is bounded above by $(\mu({\mathcal L})+r({\mathcal L})-1)\cdot N^{n-1}$. One obtains:
\begin{cor}\label{mu2}
There exists a very ample sheaf ${\mathcal L}$ on $X^{(n)}$ with
$$
\dim_k ({\rm image}\,( H^0(X^{(n)},{\mathcal L}) \to H^0(X_i,{\mathcal L}|_{X_i}))) \geq
2\cdot \mu({\mathcal L})+r+2,
$$
for $i=1, \ldots, r$.
\end{cor}
Over a field $k$ of positive characteristic we will need
a stronger technical condition. Recall that
$\Pi_{X_{\rm reg}} = \bigcup_{i=1}^r (U_i\times U_i)$, where
$U_i=X_i \cap X_{\rm reg}$ are the irreducible components of
$X_{\rm reg}$.
\begin{ass}\label{dominant}
Let $Z \subset S^d(\Pi_{X_{\rm reg}})\times
S^d(\Pi_{X_{\rm reg}})\times |{\mathcal L}|^{n-1}_0$ be the incidence variety
of points
\begin{multline*}
(((x_1,x'_1), \ldots ,(x_d,x'_d)),((x_{d+1},x'_{d+1}),
\ldots ,(x_{2d},x'_{2d})), (D_1, \ldots ,D_{n-1}))\\
\in S^d(\Pi_{X_{\rm reg}})\times S^d(\Pi_{X_{\rm reg}})\times |{\mathcal L}|^{n-1}_0
\end{multline*}
with
$x_1, \ldots, x_{2d},x'_1, \ldots, x'_{2d} \in C=D_1 \cap \cdots
\cap D_{n-1}.$
Then the projection
$$
pr_{12}'=pr_{12}|_{Z}: Z \>>> S^d(\Pi_{X_{\rm reg}})\times
S^d(\Pi_{X_{\rm reg}})
$$
is dominant.
\end{ass}
\begin{prop}\label{mu3}
There exists a very ample sheaf ${\mathcal L}$ on $X^{(n)}$ which satisfies the
assumption \ref{dominant}, for all $d \leq \mu({\mathcal L})$.
\end{prop}
\begin{proof}
Let ${\mathcal I}_i$ be the ideal sheaf of $\bigcup_{j\neq i} X_j$ on
$X^{(n)}$. In particular ${\mathcal I}_i|_{X_j}$ is zero, for $j\neq i$.
Hence if ${\mathcal F}$ is a torsion-free coherent sheaf on $X^{(n)}$
then
\begin{align*}
&H^0(X_i, {\mathcal F} \otimes {\mathcal I}_i|_{X_i}/_{\rm
torsion})=H^0(X^{(n)},{\mathcal F}\otimes {\mathcal I}_i/_{\rm torsion})\\
\mbox{and \ \ \ }&\bigoplus_{i=1}^r H^0(X_i, {\mathcal F} \otimes
{\mathcal I}_i|_{X_i}/_{\rm torsion}) \subset H^0(X^{(n)},{\mathcal F}).
\end{align*}
\begin{claim}\label{mu4}
There exists a very ample invertible sheaf ${\mathcal L}$ on $X^{(n)}$ such
that
\begin{equation}\label{mueq}
\dim_k({\rm image}\,(H^0(X_i, {\mathcal L} \otimes {\mathcal I}_i|_{X_i}) \to
H^0(C, {\mathcal L}|_C))) \geq 4\cdot \mu({\mathcal L}),
\end{equation}
for $i = 1, \ldots , r$, and for all $C \in |{\mathcal L}|^{n-1}_0$.
\end{claim}
\begin{proof}
Given a very ample invertible sheaf ${\mathcal L}$ it suffices to find a lower
bound for the dimension of the image of the composite map
$$
\tau:
H^0(X_i, {\mathcal L}^N \otimes {\mathcal I}_i|_{X_i}) \>>> H^0(X^{(n)},{\mathcal L}^N) \>>> H^0(C,
{\mathcal L}^N|_C) $$
which is independent of $C\in |{\mathcal L}^N|^{n-1}_0$ and grows like
$N^n$. If ${\mathcal J}_C$ denotes the ideal sheaf of $C$ on $X^{(n)}$, then
$$
\ker(\tau)= H^0\left(X_i, {\mathcal L}^N \otimes {\mathcal J}_C \otimes
{\mathcal I}_i|_{X_i}/_{({\rm torsion})}\right)
\subset H^0(X^{(n)}, {\mathcal L}^N \otimes {\mathcal J}_C).
$$
Hence it is sufficient to give an upper bound for
$\dim(H^0(X^{(n)}, {\mathcal L}^N \otimes {\mathcal J}_C))$ by some polynomial in $N$ of
degree $n-1$, independent of $C$.
For $j < n$ the dimension of
$H^{j}(X^{(n)}, {\mathcal L}^{-N})$ is bounded by a polynomial of
degree $n-2$. In fact, $X^{(n)}$ is a subscheme of $\P^M=\P(H^0(X^{(n)},
{\mathcal L}))$ and by \cite{Ha}, III.7.1 and III.6.9 one has, for $N$
sufficiently large,
\begin{align*}
H^j(X^{(n)}, {\mathcal L}^{- N}) &\cong {\rm Ext}^{M-j}({\mathcal O}_{X^{(n)}}\otimes
{\mathcal L}^{-N},\omega_{\P^M}) \\
&\cong H^0(\P^M,\sext^{M-j}({\mathcal O}_{X^{(n)}},\omega_{\P^M}\otimes
{\mathcal O}_{\P^M}(N)))\\
&\cong H^0(\P^M,\sext^{M-j}({\mathcal O}_{X^{(n)}},\omega_{\P^M})\otimes {\mathcal O}_{\P^M}(N)).
\end{align*}
Since $X$ is Cohen-Macaulay outside of a subscheme $T$ of
codimension $2$, the support of $\sext^{M-j}({\mathcal O}_{X^{(n)}},\omega_{\P^M})$
lies in $T$ for $M-j >M-n$.
The curve $C$ being a complete intersection of divisors in
$|{\mathcal L}^N|$, a resolution of the ideal sheaf ${\mathcal J}_C$ on $X^{(n)}$ is given by
the Koszul complex
$$
0 \to {\mathcal L}^{-(n-1)N}=\bigwedge^{n-1}\bigl( \soplus{n-1} {\mathcal L}^{-N} \bigr) \to
\ \ \cdots \ \ \to
\bigwedge^{2}\bigl( \soplus{n-1} {\mathcal L}^{-N} \bigr) \to
\soplus{n-1} {\mathcal L}^{-N} \to {\mathcal J}_C \to 0.
$$
Therefore $\dim(H^0(X^{(n)}, {\mathcal L}^N \otimes {\mathcal J}_C))$ is bounded from above
by
$$\sum_{j=0}^{n-2}
\dim_k \bigl( H^j\bigl(X^{(n)},{\mathcal L}^N\otimes \bigwedge^{j+1}\bigl( \soplus{n-1}
{\mathcal L}^{-N} \bigr)\bigr)=\sum_{j=0}^{n-2}
\dim_k \bigl( H^j\bigl(X^{(n)},{\mathcal L}^{-jN}\bigr)\bigr)
{\binom{n-1}{j+1}}.
$$
\end{proof}
Let ${\mathcal L}$ be a very ample invertible sheaf on $X^{(n)}$ which
satisfies the inequality (\ref{mueq}) in \ref{mu4}. We fix some
curve $C\in |{\mathcal L}|^{n-1}_0$ and some natural number $d \leq
\mu({\mathcal L})$.
Each irreducible component of $S^d(\Pi_{X_{\rm reg}})\times
S^d(\Pi_{X_{\rm reg}})$ is of the form
$$
S_{\underline{d}}=(S^{d_1}(U_1 \times U_1) \times \cdots \times
S^{d_r}(U_r \times U_r)) \times (S^{d_{r+1}}(U_1 \times U_1)
\times \cdots \times S^{d_{2r}}(U_r \times U_r)),
$$
for some tuple $\underline{d}=(d_1, \ldots d_{2r})$ of non-negative
integers with
$$
d_1 + \cdots + d_r = d_{r+1} + \cdots + d_{2r} =d.
$$
Given such a tuple ${\underline{d}}$, we claim that there are
(pairwise distinct) points $x_1, \ldots x_{2d},$
$x'_1 \ldots x'_{2d}$, with $x_\nu,x'_\nu \in C \cap U_i$, for
$$
\sum_{j=1}^{i-1} d_j < \nu \leq \sum_{j=1}^i d_j
\mbox{ \ \ \ and for \ \ \ }
\sum_{j=1}^{r+i-1} d_j < \nu \leq \sum_{j=1}^{r+i} d_j,
$$
such that the restriction map
\begin{equation}\label{restr}
H^0(X^{(n)},{\mathcal L}) \>>> \bigoplus_{i=1}^{2d} k_{x_i}\oplus k_{x'_i}
\end{equation}
is surjective. In fact, by the inequality (\ref{mueq}) the
dimension of the image of
$$
H^0(X_i, {\mathcal L} \otimes {\mathcal I}_i|_{X_i}) \>>>
H^0(C\cap X_i, {\mathcal L} \otimes {\mathcal I}_i|_{C\cap X_i}) \>>>
H^0(C, {\mathcal L}|_C)))
$$
is at least $4\cdot \mu({\mathcal L}) \geq 4 \cdot d \geq 2
\cdot (d_i + d_{r+i})$
and for sufficiently general points $x_1, \ldots x_{2d}, x'_1
\ldots x'_{2d} \in C$ the composite
$$
\bigoplus_{i=1}^rH^0(X_i,{\mathcal L}\otimes {\mathcal I}_i|_{X_i}) \subset
H^0(X^{(n)},{\mathcal L}) \>>> \bigoplus_{i=1}^{2d} k_{x_i} \oplus k_{x'_i}
$$
is surjective.
By construction
$$
w: =(((x_1,x'_1), \ldots ,(x_d,x'_d)),((x_{d+1},x'_{d+1}),
\ldots ,(x_{2d},x'_{2d}))) \in S_{\underline{d}}.
$$
Let $V$ denote the subspace of divisors $D \in |{\mathcal L}|$
with
$$
x_1, \ldots ,x_{2d},x'_1,\ldots,x'_{2d} \in D.
$$
The fibre ${pr'}^{-1}_{12}(w)$ of the morphism
$pr_{12}': Z \to S^d(\Pi_{X_{\rm reg}})\times S^d(\Pi_{X_{\rm reg}})$
is the intersection of $V^{n-1}$ with $|{\mathcal L}|^{n-1}_0$.
In particular, since $(w,C) \in Z$, this intersection is
non-empty.
If $\delta = \dim(|{\mathcal L}|)$, the surjectivity of the restriction
map (\ref{restr}) implies that $\dim(V)= \delta - 4 \cdot d$
and $\dim({pr'}_{12}^{-1}(w)) = (n-1)\cdot (\delta - 4 \cdot d).$
The fibres of $pr_3':Z \to |{\mathcal L}|^{n-1}_0$ are equidimensional of
dimension $4\cdot d$ and hence $Z$ is equidimensional of dimension
$(n-1)\cdot \delta + 4\cdot d$. Therefore the dimension of
$pr_{12}'(Z) \cap S_{\underline{d}}$ can not be smaller than
$$
(n-1)\cdot \delta + 4\cdot d - (n-1)\cdot (\delta - 4 \cdot d)=
n \cdot 4 \cdot d = \dim(S_{\underline{d}}).
$$
\end{proof}
\section{The algebraic construction of $A^n(X)$}
Let $X$ be a projective variety of dimension $n$, defined over
an algebraically closed field $k$.
As a first step towards the construction of $A^n(X)$ we need
to bound the dimension of the image of a regular homomorphism
$$\phi:CH^n(X)_{\deg 0} \>>> G
$$
to a smooth connected commutative algebraic group $G$.
By the theorem of Chevalley and Rosenlicht (theorems 1 and 2 in
\cite{BLR}, 9.2) there exists a unique smooth linear subgroup
$L$ of $G$ such that $G/L=A$ is an abelian variety. In addition,
$L$ is canonically isomorphic to a product of a unipotent group
and a torus. Let us write
$$
0\>>> L \>>> G \> \delta >> A \>>> 0
$$
for the extension.
\begin{lemma}\label{subgroups}
There exists a unique smooth connected algebraic subgroup $H$ of $G$, with
$\delta(H) = A$, such that every smooth connected algebraic subgroup $J$
of $G$ with $\delta(J)=A$ contains $H$. Moreover, the quotient
group $G/H$ is linear.
\end{lemma}
\begin{proof}
Given a smooth algebraic subgroup $J$ of $G$, one has the
commutative diagram of exact sequences
$$
\begin{CD}
\noarr 0 \noarr 0 \noarr 0 \\
\noarr {\mathbb V} V V {\mathbb V} V V {\mathbb V} V V\\
0 \>>> L\cap J \>>> J \>>> \delta(J) \>>> 0\\
\noarr {\mathbb V} V V {\mathbb V} V V {\mathbb V} V V\\
0 \>>> L \>>> G \>>> A \>>> 0\\
\noarr {\mathbb V} V V {\mathbb V} V V {\mathbb V} V V\\
0 \>>> L/(L\cap J) \>>> G/J \>>> A/\delta(J) \>>> 0\\
\noarr {\mathbb V} V V {\mathbb V} V V {\mathbb V} V V\\
\noarr 0 \noarr 0 \noarr 0
\end{CD}
$$
Since $A/\delta(J)$ is an abelian variety and $L/(L\cap J)$
a linear algebraic group, $\delta(J)=A$ if and only if
$G/J$ is linear. Observe further, that $\delta(J) = A$ if and
only if $\delta(J') = A$ for the connected component $J'$ of $J$
containing the identity.
Choose $H$ to be any smooth connected algebraic subgroup of $G$ with
$\delta(H)=A$ and such that $\delta(H') \neq A$ for all proper
algebraic subgroups $H'$ of $H$. For $J$ as in \ref{subgroups}
consider the commutative diagram of exact sequences
$$
\begin{CD}
0 \>>> J\cap H \>>> G \>>> G/(J\cap H) \>>> 0\\
\noarr {\mathbb V} V V {\mathbb V} \Delta V V {\mathbb V} \iota V V\\
0 \>>> J\oplus H \>>> G\oplus G \>>> G/J \oplus G/H \>>> 0
\end{CD}
$$
where $\Delta$ is the diagonal embedding. Since
$J\cap H = \Delta^{-1}(J\oplus H)$ the morphism $\iota$
is injective on closed points, and hence $G/(J\cap H)$ is a linear
algebraic group. By the choice of $H$ one obtains $J\cap H = H$.
\end{proof}
Recall that $X$ has $n$-dimensional irreducible components $X_1, \ldots
,X_r$, whose union is denoted $X^{(n)}$, and $U_i=X_{reg}\cap X_i$. Also
$X^{<n}$ is the union of the lower dimensional components of $X$.
\begin{prop}\label{bound}
Let ${\mathcal L}$ be a very ample invertible sheaf on $X^{(n)}$ which
satisfies the assumption \ref{dominant}. Let
$g=\dim_k(H^1(C,{\mathcal O}_C))$, for $C \in |{\mathcal L}|^{n-1}_0$.
Let $\phi:CH^n(X)_{\deg 0} \to G$ be a surjective regular
homomorphism to a smooth connected commutative algebraic group $G$.
Then the induced morphism (see \ref{difference})
$$
\pi^{(-)}: S^{g+\nu\cdot \mu({\mathcal L})}(\Pi_{X_{\rm reg}}) \>>>
CH^n(X)_{\deg 0} \>\phi >> G
$$
is dominant, for $\nu > 0$, and surjective, for $\nu > 1$.
In particular the dimension of $G$ is bounded by $2\cdot n \cdot
(g+\mu({\mathcal L}))$.
\end{prop}
Probably the bound for the dimension of $G$ is far from being
optimal. We will indicate in \ref{bound2} how to obtain
$\dim (G) \leq g$ in characteristic zero, under a weaker
assumption on ${\mathcal L}$.
\begin{proof}[Proof of \ref{bound}]
Let again $L$ be the largest smooth linear algebraic subgroup
and $\delta:G \mbox{$\,\>>>\hspace{-.5cm}\to\hspace{.15cm}$} A=G/L$ the projective quotient group.
Recall that $|{\mathcal L}|^{n-1}_0$ denotes the set of tuples
$(D_1, \ldots , D_{n-1})$ of divisors in the linear system
$|{\mathcal L}|$ for which $C=D_1\cap \cdots \cap D_{n-1}$ is a reduced
complete intersection curve (in $X^{(n)}$), $C\cap X^{<n}=\emptyset$, and
$C\cap X_{\rm reg}$ non-singular and dense in $C$.
\begin{claim}\label{surjective}
There exists an open dense subscheme $S \subset |{\mathcal L}|^{n-1}_0$
such that
$$
{\rm Pic}^0(C) \>\psi >> G \> \delta >> A
$$
is surjective and such that the dimension of ${\rm image}\, (\psi:
{\rm Pic}^0(C) \to G)$ is constant, for $C \in S$.
\end{claim}
\begin{proof}
Returning to the notation introduced in \ref{linearsystem}
let $S\subset |{\mathcal L}|^{n-1}_0$ be an open subvariety, and let
$$
\begin{CD}
{\mathcal C} \> \sigma >> X\\
{\mathbb V} f V V\\
S
\end{CD}
$$
denote the restriction of the universal complete intersection to
$S$. The smooth locus of $f$ is ${\mathcal C}_{{\rm sm}}=\sigma^{-1}(X_{\rm reg})$
and ${\mathcal C}_{\rm sm}$ is dominant over $X_{\rm reg}$.
Let $S' \to S$ be the finite morphism, and let
$W' \subset S^g({\mathcal C}_{\rm sm}\times_S S' /S')$
be the open connected subscheeme considered in lemma
\ref{relpic2}. Replacing $S$ and $S'$ by some open subschemes, and $S'$ by
a branched cover if necessary, we may assume that there exists a section
$\epsilon'$ of $W' \to S'$. By \ref{relpic2}
\begin{align*}
W' &\>>> {\rm Pic}({\mathcal C}/S)\times_S S'= {\rm Pic}({\mathcal C}\times_S S'/S')\\
w' &\longmapsto \vartheta_{W'}(w') - \vartheta_{W'}(\epsilon'(pr_2(w')))
\end{align*}
is an open embedding. On the other hand, one has a morphism
of schemes
$$
h: W' \>>> S^g({\mathcal C}_{\rm sm}\times_S S' /S') \>>> S^g(X_{\rm reg}),
$$
and the image of the connected scheme $W'$ lies in some
connected component, say
$S_{\underline{g}}=S^{g_1}(U_1) \times \cdots \times S^{g_r}(U_r).$
Since $\phi: CH^n(X)_{\deg 0} \to G$ is regular, the composite
$$
h^{(-)}:W'\times_{S'} W' \>>> S_{\underline{g}}\times S_{\underline{g}} \>
\theta >> G
$$
is a morphism, where
$$
\theta(\underline{x},\underline{x}')= \phi\bigl(
\sum_{i=1}^g \gamma(x_i) - \sum_{i=1}^g \gamma(x'_i)\bigr) .
$$
The morphism $h^{(-)}$ induces $S'$ morphisms
$$
h_{S'}^{(-)}: W'\times _{S'} W' \>>> G\times S' \mbox{ \ \ and \ \ }
h_{S'}^{(-)}\circ\delta:W'\times_{S'} W' \>>> A \times S'.
$$
Let $W_G$ and $W_A$ be locally closed subschemes of the images
$$
h_{S'}^{(-)}(W'\times_{S'} W')\mbox{ \ \ and \ \ }
h_{S'}^{(-)}\circ\delta(W'\times_{S'} W')
$$
respectively, dense in the closure of the images.
Choosing $S'$ and $S$ small enough, one may assume that
$S' \to S$ is surjective and that $W_G$ and $W_A$
are both equidimensional over $S'$. For $C\in S$ choose a point
$s' \in S'$ mapping to
$C \in S$ and let $W'_{s'}$ denote the fibre of $W'$ over $s'$.
Then the image of $W'_{s'} \times W'_{s'}$ in ${\rm Pic}^0(C)$ is dense and
thereby $\dim( \psi({\rm Pic}^0(C)))$ and
$\dim(\delta(\psi({\rm Pic}^0(C)))) = d'$ are both constant on $S$.
Asume that $d' < \dim(A)$. The closure of
$\delta(h^{(-)}(W'_C\times W'_C))$ is the image of ${\rm Pic}^0(C)$,
hence $\delta(h^{(-)}(W'_C\times W'_C))$ lies in some abelian
subvariety $B$ of $A$ of dimension $d'< \dim(A)$. Since $S'$ and $W'$ are
connected, and since an abelian variety $A$ does not contain
non-trivial families of abelian subvarieties, $B$ is independent
of the curve $C$ chosen.
${\mathcal C}_{\rm sm}$ being dominant over $X_{\rm reg}$ this implies
that the image $\delta\phi(CH^n(X)_{\deg 0})$ lies in $B$,
contradicting the assumptions made.
\end{proof}
In general, a commutative algebraic group $G$ can contain non-trivial
families of subgroups and the argument used above does not
extend to $G$ instead of $A$.
Let $H \subset G$ be the smallest connected algebraic subgroup
with $\delta(H) = A$,
as constructed in \ref{subgroups}. By \ref{surjective} and by
the universal property in \ref{subgroups}, for $C \in S$ the
image of $\psi({\rm Pic}^0(C))$ contains $H$.
By \ref{generators} the image
of $S^g(\Pi_{C_{\rm reg}})$ in $G$ is $\psi({\rm Pic}^0(C))$ and hence $H$
is contained in the image of $S^g(\Pi_{X_{\rm reg}})$. In order to
show that
$$
\pi^{(-)}: S^{g+\mu({\mathcal L})}(\Pi_{X_{\rm reg}}) \>>>
CH^n(X)_{\deg 0} \>\phi >> G
$$
is dominant, it suffices to verify that the
image $Y_0$ of the composite
$$
\tau^{(-)}: S^{d}(\Pi_{X_{\rm reg}}) \>>>
CH^n(X)_{\deg 0} \>\phi >> G \>>> G/H
$$
is dense, for some $d \leq \mu({\mathcal L})$. Applying claim
\ref{surjective} to $G/H$ instead of $G$ one finds a non-empty
open subscheme $S\subset |{\mathcal L}|^{n-1}_0$ such that the dimension $d$
of $\psi'({\rm Pic}^0(C))$ is constant on $S$, where $\psi':{\rm Pic}^0(C) \to
G/H$ is the natural map (see \ref{pic}). Since $G/H$ is a linear
algebraic group, we must have $d\leq \mu({\mathcal L})$, and
choosing $S$ small enough, we may assume that
\begin{equation}\label{bd}
d=\dim(\psi'({\rm Pic}^0(C))) \leq \mu(C) = \mu({\mathcal L}), \mbox{ \ \ \ \
for all \ $C \in S$.}
\end{equation}
Since $Y_0$ generates the group $G/H$, it is dense in $G/H$
if and only if its closure $Y$ is a group. By assumption
the image of the incidence variety
$$
Z \> pr'_{12} >>S^{d}(\Pi_{X_{\rm reg}})\times S^{d}(\Pi_{X_{\rm reg}})
\>\tau^{(-)}\times \tau^{(-)}>> Y \times Y
$$
defined in \ref{dominant} contains some open dense subscheme
$T$. By definition, for each $t \in T$ there exist divisors
$D_1, \ldots, D_{n-1}$ with
$C=D_1\cap \cdots \cap D_{n-1} \in S$ and with
$$
t \in {\rm image}\,( S^{d}(\Pi_{B})\times S^{d}(\Pi_{B})
\>{\vartheta}^{(-)}\times {\vartheta}^{(-)}>> Y \times Y)
$$
for $B = C\cap X_{\rm reg}$ and for the induced map $\vartheta^{(-)}$
from $\Pi_B$ to $G/H$.
By \ref{generators}
$\psi'({\rm Pic}^0(C))={\vartheta}^{(-)}(S^d(\Pi_B))\subset Y$. Since
$\psi'({\rm Pic}^0(C))$ is an algebraic subgroup of $G/H$, the image of
$t$ under the morphism
$$
{\rm diff}:G/H \times G/H \>>> G/H \mbox{ \ \ \ with \ \ \ } (g,g')
\longmapsto g-g'
$$
is contained in $\psi'({\rm Pic}^0(C))$, hence in $Y$.
Thereby $T$ is a subset of ${\rm diff}^{-1}(Y)$, and the same is
true of its closure $Y \times Y$. One obtains that
${\rm diff} (Y \times Y) \subset Y$ and $Y$ is a subgroup of
$G/H$.
Since $Y_0$ is dense in $G/H$, by lemma \ref{generators} (ii)
the image of $S^{2d}(\Pi_{X_{\rm reg}})$ is $G/H$.
\end{proof}
As indicated already, the proposition \ref{bound} can be
improved in characteristic zero.
\begin{variant}\label{bound2} Assume that ${\rm char} (k) =0$.
Let ${\mathcal L}$ be a very ample invertible sheaf on $X^{(n)}$ with
\begin{equation}\label{eqmu2}
\dim_k ({\rm image}\,( H^0(X^{(n)},{\mathcal L}) \to H^0(X_i,{\mathcal L}|_{X_i}))) \geq
2\cdot \mu({\mathcal L}) + r + 2,
\end{equation}
for $i=1,\ldots ,r$. Let $G$ be a smooth connected commutative
algebraic group, and let $\phi:CH^n(X)_{\deg 0} \to G$ be a
surjective regular homomorphism. Then there exists
an open dense subvariety $S \subset |{\mathcal L}|^{n-1}_0$ such that for
each $C \in S$ the induced homomorphism (see \ref{pic})
$$
\psi: {\rm Pic}^0(C) \>>> CH^n(X)_{\deg 0} \>\phi >> G
$$
is surjective. In particular the dimension of $G$ is bounded by
$g=\dim_k H^1(C,{\mathcal O}_C)$.
\end{variant}
\begin{proof}
The first part of the proof is the same as the one for
\ref{bound}. In particular we may assume claim \ref{surjective}
to hold true.
Let $H \subset G$ be the smallest subgroup with $\delta(H) = A$,
as constructed in \ref{subgroups}. By \ref{surjective} and by
the universal property in \ref{subgroups}, for all $C \in S$ the
image of $\psi({\rm Pic}^0(C))$ contains $H$. Hence $\psi:{\rm Pic}^0(C)
\>>> G$ is surjective if and only if
$$
{\rm Pic}^0(C) \> \psi >> G \>>> G/H
$$
is surjective. In order to prove \ref{bound2} we may assume
thereby that $G$ is linear and $A=0$.
For $C \in S$, let $\gamma_B : B=C\cap X_{\rm reg} \to CH^n(X)$
denote the natural map and let $\Gamma_B$ be the image of the
composite
$$
\vartheta^{(-)}:\Pi_B \> \gamma_B^{(-)} >> CH^n(X)_{\deg 0} \> \phi >> G.
$$
For any subset $M \subset G$ we will denote by $G(M)$ the smallest
algebraic subgroup of $G$ which contains $M$. If $M$ contains a
point of infinite order, then $\dim(G(M)) > 0$. In
characteristic zero the converse holds true, as well.
In fact, if $\dim(G(M)) > 0$ then $G(M)$ contains a subgroup
isomorphic either to ${\mathbb G}_a$ or to ${\mathbb G}_m$. In characteristic
zero, both contain points of infinite order.
Hence if the dimension of $G(\Gamma_B) = \psi({\rm Pic}^0(C))$
is larger than zero, the constructible set
$\Gamma_B$ contains a point $\alpha_1$ of infinite order and
$\dim(G(\alpha_1)) > 0$. Repeating
this for $G/G(\alpha_1,\ldots,\alpha_\nu)$ instead of
$G$, we find recursively points $\alpha_1, \ldots, \alpha_d\in
\Gamma_B$ with $G(\Gamma_B)=G(\alpha_1,\ldots, \alpha_d).$
Let us choose points $x_1, \ldots, x_d,x'_1, \ldots, x'_d \in B$
with $\alpha_j=\vartheta^{(-)}((x_j,x'_j))$, and moreover,
for each component $X_i$ of $X^{(n)}$, choose a base point $q_i \in B\cap
X_i$.
\begin{claim}\label{dominant2}
There exists a closed suscheme $Z\subset S$ such that the restriction
$$
\begin{CD}
{\mathcal C}'={\mathcal C}\times_S Z \> \sigma'=\sigma|_{{\mathcal C}'} >> X^{(n)}\>\subset>> X\\
{\mathbb V} f'=f|_{{\mathcal C}'} V V\\
Z \>\subset >> S\> \subset >>|{\mathcal L}|^{n-1}_0
\end{CD}
$$
of the universal family satisfies:
\begin{enumerate}
\item[(a)] For each point $z \in Z$ the curve $C_z={f'}^{-1}(z)$
contains the points
$$x_1, \ldots, x_d,x'_1, \ldots, x'_d, q_1,\ldots , q_r.$$
\item[(b)] $\sigma': {\mathcal C}' \to X^{(n)}$ is dominant.
\end{enumerate}
\end{claim}
\begin{proof}
For
$$
V_i=({\rm image}\,(H^0(X^{(n)},{\mathcal L}) \to H^0(X_i,{\mathcal L}|_{X_i}))-0)/k^* \subset
|({\mathcal L}|_{X_i})|
$$
consider the rational map $\tilde{p}_i:|{\mathcal L}|^{n-1} \to
V_i^{n-1}$. Since each $C\in S$ is a complete intersection
curve, the restriction $p_i:S\to V_i^{n-1}$ of $\tilde{p}_i$ is
a morphism.
For $x \in X_i\cap \sigma({\mathcal C})$ the condition ``$x
\in C_s$'' defines a multilinear subspace $\Delta_x^i$ of
$V_i^{n-1}$ of codimension $n-1$.
Let $I_i\subset\{ 1, \ldots,d\}$ denote the set of all the $\nu$
with $x_\nu, x'_\nu \in X_i$. Then the codimension of
$$
\Delta^i = \Delta^i_{q_i} \cap \bigcap_{\nu \in
I_i}(\Delta^i_{x_\nu} \cap \Delta^i_{x'_\nu})
$$
is at most $(n-1)\cdot(2\cdot \# I_i +1)$.
Let ${\mathcal C}^i \to \Delta^i$ be the intersection on $X_i$ of the
divisors in $\Delta^i \subset V_i^{n-1}$. Then the
general fibre of ${\mathcal C}^i \to X_i$ has dimension at least
$$\dim\Delta^i+1-\dim X=\dim V_i^{n-1}+1-n-{\rm codim}\Delta^i$$
$$\geq
\dim (V_i^{n-1}) +1 -n - (n-1)\cdot(2\cdot \# I_i +1) \geq
(n-1)\cdot(2\cdot(\mu({\mathcal L}) - \# I_i)+r-1).
$$
Since some $C \in S$ contains all the points $x_j, x'_j$ and
$q_i$, the intersection
$$
Z = S\cap \bigcap_{i=1}^r p_i^{-1}(\Delta^i)
$$
is non-empty. For the restriction ${\mathcal C}'$ of the universal curve
${\mathcal C}$ to $Z$ the dimension of the general fibre of $\sigma':{\mathcal C}'
\to X$ over $X_i$ has dimension larger than or equal to
\begin{align*}
& (n-1)\cdot(2\cdot(\mu({\mathcal L}) - \# I_i)+r-1)-\sum_{j\neq i}
(n-1)\cdot(2\cdot \# I_j +1)\\
&= (n-1)\cdot 2\cdot(\mu({\mathcal L}) - \sum_{j=1}^r \# I_i)
=(n-1)\cdot 2\cdot(\mu({\mathcal L}) - d).
\end{align*}
By the inequality (\ref{bd}) the last expression is larger than or
equal to $0$ and $\sigma'$ is dominant.
\end{proof}
Let $G(C_z)$ denote the image of ${\rm Pic}^0(C_z)$ in $G$.
By the choice of $Z$ the intersection $B_z=C_z\cap X_{\rm reg}$ is
non-singular and the dimension of $G(C_z)=G(\Gamma_{B_z})$ is
the same as the dimension of $G(\Gamma_B)=G(\alpha_1,\ldots, \alpha_d)$.
By \ref{dominant2} the points $\alpha_i =
\phi(\sigma'(x_i)-\sigma'(x'_i))$ are contained in
$\Gamma_{B_z}$, hence
$$
G(C_z)=G(\Gamma_{B_z}) = G(\Gamma_B)=G(C)
$$
for all $z \in Z$.
As $\sigma'$ is dominant, $\sigma'({\mathcal C}')$ contains some $V$,
open and dense in $X_{reg}$ (and hence in $X^{(n)}$). For $q\in V\cap X_i$
one finds some $z \in Z$ with $q \in C_z$. By \ref{dominant} $C_z$
contains the chosen base point $q_i$ and
$$
\phi(\gamma(q_i) - \gamma(q)) \in G(C_z)=G(C).
$$
By \ref{equidim} (i), the points $\gamma(q_i) - \gamma(q)$ (for $q\in V$)
generate $CH^n(X)_{\deg 0}$. Since $\phi$ was assumed to be surjective,
we obtain $G=G(C)$, as claimed.
\end{proof}
Using proposition \ref{bound} or its variant \ref{bound2}
the construction of $A^n(X)$
proceeds now along the lines of Lang's construction in
\cite{La} of the Albanese variety of a smooth projective variety.
\begin{thm}\label{existence}
There exists a smooth connected commutative algebraic group $A^n(X)$
and a surjective regular homomorphism $\varphi: CH^n(X)_{\deg 0} \to
A^n(X)$ satisfying the following universal property:
For any smooth commutative algebraic group $G$ and for any
regular homomorphism $\phi:CH^n(X)_{\deg 0}\to G$ there
exists a unique homomorphism $h:A^n(X)\to G$ of algebraic groups
with $\phi=h\circ\varphi$.
Moreover, if $k \subset K$ is an extension of algebraically
closed fields, then
$$
A^n(X\times_kK)=A^n(X)\times_kK.
$$
\end{thm}
\begin{proof}
By lemma \ref{reg_sur} it is sufficient to consider connected
groups $G$, and surjective regular homomorphisms $\phi$.
By \ref{mu3} there exists a very ample invertible
sheaf ${\mathcal L}$ which satisfies the assumption
\ref{dominant} and we can apply \ref{bound}.
(As we have seen in \ref{mu2} the inequality (\ref{eqmu2}) in
\ref{bound2} holds true for some ${\mathcal L}$, and if ${\rm char}(k)=0$
we can use the variant \ref{bound2}, as well.)
Let $g=\dim_k(H^1(C,{\mathcal O}_C))$, for some curve $C\in |{\mathcal L}|^{n-1}_0$ in
general position. Then for all regular homomorphisms
$\phi:CH^n(X)_{\deg 0}\to G$ to smooth connected commutative
algebraic groups $G$ the induced morphism
$$
\pi^{(-)}: S^{g+\mu({\mathcal L})}(\Pi_{X_{\rm reg}})
\>\gamma^{(-)}_{g+\mu({\mathcal L})}>>
CH^n(X)_{\deg 0} \>\phi >> G
$$
has a dense image in $G$. Hence for the product $\Pi$ of all the
different connected components of $S^{g+\mu({\mathcal L})}(\Pi_{X_{\rm reg}})$
the induced morphism $\pi' : \Pi \to G$ is dominant and $\pi'$
induces a unique embedding of function fields $k(G) \subset k(\Pi)$.
If $\phi_\nu : CH^n(X)_{\deg 0} \to G_\nu$, for $\nu = 1,2$ are
two surjective regular homomorphisms to smooth connected commutative algebraic
groups, then
$$
\phi_3:CH^n(X)_{\deg 0} \>>> G_1\times G_2
$$
is regular. Let $G_3$ be the image of $\phi_3$.
Then $\phi_\nu$ factors through the regular homomorphism
$\phi_3:CH^n(X)_{\deg 0} \to G_3$ and
$k(G_\nu) \subset k(G_3) \subset k(\Pi)$, for $\nu = 1,2$.
Hence among the smooth connected commutative algebraic groups
$G$ with a regular surjective homomorphisms from
$\phi:CH^n(X)_{\deg 0}\to G$, there is one, $A^n(X)$, for which the subfield
$k(A^n(X))$ is maximal in $k(\Pi)$ and $A^n(X)$ dominates all
the other $G$ in a unique way.
It remains to show that $A^n(X)$ satisfies base-change for
algebraically closed fields. Let us write $Z_K = Z\times_kK$,
for a variety $Z$ defined over $k$. We first show:
\begin{claim}\label{basechange1}
Let $K\supset k$ be an algebraically closed extension field of $k$.
The cycle map $\varkappa^{(-)}_K:(\Pi_{X_{\rm reg}})_K \to A^n(X)_K$
factors through a surjective homomorphism
$u_{K,k}:A^n(X_K) \to A^n(X)_K$ of algebraic groups.
\end{claim}
\begin{proof} Let $U=X_{reg}$, and let $(C', \iota)$ be an admissible pair
defined over $K$, with $B =\iota^{-1}(U_K)_{{\rm reg}}$.
Choose a rational function $f \in R(C',X_K)$ such that
$$ {\rm div} f = \sum a_i - \sum b_i $$
for $p =(a_1,b_1,\ldots,a_m,b_m)\in S^m(\Pi_B)(K)$. Choose a smooth $k$
variety $S$ with
$k(S)\subset K$, such that $C'$, $B$, $p$, $a_i$, $b_i$, $f$ come by
base-change from $k(S)$ to $K$ from
$${\mathcal C}' \to S, \ \ {\mathcal B} \to S, \ \ \pi: S \to S^m(\Pi_{{\mathcal B}/S}), \ \ \alpha_i,
\beta_i: S \to {\mathcal B}, \ \ \varphi \in k({\mathcal B})^{\times} $$
with ${\rm div} \varphi = \sum \alpha_i - \sum \beta_i$. Since $f\in
R(C',X_K)$, we can replace $S$ by a dense open subscheme, so that we can
arrange that for each $s\in S(k)$, if we specialize to ${\mathcal C}'_s = {\mathcal C} _S
\times s$, then $\pi(s)$ maps to zero in ${\rm Pic}^0({\mathcal C}'_s)$. As
\[S^m(\Pi_{{\mathcal B}_s}) \>>> S^m (\Pi_U) \times s \>>> A^n(X) \times s\]
factors through ${\rm Pic}^0({\mathcal C}'_s)$, the composite morphism
$$
S \> \pi >> S^m(\Pi_{{\mathcal B}/S}) \>>> S^m(\Pi_U)\times S
\>>> A^n(X)\times S
$$
maps all $k$-points of $S$ to the zero section. Thus it is the zero
section, and therefore
$S^m(\Pi_U)_K \to A^n(X)_K$
factors through $CH^n(X_K)$, inducing $u_{K,k}$ by lemma \ref{pic}.
\end{proof}
Since $d_K:= \dim A^n(X_K)$ is bounded by $2n(g + \mu({\mathcal L}))$
(proposition~\ref{bound}), there is an algebraically closed field $K_1$
with $d_{K_1}=d_L$ for all $L\supset K_1$ algebraically closed. Since for
any ascending chain $K_i\subset K_{i+1}$ of algebraically closed fields
with $K_i \supset K_1$ one has
$$
{\rm deg} \ u_{K_i,K_1} \leq {\rm deg} \ u_{K_{i+1},K_1} \leq
{\rm deg} \ u_{\cup K_i, K_1},
$$
one concludes that there is an algebraically closed field
$E\supset K_1$ such that $u_{E,L}$ is an isomorphism for all
algebraically closed fields $L\supset E$.
We will make use of the following lemma.
\begin{lemma}\label{descent}
Let $K$ be a field, $W$, $Y$, $Z$ be geometrically integral $K$-varieties,
such that there are $K$-morphisms $\alpha:W\to Y$,
$\beta:W\to Z$, such that $\alpha$ has dense image. Then:
\begin{enumerate}
\item[(i)] there is at most one $K$-morphism $f:Y\to Z$ such that
$\beta=f\circ\alpha$
\item[(ii)] suppose that for some extension field $L$ of $K$, there
is an $L$-morphism $h:Y_L\to Z_L$ such that $\beta_L=h\circ
\gamma_L:W_L\to Z_L$; then there is a $K$-morphism $f:Y\to Z$ as in
(i), and we have $h=f_L$.
\end{enumerate}
\end{lemma}
\begin{proof} Let $\Gamma\subset W\times_K Z$ be the graph of $\beta$, and
let $\bar{\Gamma}\subset Y\times_K Z$ be the closure of $\alpha\times
1_Z(\Gamma)$. The projection $\bar{\Gamma}\to Y$ has dense image. If there
is a $K$-morphism $f:Y\to Z$ as in (i), then $\bar{\Gamma}$ must be its
graph, and so there is at most one such morphism, which exists precisely
when $\bar{\Gamma}\to Y$ is an isomorphism. Clearly if this is an
isomorphism after base change to $L$, it is an isomorphism to begin with.
\end{proof}
There is a smooth $k$ variety $S$, with $k(S) \subset E$, together
with a smooth commutative $S$-group scheme ${\mathcal A} \to S$ with connected
fibers, such that $A^n(X_E) = {\mathcal A} \times_S {\rm Spec \,} E$. Choosing $S$ small
enough one also has natural surjective $S$-morphisms $\Pi\times S \to {\mathcal A}$
and $u_{S,k}:{\mathcal A} \to A^n(X)\times S$, where $\Pi$ is the irreducible
variety constructed in the first part of the proof.
Let $F$ be an algebraic closure of the quotient field of $E\otimes_k E$,
$p: k(S\times_k S) \hookrightarrow F$ the natural inclusion, and let
$p_i^*: k(S)\hookrightarrow k(S\times_k S)$, $i=1,2$ be the inclusions
defined by the two projections $p_i:S\times_k S\to S$. Set $q_i= p \circ
p_i^*$, and for any $S$-scheme $T$, let $q_i^*T$ be the $F$-scheme
obtained by the base change to $F$ determined by $q_i$.
The surjective $S$-morphism $\Pi\times S \to {\mathcal A}$ gives rise to the
surjections
\[\alpha_i':q_i^* (\Pi\times_kS) = \Pi_{F} \>>> q_i^*{\mathcal A}.\]
By the assumption on $E$ the two $F$-varieties $q_i^*{\mathcal A}$ are isomorphic
via
$$ u'= u_{F, E_1} \circ u^{-1}_{F,E_2}: q_1^*A^n(X_E) \>>> q_2^*A^n(X_E).
$$
where $E_i\subset F$ are the images of the two embeddings $E\hookrightarrow
E\otimes_k E\hookrightarrow F$, $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$.
By construction, $u'$ satisfies $\alpha_2'=u'\circ\alpha_1'$.
Hence by lemma~\ref{descent}, applied to the extension of fields
$k(S\times_kS)\hookrightarrow F$, the isomorphism $u'$ comes from an isomorphism
$$u: (p_1^*{\mathcal A})_{k(S\times_k S)} \>>> (p_2^*{\mathcal A})_{k(S\times_k S)},$$
Then $u$ in fact extends uniquely to an isomorphism of groups schemes
(again denoted $u$)
\[u: (p_1^*{\mathcal A})_U \>>> (p_2^*{\mathcal A})_U\]
over an open dense subset $U \subset S\times_k S$.
Replacing $S$ by some open dense subscheme, we may assume that $p_i:U\to S$
is surjective, for $i=1,2$. Further, if $\alpha_i:\Pi\times U\to
(p_i^*{\mathcal A})_U$, $i=1,2$ are the natural surjections, then $\alpha_2=u\circ
\alpha_1$.
The uniqueness statement in lemma~\ref{descent}~(i) similarly implies
that $u$ satisfies the ``cocycle condition''
\[u_{23}\circ u_{12}=u_{13}:\pi_1^*{\mathcal A}\>>>\pi_3^*{\mathcal A}\]
on the fibers over the generic point of $S\times_k S\times_k S$, and hence
(by continuity) over the open dense subset
\[\pi_{12}^{-1}(U)\cap\pi_{23}^{-1}(U)\cap \pi_{13}^{-1}(U)\subset
S\times_k S\times_k S.\]
Here $\pi_j:S\times_k S\times_k S\to S$ are the 3 projections, and
$u_{ij}=\pi_{ij}^*{u}$, for the 3 projections
$\pi_{ij}:S\times_k S\times_k S\to S\times_k S$.
Given two points $s_i \in S(k)$, one finds a third one $s
\in S(k)$ such that $(s_1,s) \in U(k)$ and $(s, s_2) \in U(k)$. The cocycle
condition implies that the induced composite isomorphism
$$ \theta_{s_1s_2}: {\mathcal A}|_{s_1} \> u|_{(s_1,s)} >> {\mathcal A}|_s \> u|_{(s,s_2)}
>> {\mathcal A}|_{s_2}$$
does not depend on the point $s\in S(k)$ chosen. Also
$u$ is compatible with the surjective morphisms $\Pi \times U \to
p_i^*{\mathcal A}$.
We claim that for each closed point $s\in S(k)$, the morphism $\Pi\times
s\to {\mathcal A}|_s$ induces a regular homomorphism $CH^n(X)_{\deg 0}\to
{\mathcal A}|_s$. Let $(C',\iota)$ be an admissible pair on $X$, defined over
$k$, with $B = \iota^{-1}(U)_{{\rm reg}}$. The morphism $(\Pi_B)_E\to
A^n(X_E)={\mathcal A}_E$, and the resulting morphism $S^g(\Pi_B)_E\to {\mathcal A}_E$ (with
$g:=\dim{\rm Pic}^0(C')$) induces a homomorphism ${\rm Pic}^0(C')_E\to {\mathcal A}_E$, since
by the defining property of $A^n(X_E)$, we have a factorization through
$CH^n(X_E)_{\deg 0)}$. Since $S^g(\Pi_B)\mbox{$\,\>>>\hspace{-.5cm}\to\hspace{.15cm}$}{\rm Pic}^0(C')$,
lemma~\ref{descent} gives a map ${\rm Pic}^0(C') \times_k k(S) \to
{\mathcal A}_{k(S)}$, compatible with the maps from $(\Pi_B)_{k(S)}$. This then
induces a map ${\rm Pic}^0(C')\times S^0\to{\mathcal A}_{S^0}$ for some open dense
subscheme $S^0\subset S$. Choosing a $k$-point $s_1\in S^0(k)$, we get
that the map $\Pi_B\times s_1\to {\mathcal A}|_{s_1}$ is compatible with a
homomorphism ${\rm Pic}^0(C')\to {\mathcal A}|_{s_1}$. The isomorphism $u$ is
compatible with the morphisms $\Pi_B \times U \to p_i^*{\mathcal A}$.
Hence, the isomorphisms $\theta_{ss_1}$ are compatible with the maps
$\Pi_B\cong \Pi_B\times s\to {\mathcal A}|_s$ and $\Pi_B\cong \Pi_B\times s_1\to
{\mathcal A}|_{s_1}$, for all $s\in S(k)$. We deduce that for any $s\in S(k)$,
the map $\Pi_B\times s\to {\mathcal A}|_s$ gives rise to a compatible morphism
${\rm Pic}^0(C') \times _k s \to {\mathcal A}|_s$. This implies that there is an induced
regular homomorphism $CH^n(X)_{\deg 0}\to {\mathcal A}|_s$ for each $s\in S(k)$.
Hence, one obtains morphisms $v_s: A^n(X) \to {\mathcal A}|_s$, verifying $ v_t =
\theta_{st}
\circ v_s$ for all $s,\ t \in S(k)$. Choosing now $s \in S(k)$,
we set $G= {\mathcal A}|_s$, and $ v= v_s$.
The surjective morphism $\Pi\times_k S \to {\mathcal A}$ induces a surjection from
$\Pi \times_k s$ onto $G$, hence $v$ is surjective. Since the composite
$$
u_{S,k}\circ v : A^n(X) \>>> G \>>> A^n(X)
$$
is an isomorphism, $v$ is an isomorphism. Thus
$u_{S, k}: {\mathcal A} \to A^n(X)\times S$ is an isomorphism when restricted to
each $t \in S(k)$, and is hence an isomorphism. By base change to $E$ one
finds that $u_{E,k}:A^n(X_E) \to A^n(X)_E$ is an isomorphism.
Now if $K \supset k$ is any algebraically closed field, we
choose an algebraically closed field $F$ with
\begin{gather*}
\hspace*{1.6cm}F\supset K \supset k\mbox{ \ \ \ and \ \ \ }F\supset
E\supset k, \mbox{ \ \ \ hence}\\
u_{K, k} \otimes {\rm id}_F \circ u_{F,K}=
u_{F, k} = u_{E, k} \otimes {\rm id}_F \circ u_{F,E},
\end{gather*}
and $u_{K,k}$ is an isomorphism as well.
\end{proof}
\section{Finite dimensional Chow groups of zero cycles}
The definition of finite dimensionality for the Chow group
of 0-cycles is a natural generalization of the definition in the
non-singular (and normal) case (see \cite{M}, \cite{S}).
\begin{defn}\label{fin-dim} $CH^n(X)$ is said to be {\em finite
dimensional} if for some $m>0$, the map
$$
\gamma_m : S^m(X_{\rm reg}) \>>> CH^n(X)_{\deg 0}
$$
(introduced in \ref{difference}) is surjective.
\end{defn}
One can see that this is also equivalent to the statement that for some
integer $m'>0$, depending only on $X$, any element
of $CH^n(X)_{\deg 0}$ is represented by a 0-cycle $\sum_{i=1}^r\delta_i$,
where for each $i$, the cycle $\delta_i$ is a difference of two effective
0-cycles of degree $m'$ supported in $X_i$.
In the proof of the next theorem we will use the notion of a
{\em regular map} $f:Z\to CH^n(X)_{\deg 0}$ from a variety $Z$.
This is a map of sets such that
\begin{points}
\item the composition $Z\to CH^n(X)_{\deg 0}\to A^n(X)$ is a
morphism \vspace{.05cm}
\item there is a surjective morphism $W\to Z$ such that
$$W\>>> Z\>{f}>> CH^n(X)_{\deg 0}$$
factors as $W\>{h}>>S^m(X_{\rm reg})\>{\gamma_m}>> CH^n(X)_{\deg 0},$
for some morphism $h$.
\end{points}
For example, let $C'$ be a reduced Cartier curve in $X$ or, more general,
let $(C',\iota)$ be an admissible pair. Then the homomorphism
$\eta:{\rm Pic}^0(C')\to CH^n(X)_{\deg 0}$
constructed in lemma~\ref{gysin} is regular. In fact, the first
condition holds true by \ref{pic} whereas the second one
follows from the dominance of $S^g(C'_{\rm reg}) \to {\rm Pic}^0(C')$,
for $g=\dim_k(H^1(C',{\mathcal O}_{C'}))$.
Recall that $k$ is called a universal domain, if its
trancendence degree over the prime field is uncountable.
\begin{thm}\label{finite} Let $X$ be a projective variety of
dimension $n$ over a universal domain $k$. Then $CH^n(X)$ is
finite dimensional if and only if
$$
\varphi:CH^n(X)_{\deg 0}\>>> A^n(X)
$$
defines an isomorphism between $CH^n(X)_{\deg 0}$ and
{\em (the closed points of)} $A^n(X)$.
\end{thm}
\noindent{\bf Proof}\ \\
Let us write $U=X_{\rm reg}$. By lemma \ref{generators} (ii) the composite
\[S^m(U)\>>> CH^n(X)_{\deg 0}\>>> A^n(X)\]
is always surjective for $m=2\cdot \dim(A^n(X))$. Hence, if
$CH^n(X)_{\deg 0}\to A^n(X)$ is an isomorphism, then $CH^n(X)$ is finite
dimensional.
So the main thrust of the theorem is the converse, that if
$CH^n(X)_{\deg 0}$ is finite dimensional, then $CH^n(X)_{\deg 0}\to
A^n(X)$ is an isomorphism. We imitate Roitman's proof of this result in
the non-singular case, and the analogous argument for the normal case in
\cite{S}; however there are extra refinements needed here, particularly in
characteristic $p>0$.
First, we note that by \cite{LW}, proposition 4.2, the ``graphs
of rational equivalence''
\[\Gamma_{r,s}=S^r(U)\times_{CH^n(X)_{\deg 0}}S^s(U)\]
decompose as a countable union of locally closed subvarieties,
for each $r,s$, and over a universal domain such a decomposition
is unique. This immediately implies that if $f_j:Z_j\to CH^n(X)_{\deg 0}$,
$j=1,2$ are regular maps, then
\[Z_1\times_{CH^n(X)_{\deg 0}}Z_2 = \{(z_1,z_2)\in Z_1\times
Z_2; \ f_1(z_1)=f_2(z_2)\}\]
is a countable union of locally closed subvarieties of $Z_1\times Z_2$.
Now arguing as in \cite{S}, lemma~(1.3), we first see that if $G$ is a
smooth connected commutative algebraic group, and $f:G\to CH^n(X)_{\deg 0}$ is
any regular map which is a group homomorphism, then there is a
well-defined connected component of the identity $G^0\subset \ker f$,
which is a connected algebraic subgroup of $G$, and which has countable
index in $\ker f$. Then the induced homomorphism
$$G/G^0\>>> CH^n(X)_{\deg 0}$$
has a countable kernel. Hence, for any such homomorphism $G\to
CH^n(X)_{\deg 0}$, we can define the {\em dimension of the image of $G$}
to be the dimension of $G/G^0$.
Next, notice that if $G_1\to CH^n(X)_{\deg
0}$ and $G_2\to CH^n(X)_{\deg 0}$ are two regular homomorphisms from
smooth connected commutative algebraic groups $G_i$ such that
${\rm image}\,G_1$ is properly contained in ${\rm image}\, G_2$, then in
fact
\[\dim\, {\rm image}\,G_1< \dim\, {\rm image}\,G_2.\]
Indeed, we may assume the maps $G_i\to CH^n(X)_{\deg 0}$ have countable
kernel, so that we wish to assert that $\dim G_1<\dim G_2$. Now
$G_3=G_1\times_{CH^n(X)_{\deg 0}}G_2$
is a subgroup of $G_1\times G_2$ which is a countable union of
locally closed subvarieties, and hence has a connected component of the
identity which is a connected algebraic group, say $H$. Then $H\to G_i$ are
homomorphisms of algebraic groups with countable, hence finite
kernels, such that $H\to G_1$ is surjective, and the image of $H$ in $G_2$
is a strictly smaller subgroup. Thus $\dim G_1=\dim H<\dim G_2$.
Now suppose $\gamma_m$ is surjective. We claim that for any homomorphism
$$
G\>>> CH^n(X)_{\deg 0}
$$
as above, with countable kernel, we have
$\dim G\leq \dim S^m(U).$
Indeed,
\[G\times_{CH^n(X)_{\deg 0}}S^m(U)\]
is a countable union of subvarieties of $G\times S^m(U)$ which projects onto
$G$, and maps to $S^m(U)$ with countable fibres. Hence some irreducible
component of this fibre product dominates $G$ under the projection, and
maps to $S^m(U)$ with finite fibres.
We now claim that we can find a finite number of reduced complete
intersection curves $C_1,\ldots,C_s$ such that the induced homomorphism
from $\oplus {\rm Pic}^0(C_j)$ to $CH^n(X)_{\deg 0}$ is surjective. Indeed, given a
finite collection of such curves, if
$$P=\bigoplus {\rm Pic}^0(C_j)\>>> CH^n(X)_{\deg 0}$$
is not surjective, we can find a curve $C$ of the same sort such that
$${\rm image}\,( {\rm Pic}^0(C)\>>> CH^n(X)_{\deg 0})$$
is not contained in the image of $P$. Then the induced map
$$P\times {\rm Pic}^0(C)\>>> CH^n(X)_{\deg 0}$$
has strictly larger
dimensional image than that of $P$. Since the dimension
of the image is bounded above by $\dim S^m(U)=mn$, this process can be
repeated at most a finite number of times.
So we may assume given a surjective regular homomorphism
$$f: A\>>> CH^n(X)_{\deg 0}$$
with countable kernel, where $A$ is a
connected smooth commutative algebraic group, and for some
Cartier curves $C_1, \ldots , C_s$ a surjective homomorphism
\begin{equation}\label{surjection}
\bigoplus_{j=1}^s {\rm Pic}^0(C_j) \> \oplus \rho_j >> A.
\end{equation}
Note that the composition $h:A\to CH^n(X)_{\deg 0}\to A^n(X)$ is
then a surjective homomorphism of algebraic groups.
We now distinguish between the case $k={\mathbb C}$, and that of a general
universal domain $k$.
\begin{proof}[Proof of \ref{finite} for $k={\mathbb C}$]
We first show that the surjective homomorphism $h:A\mbox{$\,\>>>\hspace{-.5cm}\to\hspace{.15cm}$} A^n(X)$ is an
{\em isogeny}. Clearly $h$ induces an injective homomorphism
$$h^*:\Omega(A^n(X))\to\Omega(A).$$
We will use proposition~\ref{basic} to show that
$h^*:\Omega(A^n(X))\to\Omega(A)$ is an isomorphism. Since $h^*$ is
injective, it suffices to prove that $\dim \Omega(A)\leq\dim
\Omega(A^n(X))$.
Consider the set $\Gamma=U\times_{CH^n(X)_{\deg 0}}A$.
This is a countable union of algebraic subvarieties, and maps surjectively
to $U$ under the projection. Recalling that $U=\cup_j U_j$, we can then
find irreducible varieties $\Gamma_j\subset \Gamma$ such that
$\Gamma_j$ dominates $U_j$ under the projection $\Gamma\to U$. Then
$\pi_j:\Gamma_j\to U_j$ has countable, and hence finite, fibres.
Let $d_j$ be the degree of $\pi_j$, and let $V_j\subset U_j$ be a dense
open subset such that $\pi_j:\pi_j^{-1}(V_j)\to V_j$ is an \'etale
covering of degree $d_j$. Let $c$ be the l.c.m. of the $d_j$, and let
$c=d_jc_j$. If $q:\Gamma\to A$ is the second projection, then consider the
morphism
\begin{gather*}
\mu:V=\bigcup_jV_j\>>> A,\\
\mu(x)=c_j\mathop{\sum}_{y\in \pi_j^{-1}(x)}q(y)\,\,\mbox{for $x\in
V_j$.}
\end{gather*}
One verifies at once that the diagram
\begin{equation}\label{diag-1}
\begin{CD}
\bigcup_j V_j=V \>{\mu}>> A \\
{\mathbb V} VV {\mathbb V} V f V\\
\bigcup_j U_j=U \>{c\cdot\gamma_1}>>CH^n(X)_{\deg 0}
\end{CD}
\end{equation}
commutes.
The image of $\mu(V)$ in $CH^n(X)_{\deg 0}$ generates
$CH^n(X)_{\deg 0}$ as a group, since any 0-cycle on $X$ is rationally
equivalent to a cycle supported on $V$. Hence the subgroup of $A$
generated by $\mu(V)$ has countable index, and is also a countable
increasing union of constructible subsets, namely the images of $\mu(V)^{2m}$
under the maps
\begin{align*}
\sigma_m:A^{2m}&\>>>A,\;\;\;\; m\geq 1,\\
(a_1,\ldots,a_{2m})&\longmapsto
a_1+\cdots+a_m-a_{m+1}-\cdots-a_{2m}.
\end{align*}
By dimension considerations, one of the subsets
$\sigma_m(\mu(V)^{2m})$ must be dense in $A$, and then
$\sigma_{2m}(\mu(V)^{4m})=A$. Hence the induced map on 1-forms
$$
\Omega(A)\to H^0(V^{4m},\Omega^1_{V^{4m}/{\mathbb C}})
$$
is injective. Now the action of $\sigma_{2m}$ on 1-forms is given by
\[\sigma_{2m}^*(\omega)=
(\omega,\ldots,\omega,-\omega,-\omega,\cdots,-\omega).\]
This means that the map on 1-forms
$\Omega(A)\to H^0(V,\Omega^1_{V/{\mathbb C}})$, induced by the morphism $V\to
\mu(V)\hookrightarrow A$, is injective.
We claim that ${\rm image}\,\Omega(A)\subset \Omega(A^n(X))$, so that
$\dim\Omega(A)\leq \dim\Omega(A^n(X))$. To see this, it
suffices by proposition~\ref{basic} to show that for any reduced Cartier
curve $C\subset X$ with $B=(C_{{\rm reg}})\cap V$ dense in $C$, the image of
any element of $\Omega(A)$ in $H^0(B,\Omega^1_{B/{\mathbb C}})$ lies in the image of
$H^0(C,\omega_C)$. Fixing base points in each component of $B$,
we obtain a morphism $\vartheta:C_{{\rm reg}}\to{\rm Pic}^0(C)$.
If $C_i$ is any component of $C_{{\rm reg}}$, then the two induced maps
\begin{gather*}
C_i\>>> {\rm Pic}^0(C)\>>> CH^n(X)_{\deg 0},\\
C_i\hookrightarrow U\>{\gamma_1}>> CH^n(X)_{\deg 0}
\end{gather*}
agree up to translation by a fixed element of $CH^n(X)_{\deg
0}$.
Now consider the subgroup $\Gamma_C={\rm Pic}^0(C)\times_{CH^n(X)_{\deg 0}} A$.
As before, this is a countable union of subvarieties of ${\rm Pic}^0(C)\times
A$. Hence there is a connected algebraic subgroup
$\Gamma^0_C\subset\Gamma_C$ such that $\Gamma_C/\Gamma^0_C$ is a countable
group. Further, $\Gamma_C\to{\rm Pic}^0(C)$ is surjective with countable
fibres. Hence $\Gamma^0_C\to{\rm Pic}^0(C)$ is an isogeny.
Restricting (\ref{diag-1}) one obtains a
commutative diagram
\[\begin{CD}
B \>{\mu}>> A\\
{\mathbb V} VV {\mathbb V} V f V\\
C_{{\rm reg}}\>{c\gamma_1}>> CH^n(X)_{\deg 0}
\end{CD}\]
and hence a morphism $B\to \Gamma^0_C$ such that
\begin{points}
\item for each component $C_i$ of $C_{{\rm reg}}$,
the composite $C_i\cap B\to \Gamma^0_C\to{\rm Pic}^0(C)$
equals the restriction of the composite $C_i\to {\rm Pic}^0(C)\longby{c\cdot
}{\rm Pic}^0(C)$, up to a translation (here $c\cdot$ denotes multiplication
by $c$)
\item $C_i\cap B\to \Gamma^0_C\to A$ agrees with $\mu$, up to a
translation.
\end{points}
Hence, by (ii),
$$
{\rm image}(\Omega(A)\>{\mu^*}>>\Gamma(B,\Omega^1_{C/{\mathbb C}}))
\subset {\rm image}\left(\Omega(\Gamma^0_C)\>>>
\Gamma(B,\Omega^1_{C/{\mathbb C}})\right)
$$
while by (i),
\[{\rm image}\,\Omega(\Gamma^0_C)={\rm
image}\,\Omega({\rm Pic}^0(C))={\rm image}\,\Gamma(C,\omega_C).\]
Since $C$ was arbitrary, we have verified the hypotheses of
proposition~\ref{basic}. This completes the proof that the composite
$h:A\to CH^n(X)_{\deg 0}\to A^n(X)$ is an isogeny.
In particular, $f:A\mbox{$\,\>>>\hspace{-.5cm}\to\hspace{.15cm}$} CH^n(X)_{\deg 0}$ has a finite kernel. Replacing
$A$ by $A/(\ker f)$, we may assume given a regular homomorphism $f:A\to
CH^n(X)_{\deg 0}$ which is an {\em isomorphism} of groups. Now repeating
the above arguments once more, we obtain (\ref{diag-1}) with $c=1$. By
corollary~\ref{equ-reg}, this means the group isomorphism
$f^{-1}:CH^n(X)_{\deg 0}\to A$ is a regular homomorphism, which must
factor through $\varphi:CH^n(X)_{\deg 0}\mbox{$\,\>>>\hspace{-.5cm}\to\hspace{.15cm}$} A^n(X)$. This forces
$\varphi$ to be an isomorphism of groups, as well.
\end{proof}
\begin{rmk}\label{roitman}
Over the field of complex numbers the last part of the proof of \ref{finite}
is consistent with the Roitman theorem proved in \cite{BiS}.
In fact, if
$$A\cong CH^n(X)_{\deg 0}\>>> A^n(X)$$
is surjective with finite kernel the generalization of Roitman's
theorem implies that the composite
\[A\>{\cong}>> CH^n(X)_{\deg 0}\>>> A^n(X)\>>> J^n(X)\]
is an isomorphism on torsion subgroups, so that $CH^n(X)_{\deg 0}\to
A^n(X)$ is an injection on torsion subgroups. Hence the isogeny
$A\cong CH^n(X)_{\deg 0}\to A^n(X)$ must be an isomorphism.
\end{rmk}
In the algebraic case we have to modify the arguments, in
particular since the lower horizontal morphism in the diagram
(\ref{diag-1}) need not to be surjective in characteristic $p>0$.
\begin{proof}[Proof of \ref{finite} for $k$ a universal
domain] \ \\
Let us write $B$ for the kernel of $h:A \to A^n(X)$, a
closed subgroup scheme of $A$, not necessarily reduced.
We may replace $A$ by $A/\kappa$, for any (zero dimensional) closed
subgroup scheme $\kappa$ of $B$ such that $\kappa(k)\subset\ker f$.
The group $B$ acts on $U\times_{A^n(X)}A$ with quotient
$U\times_{A^n(X)}A^n(X)=U$. The kernel ${\mathcal K}$ of the map
$A(k) \to CH^n(X)_{\deg 0}$ consists of countably many closed
points, the induced action on $U\times_{CH^n(X)_{\deg 0}}A$
is free, and the induced map on the quotient
$$
(U\times_{CH^n(X)_{\deg 0}}A)/{\mathcal K} = U\times_{CH^n(X)_{\deg
0}}(A/{\mathcal K}) \>>> U
$$
is a bijection on the closed points.
Let $V \subset U$ be an open dense subscheme, and
let $\Gamma_j$ be a locally closed irreducible subscheme of
$V\times_{A^n(X)}A$, contained in $V\times_{CH^n(X)_{\deg 0}}A$,
and dominant over the component $V_j=V\cap U_j$ of $V$ under the first
projection. For $V$ small enough, we may assume that $\Gamma_j
\to V_j$ is finite. Let $\kappa_j\subset {\mathcal K}$ be the subgroup
of elements $g$ with $g(\Gamma_j) = \Gamma_j$. Then
$\kappa_j$ is a finite group and $\Gamma_j/\kappa_j \to V_j$
is an isomorphism on the closed points. Replacing
$\Gamma_j$ by its image in $U\times_{A^n(X)}(A/\kappa_j)$ and
$A$ by $A/\kappa_j$ we may assume that $\kappa_j$ is trivial,
and thereby that $\Gamma_j \to V_j$ is purely inseparable.
Repeating this construction for the different components of $U$ we finally
reduce to the situation, where $U$ has an open dense subscheme $V$, and
where $V\times_{CH^n(X)_{\deg 0}}A$ has a closed subscheme $\Gamma$ which
is purely inseparable over $V$.
Assume that $\Gamma \to V$ is not an isomorphism, in particular,
that the characteristic of $k$ is $p>0$. The restriction of the
group action to $B\times \Gamma$ factors as
$$
B \times \Gamma \> \cong >> (V\times_{A^n(X)}A)\times_V\Gamma
\> pr_1 >> V\times_{A^n(X)}A
$$
and the preimage $S(\Gamma)$ of $\Gamma \subset V\times_{A^n(X)}A$
is isomorphic to $\Gamma \times_V \Gamma$. Thus $S(\Gamma)$ is a
subscheme of $B\times \Gamma$, supported in the zero section
$\{e\}\times \Gamma$. Hence $S(\Gamma)$ is contained in the
$\nu$-th infinitesimal neighbourhood $\{e\}_\nu \times \Gamma$
of the zero section, for some $\nu > 0$.
The kernel $\kappa^{(\nu')}$ of the $\nu'$-th geometric
Frobenius $F^{(\nu')}: B \to B^{(\nu')}$ is defined by the
sheaf of ideals in ${\mathcal O}_B$, generated by the $p^{\nu'}$-th
powers of the generators of the sheaf of ideals $\rm \bf m$
defining $\{e\} \subset B$.
For some $\nu'>0$ it is contained in ${\rm \bf m}^\nu$
and $\{e\}_\nu$ is a subscheme of $\kappa^{(\nu')}$.
Dividing $A$ by $\kappa^{(\nu')}$, we may assume that
$S(\Gamma)=\Gamma\times_V\Gamma$ is isomorphic to $\Gamma$, and
thereby that $\Gamma$ is isomorphic to $V$.
Independent of the characteristic of $k$, we have thus reduced
to the situation where $U$ has an open dense
subscheme $V$, for which
$$
pr_1:V\times_{CH^n(X)_{\deg 0}}A \to V
$$
has a section, such that on projecting to $A$ we obtain a morphism
$\mu:V\to A$ and (using the notation introduced in \ref{difference})
a commutative diagram
\begin{equation}\label{diag2}
\begin{CD}
\Pi_V \>{\mu^{(-)}}>> A \\
{\mathbb V}\subseteq VV {\mathbb V} V f V\\
\Pi_U \>{\gamma^{(-)}}>>CH^n(X)_{\deg 0}.
\end{CD}
\end{equation}
\begin{claim}\label{claim2}
There exists a surjective homomorphism $\phi:CH^n(X)_{\deg 0} \to A$
with $\mu^{(-)}=\phi \circ \gamma^{(-)}|_{\Pi_V}$. In
particular, $\phi$ is regular.
\end{claim}
\begin{proof}
Let $(C',\iota)$ be an admissible pair with $B=(\iota^{-1}(V))_{\rm reg}$
dense in $C'$. By restriction (\ref{diag2}) gives rise to a
commutative diagram
\begin{equation}\label{diag3}
\begin{CD}
\Pi_B \>{{\mu'}^{(-)}}>> A \\
{\mathbb V}= VV {\mathbb V} V f V\\
\Pi_B \>{\gamma_B^{(-)}}>>CH^n(X)_{\deg 0}
\end{CD}
\end{equation}
where $\mu'=\mu|_B$. By lemma \ref{pic} the lower horizontal
map in the diagram (\ref{diag3}) factors as
\begin{equation}\label{diag4}
\begin{CDS}
\Pi_B \>\gamma_B^{(-)}>> CH^n(X)_{\deg 0}\\
& \SE E \vartheta^{(-)} E {\mathbb A} \eta A A \\
&& {\rm Pic}^0(C').
\end{CDS}
\end{equation}
Let $\Gamma_{C'}^0$ be the connected component of
$\Gamma_{C'}={\rm Pic}^0({C'})\times_{CH^n(X)_{\deg 0}} A$ containing the origin.
$\Gamma_{C'}/\Gamma_{C'}^0$ is a countable group and $\Gamma_{C'}^0 \to
{\rm Pic}^0({C'})$ is an isogeny. Since the diagrams (\ref{diag3}) and
(\ref{diag4}) are commutative, the image of
$$
\Pi_B \> (\vartheta^{(-)}, {\mu'}^{(-)}) >> {\rm Pic}^0({C'})\times A
$$
is contained in $\Gamma_{C'}^0$. This implies that $\Gamma_{C'}^0\to
{\rm Pic}^0(C')$ must be an isomorphism. In fact, by \ref{generators2} there is
an open connected subscheme $W$ of $S^g(B)$ such that the morphism
$\vartheta_{W} : W \to {\rm Pic}^0(C)$ is an open embedding.
On the other hand, $\vartheta_W$ factors through the isogeny
$\Gamma_{C'}^0\to {\rm Pic}^0(C')$.
Hence the morphism ${\mu'}^{(-)}$ in the diagram (\ref{diag3})
is the composite
$$
\Pi_B \> \vartheta^{(-)} >> {\rm Pic}^0(C')\cong \Gamma_{C'}^0 \> pr_2
>> A,
$$
and the condition (b) in lemma \ref{pic} holds true.
Thereby the homomorphism $\phi$ in \ref{claim2} exists, and
it remains to show that $\phi$ is surjective.
Equivalently, it suffices to show that the image of $\phi$ generates $A$
as a group, which will follow if we show that $\mu^{-}(\Pi_V)$ generates
$A$. But we know that $\gamma^{-}(\Pi_V)$ generates $CH^n(X)_{\deg 0}$,
and so $\mu^{-}(\Pi_V)$ generates a subgroup of countable index in $A$.
Since $k$ is a universal domain, $\mu^{-}(\Pi_V)$ generates $A$.
\end{proof}
By claim \ref{claim2} and by the universal property for
$A^n(X)$ the regular homomorphism $\phi: CH^n(X)_{\deg 0}\to A$
factors through a homomorphism of algebraic groups $\chi:A^n(X)\to A$.
Since $\phi$ is surjective, the induced morphism $\chi$ is surjective as
well. Further, the composite $CH^n(X)_{\deg 0}\>{\phi}>> A\> f>>
CH^n(X)_{\deg 0}$ is clearly the identity, since it is so on the image of
$\Pi_V$, which is a set of generators. By the universal property of
$\varphi:CH^n(X)_{\deg 0}\to A^n(X)$, we deduce that the composite
$A^n(X)\>{\chi}>> A\>h>> A^n(X)$ is the identity. Hence $\chi$ and $h$ are
inverse isomorphisms of algebraic groups, and $f:A\to CH^n(X)_{\deg 0}$
and $\varphi:CH^n(X)_{\deg 0}\to A^n(X)$ are both isomorphisms (of groups)
as well.
\end{proof}
\bibliographystyle{plain}
|
1997-12-11T20:58:25 | 9712 | alg-geom/9712012 | en | https://arxiv.org/abs/alg-geom/9712012 | [
"alg-geom",
"math.AG"
] | alg-geom/9712012 | Misha Verbitsky | Misha Verbitsky | Hyperholomorphic sheaves and new examples of hyperkaehler manifolds | 113 pages, v. 2.0, an error in the statement of Theorem 8.15
corrected; Mathematical Physics, 12. International Press, 1999. iv+257 pp.
ISBN: 1-57146-071-3 | null | null | null | http://creativecommons.org/licenses/by/3.0/ | Given a compact hyperkaehler manifold $M$ and a holomorphic bundle B over
$M$, we consider a Hermitian connection $\nabla$ on B which is compatible with
all complex structures on $M$ induced by the hyperkaehler structure. Such a
connection is unique, because it is Yang-Mills. We call the bundles admitting
such connections hyperholomorphic bundles. A stable bundle is hyperholomorphic
if and only if its Chern classes $c_1$, $c_2$ are SU(2)-invariant, with respect
to the natural SU(2)-action on the cohomology. For several years, it was known
that the moduli space of stable hyperholomorphic bundles is singular
hyperkaehler. More recently, it was proven that singular hyperkaehler varieties
admit a canonical hyperkaehler desingularization. In the present paper, we show
that a moduli space of stable hyperholomorphic bundles is compact, given some
assumptions on Chern classes of B and hyperkaehler geometry of $M$ (we also
require $dim_C M>2$). Conjecturally, this leads to new examples of hyperkaehler
manifolds. We develop the theory of hyperholomorphic sheaves, which are
(intuitively speaking) coherent sheaves compatible with hyperkaehler structure.
We show that hyperholomorphic sheaves with isolated singularities can be
canonically desingularized by a blow-up. This theory is used to study
degenerations of hyperholomorphic bundles.
| [
{
"version": "v1",
"created": "Thu, 11 Dec 1997 19:58:43 GMT"
},
{
"version": "v2",
"created": "Sun, 9 Dec 2012 22:41:59 GMT"
}
] | 2012-12-11T00:00:00 | [
[
"Verbitsky",
"Misha",
""
]
] | alg-geom | \section{Introduction}
\label{_intro_Section_}
\hfill
For an introduction to basic results and the history of
hyperk\"ahler geometry, see \cite{_Besse:Einst_Manifo_}.
\hfill
This Introduction is independent from the rest of this paper.
\subsection{An overview}
\subsubsection{Examples of hyperk\"ahler manifolds}
A Riemannian manifold $M$ is called {\bf hyperk\"ahler}
if the tangent bundle of $M$ is equipped with an action of
quaternian algebra, and its metric is K\"ahler with respect to the
complex structures $I_\iota$, for all
embeddings ${\Bbb C} \stackrel{\iota}\hookrightarrow \Bbb H$.
The complex structures $I_\iota$ are called
{\bf induced complex structures}; the corresponding
K\"ahler manifold is denoted by $(M, I_\iota)$.
For a more formal definition of a hyperk\"ahler manifold, see
\ref{_hyperkahler_manifold_Definition_}.
The notion of a hyperk\"ahler manifold was introduced by E. Calabi
(\cite{_Calabi_}).
Clearly, the real dimension of $M$ is divisible by 4.
For $\dim_{\Bbb R} M= 4$, there are only two classes of compact
hyperk\"ahler manifolds: compact tori and K3 surfaces.
Let $M$ be a complex surface and $M^{(n)}$ be its $n$-th
symmetric power, $M^{(n)} = M^n/S_n$. The variety $M^{(n)}$
admits a natural desingularization $M^{[n]}$, called
{\bf the Hilbert scheme of points}.
The manifold $M^{[n]}$ admits a hyperk\"ahler metrics
whenever the surface $M$ is compact and hyperk\"ahler
(\cite{_Beauville_}). This way, Beauville constructed
two series of examples of hyperk\"ahler manifolds,
associated with a torus (so-called
``higher Kummer variety'')
and a K3 surface. It was conjectured that
all compact hyperk\"ahler manifolds $M$ with $H^1(M) =0$, $H^{2,0}(M)={\Bbb C}$
are deformationally equivalent to one of these examples.
In this paper, we study the deformations of coherent sheaves
over higher-dimensional hyperk\"ahler manifolds
in order to construct counterexamples to
this conjecture. A different approach to the
construction of new examples of hyperk\"ahler manifolds
is found in the recent paper of K. O'Grady, who studies
the moduli of semistable bundles over a K3 surface and resolves the
singularities using methods of symplectic geometry (\cite{_O'Grady_}).
\subsubsection{Hyperholomorphic bundles}
Let $M$ be a compact hyperk\"ahler manifold, and $I$ an induced
complex structure. It is well known that the differential
forms and cohomology
of $M$ are equipped with a natural $SU(2)$-action
(\ref{_SU(2)_commu_Laplace_Lemma_}).
In \cite{_Verbitsky:Hyperholo_bundles_}, we studied the
holomorphic vector bundles $F$ on $(M,I)$
which are compatible with a hyperk\"ahler
structure, in the sense that any of the following
conditions hold:
\begin{equation}\label{_hyperho_condi_Equation_}
\begin{minipage}[m]{0.8\linewidth}
\begin{description}
\item[(i)] The bundle $F$ admits a Hermitian connection $\nabla$
with a
curvature $\Theta\in \Lambda^2(M, \operatorname{End}(F))$ which is
of Hodge type (1,1) with respect to any of induced complex
structures.
\item[(ii)] The bundle $F$ is a direct sum of stable bundles,
and its Chern classes $c_1(F)$, $c_2(F)$ are $SU(2)$-invariant.
\end{description}
\end{minipage}
\end{equation}
These conditions are equivalent
(\ref{_inva_then_hyperho_Theorem_}). Moreover, the
connection $\nabla$ of \eqref{_hyperho_condi_Equation_} (i)
is Yang-Mills (\ref{_hyperholo_Yang--Mills_Proposition_}),
and by Uhlenbeck--Yau theorem (\ref{_UY_Theorem_}),
it is unique.
A holomorphic vector bundle satisfying any of the
conditions of \eqref{_hyperho_condi_Equation_}
is called {\bf hyperholomorphic}
(\cite{_Verbitsky:Hyperholo_bundles_}).
Clearly, a stable deformation of a hyperholomorphic bundle
is again a hyperholomorphic bundle. In
\cite{_Verbitsky:Hyperholo_bundles_}, we proved that a
deformation space of hyperholomorphic bundles is a singular
hyperk\"ahler variety. A recent development in the theory
of singular hyperk\"ahler varieties
(\cite{_Verbitsky:Desingu_},
\cite{_Verbitsky:DesinguII_}, \cite{_Verbitsky:hypercomple_})
gave a way to desingularize singular hyperk\"ahler manifolds,
in a canonical way. It was proven (\ref{_desingu_Theorem_})
that a normalization of a singular hyperk\"ahler variety
(taken with respect to any induced complex structure $I$)
is a smooth hyperk\"ahler manifold.
This suggested a possibility of constructing new examples of
compact hyperk\"ahler manifolds, obtained as deformations of
hyperholomorphic bundles. Two problems arise.
\hfill
{\bf Problem 1.} The deformation space
of hyperholomorphic bundles is {\it a priori} non-compact
and must be compactified.
\hfill
{\bf Problem 2.} The geometry of deformation spaces is notoriously
hard to study. Even the dimension of a deformation space
is difficult to compute, in simplest examples. How to
find, for example, the dimension of the deformation
space of a tangent bundle, on a Hilbert scheme of points on a K3
surface? The Betti numbers are even more difficult to compute.
Therefore, there is no easy way to distinguish
a deformation space of hyperholomorphic bundles from already
known examples of hyperk\"ahler manifolds.
\hfill
In this paper, we address Problem 1. Problem 2 can be solved
by studying the algebraic geometry of moduli spaces.
It turns out that, for a generic deformation of a complex
structure, the Hilbert scheme of points on a K3 surface has no
closed complex subvarieties (\cite{_Verbitsky:Hilbert_};
see also \ref{_no_triana_subva_of_Hilb_Theorem_}).
It is possible to find a 21-dimensional family
of deformations of the moduli space $\operatorname{Def}(B)$ of hyperholomorphic
bundles, with all fibers having complex subvarieties
(\ref{_double_Fou_embedding_Lemma_}).
Using this observation, it is possible to show that
$\operatorname{Def}(B)$ is a new example of a hyperk\"ahler manifold.
Details of this approach are given in
Subsection \ref{_new_exa_F-M_checking_Subsection_},
and the complete proofs will be given in a forthcoming paper.
It was proven
that a Hilbert scheme of a generic K3 surface
has no trianalytic subvarieties.\footnote{Trianalytic subvariety
(\ref{_trianalytic_Definition_}) is a closed subset which is
complex analytic with respect to any of induced complex structures.}
Given a hyperk\"ahler manifold $M$ and an appropriate
hyperholomorphic bundle $B$, denote the
deformation space of hyperholomorphic connections
on $B$ by $\operatorname{Def}(B)$. Then the moduli of complex structures
on $M$ are locally embedded to a moduli of complex structures
on $\operatorname{Def}(B)$ (\ref{_maps_pre_tw_curves_Claim_}).
Since the dimension of the moduli of complex
structures on $\operatorname{Def}(B)$ is equal to
its second Betti number minus 2
(\ref{_Bogomo_etale_Theorem_}), the second
Betti number of $\operatorname{Def}(B)$ is no less than the second Betti number
of $M$. The Betti numbers of Beauville's examples of
simple hyperk\"ahler manifolds are 23
(Hilbert scheme of points on a K3 surface)
and 7 (generalized Kummer variety). Therefore,
for $M$ a generic deformation of a
Hilbert scheme of points on K3, $\operatorname{Def}(B)$ is either a new
manifold or a generic deformation of a Hilbert scheme of points on K3.
It is easy to construct trianalytic subvarieties of
the varieties $\operatorname{Def}(B)$, for hyperholomorphic $B$
(see \cite{_Verbitsky:Symplectic_I_},
Appendix for details). This was the motivation of our work on trianalytic
subvarieties of the Hilbert scheme of points on a K3 surface
(\cite{_Verbitsky:Hilbert_}).
For a generic complex structure on a hyperk\"ahler
manifold, all stable bundles are hyperholomorphic
(\cite{_Verbitsky:Symplectic_I_}).
Nethertheless, hyperholomorphic bundles over
higher-dimensional hyperk\"ahler manifolds are in short supply.
In fact, the only example to work with is the
tangent bundle and its tensor powers, and their
Chern classes are not prime.
Therefore, there is no way to insure that
a deformation of a stable bundle will remain
stable (like it happens, for instance, in the case of
deformations of stable bundles
of rank 2 with odd first Chern class
over a K3 surface). Even worse, a new
kind of singularities may appear which never appears
for 2-dimensional base manifolds: a deformation
of a stable bundle can have
a singular reflexization. We study the singularities
of stable coherent sheaves over hyperk\"ahler manifolds,
using Yang-Mills theory for reflexive
sheaves developed by S. Bando and Y.-T. Siu
(\cite{_Bando_Siu_}).
\subsubsection{Hyperholomorphic sheaves}
A compactification of the moduli of hyperholomorphic bundles
is the main purpose of this paper. We require the compactification
to be singular hyperk\"ahler. A natural approach to this problem
requires one to study the coherent sheaves which are
compatible with a hyperk\"ahler structure, in the same
sense as hyperholomorphic bundles are holomorphic
bundles compatible with a hyperk\"ahler structure. Such
sheaves are called {\bf hyperholomorphic sheaves}
(\ref{_hyperho_shea_Definition_}). Our approach to the
theory of hyperholomorphic sheaves uses the notion of
admissible Yang-Mills connection on a coherent sheaf
(\cite{_Bando_Siu_}).
The equivalence of conditions \eqref{_hyperho_condi_Equation_} (i)
and \eqref{_hyperho_condi_Equation_} (ii) is based on Uhlen\-beck--\-Yau
theorem (\ref{_UY_Theorem_}), which states that evey stable bundle
$F$ with $\deg c_1(F) =0$ admits a unique Yang--Mills connection,
that is, a connection $\nabla$ satisfying $\Lambda\nabla^2=0$
(see Subsection \ref{_sta_bu_and_YM_Subsection_} for details).
S. Bando and Y.-T. Siu developed a similar approach to the
Yang--Mills theory on (possibly singular) coherent sheaves.
Consider a coherent sheaf $F$ and a Hermitian metric
$h$ on a locally trivial part of $F\restrict{U}$. Then $h$ is called
admissible (\ref{_admi_metri_Definition_}) if the curvature $\nabla^2$
of the Hermitian connection on $F\restrict{U}$ is square-integrable,
and the section $\Lambda\nabla^2\in \operatorname{End}(F\restrict{U})$ is
uniformly bounded. The admissible metric is called {\bf Yang-Mills}
if $\Lambda\nabla^2=0$ (see \ref{_Yang-Mills_sheaves_Definition_}
for details). There exists an analogue of Uhlenbeck--Yau theorem
for coherent sheaves (\ref{_UY_for_shea_Theorem_}): a stable sheaf
admits a unique admissible Yang--Mills metric, and conversely,
a sheaf admitting a Yang--Mills metric is a direct sum of stable
sheaves with the first Chern class of zero degree.
A coherent sheaf $F$ is called {\bf reflexive} if it is isomorphic
to its second dual sheaf $F^{**}$. The sheaf $F^{**}$ is always
reflexive, and it is called {\bf a reflexization} of $F$
(\ref{_refle_Definition_}).
Applying the arguments of Bando and Siu
to a reflexive coherent sheaf $F$ over a hyperk\"ahler
manifold $(M, I)$, we show that the following
conditions are equivalent (\ref{_hyperho_conne_exi_Theorem_}).
\begin{description}
\item[(i)] The sheaf $F$ is stable and its Chern classes
$c_1(F)$, $c_2(F)$ are $SU(2)$-invariant
\item[(ii)] $F$ admits an admissible Yang--Mills connection,
and its curvature is of type (1,1) with respect to all
induced complex structures.
\end{description}
A reflexive sheaf satisfying any of the these conditions is called
{\bf reflexive stable
hyperholomorphic}. An arbitrary torsion-free coherent sheaf is
called {\bf stable hyperholomorphic} if its reflexization is
hyperholomorphic, and its second Chern class is $SU(2)$-invariant,
and {\bf semistable hyperholomorphic} if it is a successive extension of
stable hyperholomorphic sheaves (see
\ref{_hyperho_shea_Definition_} for details).
This paper is dedicated to the study of hyperholomorphic sheaves.
\subsubsection{Deformations of hyperholomorphic sheaves}
By \ref{_generic_are_dense_Proposition_}, for an induced
complex structure $I$ of general type, {\bf all} coherent sheaves
are hyperholomorphic. However, the complex structures of general
type are never algebraic, and in complex analytic situation,
the moduli of coherent sheaves are, generaly speaking, non-compact.
We study the flat deformations of hyperholomorphic sheaves
over $(M,I)$, where $I$ is an algebraic complex structure.
{\it A priori}, a flat deformation of a hyperholomorphic sheaf will
be no longer hyperholomorphic. We show that
for some algebraic complex structures, called {\bf $C$-restricted
complex structures}, a flat deformation of a hyperholomorphic sheaf
remains hyperholomorphic (\ref{_sheaf_on_C_restr_hyperho_Theorem_}).
This argument is quite convoluted, and takes two sections (Sections
\ref{_cohomo_hype_Section_} and \ref{_C_restri_Section_}).
Further on, we study the local structure of stable
reflexive hyperholomorphic sheaves with isolated singularities.
We prove the Desingularization Theorem for such hyperholomorphic
sheaves (\ref{_desingu_hyperho_Theorem_}). It turns out that
such a sheaf can be desingularized by a single blow-up.
The proof of this result is parallel to the proof of
Desingularization Theorem for singular hyperk\"ahler varieties
(\ref{_desingu_Theorem_}).
The main idea of the desingularization of singular hyperk\"ahler
varieties (\cite{_Verbitsky:DesinguII_}) is the following.
Given a point $x$ on a singular hyperk\"ahler variety $M$
and an induced complex structure $I$, the complex variety
$(M, I)$ admits a local ${\Bbb C}^*$-action which preserves $x$ and
acts as a dilatation on the Zariski tangent space of $x$.
Here we show that any stable hyperholomorphic sheaf $F$ is equivariant
with respect to this ${\Bbb C}^*$-action
(\ref{_Psi_equiv_hyperho_Theorem_},
\ref{_C^*_stru_on_sge_Definition_}).
Then an elementary algebro-geometric argument
(\ref{_desingu_C^*_equi_Proposition_})
implies that $F$ is desingularized by a blow-up.
Using the desingularization of hyperholomorphic sheaves, we prove
that a hyperholomorphic deformation of a hyperholomorphic bundle
is again a bundle (\ref{_reflexi_defo_loca_trivi_Theorem_}),
assuming that it has isolated singularities.
The proof of this result is conceptual but
quite difficult, it takes 3 sections
(Sections
\ref{_twisto_tra_Section_}--\ref{_modu_hyperho_Section_}),
and uses arguments of
quaternionic-K\"ahler geometry (\cite{_Swann_},
\cite{_Nitta:Y-M_}) and twistor transform (\cite{_NHYM_}).
In our study of deformations of hyperholomorphic
sheaves, we usually assume
that a deformation of a hyperholomorphic sheaf over
$(M, I)$ is again hyperholomorphic,
i. e. that an induced complex structure $I$ is $C$-restricted,
for $C$ sufficiently big (\ref{_C_restri_Definition_}).
Since $C$-restrictness is a tricky condition, it is preferable to
get rid of it. For this purpose, we use the theory of twistor
paths, developed in \cite{_coho_announce_}, to show
that the moduli spaces of hyperholomorphic sheaves are
real analytic equivalent for different complex structures
$I$ on $M$ (\ref{_iso_Bun_exists_gene_pola_Theorem_}).
This is done as follows.
A hyperk\"ahler structure on $M$ admits a 2-dimensional sphere of induced
complex structures. This gives a rational curve in the moduli space
$Comp$ of complex structures on $M$, so-called {\bf twistor curve}.
A sequence of such rational curves connect any two points of $Comp$
(\ref{_twistor_connect_Theorem_}).
A sequence of connected twistor curves is called {\bf a twistor
path}. If the intersection points of these curves
are generic, the twistor path is called {\bf admissible}
(\ref{_admi_twi_path_Definition_}). It is known
(\ref{_admi_twi_impli_Theorem_}) that an admissible twistor
path induces a real analytic isomorphism of the moduli spaces
of hyperholomorphic bundles. There exist
admissible twistor paths connecting
any two complex structures (\ref{_admi_pa_exist_for_gene_pol_Claim_}).
Thus, if we prove that the moduli
of deformations of hyperholomorphic bundles
are compact for one generic hyperk\"ahler structure,
we prove a similar result for all generic
hyperk\"ahler structures (\ref{_iso_Bun_exists_gene_pola_Theorem_}).
Applying this argument to the moduli of deformations
of a tangent bundle, we obtain the following theorem.
\hfill
\theorem \label{_defo_tange_Hilb_compact_intro_Theorem_}
Let $M$ be a Hilbert scheme of points on a K3 surface,
$\dim_{\Bbb H}(M)>1$ and $\c H$ a generic
hyperk\"ahler structure on $M$. Assume
that for all induced complex structures $I$, except at most
a finite many of, all semistable bundle deformations of the tangent bundle
$T(M, I)$ are stable. Then, for all complex structures $J$ on $M$ and
all generic polarizations $\omega$
on $(M, J)$, the deformation space
$\c M_{J, \omega}(T(M, J))$ is singular
hyperk\"ahler and compact, and admits
a smooth compact hyperk\"ahler desingularization.
{\bf Proof:} This is \ref{_defo_tange_compact_Theorem_}.
\blacksquare
\hfill
In the course of this paper, we develop the theory of
$C$-restricted complex structures (Sections
\ref{_cohomo_hype_Section_} and \ref{_C_restri_Section_})
and another theory, which we called {\bf the Swann's formalism
for vector bundles} (Sections
\ref{_twisto_tra_Section_} and \ref{_C_equiv_twi_spa_Section_}).
These themes are of independent interest. We give a separate
introduction to $C$-restricted complex structures
(Subsection \ref{_C_restri_intro_Subsection_})
and Swann's formalism (Subsection \ref{_Swann's_intro_Subsection_}).
\subsection{$C$-restricted complex structures: an introduction}
\label{_C_restri_intro_Subsection_}
This part of the Introduction is highly non-precise.
Our purpose is to clarify the intuitive meaning
of $C$-restricted complex structure.
Consider a compact hyperk\"ahler manifold $M$,
which is {\bf simple}
(\ref{_simple_hyperkahler_mfolds_Definition_}),
that is, satisfies $H^1(M) =0$, $H^{2,0}(M) = {\Bbb C}$.
A {\bf reflexive hyperholomorphic sheaf} is by definition a semistable
sheaf which has a filtration of stable sheaves with
$SU(2)$-invariant $c_1$ and $c_2$. A {\bf hyperholomorphic
sheaf} is a torsion-free sheaf which has hyperholomorphic
reflexization and has $SU(2)$-invariant $c_2$
(\ref{_hyperho_shea_Definition_}). If the
complex structure $I$ is of general type,
all coherent sheaves are hyperholomorphic
(\ref{_generic_manifolds_Definition_},
\ref{_generic_are_dense_Proposition_}),
because all integer $(p,p)$-classes are
$SU(2)$-invariant. However, for generic
complex structures $I$, the corresponding
complex manifold $(M, I)$ is never algebraic.
If we wish to compactify the moduli
of holomorphic bundles, we need to consider algebraic
complex structures, and if we want to stay in hyperholomorphic
category, the complex structures must be generic.
This paradox is reconciled by considering the $C$-restricted complex
structures (\ref{_C_restri_Definition_}).
Given a generic hyperk\"ahler structure $\c H$, consider
an algebraic complex structure $I$ with
$Pic(M, I) = {\Bbb Z}$. The group of rational $(p,p)$-cycles
has form
\begin{equation} \label{_Pic_1_decomposi_Equation_}
\begin{split}
H^{p,p}_I(M, {\Bbb Q}) = &H^{2p}(M ,{\Bbb Q})^{SU(2)} \oplus
a \cdot H^{2p}(M ,{\Bbb Q})^{SU(2)} \\
& a^2 \cdot\oplus H^{2p}(M ,{\Bbb Q})^{SU(2)} \oplus...
\end{split}
\end{equation}
where $a$ is a generator of $Pic(M, I)\subset H^{p,p}_I(M, {\Bbb Z})$
and $H^{2p}(M ,{\Bbb Q})^{SU(2)}$ is the group of rational
$SU(2)$-invariant cycles. This decomposition follows
from an explicit description of the algebra of cohomology
given by \ref{_S^*H^2_is_H^*M_intro-Theorem_}.
Let
\[ \Pi:\; H^{p,p}_I(M, {\Bbb Q}){\:\longrightarrow\:} a \cdot
H^{2p}(M ,{\Bbb Q})^{SU(2)}\oplus a^2 \cdot H^{2p}(M ,{\Bbb Q})^{SU(2)}
\oplus ...
\]
be the projection onto non-$SU(2)$-invariant part.
Using Wirtinger's equality, we prove that a fundamental
class $[X]$ of a complex subvariety $X\subset (M, I)$ is $SU(2)$-invariant
unless $\deg \Pi([X])\neq 0$
(\ref{_Wirti_hyperka_Proposition_}). A similar result holds for
the second Chern class of a stable bundle
(\ref{_stable_shea_degree_Corollary_},).
A $C$-restricted complex structure is, heuristically,
a structure for which the decomposition
\eqref{_Pic_1_decomposi_Equation_} folds, and
$\deg a>C$. For a $C$-restricted complex structure $I$,
and a fundamental class $[X]$ of a complex subvariety $X\subset (M, I)$
of complex codimension 2, we have $\deg [X]>C$
or $X$ is trianalytic. A version of Wirtinger's inequality
for vector bundles (\ref{_stable_shea_degree_Corollary_})
implies that a stable vector bundle $B$ over $(M, I)$ is
hyperholomorphic, unless $|\deg c_2(B)| >C$.
Therefore, over a $C$-restricted $(M, I)$,
all torsion-free semistable coherent sheaves
with bounded degree of the second
Chern class are hyperholomorphic
(\ref{_sheaf_on_C_restr_hyperho_Theorem_}).
The utility of $C$-restricted induced complex structures is that
they are algebraic, but behave like generic induced complex
structures with respect to the sheaves $F$ with low
$|\deg c_2(F)|$ and $|\deg c_1(F)|$.
We prove that a generic hyperk\"ahler structure admits
$C$-restricted induced complex structures for all $C$,
and the set of $C$-restricted induced complex structures
is dense in the set of all induced complex structures
(\ref{_C_restri_dense_Theorem_}).
We prove this by studying the algebro-geometric properties
of the moduli of hyperk\"ahler structures on a given
hyperk\"ahler manifold (Subsection
\ref{_modu_and_C-restri_Subsection_}).
\subsection{Quaternionic-K\"ahler manifolds and Swann's formalism}
\label{_Swann's_intro_Subsection_}
Quaternionic-K\"ahler manifolds (Subsection \ref{_B_2_bundles_Subsection_})
are a beautiful
subject of Riemannian geometry. We are interested in these manifolds
because they are intimately connected with singularities of
hyperholomorphic sheaves. A stable hyperholomorphic sheaf
is equipped with a natural connection, which is called
{\bf hyperholomorphic connection}. By definition, a
{hyperholomorphic connection} on a torsion-free coherent sheaf
is a connection $\nabla$ defined outside of singularities of $F$,
with square-integrable curvature $\nabla^2$ which is an $SU(2)$-
invariant 2-form (\ref{_hyperholo_co_Definition_}).
We have shown that a stable hyperholomorphic sheaf admits
a hyperholomorphic connection, and conversely, a
reflexive sheaf admitting a hyperholomorphic connection is a direct
sum of stable hyperholomorphic sheaves
(\ref{_hyperho_conne_exi_Theorem_}).
Consider a reflexive sheaf $F$ over $(M, I)$
with an isolated singularity in $x\in M$. Let $\nabla$ be
a hyperholomorphic connection on $F$. We prove that $F$
can be desingularized by a blow-up of its
singular set. In other words, for $\pi:\; \tilde M {\:\longrightarrow\:} (M, I)$
a blow-up of $x\in M$, the pull-back
$\pi^* F$ is a bundle over $\tilde M$.
Consider the restriction $\pi^* F\restrict C$ of
$\pi^* F$ to the blow-up divisor \[ C = {\Bbb P} T_x M \cong{\Bbb C} P^{2n-1}. \]
To be able to deal with the singularities of $F$ effectively,
we need to prove that the bundle $\pi^* F\restrict C$
is semistable and satisfies $c_1\left(\pi^* F\restrict C\right)=0$.
The following
intuitive picture motivated our work with bundles over
quaternionic-K\"ahler manifolds.
The manifold $C = {\Bbb P} T_x M$ is has a quaternionic structure,
which comes from the $SU(2)$-action on $T_xM$.
We know that bundles which are compatible with
a hyperk\"ahler structure (hyperholomorphic bundles)
are (semi-)stable. If we were able to prove that
the bundle $\pi^* F\restrict C$ is in some way compatible
with quaternionic structure on $C$,
we could hope to prove that it is (semi-)stable.
To give a precise formulation of these heuristic arguments,
we need to work with the theory of quaternionic-K\"ahler manifolds,
developed by Berard-Bergery and Salamon (\cite{_Salamon_}).
A quaternionic-K\"ahler manifold
(\ref{_q-K_Definition_})
is a Riemannian manifold $Q$ equipped with a
bundle $W$ of algebras acting on its tangent bundle, and satisfying
the following conditions. The fibers of $W$ are
(non-canonically) isomorphic
to the quaternion algebra, the map $W\hookrightarrow \operatorname{End}(TQ)$
is compatible with the Levi-Civita connection, and the
unit quaternions $h\in W$ act as orthogonal automorphisms on $TQ$.
For each quaternion-K\"ahler manifold $Q$,
one has a twistor space $\operatorname{Tw}(Q)$ (\ref{_twi_q-K_Definition_}),
which is a total space of a spherical fibration consisting
of all $h\in W$ satisfying $h^2=-1$. The twistor space is a complex
manifold (\cite{_Salamon_}), and it is K\"ahler unless
$W$ is flat, in which case $Q$ is hyperk\"ahler.
Further on, we shall use the term ``quaternionic-K\"ahler''
for manifolds with non-trivial $W$.
Consider the twistor space $\operatorname{Tw}(M)$ of a hyperk\"ahler manifold
$M$, \\ equipped with a natural map \[ \sigma:\;\operatorname{Tw}(M) {\:\longrightarrow\:} M.\]
Let $(B, \nabla)$ be a bundle over $M$ equipped with a
hyperholomorphic connection. A pullback
$(\sigma^* B, \sigma^*\nabla)$
is a holomorphic bundle on $\operatorname{Tw}(M)$
(\ref{_autodua_(1,1)-on-twi_Lemma_}), that is,
the operator $\sigma^*\nabla^{0,1}$ is a holomorphic structure
operator on $\sigma^* B$. This correspondence
is called {\bf the direct twistor transform}. It is
invertible: from a holomorphic bundle
$(\sigma^* B, \sigma^*\nabla^{0,1})$ on $\operatorname{Tw}(M)$
it is possible to reconstruct $(B, \nabla)$, which
is unique (\cite{_NHYM_}; see also \ref{_dire_inve_twisto_Theorem_}).
A similar construction exists on quaternionic-K\"ahler manifolds,
due to T. Nitta (\cite{_Nitta:bundles_}, \cite{_Nitta:Y-M_}).
A bundle $(B, \nabla)$ on a quaternionic-K\"ahler manifold $Q$
is called {\bf a $B_2$-bundle} if its curvature $\nabla^2$ is
invariant with respect to the adjoint action of ${\Bbb H}^*$
on $\Lambda^2(M, \operatorname{End}(B))$ (\ref{_B_2_bu_Definition_}).
An analogue of direct and inverse transform exists
for $B_2$-bundles (\ref{_dire_inve_q-K_Theorem_}).
Most importantly, T. Nitta proved that on a quaternionic-K\"ahler manifold
of positive scalar curvature a twistor transform
of a $B_2$-bundle is a Yang-Mills bundle on
$\operatorname{Tw}(Q)$ (\ref{_twi_tra_YM_q-K_Theorem_}). This implies
that a twistor transform of a Hermitian $B_2$-bundle is a direct
sum of stable bundles with $\deg c_1 =0$.
In the situation described in the beginning of this Subsection,
we have a manifold
$C= {\Bbb P} T_x M \cong {\Bbb C} P ^{2n-1}$ which is a twistor
space of a quaternionic projective space
\[ {\Bbb P}_{\Bbb H}T_x M =
\bigg(T_x M\backslash 0\bigg)/{\Bbb H}^* \cong {\Bbb H} P^n.
\]
To prove that $\pi^* F\restrict C$ is stable,
we need to show that $\pi^* F\restrict C$ is obtained
as twistor transform of some Hermitian $B_2$-bundle
on ${\Bbb P}_{\Bbb H}T_x M$.
This is done using an equivalence between the category of
qua\-ter\-ni\-onic-\-K\"ah\-ler manifolds of positive scalar curvature
and the category of hyperk\"ahler manifolds
equipped with a special type of ${\Bbb H}^*$-action,
constructed by A. Swann (\cite{_Swann_}).
Given a quaternionic-K\"ahler manifold $Q$, we consider
a principal bundle $\c U(Q)$ consisting of all quaternion
frames on $Q$ (\ref{_specia_and_q-K-Subsection_}).
Then $\c U(Q)$ is fibered over $Q$ with a fiber
${\Bbb H}/\{\pm 1\}$. It is easy to show that
$\c U(Q)$ is equipped with an action of quaternion algebra
in its tangent bundle. A. Swann proved that if $Q$ has
with positive scalar curvature, then this action of quaternion
algebra comes from a hyperk\"ahler structure on $\c U(M)$
(\ref{_U(Q)_hyperk_Theorem_}).
The correspondence $Q{\:\longrightarrow\:} \c U(Q)$ induces an equivalence of
appropriately defined categories (\ref{_U(Q)_equiva_cate_Theorem_}).
We call this construction {\bf Swann's formalism}.
The twistor space $\operatorname{Tw}(\c U(Q))$ of the hyperk\"ahler manifold
$\c U(Q)$ is equ\-ipped with a holomorphic action of ${\Bbb C}^*$.
Every $B_2$-bundle corresponds to a ${\Bbb C}^*$-invariant
holomorphic bundle on $\operatorname{Tw}(\c U(Q))$ and this correspondence
induces an equivalence of appropriately defined categories,
called {\bf Swann's formalism for budnles}
(\ref{_B_2_to_C^*_equiva_Theorem_}). Applying this equivalence
to the ${\Bbb C}^*$-equivariant sheaf obtained as an associate graded
sheaf of a hyperholomorphic sheaf, we obtain a $B_2$ bundle on
${\Bbb P}_{\Bbb H}T_x M$, and $\pi^* F\restrict C$ is obtained
from this $B_2$-bundle by a twistor transform.
The correspondence between $B_2$-bundles on $Q$ and
${\Bbb C}^*$-invariant holomorphic bundles on $\operatorname{Tw}(\c U(Q))$
is an interesting geometric phenomenon which is of independent
interest. We construct it by reduction to $\dim Q=0$,
where it follows from an explicit calculation involving
2-forms over a flat manifold of real dimension 4.
\subsection{Contents}
The paper is organized as follows.
\begin{itemize}
\item Section \ref{_intro_Section_} is an introduction.
It is independent from the rest of this apper.
\item Section \ref{_basics_hyperka_Section_} is an introduction
to the theory of hyperk\"ahler manifolds. We give a compenduum of
results from hyperkaehler geometry
which are due to F. Bogomolov (\cite{_Bogomolov_}) and
A. Beauville (\cite{_Beauville_}), and give an introduction
to the results of \cite{_Verbitsky:Hyperholo_bundles_},
\cite{_Verbitsky:hypercomple_}, \cite{_Verbitsky:Symplectic_II_}.
\item Section \ref{_hyperho_shea_Section_} contains a definition
and basic properties of hyperholomorphic sheaves. We prove that
a stable hyperholomorphic sheaf admits a hyperholomorphic connection,
and conversely, a reflexive sheaf admitting a
hyperholomorphic connection is stable hyperholomorphic
(\ref{_hyperho_conne_exi_Theorem_}).
This equivalence is constructed using
Bando-Siu theory of Yang--Mills connections
on coherent sheaves.
We prove an analogue of Wirtinger's inequality for stable sheaves
(\ref{_stable_shea_degree_Corollary_}),
which states that for any
induced complex structure $J\neq \pm I$, and
any stable reflexive sheaf $F$ on
$(M,I)$, we have
\[
\deg_I\left(2c_2(F) - \frac{r-1}{r} c_1(F)^2\right) \geq
\left|\deg_J\left(2c_2(F) - \frac{r-1}{r} c_1(F)^2\right)\right|,
\]
and the equality holds if and only if
$F$ is hyperholomorphic.
\item Section \ref{_cohomo_hype_Section_}
contains the preliminary material used for the study
of $C$-restricted complex structures in Section \ref{_C_restri_Section_}.
We give an exposition of various algebraic structures on
the cohomology of a hyperk\"ahler manifold, which
were discovered in \cite{_so(5)_}
and \cite{_Verbitsky:cohomo_}.
In the last Subsection, we
apply the Wirtinger's inequality to prove that
the fundamental classes of complex subvarieties
and $c_2$ of stable reflexive sheaves satisfy
a certain set of axioms. Cohomology classes satisfying these
axioms are called CA-classes. This definition simplifies
the work on $C$-restricted complex structures in
Section \ref{_C_restri_Section_}.
\item In Section \ref{_C_restri_Section_} we define $C$-restricted
complex structures and prove the following. Consider a compact
hyperk\"ahler manifold and an $SU(2)$-invariant class $a\in H^4(M)$.
Then for all $C$-restricted complex
structures $I$, with $C> \deg a$, and all semistable sheaves
$I$ on $(M, I)$ with $c_2(F) =a$, the sheaf $F$
is hyperholomorphic
(\ref{_sheaf_on_C_restr_hyperho_Theorem_}).
This is used to show that a
deformation of a hyperholomorphic sheaf is again
hyperholomorphic, over $(M,I)$ with $I$
a $C$-restricted complex structure,
$c> \deg c_2(F)$.
We define the moduli space of hyperk\"ahler structures,
and show that for a dense set $\c C$ of hyperk\"ahler
structures, all $\c H \in \c C$ admit a dense set of
$C$-induced complex structures, for all $C\in {\Bbb R}$
(\ref{_C_restri_dense_Theorem_}).
\item In Section \ref{_desingu_she_Section_} we give a proof
of Desingularization Theorem for stable reflexive
hyperholomorphic sheaves with isolated singularities
(\ref{_desingu_hyperho_Theorem_}).
We study the natural ${\Bbb C}^*$-action on a local ring of
a hyperk\"ahler manifold defined in \cite{_Verbitsky:DesinguII_}.
We show that a sheaf $F$ admitting a hyperholomorphic
connection is equivariant with respect to this
${\Bbb C}^*$-action. Then $F$ can be desingularized by a blow-up,
because any ${\Bbb C}^*$-equivariant sheaf with an isolated
singularily can be desingularized by a blow-up
(\ref{_desingu_C^*_equi_Proposition_}).
\item Section \ref{_twisto_tra_Section_} is a primer
on twistor transform and quaternionic-K\"ahler geometry.
We give an exposition of the works of A. Swann
(\cite{_Swann_}), T. Nitta
(\cite{_Nitta:bundles_}, \cite{_Nitta:Y-M_}) on quaternionic-K\"ahler
manifolds and explain the direct and inverse twistor transform
over hyperk\"ahler and qua\-ter\-ni\-onic-\--K\"ah\-ler manifolds.
\item Section \ref{_C_equiv_twi_spa_Section_}
gives a correspondence between $B_2$-bundles
on a qua\-ter\-ni\-onic-\--K\"ah\-ler manifold, and ${\Bbb C}^*$-equivariant
holomorphic bundles on the twistor space of the corresponding
hyperk\"ahler manifold constructed by A. Swann. This is
called ``Swann's formalism for vector bundles''.
We use this correspondence to prove that an associate
graded sheaf of a hyperholomorphic sheaf is equipped with
a natural connection which is compatible with quaternions.
This implies polystability of the bundle $\pi^* F\restrict C$
(see Subsection \ref{_Swann's_intro_Subsection_}).
\item In Section \ref{_modu_hyperho_Section_},
we use the polystability of the bundle $\pi^* F\restrict C$
to show that a hyperholomorphic deformation of
a hyperholomorphic bundle is again a bundle.
Together with results on $C$-restricted complex structures
and Maruyama's compactification (\cite{_Maruyama:Si_}),
this implies that the moduli of semistable bundles
are compact, under conditions of $C$-restrictness and
non-existence of trianalytic subvarieties
(\ref{_space_semista_bu_compa_Theorem_}).
\item In Section \ref{_new_exa_Section_},
we apply these results to the hyperk\"ahler geometry.
Using the desingularization theorem for singular hyperk\"ahler
manifolds (\ref{_desingu_Theorem_}), we prove that the moduli of
stable deformations of a hyperholomorphic bundle has a compact
hyperk\"ahler desingularization
(\ref{_space_sta_bu_compa_hyperka_Theorem_}).
We give an exposition of the theory of twistor
paths, which allows one to identify the categories of stable bundles
for different K\"ahler structures on the same
hyperk\"ahler manifold
(\ref{_admi_twi_impli_Theorem_}).
These results allow one to weaken the conditions
necessary for compactness of the moduli spaces of
vector bundles. Finally, we give a conjectural exposition
of how these results can be used to obtain new examples
of compact hyperk\"ahler manifolds.
\end{itemize}
\section{Hyperk\"ahler manifolds}
\label{_basics_hyperka_Section_}
\subsection{Hyperk\"ahler manifolds}
\label{_hyperka_Subsection_}
This subsection contains a compression of
the basic and best known results
and definitions from hyperk\"ahler geometry, found, for instance, in
\cite{_Besse:Einst_Manifo_} or in \cite{_Beauville_}.
\hfill
\definition \label{_hyperkahler_manifold_Definition_}
(\cite{_Besse:Einst_Manifo_}) A {\bf hyperk\"ahler manifold} is a
Riemannian manifold $M$ endowed with three complex structures $I$, $J$
and $K$, such that the following holds.
\begin{description}
\item[(i)] the metric on $M$ is K\"ahler with respect to these complex
structures and
\item[(ii)] $I$, $J$ and $K$, considered as endomorphisms
of a real tangent bundle, satisfy the relation
$I\circ J=-J\circ I = K$.
\end{description}
\hfill
The notion of a hyperk\"ahler manifold was
introduced by E. Calabi (\cite{_Calabi_}).
\hfill
Clearly, a hyperk\"ahler manifold has a natural action of
the quaternion algebra ${\Bbb H}$ in its real tangent bundle $TM$.
Therefore its complex dimension is even.
For each quaternion $L\in \Bbb H$, $L^2=-1$,
the corresponding automorphism of $TM$ is an almost complex
structure. It is easy to check that this almost
complex structure is integrable (\cite{_Besse:Einst_Manifo_}).
\hfill
\definition \label{_indu_comple_str_Definition_}
Let $M$ be a hyperk\"ahler manifold, and $L$ a quaternion satisfying
$L^2=-1$. The corresponding complex structure on $M$ is called
{\bf an induced complex structure}. The $M$, considered as a K\"ahler
manifold, is denoted by $(M, L)$. In this case,
the hyperk\"ahler structure is called {\bf combatible
with the complex structure $L$}.
Let $M$ be a compact complex variety. We say
that $M$ is {\bf of hyperk\"ahler type}
if $M$ admits a hyperk\"ahler structure
compatible with the complex structure.
\hfill
\hfill
\definition \label{_holomorphi_symple_Definition_}
Let $M$ be a complex manifold and $\Theta$ a closed
holomorphic 2-form over $M$ such that
$\Theta^n=\Theta\wedge\Theta\wedge...$, is
a nowhere degenerate section of a canonical class of $M$
($2n=dim_{\Bbb C}(M)$).
Then $M$ is called {\bf holomorphically symplectic}.
\hfill
Let $M$ be a hyperk\"ahler manifold; denote the
Riemannian form on $M$ by $<\cdot,\cdot>$.
Let the form $\omega_I := <I(\cdot),\cdot>$ be the usual K\"ahler
form which is closed and parallel
(with respect to the Levi-Civitta connection). Analogously defined
forms $\omega_J$ and $\omega_K$ are
also closed and parallel.
A simple linear algebraic
consideration (\cite{_Besse:Einst_Manifo_}) shows that the form
$\Theta:=\omega_J+\sqrt{-1}\omega_K$ is of
type $(2,0)$ and, being closed, this form is also holomorphic.
Also, the form $\Theta$ is nowhere degenerate, as another linear
algebraic argument shows.
It is called {\bf the canonical holomorphic symplectic form
of a manifold M}. Thus, for each hyperk\"ahler manifold $M$,
and an induced complex structure $L$, the underlying complex manifold
$(M,L)$ is holomorphically symplectic. The converse assertion
is also true:
\hfill
\theorem \label{_symplectic_=>_hyperkahler_Proposition_}
(\cite{_Beauville_}, \cite{_Besse:Einst_Manifo_})
Let $M$ be a compact holomorphically
symplectic K\"ahler manifold with the holomorphic symplectic form
$\Theta$, a K\"ahler class
$[\omega]\in H^{1,1}(M)$ and a complex structure $I$.
Let $n=\dim_{\Bbb C} M$. Assume that
$\int_M \omega^n = \int_M (Re \Theta)^n$.
Then there is a unique hyperk\"ahler structure $(I,J,K,(\cdot,\cdot))$
over $M$ such that the cohomology class of the symplectic form
$\omega_I=(\cdot,I\cdot)$ is equal to $[\omega]$ and the
canonical symplectic form $\omega_J+\sqrt{-1}\:\omega_K$ is
equal to $\Theta$.
\hfill
\ref{_symplectic_=>_hyperkahler_Proposition_}
follows from the conjecture of Calabi, pro\-ven by
Yau (\cite{_Yau:Calabi-Yau_}).
\blacksquare
\hfill
Let $M$ be a hyperk\"ahler manifold. We identify the group $SU(2)$
with the group of unitary quaternions. This gives a canonical
action of $SU(2)$ on the tangent bundle, and all its tensor
powers. In particular, we obtain a natural action of $SU(2)$
on the bundle of differential forms.
\hfill
\lemma \label{_SU(2)_commu_Laplace_Lemma_}
The action of $SU(2)$ on differential forms commutes
with the Laplacian.
{\bf Proof:} This is Proposition 1.1
of \cite{_Verbitsky:Symplectic_II_}. \blacksquare
Thus, for compact $M$, we may speak of the natural action of
$SU(2)$ in cohomology.
\hfill
Further in this article, we use the following statement.
\hfill
\lemma \label{_SU(2)_inva_type_p,p_Lemma_}
Let $\omega$ be a differential form over
a hyperk\"ahler manifold $M$. The form $\omega$ is $SU(2)$-invariant
if and only if it is of Hodge type $(p,p)$ with respect to all
induced complex structures on $M$.
{\bf Proof:} This is \cite{_Verbitsky:Hyperholo_bundles_},
Proposition 1.2. \blacksquare
\subsection{Simple hyperk\"ahler manifolds}
\definition \label{_simple_hyperkahler_mfolds_Definition_}
(\cite{_Beauville_}) A connected simply connected
compact hy\-per\-k\"ah\-ler manifold
$M$ is called {\bf simple} if $M$ cannot be represented
as a product of two hyperk\"ahler manifolds:
\[
M\neq M_1\times M_2,\ \text{where} \ dim\; M_1>0 \ \ \text{and}
\ dim\; M_2>0
\]
Bogomolov proved that every compact hyperk\"ahler manifold has a finite
covering which is a product of a compact torus
and several simple hyperk\"ahler manifolds.
Bogomolov's theorem implies the following result
(\cite{_Beauville_}):
\hfill
\theorem\label{_simple_mani_crite_Theorem_}
Let $M$ be a compact hyperk\"ahler manifold.
Then the following conditions are equivalent.
\begin{description}
\item[(i)] $M$ is simple
\item[(ii)] $M$ satisfies $H^1(M, {\Bbb R}) =0$, $H^{2,0}(M) ={\Bbb C}$,
where $H^{2,0}(M)$ is the space of $(2,0)$-classes taken with
respect to any of induced complex structures.
\end{description}
\blacksquare
\subsection{Trianalytic subvarieties in hyperk\"ahler
manifolds.}
In this subsection, we give a definition and basic properties
of trianalytic subvarieties of hyperk\"ahler manifolds.
We follow \cite{_Verbitsky:Symplectic_II_},
\cite{_Verbitsky:DesinguII_}.
\hfill
Let $M$ be a compact hyperk\"ahler manifold, $\dim_{\Bbb R} M =2m$.
\hfill
\definition\label{_trianalytic_Definition_}
Let $N\subset M$ be a closed subset of $M$. Then $N$ is
called {\bf trianalytic} if $N$ is a complex analytic subset
of $(M,L)$ for any induced complex structure $L$.
\hfill
\hfill
Let $I$ be an induced complex structure on $M$,
and $N\subset(M,I)$ be
a closed analytic subvariety of $(M,I)$, $dim_{\Bbb C} N= n$.
Consider the homology class
represented by $N$. Let $[N]\in H^{2m-2n}(M)$ denote
the Poincare dual cohomology class, so called
{\bf fundamental class} of $N$. Recall that
the hyperk\"ahler structure induces the action of
the group $SU(2)$ on the space $H^{2m-2n}(M)$.
\hfill
\theorem\label{_G_M_invariant_implies_trianalytic_Theorem_}
Assume that $[N]\in H^{2m-2n}(M)$ is invariant with respect
to the action of $SU(2)$ on $H^{2m-2n}(M)$. Then $N$ is trianalytic.
{\bf Proof:} This is Theorem 4.1 of
\cite{_Verbitsky:Symplectic_II_}.
\blacksquare
\hfill
The following assertion is the key to the proof of
\ref{_G_M_invariant_implies_trianalytic_Theorem_}
(see \cite{_Verbitsky:Symplectic_II_} for details).
\hfill
\proposition \label{_Wirti_hyperka_Proposition_}
(Wirtinger's inequality)
Let $M$ be a compact hyperk\"ahler manifold,
$I$ an induced complex structure and $X\subset (M, I)$
a closed complex subvariety for complex dimension $k$.
Let $J$ be an induced complex structure, $J \neq \pm I$,
and $\omega_I$, $\omega_J$ the associated K\"ahler forms.
Consider the numbers
\[
\deg_I X:= \int_X \omega_I^k, \ \ \deg_J X:= \int_X \omega_J^k
\]
Then $\deg_I X\geq |\deg_J X|$, and the inequality is strict
unless $X$ is trianalytic.
\blacksquare
\hfill
\remark \label{_triana_dim_div_4_Remark_}
Trianalytic subvarieties have an action of quaternion algebra in
the tangent bundle. In particular,
the real dimension of such subvarieties is divisible by 4.
\hfill
\definition \label{_generic_manifolds_Definition_}
Let $M$ be a complex manifold admitting a hyperk\"ahler
structure $\c H$. We say that $M$ is {\bf of general type}
or {\bf generic} with respect to $\c H$ if all elements of the group
\[ \bigoplus\limits_p H^{p,p}(M)\cap H^{2p}(M,{\Bbb Z})\subset H^*(M)\]
are $SU(2)$-invariant.
We say that $M$ is {\bf Mumford--Tate generic}
if for all $n\in {\Bbb Z}^{>0}$, all the cohomology classes
\[ \alpha \in
\bigoplus\limits_p H^{p,p}(M^n)\cap H^{2p}(M^n,{\Bbb Z})\subset H^*(M^n)
\]
are $SU(2)$-invariant. In other words,
$M$ is Mumford--Tate generic if for all
$n\in {\Bbb Z}^{>0}$, the $n$-th power $M^n$ is
generic. Clearly, Mumford--Tate generic
implies generic.
\hfill
\proposition \label{_generic_are_dense_Proposition_}
Let $M$ be a compact manifold, $\c H$ a hyperk\"ahler
structure on $M$ and $S$
be the set of induced complex structures over $M$. Denote by
$S_0\subset S$ the set of $L\in S$ such that
$(M,L)$ is Mumford-Tate generic with respect to $\c H$.
Then $S_0$ is dense in $S$. Moreover, the complement
$S\backslash S_0$ is countable.
{\bf Proof:} This is Proposition 2.2 from
\cite{_Verbitsky:Symplectic_II_}
\blacksquare
\hfill
\ref{_G_M_invariant_implies_trianalytic_Theorem_} has the following
immediate corollary:
\corollary \label{_hyperkae_embeddings_Corollary_}
Let $M$ be a compact holomorphically symplectic
manifold. Assume that $M$ is of general type with respect to
a hyperk\"ahler structure $\c H$.
Let $S\subset M$ be closed complex analytic
subvariety. Then $S$ is trianalytic
with respect to $\c H$.
\blacksquare
\hfill
In \cite{_Verbitsky:hypercomple_},
\cite{_Verbitsky:Desingu_}, \cite{_Verbitsky:DesinguII_},
we gave a number of equivalent definitions of a singular hyperk\"ahler
and hypercomplex variety. We refer the reader to
\cite{_Verbitsky:DesinguII_} for the precise definition;
for our present purposes it suffices to say that all
trianalytic subvarieties are hyperk\"ahler varieties.
The following Desingularization Theorem is very
useful in the study of trianalytic subvarieties.
\hfill
\theorem \label{_desingu_Theorem_}
(\cite{_Verbitsky:DesinguII_})
Let $M$ be a hyperk\"ahler or a hypercomplex variety,
$I$ an induced complex structure.
Let \[ \widetilde{(M, I)}\stackrel n{\:\longrightarrow\:} (M,I)\]
be the normalization of
$(M,I)$. Then $\widetilde{(M, I)}$ is smooth and
has a natural hyperk\"ahler
structure $\c H$, such that the associated
map $n:\; \widetilde{(M, I)} {\:\longrightarrow\:} (M,I)$ agrees with $\c H$.
Moreover, the hyperk\"ahler
manifold $\tilde M:= \widetilde{(M, I)}$
is independent from the choice of induced complex structure $I$.
\blacksquare
\hfill
Let $M$ be a K3 surface, and $M^{[n]}$ be a Hilbert scheme of points on
$M$. Then $M^{[n]}$ admits a hyperk\"ahler structure
(\cite{_Beauville_}).
In \cite{_Verbitsky:Hilbert_}, we proved the following
theorem.
\hfill
\theorem\label{_no_triana_subva_of_Hilb_Theorem_}
Let $M$ be a complex K3 surface without automorphisms. Assume that
$M$ is Mumford-Tate generic with respect to some hyperka\"hler structure.
Consider the Hilbert scheme $M^{[n]}$ of points on $M$.
Pick a hyperk\"ahler structure on $M^{[n]}$ which is compatible with
the complex structure. Then $M^{[n]}$ has no proper
trianalytic subvarieties.
\blacksquare
\hfill
\subsection{Hyperholomorphic bundles}
\label{_hyperholo_Subsection_}
This subsection contains several versions of a
definition of hyperholomorphic connection in a complex
vector bundle over a hyperk\"ahler manifold.
We follow \cite{_Verbitsky:Hyperholo_bundles_}.
\hfill
Let $B$ be a holomorphic vector bundle over a complex
manifold $M$, $\nabla$ a connection
in $B$ and $\Theta\in\Lambda^2\otimes End(B)$ be its curvature.
This connection
is called {\bf compatible with a holomorphic structure} if
$\nabla_X(\zeta)=0$ for any holomorphic section $\zeta$ and
any antiholomorphic tangent vector field $X\in T^{0,1}(M)$.
If there exists a holomorphic structure compatible with the given
Hermitian connection then this connection is called
{\bf integrable}.
\hfill
One can define a {\bf Hodge decomposition} in the space of differential
forms with coefficients in any complex bundle, in particular,
$End(B)$.
\hfill
\theorem \label{_Newle_Nie_for_bu_Theorem_}
Let $\nabla$ be a Hermitian connection in a complex vector
bundle $B$ over a complex manifold. Then $\nabla$ is integrable
if and only if $\Theta\in\Lambda^{1,1}(M, \operatorname{End}(B))$, where
$\Lambda^{1,1}(M, \operatorname{End}(B))$ denotes the forms of Hodge
type (1,1). Also,
the holomorphic structure compatible with $\nabla$ is unique.
{\bf Proof:} This is Proposition 4.17 of \cite{_Kobayashi_},
Chapter I.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
This result has the following more general version:
\hfill
\proposition \label{_Newle_Nie_for_NH_bu_Proposition_}
Let $\nabla$ be an arbitrary (not necessarily Hermitian)
connection in a complex vector bundle $B$. Then
$\nabla$ is integrable
if and only its $(0,1)$-part has square zero.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
This proposition is a version of Newlander-Nirenberg theorem.
For vector bundles, it was proven by Atiyah and Bott.
\hfill
\definition \label{_hyperho_conne_Definition_}
Let $B$ be a Hermitian vector bundle with
a connection $\nabla$ over a hyperk\"ahler manifold
$M$. Then $\nabla$ is called {\bf hyperholomorphic} if
$\nabla$ is
integrable with respect to each of the complex structures induced
by the hyperk\"ahler structure.
As follows from
\ref{_Newle_Nie_for_bu_Theorem_}, $\nabla$ is hyperholomorphic
if and only if its curvature $\Theta$ is of Hodge type (1,1) with
respect to any of complex structures induced by a hyperk\"ahler
structure.
As follows from \ref{_SU(2)_inva_type_p,p_Lemma_},
$\nabla$ is hyperholomorphic
if and only if $\Theta$ is a $SU(2)$-invariant differential form.
\hfill
\example \label{_tangent_hyperholo_Example_}
(Examples of hyperholomorphic bundles)
\begin{description}
\item[(i)]
Let $M$ be a hyperk\"ahler manifold, and $TM$ be its tangent bundle
equi\-p\-ped with the Levi--Civita connection $\nabla$. Consider a complex
structure on $TM$ induced from the quaternion action. Then $\nabla$
is a Hermitian connection
which is integrable with respect to each induced complex structure,
and hence, is Yang--Mills.
\item[(ii)] For $B$ a hyperholomorphic bundle, all its tensor powers
are also hyperholomorphic.
\item[(iii)] Thus, the bundles of differential forms on a hyperk\"ahler
manifold are also hyperholomorphic.
\end{description}
\subsection{Stable bundles and Yang--Mills connections.}
\label{_sta_bu_and_YM_Subsection_}
This subsection is a compendium of the most
basic results and definitions from the Yang--Mills theory
over K\"ahler manifolds, concluding in the fundamental
theorem of Uhlenbeck--Yau \cite{_Uhle_Yau_}.
\hfill
\definition\label{_degree,slope_destabilising_Definition_}
Let $F$ be a coherent sheaf over
an $n$-dimensional compact K\"ahler manifold $M$. We define
$\deg(F)$ as
\[
\deg(F)=\int_M\frac{ c_1(F)\wedge\omega^{n-1}}{vol(M)}
\]
and $\text{slope}(F)$ as
\[
\text{slope}(F)=\frac{1}{\text{rank}(F)}\cdot \deg(F).
\]
The number $\text{slope}(F)$ depends only on a
cohomology class of $c_1(F)$.
Let $F$ be a coherent sheaf on $M$
and $F'\subset F$ its proper subsheaf. Then $F'$ is
called {\bf destabilizing subsheaf}
if $\text{slope}(F') \geq \text{slope}(F)$
A coherent sheaf $F$ is called {\bf stable}
\footnote{In the sense of Mumford-Takemoto}
if it has no destabilizing subsheaves.
A coherent sheaf $F$ is called {\bf semistable}
if for all destabilizing subsheaves $F'\subset F$,
we have $\text{slope}(F') = \text{slope}(F)$.
\hfill
Later on, we usually consider the bundles $B$ with $deg(B)=0$.
\hfill
Let $M$ be a K\"ahler manifold with a K\"ahler form $\omega$.
For differential forms with coefficients in any vector bundle
there is a Hodge operator $L: \eta{\:\longrightarrow\:}\omega\wedge\eta$.
There is also a fiberwise-adjoint Hodge operator $\Lambda$
(see \cite{_Griffi_Harri_}).
\hfill
\definition \label{Yang-Mills_Definition_}
Let $B$ be a holomorphic bundle over a K\"ahler manifold $M$
with a holomorphic Hermitian connection $\nabla$ and a
curvature $\Theta\in\Lambda^{1,1}\otimes End(B)$.
The Hermitian metric on $B$ and the connection $\nabla$
defined by this metric are called {\bf Yang-Mills} if
\[
\Lambda(\Theta)=constant\cdot \operatorname{Id}\restrict{B},
\]
where $\Lambda$ is a Hodge operator and $\operatorname{Id}\restrict{B}$ is
the identity endomorphism which is a section of $End(B)$.
Further on, we consider only these Yang--Mills connections
for which this constant is zero.
\hfill
A holomorphic bundle is called {\bf indecomposable}
if it cannot be decomposed onto a direct sum
of two or more holomorphic bundles.
\hfill
The following fundamental
theorem provides examples of Yang-\--Mills \linebreak bundles.
\theorem \label{_UY_Theorem_}
(Uhlenbeck-Yau)
Let B be an indecomposable
holomorphic bundle over a compact K\"ahler manifold. Then $B$ admits
a Hermitian Yang-Mills connection if and only if it is stable, and
this connection is unique.
{\bf Proof:} \cite{_Uhle_Yau_}. \blacksquare
\hfill
\proposition \label{_hyperholo_Yang--Mills_Proposition_}
Let $M$ be a hyperk\"ahler manifold, $L$
an induced complex structure and $B$ be a complex vector
bundle over $(M,L)$.
Then every hyperholomorphic connection $\nabla$ in $B$
is Yang-Mills and satisfies $\Lambda(\Theta)=0$
where $\Theta$ is a curvature of $\nabla$.
\hfill
{\bf Proof:} We use the definition of a hyperholomorphic
connection as one with $SU(2)$-invariant curvature.
Then \ref{_hyperholo_Yang--Mills_Proposition_}
follows from the
\hfill
\lemma \label{_Lambda_of_inva_forms_zero_Lemma_}
Let $\Theta\in \Lambda^2(M)$ be a $SU(2)$-invariant
differential 2-form on $M$. Then
$\Lambda_L(\Theta)=0$ for each induced complex structure
$L$.\footnote{By $\Lambda_L$ we understand the Hodge operator
$\Lambda$ associated with the K\"ahler complex structure $L$.}
{\bf Proof:} This is Lemma 2.1 of \cite{_Verbitsky:Hyperholo_bundles_}.
\blacksquare
\hfill
Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced
complex structure.
For any stable holomorphic bundle on $(M, I)$ there exists a unique
Hermitian Yang-Mills connection which, for some bundles,
turns out to be hyperholomorphic. It is possible to tell when
this happens.
\hfill
\theorem \label{_inva_then_hyperho_Theorem_}
Let $B$ be a stable holomorphic bundle over
$(M,I)$, where $M$ is a hyperk\"ahler manifold and $I$
is an induced complex structure over $M$. Then
$B$ admits a compatible hyperholomorphic connection if and only
if the first two Chern classes $c_1(B)$ and $c_2(B)$ are
$SU(2)$-invariant.\footnote{We use \ref{_SU(2)_commu_Laplace_Lemma_}
to speak of action of $SU(2)$ in cohomology of $M$.}
{\bf Proof:} This is Theorem 2.5 of
\cite{_Verbitsky:Hyperholo_bundles_}. \blacksquare
\subsection{Twistor spaces}
Let $M$ be a hyperk\"ahler manifold. Consider the product manifold $X = M
\times S^2$. Embed the sphere $S^2 \subset {\Bbb H}$
into the quaternion algebra
${\Bbb H}$
as the subset of all quaternions $J$ with $J^2 = -1$. For every point
$x = m \times J \in X = M \times S^2$ the tangent space $T_xX$ is
canonically decomposed $T_xX = T_mM \oplus T_JS^2$. Identify $S^2 = {\Bbb C} P^1$
and let $I_J:T_JS^2 \to T_JS^2$ be the complex structure operator. Let
$I_m:T_mM \to T_mM$ be the complex structure on $M$ induced by $J \in S^2
\subset {\Bbb H}$.
The operator $I_x = I_m \oplus I_J:T_xX \to T_xX$ satisfies $I_x \circ I_x =
-1$. It depends smoothly on the point $x$, hence defines an almost complex
structure on $X$. This almost complex structure is known to be integrable
(see \cite{_Salamon_}).
\hfill
\definition\label{_twistor_Definition_}
The complex manifold $\langle X, I_x \rangle$ is called {\it the twistor
space} for the hyperk\"ahler manifold $M$, denoted by $\operatorname{Tw}(M)$.
This manifold is equipped with a real analytic projection
$\sigma:\; \operatorname{Tw}(M){\:\longrightarrow\:} M$ and a complex analytic
projection $\pi:\; \operatorname{Tw}(M) {\:\longrightarrow\:} {\Bbb C} P^1$.
\hfill
The twistor space $\operatorname{Tw}(M)$ is not, generally speaking,
a K\"ahler manifold. For $M$ compact,
it is easy to show that $\operatorname{Tw}(M)$ does not admit
a K\"ahler metric. We consider $\operatorname{Tw}(M)$
as a Hermitian manifold with the product metric.
\hfill
\definition \label{_Li_Yau_condi_Definition_}
Let $X$ be an $n$-dimensional Hermitian manifold and let
$\sqrt{-1}\omega$ be the imaginary part of the metric on $X$. Thus $\omega$
is a real $(1,1)$-form.
Assume that the form $\omega$ satisfies the following condition
of Li and Yau (\cite{_Li_Yau_}).
\begin{equation}\label{_Li_Yau_condi_Equation_}
\omega^{n-2} \wedge d\omega = 0.
\end{equation}
Such Hermitian metrics are called {\bf metrics
satisfying the condition of Li--Yau}.
For a closed real $2$-form $\eta$ let
$$
\deg\eta = \int_X \omega^{n-1} \wedge \eta.
$$
The condition \eqref{_Li_Yau_condi_Equation_}
ensures that $\deg\eta$ depends only on the
cohomology class of $\eta$. Thus it defines a degree functional
$\deg:H^2(X,{\Bbb R}) \to {\Bbb R}$. This functional allows one to repeat verbatim the
Mumford-Takemoto definitions of stable and semistable bundles in this
more general situation. Moreover, the Hermitian Yang-Mills equations also
carry over word-by-word. Li and Yau proved a version of
Uhlenbeck--Yau theorem in this situation
(\cite{_Li_Yau_}; see also \ref{_UY_for_shea_Theorem_}).
\hfill
\proposition
Let $M$ be a hyperk\"ahler manifold and
$\operatorname{Tw}(M)$ its twistor space, considered
as a Hermitian manifold. Then $\operatorname{Tw}(M)$
satisfies the conditions of Li--Yau.
{\bf Proof:} \cite{_NHYM_}, Proposition 4.5.
\blacksquare
\section{Hyperholomorphic sheaves}
\label{_hyperho_shea_Section_}
\subsection{Stable sheaves and Yang-Mills connections}
In \cite{_Bando_Siu_}, S. Bando and Y.-T. Siu developed the
machinery allowing one to apply the methods of Yang-Mills
theory to torsion-free coherent sheaves. In the course of
this paper, we apply their work to generalise the results of
\cite{_Verbitsky:Hyperholo_bundles_}. In this Subsection,
we give a short exposition of their results.
\hfill
\definition\label{_refle_Definition_}
Let $X$ be a complex manifold, and $F$ a coherent sheaf on $X$.
Consider the sheaf $F^*:= \c Hom_{{\cal O}_X}(F, {\cal O}_X)$.
There is a natural functorial map
$\rho_F:\; F {\:\longrightarrow\:} F^{**}$. The sheaf $F^{**}$
is called {\bf a reflexive hull}, or {\bf reflexization}
of $F$. The sheaf $F$ is called {\bf reflexive} if the map
$\rho_F:\; F {\:\longrightarrow\:} F^{**}$ is an isomorphism.
\hfill
\remark
For all coherent sheaves $F$, the map
$\rho_{F^*}:\; F^* {\:\longrightarrow\:} F^{***}$ is an isomorphism
(\cite{_OSS_}, Ch. II, the proof of Lemma 1.1.12).
Therefore, a reflexive hull of a sheaf is always
reflexive.
\hfill
\claim
Let $X$ be a K\"ahler manifold, and $F$ a torsion-free coherent sheaf over
$X$. Then $F$ (semi)stable if and only if $F^{**}$
is (semi)stable.
{\bf Proof:} This is
\cite{_OSS_}, Ch. II, Lemma 1.2.4.
\blacksquare
\hfill
\definition
Let $X$ be a K\"ahler manifold, and $F$ a coherent sheaf over
$X$. The sheaf $F$ is called
{\bf polystable} if $F$ is a direct sum of stable sheaves.
\hfill
The admissible Hermitian metrics, introduced by Bando and Siu
in \cite{_Bando_Siu_}, play the role of the
ordinary Hermitian metrics for vector bundles.
\hfill
Let $X$ be a K\"ahler manifold.
In Hodge theory, one considers the operator
$\Lambda:\; \Lambda^{p, q}(X) {\:\longrightarrow\:}\Lambda^{p-1, q-1}(X)$
acting on differential forms on $X$, which is adjoint to the
multiplication by the K\"ahler form. This operator is defined
on differential forms with coefficient in every bundle.
Considering a curvature $\Theta$ of a bundle $B$
as a 2-form with coefficients in $\operatorname{End}(B)$, we define
the expression $\Lambda\Theta$ which is a section of
$\operatorname{End}(B)$.
\hfill
\definition \label{_admi_metri_Definition_}
Let $X$ be a K\"ahler manifold, and $F$ a reflexive
coherent sheaf over $X$. Let $U\subset X$ be the set of all
points at which $F$ is locally trivial. By definition,
the restriction $F\restrict U$ of $F$ to $U$ is a bundle.
An {\bf admissible metric} on $F$ is a Hermitian metric $h$
on the bundle $F\restrict U$ which satisfies the following
assumptions
\begin{description}
\item[(i)] the curvature $\Theta$ of $(F, h)$ is square integrable, and
\item[(ii)] the corresponding section $\Lambda \Theta\in \operatorname{End}(F\restrict U)$
is uniformly bounded.
\end{description}
\hfill
\definition \label{_Yang-Mills_sheaves_Definition_}
Let $X$ be a K\"ahler manifold, $F$ a reflexive
coherent sheaf over $X$, and $h$ an admissible metric on $F$.
Consider the corresponding Hermitian connection
$\nabla$ on $F\restrict U$. The metric $h$ and
the connection $\nabla$ are called {\bf Yang-Mills}
if its curvature satisfies
\[ \Lambda \Theta\in \operatorname{End}(F\restrict U) = c\cdot \operatorname{\text{\sf id}}
\]
where $c$ is a constant and $\operatorname{\text{\sf id}}$ the unit section
$\operatorname{\text{\sf id}} \in \operatorname{End}(F\restrict U)$.
\hfill
Further in this
paper, we shall only consider Yang-Mills connections
with $\Lambda \Theta=0$.
\hfill
\remark
By Gauss-Bonnet formule, the
constant $c$ is equal to $\deg(F)$, where $\deg(F)$
is the degree of $F$ (\ref{_degree,slope_destabilising_Definition_}).
\hfill
One of the main results of \cite{_Bando_Siu_}
is the following analogue of Uhlenbeck--Yau theorem
(\ref{_UY_Theorem_}).
\hfill
\theorem\label{_UY_for_shea_Theorem_}
Let $M$ be a compact K\"ahler manifold, or a compact
Hermitian manifold satisfying conditions of Li-Yau
(\ref{_Li_Yau_condi_Definition_}), and $F$ a coherent
sheaf without torsion. Then $F$ admits an admissible Yang--Mills
metric is and only if $F$ is polystable. Moreover, if $F$
is stable, then this metric is unique, up to a constant
multiplier.
{\bf Proof:} In \cite{_Bando_Siu_}, \ref{_UY_for_shea_Theorem_}
is proved for K\"ahler $M$ (\cite{_Bando_Siu_}, Theorem 3).
It is easy to adapt this proof for
Hermitian manifolds satisfying conditions of Li--Yau.
\blacksquare
\hfill
\remark
Clearly, the connection, corresponding to a metric on $F$,
does not change when the metric is multiplied by a scalar.
The Yang--Mills metric on a polystable sheaf is unique up to
a componentwise multiplication by scalar multipliers.
Thus, the Yang--Mills connection of \ref{_UY_for_shea_Theorem_}
is unique.
\hfill
Another important theorem of \cite{_Bando_Siu_} is the following.
\hfill
\theorem\label{_YM_can_be_exte_Theorem_}
Let $(F, h)$ be a holomorphic vector bundle with a
Hermitian metric $h$ defined on a K\"ahler manifold
$X$ (not necessary compact nor complete) outside a
closed subset $S$ with locally finite Hausdorff measure of
real co-dimension $4$.
Assume that the curvature tensor
of $F$ is locally square integrable on $X$. Then
$F$ extends to the
whole space $X$ as a reflexive sheaf $\c F$.
Moreover, if the metric $h$ is
Yang-Mills, then $h$ can be smoothly extended
as a Yang-Mills metric over the place where ${\cal F}$ is
locally free.
{\bf Proof:} This is \cite{_Bando_Siu_}, Theorem 2.
\blacksquare
\subsection{Stable and semistable sheaves over
hyperk\"ahler manifolds}
Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced
complex structure, $F$ a torsion-free coherent sheaf
over $(M,I)$ and $F^{**}$ its reflexization. Recall that the cohomology of
$M$ are equipped with a natural $SU(2)$-action
(\ref{_SU(2)_commu_Laplace_Lemma_}). The motivation for
the following definition is \ref{_inva_then_hyperho_Theorem_}
and \ref{_UY_for_shea_Theorem_}.
\hfill
\definition \label{_hyperho_shea_Definition_}
Assume that the first two Chern classes
of the sheaves $F$, $F^{**}$ are $SU(2)$-invariant.
Then $F$ is called {\bf stable hyperholomorphic} if
$F$ is stable, and {\bf semistable hyperholomorphic}
if $F$ can be obtained as a successive
extension of stable hyperholomorphic
sheaves.
\hfill
\remark \label{_slope_hyperho_Remark_}
The slope of a hyperholomorphic sheaf is zero, because
a degree of an $SU(2)$-invariant 2-form
is zero (\ref{_Lambda_of_inva_forms_zero_Lemma_}).
\hfill
\claim \label{_hyperho_suppo_of_F^**/F_Claim_}
Let $F$ be a semistable coherent sheaf over $(M,I)$. Then the following
conditions are equivalent.
\begin{description}
\item[(i)] $F$ is stable hyperholomorphic
\item[(ii)] Consider the support $S$ of the sheaf $F^{**}/F$
as a complex subvariety of $(M,I)$. Let $X_1$, ... , $X_n$ be
the set of irreducible components of $S$ of codimension 2.
Then $X_i$ is trianalytic for all $i$,
and the sheaf $F^{**}$ is stable hyperholomorphic.
\end{description}
{\bf Proof:} Consider an exact sequence
\[ 0 {\:\longrightarrow\:} F {\:\longrightarrow\:} F^{**} {\:\longrightarrow\:} F^{**}/ F{\:\longrightarrow\:} 0. \]
Let $[F / F^{**}]\in H^4(M)$ be the fundamental class
of the union of all co\-di\-men\-sion-2 components of support of
the sheaf $F / F^{**}$, taken with appropriate multiplicities.
Then, $c_2(F^{**}/ F) =- [F / F^{**}]$.
From the product formula for Chern classes,
it follows that
\begin{equation} \label{_c_2(F)_and_F^**_Equation_}
c_2(F)= c_2(F^{**}_i) + c_2(F^{**}/ F)
= c_2(F^{**}_i) - [F / F^{**}].
\end{equation}
Clearly, if all $X_i$ are trianalytic then
the class $[F / F^{**}]$ is $SU(2)$-invariant.
Thus, if the sheaf $F^{**}$ is hyperholomorphic
and all $X_i$ are trianalytic, then the second
Chern class of $F$ is $SU(2)$-invariant, and $F$
is hyperholomorphic. Conversely, assume that
$F$ is hyperholomorphic. We need to show that
all $X_i$ are trianalytic.
By definition,
\[ [F / F^{**}] = \sum_i \lambda_i [X_i]
\]
where $[X_i]$ denotes the fundamental class of $X_i$,
and $\lambda_i$ is the multiplicity of $F / F^{**}$
at $X_i$. By \eqref{_c_2(F)_and_F^**_Equation_},
($F$ hyperholomorphic) implies that the class
$[F / F^{**}]$ is $SU(2)$-invariant. Since
$[F / F^{**}]$ is $SU(2)$-invariant, we have
\[ \sum_i \lambda_i\deg_J(X_i) = \sum_i \lambda_i\deg_I(X_i).
\]
By Wirtinger's inequality
(\ref{_Wirti_hyperka_Proposition_}),
\[ \deg_J(X_i) \leq\deg_I(X_i),
\]
and the equality is reached only if $X_i$ is trianalytic.
By definition, all the numbers $\lambda_i$ are positive.
Therefore,
\[ \sum_i \lambda_i\deg_J(X_i) \leq \sum_i \lambda_i\deg_I(X_i).
\]
and the equality is reached only if all
the subvarieties $X_i$ are trianalytic.
This finishes the proof of
\ref{_hyperho_suppo_of_F^**/F_Claim_}.
\blacksquare
\hfill
\claim
Let $M$ be a compact hyperk\"ahler
manifold, and $I$ an induced complex
structure of general type. Then
all torsion-free coherent sheaves
over $(M, I)$ are semistable hyperholomorphic.
{\bf Proof:} Let $F$ be a torsion-free coherent sheaf
over $(M, I)$. Clearly from the definition
of induced complex
structure of general type, the sheaves
$F$ and $F^{**}$ have $SU(2)$-invariant
Chern classes. Now, all $SU(2)$-invariant
2-forms have degree zero (\ref{_Lambda_of_inva_forms_zero_Lemma_}),
and thus, $F$ is semistable.
\blacksquare
\subsection{Hyperholomorphic connection in
torsion-free sheaves}
Let $M$ be a hyperk\"ahler manifold, $I$ an induced complex
structure, and $F$ a torsion-free sheaf over $(M,I)$.
Consider the natural $SU(2)$-action in the bundle
$\Lambda^i (M,B)$ of the differential $i$-forms
with coefficients in a vector bundle $B$. Let
$\Lambda^i_{inv}(M, B)\subset \Lambda^i (M, B)$
be the bundle of $SU(2)$-invariant $i$-forms.
\hfill
\definition \label{_hyperholo_co_Definition_}
Let $X\subset (M, I)$ be a complex subvariety of
codimension at least 2, such that $F\restrict{M\backslash X}$
is a bundle, $h$ be an admissible metric on
$F\restrict{M\backslash X}$ and $\nabla$ the associated
connection. Then $\nabla$ is called {\bf hyperholomorphic} if its
curvature
\[ \Theta_\nabla =
\nabla^2 \in
\Lambda^2\left(M, \operatorname{End}\left(F\restrict{M\backslash X}\right)\right)
\]
is $SU(2)$-invariant, i. e. belongs to
$\Lambda^2_{inv}\left(M,
\operatorname{End}\left(F\restrict{M\backslash X}\right)\right)$.
\hfill
\claim\label{_singu_triana_Claim_}
The singularities of a hyperholomorphic connection form a trianalytic
subvariety in $M$.
{\bf Proof:} Let $J$ be an induced complex structure on $M$,
and $U$ the set of all points of $(M,I)$ where $F$ is non-singular.
Clearly, $(F, \nabla)$ is a bundle with admissible connection on
$(U,J)$. Therefore, the holomorphic structure on $F\restrict{(U,J)}$
can be extended to $(M,J)$. Thus, the singular set of $F$ is
holomorphic with respect to $J$. This proves
\ref{_singu_triana_Claim_}.
\blacksquare
\hfill
\proposition \label{_conne_=>_hyperho_Proposition_}
Let $M$ be a compact hyperk\"ahler manifold, $I$ an
induced complex structure and $F$
a reflexive sheaf admitting a hyperholomorphic
connection. Then $F$ is a polystable hyperholomorphic sheaf.
\hfill
{\bf Proof:} By \ref{_hyperho_co_YM_Remark_} and
\ref{_UY_for_shea_Theorem_}, $F$ is polystable. We
need only to show that the Chern classes $c_1(F)$ and $c_2(F)$
are $SU(2)$-invariant. Let $U\subset M$ be the maximal
open subset of $M$ such that $F\restrict U$ is locally
trivial. By \ref{_YM_can_be_exte_Theorem_}, the metric $h$
and the connection $\nabla$ can be extended to $U$.
Let $\operatorname{Tw} U\subset \operatorname{Tw} M$ be the corresponding twistor space,
and $\sigma:\; \operatorname{Tw} U {\:\longrightarrow\:} U$ the standard map.
Consider the bundle $\sigma^* F\restrict U$, equipped with
a connection $\sigma^* \nabla$. It is well known \footnote{See
for instance the section
``Direct and inverse twistor transform'' in
\cite{_NHYM_}.}
that $\sigma^* F\restrict U$ is a bundle
with an admissible Yang-Mills metric (we use Yang-Mills in the
sense of Li-Yau, see \ref{_Li_Yau_condi_Definition_}). By
\ref{_YM_can_be_exte_Theorem_}, $\sigma^* F\restrict U$
can be extended to a reflexive sheaf $\c F$ on $\operatorname{Tw} M$.
Clearly, this extension coincides with the push-forward of
$\sigma^* F\restrict U$.
The singular set $\tilde S$ of $\c F$ is a pull-back of the singular set
$S$ of $F$. Thus, $S$ is trianalytic. By desingularization theorem
(\ref{_desingu_Theorem_}),
$S$ can be desingularized to a hyperk\"ahler manifold
in such a way that its twistors form a desingularization of $\c S$.
{}From the exact description of the singularities of $\c S$, provided by the
desingularization theorem, we obtain that the standard projection
$\pi:\; \c S {\:\longrightarrow\:} {\Bbb C} P^1$ is flat.
By the following lemma,
the restriction of $\c F$ to the fiber $(M,I) = \pi^{-1}(\{I\})$
of $\pi$ coincides with $F$.
\hfill
\lemma \label{_exte_flat_Lemma_}
Let $\pi:\; X {\:\longrightarrow\:} Y$ be a map of complex varieties, and
$S\hookrightarrow X$ a subvariety of $X$ of codimension
at least 2, which is flat over $Y$.
Denote by $U\stackrel j \hookrightarrow X$
the complement $U = (X\backslash S)$. Let $F$ be a vector bundle over
$U$, and $j_* F$ its push-forward. Then
the restriction of
$j_* F$ to the fibers of $\pi$ is reflexive.
\hfill
{\bf Proof:} Let $Z= \pi^{-1}(\{y\})$ be a fiber of $\pi$.
Since $S$ is flat over $Y$ and of codimension at least 2, we have
$j_*({\cal O}_{Z \cap U}) = {\cal O}_{Z}$.
Clearly, for an open embedding $\gamma:\; T_1 {\:\longrightarrow\:} T_2$ and coherent sheaves
$A, B$ on $T_1$, we have $\gamma_*(A\otimes B) = \gamma_* A \otimes \gamma_* B$.
Thus, for all coherent sheaves $A$ on $U$, we have
\begin{equation} \label{_j_*_commu_tenso_Equation_}
j_* A \otimes {\cal O}_{Z} = j_*(A \otimes {\cal O}_{Z\cap U}).
\end{equation}
This implies that $j_*(F \restrict Z) = j_* F\restrict Z$.
It is well known (\cite{_OSS_}, Ch. II, 1.1.12; see also
\ref{_normal_refle_Lemma_}) that a push-forward of a
reflexive sheaf under an open embedding $\gamma$ is reflexive, provided that
the complement of the image of $\gamma$ has codimension at least 2.
Therefore, $j_* F\restrict Z$ is a reflexive sheaf over $Z$.
This proves \ref{_exte_flat_Lemma_}. \blacksquare
\hfill
Return to the proof of \ref{_conne_=>_hyperho_Proposition_}.
Consider
the sheaf $\c F$ on the twistor space constructed above.
Since $\c F$ is reflexive, its singularities have
codimension at least 3
(\cite{_OSS_}, Ch. II, 1.1.10).
Therefore, $\c F$ is flat in codimension 2, and
the first two Chern classes of
$F= \c F\restrict{\pi^{-1}(I)}$ can be obtained by restricting
the first two Chern classes of $\c F$ to the subvariety
$(M,I) = \pi^{-1}(I) \subset \operatorname{Tw}(M)$. It remains to show
that such restriction is $SU(2)$-invariant.
Clearly, $H^2((M, I)) = H^2((M, I)\backslash S)$,
and $H^4((M, I)) = H^4((M, I)\backslash S)$.
Therefore,
\[ c_1\left(\c F \restrict {(M,I)}\right)
= c_1\left(\c F \restrict {(M,I)\backslash S}\right)
\]
and
\[ c_2\left(\c F \restrict {(M,I)}\right) =
c_2\left(\c F \restrict {(M,I)\backslash S}\right).
\]
On the other hand, the restriction
$\c F \restrict {\operatorname{Tw}(M)\backslash S}$ is a bundle.
Therefore, the classes
\[
c_1\left(\c F \restrict {(M,I)\backslash S}\right), \ \
c_2\left(\c F \restrict {(M,I)\backslash S}\right)
\]
are independent from $I\in {\Bbb C} P^1$.
On the other hand, these classes are of type $(p,p)$
with respect to all induced complex structures
$I\in {\Bbb C} P^1$. By \ref{_SU(2)_inva_type_p,p_Lemma_},
this implies that the classes $c_1(\c F \restrict {(M,I)})$,
$c_1(\c F \restrict {(M,I)})$ are $SU(2)$-invariant.
As we have shown above, these two classes
are equal to the first Chern classes of $F$.
\ref{_conne_=>_hyperho_Proposition_}
is proven.
\blacksquare
\hfill
\subsection{Existence of hyperholomorphic connections}
The following theorem is the main result of this section.
\hfill
\theorem\label{_hyperho_conne_exi_Theorem_}
Let $M$ be a compact hyperk\"ahler manifold, $I$
an induced complex structure and $F$ a reflexive sheaf on
$(M,I)$. Then $F$ admits a hyperholomorphic connection if
and only if $F$ is polystable hyperholomorphic
in the sense of \ref{_hyperho_shea_Definition_}.
\hfill
\remark \label{_hyperho_co_YM_Remark_}
{}From \ref{_Lambda_of_inva_forms_zero_Lemma_}, it is clear that
a hyperholomorphic connection is always Yang-Mills. Therefore,
such a connection is unique (\ref{_UY_for_shea_Theorem_}).
\hfill
The ``only if'' part of \ref{_hyperho_conne_exi_Theorem_}
is \ref{_conne_=>_hyperho_Proposition_}.
The proof of ``if'' part of
\ref{_hyperho_conne_exi_Theorem_} takes the rest of
this subsection.
\hfill
Let $I$ be an induced complex structure.
We denote the corresponding Hodge decomposition on differential forms
by $\Lambda^*(M)= \oplus \Lambda^{p,q}_I(M)$, and
the standard Hodge operator by
$\Lambda_I:\; \Lambda^{p,q}_I(M) {\:\longrightarrow\:} \Lambda^{p-1,q-1}_I(M)$.
All these structures are defined on the differential forms
with coefficients in a bundle.
Let $\deg_I \eta:= \int_M Tr (\Lambda_I)^k(\eta)$, for
$\eta\in \Lambda^k(M, \operatorname{End} B)$.
The following claim follows from an elementary linear-algebraic
computation.
\hfill
\claim\label{_degree_2-forms-in-End(B)_line-alge_Claim_}
Let $M$ be a hyperk\"ahler manifold, $B$ a Hermitian
vector bundle over $M$, and $\Theta$ a 2-form on $M$
with coefficients in $\frak{su}(B)$. Assume that
\[ \Lambda_I \Theta =0, \ \ \ \Theta\in \Lambda^{1,1}_I(M, \operatorname{End} B)
\]
for some induced complex structure $I$.
Assume, moreover, that $\Theta$ is square-integrable.
Let $J$ be another induced complex structure, $J\neq \pm I$.
Then
\[ \deg_I \Theta^2 \geq |\deg_J \Theta^2|, \]
and the equality is reached only if $\Theta$ is
$SU(2)$-invariant.
{\bf Proof:}
The following general argument is used.
\hfill
\sublemma\label{_deg_coeff_End(B)_Sublemma_}
Let $M$ be a K\"ahler manifold, $B$ a Hermitian
vector bundle over $M$, and $\Xi$ a square-integrable
2-form on $M$ with coefficients in $\frak{su}(B)$.
Then:
\begin{description}
\item[(i)] For
\[ \Lambda_I \Xi =0, \ \ \ \Xi\in \Lambda^{1,1}_I(M, \operatorname{End} B)
\]
we have
\[ \deg_I \Xi^2 = C \int_M|\Xi|^2 \operatorname{Vol} M , \]
where $C= (4\pi^2 n (n-1))^{-1} $ $M$.
\item[(ii)] For
\[ \Xi\in \Lambda^{2,0}_I(M, \operatorname{End} B)\oplus \Lambda^{0,2}_I(M, \operatorname{End} B)
\]
we have
\[ \deg_I \Xi^2 = -C \int_M |\Xi|^2\operatorname{Vol} M, \]
where $C$ is the same constant as appeared in (i).
\end{description}
{\bf Proof:} The proof is based on a linear-algebraic computation
(so-called L\"ubcke-type argument). The same computation is used to
prove Hodge-Riemann bilinear relations.
\blacksquare
\hfill
Return to the proof of \ref{_degree_2-forms-in-End(B)_line-alge_Claim_}.
Let $\Theta= \Theta^{1,1}_J + \Theta^{2,0}_J + \Theta^{0,2}_J$
be the Hodge decomposition associated with $J$.
The following Claim shows that
$\Theta^{1,1}_J$ satisfies conditions of
\ref{_deg_coeff_End(B)_Sublemma_} (i).
\hfill
\claim
Let $M$ be a hyperk\"ahler manifold,
$I$, $L$ induced complex structures and
$\Theta$ a 2-form on $M$ satisfying
\[ \Lambda_I \Theta =0, \ \ \ \Theta\in \Lambda^{1,1}_I(M).
\]
Let $\Theta^{1,1}_L$ be the $(1,1)$-component of $\Theta$
taken with respect to $L$. Then $\Lambda_L \Theta^{1,1}_L =0$.
\hfill
{\bf Proof:} Clearly, $\Lambda_L \Theta^{1,1}_L= \Lambda_L\Theta$.
Consider the natural Hermitian structure
on the space of 2-forms. Since $\Theta$ is of type $(1,1)$ with respect
to $I$, $\Theta$ is fiberwise orthogonal to
the holomorphic symplectic form
$\Omega= \omega_J +\sqrt{-1}\: \Omega_K\in \Lambda^{2,0}_I(M)$.
By the same reason, $\Theta$ is orthogonal to $\bar\Omega$.
Therefore, $\Theta$ is orthogonal to
$\omega_J$ and $\omega_K$. Since $\Lambda_I \Theta =0$,
$\Theta$ is also orthogonal to $\omega_I$. The map
$\Lambda_L$ is a projection to the form $\omega_L$
which is a linear combination of $\omega_I$, $\omega_J$
and $\omega_K$. Since $\Theta$ is fiberwise orthogonal
to $\omega_L$, we have $\Lambda_L \Theta =0$. \blacksquare
\hfill
By
\ref{_deg_coeff_End(B)_Sublemma_},
we have
\[ \deg_J \left(\Theta^{1,1}_J\right)^2 = C \int_M|\Theta^{1,1}_J|^2
\]
and
\[ \deg_J \left(\Theta^{2,0}_J +
\Theta^{0,2}_J\right)^2 = - C \int_M|\Theta^{2,0}_J +
\Theta^{0,2}_J|^2.
\]
Thus,
\[ \deg_J \Theta^2 = C \int_M\left|\Theta^{1,1}_J\right|~2- C \int_M\left|\Theta^{2,0}_J +
\Theta^{0,2}_J\right|~2.
\]
On the other hand,
\[ \deg_I \Theta^2 = C \int_M\left|\Theta\right|~2 =
C \int_M\left|\Theta^{1,1}_J\right|~2 + C \int_M\left|\Theta^{2,0}_J +
\Theta^{0,2}_J\right|~2.
\]
Thus, $\deg_I \Theta^2> |\deg_J \Theta^2|$ unless
$\Theta^{2,0}_J +
\Theta^{0,2}_J=0$. On the other hand,
$\Theta^{2,0}_J +
\Theta^{0,2}_J=0$ means that $\Theta$ is of type $(1,1)$
with respect to $J$. Consider the standard $U(1)$-action on
differential forms associated with the complex structures
$I$ and $J$. These two $U(1)$-actions generate
the whole Lie group $SU(2)$ acting on $\Lambda^2(M)$
(here we use that $I\neq \pm J$). Since $\Theta$ is of type $(1,1)$
with respect to $I$ and $J$, this form is
$SU(2)$-invariant. This proves
\ref{_degree_2-forms-in-End(B)_line-alge_Claim_}.\blacksquare
\hfill
Return to the proof of \ref{_hyperho_conne_exi_Theorem_}.
Let $\nabla$ be the admissible Yang-Mills connection in $F$,
and $\Theta$ its curvature.
Recall that the form $Tr \Theta^2$ represents
the cohomology class
$2c_2(F) - \frac{r-1}{r} c_1(F)^2$, where
$c_i$ are Chern classes of $F$.
Since the form $Tr \Theta^2$ is square-integrable, the integral
\[ \deg_J \Theta^2= \int_M Tr \Theta^2\omega_J^{n-2} \]
makes sense. In \cite{_Bando_Siu_}, it was shown
how to approximate the connection $\nabla$ by smooth
connections, via the heat equation.
This argument, in particular, was used to show that
the value of integrals like
$\int_M Tr \Theta^2\omega_J^{n-2}$
can be computed through cohomology classes
and the Gauss--Bonnet formula
\[ Tr \Theta^2 = 2c_2(F) - \frac{r-1}{r} c_1(F)^2.\]
Since the classes $c_2(F)$, $c_1(F)$ are
$SU(2)$-invariant, we have
\[ \deg_I\Theta^2= \deg_J \Theta^2 \]
for all induced complex structures $I$, $J$.
By \ref{_degree_2-forms-in-End(B)_line-alge_Claim_},
this implies that $\Theta$ is $SU(2)$-invariant.
\ref{_hyperho_conne_exi_Theorem_} is proven.
\blacksquare
\hfill
The same argument implies the following corollary.
\hfill
\corollary\label{_stable_shea_degree_Corollary_}
Let $M$ be a compact hyperk\"ahler manifold, $I$
an induced complex structure, $F$ a stable reflexive sheaf on
$(M,I)$, and $J$ be an induced complex structure,
$J\neq \pm I$. Then
\[
\deg_I\left(2c_2(F) - \frac{r-1}{r} c_1(F)^2\right) \geq
\left|\deg_J\left(2c_2(F) - \frac{r-1}{r} c_1(F)^2\right)\right|,
\]
and the equality holds if and only if
$F$ is hyperholomorphic.
\blacksquare
\subsection{Tensor category of hyperholomorphic sheaves}
This subsection is extraneous. Further on, we do not use the
tensor structure on the category of hyperholomorphic sheaves.
However, we need the canonical idenitification of the categories
of hyperholomorphic sheaves associated with different
induced complex structures.
{}From Bando-Siu (\ref{_UY_for_shea_Theorem_})
it follows that on a compact K\"ahler manifold
a tensor product of stable reflexive sheaves is polystable. Similarly,
\ref{_hyperho_conne_exi_Theorem_} implies that a tensor product of polystable
hyperholomorphic sheaves is polystable hyperholomorphic.
We define the following
category.
\hfill
\definition
Let $M$ be a compace hyperk\"ahler manifold and $I$ an induced complex
structure. Let $\c F_{st}(M,I)$ be a category with objects
reflexive polystable
hyperholomorphic sheaves and morphisms as in category
of coherent sheaves. This category is obviously additive. The tensor product
on $\c F_{st}(M,I)$ is induced from the tensor product of coherent sheaves.
\hfill
\claim
The category $\c F_{st}(M,I)$ is abelian. Moreover, it is a Tannakian tensor
category.
{\bf Proof:} Let $\phi:\; F_1 {\:\longrightarrow\:} F_2$
be a morphism of hyperholomorphi sheaves. In
\ref{_degree,slope_destabilising_Definition_} , we introduced
{\bf a slope} of a coherent sheaf. Clearly,
$sl(F_1)\leq sl(\operatorname{im} \phi)\leq sl(F_2)$.
All hyperholomorphic sheaves have slope 0 by
\ref{_slope_hyperho_Remark_}. Thus,
$sl(\operatorname{im} \phi)=0$ and the subsheaf $\operatorname{im}\phi\subset F_2$ is
destabilizing. Since $F_2$ is polystable, this sheaf is decomposed:
\[ F_2 = \operatorname{im} \phi \oplus \operatorname{coker} \phi.\]
A similar argument proves that
$F_1 = \ker\phi \oplus \operatorname{coim} \phi$, with all summands hyperholomorphic.
This proves that $\c F_{st}(M,I)$ is abelian. The Tannakian properties
are clear. \blacksquare
\hfill
The category $\c F_{st}(M,I)$ does not depend from the choice of
induced complex structure $I$:
\hfill
\theorem\label{_equi_cate_Theorem_}
Let $M$ be a compact hyperk\"ahler
manifold, $I_1$, $I_2$ induced complex structures and
$\c F_{st}(M,I_1)$, $\c F_{st}(M,I_2)$ the associated categories
of polystable reflexive hyperholomorphic sheaves.
Then, there exists a natural equivalence of tensor categories
\[
\Phi_{I_1, I_2}:\; \c F_{st}(M,I_1){\:\longrightarrow\:} \c F_{st}(M,I_2).
\]
{\bf Proof:} Let $F\in \c F_{st}(M,I_1)$ be a reflexive
polystable hyperholomorphic sheaf
and $\nabla$ the canonical admissible Yang-Mills connection.
Consider the sheaf $\c F$ on the twistor space $\operatorname{Tw}(M)$ constructed
as in the proof of \ref{_conne_=>_hyperho_Proposition_}.
Restricting $\c F$ to $\pi^{-1}(I_2)\subset \operatorname{Tw}(M)$,
we obtain a coherent sheaf $F'$ on $(M,I_2)$. As we have shown in
the proof of \ref{_conne_=>_hyperho_Proposition_},
the sheaf $(F')^{**}$ is polystable hyperholomorphic.
Let $\Phi_{I_1, I_2}(F):= (F')^{**}$.
It is easy to check that thus constructed map
of objects gives a functor
\[
\Phi_{I_1, I_2}:\; \c F_{st}(M,I_1){\:\longrightarrow\:} \c F_{st}(M,I_2),
\]
and moreover, $\Phi_{I_1, I_2}\circ \Phi_{I_2, I_1} = Id$.
This shows that $\Phi_{I_1, I_2}$ is an equivalence.
\ref{_equi_cate_Theorem_} is proven. \blacksquare
\hfill
\definition \label{_hyperho_shea_on_M_Definition_}
By \ref{_equi_cate_Theorem_}, the category
$\c F_{st}(M,I_1)$ is independent from the choice of
induced complex structure. We call this category
{\bf the category of polystable hyperholomorphic reflexive
sheaves on $M$} and denote it by $\c F(M)$.
The objects of $\c F(M)$ are called {\bf hyperholomorphic sheaves on
$M$}. For a hyperholomorphic sheaf on $M$,
we denote by $F_I$ the corresponding
sheaf from $\c F_{st}(M,I_1)$.
\hfill
\remark
Using the same argument as proves \ref{_admi_twi_impli_Theorem_} (ii),
it is easy to check that
the category $\c F(M)$ is a deformational invariant of $M$. That is,
for two hyperk\"ahler manifolds $M_1$, $M_2$ which are
deformationally equivalent, the categories $\c F(M_i)$ are
also equivalent, assuming that $Pic(M_1)= Pic(M_2)=0$.
The proof of this result is essentially contained in
\cite{_coho_announce_}.
\hfill
\remark
As Deligne proved (\cite{_Deli:Tanna_}),
for a each Tannakian category $\c C$ equipped with a fiber functor,
there exists a natural pro-algebraic group $G$ such that
$\c C$ is a group of representations of $G$. For $\c F(M)$, there
are several natural fiber functors. The simplest one is defined for
each induced complex structure $I$ such that $(M,I)$ is algebraic
(such complex structures always exist, as proven in
\cite{_Fujiki_}; see also \cite{_Verb:alge_} and
Subsection \ref{_alge_indu_Subsection_}).
Let $\c K(M,I)$ is the space of rational functions on $(M,I)$.
For $F\in \c F_{st}(M, I)$, consider the functor
$F{\:\longrightarrow\:} \eta_I(F)$, where $\eta_I(F)$ is
the space of global sections of $F\otimes \c K(M,I)$.
This is clearly a fiber functor, which associates
to $\c F(M)$ the group $G_I$. The corresponding pro-algebraic
group $G_I$ is a deformational, that is, topological,
invariant of the hyperk\"ahler
manifold.
\section{Cohomology of hyperk\"ahler manifolds}
\label{_cohomo_hype_Section_}
This section contains a serie of preliminary results which
are used further on to define and study the $C$-restricted
complex structures.
\subsection{Algebraic induced complex structures}
\label{_alge_indu_Subsection_}
This subsection contains a recapitulation of results
of \cite{_Verb:alge_}.
\hfill
A more general version of the
following theorem was proven by
A. Fujiki (\cite{_Fujiki_}, Theorem 4.8 (2)).
\hfill
\theorem \label{_alge_dense_Theorem_}
Let $M$ be a compact simple hyperk\"ahler manifold and
$\c R$ be the set of induced complex structures
$\c R \cong {\Bbb C} P^1$. Let $\c R_{alg}\subset \c R$
be the set of all algebraic induced complex structures.
Then $\c R_{alg}$ is countable and dense in $\c R$.
{\bf Proof:} This is \cite{_Verb:alge_}, Theorem 2.2. \blacksquare
\hfill
In the proof of \ref{_alge_dense_Theorem_}, the following
important lemma was used.
\hfill
\lemma \label{_K_proje_on_R_Lemma_}
\begin{description}
\item [(i)] Let ${\cal O}\subset H^2(M, {\Bbb R})$ be the set of all
cohomology classes
which are K\"ahler with respect to some induced comples structure.
Then ${\cal O}$ is open in $H^2(M, {\Bbb R})$. Moreover, for all
$\omega\in {\cal O}$, the class $\omega$ is {\it not} $SU(2)$-invariant.
\item[(ii)]
Let $\eta\in H^2(M, {\Bbb R})$
be a cohomology class which is not $SU(2)$-invariant. Then
there exists a unique up to a sign induced complex structure
$I\in \c R/\{\pm 1\}$ such that $\eta$ belongs to $H^{1,1}_I(M)$.
\end{description}
{\bf Proof:} This statement is a form of
\cite{_Verb:alge_}, Lemma 2.3. \blacksquare
\subsection{The action of $\frak{so}(5)$ on the
cohomology of a hyperk\"ahler manifold}
\label{_so(5)_Subsection_}
This subsection is a recollection of data from
\cite{_so(5)_} and \cite{_Verbitsky:Symplectic_II_}.
\hfill
Let $M$ be a hyperk\"ahler manifold. For an induced complex structure
$R$ over $M$, consider the K\"ahler form
$\omega_R=(\cdot,R\cdot)$, where $(\cdot,\cdot)$ is the Riemannian form.
As usually, $L_R$ denotes the operator of exterior
multiplication by $\omega_R$, which is acting on the differential
forms $A^*(M,{\Bbb C})$ over $M$. Consider the adjoint operator to $L_R$,
denoted by $\Lambda_R$.
One may ask oneself, what algebra is generated by $L_R$ and
$\Lambda_R$ for all induced complex structures $R$? The answer was given
in \cite{_so(5)_}, where the following theorem was proven.
\hfill
\theorem
(\cite{_so(5)_}) Let $M, \c H$ be a hyperk\"ahler manifold,
and $\frak{a}_{\c H}$ be a Lie algebra generated by $L_R$ and $\Lambda_R$
for all induced complex structures $R$ over $M$. Then the
Lie algebra $\frak{a}_{\c H}$ is isomorphic to $\frak{so}(4,1)$.
\blacksquare
\hfill
The following facts about a structure of $\frak{a}_{\c H}$
were also proven in \cite{_so(5)_}.
Let $I$, $J$ and $K$ be three induced complex structures on
$M$, such that $I\circ J=-J\circ I=K$. For an induced complex structure
$R$, consider an operator $ad R$ on cohomology, acting on $(p,q)$-forms
as a multiplication by $(p-q)\sqrt{-1}\:$. The operators $ad R$ generate a 3-dimensional
Lie algebra ${\frak g}_{\c H}$, which is isomorphic to $\frak{su}(2)$.
This algebra coincides with the Lie algebra associated to the
standard $SU(2)$-action on $H^*(M)$.
The algebra $\frak{a}_{\c H}$ contains
${\frak g}_{\c H}$ as a subalgebra, as follows:
\begin{equation}\label{_ad_I_as_commu_Equation_}
[\Lambda_J,L_K]=[L_J,\Lambda_K]= ad\: I\;\; \text{(etc)}.
\end{equation}
The algebra $\frak{a}_{\c H}$ is 10-dimensional.
It has the following basis: $L_R,\Lambda_R$, $ad\: R$
$(R=I,J,K)$ and the element $H=[L_R,\Lambda_R]$.
The operator $H$ is a standard Hodge operator; it acts on $r$-forms
over $M$ as multiplication by a scalar $n-r$,
where $n=dim_{\Bbb C} M$.
\hfill
\definition \label{_isoty_Definition_}
Let ${\frak g}$ be a semisimple Lie algebra, $V$ its representation
and $V= \oplus V_{\alpha}$ a ${\frak g}$-invariant decompostion of $V$,
such that for all $\alpha$, $V_\alpha$ is a direct sum of
isomorphic finite-dimensional representations $W_\alpha$ of $V$, and
all $W_\alpha$ are distinct. Then the decomposition $V= \oplus V_{\alpha}$
is called {\bf the isotypic decomposition of $V$}.
\hfill
It is clear that for all
finite-dimensional representations, isotypic decomposition always
exists and is unique.
\hfill
Let $M$ be a compact hyperk\"ahler manifold.
Consider the cohomology space $H^*(M)$ equipped with the natural
action of $\frak{a}_{\c H}=
\frak{so}(5)$. Let $H_o^*\subset H^*(M)$ be the isotypic
component containing $H^0(M)\subset H^*(M)$.
Using the root system written explicitly for
$\frak{a}_{\c H}$ in \cite{_so(5)_}, \cite{_Verbitsky:cohomo_},
it is easy to check that $H^*_o(M)$ is an irreducible
representation of $\frak{so}(5)$. Let
$p:\; H^*(M) {\:\longrightarrow\:} H^*_o(M)$ be the unique
$\frak{so}(5)$-invariant projection, and
$i:\; H^*_o(M) \hookrightarrow H^*(M)$ the natural embedding.
\hfill
Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced complex
structure, and $\omega_I$ the corresponding K\"ahler form.
Consider the {\bf degree
map} $\deg_I:\; H^{2p}(M) {\:\longrightarrow\:} {\Bbb C}$,
$\eta{\:\longrightarrow\:} \int_M \eta \wedge \omega_I^{n-p}$,
where $n= \dim_C M$.
\hfill
\proposition\label{_degree_isotypic_Proposition_}
The space \[ H^*_o(M)\subset H^*(M)\] is a subalgebra of $H^*(M)$,
which is invariant under the $SU(2)$-action. Moreover,
for all induced complex structures $I$, the degree map
\[\deg_I:\; H^*(M) {\:\longrightarrow\:} {\Bbb C}\] satisfies
\[ \deg_I(\eta) =\deg_I(i(p(\eta)),
\]
where $i:\; H^*_o(M) \hookrightarrow H^*(M)$, $p:\; H^*(M) {\:\longrightarrow\:} H^*_o(M)$
are the $\frak{so}(5)$-invariant maps defined above.
And finally, the projection $p:\; H^*(M) {\:\longrightarrow\:} H^*_o(M)$
is $SU(2)$-invariant.
\hfill
{\bf Proof:} The space $H^*_o(M)$ is generated from
$\pmb 1\in H^0(M)$ by operators $L_R$, $\Lambda_R$.
To prove that $H^*_o(M)$ is closed under multiplication,
we have to show that $H^*_o(M)$ is generated (as a linear space)
by expressions of type $L_{r_1} \circ L_{R_2} \circ ... \circ \pmb 1$.
By \eqref{_ad_I_as_commu_Equation_},
the commutators of $L_R$, $\Lambda_R$ map such expressions
to linear combinations of such expressions. On the other hand,
the operators $\Lambda_R$ map $\pmb 1$ to zero. Thus,
the operators $\Lambda_R$ map expressions of type
$L_{r_1} \circ L_{R_2} \circ ... \circ \pmb 1$ to linear combinations
of such expressions. This proves that $H^*_o(M)$ is closed under multiplication.
The second statement of \ref{_degree_isotypic_Proposition_}
is clear (see, e. g. \cite{_Verbitsky:Symplectic_II_},
proof of Proposition 4.5).
It remains to show that $H^*_o(M)\subset H^*(M)$ is an $SU(2)$-invariant
subspace and that $p:\; H^*(M) {\:\longrightarrow\:} H^*_o(M)$
is compatible with the $SU(2)$-action. From
\eqref{_ad_I_as_commu_Equation_}, we obtain that
the Lie group $G_A$ associated with $\frak a_{\c H}\cong \frak{so}(1,4)$
contains $SU(2)$ acting in a standard way
on $H^*(M)$. Since the map $p:\; H^*(M) {\:\longrightarrow\:} H^*_o(M)$
commutes with $G_A$-action, $p$ also commutes with $SU(2)$-action.
We proved \ref{_degree_isotypic_Proposition_}.
\blacksquare
\subsection{Structure of the cohomology ring}
\label{_cohomo_stru_Subsection_}
In \cite{_Verbitsky:cohomo_} (see also \cite{_coho_announce_}),
we have computed explicitly the subalgebra of cohomology of $M$ generated
by $H^2(M)$. This computation can be summed up as follows.
\hfill
\theorem \label{_S^*H^2_is_H^*M_intro-Theorem_}
(\cite{_Verbitsky:cohomo_}, Theorem 15.2)
Let $M$ be a compact hyperk\"ahler manifold, $H^1(M)=0$,
$\dim_{\Bbb C} M=2n$, and $H^*_r(M)$ the subalgebra of cohomology of $M$ generated
by $H^2(M)$.
Then
\[\bigg\{\begin{array}{lr}
H^{2i}_r(M)\cong S^i H^2(M)&
\mbox{\ \ for $i\leq n$, and}\\
H^{2i}_r(M)\cong S^{2n-i} H^2(M) &
\mbox{\ \ for $i\geq n$ }
\end{array}
\]
\blacksquare
\hfill
\theorem\label{_gene_all_SU(2)_Theorem_}
Let $M$ be a simple hyperk\"ahler manifold.
Consider the group $G$ generated by a union of all $SU(2)$ for
all hyperk\"ahler structures on $M$. Then the Lie algebra of
$G$ is isomorphic to $\frak{so}(H^2(M))$,
for a certain natural integer bilinear symmetric form on $H^2(M)$,
called Bogomolov-Beauville form.
{\bf Proof:} \cite{_Verbitsky:cohomo_} (see also \cite{_coho_announce_}).
\blacksquare
\hfill
The key element in the proof of
\ref{_S^*H^2_is_H^*M_intro-Theorem_} and \ref{_gene_all_SU(2)_Theorem_}
is the following algebraic computation.
\hfill
\theorem \label{_r_proper_Theorem_}
Let $M$ be a simple hyperk\"ahler manifold,
and $\c H$ a hyperk\"ahler structure on $M$.
Consider
the Lie subalgebra
\[ {\frak a}_{\c H}\subset \operatorname{End}(H^*(M)),
\ \ {\frak a}_{\c H} \cong \frak{so}(1,4),
\]
associated with the hyperk\"ahler structure
(Sub\-sec\-tion \ref{_so(5)_Subsection_}). Let
\[ {\frak g}\subset \operatorname{End}(H^*(M)) \]
be the Lie algebra generated by subalgebras
${\frak a}_{\c H}\subset \operatorname{End}(H^*(M))$,
for all hyperk\"aher structures $\c H$ on $M$.
Then
\begin{description}
\item[(i)] The algebra ${\frak g}$ is naturally isomorphic to
the Lie algebra $\frak{so}(V\oplus \frak H)$,
where $V$ is the linear space $H^2(M, {\Bbb R})$
equipped with the Bogomolov--Beauville pairing,
and $\frak H$ is a 2-dimensional vector space
with a quad\-ra\-tic form of signature $(1, -1)$.
\item[(ii)]
The space $H_r^*(M)$ is invariant under the action of ${\frak g}$,
Moreover, \[ H_r^*(M)\subset H^*(M)\] is an isotypic
\footnote{See \ref{_isoty_Definition_} for the definition of
isotypic decomposition.} component of
the space $H^*(M)$ considered as a representation of ${\frak g}$.
\end{description}
{\bf Proof:} \cite{_Verbitsky:cohomo_} (see also \cite{_coho_announce_}).
\blacksquare
\hfill
As one of the consequences of
\ref{_S^*H^2_is_H^*M_intro-Theorem_}, we obtain the
following lemma, which will be used further on in this paper.
\hfill
\lemma\label{_p_multi_Lemma_}
Let $M$ be a simple hyperk\"ahler manifold,
$\dim_{\Bbb H} M =n$, and $p:\; H^*(M) {\:\longrightarrow\:} H^*_o(M)$ the
map defined in Subsection \ref{_so(5)_Subsection_}.
Then, for all $x, y \in H^*_r(M)$, we have
\[ p(x) p(y) = p(xy), \text{\ \ whenever\ \ }
xy \in \bigoplus\limits_{i\leq 2n} H^i(M).
\]
{\bf Proof:} Let $\omega_I$, $\omega_J$,
$\omega_K$,
$x_1$, ..., $x_n$ be an orthonormal basis in $H^2(M)$.
Clearly, the vectors $x_1$, ..., $x_k$ are $SU(2)$-invariant.
Therefore, these vectors are highest vectors of the corresponding
${\frak a}_{\c H}$-representations, with respect
to the root system and Cartan subalgebra for ${\frak a}_{\c H}$
which is written in \cite{_so(5)_} or \cite{_Verbitsky:cohomo_}.
We obtain that the monomials
\[ P_{k_1, k_2, k_3, \{n_i\} }
= \omega_I^{k_1} \omega_J^{k_2} \omega_K^{k_3} \prod x_i^{n_i},
\sum n_i = N, \ \ P_{k_1, k_2, k_3, \{n_i\} }
\in \bigoplus\limits_{i\leq 2n} H^i(M)
\]
belong to the different isotypical components for different $N$'s.
By \ref{_S^*H^2_is_H^*M_intro-Theorem_}, a product of
two such monomials $P_{k_1, k_2, k_3, \{n_i\} }$ and
$P_{k_1', k_2', k_3', \{n_i'\} }$ is equal to
$P_{k_1+k_1', k_2+k_2', k_3+k_3', \{n_i+n_i'\} }$,
assuming that
\[ P_{k_1, k_2, k_3, \{n_i\} }
P_{k_1', k_2', k_3', \{n_i'\} }\in \bigoplus\limits_{i\leq 2n} H^i(M).
\]
Thus,
the isotypical decomposition associated with the
$\frak{a}_{\c H}$-action is compatible
with multiplicative structure on $H^*(M)$,
for low-dimensional cycles.
This implies \ref{_p_multi_Lemma_}.
\blacksquare
\hfill
We shall use the following corollary of
\ref{_p_multi_Lemma_}.
\hfill
\corollary \label{_p_prods_H^2_Corollary_}
Let $M$ be a simple hyperk\"ahler manifold,
$\dim_{\Bbb H} M>1$, and $\omega_1, \omega_2 \in H^2(M)$
cohomology classes which are $SU(2)$-invariant.
Then, for all induced complex structures $I$, we have
$\deg_I(\omega_1 \omega_2) =0$.
{\bf Proof:} By definition, the classes $\omega_1, \omega_2$
satisfy $\omega_i\in \ker p$.
By \ref{_p_multi_Lemma_}, we have
$\omega_1 \omega_2\in \ker p$. By
\ref{_degree_isotypic_Proposition_},
$\deg_I\omega_1 \omega_2 =0$.
\blacksquare
\hfill
Let $\omega$ be a rational K\"ahler form. The corresponding
$\frak{sl}(2)$-action on $H^*(M)$ is clearly compatible with the
rational structure on $H^*(M)$. It is easy to see (using, for instance,
\ref{_K_proje_on_R_Lemma_}) that ${\frak g}$ is generated by
$\frak{sl}(2)$-triples associated with rational K\"ahler forms
$\omega$. Therefore,
the action of ${\frak g}$ on $H^*(M)$ is compatible with the rational structure
on $H^*(M)$. Using the isotypic decomposition, we define
a natural ${\frak g}$-invariant map $r:\; H^*(M) {\:\longrightarrow\:} H^*_r(M)$.
Further on, we shall use the following properties of this map.
\hfill
\claim \label{_r_H^*_to_H_r_Claim_}
\begin{description}
\item[(i)] The map $r:\; H^*(M) {\:\longrightarrow\:} H^*_r(M)$
is compatible with the rational structure on $H^*(M)$.
\item[(ii)] For every $x\in \ker r$, and every hyperk\"ahler
structure $\c H$, the corresponding map
$p:\; H^*(M) {\:\longrightarrow\:} H^*_o(M)$ satisfies $p(x)=0$.
\item[(iii)] For every $x\in \ker r$, every hyperk\"ahler
structure $\c H$, and every induced complex
structure $I$ on $M$, we have $\deg_I x =0$.
\end{description}
{\bf Proof:}
\ref{_r_H^*_to_H_r_Claim_} (i) is clear, because
the action of ${\frak g}$ on $H^*(M)$ is compatible with the rational structure
on $H^*(M)$.
To prove \ref{_r_H^*_to_H_r_Claim_} (ii), we notice that
the space $H^*_r(M)$ is generated from $H^0(M)$ by
the action of ${\frak g}$, and $H^*_o(M)$ is generated from $H^0(M)$ by
the action of ${\frak a}_{\c H}$. Since ${\frak a}_{\c H}$ is by definition a
subalgebra in ${\frak g}$, we have $H^*_o(M) \subset H^*_r(M)$.
The isotypic projection
$r:\; H^*(M) {\:\longrightarrow\:} H^*_r(M)$ is by definition compatible with
the ${\frak g}$-action. Since ${\frak a}_{\c H}\subset {\frak g}$, the map $r$
is also compatible with the ${\frak a}_{\c H}$-action.
Therefore, $\ker r\subset \ker p$.
\ref{_r_H^*_to_H_r_Claim_} (iii) is implied
by \ref{_r_H^*_to_H_r_Claim_} (ii) and
\ref{_degree_isotypic_Proposition_}.
\blacksquare
\hfill
Let $x_i$ be an basis in $H^2(M, {\Bbb Q})$ which is
rational and orthonormal
with respect to Bogomolov-Beauville pairing,
$(x_i, x_i)_{\c B} = \epsilon_i = \pm 1$.
Consider the cohomology class
$\theta':= \epsilon_i x_i^2\in H^4(M, {\Bbb Q})$.
Let $\theta \in H^4(M, {\Bbb Z})$ be a non-zero integer cohomology
class which is proportional to $\theta'$.
{}From results of \cite{_Verbitsky:cohomo_} (see also \cite{_coho_announce_}),
the following proposition can be easily deduced.
\hfill
\proposition\label{_theta_SU(2)_inva_Proposition_}
The cohomology class $\theta\in H^4(M, {\Bbb Z})$
is $SU(2)$-invariant for all hyperk\"ahler structures on $M$.
Moreover, for a generic hyperk\"ahler structure,
the group of $SU(2)$-invariant integer classes $\alpha\in H^4_r(M)$
has rank one, where $H^*_r(M)$ is the subalgebra of
cohomology generated by $H^2(M)$.
\hfill
{\bf Proof:} Clearly, if an integer class $\alpha$
is $SU(2)$-invariant for a generic hyperk\"ahler structure, then
$\alpha$ is $G$-invariant, where $G$ is the group defined in
\ref{_gene_all_SU(2)_Theorem_}. On the other hand, $H^4_r(M)\cong S^2(H^2(M))$,
as follows from \ref{_S^*H^2_is_H^*M_intro-Theorem_}. Clearly,
the vector
$\theta\in H^4_r(M)\cong S^2(H^2(M))$ is $\frak{so}(H^2(M))$-invariant.
Moreover, the space of $\frak{so}(H^2(M))$-invariant vectors in
$S^2(H^2(M))$ is one-dimensional. Finally, from an explicit computation
of $G$ it follows that $G$ acts on $H^4(M)$ as $SO(H^2(M))$,
and thus, the Lie algebra invariants coincide with invariants
of $G$. We found that the space of $G$-invariants in
$H^4_r(M)$ is one-dimensional and generated by $\theta$.
This proves \ref{_theta_SU(2)_inva_Proposition_}.
\blacksquare
\hfill
\remark
It is clear how to generalize \ref{_theta_SU(2)_inva_Proposition_}
from dimension 4 to all dimensions.
The space $H^{2d}_r(M)^G$ of $G$-invariants
in $H^{2d}_r(M)$ is 1-dimensional
for $d$ even and zero-dimensional for $d$ odd.
\subsection{Cohomology classes of CA-type}
Let $M$ be a compact hyperk\"ahler manifold, and $I$ an induced
complex structure.
All cohomology classes which appear as fundamental
classes of complex subvarieties of $(M, I)$
satisfy certain properties. Classes satisfying
these properties are called classes of CA-type, from
Complex Analytic. Here is the definition of CA-type.
\hfill
\definition
Let $\eta\in H^{2,2}_I(M) \cap H^4(M, {\Bbb Z})$ be an
integer (2,2)-class. Assume that
for all induced complex structures $J$,
satisfying $I\circ J = - J\circ I$,
we have $\deg_I(\eta)\geq \deg_J(\eta)$,
and the equality is reached only if
$\eta$ is $SU(2)$-invariant.
Assume, moreover, that
$\deg_I(\eta)\geq |\deg_J(\eta)|$.
Then $\eta$ is called {\bf a class
of CA-type}.
\hfill
\theorem \label{_funda_and_Chern_CA_Theorem_}
Let $M$ be a simple hyperk\"ahler manifold,
of dimension $\dim_{\Bbb H} M>1$, $I$ an induced
complex structure, and $\eta \in H^{2,2}_I(M) \cap H^4(M, {\Bbb Z})$
an integer (2,2)-class. Assume that one of the following
conditions holds.
\begin{description}
\item[(i)] There exists a complex subvariety
$X\subset (M, I)$ such that $\eta$ is the fundamental class of $X$
\item[(ii)] There exists a stable coherent torsion-free sheaf $F$
over $(M,I)$, such that the first Chern class of $F$
is zero, and $\eta=c_2(F)$.
\end{description}
Then $\eta$ is of CA-type.
\hfill
{\bf Proof:} \ref{_funda_and_Chern_CA_Theorem_} (i)
is a direct consequence of Wirtinger's inequality
(\ref{_Wirti_hyperka_Proposition_}).
It remains to prove \ref{_funda_and_Chern_CA_Theorem_} (ii).
We assume, temporarily, that $F$ is reflexive.
By \ref{_stable_shea_degree_Corollary_},
we have
\begin{equation} \label{_c_2,1_ineq_Equation_}
\deg_I(2c_2(F) - \frac{r-1}{r} c_1(F)^2) \geq
\left|\deg_J(2c_2(F) - \frac{r-1}{r} c_1(F)^2)\right|,
\end{equation}
and the equality happens only if $F$ is hyperholomorphic.
Since $c_1(F)$ is $SU(2)$-invariant, we have
$\deg_I(c_1(F)^2) = \deg_J (c_1(F)^2) =0$
(\ref{_p_prods_H^2_Corollary_}).
Thus, \eqref{_c_2,1_ineq_Equation_}
implies that
\[ \deg_I 2c_2(F) \geq |\deg_J 2c_2(F)| \]
and the inequality is strict unless $F$ is hyperholomorphic,
in which case, the class $c_2(F)$ is $SU(2)$-invariant by definition.
We have proven \ref{_funda_and_Chern_CA_Theorem_} (ii)
for the case of reflexive $F$.
\hfill
For $F$ not necessary reflexive sheaf,
we have shown in the proof of \ref{_hyperho_suppo_of_F^**/F_Claim_}
that
\[ c_2(F) = c_2(F^{**}) + \sum n_i [X_i], \]
where $n_i$ are positive integers, and $[X_i]$ are the
fundamental classes of irreducible components of support
of the sheaf $F^{**}/F$. Therefore, the class $c_2(F)$
is a sum of classes of CA-type. Clearly, a sum of
cohomology classes of CA-type is again a class of
CA-type. This proves
\ref{_funda_and_Chern_CA_Theorem_}. \blacksquare
\section{$C$-restricted complex structures on hy\-per\-k\"ah\-ler manifolds}
\label{_C_restri_Section_}
\subsection{Existence of $C$-restricted complex structures}
We assume from now till the end of this section
that the hyperk\"ahler manifold $M$ is simple
(\ref{_simple_hyperkahler_mfolds_Definition_}).
This assumption can be avoided, but
it simplifies notation.
We assume from now till the end of this section
that the hyperk\"ahler manifold $M$ is compact
of real dimension $\dim_{\Bbb R} M \geq 8$,
i. e. $\dim_{\Bbb H}M \geq 2$. This assumption is absolutely necessary.
The case of hyperk\"ahler surfaces with $\dim_{\Bbb H}M =1$ (torus and
K3 surface) is trivial and for our purposes not interesting.
It is not difficult to extend our definitions and
results to the case of a compact hyperk\"ahler manifold
which is a product of simple hyperk\"ahler manifolds
with $\dim_{\Bbb H}M \geq 2$.
\hfill
\definition\label{_C_restri_Definition_}
Let $M$ be a compact hyperk\"ahler manifold,
and $I$ an induced complex structure. As usually, we denote by
$\deg_I:\; H^{2p}(M) {\:\longrightarrow\:} {\Bbb C}$ the associated degree
map, and by $H^*(M)= \oplus H^{p,q}_I(M)$
the Hodge decomposition. Assume that $I$ is algebraic. Let $C$ be a positive
real number. We say that the induced complex structure
$I$ is {\bf $C$-restricted} if the following conditions hold.
\begin{description}
\item[(i)] For all non-$SU(2)$-invariant cohomology classes
classes $\eta\in H^{1,1}_I(M) \cap H^2(M, {\Bbb Z})$,
we have $|\deg_I(\eta)| > C$.
\item[(ii)] Let $\eta \in H^{2,2}_I(M)$ be a cohomology class
of CA-type which is not $SU(2)$-invariant.
Then $|\deg_I(\eta)| > C$.
\end{description}
\hfill
The heuristic (completely informal) meaning of
this definition is the following.
The degree map plays the role of the metric on the cohomology.
Cohomology classes with small degrees are ``small'', the rest is ``big''.
Under reasonably strong assumptions,
there are only finitely many ``small'' integer classes, and the
rest is ``big''. For each non-$SU(2)$-invariant cohomology class $\eta$
there exists at most two induced complex structures for which
$\eta$ is of type $(p,p)$. Thus, for most induced complex structures,
all non-$SU(2)$-invariant integer $(p,p)$ classes are ``big''.
Intuitively, the $C$-restriction means that
all non-$SU(2)$-invariant integer (1,1) and (2,2)-cohomology
classes are ``big''. This definition is needed for the
study of first and second Chern classes of sheaves.
The following property of $C$-restricted complex structures
is used (see \ref{_funda_and_Chern_CA_Theorem_}):
for every subvariety $X\subset (M,I)$
of complex codimension 2, either $X$
is trianalytic or $\deg_I(X)>C$.
\hfill
\definition \label{_admitti_C_restri_Definition_}
Let $M$ be a compact manifold,
and $\c H$ a hyperk\"ahler structure on $M$. We say that
$\c H$ {\bf admits $C$-restricted complex structures}
if for all $C>0$, the set of all $C$-restricted algebraic
complex structures is dense in the set $R_{\c H}= {\Bbb C} P^1$
of all induced complex structures.
\hfill
\proposition \label{_restri_for_H^11_1-dim-Proposition_}
Let $M$ be a compact simple hyperk\"ahler manifold, $\dim_{\Bbb H}(M) >1$,
and $r:\; H^4(M) {\:\longrightarrow\:} H_r^4(M)$ be the map defined in
\ref{_r_H^*_to_H_r_Claim_}. Assume that
for all algebraic induced complex structures $I$,
the group $H^{1,1}_I(M)\cap H^2(M, {\Bbb Z})$ has rank one, and
the group \[ H^{2,2}_I(M)\cap H^4(M, {\Bbb Z})/(\ker r)\] has rank 2.
Then $M$ admits $C$-restricted complex structures.
\hfill
{\bf Proof:}
The proof of \ref{_restri_for_H^11_1-dim-Proposition_} takes the
rest of this section.
Denote by $\c R$ the set $\c R\cong {\Bbb C} P^1$ of all induced complex
structures on $M$. Consider
the set $\c R/\{\pm 1\}$ of induced complex structures up to a sign
(\ref{_K_proje_on_R_Lemma_}). Let $\alpha \in H^2(M)$ be a cohomology class
which is not $SU(2)$-invariant. According to
\ref{_K_proje_on_R_Lemma_}, there exists a unique
element $c(\alpha)\in \c R/\{\pm 1\}$ such that
$\alpha \in H^{1,1}_{c(\alpha)}(M)$.
This defines a map
\[
c:\; \left( H^2(M, {\Bbb R}) \backslash H^2_{inv}(M)\right) {\:\longrightarrow\:} \c R/\{\pm 1\},
\]
where $H^2_{inv}(M)\subset H^2(M)$ is the set of all
$SU(2)$-invariant cohomology classes.
For induced complex structures $I$ and $-I$, and $\eta\in H^{2p}(M)$,
the degree maps satisfy
\begin{equation} \label{_deg_-I_Equation_}
\deg_I(\eta) = (-1)^{p}\deg_{-I}(\eta).
\end{equation}
Thus, the number $|\deg_I(\eta)|$ is independent from the sign
of $I$.
Let $\eta\in H^*(M, {\Bbb Z})$ be a cohomology class. The {\bf largest
divisor} of $\eta$ is the biggest positive integer number $k$
such that the cohomology class $\frak \eta k$ is also integer.
Let $\alpha\in H^2(M, {\Bbb Z})$ be an integer cohomology class,
which is not $SU(2)$-invariant,
$k$ its largest divisor and $\tilde \alpha:= \frak \alpha k$
the corresponding integer class. Denote by $\widetilde\deg(\alpha)$
the number
\[
\widetilde\deg(\alpha) := \left| \deg_{c(\alpha)}(\tilde \alpha)\right|.
\]
The induced complex structure $c(\alpha)$ is defined up to a sign,
but from \eqref{_deg_-I_Equation_} it is clear that $\widetilde\deg(\alpha)$
is independent from the choice of a sign.
\hfill
\lemma \label{_C_restri_from_A,d_Lemma_}
Let $M$ be a compact hyperk\"ahler manifold, and
$I$ be an algebraic induced complex structure, such that
the group $H^{1,1}_I(M)\cap H^2(M, {\Bbb Z})$ has rank one,
and the group $H^{2,2}_I(M)\cap H^4(M, {\Bbb Z})/(\ker r)$ has rank 2.
Denote by $\alpha$ the generator of $H^{1,1}_I(M)\cap H^2(M, {\Bbb Z})$.
Since the class $\alpha$ is proportional to a K\"ahler form, $\alpha$
is not $SU(2)$-invariant (\ref{_K_proje_on_R_Lemma_}, (i)).
Let $d:= \widetilde \deg\alpha$.
Then, there exists a positive real constant $A$ depending
on volume of $M$, its topology
and its dimension, such that $I$ is $d\cdot A$-restricted.
\hfill
{\bf Proof:} This lemma is a trivial calculation based on
results of \cite{_Verbitsky:cohomo_} (see also \cite{_coho_announce_}
and Subsection \ref{_cohomo_stru_Subsection_}).
Since $H^{1,1}_I(M)\cap H^2(M, {\Bbb Z})$ has rank one,
for all $\eta \in H^{1,1}_I(M)\cap H^2(M, {\Bbb Z})$, $\eta\neq 0$,
we have $| \deg_I \eta| \geq d$. This proves the first condition of
\ref{_C_restri_Definition_}.
Let $\theta$ be the $SU(2)$-invariant integer cycle $\theta\in H^4(M)$
defined in \ref{_theta_SU(2)_inva_Proposition_}.
By \ref{_SU(2)_inva_type_p,p_Lemma_},
$\theta\in H^{2,2}_I(M)$. Consider
$\alpha^2\in H^{2,2}_I(M)$, where $\alpha$ is the
generator of $H^{1,1}_I(M)\cap H^2(M, {\Bbb Z})$.
\hfill
\sublemma\label{_degrees_theta_alpha^2_Sublemma_}
Let $J$ be an induced complex structure,
$J\circ I=-J\circ I$, and $\deg_I$, $\deg_J$ the degree maps
associated with $I$, $J$.
Then
\[ \deg_I \alpha^2 >0, \deg_J\alpha^2 =0, \deg_I\theta=\deg_J\theta >0.
\]
{\bf Proof:}
Since $\alpha$ is a K\"ahler class with respect to $I$, we have
$\deg_I \alpha^2 >0$. Since the cohomology class $\theta$ is $SU(2)$-invariant,
and $SU(2)$ acts transitively on the set of induced complex structures,
we have $\deg_I\theta=\deg_J\theta$. It remains to show that
$\deg_J\alpha^2 =0$ and $\deg_J\theta >0$. The manifold $M$ is
by our assumptions simple; thus, $\dim H^{2,0}(M) =1$
(\cite{_Besse:Einst_Manifo_}). Therefore, in the natural $SU(2)$-invariant
decomposition
\begin{equation}\label{_H^2_isoty_Equation_}
H^2(M) = H_{inv}^2(M) \oplus H^2_+(M),
\end{equation}
we have $\dim H^2_+(M) = 3$. In particular, the intersection
$H^2_+(M)\cap H^{1,1}_I(M)$ is 1-dimensional. Consider the
decomposition of $\alpha$, associated
with \eqref{_H^2_isoty_Equation_}: $\alpha = \alpha_+ + \alpha_{inv}$.
Since $\alpha$ is of type $(1,1)$ with respect to $I$,
the class $\alpha_+$ is proportional to the K\"ahler class
$\omega_I$, with positive coefficient. A similar argument leads to
the following decomposition for $\theta$:
\[
\theta = \omega_I^2 + \omega_J^2 +\omega_K^2 + \sum x_i^2,
\]
where $K= I\circ J$ is an induced complex structure, and
the classes $x_i$ belong to $H_{inv}^2(M)$.
{}From \ref{_p_prods_H^2_Corollary_},
we obtain that the classes $x_i^2$ satisfy
$\deg_I(x_i^2)=0$
(here we use $\dim_{\Bbb H}(M)>1$).
Thus,
\[ \deg_I(\theta) = \deg_I(\omega_I^2 + \omega_J^2 +\omega_K^2) =
\deg_I(\omega_I^2) >0.
\]
Similarly one checks that
\[ \deg_J(\alpha^2) = \deg_J((\alpha_+ + \alpha_{inv})^2)
= \deg_J(\alpha_+^2) = \deg_J (c^2 \omega_I) =0.
\]
This proves \ref{_degrees_theta_alpha^2_Sublemma_}.
\blacksquare
\hfill
Return to the proof of \ref{_C_restri_from_A,d_Lemma_}.
Since $\deg_I\alpha^2 \neq \deg_J \alpha^2$, the class
$\alpha^2$ is {\it not}
$SU(2)$-invariant. Since $\theta$ {\it is} $SU(2)$-invariant,
$\theta$ is not collinear with $\alpha^2$.
The degrees $\deg_I$ of $\theta$ and $\alpha^2$ are non-zero;
we have $\deg_I(\theta)=\deg_J(\theta)$,
$\deg_I(\alpha^2) \neq \deg_J(\alpha^2)$ for
$J$ an induced complex structure, $J\neq \pm I$.
By \ref{_degree_isotypic_Proposition_}, no non-trivial
linear combination of $\theta$, $\alpha^2$ belongs to
$\ker p$. By \ref{_r_H^*_to_H_r_Claim_} (ii), the classes
$\theta$, $\alpha^2$ generate a 2-dimensional subspace in
$H^4(M, {\Bbb Q})/\ker r$.
By assumptions of \ref{_C_restri_from_A,d_Lemma_},
the group $H^{2,2}_I(M)\cap H^4(M, {\Bbb Z})/(\ker r)$ has rank 2. Therefore
$\omega$ and $\alpha^2$ generate the space
\[ H^{2,2}_I(M)\cap H^4(M, {\Bbb Q})/(\ker r).
\]
To prove \ref{_C_restri_from_A,d_Lemma_}
it suffices to show that for all integer classes
\[ \beta = a\alpha^2 + b \theta, \ \ a \in {\Bbb Q}\backslash 0, \ \
\deg_I \beta \geq \deg_J \beta,
\]
we have $|\deg_I \beta| >A \cdot d$,
for a constant $A$ depending only on volume, topology and dimension of $M$.
Since $\deg_I \beta \geq |\deg_J \beta|$,
and $\deg_J \alpha^2 =0$ (\ref{_degrees_theta_alpha^2_Sublemma_}),
we have
\[
\deg_I (a \alpha^2 + b \theta) \geq |\deg_J b \theta|.
\]
Therefore, either $a$ and $b$ have the same sign,
or $\deg_I (a \alpha^2) > 2\deg_I(b\theta)$.
In both cases,
\begin{equation}\label{_deg_beta_geq_deg_alpha_Equation_}
|\deg_I \beta|\geq \frac{1}{2}\deg_I (a \alpha^2).
\end{equation}
Let $x\in {\Bbb Q}^{>0}$ be the smallest positive rational value of
$a$ for which there exists an integer class
$\beta = a\alpha^2 + b \theta$. We have an integer
lattice $L_1$ in $H^4_r(M)$ provided by the products of
integer classes; the integer lattice $L_2\supset L_1$ provided by
integer cycles might be different from that one.
Clearly, $x$ is greater than determinant
$\det (L_1 /L_2)$ of $L_1$ over $L_2$,
and this determinant is determined by the topology of $M$.
Form the definition of $x$ and \eqref{_deg_beta_geq_deg_alpha_Equation_},
we have $|\deg_I \beta| > x^2 \deg_I (\alpha^2)$.
On the other hand, $\deg_I (\alpha^2)> C \deg_I(\alpha)$,
where $C$ is a constant depending on volume and
dimension of $M$. Setting $A:= x^2 \cdot C$, we obtain
$|\deg_I \beta| > x^2 \cdot C \cdot d$. This proves
\ref{_C_restri_from_A,d_Lemma_}. \blacksquare
\hfill
Consider the maps
\[ \widetilde\deg:\; H^2(M, {\Bbb Z})\backslash H^2_{inv}(M)
{\:\longrightarrow\:} {\Bbb R},
\]
\[
c:\; H^2(M)\backslash H^2_{inv}(M) {\:\longrightarrow\:} \c R/\{\pm 1\}
\]
introduced in the beginning of the proof of
\ref{_restri_for_H^11_1-dim-Proposition_}.
\hfill
\lemma \label{_dense_big_tilde_deg_Lemma_}
In assumptions of \ref{_restri_for_H^11_1-dim-Proposition_},
let \[ {\cal O}\subset H^2(M, {\Bbb R})\backslash H^2_{inv}(M) \]
be an open subset of $H^2(M, {\Bbb R})$,
such that for all $x\in {\cal O}$, $k\in {\Bbb R}^{>0}$, we have
$k\cdot x \in {\cal O}$. Assume that
${\cal O}$ contains the K\"ahler class $\omega_I$ for all
induced complex structures $I\in \c R$.
For a positive number $C\in {\Bbb R}^{>0}$,
consider the set $X_C\subset {\cal O}$
\[ X_C := \left\{ \vphantom\prod\alpha \in {\cal O} \cap H^2(M, {\Bbb Z})\;\; | \;\;
\widetilde \deg(\alpha) \geq C \right\}.
\]
Then
$c(X_C)$ is dense in $\c R/\{\pm 1\}$ for all
$C\in {\Bbb R}^{>0}$.
\hfill
{\bf Proof:}
The map $\widetilde \deg$ can be expressed in the
following wey. We call an integer cohomology class
$\alpha\in H^2(M, {\Bbb Z})$ {\bf indivisible}
if its largest divisor is 1, that is, there are no
integer classes $\alpha'$, and numbers $k\in {\Bbb Z}$, $k>1$, such that
$\alpha = k \alpha'$.
\hfill
\sublemma
Let $\alpha \in H^2(M)$ be an
non-$SU(2)$-invariant cohomology class and
$\alpha= \alpha_{inv} + \alpha_+$ be a decomposition
associated with \eqref{_H^2_isoty_Equation_}.
Assume that $\alpha$ is indivisible.
Then
\begin{equation}\label{_tilde_deg_Equation_}
\widetilde \deg (\alpha) = C\sqrt{((\alpha_+,\alpha_+)_{\c B})},
\end{equation}
where $(\cdot,\cdot)_{\c B}$ is the Bogomolov-Beauville pairing
on $H^2(M)$ (\cite{_coho_announce_}; see also
\ref{_gene_all_SU(2)_Theorem_}),
and $C$ a constant depending on $\dim M$, $\operatorname{Vol} M$.
{\bf Proof:}
By \ref{_degree_isotypic_Proposition_},
\[ \deg_I(\alpha) = \deg_I(\alpha_+)
\]
(clearly, $p(\alpha) = \alpha_+$). By definition of
$(\cdot,\cdot)_{\c B}$, we have
\[ \deg_I(\alpha_+)=(\alpha_+,\omega_{c(\alpha)})_{\c B} \]
On the other hand, $\alpha_+$ is collinear with $\omega_{c(\alpha)}$
by definition of the map $c$. Now \eqref{_tilde_deg_Equation_}
follows trivially from routine properties of bilinear forms.
\blacksquare
\hfill
Let $I$ be an induced complex structure
such that the cohomology class $\omega_I$ is irrational:
$\omega_I \notin H^2(M, {\Bbb Q})$.
\begin{equation}\label{_sequence_x_i_Equation_}
\begin{minipage}[m]{0.8\linewidth}
To prove \ref{_dense_big_tilde_deg_Lemma_},
we have to produce a sequence $x_i \in {\cal O} \cap H^2(M, {\Bbb Z})$
such that
\begin{description}
\item[(i)] $c(x_i)$
converges to $I$,
\item[(ii)] and $\lim\tilde \deg (x_i) = \infty$.
\end{description}
\end{minipage}
\end{equation}
We introduce a metric $(\cdot,\cdot)_{\c H}$ on $H^2(M, {\Bbb R})$,
\[ (\alpha, \beta)_{\c H}:= (\alpha_+,\beta_+)_{\c B} -
(\alpha_{inv},\beta_{inv})_{\c B}.
\]
It is easy to check that $(\cdot,\cdot)_{\c H}$
is positive definite (\cite{_Verbitsky:cohomo_}).
For every $\epsilon$, there exists
a rational class $\omega_\epsilon \in H^2(M, {\Bbb Q})$
which approximates $\omega_I$ with precision
\[
(\omega_\epsilon - \omega_I, \omega_\epsilon - \omega_I)_{\c H} <\epsilon.
\]
Since ${\cal O}$ is open and contains $\omega_I$, we may assume that
$\omega_\epsilon$ belongs to ${\cal O}$.
Take a sequence $\epsilon_i$ converging to $0$, and let
$\tilde x_i:= \omega_{\epsilon_i}$ be the corresponding
sequence of rational cohomology cycles. Let
$x_i:= \lambda_i\tilde x_i$ be the minimal
positive integer such that $x_i \in H^2(M, {\Bbb Z})$.
We are going to show that the sequence $x_i$
satisfies the conditions of \eqref{_sequence_x_i_Equation_}.
First of all, $\tilde x_i$ converges to $\omega_I$,
and the map
\[
c:\; H^2(M)\backslash H^2_{inv}(M) {\:\longrightarrow\:} \c R/\{\pm 1\}
\]
is continuous. Therefore, $\lim c(\tilde x_i) = c(\omega_I) = I$.
By construction of $c$, $c$ satisfies $c(x) = c(\lambda x)$, and thus,
$c(x_i) = c(\tilde x_i)$. This proves the condition (i) of
\eqref{_sequence_x_i_Equation_}. On the other hand,
since $\omega_I$ is irrational, the sequence $\lambda_i$
goes to infinity. Therefore,
\[ \lim (x_i, x_i)_{\c H} = \infty. \]
It remains to compare $(x_i, x_i)_{\c H}$ with
$\tilde \deg x_i$. By \eqref{_tilde_deg_Equation_},
\[ \tilde \deg x_i = \sqrt{((x_i)_+, (x_i)_+)_{\c B}}. \]
On the other hand, since $(x_i)_+\in H^2_+(M)$, we have
\[ ((x_i)_+, (x_i)_+)_{\c B} = ((x_i)_+, (x_i)_+)_{\c H}.\]
To prove \eqref{_sequence_x_i_Equation_} (ii),
it remains to show that
\[ \lim ((x_i)_+, (x_i)_+)_{\c H} = \lim (x_i, x_i)_{\c H}. \]
Since the cohomology class $\tilde x_i\in H^2(M, {\Bbb Q})$
$\epsilon$-approximates $\omega_I$, and $\omega_I$ belongs to
$H^2_+(M)$, we have
\[
(\tilde x_i - (\tilde x_i)_+, \tilde x_i - (\tilde x_i)_+)_{\c H} <\epsilon_i.
\]
Therefore,
\begin{equation}\label{_x_i_close_x_i_+_Equation_}
( x_i - ( x_i)_+, x_i - ( x_i)_+)_{\c H} <\lambda_i \epsilon_i.
\end{equation}
On the other hand, for $i$ sufficiently big, the cohomology
class $\tilde x_i$ approaches $\omega_I$, and
\begin{equation}\label{_x_i_bigger_Equation_}
(x_i, x_i)_{\c H} >
\frac 1 2 \lambda_i (\omega_I, \omega_I)_{\c H}
\end{equation}
Comparing \eqref{_x_i_close_x_i_+_Equation_} and
\eqref{_x_i_bigger_Equation_} and using the distance property
for the distance given by $\sqrt{(\cdot,\cdot)_{\c H}}$, we find that
\begin{equation}\label{_x_i_+_bigger_Equation_}
\sqrt{(x_i)_+, (x_i)_+} >
\sqrt{\frac 1 2 \lambda_i (\omega_I, \omega_I)_{\c H}}
- \sqrt{\lambda_i \epsilon_i} =
\sqrt{\lambda_i}\cdot
\left(\sqrt{\frac 1 2 (\omega_I, \omega_I)_{\c H}} -
\sqrt{\epsilon_i} \right).
\end{equation}
Since $\epsilon_i$ converges to 0 and $\lambda_i$ converges
to infinity, the right hand side of
\eqref{_x_i_+_bigger_Equation_} converges to infinity.
On the other hand, by
\eqref{_tilde_deg_Equation_}
the left hand side of \eqref{_x_i_+_bigger_Equation_}
is equal constant times $\tilde \deg x_i$, so
$\lim \tilde \deg x_i = \infty$. This proves the second condition of
\eqref{_sequence_x_i_Equation_}.
\ref{_dense_big_tilde_deg_Lemma_} is proven.
\blacksquare
\hfill
We use \ref{_C_restri_from_A,d_Lemma_} and
\ref{_dense_big_tilde_deg_Lemma_} in order to finish the proof of
\ref{_restri_for_H^11_1-dim-Proposition_}.
Let $M$ be a compact hyperk\"ahler manifold,
and ${\cal O}\subset H^2(M, {\Bbb R})$ be the set of all K\"ahler classes
for the K\"ahler metrics compatible with one of induced complex
structures. By \ref{_K_proje_on_R_Lemma_}, ${\cal O}$ is open in
$H^2(M, {\Bbb R})$. Applying \ref{_dense_big_tilde_deg_Lemma_} to
${\cal O}$, we obtain the following. In assumptions of
\ref{_restri_for_H^11_1-dim-Proposition_},
let $Y_C\subset \c R$ be the set of all
algebraic induced complex structures
$I$ with $\tilde \deg \alpha >C$, where
$\alpha$ is a rational K\"ahler class, $\alpha \in H^{1,1}(M) \cap H^2(M, {\Bbb Z})$.
Then $Y_C$ is dense in $\c R$.
Now, \ref{_C_restri_from_A,d_Lemma_},
implies that for all $I \in Y_C$, the
induced complex structure
$I$ is $A \cdot C$-restricted, where $A$ is the
universal constant of \ref{_C_restri_from_A,d_Lemma_}.
Thus, for all $C$ the set of $C$-restricted induced complex
structures is dense in $\c R$. This proves that
$M$ admits $C$-restricted complex structures.
We finished the proof of
\ref{_restri_for_H^11_1-dim-Proposition_}.
\blacksquare
\subsection{Hyperk\"ahler structures admitting
$C$-restricted complex structures}
\label{_modu_and_C-restri_Subsection_}
Let $M$ be a compact complex manifold admitting a hyperk\"ahler
structure $\c H$. Assume that $(M, \c H)$ is a
simple hyperk\"ahler manifold of dimension $\dim_{\Bbb H} M >1$.
The following definition of (coarse, marked) moduli space for
complex and hyperk\"ahler structures on $M$ is standard.
\hfill
\definition\label{_moduli_hyperka_Definition_}
Let $M_{C^\infty}$ be the $M$ considered as a
differential manifold, $\widetilde{Comp}$ be the set
of all integrable complex structures,
and $\widetilde{\mbox{\it Hyp}}$ be the set of all hyperk\"ahler
structures on $M_{C^\infty}$. The set $\widetilde{\mbox{\it Hyp}}$
is equipped with a natural topology. Let $\widetilde{\mbox{\it Hyp}}^0$
be a connected component of $\widetilde{\mbox{\it Hyp}}$ containing $\c H$
and $\widetilde{Comp}^0$ be a set of all complex structures
$I\in \widetilde{Comp}$ which are compatible with some hyperk\"ahler
structure $\c H_1\in \widetilde{\mbox{\it Hyp}}^0$.
Let $\mbox{\it Diff}$ be the group of diffeomorphisms
of $M$ which act trivially on the cohomology. The
coarse, marked moduli $\mbox{\it Hyp}$ of hyperk\"ahler structures on
$M$ is the quotient
$\mbox{\it Hyp}:= \widetilde{\mbox{\it Hyp}}^0/\mbox{\it Diff}$
equipped with a natural topology. The coarse, marked moduli $Comp$ of
complex structures on $M$ is defined as
$Comp:= \widetilde{Comp}^0/\mbox{\it Diff}$.
For a detailed discussion of various aspects of this definition,
see \cite{_Verbitsky:cohomo_}.
\hfill
Consider the variety
\[ X \subset {\Bbb P} H^2(M, {\Bbb C}),\] consisting of all lines
$l\in {\Bbb P} H^2(M, {\Bbb C})$ which are isotropic with respect to the
Bogomolov-Beauville's pairing:
\[
X:= \{ l \in H^2(M, {\Bbb C}) \; \; | \; (l, l)_{\c B} =0 \}.
\]
Since $M$ is simple, $\dim H^{2,0}(M, I) =1$ for all induced complex structures.
Let $P_c:\; Comp {\:\longrightarrow\:} {\Bbb P} H^2(M, {\Bbb C})$ map $I$ to the line
$H^{2,0}_I(M) \subset H^2(M, {\Bbb C})$. The map $P_c$ is
called {\bf the period map}. It is well known that
$Comp$ is equipped with a natural complex structure. From general
properties of the period map it follows that
$P_c$ is compatible with this complex structure.
Clearly from the definition
of Bogomolov-Beauville's form, $P_c(I)\in X$ for all induced complex
structures $I\in Comp$ (see \cite{_Beauville_} for details).
\hfill
\theorem \label{_Bogomo_etale_Theorem_}
\cite{_Besse:Einst_Manifo_} (Bogomolov)
The complex analytic
map \[ P_c:\; Comp {\:\longrightarrow\:} X\] is locally an etale covering.
\footnote{
The space $Comp$ is smooth, as follows from
\ref{_Bogomo_etale_Theorem_}. This space
is, however, in most cases not separable
(\cite{_Huybrechts_}). The space $\mbox{\it Hyp}$ has no
natural complex structures, and can be odd-dimensional.
}
\blacksquare
\hfill
It is possible to formulate a similar statement about hyperk\"ahler structures.
For a hyperk\"ahler structure $\c H$, consider the set $\c R_{\c H}\subset Comp$
of all induced complex structures associated with this hyperk\"ahler
structure. The subset $\c R_{\c H}\subset Comp$ is a complex analytic
subvariety, which is isomorphic to ${\Bbb C} P^1$. Let $S:= P_c(\c R_{\c H})$
be the corresponding projective line in $X$, and $\bar L(X)$ be the
space of smooth deformations of $S$ in $X$. The points of $L(X)$ correspond
to smooth rational curves of degree 2 in ${\Bbb P} H^2(M, {\Bbb C})$.
For every such curve $s$, there exists a unique 3-dimensional plane
$L(s) \subset H^2(M, {\Bbb C})$, such that $s$ is contained in ${\Bbb P} L$.
Let $Gr$ be the Grassmanian manifold of all 3-dimensional
planes in $H^2(M, {\Bbb C})$ and $Gr_0\subset Gr$ the set of all planes
$L\in Gr$ such that the restriction of the Bogomolov-Beauville
form to $L$ is non-degenerate.
Let $L(X)\subset \bar L(X)$ be the space of all rational curves
$s\in \bar L(X)$ such that the restriction of the Bogomolov-Beauville
form to $L(s)$ is non-degenerate: $L(s)\in Gr_0$.
The correspondence $s{\:\longrightarrow\:} L(s)$ gives a map
$\kappa:\; L(X) {\:\longrightarrow\:} Gr_0$.
\hfill
\lemma
The map $\kappa:\; L(X) {\:\longrightarrow\:} Gr_0$ is an isomorphism of complex
varieties.
{\bf Proof:} For every plane $L\in Gr_0$, consider the set $s(L)$
of all isotropic lines $l\in L$, that is, lines
satisfying $(l, l)_{\c B}=0$.
Since $(\cdot,\cdot)_{\c B}\restrict L$ is non-degenerate,
the set $s(L)$ is a rational curve in ${\Bbb P}L$.
Clearly, this curve has degree 2. Therefore,
$s(L)$ belongs to $X(L)$. The map $L{\:\longrightarrow\:} s(L)$
is inverse to $\kappa$. \blacksquare
\hfill
Consider the standard anticomplex involution
\[ \iota:\; H^2(M, {\Bbb C}) {\:\longrightarrow\:} H^2(M, {\Bbb C}), \ \ \ \eta{\:\longrightarrow\:} \bar \eta. \]
Clearly, $\iota$
is compatible with the Bogomolov-Beauville form.
Therefore, $\iota$ acts on $L(X)$ as an anticomplex involution.
Let $L(X)_\iota\subset L(X)$ be the set of all $S\in L(X)$ fixed
by $\iota$.
\hfill
Every hyperk\"ahler structure \[ \c H\in \mbox{\it Hyp} \]
gives a rational curve $\c R_{\c H}\subset Comp$ with points
corresponding all induced complex structures. Let
$P_h(\c H)\subset X$ be the line $P_c(\c R_{\c H})$.
Clearly from the definition,
$P_h(\c H)$ belongs to $L(X)_\iota$. We have constructed
a map $P_h:\; \mbox{\it Hyp} {\:\longrightarrow\:} L(X)_\iota$.
Let $L(Comp)$ be the space of deformations if
$\c R_{\c H}$ in $Comp$.
Denote by \[ \gamma:\; \mbox{\it Hyp} {\:\longrightarrow\:} L(Comp) \]
the map $\c H {\:\longrightarrow\:} \c R_{\c H}$.
The following result gives a hyperk\"ahler analogue
of Bogomolov's theorem (\ref{_Bogomo_etale_Theorem_}).
\hfill
\theorem \label{_hyperka_etale_Theorem_}
The map $\gamma:\; \mbox{\it Hyp} {\:\longrightarrow\:} L(Comp)$
is an embedding. The map $P_h:\; \mbox{\it Hyp} {\:\longrightarrow\:} L(X)_\iota$
is locally a covering.
{\bf Proof:} The first claim is an immediate
consequence of Calabi-Yau Theorem
(\ref{_symplectic_=>_hyperkahler_Proposition_}).
Now, \ref{_hyperka_etale_Theorem_}
follows from the Bogomolov's theorem
(\ref{_Bogomo_etale_Theorem_}) and dimension count.
\blacksquare
\hfill
Let $I\in Comp$ be a complex structure on $M$.
Consider the groups
\[
H_h^2(M, I):= H^{1,1}(M, I) \cap H^2(M, {\Bbb Z})
\]
and
\[
H_h^2(M, I):= H^{2,2}_r(M, I) \cap H^4(M, {\Bbb Z}).
\]
For a general $I$,
$H^2_h(M, I)=0$ and $H^4_h(M, I)={\Bbb Z}$ as follows from
\ref{_theta_SU(2)_inva_Proposition_}. Therefore,
the set of all $I$ with $\operatorname{rk} H^2_h(M, I) =1$,
$\operatorname{rk} H^4_h(M, I)=2$ is a union of countably
many subvarieties of codimension 1 in $Comp$.
Similarly, the set $V\subset Comp$
of all $I$ with $\operatorname{rk} H^2_h(M, I) >1$,
$\operatorname{rk} H^4_h(M, I)>2$ is a union of countably
many subvarieties of codimension more than 1.
Together with \ref{_hyperka_etale_Theorem_},
this implies the following.
\hfill
\claim\label{_hype_1_2_dense_Claim_}
Let $U\subset \mbox{\it Hyp}$ be the set of
all $\c H\in Hyp$ such that $\c R_{\c H}$ does not intersect $V$.
Then $U$ is dense in $\mbox{\it Hyp}$.
\hfill
{\bf Proof:} Consider a natural involution $i$ of $Comp$
which is compatible with the involution $\iota:\; X {\:\longrightarrow\:} X$ inder
the period map $P_c:\; Comp {\:\longrightarrow\:} X$. This involution maps
the complex structure $I$ to $-I$.
\begin{equation}\label{_Hyp_identi_Equation_}
\begin{minipage}[m]{0.8\linewidth}
By \ref{_hyperka_etale_Theorem_},
$\mbox{\it Hyp}$ is identified with an open subset in
the set $L(X)_\iota$ of real points of $L(Comp)$.
\end{minipage}
\end{equation}
Let $L_U\subset L(Comp)$ be the set of all lines
which do not intersect $V$. Since $V$ is a union of
subvarieties of codimension at least 2, a general rational
line $l \in L(Comp)$ does not intersect $V$. Therefore,
$L_U$ is dense in $L(Comp)$. Thus, the set of real points
of $L_U$ is dense $L(X)_\iota$. Using the idenitification
\eqref{_Hyp_identi_Equation_},
we obtain the statement of \ref{_hype_1_2_dense_Claim_}.
\blacksquare
\hfill
\ref{_hype_1_2_dense_Claim_}
together with \ref{_restri_for_H^11_1-dim-Proposition_}
imply the following theorem.
\hfill
\theorem\label{_C_restri_dense_Theorem_}
Let $M$ be a compact simple hyperk\"ahler manifold,
$\dim_{\Bbb H}M >1$, and $\mbox{\it Hyp}$ its coarse
marked moduli of hyperk\"ahler structures.
Let $U\subset \mbox{\it Hyp}$ be the set of
all hyperk\"ahler structures which admit $C$-restricted complex
structures (\ref{_admitti_C_restri_Definition_}).
Then $U$ is dense in $\mbox{\it Hyp}$.
\blacksquare
\subsection{Deformations of coherent sheaves over manifolds with
$C$-res\-t\-ric\-ted complex structures}
The following theorem shows that a semistable
deformation of a hyperholomorphic sheaf on $(M,I)$
is again hyperholomorphic, provided that $I$ is a
$C$-restricted complex structure and $C$ is sufficiently big.
\hfill
\theorem \label{_sheaf_on_C_restr_hyperho_Theorem_}
Let $M$ be a compact hyperk\"ahler manifold, and
$\c F \in \c F(M)$ a polystable hyperholomorphic sheaf
on $M$ (\ref{_hyperho_shea_on_M_Definition_}).
Let $I$ be a $C$-restricted induced complex structure,
for $C= \deg_I c_2(\c F)$,\footnote{Clearly, since
$\c F$ is hyperholomorphic, the class
$c_2(\c F)$ is $SU(2)$-invariant, and the number
$\deg_I c_2(\c F)$ independent from $I$.}
and $F'$ be a semistable torsion-free coherent sheaf
on $(M,I)$ with the same rank and Chern classes as
$\c F$. Then the sheaf $F'$ is hyperholomorphic.
\hfill
{\bf Proof:} Let $F_1$, ..., $F_n$ be the Jordan-H\"older
series for the sheaf $F'$. Since $\c F$ is hyperholomorphic,
we have $\text{slope}(\c F)=0$ (\ref{_slope_hyperho_Remark_}).
Therefore, $\operatorname{slope}(F_i)=0$, and $\deg_I(c_1(F_i))=0$.
By \ref{_C_restri_Definition_} (i), then, the class
$c_1(F_i)$ is $SU(2)$ invariant for all $i$.
To prove that $F'$ is hyperholomorphic
it remains to show that the classes $c_2(F_i)$, $c_2(F_i^{**})$
are $SU(2)$-invariant for all $i$.
\hfill
Consider an exact sequence
\[
0 {\:\longrightarrow\:} F_i {\:\longrightarrow\:} F^{**}_i {\:\longrightarrow\:} F_i / F_i^{**} {\:\longrightarrow\:} 0.
\]
Let $[F_i / F_i^{**}]\in H^4(M)$ be the fundamental class
of the union of all components of $Sup(F_i / F_i^{**})$
of complex codimension 2, taken with appropriate multiplicities.
Clearly,
$c_2(F_i)= c_2(F^{**}_i) + [F_i / F_i^{**}]$.
Since $[F_i / F_i^{**}]$ is an effective cycle,
$\deg_I([F_i / F_i^{**}])\geq0$. By the Bogomolov-Miyaoka-Yau
inequality (see \ref{_stable_shea_degree_Corollary_}),
we have $\deg_I (c_2(F^{**}_i)\geq 0$. Therefore,
\begin{equation}\label{_c_2_greater_Equation_}
\deg_I c_2(F_i)\geq \deg_I c_2(F^{**}_i)\geq 0.
\end{equation}
\hfill
Using the product formula for Chern classes, we obtain
\begin{equation}\label{_pro_Chern_2_Equation_}
c_2(F) = \sum_i c_2(F_i) + \sum_{i, j} c_2(F_i)\wedge c_2(F_j).
\end{equation}
By \ref{_p_prods_H^2_Corollary_},
$\deg_I(\sum_{i, j} c_2(F_i)\wedge c_2(F_j)) =0$.
Since the numbers $\deg_I c_2(F_i)$ are non-negative,
we have $\deg_I c_2(F_i)\leq \deg_I c_2(F) =C$.
By \ref{_funda_and_Chern_CA_Theorem_}, the classes
$c_2(F_i)$, $c_2(F_i^{**})$
are of CA-type. By \ref{_C_restri_Definition_} (ii),
then, the inequality $\deg_I c_2(F_i)\leq C$ implies that
the class $c_2(F_i)$ is $SU(2)$-invariant. By
\eqref{_c_2_greater_Equation_},
$\deg_I c_2(F^{**}_i)\leq\deg_I c_2(F_i)$,
so the class $c_2(F_i^{**})$ is also $SU(2)$-invariant.
\ref{_sheaf_on_C_restr_hyperho_Theorem_}
is proven.
\blacksquare
\section{Desingularization of hyperholomorphic sheaves}
\label{_desingu_she_Section_}
The aim of this section is the following theorem.
\hfill
\theorem\label{_desingu_hyperho_Theorem_}
Let $M$ be a hyperk\"ahler manifold, not necessarily
compact, $I$ an induced complex structure,
and $F$ a reflexive coherent sheaf over $(M, I)$
equipped with a hyperholomorphic connection
(\ref{_hyperholo_co_Definition_}). Assume that
$F$ has isolated singularities. Let $\tilde M\stackrel \sigma{\:\longrightarrow\:} M$
be a blow-up of $(M,I)$ in the singular set of $F$, and
$\sigma^* F$ the pullback of $F$. Then
$\sigma^* F$ is a locally trivial sheaf,
that is, a holomorphic vector bundle.
\hfill
We prove \ref{_desingu_hyperho_Theorem_}
in Subsection \ref{_desingu_she_Subsection_}.
\hfill
The idea of the proof is the following. We apply to $F$ the
methods used in the proof of Desingularization Theorem
(\ref{_desingu_Theorem_}). The main ingredient in the proof of
Desingularization Theorem is the existence of a natural ${\Bbb C}^*$-action
on the completion $\hat {\cal O}_x(M, I)$ of the local ring ${\cal O}_x(M, I)$, for
all $x\in M$. This ${\Bbb C}^*$-action identifies $\hat {\cal O}_x(M, I)$
with a completion of a graded ring. Here we show that a sheaf
$F$ is ${\Bbb C}^*$-equivariant. Therefore, a germ of $F$ at $x$ has a
grading, which is compatible with the natural ${\Bbb C}^*$-action on
$\hat {\cal O}_x(M, I)$. Singularities of such reflexive sheaves
can be resolved by a single blow-up.
\subsection{Twistor lines and complexification}
\label{_twi_lines_C^*_Subsection_}
Further on, we need the following definition.
\hfill
\definition
Let $X$ be a real analytic variety,
which is embedded to a complex variety $X_{\Bbb C}$.
Assume that the sheaf of complex-valued real analytic
functions on $X$ coincides with the restriction
of ${\cal O}_{X_{\Bbb C}}$ to $X\subset X_{\Bbb C}$. Then
$X_{\Bbb C}$ is called {\bf a complexification of $X$}.
\hfill
For more details on complexification, the reader is referred to
\cite{_GMT_}. There are the most important properties.
\hfill
\claim \label{_complexi_Claim_}
In a neighbourhood of $X$, the manifold $X_{\Bbb C}$ has an anti-complex
involution. The variety $X$ is identified with the set of
fixed points of this involution, considered as a
real analytic variety.
Let $Y$ be a complex variety, and $X$ the underlying
real analytic variety. Then the product of $Y$ and its complex
conjugate is a complexification of $X$, with embedding
$X\hookrightarrow Y \times \bar Y$ given by the diagonal.
The complexification is unique in the following weak sense.
For $X_{\Bbb C}$, $X'_{\Bbb C}$ complexifications of $C$, the complex manifolds
$X_{\Bbb C}$, $X'_{\Bbb C}$ are naturally identified in a neighbourhood
of $X$.
\blacksquare
\hfill
Let $M$ be a hyperk\"ahler manifold, $\operatorname{Tw}(M)$ its twistor space,
and $\pi:\; \operatorname{Tw}(M) {\:\longrightarrow\:} {\Bbb C} P^1$ the twistor projection.
Let $l\subset \operatorname{Tw}(M)$ be a rational curve, such that
the restriction of $\pi$ to $l$ is an identity.
Such a curve gives a section of $\pi$, and vice versa, every
section of $\pi$ corresponds to such a curve.
The set of sections of the projection $\pi$ is called
{\bf the space of twistor lines}, denoted by $\operatorname{Lin}$, or $\operatorname{Lin}(M)$.
This space is equipped with complex structure,
by Douady (\cite{_Douady_}).
Let $m\in M$ be a point. Consider a twistor line
${s_m}:\; I {\:\longrightarrow\:} (I \times m)\in {\Bbb C} P^1 \times M = \operatorname{Tw}$.
Then $s_m$ is called {\bf a horisontal twistor
line}. The space of horisontal twistor lines is a
real analytic subvariety in $\operatorname{Lin}$, denoted by $\operatorname{Hor}$, or $\operatorname{Hor}(M)$.
Clearly, the set $\operatorname{Hor}$ is naturally identified with $M$.
\hfill
\proposition\label{_Lin_is_MxM_Proposition_}
(Hitchin, Karlhede, Lindstr\"{o}m, Ro\v{c}ek)
Let $M$ be a hyperk\"ahler manifold,
$\operatorname{Tw}(M)$ its twistor space, $I$, $J\in {\Bbb C} P^1$ induced complex
structures, and $\operatorname{Lin}$ the space of twistor lines.
The complex manifolds $(M,I)$ and $(M, J)$ are naturally
embedded to $\operatorname{Tw}(M)$:
\[ (M,I) = \pi^{-1}(I),\ \ (M,J) = \pi^{-1}(J).
\]
Consider a point $s\in \operatorname{Lin}$, $s:\; {\Bbb C} P^1{\:\longrightarrow\:} \operatorname{Tw}(M)$.
Let \[ ev_{I,J}:\; \operatorname{Lin}(M) {\:\longrightarrow\:} (M, I)\times (M,J) \]
be the map defined by $ev_{I,J}(s) = (s(I), s(J))$.
Assume that $I\neq J$. Then there exists a
neighbourhood $U$ of $\operatorname{Hor}\subset \operatorname{Lin}$, such that
the restriction of $ev_{I,J}$ to $U$ is an open embedding.
{\bf Proof:} \cite{_HKLR_}, \cite{_Verbitsky:hypercomple_}.
\blacksquare
\hfill
Consider the anticomplex involution $i$ of ${\Bbb C} P^1 \cong S^2$
which corresponds to the central symmetry of $S^2$.
Let $\iota:\; \operatorname{Tw} {\:\longrightarrow\:} \operatorname{Tw}$ be the corresponding
involution of the twistor space $\operatorname{Tw}(M) = {\Bbb C} P^1\times M$,
$(x, m) {\:\longrightarrow\:} (i(x), m)$. It is clear that
$\iota$ maps holomorphic subvarieties of $\operatorname{Tw}(M)$ to
holomorphic subvarieties. Therefore, $\iota$ acts
on $\operatorname{Lin}$ as an anticomplex involution.
For $J= -I$, we obtain a local identification of
$\operatorname{Lin}$ in a neighbourhood of $\operatorname{Hor}$ with
$(M, I) \times (M, -I)$, that is, with $(M, I)$ times its complex
conjugate. Therefore, the space of twistor lines is a
complexification of $(M,I)$. The natural anticomplex involution
of \ref{_complexi_Claim_} coincides with $\iota$. This
gives an identification of $\operatorname{Hor}$ and the real analytic
manifold underlying $(M,I)$.
\hfill
We shall explain how to construct the natural ${\Bbb C}^*$-action on
a local ring of a hyperk\"ahler manifold, using the machinery
of twistor lines.
\hfill
Fix a point $x_0\in M$ and induced complex structures
$I$, $J$, such that $I\neq \pm J$. Let $V_1$, $V_2$ be neighbourhoods of
$s_{x_0}\in \operatorname{Lin}$, and $U_1$, $U_2$ be neighbourhoods of
$(x_0,x_0)$ in $(M, I)\times (M,-I)$, $(M, J)\times (M,-J)$,
such that the evaluation maps $ev_{I, -I}$, $ev_{J, -J}$
induce isomorphisms
\[ ev_{I, -I}:\; V_1 \oldtilde{\:\longrightarrow\:} U_1, \ \ \ \
ev_{J, -J}:\; V_2\oldtilde{\:\longrightarrow\:} U_2.
\]
Let $B$ be an open neighbourhood of $x_0\in M$,
such that $(B, I)\times (B,-I)\subset U_1$
and $(B, I)\times (B,-I)\subset U_2$. Denote by
$V_I\subset V_1$ be the preimage of $(B, I)\times (B,-I)$
under $ev_{I, -I}$, and by
$V_J\subset V_2$ be the preimage of $(B, J)\times (B,-J)$
under $ev_{J, -J}$. Let $p_I:\; V_I {\:\longrightarrow\:} (B, I)$ be the evaluation,
$s{\:\longrightarrow\:} s(I)$, and $e_I:\; (B, I){\:\longrightarrow\:} V_1$ the map associating
to $x\in B$ the unique twistor line passing through
$(x, x_0) \subset (B, I)\times (B,-I)$.
In the same fashion, we define $e_J$ and $p_J$.
We are interested in the composition
\[ \Psi_{I, J}:= e_I\circ p_J \circ e_J \circ p_I:\; (B_0, I) {\:\longrightarrow\:} (B,I) \]
which is defined in a smaller neighbourhood
$B_0\subset B$ of $x_0\in M$.
\hfill
The following proposition is the focal point of this Subsection:
we explain the map $\Psi_{I, J}$ of \cite{_Verbitsky:DesinguII_},
\cite{_Verbitsky:hypercomple_} is geometric terms (in
\cite{_Verbitsky:DesinguII_}, \cite{_Verbitsky:hypercomple_}
this map was defined algebraically).
\hfill
\proposition \label{_Psi_acts_on_TM_Proposition_}
Consider the map $\Psi_{I, J}:\; (B_0, I) {\:\longrightarrow\:} (B,I)$
defined above. By definition, $\Psi_{I, J}$
preserves the point $x_0\in B_0\subset B$.
Let $d\Psi_{I, J}$ be the differential of $\Psi_{I, J}$ acting
on the tangent space $T_{x_0}B_0$. Assume that
$I\neq \pm J$. Then
$d\Psi_{I, J}$ is a multiplication by a scalar
$\lambda\in {\Bbb C}$, $0<|\lambda| <1$.
\hfill
{\bf Proof:}
The map $\Psi_{I, J}$ was defined in \cite{_Verbitsky:DesinguII_},
\cite{_Verbitsky:hypercomple_} using the identifications
between the real analytic varieties underlying $(M,I)$ and $(M, J)$.
We proved that $\Psi_{I, J}$ defined this way acts on $T_{x_0}B_0$
as a multiplication by the scalar $\lambda\in {\Bbb C}$, $0<|\lambda| <1$.
It remains to show that the map $\Psi_{I, J}$ defined in
\cite{_Verbitsky:DesinguII_}, \cite{_Verbitsky:hypercomple_}
coincides with $\Psi_{I, J}$ defined above.
Consider the natural identification
\[ (B, I)\times (B,-I)\sim (B, J)\times (B,-J), \]
which is defined in a neighbourhood $B_{\Bbb C}$ of $(x_0, x_0)$.
There is a natural projection
$a_I:\; B_{\Bbb C} {\:\longrightarrow\:} (M,I)$. Consider the embedding
$b_I:\; (B, I) {\:\longrightarrow\:} B_{\Bbb C}$, $x{\:\longrightarrow\:} (x, x_0)$,
defined in a neighbourhood of $x_0\in (B,I)$.
In a similar way we define $a_J$, $b_J$.
In \cite{_Verbitsky:DesinguII_},
\cite{_Verbitsky:hypercomple_} we defined
$\Psi_{I, J}$ as a composition
$b_I\circ a_J \circ b_J \circ a_I$.
Earlier in this Subsection, we described
a local identification of $(B, I)\times (B,-I)$
and $\operatorname{Lin}(B)$. Clearly, under this identification,
the maps $a_I$, $b_I$ correspond to $p_I$, $e_I$. Therefore,
the definition of $\Psi_{I,J}$ given in this paper
is equivalent to the definition given in
\cite{_Verbitsky:DesinguII_},
\cite{_Verbitsky:hypercomple_}.
\blacksquare
\subsection[The automorphism $\Psi_{I,J}$ acting on hyperholomorphic
sheaves]{The automorphism $\Psi_{I,J}$ acting on hyperholomorphic \\
sheaves}
\label{_Psi_on_shea_Subsection_}
In this section, we prove that hyperholomorphic sheaves
are equivariant with respect to the map
$\Psi_{I, J}$, considered as an automorphism of the
local ring ${\cal O}_{x_0}(M,I)$.
\hfill
\theorem \label{_Psi_equiv_hyperho_Theorem_}
Let $M$ be a hyperk\"ahler manifold, not necessarily compact,
$x_0\in M$ a point,
$I$ an induced complex structure and $F$ a reflexive sheaf
over $(M,I)$ equipped with a hyperholomorphic connection.
Let $J\neq \pm I$ be another induced complex structure,
and $B_0$, $B$ the neighbourhoods of $x_0\in M$ for which
the map $\Psi_{I,J}:\; B_0 {\:\longrightarrow\:} B$ was defined in
\ref{_Psi_acts_on_TM_Proposition_}.
Assume that $\Psi_{I,J}:\; B_0 {\:\longrightarrow\:} B$ is an isomorphism.
Then there exists
a canonical functorial isomorphism of coherent sheaves
\[ \Psi_{I,J}^F:\; F\restrict {B_0} {\:\longrightarrow\:} \Psi_{I,J}^*(F\restrict B).
\]
{\bf Proof:}
Return to the notation introduced in Subsection
\ref{_twi_lines_C^*_Subsection_}. Let $W:= V_I \cap V_J$.
By definition of $V_I$, $V_J$, the evaluation maps
produce open embeddings
\[ ev_{I, -I}:\; \operatorname{Lin}(W) \hookrightarrow (W,I)\times (W, -I),\]
and
\[ ev_{J, -J}:\; \operatorname{Lin}(W) \hookrightarrow (W,J)\times (W, -J),\]
Let $S\subset W$ be the singular set of $F\restrict W$,
$\operatorname{Tw}(S) \subset \operatorname{Tw}(W)$ the corresponding embedding,
and $L_0\subset \operatorname{Lin}(W)$ be the set of all
lines $l\in \operatorname{Lin}(W)$ which do not intersect
$\operatorname{Tw}(S)$. Consider the maps
\[ p_I:\; L_0 \hookrightarrow
(W,I)\backslash S
\]
and
\[ p_J:\; L_0 \hookrightarrow
(W,J)\backslash S
\]
obtained by restricting the evaluation map
$p_I:\; \operatorname{Lin}(M) {\:\longrightarrow\:} (M, I)$ to $L_0\subset \operatorname{Lin}(M)$.
Since $F$ is equipped with a hyperholomorphic connection,
the vector bundle $F\restrict{(M, J)\backslash S}$ has a natural
holomorphic structure. Let
$\underline F_1:= p_I^*\left(F\restrict{(M, I)\backslash S}\right)$
and $\underline F_2:= p_J^*\left(F\restrict{(M, J)\backslash S}\right)$
be the corresponding pullback sheaves over $L_0$,
and $F_1$, $F_2$ the sheaves on $\operatorname{Lin}(W)$ obtained as direct
images of $\underline F_1$, $\underline F_2$ under the open embedding
$L_0 \hookrightarrow \operatorname{Lin}(W)$.
\hfill
\lemma \label{_F_1_=F_2_Lemma_}
Under these assumptions, the sheaves
$F_1$, $F_2$ are coherent reflexive sheaves. Moreover,
there exists a natural isomorphism of coherent sheaves
$\Psi_{1,2}:\;F_1{\:\longrightarrow\:} F_2$.
\hfill
{\bf Proof:}
The complex codimension of the singular set $S$ in $(M,I)$
is at least 3, because $F$ is reflexive
(\cite{_OSS_}, Ch. II, 1.1.10). Since $S$
is trianalytic (\ref{_singu_triana_Claim_}),
this codimension is even. Thus,
$\operatorname{codim}_{\Bbb C} (S, (M,I)) \geq 4$. Therefore,
\[ \operatorname{codim}_{\Bbb C} (\operatorname{Tw}(S), \operatorname{Tw}(M)) \geq 4. \]
Consider the set $L_S$ of all twistor lines
$l\in \operatorname{Lin}(W)$ passing through $\operatorname{Tw}(S)$.
For generic points $x,y\in \operatorname{Tw}(W)$, there exists
a unique line $l\in \operatorname{Lin}(W)$ passing through $x, y$. Therefore,
\[ \operatorname{codim}_{\Bbb C} (L_S, \operatorname{Lin}(W)) = \operatorname{codim}_{\Bbb C} (\operatorname{Tw}(S), \operatorname{Tw}(M))-1\geq 3. \]
By definition, $L_0:= \operatorname{Lin}(W)\backslash L_S$. Since
$F_1$, $F_2$ are direct images of bundles
$\underline F_1$, $\underline F_2$ over a subvariety
$L_S$ of codimension 3, these sheaves are coherent and reflexive
(\cite{_OSS_}, Ch. II, 1.1.12; see also \ref{_normal_refle_Lemma_}).
To show that they are naturally isomorphic it remains
to construct an isomorphism between $\underline F_1$
and $\underline F_2$.
Let $\c F$ be a coherent sheaf on $\operatorname{Tw}(W)$ obtained
from $F\restrict W$ as in the proof of
\ref{_conne_=>_hyperho_Proposition_}.
The singular set of $\c F$ is $\operatorname{Tw}(S)\subset \operatorname{Tw}(W)$.
Therefore, the restriction
$\c F\restrict{\operatorname{Tw}(W)\backslash \operatorname{Tw}(S)}$ is a holomorphic vector bundle.
For all horisontal twistor lines $l_x\subset \operatorname{Tw}(W)\backslash \operatorname{Tw}(S)$,
the restriction $\c F\restrict {l_x}$ is clearly a trivial vector
bundle over $l_x\cong {\Bbb C} P^1$. A small deformation of a trivial
vector bundle is again trivial. Shrinking
$W$ if necessary, we may assume that for all
lines $l\in L_0$, the restriction of $\c F$ to $l\cong {\Bbb C} P^1$
is a trivial vector bundle.
The isomorphism
$\underline \Psi_{1,2}:\;\underline F_1{\:\longrightarrow\:}\underline F_2$
is constructed as follows. Let $l\in L_0$ be a twistor line. The restriction
$\c F\restrict {l}$ is trivial. Consider $l$ as a map
$l:\; {\Bbb C} P^1 {\:\longrightarrow\:} \operatorname{Tw}(M)$. We identify ${\Bbb C} P^1$ with the set
of induced complex structures on $M$. By definition,
the fiber of $F_1$ in $l$
is naturally identified with the space $\c F\restrict {l(I)}$,
and the fiber of $F_2$ in $l$ is identifies with
$\c F\restrict {l(J)}$. Since $\c F\restrict {l}$ is trivial,
the fibers of the bundle $\c F\restrict {l}$ are naturally
identified. This provides a vector bundle isomorphism
$\underline \Psi_{1,2}:\;\underline F_1{\:\longrightarrow\:}\underline F_2$
mapping $\underline F_1 \restrict l = \c F\restrict {l(I)}$
to $\underline F_2 \restrict l = \c F\restrict {l(J)}$.
It remains to show that this isomorphism is compatible
with the holomorphic structure.
Since the bundle $\c F\restrict l$ is trivial, we have
an identification
\[ \c F\restrict {l(I)}\cong\c F\restrict{l(J)}= \Gamma(\c F\restrict l),
\]
where $\Gamma(\c F\restrict l)$ is the space of global sections
of $\c F\restrict l$.
Thus, $F_i \restrict l = \Gamma(\c F\restrict l)$, and this
identification is clearly holomorphic. This proves
\ref{_F_1_=F_2_Lemma_}. \blacksquare
\hfill
We return to the proof of \ref{_Psi_equiv_hyperho_Theorem_}.
Denote by $F_J$ the restriction of $\c F$ to
$(M, J)= \pi^{-1}(J) \subset \operatorname{Tw}(M)$.
The map $\Psi_{I, J}$ was defined as a composition
$e_I\circ p_J \circ e_J \circ p_I$.
The sheaf $p_I^* F$ is by definition isomorphic to
$F_1$, and $p_J^* F_J$ to $F_2$. On the other hand, clearly,
$e_J^* F_2= F_J$. Therefore,
$(p_J \circ e_J)^* F_2 \cong F_2$.
Using the isomorphism $F_1\cong F_2$, we obtain
$(p_J \circ e_J)^* F_1 \cong F_1$.
To sum it up, we have the following isomorphisms:
\begin{equation*}
\begin{split}
p_I^* F &\cong F_1,\\
(p_J \circ e_J)^* F_1 &\cong F_1,\\
e_I^* F_1 &\cong F.
\end{split}
\end{equation*}
A composition of these isomorphisms gives
an isomorphism
\[ \Psi_{I,J}^F:\; F\restrict {B_0} {\:\longrightarrow\:} \Psi_{I,J}^*(F\restrict B).
\]
This proves \ref{_Psi_equiv_hyperho_Theorem_}.
\blacksquare
\subsection{A ${\Bbb C}^*$-action on a local ring of a hyperk\"ahler manifold}
\label{_C^*_action_on_loca_ring_Subsection_}
Let $M$ be a hyperk\"ahler manifold, non necessarily compact,
$x\in M$ a point and
$I$, $J$ induced complex structures, $I\neq J$.
Consider the complete local ring
${\cal O}_{x, I}:= \hat {\cal O}_x(M, I)$.
Throughout this section we consider the map $\Psi_{I,J}$
(\ref{_Psi_acts_on_TM_Proposition_})
as an automorphism of the ring ${\cal O}_{x, I}$.
Let $\frak m$ be the maximal ideal of
${\cal O}_{x, I}$, and $\frak m/\frak m^2$
the Zariski cotangent space of $(M,I)$ in $x$.
\begin{equation} \label{_Psi_acts_on_cota_Equation_}
\begin{minipage}[m]{0.8\linewidth}
By \ref{_Psi_acts_on_TM_Proposition_},
$\Psi_{I,J}$ acts on $\frak m/\frak m^2$
as a multiplication by a number $\lambda\in{\Bbb C}$,
$0<|\lambda|<1$.
\end{minipage}
\end{equation}
Let $V_{\lambda^n}$ be the eigenspace corresponding to
the eigenvalue $\lambda^n$,
\[ V_{\lambda^n}:= \{ v\in {\cal O}_{x, I}\ \
|\ \ \Psi_{I,J}(v) = \lambda^n v\}.
\]
Clearly, $\oplus V_{\lambda^i}$ is a graded subring in ${\cal O}_{x, I}$.
In \cite{_Verbitsky:DesinguII_},
(see also \cite{_Verbitsky:hypercomple_}) we proved that
the ring $\oplus V_{\lambda^i}$ is dense in ${\cal O}_{x, I}$
with respect to the adic topology. Therefore, the ring
${\cal O}_{x, I}$ is identified with the adic completion of
$\oplus V_{\lambda^i}$.
\hfill
Consider at action of ${\Bbb C}^*$ on $\oplus V_{\lambda^i}$, with
$z\in {\Bbb C}^*$ acting on $V_{\lambda^i}$ as a multiplication by
$z^i$. This ${\Bbb C}^*$-action is clearly continuous, with respect to
the adic topology. Therefore, it can be extended to
\[{\cal O}_{x, I} = \widehat {\oplus V_{\lambda^i}}.\]
\hfill
\definition \label{_Psi(z)_Definition_}
Let $M$, $I$, $J$, $x$, ${\cal O}_{x, I}$ be as in the beginning
of this Subsection. Consider the ${\Bbb C}^*$-action
\[ \Psi_{I,J}(z):\; {\cal O}_{x, I} {\:\longrightarrow\:} {\cal O}_{x, I} \]
constructed as above. Then $\Psi_{I,J}(z)$ is called
{\bf the canonical ${\Bbb C}^*$-action associated with
$M$, $I$, $J$, $x$.}
\hfill
In the above notation, consider a reflexive sheaf $F$ on $(M,I)$
equipped with a hyperholomorphic connection. Denote the germ of
$F$ at $x$ by $F_x$, $F_x:= F\otimes_{{\cal O}_{(M,I)}} {\cal O}_{x, I}$.
From \ref{_Psi_equiv_hyperho_Theorem_}, we obtain an isomorphism
$F_x \cong \Psi^*_{I,J} F_x$. This isomorphism can be interpreted
as an automorphism
\[ \Psi^F_{I,J}:\; F_x{\:\longrightarrow\:} F_x\]
satisfying
\begin{equation}\label{_Psi^F_and_multi_Equation_}
\Psi^F_{I,J}(\alpha v) = \Psi_{I,J}(\alpha) v,
\end{equation}
for all $\alpha \in {\cal O}_{x, I}$, $v\in F_x$.
By \eqref{_Psi^F_and_multi_Equation_},
the automorphism $\Psi^F_{I,J}$ respects the filtration
\[ F_x \supset \frak m F_x \supset \frak m^2 F_x \supset ... \]
Thus, it makes sense to speak of $\Psi^F_{I,J}$-action
on $\frak m^i F_x /\frak m^{i+1} F_x$.
\hfill
\lemma\label{_Psi^F_on_m^iF/m^i+1F_Lemma_}
The automorphism $\Psi^F_{I,J}$ acts on
$\frak m^i F_x /\frak m^{i+1} F_x$
as a multiplication by $\lambda^i$, where
$\lambda\in {\Bbb C}$ is the number considered in
\eqref{_Psi_acts_on_cota_Equation_}.
\hfill
{\bf Proof:} By \eqref{_Psi^F_and_multi_Equation_},
it suffices to prove \ref{_Psi^F_on_m^iF/m^i+1F_Lemma_}
for $i=0$. In other words, we have to show that
$\Psi^F_{I,J}$ acts as identity on $F_x /\frak m F_x$.
We reduced \ref{_Psi^F_on_m^iF/m^i+1F_Lemma_} to the following
claim.
\hfill
\claim \label{_Psi^F_identi_on_F/mF_Claim_}
In the above assumptions,
the automorphism $\Psi^F_{I,J}$
acts as identity on $F_x /\frak m F_x$.
\hfill
{\bf Proof:}
In the course of defining the map
$\Psi^F_{I,J}$, we identified the space $\operatorname{Lin}(M)$ with
a complexification of $(M,I)$, and defined
the maps
\[ p_I:\; \operatorname{Lin}(M) {\:\longrightarrow\:} (M, I),\ \
p_J:\; \operatorname{Lin}(M) {\:\longrightarrow\:} (M, I)
\]
(these maps are smooth, in a neighbourhood of $\operatorname{Hor}\subset \operatorname{Lin}(M)$,
by \ref{_Lin_is_MxM_Proposition_}),
and
\[ e_I:\; (B, I) {\:\longrightarrow\:} \operatorname{Lin}(M),\ \
e_J:\; (B, J) {\:\longrightarrow\:} \operatorname{Lin}(M)
\]
(these maps are locally
closed embeddings). Consider $(M,J)$ as a subvariety
of $\operatorname{Tw}(M)$, $(M, J) = \pi^{-1}(J)$.
Let $\c F$ be the lift of $F$ to $\operatorname{Tw}(M)$
(see the proof of \ref{_conne_=>_hyperho_Proposition_}
for details). Denote the completion of
${\cal O}_x(M, J)$ by ${\cal O}_{x, J}$.
Let $F_J$ denote the ${\cal O}_{x, J}$-module
$\left(\c F\restrict{(M,J)} \right)\otimes_{{\cal O}_{(M,J)}} \hat {\cal O}_{x, J}$.
Consider the horisontal twistor line
$l_x\in \operatorname{Lin}(M)$. Let $\operatorname{Lin}_x(M)$ be the spectre of the
completion ${\cal O}_{x,\operatorname{Lin}}$ of the local ring of
holomorphic functions on $\operatorname{Lin}(M)$ in $l_x$.
The maps $p_I$, $p_J$, $e_I$, $e_J$ can be considered
as maps of corresponding formal manifolds:
\begin{equation*}
\begin{split}
p_I:\; \operatorname{Lin}_x(M) & {\:\longrightarrow\:} \operatorname{Spec}({\cal O}_{x, I}),\\
p_J:\; \operatorname{Lin}_x(M) & {\:\longrightarrow\:} \operatorname{Spec}({\cal O}_{x, J}),\\
e_I:\; \operatorname{Spec}({\cal O}_{x, I})& {\:\longrightarrow\:}\operatorname{Lin}_x(M),\\
e_J:\; \operatorname{Spec}({\cal O}_{x, J})& {\:\longrightarrow\:}\operatorname{Lin}_x(M),
\end{split}
\end{equation*}
As in Subsection \ref{_Psi_on_shea_Subsection_},
we consider the ${\cal O}_{x,\operatorname{Lin}}$-modules
$F_1:= p_I^* F_x$ and $F_2:= p_J^* F_J$. By \ref{_F_1_=F_2_Lemma_},
there exists a natural isomorphism $\Psi_{1,2}:\; F_1{\:\longrightarrow\:} F_2$.
Let $\frak m_{l_x}$ be the maximal ideal of ${\cal O}_{x,\operatorname{Lin}}$.
Since the morphism $p_I$ is smooth, the space
$F_1/ \frak m_{l_x} F_1$ is naturally isomorphic to
$F_x/\frak m F_x$. Similarly, the space
$F_2/ \frak m_{l_x} F_2$ is isomorphic to
$F_J / \frak m_J F_J$, where $\frak m_J$ is the
maximal ideal of ${\cal O}_{x, J}$. We have a chain of isomorphisms
\begin{equation} \label{_chain_iso_Psi_F/mF_Equation_}
\begin{split}
F_x/\frak m F_x &\stackrel{p_I^*}{\:\longrightarrow\:} F_1/ \frak m_{l_x} F_1
\stackrel {\Psi_{1,2}} {\:\longrightarrow\:} F_2/ \frak m_{l_x} F_2\\
&\stackrel{e_J^*}{\:\longrightarrow\:} F_J/ \frak m_{J} F_J
\stackrel{p_J^*}{\:\longrightarrow\:} F_2/ \frak m_{l_x} F_2\\
&\stackrel {\Psi_{1,2}^{-1}} {\:\longrightarrow\:} F_1/ \frak m_{l_x} F_1
\stackrel{e_I^*}{\:\longrightarrow\:} F_x/ \frak m F_x.
\end{split}
\end{equation}
By definition, for any $f\in F_x/\frak m F_x$,
the value of $\Psi^F_{I,J}(f)$ is given by
the composide map of \eqref{_chain_iso_Psi_F/mF_Equation_}
applied to $f$. The composition
\begin{equation} \label{_restri_F/mF_F_2_and_back_Equation_}
F_2/ \frak m_{l_x} F_2
\stackrel{e_J^*}{\:\longrightarrow\:} F_J/ \frak m_{J} F_J
\stackrel{p_J^*}{\:\longrightarrow\:} F_2/ \frak m_{l_x} F_2
\end{equation}
is identity, because the spaces
$F_2/ \frak m_{l_x} F_2$ and
$F_J / \frak m_J F_J$ are canonically identified, and this
identification can be performed via
$e_J^*$ or $p_J^*$. Thus,
the map \eqref{_chain_iso_Psi_F/mF_Equation_} is a composition
\[
F_x/\frak m F_x \stackrel{p_I^*}{\:\longrightarrow\:} F_1/ \frak m_{l_x} F_1
\stackrel {\Psi_{1,2}} {\:\longrightarrow\:} F_2/ \frak m_{l_x} F_2
\stackrel {\Psi_{1,2}^{-1}} {\:\longrightarrow\:} F_1/ \frak m_{l_x} F_1
\stackrel{e_I^*}{\:\longrightarrow\:} F_x/ \frak m F_x.
\]
This map is clearly equivalent to a composition
\[ F_x/\frak m F_x \stackrel{p_I^*}{\:\longrightarrow\:} F_1/ \frak m_{l_x} F_1
\stackrel{e_I^*}{\:\longrightarrow\:} F_x/ \frak m F_x,
\]
which is identity according to the same reasoning which proved that
\eqref{_restri_F/mF_F_2_and_back_Equation_} is identity.
We proved \ref{_Psi^F_identi_on_F/mF_Claim_}
and \ref{_Psi^F_on_m^iF/m^i+1F_Lemma_}. \blacksquare
\hfill
Consider the $\lambda^n$-eigenspaces $F_{\lambda^n}$ of $F_x$.
Consider the $\oplus V_{\lambda^n}$-submodule
$\oplus F_{\lambda^n}\subset F_x$, where
$\oplus V_{\lambda^n}\subset {\cal O}_{x, I}$ is
the ring defined in Subsection
\ref{_C^*_action_on_loca_ring_Subsection_}.
From \ref{_Psi^F_identi_on_F/mF_Claim_} and
\eqref{_Psi_acts_on_cota_Equation_} it follows that
$\oplus F_{\lambda^n}$ is dense in
$F_x$, with respect to the adic topology on
$F_x$. For $z\in {\Bbb C}^*$, let
$\Psi_{I,J}^F(z):\; \oplus F_{\lambda^n}{\:\longrightarrow\:} \oplus F_{\lambda^n}$
act on $F_{\lambda^n}$ as a multiplication by $z^n$.
As in \ref{_Psi(z)_Definition_}, we extend
$\Psi_{I,J}^F(z)$ to $F_x = \widehat{\oplus F_{\lambda^n}}$.
This automorphism makes $F_x$ into a ${\Bbb C}^*$-equivariant
module over ${\cal O}_{x, I}$
\hfill
\definition\label{_C^*_stru_on_sge_Definition_}
The constructed above ${\Bbb C}^*$-equivariant structure
on $F_x$ is called {\bf the canonical ${\Bbb C}^*$-equivariant structure
on $F_x$ associated with $J$}.
\subsection{Desingularization of ${\Bbb C}^*$-equivariant sheaves}
\label{_desingu_she_Subsection_}
Let $M$ be a hyperk\"ahler manifold, $I$ an induced complex structure
and $F$ a reflexive sheaf with isolated singularities
over $(M,I)$, equipped with a
hyperholomorphic connection. We have shown that the sheaf $F$ admits
a ${\Bbb C}^*$-equivariant structure compatible
with the canonical ${\Bbb C}^*$-action on the local ring of $(M,I)$.
Therefore, \ref{_desingu_hyperho_Theorem_}
is implied by the following proposition.
\hfill
\proposition \label{_desingu_C^*_equi_Proposition_}
Let $B$ be a complex manifold, $x\in B$ a point.
Assume that there is an action $\Psi(z)$ of ${\Bbb C}^*$ on $B$
which fixes $x$ and acts on $T_x B$ be dilatations. Let
$F$ be a reflexive coherent sheaf on $B$, which is
locally trivial outside of $x$.
Assume that the germ
$F_x$ of $F$ in $x$ is equipped with a ${\Bbb C}^*$-equivariant structure,
compatible with $\Psi(z)$.
Let $\tilde B$ be a blow-up of $B$
in $x$, and $\pi:\; \tilde B{\:\longrightarrow\:} B$ the
standard projection. Then the pullback
sheaf $\tilde F:= \pi^* F$ is locally trivial on $\tilde B$.
\hfill
{\bf Proof:} Let $C:= \pi^{-1}(x)$ be the singular
locus of $\pi$. The sheaf $F$ is locally trivial
outside of $x$. Let $d$ be the rank of
$F\restrict{B\backslash x}$. To prove that
$\tilde F$ is locally trivial, we need to show that
for all points $y\in \tilde B$, the fiber
$\pi^* F\restrict y$ is $d$-dimensional. Therefore,
to prove \ref{_desingu_C^*_equi_Proposition_}
it suffices to show that $\pi^* F\restrict C$
is a vector bundle of dimension $d$.
The variety $C$ is naturally identified
with the projectivization ${\Bbb P} T_x B$ of the
tangent space $T_xB$. The total space of
$T_x B$ is equipped with a natural action of ${\Bbb C}^*$,
acting by dilatations. Clearly, coherent sheaves on
${\Bbb P} T_x B$ are in one-to-one
correspondence with ${\Bbb C}^*$-equivariant
coherent sheaves on $T_x B$.
Consider a local isomorphism
$\phi:\; T_x B {\:\longrightarrow\:} B$ which is compatible
with ${\Bbb C}^*$-action, maps $0\in T_x B$ to $x$ and acts as
identity on the tangent space $T_0 (T_x B)=T_x B$.
The sheaf $\phi^* F$ is ${\Bbb C}^*$-equivariant.
Clearly, the corresponding sheaf on
${\Bbb P} T_x B$ is canonically isomorphic with
$\pi^*F \restrict C$. Let $l\in T_x B$ be a line passing
through $0$, and $l\backslash 0$ its complement to $0$.
Denote the corresponding point of ${\Bbb P} T_x B$ by $y$.
The restriction $\phi^* F\restrict {l\backslash 0}$
is a ${\Bbb C}^*$-equivariant vector bundle.
The ${\Bbb C}^*$-equivariant structure
identifies all the fibers of the bundle
$\phi^* F\restrict {l\backslash 0}$.
Let $F_l$ be one of these fibers.
Clearly, the fiber of $\pi^*F \restrict C$ in $y$
is canonically isomorphic to $F_l$. Therefore,
the fiber of $\pi^*F \restrict C$ in $y$ is $d$-dimensional.
We proved that $\pi^*F$ is a bundle. \blacksquare
\section{Twistor transform and quaternionic-K\"ahler geometry}
\label{_twisto_tra_Section_}
This Section is a compilation of results known from the literature.
Subsection \ref{_dire_inve_twi_Subsection_} is based on
\cite{_NHYM_} and the results of
Subsection \ref{_twi_tra_Hermi_Subsection_}
are implicit in \cite{_NHYM_}.
Subsection \ref{_B_2_bundles_Subsection_} is based
on \cite{_Salamon_}, \cite{_Nitta:bundles_}
and \cite{_Nitta:Y-M_}, and Subsection
\ref{_specia_and_q-K-Subsection_} is a recapitulation
of the results of A. Swann (\cite{_Swann_}).
\subsection{Direct and inverse twistor transform}
\label{_dire_inve_twi_Subsection_}
In this Subsection, we recall the definition and the main properties
of the direct and inverse twistor transform for bundles over
hyperk\"ahler manifolds (\cite{_NHYM_}).
\hfill
The following definition is a non-Hermitian analogue of the
notion of a hyperholomorphic connection.
\hfill
\definition
Let $M$ be a hyperk\"ahler manifold,
not necessarily compact, and
$(B, \nabla)$ be a vector bundle with a connection
over $M$, not necessarily Hermitian. Assume that
the curvature of $\nabla$ is contained in the space
$\Lambda^2_{inv}(M, \operatorname{End}(B))$ of $SU(2)$-invariant
2-forms with coefficients in $\operatorname{End}(B)$. Then
$(B, \nabla)$ is called {\bf an autodual bundle},
and $\nabla$ {\bf an autodual connection}.
\hfill
Let
$\operatorname{Tw}(M)$ be the twistor space of $M$, equipped with the standard
maps $\pi:\; \operatorname{Tw}(M) {\:\longrightarrow\:} {\Bbb C} P^1$, $\sigma:\; \operatorname{Tw}(M) {\:\longrightarrow\:} M$.
\hfill
We introduce the direct and inverse twistor
transforms which relate autodual
bundles on the hyperk\"ahler manifold $M$ and
holomorphic bundles on its twistor space $\operatorname{Tw}(M)$.
\hfill
Let $B$ be a complex vector bundle on $M$ equipped with a connection
$\nabla$. The pullback $\sigma^*B$ of $B$ to $\operatorname{Tw}(M)$ is equipped with a
pullback connection $\sigma^*\nabla$.
\hfill
\lemma \label{_autodua_(1,1)-on-twi_Lemma_}
(\cite{_NHYM_}, Lemma 5.1)
The connection $\nabla$ is autodual if and only if the connection
$\sigma^*\nabla$ has curvature of Hodge type $(1,1)$.
{\bf Proof:} Follows from \ref{_SU(2)_inva_type_p,p_Lemma_}. \blacksquare
\hfill
In assumptions of
\ref{_autodua_(1,1)-on-twi_Lemma_},
consider the $(0,1)$-part $(\sigma^*\nabla)^{0,1}$
of the connection $\sigma^*\nabla$. Since
$\sigma^*\nabla$ has curvature of Hodge type $(1,1)$,
we have \[ \left((\sigma^*\nabla)^{0,1}\right)^2=0, \] and by
\ref{_Newle_Nie_for_NH_bu_Proposition_}, this connection
is integrable. Consider $(\sigma^*\nabla)^{0,1}$
as a holomorphic structure operator on $\sigma^* B$.
\hfill
Let $\c A$ be the category of autodual bundles on
$M$, and $\c C$ the category of holomorphic vector
bundles on $\operatorname{Tw}(M)$. We have constructed a functor
\[ (\sigma^* \bullet)^{0,1}:\; \c A {\:\longrightarrow\:} \c C, \]
$\nabla {\:\longrightarrow\:} (\sigma^*\nabla)^{0,1}$.
Let $s\in \operatorname{Hor}\subset \operatorname{Tw}(M)$ be a horisontal twistor
line (Subsection \ref{_twi_lines_C^*_Subsection_}).
For any $(B, \nabla)\in \c A$,
consider corresponding holomorphic vector
bundle $(\sigma^* B, (\sigma^*\nabla)^{0,1})$.
The restriction of $(\sigma^* B, (\sigma^*\nabla)^{0,1})$
to $s\cong {\Bbb C} P^1$ is a trivial vector bundle.
A converse statement is also true. Denote by
$\c C_0$ the category of holomorphic vector bundles
$C$ on $\operatorname{Tw}(M)$, such that the restriction of
$C$ to any horisontal twistor line is trivial.
\hfill
\theorem \label{_dire_inve_twisto_Theorem_}
Consider the functor
\[ (\sigma^* \bullet)^{0,1}:\; \c A {\:\longrightarrow\:} \c C_0\]
constructed above. Then it is an equivalence of categories.
{\bf Proof:} \cite{_NHYM_}, Theorem 5.12. \blacksquare
\hfill
\definition
Let $M$ be a hyperk\"ahler manifold,
$\operatorname{Tw}(M)$ its twistor space and $\c F$ a holomorphic vector
bundle. We say that $\c F$ is {\bf compatible with twistor
transform} if the restriction of
$C$ to any horisontal twistor line
$s\in \operatorname{Tw}(M)$ is a trivial bundle on $s\cong {\Bbb C} P^1$.
\hfill
Recall that a connection $\nabla$ in a vector bundle over
a complex manifold is called $(1,1)$-connection if its
curvature is of Hodge type $(1,1)$.
\hfill
\remark \label{_cano_conne_Remark_}
Let $\c F$ be a holomorphic bundle over $\operatorname{Tw}(M)$
which is compatible with twistor transform. Then $\c F$
is equipped with a natural $(1,1)$-connection
$\nabla_{\c F}= \sigma^* \nabla$,
where $(B, \nabla)$ is the corresponding autodual bundle over $M$.
The connection $\nabla_{\c F}$ is not, generally speaking,
Hermitian, or compatible with a Hermitian structure.
\subsection{Twistor transform and Hermitian structures
on vector bundles}
\label{_twi_tra_Hermi_Subsection_}
Results of this Subsection were implicit in \cite{_NHYM_},
but in this presentation, they are new.
Let $M$ be a hyperk\"ahler manifold, not necessarily compact,
and $\operatorname{Tw}(M)$ its twistor space.
In Subsection \ref{_dire_inve_twi_Subsection_}, we have shown that certain
holomorphic vector bundles over $\operatorname{Tw}(M)$ admit a canonical
(1,1)-connection $\nabla{_\c F}$ (\ref{_cano_conne_Remark_}).
This connection can be non-Hermitian. Here we study
the Hermitian structures on $(\c F, \nabla{_\c F})$
in terms of holomorphic properties of $\c F$.
\hfill
\definition
Let $F$ be a real analytic complex vector bundle over a
real analytic manifold $X_{\Bbb R}$, and $h:\; F\times F {\:\longrightarrow\:} {\Bbb C}$ a
${\cal O}_{X_{\Bbb R}}$-linear pairing on $F$. Then $h$ is called
{\bf semilinear} if for all $\alpha \in {\cal O}_{X_{\Bbb R}}\otimes_{\Bbb R} {\Bbb C}$,
we have
\[ h(\alpha x, y) = \alpha\cdot h(x, y), \text{\ \ and\ \ }
h(x,\alpha y) = \bar \alpha\cdot h(x, y).
\]
For $X$ a complex manifold and $F$ a holomorphic vector bundle,
by a semilinear pairing on $F$ we understand a semilinear pairing
on the underlying real analytic bundle.
Clearly, a real analytic Hermitian metric is always semilinear.
\hfill
Let $X$ be a complex manifold, $I:\; TX {\:\longrightarrow\:} TX$
the complex structure operator, and $i:\; X {\:\longrightarrow\:} X$ a
real analytic map. We say that $X$ is anticomplex if
the induced morphism of tangent spaces satisfies
$i\circ I = - I \circ I$. For a complex
vector bundle $F$ on $X$, consider the complex
adjoint vector bundle $\bar F$, which coincides with
$F$ as a real vector bundle, with ${\Bbb C}$-action which is
conjugate to that defined on $F$. Clearly, for every
holomorphic vector bundle $F$, and any anticomplex map
$i:\; X {\:\longrightarrow\:} X$, the bundle
$i^* \bar F$ is equipped with a natural
holomorphic structure.
\hfill
Let $M$ be a hyperk\"ahler manifold, and $\operatorname{Tw}(M)$ its twistor space.
Recall that $\operatorname{Tw}(M)={\Bbb C} P^1\times M$
is equipped with a canonical anticomplex involution $\iota$,
which acts as identity on $M$ and as central symmetry
$I{\:\longrightarrow\:} -I$ on ${\Bbb C} P^1= S^2$.
For any holomorphic bundle $\c F$ on $\operatorname{Tw}(M)$, consider the corresponding
holomorphic bundle $\iota^* \bar{\c F}$.
\hfill
Let $M$ be a hyperk\"ahler manifold, $I$ an induced complex
structure, $F$ a vector bundle over $M$, equipped with an autodual
connection $\nabla$, and $\c F$ the corresponding holomorphic
vector bundle over $\operatorname{Tw}(M)$, equipped with a canonical connection
$\nabla_{\c F}$.
As usually, we identify $(M, I)$ and the fiber $\pi^{-1}(I)$
of the twistor projection $\pi:\; \operatorname{Tw}(M) {\:\longrightarrow\:} {\Bbb C} P^1$.
Let $\nabla_{\c F} = \nabla^{1,0}_I + \nabla^{0,1}_I$
be the Hodge decomposition of $\nabla$ with respect to $I$.
\begin{equation} \label{_nabla^1,0_as_holo_Equation_}
\begin{minipage}[m]{0.8\linewidth}
Clearly, the operator $\nabla^{1,0}_I$ can be considered
as a holomorphic structure operator on $F$, considered as
a complex vector bundle over $(M, -I)$.
\end{minipage}
\end{equation}
Then the holomorphic structure operator on $\c F\restrict{(M,I)}$
is equal to $\nabla^{0,1}_I$, and the
holomorphic structure operator on $\c F\restrict{(M,-I)}$
is equal to $\nabla^{1,0}_I$.
Assume that
the bundle $(\c F, \nabla_{\c F})$ is equipped with a
non-degenerate semilinear pairing $h$ which is compatible with
the connection. Consider the natural connection $\nabla_{\c F^*}$
on the dual bundle to $\c F$, and its Hodge decomposition
(with respect to $I$)
\[ \nabla_{\c F^*} = \nabla^{1,0}_{\c F^*} + \nabla^{0,1}_{\c F^*}. \]
Clearly, the pairing $h$ gives a $C^\infty$-isomorphism
of $\c F$ and the complex conjugate of its dual bundle, denoted as
$\bar{\c F}^*$. Since $h$ is semilinear and
compatible with the connection, it maps the holomorphic structure
operator $\nabla^{0,1}_I$ to the complex conjugate of
$\nabla^{1,0}_{\c F^*}$. On the other hand,
the operator $\nabla^{1,0}_{\c F^*}$ is a
holomorphic structure operator in
$\c F^*\restrict{(M,-I)}$, as
\eqref{_nabla^1,0_as_holo_Equation_} claims.
We obtain that the map $h$ can be considered as
an isomorphism of holomorphic vector bundles
\[ h:\; \c F {\:\longrightarrow\:} (\iota^* \bar{\c F})^*.\]
This correspondence should be thought of as a
(direct) twistor
transform for bundles with a semilinear pairing.
\hfill
\proposition\label{_twi_tra_for_semili_Proposition_}
(direct and inverse twistor
transform for bundles with semilinear pairing)
Let $M$ be a hyperk\"ahler manifold, and $\c C_{sl}$
the category of autodual bundles over $M$ equipped with
a non-degenerate semilinear pairing. Consider the category
$\c C_{hol,sl}$ of holomorphic vector bundles $\c F$
on $\operatorname{Tw}(M)$, compatible with twistor transform
and equipped with an isomorphism
\[ {\frak h}:\; \c F {\:\longrightarrow\:} (\iota^* \bar{\c F})^*.\]
Let $\c T:\; \c C_{sl}{\:\longrightarrow\:} \c C_{hol,sl}$ be the functor
constructed above. Then $\c T$ is an isomorphism of categories.
\hfill
{\bf Proof:} Given a pair
$\c F, {\frak h}:\; \c F {\:\longrightarrow\:} (\iota^* \bar{\c F})^*$,
we need to construct a non-degenerate semilinear pairing
$h$ on $\c F\restrict{(M,I)}$, compatible with a connection.
Since $\c F$ is compatible with twistor transform, it
is a pullback of a bundle $(F, \nabla)$ on $M$.
This identifies the real analytic bundles $\c F\restrict{(M,I')}$,
for all induced complex structures $I'$. Taking $I' = \pm I$,
we obtain an identification of the $C^{\infty}$-bundles
$\c F\restrict{(M,I)}$, $\c F\restrict{(M,-I)}$. Thus,
$\frak h$ can be considered as an isomorphism of
$F= \c F\restrict{(M,I)}$ and $(\bar {\c F})^*\restrict{(M,I)}$.
This allows one to consider $\frak h$ as a semilinear form $h$ on
$F$. We need only to show that $h$ is compatible with the connection
$\nabla$. Since $\nabla_{\c F}$ is an invariant of holomorphic
structure, the map ${\frak h}:\; \c F {\:\longrightarrow\:} (\iota^* \bar{\c F})^*$
is compatible with the connection $\nabla_{\c F}$. Thus, the obtained
above form $h$ is compatible with
the connection $\nabla_{\c F}\restrict{(M,I)}=\nabla$.
This proves \ref{_twi_tra_for_semili_Proposition_}.
\blacksquare
\subsection{ $B_2$-bundles on quaternionic-K\"ahler manifolds}
\label{_B_2_bundles_Subsection_}
\definition \label{_q-K_Definition_}
(\cite{_Salamon_}, \cite{_Besse:Einst_Manifo_})
Let $M$ be a Riemannian manifold. Consider a
bundle of algebras $\operatorname{End}(TM)$, where $TM$ is the tangent
bundle to $M$. Assume that $\operatorname{End}(TM)$ contains a
4-dimensional bundle of subalgebras $W\subset \operatorname{End}(TM)$,
with fibers isomorphic to a quaternion algebra
${\Bbb H}$. Assume, moreover, that
$W$ is closed under the transposition map
$\bot:\; \operatorname{End}(TM){\:\longrightarrow\:} \operatorname{End}(TM)$ and
is preserved by the Levi-Civita connection.
Then $M$ is called {\bf quaternionic-K\"ahler}.
\hfill
\example
Consider the quaternionic projective space
\[ {\Bbb H} P^n= ({\Bbb H}^n \backslash 0) / {\Bbb H}^*. \]
It is easy to see that ${\Bbb H} P^n$
is a quaternionic-K\"ahler manifold.
For more examples of quaternionic-K\"ahler manifolds,
see \cite{_Besse:Einst_Manifo_}.
\hfill
A quaternionic-K\"ahler manifold is Einstein
(\cite{_Besse:Einst_Manifo_}),
i. e. its Ricci tensor is proportional to the
metric: $Ric(M) = c \cdot g$, with $c\in {\Bbb R}$.
When $c=0$, the manifold $M$ is hyperk\"ahler,
and its restricted holonomy group is $Sp(n)$;
otherwise, the restricted holonomy is
$Sp(n)\cdot Sp(1)$. The number $c$ is called
{\bf the scalar curvature} of $M$. Further on, we
shall use the term {\it quaternionic-K\"ahler manifold}
for manifolds with non-zero scalar curvature.
The quaternionic projective space
${\Bbb H} P^n$ has positive scalar curvature.
\hfill
The quaternionic projective space is the only example of
quaternionic-K\"ahler manifold which we need, in the course of this
paper. However, the formalism of quaternionic-K\"ahler manifolds
is very beautiful and significantly
simplifies the arguments, so we state the definitions and
results for a general quaternionic-K\"ahler manifold
whenever possible.
\hfill
Let $M$ be a quaternionic-K\"ahler manifold,
and $W\subset \operatorname{End}(TM)$ the corresponding 4-dimensional
bundle. For $x\in M$, consider the set $\c R_x\subset W\restrict x$,
consisting of all $I\in W\restrict x$ satisfying
$I^2=-1$. Consider $\c R_x$ as a Riemannian
submanifold of the total space of $W\restrict x$. Clearly,
$\c R_x$ is isomorphic to a 2-dimensional sphere.
Let $\c R= \cup_x \c R_x$ be the corresponding spherical fibration
over $M$, and $\operatorname{Tw}(M)$ its total space. The manifold
$\operatorname{Tw}(M)$ is equipped with an almost complex structure,
which is defined in the same way as the almost complex
structure for the twistor space of a hyperk\"ahler manifold.
This almost complex structure is known to be integrable
(see \cite{_Salamon_}).
\hfill
\definition\label{_twi_q-K_Definition_}
(\cite{_Salamon_}, \cite{_Besse:Einst_Manifo_})
Let $M$ be a quaternionic-K\"ahler manifold.
Consider the complex manifold $\operatorname{Tw}(M)$ constructed
above. Then $\operatorname{Tw}(M)$ is called {\bf the twistor space of
$M$.}
\hfill
Note that (unlike in the hyperk\"ahler case) the space
$\operatorname{Tw}(M)$ is K\"ahler. For quaternionic-K\"ahler manifolds
with positive scalar curvature, the anticanonical bundle
of $\operatorname{Tw}(M)$ is ample, so $\operatorname{Tw}(M)$ is a Fano manifold.
\hfill
Quaternionic-K\"ahler analogue of a twistor transform
was studied by T. Nitta in a serie of papers
(\cite{_Nitta:bundles_}, \cite{_Nitta:Y-M_} etc.)
It turns out that the picture given in
\cite{_NHYM_} for K\"ahler manifolds is
very similar to that observed by T. Nitta.
\hfill
A role of $SU(2)$-invariant 2-forms is played
by so-called $B_2$-forms.
\hfill
\definition
Let $SO(TM)\subset \operatorname{End}(TM)$ be a group bundle of all
orthogonal automorphisms of $TM$, and
$G_M:= W\cap SO(TM)$. Clearly, the fibers of $G_M$
are isomorphic to $SU(2)$. Consider the action of
$G_M$ on the bundle of 2-forms $\Lambda^2(M)$.
Let $\Lambda^2_{inv}(M)\subset \Lambda^2(M)$
be the bundle of $G_M$-invariants. The bundle
$\Lambda^2_{inv}(M)$ is called {\bf the bundle
of $B_2$-forms}. In a similar fashion we define
$B_2$-forms with coefficients in a bundle.
\hfill
\definition\label{_B_2_bu_Definition_}
In the above assumptions, let $(B, \nabla)$ be a bundle
with connection over $M$. The bundle $B$ is called
{\bf a $B_2$-bundle}, and
$\nabla$ is called {\bf a $B_2$-connection},
if its curvature is a $B_2$-form.
\hfill
Consider the natural projection $\sigma:\; \operatorname{Tw}(M){\:\longrightarrow\:} M$.
The proof of the following claim is completely analogous
to the proof of \ref{_SU(2)_inva_type_p,p_Lemma_} and
\ref{_autodua_(1,1)-on-twi_Lemma_}.
\hfill
\claim\label{_B_2_=_holo_on_Tw_Claim_}
\begin{description}
\item[(i)] Let $\omega$ be a $2$-form on $M$.
The pullback $\sigma^* \omega$ is of type
$(1,1)$ on $\operatorname{Tw}(M)$ if and only if $\omega$
is a $B_2$-form on $M$.
\item[(ii)]
Let $B$ be a complex vector bundle on $M$ equipped with a
connection $\nabla$, not necessarily Hermitian.
The pullback $\sigma^*B$ of $B$ to $\operatorname{Tw}(M)$ is equipped
with a pullback connection $\sigma^*\nabla$.
Then, $\nabla$ is a $B_2$-connection if and only if
$\sigma^*\nabla$ has curvature of Hodge type $(1,1)$.
\end{description}
\blacksquare
There exists an analogue of direct and inverse twistor
transform as well.
\hfill
\theorem \label{_dire_inve_q-K_Theorem_}
For any $B_2$-connection $(B, \nabla)$,
consider the corresponding holomorphic vector
bundle \[ (\sigma^* B, (\sigma^*\nabla)^{0,1}). \]
The restriction of $(\sigma^* B, (\sigma^*\nabla)^{0,1})$
to a line $\sigma^{-1}(m)\cong {\Bbb C} P^1$ is a trivial vector bundle,
for any point $m\in M$.
Denote by $\c C_0$ the category of holomorphic vector bundles
$C$ on $\operatorname{Tw}(M)$, such that the restriction of
$C$ to $\sigma^{-1}(m)$ is trivial, for all $m\in M$,
and by $\c A$ the category of $B_2$-bundles (not necessarily
Hermitian). Consider the functor
\[ (\sigma^* \bullet)^{0,1}:\; \c A {\:\longrightarrow\:} \c C_0\]
constructed above. Then it is an equivalence of categories.
{\bf Proof:} It is easy to modify
the proof of the direct and inverse twistor transform
theorem from \cite{_NHYM_} to work in quaternionic-K\"ahler
situation. \blacksquare
\hfill
We will not use \ref{_dire_inve_q-K_Theorem_},
except for its consequence, which was proven
in \cite{_Nitta:bundles_}.
\hfill
\corollary \label{_twi_tra_q-K_Hermi_Corollary_}
Consider the functor
\[ (\sigma^* \bullet)^{0,1}:\; \c A {\:\longrightarrow\:} \c C_0\]
constructed in \ref{_dire_inve_q-K_Theorem_}.
Then $(\sigma^* \bullet)^{0,1}$ gives an injection $\kappa$
from the set of equivalence classes of Hermitian $B_2$-connections over $M$ to
the set of equivalence classes of holomorphic connections
over $\operatorname{Tw}(M)$.
\blacksquare
\hfill
Let $M$ be a quaternionic-K\"ahler manifold. The space $\operatorname{Tw}(M)$
has a natural K\"ahler metric $g$,
such that the standard map $\sigma:\; \operatorname{Tw}(M){\:\longrightarrow\:} M$ is a Riemannian
submersion, and the restriction of $g$ to the fibers $\sigma^{-1}(m)$
of $\sigma$ is a metric of constant curvature on $\sigma^{-1}(m)= {\Bbb C} P^1$
(\cite{_Salamon_}, \cite{_Besse:Einst_Manifo_}).
\hfill
\example\label{_Tw_HP^n_Fu-St_Example_}
In the case
$M= {\Bbb H}P^n$, we have $\operatorname{Tw}(M) = {\Bbb C} P^{2n+1}$, and the K\"ahler
metric $g$ is proportional to the Fubini-Study metric on
${\Bbb C} P^{2n+1}$.
\hfill
\theorem \label{_twi_tra_YM_q-K_Theorem_}
(T. Nitta)
Let $M$ be a quaternionic-K\"ahler manifold of positive scalar curvature,
$\operatorname{Tw}(M)$ its twistor space, equipped with
a natural K\"ahler structure,
and $B$ a Hermitian $B_2$-bundle on $M$.
Consider the pullback $\sigma^* B$, equipped with
a Hermitian connection. Then $\sigma^* B$
is a Yang-Mills bundle on $\operatorname{Tw}(M)$, and
$\deg c_1(\sigma^* B) =0$.
{\bf Proof:} \cite{_Nitta:Y-M_}. \blacksquare
\hfill
Let $\kappa$ be the map considered in
\ref{_twi_tra_q-K_Hermi_Corollary_}. Assume that
$M$ is a compact manifold. In \cite{_Nitta:Y-M_}, T. Nitta
defined the moduli space of Hermitian $B_2$-bundles.
By Uhlenbeck-Yau theorem, Yang-Mills bundles
are polystable. Then the map $\kappa$ provides an
embedding from the moduli of non-decomposable
Hermitian $B_2$-bundles to the moduli $\c M$
of stable bundles on $\operatorname{Tw}(M)$. The image of
$\kappa$ is a totally real subvariety in $\c M$
(\cite{_Nitta:Y-M_}).
\subsection{Hyperk\"ahler manifolds with special
${\Bbb H}^*$-action and qua\-ter\-ni\-o\-nic-\-K\"ah\-ler manifolds
of positive scalar curvature}
\label{_specia_and_q-K-Subsection_}
Further on, we shall need the following definition.
\hfill
\definition
{\bf An almost hypercomplex manifold}
is a smooth manifold $M$ with an action of quaternion
algebra in its tangent bundle
For each $L\in \Bbb H$, $L^2 = -1$, $L$ gives
an almost complex structure on $M$.
The manifold $M$ is caled {\bf hypercomplex}
if the almost complex structure $L$ is integrable,
for all possible choices $L\in \Bbb H$.
\hfill
The twistor space for a hypercomplex manifold is defined in the
same way as for hyperk\"ahler manifolds. It is also a
complex manifold (\cite{_Kaledin_}). The formalism of direct
and inverse twistor transform can be repeated for
hypercomplex manifolds verbatim.
\hfill
Let ${\Bbb H}^*$ be the group of non-zero quaternions.
Consider an embedding $SU(2)\hookrightarrow {\Bbb H^*}$.
Clearly, every quaternion $h\in {\Bbb H}^*$
can be uniquely represented as $h= |h| \cdot g_h$,
where $g_h\in SU(2)\subset {\Bbb H}^*$. This gives a
natural decomposition ${\Bbb H}^*= SU(2)\times {\Bbb R}^{>0}$.
Recall that $SU(2)$ acts naturally on the set of induced complex
structures on a hyperk\"ahler manifold.
\hfill
\definition\label{_H^*_specia_Definition_}
Let $M$ be a hyperk\"ahler manifold equipped with a free
smooth action $\rho$ of the group ${\Bbb H}^*= SU(2)\times {\Bbb R}^{>0}$.
The action $\rho$ is called {\bf special}
if the following conditions hold.
\begin{description}
\item[(i)] The subgroup $SU(2)\subset {\Bbb H}^*$
acts on $M$ by isometries.
\item[(ii)] For $\lambda\in {\Bbb R}^{>0}$, the corresponding
action $\rho(\lambda):\; M{\:\longrightarrow\:} M$ is compatible
with the hyperholomorphic structure (which is a fancy way
of saying that $\rho(\lambda)$ is holomorphic with respect to any
of induced complex structures).
\item[(iii)] Consider the smooth
${\Bbb H^*}$-action $\rho_e:\; {\Bbb H^*}\times \operatorname{End}(TM) {\:\longrightarrow\:} \operatorname{End}(TM)$
induced on $\operatorname{End}(TM)$ by $\rho$. For any $x\in M$ and
any induced complex structure $I$, the restriction
$I\restrict x$ can be considered as a point in the total
space of $\operatorname{End}(TM)$.
Then, for all induced complex structures $I$,
all $g\in SU(2)\subset {\Bbb H^*}$, and all $x\in M$,
the map $\rho_e(g)$ maps $I\restrict{x}$ to
$g(I)\restrict{\rho_e(g)(x)}$.
Speaking informally, this can be stated as
``${\Bbb H}^*$-action interchanges the induced complex
structures''.
\item[(iv)]
Consider the
automorphism of $S^2 T^*M$ induced by $\rho(\lambda)$, where
$\lambda\in {\Bbb R}^{>0}$. Then $\rho(\lambda)$
maps the Riemannian metric tensor
$s\in S^2 T^*M$ to $\lambda^2 s$.
\end{description}
\hfill
\example
Consider the flat hyperk\"ahler manifold
$M_{\rm fl}= {\Bbb H}^n \backslash 0$.
There is a natural action of ${\Bbb H^*}$
on ${\Bbb H}^n\backslash 0$. This gives a special action of
${\Bbb H^*}$ on $M_{\rm fl}$.
\hfill
The case of a flat manifold $M_{\rm fl}= {\Bbb H}^n \backslash 0$
is the only case where we apply the results of this
section. However, the general statements are just as
difficult to prove, and much easier to comprehend.
\hfill
\definition\label{_speci_equi_Definition_}
Let $M$ be a hyperk\"ahler manifold with a special action $\rho$ of
${\Bbb H^*}$. Assume that $\rho(-1)$ acts non-trivially on $M$.
Then $M/\rho(\pm 1)$ is also a hyperk\"ahler manifold with
a special action of ${\Bbb H^*}$. We say that the manifolds
$(M, \rho)$ and $(M/\rho(\pm 1), \rho)$ are
{\bf hyperk\"ahler manifolds with special action of
${\Bbb H^*}$ which are special equivalent}.
Denote by $H_{sp}$ the category
of hyperk\"ahler manifolds with a special action of
${\Bbb H^*}$ defined up to special equivalence.
\hfill
A. Swann (\cite{_Swann_}) developed an equivalence between the category of
qua\-ter\-ni\-o\-nic-\-K\"ah\-ler manifolds of positive scalar curvature and
the category $H_{sp}$. The purpose of this Subsection is to
give an exposition of Swann's formalism.
\hfill
Let $Q$ be a quaternionic-K\"ahler manifold. The restricted holonomy
group of $Q$ is $Sp(n)\cdot Sp(1)$, that is,
$(Sp(n)\times Sp(1))/\{\pm 1\}$. Consider the principal bundle $\c G$
with the fiber $Sp(1)/\{\pm 1\}$,
corresponding to the subgroup
\[ Sp(1)/\{\pm 1\}\subset (Sp(n)\times Sp(1))/\{\pm 1\}.\]
of the holonomy. There is a natural $Sp(1)/\{\pm 1\}$-action
on the space \\ ${\Bbb H^*}/\{\pm 1\}$. Let
\[ \c U(Q):= \c G\times_{Sp(1)/\{\pm 1\}}{\Bbb H^*}/\{\pm 1\}.\]
Clearly, $\c U(Q)$ is fibered over $Q$, with fibers
which are isomorphic to \\ ${\Bbb H^*}/\{\pm 1\}$.
We are going to show that the manifold $\c U(Q)$ is equipped
with a natural hypercomplex structure.
\hfill
There is a natural smooth decomposition
$\c U(Q)\cong \c G \times {\Bbb R}^{>0}$
which comes from the isomorphism
${\Bbb H^*}\cong Sp(1)\times {\Bbb R}^{>0}$.
\hfill
Consider the standard 4-dimensional bundle
$W$ on $Q$. Let $x\in Q$ be a point. The fiber
$W\restrict q$ is isomorphic to $\Bbb H$, in a non-canonical way.
The choices of isomorphism $W\restrict q\cong \Bbb H$
are called {\bf quaternion frames in $q$}.
The set of quaternion frames gives a fibration
over $Q$, with a fiber $\operatorname{Aut}({\Bbb H})\cong Sp(1)/\{\pm 1\}$.
Clearly, this fibration coincides with the principal
bundle $\c G$ constructed above. Since
$\c U(Q)\cong \c G \times {\Bbb R}^{>0}$,
a choice of $u\in \c U(Q)\restrict q$
determines an isomorphism $W\restrict q\cong \Bbb H$.
\hfill
Let $(q, u)$ be the point of $\c U(Q)$,
with $q\in Q$, $u\in \c U(Q)\restrict q$.
The natural connection in $\c U(Q)$
gives a decomposition
\[ T_{(q, u)}U(Q) = T_u \bigg(\c U(Q)\restrict q\bigg) \oplus T_q Q. \]
The space $\c U(Q)\restrict q \cong {\Bbb H^*}/\{\pm 1\}$
is equipped with a natural hypercomplex structure.
This gives a quaternion action on $T_u \bigg(\c U(Q)\restrict q\bigg)$
The choice of $u\in \c U(Q)\restrict q$ determines a
quaternion action on $T_q Q$, as we have seen above.
We obtain that the total space of $\c U(Q)$ is
an almost hypercomplex manifold.
\hfill
\proposition\label{_U(Q)_hypercomple_Proposition_}
(A. Swann) Let $Q$ be a quaternionic-K\"ahler manifold.
Consider the manifold $\c U(Q)$
constructed as above, and equipped with a
quaternion algebra action in its tangent space.
Then $\c U(Q)$ is a hypercomplex manifold.
\hfill
{\bf Proof:} Clearly, the manifold $\c U(Q)$
is equipped with a ${\Bbb H}^*$-action,
which is related with the almost hypercomplex
structure as prescribed by \ref{_H^*_specia_Definition_}
(ii)-(iii). Pick an induced complex structure
$I\in {\Bbb H}$. This gives an
algebra embedding ${\Bbb C} {\:\longrightarrow\:} {\Bbb H}$.
Consider the corresponding
${\Bbb C}^*$-action $\rho_I$ on an almost complex manifold
$(\c U(Q), I)$. This ${\Bbb C}^*$-action is compatible
with the almost complex structure. The quotient
$\c U(Q)/\rho(I)$ is an almost complex manifold, which
is naturally isomorphic to the twistor space $\operatorname{Tw}(Q)$.
Let $L^*$ be a complex vector bundle of all $(1,0)$-vectors
$v\in T (\operatorname{Tw}(Q))$ tangent to the fibers of the standard
projection $\sigma:\; \operatorname{Tw}(Q){\:\longrightarrow\:} Q$, and $L$
be the dual vector bundle. Denote by
$Tot_{\neq 0}(L)$ the complement
$\operatorname{Tot}(L)\backslash N$, where $N=\operatorname{Tw}(Q)\subset \operatorname{Tot}(L)$
is the zero section of $L$.
Using the natural connection in $L$, we obtain
an almost complex structure on $\operatorname{Tot}(L)$.
Consider the natural
projection $\phi:\; Tot_{\neq 0}(L){\:\longrightarrow\:} Q$.
The fibers $\phi^{-1}(q)$ of $\phi$ are identified with the
space of non-zero vectors in the total space
of the cotangent bundle
$T^* \sigma^{-1}(q)\cong T^*({\Bbb C} P^1)$. This space is naturally
isomorphic to
\[ \c G\restrict q \times {\Bbb R}^{>0}=
\c U(Q)\restrict q\cong {\Bbb H}^*/\{\pm 1\}.
\]
This gives a canonical isomorphism of almost complex manifolds
\[ (\c U(Q), I){\:\longrightarrow\:} Tot_{\neq 0}(L).\] Therefore, to prove
that $(\c U(Q), I)$ is a complex manifold, it suffices to
show that the natural almost complex structure on
$Tot_{\neq 0}(L)\subset \operatorname{Tot}(L)$ is integrable.
Consider the natural connection
$\nabla_L$ on $L$. To prove that $\operatorname{Tot}(L)$
is a complex manifold, it suffices to show that
$\nabla_L$ is a holomorphic connection.
The bunlde $L$ is known under the name of
{\bf holomorphic contact bundle}, and
it is known to be holomorphic
(\cite{_Salamon_}, \cite{_Besse:Einst_Manifo_}).
\blacksquare
\hfill
\remark
The result of \ref{_U(Q)_hypercomple_Proposition_}
is well known. We have given its proof because
we shall need the natural identification
$Tot_{\neq 0}(L)\cong \c U(Q)$ further on
in this paper.
\hfill
\theorem\label{_U(Q)_hyperk_Theorem_}
Let $Q$ be a quaternionic-K\"ahler manifold of positive scalar curvature,
and $\c U(Q)$ the hypercomplex manifold constructed above.
Then $\c U(Q)$ admits a unique (up to a scaling)
hyperk\"ahler metric compatible with the
hypercomplex structure.
{\bf Proof:} \cite{_Swann_}. \blacksquare
\hfill
Consider the action of ${\Bbb H}^*$ on
$\c U(M)$ defined in the
proof of \ref{_U(Q)_hypercomple_Proposition_}.
This action satisfies the conditions
(ii) and (iii) of \ref{_H^*_specia_Definition_}.
The conditions (i) and (iv) of \ref{_H^*_specia_Definition_}
are easy to check (see \cite{_Swann_} for details).
This gives a functor from the category $\c C$ of quaternionic-K\"ahler
manifolds of positive scalar curvature to
the category $H_{sp}$ of \ref{_speci_equi_Definition_}.
\hfill
\theorem \label{_U(Q)_equiva_cate_Theorem_}
The functor $Q{\:\longrightarrow\:} \c U(Q)$ from
$\c C$ to $H_{sp}$ is an equivalence of categories.
{\bf Proof:} \cite{_Swann_}. \blacksquare
\hfill
The inverse functor from $H_{sp}$ to $C$
is constructed by taking a quotient of $M$ by the action
of ${\Bbb H}^*$. Using the technique of quaternionic-K\"ahler
reduction anf hyperk\"ahler potentials
(\cite{_Swann_}), one can equip the quotient
$M/{\Bbb H}^*$ with a natural quaternionic-K\"ahler structure.
\section{${\Bbb C}^*$-equivariant twistor spaces}
\label{_C_equiv_twi_spa_Section_}
\hfill
In Section \ref{_twisto_tra_Section_}, we gave an exposition of
the twistor transform, $B_2$-bundles and Swann's formalism.
In the present Section, we give a synthesis of these theories,
obtaining a construction with should be thought
of as Swann's formalism for vector bundles.
Consider the equivalence of categories $Q{\:\longrightarrow\:} \c U(Q)$
constructed in \ref{_U(Q)_equiva_cate_Theorem_}
(we call this equivalence ``Swann's formalism'').
We show that $B_2$-bundles on $Q$ are in functorial
bijective correspondence with
${\Bbb C}^*$-equivariant holomorphic bundles on $\operatorname{Tw}(\c U(Q))$
(\ref{_B_2_to_C^*_equiva_Theorem_}).
In Subsection
\ref{_hyperho_shea_C^*_equiv_Y-M_on_blow-up_Subsection_},
this equivalence is applied
to the vector bundle $\pi^*(F)$ of
\ref{_desingu_hyperho_Theorem_}. We use it
to construct a canonical Yang-Mills connection
on $\pi^*(F)\restrict C$, where $C$ is
a special fiber of $\pi:\; \tilde M{\:\longrightarrow\:} (M, I)$
(see \ref{_desingu_hyperho_Theorem_} for details
and notation). This implies that the holomorphic
bundle $\pi^*(F)\restrict C$ is polystable
(\ref{_hyperho_blow-up_stable_Theorem_}).
\subsection[$B_2$-bundles on quaternionic-K\"ahler manifolds
and ${\Bbb C}^*$-equi\-va\-ri\-ant holomorphic bundles over twistor
spaces]{$B_2$-bundles on quaternionic-K\"ahler manifolds
and \\${\Bbb C}^*$-equi\-va\-ri\-ant holomorphic bundles over twistor
spaces}
\label{_B_2_to_C^*-invaholo_ove_twi_Subsection_}
For the duration of this Subsection, we fix a hyperk\"ahler manifold
$M$, equipped with a special ${\Bbb H}^*$-action $\rho$, and the corresponding
quaternionic-K\"ahler manifold $Q= M/{\Bbb H}^*$. Denote the natural
quotient map by $\phi:\; M{\:\longrightarrow\:} Q$.
\hfill
\lemma \label{_phi^*_B_2-forms_1,1_Lemma_}
Let $\omega$ be a 2-form
over $Q$, and $\phi^* \omega$ its pullback to $M$.
Then the following conditions are equivalent
\begin{description}
\item[(i)] $\omega$ is a $B_2$-form
\item[(ii)] $\phi^* \omega$ is of Hodge type $(1,1)$
with respect to some induced complex structure $I$ on $M$
\item[(iii)] $\phi^* \omega$ is $SU(2)$-invariant.
\end{description}
{\bf Proof:}
Let $I$ be an induced complex structure on $M$.
As we have shown in the proof of \ref{_U(Q)_hypercomple_Proposition_},
the complex manifold $(M, I)$ is idenified with an open subset
of the total space $\operatorname{Tot}(L)$ of a holomorphic line bundle $L$
over $\operatorname{Tw}(Q)$. The map $\phi$ is represented as a composition
of the projections $h:\; \operatorname{Tot}(L){\:\longrightarrow\:} \operatorname{Tw}(Q)$ and
$\sigma_Q:\; \operatorname{Tw}(Q) {\:\longrightarrow\:} Q$. Since the map $h$ is
smooth and holomorphic, the form $\phi^* \omega$ is of Hodge type $(1,1)$
if and only if $\sigma_Q^*\omega$ is of type $(1,1)$.
By \ref{_B_2_=_holo_on_Tw_Claim_} (i), this happens
if and only if $\omega$ is a $B_2$-form. This proves
an equivalence (i) $\Leftrightarrow$ (ii). Since
the choice of $I$ is arbitrary, the pullback
$\phi^* \omega$ of a $B_2$-form is
of Hodge type $(1,1)$
with respect to all induced complex structures.
By \ref{_SU(2)_inva_type_p,p_Lemma_}, this proves the implication
(i) $\Rightarrow$ (iii). The implication
(iii) $\Rightarrow$ (ii) is clear.
\blacksquare
\hfill
\proposition
Let $(B, \nabla)$ be a complex vector bundle with connection
over $Q$, and $(\phi^*B, \phi^*\nabla)$ its pullback to $M$.
Then the following conditions are equivalent
\begin{description}
\item[(i)] $(B, \nabla)$ is a $B_2$-form
\item[(ii)] The curvature of $(\phi^*B, \phi^*\nabla)$ is of
Hodge type (1,1)
with respect to some induced complex structure $I$ on $M$
\item[(iii)] The bundle $(\phi^*B, \phi^*\nabla)$ is autodual
\end{description}
{\bf Proof:} Follows from \ref{_phi^*_B_2-forms_1,1_Lemma_}
applied to $\omega = \nabla^2$. \blacksquare
\hfill
For any point $I\in {\Bbb C} P^1$,
consider the corresponding algebra embedding
${\Bbb C} \stackrel {c_I} \hookrightarrow {\Bbb H}$. Let
$\rho_I$ be the action of ${\Bbb C}^*$ on $(M, I)$ obtained
as a restriction of $\rho$ to $c_I({\Bbb C}^*)\subset {\Bbb H}^*$.
Clearly from \ref{_H^*_specia_Definition_} (ii),
$\rho_I$ acts on $(M, I)$ by holomorphic automorphisms.
Consider $\operatorname{Tw}(M)$ as a union
\[ \operatorname{Tw}(M) = \bigcup_{I\in {\Bbb C} P^1}
\pi^{-1}(I), \ \ \pi^{-1}(I)= (M, I)
\]
Gluing $\rho(I)$ together, we obtain a
smooth ${\Bbb C}^*$-action $\rho_{\Bbb C}$ on $\operatorname{Tw}(M)$.
\hfill
\claim \label{_C^*_acti_on_Tw_holo_Claim_}
Consider the action $\rho_{\Bbb C}:\; {\Bbb C}^*\times \operatorname{Tw}(M) {\:\longrightarrow\:} \operatorname{Tw}(M)$
constructed above. Then $\rho_{\Bbb C}$ is holomorphic.
{\bf Proof:} It is obvious from construction that
$\rho_{\Bbb C}$ is compatible with the complex structure
on $\operatorname{Tw}(M)$. \blacksquare
\hfill
\example
Let $M = {\Bbb H}^n \backslash 0$. Since $\operatorname{Tw}({\Bbb H}^n)$ is
canonically isomorphic to a total space
of the bundle ${\cal O}(1)^n$ over ${\Bbb C} P^1$, the twistor
space $\operatorname{Tw}(M)$ is $\operatorname{Tot}({\cal O}(1)^n)$ without zero section.
The group ${\Bbb C}^*$ acts on $\operatorname{Tot}({\cal O}(1)^n)$ by dilatation,
and the restriction of this action to $\operatorname{Tw}(M)$
coincides with $\rho_{\Bbb C}$.
\hfill
Consider the map $\sigma:\; \operatorname{Tw}(M) {\:\longrightarrow\:} M$.
Let $(B, \nabla)$ be a $B_2$-bundle over $Q$.
Since the bundle $(\phi^*B, \phi^*\nabla)$
is autodual, the curvature of $\sigma^*\phi^*\nabla$
has type $(1,1)$. Let
$(\sigma^* \phi^*B, (\sigma^*\phi^*\nabla)^{0,1})$
be the holomorphic bundle obtained from $(\phi^*B, \phi^*\nabla)$
by twistor transform. Clearly, this bundle
is ${\Bbb C}^*$-equivariant, with respect to the natural
${\Bbb C}^*$-action on $\operatorname{Tw}(M)$. It turns out that any
${\Bbb C}^*$-equivariant bundle $\c F$ on $\operatorname{Tw}(M)$
can be obtained this way, assuming that
$\c F$ is compatible with twistor transform.
\hfill
\theorem\label{_B_2_to_C^*_equiva_Theorem_}
In the above assumptions, let $\c C_{B_2}$
be the category of of $B_2$-bundles on $Q$,
and $\c C_{\operatorname{Tw}, {\Bbb C}^*}$ the category of ${\Bbb C}^*$-equivariant
holomorphic bundles on $\operatorname{Tw}(M)$ which are compatible
with the twistor transform. Consider
the functor
\[ (\sigma^* \phi^*)^{0,1}: \c C_{B_2}{\:\longrightarrow\:} \c C_{\operatorname{Tw}, {\Bbb C}^*},
\]
$(B, \nabla){\:\longrightarrow\:} (\sigma^* \phi^*B, (\sigma^*\phi^*\nabla)^{0,1})$,
constructed above. Then $(\sigma^* \phi^*)^{0,1}$
establishes an equivalence of categories.
\hfill
We prove \ref{_B_2_to_C^*_equiva_Theorem_}
in Subsection \ref{_twi_tra_H^*_Subsection_}.
\hfill
\remark
Let $Q$ be an arbitrary quaternionic-K\"ahler manifold,
and $M= \c U(Q)$ the corresponding fibration. Then
$M$ is hypercomplex, and its twistor space is equipped with
a natural holomorphic action of ${\Bbb C}^*$. This gives necessary
ingredients needed to state \ref{_B_2_to_C^*_equiva_Theorem_}
for $Q$ with negative scalar curvature.
The proof which we give for $Q$ with positive scalar curvature
will in fact work for all quaternionic-K\"ahler manifolds.
\hfill
\question
What happens with this construction when $Q$ is a hyperk\"ahler manifold?
\hfill
In this paper, we need \ref{_B_2_to_C^*_equiva_Theorem_}
only in the case $Q={\Bbb H}P^n$, $M = {\Bbb H}^n\backslash 0$,
but the general proof is just as difficult.
\hfill
\subsection{${\Bbb C}^*$-equivariant bundles and twistor transform}
\label{_C^*_equiva_and_twistor_Subsection_}
Let $M$ be a hyperk\"ahler manifold, and $\operatorname{Tw}(M)$ its twistor space.
Recall that $\operatorname{Tw}(M)={\Bbb C} P^1\times M$
is equipped with a canonical anticomplex involution $\iota$,
which acts as identity on $M$ and as central symmetry
$I{\:\longrightarrow\:} -I$ on ${\Bbb C} P^1= S^2$.
\hfill
\proposition \label{_conne_flat_along_leave_C^*_Proposition_}
Let $M$ be a hyperk\"ahler manifold, and $\operatorname{Tw}(M)$ its twistor space.
Assume that $\operatorname{Tw}(M)$ is equipped with a free holomorphic action
$\rho(z):\; \operatorname{Tw}(M){\:\longrightarrow\:} \operatorname{Tw}(M)$ of ${\Bbb C}^*$, acting along the
fibers of $\pi:\; \operatorname{Tw}(M){\:\longrightarrow\:} {\Bbb C} P^1$. Assume, moreover, that
$\iota\circ \rho(z) = \rho(\bar z) \circ \iota$,
where $\iota$ is the natural anticomplex involution of $\operatorname{Tw}(M)$.%
\footnote{These assumptions are automatically satisfied
when $M$ is equipped with a special ${\Bbb H}^*$-action,
and $\rho(z)$ is the corresponding ${\Bbb C}^*$-action on
$\operatorname{Tw}(M)$.}
Let $\c F$ be a ${\Bbb C}^*$-equivariant holomorphic vector
bundle on $\operatorname{Tw}(M)$. Assume that $\c F$ is compatible
with the twistor transform. Let $\nabla_{\c F}$ be the
natural connection on $\c F$ (\ref{_cano_conne_Remark_}).
Then $\nabla_{\c F}$ is flat along the leaves of
$\rho$.
\hfill
{\bf Proof:} First of all, let us
recall the construction of
the natural connection $\nabla_{\c F}$.
Let $\c F$ be an arbitrary bundle compatible
with the twistor transform. We construct
$\nabla_{\c F}$ in terms of the
isomorphism $\Psi_{1,2}$ defined
in \ref{_F_1_=F_2_Lemma_}.
\hfill
Consider an induced complex structure $I$.
Let $F_I$ be the restriction of $\c F$ to
$(M, I)= \pi^{-1}(I)\subset \operatorname{Tw}(M)$.
Consider the evaluation map \[ p_I:\; \operatorname{Lin}(M) {\:\longrightarrow\:} (M, I)\]
(Subsection \ref{_twi_lines_C^*_Subsection_}).
In a similar way we define the holomorphic vector bundle
$F_{-I}$ on $(M, -I)$ and the map
$p_{-I}:\; \operatorname{Lin}(M) {\:\longrightarrow\:} (M, -I)$.
Denote by $F_1$, $F_{-1}$ the sheaves
$p_I^*(F_I)$, $p_{-I}^*(F_{-I})$.
In \ref{_F_1_=F_2_Lemma_}, we constructed
an isomorphism $\Psi_{1,-1}:\; F_1 {\:\longrightarrow\:} F_{-1}$.
Let us identify $\operatorname{Lin}(M)$ with $(M, I)\times (M, I)$
(this idenitification is naturally defined in a neighbourhood
of $\operatorname{Hor}\subset \operatorname{Lin}(M)$ -- see
\ref{_Lin_is_MxM_Proposition_}). Then
the maps $p_I$, $p_{-I}$ became projections
to the relevant components. Let
\begin{equation*}
\begin{split}
\bar\partial:\; F_1 &{\:\longrightarrow\:} F_1 \otimes p_I^* \Omega^1(M, -I), \\
\partial:\; F_{-1} &{\:\longrightarrow\:} F_{-1} \otimes p_{-I}^* \Omega^1(M, I),
\end{split}
\end{equation*}
be the sheaf maps obtained as pullbacks of de Rham differentials
(the tensor product is taken in the category of coherent sheaves
over $\operatorname{Lin}(M)$).
Twisting $\partial$ by an isomorphism $\Psi_{1,-1}:\; F_1 {\:\longrightarrow\:} F_{-1}$,
we obtain a map
\[ \partial^\Psi:\; F_1 {\:\longrightarrow\:} F_1 \otimes p_I^* \Omega^1(M, I).
\]
Adding $\bar \partial$ and $\partial^\Psi$, we obtain
\[
\nabla:\; F_1 {\:\longrightarrow\:} F_1 \otimes
\bigg (p_I^* \Omega^1(M, I) \oplus p_I^* \Omega^1(M, -I)\bigg).
\]
Clearly, $\nabla$ satisfies the Leibniz rule.
Moreover, the sheaf
$p_I^* \Omega^1(M, I) \oplus p_I^* \Omega^1(M, -I)$
is naturally isomorphic to the sheaf of differentials over
\[ \operatorname{Lin}(M) = (M,I)\times (M, -I).\]
Therefore, $\nabla$ can be considered as a connection
in $F_1$, or as a real analytic connection in a
real analytic complex vector bundle underlying $F_I$.
From the definition of $\nabla_{\c F}$ (\cite{_NHYM_}),
it is clear that $\nabla_{\c F}\restrict{(M, I)}$
equals $\nabla$.
\hfill
Return to the proof of \ref{_conne_flat_along_leave_C^*_Proposition_}.
Consider a ${\Bbb C}^*$-action $\rho_I(z)$ on $(M, I)$, $(M, -I)$
induced from the natural embeddings
$(M,I)\hookrightarrow \operatorname{Tw}(M)$, $(M,-I)\hookrightarrow \operatorname{Tw}(M)$.
Then $F_I$ is a ${\Bbb C}^*$-equivariant bundle.
Since $\iota\circ \rho(z) = \rho(\bar z) \circ \iota$,
the identification $\operatorname{Lin}(M)=(M, I)\times (M, I)$
is compatible with ${\Bbb C}^*$-action.
Let ${\mathbf r}= \frac{d}{dr}$ be the holomorphic
vector field on $(M, I)$ corresponding to the ${\Bbb C}^*$-action.
To prove \ref{_conne_flat_along_leave_C^*_Proposition_},
we have to show that the operator
\[ [\nabla_{\mathbf r},\nabla_{\bar {\mathbf r}}]:\;
F_I {\:\longrightarrow\:} F_I\otimes \Lambda^{1,1}(M, I)
\]
vanishes.
Consider the equivariant structure operator
\[ \rho(z)^F: \rho_I(z)^* F_I {\:\longrightarrow\:} F_I. \]
Let $U$ be a ${\Bbb C}^*$-invariant Stein subset of $(M, I)$.
Consider $\rho(z)^F$ an an endomorphism of the
space of global holomorphic sections $\Gamma_U(F_I)$.
Let
\[ D_r(f):= \lim\limits_{\epsilon\rightarrow 0}
\frac{\rho_I(1+\epsilon)}{\epsilon},
\]
for $f\in \Gamma_U(F_I)$.
Clearly, $D_r$ is a well defined sheaf endomorphism of $F_I$,
satisfying
\[ D_r(\alpha \cdot f) =
\frac{d}{dr}\alpha \cdot f +\alpha \cdot D_r(f),
\]
for all $\alpha\in {\cal O}_{(M,I)}$.
We say that a holomorphic section $f$ of $F_I$ is {\bf ${\Bbb C}^*$-invariant}
if $D_r(f)=0$.
Clearly, the ${\cal O}_{(M,I)}$-sheaf $F_I$
is generated by ${\Bbb C}^*$-invariant sections. Therefore,
it suffices to check the equality
\[ [\nabla_{\mathbf r},\nabla_{\bar {\mathbf r}}] (f)=0
\]
for holomorphic ${\Bbb C}^*$-invariant $f\in F_I$.
Since $f$ is holomorphic, we have $\nabla_{\bar {\mathbf r}} f =0$.
Thus,
\[ [\nabla_{\mathbf r},\nabla_{\bar {\mathbf r}}] (f)
= \nabla_{\bar {\mathbf r}} \nabla_{\mathbf r}(f).
\]
We obtain that \ref{_conne_flat_along_leave_C^*_Proposition_}
is implied by the following lemma.
\hfill
\lemma \label{_conne_on_C^*_inva_Lemma_}
In the above assumptions, let $f$ be a ${\Bbb C}^*$-invariant
section of $F_I$. Then $\nabla_{\mathbf r}(f)=0$.
\hfill
{\bf Proof:} Return to the notation we used in the beginning
of the proof of \ref{_conne_flat_along_leave_C^*_Proposition_}.
Then, $\nabla(f) = \bar \partial(f) +\partial^\Psi(f)$. Since
$f$ is holomorphic, $\bar \partial(f)=0$, so we
need to show that $\partial^\Psi(f)({\mathbf r})=0$.
By definition of $\partial^\Psi$, this is equivalent to
proving that
\[ \partial\Psi_{1,-1}(f)({\mathbf r})=0.\]
Consider the ${\Bbb C}^*$-action on $\operatorname{Lin}(M)$ which is induced by
the ${\Bbb C}^*$-action on $\operatorname{Tw}(M)$. Since the maps $p_I$,
$p_{-I}$ are compatible with the ${\Bbb C}^*$-action, the sheaves
$F_1$, $F_{-1}$ are ${\Bbb C}^*$-equivariant. We can repeat the construction
of the operator $D_r$ for the sheaf $F_{-I}$. This allows
one to speak of holomorphic ${\Bbb C}^*$-invariant sections
of $F_{-I}$. Pick a ${\Bbb C}^*$-invariant Stein subset $U\subset (M, -I)$.
Since the statement of \ref{_conne_on_C^*_inva_Lemma_} is local,
we may assume that $M=U$. Let $g_1, ... , g_n$ be a set of
${\Bbb C}^*$-invariant sections of $F_I$ which generated $F_I$.
Then, the sections $p_{-I}^*(g_1), ..., p_{-I}^*(g_n)$ generate
$F_{-1}$. Consider the section $\Psi_{1,-1}(f)$ of $F_{-1}$.
Clearly, $\Psi_{1,-1}$ commutes with the natural
${\Bbb C}^*$-action. Therefore, the section $\Psi_{1,-1}(f)$
is ${\Bbb C}^*$-invariant, and can be written as
\[ \Psi_{1,-1}(f)= \sum \alpha_i p_{-I}^* (g_i), \]
where the functions $\alpha_i$ are ${\Bbb C}^*$-invariant.
By definition of $\partial$ we have
\[ \partial\left(\sum \alpha_i p_{-I}^* (g_i)\right) =
\sum \partial(\alpha_i p_{-I}^* (g_i))+\sum \alpha_i \partial (p_{-I}^* (g_i)).
\]
On the other hand, $g_i$ is a holomorphic section of $F_{-I}$,
so $\partial p_{-I}^* (g_i)=0$. We obtain
\[ \partial\left(\sum \alpha_i p_{-I}^* \cdot (g_i)\right) =
\sum \partial\alpha_i p_{-I}^* (g_i).
\]
Thus,
\[ \partial\Psi_{1,-1}(f)({\mathbf r}) = \sum \frac{\partial\alpha_i}{\partial r}
p_{-I}^* (g_i),
\]
but since the functions $\alpha_i$ are ${\Bbb C}^*$-invariant,
their derivatives along ${\mathbf r}$ vanish.
We obtain $\partial\Psi_{1,-1}(f)({\mathbf r})=0$. This proves
\ref{_conne_on_C^*_inva_Lemma_}.
\ref{_conne_flat_along_leave_C^*_Proposition_}
is proven.
\blacksquare
\subsection{Twistor transform and the ${\Bbb H}^*$-action}
\label{_twi_tra_H^*_Subsection_}
For the duration of this Subsection, we fix a hyperk\"ahler manifold
$M$, equipped with a special ${\Bbb H}^*$-action $\rho$, and the
corresponding quaternionic-K\"ahler manifold $Q= M/{\Bbb H}^*$.
Denote the natural quotient map by $\phi:\; M{\:\longrightarrow\:} Q$.
Clearly, \ref{_B_2_to_C^*_equiva_Theorem_}
is an immediate consequence of the following theorem.
\hfill
\theorem \label{_C^*_equi_cano_conne_Theorem_}
Let $\c F$ be a ${\Bbb C}^*$-equivariant holomorphic bundle
over $\operatorname{Tw}(M)$, which is compatible with the twistor
transform. Consider the natural connection
$\nabla_{\c F}$ on $\c F$. Then $\nabla_{\c F}$
is flat along the leaves of ${\Bbb H}^*$-action.
\hfill
{\bf Proof:} The leaves of ${\Bbb H}^*$-action are parametrized by
the points of $q\in Q$. Consider such a leaf
$M_q:=\phi^{-1}(q)\subset M$.
Clearly, $M_q$ is a hyperk\"ahler submanifold in
$M$, equipped with a special action of ${\Bbb H}^*$.
Moreover, the restriction of $\c F$ to $\operatorname{Tw}(M_q)\subset \operatorname{Tw}(M)$
satisfies assumptions of \ref{_C^*_equi_cano_conne_Theorem_}.
To prove that $\nabla_{\c F}$
is flat along the leaves of ${\Bbb H}^*$-action,
we have to show that ${\c F}\restrict{\operatorname{Tw}(M_q)}$
is flat, for all $q$. Therefore, it suffices to prove
\ref{_C^*_equi_cano_conne_Theorem_} for
$\dim_{\Bbb H} M=1$.
\hfill
\lemma\label{_cano_conne_flat_on_4-dim_Lemma_}
We work in notation and assumptions of
\ref{_C^*_equi_cano_conne_Theorem_}. Assume that
$\dim_{\Bbb H} M=1$. Then the connection
$\nabla_{\c F}$ is flat.
\hfill
{\bf Proof:}
Let $I$ be an induced complex structure, and
$F_I:= F\restrict{(M,I)}$ the corresponding holomorphic
bundle on $(M, I)$. Denote by $z_I$ the vector field
corresponding to the ${\Bbb C}^*$-action $\rho_I$ on $(M, I)$.
By definition, the connection $\nabla\restrict{F_I}$
has $SU(2)$-invariant curvature $\Theta_I$.
On the other hand, $\Theta_I(z_I, \bar z_I)=0$ by
\ref{_conne_flat_along_leave_C^*_Proposition_}.
Since $\nabla_{\c F}=\sigma^*\nabla$ is a pullback of an autodual connection
$\nabla$ on $M$, its curvature is a pullback of
$\Theta_I$. In particular, $\Theta= \Theta_I$ is independent from
the choice of induced complex structure $I$.
We obtain that $\Theta(z_I, \bar z_I)=0$ for all
induced complex structures $I$ on $M$.
Now \ref{_cano_conne_flat_on_4-dim_Lemma_} is implied
by the following linear-algebraic claim.
\hfill
\claim \label{_SU_2_inva_2-form_z_bar_z=0_is_zero_Claim_}
Let $M$ be a hyperk\"ahler manifold equipped with a special
${\Bbb H}^*$-action, $\dim_{\Bbb H}M=1$. Consider the vectors
$z_I$, $\bar z_I$ defined above. Let $\Theta$ be a smooth
$SU(2)$-invariant 2-form, such that for all induced complex structures,
$I$, we have $\Theta(z_I, \bar z_I)=0$. Then $\Theta=0$.
\hfill
{\bf Proof:} The proof of
\ref{_SU_2_inva_2-form_z_bar_z=0_is_zero_Claim_}
is an elementary calculation.
Fix a point $m_0\in M$.
Consider the flat hyperk\"ahler manifold ${\Bbb H}\backslash 0$,
equipped with a natural special action of ${\Bbb H}^*$.
From the definition of a special action, it is clear
that the map $\rho$ defines a covering ${\Bbb H}\backslash 0{\:\longrightarrow\:} M$,
$h{\:\longrightarrow\:} \rho(h)m_0$
of hyperk\"ahler manifolds, and this covering is compatible
with the special action. Therefore, the hyperk\"ahler manifold $M$ is flat,
and the ${\Bbb H}^*$-action is linear in the
flat coordinates.
Let
\[ \Lambda^2(M) = \Lambda^+(M) \oplus\Lambda^-(M)
\]
be the standard decomposition of $\Lambda^2(M)$ according
to the eigenvalues of the Hodge $*$ operator. Consider the
natural Hermitian metric on $\Lambda^2(M)$. Then
$\Lambda^-(M)$ is the bundle of $SU(2)$-invariant 2-forms
(see, e. g., \cite{_Verbitsky:Hyperholo_bundles_}), and
$\Lambda^+(M)$ is its orthogonal complement.
Consider the corresponding orthogonal projection
$\Pi:\; \Lambda^2(M) {\:\longrightarrow\:} \Lambda^-(M)$. Denote by
$dz_I\wedge d \bar z_I$ the differential form which is dual to
the bivector $z_I\wedge \bar z_I$.
Let $R\subset \Lambda^-(M)$ be the $C^\infty(M)$-subsheaf
of $\Lambda^-(M)$ generated by $\Pi(dz_I\wedge d \bar z_I)$,
for all induced complex structures $I$ on $M$.
Clearly, $\Theta\in \Lambda^-(M)$ and $\Theta$ is orthogonal
to $R\subset \Lambda^-(M)$. Therefore, to prove that
$\Theta=0$ it suffices to show that $R= \Lambda^-(M)$.
Since $M$ is covered by ${\Bbb H}\backslash 0$,
we may prove $R= \Lambda^-(M)$ in assumption
$M={\Bbb H}\backslash 0$.
\hfill
Let $\gamma$ be the real vector field corresponding to dilatations
of $M={\Bbb H}\backslash 0$, and $d\gamma$ the dual 1-form.
Clearly,
\[ dz_I\wedge d \bar z_I = 2\sqrt{-1}\: d\gamma\wedge I(d\gamma).
\]
Averaging $d\gamma\wedge I(d\gamma)$ by $SU(2)$, we obtain
\[ \Pi(dz_I\wedge d \bar z_I) = \sqrt{-1}\:\bigg(
d\gamma\wedge I(d\gamma) - J(d\gamma) \wedge K(d\gamma)\bigg)
\]
where $I$, $J$, $K$ is the standard triple of generators for
quaternion algebra. Similarly,
\[ \Pi(dz_J\wedge d \bar z_J) = \sqrt{-1}\:\bigg(
d\gamma\wedge J(d\gamma) + K(d\gamma) \wedge I(d\gamma)\bigg)
\]
and
\[ \Pi(dz_K\wedge d \bar z_K) = \sqrt{-1}\:\bigg(
d\gamma\wedge K(d\gamma) + I(d\gamma) \wedge J(d\gamma)\bigg)
\]
Thus, $\Pi(R)$ is a 3-dimensional sub-bundle of $\Lambda^-(M)$.
Since $\dim \Lambda^-(M) =3$, we have $\Pi(R)= \Lambda^-(M)$.
This proves \ref{_SU_2_inva_2-form_z_bar_z=0_is_zero_Claim_}.
\ref{_cano_conne_flat_on_4-dim_Lemma_} and
\ref{_C^*_equi_cano_conne_Theorem_} is proven. \blacksquare
\subsection{Hyperholomorphic sheaves and ${\Bbb C}^*$-equivariant bundles
over $M_{\rm fl}$}
\label{_hyperho_shea_C^*_equiv_Y-M_on_blow-up_Subsection_}
Let $M$ be a hyperk\"ahler manifold, $I$ an induced complex structure
and $F$ a reflexive sheaf over $(M, I)$, equipped with a hyperholomorphic
connection. Assume that $F$ has an isolated singularity
in $x\in M$. Consider the sheaf $\c F$ on $\operatorname{Tw}(M)$
corresponding to $\c F$ as in the proof of
\ref{_conne_=>_hyperho_Proposition_}.
Let $s_x\subset \operatorname{Tw}(M)$ be the horizontal twistor line
corresponding to $x$, and $\frak m$ its ideal.
Consider the associated graded sheaf of $\frak m$.
Denote by $\operatorname{Tw}^{gr}$ the spectre of this associated graded
sheaf.
Clearly, $\operatorname{Tw}^{gr}$ is naturally isomorphic to
$\operatorname{Tw}(T_x M)$, where $T_xM$ is the flat hyperk\"ahler manifold
corresponding to the space $T_xM$ with induced quaternion action.
Consider the natural ${\Bbb H}^*$-action on
$T_xM$. This provides the hyperk\"ahler manifold
$T_xM\backslash 0$ with a special ${\Bbb H}^*$-action.
Let $s_0\subset \operatorname{Tw}^{gr}$ be the horisontal twistor
line corresponding to $s_x$. The space $\operatorname{Tw}^{gr}\backslash s_0$
is equipped with a holomorphic ${\Bbb C}^*$-action
(\ref{_C^*_acti_on_Tw_holo_Claim_}).
Denote by $\c F^{gr}$ the sheaf on $\operatorname{Tw}^{gr}$ associated with
$\c F$. Clearly, $\c F^{gr}$ is ${\Bbb C}^*$-equivariant.
In order to be able to apply \ref{_B_2_to_C^*_equiva_Theorem_}
and \ref{_C^*_equi_cano_conne_Theorem_} to
$\c F^{gr}\restrict{\operatorname{Tw}^{gr}\backslash s_0}$,
we need only to show that $\c F^{gr}$
is compatible with twistor transform.
\hfill
\proposition \label{_F^gr_compa_twi_tra_Proposition_}
Let $M$ be a hyperk\"ahler manifold, $I$ an induced complex structure
and $F$ a reflexive sheaf over $(M, I)$, equipped with a hyperholomorphic
connection. Assume that $\c F$ has an isolated singularity
in $x\in M$. Let $\c F^{gr}$ be the ${\Bbb C}^*$-equivariant bundle
on $\operatorname{Tw}^{gr}\backslash s_0$ constructed above. Then
\begin{description}
\item[(i)] the bundle $\c F^{gr}$
is compatible with twistor transform.
\item[(ii)]
Moreover, the natural
connection $\nabla_{\c F^{gr}}$ (\ref{_cano_conne_Remark_})
is Hermitian.
\end{description}
{\bf Proof:} The argument is clear, but cumbersome, and
essentially hinges on taking associate graded quotients everywhere
and checking that all equations remain true. We give a simplified
version of the proof, which omits some details and notation.
Consider the bundle $\c F\restrict {M\backslash s_x}$.
This bundle is compatible with
twistor transform, and therefore, is equipped with a natural
connection $\nabla_{\c F}$. This connection is constructed
using the isomorphism $\Psi_{1,-1}:\; F_1{\:\longrightarrow\:} F_{-1}$
(see the proof of \ref{_conne_flat_along_leave_C^*_Proposition_}).
We apply the same consideration to $\c F^{gr}\restrict{(T_x M, I)}$,
and show that the resulting connection $\nabla_{\c F^{gr}}$
is hyperholomorphic. This implies that $\c F^{gr}$
admits a $(1,1)$-connection which is a pullback of
some connection on $\c F^{gr}\restrict{(T_x M, I)}$.
This argument is used to prove
that $\c F^{gr}$ is compatible with the twistor
transform.
We use the notation introduced in
the proof of \ref{_conne_flat_along_leave_C^*_Proposition_}.
Let $\operatorname{Lin}^{gr}$ be the space of twistor maps in $\operatorname{Tw}^{gr}$.
Consider the maps
$p_{\pm I}^{gr}:\; \operatorname{Lin}^{gr}{\:\longrightarrow\:} (T_x M, \pm I)$
and the sheaves $F^{gr}_{\pm1}:=(p_{\pm I}^{gr})^*\c F^{gr}_{\pm I}$
obtained in the same way as the maps $p_{\pm I}$
and the sheaves $F_{\pm1}$ from the corresponding associated graded
objects. Taking the associated graded of $\Psi_{1,-1}$
gives an isomorphism $\Psi^{gr}_{1,-1}:\; F^{gr}_1{\:\longrightarrow\:} F^{gr}_{-1}$.
Using the same construction as in the proof of
\ref{_conne_flat_along_leave_C^*_Proposition_},
we obtain a connection operator
\[
\bar \partial^{gr}+\partial^{\Psi^{gr}}=\nabla^{gr}_I:\; F_1^{gr} {\:\longrightarrow\:} F_1^{gr} \otimes
\bigg ((p^{gr}_{-I})^* \Omega^1(T_x M, I) \oplus
(p^{gr}_I)^* \Omega^1(T_x M, -I)\bigg).
\]
Since $(\bar \partial^{gr})^2 = (\partial^{\Psi^{gr}})^2=0$, the
curvature of $\nabla^{gr}_I$ has Hodge type $(1,1)$ with respect to $I$.
To prove that $\nabla^{gr}_I$ is hyperholomorphic,
we need to show that the curvature of $\nabla^{gr}_I$
has type $(1,1)$ with respect to every induced
complex structure. Starting from another induced complex
structure $J$, we obtain a connection $\nabla^{gr}_J$,
with the curvature of type $(1,1)$ with respect to $J$.
To prove that $\nabla^{gr}_J$ is hyperholomorphic
it remains to show that $\nabla^{gr}_J=\nabla^{gr}_I$.
Let $\nabla_I$, $\nabla_J$ be the corresponding operators on
$F_1$. From the construction, it is clear that
$\nabla^{gr}_I$, $\nabla^{gr}_J$ are obtained from
$\nabla_I$, $\nabla_J$ by taking the associated graded quotients.
On the other hand, $\nabla_I = \nabla_J$.
Therefore, the connections $\nabla^{gr}_I$ and $\nabla^{gr}_J$
are equal. We proved that the bundle
$\c F^{gr}\restrict{\operatorname{Tw}^{gr}\backslash s_0}$ is compatible
with the twistor structure.
To prove \ref{_F^gr_compa_twi_tra_Proposition_},
it remains to show that the natural connection
on $\c F^{gr}$ is Hermitian.
The bundle ${\c F} \restrict{\operatorname{Tw}(M\backslash{x_0})}$ is by definition
Hermitian. Consider the corresponding isomorphism
$\c F {\:\longrightarrow\:} (\iota^*\bar{\c F})^*$
(\ref{_twi_tra_for_semili_Proposition_}). Taking an associate
graded map, we obtain an isomorphism
\[ \c F^{gr}\oldtilde\rightarrow (\iota^*\bar{\c F}^{gr})^*.\]
This gives a non-degenerate semilinear form $h^{gr}$ on $\c F^{gr}$.
It remains only to show that
$h^{gr}$ is pseudo-Hermitian (i. e. satisfies
$h(x, y) = \overline{h(y,x)}$) and positive definite.
\hfill
Let $M^{gr}_{\Bbb C}$ be a complexification of $M^{gr}=T_x M$,
$M^{gr}_{\Bbb C} = \operatorname{Lin}(M^{gr})$.
Consider the corresponding complex vector bundle
$\c F^{gr}_{\Bbb C}$ over $M^{gr}_{\Bbb C}$ underlying $\c F^{gr}$. The metric
$h^{gr}$ can be considered as a semilinear form
$\c F^{gr}_{\Bbb C}\times \c F^{gr}_{\Bbb C}{\:\longrightarrow\:} {\cal O}_{M^{gr}_{\Bbb C}}$.
This semilinear form is obtained from the corresponding form
$h$ on $\c F$ by taking the associate graded quotients.
Since $h$ is Hermitian, the form $h^{gr}$ is pseudo-Hermitian.
To prove that $h^{gr}$ is positive semidefinite, we need to show that
for all $f\in \c F^{gr}_{\Bbb C}$, the function
$h^{gr}(f, \bar f)$ belongs to
${\cal O}_{M^{gr}_{\Bbb C}}\cdot\bar{\cal O}_{M^{gr}_{\Bbb C}}$,
where ${\cal O}_{M^{gr}_{\Bbb C}}\cdot\bar{\cal O}_{M^{gr}_{\Bbb C}}$
denotes the ${\Bbb R}^{>0}$-semigroup of ${\cal O}_{M^{gr}_{\Bbb C}}$
generated by $x\cdot\bar x$, for all $x\in {\cal O}_{M^{gr}_{\Bbb C}}$.
A similar property for $h$ holds, because $h$ is positive definite.
Clearly, taking associated graded quotient of the
semigroup ${\cal O}_{M_{\Bbb C}}\cdot\bar{\cal O}_{M_{\Bbb C}}$, we obtain
${\cal O}_{M^{gr}_{\Bbb C}}\cdot\bar{\cal O}_{M^{gr}_{\Bbb C}}$.
Thus,
\[ h^{gr}(f, \bar f)\in
\left({\cal O}_{M_{\Bbb C}}\cdot\bar{\cal O}_{M_{\Bbb C}}\right)^{gr} =
{\cal O}_{M^{gr}_{\Bbb C}}\cdot\bar{\cal O}_{M^{gr}_{\Bbb C}}
\]
This proves that $h^{gr}$ is positive semidefinite.
Since $h^{gr}$ is non-degenerate, this form
in positive definite. \ref{_F^gr_compa_twi_tra_Proposition_} is proven.
\blacksquare
\hfill
\remark\label{_exte_conne_conje_Remark_}
Return to the notations of \ref{_desingu_hyperho_Theorem_}.
Consider the bundle $\pi^* F\restrict C$, where $C= {\Bbb P}T_xM$
is the blow-up divisor. Clearly, this bundle corresponds to
the graded sheaf $F_I^{gr}= \c F^{gr}\restrict{(M,I)}$
on $(T_xM, I)$. By \ref{_F^gr_compa_twi_tra_Proposition_}
(see also \ref{_C^*_equi_cano_conne_Theorem_}),
the bundle $\pi^* F\restrict C$ is equipped with a natural
${\Bbb H}^*$-invariant connection and Hermitian structure.%
\footnote{As usually, coherent sheaves over projective variety $X$
correspond to finitely generated graded modules over
the graded ring $\oplus \Gamma({\cal O}_X(i))$.}
The sheaf $\pi^*F \restrict{\tilde M \backslash C}$
is a hyperholomorphic bundle over $\tilde M \backslash C\cong M\backslash x_0$.
Therefore, $\pi^*F \restrict{\tilde M \backslash C}$
is equipped with a natural metric and a hyperholomorphic
connection. It is expected that the natural connection and metric
on $\pi^* F\restrict{\tilde M\backslash C}$ can be extended to
$\pi^* F$, and the rectriction of the resulting connection and metric
to $\pi^* F\restrict C$ coincides with that given by
\ref{_F^gr_compa_twi_tra_Proposition_} and
\ref{_C^*_equi_cano_conne_Theorem_}. This will give
an alternative proof of \ref{_F^gr_compa_twi_tra_Proposition_} (ii),
because a continuous extension of a positive definite
Hermitian metric is a positive semidefinite Hermitian metric.
\subsection{Hyperholomorphic sheaves and stable bundles
on ${\Bbb C} P^{2n+1}$}
The purpose of the current Section was to prove the following
result, which is a consequence of
\ref{_F^gr_compa_twi_tra_Proposition_} and
\ref{_C^*_equi_cano_conne_Theorem_}.
\hfill
\theorem \label{_hyperho_blow-up_stable_Theorem_}
Let $M$ be a hyperk\"ahler manifold, $I$ an induced complex
structure and $F$ a reflexive sheaf on $(M,I)$ admitting
a hyperholomorphic connection. Assume that $F$ has an isolated
singularity in $x\in M$, and is locally trivial outside of $x$.
Let $\pi:\; \tilde M{\:\longrightarrow\:} (M,I)$ be the blow-up of $(M,I)$ in $x$.
Consider the holomorphic vector bundle $\pi^* F$ on $\tilde M$
(\ref{_desingu_hyperho_Theorem_}). Let $C\subset (M,I)$ be the
blow-up divisor, $C= {\Bbb P}T_xM$. Then the holomorphic
bundle $\pi^* F\restrict{ C}$ admits a
natural Hermitian connection $\nabla$ which is flat along the leaves
of the natural ${\Bbb H}^*$-action on ${\Bbb P}T_xM$.
Moreover, the connection $\nabla$ is Yang-Mills, with respect to the
Fubini-Study metric on $C= {\Bbb P}T_xM$, the degree
$\deg c_1\left(\pi^* F\restrict{C}\right)$ vanishes,
and the holomorphic vector bundle $\pi^* F\restrict{C}$
is polystable.
\hfill
{\bf Proof:} By definition, coherent sheaves on
$C= {\Bbb P}T_xM$ correspond bijectively to ${\Bbb C}^*$-equivariant
sheaves on $T_xM\backslash 0$. Let $F^{gr}$ be the
associated graded sheaf of $F$ (Subsection
\ref{_hyperho_shea_C^*_equiv_Y-M_on_blow-up_Subsection_}).
Consider $F^{gr}$ as a bundle on $T_xM\backslash 0$.
In the notation of \ref{_F^gr_compa_twi_tra_Proposition_},
$F^{gr}= \c F^{gr}\restrict{(M, I)}$. By
\ref{_F^gr_compa_twi_tra_Proposition_}, the sheaf
$\c F^{gr}$ is ${\Bbb C}^*$-equivariant and compatible with the twistor
structure. According to the Swann's formalism for bundles
(\ref{_C^*_equi_cano_conne_Theorem_}),
the bundle $F^{gr}\restrict{T_xM\backslash 0}$ is equipped
with a natural Hermitian connection $\nabla_{F^{gr}}$ which is flat along
the leaves of ${\Bbb H}^*$-action. Let
$(B, \nabla_{\Bbb H})$ be the corresponding $B_2$-bundle
on
\[ {\Bbb P}_{\Bbb H} T_x M:=
\bigg(T_xM\backslash 0\bigg)/{\Bbb H}^*\cong {\Bbb H}P^n.
\]
Then $\pi^*F\restrict C$ is a holomorphic
bundle over the corresponding twistor space
$C= \operatorname{Tw}({\Bbb P}_{\Bbb H} T_x M)$, obtained as a pullback of
$(B, \nabla_{\Bbb H})$ as in \ref{_B_2_=_holo_on_Tw_Claim_}.
The natural K\"ahler metric on
the twistor space $C= \operatorname{Tw}({\Bbb P}_{\Bbb H} T_x M)$
is the Fubini-Study metric (\ref{_Tw_HP^n_Fu-St_Example_}).
By \ref{_twi_tra_YM_q-K_Theorem_},
the bundle $\pi^*F\restrict C$ is Yang-Mills and
has $\deg c_1\left(\pi^* F\restrict{C}\right) =0$.
Finally, by Uhlenbeck-Yau theorem (\ref{_UY_Theorem_}),
the bundle $\pi^*F\restrict C$ is polystable.
\blacksquare
\section{Moduli spaces of hyperholomorphic sheaves and bundles}
\label{_modu_hyperho_Section_}
\subsection{Deformation of hyperholomorphic sheaves with
isolated singularities}
The following theorem is an elementary
consequence of \ref{_hyperho_blow-up_stable_Theorem_}.
The proof uses well known results on stability and reflexization
(see, for instance, \cite{_OSS_}).
The main idea of the proof is the following.
Given a family of hyperholomorphic sheaves with an
isolated singularity, we blow-up this singularity
and restrict the obtained family to a blow-up
divisor. We obtain a family of coherent sheaves
$\frak V_s$, $s\in S$ over ${\Bbb C} P^{2n+1}$, with fibers semistable
of slope zero. Assume that for all $s\in S$,
$s\neq s_0$, the sheaf $\frak V_s$ is trivial.
Then the family $\frak V$ is also trivial,
up to a reflexization.
\hfill
We use the following property of reflexive sheaves.
\hfill
\definition
Let $X$ be a complex manifold, and $F$ a torsion-free coherent sheaf.
We say that $F$ is {\bf normal} if
for all open subvarieties $U\subset X$,
and all closed subvarieties $Y\subset U$ of codimension 2,
the restriction \[ \Gamma_U(F) {\:\longrightarrow\:} \Gamma_{U\backslash Y}(F)\]
is an isomorphism.
\hfill
\lemma\label{_normal_refle_Lemma_}
Let $X$ be a complex manifold, and $F$ a torsion-free coherent sheaf.
Then $F$ is reflexive if and only if $F$ is normal.
{\bf Proof:} \cite{_OSS_}, Lemma 1.1.12. \blacksquare
\hfill
\theorem \label{_reflexi_defo_loca_trivi_Theorem_}
Let $M$ be a hyperk\"ahler manifol, $I$ an induced
complex structure, $S$ a complex variety
and $\frak F$ a family of coherent sheaves over
$(M, I)\times S$. Consider the sheaf
$F_{s_0}:= {\frak F}\restrict{(M, I)\times \{s_0\}}$.
Assume that the sheaf $F_{s_0}$ is equipped with a filtration
$\xi$.
Let $F_i$, $i= 1, ..., m$ denote the associated graded components of
$\xi$, and $F_i^{**}$ denote their reflexizations.
Assume that $\frak F$ is locally trivial outside
of $(x_0, s_0)\in (M, I)\times S$.
Assume, moreover, that all sheaves $F_i^{**}$,
$i= 1, ..., m$ admit a hyperholomorphic
connection. Then the reflexization
${\frak F}^{**}$ is locally trivial.
\hfill
{\bf Proof:}
Clearly, it suffices to prove
\ref{_reflexi_defo_loca_trivi_Theorem_} for
${\frak F}$ reflexive.
Let $\tilde X$ be the blow-up of
$(M, I)\times S$ in $\{x_0\} \times S$,
and $\tilde {\frak F}$ the pullback of
$\frak F$ to $\tilde X$. Clearly,
$\tilde X= \tilde M\times S$, where $\tilde M$ is a
blow-up of $(M,I)$ in $x_0$. Denote by $C\subset \tilde M$ the blow-up
divisor of $\tilde M$. Taking $S$ sufficiently small,
we may assume that the bundle
$\frak F\restrict{\{x_0\}\times(S\backslash\{s_0\})}$
is trivial. Thus, the bundle
$\tilde{\frak F}\restrict {(C\times S)\backslash (C\times \{s_0\})}$,
which is a pullback of
$\frak F\restrict{\{x_0\}\times(S\backslash\{s_0\})}$
under the natural projection
$(C\times S)\backslash (C\times \{s_0\})
{\:\longrightarrow\:}\{x_0\}\times(S\backslash\{s_0\}$
is trivial.
To prove that $\frak F$ is locally trivial, we have to show that
$\tilde {\frak F}$ is locally trivial,
and that the restriction of
$\tilde {\frak F}$ to $C\times S$
is trivial along the fibers of the natural
projection $C \times S{\:\longrightarrow\:} S$. Clearly, to show that
$\tilde {\frak F}$ is locally trivial we need only to prove
that the fiber $\tilde {\frak F}\restrict z$ has constant
dimension for all $z\in C \times S$.
Thus, $\tilde {\frak F}$ is locally trivial
if and only if $\tilde {\frak F}\restrict {C\times S}$
is locally trivial. This sheaf is reflexive, since
it corresponds to an associate graded sheaf of a reflexive sheaf,
in the sense of Footnote to \ref{_exte_conne_conje_Remark_}.
It is non-singular in codimension 2, because all
reflexive sheaves are non-singular in codimension 2
(\cite{_OSS_}, Ch. II, Lemma 1.1.10).
By \ref{_hyperho_blow-up_stable_Theorem_}, the sheaf
$\tilde {\frak F}\restrict {C\times\{s\}}$
is semistable of slope zero.
\ref{_reflexi_defo_loca_trivi_Theorem_} is implied
by the following lemma, applied to the
sheaf $\tilde {\frak F}\restrict {C\times S}$.
\hfill
\lemma \label{_F_to_blow-up_stable=>loc_triv_Lemma_}
Let $C$ be a complex projective space, $S$ a complex variety
and $\frak F$ a torsion-free sheaf over $C\times S$.
Consider an open set $U\stackrel j\hookrightarrow C\times S$,
which is a complement of $C\times\{s_0\}\subset C\times S$.
Assume that the sheaf ${\frak F}\restrict U$ is trivial:
${\frak F}\restrict U\cong {\cal O}_U^n$.
Assume, moreover, that $\frak F$ is non-singular in codimension 2,
the sheaf $\left({\frak F}\restrict{C \times\{s_0\}}\right)^{**}$
is semistable of slope zero and
\[ \operatorname{rk} {\frak F}=\operatorname{rk} {\frak F}\restrict {C\times \{s_0\}}. \]
Then the reflexization
${\frak F}^{**}$ of $\frak F$ is a trivial bundle.
\hfill
{\bf Proof:}
Using induction,
it suffices to prove \ref{_F_to_blow-up_stable=>loc_triv_Lemma_}
assuming that it is proven for all $\frak F'$ with
$\operatorname{rk} {\frak F}'<\operatorname{rk} {\frak F}$.
We may also assume that
$S$ is Stein, smooth and 1-dimensional.
\hfill
{\bf Step 1:} {\it We construct an exact sequence
\[ 0{\:\longrightarrow\:}\frak F_2{\:\longrightarrow\:}\frak F{\:\longrightarrow\:} \operatorname{im} p_{O_1} {\:\longrightarrow\:} 0 \]
of sheaves of positive rank, which, as we prove in Step 3, satisfy assumptions
of \ref{_F_to_blow-up_stable=>loc_triv_Lemma_}.}
\hfill
Consider the pushforward sheaf $j_* {\cal O}_U^n$.
From the definition of $j_*$, we obtain a
canonical map
\begin{equation}\label{_embe_tilde_F_to_j_*_Lemma_}
{\frak F}{\:\longrightarrow\:} j_* {\cal O}_U^n,
\end{equation}
and the kernel of this map is a torsion
subsheaf in ${\frak F}$.
Let $f$ be a coordinate
function on $S$, which vanishes in $s_0\in S$. Clearly,
\[
j_* {\cal O}_U^n\cong {\cal O}_{C\times S}^n\left[\frac{1}{f}\right].
\]
On the other hand, the sheaf ${\cal O}_{C\times S}\left[\frac{1}{f}\right]$
is a direct limit of the following diagram:
\[ {\cal O}_{C\times S}^n\stackrel{\cdot f}{\:\longrightarrow\:}
{\cal O}_{C\times S}^n\stackrel{\cdot f}{\:\longrightarrow\:}
{\cal O}_{C\times S}^n\stackrel{\cdot f}{\:\longrightarrow\:} ...,
\]
where $\cdot f$ is the injection given by the multiplication by $f$.
Thus, the map \eqref{_embe_tilde_F_to_j_*_Lemma_}
gives an embedding
\[ {\frak F}\stackrel p \hookrightarrow {\cal O}_{C\times S}^n,\]
which is idenity outside of $(x_0, s_0)$.
Multiplying
$p$ by $\frac{1}{f}$ if necessary, we may assume that
the restriction $p\restrict {C\times \{s_0\}}$ is non-trivial.
Thus, $p$ gives a map
\begin{equation} \label{_goth_F_to_calo_Equation_}
{\frak F}\restrict {C\times \{s_0\}}{\:\longrightarrow\:}
{\cal O}^n_{C\times \{s_0\}}.
\end{equation}
with image of positive rank.
Since both sides of \eqref{_goth_F_to_calo_Equation_}
are semistable of slope zero,
and ${\cal O}^n_{C\times \{s_0\}}$ is po\-ly\-stable,
the map \eqref{_goth_F_to_calo_Equation_}
satisfies the following conditions.
(see \cite{_OSS_}, Ch. II, Lemma 1.2.8
for details).
\hfill
\begin{minipage}[m]{0.8\linewidth}
Let $F_1:= \operatorname{im} p\restrict {C\times \{s_0\}}$,
and $F_2:= \ker p\restrict {C\times \{s_0\}}$.
Then the reflexization
of $F_1$ is a trivial bundle ${\cal O}^k_{C\times \{s_0\}}$,
and $p$ maps $F_1$ to the direct summand
$O_1'= {\cal O}^k_{C\times \{s_0\}}\subset {\cal O}^n_{C\times \{s_0\}}$.
\end{minipage}
\hfill
Let $O_1 = {\cal O}^k_{C\times S}\subset {\cal O}^n_{C\times S}$
be the corresponding free subsheaf of ${\cal O}^n_{C\times S}$.
Consider the natural projection $\pi_{O_1}$ of
${\cal O}^n_{C\times S}$ to $O_1$.
Let $p_{O_1}$
be the composition of $p$ and $\pi_{O_1}$,
$\frak F_1$ the image of $p_{O_1}$,
and $\frak F_2$ the kernel of $p_{O_1}$.
\hfill
{\bf Step 2:} {\it We show that the sheaves
$\frak F_2$ and $\frak F_1$ and non-singular
in codimension 2.}
\hfill
Consider the exact sequence
\[ Tor^1({\cal O}_{C\times \{s_0\}}, {\frak F}_1){\:\longrightarrow\:}
\frak F_2\restrict {C\times \{s_0\}}{\:\longrightarrow\:}
\frak F\restrict {C\times \{s_0\}} {\:\longrightarrow\:}
{\frak F}_1\restrict {C\times \{s_0\}} {\:\longrightarrow\:} 0
\]
obtained by tensoring the
sequence \[ 0{\:\longrightarrow\:}\frak F_2{\:\longrightarrow\:}\frak F{\:\longrightarrow\:} {\frak F}_1 {\:\longrightarrow\:} 0\]
with ${\cal O}_{C\times \{s_0\}}$. From this sequence,
we obtain an isomorphism
${\frak F}_1\restrict {C\times \{s_0\}}\cong F_1$.
A torsion-free coherent sheaf over a smooth manifold
is non-singular in codimension 1
(\cite{_OSS_}, Ch. II, Corollary 1.1.8).
Since $\frak F$ is non-singular in codimension 2,
the restriction $\frak F\restrict {C\times \{s_0\}}$
is non-singular in codimension 1. Therefore, the torsion of
$\frak F\restrict {C\times \{s_0\}}$ has support of
codimension at least 2 in $C\times \{s_0\}$.
Since the sheaf $F_2$ is a subsheaf of
$\frak F\restrict {C\times \{s_0\}}$, its torsion
has support of codimension at least 2.
Therefore, the singular set of $F_2$
has codimension at least 2 in $C\times \{s_0\}$.
The rank of $F_2$ is by definition equal to $n-k$.
Since $F_1$ has rank $k$, the singular
set of $\frak F_1$ coincides with the singular set of
$F_1$. Since the restriction
${\frak F}_1\restrict {C\times \{s_0\}}=F_1$,
is a subsheaf of a trivial
bundle of dimension $k$ on $C\times \{s_0\}$, it
is torsion-free. Therefore, the singularities of
${\frak F}_1$ have codimension at least
2 in $C\times \{s_0\}$.
We obtain that the support of
$Tor^1({\cal O}_{C\times \{s_0\}}, {\frak F}_1)$
has codimension at least 2 in $C\times \{s_0\}$. Since
the quotient sheaf
\begin{equation}\label{_F_2_quoti_Equation_}
\frak F_2\restrict {C\times \{s_0\}}\bigg /
Tor^1({\cal O}_{C\times \{s_0\}}, {\frak F}_1)\cong F_2
\end{equation}
is isomorphic to the sheaf $F_2$,
this quotient is
non-singular in codimension 1. Since we proved that
$F_2$ is non-singular in codimension 1, the sheaf
$\frak F_2 \restrict {C\times \{s_0\}}$ is also non-singular
in codimension 1, and its rank is equal to the rank
of $F_2$.
Let $R$ be the union of singular sets
of the sheaves $\frak F_2$, $\frak F$, ${\frak F}_1$.
Clearly, $R$ is contained in $C\times \{s_0\}$,
and $R$ coincides with the set of all
$x\in C\times \{s_0\}$ where the dimension of the
fiber of the sheaves
$\frak F_2$, $\frak F$, ${\frak F}_1$ is
not equal to $n-k$, $n$, $k$. We have seen that
the restrictions of $\frak F_2$, ${\frak F}_1$
to ${C\times \{s_0\}}$ have ranks $n-k$, $k$.
Therefore, the singular sets of $\frak F_2$, ${\frak F}_1$
coincide with the singular sets of
$\frak F_2\restrict{C\times \{s_0\}}$,
${\frak F}_1\restrict{C\times \{s_0\}}$. We have shown that these
singular sets have codimension at least 2 in $C\times \{s_0\}$.
On the other hand, $\frak F$ is non-singular in codimension 2,
by the conditions of \ref{_F_to_blow-up_stable=>loc_triv_Lemma_}.
Therefore, $R$ has codimension at least 3 in $C\times S$.
\hfill
{\bf Step 3:} {\it We check the assumptions of
\ref{_F_to_blow-up_stable=>loc_triv_Lemma_} applied to the sheaves
$\frak F_2$, ${\frak F}_1$.}
\hfill
Since
the singular set of ${\frak F}_1$ has codimension 2 in $C\times \{s_0\}$,
the ${\cal O}_{C\times \{s_0\}}$-sheaf
$Tor^1({\cal O}_{C\times \{s_0\}},{\frak F}_1)$ is a torsion sheaf
with support of codimension 2 in $C\times \{s_0\}$.
By \eqref{_F_2_quoti_Equation_}, the
reflexization of $\frak F_2\restrict {C\times \{s_0\}}$
coincides with the reflexization of $F_2$. Thus, the sheaf
$\left(\frak F_2\restrict {C\times \{s_0\}}\right)^{**}$
is semistable. On the other hand, outside of ${C\times \{s_0\}}$,
the sheaf $\frak F_2$ is a trivial bundle.
Thus, $\frak F_2$ satisfies assumptions of
\ref{_F_to_blow-up_stable=>loc_triv_Lemma_}.
Similarly, the sheaf $\frak F_1$ is non-singular in
codimension 2, its restriction to
$C\times \{s_0\}$ has trivial reflexization, and
it is free outside of $C\times \{s_0\}$.
\hfill
{\bf Step 4:} {\it We apply induction
and prove \ref{_F_to_blow-up_stable=>loc_triv_Lemma_}.}
\hfill
By induction assumption, the
reflexization of $\frak F_2$ is isomorphic to
a trivial bundle ${\cal O}^{n-k}_{C\times S}$.
and reflexization of $\frak F_1$ is
${\cal O}^k_{C\times S}$. We obtain an exact sequence
\begin{equation}\label{_F_2_to_F_exa_Equation_}
0 {\:\longrightarrow\:} \frak F_2{\:\longrightarrow\:} \frak F {\:\longrightarrow\:} {\frak F}_1 {\:\longrightarrow\:} 0,
\end{equation}
where the sheaves $\frak F_2$ and ${\frak F}_1$ have trivial
reflexizations.
Let $V:= C\times S\backslash R$.
Restricting the exact sequence \eqref{_F_2_to_F_exa_Equation_} to $V$,
we obtain an exact sequence
\begin{equation}\label{_F_2_to_F_restri_V_exa_Equation_}
0 {\:\longrightarrow\:} {\cal O}^{n-k}_{V} \stackrel a {\:\longrightarrow\:} {\frak F}\restrict V
\stackrel b{\:\longrightarrow\:} {\cal O}^k_{V} {\:\longrightarrow\:} 0.
\end{equation}
Since $V$ is a complement of a codimension-3 complex subvariety in
a smooth Stein domain, the first cohomology of a trivial sheaf on
$V$ vanish. Therefore, the sequence
\eqref{_F_2_to_F_restri_V_exa_Equation_}
splits, and the sheaf ${\frak F}\restrict V$
is a trivial bundle. Consider the pushforward
$\zeta_*{\frak F}\restrict V$, where $\zeta:\; V{\:\longrightarrow\:} C\times S$
is the standard map. Then $\zeta_*{\frak F}\restrict V$
is a reflexization of ${\frak F}$
(a pushforward of a reflexive sheaf over a
subvariety of codimension 2 or more is
reflexive -- see \ref{_normal_refle_Lemma_}).
On the other hand, since the sheaf ${\frak F}\restrict V$
is a trivial bundle, its push-forward over a
subvariety of codimension at least 2 is also
a trivial bundle over $C\times S$. We proved that
the sheaf ${\frak F}^{**}=\zeta_*{\frak F}\restrict V$ is a trivial bundle
over $C\times S$. The push-forward
$\zeta_*{\frak F}\restrict V$ coincides with reflexization
of $\frak F$, by \ref{_normal_refle_Lemma_}.
This proves
\ref{_F_to_blow-up_stable=>loc_triv_Lemma_} and
\ref{_reflexi_defo_loca_trivi_Theorem_}.
\blacksquare
\subsection{The Maruyama moduli space of coherent sheaves}
This Subsection is a compilation of results of Gieseker and Maruyama
on the moduli of coherent sheaves over projective manifolds.
We follow \cite{_OSS_}, \cite{_Maruyama:Si_}.
To study the moduli spaces of holomorphic bundles and coherent sheaves,
we consider the following definition of stability.
\hfill
\definition
(Gieseker--Maruyama stability)
(\cite{_Gieseker_}, \cite{_OSS_})
Let $X$ be a projective variety, ${\cal O}(1)$ the standard line bundle
and $F$ a torsion-free coherent sheaf. The sheaf $F$ is called
{\bf Gieseker--Maruyama stable} (resp. Gieseker--Maruyama
semistable) if for all coherent subsheaves $E\subset F$
with $0<\operatorname{rk} E<\operatorname{rk} F$, we have
\[ p_F(k) < p_E(k) \ \ (\text{resp.}, \ \ p_F(k) \leq p_E(k))
\]
for all sufficiently large numbers $k\in {\Bbb Z}$. Here
\[ p_F(k) = \frac{\dim \Gamma_X(F\otimes {\cal O}(k))}{\operatorname{rk} F}.
\]
\hfill
Clearly, Gieseker--Maruyama stability is weaker than the
Mum\-ford-\-Ta\-ke\-mo\-to stability. Every Gieseker--Maruyama
semistable sheaf $F$ has a so-called Jordan-H\"older filtration
$F_0\subset F_1\subset ...\subset F$ with Gieseker--Maruyama
stable successive quotients $F_i/F_{i-1}$. The corresponding
associated graded sheaf
\[ \oplus F_i/F_{i-1} \]
is independent from a choice of a filtration.
It is called {\bf the associate graded quotient of
the Jordan-H\"older filtration on $F$}.
\hfill
\definition
Let $F$, $G$ be Gieseker--Maruyama
semistable sheaves on $X$. Then
$F$, $G$ are called {\bf $S$-equivalent}
if the corresponding
associate graded quotients
$\oplus F_i/F_{i-1}$, $\oplus G_i/G_{i-1}$
are isomorphic.
\hfill
\definition
Let $X$ be a complex manifold, $F$ a torsion-free sheaf on $X$,
and $Y$ a complex variety. Consider a sheaf $\c F$ on
$X\times Y$ which is flat over $Y$. Assume that for
some point $s_0\in Y$, the sheaf $\c F\restrict {X\times \{s_0\}}$
is isomorphic to $F$. Then $\c F$ is called {\bf a deformation of
$F$ parametrized by $Y$}.
We say that a sheaf $F'$ on $X$ is {\bf deformationally
equivalent} to $F$ if for some $s\in Y$, the restriction
$\c F\restrict {X\times \{s\}}$ is isomorphic to $F'$.
Slightly less formally, such sheaves are called {\bf deformations of
$F$}. If $F'$ is a (semi-)stable bundle, it is called
{\bf a (semi-)stable bundle deformation of $F$.}
\hfill
\remark\label{_Chern_deforma_equal_}
Clearly, the Chern classes of deformationally equivalent
sheaves are equal.
\hfill
\definition \label{_coarse_modu_Definition_}
Let $X$ be a complex manifold, and $F$ a torsion-free sheaf on $X$,
and $\c M_{mar}$ a complex variety. We say that $\c M_{mar}$ is a
{\bf coarse moduli space of deformations of $F$}
if the following conditions hold.
\begin{description}
\item[(i)] The
points of $s\in \c M_{mar}$ are in bijective correspondence
with $S$-equi\-va\-lence classes of coherent sheaves $F_s$
which are deformationally
equivalent to $F$.
\item[(ii)] For any flat deformation
$\c F$ of $F$ parametrized by $Y$, there exists a unique
morphism $\phi:\; Y{\:\longrightarrow\:} \c M_{mar}$ such that
for all $s\in Y$, the restriction
$\c F\restrict {X\times \{s\}}$ is $S$-equivalent
to the sheaf $F_{\phi(s)}$
corresponding to $\phi(s)\in \c M_{mar}$.
\end{description}
Clearly, the coarse moduli space is unique.
By \ref{_Chern_deforma_equal_},
the Chern classes of $F_s$ are equal for
all $s\in \c M_{mar}$.
\hfill
It is clear how to define other kinds of moduli spaces.
For instance, replacing the word {\it sheaf} by the word
{\it bundle} throughout \ref{_coarse_modu_Definition_},
we obtain a definition of {\bf the coarse moduli space
of semistable bundle deformations of $F$}. Further on,
we shall usually omit the word ``coarse'' and say
``moduli space'' instead.
\hfill
\theorem\label{_Maruya_exists_}
(Maruyama) Let $X$ be a projective manifold and
$F$ a coherent sheaf over $X$. Then
the Maruyama moduli space $\c M_{mar}$ of deformations of $F$ exists
and is compact.
{\bf Proof:} See, e. g., \cite{_Maruyama:Si_}. \blacksquare
\hfill
\subsection{Moduli of hyperholomorphic sheaves and $C$-restricted
comples structures}
Usually, the moduli space of semistable bundle
deformations of a bundle $F$ is not compact. To compactify
this moduli space, Maruyama adds points corresponding to
the deformations of $F$ which are singular
(these deformations can be non-reflexive
and can have singular reflexizations). Using the
desingularization theorems for hyperholomorphic sheaves,
we were able to obtain \ref{_reflexi_defo_loca_trivi_Theorem_},
which states (roughly speaking) that a deformation of a
semistable hyperholomorphic bundle is again a semistable
bunlde, assuming that all its singularities are isolated. In
Section \ref{_C_restri_Section_}, we
showed that under certain conditions,
a deformation of a hyperholomorphic sheaf is
again hyperholomorphic (\ref{_sheaf_on_C_restr_hyperho_Theorem_}).
This makes it possible to prove that a deformation of a semistable
hyperholomorphic bundle is locally trivial.
\hfill
In \cite{_Verbitsky:Hilbert_}, we have shown that
a Hilbert scheme of a K3 surface has no non-trivial trianalytic
subvarieties, for a general hyperk\"ahler structure.
\hfill
\theorem\label{_space_semista_bu_compa_Theorem_}
Let $M$ be a compact hyperk\"ahler manifold
without non-trivial trianalytic subvarieties, $\dim_{\Bbb H}\geq 2$,
and $I$ an induced complex structure.
Consider a hyperholomorphic bundle $F$ on $(M, I)$
(\ref{_hyperho_shea_Definition_}). Assume that $I$ is a
$C$-restricted complex structure, $C= \deg_I c_2(F)$. Let
$\c M$ be the moduli space of semistable bundle
deformations of $F$ over $(M, I)$. Then
$\c M$ is compact.
\hfill
{\bf Proof:} The complex structure $I$ is by definition
algebraic, with unique polarization. This makes it possible
to speak of Gieseker--Maruyama stability on $(M, I)$. Denote
by $\c M_{mar}$ the Maruyama moduli of deformations of $F$.
Then $\c M$ is naturally an open subset of $\c M_{mar}$.
Let $s\in \c M_{mar}$ be an arbitrary point
and $F_s$ the corresponding coherent sheaf
on $(M, I)$, defined up to $S$-equivalence.
According to \ref{_Chern_deforma_equal_},
the Chern classes of $F$ and $F_s$ are equal.
Thus, by \ref{_sheaf_on_C_restr_hyperho_Theorem_},
the sheaf $F_s$ is hyperholomorphic. Therefore,
$F_s$ admits a filtration with hyperholomorphic stable
quotient sheaves $F_i$, $i= 1, ..., m$. By \ref{_singu_triana_Claim_},
the singular set $S$ of $F_s$ is trianalytic.
Since $M$ has no proper trianalytic subvarieties,
$S$ is a collection of points. We obtain that
$F_s$ has isolated singularities. Let $\frak F$ be a family
of deformations of $F$, parametrized by $Y$.
The points $y\in Y$ correspond to deformations
$F_y$ of $F_s$. Assume that for all $y\in Y$,
$y\neq s$, the sheaf $F_y$ is a bundle.
Since $\c M$ is open in $\c M_{mar}$, such a
deformation always exists.
The sheaf $F_s$ has isolated singularities and
admits a filtration with hyperholomorphic stable
quotient sheaves. This implies that the family $\frak F$ satisfies the
conditions of \ref{_reflexi_defo_loca_trivi_Theorem_}.
By \ref{_reflexi_defo_loca_trivi_Theorem_},
the reflexization $\frak F^{**}$ is locally trivial.
To prove that $\c M= \c M_{mar}$, we have to show that
for all $s\in \c M_{mar}$, the corresponding
coherent sheaf $F_s$ is locally trivial.
Therefore, to finish the proof of
\ref{_space_semista_bu_compa_Theorem_}, it
remains to prove the following
algebro-geometric claim.
\hfill
\claim \label{_defo_shea-w-holes-has-holes_Claim_}
Let $X$ be a compact complex manifold, $\dim_{\Bbb C} X>2$,
and $\frak F$ a torsion-free coherent sheaf over
$X\times Y$ which is flat over $Y$.
Assume that the reflexization of $\frak F$ is locally trivial,
$\frak F$ has isolated singularities, and for
some point $s\in Y$, the restriction of
$\frak F$ to the complement $(X\times Y)\backslash (X\times \{s\})$
is locally trivial. Then the reflexization
$\left(\frak F\restrict X\times\{s\}\right)^{**}$
is locally trivial.
\hfill
\remark
We say that a kernel of a map from a bundle
to an Artinian sheaf is {\bf a bundle with holes}.
In slightly more intuitive terms,
\ref{_defo_shea-w-holes-has-holes_Claim_}
states that a flat deformation of a bundle with holes
is again a bundle with holes, and cannot be smooth,
assuming that $\dim_{\Bbb C} X>2$.
\hfill
{\bf Proof of \ref{_defo_shea-w-holes-has-holes_Claim_}:}
\ref{_defo_shea-w-holes-has-holes_Claim_} is well known.
Here we give a sketch of a proof.
Consider a coherent sheaf $F_s= \frak F \restrict{X\times \{s\}}$,
and an exact sequence
\[ 0{\:\longrightarrow\:} F_s{\:\longrightarrow\:} F_s^{**} {\:\longrightarrow\:} k{\:\longrightarrow\:} 0, \]
where $k$ is an Artinian sheaf. By definition,
the sheaf $F_s^{**}$ is locally trivial. The flat deformations
of $F_s$ are infinitesimally classified by $Ext^1(F_s, F_s)$.
Replacing $F_s$ by a quasi-isomorphic complex of sheaves
$F_s^{**} {\:\longrightarrow\:} k$, we obtain a spectral sequence converging
to $Ext^\bullet(F_s, F_s)$. In the $E_2$-term of this sequence,
we observe the group
\[ Ext^1(F_s^{**}, F_s^{**})\oplus Ext^1(k, k)
\oplus Ext^2(k, F_s^{**})\oplus Ext^0(F_s^{**}, k).
\]
which is responsible for $Ext^1(F_s, F_s)$.
The term $Ext^1(F_s^{**}, F_s^{**})$ is responsible
for deformations of the bundle $F_s^{**}$,
the term $Ext^0(F_s^{**}, k)$ for the deformations
of the map $F_s^{**} {\:\longrightarrow\:} k$, and
the term $Ext^1(k, k)$ for the deformations
of the Artinian sheaf $k$. Thus,
the term $Ext^2(k, F_s^{**})$ is responsible for
the deformations of $F_s$ which change the dimension
of the cokernel of the embedding $F_s{\:\longrightarrow\:} F_s^{**}$.
We obtain that whenever $Ext^2(k, F_s^{**})=0$,
all deformations of $F_s$ are singular.
On the other hand, $Ext^2(k, F_s^{**})=0$,
because the $i$-th $Ext$ from the skyscraper to a
free sheaf on a manifold of dimension
more than $i$ vanishes (this is a basic result of
Grothendieck's duality, \cite{_Hartshorne:Grothendieck's_}).
\blacksquare
\section{New examples of hyperk\"ahler manifolds}
\label{_new_exa_Section_}
\subsection{Twistor paths}
\label{_twi_paths_Subsection_}
This Subsection contains an exposition and
further elaboration of the results of
\cite{_coho_announce_} concerning
the twistor curves in the moduli space of
complex structures on a complex manifold of hyperk\"ahler type.
Let $M$ be a compact manifold admitting a hyperk\"ahler structure.
In \ref{_moduli_hyperka_Definition_}, we
defined the coarse, marked moduli space of
complex structures on $M$, denoted by $Comp$.
For the duration of this section, we fix a
compact simple hyperk\"ahler manifold $M$, and
its moduli $Comp$.
\hfill
Further on, we shall need the following fact.
\hfill
\claim \label{_simple=hyperho_Claim_}
Let $M$ be a hyperk\"ahler manifold,
$I$ an induced complex structure of general type,
and $B$ a holomorphic vector bundle over $(M, I)$.
Then $B$ is stable if an only if $B$ is
simple.\footnote{Simple sheaves are coherent sheaves which have
no proper subsheaves}
{\bf Proof:}
By \ref{_Lambda_of_inva_forms_zero_Lemma_},
for all $\omega \in Pic(M, I)$, we have
$\deg_I(\omega)=0$. Therefore, every subsheaf
of $B$ is destabilising.
\blacksquare
\hfill
\remark
In assumptions of \ref{_simple=hyperho_Claim_},
all stable bundles are hyperholomorphic
(\ref{_inva_then_hyperho_Theorem_}).
Therefore, \ref{_simple=hyperho_Claim_}
implies that $B$ is hyperholomorphic
if it is simple.
\hfill
In Subsection \ref{_modu_and_C-restri_Subsection_},
we have shown that every hyperk\"ahler structure $\c H$
corresponds to a holomorphic
embedding \[ \kappa(\c H):\; {\Bbb C} P^1 {\:\longrightarrow\:} Comp, \ \
L {\:\longrightarrow\:} (M, L).\]
\definition
A projective line $C \subset Comp$ is called
{\bf a twistor curve} if $C= \kappa(\c H)$ for some
hyperk\"ahler structure $\c H$ on $M$.
\hfill
The following theorem was proven in \cite{_coho_announce_}.
\hfill
\theorem \label{_twistor_connect_Theorem_}
(\cite{_coho_announce_}, Theorem 3.1)
Let $I_1, I_2\in Comp$. Then there exist a sequence of
intersecting twistor curves which connect $I_1$ with $I_2$.
\blacksquare
\hfill
\definition
Let $P_0$, ..., $P_n\subset Comp$ be
a sequence of twistor curves, supplied with an intersection point
$x_{i+1}\in P_i\cap P_{i+1}$ for each $i$. We say that
$\gamma= P_0, ..., P_n, x_1, ..., x_n$ is
a {\bf twistor path}. Let $I$, $I'\in Comp$.
We say that $\gamma$ is {\bf a twistor path
connecting $I$ to $I'$} if $I\in P_0$ and $I'\in P_n$.
The lines $P_i$ are called {\bf the edges},
and the points $x_i$ {\bf the vertices}
of a twistor path.
\hfill
Recall that in \ref{_generic_manifolds_Definition_},
we defined induced complex structures which are
generic with respect to a hyperk\"ahler structure.
\hfill
Given a twistor curve $P$, the corresponding hyperk\"ahler
structure $\c H$ is unique (\ref{_hyperka_etale_Theorem_}).
We say that a point $x\in P$ is {\bf of general type},
or {\bf generic with respect
to $P$} if the corresponding complex structure is generic
with respect to $\c H$.
\hfill
\definition \label{_admi_twi_path_Definition_}
Let $I$, $J\in Comp$ and $\gamma= P_0, ..., P_n$ be a twistor path from
$I$ to $J$, which corresponds to the hyperk\"ahler structures
$\c H_0$, ..., $\c H_n$. We say that $\gamma$ is {\bf admissible}
if all vertices of $\gamma$ are of general type with respect
to the corresponding edges.
\hfill
\remark
In \cite{_coho_announce_},
admissible twistor paths were defined slightly differently.
In addition to the conditions above, we required that
$I$, $J$ are of general type with respect to $\c H_0$, $\c H_n$.
\hfill
\ref{_twistor_connect_Theorem_} proves that
every two points $I$, $I'$ in $Comp$ are connected
with a twistor path. Clearly, each twistor path
induces a diffeomorphism $\mu_\gamma:\; (M,I){\:\longrightarrow\:} (M,I')$.
In \cite{_coho_announce_}, Subsection 5.2,
we studied algebro-geometrical properties of this
diffeomorphism.
\hfill
\theorem \label{_admi_twi_impli_Theorem_}
Let $I$, $J\in Comp$, and $\gamma$ be an admissible twistor path from
$I$ to $J$. Then
\begin{description}
\item[(i)] There exists a natural
isomorphism of tensor cetegories
\[ \Phi_{\gamma}:\; Bun_I(\c H_0){\:\longrightarrow\:} Bun_J(\c H_n),\]
where $Bun_I(\c H_0)$, $Bun_J(\c H_n)$ are the categories of
polystable hyperholomorphic vector bundles on $(M, I)$,
$(M, J)$, taken with respect to $\c H_0$,
$\c H_n$ respectively.
\item[(ii)] Let $B\in Bun_I(\c H_0)$ be a stable hyperholomorphic bundle,
and \[ \c M_{I, \c H_0}(B)\] the moduli of stable deformations of $B$,
where stability is taken with respect to the K\"ahler metric induced
by $\c H_0$. Then $\Phi_{\gamma}$ maps stable bundles
which are deformationally equivalent to $B$ to the stable bundles
which are deformationally equivalent to $\Phi_\gamma(B)$.
Moreover, obtained this way bijection
\[ \Phi_\gamma:\; \c M_{I, \c H_0}(B){\:\longrightarrow\:} \c M_{J, \c H_n}(\Phi_\gamma(B))\]
induces a real analytic isomorphism of deformation spaces.
\end{description}
{\bf Proof:} \ref{_admi_twi_impli_Theorem_} (i) is
a consequence of \cite{_coho_announce_},
Corollary 5.1. Here we give a sketch of its proof.
Let $I$ be an induced complex structure
of general type. By \ref{_simple=hyperho_Claim_},
a bundle $B$ over $(M, I)$
is stable if and only if it is simple. Thus,
the category $Bun_I(\c H)$ is independent from
the choice of $\c H$ (\ref{_simple=hyperho_Claim_}).
In \ref{_equi_cate_Theorem_},
we constructed the equivalence of categories
$\Phi_{I, J}$, which gives the functor
$\Phi_\gamma$ for twistor path which
consists of a single twistor curve.
This proves \ref{_admi_twi_impli_Theorem_} (i)
for $n=1$. A composition of isomorphisms
$\Phi_{I, J}\circ \Phi_{J, J'}$ is well
defined, because the category $Bun_I(\c H)$
is independent from the choice of $\c H$.
Taking successive compositions of the maps
$\Phi_{I, J}$, we obtain an isomorphism $\Phi_\gamma$.
This proves \ref{_admi_twi_impli_Theorem_} (i).
The
variety $\c M_{I, \c H}(B)$ is singular hyperk\"ahler
(\cite{_Verbitsky:Hyperholo_bundles_}),
and the variety $\c M_{J, \c H}(B)$ is the same
singular hyperk\"ahler variety, taken with another
induced complex structure. By definition of
singular hyperk\"ahler varieties, this implies
that $\c M_{I, \c H}(B)$, $\c M_{J, \c H}(B)$
are real analytic equivalent, with equivalence
provided by $\Phi_{I, J}$. This proves
\ref{_admi_twi_impli_Theorem_} (ii).
\blacksquare
\hfill
For $I\in Comp$, denote by $Pic(M, I)$
the group $H^{1,1}(M, I)\cap H^2(M,{\Bbb Z})$, and by
$Pic(I, {\Bbb Q})$ the space
$H^{1,1}(M, I)\cap H^2(M, {\Bbb Q})\subset H^2(M)$.
Let $Q\subset H^2(M, {\Bbb Q})$ be a subspace of $H^2(M, {\Bbb Q})$,
and
\[ Comp_Q:= \{ I\in Comp \;\; | \;\; Pic(I, {\Bbb Q}) =Q\}. \]
\theorem\label{_admi_exi_Theorem_}
Let $\c H$, $\c H'$
be hyperk\"ahler structures, and
$I$, $I'$ be complex structures of general type to and induced by
$\c H$, $\c H'$. Assume that $Pic(I, {\Bbb Q}) = Pic(I', {\Bbb Q}) =Q$,
and $I$, $I'$ lie in the same connected component of $Comp_Q$.
Then $I$, $I'$ can be connected by an admissible path.
{\bf Proof:} This is \cite{_coho_announce_}, Theorem 5.2.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
For a general $Q$, we have no control over the number of connected
components of $Comp_Q$ (unless global Torelli theorem is proven),
and therefore we cannot directly apply
\ref{_admi_exi_Theorem_} to obtain results from algebraic
geometry.\footnote{Exception is a K3 surface, where Torelli holds.
For K3, $Comp_Q$ is connected for all $Q\subset H^2(M, {\Bbb Q})$.}
However, when $Q=0$, $Comp_Q$ is clearly
connected and dense in $Comp$. This is used to prove
the following corollary.
\hfill
\corollary \label{_I_conne_w_admi_Corollary_}
Let $I$, $I'\in Comp_0$. Then $I$ can be connected to $I'$
by an admissible twistor path.
{\bf Proof} This is \cite{_coho_announce_}, Corollary 5.2.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\definition\label{_gene_pola_Definition_}
Let $I\in Comp$ be a complex structure, $\omega$ be a K\"ahler form on
$(M, I)$, and $\c H$ the corresponding hyperk\"ahler
metric, which exists by Calabi-Yau theorem.
Then $\omega$ is called {\bf a generic polarization}
if any of the following conditions hold
\begin{description}
\item[(i)] For all $a\in Pic(M, I)$, the degree
$\deg_\omega(a)\neq 0$, unless $a=0$.
\item[(ii)] For all $SU(2)$-invariant integer classes
$a\in H^2(M, {\Bbb Z})$, we have $a=0$.
\end{description}
The conditions (i) and (ii) are equivalent
by \ref{_Lambda_of_inva_forms_zero_Lemma_}.
\hfill
\claim\label{_omega_gene_otho_Claim_}
Let $I\in Comp$ be a complex structure, $\omega$ be a K\"ahler form on
$(M, I)$, and $\c H$ the corresponding hyperk\"ahler structure,
which exists by Calabi-Yau theorem. Then $\omega$ is generic
if and only if for all integer classes $a\in H^{1,1}(M, I)$,
the class $a$
is not orthogonal to $\omega$ with respect to the Bogomolov-Beauville
pairing.
{\bf Proof:} Clearly, the map $\deg_\omega:\; H^2(M) {\:\longrightarrow\:} {\Bbb R}$
is equal (up to a scalar multiplier) to
the orthogonal projection onto the line ${\Bbb R}\cdot\omega$.
Then, \ref{_omega_gene_otho_Claim_}
is equivalent to \ref{_gene_pola_Definition_}, (i).
\blacksquare
\hfill
From \ref{_omega_gene_otho_Claim_}
it is clear that the set of generic polarizations is a complement
to a countable union of hyperplanes. Thus, generic
polarizations are dense in the K\"ahler cone of $(M, I)$,
for all $I$.
\hfill
\claim \label{_admi_pa_exist_for_gene_pol_Claim_}
Let $I, J\in Comp$, and $a$, $b$ be generic polarizations
on $(M, I)$. Consider the corresponding hyperk\"ahler structures
$\c H_0$ and $\c H_n$ inducing $I$ and $J$. Then there exists an
admissible twistor path starting from $I, \c H_0$
and ending with $\c H_n, J$.
\hfill
{\bf Proof:} Consider the twistor curves $P_0$,
$P_n$ corresponding to $\c H_0$, $\c H_n$. Since $a$,
$b$ are generic, the curves $P_0$,
$P_n$ intersect with $Comp_0$.
Applying \ref{_I_conne_w_admi_Corollary_},
we connect the curves $P_0$ and $P_n$ by an admissible path.
\blacksquare
\hfill
Putting together \ref{_admi_pa_exist_for_gene_pol_Claim_}
and \ref{_admi_twi_impli_Theorem_}, we obtain the following
result.
\hfill
\theorem\label{_iso_Bun_exists_gene_pola_Theorem_}
Let $I$, $J\in Comp$ be complex structures, and $a, b$ be generic
polarizations on $(M, I)$, $(M, J)$.
Then
\begin{description}
\item[(i)] There exist an
isomorphism of tensor cetegories
\[ \Phi_{\gamma}:\; Bun_I(a){\:\longrightarrow\:} Bun_J(a),\]
where $Bun_I(a)$, $Bun_J(b)$ are the categories of
polystable hyperholomorphic vector bundles on $(M, I)$,
$(M, J)$, taken with respect to
the hyperk\"ahler structures defined by
the K\"ahler classes $a$, $b$ as in
\ref{_symplectic_=>_hyperkahler_Proposition_}.
\item[(ii)] Let $B\in Bun_I(a)$ be a stable hyperholomorphic bundle,
and \[ \c M_{I, a}(B)\]
the moduli of stable deformations of $B$,
where stability is taken with respect to the
polarization $a$. Then $\Phi_{\gamma}$ maps stable bundles
which are deformationally equivalent to $B$ to the stable bundles
which are deformationally equivalent to $\Phi_\gamma(B)$.
Moreover, obtained this way bijection
\[ \Phi_\gamma:\; \c M_{I, a}(B){\:\longrightarrow\:} \c M_{J, b}(\Phi_\gamma(B))\]
induces a real analytic isomorphism of deformation spaces.
\end{description}
\blacksquare
\hfill
\lemma \label{_Phi_of_tange_Lemma_}
In assumptions of \ref{_admi_twi_impli_Theorem_},
let $B$ be a holomorphic tangent bundle of $(M, I)$. Then
$\Phi_\gamma(B)$ is a holomorphic tangent bundle of $(M, J)$.
{\bf Proof:} Clear. \blacksquare
\hfill
\corollary \label{_mod_of_tange_compa_Corollary_}
Let $I, J\in Comp$ be complex structures, and
$a$, $b$ generic polarizations on $(M, I)$, $(M, J)$. Assume that
the moduli of stable deformations $\c M_{I, a}(T(M, I))$ of
the holomorphic tangent bundle $T^{1,0}(M, I)$
is compact. Then the space $\c M_{J, b}(T(M, J))$
is also compact.
{\bf Proof:}
Let $\gamma$ be the twistor path
of \ref{_admi_pa_exist_for_gene_pol_Claim_}.
By \ref{_Phi_of_tange_Lemma_},
$\Phi_\gamma(T(M, I)) = T(M, J)$. Applying
\ref{_admi_twi_impli_Theorem_}, we
obtain a real analytic equivalence
from $\c M_{I, a}(T(M, I))$ to
$\c M_{J, b}(T(M, J))$.
\blacksquare
\subsection{New examples of hyperk\"ahler manifolds}
\theorem\label{_space_sta_bu_compa_hyperka_Theorem_}
Let $M$ be a compact hyperk\"ahler manifold
without non-trivial trianalytic subvarieties, $\dim_{\Bbb H}M\geq 2$,
and $I$ an induced complex structure.
Consider a hyperholomorphic bundle $F$ on $M$
(\ref{_hyperho_shea_on_M_Definition_}). Let
$F_I$ be the corresponding holomorphic bundle
over $(M, I)$. Assume that $I$ is a
$C$-restricted complex structure, $C= \deg_I c_2(F)$.
Assume, moreover, that all semistable bundle deformations of
$F_I$ are stable.\footnote{This may happen, for instance,
when $\operatorname{rk} F= \dim_{\Bbb C} M=n$, and the number $c_n(F)$ is prime.}
Denote by $\c M_F^I$ the moduli of stable bundle deformations of
$F_I$ over $(M, I)$. Then
\begin{description}
\item[(i)]
the normalization
$\tilde{\c M}_F^I$ is a compact and
smooth complex manifold equipped with a natural hyperk\"ahler
structure.
\item[(ii)]
Moreover, for all induced complex structures $J$ on $M$,
the the variety $\c M_F^J$ is compact, and has a smooth
normalization $\tilde{\c M}_F^J$,
which is also equipped with a natural hyperk\"ahler
structure.
\item[(iii)] Finally, the hyperk\"ahler manifolds
$\tilde{\c M}_F^J$, $\tilde{\c M}_F^I$ are naturally isomorphic.
\end{description}
{\bf Proof:} The variety $\c M_F^I$ is compact by
\ref{_space_semista_bu_compa_Theorem_}.
In \cite{_Verbitsky:Hyperholo_bundles_},
it was proven that the space $\c M_F^I$ of stable
deformations of $F$ is a singular hyperk\"ahler
variety (see also \cite{_NHYM_} for an explicit construction
of the twistor space of $\c M_F^I$).
Then \ref{_space_sta_bu_compa_hyperka_Theorem_}
is a consequence of the Desingularization Theorem
for singular hyperk\"ahler varietiess (\ref{_desingu_Theorem_}).
\blacksquare
\hfill
The assumptions of \ref{_space_sta_bu_compa_hyperka_Theorem_}
are quite restrictive. Using the technique of
twistor paths, developed in
Subsection \ref{_twi_paths_Subsection_},
it is possible to prove a more accessible form
of \ref{_space_sta_bu_compa_hyperka_Theorem_}.
\hfill
Let $M$ be a hyperk\"ahler manifold, and $I$, $J$ induced complex structures.
Given an admissible twistor
path from $I$ to $J$, we obtain an equivalence
$\Phi_\gamma$ between the category of hyperholomorphic bundles
on $(M, I)$ and $(M, J)$.
\hfill
\theorem \label{_twi_pa_space_sta_compa_Theorem_}
Let $M$ be a compact simple hyperk\"ahler manifold, $\dim_{\Bbb H}M >1$,
and $I$ a complex structure
on $M$. Consider a generic polarization $a$ on $(M, I)$.
Let $\c H$ be the corresponding hyperk\"ahler structure, and $F$
a hyperholomorphic bundle on $(M, I)$. Fix
a hyperk\"ahler structure $\c H'$ on $M$ admitting $C$-restricted
complex structures, such that $M$ has no trianalytic subvarieties
with respect to $\c H'$. Assume that for some
$C$-restricted complex structure $J$
induced by $\c H'$, $C=\deg_I c_2(F)$, all admissible twistor
paths $\gamma$ from $I$ to $J$, and all
semistable bundles $F'$ which are deformationally
equivalent to $\Phi_\gamma(F)$, the bundle $F'$ is stable.
Then the space of stable deformations of $F$ is compact.
\hfill
\remark
The space of stable deformations of $F$ is singular hyperk\"ahler
(\cite{_Verbitsky:Hyperholo_bundles_}) and its normalization
is smooth and hyperk\"ahler (\ref{_desingu_Theorem_}).
\hfill
{\bf Proof of \ref{_twi_pa_space_sta_compa_Theorem_}:}
Clearly, $F'$ satisfies assumptions
of \ref{_space_sta_bu_compa_hyperka_Theorem_},
and the moduli space of its stable deformations is compact.
Since $\Phi_\gamma$ induces a homeomorphism of moduli spaces
(\ref{_admi_twi_impli_Theorem_}),
the space of stable deformations of $F$ is also compact.
\blacksquare
\hfill
Applying \ref{_twi_pa_space_sta_compa_Theorem_}
to the holomorphic tangent bundle $T(M, I)$,
we obtain the following corollary.
\hfill
\theorem \label{_defo_tange_compact_Theorem_}
Let $M$ be a compact simple hyperk\"ahler manifold,
$\dim_{\Bbb H}(M)>1$. Assume that for a generic
hyperk\"ahler structure $\c H$ on $M$, this manifold admits
no trianalytic subvarieties.\footnote{This assumption holds for
a Hilbert scheme of points on a K3 surface.} Assume, moreover,
that for some $C$-restricted induced complex structure $I$,
all semistable bundle deformations of
$T(M, I)$ are stable, for $C> \deg_I c_2(M)$.
Then, for all complex structures $J$ on $M$ and
all generic polarizations $\omega$
on $(M, J)$, the deformation space
$\c M_{J, \omega}(T(M, J))$ is compact.
{\bf Proof:} Follows from
\ref{_twi_pa_space_sta_compa_Theorem_}
and \ref{_mod_of_tange_compa_Corollary_}.
\blacksquare
\subsection{How to check that we obtained new examples
of hyperk\"ahler manifolds?}
\label{_new_exa_F-M_checking_Subsection_}
A. Beauville \cite{_Beauville_}
described two families of compact hyperk\"ahler manifolds, one
obtained as the Hilbert scheme of points on a K3-surface,
another obtained as the Hilbert scheme of a
2-dimensional torus factorized by the free torus action.
\hfill
\conjecture\label{_anti_Beauville_Conjecture}
There exist compact simple hyperk\"ahler manifolds
which are not isomorphic to deformations of these two fundamental examples.
\hfill
Here we explain our strategy of a proof of
\ref{_anti_Beauville_Conjecture} using results on
compactness of the moduli space of hyperholomorphic bundles.
\hfill
The results of this subsection are still in writing,
so all statements below this line should be considered as conjectures.
We give an idea of a proof for each result and label it
as ``proof'', but these ``proofs'' are merely sketches.
\hfill
First of all, it is possible to prove the following theorem.
\hfill
\theorem \label{_no_rk-2-bu_on_Hilb_Theorem_}
Let $M$ be a complex K3 surface without automorphisms. Assume that
$M$ is Mumford-Tate generic with respect to some hyperk\"ahler structure.
Consider the Hilbert scheme $M^{[n]}$ of points on $M$, $n>1$.
Pick a hyperk\"ahler structure $\c H$ on $M^{[n]}$ which is compatible with
the complex structure. Let $B$ be a hyperholomorphic
bundle on $(M^{[n]}, \c H)$, $\operatorname{rk} B=2$. Then
$B$ is a trivial bundle.
{\bf Proof:} The proof of \ref{_no_rk-2-bu_on_Hilb_Theorem_}
is based on the same ideas as the proof of
\ref{_no_triana_subva_of_Hilb_Theorem_}.
\blacksquare
\hfill
For a compact complex manifold $X$ of hyperk\"ahler type, denote its
coarse, marked moduli space (\ref{_moduli_hyperka_Definition_})
by $Comp(X)$.
\hfill
\corollary\label{_no_rk-2-bu_on_def_Theorem_}
Let $M$ be a K3 surface,
$I\in Comp(X)$ an arbitrary complex structure
on $X = M^{[n]}$, $n>1$,
and $a$ a generic polarization on $(X, J)$.
Consider the hyperk\"ahler structure $\c H$ which
corresponds to ($I$, $a$)
as in \ref{_symplectic_=>_hyperkahler_Proposition_}.
Let $B$, $\operatorname{rk} B=2$ be a hyperholomorphic bundle over $(X, \c H)$.
Then $B$ is trivial.
{\bf Proof:} Follows from
\ref{_no_rk-2-bu_on_Hilb_Theorem_}
and \ref{_iso_Bun_exists_gene_pola_Theorem_}.
\blacksquare
\hfill
\corollary\label{_defo_4-dim_bu_Corollary_}
Let $M$ be a K3 surface,
$I\in Comp(X)$ an arbitrary complex structure
on $X = M^{[n]}$, $n>1$, and $a$ a generic polarization on $(X, I)$.
Consider the hyperk\"ahler structure $\c H$ which
corresponds to ($J$, $a$)
(\ref{_symplectic_=>_hyperkahler_Proposition_}).
Let $B$, $\operatorname{rk} B\leq 6$ be a stable hyperholomorphic bundle on
$(X, \c H)$. Assume that the Chern class $c_{\operatorname{rk} B}(B)$ is non-zero.
Assume, moreover, that $I$ is $C$-restricted, $C = \deg_I(c_2(B))$.
Let $B'$ be a semistable deformation of $B$ over $(X, I)$.
Then $B'$ is stable.
\hfill
{\bf Proof:} Consider the Jordan--H\"older serie for $B'$.
Let $Q_1 \oplus Q_2 \oplus ...$ be the associated graded
sheaf. By \ref{_sheaf_on_C_restr_hyperho_Theorem_},
the stable bundles $Q_i$ are hyperholomorphic.
Since $c_{\operatorname{rk} B}(B)\neq 0$, we have
$c_{\operatorname{rk} Q_i}(Q_i)\neq 0$. Therefore,
the bundles $Q_i$ are non-trivial.
By \ref{_no_rk-2-bu_on_def_Theorem_},
$\operatorname{rk} Q_i >2$. Since all the Chern classes
of the bundles $Q_i$ are $SU(2)$-invariant,
the odd Chern classes of $Q_i$ vanish
(\ref{_SU(2)_inva_type_p,p_Lemma_}).
Therefore, $\operatorname{rk} Q_i\geq 4$ for all $i$. Since
$\operatorname{rk} B\leq 6$, we have $i=1$ and
the bundle $B'$ is stable.
\blacksquare
\hfill
Let $M$ be a K3 surface, $X= M^{[i]}$, $i= 2$, $3$ be its second
or third Hilbert
scheme of points, $I\in Comp(X)$ arbitrary complex structure
on $X$, and $a$ a generic polarization on
$(X, I)$. Consider the hyperk\"ahler structure $\c H$ which
corresponds to $J$ and $a$ by
Calabi-Yau theorem (\ref{_symplectic_=>_hyperkahler_Proposition_}).
Denote by $TX$ the tangent bundle of $X$,
considered as a hyperholomorphic bundle. Let $\operatorname{Def}(TX)$ denote the
hyperk\"ahler desingularization of the moduli of stable deformations of $TX$.
By \ref{_iso_Bun_exists_gene_pola_Theorem_},
the real analytic subvariety underlying
$\operatorname{Def}(TX)$ is independent from the choice of $I$.
Therefore, its dimension is also independent from the choice of $I$.
The dimension of the
deformation space $\operatorname{Def}(TX)$ can be estimated by a direct
computation, for $X$ a Hilbert scheme. We obtain that
$\dim \operatorname{Def}(TX)> 40$.
\hfill
\claim \label{_Def_TX_compa_}
In these assumptions, the space $\operatorname{Def}(TX)$
is a compact hyperk\"ahler manifold.
{\bf Proof:} By \ref{_defo_4-dim_bu_Corollary_},
all semistable bundle deformations of $TX$ are stable.
Then \ref{_Def_TX_compa_} is implied by
\ref{_defo_tange_compact_Theorem_}.
\blacksquare
\hfill
Clearly, deforming the complex structure on $X$, we obtain a
deformation of complex structures on $\operatorname{Def}(TX)$.
This gives a map
\begin{equation} \label{_Comp_X_to_Comp_Def_Equation_}
Comp(X) {\:\longrightarrow\:} Comp(\operatorname{Def}(TX)).
\end{equation}
It is easy to check that the map
\eqref{_Comp_X_to_Comp_Def_Equation_} is complex analytic,
and maps twistor curves to twistor curves.
\hfill
\claim \label{_maps_pre_tw_curves_Claim_}
Let $X$, $Y$ be hyperk\"ahler manifolds,
and \[ \phi:\; Comp(X) {\:\longrightarrow\:} Comp(Y)\] be a holomorphic
map of corresponding moduli spaces
which maps twistor curves to twistor curves.
Then $\phi$ is locally an embedding.
{\bf Proof:} An elementary argument using the period maps, in
the spirit of Subsection \ref{_modu_and_C-restri_Subsection_}.
\blacksquare
\hfill
The following result, along with
\ref{_no_rk-2-bu_on_Hilb_Theorem_}, is the major stumbling block
on the way to proving \ref{_anti_Beauville_Conjecture}.
The other results of this Subsection are elementary or
routinely proven, but the complete proof of
\ref{_no_rk-2-bu_on_Hilb_Theorem_} and
\ref{_defo_simple_Theorem_} seems to be difficult.
\hfill
\theorem \label{_defo_simple_Theorem_}
Let $X$ be a simple hyperk\"ahler manifold
without proper trianalytic subvarieties, $B$ a hyperholomorphic
bundle over $X$, and $I$ an induced complex structure.
Denote the corresponding holomorphic bundle over $(X, I)$ by $B_I$.
Assume that the space $\c M$ of stable bundle deformations of $B$ is compact.
Let $\operatorname{Def}(B)$ be the hyperk\"ahler desingularization of $\c M$.
Then $\operatorname{Def}(M)$ is a simple hyperk\"ahler manifold.
\hfill
{\bf Proof:} Given a decomposition $\operatorname{Def}(M) = M_1\times M_2$,
we obtain a parallel 2-form on $\Omega_1$ on $\operatorname{Def}(B)$,
which is a pullback of the holomorphic symplectic form on $M_1$.
Consider the space $\c A$ of connections on $B$, which is an
infinitely-dimensional complex analytic Banach manifold.
Then $\Omega_1$ corresponds to a holomorphic
2-form $\tilde \Omega_1$ on $\c A$. Since $\Omega_1$
is parallel with respect to the natural connection
on $\operatorname{Def}(B)$, the form $\tilde \Omega_1$ is also a
parallel 2-form on the tangent space to $\c A$,
which is identified with $\Omega^1(X, \operatorname{End}(B))$.
It is possible to prove that this 2-form is obtained
as
\[ A, B {\:\longrightarrow\:} \int_{Y} \Theta\left(A\restrict Y,
B\restrict Y\right)\operatorname{Vol}(Y)
\]
where
\[ \Theta:\;\Omega^1(Y, \operatorname{End}(B)) \times \Omega^1(Y, \operatorname{End}(B)){\:\longrightarrow\:}
{\cal O}_Y
\]
is a certain holomorphic
pairing on the bundle $\Omega^1(Y, \operatorname{End}(B))$,
and $Y$ is a trianalytic subvariety of $X$.
Since $X$ has no trianalytic subvarieties,
$\tilde \Omega_1$ is obtained from a ${\cal O}_X$-linear
pairing
\[ \Omega^1(X, \operatorname{End}(B))\times \Omega^1(X, \operatorname{End}(B)) {\:\longrightarrow\:} {\cal O}_X.
\]
Using stability of $B$, it is possible to show that
such a pairing is unique, and thus, $\Omega_1$ coincides
with the holomorphic symplectic form on $\operatorname{Def}(B)$.
Therefore, $\operatorname{Def}(B) = M_1$, and this manifold is simple.
\blacksquare
\hfill
Return to the deformations of tangent bundles
on $X= M^{[i]}$, $i=2,3$. Recall that the
second Betti number of a Hilbert scheme of
points on a K3 surface is equal to $23$,
and that of the generalized Kummer variety is
7 (\cite{_Beauville_}).
Consider the map \eqref{_Comp_X_to_Comp_Def_Equation_}.
By \ref{_defo_simple_Theorem_}, the manifold
$\operatorname{Def}(TX)$ is simple. By
Bogomolov's theorem (\ref{_Bogomo_etale_Theorem_}),
we have \[ \dim Comp(\operatorname{Def}(TX)) = \dim H^2(\operatorname{Def}(TX)) -2.\]
Therefore, either $\dim H^2(\operatorname{Def}(TX))> \dim H^2(X)=23$,
or the map \eqref{_Comp_X_to_Comp_Def_Equation_} is etale.
In the first case, the second Betti number of
$\operatorname{Def}(TX)$ is bigger than that of known simple
hyperk\"ahler manifolds, and thus, $\operatorname{Def}(TX)$
is a new example of a simple hyperk\"ahler
manifold; this proves \ref{_anti_Beauville_Conjecture}.
Therefore, to prove \ref{_anti_Beauville_Conjecture},
we may assume that $\dim H^2(\operatorname{Def}(TX))=23$,
the map \eqref{_Comp_X_to_Comp_Def_Equation_} is etale,
and $\operatorname{Def}(TX)$ is a deformation of a Hilbert scheme of
points on a K3 surface.
\hfill
Consider the universal bundle $\tilde B$ over
$X\times \operatorname{Def}(TX)$. Restricting $\tilde B$ to
$\{x\} \times \operatorname{Def}(TX)$, we obtain a bundle $B$ on
$\operatorname{Def}(TX)$. Let $\operatorname{Def}(B)$ be the hyperk\"ahler desingularization
of the moduli space of stable deformations of $B$.
Clearly, the manifold $\operatorname{Def}(B)$ is independent from the
choice of $x\in X$. Taking the generic hyperk\"ahler structure
on $X$, we may assume that the hyperk\"ahler structure $\c H$ on $\operatorname{Def}(TX)$
is also generic. Thus, $(\operatorname{Def}(TX),\c H)$
admits $C$-restricted complex structures and has no trianalytic
subvarieties. In this situation,
\ref{_defo_4-dim_bu_Corollary_} implies
that the hyperk\"ahler manifold $\operatorname{Def}(B)$ is compact.
Applying \ref{_maps_pre_tw_curves_Claim_} again,
we obtain a sequence of maps
\[ Comp(X) {\:\longrightarrow\:} Comp(\operatorname{Def}(TX)){\:\longrightarrow\:} Comp(\operatorname{Def}(B)) \]
which are locally closed embeddings. By the same argument
as above, we may assume that the composition
$Comp(X) {\:\longrightarrow\:} Comp(\operatorname{Def}(B))$ is etale, and
the manifold $\operatorname{Def}(B)$ is a deformation of a Hilbert
scheme of points on K3. Using Mukai's version of Fourier
transform (\cite{_Orlov:K3_}, \cite{_BBH-R_}),
we obtain an embedding of the corresponding
derived categories of coherent sheaves,
\[ D(X) {\:\longrightarrow\:} D(\operatorname{Def}(TX)){\:\longrightarrow\:} D(\operatorname{Def}(B)).
\]
Using this approach, it is easy to prove that
\[ \dim X\leq \dim \operatorname{Def}(TX)\leq \dim \operatorname{Def}(B).
\]
Let $x\in X$ be an arbitrary point.
Consider the complex $C_x \in D(\operatorname{Def}(B))$
of coherent sheaves on $\operatorname{Def}(B)$, obtained as a composition
of the Fourier-Mukai transform maps. It is easy to check that
the lowest non-trivial cohomology sheaf of $C_x$ is
a skyscraper sheaf in a point $F(x)\in \operatorname{Def}(B)$.
This gives an embedding
\[ F:\; X{\:\longrightarrow\:} \operatorname{Def}(B).
\]
The map $F$ is complex analytic for all induced complex structure.
We obtained the following result.
\hfill
\lemma \label{_double_Fou_embedding_Lemma_}
In the above assumptions,
the embedding \[ F:\; X{\:\longrightarrow\:} \operatorname{Def}(B) \]
is compatible with the hyperk\"ahler structure.
\blacksquare
\hfill
By \ref{_double_Fou_embedding_Lemma_},
the manifold $\operatorname{Def}(B)$ has a trianalytic subvariety
$F(X)$, of dimension
$0<\dim F(X)< 40< \dim \operatorname{Def}(B)$.
On the other hand, for a hyperk\"ahler structure on $X$ generic,
the corresponding hyperk\"ahler structure on
$\operatorname{Def}(B)$ is also generic, so this manifold
has no trianalytic subvarieties. We obtained a contradiction.
Therefore, either $\operatorname{Def}(TX)$ or $\operatorname{Def}(B)$ is a new example of a
simple hyperk\"ahler manifold. This proves
\ref{_anti_Beauville_Conjecture}.
\hfill
\hfill
{\bf Acknowledegments:}
I am grateful to V. Batyrev, A. Beilinson, P. Deligne, D. Gaitsgory,
D. Kaledin, D. Kazhdan, M. Koncevich and T. Pantev for valuable
discussions. My gratitude to
D. Kaledin, who explained to me the results of \cite{_Swann_}.
This paper uses many ideas of our joint work on direct
and inverse twistor transform (\cite{_NHYM_}).
{\small
|
1999-03-08T22:37:57 | 9712 | alg-geom/9712011 | en | https://arxiv.org/abs/alg-geom/9712011 | [
"alg-geom",
"dg-ga",
"hep-th",
"math.AG",
"math.DG"
] | alg-geom/9712011 | Bong Lian | B. Lian, K. Liu, and S.T. Yau | Mirror Principle I | Typos corrected, Plain Tex 50 pages with t.o.c option | null | null | null | null | We propose and study the following Mirror Principle: certain sequences of
multiplicative equivariant characteristic classes on Kontsevich's stable map
moduli spaces can be computed in terms of certain hypergeometric type classes.
As applications, we compute the equivariant Euler classes of obstruction
bundles induced by any concavex bundles -- including any direct sum of line
bundles -- on $\P^n$. This includes proving the formula of Candelas-de la
Ossa-Green-Parkes hence completing the program of Candelas et al, Kontesevich,
Manin, and Givental, to compute rigorously the instanton prepotential function
for the quintic in $\P^4$. We derive, among many other examples, the multiple
cover formula for Gromov-Witten invariants of $\P^1$, computed earlier by
Morrison-Aspinwall and by Manin in different approaches. We also prove a
formula for enumerating Euler classes which arise in the so-called local mirror
symmetry for some noncompact Calabi-Yau manifolds. At the end we interprete an
infinite dimensional transformation group, called the mirror group, acting on
Euler data, as a certain duality group of the linear sigma model.
| [
{
"version": "v1",
"created": "Thu, 11 Dec 1997 15:15:07 GMT"
}
] | 2009-09-25T00:00:00 | [
[
"Lian",
"B.",
""
],
[
"Liu",
"K.",
""
],
[
"Yau",
"S. T.",
""
]
] | alg-geom | |
1997-12-17T13:14:09 | 9712 | alg-geom/9712019 | en | https://arxiv.org/abs/alg-geom/9712019 | [
"alg-geom",
"math.AG"
] | alg-geom/9712019 | Thomas Bauer | Thomas Bauer | On the cone of curves of an abelian variety | null | null | null | null | null | Let $X$ be a smooth projective variety over the complex numbers. One knows by
the Cone Theorem that the closed cone of curves of $X$ is rational polyhedral
whenever $c_1(X)$ is ample. For varieties $X$ such that $c_1(X)$ is not ample,
however, it is in general difficult to determine the structure of $\bar NE(X)$.
The purpose of this paper is to study the cone of curves of abelian varieties.
Specifically, the abelian varieties $X$ are determined such that the closed
cone $\bar NE(X)$ is rational polyhedral. The result can also be formulated in
terms of the nef cone of $X$ or in terms of the semi-group of effective classes
in the N\'eron-Severi group of $X$.
| [
{
"version": "v1",
"created": "Wed, 17 Dec 1997 12:14:09 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Bauer",
"Thomas",
""
]
] | alg-geom | \section{Introduction}
Let $X$ be a smooth projective variety over the
complex numbers and let $N_1(X)$ be the real vector
space
$$
N_1(X)=_{\operatorname{def}}\{\mbox{1-cycles on $X$ modulo numerical
equivalence}\}\otimes{\Bbb R} \ .
$$
As usual denote by $N\hskip-0.25em E(X)$ the {\em cone of curves} on $X$,
i.e.\ the convex cone in
$N_1(X)$ generated by the effective 1-cycles.
The {\em closed cone of curves}
${}\hskip0.25em\bar{\hskip-0.25em N\hskip-0.25em E}(X)$ is the closure of $N\hskip-0.25em E(X)$ in $N_1(X)$.
One knows by the Cone Theorem \cite{Mor82} that
it is
rational polyhedral whenever $c_1(X)$ is ample. For
varieties $X$ such that
$c_1(X)$ is not ample, however,
it is in general difficult to determine the
structure of ${}\hskip0.25em\bar{\hskip-0.25em N\hskip-0.25em E}(X)$, since it may depend in a subtle way
on the geometry of $X$ (cf.\ \cite[\S4]{CKM}).
This becomes already apparent in the
surface case, as work of Kov\'acs on K3 surfaces shows
(see \cite{Kov94}).
The purpose of this paper is to study
the cone of curves of abelian
varieties. Specifically, we focus on the problem of
determining the abelian varieties $X$
such that the closed cone ${}\hskip0.25em\bar{\hskip-0.25em N\hskip-0.25em E}(X)$ is rational polyhedral.
Attacking the question from the dual point of view, one
is lead to
consider the nef cone of $\operatorname{Nef}(X)$ or the
semi-group $\NP(X)$ of homology classes of
effective line bundles, i.e.\ the subset
$$
\NP(X)=_{\operatorname{def}}\{\lambda\inN\hskip-0.2em S(X)\mid\lambda=c_1(L)\mbox{ for some
}L\in\operatorname{Pic}(X)\mbox{ with }h^0(X,L)> 0\}
$$
of the N\'eron-Severi group of $X$.
In fact, Rosoff has studied this semi-group in
\cite{Ros81}, where he gives examples
of abelian varieties for which $\NP(X)$ is finitely generated,
as well as examples where finite generation fails.
He shows:
\begin{items}
\item[(1)]
If $X$ is a singular abelian variety, i.e.\ if
$\rkN\hskip-0.2em S(X)=(\dim X)^2$, and if $\dim X\ge 2$,
then $\NP(X)$ is not finitely generated.
\item[(2)]
For elliptic curves $E_1$ and $E_2$, $\NS^+(E_1\times
E_2)$ is finitely generated if and only if
$\rkN\hskip-0.2em S(E_1\times E_2)=2$.
\end{items}
Considering these examples it is natural to
ask if the abelian varieties $X$ for which $\NP(X)$ is
finitely generated or, equivalently, for which
${}\hskip0.25em\bar{\hskip-0.25em N\hskip-0.25em E}(X)$ is rational polyhedral,
can be characterized in a simple way.
Our main result shows that this is in fact the case:
\pagebreak
\begin{varthm*}{Theorem}
Let $X$ be an abelian variety over the field of complex
numbers. Then the following conditions are equivalent:
\begin{items}
\item[(ia)]
The closed cone of curves ${}\hskip0.25em\bar{\hskip-0.25em N\hskip-0.25em E}(X)$ is rational polyhedral.
\item[(ib)]
The nef cone $\operatorname{Nef}(X)$
is rational polyhedral.
\item[(ic)]
The semi-group $\NP(X)$ is finitely generated.
\item[(ii)]
$X$ is isogenous to a product
$$
X_1\times\dots\times X_r
$$
of mutually non-isogenous abelian varieties $X_i$
with $N\hskip-0.2em S(X_i)\cong{\Bbb Z}$ for $1\le i\le r$.
\end{items}
\end{varthm*}
Note that, since on abelian varieties the nef cone
coincides with the effective cone, the equivalence of
(ia), (ib) and (ic) follows from elementary properties of cones
and is stated here merely for the sake of completeness (see
Sect.\ \ref{sect proof}).
Observe that the theorem of course contains statement (2)
above, while statement (1) follows from the theorem plus
the fact that by \cite{ShiMit74}
a singular abelian variety is isogenous to a
product $E^n$ for some elliptic curve $E$.
\begin{varthm*}{\it Notation and Conventions}
\rm
We work throughout over the field ${\Bbb C}$ of complex numbers.
We will always
use additive notation for the tensor product of line
bundles, since this is more convenient for our purposes
(for example when working with ${\Bbb Q}$- or ${\Bbb R}$-line
bundles).
Numerical equivalence of divisors or line bundles, which
for abelian varieties coincides with algebraic equivalence,
will be denoted by $\equiv$.
If $X$ is an abelian variety and $L$ is a line bundle on
$X$, then $\phi_L$ denotes the homomorphism $X\longrightarrow\widehat X$,
$x\mapstochar\longrightarrow t_x^*L - L$, where $t_x$ is the translation map
$y\mapstochar\longrightarrow x+y$ and $\widehat X=\operatorname{Pic}^0(X)$ is the dual abelian
variety. Recall that $\phi_L$ depends only on the
algebraic equivalence class of $L$.
\end{varthm*}
\section{Effective classes on simple abelian varieties}
In this section we consider the semi-group of
effective divisor classes on
simple abelian varieties. We start by stating alternative
characterizations of $\NP(X)$ which we will use in the sequel.
While these are at least implicitly well-known, we include
a proof for the convenience of the reader.
\begin{lemma}\label{nef and effective}
Let $X$ be an abelian variety of dimension $n$ and let $A$
be an ample line bundle on $X$. Then the following
conditions on a line bundle $L\in\operatorname{Pic}(X)$ are equivalent:
\begin{items}
\item[(i)]
$L$ is algebraically equivalent to some effective
line bundle.
\item[(ii)]
$L$ is nef.
\item[(iii)]
$L^iA^{n-i}\ge 0$ for $1\le i\le n$.
\end{items}
\end{lemma}
\startproof{\it Proof. }
Condition (i) certainly implies (ii), since if $L\equiv L'$
for some effective line bundle $L'$, then a suitable
translate of an effective divisor in $|L'|$ will intersect
any given curve properly.
The implication (ii) $\Rightarrow$ (iii) is clear, since an
intersection product of nef line bundles is non-negative.
For (iii) $\Rightarrow$ (ii) it is enough to show that the
line bundle $A+mL$ is ample for all $m\ge 0$. But this
follows from
$$
(A+mL)^i A^{n-i}=A^n+\sum_{k=0}^{i-1} {i \choose k}
A^{n+k-i}L^{i-k} m^{i-k} > 0 \ ,
$$
and the version of the Nakai-Moishezon Criterion given in
\cite[Corollary 4.3.3]{LB}.
Finally, for the implication (ii) $\Rightarrow$ (i), suppose
that $L$ is nef. Then $A+mL$ is ample for all $m\ge 0$, so
that the first Chern class of $L$, viewed as a hermitian
form on $T_0X$, cannot have negative
eigenvalues. But this implies that there is a line bundle
$P\in\operatorname{Pic}^0(X)$ such that $L+P$ descends to an ample line
bundle on a quotient of $X$ and is therefore effective
(cf.\ \cite[Sect.\ 3.3]{LB} and \cite[p.\ 95]{Mum70}).
\nopagebreak\hspace*{\fill}\qedsymbol\par\addvspace{\bigskipamount}
We show next that on a simple abelian variety the existence
of two algebraically independent line bundles already
prevents $\NP(X)$ from being finitely generated:
\begin{proposition}\label{simple case}
Let $X$ be a simple abelian variety such that $\NP(X)$ is
finitely generated. Then $N\hskip-0.2em S(X)\cong{\Bbb Z}$.
\end{proposition}
\startproof{\it Proof. }
Assume to the contrary that $\rkN\hskip-0.2em S(X)>1$ and choose ample
line
bundles $L_1$ and $L_2$ whose classes are not proportional
in $\NS_{\bbQ}(X)$. Consider then the positive real number
$$
s =_{\operatorname{def}}\inf\Big\{t\in{\Bbb R}\ \Big\vert\ tL_1-L_2\mbox{ is nef }\Big\} \ .
$$
Here $tL_1-L_2$ is considered as an ${\Bbb R}$-line bundle and
nefness means that $tL_1C\ge L_2C$ for every irreducible
curve $C$ in $X$.
We assert that
\begin{equation}\label{s rational}
s \not\in{\Bbb Q} \ .
\end{equation}
Suppose to the contrary that $s$ is rational and consider
the line bundle
$$
L=sL_1-L_2 \in\operatorname{Pic}_{{\Bbb Q}}(X) \ .
$$
We choose an
integer $n$ such that $ns\in{\Bbb Z}$. The line bundle $nL$ is
then
algebraically equivalent to an effective (and integral)
line bundle. But $L$, and hence $nL$, is certainly not
ample, so that the kernel $K(nL)$ of $\phi_{nL}$ is of
positive dimension.
On the other hand, since $L_1$ and $L_2$ are not
proportional, $nL$ is not algebraically equivalent to $0$, and
hence $K(nL)$ cannot be the whole of $X$. So we find that
the neutral component of $K(nL)$ is a non-trivial abelian
subvariety of $X$, contradicting the simplicity assumption
on $X$. This establishes the assertion \eqnref{s rational}
One checks next that, since $\NP(X)$ is finitely generated,
its intersection with ${\Bbb Z}[L_1]\oplus{\Bbb Z}[L_2]$ is
finitely generated as well (cf.\ for example
\cite[Sect.\ 1.3]{Zie95}). Choose generators
$\liste N1k$ for this intersection. Let now
$0<\varepsilon\ll 1$ and fix large integers $p_1,p_2$ such
that
\begin{equation}\label{s inequality}
s < \frac{p_1}{p_2} < s+\varepsilon \ .
\end{equation}
The line bundle
$$
A=p_1L_1-p_2L_2=p_2L+p_2\(\frac{p_1}{p_2}-s\)L_1
$$
is then ample and therefore effective, so that we
have $A\equiv\sum_{i=1}^k \ell_i N_i$ with integers
$\ell_i\ge 0$. Thus, writing $N_i\equiv a_iL_1-b_iL_2$ with
$a_i,b_i\ge 0$, we get
$$
\frac{p_1}{p_2} = \frac{\sum_{i=1}^k \ell_i
a_i}{\sum_{i=1}^k \ell_i b_i} \ ,
$$
which, upon letting
$$
q =_{\operatorname{def}}\min\left\{\frac{a_i}{b_i}\ \Big\vert\ 1\le i \le
k\right\} \ ,
$$
yields the lower bound
$$
\frac{p_1}{p_2} \ge q \ .
$$
But, due to the fact that
$s$ is irrational, which implies $q>s$, and since
$\varepsilon$ can be taken arbitrarily small, this is
incompatible with \eqnref{s inequality}
\nopagebreak\hspace*{\fill}\qedsymbol\par\addvspace{\bigskipamount}
\section{Classes on products}
We study in this section effective divisor classes on the self-product
$X\times X$ of an abelian variety $X$. Suppose for a moment
that $X$ is an elliptic curve. Then, since $N\hskip-0.2em S(X\times X)$ is of
rank $\ge 3$, statement (2) of the introduction says that
$\NS^+(X\times X)$ is not finitely generated.
The argument given in \cite{Ros81} revolves around
the alternating matrices associated
with effective line bundles.
Our aim here is to prove by different methods that the
analogous statement holds in any dimension. To simplify
the proof, we only consider abelian varieties of Picard
number $1$ for now, as the general case will follow with no
effort from the proof of the theorem in Sect.\ \ref{sect proof}.
\begin{proposition}\label{product case}
Let $X$ be an abelian variety with $N\hskip-0.2em S(X)\cong{\Bbb Z}$. Then
$\NS^+(X\times X)$ is not finitely generated.
\end{proposition}
\startproof{\it Proof. }
We denote by
$\iota_1,\iota_2,\iota_3$ the closed embeddings of $X$ in
$X\times X$ given by
$$
\iota_1:x\mapstochar\longrightarrow(x,0),\
\iota_2:x\mapstochar\longrightarrow(0,x),\
\iota_3:x\mapstochar\longrightarrow(x,x) \ .
$$
Further, fix an ample line bundle $M$ whose algebraic
equivalence class
generates $N\hskip-0.2em S(X)$ and let
$n$ denote the dimension of $X$.
Supposing to the contrary that $\NS^+(X\times X)$ is finitely
generated,
our first claim is then the following
boundedness statement:
\begin{items}
\item[(*)]
There is an integer $c>0$ such that for all effective
line bundles $B$ on $X\times X$ with $\iota_1^* B\equiv M$
the inequality
$$
\(\iota_2^* B-\iota_3^* B\)^n \le c
$$
holds.
\end{items}
To prove (*), choose a finite set of generators $\liste N1k$
of $\NS^+(X\times X)$ and write
$$
B\equiv\sum_{i=1}^k b_iN_i
$$
with integers $b_i\ge 0$. Because of $N\hskip-0.2em S(X)={\Bbb Z}\cdot[M]$
we have $\iota_1^* N_i\equiv n_i M$ with integers $n_i\ge 0$ for $1\le
i\le k$. The equivalences
$$
M\equiv\iota_1^* B\equiv\sum_{i=1}^k b_i \iota_1^* N_i
\equiv\(\sum_{i=1}^k b_in_i\)M
$$
show that there is a subscript $i_0$ with the property
$$
b_i n_i =
\left\{
\begin{array}{ll}
1 & \mbox{, if $i=i_0$} \\
0 & \mbox{, if $i\ne i_0$} \ .
\end{array}
\right.
$$
If now $N$ is any effective line bundle on $X\times X$ with
$\iota_1^* N\equiv 0$, then it follows (for instance using the
Seesaw Principle) that $N$ is a multiple of $pr_2^* M$,
where $pr_2:X\times X\longrightarrow X$ is the second projection, so
that $\iota_2^* N\equiv\iota_3^* N$. In particular we therefore obtain
$$
\iota_2^* B-\iota_3^* B\equiv \sum_{i=1}^k b_i\( \iota_2^* N_i-\iota_3^* N_i\)
\equiv \iota_2^* N_{i_0} - \iota_3^* N_{i_0} \ ,
$$
so that (*) will hold, if we take the integer constant $c$
to be
$$
c =_{\operatorname{def}}\max\Big\{ \(\iota_2^* N_i-\iota_3^* N_i\)^n\ \Big\vert\
1\le i\le k\Big\}
\ .
$$
Having established (*), the idea is now to construct a
contradiction by exhibiting a sequence of nef line bundles
$B_m$, $m\ge 1$, satisfying
\begin{equation}\label{B conditions}
\iota_1^* B_m\equiv M \quad\mbox{ and }\quad \lim_{m\rightarrow\infty}\(\iota_2^*
B_m-\iota_3^* B_m \)^n =\infty \ .
\end{equation}
To this end we set
$$
L_1=pr_1^*M\ ,\
L_2=pr_2^*M\ ,\
L_3=\mu^*M\ ,
$$
where $pr_1,pr_2$ are the projections and $\mu$ is the
addition map $X\times X\longrightarrow X$. We then consider the line
bundles
$$
B_m=_{\operatorname{def}} (1-m)L_1 + (m^2-m)L_2 + mL_3 \ .
$$
One checks that with this choice of the bundles $B_m$ the
conditions \eqnref{B conditions}
are satisfied. So we will be done as soon
as have shown that $B_m$ is nef for $m\ge 1$.
Now recall that an ample line bundle $A$ on $X\times X$
defines an injective homomorphism of vector spaces
$$
\NS_{\bbQ}(X\times X)\longrightarrow\End_{\bbQ}(X\times X)\ , \
L\mapstochar\longrightarrow\phi_A^{-1}\phi_L \ ,
$$
whose image consists of the elements of $\End_{\bbQ}(X\times X)$
which are symmetric with respect to the Rosati involution
$f\mapstochar\longrightarrow f'=\phi_A^{-1}\widehat f\phi_A$. In particular,
for an endomorphism $f$ of $X\times X$, the pullback $f^*A$
corresponds to the symmetric endomorphism $f'f$. Let
now $A=L_1+L_2$. One checks that, thanks to the fact that
$A$ is a product polarization, an endomorphism
$$
f=\(\begin{array}{cc} f_1 & f_2 \\ f_3 & f_4 \end{array}\):
X\times X\longrightarrow X\times X
$$
is symmetric if and only if both $f_1$ and $f_4$ are
symmetric and $f_2'=f_3$. Therefore
the endomorphisms
$\alpha_1,\alpha_2,\alpha_3$, which are defined by
\begin{eqnarray*}
\alpha_1:(x,y)&\mapstochar\longrightarrow&(x,0) \\
\alpha_2:(x,y)&\mapstochar\longrightarrow&(0,x) \\
\alpha_3:(x,y)&\mapstochar\longrightarrow&(x+y,x+y)
\end{eqnarray*}
are symmetric and, upon using $\alpha_1^2=\alpha_1$,
$\alpha_2^2=\alpha_2$ and $\alpha_3^2=2\alpha_3$,
one finds that they correspond to the line bundles
$L_1,L_2,L_3$. This in turn shows that
the line bundle $B_m$ corresponds to the endomorphism
$$
\beta_m=(1-m)\alpha_1 + (m^2-m)\alpha_2 + m\alpha_3 \ .
$$
The point is now that $\beta_m^2=(m^2+1)\beta_m$,
so that $\beta_m|\operatorname{im}\beta_m$ is just multiplication by
$m^2+1$. Therefore, if we denote by $Y_m$ the complementary
abelian subvariety of $\operatorname{im}\beta_m$, then the differential
of $\beta_m$ at the point $0$ is the map
$$
d_0\beta_m:
T_0\operatorname{im}\beta_m\oplus T_0 Y_m
\longrightarrow
T_0\operatorname{im}\beta_m\oplus T_0 Y_m, \
(u,v)\mapstochar\longrightarrow((m^2+1)u,0) \ ,
$$
so that the analytic characteristic polynomial of $\beta_m$
is
$$
P_m(t)=t^n\(t-(m^2+1)\)^n \ ,
$$
But the alternating coefficients of $P_m(t)$ are positive
multiples of the intersection numbers $A^iB_m^{2n-i}$, so
that $B_m$ is nef, as claimed. This completes the proof of
the proposition.
\nopagebreak\hspace*{\fill}\qedsymbol\par\addvspace{\bigskipamount}
\section{The cone of curves and the nef cone
of an abelian variety}\label{sect proof}
Finally, we give in this section the proof of
the theorem stated in the introduction.
So let $X$ be an abelian variety and
denote by $N_1(X)$ the
vector space of numerical equivalence classes of
real-valued 1-cycles on $X$, and by $N\hskip-0.25em E(X)$ the convex
cone in $N_1(X)$ generated by irreducible curves. Through
the intersection product the vector space $N_1(X)$ is dual
to the N\'eron-Severi vector space
$\NS_{\bbR}(X)=N\hskip-0.2em S(X)\otimes{\Bbb R}$. The dual cone of $N\hskip-0.25em E(X)$ is
the nef cone
$$
\operatorname{Nef}(X)
=\{\lambda\in\NS_{\bbR}(X)\mid\lambda\xi\ge 0\mbox{
for all }\xi\inN\hskip-0.25em E(X)\} \ ,
$$
which in the case of abelian varieties coincides with the
effective cone (cf.\ Lemma \ref{nef and effective}).
The dual of $\operatorname{Nef}(X)$ in turn is the closed cone ${}\hskip0.25em\bar{\hskip-0.25em N\hskip-0.25em E}(X)$,
so that
one of these two cones is rational polyhedral if and only if the
other is. By Gordon's Lemma this is
equivalent to the semi-group $\NP(X)$ being finitely
generated. (See e.g.\ \cite[Theorem 14.1 and \S\S19,20]{Roc70}
for the elementary properties of cones used here).
The idea is now, given an abelian variety, to first apply
Poincar\'e`s Complete Reducibility
Theorem, i.e.\ to decompose it up to
isogenies into a product of powers of non-isogenous simple abelian
varieties, and to apply Proposition \ref{simple case} and
Proposition \ref{product case} subsequently.
One needs then that
finite generation of $\NP(X)$ is a property which is invariant
under isogenies:
\begin{lemma}[\cite{Ros81}]\label{isog}
Let $X$ and $Y$ be isogenous abelian varieties. Then
$\NS^+(X)$ is finitely generated if and only if $\NS^+(Y)$ is.
\end{lemma}
Since this observation is crucial for our approach,
let us briefly indicate a proof, before
we proceed to the proof of the theorem.
So suppose that $\NP(X)$ is finitely generated and that there
is an isogeny $f:X\longrightarrow Y$. Thanks to the fact that $f^*$
embeds $\NS^+(Y)$ into $\NP(X)$ and to the symmetry of the
situation, it is enough to show that $f^*\NS^+(Y)$ is finitely
generated. Let then $\liste N1k$ be generators for $\NP(X)$
and put for $1\le i\le k$
$$
n_i =_{\operatorname{def}}\min\{ n\in{\Bbb Z}\mid nN_i\in f^*\NS^+(Y) \} \ .
$$
(The set on the right-hand side is non-empty, since $f$ is
an isogeny.) Then $f^*\NS^+(Y)$ is generated by the elements
$n_1N_1,\dots,n_rN_r$ together with those elements
$\sum_{i=1}^k m_iN_i$, $0\le m_i<n_i$, which belong to
$f^*\NS^+(Y)$.
\nopagebreak\hspace*{\fill}\qedsymbol\par\addvspace{\bigskipamount}
Consider now for an abelian variety $X$ its decomposition
$$
X_1^{n_1}\times\dots\times X_r^{n_r}
$$
up to isogenies, where $\liste X1r$
are mutually non-isogenous
simple abelian varieties and $\liste n1r$ are positive
integers. In view of the remarks made at the beginning of
this section, the theorem stated in the introduction will
follow from
\begin{theorem}
The semi-group $\NP(X)$ is finitely generated if and only if
$N\hskip-0.2em S(X_i)\cong{\Bbb Z}$ and $n_i=1$ for $1\le i\le r$.
\end{theorem}
\startproof{\it Proof. }
Suppose first that the conditions on the factors $X_i$ and the
exponents $n_i$ are satisfied for $X$.
By Lemma \ref{isog} we may assume that $X$
is the product $X_1\times\dots\times X_r$.
Fix for $1\le i\le r$ an ample
generator $N_i$ of $\operatorname{Pic}(X_i)$ and let $A$ be the
product polarization $A=\sum_{i=1}^r pr_i^*N_i$. Due to
the fact that the $X_i$ are non-isogenous, we have
$$
\NS_{\bbQ}\(\prod_{i=1}^r X_i\)
\cong\End_{\bbQ}^s\(\prod_{i=1}^r X_i\)
\cong\bigoplus_{i=1}^r\End_{\bbQ}^s(X_i)
\cong\bigoplus_{i=1}^r\NS_{\bbQ}(X_i) \ ,
$$
where $\End_{\bbQ}^s\(\prod_{i=1}^r X_i\)$ and $\End_{\bbQ}^s(X_i)$
denote the subgroups of symmetric endomorphisms with
respect to the Rosati involutions associated with $A$ and
$N_i$ respectively. Therefore
$$
\NP(X)=\bigoplus_{i=1}^r {\Bbb Z}^+\cdot[N_i]
$$
is finitely generated.
Now suppose conversely
that $\NP(X)$ is finitely generated.
By Lemma \ref{isog} again, we may assume that
$X$ is the product $X_1^{n_1}\times\dots\times X_r^{n_r}$.
Note that if $V_1$ and $V_2$ are
varieties such that $\NS^+(V_1\times V_2)$ is finitely generated,
then $\NS^+(V_1)$ and $\NS^+(V_2)$ are finitely generated as
well. So in particular Proposition
\ref{simple case} applies to the factors $X_i$ and shows
that we have $N\hskip-0.2em S(X_i)\cong{\Bbb Z}$ for all $i$.
Further, if we had $n_i>1$ for some $i$, i.e.\ if
a multiple factor $X_i$ appeared in the product decomposition of
$X$, then $\NS^+(X_i\times X_i)$
would be finitely generated, which however is impossible
according to Proposition \ref{product case}.
This completes the proof
of the theorem.
\nopagebreak\hspace*{\fill}\qedsymbol\par\addvspace{\bigskipamount}
\begin{varthm*}{\it Acknowledgements}
\rm
This research was supported by DFG contract Ba 1559/2-1.
I would like to thank R.\ Lazarsfeld for helpful
discussions and the University of California, Los Angeles,
for its hospitality.
\end{varthm*}
|
1998-01-06T12:30:51 | 9712 | alg-geom/9712020 | en | https://arxiv.org/abs/alg-geom/9712020 | [
"alg-geom",
"math.AG"
] | alg-geom/9712020 | Carlos Simpson | Carlos Simpson | Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology | 75 pages. Correction-existence and functoriality of decomposition of
an infinite loop stack into product of Eilenberg-MacLane stacks don't hold in
general. However, what we need for the calculation is still true | null | null | null | null | If $X$ is a smooth projective variety moving in a family, we define a
secondary Kodaira-Spencer class for nonabelian Dolbeault cohomology
$Hom(X_{Dol}, T)$ of $X$ with coefficients in the complexified 2-sphere
$T=S^2\otimes \cc$ (which is a 3-stack on $Sch /\cc$). Let $Z$ be a simply
connected projective surface with $h^{2,0}\neq 0$, and let $X$ be the blow-up
of $Z$ at a point $P$. As $P$ moves in $Z$, the blow-up $X$ moves in a family
and we show that the secondary Kodaira-Spencer class is nontrivial. This
contrasts with the fact that the variations of mixed Hodge structures on the
homotopy groups of $X$ are constant. We discuss various surrounding notions,
including two appendices where we give some details about the Breen
calculations in characteristic zero and representability of simply connected
complex shapes.
| [
{
"version": "v1",
"created": "Thu, 18 Dec 1997 21:25:20 GMT"
},
{
"version": "v2",
"created": "Tue, 6 Jan 1998 11:30:50 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Simpson",
"Carlos",
""
]
] | alg-geom | \section*{Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology}
Carlos Simpson\newline
CNRS, Laboratoire Emile Picard\newline
Universit\'e Toulouse III\newline
31062 Toulouse CEDEX, France\newline
[email protected]
\bigskip
One of the nicest things about variations of Hodge structure is the
``infinitesimal variation of Hodge structure'' point of view \cite{IVHS}.
A variation of Hodge structure $(V=\bigoplus V^{p,q}, \nabla )$
over a base $S$
gives rise at any point $s\in S$ to the {\em Kodaira-Spencer map}
$$
\kappa _s: T(S)_s \rightarrow Hom (V^{p,q}_s, V^{p-1, q+1}_s).
$$
In the geometric situation of a family $X\rightarrow S$ we have
$$
V^{p,q}_s = H^q(X_s, \Omega ^p_{X_s}),
$$
and the Kodaira-Spencer map is given by cup-product with the Kodaira-Spencer
deformation class
$$
T(S)_s \rightarrow H^1(X_s, T(X_s)).
$$
The Kodaira-Spencer map is a component of the connection $\nabla$. In
particular, this implies that if $\kappa _s\neq 0$ then the connection
$\nabla$ is nontrivial with respect to the Hodge decomposition. Various
Hodge-theory facts imply that the global monodromy must be nontrivial in this
case. We can be a bit more precise: if $u\in V^{p,q}$ is a vector such that
$\kappa _s(v)(u)\neq 0$ for some tangent vector $v\in T(S)_s$, then $u$ cannot
be preserved by the global monodromy. Thus a local calculation (which actually
only depends on the first-order deformation of $X_s$) implies a global fact. In
particular this global fact would hold for any family of varieties $X'$ over
any base $S'$, such that the new family osculates to order $1$ with the
original one (say as a map from $S'$ into the moduli stack of the fibers).
A particularly nice aspect of this situation is that the Kodaira-Spencer map is
defined on the Dolbeault cohomology $H^q(X_s, \Omega ^p)$ and in particular it
is obtained involving only algebraic-geometric calculations (just a cup-product
with the deformation class)---no analytic considerations are needed.
The goal of this paper is to calculate an example showing a similar type of
behavior with a secondary Kodaira-Spencer class coming from nonabelian
cohomology with coefficients in the complexified $2$-sphere $T=S^2 \otimes {\bf C}$.
For such a $T$ (or any other coefficient stack similar in nature) we define
the {\em nonabelian Dolbeault cohomology of $X$ with coefficients in $T$},
denoted $Hom (X_{Dol}, T)$. When $X$ varies in a family parametrized by a base
$S$ then we show how to define a secondary class which is a map
$$
\bigwedge ^2T(S)_s\rightarrow
\pi _1(Hom (X_{Dol}, T)) = H^2_{Dol}(X)/(\eta )
$$
where $\eta$ is the class in $H^2_{Dol}(X)$ pulled back from the tautological
class on $T$ by the map $X_{Dol}\rightarrow T$ which we take as basepoint.
The secondary class is defined when the primary Kodaira-Spencer classes vanish.
It seems likely, although we don't show that here, that
our secondary class is just a quadruple Massey
product
$$
\alpha \wedge \beta \mapsto \{ \eta , \eta , \alpha , \beta \}
$$
for $\alpha , \beta \in H^1(X, TX)$ and $\eta \in H^2_{Dol}(X)$ with $\eta \cup
\eta = 0$. Instead of interpreting the secondary class this way, we calculate
it directly using Mayer-Vietoris arguments for nonabelian cohomology and
reducing to calculations in abelian cohomology.
Our
main purpose in the present paper is to calculate the secondary class in a
specific, somewhat instructive, example. We look at the family of varieties $X$
obtained by blowing up a point $P\in Z$ on a smooth surface $Z$. This family is
parametrized in an obvious way by $Z$. If $Z$ is simply connected, then any
standard Hodge-theoretic information related to $X$ must be independent of the
basepoint because it would vary in a variation of (mixed) Hodge structure
parametrized by $Z$, and such a variation is forcibly constant. However, we will
show that the secondary Kodaira-Spencer class for nonabelian Dolbeault
cohomology is nonzero, if $Z$ has a nonzero holomorphic $2$-form.
Before getting to the example, we will discuss some general aspects of
nonabelian cohomology with emphasis on the case of Dolbeault cohomology. We
start with a quick review of $n$-stacks in \S 1. Then we define the notion of
``connected very presentable shape''in \S 2 by looking at maps to $n$-stacks $T$
which are $0$-connected, with $\pi _1(T)$ an affine algebraic group scheme, and
$\pi _i(T)$ vector spaces, for $i\geq 2$. The next step in \S 3 is to set
$X_{Dol}$ equal to the $1$-stack over $X$ whose fiber is $K(\widehat{TX}, 1)$
where $\widehat{TX}$ is the completion of the tangent bundle of $X$ along the
zero-section. We define the {\em nonabelian Dolbeault cohomology of $X$ with
coefficients in an $n$-stack $T$} to be $Hom (X_{Dol},T)$. We look at this
particularly for connected very presentable $T$. This gives rise to the {\em
Dolbeault shape} which is the $(n+1)$-functor
$$
T\mapsto Hom (X_{Dol}, T)
$$
on connected very presentable $T$. Sections 4 and 5 treat group actions and
secondary classes in the $n$-stack situation. In \S 5 we define the secondary
Kodaira-Spencer class in general. Then in \S 6 we look at a particular case of
$T$, namely the {\em complexified $2$-sphere}, defined by the conditions that
$\pi _2=\pi _3 = {\cal O} $ and having nontrivial Whitehead product.
With these preliminary steps done, we get to our example in \S\S 7-8. The main
result is Theorem \ref{calculation}. It says that if $X$ is the blowup of a
surface $Z$ at a point $P$ then the secondary Kodaira-Spencer class for
nonabelian Dolbeault cohomology $Hom (X_{Dol}, T)$ with coefficients in
the complexified $2$-sphere $T$, is Serre dual to the evaluation map of
holomorphic $2$-forms at $P$. In particular if $H^0(Z,\Omega ^2_Z)\neq 0$ then
this class is nonzero.
At the end in two appendices, we will discuss some topics from \cite{kobe} but
in greater detail. First, the ``Breen calculations'' giving the cohomology of
$K({\cal O} , n)$ or more generally $K(V/S,n)$ for a vector sheaf $V$ over a base
scheme $S$. Then we discuss representability of certain shapes.
These topics come up in a few
places in the body of the paper, which is the reason for the appendices; for
an introduction we refer the
reader to the appendices.
Without going into great bibliographic detail, we point out here some recent
papers which seem to be somewhat related. Karpishpan \cite{Karpishpan1}
\cite{Karpishpan2} treats higher-order Kodaira-Spencer mappings, i.e. the
higher order derivatives of the period map. The same is treated by Ran
\cite{Ran}
and Esnault-Viehweg \cite{EsnaultViehweg}. Biswas
defines secondary invariants for families of Higgs bundles \cite{Biswas}. Also
Bloch and Esnault treat algebraic Cherns-Simons classes \cite{BlochEsnault},
which are types of secondary classes.
I would like to thank Mark Green for an inspiring question---albeit one which
the present paper doesn't answer. He asked whether there are examples of
families of varieties where the variation of Hodge structure on the
cohomology is constant, but where the variation of mixed Hodge structure on the
homotopy groups is nontrivial. In the absence of an answer to that question
(which is very interesting), the typical mathematician's reply is to change the
question---in this case, to look for an example where even the mixed Hodge
structures on the homotopy groups remain constant, but where the Hodge
filtration on the full homotopy type is nonconstant.
\footnote{
After the first version of this paper, Richard Hain pointed out to me the
following example answering Mark Green's original question. The example comes
from a paper of Carlson, Clemens, and Morgan (\cite{CarlsonEtAl}, p. 330). Let
$C\subset {\bf P} ^3$ be an embedded curve of positive genus. For points $p,q\in C$,
let $X_{p,q}$ be the $3$-fold obtained by first blowing up $p$ and $q$ and then
blowing up the strict transform of $C$. The family of $X_{p,q}$ parametrized by
(an open subset of) $C\times C$ has constant variation of Hodge structures on
the cohomology but, according to \cite{CarlsonEtAl} the variation of MHS on the
homotopy is nonconstant.}
\subnumero{Notation}
We always work in characteristic $0$. In order to simplify notation we use
${\bf C}$ as the ground field (i.e. $Spec ({\bf C} )$ as base scheme), but everything we
say would work equally well over any ground field of characteristic $0$. Let
$Sch /{\bf C}$ denote the site of schemes of finite type over $Spec ({\bf C} )$ with
the etale topology.
The structure sheaf ${\cal O}$ on $Sch /{\bf C}$ is the sheaf defined by
$$
{\cal O} (Y):= \Gamma (Y, {\cal O} _Y).
$$
It is represented by the affine line ${\bf A}^1$, in other words it is
represented by the $1$-dimensional vector space ${\bf C}$. A finite dimensional
vector space represents a sheaf of the form ${\cal O} ^a$.
\numero{Basic remarks about $n$-stacks}
We make some brief remarks about $n$-stacks as we shall use them in this paper.
For all details the reader is referred to the following references:
\newline
---for the history and basic notions of simplicial presheaves: Brown
\cite{Brown}, Illusie \cite{Illusie}, Jardine \cite{Jardine};
\newline
---for the history and basic notions of $1$-stacks: Artin
\cite{ArtinInventiones}, Deligne-Mumford \cite{Deligne-Mumford}, and
Laumon-Moret-Bailly \cite{LMB};
\newline
---for cohomological theory using simplicial presheaves: Thomason
\cite{Thomason};
\newline
---for a ``homotopy coherent'' approach: Cordier-Porter \cite{CordierPorter}
and also \cite{flexible};
\newline
---for $n$-categories and $n$-stacks: Grothendieck \cite{PursuingStacks},
Breen \cite{BreenAsterisque}, Gordon-Power-Street
\cite{GordonPowerStreet}, Tamsamani \cite{Tamsamani}, Baez-Dolan
\cite{Baez-Dolan}, and several papers of the author.
The first main remark is that we shall almost always be concerned with
$n$-stacks of $n$-groupoids, and following the intuition put forth in
\cite{PursuingStacks}, an $n$-groupoid is the same thing (up to homotopy) as a
topological space whose homotopy groups vanish in degrees $i>n$---we call such
a space {\em $n$-truncated}. Thus it is safe to replace $n$-groupoids
everywhere by $n$-truncated topological spaces or, again equivalently,
$n$-truncated simplicial sets. In this point of view, an $n$-stack (on the
site $Sch /{\bf C} $ which is fixed throughout) is just a presheaf of
simplicial sets
otherwise known as a {\em simplicial presheaf}, which is object-by-object
$n$-truncated. Once one has made the passage to simplicial presheaves, the
truncation condition is no longer crucial (although it often facilitates
arguments and many things in the literature are only stated in this case or
under a complementary hypothesis about vanishing cohomological dimension).
Thus, when we speak of ``$n$-stacks'', one way to read this is in terms of the
homotopy theory of simplicial presheaves, see \cite{Brown} \cite{Illusie}
\cite{Jardine} \cite{Thomason}.
Another reading would plunge directly into
the theory of $n$-categories and $n$-stacks, imposing the groupoid condition
along the way. For $n\leq 3$ (which in the end is the case we treat in the
present paper) this can be had in a relatively formulaic way in
\cite{BreenAsterisque} and \cite{GordonPowerStreet}. For arbitrary $n$, see
\cite{Tamsamani} \cite{nCAT}, but unfortunately the $n$-stack part of this
theory still needs to be worked out a bit more.
The only place where we make reference to $n$-categories which are not
$n$-groupoids is when we look at the $n+1$-category $nSTACK$ of $n$-stacks (of
groupoids). This $n+1$-category has the property of being {\em $1$-groupic},
i.e. the morphism $n$-categories are $n$-groupoids. One can safely replace the
morphism $n$-groupoids by spaces or simplicial sets, and one obtains the
notion of {\em Segal category} \cite{effective}, motivated by Segal's delooping
machine \cite{Segal}. This notion came into higher category theory in
Tamsamani's definition of $n$-category \cite{Tamsamani}. Thus $nSTACK$ may be
considered as a Segal category. This fits in relatively nicely with the
simplicial presheaf point of view; in fact this Segal category comes from the
simplicial category of fibrant and cofibrant objects in Jardine's closed model
category of simplicial presheaves. This comes up in
looking at the functoriality in $T$ of the construction $Hom (X_{Dol}, T)$.
In one other place we refer to the $n+1$-stack $n\underline{STACK}$ of
$n$-stacks, which is discussed somewhat in
\cite{nCAT} and \cite{limits}; we don't get any further into the general theory
here.
Currently, the
``simplicial presheaves'' alternative is the most accessible (it is also
historically the first, dating from \cite{Brown}). In terms of simplicial
presheaves, an {\em $n$-stack} is a simplicial presheaf on $Sch/{\bf C} $ which is
object-by-object $n$-truncated. If $X,Y$ are simplicial presheaves, then we
obtain a simplicial presheaf $Hom(X,Y)$ by first replacing $Y$ by a fibrant
object \cite{Jardine}, then looking at the internal $Hom$ of simplicial
presheaves. In particular this gives a simplicial set $Hom(X,Y)(Spec ({\bf C} ))$
and this family of simplicial sets makes the simplicial presheaves into a
simplicial category (or Segal category) which we denote $nSTACK$. When we speak
of morphisms between $n$-stacks, the above procedure is always understood, i.e.
we always replace the target by a fibrant object.
Given a presheaf of $n$-groupoids or presheaf of spaces, we often want to take
the ``associated $n$-stack''. What this means depends somewhat on the point of
view which is taken. If one works with objects in a closed model category such
as that of simplicial presheaves, then this just means to consider the object
as an element of the closed model category. One might also want to say that it
means to replace the object by a weakly equivalent fibrant object. Finally
there is an intermediate notion based on enforcing the global descent condition
but not the local fibrant condition. It doesn't really matter which point of
view we adopt, since when looking at morphisms to a given object, we always
replace it by an equivalent fibrant object anyway.
The ``yoga'' of the situation is that one can do topology with $n$-stacks
instead of spaces (or more precisely $n$-truncated spaces). In particular, all
standard constructions and results in algebraic topology carry over to
$n$-stacks. Most of these are contained somewhere in the literature referred to
above; but if not, we don't give proofs here as that would get beyond the scope
of the present paper.
There is basically only one slight ``twist'' which is not present in the
topological case: this is that the $0$-truncated objects can be topologically
nontrivial, i.e. can have cohomology. In the usual topological case, the
$0$-truncated objects are just the disjoint unions of contractible components
and these make no significant contribution to homotopy. In the case of
$n$-stacks over a site, one can have $0$-stacks, i.e. sheaves of sets, with
nontrivial cohomology (this is the case of a smooth projective variety $X$, for
example); and similarly there are sheaves of groups over $\ast$ (the site
itself) which can have nontrivial cohomology. The upshot of all this is that
when it comes to choosing basepoints for an $n$-stack $T$, one must choose first
an object $Y\in Sch /{\bf C}$ and then choose a basepoint $t\in T(Y)$. (In other
words, if we look only at basepoints in $T(Spec {\bf C} )$ we might be missing
some topology.)
The first place where the previous paragraph has an impact is in the notion of
{\em homotopy groups}. If $T$ is an $n$-stack then for any
$Y\in Sch /{\bf C}$ and $t\in T(Y)$ we obtain a presheaf, denoted in utmost
precision by
$$
\pi _i^{\rm pre}(T|_{Sch /Y}, t)
$$
but which we often shorten to $\pi _i^{\rm pre}(T,t)$. This is a presheaf of
groups over the site $Sch /Y$, abelian if $i\geq 2$. On the other hand the
presheaf $\pi _0^{\rm pre}(T)$ is defined absolutely as a presheaf of sets over
$Sch /{\bf C} $.
The definition which is fundamental to the theory is that we define
$$
\pi _i(T|_{Sch /Y}, t)
$$
to be the sheaf associated to the presheaf
$\pi _i^{\rm pre}(T|_{Sch /Y}, t)$. Similarly $\pi _0(T)$ is the sheaf of sets
associated to the presheaf $\pi _0^{\rm pre}(T)$. These, and not the presheaf
versions, are the only thing we care about. This is formalized by saying that
a morphism $f:T\rightarrow T' $ is called a {\em weak equivalence}
\cite{Illusie} if for all $Y\in Sch /{\bf C}$ and $t\in T(Y)$, the resulting
morphisms $$
\pi _i(T|_{Sch /Y}, t)\rightarrow \pi _i(T'|_{Sch /Y}, f(t))
$$
are isomorphisms of sheaves on $Sch /Y$ (resp. $\pi _0(T)\rightarrow \pi
_0(T')$ is an isomorphism of sheaves of sets on $Sch /{\bf C} $). The theory is
localized by this notion of equivalence, in other words $T$ and $T'$ are
thought of as equivalent if there is a weak equivalence between them. Jardine
constructs a closed model category which takes this into account \cite{Jardine}.
This leads, in particular, to the right notion of morphism, namely we only look
at morphisms whose target is a fibrant object; if necessary, a target object is
replaced by a weakly equivalent fibrant object. Without further mentionning
this, we make the convention that whenever we speak of morphisms between
$n$-stacks, the target object is made fibrant.
For a general site, one can have a connected stack $T$ (i.e. $\pi _0(T)=\ast$)
but where the global section space of $T$ is empty, or nonconnected. However,
in the present case we are working in the etale topology over an algebraically
closed field ${\bf C}$. In this case we have the implication
$$
\pi _0(T)= \ast \; \Rightarrow \; \pi _0(T(Spec ({\bf C} ))= \ast .
$$
Indeed, the etale coverings of $Spec ({\bf C} )$ are trivial, so there is no change
over the object $Spec ({\bf C} )$ when one passes from the presheaf $\pi _0^{\rm
pre}(T)$ to the associated sheaf.
If $T$ is connected, then, we can choose a basepoint $t\in T(Spec ({\bf C} ))$ which
is unique up to homotopy, so the sheaf of groups $\pi _1(T,t)$ is uniquely
defined up to global conjugacy. If $Y$ is any scheme and $t'\in T(Y)$ then
locally on $Y$, $t'$ is equivalent to $t|_Y$ so $\pi _1(T, t')$ is locally over
$Y$ equivalent to the restriction of $\pi _1(T,t)$ to $Y$. Thus in this case,
the fundamental group sheaf $\pi _1(T,t)$ over $Sch /{\bf C} $ gives a relatively
accurate picture of the $1$-type of $T$.
In particular, it makes sense to require that $T$ be {\em $1$-connected}, that
is that $\pi _0(T)=\ast $ and $\pi _1(T,t)=\{ 1\} $ for the
basepoint $t\in T(Spec ({\bf C} ))$. If $T$ is $1$-connected then for any scheme
$Y$ and $t'\in T(Y)$, $\pi _1(T,t')=\{ 1\}$. Furthermore, in this case
the fundamental group of $T(Spec ({\bf C} ))$ is trivial (the cohomological
contributions from the higher homotopy vanish because etale cohomology of $Spec
({\bf C} )$ is trivial). Therefore the basepoint $t\in T(Spec ({\bf C} ))$ is
well-defined up to unique homotopy.
We now describe the standard topological constructions which we shall use for
$n$-stacks. The first is the notion of {\em homotopy fiber product}. If
$A\rightarrow B \leftarrow C$ are morphisms of $n$-stacks then we obtain
the {\em homotopy fiber product}
$A\times _BC$
with a
diagram
$$
\begin{array}{ccc}
A\times _BC&\rightarrow & A\\
\downarrow && \downarrow \\
C & \rightarrow & B
\end{array}
$$
together with a homotopy of commutativity of the diagram. These data are
essentially well-defined (in the sense that they are well-defined up to
homotopy which is itself well-defined up to homotopy \ldots ). In the
simplicial presheaf theory, the homotopy fiber product is obtained by replacing
one of the two morphisms by a fibrant morphism and then taking the usual fiber
product. In the $n$-category theory, see \cite{limits}.
Suppose $f:A\rightarrow B$ is a morphism and suppose $b\in B(Spec ({\bf C} ))$. We
can think of $b$ as a morphism $b: \ast \rightarrow B$ where $\ast$ denotes the
constant presheaf with values the $1$-point topological space. Define the {\em
fiber of $f$ over $b$} to be the homotopy fiber product
$$
Fib(f,b):= \ast \times _B A.
$$
If the base $B$ is $0$-connected, then as mentionned above, the choice of
basepoint $b$ exists and is unique up to a global homotopy (i.e. a path in
$B(Spec ({\bf C} ))$.
Thus we can denote by $Fib(f)$ the fiber over this $b$, bearing in mind that it
is defined up to the conjugation action of $\pi _1(B (Spec ({\bf C} )))$.
If $B$ is $1$-connected the choice of basepoint $b$ is unique up to unique
homotopy, so $Fib(f)$ is well-defined up to homotopy. We call this the {\em
homotopy fiber of $f$}.
We say that
$$
A\rightarrow B \rightarrow C
$$
is a {\em fiber sequence} if $C$ is $1$-connected or if we are otherwise given
a basepoint $c\in C(Spec ({\bf C} ))$, if we are given a homotopy between the
composition $A\rightarrow C$ and the constant map at the basepoint $c$, and if
the map $A\rightarrow B$ together with this homotopy induce an equivalence
between $A$ and $Fib(B\rightarrow C, c)$.
A morphism $T\rightarrow R$ is said to be a {\em locally constant fibration
with fiber $F$} if for every scheme $Y\rightarrow R$, locally on $Y$ (in the
etale topology) we have $Y\times _RT \cong Y \times F$. In the usual
topological case with connected base, this is vacuous. In the case of
$n$-stacks over a site, we still have that if $R$ is $0$-connected then any
morphism $T\rightarrow R$ is a locally constant fibration. However, we are
often interested in cases where $R$ is not connected (i.e. $\pi _0(R)$ is
some nontrivial sheaf of sets). In these cases, being locally constant
condition is a nontrivial additional condition.
The next general type of operation we discuss is {\em truncation}. If $T$ is an
$n$-stack and if $m\leq n$ then we obtain an $m$-stack $\tau_{\leq m}T$
(i.e. a simplicial presheaf which is $m$-truncated, in other words has
homotopy group sheaves vanishing in degrees $>m$) together with a morphism of
$n$-stacks $$
T\rightarrow \tau_{\leq m}T
$$
which induces an isomorphism on homotopy group sheaves in degrees $i\leq m$.
The $m$-stack $\tau_{\leq m}T$ together with this morphism are essentially
well-defined. We can construct $\tau _{\leq m}T$ as the $m$-stack associated to
the presheaf of spaces
$$
(\tau _{\leq m}^{\rm pre}T)(Y):=
\tau _{\leq m}(T(Y))
$$
where the truncation on the left is just truncation of topological spaces
(also known as the coskeleton operation).
A first example of truncation is the sheaf of sets $\pi _0(T)= \tau _{\leq 0}T$.
If $T\rightarrow R$ is a morphism of $n$-stacks then there is a relative
(or ``fiberwise'')
version of the truncation denoted $\tau _{\leq m/R}(T)\rightarrow R$. This is
defined by the property that for any scheme $Y$ and morphism $Y\rightarrow R$,
$$
\tau _{\leq m/R}(T)\times _R Y = \tau _{\leq m}(T\times _RY).
$$
Using the operations of truncation and homotopy fiber products, we obtain the
{\em Postnikov tower}. If $T$ is an $n$-stack then we have morphisms
$$
T\rightarrow \ldots \rightarrow \tau _{\leq m}T
$$
$$
\rightarrow \tau _{\leq m-1}T \rightarrow \ldots \rightarrow \pi _0(T).
$$
In order to describe the stages in this tower of maps, we need a few more
notions.
Suppose $Y$ is a scheme and $L$ is a sheaf of groups over $Y$. Fix $m\leq n$ and
suppose $L$ is abelian if $m\geq 1$. Then we can construct the simplicial
presheaf $K^{\rm pre}(L,m)$ on $Sch/Y$ by the standard construction applied
to $L$; let $K(L,m)$ be the associated stack. Note that
$K(L,m)$ has a chosen basepoint section (over $Y$) which we denote by $0$, and
$\pi _i(K(L,m),0)=0$ for $i\neq m$, and it is $=L$ for $i=m$. Furthermore these
properties characterize $K(L,m)$ essentially uniquely.
The $K(L,m)$ on the site $Sch /Y$ corresponds to an $n$-stack
on $Sch /{\bf C}$ with morphism to $Y$ (where $Y$ is considered as a $0$-stack or
sheaf of sets), which we denote by $K(L/Y,m)\rightarrow Y$.
We can do the same construction relative to any $n$-stack but for this we need
to have a notion corresponding to sheaf of (abelian) groups.
If $A$ is an $n$-stack then a {\em local system of (abelian) groups)} on
$A$ is a morphism $L\rightarrow A$ with relative group structure, which is
relatively $0$-truncated (i.e. for any scheme $Y$ and map $Y\rightarrow A$, the
homotopy fiber product $Y\times _AL$ is $0$-truncated). This is equivalent to
the data for every $Y$ of a local system of (abelian) groups $L_Y$ over $A(Y)$,
together with restriction morphisms $L_Y|_{A_{Y'}}\rightarrow L_{Y'}$
for $Y'\rightarrow Y$,
satisfying the obvious associativity condition.
{\em Caution:} if $X$ is a sheaf of sets represented by a scheme, then
a local system over $X$ (according to the above terminology) is the same
thing as
a sheaf of (abelian) groups over $X$. It doesn't have anything to do with the
notion of ``flat vector bundle'' over $X$.
If $L\rightarrow A$ is a local
system of abelian groups then we obtain a morphism
$$
K(L/A,n)\rightarrow A,
$$
whose homotopy fiber over any $a\in A(Y)$ is the Eilenberg-MacLane $n$-stack
$K(L|_Y,n)$ over $Y$. For $n=1$ we can make do with any local
system of groups not necessarily abelian.
There is a standard fibration sequence relative to $A$
$$
K(L/A,m)\rightarrow A \rightarrow K(L/A, m+1),
$$
in other words
$$
K(L/A,m)= A \times _{K(L/A, m+1)}A.
$$
Using this we obtain the usual description of the stages in the Postnikov
tower: if $m\geq 2$ then, setting $A:= \tau _{\leq m-1}T$ there is a local
system $L$ of abelian groups over $A$ and a section $ob:A\rightarrow K(L/A,
m+1)$
such that the morphism in the Postnikov tower
$$
\tau _{\leq m}T \rightarrow \tau _{\leq m-1}T =A
$$
is equivalent to
$$
A\times _{K(L/A,m+1)} A\rightarrow A
$$
where the first morphism in the fiber product is $ob$ and the second is $0$.
The description of the first stage $\tau _{\leq 1}T\rightarrow \pi _0(T)$
is much more complicated and is basically the subject of Giraud's book
\cite{Giraud}.
Using the notion of local system we can define a relative version of the
homotopy group sheaves. If $T\rightarrow R$ is a morphism of $n$-stacks and $s:
R\rightarrow T$ is a section then we obtain local systems of groups $\pi
_i(T/R, s)$ over $R$.
Suppose $X\rightarrow Z$ and $Y\rightarrow Z$ are morphisms of $n$-stacks. Then
we obtain a {\em relative internal $Hom$} which is an $n$-stack with morphism
to $S$,
$$
Hom (X/Z,Y/Z)\rightarrow S.
$$
It is defined by the universal property that maps $A\rightarrow Hom (X/Z,Y/Z)$
are the same (in an essentially well-defined way) as maps $X\times
_ZA\rightarrow Y$ over $Z$. For existence, if the proof
isn't contained somewhere in the literature then one might have to apply the
techniques of \cite{limits}. If $Z=\ast$ then we get back to the usual internal
$Hom(X,Y)$.
Similarly if $Y\rightarrow X \rightarrow Z$ then we obtain the {\em relative
section stack}
$$
\Gamma (X/Z, Y)
$$
which is defined to be the fiber product
$$
Hom (X/Z,Y/Z)\times _{Hom (Y/Z,Y/Z)}Z
$$
where the second map in the fiber product is that corresponding to the identity
of $Y$, and the first map is induced by $Y\rightarrow X$. Again if $Z=\ast$ we
denote this simply by $\Gamma (X,Y)$.
We now come to one of the main types of observations, namely the relationship
between the above objects and cohomology. See for example Thomason
\cite{Thomason} for much of this. If $A$ is an $n$-stack and $L$ a local system
of abelian groups over $A$ then we define
$$
H^i(A, L):= \pi _0\Gamma (A, K(L/A,i)).
$$
It is a sheaf of abelian groups on the site $Sch /{\bf C} $. Similarly if $Z$ is an
$n$-stack, $p:A\rightarrow Z$ a morphism and $L$ a local system of abelian
groups
over $A$ then we define
$$
H^i(A/Z,L):= \tau _{\leq 0 /Z}\Gamma (A/Z, K(L/A,i))
$$
where $\tau _{\leq 0 /Z}$ is the relative version of the truncation operation
for $n$-stacks over $Z$. Note that $H^i(A/Z,L)$ is a local system of abelian
groups on $Z$. We can also denote it by $R^ip_{\ast} (L)$.
One has the result that the cohomology defined above coincides with sheaf
cohomology over simplicial objects (representing $A$ by a simplicial object in
the topos of $Sch /{\bf C}$). See \cite{Thomason} or
\cite{flexible}. In particular the notation $R^ip_{\ast} (L)$ coincides with the
usual meaning (particularly when we are looking at $A$ and $Z$ which are
represented by schemes, for example).
We have the formulae
$$
\pi _i(\Gamma (A, K(L/A,m)), 0) =H^{m-i}(A,L)
$$
and (using the relative version of homotopy groups)
$$
\pi _i(\Gamma (A/Z, K(L/A,m))/Z, 0) =H^{m-i}(A/Z,L).
$$
The usual results concerning cohomology of topological spaces hold for
cohomology as defined above. In particular, we have cup-products,
corresponding to the following operations on Eilenberg-MacLane spaces. If
$L$, $L'$ and $L''$ are local systems of abelian groups over $A$ and if
$$
L\times _AL'\rightarrow L''
$$
is a bilinear morphism (of relative abelian group objects) then we obtain
morphisms $$
K(L/A, i)\times K(L'/A, j)\rightarrow K(L'' /A, i+j).
$$
These give cup-products in cohomology which are bilinear morphisms
$$
H^i(A, L)\times H^j(A,L')\rightarrow H^{i+j}(A, L'').
$$
We also have a K\"unneth formula. The case which we use in the present paper is
as follows. Suppose $X$ and $Y$ are $n$-stacks. Then
$$
H^m(X\times Y, {\cal O} )= \bigoplus _{i+j= m} H^i(X, H^j(Y,{\cal O} )).
$$
If $H^j(Y,{\cal O} )$ are represented by finite dimensional vector spaces then we can
write the more usual formula
$$
H^m(X\times Y, {\cal O} )= \bigoplus _{i+j= m} H^i(X, {\cal O} )\otimes _{{\cal O}} H^j(Y,{\cal O}
).
$$
This extends to the relative case of morphisms $X\rightarrow S$ and
$Y\rightarrow S$ if these families are locally trivial over the etale topology
of $S$.
Finally, we have a Leray-Serre spectral sequence. See \cite{Thomason} for
one way
to set this up. If $f: X\rightarrow Y$ is a morphism of $n$-stacks and if $L$ is
a local system of abelian groups on $X$ then we obtain a ``complex'' $R^{\cdot}
f_{\ast}(L)$ on $Y$ and the cohomology of $X$ is the ``hypercohomology'' of this
complex. These terms are put in quotations because one should actually interpret
the notion of complex as being a fibration in spectra over $Y$ (the raw notion
of complex of local systems is not adapted to the higher homotopy involved if
$Y$ is not $0$-truncated and locally cohomologically trivial). In any case we
get the cohomology objects of the direct image, which are the relative
homotopy group sheaves of the fibration of spectra, denoted $R^if_{\ast}(L)$.
These are local systems over $Y$. We have the Leray-Serre spectral sequence
(cf Thomason \cite{Thomason})
$$
E^{i,j}_2 = H^i(Y, R^jf_{\ast}(L))\Rightarrow H^{i+j}(X, L).
$$
One way of approaching all of the above details is to replace any
$n$-stack i.e. simplicial presheaf (particularly those coming in as the base in
questions about cohomology) by a simplicial object whose stages are formal,
possibly infinite, disjoint unions of schemes (this technique was pointed out to
me by C. Teleman). Similarly, we can replace morphisms of $n$-stacks by
morphisms of such objects. Then questions about cohomology become just
questions about cohomology of simplicial schemes with obvious modifications
made to allow for the infinite number of components.
The $n$-stack $K({\cal O} , n)$ has an infinite loop-space structure or
$E_{\infty}$-structure, in other words it has
an infinite delooping (the $m$-fold delooping is just $K({\cal O} , m+n)$).
This structure is the homotopical analogue of an abelian group structure:
it contains a homotopy class of maps
$$
K({\cal O} , n)\times K({\cal O} , n)\rightarrow K({\cal O} ,n)
$$
but also higher homotopies of associativity, commutativity etc.
We often think of $K({\cal O} , n)$ as a homotopical group object and speak of
things such as ``principal bundles'' for it.
The infinite loop-space structure is
inherited by $Hom ({\cal F} , K({\cal O} , n))$ for any sheaf ${\cal F}$.
In the first version of the paper, it was stated that the infinite loop-space
structure provides a functorial decomposition into products of
Eilenberg-MacLane stacks. This is not true in general. For one thing, such a
decomposition may not exist, and for another it is never completely functorial.
For existence, the obstruction is in the $Ext$ between the various homotopy
group sheaves. If such a decomposition exists, then it can be functorial up to
one homotopy, but this homotopy itself will not be uniquely determined up to a
second homotopy.
The statement which we actually need for the calculation in \S 8 below, does
work, and we state it as a proposition.
\begin{proposition}
\label{decomp}
(A)\, Suppose $(Y,y)$ is a basepointed $n$-stack with infinite loop structure.
Suppose that $\pi _i(Y,y)$ are represented by finite-dimensional vector spaces.
Then there exists an equivalence
$$
\varepsilon : Y\cong Y_0 \times \ldots \times Y_n
$$
where
$$
Y_i = K( {\cal O} ^{a_i}, i) = K(\pi _i(Y,y), i).
$$
(B)\, Suppose $f:Y\rightarrow Y'$ is a map of basepointed $n$-stacks both as in
(A), and suppose $f$ is compatible with the infinite loop structure. Let
$$
\varepsilon : Y\cong Y_0\times \ldots \times Y_n,\;\;\;
\varepsilon ': Y'\cong Y'_0\times \ldots \times Y'_n
$$
denote the maps given by (A) (chosen independantly of $f$). Then there is a
homotopy making the square
$$
\begin{array}{ccc}
Y&\cong &Y_0\times \ldots \times Y_n\\
\downarrow &&\downarrow \\
Y'&\cong &Y'_0\times \ldots \times Y'_n
\end{array}
$$
commute, where the vertical arrow on the right is a product of the morphisms
of Eilenberg-MacLane stacks
$Y_i\rightarrow Y'_i$ induced by
$f_{\ast}:\pi _i(Y,y)\rightarrow \pi _i(Y', y')$.
\end{proposition}
{\em Proof:}
The short way of saying this is that the $Ext^j(V,W)$ vanish for $j>0$ for
finite dimensional vector spaces, so the obstruction to splitting vanishes.
We give the following more concrete argument (which is in a certain sense just
repeating the argument which will be given in \ref{ext} below for the vanishing
of the $Ext^j$).
For $N>n$ we are given an $N-1$-connected pointed $n+N$-stack $(Z,z)$ with
$(Y,y)= \Omega ^N(Z,z)$ (this is the {\em ad hoc} definition of ``infinite loop
structure'' which we use). In part (B) the map $f$ comes from a map
$g:Z\rightarrow Z'$. Thus for part (A) it suffices to obtain a decomposition
$$
Z\cong Z_0\times \ldots \times Z_n
$$
with
$$
Z_i = K(V_i, N+i),\;\;\; V_i := \pi _i(Y,y) = \pi _{N+i}(Z,z),
$$
and for part (B) it suffices
to obtain the homotopy of functoriality on the level of the morphism $g$.
Calculate the cohomology of $Z$ with coefficients in ${\cal O}$. Using Leray-Serre
spectral sequences for the stages in the Postnikov tower, and using the Breen
calculations \ref{bc} in view of the hypothesis that the $\pi _j$ are finite
dimensional vector spaces, we find that for $i\leq N+n$,
$$
H^{j}(Z, {\cal O} ) = Hom (\pi _j(Z,z), {\cal O} ) =V_{N-j}^{\ast}.
$$
Thus we have tautological classes
$$
\varepsilon _i \in H^{N+i}(Z, \pi _{N+i}(Z,z))= H^{N+i}(Z, V_i)
$$
which together provide us with a map
$$
\varepsilon = (\varepsilon _0,\ldots , \varepsilon _n) :
Z \rightarrow K(V_0,N)\times \ldots \times K(V_n, N+n).
$$
This map induces an isomorphism on homotopy groups (in degrees up to $N+n$).
Thus it provides the required splitting for (A).
For part (B) suppose we have a map
$$
g:Z= Z_0\times \ldots \times Z_n \rightarrow Z'_0\times \ldots \times Z'_n
$$
with $Z_i= K(V_i, N+i)$ and $Z'_i= K(W_i, N+i)$. Such a map corresponds, up to
homotopy, to a collection of classes in
$H^{N+i}(Z, W_i)$. From the K\"unneth formula (which can be seen by a
collection
of Leray-Serre spectral sequences for the projections onto the factors) we have
$$
H^{N+i}(Z, W_i) = H^{N+i}(Z_i, W_i).
$$
The other factors vanish again using the Breen calculations \ref{bc} from the
fact that $V_j$ and $W_i$ are represented by finite dimensional vector spaces.
Our map $g$ is therefore homotopic to a map given by the classes
in $H^{N+i}(Z_i, W_i)$, i.e. a map compatible with the product decomposition.
\hfill $\Box$\vspace{.1in}
The homotopy in part (B) is not unique: it can be changed by a map
$Y\rightarrow \Omega Y'$ in other words by a collection of morphisms
$Y_i\rightarrow Y'_{i+1}$.
\medskip
Suppose that ${\cal F}$ is an $n$-stack such that the $H^i({\cal F} , {\cal O} )$ are
represented by finite dimensional vector spaces. Then we can apply the above
proposition to
$$
Y:= Hom ({\cal F} , K({\cal O} ,n)).
$$
The decomposition of $Hom ({\cal F} , K({\cal O} ,n))$ into a
product of Eilenberg-MacLane spaces
$$
Hom ({\cal F} , K({\cal O} ,n)) = \prod _i K(H^{n-i}({\cal F} , {\cal O} ), i),
$$
is related to the K\"unneth formula.
A morphism $Z\rightarrow Hom ({\cal F} , K({\cal O} , n))$ corresponds (by the definition
of internal $Hom$) to a morphism
$Z\times {\cal F} \rightarrow K({\cal O} ,n)$, in other words to a class $f\in H^n(Z\times
{\cal F} , {\cal O} )$. By the above product structure this class decomposes into a
collection of classes $f_i\in H^i(Z, H^{n-i}({\cal F} , {\cal O} ))$. The $f_i$ are the
K\"unneth components of $f$.
\begin{center}
$\ast$ \hspace*{2cm}$\ast$ \hspace*{2cm}$\ast$
\end{center}
For the remainder of the paper, we look at $n$-stacks of groupoids on $Sch /{\bf C}$
and unless specified otherwise, the reader may fix any $n\geq 3$ (for the
calculation it suffices to take $n=3$.)
\numero{Connected very presentable shape}
We isolate some special $n$-stacks $T$ and then use them to measure the
``shape''
of an arbitrary $n$-stack ${\cal F}$. In other words, take a sub-$n+1$-category
${\cal P} \subset nSTACK$ of the $n+1$-category of $n$-stacks (of groupoids,
say), and look at the $n+1$-functor
$$
{\cal P} \rightarrow nSTACK
$$
$$
T\mapsto Hom ({\cal F} , T).
$$
We call this $n+1$-functor the {\em shape of ${\cal F}$ as measured by ${\cal P}$}.
There are many possible ways to choose ${\cal P}$. Some reasonable parameters
are to require that ${\cal P} \subset nSTACK$ be a full sub-$n+1$-category
(in other words that we make no limitation on the morphisms of ${\cal P}$);
and that the condition $T\in {\cal P}$ should be measured only by looking at
the homotopy group sheaves $\pi _i(T,t)$. For our present purposes we start by
requiring that $T$ be connected, i.e. $\pi _0(T)=\ast$. Among other things, this
insures that the isomorphism classes of the higher homotopy group sheaves $\pi
_i(T,t)$ be well defined.
Before describing our choice of conditions for the $\pi _i(T,t)$ we take note
of the following: any $T\in {\cal P}$ will decompose in a Postnikov tower
whose stages are $K(\pi _i , i)$. Morphisms ${\cal F} \rightarrow K(\pi _i,i)$
are classified by $H^i({\cal F} ,\pi _i)$ and more generally, one has obstruction
theory for classifying the morphisms ${\cal F} \rightarrow T$ going up in the
Postnikov tower; the obstruction classes are in $H^{i+1}({\cal F} ,\pi _i)$.
Thus, one should choose the class of possible $\pi _i$ to be a class of sheaves
such that, for the ${\cal F}$ we are interested in, the cohomology $H^j({\cal F} ,\pi _i)$
has reasonable properties.
For the topic of Dolbeault cohomology, we already know how to take nonabelian
$H^1$ with coefficients in an affine algebraic group \cite{Moduli} \cite{hbls},
and we know how to take higher Dolbeault cohomology with coefficients in ${\bf C}$
or more generally in a finite-dimensional complex vector space. This suggests
that our condtions should be that $\pi _1$ be an affine algebraic group, and
$\pi _i$ be represented by finite-dimensional vector spaces for $i\geq 2$.
Recall from \cite{RelativeLie} and \cite{GeometricN} that a {\em connected very
presentable $n$-stack $T$} is an $n$-stack of groupoids $T$ on $Sch /{\bf C}$
subject
to the following conditions:
\newline
{\bf (connectedness):} $\pi _0(T) = \ast$ as a sheaf of sets on $Sch /{\bf C}$;
\newline
{\bf (very presentability):} if $t\in T(Spec ({\bf C} ))$ is a basepoint
(which we assume exists) then $\pi _i(T,t)$ are representable by group schemes
of finite type over $Spec ({\bf C} )$, which are required to be affine for $i=1$
and vector spaces (i.e. affine unipotent abelian) for $i\geq 2$.
(This is the ``very presentability'' condition of \cite{RelativeLie} under the
additional hypothesis of connectedness; in the non-connected case the
definition is more complicated and that is basically the subject of the paper
\cite{RelativeLie}.)
We now choose ${\cal P}$ for our shape theory to be the $n+1$-category of
connected very presentable $n$-stacks of groupoids. Suppose ${\cal F}$ is an
$n$-stack on $Sch/{\bf C}$. The{\em shape of ${\cal F}$} is defined as the
$n+1$-functor from the $n+1$-category of connected very presentable $n$-stacks
$T$, to the $n+1$-category $nSTACK$ of all $n$-stacks, given by the formula
$$
Shape ({\cal F} )(T):= Hom ({\cal F} , T).
$$
This contains all information about ${\cal F}$ which one can extract by looking at
$G$-torsors over ${\cal F}$ and cohomology of
associated vector bundles.
In many cases, $Hom({\cal F} , T)$ will be a geometric or locally geometric
$n$-stack \cite{GeometricN}. For example in the case ${\cal F} = X_{Dol}$ we look at
below, $Hom(X_{Dol} , T)$ will be locally geometric. In these cases the shape
of ${\cal F}$ may be considered as an $n+1$-functor from connected very presentable
$n$-stacks to the $n+1$-category of (locally) geometric $n$-stacks, sitting
inside $nSTACK$.
\subnumero{Examples}
We explain how to understand the structure of connected very presentable $T$.
First is the simply connected case. Here $T$ is given by a Postnikov tower
where the stages are of the form $K({\cal O} ^a, m)$. The only question is how they
are put together. The fibration
$$
K({\cal O} ^a, m)\rightarrow \tau _{\leq m}T \rightarrow \tau _{\leq m-1}T
$$
is classified by a map
$$
\tau _{\leq m-1}T\rightarrow K({\cal O} ^a, m+1),
$$
in other words $\tau _{\leq m}T$ is the pullback by this map of the standard
fibration
$$
K({\cal O} ^a, m)\rightarrow \ast \rightarrow K({\cal O} ^a, m+1).
$$
We can write
$$
\tau _{\leq m}T= \tau _{\leq m-1}T\times _{K({\cal O} ^a, m+1)}\;\; \ast .
$$
The classifying map is a class in $H^{m+1}(\tau _{\leq m-1}T, {\cal O} ^a)$.
In turn, this cohomology can be ``calculated'' by the Leray-Serre spectral
sequence applied to the previous part of the Postnikov tower for
$\tau _{\leq m-1}T$. The basic pieces that we need to know are the cohomology
of the Eilenberg-MacLane spaces. These are given by the {\em Breen
calculations} \cite{Breen1} \cite{Breen2}, which we recall in Appendix I
(giving a relative version). For the present discussion the answer is that
$H^{\ast} (K({\cal O} ^a, m), {\cal O} )$
is a graded-symmetric algebra on ${\cal O}^a$ in degree $m$. Note that this answer
is the same as the classical answer for rational cohomology of rational
Eilenberg-MacLane spaces $H^{\ast} (K({\bf Q} ^a, m),{\bf Q} )$.
As we shall explain below and also in Appendix II, if $Y$ is a finite
simply-connected $CW$-complex then we obtain a $1$-connected very presentable
$T= Y\otimes {\bf C}$ whose homotopy group sheaves are $\pi _i(Y,y)\otimes _{{\bf Q}}
{\cal O}$.
We now look at connected but not simply connected very presentable $T$.
Note that
since $T$ is connected, we can choose a basepoint $t\in T(Spec ({\bf C} ))$. Let
$G:= \pi _1(T,t)$; by hypothesis it is an affine algebraic group scheme over
${\bf C}$. We have a fiber sequence
$$
T'\rightarrow T \rightarrow K(G,1)
$$
where $T'$ is simply connected very presentable. The homotopy group sheaves
$\pi _i(T', t)= \pi _i(T,t)$ are vector spaces ${\cal O} ^a$ for $i\geq 2$; but
notice also that $G$ acts on these vector spaces. The action is an action of
sheaves on the site $Sch /{\bf C} $ so it is automatically algebraic; we can write
$\pi _i(T', t)= V^i$ with $V^i$ a linear representation of $G$.
The same Postnikov tower discussion as above, works here. The only difference
is that in calculating the cohomology of the $\tau _{\leq m}T$ we may have
coefficients which are linear representations of $G$, and at the end we get
down to a step where we have to calculate $H^i(K(G,1), V)$. If $G$ is
reductive, this ``algebraic cohomology'' vanishes for $i\geq 1$, whereas if $G$
is unipotent then it is equal to the Lie algebra cohomology.
A simple example of non-simply connected $T$ may be obtained as follows. Start
with a linear algebraic group $G$ with a linear representation $V$,
corresponding to a local system $\underline{V}\rightarrow K(G,1)$.
Use the notational shorthand
$$
K(V/G; n):= K(\underline{V}/K(G,1), n).
$$
We have a fibration sequence
$$
K(V, n)\rightarrow K(V/G,n) \rightarrow K(G,1).
$$
Maps ${\cal F} \rightarrow T$ correspond
to pairs $(E, \eta )$ where $E$ is a $G$-torsor over ${\cal F}$ and
\newline
$\eta \in H^n({\cal F},
E\times ^GV)$.
An advantage of the nonabelian cohomological formulation of the
secondary Kodaira-Spencer classes we define is that the definition works for
cohomology with coefficients in a connected very presentable $T$, even
non-simply connected. However, for the calculation we will do, we look at a
particular simply-connected $T$ (the ``complexified $2$-sphere''
$S^2\otimes {\bf C}$).
\subnumero{Representability}
Under certain circumstances, basically when the shape of ${\cal F}$ is simply
connected and has reasonable cohomology sheaves, then $Shape ({\cal F} )$ is {\em
representable}. By this we mean that there is a morphism ${\cal F} \rightarrow
\Sigma$ from ${\cal F}$ to a very presentable $n$-stack $\Sigma$ such that for any
other very presentable $n$-stack $T$ we have
$$
Hom (\Sigma , T)\stackrel{\cong}{\rightarrow} Hom ({\cal F} , T).
$$
For example we have the following precise statement.
\begin{theorem}
\mylabel{representable0}
Suppose ${\cal F}$ is an $n$-stack on $Sch /{\bf C} $ such that for any affine algebraic
group $G$,
$$
K(G,1)\stackrel{\cong}{\rightarrow}Hom ({\cal F} , K(G,1)).
$$
Suppose that the $H^i({\cal F} , {\cal O} )$ are representable by finite dimensional
vector spaces. Then there is a morphism ${\cal F} \rightarrow \Sigma$ to a
$1$-connected very presentable $n$-stack $\Sigma$, such that for any connected
very presentable $n$-stack $T$ we have
$$
Hom (\Sigma , T)\stackrel{\cong}{\rightarrow} Hom ({\cal F} , T).
$$
\end{theorem}
The proof will be given in Appendix II.
On the other hand, for $n$-stacks ${\cal F}$ whose shape is not $1$-connected, the
shape will not in general be representable. For example, suppose $W$ is a
finite $CW$ complex (considered as a constant $n$-stack) such that $\pi
_1(W)=\Gamma := {\bf Z} $ and such that some $\pi _i(W)\otimes {\bf Q}$ is a ${\bf Q}
[\Gamma
]$-module which is not completely torsion (hence infinite-dimensional over
${\bf Q}$). Then $Shape (W)$ is not representable. Indeed, the infinite
dimensionality of $\pi _i(W)\otimes {\bf Q}$ is seen by the shape, since all
irreducible representations of $\Gamma$ are finite ($1$-) dimensional.
\numero{Nonabelian Dolbeault cohomology}
Suppose $X$ is a smooth quasiprojective variety. Recall that one defines the
{\em Dolbeault cohomology} of $X$ as the hypercohomology of the trivial complex
$\Omega ^{\cdot}_X$ with differential equal to $0$:
$$
H^i_{Dol}(X):= {\bf H}^i(X, \Omega ^0_X \stackrel{0}{\rightarrow} \ldots )
= \bigoplus _{p+q=i} H^q(X, \Omega ^q_X)
$$
(generally speaking this is only motivated by topology when $X$ is projective;
but we make the notation for quasiprojective $X$ too, for use in Mayer-Vietoris
arguments).
A nonabelian version for $H^1$ may be defined by setting $H^1_{Dol}(X, G)$
equal to the moduli stack of Higgs principal $G$-bundles (\cite{Hitchin0}
\cite{Hitchin}
\cite{hbls} \cite{Moduli}) $(P,\theta )$.
One can impose semistability conditions
and vanishing of Chern classes to get a version more closely related to
topology,
but we don't need that for the present algebraic discussion.
If $(P,\theta )$ is a principal Higgs bundle and if $V$ is a representation of
$G$ then we obtain an associated Higgs bundle $(E,\theta )$. Recall that we
define the {\em Dolbeault cohomology of $(E,\theta )$} as the hypercohomology of
the {\em Dolbeault complex} (cf \cite{hbls})
$$
\ldots \stackrel{\theta}{\rightarrow} E\otimes _{{\cal O} _X}\Omega ^i_X
\stackrel{\theta}{\rightarrow} \ldots ,
$$
$$
H^i_{Dol}(X, (E,\theta )):= {\bf H}^i(X,
(E\otimes _{{\cal O}_X}\Omega ^{\cdot}_X,\theta )).
$$
We present a way of unifying these definitions into a notion of {\em nonabelian
Dolbeault cohomology}. For nonabelian $H^1$ the present interpretation was
explained in \cite{SantaCruz}.
Let $\widehat{TX}$ denote the formal completion of
$TX$ along the zero section. Considered as a presheaf on $Sch /{\bf C} $ it
associates to any ${\bf C}$-scheme $Y$, the set of maps $Y\rightarrow TX$ which map
the underlying reduced subscheme $Y^{\rm red}$ to the zero-section $X\subset
TX$.
Define the
$1$-stack $X_{Dol}$ to be the relative $K(\widehat{TX}/X, 1)$, i.e. the relative
classifying stack for the group scheme $\widehat{TX} \rightarrow X$.
A variant which is technically easier to work with is
$$
X_{UDol} := K(TX/X, 1).
$$
The $U$ in the notation stands for ``unipotent'': as we shall see below
(Proposition \ref{calcDol}), a morphism $X_{UDol} \rightarrow K(G,1)$
corresponds
to a principal Higgs bundle with structure group $G$, over $X$, such that the
Higgs field is a section of unipotent elements of the Lie algebra of $G$.
However, for morphisms to simply connected $T$, we can safely replace $X_{Dol}$
by $X_{UDol}$ and for the purposes of the present paper this is what we shall
do.
If $T$ is an $n$-stack then we define the {\em
nonabelian Dolbeault cohomology of $X$ with coefficients in $T$} to be the
$n$-stack (of $n$-groupoids)
$$
Hom (X_{Dol}, T).
$$
Of course this includes the classical abelian case, when we take $T= K({\cal O} , n)$
where ${\cal O}$ denotes the structural sheaf, equal to ${\bf G}_a$, represented by
the affine line. It also includes the somewhat classical case of nonabelian
Dolbeault $H^1$ with coefficients in a group scheme $G$.
More generally we will be most interested in the case where $T$ is a connected
very presentable $n$-stack.
The calculation of the present paper involves a $1$-connected $T$, so it doesn't
refer to the case of Higgs principal $G$-bundles or Dolbeault cohomology with
coefficients in Higgs bundles. These aspects are only presented to show the
unified character of the definition.
\begin{proposition}
\mylabel{calcDol}
We have that
$$
\pi _0 (Hom (X_{Dol}, K({\cal O} , n))) = H^n_{Dol}(X, {\bf C} )
$$
is the usual Dolbeault cohomology of $X$. The same holds for
$X_{UDol}$. If $G$ is an affine algebraic group scheme then
$$
Hom (X_{Dol} , K(G,1))= {\cal M}_{Dol}(X, G)
$$
is the moduli $1$-stack of principal Higgs bundles with structure group $G$.
On the other hand,
$$
Hom (X_{UDol} , K(G,1))
$$
is the moduli $1$-stack of principal Higgs bundles $(P, \theta )$ with structure
group $G$, such that for every $x\in X$ the element $\theta _x \in ad(P)_x\cong
{\bf g} $ is a unipotent element of the Lie algebra ${\bf g}$ of $G$ (this condition
is of course independant of the isomorphism $ad(P)_x\cong
{\bf g} $ chosen).
\end{proposition}
{\em Proof:}
Let $X^{\rm fc}_{Dol}$ denote the formal category defined in \cite{SantaCruz}
which gives the $1$-stack $X_{Dol}$. We have
$$
Ob\, X^{\rm fc}_{Dol} = X,
$$
and
$$
Mor (X^{\rm fc}_{Dol}) = \widehat{TX} \rightarrow X \hookrightarrow X\times X
$$
(the morphism object lies over the diagonal in $X\times X$). The composition of
morphisms is just addition in $\widehat{TX}$ (which is a formal group scheme
over $X$). This formal groupoid gives in an obvious way a presheaf of
groupoids on $Sch /{\bf C} $, whose associated stack is $X_{Dol}$. A morphism
$X_{Dol}\rightarrow K(G,1)$ is the same thing as a $G$-torsor over
$X_{Dol}^{\rm fc}$, which in turn is the same thing as a principal $G$-bundle
$P$ over $X$ together with action of the formal group scheme $\widehat{TX}$.
The action may be interpreted as a morphism of sheaves of groups over $X$,
$$
\widehat{TX} \rightarrow Ad (P).
$$
Since the domain is formal, this is the same thing as a morphism of Lie
algebras over $X$,
$$
\theta : TX \rightarrow ad(P).
$$
This proves the second statement. For the third statement, the same proof works
but with $\widehat{TX}$ replaced by $TX$. Note that $TX$ is a unipotent group
scheme over $X$, so
$$
TX\rightarrow Ad(P)
$$
corresponds to a morphism of Lie algebras
$$
\theta : TX \rightarrow ad(P)
$$
with image in the unipotent elements of $ad(P)$.
For the first statement, one way to proceed is to notice that the cohomology of
$X_{Dol}$ with coefficients in ${\cal O}$ is the same as the cohomology of the
formal category $X^{\rm fc}_{Dol}$ as considered by Berthelot \cite{Berthelot}
and Illusie
\cite{Illusie}. From those references, one gets a generalized de Rham complex
calculating the cohomology, which in our case is seen to be exactly the
Dolbeault complex.
We also need to prove that the morphism $X_{Dol}\rightarrow X_{UDol}$ induces an
isomorphism on cohomology. For this it suffices to look locally over $X$, so we
can assume that $TX$ is trivial. Thus it suffices to prove that if $V$ is a
vector space of dimension $n$ then the morphism
$$
K(\widehat{V},1)\rightarrow K(V,1)
$$
induces an isomorphism of cohomology. This morphism has homotopy fiber the
sheaf $V_{DR}$ defined by $V_{DR}(Y)=V(Y^{\rm red})$ (cf \cite{kobe}). The
cohomology of $V_{DR}$ with coefficients in ${\cal O}$ is the algebraic de Rham
cohomology of $V$ (\cite{kobe} Theorem 6.2) which is trivial because $V$ is an
affine space.
Since the statement of the first part for $X_{UDol}$ is what we actually use,
we indicate a somewhat more elementary proof. Let $V$ be a vector space of
dimension $n=dim(X)$. We have a natural (split) extension
$$
1\rightarrow V \rightarrow G \rightarrow GL(V) \rightarrow 1
$$
which gives a fiber sequence of $1$-stacks
$$
K(V,1)\rightarrow K(G,1)\stackrel{p}{\rightarrow} K(GL(V), 1).
$$
Let $C^{\cdot} = R^{\cdot} p_{\ast}({\cal O} )$. Using the Breen calculations
\cite{Breen2} \cite{kobe},
which we recall in Theorem \ref{bc} in Appendix I below, it is easy to see that
$$
H^i(C^{\cdot}) = \bigwedge ^i(V^{\ast}).
$$
However, the fact that $GL(V)$ is a reductive group implies that all of the
algebraic group cohomology, in other words the cohomology of $K(GL(V), 1)$
with coefficients in any local system associated to a representation of
$GL(V)$, vanishes except in degree $0$. Therefore the invariants of the complex
$C^{\cdot}$ vanish so
$$
C^{\cdot} \sim \bigoplus _{i} \bigwedge ^i(V^{\ast}).
$$
Now notice that the vector bundle $TX\rightarrow X$ corresponds to a map
$X\rightarrow K(GL(V), 1)$ and
$$
X_{UDol} = K(G,1)\times _{K(GL(V),1)} X.
$$
Let $q: X_{UDol}\rightarrow X$ denote the projection; then
$R^{\cdot} q_{\ast}{\cal O} $ is the pullback of $C^{\cdot}$, hence it splits as
$$
R^{\cdot} q_{\ast}{\cal O} = \bigoplus _i\Omega ^{i}_X.
$$
The cohomology of $X_{UDol}$ with coefficients in ${\cal O}$ is equal to the
hypercohomology of $X$ with coefficients in this complex, which is the Dolbeault
cohomology.
\hfill $\Box$\vspace{.1in}
{\bf Remark:} We have the following which are sometimes useful:
$$
\pi _i (Hom (X_{Dol}, K({\cal O} , n))) = H^{n-i}_{Dol}(X, {\bf C} ).
$$
\begin{proposition}
\mylabel{calcDol2}
Suppose $(P,\theta )$ is a Higgs principal $G$-bundle on $X$ corresponding to a
map $X_{Dol}\rightarrow K(G,1)$. Suppose $V$ is a representation of $G$ and let
$(E,\theta )$ be the associated Higgs bundle. Then the cohomology of $X_{Dol}$
with coefficients in the local system $(P,\theta )^{\ast}(V)$ is naturally
isomorphic to the Dolbeault cohomology $H^n_{Dol}(X, (E,\theta ))$. The same
works for a family of principal bundles parametrized by a base scheme $S$.
\end{proposition}
{\em Proof:}
As before, we interpret $(P,\theta )$ as a $G$-torsor over the formal category
$X_{Dol}^{\rm fc}$ which defines the stack $X_{Dol}$. Associated to the
representation $V$ we get a local system over the formal category, and its
cohomology (which is that mentionned in the statement of the proposition) is
calculated by a de Rham complex (cf \cite{Berthelot} \cite{Illusie}). This de
Rham complex is exactly the Dolbeault complex for $(E,\theta )$.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{UdolDol}
Suppose $T$ is a connected very presentable $n$-stack such that $\pi _1(T)$ is
a unipotent group scheme over $Spec ({\bf C} )$. Then the morphism
$$
Hom (X_{UDol} , T)\rightarrow Hom (X_{Dol} , T)
$$
is an equivalence.
\end{corollary}
{\em Proof:}
This follows immediately from the previous propositions using a Postnikov tower
for $T$, and noting that if $G$ is a unipotent group scheme then
all principal $G$-Higgs bundles satisfy the condition that the Higgs field be a
section of unipotent elements.
\hfill $\Box$\vspace{.1in}
\subnumero{Cohomology classes of subschemes}
A particular detail of (usual) Dolbeault cohomology which enters into our
calculations is the cohomology class of a subscheme. Suppose $X$ is a smooth
projective variety and suppose that $Z\subset X$ is a subscheme of codimension
$d$. Then we obtain the class
$$
[Z] \in H^d(X, \Omega ^d_X)\subset H^{2d}_{Dol}(X).
$$
In fact this class has a canonical trivialization over $U:=X-Z$, in other words
we are actually given a lifting to
$$
[Z] \in H^d((X,U), \Omega ^d_X)\subset H^{2d}((X_{Dol}, U_{Dol}),{\cal O} ).
$$
Suppose for example $d=1$.
Take an open covering $X=\bigcup U_i$ with $U_0=U=X-Z$, and the
remaining $U_i$ affine with defining equations $z_i\in {\cal O} (U_i)$ having zeros
of order one along $Z$ and nonvanishing elsewhere. Put
$$
g_{ij}:= \frac{dz_i}{z_i} - \frac{dz_j}{z_j}.
$$
This gives a $1$-cocycle which determines the cohomology class $[Z]$. Since
$U_0=X-Z$ is a part of the covering, it gives a class in the cohomology of the
pair $(X,X-Z)$.
The case of higher codimension is treated similarly. The only thing we need to
know is that if $\lambda , \lambda ' \in {\cal O}
(X)$ are regular functions such that $\lambda |_Z= \lambda '|_Z$, then $\lambda
[Z] = \lambda '[Z]$. In particular if $P$ is a point then $\lambda [P]=\lambda
(P)[P]$.
\numero{Group actions}
Suppose $W$ is a sheaf of groups on $Sch /{\bf C} $, and suppose $R$ is an
$n$-stack on $Sch /{\bf C} $. Then an {\em action of $W$ on $R$} is a morphism
$$
\rho : A \rightarrow K(W, 1)
$$
together with an identification of the fiber $\rho ^{-1}(0)$ (by which
we mean the homotopy fiber product $\{ 0\} \times _{K(W, 1)} A$) with $R$,
$$
R\cong \rho ^{-1}(0).
$$
To put this more briefly, an action of $W$ on $R$ is a fiber sequence
$$
R\rightarrow A \rightarrow K(W,1).
$$
If $T$ is another $n$-stack, and if $\rho$ is an action of $W$ on $R$
then we obtain an associated action $Hom (\rho , T)$ of $W$ on $Hom (R,T)$.
It is given by the relative $Hom$,
$$
Hom (A/K(W,1), T)\rightarrow K(W,1),
$$
whose fiber is naturally identified with $Hom(R,T)$.
{\bf Example 1:} If $W$ acts on a sheaf of sets $R$ in the usual sense,
then this
can be interpreted as an action in the above sense. The stack $A$
(which in this case is a $1$-stack) is the
stack-theoretical quotient $R/W$ with its canonical principal $W$-bundle
$R\rightarrow R/W$ which corresponds to a morphism $R/W \rightarrow K(W,1)$.
{\bf Example 2:} The group ${\bf G}_m$ acts on $\widehat{TX}$ (resp. $TX$) over $X$ by
scalar multiplication. Therefore ${\bf G}_m$ acts on $X_{Dol}$ (resp. $X_{UDol}$)
and on $Hom (X_{Dol}, T)$ (resp. $Hom(X_{UDol}, T)$). In the case $T=K(G,1)$
this is the usual action of ${\bf G}_m$ on the moduli stack of principal Higgs bundles
\cite{Hitchin} \cite{hbls} \cite{Moduli}. In the case $T=K({\cal O} , n)$ this
action gives the decomposition of $H^i_{Dol}(X)$ into pieces $H^q(X, \Omega
^p_X)$.
{\bf Example 3:} A different example, more closely related to what we are
interested in, is the following. Suppose
$$
1\rightarrow V \rightarrow E \rightarrow W \rightarrow 1
$$
is an exact sequence of sheaves of groups on $Sch /{\bf C}$. This gives a fibration
sequence
$$
K(V,1)\rightarrow K(E,1)\rightarrow K(W,1),
$$
hence by definition it is an action of $W$ on $K(V,1)$.
More generally if $X$ is a sheaf of sets and if $V\rightarrow X$
is a sheaf of groups over $X$, then we can look at the relative
Eilenberg-MacLane stack $K(V/X, 1)\rightarrow X$.
Suppose
$$
1\rightarrow V \rightarrow E \rightarrow p^{\ast}(W)\rightarrow 1
$$
is an exact sequence of sheaves of groups on $X$ (where $p:X\rightarrow \ast$
denotes the projection). Then we obtain a fibration sequence
$$
K(V/X, 1)\rightarrow K(E/X, 1)\rightarrow K(W,1),
$$
the latter map being the composition of the map induced by
the second map in the exact sequence, with the projection
$$
K(p^{\ast}W /X,1)= K(W\times X/X, 1)= K(W, 1)\times X \rightarrow K(W,1).
$$
Thus, again by definition, our exact sequence corresponds to an action of $W$
on $K(V/X,1)$ (lying over the trivial action of $W$ on $X$).
\subnumero{Secondary classes}
We first discuss classifying spaces.
If $R$ is an $n$-stack then $Aut (R)$ is an $n$-stack with ``group'' structure,
more precisely with the structure of a loop space.
To make this statement precise, we construct a pointed $n+1$-stack
$B\, Aut(R)$ with basepoint denoted $0$, with an equivalence
$$
\Omega (B\, Aut(R), 0) \cong Aut (R).
$$
Construct the presheaf of spaces $B^{\rm pre}\, Aut(R)$ as the realization of a
simplicial $n$-stack $B_{\cdot}Aut(R)$ defined as follows: let
$\overline{I}^{(i)}$ be the $1$-category with $i+1$
isomorphic objects $0,\ldots , i$ (which we shall call ``vertices''); then
set $B_iAut(R)$ equal to the
$n$-stack of morphisms $\overline{I}^{(i)}\rightarrow n\underline{STACK}$
sending
the vertices to the object $R\in nSTACK$. Note that the
component $B_iAut(R)$ is homotopic to $Aut(R)\times \ldots \times Aut(R)$.
This simplicial $n$-stack may be interpreted as a presheaf of $n+1$-categories.
Note that $Aut(R)$ is defined
to be the $n$-stack of morphisms
$\overline{I}\rightarrow n\underline{STACK}$ sending $0,1$ to $R$, and it is a
stack of $n$-groupoids. It may safely be confused with a presheaf of
$n$-truncated spaces, and $B_{\cdot}\, Aut(R)$ becomes a presheaf of
simplicial spaces. The component simplicial spaces satisfy Segal's condition (cf
\cite{Segal} \cite{Tamsamani} \cite{nCAT} \cite{effective}) so if we set $B^{\rm
pre}\, Aut(R)$ equal to the realization into a presheaf of spaces then Segal's
Proposition 1.5 \cite{Segal} implies that the natural map
$$
Aut(R) = B_1Aut(R) \rightarrow \Omega (B^{\rm pre}\, Aut (R), 0)
$$
is an equivalence (object-by-object).
Finally, let $B\, Aut(R)$ be the $n+1$-stack associated to $B^{\rm pre}Aut(R)$.
If $R$ is an $n$-stack of groupoids,
the locally constant fibrations with fiber $R$
$$
R \rightarrow E \rightarrow S
$$
are classified exactly by maps $S\rightarrow B\, Aut(R)$. In other words, given
$S$ and $R$, the $n+1$-category of such fiber sequences is equivalent to the
$n+1$-category $Hom (S, B\, Aut(R))$.
In particular, an action of a sheaf of groups $W$ on an $n$-stack of groupoids
$R$ is the same thing as a morphism
$$
f:K(W,1)\rightarrow B\, Aut (R).
$$
This leads to the notion of {\em characteristic classes} for the action:
if ${\cal G}$ is a sheaf of groups and
if $c\in H^i(B\, Aut (R), {\cal G} )$ then for any action we can pull back $c$ to
obtain a class in $H^i(K(W,1), {\cal G} )$.
One can also obtain {\em secondary invariants}, which only become defined when
some primary invariants vanish. For example, suppose
$c\in H^i(B\, Aut (R), {\cal G} )$ is a cohomology class corresponding to a map
$$
c: B\, Aut(R) \rightarrow K({\cal G} , i).
$$
Suppose that $f: K(W, 1)\rightarrow B\, Aut(R)$ is an action of $W$ on $R$,
such that $c\circ f : K(W, 1)\rightarrow K({\cal G}, i)$ is trivial, i.e. homotopic
to the constant map at the basepoint. Let $Fib(c)$ denote the homotopy fiber
of the morphism $c$ over $0$. A choice of trivialization $\psi$ of $c\circ f$
gives rise to a morphism
$$
f_{\psi} : K(W, 1)\rightarrow Fib(c).
$$
The choices of trivialization $\psi$ are, up to homotopy, classified by
$H^{i-1}(K(W,1), {\cal G} )$. If $c'$ is a class in $H^j(Fib(c), {\cal G} ')$
then the composition $c'\circ f_{\psi}$ gives a class in $H^j(K(W, 1), {\cal G} ')$.
This is (a typical example of) a secondary characteristic class for the action.
We don't get any further into the general theory of secondary classes, because
insofar as the formalism is concerned, there is no difference between the
present
case of $n$-stacks and the classical topological version of the theory.
Instead, we give a more explicit description of the secondary classes that we
are interested in. These concern only classes of degrees $1$ and $2$ on $B\,
Aut (R)$, so a more concrete discussion is possible.
The first step is to replace $R$ by its truncation down to a $1$-stack,
denoted $\tau _{\leq 1}R$. We fix a point $r\in R (Spec ({\bf C} ))$ (and denote
also by $r$ its image in $\tau _{\leq 1}(R)$). Note that the ``group''
$Aut(R)$
acts on $\tau _{\leq 1}R$. The first of the primary classes comes from a
morphism
$$
Aut(R)\rightarrow Aut(\pi _0R).
$$
On the left is an actual sheaf of groups ${\cal G}$, and this map corresponds
to a class in $H^1(B\, Aut(R), {\cal G} )$. Suppose that we have an action by $W$
such that this class vanishes. This implies that $W$ fixes the point $r\in \pi
_0(R)(Spec ({\bf C} ))$. (Vanishing of this class actually implies that $W$ fixes
every point but for what follows we only need that it fixes $r$). Let
$R^r$ denote the component of $R$ containing $r$, more precisely it is the
fiber product
$$
R^r:= \ast \times _{\pi _0(R)} R.
$$
Under our preliminary vanishing hypothesis, we obtain an action of $W$ on $R^r$.
Now $\tau _{\leq 1}(R^r)= K(G, 1)$ where $G= \pi _1(R,r)$ (the point $r$ is a
basepoint defined over $Spec ({\bf C} )$ giving this trivialization of the $1$-gerb
$\tau _{\leq 1}(R^r)$---the basepoint $r$ corresponds to $0\in K(G,1)$).
For any group $G$ we have
a fibration sequence
$$
K(Center(G), 2)\rightarrow
B\, Aut (K(G,1)) \rightarrow K(Out(G), 1).
$$
In our case above, $W$ acts on $R^r$, hence on $\tau _{\leq 1}R^r=K(G,1)$.
The map
$$
K(W,1)\rightarrow B\, Aut(\tau _{\leq 1}R^r)= B\, Aut (K(G,1))
$$
thus projects first of all to a morphism of groups
$$
W\rightarrow Out \left( \pi _1(R, r )\right) .
$$
This is in a certain sense again a primary invariant. Suppose that this
invariant vanishes. Then, given that
$$
K(W,1)\rightarrow K(Out(G), 1)
$$
is a pointed map, there is a canonical homotopy of this map to the constant map
at the basepoint, so we canonically can identify our map
$$
K(W,1)\rightarrow B\, Aut(\tau _{\leq 1}R^r)
$$
as a map
$$
K(W,1)\rightarrow K(Center \, \pi _1(R, r) , 2),
$$
in other words as a class in $H^2(K(W,1), Center)$ where $Center$ is the
center of $\pi _1(R, r)$. This is the secondary class we will be interested in
calculating below.
We can describe the secondary class a bit more concretely in the following way.
Suppose $\alpha , \beta$ are elements of $W$ thought of as paths in $K(W,1)$.
Choose trivializations of the fibration
$$
R\rightarrow A\rightarrow K(W,1)
$$
above the paths $\alpha$ and $\beta$. These trivializations lead in particular
to liftings of our paths starting with the basepoint $r$, and ending at points
we denote $\alpha ^{\ast}(r)$ and $\beta ^{\ast}(r)$. By the hypothesis of
vanishing of primary classes, $\alpha ^{\ast}(r)$ and
$\beta ^{\ast}(r)$ are homotopic to $r$. Thus we can choose homotopies which we
denote $h_{\alpha}$ and $h_{\beta}$ (these are paths in $R$ joining
$\alpha ^{\ast}(r)$ resp. $\beta ^{\ast}(r)$ to $r$). Applying the
trivializations chosen above to these paths we obtain paths in $R$
$$
\alpha ^{\ast}(h_{\beta}): \alpha^{\ast} \beta^{\ast} (r) \rightarrow
\alpha ^{\ast} (r),
$$
$$
\beta ^{\ast}(h_{\alpha}): \beta^{\ast} \alpha^{\ast} (r) \rightarrow
\beta ^{\ast} (r).
$$
Finally, the commutativity of $W$ means that there is a torus obtained by
attaching a $2$-cell along the commutator of $\alpha$ and $\beta$. Lifting this
$2$-cell (more precisely, trivializing the family above this $2$-cell)
provides a
path between $\alpha^{\ast} \beta^{\ast} (r)$ and $\alpha^{\ast} \beta^{\ast}
(r)$. Combining these all together we get a $5$-sided loop based at $r$; this is
the secondary class evaluated on $\alpha \wedge \beta$. The loop is in the
center of $\pi _1(R,r)$ because of the fact that we chose trivializations of
the family over our paths (or $2$-cell)
rather than just liftings starting at the basepoint.
A slightly different and more geometric way of looking at this is to look at
the torus in $K(W,1)$ given by the commutator $2$-cell for $\alpha$ and
$\beta$. Attaching two $2$-cells to the torus, one along $\alpha$ and the other
along $\beta$, gives a $2$-sphere. The vanishing of the primary classes
means that the fibration $A\rightarrow K(W,1)$ can be extended across these new
$2$-cells, so we get a family over $S^2$. Trivializing over the northern and
southern hemispheres, the family is determined by a morphism from the equator
to $Aut(R)$: this element of $\pi _1(Aut(R))=Center$ is the image of $\alpha
\wedge \beta$ under the secondary class.
\numero{The secondary Kodaira-Spencer map}
We now come to the situation which gives a ``secondary Kodaira-Spencer map''.
Suppose $f:X_S\rightarrow S$ is a smooth projective morphism. Fix a basepoint
$s\in S$ and denote by $X$ the fiber of $X_S$ over $s$.
Define
$$
W:= T(S)_s
$$
which is a vector space, thus unipotent abelian group scheme, considered as a
sheaf of groups on $Sch /{\bf C} $.
We have an exact
sequence (of unipotent abelian group schemes i.e. vector bundles over $X$)
$$
0\rightarrow TX \rightarrow T(X_S)|_X \rightarrow f^{\ast}(T(S)_s)\rightarrow
0.
$$
This is an action of $W=T(S)_s$ on the stack $K(TX/X, 1)= X_{UDol}$. In
particular, for any $n$-stack $T$ we obtain an action of $W$ on
$Hom (X_{UDol}, T)$.
The primary invariant in this situation is an action of $W$ on the sheaf of
sets \linebreak $\pi _0Hom (X_{UDol}, T)$. For example,
if $T=K({\cal O} , n)$ then by Proposition \ref{calcDol}
$$
\pi _0Hom (X_{UDol}, T)= H^n_{Dol}(X)= \bigoplus _{p+q=n}H^q(X, \Omega ^p_X)
$$
and we obtain an action of $W=T(S)_s$ on $H^n_{Dol}(X)$. This action is of
course just the usual Kodaira-Spencer map which decomposes into components
$$
T(S)_s \rightarrow Hom \left( H^q (X, \Omega ^p_X), H^{q+1}(X, \Omega
^{p-1}_X) \right) .
$$
To obtain secondary invariants, we proceed as described above, using vanishing
of the primary invariants if we want to
(but bearing in mind that the secondary invariants will then only be defined
when the primary invariants vanish). For example, suppose that $T$ is an
$n$-stack and suppose $\eta \in Hom (X_{UDol}, T)$ such that the point $[\eta ]
\in \pi _0Hom (X_{UDol}, T)$ is fixed by the action of $W$. Then $W$ acts on the
connected $n$-stack $Hom ^{\eta}(X_{UDol}, T)$ which is the connected component
containing $\eta$. We obtain a morphism
$$
K(W,1)\rightarrow B\, Aut (Hom ^{\eta}(X_{UDol}, T)),
$$
and as remarked above, cohomology classes on the right can be pulled back to
give classes on $K(W,1)$. At this point we refer to the Breen
calculations \cite{Breen2}. In Appendix II we prove a relative version in
characteristic zero which was stated in \cite{kobe}; the reader may refer there
for the general statement.
In our case, as $W$ is represented by a
finite-dimensional vector space (in particular, $W\cong {\cal O} ^a$) we have
$$
H^i(K(W,1), {\cal O} ) = \bigwedge ^iW^{\ast}
$$
where $W^{\ast} = Hom (W, {\cal O} )$ is the sheaf represented by the dual vector
space.
To get down to the concrete example we would like to consider, we boil things
down a bit farther, following the discussion at the end of the previous section.
Namely instead of looking at the full $Hom ^{\eta}(X_{UDol}, T)$ we truncate it
down to a $1$-stack (which is connected, too) by looking at
$$
\tau _{\leq 1}Hom ^{\eta}(X_{UDol}, T).
$$
In the very presentable case, this automatically has a base point over $Spec
({\bf C} )$ (but also we have chosen a basepoint $\eta$) so it is equivalent to
something of the form $K(G,1)$ with
$$
G= \pi _1(Hom (X_{UDol}, T), \eta ).
$$
Again,
in the very presentable case (i.e. if $T$ is very presentable which implies the
same for $Hom ^{\eta}(X_{UDol}, T)$) then $G$ is an affine algebraic group.
In our
example below $G$ will itself be a vector space. The first invariant is
$$
W\rightarrow Out \left( \pi _1(Hom (X_{UDol}, T), \eta )\right) .
$$
Suppose that this
invariant vanishes. Then we get a map
$$
K(W,1)\rightarrow K(Center \, \pi _1(Hom (X_{UDol}, T), \eta ) , 2),
$$
which may be interpreted as a class in $H^2(K(W,1), Center)$ where $Center$ is
the center of $\pi _1(Hom (X_{UDol}, T), \eta )$. If $T$ is very presentable
then this center will be an affine abelian group scheme. This class in $H^2$ is
the class we are interested in calculating below.
The secondary class can be described concretely by choosing homotopies
trivializing the primary classes and combining them together using the
commutativity homotopy for the action of $W$, as at the end of the previous
section.
\numero{The complexified $2$-sphere}
We discuss in more detail the example of a $3$-stack $T$ for which we make our
calculation. Recall from ``standard topology'' how to describe the
rational homotopy type of $S^2$. The only nontrivial stages in the Postnikov
tower are $K({\bf Q} , 2)$ and $K({\bf Q} , 3)$. Thus the rational homotopy type is
described by the fibration sequence
$$
K({\bf Q} , 3) \rightarrow S^2 \otimes {\bf Q} \rightarrow K({\bf Q} , 2).
$$
In turn, the classifying space for fibrations with fiber $K({\bf Q} , 3)$ is the
base for the universal fibration
$$
K({\bf Q} , 3)\rightarrow \ast \rightarrow K({\bf Q} , 4).
$$
Thus the above fibration is determined by a morphism
$$
K({\bf Q} , 2)\rightarrow K({\bf Q} , 4),
$$
in other words a class $\sigma \in H^4(K({\bf Q} , 2), {\bf Q} )$.
The classical calculations give
$$
H^{2m}(K({\bf Q} , 2), {\bf Q} ) = Sym ^m({\bf Q} ).
$$
In particular
$$
H^4(K({\bf Q} , 2), {\bf Q} )\cong Sym ^2({\bf Q} ) = {\bf Q} .
$$
Up to change of basis element for $\pi _3$, there are only two possibilities:
either $\sigma = 0$ or $\sigma \neq 0$. The case $\sigma = 0$ corresponds to
the direct product $K({\bf Q} , 2)\times K({\bf Q} , 3)$ but in this case $H^3$
would be nonzero, whereas $H^3(S^2\otimes {\bf Q} , {\bf Q} )=0$. Therefore we must
have $\sigma \neq 0$, in other words $\sigma$ is the cup product
$\eta \cup \eta$ of the canonical class $\eta \in H^2(K({\bf Q} , 2), {\bf Q} )$
with itself.
In view of this discussion, we can do the same with very presentable stacks. We
will construct $T= S^2\otimes {\bf C}$ with two stages $K({\cal O} , 2)$ and $K({\cal O} ,
3)$ in the Postnikov tower. We construct it as the pullback of the universal
fibration
$$
K({\cal O} , 3)\rightarrow \ast \rightarrow K({\cal O} , 4)
$$
by a morphism
$$
\sigma : K({\cal O} , 2) \rightarrow K({\cal O} , 4).
$$
The Breen calculations say that the cohomology of $K({\cal O} , n)$
with coefficients in ${\cal O}$ has the same answer as the cohomology of $K({\bf Q} , n)$
with coefficients in ${\bf Q}$. In other words,
$$
H^{2m} (K({\cal O} , 2), {\cal O} ) = Sym ^m_{{\cal O}} ({\cal O} ) \cong {\cal O}
$$
and
$$
H^4(K({\cal O} , 2), {\cal O} )= {\cal O} .
$$
Let $\sigma$ be the generator of ${\cal O} (Spec \, {\bf C} ) = {\bf C}$. This gives a map
$$
\sigma : K({\cal O} , 2)\rightarrow K({\cal O} , 4).
$$
In terms of cohomology operations, a morphism ${\cal F} \rightarrow K({\cal O} , 2)$
corresponds to a cohomology class $\eta \in H^2({\cal F} , {\cal O} )$ and the
composition of such a map with $\sigma$ corresponds to the cup-product square
$\eta \cup \eta \in H^4({\cal F} , {\cal O} )$.
Now set
$$
T:= S^2\otimes {\bf C} := K({\cal O} , 2) \times _{K({\cal O} , 4)} \ast ,
$$
(this choice of $T$ shall be in vigor for the rest of the paper unless
explicitly mentionned otherwise). For any $n$-stack ${\cal F}$, a morphism
$$
{\cal F} \rightarrow T
$$
corresponds to a pair $(\eta , \varphi )$ where $\eta : {\cal F} \rightarrow K({\cal O} ,
2)$ and $\varphi$ is a homotopy between
$$
\sigma \circ \eta = \eta \cup \eta : {\cal F} \rightarrow K({\cal O} , 4)
$$
and the constant map at the basepoint
$$
\underline{o}: {\cal F} \rightarrow K({\cal O} , 4).
$$
Alternatively, $\varphi$ may be thought of as a section of the fibration
$$
K({\cal O} , 3) \rightarrow {\cal F} \times _{K({\cal O} , 4)} \ast \rightarrow {\cal F} .
$$
A first remark is that for a given class $\eta \in H^2({\cal F} , {\cal O} )$ there
exists a lifting to a map ${\cal F} \rightarrow T$ if and only if $\eta \cup \eta =
0$ in $H^4({\cal F} , {\cal O} )$. If this cup-product is zero so that there exists one
lifting, then the fiber of the map
$$
Hom ({\cal F} , T) \rightarrow Hom ({\cal F} , K({\cal O} , 2))
$$
over the point $\eta$ is equivalent to $Hom ({\cal F} , K({\cal O} , 3))$
This is because if there exists one lifting then we can choose a lifting to
trivialize the fibration
$$
{\cal F} \times _{K({\cal O} , 4)} \ast \rightarrow {\cal F}
$$
i.e. to make this fibration equivalent to
$$
{\cal F} \times K({\cal O} , 3) \rightarrow {\cal F}
$$
and then the stack of other liftings is just $Hom ({\cal F} , K({\cal O} , 3))$.
{\bf Example:} We give an example of a topological space $Y$ which admits no
nonconstant maps to the actual $2$-sphere $S^2$ but which admits maps to the
complexified $2$-sphere $S^2\otimes {\bf C}$. Construct $Y$ by taking a wedge of
two $2$-spheres and adding on a $4$-cell via an attaching map
$$
f: S^3\rightarrow S^2 \vee S^2.
$$
Note that
$$
\pi _3(S^2 \vee S^2)= Sym ^2 \pi _2(S^2\vee S^2) = {\bf Z} ^3,
$$
and the class of the attaching map $f$ determines the cup product. We can think
of the class of $f$ as a symmetric $2\times 2$ matrix, which can be chosen
arbitrarily. Choose the matrix to be diagonal with $(r,s)$ on the diagonal.
The two obvious classes $e,f\in H^2(S^2\vee S^2,{\bf Q} )$ persist as classes in
$H^2(Y,{\bf Q} )$. Note that $H^4(Y,{\bf Q} ) \cong {\bf Q} $ and we have the formulae
$$
e\cup e = r, \;\; f\cup f = s,\;\; e\cup f = 0.
$$
Now a map $Y\rightarrow S^2$ (or even to $S^2\otimes {\bf Q}$) corresponds to a
class
$$
\eta = ae +bf \in H^2(Y,{\bf Q} )
$$
with $a,b\in {\bf Q} $, such that $\eta \cup \eta =0$. The lifting to a map into
$S^2$ is unique because in our case $H^3(Y,{\bf Q} )=0$. However, if $r$ and $s$
are chosen to be relatively prime and having no square prime factors,
then the equation
$$
\eta \cup \eta = a^2r + b^2s = 0
$$
doesn't have any nonzero solutions with $a,b\in {\bf Q}$ (the same holds if $r$ and
$s$ have the same sign). Thus there are no nontrivial maps $Y\rightarrow S^2$.
On the other hand, the above equation always has nonzero complex solutions,
so there is always a nontrivial map $Y\rightarrow S^2\otimes {\bf C} $.
\numero{Our example}
We now come down to the example which we would like to calculate. Fix $T=
S^2\otimes {\bf C}$ as defined above. Let $Z$ be a smooth projective surface with
$H^1(Z, {\cal O} ) = 0$ (hence $H^1(Z, {\bf C} )= H^3(Z, {\bf C} )=0$). Let $P\in Z$ and
let $X$ be the blow-up of $Z$ at $P$. Let $W\subset H^1(X, TX)$ be the rank two
subspace of deformations of $X$ corresponding to moving the point $P$ which is
blown up (thus canonically $W\cong T(Z)_P$). Then $W$ acts on $X_{UDol}$, via
the exact sequence
$$
0\rightarrow TX\rightarrow T(Tot)|_X \rightarrow W\otimes _{{\bf C}}{\cal O} _X
\rightarrow 0
$$
where $Tot$ refers to the total space of the family.
Let
$$
R:= Hom (X_{UDol}, T).
$$
Then $W$ acts on $R$ and we will look at secondary classes for this action.
Because of our hypothesis $H^3_{Dol}(X)=0$, the map $T\rightarrow K({\cal O} , 2)$
induces an injection
$$
\pi _0(R) \hookrightarrow H^2_{Dol}(X)
$$
with the image the sheaf represented by the subscheme defined by the equation
$\eta \cup \eta = 0$. The variation of Hodge structure of $H^2(X, {\bf C} )$
parametrized by $P\in Z$ as we move the point which is blown up, is trivial.
Thus the Kodaira-Spencer class is trivial, in other words $W$ acts trivially on
$H^2_{Dol}(X)$ and hence it acts trivially on $\pi _0(R)$. If we fix
$\rho = (\eta , \varphi )\in R(Spec \, {\bf C} )$ then we obtain a secondary class
$$
\kappa \in H^2(K(W, 1), \pi _1(R, \rho )).
$$
The first task is to calculate $\pi _1(R, \rho )$ (which will be a group scheme
over $Spec ({\bf C} )$ since $\rho$ is defined over $Spec ({\bf C} )$). Recall
that we have the fibration
$$
R\rightarrow Hom (X_{UDol}, K({\cal O} , 2))
$$
whose fiber over a point in the target is either empty or else equivalent to
\newline
$Hom (X_{UDol}, K({\cal O} , 3))$. Thus the long exact sequence for this fibration
gives
$$
{\bf C} = H^0_{Dol}(X) \rightarrow H^2_{Dol}(X)\rightarrow \pi _1(R, \rho )
\rightarrow H^1_{Dol}(X) = 0.
$$
the first term being $\pi _2Hom (X_{UDol}, K({\cal O} , 2))$, the
second term being
\newline
$\pi _1Hom (X_{UDol}, K({\cal O} , 3))$, and so on.
{\bf Claim:} The connecting morphism $H^0_{Dol}(X) \rightarrow H^2_{Dol}(X)$
in the above exact sequence is multiplication by a nonzero multiple of $\eta$.
{\em Proof:} For any $n$-stack $Y$ we can define $Hom (Y, T)$ and look at the
long exact sequence for the fibration
$$
Hom(Y,T)\rightarrow Hom(Y,K({\cal O} , 2)).
$$
The connecting morphism will be functorial in $Y$. Apply this to $Y=T$ itself;
then for any other $Y$ (such as $X_{UDol}$ considered in the claim) the
connecting morphism for the long exact sequence at a basepoint $Y\rightarrow T$
is obtained by pulling back the connecting morphism for $Hom (T,T)$ at the
identity map.
In other words, looking at $Y=T$ gives a universal version of the connecting
morphism. On the other hand, note that
$$
\pi _1 Hom (T, K({\cal O} , 3))=H^2(T, {\cal O} )={\cal O}
$$
and
$$
\pi _2Hom (T,K({\cal O} , 2)) = H^0(T,{\cal O} )={\cal O} .
$$
Thus the universal connecting morphism is a scalar constant $C$, and for any
$\rho : Y\rightarrow T$ the connecting morphism for $Hom (Y,T)$ based at $\rho$
fits into the diagram
$$
\begin{array}{ccc}
H^0(T,{\cal O} ) & \rightarrow & H^2(T, {\cal O} )\\
\downarrow && \downarrow \\
H^0(Y,{\cal O} ) & \rightarrow & H^2(Y, {\cal O} ).
\end{array}
$$
The vertical maps are those induced by $\rho$. It follows that the connecting
map for $Hom (Y,T)$ based at $\rho$ is the same constant $C$ multiplied by the
class $\eta$ which is the image of $\rho$ in $H^2(Y,{\cal O} )$. For the claim, apply
this to $Y=X_{UDol}$. To finish proving the claim we just have to show that $C$
is nonzero. But if $C$ were zero then the generator for $\pi _2(Hom (T,K({\cal O} ,
2)), 1_T)$ would lift to an element of $\pi _2(Hom(T,T), 1_T)$. This would give
a map $$
S^2 \times T \rightarrow T
$$
(where $S^2$ denotes the constant presheaf with values $S^2$),
which is nontrivial on $S^2\times \{ 0\} $ and $\{ 0\} \times T$. This map
would correspond to an element
$\mu \in H^2(S^2\times T, {\cal O} )$ with $\mu \cup \mu = 0$. But we have
$$
H^2(S^2\times T, {\cal O} )= H^2(S^2, {\cal O} )\oplus H^2(T, {\cal O} ) = {\cal O} \oplus {\cal O}
$$
and the cup product of the two components is nontrivial (by K\"unneth).
Therefore it is impossible to have a class $\mu$ which is nontrivial in both
components but with $\mu \cup \mu = 0$. This contradiction implies that $C\neq
0$, giving the claim.
\hfill $\Box$\vspace{.1in}
With the claim we obtain
$$
\pi _1(R,\rho ) \cong H^2_{Dol}(X)/(\eta ).
$$
Thus our characteristic class $\kappa $ becomes a map
$$
\kappa : \bigwedge ^2(W) \rightarrow H^2_{Dol}(X)/(\eta ).
$$
Next, we choose $\eta$ (which fixes the choice of $\rho$ up to homotopy).
Let $E$ be the exceptional divisor on $X$ and let $H$ denote the pullback of an
ample divisor on $Z$ not meeting the point $P$. Let $[E]$ and $[H]$ denote
their Chern classes in $H^1(X, \Omega ^1_X)\subset H^2_{Dol}(X)$.
We set
$$
\eta = m[E] + n[H] \in H^2_{Dol}(X).
$$
We have to choose $n$ and $m$ so that $\eta \cup \eta = 0$. Note that
$H^4_{Dol}(X)\cong {\bf C} $ with natural morphism given by the residue map,
normalized so that the cohomology class of a point is equal to $1$. Via this
isomorphism, $[E] \cup [E] = E.E = -1$ and $[H] \cup [H] = H.H\in {\bf Z} $.
Note that $[E]\cup [H]=0$ since the two divisors don't intersect.
Thus
$$
\eta \cup \eta = n^2H.H - m^2.
$$
We choose $m,n$ so that this is equal to $0$.
{\bf Remark:} If $H.H$ is not the square of an integer, then the pair
$(m,n)$ can not be chosen in ${\bf Q} ^2$, and in particular our map will not exist
as a topological map $X^{\rm top} \rightarrow S^2$. However, our map will
exist as a map from the constant
presheaf with values $X^{\rm top}$, to $S^2\otimes {\bf C} $. (This is a heuristic
remark since in the present paper we don't treat the question of the
relationship between Betti cohomology and Dolbeault cohomology).
Here is the result of the calculation which will be done below.
\begin{theorem}
\mylabel{calculation}
With the above choices of $T$, $X$,
$R:= Hom (X_{UDol}, T)$, $\eta = m[E] + n[H]$ ($\eta \neq 0$), and
hence $\rho \in R(Spec \, {\bf C} )$, the secondary Kodaira-Spencer class
$$
\kappa : \bigwedge ^2 W \rightarrow \pi _1(R, \rho )=
H^2_{Dol}(X)/(\eta )
$$
lands in $H^2(X, {\cal O} ) \subset H^2_{Dol}(X)/(\eta )$ and the map
$$
\bigwedge ^2 W \rightarrow H^2(X,{\cal O} )
$$
is dual (using Serre duality,
the isomorphism $H^0(X, \Omega ^2_X)\cong H^0(Z, \Omega ^2_Z)$, and
the isomorphism $W^{\ast} \cong T^{\ast}(Z)_P$)
to $m^2ev_P$ where
$$
ev_P:H^0(Z, \Omega ^2_Z) \rightarrow \bigwedge ^2T^{\ast}(Z)_P
$$
is the evaluation morphism.
In particular, $\kappa \neq 0$ if $h^{2,0}(X)=h^{2,0}(Z) > 0$ and $m\neq 0$.
\end{theorem}
Before getting on with the proof in \S 8 below, we make a few general remarks
about this result. A similar thing can be stated for the ``Dolbeault
homotopy type of $X$''. One way of defining this (which wouldn't be the
historical way, though) is as the $1$-connected very presentable $n$-stack
$\Sigma$
representing the very presentable shape of $X_{Dol}$ (cf Theorem
\ref{representable1} in Appendix II below). In this point of view,
we get an action of $W$ on the Dolbeault homotopy type. The theorem says
that this action of $W$ is nontrivial.
Note however that the action of $W$ on the homotopy group sheaves
(which are the homotopy groups of $X$ tensored with ${\bf C}$) will be
trivial. It is certainly possible to define the action of $W$, and to
make the same calculation as below to show that the action is nontrivial, using
the algebra of forms $$
A^{\cdot}
_{Dol}(X)= (\bigoplus _{p,q} A^{p,q}(X), \overline{\partial} ).
$$
In fact, my first heuristic version of the calculation was done using forms.
However, the technical details relating a differential-forms version of
nonabelian cohomology, with the version presented here, seem for the moment
somewhat difficult, so we restrict in the present paper to an algebraic
version of the calculation.
The secondary class $\kappa$ is a natural map, so it doesn't really have any
choice other than to be a multiple of the dual of the evaluation map $ev_P$. The
only question is whether this multiple is nonzero or not. Here is a heuristic
global argument to see why, in principle, the constant should be nonzero.
Let ${\cal X} \rightarrow Z$ be the total space of the family of blow-ups of points
moving in $Z$. It is obtained by blowing up the diagonal in $Z\times Z$. The
secondary Kodaira-Spencer class we calculate here is (or should be, at least)
the $(2,0)\times (0,2)$ Hodge component
(i.e. the component of type $(2,0)$ on the base and $(0,2)$ on the fiber) of the
following globally defined invariant. Fix $\eta \in H^2(X,{\bf C} )$, which is
invariant under the monodromy since the monodromy is trivial (the base $Z$ being
simply connected); the degeneration of the Leray spectral sequence says that
$\eta$ comes from restriction of a global class $\tilde{\eta} \in H^2({\cal X} , {\bf C}
)$; the cup product $\tilde{\eta} \cup \tilde{\eta}$ restricts to zero on the
fibers, so it lies in the next step for the Leray filtration, which in our case
is $H^2(Z, H^2(X, {\bf C} ))$. This cup product is therefore a globally defined
class. It is the obstruction to extending $\rho : X^{\rm top}\rightarrow T$ to a
map ${\cal X} ^{\rm top}\rightarrow T$. The $(2,0)\times (0,2)$ component of this
class, which is a holomorphic $2$-form on $Z$ with coefficients in
$H^2(X, {\cal O} )$, {\em should} give $\kappa$ when evaluated at the point $P\in Z$.
I don't currently have a proof of this, though.
In our example, it is relatively easy to see by looking at the
cohomology class of the diagonal that the
$(2,0)\times (0,2)$ component of the global class is
nonzero, and in fact it is the identity matrix (via Serre duality).
Thus if one could prove the above statement that the global class gives $c$
under evaluation at $P\in Z$, then this would prove that $\kappa\neq 0$ when
$H^2(X, {\cal O} )\neq 0$.
With the previous paragraphs as heuristic argument, the result that $\kappa\neq
0$ doesn't look all that surprising. Still, it means that the ``variation of
nonabelian Hodge structure''
\footnote{
This terminology is put in quotes because the current discussion of Dolbeault
cohomology is only a first step towards defining what a
``variation of
nonabelian Hodge structure'' is.}
on the family of $Hom (X^{\rm top}, T)$,
when $X$ is a variable fiber in the family ${\cal X} \rightarrow Z$, is nontrivial,
and even nontrivial for infinitesimal reasons. The base of this ``variation'' is
$Z$ which is simply connected. In particular, the variations of mixed Hodge
structure on the homotopy groups (or anything else you could think of) are
trivial. From a topological point of view, it is never a surprise to find a
family where the homotopy groups are constant but the family nontrivial. On the
other hand, this goes against the commonly held intuition for
projective algebraic varieties that ``formality means that everything is
determined by the cohomology ring'': in the example ${\cal X} \rightarrow Z$, the
locally constant family---parametrized by $Z^{\rm top}$---of cohomology rings of
the fibers $X$ is trivial. What remains true of course is that the topology of
the family is determined by the cohomology ring of the total space ${\cal X}$. Our
secondary Kodaira-Spencer invariant is a local invariant which
contributes to nontriviality of the global cohomology ring of the total space of
the family.
Our class detects the motion of a point $P\in Z$ exactly when $H^0(Z,\Omega
^2_Z)\neq 0$. This seems to fit in with the standard intuition that
$H^0(Z,\Omega ^2_Z)\neq 0$ causes the class group of zero cycles to be big
(Mumford's and Clemens' results, Bloch conjecture etc. cf
Voisin \cite{Voisin}). I don't see a precise connection, though.
\numero{The calculation}
We keep the above notations $T$, $P\in Z$, $X$, $E$, $H$, $\eta$, $\rho$.
We establish some more: let $N$ be an affine neighborhood of $P$ in $Z$
such that $TZ$ is trivialized over $N$. Assume that $N$ doesn't meet the
divisor image of $H$ in $Z$. Let $\alpha$ and $\beta$ denote basis sections in
$TZ(N)$. Let $B$ be the inverse image of $N$ in $X$, and let $C=X-E \cong Z-\{
P\}$. Then put $A:= B\cap C$. Note that $\{ B,C\}$ is an open covering of $X$.
We can write $$
X= B\cup ^AC
$$
(this is true as a pushout of sheaves of sets on $Sch/{\bf C} $). Similarly we have
$$
X_{UDol}= B_{UDol} \cup ^{A_{UDol}}C_{UDol}.
$$
The basis vectors $\alpha , \beta$ give cocycles for elements in $H^1(X, TX)$
(actually in the \v{C}ech cohomology relative to our covering) and it is easy
to see that these cocycles project to basis elements of our
$2$-dimensional space $W$.
We denote the basis vectors of $W$ also by $\alpha$ and $\beta$.
We can describe the action of $W$ on $X_{UDol}$ concretely in the following
way. Set $K:= K(W, 1)$ with basepoint denoted $0\in K$. Then the trivialization
of $TZ|_N$ gives an equivalence
$$
A_{UDol} \cong A\times K.
$$
There is a group structure on $K$, that is a morphism $K\times K\rightarrow K$
corresponding to the addition on $W$, and this gives
$$
A\times K\times K\rightarrow A\times K,
$$
which we can rewrite as
$$
\mu : A_{UDol}\times K \rightarrow A_{UDol}.
$$
Putting the identity $1_K$ in the second variable we get a map
$$
\Phi: A_{UDol} \times K \rightarrow A_{UDol} \times K
$$
which is an equivalence. Note also that
$\Phi |_{A_{UDol} \times 0}$ is the identity of $A_{UDol}$.
Let $i,j$ be the
inclusions from $A_{UDol}$ to $B_{UDol}$ and $C_{UDol}$ respectively. Then use
the inclusions
$$
(i\times 1_K)\circ \Phi : A_{UDol}\times K \rightarrow B_{UDol}\times K
$$
and
$$
j\times 1_K: A_{UDol}\times K \rightarrow C_{UDol}\times K
$$
to construct the pushout
$$
P:= B_{UDol}\times K\cup ^{A_{UDol}\times K}C_{UDol}\times K.
$$
This comes equipped with a morphism
$$
P\rightarrow K
$$
and the fiber over the basepoint $0\in K$ is just
$$
B_{UDol}\cup ^{A_{UDol}}C_{UDol} = X_{UDol}.
$$
Therefore according to our definition, $P\rightarrow K$ is an action
of $W$ on $X_{UDol}$.
The corresponding action of $K$ on $R:= Hom (X_{UDol}, T)$ is by definition
$$
Hom (P/K, T)\rightarrow K.
$$
Using the Mayer-Vietoris principle we get
$$
Hom (P/K, T) =
$$
$$
Hom (B_{UDol}\times K/K,T) \times _{Hom (A_{UDol}\times K/K,T)}
Hom (C_{UDol}\times K/K,T).
$$
However, note that
$$
Hom (B_{UDol}\times K/K,T) = Hom (B_{UDol}, T) \times K
$$
and similarly for the other factors. The morphism
$$
Hom (C_{UDol}, T) \times K\rightarrow
Hom (A_{UDol}, T) \times K
$$
induced by $j\times 1_K$ is just the product of the morphism induced by $j$,
with $1_K$. (On the other hand, the same is not true of the first morphism in
the fiber product, as it is induced by $(i\times 1_K)\circ \Phi$.) We can now
write
$$
Hom (P/K, T) =
$$
$$
(Hom (B_{UDol},T)\times K) \times _{Hom (A_{UDol},T)}
Hom (C_{UDol},T).
$$
The first morphism in the fiber product is the composition of the
product-compatible morphism
$$
(j^{\ast} \times 1_K): Hom (B_{UDol},T)\times K
\rightarrow Hom (A_{UDol},T)\times K,
$$
with the morphism
$$
\Psi : Hom (A_{UDol},T)\times K\rightarrow Hom (A_{UDol}, T).
$$
This map is equivalent (by the definition of internal $Hom$) to
$$
Hom (A_{UDol},T)\rightarrow Hom (K, Hom (A_{UDol},T)),
$$
which in turn is equivalent to a map
$$
Hom (A_{UDol},T)\rightarrow Hom (A_{UDol}\times K, T),
$$
this latter being induced by our action $\mu : A_{UDol}\times K \rightarrow
A_{UDol}$.
The first step is to notice that the map $\rho |_{A_{UDol}}$ from $A_{UDol}$ to
$T$ factors through a map
$$
h: A_{UDol} \rightarrow K({\cal O} , 3)\rightarrow T.
$$
This factorization is given by the fact that $\eta$ is a class in
$H^2((X_{Dol}, A_{Dol}), {\cal O} )$, in other words we are given a trivialization
of $\eta$ over $A_{Dol}$.
Recall that we write $\rho = (\eta , \varphi )$ where $\eta :
X_{UDol}\rightarrow K({\cal O} , 2)$ and $\varphi$ is a section of the pullback
bundle
$$
L_{\eta}:= X_{UDol} \times _{K({\cal O} , 4)} \ast
$$
which is a bundle with fiber $K({\cal O} , 3)$ over $X_{UDol}$. The section
$\varphi$ determines a trivialization
$$
L_{\eta} \cong X_{UDol}\times K({\cal O} , 3)
$$
such that $\varphi$ corresponds to the $0$-section. This trivialization is
uniquely determined by the condition that it be compatible with the structure
of ``principal bundle'' under the ``group'' $K({\cal O} , 3)$.
We adopt the following strategy for calculating the secondary class. We
will look
at new $n$-stacks $P^i\rightarrow K$ with $K$-maps $P^i \rightarrow P$, and
points $\rho ^i\in P^i_0$ (where $P^i_0$ means the fiber of $P^i$ over
$0\in K$), such that $\rho ^i$ maps to $\rho$. We have to arrange so that
the primary class is trivial, in other words that the class of $\rho ^i$ in $\pi
_0(P^i_0)$ should be invariant under the action of $W=\pi _1(K)$.
We also have to insure that the other primary class, the action of $W$ on $\pi
_1(P^i_0, \rho ^i)$ by outer automorphisms, should be trivial. In this case,
we obtain a secondary class for $\rho ^i$, which is an element of
$$
H^2(K, \pi _1(P^i_0, \rho ^i))
$$
and this secondary class maps to our class for $P$.
First of all, let $\ast \rightarrow Hom (B_{UDol}, T)$ be the morphism
corresponding to the point $\rho |_{B_{UDol}}$. We get
$$
K= \ast \times K \rightarrow Hom (B_{UDol}, T)\times K.
$$
Thus we obtain a morphism
$$
P^1:= K\times _{Hom (A_{UDol}, T)}Hom (C_{UDol}, T) \rightarrow
$$
$$
(Hom (B_{UDol}, T)\times K)
\times _{Hom (A_{UDol}, T)}Hom (C_{UDol}, T)
=P.
$$
This morphism is compatible with the projections to $K$.
The first morphism
$$
u: K\rightarrow Hom (A_{UDol}, T)
$$
in the fiber product is obtained by the composition
$$
K \rightarrow Hom (B_{UDol}, T)\times K\stackrel{(i\times 1_K)\circ
\Phi}{\rightarrow}
$$
$$
Hom (A_{UDol}, T)\times K \stackrel{p_1}{\rightarrow}
Hom (A_{UDol}, T).
$$
Thus $u$ corresponds to the map $K\times A_{UDol} \rightarrow T$ obtained by
composing
$$
K\times A_{UDol} \stackrel{\mu }{\rightarrow}
A_{UDol} \stackrel{\rho}{\rightarrow} T.
$$
The map $u$
factors through a morphism
$$
K\rightarrow \Gamma (A_{UDol}, L_{\eta})\rightarrow Hom (A_{UDol}, T)
$$
(technically speaking what should enter into the above
notation is $L_{\eta}|_{A_{UDol}}$ but for brevity we omit the restriction
since it is implicitly determined by the notation $\Gamma (A_{UDol}, -)$.)
The factorization comes about because the composition
$$
A_{UDol} \stackrel{\rho}{\rightarrow} T \rightarrow K({\cal O} , 2),
$$
is given as the constant map at the basepoint. Thus pulling back by
$$
A_{UDol}\times K \rightarrow A_{UDol} \rightarrow K({\cal O} , 2)
$$
is again constant at the basepoint so the map
$$
K\rightarrow Hom( A_{UDol}, T)
$$
factors through a map
$$
\tilde{u}:K\rightarrow Hom( A_{UDol}, K({\cal O} , 3)).
$$
Following above, $\tilde{u}$ corresponds to the composition
$$
K\times A_{UDol}\stackrel{\mu}{\rightarrow} A_{UDol} \stackrel{h}{\rightarrow }
K({\cal O} , 3),
$$
which we can write as $\tilde{u} = \mu ^{\ast}(h)$.
Now we obtain a morphism (over $K$)
$$
P^2:= K\times _{\Gamma (A_{UDol}, L_{\eta})}\Gamma (C_{UDol}, L_{\eta})
$$
$$
\rightarrow
K\times _{Hom (A_{UDol}, T)}Hom (C_{UDol}, T) = P^1.
$$
We have used the section $\varphi$ of $L_{\eta}$ to obtain a trivialization
$$
L_{\eta} \cong X_{UDol} \times K({\cal O} , 3).
$$
Via this equivalence
the section corresponding to $\rho$ (that is, the
section $\varphi$) corresponds to the zero-section.
Using the trivialization given by $\varphi$ we can write
$$
\Gamma (A_{UDol}, L_{\eta})\cong Hom (A_{UDol}, K({\cal O} , 3))
$$
({\em Caution:} this is not the same trivialization as given by saying that
$A_{UDol}$ maps to the basepoint of $K({\cal O} , 2)$ so the pullback fibration
$L_{\eta}$ is trivial over $A_{UDol}$; these two trivializations differ by
translation by $h$, a point which will come up below);
and
$$
\Gamma (C_{UDol}, L_{\eta})\cong Hom (C_{UDol}, K({\cal O} , 3)).
$$
These are compatible with the restriction $j^{\ast}$ so we can write
$$
P^2 = K\times _{Hom (A_{UDol}, K({\cal O} , 3))} Hom (C_{UDol}, K({\cal O} , 3)),
$$
with the second morphism in the fiber product being the restriction $j^{\ast}$
acting on maps to $K({\cal O} , 3)$.
We have to re-calculate the first morphism in the fiber product
$$
a:K\rightarrow Hom (A_{UDol}, K({\cal O} , 3)).
$$
It is no longer equal to $\tilde{u}=\mu ^{\ast}(h)$, because when we set
the section $\varphi$ of $L_{\eta}$ equal to the zero-section to get an
equivalence between $L_{\eta}$ and $K({\cal O} , 3)$, this made a translation on
$L_{\eta} |_{A_{UDol}}$---which was already trivial due to the fact that $\eta
|_{A_{UDol}}=0$---this translation has the effect of setting $h$ equal to the
$0$-section. This translation gives us the formula
$$
a = \mu
^{\ast}(h)-p_2^{\ast}(h) $$ where $p_2^{\ast}(h)$ is the map $K\rightarrow Hom
(A_{UDol}, K({\cal O} , 3))$ corresponding to the composition
$$
K\times A_{UDol} \stackrel{p_2}{\rightarrow}
A_{UDol} \stackrel{h}{\rightarrow} K({\cal O} , 3).
$$
Note that $p_2$ denotes the second projection. The minus sign in the equation
for $a$ is subtraction using the ``abelian group'' (i.e. $E_{\infty}$) structure
of $Hom(A_{UDol}, K({\cal O} , 3))$ induced by the ``abelian group'' structure of
$K({\cal O} , 3)$.
We now turn back to the second morphism in the fiber product,
$$
j^{\ast} : Hom( C_{UDol}, K({\cal O} , 3))\rightarrow Hom( A_{UDol}, K({\cal O} , 3)).
$$
The infinite loop space structure of $K({\cal O} , 3)$ which is inherited
by $Hom ({\cal F} , K({\cal O} , 3))$ for ${\cal F} = C_{UDol}$ and ${\cal F} = A_{UDol}$.
By Proposition \ref{decomp}, this delooping structure gives a
decomposition of
$Hom ({\cal F} , K({\cal O} , 3))$ into a product of Eilenberg-MacLane stacks.
The restriction morphism $j^{\ast}$ above is compatible with the
delooping structures, so by Proposition \ref{decomp} (B), it is homotopic
to a map
compatible with the decomposition into a product of Eilenberg-MacLane stacks.
Recall that
$$
\pi _i(Hom (C_{UDol}, K({\cal O} , 3)), 0) = H^{3-i}(C_{UDol}, {\cal O} )=
H^{3-i}_{Dol}(C)
$$
and
$$
\pi _i(Hom (A_{UDol}, K({\cal O} , 3)), 0) = H^{3-i}(A_{UDol}, {\cal O} )=
H^{3-i}_{Dol}(A).
$$
From the splitting given by the delooping structures via Proposition
\ref{decomp} with homotopy of functoriality of part (B) of that proposition
(choose one), we
obtain a homotopy-commutative diagram
$$
\begin{array}{ccc}
K(H^2_{Dol}(C), 1)& \rightarrow & K(H^2_{Dol}(A), 1) \\
\downarrow && \downarrow \\
Hom( C_{UDol}, K({\cal O} , 3))&\rightarrow &Hom( A_{UDol}, K({\cal O} , 3)).
\end{array}
$$
All maps are infinite loop maps.
Using this diagram, we get the map
$$
P^3:= K\times _{Hom( A_{UDol}, K({\cal O} , 3))}K(H^2_{Dol}(C), 1)
\rightarrow
$$
$$
K\times _{Hom (A_{UDol}, K({\cal O} , 3))} Hom (C_{UDol}, K({\cal O} , 3))=P^2.
$$
Set $\rho ^3:= (0,0)$ in $P^3_0$. It maps to $\rho ^2$ (a point which we didn't
specify because it was always obviously given by $\rho$), because we have
normalized so that $\varphi$ becomes the zero-section. The map $P^3\rightarrow
P^2$ (obviously a map over $K$) takes $\rho ^3$ to $\rho ^2$.
We have
$$
H^2_{Dol}(C)= H^2(C, {\cal O} _C)\oplus H^1(C, \Omega ^1_C)\oplus H^0(C, \Omega
^2_C),
$$
and similarly
$$
H^2_{Dol}(A)= H^2(A, {\cal O} _A)\oplus H^1(A, \Omega ^1_A)\oplus H^0(A, \Omega
^2_A).
$$
Recall that $C$ and $A$ are isomorphic to open subsets of $Z$: together $N$ and
$C$ form a covering of $Z$ and $N\cap C= A$. On the other hand,
cohomology of coherent sheaves on $N$ vanishes because $N$ is affine. Therefore
Mayer-Vietoris gives an exact sequence
$$
H^1(C,\Omega ^1_C) \rightarrow H^1(A, \Omega ^1_A) \rightarrow H^2(Z, \Omega
^1_Z).
$$
We are supposing that $H^1_{Dol}(Z)=0$ so by duality $H^3_{Dol}(Z)=0$.
Thus the term on the right is zero, and the morphism of vector spaces
$$
H^1(C,\Omega ^1_C) \rightarrow H^1(A, \Omega ^1_A)
$$
is surjective. Choose a splitting. This gives a morphism
$$
K(H^1(A, \Omega ^1_A), 1)\rightarrow K(H^2_{Dol}(C), 1)
$$
such that the projection into $K(H^2_{Dol}(A), 1)$ is equal to the inclusion
from the Dolbeault direct sum decomposition for $A$. Using our choice of
splitting we obtain a map
$$
P^4:= K\times _{Hom( A_{UDol}, K({\cal O} , 3))}K(H^1(A, \Omega ^1_A), 1)
\rightarrow
$$
$$
K\times _{Hom( A_{UDol}, K({\cal O} , 3))}K(H^2_{Dol}(C), 1) = P^3.
$$
Again set $\rho ^4 = (0,0)\in P^4_0$, which maps to $\rho ^3$.
Now $P^4$ is defined entirely in terms of $A$.
We decompose things a bit more. Using the infinite loop-space
structure of $Hom( A_{UDol}, K({\cal O} , 3))$ and again Proposition \ref{decomp}
we get the decomposition into a product of Eilenberg-MacLane spaces
$$
Hom( A_{UDol}, K({\cal O} , 3))\cong
J^0\times J^1 \times J^2 \times J^3
$$
where
$$
J^i = K(H^{3-i}_{Dol}(A), i).
$$
The second map in the fiber product defining $P^4$ is homotopic (choose a
homotopy) to one which is compatible with this decomposition, coming from the
morphism
$$
K(H^1(A, \Omega ^1_A), 1)\rightarrow J^1= K(H^2_{Dol}(A), 1)
$$
(the other components are the maps sending $\ast$ to the basepoints $0\in J^i$,
$i\neq 2$).
Thus we can write
$$
P^4 = Q^0\times Q^1 \times Q^2 \times Q^3
$$
where
$$
Q^i= K\times _{J^i}\ast = K\times _{K(H^{3-i}_{Dol}(A), i)}\ast
$$
for $i\neq 1$ and where
$$
Q^1 = K\times _{K(H^{2}_{Dol}(A), 1)}K(H^1(A, \Omega ^1_A), 1).
$$
This means that the action of $W$ on $P^4_0$ decomposes into a
product of actions
on the $Q^i_0$. Note that the $Q^i_0$ are themselves Eilenberg-MacLane spaces,
$$
Q^i_0 = \Omega J^i = K( H^{3-i}_{Dol}(A), i-1)
$$
for $i\geq 2$, and
$$
Q^1_0 = K(H^{2}_{Dol}(A)/H^1(A, \Omega ^1_A), 0).
$$
The classifying maps for the actions of $W$ on the components $Q^i_0$
factor as
$$
K\rightarrow J^i \rightarrow B\, Aut (Q^i_0)
$$
for $i\geq 2$, and
$$
K\rightarrow K(H^{2}_{Dol}(A)/H^1(A, \Omega ^1_A), 1)\rightarrow
B\, Aut (Q^1_0)
$$
for $i=1$. Thus the classifying map for the action
$K\rightarrow B\, Aut (P^4_0)$
factors through our above map
$$
a: K\rightarrow
K(H^{2}_{Dol}(A)/H^1(A, \Omega ^1_A), 1)\times J^2\times J^3.
$$
We use this to calculate the characteristic classes for the action:
the primary invariant is the first component, corresponding to a map
$$
W\rightarrow H^{2}_{Dol}(A)/H^1(A, \Omega ^1_A).
$$
The
other primary invariant giving the action of $W$ on $\pi _1$ by outer
automorphisms, is trivial: it is the action on $\Omega J^2$ induced by the
classifying map $K\rightarrow J^2$, and $J^2$ is simply connected.
The secondary invariant which we are interested in is the map
$K\rightarrow J^2$ which corresponds to a class in $H^2(K, H^1_{Dol}(A))$.
A preliminary remark is that the formula $a= \mu ^{\ast}(h)-p_2^{\ast}(h)$
means that the component of $a$ in $J^0$ is equal to $0$. In fact,
$p_2^{\ast}(h)$ is exactly the $J^0$-component of $\mu ^{\ast}(h)$.
On the other hand, the remaining components of $a$ are the same
as those of $\mu
^{\ast}(h)$; these are the K\"unneth components of
$$
h\circ \mu : K\times A_{UDol} \rightarrow K({\cal O} , 3).
$$
The first thing to check is that
the primary invariant is trivial for $(P^4, \rho ^4)$.
As we have said above, it is the map
$$
K(W, 1)\rightarrow K(H^{2}_{Dol}(A)/H^1(A, \Omega ^1_A), 1).
$$
To check that it is trivial, it suffices to prove the
\newline
{\bf Claim} ${\bf (\ast )}:$
the map
$$
W\rightarrow \pi _1(Hom( A_{UDol}, K({\cal O} , 3)), 0)= H^2_{Dol}(A)
$$
takes $W$ into the component
$H^1(A, \Omega ^1_A)$. For this we must again get back to the description of
the map
$$
\mu ^{\ast}(h): K\rightarrow Hom (A_{UDol}, K({\cal O} , 3))
$$
(note as remarked above that all of the components except the $J^0$ component,
are the same for $\mu ^{\ast}(h)$ or for $a$).
This map, which is equivalent to a
map
$$
A_{UDol}\times K \rightarrow K({\cal O} , 3),
$$
is the pullback of
$$
h: A_{UDol}\rightarrow K({\cal O} , 3)
$$
by the map
$$
\mu : A_{UDol}\times K \rightarrow A_{UDol}.
$$
Now we have
$$
A_{UDol} = A\times K
$$
so
$$
H^3(A_{UDol}, {\cal O} ) = H^3(A, {\cal O} )\oplus H^2(A, H^1(K, {\cal O} ))
\oplus H^1(A, H^2(K, {\cal O} ) ) \oplus H^3(K , {\cal O} ).
$$
The last term $H^3(K , {\cal O} )$ vanishes because it would be
$\bigwedge ^3W^{\ast}$ but $W$ is $2$-dimensional. Similarly, one can arrange
(by an appropriate choice of $N$) so that $A$ has an open covering by two
affine open sets. Thus $H^3(A, {\cal O} ) = H^2(A, {\cal O} )= 0$. The only remaining
term is
$$
H^3(A_{UDol}, {\cal O} ) = H^1(A, H^2(K, {\cal O} ))= H^1(A, \bigwedge ^2W^{\ast})
= H^1(A, \Omega ^2_A).
$$
Our map
$$
A_{UDol}\times K \rightarrow K({\cal O} , 3)
$$
is obtained by pulling back the above, using the map $K\times K\rightarrow K$.
Pullback for this map is
$$
\bigwedge ^2W^{\ast} \rightarrow
$$
$$
H^2(K,{\cal O})\otimes _{{\cal O}}{\cal O} \oplus H^1(K,{\cal O} )\otimes _{{\cal O}}H^1(K, {\cal O} ) \oplus
{\cal O} \otimes _{{\cal O}}H^2(K,{\cal O} ).
$$
Each of the factors is nontrivial, with the middle being (up to a multiple
which depends on normalizations for notation in exterior products)
the standard map
$$
\bigwedge ^2 W^{\ast} \rightarrow W^{\ast}\otimes _{{\cal O}}W^{\ast}.
$$
The morphism induced by pulling back $h$ to $A_{UDol}\times K$ thus decomposes
into K\"unneth components
$$
H^1(A, \Omega ^2_A)\otimes _{{\cal O}}{\cal O} \; \; \oplus \; \;
H^1(A, \Omega ^1_A)\otimes _{{\cal O}}W^{\ast} \; \; \oplus \; \;
H^1(A, {\cal O} _A)\otimes _{{\cal O}}\bigwedge ^2W^{\ast}.
$$
Each component is induced by $h\in H^1(A,\Omega ^2_A)$.
The map
$$
\pi _1(K)=W\rightarrow \pi _1(Hom (A_{UDol}, K({\cal O} , 3))
= H^2_{Dol}(A)
$$
corresponds to the middle component, which is in fact a map
$$
W\rightarrow H^1(A, \Omega ^1_A).
$$
This was exactly the claim ${\bf (\ast )}$ we needed to prove to show that the
primary invariant for $(P^4, \rho ^4)$ was trivial.
The other primary invariant, the action of $W$ on $\pi _1(P^4_0, \rho ^4)$, is
trivial as remarked above. To restate the argument, the map $a'$ induces an
injection on $\pi _1$, therefore the $\pi _1$ of the fiber is the image of $\pi
_2$ of the base; but since the base is an infinite loop space, the action of
$\pi _1$ of the base on $\pi _2$ of the base is trivial; thus the action of $\pi
_1$ of the base and in particular of $W$ on $\pi _1(P^4_0, \rho ^4)$ is
trivial.
We can now look at the secondary invariant for $(P^4, \rho ^4)$.
It is the map
$$
K\rightarrow J^2 = K(H^1_{Dol}(A), 2),
$$
which is the next K\"unneth component of the pullback of $h$ to
$A_{UDol}\times K$,
$$
\bigwedge ^2W \rightarrow H^1(A, {\cal O} _A)\subset H^1_{Dol} (A) = \pi _2(Hom
(A_{UDol}, K({\cal O} , 3))).
$$
This claim tells us that the secondary class for $P^4$ is just $h$ considered
as an element of
$$
H^1(A, \Omega ^2_A)= H^1(A, \bigwedge ^2W^{\ast})= H^1(A,{\cal O} _A)\otimes _{{\cal O}}
\bigwedge ^2W^{\ast}.
$$
Recalling that
$$
P^4_0 = \ast \times _{Hom (A_{UDol}, K({\cal O} , 3))}K(H^1(A, {\cal O} _A), 1)
$$
and $\rho ^4= (0,0)$,
we have that $P^4_0$ is just the homotopy fiber of the second morphism.
The long exact sequence for the fibration gives
$$
0\rightarrow \pi _2(Hom (A_{UDol}, K({\cal O} , 3)))\rightarrow
\pi _1(P^4_0, \rho ^4) \rightarrow 0
$$
(the morphism from $\pi _1$ of the total space to $\pi _1$ of the base being
injective, and $\pi _2$ of the total space being zero).
This long exact sequence persists under the maps
$$
P^4_0\rightarrow P^3_0\rightarrow P^2_0\rightarrow P^1_0
$$
and furthermore, even into $P_0$ where the long exact sequence for the fiber of
a morphism is replaced by a long exact sequence for the homotopy fiber product.
It follows that the secondary class for $(P, \rho )$ is the image of our above
class (basically $h$)
$$
\bigwedge ^2W \rightarrow H^1(A, {\cal O} _A)
$$
under composition with the map
$$
H^1(A, {\cal O} _A)=\pi _2(Hom (A_{UDol}, K({\cal O} , 3)), 0)
\rightarrow
\pi _2(Hom (A_{UDol}, T), \rho |_{A_{UDol}})
$$
$$
\rightarrow
\pi _1(Hom (X_{UDol}, T), \rho )=H^2_{Dol}(X)/(\eta ).
$$
One slight twist to notice is that the morphism
$K({\cal O} , 3)\rightarrow T$ in question (over $A_{UDol}$) is shifted
by $h$. This shift is recovered in the last equality,
where we undo a shift by $\varphi$.
This morphism from the long exact sequence for the fiber product (of $Hom$'s)
is equal to the connecting morphism
$$
H^1(A, {\cal O} _A)\rightarrow H^2(X, {\cal O} _X) \subset H^2_{Dol}(X)/(\eta ).
$$
Finally, we have concluded that our secondary class is the composition
$$
\bigwedge ^2W^{\ast} \rightarrow H^1(A, {\cal O} _A) \rightarrow H^2(X, {\cal O} _X)
$$
where the first map is $h$ (which depends on $\eta$ and which we investigate
below) and the second map is the connecting morphism.
The remaining problem is to calculate
$h\in H^1(A, \Omega ^2_A)$ or more precisely its image by the connecting
morphism.
We have the exact sequence of the cohomology of the pair $(X, A)$ with
coefficients in $\Omega ^2_X$:
$$
0= H^1(X, \Omega ^2_X) \rightarrow
H^1(A, \Omega ^2_A) \rightarrow
$$
$$
H^2((X, A), \Omega ^2_X) \rightarrow
H^2(X, \Omega ^2_X).
$$
The class $\eta$ may be considered as lying in
$H^1((X, A), \Omega ^1_X)$, which is the statement that our map to $T$ factors,
over $A$, through a map to $K({\cal O} , 3)$. Therefore the
cup product $\eta \cup \eta$ can be
considered as lying in $H^2((X, A), \Omega ^2_X)$ and mapping to zero in
$H^2(X, \Omega ^2_X)$. The class $h\in H^1(A, \Omega ^2_A)$ is the preimage of
$\eta \cup \eta$ in the exact sequence for the pair $(X,A)$.
Now recall that $X = B\cup C$. This means that the pair $(X,A)$ decomposes
as a ``disjoint union'' of the pairs $(X,B)$ and $(X,C)$ (after applying
excision). Thus we can write
$$
H^2((X, A), \Omega ^2_X) =
H^2((X, B), \Omega ^2_X) \oplus H^2((X, C), \Omega ^2_X) .
$$
Our class $h\in H^1(A, \Omega ^2_A)$ corresponds
to $b+c$ with
$$
b= -n^2 [H]^2\in H^2((X, B), \Omega ^2_X)
$$
and
$$
c= m^2 [E]^2\in H^2((X, C), \Omega ^2_X).
$$
We now do the same thing for $Z= N\cup C$. The class $h$ corresponds here to an
element of
$$
H^2((Z, N), \Omega ^2_Z) \oplus H^2((Z, C), \Omega ^2_Z) .
$$
This is again of the form $b'+c'$ but now with
$$
b'= -n^2 [H]^2\in H^2((Z, N), \Omega ^2_Z)
$$
and
$$
c'= m^2 [P] \in H^2((Z, C), \Omega ^2_Z).
$$
From the long exact sequence of the pair $(Z, N)$ and the fact that $N$ is
affine we find that
$$
H^2((Z, N), \Omega ^2_Z) \cong H^2(Z, \Omega ^2_Z)={\bf C} .
$$
To calculate our secondary class we have to contract $h$ with $\alpha \wedge
\beta$ to obtain a class in $H^1(A, {\cal O} _A)$ and then take its image by the
connecting map in $H^2(X,{\cal O} _X)$ or equivalently in $H^2(Z, {\cal O} _Z)$.
To measure this image, use Serre duality: we will choose a form $\omega \in
H^0(Z, \Omega ^2_Z)$ and take the cup-product to end up with a class in $H^2(Z,
\Omega ^2_Z)$ (of which we then take the residue to end up in ${\bf C}$).
The result of this procedure is the same as if we first contract
$\omega$ with $\alpha \wedge \beta$ and then multiply this section of
$H^0(A, {\cal O} _A)$ by $h$ getting a class in $H^1(A, \Omega ^2_A)$. Then take the
image of this class by the connecting map and take its residue.
Note that the contraction of $\omega$ with $\alpha \wedge \beta$ is
defined over all of the neighborhood $N$. Call this section $\lambda \in {\cal O}
(N)$. We are now reduced to the problem of calculating the
image
in $H^2(Z, \Omega ^2_Z)$ under the connecting map for the covering $Z=N\cup C$,
of $\lambda h \in H^1(A, \Omega ^2_A)$.
We have written that the image of $h$ in $H^2((Z, A), \Omega ^2_Z)$
(which we shall denote $[h]$)
is equal to $b'+c'$ where $b'\in H^2((Z, N), \Omega ^2_Z)$ and
$c'\in H^2((Z, C),\Omega ^2_Z)$. The components $b'$ and $c'$ are obtained by
residue maps for the class $h$, along respectively $H$ and $P$
(noting that $H= Z-N$ and $P= Z-C$). The form of the residue map is not
important for us, just the fact that the classes have poles of order $1$; it
follows that if $\lambda$ is a regular function on $N$ (a neighborhood of $P$)
then the residue of $\lambda h$ at $P$, is equal to $\lambda $ times the
residue of $h$ at $P$. Therefore we can write
$$
[\lambda h] = b'' + \lambda c',
$$
with
$$
b''\in H^2((Z,N), \Omega ^2_Z)
$$
and
$$
\lambda c'\in H^2((Z,C), \Omega ^2_Z)
$$
both being obtained by excision.
The value of $b''\in H^2((Z,N), \Omega ^2_Z)\cong {\bf C}$ is determined by the
condition that the image of $[\lambda h]$ in $H^2(Z, \Omega ^2_Z)$ be
zero (from the long exact sequence for the pair $(Z,A)$).
We now look at the image of $\lambda h$ by the connecting map
in the long exact sequence of the covering $Z= N\cup C$,
$$
H^1(N, \Omega ^2_N)\oplus H^1(C, \Omega ^2_C) \rightarrow
H^1(A, \Omega ^2_A)
$$
$$
\rightarrow H^2(Z, \Omega ^2_Z)\rightarrow \ldots .
$$
We can decompose this connecting map as a composition
$$
H^1(A, \Omega ^2_A) \rightarrow H^2((Z, A), \Omega ^2_Z)
\rightarrow H^2((Z, C), \Omega ^2_Z) \rightarrow H^2(Z, \Omega ^2_Z),
$$
where the second arrow is the projection onto the first factor in the excision
decomposition
$$
H^2((Z, A), \Omega ^2_Z) = H^2((Z, C), \Omega ^2_Z) \oplus
H^2((Z, N), \Omega ^2_Z)
$$
One could equally well use the second factor, with a sign change; our
calculations are not accurate insofar as signs are concerned.
Thus the image of $\lambda h$ by the connecting map
for the covering $Z= N\cup C$, is equal to the class of either $\lambda c'$ or
of $-b''$. We don't know how to calculate $b''$ so we use the representation as
$\lambda c'$. This image is then equal to
$$
\lambda \cdot (m^2 [P])
$$
which is just $m^2\lambda (P)$ (because as noted above, $[P]$ is represented
by cocycles with poles of order $1$).
We have established the formula that the image of
$\alpha \wedge \beta $ under the map
$$
\bigwedge ^2W \rightarrow H^2(Z, {\cal O} _Z)
$$
is a class which, when paired with a form $\omega \in H^0(Z, \Omega ^2_Z)$,
gives
$$
m^2\omega (\alpha \wedge \beta )(P).
$$
This completes the proof of Theorem \ref{calculation}.
\hfill $\Box$\vspace{.1in}
\numero{APPENDIX I: Relative Breen calculations in
characteristic $0$}
Crucial to the reasonable working of a theory of nonabelian cohomology is the
possibility of calculating the invariants in the Postnikov tower of the spaces
which measure the ``shape''. In our setup, this means that we would like to
calculate $H^i(K({\cal O} , m), {\cal O} )$. This calculation is the algebraic analogue
of the classical Eilenberg-MacLane calculations. The algebraic version is the
subject of Breen's work \cite{Breen1}, \cite{Breen2}. His motivation came
mostly from arithmetic geometry, so he concentrated on the case of
characteristic $p$ in \cite{Breen2}. The characteristic $0$ version, while
not explicitly stated in \cite{Breen2}, is implicit there because it is strictly
easier than the characteristic $p$ case: there are no new classes
coming from Frobenius.
These calculations for
the case of base scheme $Spec ({\bf C} )$ are sufficient for the purposes of the
present paper, but eventually a relative version will also be useful. In the
context of calculation of $Ext$ sheaves (i.e. the stable part of the
calculation) this relative version was already evoked in \cite{Breen1}, where
Breen states that the $Ext ^i(G,\cdot )$ for representable group schemes
$G$ can always be calculated. This part of the topic was not really taken up
afterward, probably for lack of a reasonable category of sheaves over a base
scheme $S$.
Such a category of sheaves is provided by Hirschowitz's notion of {\em
$U$-coherent sheaf} \cite{Hirschowitz}, see also Jaffe's recent paper
\cite{Jaffe}.
\footnote{
Hirschowitz's notion is similar to, but not quite the same as Auslander's
theory of ``coherent functors'' developed in the 1960's. Jaffe's paper
\cite{Jaffe} views $U$-coherent sheaves as a modification or generalization of
Auslander's theory---one looks at functors of algebras rather than functors of
modules. Jaffe, who seems to have been unaware of Hirschowitz's paper, cites
an unpublished letter from Artin to Grothendieck, dating from the 1960's, as a
reference for the generalized version of Auslander's theory. In order to
straighten out the history of this notion, one would have to compare Artin's
letter with \cite{Hirschowitz}.}
We change Hirschowitz's
notation and call these objects {\em vector sheaves}. The category of vector
sheaves over a base $S$ is defined in \cite{Hirschowitz} as the smallest abelian
subcategory of sheaves of ${\cal O}$-modules on the big site $Sch /S$, containing
${\cal O}$ and stable under localization of the base. Thus, locally over $S$ vector
sheaves are obtained starting with ${\cal O}$ by repeated applications of taking
direct sums, kernels and cokernels. The abelian category of vector sheaves has
several nice properties \cite{Hirschowitz}. The coherent sheaves on $S$, which
are defined as cokernels $$ {\cal O} ^a \rightarrow {\cal O} ^b \rightarrow {\cal F}
\rightarrow 0, $$
are vector sheaves. Coherent sheaves are injective objects. Their duals,
which we
call {\em vector schemes}, are the group-schemes with vector space structure
(but not necessarily flat) over $S$. These admit dual presentations as
kernels of maps ${\cal O} ^b\rightarrow {\cal O} ^a$. They are projective objects (at
least
if the base $S$ is affine). If $S$ is affine, then any vector sheaf $U$ admits
resolutions $$ 0\rightarrow V\rightarrow V' \rightarrow V'' \rightarrow U
\rightarrow 0 $$
with $V$, $V'$ and $V''$ vector schemes; and
$$
0\rightarrow U \rightarrow {\cal F} \rightarrow {\cal F} '\rightarrow {\cal F} '' \rightarrow 0
$$
with ${\cal F}$, ${\cal F} '$, ${\cal F} ''$ coherent sheaves.
{\bf Example:} The motivating example for the definition of vector sheaf in
\cite{Hirschowitz} was the following example. If $E^{\cdot}$ is a complex of
vector bundles over $S$ then the cohomology sheaves defined on the big site
$Sch /S$ are vector sheaves. Indeed they are vector sheaves of a special type
which Hirschowitz calls
``cohomologies'': quotients of vector schemes by coherent sheaves. This example
is important because it arises from the cohomology of flat families of coherent
sheaves parametrized by $S$: if $f:X\rightarrow S$ is a projective morphism and
${\cal F}$ is a coherent sheaf on $X$ flat over $S$ then a classical result says
that the higher direct image complex $R^{\cdot} f_{\ast}({\cal F} )$ (calculated on
the big site) is quasiisomorphic to a complex of vector bundles. The notion of
vector sheaf (``$U$-coherent sheaf'' in \cite{Hirschowitz}) thus keeps track of
the jumping of cohomology of flat families of cohoerent sheaves.
\footnote{
This type of example comes up in relation with Dolbeault cohomology: for example
let $$
M:= {\cal M}_{Dol}(X,G) = Hom (X_{Dol}, K(G,1))
$$
be the moduli stack of principal
Higgs $G$-bundles. If $V$ is a representation of $G$ then we obtain
$$
T:= K(V/G, n) \rightarrow K(G,1)
$$
with fiber $K(V,n)$. There is a universal local system $E$ on $X_{Dol}\times
M$, and
$$
\pi _i(Hom (X_{Dol}, T)/M, 0) = H^i(X_{Dol}\times M/M, E).
$$
This is calculated by a Dolbeault complex for $E$ on $X/M$, and the general
discussion of cohomology in flat families applies. Therefore the
$H^i(X_{Dol}\times M/M, E)$ are vector sheaves over $M$ (here $M$ is an
algebraic stack; the condition of being a vector sheaf means that the
pullback to any scheme $Y\rightarrow M$ is a vector sheaf on $Y$). }
The most surprising property from \cite{Hirschowitz} is
that the duality functor $U^{\ast} := Hom (U,{\cal O} )$ is exact, and is an
involution. This is due to the fact that we take the big site $Sch /S$ rather
than the small Zariski or etale sites.
Another interesting point is that there are two different types of tensor
products of vector sheaves: the {\em tensor product}
$$
U\otimes _{{\cal O}} V := Hom (U, V^{\ast} )^{\ast},
$$
and the {\em cotensor product}
$$
U\otimes ^{{\cal O}} V := Hom (U^{\ast}, V).
$$
These are not the same (although they coincide for coherent sheaves cf Lemma
\ref{cohtensor} below) and in particular they don't have the same exactness
properties. Neither of them is equal to the tensor product of sheaves of
${\cal O}$-modules. See \cite{kobe} and \cite{RelativeLie} for further discussion.
The above facts work in any characteristic and depend on the
${\cal O}$-module structure. However, in characteristic zero vector sheaves have the
additional property that the morphisms $U\rightarrow V$ of sheaves of abelian
groups over $Sch /{\bf C} $ are automatically morphisms of ${\cal O}$-modules, see
\cite{kobe} \cite{RelativeLie}. Similarly, extensions of sheaves of abelian
groups, between two vector sheaves, are again vector sheaves. These properties
persist for the higher $Ext^i$, see Corollary \ref{ext} below. These properties
do not remain true in characteristic $p$, as shown precisely by Breen's
calculations of \cite{Breen2}. The basic problem is that Frobenius provides a
morphism ${\cal O} \rightarrow {\cal O}$ of sheaves of abelian groups, which is not a
morphism of sheaves of ${\cal O}$-modules. This difficulty in characteristic $p$
seems to be the main obstacle to realizing a reasonable analogue of rational
homotopy theory, for homotopy in characteristic $p$. So we stick to
characteristic $0$!
We don't give a detailed introduction to vector sheaves, rather we refer the
reader to \cite{Hirschowitz}, \cite{kobe} and \cite{RelativeLie}. However, we
do take this opportunity to correct an omission from
\cite{kobe} and \cite{RelativeLie}.
Without the following lemma, the discussion in those references often seems
contradictory, as tensor products and cotensor products are interchanged when
the coefficients are coherent sheaeves.
For example,
in the statement of Corollary 3.9 of \cite{kobe} (which we restate as Theorem
\ref{bc} and prove in more detail below), all terms occuring are coherent
sheaves. Thus the tensor product which appears in the notation is also
equal to the cotensor product. It is the cotensor product which appears
most naturally in that situation. Indeed, Lemma \ref{cohtensor} below explains
(i.e. justifies) the seemingly erroneous statement ${\cal F} \otimes _{{\cal O}} {\cal G}
=\underline{Hom}({\cal F} ^{\ast}, {\cal G} )$ in the proof of Corollary 3.9 of
\cite{kobe}.
\begin{lemma}
\mylabel{cohtensor}
Suppose ${\cal F}$ and ${\cal G}$ are coherent sheaves. Then the tensor product ${\cal F}
\otimes _{{\cal O}} {\cal G}$ and the cotensor product ${\cal F} \otimes ^{{\cal O}}{\cal G}$ coincide.
\end{lemma}
{\em Proof:}
Choose a presentation
$$
{\cal O} ^a \rightarrow {\cal O} ^b \rightarrow {\cal G} \rightarrow 0.
$$
Now ${\cal F} ^{\ast} := Hom ({\cal F} , {\cal O} )$ is a vector scheme, in particular it is a
scheme affine over $S$. Therefore the functor
$$
U\mapsto Hom ({\cal F} ^{\ast} , U)
$$
is exact in $U$. Applying this functor to the above presentation we obtain
$$
{\cal F} ^a \rightarrow {\cal F} ^b \rightarrow Hom ({\cal F} ^{\ast} , {\cal G} )\rightarrow 0.
$$
The term $Hom ({\cal F} ^{\ast} , {\cal G} )$ is by definition the cotensor product ${\cal F}
\otimes ^{{\cal O}} {\cal G}$.
Taking the dual of the above presentation we obtain
$$
0\rightarrow {\cal G} ^{\ast} \rightarrow {\cal O} ^b \rightarrow {\cal O} ^a.
$$
Applying the functor
$$
U\mapsto Hom ({\cal F} , U)
$$
which is exact on the left, we obtain
$$
0\rightarrow Hom ({\cal F} , {\cal G} ^{\ast})\rightarrow ({\cal F} ^{\ast})^a
\rightarrow ({\cal F} ^{\ast})^b.
$$
Taking the dual we get
$$
{\cal F} ^a \rightarrow {\cal F} ^b \rightarrow Hom ({\cal F} , {\cal G} ^{\ast})^{\ast}\rightarrow
0.
$$
This time the term $Hom ({\cal F} , {\cal G} ^{\ast})^{\ast}$ is by definition the
tensor product ${\cal F} \otimes _{{\cal O}} {\cal G}$.
In general there is a natural morphism
$$
U\otimes _{{\cal O}} V = Hom (U,V^{\ast})^{\ast} \rightarrow Hom (U^{\ast}, V)=
U\otimes ^{{\cal O}} V.
$$
To define this map we define a trilinear morphism
$$
Hom (U,V^{\ast})^{\ast}\times U^{\ast} \times V^{\ast} \rightarrow {\cal O} ,
$$
by $(\lambda , \mu , \nu )\mapsto \lambda (\nu \mu )$, the product
$\nu \mu$ being the composed morphism
$$
U\stackrel{\mu}{\rightarrow} {\cal O} \stackrel{\nu}{\rightarrow} V^{\ast}.
$$
This trilinear map gives a morphism
$$
Hom (U,V^{\ast})^{\ast}\rightarrow Hom (U^{\ast}, (V^{\ast})^{\ast})
$$
then note that $(V^{\ast})^{\ast}=V$.
The above presentations for ${\cal F} \otimes ^{{\cal O}} {\cal G}$ and
${\cal F} \otimes _{{\cal O}} {\cal G}$ (which are the same) are compatible with this natural
morphism, so the natural morphism is an isomorphism
$$
{\cal F} \otimes ^{{\cal O}} {\cal G}\cong {\cal F} \otimes ^{{\cal O}} {\cal G} .
$$
\hfill $\Box$\vspace{.1in}
We now come to the statement of the ``relative Breen calculations in
characteristic $0$''. The case $S=Spec (k)$ for $k$ a field is due to
\cite{Breen2}, and the relative case for a group scheme was suggested in
\cite{Breen1} (where the case of cohomology with coefficients in the
multiplicative group scheme was treated).
\begin{theorem}
\mylabel{bc}
Suppose $S$ is a scheme over $Spec ({\bf Q} )$.
Suppose $V$ is a vector scheme over $X$ and suppose ${\cal F}$ is a coherent sheaf
over $S$. Then for $n$ odd we have
$$
H^i(K(V/S, n)/S, {\cal F} ) = {\cal F} \otimes _{{\cal O}} \bigwedge _{{\cal O}}^{i/n} (V^{\ast}).
$$
For $n$ even we have
$$
H^i(K(V/S, n)/S, {\cal F} ) = {\cal F} \otimes _{{\cal O}} Sym _{{\cal O}}^{i/n} (V^{\ast}).
$$
In both cases the answer is $0$ of $i/n$ is not an integer. The multiplicative
structures on the left sides, in the case ${\cal F} = {\cal O}$, coincide with the
obvious ones on the right sides. In the case of arbitrary ${\cal F}$, the
natural structures
of modules over the cohomology with coefficients in ${\cal O}$, on both sides,
coincide. \end{theorem}
We first recall the following result.
\begin{proposition}
\mylabel{somecomplexes}
Suppose $S$ is a scheme and $V$ is a vector sheaf on $S$. Then the complexes
$$
\ldots {\cal F} \otimes _{{\cal O}}\bigwedge _{{\cal O}} ^j \otimes _{{\cal O}} Sym _{{\cal O}}^k V
\rightarrow {\cal F} \otimes _{{\cal O}}\bigwedge _{{\cal O}} ^{j-1}
\otimes _{{\cal O}} Sym _{{\cal O}}^{k+1} V
\ldots $$
and
$$
\ldots {\cal F} \otimes _{{\cal O}}\bigwedge _{{\cal O}} ^j
\otimes _{{\cal O}} Sym _{{\cal O}}^k V \rightarrow
{\cal F} \otimes _{{\cal O}}\bigwedge _{{\cal O}} ^{j+1}
\otimes _{{\cal O}} Sym _{{\cal O}}^{k-1} V \ldots
$$
are exact as sequences of vector sheaves (i.e. as sequences of sheaves on the
site $Sch /{\bf C} $).
\end{proposition}
{\em Proof:} For ${\cal F} = {\cal O}$ this is Proposition 3.8 of \cite{kobe}. The proof
is easy, obtained by taking the graded symmetric powers of the cohomologically
trivial complex $V\rightarrow V$ (placing this complex starting in odd or even
degrees, leads to the two cases of the statement).
For a general ${\cal F}$ note that the cotensor product with a coherent sheaf
is exact---indeed, for any vector sheaf $U$, $Hom ({\cal F} ^{\ast}, U)$ is exact in
$U$ because ${\cal F} ^{\ast}$ is represented by a scheme, i.e. an element of the
site $Sch /{\bf C} $. The tensor product is equal to the cotensor product because
both sides are coherent sheaves (here is where we use the hypothesis that $V$
is a vector scheme, i.e. $V^{\ast}$ is a coherent sheaf).
\hfill $\Box$\vspace{.1in}
\subnumero{Proof of Theorem \ref{bc}}
Now we start the proof of Theorem \ref{bc}. Suppose that it is
true for $n\leq m-1$, and we prove it for $n=m$. (The case $m=1$ to start the
induction will be treated separately at the end.) Look at the fiber sequence $$
K(V/S, m-1)\rightarrow \ast \stackrel{p}{\rightarrow} K(V/S, m).
$$
We will look at the Leray spectral sequence for the morphism $p$, for cohomology
with coefficients in ${\cal F}$.
We may assume that $S$ is affine
Let $A_m$ denote the algebra $H^{\ast}(K(V/S, m)/S, {\cal O} )$ (this is a sheaf of
algebras over $S$) and let $A_m({\cal F} )$ denote the $A_m$-module
$H^{\ast}(K(V/S,
m)/S, {\cal F} )$. Note that if ${\cal F}$ itself is an algebra-object then $A_m({\cal F} )$ is
an $A_m$-algebra (graded-commutative). By the inductive hypothesis, the $A_k$
and $A_k({\cal F} )$ are direct sums of coherent sheaves (coherent in each degree)
for $k\leq m-1$.
The $E_2$ term of our spectral sequence is
$$
H^i(K(V/S, m)/S, H^j(K(V/S, m-1)/S, {\cal F} ))\Rightarrow H^{i+j}(\ast , {\cal F} ).
$$
In the case ${\cal F} = {\cal O}$ the $E_2$ term has a structure of algebra, and for
arbitrary ${\cal F}$, a structure of module over that algebra. These are
respectively
$$
A_m(A_{m-1})
$$
and
$$
A_m(A_{m-1}({\cal F} )).
$$
By induction we know that
$$
A_{m-1}({\cal F} ) = A_{m-1}\otimes _{{\cal O}} {\cal F} .
$$
The first possible nonzero differential in the spectral sequence is
$$
H^i(K(V/S, m)/S, H^j(K(V/S, m-1)/S, {\cal F} ))\rightarrow
$$
$$
H^{i+m}(K(V/S, m)/S, H^{j+1-m}(K(V/S, m-1)/S, {\cal F} )).
$$
We prove by a second induction on $k$, that for all ${\cal F}$ the answer is as given
in the theorem, for $H^i(K(V/S, m)/S,{\cal F} )$ for all $i\leq k$. Suppose this is
true for $i\leq k-1$. Then the elements of the diagonal complex for the above
differential, ending at $(i,j)= (k, 0)$, are all in the region $i\leq k-1$,
except for the term $(k,0)$. By our second inductive hypothesis (applied to the
cohomology of $K(V/S, m)$ with coefficients in the
coherent sheaves $H^j(K(V/S, m-1), {\cal F} )$, this complex coincides
with one of the two complexes appearing in Proposition
\ref{somecomplexes}, except maybe for the
last term. However, the complex must be exact at the last stage because
otherwise, what is left over would persist into $E_{\infty}$ contradicting the
answer of the spectral sequence (which must be ${\cal F}$ in degree $0$ and $0$
otherwise). Proposition \ref{somecomplexes} gives exactness of the complexes
appearing there. Therefore the $E^{k,0}_2$-term of our spectral sequence must
also coincide with the last term of the complex from Propositon
\ref{somecomplexes}.
For example in the case where $m$ is odd,
the end of the spectral sequence is
$$
\ldots \rightarrow
{\cal F} \otimes _{{\cal O}}Sym ^2_{{\cal O}}(V^{\ast})\otimes_{{\cal O}}\bigwedge _{{\cal O}}
^{(k-2m)/m} (V^{\ast})
$$
$$
\rightarrow
{\cal F} \otimes _{{\cal O}}V^{\ast} \otimes_{{\cal O}}\bigwedge _{{\cal O}} ^{(k-m)/m}
(V^{\ast})
$$
$$
\stackrel{d}{\rightarrow} E^{k,0}_2({\cal F} ) \rightarrow 0
$$
(where we denote by $d$ the last differential),
whereas the end of the complex of
Proposition \ref{somecomplexes} is
$$
\ldots \rightarrow
{\cal F} \otimes _{{\cal O}}Sym ^2_{{\cal O}}(V^{\ast})\otimes_{{\cal O}}\bigwedge _{{\cal O}}
^{(k-2m)/m} (V^{\ast})
$$
$$
\rightarrow
{\cal F} \otimes _{{\cal O}}V^{\ast} \otimes_{{\cal O}}\bigwedge _{{\cal O}} ^{(k-m)/m}
(V^{\ast})
$$
$$
\rightarrow {\cal F} \otimes _{{\cal O}}\bigwedge _{{\cal O}} ^{k/m} (V^{\ast})
\rightarrow 0.
$$
Therefore if $m$ is odd,
$$
E^{k,0}_2 =
{\cal F} \otimes _{{\cal O}} \bigwedge _{{\cal O}}^{k/n} (V^{\ast}).
$$
The same holds with symmetric power instead of exterior power if $m$ is even.
Cup product gives a bilinear morphism
$$
\mu : E^{k-m,0}_2\times H^m(K(V/S, m), {\cal O} ) =
E^{k-m,0}_2\times V^{\ast} \rightarrow E^{k,0}_2 .
$$
Let
$$
d':
{\cal F} \otimes _{{\cal O}}V^{\ast} \otimes_{{\cal O}}\bigwedge _{{\cal O}} ^{(k-2m)/m}
(V^{\ast})
\rightarrow E^{k-m,0}_2({\cal F} )
$$
denote the previous differential. We know by induction that $d'$ establishes an
isomorphism between $E^{k-m,0}_2({\cal F} ) $ and
$$
{\cal F} \otimes _{{\cal O}}\bigwedge _{{\cal O}} ^{k/m} (V^{\ast})
$$
where this latter is considered as a quotient of the range of $d'$ via
Proposition \ref{somecomplexes}.
Refering to the cup-product morphism $\mu$ considered above and its precursor
$$
\mu ': E^{k-2m,m-1}_2\times V^{\ast} \rightarrow E^{k-m,m-1}_2 ,
$$
we
have the Leibniz formula
$$
d\mu '(a,v) = \mu (d'(a), v)
$$
noting that the term $V^{\ast}$ appearing in the formulas is $E^{m,0}_2$
so the differential acts trivially on the variable $v$.
We have exactly the same formula when the terms $E^{i,j}_2$ are replaced by
their counterparts from the sequences of Proposition \ref{somecomplexes}.
Denote
the multiplication in these counterparts by $\nu$ and $\nu '$ and the
differentials by $\delta$ and $\delta '$.
Call the
isomorphism established by $d$,
$$
\psi ^{k,0}: {\cal F} \otimes _{{\cal O}}\bigwedge _{{\cal O}} ^{k/m} (V^{\ast}) \cong
E^{k,0} _2
$$
and similarly we have isomorphisms (by the inductive hypothesis)
$\psi ^{k-m, 0}$, $\psi ^{k-m, m-1}$ and $\psi ^{k-2m, m-1}$.
The definition of $\psi^{k,0}$ is given by the equation
$$
\psi ^{k,0}(\delta (b))= d \psi ^{k-m, m-1} (b).
$$
Similarly we have
$$
\psi ^{k-m,0}(\delta '(b))= d' \psi ^{k-2m, m-1} (b).
$$
We get
$$
\psi ^{k,0}(\nu (\delta '(a), v))=
\psi ^{k,0}(\delta \nu '(a,v)) = d\psi ^{k-m, m-1} (\nu '(a,v)).
$$
On the other hand, in this region we know by induction that the $\psi$ are
compatible with products. Therefore we get
$$
d\psi ^{k-m, m-1} (\nu '(a,v)) = d \mu '(\psi ^{k-2m,m-1}(a), v)
$$
$$
= \mu (d' \psi ^{k-2m,m-1}(a), v)
$$
$$
=\mu (\psi ^{k-m,0}(\delta '(a)), v).
$$
In all we have established the formula
$$
\psi ^{k,0}(\nu (b, v))
=\mu (\psi ^{k-m,0}(b), v)
$$
for any $b= \delta '(a)$. But $\delta '(a)$ is surjective. This establishes the
compatibility of our isomorphism $\psi ^{k,0}$ with products (given already the
compatibility of $\psi ^{k-m, 0}$).
We note in the above proof that elements of the tensor products are always
(locally on $S$)
finite sums of tensors. This can be seen for example from the proof of Lemma
\ref{cohtensor}. Thus to check compatibility with products, for example, it
suffices to check it on elementary tensors as we have done above.
This completes the proof of the theorem, modulo the case $m=1$ which we now
treat.
We have the fiber sequence
$$
V\rightarrow S \stackrel{p}{\rightarrow} K(V,1).
$$
The higher direct images vanish for coefficients in a coherent sheaf ${\cal F}$ so
the
Leray spectral sequence implies that
$$
R^ip_{\ast}({\cal F} ) = 0, \;\; i> 0
$$
and
$R^0p_{\ast}({\cal F} ) $ is a local system on $K(V,1)$ which when restricted to $S$
gives
$$
{\cal F} [V] \cong {\cal F} \otimes _{{\cal O}} Sym ^{\cdot}_{{\cal O}} (V^{\ast}).
$$
Now we use the complex given in Proposition \ref{somecomplexes}, which is
basically a de Rham complex in our situation:
$$
0\rightarrow {\cal O} \rightarrow Sym ^{\cdot}_{{\cal O}} (V^{\ast})
\ldots \rightarrow Sym ^{\cdot}_{{\cal O}} (V^{\ast})\otimes _{{\cal O}}
\bigwedge ^i_{{\cal O}}
(V^{\ast})\ldots
$$
which can then be tensored by ${\cal F}$ and remains exact. We can define the
translation action of $V$ on all of the terms, and the exact sequence remains
an exact sequence of sheaves with action of $V$. All of the terms
except for the first one are acyclic by the previous result. Therefore
the cohomology of $K(V,1)$ with coefficients in ${\cal F}$ is equal to the
cohomology of the complex
$$
{\cal F} \rightarrow
\ldots \rightarrow {\cal F} \otimes _{{\cal O}}
\bigwedge ^i_{{\cal O}}
(V^{\ast})\ldots .
$$
One can check that the differentials are actually zero, so the cohomology is as
desired.
One should check that the cup-product is equal to the obvious product structure
on the exterior-algebra side of the answer. Instead of doing this (which
as such would seem to
be a difficult task) we proceed as follows. The above construction is functorial
(contravariantly) for morphisms $V\rightarrow V'$. It is easy to see that if one
considers an injection $V\hookrightarrow {\cal O} ^a$ (which exists locally on $S$
by the definition of vector scheme) then the morphism of functoriality induces a
surjection on cohomology, coming from the surjection ${\cal O} ^a \rightarrow
V^{\ast}$. Thus to establish a formula for cup-products in cohomology, it
suffices to establish the formula for the case $V={\cal O} ^a$. In that case we
can apply the K\"unneth formula, or more precisely remark that the same
K\"unneth formula holds for the cohomology as for the exterior algebra, and that
these formulas are compatible via the above isomorphism. The K\"unneth formulae
are both compatible with cup-products. Thus we can reduce to the case
$V={\cal O}$, but here the cohomology is concentrated in degrees $0$ and $1$ so there
are no cup-products to verify (excepting the product with a degree $0$ class
but this case is easy).
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{cohVPisVS}
Suppose $S$ is a scheme and $T\rightarrow S$ is a relatively $1$-connected very
presentable $n$-stack. Then for any vector sheaf $V$ on $S$,
$$
H^i(T/S, V)
$$
is a vector sheaf.
\end{corollary}
{\em Proof:}
We use systematically (without further mention) the fact that the category of
vector sheaves is closed under kernels, cokernels and extensions cf \cite{kobe}
\cite{RelativeLie}. The case of $T= K(U/S,n)$ for $U$ a vector scheme and $V$ a
coherent sheaf is given by Theorem \ref{bc}. The case of coefficients in any
vector sheaf $V$ is obtained by taking a resolution of $V$ by coherent sheaves
and using the long exact sequence of cohomology. The case of $T=K(U/S,n)$ for
any vector sheaf $U$ is obtained by taking a resolution of $U$ by vector
schemes (divided into two short exact sequences which give rise to two
fibration sequences) and then applying the Leray spectral sequence. Finally, any
relatively $1$-connected very presentable $T$ has a Postnikov tower (relative
to $S$) whose stages are $K(U/S,n)$. Repeated application of the Leray
spectral sequence gives the result.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{homVP}
If $S$ is a scheme and $T\rightarrow S$ and $T'\rightarrow S$ are relatively
$1$-connected very presentable $n$-stacks then $Hom (T/S, T'/S)$ is very
presentable.
\end{corollary}
For this one has to use the fact that $Aut(V)$ is a very presentable group
sheaf when $V$ is a vector sheaf, see \cite{RelativeLie}.
\hfill $\Box$\vspace{.1in}
The following application was the original motivation for
Breen's calculations of the cohomology of the Eilenberg-MacLane
sheaves \cite{Breen1} \cite{Breen2}. From our version Theorem \ref{bc}, we
obtain
the corresponding result in the relative case in characteristic zero. Similar
corollaries were stated for example for cohomology with coefficients in the
multiplicative group ${\bf G}_m$, in \cite{Breen1}.
\begin{corollary}
\mylabel{ext}
{\rm (\cite{kobe} Corollary 3.11)}
Suppose $U,V$ are vector sheaves over a scheme $S$. Let $Ext^i_{\rm gp} (U,V)$
denote the $Ext$ sheaves between $U$ and $V$ considered as sheaves of abelian
groups on $Sch /S$, let $Ext^i_{\rm vs}(U,V)$ denote the $Ext$ sheaves between
$U$ and $V$ considered as vector sheaves on $S$. Then the natural morphisms are
isomorphisms
$$
Ext^i_{\rm vs} (U,V)
\stackrel{\cong}{\rightarrow}Ext^i_{\rm gp} (U,V).
$$
The $Ext^i$ vanish for $i>2$.
\end{corollary}
{\em Proof:}
Let $K_{\cdot}(U,n)$ denote the simplicial presheaf
$$
Y\mapsto K_{\cdot}(U(Y), n )
$$
given by the standard
simplicial Eilenberg-MacLane construction (i.e. Dold-Puppe applied to the
complex with $U$ placed in degree $n$). We don't take the associated stack
(as doing this or not doesn't affect the morphisms to an $m$-stack). Let
${\bf Z} K_{\cdot}(U,n)$ denote the associated presheaf of simplicial free abelian
groups. Finally let $N{\bf Z} K_{\cdot}(U,n)$ be the presheaf of normalized
complexes (in the homology direction i.e. with differential of degree $-1$) of
this simplicial abelian group. For each $Y$,
$$
N{\bf Z} K_{\cdot}(U,n)(Y)
$$
is a complex with homology group $U(Y)$ in degree $n$, and with all other
cohomology groups equal to $0$ in degrees $<2n$. Thus if ${\cal F}$ is an injective
sheaf of groups then the (cohomological) complex of sheaves
$$
Hom (N{\bf Z} K_{\cdot}(U,n), {\cal F} )
$$
has homology sheaf $Hom (U, {\cal F} )$ in degree $n$ and zero homology in all other
degrees $< 2n$.
It follows that if ${\cal F}$ is any sheaf of groups then for $i<n$,
$$
H^{n+i}Hom (N{\bf Z} K_{\cdot}(U,n), {\cal F} ) = Ext ^i_{\rm gp}(U, {\cal F} ).
$$
On the other hand, this complex of sheaves also calculates
$H^{\cdot}(K(U, n), {\cal F} )$. Thus we find that
$$
H^{n+i}(K(U,n), {\cal F} ) = Ext ^i_{\rm gp}(U,{\cal F} ), \;\; i<n.
$$
This holds true for any sheaves of groups $U$ and ${\cal F}$. This is one of the
motivating observations of Breen's paper \cite{Breen2}---we have repeated the
proof here for the reader's convenience.
Now suppose that $U$ and ${\cal F}$ are vector sheaves. If $U$ is a vector scheme
and ${\cal F}$ is a coherent sheaf then, via the above observation, the relative
Breen
calculations (Theorem \ref{bc}) show that $Ext ^i_{\rm gp}(U,{\cal F} )=0$ for $i>0$.
A coherent sheaf ${\cal F}$ is an injective object in the category of vector
sheaves, and the functor $Hom (\cdot , {\cal F} )$ is exact (cf the discussion of
vector sheaves in \cite{RelativeLie} for example). Thus if $U$ is any vector
sheaf, we can (locally on $S$) resolve it by vector schemes and apply the
previous paragraph. The functor $Ext ^0(\cdot , {\cal F} )= Hom (\cdot , {\cal F}
)$ is exact (recall from \cite{kobe} Lemma 3.2 or \cite{RelativeLie} Lemma
4.5 that morphisms of sheaves of abelian groups are the same as morphisms of
vector sheaves so the $Hom$ is the same in the two categories). Using this
exactness we get that $Ext ^i_{\rm gp}(U,{\cal F} )=0$ for $i> 0$. Finally, if $V$ is
any vector sheaf then we can resolve it by coherent sheaves, which is an
injective resolution in the category of vector sheaves. This is also an acyclic
resolution for $Ext$ in the category of sheaves of abelian groups, so we obtain
the isomorphism
$$
Ext^i_{\rm vs} (U,V)
\stackrel{\cong}{\rightarrow}Ext^i_{\rm gp} (U,V).
$$
The vanishing of the $Ext^i$ for $i>2$ comes from the fact that any vector
sheaf $V$ has a resolution of length $2$ (i.e. with terms in degrees $0,1,2$)
by coherent sheaves (cf \cite{kobe} \cite{RelativeLie}.
\hfill $\Box$\vspace{.1in}
We can apply \ref{ext} to the example discussed at the start of the appendix.
Suppose $S$ is a scheme and suppose $E^{\cdot}$ is a complex of vector bundles
on $S$. The cohomology sheaves $V^i= {\cal H}^i(E^{\cdot})$ are vector sheaves.
In general, a complex with given cohomology objects is determined by higher
extension classes in $Ext^i$ for all values of $i\geq 2$. However, by virtue of
the above theorem the $Ext ^i(V^j, V^k)$ vanish for $i\geq 3$. Thus the complex
$E^{\cdot}$ is determined completely by the successive extension classes
$$
\delta _{j,j+1} \in Ext ^2(V^{j+1}, V^j).
$$
The same is true for any complex of vector sheaves with $V^j$ as cohomology
objects.
{\bf Problem:} describe the conditions which must be satisfied by the classes
$\delta _{j,j+1}$ for the complex determined by these classes to be
(quasiisomorphic to) a complex of vector bundles.
\numero{APPENDIX II: Representability of very presentable shape}
The following result was stated without proof in
(\cite{kobe}, the discussion above Theorem 5.7). Since we refer anew to this
result in our discussion after Theorem \ref{calculation} of the present paper, I
felt it to be an opportune time to give a proof.
\begin{theorem}
\mylabel{representable1}
Suppose ${\cal F}$ is a connected $n$-stack on $Sch /{\bf C} $ such that
the cohomology sheaves $H^i({\cal F} , {\cal O} )$ are represented by finite dimensional
vector spaces.
Suppose furthermore that
$H^0({\cal F} , {\cal O} ) = {\cal O} $ and $H^1({\cal F} , {\cal O} )= 0$.
Then the $n+1$-functor
$$
T\mapsto Hom ({\cal F} , T)
$$
from $1$-connected very presentable $n$-stacks of groupoids $T$ to the same,
is representable by a morphism ${\cal F} \rightarrow \Sigma$, with $\Sigma$ being a
$1$-connected very presentable $n$-stack.
\end{theorem}
{\em Proof:}
It suffices to have a morphism ${\cal F} \rightarrow \Sigma$
which induces an isomorphism
$$
H^i(\Sigma ,{\cal O} )\stackrel{\cong}{\rightarrow}
H^i({\cal F} ,{\cal O} )
$$
for any $i$.
We say that a morphism ${\cal F} \rightarrow \Sigma _m$ is {\em $m$-arranged}
if the induced morphisms on cohomology with coefficients in ${\cal O}$ are
isomorphisms for $k\leq i < m$ and injective for $k\leq i = m$. Note that the
morphism ${\cal F} \rightarrow \ast$ is $1$-arranged because of the hypothesis that
$H^1({\cal F} , {\cal O} )=0$. The strategy of the proof (taken from E. Brown
\cite{EBrown}) will be to suppose that we have constructed ${\cal F} \rightarrow
\Sigma _m$ which is $m$-arranged. Then we will construct a factorization
$$
{\cal F} \rightarrow \Sigma _{m+1} \rightarrow \Sigma _m
$$
where the first morphism is $m+1$-arranged. By induction this suffices to prove
the theorem (we can stop as soon as we get to $m>n$).
So start with the situation of ${\cal F} \rightarrow \Sigma _m$, $m$-arranged,
$m\geq 1$. Let
$$
{\cal C} := Cone ({\cal F} \rightarrow \Sigma _m)
$$
so we have a map $\Sigma _m \rightarrow {\cal C}$ which restricted to ${\cal F}$ gives a
map homotopic to the basepoint $\ast \rightarrow {\cal C}$ (this basepoint is
included in the definition of $Cone$---it is the vertex of the cone over ${\cal F}$).
It is easy to see using the $m$-arrangedness
of our map, that the cohomology of ${\cal C}$ with coefficients in ${\cal O}$ vanishes in
degrees $\leq m$. Furthermore, the $m+1$-st cohomology fits into a long exact
sequence with those of ${\cal F} $ and $\Sigma _m$:
$$
0\rightarrow H^m(\Sigma _m,{\cal O} )\rightarrow H^m({\cal F} , {\cal O} )
$$
$$
\rightarrow H^{m+1}({\cal C} ,{\cal O} ) \rightarrow H^{m+1}(\Sigma _m, {\cal O} )
$$
$$
\rightarrow H^{m+1}({\cal F} , {\cal O} ) \rightarrow H^{m+2}({\cal C} , {\cal O} )
$$
$$
\rightarrow
H^{m+2}(\Sigma _m, {\cal O} )\rightarrow \ldots .
$$
The cohomology of $\Sigma _m$ with coefficients in ${\cal O}$ is a finite
dimensional vector space, by Theorem \ref{bc} (this case is contained in the
original characteristic $0$ version obtainable from \cite{Breen2}). The
property of being represented by a finite dimensional vector space is
closed under extensions, kernels and cokernels (\cite{kobe} Theorem 3.3
and \cite{RelativeLie} Corollary 4.10 and Theorem 4.11). Therefore the
cohomology
of ${\cal C}$ with coefficients in ${\cal O}$ is again a (sheaf represented by a) finite
dimensional vector space.
Now let
$$
W^{\ast} := H^{m+1} ({\cal C} , {\cal O} )
$$
define the finite dimensional vector space $W$. We get a morphism
${\cal C} \rightarrow K(W , m+1)$ which is universal for morphisms to $K(U, m+1)$
with $U$ a finite dimensional vector space.
In particular it induces an isomorphism
$$
H^{m+1}(K(W, m+1), {\cal O} )\stackrel{\cong}{\rightarrow} H^{m+1}({\cal C} , {\cal O} ).
$$
We will compare the previous long exact sequence with the long exact
sequence that occurs at the start of the Leray-Serre spectral sequence for the
morphism $p:\Sigma _m\rightarrow K(W, m+1)$.
Set
$$
Fib:=Fib(\Sigma _m \rightarrow K(W , m+1) ).
$$
Note that we have a morphism ${\cal F} \rightarrow Fib$.
The higher direct images
occuring in the Leray-Serre spectral sequence for $p$ are constant local systems
over the base, because $K(W,m+1)$ is $1$-connected.
In other words,
$$
R^ip_{\ast} {\cal O} = H^i(Fib ,{\cal O} ) \times K(W,m+1)
\rightarrow K(W,m+1) .
$$
Do a standard type of spectral sequence argument. First of all, for
$k<m$ we prove by induction on $k$ that for all $i\leq k$, the
$H^i(Fib,{\cal O} ) $ are finite dimensional vector
spaces. Suppose we know this for $k-1$. Then in view of the vanishing of the
cohomology of $K(W, m+1)$ with vector space coefficients (the sheaves
represented
by vector spaces are ${\cal O} ^a$) the terms $E^{i,j}_2$ with $1\leq i\leq m$ and
$j<k$ vanish; whereas for $j=0$ we have
$$
E^{i,0}_2 = H^i(Fib, {\cal O} )
$$
because of the fact that $K(W,m+1)$ is $1$-connected. Therefore the terms
$E^{k,0}_2$ persist to $E_{\infty}$ and we have
$$
H^k(Fib, {\cal O} ) = H^k(\Sigma _m , {\cal O} )= H^i({\cal F} , {\cal O} ).
$$
In view of the hypothesis of the theorem, this proves the induction step of the
first part of the argument. Incidentally we get that the morphism
${\cal F} \rightarrow Fib$ induces an isomorphism on cohomology with coefficients in
${\cal O}$ in degrees $k< m$. Now look at the term $E^{m,0}_2= H^m(Fib, {\cal O} )$. The
only differential concerning it is
$$
d_{m+1}: H^m(Fib , {\cal O} )\rightarrow H^{m+1}(K(W,m+1) , {\cal O} )
$$
(noting that we already have $H^0(Fib , {\cal O} ) = {\cal O} $).
From our hypothesis which implies that $H^1(Fib, {\cal O} )=0$ we get, similarly,
that the only differential concerning the term $E^{m+1,0}_2$ is
$$
d_{m+2}: H^{m+1}(Fib , {\cal O} )\rightarrow H^{m+2}(K(W,m+1) , {\cal O} ).
$$
From these and the fact that the spectral sequence abuts to the cohomology of
$\Sigma _m$, we get the long exact sequence
$$
0\rightarrow H^m(\Sigma _m,{\cal O} )\rightarrow H^m(Fib, {\cal O} )
$$
$$
\rightarrow H^{m+1}(K(W,m+1) ,{\cal O} ) \rightarrow H^{m+1}(\Sigma _m, {\cal O} )
\rightarrow
$$
$$
H^{m+1}(Fib , {\cal O} ) \rightarrow H^{m+2}(K(W,m+1) , {\cal O} )\rightarrow
H^{m+2}(\Sigma _m, {\cal O} ).
$$
Remark that $H^{m+2}(K(W, m+1),{\cal O} )=0$---this comes from Theorem \ref{bc}
and it is here where we use $m\geq 1$.
In particular, the morphism
$$
H^{m+2}(K(W, m+1), {\cal O} )\rightarrow H^{m+2}({\cal C} ,{\cal O} )
$$
is injective for the trivial reason that the left side is $0$.
Recall that the same induced morphism in degree $m+1$ was an isomorphism
(by the construction of $W$).
Therefore, comparing with the previous long exact sequence and
using the $5$-lemma, we get that the morphism
$$
{\cal F} \rightarrow Fib
$$
induces isomorphisms on cohomology in degrees $\leq m$ and an injection in
degree $m+1$. In other words this morphism is $m+1$-arranged. Thus we can set
$$
\Sigma _{m+1}:= Fib
$$
and we have completed our inductive construction to prove the theorem.
\hfill $\Box$\vspace{.1in}
{\em Definition:} If ${\cal F}$ satisfies the condition of
Theorem \ref{representable1} then we obtain the representing $1$-connected
very presentable $\Sigma ({\cal F} )$ with universal morphism
$$
{\cal F} \rightarrow \Sigma ({\cal F} ).
$$
We define
(for any basepoint $f:Y\rightarrow {\cal F}$)
$$
\pi ^{\rm vp}_i({\cal F} \times Y/Y, f):= \pi _i(\Sigma \times Y/Y, f).
$$
In the latter case we usually just take a basepoint $f\in {\cal F} (Spec \, {\bf C} )$
and then denote this by $\pi ^{\rm vp}_i({\cal F} , f)$.
\begin{theorem}
\mylabel{fibration}
Suppose ${\cal F}$ and ${\cal G}$ are $n$-stacks with basepoint $g\in {\cal G} (Spec
({\bf C} )$, which satisfy the criterion of Theorem \ref{representable1} so their
shapes are representable. Suppose
$$
f:{\cal F} \rightarrow {\cal G}
$$
is a morphism of $n$-stacks with the following property (we denote by ${\cal H}$ the
fiber over $g$): the local systems $R^if_{\ast} ({\cal O} )$ are
isomorphic to ${\cal O} ^{a_i}$ on ${\cal G}$, and that the morphisms
$$
R^if_{\ast}({\cal O} ) |_g \rightarrow H^i({\cal H} , {\cal O} )
$$
are isomorphisms. Suppose that
$$
H^1({\cal F}, {\cal O} ) =H^1({\cal G} ,{\cal O} ) = H^1({\cal H} ,{\cal O} )={\cal O}
$$
and
$$
H^1({\cal F}, {\cal O} ) =H^1({\cal G} ,{\cal O} ) = H^1({\cal H} ,{\cal O} )=0.
$$
Then we have a fiber sequence for the representing objects
$$
\Sigma ({\cal H} )\rightarrow \Sigma ({\cal F} )\rightarrow \Sigma ({\cal G} ).
$$
\end{theorem}
{\em Proof:}
The Leray spectral sequence for $f$ is
$$
H^i({\cal G} , R^jf_{\ast}({\cal O} ))\Rightarrow H^{i+j}({\cal F} , {\cal O} ).
$$
In view of the hypothesis, this becomes
$$
H^i({\cal G} , {\cal O} )\otimes _{ {\cal O} } H^j({\cal H} , {\cal O} )\Rightarrow H^{i+j}({\cal F} , {\cal O} ).
$$
On the other hand, we obtain a morphism of representing shapes
$$
\Sigma ({\cal F} )\rightarrow \Sigma ({\cal G} )
$$
(by the universal property of $\Sigma ({\cal F} )$). Let $Fib$ denote the fiber of
$\Sigma ({\cal F} )$ (over the image of the point $g$).
Note that the $R^i\Sigma (f)_{\ast} ({\cal O} )$ are constant on $\Sigma
({\cal G} )$ because $\Sigma ({\cal G} )$ is $1$-connected. In particular
$$
R^if_{\ast}({\cal O} ) |_g \stackrel{\cong}{\rightarrow} H^i(Fib , {\cal O} ).
$$
We obtain the spectral sequence
$$
H^i(\Sigma ({\cal G} ) , {\cal O} )\otimes _{{\cal O}} H^j(Fib , {\cal O} )
\Rightarrow H^{i+j}
(\Sigma ({\cal F} ), {\cal O} ).
$$
Note that the $\pi _i(Fib )$ are finite dimensional vector spaces (using the
long exact sequence of homotopy groups of a fibration, and \cite{RelativeLie}
Theorem 4.11 applied to the case of trivial base $S=\ast $). Thus
$H^j(Fib , {\cal O} )$ are finite dimensional vector spaces.
The composition
$$
{\cal H} \rightarrow \Sigma ({\cal F} )\rightarrow \Sigma ({\cal G} )
$$
is homotopic to the constant map at the basepoint, so we get a map
$$
{\cal H} \rightarrow Fib.
$$
This gives maps $H^i(Fib, {\cal O} )\rightarrow H^i({\cal H} , {\cal O} )$. We claim that
these
are isomorphisms, which would imply that ${\cal H} \rightarrow Fib$ is a map
representing the very presentable shape of ${\cal H}$, in other words $\Sigma ({\cal H}
)\cong Fib$, thus giving the desired result.
To prove the claim, note that the maps in question are compatible via the
previous identifications, with the maps
$$
R^i\Sigma ({\cal F} )_{\ast} {\cal O} \rightarrow R^if_{\ast} ({\cal O} ).
$$
These in turn fit into a morphism of Leray spectral sequences. We show using
a spectral sequence argument, by induction on $k$, that for all $i\leq k$ we
have
$$
H^i(Fib, {\cal O} )\stackrel{\cong}{\rightarrow} H^i({\cal H} , {\cal O} )
$$
or equivalently
$$
R^i\Sigma ({\cal F} )_{\ast} {\cal O} \stackrel{\cong}{\rightarrow} R^if_{\ast} ({\cal O} ).
$$
Suppose this is true for $k-1$. Then look at the term
$E^{0,k}_2 = H^k({\cal H} , {\cal O} )$. When we look at the $r$th differential
$$
d^{0,k}_r:E^{0,k}_r\rightarrow E^{r,k+1-r}_r
$$
the term $E^{r,k+1-r}_r$ has not yet been touched by any term $E^{i,j}$ with
$j\geq k$, and after this differential, the term $E^{r,k+1-r}$ is no longer
touched by any further differentials. We have a morphism of spectral sequences
(the above remarks apply to both) which induces an isomorphism on the
abuttments. It follows that the morphism between spectral sequences induces an
isomorphism on images of $d^{0,k}_r$. Furthermore, the morphism induces an
isomorphism on the intersection (for all $r$) of the kernels of the $d^{0,k}_r$.
This implies that the morphism induces an isomorphism on $E^{0,k}_2$ and we
obtain the inductive step for $k$. This proves the claim and hence the theorem.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{complexify}
Suppose $Y$ is a simply connected finite CW complex. Let ${\cal F} $
be the constant $n$-stack associated to the constant prestack with values
$Y$ (or
more precisely, with values the $n$-type $\tau _{\leq n}Y$). Fix a basepoint
$y\in Y$ which also gives a basepoint section of ${\cal F}$. Let ${\cal F} \rightarrow
\Sigma$ be the morphism representing the shape of ${\cal F}$. The morphisms
induced by $Y\rightarrow \Sigma (Spec {\bf C} )$,
$$
\pi _i(Y,y)\rightarrow \pi _i(\Sigma , y)
$$
induce isomorphisms
$$
\pi _i(Y,y)\otimes _{{\bf Z}} {\cal O} \cong \pi _i(\Sigma , y).
$$
\end{corollary}
{\em Proof:}
Using the previous theorem, we can reduce by the Postnikov tower to the case $Y=
K(A, n)$ for a finitely generated abelian group $A$. Then the Breen
calculations imply that the morphism
$$
K(A, n)\rightarrow K(A\otimes _{{\bf Z}} {\cal O} , n)
$$
induces an isomorphism on cohomology with coefficients in ${\cal O}$. This implies
that
$$
\Sigma (K(A,n))= K(A\otimes _{{\bf Z}}{\cal O} , n),
$$
which gives the statement of the corollary.
\hfill $\Box$\vspace{.1in}
{\bf Definition:} Fix $n$. If $Y$ is a $1$-connected finite CW complex, then we
define the {\em complexification of $Y$} denoted $Y\otimes {\bf C}$ to be the
$n$-stack $\Sigma ({\cal F} )$ representing the very presentable shape of the
constant
$n$-stack ${\cal F}$ with values $\tau _{\leq n}Y$. Note that this notion depends
on $n$ because we have chosen not to treat the questions arising if we try to
take $n=\infty$.
{\bf Example:}
If we apply this to $Y=S^2$ then we
obtain $\Sigma = S^2\otimes {\bf C} $ as defined in \S 6 above. This is
easy to see because, using the previous theorem, the homotopy sheaves of
$\Sigma$ are ${\cal O}$ in degrees $2$ and $3$; then there are only two
possibilities for $\Sigma$ and they are distinguished by the vanishing or
nonvanishing of the Whitehead product. As the Whitehead product is nonzero for
$S^2$ and the isomorphisms of the previous theorem are compatible with the
Whitehead product (exercise), this implies that the Whitehead product for
$\Sigma$ is nontrivial, therefore $\Sigma$ is equal to the $T$ defined in \S 6.
We propose the above results as a way of interpreting what it means to look
at the ``complexified homotopy type of a space $Y$''. We could do the same
thing over the ground field ${\bf Q}$, and then we propose that this is what it
means to look at the ``rational homotopy type'' of $Y$. This notion is
preserved by base extension of the ground field.
Of course this should all be related to the usual definitions of Quillen,
Sullivan, Morgan, Hain et.al. which refer (excepting Quillen) to algebras of
differential forms. In those theories, base extension is obtained by tensoring
the algebra of forms with the field extension. It has always been somewhat
unclear what geometric interpretation to put on this base-extension process,
and we propose the above theory as a way of obtaining a reasonable
interpretation. We don't, however, get into details of the
relationship between the above theory and the differential-forms theories.
One advantage of the present formulation is that it explains what is going on
in the non-simply connected case: the shape of the constant
sheaf ${\cal F}=Const(Y)$ is no longer representable by a very presentable object
(except in fairly restricted cases such as finite fundamental group). Thus, the
object which carries the ``rational homotopy'' information of $Y$ is the shape
itself, rather than the representing object which may not exist. The shape, i.e.
the functor
$$
T\mapsto Hom ({\cal F} , T)
$$
exists even when $Y$ is not simply connected.
\subnumero{Proof of Theorem \ref{representable0}}
The statement of Theorem \ref{representable0} from \S 2 is very slightly
different from the statement \ref{representable1} given above. We indicate
here how to get \ref{representable0}. Suppose that ${\cal F}$ is an $n$-stack on
$Sch /{\bf C}$ such that for any affine algebraic group $G$,
$$
K(G,1)\stackrel{\cong}{\rightarrow} Hom ({\cal F} , K(G,1)).
$$
In particular this implies that $H^0({\cal F} , {\cal O} )={\cal O} $ and $H^1({\cal F} , {\cal O} )=0$.
With the hypothesis that $H^i({\cal F} , {\cal O} )$ are represented by finite
dimensional vector spaces, we can apply Theorem \ref{representable1} to get a
morphism
$$
{\cal F} \rightarrow \Sigma ({\cal F} )
$$
universal for morphisms to $1$-connected very prepresentable $T$. Note that
$\Sigma ({\cal F} )$ is $1$-connected. We have to show that it is also universal for
morphisms to $0$-connected very presentable $T$; suppose that $T$ is one such.
We may choose a basepoint $t$, and let $G= \pi _1(T,t)$ (which is an affine
algebraic group). We have a fiber sequence
$$
T'\rightarrow T \rightarrow K(G,1).
$$
This gives a diagram
$$
\begin{array}{ccccc}
Hom (\Sigma ({\cal F} ), T')& \rightarrow &Hom (\Sigma ({\cal F} ), T)&\rightarrow &Hom
(\Sigma ({\cal F} ), K(G,1))\\
\downarrow & & \downarrow && \downarrow \\
Hom ({\cal F} , T')&\rightarrow & Hom ({\cal F} ,T) & \rightarrow & Hom ({\cal F} , K(G,1)),
\end{array}
$$
where the horizontal sequences are fiber sequences. Since $\Sigma ({\cal F} )$ is
$1$-connected we have $$
K(G, 1) \stackrel{\cong}{\rightarrow } Hom (\Sigma ({\cal F} ), K(G,1)),
$$
and the same holds for ${\cal F}$ by hypothesis. Therefore the vertical map on the
right is an equivalence between $K(G,1)$ and our diagram becomes
$$
\begin{array}{ccccc}
Hom (\Sigma ({\cal F} ), T')& \rightarrow &Hom (\Sigma ({\cal F} ), T)&\rightarrow
&K(G,1) \\
\downarrow & & \downarrow && \downarrow =\\
Hom ({\cal F} , T')&\rightarrow & Hom ({\cal F} ,T) & \rightarrow & K(G,1).
\end{array}
$$
Now note that $T'$ is a $1$-connected very presentable $n$-stack, so the
vertical arrow on the left is an equivalence. Since the base $K(G,1)$ is
$0$-connected, we can use the long exact sequences of homotopy for these
fibrations to conclude that the vertical morphism in the middle is an
equivalence. This is what we needed to know to establish the universal property
of ${\cal F} \rightarrow \Sigma ({\cal F} )$ for Theorem \ref{representable0}.
\hfill $\Box$\vspace{.1in}
\subnumero{A relative version}
While we are on the subject, we give a relative version of Theorem
\ref{representable1}.
Recall \cite{kobe} \cite{RelativeLie} that if $Y$ is a scheme then a
$1$-connected $n$-stack ${\cal F} \rightarrow Y$ (which can also be thought of as a
$1$-connected $n$-stack on the site $Sch /Y$ of schemes over $Y$) is said to
be {\em very presentable} if for any basepoint section $f: Y'\rightarrow {\cal F}$
for a scheme $Y'\rightarrow Y'$,
the homotopy group sheaves $\pi _i({\cal F} |_{Y'}, f)$ are vector sheaves on $Y'$.
Since (for the present discussion) we have assumed ${\cal F}$ to be relatively
$1$-connected, the homotopy group sheaves don't depend on the choice of
basepoint (indeed, the choice of basepoint is locally unique up
to homotopy which itself is unique up to---nonunique---homotopy).
Therefore they descend to sheaves of abelian groups $\pi _i({\cal F} /Y)$ on $Y$.
For ${\cal F}$ to be very presentable, it is equivalent to require that these be
vector sheaves.
We introduce the following terminology. We say that a covariant endofunctor $F$
from the category of vector sheaves on $Y$ to itself, is {\em anchored} if
the natural map
$$
F(U)\rightarrow Hom (Hom (F({\cal O} ),{\cal O} ), U)
$$
is an isomorphism for any coherent sheaf $U$ (recall that the coherent sheaves
are the injective objects in the category of vector sheaves). The above natural
map comes from the trilinear map
$$
F(U)\times Hom (F({\cal O} ), {\cal O} ) \times Hom (U,{\cal O} ) \rightarrow {\cal O}
$$
defined by $(f,g,h)\mapsto g( F(h)(f))$.
\begin{lemma}
\mylabel{anchored1}
(A) If
$$
0\rightarrow F' \rightarrow F \rightarrow F'' \rightarrow 0
$$
is a short exact sequence of natural transformations between covariant
endofuncturs on the category of vector sheaves over $Y$, then if any two of the
three endofunctors is anchored, so is the third.
\newline
(B) If $F$ is an anchored endofunctor which is also left exact, then $F$ is
representable $F(V)= Hom (W, V)$ for a vector sheaf $W=Hom (F({\cal O} ), {\cal O} )$.
\end{lemma}
{\em Proof:}
(A) follows from the $5$-lemma. For (B) suppose $F$ is a left exact anchored
endofunctor.
Set $W:= Hom (F({\cal O} ), {\cal O} )$. The natural map $F (U)\rightarrow Hom (W, U)$
is an isomorphism for coherent sheaves $U$. On the other hand, both sides are
left exact in $U$. Suppose
$$
0\rightarrow U \rightarrow U' \rightarrow U''
$$
is an exact sequence with $U'$ and $U''$ being coherent sheaves. Then we obtain
exact sequences
$$
0\rightarrow F(U) \rightarrow F(U' )\rightarrow F(U'')
$$
and
$$
0\rightarrow Hom(W,U) \rightarrow Hom(W,U' )\rightarrow Hom(W,U''),
$$
and our natural map is a morphism between these exact sequences inducing
isomorphisms on the last two terms. Thus $F(U)\rightarrow Hom (W,U)$ is an
isomorphism. This completes the proof in view of the fact that (locally on $S$)
any vector sheaf $U$ fits into such a short exact sequence.
\hfill $\Box$\vspace{.1in}
\begin{lemma}
\mylabel{anchored2}
Suppose $V$ is a vector sheaf. Then the endofunctor on the category of vector
sheaves defined by
$$
U\mapsto H^i(K(V,m), U)
$$
is anchored.
\end{lemma}
{\em Proof:}
This follows immediately from Theorem \ref{bc} if $V$ is a vector scheme.
Now suppose that we have an exact sequence
$$
0\rightarrow V' \rightarrow V'' \rightarrow V \rightarrow 0
$$
where $V''$ is a vector scheme, and where we know the lemma for $V'$.
This gives a fibration sequence
$$
K(V'' , m)\rightarrow K(V,m)\rightarrow K(V', m+1),
$$
and taking the cohomology with coefficients in a coherent sheaf $U$ leads to a
Leray spectral sequence
$$
H^i(K(V', m+1), H^j(K(V'', m),U))\Rightarrow H^{i+j}(K(V,m), U).
$$
The cohomology
of the fiber are again coherent sheaves by Theorem \ref{bc}, so by the lemma for
$V'$ the natural map occuring in the definition of ``anchored'' induces an
isomorphism on the $E_2$ terms of the spectral sequence. Since the property of
being anchored is preserved by kernels, cokernels and extensions, we get that
the cohomology of $K(V,m)$ is anchored.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{anchored3}
Suppose $T$ is a relatively $1$-connected very presentable $n$-stack over
a scheme $S$. Then the endofunctor
$$
U\mapsto H^i(T/S, U)
$$
is anchored.
\end{corollary}
{\em Proof:}
Decompose $T$ into a Postnikov tower where the pieces are of the form $K(V,m)$
for vector sheaves $V$
\hfill $\Box$\vspace{.1in}
\begin{theorem}
\mylabel{representable2}
Suppose $S$ is a scheme and ${\cal F} \rightarrow S$ is a morphism of $n$-stacks on
$Sch /{\bf C}$. Suppose that for any vector sheaf $V$ over $S$, the cohomology
$H^i({\cal F} /S, V)$
is again a vector sheaf over $S$. Suppose furthermore that
$H^0({\cal F} /S, V) = V$ and $H^1({\cal F} /S, V)= 0$ for any vector sheaf $V$.
Finally suppose that the functors $V\mapsto H^i({\cal F} /S, V)$ are anchored.
Then
the functor
$$
T\mapsto Hom ({\cal F} /S , T/S )
$$
from relatively $1$-connected very presentable $n$-stacks of
groupoids $T\rightarrow S$ to the same, is represented by a morphism
${\cal F} \rightarrow \Sigma$ over $S$, with $\Sigma \rightarrow S$
being a relatively $1$-connected and very presentable $n$-stack over $S$.
\end{theorem}
{\em Proof:}
Follow the same outline as for the proof of Theorem \ref{representable1}.
We try to find a relatively $1$-connected very presentable $\Sigma
\rightarrow Y$ with a morphism
$$
{\cal F} \rightarrow \Sigma
$$
inducing an isomorphism on cohomology with coefficients in any coherent sheaf
$U$ on $S$ (the isomorphism for coefficients in any vector sheaf $U$ then follows
by resolving $U$ by coherent sheaves).
We say that a morphism ${\cal F} \rightarrow \Sigma _m$ is {\em $m$-arranged}
if the induced morphisms on cohomology with coefficients in any coherent sheaf
$U$ on $S$ are isomorphisms for $k\leq i < m$ and injective for $k\leq i = m$.
Note that the morphism ${\cal F} \rightarrow S$ is $1$-arranged because of the
hypothesis that $H^1({\cal F} /S , U )=0$.
Thus we may take $\Sigma _1 := S$.
The strategy of the proof will be to
suppose for some $m\geq 1$ that we have constructed ${\cal F} \rightarrow \Sigma _m$
which is $m$-arranged. Then we will construct a factorization
$$
{\cal F} \rightarrow \Sigma _{m+1} \rightarrow \Sigma _m
$$
where the first morphism is $m+1$-arranged. By induction this suffices to prove
the theorem.
Make the same constructions, using the same notations (which we won't repeat
here) as in the proof of Theorem \ref{representable1}. Along the way,
replace the
cohomology with coefficients in ${\cal O}$ (and the higher direct images of ${\cal O}$
etc.) by cohomology with coefficients in any coherent sheaf $U$ on $S$.
We obtain the first long exact sequence (actually valid for any vector sheaf
$U$ as coefficients)
$$
0\rightarrow H^m(\Sigma _m/S,U )\rightarrow H^m({\cal F} /S, U )
$$
$$
\rightarrow H^{m+1}({\cal C} /S,U) \rightarrow H^{m+1}(\Sigma _m/S, U)
$$
$$
\rightarrow H^{m+1}({\cal F} /S, U ) \rightarrow H^{m+2}({\cal C} /S, U )
$$
$$
\rightarrow
H^{m+2}(\Sigma _m/S, U )\rightarrow \ldots .
$$
The cohomology of $\Sigma _m$ with coefficients in a vector sheaf is again a
vector sheaf, by Corollary \ref{cohVPisVS} above. The property of being
represented by a finite dimensional vector space is closed under extensions,
kernels and cokernels (\cite{kobe} Theorem 3.3 and \cite{RelativeLie} Corollary
4.10 and Theorem 4.11). Therefore the cohomology of ${\cal C}$ with coefficients in a
vector sheaf $U$ is again a vector sheaf.
When we come to the construction of $W$ we need to say something
more---this is the reason for introducing the notion of ``anchored'' above.
The functor
$$
U\mapsto H^{m+1}({\cal C} , U)
$$
is anchored. This comes from the facts that the cohomology of $\Sigma _m$ is
anchored by Corollary \ref{anchored3}, that the cohomology of ${\cal F}$ is anchored
by hypothesis, and the fact that being anchored is preserved by kernels,
cokernels and extensions (Lemma \ref{anchored1}).
On the other hand, the fact that the cohomology of ${\cal C}$ vanishes in degrees
$0<i\leq m$ (note that $m\geq 1$) implies that the above functor is left-exact.
Therefore by Lemma \ref{anchored1} (B) it is representable by a vector
sheaf $W$:
we have
$$
H^{m+1}({\cal C} , U) = Hom (W, U).
$$
In particular there is a tautological class in $H^{m+1}({\cal C} , W)$
corresponding to a morphism ${\cal C} \rightarrow K(W, m+1)$, and this morphism is
universal for morphisms from ${\cal C}$ to things of the form $K(U,m+1)$.
In particular it induces an isomorphism
$$
H^{m+1}(K(W, m+1), U )\stackrel{\cong}{\rightarrow} H^{m+1}({\cal C} , U ).
$$
Again set
$$
Fib:=Fib(\Sigma _m \rightarrow K(W , m+1) ).
$$
Note that we have a morphism ${\cal F} \rightarrow Fib$.
Compare the first long
exact sequence with the long exact sequence that occurs at the start of the
Leray-Serre spectral sequence for the morphism $p:\Sigma _m\rightarrow K(W,
m+1)$, using the same argument as in the proof of Theorem
\ref{representable1}. We need to know
that the morphism
$$
H^{m+2}(K(W, m+1), U )\rightarrow H^{m+2}({\cal C} ,U )
$$
is injective for cohomology with coefficients in a coherent sheaf $U$ (recall
that only coherent sheaves occur as coefficients for the cohomology in the
definition of arrangedness---one goes back to the general case after the
induction on $m$ is finished). To prove this we again show that
$H^{m+2}(K(W, m+1), U )=0$ (note that this wouldn't be true if $U$ were not a
coherent sheaf and that is the reason why we restrict to coherent sheaves in
the definition of arrangedness).
In fact, using that $m\geq 1$ and following the argument of
\ref{representable1} we get
that
$$
H^{m+2}(K(W, m+1), U )=Ext ^1(W, U).
$$
However, a coherent sheaf $U$ is an injective object in the category of vector
sheaves \cite{RelativeLie} Lemma 4.17, so $Ext ^1(W, U)=0$. This gives a
proof of
the desired statement. Alternatively one can obtain a proof using a spectral
sequence argument with a resolution
$$
0\rightarrow V \rightarrow V' \rightarrow V'' \rightarrow W \rightarrow 0
$$
of $W$ by vector schemes (decompose this into two short exact sequences and use
a Leray spectral sequence argument for each of the corresponding fibration
sequences).
After that the rest of the argument works exactly the same
way as in Theorem \ref{representable1} (calling upon Theorem \ref{bc} in the
relative case as necessary). We don't repeat this.
\hfill $\Box$\vspace{.1in}
|
1997-11-30T02:08:36 | 9712 | alg-geom/9712001 | en | https://arxiv.org/abs/alg-geom/9712001 | [
"alg-geom",
"math.AG"
] | alg-geom/9712001 | Richard Mayer | Richard Mayer | Coupled Contact Systems and Rigidity of Maximal Dimensional Variations
of Hodge Structure | 24 pages, latex2e uses amsart.cls | null | null | null | null | In this article we prove that locally Griffiths' horizontal distribution on
the period domain is given by a generalized version of the familiar contact
differential system. As a consequence of this description we obtain strong
local rigidity properties of maximal dimensional variations of Hodge structure.
For example, we prove that if the weight is odd then there is a unique germ of
maximal dimensional variation of Hodge stucture though every point of the
period domain. Similar results hold if the weight is even with the exception of
one case.
| [
{
"version": "v1",
"created": "Sun, 30 Nov 1997 01:08:37 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Mayer",
"Richard",
""
]
] | alg-geom | \section{Introduction}
Variations of Hodge structure are integral manifolds of Griffiths' horizontal distribution on the period domain \(D\) of a fixed integral lattice which are stable under the action of a discrete subgroup of the isometry group of \(D\). This distribution is defined in terms of the filtrations \(\{F^p\}\) by
\[
\frac{\partial F^p}{\partial z_i}\subset F^{p-1}.
\]
The horizontal distribution is not completely integrable in general if the weight is greater than one, so integral manifolds must have lower dimension than the distribution. There are two fundamental questions for not completely integrable differential systems:
\begin{itemize}
\item What is the maximal dimension an integral manifold can have? \\
\item What is the structure of the integral manifolds that attain the dimension bound?
\end{itemize}
We shall examine these two questions for the horizontal system in detail. The tool that makes this possible is the fact that the horizontal differential system is locally given by coupled matrix valued contact systems. (See Section 4.) We derive two consequences of this description. The first one is a new proof of the main result of \cite{C-K-T} which says that there are explicit quadratic functions \(q^{even}_1, q^{even}_2, q^{even}_3\) (weight even) and \(q^{odd}_1, q^{odd}_2\) (weight odd) of the Hodge numbers \(h^{i,j}\) which give sharp upper bounds for the dimension of a variation of Hodge structure.
(Theorem~\ref{dimboundthm}). In fact, we cover a case which is missing from \cite{C-K-T}, this will be explained in Section 4. The second consequence is the main result of this work which generalizes Carlson's rigidity theorem for weight two variations (cf. \cite{C1}) to arbitrary weights. The precise statement is as follows:
\begin{thm}\label{mainthm}
Let \(D\) denote the period domain and let \(w\) be the weight.
\begin{enumerate}
\item Assume that one of the following holds:
\begin{enumerate}
\item \(w=2k+1\), \(h^{k,k+1}>2\) and all the other Hodge numbers are greater than one
\item \(w=2k\), all the Hodge numbers are greater than one and the maximum dimension is \(q^{even}_1\)
\end{enumerate}
then there is a unique maximal dimensional germ of variations of Hodge structure through each point of \(D\), i.e., there is a unique maximal dimensional foliation the leaves of which are the maximal dimensional variations of Hodge structure.
\item Let \(S_1, S_2\subset D\) be two maximal dimensional variations of Hodge structure. Assume that one of the following holds:
\begin{enumerate}
\item \(w=2k\), \(dim(S_1)=dim(S_2)=q^{even}_2\), \(h^{k+1}>2\), the other Hodge numbers are greater than one and \(h^k\geq 4\) is even
\item \(w=2k\), \(dim(S_1)=dim(S_2)=q^{even}_3\) and \(h^{k+1}>2\)
\end{enumerate}
then there is an element \(g\in Aut(D)\) such that \(g\cdot S_1=S_2\) holds in a neighborhood.
\item Let \(w=2k\) and let \(E\) denote a maximal dimensional integral element. If \(dim(E)=q^{even}_2\) and \(h^k\) is odd then there is an infinite dimensional family of germs of maximal dimensional integral manifolds tangent to \(E\), i.e. flexibility holds.
\end{enumerate}
\end{thm}
In the remaining special cases when the Hodge numbers are small (i.e., they do not satisfy the requirements of the theorem) rigidity does not hold. In fact, in these cases we have the behavior of the classical contact system explained in Section~\ref{contactsection}.
The outline of the paper is as follows. In Section 2 we give the necessary definitions of exterior differential systems and treat the case of the classical contact system. In Section 3 we introduce local coordinate systems on the period domain. Using these coordinates we show in Section 4 that the horizontal system is locally equivalent to a system given by analogs of the classical contact system. Analyzing this new system leads to the proof of the main theorem in Section 5.
The author would like to thank his advisor James A. Carlson for suggesting this problem to him and for the support throughout his graduate studies.
\section{Exterior Differential Systems}
The main reference for this section is \cite{B}. Let \(X\) be a manifold, let \(T^*\) denote the cotangent sheaf of \(X\) and \(\bigwedge ^* T^*\) the associated deRham algebra. We will be concerned with complex manifolds and the holomorphic cotangent sheaf, but most of the results remain true even if we consider \(C^\infty \) real manifolds.
\subsection{Integral Elements and Manifolds}
\begin{defi}
A differential system is defined by an ideal \(\mathcal{I} \subset \bigwedge ^* T^*\) which is closed under exterior differentiation, i.e., \(d\mathcal{I}\subset\mathcal{I}\). An integral manifold of \(\mathcal{I}\) is a holomorphic mapping \(i\):~ \(S \longrightarrow X\) such that \(i^*\omega = 0\) for each germ \(\omega\) of \(\mathcal{I}\).
\end{defi}
Often we will take \(i\) to be the inclusion and we will talk about the {\it germ of an integral manifolds} by which we will mean an equivalence class of integral manifolds where two integral manifolds are equivalent if their images agree in a neighborhood of a point.
Let \(i\):~ \(S \longrightarrow X\) be an integral manifold of \(\mathcal{I}\). If \(s \in S \) and \(E = T_s S \subset T_s M\) is the tangent space to \(S\) at \(s\) then for each \(\omega \in \mathcal{I}\) \(\omega_E = 0\) where \(\omega_E\) denotes the restriction of \(\omega\) to \(E\). It is clear that the vanishing of \(i^*\omega\) at each point depends only on the tangent space of \(S\). This leads to the following definition.
\begin{defi}
A linear subspace \(E \subset M\) is an integral element of \(\mathcal{I}\) if \(\omega_E = 0\) for each \(\omega \in \mathcal{I}\).
\end{defi}
Note that in general an integral element is not necessarily tangent to an integral manifold, see \cite{B} for examples.
Suppose now that \(\mathcal{I}\) is generated algebraically by the
1-forms \(\{\omega^1, \ldots, \omega^{n-k}\}\) where \(n\) is the
dimension of \(M\) and let \({\mathcal{D}}_{v} \subset T_{v} M\) denote
the linear space of tangent vectors that are annihilated by \(\mathcal{I}\). If \({\mathcal{D}}_{v}\) has the same dimension for each \(v\in M\) then these subspaces define a {\it distribution} in the tangent bundle. The dimension of the distribution \(\mathcal{D}\) is the dimension of \({\mathcal{D}}_{v}\). From now on we will consider differential systems that are defined by 1-forms of this type and we will use the words differential system and distribution interchangeably.
The condition that \(\mathcal{I}\) is closed means that \(d\omega^i\) is
given as an algebraic combination of \(\{\omega^1, \ldots,
\omega^{n-k}\}\) for each \(i\). This condition \((F)\) is called the
{\it Frobenius condition}. It is not hard to see (cf. \cite{W}
Proposition 2.30) that the Frobenius condition for differential forms is equivalent to the following for a distribution \(\mathcal{D}\): for any two holomorphic vector fields \(X\) and \(Y\) lying in \(\mathcal{D}\), the Lie bracket [\(X,Y\)] also lies in \(\mathcal{D}\). The fundamental integrability result for such systems is the following theorem of Frobenius.
\begin{thm}[Frobenius]\label{frobenius}
Let \(\mathcal{I}\) be a differential ideal generated algebraically by the linearly independent 1-forms \(\{\omega^1, \ldots, \omega^{n-k}\}\). There is a local coordinate system \((y^1,\ldots ,y^n)\) at each point of \(M\) such that \(\mathcal{I}\) is generated by \(\{dy^{k+1}, \ldots, dy^{n}\}\) if and only if the condition \((F)\) holds.
\end{thm}
If such coordinate system exists then clearly there is a \(k\)-dimensional integral manifold through each point of \(M\). For this reason such differential systems are called {\it completely integrable}. In this work we will be interested in differential systems that are not completely integrable. If this is the case then the maximal possible dimension of an integral manifold must be strictly less than the dimension of the distribution. It is an interesting question to determine this dimension.
\subsection{Contact Differential Systems}\label{contactsection}
Let us start with a differential system defined by a single 1-form.
\begin{defi}
Let \(M=\C^{2n+1}\) with coordinates \((x_1,\ldots,x_n,y_1,\ldots,y_n,z)\).
The differential system whose differential ideal is generated by the 1-form
\[ \omega = dz - \sum_{i=1}^{n}x_idy_i \]
is called the contact system.
\end{defi}
\begin{rem}
Sometimes we will refer to this system as the classical contact system
in order to distinguish it from matrix valued analogs that will be
discussed later.
\end{rem}
Let us observe that the distribution defined by this system has dimension \(2n\). However, the system is not completely integrable (as we will see in a moment) so any integral manifold must have dimension less than \(2n\). An easy computation shows that
\[\omega \wedge (dw)^{\wedge n} = \pm dz\wedge dx_1\wedge \ldots \wedge dx_n \wedge dy_1 \wedge \ldots \wedge dy_n \]
so \(d\omega \wedge \omega\) is clearly nonzero which implies that the Frobenius condition can not hold for the contact system. In light of this, our next task is going to be to determine the maximal dimension an integral manifold can have.
In what follows the notation \(f(x_I,y_J)\) means that the function \(f\) depends on the variables \(x_i, i \in I\) and \(y_j, j \in J\).
\begin{prop}\label{contactmax}
Let \(S \subset {\C}^{2n+1}\) be a \(k\)-dimensional integral manifold
of the contact system going through the point \(s\). Then there exist
two disjoint sets of indices \(I,J\subset \{1,\ldots,n\}\) of total
length \(k\) such that in a neighborhood of \(s\), \(S\) is
parameterized by a set of holomorphic functions
\((g_1(x_I,y_J),\ldots,g_{2n+1}(x_I,y_J))\) where the variables
\(x_{i}\) and \(y_{j}\) \(({i\in I},{j \in J})\) are independent on \(S\).
\end{prop}
\begin{rem}
By definition the variables \(x_{i}\), \(y_{j}\) \(({i\in I},{j \in
J})\) are independent on \(S\) if the differential 1-forms
\(dx_{i}\) and \(dy_{j}\) \(({i\in I},{j \in J})\) are linearly independent at each point of a
neighborhood of
\(S\). This also means that independent variables form part of a
coordinate system in a neighborhood.
\end{rem}
\begin{proof}
In a neighborhood \(U\) of \(s\) the manifold \(S\) is given by a set
of holomorphic functions \((g_1,\ldots,g_{2n+1})\). Let \(L\subset \{x_i, y_j,z\}\) denote
the set of {\em independent} coordinates that the functions \(g_i\) depend on,
(\(|L|=k\)). Let \(I\subset L\), \(J\subset L\) be the sets consisting of
variables \(x_{i}\), \(y_{j}\) respectively. Since \(S\) is an integral manifold, \(\omega\) and \(d\omega\) must vanish on \(S\). This implies that the coordinate \(z\) can be expressed as a function of the other variables so we can assume that the functions \(g_i\) do not depend on \(z\), i.e., \(z\notin L\).
Suppose that there is a pair of coordinates \((x_i,y_i)\), \(x_i,y_i\in L\) such that \(dx_i \wedge dy_i \) does not vanish on \(U\). Since
\[
0 = d\omega = \sum_{l=1}^{n}dx_l \wedge dy_l \;\;\;\;\;\;\mathrm{on}\;\;\;S
\]
there must be another index \(j\neq i\) such that \(dx_j \wedge dy_j
\) does not vanish on \(U\) and the variables \(x_j,y_j\) are not in \(L\) since they depend on \(x_i,y_i\). Now
\[0 \neq dx_j\wedge dy_j = (\frac{\partial x_j}{\partial x_i} \frac{\partial y_j}{\partial y_i} - \frac{\partial y_j}{\partial x_i} \frac{\partial x_j}{\partial y_i})dx_i\wedge dy_i + \eta\]
where \(\eta\) is a 2-form that does not contain \(dx_i\wedge dy_i\). This means that one of the coefficient functions, say \(\frac{\partial x_j}{\partial x_i}\), must be nonvanishing so we can introduce the coordinate change \(x_i \longleftrightarrow x_j\) in a neighborhood of \(s\), by which we mean that we replace the coordinate \(x_i\) by \(x_j\) and leave the other coordinates intact. This means that we replaced the independent pair of variables \((x_i,y_i)\) by another independent pair \((x_j,y_i)\) thereby decreasing \(|I\cap J|\). Continuing in the same manner, after at most \(k\) steps we arrive at two {\em disjoint} sets of indices \(I,J\subset \{1,\ldots,n\}\) that satisfy the requirements of the proposition.
\end{proof}
\begin{rem}\label{fixj}
It is clear from the proof that if we choose the
set \(J\) to be maximal in the sense that no other independent set of variables that \(S\) depends on contains more \(y_j\)'s then we
can find the set \(I\) with \(|I\cap J|=0\) without changing the set
\(J\). This simple remark will be used quite frequently later,
sometimes without explicit reference to it.
\end{rem}
\begin{cor}
Let \(S \subset {\C}^{2n+1}\) be an integral manifold of the contact system. Then \(dim(S) \leq n\).
\end{cor}
\begin{proof}
By Proposition~\ref{contactmax}, \(S\) is locally defined by functions \(g_i\) depending on at most \(n\) independent variables. This clearly implies the claim.
\end{proof}
The following
result of \cite{A} (Appendix 4) gives a complete characterization of maximal dimensional
integral manifolds of the contact system.
\begin{thm}\label{arnold}
For any partition \(I+J\) of the set of indices \(\{1,\ldots,n\}\) into two disjoint subsets and for any function \(f(x_I,y_J)\) of \(n\) variables \(x_i,y_j\) \((i\in I, j\in J)\), the formulas
\[y_i = \frac{\partial f}{\partial x_i} \; (i\in I),\; \; \; x_j = -\frac{\partial f}{\partial y_j}\; (j\in J),\; \; \; z = f - \sum_{i\in I}x_i\frac{\partial f}{\partial x_i} \]
define an \(n\)-dimensional integral manifold of the contact system.
Conversely, every \(n\)-dimensional integral manifold is defined in a neighborhood of every point by these formulas for a choice of the subset \(I\) and for some generating function \(f\).
\end{thm}
\begin{proof}
A simple calculation shows that the manifold defined by the formulas in the first part of the Theorem is indeed an integral manifold of the contact system.
Assume now that \(S\) is an \(n\)-dimensional integral manifold. By Proposition~\ref{contactmax} we can find \(n\) {\em independent} variables \(x_I,y_J\) such that \(S\) is defined by functions of these variables. Let
\[f = z + \sum_{i\in I} x_iy_i\]
and notice that
\[df = \sum_{i\in I}y_idx_i - \sum_{j\in J}x_jdy_j \]
since \(0 = \omega = dz + \sum x_ldy_l\) on \(S\). Also, the differential forms \(dx_i,\, dy_j\) are independent so it follows from the defining equation of \(f\) that
\[y_i = \frac{\partial f}{\partial x_i} \; (i\in I),\; \; \; x_j = -\frac{\partial f}{\partial y_j}\; (j\in J)\]
which finishes the proof.
\end{proof}
\begin{rem}\label{contactflexible}
Let us mention here that if we fix an \(n\)-dimensional integral element at the point \(s\), then there exists an integral manifold that is tangent to the given integral element (see Theorem~\ref{exptrick}) but this integral manifold is by no means unique. This follows from the fact that we can choose \(f\) in the previous theorem to be an arbitrary holomorphic function and fixing an integral element specifies only the linear part of \(f\). This means that to any integral element of dimension \(n\) there is an infinite dimensional family of germs of integral manifolds that are tangent to the integral element at the point \(s\).
\end{rem}
\section{Variations of Hodge Structure}
In this section we will consider variations of Hodge structure as integral manifolds of the horizontal distribution. First let us recall the necessary definitions.
\subsection{Hodge Structures and Classifying Spaces}
For a more detailed discussion, proofs of the basic results, as well as geometric motivation for some of the following definitions, see \cite{G1} and \cite{G2}.
Let us fix a finite dimensional real vector space \(H_{\mathbb R}\) and a lattice \(H_{\Z}\subset H_{\mathbb R}\) together with an integer \(w\), (\(w\) will be referred to as the weight). Suppose that we are given a non-degenerate bilinear form \(Q\) on \(H_{\mathbb R}\) which satisfies \(Q(x,y) = (-1)^w Q(y,x)\) for all \(x,y\in H_{\mathbb R}\) and takes rational values on \(H_{\Z}\). Let \(H_{\C} = H_{\mathbb R} \otimes_{\mathbb R} \C\) denote the complexification of \(H_{\mathbb R}\).
\begin{defi}
A Hodge structure of weight \(w\) on \(H_{\mathbb R}\) is a decomposition
\[H_{\C} = \bigoplus_{p+q = w}H^{p,q}\]
such that \(H^{p,q}\) and \(H^{q,p}\) are complex conjugate to each other with respect to \(H_{\mathbb R}\). The integers \(h^{p,q} = dim\,H^{p,q}\) are the Hodge numbers.
\end{defi}
Let \(h(x,y) = (-1)^{w(w-1)/2}Q(x,\overline{y})\) and let \(C\) denote the Weil operator that acts on \(H^{p,q}\) by multiplication by \(i^{p-q}\).
\begin{defi}
The Hodge structure is weakly polarized if
\[Q(H^{p,q},H^{r,s})=0 \; \; \; \;\rm{for} \;\;\;\; (r,s) \neq (q,p).\]
It is strongly polarized if, in addition, the form \(h_C(x,y) = h(Cx,y)\) is positive hermitian.
\end{defi}
To each Hodge structure of weight \(w\) we can assign the {\it Hodge filtration}
\[H_{\C} \supset \ldots \supset F^{p-1} \supset F^{p} \supset F^{p+1}\supset \ldots \supset 0\]
where
\[F^{p} = \bigoplus_{i\geq p}H^{i,w-i}.\]
This filtration has the property
\[H_{\C} = F^{p} \oplus \bar{F}^{w-p+1} \; \; \; \rm{for}\; \;\rm{each} \; \;p\]
where the bar denotes complex conjugation.
If the Hodge structure is strongly polarized then the associated Hodge filtration satisfies \(Q(F^p,F^{w-p+1})=0\) for all \(p\), and the form \(h_C(x,y)\) is positive hermitian. It is not hard too see that a Hodge filtration with the listed properties determines a strongly polarized Hodge structure so these two notions are in fact equivalent.
Let \(\check{D}\) denote the set of weakly polarized Hodge structures (or Hodge filtrations) with fixed Hodge numbers \(h^{p,q}\), \((p+q=w)\) and let \(D \subset \check{D}\) be the subset of strongly polarized Hodge structures. There is a natural complex structure on the set \(\check{D}\) which can be described as follows. Since a Hodge structure is determined by its associated filtration, we can view \(\check{D}\) as a subset of a product of Grassmannian manifolds. The conditions \(F^{p} \supset F^{p+1}\) and \(Q(F^p,F^{w-p+1})=0\) are algebraic so \(\check{D}\) is a subvariety of a product of Grassmannians. As such it has the structure of a complex projective variety. It can be checked that the special orthogonal group \(G_{\C} = SO(Q,\C)\) acts transitively on \(\check{D}\), in particular \(\check{D}\) is a complex manifold. The subset \(D\subset \check{D}\) is open in the complex topology and it is homogeneous for the corresponding real group \(G_{\mathbb R}\). The classifying space \(D\) is sometimes referred to as the {\it period domain} of Hodge structures.
\subsection{Canonical Coordinates on the Period Domain}
In this section we will describe two sets of local coordinate systems on the period domain. These coordinate systems will be used to investigate the local properties of differential systems on the period domain. The two coordinate systems are equivalent to each other and the reason they will both be used is that certain results can be described more conveniently in one of these systems.
\subsubsection{Lie algebra coordinates}
Let \(\mathfrak{g}_{\C}\) denote the Lie algebra of \(G_{\C}\) and let us fix a reference Hodge structure \(H\in D\). \(H\) defines a Hodge structure of weight 0 on \(\mathfrak{g}_{\C}\) (cf. \cite{S}) where
\[\mathfrak{g}^{p,-p} = \{\phi \in \mathfrak{g}_{\C}\,|\, \phi(H^{r,s})\subset H^{r+p,s-p}\; \rm{for \;all} \;(r,s)\}.\]
\(\mathfrak{g}_{\C}\) can be decomposed as
\[\mathfrak{g}_{\C} = \mathfrak{g}^{-}\oplus \mathfrak{g}^{0}\oplus \mathfrak{g}^{+}\]
where the subalgebras \(\mathfrak{g}^{-}\), \(\mathfrak{g}^{0}\) and \(\mathfrak{g}^{+}\) are defined as
\[\mathfrak{g}^{-} = \bigoplus_{p<0}\mathfrak{g}^{p,-p},\;\;\;\;
\mathfrak{g}^{0} = \mathfrak{g}^{0,0},\;\;\;\;
\mathfrak{g}^{+} = \bigoplus_{p>0}\mathfrak{g}^{p,-p}.\]
Recall that \(\check{D} \cong G_{\C}/B\) as a homogeneous space where
\[B = \{g\in G_{\C} \,|\, g(F^{p}) \subset F^{p}\}\]
is the isotropy subgroup of the reference Hodge structure. The Lie algebra of \(B\) is
\[\mathfrak{b} = \mathfrak{g}^{0}\oplus \mathfrak{g}^{+}\]
so the complement \(\mathfrak{g}^{-}\) can be identified with the holomorphic tangent space of \(\check{D}\). Note that the Lie bracket is compatible with the Hodge decomposition, i.e.,
\begin{equation}\label{liecompat}
[\mathfrak{g}^{p,q},\mathfrak{g}^{r,s}]\subset \mathfrak{g}^{p+r,q+s}.
\end{equation}
\begin{defi}\label{algebracoord}
The local Lie algebra coordinates in a neighborhood of the reference Hodge structure \(H\in D\) are given by the map
\[\Phi:\mathfrak{g}^{-}\longrightarrow D,\;\;\;\; N\longmapsto (exp(N))\cdot H\]
\end{defi}
Note that the image will lie in \(D\) if the norm of \(N\) is small since \(D\subset \check{D}\) is an open subset. Also, the differential of the map is the identity so \(\Phi\) defines local coordinates in a neighborhood. The actual local coordinates will be chosen by specifying a convenient basis in \(\mathfrak{g}^{-}\). This is our next task (cf. \cite{C-K-T}).
\subsubsection{Hodge Frames and Block Decomposition}
\begin{defi}
A Hodge frame for \(H\) is a set of bases
\(B^{p,q} = \{B^{p,q}_j \,|\, j=1,\ldots,h^{p,q}\}\)
such that
\begin{enumerate}
\item[(i)] \(B^{p,q}\) is an \(h_C\)-unitary basis of \(H^{p,q}\)
\item[(ii)] \(B^{p,q} = \overline{B^{q,p}}.\)
\end{enumerate}
\end{defi}
It is clear that the matrix of the bilinear form \(Q\) relative to a Hodge frame has a block decomposition such that the only nonzero blocks are on the antidiagonal and these blocks are identity matrices up to a sign. Let us denote by \(M[i,j]\) the matrix whose only nonzero block is the matrix \(M\) of size \(h^{w-i,i}\times h^{w-j,j}\) in the \((i,j)\) position. Then the matrix of the polarization \(Q\) can be written as
\[Q=\sum_{k=0}^{w} (-1)^kI[k,w-k]\]
where \(I\) stands for the identity matrix.
Similarly, the matrix of an endomorphism \(X\) of \(H_{\C}\) relative to a Hodge frame also has a block decomposition \(\sum X_{i,j}[i,j]\) where the matrix \(X_{i,j}\) represents an endomorphism from \(H^{w-j,j}\) to \(H^{w-i,i}\). Now let us consider the block decomposition of a general element \(X\in \mathfrak{g}^{-}\). Clearly the matrix of \(X\) is strictly lower triangular and because it is and element of the orthogonal Lie algebra it satisfies the relation \(X^tQ+QX=0\). In terms of the \(X_{i,j}\) we have
\begin{equation}\label{ortcond}
(-1)^{w-i}(X_{i,j})^t+(-1)^{w-j}X_{w-j,w-i}=0\;\;\;\;i,j\in \{0,\ldots ,w\}.
\end{equation}
\begin{ex}
To illustrate the definitions above let us consider the weight two case. The matrix of \(Q\) will be
\[Q=
\begin{pmatrix}
0&0&I_{h^{2,0}}\\
0&-I_{h^{1,1}}&0\\
I_{h^{2,0}}&0&0
\end{pmatrix}
\]
and a general element \(X\in \mathfrak{g}^{-}\) can be written as
\[X=
\begin{pmatrix}
0&0&0\\
X_{1,0}&0&0\\
X_{2,0}&X_{2,1}&0
\end{pmatrix}
\]
where \((X_{1,0})^t=X_{2,1}\) and \((X_{2,0})^t=-X_{2,0}\) because of Equation~\ref{ortcond}. In this case \(dim(D) = h^{2,0}h^{1,1} + \frac{1}{2}h^{2,0}(h^{2,0}-1)\) since this is the number of coordinates in the Lie algebra coordinate system given by the entries of the matrices \(X_{i,j}\).
\end{ex}
\subsubsection{Lie Group Coordinates}\label{groupcord}
Now it is easy to describe the other coordinate system. Let \(G^-\) denote the unipotent Lie group which corresponds to the Lie algebra \(\mathfrak{g}^{-}\).
\begin{defi}\label{groupcoord}
The local Lie group coordinates in a neighborhood of the reference Hodge structure \(H\in D\) are given by the map
\[\Psi:G^{-}\longrightarrow D,\;\;\;\; g\longmapsto g\cdot H\]
\end{defi}
It follows from the discussion of the Lie algebra coordinates that this is a coordinate system in a neighborhood of \(H\). If we fix a Hodge frame of \(H\) then an element \(Y\in G^-\) has a block decomposition \(\sum Y_{i,j}[i,j]\) which is induced by the corresponding Lie algebra element \(log(Y)\).
\begin{rem}
Let us note that the blocks \(Y_{i,j}\) will satisfy the condition
corresponding to Equation~\ref{ortcond} and we will analyze this
later.
\end{rem}
\begin{rem}
To visualize the matrices in the Lie group coordinate system see Example~\ref{visex}.
\end{rem}
\subsection{Variations of Hodge Structure}
\subsubsection{Horizontal Distribution}
In order to define a distribution we have to specify a fixed dimensional linear subspace of the holomorphic tangent space at each point of \(\check{D}\). The holomorphic tangent space was identified by \(\mathfrak{g}^{-}\) so we can proceed as follows.
\begin{defi}
The horizontal distribution on \(\check{D}\) is given by the vector space \(\mathfrak{g}^{-1,1}\subset \mathfrak{g}^{-}\) at each point of \(\check{D}\).
\end{defi}
This distribution is holomorphic and homogeneous as can be seen by elementary Lie group theory (cf. \cite{S}). Integral manifolds of the horizontal distribution are sometimes referred to as {\it horizontal maps}. We have the following
\begin{prop}\label{nonintegrable}
The horizontal differential system is not completely integrable provided that \(\mathfrak{g}^{-2,2}\neq 0\).
\end{prop}
\begin{proof}
Since \([\mathfrak{g}^{-1,1},\mathfrak{g}^{-1,1}]=\mathfrak{g}^{-2,2}\), the Proposition follows from Theorem~\ref{frobenius}.
\end{proof}
\begin{defi}
Let \(\Gamma\) denote a properly discontinuous group of automorphisms of \(D\) and let \(M\) be a complex manifold. A holomorphic map \(\Phi : M \longrightarrow \Gamma~\backslash~ D\) is a variation of Hodge structure if the map \(\Phi\) is locally liftable to \(D\) and the local liftings are horizontal (i.e., integral manifolds of the horizontal differential system).
\end{defi}
In this work we will be interested in the local properties of variations of Hodge structure.
\begin{defi}
A germ of a variation of Hodge structure is an equivalence class of integral manifolds \(\Phi :U \longrightarrow D\) of the horizontal system, where two integral manifolds are equivalent if they have the same image in a neighborhood of a point of \(D\).
\end{defi}
\subsubsection{Local Description of the Horizontal Differential System}
Let us consider the description of the horizontal differential system in terms of the Lie algebra and Lie group coordinate systems.
At the reference Hodge structure the horizontal system is defined by
the differential 1-form entries of the matrices that are at least two
steps below the diagonal
\begin{equation}\label{systemorigin}
dX_{i,j}\;\;\;\;\rm{for}\;\;\;\; i\geq j+2.
\end{equation}
This is clear since at the reference point the horizontal distribution is defined by the subspace \(\mathfrak{g}^{-1,1}\) and the differential forms in Equation~\ref{systemorigin} give the dual of this subspace. Let us now determine the differential forms that define the horizontal differential system in a neighborhood of the reference structure. Since the horizontal system is homogeneous it is given by the left invariant extensions of the differential forms in Equation~\ref{systemorigin}. Let \(\Omega\) denote the Maurer-Cartan matrix, which can be written as \(\Omega = exp(-X)d\,exp(X)\) or \(\Omega = Y^{-1}d\,Y\) in the Lie algebra and Lie group coordinate systems, respectively. Then, in light of the above discussion, we have the following.
\begin{prop}\label{maurer}
The horizontal differential system in a neighborhood of the reference Hodge structure is given by the differential form entries of the matrices
\begin{equation}\label{maurerequation}
\Omega_{i,j}\;\;\;\;\rm{for}\;\;\;\; i\geq j+2.
\end{equation}
\end{prop}
\subsubsection{Integral Elements of the Horizontal Differential System}
Now that we have a local description of the horizontal system we can examine the question: for which integral elements can we find an integral manifold tangent to it? First let us give a characterization of the integral elements.
\begin{prop}\label{commutative}
Let \(E\subset \mathfrak{g}^{-1,1}\) be a linear subspace. Then \(E\) is an integral element of the horizontal differential system if and only if it is an abelian subalgebra.
\end{prop}
\begin{proof}
See \cite{C-K-T} Proposition 2.2 or Section~\ref{proofcom} for another proof.
\end{proof}
Using this result we find the following theorem.
\begin{thm}\label{exptrick}
Let \(E\subset \mathfrak{g}^{-1,1}\) be an integral element at \(v\in D\). Then there is a germ of an integral manifold through \(v\) which is tangent to \(E\).
\end{thm}
\begin{proof}
Let \((e_1,\ldots,e_k)\) be a basis of \(E\). According to Proposition~\ref{commutative} the Lie brackets
\begin{equation}\label{bracketcomm}
[e_i,e_j]=0\;\;\;\; \rm{for} \;\;\;\; i,j\in (1,\ldots,k)
\end{equation}
and since \((e_1,\ldots,e_k)\subset \mathfrak{g}^{-1,1}\) the basis elements \(e_i\) are represented by matrices with all blocks zero except the ones one step below the diagonal. Our aim is to show that the following map \(\varphi\) is an integral manifold in a neighborhood \(U\) of the origin of \(\C^{k}\).
\begin{equation}
\varphi : \C^{k} \longrightarrow D,\;\;\;\;
(x_1,\ldots,x_k)\longmapsto \Phi (x_1\cdot e_1,\ldots,x_k\cdot e_k)
\end{equation}
where the map \(\Phi\) is the Lie algebra coordinate system given in Definition~\ref{algebracoord}. Note that the map \(\varphi\) is well defined and injective in a neighborhood of the origin and its tangent space at the origin is \(E\). What remains to be shown is that \(\varphi\) is horizontal. By Proposition~\ref{maurer} it needs to be checked that the entries \(\Omega_{i,j}\;\;\rm{for}\;\; i\geq j+2\) of the Maurer-Cartan matrix vanish along the image of \(\varphi\). Now
\[\Omega |\varphi (x_1,\ldots,x_n) = exp(-\sum x_i\cdot e_i)\;d\,exp(\sum x_i\cdot e_i)=\]
\[=\prod exp(-x_i\cdot e_i)\;d\,\prod exp(x_i\cdot e_i)=e_1dx_1+\ldots +e_kdx_k\]
where the second equality holds because of Equation~\ref{bracketcomm}. Since the blocks \((e_l)_{i,j} = 0\) for \(i\geq j+2\) the proposition follows.
\end{proof}
\section{Coupled Contact Systems and Dimension Bounds}
In the previous section we gave a local description of the horizontal differential system and we saw that it is given by the 1-form entries in those blocks of the Maurer-Cartan matrix that are at least two steps below the diagonal (Equation~\ref{maurerequation}). Now we will see that this system can be replaced locally by another equivalent system which is easier to deal with. A closer examination of this system will lead to an upper bound on the dimension of integral manifolds.
\subsection{Coupled Contact Differential Systems}
It will be more convenient to work in the coordinate system provided by the Lie group coordinates \(Y\in G^-\) of Section~\ref{groupcord}. In this coordinate system the identity matrix corresponds to the reference Hodge structure \(H\).
Recall that the matrix \(Y\) has a block decomposition \(\sum
Y_{i,j}[i,j]\) where the block \(Y_{i,j}\) at position \((i,j)\) has
size \(h^{w-i,i}\times h^{w-j,j}\). Note that the matrices \(Y_{i,j}\)
must satisfy the Lie group coordinate version of
Equation~\ref{ortcond}, namely
\begin{equation}\label{grouportcond}
(log(Y))^tQ+Qlog(Y)=0
\end{equation}
As a first step let us write the generators of the differential ideal of the horizontal differential system in terms of the entries of the blocks \(Y_{i,j}\).
\begin{prop}\label{firstreduction}
The differential ideal of the horizontal system is generated algebraically by the 1-form entries of the matrices
\begin{equation}\label{almostcontact}
dY_{i,j}-Y_{i,j+1}dY_{j+1,j}\;\;\;\;\rm{for}\;\;\;\;i\geq j+2
\end{equation}
\end{prop}
\begin{proof}
By Proposition~\ref{maurer} we have to show that the entries of the matrices \(\Omega_{i,j}\) for \(i\geq j+2\) can be generated by the 1-forms in Equation~\ref{almostcontact}. We will proceed by induction on \(i-j\). If \(i-j=2\) then an easy computation shows that
\begin{equation}\label{contactfirst}
\Omega_{i,j}=(Y^{-1}dY)_{i,j}=dY_{i,j}-Y_{i,j+1}dY_{j+1,j}
\end{equation}
so the result holds in this case. Let us now assume that the entries of \(\Omega_{i,j}\) are generated by the forms in Equation~\ref{almostcontact} for \(i-j<k\) and consider \(\Omega_{a,b}\) with \(a-b=k\). Let us compute the \((a,b)\) blocks of both sides of the the defining equation of the Maurer-Cartan matrix
\[dY=Y\cdot\Omega .\]
We get
\[dY_{a,b}=\sum_{l=0}^{a}Y_{a,l}\Omega_{l,b}=Y_{a,b+1}dY_{b+1,b}+\sum_{l=b+2}^{a-1}Y_{a,l}\Omega_{l,b}+\Omega_{a,b}.\]
So
\[\Omega_{a,b}=dY_{a,b}-Y_{a,b+1}dY_{b+1,b}-\sum_{l=b+2}^{a-1}Y_{a,l}\Omega_{l,b}\]
and since in the sum we only have \(\Omega_{l,b}\) with \(l-b\leq a-1-b=k-1\) the right hand side of the above equation is generated by forms in Equation~\ref{almostcontact} by the induction hypothesis and this implies the result.
\end{proof}
As the next step let us define a new differential system on a submanifold of the period domain in a neighborhood of the reference Hodge structure. Let \(W\subset D\) denote the submanifold defined by the equations
\begin{equation}
Y_{i,j}=0 \;\;\;\;\rm{for}\;\;\;\;i>j+2.
\end{equation}
\begin{defi}\label{contactdef}
The coupled contact system on the submanifold \(W\) is given by the differential 1-form entries of the matrices
\begin{equation}\label{contactsystem}
(Y^{-1}d\,Y)_{i,j}=dY_{i,j}-Y_{i,j+1}dY_{j+1,j}\;\;\;\;\rm{for}\;\;\;\;i=j+2.
\end{equation}
\end{defi}
\begin{rem}
Let us emphasize that the blocks \(Y_{i,j}\) in the previous definition are of course subject to the condition specified by Equation~\ref{grouportcond}.
\end{rem}
\begin{rem}\label{hhell}
Note that the matrices in the definition are exactly the blocks \(\Omega_{i,j}\) for \(i=j+2\) of the Maurer-Cartan matrix written in terms of the Lie group coordinates, i.e., the blocks which are two steps below the diagonal. In other words we defined a differential systems which is the projection of the horizontal system to the submanifold \(W\) (cf. Equation~\ref{almostcontact}). What is remarkable is that this system is locally equivalent to the horizontal system as we will see shortly.
\end{rem}
\begin{rem}
Each of the equations in Definition~\ref{contactdef} is a matrix
valued contact system. In fact, in the weight two case
when \(h^{2,0}=2\) this system is the classical contact system. Note
furthermore that each matrix \(Y_{i,j}\) appears in two
consecutive matrix valued contact systems and so these systems are
coupled through these matrices, hence the name in the definition.
\end{rem}
\begin{thm}\label{contactequiv}
There is a one--to--one dimension preserving correspondence between germs of integral manifolds of the horizontal differential system on \(D\) and germs of integral manifolds of the coupled contact system on \(W\). The correspondence also identifies integral elements of the two systems.
\end{thm}
\begin{rem}
The theorem implies that instead of studying the uniqueness and dimension properties of germs of variations of Hodge structure we can study the corresponding properties of integral manifolds of the coupled contact system. The results we arrive at will remain true for variations of Hodge structure.
\end{rem}
\begin{proof}
By Proposition~\ref{firstreduction} the horizontal system is given by the differential form entries of the matrices
\begin{equation}\label{variation}
dY_{i,j}-Y_{i,j+1}dY_{j+1,j}\;\;\;\;\rm{for}\;\;\;\; i\geq j+2
\end{equation}
and by Definition~\ref{contactdef} the coupled contact system is given by
\begin{equation}\label{contact}
dY_{i,j}-Y_{i,j+1}dY_{j+1,j}\;\;\;\;\rm{for}\;\;\;\; i=j+2.
\end{equation}
Let \(S\subset D\) be a germ of an integral manifold of the horizontal system. Then because of the above equations the projection \(pr_{W}(S)\) to the submanifold \(W\) defines an integral manifold of the contact system. To prove the theorem we have to show that we can go the other way, namely given an integral manifold \(T\subset W\) of the contact system we need to find a {\it locally unique} extension of \(T\) to an integral manifold \(\bar{T}\subset D\) of the horizontal system. This amounts to showing that the differential equations
\begin{equation}\label{needtosolve}
dY_{i,j}-Y_{i,j+1}dY_{j+1,j}=0\;\;\;\;\rm{for}\;\;\;\; i>j+2
\end{equation}
have unique local solutions, provided that we have a solution of
\begin{equation}\label{given}
dY_{i,j}-Y_{i,j+1}dY_{j+1,j}=0\;\;\;\;\rm{for}\;\;\;\; i=j+2.
\end{equation}
Since \(T\) is an integral manifold we also have the equations
\begin{equation}\label{givenwedge}
dY_{i,j+1}\wedge dY_{j+1,j}=0\;\;\;\;\rm{for}\;\;\;\; i=j+2.
\end{equation}
Let us proceed by induction on \(i-j\). Consider the equation
\begin{equation}\label{solve1}
dY_{a,b}=Y_{a,b+1}dY_{b+1,b}\;\;\;\;\rm{where}\;\;\;\; a-b=3.
\end{equation}
Note that the right hand side of Equation~\ref{solve1} consists of matrices which are already determined at this point since they appear in Equation~\ref{given}. This implies that if we can solve \ref{solve1} for \(Y_{a,b}\) then the solution will be unique. By the Poincar\'e Lemma there is a local solution of \ref{solve1} if and only if the 1-form entries of the right hand side are {\it closed} 1-forms, i.e., it needs to be checked that
\begin{equation}
dY_{a,b+1}\wedge dY_{b+1,b}=0\;\;\;\;\rm{for}\;\;\;\; a-b=3
\end{equation}
holds. Now, since \(a-(b+1)=2\) we have \(dY_{a,b+1}=Y_{a,b+2}dY_{b+2,b+1}\) by Equation~\ref{given} so
\[
dY_{a,b+1}\wedge dY_{b+1,b}=(Y_{a,b+2}dY_{b+2,b+1})\wedge dY_{b+1,b}=Y_{a,b+2}(dY_{b+2,b+1}\wedge dY_{b+1,b})=0
\]
where the last equality holds by Equation~\ref{givenwedge}. This completes the first step of the induction.
Assume now that the equations
\begin{equation}\label{givengeneral}
dY_{i,j}=Y_{i,j+1}dY_{j+1,j}
\end{equation}
are solvable for \(i-j<k\). This also implies that
\begin{equation}\label{wedgegeneralgiven}
dY_{i,j+1}\wedge dY_{j+1,j}=0\;\;\;\;\rm{for}\;\;\;\;i-j<k.
\end{equation}
By the same argument as above we see that to solve
\begin{equation}\label{solve2}
dY_{a,b}=Y_{a,b+1}dY_{b+1,b}\;\;\;\;\rm{where}\;\;\;\; a-b=k
\end{equation}
we need that
\begin{equation}\label{wedgegeneral}
dY_{a,b+1}\wedge dY_{b+1,b}=0\;\;\;\;\rm{for}\;\;\;\; a-b=k.
\end{equation}
Note again that the matrices on the right hand side of \ref{solve2} have already appeared in Equation~\ref{givengeneral} since \(a-(b+1)=k-1<k\) so uniqueness holds. To check \ref{wedgegeneral}, note that \(dY_{a,b+1}=Y_{a,b+2}dY_{b+2,b+1}\) by \ref{givengeneral} since \(a-(b+1)<k\) so
\[
dY_{a,b+1}\wedge dY_{b+1,b}=Y_{a,b+2}(dY_{b+2,b+1}\wedge dY_{b+1,b})=0
\]
by Equation~\ref{wedgegeneralgiven}.
\end{proof}
\subsection{Integral Elements and Abelian Lie Algebras}\label{proofcom}
In this section we will prove Proposition~\ref{commutative}. Let us recall the statement.
\begin{prop}
Let \(E\subset \mathfrak{g}^{-1,1}\) be a linear subspace. Then \(E\) is an integral element of the horizontal differential system if and only if it is an abelian subalgebra.
\end{prop}
\begin{proof}
By definition \(E\) is an integral element if and only if \(\omega_E = 0\) for each \(\omega\) in the differential ideal. By Theorem~\ref{contactequiv} this is equivalent to the vanishing of the two forms \(\varphi_{i,j}=dY_{i,j+1}\wedge dY_{j+1,j}\) on \(E\) for \(i=j+2\). Now let \(X^1,X^2\in E\) be two general tangent vectors. Then
\begin{equation}
0=\varphi_{i,j}(X^1,X^2)=dY_{i,j+1}\wedge dY_{j+1,j}(X^1,X^2)=X^1_{i,j+1}X^2_{j+1,j}-X^2_{i,j+1}X^1_{j+1,j}
\end{equation}
which is exactly the condition for the commutator \([X^1,X^2]\) to vanish since the matrices \(X^1,X^2\) have nonzero blocks only at the \(i=j+1\) positions.
\end{proof}
\begin{rem}
In the language of Hodge theory integral elements correspond to infinitesimal variations of Hodge structure and the above commutativity condition is part of the definition of an infinitesimal variation of Hodge structure (see \cite{C1}). In this context Proposition~\ref{exptrick} says that to every infinitesimal variation of Hodge structure there is a germ of a variation of Hodge structure tangent to it. Apparently this simple result was not well known.
\end{rem}
\subsection{Dimension Bound for Variations of Hodge Structure}\label{dimbound}
In this section we will use the contact differential system to give sharp upper bounds for the dimension of variations of Hodge structure. This is the main result of \cite{C-K-T}. The proof is different, however, and less elementary than the one in this section. Also a case is not covered in \cite{C-K-T} so it seems worthwhile to reprove this result here.
Up to this point we did not need to use the conditions that Equation~\ref{grouportcond} specifies in terms of the blocks \(Y_{i,j}\) but to give the upper bound we will have to compute these at least for the contact system. By doing this it is possible to further reduce the contact system to an equivalent differential system.
\begin{prop}\label{realreduced}
a) If the weight \(w=2k+1\) is odd then there is a one-to-one correspondence between germs of integral manifolds of the coupled contact differential system and the system given by the 1-form entries of the matrices
\begin{equation}\label{reducedcontactodd}
dY_{i,j}-Y_{i,j+1}dY_{j+1,j}\;\;\;\;\rm{for}\;\;\;\;i=j+2\;\;\;\;\rm{with}\;\;\;\;j<k.
\end{equation}
where the matrix \(Y_{k+1,k}\) is symmetric.
b) If \(w=2k\) is even then the above correspondence is between the coupled contact system and the system given by the entries of
\begin{equation}\label{reducedcontacteven}
dY_{i,j}-Y_{i,j+1}dY_{j+1,j}\;\;\;\;\rm{for}\;\;\;\;i=j+2\;\;\;\;\rm{with}\;\;\;\;j<k-1 \;\;\;\;\rm{and}
\end{equation}
\begin{equation}\label{weight2add}
dY_{k+1,k-1}-Y_{k,k-1}^tdY_{k,k-1}
\end{equation}
and we also have
\begin{equation}\label{weight2cond}
Y_{k+1,k-1}+Y_{k+1,k-1}^t=Y_{k,k-1}^tY_{k,k-1}.
\end{equation}
\end{prop}
\begin{rem}
The content of this proposition is that we need to consider only blocks of the decomposition that are above the main antidiagonal and we have an explicit description of the dependencies between the entries of the matrices (e.g., the condition that \(Y_{k+1,k}\) is symmetric when the weight is odd). To make this easier to visualize let us consider the \(w=3\) and \(w=4\) cases.
\end{rem}
\begin{ex}\label{visex}(\(w=3\))
In this case a general element \(Y\in G^-\) has the form
\[Y=
\begin{pmatrix}
I & 0 & 0 & 0\\
Y_{1,0} & I & 0 & 0\\
Y_{2,0}& Y_{2,1} & I &0\\
Y_{3,0}& Y_{3,1} & Y_{3,2} &I
\end{pmatrix}
\]
where the blocks satisfy Equation~\ref{grouportcond}. Now the proposition says that the contact differential system is determined by the 1-forms in
\[ dY_{2,0}-Y_{2,1}dY_{1,0}\]
where \(Y_{2,1}\) is symmetric and this is the only dependence between the entries of the matrices giving local coordinates. So to exhibit local variations of Hodge structure in the weight three case it is enough to solve the differential equations in \(dY_{2,0}=Y_{2,1}dY_{1,0}\).
\end{ex}
\begin{ex}(\(w=4\))
Now a general element \(Y\in G^-\) has the form
\[Y=
\begin{pmatrix}
I & 0 & 0 & 0 & 0\\
Y_{1,0} & I & 0 & 0 & 0\\
Y_{2,0}& Y_{2,1} & I &0 & 0\\
Y_{3,0}& Y_{3,1} & Y_{3,2} &I & 0\\
Y_{4,0}& Y_{4,1} & Y_{4,2} & Y_{4,3}& I
\end{pmatrix}
\]
where the blocks satisfy Equation~\ref{grouportcond}. By the proposition this means that the contact system is determined by
\[ dY_{2,0}-Y_{2,1}dY_{1,0}\;\;\;\;\rm{and}\;\;\;\;dY_{3,1}-Y_{2,1}^tdY_{2,1}\]
where there is one more restriction of the form
\[ Y_{3,1}^t + Y_{3,1}=Y_{2,1}^tY_{2,1}\]
\end{ex}
\begin{proof}
Since the contact system involves blocks that are one or two steps below the main diagonal, to compute these blocks of \(log(Y)\) only the first two terms of the power series expansion will have to be considered: \((Y-I)-(Y-I)^2/2\).
If we consider the blocks that are one step below the diagonal it is immediate from Equation~\ref{ortcond} that
\begin{equation}\label{symmetry}
Y_{i,i-1}^t=Y_{w-i+1,w-i}\;\;\;\;i\in \{0,\ldots ,w\}.
\end{equation}
If \(w\) is odd this implies that \(Y_{k+1,k}\) is symmetric.
Now we have to show that an equation consisting of blocks below the antidiagonal is automatically solved if we have solutions for the equations above the antidiagonal. This is a simple consequence of Equation~\ref{ortcond}. In detail, let us consider the equation
\begin{equation}
dY_{j+2,j}=Y_{j+2,j+1}dY_{j+1,j}\;\;\;\;\rm{for}\;\;\;\;j>k\;\;\;\;w=2k+1\;\;\;\;\rm{or}\;\;\;\;j>k-1\;\;\;\;w=2k.
\end{equation}
Computing \(log(Y)_{j+2,j}\) and substituting it to Equation~\ref{ortcond} we get
\begin{equation}\label{reducedcontact}
Y_{j+2,j} = Y_{j+2,j+1}Y_{j+1,j}-Y_{w-j,w-j-2}^t
\end{equation}
from which
\begin{equation}
dY_{j+2,j}=Y_{j+2,j+1}dY_{j+1,j}+dY_{j+2,j+1}Y_{j+1,j}-Y_{w-j,w-j-2}^t=Y_{j+2,j+1}dY_{j+1,j}
\end{equation}
by Equation~\ref{symmetry} and Equation~\ref{reducedcontactodd} (or Equation~\ref{reducedcontacteven}).
\end{proof}
Using this reduced form of the contact system we will proceed to give upper bounds for the dimension of variations of Hodge structure. Let us define the following quadratic functions that depend on the Hodge numbers \(h^{i,j}\). Let \(h^l\) stand for \(h^{l,w-l}\). If the weight is odd \((w=2k+1>1)\) let
\begin{eqnarray*}
q^{odd}_1& = &\sum_{i=0}^{\infty}h^{k+2+2i}h^{k+3+2i}\cr
q^{odd}_2& = &\frac{1}{2}h^{k+1}(h^{k+1}+1)+\sum_{i=0}^{\infty}h^{k+3+2i}h^{k+4+2i}.
\end{eqnarray*}
If the weight is even \((w=2k)\) let
\begin{eqnarray*}
q^{even}_1& =&\sum_{i=0}^{\infty}h^{k+1+2i}h^{k+2+2i}\cr
q^{even}_2& = &\overline{q_2}+\sum_{i=0}^{\infty}h^{k+2+2i}h^{k+3+2i}\cr
q^{even}_3& = &h^k+h^{k+2}(h^{k+1}-1)+\sum_{i=0}^{\infty}h^{k+3+2i}h^{k+4+2i}\;\;\;\;\rm{if}\;\;\;\;w\geq 4.
\end{eqnarray*}
where
\begin{equation}
\overline{q_2}=
\left\{\begin{array}{ll}
h^k & \mbox{if \(h^{k+1}=1\),}\cr
\frac{1}{2}h^{k+1}h^k& \mbox{if \(h^k\) is even and \(h^{k+1}>1\),}\cr
\frac{1}{2}h^{k+1}(h^k-1)+1& \mbox{if \(h^k\) is odd and \(h^{k+1}>1\).} \end{array} \right.
\end{equation}
\begin{rem}
The sums are of course finite since \(h^i=0\) if \(i>w\). \(q^{even}_3\) is defined only if the weight is at least four.
\end{rem}
\begin{thm}\label{dimboundthm}
Let \(S\subset D\) be a variation of Hodge structure.
\begin{enumerate}
\item
If \(w=2k+1\) then \(dim(S)\leq max\{q^{odd}_1,q^{odd}_2\}\).
\item
If \(w=2k\) then \(dim(S)\leq max\{q^{even}_1,q^{even}_2,q^{even}_3\}\).
\end{enumerate}
\end{thm}
\begin{proof}1) According to Proposition~\ref{realreduced} the contact system is equivalent to the system defined by
\begin{equation}\label{dimproofcontact} dY_{i,j}-Y_{i,j+1}dY_{j+1,j}\;\;\;\;\rm{for}\;\;\;\;i=j+2\;\;\;\;\rm{with}\;\;\;\;j<k.
\end{equation}
with \(Y_{k+1,k}\) symmetric. We will analyze this system to derive the dimension bound.
In a neighborhood of the reference Hodge structure \(S\) is defined by holomorphic functions depending on a set \(L\) of independent coordinate variables. Clearly \(dim(S)\leq |L|\) where \(|L|\) denotes the cardinality of the set \(L\). We will give a bound for \(|L|\). At the origin the system is defined by the vanishing of the differential form entries of \(dY_{j+2,j}\) \((j<k)\) so the coordinate entries of the matrices \(Y_{j+2,j}\) are not elements of \(L\), i.e., all elements of \(L\) appear in the matrices \(dY_{j+1,j}\). Let \(y_{1,0}\) denote the maximal number of the coordinates in the set \(L\) that appear in a column of the matrix \(Y_{1,0}\). Then, clearly, \(Y_{1,0}\) contains at most \(y_{1,0}\cdot h^{w}\) elements of \(L\) since \(Y_{1,0}\) has \(h^{w}\) columns.
Notice that a single entry of the matrix valued differential form \(dY_{2,0}-Y_{2,1}dY_{1,0}\) defines a classical contact system so we can use Proposition~\ref{contactmax} and Remark~\ref{fixj} to partition the variables of this classical contact system into two disjoint subsets. More precisely, suppose that the \(l^{th}\) column in the matrix \(Y_{1,0}\) contains \(y_{1,0}\) elements of \(L\), i.e., it is one of the columns containing the maximal possible number of elements of \(L\). Consider the classical contact system defined by the \((i,l)\) entry of the system \(dY_{2,0}-Y_{2,1}dY_{1,0}\) for every \(i\in \{0,\ldots,h^{w-2}\}\):
\begin{equation}\label{classcont}
(dY_{2,0})^{i,l}-\sum_{t}(Y_{2,1})^{i,t}(dY_{1,0})^{t,l}
\end{equation}
where superscripts denote matrix entries. By
Proposition~\ref{contactmax} the \(i^{th}\) row of the matrix
\(Y_{2,1}\) can contain at most \(h^{w-1}-y_{1,0}\) elements of
\(L\). This is because the classical contact system (\ref{classcont}) already contains
\(y_{1,0}\) independent variables coming from the \(l^{th}\) column of
\(Y_{1,0}\), (the set \(J\) in Proposition~\ref{contactmax}), so since
\(I\) and \(J\) are disjoint we get \(|I|\leq
h^{w-1}-y_{1,0}\). This implies that there are at most
\(h^{w-1}-y_{1,0}\) columns of the matrix \(Y_{2,1}\) that contain
elements of \(L\) so if \(y_{2,1}\) denotes the maximal number of the
coordinates in the set \(L\) that appear in a column of the matrix
\(Y_{2,1}\) then this matrix can contain at most
\(y_{2,1}\cdot (h^{w-1}-y_{1,0}) \) elements of \(L\).
Applying the same argument to all of the matrix valued contact systems
in Equation~\ref{dimproofcontact} and taking into consideration that
\(Y_{k+1,k}\) is symmetric, we arrive at the following upper bound for \(|L|\):
\begin{eqnarray*}\label{boundd}
|L|& \leq & y_{1,0}\cdot h^{w}+y_{2,1}\cdot
(h^{w-1}-y_{1,0})+\ldots+\\
& + & y_{k,k-1}\cdot (h^{w-(k-1)}-y_{k-1,k-2})+\frac{1}{2}(h^{w-k}-y_{k,k-1})(h^{w-k}-y_{k,k-1}+1)
\end{eqnarray*}
The next step is to determine the maximum value of the right hand side
of the above equation for \(y_{j+1,j}\in [1,h^{w-(j+1)}]\)
(\(j<k)\). This is a quadratic programming problem in a given
rectangle that can be solved as follows. Let \(f(y_{i,j})\) denote
the right hand side of this equation. Then the function \(f(y_{i,j})\) can not have interior maximum because its Hessian matrix is not negative definite and it is nowhere negative semidefinite. If we restrict \(f\) to a face than we get a function which is linear or has a non-negative Hessian. Applying this repeatedly we can see that the maximum must occur at a vertex and examining the function \(f(vertex)\) we conclude the theorem in the odd weight case.
2) First let us consider the \(w=2\) case. By Proposition~\ref{realreduced} the contact system is defined by the equation
\begin{equation}\label{w2}
dY_{2,0}=Y_{1,0}^tdY_{1,0}
\end{equation}
which is subject to the condition
\begin{equation}\label{weight2cond2}
Y_{2,0}+Y_{2,0}^t=Y_{1,0}^tY_{1,0}.
\end{equation}
\(Y_{1,0}\) determines the symmetric part of \(Y_{2,0}\) by Equation~\ref{weight2cond2} so it is enough to consider the 1-form entries \((i,j)\) of Equation~\ref{w2} for which \(i<j\). The rest of the equations are automatically satisfied as it can be seen by taking the exterior derivative of Equation~\ref{weight2cond2}.
Let the set \(L\) be as in the first part of the proof and let \(y_{1,0}\) denote the maximal number of the coordinates in the set \(L\) that appear in a column of the matrix \(Y_{1,0}\). (Just like above we can assume that the entries of the matrix \(Y_{2,0}\) do not appear in \(L\)). Then we have the following bounds for \(L\):
\begin{equation}
|L|\leq y_{1,0}\cdot h^{2,0}
\end{equation}
since the matrix \(Y_{1,0}\) has \(h^{2,0}\) columns. We also have
\begin{equation}
|L|\leq y_{1,0}+(h^{1,1}-y_{1,0})\cdot (h^{2,0}-1)
\end{equation}
which follows by fixing a column having \(y_{1,0}\) coordinate entries in \(L\) and applying the argument of Proposition~\ref{contactmax} to each of the classical contact systems determined by this fixed column and the other columns of \(Y_{1,0}\). Then in the fixed column we have \(y_{1,0}\) independent variables and each of the other columns can have at most \(h^{1,1}-y_{1,0}\) of them. The above two bounds for \(L\) imply that the maximum occurs if \(y_{1,0}=\frac{1}{2}h^{1,1}\). If \(h^{1,1}\) is even then this implies that \(|L|\leq \frac{1}{2}h^{1,1}\cdot h^{2,0}\). If \(h^{1,1}\) is odd then an easy computation shows that \(\frac{1}{2}h^{2,0}(h^{1,1}-1)+1\) is the largest value satisfying both of the bounds. This proves the theorem in the weight two case.
Assume now that \(w\geq 4\). The argument will be the same as in the odd weight case except that when considering the last of the matrix valued differential forms in Equation~\ref{reducedcontacteven} it will be necessary to use similar arguments as we did in the weight two case. Let us consider the last two equations:
\begin{equation}\label{4one}
dY_{k,k-2}=Y_{k,k-1}dY_{k-1,k-2}
\end{equation}
and
\begin{equation}\label{4two}
dY_{k+1,k-1}=Y_{k,k-1}^tdY_{k,k-1}.
\end{equation}
As in the weight two case we need to consider only the antisymmetric part of Equation~\ref{4two}. Using the same arguments as above we have the following bounds for the number of independent variables in the matrix \( Y_{k,k-1}\), (\(dim(Y_{k,k-1})\) will denote this number):
\begin{equation}
dim(Y_{k,k-1})\leq y_{k,k-1}\cdot (h^{k+1}-y_{k-1,k-2}).
\end{equation}
which follows from the fact that by Equation~\ref{4one} \( Y_{k,k-1}\) can have at most \(h^{k+1}-y_{k-1,k-2}\) columns that contain elements of \(L\) and by definition one column can contain at most \(y_{k,k-1}\) independent variables. We also have
\begin{equation}
dim(Y_{k,k-1})\leq y_{k,k-1} + (h^{k+1}-y_{k-1,k-2}-1)(h^{k}-y_{k,k-1})
\end{equation}
from Equation~\ref{4two} by applying the argument we had in the weight two case.
Now we have to distinguish two cases.\newline
a) If \(h^{k+1}-y_{k-1,k-2}-1\neq 0\) then from the above two bounds we get that the maximum occurs at \(y_{k,k-1}=\frac{1}{2}h^{k}\) which means that
\begin{equation}
dim(Y_{k,k-1})\leq \frac{1}{2}h^{k}(h^{k+1}-y_{k-1,k-2})\;\;\;\;\rm{if}\;\;\;\;h^{k}\;\;\;\;\rm{even}
\end{equation}
\begin{equation}
dim(Y_{k,k-1})\leq \frac{1}{2}(h^{k}-1)(h^{k+1}-y_{k-1,k-2})+1\;\;\;\;\rm{if}\;\;\;\;h^{k}\;\;\;\;\rm{odd}.
\end{equation}
Applying arguments as in the odd weight case to the remaining matrix valued contact systems we arrive at the following upper bound for \(L\):
\begin{eqnarray*}
|L| & \leq & y_{1,0}\cdot h^{w}+y_{2,1}\cdot
(h^{w-1}-y_{1,0})+\ldots+\\
& + & y_{k-1,k-2}\cdot (h^{w-(k-2)}-y_{k-2,k-3})+dim(Y_{k,k-1})
\end{eqnarray*}
Now the usual quadratic programming argument applies to this function to give the result in this case.\newline
b) If \(h^{k+1}-y_{k-1,k-2}-1= 0\) then
\begin{equation}
dim(Y_{k,k-1})\leq h^{k}
\end{equation}
since there can only be one column in \(Y_{k,k-1}\) that contains elements of \(L\). In this case we get the following upper bound:
\begin{eqnarray*}
|L| & \leq & y_{1,0}\cdot h^{w}+y_{2,1}\cdot
(h^{w-1}-y_{1,0})+\ldots+\\
& + & (h^{k+1}-1)\cdot (h^{w-(k-2)}-y_{k,k-1})+h^{k}
\end{eqnarray*}
and applying quadratic programming again we get the claim of the theorem.
\end{proof}
\begin{rem}
Let \(\Phi : M \longrightarrow \Gamma~\backslash~ D\) be a global variation of Hodge structure. Then \(rank(\Phi)\) also satisfies the dimension bound given in the theorem.
\end{rem}
\begin{rem}
For sharpness of this result see \cite{C-K-T}. Note that the case when the maximal dimension is \(q^{even}_3\) is missing from the main theorem in \cite{C-K-T}. It is easy to see that there are Hodge numbers for which \(q^{even}_3\) is indeed the maximal dimension so this function is needed in the proper form of the dimension bound.
\end{rem}
\section{Rigidity of Maximal Dimensional Variations of
Hodge Structure}
In this section we will examine maximal dimensional variations. We will proceed in two steps. First, we fix a maximal dimensional integral element of the horizontal system and investigate the nature of germs of integral manifolds tangent to this integral element. It turns out that in most cases there is a unique germ of an integral manifold tangent to the given integral element. This is sometimes referred to as local rigidity. In the second step we will fix a point and investigate the nature of integral elements through the given point hence obtaining infinitesimal rigidity. These two steps together lead to the main rigidity results.
\subsection{Germs of Integral Manifolds}
Let \(E\) be a maximal dimensional integral element of the horizontal differential system going through the point \(p\). By Theorem~\ref{exptrick} there is a maximal dimensional integral manifold tangent to \(E\) so the only question we have to investigate is uniqueness of such integral manifolds. The following result answers this question.
\begin{thm}\label{localrigid}
Let \(E\) be a maximal dimensional integral element of the horizontal differential system.
\begin{enumerate}
\item
Let \(w=2k+1\). If \(h^{k,k+1}>2\) and all the other Hodge numbers are greater than one then there is a unique germ of an integral manifold whose tangent space at \(x\) is \(E\).
\item Let \(w=2k\). Assume that one of the following conditions holds:
\begin{enumerate}
\item
\(dim(E)=q^{even}_1\) and all the Hodge numbers are greater than one
\item \(dim(E)=q^{even}_2\), \(h^{k+1}>2\), the other Hodge numbers are greater than one and \(h^k\geq 4\) is even,
\item \(dim(E)=q^{even}_3\) and \(h^{k+1}>2\)
\end{enumerate}
then there is a unique germ of an integral manifold whose tangent space at \(x\) is \(E\).
\item If \(dim(E)=q^{even}_2\) and \(h^k\) is odd then there is an infinite dimensional family of germs of maximal dimensional integral manifolds tangent to \(E\).
\end{enumerate}
\end{thm}
\begin{proof}
1) Assume first that the maximum is \(q^{odd}_1\) i.e., \(dim(E)=q^{odd}_1\). Consider the first matrix valued differential form of Equation~\ref{reducedcontactodd}
\begin{equation}\label{egy}
dY_{k+1,k-1}=Y_{k+1,k}dY_{k,k-1}
\end{equation}
and its exterior derivative
\begin{equation}\label{ketto}
0=dY_{k+1,k}\wedge dY_{k,k-1}
\end{equation}
From the proof of Theorem~\ref{dimboundthm} we see that in this case the entries of the matrix \(Y_{k,k-1}\) are independent variables (elements of the set \(L\)). We would like to see that the entries of the matrix \(Y_{k+1,k}\) are at most linear functions of the variables in \(L\) since this implies that they are uniquely determined by fixing the tangent space \(E\). To this end, consider an entry of Equation~\ref{ketto} (e.g., the \((1,1)\) entry):
\begin{equation}\label{harom}
0=\sum_{i}(dY_{k+1,k})^{1,i}\wedge (dY_{k,k-1})^{i,1}.
\end{equation}
This equation involves the first row of \(dY_{k+1,k}\) and the first
column of \(dY_{k,k-1}\). Since all the entries of \(dY_{k,k-1}\) are
independent, Cartan's lemma (\cite{W} Exercise 2.16) implies that
\begin{equation}
(dY_{k+1,k})^{1,i}=\sum_jA_{i,j}(dY_{k,k-1})^{j,1}\;\;\;\;\rm{for}\;\;\;\;\rm{each}\;\;\;\;i
\end{equation}
for some functions \(A_{i,j}\). Since the matrix \(Y_{k,k-1}\) has at least two columns because of the condition on the Hodge numbers in the theorem, we can do the same for the (1,2) entry of Equation~\ref{ketto}. The same argument shows that
\begin{equation}
(dY_{k+1,k})^{1,i}=\sum_jB_{i,j}(dY_{k,k-1})^{j,2}\;\;\;\;\rm{for}\;\;\;\;\rm{each}\;\;\;\;i
\end{equation}
for some functions \(B_{i,j}\). Since the entries in the first and
second column of the matrix \(Y_{k,k-1}\) are independent from each
other this is possible only if all the functions \(A_{i,j}\) and \(B_{i,j}\) are zero. This implies that entries in the first row of the matrix \(Y_{k+1,k}\) must be constant functions. The same argument applies to all the rows of this matrix so we conclude that all the entries of the matrix \(Y_{k+1,k}\) are constant functions.
Consider now the next matrix valued differential form of Equation~\ref{reducedcontactodd}
\begin{equation}\label{egyy}
dY_{k,k-2}=Y_{k,k-1}dY_{k-1,k-2}.
\end{equation}
Notice that this equation contains the matrix \(Y_{k,k-1}\) which has independent entries so applying the above argument we conclude that the entries of the matrix \(dY_{k-1,k-2}\) are constant functions. Continuing in the same manner we see that the matrices involved in Equation~\ref{reducedcontactodd} are either constant matrices or contain entries that are independent coordinate functions on the integral manifold. This implies the claim of the theorem showing that there is a unique germ of an integral manifold tangent to the fixed integral element \(E\).
If the maximum is \(q^{odd}_2\) (i.e., \(dim(E)=q^{odd}_2\)) the above argument applies but we have to be careful about the equation containing the symmetric matrix \(Y_{k+1,k}\). Consider this equation
\begin{equation}\label{hello}
dY_{k+1,k-1}=Y_{k+1,k}dY_{k,k-1}
\end{equation}
and its exterior derivative
\begin{equation}
0=dY_{k+1,k}\wedge dY_{k,k-1}.
\end{equation}
From Theorem~\ref{dimboundthm} we see that the entries of the symmetric matrix \(Y_{k+1,k}\) are independent variables and we would like to conclude that this implies that \(Y_{k,k-1}\) has constant entries. Consider the first column of \(Y_{k,k-1}\). As before from the \((1,1)\) entry of Equation~\ref{hello} we have
\begin{equation}
(dY_{k,k-1})^{i,1}=\sum_jA_{i,j}(dY_{k+1,k})^{1,j}\;\;\;\;\rm{for}\;\;\;\;\rm{each}\;\;\;\;i
\end{equation}
and from the \((2,1)\) entry we have
\begin{equation}
(dY_{k,k-1})^{i,1}=\sum_jB_{i,j}(dY_{k+1,k})^{2,j}\;\;\;\;\rm{for}\;\;\;\;\rm{each}\;\;\;\;i.
\end{equation}
Since \(Y_{k+1,k}\) is symmetric this implies only that the entries in the first column of \(Y_{k,k-1}\) are functions of the \((1,2)\) entry of \(Y_{k+1,k}\). On the other hand considering the \((1,1)\) and \((1,3)\) entries of Equation~\ref{hello} (\(h^{k,k+1}>2\)) we conclude that these same entries are functions of the \((1,3)\) entry of \(Y_{k+1,k}\). These two facts together imply that the entries must be constant functions. The same argument applies to all the remaining entries and to all of the remaining matrix valued contact equations and this implies the claim of the theorem in this case.
2)
a) The difference between the odd weight and even weight cases is the appearance of the equation
\begin{equation}\label{extraeq}
dY_{k+1,k-1}=Y_{k,k-1}^tdY_{k,k-1}.
\end{equation}
However, if the maximum dimension occurs at \(dim(E)=q^{even}_1\) then there are no independent variables among the entries of matrix \(Y_{k,k-1}\); consequently the same argument applies as in the odd weight, \(q^{odd}_1\) case.
b) Assume now that \(dim(E)=q^{even}_2\). According to Theorem~\ref{dimboundthm} this implies that each column of the matrix \(Y_{k,k-1}\) has exactly \(\frac{1}{2}h^k\) independent entries. Without loss of generality we can assume that the first \(\frac{1}{2}h^k\) entries in the first column (\((Y_{k,k-1})^{l,1}\) with \(l\leq \frac{1}{2}h^k\)) are independent. Consider the \((1,2)\) entry in Equation~\ref{extraeq}:
\begin{equation}
(dY_{k+1,k-1})^{1,2}=\sum_l(Y_{k,k-1})^{l,1}(dY_{k,k-1})^{l,2}.
\end{equation}
Applying Proposition~\ref{contactmax} to this classical contact system it follows that the independent entries in the second column are
\begin{equation}
(Y_{k,k-1})^{l,2}\;\;\;\;\rm{with}\;\;\;\;l>\frac{1}{2}h^k.
\end{equation}
Furthermore, by Cartan's lemma, the entries \((dY_{k,k-1})^{l,1}\) with \(l>\frac{1}{2}h^k\) can be expressed in terms of the independent 1-forms \((dY_{k,k-1})^{l,1}\) with \(l\leq \frac{1}{2}h^k\) and \((dY_{k,k-1})^{l,2}\) with \(l>\frac{1}{2}h^k\). Considering the \((1,3)\) entry of Equation~\ref{extraeq} we can similarly conclude that the same entries can be expressed in terms of \((dY_{k,k-1})^{l,1}\) with \(l\leq \frac{1}{2}h^k\) and \((dY_{k,k-1})^{l,3}\) with \(l>\frac{1}{2}h^k\). This implies that these entries depend only on \((dY_{k,k-1})^{l,1}\) with \(l\leq \frac{1}{2}h^k\). Similarly, the entries \((dY_{k,k-1})^{l,2}\) with \(l\leq \frac{1}{2}h^k\) depend only on \((dY_{k,k-1})^{l,2}\) with \(l>\frac{1}{2}h^k\). It follows from this and the \((1,2)\) entry of the exterior derivative of Equation~\ref{extraeq}
\begin{equation}
0=\sum_l(dY_{k,k-1})^{l,1}\wedge (dY_{k,k-1})^{l,2}
\end{equation}
that the entries \((Y_{k,k-1})^{l,1}\) with \(l>\frac{1}{2}h^k\) can be at most linear functions which is what we wanted to prove.
Consider now the exterior derivative of the next matrix valued contact system in Equation~\ref{reducedcontacteven}
\begin{equation}
0=dY_{k,k-1}\wedge dY_{k-1,k-2}
\end{equation}
By applying changes of coordinates we can assume that the independent variable entries in matrix \(Y_{k,k-1}\) are the first \(\frac{1}{2}h^k\) entries in each column. This implies that at least the first two rows of matrix \(Y_{k,k-1}\) consist of independent variables, which in turn implies that the entries of the matrix \(Y_{k-1,k-2}\) must be constant functions by applying the usual Cartan's lemma type argument. All the remaining matrix valued contact systems in Equation~\ref{reducedcontacteven} can be treated exactly the same way as in the odd weight case so we conclude that the entries in these matrices which are not independent variables must be constants. Together with the above results this concludes the proof for this case.
c) In this case \(w\geq 4\) and \(dim(E)=q^{even}_3\). By Theorem~\ref{dimboundthm} the matrix \(Y_{k,k-1}\) has exactly one column that consists entirely of independent variables and the other columns do not contain independent variables at all. We can assume that the entries of the first column are independent. Considering the entries of Equation~\ref{weight2add} it follows that the remaining entries of \(Y_{k,k-1}\) can be expressed as functions of the variables in the first column. In fact, from the \((1,2)\) entry of the exterior derivative of Equation~\ref{weight2add}
\begin{equation}
0=\sum_l(dY_{k,k-1})^{l,1}\wedge (dY_{k,k-1})^{l,2}
\end{equation}
it follows that the entries \((Y_{k,k-1})^{l,2}\) depend only on the independent variables \((Y_{k,k-1})^{l,1}\). Similarly, the same holds for the other columns.
In this case the matrix \(Y_{k-1,k-2}\) consists of independent variables except for its first row (cf. proof of Theorem~\ref{dimboundthm}). From the \((1,1)\) entry of the exterior derivative of Equation~\ref{4one}
\begin{equation}
0=\sum_l(dY_{k,k-1})^{1,l}\wedge (dY_{k-1,k-2})^{l,1}
\end{equation}
we can conclude that the entries \((Y_{k,k-1})^{1,l}\) for \(l\geq 2\)
depend only on \((Y_{k,k-1})^{1,1}\) and in fact, they must be linear
functions in this variable. Similarly, the entry
\((Y_{k-1,k-2})^{1,1}\) must be a linear function of the variables
\((Y_{k-1,k-2})^{1,l}\) \((l\geq 2)\). These claims easily follow from
the fact that the differential 1-forms \((dY_{k,k-1})^{1,1}\) and
\((dY_{k-1,k-2})^{1,l}\), \((l\geq 2)\) are linearly
independent. Considering the other entries we conclude that the
entries that are not independent variables must be linear functions. In fact, let us remark here that the coefficients of the linear functions in a given row of the matrix \(Y_{k,k-1}\) must be the same for each entry since for example in the second column each coefficient is equal to \(\frac{\partial((Y_{k-1,k-2})^{1,1})}{\partial((Y_{k-1,k-2})^{2,1})}\). It also follows that the coefficients of the linear functions \((Y_{k-1,k-2})^{1,i}\) are the same for each \(i\).
The remaining matrix valued contact systems in Equation~\ref{reducedcontacteven} can now be treated the same way as in the odd weight case which concludes the proof of the theorem in this case.
3) What remains to be considered to complete the proof of the theorem is the case when \(h^k\) is odd. We will exhibit an infinite dimensional family of germs of maximal dimensional integral manifolds tangent to an integral element \(E\). To this end let us specify the entries of the matrices in Equation~\ref{reducedcontacteven}. To give a maximal dimensional integral manifold we must specify \(\frac{1}{2}h^{k+1}(h^k-1)+1\) independent variables among the entries of \(Y^{k-1,k}\), according to Theorem~\ref{dimboundthm}. Let us choose the first \(\frac{1}{2}(h^k-1)\) variables in each column of \(Y^{k-1,k}\) to be independent and let the next \(\frac{1}{2}(h^k-1)\) entries in each column to be \(\sqrt{-1}\) times the first \(\frac{1}{2}(h^k-1)\) independent entries. This leaves the last row to be considered. Let the first element of the last row be an independent variable and let the remaining entries of the last row be arbitrary holomorphic functions of of the first entry.
For example, if \(h^k=5\) and \(h^{k+1}=4 \) then
\[
Y_{k-1,k}=
\begin{pmatrix}
y_{1,1}& y_{1,2} & y_{1,3} & y_{1,4}\\
y_{2,1}& y_{2,2} & y_{2,3} & y_{2,4}\\
i\cdot y_{1,1}& i\cdot y_{1,2} & i\cdot y_{1,3} & i\cdot y_{1,4}\\
i\cdot y_{2,1}& i\cdot y_{2,2} & i\cdot y_{2,3} & i\cdot y_{2,4}\\
y_{5,1}& f_{5,2} & f_{5,3} & f_{5,4}
\end{pmatrix}
\]
where the functions \(f_{a,b}\) are arbitrary holomorphic functions of the single variable \(y_{5,1}\) and \(i=\sqrt{-1}\).
Furthermore, let the matrices \(Y_{k-3-2j,k-2-2j}\) (for each \(j\)) consist entirely of independent variables and let the remaining matrices \(Y_{k-2-2j,k-1-2j}\) (for each \(j\)) be zero. This means that we have \(q^{even}_2\) independent variables. It is an easy computation to verify that all the equations in (\ref{reducedcontacteven}) are satisfied so we have defined a maximal dimensional integral manifold. Fixing an integral element to which this integral manifold has to be tangent to can determine only the linear parts of the functions \(f_{a,b}\) which implies that by varying these functions we can exhibit an infinite dimensional family of germs of integral manifolds tangent to a fixed integral element.
\end{proof}
\subsection{Maximal Dimensional Integral Elements}
In this section we examine how maximal dimensional integral elements are related to each other. First let us recall a theorem of Carlson (\cite{C1}) that will be used in this section.
\begin{thm}[Carlson]
Let \(D\) be the period domain of weight two Hodge structures with \(h^{2,0}>2\) and \(h^{1,1}\) even. Let \(E_1\) and \(E_2\) be maximal dimensional integral elements of the horizontal distribution. Then there is an element \(g\in Aut(D)\) such that \(g\cdot E_1=E_2\).
\end{thm}
We will consider the same question for higher weight, namely what can be said about the relationship between maximal dimensional integral elements of the horizontal differential system.
\begin{thm}\label{elementrigid}
Let \(D\) denote the period domain.
\begin{enumerate}
\item Assume that one of the following holds:
\begin{enumerate}
\item \(w=2k+1\), \(h^{k,k+1}>2\) and all the other Hodge numbers are greater than one
\item \(w=2k\), all the Hodge numbers are greater than one and the maximum dimension is \(q^{even}_1\)
\end{enumerate}
then there is a unique maximal dimensional integral element of the horizontal system through each point of \(D\).
\item Let \(E_1\), \(E_2\) be two maximal dimensional integral elements. Assume that one of the following holds:
\begin{enumerate}
\item \(w=2k\), \(dim(E_1)=dim(E_2)=q^{even}_2\), \(h^{k+1}>2\), the other Hodge numbers are greater than one and \(h^k\geq 4\) is even
\item \(w=2k\), \(dim(E_1)=dim(E_2)=q^{even}_3\) and \(h^{k+1}>2\)
\end{enumerate}
then there is an element \(g\in Aut(D)\) such that \(g\cdot E_1=E_2\).
\end{enumerate}
\end{thm}
\begin{proof}
1) a) and b) The horizontal system is homogeneous so we can always assume that the integral elements are at the reference Hodge structure. In Theorem~\ref{localrigid} we proved that in cases 1.a and 1.b the matrix entries are either independent variables or constants, and these constants are determined be the point of the period domain. If we are at the reference Hodge structure then all these constants are zero. To compute the tangent space to the integral manifold we have to take partial derivatives of the matrix entries by the independent variables. From this it is clear that there can be only one maximal dimensional integral element through each point of \(D\).
2) a) In this case it follows from the proof of Theorem~\ref{localrigid} that the matrices besides \(Y_{k,k-1}\) consist of independent variables or constant functions, so the subspace of the tangent space coming from these matrices is unique. Now, it is clear that the equations \(Y_{k+1,k}\) has to satisfy are exactly the weight two equations so Carlson's theorem applies to conclude the theorem in this case.
2) b) Again, from the proof of Theorem~\ref{localrigid} we see that the subspace of the tangent space coming from the matrices besides \(Y_{k,k-1}\) and \(Y_{k-1,k-2}\) is unique, so we need to consider these two matrices. About these matrices we proved that they contain linear entries such that the coefficients of the columns are the same. This implies that we can conjugate the possible tangent spaces into each other by multiplying by elements of \(Aut(D)\).
\end{proof}
\begin{rem}
It is true that result 2) of the above theorem holds even if \(h^{k}\) is odd. However, the author does not know a simple proof of this along the lines of these other results. Since there is an infinite dimensional family of integral manifolds to a given integral element in this case, we could not use this result to conclude further rigidity properties; hence it is not proved here.
\end{rem}
\begin{rem}
Theorem~\ref{localrigid} and Theorem~\ref{elementrigid} immediately imply our main result Theorem~\ref{mainthm}. Note that if the Hodge numbers are smaller than what is required for these results, then we are reduced to the classical contact case and so flexibility holds by Theorem~\ref{arnold}.
\end{rem}
\begin{rem}
It is not known whether the maximal dimensional variations are geometric if the weight is bigger than two. In the weight two, \(h^{1,1}\) even case this is true by \cite{C-S}.
It is an interesting question in the author's opinion whether the reduction of the horizontal system to the much smaller coupled contact system is merely a local possibility or there is some global geometric reason that would explain this result.
\end{rem}
|
1997-12-03T00:45:43 | 9712 | alg-geom/9712003 | en | https://arxiv.org/abs/alg-geom/9712003 | [
"alg-geom",
"math.AG"
] | alg-geom/9712003 | Janos Kollar | J\'anos Koll\'ar | Real Algebraic Surfaces | LATEX2e, 27 pages | null | null | null | null | These are the notes for my lectures at the Trento summer school held
September 1997. The aim of the lectures is to provide an introduction to real
algebraic surfaces using the minimal model program. This leads to a fairly
complete understanding of real rational surfaces and to a complete topological
classification of real Del Pezzo surfaces. Almost all the results are contained
in the works of Comessatti and Silhol.
| [
{
"version": "v1",
"created": "Tue, 2 Dec 1997 23:45:42 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Kollár",
"János",
""
]
] | alg-geom | \section{Minimal models of real algebraic surfaces}
\begin{defn} Let $X$ be a variety over a field $k$.
A {\it 1--cycle} on $X$ is a formal linear combination $C=\sum c_iC_i$,
where the $C_i\subset X$ are irreducible, reduced and proper curves. A
1--cycle is called {\it effective} if $c_i\geq 0$ for every $i$.
Two 1--cycles $C,C'$ are {\it numerically equivalent} if
$(C\cdot D)=(C'\cdot D)$ for every Cartier divisor $D$ on $X$.
1--cycles with real coefficients modulo numerical equivalence form a
vectorspace, denoted by $N_1(X)$. $N_1(X)$ is finite dimensional by the
Theorem of the base of N\'eron--Severi (cf.
\cite[p.447]{Hartshorne77}). Its dimension, denoted by $\rho(X)$, is
called the {\it Picard number} of $X$.
Effective 1--cycles generate a cone $NE(X)\subset N_1(X)$. Its closure
in the Euclidean topology $\overline{NE}(X)\subset N_1(X)$ is called
the {\it cone of curves} of $X$.
If $K_X$ is Cartier (or at least some multiple of $K_X$ is Cartier) then
set
$$
\overline{NE}(X)_{K\geq 0}:=\{z\in \overline{NE}(X)\vert (z\cdot
K_X)\geq 0\}.
$$
Let $V\subset \r^n$ be a closed convex cone. For $v\in V$, a ray
$\r^{\geq 0}v\subset V$ is called {\it extremal} if $u,u'\in V$, $u+u'\in
\r^{\geq 0}v$ implies that $u,u'\in\r^{\geq 0}v$. Intuitively: $\r^{\geq 0}v$
is an edge of
$V$.
An extremal ray $\r^{\geq 0}z\subset \overline{NE}(X)$ is called {\it
$K_X$-negative} if $(z\cdot K_X)<0$. This does not depend on the choice
of $z$ in the ray.
Let $R\subset \overline{NE}(X)$ be a ray. A {\it
contraction} of
$R$ is a morphism $f_R:X\to X'$ such that $(f_R)_*\o_X=\o_{X'}$ and a
curve $C\subset X$ is mapped to a point iff $[C]\in R$.
\end{defn}
\begin{exrc}\label{er.noncontr.ex}
Show that the contraction of a ray is unique
(if it exists). Also, if $X'$ is projective then $R$ is an extremal
ray. Find examples of extremal rays which can not be contracted.
\end{exrc}
\begin{exrc}\label{int.cone.exrc}
Let $F$ be a smooth projective surface and $z\in
\overline{NE}(F)$ a 1--cycle such that $(z^2)>0$.
Then $z$ or $-z$ is in the interior of $\overline{NE}(F)$.
Thus if $\r^{\geq 0}z$ is extremal and $(z^2)>0$ then
$\overline{NE}(F)$ is 1--dimensional.
\end{exrc}
We use the following description of the cone of curves of smooth
surfaces over $\c$. The result is essentially equivalent to the theory
of minimal models of surfaces developed around the turn of the 20th
century. This formulation (and its higher dimensional generalization)
is due to \cite{Mori82}. See also \cite{koll96, km98} for proofs.
\begin{thm}[Cone Theorem]\label{cone.thm.c}
Let $F$ be a smooth projective surface over an
algebraically closed field. Then there are curves $C_i\subset F$
such that
$$
\overline{NE}(F)=\overline{NE}(F)_{K\geq 0}+\sum \r^{\geq 0}[C_i],
$$
and the $\r^{\geq 0}[C_i]$ are $K_F$-negative extremal rays of
$\overline{NE}(F)$.
Moreover, we can assume that each
$C_i\subset F$ is a smooth rational curve and
$(C_i^2)\in\{-1,0,1\}$. If $(C_i^2)=1$ (resp. $(C_i^2)=0$) for
some $i$ then $F\cong
\p^2$ (resp. $F$ is a minimal ruled surface over a curve). \qed
\end{thm}
Let now $F$ be a smooth projective surface over $\r$. If $C$ is a
1--cycle on $F_{\c}$ then $C+\bar C$ is a 1--cycle on $F$, and
every 1--cycle on $F$ arises this way, at least if we use rational or
real coefficients. Thus (\ref{cone.thm.c}) immediately gives:
\begin{thm}[Cone Theorem over $\r$]\label{cone.thm.r}
Let $F$ be a smooth projective surface over $\r$. Then there are
smooth rational curves
$C_i\subset F_{\c}$ with $(C_i^2)\in\{-1,0,1\}$ such that
$$
\overline{NE}(F)=\overline{NE}(F)_{K\geq 0}+\sum \r^{\geq 0}[C_i+\bar C_i],
$$
and the $\r^{\geq 0}[C_i+\bar C_i]$ are $K_F$-negative extremal rays of
$\overline{NE}(F)$.
\end{thm}
Proof. There is one point that we need to be careful about.
Namely, it happens frequently that $C_i$ gives an extremal ray but
$C_i+\bar C_i$ does not. So we have to throw away some of the $C_i$
appearing in (\ref{cone.thm.c}). \qed
\medskip
\begin{say}[Geometric irreducibility] Let $X\subset \p^n$ be a variety
over
$\c$ and $\bar X$ the variety defined by conjugate equations. The
disjoint union of $X$ and $\bar X$ is invariant under conjugation, and
so there is a real variety $Y_{\r}$ such that $Y_{\c}\cong X\cup \bar
X$. Such real varieties are not particularly interesting since the
theory of $Y_{\r}$ over $\r$ is equivalent to the theory
of $X$ over $\c$. Thus it is reasonable to restrict our attention to
real varieties $Y$ such that $Y_{\c}$ is irreducible, that is, $Y$ is
geometrically irreducible.
Of course, during a proof we may run into a subvariety of $Y_{\r}$
which is geometrically reducible, and these have to be dealt with
appropriately. Thus we can not ignore such varieties completely.
\end{say}
\begin{defn} Let $S$ be a smooth projective surface over a field $k$.
$S$ is called a {\it Del Pezzo surface} if $S$ is geometrically
irreducible and
$-K_S$ is ample. It is called {\it minimal} (over $k$) if $\rho(S)=1$.
$S$, together with a morphism to a smooth curve $f:S\to B$ is called a
{\it conic bundle} if every fiber is isomorphic to a plane conic.
A conic bundle is called {\it minimal} if $\rho(S)=2$.
\end{defn}
The geometric description and meaning of the extremal rays occurring in
(\ref{cone.thm.r}) is given in the next result:
\begin{thm}\label{mmp.over.R}
Let $F$ be a smooth projective geometrically irreducible surface over
$\r$ and $R\subset \nec{F}$ a $K_F$-negative extremal ray. Then $R$ can
be contracted
$f:F\to F'$, and we obtain one of the following cases:
\begin{enumerate}
\item[(B)] (Birational) $F'$ is a smooth projective surface over
$\r$ and $\rho(F')=\rho(F)-1$.
$F$ is the blow up of $F'$ at a closed point $P$. We have two
cases:
\begin{enumerate}
\item $P\in F'(\r)$, or
\item $P$ is a pair of conjugate points.
\end{enumerate}
\item[(C)] (Conic bundle) $B:=F'$ is a smooth curve, $\rho(F)=2$ and
$F\to B$ is a conic bundle. The fibers $f^{-1}(P): P\in B(\c)$ are
smooth, except for an even number of ponts $P_1,\dots,P_{2m}\in B(\r)$.
$(K_F^2)=8(1-g(B))-2m$.
\item[(D)] (Del Pezzo surface) $F'$ is a point, $\rho(F)=1$, $-K_F$ is
ample and we have one of the following cases
\begin{enumerate}
\item $(K_F^2)=9$ and $F\cong \p^2$.
\item $(K_F^2)=8$ and $F\cong (x_0^2+x_1^2+x_2^2-x_3^2=0)\subset
\p^3$.
\item $(K_F^2)=2$.
\item $(K_F^2)=1$.
\end{enumerate}
\end{enumerate}
\end{thm}
Proof. By (\ref{mmp.over.R}),
there is a curve $C\cong \p^1$ over $\c$ such that
$C+\bar
C$ generates $R$ and $(C^2)\in \{-1,0,1\}$.
We consider various possibilities.
Assume first that $(C^2)=-1$. If $C=\bar C$ then
the contraction of $C$ in $F_{\c}$ is defined over $\r$, thus
$F$
is the blow up of a surface at a real point.
(This is Castelnuovo's contraction theorem, cf.
\cite[V.5.7]{Hartshorne77}.) If $C$ and $\bar C$ are disjoint, then we
can contract them simultaneously over
$\r$ to obtain
$f:F\to F'$ which is an isomorphism near $F(\r)$.
If $(C^2)=-1$ and $(C\cdot \bar C)=1$ or $(C^2)= (C\cdot \bar
C)=0$, then $C+\bar C$ has selfintersection 0.
From Riemann--Roch we obtain that
$$
h^0(F, \o_F(m(C+\bar C))
\geq \chi (F, \o_F(m(C+\bar C))= m+\chi(\o_F),
$$
thus $m(C+\bar C)$ moves in a linear system for $m\gg 1$. It's moving
part is base point free by (\ref{bpf.exer}).
Let $f:F\to B$ be the Stein factorization of the resulting morphism.
Let $A\subset F$ be an irreducible fiber. If $A$ is a multiple
fiber, write it as $A=mA_1$. Since $[A_1]\in R$,
$$
2g(A_1)-2=(A_1^2)+(A_1\cdot K_F)=(A_1\cdot K_F)<0.
$$
Thus $A_1$ is isomorphic to a smooth conic over $\r$ (\ref{conics.r})
and $(A_1\cdot K_F)=-2$. The generic fiber $A_g$ is not multiple, so
$(mA_1\cdot K_F)=(A_g\cdot K_F)=-2$ which shows that there are no
multiple fibers.
Let $A_1+ A_2=f^{-1}(b)$ be a reducible fiber over $\c$, where $A_1$
is an irreducible and reduced curve. In particular, $(A_1^2)<0$.
$\bar A_1+ \bar A_2=f^{-1}(\bar b)$ is also a fiber.
If $b\neq \bar b$ then $A_1$ is disjoint from $\bar A_1$, thus
$((A_1+\bar A_1)^2)=(A_1^2)+(\bar A_1^2)<0$.
$[A_1+\bar A_1]+[A_2+\bar A_2]\in R$ and
$[A_1+\bar A_1]\not\in R$, a contradiction. Thus all singular fibers
lie over real points $P_1,\dots, P_r\in B(\r)$.
Therefore, $A_1+\bar A_1\subset f^{-1}(b)$. Every fiber of $f$ is
irreducible over $\r$, thus $A_1+\bar A_1= f^{-1}(b)$.
We get that $(A_1\cdot K_X)=-1$ and so $A_1$ is a $-1$-curve
and $A_1+\bar A_1$ is isomorphic to a pair of conjugate lines in
$\p^2$.
$F$ is a conic bundle over $B$ by (\ref{conic.bund}).
Over $\c$ we can contract one of the components of every singular fiber
to obtain a minimal ruled surface. The selfintersection number of the
canonical class of a minimal ruled surface is $8(1-g(B))$, and each
singular fiber drops this number by 1. We see in (\ref{mmp.over.R.top})
that the number of singular fibers is even.
Assume next that $(C^2)=-1$ and
$r:=(C\cdot \bar C)\geq 2$. Then
$$
((C+\bar C)^2)=-2+2(C\cdot \bar C)=2r-2>0.
$$
By (\ref{int.cone.exrc})
this implies that $\nec{F}$ is 1-dimensional, hence $-K_F\equiv
a(C+\bar C)$ for some $a>0$.
$(-K_F\cdot (C+ \bar C))=2$, thus
$$
(C+\bar C)\equiv (1-r)K_F\qtq{and} 2r-2=(1-r)^2(K_F^2).
$$
This gives the possibilities $r=2, (K_F^2)=2$ or $r=3, (K_F^2)=1$.
If $(C^2)=0$ and
$r:=(C\cdot \bar C)\geq 1$ then a computation as above gives that
$8=r(K_F^2)$, which allows too many cases. It is better to consider
this geometrically.
By (\ref{cone.thm.c}),
$C$ is a fiber of a $\p^1$-bundle $g:F_{\c}\to D$ over $\c$
and $\bar C$ is a (possibly multiple) section of $g$.
Thus $D$ is rational. By the classification of minimal ruled surfaces,
either $F_{\c}\cong \p^1\times \p^1$ and we are done by
(\ref{quadric.lem}), or
$g$ has a unique section
$E$ with negative selfintersection. $E$ is then defined over $\r$, thus
$\rho(F)=2$, a contradiction.
We are left with the case when $F_{\c}\cong \c\p^2$
and $C$ is a line in $\c\p^2$. Then $\bar C$ is another line
and $C$ and $\bar C$ intersect in a unique point, which is therefore
real. We can get another real point, and so also a real line. Thus
$\o_F(1)$ is defined over
$\r$ and
$F\cong
\r\p^2$. \qed
\medskip
As a consequence we obtain the minimal model program (MMP for short)
for real algebraic surfaces:
\begin{thm}[MMP for surfaces]\label{mmp.surf.thm}
Let $F$ be a smooth projective geometrically irreducible surface over
$\r$. Then there is a sequence of morphisms
$$
F=F_0\stackrel{f_0}{\to} F_1 \to \cdots F_{m-1}\stackrel{f_{m-1}}{\to}
F_m=F^*
$$
such that each $f_i:F_i{\to} F_{i+1}$ is a birational contraction as in
(\ref{mmp.over.R}.B) and $F^*$ satisfies precisely one of the following
properties:
\begin{enumerate}
\item[(M)] (Minimal model) $K_{F^*}$ is nef. (That is, it has
nonnegative intersection number with every curve in $F^*$.)
\item[(C)] (Conic bundle) $F^*$ is a conic bundle
over a curve
$f:F^*\to B$. In particular, $\rho(F^*)=2$
\item[(D)] (Del Pezzo surface) $\rho(F^*)=1$, $-K_{F^*}$ is ample
and $F^*$ is among those listed in (\ref{mmp.over.R}.D).
\end{enumerate}
\end{thm}
Proof. We do the steps of (\ref{mmp.over.R}.B) as long as we can.
$\rho(F_{i+1})=\rho(F_i)-1$, so eventually we reach $F^*=F_m$ where we
can not perform a contraction as in (\ref{mmp.over.R}.B).
If $K_{F^*}$ is nef then we have a minimal model.
If $K_{F^*}$ is not nef, then by (\ref{mmp.over.R}) we can perform a
contraction as in (\ref{mmp.over.R}.C--D). This gives our last two
cases.\qed
\begin{exrc}\label{conics.r} Let $k$ be a field and $C$ a smooth
projective curve over $k$ such that $C_{\bar k}\cong \p^1$. Show that
$C$ is isomorphic to a smooth conic over $k$. Also, $C\cong \p^1_k$
iff $C(k)\neq\emptyset$.
\end{exrc}
\begin{exrc}\label{conic.bund} Let $S$ be a smooth projective surface
over $\c$ and $f:S\to B$ a morphism to a smooth curve. Assume that
$f^{-1}(b)\cong \p^1$ for some $b\in B$ and every fiber has at most 2
irreducible components. Show that
$-K_F$ is very ample on the fibers, $f_*\o_F(-K_F)$ is a rank 3
vector bundle over $B$ and we have an injection
$$
F\DOTSB\lhook\joinrel\rightarrow \proj_B f_*\o_F(-K_F).
$$
Under this injection the fibers of $f$ become conics. Such a surface is
called
a {\it conic bundle} over $B$.
\end{exrc}
\begin{exrc}\label{projspace.ex}
Let $X$ be a variety over $\r$ such that $X_{\c}\cong
\p^n$. Show that $X\cong \p^n$ if $n$ is even, but not necessarily if
$n$ is odd.
\end{exrc}
\begin{exrc}[Cohomology commutes with base change]\label{coh.b.c}
Let $X$ be a variety over $\r$ and $F$ a coherent sheaf on $X$. Show
that
$$
H^i(X,F)\otimes_{\r}\c\cong H^i(X_{\c}, F_{\c}).
$$
\end{exrc}
\begin{exrc}\label{bpf.exer} Let $F$ be a smooth projective surface and
$D\subset F$ an irreducible curve such that $(D^2)=0$. Then the moving
part of $|mD|$ is either empty or base point free.
\end{exrc}
\begin{notation} $Q^{r,s}$ denotes the quadric hypersurface
$(x_1^2+\cdots +x_r^2-x_{r+1}^2-\cdots-x_{r+s}^2=0)$.
$Q^{2,1}$ is isomorphic to $\p^1$ by (\ref{conics.r}).
\end{notation}
\begin{lem}\label{quadric.lem}
Let $F$ be a smooth projective surface over $\r$ such that
$F_{\c}\cong \p^1\times \p^1$. Then
one of the
following holds:
\begin{enumerate}
\item $F\cong Q^{2,2}\cong Q^{2,1}\times Q^{2,1}$, $\rho(F)=2$ and
$F(\r)\sim S^1\times S^1$,
\item $F\cong Q^{3,1}$, $\rho(F)=1$ and $F(\r)\sim S^2$,
\item $F\cong Q^{4,0}\cong Q^{3,0}\times Q^{3,0}$, $\rho(F)=2$ and
$F(\r)=\emptyset$,
\item $F\cong Q^{3,0}\times \p^1$, $\rho(F)=2$ and
$F(\r)= \emptyset$.
\end{enumerate}
\end{lem}
Proof. Let $C\subset F_{\c}$ be one of the rulings. Then $\bar C$ is
another ruling, thus either $(C\cdot \bar C)=0$ or $(C\cdot \bar C)=1$.
If $(C\cdot \bar C)=0$ then the linear system $|C+\bar C|$ is defined
over $\r$ and maps $F$ onto a conic. Similarly for the other rulings,
thus $F$ is the product of two conics.
All 3 possibilities are listed.
If $(C\cdot \bar C)=1$ then $\o_F(C+\bar C)$ is a line bundle on $F$
which is of type
$\o_{F_{\c}}(1,1)$ over $\c$. Thus its global sections embed $F$ as a
quadric.
$Q^{3,1}$ is the only quadric not yet accounted for.\qed
\medskip
In the above proofs we had to establish several times that certain line
bundles on $F_{\c}$ are defined over $\r$. This is frequently a quite
subtle point. Some aspects of it are treated in the next exercise.
\begin{exrc}\label{pic=gal.inv} Let $X$ be a scheme over $\r$ and
$L$ a line bundle on $X$. Then $L_{\c}$ is a line bundle on $X_{\c}$
and $L_{\c}\cong \bar L_{\c}$.
Thus if $M$ is a line bundle on $X_{\c}$ and $M\not\cong \bar M$,
then $M$ is not the complexification of a real line bundle.
Find a curve $C$ over $\r$ and a line bundle $M$ on $C_{\c}$ such that
$M\cong \bar M$ but $M$ is not the complexification of a real line
bundle.
Let $X$ be a scheme over $\r$ and $M$ a line bundle on $X_{\c}$ such
that
$M\cong \bar M$. Show that $M^{\otimes 2}$ is the complexification of a
real line bundle. If $X$ is connected, reduced and $X(\r)\neq \emptyset$
then $M$ itself is the complexification of a real line
bundle.
More generally, let $X_K$ be an integral scheme defined
over a field
$K$ and
$L\supset K$ a Galois extension with Galois group $G$.
Show that if $X_K$
has a $K$-point then $\pic (X_K)=\pic (X_L)^G$.
\end{exrc}
\section{The topology of $F(\r)$}
In this section we study the MMP from the topological point of view.
The main results of this section are already in \cite{Comessatti14}.
\begin{notation} $M\uplus N$ denotes the disjoint union of $M$ and $N$.
$\uplus rN$ denotes the disjoint union of $r$ copies of $N$.
$M\# N$ denotes the connected sum of two manifolds $M$ and $N$ (which
are assumed to have the same dimension).
$\# rN$ denotes the connected sum of $r$ copies of $N$.
(By definition, $\#0M=S^{\dim M}$.) $M\sim N$ denotes
that $M$ and $N$ are homeomorphic.
\end{notation}
One can give a complete topological description of the
various contractions in (\ref{mmp.over.R}):
\begin{thm}\label{mmp.over.R.top}
Let $F$ be a smooth projective geometrically irreducible surface over
$\r$ and $R\subset \nec{F}$ a $K_F$-negative extremal ray. The following
is the topological description of the corresponding contraction:
\begin{enumerate}
\item[(B)] (Birational) $F$ is the blow up of $F'$ at a closed point
$P$. We have two cases:
\begin{enumerate}
\item If $P\in F'(\r)$ then $F(\r)\sim F'(\r)\#\r\p^2$.
\item If $P$ is a pair of conjugate points then $F(\r)\sim F'(\r)$.
\end{enumerate}
\item[(C)] (Conic bundle) $f:F\to B $ is a conic bundle with singular
fibers
$f^{-1}(P_1),\dots,f^{-1}(P_{2m})$. Then
$$
F(\r)\sim \uplus mS^2\uplus N_1\uplus\cdots \uplus N_b,
$$
where $b$ is the number of connected components of $B(\r)$
which do not contain any of the points $P_i$ and each $N_i$ is either a
torus or a Klein bottle.
\item[(D)] (Del Pezzo surface) There are 4 cases:
\begin{enumerate}
\item If $(K_F^2)=9$ then $F(\r)\sim \r\p^2$.
\item If $(K_F^2)=8$ then $F(\r)\sim S^2$.
\item If $(K_F^2)=2$ then $F(\r)\sim \uplus 4S^2$.
\item If $(K_F^2)=1$ then $F(\r)\sim \r\p^2\uplus 4S^2$.
\end{enumerate}
\end{enumerate}
\end{thm}
Proof. Blowing up replaces a point with all tangent directions
through that point. So we remove a disc and put in an interval bundle
over $S^1$ whose boundary is connected. This is a M\"obius strip and so
$F(\r)\sim
F'(\r)\#\r\p^2$.
In the conic bundle case,
let $M\sim S^1$ be a connected component of $B(\r)$. If none of the
$P_i$ lie on $M$ then $F(\r)\to B(\r)$ is a smooth $S^1$-bundle over
$M$, this gives either a
torus or a Klein bottle. If $k$ of the points
$P_1,\dots,P_k\in M\sim S^1$ correspond to singular
fibers then, after reindexing, they divide $M$ into $k$ intervals
$[P_i,P_{i+1}]$ (subscript
$\mod k$). $F(\r)$ is alternatingly empty or a copy of $S^2$ over
the intervals. Thus $k$ is even.
In the Del Pezzo case we are done if $(K_F^2)=9,8$. The cases
$(K_F^2)=2,1$ are considerably harder. They follow from
(\ref{deg2.equiv.thm}) and (\ref{deg1.equiv.thm}).\qed
\medskip
Using (\ref{mmp.over.R.top}) it is easy to determine which
2-manifolds occur as $F(\r)$ for geometrically rational surfaces $F$.
The conclusion is that orientable surfaces of genus $>1$ do not
occur. This is the main result of
\cite{Comessatti14}.
\begin{thm}\label{c-rat.top}
Let $F$ be a smooth,
projective surface over
$\r$ such that $F_{\c}$ is rational.
Then one of the following holds:
\begin{enumerate}\setcounter{enumi}{-1}
\item $F(\r)=\emptyset$.
\item $F(\r)\sim S^1\times S^1$.
\item $F(\r)\sim \#r_1\r\p^2\uplus \cdots \uplus \#r_m\r\p^2$ for some
$r_1,\dots,r_m\geq 0$.
\end{enumerate}
All these cases do occur.
\end{thm}
Proof. Apply the MMP over $\r$ to get $F=F_1\to F_2\to\cdots$.
We prove the theorem by induction on the number of blow ups in the
sequence. If $F_i\to F_{i+1}$ is the inverse of the blowing up
of a real point, then $F_i(\r)\sim F_{i+1}(\r)\#\r\p^2$. If
$F_i\to F_{i+1}$ is the inverse of the blowing up
of a pair of conjugate points, then $F_i(\r)\sim F_{i+1}(\r)$.
The induction works since $(S^1\times S^1)\#\r\p^2\sim \#3\r\p^2$.
Thus we are reduced to
one of the following two cases:
\begin{enumerate}
\item $F$ has a conic bundle structure $F\to B$, or
\item $F$ is Del Pezzo and $\rho(F)=1$.
\end{enumerate}
In the first case, $B_{\c}\cong \c\p^1$ since $F_{\c}$ is rational.
Thus either
$B(\r)=\emptyset$ and so $F(\r)=\emptyset$, or $B\cong \r\p^1$.
Thus $F(\r)$ is the torus or the Klein bottle if there are no singular
fibers and
$F(\r)\sim \uplus mS^2$ if there are $2m>0$ singular fibers
by (\ref{mmp.over.R.top}.C). Note that $S^2=\#0\r\p^2$ by convention.
In the second case we use (\ref{mmp.over.R.top}.D). \qed
\begin{exrc}\label{comp.bir.inv}
Let $X$ and $Y$ be smooth projective varieties over $\r$.
Assume that $X$ and $Y$ are birational to each other (over $\r$). Show
that $X(\r)$ and $Y(\r)$ have the same number of connected components.
\end{exrc}
\begin{say}[Vector bundles over real varieties] Let $X$ be a veriety
over $\r$ and $p:V\to X$ a vector bundle of rank $n$. Locally
$V$ is like
$U\times {\mathbb A}_{\r}^n\to U$ where $U\subset X$ is Zariski open
and ${\mathbb A}_{\r}^n=\spec_{\r}\r[t_1,\dots,t_n]$ is affine $n$-space
over $\r$. (Which should {\em not} be identified with $\r^n$!)
As usual, to $V$ one can associate a vector bundle $p_{\c}:V_{\c}\to
X_{\c}$ and also a real vector bundle
$p(\r):V(\r)\to X(\r)$ which is obtained by taking the $\r$-valued
points of ${\mathbb A}_{\r}^n$ which is exactly $\r^n$.
(To complete the picture, any real vector bundle on a manifold can be
complexified, and $V(\r)\otimes_{\r}\c\cong V_{\c}|_{X(\r)}$.)
\end{say}
\begin{say}[Degrees of line bundles over $\r$ and $\c$]
\label{dolb.exrc}{\ }
Let $B$ be a smooth projective curve over $\c$ and $L$ a line
bundle on $B$. Let $s$ be a nonzero meromorphic section of
$L$. The number of zeros minus the number of poles of $s$ on $B$
(counted with multiplicity) is called the {\it degree of $L$}. Let $Y$
be a smooth projective variety over
$\c$ and
$L$ a line bundle on $Y$. For any curve $B\subset Y$
the degree of $L|_B$ is defined. It is
also
called the {\it intersection number} of $B$ and $L$ and denoted by
$(B\cdot L)$.
Let $A\sim S^1$ be a compact 1--dimensional manifold and $L$ a real
line bundle on $M$. Let $s$ be a nonzero section of
$L$. The number of zeros of $s$ on $A$
(counted with multiplicity) makes sense only mod 2. If $M$ is a
compact manifold
and
$L$ a real line bundle on $M$ then for any 1-cycle $A\subset M$
we obtain the $\z_2$-valued {\it intersection number} of $A$ and $L$.
It is denoted by
$(A\cap L)$. (To be precise, I should write
$(A\cap w_1(L))$ where $w_1(L)$ stands for the first
Stiefel--Whitney class of $L$. This is a class
in $H^1(X(\r),\z_2)$ analogous to the first Chern class of a
complex line bundle, cf. \cite[Sec. 4]{milnor-s74}.)
Let now $X$ be a smooth projective variety over $\r$, $C\subset X$ a
curve and $L$ a line bundle on $X$.
We obtain two numbers:
$$
(L(\r)\cap C(\r)) \qtq{and} (C_{\c}\cdot L_{\c}).
$$
What is the relationship between them?
To answer this, take a real meromorphic section
$s$ of $L$ which has only finitely many zeros and poles on $C$.
When we count the real zeros and poles of $s$ on $C(\r)$, we miss the
complex zeros and poles of $s$ on $C_{\c}$. Since $s$ is real, the
complex zeros and poles come in conjugate pairs. Thus we conclude that
$$
(C(\r)\cap L(\r))\equiv (C_{\c}\cdot L_{\c}) \mod 2,
$$
which is best possible since the left hand side is defined only mod 2
anyhow.
\end{say}
\begin{say}[Orientability of $X(\r)$ and the canonical class]{\ }
Let $M$ be a differentiable manifold,
$0\in M$ a point and $x_1,\dots,x_n$ local coordinates.
A {\it local orientation} of $M$ at $0$ is a choice of an $n$-form
$f(x)dx_1\wedge \dots\wedge dx_n$ with $f(0)\neq 0$ up to multiplication
by a positive function. An {\it orientation} of $M$ is a nowhere zero
global $n$-form on $M$, up to multiplication
by a positive function. $n$-forms are sections of the real line bundle
$\det T^*_M$.
If $S^1\sim A\subset M$ is a loop then one can choose a consistent
oreintation of $M$ along $A$ $\Leftrightarrow$ $\det T^*_M$ has a
nowhere zero section along $A$ $\Leftrightarrow$ $(\det T^*_M\cap A)=0$.
If $X$ is a smooth variety over $\r$ then $n$-forms appear as
sections of the canonical line bundle. This proves that
$$
\det T^*_{X(\r)} \cong K_X(\r).
$$
In many cases this gives a way to decide if $X(\r)$ is orientable or
not.
\end{say}
\begin{cor} Let $X$ be a smooth projective variety over $\r$. Assume
that there is a curve $C\subset X$ such that $(C\cdot K_X)$ is odd.
Then $X(\r)$ is not orientable.
\end{cor}
Proof. We have proved above that
$$
(\det T^*_{X(\r)}\cap C(\r))\equiv (C\cdot K_X) \equiv 1\mod 2.
$$
$C(\r)$ may have several components, but along one of them
$\det T^*_{X(\r)}$ has odd degree, so we can not choose a consistent
orientation along that component.\qed
\begin{exrc} Show that $\r\p^n$ is orientable iff $n$ is odd.
Let $X\subset \p^n$ be a smooth hypersurface of degree $d$.
Show that $X(\r)$ is orientable if $n-d$ is odd.
Show that $X(\r)$ is not orientable if $n$ and $d$ are both odd.
If $n$ and $d$ are both even, then $X(\r)$ may or may not be
orientable.
\end{exrc}
If $(C\cdot K_X)$ is even, then it can happen that $X(\r)$ is not
orientable along an even number of components of $C(\r)$. In some cases
we are still able to conclude orientability of $X(\r)$ using
stronger assumptions:
\begin{exrc}\label{gen.orient.ex}
Let $X$ be a smooth projective variety over $\r$. Assume
that $K_X\cong L^{\otimes 2}$ for a real line bundle $L$.
Show that $X(\r)$ is orientable.
More generally, assume that $K_X\cong \o_X(2D+D')$ where $D,D'$ are
divisors over $\r$ and $D'(\r)$ has codimension at least 2 in $X(\r)$.
(This is equivalent to assuming that every irreducible component of
$D'$ is geometrically reducible.) Show that $X(\r)$ is orientable.
\end{exrc}
\section{Birational classification}
\begin{defn} Let $F$ be a smooth real algebraic surface. A surface
obtained from $F$ by blowing up $a$ real points and $b$ pairs of
conjugate complex points (possibly infinitely near) is denoted by
$(F,a,2b)$.
Given $F$ and $a,b$, the surfaces of the form $(F,a,2b)$
consitute a connected family if $F(\r)$ and $F_{\c}$ are both connected.
\end{defn}
\begin{lem}\label{elem.bir.lem}
We have the following elementary birational equivalences
between the minimal models in (\ref{mmp.over.R}).
\begin{enumerate}
\item $(\p^2,2,0)\cong (Q^{2,2},1,0)$.
\item $(\p^2,0,2)\cong (Q^{3,1},1,0)$.
\item $(Q^{4,0},0,2)$ is isomorphic to the blow up of $Q^{3,0}\times
\p^1$ at a pair of conjugate points on the same section
$Q^{3,0}\times P$, $P\in \r\p^1$.
\item Any minimal conic bundle over a rational curve with $2$ singular
fibers is isomorphic to $(Q^{3,1},0,2)$.
\end{enumerate}
\end{lem}
Proof. In the first two cases we blow up the 2 points in $\p^2$ and
then contract the line through them to get a quadric.
$Q^{4,0}\cong Q^{3,0}\times Q^{3,0}$, let $\pi_1$ be the first
projection. The pencil of planes through the 2 points gives a map
$p:Q^{4,0}\map \p^1$.
$$
(\pi_1,p):Q^{4,0}\map Q^{3,0}\times \p^1 \qtq{is birational}
$$
and becomes a morphism after blowing up the 2 points.
Finally assume that $F\to B$ is a minimal conic bundle over a rational
curve with $2$ singular fibers. By (\ref{mmp.over.R}.C),
$B(\r)\neq\emptyset$, thus $B\cong \p^1$. $F_{\c}$ is the blow up of a
minimal ruled surface $F''$ at 2 points. We can even assume that
$F''$ has a section $E$ with negative selfintersection $(E^2)=-k$
and the two points are not on $E$. If $k\geq 2$ then all other
sections of $F''$ have selfintersection at least 2, so $E\subset F$ is
the unique section with negative selfintersection. Thus $E$ is
defined over $\r$ and $F\to B$ is not minimal.
Thus $k=1$ and there is a unique section $E'\subset F''$ such that
${(E'}^2)=1$ and $E'$ passes through the two blown up points.
Let $\bar E\subset F$ be the birational transform. Then $E$ and $\bar
E$ have to be conjugate. Contracting them gives the quadric
$Q^{3,1}$.\qed
\begin{lem}\label{minruled.lem}
Let $F$ be a smooth projective surface over $\r$ such that
$F_{\c}$ is a minimal ruled surface over $\c\p^1$. Then
one of the
following holds:
\begin{enumerate}
\item $F_{\c}\cong \p^1\times \p^1$ (these cases were enumerated in
(\ref{quadric.lem})), or
\item $F\cong \proj_B(\o_B+\o_B(-r))$ is a minimal ruled surface over a
smooth real conic $B$ for some $r>0$.
\end{enumerate}
\end{lem}
Proof. By the classification of minimal ruled surfaces,
either $F_{\c}\cong \p^1\times \p^1$, or $F_{\c}$
has a unique irreducible curve
$E$ with negative selfintersection $-r$. $E$ and the ruling $g:F\to B$
are then defined over
$\r$ and $g_*\o_F(E)\cong \o_B+\o_B(-r)$. \qed
\medskip
As a corollary, we obtain the following birational classification
of real surfaces such that $F_{\c}$ is rational:
\begin{cor}\label{birclass.over.R}
Let $F$ be a smooth real projective surface
such that $F_{\c}$ is rational. Then $F$ is birationally equivalent
over $\r$ to a surface in exactly one of the following classes:
\begin{enumerate}
\item[1.] $Q^{3,0}\times \p^1$. In this case $F(\r)=\emptyset$.
\item[2.] $\p^2$. In this case $F(\r)$ is connected.
\item[3$_m$.] Minimal conic bundle with $2m$ ($m\geq 2$) singular
fibers. In this case
$F(\r)$ has $m$ connected components.
\item[4.] Degree 2 minimal Del Pezzo surface.
\item[5.] Degree 1 minimal Del Pezzo surface.
\end{enumerate}
\end{cor}
Remark. In (\ref{deg2.equiv.thm}) and (\ref{deg1.equiv.thm})
we prove that $F(\r)$ has 4 (resp. 5) connected components if $F$ is
a minimal Del Pezzo surface of degree 2 (resp. 1).
\medskip
Proof. Let $F\to F^*$ be the minimal model of $F$. By (\ref{mmp.over.R})
$F^*$ is either one of those listed above, or $F^*$ is a conic bundle
with $0$ or $2$ singular fibers.
The former are treated in (\ref{minruled.lem}).
The latter are birational to
$Q^{3,0}\times \p^1$ by (\ref{elem.bir.lem}).
The number of connected components of the real part is a birational
invariant (\ref{comp.bir.inv}), hence the cases (1--3$_m$) are all
different birationally.
The cases (4--5) differ
birationally from the other ones by (\ref{seg-man.deg123}).\qed
We use, without proof, the following result about the birational
classification of low degree Del Pezzo surfaces over any field.
Lectures 2--3 of \cite{KS97} serve as a good introduction.
\begin{thm}\cite{Segre51, Manin66}\label{seg-man.deg123}
Let $k$ be a field (of characteristic zero) and $F$ a minimal Del
Pezzo surface of degree 1,2 or 3 over $k$. Then
\begin{enumerate}
\item $F$ is not rational (over $k$),
\item $F$ is not birational (over $k$) to any conic fibration,
\item $F$ is birational to another minimal Del
Pezzo surface $F'$ of degree 1,2 or 3 over $k$ iff $F$ is isomorphic to
$F'$. \qed
\end{enumerate}
\end{thm}
This theorem, (\ref{birclass.over.R}) and (\ref{comp.bir.inv}) imply the
following:
\begin{cor}\label{rat.char.over.R}\cite[VI.6.5]{Silhol89}
Let $F$ be a smooth projective surface over $\r$.
The following are equivalent:
\begin{enumerate}
\item $F$ is birational to $\p^2$ over $\r$.
\item $F_{\c}$ is birational to $\c\p^2$ and $F(\r)$ is connected. \qed
\end{enumerate}
\end{cor}
\section{Birational Classification of Conic Fibrations}
\begin{defn} Let $F$ be a smooth projective surface over a field $k$. A
morphism
$f:F\to B$ to a smooth curve is called a {\it conic fibration} if the
generic fiber is isomorphic to a plane conic (over $k(B)$).
By (\ref{conics.r}) this is equivalent to assuming that $f^{-1}(b)\cong
\p^1_{\bar k}$ for a general $b\in B(\bar k)$.
\end{defn}
In this section we discuss the birational classification
of those surfaces over $\r$ which admit a
conic fibration. This covers all surfaces where the MMP ends with the
case (\ref{mmp.surf.thm}.C).
This is done in two steps. First we consider those
birational maps which preserve the conic fibration. To be precise:
\begin{defn} Two conic fibrations $f:F\to B$ and $f':F'\to B'$ are
called {\it birational} if there is a birational map
$\phi:F\map F'$ and an isomorphism $\tau: B \cong B'$
(both over $k$) which give a commutative diagram
$$
\begin{array}{ccc}
F & \stackrel{\phi}{\map} & F'\\
f\downarrow{\ } & & {\ }\downarrow f'\\
B & \stackrel{\tau}{\map} & B'
\end{array}
$$
\end{defn}
The second step is to understand the birational maps between $F$ and
$F'$ which do not preserve the conic fibration. Fortunately, in many
cases there are no such maps. (For a proof see
\cite[V--VI]{Silhol89} or the original paper of
\cite{Iskovskikh67}.)
\begin{thm}\label{cb.unique.thm}
Let $k$ be a field and $f:F\to B$ a relatively minimal
conic bundle over $B$. Let $f':F'\to B'$ be
any conic fibration and $\phi:F\map F'$ any birational map
(over $k$). Then
\begin{enumerate}
\item If $(K_F^2)\leq 0$ then $\phi$ is a birational map of the
conic fibrations.
\item If $(K_F^2)=2$ then $F$ and $F'$ are birational
conic fibrations (though $\phi$ itself need not respect the fibration
structure).\qed
\end{enumerate}
\end{thm}
\begin{defn} Let $f:F\to B$ be a conic fibration over $\r$.
The image of the set of real points
$f(F(\r))\subset B(\r)$ is a union of finitely many closed intervals.
Let us denote it by $I(F)$.
\end{defn}
The main theorem of the section shows that $I(f)$ characterizes $f$:
\begin{thm} Two conic fibrations $f:F\to \p^1$ and $f':F'\to \p^1$
over $\r$ are
birational iff there is an isomorphism $\tau:\p^1 \cong \p^1$
such that $\tau(I(f))=I(f')$.
\end{thm}
Proof. Let $\phi:F\map F'$ and $\tau:\p^1 \cong \p^1$ be a
birational map of the two
conic fibrations. Then
$F'(\r)$ and $\phi(F(\r))$ agree outside finitely many fibers, thus
$I(f')$ and $\tau(I(f))$ differ only at finitely many points.
Unions of closed intervals can not differ at finitely many points only,
thus in fact $\tau(I(f))=I(f')$.
The converse is established by bringing each conic fibration to a
normal form. (The roots $a_i$ in (\ref{cb.standard.form.thm})
are the boundary points of $I(f)$. This leaves
two choices for $I(f)$ itself, corresponding to the two choices of
the sign on the right hand side.)
\qed
\begin{thm}\label{cb.standard.form.thm}
Let $f:F\to \p^1$ be a conic fibration over $\r$. Then
$f$ is birational to a conic fibration $f':F'\to \p^1$ with affine
equation
$$
x^2+y^2=\pm \prod_{i=1}^{2m}(z-a_i)\subset {\mathbb A}^3,
$$
where the $a_i$ are distinct real numbers.
\end{thm}
The proof rests on the following simple lemma about quadratic forms:
\begin{lem}\label{cb.standard.form.lem} Let $k$ be a field (of
characteristic different from 2) and $Q(x_0,\dots,x_n)$ a quadratic form
over $k$ which is anisotropic (that is $Q=0$ has no nontrivial solution
over $k$). For any $a\in k$ the following are equivalent
\begin{enumerate}
\item $Q=0$ has a nontrivial solution over $k(\sqrt{a})$.
\item After a suitable coordinate change, $Q$ can be written as
\noindent $b(y_0^2-ay_1^2)+Q'(y_2,\dots,y_n)$.
\end{enumerate}
\end{lem}
Proof. (2) $\Rightarrow$ (1) is shown by the substitution
${\bf y}:=(\sqrt{a}:1:0\cdots:0)$.
Conversely, assume that ${\bf v}\in k(\sqrt{a})^{n+1}$
satisfies $Q({\bf v})=0$. Let $\bar {\bf v}$ denote the
conjugate of ${\bf v}$. Then $Q(\bar{\bf v})=0$.
${\bf v}$ and $\bar{\bf v}$ span a 2-dimensional linear
subspace of $k(\sqrt{a})^{n+1}$ which is defined over $k$. That is,
there is a linear subspace $V\subset k^{n+1}$ such that
$V\otimes_kk(\sqrt{a})=\langle {\bf v}, \bar{\bf
v}\rangle$. $Q$ is nondegenerate on $V$ (since $Q=0$ has no solutions
in $k$), thus $k^{n+1}=V+V^{\perp}$ where $V^{\perp}$ is the orthogonal
complement of $V$ with respect to $Q$. Let $y_2,\dots,y_n$ be
coordinates on $V^{\perp}$ and choose coordinates $y_0,y_1$ on $V$
such that
$$
{\bf v}=(\sqrt{a},1)\qtq{and} \bar{\bf v}=(-\sqrt{a},1).\qed
$$
\begin{say}[Proof of (\ref{cb.standard.form.thm})]{\ }
Let $k:=\r(t)$ be the quotient field of $\p^1_{\r}$. The generic fiber
of $F\to \p^1$ is birational to a plane conic $C_k$ over $k$
(\ref{conics.r}).
If $C_k$ has a $k$-point (equivalently, if $F\to \p^1$ has a section)
then $F$ is birational to $\p^1\times \p^1$ by (\ref{conics.r}). $C_k$
has a point over
$k(\sqrt{-1})=\c(t)$ (equivalently, $F\to \p^1$ has a section over
$\c$). Thus by (\ref{cb.standard.form.lem}), in suitable coordinates
the equation of $C_k$ becomes $x_0^2+x_1^2=g(t)x_2^2$
for some $g(t)\in \r(t)$.
We can multiply through with the square of the denominator of $g$,
thus we may assume that $g(t)\in \r[t]$.
Write $g=f(t)g_1(t)^2\prod_a (z-a)(z-\bar a)$ where $f$ has only simple
real roots and the $a$ are nonreal complex numbers. We can divide by
$g_1(t)^2$. If $a=u+iv$ then $(z-a)(z-\bar a)=(z-u)^2+v^2$.
Note that
\begin{eqnarray*}
g(t)(h_0^2+h_1^2)x_2^2&=&x_0^2+x_1^2\qtq{is equivalent to}\\
g(t)x_2^2&=&
\left(\frac{x_0h_0-x_1h_1}{h_0^2+h_1^2}\right)^2+
\left(\frac{x_0h_1+x_1h_0}{h_0^2+h_1^2}\right)^2.
\end{eqnarray*}
Using this, we can get rid of the complex factors $\prod_a (z-a)(z-\bar
a)$ one at a time. At the end we obtain the required normal form,
except that we may have an odd number of factors on the right hand
side:
$$
x^2+y^2=\pm \prod_{i=1}^{2m-1}(z-a_i).
$$
In this case we first apply a
translation to ensure that $0$ is not among the $a_i$ and then make a
substitution
$(x,y,z)\mapsto (xz^{-n},yz^{-n}, z^{-1})$ to get the equation
$$
x^2+y^2=\pm \prod_{i=1}^{2m}(z-a'_i),
$$
where $a'_i=a_i^{-1}$ for $i<2m$ and $a'_{2m}=0$.\qed
\end{say}
Putting things together, we obtain the following criterion for
birational equivalence of conic fibrations.
The result corrects a slight inaccuracy in \cite[VI.3.15]{Silhol89}.
\begin{exrc}\label{cb.isom.ex} Two conic bundles
\begin{eqnarray*}
F & = & (x^2+y^2= c\prod_{i=1}^{2m}(z-a_i))\to \p^1\qtq{and}\\
F' & = & (x^2+y^2= c'\prod_{i=1}^{2m}(z-a'_i))\to \p^1
\end{eqnarray*}
are birational to each other iff there is a permutation $\sigma\in
S_{2m}$ and a matrix
$$
\left(\begin{array}{cc}
\alpha & \beta \\
\gamma & \delta
\end{array}
\right)\in GL(2,\r)
\qtq{such that}
a'_{\sigma(i)}=\frac{\alpha a_i+\beta }{\gamma a_i+\delta}, \qtq{and}
$$
$c'$ and $c\prod_i(\gamma a_i+\delta)$ have the same sign.
\end{exrc}
\begin{exrc}\label{geom.cb.ex}
Using elementary transformations of conic bundles, give a
geometric proof of the results in this section.
\end{exrc}
\begin{say}[Moduli of conic fibrations]{\ }
Let $F$ be a smooth projective surface over $\r$ which admits a conic
fibration $f:F\to \p^1$. We proved that if $I(f)\subset \r\p^1$
has at least 3 components then $I(f)$ (modulo the action of $GL(2,\r)$)
determines $F$ up to birational equivalence (over $\r$).
The space of $m$ disjoint closed intervals in $\r\p^1$ is a connected
manifold of real dimension $2m$. The quotient by the $GL(2,\r)$ action
gives a $2m-3$ dimensional topological space (it has some quotient
singularities). With some more care, we could even realize this space
as the set of real points of a $(2m-3)$-dimensional algebraic variety.
For $m=0$ there are 2 conic fibrations up to birational equivalence:
$\p^1\times \p^1$ gives $I(f)=\r\p^1$ and $Q^{3,0}\times \p^1$ gives
$I(f)=\emptyset$. For $m=1$ we have only one birational equivalence
class by (\ref{elem.bir.lem}.4).
For $m=2$ we see in (\ref{deg4.cb.ex}) that all such surfaces are
birational to each other (though they are not birational as conic
fibrations).
\end{say}
\section{Del Pezzo Surfaces of Degree $\geq 3$}
In this section we describe all Del Pezzo surfaces of degree $d\geq 3$
over
$\r$.
\begin{exrc}\label{amp.image.ex}
Let $g:S\to S'$ be a birational morphism of smooth
surfaces. Show that if $H$ is ample on $S$ then $f(H)$ is ample on
$S'$. Thus if $S$ is Del Pezzo then $S'$ is also Del Pezzo.
\end{exrc}
\begin{prop}\label{bireg.empty}
Let $F$ be a smooth real Del Pezzo surface which is
birational to $Q^{3,0}\times \p^1$.
Set $d:=(K_F^2)$. Then $d\in\{8,6,4,2\}$.
If $d=8$ then
$F$ is isomorphic to
either $Q^{3,0}\times Q^{3,0}$ or to $Q^{3,0}\times \p^1$.
If $d<8$ then
$F$ is isomorphic to
to $Q^{3,0}\times \p^1$ blown up in $\frac12(8-d)$ pairs of conjugate
points.
Therefore, for $d\in\{6,4,2\}$ such surfaces form a connected family.
\end{prop}
Proof. Apply the MMP over $\r$ to obtain $F\to \cdots\to F^*$.
If $F$ is Del Pezzo then so is $F^*$ by (\ref{amp.image.ex}),
and $(K_{F^*}^2)\geq (K_F^2)$. Hence
in our case $F^*$ is either
$Q^{3,0}\times Q^{3,0}$ or $Q^{3,0}\times \p^1$.
By (\ref{elem.bir.lem}.3) any blow up of $Q^{3,0}\times Q^{3,0}$ at
a pair of conjugate points is also a blow up of
$Q^{3,0}\times \p^1$. \qed
\begin{prop}\label{bireg.rtl} Let $F$ be a smooth real Del Pezzo
surface which is birational to $\p^2$.
Set $d:=(K_F^2)$. Then $9\geq d\geq 1$ and we have one of the following
cases:
If $d=9$ then
$F$ is isomorphic to $\p^2$.
If $d=8$ then
$F$ is isomorphic to
either $Q^{3,1}$ or to $Q^{2,2}$ or to
$\p^2$ blown up at a real point.
If $d<8$ then
$F$ is isomorphic to one of the following:
\begin{enumerate}
\item
$\p^2$ blown up at $a\geq 0$ real points and $b\geq 0$ pairs of
conjugate points for some $a+2b=9-d$. Thus $F(\r)\sim \#(a+1)\r\p^2$.
\item $Q^{3,1}$ blown up at $b=\frac12(8-d)$ pairs of
conjugate points (so $d$ is even). Thus $F(\r)\sim S^2$.
\item $Q^{2,2}$ blown up at $b=\frac12(8-d)$ pairs of
conjugate points (so $d$ is even). Thus $F(\r)\sim S^1\times S^1$.
Therefore, for any $d<8$, such surfaces with a given topological
type $F(\r)$ form a connected family.
\end{enumerate}
\end{prop}
Proof. The minimal model of such a surface is either $\p^2$,
$Q^{3,1}$ or $Q^{2,2}$.
By (\ref{elem.bir.lem}.1--2) any blow up of $Q^{3,1}$ or to $Q^{2,2}$ at
a real point is also a blow up of
$\p^2$. \qed
\medskip
The two propositions above account for all Del Pezzo surfaces of degrees
$d\geq 5$. The results are summarized in the next statement:
\begin{cor}\label{d>4.dp.top} The following table lists all topological
types of the real points of Del Pezzo surfaces of degrees $9\geq d\geq
5$. All surfaces of a fixed degree and topological type form a
connected family, except for $d=8$ and $F(\r)=\emptyset$ when there are
2 such surfaces.
$$
\begin{tabular}{|c|l|}
\hline degree & \qquad\qquad topological types\\
\hline 9 & $\r\p^2$\\
\hline 8 & $S^2$ or $S^1\times S^1$ or $\r\p^2\#\r\p^2$ or $\emptyset$ \\
\hline 7 & $\r\p^2$ or $\#3\r\p^2$ \\
\hline 6 & $S^2$ or $S^1\times S^1$ or $\r\p^2\#\r\p^2$ or
$\#4\r\p^2$ or $\emptyset$\\
\hline 5 & $\r\p^2$ or $\#3\r\p^2$ or $\#5\r\p^2$ \\
\hline
\end{tabular}
$$
\end{cor}
The following result shows that odd degree Del Pezzo surfaces over $\r$
are relatively easy to understand:
\begin{lem}\label{d1=d2.blowup}
Every degree $2d-1$ Del Pezzo surface $F$ over $\r$ with
$\rho(F)\geq 2$ is the blow up of a degree $2d$ Del Pezzo surface at a
real point.
\end{lem}
Proof. Since $(K_F^2)$ is odd, $F$ is not a minimal conic bundle.
Thus $F$ is either the
blow up of a degree $2d$ Del Pezzo surface at a real point
or the blow up of a degree $2d+1$ Del Pezzo $F'$ surface at a conjugate
pair of complex points $P+\bar P$.
If $F'\cong \p^2$ then let $L\subset \p^2$ be the line through the two
points. Its birational transform on $F$ is a line.
Otherwise, $F'$ is again not minimal by (\ref{mmp.over.R}), hence
$F'$ contains a line $L$ over $\r$ by induction.
$P,\bar P\not\in L$ since otherwise the birational transform of $L$ on
$F$ would have a nonnegative intersection number with $K_F$.
Thus $L$ gives a real line on $F$.
Contracting a real line on $F$ we get a degree $2d$ Del
Pezzo surface.\qed
\medskip
This shows that the study of degree 3 Del Pezzo surfaces
is reduced to the study of degree 4 cases.
The classification of these two classes is summarized next.
These results were obtained by
\cite{Schlafli1863}, who actually worked directly with cubic surfaces.
\begin{cor}\label{d=4,3.dp.top} The following table lists all topological
types of the real points of Del Pezzo surfaces of degrees 4 and 3. All
surfaces of a fixed degree and topological type form a connected family.
$$
\begin{tabular}{|c|l|}
\hline degree & \qquad\qquad topological types\\
\hline 4 & $S^2$ or $S^1\times S^1$ or $\r\p^2\#\r\p^2$ or
$\#4\r\p^2$ or $\emptyset$ or $S^2\uplus S^2$\\
\hline 3 & $\r\p^2$ or $\#3\r\p^2$ or $\#5\r\p^2$ or $\#7\r\p^2$
or $S^2\uplus \r\p^2$ \\
\hline
\end{tabular}
$$
Moreover, in the $S^2\uplus S^2$ case the
monodromy interchanges the two components.
\end{cor}
Proof. As we noted above, it is sufficient to describe
all degree 4
Del Pezzo surfaces.
If a degree 4
Del Pezzo surface $F$ is obtained from a higher degree surface by
blowing up then we are reduced to (\ref{d>4.dp.top}). Otherwise
$F$ is a conic bundle over $\p^1$ with 4 singular fibers.
These 4 singular fibers give 8 lines on $F$. By looking at the set of
all lines over $\c$, we see that the remaining 8 lines again form 4
pairs and determine another morphism to $\p^1$. Thus $F$ is a double
cover of $\p^1\times \p^1$ ramified along a curve $D\subset \p^1\times
\p^1$ of type
$(2,2)$. $D$ has 4 horizontal and 4 vertical tangents and the 16 lines
are sitting over these tangents.
The rest is a special case of (\ref{(2,2)-curves}). \qed
\begin{exrc}\label{(2,2)-curves}
Show that the space of all smooth real curves of type $(2,2)$ on
$\p^1\times
\p^1$ has 7 connected components. They are determined by the homotopy
classes of the components of $D(\r)$: $\emptyset$ or $(0,0)$ or
$(0,0)\uplus (0,0)$ or $(1,1)\uplus (1,1)$ or $(1,-1)\uplus (1,-1)$ or
$(1,0)\uplus (1,0)$ or $(0,1)\uplus (0,1)$.
\end{exrc}
\begin{exrc}\label{deg4.cb.ex}\cite[VI.3.5]{Silhol89}
Using the correspondence between $(2,2)$-curves
on
$\p^1\times\p^1$ and degree 4 Del Pezzo surfaces
show that any two minimal conic bundles $F,F'$ with 4 singular fibers
are birational, by producing examples $S\to \p^1\times \p^1$ such that
one conic bundle structure of $S$ is birational to $F$ and the other
to $F'$.
(This should be easier after the next section.)
\end{exrc}
\begin{exrc}\cite{Schlafli1863}\label{cubic.lines.ex}
Show that a smooth cubic over $\r$ has 27,15,7 or 3 real
lines.
\end{exrc}
\section{Del Pezzo Surfaces of Degree 2 and 1}
\begin{notation} Let $D\subset \p^2$ be a degree 4 smooth real curve.
$D(\r)$ divides
$\r\p^2$ into connected open sets and precisely one of these is
nonorientable (denoted by $U_D$). We choose an equation $f(x,y,z)\in
\r[x,y,z]$ of $D$ such that $f$ is negative
on $U_D$.
We can associate two different degree 2 Del Pezzo surfaces to $D$.
One is $F^+_D:=(u^2=f(x,y,z)\subset \p^3(1,1,1,2)$
and the other $F^-_D:=(u^2=-f(x,y,z)\subset \p^3(1,1,1,2)$.
The correspondence $F^+_D\leftrightarrow F^-_D$
is a natural involution on the space of degree 2 real Del Pezzo
surfaces.
$D$ has 28 bitangents over $\c$ and over each bitangent of $D$
we get a pair of lines on $F^{\pm}_D$. This gives a total of 56 lines.
\end{notation}
The topological classification of degree 4 plane curves over $\r$ is
very old, it is already contained in \cite{Plucker1839}.
(See \cite{Viro90} for a recent survey of the study of low degree
real plane curves.)
This implies the topological classification of degree
2 real Del Pezzo surfaces. The following proposition summarizes
these results.
\begin{prop}\label{deg2.dp.top}
There are 6 topological types of degree 4 smooth real plane
curves. Correspondingly
there are 12 topological types of degree 2 real Del Pezzo surfaces.
The following table gives the complete list.
The types
in the same row correspond to each other under
$D \leftrightarrow F^+_D\leftrightarrow F^-_D$.
$$
\begin{tabular}{|c|c|c|}
\hline $D(\r)$ & $F^+_D(\r)$ & $F^-_D(\r)$\\
\hline $\bigcirc\bigcirc\bigcirc\bigcirc{}$ & $\uplus 4S^2$ &
$\#8\r\p^2$\\
\hline $\bigcirc\bigcirc\bigcirc$ & $\uplus 3S^2$ & $\#6\r\p^2$\\
\hline $\bigcirc\bigcirc{}$ & $S^2\uplus S^2$ & $\#4\r\p^2$\\
\hline $\bigcirc$ & $S^2$ & $ \#2\r\p^2$\\
\hline $\emptyset$ & $\emptyset$ & $\r\p^2\uplus \r\p^2$\\
\hline $\bigcirc \!\!\!\!\!\circ$ & $S^1\times S^1$ & $S^2\uplus
\#2\r\p^2$\\
\hline
\end{tabular}
$$
\end{prop}
\cite{Zeuthen1874} studied the bitangents of degree 4 plane curves.
He proved the equivalence of (\ref{deg2.equiv.thm}.1) and
(\ref{deg2.equiv.thm}.5). He understood the relationship between
degree 4 plane curves and cubic surfaces.
(Projecting a cubic surface from one of its points, the branch curve
is a plane quartic. Equivalently, blowing up the cubic at a point we
get a degree 2 Del Pezzo surface.)
This is, however, not the
natural thing to do from the modern viewpont.
Most of (6.3) is proved in \cite{Comessatti13}.
\begin{thm} \label{deg2.equiv.thm}
Let $D\subset \p^2$ be a degree 4 smooth real curve. The
following are equivalent:
\begin{enumerate}
\item All 28 bitangents of $D$ are real.
\item All 56 lines of $F^-_D$ are real.
\item $F^-_D$ is isomorphic to $\p^2$ blown up in 7 real points.
\item $F^-_D(\r)\sim \#8\r\p^2$.
\item $D(\r)\sim \uplus 4S^1$.
\item $F^+_D(\r)\sim \uplus 4S^2$.
\item $F^+_D$ has Picard number 1 over $\r$.
\end{enumerate}
\end{thm}
Proof. (1) $\Rightarrow$ (2): A neighborhood of a line in $\r\p^2$ is
not orientable, thus any bitangent is contained in $U_D$ (except for the
points of tangency). $f$ is negative on any bitangent and so
$u^2=-f$ has real solutions, giving 56 real lines on
$F^-_D$.
(2) $\Rightarrow$ (3): Over $\c$, $F^-_D$ is the blow up of $\p^2$ at 7
points, hence it has 7 disjoint lines. If all lines are real, we have 7
disjont real lines. Contracting these we get a Del Pezzo surface of
degree 9 over
$\r$. By (\ref{mmp.over.R}.D) it is $\p^2_{\r}$.
(3) $\Rightarrow$ (4): Topologically, each blowing up is connected sum
with
$\r\p^2$.
(4) $\Rightarrow$ (5): This follows from (\ref{deg2.dp.top}).
(5) $\Rightarrow$ (6): This also follows from (\ref{deg2.dp.top}).
(6) $\Rightarrow$ (7): Assume to the contrary that
$F^+_D$ has Picard number $\geq 2$ over $\r$. By (\ref{mmp.over.R})
we have one of 2 cases:
\begin{enumerate}
\item $F^+_D$ is a minimal conic bundle with 6 singular fibers. In this
case $F^+_D(\r)\sim \uplus 3S^2$, a contradiction.
\item $F^+_D$ is the blow up of a Del Pezzo surface of degree 3
or 4 over
$\r$. By (\ref{d=4,3.dp.top})
$F^+_D(\r)$ has at most 2 connected components, a contradiction.
\end{enumerate}
(7) $\Rightarrow$ (1): Assume that $D$ has a complex bitangent $L$.
Its conjugate $\bar L$ is again a bitangent. Let $C\subset F^+_D$ be a
complex line over $L$. Its conjugate $\bar C$ lies over $\bar L$.
Then $(C\cdot \bar C)\leq 1$ (the only possible intersection point lies
over $L\cap \bar L$). Thus $F^+_D$ has either a disjoint pair of
conjugate lines or a conic bundle structure, a contradiction.\qed
\begin{prop}\label{deg2.dp.top.conn}\cite{Klein1876}
The space of degree 4 smooth real plane
curves has 6 connected components corresponding to the
6 topological types in (\ref{deg2.dp.top}). The space of degree 2
real Del Pezzo surfaces has 12 connected components corresponding to the
12 topological types in (\ref{deg2.dp.top}).
\end{prop}
Proof. The two parts are equivalent and it is sufficient to treat the
$F^-_D$ cases. $D$ always has a real bitangent (\ref{deg.4.bitang}),
thus
$F^-_D$ contains a real line. So $F^-_D$ is obtained by blowing up a
degree 3 Del Pezzo surface at a real point.
(\ref{deg2.dp.top.conn}) now follows from (\ref{d=4,3.dp.top}).\qed
\begin{exrc}\cite{Zeuthen1874}\label{deg.4.bitang}
Let $C$ be a smooth real plane curve
of degree 4. Let $d$ be the number of ovals of $D(\r)$
which are not contained in another oval.
Show that $C$ has $4+2d(d-1)$ real bitangents.
\end{exrc}
\begin{say}[Degree 1 Del Pezzo surfaces]\label{deg1.dp.say}{\ }
Let $F$ be a degree 1 Del Pezzo surface over any field $k$. $|-K_F|$ is
a pencil with exactly one base point. So this is a $k$-point and
$F(k)\neq
\emptyset$. $|-2K_F|$ is base point free and exhibits $F$ as a double
cover of a quadric cone $Q\subset \p^3$, ramified along a curve
$D\subset Q$ which is a complete
intersection of $Q$ with a cubic surface with equation $(f=0)$.
$D$ does not pass through the vertex of the cone.
$F_{\bar k}$ contains 240
lines (that is $-1$-curves); cf. \cite[IV.4.3]{Manin72}. We obtain these
as follows. Take a plane $H\subset \p^3$ which is tangent
to
$D$ at 3 points. The preimage of $H\cap Q$ in $F$ has 2 irreducible
components, each is a line. Thus we conclude that there are 120 planes
which are tangent to $D$ at 3
points.
Assume now that $k=\r$.
Since $Q(\r)\neq \emptyset$, we can write
$Q=(x^2+y^2=1)$
in suitable affine coodinates
$(x,y,z)$ on ${\Bbb A}^3$. That is, $Q(\r)$ is a cylinder with a singular
point at infinity.
As in the degree 2 case, for each (nonhomogeneous) cubic $f(x,y,z)$ we
obtain two degree 1 Del Pezzo surfaces, given by affine equations
$$
F^{\pm}_f:=(x^2+y^2-1=u^2\mp f(x,y,z)=0)\subset {\Bbb A}^4.
$$
There are 2 types of simple closed loops on a cylinder: null
homotopic ones (I call them ovals) and those homotopic to a plane
section (I call them big circles).
Since $D(\r)$ is the intersection of a cylinder with a cubic, it has 3
or 1 intersection points with any ruling line of the cylinder.
Thus $D(\r)$ contains either 3 big circles (and no ovals) or 1 big
circle. $D$ has genus 4, hence by Harnack's theorem
(cf. \cite[VII.4]{Shafarevich72}), $D(\r)$ has at most 5 connected
components. An oval can not be inside another oval since this would give
4 points on a ruling. Furthermore, we can not have an oval on one side
the big circle and at least two ovals on the other side. Indeed, choosing
points
$P_1,P_2,P_3$ inside the 3 ovals, there is a plane $H$ through them.
Then $H$ intersects each oval in at least 2 points, and also the big
circle. So $(H\cdot D)\geq 8$, but $D$ has degree 6, a contradiction.
If all the ovals are on the same side of the big circle, we can
normalize $f$ so that it is positive on the other side.
The other cases are symmetrical and it makes little sense to normalize
$f$.
We can summarize these results in the following table:
\end{say}
\begin{prop}\label{deg1.dp.top}
There are 7 topological types of degree 6 smooth real complete
intersection curves on the cylinder $(x^2+y^2=1)$, not passing through
the point at infinity. Correspondingly there are 11 topological types of
degree 1 real Del Pezzo surfaces. The following table gives the complete
list. The types
in the same row correspond to each other under
$D=(f=0)\cap Q \leftrightarrow F^+_f\leftrightarrow F^-_f$.
$$
\begin{tabular}{|c|c|c|}
\hline $D(\r)$ & $F^+_f(\r)$ & $F^-_f(\r)$\\
\hline 1 big circle + 4 ovals & $\r\p^2\uplus 4S^2$ &
$ \#9\r\p^2$\\
\hline 1 big circle + 3 ovals & $\r\p^2\uplus 3S^2$ &
$ \#7\r\p^2$\\
\hline 1 big circle + 2 ovals & $\r\p^2\uplus 2S^2$ &
$ \#5\r\p^2$\\
\hline 1 big circle + 1 oval & $\r\p^2\uplus S^2$ &
$ \#3\r\p^2$\\
\hline 1 big circle + 0 oval & $\r\p^2$ & $\r\p^2$\\
\hline 1 big circle + 1+1 ovals & $ \#3\r\p^2\uplus S^2$ &
$ \#3\r\p^2\uplus S^2$\\
\hline 3 big circles & $\r\p^2\uplus \#2\r\p^2$ &
$\r\p^2\uplus \#2\r\p^2$\\
\hline
\end{tabular}
$$
\end{prop}
The following theorem, due to \cite{Comessatti13}, is the degree 1
version of (\ref{deg2.equiv.thm}). I thank F. Russo for checking the
arguments of Comessatti.
\begin{thm} \label{deg1.equiv.thm}
Let $D=(f=0)\subset Q$ be a degree 6 smooth real complete
intersection curve on the cylinder $Q=(x^2+y^2=1)$. The following are
equivalent:
\begin{enumerate}
\item All 120 triple tangents of $D$ are real and $f$ is negative on
all of them.
\item All 240 lines of $F^-_f$ are real.
\item $F^-_f$ is isomorphic to $\p^2_{\r}$ blown up in 8 real points.
\item $F^-_f(\r)\sim \#9\r\p^2$.
\item $D(\r)\sim \uplus 5S^1$.
\item $F^+_f(\r)\sim \r\p^2\uplus 4S^2$.
\item $F^+_f$ has Picard number 1 over $\r$.
\end{enumerate}
\end{thm}
Proof. (1) $\Rightarrow$ (2): If $f$ is negative on a triple tangent
then
$u^2=-f$ has real solutions, giving a pair of real lines on
$F^-_f$.
(2) $\Rightarrow$ (3): Over $\c$, $F^-_f$ is the blow up of $\p^2$ at 8
points. Thus it has 8 disjoint lines. If all lines are real, we have 8
disjont real lines. Contracting these we get a Del Pezzo surface of
degree 9 over
$\r$. By (\ref{mmp.over.R}.D) it is $\p^2_{\r}$.
(3) $\Rightarrow$ (4): Topologically, each blowing up is connected sum
with
$\r\p^2$.
(4) $\Rightarrow$ (5): This follows from (\ref{deg1.dp.top}).
(5) $\Rightarrow$ (6): This also follows from (\ref{deg1.dp.top}).
(6) $\Rightarrow$ (7): Assume to the contrary that
$F^+_f$ has Picard number $\geq 2$ over $\r$.
$F^+_f$ can not be a minimal conic bundle since $(K^2)$ is odd.
Thus $F^+_f$ is the blow up of Del Pezzo surface of degree 2 or 3
over
$\r$. By (\ref{deg2.dp.top}, \ref{d=4,3.dp.top})
$F^+_f(\r)$ has at most 4 connected components, a contradiction.
(7) $\Rightarrow$ (1): If $D$ has a complex triple tangent, we can
argue as in (\ref{deg2.equiv.thm}.(7) $\Rightarrow$ (1)) that $F^+_f$
contains a conjugate pair of lines $C,\bar C$ such that $(C\cdot \bar
C)\leq 2$. $(C+\bar C)\equiv rK$ for some $r\in \z$, thus
$2(C\cdot \bar C)-2=r^2$. This is impossible.
If there is a real triple tangent such that $f$ is positive on it then
as in (1) $\Rightarrow$ (2) we get real lines on $F^+_f$.
\qed
\begin{prop}\label{deg1.dp.top.conn}
The space of degree 1
real Del Pezzo surfaces has 11 connected components corresponding to the
11 topological types in (\ref{deg1.dp.top}).
\end{prop}
Proof. For the nonminimal ones, this follows from
(\ref{deg1.dp.top.conn}) and (\ref{d1=d2.blowup}).
The minimal ones are in one--to-one correspondence with the blow ups of
$\p^2$ at 8 real points, this is again a connected space.\qed
\begin{ack} I thank my audience at the Trento sumer school, especially
L. Bonavero, S. Cynk and S. Endrass for numerous comments and
improvements. A. Marin directed me to several 19th century references.
F. Russo checked the arguments of Comessatti about degree 1 and 2
Del Pezzo surfaces and pointed out some misunderstandings on my part.
Partial financial support was provided by the NSF
under grant number DMS-9622394.
\end{ack}
\section{Hints to selected exercises}
(\ref{er.noncontr.ex}) Blow up $\geq 10$ general points (over $\c$) on a
smooth plane cubic. The birational transform of the cubic generates an
extremal ray which is not contractible. To see this show that $\pic$ of
the blown up surface injects into $\pic$ of the cubic.
\medskip
(\ref{int.cone.exrc}) Write down Riemann--Roch for $mz$ and $-mz$ to
get the first part. Then use this for $z+\epsilon z'$ for any $z'\in
N_1(F)$.
(cf. \cite[V.1.8]{Hartshorne77}.)
\medskip
(\ref{conics.r}) $|-K|$ embeds $C$ as a conic.
\medskip
(\ref{projspace.ex}) Let $H\in X_{\c}$ be a hyperplane. Show that
$H\cap \bar H$ is real and use induction.
Even degree symmetric powers of the empty conic give examples.
\medskip
(\ref{coh.b.c}) This is a special case of
\cite[III.9.3]{Hartshorne77}.
\medskip
(\ref{pic=gal.inv}) The empty conic gives a good example.
For the rest, the key point is to understand that we know more than the
existence of an isomorphism $\tau:L_{\c}\cong \bar L_{\c}$. Namely, by
conjugation this induces
$\bar\tau:\bar L_{\c}\cong \bar{\bar L_{\c}}\cong L_{\c}$, and the
composite of these two gives the identity of $L_{\c}$ (and not just an
isomorphism of
$L_{\c}$ to itself). Thus we have to choose a specific
isomorphism $\tau:M\cong \bar M$.
If $X$ has a real point $P$, then on the fiber over $P$ we can choose
$\tau$ to be conjugation (and not just some constant times conjugation).
Once things are set up right, the real sections of $M$ are those
sections $s$ of $M$ such that $\tau(s)=\bar s$.
\medskip
(\ref{comp.bir.inv}) Use the fact that a birational map between
projective varieties is defined outside a codimension 2 subset.
\medskip
(\ref{gen.orient.ex}) Let $S^1\sim A\subset X(\r)$ be any loop.
Perturb it to achieve that $A$ intersects $D(\r)$ transversally at
smooth points and is disjoit from $D'(\r)$.
\medskip
(\ref{cb.isom.ex}) $z$ is transformed by the inverse of the matrix.
Then do the explicit computation.
\medskip
(\ref{geom.cb.ex}) Let $F\to \p^1$ be a minimal conic bundle.
There are 2 types of elementary transformations: blow up a real point
in a fiber and then contract the fiber, or blow up conjugate points
in conjugate fibers and then contract the fibers.
Pick any
section $C$ over $\c$. Using elementary transformations get to the
situation when $C$ and $\bar C$ are disjoint. Show that $(C^2)=(\bar
C^2)=-m$ if there are $2m$ singular fibers. The normal form is an
affine piece of representing $F$ as a conic bundle in $\proj
f_*\o_F(C+\bar C)$.
\medskip
(\ref{(2,2)-curves})
The case when $D(\r)=\emptyset$ is easy.
There are many ways to study the remaining cases.
For instance, pick a point $P\in D(\r)$ and blow it up. By contracting
the birational transforms of the two sections through $P$, we obtain a
correspondance between pairs $(P\in D(\r))$ and triplets $(Q_1,Q_2\in
E(\r))$ where $E\subset \p^2$ is an elliptic curve.
One has to be a little careful since $\r\p^2$ is not orientable.
If we fix the orientations of the two copies of $\r\p^1$
in $\r\p^1\times \r\p^1$ then they give
local orientations of $\r\p^2$ at the points $Q_1,Q_2$. There is an
ambiguity of changing both orientations (since this does not change the
orientation of $\r\p^1\times \r\p^1$).
We have to study various cases according to the location of $Q_1,Q_2$ on
$E(\r)$. Moreover, we have to see how the local orientations match up
if we move from $Q_1$ to $Q_2$ along $E(\r)$.
If $Q_1,Q_2$ are both on a pseudo line, then the two local orientations
are consistent if we move in one direction and
inconsistent in the other direction. However, if
$Q_1,Q_2$ are both on an oval, then the two local orientations
are either consistent in both directions or
inconsistent if both directions. This gives 2 cases.
\medskip
(\ref{deg4.cb.ex}) Given a ramification curve $D\subset \p^1\times
\p^1$ we get 2 different degree 4 Del Pezzo surfaces. We want one
surface $S^+$ where none of the lines are real. Then, in the other
surface $S^-$, all lines are real. One can construct $S^-$ as
follows. Start with $\p^1\times
\p^1$, points $P_1,\dots,P_4$ in the first factor and
$P'_1,\dots,P'_4$ in the second factor. Blow up the 4 points
$(P_i,P'_i)$. Show that we get a degree 4 Del Pezzo surface iff there
is no isomorphism $\tau:\p^1\times \p^1$ such that $\tau(P_i)=P'_i$.
Projecting to the first factor gives a conic fibration with singular
fibers over $P_i$. Projection to the second factor is not the right
thing to do. Instead, the second conic fibration is given by the linear
system of curves of type $(2,2)$ passing through the 4 points
$(P_i,P'_i)$.
Also keep in mind that we have to take care not only of the 4 points
but also the set $I(f)$.
\medskip
(\ref{cubic.lines.ex}) and (\ref{deg.4.bitang}) can both be seen from
the classification. One has to prove that we can not blow up a point
on a line. \cite{Schlafli1863} proved first that a cubic can be written
as $C_1-C_2=0$ where $C_i$ are products of linear factors and then
studied the various cases when the linear factors are all real or some
are conjugates. \cite{Zeuthen1874} notes that 2 ovals have 4 tangents,
thus we need to show that there are 4 more which are either tangents to
the same oval or at complex points. He does this by a continuity
argument. This is a bit tricky since these 4 tangents are not
invariant under deformations of the curve, just their number is.
\medskip
|
1997-12-15T00:53:17 | 9712 | alg-geom/9712013 | en | https://arxiv.org/abs/alg-geom/9712013 | [
"alg-geom",
"math.AG"
] | alg-geom/9712013 | Christopher Thomas Woodward | Sharad Agnihotri (Amsterdam) and Chris Woodward (Harvard) | Eigenvalues of products of unitary matrices and quantum Schubert
calculus | 18 pages, uses amssymb | null | null | null | null | We describe the inequalities on the possible eigenvalues of products of
unitary matrices in terms of quantum Schubert calculus. Related problems are
the existence of flat connections on the punctured two-sphere with prescribed
holonomies, and the decomposition of fusion product of representations of
SU(n), in the large level limit. In the second part of the paper we investigate
how various aspects of the problem (symmetry, factorization) relate to
properties of the Gromov-Witten invariants.
| [
{
"version": "v1",
"created": "Sun, 14 Dec 1997 23:53:17 GMT"
}
] | 2016-08-30T00:00:00 | [
[
"Agnihotri",
"Sharad",
"",
"Amsterdam"
],
[
"Woodward",
"Chris",
"",
"Harvard"
]
] | alg-geom | \section{Introduction}
Beginning with Weyl \cite{we:ei}, many mathematicians have been
interested in the following question: given the eigenvalues of two
Hermitian matrices, what are the possible eigenvalues of their sum?
In a recent preprint \cite{kl:sb}, Klyachko observes that a complete
solution to this problem is given by an application of Mumford's
criterion in geometric invariant theory. The eigenvalue inequalities
are derived from products in Schubert calculus. In particular, Weyl's
inequalities correspond to Schubert calculus in projective space. The
necessity of these conditions is due to Helmke and Rosenthal
\cite{hr:ei}.
One of the fascinating points about the above problem are several
equivalent formulations noted by Klyachko. For instance, the problem
is related to the following question in representation theory: Given a
collection of irreducible representations of $SU(n)$, which
irreducibles appear in the tensor product? A second equivalent
problem involves toric vector bundles over the complex projective
plane.
In this paper we investigate the corresponding problem for {\em
products} of {\em unitary} matrices. This question also has a
relationship with a representation-theoretic problem, that of the
decomposition of the fusion product of representations. The solution
to the multiplicative problem is also derived from geometric invariant
theory, namely from the Mehta-Seshadri theory of parabolic bundles
over the projective line. The main result of this paper, Theorem
\ref{final}, shows that the eigenvalue inequalities are derived from
products in {\em quantum} Schubert calculus. This improves a result
of I. Biswas \cite{bi:ex}, who gave the first description of these
inequalities. A similar result has been obtained independently by
P. Belkale \cite{bl:ip}.
The proof is an application of the Mehta-Seshadri theorem. A set of
unitary matrices $A_1,\ldots ,A_l$ such that each $A_i$ lies in a
conjugacy class $\mathcal{C}_i$ and such that their product is the identity is
equivalent to a unitary representation of the fundamental group of the
$l$ times punctured sphere, with each generator $\gamma_i$ being mapped to
the conjugacy class $\mathcal{C}_i.$ By the Mehta-Seshadri theorem such
a representation exists if and only if there exists a semi-stable
parabolic bundle on $\P^1$ with $l$ parabolic points whose parabolic
weights come from the choice of conjugacy classes $\mathcal{C}_i.$ This last
interpretation of the original eigenvalue problem can be related to
the Gromov-Witten invariants of the Grassmannian and this is done in
Section 5 below.
In Sections 6 and 7 we investigate how factorization and hidden
symmetries of these Gromov-Witten invariants relate to the
multiplicative eigenvalue problem.
\section{Additive inequalities (after Klyachko and Helmke-Rosenthal)}
Let $\lie{su}(n)$ denote the Lie algebra of $SU(n)$, and
$$\t = \{ (\lambda_1,\ldots,\lambda_n) \in \mathbb{R}^n \ | \
\sum \lambda_i = 0 \} $$
its Cartan subalgebra. Let
$$ \t_+ = \{ (\lambda_1,\ldots,\lambda_n) \in \t \ | \
\lam_i \ge \lam_{i+1}, \ \ i=1,\ldots,n-1 \} $$
be a choice of closed positive Weyl chamber. For any matrix $A \in
\lie{su}(n)$ let
$$ \lam(A) = (\lam_1(A),\lam_2(A),\cdots,\lam_n(A)) \in \t_+ $$
be the eigenvalues of the Hermitian matrix $-i A$ in non-increasing
order. Let $\Delta(l) \subset (\t_+)^l$ denote the set
$$ \Delta(l) = \{ (\lam(A_1),\lam(A_2),\ldots,\lam(A_l)) \ \vert \
A_1,\ldots,A_l \in \lie{su}(n), \ A_1 + A_2 + \ldots + A_l = 0 \} .$$
Define an involution
$$ *: \ \t_+ \cong \t_+, \ \ (\lambda_1,\ldots,\lambda_n)
\mapsto (-\lambda_n, \ldots, -\lambda_1) .$$
For any $A \in \lie{su}(n)$ the matrix $-A$ has eigenvalues $ \lam(-A) =
*\lam(A)$. The set $\Delta(l)$ is invariant under the map
$$ *^l:\ (\t_+)^l \to (\t_+)^l, \ \ (\xi_1,\ldots,\xi_l) \mapsto
(*\xi_1,\ldots,*\xi_l) $$
and also under the action of the symmetric group $S_l$ on
$(\t_+)^l$.
The set $\Delta(l)$ has interesting interpretations in symplectic
geometry and representation theory. Consider the cotangent bundle $
T^*SU(n)^{l-1} $ with the action of $SU(n)^l$ given by $SU(n)$ acting
diagonally on the left and $SU(n)^{l-1}$ on the right. The moment
polytope of this action may be identified with $\Delta(l)$ (see Section
5.) From convexity theorems in symplectic geometry (see e.g.
\cite{sj:co} and \cite{le:co}) it follows that $\Delta(l)$ is a
finitely-generated convex polyhedral cone. In particular there are a
finite number of inequalities defining $\Delta(l) $ as a subset of the
polyhedral cone $(\t_+)^l$.
The set $\Delta(l) $ may also be described in terms of the tensor
product of representations. Let
$$(\ ,\ ) : \ \lie{su}(n) \times \lie{su}(n) \to \mathbb{R}, \ \ \ (A,B) \mapsto - \Tr(AB) $$
denote the basic inner product on $\lie{su}(n)$, which induces an
identification $\lie{su}(n) \cong \lie{su}(n)^*$. Let $ \Lambda = \mathbb{Z}^n \cap \t$
denote the integral lattice and $ \Lambda^* \subset \t$ its dual, the
weight lattice. For each $\lam \in \Lambda^*\cap \t_+$, let $V_\lam$
denote the corresponding irreducible representation of $SU(n)$. We
will see in equation \eqref{inv_eqn} that $\Delta(l) \cap \mathbb{Q}^l$ is the
set of $(\lam^1,\ldots,\lam^l)$ such that for some $N$ such that
$N\lam^1,\ldots, N \lam^l \in \Lambda^*$, we have
$$V_{N \lam^1} \otimes \ldots \otimes V_{N\lam^{l-1}} \supset V^*_{N
\lam^l},$$
that is, $V_{N \lam^1} \otimes \ldots \otimes V_{N \lam^l}$ contains
a non-zero invariant vector.
The work of Klyachko and Helmke-Rosenthal gives a complete set of
inequalities describing $\Delta(l)$ in terms of Schubert calculus. Let
$$ \mathbb{C}^n = F_n \supset F_{n-1} \supset \ldots \supset F_0 = \{ 0 \} $$
be a complete flag in $\mathbb{C}^n$, $G(r,n)$ the Grassmanian of $r$-planes
in $\mathbb{C}^n$, and for any subset $I = \{i_1,\ldots,i_r \} \subset \{1,
\ldots n \}$ let
$$ \sig_I = \{ W \in G(r,n) \ \vert \ \dim(W \cap F_{i_j}) \ge j, \
j = 1,\ldots,r \} $$
denote the corresponding Schubert variety. The Schubert cell $C_I
\subset \sigma_I$ is defined as the complement of all lower-dimensional Schubert varieties contained in $\sig_I$:
$$ C_I = \sigma_I \backslash \bigcup_{\sigma_J \subset \sigma_I}
\sigma_J.
$$
We say that $W$ is in {\em position $I$} with respect to the flag
$F_*$ if $W \in C_I$.
The homology classes $[ \sig_I ]$ form a basis of $H_*(G(r,n),\mathbb{Z})$.
Given two Schubert cycles $\sig_I,\sig_J$, we can expand the
intersection product $[\sig_I] \cap [\sig_J]$ in terms of this basis.
We say $[\sig_I] \cap [\sig_J]$ contains $[\sig_K]$ if $[\sig_K]$
appears in this expansion with non-zero (and therefore positive)
coefficient. Equivalently, let
$$*K = \{n + 1 - i_r,n+ 1 - i_{r-1},\ldots, n+1-i_1 \},$$
so that $[\sig_{*K}]$ is the Poincare dual of $[\sig_K]$. Then
$[\sig_I] \cap [\sig_J]$ contains $[\sig_K]$ if and only if the
intersection of general translates of the Schubert cycles
$\sig_I,\sig_J,\sig_{*K}$ is non-empty and finite.
\begin{theorem}[Klyachko, resp. Helmke-Rosenthal] \labell{klyachko}
A complete (resp. necessary) set of inequalities
describing $\Delta(l)$ as a subset of $(\t_+)^l$ are
\begin{equation} \labell{add_ineq}
\sum_{i \in I_1} \lam_i(A_1) + \sum_{i \in I_2} \lam_i(A_2)
+ \ldots + \sum_{i \in I_l} \lam_i(A_l) \leq 0,
\end{equation}
where $I_1,\ldots,I_l$ are subsets of $\{ 1 ,\ldots, n \}$ of the same
cardinality $r$ such that $[\sig_{I_1}] \cap \ldots \cap
[\sig_{I_{l-1}}] \supset [\sig_{*I_l}]$, and $r$ ranges over all
values between $1$ and $n-1$.
\end{theorem}
Note that the cases $l =1,2$ are trivial: $\Delta(1) = \{ 0 \},$ and $
\Delta(1) = \{ (\mu,*\mu) \ |\ \mu \in \t_+ \}.$ Klyachko also claims
that these inequalities are independent. From Theorem \ref{klyachko}
follows a complete set of inequalities for the possible eigenvalues of
a sum of skew-Hermitian matrices. For instance, for $l=3$ one obtains
the inequalities
\begin{equation} \labell{add_ineq2}
\sum_{i \in I}
\lam_i(A) + \sum_{j \in J} \lam_j(B) \le \sum_{k \in K} \lam_k(A + B),
\end{equation}
where $I,J,K \subset \{ 1 ,\ldots, n \}$ range over subsets such that
$[\sig_I] \cap [\sig_J]$ contains $[\sig_K]$.
\begin{example} Let $r=1$ so that $G(r,n) \cong \P^{n-1}$ and
$I = \{ n - i + 1 \}, \ J = \{ n - j + 1 \}$. Then $\sigma_I \cong
\P^{n-i},\ \sigma_J \cong \P^{n-j}$ so that $[\sigma_I] \cap
[\sigma_J] \cong \P^{n-i-j+1} = \sigma_K$ where $K = \{n - i -j +
2\}$. One obtains
\begin{equation} \labell{dual_Weyl}
\lam_{n-i+1}(A) + \lam_{n-j+1}(B) \le \lam_{n-i-j+2}(A + B) .
\end{equation}
\end{example}
\subsection{Duality} Let $A_1,\ldots, A_l \in \lie{su}(n)$.
From \eqref{add_ineq2} applied to $-A_1,\ldots ,- A_l$ one obtains
\begin{equation} \labell{dual_ineq}
- \sum_{i \in *I_{1}} \lam_i(A_1) -\ldots - \sum_{i \in *I_{l}} \lam_i(A_l)\leq 0
\end{equation}
or equivalently $ \sum_{i \in *I_{1}} \lam_i(A_1) +\ldots + \sum_{i \in *I_{l}} \lam_i(A_l)\geq 0. $
By the trace condition, \eqref{dual_ineq} is equivalent to
$$ \sum_{i \notin *I_{1}} \lam_i(A_1) +\ldots + \sum_{i \notin *I_{l}} \lam_i(A_l)\leq 0 .$$
Let $I_i^c = \{ 1,\ldots,n \} \backslash * I_i$. Then
$[\sigma_{I_i^c}]$ is the image of $[\sigma_{I_{i}}]$ under the
isomorphism of homology induced by $G(r,n) \cong G(n-r,n)$ (see page
197 onwards of Griffiths and Harris \cite{gr:pr}). Thus the
appearance of \eqref{dual_ineq} in \eqref{add_ineq} corresponds to a
product in the Schubert calculus of $G(n-r,n)$.
\begin{example} \labell{Weyl_example}
The dual equation to \eqref{dual_Weyl} is Weyl's 1912 \cite{we:ei}
inequality
\begin{equation} \labell{Weyl}
\lam_i(A) + \lam_j(B) \ge \lam_{i + j -1}(A + B).
\end{equation}
\end{example}
\section{Multiplicative Inequalities}
Let $\lie{A} \subset \t_+$ be the fundamental alcove of $SU(n)$:
$$\lie{A} = \{ \lam \in \t_+ \ | \ \lam_1 - \lam_n \leq 1 \}.$$
Let $A \in SU(n)$ be a unitary matrix with determinant $1$. Its
eigenvalues may be written
$$ e^{2\pi i \lam_1(A)}, e^{2\pi i \lam_2(A)}, \ldots ,e^{2\pi i \lam_n(A)} $$
where $\lam(A) = (\lam_1(A),\ldots,\lam_n(A)) \in \lie{A}$. The map $A
\mapsto \lam(A)$ induces a homeomorphism
$$ \lie{A} \cong SU(n) / \Ad(SU(n)) .$$
Let $\Delta_q(l) \subset \lie{A}^l$ ($q$ for quantum) denote the set
$$ \Delta_q(l) = \{ (\lam(A_1),\ldots,\lam(A_l)) \ \vert \
A_1,\ldots,A_l \in SU(n), \ A_1A_2\ldots A_l = I \}. $$
As before, $\Delta_q(l)$ is invariant under the involution, $*^l: \lie{A}^l
\to \lie{A}^l$, and the action of the symmetric group $S_l$ on $\lie{A}^l$.
The set $\Delta_q(l)$ has an interpretation as a moment polytope. Let
$\M$ be the space of flat $SU(n)$-connections on the trivial $SU(n)$
bundle over the $l$-holed two-sphere, modulo gauge transformations
which are the identity on the boundary (see \cite{me:lo}). The gauge
group of the boundary acts on $\M$ in Hamiltonian fashion and the set
$\Delta_q(l)$ is the moment polytope for this action. By \cite[Theorem
3.19]{me:lo}, $\Delta_q(l)$ is a convex polytope. In fact, an analogous
statement holds for arbitrary compact, simply-connected Lie groups.
In particular, a finite number of inequalities describe $\Delta_q(l)$.
In the case $n=2$, these inequalities were given explicitly for $l=3$
in Jeffrey-Weitsman \cite{jw:bs} and for arbitrary numbers of marked
points in Biswas \cite{bi:r2}. A description of the inequalities in
the arbitrary rank case was given in \cite{bi:ex} but the description
given there does not seem to be computable.
There is also an interpretation of $\Delta_q(l)$ in terms of fusion
product. Let $\fus_N$ denote the fusion product on the Verlinde
algebra $R(SU(n)_N)$ of $SU(n)$ at level $N$. Then $\Delta_q(l) \cap
\mathbb{Q}^l$ is the set of $(\lam^1,\ldots,\lam^l) \in \lie{A} \cap \mathbb{Q}^l$ such
that for some $N$ such that $N\lam^1,\ldots, N \lam^l \in \Lambda^*$,
we have
\begin{equation} \labell{fus}
V_{N \lam^1} \fus_N \ldots \fus_N V_{N\lam^{l-1}} \supset
V_{N * \lam^l}.\end{equation}
See Section \ref{Verlinde}.
\subsection{Quantum Schubert calculus}
Quantum cohomology is a deformation of the ordinary cohomology ring
that was introduced by the physicists Vafa and Witten. Quantum
cohomology of the Grassmannian (quantum Schubert calculus) was put on
a rigorous footing by Bertram \cite{be:qs}. Recall that the degree of
a holomorphic map $\varphi: \ \P^1 \rightarrow G(r,n)$ is the homology class
$[\varphi] \in H^2(G(r,n),\mathbb{Z}) \cong \mathbb{Z}$. Let $p_1,\ldots,p_l$ be
distinct marked points in $\P^1$. The quantum intersection product
$\star$ on $H_*(G(r,n),\mathbb{C}) \otimes \mathbb{C}[q]$ is defined by
$$ [\sigma_{I_1} ] \star \ldots \star [\sigma_{I_l}] = \sum_{J}
\langle[\sig_{I_1}],\ldots, [\sigma_{I_l}], [\sig_J] \rangle_d \
[\sigma_{*J}] q^d,$$
where the Gromov-Witten invariant $\langle[\sig_{I_1}],\ldots,
[\sigma_{I_l}], [\sig_J] \rangle_d$ is equal to the number of
holomorphic maps $\P^1 \mapsto G(r,n)$ sending $p_1,\ldots,p_l,p$ to
general translates of $\sigma_{I_1},\ldots,\sigma_{I_l},\sigma_J$ if
this number is finite, and is otherwise zero.
Our main result is the following description of $\Delta_q(l)$:
\begin{theorem}\labell{final} A complete set of inequalities for
$\Delta_q(l)$ are given by
\begin{equation}
\sum_{i \in I_1} \lam_i(A_1) + \sum_{i \in I_2} \lam_i(A_2)
+ \ldots + \sum_{i \in I_l} \lam_i(A_l) \leq d
\end{equation}
for $(I_1,\ldots,I_l,d)$ such that $\l [\sig_{I_1}] \star \ldots \star
[ \sig_{I_l} ] \r_d \neq 0,$ that is, $\ [\sig_{I_1}] \star \ldots
\star [ \sig_{I_{l-1}}] \supset q^d [\sig_{*I_l}] .$
\end{theorem}
In the last few years several techniques have been developed for
computing the coefficients of quantum Schubert calculus. See for
instance Bertram, Ciocan-Fontanine, Fulton \cite{be:qm}. Therefore
the above theorem makes the question of which inequalities occur
computable in practice.
One recovers the inequalities for $\Delta(l)$ from the degree $0$
Gromov-Witten invariants. This shows that $\Delta(l) $ is
the cone on $\Delta_q(l)$ at the $0$-vertex, i.e.
$$ \Delta(l) = \mathbb{R}_+ \cdot \Delta_q(l) $$
This may be verified by several alternative methods, e.g.
Remark \ref{small}.
The simplest example of a positive degree inequality is given by the
following:
\begin{example} \labell{quantum_Weyl_example}
Let $r = 1$ so that $G(r,n) = \P^{n-1}$, and $U,V,W \subset \mathbb{C}^n$ be
subspaces in general position of dimensions $i,j,n+1-i-j$. There is a
unique degree $1$ map $\P^1 \rightarrow \P^{n-1}$ mapping $p_1,p_2,p_3$ to
$\P(U),\P(V),\P(W)$ respectively. Together with the degree $0$
inequality mentioned before, this gives
\begin{equation} \labell{quant_Weyl}
\lam_{i+j-1}(AB) \le \lam_i(A) + \lam_j(B) \le \lam_{i+j}(AB) + 1.
\end{equation}
We will see in Section \ref{symmetry} that these inequalities are
related by a symmetry of $\Delta_q(l)$.
\end{example}
\vskip .1in {\noindent \em Question:} Are the inequalities in Theorem
\ref{final} independent? \vskip .1in
\section{Moduli of flags and Mumford's criterion}
As a warm-up we review some of the ideas involved in Klyachko's
proof. For any $\xi \in \t_+$ let
$$\O_\xi = SU(n) \cdot \xi = \{ A \in \lie{su}(n) \ | \ \lam(A) = \xi \} $$
denote the corresponding adjoint orbit. Via the identification
$\lie{su}(n) \cong \lie{su}(n)^*$, $\O_\xi$ inherits a canonical symplectic
structure, called the Kostant-Kirillov-Souriau two-form, and the
action of $SU(n)$ on $\O_\xi$ is Hamiltonian with moment map given by
inclusion into $\lie{su}(n)$.
The diagonal action of $SU(n)$ on $\O_{\xi_1} \times \ldots \times
\O_{\xi_l}$ has moment map given by
$$ (A_1,\ldots,A_l) \mapsto A_1 + A_2 + \ldots + A_l.$$
The symplectic quotient $\mathcal{N}({\xi_1,\ldots,\xi_l}) = \O_{\xi_1} \times
\ldots \times \O_{\xi_l} \qu SU(n)$ is given by
$$ \mathcal{N}({\xi_1,\ldots,\xi_l}) = \{(A_1,\ldots,A_l) \in \O_{\xi_1} \ldots
\O_{\xi_l} \ | \ A_1 + A_2 + \ldots + A_l = 0 \} / SU(n) .$$
For generic $(\xi_1,\ldots,\xi_l),$ that is, values where the moment
map has maximal rank, the quotient $\mathcal{N}({\xi_1,\ldots,\xi_l})$ is a
symplectic manifold. The $l$-tuple $(\xi_1,\ldots,\xi_l)$ lies in
$\Delta(l) $ if and only if $\mathcal{N}({\xi_1,\ldots,\xi_l})$ is non-empty.
The quotients $\mathcal{N}({\xi_1,\ldots,\xi_l})$ may be viewed as symplectic
quotients of the cotangent bundle $ T^*SU(n)^{l-1} $. Indeed, the
symplectic quotient
$$ (T^*SU(n) \times \O_\xi) \qu SU(n) \cong \O_\xi .$$
Therefore, the quotient of $T^*SU(n)^{l-1}$ by the right action of
$SU(n)^{l-1}$ and the diagonal left action of $SU(n)$ is
$$ (T^*SU(n)^{l-1} \times \O_{\xi_1} \times \ldots \times \O_{\xi_{l}})
\qu SU(n)^{l} \cong \mathcal{N}({\xi_1,\ldots,\xi_l}).$$
It follows that $\Delta(l)$ is the moment polytope of the action of
$SU(n)^l$ on $ T^*SU(n)^{l-1}$.
One can determine whether $\mathcal{N}({\xi_1,\ldots,\xi_l})$ is empty by
computing its symplectic volume. This is given by a formula derived
from the Duistermaat-Heckman theorem due to Guillemin-Prato (see
\cite{gu:he} or \cite[(4)]{me:wi}). Unfortunately the formula involves
cancelations and it is not apparent what the support of the volume
function is, or even that the support is a convex polytope.
The manifolds $\O_{\xi_i}$ have canonical complex structures (induced
by the choice of positive Weyl chamber) and are isomorphic to
(possibly partial) flag varieties. Suppose that $\xi_1,\ldots,\xi_l$
lie in the weight lattice $\Lambda^*$, so that there exist pre-quantum
line bundles $L_{\xi_i} \to \O_{\xi_i}$; i.e., equivariant line
bundles with curvature equal to $2 \pi i$ times the symplectic
form. The sections of $L_{\xi_1}\boxtimes \ldots \boxtimes L_{\xi_l}$
define a K\"ahler embedding
$$\O_{\xi_1} \times \ldots \times
\O_{\xi_l} \to \P(V_{\xi_1}^*) \times \ldots \times \P(V_{\xi_l}^*),$$
where $V_{\xi_1},\ldots,V_{\xi_l}$ are the irreducible representations
with highest weights $\xi_1,\ldots,\xi_l$. By an application of a
theorem of Kirwan and Kempf-Ness (which holds for arbitrary smooth
projective varieties, see \cite[page 109]{ki:coh}) the symplectic
quotient is homeomorphic to the geometric invariant theory quotient
$$\mathcal{N}({\xi_1,\ldots,\xi_l}) \cong \O_{\xi_1} \times \ldots \times \O_{\xi_l} \qu
SL(n,\mathbb{C}).$$
By definition, $\O_{\xi_1} \times \ldots \times \O_{\xi_l} \qu
SL(n,\mathbb{C}) $ is the quotient of the set of semi-stable points in
$\O_{\xi_1} \times \ldots \times \O_{\xi_l}$ by the action of
$SL(n,\mathbb{C})$, where $(F_{\xi_1},\ldots,F_{\xi_l}) \in \O_{\xi_1} \times
\ldots \times \O_{\xi_l}$ is called semi-stable if and only if for
some $N$ there is an invariant section of $(L_{\xi_1} \boxtimes \ldots
\boxtimes L_{\xi_l})^{\otimes N} $ which is non-vanishing at
$(F_{\xi_1},\ldots,F_{\xi_l})$. The quotient $\mathcal{N}({\xi_1,\ldots,\xi_l})$
is therefore non-empty if and only if there exists a non-zero
$SU(n)$-invariant vector in
\begin{equation} \labell{inv_eqn}
H^0((L_{\xi_1} \boxtimes \ldots \boxtimes L_{\xi_l})^{\otimes N}) =
V_{N \xi_1} \otimes \ldots \otimes V_{N \xi_l}.\end{equation}
This explains the representation-theoretic interpretation of $\Delta(l)
$ alluded to in the introduction.
In order to obtain the inequalities in Theorem \ref{klyachko}, one
applies the criterion of Mumford, which says that {\em a point is
semi-stable if and only if it is semi-stable for all one-parameter
subgroups} \cite[Chapter 2]{mu:ge}, see also \cite[Lemma 8.8]{ki:coh}.
Let us assume that the $\xi_i$ are generic. An application of the
criterion gives that an $l$-tuple of complete flags
$(F_{1},\ldots,F_{l}) \in \O_{\xi_1} \times \ldots \times \O_{\xi_l}$
is semi-stable if and only if for all subspaces $W \subset \mathbb{C}^n$, one
has
$$ \sum_{i \in I_1} \xi_{1,i} + \ldots + \sum_{i \in I_l} \xi_{l,i} \le 0, $$
where $I_j$ is the position of $W$ with respect to the flag $F_{j}$.
The proof similar to that for Grassmannians given in Section 4.4 of
\cite{mu:ge}.
The set of semi-stable points is dense if non-empty. It follows that
$\mathcal{N}({\xi_1,\ldots,\xi_l})$ is non-empty if and only if the above
inequality holds for every intersection $\sigma_{I_1} \cap \ldots \cap
\sigma_{I_l}$ of Schubert cycles in general position. Any inequality
corresponding to a positive dimensional intersection must be
redundant. Indeed, since the intersection is a projective variety, it
cannot be contained in any of the Schubert cells. The boundary of
$\sigma_{I_l}$ consists of Schubert varieties $\sigma_J$ with $J$ such
that $j_k \le i_k$ for $k = 1,\ldots,r$, where $i_1,\ldots,i_r$ and
$j_1,\ldots, j_r$ are the elements of $I_l$ and $J$ in increasing
order. The inequality obtained from an intersection $\sigma_{I_1}
\cap \ldots \cap \sigma_{I_{l-1}} \cap \sigma_J \neq \emptyset$
therefore implies the inequality obtained from $\sigma_{I_1} \cap
\ldots \cap \sigma_{I_l} \neq \emptyset$.
\section{Application of the Mehta-Seshadri theorem}
For any $\xi \in \lie{A}$, let
$$\mathcal{C}_{\xi} = \{ A \in SU(n) \ | \ \lam(A) = \xi \} $$
denote the corresponding conjugacy class. The mapping $A \mapsto
\lam(A)$ induces a homeomorphism $SU(n)/\Ad(SU(n)) \cong \lie{A} $.
Let $p_1,\ldots,p_l \in \P^1$ be distinct marked points and
$\M(\xi_1,\ldots,\xi_l)$ the moduli space of flat $SU(n)$-connections
on $\P^1 \backslash \{p_1,\ldots, p_l \} $ with holonomy around $p_i$
lying in $\mathcal{C}_{\xi_i}$. Since the fundamental group of $\P^1
\backslash \{p_1,\ldots, p_l\}$ has generators the loops $\gamma_1,
\ldots, \gamma_l$ around the punctures, with the single relation
$\gamma_1 \cdot \ldots \cdot \gamma_l=1$,
$$ \M({\xi_1,\ldots,\xi_l}) \cong \{ (A_1,\ldots,A_l) \in \mathcal{C}_{\xi_1}
\times \ldots \times \mathcal{C}_{\xi_l} \ | \ A_1A_2 \cdots A_l = I \} /
SU(n) .$$
In particular $\M({\xi_1,\ldots,\xi_l})$ is non-empty if and only if
$(\xi_1,\ldots,\xi_l) \in \Delta_q(l)$. In theory one can determine if
$\M({\xi_1,\ldots,\xi_l})$ is non-empty by computing its symplectic
volume by the formulae stated in Witten \cite[(4.11)]{wi:tw}, Szenes
\cite{sz:vo}, and \cite[Theorem 5.2]{me:co}.
For rational $\xi_1,\ldots,\xi_l$ the space $\M({\xi_1,\ldots,\xi_l})$
has an algebro-geometric description due to Mehta-Seshadri
\cite{ms:pb}. Let $C$ be a Riemann surface with marked points
$p_1,\ldots,p_l \in C$ and let $\cE \rightarrow C$ be a holomorphic bundle. A
parabolic structure without multiplicity on $\cE$ consists of the
following data at each marked point $p_i$: a complete ascending flag
$$ 0 = \cE_{p_i,0} \subset \cE_{p_i,1} \subset \cE_{p_i,2} \ldots \subset \cE_{p_i,n} = \cE_{p_i} $$
in the fiber $\cE_{p_i}$ and a set of {\em parabolic weights}
$$ \lam_{i,1} > \lam_{i,2} > \ldots > \lam_{i,n} $$
satisfying $\lam_{i,1} - \lam_{i,n} \leq 1$. In \cite{ms:pb} the
weights are required to lie in the interval $[0,1)$, but the
definitions work without this assumption. A parabolic bundle is a
holomorphic bundle with a parabolic structure. Recall that the degree
$\deg(\cE)$ of $\cE$ is the first Chern class $c_1(\cE) \in H^2(C,\mathbb{Z})
\cong \mathbb{Z}$. The parabolic degree $\pardeg(\cE)$ is defined by
$$ \pardeg(\cE) = \deg(\cE) + \sum_{i=1,j=1}^{l,n} \lam_{i,j} .$$
The parabolic slope $\mu(\cE)$ is
$$ \mu(\cE) = \frac{\pardeg(\cE)}{\rk(\cE)} .$$
Given a holomorphic sub-bundle $\mathcal{F} \subset \cE$ of rank $r$ one
obtains a parabolic structure on $\mathcal{F}$ as follows. An ascending flag
in the fiber $\mathcal{F}_{p_i}$ at each marked point $p_i$ is obtained by
removing from
$$ \mathcal{F}_{p_i} \cap \cE_{p_i,1} \subseteq \mathcal{F}_{p_i} \cap \cE_{p_i,2}
\subseteq \ldots \subseteq \mathcal{F}_{p_i} \cap \cE_{p_i,n} = \mathcal{F}_{p_i} $$
those terms for which the inclusion is not strict. The parabolic
weights for $\mathcal{F}$ are
$$ \mu_{i,j} = \lam_{i,k_j}, $$
where $k_j$ is the minimal index such that $\mathcal{F}_{p_i,j} \subseteq
\cE_{p_i,k_j}$. Let $K_i = \{ k_1, \ldots, k_r \}$. The fiber
$\mathcal{F}_{p_i}$ may be viewed as a element of the Grassmannian of
$r$-planes in $\cE_{p_i}$, and $K$ is the position of $\mathcal{F}_{p_i}$ with
respect to the flag $\cE_{p_i,*}$. The parabolic degree of $\mathcal{F}$ is
$$ \pardeg(\mathcal{F}) = \deg(\mathcal{F}) + \sum_{i,\ k \in K_i} \lam_{i,k} .$$
A parabolic sub-bundle of $\cE$ is a holomorphic sub-bundle $\mathcal{F}
\subset E$ whose parabolic structure is the one induced from the
inclusion. A parabolic bundle $\cE \to C$ is called parabolic
semi-stable if $\mu(\mathcal{F}) \le \mu(\cE)$ for all parabolic sub-bundles
$\mathcal{F} \subset \cE$. There is a natural equivalence relation on
parabolic bundles: Two bundles are said to be grade equivalent if the
associated graded bundles are isomorphic as parabolic bundles. See
\cite{ms:pb} for more details.
\begin{theorem}[Mehta-Seshadri] Suppose the parabolic weights $\lam_{i,j}$
are rational and lie in the interval $[0,1)$. Then the moduli space
$\M(\lam_1, \ldots,\lam_l)$ of grade equivalence classes of
semi-stable parabolic bundles with parabolic weights $\lam_{i,j}$ and
parabolic degree $0$ is a normal, projective variety, homeomorphic to
the moduli space of flat unitary connections over $C \backslash \{
p_1,\ldots,p_r \}$ such that the holonomy of a small loop around $p_i$
lies in $\mathcal{C}_{\lam_{i}}$.
\end{theorem}
In fact, the Mehta-Seshadri theorem also holds without the assumption
that the parabolic weights lie in $[0,1)$. One can see this either
through the theory of elementary transformations, or through the
extension of the Mehta-Seshadri theorem to non-zero parabolic degree
given in Boden \cite{bo:re}.
The explanation using elementary transformations goes as follows. Let
$\cQ$ denote the skyscraper sheaf with fiber $\cE_{p_i}/\cE_{p_i,n-1}$
at $p_i$. One has an exact sequence of sheaves
$$ 0 \to \cE' \to \cE \to \cQ \to 0 .$$
The kernel $\cE'$ is a sub-sheaf of a locally free sheaf and therefore
locally free. Since degree is additive in short exact sequences
$\deg(\cE') = \deg(\cE)-1$. One calls the $\cE'$ an elementary
transformation of $\cE$ at $p_i$. There is a canonical line
$\cE'_{p_i,1}$ in the fiber $\cE'_{p_i}$ which is the kernel of the
fiber map $\pi: \cE'_{p_i} \to \cE_{p_i}$. One extends the canonical
line to a complete flag by taking $\cE'_{p_i,j} =
\pi^{-1}(\cE'_{p_i,j-1})$ for $j>1$. Finally one takes as parabolic
weights at $p_i$ the set $ \lam_{i,n} + 1,\lambda_{i,1},
\ldots,\lam_{i,n-1}$. With this parabolic structure the bundle $\cE'$
is parabolic semi-stable of the same parabolic degree as $\cE$. Details,
in a slightly different form, can be found in Boden and Yokogawa
\cite{bo:ra}.
The following is the key lemma in the derivation of Theorem
\ref{result} from Mehta-Seshadri. Let $d = \deg(\cE) = -\sum
\lam_{i,j}$ denote the degree of any element $\cE \in \M(\lam_1,
\ldots,\lam_l)$.
\begin{lemma} \labell{key} Suppose that there is some ordinary semi-stable
bundle on $C$ of degree $d$. Then the set of equivalence classes of
parabolic semi-stable bundles of parabolic degree $0$ whose underlying
holomorphic bundle is ordinary semi-stable is Zariski dense in
$\M(\lam_1,\ldots,\lam_l)$.
\end{lemma}
\begin{proof}
Recall from the construction of $\M(\lam_1,\ldots,\lam_l)$ in \cite{ms:pb} that for
some integer $N$ there exists an $SL(N)$-equivariant bundle
$ \tilde{R} \stackrel{\pi}{\to} R $
whose fibers are products of $l$ complete flag varieties, such that the
geometric invariant theory quotients of $\tilde{R},R$ are
$$ \tilde{R} \qu SL(N) = \M(\lam_1,\ldots,\lam_l) , \ \ \ \ R \qu SL(N) = \M,$$
where $\M$ denotes the moduli space of ordinary semi-stable bundles on
$C$ of degree $d$. By definition,
$$ \tilde{R} \qu SL(N) = \tilde{R}^{\ss} / SL(N), \ \ \ \ R \qu SL(N) =
R^{\ss} / SL(N) $$
where $\tilde{R}^{\ss},R^{\ss}$ denote the Zariski dense set of
semi-stable points in $\tilde{R},R$ respectively. The inverse
image $\pi^{-1}(R^{\ss}) \cap \tilde{R}^{\ss} / SL(N)$ is therefore dense
in $\tilde{R}^{\ss} / SL(N) = \M(\lam_1,\ldots,\lam_l)$.
\end{proof}
Now we specialize to the case $C = \P^1$ with $l$ marked points
$p_1,p_2,\ldots,p_l$. Let $\xi_1,\ldots,\xi_l \in \lie{A}^l \cap \mathbb{Q}^l$.
By Lemma \ref{key}, $\M({\xi_1,\ldots,\xi_l})$ is non-empty if and
only there exists a parabolic semi-stable $\cE$ with parabolic degree $0$
and weights $\xi_1,\ldots,\xi_l$ whose underlying holomorphic bundle
is semi-stable. Since the sum of the parabolic weights is zero, the
degree of $\cE$ is also zero. By Grothendieck's theorem, $\cE$ is
holomorphically trivial. A sub-bundle $\cF \subset \cE$ of rank $r$
is given by a holomorphic map
$$ \varphi_{\cF}: \ \P^1 \to G(r,n).$$
Since $\varphi_{\cF}$ is the classifying map of the quotient
$\cE/\cF$, the degree of $\cF$ is minus the degree of $\varphi_{\cF}$.
The parabolic slope of $\cF$ is given by
$$ \mu(\cF) = -\deg(\varphi_{\cF}) + \sum_{i \in I_1(\varphi)} \xi_{1,i} +
\ldots + \sum_{i \in I_1(\varphi)} \xi_{l,i}, $$
where $I_i(\varphi)$ is the position of the subspace $\varphi(p_i)
\subset \cE_{p_i}$ with respect to the flag $\cE_{p_i,* }$ above.
The parabolic bundle $\cE$ is called parabolic semi-stable if and only
if for all such $F$, $ \mu(F) \leq 0 $, that is,
$$ \sum_{i \in I_1(\varphi)} \xi_{1,i} + \ldots + \sum_{i \in
I_1(\varphi)} \xi_{l,i} \leq \deg(\varphi) $$
for all maps $\varphi:\P^1 \to G(r,n)$.
The following result was obtained independently by P. Belkale \cite{bl:ip}.
\begin{theorem} \labell{result}
A complete set of inequalities for $\Delta_q(l)$ as a subset of $\lie{A}^l$
is given by
\begin{equation} \labell{mult_ineq}
\sum_{i \in I_1} \lam_i(A_1) + \sum_{i \in I_2} \lam_i(A_2)
+ \ldots + \sum_{i \in I_l} \lam_i(A_l) \leq d
\end{equation}
for subsets $I_1,\ldots,I_l \subset \{ 1 ,\ldots, n \}$ of the same
cardinality $r$ and non-negative integers $d$ such that there exists a
rational map $\P^1 \to G(r,n)$ of degree $d$ mapping $p_1,\ldots,p_l$
to the Schubert cells $C_{I_1},\ldots,C_{I_l}$ in general position.
\end{theorem}
\begin{proof}
If $\M(\xi_1,\ldots,\xi_l)$ is non-empty, then a trivial bundle with a
general choice of flags will be parabolic semi-stable. Indeed, by the
above discussion the fiber $\operatorname{Flag}^l$ of $\pi :\tilde{R} \to R$ over a
trivial bundle intersects $\tilde{R}^{ss}$, so $\tilde{R}^{ss}\cap
\operatorname{Flag}^{l}$ is open in $\operatorname{Flag}^{l}$. Therefore,
$\M(\xi_1,\ldots,\xi_l)$ is non-empty if and only if
$$ \sum_{i \in I_1} \xi_{1,i} + \ldots + \sum_{i \in
I_l} \xi_{l,i} \leq d $$
for all subsets $I_1,\ldots,I_l$ and integers $d$ such that there
exists a degree $d$ map sending $p_1,\ldots,p_l$ to general translates
of the Schubert cells $C_{I_1},\ldots,C_{I_l}$.
\end{proof}
\begin{remark} \labell{small} For sufficiently small parabolic weights
$\lam_{i,j}$ any parabolic semi-stable bundle on $\P^1$ is
necessarily ordinary semi-stable of degree $0$, and therefore
trivial. It follows that the moduli spaces
$\M(\lam_1,\ldots,\lam_l)$ and $\mathcal{N}(\lam_1,\ldots,\lam_l)$ are
isomorphic. This shows that Klyachko's result is implied by Theorem
\ref{result}.
\end{remark}
We now show that the existence of the maps described in Theorem
\ref{result} may be detected by Gromov-Witten invariants.
Let $\sig_{I_1},\ldots,\sig_{I_l}$ be some collection of Schubert
varieties, and consider the expansion
$$ [\sig_{I_1}] \star [\sig_{I_2}] \ldots \star [\sig_{I_l}] =
\sum_i q^i \alpha_i $$
where $\alpha_i \in H_*(G(r,n))$. (Question: is this product always
non-zero?) We say that $q^d $ divides $ [\sig_{I_1}] \star
[\sig_{I_2}] \ldots \star [\sig_{I_l}]$ if $\alpha_i = 0$ for all $i <
d$. The following lemma is stated in Ravi \cite{ra:in}.
\begin{lemma} \labell{compute} Let $d$ be the lowest degree of a map $\P^1
\to G(r,n)$ sending $p_1,\ldots,p_l$ to general translates of
$\sig_{I_1},\ldots,\sig_{I_l}$ respectively. Then $q^d$ is the
maximal power of $q$ dividing $[\sig_{I_1}] \star
\ldots \star [\sig_{I_l}]$.
\end{lemma}
\begin{proof}
Let $\M_d$ denote the space of maps $\P^1 \to G(r,n)$ of degree $d$,
$\operatorname{ev}^l: \, \M_d \to G(r,n)^l$ the evaluation map, and
$\sig_{I_*}(p_*) = (\operatorname{ev}^l)^{-1}( \sigma_{I_*}) $ the subset of
maps sending $p_j$ to $\sigma_{I_j}$ for $j=1,\ldots,l$. By
\cite[Moving Lemma 2.2A]{be:qs}, $ \sig_{I_*}(p_*) $ is a
quasi-projective variety, of the expected codimension in $\M_d$. By
choosing enough additional marked points $p_1',\ldots,p_m'$, we can
insure that the corresponding evaluation map $\operatorname{ev}^m: \ \M_d \to
G(r,n)^m$ is injective when restricted to $\sig_{I_*}(p_*).$ Let $Y
\subset G(r,n)^{l}\times G(r,n)^{m}$ be the closure of
$(\operatorname{ev}^l \times \operatorname{ev}^m)(\M_{d})$, and let $\phi: \
G(r,n)^{l}\times G(r,n)^{m} \to G(r,n)^m $ be the projection.
Since the homology class $[\phi(Y \cap \sigma_{I_*})]$ is non-trivial
\cite[page 64]{gr:pr}, $\phi(Y \cap \sigma_{I_*})$ must intersect
some Schubert variety
$$
\sigma_{J_*} = \sigma_{J_1} \times \sigma_{J_2} \times \ldots
\times \sigma_{J_m} \subset G(r,n)^m $$
of complementary dimension. By Kleiman's lemma, \cite[Theorem 10.8
page 273]{ha:al}, the singular locus of $\phi(Y \cap \sigma_{I_*})$
does not intersect a general translate of $\sigma_{J_*}$, and
similarly the singular locus of $\sigma_{J_*}$ does not intersect
$\phi(Y \cap \sigma_{I_*})$. Therefore the intersection occurs in the
smooth loci of $\phi(Y \cap \sigma_{I_*})$ and $\sigma_{J_*}$, and
another application of the lemma implies that the intersection is
finite.
For generic translates of $\sigma_{J_*}$, the intersection is
contained in $\operatorname{ev}^m(\sigma_{I_*}(p_*))$. Indeed, let $
{\overline{\sig}}_{I_*}(p_*) $ be the compactification of $\sig_{I_*}(p_*)$
given in \cite{be:qs}, and $\Gamma \subset \overline{ \sig}_{I_*}(p_*)
\times G(r,n)^m$ the closure of the graph of $\operatorname{ev}^m$. Let $Z
\subset \Gamma$ be the complement of the graph of $\operatorname{ev}^m$. The
projection $\pi(Z)$ of $Z$ in $ G(r,n)^m$ is a closed sub-variety of
$\phi(Y \cap \sigma_{I_*})$. By Kleiman's lemma, for generic
translates of $\sig_{J_*}$ the intersection of $\pi(Z)$ and
$\sig_{J_*}$ is empty, so the intersection is contained in
$\operatorname{ev}^m(\sigma_{I_*}(p_*))$.
Because $\operatorname{ev}^m \ | \ \sigma_{I_*}(p_*)$ is injective, the
intersection $\sig_{I_*}(p_*) \cap \sig_{J_*}(p_*') $ is finite and
non-empty. Since the homology class $[\phi(Y \cap \sigma_{I_*})]$ is
independent of the choice of general translate of $\sigma_{I_*}$, the
above intersection is finite and non-empty for general translates of
the $\sig_{I_i}$ and $\sig_{J_j}$. This implies that Gromov-Witten
invariant
$$
\l
[\sigma_{I_1}],\ldots,[\sig_{I_l}],[\sig_{J_1}],\ldots,[\sig_{J_m}]
\r_d \neq 0 .$$
In terms of the quantum product
$$
[\sigma_{I_1}] \star \ldots \star [\sig_{I_l}] \star [\sig_{J_1}]
\star \ldots \star [\sig_{J_{m-1}}] \supset q^d \sig_{*J_m} $$
which implies that $ [\sigma_{I_1}] \star \ldots \star [\sig_{I_l}]$
contains a term with coefficient $q^i$ with $i \leq d$.
That is, for some Schubert variety $\sigma$,
$$
\l [\sig_{I_1}], [\sig_{I_2}], \ldots , [\sig_{I_l}],[\sigma] \r_i
\neq 0 .$$
To prove the lemma it suffices to show that $i=d$.
By \cite[Moving Lemma 2.2]{be:qs}, for general translates of the
Schubert varieties the degree $i$ moduli space $\sig_{I_1}(p_1) \cap
\ldots \cap \sig_{I_l}(p_l) \cap \sigma(p)$ is finite and consists of
maps sending $p_1,\ldots,p_l,p$ to the corresponding Schubert cells.
Since $d$ is minimal, $i = d$.
\end{proof}
\section{Factorization}
In this section we show that a relationship between the polytopes for
different numbers of marked points is related to factorization of
Gromov-Witten invariants (i.e. associativity of quantum
multiplication). A similar, easier, discussion holds for the additive
polytopes $\Delta(l)$. A consideration of a ``trivial'' factorization
completes the proof of Theorem \ref{final}.
Suppose that $l$ can be written $l = j + k -2 $ for
positive integers $j,k \ge 2$. It is easy to see that $\Delta_q(l)$ are
projections of a section of $\Delta_q(j) \times \Delta_q(k)$\footnote{In fact, the volume functions
satisfy the factorization properties
$$ \on{Vol}(\mathcal{N}(\mu_1,\ldots,\mu_{j-1},\nu_1,\ldots,\nu_{k-1}))
= \int_{\t_+} \on{Vol}(\mathcal{N}(\mu_1,\ldots,\mu_{j-1},*\lam))
\on{Vol}(\mathcal{N}(* \lam, \nu_1,\ldots,\nu_{k-1}) )\d \lam $$
$$ \on{Vol}(\M(\mu_1,\ldots,\mu_{j-1},\nu_1,\ldots,\nu_{k-1}))
= \int_{\lie{A}} \on{Vol}(\M(\mu_1,\ldots,\mu_{j-1},*\lam) )
\on{Vol}(\M(* \lam, \nu_1,\ldots,\nu_{k-1})) \d \lam .$$
The second formula is implicit in Witten \cite[p.51]{wi:tw}, proved in
\cite{jw:va}, and generalized in \cite{me:lo}.}.
$$ \Delta_q(l) = \{ (\mu_1,\ldots,\mu_{j-1},\nu_1,\ldots,\nu_{k-1}) \ | \
(\mu,\nu) \in \Delta_q(j) \times \Delta_q(k), \ \ \mu_j = *\nu_k \}
$$
To show the forward inclusion, note that if $A_1 A_2 \ldots A_l = I$
then letting $ B = A_j A_{j+1} \ldots A_l$ we have
$$(\lam(A_1),\ldots,\lam(A_{j-1}),\lam(B)) \in
\Delta_q(j), $$
$$ (\lam(B^{-1}),\lam(A_j), \ldots, \lam(A_l) ) \in \Delta(k).$$
In particular this means that any face of $\Delta_q(l)$ is a projection
of a face (usually not of codimension $1$) of $\Delta_q(j) \times
\Delta_q(k)$. Any face is the intersection of codimension $1$ faces.
This shows that any defining inequality of $\Delta_q(l)$ is implied by a
finite set of defining inequalities for $\Delta_q(j)$ and $\Delta_q(k)$.
Using associativity of quantum cohomology one can be more specific
about which inequalities for $\Delta_q(j),\Delta_q(k)$ are needed to imply
an inequality for $\Delta_q(l)$. Suppose that a Gromov-Witten invariant
$ \l \sigma_{I_1},\ldots,\sigma_{I_l}, \sigma_J \r_d \neq 0 $
so that one has an inequality for $\Delta(l)$ given by
\begin{equation} \labell{desired}
\sum_{i \in I_1} \lam_{1,i} + \ldots + \sum_{i \in I_{l}}
\lam_{l,i} \leq d .\end{equation}
Associativity of quantum multiplication says that
$$ \l \sigma_{I_1},\ldots,\sigma_{I_l}, \sigma_J \r_d
= \sum_{d_1 + d_2 = d, \ \ |K|=r }
\l \sigma_{I_1},\ldots,\sigma_{I_{j-1}}, \sigma_K \r_{d_1}
\l \sigma_{*K}, \sigma_{I_j},\ldots,\sigma_{I_l}, \sigma_J \r_{d_2} .$$
In particular there exist some $d_1,d_2$ with $d_1 + d_2 = d$ and
some Schubert variety $\sigma_K$ such that
$$ \l \sigma_{I_1},\ldots,\sigma_{I_{j-1}}, \sigma_K \r_{d_1} \neq 0, \ \ \
\l \sigma_{*K} , \sigma_{I_j},\ldots,\sigma_{I_l}, \sigma_J \r_{d_2} \neq 0 .$$
From the non-vanishing of these Gromov-Witten invariants one deduces the
inequalities for $\Delta_q(j),\Delta_q(k):$
\begin{equation} \labell{add1}
\sum_{i \in I_1} \mu_{1,i} + \ldots + \sum_{i \in I_{j-1}}
\mu_{j-1,i} + \sum_{k \in K} \mu_{j,k} \leq d_1 ;\end{equation}
\begin{equation} \labell{add2}
\sum_{k \in *K} \nu_{1,k} + \sum_{i \in I_{j}} \nu_{2,i} +
\ldots + \sum_{i \in I_{l}} \nu_{k,i} \leq d_2 .\end{equation}
Restricting to the section $\mu_j = *\nu_1$ one has that
$$ \sum_{k \in *K} \nu_{1,k} = - \sum_{k \in K} (* \nu_1)_k =
- \sum_{k \in K} \mu_{j,k}, $$
so by adding the two inequalities one obtains \eqref{desired}.
Using the trivial factorization $l = (l + 2) - 2$ we complete the
proof of Theorem \ref{final}.
\begin{lemma} \labell{factor}
Any inequality for $\Delta_q(l)$ corresponding to a Gromov-Witten
invariant
$$ \l [\sigma_{I_1}],...,[\sigma_{I_l}],[\sigma_K]\r_d \neq 0 $$
is a consequence of an inequality corresponding to a Gromov-Witten
invariant of the form
$$ \l [\sigma_{I_1}],...,[\sigma_{I_{l-1}}],[\sigma_J]\r_{d_{1}} \neq 0 $$
for some $J \subset \{ 1,\ldots, n\}$ and $d_1\leq d.$
\end{lemma}
\begin{proof} Suppose that
$$ \l [\sigma_{I_1}],...,[\sigma_{I_l}],[\sigma_K]\r_d \neq 0 .$$
Taking $k=2$ we obtain that for some $J$ and $d_1\leq d$
$$ \l [\sigma_{I_1}],...,[\sigma_{I_{l-1}}],[\sigma_J]\r_{d_{1}}
\l [\sigma_{*J}],[\sigma_{I_l}],[\sigma_K]\r_{d_2} \neq 0 .$$
Thus the inequality
\begin{equation} \labell{want}
\sum_{i \in I_1} \lam_{1,i} + \ldots + \sum_{i \in I_{l}}
\lam_{l,i} \leq d .\end{equation}
follows from the inequalities
\begin{equation} \labell{first}
\sum_{i \in I_1} \lam_{1,i} + \ldots + \sum_{i \in I_{l-1}}
\lam_{l-1,i} + \sum _{j\in J} \lambda_{l,j} \leq d_1 \end{equation}
for $\lam \in \Delta_q(l)$ and
\begin{equation} \labell{last}
\sum _{j\in *J}
(*\lam_l)_j + \sum _{i\in I_l} (\lam_l)_i \leq d_2.
\end{equation}
The last equation is a tautology for $\lam_l \in \lie{A}$ by the $l=2$
case of Theorem \ref{result}. In other words, \eqref{last} is implied
by the equations $\lam_{l,i} \ge \lam_{l,i+1}, \ \lam_{l,1} -
\lam_{l,n} \le 1$. Thus \eqref{want} follows from \eqref{first} and
the inequalities defining $\lie{A}^l$.
\end{proof}
\section{Hidden symmetry}
\labell{symmetry}
An interesting aspect of the multiplicative problem is that it
possesses a symmetry not present in the additive case, related to the
symmetry of the fundamental alcove $\lie{A}$ of $SU(n)$. Let
$Z \cong \mathbb{Z}/n\mathbb{Z}$ denote the center of $SU(n)$, with generator $c \in SU(n)$
the unique element of $SU(n)$ with
$$ \lambda(c) = (1/n,1/n, \ldots, 1/n, (1-n)/n) .$$
The action of $Z$ on $SU(n)$ induces an action on $\lie{A} \cong SU(n) /
\Ad(SU(n))$, given by
$$ c \cdot (\lambda_1,\ldots,\lambda_n) = (\lambda_2 + 1/n, \lambda_3
+ 1/n,\ldots, \lambda_n + 1/n, \lambda_1 - (n-1)/n) .$$
Let $C(l) \subset SU(n)^l$ denote the subgroup
$$ C(l) = \{ (z_1,\ldots,z_l) \subset Z^l \ | \ z_1z_2 \ldots z_l = 1 \}
\cong Z^{l-1}.$$
The action of $C(l)$ on $\lie{A}^l$ leaves the polytope $\Delta_q(l)$
invariant.
This symmetry of the polytope $\Delta_q(l)$ implies a symmetry on the
facets of $\Delta_q(l)$. Let $c$ act on subsets of $\{ 1,2,\ldots,n \}$
via the action of $(12\ldots n)^{-1} \in S_n$:
$$ c^m \{ i_1, \ldots, i_r \} = \{i_{s+1} - m, \ldots, i_r - m,
i_1 - m + n, \ldots, i_s - m + n\} $$
where $s$ is the largest index for which $i_s - m \ge 1$
Suppose an $l+1$-tuple $(I_1,\ldots,I_l,d)$ defines a facet of
$\Delta_q(l)$ via the inequality \eqref{mult_ineq}. Under the action of
$ (c^{m_1},\ldots,c^{m_l}) \in C(l)$, \eqref{mult_ineq}
becomes the inequality corresponding to $(c^{m_1}I_1,\ldots,c^{m_l}I_l,d')$
where $d'$ is defined by
\begin{equation} \labell{4ac}
\sum_{i=1}^l | c^{m_i}I_i | + nd' = \sum_{i=1}^l | I_i | + nd.
\end{equation}
\begin{example} From the degree $0$ inequality $\lambda_n(A) +
\lambda_n(B) \le \lambda_n(AB)$ we obtain by the action of
$(c^{-i},c^{-j},c^{i+j}), \ i + j \le n$ the degree $1$ inequality
\eqref{quant_Weyl}.
\end{example}
Equation \eqref{4ac} defines a $C(l)$ action on the set of
$l+1$-tuples $(I_1,\ldots,I_l,d)$ defining facets of $\Delta_q(l)$. It
is an interesting fact that the Gromov-Witten invariants $\l
\sig_{I_1},\ldots,\sig_{I_l}\r_d$ are invariant under this action:
\begin{proposition} Let $(c^{m_1},\ldots,c^{m_l}) \in C(l)$. Then
$\l \sig_{I_1},\ldots,\sig_{I_l} \r_d = \l \sigma_{c^{m_1}
I_1}, \ldots, \sigma_{c^{m_l} I_l} \r_{d'}.$
\end{proposition}
\begin{proof} Let $\sigma_c = \sigma_{r,r+1,\ldots,n-1}$ denote the
Schubert variety isomorphic to the Grassmannian $G(r,n-1)$ of
$r$-planes contained in $n-1$-space. We claim that quantum
multiplication by $\sigma_c$ is given by the following formula:
\begin{equation} \labell{Cox_mult}
[ \sigma_c ] \star [ \sigma_I ] = q^{(|cI| + r - |I|)/n} [ \sigma_{cI} ] .
\end{equation}
The exponent $(|cI| + r - |I|)/n$ equals $1$ if $1 \in I$, and equals
$0$ otherwise. In particular $[\sig_c]^{\star n}=q^r.$
The lemma then follows by associativity of the quantum product.
Without loss of generality it suffices to show that the Gromov-Witten
invariants are invariant under an element of the form
$(c,c^{-1},1,\ldots,1) \in C(l)$.
Given that
$$
[\sig _{I_1}]\star \ldots \star [\sig _{I_{l-1}}]\supset
\l \sig_{I_1},\ldots,\sig_{I_l} \r_d
[\sigma_{*I_{l}}]q^d$$
multiplying by $[\sig_c]$ on both sides yields
$$[\sig _{cI_1}]\star \ldots \star [\sig _{I_{l-1}}]\supset \l
\sig_{I_1},\ldots,\sig_{I_l} \r_d [\sigma_{c(*I_{l})}]q^{d'}=\l
\sig_{I_1},\ldots,\sig_{I_l} \r_d [\sig_{*c^{-1}I_l}]q^{d'}.
$$
The formula \eqref{Cox_mult} may be proved using either the canonical
isomorphism of quantum Schubert calculus with the Verlinde algebra of
$U(r)$,
$$ QH^*(G(r,n))/(q=1) \cong R(U(r)_{n-r,n}).$$
given a mathematical proof in Agnihotri \cite{ag:th}, or
using the combinatorial formula of Bertram, Ciocan-Fontanine and
Fulton \cite{be:qm}. $R(U(r)_{n-r,n})$ denotes the Verlinde
algebra of $U(r)$ at $SU(r)$ level $n-r$ and $U(1)$ level $n$, and is
the quotient of the tensor algebra $R(U(r))$ by the relations
$$ V_{\lam} \sim (-1)^l(w) V_{w(\lam+ \rho)-\rho}, w \in \Waff $$
and if $\lam_1 - \lam_r \le n -r $ then
$$ V_{(\lam_1,\ldots,\lam_r)} \sim V_{(\lam_2-1,\lam_3-1,
\ldots,\lam_r-1,\lam_1 - (n-r+1)} .$$
Here $\Waff$ acts on $\Lambda^*$ at level $n$, and $\rho$ is the
half-sum of positive roots. The Verlinde algebra $R(U(r)_{n-r,n})$
has as a basis the (equivalence classes of the) representations
$V_{\lam}$, where $\lam = (\lambda_1,\ldots,\lambda_r) \in \mathbb{Z}^r, \ 0
\leq \lambda_i \leq n- r $ are dominant weights of $U(r)$ at level
$n-r$.
The canonical isomorphism is given by $\sigma_I \mapsto V_{\lambda}$,
where $\lambda$ is defined by
$$ \lambda_j = n-r + j - i_j .$$
The key point is that the sub-algebra $R(U(1)) \subset R(U(r))$
descends to a sub-algebra $ R(U(1)_n) \subset R(U(r)_{n-r,r})$
generated by the representation $V_c := V_{(1,1,\ldots,1)}$, which
maps under the isomorphism to the Schubert variety $\sigma_c$.
From the description of the algebra given above one sees
that
$ V_c \fus V_{\lam} = V_{\lambda'} $
where
$$ \lambda' =
\begin{array}{cl}
(\lambda_1 + 1,\lambda_2 +1,\ldots, \lambda_r + 1) & \hbox{\ if\ }\lambda_1 < n-r
\\ (\lambda_2,\ldots,\lambda_r,\lambda_1 - n +r) & \hbox{\ if\ }\lambda_1 = n-r
\end{array} .$$
Since $V_{\lambda'}$ maps to $\sigma_{cI}$ under the canonical
isomorphism, this proves \eqref{Cox_mult}.
Alternatively, \eqref{Cox_mult} can be derived from the combinatorial
rim-hook formula of \cite[p. 8]{be:qm}. Let $\lambda^t$ denote the
transpose of $\lambda$, so that $\sig_{\lambda^t}$ is the image of
$\sig_\lambda$ under the isomorphism $G(r,n) \cong G(n-r,n)$. The
ordinary (resp. quantum) Littlewood-Richardson numbers are invariant
under transpose
$$ N_{\lam^t \mu^t}^{\rho^t} = N_{\lam \mu}^{\rho}, \ \ \
N_{\lam^t \mu^t}^{\rho^t}(n-r,r) = N_{\lam \mu}^{\rho}(r,n-r).
$$
It follows from \cite[Corollary]{be:qm} that
$$ N_{\lam \mu}^{\rho}(r,n-r) =\sum \eps(\rho^t/\nu^t)
N_{\lam \mu}^{\rho} $$
where $\rho$ ranges over all diagrams of height $\leq r$ that can be
obtained by adding $m$ rim-hooks.
If $\mu = (1,1,\ldots,1)$ then
$$ V_\mu \otimes V_{\lam} = V_\rho, \ \ \rho =
(\lam_1 + 1, \ldots, \lam_r + 1) .$$
If $\lam_1 < n-r$, then since the height of $\rho$ is $\leq r$, there
are no rim $n$-hooks in $\rho$. On the other hand, if $\lam_1 = n-r$,
then it is easy to see that there is a unique rim $n$-hook in $\rho$,
whose complement is $\lam'$ above.
We have learned from A. Postnikov that formula similar to
\eqref{Cox_mult} holds for the full flag variety \cite{po:hs}. A
deeper reason for the appearance of symmetry is given by Seidel
\cite{se:pi}.
\end{proof}
This symmetry simplifies the computation of many Gromov-Witten
invariants. For sufficiently small $n$ and $l$ all Gromov-Witten
invariants are equivalent to degree $0$ ones. An example of a
Gromov-Witten invariant not equivalent via symmetry to a degree $0$
invariant is the degree $1$ invariant for $G(5,10)$
$$ \l \sigma_{\{ 2,4,6,8,10 \}}, \sigma_{\{ 2,4,6,8,10 \}}, \sigma_{\{
1,3,5,7,9 \}} \r_1 $$
which may be computed using the formula of \cite{be:qm}. That is, for
sufficiently large $r,n$, not all of the inequalities are related to
``classical'' inequalities via symmetry.
\section{Verlinde algebras}
\labell{Verlinde}
Finally we want to explain the representation-theoretic interpretation
of $\Delta_q(l)$ in terms of the Verlinde algebra of $SU(n)$. Denote by
$\Lambda^*_N$ the set of dominant weights of $SU(n)$ at level $N$:
$$ \Lambda^*_N = \{ (\lambda_1,\ldots,\lambda_n) \in (\mathbb{Z}/n)^n \ | \
\lam_i - \lam_{i+1} \in \mathbb{Z}_{\ge 0},\ \ \lam_1 - \lam_n \le N \} .$$
The Verlinde algebra $R(SU(n)_N)$ is the free group $\mathbb{Z}[\Lambda_N^*]$
on the generators $V_{\xi}, \xi \in \Lambda_N^*$. The algebra
structure is given by ``fusion product''
$$ V_{\xi_1} \fus_N \ldots \fus_N \mathcal{V}_{\xi_l} = \sum_{\nu \in
\Lambda^*_N} m^N({\xi_1,\ldots,\xi_l,\nu}) \ V_{* \nu} $$
where the coefficients $m^N(\xi_1,\ldots,\xi_l)$ are defined as
follows. There is a positive line bundle $L^N({\xi_1,\ldots,\xi_l})
\to\M(\xi_1/N ,\ldots ,\xi_l/N)$ which descends from the polarizing
line bundle on $\tilde{R}$ (see Pauly \cite[Section 3]{pa:em}). The
coefficient $m^N({\xi_1,\ldots,\xi_l})$ is defined by
$$ m^N({\xi_1,\ldots,\xi_l}) = \dim(H^0(L^N(\xi_1,\ldots,\xi_l)) .$$
Since $L^N$ is positive, $\M(\xi_1/N,\ldots,\xi_l/N)$ is non-empty if
and only if for some $k$
$$\dim(H^0(L^k(\xi_1,\ldots,\xi_l)^{\otimes
N})=\dim(H^0(L^{kN}(k\xi_1,\ldots,k\xi_l))\neq 0,$$
that is, $m^{kN}({k\xi_1,\ldots,k\xi_l}) \neq 0.$
|
1999-06-25T17:31:55 | 9712 | alg-geom/9712026 | en | https://arxiv.org/abs/alg-geom/9712026 | [
"alg-geom",
"math.AG"
] | alg-geom/9712026 | Dr G. K. Sankaran | K. Hulek, I. Nieto, G. K. Sankaran | Degenerations of (1,3) abelian surfaces and Kummer surfaces | LaTeX, 16 pages, 2 figures. Final version, with minor corrections: to
appear in Algebraic Geometry - Hirzebruch 70 (AMS Contemporary Mathematics) | null | null | null | null | We continue our study of the geometry of Nieto's quintic threefold, looking
at degenerate surfaces that correspond to certain loci and showing how they
arise from a toroidal compactification of a suitable moduli space.
| [
{
"version": "v1",
"created": "Fri, 19 Dec 1997 22:02:57 GMT"
},
{
"version": "v2",
"created": "Fri, 25 Jun 1999 15:31:55 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Hulek",
"K.",
""
],
[
"Nieto",
"I.",
""
],
[
"Sankaran",
"G. K.",
""
]
] | alg-geom | \section{Theta functions}
In this section we will give an explicit description of a basis of the
space $H^0({\mathcal L}^{\otimes 2})^-$ which defines the map from $A$ to
${\mathbb{P}}^3$, factoring through the Kummer surface, in terms of theta functions.
Our standard reference for theta functions is Igusa's book \cite {I}. We
shall denote points of the Siegel upper half plane ${\mathbb{H}}_2$ by $\tau=\left(
\begin{array}{cc}
\tau_1 & \tau_2\\
\tau_2 & \tau_3
\end{array}
\right),$ and $z=(z_1, z_2)$ will denote the coordinates on ${\mathbb{C}}^2$. For
every pair $(m', m'')\in{\mathbb{R}}^2\times{\mathbb{R}}^2$ we define the theta function
$$
\Theta_{m' m''}(\tau, z)=\sum\limits_{q\in{\mathbb{Z}}^2} e^{2\pi i[\frac 12
(q+m')\tau^t(q+m')+(q+m')^t(z+m'')]}.
$$
Given a point $\tau=\left(
\begin{array}{cc}
\tau_1 & \tau_2\\
\tau_2 & \tau_3
\end{array}
\right)\in {\mathbb{H}}_2$ we associate to it a period matrix
$$
\Omega_{\tau}=\left(
\begin{array}{cccc}
2\tau_1 & 2\tau_2 & 2 & 0\\
2\tau_2 & 2\tau_3 & 0 & 6
\end{array}
\right)
$$
and the lattice
$$
L_{\tau}={\mathbb{Z}}^4\Omega_{\tau}={\mathbb{Z}} e_1+{\mathbb{Z}} e_2 + {\mathbb{Z}} e_3 +{\mathbb{Z}} e_4
$$
generated by the columns $e_i$ of the period matrix $\Omega_{\tau}$. The
abelian surface
$$
A_{\tau}={\mathbb{C}}^2/L_{\tau}
$$
has a $(1,3)$--(and hence also a $(2,6)$--)polarization. Normally $0\in
A_{\tau}$ is chosen as the origin and the involution given by taking the
inverse is $\iota:x\mapsto -x$. For reasons which will become apparent
later we shall want to define the origin of $A_{\tau}$ as the image of the
point
$$
\omega=\frac 12(1,1)\left(
\begin{array}{cc}
\tau_1 & \tau_2\\
\tau_2 & \tau_3
\end{array}
\right)=\frac 12 (\tau_1+\tau_2, \tau_2+\tau_3).
$$
Note that with respect to $0$ this is a $4$-torsion point. Then the
involution with respect to $\omega$ is given by
$$
\iota_{\omega}(z)=-z+2\omega.
$$
Finally we set
$$
\tau'=\left(
\begin{array}{cc}
\tau_1/2 & \tau_2/6\\
\tau_2/6 & \tau_3/18
\end{array}
\right),\quad
z'=(z_1/2,z_2/6).
$$
The main objects of this section are the functions
$$
{\widehat{\Theta}}_{\alpha \beta}(\tau,z): =
\Theta_{00\frac{\alpha}{2}\frac{\beta}{6}}(\tau',z'-{\omega}'),
\quad \alpha=0,1;\
\beta=0,\ldots,5.
$$
Note that ${\widehat{\Theta}}_{\alpha+2, \beta}={\widehat{\Theta}}_{\alpha
\beta}$ and $\widehat{\Theta}_{\alpha, \beta+6}=\widehat{\Theta}_{\alpha
\beta}$
so that we can read the indices cyclically.
\begin{lemma}\label{lem11}
{\rm(i)} The functions ${\widehat{\Theta}}_{\alpha\beta}$ are
all sections of the same line bundle ${\mathcal L}_{\tau}$ on $A_{\tau}$.\\
{\rm(ii)} ${\mathcal L}_{\tau}$ represents a polarization of type $(2,6)$.
\end{lemma}
\begin{proof}
(i)\ We must prove that the automorphy factor of the functions
${\widehat{\Theta}}_{\alpha \beta}$ with respect to $z\mapsto z+e_i$ does
not depend on $(\alpha, \beta)$. This follows immediately from the
formulae $(\Theta 1)-(\Theta 5)$ of \cite[pp. 49, 50]{I}.\\
\noindent
(ii)\ Since the type of a polarization is constant in families it is enough
to prove the statement for $\tau_2=0$ where
$$
A_{\tau}=E({\tau_1}) \times E({\tau_3})
$$
with
$$
E(\tau_1)={\mathbb{C}}/({\mathbb{Z}} 2\tau_1+{\mathbb{Z}} 2),\ E(\tau_3)={\mathbb{C}}/({\mathbb{Z}} 2\tau_3+{\mathbb{Z}} 6).
$$
In this case
$$
{\widehat{\Theta}}_{\alpha
\beta}(\tau,z)=\vartheta_{0\frac{\alpha}{2}}(\tau_1/2, z_1/2-{\omega_1}/2)\
\vartheta_{0\frac{\beta}{6}}(\tau_3/18, z_2/6-{\omega_2}/6)
$$
where we use $\vartheta$ to denote theta functions in one variable. We
claim that the degree on $E(\tau_1)$ is $2$ and that the degree on
$E(\tau_3)$ is $6$. Indeed the first claim follows since
$$
\begin{array}{rcl}
\vartheta_{0\frac{\alpha}{2}}(\tau_1/2, z_1/2-{\omega_1}/2)=0 &
\Leftrightarrow &
z_1/2 \in ({\mathbb{Z}} \tau_1/2 +{\mathbb{Z}})-\alpha/2+{\omega_1}/2\\
&\Leftrightarrow & z_1\in ({\mathbb{Z}} \tau_1+{\mathbb{Z}} 2)-\alpha+{\omega_1}.
\end{array}
$$
This means that $\vartheta_{0\frac{\alpha}{2}}(\tau_1/2,
z_1/2-{\omega_1}/2)$ has two zeroes on $E(\tau_1)$. The other claim follows
in exactly the same way. \hfill
\end{proof}
We shall denote the sections of ${\mathcal L}_{\tau}$ defined by
${\widehat{\Theta}}_{\alpha\beta}(\tau, z)$ by ${\widehat{s}}_{\alpha
\beta}$. By general theory the twelve sections
${\widehat{s}}_{\alpha\beta};\quad \alpha=0,1,\ \beta=0,\ldots,5$ form a
{\em basis} of $H^0({\mathcal L}_{\tau})$. (Cf. \cite[p.75]{I} for an analogous
statement.)
We now want to describe the symmetry properties of the line bundle ${\mathcal
L}_{\tau}$ and the sections ${\widehat{s}}_{\alpha\beta}$. The kernel of
the map
$$
\begin{array}{rcl}
\lambda:\ A_{\tau} & \rightarrow & \mbox{Pic}^0A_{\tau}\\
x & \mapsto & t_x^*{\mathcal L}_{\tau} \otimes{\mathcal L}_{\tau}^{-1}
\end{array}
$$
where $t_x$ is translation by $x$ is equal to
$$
\operatorname{ker } \lambda =({\mathbb{Z}} \frac{e_1}{2}
+{\mathbb{Z}}\frac{e_2}{6}+{\mathbb{Z}}\frac{e_3}{2}+{\mathbb{Z}}\frac{e_4}{6}) L_{\tau}\cong({\mathbb{Z}}/2)^2
\times({\mathbb{Z}}/6)^2.
$$
\noindent
We set $\rho_6:=e^{2\pi i/6}$.
\begin{proposition}\label{prop12}
{\rm{(i)}} The group $\operatorname{ker} \lambda$ acts on the sections
${\widehat{s}}_{\alpha \beta}$ as follows
$$
\begin{array}{rlcllclcl}
e_1 /2 & : & {\widehat{s}}_{\alpha\beta} & \mapsto &
(-1)^{\alpha}\ {\widehat{s}}_{\alpha\beta}\ , & e_2 /6 & :
{\widehat{s}}_{\alpha\beta} & \mapsto & \rho_6^{-\beta}\
{\widehat{s}}_{\alpha\beta}\\
e_3 /2 & : & {\widehat{s}}_{\alpha\beta} & \mapsto &
{\widehat{s}}_{\alpha+1, \beta}\ , & e_4 /6 & :
{\widehat{s}}_{\alpha\beta} & \mapsto & {\widehat{s}}_{\alpha, \beta+1}.
\end{array}
$$
{\rm{(ii)}} The involution $\iota_{\omega}$ acts on the sections
${\widehat{s}}_{\alpha\beta}$ by
$$
\iota_{\omega}:{\widehat{s}}_{\alpha\beta}\mapsto {\widehat{s}}_{-\alpha,
-\beta}.
$$
\end{proposition}
\begin{proof}
(i)\ We shall prove this for $e_1/2$, the other cases being similar.
Again using \cite[pp. 49, 50]{I} we find
$$
\begin{array}{rcl}
{\widehat{\Theta}}_{\alpha\beta}(\tau, z+\frac{e_1}{2}) & =
& \Theta_{00\frac{\alpha}{2}\frac{\beta}{6}}(\tau',
z'-{\omega}'+(\tau_1/2, \tau_2/6))\\[2mm]
& {\begin{array}{c}
(\Theta 3)\\
{\sim}
\end{array}}
& e^{2\pi i (-(1,0)({\begin{array}{c}\alpha/2\\ \beta/6
\end{array})})} \Theta_{10\frac{\alpha}{2}\frac{\beta}{6}}(\tau',
z'-{\omega}')\\[2mm]
& {\begin{array}{c}
(\Theta 1)\\
=
\end{array}} & (-1)^{\alpha} \Theta_{00\frac{\alpha}{2}\frac{\beta}{6}}(\tau',
z'-{\omega}')\\ [2mm]
& = & (-1)^{\alpha}{\widehat{\Theta}}_{\alpha\beta}(\tau, z).
\end{array}
$$
\noindent
Here $\sim$ denotes equality up to a nowhere vanishing function which is
independent of $\alpha$ and $\beta$.\\
\noindent (ii) Here we have that
$$
\begin{array}{rcl}
{\widehat{\Theta}}_{\alpha\beta}(\tau, -z+2\omega) & = &
\Theta_{00\frac{\alpha}{2}\frac{\beta}{6}}(\tau', -z'+{\omega}')\\
& {\begin{array}{c}
(\Theta 1)\\
=
\end{array}} & \Theta_{00-\frac{\alpha}{2}-\frac{\beta}{6}}(\tau',
z'-{\omega}')\\ & = & {\widehat{\Theta}}_{-\alpha -\beta}(\tau, z)
\end{array}
$$
where indices are to be read cyclically.\hfill
\end{proof}
\begin{remark}\label{rem13}
{\rm{(i)}} Part (i) of the above proposition gives an explicit
description of the lifting of the group $({\mathbb{Z}} /2)^2 \times ({\mathbb{Z}} /6)^2$ to
the Heisenberg group $H_{26}$.\\
{\rm{(ii)}} Note that part (ii) of the above proposition is true for any
choice of the point $\omega$ and hence in particular also for the
involution $\iota$ itself.
\end{remark}
We can now describe a basis of the eigenspaces
$H^0({\mathcal L}_\tau)^+$ and
$H^0({\mathcal L}_\tau)^-$ as follows:
$$
\begin{array}{ll}
{\widehat{u}}_{\alpha\beta}={\widehat{s}}_{\alpha\beta}+{\widehat{s}}
_{-\alpha,-\beta}\in H^0({\mathcal L}_\tau)^+; &
\alpha \in \{0,1\}, \beta \in \{0,1,2,3\}\\
{\widehat{t}}_{\alpha\beta}={\widehat{s}}_{\alpha\beta}-{\widehat{s}}
_{-\alpha,-\beta}\in H^0({\mathcal L}_\tau)^-; &
(\alpha,\beta)= (0,1),(0,2),(1,1),(1,2).
\end{array}
$$
\noindent
For our purposes it is, however, better to work with a different basis of
$H^0({\mathcal L}_\tau)^-$.
$$
\begin{array}{ccrclclcl}
{\widehat{g}}_0 & := & {\widehat{t}}_{01} & + &{\widehat{t}}_{11}& - &
{\widehat{t}}_{02} & - & {\widehat{t}}_{12}\\
{\widehat{g}}_1 & := & -{\widehat{t}}_{01} & - &{\widehat{t}}_{11}& - &
{\widehat{t}}_{02} & - & {\widehat{t}}_{12}\\
{\widehat{g}}_2 & := & {\widehat{t}}_{01} & - &{\widehat{t}}_{11}& - &
{\widehat{t}}_{02} & + & {\widehat{t}}_{12}\\
{\widehat{g}}_3 & := & -{\widehat{t}}_{01} & + &{\widehat{t}}_{11}& - &
{\widehat{t}}_{02} & + & {\widehat{t}}_{12}.
\end{array}
$$
Recall the Heisenberg group $H_{22}$ from \cite {BN}. The group $H_{22}$ has
order~$32$ and
$$
H_{22}/\mbox{ centre } \cong ({\mathbb{Z}}/2)^4
$$
is the group generated by the elements
$$
\begin{array}{cclcl}
\sigma_1 & : & (z_0: z_1: z_2: z_3) & \mapsto & (z_2: z_3: z_0: z_1)\\
\sigma_2 & : & (z_0: z_1: z_2: z_3) & \mapsto & (z_1: z_0: z_3: z_2)\\
\tau_1 & : & (z_0: z_1: z_2: z_3) & \mapsto & (z_0: z_1: -z_2:-z_3)\\
\tau_2 & : & (z_0: z_1: z_2: z_3) & \mapsto & (z_0:-z_1: z_2: -z_3).
\end{array}
$$
The group ${}_2A_{\tau}$ of $2$-torsion points of $A_{\tau}$ is
contained in $\operatorname{ker} \lambda$. Here we identify ${}_2A_{\tau}$
with translations of $A_{\tau}$ of order~$2$. Using the translations
$x\mapsto x+e_i/2$ as generators we obtain an identification of
${}_2A_{\tau}$ with $({\mathbb{Z}}/2)^4$. A straightforward calculation using
Proposition \ref{prop12} and the definition of the basis
${\widehat{g}}_0,\ldots,{\widehat{g}}_3$ shows that
$$
\begin{array}{clclll}
e_1/2 : &(\widehat{g}_0: \widehat{g}_1: \widehat{g}_2:
\widehat{g}_3) & \mapsto &
(\widehat{g}_2: \widehat{g}_3: \widehat{g}_0: \widehat{g}_1) & = &
\sigma_1 (\widehat{g}_0:
\widehat{g}_1: \widehat{g}_2: \widehat{g}_3)\\
e_2/2 : &(\widehat{g}_0: \widehat{g}_1: \widehat{g}_2:
\widehat{g}_3) & \mapsto &
(\widehat{g}_1: \widehat{g}_0: \widehat{g}_3: \widehat{g}_2) & = & \sigma_2
(\widehat{g}_0:
\widehat{g}_1: \widehat{g}_2: \widehat{g}_3)\\
e_3/2 : &(\widehat{g}_0: \widehat{g}_1: \widehat{g}_2:
\widehat{g}_3) & \mapsto &
(\widehat{g}_0: \widehat{g}_1: -\widehat{g}_2: -\widehat{g}_3) & = & \tau_1
(\widehat{g}_0:
\widehat{g}_1: \widehat{g}_2: \widehat{g}_3)\\
e_4/2 : &(\widehat{g}_0: \widehat{g}_1: \widehat{g}_2:
\widehat{g}_3) & \mapsto &
(\widehat{g}_0: -\widehat{g}_1: \widehat{g}_2: -\widehat{g}_3) & = & \tau_2
(\widehat{g}_0:
\widehat{g}_1: \widehat{g}_2: \widehat{g}_3)
\end{array}
$$
We can, therefore, summarize our results as follows:
\begin{theorem}\label{theo13}
{\rm(i)} The basis $\widehat{g}_0,\ldots,\widehat{g}_3 \in H^0({\mathcal
L}_{\tau})^-$ defines a rational map from $A_{\tau}$ to ${\mathbb{P}}^3$ which
factors through $\widetilde{\operatorname{Km}}(A_{\tau})$. This map is
equivariant with respect to the action of ${}_2A_{\tau}\cong ({\mathbb{Z}}/2)^4$
on $A_{\tau}$ and of $H_{22}/\mbox{centre }\cong ({\mathbb{Z}}/2)^4$ on ${\mathbb{P}}^3$.
In particular the image is $H_{22}$-invariant.\\
{\rm(ii)} The Kummer surface $\widetilde{\operatorname{Km}}(A_{\tau})$ is
embedded as a smooth quartic surface if and only if $A_{\tau}$ is neither
a product nor a bielliptic abelian surface. If $A_{\tau}$ is bielliptic
then $\widetilde{\operatorname{Km}}(A_{\tau})$ is mapped to a quartic
with four nodes; if $A_{\tau}$ is a product, then
$\widetilde{\operatorname{Km}}(A_{\tau})$ is mapped $2:1$ onto a quadric.
\end{theorem}
\begin{proof}
{\rm(i)} Follows immediately from our above calculations.\\
{\rm(ii)} This was shown in \cite{HNS}.
\hfill
\end{proof}
\begin{remark}
${\mathcal L}_{\tau}$ is the unique totally symmetric line bundle with
respect to the involution ${\iota}_{\omega}$.
\end{remark}
\section{Degenerations}
In this section we construct degenerations of $(1,3)$--polarized abelian
surfaces which correspond to points on the S-planes. The construction of
degenerating families of abelian varieties is in general technically
complicated (see e.g. \cite{FC}, \cite{AN}). Although we cannot avoid these
technicalities entirely, we have tried to present our construction in a way
which uses only a minimum of technical steps. These, however, cannot be
avoided.
We consider the group
$$
P=\left\{
\left(
\begin{array}{c|c}
\left.{\bf 1}\right. &
\begin{array}{rr}
2 {\mathbb{Z}} & 6{\mathbb{Z}}\\
6 {\mathbb{Z}} & 18 {\mathbb{Z}}
\end{array}\\
\hline
0 & {\bf 1}
\end{array}
\right)
\right\} \subset \mbox{Sp}(4,{\mathbb{Z}}).
$$
Note that this is the lattice contained in the parabolic subgroup of
$\Gamma_{1,3}(2)\cap \Gamma_{1,3}^{\mbox{\scriptsize lev}} $ which fixes
the isotropic plane $h=(0, 0, 1, 0)\wedge (0, 0, 0, 1)$. Here
$\Gamma_{1,3}(2)$ is the group which defines the moduli space of abelian
surfaces with a $(1,3)$--polarization and a level-$2$ structure, whereas
$\Gamma_{1,3}^{\mbox{\scriptsize lev}}$ belongs to the moduli space of
$(1,3)$--polarized abelian surfaces with a canonical level structure (cf.
\cite[I.1] {HKW}). There are two reasons for considering this group. One is
that we can then make use of the constructions in \cite{HKW} which from our
point of view is the most economical way to construct the degenerations
which we are interested in; the second reason is that, at least with the
known constructions of degenerations of abelian surfaces, the presence of a
canonical level structure is necessary. We could also use the method of
Alexeev and Nakamura \cite{AN} which likewise goes back to Mumford's
construction \cite{M}, and \cite{Nak}, \cite{Nam}. For the surfaces which
we are interested in it makes, however, little difference which of these
methods we choose.
The group $P$ acts on ${\mathbb{H}}_2$ by
$$
\left(
\begin{array}{cc}
\tau_1 & \tau_2\\
\tau_2 & \tau_3
\end{array}
\right)\mapsto\left(
\begin{array}{cc}
\tau_1+2{\mathbb{Z}} & \tau_2+6{\mathbb{Z}}\\
\tau_2+6{\mathbb{Z}} & \tau_3+18{\mathbb{Z}}
\end{array}\right).
$$
The partial quotient of ${\mathbb{H}}_2$ by $P$ is given by
$$
\begin{array}{ccl}
{\mathbb{H}}_2 & \rightarrow & {\mathbb{H}}_2/P\subset ({\mathbb{C}}^*)^3\\
\left(
\begin{array}{cc}
\tau_1 & \tau_2\\
\tau_2 & \tau_3
\end{array}
\right) & \mapsto &
(e^{2\pi i \tau_1/2}, e^{2\pi i \tau_2/6},e^{2\pi i \tau_3/18})=(t_1,t_2,t_3).
\end{array}
$$
Recall that
$$
A_{\tau}={\mathbb{C}}^2/L_{\tau}
$$
where
$$
L_{\tau}={\mathbb{Z}}
\left(
\begin{array}{l}
2\tau_1\\
2\tau_2
\end{array}
\right)+
{\mathbb{Z}}
\left(
\begin{array}{l}
2\tau_2\\
2\tau_3
\end{array}
\right)+
{\mathbb{Z}}
\left(
\begin{array}{c}
2\\
0
\end{array}
\right)+
{\mathbb{Z}}
\left(
\begin{array}{c}
0\\
6
\end{array}
\right)=L'_{\tau}+L''.
$$
Here $L_{\tau}'$ is spanned by the first two columns of $\Omega_{\tau}$
and $L''$ by the last two. Obviously $L''$ does not depend on $\tau$ and
$$
{\mathbb{C}}^2/L''=({\mathbb{C}}^*)^2.
$$
We shall use the coordinates
$$
w_1=z_1/2,\quad w_2=z_2/6
$$
on $({\mathbb{C}}^*)^2$. The lattice $L'_{\tau}$ acts on the trivial torus bundle
${\mathbb{H}}_2/P\times({\mathbb{C}}^*)^2$ by
$$
(m, n): (t_1, t_2, t_3; w_1, w_2)\mapsto (t_1, t_2, t_3;\ t_1^{2m}\ t_2^{6n}
w_1, t_2^{2m}\ t_3^{6n} w_2).
$$
We have to extend the trivial bundle ${\mathbb{H}}_2/P\times({\mathbb{C}}^*)^2$ to the
boundary in such a way that the action of $L'_{\tau}$ also extends. The
general theory of toroidal compactifications of moduli spaces of abelian
surfaces (the material which is relevant in our situation can be found in
\cite{HKW}) leads us to consider first the map
$$
\begin{array}{ccl}
({\mathbb{C}}^*)^3 & \rightarrow & {\mathbb{C}}^3\\
(t_1, t_2, t_3) & \mapsto & (t_1 t_2, t_2 t_3, t_2^{-1})=(T_1, T_2, T_3).
\end{array}
$$
Let
$$
B:=\overset{\circ}{(\overline{{\mathbb{H}}_2/P})}
$$
be the interior of the closure of ${\mathbb{H}}_2/P$ in ${\mathbb{C}}^3$ in the
${\mathbb{C}}$--topology. (What we have considered here is an open part of the
partial compactification in the direction of the cusp corresponding to
$h=(0,0,1,0) \wedge (0,0,0,1)$. The surfaces $B\cap\{T_i=0\}$ are mapped to
boundary surfaces in the Igusa compactification of the moduli space ${\mathcal
A}_{1,3}^{\mbox{\scriptsize lev}}(2)$ of $(1,3)$--polarized abelian
surfaces with both a level-$2$ and a canonical level structure.) In terms
of the coordinates $T_i$ the action of $L'_{\tau}$ is now given by
$$
(m, n): (T_1, T_2, T_3; w_1, w_2)\mapsto (T_1, T_2, T_3;\ T_1^{2m} T_3^{2m-6n}
w_1, T_2^{6n} T_3^{6n-2m} w_2).
$$
Here we are particularly interested in degenerations which are given by
$\tau_1\rightarrow i\infty$. This is equivalent to $t_1=0$ and hence
corresponds to points on the surface $T_1=0$.
We now consider the space
$$
{\tilde {\mathcal P}}=\mbox{Proj }
R_{\Phi,\Sigma}\rightarrow \mbox{Spec } {\mathbb{C}}[T_1, T_2,
T_3]\cong {\mathbb{C}}^3
$$
which was defined in \cite[p.210]{HKW}. Let
$$
{\mathcal P}:={\tilde {\mathcal P}}|_{B}.
$$
Then ${\mathcal P}$ is a partial compactification of the trivial torus
bundle ${\mathbb{H}}_2/P\times({\mathbb{C}}^*)^2$ over ${\mathbb{H}}_2/P\subset B$. Moreover the
action of $L_{\tau}$ on the trivial torus bundle extends to an action on
${\mathcal P}$. The construction of ${\tilde {\mathcal P}}$ is originally due to
Mumford \cite[final example]{M}. Let
$$
{\bar A}:={\mathcal P}/L_{\tau}.
$$
Then we have a diagram
$$
\begin{array}{ccc}
A & \subset & {\bar A}\\
\pi\downarrow & & \downarrow\pi\\
{\mathbb{H}}_2/P & \subset & B
\end{array}
$$
where $A=({\mathbb{H}}_2/P\times({\mathbb{C}}^*)^2)/L_{\tau}$ is the universal family. In
particular ${\bar A}$ extends the universal family $A$ to the boundary. The
fibres
$$
{\bar A}_{u}=\pi^{-1}(u)
$$
over ``boundary points'' $u\in B\backslash({\mathbb{H}}_2/P)$ are degenerate
abelian surfaces. We are interested in the fibres ${\bar A}_u$ over points
$u=(0, T_2, T_3)$ with $T_2 T_3\neq 0$. These are the corank~$1$
degenerations associated to the boundary component given by
$\tau_1\rightarrow i\infty$. Note that if $T_2T_3\neq 0$ then this gives
$t_2=T_3^{-1}$ and $t_3=T_2T_3$. In particular the point $u$ determines a
point $(\tau_2, \tau_3)\in{\mathbb{C}}\times {\mathbb{H}}_1$ where $\tau_2$ and $\tau_3$
are uniquely defined up to $6{\mathbb{Z}}$ and $18{\mathbb{Z}}$ respectively.
We can now formulate the main result of this section.
\begin{theorem}\label{theo21}
Let $u=(0, T_2, T_3)\in B$. Then ${\bar A}_u$ is a degenerate abelian
surface with the following properties:\\
{\rm(i)} ${\bar A}_u$ is a corank $1$ degeneration. More precisely ${\bar
A}_u$ is a chain of two elliptic ruled surfaces $A_{u,1}, A_{u,2}$ i.e.
there exists an elliptic curve $E$ and a line bundle ${\mathcal M}_u\in
\operatorname{Pic}^{0}(E)$ such that $A_{u,i}={\mathbb{P}}({\mathcal O}_E\oplus {\mathcal
M}_u), \ i=1,2$. The surfaces $A_{u,i}$ are glued with
a glueing parameter $e$ as shown below in Figure~$1$.\\
{\rm(ii)} The base curve $E\cong E(\tau_3)={\mathbb{C}}/({\mathbb{Z}} 2\tau_3 + {\mathbb{Z}} 6)$.\\
{\rm(iii)} The line bundle ${\mathcal M}_u={\mathcal
O}_{E(\tau_3)}(6[\tau_2]-6[0])$ where $[\tau_2],[0]$
are the points of $E(\tau_3)$ given by $\tau_2$ and~$0$.\\
{\rm(iv)} The glueing parameter $e=[2\tau_2] \in E(\tau_3).$
\unitlength1cm
\begin{figure}[htb]
\begin{picture}(13.5,8.5)
\put(-0.7,0){\includegraphics{bild13.eps}}
\end{picture}
\caption{Glueing of the surface $\bar A_u$}
\end{figure}
\end{theorem}
\begin{proof}
We can derive this from \cite[part II]{HKW}. There the quotient ${\hat
A}={\mathcal P}/{\hat L}$ was considered where ${\hat L}\cong {\mathbb{Z}}^2$ acts on
the trivial torus bundle ${\mathbb{H}}_2/P\times({\mathbb{C}}^*)^2$ by
$$
(m, n): (T_1, T_2, T_3; w_1, w_2) \mapsto (T_1, T_2, T_3; T_1^{m} T_3^{m-n}
w_1, T_2^{n} T_3^{n-m} w_2).
$$
Hence $L'_{\tau}$ is a subgroup of ${\hat L}$ with ${\hat
L}/L_{\tau}'\cong({\mathbb{Z}}/2)\times({\mathbb{Z}}/6)$. This means that we can use the
description of ${\hat A}$ given in \cite[part II]{HKW} to give a
description of ${\bar A}$. In the terminology of \cite{HKW} the group
$L_{\tau}'=<s^{-2}, r^{-6}>$. The statements (i) and (ii) now follow
exactly as in the proof of \cite [Theorem II.3.10]{HKW}. In particular, the
fact that $s^2\in L_{\tau}'$, but $s \not\in L_{\tau}'$ implies that ${\bar
A}_u$ has two irreducible components. The statement about the base curve
$E$ follows from diagram \cite[II.3.13]{HKW}. Statements (iii) and (iv) are
an immediate consequence of the proof of \cite[Proposition (II.3.20)]{HKW}.
\end{proof}
Our next task is to study the involutions $\iota$ and $\iota_{\omega}$ on
$A$ and their extensions to ${\bar A}$. If we choose $0\in
A_{\tau}={\mathbb{C}}^2/L_{\tau}$ as the origin, then this defines a section of $A$
which extends to a section of ${\bar A}$. Moreover, the involution
$\iota:z\mapsto -z$ defines an involution of $A$ which extends to ${\bar
A}$ (this is the involution given by \cite [Lemma (II.2.9)(ii)]{HKW}. But
we said in section 1 that we wanted to choose $\omega=[(\tau_1+\tau_2)/2,
(\tau_2 + \tau_3)/2]$ as the origin. This point is a $4$-torsion point of
$A_{\tau}$ if we choose~$0$ as the origin. This choice of origin will be
necessary for what follows, but at this point it has the disadvantage that
it only defines a multisection of $A$, not a section. Nevertheless this
multisection extends to ${\bar A}$. We also claim that the involution
$\iota_{\omega}(z)=-z+2\omega$ extends to ${\bar A}$. Since $z\mapsto -z$
is defined on ${\bar A}$ it is enough to show that the translation
$z\mapsto z+2\omega$ is defined on $A$ and extends to ${\bar A}$. This is
easy to see, since $z\mapsto z+2\omega$ in terms of the coordinates $w_1,
w_2$ is given by
$$
(w_1, w_2)\mapsto(t_1 t_2^3 w_1, t_2 t_3^3 w_2)=(T_1 T_3^{-2} w_1, T_2^3 T_3^2
w_2).
$$
This is the element $s^{-1} r^{-3} \in {\hat L}$ and hence acts on
${\mathcal P}$ and on ${\bar A}={\mathcal P}/L'_{\tau}$. Since $s^{-2} r^{-6}\in
L'_{\tau}$, this is an involution. In particular $\iota_{\omega}$ defines
an involution on the fibres ${\bar A}_u$ of ${\bar A}$. Recall that for
$u=(0, T_2, T_3)$ with $T_2 T_3\neq 0$ the surface ${\bar A}_u$ has two
irreducible components $A_{u,i},\ i=1,2$ and that the singular locus of
${\bar A}_u$ consists of two disjoint elliptic curves $E_1$ and $E_2$ with
$E_i \cong E(\tau_3)={\mathbb{C}}/({\mathbb{Z}} 2\tau_3+{\mathbb{Z}} 6)$.
\begin{proposition}\label{prop22}
The involution $\iota_{\omega}$ interchanges the two components $A_{u,1}$
and $A_{u,2}$ of ${\bar A}_u$ and induces an involution on each of the
two curves $E_1$ and $E_2$ with four fixed points on each of these curves.
\end{proposition}
\begin{proof}
The involution $\iota$ fixes each of the surfaces $A_{u,1}$ and $A_{u,2}$
and interchanges $E_1$ and $E_2$. Addition by the $2$-torsion point
$2\omega$ also interchanges $A_{u,1}$ and $A_{u,2}$ as well as $E_1$ and
$E_2$. Hence $\iota_{\omega}$ interchanges $A_{u,1}$ and $A_{u,2}$ but
induces non-trivial involutions on $E_1$ and $E_2$. In order to
determine the fixed points of $\iota_{\omega}$ it is sufficient to
compute the limit of the fixed points of $\iota_{\omega}$ in $A_{\tau}$
as $\tau_1\rightarrow i\infty$. The sixteen fixed points of $\iota_{\omega}$
on $A_{\tau}$ are given by
$$
\left[(\tau_1+\tau_2,
\tau_2+\tau_3)/2+\varepsilon_1(\tau_1,\tau_2)+\varepsilon_2(\tau_2,\tau_3)+
\varepsilon_3
(1,0)+ \varepsilon_4(0,3)\right]\in A_{\tau}
$$
where $\varepsilon_i=0$ or $1$. As $\tau_1\rightarrow i\infty$ these 16
points come together in pairs; more precisely any two points which only
differ by $\varepsilon_3$ have the same limit. This gives us eight points of
which four lie on each of the curves $E_i$ (depending on whether
$\varepsilon_1=0$ or $1$). These points are given by
$$
\begin{array}{ll}
\left[(\tau_2+\tau_3)/2+\varepsilon_2\tau_3+\varepsilon_4 3\right] \in
E(\tau_3) & (\varepsilon_1=0)\\[2mm]
\left[(\tau_2+\tau_3)/2+\tau_2+\varepsilon_2\tau_3+\varepsilon_4
3\right]\in E(\tau_3) & (\varepsilon_1=1).
\end{array}
$$
\hfill
\end{proof}
Figure~$2$ indicates the action of $\iota_{\omega}$ and the position of the
eight fixed points on $\bar A_u$.
\unitlength1cm
\begin{figure}[htb]
\begin{center}
{\includegraphics[width=0.5\columnwidth]{bild16.eps}}
\end{center}
\caption{The involution $\iota_{\omega}$}
\end{figure}
Note that two fixed points lie on one ruling if and only if $[\tau_2]$ is a
$2$-torsion point on $E(\tau_3)$, i.e. if and only if the glueing parameter
$e=[2\tau_2]=0$.
The next step is to extend the polarization to the degenerate abelian
surfaces. Ideally we would like to glue the line bundle ${\mathcal L}_{\tau}$
on $A_{\tau}$ to a line bundle ${\mathcal L}$ on $A$ and to extend this line
bundle to ${\bar A}$ in such a way that the sections ${\widehat s}_{\alpha
\beta}$ as well as the action of the symmetry group (see Proposition \ref
{prop12}) extend. At this point, however, we encounter a fundamental
difficulty. We have seen that it is possible to extend $A$ to ${\bar A}$ in
such a way that the symmetries, and here in particular the involution
$\iota_{\omega}$, extend. It is also possible to define a suitable line
bundle ${\mathcal L}$ and extend it to a line bundle ${\bar {\mathcal L}}$ on
${\bar A}$ (see \cite[II.5]{HKW}). But it is not possible to do this in
such a way that the action of the symmetry group also extends to
${\bar{\mathcal L}}$. (This leads in particular to a numerical contradiction on
the fibre over the origin $0\in B$.) For this reason we shall now restrict
ourselves to taking the partial quotient with respect to the group
$$
P'=\left\{
\left(
\begin{array}{c|c}
{\bf 1} &
\begin{array}{cc}
2{\mathbb{Z}}\ & 0\\
0 & 0
\end{array}\\
\hline
0 & {\bf 1}
\end{array}
\right)
\right\}
\subset \mbox {Sp}(4,{\mathbb{Z}}).
$$
This is the lattice contained in the stabilizer of the line generated by
$l_0=(0, 0, 1, 0)$ in the group $\Gamma_{1,3}(2)\cap
\Gamma_{1,3}^{\mbox{\scriptsize lev}} $. The partial quotient defined by
this group is given by the map
$$
\begin{array}{rcl}
{\mathbb{H}}_2 & \rightarrow &
{\mathbb{C}}^*\times{\mathbb{C}}\times{\mathbb{H}}_1\subset{\mathbb{C}}\times{\mathbb{C}}\times{\mathbb{H}}_1\\[2mm]
\left(\begin{array}{cc}
\tau_1 & \tau_2\\
\tau_2 & \tau_3
\end{array}\right)
& \mapsto & (t_1=e^{2\pi i\tau_1/2}, \tau_2, \tau_3).
\end{array}
$$
Partial compactification of ${\mathbb{H}}_2/P'$ in ${\mathbb{C}}\times{\mathbb{C}}\times{\mathbb{H}}_1$ is given by
$$
B':=\overset{\circ}{(\overline{{\mathbb{H}}_2/P'})}
$$
The two partial quotients with respect to $P$ and $P'$ are related by
the glueing map
$$
\begin{array}{rcl}
\varphi:\quad B' & \rightarrow & B\\
(t_1, \tau_2, \tau_3) & \mapsto & (t_1 t_2, t_2 t_3, t_2^{-1})
\end{array}
$$
where $t_2=e^{2\pi i\tau_2/6}$ and $t_3=e^{2\pi i \tau_3/18}$. The image
of $\varphi$ is $B\backslash (B\cap\{T_2T_3=0\})$ and the map $\varphi$ is
unramified onto its image. We can pull the family ${\bar A}$ over $B$ back
to $B'$ via $\varphi$ and we shall denote the resulting family by ${\bar
A'}$. This family extends the universal family $A'$ over ${\mathbb{H}}_2/P'$. We
shall denote the projection from ${\bar A'}$ to $B'$ by $\pi'$.
\begin{proposition}\label{prop23}
{\rm(i)} The line bundles ${\mathcal L}_{\tau}$ on $A_{\tau}$ glue to a line
bundle ${\mathcal L}'$ on~$A'$.\\
{\rm(ii)} The line bundle ${\mathcal L}'$ can be extended to a line bundle
${\bar{\mathcal L'}}$ on ${\bar A'}$ in such a way that the sections
${\widehat s}_{\alpha \beta}$ as well as the action of the Heisenberg
group $H_{2 6}$ and the involution $\iota_{\omega}$ extend.
\end{proposition}
\begin{proof}
A straightforward computation shows that with respect to the coordinates
$t_1=e^{2\pi i\tau_1/2}$ and $w_1=e^{2\pi i z_1/2}, w_2=e^{2\pi i z_2/6}$
the functions ${\widehat\Theta}_{\alpha \beta}(\tau, z)$ are given by:
$$
\begin{array}{rcl}
{\widehat\Theta}_{\alpha \beta}(\tau, z) & = &
{\Theta}_{00}\frac{\alpha}{2}\frac{\beta}{6}(\tau', z' -{\omega}')\\[2mm]
& = & \sum\limits_{q\in{\mathbb{Z}}^2} t_1^{\frac 12 q_1(q_1-1)} \exp\{9\pi i
q_2(q_2-3)\tau_3\}\\[3mm]
& & \qquad\cdot\exp\{6\pi i(2q_1 q_2 -3 q_1 -q_2)\tau_2\}w_1^{q_1}
w_2^{q_2} (-1)^{\alpha q_1}\rho_6^{\beta q_2}.
\end{array}
$$
In particular this shows that we can consider these functions as
functions on $({\mathbb{H}}_2/P')\times ({\mathbb{C}}^*)^2$. Similarly we find that with
respect to the lattice $L_{\tau}$ the functions ${\widehat\Theta}_{\alpha
\beta}(\tau, z)$ have the following transformation behaviour. For $(k,
l)\in {\mathbb{Z}}^2$:
$$
{\widehat\Theta}_{\alpha\beta} (\tau,
z+(2k,6l))={\widehat\Theta}_{\alpha\beta}(\tau,z).
$$
For $(m,n)\in {\mathbb{Z}}^2$:
$$
\begin{array}{l}
{\widehat\Theta}_{\alpha\beta} \left(\tau, z+(m,n)\left(
\begin{array}{cc}
2\tau_1 & 2\tau_2\\
2\tau_2 & 2\tau_3
\end{array}
\right)\right) \\
\qquad= \Theta_{00\frac{\alpha}{2}\frac{\beta}{6}}\left(\tau',
z'-{\omega}'+(2m,6n)
\left(
\begin{array}{cc}
\tau'_1 & \tau'_2\\
\tau'_2 & \tau'_3
\end{array}\right)\right)\\[2mm]
\qquad=\exp\left\{2\pi i\left[-\frac 12 (2m, 6n)\left(
\begin{array}{cc}
\tau'_1 & \tau'_2\\
\tau'_2 & \tau'_3
\end{array}\right)\left(\begin{array}{c} 2m\\ 6n\end{array}\right)-(2m,
6n)(z'-{\omega}')\right]\right\}\\[2mm]
\qquad\qquad\cdot\Theta_{2m, 6n,\frac {\alpha}{2},\frac{\beta}{6}} (\tau',
z'-{\omega}')\\[2mm]
\qquad= t_1^{-2m^2+m} w_1^{-2m} w_2^{-6n} e^{2\pi
i[(-2mn+\frac{m^2}{2})\tau_2+(-n^2+\frac n2)\tau_3]}
{\widehat\Theta}_{\alpha \beta}(\tau, z).
\end{array}
$$
\noindent
These calculations show claim (i). To prove (ii) we have to consider the
limit as $t_1\rightarrow 0$. Here we find
$$
\begin{array}{rcl}
\lim\limits_{t_1\rightarrow 0}{\widehat\Theta}_{\alpha \beta}(\tau,
z) & = &\sum\limits_{q_2\in {\mathbb{Z}}} e^{9\pi i q_2(q_2-3)} e^{6\pi i
(-q_2\tau_2)}
w_2^{q_2} \rho_6^{\beta q_2}\\
&&\quad+(-1)^{\alpha} w_1\sum\limits_{q_2\in{\mathbb{Z}}} e^{9\pi i q_2(q_2-3)}
e^{2\pi i
(q_2-3)\tau_2} w_2^{q_2} \rho_6^{\beta q_2}\\
& = &\vartheta_{0\frac k6} \left(\tau_3 /
6,(z_2-\tau_3/2-\tau_2/2)/6\right )
\\
&&\quad+(-1)^{\alpha} w_1 e^{2\pi
i\left(-\frac{\tau_2}{4}\right)}\vartheta_{0\frac k6}\left(\tau_3
/6,(z_2-\tau_3/2+\tau_2/2)/6
\right).
\end{array}
$$
To prove that ${\mathcal L}'$ can be extended to a line bundle ${\bar{\mathcal
L}'}$ on $\bar A'$ we can argue exactly as in the proof of
\cite[Proposition (II.5.13)]{HKW}, the only difference being that we took
the partial quotient with respect to a smaller group. (Note that if we take
the quotient with respect to $P$ we no longer obtain integer exponents of
$t_2$.) The extension of the action of the symmetry group follows as in the
proof of \cite[Proposition (II.5.41)]{HKW}. \hfill
\end{proof}
\section {The map to ${\mathbb{P}}^3$}
We consider boundary points $u'=(0,\tau_2, \tau_3) \in B'$ and
$u=(0,t_2t_3, t_2^{-1})\in B$ (so $u=\varphi(u')$) and the associated
degenerate abelian surfaces
$$
{\bar A'}_{u'}={\bar A}_u=A_{u,1}\cup A_{u,2}
$$
where $A_{u,1}=A_{u,2}$ is an elliptic ruled surface. We gave a precise
description of the surfaces $A_{u,i}$ and the way they are glued in Theorem
\ref{theo21}. Recall that $A_{u,i}$ is an elliptic ruled surface with two
disjoint sections $E_i, i=1,2$ of self-intersection number $E_i^2=0$. We
shall denote the fibre over a point $P$ of the base curve by $f_P$. Recall
also the line bundle ${\bar{\mathcal L}}'$ on ${\bar A}'$. We set
$$
{\bar{\mathcal L}}_u:={\bar{\mathcal L}}'|_{{\bar A}'_u}={\bar{\mathcal L}}'|_{{\bar A}_u},
\quad {\mathcal
L}_{u,i}:={\bar{\mathcal L}}'|_{A_{u,i}}.
$$
\noindent
In the proof of Proposition \ref{prop23} we computed that
$$
\begin{array}{rl}
\lim\limits_{t_1\rightarrow 0}
{\widehat\Theta}_{\alpha\beta}(\tau,z)=&
\vartheta_{0\frac{k}{6}}(\tau_3/6,(z_2-\tau_3/2-\tau_2/2)/6)\\[2mm]
&+w^{-1}\vartheta_{0\frac{k}{6}}(\tau_3/6,(z_2-\tau_3/2+\tau_2/2)/6).
\end{array}
$$
The theta function $\vartheta_{0\frac k6}(\tau_3/6,
(z_2-\tau_3/2+\tau_2/2)/6)$ has six zeroes on the elliptic curve
$E(\tau_3)={\mathbb{C}}/({\mathbb{Z}} 2\tau_3+\ZZ6)$. Hence
$$
\operatorname{deg} {\mathcal L}_{u,i}|_{E_i}=6.
$$
Since the exponent of $w$ is $-1$ it follows that
$$
\operatorname{deg} {\mathcal L}_{u,i}|_{f_P}=1.
$$
(See \cite[Proposition(II.5.35)]{HKW} for similar considerations in
the $(1,p)$ case.) Hence
$$
{\mathcal L}_{u,i}={\mathcal O}_{A_{u,i}}(E_1+6 f_{P_i})
$$
for a suitable point $P_i\in E(\tau_3)$. (The point $P_i$ can be
computed from $\lim\limits_{t_{1\rightarrow
0}}{\widehat\Theta}_{\alpha\beta}(t,z)$ and the normal bundle of $E_1$ in
$A_{u,i}$, but we shall not need this later.) Standard arguments using
Riemann-Roch show that
$$
h^0(A_{u,i}, {\mathcal L}_{u,i})=h^0(A_{u,i}, {\mathcal O}_{A_{u,i}}(E_1+6f_{P_i}))=12.
$$
\begin{proposition}\label{prop31}
The restriction ${\bar{\mathcal L}}_u={\bar{\mathcal L}}'|_{{\bar A}'_{u'}}
={\bar{\mathcal L}}'|_{{\bar A}_{u}}$ of ${\bar{\mathcal L}}'$ to the degenerate
abelian surface ${\bar A}_u$ has the
following properties:\\
{\rm (i)} $h^0({\bar A}_u,{\bar{\mathcal L}}_u)=12,$\\
{\rm (ii)} The restriction map $\operatorname{rest}: H^0({\bar
A}',{\bar{\mathcal
L}}')\rightarrow H^0({\bar A}_u, {\bar{\mathcal L}}_u)$ is surjective.\\
{\rm (iii)} The restriction map $\operatorname{rest}: H^0({\bar
A}_u,{\bar{\mathcal L}}_u)\rightarrow H^0 (A_{u,i} {\mathcal L}_{u,i})$ is an
isomorphism.
\end{proposition}
\begin{proof}
We consider the space $V\subset H^0({\bar A}',{\bar{\mathcal L}}')$ which is
spanned by the twelve sections ${\widehat s}_{\alpha \beta}$. Since these
sections are a basis of $H^0(A_{\tau},{\mathcal L}_{\tau})$ for every $\tau$
the space $V$ has dimension~$12$. We claim that the restriction map
$$
\mbox{rest}: V\rightarrow H^0({\bar A}_u, {\bar{\mathcal L}}_u)
$$
is injective. By our computation of $\lim\limits_{t\rightarrow 0}
{\widehat\Theta}_{\alpha \beta}(\tau, z)$ it follows that this map is not
identically zero. It is also $H_{26}$-equivariant and hence our claim
follows if we can show that $V$ is irreducible as an $H_{26}$--module. But
this is easy to see: As an $H_6$-module $V=V_0\oplus V_1$ where
$V_i=\mbox{span } (\widehat s_{i\beta}, \beta=0,\ldots, 5)$. The
$H_6$--modules $V_0$ and $V_1$ are irreducible, Moreover addition by
$e_3/2$ interchanges $V_0$ and $V_1$. Hence $h^0({\bar A}_u, {\bar{\mathcal
L}}_u)\ge 12$.
We have already remarked that $h^0(A_{u,i},{\mathcal
O}_{A_{u,i}}(E_1+6f_{P_i}))=12$. Our next claim is that the map
$$
\mbox{rest }: H^0(A_{u,i}, {\mathcal L}_{u,i})\rightarrow H^0(E_1,{\mathcal
L}_{u,i}|_{E_1})\oplus H^0(E_2,{\mathcal L}_{u,i}|_{E_2})
$$
is an isomorphism. Since the vector spaces on both side have the same
dimension, namely~$12$, it is enough to prove injectivity. This follows
from $h^0(A_{u,i},{\mathcal O}_{A_{u,i}}(-E_2+6f_{P_i}))=0$. But now this
implies that glueing sections on $A_{u,1}$ and $A_{u,2}$ along $E_1$ and
$E_2$ gives at least $2\times 6=12$ conditions. Hence $h^0({\bar A}_u,
{\bar{\mathcal L}}_u)\le 12$. With our previous argument this shows that
$h^0({\bar A}_u, {\bar{\mathcal L}}_u)=12$ and hence both (i) and (ii) are
proved. This also shows that
$$
\mbox{rest}: H^0({\bar A}_u,{\bar{\mathcal L}}_u)\rightarrow H^0(A_{u,i}, {\mathcal
L}_{u,i})
$$
is an isomorphism and hence we have shown (iii).\hfill
\end{proof}
Since we are interested in the map to ${\mathbb{P}}^3$ given by $H^0({\mathcal L})^-$ we
consider the subspace
$$
V^-=\langle {\widehat g}_0, {\widehat g}_1, {\widehat g}_2, {\widehat
g}_3\rangle\subset V
\subset H^0({\bar A}', {\bar{\mathcal L}}')
$$
and
$$
\begin{array}{lcl}
V^-_{u}&=&\mbox{ rest } (V^-\rightarrow H^0 ({\bar A}_u, {\bar {\mathcal
L}}_u)),\\[2mm]
V^-_{u,i}&=&\mbox{ rest } (V^-\rightarrow H^0 (A_{u,i},
{\mathcal L}_{u,i})).
\end{array}
$$
The spaces $V^-_u$ and $V^-_{u,i}$ are $4$--dimensional. We want to
study the map
$$
\varphi_{V^-_u}:{\bar A}_u--\rightarrow {\mathbb{P}}^3.
$$
The sections ${\widehat g}_i$ vanish at the eight ``$2$-torsion'' points
$P_1,\ldots ,P_8$ on ${\bar A}_u$ and hence
$$
V^-_u\subset H^0(A_{u,i}, {\mathcal O}_{A_{u,i}}(E_1+6f_{P_i}-\sum\limits^8_{j=1}
P_j)).
$$
Again using restriction to $E_1$ and $E_2$ it follows that the vector
space on the right hand side has dimension~$4$ and hence
$$
V^-_u=H^0(A_{u,i},{\mathcal O}_{A_{u,i}}( E_1+6f_{P_i}-\sum\limits^8_{j=1} P_j)).
$$
We shall first consider the {\em product case}, i.e. $e=[\tau_2]=0$. In
this case $A_{u,i}=E(\tau_3)\times {\mathbb{P}}^1$ and there are four rulings which
contain two of the points $P_j$ each. These four rulings are, therefore, in
the base locus of the linear system $|V^-_u|$. Removing this base locus we
obtain the complete linear system of a line bundle on $A_{u,i}$ which has
degree~$2$ on the sections $E_i$ and degree $1$ on the fibres. This maps
$A_{u,i}$ $2:1$ onto a quadric. Since the map ${\bar A}_u--\rightarrow
{\mathbb{P}}^3$ factors through $A_{u,i}$ this shows that the ``Kummer surface''
${\bar A}_u /\iota_{\omega}$ is mapped $2:1$ onto a quadric. It should be
noted that double quadrics arise not only from degenerations of abelian
surfaces, but also from special abelian surfaces, namely products (cf.
Theorem \ref{theo13}). In fact what happens is that the map from the moduli
space ${\mathcal A}_{1,3}(2)$ (or its extension to a toroidal
compactification) contracts each Humbert surface parametrizing product
surfaces to a double point in $N$ corresponding to a quadric.
{}From now on we shall assume $e\neq 0$. Then no fibre of the ruling
contains two of the points $P_i$. We have to recall the notion of an {\em
elementary transformation} of a ruled surface $S$ at a point $P$. This
consists of first blowing up $S$ in $P$ and then blowing down the strict
transform of the fibre through $P$. The result is again a ruled surface
$\mbox{elm}_{P}S$. Let
$$
{\hat A}_u:=\mbox{elm}_{P_1,\ldots,P_8}(A_{u,i}).
$$
Then ${\hat A}_u$ has again two disjoint sections $E_1$ and $E_2$ with
$E_i^2=0$. In particular
$$
{\hat A}_u={\mathbb{P}}({\mathcal O}_{E(\tau_3)}\oplus{\widehat{\mathcal M}}_u) \mbox { for
some }
{\widehat{\mathcal M}}_u\in \mbox{Pic}^{0}(E(\tau_3)).
$$
Note that ${\widehat{\mathcal M}}_u$ and ${{\widehat{\mathcal M}}_u}^{-1}$
define the same ${\mathbb{P}}_1$-bundle. It is straightforward to compute the normal
bundle of the sections $E_1$ and $E_2$ in $A_{u,i}$ (cf. \cite[p. 229]
{HKW}. Since we can control the self-intersection of a section under
blowing up and blowing down and since we know the points $P_j$ it is
straightforward to show that ${{\widehat{\mathcal M}}_u}^{\pm 1}={\mathcal
O}_{E(\tau_3)}(2[\tau_2]-2[0])$, and hence
$$
{\hat A}_u={\mathbb{P}}({\mathcal O}_{E(\tau_3)}\oplus {\mathcal
O}_{E(\tau_3)}(2[\tau_2]-2[0])).
$$
Consider the diagram
$$
\unitlength1pt
\begin{picture}(100,60)(0,0)
\put(45,55){$\tilde A_u$}
\put(16,35){$\pi_1$}
\put(75,35){$\pi_2$}
\put(43,51){\vector(-1,-1){35}}
\put(53,51){\vector(1,-1){35}}
\put(0,5){$A_u$}
\put(90,5){$\hat A_u$}
\end{picture}
$$
where $A_u=A_{u,i}$ and ${\tilde A}_u$ is $A_u$ blown up in $P_1,\ldots,
P_8$. We denote the exceptional divisors over $P_1,\ldots, P_8$ by
$E^1,\ldots, E^8$. The line bundle $\pi^*_1{\mathcal L}_{u,i}\otimes{\mathcal
O}_{{\tilde A}_u}(-E^1-\ldots-E^8)$ has degree $0$ on the strict transforms
of the fibres through the points $P_j$. Hence
$$
{\hat{\mathcal L}}_u:=\pi_{2*}(\pi^*_1{\mathcal L}_{u,i}\otimes {\mathcal O}_{{\tilde
A}_u}(-E^1-\ldots-E^8))\in \mbox{Pic } {\hat A}_u
$$
is a line bundle on ${\hat A}_u$. The degree of ${\hat{\mathcal L}}_u$ is
$1$ on a ruling and $2$ on the sections $E_i$. Hence
$$
{\hat{\mathcal L}}_u={\mathcal O}_{{\hat A}_u}(E_1+2f_Q)
$$
for a suitable point $Q$ on the base curve. (Clearly $Q$ can be computed
explicitly, but this is immaterial for our purposes.) By the usual
arguments
$$
h^{0}({\hat A}_u, {\hat{\mathcal L}}_u)=4
$$
and the rational map from $A_u$ to ${\hat A}_u$ defines an isormorphism
$$
\pi_{2*} \pi^*_1:V^-\cong H^{0}({\hat A}_u, {\hat{\mathcal L}}_u).
$$
\begin{proposition}\label{prop32}
Let $e\neq 0$. The linear system $|V^-|$ on ${\hat A}_u$ has the
following
properties:\\
{\rm(i)} $|V^-|$ is base point free.\\
{\rm(ii)} $|V^-|$ maps the two sections $E_i$ each $2:1$ onto two skew
lines.\\
{\rm(iii)} $|V^-|$ is very ample outside the sections $E_i$. More
precisely, if a cluster $\zeta$ of length~$2$ (i.e. two points or a point
and a tangent direction) is not embedded, then $\zeta$ is contained in
$E_1$ or $E_2$.
\end{proposition}
\begin{proof}
This follows easily from Reider's theorem. We write
$$
|V^-|=|E_1+2f_Q|=|K+L|
$$
where the canonical divisor $K=-E_1-E_2$ and $L=2E_1+E_2+2f_Q$. Then
$L^2=12$. If $|V^-|$ is not base point free, then there exists a curve $D$
with $L.D=0$ or $1$. Clearly such a curve cannot exist. If $|V ^-|$ fails
to embed a cluster $\zeta$ then there exists a curve $D\supset\zeta$ with
$L.D=2$ and $D^2=0$. Then $D$ must be a section with $D.E_i=0$. Since
${\widehat{\mathcal M}}_u\neq{\mathcal O}_{E(\tau_3)}$ (here we use
$e=2[\tau_2]\neq 0)$ it follows that $D=E_1$ or $D=E_2$. Finally note that
the restriction of $|V^-|$ to the elliptic curves $E_i$ gives a complete
linear system of degree~$2$. Hence these curves are mapped $2:1$ to lines.
Since pairs $(x,y)$ with $x\in E_1$ and $y\in E_2$ are separated, there
lines are skew. \hfill
\end{proof}
We can now summarize our results as follows.
\begin{theorem}\label{theo33}
Let ${\bar A}_u$ be a corank~$1$ degenerate abelian surface over a point
$u=(0,T_2,T_3)\in B$ with $T_2T_3\neq0$ and consider the Kummer map given
by the linear system $|V^-|$:
$$
\phi_{|V^-|}:{\bar A}_u--\rightarrow{\mathbb{P}}^3.
$$
{\rm(i)} If $e=0$ then ${\bar A}_u$ is mapped $4:1$ onto a smooth quadric.\\
{\rm(ii)} If $e\neq 0$ then there is a commutative diagram.
$$
\unitlength1pt
\begin{picture}(100,60)(0,0)
\put(0,46){$\bar A_u$}
\put(13,50){\line(1,0){15}}
\put(43,50){\line(1,0){15}}
\put(73,50){\vector(1,0){20}}
\put(100,46){${\mathbb{P}}^3$}
\put(13,45){\line(1,-1){10}}
\put(27,31){\line(1,-1){10}}
\put(12,20){$2:1$}
\put(41,17){\vector(1,-1){12}}
\put(54,-3){$\hat A_u$}
\put(67,8){\vector(1,1){30}}
\end{picture}
$$
The image of ${\bar A}_u$ is an elliptic ruled surface which has double
points along two skew lines but no other singularities.
\end{theorem}
\begin{proof}
By construction the Kummer map ${\bar A}_u--\rightarrow {\mathbb{P}}^3$ factors
through ${\bar A}_u/{\iota_{\omega}}=A_{u,i}$. All other statements
follow from our above discussion of the map $\phi_{|V^-|}: {\hat A}_u \to
{\mathbb{P}}^3$ given by the linear system $|V^-|$. \hfill
\end{proof}
This theorem explains the irreducible singular quartic surfaces which are
parametrized by the S-planes, appearing already in \cite[Section 5-2]{Ni}.
We want to conclude this paper with some remarks. We have already seen that
the degenerate surfaces with $e=0$ correspond to products. The limits of
{\em bielliptic} abelian surfaces are characterised by $2e=0, e\neq 0$.
Geometrically this means that the surfaces ${\bar A}_u$ contain degenerate
elliptic curves which are $4$-gons, i.e. cycles consisting of $4$ rulings.
The degenerate abelian surfaces ${\bar A}_u$ where $u$ is on another
boundary component or where more than one of the $T_i$ vanishes can also be
described. If $T_3=0$ and $T_1 T_2\neq 0$ then ${\bar A}_u$ is a chain
of~$6$ elliptic ruled surfaces. If two of the $T_i$ are zero, then ${\bar
A}_u$ consists of $12$ quadrics, whereas ${\bar A}_0$ has $36$ components,
of which $24$ are ${\mathbb{P}}^2$ and $12$ are ${\mathbb{P}}^2$ blown up in three points.
Limits of polarizations of type $(2,6)$ exist on these surfaces, but as we
pointed out before, there is no possibility of defining the Kummer map
globally over $B$. On the other hand it is easy to construct degenerations
of the ruled surfaces $A_{u,i}$ which lead to a union of two quadrics
intersecting along a quadrangle or to a tetrahedron.
Finally we want to comment on the boundary components of the Igusa
compactification of the moduli space ${\mathcal A}_{1,3}(2)$. These are
enumerated by the {\em Tits building} of the group $\Gamma_{1,3}(2)$, i.e.
by the equivalence classes modulo $\Gamma_{1,3}(2)$ of the lines and
isotropic planes in ${\mathbb{Q}}^4$. The Tits building was calculated by Friedland
in \cite{F}: for details, and for some other cases, see \cite{FS}.
There are $30$ equivalence classes of lines. These correspond to the $15$
equivalence classes of short, respectively long vectors. Each set of $15$
lines is naturally parametrized by $({\mathbb{Z}}/2)^4 \setminus
\{0\}={\mathbb{P}}^3({\mathbb{F}}_2)$. The $15$ planes are parametrized by the $15$ isotropic
planes in $\operatorname{Gr}(1,{\mathbb{P}}^3({\mathbb{F}}_2))$. The isotropic planes are a
hyperplane section of $\operatorname{Gr}(1,{\mathbb{P}}^3({\mathbb{F}}_2))$ embedded as a
quadric via the Pl\"ucker embedding. The $15$ short and the $15$ long
vectors as well as the $15$ planes are identified under the group
$\Gamma_{1,3}/\Gamma_{1,3}(2)\cong \operatorname{Sp}(4,{\mathbb{F}}_2)\cong S_6$.
That is, there are two equivalence classes of lines modulo $\Gamma_{1,3}$
and one plane (see also \cite [Theorem(I.3.40)]{HKW}). Finally the
involution $V_3$ of the maximal arithmetic subgroup $\Gamma^*_{1,3}$
identifies short and long vectors (see \cite[Folgerung 3.7]{G} and
\cite[Section 2]{HNS}). In our computations above the boundary component
given by $T_1=0$ corresponds to a short vector, whereas the boundary
component given by $T_3=0$ corresponds to a long vector. We described the
degenerate abelian surfaces associated to points on a boundary component
correspronding to a short vector. The matrix $V_3$ (and similarly any
involution $gV_3$ where $g$ is an element of $\Gamma_{1,3}$ -- cf.
\cite[Theorem 2.4]{HNS}) interchanges boundary components associated to
short vectors with boundary components associated to long vectors. It
should, however, be pointed out that the induced action of $V_3$ on the
Igusa compactification ${\mathcal A}_{1,3}^*$ is only a rational map, not a
morphism. This follows since the boundary components associated to long,
and short vectors are not isomorphic: although their open parts (i.e.
away from the corank-$2$ boundary components) are isomorphic (namely to the
open Kummer modular surface $K^{0}(1)$), they contain different
configurations of rational curves in the corank-$2$ boundary components.
This follows from \cite[Satz III.5.19]{B} and \cite[Theorem 4.13]{W}. The
degenerate abelian surfaces belonging to points on $T_3=0$ are different
from those associated to points on $T_1=0$: they are a cycle of six
elliptic ruled surfaces rather than two. At first this looks like a
contradiction to \cite[Theorem 2.4]{HNS}, but this is not the case. The
polarization on each of the six components of the surface $A_P$ where $P\in
\{T_3=0\}$ is of the form ${\mathcal O}(E_1+2f_P)$. On four of the six
components the linear system $|V^-|$ has a base locus consisting of a
section. These four components are contracted. The other two components are
identified under $|V^-|$ and are mapped to a quartic which is an elliptic
ruled surface singular along two skew lines. In this way we find the same
images in ${\mathbb{P}}^3$ as in the case $T_1=0$.
\bibliographystyle{amsalpha}
|
1997-12-03T00:51:33 | 9712 | alg-geom/9712005 | en | https://arxiv.org/abs/alg-geom/9712005 | [
"alg-geom",
"math.AG"
] | alg-geom/9712005 | Janos Kollar | J\'anos Koll\'ar | Real Algebraic Threefolds II: Minimal Model Program | LATEX2e, 61 pages | null | null | null | null | This is the second of a series of papers studying real algebraic threefolds
using the minimal model program. The main result is the following. Let $X$ be a
smooth projective real algebraic 3-fold. Assume that the set of real points is
an orientable 3-manifold (this assumption can be weakened considerably). Then
there is a fairly simple description on how the topology of real points changes
under the minimal model program. The first application is to study the topology
of real projective varieties which are birational to projective 3-space (Nash
conjecture). The second application is a factorization theorem for birational
maps.
| [
{
"version": "v1",
"created": "Tue, 2 Dec 1997 23:51:33 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Kollár",
"János",
""
]
] | alg-geom | \section{Introduction}
In real algebraic geometry, one of the main directions of
investigation is the topological study of the set of real solutions
of algebraic equations. The first general result was proved in
\cite{Nash52}, and later developed by many others (see
\cite{AK92} for some recent directions). One of these theorems says
that every compact differentiable manifold can be realized as the set
of real points of an algebraic variety.
\cite{Nash52} posed the problem of obtaining similar results using a
restricted class of varieties, for instance rational varieties. For
real algebraic surfaces this question was settled in
\cite{Comessatti14}.
The aim of this series of papers is to utilize the theory of minimal
models to investigate this question for real algebraic threefolds.
This approach is very similar in spirit to the one employed by
\cite{Comessatti14}. (See \cite{Silhol89, ras} for introductions to
real algebraic surfaces from the point of view of the minimal model
program.)
For algebraic threefolds over $\c$, the minimal model program (MMP for
short) provides a very powerful tool. The method of the program is
the following. (See \cite{koll87, CKM88} or \cite{KM98} for
introductions.)
Starting with a smooth projective 3-fold $X$, we perform a series of
``elementary" birational transformations
$$
X=X_0\map X_1\map \cdots \map X_n=:X^*
$$
until we reach a variety $X^*$ whose global structure is ``simple".
(Neither the intermediate steps $X_i$ nor the final
$X^*$ are uniquely determined by $X$.) In essence the minimal model
program allows us to investigate many questions in two steps: first
study the effect of the ``elementary" transformations and then
consider the ``simple" global situation.
In practice both of these steps are frequently rather difficult. For
instance, we still do not have a complete list of all possible
``elementary" steps, despite repeated attempts to obtain it.
A somewhat unpleasant feature of the theory is that the varieties
$X_i$ are not smooth, but have so called terminal singularities. This
means that $X_i(\r)$ is not necessarily a manifold. In developing
the theory of minimal models for real algebraic threefolds, we again
have to understand the occurring terminal singularities. This was done
in the first paper of this series \cite{rat1}.
If $X$ is defined over a field $K$, then there is a variant of the MMP
where the intermediate varieties $X_i$ are also defined over
$K$. I refer to this as the MMP over $K$. This suggests the following
two step approach to understand the topology of
$X(\r)$:
\begin{enumerate}
\item Study the topological effect of the ``elementary"
transformations.
\item Investigate the topology of $X^*(\r)$.
\end{enumerate}
\noindent The aim of this paper is to complete the first of these two
steps.
I am unable to say much about this question in general. There are
serious problems coming from algebraic geometry and also from
3-manifold topology. Some of these are discussed in section 4. My aim
is therefore more limited: find reasonable conditions which ensure
that the steps of the MMP can be described topologically.
The simplest case to study is contractions
$f:X\to Y$ where $X$ is smooth. Over $\c$ the complete list of such
contractions is known \cite{Mori82}, and it is not hard to obtain a
complete list over $\r$. From this list one can see that in all such
examples where $X(\r)\to Y(\r)$ is complicated,
$X(\r)$ contains a special surface of nonnegative Euler
characteristic. This turns out to be a general pattern, though the
proof presented here relies on a laborious case analysis. The
precise technical theorem is stated in (\ref{int.nonorient.thm}).
None of the complicated examples occur if $X(\r)$ is orientable, and
this yields the following:
\begin{thm}\label{int.orient.thm}
Let $X$ be a smooth, projective, real algebraic $3$-fold and $X^*$
the result of the MMP over $\r$. Assume that $X(\r)$ is orientable.
Then the topological normalization $\overline{X^*(\r)}$ of $X^*(\r)$
is a PL-manifold, and
$X(\r)$ can be obtained from
$\overline{X^*(\r)}$ by repeated application of the following
operations:
\begin{enumerate}
\setcounter{enumi}{-1}
\item throwing away all isolated points of $\overline{X^*(\r)}$,
\item taking connected sums of connected components,
\item taking connected sum with $S^1\times S^2$,
\item taking connected sum with $\r\p^3$.
\end{enumerate}
\end{thm}
\begin{rem} $X^*$ uniquely determines (\ref{int.orient.thm}.0) and
also (\ref{int.orient.thm}.1). The latter can be seen by analyzing
real analytic morphisms $h:[0,1]\to X^*(\r)$ where the endpoints map
to different connected components of
$\overline{X^*(\r)}$. In practice this may be quite hard, and it could
be easier to work through the MMP backwards.
$X^*$ contains some information about the steps
(\ref{int.orient.thm}.2--3), but these are by no means unique. Even if
$X^*$ is smooth, both of these steps are possible, as shown by the
next example.
\end{rem}
\begin{exmp}\label{connsum.exmp}
It is well known how to create connected sum with $\r\p^3$
algebraically. Let $X$ be a smooth 3-fold over $\r$ and $0\in X(\r)$ a
real point. Set $Y=B_0X$. Then
$Y(\r)\sim X(\r)\ \#\ \r\p^3$. (The connected sum of two
nonoriented manifolds is, in general, not unique. It is, however,
unique if one of the summands has an automorphism with an isolated
fixed point which reverses local orientation there.)
Connected sum with $S^1\times S^2$ is somewhat harder. Let $X$ be a
smooth 3-fold over $\r$ and
$D\subset X$ a real curve which has a unique real point
$\{0\}= D(\r)$. Assume furthermore that near $0$ the curve is given by
equations
$(z=x^2+y^2=0)$. Set
$Y_1=B_DX$. $Y_1$ has a unique singular point $P$; set $Y=B_PY_1$. It
is not hard to see that $Y$ is smooth and
$Y(\r)\sim X(\r)\ \#\ (S^1\times S^2)$.
\end{exmp}
\begin{rem}\label{1.basic.top.facts}
As (\ref{int.orient.thm}) already shows, we have to move between
topological, PL and differentiable manifolds. In dimension 3 every
compact topological 3--manifold carries a unique PL--manifold
structure (cf.\ \cite[Sec.\ 36]{Moise77}) and also a unique
differentiable structure (cf.\ \cite[p.3]{Hempel76}). I mostly use the
PL--structure since most algebraic constructions are natural in the
PL--category. For instance, $\r^1\to \r^2$ given by $t\mapsto
(t^2,t^3)$ is a PL--embedding but not a differentiable embedding in the
natural differentiable structures.
In dimension 3 the PL--structure behaves very much like a
differentiable structure. For instance, let $M^3$ be a PL 3--manifold,
$N$ a compact PL--manifold of dimension 1 or 2 and $g:N\DOTSB\lhook\joinrel\rightarrow M$ a
PL--embedding. Then a suitable open neighborhood of $g(N)$ is
PL--homeomorphic to a real vector bundle over $N$ (cf.\ \cite[Secs.\
24 and 26]{Moise77}). (Note that a similar result fails for
topological 3--manifolds (cf.\ \cite[Sec.\ 18]{Moise77}), and it also
fails for PL 4--manifolds: take any nontrivial knot in $S^3$ and
suspend it in $S^4$.)
\end{rem}
\begin{say}[Surfaces in 3--manifolds]\label{surf.in.3-man}
Let $M$ be a PL 3--manifold without boundary,
$N$ a compact PL 2--manifold without boundary and $g:N\DOTSB\lhook\joinrel\rightarrow M$ a
PL--embedding. As we noted above, a neighborhood of $N$ is an
$\r$-bundle over $N$. $\r$-bundles over $N$ are classified by group
homomorphisms $\rho:\pi_1(N)\to \{\pm 1\}$. If $\rho $ is trivial then
$N$ is 2--sided in $M$, otherwise it is 1--sided. We also allow self
homeomorphisms of $N$, thus we get the following possibilities when
$N$ has nonnegative Euler characteristic:
\begin{description}
\item[$S^2$] Always 2--sided, many such surfaces in every $M^3$.
\item[$\r\p^2$] $M^3$ is not orientable in the 2--sided case. Such
manifolds are called $\p^2$-reducible (cf.\
\cite[p.88]{Hempel76}). In the 1--sided case the boundary of a regular
neighborhood is $S^2$, thus $M\sim M'\ \#\ \r\p^3$ for some 3--manifold
$M'$. Most 3--manifolds do not contain any $\r\p^2$.
\item[{\rm Torus}] The 2--sided case occurs in any 3--manifold as the
boundary of a regular neighborhood of any $S^1$ along which $M$ is
orientable. There is a unique 1--sided case. For these $M$ is not
orientable. Most nonorientable 3--manifolds do not contain 1--sided
tori, see section 12.
\item[{\rm Klein bottle}] $M$ is nonorientable in the 2--sided case.
The boundary of a regular neighborhood of any $S^1$ along which $M$
is nonorientable is such. There are two different 1--sided cases,
depending on whether $M$ is orientable near $N$ or not. These are
again rare, see section 12.
\end{description}
\noindent This shows that there are many 3--manifolds which do not
contain
$\r\p^2$, 1--sided tori or Klein bottles. These correspond to 6
different cases on the above list. It turns out that we need to exclude
only 3 of these for our main theorem.
\end{say}
\begin{condition}\label{int.no.cond}
Let $M$ be a PL 3-manifold without boundary. Consider the following
properties:
\begin{enumerate}
\item $M$ does not contain a 2-sided $\r\p^2$,
\item $M$ does not contain a 1-sided torus,
\item $M$ does not contain a 1-sided Klein bottle with nonorientable
neighborhood.
\end{enumerate} Failure of any of these properties implies that $M$
is not orientable, but there are many nonorientable 3-manifolds which
do satisfy all 3 of the above conditions. For instance, this holds
if
$M$ is hyperbolic (\ref{hyp.doesnotcont.thm}).
\end{condition}
\begin{thm}\label{int.nonorient.thm}
Let $X$ be a smooth, projective, real algebraic $3$-fold and $X^*$
the result of the MMP over $\r$. Assume that $X(\r)$ satisfies
the 3 conditions (\ref{int.no.cond}.1--3).
Then the conclusions of (\ref{int.orient.thm}) hold.
\end{thm}
\begin{rem} It would seem that we also need to allow connected sum
with
$S^1\tilde{\times} S^2$ (cf.\ (\ref{5.notation})), corresponding to
attaching a nonorientable 1--handle. This, however, would give a
1--sided torus which we excluded.
All 3 conditions (\ref{int.no.cond}.1--3) are necessary for the
theorem to hold. My feeling is that essentially nothing can be said
without
(\ref{int.no.cond}.1) or (\ref{int.no.cond}.3).
(\ref{int.no.cond}.2) has a twofold role in the proof. First, it
ensures that $X$ is not obtained as a blow up of a smooth 3-fold $Y$
along a curve. This in itself would not be a problem, but it may
happen that $Y(\r)$ contains a 2-sided
$\r\p^2$ but $X(\r)$ does not. It seems to me that this leads to
rather complicated topological questions. Still, a suitable
reformulation of the theorem may get around this problem. Second,
(\ref{int.no.cond}.2) is also used to exclude a few singularities on
the $X_i$. These cases are of index 1 and they can be described very
explicitly. It should be possible to work with them.
\end{rem}
The technical heart of the proof is a listing of the possible
singularities that occur in the course of the MMP and a fairly detailed
description of the steps of the MMP. The final result is relatively
easy to state but the proof is a case-by-case examination.
\begin{thm}\label{int.mmp.sings}
Let $X$ be a smooth, projective, real algebraic $3$-fold and assume
that $X(\r)$ satisfies the 3 conditions (\ref{int.no.cond}.1--3).
Let $X_i$ be any of the intermediate steps of the MMP over $\r$
starting with $X$ and $0\in X_i(\r)$ a real point. Then a neighborhood
of $0\in X_i$ is real analytically equivalent to one of the following
standard forms:
\begin{enumerate}
\item ($cA_0$) Smooth point.
\item ($cA_{>0}^+$)
$(x^2+y^2+g_{\geq 2}(z,t)=0)$, where $g$ is not everywhere negative
in a punctured neighborhood of $0$.
\item ($cE_6$)
$(x^2+y^3+(z^2+t^2)^2+ yg_{\geq 4}(z,t)+g_{\geq 6}(z,t)=0)$.
\end{enumerate}
\end{thm}
\begin{rem} The symbol $g_{\geq m}$ denotes a power series of
multiplicity at least $m$.
The name of the cases is explained in \cite{rat1}.
The above points of type $cE_6$
form a codimension 7 family in the space of all $cE_6$ singularities.
They all occur, even if $X(\r)$ is orientable. Points of type
$cA_{>0}^+$ occur for many choices of $g$. Section 10 gives an
algorithm to decide which cases of $g$ do occur, but I was unable to
write the condition in closed form. For the applications this does not
seem to matter.
\end{rem}
Using \cite[4.3, 4.4, 4.9]{rat1}, this immediately implies:
\begin{cor}\label{int.mmp.sings.top} Notation and assumptions as in
(\ref{int.mmp.sings}). Then
$\overline{X_i(\r)}\setminus\{\mbox{isolated points}\}$ is a compact
PL 3-manifold without boundary.\qed
\end{cor}
The next step is to understand the ``elementary" steps of the MMP
over $\r$. (\ref{int.nonorient.thm}) turns out to be a consequence of
(\ref{int.mmp.steps}). (See (\ref{8.wbup}) for the definition of
weighted blow-ups.)
\begin{thm}\label{int.mmp.steps}
Let $X$ be a smooth, projective, real algebraic $3$-fold such that
$X(\r)$ satisfies the conditions (\ref{int.no.cond}.1--3).
Let $f_i:X_i\map X_{i+1}$ be any of the intermediate steps of the MMP
over
$\r$ starting with $X$. Then the induced map $f_i:X_i(\r)\to
X_{i+1}(\r)$ is everywhere defined and the following is a complete
list of possibilities for $f_i$:
\begin{enumerate}
\item ($\r$-trivial) $f_i$ is an isomorphism in a (Zariski)
neighborhood of the set of real points.
\item ($\r$-small) $f_i:X_i(\r)\to X_{i+1}(\r)$ collapses a 1-complex
to points and there are small perturbations $\tilde f_i$ of
$f_i$ such $\tilde f_i: \overline{X_i(\r)}\to \overline{X_{i+1}(\r)}$
is a PL-homeomorphism.
\item (smooth point blow up) $f_i$ is the inverse of the blow up of a
smooth point $P\in X_{i+1}(\r)$.
\item (singular point blow up) $f_i$ is the inverse of a (weighted)
blow up of a singular point $P\in X_{i+1}(\r)$. There are two cases:
\begin{enumerate}
\item ($cA_{>0}^+$, $\mult_0g$ even) Up to real analytic equivalence
near $P$,
$X_{i+1}\cong (x^2+y^2+g_{\geq 2m}(z,t)=0)$ where $g_{2m}(z,t)\neq 0$,
$m\geq 1$ and $X_i$ is the weighted blow up
$B_{(m,m,1,1)}X_{i+1}$.
\item ($cA_{>0}^+$, $\mult_0g$ odd)
Up to real analytic equivalence near $P$,
$X_{i+1}\cong (x^2+y^2+g_{\geq 2m+1}(z,t)=0)$ where $m\geq 1$,
$z^{2m+1}\in g$ and
$z^it^j\not\in g$ for $2i+j< 4m+2$. $X_i$ is the weighted blow up
$B_{(2m+1,2m+1,2,1)}X_{i+1}$.
\end{enumerate}
\end{enumerate}
\end{thm}
\begin{rem} The more precise results in sections 9--11 give a
description of the various cases when $f_i$ is $\r$-small (though so
far I have not excluded some cases).
The $\r$-trivial steps do not change anything in a neighborhood of the
real points, but it is in these steps that the full complexity of the
MMP appears. All the difficulties involving higher index terminal
singularities and flips are present, but they always appear in
conjugate pairs.
For the topological questions these have no effect, but in other
applications of (\ref{int.mmp.steps}) this should be taken into
account.
\end{rem}
\begin{rem} The lists in (\ref{int.mmp.sings}) and
(\ref{int.mmp.steps}) are fairly short, but I do not see a simple
conceptual way of stating the results, let alone proving them by
general arguments. The appearence of the singularities of type
$cE_6$ in (\ref{int.mmp.sings}) was rather unexpected for me.
The formulations also hide the cicumstance that there does not seem to
be a single method of excluding all other a priori possible cases. The
algebraic method of the proof of (\ref{int.mmp.sings}) ends with a
much longer list (\ref{ge.gwextr.thm}). The topological method excludes
many of these right away, but in a few cases several steps of the MMP
need to be analyzed.
\end{rem}
\begin{say}[Method of the proof of (\ref{int.mmp.steps})]{\ }
The proof relies on rather extensive computations. The first step is
a classification of all 3--dimensional terminal singularities over
$\r$ and the study of their topological properties. This was carried
out in
\cite{rat1}. The next step is to gain a good understanding of the
resolutions of these singularities. More precisely, we need to
understand the ``simplest" exceptional divisors in these resolutions.
(Simplicity is measured by the discrepancy, cf.\
(\ref{mmp.discr.def}).) Over $\c$ the first step in this direction is
\cite{Markushevich96}. A much more detailed study of such exceptional
divisors was completed by \cite{Hayakawa97}. Our main emphasis is over
$\r$, and it turns out that there is very little overlap between the
computations of \cite{Hayakawa97} and those in sections 9--11.
Nonetheless, the basic underlying principles are exactly the same.
\end{say}
\begin{ack} I thank M. Bestvina, S. Gersten, M. Kapovich and G.
Mikhalkin for answering my numerous questions about 3-manifold
topology and real algebraic geometry. The existence of
$cE_6$ type points in (\ref{int.mmp.sings}) was established with the
help of V. Alexeev. I have received helpful comments and questions
from A. Bertram, M. Fried, L. Katzarkov and B. Mazur.
Partial financial support was provided by the NSF under grant number
DMS-9622394.
\end{ack}
\section{Applications and Speculations}
\subsection{Factorization of Birational Morphisms}
Let $f:Y\to X$ be a birational morphism between smooth and projective
varieties. It is a very old problem to factor $f$ as a composite of
``elementary" birational morphisms. In dimension 2 this is easy to do:
$f$ is the composite of blow ups of points. In dimension 3 and over
$\c$, the MMP factors $f$ as a composition of divisorial contractions
and flips, but these intermediate steps are rather complicated and
not too well understood.
If $f:Y\to X$ is a birational morphism between smooth and projective
threefolds over $\r$, then one would like to get a factorization where
the intermediate steps are also defined over $\r$. It turns out that
if $Y(\r)$ is orientable, the answer is very simple. As with minimal
models in general, the intermediate steps involve singular varieties
though in this case the real singularites are very mild.
\begin{defn} A real 3--fold $X$ is said to have a $cA_1$ singularity at
$0\in X(\r)$ if in suitable real analytic cordinates $X$ can be given
by an equation $(\pm x^2\pm y^2\pm z^2\pm t^m=0)$ for a suitable
choice of signs and $m\geq 1$.
\end{defn}
\begin{thm}\label{bir.morph.factor} Let $f:Y\to X$ be a birational
morphism between smooth and projective threefolds over $\r$. Assume
that
$Y(\r)$ satisfies the conditions (\ref{int.no.cond}). Then $f$ can be
factored as
$$
f: Y=X_n\stackrel{f_n}{\to} X_{n-1}\to \cdots\to X_1
\stackrel{f_1}{\to} X_0=X,
$$ where each $X_i$ has only $cA_1$ singularities at real points and
the following is a complete list of possibilities for the $f_i$:
\begin{enumerate}
\item (smooth point blow up) $f_i$ is the blow up of a smooth point
$P\in X_{i-1}(\r)$.
\item (singular point blow up) $f_i$ is the blow up of a singular
point $P\in X_{i-1}(\r)$.
\item (curve blow up) $f_i$ is the blow up of a real curve $C\subset
X_{i-1}$. $C$ has only finitely many real points, $X_{i-1}$ is smooth
at each of these and in suitable real analytic coordinates $C$ can be
written as $(z=x^2+y^{2m}=0)$.
\item ($\r$-trivial) $f_i$ is an isomorphism in a (Zariski)
neighborhood of the set of real points.
\end{enumerate}
\end{thm}
\begin{rem} As in (\ref{int.mmp.steps}), it is in the $\r$-trivial
steps that the full complexity of the MMP appears. In particular,
the $\r$-trivial steps may be flips where the flipping curve has no
real points.
\end{rem}
Proof. For purposes of induction we consider the more general case
when $X$ is allowed to have $cA_1$-type singularities at real points
and terminal singularities at complex points. We assume that $X$ is
$\q$-factorial (that is, a suitable multiple of every Weil divisor is
Cartier). Run the real MMP for $Y$ over $X$ to obtain
$$
f: Y=X_n\stackrel{f_n}{\map} X_{n-1}\map \cdots\map X_1
\stackrel{f_1}{\to} X_0=X.
$$ The proof is by induction on the number of steps it takes the MMP
to reach $X$.
The last step, $f_1:X_1\to X_0=X$, is a contraction since we work over
$X$.
The possibilities for $f_1$ are described in (\ref{int.mmp.steps}).
We are done by induction if $f_1$ is
$\r$-trivial or a smooth point blow up. Assume that $f_1$ is a
singular point blow up. Since $X_0$ has only $cA_1$ points, we are in
case (\ref{int.mmp.steps}.4a) with $m=1$. $f_1$ is the ordinary blow up
and by explicit computation we see that $X_1$ still has only
$cA_1$ singularities.
The case when $f_1$ is $\r$-small (\ref{int.mmp.steps}.2) needs to be
studied in greater detail. $f_1$ can not be a g--extraction
(\ref{gw.g-e.def}) since $cA_1$ type points do not have g--extractions
other than the one listed above by (\ref{ge.gwextr.thm}). Thus $f_1$
is the blow up of a curve $C\subset X_0$. Moreover,
$X_0$ is smooth along $C(\r)$ and
$C$ is locally planar along $C(\r)$ by (\ref{sm.thm}).
$C(\r)$ is finite since $f_1$ is $\r$-small.
Pick any point $P\in C(\r)$ and assume that $C$ is given by real
analytic equations $(z=g(x,y)=0)$. By explicit computation, $B_CX_0$
has a unique singular point with equation
$(st-g(x,y)=0)$ which is equivalent to $(u^2-v^2-g(x,y)=0)$.
$X_1$ is an intermediate step of an MMP starting with $Y$, hence its
singularities are among those listed (\ref{int.mmp.sings}). Thus $g$
has multiplicity 2 and so it can be written as
$\pm x^2\pm y^r$. Since $(g=0)$ has only the origin as its real
solution, $g=\pm(x^2+y^{2m})$. \qed
\subsection{Application to the Nash Conjecture}
The main conclusion of (\ref{int.orient.thm}) and
(\ref{int.nonorient.thm}) is that if we want to understand the topology
of $X(\r)$ (say when it is orientable), it is sufficient to study the
topology of $X^*(\r)$ instead. $X^*$ has various useful properties,
depending on the conditions imposed on
$X$.
Consider, for instance, the original Nash question: what happens if $X$
is rational. Since the fifties it has been understood that being
rational is a very subtle condition and it is very hard to work with.
\cite{KoMiMo92} introduced the much more general notion of being {\it
rationally connected}. A $X$ is rationally connected if two general
points of $X(\c)$ can be connected by an irreducible rational curve.
The lines show that $\p^n$ is rationally connected.
The structure theory of \cite{KoMiMo92} implies that a 3-fold $X$ is
rationally connected iff
$X^*$ falls in one of 3 classes:
\begin{enumerate}
\item (Conic fibrations) There is a morphism (over $\r$) $g:X^*\to S$
onto a surface such that the general fiber is a conic. Correspondingly
there is a morphism $X^*(\r)\to S(\r)$ whose general fiber is $S^1$ or
empty. These cases will be studied in a subsequent paper.
\item (Del Pezzo fibrations) There is a morphism (over $\r$) $g:X^*\to
C$ onto a curve such that the general fiber is a Del Pezzo surface. If
$X(\r)$ is orientable, then this induces
a morphism $X^*(\r)\to C(\r)$ whose general fiber is a torus or a
union of some copies of $S^2$. These cases will be studied later.
\item (Fano varieties) The anticanonical bundle of $X^*$ is ample.
There is a complete list of such varieties if $X^*$ is also smooth
\cite{Iskovskikh80}. Even if $X^*$ is known rather explicitly, a
topological description of
$X^*(\r)$ may not be easy. It would be interesting to work out at least
some of the cases, for instance hypersurfaces of degree 3 or 4 in
$\p^4$. (Mikhalkin pointed out that the degree 3 cases can be
understood using the classification of degree 4 real surfaces in
$\r\p^3$
\cite{Kharlamov76}.)
In general it is known that there are only finitely many families of
singular Fano varieties in dimension 3
\cite{Kawamata92}. Thus we can get only finitely many different
topological types for
$X^*(\r)$ in this case.
\end{enumerate}
\subsection{Homology Spheres}
It is interesting to consider if we can get further simplifications of
the real MMP if we pose further restrictions on $X(\r)$. We may assume,
for instance, that $X(\r)$ is a homology sphere. This was in fact the
assumption I considered first. One can ask if under this assumption
$X(\r)\to X^*(\r)$ is a homeomorphism.
Unfortunately this is not the case. Consider for instance the singular
real threefold $X^*$ given by affine equation
$$ x^2+y^2+z^2+(t-a_0)(t-a_m)\prod_{i=1}^{m-1}(t-a_i)^{2r}=0,
$$ where $a_0<a_1<\cdots<a_m$ are reals. This has $m-1$ singular points
of the form $x^2+y^2+z^2-u^{2r}=0$, which can be resolved by
$r$ successive blow ups. Resolving all singular points we obtain the
3--fold $X$. One can easily see that $X(\r)\sim S^3$, but
$\overline{X^*(\r)}$ is the disjoint union of $m$ copies of $S^3$.
One may also study the types of singularities that occur if we pose
stronger restrictions on $X(\r)$. It seems to me that the best one
can get is the following:
\begin{conj}\label{no.E6.conj}
Let $X$ be a smooth projective 3--fold over $\r$. Assume that $X(\r)$
satisfies the conditions (\ref{int.no.cond}) and $X(\r)$ can not be
written as a connected sum
with $S^1\times S^2$.
Let $X_i$ be any of the intermediate steps of the MMP over $\r$
starting with $X$ and $0\in X_i(\r)$ a real point. Then a neighborhood
of $0\in X_i$ is real analytically equivalent to one of the following
standard forms:
\begin{enumerate}
\item ($cA_0$) Smooth point.
\item ($cA_{>0}^+$)
$(x^2+y^2+g_{\geq 2}(z,t)=0)$, where $\mult_0g$ is even and $g$ is
not everywhere negative in a punctured neighborhood of $0$.
\end{enumerate}
\end{conj}
In fact, most $cA_{>0}^+$-type singularities should not occur.
It is possible that one can write down a complete list. Also, one can
be more precise about how the singular points separate $X_i(\r)$.
The results in sections 8--11 come close to proving (\ref{no.E6.conj}),
but two points remain unresolved. In order to exclude $cE_6$ type
points, one needs to show that the only possible g--extraction is the
one described in (\ref{cE6.g--extr.exist}). This should be a feasible
computation. The main problem is that in (\ref{cA+.multg-odd}) I could
not exclude certain $\r$-small contractions. I do not see how to deal
with this case.
\subsection{Beyond the Nash Conjecture}
One can refine the 3--dimensional Nash conjecture in two ways.
\medskip
First, one can study the topology of $X(\r)$ for other classes of real
algebraic varieties. The simplest cases may be those whose
minimal models admit a natural fibration. This should be very helpful
in their topological study. One such class is elliptic threefolds,
where we have a morphism $X^*\to S$ whose general fiber is an elliptic
curve. A study of the singular fibers occurring in codimension 1 was
completed by \cite{Silhol84}.
Another, probably more difficult class are Calabi--Yau 3--folds. It
would be very interesting to find some connection between the topology
of $X(\r)$ and mirror symmetry.
The following question is consistent with the examples that I know:
\begin{question} Let $X$ be a smooth projective real 3--fold. Asume
that $X(\r)$ is hyperbolic. Does this imply that $X$ is of general
type?
\end{question}
\medskip
One can also start with a 3--manifold $M$ and look for a ``simple" real
projective 3--fold $X$ such that $X(\r)\sim M$. Ideally one would like
to find a solution where certain topological structures on $M$ are
reflected by the algebraic properties of $X$.
There are hyperbolic 3--manifolds which embedd into
$\r^4$. Ths implies that they can be realized by real algebraic
hypersurfaces in $\r^4$. It would be interesting to find such examples.
\medskip
The methods of this paper require a very detailed study of the steps of
the MMP, which is currently feasible only in dimension 3. It would
be, however, interesting to develop some examples in higher dimensions.
Example (\ref{connsum.exmp}) describing connected sum with $S^1\times
S^2$ should have interesting higher dimensional versions. There
may be other, more complicated examples as well.
The first steps of the 4--dimensional MMP over $\c$ have been recently
classified by \cite{andwis}. It should be possible to obtain the
complete list over $\r$ and to study their topology.
\section{The Minimal Model Program over $\r$}
This section is intended to provide a summary of the MMP over $\r$.
More generally, I discuss the MMP over an arbitrary field $K$ of
characteristic zero, since there is no difference in the general
features. Conjecturally the whole program works in all dimensions but
at the moment it is only established in dimensions $\leq 3$.
\cite{koll87, koll90} provide general introductions. The minimal model
program for real algebraic surfaces is explained in detail in
\cite{ras}.
For more comprehensive treatments (mostly over $\c$) see \cite{CKM88,
kolletal92, KoMo98}.
One of the special features of the 3-dimensional MMP is that we have
to work with certain singular varieties in the course of the program.
\begin{defn}\label{mmp.qf.def}
Let $X$ be a normal variety defined over a field $K$. A {\it (Weil)
divisor} over $K$ is a formal linear combination
$D:=\sum a_iD_i$ ($a_i\in \z$) of codimension 1 subvarieties, each
defined and irreducible over $K$. A {\it $\q$-divisor} is defined
similarly, except we allow $a_i\in \q$. A divisor $D$ is called {\it
Cartier} if it is locally definable by one equation and {\it
$\q$-Cartier} if $mD$ is Cartier for some $m\in\n$. The smallest such
$m>0$ is called the {\it index} of $D$.
We say that $X$ is {\it factorial} (resp. {\it $\q$-factorial}) if
every Weil divisor is
Cartier (resp. $\q$-Cartier).
A divisor $D$ defined over $K$ is Cartier (resp. $\q$-Cartier) iff
it is Cartier (resp. $\q$-Cartier) after some field extension.
However, a variety may be $\q$-factorial over $K$ and not
$\q$-factorial over $\bar K$. For instance, the cone
$x^2+y^2+z^2-t^2$ is factorial over $\r$ but not over $\c$. (For
instance, $(x-\sqrt{-1}y=z-t=0)$ is a not $\q$-Cartier.)
\end{defn}
\begin{defn}\label{mmp.KX.def}
For a normal variety $X$, let
$K_X$ denote its {\it canonical class}. $K_X$ is a linear equivalence
class of Weil divisors. The corresponding reflexive sheaf $\o_X(K_X)$
is isomorphic to the {\it dualizing sheaf}
$\omega_X$ of
$X$.
The index of $K_X$ is called the {\it index } of $X$.
\end{defn}
\begin{defn}\label{mmp.discr.def} Let $X,Y$ be normal varieties and
$f:Y\to X$ a birational morphism with exceptional set $\ex(f)$. Let
$E_i\subset \ex(f)$ be the exceptional divisors. If
$mK_X$ is Cartier, then
$f^*\o_X(mK_X)$ is defined and there is a natural isomorphism
$$ f^*\o_X(mK_X)|(Y\setminus \ex(f))\cong
\o_Y(mK_Y)|(Y\setminus \ex(f)).
$$
Hence there are integers $b_i$ such that
$$
\o_Y(mK_Y)\cong f^*\o_X(mK_X)(\sum b_iE_i).
$$ Formally divide by $m$ and write this as
$$ K_Y\equiv f^*(K_X)+\sum a(E_i, X)E_i,\qtq{where $a(E_i,X)\in \q$.}
$$ The rational number $a(E_i,X)$ is called the {\it discrepancy} of
$E_i$ with respect to $X$.
The closure of $f(E_i)\subset X$ is called the {\it center} of $E_i$ on
$X$. It is denoted by $\cent_XE_i$.
If $f':Y'\to X$ is another birational morphism and $E'_i:=(f'\circ
f^{-1})(E_i)\subset Y'$ is a divisor then $a(E'_i,X)=a(E_i,X)$ and
$\cent_XE_i=\cent_XE'_i$. Thus the discrepancy and the center depend
only on the divisor up to birational equaivalence, but not on the
particular variety where
the divisor appears.
\end{defn}
\begin{defn} Let $X$ be a normal variety such that $K_X$ is
$\q$-Cartier. We say that $X$ is {\it terminal} (or that it has {\it
terminal singularities}) if for every $f:Y\to X$, the discrepancy of
every exceptional divisor is positive.
\end{defn}
The following result makes it feasible to decide if $X$ is terminal
or not.
\begin{lem}\label{mmp.term.lem} For a normal variety $X$ the following
are equivalent:
\begin{enumerate}
\item $X$ is terminal,
\item $a(E,X)>0$ for every resolution of singularities $f:Y\to X$
and for every exceptional divisor $E\subset \ex(f)$.
\item There is a resolution of singularities
$f:Y\to X$ such that $a(E,X)>0$ for every exceptional divisor
$E\subset \ex(f)$.\qed
\end{enumerate}
\end{lem}
\begin{exmp}\label{mmp.comp.discr.exmp}
It is frequently not too hard to compute discrepancies. Assume for
instance that $X$ is a hypersurface defined by
$(F(x_1,\dots,x_n)=0)$. A local generator of $\o_X(K_X)$ is given by
any of the forms
$$
\eta_i:=\frac1{\partial F/\partial x_i}dx_1\wedge\cdots\wedge
dx_{i-1}\wedge dx_{i+1}\wedge\cdots\wedge dx_n.
$$ Let $f:Y\to X$ be a resolution of singularities and $P\in Y$ a point
with local coordinates $y_1,\dots,y_{n-1}$. $f$ is given by coordinate
functions
$x_i=f_i(y_1,\dots,y_{n-1})$ and so we can write
\begin{eqnarray*} f^*\eta_n&=&f^*\left(\frac1{\partial F/\partial
x_n}\right)
\operatorname{Jac} dy_1\wedge\cdots\wedge dy_{n-1},
\qtq{where}\\
\operatorname{Jac}&=& \operatorname{Jac}
\left(\frac{f_1,\dots,f_{n-1}}{x_1,\dots,x_{n-1}}\right)
\end{eqnarray*} denotes the determinant of the Jacobian matrix.
Hence the discrepancies can be computed as the order of vanishing of
the Jacobian minus the order of vanishing of
$f^*(\partial F/\partial x_n)$.
If $X$ is smooth then we conclude that $a(E,X)\geq 1$ for every
exceptional divisor. Thus smooth varieties are terminal.
\end{exmp}
Next we define various birational maps which have special role in the
MMP.
\begin{defn}\label{mmp.extremal.def}
Let $X$ be a variety over $K$ and assume that $K_X$ is
$\q$-Cartier. A proper morphism $g:X\to Y$ is called an {\it extremal
contraction} if the following conditions hold:
\begin{enumerate}
\item $g_*\o_X=\o_Y$,
\item $X$ is $\q$-factorial,
\item Let $C\subset X$ be any irreducible curve such that
$g(C)=\mbox{point}$. Then a
$\q$-divisor $D$ on $X$ is the pull back of a $\q$-Cartier
$\q$-divisor $D'$ on $Y$ iff $(D\cdot C)=0$. (Necessarily,
$D'=g_*(D)$.)
\end{enumerate}
\end{defn}
\begin{defn}\label{mmp.excontypes.def}
Let $g:X\to Y$ be an extremal contraction.
We say that $g$ is of {\it fiber type} if $\dim Y<\dim X$.
We say that $g$ is a {\it divisorial} contraction if the exceptional
set $\ex(g)$ is the support of $\q$-Cartier divisor. In this case
$\ex(g)$ is irreducible over $K$.
We say that $g$ is a {\it small} contraction if
$\dim \ex(g)\leq \dim X-2$.
One can see that every extremal contraction is in one of these 3
groups.
\end{defn}
\begin{defn}\label{mmp.KXneg.def}
A proper morphism $f:X\to Y$ is called {\it $K_X$-negative} if $-K_X$
is $f$-ample.
\end{defn}
\begin{defn}\label{mmp.flip.def} Let
$f: X \to Y$ be a small $K_X$-negative extremal contraction.
A variety $X^+$ together with a proper birational morphism
$f^+: X^+
\to Y$
is called a {\it flip} of $f$ if
\begin{enumerate}
\item $K_{X^+}$ is $\q$-Cartier,
\item $K_{X^+}$ is $f^+$-ample, and
\item the exceptional set $\ex(f^+)$ has codimension at least two
in $X^+$.
\end{enumerate} By a slight abuse of terminology, the rational map
$\phi: X \map X^+$ is also
called a flip. A flip gives the following diagram:
$$
\begin{array}{rcl} X &\stackrel{\phi}{\map} &X^+\\
\mbox{$-K_{X}$ is $f$-ample} &\searrow \quad\swarrow &
\mbox{$K_{X^+}$ is $f^+$-ample}\\ &Y&
\end{array}
$$ It is not hard to see that a flip is unique and the main question
is its existence.
\end{defn}
We are ready to state the 3-dimensional MMP over an arbitrary field:
\begin{thm}[MMP over $K$]\label{mmp.mmp.thm}
Let $X$ be a smooth projective 3-fold defined over a field $K$ (of
characteristic zero). Then there is a sequence
$$
X=X_0\stackrel{f_0}{\map} X_1\map \cdots \map X_i
\stackrel{f_i}{\map} X_{i+1}\map \cdots \stackrel{f_{n-1}}{\map}
X_n=:X^*
$$ with the following properties
\begin{enumerate}
\item Each $X_i$ is a terminal projective 3-fold over $K$ which is
$\q$-factorial over $K$.
\item Each $f_i$ is either a $K_X$-negative divisorial extremal
contraction or the flip of a $K_X$-negative small extremal
contraction.
\item One of the following holds for $X^*$:
\begin{enumerate}
\item
$K_{X^*}$ is nef (that is $(C\cdot K_{X^*})\geq 0$ for any curve
$C\subset X^*$), or
\item there is a fiber type extremal contraction $X^*\to Z$.
\end{enumerate}
\end{enumerate}
\end{thm}
\begin{rem} For the purposes of this paper one can handle the MMP as
a black box. It is sufficient to know that it works, but I will use
very few of its finer properties. In particular, there is no need to
know anything about flips beyond believing their existence.
The rest of the section is devoted to explicitly stating all further
results from minimal model theory that I use later. The most
significant among these is the classification of terminal 3-fold
singularities over nonclosed fields, established in \cite{rat1}.
\end{rem}
\begin{notation}\label{mmp.ps.not}
For a field $K$ let
$K[[x_1,\dots,x_n]]$ denote the ring of formal power series in $n$
variables over
$K$. For $K=\r$ or
$K=\c$, let $K\{x_1,\dots,x_n\}$ denote the ring of those formal power
series which converge in some neighborhood of the origin.
For a power series $F$, $F_d$ denotes the degree $d$ homogeneous part.
The multiplicity, denoted by $\mult_0F$, is the smallest $d$ such that
$F_d\neq 0$. If we write a power series as $F_{\geq d}$ then it is
assumed that its multiplicity is at least $d$.
For $F\in \r\{x_1,\dots,x_n\}$ let $(F=0)$ denote the germ of its
zero set in $\c^n$ with its natural real structure. I always think of
it as a complex analytic germ with a real structure and not just as a
real analytic germ in $\r^n$.
$(F=0)/\frac1{n}(a,b,c,d)$ means the following. Define a
$\z_n$-grading of $\c\{x,y,z,t\}$ by $x\mapsto a, y\mapsto b, z\mapsto
c,t\mapsto d$. If $F$ is graded homogeneous, then
$(F=0)/\frac1{n}(a,b,c,d)$ denotes the germ whose ring of holomorphic
functions is the ring of grade zero elements of
$\c\{x,y,z,t\}/(F)$.
If $(F=0)$ is terminal then $n$ coincides with the index
(\ref{mmp.KX.def}) of the singularity.
\end{notation}
\begin{exmp} In case $X=(x^2+y^2+z^2+ t^2=0)/\frac12(1,1,1,0)$
the ring is
$$
\o_X=\c\{x^2,y^2,z^2,t,xy,yz,zx\}/(x^2+y^2+z^2+ t^2),
$$ with the natural real structure.
$X$ can also be realized as the image of the hypersurface
$(x^2+y^2+z^2+ t^2=0)$ under the map
$$
\phi: \c^4\to \c^7:\quad (x,y,z,t)\mapsto (x^2,y^2,z^2,t,xy,yz,zx),
$$ which has degree 2 over its image.
Although
$(x^2+y^2+z^2+ t^2=0)$ has only the origin as its real solution,
$X$ has plenty of real points. Indeed, any real solution of
$x^2+y^2+z^2- t^2=0$ gives a {\it real} point
$P=\phi(\sqrt{-1}x,\sqrt{-1}y, \sqrt{-1}z, t)\in X(\r)$.
$\phi^{-1}(P)$ is a pair of conjugate points on the hypersurface
$(x^2+y^2+z^2+ t^2=0)$. All the real elements of
$\o_X$ take up real values at $P$.
This way we see that $X(\r)$ is a cone over 2 copies of $\r\p^2$.
\end{exmp}
The following is a summary of the classification of terminal
singularities obtained in \cite{rat1}. As it turns out, the
classification closely follows the earlier results over algebraically
closed fields. The choice of the subdivison into cases is dictated by
the needs of the proof in sections 9--11, rather than the internal
logic of the classification.
\begin{thm}\label{mmp.ts.thm} Let $X$ be a real algebraic or analytic
3-fold and
$0\in X(\r)$ a real point. Then $X$ has a terminal singularity at $0$
iff a neighborhood of
$0\in X$ is real analytically equivalent to one of the following:
$$
\begin{tabular}{ll} name & \qquad equation \\
$cA_0$ &$(t=0)$\\
$cA_1$ &$(x^2+y^2\pm z^2\pm t^m=0)$\\
$cA_{>1}^+$ &$(x^2+y^2+g_{\geq 3}(z,t)=0)$ \\
$cA_{>1}^-$ &$(x^2-y^2+g_{\geq 3}(z,t)=0)$ \\
$cD_4$ &$(x^2+f_{\geq 3}(y,z,t)=0)$, where $f_3\neq l_1^2l_2$ for
linear forms $l_i$\\
$cD_{>4}$ &$(x^2+y^2z+f_{\geq 4}(y,z,t)=0)$,\\
$cE_6$ &$(x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq 4}(z,t)=0)$, where
$h_4\neq 0$\\
$cE_7$ &$(x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq 5}(z,t)=0)$, where
$g_3\neq 0$\\
$cE_8$ &$(x^2+y^3+yg_{\geq 4}(z,t)+h_{\geq 5}(z,t)=0)$, where
$h_5\neq 0$\\
$cA_0/n$ & $(t=0)/\frac1{n}(r,-r,1,0)$ where $n\geq 2$ and
$(n,r)=1$\\
$cA_1/2$ & $(x^2+y^2\pm z^n\pm t^m=0)/\frac12(1,1,1,0)$ where
$\min\{n,m\}=2$\\
$cA_{>1}^+/2$ & $(x^2+y^2+f_{\geq 3} (z,t)=0)/\frac12(1,1,1,0)$\\
$cA_{>1}^-/2$ & $(x^2-y^2+f_{\geq 3}(z,t)=0)/
\frac12(1,1,1,0)$\\
$cA/n$ & $(xy+f(z,t)=0)/\frac1{n}(r,-r,1,0)$ where $n\geq 3$ and
$(n,r)=1$\\
$cAx/2$ & $(x^2\pm y^2 +f_{\geq 4} (z,t)=0)/\frac12(0,1,1,1)$\\
$cAx/4$ & $(x^2\pm y^2 +f_{\geq 2}(z,t)=0)/\frac14(1,3,1,2)$
where $f_2(0,1)=0$\\
$cD/2$ &$(x^2+f_{\geq 3}(y,z,t)=0)/\frac12(1,1,0,1)$\\
$cD/3$ &$(x^2+f_{\geq 3}(y,z,t)=0)/\frac13(0,1,1,2)$ where
$f_3(0,0,1)\neq 0$\\
$cE/2$ &$(x^2+y^3+f_{\geq 4}(y,z,t)=0)/\frac12(1,0,1,1)$\\
\end{tabular}
$$
\end{thm}
\section{The Topology of Real Points and the MMP}
Starting with a projective variety
$X$ over $\r$, let us run the MMP over $\r$. We obtain a sequence of
birational maps
$$ X=X_0\map X_1\map \cdots\map X_i\stackrel{f_i}{\map}
X_{i+1}\map\cdots\map X^*.
$$ These in turn induce (not necessarily everywhere defined) maps
between the sets of real points
$$ X(\r)=X_0(\r)\map \cdots\map X_i(\r)\stackrel{f_i}{\map}
X_{i+1}(\r)\map\cdots\map X^*(\r).
$$ Our aim is to see if there is a way of describing $X(\r)$ in terms
of
$X^*(\r)$ and a local description of the maps
$X_i(\r)\map X_{i+1}(\r)$ in a neighborhood of their exceptional sets.
\begin{prop}\label{mmpt.cases.prop}
Every step $f_i$ of the MMP over
$\r$ is among the following five:
\begin{enumerate}
\item (divisor--to--point) $f_i$ contracts a geometrically irreducible
divisor $E_i\subset X_i$ to a point $P_{i+1}\in X_{i+1}(\r)$.
\item (divisor--to--curve) $f_i$ contracts a geometrically irreducible
divisor $E_i\subset X_i$ to a real curve $C_{i+1}\subset X_{i+1}$.
\item ($\r$-small) $f_i:X_i(\r)\to X_{i+1}(\r)$ collapses a 1-complex
to points and is a homeomorphism elsewhere.
\item (flip) $f_i$ is the flip of a curve $C_i\subset X_i$.
\item ($\r$-trivial) $f_i$ is an isomorphism in a (Zariski)
neighborhood of the set of real points.
\end{enumerate}
\end{prop}
Proof. If $f_i$ is a flip then we have case (4). Thus we may assume
that $f_i$ is the contraction of a divisor
$E_i\subset X_i$ and $E_i$ is irreducible over $\r$. If $E_i$ is
irreducible over $\c$ then we have one of the cases (1--2). If $E_i$
is reducible over $\c$ then $E_i(\r)$ is a 1-complex by
(\ref{mmpt.red.small.lem}) and so we are in case (3).
Any of the above cases can also be of type (5).\qed
\begin{lem}\label{mmpt.red.small.lem}
Let $X$ be an n-dimensional scheme over $\r$ (that is, an algebraic
variety possibly with several irreducible components and with
singularities). Assume that if $X_i\subset X$ is any $\r$-irreducible
component then
$X_i$ is reducible over $\c$. Then
$X(\r)= (\sing X)(\r)$, that is, every real point is singular. In
particular, $\dim X(\r)\leq n-1$.
\end{lem}
Proof. Assume that $P\in X(\r)$ is a smooth real point. Then $P$ lies
on a unique irreducible component $Y\subset X_{\c}$, thus $Y$ is
invariant under complex conjugation. So
$Y$ is an irreducible real component which stays irreducible over
$\c$, a contradiction.\qed
\medskip
Each of the 5 steps $X_i(\r)\map X_{i+1}(\r)$ have different
topological behaviour. The following informal discussion intends to
emphasize their main features.
\begin{say}[Divisor--to--point] Let $M=X_i(\r)$ be a 3-complex
(with only finitely many singular points) and
$F=E_i(\r)\subset M$ a 2-complex. We collapse $F$ to a point:
$$
\begin{array}{ccc} F& \subset & M\\
\downarrow && \ \ \downarrow f\\ P&\in &N.
\end{array}
$$ In practice we are frequently able to describe a regular
neighborhood
$F\subset U\subset M$ (this is a local datum) and by assumption we
know a regular neighborhood $P\in V\subset N$. Thus we see that M is
obtained from $U$ and $N\setminus
\inter V$ by gluing them together along the boundaries
$\partial U$ and $\partial V$.
The gluing is determined by a PL-homeomorphism $\phi:
\partial U\to \partial V$. Thus, besides knowing
$U$ and
$N$, we also need to know $\phi$ up to PL-isotopy. If one of the
connected components of $\partial U$ has genus at least 2, this is a
very hard problem. In fact, as the example of Heegard splittings shows
(cf.
\cite[Ch.2]{Hempel76}), the choice of $\phi$ is usually the most
significant information. Unfortunately, $\phi$ can be described
only in terms of global data.
If $\partial U$ is a union of $m$ copies of $S^2$, then
$\phi$ is classified by an element of the symmetric group on
$m$ elements (which $S^2$ maps where) and a sign for each
$S^2$ (describing whether the map is orientation preserving or
reversing on that $S^2$). Hence, knowing $U$ and $N$, we can
determine $M$ up to finite ambiguity.
In many cases $U$ is so simple that different choices of
$\phi$ give the same $M$, giving even fewer possibilities for $M$.
If $P\in N$ is an isolated singular point, then $\partial V$ is a
union of spheres iff $\bar N$ (the topological normalization of $N$)
is a manifold.
The situation is similarly simple if $\partial U$ is a union of
copies of $\r\p^2$ and of $S^2$, and still manageable if $\partial
U$ also contains tori and Klein bottles. For us these more general
cases do not come up.
\end{say}
\begin{say}[Divisor--to--curve] This time we construct it bottom up.
Assume for simplicity that
$N=X_{i+1}(\r)$ is a 3-manifold and
$L=C_{i+1}(\r)\subset N$ a link. The projectivized normal bundle is
an $S^1$-bundle $S\to L$. The blow up of $L$ in $N$ replaces $L$ by
$S$ to obtain:
$$
\begin{array}{ccc} S& \subset & M\\
\downarrow && \ \ \downarrow f\\ L&\in &N.
\end{array}
$$ (In general $N$ may have finitely many singular points and $L$ is
only a 1-complex, but I believe that a similar description is
possible in all cases.)
Here $M$ is uniquely determined, once we know $N$ and $L$. By
assumption we know $N$ but $L$ is a free choice. The
Jaco--Johannson--Shalen decomposition (cf. \cite[p.483]{Scott83})
shows that in most cases $B_LN$ determines $M\setminus L$. Thus the
description of all possible $B_LN$ is essentially equivalent to the
description of all links.
For us
$L$ has to come from an algebraic curve, thus we are led to the
question: Which links in a real algebraic 3-fold can be realized by
algebraic curves? In some cases every link is realized (cf.
\cite{AK81}), thus we again run into a hard topological problem.
So $M$ can be described in terms of $N$, though the answer depends
on the choice of a link, which is a very complicated object.
\end{say}
\begin{say}[$\r$-small contraction] $N=X_{i+1}(\r)$ is obtained
from $M=X_i(\r)$ by collapsing a 1-complex
$C=(\sing E_i)(\r)=E_i(\r)$ to a point:
$$
\begin{array}{ccc} C& \subset & M\\
\downarrow && \ \ \downarrow f\\ P&\in &N.
\end{array}
$$ If the normalizations $\bar M$ and $\bar N$ are manifolds, then
we see in (\ref{top.alg.homeo.lem}) that a suitable small perturbation
of
$f$ is a homeomorphism between $\bar M$ and $\bar N$. Thus this
step (which is actually more complicated from the point of view of
algebraic geometry than the previous two cases) is easy to analyze
topologically.
\end{say}
\begin{say}[Flip] Assume for simplicity that
$M=X_i(\r)$ is an orientable 3-manifold and
$C(\r)\sim S^1$. $N=X_{i+1}(\r)$ is obtained from $M$ by a surgery
along $S^1$. The boundary of a regular neighborhood of $S^1$ is
$S^1\times S^1$, and the surgery is determined by a diffeomorphism of
$S^1\times S^1$ up to isotopy. These are classified by $SL(2,\z)$. (In
general $M$ may have finitely many singular points and $C(\r)$ is a
1-complex, but I believe that a similar description is possible in
all cases.)
A complete classification of flips is known \cite{KoMo92}, thus it
should be possible to compute the resulting diffeomorphism of
$S^1\times S^1$.
Here again we run into a global problem. $S^1\subset M$ may be very
knotted, and the result of the surgery depends mostly on the knot
$S^1\subset M$. The usual descriptions of flips characterize a complex
analytic neighborhood of $C$, thus they say nothing about how its real
part is knotted. From the point of view of algebraic geometry, this
is a global invariant.
We have the additional problem that flipping curves are rigid
objects, thus we can not hope to get a flipping curve by
approximating a real curve algebraically. Furthermore, it is very
hard to determine which curves are obtained by a flip. (Even if $Z$ is
a smooth complex 3-fold and $C\subset Z$ a smooth
$\c\p^1$, I know of no practical way of determining if
$C\subset Z$ is obtained as a result of a flip.)
\end{say}
\begin{say}[Conclusion]\label{mmpt.concl}
Start with a projective 3-fold
$X$ over $\r$ and run the MMP over $\r$:
$$ X=X_0\map X_1\map \cdots\map X_i\stackrel{f_i}{\map}
X_{i+1}\map\cdots\map X^*.
$$ If we would like to understand the topology of
$X(\r)$ in terms of $X^*(\r)$, then we have to ensure that the MMP
has the following properties:
\begin{enumerate}
\item
$\overline{X_i(\r)}$ is a manifold for every $i$.
\item Each $f_i$ is either $\r$-trivial or $\r$-small or a
divisor--to--point contraction.
\end{enumerate}
\end{say}
(\ref{int.mmp.steps}) asserts that both of these conditions can be
satisfied by imposing certain mild conditions on the topology of
$X(\r)$.
\section{The Topology of Divisorial Contractions}
The aim of this section is to describe some examples where the change
of the topology of a real algebraic variety under a divisorial
contraction can be readily understood by topological methods.
\begin{notation}\label{5.notation}
The disjoint union of two topological spaces is denoted by $M\uplus
N$. Direct product is denoted by $M\times N$. The unique nontrivial
$S^2$-bundle over $S^1$ is denoted by
$S^1\tilde{\times} S^2$. This is obtained from $[0,1]\times S^2$ by
indentifying the 2 ends via an orientation reversing homeomorphism.
Homeomorphism of two topological spaces is denoted by $M\sim N$.
\end{notation}
We start with the study of $\r$-small contractions:
\begin{lem}\label{top.homeo.lem}
Let $f:M\to N$ be a proper PL-map between PL-manifolds of dimension
$n\geq 3$. Assume that there is a 1-complex
$C\subset M$ and a finite set of points $P\subset N$ such that
$f:M\setminus C\to N\setminus P$ is a PL-homeomorphism.
Then $M$ and $N$ are PL-homeomorphic (by a small perturbation of $f$).
\end{lem}
Proof. If $C$ is collapsible to points, then a regular neighborhood
of $C$ in $M$ is a union of disjoint $n$-cells
\cite[1.8]{Hempel76} and we are done.
In order to see that $C$ is collapsible to points, we may assume that
$P$ is a point and
$N=S^n$. Thus $M$ is also a compact PL-manifold.
$M$ is orientable outside the codimenison $\geq 2$ subset
$C$, hence it is orientable. Consider the exact homology sequences
$$
\begin{array}{ccccccc} H_i(C)&\to &H_i(M)&\to & H_i(M,C)&\to &
H_{i-1}(C)\\
\downarrow &&\downarrow &&\downarrow &&\downarrow \\ H_i(P)&\to
&H_i(S^n)&\to & H_i(S^n,P)&\to & H_{i-1}(P)
\end{array}
$$ We compute $H_i(M,C)=H_i(S^n,P)$ from the second sequence and
substitute into the first to obtain that
$$ H_1(C)\cong H_1(M), \qtq{and} 0=H_{n-1}(C)\cong H_{n-1}(M).
$$ By Poincar\'e duality we conclude that $H_1(C)=0$, thus
$C$ is contractible.
\qed
\begin{cor}\label{top.alg.homeo.lem}
Let $f:X\to Y$ be a morphism a $n$-dimensional real algebraic
varieties,
$n\geq 3$. Assume that
\begin{enumerate}
\item $\overline{X(\r)}=M\uplus R$ and $\overline{Y(\r)}=N\uplus R'$
where $M,N$ are PL-manifolds and $\dim R,\dim R'<n$.
\item $f$ induces an isomorphism $R\cong R'$.
\item $\ex(f)(\r)$ is a 1-complex.
\end{enumerate}
\noindent Then $\overline{X(\r)}$ is PL-homeomorphic to
$\overline{Y(\r)}$.
\end{cor}
Proof. Set $C=\ex(f)(\r)$ and $\bar C\subset M\subset
\overline{X(\r)}$ its preimage. Since $\bar C$ has dimension 1, there
is a one--to--one correspondence between the connected components of
$\bar C$ and the connected components of the boundary of a regular
neighborhood of $\bar C$. Hence $f$ lifts to a morphism $\bar f:
\overline{X(\r)}\to
\overline{Y(\r)}$.
Thus (\ref{top.homeo.lem}) implies (\ref{top.alg.homeo.lem}), and the
homeomorphism is given by a small perturbation of
$\bar f$.\qed
\begin{rem} A real algebraic curve is a union of copies of
$S^1$. The proof of (\ref{top.homeo.lem}) shows that the preimage of
$\ex(f)(\r)$ in $\overline{X(\r)}$ is contractible. Thus if
$f$ itself is not a homeomorphism, then $X(\r)$ is not a manifold.
\end{rem}
Next we look at divisor--to--point contractions.
\begin{prop}\label{top.normsurf.nbd.boundary}
Let $M$ be a 3-dimensional PL-manifold and $F\subset M$ a connected
2-complex with only finitely many singular points. Let
$F\subset \inter U\subset M$ be a regular neighborhood of
$F$. Then
$$
\dim H_1(\bar F,\q)\leq \dim H_1(\partial U,\q),
$$ and strict inequality holds unless every connected component of
$\bar F$ is one of the following:
\begin{enumerate}
\item $S^2$ or $\r\p^2$,
\item a one-sided $S^1\times S^1$,
\item a one-sided Klein bottle whose neighborhood is not orientable.
\end{enumerate}
\end{prop}
Proof. Let $F$ be a compact 2-complex with only finitely many
singular points. Its normalization $\bar F$ can be written as
$F^{(2)}\uplus F^{(1)}$ where $F^{(2)}$ is a compact 2-manifold and a
$ F^{(1)}$ is a 1-complex.
Pick a point $P\in F$ whose link in $F$ consists of at least 2
circles. Locally $F$ looks like the cone over parallel plane
sections $(z=a_i)\cap (x^2+y^2+z^2=1)$ of the unit sphere in
$\r^3$ (plus a few 1-cells). By a homotopy we can replace this by the
parallel plane sections of the unit ball
$(z=a_i)\cap (x^2+y^2+z^2\leq 1)$
and add the interval $[\min_i\{a_i\},\max_i\{a_i\}]$ on the
$z$-axis. This does not change the boundary of the regular
neighborhood. Thus we may assume that $F^{(2)}\to M$ is an embedding.
Let us take a point or a 1-cell $e$ in $F^{(1)}$. If $e$ does not
intersect the rest of $F$, then a regular neighborhood of
$e$ is a 3-cell. $e$ can be deleted from $F$ without changing the
inequality.
If $e$ intersects the rest of $F$ in one endpoint only, then we can
delete $e$ from $F$ without changing the regular neighborhood.
If $e$ intersects the rest of $F$ at both endpoints, then removing
$e$ creates a new 2-complex $F'$, and $F^{(2)}={F'}^{(2)}$. Let
$F'\subset\inter U'$ be its regular neighborhood.
$\partial U$ is obtained from $\partial U'$ by attaching a handle
$[0,1]\times S^1$. Thus $H_1(\partial U)\geq H_1(\partial U')$, and it
is sufficient to verify our inequality for $F'$.
At the end we are reduced to the situation when $F$ is the disjoint
union of embedded 2-manifolds, and it is sufficient to check the
ineqality for each connected component of $F$ separately.
$\partial U \to F$ is a 2 sheeted cover, thus
$H_1(\partial U)\geq H_1(F)$ with equality only if
$F\sim S^2, F\sim \r\p^2, F\sim S^1\times S^1\sim \partial U$ or
$F$ and $\partial U$ are both Klein bottles.\qed
\begin{prop}\label{top.normsurf.collapse}
Let $M$ be a 3-dimensional PL-manifold and $F\subset M$ a compact
2-complex with only finitely many singular points. Let $0\in N$ be
obtained from $M$ by collapsing $F$ to a point. Assume that
$\bar N$ is a 3-manifold. Then
$M$ can be obtained from $\bar N$ by repeated application of the
following operations:
\begin{enumerate}
\item taking connected sums of connected components,
\item taking connected sum with $S^1\times S^2$,
\item taking connected sum with $S^1\tilde{\times} S^2$, or
\item taking connected sum with $\r\p^3$.
\end{enumerate}
\end{prop}
Proof. We use the notation of (\ref{top.normsurf.nbd.boundary}) and
of its proof. Let
$F\subset \inter U\subset M$ and $0\in \inter V\subset N$ be regular
neighborhoods such that $U=f^{-1}(V)$. Then $\partial U=\partial V$.
Since
$\bar N$ is a manifold, this implies that $\partial U$ is a union of
2-spheres. We also see that $\bar N$ is obtained from
$M\setminus\inter U$ by attaching a 3-ball to each
$S^2$ in $\partial U$.
As in the proof of (\ref{top.normsurf.nbd.boundary}) we may assume that
$F^{(2)}\to M$ is an embedding.
If $e$ is a point or a 1-cell in $F^{(1)}$ which intersects the
rest of
$F$ in zero or one point only, then we can delete $e$ from $F$.
If $e$ intersects the rest of $F$ at both endpoints, then removing
$e$ creates a new 2-complex $F'$ such that $\bar F=\bar F'$. Let
$F'\subset\inter U'$ be its regular neighborhood.
$\partial U$ is obtained from $\partial U'$ by attaching a handle
$[0,1]\times S^1$. The two ends $\{0\}\times S^1$ and
$\{1\}\times S^1$ can not attach to the same connected component of
$\partial U'$ since that would create a torus or a Klein bottle in
$\partial U$. Thus $\partial U'$ has one more copies of
$S^2$ than
$\partial U$.
$\bar N$ is obtained from $\bar N'$ by collapsing the image of
$e$ to a point, hence
$\bar N$ and $\bar N'$ are homeomorphic by (\ref{top.homeo.lem}).
At the end we are reduced to the situation when $F$ is the disjoint
union of embedded copies of $S^2$ and
$\r\p^2$. An $S^2$ is necessarily 2-sided. Removing it from $F$
corresponds to taking connected sums of connected components (if $S^2$
separates $M$) or to taking connected sum with $S^1\times S^2$ or
$S^1\tilde{\times} S^2$ (if $S^2$ does not separate
$M$) (cf. \cite[Chap.\ 3]{Hempel76}).
If $\r\p^2$ is 2-sided, then the boundary of its regular neighborhood
consists of two copies of $\r\p^2$, so this can not happen. A 1-sided
$\r\p^2$ corresponds to taking connected sum with
$\r\p^3$. \qed
\begin{cor}\label{top.alg.normsurfup.lem}
Let $f:X\to Y$ be a morphism a real algebraic $3$-folds. Assume that
\begin{enumerate}
\item $\overline{X(\r)}$ and $\overline{Y(\r)}$ are PL-manifolds, and
\item $\ex(f)$ is a geometrically irreducible normal surface which is
contracted to a point.
\end{enumerate}
\noindent Then $\overline{X(\r)}$ can be obtained from
$\overline{Y(\r)}$ by repeated application of the following
operations:
\begin{enumerate}
\setcounter{enumi}{2}
\item removing an isolated point from $\overline{Y(\r)}$,
\item taking connected sums of connected components,
\item taking connected sum with $S^1\times S^2$,
\item taking connected sum with $S^1\tilde{\times} S^2$, or
\item taking connected sum with $\r\p^3$.
\end{enumerate}
\end{cor}
Proof. If $\ex(f)(\r)=\emptyset$ then the image of $\ex(f)$ is an
isolated real point of $Y(\r)$ which has to be thrown away to obtain
$X(\r)$. If $\ex(f)(\r)\neq\emptyset$, then isolated points of
$X(\r)$ correspond to isolated points of $Y(\r)$, hence they can be
ignored.
Let $M$ be the topological normalization of
$X(\r)\setminus(\mbox{isolated points})$,
$N$ the topological normalization of $Y(\r)\setminus(\mbox{isolated
points})$ and
$F$ the preimage of $\ex(f)(\r)$ in $M$. $F$ is a 2-complex with
isolated singularities since $\ex(f)$ is normal.
Thus (\ref{top.alg.normsurfup.lem}) follows from
(\ref{top.normsurf.collapse}).\qed
\begin{complement} It is worthwhile to note that condition
(\ref{top.alg.normsurfup.lem}.2) can be weakened to:
\begin{enumerate}
\item[2'] $\ex(f)$ contains a unique geometrically irreducible
surface
$S$. $S$ has only isolated singularities and $S$ is contracted to a
point by $f$.
\end{enumerate}
\end{complement}
It would be very useful to have a version of
(\ref{top.alg.normsurfup.lem}) which works if $\ex(f)$ is an
irreducible but nonnormal surface.
In the topological version (\ref{top.normsurf.collapse}) esentially
nothing can be said if $F$ is allowed to become an arbitrary compact
2-complex. For instance, let $M$ be an arbitrary compact 3-manifold
and
$F$ the 2-skeleton of a triangulation of $M$. Then $\bar N$ is the
union of some copies of $S^3$ (one for each 3-simplex).
This example usually can not arise as the real points of an algebraic
surface, but it is not hard to modify this example by approximating
each simplex with a sphere to get the following. (This is not used in
the sequel and so no proof is given here.)
\begin{prop}\label{top.exmp}
Let $M$ be a compact differentiable manifold of dimension
$n$. Then there is a smooth real algebraic variety $X$ and a morphism
$f:X\to Y$ with the following properties:
\begin{enumerate}
\item $X(\r)\sim M$,
\item $\overline{Y(\r)}$ is a disjoint union of copies of $S^n$,
\item $\ex(f)$ is
a geometrically irreducible divisor and $\ex(f)(\r)$ is a union of
copies of $S^{n-1}$ intersecting transversally.
\end{enumerate}
\end{prop}
\section{The Gateway Method}
At the beginnings of the MMP, divisorial contractions were considered
to be the easily understandable part of the program and flips the hard
part. Lately, however, more and more questions require a detailed
understanding of all the steps of the MMP. A fairly complete
description of all flips is known
\cite{KoMo92}, but it seems very difficult to obtain a
list of all divisorial contractions. One can try to study the MMP in
two basic ways:
\begin{say}[Analysis of the MMP] Starting with a projective variety
$X$, let us run the MMP. We obtain a sequence of birational maps
$$ X=X_0\map X_1\map \cdots\map X_i\map X_{i+1}\map\cdots\map X^*.
$$ Assume that $X$ has some nice property that we would like to
preserve. We need some way of proving that
$X^*$ also has this property, at least under some additional
assumptions. One way is to prove this directly, by analyzing each step
of the MMP. This would sometimes require knowing each step of the
MMP, and even in dimension 3 the list is not yet available. Still there
are many results that can be established this way, for instance the
existence of the MMP itself. In this approach one starts with a
variety $X$ and tries to understand every possible way an MMP can {\it
start} with $X$. This is oftentimes manageable if $X$ has only mild
singularities.
Another way is to look at each step of the MMP backwards. In
dimension 3 we have a pretty good description of the possible
singularities that arise in the course of an MMP. Thus we can start
with a variety
$Y$ and try to understand every possible way an MMP can {\it end}
with $Y$. This also seems rather hard. Even the case when
$Y$ is smooth is not at all understood, but in some other cases this
approach has been carried through
\cite{Kawamata9?}. It seems that this method is easier to apply when
$X$ is fairly singular.
The {\it gateway method} attempts to solve the original problem in an
intermediate way. In the above chain of maps there is a smallest index
$i$ such that
$X_i$ is still ``nice" but $X_{i+1}$ is not. Hence
$X_i\map X_{i+1}$ is a ``gateway" through which the process leaves the
set of ``nice" varieties. Analysing these ``gateways" should be
easier since the direct approach tends to work for the nice variety
$X_i$ and the backwards method tends to work for more complicated
singularities of
$X_{i+1}$.
Once such a list of ``gateways" is obtained, it is a matter of
checking the list to see if some additional properties ensure that
this step does not happen.
\end{say}
One of the simplest examples where these ideas yield a nontrivial
result is the following.
\begin{exmp} Assume that we want to stay within the class of varieties
of index 1. In this case there is only one gateway:
{\bf Proposition.} {\it Let $f:X\map X'$ be a step of the
3-dimensional MMP where $X$ has index 1 but $X'$ has higher index.
Then $f$ is the contraction of a divisor
$E\subset X$ to a point. Furthermore,
$E\cong \p^2$, $X$ is smooth along $E$ and $E$ has normal bundle
$\o_E(-2)$.}
This result is a special case of \cite{Mori88} and
\cite{Cutkosky88}, though they did not approach this from the point of
view of gateways. A proof along the lines suggested by the gateway
method is not hard to construct, but this is not any shorter then the
direct proofs.
As a consequence we obtain:
{\bf Corollary} {\it Let $X$ be a projective 3-fold with index 1
terminal singularities. Assume that $X$ does not contain any surface
$S\subset X$ which admits a birational morphism onto $\p^2$. Then
each step of the MMP starting with $X$ is a projective 3-fold with
index 1 terminal singularities.}
Unfortunately the above condition needs to be checked for every
surface $S$, even for very singular ones. Thus in practice this does
not seem to be a useful observation.
\end{exmp}
\begin{say} Our aim is to develop a similar theory for real algebraic
threefolds. Thus we have to decide which varieties are ``nice" and
then describe all possible gateways through which the MMP can leave
the class of ``nice" varieties.
(\ref{mmpt.concl}) naturally suggest a topological choice: $X$ is
``nice" if $X(\r)$ or $\overline{X(\r)}$ is a 3-manifold, maybe with
some additional properties. This was my first attempt, but I was
unable to make it work. The main problem seems to be that, as the
computations of \cite{rat1} show, there is basically no relationship
between the algebraic complexity of a terminal singularity $0\in X$
and the topological complexity of its real points $X(\r)$.
Eventually I settled at a completely algebraic choice: $X$ is nice if
it has index 1 along $X(\r)$. There are two main reason for adopting
this definition:
\begin{enumerate}
\item Most complications of 3-dimensional birational geometry come
from the appearance of points of index $>1$. Hence this is likely to
be the right choice algebraically.
\item One of the first things I realized was that under this condition
there would be no flips. Indeed, flips need higher index singular
points to exist. If we have only index 1 points along $X(\r)$, then
all higher index points appear in conjugate pairs. A look at the
list of flips
\cite{KoMo92} shows that the singularities appearing along a flipping
curve are {\it always} asymmetrical.
\end{enumerate}
Thus our task is to get a list of all steps $f:Y\to X$ of the MMP over
$\r$ such that $Y$ has index 1 along $Y(\r)$. The case of
divisor--to--curve contraction is relatively easy. Most of the work is
devoted to studying the divisor--to--point contractions. Let $0\in
X(\r)$ be the point in question. The existence of $f$ is local in
the Euclidean topology. I will go through the classification (up to
real analytic equivalence) of 3-dimensional terminal singularities
over $\r$ and for each describe all possible
$f:Y\to X$.
There is one subtle point here: the condition of
$\q$-factoriality is not preserved under analytic equivalence. Thus
first we need to develop a notion of ``extremal contraction without
$\q$-factoriality".
\end{say}
\begin{defn}\label{elem.extr.defn}
Let $X$ be a normal variety over a field $K$ such that
$K_X$ is $\q$-Cartier. A proper birational morphism
$f:Y\to X$ is called an {\it elementary extraction} of
$X$ if
\begin{enumerate}
\item $Y$ is normal and $K_Y$ is $\q$-Cartier.
\item The exceptional set $\ex(f)$ contains a unique
$K$-irreducible divisor
$E$.
\item $-K_Y$ is $f$-ample.
\end{enumerate}
If we start with $Y$ and construct $f:Y\to X$ then $f$ is usually
called an {\it elementary contraction} of $Y$.
We can write $K_Y\equiv f^*K_X+a(E,X)E$ where $a(E,X)$ is the
discrepancy of $E$. Thus $-a(E,X)E$ is $f$-ample. An exceptional
divisor can never be relatively ample (or nef) (cf.
\cite[3.35]{KoMo98}), thus $a(E,X)>0$ and so $-E$ is
$f$-ample. This implies that
$\ex(f)=\supp E$.
$f(E)$ is also called the {\it center} of $f$ on $X$.
\end{defn}
A crucial property of elementary extractions is that they are
determined by their exceptional divisors:
\begin{prop}\label{gw.mats-mumf.lem} Let $X$ be a normal variety over
a field
$K$ such that
$K_X$ is $\q$-Cartier. Let $f_i:Y_i\to X$ be elementary extractions
with exceptional divisors $E_i\subset Y_i$ for $i=1,2$. Assume that
$E_1$ and $E_2$ correspond to each other under the birational map
$f_2^{-1}\circ f_1:Y_1\map Y_2$. Then $Y_1$ and $Y_2$ are isomorphic
(over
$X$).
\end{prop}
Proof. Let $\phi:f_2^{-1}\circ f_1:Y_1\map Y_2$ be the composition.
$\phi$ is birational, and $\ex(\phi)$, $\ex(\phi^{-1})$ have
codimension at least 2. Furthermore, $K_{Y_1}$ and
$K_{Y_2}=\phi_*(K_{Y_1})$ are relatively ample. Thus
$\phi$ is an isomorphism by an argument of
\cite[p.671]{Matsusaka-Mumford64}.
\qed
\medskip
In some sense this gives a way of enumerating all elementary
extractions of
$X$. We try to list all exceptional divisors over $X$ and
for each construct the corresponding unique elementary
extraction. Usually there are infinitely many elementary extractions
for a given $X$ and there does not seem to be an easy way to predict
for which divisors does the corresponding elementary extraction exist.
The next definition singles out a special class of
elementary extractions, by restricting the singularities allowed on
$Y$. The aim is to formalize a special case of the gateway method: we
assume that $Y$ is ``nice".
\begin{defn}\label{gw.g-e.def}
Let $X$ be a normal variety over a field $K$ such that
$K_X$ is $\q$-Cartier. A proper birational morphism
$f:Y\to X$ is called a {\it gateway--extraction} or {\it
g--extraction} if
\begin{enumerate}
\item $f$ is an elementary extraction with exceptional divisor
$E\subset Y$.
\item $Y$ has terminal singularities.
\item $K_X$ and $E$ are Cartier at the generic point of every
geometrically irreducible $K$-subvariety of
$\ex(f)$.
\end{enumerate}
\noindent In dimension three $Y$ has only isolated singularities,
hence (3) is equivalent to the apparently weaker condition:
\begin{enumerate}
\item[3'] $K_X$ and $E$ are Cartier at every $K$-point of
$\ex(f)$.
\end{enumerate}
If we start with $Y$ and construct $f:Y\to X$ then $f$ is usually
called a {\it g--contraction} of $Y$.
\end{defn}
The main technical aim of this article is to obtain a list of
g--extractions for threefolds with terminal singularities. The
project turns out to be feasible since the discrepancy
$a(E,X)$ is always quite small. I have no a priori proof of this, but
in every case the study of low discrepancy divisors leads to a
description of all g--extractions.
The relationship between low discrepancy divisors and g--extractions
rests on the
following easy observation:
\begin{prop}\label{gw.d-ineq.prop}
Let $X$ be a normal variety over a field $K$ such that
$K_X$ is $\q$-Cartier. Let
$f:Y\to X$ be a g--extraction with exceptional divisor
$F\subset Y$. Let $E$ be a geometrically irreducible
$K$-divisor over $X$ such that $\cent_XE\subset
\cent_XF$. Then
$$ a(E,X)\geq a(E,Y)+a(F,X).
$$
\end{prop}
Proof. Let $g:Z\to X$ be a proper birational morphism such that
$\cent_ZE$ is a divisor on $Z$. We may assume that the induced
rational map $h:Z\map Y$ is a morphism.
$h(E)$ is a geometrically irreducible $K$-subvariety of
$Y$ which is contained in
$\ex(f)$. Write
$$
\begin{array}{rcl} K_Z&\equiv& g^*K_X+a(E,X)E+(\mbox{other exceptional
divisors}),\\ K_Z&\equiv& h^*K_Y+a(E,Y)E+(\mbox{other exceptional
divisors}),\\ K_Y&\equiv& h^*K_X+a(F,Y)F, \qtq{and}\\ h^*F&\equiv&
cE+(\mbox{other exceptional divisors}),
\end{array}
$$ where $c>0$ since $h(E)\subset \ex(f)=\supp F$ and $c$ is an
integer by (\ref{gw.g-e.def}.3). Making the substitutions we obtain
that
$a(E,X)= a(E,Y)+c\cdot a(F,X)\geq a(E,Y)+a(F,X)$. \qed
\medskip
The same method also proves the following result:
\begin{prop}\label{gw.d-ineq.prop.cor}
Let $X$ be a normal variety over a field $K$ such that
$K_X$ is $\q$-Cartier. Let
$f:Y\to X$ be a morphism with exceptional divisor
$F=\cup F_i\subset Y$. Assume that $Y$ has terminal singularities and
$K_X$ and $F$ are Cartier at the generic point of every geometrically
irreducible $K$-subvariety of
$\ex(f)$.
Let $E$ be a geometrically irreducible
$K$-divisor over $X$ such that $\cent_XE\subset
\cup_i\cent_XF_i$. Then
$$ a(E,X)\geq a(E,Y)+\min_i\{a(F_i,X)\}.\qed
$$
\end{prop}
\begin{cor}\label{gw.discr1.cor}
Let $X$ be a normal variety over a field $K$ such that
$K_X$ is $\q$-Cartier. Let
$f:Y\to X$ be an elementary extraction with exceptional divisor
$E\subset Y$. Assume that $E$ is geometrically irreducible and
$a(E,X)\leq 1$.
Then either $f:Y\to X$ is a g--extraction, or $X$ has no
g--extractions whose center contains $f(E)$.
\end{cor}
Proof. Let $g:Z\to X$ be a g--extraction of $X$ whose center
contains $f(E)$. Let $F\subset Z$ be the exceptional divisor. Then
$a(E,X)\geq a(E,Z)+a(F,X)$. If
$a(E,Z)=0$ then
$\cent_ZE$ is a divisor which is contained in $F$. Since
$F$ is an irreducible divisor, $\cent_ZE=F$, hence $Y=Z$ by
(\ref{gw.mats-mumf.lem}). Otherwise $a(E,Y)\geq 1$ which would force
$a(F,X)\leq 0$. This contradicts (\ref{gw.g-e.def}.2).
\qed
\begin{rem} This corollary gives a very efficient way of finding all
g--extractions of a given $X$ in some cases. We have to find {\it
one}
geometrically irreducible divisor $E$ such that
$a(E,X)\leq 1$ and construct the corresponding elementary extraction
$f:Y\to X$. Then it is usually easy to determine the singularities of
$Y$.
\cite{Markushevich96} proved that if $0\in X$ is a terminal threefold
singularity which is not smooth, then there is a divisor $E$ over
$\bar K$ with
$\cent_XE=\{0\}$ and $a(E,X)\leq 1$. Thus there is always such an
irreducible $K$-divisior, but it may not be geometrically irreducible.
Still, in many cases we are able to apply (\ref{gw.discr1.cor})
directly.
In the remaining cases we show that there is always a geometrically
irreducible divisor $E$ with
$\cent_XE=\{0\}$ and $a(E,X)\leq 3$. This is still very useful, thanks
to the following:
\end{rem}
\begin{cor}\label{gw.discr2.cor}
Let $X$ be a normal variety over a field $K$ such that
$K_X$ is Cartier. Let
$f:Y\to X$ be an elementary extraction with exceptional divisor
$E\subset Y$. Assume that $E$ is geometrically irreducible. Let
$g:Z\to X$ be any g--extraction with exceptional divisor $F$ whose
center contains $f(E)$.
Then either $g=f$ or $a(F,X)\leq a(E,X)-1$.
\end{cor}
Proof. By (\ref{gw.d-ineq.prop}), $a(E,X)\geq a(E,Z)+a(F,X)$. If
$a(E,Z)=0$ then $E$ and
$F$ correspond to each other, hence $Y=Z$ by
(\ref{gw.mats-mumf.lem}). Otherwise $a(E,Y)\geq 1$, thus
$a(F,X)\leq a(E,X)-1$. \qed
\section{Small and Divisor--to--Curve Contractions}
In this section we look at those steps $f:X\to Y$ of the MMP over $\r$
which are either small contractions or contract a divisor to a curve.
The two cases can be treated together in the following setting:
\begin{notation}\label{sm.not}
Let $K$ be a field of characteristic 0. Let $X$ be a 3-fold over $K$
with terminal singularities and $f:X\to Y$ a proper birational
morphism over $K$ such that $-K_X$ is $f$-ample and $f_*\o_X=\o_Y$.
Let $0\in Y(K)$ be a closed point such that $\dim f^{-1}(0)=1$.
We will need that under these assumptions
$R^1f_*\o_X=R^1f_*\o_X(K_X)=0$ by the generalized
Grauert--Riemenschneider vanishing theorem (see, for instance,
\cite[8.8]{CKM88} or \cite[2.65]{KoMo98}).
\end{notation}
In keeping with the principles of the gateway method, we are interested
in the case when
$X$ has index 1 at all points of $X(K)$. The following theorem gives a
complete description of such contractions:
\begin{thm}\label{sm.thm}
Notation and assumptions as in (\ref{sm.not}). Assume in addition
that $X$ has index 1 at all points of $X(K)$. Then $Y$ is smooth at
$0$ and one can choose local (analytic or formal) coordinates
$(x,y,z)$ at $0\in Y$ such that $X$ is the blow up of the curve
$(z=g(x,y)=0)\subset Y$ for some $g\in K[[x,y]]$.
In particular, $f$ can not be small.
\end{thm}
This theorem has some very useful consequences for the MMP over $\r$:
\begin{cor}\label{sm.mmp.cor1} Starting with a projective variety
$X$ over $\r$, let
$$ X=X_0\map X_1\map \cdots\map X_{i}\stackrel{f_{i}}{\map} X_{i+1}
$$ be the beginning of an MMP over $\r$. Assume that $X_j$ has
index 1 at all points of $X_j(\r)$ for
$j\leq i$. Then the induced maps between the sets of real points
$$ X(\r)=X_0(\r)\to \cdots\to X_{i}(\r)\stackrel{f_{i}}{\to}
X_{i+1}(\r)
$$ are everywhere defined.
\end{cor}
Proof. The only steps of the MMP over $\r$ which are not everywhere
defined are the flips of small contractions (\ref{mmpt.cases.prop}).
By (\ref{sm.thm}) there are no flips in the sequence.\qed
\medskip
The topological behavior of divisor--to--curve contractions can also
be determined using (\ref{sm.thm}):
\begin{thm}\label{sm.top.thm} Let $X$ be a proper 3-fold over $\r$
with terminal singularities such that $X$ has index 1 at all points of
$X(\r)$ and $\overline{X(\r)}$ is a 3-manifold. Let $f:X\to Y$ be a
proper birational morphism over $\r$ such that $-K_X$ is $f$-ample and
$f_*\o_X=\o_Y$. Assume that $\dim f^{-1}(y)\leq 1$ for every $y\in
Y$. Then either
\begin{enumerate}
\item $f$ is $\r$-small, or
\item $\overline{X(\r)}$ contains a 1-sided torus or Klein bottle with
nonorientable neighborhood.
\end{enumerate}
\end{thm}
Proof. By (\ref{sm.thm}), there is a real curve $D\subset Y$ such
that
$Y$ is smooth along $D$ and $X=B_DY$ (at least in a neighborhood of
$Y(\r)$). Pick $0\in D(\r)$ and let $(z=g(x,y)=0)$ be a local
equation of
$D$. By (\ref{sm.ci.blowup}), either $D$ is smooth at $0$ or
$X$ has a unique singular point over $0$ with local equation
$st=g(x,y)$, which is equivalent to
$s^2-t^2-g(x,y)=0$. These are of type
$cA_{>1}^-$ or $cA_1$ in the classification of \cite{rat1}. If $g$
does not change sign on the $(x,y)$-plane then
$X(\r)\setminus f^{-1}(0)\to Y\setminus\{0\}$ is one--to--one near $0$,
hence
$f$ is $\r$-small near $0$. If $g$ does change sign on the
$(x,y)$-plane, then from \cite[sec. 4]{rat1} we see that (after a
coordinate change) $g=\pm(x^2+y^{2r+1})$ and
$X(\r)$ is a manifold near $f^{-1}(0)$. In particular, $D(\r)$ is the
disjoint union of some isolated points and some copies of $S^1$.
If $D(\r)$ is finite then
$f$ is $\r$-small. Otherwise $D(\r)$ has a connected component
$M\sim S^1$. Let $E\subset X$ be the exceptional divisor of
$f$. By explicit computation we see that
$E(\r)\to D(\r)$ is an $S^1$-bundle. Hence there is a unique connected
component $N\subset E(\r)$ such that $N$ is an $S^1$-bundle over
$M$. Thus
$N$ is either a torus or a Klein bottle. $N$ is 1-sided with
nonorientable neighborhood, since these hold locally for the blow up of
a smooth curve in a smooth 3-fold.\qed
\begin{exmp}\label{sm.ci.blowup}
Set $Y={\Bbb A}^3$ with
coordinates
$(x,y,z)$. Let $X$ be the blow up of the curve
$(z=g(x,y)=0)\subset Y$. Then $X$ has a unique singular point which is
given by an equation $st-g(x,y)=0$.
\end{exmp}
\begin{cor}\label{sm.mmp.cor2}
Starting with a projective variety
$X$ over $\r$, let
$$ X=X_0\map X_1\map \cdots\map X_{i}\stackrel{f_{i}}{\map} X_{i+1}
$$ be the beginning of an MMP over $\r$. Assume that
\begin{enumerate}
\item $X_j$ has index 1 at all points of $X_j(\r)$ for
$j\leq i$,
\item $\overline{X_j(\r)}$ is a PL-manifold for
$j\leq i$,
\item $\overline{X(\r)}$ satisfies the conditions (\ref{int.no.cond}).
\end{enumerate}
\noindent Then:
\begin{enumerate}
\setcounter{enumi}{3}
\item The induced maps between the sets of real points
$f_j: X_{j}(\r)\to X_{j+1}(\r)$ are everywhere defined for $j\leq
i$,
\item For every $j\leq i+1$, there is a finite set
$S_j\subset X_{j}(\r)$ such that
$\overline{X_{j}(\r)}\setminus S_j$ is homeomorphic to an open subset
of $\overline{X(\r)}$,
\item The smooth part of $\overline{X_{i+1}(\r)}$ also satisfies the
conditions (\ref{int.no.cond}).
\end{enumerate}
\end{cor}
Proof. The steps of an MMP are everywhere defined by
(\ref{sm.mmp.cor1}).
If $g:U\to V$ is any divisorial contraction over $\r$ then
$\ex(g^{-1})(\r)$ is finite unless $g$ is a divisor--to--curve
contraction which is not $\r$-small.
Let $f_j$ be the first divisor--to--curve contraction in the sequence
which is not $\r$-small. By the above remark, (5) holds for
$j$. By (\ref{sm.top.thm}),
$X_j(\r)$ contains a surface $F$ which is either a 1-sided torus or
Klein bottle with nonorientable neighborhood. We can move $F$ away from
any finitely many points, thus by (5)
$X(\r)$ also contains a 1-sided torus or Klein bottle with
nonorientable neighborhood. This is a contradiction. Hence among the
steps there is no divisor--to--curve contraction which is not
$\r$-small. This gives (5) and (6).\qed
\medskip
The proof of (\ref{sm.thm}) relies on two results:
\begin{prop}\cite[Thm. 4]{Cutkosky88} \label{sm.cut} (\ref{sm.thm})
holds if $K$ is algebraically closed and $C$ is irreducible.\qed
\end{prop}
\begin{lem}\label{sm.tree}(cf. \cite[1.14]{Mori88}) Let $f:X\to Y$
be a proper morphism and
$0\in Y$ a closed point such that $\dim f^{-1}(0)=1$. Set $\red
f^{-1}(0)=C=\cup C_i$.
\begin{enumerate}
\item If $R^1f_*\o_X=0$ then $C$ is a tree of smooth rational curves.
\item Let $D$ be a $\q$-Cartier Weil divisor on $X$ such that $D$ is
Cartier at all but finitely many points of $f^{-1}(0)$. Assume that
$(D\cdot C_i)<0$ for every $i$ and $R^1f_*\o_X(D)=0$. Then
$-1\leq (D\cdot C_i)<0$ for every $i$ and
$D$ is not Cartier at the singular points of $C$.
\end{enumerate}
\end{lem}
Proof. By replacing $Y$ with a neighborhood of $0$, we may assume that
every fiber of $f$ has dimension at most 1.
Let $G$ be a sheaf on $X$ such that $R^1f_*G=0$ and $Q=G/F$ a quotient
of
$G$ whose support is in $f^{-1}(0)$. We get an exact sequence
$$ R^1f_*G\to R^1f_*Q\to R^2f_*F.
$$ The left hand side is zero by assumption and the right hand side is
zero since every fiber of $f$ has dimension at most 1. Thus
$R^1f_*Q=0$.
Applying this with $G=\o_X$ and $Q=\o_C$ we conclude that
$H^1(C,\o_C)=R^1f_*\o_C=0$, hence $C$ is a tree of smooth rational
curves. This proves (1).
In order to see the second part, we may assume that the residue field
of
$0$ is algebraically closed. Then a point $P\in C$ is singular iff
there are at least 2 irreducible components through $P$.
$\o_X(D)\otimes \o_{C_i}$ is a rank one locally free sheaf except
possibly at the ponts where $D$ is not Cartier. Let $L_i$ denote its
quotient by the torsion subsheaf. Then $L_i$ is an invertible sheaf
and we have a surjection
$\o_X(D)\to L_i$. Applying $R^1f_*$ we obtain as above that
$H^1(C_i,L_i)=0$. Thus $\deg L_i\geq -1$.
On the other hand, for every
$m>0$ we have an injection
$$ L_i^m\cong (\o_X(D)^{\otimes m}\otimes \o_{C_i})/(\mbox{torsion})
\DOTSB\lhook\joinrel\rightarrow (\o_X(mD)\otimes \o_{C_i})/(\mbox{torsion}).
$$ If $mD$ is Cartier then the right hand side has negative degree,
thus
$L_i^m$ has negative degree. Therefore $\deg L_i=-1$ for every $i$.
Furthermore, $m(D\cdot C_i)\geq m\deg L_i=-m$, so
$(D\cdot C_i)\geq -1$.
Set $M:=(\o_X(D)\otimes \o_{C})/)\mbox{torsion})$.
$H^1(C,M)=0$ as above. We have an exact sequence
$$ 0\to M\to \sum L_i\to Q\to 0,
$$ where $Q$ is supported at the singular points of $C$. Taking
cohomologies, we conclude that $H^0(C,Q)=0$ thus $Q=0$.
If $D$ is Cartier at a singular point $P$ of $C$ then $M$ is locally
free at $P$ and $M\to \sum L_i$ can not be surjective at
$P$ (it is not even surjective when tensored with the residue field at
$P$).\qed
\begin{cor}\label{sm.tree2}
Notation and assumptions as in (\ref{sm.not}). Then
$C$ is a tree of smooth rational curves and $K_X$ is not Cartier at
the singular points of $C$.
\end{cor}
Proof. Apply (\ref{sm.tree}) with $D=K_X$.
$R^1f_*\o_X=R^1f_*\o_X(K_X)=0$ by (\ref{sm.not}). \qed
\begin{say}[Proof of (\ref{sm.thm})]
The assumptions and conclusions are local near
$0$, thus we may replace $Y$ by a suitable analytic or formal
neighborhood of
$0$.
By (\ref{sm.tree2}), $C$ is a connected tree of smooth rational
curves. $\gal(\bar K/K)$ acts on $C$, thus $C$ either has a
singular $K$-point or a geometrically irreducible component defined
over
$K$.
If $P\in C$ is a singular point then $K_X$ is not Cartier at $P$ by
(\ref{sm.tree2}), but if $P\in X(K)$ then $K_X$ is Cartier at $P$ by
assumption. Thus $C$ can not have a singular $K$-point.
Let $C_0\subset C$ be a geometrically irreducible component defined
over
$K$. Let $H\subset X$ be a divisor defined over $K$ which intersects
all irreducible components of $C\setminus C_0$ transversally but is
disjoint from $C_0$. A large multiple of $H$ defines a morphism $X\to
Y'\to Y$ such that
$C_0$ is contracted to a point in $Y'$. If (\ref{sm.thm}) holds for
$X\to Y'$, then $X$ has index one along $C_0$. By (\ref{sm.tree2}) this
implies that $C_0$ is a connected component of
$C$. On the other hand, $C$ is connected since $f_*\o_X=\o_Y$. Thus
$C=C_0$ and $Y'=Y$.
Therefore it is sufficient to prove (\ref{sm.thm}) under the additional
assumption that $C$ is geometrically irreducible.
First we show that (\ref{sm.thm}) holds if $X$ has only index 1 points
along
$C$. By (\ref{sm.cut}), $Y$ is smooth at $0$ and $X=B_DY$ where
$D\subset Y$ is a curve of embedding dimension 2.
$D$ is the image of the exceptional divisor of $f$, hence $D$ is
defined over $K$. Since
$D$ has embedding dimension 2, its ideal is of the form $(z,g(x,y))$.
Finally we show that $X$ has only index 1 points along $C$. We start
with the case when $K=\r$. Let $P_1,\bar P_1, \dots, P_k,\bar P_k$ be
all the conjugate pairs of points of index $>1$. At each $P_i$ pick a
local member $D_i\in |K_X|$ such that $C\cap D_i=P_i$. (In order to
do this, we may need to replace $X_{\bar K}$ with a smaller analytic
neighborhood of $C$.) Let $\bar D_i$ be the conjugates. Set
$D=\sum D_i$. Let $m>1$ be the smallest natural number such that $mD$
is Cartier.
$D-\bar D$ is a Weil divisor and $\o_X(m(D-\bar D))\cong \o_X$ since
the Picard group of a neighborhood of $C$ is isomorphic to
$H^2(C(\c),\z)$ (cf. \cite[4.13]{KoMo98}).
Corresponding to $1\in H^0(X,\o_X)$ we obtain an $m$-sheeted cyclic
cover $\pi: \tilde X\to X$ which is unramified outside the points of
index
$>1$. Thus $K_{\tilde X}=\pi^*K_X$ and $\tilde X$ has index 1 terminal
singularities. Let $\tilde f:\tilde X\to \tilde Y$ be the Stein
factorization of $\tilde X\to Y$. By the already discussed index 1
case,
$\tilde Y$ is smooth and one can choose local analytic coordinates
$(x,y,z)$ at $0\in \tilde Y$ such that $\tilde X$ is the blow up of the
curve
$(z=g(x,y)=0)\subset \tilde Y$.
The group of $m^{th}$ roots of unity (denoted by $\z_m$) acts on
$\tilde f:\tilde X\to \tilde Y$ and the quotient is $ f: X\to Y$. If
$\mult_0g\geq 2$ then $\tilde X$ has a unique singular point
(\ref{sm.ci.blowup}), which is necessarily fixed by the
$\z_m$-action. Thus $X$ would have a unique point (of index $m$)
which is the quotient of a singular point. On the other hand, the
index $>1$ singularities of
$X$ come in conjugate pairs. Therefore
$\tilde X$ is smooth and $\tilde f:\tilde X\to \tilde Y$ is the blow up
of a smooth curve $(z=y=0)\subset \tilde Y$.
We can choose local coordinates $(x,y,z)$ on $\tilde Y$ such that the
action is
$$ (x,y,z)\mapsto (\epsilon^a x,\epsilon^by, \epsilon^cz)
$$ where $\epsilon$ is a primitive $m^{th}$ root of unity. The
corresponding action on $\tilde X$ has two fixed points (or a fixed
curve) and the corresponding quotients are
$$
\c^3/{\textstyle \frac1{m}}(a,b-c,c)\qtq{and}
\c^3/{\textstyle \frac1{m}}(a,b,c-b).
$$ These are both of type $cA_0/n$ on the list (\ref{mmp.ts.thm}). A
simple checking shows that both of these can not be simultaneously
terminal.
If $K$ is arbitrary, we can still proceed as above if we can find local
divisors $D_i\in |K_X|$ at the index $>1$ points such that $(C\cdot
\sum D_i)=0$. Finding the $D_i$ needs a little case by case analysis,
and sometimes it can be done only after first taking an auxiliary
cover. It is probably easier to observe that there can be at most 2
points of index $>1$ along $C$ (see, for instance,
\cite[14.5.5]{CKM88}), thus in fact the only case we need to handle
is when there is precisely one pair of conjugate points of index $>1$.
\qed
\end{say}
\section{Proof of the Main Theorems}
The determination of all divisor--to--point g--extractions is rather
technical and lengthy. In this section I state
a summary of the list of all g--extractions, and then use it to prove
the main theorems stated in the introduction. The proofs of
(\ref{ge.gwsing.thm}) and of (\ref{ge.gwextr.thm}) are given in
sections 9--11.
\begin{notation}\label{7.notation}
Let $g(x_1,\dots,x_m)$ be a polynomial or power series and let $M$ be
a monomial in the $x_i$. $M\in g$
means that $M$ appears in $g$ with nonzero coefficient.
\end{notation}
\begin{thm}\label{ge.gwsing.thm}
Let $0\in X$ be a three dimensional terminal singularity over
$\r$. If $X$ has a g--extraction then $0\in X$ is one of the
following (up to real analytic equivalence near $0$).
\begin{enumerate}
\item ($cA_0$) Smooth point.
\item ($cA_0/2$) Quotient of a smooth point by the
$\z_2$-action
$(x,y,z)\mapsto (-x,-y,-z)$.
\item ($cA_{>0}^+$) Given as
$(x^2+y^2+g_{\geq m}(z,t)=0)$, where $g_m(z,t)\neq 0$ and
$m\geq 2$.
\item ($cA_{>0}^+/2$) Given as
$(x^2+y^2+g_{\geq m}(z,t)=0)/\z_2$, where $g_m(z,t)\neq 0$, $m\geq 2$
and the
$\z_2$-action is $(x,y,z,t)\mapsto (-x,-y,-z,t)$.
Furthermore, one of the following two conditions has to be satisfied:
\begin{enumerate}
\item $m$ is divisible by $4$ and $z^m,t^m\in g$, or
\item\label{7.bad.index.2} $m$ is odd.
\end{enumerate}
\item ($cE_6$) Given as
$(x^2+y^3+(z^2+t^2)^2+ yg_{\geq 4}(z,t)+h_{\geq 6}(z,t)=0)$.
\end{enumerate}
\end{thm}
\begin{complement} All the singularities on the above list have
g--extractions, with the possible exception of types
(\ref{7.bad.index.2}). These singularities have not been analyzed
completely.
\end{complement}
\begin{thm}\label{ge.gwextr.thm}
Let $0\in X$ be a three dimensional terminal singularity over
$\r$ and $f:Y\to X$ a g--extraction with exceptional divisor
$E=\red f^{-1}(0)$. If
$E$ is geometrically irreducible then $f:Y\to X$ is on the following
list (up to real analytic equivalence near $0$).
\begin{enumerate}
\item ($cA_0$, point blow up) $B_0{\Bbb A}^3\to {\Bbb A}^3$,
$E\cong
\p^2$.
\item ($cA_0$, curve blow up) $B_C{\Bbb A}^3\to {\Bbb A}^3$ where
$C\subset {\Bbb A}^3$ is a geometrically irreducible, real and
locally planar curve.
\item ($cA_0/2$) $B_0{\Bbb A}^3/\z_2\to {\Bbb A}^3/\z_2$, where the
$\z_2$-action on ${\Bbb A}^3$ is
$(x,y,z)\mapsto (-x,-y,-z)$.
$E\cong \p^2$.
\noindent Furthermore, in this case there are no other g--extractions
whose center contains the origin.
\item ($cA_{>0}^+$, $\mult_0g$ even)
$X=(x^2+y^2+g_{\geq 2m}(z,t)=0)$ where $g_{2m}(z,t)\neq 0$ and $m\geq
1$.
$Y=B_{(m,m,1,1)}X$
and
$E=(x^2+y^2+g_{2m}(z,t)=0)\subset \p^3(m,m,1,1)$.
\noindent Furthermore, in this case there are no other g--extractions
whose center contains the origin.
\item ($cA_{>0}^+$, $\mult_0g$ odd)
$X=(x^2+y^2+g_{\geq 2m+1}(z,t)=0)$ where $g_{2m+1}(z,t)\neq 0$ and
$m\geq 1$. This case occurs only if there is a linear change of the
$(z,t)$-coordinates such that
$t^{2m+1}\in g$ and
$z^it^j\not\in g$ for $i+2j< 4m+2$. In this coordinate system,
$Y=B_{(2m+1,2m+1,1,2)}X$ and
$E=(x^2+y^2+g_{2m+1}(z,t)=0)\subset \p^3(2m+1,2m+1,1,2)$.
\item ($cA_{>0}^+/2$, $\mult_0g$ even)
$X=(x^2+y^2+g_{\geq 2m}(z,t)=0)/\z_2$, where $g_{2m}(z,t)\neq 0$,
$m\geq 1$ and the
$\z_2$-action is $(x,y,z,t)\mapsto (-x,-y,-z,t)$. This case occurs
only if $m$ is even and $z^{2m},t^{2m}\in g$. Then
$Y=B_{(m,m,1,1)}\tilde X/\z_2$ and $E=\tilde E/\z_2$, where
\noindent
$\tilde X=(x^2+y^2+g_{\geq 2m}(z,t)=0)$ and
\noindent
$\tilde E=(x^2+y^2+g_{2m}(z,t)=0)\subset \p^3(m,m,1,1)$.
Furthermore, in this case there are no other g--extractions whose
center contains the origin.
\item ($cA_{>0}^+/2$, $\mult_0g$ odd)
$X=(x^2+y^2+g_{\geq 2m+1}(z,t)=0)/\z_2$ where $g_{2m+1}(z,t)\neq 0$,
$m\geq 1$ and the
$\z_2$-action is $(x,y,z,t)\mapsto (-x,-y,-z,t)$. In this case I do
not have a complete list.
\end{enumerate}
\end{thm}
\begin{cor}\label{ge.Enorm.cor}
Let $0\in X$ be a three dimensional terminal singularity over
$\r$ and $f:Y\to X$ a g--extraction with exceptional divisor
$E=\red f^{-1}(0)$. Assume that we are not in case
(\ref{ge.gwextr.thm}.7). If $E$ is geometrically irreducible then
$E$ is normal.
\end{cor}
Proof. Equations for $E$ are given in (\ref{ge.gwextr.thm}).
$E\cong
\p^2$ in the first two cases. In the remaining cases $E$ is (or is
the quotient of) a surface of the form
$$ F:=(x^2+y^2+p(z,t)=0)\subset \p^3(r,r,1,s).
$$
All the singularities of $F$ are contained in the
$(x=y=0)$ line. Thus we get only finitely many singularities if $p$ is
not identically zero, which is always the case in
(\ref{ge.gwextr.thm}).\qed
\begin{say}[Proof of (\ref{ge.gwsing.thm}) and (\ref{ge.gwextr.thm})
$\Rightarrow$ (\ref{int.mmp.sings}) and (\ref{int.mmp.steps})]{\ }
Under the additional assumption that $X(\r)$ satisfies the conditions
(\ref{int.no.cond}), we need to exclude the cases
(\ref{ge.gwsing.thm}.2), (\ref{ge.gwsing.thm}.4),
(\ref{ge.gwextr.thm}.2) and in (\ref{ge.gwsing.thm}.3) we need to show
that $g$ is not everywhere negative in a punctured neighborhood of
$0$.
Starting with $X$, let us run the MMP over $\r$. We get a sequence
$$ X=X_0\map X_1\map \cdots\map X_{i}\stackrel{f_{i}}{\map} X_{i+1}.
$$ Assume by induction that (\ref{int.mmp.sings}) holds for $X_j$
for $j\leq i$ and (\ref{int.mmp.steps}) holds for
$f_j:X_j\map X_{j+1}$ for $j\leq i-1$.
We need to show that (\ref{int.mmp.sings}) holds for $X_{i+1}$ and
(\ref{int.mmp.steps}) holds for
$f_i:X_i\map X_{i+1}$.
By (\ref{sm.mmp.cor2}) $f_j:X_j(\r)\to X_{j+1}(\r)$ are everywhere
defined for $j\leq i-1$ and $\overline{X_{i}(\r)}$ does not contain
a 1-sided torus or Klein bottle with nonorientable neighborhood.
Furthermore, by (\ref{sm.top.thm}), $f_i$ is either $\r$-small or a
divisor--to--point contraction.
By induction
$X_i$ has index 1 along $X_i(\r)$, thus $f_i$ is a g--extraction,
hence it is one of the cases listed in (\ref{ge.gwsing.thm}). We
excluded several cases one at a time.
\medskip {\it Excluding (\ref{ge.gwsing.thm}.2).} By
(\ref{ge.gwextr.thm}.3)
$X_i\to X_{i+1}$ is the blow up of the singular point $\a^3/\z_2$.
This gives a 1--sided $\r\p^2$ in $X_i(\r)$. This is a contradiction
by (\ref{sm.mmp.cor2}).
\medskip {\it Excluding (\ref{ge.gwextr.thm}.2).} In this case
$X_i$ contains a 1-sided torus or Klein bottle with nonorientable
neighborhood by (\ref{sm.top.thm}). This is again a contradiction by
(\ref{sm.mmp.cor2}).
\medskip {\it Excluding (\ref{ge.gwsing.thm}.3) with $g<0$.} That is,
we consider the case when $0\in X_{i+1}$ is of the form
$(x^2+y^2+g_{\geq m}(z,t)=0)$ and $g$ is everywhere negative in a
punctured neighborhood of $0$. (These are called $cA^+_{>0}(0,-)$ in
\cite{rat1}.) By \cite[4.4]{rat1} the link of $0\in X_{i+1}(\r)$ is a
torus. This gives only a 2-sided torus in $X(\r)$ which is allowed. I
proceed to prove, however, that we still get a 1-sided torus in
$X(\r)$ coming from the exceptional divisor of
$X(\r)\to X_{i+1}(\r)$. This contradicts (\ref{int.no.cond}.2).
$m=\mult_0g(z,t)$ is necessarily even, say
$m=2r$. By (\ref{ge.gwextr.thm}.4) the only g--extraction is the
$(r,r,1,1)$-blow up. Thus $X_i\to X_{i+1}$ is this blow up. We
distinguish two cases:
General case: $g_{2r}(z,t)$ is
negative on $\r^2\setminus\{0\}$. The exceptional divisor $E$ of the
above g--extraction is the weighted hypersurface
$$ E=(x^2+ y^2+g_{2r}(z,t)=0)\subset \p(r,r,1,1).
$$ Its canonical divisor is $K_E=\o_E(-2)$, thus $E$ is orientable.
The projection $(x:y:z:t)\mapsto (z:t)$ exhibits $E$ as an
$S^1$-bundle over $\r\p^1$, thus
$E\sim S^1\times S^1$. $L(0\in X_{i+1}(\r))$ is connected, thus
$E(\r)\subset X_i(\r)$ is a 1-sided torus. By (\ref{sm.mmp.cor2}),
$X(\r)$ also contains a 1-sided torus, a contradiction.
Special case: $g_{2r}(z,t)$ is not
negative on $\r^2\setminus\{0\}$.
$g_{2r}(z,t)$ is the leading term of
$g_{\geq m}(z,t)$, which is
negative on $\r^2\setminus\{0\}$. Thus
$g_{2r}(z,t)$ is nonpositive
on $\r^2\setminus\{0\}$.
The $t$-chart on $X_i\cong B_{(r,r,1,1)}X_{i+1}$ is
$x_1^2+ y_1^2+ t_1^{-2m}g(z_1t_1,t_1)$. Set
$g'(z_1,t_1):=t_1^{-2m}g(z_1t_1,t_1)$. Then
$g'(z_1,t_1)$ is strictly negative outside the
$z_1$-axis, and is not identically zero on the
$z_1$-axis. Thus $g'(z_1,t_1)$ is everywhere nonpositive with only
finitely many zeros. Thus at each zero of
$g'(z_1,t_1)$, $X_i$ has a singular point of type
$cA^+_{>0}(0,-)$. This contradicts the inductive assumption.
Thus $X_{i+1}$ does not contain any
$cA^+_{>0}(0,-)$ type points.
\medskip {\it Excluding (\ref{ge.gwsing.thm}.4).}
\cite[5.9]{rat1} shows that the link of a singularity of type
$cA^+/2$ contains a connected component homeomorphic to
$\r\p^2$, except when we can write the singularity as
$(x^2+y^2+g_{\geq m}(z,t)=0)/\z_2$, where the
$\z_2$-action is $(x,y,z,t)\mapsto (-x,-y,-z,t)$ and $g$ is
everywhere negative in a punctured neighborhood of $0$.
If the link of a singularity of $X_{i+1}(\r)$
contains a connected component homeomorphic to
$\r\p^2$, then $X_i(\r)$ contains a 2-sided
$\r\p^2$. Hence by (\ref{sm.mmp.cor2}),
$X(\r)$ also contains a 2-sided
$\r\p^2$ which is excluded by (\ref{int.no.cond}.1).
Thus we are reduced to the case
$(x^2+y^2+g_{\geq m}(z,t)=0)/\z_2$, where the
$\z_2$-action is $(x,y,z,t)\mapsto (-x,-y,-z,t)$ and $g$ is
everywhere negative in a punctured neighborhood of $0$. As in the
previous case, $m=\mult_0g(z,t)$ is necessarily even, say
$m=2r$ and by (\ref{ge.gwextr.thm}.6) the only g--extraction is the
$(r,r,1,1)$-blow up. We proceed to prove that the exceptional
divisor of
$X_i(\r)\to X_{i+1}(\r)$ contains a 1-sided Klein bottle with
nonorientable neighborhood, contradicting (\ref{int.no.cond}.3). We
again distinguish two cases:
General case: $g_{2r}(z,t)$ is
negative on $\r^2\setminus\{0\}$. The exceptional divisor $E$ is the
$\z_2$ quotient of the weighted hypersurface
$$
\tilde E=(x^2+ y^2+g_{2r}(z,t)=0)\subset \p(r,r,1,1).
$$ We already determined that $\tilde E(\r)$ is a torus and
a choice of orientation is given by
$(dy\wedge dz)/x$. The $\z_2$-action sends this to
$(d(-y)\wedge d(-z))/(-x)=-(dy\wedge dz)/x$, hence
$\tilde E(\r)/\z_2$ is not orientable. We conclude that one of the
connected components of $E(\r)$ is a Klein bottle. (There may be
other connected components.) The Klein bottle is 1-sided since
$\tilde E(\r)$ is 1-sided. The regular
neighborhood is nonorientable since its boundary, the link of $0\in
X(\r)$, is again a Klein bottle.
Special case: $g_{2r}(z,t)$ is not negative on $\r^2\setminus\{0\}$.
The same computation as above shows that this leads to a $cA^+/2$
point of the same type that we started with on
$X_i$, which contradicts the inductive assumption.
Thus we conclude that $X_{i+1}$ does not contain any
$cA^+/2$ type points.\qed
\end{say}
\begin{say}[Proof of (\ref{int.orient.thm}) and
(\ref{int.nonorient.thm})]{\ }
We follow the steps of an MMP over $\r$, using (\ref{int.mmp.steps}).
$f_i:X_i(\r)\to X_{i+1}(\r)$ is a homeomorphism in cases
(\ref{int.mmp.steps}.1--2) while (\ref{int.mmp.steps}.3) gives a
connected sum with $\r\p^3$.
In the cases (\ref{int.mmp.steps}.4) the exceptional divisor is normal
by (\ref{ge.Enorm.cor}), hence we get various cases of
(\ref{int.orient.thm}) by (\ref{top.alg.normsurfup.lem}). \qed
\end{say}
\begin{exmp}\label{ge.2extr.exmp} Consider the singularity
$X:=(x^2+y^2+z^{2m+1}+t^{4m+2}=0)$. The
$(2m+1,2m+1,2,1)$ blow up $X_1\to X$ is a g--extraction which is
smooth along the $\r$-points.
The $(m,m,1,1)$-blow up is another g--extraction whith one singular
point
$(x_1^2+y_1^2+z_1^{2m+1}t_1+t_1^{2m+2}=0)$ on the
$t$-chart. After the $(m+1,m+1,1,1)$-blow up we obtain a variety
$X_2\to X$ which is smooth along its $\r$-points.
These two resolutions are indeed quite different. Using the methods of
section 5, we see that
$X_1(\r)\sim X(\r)\ \#\ \r\p^3$ and
$X_2(\r)\sim X(\r)\ \#\ S^1\times S^2$.
\end{exmp}
\section{$cAx$ and $cD$-type Points}
In this section we begin to classify g--extractions (\ref{gw.g-e.def})
of terminal singularities over any field. The classification of
3--fold terminal singularities over nonclosed fields is done in
\cite{rat1}. The results are summarized in (\ref{mmp.ts.thm}). We work
through the list of the singularities. In most cases it is easy to see
that there are no g--extractions. This is done by exhibiting an
elementary extraction (\ref{elem.extr.defn}) which is not a
g--extraction. If the discrepancy of the exceptional divisor is
$\leq 1$ then there are no g--extractions by (\ref{gw.discr1.cor}).
In this section we deal with the cases $cAx/2, cAx/4, cD, cD/2, cD/3$.
Among terminal singularities these are somewhat esoteric but the
proofs work well for them: in each case (\ref{gw.discr1.cor}) applies.
The remaining terminal singularities are considered in the next 2
sections. In some cases much more complicated arguments are needed to
classify all g--extractions.
\begin{defn}\label{8.wbup}[Weighted blow-ups]{\ }
Let $x_1,\dots,x_n$ be coordinates on $\a^n$. The usual blow up of
the origin is patched together from affine charts with morphisms of
the form
$$ x_j=x'_jx'_i\qtq{if} j\neq i\qtq{and} x_i=x'_i.
$$ I refer to this as the {\it $x_i$-chart}.
Let $a_1,\dots,a_n$ be a sequence of positive integers. For every
$1\leq i\leq n$ we can define a morphism $\Pi_i:\a^n\to \a^n$ by
$$ x_j=x'_j(x'_i)^{a_j}\qtq{if} j\neq i\qtq{and} x_i=(x'_i)^{a_i}.
$$ This morphism is birational iff $a_i=1$ and has degree $a_i$ in
general. One can easily notice that $\Pi_i$ is invariant under the
action
$$
\a^n(x'_1,\dots,x'_n)/\textstyle{\frac1{a_i}}
(-a_1,\dots,-a_{i-1},1,-a_{i+1},\dots,-a_n)
$$ and it descend to a birational morphism $\pi_i$
$$
\Pi_i:\a^n(x'_1,\dots,x'_n)\to \a^n(x'_1,\dots,x'_n)/\z_{a_i}
\stackrel{\pi_i}{\longrightarrow} \a^n(x_1,\dots,x_n).
$$ Furthermore, these charts patch together to give a projective
morphism
$$
\pi:B_{(a_1,\dots,a_n)}\a^n\to \a^n.
$$ This is called the {\it weighted blow up} of $\a^n$ with weights
$a_1,\dots,a_n$.
\end{defn}
\begin{notation}\label{8.notation}
In the proofs in sections 9--11 I use the following conventions.
Firts I state the name of the singularity $X$ from (\ref{mmp.ts.thm})
and possibly some other restrictions. Then I write down the normal
form of the equation $X=F(x,y,z,t)/\frac1{r}(b_x,b_y,b_z,b_t)$. Any
restrictions on $F$ are explained in detail here.
Then I specify the weights $(a_x,a_y,a_z,a_t)$ for a weighted blow up
and write down the equation of the birational transform of $X$ on one
of the charts on the weighted blow up. Before taking quotients, this
has the form
$t_1^{-m}F(x_1t_1^{a_x}, y_1t_1^{a_y}, z_1t_1^{a_z}, t_1^{a_t})$ if I
use the
$t$-chart. This is denoted by $B\tilde X$.
I need to take quotient by 2 actions. First is the
$\frac1{a_t}(-a_x,-a_y,-a_z,1)$-action coming from the weighted blow
up. Second, the $\frac1{r}(b_x,b_y,b_z,b_t)$-action needs to be lifted
to the $(x_1,y_1,z_1,t_1)$-space. In some cases this lifts as a
$\z_r$-action but in other cases the actions combine into a
$\z_{(ra_t)}$-action. The quotient of $B\tilde X$ by these 2 actions
is a chart on the weighted blow up of $X$; it
is denoted by $BX$.
All these can be done in 4 different charts. I chose the chart where
the singularities are most visible or the discrepancy computation is
the clearest.
Finally I compute the exceptional divisor of the blow up, the
singularities of $BX$ and the discrepancy of the exceptional divisor.
\end{notation}
\begin{say}[$cAx/2$]
{\ } \newline Normal form: $ax^2+by^2+g_{\geq 4}(z,t)/{\textstyle
\frac12(1,0,1,1)}$, where $ab\neq 0$.
Weights for blow-up: (1,1,1,1)
$t$-chart: $ax_1^2+by_1^2+t_1^{-2}g_{\geq 4}(z_1t_1,t_1)/{\textstyle
\frac12(0,1,0,1)}$
Exceptional divisor: $(t_1=ax_1^2+by_1^2=0)$. Over $\bar K$ this is
reducible and the two irreducible components are
$(t_1=\sqrt{a}x_1\pm \sqrt{-b}y_1=0)$. The $\z_2$-action interchanges
these two, so on the quotient we get a geometrically irreducible
exceptional divisor.
Singularity: The $\z_2$-action has a fixed curve on
$B\tilde X$: the intersection with the $(x_1=z_1=0)$-plane. Thus we
get a curve of nonterminal singularities on $BX$.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= t_1\frac{dy_1\wedge
dz_1\wedge dt_1}{x_1}$, so $a(E,X)=1$.
\end{say}
\begin{say}[$cAx/4$]
{\ } \newline Normal form: $ax^2+by^2+g_{\geq 2}(z,t)/{\textstyle
\frac14(1,3,1,2)}$, where $ab\neq 0$ and $g_2(0,1)=0$ for weight
reasons.
Weights for blow-up: (1,1,1,1)
$t$-chart: $ax_1^2+by_1^2+t_1^{-2}g_{\geq 2}(z_1t_1,t_1)/{\textstyle
\frac14(3,1,3,2)}$
Exceptional divisor: $\tilde E:=(t_1=ax_1^2+by_1^2+g_2(1,0)z_1^2=0)$.
$\tilde E$ is geometrically irreducible if $g_2(1,0)\neq 0$. If
$g_2(1,0)= 0$, then $\tilde E$ is reducible over $\bar K$, and the
two irreducible components are
$(t_1=\sqrt{a}x_1\pm \sqrt{-b}y_1=0)$. The $\z_4$-action interchanges
these two, so on the quotient we get a geometrically irreducible
exceptional divisor $E$.
Singularity: The origin is on $B\tilde X$ since
$g_2(0,1)=0$ and it is a fixed point. We get an index 4 point on $BX$
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= t_1\frac{dy_1\wedge
dz_1\wedge dt_1}{x_1}$, so $a(E,X)=1$.
\end{say}
\begin{say}[$cD_4$ main series]
{\ } \newline Normal form: $x^2+f_{\geq 3}(y,z,t)$, where we assume
that $f_3(y,z,t)$ is irreducible over $\bar K$.
Weights for blow-up: (2,1,1,1)
$x$-chart: $x_1+x_1^{-3}f_{\geq 3}(y_1x_1, z_1x_1,t_1x_1)/{\textstyle
\frac12(1,1,1,1)}$
Exceptional divisor: $\tilde E:=(x_1=f_3(y_1,z_1,t_1)=0)$.
$\tilde E$ is geometrically irreducible by our assumption.
Singularity: The origin is a fixed point on $B\tilde X$, hence we get
an index 2 point on $BX$
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= x_1\cdot dy_1\wedge
dz_1\wedge dt_1$, so $a(E,X)=1$.
\end{say}
\begin{say}[$cD_4/2$ main series]
{\ } \newline Normal form: $x^2+f_{\geq 3}(y,z,t)/{\textstyle
\frac12(1,1,0,1)}$, where we assume that $f_3(y,z,t)$ is irreducible
over $\bar K$. However, for weight reasons
$z|f_3(y,z,t)$, so this can not happen.
\end{say}
\begin{say}[$cD/3$]
{\ } \newline Normal form: $x^2+f_{\geq 3}(y,z,t)/{\textstyle
\frac13(0,1,1,2)}$, where $f_3(0,0,t)\neq 0$. Since this is not a
$cE$ point and for weight reasons, also $f_3(y,z,0)\neq 0$. We can
write $f_3=t^3+f_3(y,z,0)$.
Weights for blow-up: (2,1,1,1)
$x$-chart: $x_1+x_1^{-3}f_{\geq 3}(y_1x_1, z_1x_1,t_1x_1)/{\textstyle
\frac12(1,1,1,1)}$, and then take the $\z_3$-action.
Lifting of the $\z_3$-action: It lifts to
$\frac16(3,5,5,1)$.
Exceptional divisor: $\tilde E:=(x_1=f_3(y_1,z_1,t_1)=0)$.
$\tilde E$ is geometrically irreducible if $f_3(y,z,0)$ is not a
cube. If $f_3(y,z,0)=-L(y,z)^3$ over
$\bar K$, then $\tilde E$ has three geometrically irreducible
components
$(x_1=t_1-\eta L(y_1,z_1)=0)$ where $\eta^3=1$. The
$\z_6$-action permutes these, so on $BX$ we get a geometrically
irreducible exceptional divisor.
Singularity: The origin is a $\z_6$-fixed point which has multiplicity
1 on $B\tilde X$. $BX$ has a terminal quotient singularity of index
6.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= x_1\cdot dy_1\wedge
dz_1\wedge dt_1$, so $a(E,X)=1$.
\end{say}
\begin{say}[$cD_{>4}$ and special $cD_4$]
{\ } \newline Normal form: $x^2+Q_2(y,z,t)z+g_{\geq 4}(y,z,t)$, where
$Q_2(y,0,t)\neq 0$. In the $cD_{>4}$ we always have this form (with
$Q_2(y,z,t)=y^2$). In the
$cD_4$-case we can achieve this form iff $f_3(y,z,t)$ has a simple
linear factor over $K$.
Weights for blow-up: (2,1,2,1)
$z$-chart: $x_1^2+Q_2(y_1,z_1,t_1)+z_1^{-4}g_{\geq 4}(y_1z_1,
z_1^2,t_1z_1)/{\textstyle
\frac12(0,1,1,1)}$
Exceptional divisor: $\tilde
E:=(z_1=x_1^2+Q_2(y_1,0,t_1)+g_4(y_1,0,t_1)=0)$. $\tilde E$ is
geometrically irreducible iff $Q_2(y_1,0,t_1)$ is not a square over
$\bar K$ or
$g_4(y_1,0,t_1)\neq 0$. If
$Q_2(y_1,0,t_1)=-L_1(y_1,t_1)^2$ (over $\bar K$) and
$g_4(y_1,0,t_1)=0$ then $\tilde E$ is reducible over $\bar K$, and
the two irreducible components are
$(z_1=x_1\pm L_1(y_1,t_1)=0)$. The $\z_2$-action interchanges these
two, so $E\subset BX$ is geometrically irreducible.
Singularity: The origin is a fixed point on $B\tilde X$, hence we get
an index 2 point on $BX$
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}=
2z_1\frac{dy_1\wedge dz_1\wedge dt_1}{x_1}$, so $a(E,X)=1$.
\end{say}
\begin{say}[$cD_{>4}/2$ and special $cD_4/2$]
{\ } \newline Normal form: $x^2+Q_2(y,z,t)z+g_{\geq
4}(y,z,t)/{\textstyle
\frac12(1,1,0,1)}$.
Weights for blow-up: (2,1,2,1)
$z$-chart: $x_1^2+Q_2(y_1,z_1,t_1)+z_1^{-4}g_{\geq 4}(y_1z_1,
z_1^2,t_1z_1)/{\textstyle
\frac12(0,1,1,1)}$ and then take the $\z_2$-action.
Lifting of the $\z_2$-action: We get a pair of commuting
$\z_2$-action on
$B\tilde X$, given by
$\frac12(0,1,1,1)$ and $\frac12(1,0,1,0)$.
Singularity: The second action has a fixed curve on $B\tilde X$, so
$BX$ is singular along a curve.
Exceptional divisor and discrepancy: as in the $cD_{>4}$-case.
\end{say}
\begin{say}[$cD$-cases, conclusion]
{\ } \newline We have settled all the $cD_{>4}$, $cD/2$ and
$cD/3$ cases, they have no g--extractions.
In the $cD_4$ cases there are no g--extractions if $f_3$ is
irreducible or if it has a simple linear factor over
$K$. The only remaining case is when $f_3$ is the product of 3 linear
factors which are conjugate over $K$.
This can not happen when $K=\r$, so over $\r$ points of type
$cD, cD/2$ and $cD/3$ do not have g--extractions.
The situation is more complicated over fields which do have cubic
extensions, as the following example shows. I have not classified all
cases.
\end{say}
\begin{exmp} Consider $x^2+y^3+az^3+t^6$, where $a\in K$ is not a
cube. The exceptional divisor of the $(3,2,2,1)$-blow up is
irreducible and has discrepancy 1. It has three points of index 2
which are conjugate over $K$, and no other singularities. Hence this
is a g--extraction.
\end{exmp}
\begin{exmp} We obtain an interesting example from the equation
$x^2+(y^2+z^2)z+t^5$. The $(2,1,2,1)$ blow up has terminal
singularities (one with index 2). The exceptional divisor $E$ is
singular along a curve.
\end{exmp}
\section{$cA$-type Points}
In this section we study g--extractions of $cA$ type terminal
singularities. The conventions of (\ref{7.notation}), (\ref{8.wbup})
and of (\ref{8.notation}) are used throughout.
\begin{say}[$cA_0$] (That is, smooth points.) {\ } \newline Normal
form: ${\Bbb A}^3$.
The blow up of the origin is smooth with exceptional divisor
$E\cong \p^2$. $a(E,X)=2$, and by (\ref{gw.d-ineq.prop})
$a(F,X)\geq 2$ for every exceptional divisor
$F$ with $\cent_XF=\{0\}$. Therefore by (\ref{gw.discr2.cor}), the
blow up of the origin is the only g--extraction.
The exceptional divisor is $E\cong \p^2$ with normal bundle
$\o_{\p^2}(-1)$.
\end{say}
\begin{say}[$cA_0/n,\ n\geq 2$]
{\ } \newline Normal form: ${\Bbb A}^3/{\textstyle
\frac1{n}(r,-r,1)}$, where $(r,n)=1$ and $1\leq r\leq n-1$.
Weights for blow-up: (r,n-r,1).
$x$-chart: ${\Bbb A}^3(x_1,y_1,z_1)/{\textstyle
\frac1{r}(1,-n,-1)}$.
Exceptional divisor: $\tilde E:=(x_1=0)$. Geometrically irreducible
and invariant under the
$\z_r$-action.
Lifting of the $\z_n$-action: The $\z_n$-action lifts to
$\frac1{n}(1,0,0)$. Its invariants are $x_2:=x_1^n$ and
$y_1,z_1$. The $\z_r$-action descends to the quotient of the
$\z_n$-action as
${\Bbb A}^3(x_2,y_1,z_1)/{\textstyle
\frac1{r}(n,-n,-1)}$.
Singularity: We obtain an index $r$ point on the
$x$-chart, and similarly an index $n-r$ point on the
$y$-chart.
Discrepancy: $\pi^*dx\wedge dy\wedge dz= rx_1^ndx_1\wedge dy_1\wedge
dz_1=
\frac{r}{n}x_1dx_2\wedge dy_1\wedge dz_1$. Since
$x_1=x_2^{1/n}$, we obtain that
$a(E,X)=1/n$.
Conclusion: The above blow up is the only possible g--extraction. If
$n\geq 3$ then either $r\geq 2$ or
$n-r\geq 2$, and we obtain a singular point of index
$\geq 2$ on $BX$.
If $r=2$ then $BX$ is smooth, the exceptional divisor is
$E\cong \p^2$ with normal bundle $\o_{\p^2}(-2)$. $BX\to X$ is the
unique g--extraction.
\end{say}
\begin{say}[$cA_1$]
{\ } \newline Normal form: $ax^2+by^2+cz^2 +dt^m$, where
$abcd\neq 0$.
Weights for blow-up: (1,1,1,1)
$t$-chart: $ax_1^2+by_1^2+cz_1^2 +dt_1^{m-2}$.
Exceptional divisor: $E:=(t_1=ax_1^2+by_1^2+cz_1^2=0)$ for
$m\geq 3$ and
$(t_1=ax_1^2+by_1^2+cz_1^2+d=0)$ for $m=2$.
$E$ is geometrically irreducible.
Singularity: $BX$ has exactly one singular point for
$m\geq 4$, it lies on the
$t$-chart. $BX$ is smooth for $m=2,3$.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}=
t_1\frac{dy_1\wedge dz_1\wedge dt_1}{x_1}$, so $a(E,X)=1$.
Conclusion: The only g--extraction is this blow up. The singularities
can be resolved by repeatedly blowing up the unique singular point.
\end{say}
\begin{say}[$cA_1/2$]
{\ } \newline Normal form: $ax^2+by^2+cz^n +dt^m/{\textstyle
\frac12(1,1,1,0)}$, where
$abcd\neq 0$ and $\min\{n,m\}=2$.
Weights for blow-up: (1,1,1,1)
$z$-chart: $ax_1^2+by_1^2+cz_1^{n-2} +dt_1^mz_1^{m-2}$.
Exceptional divisor: $\tilde E:=(z_1=ax_1^2+by_1^2+c=0)$ for
$m\geq 3$,
$(z_1=ax_1^2+by_1^2+dt_1^2=0)$ for $n\geq 3$ and
$(z_1=ax_1^2+by_1^2+c+dt_1^2=0)$ for $n=m=2$. $E$ is geometrically
irreducible.
Singularity: The $\frac12(1,1,1,0)$ action lifts to a
$\frac12(0,0,1,1)$ action. Thus we get a fixed curve, where the blow
up intersects the plane $(z_1=t_1=0)$.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= z_1\frac{dy_1\wedge
dz_1\wedge dt_1}{x_1}$, so $a(E,X)=1$.
Conclusion: The only possible g--extraction is this blow up. It has
nonterminal singularities, so this does not occur.
\end{say}
\begin{say}[$cA_1/n,\ n\geq 3$]
{\ } \newline Normal form: $xy+cz^{pm}+dt^2/{\textstyle
\frac1{n}(r,-r,1,0)}$, where $(r,n)=1$ and $cd\neq 0$.
Weights for blow-up: (1,1,1,1)
$z$-chart: $x_1y_1+cz_1^{pm-2}+dt_1^2/{\textstyle
\frac1{n}(r-1,1-r,1,-1)}$
Exceptional divisor: $\tilde E:=(z_1=x_1y_1+dt_1^2=0)$, it is
geometrically irreducible.
Singularity: The $\z_n$-action has an isolated fixed point at the
origin on $B\tilde X$. Thus $BX$ has an index
$n$ point.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= z_1\frac{dy_1\wedge
dz_1\wedge dt_1}{x_1}$, so $a(E,X)=1$.
Conclusion: The only possible g--extraction is this blow up. It has a
higher index point, so this does not occur.
\end{say}
\begin{say}[$cA_{>1}^-$]
{\ } \newline Normal form: $xy+g_{\geq 3}(z,t)$.
Weights for blow-up: (1,1,1,1)
$t$-chart: $x_1y_1+t_1^{-2}g_{\geq 3}(z_1t_1,t_1)$.
Exceptional divisor: $E:=(t_1=x_1y_1=0)$. It has two geometrically
irreducible components.
Singularity: Not important
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= t_1\frac{dy_1\wedge
dz_1\wedge dt_1}{x_1}$, so $a(E,X)=1$.
Conclusion: At least 2 geometrically irreducible divisors with
discrepancy
$\leq 1$.
\end{say}
\begin{say}[$cA_{>1}/n,\ n\geq 3$ and $cA_{>1}^-/2$]
{\ } \newline Normal form: $xy+g_{\geq 3}(z,t)/{\textstyle
\frac1{n}(r,-r,1,0)}$, where $(r,n)=1$.
Weights for blow-up: (1,1,1,1)
$t$-chart: $x_1y_1+t_1^{-2}g_{\geq 3}(z_1t_1,t_1)/{\textstyle
\frac1{n}(r,-r,1,0)}$
Exceptional divisor: $\tilde E:=(t_1=x_1y_1=0)$. It is reducible and
both irreducible components are geometrically irreducible and
invariant under the
$\z_n$-action.
Singularity: Not important
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= z_1\frac{dy_1\wedge
dz_1\wedge dt_1}{x_1}$, so $a(E,X)=1$.
Conclusion: At least 2 geometrically irreducible divisors with
discrepancy
$\leq 1$.
\end{say}
\begin{say}[$cA_{>1}^+$, $\mult_0g$ even]
{\ } \newline Normal form: $ax^2+by^2+g_{\geq 2m}(z,t)$, where $m\geq
2$, $-ab$ is not a square and
$g_{2m}\neq 0$.
Weights for blow-up: (m,m,1,1)
$t$-chart: $ax_1^2+by_1^2+t_1^{-2m}g_{\geq 2m}(z_1t_1,t_1)$.
Exceptional divisor:
$E:=(t_1=ax_1^2+by_1^2+g_{2m}(z_1,1)=0)$, it is geometrically
irreducible.
Singularity: The $t$-chart on $BX$ is singular only at points $P$
corresponding to the multiple roots of
$g_{2m}(z,1)$. The singularity at $P$ again has type
$cA_{>1}^+$, but the multiplicity of the corresponding
$g^P(z_1,t_1)$ is not necessarily even. The
$z$ chart is similar.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= t_1\frac{dy_1\wedge
dz_1\wedge dt_1}{x_1}$, so $a(E,X)=1$.
$x$-chart: $a+by_1^2+x_1^{-2m}g_{\geq 2m}(z_1x_1,t_1x_1)/ {\textstyle
\frac1{m}(1,0,-1,-1)}$.
Singularity: The fixed points of the $\z_m$-action are along the
$y_1$-axis, this itersects $B\tilde X$ in two points
$(0,\sqrt{-a/b},0,0)$ which are conjugate over $K$. Thus
$BX$ has 2 index $m$ terminal singularities which are conjugate over
$K$.
No other new singular points. The $y$-chart is similar.
Conclusion: The only g--extraction is the above weighted blow up.
The exceptional divisor is geometrically irreducible with a pair of
conjugate index
$m$-points. The other singular $K$-points of $BX$ are again of type
$cA_{>1}^+$ or $cA_1$.
\end{say}
\begin{say}[$cA^+_{>1}/2$, $\mult_0g$ even]
{\ } \newline Normal form: $ax^2+by^2+ g_{\geq 2m}(z,t)/{\textstyle
\frac12(1,1,1,0)}$, where $-ab$ is not a square and
$g_{2m}\neq 0$.
Weights for blow up: $(m,m,1,1)$.
$z$-chart: $ax_1^2+by_1^2+z_1^{-2m}g_{\geq 2m}(z_1,t_1z_1)
/{\textstyle \frac12(1-m,1-m,1,1)}$.
Exceptional divisor: $\tilde E:=(z_1=ax_1^2+by_1^2+g_{2m}(1,t_1)=0)$
is geometrically irreducible.
Singularities: If $m$ is odd then $(z_1=t_1=0)$ intersects $B\tilde
X$ in a fixed curve of the
$\z_2$-action, thus we get a singular curve on $BX$.
If $m$ is even, then on the $z$-chart the only
$\z_2$-fixed point is the origin. This is not on $B\tilde X$ iff
$z^{2m}\in g$.
$t$-chart: $ax_1^2+by_1^2+t_1^{-2m}g_{\geq 2m}(z_1t_1, t_1)
/{\textstyle \frac12(1,1,1,0)}$.
Singularities: On the $t$-chart the fixed point set is the
$t_1$-axis. This intersects the exceptional divisor at the origin.
This is not on the blow up iff $t^{2m}\in g$.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= t_1\frac{dy_1\wedge
dz_1\wedge dt_1}{x_1}$, so $a(E,X)=1$.
$x$-chart: $a+by_1^2+x_1^{-2m}g_{\geq 2m}(z_1x_1, t_1x_1) /{\textstyle
\frac1{m}(1,0,-1,-1)}$ and we also need to take the quotient by the
$\z_2$-action.
Lifting the $\z_2$-action: $a+by_1^2+x_1^{-2m}g_{\geq 2m}(z_1x_1,
t_1x_1) /{\textstyle \frac1{2m}(1,0,m-1,-1)}$
Singularities: On the $x$-chart the fixed point set is the
$y_1$-axis. This intersects $B\tilde X$ at two points
$(0,\pm\sqrt{-a/b},0,0)$. We get a conjugate pair of terminal
singularities of index $2m$ on $BX$.
$y$-chart: Similar to the $x$-chart.
Conclusion: $ax^2+by^2 +g_{\geq 2m}(z,t)/{\textstyle
\frac12(1,1,1,0)}$ where $-ab$ is not a square has a g--extraction
iff $m$ is even and $z^{2m},t^{2m}\in g_{2m}(z,t)$. Under these
assumptions, the unique g--extraction is the $(m,m,1,1)$-blow up.
\end{say}
\begin{say}[$cA_{>1}^+$, $\mult_0g$ odd]\label{cA+.multg-odd}
{\ } \newline Normal form: $ax^2+by^2+g_{\geq 2m+1}(z,t)$, where
$m\geq 1$, $-ab$ is not a square and
$g_{2m+1}\neq 0$.
Weights for blow-up: $(s,s,1,1)$ for $1\leq s\leq m$,
giving $B_sX\to X$.
$t$-chart: $ax_1^2+by_1^2+t_1^{-2s}g_{\geq 2m+1}(z_1t_1,t_1)$.
Exceptional divisor:
$E:=(t_1=ax_1^2+by_1^2=0)$, it is irreducible over $K$ but
geometrically reducible.
Singularity: The exceptional divisor itself has only smooth or normal
crossing points, thus $B_sX$ has only
$cA$ type points. The
$(x_1=y_1=0)$ line is singular if $s<m$ and generically smooth for
$s=m$.
$B_mX$ is terminal.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= t_1\frac{dy_1\wedge
dz_1\wedge dt_1}{x_1}$, so $a(E,X)=1$.
Divisors with discrepancy 1: Take the $(1,1,1,1)$-blow up.
$BX$ is singular along a line with an $A_{2m-2}$ transversal section.
We can blow up the line
$(m-1)$-times. At each time the exceptional divisor is a pair of
transversally intersecting planes, thus we have only $cA$ type
singularities. After $(m-1)$ blow ups we obtain $g:Y\to X$ and $Y$ has
only isolated $cA$ points, hence terminal. By
(\ref{gw.d-ineq.prop.cor}), all the exceptional divisors over $0\in
X$ with discrepancy $1$ are birational to divisors on $Y$. They all
come in conjugate pairs and have been enumerated by the above
$(s,s,1,1)$ blow ups.
Conclusion: There is a unique g--extraction whose exceptional divisor
has discrepancy 1. It is the
$(m,m,1,1)$-blow up $B_mX\to X$. Its exceptional divisor is
geometrically reducible, so we need to look further.
\medskip
Divisors with discrepancy 2: Let $F$ be a geometrically irreducible
exceptional divisor over $0\in X$ with discrepancy $2$. Then
$\cent_{B_mX}F$ is real. The center can not be the whole
$(x_1=y_1=0)$ line or a smooth point on it since both would give
$a(F,X)\geq 3$. Thus it is one of the singular points, corresponding
to a linear factor of $g_{2m+1}$.
By a linear change of the $z,t$-coordinates we may assume that this
linear factor is $z$. Thus $\cent_{B_mX}F$ is the origin of the $t$
chart, where $B_mX$ has equation
$ax_1^2+by_1^2+t_1^{-2m}g_{\geq 2m+1}(z_1t_1,t_1)$. This is again a
$cA_{>1}^+$ type point, ($\mult_0g$ can be even or odd) and
$a(F,B_mX)=1$. We have already enumerated all these cases, and we
know that $F$ is obtained by an
$(r,r,1,1)$-blow up. Putting the two steps together, we see that $F$
is obtained from $X$ by an
$(m+r,m+r,2,1)$-blow up. Next we compute these.
Normal form: $ax^2+by^2+g_{\geq 2m+1}(z,t)$, where
$m\geq 1$, $-ab$ is not a square,
$g_{2m+1}\neq 0$ and $\mult_0g(Z^2,T)\geq 2(m+r)$.
Weights for blow-up: $(m+r,m+r,2,1)$, giving
$B^rX\to X$.
$z$-chart:
$ax_1^2+by_1^2+z_1^{-2(m+r)}g_{\geq 2m+1}(z_1^2,t_1z_1)/{\textstyle
\frac12(m+r,m+r,1,1)}$.
Singularity: If $m+r$ is even, then the action has a fixed curve on
$B^r\tilde X$, so $B^rX$ is not terminal. If $m+r$ is odd and
the origin is in $B^r\tilde X$, then we get an index 2 point.
$z_1^{-2(m+r)}g_{\geq 2m+1}(z_1^2,t_1z_1)$ does not vanish at the
origin iff $z^{m+r}\in g_{\geq 2m+1}(z,t)$. This implies that $r\geq
m+1$. But
$g_{2m+1}(z_1^2,t_1z_1)$ itself is not divisible by
$z_1^{4m+3}$, hence $r=m+1$.
Conclusion: Assume that there is a linear change of the
$(z,t)$-coordinates such that
$$ g_{\geq 2m+1}(z,t)=\sum_{2i+j\geq 2m+2r}
\gamma_{ij}z^it^j,\qtq{and
$\gamma_{ij}\neq 0$ for some $2i+j=2m+2r$.}
$$ In this coordinate system, the $(m+r,m+r,2,1)$ blow-up gives
an elementary extraction whose exceptional divisor is geometrically
irreducible and has discrepancy 2.
Thus the only possible g--extractions are this weighted blow up and
the $(m,m,1,1)$ blow up found earlier.
The $(m+r,m+r,2,1)$ blow up is a g--extraction only in the $r=m+1$
case:
$$ g_{\geq 2m+1}(z,t)=\sum_{2i+j\geq 4m+2}
\gamma_{ij}z^it^j,\qtq{and}
\gamma_{2m+1,0}\neq 0.
$$ In some cases (cf. (\ref{cA.d=3.ex})), we do not have any
geometrically irreducible exceptional divisor over $0\in X$ with
discrepancy $2$. Then we have to compute further with discrepancy 3.
Fortunately, we can stop there.
\medskip
Divisors with discrepancy 3:
Normal form: $ax^2+by^2+g_{\geq 2m+1}(z,t)$, where
$m\geq 1$, $-ab$ is not a square and
$g_{2m+1}\neq 0$.
Weights for blow-up: $(2m+1,2m+1,2,2)$, giving $Y\to X$.
$z$-chart:
$ax_1^2+by_1^2+z_1^{-4m-2}g_{\geq 2m+1}(z_1^2,t_1z_1^2)/{\textstyle
\frac12(1,1,1,0)}$.
Exceptional divisor:
$E:=(t_1=ax_1^2+by_1+g_{2m+1}(1,t_1)=0)$, it is geometrically
irreducible.
Singularity: We get an index 2 point corresponding to the linear
factors of
$g_{2m+1}(z,t)$. Thus over $\r$ there is always an index 2 point.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}=
2z_1^3\frac{dy_1\wedge dz_1\wedge dt_1}{x_1}$, so $a(E,X)=3$.
\medskip
Final conclusion: These singularities always have g--extractions. One
is the
$(m,m,1,1)$-blow up. Its exceptional divisor is geometrically
reducible. This is the only g--extraction with discrepancy 1.
In some cases after a suitable coordinate change we can also perform
the
$(2m+1,2m+1,2,1)$ blow up. This is the only g--extraction whose
exceptional divisor is geometrically irreducible and has discrepancy
2.
If $g_{2m+1}(z,t)$ has no linear factors over $K$, then the
$(2m+1,2m+1,2,2 )$ blow up is a g--extraction whose exceptional
divisor is geometrically irreducible and has discrepancy 3. This is
the only one such. This case never happens over $\r$.
There may be other g--extractions whose exceptional divisor is
geometrically reducible and has discrepancy 2. I have no such examples.
\end{say}
\begin{exmp}\label{cA.d=3.ex} Consider the singularity
$X:=(x^2+y^2+z^m+t^n=0)$ for $m,n$ odd and $m+2\leq n\leq 2m-1$. The
above computations show that there is no geometrically irreducible
exceptional divisor over $0\in X$ with discrepancy $\leq 2$.
\end{exmp}
\begin{say}[$cA^+_{>1}/2$, $\mult_0g$ odd] {\ } \newline Normal
form: $ax^2+by^2+ g_{\geq 2m+1}(z,t)/{\textstyle
\frac12(1,1,0,1)}$, where $-ab$ is not a square and
$g_{2m+1}\neq 0$.
Weights for blow up: For weight reasons, only even powers of $t$
appear in
$g$. Thus we can define an integer $r$ by
$2m+2r=\mult_0g(Z^2,T)$.
$r\leq m+1$ since $g_{2m+1}\neq 0$. We consider the
$(m+r,m+r,2,1)$ blow up.
$z$-chart: $ax_1^2+by_1^2+z_1^{-2m-2r}g_{\geq 2m+1}(z_1^2,t_1z_1)
/{\textstyle \frac12(m+r,m+r,1,1)}$ and then we have to take the
quotient by the
$\frac12(1,1,0,1)$-action. This lifts to a
$\frac12(1,1,0,1)$-action on $B\tilde X$. We get a pair of commuting
$\z_2$-actions.
Exceptional divisor: $\tilde E:=(z_1=ax_1^2+by_1^2+\sum_{2i+j=2m+2r}
\gamma_{ij}t_1^j=0)$ is geometrically irreducible.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}=
2z_1^2\frac{dy_1\wedge dz_1\wedge dt_1}{x_1}$, so
$a(\tilde E,X)=2$.
Singularities: If $m+r$ is even then the $\z_2\times
\z_2$-action is free in codimension one. One of the elements acts by
$(0,0,1,1)$, thus we get a singular curve in $BX$.
If $m+r$ is odd then one of the elements acts by
$(0,0,1,0)$. Coordinates on the quotient are given by
$x_1,t_1, z_2=z_1^2,t_1$ and we get the equation
$$ ax_1^2+by_1^2+z_2^{-m-r}h_{\geq m+r}(z_2,t_1^2z_2) /{\textstyle
\frac12(1,1,0,1)}
$$ where $h(Z,T^2)=g(Z,T)$. At the origin we get a
$\z_2$-fixed point unless
$z^{m+r}\in g$. Thus $r\geq m+1$. On the other hand
$r\leq m+1$, thus $r=m+1$. Computing the $t$-chart shows that $BX$
has an index 2 point unless $t^{4m+2}\in g$.
Discrepancy: From this we see that
$a(E,X)=1/2$ if $m+r$ is odd and $a(E,X)=2$ if $m+r$ is even.
Conclusion: If $m+r$ is odd then a g--extraction exists iff
$$ g_{\geq 2m+1}(z,t)=\sum_{2i+j\geq 4m+2}
\gamma_{ij}z^it^j,\qtq{and}
\gamma_{2m+1,0}\neq 0 \neq \gamma_{0,4m+2}.
$$ If this holds then the $(2m+1,2m+1,2,1)$ blow-up is the unique
g--extractions. It has a geometrically irreducible exceptional
divisor with discrepancy $1/2$.
If $m+r$ is even then there may exist g--extractions with discrepancy
1/2 or 1. These can be determined by classifying all $\z_2$-invariant
divisors of disrepancy 1
and pointwise
$\z_2$-fixed divisors of disrepancy 2 or 3 over $\tilde X$. The first
task is easy, and we never get any g--extractions this way. The
second task is harder and it seems to require separate consideration
of about a dozen cases; I have not done all of them. Fortunately,
these singularities can be easily excluded in the main theorems using
topological considerations.
\end{say}
\section{$cE$-type Points}
In this section we study g--extractions of $cE$ type terminal
singularities. The conventions of (\ref{7.notation}), (\ref{8.wbup})
and of (\ref{8.notation}) are used throughout.
\begin{say}[$cE_6$ main series]
{\ } \newline Normal form:
$x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq 4}(z,t)$.
Weights for blow-up: (2,2,1,1)
$y$-chart:
$x_1^2+y_1^2+h_{4}(z_1,t_1)+y_1\Phi(y_1,z_1,t_1)/{\textstyle
\frac12(0,1,1,1)}$.
Exceptional divisor:
$\tilde E:=(y_1=x_1^2+h_{4}(z_1,t_1)=0)$.
$\tilde E$ is geometrically irreducible iff $h_4$ is not a square
over $\bar K$. If $-h_4$ is a square over
$K$, then $\tilde E$ has 2 geometrically irreducible components. In
the other cases $\tilde E$ is irreducible over $K$ but reducible over
$\bar K$. Both of the components are fixed by the
$\z_2$-action, so the same 3 cases happen for $E$.
Singularity: The origin is a fixed point of the
$\z_2$-action which is on
$B\tilde X$. So we get an index 2 point on $BX$.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}=
2y_1\frac{dy_1\wedge dz_1\wedge dt_1}{x_1}$, so
$a(E,X)=1$.
Conclusion: If $-h_4$ is a square over $K$ then there are 2
geometrically irreducible divisors with discrepancy 1, so no
g--extractions. If
$h_4$ is not a square over $\bar K$ then we get an index 2 point, so
again there are no g--extractions.
\end{say}
\begin{say}[$cE/2$]
{\ } \newline Normal form:
$x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq 4}(z,t)/{\textstyle
\frac12(1,0,1,1)}$. By weight considerations $g_3=0$ and $h_5=0$.
$h_4\neq 0$ since otherwise we would not have a terminal point. This
is a $cE_6/2$ point.
Weights for blow-up: (2,2,1,1)
$y$-chart:
$x_1^2+y_1^2+h_{4}(z_1,t_1)+y_1\Phi(y_1,z_1,t_1)/{\textstyle
\frac12(0,1,1,1)}$.
Lifting of the $\z_2$-action. The $\z_2$-action lifts to
$\frac12(1,1,0,0)$. Thus on $B\tilde X$ we have two commuting
$\z_2$-actions.
Exceptional divisor:
$\tilde E:=(y_1=x_1^2+h_{4}(z_1,t_1)=0)$. It is geometrically
irreducible iff $h_4$ is not a square over $\bar K$. If
$h_4=-Q_2(z_1,t_1)^2$ then $\tilde E$ has 2 geometrically irreducible
components
$(y_1=x_1\pm Q_2(z_1,t_1)=0)$. The $\frac12(1,1,0,0)$ action
interchanges the 2 components, thus $E\subset BX$ is geometrically
irreducible.
Singularity: The $\frac12(1,1,0,0)$ action has a fixed curve, thus we
get a nonterminal singular curve on
$BX$.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}=
2y_1\frac{dy_1\wedge dz_1\wedge dt_1}{x_1}$, so
$a(E,X)=1$.
\end{say}
\begin{say}[$cE_7$ main series]\label{cE7.main.ser}
{\ } \newline Normal form:
$x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq 5}(z,t)$.
Weights for blow-up: (3,2,1,1)
$x$-chart:
$x_1+y_1^3x_1+y_1g_{3}(z_1,t_1)+h_{5}(z_1,t_1)+x_1\Phi(y_1,z_1,t_1)/{\textstyle
\frac13(1,1,2,2)}$.
Exceptional divisor:
$\tilde E:=(x_1=y_1g_{3}(z_1,t_1)+h_{5}(z_1,t_1)=0)$. It is
geometrically irreducible iff $g_3$ and $h_5$ have no common factors.
Singularity: The origin is a fixed point of the
$\z_3$-action which is on
$B\tilde X$. So we get an index terminal 3 point on
$BX$. In fact, it is the index 3 terminal point ${\Bbb
A}^3/\frac13(1,1,2)$.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= 2x_1\cdot
dy_1\wedge dz_1\wedge dt_1$, so
$a(E,X)=1$.
Conclusion: If $g_3$ and $h_5$ have no common factors then $E$ is
irreducible and there are no g--extractions.
\end{say}
\begin{say}[$cE$ with common linear factors]\label{cE.comm.l.f}
{\ } \newline Normal form:
$$ x^2+y^3+yzG_2(z,t)+z^2Q_2(z,t)+zH_4(z,t)+yg_{\geq 4}(z,t)+h_{\geq
6}(z,t).
$$ The following cases are of this form:
$cE_8$: $x^2+y^3+yg_{\geq 4}(z,t)+h_{\geq 5}(z,t)$, if
$h_5$ has a linear factor over $K$, which we can call $z$.
$cE_7$:
$x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq 5}(z,t)$, if
$g_3$ and $h_5$ have a common linear factor over $K$, which we can
call $z$.
$cE_6$:
$x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq 4}(z,t)$ if there is a linear
factor over $K$, which we can call $z$, such that
$z^2|h_4,\ z|g_3$ and $z|h_5$.
Weights for blow-up: (3,2,2,1)
$z$-chart:
$$
\begin{array}{rl} x_1^2+y_1^3&+y_1G_2(0,t_1)+Q_2(0,t_1)+H_4(0,t_1)\\
&+y_1g_4(0,t_1)+h_6(0,t_1)+z_1\Phi(y_1,z_1,t_1)/{\textstyle
\frac12(1,0,1,1)}.
\end{array}
$$
Exceptional divisor $\tilde E$ is geometrically irreducible:
$$ (z_1=
x_1^2+y_1^3+y_1G_2(0,t_1)+Q_2(0,t_1)+H_4(0,t_1)+y_1g_4(0,t_1)+h_6(0,t_1)=0).
$$
Singularity: The origin is a fixed point of the
$\z_2$-action which is on
$B\tilde X$. So we get an index 2 point on $BX$.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}=
2z_1\frac{dy_1\wedge dz_1\wedge dt_1}{x_1}$, so
$a(E,X)=1$.
\end{say}
\begin{say}[$cE_6$ with $h_4$ a square]
{\ } \newline Normal form:
$x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq 4}(z,t)$.
Weights for blow-up: (1,1,1,1)
$t$-chart:
$$ x_1^2+y_1^3t_1+y_1t_1^2g_3(z_1,1)+y_1t_1^3g_4(z_1,1)+
t_1^2h_{4}(z_1,1)+t_1^3h_5(z_1,1)+t_1^4\Phi(y_1,z_1,t_1).
$$ The $z$-chart is similar.
Exceptional divisor:
$E:=(t_1=x_1=0)$, and the scheme theoretic exceptional divisor is
$2E$.
Singularity: On the $t$-chart the singular set is the line
$L:=(x_1=y_1=t_1=0)$. We determine the singularities along this line.
For a fixed value
$z_1=b\in \bar K$ we get a $cA$-point if $h_4(b,1)\neq 0$. If
$h_4(z,1)$ has a simple root at $b$ then we get a
$cD$-point. If $h_4(z,1)$ has a multiple root at $b$ then we still get
a $cD$ point if $g_3(b,1)\neq 0$ and a
$cE$-point if $h_5(b,1)\neq 0$.
Hence, if $b\in K$ and we do not have a cDV point, then $h_4$ has a
multiple linear factor which also divides $g_3$ and $h_5$. This case
was settled in (\ref{cE.comm.l.f}). Assuming that this is not the
case, we obtain that $BX$ has $cDV$ points along $L$.
The $z$-chart is similar and easy computations show that the $x$ and
$y$-charts are smooth along $E$.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}= t_1\frac{dy_1\wedge
dz_1\wedge dt_1}{x_1}$, so
$a(E,X)=2$.
First conclusion: $BX$ is not a g--extraction since it has a singular
curve.
$E$ is geometrically irreducible and $a(E,X)=2$, thus if
$g:Z\to X$ is a g--extraction with exceptional divisor
$F$ then $a(F,X)=1$ by (\ref{gw.discr2.cor}).
Computations: Here we determine all divisors $F$ over
$0\in X$ with
$a(F,X)=1$. If $\cent_{BX} F$ is not on $L$ then
$a(F,X)\geq 3$, and if $\cent_{BX} F$ is a point on $L$ then
$a(F,X)\geq 2$. Thus if $a(F,X)=1$ then
$\cent_{BX}F=L$ and $a(F,BX)=0$.
Along $L$ the threefold $BX$ has transversal type
$A_5$ whose singularity is resolved by blowing up the line 3-times. By
explicit computation we see that only the first of these produces an
exceptional divisor $F$ with
$a(F,X)=1$. This is the same divisor that we encountered in the
$(2,2,1,1)$-blow up and so it was already accounted for.
Final conclusion: There is no g--extraction except possibly when there
is a $b\in \bar K\setminus K$ such that
$(z-bt)^2|h_4, (z-bt)|g_3,(z-bt)|h_5$. In these cases the same
divisibilities hold if we replace $b$ by its conjugates over $K$. Thus
$b$ is quadratic over $K$, a root of $Q_2(z,1)$. If $F$ is any
divisor over $0\in X$ with $a(F,X)=1$ then its center in $BX$ is
$(z_1-b=0)\in L$ or its conjugate. Thus $F$ is geometrically
reducible.
\end{say}
\begin{say}[$cE_6$ last case]\label{cE6.g--extr.exist}
{\ } \newline Normal form:
$$ x^2+y^3+cQ_2(z,t)^2+yL_1(z,t)Q_2(z,t)+C_3(z,t)Q_2(z,t)+ yg_{\geq
4}(z,t)+h_{\geq 6}(z,t),
$$ where $Q_2$ is a quadratic form which is irreducible over $K$ and
$-c$ is not a square in $K$. By a coordinate change as in
\cite[I.12.6]{AGV85} we can bring this to the simpler form
$$ x^2+y^3+cQ_2(z,t)^2+ yg_{\geq 4}(z,t)+h_{\geq 6}(z,t),
$$ though this is not important.
Normal form and topology over $\r$: We can choose $Q_2$ to be positive
definite and diagonalize it. $-c\in \r$ is not a square, so we can
choose $c=1$. Thus we get the normal form
$$ x^2+y^3+(z^2+t^2)^2+ yg_{\geq 4}(z,t)+h_{\geq 6}(z,t).
$$ By (\cite[4.9]{rat1}) we obtain that $X(\r)$ is homeomorphic to
$\r^3$.
g--extractions: As we discussed above, all the g--extractions of $X$
have geometrically reducible exceptional divisors.
Construction of g--extractions: It turns out that in these cases there
is a g--extraction. By above remark we do not need to know this for
certain to understand the topology over $\r$, thus I only outline the
construction.
Basic constructions of toric geometry are used without reference; see
\cite{Fulton93} for an introduction.
Over $\bar K$ we can bring the equation to the form
$$ x^2+y^3+z^2t^2+ yg_{\geq 4}(z,t)+h_{\geq 6}(z,t).
$$ Let $e_x,e_y,e_z,e_t$ be a basis of $\r^4$. Consider the vectors
$w_z=\frac18(3,2,2,1)$ and $w_t=\frac18(3,2,1,2)$. These vectors give
a triangulation of the simplex with vertices $e_x,e_y,e_z,e_t$ where
the edges are
$$ (e_x,w_z), (e_x,w_t), (e_y,w_z), (e_y,w_t), (e_z,w_z), (e_t,w_t).
$$ Let us take the corresponding toric blow up. One can check by a
rountine but tedious computation that all singularities of $BX$ are
terminal and we get two index 3 points on the chart corresponding to
the simplex
$(e_x,e_y,w_z,w_t)$.
Note that the above construction is symmetric in $z$ and
$t$. Thus if we start with a quadratic form
$Q_2=z^2+qt^2$ and introduce new coordinates
$z'=z+\sqrt{q}t$ and $t'=z-\sqrt{q}t$ then
$Q_2=z't'$ and any blow up which is symmetric in $z',t'$ can be
transformed back to a blow up of $X$ defined over $K$. We need to
check that the two index 3 points become conjugates over $K$, but this
is easy to see from the explicit equations.
\end{say}
\begin{say}[$cE_7$ with common nonlinear factor]
{\ } \newline Normal form:
$x^2+y^3+yg_{\geq 3}(z,t)+h_{\geq 5}(z,t)$, where we assume that the
greatest common divisor of
$g_3$ and $h_5$ is $K$-irreducible (and nonconstant). We write
$g_3=Q(z ,t)G(z,t)$ and
$h_5=Q(z,t)H(z,t)$. ($Q$ is allowed to be linear, though this case is
treated already.)
Weights for blow-up: (3,2,1,1)
$x$-chart:
$x_1+y_1^3x_1+y_1g_{3}(z_1,t_1)+h_{5}(z_1,t_1)+x_1\Phi(y_1,z_1,t_1)/{\textstyle
\frac13(1,1,2,2)}$.
Exceptional divisor: It has two irreducible components:
\begin{eqnarray*}
\tilde E&:=&(x_1=y_1G(z_1,t_1)+H(z_1,t_1)=0),\qtq{and}\\
\tilde F&:=&(x_1=Q(z_1,t_1)=0).
\end{eqnarray*}
$\tilde E$
is geometrically irreducible, $\tilde F$ is irreducible but
geometrically reducible if $Q$ is not linear.
Discrepancy: $\pi^*\frac{dy\wedge dz\wedge dt}{x}=
3x_1\frac{dy_1\wedge dz_1\wedge dt_1}{x_1}$, so
$a(E,X)=1=a(F,X)$. (The latter equality uses that $Q$ is not a
multiple factor.)
Further aim: We would like to construct a birational morphism $g:Z\to
X$ whose exceptional divisor corresponds to $E$, and determine the
singularities of
$Z$. Thus in $BX$ we have to contract $F$. $F$ is not
$\q$-Cartier in $BX$ and $F$ can not be contracted in
$BX$. First we have to correct this problem.
Singularities of $BX$: I claim that $BX$ has only canonical
singularities. This can be done 2 ways. One can compute each chart
explicitly, which is rather tedious. I found it easier to use a
degeneration argument as follows. Let $F$ be the normal form of the
equation as above. We may assume that
$g_3(1,0)=1$. Consider the substitution
$$ F(x,y,z,t)\mapsto
\epsilon^{-24}F(\epsilon^{12}x,\epsilon^{8}y,\epsilon^{6}z,
\epsilon^{7}t).
$$ The exponents are chosen so that for $\epsilon\to 0$ the limit is
$X_0:=(x^2+y^3+yz^3=0)$. The $(3,2,1,1)$-blow up $BX_0$ is easy to
compute. We find an index 3 terminal point
${\Bbb A}^3/\frac13(1,1,2)$, a curve of $cA$-points and a curve of
$cE_7$-points corresponding to the
$t$-axis. Thus $BX$, as a small deformation of $BX_0$, has an index 3
point at the origin and some $cDV$ singularities. (These turn out to
be isolated points but we do not need this.) As in
(\ref{cE7.main.ser}) we see that the index 3 point is at the origin of
the $x$-chart and it is
${\Bbb A}^3/\frac13(1,1,2)$. In particular it is
$\q$-factorial.
Small blow up: Let $p:Y\to BX$ be the blow up of $F$ in
$BX$. Let $F'\subset Y$ denote the birational transform of $F$. Away
from the index 3 point
$BX$ is locally isomorphic to $B\tilde X$.
$\tilde F$ is defined by 2 equations $(x_1=Q(z_1,t_1)=0)$, thus
$p:Y\to BX$ is small and is an isomorphism at all points where $F$ is
$\q$-Cartier. The index 3 point is
$\q$-factorial, so $F$ is $\q$-Cartier there. Thus
$p:F'\to F$ is an isomorphism.
Contracting $F'$: $F$ is a cone over a $K$-irreducible curve, hence
its cone of curves over $K$ is 1-dimensional. If $C\subset F'$ is a
general curve then
$(C\cdot K_Y)=(p(C)\cdot K_{BX})<0$ and
$(C\cdot F')=(p(C)\cdot F)<0$. Thus the curves in $F'$ generate a
$K_Y$-negative extremal ray of $Y/X$, which can be contracted. We
obtain
$f: Y\to Z$ and $g:Z\to X$.
$P:=f(F')$ is a $K$-point since $F'$ is connected.
Conclusion: $g:Z\to X$ has a geometrically irreducible exceptional
divisor corresponding to $E$ and it has discrepancy 1. Furthermore, by
(\ref{gw.discr2.cor}) the index of
$P$ can not be one since $a(F,X)=1$. Hence there are no
g--extractions.
\end{say}
\begin{say}[Conclusion] The $cE_8$ case is settled if
$h_5(z,t)$ has a linear factor over $K$. This always holds if $K=\r$,
hence at least in this case there are no g--extractions. I do not
know what happens if $K\neq
\r$.
The $cE_7$ case is settled if $g_3(z,t)$ and
$h_5(z,t)$ have no common factor, or if they have a common linear
factor over
$K$ or if they have a unique common factor over $K$. This accounts for
all the possibilities, hence there are no g--extractions.
The $cE_6$ case is settled if $h_4(z,t)$ is not a square over $\bar K$,
if $-h_4(z,t)$ is a square over $K$ or if $h_4(z,t)$ is divisible by
the square of a linear form over $K$. In these cases there are no
g--extractions.
The remaining case is treated in (\ref{cE6.g--extr.exist}) and the
unique g--extraction is written down explicitly. For the applications
in this paper the existence is not crucial.
\end{say}
\begin{exmp} Let $X$ be the $cE_7$ type singularity
$x^2+y^3+yg_3(z,t)+h_5(z,t)$, where $g_3$ and $h_5$ do not have a
common factor. It is not hard to see that
$X$ is an isolated singular point and its
$(3,2,1,1)$-blow up has only terminal singularities. As in
(\ref{cE7.main.ser}), the $y$ chart on the blow up gives the
exceptional divisor
$$ E=(g_3(z,t)+h_5(z,t)=0)/{\textstyle
\frac12(1,1,1,1)}.
$$ This gives examples of extremal contractions whose exceptional
divisor $E$ has a quite complicated singularity along the
$(z=t=0)$-line.
\begin{enumerate}
\item $x^2+y^3+yz^3+t^5$. $E$ is singular along
$(z=t=0)$, with a transversal singularity type
$z^3+t^5$, that is $E_8$.
\item $x^2+y^3+y(z-at)(z-bt)(z-ct)+t^5$. $E$ has triple
selfintersection along
$z=t=0$.
\end{enumerate}
\end{exmp}
\section{Hyperbolic 3--manifolds}
The aim of this section is to show that every hyperbolic 3--manifold
satisfies the conditions (\ref{int.no.cond}).
\begin{thm}\label{hyp.doesnotcont.thm} Let $M$ be a compact
hyperbolic 3--manifold. Then $M$ does not contain any PL submanifold
of the following types:
\begin{enumerate}
\item $\r\p^2$
\item 1--sided $S^1\times S^1$
\item 1--sided Klein bottle.
\end{enumerate}
\end{thm}
We use two facts about hyperbolic 3--manifolds. First, that their
universal cover is homeomorphic to $\r^3$. Second, that their
fundamental group does not contain a subgroup isomorphic to $\z^2$
(see, for instance, \cite[4.6]{Scott83}).
\medskip
More generally, we see how these conditions fit in the framework of
Thurston's geometrization conjecture. This version was pointed out to
me by Kapovich.
\begin{thm}\label{gen.doesnotcont.thm} Let $M$ be a compact
3--manifold. Assume that $M=M_1\ \#\ \cdots \ \#\ M_k$, where
\begin{enumerate}
\item[(i)] each $M_i$ is aspherical, and
\item[(ii)] the Seifert fibered part of the Jaco--Shalen--Johannson
decomposition of $M_i$ is orientable.
\end{enumerate} Then $M$ does not contain any PL submanifold
of the following types:
\begin{enumerate}
\item $\r\p^2$
\item 1--sided $S^1\times S^1$
\item 1--sided Klein bottle.
\end{enumerate}
\end{thm}
We consider the 3 types of submanifolds separately. Condition
(\ref{int.no.cond}.1) is closely related to the notion of
$\p^2$-irreducibility (cf.\ \cite[p.88]{Hempel76}).
\begin{lem} Let $M$ be a 3--manifold with universal cover $\tilde M$.
\begin{enumerate}
\item If $M\sim M_1\ \#\ M_2$, then $M$ contains a 2--sided $\r\p^2$
iff one of the summands does.
\item Assume that $\tilde M$ is homeomorphic to
$\r^3$. Then $M$ does not contain an $\r\p^2$ and $M$ can not be
written as a nontrivial connected sum.
\end{enumerate}
\end{lem}
Proof. Assume that $F\subset M$ is a 2--sided $\r\p^2$. We may
assume that $F$ is transversal to the gluing $S^2$. Thus $C=F\cap S^2$
is an embedded curve in $F$. Assume first that
$F$ has a connected component $C_1\subset C$ which is not null
homotopic in
$F$. Then $F$ is not orientable along $C_1$, and the same holds for
$M$ along $C_1$. But $M$ is orientable along $S^2$, a contradiction.
Take any connected component $C_i\subset C$ such that
$C_i\subset S^2$ bounds a disc $D_i$ which is disjoint from $C$.
$C_i$ also bounds a disc $D'_i$ in $F$ (since it is null homotopic in
$F$). Thus we can change the embedding $\r\p^2\to M$ by replacing
$D'_i$ with
$D_i$ and then pushing it to one side. The new embedding is still
2--sided. Repeating if necessary, we eventally get an embedding which
is disjoint from $S^2$, proving (1).
$\r\p^2$ can not be embedded into $\r^3$ (cf. \cite[27.11]{GrHa81}),
thus the preimage of
$\r\p^2$ in $\r^3$ is a union of copies of $S^2$. Fix one of these and
call it $N$. By the Schoenflies theorem (cf.\ \cite[Sec. 17]{Moise77})
$N$ bounds a 3--ball $B^3$. At least one element of $\pi_1(M)$ maps
$N$ to itself. It can not map the inside of
$N$ to its outside since these are not homeomorphic. If it maps
$B^3$ to itself, then by the Borsuk--Ulam theorem (cf.\
\cite[23.20]{Fulton95}) we have a covering transformation with a
fixed point, a contradiction.
Assume that we have $S^2\sim N'\subset M$ and let $S^2\sim N\subset
\tilde M$ be one of the preimages. Then $N$ bounds a 3--ball and so
does $N'$.
\qed
\medskip
In order to study the conditions (\ref{int.no.cond}.2--3) we have to
distinguish two cases.
\begin{say}[Incompressible case]\label{12.incompr.say}
Let $M$ be a compact 3--manifold and $S\subset M$ a compact 1--sided
torus or Klein bottle. Assume that $\pi_1(S)\DOTSB\lhook\joinrel\rightarrow \pi_1(M)$. Let
$\partial U$ be the boundary of a regular neighborhood of $S$. Then
$\partial U$ is a 2--sided torus or Klein bottle and
$\pi_1(\partial U)\DOTSB\lhook\joinrel\rightarrow \pi_1(S)\DOTSB\lhook\joinrel\rightarrow \pi_1(M)$ is an injection. This
implies that $\partial U$ is incompressible in $M$ (cf.\
\cite[pp.88-89]{Hempel76}). Thus $U$ is one of the pieces of the
Jaco--Shalen--Johannson decomposition of $M$ (cf.\
\cite[p.483]{Scott83}). We have to be a little more careful since
$U$ is Seifert fibered, thus it may sit inside one of the Seifert
fibered components.
The fundamental group of a hyperbolic 3--manifold does not contain a
subgroup isomorphic to $\z^2$ (see, for instance,
\cite[4.6]{Scott83}), hence the incompressible case does not happen
for hyperbolic 3--manifolds.
\end{say}
\begin{say}[Compressible case]\label{12.compr.say}
In this case we show that $M$ can be written as a connected sum with
a very special summand.
\begin{prop}\label{1-s.torus}
Let $M$ be a compact 3--manifold. Then $M$ contains a 1--sided torus
$T$ such that $\pi_1(T)\to \pi_1(M)$ is not an injection iff
$M\sim N\ \#\ (S^1\tilde{\times}S^2)$ or $M\sim N\ \#\ (S^1
\times\r\p^2)$
\end{prop}
Proof. Let $T\subset U\subset M$ be a regular neighborhood. Set
$V=M\setminus U$. Then $\partial U=\partial V\sim S^1\times S^1$. We
know that
$\pi_1(\partial U)$ injects into $\pi_1(U)$. If $\pi_1(\partial
U)\DOTSB\lhook\joinrel\rightarrow \pi_1(V)$, then
$\pi_1(\partial U)\DOTSB\lhook\joinrel\rightarrow \pi_1(U)\DOTSB\lhook\joinrel\rightarrow \pi_1(M)$
by Schreier's theorem (cf. \cite[IV.2.6]{Lyndon-Schupp77}).
$\pi_1(\partial U)$ is an index 2 subgroup of $\pi_1(T)$ an $\pi_1(T)$
is torsion free. Thus
$\pi_1(T)\to \pi_1(M)$ is also an injection, a contradiction.
Therefore, by the Loop theorem (cf.\ \cite[4.2]{Hempel76}), there is an
embedding of the disc $j:(B,\partial B)\DOTSB\lhook\joinrel\rightarrow (V,\partial V)$ such that
the image of $j(\partial B)$ is not contractible in $\partial V$.
Let us cut $V$ along $j(B)$ to get $W$.
The boundary of $W$ is $\partial V$ cut along
$j(\partial B)$ (which is a cylinder) with two copies of $B$ pasted to
the ends. That is, $\partial W\sim S^2$. Therefore $M$ is obtained by
pasting $W$ to a 3--manifold (with boundary)
$K$, which is obtained from
$U$ by attaching a 2--handle.
There are two cases corresponding to whether $j(\partial B)$ is a
primitive element of $\pi_1(U)\cong \z^2$ (hence $\pi_1(K)\cong \z$)
or is contained in $2\z^2$ (hence $\pi_1(K)\cong \z+\z_2$).\qed
\begin{prop}\label{1-s.kb}
Let $M$ be a compact 3--manifold which does not contain a 2--sided
$\r\p^2$. Then $M$ contains a 1--sided Klein bottle
$K$ such that $\pi_1(K)\to \pi_1(M)$ is not an injection iff
$M\sim N\ \#\ (S^1\tilde{\times}S^2)$ or $ M\sim N\ \#\ (\r\p^3\ \#\
\r\p^3)$.
\end{prop}
Proof. Let $K\subset U\subset M$ be a regular neighborhood. As in the
proof of (\ref{1-s.torus}) we obtain
an embedding of the disc $j:(B,\partial B)\DOTSB\lhook\joinrel\rightarrow (V,\partial V)$ such
that the image of $j(\partial B)$ is not contractible in $\partial V$.
We again cut $V$ along $j(B)$ to get $W$. Let $\partial V^*$ denote
$\partial V$ cut along
$j(\partial B)$.
There are 3 cases to consider corresponding to what $\partial V^*$ is:
\begin{enumerate}
\item ($\partial V^*$ is a cylinder). Then we obtain a connected sum
decomposition as in (\ref{1-s.torus}).
\item ($\partial V^*$ consists of two Moebius bands). Then $\partial W$
is two disjoint projective planes, hence $M$ contains a 2--sided
projective plane. This can not happen by assumption.
\item ($\partial V^*$ is a Moebius band). In this case
$j(\partial B)$ is 1--sided in $\partial V$, thus $M$ is not orientable
along $j(\partial B)$. Then $j(\partial B)$ can not be the boundary of
an embedded disc.\qed
\end{enumerate}
\end{say}
\begin{rem} So far we have excluded Seifert fiber spaces from
considerations. Many Seifert fiber spaces do contain 1--sided tori
or Klein bottles.
If $p:M\to F$ is a Seifert fiber space and $C\subset F$ a 1--sided
curve not passing through any critical value, then $p^{-1}(C)\subset
M$ is a 1--sided torus or Klein bottle. Another example can be
obtained as follows. Let $x,x'\in F$ be two points such that the
fibers over them have multiplicity 2. Let $I\subset F$ be a simple
path connecting $x$ and
$x'$. Then $p^{-1}(I)$ is a 1--sided Klein bottle.
It is not hard to see that if $T\subset M$ is a
1--sided torus or Klein bottle such that $p(T)$ is 1--dimensional
(these are called vertical) then $T$ is obtained by one of the
above constructions.
Assume now in addition that $M$ has a geometry modelled on ${\mathbb
H}^2\times \r$ (cf.\ \cite[p.459]{Scott83}). Then by
\cite[5.6]{Johannson79}, every 1--sided torus or Klein bottle in $M$ is
isotopic to a vertical one.
This way we obtain many examples of nonorientable Seifert fiber
spaces which satisfy the conditions (\ref{int.no.cond}).
\end{rem}
|
1998-01-02T06:55:25 | 9712 | alg-geom/9712008 | en | https://arxiv.org/abs/alg-geom/9712008 | [
"alg-geom",
"math.AG"
] | alg-geom/9712008 | Bumsig | Bumsig Kim (University of California - Davis) | Quantum Hyperplane Section Theorem For Homogeneous Spaces | 24 pages, LaTeX 2e. The presentation is improved a lot | null | null | null | null | We formulated a mirror-free approach to the mirror conjecture, namely,
quantum hyperplane section conjecture, and proved it in the case of nonnegative
complete intersections in homogeneous manifolds. For the proof we followed the
scheme of Givental's proof of a mirror theorem for toric complete
intersections.
| [
{
"version": "v1",
"created": "Fri, 5 Dec 1997 21:07:48 GMT"
},
{
"version": "v2",
"created": "Tue, 16 Dec 1997 23:46:55 GMT"
},
{
"version": "v3",
"created": "Fri, 2 Jan 1998 05:55:25 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Kim",
"Bumsig",
"",
"University of California - Davis"
]
] | alg-geom | \section{Introduction}
Quantum cohomology of a symplectic manifold
is a certain deformed ring of the ordinary
cohomology ring with parameter space given by the second cohomology
group. It encodes enumerative geometry
of rational curves on the manifold. In general it is difficult to compute the
quantum cohomology structure. On the other hand,
mirror symmetry predicts an answer to
traditional questions of counting the virtual numbers of
rational curves of a given degree on a three-dimensional Calabi-Yau
manifold, which amounts to knowing the
structure of the quantum cohomology.
In the large class of Calabi-Yau manifolds, the complete
intersections
in toric manifolds or homogeneous spaces, this mirror symmetry prediction
\cite{Ca, BV, BCKV1, BCKV2}
can be interpreted as a quantum cohomology counterpart of the
weak Lefschetz hyperplane section theorem relating
cohomology algebras of the ambient manifolds and their hyperplane sections.
As it is mentioned in \cite{GS}, \lq\lq quantum hyperplane section conjecture"
can be formulated in intrinsic terms of Gromov-Witten
theory on the ambient manifold and does not require a reference
to its mirror partner.
In this paper we formulate and prove the conjecture for homogeneous spaces.
It would be one of the highly nontrivial functorial properties
enjoyed by quantum cohomology algebras. One
can compute the virtual numbers of rational curves on a Calabi-Yau
3-fold complete intersection, provided one knows the
quantum cohomology algebra of the ambient space. In fact, one needs to know the
quantum differential equations of the space, which are certain
linear differential equations arising from the flat
connection in the quantum cohomology algebra.
The mirror symmetry prediction is that
the quantum differential equations of a Calabi-Yau manifold are
equivalent (in a sense) to the Picard-Fuchs differential
equations of another Calabi-Yau manifold. In contrast,
the proposed conjecture is that
there is a certain relation between quantum differential
equations of a manifold and those of a certain complete
intersection.
When the ambient space is a symplectic toric manifold, the conjecture is
a corollary of the Givental mirror theorem \cite{GT}.
\bigskip
Let $X$ be a compact
homogeneous space of a semi-simple complex Lie group
and let $V$ be a vector bundle over $X$.
Suppose $V'_\beta :=\pi _*e_1^*V$ becomes a vector orbi-bundle
over Kontsevich moduli space $\overline{M}_{0,0}(X,\beta)$
where $e_1$ is the evaluation map at the (first) marked point
from $\overline{M}_{0,1}(X,\beta )$ to $X$ and
$\pi$ is the map from $\overline{M}_{0,1}(X,\beta)$
to $\overline{M}_{0,0}(X,\beta)$ associated with \lq\lq forgetting
the marked point" \cite{Ko}.
Then one might want compute
\[ \int _{\overline{M}_{0,0}(X,\beta)}Euler(V'_{\beta}). \]
Introduce a formal parameter $\hb$. Then
it turns out that the classes
\[
G_{\beta}^V
:=(e_1)_*\frac{Euler(V_{\beta})}{\hb (\hb -c)}\]
would be better considered \cite{GE}, where
$V_\beta =\pi ^*(V'_\beta )$ and $c$ (depending on $\beta$)
are the first Chern classes
of the universal cotangent line bundles.
The classes are in $H^*(X) [\hb ^{-1}]$.
They recover the original integrals which we want:
\[ \int _X G^V_\beta =
\frac{-2}{\hb ^3}\int _{\overline{M}_{0,0}(X,\beta)}Euler(V'_{\beta})
+o(\hb ^{-3}).\]
Consider the classes \[ G_{\beta}^X:=(e_1)_*
\frac{1}{\hb(\hb -c)}\]
corresponding to $X$ itself (without $V$).
When $V$ is a convex, decomposable, vector bundle $\oplus L_j$ of
line bundles $L_j$,
the main result of this paper proves some explicit
relationship between $A:=\{ G_{\beta}^V |\ \beta\in H_2(X,\ZZ ) \} $
and $B:=\{ H^V_{\beta}\cup G_{\beta}^X | \ \beta\in H_2(X,\ZZ ) \}$, where
\[ H^V_\beta = \prod _j\prod _{m=0}^{<c_1(L_j) ,\beta >}
(c_1(L_j) + m\hb ) \]
which is the key object introduced in this sequel.
\bigskip
We now formulate the precise result of this paper.
Let
$\{ p_i \}_{i=1}^k$ denote the $\ZZ _+$ basis of the
closed integral K\"ahler (ample) cone of $X$.
Let us introduce formal parameters $q_i$, $i=1,...,k$,
and the ring $\QQ [[ q_1,...,q_k]]$ of formal power series of $q_i$.
Denote by $q^\beta$
\[ \prod _{i=1}^k q_i^{<p_i,\beta >}.\]
For simplicity, let $G^X_0 =1$ and $G^V_0=Euler(V)$.
We want to compare generating functions $J^V$ and $I^V$ from $A$ and $B$,
respectively:
\begin{eqnarray*} S^V &:=& \sum _{\beta} q^\beta G^V_\beta \\
\Phi ^V &:=& \sum _{\beta }q^\beta H_\beta ^V \cup G^X_\beta .\end{eqnarray*}
We prove that one can be transformed to another by a
unique \lq\lq mirror" transformation. To describe the transformation, let
\[ q_i = e^ {t_i}, \text{ for } i=1,...,k, \]
and introduce another formal variable $t_0$. Define degree
of $q_i$ by
\[ c_1(TX )
-c_1(V)=\sum (\deg q_i) p_i .\]
Let
\begin{eqnarray*} J^V (t_0,...,t_k) &:=& e^{(t_0+\sum _i p_it_i)/\hb }S^V \end{eqnarray*}
and
\begin{eqnarray*}
I^V (t_0,...,t_k)&:=& e^{(t_0+\sum _i p_it_i)/\hb }\Phi ^V ,\end{eqnarray*}
which are formal power series of $t_1,...,t_k,
e^{t_0},...,e^{t_k}$ over $H^*(X)[\hb ^{-1}]$.
{\theorem\label{thmmain} Assume that $\deg q_i\ge 0$ for all $i$. Then
$J^V$ and $I^V$ coincide
up to a unique weighted homogeneous change of variables:
$t_0\mapsto t_0+f_0\hb +f_{-1}$
and $t_i\mapsto t_i +f_i$,
where $f_{-1},...,f_k$ are power
series of $q_1,...,q_k$ over $\QQ$ without constant terms,
$\deg f_i=0$, $i=0,...,k,$ and $\deg f_{-1}=1$.}
\bigskip
{\it Remarks:}
0. $J^V$ will be shown to be the cohomological expression
of solutions to quantum differential equations associated
to $(X,V)$, which is closely related to the quantum differential
equations of the smooth zero locus of $V$.
The theorem can be extended to the case of decomposable
concavex vector bundles $V$.
1. The change of variables is uniquely determined by
coefficients of $1=(\frac{1}{\hb})^0$ and $\frac{1}{\hb}$
in the expansions of $J^V$ and $I^V$ as power
series of $\frac 1\hb$.
2. In the case of a symplectic toric manifold $X$
the similar statement is a corollary of a mirror theorem in \cite{GT},
where $\Phi ^X$ is explicitly known.
3. For the proof of \ref{thmmain} we follow the scheme of Givental's
proof \cite{GE, GT} of the mirror theorem for nonnegative complete
intersections in toric manifolds.
4. The theorem verifies the prediction \cite{BCKV1} of virtual numbers in
Calabi-Yau 3-fold complete intersections in Grassmannians.
5. A mirror construction is established for complete intersections
in partial flag manifolds \cite{BCKV1, BCKV2}.
Because of the known quantum cohomology structure \cite{Ionut},
in principle there is no essential difficulty in finding $G^X_\beta$ for
each partial flag manifold $X$, even though a general formula of it is
unknown.
6. The quantum hyperplane section principle is applied to a nonconvex
manifold in \cite{To}.
\smallskip
{\it Notation:}
$X$ will always be a generalized flag manifold $G/P$,
where $G$ is a complex semi-simple Lie group and $P$ is
a parabolic subgroup.
Let $T$ be a maximal torus of $G$ in $P$ and let $T$ act on $X$ on
the left.
Let a complex torus $T'$ act on $X$ trivially and let
$V$ be a $T\ti T'$-equivariant convex vector bundle over $X$.
Consider $E$ a multiplicative class and suppose that
$E(V)\in H_{T\ti T'}(X)$ is invertible in
$H_{(T\ti T')}(X):= H_{T\ti T'}(X)\ot H_{T\ti T'}$,
where $H_{(T\ti T')}$ is the quotient field of $H_{T\ti T'}(pt)$.
In section 2, we will not consider $T$-action on $X$.
In section 6, additionally we will assume that $V$ is decomposable.
The convexity of $V$ is by definition that
$H^1(\PP ^1, f^*V)=0$ for any morphism $f:\PP ^1\ra X$.
Let $T\ti T'$ equivariant line bundles $U_i$, $i=1,...,k$,
form ample basis of ordinary Picard group.
We denote $\int _X ABE(V)$
by $<A,B>^V_{0}$, for $A, B\in H^*_{(T\ti T')}(X)$
and also we use $\int _V A:=\int AE(V)$ (equivariant push forwards).
The Mori cone of $X$ will be denote by $\Lambda$, which
can be identified with $\ZZ _+^k$ with respect to
coordinates $p_i:=c_1(U_i)$.
On the additive group $\ZZ ^{k}$
we will give the standard partial ordering, so
that $d:=(d_1,...,d_k)\ge 0$ means $d_i\ge 0$.
Let $\phi _v$ denote the equivariant pushforward
of $1$ under the embedding $i_v$ of
the fixed point $v$ to $(X,V,E)$; this $(X,V,E)$ has the
Frobenius structure by pairing $<,>^V_0$, so that
$A_v:=<A,\phi _v>_0^V=i_v^*(A)$ for $A\in H^*_{(T\ti T')}(X)$.
For a $G$-manifold $M$, let
$M^G$ denote the set of $G$-fixed points of $M$.
We will say simply degree and dimension for
complex degree and complex dimension, respectively.
Let $\sum _a T_a\ot T^a$ be the equivariant diagonal class of $(X,V,E)$
in $X\ti X$. That is, $<T_a, T^b>^V_0=\delta _{a,b}$.
In the paper we will consider various rings
$H^*_{T\ti T'}[[\hb ^{-1}]][[q]]$ formal power series ring
of $\hb ^{-1}, q$ over $H^*_{T\ti T'}$, $H^*_{(T\ti T')}[[\hb ^{-1}]][[q]]$
formal power series ring
of $\hb ^{-1}, q$ over $H^*_{(T\ti T')}$, and
$H^*_{(T\ti T')}(\hb )[[q]]$ formal power series ring
of $q$ over quotient field of $H^*_{T\ti T'}[\hb ]$.
\smallskip
{\it Structure of the paper:}
In section 2, we recall a general theory of Gromov-Witten invariants
and quantum cohomology. We introduce the Givental Correlators $S^V$.
In section 3, we show that the equivariant correlators satisfy certain
\lq\lq almost recursion relations."
In section 4, we introduce the double construction and
show that the correlators satisfy certain polynomiality
in the double construction.
In section 5, we introduce certain class $\mathcal{P}(X,V,E)$
of series of $q=(q_1,...,q_k), \hb ^{-1}$ over $H^*_{T\ti T'}(X)$,
where a \lq mirror' group acts freely and transitively.
In section 6, we introduce a modified correlator of $S^X$.
It will also
belong to the class $\mathcal P (X,V,E)$.
The modification is given by the hypergeometric correcting Euler classes
$H^V_\beta$ according to the decomposition type of $V$.
In sections 7 and 8, we analyze the torus $T$ action on a generalized
flag manifold and its one dimensional orbits, the representations of
the section spaces of equivariant line bundles restricted to the orbits.
The analysis would be useful to find the explicit expression of $\Phi ^X$.
{\it Acknowledgments:}
I am grateful to A. Givental and Y.-P. Lee for helping me
to understand the paper \cite{GE}; and
V. Batyrev, I. Ciocan-Fontaine, B. Fulton, B. Kreu\ss ler,
E. Tj\o tta, K. Wirthm\"uller for useful discussions
on the papers \cite{GE, GT}.
Also, I would like to thank Institut Mittag-Leffler
for the financial support during
the year-long program, \lq\lq enumerative
geometry and its interactions
with theoretical physics" in 1996/1997.
My special thank goes to D. van Straten for numerous comments and
help to improve the clarity of the paper.
\section{Mirror Symmetry}\label{setup}
\subsection{The moduli space of
stable maps}
To fix notation we recall the definition of
stable maps and some elementary properties of
the moduli spaces of stable maps to $X$ \cite{Ko, FuP, BM}.
The notion of stable maps is due to M. Kontsevich.
We recommend the (survey)
paper of W. Fulton and R. Pandharipande \cite{FuP}.
A prestable rational curve $C$ is
a connected arithmetic genus $0$
projective curve with possibly nodes.
The curve is not necessary irreducible.
A prestable map $(f,C; x_1,...,x_n )$ is a morphism $f$ from $C$ to
$X$ with fixed ordered $n$-many marked distinct smooth points $x_i\in C$.
We will identify $(f,C; \{ x_i\} )$ with $(f',C',\{ x_i'\})$
if there is an isomorphism $h$ from $C$ to $C'$ preserving
the configuration of marked points such that $f=f'\circ h$.
A stable map $(f,C; \{ x_i\} )$ is a prestable map
with only finitely many automorphisms.
Let $\overline{M}_{0,n}(X,\beta )$ be the (coarse moduli)
space of all stable maps $(f, C; \{ x_i \} _{i=1}^n)$
with the fixed homology type
$\beta = f_*([C])\in H_2(X,\ZZ )$.
Whenever it is nonempty, the moduli space
is a connected\footnote{For a proof of the connectedness see \cite{Th}.}
compact complex {\it orbifold}
with complex dimension
$\dim X + <c_1(TX),\beta > + n -3$.
\bigskip
More precisely,
locally near a stable map the moduli space has data of a quotient
of a holomorphic domain by the (finite)
group action of all automorphisms of the stable map.
In the paper \cite{FuP} are
constructed smooth open complex domains $V$ with
finite groups $\Gamma$ which act on $V$ such that $V/\Gamma$ are
naturally glued together in the moduli space of stable maps.
Let $X\subset _i \PP ^N$, $\beta\ne 0$, and $(X,i_*(\beta ))\ne
(\PP ^1, \kline )$. Here $\kline$ denotes the line class of $H_2(\PP ^N)$.
Given a stable map $(f,C)$ (without marked points for
simplicity), choose hyperplanes $H_j$ in $\PP ^N$
satisfying that $\{ H_j\}$ gives rise to
a basis of $H^0(\PP ^N ,{\mathcal O}(1))$,
$f$ is transversal to the hyperplanes,
and their inverse images $\{ x_{i,j} \} _i=f^{-1}(H_j)$
contain no nodes of $C$.
Then the data $(C;\{ x_{i,j}\})$ determines a point in
the moduli space of marked stable curves. Conversely, a point
in a suitable closed subvariety of an
open smooth domain of the moduli space of marked stable curves
naturally determines a stable map $f$ with the extra
choices of elements in $(\CC ^\ti )^N$.
If $G$ is the product of the symmetric group of the
elements of the each group $\{ x_{i,j}\}_i$, then
this
$G$ has an action sending the data $(f,C; \{ x_{i,j}\})$
to another by permutations of the new marked points.
A $(\CC ^\ti )^N$ - bundle of the smooth closed subvariety is
an algebraic local chart of the moduli space of stable maps at $f$ with the
induced $G$ action.
\smallskip
{\it Example: }
Let $X=\PP ^2$ and $f$ be a stable map without marked points
such that $f$ is transversal to the hyperplanes $x=0$, $y=0$, and $z=0$.
Assume no singular points of $C$ are mapped into the
hyperplanes, and $f_*[C]=2[\text{line}]$.
Consider their inverse images (Cartier divisors),
$a_1, a_2$, $b_1, b_2$, $c_1, c_2$ in $C$.
This information $(C; a_1,...,c_2)$ as a stable curve
will determine $f$ uniquely with
$(\CC ^\ti )^2$ ambiguity. This $(\CC ^\ti )^2$-bundle
over some open subset of the smooth space
$\overline{M}_{0,6}$ is the local smooth
chart. Notice that for instance,
$(C; a_2, a_1, b_1, b_2, c_2, c_1)$ gives rise to the
same $f$ up to isomorphism.
Thus we have to take account of the quotient by the finite
group permuting the elements of sets
$\{ a_1, a_2 \}$, $\{ b_1, b_2 \}$ and
$\{ c_1, c_2\}$.
{\it Claim:} The stabilizer subgroup $G _{(C;\{ x_{i,j}\} )}$ of
$G$ is
exactly the automorphism group $Aut(f,C)$ of $(f,C)$.
{\it Proof:} We shall construct a correspondence between
$G _{(C;\{ x_{i,j}\})}$
and $Aut(f,C)$. Let $g\in G _{(C;\{ x_{i,j}\} )}$
which is given by one of the suitable permutations of $x_{i,j}$.
So, $g(C;\{ x_{i,j}\} ) = (C; \{ g(x_{i,j})\} )$.
Since the permutation does not change the
stable curve $(C;\{ x_{i,j}\} )$,
there is an isomorphism $h$ from
$(C;\{ x_{i,j}\})$ to $(C;\{ g(x_{i,j})\} )$.
The isomorphism $h$ is unique since
there is no nontrivial automorphism in the stable curve of genus
$0$.
Of course this $h$ gives rise to an automorphism of $(f,C)$.
Conversely, if $h$ is an automorphism of $(f,C)$, then
it induces an isomorphism from $(C;\{ x_{i,j}\} )$
to $(C;\{ g(x_{i,j})\})$ for
a unique permutation $g$ which we allow.
Thus we established 1-1 correspondence,
which can be easily seen to be a group homomorphism.
{\it Remark:}
The action of $Aut(f,C)$ may not be effective in
general. For instance,
see $\overline{M}_{0,0}(\PP ^1,2\kline )$.
\subsection{Gromov-Witten Invariants and $QH_{(T')}^*(V)$}
There are natural morphisms on the moduli spaces,
namely, evaluation maps $e_i$
at the $i$-th marked points and forgetting-marked-point maps $\pi$:
\[\begin{CD}
\overline{M}_{0,n+1}(X,\beta ) @> e_{n+1} >> X \\
@V\pi VV \\
\overline{M}_{0,n}(X,\beta ). \end{CD} \]
If $s_i$ are the universal sections for the marked points, then
$e_i=e_{n+1}\circ s_i$ (here we assume that $\pi$ is the
forgetful map of the last marked point).
In the orbifold charts,
$\pi$ gives the universal family of stable maps as a fine
moduli space.
Consider, for a second homology class $\beta \ne 0$ and
an integer $n\ge 0$, the vector orbi-bundle
$V_{\beta}=\pi _*(e_{n+1}^*(V ))$.
Here $\pi$ is a flat morphism in the level of orbifold charts.
Thus indeed, $V_{\beta}$ is vector orbi-bundle
with the fiber $H^0(C, f^*(V ))$ at
$(f, C; \{ x_i\})$.
Notice that $V_{\beta} =\pi ^*(V_{\beta})$
(it has nothing to do with marked points).
\bigskip
{\bf Notation:} for $A_i\in H_{(T')}^*(X)$,
\begin{eqnarray*}
V_0&:=& V \\
\overline{M}_{0,i}(X,0)&:=& X \ \ \text{ for } i=0,1,2 \\
<A_1,...,A_N>^V_{\beta} &:=& \int _{\overline{M}_{0,N}(X,\beta )}e_1^*(A_1)
\cup ....\cup e_N^*(A_N) \cup E(V_\beta )
\end{eqnarray*}
Then one can show that for all $\beta$
$$\sum _{\beta _1+\beta _2=\beta}\sum _a <A_1,A_2,T_a>^V<T^a,A_3,A_4>^V $$
are totally symmetric in $A_i$. This property will be
equivalent to the associativity of the quantum cohomology of $QH_{(T')}^*(V)$
which we define in the below.
\bigskip
Let us choose a basis $\{ p_i \}_{i=1}^k$
of $H^2(X)$ by classes in the closed K\"ahler cone.
\bigskip
{\bf Notation:}
\begin{eqnarray*} q^\beta &:=& \prod _i q_i^{<p_i,\beta>} \\
<A_1,...,A_N>^V &:=& \sum q^\beta <A_1,....,A_N>^V_\beta \end{eqnarray*}
\bigskip
The quantum multiplication $\circ$
is defined by the following simple requirement;
for $A, B ,C\in H^*_{(T')}(X)$
$$<A\circ B,C >_0^V = <A,B,C>^V$$
which is a formal power series of parameters $q_i$.
Thus our quantum cohomology $QH^*_{(T')}(V)$
is defined as $H^*_{(T')}(X)\ot _{\QQ }\QQ [[q_1,...,q_k]]$ with a product
structure.
\subsection{Givental's Correlators}
We review the topic after \cite{Du, GE}.
\subsubsection{The flat connections and the
fundamental solutions}\label{fund}
Now let $q_i=e^{t_i}$ with the formal parameters $t_i$.
We have a one-parameter family
of the formal $\mathcal D$-module structures on $QH^*_{(T')}(V)$ by
giving a flat connection
$\nabla _i= \hb\frac{\partial}{\partial t_i}-p_i\circ$
for any nonzero $\hb$, $i=1,...,k$.
For the fundamental solutions we introduce $c_i\in H^*_{T'}(
\overline{M}_{0,N}(X,\beta ))$, so-called
Gravitational descendents. These $c_i$ are the first Chern classes of
the universal cotangent line bundles at the $i$-th marked
points.
The line bundles are, by definition,
the dual of the normal bundle of $s_i(\overline{M}_{0,N}(X,\beta ))$
in $\overline{M}_{0,N+1}(X,\beta )$.
\bigskip
{\bf Notation:}
Let $f_i(x)\in H^*_{(T')} [y][[x]]$ for indeterminant $x,y$.
Through out the paper,
\begin{eqnarray*} <A_1f_1(c),...,A_Nf_N(c);B>_{\beta }^X
&:= &\int _{\overline{M}_{0,N}(X,\beta )}
e_1^*(A_1)f_1(c_1) ... e_N^*(A_N)f_N(c_N) B \\
<A_1f_1(c),...,A_Nf_N(c);B>_{\beta }^V
&:=& <A_1f_1(c),...,A_Nf_N(c);BE(V_\beta )>_{\beta}^X \\
<A_1f_1(c),...,A_Nf_N(c);B>^X &=& \sum _{\beta} q^{\beta}
<A_1f_1(c),...,A_Nf_N(c);B>_{\beta }^X \\
<A_1f_1(c),...,A_Nf_N(c);B>^V
&:=& \sum _{\beta} q^{\beta}<A_1f_1(c),...,A_Nf_N(c);B>_{\beta }^V
\end{eqnarray*}
where $A_i\in H^*_{(T')}(X),$ $B\in H^*_{(T')}(\overline{M}_{0,N}(X,\beta ))$.
\bigskip
The system of the first order equations
$\nabla _i \vec{s}=0$, $i=1,...,k$, has
the following complete set of ($\dim H^*(X)$)-many
solutions \cite{GE},
\begin{eqnarray*} \vec{s}_a &:=&
\sum _b<\frac{e^{pt/\hb}T_a}{\hb -c}, T_b>^VT^b ,
\end{eqnarray*}
where $pt$ denotes $\sum _{i=1}^k p_it_i$ and
$\hb$ is a formal variable (but when $\beta =0$, set $\hb =1$).
The following two formulas show that $\vec{s}_a$ are indeed
solutions to the quantum differential system $\nabla _i\vec{s}=0$.
Using that $c_i-\pi ^*(c_i )$ is the fundamental
class $\Delta _i$ represented by the section $s_i:
\overline{M}_{0,n}(X,\beta )\ra
\overline{M}_{0,n+1}(X,\beta )$ and $c_i\cup\Delta _i =0$ (the
image of $s_i$ is isomorphic to
$\overline{M}_{0,3}(X,0)\ti _X\overline{M}_{0,n}(X,\beta )$), it is easy to
derive so-called
the fundamental class axiom and the divisor axiom \cite{W, GE}.
Let $f_i(x)$ be polynomial with coefficients in
$\pi ^*(H^*_{(T')}(\overline{M}_{0,n}(X,d)))$. Let $D$ be a divisor class
in $H^*_{(T')}(X)$. Then (for $n>0$)
\begin{eqnarray*}
<f_1(c),...,f_n(c),1>^V_{\beta}
&=&\sum _i<f_1(c),...,\frac{f_i(c)-f_i(0)}{c},...,f_n(c)>^V_{\beta} ,
\end{eqnarray*}
(where we abuse the notation \lq$f_i(\pi ^*(c))=f_i(c)$',)
and
\begin{eqnarray*}
<f_1(c), ..., f_n(c), D>^V_{\beta}
&=&
<D,\beta >
<f_1(c), ...,f_n(c)>^V_{\beta} \\
+
\sum _i
<f_1(c),...,f_{i-1}(c), &
D\frac{f_i(c)-f_i(0)}{c} &,f_{i+1}(c),..., f_n(c)>^V_{\beta}.
\end{eqnarray*}
\bigskip
Consider
\[e^{pt/\hb }S^V :=
\sum _a<\vec{s}_{a},1>_0^VT^a=
e^{pt/\hb }\sum _a <\frac{T_a}{\hb(\hb -c)}>^VT^a=e^{pt/\hb }(1+o(1/\hb )). \]
which is the main object in this paper. This $S^V$
will be called the Givental's correlator for $(X,V,E)$.
It is an element in $H^*_{(T')}(X)[\hb ^{-1}][[q_1,...,q_k]]$.
Notice that $S^V$ for $(X,V,Euler)$ is homogeneous of degree $0$
if we let $\sum (\deg q _i)p_i= c_1(TX)-c_1(V)$, $\deg \hb =1$,
and $\deg A=b$ if $A\in H^{2b}_{T'}(X)$.
\bigskip
The quantum $\mathcal D$-module of $QH^*_{(T')}(V)$ is defined
by the $\mathcal D$-module generated by $<\vec{s},1>^V_{0}$ for all
flat sections $\vec{s}$.
When there is no $V$ considered, we denote by $QH^*(X)$ the
quantum cohomology. That is, using $<...>^X$, we define
$QH^*(X)$.
\bigskip
{\it Remark:}
Suppose a differential operator $P(\hbar\frac{\partial}{\partial t_i} ,e^{t_i} ,\hb )$ with
coefficients in $H^*_{(T')}$
annihilates $<\vec{s},1>^V_0$ for all flat sections $\vec{s}$,
then $P(p_1,...,p_k,q_1,...q_k,0)$
holds in $QH^*_{(T')}(V)$ \cite{GE}.
\subsubsection{Examples}
The projective space $\PP ^{n}$: It is
well known that in the quantum cohomology ring $QH^*(\PP ^n)$,
$(p\circ )^{n+1}=q$, where $p=c_1({\mathcal O}(1))$
and $q$ is given with respect to the line class dual to $p$.
The corresponding operator is $(\hb\frac d{dt})^{n+1} - e^t$ .
The solutions are
explicitly known in \cite{GH}. $S^X$ is
\[ 1+ \sum _{d>0}e^{dt}
\frac 1{((p+\hb )(p+2\hb )...(p+d\hb ))^{n+1}}.\]
The complete flag manifolds $F(n):$ Let $F(n)$ be the set of all
complete flags $(\CC ^1\subset ...\subset \CC ^n)$ in $\CC ^n$.
The usual cohomology ring is
$\QQ [x_1, x_2,..., x_n]/(I_1, ..., I_n)$
where $x_i$ are the Chern classes of $(S_i/S_{i-1})^*$,
$S_i$ are the universal subbundles with fibers $\CC ^i$ and
$I_i$ are the $i$-th elementary symmetric polynomials of
$x_1,...,x_n$.
Let us use as a basis of $H_2(F(n),\ZZ )$ duals of
the first Chern classes of $(S_i)^*$, $i=1,...,n-1$.
They are in the edges of the closed K\"ahler cone.
Let $A(x_i)$ be a matrix
\[\left( \begin{array}{cccccc}
x_1 & q_1 & 0 & 0 & ... & 0 \\
-1 & x_2 & q_2 & 0 & ... & \\
& ... & & & ... & \\
0 & ... & & -1 & x_{n-1} & q_{n-1} \\
0 & ... & & 0 & -1 & x_n
\end{array}\right).\]
Then the quantum relations are generated
by the coefficients of the characteristic
polynomial of the matrix $A(x_i)$.
The corresponding differential operators turn out to be
obtained by the same method using $A(x_i)$ with arguments
$\hb\frac{\partial}{
\partial t_1}$ instead of $x_1$,
$\hb\frac{\partial}{
\partial t_{i}}-\hb\frac{\partial}{
\partial t_{i-1}}$ instead of $x_i$, and
$-\hb\frac{\partial}{
\partial t_{n-1}}$ instead of $x_n$ \cite{GS, KT}.
These differential operators are the integrals
of the quantized Toda lattices.
The quadratic differential operator of them
can be easily derived.
In fact, given a quantum relation of $F(n)$ between
the divisors $x_i$, there is a unique operator satisfying that
its symbol becomes the relation and it
annihilates $<\vec{s},1>_0$ for all flat sections $\vec{s}$.
In general, the explicit cohomological expression $S^X$ of
solutions to the quantum differential operators are not known.
\subsubsection{The general quintic hypersurface
in $\PP ^4$} Let $Y$ be
a smooth degree $5$ hypersurface in $\PP ^4$.
$Y$ is not a homogeneous space. However, using virtual
fundamental class $[\overline{M}_{0,n}(Y,\beta )]$ \cite{Be, BF,
LT}, one can
define also the quantum cohomology $QH^*(Y)$ of $Y$.
It is expected that
\[ <A_1f_1(c),...,A_Nf_N(c)>^Y = <A_1f_1(c),...,A_Nf_N(c)>^{{\mathcal O}(5)}. \]
Let $p$ be the induced class of
the hyperplane divisor in $\PP ^4$. The quantum
relation is $(p\circ ) ^4=0$.
The corresponding operator is,
however, {\it not} $(\hb\frac d{dt})^4$,
but $$(\hb\frac d{dt})^2
\left( \frac{(\hb\frac d{dt})^2}{5+f(q)}\right),$$ where
$<p\circ p,p>^Y_{0} = 5+f(q)$.
Notice that in this $Calabi-Yau$ 3-fold case,
we lose the whole information of quantum cohomology when one
concerns only the quantum relation, $(p\circ )^4=0$.
The unknown $f(q)$ was conjectured by physicists
\cite{Ca}. The general idea of the prediction
is the following. Roughly speaking,
in theoretical physics, there are quantum
field theories
associated to Calabi-Yau three-folds by $A$-model and $B$-model.
What we have constructed so far are $A$-model objects for
Calabi-Yau three-folds. On the other hand using a
family of the so-called mirror manifolds which are also Calabi-Yau
three-folds, conjecturally
one may construct the equivalent quantum field theory by $B$-model.
The corresponding mirror partner of a quantum differential
equation / quantum ${\mathcal D}$ module
is the Picard-Fuchs differential equation
/ Gauss-Manin connection of the
mirror family. It was predicted that
they are equivalent by a certain transformation.
In \cite{Ca} are obtained the conjectural mirror family of quintics,
the Picard-Fuchs differential equation and the transformation.
That is how the prediction is made. The prediction is now proven to be correct
by Givental \cite{GE}.
\subsection{The idea of the proof of theorem \ref{thmmain}}
To describe the idea, let us
notice that Givental's proof \cite{GT} of the mirror
conjecture for the nonnegative toric complete intersections
can be divided into three parts. (He shows in the paper that
the mirror phenomenon occurs also in non-Calabi-Yau manifolds.)
Let $X$ be a Fano toric manifold
with a big torus $T$,
and $V$ be a $T\ti T'$-equivariant decomposable convex vector bundle over $X$,
where $T'$ acts on $X$ trivially.
\begin{enumerate}
\item
In $A$-part, it is proven that
\begin{enumerate}
\item
the $T\ti T'$-equivariant
solution vector $S^V\in H^*_{(T\ti T')}(X)[[q,\hb ^{-1}]]$ has an
\lq\lq almost recursion relation,"
\item
it satisfies the polynomiality in the so-called
\lq\lq double construction," and
\item
it is uniquely determined by the above two properties with
the aymptotical behavior $S^V=1+o(\frac 1{\hb})$.
\end{enumerate}
\item In $B$-part, another
($T\ti T'$-equivariant hypergeometric) vector $\Phi ^V$, presumably
given by the mirror symmetry conjecture, is constructed.
It is verified that it also satisfies (a) and (b)
using a toric (naive) compactification of holomorphic maps from
$\PP ^1$ to $X$.
\item
When $c_1(X)-c_1(V)$ is nonnegative and $E=Euler$,
there is a suitable equivalence transformation between
$\Phi ^V$ and $S^V$.
\end{enumerate}
In this paper, for a $T\ti T'$ equivariant decomposable
convex vector bundle $V$ over
any compact homogeneous $X$ of a semi-simple complex
Lie group $G$, we will show that $S^V$
satisfies property 1 above. In this case, $T$ is
a maximal torus of $G$.
We define $\Phi ^V$ which corresponds $\Phi ^V$ of the toric case
in property 2:
Let $\Phi ^X=S^X=\sum _d\Phi ^X_{d}q^d$.
For $\Phi ^V$, we will find
a modification $H'_d\in H^*_{T\ti T'}(X)[\hb ]$
(depend on $V$ and $d$) such that
if $\Phi ^V :=\sum _d \Phi ^X_{d}H'_dq^d$,
then
(A) $\Phi ^V$ (after the restriction to
the fixed points) has the almost recursion relation exactly like $S^V$
and
(B) $\Phi ^V$ has
the polynomial property in the double construction.
In fact, we design $H'_d$ to satisfy (A) and (B).
Finally, when $E=Euler$ and $c_1(TX)-c_1(V)$ is nonnegative,
we will prove that a certain operation
will transform $S^V$ to $\Phi ^V$, since they satisfy
the same almost
recursion relation and the polynomiality of the double construction.
\section{The almost recursion relations}\label{sectionre}
As in section \ref{setup}
let $X$ be a homogeneous manifold $G/P$ where $G$ is a complex semi-simple
Lie group and $P$ is a parabolic subgroup. Let $T$ be a maximal torus.
The $T$ action has only isolated fixed points $\{ v,w,... \}$.
The one dimensional invariant orbit of $T$ is analyzed in detail in
section \ref{grass} and \ref{flag}. For a moment we need the fact
that the closures of orbits are {\em finite}
$\PP ^1$'s connecting a fixed point $v$ to another
fixed point $w$.
For a given equivariant vector bundle $W$
over a $T\ti T'$-space $M$, we use $[W]$ which denote the element in
the $K$-group
$K^0_{T\ti T'}(M)$ corresponding to the $T\ti T'$ vector bundle $W$.
\bigskip
The torus action on $X$ induces the natural action
on the moduli space of stable maps by the functorial property.
Since the evaluation maps are $T\ti T'$-equivariant,
the pullbacks of $T\ti T'$-bundles have natural actions
in the orbifold sense.
In turn, $V_\beta$ has
the induced $T\ti T'$-action.
{\it All ingredients in section 2
are from now on the equivariant ones.}
We would like to evaluate $S^V$ as a specialization
of the equivariant one corresponding to $S^V$.
We use the same notation
$S^V\in H^*_{(T\ti T')}(X)[[\hb ^{-1}]][[q_1,...,q_k]]$
for the equivariant one.
Notice that $S^V$ might have power series of $\hb ^{-1}$ in each coefficient
of $q^d$, since $c$ are not anymore nilpotent.
Using the localization theorem,
we shall find an \lq\lq almost recursion relation" on the equivariant
Givental correlator.
To begin with,
we summarize the fixed points of the induced action on
the moduli space of stable maps.
If a stable map represents a fixed point in the moduli
space, the image of the map should lie in the
closure of the 1-dimensional
orbits. The special points are mapped to isolated fixed points.
Let us denote by $\ka _{v,w}$
the character of the tangent space of a
1-dimensional orbit connecting an isolated fixed point $v$ to another $w$.
Then, $-\ka _{v, w}$ is the character of the tangent
line of the 1-dimensional
orbit $o(v,w)$ at $w$. We use $\beta _{v,w}$ to stand for
the second homology class represented by the ray.
Denote by $o(v)$ the set of all fixed points $w\ne v$ which
can be connected by a one-dimensional orbit with $v$.
\bigskip
{\lemma\label{lemmare} {\bf Recursion Lemma} {\em (\cite {GE})}
Denote by $\phi _{v}$ the equivariant classes $i_*(1)$ at $v$,
here $i_v$ denotes the $T\ti T'$-equivariant
inclusion of the point $v$ into $(X,V)$.
Then $S^V\in H^*_{T\ti T'}(X)[[\hb ^{-1}]][[q_1,...,q_k]]$
has an \lq\lq almost" recursion relation, namely, for any $v\in X^{T}$,
0) $S^V_v(q,\hb ):=<S^V,\phi _v>^V_0 \in H^*_{(T\ti T')}(\hb )[[q]]$
and the substitution
$S_w(q,-\ka _{v,w}/m)$ of $\hb$ with $-\ka _{v,w}$ in $S_w(q,\hb )$
is well-defined,
1) The difference $R_v$ of $S^V_v(q,\hb )$ and the \lq\lq recursion part"
is a power series of $q$
over the {\em polynomial} ring of $1/\hb$, that is,
\[ R_v:=
S^V_v(q,\hb ) -
\sum _{w\in o(v),\ m>0} q^{m\beta _{v,w}}
\frac {(-\ka _{v,w})/m}{\hb (\ka _{v,w}+m\hb)}
\frac{E(V_{v, w,m})i_v^*(\phi _v)}
{Euler(N_{v,w,m})}S_w(q,-\ka _{v,w}/m),
\]
is in $H^*_{(T\ti T')}[\hb ^{-1}][[q]]$, where
$V_{v,w,m}$ is $T\ti T'$ representation space $H^0(\PP ^1,f^*V)$,
here $f$ is the totally ramified $m$-fold map onto $o(v,w)$
over $v$ and $w$; and
$N_{v,w,m}$ is the $T\ti T'$-representation space $[H^0(\PP ^1,f^*TX)]-[0]$;
and
2) furthermore, for $S^X$ itself, the first term $R_v$
is $1$.}
\bigskip
We will say that the statement 1) reveals the almost recursion relation
of $S^V$.
The statement 2) shows that $S_v^X$ have recursion relations in the
ordinary sense.
\bigskip
{\bf Proof.}
First of all, using the short exact sequence
\[
0 \ra Ker \ra V_d \ra e_1^*(V) \ra 0 \]
over $\overline{M}_{0,1}(X,d)$, we see that
$S^V$ is indeed in $H^*_{T\ti T'}[[\hb ^{-1}]][[q_1,...,q_k]]$.
(The last map in the sequence is given by the evaluation
of global sections at the marked point.)
A connected component of the $T$-fixed loci of
the moduli space $X_d:=\overline{M}_{0,1}(X,d)$ is isomorphic to a product of
Deline-Mumford spaces with marked points from
the special points
of the inverse image $f^{-1}(v)$ of the generic $f$
in the component for all $v\in X^{T}$.
Now fix a $v$ and consider $S^V_v$.
It is enough to count the fixed locus $F^{d,v}$ where
the marked point $x$ should be mapped to the
fixed point $v$ since $\phi _v$ can be supported only
near the point. For a stable map
$(f,C;x)$ denote by $C_1$
the irreducible component of $C$ containing the marked
point $x$.
Then $F^{d,v}$ is the disjoint union of
\[ F^{d,v}_1:=\{ (f,C;x)\in F^{d,v} \ | \ f(C_1)=v \}\]
and
\[ F^{d,v}_2:=
\bigcup _{w\in o(v), m=1,...,m\beta _{v,w}\le d} F^{d,v,w,m}, \]
where $F^{d,v,w,m}$ is
\[ \{ (f,C;x)\in F^{d,v} \ | \
w\in f(C_1),\ \deg f|_{C_1}=m \}. \]
$S^V_v$ is an integral over $F^{d,v}$'s by a
localization theorem for orbifolds.
We claim that the integral of
\[ \frac{E(V_d)e_1^*(\phi _v)}{\hb (\hb -c )} \]
over $F^{d,v}_1$ is in $H^*_{(T\ti T')}[\hb ^{-1}]$.
The reason is that
the universal cotangent line bundle over $F^{d,v}_1$
in the moduli space has the trivial action. It implies that
the equivariant class $c$ restricted to $F^{d,v}_1$ is nilpotent.
\smallskip
Now we shall obtain the \lq almost recursion relation'
from the contribution of the fixed loci $F^{d,v}_2$.
Denote $d-m\beta {v,w}$ by $d'$.
Since $C_1$ is always one end of $C$ for any $(f,C;x)\in F^{d,v}_2$,
we can have a natural isomorphism from
$F^{d',w}$ to $F^{d,v,w,m}$, where $F^{d',w}$ are
fixed loci in $X_{d'}:=
\overline{M}_{0,1}(X,d')$, consisting
of the stable maps sending the marked points to $w$.
We obtain the morphism, joining the $m$-covering of $o(v,w)$ to stable
maps in $F^{d',w}$. By the $m$-covering of $o(v,w)$, we
mean a totally $m$-ramified map from $\PP ^1\cong C_1$
to $o(v,w)$ over $v$ and $w$. Let $x'=f^{-1}(w)\cap C_1$.
\bigskip
We claim that the normal bundles as in
$K^0(F^{d, v,w,m}\cong F^{d',w})$ satisfy the equality
\begin{eqnarray}
[N_{X_d/F^{d, v,w,m}}]-[N_{X_{d'}/F^{d',w}}] &=&
[N_{v,w,m}] -[T_wX]
+[T_{x'}C_1\ot L|_{F^{d',w}}] \label{normal}
\end{eqnarray}
where $L$ is the universal tangent line bundle
over $X_{d'}$.
The reason of the claim is as follows:
Recall that each fixed component is isomorphic to the product
of moduli space of stable curves (see section 3 in \cite{Ko} for detail).
Hence, we conclude that
$[N_{X_d/F^{d, v,w,m}}]-[N_{X_{d'}/F^{d',w}}]
-[L|_{F^{d',w}}]$ (over each
fixed components)
is equal to a trivial bundle with nontrivial actions.
The twister by action can be computed by study of action
on normal spaces at $(f,C_1\cup C_2 ; x)\in F^{d,v,w,m}$.
Let $N_1$ be the normal space of $F^{d,v,w,m}$ at $(f,C_1\cup C_2 ;x )$
and $N_2$ be the normal space of $F^{d',w}$
at $(f|_{C_2}, C_2; x':=C_1\cap C_2)$. Then
as representation spaces
\begin{eqnarray*} [N_1] &= &[N_2]+([H^0(C_1, f|_{C_1}^*TX)]-[H^0(C_1,TC_1)]) -[T_{w}X] \\
&+& [T_{x'}C_1\ot T_{x'}C_2] +[T_{x'}C_1]+[T_xC_1].\end{eqnarray*}
Hence we conclude the claim (\ref{normal}) after canceling of $[H^0(C_1,TC_1)]
=[0]+[T_{x'}C_1]+[T_xC_1]$.
\bigskip
On the other hand,
the direct sum of
the fiber of $V_d$ at $(f,C_1\cup C_2 ; x)\in F^{d,v,w,m}$
and $V|_w$
is equal to the direct sum of the fiber of
$V_{d'}$ at $(f|_{C_2}, C_2;x')$
and $H^0(C_1, (f|_{C_1})^*V)$.
Thus, applying the localization theorem we obtain
\begin{eqnarray*}
&&\int _{X_d}\frac{E(V_d)e_1^*(\phi _v)}{\hb(\hb -c)}=I+
\sum _{w\in o(m), 0<m; m\beta _{v,w}\le d}
\frac{E(V_{v,w,m})i^*_v(\phi _v)(-\ka _{v,w}/m)}{m\hb (\ka _{v,w}/m +\hb )
Euler(N_{v,w,m})} \\
&& \ti \int _{X_{d-m\beta _{v,w}}}
\frac{E(V_{d-m\beta _{v,w}})e_1^*(\phi _w)}{(-\ka _{v,w}/m)(-\ka _{v,w}/m - c)},
\end{eqnarray*}
where $I$ is the integral over $F^{d,v}_1$.
The factor $m$ in $m\hb (\ka _{v,w}/m +\hb ) $ comes from the nature of
orbifold localization theorem. (There are $m$ automorphisms of $f|_{C_1}$.)
\bigskip
Using induction on $|d|=\sum d_i$,
we may assume that the integral factors in the second term
are well-defined and belong to $H^*_{(T\ti T')}$.
(Localization theorem itself also explains them.)
So, statements 0) and 1) in the lemma are proven.
\smallskip
Now let us prove statement 2).
Since
$<c_1(TX),\beta >\ge 2$ for all $\beta$, by
degree counting we see that there are no contributions from
the integral over $F^{d,v}_1$. The reason is that
$\dim \overline{M}_{0,\sum d_i+1} =(\sum d_i)-2$ is less than
$2(\sum d_i )-2$ if $(d_1,..,d_k)\ne 0$ and
$\dim \overline{M}_{0,1}(X,d)\ge 2\sum d_i+\dim X -2$.
So, in the case of $S^X$, $R_v =1$.
\section{The double construction}
{\lemma\label{lemmado}{\bf Double Construction Lemma}
The double construction
\[ W(S^V):=\int _{V} S^V (qe^{\hb z},\hb )e^{\sum p_iz_i}S^V (q,-\hb )\]
is a power series of $q_1,...,q_k$ and $z_1,...,z_k$ with
coefficients in $H_{T\ti T'}^*[\hb ]$.}
\bigskip
A priori $W(S^V)$ has coefficients in Laurent power series ring of
$\hb ^{-1}$ over $H_{T\ti T'}^*$.
For the proof we will make use of graph spaces
and universal classes defined in the below.
\subsection{The main lemma}
Let $L_d$ be the projective space of the
collection of all $(f_0,...,f_N)$ such that
$f_i(z_0,z_1)$ are homogeneous polynomials of degree $d$.
$L_d$ is isomorphic to $\PP ^{(d+1)(N+1)-1}$.
Given a stable map of degree $(d,1)$ from a prestable curve
$C$ to $\PP ^N\ti \PP ^1$,
there is a special irreducible component $C_0$ of $C$ such that $C_0$
has degree $(d_0,1)$ under the stable map. This special
component $C_0$ is parameterized by $\PP ^1$ in the target space.
Thus we can identify $C_0$ with $\PP ^1$ and keep track where
the other components intersect. Suppose the other
connected components $C_1,...,C_l$ of $C-C_0$
intersect with $C_0=\PP ^1$ at $[x_1: y_1],...,[x_l: y_l]$.
If the degrees of $C_i$ are $d_i$ under the stable map,
we now associate the stable map to
$$\prod _{i=1}^l(y_iz_0 -x_iz_1)^{d_i}(f^0_0,...,f_N^0),$$
where $(f^0_0,...,f_N^0)$ are the polynomials coming from
the data of the restriction of $f$ to $C_0$.
\bigskip
{\bf Main lemma: } (Givental \cite{GE})
{\em The above \lq\lq polynomial"
mapping from $G_d(\PP ^N):=\overline{M}_{0,0}(
\PP ^N\ti \PP ^1, (d,1))$ to $L_d$ is a
$(\CC ^\ti)^N\ti \CC ^\ti$-equivariant morphism,
where $\PP ^N$ has the diagonal $(\CC ^\ti)^N$ action
and $\PP ^1$ has the $\CC ^\ti$ action by $[z_0:z_1]\mapsto
[tz_0:z_1]$ for $t\in \CC ^\ti$.}
\bigskip
Notice that the $\CC^\ti$ action on $L_d$ is given by
\[ [f_0(z_0,z_1):...:f_N(z_0,z_1)]
\mapsto [f_0(t^{-1}z_0,z_1):...:f_N(t^{-1}z_0,z_1)]\]
for $t\in \CC ^\ti$.
\subsection{The universal class}
The $T\ti T'$-equivariant spanned line bundle $U_i$ over $X$
gives rise to the $T\ti T'$-equivariant morphism $\mu ^i_0: X\ra\PP ^N$,
and
so we obtain:
\[
\begin{CD}
(\mu _d^i) ^*({\mathcal O}(1)) @. @. {\mathcal O}(1) \\
@VVV @. @VVV \\
G_d(X)@>>> G_{d_i}(\PP ^N)@>>> L_{d_i} \\
\end{CD},
\]
where $G_d(X)$ is the graph
space $\overline{M}_{0,0}(X\ti \PP ^1, (d,1))$,
and $\mu _d^i$ is the $T\ti T'\ti \CC ^\ti$-equivariant
map from $G_d(X)$ to $L_{d_i}$.
On ${\mathcal O}(1)$ we choose the lifted $\CC ^{\ti}$-action coming from
the action on the vector space of $N+1$ $d_i$-homogeneous polynomials by
\[ [f_0(z_0,z_1):...:f_N(z_0,z_1)]
\mapsto [f_0(z_0,tz_1):...:f_N(z_0,tz_1)]\]
for $t\in \CC ^\ti$.
Denote by $P_i=c_1((\mu ^i_d) ^*{\mathcal O}(1))$, the $T\ti T'\ti
\CC ^\ti $-equivariant
Chern class. It is said to be
a universal class in the paper \cite{GT}.
Denote by $W_d$ the vector orbi-bundle
over $G_d(X)$ with the fiber $H^0(C,\psi ^*\pi _1^*V)$
at $(C,\psi )$: Consider
\[\begin{CD}
G_{d,1}(X)@>>{e_1}> X\ti \PP ^1 \\
@V\pi VV @AAA \\
G_d(X) @. \pi _1^*V,
\end{CD}\]
where $G_{d,1}(X)$ denotes the graphs space with one marked point
and $\pi _1$ is the projection of $X\ti \PP ^1$ to
the first factor $X$. Then $W_d:=\pi _* e_1^*\pi _1^*V$.
\subsection{Proof of Lemma \ref{lemmado}}\label{pfdo}
It is enough to show the equality
\[\sum _dq^d\int _{G_d(X)}e^{Pz}E(W_d)
=\int _{V} S^V (q,\hb )e^{pz}S^V (qe^{-\hb z},-\hb ).\]
The left integral is a $T\ti T'\ti \CC ^{\ti}$-equivariant
push forward with $\hb$ as $c_1({\mathcal O}(1))$
over $\PP ^{\infty}$ and the right one is a $T\ti T'$-equivariant push forward
with a formal variable $\hb$.
\bigskip
We will apply localization theorem.
Let us analyze the $\CC ^\ti$-action fixed loci
$G_d(X)^{\CC ^\ti}$ of $G_d(X)$.
$G_d(X)^{\CC ^\ti}$ is isomorphic to $\sum _{d^{(1)}+d^{(2)}=d}
\overline{M}_{0,1}(X,d^{(1)})\ti _X\overline{M}_{0,1}(X,d^{(2)})$.
\bigskip
Suppose $|d^{(1)}|+|d^{(2)}|\ne 0$. The normal bundle is as follows:
When $|d^{(1)}||d^{(2)}|=0$: The codimension is 2 (one from
the nodal condition and the other from the condition
of the image of the nodal point).
Then the Euler class of the normal bundle is $\hb (\hb -c_0)$,
or $-\hb (-\hb -c_\infty)$, where $c_0$ and $c_\infty$ are
the Chern classes of universal
cotangent line bundles of the first marked point
over $\overline{M}_{0,1}(X, d^{(1)})$ and
$\overline{M}_{0,1}(X, d^{(2)})$, respectively.
Here we assume the following convention:
$0=[0:1], \infty =[1:0]$, the associated equivariant line bundle
to the character 1 of the group $\CC ^\ti$ has $\hb$ as its equivariant
Chern class.
When $|d^{(1)}||d^{(2)}|\ne 0$: The codimension is $4$ and
the Euler class is $\hb (\hb -c_0)(-\hb )(-\hb -c_\infty )$.
Here, for instance, $c_0\in
H^2(\overline{M}_{0,1}(X,d^{(1)})\ti _X\overline{M}_{0,1}(X,d^{(2)}))$
is the pull-back of
the Chern class of the universal cotangent line bundle
of the first factor of
$\overline{M}_{0,1}(X,d^{(1)})\ti _X\overline{M}_{0,1}(X,d^{(2)})$.
\bigskip
Let us analyze $P_i$ restricted to $G_d(X)^{\CC ^\ti}$.
Consider the commutative diagram,
\[\begin{CD}
G_{d^{(1)},d^{(2)}}(X):=\overline{M}_{0,1}(X,d^{(1)})\ti _X
\overline{M}_{0,1}(X,d^{(2)})
@>>\mu _d^i > L_{d_i} @.
\ni z_0^{d^{(1)}_i}z_1^{d^{2)}_i}[x_0:...:x_N]\\
@V{\pi _2} VV @AAA @AAA\\
\overline{M}_{0,1}(X,d^{(2)})
@>>{\mu ^i_0\circ e_1}> \PP ^N @. \ni [x_0:...:x_N],
\end{CD}\]
where the first vertical map $\pi _2$ is the projection and
under the second vertical map
$\PP ^N$ is embedded into $L_{d_i}$ as the $\CC ^\ti$
-action fixed locus of the part
$\{ z_0^{d^{(1)}}z_1^{d^{(2)}}[x_0:...:x_N] | [x_0:...:x_N]\in
\PP ^N\} $.
One concludes that
$e_1^*\circ (\mu _0^i)^*
(c_1({\mathcal O}(1)|_{\PP ^N}))=
e_1^*(p_i)-d_i^{(2)}\hb $ and so
\[ \sum P_iz_i|_{G_{d^{(1)},d^{(2)}}}
=\sum (\pi _2^*e_1^*(p_i) -d^{(2)}_i\hb )z_i. \]
Since
\begin{eqnarray*}
&& S^V(q,\hb )e^{pz} S^V(qe^{-\hb z},-\hb) \\
&=& \sum _{a ,b,d^{(1)},d^{(2)} } <\frac{T_{a}}{\hb (\hb -c)}>^V_{d^{(1)}}
T^{a}q^{d^{(1)}}
e^{pz}<\frac{T^{b}}{-\hb (-\hb -c)}>^V_{d^{(2)}}T_{b}q^{d^{(2)}}e^{-d^{(2)}\hb z},
\end{eqnarray*}
we see that
\begin{eqnarray*}
& & \int _{V} S^V(q,\hb)e^{pz}S^V(qe^{-\hb z},-\hb) \\
&=& \sum _{a ,d^{(1)},d^{(2)}}<\frac{T_{a}}{\hb (\hb -c)}>^V_{d^{(1)}}
<\frac{T^{a}e^{pz-d^{(2)}\hb z}}{-\hb (-\hb -c)}>^V_{d^{(2)}}q^{d^{(1)}+d^{(2)}} \\
&=& \sum _d q^d\int _{G_{d^{(1)},d^{(2)}}(X)}
\frac{e^{(\pi _2^*e_1^*p-d^{(2)}\hb)z}E(W_d)}
{[N_{G_d(X)/G_{d^{(1)},d^{(2)}}(X)}]} \\
&=& \sum _d q^d\int _{G_d(X)}e^{Pz}E(W_d),
\end{eqnarray*}
after applying the localization theorem only for $\CC ^\ti$ action
on $G_d(X)$.
\section{The class ${\mathcal P}({\mathcal C})$
and mirror transformations}\label{sectiontr}
\subsection{The class $\mathcal P({\mathcal C})$}
Let
${\mathcal C}$ be the collection of
given data of
$C_{v,w,m}\in H^*_{(T\ti T')}$, $\ka _{v,w}\in H^*_{T\ti T'}$, and
$\beta _{v,w}\in \Lambda -0$, for all
$(v,w,m)\in X^{T}\ti X^{T}\ti \NN$ with $v\in o(w)$. Here $\NN$ is
the set of positive integers.
Assume that $(p_i)_w-(p_i)_v=-<p_i,\beta _{v,w}>\ka _{v,w}$ for all $i=1,...,k$.
Define degree of $\hb$ as 1. Let $q_1,...,q_k$ be formal
parameters with
some given nonnegative degrees. Define the degree of a homogeneous
class of $H_{T\ti T'}^b(X)$ as $b/2$.
Let $\mathcal P({\mathcal C})$ be the class of all
$Z(q,\hb )\in H^*_{T\ti T'}(X)[[\hb ^{-1}, q]]$ of
homogeneous degree 0 such that
\begin{description}
\item[a)] $Z(0,\hb )=1$,
$Z_v(q,\hb ):=<Z,\phi _v>^V_0$ (this is not depend on $V$)
is in $H^*_{(T\ti T')}(\hb )[[q]]$ for any fixed
point $v$, and $Z_w(q,-\ka _{v,w}/m )$ are well-defined
for all $v\in o(w),m>0$ ($m$ are
positive integers),
\item[b)] the almost recursion relation for each fixed point
$v$ holds, that is by definition,
\begin{eqnarray*} R_v:= Z_v(q, \hb ) -
\sum _{m>0,w\in o(v)}
q^{m\beta _{v,w}}\frac{C_{v,w,m}}{\hb (\ka _{v,w} +m\hb )}Z_w(q,-\ka _{v,w}/m),
\end{eqnarray*}
is in $H^*_{(T\ti T')}[\hb ^{-1}][[q]]$, where
$$ q^{m\beta _{v,w}}:=\prod _i q_i^{m<p_i,\beta_{v,w}>}$$; and
\item[c)] in the double construction
\[ W(Z)(q,z):=\int _{V}Z(qe^{\hb z}, \hb )e^{\sum p_iz_i}Z(q,-\hb ), \]
is in $H^*_{T\ti T'}[\hb][[q,z]]$.
(We use the multi-index notation for $z=(z_1,...,z_k)$ and $q=(q_1,...,q_k)$.)
\end{description}
\bigskip
Whenever the data $\mathcal C$ comes from $(X,V,E)$ as in lemma
\ref{lemmare}, we denote the class by ${\mathcal P}(X,V,E)$.
So, in the case
\[ C_{v,w,m} :=C_{v,w,m}^V:= \frac{(-\ka _{v,w})/m\ E(V_{v, w,m})i_v^*(\phi _v)}
{Euler(N_{v,w,m})},
\]
$\ka _{v,w}$ is the character of $T_vo(v,w)$,
$\beta _{v,w}=[o(v,w)]\in H_2(X,\ZZ )$,
and $$c_1(TX)-c_1(V)=\sum _{i=1,...,k}(\deg q_i) p_i.$$
So far, we proved that $S^V$ for $E=Euler$ is in class
${\mathcal P}(X,V,Euler)$.
\bigskip
In the below we introduce
on ${\mathcal P}({\mathcal C})$
a transformation group generated by the following three types of operations.
\begin{description}
\item[1) Multiplication by $f(q)$]
Let $f(q)=\sum _{d\ge 0}f_dq^d$, where $f_d\in \QQ$,
$f(q)$ is homogeneous of degree $0$, and $f(0)=1$.
Then $f(q)Z\in {\mathcal P}({\mathcal C})$.
\item[2) Multiplication by $\mathrm{exp} (f(q)/\hb )$]
Let $f(q)=\sum _{d>0}f_d q^d$,
where $f_d$ are in $H^*_{T\ti T'}$.
Suppose $\deg (f(q))=1$.
Then $Z^{new}:=\exp (f(q)/\hb )Z$ is still in $\mathcal P({\mathcal C})$.
\item[3) Coordinate changes]
Consider a transformation: $$Z\ra Z^{new}:=
\exp (\sum _if_i(q)p_i/\hb )Z(q\exp (f(q)), \hb )$$
where $f_i(q)=\sum _{d>0} f^{(d)}_iq^d$ of homogeneous degree $0$,
$f_i^{(d)}\in \QQ$, and $q\exp (f(q))=(q_1\exp (f_1(q)),...,q_k\exp (f_k(q)))$.
Then $Z^{new}$ is still in $\mathcal P({\mathcal C})$.
\end{description}
Let us call the transformation group the mirror group.
{\theorem\label{thmtr} {(\em \cite{GT})}
Suppose $\deg q $ are nonnegative
and there is at least one element
of form $1+o(\hb ^{-1} )$ in the class ${\mathcal P}({\mathcal C})$.
Then the mirror group action on ${\mathcal P}({\mathcal C})$ is free and
transitive.}
\bigskip
First, we will check 1), 2) and 3); and prove the so-called
uniqueness lemma and then theorem above.
{\it Proof of 1).}
First, $Z^{new}:=fZ$ is homogeneous of degree $0$, $f(0)Z(0,\hb )=1$,
and $fZ_v$ are in
$H_{(T\ti T')}^*(\hb )[[q]]$, and of course $Z^{new}_w(q,-\ka _{v,w}/m)$ are
well-defined.
Second,
\begin{eqnarray*} Z_v^{new} &=&
f(q)R_v
+ \sum q^{m\beta _{v,w}}\frac{C_{v,w,m}}{\hb (\ka _{v,w} +m\hb )}
Z^{new}_w(q,-\ka _{v,w}/m).
\end{eqnarray*}
Thus $fZ$ has the almost recursion relation.
Finally,
\begin{eqnarray*} W^{new}
&:=&\int _{V} Z^{new}(qe^{\hb z},\hb )e^{pz}Z^{new}(q,-\hb ) \\
&=& f(qe^{\hb z})f(q)W,
\end{eqnarray*}
which still has the polynomial coefficients in $H^*_{T\ti T'}[\hb ]$.
{\it Proof of 2).}
The new $Z^{new}$ is homogeneous of degree $0$, $Z^{new}(0,\hb)=1$,
$Z^{new}_v$ are in $H^*_{(T\ti T')}(\hb )[[q]]$,
and $Z^{new}_w(q,-\ka _{v,w}/m)$ are well-defined.
Since $\exp (\frac{f(q)}{\hb }+\frac{mf(q )}{\ka _{v,w}})
=1+(\ka _{v,w}+m\hb ) g_{\ka _{v,w},m}$ and
$g_{\ka _{v,w},m}$ is a
$q$-series with polynomial coefficients in $H^*_{(T\ti T')}[\hb ^{-1}]$,
$Z^{new}$ has the almost recursion relation.
Once again,
\begin{eqnarray*}
W^{new}
&=& \exp(\frac {1}{\hb} (f(qe^{\hb z})-f(q)))W .\end{eqnarray*}
But $f(qe^{\hb z} )-f(q ) =\sum_{d>0} f_d((e^{\hb z})^d -1) q^d$
is a $(z,q)$-series with polynomial coefficients
in $\hb H^*_{T\ti T'}[\hb ]$.
{\it Proof of 3).}
The $Z^{new}$ is homogeneous of degree $0$, $Z^{new}(0,\hb )=1$,
$Z^{new}_v$
are in $H^*_{(T\ti T')}( \hb )[[q]]$, and $Z_w^{new}(q,-\ka _{v,w}/m)$
make sense.
Since $(p_i)_w-(p_i)_v=-<p_i,\beta _{v,w}>\ka _{v,w}$,
\begin{eqnarray*} && \sum _if_i(q)(p_i)_v/\hb
= \sum _i f_i(q)(p_i)_w/(-\ka _{v,w}/m) \\
&& - m\sum _i <p_i,\beta _{v,w}>f_i(q)
+ \sum _i\frac{f_i(q)(p_i)_v }{\ka _{v,w}\hb }(m\hb +\ka _{v,w}) .
\end{eqnarray*}
The exponential of the last term on the right
can be denoted by $1+(\ka _{v,w}+m\hb )g_{\ka _{v,w},m}$
where $g_{\ka _{v,w},m}$ is a $q$-series with coefficients which
are in $H^*_{(T\ti T')}[\hb ^{-1}]$.
$Z^{new}$ satisfies the almost recursion relation.
Consider the double construction
\begin{eqnarray*}
W^{new}(q,z) &=&
\int _{V}e^{f(qe^{\hb z})p/\hb } Z(qe^{\hb z}e^{f(qe^{\hb z})},\hb )
e^{pz}e^{-f(q)p/\hb }Z(qe^{f(q)},-\hb ) \\
&=& W(qe^{f(q)}, z+\frac{f(qe^{\hb z})-f(q)}{\hb }) . \end{eqnarray*}
But since $f(qe^{\hb z})-f(q)$ is divisible by $\hb $, $W^{new}$
is a polynomial $(q,z)$-series.
\bigskip
{\lemma\label{lemmaun} {\bf Uniqueness Lemma}
Let $Z=\sum _{d\ge 0}Z_dq^d$ and $Z'=\sum _{d\ge 0} Z'_dq^d$ be series in
${\mathcal P}({\mathcal C})$.
Suppose $Z\equiv Z'$ modulo $(\frac 1\hb )^2$.
Then $Z'=Z$.}
\bigskip
{\bf Proof.}
We may suppose that
$Z'_{d}=Z_{d}$ for all $0\le d <d_0$ for some $d_0\ge 1$.
Let \begin{eqnarray*}
D(\hb ):=Z'_{d_0}-Z_{d_0}
&=& A\hb ^{-2r-1}+B\hb ^{-2r}+....\\
&=&\hb ^{-2r}(A/\hb +B + O(\hb )) ,\end{eqnarray*}
where $A, \ B \in H^*_{T\ti T'}(X)$. ($A$ might be $0$.)
This is possible since $<D,\phi _v>_0$ for all $v$ are polynomials
of $1/\hb$ over $H^*_{(T\ti t')}$
and so $D$ is a polynomial of $1/\hb$ over $H^*_{T\ti T'}(X)$.
Consider the coefficient of $q^{d_0}$ in $W(Z')-W(Z)$, which can
be set $\delta (D)
=\int _{V}e^{(p+d_0\hb )z}D(\hb )+e^{pz}D(-\hb ) $.
If $r=0$, then $D=0$ since $D\equiv 0$ modulo $(1/\hb )^2$.
Assume $r\ge 1$. We shall show that $A=0=B$, which
implies by induction that $D=0$.
Notice that, since $\kd (D)$ is a polynomial of $\hb $,
\begin{eqnarray*}
O(\hb ^2)=\hb ^{2r}\kd (D) &=&\int _{V}e^{(p+d_0\hb )z} (A/\hb + B+ O(\hb ))
+e^{pz}(-A/\hb +B +O(\hb )) \\
&=&\int _{V}e^{pz}Ad_oz +2Be^{pz} +O(\hb ).
\end{eqnarray*}
So,
\begin{eqnarray*}
0 &=& d_0z\int _{V}e^{pz}A + 2 \int _{V}Be^{pz} \\
&=& \sum _{v\in X^T}(d_0ze^{p_vz}A_v+ 2 e^{p_vz}B_v)\frac {1}{i_v^*(\phi _v)},
\end{eqnarray*}
where $A_v$, $B_v$, and $Euler(V)_v$
are the restrictions of $A$, $B$, and $Euler(V)$ to the fixed
point $v$, respectively.
Since
$p_vz$ are different as $v$ are different (this can be seen in
section \ref{grass} and \ref{flag}),
$e^{p_v z}$ and $ze^{p_vz}$
are independent over $H^*_{(T\ti T')}$.
So we conclude that
$A_v=0=B_v$ for all $v$, and hence $A=0=B$.
\subsection{Proof of theorem \ref{thmtr}}
It suffices to show the transitivity of the action.
Let $Z_1$ and $Z_2$ be in class ${\mathcal P}({\mathcal C})$
and let $Z_1=1+o(1/\hb )$.
Since $\deg q\ge 0$, we may let
$$ Z_2=Z_2^{(0)} +Z_2^{(1)}\frac 1\hb + o(\frac 1\hb ),$$ where
$Z_2^{(0)}\in H^*_{T\ti T'}(X)[[q]]$
is of homogeneous of degree $0$ and $Z_2^{(1)}$
is homogeneous of degree $1$. Furthermore,
$Z_2^{(0)}(q)\in H^*_{T\ti T'}(X)[[q]]$ is a $q$-series with coefficients
in $\QQ$, $Z_2^{(0)}(0)=1$, and
$Z_2^{(1)}(q)$ is a $q$-series with coefficients in $H^*_{T\ti T'} [p]$
by degree counting.
We may let $$\frac{Z_2^{(1)}(q )}{Z _2^{(0)}(q)}=
\sum _i (f_i(q)\cdot p_i) + g(q),$$
where $f_i(q)$
are pure $q$-series over $\QQ$ of degree 0 and $g(q )$ are degree 1
in $H^*_{T\ti T'}[[q]]$.
In addition, $f_i(0)=0=g(0)$.
Now, consider operations on $Z_1$: first, coordinate changes,
\[ Z_1'=\exp (f(q)p/\hb )Z_1(q\exp (f(q)), \hb )
= 1+ f(q)p/\hb +o(1/\hb ), \]
second, multiplication by $\exp (g(q)/\hb )$,
\[Z_1''=\exp (g(q) /\hb ) Z_1'
= 1 + \frac 1\hb (f(q)p + g(q)) + o(1/\hb ), \]
finally, multiplication by $Z_2^{(0)}(q)$,
\[ Z_1''' = Z_2 ^{(0)}(q)Z_1''
=Z_2 ^{(0)} +\frac 1\hb Z_2 ^{(1)} + o (1/\hb ).\]
According to the uniqueness lemma, the last one $Z_1'''$ must be equal to
$Z_2$ since
$Z_1'''\cong Z_2$ modulo $(1/\hb )^2$.
\subsection{Transformation from $J^V$ to $I^V$}
We explain the transformation introduced in the introduction.
Let $\tilde{Z}$ be the nonequivariant specialization of $Z$.
Let $Z_1$ and $Z_2$ be in class ${\mathcal P}({\mathcal C})$
and let $Z_1=1+o(1/\hb )$.
Now let us specialize the equivariant setting to nonequivariant one.
Let $J^V=e^{(t_0+pt)/\hb }\tilde{Z_1}(q)$ and
$I^V=e^{(t_0+pt)/\hb }\tilde{Z_2}(q)$.
Then, they are equivalent up to the unique coordinate
change $t_0\mapsto t_0 + f_0(q)\hb + f_{-1}(q)$
and $t\mapsto t_i + f_i(q)$, $i=1,...,k$, where
$f_j\in \QQ [[q]]$ for all $j$,
$f_0$ and $f_i$ ($i=1,...,k$) have degree $0$, $f_{-1}$ has degree $1$; and
$f_j(0)=0$ for all $j$.
\section{The modified B series}
Let $X$ be a homogeneous manifold with
the torus $T\ti T'$ action. From now on
let $V=L_1\oplus ...\oplus L_l$ be an equivariant decomposable
convex vector bundle over $X$, where $L_i$ are
line bundles.
\subsection{The correcting Euler classes}
Let $x=(x_1,...,x_l)$ be indeterminant.
Define a polynomial of $x$ over $\ZZ [\hb ]$ for $\beta\in \Lambda$:
$$ H_{\beta}(x,\hb )
:=\prod _{i=1}^{l}\prod _{m=0}^{<c_1(L_i),\beta >}(x_i+m\hb ).$$
Set $$H'_{\beta}(x,\hb ):=\frac{H_{\beta}(x,\hb )}{\prod x_i}. $$
We treat each linear factor $(x_i+m\hb)$ of $H_\beta$ as a Chern character.
Define \[ \Phi ^V (q,\hb ):=\sum _{d\in\Lambda }\sum _a
q^d<\frac{T_a}{\hb (\hb -c)}>^X_dT^a
E(H'_d(x,\hb ))(c_1(L),\hb ),\]
where $c_1(L)=(c_1(L_1),...,c_1(L_k))$.
\bigskip
{\it Claim}
1. $(p_i)_w = (p_i)_v-<p_i, \beta _{v,w}>\ka _{v,w}$,
2. $c_1(L)_w = c_1(L)_v - < c_1(L), \beta _{v,w}>\ka _{v,w}$
3. $E(V_{v,w,m})=E(H_{m\beta _{v,w}})(c_1(L)_v,-\frac{\ka _{v,w}}{m}).$
{\it Proof:} Let $U$ be any equivariant convex line bundle.
On the ray $o(v,w)$ ($\cong \PP ^1$),
we have a homogeneous coordinate $[z_0:z_1]$ such that the induced action
on the ray is linear (because of the equivariant embedding theorem).
We have also global sections $z_0^n,z_0^{n-1}z_1,...,z_1^n$ of the restriction
$U|_{\PP ^1}$ of $U$ to the ray,
where $[1:0]=w$, $[0:1]=v$ and $n=<c_1(U), \beta _{v,w}>$.
We know that $z_0^n$, $z_1^n$, $z_0/z_1$ have the characters $c_1(U)_w$,
$c_1(U)_v$ and $-\ka _{v,w}$, respectively. This concludes the proof.
\bigskip
The first one in the claim shows that we have the well-defined classes
${\mathcal P}(X,V,E)$.
(Otherwise, the mirror group transformation may not
preserve the class ${\mathcal P}(X,V,E)$.)
\bigskip
{\theorem\label{thmmo}
Suppose $c_1(TX)-c_1(V)$ is in ample cone. Then
$\Phi ^V$ is in the class ${\mathcal P}(X,V,Euler)$. }
\bigskip
Notice that for $\beta ' \le \beta$
\begin{eqnarray}
H_\beta (x-<c_1(L), \beta ' >\hb , \hb )
&= &H_{\beta '}(x,-h)H'_{\beta -\beta '}(x,\hb ) \label{Hdo} \\
H'_\beta (x, \hb )
&=&H'_{\beta '}(x,h)H'_{\beta -\beta '}(x+<c_1(L),\beta '>\hb ,\hb ) ,
\label{Hre}
\end{eqnarray}
which will show the polynomiality of double construction
and the almost recursion relation for $\Phi ^V$, respectively.
\subsection{The proof of theorem}
The homogeneous of $\Phi ^V$ is clear when $E=Euler$
and the rest properties will be proven for general $E$.
First of all, it is easy check to see
\[ \Phi ^V \in H^*_{T\ti T'}[[\hb ^{-1}]][[q]] .\]
For the polynomiality, consider
\begin{eqnarray}
&& \int _V \Phi ^V(q,\hb ) e^{pz}\Phi ^V(qe^{-\hb z},-\hb ) \label{Hdoo} \\
&=& \sum _d \sum _{d^{(1)}+d^{(2)}=d, a} q^{d^{(1)}}
<\frac{T_a E(H_{d^{(1)}})(c_1(L),\hb )}{E(V)\hb (\hb -c)}>^X_{d^{(1)}}
\nonumber \\
&&
q^{d^{(2)}}<\frac{T^ae^{(p-d^{(2)})z} E(H_{d^{(2)}})(c_1(L),-\hb )}{
-\hb (-\hb -c)}>^X_{d^{(2)}}, \nonumber \end{eqnarray}
where $<T_a, T^b>_0^X=\delta _{a,b}$.
Let us use the notation and facts in \ref{pfdo}.
Since \begin{eqnarray*} && E(H_{d^{(1)}})(c_1(L),\hb )E(H_{d^{(2)}})(c_1(L),-\hb ) \\ &=&
E(H_d(x-<c_1(L),d^{(2)}>\hb,\hb ))(c_1(L),\hb )E(V) \end{eqnarray*}
from (\ref{Hdo}),
the universal class $U(c_1(L))$ corresponding to $c_1(L)$ restricted to
$G_{d_1,d_2}(X)$ is
\[ c_1(L)-<c_1(L),d^{(2)}>\hb ,\]
and $e_1\circ \pi _1 = e_1\circ \pi _2$,
(\ref{Hdoo}) is equal to
\begin{eqnarray*}
&&
\sum _d q^d\int _{G_{d^{(1)},d^{(2)}}(X)}
\frac{ e^{(\pi _2e^*_1p-d^{(2)}\hb)z}E(H_d)(U(c_1(L)),\hb )}
{[N_{G_d(X)/G_{d^{(1)},d^{(2)}}(X)}]} \\
&=& \sum _d q^d \int _{G_d(X)} e^{Pz}E(H_d)(U(c_1(L)),\hb ), \end{eqnarray*}
which shows the polynomiality.
Now let us check the almost recursion relation.
Let
$$S _v^X(q,\hb ):=
<S ^X,\phi ^X_v>^X_0=\sum _d S _{v,d}^X(\hb )q^d.$$ Since (if $d\ne 0$)
$$S _{v,d}^X(\hb )=\sum _{w\in o(v), 0<m; m\beta _{v,w}\le d}
\frac {C^X_{v,w,m}}{\hb (\ka _{v,w} + m\hb )}
S ^X_{w,d-m\beta _{v,w}}( -\frac{\ka _{v,w}}{m}) $$
and
\[ E(H'_{\beta})(c_1(L)_v,-\frac{\ka _{v,w}}{m})
=\frac{E(V_{v, w,m})}{E(V)_v}
E(H'_{\beta -m\beta _{v,w}})(c_1(L)_w, -\frac{\ka _{v,w}}{m}) \]
from (\ref{Hre}) and the Claim,
we obtain that
\begin{eqnarray*}
&&\Phi ^{V}_{v,d}(\hb ):=<\Phi ^V_d(\hb ),\phi _v>^V_0
= R_{v,d} \\ &+&\sum _{w\in o(v), 0<m; m\beta _{v,w}\le d}
\frac{C^X_{v,w,m}E(V_{v,w,m})}
{E(V)_v\hb (\ka _{v,w} +m \hb )} \\
&& \ti \Phi ^X_{w,d-m\beta _{v,w}}
(-\frac{\ka _{v,w}}{m})
E(H'_{d-m\beta _{v,w}})(c_1(L)_w,-\frac{\ka _{v,w}}{m}),
\end{eqnarray*}
where $R_{v,d}$ is indeed
a polynomial of $1/\hb $ over $H^*_{(T\ti T')}$.
However, since
$$ C^V_{v,w,m} = \frac{C^X_{v,w,m}E(V_{v,w,m})}{E(V)_v},$$
$\Phi ^V_v(q,\hb )$ has the same almost recursion coefficients
$C^V_{v,w,m}$ with $S^V$.
\subsection{Proof of main theorem \ref{thmmain}}
Recursion lemma \ref{lemmare} and double construction lemma \ref{lemmado}
show that $S^V$ is in class ${\mathcal P}(X,V,Euler)$.
Certainly $S^V$ is form of
$1+o(\hb ^{-1})$. According to theorem \ref{thmmo}, $\Phi ^V$ also belongs to
${\mathcal P}(X,V,Euler)$. Then theorem \ref{thmtr} concludes the proof.
(We use the condition that $E=Euler$, in order to make sure that
$S^V$ and $\Phi ^V$ are homogeneous of degree 0.)
\section{Grassmannians}\label{grass}
\subsection{Notation}
Let $e_1,...,e_n$ form the standard basis of $\CC ^n$,
$T=(\CC ^\ti )^n$ the complex torus, and
$X:=Gr(k,n)$ the Grassmannian, the set of all
$k$-subspaces in $\CC ^n$.
As usual, let
$T$ act on $Gr(k,n)$ by the diagonal action.
The fixed points $v=(i_1,...,i_k)$ are then the $k$-planes generated
by vectors $e_{i_1},...,e_{i_k}$.
Denote by $\CC ^n\ti X$ the trivial vector bundle with the
standard action.
Then we may consider
$L$, the determinant of the bundle dual to
the $T$-equivariant universal $k$-subbundle of $\CC ^n\ti X$.
Define
$V=L^{\ot l}$, $l>0$.
Denote
by $p$ the equivariant class $c_1(L)$.
We may identify $H^*(BT)$ with $\QQ [\ke _1,...,\ke _n]$
by the correspondence that $\ke _i$ is also
denoted the equivariant Chern class of the line
bundle over a point equipped with $T$ action as
the representation of the character $\ke _i$.
With respect to the Chern class of $L$,
we shall write $d\in \ZZ=H_2(X, \ZZ )$.
\subsection{A series}
\subsubsection{Fixed points}
Let $v$ be, say, $(1,2,..,k)$. Then around the point,
a local chart can be described by
\[\left( \begin{array}{cccc}
1 & 0 & ... & 0\\
0 & 1 & & 0\\
0 & 0 & ...& 1 \\
x_{1,1} & x_{1,2} & ...& x_{n-k,k} \\
& ... & \\
x_{n-k,1} & x_{n-k,2} & ... & x_{n-k,k}
\end{array}\right).\]
For each complex value $(x_{i,j})$ the column vectors
in the matrix span a $k$-plane which stands for
a point in $Gr(k,n)$.
Then in the chart the action by $(t_1,...,t_n)\in T$ is described as
follows:
\[\left(\begin{array}{cccc}
x_{1,1} & x_{1,2} & ...& x_{n-k,k} \\
& ... & \\
x_{n-k,1} & x_{n-k,2} & ... & x_{n-k,k}
\end{array}\right)
\mapsto \left(\begin{array}{cccc}
t_1^{-1}t_{k+1} x_{1,1} & t_2 ^{-1}t_{k+1} x_{1,2}
& ...& t_k^{-1}t_{k+1} x_{1,k}\\
& & ...& \\
t_1^{-1}t_n x_{n-k,1} & t_2 ^{-1}t_{n} x_{n-k,2} & ...
& t_k^{-1}t_{n}x_{n-k,k}
\end{array}\right).\]
In each isolated fixed point of the Grassmannian there is
$\dim Gr(k,n)$-many 1-dimensional orbit (ray) passing through
the point. For instance, if $v=(1,2,...,k)$, then
there is only one ray $(v,w)$
from $v$ to $w=(...,\hat{i},..., j)$
for any $i\le k <j$. These rays have
degree $1\in H_2(X, \ZZ )$.
\subsubsection{The Euler classes}
Notice that the tangent space at $v=(1,2,...,k)$ of the ray
connecting $v$ to $w=(...,\hat {i},...,j)$
has the character $\ka (v,w)=\ke _j - \ke _i$, where $j>k\ge i$.
Similarly one can find out the characters for the other
cases.
Let $f:\PP ^1\ra Gr(k,n)$ be a $m$-fold morphism totally ramifying
the ray over $v$ and $w$.
The $T$ representation space $H^0(f^*L^{\ot l})$ has
the orbi-characters
\begin{eqnarray*}
\frac {ap_v + b p_w}{m}
= lp_v -\frac{\ke _j-\ke _i}{m}b,
\text{ for } a+b=lm, \ a\ge 0,\ b \ge 0,
\end{eqnarray*}
where $p_v=-(\ke _1+...+\ke _k)$
and $p_w=-(\ke _1+...+\hat{\ke _i} +....+\ke _k + \ke _j)$
are $p=c_1(L)$ restricted to the fixed
points $v$ and $w$, respectively.
\section{The flag manifolds}\label{flag}
We analyze
fixed points of the maximal torus actions and the invariant
curves connecting two fixed points. This explicit description would be
useful also to find $S^X$ explicitly.
\subsection{The complete flag manifolds}
Let $X$ be the set of all Borel subgroups of a simply connected
semi-simple Lie group $G$. It is a homogeneous space with
the $G$-action by conjugation.
Then the maximal torus $T$-action (---fix one---)
has isolated fixed points. They are exactly Borel subgroups
containing $T$. The fixed points are
naturally one-to-one corresponding to the set of
Weyl chambers. Each Borel subgroup containing $T$ gives rise to
a negative roots (---our convention---)
of $B$ and so a chamber associated to the positive
roots. Let $C$ be the set of chambers.
The tangent line subspace associated to
the positive root $\ka$ has the character $\ka$.
There is, if one fix a fixed point $v$,
a natural correspondence between the
$H^2(X,\ZZ )$ and the characters of $T$.
Then the K\"ahler cone is exactly the
positive Weyl chamber $v$. Notice that
the fundamental roots span the K\"ahler cone.
Consider co-roots $\ka ^\vee$. They span
the Mori cone. We can identify the Mori cone $\Lambda$
with the non-negative integer span of
co-roots.
\subsection{The generalized flag manifolds}
Let $X$ be the set of all parabolic subgroups with a
given conjugate type. Let $T$ be a maximal torus of $G$.
Then the fixed loci are isolated fixed points
consisting of parabolic subgroups containing $T$.
\subsubsection{Rays}\label{ray}
Let us choose a fixed point $P\supset T$. Then the rays
at the fixed points are described by the following way.
(The rays are by definition the 1-dimensional orbits of $T$
passing through $P$.)
Fix $B$ a Borel subgroup in $P$ containing $T$.
First consider the $T$-equivariant fibration,
$G/B \ra G/P$ and
the rational map
to $G/B$ by $\exp (zX_\ka )\in G$, $z\in \CC$,
where $X_\ka$ is an eigenvector of the positive root
$\ka$.
Since $\exp H\exp (zX_\ka )\exp -H
=\exp (z\exp\ad H (X_\ka ))=\exp (z\exp (\ka (H)) X_\ka )$ for
$H\in \Lie T$,
we conclude that it is a
$T$ - invariant stable map.
By the composition of the fibration, we obtain all
the rays. They are effectively labeled by the positive roots which
are not roots of $P$. So there are exactly $\dim X$-many
rays at each fixed point. The tangent line at the ray
has the character $\ka$.
\subsubsection{The K\"ahler cone}
Here we need the Levi-decomposition of $P$ and then
consider simple roots $\{
\ka _i\} _{i\in P(\Delta )}$ which
are not roots of the semi-simple part of $P$.
Then the fundamental roots with respect to $P$ is, by definition,
$\{ \kl _i \} _{i\in P(\Delta )}$, where $\kl _i$ are dual to
$\ka _i^\vee $.
Choose a fixed point $P$.
We may identify $H^2(X,\ZZ )$ with the set of
integral weights according to Borel-Weil theorem.
Then the K\"ahler cone is the set of all
dominant integral weights {\em with respect to} $P$.
\subsubsection{Homogeneous line bundles}
One can produces all very ample line bundles by homogeneous
line bundles associated to irreducible representations of $P$
with highest weights $\kl$. The weights corresponding to
the very ample line bundles are exactly the positive integral combination
of the fundamental weights with respect to $P$.
We shall denote by ${\mathcal O}(\kl )$
the homogeneous line bundle associated to
the (1-dimensional) highest weight $\kl =\sum _{i\in P(\Delta )}
a_i\kl _i$ representation of $P$.
It is a very ample bundle if and only if
$a_i >0$ for all $i$.
This also shows that the ray $\PP ^1$ associated to $\ka$ has
the homology class \lq\lq $\ka ^\vee$," in the sense that
$<\PP ^1,c_1({\mathcal O}(\kl )>=(\ka ^\vee ,\kl )$.
{\em We shall use $\ka ^\vee$ to denote the homology class.}
\subsubsection{$\sum (p_i)_vz_i$ are different for different $v$.}
Consider a line bundle $L$ associated
to $\kl =\sum _{i\in P(\Delta )}a_i\kl _i$. (Here in advance,
we have to fix $P\supset T$.)
Let $S_{\ka}$ denote the Weyl group element of the
reflection associated to the root $\ka$.
Then the line bundle
is $L={\mathcal O}(S_{\ka}(\kl ))$ if one look at it
with respect to another \lq\lq origin "
$P'=\exp (\frac{\pi}{2}(X_\ka -Y_\ka))
P \exp (-\frac{\pi}{2}(X_\ka -Y_\ka))$ where
$[X_\ka ,Y_\ka]=H_\ka$, $[H_\ka ,X_\ka ]=2X_\ka$ and
$[H_\ka ,Y_\ka]=-2Y_\ka$.
This $P'$ is the other $T$-fixed point lies in the ray
associated to $\ka$ which passes through $P$.
(Because of the $SL(2,\CC )$-equivariant map from
$\PP ^1$ to the ray, it is enough to check it
when $G=SL(2,\CC )$, which is obvious.)
\subsubsection{$V_{v,w,m}$ and $N_{v,w,m}$}
Let $V={\mathcal O}(\kl )$.
Let $\psi :\PP ^1\ra X$ be a stable map totally ramifying one of
rays, passing through $P\supset T$. Suppose the
ray is associated to a positive root $\ka $ with respect to $P$
and $f$ is a $m$-multiple branched
cover representing an isolated
$T$-fixed point of $\overline{M}_{0,0}(X,m\ka ^\vee)$.
Then the $T$-representation
space $H^0(\PP ^1, f^*({\mathcal O}(\kl ))$ has the characters,
\[ \kl - a\frac{\ka}{m}, \ a=0,...,m(\kl ,\ka ^\vee). \]
To see it, use the coordinate $z\in \CC$ around the fixed
point and $\exp H\exp (zE_\ka )\exp -H
=\exp (z\exp\ad H (E_\ka ))=\exp (z\exp (\ka (H)) E_\ka )$.
Similarly,
$N_{v,w,m}$ has characters
\begin{eqnarray*}
\delta -a\frac{\ka }{m} && \text{ for } \ka \ne \delta >0,
a=0,...,m(\delta ,\ka ^\vee ), \\
\ka - a\frac{\ka }{m} && \text{ for } a=0,...,\hat{m},...,2m,
\end{eqnarray*}
where $\delta >0$ means $\delta$ is a positive root with respect to $P$.
|
1997-12-29T21:48:21 | 9712 | alg-geom/9712030 | en | https://arxiv.org/abs/alg-geom/9712030 | [
"alg-geom",
"math.AG"
] | alg-geom/9712030 | Terrence Napier | T. Napier, M. Ramachandran | The L^2 dbar method, weak Lefschetz theorems, and the topology of Kahler
manifolds | 30 pages | null | null | null | null | A new approach to Nori's weak Lefschetz theorem is described. The new
approach, which involves the dbar-method, avoids moving arguments and gives
much stronger results. In particular, it is proved that if X and Y are
connected smooth projective varieties of positive dimension and f is a
holomorphic immersion of Y into X with ample normal bundle, then the image of
the fundamental group of Y in that of X is of finite index. This result is
obtained as a consequence of a direct generalization of Nori's theorem. The
second part concerns a new approach to the theorem of Burns which states that a
quotient of the unit ball in complex Euclidean space (of dimension at least 3)
by a discrete group of automorphisms which has a strongly pseudoconvex boundary
component has only finitely many ends. The following generalization is
obtained. If a complete Hermitian manifold X of dimension at least 3 has a
strongly pseudoconvex end E and the Ricci curvature of X is bounded above by a
negative constant, then, away from E, X has finite volume.
| [
{
"version": "v1",
"created": "Mon, 29 Dec 1997 20:48:21 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Napier",
"T.",
""
],
[
"Ramachandran",
"M.",
""
]
] | alg-geom | \section{Introduction} \label{intro}
In~[No], Nori studied the fundamental group of complements of nodal
curves with ample normal bundle in smooth projective surfaces.
The main tool was the following weak Lefschetz theorem:
\begin{norithm}
Suppose $\Phi : U @>>> X$
is a local biholomorphism from a connected complex manifold~$U$ into a connected
smooth projective variety~$X$ of dimension at least~$2$ and $U$
contains a connected effective divisor~$Y$ with compact support and ample normal
bundle. Then, for every Zariski open subset~$Z$
of~$X$, the image of $\pi _1(\Phi ^{-1} (Z))$ in $\pi _1(Z)$ is of finite
index.
\end{norithm}
For $X$ a surface, he obtained sharp bounds for the index
using the Hodge index theorem.
A striking corollary of this result is the following:
\begin{noricor}
If $X$ and $Y$ are connected smooth projective
varieties with
$$
\dim X=\dim Y +1>1
$$
and
$f : Y @>>> X$ is a holomorphic immersion with ample normal bundle,
then the image of $\pi_1(Y)$ in $\pi _1(X)$ is of finite index.
\end{noricor}
Nori's proof of these results depends heavily on deformations.
The first step
is to show that a large multiple of the divisor~$Y$ in the theorem moves
in a family in which the general member is irreducible and meets~$Y$
and the union of these members contains an open subset of~$U$.
Unfortunately, moving arguments do not seem to apply in the
higher codimensional case, because Fulton and Lazarsfeld~[FL2]
have observed that for a certain smooth projective $4$-fold and a smooth
surface~$Y$
in~$X$ with ample normal bundle constructed by Gieseker,
no multiple of~$Y$ in~$X$ moves.
Given the existence of a sufficiently large number of
deformations, the rest of the proof of Nori's
weak Lefschetz theorem has been streamlined by
Campana~[C1] and Koll\'ar~[K]. In~[NR],
another proof of Nori's theorem
was given when $Z=X$ using harmonic functions ,
but it was the same in spirit as the earlier arguments.
A survey on Lefschetz type theorems can be found in Fulton~[F].
In this paper we introduce a new approach which avoids moving arguments and
which gives much stronger results. In particular, the new approach allows
one to address the case of higher codimension. Before giving precise
statements, we recall some terminology. Let $Y$ be a complex analytic
subspace of complex space~$U$. We denote the structure sheaf of~$U$
by ${\cal O} _U$ and the ideal sheaf of~$Y$ in~$U$ by~$\cal I_Y$. The
{\it formal completion}~$\widehat U $~{\it of}~$U$ {\it with respect to }~$Y$
is the ringed space
$$
(\widehat U , {\cal O} _{\widehat U })=(Y, \lim _{ @<<< }{\cal O} _U/\cal I_Y^n).
$$
If $\cal F$ is an analytic sheaf on~$U$ we denote by $\widehat {\cal F}$
the associated analytic sheaf on~$\widehat U $ given by
$$
\widehat {\cal F} =
\lim _{ @<<< }(\cal F \otimes {\cal O} _U/\cal I_Y^n).
$$
If $\cal F$ is coherent, then $\widehat {\cal F}$ is also coherent
over~${\cal O} _{\widehat U }$. The main
result is the following generalization of Nori's weak Lefschetz theorem:
\begin{thm}
\it
Suppose $\Phi : U @>>> X$
is a holomorphic map from a connected complex manifold~$U$ into a connected
smooth projective variety~$X$ of dimension at least~$2$
which is a submersion at some point. Let~$Y\subset U$ be a
connected compact analytic subspace such that
$\dim H^0(\widehat U , \widehat {\cal L}) <\infty $
for every locally free analytic sheaf~$\cal L$ on~$U$.
Then, for every Zariski open subset~$Z$
of~$X$, the image of $\pi _1(\Phi ^{-1} (Z))$ in $\pi _1(Z)$ is of finite
index.
\end{thm}
\begin{rems} 1. For example, by a theorem of Hartshorne~[H]
(and Grothendieck~[Gr]),
$H^0(\widehat U , \widehat {\cal L})$ is finite dimensional
when $Y$ is a connected compact analytic subspace which is locally a complete
intersection and which has ample normal bundle
(or even $k$-ample normal bundle
in the sense of Sommese~[So] where $k=\dim Y-1$).
\noindent 2. Theorem~0.1 also holds for $U$ irreducible and reduced
and $X$ normal and projective. Moreover, as will be shown in
Sect.~3 (Corollary~3.4), in the smooth case one only
needs finite dimensionality for $\cal L$ the analytic pullback
of an invertible sheaf on~$X$.
\noindent 3. As a consequence of Theorem~0.1, one can remove the dimension
restriction on the subspace~$Y$ in the corollary to Nori's theorem.
More precisely, we get the following:
\end{rems}
\begin{cor}
If $X$ and $Y$ are connected smooth projective
varieties of positive dimension and
$f : Y @>>> X$ is a holomorphic immersion with ample normal bundle,
then the image of $\pi_1(Y)$ in $\pi _1(X)$ is of finite index.
\end{cor}
Hironaka and Matsumura~[HM] proved
the analogous result
for algebraic fundamental groups when $f$ is an inclusion
with ample normal bundle. However, the result
for topological fundamental groups (as stated in the
above corollary)
is new (provided $\dim X > \dim Y +1$)
even for $f$ an inclusion.
Moreover, simple examples show that,
if $\dim X > \dim Y +1$, then, even if $f$ is an inclusion
(with ample normal bundle), the
map $\pi_1(Y) @>>> \pi _1(X)$ is not necessarily surjective.
The idea of the proof of Theorem~0.1 is to form
a covering space~$\widetilde Z @>>> Z$ with fundamental group equal to
the image~$G$ of $\pi _1(\Phi ^{-1} (Z))$ and then
construct $L^2$ holomorphic sections of a
suitable line bundle which separate the sheets of the covering.
This construction is a standard application of the
$L^2$~$\bar \partial $-method
(Andreotti-Vesentini~[AV],
H\"ormander~[Ho], Skoda~[Sk], Demailly~[D1]).
Pulling these sections back to $\Phi ^{-1} (Z)$ by a lifting of~$\Phi $,
the finite dimensionality of the space of holomorphic
sections on the formal completion
gives a bound on the dimension of the space of sections on~$\widetilde Z $
and hence a bound on the degree of the covering space
(i.e.~on the index of~$G$).
\begin{rem}
Campana~[C2] has independently applied $L^2$-methods
to study exceptional curves on coverings of surfaces.
\end{rem}
The second main result of this paper generalizes a theorem of
Burns~[B]
which states that a quotient of the unit ball in~$\C ^n$ ($n\geq 3$)
by a discrete group of automorphisms which has a strongly pseudoconvex
boundary component has only finitely many ends. The main tools are
a theorem of Lempert on the compactification of a pseudoconvex
boundary from the pseudoconcave side~[L], a finiteness theorem
of Andreotti for pseudoconcave manifolds~[A], and the $L^2$ Riemann-Roch
inequality of Nadel and Tsuji~[NT]. The precise statement is
as follows:
\begin{thm}
If a complete Hermitian manifold~$X$ of (complex) dimension
at least~$3$ has a strongly pseudoconvex end and
$\text {Ricci}\, (X) \leq -C$ for some positive constant~$C$, then, away
from the strongly pseudoconvex end, the manifold has finite volume.
\end{thm}
As in the proof of Theorem~0.1, the idea is to apply finite dimensionality
of the space of holomorphic sections of a line bundle. By Lempert's theorem,
one can cap off the strongly pseudoconvex end by a domain in a smooth
projective variety. Andreotti's finiteness theorem applied to the
resulting pseudoconcave manifold gives finite dimensionality of the space
of holomorphic sections of a suitable line bundle. Finally, the $L^2$ Riemann-Roch
inequality of Nadel and Tsuji gives a (finite) upper bound for the volume
in terms of the dimension of this space of sections.
\begin{rem}
One natural question which arises is might
there be
an improved version of the $L^2$ Riemann-Roch inequality which would
give improved bounds for the volume in Theorem~0.3
as well as the index in~Theorem~0.1? Also,
for $X$ a surface in the corollary to Nori's weak Lefschetz
theorem,
Nori~[No] found bounds for the index in terms of certain intersection
numbers. It is therefore
natural to look for analogous bounds in more general cases.
\end{rem}
Sect.~1 begins with a proof of Theorem~0.1 in the case where~$\Phi $
is a local biholomorphism. The main idea of the
new approach is easy to
see in this context, and, although a few technicalities
arise in the general case, the proof is essentially the same.
The proof of Theorem~0.1 is then given.
The required result from the $L^2$ \ $\bar \partial $-method is discussed
in Sect.~2. Further generalizations of the weak Lefschetz theorem
for $X$ not necessarily projective are considered in Sect.~3.
Theorem~0.3 is proved in Sect.~4, which may be read independently
of Sects.~1--3.
\noindent {\it Acknowledgements}.
Madhav Nori suggested we reformulate
Theorem~0.1 in terms of formal completions, which considerably widened
its scope.
Charles Epstein
told us about Lempert's result. For this and other useful advice, we would
like to thank them both. We would also like to thank Alan Nadel for
bringing the $L^2$ Riemann-Roch inequality to our attention,
Dan Burns for useful discussions on his theorem, and
Raghavan Narasimhan for his interest in this work.
Finally, we would like to thank the referee for helpful
suggestions.
\section{Weak Lefschetz theorems for a projective variety}
This section contains the proof Theorem~0.1. We first
prove the theorem for the case of a local biholomorphism. This is a direct
generalization
to immersed complex spaces of {\it arbitrary} codimension
(Nori proved the theorem stated below for $Y$ an ample divisor in~$U$).
More general versions
will be stated later. Aside
from a few minor technical problems, however, the proofs of all of
the generalizations are the same in spirit as the
proof of this special case.
\begin{thm}
Let $U$ be a connected complex manifold, let~$X$ be
a connected smooth projective variety of dimension~$n>1$, let
$\Phi : U @>>> X$ be a holomorphic map, let~$Y$ be a connected compact analytic subspace
(not necessarily reduced) of~$U$, and let $\widehat U $ be the formal completion
of~$U$ with respect to~$Y$. Assume that
\begin{enumerate}
\item[(i)] $\Phi $ is locally biholomorphic , and
\item[(ii)] $\dim H^0(\widehat U , \widehat {{\cal O} (\Phi ^*L)})<\infty $
for every holomorphic line bundle~$L$ on~$X$.
\end{enumerate}
Then there is a positive constant~$b$ depending only on the mapping
$\Phi :U @>>> X$ and the subspace~$Y\subset U$ such that , if $R\subset X$
is a nowhere dense analytic subset of~$X$ and~$V$ is a connected neighborhood of~$Y$ in~$U$,
then the image~$G$ of
$\pi _1(V\setminus \Phi ^{-1} (R)) @>>> \pi _1(X\setminus R)$
is of index at most~$b$ in~$\pi _1(X\setminus R)$. Moreover, if
$\Phi (Y)\cap R=\emptyset $, then the image of
$\pi _1(Y) @>>> \pi _1(X\setminus R)$ is also of index at most~$b$.
\end{thm}
\begin{pf}
Given $R$,$V$, and~$G$ as in the statement of the theorem,
let $S=\Phi ^{-1} (R)$,
let $M=X\setminus R$, let $W=V\setminus S$,
and let $\pi : \widetilde M @>>> M$ be a connected covering space
with $\pi _* (\pi _1(\widetilde M ))=G$. Thus $\pi : \widetilde M @>>> M$ has
degree $d=[\pi _1(M):G]$ and we have the
following commutative diagram of holomorphic mappings:
\begin{center}\begin{picture}(250,80)
\put(5,10){$W=V\setminus S\subset V $}
\put(90,14){\vector(1,0){50}}
\put(145,10){$X \supset X\setminus R=M$}
\put(110,3){$\Phi $}
\put(125,55){\vector(3,-1){90}}
\put(110,60){$\widetilde M $}
\put(17,25){\vector(3,1){90}}
\put(57,45){$\tilde \Phi $}
\put(170,45){$\pi $}
\end{picture}
\end{center}
Since $X$ is projective,
there exists a Hermitian holomorphic line bundle~$(L,h)$ with
positive curvature and a K\"ahler metric~$g$ on~$X$.
As will be shown in Sect.~2 (see Corollary~2.3),
the $L^2$~$\bar \partial $-method, in the form given by
Skoda~[Sk] and Demailly~[D1], enables one to prove that there is
a positive integer~$\nu $ independent
of~$R$~and~$V$ such that
$$
d\leq \dim H^0_{L^2}(\widetilde M , {\cal O} (\pi ^* (L^{\nu }\otimes K_M)));
$$
where $K_M$ is the canonical bundle on~$M$ and
$H^0_{L^2}(\widetilde M , {\cal O} (\pi ^* (L^{\nu }\otimes K_M)))$ is the space
of holomorphic sections of $\pi ^* (L^{\nu }\otimes K_M)$ which are
in~$L^2$ with respect to the Hermitian metrics $\pi ^*(h\otimes g^*)$
on $\pi ^* (L^{\nu }\otimes K_M)$ and~$\pi ^* g$ on~$\widetilde M $.
If
$s\in H^0_{L^2}(\widetilde M , {\cal O} (\pi ^* (L^{\nu }\otimes K_M)))$, then
$\tilde \Phi ^* s$ is a holomorphic section of $\Phi ^* (L^{\nu }\otimes K_X)$
on $W$. Given a point $x_0\in S\cap V$, $\Phi $ maps a neighborhood~$Q$
of~$x_0$ in~$V$ biholomorphically onto $\Phi (Q) \subset X$. Hence $\tilde \Phi $
maps $Q\setminus S$ biholomorphically onto its image in~$\widetilde M $ and,
therefore, $\tilde \Phi ^*s$ is in~$L^2$ on $Q\setminus S$ with respect
to the Hermitian metrics $\Phi ^*(h\otimes g^*)$ in
$\Phi ^* (L^{\nu }\otimes K_X)$ and $\Phi ^*g$ on~$U$. Since these metrics
are defined over the entire set~$U$ and a square integrable function
which is holomorphic outside a nowhere dense analytic set in a manifold extends
holomorphically past the analytic set, $\tilde \Phi ^*s$
extends to a holomorphic section of $\Phi ^* (L^{\nu }\otimes K_X)$ on~$V$.
Therefore
$$
d\leq \dim H^0(V, {\cal O} (\Phi ^* (L^{\nu }\otimes K_X))).
$$
On the other hand,
by a general fact
about formal completions, if $\cal F$ is a coherent analytic sheaf on~$V$,
then the kernel of the mapping
$$
H^0(V,\cal F) @>>> H^0(\widehat V ,\widehat {\cal F})=H^0(\widehat U ,\widehat {\cal F})
$$
consists of all of the sections of~$\cal F$ on~$V$ which vanish
on a neighborhood of~$Y$ in~$V$ (see [BS,
Proposition VI.2.7]). In particular, if~$\cal F$ is locally
free, then this mapping is injective.
Therefore, taking
$\cal F={\cal O} (\Phi ^*(L^{\nu }\otimes K_X))$, we get
$$
d\leq \dim H^0(V,\cal F) \leq \dim
H^0(\widehat U ,\widehat {\cal F})<\infty .
$$
Thus $b=\dim H^0(\widehat U ,\widehat {\cal F})$ is a uniform bound
for~$d$ independent of~$R$ and~$V$.
Finally, if $\Phi (Y)\cap R=\emptyset $, then we may choose the neighborhood~$V$
so that $V\subset U\setminus S$ and the map $\pi _1(Y) @>>> \pi _1(V)$
is a surjective isomorphism. Hence the image of $\pi _1(Y)$ in
$\pi _1(M)$ is equal to the image of $\pi _1(V)=\pi _1(V\setminus S)$ and
therefore is of index at most~$b$.
\end{pf}
We now consider generalizations.
If in the above theorem one assumes only that~$\Phi $ is a generic
submersion (or a generic local biholomorphism), then a slight technical problem
arises. While (as one may easily check) the section $\tilde \Phi ^*s$
of $\Phi ^*(L^\nu \otimes K_X)$ extends holomorphically past~$S$
near points at which~$\Phi $ is submersive, $\tilde \Phi ^*s$
need not extend near points where $\text {rank} \, \Phi _* < n$.
However, as we will see, $\tilde \Phi ^*s$ does
extend as a holomorphic $n$-form with values in~$\Phi ^*L^\nu $.
A simple illustration is given by
$$
U=\Delta \ni z \overset {\Phi } \mapsto \zeta =z^2 \in \Delta , \quad
\widetilde M =\Delta ^* \ni z \overset {\pi } \mapsto \zeta =z^2 \in \Delta ^*=M,
\quad \text {and } s=z ^{-1} \pi ^* d\zeta .
$$
Here, $\tilde \Phi ^*s$ does not extend as a section of the pullback
of the canonical bundle, but the corresponding holomorphic $1$-form~$2dz$
does extend.
In fact, by passing to desingularizations, one also gets this
extension property for $U$ and $X$ singular.
Given an irreducible reduced
complex space~$A$ and a positive integer~$n$, we denote by~$\Omega ^n_A$
the coherent analytic sheaf on~$A$ obtained by forming a desingularization
$\check A @>>> A$ of~$A$ and taking the direct image of $\Omega ^n_{\check A}$.
By the following lemma, this sheaf is independent of the choice of the
desingularization.
\begin{lem}[Grauert and Riemenschneider
[GR, Sect.~2.1{]}]
Let $A$ be an irreducible reduced complex space of dimension~$m$
and let~$n$ be a positive integer.
Suppose that, for $i=1,2$, $B_i$ is a
connected complex manifold of dimension~$m$ and
$\Psi _i : B_i @>>> A$ is a proper modification.
Then
$(\Psi _1)_*\Omega ^n_{B_1}=(\Psi _2)_*\Omega ^n_{B_2}$.
\end{lem}
The proof is similar to the proof for $\dim A=n$ given
in~[GR]. The main point is that if $A$
is smooth, then $\Psi _1$ is biholomorphic outside an analytic
set of codimension at least~$2$ in~$A$. For the general
case, one passes to a common proper modification of
$B_1$~and~$B_2$.
We may now state the extension property as follows:
\begin{lem}
\it
Let $\Phi :U @>>> X$ be a holomorphic mapping of irreducible
reduced complex spaces $U$~and~$X$ of dimensions $m$~and~$n$, respectively,
such that $\Phi (U)$ has nonempty interior. Suppose
\begin{center}\begin{picture}(250,80)
\put(5,10){$W=U\setminus S\subset U$}
\put(90,14){\vector(1,0){50}}
\put(145,10){$X \supset X\setminus R=M$}
\put(110,3){$\Phi $}
\put(125,55){\vector(3,-1){90}}
\put(110,60){$\widetilde M $}
\put(17,25){\vector(3,1){90}}
\put(57,45){$\tilde \Phi $}
\put(170,45){$\pi $}
\end{picture}
\end{center}
is a commutative diagram of holomorphic mappings where
$R\subset X$ is a nowhere dense analytic subset which contains~$\sing X$,
$S=\Phi ^{-1} (R)$, and
$\pi : \widetilde M @>>> M$ is a connected holomorphic covering space.
Let $L$ be a holomorphic line bundle on~$X$ and let $\theta $ be a holomorphic
$n$-form with values in $\pi^*L$ on $\widetilde M $ which is in $L^2$
with respect to the liftings
of a Hermitian metric~$h$ in~$L$ on~$X$ and a Hermitian metric~$g$ on~$M$.
Then the pullback
$(\tilde \Phi | _{\reg W})^*\theta$ of~$\theta$ to a holomorphic
$n$-form with values in $\Phi ^*L$ on $\reg W$ extends to a (unique)
section in $H^0(U, {\cal O} (\Phi ^*L) \otimes \Omega ^n_U)$.
\end{lem}
The proof uses standard methods but will be postponed until
the end of this section (see also Sakai~[S]). We may now apply the
argument given in the proof
of Theorem~1.1 to get Theorem~0.1 of the introduction.
In fact, we get the following:
\begin{thm} Let $U$
be an irreducible reduced complex space, let $X$ be a connected normal projective variety
of dimension~$n>1$,
let $\Phi :U @>>> X$ be a holomorphic map, let~$Y$ be a connected compact analytic
subspace (not necessarily reduced) of~$U$, and
let $\widehat U $ be the formal completion of~$U$ with respect to~$Y$.
Assume that
\begin{enumerate}
\item[(i)] $\Phi (U)$ has nonempty interior, and
\item[(ii)]
$\dim H^0(\widehat U , \widehat {{\cal O} (\Phi ^*L)}\otimes \widehat {\Omega ^n_U})
<\infty $ for every holomorphic line bundle~$L$ on~$X$.
\end{enumerate}
Then there exists a positive
constant~$b$ depending only on the mapping $\Phi : U @>>> X$ and the
subspace~$Y$ such that ,
if~$R\subset X$ is a nowhere dense analytic subset and
$V$ is a connected neighborhood of~$Y$ in~$U$,
then the image of
$\pi _1(V\setminus \Phi ^{-1} (R)) @>>> \pi _1 (X\setminus R)$
is of index at most~$b$. Moreover, if
$\Phi (Y)\cap R=\emptyset $, then the image of~$\pi _1(Y)$ in
$\pi _1(X\setminus R)$ is also of index at most~$b$.
\end{thm}
\begin{pf} Given $R$ and $V$ as in the statement of the
theorem, we get a commutative diagram
\begin{center}\begin{picture}(250,80)
\put(5,10){$W=V\setminus S\subset V$}
\put(90,14){\vector(1,0){50}}
\put(145,10){$X \supset X\setminus R=M$}
\put(110,3){$\Phi $}
\put(125,55){\vector(3,-1){90}}
\put(110,60){$\widetilde M $}
\put(17,25){\vector(3,1){90}}
\put(57,45){$\tilde \Phi $}
\put(170,45){$\pi $}
\end{picture}
\end{center}
as in the proof of Theorem~1.1.
Since $X$ is normal, the map
$\pi _1(M\setminus \sing X) @>>> \pi _1(M)$ is surjective.
Therefore, by
replacing~$R$ by $R\cup\sing X$, we may assume that $\sing X\subset R$;
i.e.~that $M$ is a complete K\"ahler manifold.
If
$s\in H^0_{L^2}(\widetilde M , {\cal O} ((\pi ^*L^\nu )\otimes K_{\widetilde M }))$
for some~$\nu $ (with respect to metrics lifted
from the base), then, by Lemma~1.3, the pullback to $\reg W$
{\it as a holomorphic $n$-form
with values in~$\Phi ^*L^\nu $}
extends to a
unique section in $H^0(V, {\cal O} (\Phi ^*L^\nu ) \otimes \Omega ^n_U)$.
By applying Corollary~2.3 as in the proof of Theorem~1.1, one now
gets the required bound on the index.
\end{pf}
A finiteness theorem of
Hartshorne~[H, Theorem~III.4.1]
and Grothendieck~[Gr] and the above theorem together
imply immediately that,
in Nori's weak Lefschetz theorem, one may take the mapping to
be a generic submersion and the
subvariety to be of arbitrary codimension. More precisely, we have
the following:
\begin{cor}
Let $U$ be a connected complex manifold, let~$X$ be
a connected normal projective variety of dimension~$n>1$, let
$\Phi : U @>>> X$ be a holomorphic map, and let~$Y$ be a
positive dimensional connected compact analytic subspace
(not necessarily reduced) of~$U$. Assume that
\begin{enumerate}
\item[(i)] $\Phi (U)$ has nonempty interior,
\item[(ii)] $Y$ is locally a complete intersection in~$U$, and
\item[(iii)] The normal bundle $N_{Y/U}$ is ample.
\end{enumerate}
Then there is a positive constant~$b$ depending only on the mapping
$\Phi :U @>>> X$ and the subspace~$Y\subset U$ such that ,
if~$Z$ is a nonempty Zariski open subset of~$X$ and
$V$ is a connected neighborhood of~$Y$ in~$U$,
then the image of
$\pi _1(V\cap \Phi ^{-1} (Z)) @>>> \pi _1 (Z)$
is of index at most~$b$ in~$\pi _1(Z)$. Moreover, if
$\Phi (Y)\subset Z$, then the image of~$\pi _1(Y)$ in
$\pi _1(Z)$ is also of index at most~$b$.
\end{cor}
\begin{rems}
1. The approach of considering sections of
vector bundles
on formal completions
fits well with Grothendieck's approach to the Lefschetz theorems~[Gr]
(see also~[H]). In a sense, the results of this paper extend
to the topological fundamental group
Grothendieck's Lefschetz theorems concerning the algebraic fundamental group.
\noindent 2. Further generalizations in which $X$ is not necessarily
projective will be stated and proved in Sect.~3. A slightly more precise
bound for the index in terms of the dimension of a
space of sections will also be obtained.
\end{rems}
We conclude this section with the proof of the extension property.
\begin{pf*}{Proof of Lemma~1.3}
We first observe that we may assume that
$U$~and~$X$ are smooth and that $R$ is a divisor with normal
crossings by passing to desingularizations. More precisely,
we may form a commutative diagram
\begin{center}\begin{picture}(250,90)
\put(60,10){$U$}
\put(90,14){\vector(1,0){50}}
\put(150,10){$X$}
\put(115,0){$\Phi $}
\put(65,55){\vector(0,-1){30}}
\put(50,60){$U\times _XX'$}
\put(45,35){$\text{pr}_U$}
\put(100,64){\vector(1,0){40}}
\put(150,60){$X'$}
\put(115,75){$\text {pr}_{X'}$}
\put(160,35){$\beta $}
\put(155,55){\vector(0,-1){30}}
\end{picture}
\end{center}
where $X'$ is a connected complex manifold, $R'=\beta ^{-1} (R)$ of $R$
is a divisor with normal crossings, and
$\beta : X' @>>> X$ is a proper modification which maps
$M'=X'\setminus R'$ biholomorphically onto $M=X\setminus R$.
Since
$\text {pr}_U ^{-1} (W)=W\times _MM'$ is just the graph of the restriction
of~$\Phi $ to a mapping $W @>>> M'=M$ and $U$ is irreducible ,
$\text {pr}_U ^{-1} (W)$ is an open irreducible subset
of $U\times _XX'$ which is mapped isomorphically onto~$W$. In particular,
$\text {pr}_U ^{-1} (W)$ lies in a unique irreducible component~$C$ of $U\times _XX'$; and,
since $\text {pr}_U$ is a proper mapping,
we must have $\text {pr}_U(C)=U$. Passing to a desingularization of~$C$, we
get a
commutative diagram of holomorphic mappings
\begin{center}\begin{picture}(250,90)
\put(70,10){$U$}
\put(95,14){\vector(1,0){50}}
\put(155,10){$X$}
\put(115,0){$\Phi $}
\put(75,55){\vector(0,-1){30}}
\put(70,60){$U'$}
\put(60,35){$\alpha $}
\put(95,64){\vector(1,0){50}}
\put(155,60){$X'$}
\put(115,72){$\Phi '$}
\put(165,35){$\beta $}
\put(160,55){\vector(0,-1){30}}
\end{picture}
\end{center}
where $U'$ is a connected complex manifold of dimension~$m$,
$\alpha : U' @>>> U$ is a proper modification,
$S'\equiv \alpha ^{-1} (S)=(\Phi ') ^{-1} (R')$, and, if
$W'=U'\setminus S'=\alpha ^{-1} (W)$, then $\alpha $ maps the set
$W'\setminus \alpha ^{-1} (\sing U)$ biholomorphically
onto~$\reg W$. We also get a connected
covering space
$\pi '=(\beta |_{M'}) ^{-1} \circ \pi : \widetilde M @>>> M'$ and a
lifting $\tilde \Phi ' =
\tilde \Phi \circ (\alpha | _{W'}) : W' @>>> \widetilde M $ of
$\Phi '| _{W'}$.
Therefore, if $L'=\beta ^*L$, then $\theta $ is a holomorphic $n$-form with values in
$\pi^*L=(\pi ')^*L'$
which is in~$L^2$ with respect to the metrics $\pi ^*h=(\pi ')^*\beta ^*h$
in $(\pi ')^*L'$ and
$\pi ^*g=(\pi ')^*\beta ^*g$ on~$\widetilde M $. Suppose the pullback of $\theta $
to~$W'$ extends to
a section
$$
\eta \in H^0(U', {\cal O} ((\Phi')^*L')\otimes \Omega ^n_{U'}).
$$
Since $(\Phi')^*L'=\alpha ^*\Phi ^*L$ and $\alpha :U' @>>> U$ is a proper
modification,
we have (by the definition of $\Omega ^n_U$ and Lemma~1.2)
$$
\alpha _*\bigl( {\cal O} ((\Phi')^*L')\otimes \Omega ^n_{U'}\bigr)
={\cal O} (\Phi^*L)\otimes \Omega ^n_{U}.
$$
Hence $\eta $ determines an extension
of $(\tilde \Phi | _{\reg W})^*\theta $ to a section in
$H^0(U, {\cal O} (\Phi^*L)\otimes \Omega ^n_{U})$.
Thus we may assume that $U$~and~$X$ are smooth and that $R$ is a divisor
with normal
crossings in~$X$. In particular, $S=\Phi ^{-1} (R)$ is a divisor in~$U$.
Since the lemma is entirely local, it suffices to extend the section near each
point $x_0\in S$ and we may assume that $U=\Delta ^m$ is the unit polydisk centered
at~$x_0=0$ in~$\C ^m$, that
$X=\Delta ^n$ is the unit polydisk centered at~$\Phi (x_0)=0$ in~$\C ^n$,
that $L$ is the trivial line bundle with the trivial metric on~$X$
(since all metrics are comparable
on relatively compact subsets), and that $g$ is the restriction of the Euclidean metric
$g_{\C ^n}$ to~$M$ (since the $L^2$ condition on forms of type $(n,0)$ is
independent of the choice of the metric on an $n$-dimensional manifold).
We denote the
coordinates in~$\C ^m$ by $z=(z_1,\dots , z_m)$,
the
coordinates in~$\C ^n$ by $\zeta =(\zeta _1,\dots , \zeta _n)$, and
the coordinate functions of the mapping by $\Phi =(\Phi _1, \dots , \Phi _n)$.
Thus $\theta =fd(\zeta _1\circ \pi ) \wedge \dots \wedge
d(\zeta _n\circ \pi )$ for some holomorphic
function~$f$ which is square integrable on~$\widetilde M $ with respect to
$\pi ^*g_{\C ^n}$ and
$\tilde \Phi ^*\theta =(f\circ \tilde \Phi )d\Phi _1\wedge \dots \wedge d\Phi _n$ on $W$.
Since holomorphic sections extend past analytic sets of codimension at least~$2$,
we may assume that~$x_0\in \reg S$ and hence that~$S$ is
the zero set of~$z_1$. Since $R$ is a
divisor with
normal crossings, we may also assume that $R$ is the zero set of
$\zeta _1\cdots \zeta _k$. Finally,
if $\setof {x\in S}{\Phi _j(x)=0}$ is nowhere dense in~$S$ for some~$j$,
then, again, it suffices to
consider a point~$x_0$ which avoids this zero set. Thus we may assume that
$$
S=\Phi ^{-1} (R)=\{ \Phi _j=0 \} \text { for } j=1, \cdots , k.
$$
We now show that
$(\Phi _1\cdots \Phi _k) \cdot (f\circ \tilde \Phi )$ extends to a holomorphic
function which vanishes along $S$.
If~$x=(x_1,\dots ,x_m)$ is a point
in~$W=U\setminus S=\Delta ^*\times \Delta ^{m-1}$
near~$x_0=0$ and $y=\Phi (x)=(y_1, \dots ,y_n)$, then we have
$r_j=|y_j|< 1/2$ for $j=1,\dots , n$ and $r_j>0$
for $j=1, \dots ,k$. Thus the polydisk
$$
P =\Delta (y_1;r_1)
\times \dots \times \Delta (y_k;r_k)
\times \Delta (y_{k+1};1/2)
\times \Delta (y_n;1/2)
$$
centered at~$y$ is contained in $M=(\Delta ^*)^k\times \Delta ^{n-k}$
and is therefore evenly covered by $\pi : \widetilde M @>>> M$. Hence~$\pi $
maps the connected component~$\widetilde P $ of~$\pi ^{-1} (P)$ containing
$\tilde y =\tilde \Phi (x)$ isomorphically onto~$P$. The $L^{\infty }/L^2$-estimate
now gives
$$
|f(\tilde y )|^2\leq (\text {\rm vol} \, (\widetilde P )) ^{-1}
\int _{\widetilde P }
|f|^2 \, dV_{\pi ^* g_{\Cn }}.
$$
As $x$ approaches a point~$x_1$ in~$S$ near~$x_0$, $\text {\rm vol} \, (\widetilde P )=\text {\rm vol} \, (P)$
will approach~$0$. Therefore, after multiplying both sides of the above
inequality by $(r_1\cdots r_k )^2$ we get, since $|f|^2$ is integrable
on~$\widetilde M $,
$$
|\Phi _1(x)\cdots \Phi _k(x) f(\tilde \Phi (x))|^2=(r_1\cdots r_k)^2|f(\tilde y )|^2
\leq
\pi ^{-n}4^{(n-k)}\int _{\widetilde P }
|f|^2 \, dV_{\pi ^* g_{\Cn }} @>>> 0
$$
and the claim follows.
For each $j=1,\dots ,k$, we have $\Phi _j=z_1^{\mu _j}h_j$ where
$\mu _j =\text {ord}_S\Phi _j$ and $h_j$ is a unit. Therefore,
setting $\mu =\mu _1+\dots + \mu _k $ and
$\psi =d\Phi _{k+1}\wedge \dots \wedge d\Phi _n$, we get
\begin{align*}
d\Phi _1\wedge \dots \wedge d\Phi _n
&=z_1^\mu dh_1\wedge \dots \wedge dh_k \wedge \psi \\
&\quad+z_1^{\mu -1}
\sum _{j=1}^k \mu _j h_j (-1)^{j-1}dz_1 \wedge
dh_1\wedge \dots \wedge \widehat {dh_j}\wedge
\dots \wedge dh_k \wedge \psi .
\end{align*}
Since $z_1^{\mu }(f\circ \tilde \Phi )$ extends to a holomorphic function
which vanishes along~$S$, it follows that
the $n$-form $\tilde \Phi ^*\theta =(f\circ \tilde \Phi )d\Phi _1\wedge \dots
\wedge d\Phi _n$
also extends holomorphically as claimed.
\end{pf*}
\section{Results from the $L^2$~$\bar \partial $-method}
As described in Sect.~1, the proofs of the weak Lefschetz theorems rely
on a consequence of the $L^2$~$\bar \partial $-method
(Andreotti-Vesentini~[AV],
H\"ormander~[Ho], Skoda~[Sk], Demailly~[D1])
which will be described in this section.
Given a real-valued function~$\varphi $ of class~$C^2$ on a complex manifold~$M$
of dimension~$n$, the {\it Levi form} $\lev \varphi $ of~$\varphi $ is the
Hermitian tensor defined by
$$
\lev \varphi = \sum _{i,j=1}^n
\frac {\partial ^2\varphi }{\partial z_i\partial \bar z_j} dz_id\bar z_j
$$
in local holomorphic coordinates $(z_1,\dots ,z_n)$. The function $\varphi $ is
said to be plurisubharmonic if $\lev \varphi \geq 0$ and strictly plurisubharmonic if
$\lev \varphi > 0$. If $(L,h)$ is a
Hermitian holomorphic line bundle on a complex manifold $M$, then the
{\it curvature tensor}~$\cal C (L,h)$ of~$(L,h)$ is given by
$$
\cal C(L,h)= \lev {-\log |s|^2}
$$
for any nonvanishing local holomorphic section~$s$ of~$L$.
We will need the following
special case of a theorem of Demailly~[D1, Theorem~5.1]
concerning the $\bar \partial $-method for singular metrics with semi-positive
curvature.
\begin{thm}[Demailly] \it
Let $(E,h)$ be a Hermitian holomorphic line
bundle with semi-positive curvature (i.e.~$\cal C(L,h)\geq 0$) on a complete
K\"ahler manifold~$(M,g)$ of dimension~$n$. Suppose
$\varphi : M @>>> [-\infty ,0]$
is a function which is of class~$C^{\infty } $ outside a discrete
subset~$S$ of~$M$ and, near each point $p\in S$,
$\varphi (z)=A_p\log |z|^2$
where $A_p$ is a positive constant and $z=(z_1,\dots ,z_n)$ are local holomorphic
coordinates centered at~$p$. Assume that
$\cal C (E,he^{-\varphi })=\cal C (E,h)+\lev {\varphi }\geq 0$
on~$M\setminus S$
(and hence on~$M$ as the curvature of a singular metric) and let
$\lambda : M @>>> [0,1]$ be a continuous function such that
$\cal C (E,h)+\lev {\varphi }\geq \lambda g$
on~$M\setminus S$. Then, for every $C^{\infty } $~form~$\theta $ of type~$(n,1)$
with values in~$E$ on~$M$ which satisfies
$$
\bar \partial \theta =0 \quad \text {and}
\quad \int _M\lambda ^{-1} |\theta |^2_{h\otimes g^*}e^{-\varphi }
\, dV_g <\infty,
$$
there exists a $C^{\infty } $ form~$\eta $ of type~$(n,0)$ with values in~$L$ on~$M$
such that
$$
\bar \partial \eta =\theta \quad \text {and}
\quad \int _M |\eta |^2_{h\otimes g^*}e^{-\varphi }\, dV_g
\leq \int _M\lambda ^{-1} |\theta |^2_{h\otimes g^*}e^{-\varphi }
\, dV_g .
$$
\end{thm}
\begin{rem}
Demailly's theorem is much stronger than the above
special case. This special case also follows
from Theorem~4.1 of~[D1], since one can approximate~$\varphi (z)$ by
functions which locally have the form~$A_p\log (|z|^2+\epsilon )$ near the
nonsmooth points; or one can complete the metric on~$M\setminus S$.
\end{rem}
A well-known technique for producing sections with prescribed values on
a discrete set gives the following:
\begin{thm}
Suppose $(L,h)$ is a Hermitian
holomorphic line bundle on
an irreducible reduced complex space~$X$ of dimension~$n$ and the curvature of~$h$ is
semipositive on~$X$ and positive at some point in~$X$. Then there exist a
positive integer~$\nu _0$ and a positive constant~$c_0$ which depend only
on~$X$ and~$\cal C(L,h)$ and which have the following property. If~$\nu $ is an
integer with $\nu \geq \nu _0$, $R$ is a nowhere dense analytic subset of~$X$ whose
complement~$M=X\setminus R$ is smooth and admits a complete K\"ahler metric,
$(F,k)$ is a Hermitian holomorphic line bundle on~$X$ with semi-positive curvature,
$E_{\nu }=L^{\nu }\otimes F$,
and $\pi : \widetilde M @>>> M$ is a connected covering space of degree~$d$
($1\leq d\leq \infty $), then
$$
c_0\nu ^nd \leq
\dim H^0_{L^2}(\widetilde M , {\cal O} (\pi ^* (E_{\nu } \otimes K_M))).
$$
The $L^2$ condition is taken with respect to the Hermitian metric
$\pi ^*(h^{\nu }\otimes k)$ in $\pi ^*E_{\nu }$
and, for any choice of a Hermitian metric~$g$ on~$\widetilde M $,
with respect to the Hermitian metric
$g^*$~in~$K_{\widetilde M }=\pi ^*K_M$ and~$g$ on~$\widetilde M $
(the $L^2$-norm of an $(n,0)$-form does not depend on the choice of the
metric on the manifold).
\end{thm}
\begin{rems}
1. The curvature condition on~$L$
means that if $s$ is a nonvanishing
holomorphic section of~$L$ on an open set~$W$, then $-\log |s|^2_h$ is plurisubharmonic
and, for some choice of~$W\neq \emptyset $, $-\log |s|^2_h$ is strictly plurisubharmonic .
\noindent 2. The proof will also show that
\begin{align*}
c_0\nu ^nd
&\leq c_0\nu ^n(d-1)+\dim H^0(X,{\cal O} (E_{\nu })\otimes \Omega ^n_X) \\
&\leq
\dim \biggl(
H^0_{L^2}(\widetilde M , {\cal O} (\pi ^* (E_{\nu } \otimes K_M)))
+\pi ^*H^0(X,{\cal O} (E_{\nu }) \otimes \Omega ^n_X)
\biggl);
\end{align*}
where the sum in the last expression takes place in
$H^0(\widetilde M ,{\cal O} (\pi ^*(E_{\nu }\otimes K_M)))$.
\noindent 3. By a theorem of Demailly~[D1], $M=X\setminus R$ admits
a complete K\"ahler metric if, for example, $X$ is a complete
K\"ahler manifold and~$R$ is a compact analytic subset. In particular, any smooth
quasiprojective variety admits a complete K\"ahler metric. Thus we get as
a special case the following:
\end{rems}
\begin{cor}
Suppose $(L,h)$ is a positive Hermitian
holomorphic line bundle on an irreducible reduced projective variety~$X$
of dimension~$n$. Then there exist a
positive integer~$\nu _0$ which depends
on~$X$ and~$\cal C(L,h)$ and which has the following property. If~$\nu $ is an
integer with $\nu \geq \nu _0$, $R$ is a nowhere dense analytic subset
of~$X$ with smooth complement~$M=X\setminus R$, and
$\pi : \widetilde M @>>> M$ is a connected covering space of degree~$d$, then
$$
d\leq \dim H^0_{L^2}(\widetilde M , {\cal O} (\pi ^* (L^{\nu } \otimes K_M))).
$$
The $L^2$ condition is taken with respect to the Hermitian metric
$\pi ^*h^{\nu }$ in $\pi ^*L^{\nu }$
and, for any choice of a Hermitian metric~$g$ on~$\widetilde M $,
with respect to the Hermitian metric
$g^*$~in~$K_{\widetilde M }=\pi ^*K_M$ and~$g$ on~$\widetilde M $.
\end{cor}
\begin{pf*}{Proof of Theorem~2.2} By hypothesis, $\cal C(L,h) \geq 0$ on~$X$ and
$\cal C(L,h)>0$ on
some relatively compact open subset~$W$ of~$X$. We may assume that $W\subset \reg X$
and that there exist
holomorphic coordinates $z=(z_1,\dots ,z_n)$ with $|z|<1/2$ on~$W$.
Fix a nonempty relatively compact
open subset~$V$ of~$W$ and a~$C^{\infty } $ function~$\rho $ with compact support
in~$W$ such that $\rho \equiv 1$ on a neighborhood of~$\overline V$, and, for
each point~$p\in V$, let~$\varphi _p$ be the~$C^{\infty } $ function on~$X$ defined
by
$$
\varphi _p(x) =
\left\{
\begin{alignedat}{2}
&\rho (x)\log (|z(x)-z(p)|^2)& \quad \text { if } x\in W \\
&0&\quad \text { if } x\in X\setminus W
\end{alignedat}
\right.
$$
Then $\text {supp}\, \varphi _p =\text {supp}\, \rho \subset W$,
$\varphi _p =\log (|z-z(p)|^2)$ (a plurisubharmonic function ) on~$V$, and there is a
positive constant~$a_0$ which does not depend on~$p$ such that
$a_0\cal C(L,h)+\lev {\varphi _p}$ is semipositive on~$X\setminus \{ p \} $
and
positive on~$W\setminus \{ p \} $. Fix an integer~$\nu _0 > na_0$
(later, we will also choose $c_0$ to depend only
on~$a_0$ and~$n$).
Let $\sing X\subset R\subset X$, $\pi : \widetilde M @>>> M=X\setminus R$, and d be as in the
statement of the theorem and fix a point~$p$ in the nonempty open
set~$V\setminus R$. Given a multi-index
$\alpha =(\alpha _1,\dots ,\alpha _n)\in \Bbb Z _{\geq 0}^n$,
a nonnegative integer~$\nu $, and a $C^{\infty } $ section~$s$ of
$E_{\nu }\otimes K_X=L^{\nu }\otimes F\otimes K_X$ on a neighborhood of~$p$, we
denote by $\partial ^{|\alpha |}s/\partial z^{\alpha }$ the corresponding
multiple derivative of~$s$ with respect to some fixed trivialization
in~$L$~and~$F$ on a neighborhood of~$p$ and the trivialization in~$K_X$ induced by the
holomorphic coordinates $z=(z_1,\dots ,z_n)$ on~$W$. Similarly, if~$s$ is a~$C^{\infty } $
section of $\pi ^* (E_{\nu }\otimes K_M)=\pi ^*E_{\nu } \otimes K_{\widetilde M }$
on a neighborhood of a point $q\in \pi ^{-1} (p)$, then we denote by
$\partial ^{|\alpha |}s/\partial z^{\alpha }$ the corresponding
multiple derivative of~$s$ with respect to the trivialization and local
coordinates lifted from~$X$.
We will now apply the $\bar \partial $-method to show that if $\nu \geq \nu _0$,
$\alpha =(\alpha _1,\dots ,\alpha _n)$ is a multi-index with
$|\alpha |=\sum \alpha _j \leq (\nu /a_0)-n$, and $q\in \pi ^{-1} (p)$, then
there exists a section
$$
s\in H^0_{L^2}(\widetilde M , {\cal O} (\pi ^* (E_{\nu } \otimes K_M)))
=H^0_{L^2}(\widetilde M , {\cal O} (\pi ^* E_{\nu }\otimes K_{\widetilde M }))
$$
such that , for every multi-index~$\beta $ with
$|\beta | \leq (\nu /a_0)-n$ and for every point $r\in \pi ^{-1} (p)$, we have
$$
\frac {\partial ^{|\beta |}s}{\partial z^{\alpha }}(r)=
\left\{
\begin{alignedat}{2}
&1& \quad \text { if } \beta =\alpha \text { and } r=q \\
&0&\quad \text { otherwise}
\end{alignedat}
\right.
$$
By hypothesis, there exists a complete K\"ahler metric~$g$ on $M=X\setminus R$.
Let $\widetilde E _{\nu }=\pi ^*E_{\nu }$ for each~$\nu $, let $\tilde h =\pi ^*h$,
let $\tilde k =\pi ^*k$, let $\tilde g =\pi ^*g$, and let
$\tilde \varphi _p=\pi ^*\varphi _p$. We may choose a relatively compact neighborhood~$U$
of~$q$ in $\pi ^{-1} (V) \setminus (\pi ^{-1} (p)\setminus \{ q \} )$ and
a~$C^{\infty } $ section~$u$ of $\widetilde E _\nu \otimes K_{\widetilde M }$ with compact
support in~$U$ such that $u$ is holomorphic on a neighborhood of~$q$ in~$\widetilde M $ and, for
every multi-index~$\beta $,
$$
\frac {\partial ^{|\beta |}u}{\partial z^{\alpha }}(q)=
\left\{
\begin{alignedat}{2}
&1& \quad \text { if } \beta =\alpha \\
&0&\quad \text { if } \beta \neq \alpha
\end{alignedat}
\right.
$$
Hence the form $\theta =\bar \partial u$ is a~$C^{\infty } $~$\bar \partial $-closed
$(n,1)$-form with values in~$\widetilde E _\nu $ and the support of~$\theta $
is a compact subset of $U\setminus \pi ^{-1} (p)$ (since $\bar \partial u=0$ near~$q$).
By construction, there is also a continuous function $\lambda : \widetilde M @>>> [0,1]$
such that $\lambda >0$ on~$\pi ^{-1} (V)$ and
$$
\cal C(\widetilde E _\nu ,{\exp ({-\frac {\nu }{a_0}\tilde \varphi _p})
\tilde h ^\nu \otimes \tilde k })
=
\nu \cal C(\pi ^*L,\tilde h ) + \cal C(\pi ^*F,\tilde k )
+\frac {\nu }{a_0}\lev {\tilde \varphi _p}
\geq \lambda \tilde g
$$
on~$\widetilde M \setminus \pi ^{-1} (p) $.
Moreover,
$$
\int _{\widetilde M } \lambda ^{-1}
|\theta |^2_{\tilde h ^\nu \otimes \tilde k \otimes \tilde g ^*}
e^{-\frac {\nu }{a_0}\tilde \varphi _p} \, dV_{\tilde g } < \infty ,
$$
because $\theta $ has compact support in
$U\setminus \pi ^{-1} (p)$, $\lambda >0$ on~$U$, and $\tilde \varphi _p$ is
smooth on $U\setminus \{ q \}= U\setminus \pi ^{-1} (p)$.
Applying Demailly's theorem (Theorem~2.1), one gets a $C^{\infty } $ form~$\eta $
of type~$(n,0)$ with values in $\widetilde E _\nu $ on~$\widetilde M $ such that
$$
\bar \partial \eta =\theta \quad \text {and} \quad
\int _{\widetilde M }
|\eta |^2_{\tilde h ^\nu \otimes \tilde k \otimes \tilde g ^*}
e^{-\frac {\nu }{a_0}\tilde \varphi _p} \, dV_{\tilde g } < \infty .
$$
In particular, the $(n,0)$-form $s=u-\eta $ is in
$H^0_{L^2}(\widetilde M , {\cal O} (\widetilde E_{\nu }\otimes K_{\widetilde M }))$
because $\bar \partial s=0$ and $\tilde \varphi _p\leq 0$.
Since $u$ is holomorphic near each point $r\in \pi ^{-1} (p)$, so
is~$\eta $. Moreover, in suitable local holomorphic coordinates
$w=(w_1,\dots ,w_n)$ centered at~$r$ in a neighborhood ~$Q$, we have
$\tilde \varphi _p(w)=\log |w|^2$ and hence
$$
\int _Q|\eta |^2|w|^{-2\nu/a_0} \, dV<\infty ;
$$
where the notation for the metrics has been suppressed.
Therefore $\eta $ vanishes at~$r$ to an order greater than $(\nu /a_0)-n$.
Thus, if $\beta $ is a multi-index with $|\beta |\leq (\nu /a_0)-n$,
then $\partial ^{|\beta |}\eta /z^\beta $ vanishes at each point
$r\in \pi ^{-1} (p)$ and the claim follows.
The claim implies that if, for each $c\geq 0$,
$b_c=\binom {[c]+n}{n}$ denotes the number of
multi-indices~$\alpha $ satisfying $|\alpha | \leq c $, then we have
$$
\dim H^0_{L^2}(\widetilde M , {\cal O} (\widetilde E _{\nu }\otimes K_{\widetilde M }))
\geq b_{(\nu /a_0)-n}\cdot d
$$
for each integer $\nu \geq \nu _0$. It is easy to see that
$b_{(\nu /a_0)-n}\geq c_0\nu ^n$ for some positive constant~$c_0$ depending
only on $a_0$~and~$n$ and the theorem now follows.
\end{pf*}
\begin{rem} To obtain the inequalities given in the remark~2,
we fix a point $r\in \pi ^{-1} (p)$ and, for each point
$q\in \pi ^{-1} (p)\setminus \{ r \} $ and each multi-index~$\alpha $
with $|\alpha |\leq (\nu /a_0)-n$, we form a section in
$H^0_{L^2}(\widetilde M , {\cal O} (\widetilde E _{\nu }\otimes K_{\widetilde M }))$
as in the above proof. We then get a collection of
$b_{(\nu /a_0)-n}\cdot (d-1)$
linearly independent sections and the span of this collection
meets $\pi ^*H^0(X,{\cal O} (E_\nu ) \otimes \Omega ^n_X)$ only in the zero
section.
Therefore
\begin{align*}
c_0\nu ^n(d-1)&+\dim H^0(X,{\cal O} (E_{\nu }) \otimes \Omega ^n_X) \\
&\leq
\dim \biggl(
H^0_{L^2}(\widetilde M , {\cal O} (\widetilde E _{\nu } \otimes K_{\widetilde M }))
+
\pi ^*H^0(X,{\cal O} (E_{\nu }) \otimes \Omega ^n_X)\biggl) .
\end{align*}
Finally, observe that if we take $\widetilde M =M$ and
$s\in H^0_{L^2}(M , {\cal O} (E _{\nu }\otimes K_M))$, then, since the
$L^2$~condition in the canonical bundle is independent of the choice of
the metric on the base manifold (provided one also takes the
associated metric in the canonical bundle), the
pullback~$s'$
of $s$ to a desingularization~$X'$ of~$X$ is locally
in~$L^2$ with respect to a metric on~$X'$. Therefore $s'$ extends to
a section in~$H^0(X', \Omega ^n_{X'})$ and hence $s$ extends to
a section in~$H^0(X, \Omega ^n_X)$. It follows that
$$
c_0\nu ^n \leq \dim H^0(X,{\cal O} (E_{\nu }) \otimes \Omega ^n_X).
$$
Thus we get all of the desired inequalities.
\end{rem}
\section{Further generalizations of the weak Lefschetz theorem}
Theorem~2.2 and the arguments given in the proofs of Theorem~1.1
and Theorem~1.4
now give the following generalization:
\begin{thm}
Let $U$ be an irreducible reduced complex space, let $X$ be
connected normal complex space of dimension $n>1$,
let $\Phi :U @>>> X$ be a holomorphic mapping, and let
$(L,h)$ be a Hermitian holomorphic line bundle on~$X$. Assume that
\begin{enumerate}
\item[(i)] $\Phi (U)$ has nonempty interior, and
\item[(ii)] The curvature of~$(L,h)$ is semipositive everywhere on~$X$ and
positive at some point in~$X$.
\end{enumerate}
Then there exist a positive integer~$\nu _0$ and a positive constant~$c_0$
which depend only on~$X$ and (the curvature of)~$(L,h)$
such that , if~$R$ is a nowhere dense analytic subset of~$X$
whose complement~$\reg{X}\setminus R$ in $\reg{X}$ admits a complete K\"ahler
metric, $V$~is a (nonempty) domain in~$U$,
and~$\nu $~is an integer with $\nu \geq \nu _0$,
then
\begin{align*}
c_0\nu ^n d &\leq
\dim H^0(V, {\cal O} (\Phi ^*L^\nu )\otimes \Omega ^n_U);
\end{align*}
where $d$ is the index of the
image~$G$ of $\pi_1(V\setminus \Phi ^{-1} (R)) @>>> \pi _1(X\setminus R)$.
In particular, if
$H^0 (V, {\cal O} (\Phi ^*(L^\nu )) \otimes \Omega ^n_U)$
is finite dimensional for some choice of a sufficiently large~$\nu $,
then $G$ is of finite index.
\end{thm}
Next, we show that for $U$ and $X$ smooth, there
exists a bound on the index in terms of the dimension of the
space of sections of an {\it invertible} sheaf. We first prove
an elementary fact which relates sections of the pullback of the
canonical bundle to holomorphic $n$-forms (see also Sakai~[S]).
\begin{lem}
Suppose
$\Phi =(\Phi _1,\dots ,\Phi _n) : \Delta ^m @>>> \C ^n$ is a holomorphic mapping
and $\Phi _*$ has rank~$n$ at each point in $\Delta ^*\times\Delta ^{m-1}$.
We denote
the coordinates in~$\C ^m$ and the coordinates in~$\Cn$ by
$z=(z_1,\dots ,z_m)$
and
$\zeta =(\zeta _1,\dots ,\zeta _n)$, respectively. Let~$l\geq 0$ be the order
of vanishing of the holomorphic $n$-form $d\Phi _1\wedge \dots \wedge d\Phi _n$
along $\{ 0 \} \times \Delta ^{m-1}$. Then the mapping
${\cal O} (\Phi ^*K_{\C ^n}) @>>> \Omega ^n_{ \Delta ^m }$
given by
$$
s =f\Phi ^* (d\zeta _1\wedge \dots \wedge d\zeta _n)
\mapsto fz_1^{-l}d\Phi _1\wedge \dots \wedge d\Phi _n
$$
maps ${\cal O} (\Phi ^*K_{\C ^n}) $ isomorphically onto the sheaf of
holomorphic $n$-forms~$\theta $ such that
$\theta _z\in \C (d\Phi _1\wedge \dots \wedge d\Phi _n)_z$ for each
point $z\in \Delta ^* \times \Delta ^{m-1}$ at which~$\theta $
is defined.
\end{lem}
\begin{rem}
Here $\Phi ^* (d\zeta _1\wedge \dots \wedge d\zeta _n)$
denotes the pullback of $d\zeta _1\wedge \dots \wedge d\zeta _n$
as a section of $\Phi ^*K_{\Cn }$ while
$d\Phi _1\wedge \dots \wedge d\Phi _n$ is the pullback as
a form of type~$(n,0)$.
\end{rem}
\begin{pf} Clearly, $z_1^{-l}d\Phi _1\wedge \dots \wedge d\Phi _n$ is
a holomorphic $n$-form on~$\Delta ^m$, so we get an injective mapping
as described above. Conversely, suppose~$\theta $ is a holomorphic $n$-form on
an open set $V\subset \Delta ^m$ and there exists a holomorphic function~$h$ on
$V\cap (\Delta ^* \times \Delta ^{m-1})$ with
$\theta =hd\Phi _1\wedge \dots \wedge d\Phi _n$ on
$V\cap (\Delta ^* \times \Delta ^{m-1})$. We have
$$
\theta =\Sigma ' \theta _Idz_I \text { on } V \quad
\text {and} \quad
d\Phi _1\wedge \dots \wedge d\Phi _n
=\Sigma ' \beta _Idz_I \text { on } \Delta ^m;
$$
where $\sum '$ denotes the sum over increasing multi-indices. In particular,
$$
l=\min _I (\text {ord} _{ \{ 0 \} \times \Delta ^{m-1} }\beta _I )
=\text {ord} _{ \{ 0 \} \times \Delta ^{m-1} }\beta _{I_0}
$$
for some multi-index~$I_0$, and, for each nonzero coefficient~$\beta _I$,
we have $h=\theta _I/ \beta _I$ on
$V\cap (\Delta ^* \times \Delta ^{m-1})$.
Therefore $h$ is a
meromorphic function on~$V$ with pole set contained in
$\{ 0 \} \times \Delta ^{m-1}$ and
$z_1^lh= \theta _{I_0}/ (\beta _{I_0}/z_1^l)$. Since the intersection of the
zero set of $\beta _{I_0}/z_1^l$ and $\{ 0 \} \times \Delta ^{m-1}$
has codimension at least~$2$ in~$\Delta ^m$ and since the
the pole set of $z_1^lh$ lies in this intersection, the pole set must
be empty. Therefore $z_1^lh$
is holomorphic on~$V$ and the holomorphic section
$s=z_1^lh\Phi ^*(d\zeta _1\wedge \dots \wedge d\zeta _n)$
of $\Phi ^*K_{\C ^n}$ maps to $\theta $
(on $V\cap (\Delta ^* \times \Delta ^{m-1})$ and hence on~$V$).
\end{pf}
\begin{thm}
Let $U$~and~$X$ be connected complex manifolds of dimensions~$m$ and
$n>1$, respectively, let $\Phi :U @>>> X$ be a
holomorphic mapping, and let~$(L,h)$ be a Hermitian holomorphic line bundle on~$X$.
Assume that
\begin{enumerate}
\item[(i)] $\Phi $ has rank~$n$ at some point
(i.e.~$\Phi (U)$ has nonempty interior), and
\item[(ii)] The curvature of $(L,h)$ is semipositive everywhere on~$X$ and
positive at some point in~$X$.
\end{enumerate}
Then
there exist a positive integer~$\nu _0$ and a positive constant~$c_0$ which
depend only on~$X$ and (the curvature of)~$(L,h)$ and there exists
an effective divisor~$D_0$ in~$U$ which depends only on the mapping
$\Phi : U @>>> X$ such that , if~$R$ is a nowhere dense analytic subset of~$X$
whose complement~$X\setminus R$ admits a complete K\"ahler metric,
$(F,k)$ is a Hermitian holomorphic line bundle on~$X$
with semipositive curvature, $V$~is a (nonempty) domain in~$U$,
$\nu $~is an integer with $\nu \geq \nu _0$, $E_\nu =L^\nu \otimes F$,
and $d$ is the index of the
image~$G$ of $\pi_1(V\setminus \Phi ^{-1} (R)) @>>> \pi _1(X\setminus R)$,
then we have the estimates
\begin{align*}
c_0\nu ^nd
&\leq c_0\nu ^n (d-1) +\dim H^0 (X, {\cal O} (E_\nu \otimes K_X)) \\
&\leq
\dim H^0 (V, {\cal O} (\Phi ^*(E_\nu \otimes K_X)\otimes [D_0])) \tag 1
\\
&\leq
\dim H^0(V, {\cal O} (\Phi ^*E_\nu )\otimes \Omega ^n_U).
\end{align*}
\end{thm}
\begin{rems}
1. If, for some positive integer~$k$,
$L^k\otimes K ^{-1} _X$ is semipositive,
then we may take $F=L^k\otimes K ^{-1} _X$ and we get estimates which
do not involve the canonical bundle~$K_X$.
\noindent 2. The divisor~$D_0$ which will be constructed is probably not
the optimal choice.
\end{rems}
\begin{pf*}{Proof of Theorem~3.3}
Guided by Lemma~3.2, we first
describe~$D_0$.
The set
$$
B=\setof {x\in U}{\text {rank}\, (\Phi _*)_x < n }
$$
is a nowhere dense analytic subset of~$U$. Let $\{ A_i \} $ be the collection of all of
the irreducible components of~$B$ of dimension~$m-1$ whose image $\Phi (A_i)$ lies in some
nowhere dense analytic subset of~$X$, let $A=\cup _i A_i$, and, for each~$i$,
let~$l_i$ be the minimal order of vanishing along~$A_i$ of the
$(n\times n)$-minor determinants of~$\Phi _*$. In other words, if
$\Phi =(\Phi _1,\dots , \Phi _n)$ and
$d\Phi _1\wedge \dots \wedge d\Phi _n=\sum 'a_Jdz_J$ with respect to local
coordinates $(z_1,\dots ,z_m)$ near $x_0\in A_i$ in~$U$ and
$(\zeta _1,\dots , \zeta _n)$ near~$\Phi (x_0)$ in~$X$, then
$$
l_i=\min _J (\text {ord}_{A_i} a_J).
$$
We define
$$
D_0=\sum l_iA_i.
$$
Given a holomorphic line bundle~$E$ on~$X$,
Lemma~3.2 implies that we have an injective linear mapping
$$
H^0(U, {\cal O} ((\Phi ^* (E\otimes K_X))\otimes [D_0]))
@>>>
H^0(U, {\cal O} (\Phi ^* E)\otimes \Omega ^n_U)
$$
given as follows. Let~$t$ be a global defining section for~$[D_0]$
on~$U$. To each section
$s\in H^0(U, {\cal O} ((\Phi ^* (E\otimes K_X))\otimes [D_0]))$, we may
associate a holomorphic $n$-form~$\theta $ with values in
$(\Phi ^* E)\otimes [D_0]$ and $\theta /t$ is a holomorphic $n$-form
on~$U\setminus A$ with values in $\Phi ^* E$. But the lemma implies that,
near points of $A\setminus \sing B$, $\theta /t$ extends holomorphically
past~$A$. Thus $\theta /t$ extends to a
holomorphic $n$-form on~$U\setminus (A\cap \sing B)$ with values in~$\Phi ^* E$.
Since co$\dim \sing B \geq 2$, $\theta /t$ extends holomorphically to the
entire manifold~$U$. Thus we get a mapping $s \mapsto \theta /t$
(similarly, this mapping surjects onto the space of holomorphic $n$-forms
with values in~$\Phi ^* E$ whose restriction to $U\setminus A$ comes
from a section of $\Phi ^*(E\otimes K_X)$). In particular,
the third of the inequalities in~(1) holds.
Let
\begin{center}\begin{picture}(250,80)
\put(5,10){$W=V\setminus S\subset V $}
\put(90,14){\vector(1,0){50}}
\put(145,10){$X \supset X\setminus R=M$}
\put(110,3){$\Phi $}
\put(125,55){\vector(3,-1){90}}
\put(110,60){$\widetilde M $}
\put(17,25){\vector(3,1){90}}
\put(57,45){$\tilde \Phi $}
\put(170,45){$\pi $}
\end{picture}
\end{center}
be a commutative diagram as in the proof of Theorem~1.1.
We will show that if
$s\in H^0_{L^2}(\widetilde M , {\cal O} ((\pi ^*E_\nu )\otimes K_{\widetilde M }))$
for some~$\nu $ (with respect to metrics lifted
from the base), then $(\tilde \Phi ^*s)\otimes t$ extends to a unique
holomorphic section of $(\Phi ^* (E_\nu \otimes K_X))\otimes [D_0]$ on~$V$.
By Lemma~1.3, the pullback of $s$ {\it as an $E_\nu $-valued holomorphic $n$-form}
extends to a $\Phi ^*E_\nu $-valued holomorphic $n$-form on~$V$.
In particular, $\tilde \Phi ^*s$
extends holomorphically as a section of
$\Phi ^*(E_\nu \otimes K_M)$ near each point at which
$\Phi _*$ is of maximal rank (by Lemma~3.2 with $l=0$). Moreover,
$\tilde \Phi ^*s$ extends holomorphically
past analytic sets of codimension at least~$2$. Therefore, it suffices to
show that $(\tilde \Phi ^*s)\otimes t$ extends holomorphically near each point
$x_0\in \reg S\cap \reg B$ at which $S\cap B$ is of dimension~$m-1$.
An irreducible component of~$B$ containing such a point~$x_0$ must also be an
irreducible component of~$S=\Phi ^{-1} (R)$ and must therefore be one of the irreducible components~$A_i$
of the support~$A$ of~$D_0$. Since the pullback of $s$ as an $E_\nu $-valued
holomorphic $n$-form extends to~$V$, Lemma~3.2 and the definition of $D_0$ and $t$
now imply the claim.
Clearly, if $s$ is a holomorphic section of $E_\nu \otimes K_X$ on $X$,
then $(\Phi ^*s)\otimes t$ is a holomorphic section whose restriction to~$V$
is an extension of $(\tilde \Phi ^*\pi ^* s)\otimes t$. Thus we get an
injective linear mapping of the subspace
$$
\cal S=H^0_{L^2}(\widetilde M , {\cal O} ((\pi ^*E_\nu )\otimes K_{\widetilde M }))+
\pi ^* H^0(X, {\cal O} (E_\nu \otimes K_X))
$$
of
$H^0(\widetilde M , {\cal O} ((\pi ^*E_\nu )\otimes K_{\widetilde M }))$
into $H^0(V,{\cal O} ((\Phi ^*(E_\nu \otimes K_X))\otimes [D_0]))$.
We have, therefore
\begin{align*}
\dim \cal S &\leq
\dim H^0 (V, {\cal O} (\Phi ^*(E_\nu \otimes K_X)\otimes [D_0])) \\
&\leq
\dim H^0(V, {\cal O} (\Phi ^*E_\nu )\otimes \Omega ^n_U).
\end{align*}
The second remark following Theorem~2.2 now gives the inequalities~(1) for
$\nu $ sufficiently large
and for some constant~$c_0$ (both depending only on $(L,h)$ and $X$).
\end{pf*}
\begin{rems}
1. The proofs of Lemma~1.3 and Theorem~3.3 show that one can form a
divisor $D_R$
which depends on~$R$, but which satisfies $D_R\leq D_0$ and gives
a sharper estimate for the index. For example, it suffices to include only
those irreducible components $A_i$ which are contained in $S=\Phi ^{-1} (R)$, so one may choose
$D_R$ to have support contained in~$S$. Moreover, the proof of Lemma~1.3 shows
that if $\Phi (A_i)$ contains a point~$p\in R$ at which $R$ is a divisor with
normal crossings and $u$ is a defining function for $R$ near~$p$, then one
may take the coefficient of~$A_i$ to be $-1+ \text {ord}_{A_i}(u\circ \Phi )$.
The proof also shows that, by choosing~$p$ so that this coefficient
is minimal, we get $D_R\leq D_0$.
\noindent 2. Similarly, for $U$ a normal neighborhood of a compact complex space~$Y$
and $X$ a smooth projective variety, one can
find a uniform bound on the index
in terms of the dimension of a space of sections of a line
bundle pulled back from $X$ as in Theorem~1.1. More precisely, we have
the following:
\end{rems}
\begin{cor} Let $\Phi :U @>>> X$ be a holomorphic mapping of a
connected normal complex space~$U$ into a connected smooth projective variety~$X$
of dimension~$n>1$, let~$Y$ be a connected compact analytic
subspace (not necessarily reduced) of~$U$, and
let $\widehat U $ be the formal completion of~$U$ with respect to~$Y$.
Assume that
\begin{enumerate}
\item[(i)] $\Phi (U)$ has nonempty interior, and
\item[(ii)]
$\dim H^0(\widehat U , \widehat {{\cal O} (\Phi ^*L)})
<\infty $ for every holomorphic line bundle~$L$ on~$X$.
\end{enumerate}
Then there is a positive constant~$b$ depending only on the mapping
$\Phi :U @>>> X$ and the subspace~$Y\subset U$ such that , if $R\subset X$
is a nowhere dense analytic subset of~$X$ and~$V$ is a connected neighborhood of~$Y$ in~$U$,
then the image~$G$ of
$\pi _1(V\setminus \Phi ^{-1} (R)) @>>> \pi _1(X\setminus R)$
is of index at most~$b$ in~$\pi _1(X\setminus R)$.
\end{cor}
\begin{pf*}{Sketch of the proof} First suppose $U$ is smooth and let
$D_0=\sum l_iA_i$ be the associated divisor in~$U$ as in the proof
of Theorem~3.3. By construction, each of the sets $\Phi (A_i)$ is contained
in some nowhere dense analytic subset of~$X$. By replacing $U$ by a relatively compact
neighborhood of~$Y$,
we may assume that there is a nowhere dense analytic subset~$C$ in~$X$
which contains all of these sets and that the collection of coefficients
$\seq li$ is bounded. Hence we may choose a positive holomorphic
line bundle~$L$ on $X$ and a holomorphic section $t$ of~$L$ such that the
divisor $D_1$ of the section $\Phi ^* t$ satisfies $D_1\geq D_0$.
Now let $R$, $V$, and $G$ be as in the statement of the corollary
and let
\begin{center}\begin{picture}(250,80)
\put(5,10){$W=V\setminus S\subset V $}
\put(90,14){\vector(1,0){50}}
\put(145,10){$X \supset X\setminus R=M$}
\put(110,3){$\Phi $}
\put(125,55){\vector(3,-1){90}}
\put(110,60){$\widetilde M $}
\put(17,25){\vector(3,1){90}}
\put(57,45){$\tilde \Phi $}
\put(170,45){$\pi $}
\end{picture}
\end{center}
be a commutative diagram as in the proof of Theorem~1.1.
By the proof of Theorem~3.3 and the above remarks,
if $s\in H^0_{L^2}(\widetilde M ,{\cal O} (\pi ^* L\otimes K_{\widetilde M }))$,
then $(\tilde \Phi ^*s)\otimes (\Phi ^* t)$ extends to a
holomorphic section of $\Phi ^*(L^2\otimes K_X)$ on~$V$.
If $U$ is connected and normal (but not necessarily smooth),
then we may form a desingularization $\alpha : U' @>>> U$ of~$U$
and a commutative diagram
\begin{center}\begin{picture}(250,80)
\put(70,10){$U$}
\put(85,14){\vector(1,0){65}}
\put(155,10){$X$}
\put(115,0){$\Phi $}
\put(75,55){\vector(0,-1){30}}
\put(70,60){$U'$}
\put(60,35){$\alpha $}
\put(85,60){\vector(2,-1){70}}
\put(115,50){$\Phi '$}
\end{picture}
\end{center}
We may associate to $\Phi ' : U' @>>> X$ (after shrinking~$U$)
a line bundle~$L$ and
a section~$t$ as above, and we get the extension property for pullbacks
of $L^2$ sections as described. On the other hand,
$U$ is normal, so
$\alpha _*{\cal O} ((\Phi ')^*(L^2\otimes K_X))={\cal O} (\Phi ^*(L^2\otimes K_X))$.
Therefore the extension property also holds in~$U$, and
the usual argument now applies.
\end{pf*}
We close this section by observing that
Theorem~3.1 has immediate consequences for pseudoconcave spaces.
An open subset $\Omega $ of a complex space~$X$ is said to have
{\it pseudoconcave boundary in the sense of Andreotti}~[A]
if each point~$x_0\in \partial \Omega $ admits a fundamental system of
neighborhoods~$W$ in~$X$ such that $x_0$ is an interior point of
$$
\widehat {(W\cap \Omega )}_X
=\setof {x\in X}{|f(x)|\leq \sup _{W\cap \Omega }|f|\quad \forall \,
f\in {\cal O} (X) }.
$$
For example, by Proposition~10 of~[A],
if each irreducible component of~$X$ has dimension at least~$k>1$ and,
for each point~$x_0\in \partial \Omega $, there is a
$C^{\infty } $~$(k-1)$-convex function~$\varphi $ on a neighborhood~$W$ of~$x_0$ in~$X$
such that
$$
\Omega \cap W =\setof {x\in W}{\varphi (x) >0},
$$
then $\Omega $ has pseudoconcave boundary in the sense of Andreotti.
A connected complex space~$X$ is said to be
{\it pseudoconcave in the sense of Andreotti}~[A] if there
exists a nonempty relatively compact open subset~$\Omega $ which has
pseudoconcave boundary in the sense of Andreotti and which meets each
irreducible component of~$X$. By a finiteness theorem of
Andreotti~[A, Theorem~1], if $\cal F$ is a
torsion-free coherent analytic sheaf on a locally irreducible connected complex space~$X$
and $X$ is pseudoconcave in the sense of Andreotti, then
$\dim H^0(X,\cal F) < \infty $
(the case in which $X$ admits a $C^{\infty } $~$(k-1)$-convex exhaustion
function, where the dimension of~$X$ is at least $k>1$ at each point, is due
to Andreotti and Grauert~[AG]).
Theorem 3.1 and Andreotti's finiteness theorem together give
the following:
\begin{cor}
Let $U$ be an irreducible reduced complex space,
let $X$ be a connected normal projective variety of dimension~$n>1$, and let
$\Phi :U @>>> X$ be a holomorphic mapping. Assume that
$\Phi (U)$ has nonempty interior and that $U$ is pseudoconcave in the sense of
Andreotti.
Then there is a positive
constant~$b$ depending only on the mapping $\Phi : U @>>> X$ such that ,
if~$Z$ is a nonempty Zariski open subset of~$X$,
then the image of
$\pi _1(\Phi ^{-1} (Z)) @>>> \pi _1 (Z)$
is of index at most~$b$ in~$\pi _1(Z)$.
\end{cor}
\begin{rems}
1. Clearly, Theorem~3.1 also gives a version of the above theorem in which $X$
is not necessarily projective.
2. There are many results concerning when
a compact analytic subset~$Y$ of an $m$-dimensional complex space~$U$
admits a strongly $(m-1)$-concave neighborhood ,
and hence when one may apply Andreotti's [A] (or Andreotti and
Grauert's~[AG]) finiteness theorem as above.
For example, Okonek~[O] proved that
$Y$ admits a fundamental system of such neighborhoods if $N_{Y/U}$
is Finsler-$q$-positive, where $q=\dim Y$.
\end{rems}
\section{Burns' theorem}
The goal of this section is the following theorem:
\begin{thm}
Let $(X,g)$ be a connected complete Hermitian manifold
and let $M\subset X$ be a domain with
nonempty smooth compact boundary~$\partial M$ in~$X$.
Assume that
\begin{enumerate}
\item[(i)] $M$ is strongly pseudoconvex at each point of~$\partial M$;
\item[(ii)] There exists a Hermitian metric~$a$ in~$K_M$ and a
constant $c>0$ such that $\cal C(K_M,a)\geq cg$ on~$M$; and
\item[(iii)] $X$ has dimension~$n\geq 3$.
\end{enumerate}
Then
$\text {\rm vol} \, _g(M)<\infty $.
\end{thm}
\begin{rems}
1. Since $M$ admits a complete
K\"ahler metric, $\partial M$ is necessarily connected
(see, for example, Proposition~4.4 below).
\noindent 2. If, for example, the Ricci curvature of~$g$ is bounded
above by $-c$ on~$M$, then the associated metric $a=g^*$ in~$K_M$
satisfies the condition~(ii) since
$$
\cal C(K_M,g^*)=-\text {Ric} \, (g) \geq cg.
$$
\noindent 3. Clearly, it is not necessary to assume that $M$ is a
domain in some larger manifold~$X$. The conclusion also holds if
$M=X$ and $M$ admits a $C^{\infty } $
function which, along some end, is strictly plurisubharmonic and exhaustive; since one can then
replace $M$ by a suitable sublevel set of the function. It will, however,
be more convenient to have Theorem~4.1 stated for a domain as above.
\noindent 4. As in the proofs of the weak Lefschetz theorems,
the idea is to apply finite dimensionality of a space of holomorphic
sections of a line bundle to obtain a result about
the manifold.
\end{rems}
Theorem~4.1 and an analysis of the thick-thin decomposition as
in~[BGS] together give as a conclusion the following theorem:
\begin{thm}[Burns~[B{]}]
Let $\Gamma $ be a torsion-free discrete group of
automorphisms
of the unit ball~$B$ in~$\C ^n$ with $n \geq 3$
and let $M =\Gamma \setminus B$. Assume that the
limit set~$\Lambda $ is a proper
subset of~$\partial B$ and that the quotient
$\Gamma \setminus ((\partial B)\setminus \Lambda )$
has a compact component~$A$.
Then $M$ has only finitely many ends; all of which, except for the
(unique) end corresponding to~$A$,
are cusps. In fact, $M$ is
diffeomorphic to a compact manifold with boundary.
\end{thm}
\begin{rem}
By applying a theorem of Lempert~[L]
as in the proof of Theorem~4.1 below and the argument given by
Siu and Yau~[SY], one can close up the cusps projectively.
In other words,
$M\cong \Omega \setminus D$, where~$\Omega $ is a
strongly pseudoconvex domain in a smooth projective variety and~$D$ is
a (compact) divisor contained in~$\Omega $. The
boundary component~$A$
corresponds to~$\partial \Omega $.
\end{rem}
The main tool in the proof of Theorem~4.1 is
Nadel and Tsuji's~[NT] $L^2$ version of
Demailly's~[D2]
asymptotic Riemann-Roch inequality.
\begin{thm}[Nadel-Tsuji~[NT{]}]
Suppose $(X,g)$ is a connected complete K\"ahler manifold
of dimension~$n$ and $(L,h)$ is a Hermitian holomorphic line bundle on~$X$
such that
$$
\cal C(L,h)\geq cg
$$
for some constant $c>0$. Then
$$
\liminf _{\nu @>>> \infty } \nu ^{-n}
\dim H^0_{L^2}(X, {\cal O} (K_X\otimes L^\nu )) \geq
\frac {1}{n!}\int _X \bigl( c_1(L,h)\bigr) ^n.
$$
\end{thm}
\begin{rems}
1. The Chern form $c_1(L,h)$ is the real
form of type~$(1,1)$ (associated to the Hermitian tensor $\cal C(L,h)$)
given by
$$
c_1(L,h)=-\frac {\sqrt {-1}}{2\pi }\partial \bar \partial \log |s|^2_h
$$
for any local nonvanishing holomorphic section~$s$ of~$L$.
\noindent 2. As Nadel and Tsuji observed (see~[NT, Lemma~2.5]),
if, in particular, $X$ is pseudoconcave in the sense of
Andreotti~[A] (see Sect.~3),
then it follows that~$X$ has finite volume.
\noindent 3. The theorem is only stated in~[NT] for~$L$
the canonical bundle, but the proof of the general case is the same.
The first point is that,
for a smooth relatively compact domain~$\Omega $ in~$X$ and for $\lambda >0$,
one has Demailly's~[D2]
generalization of Weyl's asymptotic formula for the
number of eigenvalues~$N_\Omega (\lambda )$ less than or equal
to~$\nu \lambda $ for the Dirichlet problem for the Laplacian
in
$K_X\otimes L^\nu $:
$$
\liminf _{\nu @>>> \infty } \nu ^{-n}
N_\Omega (\lambda )\geq
\frac {1}{n!}\int _\Omega \bigl( c_1(L,h)\bigr) ^n.
$$
The second point is that for a $C^{\infty } $ compactly supported form~$\alpha $
of type~$(n,1)$ with values in~$L^\nu $, the Bochner-Kodaira formula implies
that
$$
\| \bar \partial \alpha \| _{L^2}^2+\| \bar \partial ^*\alpha \| _{L^2}^2
\geq c\nu \| \alpha \| _{L^2}^2.
$$
With these slight changes in mind, the proof given in~[NT]
goes through.
\end{rems}
We will also apply the following Hartogs type extension property:
\begin{prop}
Let $(X,g)$ be a connected complete Hermitian manifold
of dimension~$n>1$ and let $M\subset X$ be a domain with nonempty
smooth compact strongly pseudoconvex boundary. Assume that
the restriction $g| _M$ of~$g$ to~$M$ is K\"ahler.
Suppose~$f$ is a holomorphic function on $U\cap M$ for some
neighborhood~$U$ of~$\partial M$ in~$X$.
Then there exists a holomorphic function~$h$ on~$M$ such that $h=f$ near~$\partial M$. In
particular, $\partial M$ is connected .
\end{prop}
\begin{pf} We may assume that $M=\setof {x\in X}{\varphi (x) <0}$ for some
$C^{\infty } $ function~$\varphi $ on~$X$ which is strictly plurisubharmonic on a neighborhood
of~$X\setminus M$ in~$X$.
Since $g| _M$ is K\"ahler and $g$~is complete on~$X$, a theorem
of Nakano~[N] and of Demailly~[D1] implies that~$M$
admits a complete K\"ahler metric~$g'$. Moreover, the existence of~$\varphi $
implies that $(M,g')$ admits a positive Green's function~$G$ which vanishes
along~$\partial M$. We normalize~$G$ so that, for each point $x_0\in M$,
$$
\Delta _{\text {distr.}}G(\cdot , x_0) =-(2n-2)\sigma _{2n-1}\delta _{x_0};
$$
where $n=\dim X$, $\sigma _{2n-1}=\text {\rm vol} \, (S^{2n-1})$, and
$\delta _{x_0}$ is the Dirac function at~$x_0$.
Fix a $C^{\infty } $ function~$\lambda $ with compact support in~$U$ such that
$\lambda \equiv 1$ on a neighborhood of~$\partial M$ and let~$\alpha $ be the
$\bar \partial $-closed compactly supported form of type~$(0,1)$ on~$M$ given by
$\alpha =\bar \partial (\lambda f)$ (extended by~$0$ to~$M$). Then the function~$\beta $
defined by
$$
\beta (x) =-\frac {1}{(2n-2)\sigma _{2n-1}}
\int _M G(x,y) \bar \partial ^* \alpha (y)\, dV_{g'}(y)
$$
is a $C^{\infty } $ bounded function with finite energy
(i.e.~$\int _M |\nabla \beta | ^2 \, dV_{g'} <\infty $),
$\Delta \beta = \bar \partial ^* \alpha $, and $\beta $ vanishes on~$\partial M$. Hence
$\gamma \equiv\alpha -\bar \partial \beta $
is an $L^2$ harmonic form of type~$(0,1)$ and
the Gaffney theorem~[G] implies that~$\gamma $ is closed
(and coclosed). In particular, $\bar \gamma $ is a holomorphic $1$-form on~$M$ and
$\beta $~is pluriharmonic on $W\cap M$ for some neighborhood~$W$
of~$X\setminus M$ in~$X$.
We will show that~$\beta $ vanishes near~$\partial M$. Fix $a<0$ so
close to~$0$ that~$\varphi $ is strictly plurisubharmonic on
$V=\setof {x\in M}{\varphi (x) >a}$ and $V\subset\subset W$. If
$\rho $ is the real part or the imaginary part of~$\beta $ and $\rho $
does not vanish identically near~$\partial M$, then we may choose
a nonzero regular value~$b$ of~$\rho $ contained in~$\rho (V)$.
Since $b\neq 0$ and $\rho $ vanishes on~$\partial M$, $\rho ^{-1} (b)$
avoids~$\partial M$. Thus the restriction of~$\varphi $ to~$\rho ^{-1} (b)$
assumes its maximum at some point~$x_0\in V\subset W\cap M$ (with $\varphi (x_0)>a$).
But the leaf~$L$ through~$x_0$ of the foliation determined by the
holomorphic $1$-form~$\partial \rho $ on~$V\cap M$ is contained in~$\rho ^{-1} (b)$,
so $\varphi | _L$ also assumes its maximum at~$x_0$.
Since~$\varphi $ is strictly plurisubharmonic on~$V$, we have arrived at a contradiction.
Therefore $\beta $ vanishes near~$\partial M$. Hence
$\gamma =\alpha -\bar \partial \beta $ vanishes near~$\partial M$ and, therefore,
on all of~$M$, since~$\bar \gamma $ is a holomorphic $1$-form. Thus the
function~$h\equiv \lambda f-\beta $ is holomorphic on~$M$ (since $\bar \partial h=\gamma =0$)
and equal to~$f$ near~$\partial M$. In particular, since one can
take $f$ to be a
locally constant function which separates distinct components of~$\partial M$,
$\partial M$ is connected .
\end{pf}
\begin{pf*}{Proof of Theorem~4.1} Since $n\geq 3$, one can apply a
theorem of Rossi~[R] to
``fill in the holes'' and obtain a connected Stein space~$Y$ with isolated
singularities, a relatively compact pseudoconvex domain~$N$ in~$Y$
containing~$\sing Y$, and a biholomorphic mapping $\Phi : U @>>> V$ of a neighborhood~$U$
of~$\partial M$ in~$X$ onto a neighborhood~$V$ of~$\partial N$ in~$Y$ such that
$\Phi (U\cap M)=V\cap N$. Since $N$ may be embedded into a Euclidean space,
Proposition~4.4 implies that~$\Phi $ extends to a holomorphic
mapping $M\cup U @>>> Y$, which we also denote by~$\Phi $,
and $\Phi (M)\subset N$. Next, by a theorem
of Lempert~[L], one can form a ``cap'' on~$N$.
That is, we may assume that~$Y$ is an affine algebraic variety.
By forming the closure~$\overline {Y}$ of~$Y$ in a projective space and
desingularizing~$\overline {Y}$ at infinity, we get a projective
variety~$Z$ with isolated singularities such that
$\sing Z\subset N\cup V \subset Z$. Finally, by replacing~$X$ by
$$
(M\cup U) \cup (V\cup (Z\setminus \overline N))\bigg/
x\in U \sim \Phi (x) \in V
$$
and by replacing the metric $g$ by any extension of $g| _M$ to the
new manifold, we may assume that we have a holomorphic mapping
$\Phi : X @>>> Z$ such that $\Phi (M) \subset N$ and $\Phi $ maps
$(X\setminus M)\cup U$ biholomorphically onto $(Z\setminus N)\cup V$.
In particular, since $X\setminus \overline M \subset\subset X$, it follows
that $X$ is pseudoconcave in the sense of Andreotti.
Now let~$H$ be a positive Hermitian holomorphic line bundle on~$Z$. Then $\Phi ^*H$
is semipositive on~$X$ and positive on $(X\setminus M)\cup U$. On the
other hand, by shrinking~$M$ slightly and extending the Hermitian
metric~$a$, we may assume that $K_X$ admits a Hermitian metric whose
curvature is greater than or equal to $cg$ at each point of~$M$.
It follows that if~$m$ is a
sufficiently large positive integer and
$L=K_X\otimes \Phi ^*H^m$,
then~$L$ admits a Hermitian metric~$h$ such that
$\cal C(L,h)\geq cg$
on~$X$. In particular, $g'=\cal C(L,h)$ is a complete K\"ahler metric on~$X$.
Therefore, by the $L^2$ Riemann-Roch inequality of Nadel and
Tsuji (Theorem~4.3), we have, for every sufficiently large positive
integer~$\nu $,
$$
1+ \, \nu ^{-n}
\dim H^0_{L^2}(X, {\cal O} (K_X\otimes L^\nu )) \geq
\frac {1}{n!}\int _X \bigl( c_1(L,h)\bigr) ^n
\geq c^n\pi ^{-n} \int _X \, dV_{g};
$$
(where the Hermitian metric in $K_X\otimes L^\nu $ is
$(g')^*\otimes h^\nu $). Since, by
Andreotti's finiteness theorem~[A]
(or by~[AG]), the left-hand side is finite, we get
$$
\text {\rm vol} \, _g(M)\leq \text {\rm vol} \, _g(X) <\infty .
$$
\end{pf*}
\begin{rem}
By a version of the $L^2$~Riemann-Roch
inequality due to Takayama~[T], it is only necessary to assume
in the hypothesis~(ii) that $\cal C(K_M,a)\geq cg$ outside a relatively
compact neighborhood of $\partial M$ in $X$.
\end{rem}
\bibliographystyle{amsplain}
|
1997-12-17T19:30:53 | 9712 | alg-geom/9712002 | en | https://arxiv.org/abs/alg-geom/9712002 | [
"alg-geom",
"math.AG"
] | alg-geom/9712002 | Yuri Tschinkel | Victor V. Batyrev and Yu. Tschinkel | Tamagawa numbers of polarized algebraic varieties | 54 pages, minor corrections | null | null | null | null | Let ${\cal L} = (L, \| \cdot \|_v)$ be an ample metrized invertible sheaf on
a smooth quasi-projective algebraic variety $V$ defined over a number field.
Denote by $N(V,{\cal L},B)$ the number of rational points in $V$ having ${\cal
L}$-height $\leq B$. We consider the problem of a geometric and arithmetic
interpretation of the asymptotic for $N(V,{\cal L},B)$ as $B \to \infty$ in
connection with recent conjectures of Fujita concerning the Minimal Model
Program for polarized algebraic varieties.
We introduce the notions of ${\cal L}$-primitive varieties and ${\cal
L}$-primitive fibrations. For ${\cal L}$-primitive varieties $V$ over $F$ we
propose a method to define an adelic Tamagawa number $\tau_{\cal L}(V)$ which
is a generalization of the Tamagawa number $\tau(V)$ introduced by Peyre for
smooth Fano varieties. Our method allows us to construct Tamagawa numbers for
$Q$-Fano varieties with at worst canonical singularities. In a series of
examples of smooth polarized varieties and singular Fano varieties we show that
our Tamagawa numbers express the dependence of the asymptotic of $N(V,{\cal
L},B)$ on the choice of $v$-adic metrics on ${\cal L}$.
| [
{
"version": "v1",
"created": "Mon, 1 Dec 1997 17:38:41 GMT"
},
{
"version": "v2",
"created": "Wed, 17 Dec 1997 18:30:53 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Batyrev",
"Victor V.",
""
],
[
"Tschinkel",
"Yu.",
""
]
] | alg-geom | \section{Introduction}
\bigskip
Let $F$ be a number field (a finite extension
of ${ \bf Q }$), ${\rm Val}(F)$ the set of all valuations of $F$,
$F_v$ the $v$-adic completion of $F$
with respect to $v \in {\rm Val}(F)$, and
$|\cdot |_v\, : \, F_v \rightarrow {\bf R }$ the $v$-adic norm on $F_v$ normalized by
the conditions $| x|_v = |N_{F_v/{\bf Q}_p}(x)|_p$ for $p$-adic
valuations $v \in {\rm Val}(F)$.
Consider a projective space ${ \bf P }^m$
with standard homogeneous
coordinates $(z_0,...,z_m)$ and a locally closed
quasi-projective
subvariety $V \subset { \bf P }^m$ defined over $F$ (we want to stress that
$V$ is not assumed to be projective).
Let $V(F)$ be the set of points in $V$ with
coordinates in $F$.
A {\bf standard height function} $H\,:\, { \bf P }^m(F) \rightarrow {\bf R }_{>0}$
is defined as follows
$$
H(x):=\prod_{v \in {\rm Val}(F)} \max_{j=0, \ldots, m} \{ |z_j(x)|_v \}.
$$
A basic fact about the standard height function $H$ claims that
the set
\[ \{x\in { \bf P }^m(F)\,:\, H(x)\le B\} \]
is finite for any real number $B$ \cite{lang}.
We set
$$
N(V,B)=\#\{x\in V(F)\; : \; H(x)\le B\}.
$$
It is an experimental fact that whenever one succeeds
in proving an asymptotic formula for the
function $N(V,B)$ as
$B\rightarrow \infty$, one obtains the asymptotic
\begin{equation}
N(V,B)= c(V) B^{a(V)}(\log B)^{b(V)-1}(1+o(1))
\label{formula}
\end{equation}
with some constants $a(V) \in {\bf Q},$ $ b(V)
\in \frac{1}{2}{\bf Z}$, and
$c(V) \in {\bf R}_{>0}$.
We want to use this observation as our
starting point. It seems natural
to ask the following:
\medskip
{\bf Question A.} {\em For which
quasi-projective subvarieties
$V \subset {\bf P}^m$ defined over
$F$ do there exist constants
$a(V) \in {\bf Q},$ $ b(V) \in \frac{1}{2}{\bf Z}$ and
$c(V) \in {\bf R}_{>0}$ such that the asymptotic
formula $(1)$ holds? }
\medskip
{\bf Question B.} {\em Does there exist a
quasi-projective variety
$V$ over $F$ with an asymptotic
which is different from {\rm (\ref{formula})}?}
\medskip
In this paper we will be interested not in
Questions $A$ and $B$ themselves but in a
related to them another
natural question:
\bigskip
\noindent
{\bf Question C.} {\em Assume that
$V$ is an irreducible quasi-projective
variety over a number field $F$
such that the
asymptotic formula {\rm (\ref{formula})}
holds. How to compute the constants $a(V),b(V)$ and
$c(V)$ in this formula via some
arithmetical properties
of $V$ over $F$ and geometrical properties
of $V$ over ${\bf C}$?
}
\bigskip
To simplify our terminology, it will be convenient for
us to postulate:
\bigskip
\noindent
{\bf Assumption.} For all quasi-projective $V', V$ with
$V'\subset V\subset { \bf P }^m$ and $|V(F)|=\infty $
there exists the limit
$$
\lim_{B\rightarrow \infty} \frac{N(V',B)}{N(V,B)}.
$$
\bigskip
The following definitions have been useful to us:
\bigskip
\noindent
{\bf Definition ${\bf S_1}$.} A smooth
irreducible quasi-projective
subvariety $V \subset {\bf P}^m$ over
a number field $F$
is called {\bf weakly saturated},
if $|V(F)| = \infty$ and if
for any locally closed
subvariety $W \subset V$ with
${\rm dim}\, W < {\rm dim}\, V$ one has
\[ {\lim}_{B \rightarrow \infty}
\frac{N(W,B)}{N(V,B)} < 1. \]
\bigskip
It is important to
remark that Question C really makes sense {\em only for
weakly saturated} varieties. Indeed, if
there were a locally closed
subvariety $W \subset V$ with ${\rm dim}\,
W < {\rm dim}\, V$ and
\[ {\lim}_{B \rightarrow \infty}
\frac{N(W,B)}{N(V,B)} = 1, \]
then it would be enough to answer Question
C for each irreducible
component of $W$ and for all possible
intersections of these
components (i.e., one could forget about
the existence of $V$ and
reduce the situation to a lower-dimensional case).
In general, it is not
easy to decide whether or not a
given locally closed subvariety
$V \subset {\bf P}^m$ is weakly saturated.
We expect (and our assumption implies this) that
the orbits of connected subgroups
$G \subset PGL(m+1)$ are
examples of weakly saturated varieties
$V \subset {\bf P}^m$ (see \ref{equiv-sat}).
\bigskip
\noindent
{\bf Definition} ${\bf S_2.}$
A smooth irreducible quasi-projective subvariety
$V \subset {\bf P}^m$ with $|N(V,B)| = \infty$ is
called {\bf strongly saturated}, if
for all dense Zariski open subsets
$U \subset V$, one has
\[ {\lim}_{B \rightarrow \infty}
\frac{N(U,B)}{N(V,B)} = 1. \]
\bigskip
First of all, if $V \subset {\bf P}^m$
is a strongly saturated subvariety,
then for any locally closed
subvariety $W \subset V$ with ${\rm dim}\,
W < {\rm dim}\, V$, one has
\[ {\lim}_{B \rightarrow \infty}
\frac{N(W,B)}{N(V,B)} =0, \]
i.e., $V$ is weakly saturated.
On the other hand, if $V \subset {\bf P}^m$
is weakly saturated, but not strongly saturated,
then there must be an
infinite sequence $W_1, W_2, \ldots $ of
pairwise different
locally closed irreducible
subvarieties $W_i \subset V$ with ${\rm dim}\,
W_i < {\rm dim}\, V$
and $|W_i(F)| = \infty$
such that for an arbitrary positive integer $k$ one has
\[ 0 < {\lim}_{B \rightarrow \infty}
\frac{N(W_1 \cup \cdots
\cup W_k,B)}{N(V,B)} < 1. \]
Moreover, in this situation one can always
choose the varieties
$W_i$ to be strongly saturated
(otherwise one could find $W_i' \subset W_i$
with ${\rm dim}\, W_i' < {\rm dim}\,
W_i$ with the same properties as
$W_i$ etc.).
The strong saturatedness of each $W_i$ implies that
\[ {\lim}_{B \rightarrow \infty}
\frac{N(W_{i_1} \cap \cdots
\cap W_{i_l},B)}{N(V,B)} = 0 \]
for all pairwise different $i_1, \ldots, i_l$
and $l \geq 2$. In particular,
one has
\[ \sum_{i=1}^k {\lim}_{B \rightarrow \infty}
\frac{N(W_i,B)}{N(V,B)} =
{\lim}_{B \rightarrow \infty}
\frac{N(W_1 \cup \cdots \cup W_k,B)}{N(V,B)}
< 1\;\; \forall k >0. \]
\medskip
\noindent
{\bf Definition} ${\bf F.}$ Let $V$ be a
weakly saturated quasi-projective
variety in ${\bf P}^m$ and $W_1, W_2, \ldots$
an infinite sequence
of strongly saturated irreducible subvarieties
$W_i$ having the property
\[ 0 < \theta_i :=
{\lim}_{B \rightarrow \infty}
\frac{N(W_i,B)}{N(V,B)} < 1\;\; \forall i > 0. \]
We say that the set $\{W_1, W_2, \ldots \}$
forms an {\bf
asymptotic arithmetic fibration} on $V$,
if the following equality holds
\[ \sum_{i=1}^{\infty} \theta_i = 1. \]
\medskip
The main purpose of this paper is to explain some geometric and arithmetic
ideas concerning weakly saturated varieties and their asymptotic
arithmetic fibrations by strongly saturated subvarieties.
It seems that the cubic
bundles considered in \cite{BaTschi4} are
examples of such a fibration.
We want to remark that most of the
above terminology grew out of our
attempts to restore a conjectural picture of
the interplay between the
geometry of algebraic varieties and the
arithmetic of the distribution
of rational points on them
after we have found in \cite{BaTschi4}
an example which contradicted
general expectations formulated in \cite{BaMa}.
\medskip
In section 2 we consider smooth quasi-projective
varieties $V$ over
${\bf C}$ together with a polarization
${\cal L} = (L, \| \cdot \|_h)$ consisting of
an ample line bundle
$L$ on $V$ equipped with a positive
hermitian metric $\| \cdot \|_h$.
Our main interest in this section
is a discussion of geometric properties of
$V$ in connection
with the Minimal Model Program \cite{KMM} and
its version
for polarized algebraic varieties suggested by Fujita
\cite{fujita0,fujita01,fujita1}. We introduce
our main geometric invariants $\alpha_{\cal L}(V)$,
${\beta}_{\cal L}(V)$, and ${\delta}_{\cal L}(V)$
for an arbitrary ${\cal L}$-polarized variety $V$.
It is important to remark
that we will be only interested in the case
$\alpha_{\cal L}(V) > 0$.
The number $\alpha_{\cal L}(V)$ was first introduced in
\cite{BaMa,Ba}, it equals to the opposite
of the so called {\em Kodaira energy}
(investigated by Fujita in
\cite{fujita0,fujita01,fujita1}).
Our basic geometric notion in the study
of ${\cal L}$-polarized varieties $V$ with
$\alpha_{\cal L}(V) >0$ is the notion
of an ${\cal L}$-{\em primitive} variety.
In Fujita's program
for polarized varieties with negative
Kodaira energy ${\cal L}$-{primitive}
varieties play the same role as
${\bf Q}$-Fano varieties in Mori's program
for algebraic varieties with negative
Kodaira dimension. In particular,
one expects the existence of so called
${\cal L}$-{\em primitive fibrations},
which are analogous
to ${\bf Q}$-Fano fibrations in Mori's program.
We show that
on ${\cal L}$-primitive varieties there exists a
canonical volume measure. Moreover, this
measure allows us to construct
a descent of hermitian metrics to the base of ${\cal L}$-primitive
fibrations. Many geometric ideas of this section
are inspired by \cite{BaMa,Ba}.
In section 3 we introduce our main arithmetic notions of
{\em weakly} and {\em strongly
${\cal L}$-saturated} varieties.
Our first main diophantine conjecture claims that
if an adelic ${\cal L}$-polarized
quasi-projective algebraic variety $V$
over a number field $F$ is strongly
${\cal L}$-saturated, then
the corresponding ${\cal L}$-polarized
complex algebraic variety
$V({\bf C})$ is ${\cal L}$-primitive.
Moreover, we conjecture that
if an adelic ${\cal L}$-polarized
quasi-projective algebraic variety $V$
over a number field $F$ is weakly ${\cal L}$-saturated, then
the corresponding ${\cal L}$-polarized complex algebraic variety
$V({\bf C})$ admits an ${\cal L}$-primitive fibration having infinitely
many fibers $W$ defined over $F$ which form an asymptotic arithmetic
fibration.
These conjectures allow us to establish
a connection between the geometry of $V({\bf C})$ and the arithmetic
of $V$. Following this idea, we explain
a construction of an adelic measure on an arbitrary ${\cal L}$-primitive
variety $V$ with ${\alpha}_{\cal L}(V) > 0$
and of the corresponding Tamagawa number $\tau_{\cal L}(V)$
as a regularized adelic integral of this measure.
Our construction generalizes the definition
of Tamagawa measures associated with a metrization of
the canonical line bundle due to Peyre \cite{peyre}.
We expect that for strongly
${\cal L}$-saturated varieties $V$
the number $\tau_{\cal L}(V)$ reflects the dependence of
the constant $c(V)$ in the asymptotic formula (\ref{formula})
on the adelic metrization of the ample line bundle ${L}$. We discuss the
natural question about the behavior of the adelic constant
$\tau_{\cal L}(W)$ for fibers $W$ in
${\cal L}$-primitive fibrations on weakly
${\cal L}$-saturated varieties.
In section 4 we show that our diophantine
conjectures agree
with already known examples of asymptotic
formulas established for polarized algebraic varieties
through the study of analytic properties of
height zeta functions.
In Section 5 we illustrate our expectations
for the constants $a(V), b(V)$ and $c(V)$
in the asymptotics
of $N(V,B)$ on some examples of smooth Zariski
dense subsets $V$
in Fano varieties with singularities.
We would like to thank J.-L. Colliot-Th\'el\`ene
for his patience and encouragement.
We are very grateful to B. Mazur,
Yu. I. Manin, L. Ein and A. Chambert-Loir
for their comments and suggestions.
We thank the referee for several useful remarks.
\section{Geometry of ${\cal L}$-polarized varieties}
\subsection{${\cal L}$-closure }
Let $V$ be a smooth irreducible
quasi-projective algebraic variety over ${\bf C}$, $V({\bf C})$ the
set of closed points of $V$,
$L$ an ample invertible sheaf on $V$, i.e., $L^{\otimes k}
= i^* {\cal O}_{{\bf P}^m}(1)$ for some $k >0$ and some embedding
$i \,: \, V \hookrightarrow {\bf P}^m$. Since we don't assume
$V$ to be compact, the invertible sheaf $L$ on
$V$ itself contains
too little information about the embedding $i \,: \,
V \hookrightarrow {\bf P}^m$. For instance,
let $V$ be an affine variety
of positive dimension. Then the space of
global sections of $L$
is infinite dimensional and we don't know
anything about the projective
closure of $V$ in ${\bf P}^m$ even though we know that
the invertible sheaf $L^{\otimes k}$ is isomorphic to
$i^*{\cal O}_{{\bf P}^m}(1)$. This situation changes if
one considers $L$ together with a
positive hermitian metric, i.e.,
an ample metrized invertible
sheaf ${\cal L}$ associated with $L$.
Let us choose a positive hermitian metric
$h$ on ${\cal O}_{{\bf P}^m}(1)$
(e.g. Fubini-Study metric) and denote by $\|\cdot \|_h$ the induced
metric on $L^{\otimes k}$. Thus we obtain a metric $\| \cdot \|$
on $L$ by putting $\|s(x)\|: = \| s^k(x) \|_h^{1/k}$ for any $x \in
V({\bf C})$ and any section
$s \in { \rm H }^0(U, L)$ over an open subset $U \subset V$.
\begin{dfn}
{\rm We call a pair ${\cal L} = ( L, \|\cdot \|)$ {\bf an ample
metrized invertible sheaf} associated with $L$. We denote by
${\cal L}^{\otimes \nu}$ the pair $( L^{\otimes \nu}, \|\cdot \|^{\nu })$.}
\end{dfn}
Our next goal is to show that an ample metrized invertible sheaf
contains almost complete information about the projective closure
of $V$ in ${\bf P}^m$.
\begin{dfn}
{\rm Let ${\cal L} =( L, \|\cdot \|) $
be an ample metrized invertible sheaf on a complex
irreducible quasi-projective variety $V$. We denote by
$$
{ \rm H }^0_{\rm bd}(V, {\cal L})
$$
the subspace of ${ \rm H }^0(V, L)$
consisting of those global sections $s$ of $L$ over $V$
such that the corresponding continuous
function $x \mapsto \|s(x) \|$ $(x \in V({\bf C}))$
is globally bounded on $V({\bf C})$ from above by a positive constant
$C(s)$ depending only on
$s$. We call ${ \rm H }^0_{\rm bd}(V, {\cal L})$ the {\bf space of globally
bounded sections} of ${\cal L}$. }
\end{dfn}
\begin{prop}
Let $\overline{V}$ be the normalization of the projective closure
of $V$ with respect to the embedding $i\; : \;
V \hookrightarrow {\bf P}^m$ with
${L}^{\otimes k} = i^*{\cal O}_{{\bf P}^m}(1)$. Denote by
${e}\, : \,\overline{V} \rightarrow {\bf P}^m$ the corresponding
finite projective morphism. Then
one has a natural isomorphism
$$
{ \rm H }^0_{\rm bd}(V, {\cal L}^{\otimes k}) \cong { \rm H }^0(\overline{V},
{e}^*{\cal O}_{{\bf P}^m}(1)).
$$
\label{l-bd}
\end{prop}
\noindent
{\em Proof.}
Since $\overline{V}({\bf C})$ is compact, the
continuous function $x \mapsto \|s(x)\|$ is globally bounded on
$\overline{V}({\bf C})$ for any
$s \in { \rm H }^0(\overline{V}, {e}^*{\cal O}_{{\bf P}^m}(1))$.
Therefore, we obtain that
${ \rm H }^0(\overline{V}, {e}^*{\cal O}_{{\bf P}^m}(1))$ is a subspace
of ${ \rm H }^0_{\rm bd}(V, {\cal L}^{\otimes k})$.
Now let $f \in
{ \rm H }^0_{\rm bd}(V, {\cal L}^{\otimes k})$ be a globally bounded on $V({\bf C})$
section of $i^*{\cal O}_{{\bf P}^m}(1)$. Since
${ \rm H }^0_{\rm bd}(V, {\cal L}^{\otimes k})$ is a subspace of
${ \rm H }^0(V, L^{\otimes k})$, the section $f$ uniquely
extends to a global meromorphic section $\overline{f}
\in { \rm H }^0(\overline{V}, {e}^*{\cal O}_{{\bf P}^m}(1))$.
Since a bounded meromorphic function is holomorphic,
$\overline{f}$ is a global regular section of
${e}^*{\cal O}_{{\bf P}^m}(1)$ (we apply the theorem of Riemann to
some resolution of singularities $\rho\, : \, X \rightarrow \overline{V}$
and use the fact that
$\rho_* {\cal O}_X = {\cal O}_{\overline{V}}$). Thus we have
$$
{ \rm H }^0_{\rm bd}(V, {\cal L}^{\otimes k}) \subset
{ \rm H }^0(\overline{V}, {e}^*{\cal O}(1)).
$$
\hfill $\Box$
\begin{dfn}
{\rm We define a the graded ${\bf C}$-algebra
$$ {\rm A}(V, {\cal L}) = \bigoplus_{ \nu \geq 0}
{ \rm H }^0_{\rm bd}(V, {\cal L}^{\otimes \nu } ). \]
}
\end{dfn}
Using \ref{l-bd}, one immediately obtains:
\begin{coro}
The graded algebra $ {\rm A}(V, {\cal L})$
is finitely generated.
\end{coro}
\begin{dfn}
{\rm We call the normal projective variety
\[ \overline{V}^{\cal L} = {\rm Proj}\,
{\rm A}(V, {\cal L}). \]
the ${\cal L}$-{\bf closure} of $V$ with respect to
an ample metrized invertible sheaf ${\cal L}$.
}
\end{dfn}
\begin{rem}
{\rm By \ref{l-bd}, $\overline{V}^{\cal L}$
is isomorphic to
$\overline{V}$. Therefore,
we have obtained a way to define
the normalization of the projective closure
of $V$ with respect
to an $L^{\otimes k}$-embedding via a notion
of an ample metrized invertible sheaf
${\cal L}$ on $V$.}
\end{rem}
\subsection{Kodaira energy and $\alpha_{\cal L}(V)$}
Let $X$ be a normal irreducible algebraic
variety of dimension
$n$. We denote by ${\rm Div}(X)$ (resp. by
${\rm Z}_{n-1}(X)$)
the group of Cartier divisors
(resp. Weil divisors) on $X$.
An element of ${\rm Div}(X) \otimes {\bf Q}$
(resp.${\rm Z}_{n-1}
(X) \otimes {\bf Q}$) is called a
${\bf Q}$-Cartier divisor (resp. a ${\bf Q}$-divisor).
By $K_X$ we denote a divisor of a meromorphic
differential $n$-form
on $X$, where $K_X$ is considered as an
element of ${\rm Z}_{n-1}(X)$.
\begin{dfn}
{\rm Let $X$ be a projective variety and
$L$ be an invertible sheaf on
$X$. The {\bf Iitaka-dimension}
$\kappa(L)$ is defined as
\[ \kappa(L) = \left\{ \begin{array}{ll}
- \infty & \mbox{\rm if ${ \rm H }^0(X, L^{\otimes \nu}) = 0$ for all $\nu > 0$} \\
{\rm Max}\, \dim \phi_{L^{\otimes \nu}}(X) : &
{ \rm H }^0(X, L^{\otimes \nu}) \neq 0
\end{array} \right.
\]
where $\phi_{L^{\otimes \nu}}(X)$
is the closure of the image of
$X$ under the rational map
\[ \phi_{L^{\otimes \nu}} \, : \,
X \rightarrow {\bf P}({ \rm H }^0(X, L^{\otimes \nu})). \]
A Cartier divisor $L$ is called
{\bf semi-ample} (resp. {\bf effective}),
if $L^{\otimes \nu}$ is
generated by global sections for some
$\nu > 0$ (resp. $\kappa(L) \geq 0)$.
}
\end{dfn}
\begin{rem}
{\rm The notions of Iitaka-dimension,
ampleness and semi-ample\-ness
obviously extend to
${\bf Q}$-Cartier divisors.
Let $L$ be a Cartier divisor. Then
for all $ \kappa_1, k_2 \in {\bf N}$ we set
\[ \kappa(L^{\otimes k_1/k_2} ): = \kappa(L),\]
\[
\mbox{\rm $L^{\otimes k_1/k_2}$ is ample} \, \Leftrightarrow
\, \mbox{\rm $L$ is ample}, \]
and
\[ \mbox{\rm $L^{\otimes k_1/k_2}$ is semi-ample} \, \Leftrightarrow
\, \mbox{\rm $L$ is semi-ample}. \]
}
\end{rem}
\begin{dfn}
{\rm Let $X$ be a smooth projective variety. We denote by
${ \rm NS}(X)$ the group of divisors on $X$ modulo
numerical equivalence and set ${ \rm NS}(X)_{\bf R} = { \rm NS}(X) \otimes {\bf R}$.
By $[L]$ we denote the class of a divisor $L$ in ${ \rm NS}(X)$.
The {\bf cone of effective divisors} $\Lambda_{\rm eff}(X) \subset
{ \rm NS}(X)_{\bf R}$ is defined as the closure of the subset
\[ \bigcup_{ \kappa(L) \geq 0} {\bf R}_{\geq 0} [L] \subset
{ \rm NS}(X)_{\bf R}. \]
}
\end{dfn}
\begin{dfn}
{\rm Let $V$ be a smooth quasi-projective algebraic variety
with an ample metrized invertible sheaf ${\cal L}$,
$\overline{V}^{\cal L}$ the ${\cal L}$-closure of $V$
and $\rho$ some resolution of
singularities
\[ \rho \; : \; X \rightarrow \overline{V}^{\cal L}. \]
We define the number
\[ \alpha_{\cal L}(V) = \inf \{ t \in {\bf Q}\; : \;
t [ \rho^*L] + [ K_X] \in \Lambda_{\rm eff}(X) \}. \]
and call it the ${\cal L}$-{\bf index} of $V$. }
\end{dfn}
\begin{rem}
{\rm It is easy to see that the ${\cal L}$-{index} does not depend on
the choice of $\rho$.}
\end{rem}
\begin{rem}
{\rm The ${\cal L}$-index
$\alpha_{\cal L}(V)$ for smooth projective
varieties $V$ was first introduced in \cite{BaMa}
and \cite{Ba}. We remark that
the opposite number $- \alpha_{\cal L}(V)$
coincides with the notion
of {\bf Kodaira energy} introduced and investigated by Fujita in
\cite{fujita0,fujita01,fujita1}:
\[ \kappa\epsilon (V, L) = - \alpha_{\cal L}(V) =
- \inf \{ t \in {\bf Q}\; : \;
\kappa ( (L)^{\otimes t} \otimes K_V) \geq 0 \}. \]
From the viewpoint of our diophantine
applications it is much more natural to consider $\alpha_{\cal L}(V)$
instead of its opposite $-\alpha_{\cal L}(V)$. The only reason that
we could see for introducing
the number $-\alpha_{\cal L}(V)$ instead
of $\alpha_{\cal L}(V)$ is some kind of compatibility
between the notions of {\em Kodaira energy} and {\em Kodaira dimension},
e.g. Kodaira energy must be positive (resp. negative) iff the
Kodaira dimension is positive (resp. negative). }
\end{rem}
\noindent
The following statement was conjectured in \cite{BaMa} (see also
\cite{Ba,fujita00,fujita000}):
\begin{conj}
{\sc (Rationality)} Assume that $\alpha_{\cal L}(V)> 0$. Then
$\alpha_{\cal L}(V)$ is rational.
\end{conj}
\begin{rem}
{\rm
It was shown in \cite{Ba} that this
conjecture follows from the Minimal Model Program.
In particular, it holds for ${\rm dim}\,V \leq 3$.
If ${\rm dim}\, V
=1$, then the only possible values of
$\alpha_{\cal L}(V)$ are
numbers $2/k$ with $(k \in {\bf N})$. If
${\rm dim}\, V =2$, then
$\alpha_{\cal L}(V) \in \{ 2/k, 3/l \}$
with $( k, l \in {\bf N})$. }
\label{a-values}
\end{rem}
\begin{dfn}
{\rm A normal irreducible algebraic variety $W$ is said to have
at worst {\bf canonical} (resp. {\bf terminal}) singularities
if $K_W$ is a ${\bf Q}$-Cartier divisor
and if for some (or every) resolution of singularities
\[ \rho \; : \; X \rightarrow W \]
one has
\[ K_X = \rho^*(K_W)\otimes {\cal O}(D) \]
where $D$ is an effective ${\bf Q}$-Cartier divisor (resp.
the support of the effective divisor
$D$ coincides with the exceptional locus of
$\rho$). Irreducible components of the exceptional
locus of $\rho$ which are not contained in the support of
$D$ are called {\bf crepant divisors} of the resolution $\rho$.}
\end{dfn}
\begin{dfn}
{\rm A normal irreducible algebraic variety $W$ is called
a {\bf ca\-no\-nical} ${\bf Q}$-{\bf Fano} variety, if $W$ has
at worst { canonical} singularities
and $K_W^{-1}$ is
an ample ${\bf Q}$-Cartier divisor. A maximal positive
rational number $r(W)$ such that $K_W^{-1} = L^{\otimes r(W)}$
for some Cartier divisor $L$ is called the {\bf index} of
a canonical ${\bf Q}$-Fano variety $W$ (obviously, one has
$r(W) = \alpha_{\cal L}(W)$ for some positive
metric on $L$). }
\label{fano-c}
\end{dfn}
The following conjecture is due to Fujita \cite{fujita0}:
\begin{conj}
{\sc (Spectrum Conjecture)} Let $S(n)$ be the set all possible
values of $\alpha_{\cal L}(V)$ for smooth quasi-projective algebraic
varieties $V$ of dimension $\leq n$
with an ample metrized invertible sheaf ${\cal L}$. Then
for any $\varepsilon > 0$ the set
\[ \{ \alpha_{\cal L}(V) \in S(n) \; : \; \alpha_{\cal L}(V) >
\varepsilon \} \]
is finite.
\end{conj}
This conjecture follows from the Minimal Model Program \cite{KMM} and
from the following conjecture on the
boundedness of index for Fano varieties with canonical singularities:
\begin{conj}
{\sc (Boundedness of Index)} The set of possible values of index
$r(W)$ for canonical ${\bf Q}$-Fano varieties $W$ of dimension $n$
is finite.
\end{conj}
In particular, both conjectures are true for
${\bf Q}$-Fano varieties of dimension
$n \leq 3$ \cite{A,fujita0,fujita01,fujita1,Ka,shin}.
\subsection{${\cal L}$-primitive varieties}
\begin{dfn}
{\rm Let $X$ be a projective algebraic variety.
We call an effective ${\bf Q}$-divisor $D$ {\bf rigid}, if
$\kappa(D) = 0$.
}
\end{dfn}
\begin{prop}
{\rm An effective ${\bf Q}$-divisor
$D$ on $X$ is rigid if and only if there exist finitely many
irreducible subvarieties $D_1, \ldots, D_l \subset X$ $(l \geq 0)$
of codimension $1$ such that
$D =r_1 D_1 + \ldots + r_l D_l$ with $r_1, \ldots, r_l \in {\bf Q}_{>0}$
and
\[ \mbox {\rm dim}\, { \rm H }^0 (X,
{\cal O}(n_1 D_1 + \ldots + n_l D_l)) =1 \;\; \forall\;
(n_1, \ldots, n_l ) \in {\bf Z}^l_{\geq 0}. \]
}
\label{rigid2}
\end{prop}
\noindent
{\em Proof.} Let $D$ be rigid. Take a positive integer $m_0$
such that $m_0D$ is a Cartier divisor and
${\rm dim}\, { \rm H }^0(X, {\cal O}(m_0D) ) =1$. Denote
by $D_1, \ldots, D_l$ the irreducible components of the divisor $(s)$ of
a non-zero section $s \in { \rm H }^0(X, {\cal O}(mD) )$.
One has
\[ (s) = m_1 D_1 + \cdots + m_l D_l\;\;\; m_1, \ldots, m_l \in {\bf N}. \]
Since ${\cal O}(D_i)$ admits at least one global non-zero section we obtain that
\[ \mbox {\rm dim}\, { \rm H }^0 (X,
{\cal O}(n_1' D_1 + \ldots + n_l' D_l)) \geq
\mbox {\rm dim}\,
{ \rm H }^0 (X, {\cal O}(n_1 D_1 + \ldots + n_l D_l)), \]
whenever $n_1 \geq n_1', \ldots,
n_l \geq n_l'$ for $(n_1', \ldots, n_l' ), \;
(n_1, \ldots, n_l ) \in {\bf Z}^l_{\geq 0}$.
This implies that
\[ \mbox {\rm dim}\, { \rm H }^0 (X,
{\cal O}(n_1 D_1 + \ldots + n_l D_l)) \geq
1\;\; \forall\;
(n_1, \ldots, n_l ) \in {\bf Z}^l_{\geq 0}. \]
On the other hand, for any $(n_1, \ldots, n_l )
\in {\bf Z}^l_{\geq 0}$
there exists a positive integer $n_0$ such that
$n_0m_1 \geq n_1, \ldots, n_0m_l \geq n_l$. Therefore,
\[ \mbox {\rm dim}\, { \rm H }^0 (X,
{\cal O}(n_1 D_1 + \ldots + n_l D_l)) \leq
\mbox {\rm dim}\, { \rm H }^0 (X,
{\cal O}(n_0m_0D) ) =1, \]
since $\kappa(n_0m_0D) =\kappa(D) =0$.
\hfill $\Box$
\begin{coro}
Let $D_1, \ldots, D_l \subset X$ be
all irreducible components of the
support of a rigid ${\bf Q}$-Cartier divisor $D$. Then
a linear combination
\[ n_1 D_1 + \ldots + n_l D_l, \; \; n_1, \ldots, n_l \in {\bf Z} \]
is a principal divisor, iff $n_1 = \cdots = n_l = 0$.
\label{rigid-l}
\end{coro}
\noindent
{\em Proof.}
Assume that $n_1 D_1 + \ldots + n_l D_l$ is linearly
equivalent to $0$. Then the effective Cartier divisor
$D_0 = \sum_{n_i \geq 0} n_i D_i$ is linearly equivalent
to the effective Cartier
divisor $D_0' = \sum_{n_j <0} (-n_j) D_j$.
Since $D_0$ and $D_0'$ have different supports we have
${\rm dim}\, { \rm H }^0(X, {\cal O}(D_0)) \geq 2$.
Contradiction to \ref{rigid2}.
\hfill $\Box$
\begin{dfn}
{\rm Let $V$ be a smooth quasi-projective algebraic variety
with an ample metrized invertible sheaf ${\cal L}$ and
$\overline{V}^{\cal L}$ the projective ${\cal L}$-closure of $V$.
The variety $V$ is called ${\cal L}$-{\bf primitive}, if
the number $\alpha_{\cal L}(V)$ is rational and if for some resolution of
singularities
\[ \rho \; : \; X \rightarrow \overline{V}^{\cal L} \]
one has $\rho^*(L)^{\otimes \alpha_{\cal L}(V)} \otimes K_X =
{\cal O}(D)$, where $D$ is a rigid
effective ${\bf Q}$-Cartier divisor on $X$.
}
\end{dfn}
\begin{rem}
{\rm It is easy to see that the notion of an ${\cal L}$-primitive
variety doesn't depend
on the choice
of a resolution of singularities $\rho$. Since $V$ is smooth,
we can always assume that
the natural mapping
$$\rho\; : \; \rho^{-1}(V) \rightarrow V$$
is an isomorphism.
}
\end{rem}
\begin{exam}
{\rm Let $V_1$ and $V_2$ be two smooth quasi-projective
varieties with ample metrized invertible sheaves
${\cal L}_1$ and ${\cal L}_2$ (resp. on $V_1$ and $V_2$).
Assume that $V_1$ (resp. $V_2$ ) is ${\cal L}_1$-primitive
(resp. ${\cal L}_1$-primitive). Then the product
$V = V_1 \times V_2$ is ${\cal L}$-primitive, where
${\cal L} = \pi_1^*{\cal L}_1 \otimes \pi_2^*{\cal L}_2$. }
\end{exam}
\noindent
Our main list of examples of ${\cal L}$-primitive varieties is obtained
from canonical ${\bf Q}$-Fano varieties:
\begin{exam}
{\rm Let $V$ be the set of nonsingular
points of a canonical
${\bf Q}$-Fano variety $W$ with an ample metrized
invertible sheaf ${\cal L} = (L,\|\cdot\|)$
such that $K_W^{-1} = L^{\otimes r(W)}$ ( $W =
\overline{V}^{\cal L} $). Then $V$ is
an ${\cal L}$-primitive variety with
${\cal L}$-index $r(W)$. Indeed, let $\rho\, : \, X \rightarrow W$ be a resolution
of singularities. By \ref{fano-c}, we have
$\rho^*(L)^{\otimes r(W)} \otimes
K_X = {\cal O}(D)$, where
$D$ is an effective
${\bf Q}$-Cartier divisor. Since the
support of $D$ consists of exceptional
divisors with respect to $\rho$, $D$ is rigid
(see \ref{rigid2}). }
\end{exam}
We expect that the above examples cover
all ${\cal L}$-primitive varieties:
\begin{conj}
{\sc (Canonical ${\bf Q}$-Fano contraction)}
Let $V$ be an ${\cal L}$-primitive variety with
$\alpha_{\cal L}(V) >0$. Then there exists
a resolution of singularities $\rho\, : \, X \rightarrow \overline{V}^{\cal L}$
and a birational projective morphism $\pi\, : \, X \rightarrow W$ to a
canonical ${\bf Q}$-Fano variety $W$ such that
$\pi^*K_W^{-1} \cong \rho^*(L)^{\alpha_{\cal L}(V)}$
(i.e., $\alpha_{\cal L}(V) = r(W)$)
and the support of $D$
($\rho^*(L)^{\otimes r(W)} \otimes
K_X = {\cal O}(D)$) is contained
in the exceptional locus of $\pi$.
\label{conj-cont}
\end{conj}
The above conjecture is expected to follow
from the Minimal Model Program
using the existence and termination of
flips (in particular, it holds for toric varieties).
The following statement will be important in
our construction of Tamagawa numbers
for ${\cal L}$-primitive varieties
defined over a number field:
\begin{conj} {\sc (Vanishing)}
{\rm For $V$ an ${\cal L}$-primitive variety such that
$\alpha_{\cal L}(V) >0$ we have
\[ {\rm h}^i(X, {\cal O}_X) = 0 \;\; \forall \; i > 0 \]
for any resolution of singularities
$$
\rho \; : \; X \rightarrow \overline{V}^{\cal L}
$$
such that the support of the ${\bf Q}$-Cartier divisor
$\rho^*(L)^{\otimes \alpha_{\cal L}(V) }
\otimes K_X$ is a ${\bf Q}$-Cartier divisor with normal crossings.
In particular, $ {\rm Pic}(X)$ is a finitely generated
abelian group and one has a canonical isomorphism
\[ {\rm Pic}(X) \otimes {\bf Q} \cong { \rm NS}(X) \otimes {\bf Q}. \]
\label{vanish}
}
\end{conj}
\begin{rem}
{\rm Theorem 1-2-5 in \cite{KMM} implies the vanishing for
the structure sheaf for ${\bf Q}$-Fano varieties with canonical
singularities (even with log-terminal singularities).
All canonical (and log-terminal singularities) are rational and it
follows that the higher cohomology of the structure sheaf
on any desingularization of a canonical
or a log-terminal Fano variety must also vanish
(by Leray spectral sequence).
Therefore, we would obtain the vanishing \ref{vanish} for all
${\cal L}$-primitive varieties
which are birationally equivalent to a
Fano variety with at worst log-terminal
singularities. The existence of a canonical ${\bf Q}$-contraction
\ref{conj-cont} would insure this.
\label{vanish-rem}
}
\end{rem}
\begin{dfn}
{\rm Let $V$ be an ${\cal L}$-primitive variety with
$\alpha_{\cal L}(V) >0$, $\rho \, : \, X \rightarrow
\overline{V}^{\cal L}$ any resolution of singularities,
$D_1, \ldots, D_l$ irreducible components of the support of the
rigid effective
${\bf Q}$-Cartier divisor $D$ with ${\cal O}(D) =
\rho^*(L)^{\otimes \alpha_L(V)} \otimes K_X$.
We shall call
$$
{\rm Pic}(V, {\cal L}) : = {\rm Pic}(X \setminus \bigcup_{i =1}^l D_i)
$$
the ${\cal L}$-{\bf Picard group} of
$V$. The number
\[ \beta_{\cal L}(V): = {\rm rk}\, {\rm Pic}(V, {\cal L}) \]
will be called the ${\cal L}$-{\bf rank} of $V$.
We define the ${\cal L}$-{\bf cone of effective divisors}
$\Lambda_{\rm eff}(V, {\cal L}) \subset
{\rm Pic}(V,{\cal L})\otimes {\bf R}$ as the image
of $\Lambda_{\rm eff}(X) \subset { \rm NS}(X)_{\bf R} =
{\rm Pic}(X) \otimes {\bf R}$ under the natural surjective
${\bf R}$-linear mapping
\[ \tilde{\rho}\; : \;{\rm Pic}(X) \otimes {\bf R} \rightarrow
{\rm Pic}(V,{\cal L})\otimes {\bf R}. \]
}
\end{dfn}
\begin{rem}
{\rm
By \ref{rigid-l}, one obtains the
exact sequence
\begin{equation}
0 \rightarrow {\bf Z}[D_1] \oplus \cdots \oplus
{\bf Z}[D_l] \rightarrow {\rm Pic}(X) \stackrel{\tilde{\rho}}{\rightarrow}
{\rm Pic}(V, {\cal L}) {\rightarrow} 0
\end{equation}
and therefore
\[ \beta_{\cal L}(V)=
{\rm rk}\, {\rm Pic} (X) - l. \]
Using these facts, it is easy to show that the group
$ {\rm Pic}(V, {\cal L})$ and the cone
$\Lambda_{\rm eff}(V, {\cal L})$ do
not depend on the choice of
a resolution of singularities
$\rho \, : \,X \rightarrow \overline{V}^{\cal L}$. }
\end{rem}
\noindent
The above conjecture holds in
dimension $n \leq 3$ as a consequence
of the Minimal Model Program. More precisely,
it is a consequence
of Conjecture \ref{conj-cont} and the
following weaker statement:
\begin{conj}
{\sc (Polyhedrality)}
Let $V$ be an ${\cal L}$-primitive variety with
$\alpha_{\cal L}(V) >0$. Then
$\Lambda_{\rm eff}(V, {\cal L})$
is a rational finitely generated polyhedral cone.
\label{polyhed}
\end{conj}
\medskip
\begin{dfn}{\rm
Let $ (A,A_{{\bf R }}, \Lambda )$ be a triple consisting
of a finitely generated
abelian group
$A$ of rank $k$, a $k$-dimensional real vector space $A_{{\bf R }}=A\otimes {\bf R }$
and a convex $k$-dimensional finitely generated polyhedral cone
$ \Lambda \in A_{{\bf R }}$ such that $ \Lambda \cap - \Lambda =0\in A_{{\bf R }}$.
For ${\rm Re}({\bf s})$ contained in the interior of the cone $ \Lambda $
we define the ${\cal X}$-function of $ \Lambda $ by
the integral
$$
{\cal X}_{ \Lambda }({\bf s}):= \int_{ \Lambda ^*}e^{-<{\bf s},{\bf y}>}{\bf d}{\bf y}
$$
where $ \Lambda ^*\in A^*_{{\bf R }}$ is the dual cone to $ \Lambda $ and
${\bf d}{\bf y}$ is the Lebesgue measure on $A^*_{{\bf R }}$
normalized by the dual lattice $A^*\subset A^*_{{\bf R }}$ where
$A^* := {\rm Hom}(A, {\bf Z})$.
}
\end{dfn}
\begin{rem}{\rm
If $ \Lambda $ is a finitely generated
rational polyhedral cone the function ${\cal X}_{ \Lambda }({\bf s})$
is a rational function in ${\bf s}$. However, the explicit
determination of this function might pose serious computational
problems.
}
\end{rem}
\begin{dfn}
{\rm Let $V$ be an
${\cal L}$-primitive smooth quasi-projective algebraic variety
with a metrized invertible sheaf ${\cal L}$ and $ \alpha _{\cal L}(V)>0$.
Let $X$ be any resolution of singularities
$\rho\;:\; X\rightarrow \overline{V}^{\cal L}$.
We consider the triple
$$
({\rm Pic}(V,{\cal L}),{\rm Pic}(V,{\cal L})_{{\bf R }},
\Lambda _{\rm eff}(V,{\cal L}))
$$
and the corresponding ${\cal X}$-function.
Assuming that ${\rm Pic}(V, {\cal L})$ is a finitely generated
abelian group (cf. \ref{vanish}) and that
$ \Lambda _{\rm eff}(V,{\cal L})$ is a polyhedral
cone (cf. \ref{polyhed}), we define the constant $\gamma_{\cal L}(V)\in { \bf Q }$ by
$$
\gamma_{\cal L}(V):=
{\cal X}_{ \Lambda _{\rm eff}(V,{\cal L})}(\tilde{\rho}(-[K_X])).
$$
\label{gamma-dfn}
}
\end{dfn}
\subsection{${\cal L}$-primitive fibrations and descent of metrics}
Let $V$ be an ${\cal L}$-primitive variety of dimension $n$.
We show that there exists a canonical measure
on $V({\bf C})$ which is
uniquely defined up to a positive constant.
In order to construct this measure we choose a
resolution of singularities
$\rho \; : \; X \rightarrow \overline{V}^{\cal L}$ and
a positive integer $k_2$ such that
$k_2D$ is a Cartier divisor, where ${\cal O}(D) \cong
(\rho^*L)^{\otimes \alpha_{\cal L}(V) }
\otimes K_X$. Then $k_1 = k_2 \alpha_{\cal L}(V)$ is
a positive integer. Let $g \in { \rm H }^0(X, {\cal O}(k_2D))$ be
a non-zero global section (by \ref{rigid2},
it is uniquely defined up to
a non-zero constant).
We define a measure ${\bf \omega}_{{\cal L}}(g)$
on $X({\bf C})$ as follows.
Choose local complex analytic coordinates
$z_{1}, \ldots, z_{n}$ in
some open neighborhood $U_x \subset X({\bf C})$ of a point
$x \in V({\bf C})$.
We write the restriction of the global section $g$ to $U_x$ as
\[ g = s^{k_2\alpha_{\cal L}(V)}
( dz_{1} \wedge \cdots \wedge dz_{n})^{\otimes k_2} =
s^{k_1} ( dz_{1} \wedge \cdots \wedge dz_{n})^{\otimes k_2}, \]
where $s$ is a local section of $L$ . Then we set
\[ {\bf \omega}_{{\cal L}}(g) : = \left(\frac{\sqrt{-1}}{2} \right)^n
\|s\|^{ \alpha_{\cal L}(V)}
(dz_1 \wedge d\overline{z}_1) \wedge
\cdots \wedge (dz_n \wedge d\overline{z}_n). \]
By a standard argument, one obtains that ${\bf \omega}_{{\cal L}}(g)$
doesn't depend on the choice
of local coordinates in $U_x$ and
that it extends to the whole complex space
$X({\bf C})$. It remains to notice that the restriction of
the measure ${\bf \omega}_{{\cal L}}(g)$ to $V({\bf C}) \subset X({\bf C})$
does not depend on the choice of $\rho$. So we obtain a well-defined
measure on $V({\bf C})$.
\begin{rem}
{\rm We note that the measure $\omega_{\cal L}(g)$ depends on the
choice of $g \in { \rm H }^0(X, {\cal O}(k_2D))$. More precisely, it multiplies
by $|c|^{1/k_2}$ if we multiply $g$ by some non-zero complex number $c$.
Thus we obtain that
the mapping
\[ \| \cdot \|_{\cal L} \; :
\; { \rm H }^0(X, {\cal O}(k_2D)) \rightarrow {\bf R}_{\geq 0} \]
\[ g \mapsto \int_{X({\bf C})} \omega_{\cal L}(g) =
\int_{V({\bf C})} \omega_{\cal L}(g) \]
satisfies the property
\[ \| c g \|_{\cal L} = |c|^{1/k_2} \| g \|_{\cal L}\;\; \forall
c \in {\bf C}^*. \]
}
\end{rem}
\begin{dfn}
{\rm Let $V$ be a smooth quasi-projective variety with
an ample metrized invertible sheaf ${\cal L}$,
$\rho \, : \,X \rightarrow \overline{V}^{\cal L}$ a resolution
of singularities. A regular
projective morphism $\pi\, :\, X \rightarrow Y$ to a projective
variety $Y$ $({\rm dim}\, Y < {\rm dim}\, X)$
is called an ${\cal L}$-{\bf primitive fibration on} $V$
if there exists a Zariski dense open subset
$U \subset Y$ such that the following conditions are satisfied:
{(i)} for any point $y \in U({\bf C})$ the fiber
$V_y = \pi^{-1}({y}) \cap V$ is a smooth quasi-projective
${\cal L}$-primitive subvariety;
(ii) $\alpha_{\cal L}(V) = \alpha_{\cal L}(V_y) > 0$ for all
$y \in U({\bf C})$;
(iii) for any $k \in {\bf N}$ such that
$k\alpha_{\cal L}(V) \in {\bf Z}$,
\[ L_k := {\rm R}^0\pi_* \left(\rho^*(L)^{\otimes \alpha_L(V)} \otimes K_X
\right)^{\otimes k} \]
is an ample invertible sheaf on $Y$.
\label{prim-fb}}
\end{dfn}
\noindent
We propose the following
version of the Fibration Conjecture of
Fujita (see \cite{fujita0}):
\begin{conj}
{\sc (Existence of Fibrations)}
Let $V$ be an arbitrary smooth quasi-projective variety with
an ample metrized invertible sheaf ${\cal L}$ and $\alpha_{\cal L}(V)> 0$.
Then there exists a resolution of singularities
$\rho \, : \,X \rightarrow \overline{V}^{\cal L}$
such that $X$ admits an ${\cal L}$-primitive
fibration $\pi\, :\, X \rightarrow Y$ on some dense Zariski open subset
$V' \subset V$.
\label{conj-fb}
\end{conj}
\begin{rem}
{\rm From the viewpoint of the Minimal Model Program,
Conjecture \ref{conj-fb}
is equivalent to the statement about the conjectured
existence of ${\bf Q}$-Fano fibrations
for algebraic varieties of negative Kodaira-dimension
(cf. \cite{KMM}). The existence of
an ${\cal L}$-primitive fibration is equivalent to
the fact that
the graded algebra
\[ {\rm R}(V, {\cal L}) =
\oplus_{\nu \geq 0} { \rm H }^0(X, M^{k \nu}), \;\;
M:= \rho^*(L)^{\otimes \alpha_L(V)} \otimes K_X \]
is finitely generated (cf. 2.4 in \cite{BaMa}). One can
define $Y$ as ${\rm Proj}\,{\rm R}(V, {\cal L})$ and
$X$ as a common resolution
of singularities of $\overline{V}^{\cal L}$ and of the
indeterminacy locus and generic fiber of the natural rational map
$\overline{V}^{\cal L} \rightarrow Y$ (cf. \cite{H}).
}
\label{proj-fb}
\end{rem}
It is important to observe
that a metric $\| \cdot \|$ on $L$ induces natural
metrics on all ample invertible sheaves
$L_k$ on $Y$:
\begin{dfn}
{\rm Let $L_k$ be an ample invertible
sheaf on $Y$ as above.
We define a metric $\| \cdot \|_{{\cal L}, k}$
on $L_k$ as follows.
Let $y \in Y({\bf C})$ be a closed point,
$U \subset Y$ be a Zariski
open subset containing $y$, and
$s \in { \rm H }^0(U, L_k)$ is a section with
$s(y) \neq 0$. Then we set
\[ \|s(y)\|_{{\cal L},k} :=
\left( \int_{V_y({\bf C})} \omega_{\cal L}(\pi^* s) \right)^k , \]
where $V_y({\bf C})$ is the fiber over $y$ of the ${\cal L}$-primitive
fibration $\pi^*$, $\pi^* s$ the $\pi$-pullback of $s$ restricted to
${\cal L}$-primitive variety $V_y({\bf C})$, and
$\omega_{\cal L}(\pi^* s)$ the corresponding to $\pi^* s$ volume
measure on $V_y({\bf C})$.
We call $\| \cdot \|_{{\cal L}, k}$ a $k$-{\bf adjoint descent} to $Y$
of a metric $\| \cdot \|$ on $L$. }
\end{dfn}
\section{Heights and asymptotic formulas}
\subsection{Basic terminology and notations}
Let $F$ be a number field, ${\cal O}_F \subset F$ the
ring of integers in $F$, ${\rm Val}(F)$ the set of all valuations of $F$,
$F_v$ the completion of $F$ with respect to a valuation $v \in {\rm Val}(F)$,
${\rm Val}(F)_{\infty} = \{ v_1, \ldots, v_r \}$ the set of all archimedean
valuations of $F$.
For any algebraic variety $X$ over
a field $F$ we denote by
$X(F)$ the set of its $K$-rational points.
\begin{dfn}
{\rm Let $E$ be a vector space of dimension $m+1$ over $F$,
${\cal O}_E \subset E$ a projective ${\cal O}_F$-module of rank $m+1$
and $\| \cdot \|_{v_1}, \ldots, \| \cdot \|_{v_r}$ the set
of Banach norms on the real or complex vector spaces
$E_{v_i} = E \times_F F_{v_i}$ corresponding to elements of
${\rm Val}(F)_{\infty} = \{ v_1, \ldots, v_r \}$. It is well-known that
the above data for $E$ define a
family $\{ \| \cdot \|_v, \; v \in {\rm Val}(F) \}$ of $v$-adic metrics
for a standard invertible sheaf ${\cal O}(1)$ on ${\bf P}(E)$.
If $x \in X(F)$ is a point and
$s \in { \rm H }^0(U, {\cal O}(1))$ is a section over an open subset
$U \subset {\bf P}(E)$
containing $x$, then we denote
by $\|s(x)\|_v$ the corresponding
$v$-adic norm of
$s$ at $x$. We set
$\tilde{\cal O}(1) =
( {\cal O}(1), \|\cdot \|_v) $ to be the
standard invertible sheaf ${\cal O}(1)$ on ${\bf P}(E)$
together with a family $v$-adic metrics
$\|\cdot \|_v$ defined
by the above data and we
call $\tilde{\cal O}(1)$ the
{\bf standard ample metrized invertible
sheaf} on ${\bf P}(E)$. }
\end{dfn}
\begin{dfn}
{\rm Let $X$ be an algebraic variety over $F$. For any point
$x \in X(F)$ and any regular function
$f \in { \rm H }^0(U, {\cal O}_X)$ on an open subset
$U \subset {\bf P}(E)$
containing $x$, we define the $v$-adic norm
$\|f(x)\|_v := |f(x)|_v$. We call
this family of $v$-adic metrics on ${\cal O}_X$
the {\bf canonical metrization} of the structure sheaf ${\cal O}_X$. }
\end{dfn}
\begin{dfn}
{\rm Let $X$ be a quasi-projective algebraic variety,
$L$ a very ample invertible sheaf on $X$,
$i \; :\; X \hookrightarrow {\bf P}(E)$ an embedding with
$L = i^* {\cal O}(1)$. We denote by ${\cal L} = (L, \|\cdot \|_v )$
the sheaf $L$ together with
$v$-adic metrics induced from a
family of $v$-adic metrics on
the standard ample metrized
invertible sheaf $\tilde{\cal O}(1)$ on
${\bf P}(E)$. In this situation we call ${\cal L}$
{\bf a very ample metrized sheaf} on $X$ and write
${\cal L} = i^*\tilde{O}(1)$.
}
\end{dfn}
\begin{dfn}
{\rm Let ${\cal L} =( L, \|\cdot \|_v) $ be
a very ample metrized invertible sheaf on $X$. Then
for any point $x \in X(F)$, the ${\cal L}$-{\bf height} of
$x$ is defined as
\[ H_{\cal L}(x) = \prod_{v \in {\rm Val}(F)} \|s(x)\|_v^{-1}, \]
where $s \in \Gamma(U, L)$ is a nonvanishing at $x$
section of $L$ over some open
subset $U \subset X$. }
\end{dfn}
\begin{rem}
{\rm Using a canonical metrization of the structure sheaf ${\cal O}_X$,
the linear mapping
\[ S^k(\Gamma(U, L)) \rightarrow \Gamma(U, L^{\otimes k}) \;\; (k > 0), \]
and the $F$-bilinear mapping
\[ \Gamma(U, L^{\otimes k}) \times \Gamma(U, L^{\otimes -k}) \rightarrow
\Gamma(U, {\cal O}_X), \]
one immediately sees that a family of $v$-adic metrics
on an invertible sheaf $L$ allows to define a family of $v$-adic
metrics on $L^{\otimes k}$ and on
any invertible sheaf $M$ such that there exist
integers $k_1, k_2$ $(k_2 \neq 0)$
with $L^{\otimes k_1} = M^{\otimes k_2}$.
In this situation we write
$$
{\cal M} = (M, \|\cdot \|_v) := (L^{\otimes{k_1/k_2}},
\|\cdot \|_v^{k_1/k_2}),
$$
or simply $ {\cal M} ={\cal L}^{k_1/k_2}$.
Obviously,
one obtains
\[ H_{\cal M}(x) = (H_{\cal L}(x))^{k_1/k_2}\; \; \;\forall \;
x \in X(F). \]}
\label{fr-norms}
\end{rem}
\begin{dfn}
{\rm Let $L$ be an ample invertible sheaf on
a quasi-projective variety $X$
and $k$ a positive integer such that $L^{\otimes k}$
is very ample.
We define an
{\bf ample metrized invertible sheaf} ${\cal L}=( L, \|\cdot \|_v)$
on $X$ associated with $L$ by considering
${\cal L}^{\otimes k}: = (L^{\otimes k}, \| \cdot \|_v^k)$ as
a very ample metrized invertible sheaf on $X$.
}
\end{dfn}
\subsection{Weakly and strongly ${\cal L}$-saturated varieties}
Let $V$ be an arbitrary quasi-projective algebraic variety over $F$ with
an ample metrized invertible sheaf ${\cal L}$. We always assume
that $V(F)$ is infinite and set
$$
N(V,{\cal L}, B):=\#\{x\in V(F)\; : \; H_{\cal L}(x)\le B\}.
$$
Here and in 3.4 we will work under the following
\begin{assume}
{\rm For all quasi-projective $V',V$ with $V'\subset V$
and $|V(F)|=\infty$ there exists the limit
$$
\lim_{B\rightarrow \infty} \frac{N(V',{\cal L}, B)}{N(V,{\cal L}, B)}.
$$
}
\label{assumption}
\end{assume}
\begin{dfn}
{\rm We call an irreducible
quasi-projective algebraic variety $V$
with an ample metri\-zed
invertible sheaf ${\cal L}$
{\bf weakly ${\cal L}$-saturated} if
for any Zariski locally closed subset $W \subset V$ with
${\rm dim}\, W < {\rm dim}\, V$, one has
\[ {\lim}_{B \rightarrow \infty}
\frac{N(W,{\cal L},B)}{N(V,{\cal L},B)} < 1. \]
}
\end{dfn}
\begin{dfn}
{\rm We call an irreducible
quasi-projective algebraic variety $V$
with an ample metri\-zed
invertible sheaf ${\cal L}$
{\bf strongly ${\cal L}$-saturated} if
for any dense Zariski open subset $U \subset V$, one
has
\[ {\lim}_{B \rightarrow \infty}
\frac{N(U,{\cal L},B)}{N(V,{\cal L},B)} = 1. \]
}
\end{dfn}
\begin{dfn}
{\rm Let $V$ be a weakly ${\cal L}$-saturated
variety, $W \subset V$ a locally
closed strongly saturated
subvariety of smaller dimension.
Then we call $W$ an ${\cal L}$-{\bf target} of $V$, if
$$
0 < {\lim}_{B \rightarrow \infty}
\frac{N(W,{\cal L},B)}{N(V,{\cal L},B)} < 1.
$$
}
\end{dfn}
\begin{theo}
Let $V$
be an arbitrary quasi-projective
algebraic variety with an
ample metrized invertible sheaf ${\cal L}$. Assume that
$|V(F)|= \infty$ and that \ref{assumption} holds. Then
we have:
{\rm (i)} if $V$ is strongly
${\cal L}$-saturated then
$V$ is weakly ${\cal L}$-saturated;
{\rm (ii)} $V$ contains finitely many
weakly ${\cal L}$-saturated subvarieties
$W_1, \ldots, W_k$ with
\[ {\lim}_{B \rightarrow \infty}
\frac{N(W_1 \cup \cdots
\cup W_k,{\cal L}, B)}{N(V,{\cal L},B)} = 1. \]
{\rm (iii)} $V$
contains a strongly ${\cal L}$-saturated
subvariety $W$ having the property
\[ 0 < {\lim}_{B \rightarrow \infty}
\frac{N(W,{\cal L},B)}{N(V,{\cal L},B)} < 1. \]
{\rm (iv)} if $V$ is weakly saturated and if
it doesn't contain a dense Zariski open
subset $U \subset V$ which is
strongly saturated then $V$ contains infinitely many
${\cal L}$-targets.
\label{ws-sat}
\end{theo}
\noindent
{\em Proof.} (i) Let $W \subset V$ be a Zariski closed
subset with ${\rm dim}\, W < {\rm dim}\, V$ and $U = V \setminus W$. Then
\[ {\lim}_{B \rightarrow \infty}
\frac{N(W,{\cal L},B)}{N(V,{\cal L},B)} +
{\lim}_{B \rightarrow \infty}
\frac{N(U,{\cal L},B)}{N(V,{\cal L},B)} =
{\lim}_{B \rightarrow \infty}
\frac{N(V,{\cal L},B)}{N(V,{\cal L},B)} =1.
\]
Since $V$ is strongly ${\cal L}$-saturated, we have
\[ {\lim}_{B \rightarrow \infty}
\frac{N(U,{\cal L},B)}{N(V,{\cal L},B)} =1 \]
and therefore
\[ {\lim}_{B \rightarrow \infty}
\frac{N(W,{\cal L},B)}{N(V,{\cal L},B)} =0 <1. \]
(ii) Let $W \subset V$ be a
minimal Zariski closed subset such that
\[ {\lim}_{B \rightarrow \infty}
\frac{N(W,{\cal L},B)}{N(V,{\cal L},B)} = 1 \]
and $W_1, \ldots, W_k$ irreducible components of $W$.
It immediately follows from the minimality of $W$ that
each $W_i$ is weakly saturated.
(iii) Let $W \subset V$ be an
irreducible Zariski closed subset of
minimal dimension
such that
\[ {\lim}_{B \rightarrow \infty}
\frac{N(W,{\cal L},B)}{N(V,{\cal L},B)} =1. \]
The minimality of $W$ implies that
$W$ is weakly saturated.
(iv) By (iii) the set of ${\cal L}$-targets is nonempty.
Assume that the set of all ${\cal L}$-targets is finite:
$\{W_1, \ldots, W_k \}$.
The strong saturatedness of each $W_i$ implies that
\[ {\lim}_{B \rightarrow \infty} \frac{N(W_{i_1} \cap \cdots
\cap W_{i_l},{\cal L}, B)}{N(V,{\cal L}, B)} = 0 \]
for all pairwise different $i_1, \ldots, i_l \in \{1,\ldots, k\}$
and $l \geq 2$. In particular,
one has
\[ \sum_{i=1}^k {\lim}_{B \rightarrow \infty}
\frac{N(W_i,{\cal L},B)}{N(V,{\cal L}, B)} =
{\lim}_{B \rightarrow \infty}
\frac{N(W_1 \cup \cdots \cup W_k,{\cal L},B)}{N(V,{\cal L},B)} < 1. \]
We set $U:= V \setminus (W_1 \cup \cdots \cup W_k)$. Then
\[ {\lim}_{B \rightarrow \infty}
\frac{N(U,{\cal L},B)}{N(V,{\cal L},B)} > 0. \]
Since $U$ is not strongly saturated,
there exists an irreducible Zariski closed subset $W_0 \subset V$
of minimal dimension $< {\rm dim}\, V$ such that
\[ {\lim}_{B \rightarrow \infty}
\frac{N(W_0\cap U,{\cal L},B)}{N(U,{\cal L},B)} > 0. \]
It follows from the minimality of $W_0$ that $W_0$ is strongly
saturated.
On the other hand, one has
\[ {\lim}_{B \rightarrow \infty}
\frac{N(W_0,{\cal L},B)}{N(V,{\cal L},B)} \geq
{\lim}_{B \rightarrow \infty}
\frac{N(W_0\cap U,{\cal L},B)}{N(V,{\cal L},B)} > 0, \]
i.e., $W_0$ is an ${\cal L}$-target and $W_0 \not\in
\{ W_1, \ldots, W_k \}$. Contradiction.
\hfill $\Box$
\begin{dfn}
{\rm Let $V$ be a weakly ${\cal L}$-saturated variety
and $W_1, W_2, \ldots$ an infinite sequence
of strongly saturated irreducible subvarieties
$W_i$ having the property
\[ 0 < \theta_i :=
{\lim}_{B \rightarrow \infty}
\frac{N(W_i,{\cal L},B)}{N(V,{\cal L},B)} < 1\;\; \forall i > 0. \]
We say that the set $\{W_1, W_2, \ldots \}$
forms an {\bf
asymptotic arithmetic ${\cal L}$-fibration} on $V$,
if the following equality holds
\[ \sum_{i=1}^{\infty} \theta_i = 1. \] }
\end{dfn}
\medskip
We expect that the main source of
examples of weakly and strongly
saturated varieties should come
from the following situation:
\begin{prop}
{\rm Assume \ref{assumption}
and let $G \subset PGL(n+1)$
be a connected linear
algebraic group acting on ${ \bf P }^n$ and
$V :=Gx \subset { \bf P }^n$ a
$G$-orbit of a point $x \in { \bf P }^n(F)$.
Then $V$ is weakly $\tilde{O}(1)$-saturated.
\label{sat1}
}
\end{prop}
\noindent
{\em Proof.} Let $W \subset V$ be an arbitrary
locally closed
subset with ${\rm dim}\, W < {\rm dim}\, V$,
$\overline{W} \subset V$
its Zariski closure in $V$ and
$ U := V \setminus \overline{W} \subset V$
the corresponding
dense Zariski open
subset of $V$. Then $V$ is covered by the
open subsets $gU$,
where $g$ runs over all elements in $G(F)$
(this follows from the fact
that $G$ is unirational
and that $G(F)$ is Zariski dense in $G$ \cite{borel}).
Therefore, the orbit of $x\in V(F)$
under $G(F)$ is Zariski dense in $V$.
Since the Zariski topology is noetherian we
can choose a finite subcovering:
$V = \bigcup_{i =1 }^k g_iU$ ($g_i$ in $G(F)$).
Considering $g_i\in G(F)$ as matrices
in $PGL(n+1)$ and using standard
properties of heights \cite{lang},
one obtains positive constants
$c_i$ such that
$$
H_{\cal L}(g_i(x)) \le c_iH_{\cal L}(x)
$$
for all $x\in { \bf P }^n(F)$.
It is clear that
for $c_0: =\sum_{i =1}^k c_i$ we have
$$
N(U,{\cal L},B) \le N(V, {\cal L},B) \le c_0 N(U, {\cal L},B).
$$
It follows that
$$
\frac{1}{c_0} \leq
{\lim}_{B \rightarrow \infty}
\frac{N(U, {\cal L},B)}{N(V,{\cal L},B)} \leq 1.
$$
Hence
$$
{\lim}_{B \rightarrow \infty}
\frac{N(W, {\cal L},B)}{N(V,{\cal L},B)} \leq
{\lim}_{B \rightarrow \infty}
\frac{N(\overline{W}, {\cal L},B)}{N(V,{\cal L},B)}
\leq 1 -\frac{1}{c_0} <1.
$$
\vskip 0,5cm
\noindent
We can reformulate the statement of \ref{sat1} as follows:
\begin{prop} {\rm
Let $G$ be a connected linear algebraic group, $H \subset G$ a
closed subgroup and $V : = G/H$. If
\ref{assumption} holds then $V$ is weakly saturated
with respect to
any $G$-equivariant projective embedding of $V$.
\label{equiv-sat}
}
\end{prop}
\noindent
It is easy to see that $V = G/H$ is not necessarily weakly saturated
with respect to projective embeddings which are not
$G$-equivariant:
\begin{exam}
{\rm Let $S \subset { \bf P }^8$ be the anticanonically embedded
Del Pezzo surface which is a blow up of a
rational point in ${ \bf P }^2$. Denote by ${\cal L}$ the metrized anticanonical
sheaf on $S$.
The unique exceptional curve $C \subset S$ is
contained in the union of two open subsets $U_0, U_1 \subset S$
where $U_0 \cong U_1 \cong {\bf A}^2$. Therefore, $S$ can be
considered as a projective compactification of the algebraic
group ${\bf G}_a^2$ (after an identification of ${\bf G}_a^2$ with
$U_0$ or $U_1$). This compactification is not ${\bf G}_a^2$-equivariant.
One has
$$
a_{\cal L}(U_0) = a_{\cal L}(U_1) = a_{\cal L}(C) = 2,
$$
but
$$
a_{\cal L}(U_0 \setminus C) = a_{\cal L}(U_1 \setminus C) =1.
$$
Hence, $U_0$ and $U_1$ are not weakly
${\cal L}$-saturated.
}
\end{exam}
It is easy to show that an equivariant compactification of $G/H$
is not necessarily strongly ${\cal L}$-saturated:
\begin{exam}
{\rm Let $V = {\bf P}^1 \times {\bf P}^1$. Then $V$ is a $G$-homogeneous
variety with $G = GL(2) \times GL(2)$. However, $V$ is not strongly
${\cal L}$-saturated for $L := \pi_1^*{\cal O}(k_1) \otimes
\pi_2^*{\cal O}(k_2)$ ($k_1,k_2 \in {\bf N}$), if $k_1 \neq k_2$. }
\end{exam}
\subsection{Adelic ${\cal L}$-measure and $\tau_{\cal L}(V)$}
Now we define an adelic
measure ${\bf \omega}_{\cal L}$ corresponding to an
ample metrized invertible sheaf ${\cal L}$ on an ${\cal L}$-primitive
variety $V$ with $\alpha_{\cal L}(V) >0$ which satisfies
the assumption \ref{vanish}. This is a generalization of
a construction due to Peyre (\cite{peyre}) for $V$
being a smooth projective variety and ${\cal L}$ the metrized canonical
line bundle, which in its turn is a generalization of the classical
construction of Tamagawa measures on the adelic points of algebraic groups.
Let $V$ be an ${\cal L}$-primitive variety of dimension $n$,
$\rho \, : \, X \rightarrow \overline{V}^{\cal L}$ a resolution of
singularities, $k_2$ a positive integer such that $k_1 =
k_2 \alpha_{\cal L}(V) \in {\bf Z}$ and
$$
( \rho^*(L)^{\alpha_{\cal L}(V)} \otimes K_X)^{\otimes k_2} \cong
{\cal O}(D),
$$
where $D$
is a rigid effective Cartier divisor
on $X$.
\begin{dfn}
{\rm Let $g$ be a non-zero element of the
$1$-dimensional $F$-vector space ${ \rm H }^0(X, {\cal O} (D))$
($g$ is defined uniquely up to an element of $F^*$).
Let $v \in {\rm Val}(F)$. We define a measure ${\bf \omega}_{{\cal L},v}(g)$
on $V(F_v)$ as follows.
Choose local $v$-analytic coordinates
$x_{1,v}, \ldots, x_{n,v}$ in
some open neighborhood $U_x \subset X(F_v)$ of a point
$x \in X(F_v)$.
We write the restriction of the global section $g$ to $U_x$ as
\[ g = s^{k_1} ( dx_{1,v} \wedge \cdots \wedge dx_{n,v})^{k_2} \]
where $s$ is a local section of $L$. Define a $v$-adic measure on $U_x$
as
\[ {\bf \omega}_{{\cal L},v}(g) : = \|s\|_v^{k_1/k_2} dx_{1,v}
\cdots dx_{n,v} = \|s\|_v^{ \alpha_{\cal L}(V)} dx_{1,v}
\cdots dx_{n,v}, \]
where $dx_{1,v}\cdots dx_{n,v}$ is the usual normalized
Haar measure on $F_v^n$.
By a standard argument, one obtains that
${\bf \omega}_{{\cal L},v}(g)$
doesn't depend on the choice
of local coordinates in $U_x$ and
that it extends to the whole $v$-adic space
$X(F_v)$. The restriction of
${\bf \omega}_{{\cal L},v}(g)$ doesn't depend
on the choice of $\rho$. So we obtain a
well-defined $v$-adic measure
on $V(F_v)$. }
\end{dfn}
\begin{rem}
{\rm We remark that ${\bf \omega}_{{\cal L},v}(g)$
depends on the choice of a global section
$g \in { \rm H }^0(X, {\cal O} (D))$:
if $g' = cg$ $( c\in F^*)$ is another
global section, then
\[ {\bf \omega}_{{\cal L},v}(g') =|c|_v^{1/k}
{\bf \omega}_{{\cal L},v}(g). \]
}
\end{rem}
\noindent
Our next goal is to obtain an
explicit formula for the integral
\[ d_v(V): = \int_{V(F_v)}{\bf \omega}_{{\cal L},v}(g) =
\int_{X(F_v)}{\bf \omega}_{{\cal L},v}(g) \]
for almost all $v \in {\rm Val}(F)$. We use the $p$-adic integral formula
of Denef (\cite{denef} Th. 3.1) in the same way as the $p$-adic formula
of A. Weil (\cite{weil} Th. 2.2.3) was used by Peyre in \cite{peyre}.
\begin{rem}
{\rm By \cite{AW,H}, we can choose $\rho$ in such a way that
$\rho$ is defined over $F$ and all irreducible components $D_1, \ldots,
D_l$ $(l \geq 0)$ of the support
of $D = \sum_{i =1}^l m_i D_i$ are smooth divisors
with normal crossings over the algebraic closure
$\overline{F}$.}
\end{rem}
\begin{dfn}
{\rm Let $G$ be the image of ${\rm Gal}(\overline{F}/F)$
in the symmetric group $S_l$ that acts by permutations on $D_1, \ldots,
D_l$. We set $I: = \{ 1, \ldots, l\}$ and
denote by $I/G$ the set of
all $G$-orbits in $I$.
For any $J \subset I$ we set
\[ D_J : = \left\{ \begin{array}{ll}
\bigcap_{j \in J} D_j, \; & \; \mbox{if $J \neq \emptyset$} \\
X \setminus \bigcup_{j \in I} D_j, \; & \; \mbox{if $J = \emptyset$}
\end{array} \right. , \]
\[ D_J^{\circ} = D_J \setminus \bigcup_{j \not\in J} D_j. \]
($D_J$ is defined over $F$ if and only if $J$ is
a union of some of $G$-orbits in $I/G$.). }
\end{dfn}
We can extend $X$ to a projective
scheme ${\cal X}$ of finite type over ${\cal O}_F$ and divisors
$D_1, \ldots, D_l$ to codimension-$1$ subschemes ${\cal D}_1,
\ldots, {\cal D}_l$ in ${\cal X}$ such that
for almost all non-archimedean $v \in {\rm Val}(F)$
the reductions of $X$ and ${\cal D}_1, \ldots, {\cal D}_l$
modulo $\wp_v \subset {\cal O}_F$ are smooth projective varieties
${\cal X}_v$ and ${\cal D}_{v,1}, \ldots, {\cal D}_{v,l}$ over
the algebraic closure $\overline{k_v}$ of the
residue field $k_v$ with ${\cal D}_{v,i} \neq {\cal D}_{v,j}$
for $i \neq j$. Moreover, we can assume that
$$
{\cal D}_{v,J} : = \left\{ \begin{array}{ll}
\bigcap_{j \in J} {\cal D}_{v,j}, \; & \; \mbox{if $J \neq \emptyset$} \\
X \setminus \bigcup_{j \in I} {\cal D}_{v, j}, \; & \;
\mbox{if $J = \emptyset$}
\end{array} \right.
$$
are also smooth over $\overline{k_v}$.
\begin{dfn}
{\rm A non-archimedean valuation $v \in {\rm Val}(F)$
which satisfies all the
above assumptions will be called
a {\bf good valuation} for the pair $({\cal X},
\{ {\cal D}_i \}_{i \in I})$. }
\end{dfn}
\begin{dfn}
{\rm Let $G_v \subset G$ be a cyclic subgroup generated by a representative
of the Frobenius element in ${\rm Gal}(\overline{k_v}/k_v)$. We denote
by $I/G_v$ the set of all $G_v$-orbits in $I$. If $j \in I/G_v$, then
we set $b_j$ to be the length of the corresponding $G_v$-orbit
and put $r_j = m_j/k_2$, where $m_j$ is the multiplicity of irreducible
components of $D$ corresponding to the $G_v$-orbit $j$. }
\end{dfn}
The following theorem is a slightly generalized version of
Th. 3.1 in \cite{denef}:
\begin{theo}
Let $v \in {\rm Val}(F)$ be a good non-archimedean valuation
for $({\cal X}, \{ {\cal D}_i \}_{i \in I})$. Then
\[ d_v(V) = \int_{X(F_v)} {\bf \omega}_{{\cal L},v}(g) =
\frac{c_{\emptyset}}{q_v^n} + \frac{1}{q_v^n}
\sum_{\emptyset \neq J \subset I/{G_v} } c_J \prod_{j \in J}
\left( \frac{q_v^{b_j} -1}{q_v^{b_j(r_j + 1)} -1} \right), \]
where $q_v$ is the cardinality of $k_v$, and
$c_J$ is the cardinality of the set of $k_v$-rational points
in ${\cal D}^{\circ}_J$.
\end{theo}
Let us consider an exact sequence of
Galois ${\rm Gal}(\overline{F}/F)$-modules:
\begin{equation}
0 \rightarrow {\bf Z}[{\cal D}_1] \oplus \cdots \oplus
{\bf Z}[{\cal D}_l] \rightarrow {\rm Pic}({\cal X})
\stackrel{\tilde{\rho}}{\rightarrow} {\rm Pic}({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i ) {\rightarrow} 0
\label{sh-3}
\end{equation}
\begin{theo}
Assume that $X$ has the property ${\rm h}^1(X, {\cal O}_X) =0$.
Then
\[ d_v(V) = 1 + \frac{1}{q_v} {\rm Tr}(\Phi_v | {\rm Pic}({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i )\otimes {\bf Q}_l) +
O\left(\frac{1}{q_v^{1+ \varepsilon}} \right), \]
where
\[ \varepsilon = \min \{ 1/2, r_1, \ldots, r_l \} \]
and $\Phi_v$ is the Frobenius morphism.
\end{theo}
\noindent
{\em Proof.}
By conjectures of Weil proved
by Deligne \cite{deligne}, one has
\[ \frac{c_J}{q_v^n} = O\left(\frac{1}{q_v} \right) \; J \neq \emptyset \]
(since ${\rm dim}\, D_J \leq n-1$ for $J \neq \emptyset$)
and
\[ \frac{c_{\emptyset}}{q_v^n}
= \sum_{k =0}^{2n} (-1)^k {\rm Tr}(\Phi_v | { \rm H }^k_c({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i, {\bf Q}_l)), \]
where ${ \rm H }^k_c(\cdot , {\bf Q}_l)$ denotes the \'etale cohomology group
with compact supports.
Using long cohomology sequence of the pair $({\cal X}, \bigcup_i {\cal D}_i)$,
one obtains isomorphisms
\[ { \rm H }^{k}_c({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i, {\bf Q}_l) = { \rm H }^{k}_c({\cal X}, {\bf Q}_l)\;\;
\mbox{\rm for $k = 2n, 2n-1$} \]
and the short exact sequence
\[ 0 \rightarrow { \rm H }^{2n-2}_c({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i, {\bf Q}_l) \rightarrow
{ \rm H }^{2n-2}_c({\cal X}, {\bf Q}_l) \rightarrow \bigoplus_{i =1}^l
{ \rm H }^{2n-2}_c({\cal D}_i, {\bf Q}_l) \rightarrow 0. \]
Using isomorphisms
\[ { \rm H }^{2n-2}_c({\cal D}_i, {\bf Q}_l(n-1)) \cong {\bf Q}_l,
{ \rm H }^{2n}_c({\cal X}, {\bf Q}_l(n)) \cong {\bf Q}_l, \]
Poincar\'e duality
\[ { \rm H }^{2n-2}_c({\cal X}, {\bf Q}_l(n-1))
\times {\rm Pic}({\cal X})
\stackrel{\sim}{\rightarrow} {\bf Q}_l \]
and the vanishing property
${\rm h}^1(X, {\cal O}_X) =0$, we obtain
\[ { \rm H }^{2n-1}_c({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i, {\bf Q}_l) = 0 \]
and
\[ \frac{c_{\emptyset}}{q_v^n} = 1 +
\frac{1}{q_v} {\rm Tr}(\Phi_v |
{\rm Pic}({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i )\otimes {\bf Q}_l)
\]
\[ + \sum_{k =0}^{2n-3} \frac{ (-1)^k}{q_v^n}
{\rm Tr}(\Phi_v | { \rm H }^k_c({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i, {\bf Q}_l)) \]
\[ = 1 + \frac{1}{q_v} {\rm Tr}(\Phi_v |
{\rm Pic}({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i )\otimes {\bf Q}_l) +
O\left(\frac{1}{q_v^{3/2}} \right). \]
On the other hand, for $J \neq \emptyset$ one has
\[ \prod_{j \in J}
\left( \frac{q_v^{b_j} -1}{q_v^{b_j(r_j + 1)} -1} \right)
= O\left(\frac{1}{q_v^{ 1 + \varepsilon_0}} \right), \]
where $\varepsilon_0 = \min \{ r_1, \ldots, r_l \}$.
\hfill $\Box$
\begin{dfn}
{\rm We define the convergency factors
\[ \lambda_v : =
\left\{ \begin{array}{ll} L_v(1, {\rm Pic}({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i )), \;& \; \mbox{\rm if $v$ is good} \\
1 \; & \; \mbox{\rm otherwise} \end{array} \right. \]
where $L_v$ is the local factor of the Artin $L$-function
corresponding to the $G$-module ${\rm Pic}({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i)$
and we set
\[ \omega_{{\cal L},S} : = \sqrt{|{\rm disc}(F)|}^{-n} \prod_{v \in {\rm Val}(F)}
\lambda_v^{-1} {\bf \omega}_{{\cal L},v}(g), \]
where ${\rm disc}(F)$ is the
absolute discriminant of ${\cal O}_F$
and $S$ is the set of bad valuations.
}
\end{dfn}
By the product formula,
$\omega_{{\cal L},S}$ doesn't
depend on the choice of
$g$.
\begin{dfn}
{\rm
Denote by ${\bf A}_F$ the adele ring of $F$.
Let $\overline{X(F)}$ be the closure of $X(F)$ in
$X({\bf A}_F)$ (in direct product topology).
Under the vanishing assumption ${\rm h}^1(X,{\cal O}_X) =0$,
we define the constant
\[ \tau_{\cal L}(V) =
\lim_{s \rightarrow 1}(s-1)^{{\beta}_{\cal L}(V)} L_S(s, {\rm Pic}({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i )) \int_{\overline{X(F)}} \omega_{{\cal L},S}, \]
where ${\beta}_{\cal L}(V)$ is the rank of the submodule
of ${\rm Gal}(\overline{F}/F)$-invariants of the module
${\rm Pic}({\cal X} \setminus
\bigcup_{i =1}^l {\cal D}_i )$.
\label{tau-dfn}
}
\end{dfn}
\subsection{Main strategy}
Now we proceed to discuss our
main strategy in understanding the asymptotic
for the number $N(V, {\cal L}, B)$ as
$B \to \infty$ for an arbitrary
${\cal L}$-polarized quasi-projective variety.
Again, we shall make the assumption \ref{assumption}.
Our approach consists
in $4$ steps including $3$ subsequent simplifications
of the situation:
\medskip
\noindent
{\bf Step 1 (reduction
to weakly ${\cal L}$-saturated varieties):}
By \ref{ws-sat} (ii),
every quasi-projective ${\cal L}$-polarized
variety $V$ contains a finite number
of weakly ${\cal L}$-saturated
varieties $W_1, \ldots W_k$ such that
\[
{\lim}_{B \rightarrow \infty}
\frac{N(W,{\cal L},B)}{N(V,{\cal L},B)} = 1.
\]
Therefore, it would be enough to understand
separately the asymptotics of
$N(W_i, {\cal L}, B)$, $i \in \{1, \ldots, k\}$
modulo the asymptotics
$N(W_{i_1} \cap \cdots \cap W_{i_l}, {\cal L}, B)$
for low-dimensional
subvarieties $W_{i_1} \cap \cdots \cap W_{i_l}$, where
$\{i_1, \ldots, i_l\} \subset \{1, \ldots, k\}$
are subsets of pairwise different
elements with $l \geq 2$.
\medskip
For our next reduction step, we need:
\begin{conj}
Let $V$ be a weakly ${\cal L}$-saturated
variety which doesn't contain an open Zariski
dense and
strongly ${\cal L}$-saturated subset $U \subset V$.
Then the
set of ${\cal L}$-targets of $V$ forms
an asymptotic arithmetic
${\cal L}$-fibration.
\label{aaf}
\end{conj}
\noindent
{\bf Step 2 (reduction to strongly ${\cal L}$-saturated varieties):}
Let $V$ be an arbitrary weakly ${\cal L}$-saturated variety. Then
either $V$ contains a strongly ${\cal L}$-saturated Zariski open subset
or, according to \ref{aaf},
we obtain an asymptotic arithmetic
${\cal L}$-fibration of $V$ by ${\cal L}$-targets.
In the first situation,
it is
enough to understand the asymptotic of
$N(U, {\cal L}, B)$ for the
strongly ${\cal L}$-saturated variety $U$
(we note that the complement $V \setminus U$
consists of low-dimensional irreducible components).
In the second situation,
it is
enough to understand the asymptotic of
$N(W_i, {\cal L}, B)$ for each of the
${\cal L}$-targets $W_i \subset V$.
\medskip
For our next reduction step, we need:
\begin{conj}
Let $V$ be a smooth strongly
${\cal L}$-saturated quasi-projective
variety. Then the complex analytic
variety $V({\bf C})$ is ${\cal L}$-primitive.
\end{conj}
\noindent
{\bf Step 3
(reduction to ${\cal L}$-primitive varieties):}
Every quasi-projective algebraic
variety $V$ is a disjoint union of finitely many
locally closed smooth subvarieties $V_i$.
Therefore, if one knows
the asymptotic for each $N(V_i, {\cal L}, B)$
then one immediately
obtains the asymptotic for $N(V, {\cal L}, B)$.
\begin{dfn}
{\rm Let $V$ be an ${\cal L}$-primitive algebraic variety over
a number field $F$,
$\rho\,:\, X \rightarrow \overline{V}^{\cal L}$ a desingularization over $F$
of the closure of $V$ with the exceptional locus consisting of
smooth irreducible divisors
$D_1,\ldots, D_l$. We consider ${\rm Pic}(X)$ and ${\rm Pic}(V,{\cal L})$ as
${\rm Gal}(\overline{F}/F)$-modules
and we denote by $\beta_{\cal L}(V)$
the rank of ${\rm Gal}(\overline{F}/F)$-invariants
in ${\rm Pic}(V,{\cal L})$ and by
$\delta_{\cal L}(V)$ the cardinality of the
cohomology group
\[ { \rm H }^1({\rm Gal}(\overline{F}/F),
{\rm Pic}(V, {\cal L})). \]}
\label{kohom}
\end{dfn}
\begin{rem}
{\rm Using the long exact Galois-cohomology sequence associated with
(\ref{sh-3}), one immediately obtains that $\beta_{\cal L}(V)$
and $\delta_{\cal L}(V)$
do not depend on the choice of the resolution $\rho$.
}
\end{rem}
\noindent
{\bf Step 4 (expected asymptotic formula): }
Let $V$ be a strongly ${\cal L}$-saturated (and ${\cal L}$-primitive)
smooth quasi-projective variety. Assume that $a_{\cal L}(V) >0$.
Then we expect that the following asymptotic formula holds:
$$
N(V, {\cal L}, B) = c_{\cal L}(V)B^{\alpha_{\cal L}(V)}
(\log B)^{ \beta_{\cal L}(V)-1} \left( 1 + o(1) \right),
$$
where
$$
c_{\cal L}(V) := \frac{\gamma_{\cal L}(V)}{
\alpha_{\cal L}(V)(\beta_{\cal L}(V)-1)!}
\delta_{\cal L}(V)\tau_{\cal L}(V),
$$
$\gamma_{\cal L}(V)$ is an invariant of the triple
$({\rm Pic}(V,{\cal L}), {\rm Pic}(V,{\cal L})_{{\bf R }},
\Lambda_{\rm eff}(V,{\cal L}))$ (\ref{gamma-dfn}),
$\delta_{\cal L}(V)$ is a cohomological invariant of the
${\rm Gal}(\overline{F}/F)$-module
${\rm Pic}(V, {\cal L})$ (\ref{kohom}) and
$\tau_{\cal L}(V)$ is an adelic
invariant of a family of $v$-adic
metrics $\{ \|\cdot \|_v \}$ on $L$ (\ref{tau-dfn}).
\medskip
In sections 4 and 5 we discuss some examples which show
how the constants
$$
\alpha _{\cal L}(V), \beta _{\cal L}(V),\delta_{\cal L}(V),
\gamma_{\cal L}(V),\tau_{\cal L}(V)
$$
appear in asymptotic formulas for the number of
rational points of bounded ${\cal L}$-height
on algebraic varieties. Naturally, we expect
that the exhibited behavior is typical.
However, we also feel that one should collect
more examples which could help
to clarify the general situation.
\subsection{${\cal L}$-primitive
fibrations and $\tau_{\cal L}(V)$}
We proceed to discuss our
observations concerning the arithmetic
conjecture \ref{aaf}
at its relation to the geometric
conjecture \ref{conj-fb}.
Let $V$ be a weakly ${\cal L}$-saturated
smooth quasi-projective
variety with $a_{\cal L}(V) >0$ which is not strongly saturated
and which doesn't contain Zariski open dense strongly saturated
subvarieties. We distinguish the following
two cases:
\noindent
{\bf Case 1. $V$ is not ${\cal L}$-primitive. }
In this case we expect that some Zariski open dense subset
$U \subset V$ admits an
${\cal L}$-primitive fibration which is defined by
a projective regular
morphism $\pi\, : \, X \rightarrow Y$ over $F$
to a low-dimensional
normal irreducible projective variety $Y$
satisfying the conditions
(i)-(iii) in \ref{prim-fb}, for an appropriate
smooth projective compactification
$X$ of $U$ (see \ref{conj-fb}).
It seems natural to expect that all fibers satisfy the vanishing
assumption \ref{vanish}.
Thus we see that for any $y \in Y(F)$ such that
$V_y = \pi^{-1}(y) \cap V$ is
${\cal L}$-primitive we can define
the adelic number $\tau_{\cal L}(V_y)$.
Furthermore, we expect
that every ${\cal L}$-target $W$ is contained
in an appropriate
${\cal L}$-primitive subvariety $V_y$ which is a fiber
of the ${\cal L}$-primitive fibration
$\pi \; :\; V \rightarrow U$ on $V$.
In particular, Step 4 of our main strategy implies that if every
${\cal L}$-target $W$ {\em coincides} with a
suitable ${\cal L}$-primitive fiber $V_y$
then one should expect the asymptotic
$$
N(V, {\cal L}, B) = c_{\cal L}(V)B^{\alpha_{\cal L}(V)}
(\log B)^{ \beta_{\cal L}(V)-1} \left( 1 + o(1) \right),
$$
where the numbers $\alpha_{\cal L}(V)$
(resp. $\beta_{\cal L}(V)$)
coincide with the numbers $\alpha_{\cal L}(V_y)$
(resp. $\beta_{\cal L}(V_y)$) for the corresponding ${\cal L}$-targets
$V_y$ and the constant $c_{\cal L}(V)$ is equal to the sum
$$
\sum_{y} c_{\cal L}(V_y) =
\sum_{y} \frac{\gamma_{\cal L}(V_y)}{
\alpha_{\cal L}(V_y)(\beta_{\cal L}(V_y)-1)!}
\delta_{\cal L}(V_y)\tau_{\cal L}(V_y),
$$
where $y$ runs over all points in $Y(F)$ such that $V_y$ is an
${\cal L}$-target of $V$.
It is natural to try to understand
the dependence of $\tau_{\cal L}(V_y)$ on the choice of a point
$y \in Y(F)$. We expect that the number
$\tau_{\cal L}(V_y)$ can be
interpreted as a ``height'' of $y$.
More precisely, the examples we considered
suggest the following:
\begin{conj}
There exist a family of $v$-adic metrics on $K_Y$ and two positive
constants $c_2 > c_1 > 0$ such that
\[ c_1 H_{\cal F}(y) \leq \tau_{\cal L}(V_y) \leq c_2
H_{\cal F}(y) \;\; \forall y \in Y(F) \cap U, \]
where $U \subset Y$ is some dense Zariski open subset and
${\cal F}$ is a metrized ${\bf Q}$-invertible sheaf
associated with the ${\bf Q}$-Cartier divisor
$L_1^{-1} \otimes K_Y$
(recall that $L_1$ is the tautological ample
${\bf Q}$-Cartier divisor
on $Y$ defined by the graded
ring ${\rm R}(V, {\cal L})$ $($see
\ref{proj-fb}$))$.
\end{conj}
\noindent
{\bf Case 2. $V$ is ${\cal L}$-primitive
(but not strongly saturated!).}
We don't know examples of a precise
asymptotic formula in this situation.
\begin{exam}
{\rm Let $V$ be a Fano diagonal cubic bundle over
${\bf P}^3$ with the homogeneous coordinates $(X_0:X_1:X_2:X_3)$
defined as a hypersurface in ${\bf P}^3 \times {\bf P}^3$
by the equation
\[ X_0Y_0^3 + X_1Y_1^3 + X_2Y_2^3+ X_3Y_3^3= 0 \]
in ${\bf P}^3 \times {\bf P}^3$ (see \cite{BaTschi4}).
We expect that $V$
is not strongly saturated
with respect to a metrized
anticanonical sheaf $L:= {\cal O}(3,1)$
and that the corresponding
${\cal L}$-targets are the splitting diagonal
cubics in fibers of
the natural projection $\pi\,: \, V \to {\bf P}^3$
(this leads to the failure of the expected
asymptotic formula in Step 4 for this example).
}
\end{exam}
The next example
was suggested to us by Colliot-Th\'el\`ene:
\begin{exam}
{\rm Let $V$ be an analogous diagonal quadric bundle over
${\bf P}^3$
defined as a hypersurface in ${\bf P}^3 \times {\bf P}^3$
by the equation
\[ X_0Y_0^2 + X_1Y_1^2 + X_2Y_2^2 + X_3Y_3^2 = 0. \]
For infinitely many fibers $V_x = \pi_1^{-1}(x)$ ($x \in {\bf P}^3(F)$)
we have ${\rm rk}\,{\rm Pic}(V_x) = 2$. At the same time, we have
also ${\rm rk}\,{\rm Pic}(V) = 2$. We consider the height
function associated to some metrization of the line bundle
$L:= {\cal O}(3,2)$.
On the one hand, we think that the asymptotic on the whole variety
is $c(V)B\log B(1+o(1))$ for $B\rightarrow \infty$ with some $c(V)>0$.
On the other hand, if $X_0 X_1 X_2 X_3$ is a square in $F$
we get already about $B\log B$ solutions.
Another important observation
is the {\em expected} convergency of the series
\[ \sum_{x \in {\bf P}^3(F) : V_x \cong {\bf P}^1 \times {\bf P}^1}
c(V_x). \]
The latter would be a consequence of the following two facts.
First, the condition $V_x \cong {\bf P}^1 \times {\bf P}^1$
($x = (X_0:X_1:X_2:X_3)$) is equivalent to the conditions that $V_x$
contains an $F$-rational point and that the product
$X_0 X_1 X_2 X_3$ is a square in $F$. The number of $F$-rational points
$x'= (X_0:X_1:X_2:X_3:Z)$ with $H_{{\cal O}(1)}(x') \leq B$
lying on the hypersurface with the
equation $X_0X_1X_2X_3 = Z^2$ in the weighted projective space
${\bf P}^4(1,1,1,1,2)$ can be estimated from above by $B^2
(\log B)^3( 1 + o(1))$.
\noindent
Secondly, we expect
$$
c(V_x) = H^{-1}_{{\cal O}(4)}(x)
( 1 + o(1) ),
$$
{\em uniformly} over the base ${\bf P}^3(F)$.
This would imply the claimed convergency.
}
\end{exam}
\section{Height zeta-functions}
\subsection{Tauberian theorem}
One of the main techniques in the proofs of asymptotic formulas
for the counting function
$$
N(V,{\cal L},B):= \#\{x\in V(F)\,\,:\,\, H_{\cal L}(x)\le B\,\}
$$
has been the use of {\em height zeta functions}. Let
${\cal L}$ be an ample metrized invertible sheaf
on a smooth quasi-projective
algebraic variety $X$. We define the height zeta function
by the series
$$
Z(X,{\cal L},s):=\sum_{x\in X(F)}H_{\cal L}(x)^{-s}
$$
which converges absolutely for ${\rm Re}(s)\gg 0$.
After establishing the analytic properties
of $Z(X,{\cal L},s)$ one uses the
following version of a Tauberian theorem:
\begin{theo}(\cite{delange}) Suppose that
there exist an $ \varepsilon >0$ and a real number
$\Theta({\cal L})>0$ such that
$$
Z(X,{\cal L},s)=\frac{\Theta({\cal L})}{(s-a)^{b}}
+ \frac{f(s)}{(s-a)^{b-1}}
$$
for some $a>0$, $b\in { \bf N }$ and some function $f(s)$
which is holomorphic for ${\rm Re}(s)>a- \varepsilon $.
Then we have the following asymptotic formula
$$
N(X,{\cal L}, B)= \frac{\Theta({\cal L})}{a\cdot (b-1)!}B^a(\log B)^{b-1}(1+o(1))
$$
for $B\rightarrow \infty$.
\end{theo}
\subsection{Products}
Let $X_1$ and $X_2$ be two smooth quasi-projective
varieties with ample metrized invertible sheaves
${\cal L}_1$ and ${\cal L}_2$ (resp. on $X_1$ and $X_2$).
Denote by $X=X_1\times X_2$ the product and by ${\cal L}$
the product of ${\cal L}_1$ and ${\cal L}_2$ (with the obvious
metrization). Clearly,
\[ Z(X,{\cal L}, s)= Z(X_1,{\cal L}_1, s)\cdot Z(X_2,{\cal L}_2, s). \]
Assume that for $i=1,2$ we have
$$
Z(X_i,{\cal L}_i,s)= \frac{\Theta({\cal L}_i)}{
(s-\alpha_{{\cal L}_i}(X_i))^{ \beta _{{\cal L}_i}(X_i)}}
+ \frac{f_i(s)}{(s-\alpha_{{\cal L}_i}(X_i))^{ \beta _{{\cal L}_i}(X_i)}}
$$
with
some functions $f_i(s)$ which are holomorphic in the domains
${\rm Re}(s_i)>\alpha_{{\cal L}_i}(X_i)- \varepsilon $
for some $ \varepsilon >0$. There are two possibilities:
\smallskip
Case 1: $\alpha_{{\cal L}_1}(X) = \alpha_{{\cal L}_2}(X)$.
In this situation the constant $\Theta({\cal L})$ at the pole of
highest order $ \beta _{{\cal L}_1}(X_1)+ \beta _{{\cal L}_2}(X_2)$ is
given by $\Theta({\cal L})=\Theta({\cal L}_1)\Theta({\cal L}_2)$.
\medskip
Case 2:
$\alpha_{{\cal L}_1}(X) < \alpha_{{\cal L}_2}(X)$. In this situation
the constant is a sum
$$
\Theta({\cal L}) = \sum_{x \in X_1(F)}
H_{{\cal L}_1}^{-\alpha_{{\cal L}_2}}(x)\Theta({\cal L}_2).
$$
Consider the projection $X\rightarrow X_1$ and denote by $V_x$ the fiber over
$x\in X_1(F)$. We notice
$$
\tau_{\cal L}(V_x) = H_{{\cal L}_1}^{-\alpha_{{\cal L}_2}}(x)
\tau_{{\cal L}_2}(X_2).
$$
We denote by $M$ the ${\bf Q}$-Cartier divisor
$\pi_1^*{\cal L}_1^{\alpha_{{\cal L}_2}} \otimes K_{V_1}$.
We obtain that
$\pi_1^*{\cal L}_1^{\alpha_{{\cal L}_2}} = M \otimes K_{V_1}^{-1}$.
So we have
$\tau_{\cal L}(V_x) \sim H_{M \otimes K_{V_1}^{-1}}^{-1}$.
We observe that Tamagawa numbers of fibers
depend on the height of the points on the base.
\subsection{Symmetric product of a curve}
Let $C$ be a smooth irreducible curve of genus $g\ge 2$ over $F$.
We denote by $X=C^{(m)}$ the $m$-th symmetric product of $C$
and by $Y:={\rm Jac}(C)$ the Jacobian of $C$.
We fix an $m>2g-2$. We have a fibration
$$
\pi\, :\; C^{(m)} \,\rightarrow \, Y,
$$
with ${ \bf P }^{m-g}$ as fibers. We denote by $V_y$ a fiber over $y\in Y(F)$.
Let $\tilde{C}\rightarrow {\rm Spec}({\cal O}_F)$ be a smooth
model of $C$ over the integers and ${\cal L}$ an ample hermitian
line bundle on $\tilde{C}$. It defines a height function
$$
H_{\cal L}\,:\,C(F)\rightarrow {\bf R }_{>0}
$$
which extends to
$X(F)$.
Observe that $ \alpha _{\cal L}(X)=(m+1-g)/d$, where
$d:={\rm deg}_{{ \bf Q }}({\cal L})$.
Consider the height
zeta function
$$
Z(X,{\cal L},s):=\sum_{x\in X(F)}H_{\cal L}(x)^{-s}.
$$
This function was introduced by Arakelov in \cite{arakelov}.
\begin{theo}\cite{faltings}
Let ${\cal L}$ be an ample hermitian line bundle on $\tilde{C}$.
There exist an $ \varepsilon ({\cal L})>0$ and a real number
$\Theta({\cal L})\neq 0$ such that the height zeta function
has the following representation
$$
Z(X,{\cal L},s)=\frac{\Theta({\cal L})}{
(s- \alpha _{\cal L}(X))} +f(s)
$$
with some function $f(s)$ which is holomorphic
for ${\rm Re}(s)> \alpha _{\cal L}(X)- \varepsilon ({\cal L})$.
\end{theo}
This is Theorem 8 in (\cite{faltings}, p. 422).
Arakelov gives an explicit expression for the constant
$\Theta({\cal L})$ (\cite{arakelov}). We are very grateful to
J.-B. Bost for pointing out to us that Arakelov's formula is
not correct and for allowing us to use his notes on the
Arakelov zeta function.
\begin{theo} With the notations above we have
$$
\Theta({\cal L})=\sum_{y\in Y(F)}\tau_{\cal L}(V_y).
$$
\end{theo}
\noindent
{\em Proof.} We outline the proof for
$F={ \bf Q }$. For ${\rm Re}(s)\gg 0$ one
can rearrange the order of summation
and one obtains
$$
Z(X,{\cal L},s):=\sum_{y\in Y(F)}
\sum_{x\in V_y(F)}H_{\cal L}(x)^{-s}.
$$
It is proved in (\cite{faltings}, p. 420-422)
that the sums
$$
Z(V_y,{\cal L},s):=\sum_{x\in V_y(F)}H_{\cal L}(x)^{-s}
$$
have simple poles at $s= \alpha _{\cal L}(X)$
with non-zero residues
and that one can ``sum'' these expressions to
obtain a function with a simple pole at $s= \alpha _{\cal L}(X)$ and
meromorphic continuation to ${\rm Re}(s)> \alpha _{\cal L}(X)- \varepsilon ({\cal L})$
for some $ \varepsilon ({\cal L})>0$.
Moreover, the residue at this pole is
obtained as a sum over $y\in Y(F)$ of the residues of
$Z(V_y,{\cal L},s)$.
Choosing an element $z(y)$ in the class of $y\in {\rm Jac}(C)(F)$,
and denoting by $E(y):=\Gamma (C,{\cal O}(z(y)))$
one can identify the fiber as
$V_y= { \bf P }(E(y))$,
where $f\in E(y)= \Gamma (C,{\cal O}(z(y)))$ is mapped to
$z(y)+{\rm div}(f)$.
The height is given by the formula
$$
H_{\cal L}(z(y)+{\rm div}(f)):= H_{\cal L}(z(y))
\exp(\int_{C({\bf C })}\log |f|c_1({\cal L})).
$$
This defines a metrization of the anticanonical
line bundle on $V_y= { \bf P }(E(y))$. Assuming the Tamagawa number
conjecture for ${ \bf P }^n$ (for suitable metrizations
of the line bundle ${\cal O}(1)$ on ${ \bf P }^n$) we obtain
$$
\lim_{s\rightarrow \alpha _{\cal L}(X)}(s- \alpha _{\cal L}(X))Z(V_y,{\cal L},s)=
\tau_{\cal L}(V_y).
$$
\hfill $\Box$
\medskip
\noindent
One can write down an explicit formula for
$\tau_{\cal L}(V_y)$.
For $f\in E(y)_{{\bf C }}\backslash \{0\}$
we define
$$
\Phi(f):=\exp(\frac{1}{d}\int_{C({\bf C })}\log |f|c_1({\cal L}))
$$
and we put $\Phi(0)=0$.
It follows that
$$
\tau_{\cal L}(V_y)=
\frac{1}{2} \alpha _{\cal L}(X)H_{\cal L}(y)^{- \alpha _{\cal L}(X)}
\cdot
\frac{{\rm vol}(\{ f\in E(y)_{{\bf R }} \,|\, \Phi(f)\le 1\})}{
{\rm vol}(E(y)_{{\bf R }}/ E(y))},
$$
where the volumes are calculated with respect to some
Lebesgue measure on $E(z(y))_{{\bf R }}$.
Arakelov relates this last expression to the Neron-Tate height of
$y\in {\rm Jac}(C)$.
A detailed calculation due to Bost indicates that Arakelov's formula
is correct only up-to $O(1)$.
\subsection{Homogeneous spaces $G/P$}
Let $G$ be a split semisimple linear algebraic group defined over a
number field $F$. It contains a Borel subgroup
$P_0$ defined over $F$ and a maximal torus which is
split over $F$. Let $P$ be a standard parabolic.
Denote by $Y_P=P\backslash G$ (resp. $X=P_0\backslash G $)
the corresponding flag variety.
A choice of a
maximal compact subgroup ${\bf K}$ such that
$G({\bf A}_F)=P_0({\bf A}_F){\bf K}$ defines a metrization on
every line bundle $L$ on the flag varieties $Y_P$ (\cite{FMT}, p. 426).
We will denote by
$$
H_{\cal L}\,:\, P(F)\backslash G(F)\rightarrow {\bf R }_{>0}
$$
the associated height. We consider the height zeta function
$$
Z(X,{\cal L},s):=
\sum_{x \in X(F)} H_{\cal L}(x)^{-s}.
$$
\begin{theo} Let ${\cal L}$ be an ample metrized line bundle on $X$.
There
exist an $ \varepsilon ({\cal L})>0$ and a real number
$\Theta({\cal L})\neq 0$ such that the height zeta function
has the following representation
$$
Z(X,{\cal L},s)=\frac{\Theta({\cal L})}{(s-
\alpha _{\cal L}(X))^{ \beta _{\cal L}(X)}} +
\frac{f(s)}{(s- \alpha _{\cal L}(X))^{ \beta _{\cal L}(X)-1}}
$$
with some function $f(s)$ which is holomorphic
for ${\rm Re}(s)> \alpha _{\cal L}(X)- \varepsilon ({\cal L})$.
\end{theo}
This theorem follows from
the identification of the height zeta function
with an Eisenstein series and from the work of Langlands.
The formula (2.10) in (\cite{FMT}, p. 431) provides
an expression for $\Theta({\cal L})$ which we will now
analyze.
There is a canonical way to identify the
faces of the closed cone of effective
divisors $ \Lambda _{\rm eff}(X)\subset
{\rm Pic}(X)_{{\bf R }}$
with $ \Lambda _{\rm eff}(Y_P)$ as $P$ runs through
the set of standard parabolics.
A line bundle $L$ such that its class is
contained in the interior of the cone
$\in \Lambda _{\rm eff}(X)$ defines a line
bundle
$$
[L_Y]:= \alpha _{\cal L}(X)[L]+ [K_{X}]
$$
which is contained in the interior of the
face $ \Lambda _{\rm eff}(Y)\subset
\Lambda _{\rm eff}(X)$ for some $Y=Y_P$.
We have a fibration
$\pi_{\cal L}\,: X\rightarrow Y$
with fibers isomorphic to the flag variety $V:=P_0\backslash P$.
A fiber over $y\in Y(F)$ will be denoted by $V_y$.
Denote by ${\cal K}_Y$ the canonical line bundle on
$Y$ with the metrization defined above.
\begin{theo}
We have
$$
\Theta({\cal L})= \sum_{y\in Y(F)}\gamma_{\cal L}(X)\tau_{\cal L}(V_y).
$$
\end{theo}
\noindent
{\em Proof.}
In the domains of absolute
and uniform convergence we can rearrange the order of summation and
we obtain
$$
Z(X,{\cal L},s)=\sum_{y\in Y(F)}
\sum_{x\in P_0(F)\backslash P(F)} H_{\cal L}(yx)^{-s}.
$$
One can check that the sums
$$
Z(V_y,{\cal L},s):=\sum_{x\in V_y(F)}H_{\cal L}(x)^{-s}
=\sum_{x\in P_0(F)\backslash P(F)} H_{\cal L}(yx)^{-s}
$$
have poles at $s= \alpha _{\cal L}(X)$ of order $ \beta _{\cal L}(X)$
with non-zero residues, and that they admit meromorphic
continuation to $ \alpha _{\cal L}(X)- \varepsilon ({\cal L})$ for some
$ \varepsilon ({\cal L})>0$. Moreover,
the constant $\Theta({\cal L})$ is
obtained as a sum over $y\in Y(F)$ of the residues of
$Z(V_y,{\cal L},s)$. From the Tamagawa number
conjecture for $P_{0}\backslash P$ (with varying metrizations
of the anticanonical line bundle) we obtain
$$
\lim_{s\rightarrow \alpha _{\cal L}(X)}
(s- \alpha _{\cal L}(X))^{ \beta _{\cal L}(X)}Z(V_y,{\cal L},s)=\gamma_{\cal L}(X)
\tau_{\cal L}(V_y).
$$
The sum
$$
\sum_{y\in Y(F)}\gamma_{\cal L}(X)
\tau_{\cal L}(V_y)
$$
converges for $[L_Y]$ contained in the interior of
$ \Lambda _{\rm eff}(Y)$.
\hfill $\Box$
\medskip
\noindent
Let us recall the explicit formula for $\tau_{\cal L}(V_y)$
(see (2.9) in \cite{FMT}, p. 431):
$$
\lim_{s\rightarrow \alpha _{\cal L}(X)}(s- \alpha _{\cal L}(X))^{
\beta _{\cal L}(X)} \sum_{x\in P_0(F)\backslash P(F)}
H_{\cal L}(xy)^{-s}
= \gamma_1 c_P H_{{\cal L}_Y^{-1}\otimes {\cal K}_{Y}}(y)
$$
where $ \gamma_1\in { \bf Q }$
is an explicit constant
and the constant $c_P$ is defined in (\cite{FMT}, p. 430).
It follows that
$$
\Theta({\cal L})=
\gamma_1 c_P\sum_{y\in Y(F)}
H_{{\cal L}_Y^{-1}\otimes {\cal K}_Y}(y).
$$
Next we observe that there
is an explicit constant $ \gamma_2\in { \bf Q }$ such that we have
$$
\gamma_2\tau_{\cal L}(V_{y})=
c_P\cdot H_{{\cal L}^{-1}_Y\otimes {\cal K}_Y}(y)
$$
for all $y\in Y(F)$.
To see this, we first identify
$c_P= \gamma_2\tau_{\cal L}(V)$, this is done by a computation
of local factors of intertwining operators (\cite{peyre}, p.160-161).
The next step involves the comparison of Tamagawa measures on
$V_{y}$ for varying $y\in Y(F)$.
Finally, we have $ \gamma_{\cal L}(X)= \gamma_1 \gamma_2$.
\subsection{Toric varieties}
There are many equivalent ways to describe a toric variety $X$
over a number field $F$ together with
some projective embedding (see, for example, \cite{cox,fulton,danilov}).
For us, it will be useful to view
$X=X_{ \Sigma }$ as
a collection of the following data (\cite{BaTschi1}):
1. A splitting field $E$ of the algebraic torus $T$ and the Galois group
$G= {\rm Gal} (E/F)$.
2. The lattice of $E$-rational characters of $T$, which we denote by $M$
and its dual lattice $N$.
3. A $G$-invariant complete
fan $ \Sigma $ in $N_{{\bf R }}=N\otimes {\bf R }$.
There is an isomorphism between the group of
$G$-invariant integral piecewise linear functions $\varphi\in PL( \Sigma )^G$
and classes of $T$-linearized line bundles on $X_{ \Sigma }$.
For $\varphi\in PL( \Sigma )^G$
we denote the corresponding line bundle by $L(\varphi)$.
We define metrizations of line bundles as follows.
Let $G_v\subset G$ be the decomposition group at $v$.
We put $N_v=N^{G_v}$
for the lattice of $G_v$-invariants of $N$
for non-archimedean valuations $v$
and $N_v=N^{G_v}_{{\bf R }}$ for archimedean $v$.
We have the logarithmic map
$$
T(F_v)/T({\cal O}_v)\rightarrow N_v
$$
which is an embedding of finite index for
all non-archimedean $v$, an isomorphism of lattices for
almost all non-archimedean valuations
and an isomorphism of real vector spaces for archimedean valuations.
We denote by $\overline{t}_v$ the image of $t_v\in T(F_v)$ under
this map.
\begin{dfn}(\cite{BaTschi1}, p. 607)
{\rm
For every $\varphi\in PL( \Sigma )^G$ and
$t_v\in T(F_v)$ we define the local height
function
$$
H_{ \Sigma ,v}(t_v,\varphi):=e^{\varphi(\overline{t}_v)\log q_v}
$$
where $q_v$ is the cardinality of the residue field of $F_v$ for
non-archimedean valuations and $\log q_v=1 $ for archimedean
valuations. For $t\in T({\bf A}_F)$ we define
the global height function as
$$
H_{ \Sigma }(t,\varphi):=\prod_{v\in {\rm Val}(F)}H_{ \Sigma ,v}(t_v,\varphi).
$$
}
\end{dfn}
\noindent
We proved in (\cite{BaTschi1}, p. 608) that this pairing
can be extended to a pairing
$$
H_{ \Sigma }\,:\, T({\bf A}_F)\times PL( \Sigma )^G_{{\bf C }}\rightarrow {\bf C }
$$
and that it
defines a simultaneous metrization of $T$-linearized line bundles
on $X$. We will denote such metrized line bundles
by ${\cal L}={\cal L}(\varphi)$.
We consider the height zeta function
$$
Z(T,{\cal L},s)=\sum_{t\in T(F)}H_{\cal L}(t)^{-s}.
$$
\begin{theo}(\cite{BaTschi3})
Let ${\cal L}$ be an invertible sheaf on $X$
(with the metrization introduced above) such that its class
$[L]$ is contained in the interior of $ \Lambda _{\rm eff}(X)$.
There exist an $ \varepsilon ({\cal L})>0$ and a $\Theta({\cal L})>0$
such that the height zeta function
has the following representation
$$
Z({\cal L},T,S)=\frac{\Theta({\cal L})}{
(s- \alpha _{\cal L}(X))^{ \beta _{\cal L}(X)}}+
\frac{f(s)}{(s- \alpha _{\cal L}(X))^{ \beta _{\cal L}(X)-1}}
$$
where $f(s)$ is a function which is holomorphic for
${\rm Re}(s)> \alpha _{\cal L}(X)- \varepsilon ({\cal L})$.
\end{theo}
\begin{rem}{\rm
The computation of the constants
$ \alpha _{\cal L}(X)$ and $ \beta _{\cal L}(X)$
in specific examples is
a problem in linear programming.
For the anticanonical line
bundle on a smooth toric variety $X$ we have
$ \alpha _{{\cal K}^{-1}}(X) =1$ and
$ \beta _{{\cal K}^{-1}}(X) =
\dim PL( \Sigma )^G_{{\bf R }}-\dim M^G_{{\bf R }}$.
}
\end{rem}
Our goal is to identify the constant $\Theta({\cal L})$.
Let us recall some properties of toric
varieties and introduce more notations (see \cite{BaTschi1}).
The cone of effective divisors $ \Lambda _{\rm eff}(X)$ is generated by
the classes of irreducible components of $X\backslash T$ which we
denote by $[D_1],...,[D_r]$. These divisors correspond to
Galois orbits $ \Sigma _1(1),..., \Sigma _r(1)$
on the set of $1$-dimensional cones in $ \Sigma $.
The line bundle ${\cal L}$ defines a face $ \Lambda ({\cal L})$ of $ \Lambda _{\rm eff}(X)$.
We denote by $J=J({\cal L})$ the
maximal set of indices $J\subset [1,...,r]$
such that we have
$$
\alpha _{\cal L}(X)[L]+[K_X]=\sum_{j\in J}r_j[D_j]
$$
with $[D_j]\in \Lambda ({\cal L})$ and some $r_j\in { \bf Q }_{>0}$.
We denote by $I=I({\cal L})$ the set of indices
$i\not\in J({\cal L})$.
We denote by $M_J$ the lattice given by
$$
M_J:=\{ m\in M\,|\, <e,m>=0\hskip 0,3cm
{\bf R }_{\ge 0}e\in \cup_{i\in I } \Sigma _i(1) \},
$$
We denote by $M_I:=M/M_J$ and by $N_*$ the corresponding dual
lattices. We have an exact sequence of
algebraic tori
$$
1\rightarrow T_I\rightarrow T\rightarrow T_J\rightarrow 1
$$
which induces a map
$\pi_{\cal L}\,:\, T(F)\rightarrow T_J(F)$
with finite cokernel
and an exact sequence of lattices
$$
0\rightarrow N_I\rightarrow N\rightarrow N_J\rightarrow 0.
$$
The restriction of the fan $ \Sigma \subset N_{{\bf R }}$ to
$N_{I,{\bf R }}$ will be denoted by $ \Sigma _I$. It is again
a $G$-invariant fan and it
will define an equivariant compactification $X_I$ of $T_I$.
The class of the piecewise linear function $\varphi_I\in PL( \Sigma _I)^G$ with
$\varphi_I(e)=1$ for $e\in \cup_{i\in I} \Sigma _i$ corresponds to the class of
the anticanonical line bundle $[-K_I]\in {\rm Pic}(X_I)$.
The line bundle ${\cal L}$ defines a fibration of
varieties $\pi_{\cal L}\,:\, X_{ \Sigma }\rightarrow Y$ with fibers isomorphic to $X_I$,
which, when restricted to $T$,
gives rise to the exact sequence of tori above. We denote the
fiber over $y\in T_J(F)$ by $X_{I,y}$.
\begin{theo} We have
$$
\Theta({\cal L})= \gamma_{\cal L}(X)
\delta _{\cal L}(X)\sum_{y\in \pi_{\cal L}(T(F))}\tau_{\cal L}(X_{I,y}).
$$
\end{theo}
{\em Proof.} Let ${\cal L}={\cal L}(\varphi)$ be
an invertible sheaf on $T$ with the metrization introduced above.
In the domain of absolute and uniform convergence
we can rearrange the order of summation
and we obtain
$$
Z(T,{\cal L},s)=\sum_{y\in \pi_{\cal L}(T(F))}
\sum_{x\in T_{I,y}(F)} H_{ \Sigma }(yx,\varphi)^{-s}.
$$
From the proof of our main theorem in \cite{BaTschi3}
it follows that the sums
$$
Z(T_{I,y},{\cal L},s)=\sum_{x\in T_{I}(F)} H_{ \Sigma }(yx,\varphi)^{-s}
$$
have a pole at $ \alpha _{\cal L}(X)$ of order
$ \beta _{\cal L}(X)$.
Moreover, the constant $\Theta({\cal L})$ is obtained
as a sum over $y\in \pi_{\cal L}(T(F))$ of residues of
$Z(T_{I,y},{\cal L},s)$.
Now we want to use the Tamagawa number
conjecture for the anticanonical line bundle
(with varying metrizations) on the
toric variety $X_I$ to conclude the proof.
In \cite{BaTschi2} we proved this conjecture
for a specific metrization and under
the assumption that the fan $ \Sigma $ is {\em regular}.
We want to demonstrate that our proof goes through in the general case
needed above.
\smallskip
Our main idea was to use the Poisson summation formula on the
adelic group $T({\bf A}_F)$ and to obtain an integral
representations for the height zeta function.
We denote by
${\cal A}_I=(T_I({\bf A}_F)/{\bf K}T_I(F))^*$ the group of
unitary characters of $T_I({\bf A}_F)$ which are
trivial on $T_I(F)$ and on the maximal compact subgroup
${\bf K}\subset T_I({\bf A}_F)$. Using the
the adelic definition of the height
function we obtain
$$
Z(T_{I,y},{\cal L},s)=\sum_{t\in T_I(F)}H_{\cal L}(yt)^{-s}
=\int_{{\cal A}_I}d\chi \int_{T_I({\bf A}_F)} H_{\cal L}(yt)^{-s}\chi(t)d\mu,
$$
where $d\mu$ is a Haar measure
on $T({\bf A}_F)$ and $d\chi$ is the orthogonal
Haar measure on ${\cal A}_I$.
To apply our technical theorem in \cite{BaTschi2} about the
analytic continuation and the residues of such integrals
we need to know that
$$
\int_{T_I({\bf A}_F)} H_{\cal L}(yt)^{-s}\chi(t)d\mu =\prod_{i\in I}
L_i(\chi_i,s) \cdot \zeta_{ \Sigma _I}(\chi,s)\cdot \zeta_{\infty}(\chi,s)
$$
where
$$
\zeta_{ \Sigma _I}(\chi,s)=\prod_{v\in {\rm Val}(F)}\zeta_{ \Sigma _I,v}(\chi,s)
$$
is an absolutely convergent
Euler product for ${\rm Re}(s)> \alpha _{\cal L(X)}- \varepsilon ({\cal L})$,
$L_i(\chi_i,s)$ are Artin $L$-functions
(with some induced characters $\chi_i$)
and $\zeta_{\infty}(\chi,s)$ satisfies certain growth conditions.
First we observe that it is unnecessary to assume that
the fan $ \Sigma _I$ is regular. Using our definition of the height function
we see that the calculation of the Fourier transform (see \cite{BaTschi1})
reduces to summations of the function
$q_v^{{\varphi}(\overline{t_v}) +im_v}$
($m_v\in M_{I,{\bf R }}$) over the lattice $N_{I,v}$
(resp. to integrations in cones
for archimedean valuation). A piecewise linear function
$\varphi$ induces a piecewise linear function
on any subdivision of the fan. Clearly, the result of such summations
and integrations does not depend on any subdivisions.
Next we see that for a fixed $y\in T_J(F)$ we have
$H_{{\cal L},v}(yt)=H_{{\cal L},v}(t)$ for almost
all $v$ and all $t\in T_I({\bf A}_F)$.
Now we can refer to lemma 5.10 in \cite{BaTschi3} which proves
the required statement.
The local integrals
for the remaining finitely many non-archimedean valuations
will be absorbed into $\zeta_{ \Sigma _I}(\chi,s)$.
And finally, we need to check that the estimates
of the Fourier transform of $H_{{\cal L},v}(yt)$
at archimedean valuations are still satisfied for any $y\in T_J(F)$.
This is straightforward.
We can now apply the main technical theorem of \cite{BaTschi2}
and obtain
$$
\lim_{s\rightarrow \alpha _{\cal L}}(s- \alpha _{\cal L}(X))^{
\beta _{\cal L}(X)}Z(T_{I,y},{\cal L},s)
= \gamma_{\cal L}(X) \delta_{\cal L}(X)\tau_{\cal L}(X_{I,y}).
$$
\hfill $\Box$
\begin{rem}
{\rm
It is possible to compute $\tau_{\cal L}(X_{I,y})$ and to observe
that it is related to the height of the point $y\in T_J(F)$.
}
\end{rem}
\begin{rem}
{\rm
Similar statements hold for equivariant compactifications
of homogeneous spaces $G/U$ where $G$ is a split
reductive group and $U$ is its maximal
unipotent subgroup \cite{StrTschi}.
We hope that these results can
be extended to equivariant
compactifications of other homogeneous spaces, in particular,
to equivariant compactifications of reductive and non-reductive groups.
}
\end{rem}
\section{Singular Fano varieties}
\subsection{Weighted projective spaces}
Let $W:= {\bf P}(w) = {\bf P}(w_0, \ldots, w_n)$ be a weighted
projective space of dimension $n$ with weights
$w = (w_0, \ldots, w_n)$. We remark that $W$ is a rational variety
over ${\bf Q}$
with ${\rm Pic}(W) \cong {\bf Z}$. Moreover, the anticanonical
class $K_W^{-1}$ is an ample ${\bf Q}$-Cartier divisor. So $W$
is a (singular) Fano variety of index
\[ r = \frac{ w_0 + \cdots + w_n} { l.c.m.\{ w_0, \ldots , w_n \}}. \]
One could try to generalize the method of Schanuel \cite{schan}
for counting ${\bf Q}$-rational points of bounded height on
usual projective spaces to the case of weighted projective spaces.
Let $z_0, z_1, \ldots, z_n$ be homogeneous coordinates on
$Y$. Then a first approximation to counting points of
bounded height would be a counting of
all $(n+1)$-tuples
$(x_0, x_1, \ldots, x_n) \in {\bf Z}^n \setminus \{0 \}$
satisfying the conditions
\[ |x_i| \leq
B^{\frac{w_i}{w_0 + w_1 + \cdots + w_n}}\;\; i =0, \ldots, n. \]
Since the volume of the domain restricted by these inequalities is
\[ B = \prod_{i =0}^n B^{\frac{w_i}{w_0 + w_1 + \cdots + w_n}}, \]
one could expect that the asymptotic number
of solutions of these inequalities
agrees with ``expected''
linear growth for the anticanonical height.
However, this ``intuition''
turns out to be wrong, in general, because
the singularities of $Y$ could be even worse
than canonical.
A typical class of singularities that appear on $Y$ are
so called log-terminal
singularities introduced by Kawamata \cite{Ka}.
We give below a simple example of a Del Pezzo surface
with a log-terminal singularity
and we show that for every dense Zariski open subsets $U \subset W$ the
number $N(U,B)$ of $F$-rational points of anticanonical height $\leq B$
in $U$ has more than linear growth:
\[ N(U, B) = c(U)B^{2 - \frac{4}{m+2}}(1 + o(B)). \]
Moreover, there are no
dense Zariski open subsets
$U' \subset X$ such that the adelic
term in the constant $c(U)$
in the asymptotic formula for $N(U,B)$
would be independent of
$U$ for $U \subset U'$.
\begin{exam}
{\rm {\sc (Del-Pezzo surface with a log-terminal singularity)}
Let $W = {\bf P}(1,1,m)$ be a singular weighted projective plane
with weights $(1,1,m)$, $m \geq 2$. Then the anticanonical class of
$W$ is an ample ${\bf Q}$-Cartier divisor (i.e., $W$ is Del Pezzo
surface) and $p = (0:0:1)$ is
the unique singularity of $W$.
Let $X \rightarrow W$ be the minimal resolution
of the singularity at $p \in W$. Then $X$ is isomorphic
to a ruled surface
${\bf F}_m = {\bf P}({\cal O}_{{\bf P}^1} \oplus
{\cal O}_{{\bf P}^1}(m))$ and the exceptional divisor $E =
f^{-1}(p)$ is a smooth rational curve which is a section
of the ${\bf P}^1$-bundle over ${\bf P}^1$ and
$\langle E, E \rangle = -m$. Then we have
\[ K_X = f^*K_W + \frac{2-m}{m} E. \]
Therefore, $p$ is canonical $\Leftrightarrow$
$m = 2$ and $p$ is log-terminal $\Leftrightarrow$ $m \geq 3$.
The group ${\rm Pic}(X)$ is isomorphic to ${\bf Z}^2$ where $\{ [E],
[C] \}$ is a ${\bf Z}$-basis. Moreover, $ [E], [C]$
are generators for the cone $\Lambda_{\rm eff}(X)$
of effective divisors in ${\rm Pic}(X)_{\bf R}$. We have
\[ K_X = -2E - (m+2 ) C, \]
\[ L := f^*(-K_X) = \frac{m + 2}{m}E + (m + 2)C \]
and
\[ {\alpha}_L(W) = \inf \{ t \in {\bf Q} \; : \; t[L] +[K_X]
\in \Lambda_{\rm eff}(X) \} =
\frac{2m}{m+2}. \]
}
\end{exam}
Since $X$ is a smooth toric variety, we can apply our main result
in \cite{BaTschi3} and obtain the following:
\begin{theo}
Let $\pi: W \setminus p \rightarrow {\bf P}^1$ be the natural projection,
$C_x$ the fiber of $\pi$ over $x \in {\bf P}^1({\bf Q})$. Then
for any dense Zariski open subset in $W \setminus
p$, one has
\[ N(U,B) = c(U)B^{2 - \frac{4}{m+2}}(1 + o(B)) \]
Moreover,
\[ c(U) = \sum_{x \in {\bf P}^1({\bf Q})\cap \pi(U)} c(C_x). \]
\end{theo}
\subsection{Vaughan-Wooley cubic}
\begin{exam}
{\rm
Let $Y \subset {\bf P}^5$ be a singular cubic defined by the
equation $z_0z_1z_2 - z_3z_4z_5 = 0$, $X$ the intersection of
$Y$ with the linear subspace in ${\bf P}^5$ with the equation:
\[ z_0 + z_1 + z_2 - z_3 - z_4 - z_5 = 0. \]
Vaughan and Wooley proved \cite{VW}:
\begin{theo}
Let $U \subset X$ be the complement in $X$
to the following $15$ divisors
$D_{i_1 i_2 i_3}, D_{ij}
\subset X$ $(\{ i_1, i_2, i_3 \} = \{ 0,1,2 \}, \;
i \in \{ 0,1,2 \},\; j \in \{ 3,4,5\})$, where
\[ D_{i_1 i_2 i_3} = \{ (z_0: \ldots : z_5) \in {\bf P}^5\; : \;
z_{i_1} = z_3,
\, z_{i_2} = z_4, \, z_{i_3} = z_5 \} \]
\[ D_{ij} = \{ (z_0: \ldots : z_5) \in X\; : \;
z_i = z_j = 0 \}.\]
If $N(U,B)$ is the number of ${\bf Q}$-rational points
in $U$ of the anticanonical height $\leq B$, then
there exist some constants $c_1 > c_2 > 0$ such that
\[ c_2 B^2 (\log B)^5 \leq N(U,B) \leq c_1 B^2 (\log B)^5. \]
\end{theo}
We want to show that this result is compatible with predictions
in \cite{BaMa}. First of all we note that $Y$ is a
$4$-dimensional toric Fano
variety: an equivariant compactification of a $4$-dimensional algebraic
torus $T$ with respect to a $4$-dimensional polyhedron $\Delta$
with $6$ lattice vertices
\[ v_0 = (0,0,0,0), \, v_1 = (1,0,0,0), \, v_2 = (0,1,0,0), \]
\[ v_3 = (0,0,1,0), \, v_4 = (0,0,0,1), \, v_5 = (1,1,-1,-1) \]
($\Delta$ is the support of global
sections of a very ample divisor $Y$ corresponding to the embedding
$Y \hookrightarrow {\bf P}^4$).
The polyhedron $\Delta$ has $9$ faces $\Theta_{ij}$ of codimension $1$:
\[ \Theta_{ij} = {\rm Conv}(\{ v_0,v_1,v_2,v_3, v_4,v_5 \} \setminus
\{v_i, v_j \} ). \]
Each face $\Theta_{ij}$ defines an torus invariant divisor $Y_{ij} \subset
Y \setminus T$
such that $D_{ij} = Y_{ij} \cap X$.
It is easy to check that all singularities of $Y$ are at worst terminal and
that the hypersurface
$X \subset Y$ intersects all strata $Y_{ij}$ transversally.
From these facts we obtain that the only
exceptional divisors with the discrepancy $0$
that appear in a resolution of singularities of $X$ come from
singularities in $X \cap T$. We write down the affine equation of $X
\cap T$ as
\[ 1 + x + y - z -t - \frac{xy}{zt}, \]
where $T = {\rm Spec}\, {\bf Q}[x^{\pm 1}, y^{\pm 1},z^{\pm 1}, t^{\pm 1}]$.
From this equation one immediately
sees that the only singularities
in $X \cap T$ are the $A_1$-double points
lying on the curve $C\;:\; x=y=z=t$.
Therefore, we obtain ${\rm rk} \, {\rm Cl}(X) = {\rm rk}\,
{\rm Cl}(Y) = 9 - {\rm dim}\,T = 5$.
Moreover, there exists exactly one crepant divisor (over $C$) in a resolution
of singularities of $X$. So the predicted power of $\log B$ in the asymptotic
formula for $N(U, B)$ is $({\rm rk}\, {\rm Cl}(X) -1) + 1 = 5$.
}
\end{exam}
\subsection{Cubic $xyz=u^3$}
We consider the singular cubic surface $X\subset { \bf P }^3$ over ${ \bf Q }$
given by the homogeneous equation $xyz=u^3$.
This is a toric variety, an
equivariant compactification of the torus
$$
T = {\rm Spec}
{\bf Q}[x,y,z]/(xyz - 1)
$$
given by the condition $u\neq 0$. We can fix an
isomorphism $T \cong {\bf G}_m^2$ by choosing $\{ x, y\} $ as
a basis of the group of algebraic characters of $T$.
Consider the problem of the computation of
the asymptotic of
$$
N(T,B) = \mbox{\rm Card}\{ (x,y) \in ({ \bf Q }^*)^2\,\,:\,\,
H(x,y) \le B\,\,\}
$$
for $B\rightarrow \infty$,
where
\[ H(x,y) = \prod_{v \in {\rm Val}({\bf Q})}
\max \{\|x\|_v,\|y\|_v,\|(xy)^{-1}\|_v,\|1\|_v \} \]
This problem is addressed in \cite{fouvry}.
We would like to use this problem as a down-to-earth illustration of
our general theory of
height zeta functions of toric varieties.
First of all we note that the relation $\|x\|_v \|y\|_v \|(xy)^{-1}\|_v =
\| 1 \|_v =1$ implies that
$$
H_v(x,y): =
\max \{\|x\|_v,\|y\|_v,\|(xy)^{-1}\|_v\}.
$$
Since $l_1 =\log \|x\|_v$, $l_2 = \log \|y\|_v$, $l_3 = \log
\|(xy)^{-1}\|_v$ are linear functions
on the logarithmic space $N_{{\bf R},v} \cong {\bf R}^2$:
$$
N_{{\bf R},v} =
\left\{ \begin{array}{ll} T({\bf Q}_v) /T({\cal O}_v) \otimes_{\bf Z}
{\bf R},
& \mbox{\rm if $v = p \in {\rm Spec}\,{\bf Z}$} \\
T({\bf Q}_v) /T({\cal O}_v),
& \mbox{\rm if $v = \infty$,}
\end{array} \right.
$$
we can consider $\log h_v(x,y)$ as a piecewise linear function on
$N_{{\bf R},v}$. Let $e_1 = (-2,1)$, $e_2 = (1,-2)$ and
$e_3 = (1,1)$ be lattice vectors in ${\bf Z}^2 \subset {\bf R}^2$.
We define the following $3$ convex cones in ${\bf R}^2$:
\[ \sigma_1 = {\bf R}_{\geq 0} e_2 + {\bf R}_{\geq 0} e_3, \]
\[ \sigma_2 = {\bf R}_{\geq 0} e_1 + {\bf R}_{\geq 0} e_3, \]
\[ \sigma_3 = {\bf R}_{\geq 0} e_1 + {\bf R}_{\geq 0} e_2. \]
Then $N_{{\bf R},v} = \bigcup_{i =1}^3 \sigma_i$ and the restriction
of $\log H_v(x,y)$ to $\sigma_i$ coincides with the linear function $l_i$.
Let
\[ A := \bigoplus_{v \in {\rm Val}({\bf Q}}
T({\bf Q}_v) /T({\cal O}_v) \]
be the logarithmic adelic group.
In order to compute the height zeta function
\[ Z(s) = \sum_{(x,y) \in {({\bf Q}^*)^2 = T({\bf Q})}} H(x,y)^{-s} \]
we use the natural homomorphism $Log$ of $T({\bf Q})$
to $A$. Denote by $B$ the subgroup $Log(T({\bf Q})) \subset A$.
We remark that the kernel of $Log$ consists of $4$ elements of
finite order in $({\bf Q}^*)^2$ and the quotient
$A/B$ is isomorphic to ${\bf R}^2$. Moreover the functions
$H_v(x,y)^{-s}$ on each $T({\bf Q}_v) /T({\cal O}_v)$ define a natural
extension of $h(x,y)^{-s}$ to a function on $A$. So we obtain:
\begin{equation}
Z(s) = 4 \sum_{ b \in B} \prod_{v \in {\rm Val}({\bf Q})} H_v(b_v)^{-s}
\label{poiss1}
\end{equation}
The main idea of our proof in
\cite{BaTschi1} is to apply the Poisson summation
formula on the group $A$ and to express the
height zeta function $Z(s)$ as an integral
$$
Z(s)=\frac{4}{(2\pi )^2}
\int_{{{\bf R }}^2} \left(
\prod_{p} Q_p(s,i{\bf m})\cdot Q_{\infty}(s,i{\bf m}) \right)
{\bf d}{\bf m},
$$
where ${\bf m} =(m_1, m_2) \in {\bf R}^2$, ${\bf dm} = dm_1dm_2$,
$$
Q_{p}(s,i{\bf m}) = \sum_{(b_{1,p},b_{2,p}) \in {\bf Z}^2}
h_p(b_p)^{-s} p^{ i<b, {\bf m}>},
$$
and
$$
Q_{\infty}(s,i{\bf m}) = \int_{{\bf R}^2} H_{\infty}(b)^{-s}
\exp ( i<b, {\bf m}> ).
$$
An exact computation of $Q_{p}(s,i{\bf m})$ and
$Q_{\infty}(s,i{\bf m})$ can be obtained by a subdivision
of each of the cones $\sigma_1, \sigma_2, \sigma_3$ into
a union of $3$ subcones generated by a basis
of the lattice ${\bf Z}^2 \subset {\bf R}^2$.
(From the viewpoint of toric geometry this means that we
reduce the
counting problem for rational points in a torus
with respect to a singular compactification
to
a counting problem for rational points in a torus
with respect to the minimal resolution of
singularities of this compactification).
This calculation is done
in \cite{BaTschi1}, Section 2 for arbitrary smooth toric varieties.
What remains is the analytic continuation of the integral
and the identification of the constant at the leading pole.
For this it is necessary to work
on the whole complexified space $PL( \Sigma )_{{\bf C }}$ and to invoke the
technical theorems in Section 6 in \cite{BaTschi2}.
Applying the main theorem
of \cite{BaTschi2}, we obtain
$$
N(T,B)=
\frac{
\gamma_{{\cal K}^{-1}}(X)
\delta_{{\cal K}^{-1}}(X) \tau_{{\cal K}^{-1}}(X)}{6!}
B (\log B)^6(1+o(1))
$$
for $B\rightarrow \infty$. The constants are as follows:
$ \gamma_{{\cal K}^{-1}} (X)=1/36$, $\delta(X)=1$
and $\tau_{{\cal K}^{-1}}(X)=\tau_{{\cal K}^{-1}}(X)_{\infty}
\prod_p\tau_{{\cal K}^{-1}}(X)_p$
where
$$
\tau_{{\cal K}^{-1}}(X)_p=
\left(1 + \frac{7}{p} + \frac{1}{p^2} \right)\cdot (1-1/p)^{7}
$$
for all primes $p$ and $\tau_{{\cal K}^{-1}}(X)_{\infty}=9\cdot 4$.
Similar statements hold over any number field.
One can compute the constant
$\gamma_{{\cal K}^{-1}}(X)$ by observing that
$\Lambda_{\rm eff}^*(X)$ (the dual cone to the cone of
effective divisors) is a union of two
simplicial cones.
|
1997-12-10T12:01:51 | 9712 | alg-geom/9712010 | fr | https://arxiv.org/abs/alg-geom/9712010 | [
"alg-geom",
"math.AG"
] | alg-geom/9712010 | Francois Ducrot | Francois Ducrot | Structures du cube et fibres d'intersection | 37 pages, latex2e with XYPic | null | null | UPRESA6093/46 | null | We define the notion of a hypercube structure on a functor between two
strictly commutative Picard categories which generalizes the notion of a cube
structure on a $G_m$-torsor over an abelian scheme. We use this notion to
define the intersection bundle of $n+1$ line bundles on a relative scheme $X/S$
of relative dimension $n$ and to construct an additive structure on the functor
$I_{X/S}:PIC(X/S)^{n+1}\F PIC(S)$. Finally, we study a section of
$I_{X/S}(L_1,...,L_{n+1})$ which generalizes the resultant of $n+1$ polynomials
in $n$ variables and we interprete some classical formulas with this formalism.
| [
{
"version": "v1",
"created": "Wed, 10 Dec 1997 11:01:51 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Ducrot",
"Francois",
""
]
] | alg-geom | \section{Introduction}
\label{a}
\subsection{}
Soit $G$ un groupe alg{\'e}brique commutatif et $L$ un $G_m$-torseur sur $G$
, le classique th\'eor\`eme du cube
(\cite{M3},p58) affirme
que le $G_m$-torseur suivant sur $G^{3}$ est trivial
\begin{equation}
\label{cub1}
\t (L) \; = \; m^{\ast }L \wedge
(m_{12}^{\ast }L)^{-1} \wedge (m_{23}^{\ast }L)^{-1} \wedge (m_{31}^{\ast
}L)^{-1} \wedge p_{1}^{\ast }L \wedge p_{2}^{\ast }L \wedge p_{3}^{\ast }L
\end{equation}
o\`u $m$ , $m_{ij}$ , $p_{i}$ sont les applications $G\times G
\times G \longrightarrow G $ d\'efinies par
$
m(x_{1},x_{2},x_{3})=x_{i}+x_{j}+x_{k}
$,
$
m_{ij}(x_{1},x_{2},x_{3})=x_{i}+x_{j}
$
et
$
p_{i}(x_{1},x_{2},x_{3})=x_{i}
$.
\subsection{} En r{\'e}alit{\'e} si $G$ est une vari{\'e}t{\'e} ab{\'e}lienne, une
trivialisation du
$G_m$-torseur $\t (L)$ devra v{\'e}rifier des conditions de compatibilit{\'e}
qui s'expriment en termes de biextensions (notion introduite par
\textsc{Mumford} pour l'{\'e}tude
des groupes formels et {\'e}tudi{\'e}e
par \textsc{Grothendieck} dans un cadre plus g{\'e}n{\'e}ral).
Consid{\'e}rons en effet le $G_m$-torseur $\L =\L (L)$ sur $G \times G$ d{\'e}fini
par:
\[
\L (L) = m^{\ast }L \wedge ( p_1^{\ast }L)^{-1} \wedge ( p_2^{\ast
}L)^{-1}
\]
Une trivialisation $t$ de $\t (l)$ induit deux lois (afin d'all{\'e}ger les
notations on consid{\'e}rera les fibres de $\L$ au dessus d'un point
g{\'e}n{\'e}rique de $G\times G$):
\[
\begin{array}{lcl}
*_1 : \L _{x,y} \wedge \L _{x,z}& \longrightarrow &\L _{x,y+z}\\
*_2 : \L _{x,z} \wedge \L _{y,z}& \longrightarrow &\L _{x+y,z}
\end{array}
\]
Ces deux lois v{\'e}rifient alors des propri{\'e}t{\'e}s
d'associativit{\'e}, de
commutativit{\'e} (que le lecteur pourra {\'e}crire sans
difficult{\'e}) et et de
compatibilit{\'e} entre elles (qui traduit l'{\'e}galit{\'e} des deux
fa\c{c}ons de d{\'e}velopper $ \L _{x+y,x^{\prime} + y^{\prime} }$ ).
On dit alors que $\L$ est une biextension de $G\times G$ par $G_m$.\\
De plus $\L $ est muni d'un isomorphisme de sym{\'e}trie $s: \tau ^{\ast } \L
\simeq \L$ ; on parle alors de biextension sym{\'e}trique.
\subsection{} La notion de structure du cube, introduite par \textsc{Breen}
dans \cite{B2},
explicite les notions pr{\'e}c{\'e}dentes. Une struture du cube sur un
$G_m$-torseur $L$ sur $G$ est la donn{\'e}e d'une trivialisation $t$ du
$G_m$-torseur $\t (L)$ sur $G^3$ telle que les lois partielles $\ast
_1$ et $\ast _2$ induites par $t$ sur $\L (L)$ font de $\L (L)$ une
biextension sym{\'e}trique.
\subsection{} Si $\pi : X \longrightarrow S $ est une courbe relative lisse et $J$
d{\'e}signe la composante de degr{\'e} $0$ du sch{\'e}ma de Picard relatif
$\text{PIC} (X/S)$, \textsc{Moret-Bailly} d{\'e}duit de l'existence d'une
structure
de biextension sur le faisceau de Poincar{\'e} $\mathcal{P}$ sur $J \times
J^{\vee }$ et de l'existence d'une polarisation canonique sur $J$
l'existence d'une biextension $\mathcal{B}$ canonique de $J\times J$
par $G_m$ et il montre ensuite l'existence d'isomorphismes canoniques
\begin{equation} \label{pairing}
\mathcal{B} _{\text{cl} (L) ,\text{cl} (M)}
\simeq
\det \text{R}\pi \lst (L\otimes _{\ox} M)^{-1} \otimes _{\os} \det \text{R}\pi \lst (L) \otimes _{\os} \det \text{R}\pi \lst (M) \otimes _{\os} \det \text{R}\pi \lst (\mathcal{O}_{X} )
^{-1}
\end{equation}
\subsection{}\textsc{Deligne} propose dans \cite{d3} un programme dont une
{\'e}tape est la construction pour tout morphisme projectif plat de
dimension $d$ : $\pi : X \longrightarrow S$, du fibr{\'e} d'intersection relativement
{\`a} $S$ , de $d+1$ faisceaux inversibles $L_0 ,
\cdots , L_d$ sur $X$. Il s'agit de construire un $\mathcal{O}_{S}$-module inversible
$ I_{X/S} (L_1 , \cdots , L_{n+1})$ , dont la construction est
fonctorielle en les faisceaux $L_i$ (pour les isomorphismes de
faisceaux) et compatible aux changements de base et qui est
multiplicatif en les faisceaux $L_i$. De fa\c{c}on pr{\'e}cise, on veut
construire un syst{\`e}me d'isomorphismes:
\begin{multline}
I_{X/S} (L_1 , \cdots , L_i \otimes _{\ox} L_i ^{\prime} , \cdots , L_{n+1})
\simeq \\
I_{X/S} (L_1 , \cdots , L_i , \cdots , L_{n+1})
\otimes _{\os}
I_{X/S} (L_1 , \cdots , L_i ^{\prime} , \cdots , L_{n+1})
\end{multline}
munis de donn{\'e}es de commutativit{\'e}, d'associativit{\'e} et de
compatibilit{\'e}
entre ces diff{\'e}rentes lois partielles. Un tel faisceau d'intersection
est construit par \textsc{Deligne} dans \cite{De} dans le cas d'une courbe
relative lisse et cette m{\'e}thode est {\'e}tendue par \textsc{Elkik} dans
\cite{elkik1} au cas des
morphismes plats de Cohen-Macaulay purement de dimension $n$.\\
\textsc{Moret-Bailly} propose une autre m{\'e}thode dans \cite{MB}
dans le cas des courbes lisses, construisant le
faisceau d'intersection par la formule (\ref{pairing}), bas{\'e}e sur
le fibr{\'e}
d{\'e}terminant, et montrant ensuite la multiplicativit{\'e} de cette
construction en utilisant la propri{\'e}t{\'e} d'autodualit{\'e} de la
jacobienne. Il explique ensuite comment {\'e}tendre cette construction au
cas de courbes de Cohen-Macaulay.\\
Enfin une autre construction est propos{\'e}e par \textsc{Deligne} dans
\cite{d3}, bas{\'e}e sur des symboles. Cette id{\'e}e est utilis{\'e}e par
\textsc{Aitken} \cite{a} dans le
cas d'une courbe singuli{\`e}re quelconque sur une base r{\'e}duite.
\subsection{}
Dans ce travail on introduit une notion de structure du $p$-cube sur
un foncteur entre deux cat{\'e}gories de Picard strictement commutatives,
qui {\'e}tend les d{\'e}finitions de \textsc{Breen} {\`a} un
cadre l{\'e}g{\`e}rement plus g{\'e}n{\'e}ral (pour $p=3$, si $G$ est un groupe
alg{\'e}brique commutatif sur un corps $k$ et $L$ est un $G_m$-torseur
sur $G$, une structure du 3-cube sur le
foncteur $\gd :G\longrightarrow\text{Vect}_k,g\mapsto L_g$ coincide avec la
d{\'e}finition, donn{\'e}e par \textsc{Breen}, d'une structure du cube sur
$L$). La donn{\'e}e d'une structure du $p$-cube sur un foncteur $\gd
:\mathcal{C}\longrightarrow\mathcal{D}$ entre cat{\'e}gories de Picard strictement commutatives munit
la "dif{\'e}rence sym{\'e}trique $(p-1)$-i{\`e}me
de $\gd$", qui est un foncteur $\mathcal{C} ^{p-1}\longrightarrow \mathcal{D}$, de donn{\'e}es
d'additivit{\'e} en chaque variables.
On applique ces notions au cas $X/S$ est un sch{\'e}ma relatif projectif
de dimension $n$ quelconque sur une base localement noeth{\'e}rienne
et $\gd$ est le foncteur d{\'e}terminant de l'image directe d{\'e}riv{\'e}e
$PIC (X/S)\longrightarrow PIC (S)$. Le th{\'e}or{\`e}me principal de ce travail montre
alors alors l'existence d'une structure du $(n+2)$-cube canonique sur
$\gd$. Sa d{\'e}monstration est bas{\'e}e sur une r{\'e}currence sur
la dimension des sch{\'e}mas consid{\'e}r{\'e}s, qui construit une
structure du $(n+2)$-cube sur le foncteur d{\'e}terminant
associ{\'e} {\`a} un sch{\'e}ma relatif de dimension $n$ {\`a}
partir de la donn{\'e}e de structures du $(n+1)$-cube sur le foncteur
d{\'e}terminant de tout sous-sch{\'e}ma relatif de $X$ de dimension $n-1$. Une
telle r{\'e}currence impose donc, m{\^e}me si on veut montrer le r{\'e}sultat
uniquement pour un
sch{\'e}ma relatif lisse, de savoir le montrer pour des sous-sch{\'e}mas qui
n'ont aucune raison d'{\^e}tre lisses ou m{\^e}me simplement
r{\'e}duits et donc de travailler avec des sch{\'e}mas $X/S$ assez
g{\'e}n{\'e}raux.
Ceci nous permet de montrer que le fibr{\'e} d'intersection
$I_{X/S}:PIC (X/S)^{n+1}\longrightarrow PIC (S)$, d{\'e}fini, en
suivant la m{\'e}thode de \textsc{Moret-Bailly}, comme la
diff{\'e}rence sym{\'e}trique $(n+1)$-i{\`e}me du foncteur
d{\'e}terminant est bien muni de donn{\'e}es
d'additivit{\'e} en chaque variable. On {\'e}tudie alors une section
canonique de $I_{X/S}(L_1,\cdots ,L_n)$ qui g{\'e}n{\'e}ralise la notion de
r{\'e}sultant de $n$ polyn{\^o}mes.
On a fait appel de fa\c{c}on syst{\'e}matique dans les raisonnements {\`a} une
notion de $n$-cube dans une cat{\'e}gorie de Picard
strictement commutative. Cette notion est de nature est de nature
combinatoire et permet de repr{\'e}senter de mani{\`e}re commode des
syst{\`e}mes d'isomorphismes de la forme $a\otimes b \stackrel{\sim }{\F } c\otimes
d$. Son seul int{\'e}r{\^e}t est de permettre une repr{\'e}sentation
graphique des raisonnements de r{\'e}currence sur la dimension. Le d{\'e}faut
de cette approche est qu'il cache l'aspect g{\'e}om{\'e}trique
li{\'e} {\`a} la notion de multiextension.
\subsection{Plan de l'article}
\begin{enumerate}
\item Un exemple des m{\'e}thodes utilis{\'e}es: Les
propri{\'e}t{\'e}s d'additivit{\'e} en chaque variable du "nombre
d'intersection" de plusieurs diviseurs.
\item Pr{\'e}liminaires techniques sur les diviseurs de Cartier relatifs
et introduction d'une notion ad hoc de faisceau inversible
suffisamment positif.
\item Rappels sur les cat{\'e}gories de Picard et introduction de la
notion de structure du cube.
\item Pr{\'e}sentation des cat{\'e}gories et foncteurs utilis{\'e}s.
\item Th{\'e}or{\`e}me pricipal: l'existence d'une structure du cube sur le
fibr{\'e} d{\'e}terminant.
\item Applications au fibr{\'e} d'intersection. Construction et {\'e}tude du
r{\'e}sultant.
\end{enumerate}
\subsection{Remerciements}
On reconna{\^\i}tra dans ce travail l'influence de Larry
\textsc{Breen}, qui m'a
introduit dans ce domaine et qui m'a expliqu{\'e} avec patience les
subtilit{\'e}s des structures du cube. Je l'en remercie vivement.
\section{Un exemple introductif}
Soit $X$ un sch{\'e}ma projectif. Pour tout entier $p$ et tous faisceaux
inversibles $L_1 , \cdots , L_p$ sur X, posons:
\[<L_1 , \cdots , L_p>_X =
(-1)^p \chi (\mathcal{O}_{X} ) +
\sum_{k=1}^p (-1)^{p-k}\sum_{1 \leq i_1< \cdots < i_k\leq p}
\chi (L_{i_1} \otimes \cdots \otimes
L_{i_k})
\]
Si $p$ est {\'e}gal {\`a} la dimension $n$ de $X$, on parlera alors de {\em
nombre d'intersection} de $L_1 , \cdots , L_n$.\\
Notons d'abord que la d{\'e}finition du nombre d'intersection est
sym{\'e}trique en les $L_i$ et que pour tout entier $k$ et tous
faisceaux inversibles
$L_1 , \cdots , L_k,L,M$ , on a:
\begin{multline}
<L_1 , \cdots , L_{n-1},L, M>_X =\\
<L_1 , \cdots , L_{n-1},L>_X
+<L_1 , \cdots , L_{n-1}, M>_X
-<L_1 , \cdots , L_{n-1},L \otimes M>_X \; .
\end{multline}
Le r{\'e}sultat suivant est bien connu (cf. par exemple
\cite{beauville},Th 1.4 pour le cas de la dimension 2), mais sa
d{\'e}monstration introduit dans un cadre simple les id{\'e}es
utilis{\'e}es dans ce travail et il sera utilis{\'e}, dans sa reformulation
(\ref{caracteristique2}) pour la d{\'e}monstration du th{\'e}or{\`e}me
principal.
\begin{lemme} \label{caracteristique}
Soit $X$ un sch{\'e}ma projectif de dimension $n$, alors:
\begin{enumerate}
\item Le morphisme {\em nombre d'intersection}: $PIC(X)^n\longrightarrow \mathbb{Z}$ est
$n$-lin{\'e}aire.
\item Si $(\sigma _1,\cdots ,\sigma _n)$ est une suite r{\'e}guli{\`e}re de sections
de $L_1 , \cdots , L_n$, alors $<L_1 , \cdots , L_n>_X$ est {\'e}gal {\`a} la
longueur du sch{\'e}ma $Z$, de dimension 0, des z{\'e}ros de la section
$\sigma =\sum_{i=1}^n \sigma _i:\mathcal{O}_{X}\longrightarrow\bigoplus_{i=1}^{n}L_i$.
\end{enumerate}
\end{lemme}
\begin{proof}[Preuve]
1. Il suffit de montrer par r{\'e}currence sur $n$
que si $X$ est un sch{\'e}ma de dimension $n$ et si $L_1 , \cdots , L_{n+1}$
sont $n+1$ faisceaux inversibles sur
X, on a: $<L_1 , \cdots , L_{n+1}>_X =0$.\\
Cette assertion est {\'e}vidente dans le cas d'un sch{\'e}ma de dimension 0
puisque dans ce cas,
pour tout faisceau inversible $L$, on a $\chi (L)= \text{long}(X)$.\\
Supposons maintenant l'assertion v{\'e}rifi{\'e}e pour tout sch{\'e}ma de
dimension inf{\'e}rieure ou {\'e}gale {\`a} $n$ et consid{\'e}rons un
sch{\'e}ma projectif
$X$ de dimension $n$. Pour des faisceaux inversibles
$L_1 , \cdots , L_n$ et un diviseur effectif $D$, on a:
\begin{multline}
<L_1 , \cdots , L_n,\mathcal{O}_{X} (D)>_X =(-1)^n (\chi (\mathcal{O}_{X} (D)) - \chi (\mathcal{O}_{X} )\\ +
\sum_{k=1}^n (-1)^{n-k}\sum_{1 \leq i_1< \cdots < i_k\leq n}
(\chi (L_{i_1} \otimes \cdots \otimes L_{i_k}\otimes \mathcal{O}_{X} (D))
-\chi (L_{i_1} \otimes \cdots \otimes L_{i_k}))
\end{multline}
En appliquant la propri{\'e}t{\'e} d'additivit{\'e} de la caract{\'e}ristique
d'Euler-Poincar{\'e} {\`a}
des suites exactes de la forme
$
0 \longrightarrow L\longrightarrow L(D)\longrightarrow L(D)/L \longrightarrow 0
$
et en identifiant $\chi _X (L(D)/L)$ et $\chi_D (L(D)|_D)$, on en d{\'e}duit:
\[
<L_1 , \cdots , L_n,\mathcal{O}_{X} (D)>_X =
<L_1(D)|_D , \cdots , L_n(D)|_D>_D
\]
En appliquant l'hypoth{\`e}se de r{\'e}currence au diviseur effectif $D$, on
obtient donc:
\begin{equation}
<L_1 , \cdots , L_n,\mathcal{O}_{X} (D)>_X =0
\end{equation}
On en d{\'e}duit donc, pour tout diviseur
effectif $D$, les {\'e}galit{\'e}s:
\begin{equation}
\label{add1}
<L_1 , \cdots , L_{n-1},L_n(D)>_X=
<L_1 , \cdots , L_{n-1},L_n>_X
+
<L_1 , \cdots , L_{n-1},\mathcal{O}_{X} (D)>_X
\end{equation}
et
\begin{multline}
\label{add2}
<L_1 , \cdots ,L_i \otimes L_i^{\prime} ,\cdots , L_{n-1},\mathcal{O}_{X} (D)>_X\\
=
<L_1 , \cdots , L_i^{\prime} ,\cdots , L_{n-1},\mathcal{O}_{X} (D)>_X
+
<L_1 , \cdots ,L_i ,\cdots , L_{n-1},\mathcal{O}_{X} (D)>_X
\end{multline}
Comme $X$ est projectif, si $L_n$ est un
faisceau inversible sur $X$, il peut s'{\'e}crire $\mathcal{O}_{X} (D-E)$, o{\`u} $D$ et
$E$ sont des diviseurs effectifs. On obtient alors en appliquant (\ref{add1}):
\[
<L_1 , \cdots , L_{n-1},L_n>_X=
<L_1 , \cdots , L_{n-1},\mathcal{O}_{X} (D)_X>-
<L_1 , \cdots , L_{n-1},\mathcal{O}_{X} (E)>_X
\]
En appliquant (\ref{add2}) {\`a} chacun des termes de droite de
l'{\'e}galit{\'e} pr{\'e}c{\'e}dente, on obtient l'additivit{\'e} de
$<L_1 , \cdots ,L_{n-1},L_n>_X$ en chacune des variables $L_1 , \cdots ,
L_{n-1}$, et donc, par sym{\'e}trie, en toutes les variables.\\
2. Consid{\'e}rons le
complexe de Koszul:
\[
K_{\bullet}:\;
0 \longrightarrow \L ^n (E) \longrightarrow \L ^{n-1}(E) \longrightarrow \cdots \longrightarrow E \longrightarrow \mathcal{O}_{X} \longrightarrow \mathcal{O}_{X} /I_Z \longrightarrow 0
\]
associ{\'e} au morphisme $\sigma^{\vee } : E=\bigoplus_{i=1}^{n}L_i ^{-1} \longrightarrow
\mathcal{O}_{X}$. Par hypoth{\`e}se, $K_{\bullet}$ est exact, donc $\chi
(K_{\bullet})=0$ et le r{\'e}sultat s'en d{\'e}duit, compte tenu de
l'isomorphisme
\[
\L ^p (E) \simeq \bigoplus_{1 \leq i_1< \cdots < i_k\leq p}
L_{i_1} \otimes \cdots \otimes
L_{i_k}.
\]
\end{proof}
Dans la suite de ce travail on {\'e}tudiera, non plus une application
ensembliste de $Pic(X)^n$ dans $\mathbb{Z}$, mais un foncteur de la cat{\'e}gorie
de Picard $\text{PIC}(X)$ dans une autre cat{\'e}gorie de Picard, et les
difficult{\'e}s proviennent de la n{\'e}cessit{\'e} de faire des constructions
fonctorielles.
\section{pr{\'e}liminaires techniques}
Dans cette partie nous d{\'e}taillons quelques propri{\'e}t{\'e}s des sections
d'un faisceau inversible sur un sch{\'e}ma relatif et nous introduisons
une notion technique de faisceau suffisamment positif, utile par la
suite. $S$ est ici un sch{\'e}ma localement noeth{\'e}rien et $\pi : X \longrightarrow S$
est un morphisme projectif et plat.
\subsection{Diviseur relatif d{\'e}fini par une section d'un fibr{\'e}}
\label{div}
Soit $L$ un faisceau inversible sur le sch{\'e}ma relatif $X/S$, une
section $\sigma :
\mathcal{O}_{X} \longrightarrow L$ de $L$ sera dite $\pi$-r{\'e}guli{\`e}re si elle d{\'e}finit elle
d{\'e}finit un diviseur de Cartier relatif effectif de
$X/S$. $\sigma$ est $\pi$-r{\'e}guli{\`e}re si et seulement si pour tout point
$x\in X$, $\sigma (\mathcal{O} _{X,x})
\subset L_x$ est un $\mathcal{O} _{X,x}$-module plat et le quotient
$(L \otimes _{\ox} \mathcal{O} _{X,x}) / \sigma (\mathcal{O} _{X,x})$ est un $\mathcal{O}_{S,\pi (x)}$-module
plat.\\
L'ensemble $U^{\prime}$ des points $x$ de $X$ v{\'e}rifiant ces deux
propri{\'e}t{\'e}s
apparait comme un ensemble de platitude et est donc un ouvert de
$X$ par (\cite{EGA4}, 11.1.1). Notons $Z^{\prime}$ le compl{\'e}mentaire de
$U^{\prime}$. Comme $\pi$ est projectif, $Z= \pi (Z^{\prime} )$ est un ferm{\'e} de $S$
dont nous noterons $U$ le compl{\'e}mentaire.\\
Le th{\'e}or{\`e}me \cite{ma},22.5 entra{\^\i}ne que si $A\longrightarrow B$ est un
morphisme d'anneaux locaux et $k$ d{\'e}signe le corps r{\'e}siduel de $A$,
pour tout $u \in B$, on a l'{\'e}quivalence
des assertions:
\begin{enumerate}
\item $u$ est non diviseur de z{\'e}ro dans $B$ et $B/uB$ est un
$A$-module plat.
\item $u \otimes _A 1 $ est non diviseur de z{\'e}ro dans $B \otimes _A
k$.
\end{enumerate}
On d{\'e}duit de ceci qu'un point $s$ de $S$ est dans $U$ si et seulement
si $\sigma \mid _{X_s}$ d{\'e}finit un diviseur de Cartier sur $X_s$, soit
encore si $\sigma $ ne s'annulle sur aucun point associ{\'e} de $X_s$.
\subsection{Suites r{\'e}guli{\`e}res}
Soient $L_1,\cdots , L_p$ des faisceaux inversibles sur $X$,
muni de sections $\sigma_i$.
Notons $D_i$ le lieu des z{\'e}ros de $\sigma _i$. On
dira que que la suite (ordonn{\'e}e) $(\sigma _1,\cdots ,\sigma _p)$ est une
{\em suite $\pi$-r{\'e}guli{\`e}re}
si les deux conditions suivantes sont v{\'e}rifi{\'e}es:
\begin{enumerate}
\item $D_1\cap\cdots\cap D_p \neq \emptyset$.
\item Pour tout entier $i$ compris entre 1 et $p$, $\sigma _i$ d{\'e}finit
une section
$\pi$-r{\'e}guli{\`e}re sur le sch{\'e}ma relatif
$D_1\cap\cdots\cap D_{i-1}/S$,
si $i>1$, ou sur $X/S$, si $i=1$.
\end{enumerate}
On notera que \begin{enumerate}
\item Si $(\sigma _1,\cdots ,\sigma _p)$ est une suite
$\pi$-r{\'e}guli{\`e}re, alors pour
tout $1\leq i \leq p$, $D_1\cap\cdots\cap D_i$ est plat sur $S$.
\item Soit $(\sigma _1,\cdots ,\sigma _p)$ une suite de sections de $(L_i)$,
le sous-ensemble de $S$ au dessus duquel
$(\sigma _1,\cdots ,\sigma _p)$ est une suite r{\'e}guli{\`e}re est un
ouvert de $S$.
\end{enumerate}
\subsection{Faisceaux inversibles suffisamment positifs}
On dira qu'un faisceau inversible $L$ sur $X$ est
{\em suffisamment positif}
(on notera $L \gg 0$) si $L$ est tr{\`e}s ample relativement {\`a} $\pi$ et si
pour tout $i>0$, on a $R^i \pi _{\ast} L =0$. Ces faisceaux v{\'e}rifient
les propri{\'e}t{\'e}s suivantes:
\subsubsection{} La propri{\'e}t{\'e} pour un faisceau inversible un
faisceau inversible d'{\^e}tre suffisamment positif est conserv{\'e}e par
tout changement de base
$f:T\longrightarrow S$. Ceci r{\'e}sulte de l'invariance
par changement de base de la notion de faisceau relativement tr{\`e}s ample
(\cite{EGA2}, 4.4.10,iii ) et
du th{\'e}or{\`e}me de changement de base dans la cohomologie
(\cite{Ha},12.11).
\subsubsection{} Si $L_1 , \cdots , \L_k$ sont des
faisceaux inversibles sur $X$,
il existe des faisceaux inversibles suffisamment positifs
$M_1 , \cdots , M
_k$ tels que les $L_i \otimes _{\ox} M_i$ sont tous isomorphes
et suffisamment positifs. En effet si $M$ est un faisceau inversible
tr{\`e}s ample relativement {\`a} $\pi$, pour tout faisceau inversible $L$,
il existe un entier $N$ tel que pour tout $n \geq N$, on a
$L \otimes _{\ox} M^{\otimes n} \gg 0$ (\cite{EGA2},4.4.10.{\em ii} et
\cite{Ha},th III.8.8.c). En appliquant ceci
aux faisceaux
$\bigotimes_{j\neq i}L_j$ pour $1\leq i\leq k$
et $ L_1 \otimes \cdots \otimes L_k $ on trouve
un entier $n$ tel que les faisceaux
$\left( \bigotimes_{j\neq i}L_j \right) \otimes M^{\otimes n}$ et
$\left( \bigotimes_{1\leq j \leq k}L_j \right) \otimes M^{\otimes n}$ soit
suffisamment positifs. Il suffit alors de prendre
$M_i =\left( \bigotimes_{j\neq i}L_j
\right) \otimes M^{\otimes n}$.
\subsubsection{} \label{cb} Si $L$ est suffisamment positif, $\pi _{\ast} L$
est un faisceau
localement libre sur $S$ et pour tout $s\in S$, la fibre $(\pi _{\ast}
L)_s$ est isomorphe {\`a} $H^0 (X_s ,L)$ (ceci r{\'e}sulte du
th{\'e}or{\`e}me de
changement de base \cite{Ha},12.11). Si de plus $X$ est {\`a} fibres de
dimension au moins 1, $\pi _{\ast} L$ est de rang au moins 2.
\subsection{Le diviseur universel d'un faisceau suffisamment positif}
\label{div2}
Si $L$ est suffisamment positif, $E=(\pi_{\ast } L)^{\vee }$ est un
$\mathcal{O}_{X}$-module localement libre. Consid{\'e}rons alors le fibr{\'e}
projectif $P_L = \mathbb{P}(E)$ et effectuons le changement de base:
\[
\begin{CD}
X_{P_L} @>g>> X \\
@V\pi VV @VV\pi V\\
P_L @>f>> S
\end{CD}
\]
Par d{\'e}finition de $P_L$, on a un morphisme surjectif
$\xymatrix{f^{\ast } E \ar@{->>}[r] &{\mathcal{O}}_{P_L}(1)}$,
qui induit un morphisme
$\xymatrix{{\mathcal{O}}_{P_L}(-1) \ar@{^{(}->}[r] & (f^{\ast } E)^{\vee }}$. Or on a
$(f^{\ast } E)^{\vee } = f^{\ast }(E^{\vee })=f^{\ast }(\pi_{\ast } L)=\pi_{\ast }(g^{\ast } L)$, o{\`u} la
derni{\`e}re {\'e}galit{\'e} provient de l'hypoth{\`e}se $L\gg 0$.
Le morphisme
$\xymatrix{{\mathcal{O}}_{P_L}(-1) \ar@{^{(}->}[r] & \pi_{\ast }(g^{\ast } L)}$
ainsi obtenu induit par adjonction un morphisme
$\pi^{\ast }\mathcal{O}_{P_L}(-1) \longrightarrow g^{\ast } L$. On obtient donc finalement une section
canonique $\sigma_L$ de $g^{\ast } L \otimes \pi^{\ast }\mathcal{O}_{P_L}(1)$.
En appliquant au sch{\'e}ma
relatif $X_{P_L}/P_L$ les constructions de la section (\ref{div}), on
construit un ouvert $U_L$ de $P_L$, au
dessus duquel $\sigma _L$ d{\'e}finit un diviseur de Cartier relatif $D_L$ et
on note $Z_L$ son compl{\'e}mentaire. On obtient ainsi un isomorphisme
canonique $(L\otimes\pi^{\ast }\mathcal{O}_{P_L}(1))| _{X _{U_L}} \simeq
\mathcal{O} (D_L )$. La situation est d{\'e}crite par le diagramme suivant:
\[
\xymatrix{
&(L\otimes\pi^{\ast }\mathcal{O}_{P_L}(1)\simeq \mathcal{O} (D_L))\ar@{.}[d]
&(\mathcal{O}_{X} \stackrel{\sigma_L}{\longrightarrow} L\otimes\pi^{\ast }\mathcal{O}_{P_L}(1))\ar@{.}[d]
&L\ar@{.}[d] \\
D_L \ar[dr] \ar@{^{(}->} [r] &X_{U_L} \ar[r] \ar[d] &X_{P_L} \ar[r]
\ar[d]
& X\ar[d]_{\pi} \\
&U_L \ar@{^{(}->}[r] &P_L \ar[r] &S
}
\]
\begin{lemme} \label{genericite1}
Pour tout point $s$ de $S$, la fibre de $Z_L$ au dessus de $s$
est une union finie de sous-espaces
lin{\'e}aires propres de $P_{L,s}$.
\end{lemme}
\begin{proof}[Preuve]
En effet, soit $s \in S$, on d{\'e}duit de (\ref{div}) l'{\'e}galit{\'e}:
\[
Z_{L,s} = \bigcup_{x \in \text{Ass}(X_s)}
\mathbb{P} \left( \left\{\sigma \in H^0 (X_s ,L) | \sigma (x)=0 \right\}
\right)
\subset \mathbb{P} \left( H^0 (X_s ,L) \right)
= (P_L)_s
\]
L'ensemble des points associ{\'e}s de $X_s$ est fini puisque $X_s$ est
projectif et comme $L\mid _{X_S}$ est tr{\`e}s ample, pour tout
$x \in \text{Ass}(X_s)$, l'inclusion
$ \left\{\sigma \in H^0 (X_s ,L) | \sigma (x)=0 \right\} \subset H^0 (X_s
,L) $
est stricte.
\end{proof}
\addtocounter{subsubsection}{1}
\subsubsection{}
Donnons nous de plus une suite $\pi$-r{\'e}guli{\`e}re de sections
$(\sigma_i)_{i=1,\cdots ,p}$ de
faisceaux inversibles $L_i$ sur $X$. Soit $V$
l'ouvert de $U_L$ au dessus duquel $(\sigma_1,\cdots,\sigma_p,\sigma_L)$ est une
suite $\pi$-r{\'e}guli{\`e}re et notons $Y$ son compl{\'e}mentaire.
\addtocounter{theo}{1}
\begin{lemme}\label{genericite2}
Pour tout point $s$ de $S$, la fibre de $Y$ au dessus de $s$
est contenue dans une union finie de sous-espaces
lin{\'e}aires propres de $(P_L)_s$.
\end{lemme}
\begin{proof}[Preuve]
Soit $D$ le lieu des z{\'e}ros de la section $\bigoplus\sigma_i$ de
$\bigoplus L_i$.
Un point $u\in U_L$ est {\'e}l{\'e}ment de $V$ si et
seulement si $\sigma |_{D_u}$ d{\'e}finit un diviseur de Cartier effectif sur
$D_u$. On raisonne donc comme pour le lemme pr{\'e}c{\'e}dent en {\'e}crivant:
\[
Y_s = \bigcup_{x\in \text{Ass}(D_s)}
\mathbb{P}\{ \sigma \in H^0 (X_s ,L) |\sigma (x)=0 \}
\]
\end{proof}
\section{Structure du cube sur des cat{\'e}gories de Picard}
Si $(\mathcal{C},\otimes )$ et $(\mathcal{D},\otimes ) $ sont des
cat{\'e}gories de Picard strictement commutatives et $\gd $ est un
foncteur de $\mathcal{C}$ vers
$\mathcal{D}$, on d{\'e}finit ici la notion de structure du cube sur $\gd
$.
\subsection{Rappels sur les cat{\'e}gories de Picard commutatives}
\subsubsection{D{\'e}finitions}
Une {\em cat{\'e}gorie de Picard} est une cat{\'e}gorie $\mathcal{C}$ dont toutes les
fl{\`e}ches sont des isomorphismes, et qui est munie d'un bifoncteur
$\otimes : \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C}$ tel que, pour tout $L \in
\text{ob} (\mathcal{C} )$, les foncteurs
$X \mapsto L\otimes X $ et $X \mapsto X\otimes L $ soient des
{\'e}quivalences de cat{\'e}gories, et munie de plus de
donn{\'e}es d'associativit{\'e} pour
$\otimes$, c'est {\`a} dire d'un syst{\`e}me fonctoriel d'isomorphismes
\[
(L \otimes M) \otimes N \simeq L \otimes (M \otimes N)
\]
v{\'e}rifiant des conditions de compatibilit{\'e} d{\'e}crites par l'axiome du
pentagone.\\
Une cat{\'e}gorie de Picard $\mathcal{C}$ est dite {\em commutative} si
elle est munie
de plus de donn{\'e}es de commutativit{\'e}, c'est {\`a} dire d'un syst{\`e}me
fonctoriel d'isomorphismes
\[
L\otimes M \stackrel{\sim }{\F } M \otimes L
\]
compatible avec les donn{\'e}es d'associativit{\'e} (axiome de
l'hexagone). On prendra garde que l'isomorphisme de commutativit{\'e}
$L\otimes L \stackrel{\sim }{\F } L \otimes L$ n'est en g{\'e}n{\'e}ral pas le morphisme
identit{\'e} (quand c'est le cas pour toul $L$, on dit que la
cat{\'e}gorie est {\em
strictement commutative}).\\
On d{\'e}duit des axiomes d'une cat{\'e}gorie de Picard commutative
l'existence d'un objet unit{\'e} $\mathcal{O} $, munis de morphismes
$L \otimes \mathcal{O} \longrightarrow L \longleftarrow \mathcal{O} \otimes L$ compatibles aux
contraintes de commutativit{\'e}.\\
On obtient de m{\^e}me l'existence {\`a} isomorphismes uniques
pr{\`e}s d'objets
inverses $L^{\vee }$ munis de morphismes
$L \otimes L^{\vee } \longrightarrow \mathcal{O} \longleftarrow L^{\vee } \otimes L$.
\subsubsection{Produit d'une famille index{\'e}e par un ensemble fini}
Soit $(L_i)_{i\in I}$ une famille d'objets d'une cat{\'e}gorie de Picard
$\mathcal{C}$ index{\'e}s par un ensemble fini $I$. Si $I$ est muni d'un ordre
total $<$, les donn{\'e}es
d'associativit{\'e} de $\mathcal{C}$ permettent de d{\'e}finir de fa\c{c}on
fonctorielle un objet
$\bigotimes_{I,<} L_i$ de $\mathcal{C}$.\\
Supposons maintenant que $\mathcal{C}$ est une cat{\'e}gorie de Picard
commutative. Si $<_1$ et $<_2$ sont deux ordres totaux sur $I$, les
donn{\'e}es de commutativit{\'e} de $\mathcal{C}$ d{\'e}terminent un isomorphisme
$\bigotimes_{I,<_1} L_i \stackrel{\sim }{\F } \bigotimes_{I,<_2} L_i$. On peut alors
d{\'e}finir $\bigotimes_{i\in I}L_i$ comme la limite inductive (ou
projective) des $\bigotimes_{I,<} L_i$ sur tous les ordres totaux $<$
sur $I$. Pour d{\'e}finir un morphisme
$\bigotimes_{i\in I}L_i \longrightarrow M $ dans $\mathcal{C}$, il suffira donc de choisir
un ordre $<$ sur $I$ et de d{\'e}finir un morphisme
$\bigotimes_{I,<} L_i \longrightarrow M$. Par ailleurs pour deux ensembles
d'indices disjoints $I$ et $J$, on un isomorphisme canonique
$\bigotimes_{i\in I}L_i \otimes \bigotimes_{i\in J}L_i \simeq
\bigotimes_{i\in I\cup J}L_i$, obtenu en consid{\'e}rant un ordre sur
$I\cup J$ tel que $I<J$ et les ordres induits sur $I$ et $J$.
\subsubsection{Foncteur additif.}
Soit $F: \mathcal{C} \longrightarrow \mathcal{D}$ un foncteur additif entre deux cat{\'e}gories de
Picard, une donn{\'e}e d'additivit{\'e} $\mu$ pour $F$ sera la
donn{\'e}e pour tout
couple d'objets $L,M \in \mathcal{C}$ d'un isomorphisme fonctoriel en $L$ et
$M$:
\[
\mu _{L,M}: F(L) \otimes F(M) \longrightarrow F(L\otimes M)
\]
La donn{\'e}e d'additivit{\'e} $\mu$ sera dite compatible aux donn{\'e}es
d'associativit{\'e} de $\mathcal{C}$ et $\mathcal{D}$ si le diagramme suivant, dont les
fl{\`e}ches verticales sont donn{\'e}es par les morphismes
d'associativit{\'e} de
$\mathcal{C}$ et $\mathcal{D}$
\[
\begin{CD}
F(L) \otimes (F(M) \otimes F(N)) @>{\text{Id}\otimes\mu}>>F(L) \otimes
F(M \otimes N)
@>{\mu}>>F(L\otimes (M \otimes N)) \\
@VVV @. @VVV \\
(F(L) \otimes F(M)) \otimes F(N) @>{\mu\otimes\text{Id}}>>
F(L \otimes M) \otimes F(N)
@>{\mu}>> F((L\otimes M) \otimes N)
\end{CD}
\]
est commutatif.\\
Si de plus $\mathcal{C}$ et $\mathcal{D}$ sont des cat{\'e}gories de Picard commutatives,
$\mu$ sera dite compatible aux donn{\'e}es de commutativit{\'e} de $\mathcal{C}$ et
$\mathcal{D}$ si le diagramme
\[
\begin{CD}
F(L) \otimes F(M) @>>> F(L \otimes M)\\
@VVV @VVV \\
F(M) \otimes F(L) @>>> F(M \otimes L)
\end{CD}
\]
est commutatif (les
fl{\`e}ches verticales sont donn{\'e}es par les morphismes de
commutativit{\'e} de $\mathcal{C}$ et $\mathcal{D}$).
\subsection{n-Cube dans une cat{\'e}gorie de Picard strictement
commutative}
On introduit ici une notion de nature combinatoire, qui traduit des
calculs de fractions, du genre $ab ^{-1}= cd ^{-1}$, dans une
cat{\'e}gorie de Picard strictement
commutative $\mathcal{C}$, en indexant des objets de $\mathcal{C}$ par les sommets de
diff{\'e}rents hypercubes de $\mathbb{R} ^n$, ce qui nous permettra dans la suite
de repr{\'e}senter graphiquement certains raisonnements.
\subsubsection{Cubes standard de $\mathbb{R}^n$}
Pour tout entier positif $n$, on consid{\`e}re l'ensemble $C_n=\{0,1\}^n$ des
sommets du n-cube standard de $\mathbb{R} ^n$.\\
On consid{\'e}rera pour tout entier $i\leq n$,
les inclusions de $C_{n-1}$ dans $C_n$: $\phi _i^{\prime} : (s_1,\cdots
,s_{n-1}) \mapsto (s_1,\cdots ,s_{i-1} ,0,s_{i+1},\cdots ,s_n)$ et
$\phi _i\sec : (s_1,\cdots
,s_{n-1}) \mapsto (s_1,\cdots ,s_{i-1} ,1,s_{i+1},\cdots ,s_n)$.
L'image de $C_{n-1}$ par $\phi _i^{\prime}$ (resp. $\phi _i\sec$) sera
appel{\'e}e la i-{\`e}me $(n-1)$-face arri{\`e}re (resp. avant) de
$C_n$. De m{\^e}me,
soient $\ge_1,\cdots ,\ge_k \in \{ 0,1 \} $ et des indices distincts
$i_1,\cdots,i_k\leq n$, on peut consid{\'e}rer l'inclusion
$\phi_{i_1=\ge_1,\cdots,i_k=\ge_k}$ de $C_{n-k}$ dans $C_n$.\\
Pour tout sommet
$ s=(s_1 , \cdots , s_n)\in C_n$, on notera
$\ge (s)= (-1)^{n- \sum s_i}$.\\
Enfin il sera commode d'introduire un ordre total sur les sommets de
$C_n$ par:
\[
(s \leq s^{\prime} )
\Leftrightarrow
\begin{cases}
\sum s_i < \sum s_i^{\prime} &\\
\text{ou}&\\
(\sum s_i = \sum s_i^{\prime} ) &
\text{et}\; ( \exists i\leq n ,(s_i > s_i^{\prime} )\; \text{et} \;
(\forall j<i\, , \, s_i=s_i^{\prime} ))
\end{cases}
\]
On notera que l'ordre induit par $\leq $ sur une k-face du cube $C_n$
est encore l'ordre $\leq$ sur $C_k$.
\subsubsection{Arrangements cubiques} On appellera alors {\em
n-arrangement cubique} dans une cat{\'e}gorie de Picard
commutative $\mathcal{C}$
la donn{\'e}e de $2^n$ objets $K_s$ index{\'e}s par les sommets de $C_n$.\\
Toute permutation $\sigma\in S_n$ agit sur $\mathbb{R}^n$ par
permutation des coordonn{\'e}es, et induit donc $\sigma : C_n \longrightarrow C_n$. Pour
tout $n$-arrangement cubique $K$ dans $\mathcal{C}$, on
notera alors $\sigma^{\ast } K$ le compos{\'e} de $K$ avec la permutation de
$C^n$ induite par $\sigma$.\\
Pour tout n-arrangement cubique $K$ et tout entier $1\leq i\leq n$,
on consid{\'e}rera alors les $(n-1)$-arrangements cubiques $\phi^{\prime\ast}_i K$ et
$\phi^{\prime \prime\ast}_i K$ (i-{\`e}me face arri{\`e}re et avant de $K$). Si $A$ et $B$ sont
deux $(n-1)$-arrangements cubiques, on notera pour tout indice $1\leq
i\leq n$, $(\xymatrix{A \ar@{-}[r]_-i&B})$ le n-arrangement cubique
tel que $\phi^{\prime\ast}_i K=A$ et $\phi^{\prime \prime\ast}_i K=B$.
De m{\^e}me, si $A_{00}$, $A_{01}$, $A_{10}$ et
$A_{11}$ sont des $(n-2)$-arrangements cubiques, on utilisera, pour deux
indices distincts $1\leq i,j \leq n$, la notation
$K=
\left(
\begin{array}{c}
\xymatrix{
A_{01}\ar@{-}[d]_-j \ar@{-}[r] &A_{11}\ar@{-}[d] \\
A_{00} \ar@{-}[r]_-i &A_{10}
}
\end{array}
\right)$ pour d{\'e}signer le $n$-arrangement cubique $K$ tel que
$\phi_{i=\ge ,j=\ge^{\prime}}=A_{\ge ,\ge^{\prime}}, \forall \ge,\ge^{\prime}\in \{0,1\}$.\\
Soient $A$ et $B$ deux $n$-arrangements cubiques tels que
$\phi^{\prime \prime\ast}_iA=\phi^{\prime\ast}_iB$. Ecrivons les alors sous la forme:
$A=
(\xymatrix{U \ar@{-}[r]_-i&V})$ et $B=(\xymatrix{V
\ar@{-}[r]_-i&W})
$ et posons:
\[
(\xymatrix{U \ar@{-}[r]_-i&V})
\ast_i
(\xymatrix{V\ar@{-}[r]_-i&W})
=(\xymatrix{U\ar@{-}[r]_-i&W})
\]
On dira que $A\ast_iB$ est obtenu par recollement de $A$ et $B$ le
long de leur i-{\`e}me face.
\subsubsection{}
Soit $\gd$ un foncteur de $\mathcal{C}$ dans une cat{\'e}gorie de Picard
commutative $\mathcal{D}$, pour tout $n$-arrangement cubique $K$ dans $\mathcal{C}$,
on pose
\[
\gd (K) = \bigotimes_{s\in C_n} \gd (K_s)^{\ge (s)}
\]
Si $A$ et$B$ sont des $n$-arrangements cubiques recollables dans la
i-{\`e}me direction, les isomorphismes de commutativit{\'e} et de contraction
dans $\mathcal{D}$ induisent un isomorphisme canonique
\[
\gd (A\ast_iB)\stackrel{\sim }{\F } \gd(A) \otimes \gd (B)
\]
Pour toute permutation $\sigma\in S_n$, les isomorphismes de commutativit{\'e}
dans $\mathcal{D}$ induisent de m{\^e}me:
\[
\gd (\sigma^{\ast } A) \stackrel{\sim }{\F } \sigma (A)
\]
Ces deux isomorphismes sont compatibles entre eux, ce qu'on exprime en
disant que le diagramme suivant est commutatif:
\[
\xymatrix{
\gd (\sigma^{\ast }( A\ast_iB))\ar[d]&
\gd (A\ast_i B)\ar[l] \ar[r] &
\gd (A)\otimes \gd (B)\ar[d]\\
\gd (\sigma^{\ast } A\ast_{\sigma^{-1}(i)} \sigma^{\ast } B)\ar[rr]
&& \gd (\sigma^{\ast } A)\otimes\gd (\sigma^{\ast } B)
}
\]
\subsubsection{Cubes dans la cat{\'e}gorie de Picard strictement
commutative $\mathcal{C}$}
\label{def-cube}
On appellera $1$-cube dans $\mathcal{C}$ un 1-arrangement cubique quelconque
$(\xymatrix{L\ar@{-}[r]&M})$.\\
On appellera $2$-cube (carr{\'e}) dans $\mathcal{C}$ la donn{\'e}e d'un 2-arrangement
cubique $K$ et d'un isomorphisme
$m_K: \mathcal{O} \stackrel{\sim }{\F } \otimes_{s\in C_2} K_s^{\ge (s)}$.\\
On appellera $3$-cube dans $\mathcal{C}$ la donn{\'e}e d'un 3-arrangement
cubique
\[
K=\left(
\begin{array}{c}
\xymatrix{
&K_{001}\ar@{-}[dl]\ar@{-}'[d][dd]\ar@{-}[rr]&
&K_{011}\ar@{-}[dl]\ar@{-}[dd] \\
K_{101}\ar@{-}[dd]\ar@{-}[rr]&&K_{111}\ar@{-}[dd]&\\
&K_{000}\ar@{-}[dl]\ar@{-}'[r][rr]&&K_{010}\ar@{-}[dl]\\
K_{100}\ar@{-}[rr]&&K_{110}
}
\end{array}
\right) ,
\]
et pour chacune des six 2-faces $F$ de $K$, d'un isomorphisme
$m_F: \mathcal{O} \stackrel{\sim }{\F } \otimes_{s\in F} K_s^{\ge (s)}$ qui v{\'e}rifient la
condition de compatibilit{\'e} suivante:
Si $F$ d{\'e}signe l'une des faces de $K$ et $F^{\prime}$ d{\'e}signe la face
oppos{\'e}e, on dispose alors d'un morphisme:
\[
\begin{CD}
\mathcal{O} @>>> \mathcal{O} \otimes \mathcal{O} @>m_F \otimes m_{F^{\prime} }>>
\bigotimes _{s\in F} K_s^{\ge (s)}
\otimes
\bigotimes _{s\in {F^{\prime}}} K_s^{\ge (s)}
@>>> \bigotimes _{s\in K} K_s^{\ge (s)}
\end{CD}
\]
On impose que le morphisme ainsi obtenu soit ind{\'e}pendant du choix de la
face $F$.\\
Pour $n\geq 3$, un {\em n-cube} dans $\mathcal{C}$ sera la donn{\'e}e d'un
n-arrangement cubique $K$ et d'un isomorphisme $m_F$ associ{\'e} {\`a} chaque
2-face $F$ de $K$, tel que tout sous 3-arrangement cubique de $K$ est un cube.
\subsubsection{Construction de cubes}
(a) Pour tout $n$-cube $K$ dans $\mathcal{C}$, tout objet $L$ de $\mathcal{C}$ et tout indice
$1\leq i\leq n$, on munit le
$(n+1)$-arrangement cubique $(\xymatrix{K \ar@{-}[r]_-i&K\otimes
L})=A$ d'un syst{\`e}me de morphismes $m_F$ associ{\'e}s {\`a}
chaque 2-face de
$A$, qui en fait un n-cube:\\
Si $F$ n'est pas parall{\`e}le {\`a} la direction $i$, elle
appartient {\`a} l'un
des deux $n$-cubes $K$ ou de $K\otimes L$
et est donc d{\'e}ja muni d'un morphisme $m_F$. Sinon elle s'{\'e}crit
$F=
\left(
\begin{array}{c}
\xymatrix{Y\ar@{-}[d]_-j\ar@{-}[r] &Y\otimes L\ar@{-}[d] \\
X\ar@{-}[r]_-i &X\otimes L
}
\end{array}
\right) $ avec $X,Y\in\text{ob}(\mathcal{C} )$ et les
morphismes de commutativit{\'e} et d'associativit{\'e} de $\mathcal{C}$ induisent un
morphisme
$
m_F:(X\otimes L)\otimes Y\stackrel{\sim }{\F } X\otimes (Y\otimes L)
$.
On v{\'e}rifie sans peine, en se ramenant au cas o{\`u} $K$ est un
cube, que les relations de compatibilit{\'e} de
(\ref{def-cube}) entre les $m_F$ sont v{\'e}rifi{\'e}es.\\
(b) Pour tout $n>0$, construisons un $n$-cube
$K \stackrel{\sim }{\F } K_{L_0}(L_1, \cdots ,L_n)$ de la mani{\`e}re suivante:
On pose $K_{L_0}(L_1)= (\xymatrix{L_0 \ar@{-}[r]_-1&L_0\otimes L_1})$
et par r{\'e}currence
\[
K_{L_0}(L_1, \cdots ,L_{n+1})=
(\xymatrix{K_{L_0}(L_1, \cdots ,L_n) \ar@{-}[r]_-{n+1}&
K_{L_0}(L_1, \cdots ,L_n)\otimes L_{n+1}}).
\]
On a alors, pour tout $s \in C_n$ :
\[
( K_{L_0}(L_1, \cdots ,L_n))_s = L_0 \otimes
\left(
\bigotimes_{i=1}^n L_i^{s_i}
\right) .
\]
(c) Notons enfin que si $K$ est un $n$-cube, si $K_0$ d{\'e}signe le
$(n-1)$-cube
$(\phi _i^{\prime} )^{\ast } K$, il existe un objet $L$ de $\mathcal{C}$ et un
isomorphisme $K \stackrel{\sim }{\F } (\xymatrix{K_0\ar@{-}[r]_i&K_0\otimes L})$, uniques
{\`a} isomorphisme unique pr{\`e}s. On en d{\'e}duit donc qu'il
existe des objets
$L_0, \cdots ,L_n$ et un isomorphisme
$K \stackrel{\sim }{\F } K_{L_0}(L_1, \cdots ,L_n)$, uniques {\`a} isomorphisme unique
pr{\`e}s. On dira que l'objet $L_i$ est la {\em i-{\`e}me ar{\^e}te} de K.
\subsubsection{Recollement de cubes}
Consid{\'e}rons deux $n$-cubes $K$ et $K^{\prime}$ dans $\mathcal{C}$ tels que les deux
sous-cubes $\phi^{\prime \prime\ast}_i (K)$ et $\phi^{\prime\ast}_i (K^{\prime} )$ sont {\'e}gaux en tant que
cubes. Munissons le n-arrangement cubique $K\ast _i K^{\prime}$ d'un syst{\`e}me
de morphismes $m_F$, pour chaque 2-face $F$ de $K\ast _i K^{\prime}$, en faisant un
un $n$-cube:
Si $F$ n'est pas parall{\`e}le {\`a} la direction $i$, $F$ est une 2-face de
l'un des deux $n$-cubes $K$ ou $K^{\prime}$ et on prend le $m_F$
correspondant.
Si $F$ est parall{\`e}le {\`a} la direction $i$, elle est obtenue en recollant
une 2-face de $K$ et une de $K^{\prime}$:
\[
\xymatrix{
c\ar@{-}[d] \ar@{-}[r] &d\ar@{}[dr]|{\ast _i}
&d\ar@{-}[d] \ar@{-}[r] &f\ar@{}[dr]|{=}
&c\ar@{-}[d] \ar@{-}[r] \ar@{}[rd]|{F} &f\\
a&b\ar@{-}[u] \ar@{-}[l]_-i &b&e\ar@{-}[u] \ar@{-}[l]_-i
&a&e\ar@{-}[u] \ar@{-}[l]_-i
}
\]
et le morphisme $m_F$ est d{\'e}fini par:
\[
\mathcal{O} \stackrel{\sim }{\F } \mathcal{O} \otimes \mathcal{O}
\stackrel{\sim }{\F }
(a \otimes b^{\vee } \otimes c^{\vee } \otimes d )
\otimes
(b \otimes e^{\vee } \otimes d^{\vee } \otimes f)
\stackrel{\sim }{\F }
(a \otimes c^{\vee } \otimes c^{\vee } \otimes f)
\]
Pour montrer que ces $m_F$ v{\'e}rifient les relations de compatibilit{\'e},
il suffit de le faire dans le cas d'un recollement de deux 3-cubes $K$
et $K^{\prime}$ le
long d'une 2-face commune $F$. Cela provient alors imm{\'e}diatement du
fait que, par hypoth{\`e}se, $K$ et $K^{\prime}$ sont des cubes et les morphismes
$m_F$ et $m_F^{\prime}$ associ{\'e}s {\`a} $F$, vu comme face de $K$ et $K^{\prime}$, sont
les m{\^e}mes.
\subsection{Structure de n-cube}
Soient $\mathcal{C}$ et $\mathcal{D}$ des cat{\'e}gories de Picard strictement
commutatives et $\gd :\mathcal{C}\longrightarrow\mathcal{D}$ un foncteur.
\begin{Def}
Une structure du n-cube sur le foncteur $\gd$ est la donn{\'e}e pour tout
n-cube $K$ de $\mathcal{C}$ d'un morphisme $\psi _K:\mathcal{O} \stackrel{\sim }{\F } \gd (K)$ dans
$\mathcal{C}$
v{\'e}rifiant les propri{\'e}t{\'e}s suivantes:
\begin{enumerate}
\item {\em Fonctorialit{\'e}.} Pour tout isomorphisme de n-cubes $f: K
\stackrel{\sim }{\F } K^{\prime}$, le diagramme induit
\[
\xymatrix{
& \gd (K)\ar[dd]^-{\gd (f)} \\
{\mathcal{O}} \ar[ur]^-{\psi _K} \ar[dr]_-{\psi _{K^{\prime}}} \\
& \gd (K^{\prime} )
}
\]
est commutatif.
\item {\em Recollements de cubes.} Soient $K$ et $K^{\prime}$ deux n-cubes
ayant leur i-{\`e}me (n-1)-face en commun,
l'isomorphisme naturel
$\gd (K) \otimes \gd (K^{\prime} ) \stackrel{\sim }{\F } \gd (K \ast _i K^{\prime} )$ induit un
diagramme commutatif:
\[
\xymatrix{
& \gd (K) \otimes \gd (K^{\prime} ) \ar[dd] \\
{\mathcal{O}} \ar[ur]^-{\psi _K \otimes\psi _{K^{\prime}} }
\ar[dr]_-{\psi _{(K \ast _i K^{\prime} )}} & \\
&\gd (K \ast _i K^{\prime} )
}
\]
\item {\em Propri{\'e}t{\'e} de sym{\'e}trie.} Pour tout
{\'e}l{\'e}ment $\sigma $ de $S_n$, le diagramme suivant, dont la
fl{\`e}che verticale
est donn{\'e}e par les isomorphismes de commutativit{\'e} de $\mathcal{D}$,
est commutatif:
\[
\xymatrix{
& \gd (K) \ar[dd] \\
{\mathcal{O}} \ar[ur]^-{\psi _K} \ar[dr]_-{\psi _{\sigma ^{\ast } K}} & \\
&\gd (\sigma ^{\ast } K )
}
\]
\end{enumerate}
\end{Def}
\begin{ex}
\label{caracteristique2}
Soit $X$ un sch{\'e}ma projectif de dimension $n$, consid{\'e}rons les
cat{\'e}gories $\mathcal{C} = PIC (X)$ et $\mathcal{D} = \mathbb{Z}$ (cat{\'e}gorie discr{\`e}te) et le
foncteur $ \gd : \mathcal{C} \longrightarrow \mathcal{D} , L \mapsto \chi (L)$. On a vu (Lemme
\ref{caracteristique}) que
$\gd$ est muni d'une structure du $(n+1)$-cube.
\end{ex}
\subsection{Le n-foncteur multilin{\'e}aire associ{\'e} {\`a} une structure du
(n+1)-cube}
\subsubsection{}
Consid{\'e}rons un foncteur $\gd : \mathcal{C} \longrightarrow \mathcal{D}$ entre deux cat{\'e}gories de
Picard strictement commutatives.
Pour tout (n-1)-cube $K$ dans $\mathcal{C}$ et tout
entier $1\leq
i\leq n$, d{\'e}finissons un foncteur:
\[
\L _{K,i}: \mathcal{C} \longrightarrow \mathcal{D} , L \mapsto \gd
(\xymatrix{K\ar@{-}[r]_-i&K\otimes L}).
\]
De m{\^e}me, pour tout objet $L$ de $\mathcal{C}$, d{\'e}finissons un $n$-foncteur:
\[
\L _L : \mathcal{C} ^n \longrightarrow \mathcal{D} , (L_1, \cdots , L_n) \mapsto \gd
(K_L (L_1, \cdots , L_n)).
\]
\addtocounter{theo}{1}
\begin{rem} $\L _L $ est canoniquement muni de donn{\'e}es de sym{\'e}trie
$\L _L \stackrel{\sim }{\F } \sigma ^{\ast } \L _L$, pour tout $\sigma \in S^n$.
\end{rem}
\addtocounter{subsubsection}{1}
\begin{rem}\label{identification}
Pour tout entier $1\leq i\leq n$ on peut identifier canoniquement:
\[
\L _L (L_1, \cdots , L_n)
\stackrel{\sim }{\F }
\L _{K_L(L_1,\cdots ,L_{i-1},L_{i+1},\cdots ,L_n),i} (L_i)
\]
\end{rem}
\addtocounter{subsubsection}{1}
\subsubsection{}
\label{def-multifonct}
\addtocounter{theo}{1}
Donnons nous une structure de $(n+1)$-cube $S$ sur $\gd$. Remarquons d'abord
que $S$ induit, pour tous $L,M \in \text{Ob} (\mathcal{C})$,
un isomorphisme canonique de foncteurs $ \L_L \stackrel{\sim }{\F } \L_M$. En effet,
pour tous $L, L_1, \cdots , L_n \in \text{Ob}(\mathcal{C} )$, on a
$
K_L (L_1, \cdots , L_n) \simeq K_{\mathcal{O}} (L_1, \cdots , L_n)\otimes L
$
et la structure du cube $S$, appliqu{\'e}e au $(n+1)$-cube
$
(\xymatrix{
K_{\mathcal{O}} (L_1, \cdots , L_n)
\ar@{-}[r]_-{n+1} &
K_{\mathcal{O}} (L_1, \cdots , L_n)\otimes L})
$
{\'e}tablit donc un isomorphisme canonique:
\[
\gd ( K_{\mathcal{O}} (L_1, \cdots , L_n))
\stackrel{\sim }{\F }
\gd ( K_L (L_1, \cdots , L_n))
\]
On peut donc d{\'e}sormais parler du foncteur $\L$ (en ommettant l'indice
$L$).
\subsubsection{}
\addtocounter{theo}{1}
La donn{\'e}e de la structure du cube sur $\gd$ permet de munir chaque
foncteur $\L _{K,i}$ d'une donn{\'e}e d'additivit{\'e}:
\[
\mu _{K,i}:\L _{K,i}(L) \otimes \L _{K,i}(M) \stackrel{\sim }{\F } \L _{K,i}(L\otimes M)
\]
d{\'e}finie par:
\begin{multline} \label{linearite}
\L _{K,i}(L) \otimes \L _{K,i}(M) =
\gd (\xymatrix{K\ar@{-}[r]_-i&K\otimes L})
\otimes
\gd (\xymatrix{K\ar@{-}[r]_-i&K\otimes M}) \stackrel{\sim }{\F } \\
\gd (\xymatrix{K\ar@{-}[r]_-i&K\otimes L})
\otimes
\gd (\xymatrix{K\ar@{-}[r]_-i&K\otimes M})
\otimes
\gd \left(
\begin{array}{c}
\xymatrix{K \otimes M \ar@{-}[d]_-{i+1}\ar@{-}[r]
&K\otimes (L \otimes M) \ar@{-}[d] \\
K\ar@{-}[r]_-i &K\otimes L }
\end{array}
\right)\\
\stackrel{\sim }{\F } \gd (\xymatrix{K\ar@{-}[r]_-i&K\otimes (L\otimes M)})
=\L _{K,i}(L\otimes M)
\end{multline}
En utilisant l'identification \ref{identification}, on voit que $S$
munit $\L$ de $n+1$ donn{\'e}es d'additivit{\'e} partielles:
\begin{multline*}
*_i :\L (L_1, \cdots ,L_{i-1} ,L_i ,L_{i+1}, \cdots ,L_n)
\otimes
\L (L_1, \cdots ,L_{i-1} ,L_i^{\prime} ,L_{i+1}, \cdots ,L_n)\\
\longrightarrow
\L (L_1, \cdots ,L_{i-1} ,L_i \otimes L_i^{\prime}
,L_{i+1}, \cdots ,L_n).
\end{multline*}
\begin{rem} \label{strict.com}
On notera que la cat{\'e}gorie de Picard $\mathcal{D}$ {\'e}tant
{\em strictement} commutative, le diagramme:
\[
\xymatrix{
(A^{\vee } \otimes A) \otimes A^{\vee } \ar[dr] \ar[dd]\\
& A^{\vee } \\
A^{\vee } \otimes (A \otimes A^{\vee } ) \ar[ur]
}
\]
est commutatif, ce qui permet de ne pas pr{\'e}ciser comment on effectue
les contractions dans (\ref{linearite}).
\end{rem}
\begin{lemme}\label{comm-assoc}
Les donn{\'e}es d'additivit{\'e} $\mu _{K,i}$ pour $\L _{K,i}$ sont
compatibles aux donn{\'e}es d'as\-so\-cia\-ti\-vi\-t{\'e} et de
commutativit{\'e} de $\mathcal{C}$
et $\mathcal{D}$.
\end{lemme}
\begin{proof}[Preuve]
Ces deux assertions se traduisent en disant que si $L$, $M$ et $N$ sont des
objets de $\mathcal{D}$, les isomorphismes canoniques
\[
\gd \left(
\begin{array}{c}
\xymatrix{K \otimes M \ar@{-}[d]_-{i+1}\ar@{-}[r]
&K\otimes (L \otimes M) \ar@{-}[d] \\
K\ar@{-}[r]_-i &K\otimes L }
\end{array}
\right)
\stackrel{\sim }{\F }
\gd \left(
\begin{array}{c}
\xymatrix{K \otimes L \ar@{-}[d]_-{i+1}\ar@{-}[r]
&K\otimes (M \otimes L) \ar@{-}[d] \\
K\ar@{-}[r]_-i &K\otimes M }
\end{array}
\right)
\]
et
\begin{multline} \label{associativite1}
\gd \left(
\begin{array}{c}
\xymatrix{K \otimes M \ar@{-}[d]_-{i+1}\ar@{-}[r]
&K\otimes (L \otimes M) \ar@{-}[d] \\
K\ar@{-}[r]_-i &K\otimes L }
\end{array}
\right)
\otimes
\gd \left(
\begin{array}{c}
\xymatrix{K \otimes N \ar@{-}[d]_-{i+1}\ar@{-}[r]
&K\otimes (L \otimes M\otimes N) \ar@{-}[d] \\
K\ar@{-}[r]_-i &K\otimes (L\otimes M) }
\end{array}
\right)\\
\stackrel{\sim }{\F }
\gd \left(
\begin{array}{c}
\xymatrix{K \otimes N \ar@{-}[d]_-{i+1}\ar@{-}[r]
&K\otimes (M \otimes N) \ar@{-}[d] \\
K\ar@{-}[r]_-i &K\otimes M }
\end{array}
\right)
\otimes
\gd \left(
\begin{array}{c}
\xymatrix{K \otimes (M\otimes N) \ar@{-}[d]_-{i+1}\ar@{-}[r]
&K\otimes (L \otimes M\otimes N) \ar@{-}[d] \\
K\ar@{-}[r]_-i &K\otimes L }
\end{array}
\right)
\end{multline}
identifient les trivialisations de chacun des termes, donn{\'e}es par la
structure du (n+1)-cube.\\
Le premier point provient de l'hypoth{\`e}se que la structure du cube est
sym{\'e}trique et de la remarque (\ref{strict.com}).\\
Pour l'associativit{\'e}, remarquons d'abord que chacun des membres de
\ref{associativite1} est isomorphe {\`a}
\begin{multline} \label{associativite3}
\gd \left(
\begin{array}{c}
\xymatrix{K \otimes N \ar@{-}[d]_-{i+1}\ar@{-}[r]
&K\otimes (M \otimes N) \ar@{-}[d] \\
K\ar@{-}[r]_-i &K\otimes M }
\end{array}
\right)
\otimes
\gd \left(
\begin{array}{c}
\xymatrix{K \otimes (M\otimes N) \ar@{-}[d]_-{i+1}\ar@{-}[r]
&K\otimes (L \otimes M\otimes N) \ar@{-}[d] \\
K\otimes M \ar@{-}[r]_-i &K\otimes (L\otimes M) }
\end{array}
\right)\\
\otimes
\gd \left(
\begin{array}{c}
\xymatrix{K \otimes M \ar@{-}[d]_-{i+1}\ar@{-}[r]
&K\otimes (L \otimes M) \ar@{-}[d] \\
K\ar@{-}[r]_-i &K\otimes L }
\end{array}
\right).
\end{multline}
Consid{\'e}rons alors l'empilement de (n+1)-cubes suivant:
\begin{equation}
\label{associativite2}
\xymatrix{
K \otimes N \ar@{-}[d]_-{i+1}\ar@{-}[r]
&K\otimes (M \otimes N) \ar@{-}[d]_-{i+1} \ar@{-}[r]
&K\otimes (L\otimes M \otimes N) \ar@{-}[d] \\
K \ar@{-}[r]_-i
&K\otimes M \ar@{-}[d] \ar@{-}[r]_-i \ar@{-}[d]_-{i+1}
&K\otimes (L\otimes M ) \ar@{-}[d] \\
&K \ar@{-}[r]_-i &K\otimes L
}
\end{equation}
Il r{\'e}sulte alors de la propri{\'e}t{\'e} 2. d'une structure du cube,
appliqu{\'e}e aux deux fa\c{c}ons d'effectuer des recollements dans
(\ref{associativite2}) que les
trivialisations
des membres de (\ref{associativite1}) sont identifi{\'e}es aux
trivialisations de (\ref{associativite3}).
\end{proof}
De ces diff{\'e}rents r{\'e}sultats, on d{\'e}duit la:
\begin{prop}
La donn{\'e}e d'une structure du (n+1)-cube $S$ sur $\gd: \mathcal{C} \longrightarrow \mathcal{D}$
munit le $n$-foncteur associ{\'e} $\L : \mathcal{C} ^n \longrightarrow \mathcal{D}$ de donn{\'e}es
d'additivit{\'e} $\ast _i$ en chacune des $n$ variables. Ces donn{\'e}es
sont compatibles aux donn{\'e}es d'associativit{\'e} et de
commutativit{\'e} de
$\mathcal{C}$ et $\mathcal{D}$ ainsi qu'aux donn{\'e}es de sym{\'e}trie
de $\L$ et sont compatibles entre elles.
\end{prop}
\begin{proof}[Preuve]
Les questions d'associativit{\'e} et de commutativit{\'e} proviennent des
r{\'e}sultats analogues pour $\L _{K,i}$ (lemme \ref{comm-assoc}).\\
Pour simplifier les notations exprimons la compatibilit{\'e} des $\ast _i$
entre elles dans le cas $n=2$. Cela se traduit par la commutativit{\'e} du
diagramme:
\[
\begin{CD}
\L (L,N) \otimes \L (L,P) \otimes \L (M,N) \otimes \L (M,P)
@>{\ast _1 \otimes \ast _1}>>
\L (LM,N) \otimes \L (LM,P)\\
@V{\ast _2 \otimes \ast _2}VV @V{\ast _2 }VV \\
\L (L,NP) \otimes \L (M,NP)
@>{\ast _1}>>
\L (LM,NP)
\end{CD}
\]
\textsc{Breen} montre dans \cite{B2},2.5 que la commutativit{\'e} de ce
diagramme est
une cons{\'e}quence de l'associativit{\'e} de $\ast _1$ et $\ast
_2$. En effet,
cela se traduit en disant que l'isomorphisme canonique d{\'e}duit des
morphismes de contraction:
\begin{multline*}
\ga : \gd (K(LM,N,P)) \otimes \gd (K(L,M,N)) \otimes \gd (K(L,M,P))\\
\stackrel{\sim }{\F }
\gd (K(L,M,NP)) \otimes \gd (K(L,N,P)) \otimes \gd (K(M,N,P))
\end{multline*}
identifie les trivialisations de chacun des deux termes qui sont
d{\'e}duites de la structure du cube. Cette assertion provient alors du
fait que $\ga$ se d{\'e}compose en
\[
\begin{CD}
\gd (K(LM,N,P)) \otimes \gd (K(L,M,N)) \otimes \gd (K(L,M,P))\\
@VVV\\
\gd (K(L,MN,P)) \otimes \gd (K(M,N,P)) \otimes \gd (K(L,M,N))\\
@AAA\\
\gd (K(L,M,NP)) \otimes \gd (K(L,N,P)) \otimes \gd (K(M,N,P))
\end{CD}
\]
provenant des morphismes d'associativit{\'e} de $\ast _1$ et $\ast _2$.\\
Pour traduire la compatibilit{\'e} des donn{\'e}es d'additivit{\'e} avec les
donn{\'e}es de sym{\'e}trie, consid{\'e}rons des objets $L_1, \cdots ,
L_n,L,L^{\prime}$ de
$\mathcal{C}$ et $\sigma \in S_n$ et notons
$\underline{L}_i= (L_1,\cdots,L_{i-1},L,L_{i+1},\cdots ,L_n)$,
$\underline{L^{\prime}}_i= (L_1,\cdots,L_{i-1},L^{\prime},L_{i+1},\cdots ,L_n)$ et
$\underline{L\sec}_i= (L_1,\cdots,L_{i-1},L\otimes L^{\prime}
,L_{i+1},\cdots ,L_n)$ et $j=\sigma ^{-1} (i)$.
On veut montrer que le diagramme suivant, dont les fl{\`e}ches verticales
proviennent des donn{\'e}es de sym{\'e}trie de $\L$
\[
\begin{CD}
\L (\sigma ^{\ast } \underline{L}_i )
\otimes
\L (\sigma ^{\ast } \underline{L^{\prime}}_i ) @>{\ast _j}>>
\L (\sigma ^{\ast } \underline{L\sec}_i ) \\
@VVV @VVV \\
\L ( \underline{L}_i )
\otimes
\L ( \underline{L^{\prime}}_i ) @>{\ast _j}>>
\L ( \underline{L\sec}_i )
\end{CD}
\]
est commutatif. Ceci provient de la propri{\'e}t{\'e} de sym{\'e}trie de la
structure de $(n+1)$-cube, appliqu{\'e}e au $(n+1)$-cube
$C= (K(L_1,\cdots,L_{i-1},L, L^{\prime},L_{i+1},\cdots ,L_n)$ et {\`a} la
permutation $\tau \in S_{n+1} $ telle que
$\tau ^{\ast } C =
K(L_{\sigma (1)},\cdots,L_{\sigma (i-1)},L, L^{\prime},L_{\sigma (i+1)},\cdots
,L_{\sigma (n)})$.
\end{proof}
\section{Les cat{\'e}gories de Picard consid{\'e}r{\'e}es}
Dans ce chapitre, on consid{\`e}re un morphisme projectif et plat $\pi :
X \longrightarrow S$ sur un sch{\'e}ma localement noeth{\'e}rien et on introduit les
cat{\'e}gories de Picard qui nous serviront dans la suite.
\subsection{}
$\mathcal{C}$ d{\'e}signera la
cat{\'e}gorie $PIC (X)$ dont les objets sonts les faisceaux inversibles
sur $X$, les fl{\`e}ches sont les isomorphismes de $\mathcal{O}_{X}$-modules,
$\otimes$ d{\'e}signe le produit tensoriel usuel de deux $\mathcal{O}_{X}$-modules et
les morphismes d'associativit{\'e} et de commutativit{\'e} sont ceux
usuels. Les objets unit{\'e}s et inverses seront simplement le faisceau
structural $\mathcal{O}_{X}$ et le faisceau dual $L^{-1}$.
\subsection{}
$\mathcal{D}$ d{\'e}signera la
cat{\'e}gorie $PICgr (S)$ dont les objets sont les couples $(L,d)$ form{\'e}s
d'un faisceau inversible $L$
sur $S$ et d'une application localement constante $d: X \longrightarrow \mathbb{Z}$, les
fl{\`e}ches entre deux objets $(L,d)$ et $(M,e)$ n'existent que si $d=e$
et sont dans ce cas les isomorphismes entre les $\mathcal{O}_{S}$-modules $L$ et
$M$. Le produit tensoriel sera alors d{\'e}fini par
\[
(L,d) \otimes (M,e) = (L\otimes _{\ox} M , d+e).
\]
Les morphismes de commutativit{\'e}
$\psi : (L,d) \otimes (M,e) \longrightarrow (M,e) \otimes (L,d)$
sont donn{\'e}s par:
$\psi : L \otimes _{\os} M \longrightarrow M \otimes _{\os} L : l \otimes m \mapsto (-1)^{d.e} m\otimes
l$. On notera que la cat{\'e}gorie $PICgr (S)$ n'est pas strictement
commutative.\\
L'objet unit{\'e} de $\mathcal{D}$ sera alors $\mathcal{O} = (\mathcal{O}_{S} ,0)$ et l'inverse de
$(L,d)$ sera $(L^{-1} ,-d)$. Le morphisme d'{\'e}valuation
$(L,d) \otimes (L^{-1} ,-d)\longrightarrow \mathcal{O}$ est donn{\'e} par l'{\'e}valuation usuelle
$L\otimes _{\os} L^{-1} \simeq \mathcal{O}_{S}$. On prendra garde
que le morphisme
$(L^{-1} ,-d) \otimes (L,d)\longrightarrow \mathcal{O}$ est donn{\'e} par le morphisme usuel
d'{\'e}valuation multipli{\'e} par $(-1)^d$.\\
On consid{\`e}rera enfin le foncteur $\gd : \mathcal{C} \longrightarrow \mathcal{D}$ donn{\'e} par le
d{\'e}terminant de l'image directe d{\'e}riv{\'e}e dont l'existence,
annonc{\'e}e par \textsc{Grothendieck}, est
montr{\'e}e par \textsc{Knudsen} et \textsc{Mumford} dans \cite{KM}.
$\gd$ associe {\`a} tout $\mathcal{O}_{X}$-module inversible
$L$, le $\mathcal{O}_{S}$-module $\det \text{R}\pi \lst L $, gradu{\'e}
par la fonction localement constante $ s \mapsto \chi (L |
_{X_s})$.\\
En r{\'e}alit{\'e}, le foncteur $\gd$ est d{\'e}fini
sur la cat{\'e}gorie plus {\'e}tendue $COH (X/S)$ des $\mathcal{O}_{X}$-modules
coh{\'e}rents
et plats sur $S$. Il v{\'e}rifie les propri{\'e}t{\'e}s suivantes:
\subsubsection{}
Toute suite exacte dans $COH (X/S)$:
\[
0 \longrightarrow E \longrightarrow F \longrightarrow G \longrightarrow 0
\]
induit un isomorphisme canonique de multiplicativit{\'e}
$\gd (E) \otimes \gd (G) \stackrel{\sim }{\F } \gd (F)$
et pour tout diagramme de suites exactes courtes:
\[
\begin{CD}
@. 0 @. 0 @. 0 \\
@.@VVV @VVV @VVV\\
0 @>>> E^{\prime} @>>>F^{\prime} @>>> G^{\prime} @>>>0\\
@. @VVV @VVV @VVV\\
0 @>>> E @>>>F @>>> G @>>>0\\
@. @VVV @VVV @VVV\\
0 @>>> E\sec @>>>F\sec @>>> G\sec @>>>0\\
@. @VVV @VVV @VVV\\
@. 0 @. 0 @.0
\end{CD}
\]
le diagramme
\begin{equation}
\label{diagramme-des-neuf}
\begin{CD}
\gd (E^{\prime} ) \otimes \gd (E\sec ) \otimes \gd (G^{\prime} )\otimes \gd (G\sec
)
@>>> \gd (E) \otimes \gd (G)\\
@VVV @VVV \\
\gd (F^{\prime} ) \otimes \gd (F\sec ) @>>> \gd (F)
\end{CD}
\end{equation}
qu'on en d{\'e}duit par application des morphismes de multiplicativit{\'e} et
de commutativit{\'e} est commutatif
(\cite{KM},prop.1).
Notons que c'est la d{\'e}finition des isomorphismes de commutativit{\'e} dans
$\mathcal{D}$ qui rend possible la commutativit{\'e} du diagramme pr{\'e}c{\'e}dent.
\subsubsection{}Si $E$ est un $\mathcal{O}_{X}$-module coh{\'e}rent et plat sur $S$, {\`a}
support dans un sous-sch{\'e}ma $Y$ de $X$, on a: $\det \text{R}\pi \lst (E) = \det
\text{R} (\pi _{Y/S}) _{\ast } (E|_Y)$.
\subsubsection{} La formation de $\det \text{R}\pi \lst $ commute aux changements de base.
\subsection{} On consid{\`e}rera enfin la cat{\'e}gorie de Picard
strictement
commutative $\mathcal{D}^{\prime} =PIC (S)$ et on notera $\delta ^{\prime}$ le compos{\'e} du
foncteur $\gd = \det \text{R}\pi \lst : \mathcal{C} \longrightarrow \mathcal{D}$ avec le foncteur oubli de la
graduation $\mathcal{D} \longrightarrow \mathcal{D}^{\prime}$. On notera qu'en travaillant dans cette
cat{\'e}gorie $\mathcal{D}^{\prime}$ on gagne le fait qu'elle est strictement
commutative, mais
on perd la possibilit{\'e} d'{\'e}crire certains diagrammes
commutatifs du
paragraphe pr{\'e}c{\'e}dent, comme par exemple (\ref{diagramme-des-neuf}),
qui ne s'expriment naturellement que gr{\^a}ce au foncteur $\gd$.
\subsection{Structure du cube dans la cat{\'e}gorie des faisceaux
inversibles gradu{\'e}s}
Pour montrer l'existence {\'e}ventuelle d'un structure du $p$-cube sur le
foncteur $\delta ^{\prime} : \mathcal{C} \longrightarrow \mathcal{D}^{\prime}$ entre
cat{\'e}gories de Picard strictement commutatives, il sera n{\'e}cessaire de
passer par l'interm{\'e}diaire du foncteur $\gd : \mathcal{C} \longrightarrow \mathcal{D}$ {\`a}
valeurs
dans une cat{\'e}gorie de Picard non strictement commutative. Examinons
ici comment les axiomes d'une structure de $p$-cube sur $\delta ^{\prime}$ se
traduisent en termes de $\gd$.
\begin{nota}
Si $K$ est un $p$-cube dans $\mathcal{C}$, pour tout couple d'indices
distincts $i$ et $j$, les isomorphismes de commutativit{\'e} dans $\mathcal{D}$
induisent un isomorphisme $\gd (\sigma _{ij}^{\ast } K)\longrightarrow \gd (K)$ dans
$\mathcal{D}$, qui induit, par oubli de la graduation, un isomorphisme
$\delta ^{\prime} (\sigma _{ij}^{\ast } K)\longrightarrow \delta ^{\prime} (K)$ dans
$\mathcal{D}^{\prime}$. Celui ci diff{\`e}re de celui induit par les isomorphismes de
commutativit{\'e} dans $\mathcal{D}^{\prime}$ par un signe, que l'on notera $\ge _{ij}(K)$.
\end{nota}
\begin{rem}
Si $X\longrightarrow S$ est {\`a} fibres de dimension $n$, pour tout $(n+2)$-cube
\begin{equation}
\label{ecriture-cube}
K=\left(
\begin{array}{c}
\xymatrix{C\ar@{-}[d]_-j\ar@{-}[r]
&D \ar@{-}[d] \\
A\ar@{-}[r]_-i &B }
\end{array}
\right) ,
\end{equation}
on a
$\chi (A) = \chi (B) =
\chi (C) = \chi (D)$
(d'apr{\`e}s l'exemple
\ref{caracteristique2}) et de plus
$\ge _{ij}(K)=(-1)^{\chi (A)}$.
\end{rem}
\addtocounter{subsubsection}{2}
\subsubsection{}
\addtocounter{theo}{1}
Soit $K$ un $p$-cube dans $\mathcal{C}$ et soit $t: \mathcal{O}_{S} \stackrel{\sim }{\F } \delta ^{\prime} (K)$ une
trivialisation de $\delta ^{\prime} (K)$. Pour tout entier $1\leq i\leq p$, $t$
induit un isomorphisme
$t_i: \delta ^{\prime} (\phi^{\prime\ast}_i K) \stackrel{\sim }{\F } \delta ^{\prime} (\phi^{\prime \prime\ast}_i K)$ dans
$\mathcal{D}^{\prime}$.
Le choix de l'ordre usuel sur les sommets de
$\phi^{\prime\ast}_i K$ et $\phi^{\prime \prime\ast}_i K$
associe {\`a} $t_i$ un
isomorphisme
$s_i: \gd (\phi^{\prime\ast}_i K) \stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i K)$
dans $\mathcal{D}$. Exprimons alors, {\`a} l'int{\'e}rieur de la
cat{\'e}gorie $\mathcal{D}^{\prime}$ les
relations entre $s_i$ et $s_j$, pour $i\neq j$. On peut associer
{\`a} $s_i$ le
morphisme dans $\mathcal{D}$:
\[
\begin{CD}
\overline{s_i}:\mathcal{O} @>>>
(\gd (\phi^{\prime\ast}_i K))^{-1} \otimes \gd ( \phi^{\prime\ast}_i K)
@>{\text{Id}\otimes s_i}>>
(\gd (\phi^{\prime\ast}_i K))^{-1} \otimes \gd ( \phi^{\prime \prime\ast}_i K)
@>>>
\gd (K)
\end{CD}
\]
\begin{lemme}
La donn{\'e}e d'une trivialisation $t$ de $\delta ^{\prime}(K)$ est
{\'e}quivalente {\`a} la donn{\'e}e d'une collection $(s_i)_{1\leq i\leq p}$
d'isomorphismes
$s_i: \gd (\phi^{\prime\ast}_i K) \stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i K)$ dans $\mathcal{D}$ telle que, pour
deux indices $i$ et $j$ distincts, les trivialisations
$\overline{s_i}$ et $\overline{s_j}$
de $\gd (K)$ induites par $s_i$ et $s_j$
diff{\`e}rent d'un signe $\ge _{ij}(K)$.
\end{lemme}
\addtocounter{subsubsection}{1}
\begin{proof}[Preuve]
Reprenons la notation (\ref{ecriture-cube}). Le
diagramme, dont les fl{\`e}ches verticales sont
donn{\'e}es par les isomorphismes les isomorphismes structuraux de $\mathcal{D}^{\prime}$
est commutatif:
\[
\begin{CD}
(\delta ^{\prime} (A) \otimes _{\os} \delta ^{\prime} (C)^{\vee } ) \otimes _{\os} (\delta ^{\prime} (C)\otimes _{\os}\delta ^{\prime} (D))
@>{t_i\otimes \text{id} }>>
(\delta ^{\prime} (B) \otimes _{\os} \delta ^{\prime} (D)^{\vee } ) \otimes _{\os} (\delta ^{\prime} (C)\otimes _{\os}\delta ^{\prime} (D))\\
@VVV @VVV \\
(\delta ^{\prime} (A) \otimes _{\os} \delta ^{\prime} (D) ) \otimes _{\os} (\delta ^{\prime} (C)\otimes _{\os}\delta ^{\prime} (C)^{\vee } )
@.
(\delta ^{\prime} (B) \otimes _{\os} \delta ^{\prime} (C)^{\vee } ) \otimes _{\os} (\delta ^{\prime} (D)\otimes _{\os}\delta ^{\prime} (D)^{\vee } )\\
@VVV @VVV \\
(\delta ^{\prime} (A) \otimes _{\os} \delta ^{\prime} (D) ) @>t>> (\delta ^{\prime} (B) \otimes _{\os} \delta ^{\prime} (C) ) \\
@AAA @AAA \\
(\delta ^{\prime} (A) \otimes _{\os} \delta ^{\prime} (D) ) \otimes _{\os} (\delta ^{\prime} (B)\otimes _{\os}\delta ^{\prime} (B)^{\vee } )
@.
(\delta ^{\prime} (C) \otimes _{\os} \delta ^{\prime} (B)^{\vee } ) \otimes _{\os} (\delta ^{\prime} (D)\otimes _{\os}\delta ^{\prime} (D)^{\vee } )\\
@AAA @AAA \\
(\delta ^{\prime} (A) \otimes _{\os} \delta ^{\prime} (B)^{\vee } ) \otimes _{\os} (\delta ^{\prime} (B)\otimes _{\os}\delta ^{\prime} (D))
@>{t_j\otimes \text{id} }>>
(\delta ^{\prime} (C) \otimes _{\os} \delta ^{\prime} (D)^{\vee } ) \otimes _{\os} (\delta ^{\prime} (B)\otimes _{\os}\delta ^{\prime} (D))
\end{CD}
\]
Consid{\'e}rons le diagramme obtenu en rempla\c{c}ant dans le
pr{\'e}c{\'e}dent $\delta ^{\prime}$ par
$\gd$ et en utilisant les morphismes structuraux de $\mathcal{D}^{\prime}$. Son circuit
ext{\'e}rieur est donc
commutatif {\`a} un signe $(\ge _{ij})^k$ pr{\`e}s, o{\`u} $k$ est le nombre de
transpositions apparaissant dans le
diagramme. On constate alors que ce nombre est impair.\\
R{\'e}ciproquement, si $(s_i)_{1\leq i\leq p}$ est une telle collection de
morphismes, chaque $s_i$ induit un isomorphisme
$t_i:\delta ^{\prime} \left(\phi^{\prime\ast}_i K \right)
\stackrel{\sim }{\F }
\delta ^{\prime}\left( \phi^{\prime \prime\ast}_i K \right)$ dans $\mathcal{D}^{\prime}$. La trivialisation de $\delta ^{\prime}
(K)$ dans $\mathcal{D}^{\prime}$ est alors ind{\'e}pendante de $i$.
\end{proof}
\subsubsection{Recollements de cubes}
\addtocounter{theo}{1}
Soient $K$ et $K^{\prime}$ deux $p$-cubes dans $\mathcal{C}$ et $i$ un indice tel que
$\phi^{\prime\ast}_i K = \phi^{\prime \prime\ast}_i K$. Si $t , t^{\prime}$ sont des
trivialisations de $\delta ^{\prime} (K)$ et $\delta ^{\prime} (K^{\prime} )$, elles induisent une
trivialisation
$t \ast_i t^{\prime}: \mathcal{O}_{X} \stackrel{\sim }{\F } \mathcal{O}_{X} \otimes \mathcal{O}_{X}
\stackrel{\sim }{\F } \delta ^{\prime} (K) \otimes \delta ^{\prime} (K^{\prime} ) \stackrel{\sim }{\F } \delta ^{\prime} (K\ast_i K^{\prime} )$.
Notons
$s_i: \gd (\phi^{\prime\ast}_i K) \stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i K)$,
$s^{\prime}_i: \gd (\phi^{\prime\ast}_i K^{\prime} ) \stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i K^{\prime} )$ et
$s\sec_i: \gd (\phi^{\prime\ast}_i (K\ast _i K^{\prime} ))
\stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i
(K\ast _i K^{\prime} ))$ les isomorphismes dans $\mathcal{D}^{\prime}$ associ{\'e}s {\`a} $t$, $t^{\prime}$
et $t \ast_i t^{\prime}$. On a alors $s\sec_i =s^{\prime}_i \circ s_i$.
On peut alors regrouper les r{\'e}sultats de ce paragraphe dans la
\begin{prop} \label{cubebis}
Une structure de $p$-cube sur $\delta ^{\prime}$ est {\'e}quivalente {\`a} la donn{\'e}e,
pour tout $p$-cube $K$ dans $\mathcal{C}$, de $p$ isomorphismes dans $\mathcal{D}$:
\[
s_{K,i}:\gd \left(\phi^{\prime\ast}_i K \right)
\stackrel{\sim }{\F }
\gd\left( \phi^{\prime \prime\ast}_i K \right)
\]
tels que:
\begin{enumerate}
\item Pour tout isomorphisme de $p$-cubes $f: K \stackrel{\sim }{\F } K^{\prime}$, le
diagramme induit
\[
\begin{CD}
\gd \left( \phi^{\prime\ast}_i K \right)
@>>>
\gd\left( \phi^{\prime \prime\ast}_i K \right)\\
@VVV @VVV\\
\gd \left( \phi^{\prime\ast}_i K^{\prime} \right)
@>>>
\gd\left( \phi^{\prime \prime\ast}_i K^{\prime} \right)
\end{CD}
\]
est commutatif.
\item Soient $K$ et $K^{\prime}$ deux $p$-cubes ayant leur i-{\`e}me $(p-1)$-face
en commun, on a l'{\'e}galit{\'e}:
\[
s_{K\ast _i K^{\prime} ,i} = s_{ K^{\prime} ,i} \circ s_{K ,i}
\]
\item Les trivialisations de $\gd (K)$ induites par $s_{K,i}$ et $s_{K,j}$
diff{\`e}rent d'un signe $\ge _{ij}(K)$.
\item Pour toute permutation $\sigma _{ij} \in S_p$ de deux indices
distincts $i$ et $j$, on a:
\[
s_{\sigma _{ij}^{\ast } (K),i} = s_{K,i}
\]
o{\`u} d{\'e}signe la permutation des indices $i$ et $j$.
\end{enumerate}
\end{prop}
\section{Constructions de structures du cube}
\subsection{Cas de la dimension 0} \label{norme}
Si $\pi : X \longrightarrow S$ est un morphisme fini et plat, le foncteur $\gd =\det \text{R}\pi \lst $ se
r{\'e}duit au foncteur:
\[
\gd : PIC (X) \longrightarrow PICgr (S)\; ,\; L
\mapsto
\left(\det (\pi _{\ast } L), \deg \pi \right)
.
\]
et $\gd ^{\prime} : PIC (X) \longrightarrow PIC (S)$ est simplement le foncteur:
$ L\mapsto \det (\pi _{\ast } L)$.
\subsubsection{Norme }
Rappellons quelques propri{\'e}t{\'e}s de la norme (cf \cite{EGA2},6.5 et
\cite{FD1},3.1) pour
un morphisme fini et plat.
Notons d'abord que, comme $X$ est fini sur $S$, pour tout faisceau
inversible $L$ sur $X$ et tout $s\in S$, il
existe un ouvert
$U$ de $S$ contenant $s$ tel que $L$ est trivial sur $X_U$. Si $\ga$
est une section inversible de $\mathcal{O}_{X}$, c'est {\`a} dire un automorphisme de
$\mathcal{O}_{X}$, la norme $N_{X/S} (\ga )$ est le d{\'e}terminant de l'automorphisme
de $\det \pi _{\ast } \mathcal{O}_{X}$ induit par $\ga$. Enfin la norme du faisceau
inversible $L$ est par d{\'e}finition le $\mathcal{O}_{S}$-module inversible
\[
N_{X/S} (L) =
(\det \pi _{\ast } L) \otimes _{\os} (\det \pi _{\ast } \mathcal{O}_{X} )^{-1}
\]
Si $(U_i)_{i\in I}$ est un recouvrement ouvert de $S$ tel que $L$ est
trivial sur $X_{U_i}$ avec des fonctions de transition $(g_{ij})_{i,j
\in I}$, alors
$N_{X/S} (L)$ est trivial sur chaque $U_i$ et a pour fonctions de
transition les $N_{X/S}(g_{ij})$.
\subsubsection{}
Construisons une structure du 2-cube (carr{\'e}), qui traduit les
propri{\'e}t{\'e}s de la norme pour un morphisme fini et
plat:
Il s'agit de construire, pour tous faisceaux inversiblest $L$, $M$,
$N$ et $P$ sur $X$ et tout
isomorphisme $\phi : L \otimes _{\ox} M \stackrel{\sim }{\F } N \otimes _{\ox} P$, un isomorphisme
\[
(\det \pi _{\ast } L) \otimes _{\os} (\det \pi _{\ast } M)
\stackrel{\sim }{\F }
(\det \pi _{\ast } N) \otimes _{\os} (\det \pi _{\ast } P) .
\]
D'apr{\`e}s les remarques de la
section pr{\'e}c{\'e}dente, il suffit de construire pour tout isomorphisme
$\phi : \mathcal{O}_{X} \otimes _{\ox} \mathcal{O}_{X} \stackrel{\sim }{\F } \mathcal{O}_{X} \otimes _{\ox} \mathcal{O}_{X}$ un isomorphisme
$ \gd _{\phi}:\det (\pi _{\ast } \mathcal{O}_{X} ) \otimes _{\os} \det (\pi _{\ast } \mathcal{O}_{X} )
\stackrel{\sim }{\F }
\det (\pi _{\ast } \mathcal{O}_{X} ) \otimes _{\os} \det (\pi _{\ast } \mathcal{O}_{X} )$ tel que si $\phi$ et
$\psi$ sont deux tels isomorphismes et
$\ga$, $\gb$, $\gc$ et $\gd$ des automorphismes de $\mathcal{O}_{X}$ tels que le
diagramme
\[
\begin{CD}
\mathcal{O}_{X} \otimes _{\ox} \mathcal{O}_{X} @>\phi >> \mathcal{O}_{X} \otimes _{\ox} \mathcal{O}_{X} \\
@V{\ga \otimes \gb}VV @VV{\gc \otimes \gd}V\\
\mathcal{O}_{X} \otimes _{\ox} \mathcal{O}_{X} @>\psi >> \mathcal{O}_{X} \otimes _{\ox} \mathcal{O}_{X}
\end{CD}
\]
est commutatif, alors le diagramme induit:
\[
\begin{CD}
(\det \pi _{\ast } \mathcal{O}_{X} ) \otimes _{\os} (\det \pi _{\ast } \mathcal{O}_{X} )
@>{\det \pi _{\ast }\phi}>>
(\det \pi _{\ast } \mathcal{O}_{X} ) \otimes _{\os} (\det \pi _{\ast } \mathcal{O}_{X} ) \\
@V{N_{X/S} (\ga ) \otimes N_{X/S}(\gb )}VV
@VV{N_{X/S}(\gc ) \otimes N_{X/S}(\gd )}V \\
(\det \pi _{\ast } \mathcal{O}_{X} ) \otimes _{\os} (\det \pi _{\ast } \mathcal{O}_{X} )
@>{\det \pi _{\ast }\psi}>>
(\det \pi _{\ast } \mathcal{O}_{X} ) \otimes _{\os} (\det \pi _{\ast } \mathcal{O}_{X} )
\end{CD}
\]
est commutatif.
$\phi$ est un automorphisme de $ \mathcal{O}_{X} \otimes _{\ox} \mathcal{O}_{X}$, c'est {\`a} dire une
section inversible de $\mathcal{O}_{X}$. On peut consid{\'e}rer la section inversible
$N_{X/S}(\phi )$ de $\mathcal{O}_{S}$, qui d{\'e}finit donc l'automorphisme recherch{\'e}
de $(\det \pi _{\ast } \mathcal{O}_{X} ) \otimes _{\os} (\det \pi _{\ast } \mathcal{O}_{X} )$. La condition de
fonctorialit{\'e} se traduit en disant que, si $\ga , \gb , \gc ,\gd , \phi
,\psi$ sont des sections inversibles de $\mathcal{O}_{X}$ telles que
$\ga\gb\psi =\gc\gd\phi$, alors
\[
N_{X/S}(\ga ) N_{X/S}(\gb) N_{X/S}(\psi )
= N_{X/S}(\gc ) N_{X/S}(\gd ) N_{X/S}(\phi )\; ,
\]
ce qui est simplement la
mutiplicativit{\'e} de la norme.\\
Le lemme suivant est une cons{\'e}quence imm{\'e}diate de la
d{\'e}finition de la
norme d'un faisceau inversible et des propri{\'e}t{\'e}s de
multiplicativit{\'e}
de la norme.
\addtocounter{theo}{2}
\begin{lemme}
\begin{enumerate}
\item La construction pr{\'e}c{\'e}dente d{\'e}termine une structure
du carr{\'e} sym{\'e}trique sur $\delta ^{\prime}$.
\item Le foncteur lin{\'e}aire $PIC(X) \longrightarrow PIC (S)$ d{\'e}duit de cette
structure est la norme relativement au morphisme fini et plat $\pi$.
\end{enumerate}
\end{lemme}
\begin{rem}
\label{carre}
Consid{\'e}rons un carr{\'e}
$
\left(
\begin{array}{c}
\xymatrix{N\ar@{-}[r]\ar@{-}[d]&P\ar@{-}[d]\\
L\ar@{-}[r]&M
}
\end{array}
\right)
$
dans $PIC(X)$ correspondant {\`a} un isomorphisme
$\phi: L\otimes P \stackrel{\sim }{\F } M \otimes N$ et soient des isomorphismes
$\ga : L \stackrel{\sim }{\F } M$ et $\gb : N \stackrel{\sim }{\F } P$ tels que le diagramme
\[
\begin{CD}
L\otimes P @>{\phi}>> M \otimes N \\
@V{\ga \otimes \text{id}}VV @V{\text{id}\otimes \gb}VV \\
M \otimes P @= M \otimes P
\end{CD}
\]
soit commutatif, alors le diagramme suivant, obtenu par application de
$\det \pi _{\ast }$ l'est aussi:
\[
\begin{CD}
\det\pi_{\ast } L\otimes \det\pi_{\ast } P @>{\gd_{\phi}}>>
\det\pi_{\ast } M \otimes \det\pi_{\ast } N \\
@V{\det (\ga )\otimes \text{id}}VV @V{\text{id}\otimes \det (\gb )}VV \\
\det\pi_{\ast } M \otimes \det\pi_{\ast } P @=
\det\pi_{\ast } M \otimes \det\pi_{\ast } P
\end{CD}
\]
\end{rem}
\subsection{Restriction {\`a} un diviseur effectif}
\label{restriction}
Si $\pi : X\longrightarrow S$ un morphisme projectif et plat sur $S$ localement
noeth{\'e}rien et $D$ est un diviseur
relatif sur $X$, si $K$ est un p-cube dans $\mathcal{C} = PIC(X)$, consid{\'e}rons
le (p+1)-cube
$A =( \xymatrix{K \ar@{-}[r]_-i &K\otimes \mathcal{O} (D) })$. On a alors un
isomorphisme canonique
$r_i: \gd (K^{\prime} ) \stackrel{\sim }{\F } \gd (K\otimes \mathcal{O} (D)|_D)$ dans $\mathcal{D} = PICgr (S)$,
qu'on appellera
{\em isomorphisme de restriction}. Il est donn{\'e} en appliquant, pour
chaque sommet
$L$ de $K$, le foncteur $\gd$ {\`a} la suite exacte:
\[
0 \longrightarrow L \longrightarrow L(D) \longrightarrow L(D)|_D \longrightarrow 0
\]
\subsubsection{Application au cas des courbes}
\label{cas-des-courbes}
Soit $\pi :X\longrightarrow S$ un morphisme projectif et plat, {\`a} fibres de dimension
1, sur un sch{\'e}ma localement noeth{\'e}rien $S$. Soient $D$ et $E$ deux
diviseurs de Cartier relatifs effectifs et $L$ un
faisceau inversible sur $X$, consid{\'e}rons alors le carr{\'e}
$A=K_L(\mathcal{O}_{X} (D),\mathcal{O}_{X} (E))$. Les consid{\'e}rations pr{\'e}c{\'e}dentes
nous donnent un isomorphisme
$
r_1:\gd(A) \stackrel{\sim }{\F } \gd(
\xymatrix{
L(D)|_D \ar@{-}[r]_-1&L(D+E)|_D
})
$ et, comme $D$ est fini et plat sur $S$, on obtient un isomorphisme
\begin{equation}
\label{isom-dim1}
\gd (A)\stackrel{\sim }{\F }
(\det \pi _{\ast } L(D)|_D)^{-1} \otimes (\det \pi _{\ast } L(D+E)|_D).
\end{equation}
La section canonique de $\mathcal{O}_{X}(E)$ induit un morphisme
$\pi _{\ast } L(D)|_D \longrightarrow \pi _{\ast } L(D+E)|_D$, dont le d{\'e}terminant d{\'e}finit
donc une section $s_L(D,E)$ de $\gd(A)$.
\addtocounter{theo}{1}
\begin{lemme}
\label{indep2}
L'isomorphisme canonique
$\gd(K_L(\mathcal{O}_{X} (D),\mathcal{O}_{X} (E))) \stackrel{\sim }{\F } \gd(K_L(\mathcal{O}_{X} (E),\mathcal{O}_{X} (D)))$, donn{\'e} par
les isomorphismes de commutativit{\'e} dans $PICgr(S)$, {\'e}change
les sections $s_L(D,E)$ et $s_L(E,D)$ de ces deux faisceaux.
\end{lemme}
\begin{proof}[Preuve]
D{\'e}crivons l'isomorphisme (\ref{isom-dim1}). Consid{\'e}rons le diagramme
commutatif de suites exactes:
\[
\xymatrix{
&L(E)|_E \ar@{^{(}->}[rr]\ar@{}[rd]|{B}
&&L(D+E)|_{D+E} \ar@{->>}[rr]
&&L(D+E)|_D \\
0\ar[ru]\ar[rr]&&L(D)|_D\ar[ru]\ar@{=}[rr]&&L(D)|_D \ar[ru]\\
&L(E)\ar@{^{(}->}'[r][rr]\ar@{->>}'[u][uu]\ar@{}[rd]|{A}
&&L(D+E)\ar@{->>}'[r][rr]\ar@{->>}'[u][uu]
&&L(D+E)|_D\ar[uu] \\
L\ar@{^{(}->}[rr]\ar[ru]\ar[uu]
&&L(D)\ar@{->>}[rr]\ar[ru]\ar@{->>}[uu]
&&L(D)|_D\ar[uu]\ar[ru] \\
&L\ar@{=}'[r][rr]\ar@{^{(}->}'[u][uu]&&L\ar@{^{(}->}'[u][uu]\\
L\ar@{=}[rr]\ar@{=}[ru]\ar@{=}[uu]&&L\ar@{^{(}->}[uu] \ar@{=}[ru]
}
\]
On en d{\'e}duit que l'isomorphisme (\ref{isom-dim1}) se d{\'e}compose en
$\gd(A)\stackrel{\sim }{\F }\gd(B)\stackrel{\sim }{\F }
(\det \pi _{\ast } L(D)|_D)^{-1} \otimes (\det \pi _{\ast } L(D+E)|_D)$, o{\`u} le
premier isomorphisme est donn{\'e} par les suites exactes verticales et le
second isomorphisme est obtenu en appliquant le foncteur d{\'e}terminant
au diagramme suivant de $\mathcal{O}_{S}$-modules localement libres
\[
\begin{CD}
\pi _{\ast } L(E)|_E @>{\gb}>> \pi _{\ast } L(D+E)|_E @. \\
@| @AAA @. \\
\pi _{\ast } L(E)|_E @>>> \pi _{\ast } L(D+E)|_{D+E} @>>> \pi _{\ast } L(D+E)|_D\\
@AAA @AAA @A{\ga}AA\\
0@>>> \pi _{\ast } L(D)|_D @= \pi _{\ast } L(D)|_D
\end{CD}
\]
On traduit l'assertion du lemme en disant que la suite d'isomorphismes
dans $PICgr(S)$
\begin{multline*}
\det( \pi _{\ast } L(D)|_D)^{\vee } \otimes \det( \pi _{\ast } L(D+E)|_D)\stackrel{\sim }{\F } \\
\det( \pi _{\ast } L(D)|_D)^{\vee } \otimes
\det( \pi _{\ast } L(E)|_E)^{\vee } \otimes\det( \pi _{\ast } L(D+E)|_{D+E}) \stackrel{\sim }{\F }
\\
\det( \pi _{\ast } L(E)|_E)^{\vee } \otimes\det( \pi _{\ast } L(D)|_D)^{\vee } \otimes
\det( \pi _{\ast } L(D+E)|_{D+E}) \stackrel{\sim }{\F } \\
\det( \pi _{\ast } L(E)|_E)^{\vee } \otimes \det( \pi _{\ast } L(D+E)|_E)
\end{multline*}
fait correspondre les sections $\det(\ga)$ et $\det(\gb)$ des deux
termes extr{\`e}mes, ce qui provient des propri{\'e}t{\'e}s du
d{\'e}terminant des $\mathcal{O}_{S}$-modules localement libres.
\end{proof}
\addtocounter{ptheo}{2}
\begin{ptheo}
\addtocounter{subsection}{1}
\label{th-prin}
Soit $S$ un sch{\'e}ma localement noeth{\'e}rien.
Pour tout morphisme projectif et plat $\pi : X \longrightarrow S$ {\`a} fibres de
dimension n, il existe une structure du (n+2)-cube
canonique sur le foncteur
$\delta ^{\prime} : PIC(X) \longrightarrow PIC (S)$ telle que:
\begin{enumerate}
\item Si $\pi$ est un morphisme fini, la structure du carr{\'e}
correspondante est simplement celle donn{\'e}e par la norme
(\ref{norme}).
\item Si $D$ est un diviseur relatif sur $X$, les structures du
(n+2)-cube sur $\delta ^{\prime} _X: PIC(X)\longrightarrow PIC(S)$ et du (n+1)-cube sur
$\delta ^{\prime} _D: PIC(D) \longrightarrow PIC(S)$ sont
compatibles aux isomorphismes de restriction.
\end{enumerate}
\end{ptheo}
\begin{proof}[Preuve]
La preuve s'effectue par r{\'e}currence sur $n$.
Le cas de la dimension 0 a {\'e}t{\'e} trait{\'e} dans (\ref{norme}).
Soit n un entier strictement positif, supposons que, pour tout
morphisme projectif et plat $Y\longrightarrow S$ {\`a} fibres de dimension $p<n$,
on sache
construire une structure du (p+2)-cube sur $\delta ^{\prime}_Y$ verifiant les
propri{\'e}t{\'e}s 1 et 2 du th{\'e}or{\`e}me.
Soit $\pi :X\longrightarrow S$ projectif et plat {\`a} fibres de
dimension n, construisons une structure du (n+2)-cube sur
$\delta ^{\prime}_X$. D'apr{\`e}s la proposition (\ref{cubebis}), on doit donc
construire, pour tout $(n+2)$-cube $A$ dans $PIC(X)$, des
isomorphismes
$
s_{A,i} :
\gd ((\phi ^{\prime}_i)^{\ast } A)
\stackrel{\sim }{\F }
\gd ((\phi \sec_i)^{\ast } A
$
v{\'e}rifiant des propri{\'e}t{\'e}s de sym{\'e}trie et de
compatibilit{\'e} aux recollement de $(n+2)$-cubes. Cette
construction, qui occupe les paragraphes suivants sera
d{\'e}coup{\'e}e de la fa\c{c}on suivante:
\begin{itemize}
\item Si un $(n+2)$-cube $A$ poss{\`e}de une ar{\^e}te $L_j$ ayant
une section
r{\'e}guli{\`e}re $\sigma_j$, on construit un isomorphisme
$
s_{A,i,\sigma_j} :
\gd ((\phi ^{\prime}_i)^{\ast } A)
\stackrel{\sim }{\F }
\gd ((\phi \sec_i)^{\ast } A
$
d{\'e}pendant du choix de $\sigma_j$.
\item On s'affranchit de la d{\'e}pendance en la section $\sigma_j$, sous
l'hypoth{\`e}se que $A$ poss{\`e}de deux ar{\^e}tes $L_j$ et $L_k$
suffisamment
positives (on dira dans ce cas que $A$ est suffisamment positif dans
les directions $i$ et $j$).
\item On {\'e}limine enfin cette hypoth{\`e}se de positivit{\'e};
\end{itemize}
\subsection{Construction et propri{\'e}t{\'e}s de $s_{A,i,\sigma_j}$ }
\label{constr-avec-div}
\begin{constr} \label{constr1}
Une section r{\'e}guli{\`e}re
$\sigma_j$ de la $j$-i{\`e}me ar{\^e}te $L_j$ de $A$ d{\'e}finit un diviseur de
Cartier relatif $D$ et $A$ est
isomorphe {\`a} un $(n+2)$-cube
$( \xymatrix{K \ar@{-}[r]_-j& K\otimes \mathcal{O}_{X} (D)})$. On dispose donc,
d'apr{\`e}s
(\ref{restriction}),
d'isomorphismes de restriction
$
\gd ((\phi ^{\prime} _i)^{\ast } A) \stackrel{\sim }{\F } \gd ((\phi ^{\prime} _i)^{\ast } K(D)|_D)
$
et
$
\gd ((\phi \sec _i)^{\ast } A) \stackrel{\sim }{\F } \gd ((\phi \sec _i)^{\ast } K(D)|_D)
$,
provenant du diagramme de suites exactes
\[
\begin{CD}
0 @>>> (\phi ^{\prime} _i)^{\ast } K @>>>
(\phi ^{\prime} _i)^{\ast } K(D) @>>>
(\phi ^{\prime} _i)^{\ast } K(D)|_D @>>> 0
\end{CD}
\]
et du diagramme analogue faisant intervenir $(\phi \sec _i)^{\ast } K$.
L'hypoth{\`e}se de r{\'e}currence, appliqu{\'e}e au $(n+1)$-cube
$K(D)|_D$ sur le
sch{\'e}ma relatif $D/S$ de dimension $n-1$ entra{\^\i}ne l'existence d'un
isomorphisme
\[
s_{K(D)|_D,i} :
\gd ((\phi ^{\prime} _i)^{\ast } K(D)|_D )
\stackrel{\sim }{\F }
\gd ((\phi \sec _i)^{\ast } K(D)|_D).
\]
En composant $s_{K(D)|_D,i}$ avec les isomorphismes pr{\'e}c{\'e}dents, on
obtient $s_{A,i,\sigma _j}$.
\end{constr}
\addtocounter{subsubsection}{1}
\subsubsection{Comportement de $s_{A,i,\sigma _j}$ par recollement.}
\label{recol_div}
Si deux $(n+2)$-cubes $A$ et $B$ sont recollables le long de leurs
i-{\`e}me face, ils ont alors m{\^e}me j-i{\`e}me ar{\^e}te $L_j$. Si
$L_i$ poss{\`e}de
une section r{\'e}guli{\`e}re $\sigma_j$ d{\'e}finissant un diviseur
relatif $D$, on
peut alors {\'e}crire $A\simeq(\xymatrix{K\ar@{-}[r]_-j &K\otimes\mathcal{O}(D)})$
et $B=\simeq(\xymatrix{K^{\prime} \ar@{-}[r]_-j &K^{\prime}\otimes\mathcal{O}(D)})$, avec
$K=( \xymatrix{K_1 \ar@{-}[r]_-i &K_2 })$ et
$K^{\prime} =( \xymatrix{K_2 \ar@{-}[r]_-i &K_3 })$. En {\'e}crivant le diagramme
suivant, dont chaque ligne est exacte:
\[
\xymatrix{
0 \ar[r] & K_3 \ar@{-}[d] \ar[r]
\ar@{}[dr]|{B} &
K_3(D) \ar@{-}[d] \ar[r]
& K_3(D)|_D \ar@{-}[d] \ar[r]
&0 \\
0 \ar[r] & K_2 \ar@{-}[d]_-i \ar[r]
\ar@{}[dr]|{A}&
K_2(D) \ar@{-}[d] \ar[r]
& K_2(D)|_D \ar@{-}[d] \ar[r]
&0 \\
0 \ar[r] & K_1 \ar[r]_-j
& K_1(D) \ar[r]
& K_1(D)|_D \ar[r]
&0
}
\]
et en utilisant que, d'apr{\`e}s l'hypoth{\`e}se de r{\'e}currence,
\[
s_{(K(D)|_D \ast _i K^{\prime}(D)|_D),i}
=
s_{ K^{\prime}(D)|_D,i} \circ s_{K(D)|_D ,i}\; ,
\]
on obtient:
\[
s_{A \ast _i B,i,\sigma _j}
=
s_{B,i,D} \circ s_{A,i,\sigma _j}.
\]
\begin{lemme}[Lien entre $s_{A,i,\sigma _j}$ et $s_{A,i,\sigma _k}$]
\label{indep}
Supposons que les ar{\^e}tes $L_j$ et $L_k$ du $n+2$-cube $A$ poss{\`e}dent
des sections r{\'e}guli{\`e}res $\sigma _j$ et $\sigma _k$ et soit $i\neq j,k$. Si
l'une des deux hypoth{\`e}ses suivantes est v{\'e}rifi{\'e}e:
\begin{enumerate}
\item $X/S$ est {\`a} fibres de dimension 1 et $Z(\sigma_j)\cap
Z(\sigma_k)=\emptyset$.
\item $X/S$ est {\`a} fibres de dimension $n>1$ et les suites
$(\sigma _j,\sigma _k)$ et $(\sigma _k,\sigma _j)$ sont $\pi$-r{\'e}guli{\`e}res
\end{enumerate}
On a alors $s_{A,i,\sigma _j}= s_{A,i,\sigma _k}$.
\end{lemme}
\begin{proof}[Preuve]
Soient $D$ et $E$ les diviseurs de Cartier relatifs effectifs d{\'e}finis par
$s_{A,i,\sigma _j}$ et $s_{A,i,\sigma _k}$, on {\'e}crit alors $A$ sous la forme
$
A=\left(
\begin{array}{c}
\xymatrix{
K(E) \ar@{-}[r]\ar@{-}[d]_-k &K (D+E)\ar@{-}[d] \\
K \ar@{-}[r]_-j &K (D)
}
\end{array}
\right)
$, o{\`u} $K$ est un $n$-cube. Ecrivons le diagramme suivant,
dont les lignes et les colonnes sont des suites exactes
courtes, et qui
relie le $(n+2)$-cube $A$ sur X aux $(n+1)$-cubes $A^{\prime}$ et $A\sec$
sur $E$ et $D$ et au $n$-cube $A ^{\prime\prime\prime}$ sur $D\cap E$:
\begin{equation}
\label{double-restriction}
\xymatrix{A\sec &
K(E)|_E \ar@{^{(}->}[r]_-j&K(D+E)|_E\ar@{->>}[r]
&K(E)|_{D\cap E}
&A ^{\prime\prime\prime} \\
A&
K(E)\ar@{^{(}->}[r]\ar@{->>}[u] &K(D+E)\ar@{->>}[r]\ar@{->>}[u]
&K(D+E)|_D\ar@{->>}[u]
&A^{\prime}\\
&K\ar@{^{(}->}[r]_-j\ar@{^{(}->}[u]_-k &
\hspace{4mm} K(D) \hspace{4mm}
\ar@{^{(}->}[u]\ar@{->>}[r]&K(D)|_D\ar@{^{(}->}[u]_-k
\save "2,2"."3,3"*[F-]\frm{}\ar@{.}"2,1"\restore
\save "1,2"."1,3"*[F-]\frm{}\ar@{.}"1,1"\restore
\save "2,4"."3,4"*[F-]\frm{}\ar@{.}"2,5"\restore
\save "1,4".*[F-]\frm{}\ar@{.}"1,5"\restore
}
\end{equation}
{\em Cas 1}:\\
$K$ est ici un 1-cube $(\xymatrix{L\ar@{-}[r]&M})$ et le diagramme
(\ref{double-restriction}) se r{\'e}duit alors {\`a}:
\[
\xymatrix{
&M(E)|_E\ar[rr] \ar @{} [dr] |{A\sec}
&& M(D+E)|_E \\
L(E)|_E\ar[rr]\ar@{-}[ur]
&& L(D+E)|_E\ar@{-}[ur]&\\
&M(E)\ar@{^{(}->}'[r][rr]\ar@{->>}'[u][uu]
&& M(D+E)\ar@{->>}[uu]\ar@{->>}[rr]&&M(D+E)|_D \\
L(E)\ar@{^{(}->}[rr]\ar@{->>}[uu]\ar@{-}[ur]
&& L(D+E)\ar@{->>}[uu]\ar@{-}[ur]\ar@{->>}[rr]
&&L(D+E)|_D\ar@{-}[ur]\ar @{} [rd] |{A^{\prime}}&\\
&M\ar@{^{(}->}'[r][rr]\ar@{^{(}->}'[u][uu] \ar @{} [ur] |{A}
&& M(D)\ar@{->>}'[r][rr]\ar@{^{(}->}'[u][uu]
&&M(D)|_D\ar[uu] \\
L\ar@{^{(}->}[rr]_-j\ar@{-}[ur]_-i\ar@{^{(}->}[uu]_-k &&
L(D)\ar@{-}[ur]\ar@{->>}[rr]\ar@{^{(}->}[uu]&&L(D)|_D\ar[uu]\ar@{-}[ur]&
}
\]
o{\`u} $\{i,j,k\} =\{1,2,3\}$. On peut alors {\'e}crire un diagramme
d'isomorphismes:
\[
\xymatrix{
\gd(\phi^{\prime\ast}_i(A\sec)) \ar[dd]^-{s_{A\sec,i}}
&\gd(\phi^{\prime\ast}_i(A)) \ar[l] \ar[r]
&\gd(\phi^{\prime\ast}_i(A^{\prime})) \ar[dd]_-{s_{A^{\prime},i}}\\
& {\mathcal{O}_{S}}\ar[ru] \ar[rd]\ar[lu] \ar[ld] \\
\gd(\phi^{\prime \prime\ast}_i(A\sec))
&\gd(\phi^{\prime \prime\ast}_i(A)) \ar[l] \ar[r]
&\gd(\phi^{\prime \prime\ast}_i(A^{\prime}))
}
\]
dont les lignes sup{\'e}rieures et inf{\'e}rieures proviennent respectivement
des faces avant et arri{\`e}res du diagramme pr{\'e}c{\'e}dent et les
fl{\`e}ches
obliques sont les isomorphismes d{\'e}crits dans
(\ref{cas-des-courbes}). Il s'agit de montrer
que le trac{\'e} ext{\'e}rieur est commutatif, ce qui provient du fait que les
deux triangles lat{\'e}raux sont commutatifs par la remarque (\ref{carre})
et que les deux
autres triangles sont commutatis par le lemme (\ref{indep}).\\
{\em Cas 2}\\
Le diagramme (\ref{double-restriction}) induit un diagramme commutatif
d'isomorphismes
\[
\begin{CD}
\gd (A\sec ) @>{\gd}>>\gd (A ^{\prime\prime\prime} )\\
@A{\gc}AA @A{\gb}AA \\
\gd (A) @>{\ga}>>\gd (A^{\prime} )
\end{CD}
\]
Par construction, $\ga $ identifie $s_{A^{\prime} ,i}$ {\`a} $s_{A,i,\sigma_j}$ et
$\gc $ identifie $s_{A\sec ,i}$ {\`a} $s_{A,i,\sigma_k}$. Par l'hypoth{\`e}se de
r{\'e}currence appliqu{\'e}e {\`a} $\gd _D$, $\gd _E$ et $\gd
_{D\cap E}$, les
isomorphismes $\gb$ et $\gd$ identifient respectivement $s_{A^{\prime} ,i}$
et $s_{A\sec ,i}$ {\`a} $s_{A ^{\prime \prime\prime} ,i}$, ce qui prouve
l'assertion dans le cas 2.
\end{proof}
\subsubsection{Lien entre $s_{A,i,\sigma_j}$ et $s_{A,k,\sigma_j}$. }
\label{signe_div}
Soient $A$ un $(n+2)$-cube sur $X$ dont la $j$-i{\`e}me ar{\^e}te poss{\`e}de
une section $\pi$-r{\'e}guli{\`e}re $\sigma_j$ et soent $i,k$ deux indices
distincts et distincts de $j$, montrons que
les trivialisations de $\gd (A)$ induites par $s_{A,i,\sigma_j}$ et
$s_{A,k,\sigma_j}$ diff{\`e}rent d'un signe {\'e}gal {\`a} $\ge _{ik}
(A)$ (introduit
dans la proposition (\ref{cubebis})).\\
A cet effet, {\'e}crivons $K$ sous la forme
$\left(
\begin{array}{c}
\xymatrix{
K_2 \ar@{-}[r]\ar@{-}[d]_-k &K_3\ar@{-}[d] \\
K_0 \ar@{-}[r]_-i &K_1
}
\end{array}
\right)
$
et notons $D$ le diviseur de Cartier relatif d{\'e}fini par $\sigma_j$.
Consid{\'e}rons le diagramme suivant dont les lignes sont des suites
exactes courtes:
\[
\xymatrix{
&K_3\ar@{^{(}->}[rr]\ar@{-}'[d][dd]\ar@{-}[dl]&&
K_3(D)\ar@{->>}[rr]\ar@{-}'[d][dd]\ar@{-}[dl]
&&K_3(D)|_D \ar@{-}[dl]\ar@{-}[dd]\\
K_2\ar@{^{(}->}[rr] \ar@{-}[dd]&&
K_2(D)\ar@{->>}[rr]\ar@{-}[dd]&&K_2(D)|_D\ar@{-}[dd]& \\
&K_1\ar@{^{(}->}'[r][rr]\ar@{-}[dl]&&K_1(D)\ar@{-}[dl]\ar@{->>}'[r][rr]
&&K_1(D)|_D\ar@{-}[dl] \\
K_0\ar@{^{(}->}[rr]&&K_0(D)\ar@{->>}[rr]&&K_0(D)|_D&
}
\]
le cube de gauche de ce diagramme est pr{\'e}cis{\'e}ment $A$ et les suites
exactes donnent un isomorphisme entre $\gd (A) $ et
$\gd \left(
\begin{array}{c}
\xymatrix{
K_2(D)|_D \ar@{-}[r]\ar@{-}[d]_-j &K_3(D)|_D\ar@{-}[d] \\
K_0(D)|_D \ar@{-}[r]_-i &K_1(D)|_D
}
\end{array}
\right)
$.
Cet isomorphisme identifie
les trivialisations de $\gd (A)$ induites par $s_{A,i,\sigma_j}$ et
$s_{A,k,\sigma_j}$ aux trivialisations de $\gd (K(D)|_D)$
induites par $s_{K(D)|_D,i}$ et $s_{K(D)|_D,k}$. Or, par l'hypoth{\`e}se de
r{\'e}currence, interpr{\'e}t{\'e}e {\`a} la lumi{\`e}re de la proposition
(\ref{cubebis}), ces deux
trivialisations diff{\`e}rent d'un signe $\ge _{ik}(K(D)|_D)$. On conclut
en affirmant que $\ge _{ik}(A) = \ge _{ik}(K(D)|_D)$. En effet, par
l'additivit{\'e} de la caract{\'e}ristique d'Euler-Poincar{\'e}, on a:
\[
\chi_{D/S} (K_0 (D)|_D) =\chi_{X/S} (K_0(D))-\chi_{X/S} (K_0)
=\chi_{X/S}(\xymatrix{K_0 \ar@{-}[r] &K_0 (D)})
\]
et donc:
\[
\ge _{ik}(A) = (-1)^{\chi_{X/S}(K_0)-\chi_{X/S}(K_0 (D))}
=(-1)^{\chi_{D/S}(K_0(D)|_D)}
= \ge_{ik} (K(D)|_D)
\]
\subsection{Elimination des hypoth{\`e}ses de diviseurs effectifs}
Pour tout $(n+2)$-cube $A$ suffisamment positif dans les directions $i$
et $j$ et tout indice $i\neq j,k$, on construit dans
cette section un isomorphisme
\[
s_{A,i}: \gd (\phi^{\prime\ast}_i A) \stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i A),
\]
fonctoriel en les isomorphismes de $(n+2)$-cubes et compatible aux
empilements de cubes dans la direction $i$ (conditions 1 et 2 de la
proposition (\ref{cubebis})).
\subsubsection{Premi{\`e}re r{\'e}duction}
\label{reduction}
Soit $A$ un $(n+2)$-cube suffisamment positif dans les directions $i$
et $j$. $A$ est alors isomorphe un $(n+2)$-cube
$
\left(
\begin{array}{c}
\xymatrix{
K\otimes M\ar@{-}[r]\ar@{-}[d]_-k &K\otimes (L\otimes M)\ar@{-}[d]\\
K \ar@{-}[r]_-j &K\otimes L
}
\end{array}
\right)
$
avec $L,M \gg 0$.
Consid{\'e}rons les fibr{\'e}s projectifs $P_L$ et $P_M$ sur $S$ et effectuons
le changement de base
\[
\begin{CD}
X_{P_L \times P_M} @>g>> X\\
@V{\pi}VV @V{\pi}VV \\
P_L \times P_M @>f>> S
\end{CD}
\]
Notons encore $K$, $L$ et $M$ les images r{\'e}ciproques de $K$, $L$ et
$M$ par $g$
et introduisons les faisceaux inversibles
$L^{\prime} = L\otimes \pi_{P_L}^{\ast } \mathcal{O}_{P_L}(1)$
et
$M^{\prime} = M\otimes \pi_{P_M}^{\ast } \mathcal{O}_{P_M}(1)$ et consid{\'e}rons le
$(n+2)$-cube
$A^{\prime} =
\left(
\begin{array}{c}
\xymatrix{
K\otimes M^{\prime}\ar@{-}[r]\ar@{-}[d]_-k &K\otimes (L^{\prime}\otimes M^{\prime})\ar@{-}[d]\\
K \ar@{-}[r]_-j &K\otimes L^{\prime}
}
\end{array}
\right)
$
sur $X_{P_L \times P_M}$. On a des isomorphismes canoniques:
$
\gd(K\otimes L^{\prime}) \stackrel{\sim }{\F } f^{\ast }\gd(K\otimes L)
\otimes
(\mathcal{O}_{P_L}(1))^{\chi_{X/S}(K\otimes L)}
$,
$
\gd(K\otimes M^{\prime}) \stackrel{\sim }{\F } f^{\ast }\gd(K\otimes M)
\otimes
(\mathcal{O}_{P_M}(1))^{\chi_{X/S}(K\otimes M)}
$ et
$
\gd(K\otimes L^{\prime}\otimes M^{\prime})
\stackrel{\sim }{\F } f^{\ast }\gd(K\otimes L\otimes M)
\otimes
(\mathcal{O}_{P_L}(1)\otimes\mathcal{O}_{P_M}(1))^{\chi_{X/S}(K\otimes L\otimes M)}
$. L'{\'e}galit{\'e}
$\chi_{X/S}(K\otimes L\otimes M)=
\chi_{X/S}(K\otimes L)=\chi_{X/S}(K\otimes M)$, d{\'e}duite de
(\ref{caracteristique2}),
entraine alors l'existence d'un isomorphisme canonique
\[
\gd(A^{\prime}) \stackrel{\sim }{\F } f^{\ast } \gd(A)
\]
Comme on a
$f_{\ast } f^{\ast } \mathcal{O} _{P_L \times _S P_M} = \mathcal{O} _S$, on en d{\'e}duit que la
donn{\'e}e d'un isomorphisme
$s_{A,i}: \gd (\phi^{\prime\ast}_i A) \stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i A)$ sur $S$ est
{\'e}quivalente {\`a} la donn{\'e}e d'un isomorphisme
$s_{A^{\prime},i}: \gd (\phi^{\prime\ast}_i A^{\prime}) \stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i A^{\prime})$ sur
$P_L\times _S P_M$.
\subsubsection{Construction de $s_{A^{\prime},i}$ sur des ouverts}
Consid{\'e}rons les ouverts
$U_L = P_L \setminus Z_L$ et $U_M = P_M \setminus Z_M$
introduits en (\ref{div2}).
Sur $X_{U_L \times P_M}$, on dispose d'un diviseur relatif $D_L$ et
d'un isomorphisme canonique
$L^{\prime} \stackrel{\sim }{\F } \mathcal{O} (D_L)$. Le cube $A^{\prime}$ est donc
canoniquement isomorphe sur $X_{U_L \times P_M}$ {\`a} un cube de la forme
$(\xymatrix{ K_0 \ar@{-}[r] & K_0 (D_L)})$. En appliquant la
construction (\ref{constr1}) {\`a} cette situation, on obtient un
isomorphisme canonique $s^{\prime} :\gd (\phi^{\prime\ast}_i A)
\stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i A)$ d{\'e}fini sur $U_L\times _S P_M$. La m{\^e}me
construction, effectu{\'e}e en intervertissant les r{\^o}les de $L$ et $M$,
nous donne un autre isomorphisme $s\sec$ entre les m{\^e}mes faisceaux et
d{\'e}fini sur $P_L\times _S U_M$.
\addtocounter{theo}{2}
\addtocounter{subsubsection}{1}
\begin{lemme}
\label{coincid}
$s^{\prime}$ et $s\sec$ coincident sur $U_L \times _S U_M$.
\end{lemme}
\addtocounter{subsubsection}{1}
\begin{proof}[Preuve]
Soit $p$ la projection $U_L \times _S U_M \longrightarrow U_M$. Le lemme
(\ref{genericite2}) montre l'existence d'un ferm{\'e} $Z$ de $U_L \times
_S U_M$ tel que
\begin{enumerate}
\item $\forall u \in U_M, \forall x \in Z_u,
\text{Prof}(\mathcal{O}_{p ^{-1}(u),x}) \geq 1$.
\item Au dessus de $V=(U_L \times _S U_M) \setminus Z$, les diviseurs
$D_L$ et $D_M$ sont en position d'intersection compl{\`e}te et
$(D_L \cap D_M)_V$ est plat sur $V$.
\end{enumerate}
Au dessus de $V$, $s^{\prime}$ et $s\sec$ coincident d'apr{\`e}s
(\ref{indep}). On en d{\'e}duit donc, d'apr{\`e}s (\cite{EGA4},19.9.8), en
utilisant la
propri{\'e}t{\'e} 1 que $s^{\prime}$ et $s\sec$ coincident sur $U_L \times _S U_M$.
\end{proof}
\subsubsection{Extension de $s^{\prime}$ et $s\sec$ {\`a} $P_L \times _S P_M$.}
Consid{\'e}rons le ferm{\'e} $Y=Z_L \times _S Z_M$ du $s$-sch{\'e}ma
$P_L\times _S P_M$. Le lemme (\ref{genericite1}) entra{\^\i}ne que
pour tout $s\in S$ et
tout $y\in Y_s$, on a $\text{Prof}(\mathcal{O}_{(P_L \times _S P_M)_s,y}) \geq
2$. les isomorphismes $s^{\prime}$ et $s\sec$ sont d{\'e}finis respectivement
sur $U_L \times _S P_M$ et $P_L \times _S U_M$ et coincident sur
$U_L \times _S U_M$. Ils d{\'e}finissent donc un isomorphisme
$s: \gd ((\phi_i^{\prime})^{\ast } A) \stackrel{\sim }{\F } \gd ((\phi_i\sec ) ^{\ast } A)$ sur
l'ouvert $U= (P_L \times _S P_M)\setminus Y$. Par
(\cite{EGA4},19.9.8), $s$ se prolonge alors (de mani{\`e}re unique) {\`a}
$P_L \times _S P_M$ en un isomorphisme
$s_{A^{\prime},i}^{j,k} : \gd (\phi^{\prime\ast}_i A^{\prime}) \stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i
A^{\prime})$
d{\'e}pendant a priori du choix
des directions $j$ et $k$.\\
Par la r{\'e}duction (\ref{reduction}), on d{\'e}duit de cette construction un
isomorphisme
\[
s_{A^{\prime},i}^{j,k} : \gd (\phi^{\prime\ast}_i A^{\prime}) \stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i
A^{\prime})
\]
d{\'e}fini maintenant sur $S$. Il reste {\`a} montrer le
\addtocounter{theo}{1}
\begin{lemme}
$s_{A,i}^{j,k}$ est ind{\'e}pendant du choix des directions $j$ et $k$.
\end{lemme}
\begin{proof}[Preuve]
Supposons en effet que $A$ soit suffisamment positif dans trois
directions diff{\'e}rentes $j,k,l \neq i$. $A$ s'{\'e}crit donc de trois
fa\c{c}ons diff{\'e}rentes
$(\xymatrix{K_j \ar@{-}[r]_-j & K_j \otimes L})$,
$(\xymatrix{K_k \ar@{-}[r]_-k & K_k \otimes M})$ et
$(\xymatrix{K_l \ar@{-}[r]_-l & K_l \otimes N})$,
avec $L,M,N \gg 0$. Montrons alors $s_{A,i}^{j,k}=s_{A,i}^{k,l}$.\\
Il suffit de le montrer apr{\`e}s changement de base par $P_M \longrightarrow S$, et
donc apr{\`e}s changement de base par $U_M \longrightarrow S$ (par le m{\^e}me argument que
dans la preuve du lemme (\ref{coincid}) ). Mais, par construction, au
dessus de $U_M$ les
isomorphismes $s_{A,i}^{j,k}$ et $s_{A,i}^{k,l}$ proviennent tous
deux de la structure du cube associ{\'e}e au sch{\'e}ma relatif $D_M/S$, et
sont donc {\'e}gaux.
\end{proof}
\addtocounter{subsubsection}{1}
\subsubsection{Propri{\'e}t{\'e}s des isomorphismes ainsi construits} On a
donc construit, pour tout $(n+2)$-cube $A$, qui
est suffisamment positif dans deux directions et pour toute direction
$i$ diff{\'e}rente des pr{\'e}c{\'e}dentes, un isomorphisme
\[
s_{A,i} : \gd (\phi^{\prime\ast}_i A) \stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i
A)
\]
les $s_{A,i}$ v{\'e}rifient les propri{\'e}t{\'e}s suivantes:
\begin{enumerate}
\item Si les deux $(n+2)$-cubes $A$ et $B$ sont suffisamment positifs
dans deux directions diff{\'e}rentes de $i$ et ont leur i-{\`e}me face en
commun, alors:
\[
s_{B,i} \circ s_{A,i} = s_{A\ast _i B,i}
\]
\item Si $s_{A,i}$ et $s_{A,j}$ sont tous les deux d{\'e}finis, alors les
trivialisations de $\gd (A)$ qu'elles induisent diff{\`e}rent d'un signe
{\'e}gal {\`a} $\ge _{ij} (A)$.
\item Si $A$ est suffisamment positif dans deux directions diff{\'e}rentes
de $i$, pour toute permutation $\sigma \in S_n$ on a:
\[
s_{\sigma ^{\ast } A,i} = s_{A, \sigma (i)}
\]
\end{enumerate}
Pour montrer les propri{\'e}t{\'e}s 1 et 2, on choisit une direction $k$
dans laquelle $A$ est suffisamment positif, et on {\'e}crit
$A=(\xymatrix{K \ar@{-}[r]_-k &K \otimes L})$ avec $L\gg 0$ et on
effectue, comme pr{\'e}c{\'e}demment,
un changement de base $U_L \subset P_L \longrightarrow S$, ce qui permet de disposer
d'un diviseur effectif $D_L$. La propri{\'e}t{\'e} 1 provient alors de
la propri{\'e}t{\'e} de
recollement (\ref{recol_div}) et la propri{\'e}t{\'e} 2 provient du lien
entre $s_{A,i,D}$
et $s_{A,k,D}$ (\ref{signe_div}), montr{\'e}s tous les deux sous
l'hypoth{\`e}se de l'existence d'un diviseur effectif.\\
Pour la propri{\'e}t{\'e} 3, il
suffit de montrer l'{\'e}galit{\'e} $s_{\sigma ^{\ast } A,i} = s_{A, \sigma (i)}$ dans
le cas o{\`u} $\sigma$ est une
transposition $\sigma _{rs}$ et o{\`u} $A$ est suffisamment positif dans au
moins une direction $k$ diff{\'e}rente de $r$ et de $s$. On {\'e}crit alors
$A=(\xymatrix{K \ar@{-}[r]_-k &K \otimes L})$ avec $L\gg 0$ et on
effectue le changement de base $U_L \subset P_L \longrightarrow S$. L'{\'e}galit{\'e}
provient alors de l'{\'e}galit{\'e} analogue pour le $(n+1)$-cube $K\otimes L
|_{D_L}$, qui est v{\'e}rifi{\'e}e d'apr{\`e}s l'hypoth{\`e}se de
r{\'e}currence.
\subsection{Elimination des hypoth{\`e}ses de positivit{\'e}.}
On veut {\'e}tendre ici la d{\'e}finition de $s_{A,i}$ {\`a} un $(n+2)$-cube
quelconque $A$ dans $PIC(X)$.
\begin{rem}
\label{compar-sign}
Si $A$ et $B$ sont deux $(n+2)$-cubes dans $PIC(X)$, recollables le
long de leur j-i{\`e}me face (notons $C= A\ast _j B$) et tels que
$s_{A,i}$ et $s_{B,i}$ sont bien
d{\'e}finis. On peut alors d{\'e}finir un isomorphisme
$s_{A,i}\otimes s_{B,i}:\gd (\phi^{\prime\ast}_i C) \stackrel{\sim }{\F }
\gd (\phi^{\prime \prime\ast}_i C)$ par:
\[
\gd (\phi^{\prime\ast}_i C)
\stackrel{\sim }{\F }
\gd (\phi^{\prime\ast}_i A) \otimes \gd (\phi^{\prime\ast}_i B)
\longrightarrow
\gd (\phi^{\prime \prime\ast}_i A) \otimes \gd (\phi^{\prime \prime\ast}_i B)
\stackrel{\sim }{\F }
\gd (\phi^{\prime \prime\ast}_i C)
\]
Notons que si $s_{C,i}$ est aussi d{\'e}fini, on a la relation:
\[
s_{C,i} =\ge _{ij}(A) s_{A,i}\otimes s_{B,i}\; ,
\]
r{\'e}sultant de la comparaison entre $s_{A,i}$ et $s_{A,j}$ et de la
relation $s_{C,j} =s_{A,j}\otimes s_{B,j}$.
\end{rem}
On peut alors {\'e}noncer le
\begin{lemme}
Soit $i$ un indice compris entre 1 et $n+2$ et soit $A$ un
$(n+2)$-cube suffisamment positif dans {\em une} direction $j\neq i$.
\begin{enumerate}
\item Pour toute direction $k\neq i,j$, il existe un $(n+2)$-cube $B$
recollable avec $A$ dans la direction $k$ tel que $B$
et $A\ast _k B$ soient tous les deux suffisamment positifs dans la
direction $k$.
\item Soient $k$ et $l$ deux directions eventuellement {\'e}gales mais
distinctes de $i$ et $j$ et soient $B$ (resp. $B^{\prime}$) deux $(n+2)$-cubes
recollables avec $A$ dans les directions $k$ (resp. $l$)
tels que $B$ et $A\ast _k B$ sont suffisament positifs dans la
direction $k$ et $B^{\prime}$ et $A\ast _k B^{\prime}$ sont suffisamment positifs dans la
direction $l$. Alors les deux isomorphismes:
\[
s_{A\ast_k B ,i} \otimes s_{B ,i}^{-1}
\; \text{et} \;
s_{A\ast_l B^{\prime} ,i}\otimes s_{B^{\prime} ,i}^{-1}:
\gd (\phi^{\prime\ast}_i A) \stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i A)
\]
sont {\'e}gaux.
\end{enumerate}
\end{lemme}
\begin{proof}[Preuve]
Le premier point r{\'e}sulte du fait que pour tout faisceau inversible $L$
sur $X$, on peut trouver un faisceau inversible $M\gg 0$ tel que
$L\otimes M \gg 0$.\\
Montrons d'abord le second point quand $k=l$. Dans ce cas, on peut
trouver des $(n+2)$-cubes $C$ et $C^{\prime}$, suffisamment positifs dans la
direction $k$ , tels que
$A\ast _k B \ast _k C = A\ast _k B^{\prime} \ast _k C^{\prime}$ et $ B \ast _k C$,
$B^{\prime} \ast _k C^{\prime}$ et $A\ast _k B \ast _k C$ sont suffisamment
positifs dans la direction $k$. On a alors en utilisant la remarque
(\ref{compar-sign}:
\[
s_{A\ast _kB,i} \otimes s_{B,i}^{-1} =
s_{A\ast _kB \ast _k C,i}
\otimes s_{B \ast _k C,i}^{-1} =
s_{A\ast _kB^{\prime} \ast _k C^{\prime} ,i}
\otimes s_{B^{\prime} \ast _k C^{\prime} ,i}^{-1} =
s_{A\ast _kB^{\prime} ,i} \otimes s_{B^{\prime} ,i}^{-1}
\]
Dans le cas o{\`u} $k\neq l$, on introduit un $(n+2)$-cube $C$
d{\'e}termin{\'e}
uniquement par les conditions suivantes:
$C$ est recollable avec $B$ dans la direction $l$ et avec $B^{\prime}$ dans la
direction $k$, comme le d{\'e}crit le diagramme suivant:
\begin{center}
\begin{tabular}{cc}
$l \uparrow$ &\begin{tabular}{|c|c|}
\hline
$B^{\prime}$&$C$\\ \hline
$A$&$B$ \\ \hline
\end{tabular}
\\
&$\xrightarrow{k}$
\end{tabular}
\end{center}
En notant $D$ le cube total, on {\'e}crit alors:
\[
s_{A\ast _kB,i} \otimes s_{B,i}^{-1} =
s_{C,i} \otimes s_{B^{\prime} \ast _kC,i}^{-1} \otimes s_{B,i}^{-1}=
s_{C,i} \otimes s_{B \ast _lC,i}^{-1} \otimes s_{B^{\prime} ,i}^{-1}=
s_{A\ast _lB^{\prime} ,i} \otimes s_{B^{\prime} ,i}^{-1}
\]
\end{proof}
Ce lemme permet donc de d{\'e}finir
$s_{A,i}:\gd (\phi^{\prime\ast}_i A) \stackrel{\sim }{\F } \gd (\phi^{\prime \prime\ast}_i A)$
d{\'e}s que $A$ est suffisamment positif dans {\em une} direction $j\neq
i$ par la formule:
\[
s_{A,i}=s_{B ,i}^{-1} \circ s_{A\ast_k B ,i}
\]
En r{\'e}it{\'e}rant cette argument, on construit $s_{A,i}$ pour tout
$(n+2)$-cube $A$ dans $PIC(X)$, sans hypoth{\`e}se de positivit{\'e} sur $A$.
\begin{lemme}
Les isomorphismes $s_{A,i}$ ainsi construits v{\'e}rifient les
propri{\'e}t{\'e}s de la proposition (\ref{cubebis}).
\end{lemme}
Ce qui conclut la preuve du th{\'e}or{\`e}me principal.
\end{proof}
\section{Fibr{\'e} d'intersection et r{\'e}sultant}
\subsection{Fibr{\'e} d'intersection}
Pour tout morphisme projectif et plat $\pi:X \longrightarrow S$, {\`a} fibres de
dimension $n$, sur un sch{\'e}ma $S$ localement noeth{\'e}rien,
d{\'e}finisssons le $(n+1)$-foncteur:
\[
\begin{array}{rlcl}
I_{X/S}\; :& PIC^{n+1}(X) &\longrightarrow &PIC(S)\\
& (L_1, \cdots , L_{n+1}) &\mapsto &
{\displaystyle \bigotimes_{k=0}^{n+1}
\left(
\bigotimes_{i_1<\cdots <i_k} \gd (L_{i_1} \otimes \cdots \otimes L_{i_k})
\right)
^{(-1)^{n+1-k}}}
\end{array}\; ,
\]
appel{\'e} {\em foncteur fibr{\'e} d'intersection} pour $X/S$.
\begin{prop}
\label{propr-inter}
Le foncteur $I_{X/S}$ est un $(n+1)$-foncteur additif en chaque
variable, v{\'e}rifiant les propri{\'e}t{\'e}s suivantes:
\begin{enumerate}
\item La formation de $I_{X/S}$ commute aux changements de bases.
\item $I_{X/S}$ est muni de donn{\'e}es de sym{\'e}trie, compatibles avec les
donn{\'e}es d'additivit{\'e} en chaque variable.
\item Si $\pi : X\longrightarrow S$ est fini et plat, le foncteur
$I_{X/S}:PIC(X)\longrightarrow PIC(S) $ est simplement le foncteur norme $N_{X/S}$
et les contraintes d'additivit{\'e} pour $I_{X/S}$ sont les isomorphismes
usuels $N_{X/S} (L\otimes L^{\prime} ) \stackrel{\sim }{\F } N_{X/S} (L)
\otimes N_{X/S} (L^{\prime} )$.
\item Soient $L_1,\cdots ,L_{n+1}$ des faisceaux inversibles sur
$X$ et $\sigma_{n+1}$ une section $\pi$-r{\'e}guli{\`e}re de $L_{n+1}$
d{\'e}finissant un diviseur de Cartier relatif effectif $D$.
Il existe un isomorphisme canonique:
\[
\rho _D:
I_{X/S} (L_1, \cdots ,L_{n+1})
\stackrel{\sim }{\F }
I_{D/S} (L_1|_D, \cdots ,L_n|_D)
\]
qui est fonctoriel en les isomorphismes $L_i \stackrel{\sim }{\F } L_i^{\prime}$ pour $1\leq
i\leq n$ et
compatible avec les donn{\'e}es d'additivit{\'e} et de sym{\'e}trie de
$I_{X/S}$
en les $L_1,\cdots ,L_n$ et de $I_{D/S}$ en les
$L_1|_D,\cdots ,L_n|_D$.
\item Si, en plus des donn{\'e}es de 4, on dispose d'une section $\sigma_n$
de $L_n$ telle que $(\sigma_n,\sigma_{n+1})$ et $(\sigma_{n+1},\sigma_n)$ sont
des suites $\pi$-r{\'e}guli{\`e}res, notons $E$ le diviseur d{\'e}fini par
$\sigma_n$. Le diagramme suivant est alors commutatif:
\[
\begin{CD}
I_{X/S} (L_1, \cdots ,L_{n+1}) @>{\rho_D}>> I_{D/S} (L_1|_D, \cdots ,L_n|_D)\\
@V{\rho_E}VV @VV{\rho_E}V\\
I_{E/S} (L_1|_E, \cdots ,L_{n-1}|_E,L_{n+1}|_E)
@>{\rho_D}>>
I_{D\cap E/S} (L_1|_{D\cap E}, \cdots ,L_{n-1}|_{D\cap E})
\end{CD}
\]
\end{enumerate}
\end{prop}
\begin{proof}[Preuve]
Notons que, par d{\'e}finition,
\[
I_{X/S}(L_1,\cdots ,L_n) =
\gd _{X/S}( K_{\mathcal{O}_{X}}(L_1,\cdots ,L_n)) \; .
\]
C'est donc le $(n+1)$-foncteur multi-additif et sym{\'e}trique
associ{\'e} {\`a} la
structure du cube sur le foncteur $\gd :PIC(X) \longrightarrow PIC (S)$.
La propri{\'e}t{\'e} 1 provient de la compatibilit{\'e} du foncteur $\gd$ aux
changements de bases. L'existence de donn{\'e}es d'additivit{\'e} en chaque
variable et de donn{\'e}es de sym{\'e}trie compatibles {\`a} l'additivit{\'e}
provient de l'existence d'une structure de $(n+2)$-cube sur $\gd$.
L'identification 3 de $I_{X/S}$ et $N_{X/S}$ dans
le cas d'un sch{\'e}ma relatif fini et plat est simplement la d{\'e}finition
de la structure du carr{\'e} dans le cas de dimension relative 0.\\
Il reste {\`a} construire l'isomorphisme $\rho _D$:
Les isomorphismes de restrictions
d{\'e}crits en (\ref{th-prin})
induisent un isomorphisme canonique:
\[
\gd _{X/S} (K_{\mathcal{O}_{X}} (\mathcal{O}_{X} (D),L_1,\cdots ,L_n))
\stackrel{\sim }{\F }
\gd _{D/S} (K_{\mathcal{O}_{X} (D)|_D}(L_1 (D)|_D, \cdots ,L_n (D)|_D))
\]
Par ailleurs, l'{\'e}galit{\'e}
\[
K_{\mathcal{O}_{X} (D)|_D}(L_1(D)|_D,\cdots ,L_n(D)|_D)
=
K_{\mathcal{O} _D}(L_1|_D,\cdots ,L_n|_D) \otimes_{\mathcal{O}_D} \mathcal{O}_{X} (D)|_D
\]
induit, d'apr{\`e}s (\ref{def-multifonct}), un isomorphisme
\[
\gd _{D/S} (K_{\mathcal{O}_{X} (D)|_D}(L_1 (D)|_D, \cdots ,L_n (D)|_D))
\stackrel{\sim }{\F }
I_{D/S} (L_1 |_D, \cdots ,L_n |_D).
\]
D'apr{\`e}s les propri{\'e}t{\'e}s des structures du cube pour $\gd _{D/S}$ et
$\gd _{X/S}$,
ces deux isomorphismes sont fonctoriels et additifs en chaque $L_i$
et compatibles aux donn{\'e}es de sym{\'e}tries. Notons $\rho _D$ leur
compos{\'e}; il v{\'e}rifie donc les propri{\'e}t{\'e}s demand{\'e}es.
\end{proof}
\begin{cor}
Soient $L_1,\cdots ,L_n$ des faisceaux inversibles sur $X$ et
$(\sigma_1,\cdots ,\sigma_n)$ une suite $\pi$-r{\'e}guli{\`e}re de sections des
$L_i$. Notons $D_i$ le lieu des z{\'e}ros de $\sigma_i$. Pour tout faisceau
inversible $L$ sur $X$, les
isomorphismes de restriction successifs de
$\cap_{i=1}^p D_i$ {\`a} $\cap_{i=1}^{p+1} D_i$
induisent un
isomorphisme canonique:
\[
I_{X/S} (L_1, \cdots ,L_n,L)
\stackrel{\sim }{\F }
N_{D_1\cap \cdots \cap D_n/S} (L|_{D_1\cap \cdots \cap D_n})\; ,
\]
fonctoriel en les isomorphismes $L\stackrel{\sim }{\F } L^{\prime}$ et additif en $L$.
\end{cor}
\begin{proof}[Preuve]
Remarquons
que, par hypoth{\`e}se, pour tout entier $p$
tel que $1\leq p \leq n$, $D_1 \cap \cdots \cap D_p \longrightarrow S$ est un morphisme
projectif et plat {\`a} fibres de dimension $n-p$ et
$D_1 \cap \cdots \cap D_p\cap D_{p+1}$ est un diviseur de Cartier relatif
effectif de $D_1 \cap \cdots \cap D_p /S$.
On peut donc it{\'e}rer la construction pr{\'e}c{\'e}dente, et on obtient un
isomorphisme
\[
I_{X/S} (\mathcal{O}_{X} (D_1), \cdots , \mathcal{O}_{X} (D_n),L)
\stackrel{\sim }{\F }
I_{D_1\cap \cdots \cap D_n/S} (L|_{D_1\cap \cdots \cap D_n})\; .
\]
On conclut alors en utilisant le fait que $D_1\cap \cdots \cap D_n \longrightarrow
S$ est fini et plat, ce qui entra{\^\i}ne que
$I_{D_1\cap \cdots \cap D_n/S} (L|_{D_1\cap \cdots \cap D_n})
\simeq
N_{D_1\cap \cdots \cap D_n/S} (L|_{D_1\cap \cdots \cap D_n})$. La
fonctorialit{\'e} et l'additivit{\'e} de l'isomorphisme obtenu se
v{\'e}rifient {\`a} chaque {\'e}tape de l'it{\'e}ration.
\end{proof}
\subsection{Sections du fibr{\'e} d'intersection}
\subsubsection{Cas de la dimension 0}
Soit $\pi :Y\longrightarrow T$ est un morphisme fini et plat, et soit $L$ un
faisceau inversible sur $Y$, muni d'une section $\sigma$ qui ne s'annulle
pas au dessus d'un ouvert $U=T\setminus Z$ de $T$. La section $\sigma$
induit un isomorphisme $\mathcal{O} |_{Y_U} \stackrel{\sim }{\F } L|_{Y_U}$, qui induit donc
par fonctorialit{\'e} un isomorphisme
$N_{Y_U/U} (\sigma ):\mathcal{O} _U \stackrel{\sim }{\F } (N_{Y/T} L)|_U$. Celui ci se prolonge en
une section $s:\mathcal{O} _T \stackrel{\sim }{\F } N_{Y/T} L$, non nulle
en dehors de $Z$ et d{\'e}finie par le morphisme
\[
\det \pi _{\ast } s: \det \pi _{\ast } \mathcal{O} _Y \longrightarrow \det \pi _{\ast } L \; .
\]
\subsubsection{Construction d'une section sur un ouvert}
Soit $\pi : X\longrightarrow S$ un morphisme projectif et plat,
{\`a} fibres de dimension $n$ et $L_1,\cdots ,L_{n+1}$ des faisceaux
inversibles sur $X$.
Supposons que l'on dispose de sections $\sigma_i$ de chaque $L_i$ telles
que $(\sigma_1,\cdots ,\sigma_n)$ est une suite $\pi$-r{\'e}guli{\`e}re. Notons
$D_i$ le lieu des z{\'e}ros de $\sigma_i$ et $U$
l'ouvert de $S$ au dessus duquel
$D_1\cap \cdots \cap D_{n+1} = \emptyset$ , on peut appliquer ce
qui pr{\'e}c{\`e}de au morphisme fini et plat $D_1\cap \cdots \cap D_n \longrightarrow S$,
au faisceau inversible $L_{n+1}|_{D_1\cap \cdots \cap D_n}$
et {\`a} sa section $\sigma_{n+1}|_{D_1\cap \cdots \cap D_n}$. On obtient
ainsi une section $\mathcal{O}_{S} \longrightarrow I_{X/S}(L_1,\cdots ,L_{n+1})$,
dont la restriction {\`a} $U$ est un isomorphisme.
\subsection{Construction du r{\'e}sultant.}
Soit $\pi : X\longrightarrow S$ un morphisme projectif et plat,
{\`a} fibres de dimension $n$ et $L_1,\cdots ,L_{n+1}$ des faisceaux
inversibles sur $X$. Effectuons le changement de base:
\[
\xymatrix{X_P \ar[r]^-g\ar[d]_-{\pi} &X \ar[d]^-{\pi}\\
P= P_{L_1}\times _S \cdots \times _S P_{L_n} \ar[r]^-f & S
}
\]
Pour tout entier $i$, notons $p_i$ la projection de $P$ sur $P_i$ et
$\pi_i$ la compos{\'e}e $X\longrightarrow P\longrightarrow P_i$, et
consid{\'e}rons le faisceau inversible
$L^{\prime}_i=g^{\ast } L_i \otimes \pi_i^{\ast } \mathcal{O}_{P_i}(1)$. Consid{\'e}rons alors le
faisceau inversible sur $P$:
\[
Res (L_1,\cdots ,L_{n+1}) = I_{X_P/P}(L_1^{\prime},\cdots ,L_{n+1}^{\prime})
\]
\begin{lemme}
\label{isom-resultant}
$Res (L_1,\cdots ,L_{n+1})$ est canoniquement isomorphe {\`a}
$
f^{\ast } I_{X/S}(L_1,\cdots ,L_{n+1}) \otimes
{\displaystyle \bigotimes_{i=1}^{n+1}p_i^{\ast } \mathcal{O}_{P_i}(k_i)}
$, o{\`u} $k_i:S\longrightarrow\mathbb{Z}$ est la fonction localement constante qui {\`a} $s\in S$
associe le nombre d'intersection sur $X_s$ des restrictions {\`a} $X_s$
des faisceaux $L_j$ pour $j\neq i$.
\end{lemme}
\begin{proof}[Preuve]
Rappelons que, si $K$ est un $p$-cube dans $PIC(X)$, pour tout faisceau
inversible $L$ sur $S$, on a un isomorphisme canonique
$\gd(K\otimes\pi^{\ast } L) \stackrel{\sim }{\F } \gd(K)\otimes L^{\otimes\chi_{X/S}(K)}$.
Soit $1\leq i \leq n+1$, notons $i_1,\cdots,i_n$ les $n$ entiers
compris entre 1 et $n+1$ et distincts de $i$, class{\'e}s par ordre
croissant. Si $M_{i_1},\cdots,M_{i_n}$ sont des faisceaux inversibles
sur $P$, on a alors:
\begin{multline*}
\chi_{X_P/P}(K_{\mathcal{O}_{X}}(g^{\ast } L_{i_1}\otimes \pi^{\ast } M_{i_1} ,
\cdots ,g^{\ast } L_{i_n}\otimes \pi^{\ast } M_{i_n})=\\
\chi_{X_P/P}(K_{\mathcal{O}_{X}}(g^{\ast } L_{i_1},\cdots ,g^{\ast } L_{i_n})=
\chi_{X/S}(K_{\mathcal{O}_{X}}(L_{i_1},\cdots ,L_{i_n}))=k_i
\end{multline*}
On peut alors {\'e}crire:
\begin{multline*}
Res (L_1,\cdots ,L_{n+1})
=
I_{X_P/P}(L^{\prime}_1,\cdots ,L^{\prime}_n,
g^{\ast } L_{n+1}\otimes \pi_i^{\ast } \mathcal{O}_{P_{n+1}}(1))\\
\simeq
\gd(K)^{-1}\otimes \gd(K\otimes
g^{\ast } L_{n+1}\otimes \pi_i^{\ast } \mathcal{O}_{P_{n+1}}(1))
\simeq
\gd(K)^{-1}\otimes \gd(K\otimes
g^{\ast } L_{n+1}) \otimes( p_i^{\ast } \mathcal{O}_{P_{n+1}}(1))^{k_{n+1}}\\
\simeq\gd(
\xymatrix{K\ar@{-}[r]_-{n+1}&K\otimes
g^{\ast } L_{n+1}})\otimes p_i^{\ast } \mathcal{O}_{P_{n+1}}(k_{n+1}),
\end{multline*}
o{\`u} l'on a pos{\'e} $K=K_{\mathcal{O}_{X}}(L_1^{\prime},\cdots,L_n^{\prime})$.
En effectuant la m{\^e}me op{\'e}ration sur chacun des $L_i$, on obtient
finalement:
\begin{align*}
Res (L_1,\cdots ,L_{n+1})&
\simeq
I_{X_P/P}(g^{\ast } L_1,\cdots,g^{\ast } L_{n+1})
\otimes
\bigotimes_{i=1}^{n+1} p_i^{\ast } \mathcal{O}_{P_i}(k_i)\\
&\simeq f^{\ast } I_{X/S}(L_1,\cdots,L_{n+1})
\otimes
\bigotimes_{i=1}^{n+1} p_i^{\ast } \mathcal{O}_{P_i}(k_i)
\end{align*}
\end{proof}
Pour tout $i$, consid{\'e}rons la section canonique $\sigma_i$ des $L_i$ sur
$X_P$ et notons $D_i$ le lieu des z{\'e}ros de $\sigma_i$.
On va construire une section canonique
$Res(\sigma_1,\cdots ,\sigma_{n+1})$ de
$Res(L_1^{\prime}, \cdots , L_{n+1}^{\prime})$, qu'on appellera
{\em r{\'e}sultant} des $\sigma_i$ .\\
Pour toute permutation $\phi\in S_{n+1}$, notons $U_{\phi}$ l'ouvert
de $P$ au dessus duquel $(\sigma _{\phi (1)},\cdots ,\sigma _{\phi (n)})$
est une suite $\pi$-r{\'e}guli{\`e}re.
\begin{lemme}
\label{genericite-suite}
Soient $U=\bigcup_{\phi\in S_{n+1}}U_{\phi}$ et
$V=\bigcap_{\phi\in S_{n+1}}U_{\phi}$ et soient $Z$ et $Z^{\prime}$ les
ferm{\'e}s compl{\'e}mentaires. Si $f:P\longrightarrow S$ d{\'e}signe le morphisme
structural, on a:
\begin{enumerate}
\item
Pour tout $z\in Z$, on a: $\text{Prof} (\mathcal{O}_{f^{-1}(f(z)),z}) \geq 2$.
\item
Pour tout $z\in Z^{\prime}$, on a: $\text{Prof}(\mathcal{O}_{f^{-1}(f(z)),z})\geq
1$.
\end{enumerate}
\end{lemme}
\begin{proof}[Preuve]
Soit $z\in Z$, soit $k<n$ la longueur maximale d'une suite
$\pi$-r{\'e}guli{\`e}re prise parmi les $\sigma_i$ au dessus de $z$, soit
$(\sigma_{\phi (1)},\cdots ,\sigma_{\phi (k)})$ une telle suite et soit
$P^{\prime}= P_{\phi (1)}\times _S\cdots\times _SP_{\phi (k)}$. D{\'e}composons
$f:P\longrightarrow S$ en
\[
\xymatrix{
&P^{\prime}\times _SP_{\phi (k+1)}\ar[rd]^-{q^{\prime}} \\
P\ar[rr]^-p \ar[ru]^-{p^{\prime}} \ar[rd]^-{p\sec}
&&P^{\prime} \ar[r]^-r&S\\
&P^{\prime}\times _SP_{\phi (k+2)}\ar[ru]^-{q\sec}
}
\]
Par le lemme (\ref{genericite2}), $p^{\prime} (z)$ est contenu dans un
hyperplan de la fibre $(q^{\prime} )^{-1} (p(z))$ et, de m{\^e}me,
$p\sec (z)$ est contenu dans un
hyperplan de la fibre $(q\sec )^{-1} (p(z))$. Donc $z$ est contenu
dans l'intersection de deux hyperplans de la fibre $p ^{-1}(p(z))$ et
donc dans l'intersection de deux hyperplans de $f ^{-1}(f(z))$, ce qui
montre le r{\'e}sultat.\\
La deuxi{\`e}me assertion se montre de mani{\`e}re analogue.
\end{proof}
Pour tout $\phi\in S_{n+1}$, on a un isomorphisme d{\'e}fini sur l'ouvert
$U_{\phi}$:
\begin{equation}
\label{isom-recur}
I_{X/S} (L_1,\cdots ,L_{n+1})
\stackrel{\sim }{\F }
N_{D_{\phi(1)}\cap\cdots\cap D_{\phi(n)}/S} (L_{\phi(n+1)})
\end{equation}
et la section
$N_{D_{\phi(1)}\cap\cdots\cap D_{\phi(n)}/S} (\sigma_{\phi(n+1)})$
d{\'e}finit
donc une section $s_\phi$ de $I_{X/S}|_{U_\phi}$.
\begin{lemme}
Pour tous $\phi,\psi\in S_n$, on a
$s_\phi|_V=s_\psi|_V$.
\end{lemme}
\begin{proof}[Preuve]
Si $\phi(n+1)=\psi(n+1)$, ceci
provient de l'assertion 5 de la proposition (\ref{propr-inter}).
Il suffit donc de montrer l'assertion
dans le cas o{\`u} $\phi(n)=\psi(n+1)$ et $\phi(n+1)=\psi(n)$.
Notons alors $Y=D_{\phi(1)}\cap\cdots\cap D_{\phi(n-1)}$. C'est un
sch{\'e}ma relatif sur $S$ {\`a} fibres de dimension 1 et
les isomorphismes (\ref{isom-recur}) correspondant {\`a} $\phi$ et $\psi$
se factorisent en
\[
I_{X/S} (L_1,\cdots ,L_{n+1})
\stackrel{\sim }{\F }
I_{Y/S}(L_{\phi(n)},L_{\phi(n+1)})
\stackrel{\sim }{\F }
N_{Y\cap D_{\phi(n)}/S} (L_{\phi(n+1)})
\]
et
\[
I_{X/S} (L_1,\cdots ,L_{n+1})
\stackrel{\sim }{\F }
I_{Y/S}(L_{\phi(n)},L_{\phi(n+1)})
\stackrel{\sim }{\F }
N_{Y\cap D_{\phi(n+1)}/S} (L_{\phi(n)})
\]
Le lemme (\ref{indep2}) entraine que le diagramme
\[
\xymatrix{
&N_{Y\cap D_{\phi(n)}/S} (L_{\phi(n+1)})\ar[r]^-{\sim}&
I_{Y/S}(L_{\phi(n)},L_{\phi(n+1)})\ar[dd] \\
{\mathcal{O}_{S}} \ar[ur]^-{N_{Y\cap D_{\phi(n)}/S} (\sigma_{\phi(n+1)})}
\ar[dr]_-{N_{Y\cap D_{\phi(n+1)}/S} (L_{\phi(n)})} \\
&N_{Y\cap D_{\phi(n+1)}/S} (L_{\phi(n)})\ar[r]^-{\sim}&
I_{Y/S}(L_{\phi(n)},L_{\phi(n+1)})
}
\]
est commutatif, ce qui montre l'assertion.
\end{proof}
En utilisant l'assertion 2 du lemme (\ref{genericite-suite}), on en
d{\'e}duit que $s_\phi$ et $s_\psi$ coincident sur $U_\phi\cap U_\psi$.
Les $s_\phi$ d{\'e}finissent donc une section de $I_{X_P/P}$
sur l'ouvert $U$ de $P$. D'apr{\`e}s l'assertion 1 du lemme
(\ref{genericite-suite}), une telle section
se prolonge de mani{\`e}re unique en une section de $I_{X_P/P}$ sur
$P$.
\begin{Def}
On appelle r{\'e}sultant de $\sigma_1,\cdots ,\sigma_{n+1}$ et on note
$Res(\sigma_1,\cdots ,\sigma_{n+1})$ l'uni\-que section de
$I_{X_P/P}$ qui coincide avec $s_\phi$ sur l'ouvert $U_\phi$ pour
tout $\phi \in S_{n+1}$.
\end{Def}
Les r{\'e}sultats suivants justifient la terminologie "r{\'e}sultant":
\begin{lemme}
Soit $R\subset P$ l'image par $\pi$ du lieu des z{\'e}ros de la section
$\sigma =\bigoplus\sigma_i$ de $\bigoplus L^{\prime}_i$. $R$ est un ferm{\'e} de $P$ et
pour tout point $p\in P$ tel que $\text{Prof}(\mathcal{O}_{P_{f(p)},p})\leq 1$,
$p$ est {\'e}l{\'e}ment de $R$ si et seulement si $Res(\sigma_1,\cdots
,\sigma_{n+1})(p)=0$.
\end{lemme}
\begin{proof}[Preuve]
D'apr{\`e}s la partie 2 du lemme (\ref{genericite-suite}), il suffit de
montrer que, pour tout point $p$ de l'ouvert
$U=\bigcup_{\phi\in S_{n+1}}U_{\phi}$, on a
$Res(\sigma_1,\cdots ,\sigma_{n+1})(p)=0$ si et
seulement si $p\in R$. Soit donc $p\in U$ et $\phi\in S_{n+1}$ tel
que $p\in U_\phi$. Sur $U_\phi$, $Res(\sigma_1,\cdots ,\sigma_{n+1})$
coincide avec
$N_{D_{\phi(1)}\cap\cdots\cap D_{\phi(n)}} (\sigma_{\phi(n+1)})$ et cette
section s'annulle si et seulement si $\sigma_{\phi(n+1)}$ s'annulle en
point de la fibre $(D_{\phi(1)}\cap\cdots\cap D_{\phi(n)})_p$, ce qui
montre le r{\'e}sultat.
\end{proof}
\begin{lemme}
\label{sym-resultant}
Pour toute permutation $\phi\in S_{n+1}$, l'isomorphisme
de sym{\'e}trie
\[
Res(L_{\phi(1)},\cdots,L_{\phi(n+1)})
\stackrel{\sim }{\F }
Res(L_1,\cdots,L_{n+1})
,
\]
provenant des isomorphismes de sym{\'e}trie
du fibr{\'e} d'intersection, {\'e}change leurs sections respectives
$Res(\sigma_{\phi(1)},\cdots,\sigma_{\phi(n+1)})$
et
$Res(\sigma_1,\cdots,\sigma_{n+1})$.
\end{lemme}
\begin{proof}[Preuve]
Il suffit de montrer cette {\'e}galit{\'e} sur l'ouvert
$V=\bigcap_{\phi\in S_{n+1}}U_{\phi}$ et de plus il suffit
de consid{\'e}rer le cas o{\`u} $\phi$ est une transposition de deux indices
$i$ et $j$.
Si $n=1$, il suffit alors d'appliquer le lemme (\ref{indep2}). Si
$n>1$, choisissons une permutation $\psi\in S_n$ telle que
$\psi(n+1)\notin \{ i,j\} $. Comme, sur $V$, la section
$Res(\sigma_1,\cdots,\sigma_{n+1})$ se factorise en:
\[
\mathcal{O}_{S} \longrightarrow N_{D_{\psi(1)}\cap\cdots\cap D_{\psi(n)}/S} (L_{\psi(n+1)})
\stackrel{\sim }{\F } I_{X/S}(L_1,\cdots,L_{n+1})\; ,
\]
le r{\'e}sultat en r{\'e}sulte, puisque $D_{\psi(1)}\cap\cdots\cap
D_{\psi(n)}$ est invariant par la transposition $\phi$.
\end{proof}
\addtocounter{subsubsection}{6}
\subsubsection{Multiplicativit{\'e} du r{\'e}sultant}
\addtocounter{theo}{1}
Soient $L_1^{\prime},L_1\sec,L_2,\cdots,L_{n+1}$ des faisceaux inversibles
suf\-fisamment positifs sur $X$ et supposons de plus que
$L_1^{\prime}\otimes L_1\sec\gg 0$. On notera alors
$P^{\prime}=P_{L_1^{\prime}}\times_S\cdots\times_SP_{L_{n+1}}$,
$P\sec =P_{L_1\sec}\times_S\cdots\times_SP_{L_{n+1}}$, $P=P^{\prime} \times_S
P\sec$ et $Q=P_{L_1^{\prime}\otimes L_1\sec}\times_S\cdots\times_SP_{L_{n+1}}$. Le
morphisme canonique de multiplication des sections
$m:P_{L_1^{\prime}}\times_SP_{L_1\sec}\longrightarrow P_{L_1^{\prime}\otimes L_1\sec}$ induit un
morphisme not{\'e} toujours $m: P\longrightarrow Q$.
Consid{\'e}rons alors le diagramme suivant
\begin{equation}
\label{diagramme-resultant}
\xymatrix{
&X_P\ar[d]_-{\pi}\ar[rr]_-n \ar@/_/[ddl]_-{\pi^{\prime}}
\ar@/^/[ddr]^-{\pi\sec} \ar@/^1pc/[rrr]^g
&&X_Q\ar[d]_-{\pi}\ar[r]_-f\ar@/^1pc/[ddd]^{\pi_1}
&X\ar[d]_-{\pi} \\
&P \ar[rr]_-m \ar[dl]^-{p^{\prime}}\ar[dr]_-{p\sec}
&&Q\ar[r] \ar[dd]_-{p_1}&S\\
P^{\prime} \ar[d]_-{p_1^{\prime}}&&P\sec \ar[d]_-{p_1\sec}\\
P_{L_1^{\prime}} &&P_{L_1\sec}& P_{L_1^{\prime}\otimes L_1\sec}
}
\end{equation}
On notera enfin $p_i$ les diff{\'e}rentes
projections de $P$, $P^{\prime}$ ou $Q$ sur $P_{L_i}$ et $\pi_i$ leur
compos{\'e}e avec $\pi$.
\begin{lemme}
\label{mult-resultant}
Il existe un isomorphisme
canonique de faisceaux inversibles sur $P^{\prime}\times_SP\sec$
\[
m^{\ast } Res(L_1^{\prime}\otimes L_1\sec,L_2,\cdots,L_{n+1})
\stackrel{\sim }{\F }
(p_1^{\prime})^{\ast } Res(L_1^{\prime},L_2,\cdots,L_{n+1})
\otimes
(p_1\sec)^{\ast } Res(L_1\sec,L_2,\cdots,L_{n+1})
\]
qui induit une {\'e}galit{\'e} sur les sections:
\[
m^{\ast } Res(\tau,\sigma_2,\cdots,\sigma_{n+1})
=
(p_1^{\prime})^{\ast } Res(\sigma_1\sec,\sigma_2,\cdots,\sigma_{n+1})
\otimes
(p_1\sec)^{\ast } Res(\sigma_1^{\prime},\sigma_2,\cdots,\sigma_{n+1})
\]
o{\`u} $\tau$ est la section canonique de
$f^{\ast } (L_1\otimes L_1^{\prime})\otimes
\pi^{\ast }_1 \mathcal{O}_{P_{L_1^{\prime} \otimes L_1\sec}}(1)$.
\end{lemme}
\begin{proof}[Preuve]
Il existe un isomorphisme canonique de faisceau inversibles sur
$P^{\prime}\times_SP\sec$:
\[
m^{\ast } p_1^{\ast }\mathcal{O}_{P_{L_1^{\prime}\otimes L_1\sec}}(1)
\simeq
(p_1^{\prime})^{\ast }\mathcal{O}_{P_{L_1^{\prime}}}(1)
\otimes
(p_1\sec)^{\ast }\mathcal{O}_{P_{L_1\sec}}
\]
En prenant l'image inverse par $\pi$ et en tensorisant avec
$g^{\ast }(L_1^{\prime}\otimes L_1\sec)$, on obtient sur
$X_{P^{\prime}\times_SP\sec}$ un isomorphisme:
\[
n^{\ast } \pi^{\ast }\mathcal{O}_{P_{L\otimes M}}(1) \otimes g^{\ast }(L_1^{\prime}\otimes L_1\sec)
\simeq
\left(
(\pi_1^{\prime})^{\ast } \mathcal{O}_{P_{L_1^{\prime}}}(1) \otimes g^{\ast } L_1^{\prime}
\right)
\otimes
\left(
(\pi_1\sec)^{\ast } \mathcal{O}_{P_{L_1\sec}}(1) \otimes g^{\ast } L_1\sec
\right)
\]
qui identifie $n^{\ast }\tau$ avec $\sigma_1^{\prime} \otimes \sigma_1\sec$.
La prori{\'e}t{\'e} de multiplicativit{\'e} du fibr{\'e}
d'intersection $I_{X_Q/Q}$
induit donc un isomorphisme:
\begin{multline*}
m^{\ast } I_{X_Q/Q}
(f^{\ast } (L_1^{\prime}\otimes L_1\sec) \otimes \pi_1^{\ast }
\mathcal{O}_{P_{L_1^{\prime}\otimes L_1\sec}}(1),\cdots,
f^{\ast } L_{n+1}\otimes \pi_n^{\ast } \mathcal{O}_{P_{L_{n+1}}}(1))
\stackrel{\sim }{\F } \\
I_{X_Q/Q}(g^{\ast } L_1^{\prime}\otimes g^{\ast } L_1\sec \otimes (\pi_1^{\prime})^{\ast }
\mathcal{O}_{P_{L_1^{\prime}}}(1)
\otimes (\pi_1\sec)^{\ast }
\mathcal{O}_{P_{L_1\sec}}(1)
,\cdots,
g^{\ast } L_{n+1}\otimes \pi_n^{\ast } \mathcal{O}_{P_{L_{n+1}}}(1))
\\
\stackrel{\sim }{\F }
(p_1^{\prime})^{\ast } Res(L_1^{\prime},\cdots,L_{n+1})
\otimes
(p_1\sec)^{\ast } Res(L_1\sec,\cdots,L_{n+1})
\end{multline*}
qui identifie les sections correspondantes.
\end{proof}
\subsection{Lien avec la d{\'e}finition classique du r{\'e}sultant}
Consid{\'e}rons le sch{\'e}ma $X=\mathbb{P}^n_k$ sur $S=Spec(k)$. Dans tout ce
qui suit, on {\'e}crira explicitement $X$ sous la forme
$Proj(k[X_1,\cdots,X_{n+1}]$, et on consid{\`e}rera les $n+1$ hyperplans
$H_i$ de $X$ d'{\'e}quations $X_i=0$. Fixons des
entiers strictement positifs $d_1,\cdots,d_{n+1}$ et consid{\'e}rons les
faisceaux $L_i=\mathcal{O}_{X}(d_i)$, qu'on identifiera {\`a} $\mathcal{O}_{X}(d_iH_i)$, en notant
$\tau_i$ la section canonique correspondante. Les $L_i$
sont suffisamment positifs et on regarde les espaces
projectifs $P_i$ correspondants sont de dimension
$\binom{d_i+n}{n}-1$. Notons enfin $V_i=H^0(X,L_i)$.\\
L'isomorphisme canonique (\ref{isom-resultant}) s'{\'e}crit ici:
\[
Res (L_1,\cdots,L_{n+1})
\stackrel{\sim }{\F }
\bigotimes_{i=1}^{n+1} p_i^{\ast } \mathcal{O}_{P_i}(k_i)
\otimes
I_{X/S}(L_1,\cdots,L_{n+1})
\]
avec $k_i=\prod_{k\neq i}d_k$. Comme
$\bigcap_{i=1}^{n+1}H_i=\emptyset$, $I_{X/S}(\tau_1,\cdots,\tau_{n+1})$
d{\'e}finit une trivialisation de $ I_{X/S}(L_1,\cdots,L_{n+1})$. On en
d{\'e}duit donc un
isomorphisme, associ{\'e} canoniquement au choix des coordonn{\'e}es $X_i$:
\[
Res (L_1,\cdots,L_{n+1})
\stackrel{\sim }{\F }
\bigotimes_{i=1}^{n+1} p_i^{\ast } \mathcal{O}_{P_i}(k_i)
\]
La section canonique
$Res (\sigma_1,\cdots,\sigma_{n+1})$ de $Res (L_1,\cdots,L_{n+1})$
d{\'e}finit donc une section de
$\bigotimes_{i=1}^{n+1} p_i^{\ast } \mathcal{O}_{P_i}(k_i)$
sur le produit
$P=P_1\times\cdots\times P_{n+1}$, c'est {\`a} dire un polyn{\^o}me
quasi-homog{\`e}ne sur $V=\bigoplus_{i=1}^{n+1}V_i$, de degr{\'e}
$k_i$ relativement {\`a} $V_i$. C'est le r{\'e}sultant {\'e}tudi{\'e} par
\textsc{Jouanolou} dans \cite{jouanolou1}. On le notera alors
$\underline{Res}(\sigma_1,\cdots,\sigma_{n+1})$. Notons alors que les lemmes
(\ref{sym-resultant}) et (\ref{mult-resultant}) entrainent que ce
polyn{\^o}me est sym{\'e}trique etmultiplicatif en chaque groupe de variables.
\subsubsection{La formule de Poisson}
Dans cette section, on interpr{\`e}te la formule de Poisson pour les
polyn{\^o}mes r{\'e}sultants (\cite{jouanolou1}, Prop 2.7,p124), qui permet de
calculer le
r{\'e}sultant $\underline{Res}(\sigma_1,\cdots,\sigma_{n+1})$ par r{\'e}currence sur
la dimension, en l'exprimant en fonction
d'un r{\'e}sultant $\underline{Res}(\sigma_1|_H,\cdots,\sigma_n|_H)$
associ{\'e} {\`a} un
hyperplan $H$, et de la norme d'une fonction
d{\'e}finie sur l'ouvert affine, compl{\'e}mentaire de $H$.\\
On note ici $H$ l'hyperplan $H_{n+1}$
de $X$ et $Q=P_1\times\cdots\times P_n$ de telle sorte que $P=Q\times
P_{n+1}$. Soit $Z\subset X_P$ le lieu des z{\'e}ros communs de
$\sigma_1,\cdots ,\sigma_n$, consid{\`e}rons alors les ouverts
$U_0=\{ q\in Q| Z\cap H_{n+1}\cap X_q=\emptyset\}$ et
$U=U_0\times P_{n+1}\subset P=Q\times P_{n+1}$. $H$ est un espace
projectif de dimension $n-1$, muni de coordonn{\'e}es $X_1,\cdots,X_n$, et
on peut donc d{\'e}finir un r{\'e}sultant
$Res(L_1|_H,\cdots,L_n|_H)$ sur $P_1^{\prime}\times\cdots P_n^{\prime}$, o{\`u}
$P_i^{\prime}=\mathbb{P} (H^0(H,L_i)^{\vee })$.
R{\'e}sumons d'abord dans un diagramme la situation:
\[
\xymatrix{&&Res(\sigma_1,\cdots,\sigma_{n+1}) \ar@{.}[dr]
&X_U\subset X_P \ar[d]
\\
Res(\sigma_1|_H,\cdots,\sigma_n|_H)\ar@{.}[dr] &
H_{Q^{\prime}}\ar[d] &X_{U_0}\subset X_Q \ar[d]
& P \ar[dl]_-{\pi_Q}\ar[dr]_-{\pi_{n+1}}
& X_{P_{n+1}}\ar[d]\ar[r]_-p &X \\
&Q^{\prime} \ar[d]_-{\pi^{\prime}_i}
&U_0\subset Q\ar[l]_-{\pi_{Q^{\prime}}}\ar[d]_-{\pi_i} &&P_{n+1} \\
&P_i^{\prime} &P_i
}
\]
Sur l'ouvert $X\setminus H$, on dispose d'un isomorphisme
$L_{n+1} \stackrel{\sim }{\F } \mathcal{O}_{X\setminus H}$. La restriction $\Tilde{\sigma}_{n+1}$
de $\sigma_{n+1}$ {\`a}
$P\times (X\setminus H)$ est donc une section de
$p_n^{\ast }(\mathcal{O}(1))$. Comme $Z_U$ est contenu dans
$(X\setminus H)\times U$, la restriction de $\Tilde{\sigma}_{n+1}$ {\`a}
$Z_U$ est bien d{\'e}finie et sa norme $N_{Z_U/U}(\Tilde{\sigma}_{n+1})$ est
une section de
$\pi_{n+1}^{\ast } \mathcal{O}(k_{n+1})$. Notons que ces consid{\'e}rations montrent
l'existence d'un isomorphisme canonique
\begin{equation}
\label{poisson}
Res(L_1,\cdots,L_{n+1})|_U \stackrel{\sim }{\F }
\pi_{n+1}^{\ast } \mathcal{O}_{P_{n+1}}(k_{n+1})
\end{equation}
qui identifie les sections $Res(\sigma_1,\cdots,\sigma_{n+1}|_U$ et
$N_{Z_U/U}(\Tilde{\sigma}_{n+1})$.
\begin{prop}[Formule de Poisson]
Sur l'ouvert $U$, on a l'{\'e}galit{\'e} entre polyn{\^o}mes de
$\Gamma(U_0,\mathcal{O}_{U_0})[V_{n+1}]$ :
\[
\underline{Res}(\sigma_1,\cdots,\sigma_{n+1})|_U
=
\pi_{Q^{\prime}}^{\ast } \underline{Res}(\sigma_1|_H,\cdots,\sigma_n|_H)|_{U_0}
.
N_{Z_U/U}(\Tilde{\sigma}_{n+1})
\]
\end{prop}
\begin{proof}[Preuve]
Utilisons la d{\'e}finition du faisceau r{\'e}sultant et la
multiplicativit{\'e}
du fibr{\'e} d'inter\-sec\-tion pour {\'e}crire le diagramme d'isomorphismes:
\[
\begin{array}{ccccc}
Res(L_1,\cdots,L_{n+1}) &
\stackrel{\sim }{\F } &
I_{X/S}(L^{\prime}_1,\cdots,L_n^{\prime},p^{\ast } \mathcal{O}_{X}(d_{n+1}H))&
\otimes &
I_{X/S}(L^{\prime}_1,\cdots,L_n^{\prime},\pi_{n+1}^{\ast }\mathcal{O}_{P_{n+1}}(1))\\
\| &&\downarrow\wr && \downarrow\wr\\
Res(L_1,\cdots,L_{n+1})&&
\left( I_{H/S}(L^{\prime}_1|_H,\cdots,L^{\prime}_n|_H)\right)^{d_{n+1}} &&
N_{Z/P}(\pi_{n+1}^{\ast }\mathcal{O}_{P_{n+1}}(1))\\
\downarrow\wr&&\downarrow\wr && \downarrow\wr\\
\begin{array}{c}
\bigotimes_{i=1}^{n+1} \pi_i^{\ast }\mathcal{O}_{P_i}(k_i)\\
\otimes\\
I_{X/S}(L_1,\cdots,L_{n+1})
\end{array} &&
\begin{array}{c}
\bigotimes_{i=1}^n \pi_i^{\ast }\mathcal{O}_{P_i}(k_i)\\
\otimes \\
\left( I_{H/S}(L_1|_H,\cdots,L_n|_H)\right)^{d_{n+1}}
\end{array} &&
\begin{array}{ccc}
\pi_{n+1}^{\ast } \mathcal{O}_{P_{n+1}}(k_{n+1})\\
{}\\
{}
\end{array}
\end{array}
\]
La deuxi{\`e}me ligne de ce diagramme donne un isomorphisme
\begin{equation}
\label{poisson2}
Res(L_1,\cdots,L_{n+1})\stackrel{\sim }{\F }
\left( I_{H/S}(L^{\prime}_1|_H,\cdots,L^{\prime}_n|_H)\right)^{d_{n+1}} \otimes
N_{Z/P}(\pi_{n+1}^{\ast }\mathcal{O}_{P_{n+1}}(1))
\end{equation}
En combinant la restriction de (\ref{poisson2}) {\`a} $U$ et
l'isomorphisme
$Res(\sigma_1|_H,\cdots,\sigma_n|_H)|_U:
\mathcal{O}_U\stackrel{\sim }{\F } I_{H/S}(L^{\prime}_1|_H,\cdots,L^{\prime}_n|_H)|_U$, on obtient
l'isomorphisme (\ref{poisson}).
La restriction de (\ref{poisson2}) {\`a} $U$ identifie donc
$Res(\sigma_1,\cdots,\sigma_{n+1})|_U$ {\`a}
=
$\pi_{Q^{\prime}}^{\ast } Res(\sigma_1|_H,\cdots,\sigma_n|_H)|_{U_0}
\otimes
N_{Z_U/U}(\Tilde{\sigma}_{n+1})$. L'{\'e}galit{\'e} polyn{\^o}miale
recherch{\'e}e s'en
d{\'e}duit en utilisant la d{\'e}finition des polyn{\^o}mes
r{\'e}sultants et les
isomorphismes de la
partie inf{\'e}rieure du diagramme.
\end{proof}
|
1997-12-21T20:47:45 | 9712 | alg-geom/9712027 | en | https://arxiv.org/abs/alg-geom/9712027 | [
"alg-geom",
"math.AG"
] | alg-geom/9712027 | Ron Donagi | Ron Donagi and Ron Livne | The arithmetic-geometric mean and isogenies for curves of higher genus | Latex, 18 pages | null | null | null | null | Computation of Gauss's arithmetic-geometric mean involves iteration of a
simple step, whose algebro-geometric interpretation is the construction of an
elliptic curve isogenous to a given one, specifically one whose period is
double the original period. A higher genus analogue should involve the explicit
construction of a curve whose jacobian is isogenous to the jacobian of a given
curve. The doubling of the period matrix means that the kernel of the isogeny
should be a lagrangian subgroup of the group of points of order 2 in the
jacobian. In genus 2 such a construction was given classically by Humbert and
was studied more recently by Bost and Mestre. In this article we give such a
construction for general curves of genus 3. We also give a similar but simpler
construction for hyperelliptic curves of genus 3. We show that the
hyperelliptic construction is a degeneration of the general one, and we prove
that the kernel of the induced isogeny on jacobians is a lagrangian subgroup of
the points of order 2. We show that for g at least 4 no similar construction
exists, and we also reinterpret the genus 2 case in our setup. Our construction
of these correspondences uses the bigonal and the trigonal constructions,
familiar in the theory of Prym varieties.
| [
{
"version": "v1",
"created": "Sun, 21 Dec 1997 19:47:44 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Donagi",
"Ron",
""
],
[
"Livne",
"Ron",
""
]
] | alg-geom | \section{Introduction}
It is well-known that computation of Gauss's
arithmetic-geometric mean involves iteration of a simple
step, whose algebro-geometric interpretation is the
construction of an elliptic curve isogenous to a given
one, specifically one whose period is double the original
period (for a modern survey see \cite{cox}). A higher
genus analogue should involve the explicit
construction of a curve whose jacobian is isogenous to
the jacobian of a given curve. The doubling of the
period matrix means that the kernel of the isogeny should
be a lagrangian subgroup of the group of points of order
$2$ in the jacobian. In genus $2$ such a construction was
given classically by Humbert \cite{hum} and was studied
more recently by Bost and Mestre \cite{bome}. In this article
we give such a construction for general curves of
genus $3$. We also give a similar but simpler
construction for hyperelliptic curves of genus $3$.
We show that the hyperelliptic construction is
a degeneration of the general one, and we prove that the kernel
of the induced isogeny on jacobians is a lagrangian subgroup
of the points of order $2$. We show that for $g \geq 4$ no similar
construction exists, and we also reinterpret the
genus $2$ case in our setup.
To construct these correspondences we use the bigonal
and the trigonal constructions, familiar in the
theory of Prym varieties (\cite{don}). In genus $2$
Bost and Mestre note that Humbert's construction induces
on jacobians an isogeny whose kernel is of type
$(\ZZ/2\ZZ)^2$. We show that Humbert's construction
is an instance of the bigonal construction, and prove
that the above kernel is a lagrangian subgroup of the
points of order $2$. In fact Bost and Mestre use Humbert's
construction to give a variant of Richelot's genus
$2$ arithmetic-geometric mean.
In light of the clear analogy, in particular the
fact that a generic principally polarized abelian
threefold is a jacobian, one might hope that our
construction could be used in a similar way.
We work throughout over an algebraically closed field of
characteristic $0$. However, our methods clearly extend more
generally. For example, the results of Section~\ref{genustwo}
hold if the characteristic is not $2$, and those of
Sections~\ref{hyperelliptic} and \ref{genusthree} if it is $>3$.
The first author thanks the Hebrew University of Jerusalem
and the Institute for Advanced Studies in Princeton for their
hospitality during the time this work was done.
The second author thanks the University of Pennsylvania for
its hospitality while this article was being written.
\section{Preliminaries} \label{prel}
\noindent {\bf Polarizations.} For an abelian variety
$A$ denote by $A[n]$ the kernel of multiplication by $n$.
In the sequel we will need the following standard facts and
notation.
\begin{enumerate}
\item A polarization $\Theta$ on an abelian variety $A$
induces by restriction a polarization $\Theta_B$ on any
abelian subvariety $B$ of $A$.
\item Recall that the type of a polarization $\Theta$ on a
$g$\/-dimensional abelian variety is a g\/-tuple of
positive integers $d_g|\dots|d_2|d_1$. We say that
$\Theta$ is a principal polarization if it is of type
$1^g = (1,\dots,1,1)$ ($g$\/ times). In that case, suppose
that $p$ is a prime, and that $K$ is a subgroup of $A[p]$
isomorphic to $({\bbB Z}/p{\bbB Z})^r$ and isotropic for the Weil
pairing ${\rm w}_p$. Then $\Theta$ induces a polarization
on $A/K$, characterized by the property that its pull back
to $A$ is $p\Theta$. Its type is then $p^{g-r}\cdot 1^r$.
In this situation we will say that $K$ is a lagrangian
subgroup of $A[p]$ if $r=g$.
\item The type of a polarization is preserved under continuous
deformations.
\end{enumerate}
\vspace*{0.1cm}
\noindent {\bf Double covers.} Given a double cover, i.e. a
finite morphism $\pi: \tilde{C} \rightarrow C$ of degree $2$ between smooth
projective curves, the Prym variety ${\rm Prym}\,(\tilde{C}/C)$ is defined
to be the connected component of the kernel of the norm map
\[ \pi_*: {\rm Jac}\,(\tilde{C}) \rightarrow {\rm Jac}\,(C). \]
It is an abelian variety, and it has a natural principal
polarization when $\pi$ is unramified, namely one half of
the polarization induced on it as an abelian subvariety of
${\rm Jac}\,(\tilde{C})$ (\cite{mum2}). This definition extends to
singular curves $C$, $\tilde{C}$, if we interpret ${\rm Jac}\,$ as
the (not necessarily compact) generalized jacobian. This was
studied by Beauville \cite{bea}.
Particularly important for us will be the cases when
1. $C$, $\tilde{C}$ have only ordinary double points,
2. $\pi^{-1}(C_{\rm sing}) = \tilde{C}_{\rm sing}$, and
3. for each $x\in C_{\rm sing}$ the inverse
image $\pi^{-1}(x)$ consists of a single point,
and each branch of $\pi^{-1}(x)$ maps to a different branch
of $x$ and is ramified over it.
(We shall then say that $\pi$ is of Beauville type at $x$.)
In such cases ${\rm Prym}\,(\tilde{C}/C)$ is compact, and the following
three conditions are equivalent:
\begin{enumerate}
\item $\pi$ is unramified away from $C_{\rm sing}$.
\item The arithmetic genera satisfy $g(\tilde{C}) = 2g(C)-1$.
\item The cover $\tilde{C}/C$ is a flat limit of smooth unramified
double covers.
\end{enumerate}
We shall call a cover satisfying these conditions allowable;
from the third condition we see that the ${\rm Prym}\,$ is prinicipally
polarized in such a case.
Let $C$ be a curve having only ordinary double points as
singularities, and let $\nu_x: N_x \rightarrow C$ be the normalization
map of exactly one such singular point $x$. We denote by $L(x)$
the line bundle of order $2$ in ${\rm Ker}\,\nu_x^\ast$. (It is
obtained from the trivial line bundle on $N_x$ by gluing the
fibers over the two inverse images of $x$ with a twist of $-1$
relative to the natural identification.)
\begin{lemma} \label{polar}
Let $\pi: \tilde{C} \rightarrow C$ be an allowable double cover, $\nu \pi:
\nu \tilde{C} \rightarrow \nu C$ its (partial) normalization at $r \geq 1$
ordinary double points $x_1,\dots,x_r$. Let $g$ be the (arithmetic)
genus of the partial normalization $\nu C$, so the
arithmetic genus of $C$ is $g+r$. Then ${\rm Prym}\, (\tilde{C}/C)$
has a principal polarization,
${\rm Prym}\,(\nu \tilde{C}/ \nu C)$ has a polarization of type
$2^{g}1^{r-1}$, and the pullback map
$\nu^*:{\rm Prym}\,(\tilde{C}/C) \rightarrow {\rm Prym}\,(\nu\tilde{C}/\nu C)$ is an isogeny
of degree $2^{r-1}$. The kernel of $\nu^*$ is the subgroup
of ${\rm Prym}\,(\tilde{C}/C)[2]$ generated by the pairwise differences
of the line bundles $L(x_i)$ defined above. This subgroup is
isotropic for the mod $2$ Weil pairing $w_2$.
\end{lemma}
\begin{pf} The generalized jacobians fit in short exact
sequences
\[\begin{array}{ccccccccc}
0 & \rightarrow & \GG_m^r & \rightarrow & {\rm Jac}\,(\tilde{C}) & \rightarrow & {\rm Jac}\,(\nu \tilde{C}) &\rightarrow & 0 \\
& & \downarrow & & \downarrow & & \downarrow & & \\
0 & \rightarrow & \GG_m^r & \rightarrow & {\rm Jac}\,(C) & \rightarrow & {\rm Jac}\,(\nu C) &\rightarrow & 0
\end{array} \]
where the vertical maps are the norm maps induced by $\pi$
and by $\nu\pi$. We
compare the kernels: to begin with, the kernel of the norm
map is connected for ramified double covers (in particular
for $\nu \tilde{C}/\nu C$), and has two components for unramified
covers. This is shown in \cite{mum2} in the nonsingular case,
and so by continuity this holds also for allowable singular
covers (in particular for $\tilde{C}/C$). The multiplicative
groups parametrize extension data
and the norm is the squaring map. So the short exact sequence
of kernels gives
\[0 \rightarrow (\ZZ/2\ZZ)^r \rightarrow \Prym(\Ctil/C) \times \ZZ/2\ZZ \stackrel{\nu^*}{\rightarrow}
\Prym(\nu \Ctil/\nu C) \rightarrow 0,\]
and the first part of the lemma follows.
To prove that the subgroup $(\ZZ/2\ZZ)^{r-1}$ of
$\Prym(\Ctil/C)[2]$ is isotropic for ${\rm w}_2$, notice that
its generators are reductions modulo $2$ of the
vanishing cycles for $\tilde{C}$, and vanishing cycles for distinct
ordinary double points are disjoint. Therefore these vanishing
cycles have $0$ intersection number in ${\bbB Z}$\/- (or ${\bbB Q}$\/-)
homology. By the definition of the polarization of $\Prym(\Ctil/C)$
in Section~\ref{prel}, and the well-known expression for the
Weil pairing in terms of the intersection (or cup product)
pairing (see e.g. \cite[theorem 1, Ch. 23]{mum}), the
rest of the lemma follows.
\end{pf}
\section{The bigonal and the trigonal constructions}
\label{bigonal}
There are several elementary constructions which associate a
double cover of some special kind with another cover (or curve)
with related Prym (of Jacobian).
We now review the bigonal and the trigonal constructions,
following (\cite{don}). Assume we
are given smooth projective curves $\tilde{C}$, $C$ and $K$ and
surjective maps $f:C\rightarrow K$ and $\pi:\tilde{C}\rightarrow C$, so that
$\deg\pi = \deg f=2$ over any component. The bigonal
construction associates new curves and maps of the same
type $\tilde{{C'}}\sra{\pi'} C'\sra{f'}K$ as follows. Let
$U\subset K$ be the maximal open subset over which $f\pi$
is unramified. Then $\tilde{{C'}}$ represents over $U$ the sheaf of
sections, in the complex or the \'etale topology, of
$\pi:(f\pi)^{-1}U\rightarrow f^{-1}U$. It is a $4$\/-sheeted
cover of $U$. We then view $\tilde{{C'}}_{|U}$ as a locally
closed subvariety of $\tilde{C} \times \tilde{C}$ and define
$\tilde{{C'}}$ as the closure.
The projection to $U$ extends to a morphism $\tilde{{C'}}\rightarrow K$,
and the involution $\iota$ of $\tilde{{C'}}_{|U}$
which sends a section to the complementary section extends
to $\tilde{{C'}}$. We define $C'=\tilde{{C'}}/\iota$ and $f'$ and
$\pi'$ as the quotient maps.
We will need to extend this construction to allowable covers
of curves with ordinary double points; however in a family
acquiring a singularity of Beauville type the arithmetic genus
of the resulting $\tilde{{C'}}$ is not locally constant.
More technically, the naive construction as the
closure of $\tilde{{C'}}_{|U}$ in $\tilde{C}\times\tilde{C}$ is not
flat in families, which is not adequate for our purposes: for
example, we want the bigonal construction to be symmetric.
To achieve this, we define the bigonal construction for
singular allowable covers by {\em choosing} a flat family
of smooth covers whose limit is our allowable cover, and
{\em defining} the construction to be the limit of the
construction for the nonsingular fibers. Beauville's results
imply that this is well defined, and does give a symmetric
construction: this is more or less clear except at a
singularity of Beauville type. There the problem reduces to
a local calculation whose answer, which we record in
\ref{type}. below, is visibly symmetric.
We will need a few properties of this construction (see
\cite[Section 2.3]{don})
\begin{enumerate}
\item As we said, the construction over $U$ is symmetric:
starting with $\tilde{{C'}},\dots,f'$ gives back $\tilde{C},\dots,f$.
\item
\label{type}
Denote the type of $\tilde{C}/C$ at a point $k\in K$ by
\begin{itemize}
\item $\ssub \seq \sslash \smin\smin $ if $C$ is unramified over $k$ and $\tilde{C}$ is
ramified over exactly one point in $f^{-1}(k)$;
\item $\stackrel{\subset}{\ssub} \sslash \ssub $ if $C$ is ramified
over $k$ but $\tilde{C}$ is unramified over the point
$f^{-1}(k)$;
\item $\ssub \ssub \sslash \smin\smin $ if $C$ is
unramified over $k$ and $\tilde{C}$ is ramified over both
branches of $C$ over $k$;
\item $\btf$ if both $C$,
$\tilde{C}$ have ordinary double points above it, and $\tilde{C}/C$
is of Beauville type there.
\end{itemize}
If $\tilde{C}/C$ is of type $\ssub \seq \sslash \smin\smin $, $\stackrel{\subset}{\ssub} \sslash \ssub $, $\ssub \ssub \sslash \smin\smin $, $\btf$ at
$k$ then $\tilde{{C'}}/C'$ is respectively of type
$\stackrel{\subset}{\ssub} \sslash \ssub $, $\ssub \seq \sslash \smin\smin $, $\btf$,$\ssub \ssub \sslash \smin\smin $ there.
Notice that normalization takes type $\btf$ to type $\ssub \ssub \sslash \smin\smin $.
\item The natural $2$-$2$ correspondence between $\tilde{C}$
and $\tilde{{C'}}$ induces an isogeny
$\Prym(\Ctil/C)\rightarrow\Prym(\tilde{{C'}}/C')$, whose kernel is the same as the kernel
of the natural isogeny
$\Prym(\Ctil/C)\rightarrow\Prym(\Ctil/C)^\vee$ induced by the polarization from
$\Prym(\Ctil/C)$ to its dual abelian variety $\Prym(\Ctil/C)^{\vee}$.
In other words we get an isomorphism
$\Prym(\Ctil/C)^\vee\sra{\sim}\Prym(\tilde{{C'}}/C')$ (cf. Pantazis \cite{pan},
at least
when $K={\bbB P}^1$ which is all we need). As a check, let
$a$, $b$, $c$, and $d$
be the numbers of points where $\tilde{C}/C$ is of type
$\ssub \seq \sslash \smin\smin $, $\stackrel{\subset}{\ssub} \sslash \ssub $, $\ssub \ssub \sslash \smin\smin $, and $\btf$ respectively.
Then by Lemma~\ref{polar} the polarization type
for $\Prym(\Ctil/C)$ is $1^{\frac{a+2c}{2}-1}2^{\frac{b+2d}{2}-1}$.
Similarly the polarization type for
$\Prym(\tilde{{C'}}/C')$ is obtained by interchanging $a$ with $b$ and
$c$ with $d$, and this gives exactly the type dual to the
one of $\Prym(\Ctil/C)$.
\end{enumerate}
For Recillas's trigonal construction start with
$K$, $C$, $\tilde{C}$, $\pi$, and $f$ as before except that $f$
now has degree $3$. We get a cover $g:X\rightarrow K$
of degree $4$ by making over the smooth unramified part
$U$, defined as before, a
construction analogous to what we previously did to get $C'$.
Namely, let $\tilde{X}/U$ represent the sheaf of sections of
$\pi: (f\pi)^{-1}U\rightarrow f^{-1}U$, and define $X/U$ as the
quotient of $\tilde{X}$ divided by $\iota$ (which is
defined as before). In the nonsingular case we define $X$
as the closure of $X/U$ in $X\times X\times X$, and in the
general allowable case by taking a flat limit of the construction
for smooth, unramified covers. Here we have
(\cite[Section 2.4]{don})
\begin{enumerate}
\item Over $U$ the construction is reversible: $C_{|U}$
represents the sheaf of partitions of $X_{|U}$ to two pairs of
sections, and $\tilde{{C'}}$ represents the choice of one of these
pairs.
\item Denote the type of $\tilde{C}/C$ at a point $k\in K$
by
\begin{itemize}
\item $\tto$ for $C$, $\tilde{C}$ if $C$ has exactly one simple
branch point over $K$ and $\pi$
is unramified over $f^{-1}(k)$;
\item $\ssub\ssub\seq \sslash \smin\smin\smin $ if $f$ is unramified at $k$ and $\pi$ is branched
over two of the branches of $f$ and unramified over the third;
\item $\tttp$ if two branches of
$C$ over $K$ cross normally, the third is unramified, and
moreover, if $\tilde{C}/C$ is of Beauville type over the double
point and unramified over the unramified branch.
\end{itemize}
Then $X$ has exactly one simple branch point at a point
$k\in K$ of type $\tto$ for $\tilde{C}/C$, and we denote
by $\ssub \smin\smin $ the type of $X$ over $k$. Conversely, if $X$
is of type $\ssub \smin\smin $ at $k$
then $\tilde{C}/C$ is of type $\tto$ there. If $\tilde{C}/C$ is
of type $\ssub\ssub\seq \sslash \smin\smin\smin $ at $k$ then $X$ has two simple branch
points over $k$, which we denote by type $\ssub \ssub $.
Here the situation is not reversible:
if $X$ is of type $\ssub \ssub $ at $k$ then $\tilde{C}/C$ is of
type $\tttp$ there. Notice that normalization takes type
$\tttp$ to type $\ssub\ssub\seq \sslash \smin\smin\smin $.
\item If $K\simeq{\bbB P}^1$ and $\tilde{C}/C$ is allowable, then
$X$ is smooth and ${\rm Jac}\,(X)\simeq{\rm Prym}\,(\tilde{C}/C)$. This is
due to Recillas when $\tilde{C}/C$ is smooth unramified, and
again limiting arguments imply this in general.
\end{enumerate}
\section{The genus $2$ case}
\label{genustwo}
Humbert's correspondence of curves of genus $2$ was studied
by Bost and Mestre (see \cite{hum},
\cite{bome}). We shall show how to make this correspondence
via the bigonal construction, and use this to determine the
type of the isogeny.
Humbert's construction starts with a conic $C$ in ${\bbB P}^2$
with $6$ general points on it (see Remark~\ref{gen} below),
which are given as $3$
unordered pairs $\{P'_i,P_i''\}$, $i=1,2,3$. It associates
to these $3$ new unordered
pairs of points, all distinct, on $C$ as follows. Let
$\overline{P'_iP_i''}$ be the $3$ lines joining paired points,
and let $l_k$ be the intersection of $\overline{P'_iP_i''}$ and
$\overline{P'_jP_j''}$ if $\{i,j,k\}=\{1,2,3\}$. The new
$3$ unordered pairs of points on $C$ are then the pairs of
points of tangency to $C$ from the $l_k$\/'s.
For our purposes it is more convenient to view the new points
as lying on the conic $C^*$ dual to $C$ in the dual plane
${\bbB P}^{2*}$. A point of ${\bbB P}^{2*}$ is a line in ${\bbB P}^2$; it
is in $C^*$ if and only if this line is tangent to $C$. Let
$\phi:{\bbB P}^2\rightarrow{\bbB P}^{2*}$ be the isomorphism defined by $C$;
namely, for $P\not\in C$ there are two tangents to $C$
through $P$, and
$\phi(P)$ is the line joining their points of tangency. For
$P\in C$, $\phi(P)$ is the tangent to $C$ at $P$. Under the
isomorphism $\phi_{|C}:C\sra{\sim} C^*$, Humbert's new pairs
go to the pairs $L'_k,L''_k$ of tangents to $C$ through $l_k$.
\begin{theorem}
Let $\pi:H\rightarrow C$ and $\pi^*:H^*\rightarrow C^*$ be double covers
branched over the old and new sets of points respectively. Then
there is an isogeny ${\rm Jac}\,(H)\rightarrow{\rm Jac}\,(H^*)$ whose kernel
is a lagrangian subgroup of ${\rm Jac}\,(H)[2]$.
\end{theorem}
\begin{pf} Choose some $k\in\{1,2,3\}$. The set $L^* = L_k^*$ of
lines through $l_k$ is the line in ${\bbB P}^{2*}$ dual to $l_k$.
Let $f:C\rightarrow L^*$ be the ``projection'' sending each point of
$C$ to the line joining it to $l_k$.
Dually, let $f^*:C^*\rightarrow L = \overline{P_k'P_k''}$ be
the ``projection'' sending each tangent line of $C$ to its
intersection with $L$. Let $\psi:L\rightarrow L^*$ be the
isomorphism sending a line through $l_k$ to its intersection
with $L$. The maps $f$ and $f^*$ have degree $2$, and hence
also $g=\psi f^*$ has degree $2$. Both coverings
$H\sra{\pi}C\sra{f}L^*$ and $H^*\sra{\pi^*}C^*\sra{g}L^*$
are unramified over the complement in $L^*$ of the six points
\begin{itemize}
\item The tangents $L'_k,L''_k$ to $C$ through $l_k$; there
$H/C$ is of type $\stackrel{\subset}{\ssub} \sslash \ssub $ and $H^*/C^*$ is of type $\ssub \seq \sslash \smin\smin $.
\item The lines $\overline{P_i'P''_i}$ and $\overline{P_j'P''_j}$
whose intersection defines $l_k$; there both $H/C$ and
$H^*/C^*$ are of type $\ssub \ssub \sslash \smin\smin $.
\item The lines $\overline{P'_kl_k}$ and $\overline{P''_kl_k}$;
there $H/C$ is of type $\ssub \seq \sslash \smin\smin $ and $H^*/C^*$ is of type
$\stackrel{\subset}{\ssub} \sslash \ssub $.
\end{itemize}
It follows that if we perform the bigonal construction
on $H\rightarrow C\rightarrow L^*$, the two points $\overline{P_i'P''_i}$ and
$\overline{P_j'P''_j}$ of $L^*$ are of Beauville type for the
resulting cover $H'\rightarrow C' \rightarrow L^*$ and there are no
other singularities (see Section~\ref{bigonal}). The
preceding analysis of the ramification of $H^*/C^*$,
combined with the one for the bigonal construction
$H'/C'$ in Section~\ref{bigonal}
shows that the normalization of $H'/C'$ is isomorphic to
$H^*/C^*$. It remains to determine the kernel of the induced
isogeny on jacobians; by Pantazis's result recalled above,
it factors as
\[ \begin{array}{rcl}
{\rm Jac}\, H & \simeq & {\rm Prym}\,(H/C) \sra{\sim} {\rm Prym}\,(H/C)^{\vee}
\sra{\sim} {\rm Prym}\,(H'/C') \\
& \sra{\nu^{\ast}} & {\rm Prym}\,(H^*/C^*) \simeq {\rm Jac}\, H^* \,.
\end{array}\]
To compute the kernel of $\nu^{\ast}$ we cannot use
Lemma~\ref{polar} directly, since $H'/C'$ is
not allowable, being ramified over two points
$x'\in g^{-1}(L'_k)$, $x''\in g^{-1}(L''_k)$. Instead
glue $x'$ to $x''$ to obtain a curve with one more double
point $C''$ and glue their inverse images in $H'$ to get
a curve $H''$, which is now an allowable cover of $C''$
($H''$ is obtained from $H$ by gluing the Weierstrass
points in pairs). We have maps of covers $H^*/C^*\rightarrow
H'/C'\rightarrow H''/C''$ inducing maps of Prym varieties. Applying
Lemma~\ref{polar} twice now gives that the kernel of
${\rm Prym}\,(H''/C'') \rightarrow {\rm Prym}\,(H/C)$ is an isotropic subgroup
isomorphic to $(\ZZ/2\ZZ)^2$ and that ${\rm Prym}\,(H''/C'')$ is isomorphic
to ${\rm Prym}\,(H^*/C^*)$. This implies that ${\rm Ker}\,\nu^{\ast}$ is as
asserted, completing the proof of the Theorem.
\end{pf}
\begin{remark}
\label{gen} {\rm
The points $\{P'_i,P''_i\}$ are assumed general only to
guarantee that they are distinct and that the resulting
new $6$ points are also distinct (for which it suffices
that the tangents to $C$ from $l_k$ in the proof do not
touch $C$ at $P'_k$ nor at $P''_k$). In the case considered
in \cite{bome} this holds, because they assume that
$C$ and the points are real and satisfy some ordering
relations. }
\end{remark}
\section{The hyperelliptic genus $3$ case}
\label{hyperelliptic}
In this section we will solve our problem in the hyperelliptic
case: we will construct a correspondence between
the generic hyperelliptic curve of genus $3$ and a certain
non-generic curve of genus $3$ (which is not hyperelliptic).
Let $H$ be a hyperelliptic curve of genus $3$ and
let $\pi_1:H\rightarrow{\bbB P}^1$ be the hyperelliptic double cover.
Choose a grouping in pairs of the $8$ branch points
$w_1,\dots,w_8\in{\bbB P}^1$ of $\pi_1$. We claim that there
exists a map $g_1:{\bbB P}^1\rightarrow{\bbB P}^1$, of degree $3$, which
identifies paired points. This can be seen in several
ways. Firstly, let $T$ be the curve obtained from ${\bbB P}^1$
by identifying paired points to ordinary double points.
We think of $T$ as a curve of genus $4$ and take its
canonical embedding to ${\bbB P}^3$. As in the nonsingular
case, the canonical map is well behaved, and in particular the
canonical image of $T$ lies on a unique, generically
nonsingular quadric by the Riemann-Roch theorem. Projecting
via either of the two ruling of this quadric will give the
desired map $g_1$. Notice that by its construction $g_1$
factors as ${\bbB P}^1\sra{\nu} T \sra{g} {\bbB P}^1$, where $\nu$ is
a normalization map.
Another way to get $g_1$ is to embed ${\bbB P}^1$ in ${\bbB P}^3$
as a rational normal curve. We look for a projection
from ${\bbB P}^3$ to ${\bbB P}^1$ which identifies paired points.
The center of this projection is a line $L$ which
must meet the $4$ lines joining the pairs. The
grassmanian $G(1,{\bbB P}^3)$ of lines in ${\bbB P}^3$ is
naturally a quadric in ${\bbB P}^5$ and the condition to meet
a line is a linear condition. We see again that there is
always at least one such $L$, and generically two.
We now perform the trigonal construction. This gives a map
of degree $4$ $f:C\rightarrow{\bbB P}^1$ sitting in a diagram
\begin{equation}
\label{hyp1}
\begin{array}{rcl}
&&H \\
&&\downarrow \! \mbox{$\scriptstyle{\pi_1}$} \\
C&&{\bbB P}^1\\
&\sf\!\!\searrow\;\;\swarrow\!\!\mbox{$\scriptstyle{g_1}$}&\\
&{\bbB P}^1& \end{array}
\end{equation}
Let $w_{12},\dots,w_{78}$ be the $4$ images of the $w_i$\/'s
under $g_1$, with the indices indicating the grouping. By
the Riemann-Hurwitz formula there are generically $4$ points
$a_1,\dots,a_4$ in ${\bbB P}^1$ over which $g_1$ is branched, with
a simple branch point over each. Hence $H/{\bbB P}^1$ is of type
$\tto$ at each $a_i$ and of type $\ssub\ssub\seq \sslash \smin\smin\smin $ at each $w_{2i-1,2i}$.
>From the properties of the trigonal construction we get
$2-2g(C)=8-8-4$, so that $C$ has genus $3$.
The trigonal construction gives a birational correspondence between
\begin{itemize}
\item The moduli of the data
$(H \sra{\pi_1}{\bbB P}^1 \sra{g_1} {\bbB P}^1)$ with
$4$ points of type $\tto$ and $4$ points of type $\ssub\ssub\seq \sslash \smin\smin\smin $.
\item A component of the Hurwitz scheme parametrizing
$4$-sheeted covers $f:C\rightarrow {\bbB P}^1$ with $4$ simple branch
points and $4$ double branch points.
\end{itemize}
Each of these moduli spaces is $5$ dimensional. (Another
component of this Hurwitz scheme parametrizes bielliptics,
namely maps $f:C\rightarrow {\bbB P}^1$ which factor through a double cover
$E \rightarrow {\bbB P}^1$ where $E$ is elliptic. Curves in this latter
component are taken by the trigonal construction to towers
$H\rightarrow \overline{H} \rightarrow {\bbB P}^1$ where $\overline{H}=A \cup B$
is reducible, with $A$, $B$ of degrees $1$, $2$ respectively
over ${\bbB P}^1$. We shall not need this component in what follows.)
The key point for us is that the trigonal construction induces
an isogeny ${\rm Jac}\,(C)\rightarrow{\rm Jac}\,(H)$ whose kernel is lagrangian
in ${\rm Jac}\,(C)[2]$. More precisely we have the following
\begin{proposition}
Let ${\bbB P}^1\sra{\nu}T\sra{g}{\bbB P}^1$ be as before, and let $\tilde{T}$
be the curve obtained by identifying the Weierstrass points in
$H$ to ordinary double points with the same grouping as the one
we chose to get $T$. Then
\begin{enumerate}
\item Diagram (\ref{hyp1}) extends to
\[ \begin{array}{rcccl}
&&\tilde{T}&\sla{\tilde{\nu}}&H\\
&&\mbox{$\scriptstyle{\pi}$}\!\downarrow\;&&\;\downarrow\!\mbox{$\scriptstyle{\nu}$}\mbox{$\scriptstyle{\pi}$}\\
C&&T&\sla{\nu}&{\bbB P}^1\\
&\sf\!\!\searrow\;\;\swarrow\!\!\mbox{$\scriptstyle{g}$}&\\
&{\bbB P}^1& \end{array}\,. \]
Here $\tilde{\nu}:H\rightarrow\tilde{T}$ is the normalization map, and we view
$\nu\pi:=\pi_1:H\rightarrow {\bbB P}^1$ as the normalization of
$\pi:\tilde{T}\rightarrow T$.
\item $\tilde{\nu}$ induces an isogeny of polarized abelian varieties
$\tilde{\nu}^*:{\rm Prym}\,(\tilde{T}/T)\rightarrow{\rm Jac}\,(H)$ whose kernel is
lagrangian in ${\rm Prym}\,(\tilde{T}/T)[2]$.
\item Let $\phi:{\rm Jac}\,(C)\rightarrow{\rm Jac}\,(H)$ be the isogeny obtained
by composing $\tilde{\nu}^*$ with the isomorphism
${\rm Jac}\,(C)\simeq{\rm Prym}\,(\tilde{T}/T)$.
Then the kernel of $\phi$ is lagrangian in ${\rm Jac}\,(C)[2]$, and the
kernel of the dual isogeny $\phi^*:{\rm Jac}\,(H)\rightarrow{\rm Jac}\,(C)$ is the
lagrangian subgroup of ${\rm Jac}\,(H)[2]$ generated by the differences of
identified Weierstrass points.
\end{enumerate}
\end{proposition}
\begin{pf}
Part 1. holds because $\tilde{T}/T$ is allowable. The pairs
of points of $H$ identified by $\nu$ lie over points of
type $\tttp$ for $T$, $\tilde{T}$. Hence they are branch points
for $\pi$, namely Weierstrass points. The rest follows from
Lemma~\ref{polar}.
\end{pf}
\section{The generic genus $3$ case}
\label{genusthree}
Let $C$ be a generic curve of genus $3$. In this section we shall
give a construction of a curve $C'$ of genus $3$ and an isomorphism
${\rm Jac}\, (C)/L \simeq {\rm Jac}\, (C')$ where $L$ is a lagrangian subgroup of
${\rm Jac}\, (C)[2]$. Let $f:C\rightarrow{\bbB P}^1$
be a map of degree $4$, and let $b_1$, $b_2$ be points in
${\bbB P}^1$ such that $f$ has two simple branch points over each
$b_i$. It is easy to show such $f$, $b_1,b_2$ exist, and in fact we will
parametrize the space of such $f$\/'s in the end of this section.
We perform the trigonal construction on $f$. This gives curves
$T$, $\tilde{T}$ and maps $g:T\rightarrow{\bbB P}^1$ and $\pi:\tilde{T}\rightarrow T$, with
$\deg g = 3$ and $\deg \pi=2$. Let $\tilde{\nu}:\nu \tilde{T}\rightarrow\tilde{T}$
and $\nu:\nu T\rightarrow T$ be normalization maps and let
$\nu\pi:\nu\tilde{T}\rightarrow\nu T$ be the map induced by $\pi$. The
properties of the trigonal construction show the following.
Firstly, $T$ and $\tilde{T}$ have each two ordinary double points,
one over each $b_i$, and no other singularities. Next, the
map $g\nu:\nu T\rightarrow{\bbB P}^1$ has exactly $8$ branch points, all
simple, one over each $a_i$. It follows that the genus
$g(\nu T)$ is $2$ and therefore the arithmetic genus $g(T)$
is $4$. The map $\nu\pi$ has exactly $4$ ramification points
$P_i$, $Q_i$, two over each $b_i$ for $i = 1,2$, and hence
$g(\nu\tilde{T})=5$ and $g(\tilde{T})=7$.
Since $\nu T$ has genus $2$, it is hyperelliptic. Let
$h:\nu T\rightarrow {\bbB P}^1$ be the hyperelliptic double cover, and
let $w_1,\dots,w_6\in{\bbB P}^1$ be the branch points of $h$.
The bigonal
construction gives curves and maps of degree $2$
$\nu\tilde{T}'\sra{\nu\pi'}\nu T'\sra{h'}{\bbB P}^1$. The points
in ${\bbB P}^1$ over which $h\nu\pi$ is not \'etale are the $6$
$w_i$\/'s, which are of type $\stackrel{\subset}{\ssub} \sslash \ssub $ for $\nu T$ and $\nu\tilde{T}$,
and the $4$ points $h(P_i)$\/, $h(Q_i)$\/, $i=1,2$, which
are of type $\ssub \seq \sslash \smin\smin $. The types get reversed for $\nu T'$ and
$\nu\tilde{T}'$, and in particular $\nu T'$ is ramified exactly
over the $h(P_i)$\/'s and the $h(Q_i)$\/'s. It follows that
$g(\nu T')=1$. We also see that $\nu\pi'$ has $6$ branch points,
say $w'_1,\dots,w'_6$, one over each of the $w_i$\/'s, and
hence $g(\nu\tilde{T}')=4$. The curves $\nu T'$ and $\nu\tilde{T}'$
are nonsingular.
Choose a grouping of the $w_i$\/'s in $3$ pairs. Identify
the corresponding $w'_i$\/'s in $\nu T'$ to get a curve
$T'$ with $3$ ordinary double points, say $w'_{12}$, $w'_{34}$,
$w'_{56}$, the indices indicating the groupings. $T'$ has
arithmetic genus $4$. Likewise identify the corresponding
points above the $w'_i$\/'s on $\nu\tilde{T}'$ to obtain a curve
$\tilde{T}'$ with $3$ ordinary double points and arithmetic
genus $7$.
As in the nonsingular case, the canonical embedding sends $T'$
to ${\bbB P}^3$ and the image sits on a unique, generically smooth
quadric. Choosing one of the two rulings of this quadric gives
a map $g':T'\rightarrow{\bbB P}^1$. This map is of degree $3$, because the
canonical curve is a curve of type $(3,3)$ on the quadric. The
map $g'\nu':\nu T'\rightarrow{\bbB P}^1$ is ramified over $n=6$ points, since
$2-2g(\nu T')=0=3(2-2g({\bbB P}^1))-n$. Over these the pair $\nu T'$,
$\nu\tilde{T}'$ is of type $\tto$. There are also $3$ points of
type $\ssub\ssub\seq \sslash \smin\smin\smin $, the images under $g'$ of the identified pairs $w'_{12}$,
$w'_{34}$, $w'_{56}$. The
trigonal construction performed on
$\nu\tilde{T}'\slra{\nu\pi'}\nu T'\slra{g'\nu'}{\bbB P}^1$ gives a
curve $C'$ and a map $f':C'\rightarrow{\bbB P}^1$ of degree $4$. We
readily see it has genus $3$. The following diagram summarizes
the procedure:
\[ \begin{array}{rcccccccccl}
&&{}_7\tilde{T}&\sla{\mbox{$\scriptstyle{\tilde{\nu}}$}}&{}_5\nu\tilde{T}&&{}_4\nu\tilde{T}'&
\sra{\mbox{$\scriptstyle{\tilde{\nu}}$}'}& {}_7\tilde{T}'&&\\
&&\mbox{$\scriptstyle{\pi}$}\!\downarrow&&\downarrow\mbox{$\scriptstyle{\nu}$}\mbox{$\scriptstyle{\pi}$}&&\mbox{$\scriptstyle{\nu}$}\mbox{$\scriptstyle{\pi}$}'\!\downarrow&&\downarrow\mbox{$\scriptstyle{\pi}$}'&&\\
{}_3 C&&{}_4 T&\sla{\nu}&{}_2\nu T&&{}_1\nu T'&\sra{\nu'}&
{}_4 T'&&{}_3 C'\\
&\sf\!\searrow\;\swarrow\!\mbox{$\scriptstyle{g}$}&&&& \mbox{$\scriptstyle{h}$}\!\searrow\;\swarrow\!\mbox{$\scriptstyle{h}$}'&&&&
\mbox{$\scriptstyle{g}$}'\!\searrow\;\swarrow\!\sf'&\\
&{\bbB P}^1&&&&{\bbB P}^1&&&&{\bbB P}^1&
\end{array}\,. \]
Before stating our main result we need to discuss the choices
made in the construction. Writing $f^{-1}(b_i) = 2(P_i+Q_i)$,
we obtain a point of order $2$
\[ \alpha = \alpha(f) = P_1 + Q_1 - P_2 - Q_2. \]
in ${\rm Jac}\,(C)$. The trigonal isomorphism
${\rm Jac}\,(C) \simeq {\rm Prym}\,(\tilde{T}/T)$ maps $\alpha$ to the
difference $L(b_1) -L(b_2)$ (defined in the discussion
preceeding Lemma \ref{polar}), which is the nontrivial
element in ${\rm Ker}\,\nu^\ast$. Moreover the only choice
made other than $f$ is the grouping of $w_1,\dots,w_6$
under $\nu'$. The differences of the corresponding
paired points in $\nu T$ are the nonzero elements of a
lagrangian subgroup $L_0$ of ${\rm Jac}\,(\nu T)[2]$.
Now observe that the pullback to $\nu\tilde{T}$ by $\nu\pi$ of a
line bundle of order $2$ on $\nu T$ is in the kernel of the
norm map to $\nu T$. This gives a symplectic embedding
\[ \iota:{\rm Jac}\,(\nu T)[2] \hookrightarrow {\rm Prym}\,(\nu\tilde{T}/\nu T)[2]\,. \]
The image ${\rm Im}\,(\iota)$ of $\iota$ can be described in two ways.
On the one hand, it is the kernel of the polarization map
${\rm Prym}\,(\nu\tilde{T}/\nu T) \rightarrow {\rm Prym}\,(\nu\tilde{T}/\nu T)^\vee$.
(Observe that ${\rm Prym}\,(\nu\tilde{T}/\nu T)$ has a polarization
of type $221$ by Lemma~\ref{polar}, whose kernel is then
isomorphic to $({\bbB Z}/2{\bbB Z})^4$.) On the other hand, let
$\alpha^\perp$ denote the orthogonal complement to (the image
of) $\alpha$ in ${\rm Prym}\,(\tilde{T}/T)$ for the Weil
pairing $w_2$. Then ${\rm Im}\,(\iota)$ is
also the pullback of $\alpha^\perp$ by the normalization
map. Indeed, for $u\in {\rm Jac}\,(\nu T)[2]$ we have
$\iota(u)\in\alpha^\perp$ because
\[ w_2(\iota(u),\alpha) = w_2(u,{\rm Nm}\,_{\tilde{T}/T}(\alpha)) = 0\,, \]
and as both groups have cardinality $16$ they coincide. Hence
this image is isomorphic to
$\alpha^\perp / \langle\alpha\rangle$. In particular, the
inverse image $L$ of $L_0$ in ${\rm Jac}\,(C)$ is a lagrangian
subgroup of ${\rm Jac}\,(C)[2]$ containing $\alpha$.
Conversely, let $L\subset {\rm Jac}\,(C)[2]$ be a lagrangian
subgroup. We will say that the choices $f$, $\nu'$ made
in the course of the construction are compatible with $L$
if $\alpha = \alpha(f)$ is in $L$ and $\nu'$ corresponds
to $L/\langle \alpha \rangle$ as above.
We can now formulate our main theorem, to which we shall
give two proofs:
\begin{theorem}
\label{main}
Let $C'$ be the result of the construction applied to a curve
$C$ of genus $3$ compatibly with a lagrangian subgroup
$L\subset {\rm Jac}\,(C)[2]$. Then there is an induced isomorphism
$Jac(C)/L \sra{\sim} {\rm Jac}\,(C')$. In particular $C'$ is
independent of the (compatible) choices made in the construction.
\end{theorem}
\begin{pf}
The construction induces isogenies whose degrees are marked
below:
\[
\begin{array}{rcl}
{\rm Jac}\,(C) & \simeq & {\rm Prym}\,(\tilde{T}/T) \sbth{\nu^\ast}{\longrightarrow}{2}
{\rm Prym}\,(\nu\tilde{T}/\nu T) \sbth{\delta}{\longleftarrow}{4}
{{\rm Prym}\,(\nu \tilde{T}'/\nu T')}\\
&\sbth{{\nu'}^{\ast}}{\longleftarrow}{4}& {\rm Prym}\,(\tilde{T}'/T')
\simeq {\rm Jac}\,(C')\,.
\end{array}
\]
Here the middle step $\delta$ is identified with the
polarization map from an abelian variety of polarization
type $211$ to its dual. As before we identify $\nu^\ast$
with the quotient by $\alpha$, so to construct our isomorphism
${\rm Jac}\,(C)/L \simeq {\rm Jac}\,(C')$ it would suffice to produce
a natural map
$\epsilon: {\rm Prym}\,(\nu\tilde{T}/\nu T) \rightarrow {\rm Prym}\,(\tilde{T}{}'/ T')$
whose kernel is the subgroup $L/\langle \alpha \rangle$ of
${\rm Prym}\,(\nu\tilde{T}/\nu T)$.
One way to do this is to define $\epsilon$ as the dual map
of $\nu'{}^\ast$, using $\delta$ to identify the dual of
${\rm Prym}\,(\nu\tilde{T}'/\nu T')$ with ${\rm Prym}\,(\nu\tilde{T}/\nu T)$, and
using the principal polarization on ${\rm Prym}\,(\tilde{T}'/T')$ to
view it as its own dual. Tracing through the definitions
one verifies that ${\rm Ker}\,(\epsilon)$ is indeed
$L/\langle \alpha \rangle$ as asserted.
An alternative, and more geometric approach, is to show that
the hyperelliptic case treated in Section~\ref{hyperelliptic}
is a specialization of our present general construction. In
fact the hyperelliptic case is obtained when the $4$-sheeted
cover $f: C\rightarrow {\bbB P}^1$ happens to have $4$, rather than the
generic $2$, double branch points. We shall see that this
determines a preferred gluing $\nu'$. In going to this
special case we have to note that the limits of the curves
$\nu T$, $\nu \tilde{T}$ (which we continue to denote with the
same symbols) are no longer non-singular: they are now
only partial normalizations of $T$, $\tilde{T}$, and the map
$\nu \tilde{T} \rightarrow \nu T$ now has $2$ points of Beauville type,
at the singularities which were not normalized. The full
normalizations, say $\nu \nu T$ and $\nu \nu \tilde{T}$, now have
genera $0$ and $3$ respectively, and the resulting diagram
\[\begin{array}{rcl}
&& \nu\nu\tilde{T} \\
&& \mbox{$\scriptstyle{\pi}$}\!\downarrow \\
C && \nu\nu T \\
&\sf\!\!\searrow\;\!\swarrow\!\!\mbox{$\scriptstyle{g}$}' &\\
& {\bbB P}^1 &
\end{array} \]
clearly coincides with diagram~(\ref{hyp1}).
Reagardless of the singularities of the intermediate curves,
we will see that each of the abelian varieties in the
diagram specializes to an abelian variety. In particular,
the limit of ${\rm Prym}\,(\nu\tilde{T}/\nu T)$ is, by Lemma~\ref{polar},
a $4$\/-sheeted cover of
${\rm Prym}\,(\nu\nu\tilde{T}/\nu\nu T) \simeq {\rm Jac}\,(H)$, whose kernel is
$L/\langle \alpha \rangle$. Below we will also
identify the limit of ${\rm Prym}\,(\tilde{T}{}'/T')$ with ${\rm Jac}\,(H)$.
This will produce the desired map $\epsilon$ in this special
case, and hence in general.
For this, we note that the bigonal data
$\nu \tilde{T} \rightarrow \nu T \rightarrow {\bbB P}^1$ has $4$, $2$, $0$ and $2$
points of types $\ssub \seq \sslash \smin\smin $, $\stackrel{\subset}{\ssub} \sslash \ssub $, $\ssub \ssub \sslash \smin\smin $, and $\btf$
respectively, which turn into points of types $\stackrel{\subset}{\ssub} \sslash \ssub $, $\ssub \seq \sslash \smin\smin $,
$\btf$ and $\ssub \ssub \sslash \smin\smin $, respectively, for
$\nu \tilde{{T'}} \rightarrow \nu T' \rightarrow {\bbB P}^1$.
To obtain $\tilde{{T'}} \rightarrow T'$ we need to pair the $6$ ramification
points of $\nu \tilde{{T'}} \rightarrow \nu T'$. There are $15$ ways to do
this, of which one is distinguished: each pair of Beauville
branches gets paired, as do the remaining two ramification
points. Let $T'_{h}$ be the intermediate object, obtained by
gluing only the Beauville branches but not the remaining
pair. It is a singular hyperelliptic curve of genus $3$, and
is a partial normalization of $T'$ at the double point $p$.
Let $\tilde{\Tph}$ be the corresponding $1$-point partial
normalization of $\tilde{{T'}}$, of arithmetic genus $6$.
To continue our construction, we need to identify the two
$g_3^1$'s on $T'$: these two turn out to coincide, and the
unique $g_3^1$ is in fact given by the $g_2^1$ on $T'_{h}$
plus a base point at the double point $p$. To see this we
examine what happens to our general construction of the
$g_3^1$ in this case. The unique quadric surface through
the canonical model of $T'$ is now a quadric cone, with
vertex at (the image of) $p$, because projection from
$p$ gives the canonical image of the hyperelliptic $T'_{h}$,
which is the double cover of a conic. Therefore the two
rulings, hence the two $g_3^1$'s, coincide and have a
base point at $p$, as asserted.
At this point we need to turn the $g_3^1$ into a morphism,
which requires us to blow up the point $p$. This
results in a reducible trigonal curve $T'_{t} := T'_{h} \cup P$,
where $P$ is a copy of ${\bbB P}^1$ intersecting $T'_{h}$ in the two
inverse images $p_1$, $p_2$ of $p$ in $T'_{h}$. The trigonal
map has degrees $2$ and $1$ respectively on the two
components $T'_{h}$ and $P$. This curve is indeed a flat limit,
in the family of triple covers of ${\bbB P}^1$, of the trigonal
curves encountered in the non-hyperelliptic situation. The
corresponding double cover $\tilde{\Tpt} \rightarrow T'_{t}$ is of Beauville
type at all $4$ of the singular points (the two singularities
of $T'_{h}$ plus $p_1$, $p_2$). Here
$\tilde{\Tpt} = \tilde{\Tph} \cup \tilde{P}$, where $\tilde{P}$ is another copy
of ${\bbB P}^1$, double cover of $P$ branched at the points glued
to $p_1$ and $p_2$.
Now that we have identified the trigonal data, we can complete
the construction. By example~2.10(iii) of \cite{don}, or by
inspection, we see that the result $C'$ of applying the
trigonal construction to the reducible trigonal data
$(\tilde{\Tph} \cup \tilde{P}) \rightarrow (T'_{h} \cup P) \rightarrow {\bbB P}^1$ is the
$4$-sheeted cover of ${\bbB P}^1$ obtained by applying the bigonal
construction to $\tilde{\Tph} \rightarrow T'_{h} \rightarrow {\bbB P}^1$. But since the
bigonal construction is reversible this is nothing but the
hyperelliptic curve $H= \nu\nu \tilde{T}$ which resulted from
the construction of Section~\ref{hyperelliptic}, as claimed.
The degeneration just described involves a flat family of
abelian varieties, so the polarization type and the type of
the kernel of the isogeny on jacobians remain constant. From
the hyperelliptic case we now see that $L$ is the kernel of
our isogeny in the general case. By Torelli's theorem, $C'$
is determined by its polarized jacobian, which is ${\rm Jac}\,(C)/L$.
Hence $C'$ is indeed independent of the choices (compatible
with $L$) made during the construction. This concludes the
proof of Theorem~\ref{main}.
\end{pf}
We now make some further comments on the
choices we made in the course of the construction.
Starting on the left, we fix the curve C, the Lagrangian
subgroup L and an element $\alpha \in L$. Our
$g^1_4$\/'s $f: C \rightarrow {\bbB P}^1$ with $\alpha =\alpha(f)$
are determined by a divisor class in the intersection
\[Z = Z_\alpha =
\Theta_C \cap(\alpha + \Theta_C) \subset {\rm Pic}\,^2(C)\,.\]
More accurately $Z$ parametrizes the family of $g^1_4$'s
(with the specified $\alpha$), together with a marking of
the two singular points $b_1$, $b_2$. Interchanging these
two points gives an involution $i$ of $Z$ induced by the
involution $x \mapsto x+ \alpha$ of ${\rm Pic}\,^2(C)$, and it
is the quotient of $Z$ by $i$ which parametrizes the
$g^1_4$'s alone. $Z$ is also invariant under the involution
$j: x \mapsto K_C- x$ (where $K_C$ is the canonical class)
and $i$ and $j$ commute. In addition, since $\Theta_C$ is
an ample divisor, $Z$ is connected.
Counting fixed points shows that the respective quotients of
$Z$ by $i$, $j$, $k=ij$ have genera $4$, $1$, $4$, and that the
common quotient $\overline{\overline{Z}}:=Z/\langle i,j\rangle$ has genus $1$.
These quotients clearly have the following interpretations as
parameter spaces:
\begin{enumerate}
\item $Z$ parametrizes the $g^1_4$\/'s $f:C\rightarrow {\bbB P}^1$ (equivalently,
via the trigonal construction,
towers of double covers $\tilde{T} \rightarrow T \rightarrow {\bbB P}^1$ of the indicated type),
{\rm with} a choice of a double branch point $b_1$.
\item $Z/i$ parametrizes the $g^1_4$\/'s $f:C\rightarrow {\bbB P}^1$ (equivalently,
towers of double covers $\tilde{T} \rightarrow T\rightarrow{\bbB P}^1$ of the indicated type).
\item $Z/j$ parametrizes the double covers $\tilde{T}\rightarrow T$ of the indicated
type together with a singular point of $T$.
\item $\overline{\overline{Z}}$ parametrizes the double covers $\tilde{T}\rightarrow T$ of the
indicated type, hence it also parametrizes their normalizations, as well
as the maps $\nu\pi': \nu\tilde{T}' \rightarrow \nu T'$.
\end{enumerate}
We will now discuss what choices we make when we perform the
construction in reverse order, and how the choices from the
two directions are related.
Starting with the genus $3$ curve $C'$, we now assume given a lagrangian
subgroup $L'$\/($={\rm Jac}\,(C)[2]/L)$ of ${\rm Jac}\,(C')[2]$, and a subgroup
$G \subset L'$ of order 4 (which corresponds to
$\alpha^\perp/L$). A marking
of the three double branch points $b'_i$ of $f'$ is equivalent to a
choice of a basis $\beta' = P'_2 + Q'_2 - P'_1 -Q'_1$ and
$\gamma' = P'_3 + Q'_3 - P'_1 -Q'_1$ of $G$. Let
$\Theta' \subset {\rm Pic}\,^2(C')$ be the theta divisor of $C'$, and for
a class $u\in{\rm Jac}\,(C')$ let $\Theta'_u$ denote the translation of
$\Theta'$ by $u$. Consider a line bundle $L$ in the intersection
$S = \Theta \cap \Theta_{\beta'} \cap \Theta_{\gamma'}$.
Since the canonical bundle is the only degree 4 bundle on $C'$
with $h^0>2$, there are only two possibilities: either $L^{\otimes 2}$ gives a
$g^1_4$\/ $f':C'\rightarrow {\bbB P}^1$ with three marked
double branch points $b'_i\in {\bbB P}^1$, $i=1,\dots,3$, or else $L$
must be a theta characteristic on $C'$. We claim
the following:\\
\noindent (1) $S$ consists of six points \\
\noindent (2) $S$ is closed under $v\rightarrow K_{C'} - v$.\\
\noindent (3) Four of the points of $S$ are theta characteristics, and two are not.\\
\begin{pf}
(1) holds because $6=g!$. For (2), suppose that $f',f'':C' \rightarrow {\bbB P}^1$
correspond to $2v$, $2K_{C'} - 2v$ respectively. Then for each double
ramification point $P'_i$, $Q'_i$ of $f'$ we get a unique double
ramification point $P''_i$, $Q''_i$ for $f''$ by imposing the condition
$P''_i + Q''_i + P'_i + Q'_i = K_{C'}$.
For (3) , one checks that there is a unique coset $G'$ of
$G$ in the set of odd theta characteristics
on $C'$: indeed, in coordinates we may take the set of theta characteristics
to be $V = ({\bbB Z}/2{\bbB Z})^6$ with coordinates $x_1,\dots,x_6$, and we may suppose that $h^0(C',O_{C'}(x))$ mod 2 for $x = (x_1,\dots,x_6)$ is given by
$q(x) = x_1x_2 + x_3x_4 + x_5x_6$. Also we may simultaneously identify
${\rm Jac}\,(C')[2]$ with $V$, with the Weil pairing given by
$w_2(x,y) = q(x+y)-q(x) -q(y)$. Without loss of generality we can also
take $\beta' = e_1$ and $\gamma' = e_3$, with $e_i$ the standard $i$\/th
unit vector. Then $G' = \{(a,0,b,0,1,1)\}$.
Part (3) is now clear: $G'$ is contained in $S$, and no other theta characteristics
appear in $S$. This establishes our claim.
\end{pf}
We can now describe all the choices made when we start from the right side.
Our data $C',L',G$ determines a complementary pair of maps $f',f"$. These
determine the data
$\tilde{T}'\sra{\pi'} T'$ uniquely (the two resulting maps $g',g"$ are the
usual two $g^1_3$'s on the genus 4 curve $T'$). The normalization
$\nu\pi': \nu\tilde{T}' \rightarrow \nu T'$ is therefore also uniquely
determined. So the {\em only} choice made is that of $h'$, given by
an arbitrary point of
$ Pic^2(\nu T') \approx \nu T'$. Comparing with what we found starting
from the left, we discover that $\nu T'$ is precisely identified with the double
quotient $\overline{\overline{Z}}$.
\section{The case of genus$\geq 4$.}
One might try to generalize our
construction to higher genus by finding, for a generic curve
$C$ of genus $g$, a correspondence with another generic curve
$C'$ of genus $g$ such that ${\rm Jac}\,(C')\simeq{\rm Jac}\,(C)/K$, with
$K$ a lagrangian subgroup of ${\rm Jac}\,(C)[2]$. We shall show
that this is not possible.
\begin{theorem}
Let $K$ be a lagrangian subgroup in ${\rm Jac}\,(C)[p]$, where $C$ is a
generic curve of genus $g\geq 4$ and $p$ is a prime. Then
${\rm Jac}\,(C)/K$ with its induced principal polarization is not a jacobian.
\end{theorem}
\begin{pf}
Let ${\cal T}$, ${\cal S}$, ${\cal M}$ and ${\cal A}$ denote respectively the
Teichm\"uller space, the Siegel space, the moduli space of
curves and the moduli space of principally polarized
abelian varieties, all of genus $g$. The mapping class group
$M=M(g)$ acts on ${\cal T}$ with quotient ${\cal M}$ and the modular
group $\Gamma=\Sp(2g,{\bbB Z})$ acts on ${\cal S}$ with quotient ${\cal A}$.
Moreover $\Gamma$ is naturally a quotient of $M$, because
$M$ acts on symplectic bases for $H_1(C,{\bbB Z})$ through its
action on $\pi_1(C)$, and the period map $\tau:{\cal T}\rightarrow{\cal S}$
is $M$-equivariant for these actions. Passing to the
quotient, we get Torelli's map $\ov{\tau}:{\cal M}\rightarrow{\cal A}$, which is
injective (Torelli's theorem) and exhibits ${\cal M}$ as a locally closed
subvariety of ${\cal A}$. Since ${\cal T}$ is irreducible it follows
that the {\em Torelli space} $\ov{{\cal T}}=\tau({\cal T})$ is a locally closed
irreducible analytic subvariety of ${\cal S}$.
\[ \begin{array}{rcl}
{\cal T} & \slra{\tau} & {\cal S} \supset \overline{{{\cal T}}}=\tau ({\cal T}) \\
M \downarrow && \downarrow \Gamma \\
{\cal M} & \slra{\overline{{\tau}}} & {\cal A}
\end{array} \]
Let $W$ be the finite cover
of ${\cal S}$ obtained by taking over each marked abelian
variety $A$ the lagrangian subgroups of $A[p]$. Since $W$ is
unramified over the contractible space ${\cal S}$, it is in fact a
union of copies of ${\cal S}$. Our generic isogeny
${\rm Jac}\,(C)\rightarrow{\rm Jac}\,(C)/K$ translates to the following data. The
curve $C$ lives over an open subset of ${\cal M}$, hence of $\ov{{\cal T}}$.
The subgroup $K$ corresponds to a sheet of $W_{|\ov{{\cal T}}}$.
Therefore our isogeny extends to the quotient map by
the subgroup, still denoted $K$, corresponding to the
``same'' sheet over all of ${\cal S}$. Now recall that ${\cal S}$
is the space of symmetric
$g\times g$ complex matrices $\Omega$ with positive imaginary
part, and the abelian variety over $\Omega$ is
$A_\Omega = {\bbB C}^g/({\bbB Z}^g+\Omega{\bbB Z}^g)$. Since monodromy
(i.e. $\Gamma$) acts transitively on the lagrangian subgroups of
$A_\Omega$, we may
take $K = (\frac{1}{p}{\bbB Z}/{\bbB Z})^g$ for convenience. Then
$A_\Omega/K \simeq A_{p\Omega} = A_{s\Omega}$, with
\[ s = \matr{pI_{g\times g}}{0}{0}{I_{g\times g}}
\in\Sp(2g,{\bbB R})\,.\]
If ${\rm Jac}\,(C)/K$, with its principal polarization, were a jacobian,
it would follow that the Torelli locus $\ov{{\cal T}}$ was invariant
under the subgroup $\Delta$ of $\Sp(2g,{\bbB R})$ generated by
$\Gamma$ and by $s$. We claim that $\Delta$ is dense
in $\Sp(2g,{\bbB R})$. Indeed, consider the subgroup $N_+$ of $\Sp(2g, {\bbB R})$
consisting of the matrices
$n(x) = \matr{I_{g\times g}}{x}{0}{I_{g\times g}}$, where
$x$ runs over the real symmetric $g\times g$ matrices. Then
$\Delta$ contains $s^i n(x) s^{-i}=n(p^{-i}x)$ for all integral
symmetric matrices $x$ and integers $i$. These are dense in $N_+$,
and $\Delta$ likewise contains a
dense subgroup of $N_-={}^t N_+$. It is
well-known (and easy) that $N_+$ and $N_-$ generate $\Sp(2g,{\bbB R})$, so
$\Delta$ is indeed dense in $\Sp(2g, {\bbB R})$.
Therefore, under our assumption, $\ov{{\cal T}}$ would be dense in
${\cal S}$ (in the complex topology),
so that ${\cal M}$ would be dense in ${\cal A}$. This is a contradiction
when $g>3$, because for dimension reasons
${\cal M}$ is not
dense in $A$ even for the Zariski topology then,
and the theorem follows.
\end{pf}
|
1998-01-01T01:35:47 | 9712 | alg-geom/9712035 | en | https://arxiv.org/abs/alg-geom/9712035 | [
"alg-geom",
"math.AG"
] | alg-geom/9712035 | Jun Li | Jun Li and Gang Tian | Comparison of the algebraic and the symplectic Gromov-Witten invariants | 45 pages, Latex | null | null | null | null | We show that the algebraic and the symplectic GW-inivariants of smooth
projective varieties are equivalent.
| [
{
"version": "v1",
"created": "Thu, 1 Jan 1998 00:35:47 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Li",
"Jun",
""
],
[
"Tian",
"Gang",
""
]
] | alg-geom | \section{Introduction}
As Witten suggested in [W1], [W2], the GW-invariants for
a symplectic manifold $X$ are multi-linear maps
\begin{equation}
\gamma_{A,g,n}^X: H^{\ast}(X;{\mathbb Q})^{\times n}\times H^{\ast}(\overline{\M}_{g,n};{\mathbb Q})
\lra {\mathbb Q},
\label{eq:0.1}
\end{equation}
where $A\in H_2(X,{\mathbb Z})$ is any homology class,
$n$, $g$ are two non-negative integers, and $\overline{\M}_{g,n}$ is the Deligne-Mumford compactification of $\M_{g,n}$, the space of smooth $n$-pointed genus $g$ curves.
The basic idea of defining these invariants is to enumerate
holomorphic maps from
Riemann surfaces to the manifolds. To illustrate this, we let
$X$ be a smooth projective manifold and form the moduli space $\M_{g,n}(X,A)$
of all holomorphic maps $f\!:\!\Sigma\to X$ from smooth $n$-pointed
Riemann surfaces $(\Sigma;x_1,\ldots,x_n)$ to $X$ such that
$f_{\ast}([\Sigma])=A$. $\M_{g,n}(X,A)$ is a quasi-projective scheme and its
expected dimension can be calculated using the Riemann-Roch theorem. We
will further elaborate the notion of expected dimension later, and for the
moment we will denote it by $r_{\rm exp }$.
Note that it depends implicitly on the choice of
$X$, $A$, $g$ and $n$. When $r_{\rm exp }=0$, then $\M_{g,n}(X,A)$ is expected to
be discrete. If $\M_{g,n}(X,A)$ is discrete, then the degree of $\M_{g,n}(X,A)$, considered as
a $0$-cycle, is a GW-invariant of $X$. We remark that we have and will
ignore the issue of non-trivial automorphism groups of maps in $\M_{g,n}(X,A)$
in the introduction.
When $r_{\rm exp }>0$, then $\M_{g,n}(X,A)$ is expected to have pure dimension $r_{\rm exp }$. If
it does, then we pick $n$ subvarieties of $X$, say $V_1,\ldots,V_n$, so that
their total codimension is $r_{\rm exp }$. We then form a subscheme of
$\M_{g,n}(X,A)$ consisting of maps $f$ so that $f(x_i)\in V_i$. This subscheme
is expected to be discrete. It it does, then its degree is the GW-invariant
of $X$. Put them together, we can define the GW-invariants
$\gamma_{A,g,n}^X$ of $X$. This is similar to
construction of the Donaldson polynomial invariants for 4-manifolds.
Here are the two big {\sl ifs} in carry out this program are
{\bf Question I}: Whether the moduli scheme $\M_{g,n}(X,A)$ has pure dimension $r_{\rm exp }$.
{\bf Question II}: Whether the subschemes of $\M_{g,n}(X,A)$ that satisfy certain
incidence relations have the expected dimensions.
Similar to Donaldson polynomial invariants, the affirmative answer
to the above two questions are in general not guaranteed.
One approach to overcome this difficulty,
beginning with Donaldson's invariants of 4-manifolds,
is to ``deform'' the moduli problems and hope that the answers to the
``deformed'' moduli problems are affirmative. In the case of GW-invariants, one
can deform the complex structure of the smooth variety $X$ to not
necessary integrable almost complex structure $J$ and study the same
moduli problem by replacing holomorphic maps with pseudo-holomorphic
maps. This was investigated by Gromov in [Gr], Ruan [Ru],
in which he constructed certain GW-invariants of rational type for
semi-positive symplectic manifolds. The first mathematical theory
of GW-invariants came from the work of Ruan and the second author,
in which they found that the right set up of GW-invariants for
semi-positive manifolds can be provided by using the moduli of
maps satisfying non-homogeneous Cauchy-Riemann equations.
In this set up, they constructed the GW-invariants of all semi-positive
symplectic manifolds and proved fundemantal properties of
these invariants. All Fano-manifolds and Calabi-Yau
manifolds are special examples of semi-positive symplectic
manifolds. Also any symplectic manifold of complex dimension less than $4$
is semi-positive.
Attempts to push this to cover general symplectic manifolds so
far have failed. New approaches are needed in order
to get a hold on the
GW-invariants of general varieties (or symplectic manifolds).
The first step
is to convert the problem of counting mappings, which essentially is homology
in nature, into the frame work of cohomology theory of the moduli problem.
More precisely, we first compactify the moduli space $\M_{g,n}(X,A)$ to, say,
$\overline{\M}_{g,n}(X,A)$. We require that the obvious evaluation map
$$e: \M_{g,n}(X,A)\lra X^n
$$
that sends $(f;\Sigma;x_1,\ldots,x_n)$ to $(f(x_1),\ldots,f(x_n))$ extends to
$$\bar e: \overline{\M}_{g,n}(X,A)\lra X^n.
$$
We further require that if $\M_{g,n}(X,A)$ has pure dimension $r_{\rm exp }$, then
$\overline{\M}_{g,n}(X,A)$ supports a fundamental class
$$[\overline{\M}_{g,n}(X,A)]\in
H_{2r_{\rm exp }}(\overline{\M}_{g,n}(X,A);{\mathbb Q}).
$$
Then the GW-invariants of $X$ are multi-linear
maps
\begin{equation}
\gamma^X_{A,g,n}: H^{\ast}(X)^{\times n}\times H^{\ast}(\overline{\M}_{g,n})\lra {\mathbb Q}
\label{eq:0.4}
\end{equation}
that send $(\alpha,\beta)$ to
$$\gamma^X_{A,g,n}(\alpha,\beta)=\int_{[\overline{\M}_{g,n}(X,A)]}{\bar e}^{\ast}(\alpha)
\cup\pi^{\ast}(\beta).
$$
where $\pi\!:\!\overline{\M}_{g,n}(X,A)\to\overline{\M}_{g,n}$ is the forgetful map. Note that in such cases
the GW-invariants are defined without reference to the answer to
{\sl question II}.
Even when the answer to {\sl question I} is negative, we can still define
the GW-invariants if a virtual moduli cycle
$$[\overline{\M}_{g,n}(X,A)]^{{\rm vir}}\in H_{2r_{\rm exp }}(\overline{\M}_{g,n}(X,A);{\mathbb Q})
$$
can be found that function as the fundamental cycle
$[\overline{\M}_{g,n}(X,A)]$ should the dimension of $\M_{g,n}(X,A)$
is $r_{\rm exp }$. In this case, we simply
define
$\gamma^X_{A,g,n}$ as before with $[\overline{\M}_{g,n}(X,A)]$ replaced by $[\overline{\M}_{g,n}(X,A)]^{{\rm vir}}$.
The standard compactification of $\M_{g,n}(X,A)$ is the moduli space of stable
morphisms from $n$-pointed genus $g$ curves, possibly nodal, to $X$
of the prescribed fundamental class. This was first studied for
pseudo-holomorphic maps by T. Parker and J. Wolfson \cite{PW} and
in algebraic geometry
by Kontsevich \cite{Ko}.
Because points of the compactification $\overline{\M}_{g,n}(X,A)$ are maps $f$ whose domains
have $n$-marked points $x_1,\ldots,x_n$, the evaluation map
$e$ extends canonically to $\bar
e$ that sends such map $f$ to $(f(x_1),\ldots,f(x_n))$.
The virtual moduli cycles $[\overline{\M}_{g,n}(X,A)]^{{\rm vir}}$ for projective variety $X$ were first
constructed by the authors. Their idea is to construct a virtual
normal cone embedded in a vector bundle based on the obstruction theory of
stable morphisms \cite{LT1}. An alternative construction of such cones
was achieved by Behrend and Fantechi \cite{BF,
Be}. For general symplectic manifolds, such virtual moduli cycles were
constructed by the authors,
and independently, by Fukaya and Ono \cite{FO, LT2}. Shortly after
them, B. Siebert [Si] and later, Y. Ruan [Ru2]
gave different constructions of such virtual moduli cycles.
Both Siebert and Ruan's approach needs to construct
global, finite-dimensional resolutions of so called cokernel bundles
(cf. \cite{Si} and \cite{Ru2}, Appendix).
However, one question remains to be investigated. Namely, if $X$ is
a smooth projective variety then on one hand we have the algebraically
constructed GW-invariants, and on the other hand, by viewing $X$ as a
symplectic manifold using the K\"ahler form on $X$, we have the
GW-invariants constructed using analytic method. These two approaches
are drastically different. One may expect, although far from clear, that
for smooth projective varieties the
algebraic GW-invariants and their symplectic counterparts are identical.
The main goal of this paper is to prove what was expected is indeed true.
\begin{theo}
Let $X$ be any smooth projective variety with a K\"ahler form $\omega$.
Then the algebraically constructed GW-invariants of $X$ coincide with
the analytically constructed GW-invariants of the symplectic
manifold $(X^{{\rm top}},\omega)$.
\end{theo}
This result was first announced in [LT2].
Its proof was outlined in [LT3].
During the preparation of the paper, we learned
from B. Siebert that he was able to prove a similar result.
We now outline the proof of our Comparison Theorem. We begin with a few
words on the algebraic construction of the virtual moduli cycle.
Let $w\in\overline{\M}_{g,n}(X,A)$ be any point associated to the stable morphism
$f\!:\! \Sigma\to X$. It follows
from the deformation theory of stable morphisms that there is a complex
${\mathcal C}_w$, canonical up to quasi-isomorphisms, such that its first cohomology
${\mathcal H}^1({\mathcal C}_w)$ is the space of the first order deformations of the map $w$,
and its second cohomology ${\mathcal H}^2({\mathcal C}_w)$ is the
obstruction space to deformations of the map $w$.
Let $\varphi_w$ be a Kuranishi map of the obstruction theory of $w$.
Note that $\varphi_w$ is the germ of a holomorphic map from
a neighborhood of the origin $o\in{\mathbb C}^{m_1}$ to ${\mathbb C}^{m_2}$,
where $m_i=\dim{\mathcal H}^i({\mathcal C}_w)$.
Let $\hat o$ be the formal completion of ${\mathbb C}^{m_1}$ along $o$
and let $\hat w$ be the subscheme of $\hat o$ defined by the
vanishing of $\varphi_w$. Note that $\hat w$ is isomorphic to the
formal completion of $\overline{\M}_{g,n}(X,A)$ along $w$
(Here as before we will ignore the issue of non-trivial automorphism
groups of maps in $\overline{\M}_{g,n}(X,A)$).
This says that ``near'' $w$, the scheme $\overline{\M}_{g,n}(X,A)$
is a ``subset'' of ${\mathbb C}^{m_1}$ defined by the vanishing of $m_2$-equations.
Henceforth, it these equations are in general position, them
$\dim\hat w=m_1-m_2$,
which is the expected dimension $r_{\rm exp }$ we mentioned before. The case where
$\overline{\M}_{g,n}(X,A)$ has dimension bigger than $r_{\rm exp }$ is exactly when the
vanishing locus of these $m_2$- equations in
$\varphi_w$ do not meet properly near $o$.
Following the excess intersection theory of Fulton and MacPherson \cite{Fu},
the ``correct'' cycle should come from first constructing the normal cone
$C_{\hat w/\hat o}$ to $\hat w$ in $\hat o$, which is canonically a subcone
of $\hat w\times {\mathbb C}^{m_2}$, and then intersect the cone with
the zero section of $\hat w\times{\mathbb C}^{m_2}\to\hat w$.
The next step is to patch these cones
together to form a global cone over $\M_{g,n}(X,A)$. The main
difficulty in doing so comes from the fact that the
dimensions
${\mathcal H}^2({\mathcal C}_w)$ can and do vary as $w$ vary, only
$\dim{\mathcal H}^1({\mathcal C}_w)-\dim{\mathcal H}^2({\mathcal C}_w)$ is a topological number.
This makes the cones
$C_{\hat w/\hat o}$ to sit inside bundles of varying ranks. To overcome this
difficulty, the authors came with the idea of finding a global ${\mathbb Q}$-vector
bundle $E_2$ over $\overline{\M}_{g,n}(X,A)$ and a subcone
$N$ of $E_2$ such that near fibers over $w$,
the cone $N$ is a fattening of the cone
$C_{\hat w/\hat o}$ (See section 3 or \cite{LT1} for more details).
In the end, we let $j$ be the zero section of $E_2$ and let $j^{\ast}$
be the Gysin map
$$ A_{\ast} E_2\lra A_{\ast}\overline{\M}_{g,n}(X,A),
$$
where $A_{\ast}$ denote the Chow-cohomology group (see \cite{Fu}).
Then the algebraic virtual moduli cycle is
$$[\overline{\M}_{g,n}(X,A)]^{{\rm vir}}=j^{\ast}([N])\in A_{r_{\rm exp }}\overline{\M}_{g,n}(X,A).
$$
Now let us recall briefly
the analytic construction of GW-invariants of
symplectic manifolds. Let $(X,\omega)$ be any smooth symplectic manifold
with $J$ a tamed almost complex structure. For $A$, $g$ and $n$ as before, we can form
the moduli space of $J$-holomorphic maps $f\!:\! \Sigma\to X$ where $\Sigma$
are $n$-pointed smooth Riemann surfaces such that $f_{\ast}([\Sigma])=A$.
We denote this space by $\M_{g,n}(X,A)^{J}$.
It is a finite dimensional topological space.
As before, we compactify it to include all
$J$-holomorphic maps whose domains are possibly with nodal singularities.
We denote the compactified space by $\overline{\M}_{g,n}(X,A)^{J}$. To proceed, we will embed $\overline{\M}_{g,n}(X,A)^{J}$
inside an ambient space ${\mathbf B}$ and realize it as the vanishing locus of a section of a
``vector bundle''.
Without being precise, the space ${\mathbf B}$ is the space of all {\sl smooth} maps $f
\in{\mathbf B}$ from
possibly nodal $n$-pointed Riemann surfaces to $X$, the fiber of the bundle over $f$
are all $(0,1)$-forms over domain$(f)$ with values in $f^{\ast} T_X$ and the section is the
one that sends $f$ to $\bar\partial f$.
We denote this bundle by ${\mathbf E}$ and the section by $\Phi$.
Clearly, $\Phi^{-1}(0)$ is homeomorphic to $\overline{\M}_{g,n}(X,A)^{J}$.
Defining the GW-invariants of $(X,\omega)$ is essentially about constructing the Euler
class of $[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$.
This does not make much sense since ${\mathbf B}$ is an
infinite dimensional topological space.
Although at each $w\in\Phi^{-1}(0)$ the formal differential $d\Phi(w)\!:\! T_w{\mathbf B}\to{\mathbf E}_w$
is Fredholm, which has real index $2r_{\rm exp }$, the conventional perturbation scheme does not
apply directly since near maps in ${\mathbf B}$
whose domains are singular the space ${\mathbf B}$ is not smooth and
${\mathbf E}$ does not admit local trivializations.
To overcome this difficulty, the authors introduced the notion of
weakly ${\mathbb Q}$-Fredholm
bundles, and showed in \cite{LT2} that $[\Phi\!:\! {\mathbf B}\to{\mathbf E}]$ is a weakly ${\mathbb Q}$-Fredholm
bundle and that any weakly ${\mathbb Q}$-Fredholm bundle admits an Euler class.
Let
$$
e[\Phi\!:\!{\mathbf B}\to{\mathbf E}]\in H_{2r_{\rm exp }}({\mathbf B};{\mathbb Q})
$$
be the the Euler class of $[\Phi\!:\! {\mathbf B}\to{\mathbf E}]$.
Since the evaluation map of $\M_{g,n}(X,A)^{J}$
extends to an evaluation map ${\mathbf e}\!:\!{\mathbf B}\to X^n$,
the Euler class, which will
also be referred to as the symplectic virtual cycle of $\overline{\M}_{g,n}(X,A)^{J}$, defines a
multi-linear map
$\gamma_{A,g,n}^{X,J}$ as in \eqref{eq:0.1}.
We will review the notion of weakly smooth Fredholm bundles in section 2. Here
to say the least, $[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$ is weakly Fredholm means that near each point of
$\Phi^{-1}(0)$ we can find a finite rank subbundle $V$ of ${\mathbf E}$ such that
$W=\Phi^{-1}(V)$ is a smooth finite dimensional manifold, $V|_W$ is a smooth vector
bundle and the lift $\phi\!:\! W\to V|_W$ of $\Phi$ is smooth.
(Note that the rank of $V$ may vary but $\dim_{{\mathbb R}}W-\rank_{{\mathbb R}} V=2r_{\rm exp }$).
For such finite models
$[\phi\!:\! W\to V|_W]$, which are called weakly smooth approximations, we can perturb
$\phi$ slightly to obtain
$\phi^{\prime}$ so that $\phi^{\prime-1}(0)$ are smooth manifolds in $W$. To construct the
Euler class, we first cover a neighborhood of $\Phi^{-1}(0)$ in ${\mathbf B}$ by finitely many
such approximations that satisfy certain compatibility condition. We then perturb
each section in the approximation and obtain a collection of locally closed
${\mathbb Q}$-submanifolds of ${\mathbf B}$ of dimension $2r_{\rm exp }$. By imposing certain compatibility
condition on the perturbations, this collection of ${\mathbb Q}$-submanfolds patch
together to form a $2r_{\rm exp }$-dimensional cycle in ${\mathbf B}$, which represents a homology
class in
$H_{2r_{\rm exp }}({\mathbf B};{\mathbb Q})$. This is the Euler class of $[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$.
Now we assume that
$X$ is a smooth projective variety and $\omega$ is a K\"ahler
form of $X$. Let $J$ be the complex structure of $X$. Then $\overline{\M}_{g,n}(X,A)$ is
homeomorphic to $\overline{\M}_{g,n}(X,A)^{J}$. Hence the two GW-invariants $\gamma_{A,g,n}^X$
and $\gamma_{A,g,n}^{X,J}$ are identical if the homology classes
$[\overline{\M}_{g,n}(X,A)]^{{\rm vir}}$ and $e[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$ will be
identical. Here we view $[\overline{\M}_{g,n}(X,A)]^{{\rm vir}}$ as a class in
$H_{\ast}({\mathbf B};{\mathbb Q})$ using
$$\overline{\M}_{g,n}(X,A)\sim_{{\rm homeo}}\overline{\M}_{g,n}(X,A)^{J}\subset{\mathbf B}.
$$
To illustrate why these two classes are equal, let us first look at
the following simple
model. Let $Z$ be a compact smooth variety and let $E$ be a holomorphic
vector bundle over $Z$ with a holomorphic section $s$.
There are two ways to construct the Euler classes of
$E$. One is to perturb $s$ to a smooth section $r$ so that the graph of $r$ is
transversal to the zero section of $E$, and then define the Euler class of
$E$ to be the
homology class in $H_{\ast}(Z;{\mathbb Q})$ of $r^{-1}(0)$. This is
the topological construction of the Euler class of $E$. The algebraic
construction is as follows. Let $t$ be a large scalar and let
$\Gamma_{ts}$ be the graph of
$ts$ in the total space of $E$. Since $s$ is an algebraic section, it follows
that the limit
$$\Gamma_{\infty s} =\lim_{t\to\infty}\Gamma_{ts}
$$
is a complex dimension $\dim Z$ cycle supported on union of
subvarieties of
$E$. We then let $r$ be a smooth section of $E$
in general position and let $\Gamma_{\infty s}\cap
\Gamma_r$ be their intersection. Its image in $Z$ defines a
homology class, which is the image of the Gysin map $j^{\ast}([C])$, where $j$
is the zero section of $E$. The reason that
$$e(E)=[r^{-1}(0)]=j^{\ast}([\Gamma_{\infty s}])\in H_{\ast}(Z;{\mathbb Q})
$$
is that if we choose $r$ to be in
general position, then
$$[r^{-1}(0)]=[\Gamma_r\cap\Gamma_0]=[\Gamma_r\cap\Gamma_s],
$$
and the family $\{\Gamma_{ts}\cap\Gamma_r\}_{t\in[1,\infty]}$
forms a homotopy of the cycles $\Gamma_s\cap\Gamma_r$ and
$\Gamma_{\infty s}\cap \Gamma_r$. One important remark is that
the cone $\Gamma_{\infty s}$ is contained in $E|_{s^{-1}(0)}$ and
the intersection of $\Gamma_{\infty s}$ and $\Gamma_r$ in
$E$ is the same as their
intersection in $E|_{s^{-1}(0)}$.
Back to our construction of GW-invariants, the analytic construction of
GW-invariants, which was based on perturbations of sections
in the finite models
(weakly smooth approximations) $[\phi\!:\! W\to V|_W]$, is clearly a
generalization of the topological construction of the Euler classes
of vector bundles.
As to the algebraic construction of GW-invariants, it is based on a
cone in a ${\mathbb Q}$-vector bundle over $\overline{\M}_{g,n}(X,A)$. Comparing to the algebraic
construction of the Euler class of $E\to Z$, what is missing is the
section $s$ and that the cone is the limit of the graphs of the
dilations of $s$. Following \cite{LT1}, the cone $\Gamma_{\infty s}$
only relies on the restriction of $s$ to an ``infinitesimal'' neighborhood of
$s^{-1}(0)$ in $Z$, and can also be reconstructed using the Kuranishi maps
of the obstruction theory to deformations of points in $s^{-1}(0)$ induced
by the defining equation $s=0$. Along this line, to each finite model
$[\phi\!:\! W\to V|_W]$ we can form a cone $\Gamma_{\infty \phi}=\lim
\Gamma_{t\phi}$ in $V|_{\phi^{-1}(0)}$. Hence to show that the two virtual
moduli cycles coincide, it suffices to establish a relation, similar to
quasi-isomorphism of complexes, between the cone
$N$ constructed based the obstruction theory of $\overline{\M}_{g,n}(X,A)$ and the collection
$\{\Gamma_{\infty\phi}\}$. In the end, this
is reduced to showing that the
obstruction theory to deformations of maps in $\overline{\M}_{g,n}(X,A)$ is identical to the
obstruction theory to deformations of elements in $\phi^{-1}(0)$ induced by
the defining equation $\phi$. This identification of two obstruction
theories follows from the canonical isomorphism of
the C\v{e}ch cohomology and
the Dolbeault cohomology of vector bundles.
The layout of the paper is as follows. In section two, we will recall the analytic
construction of the GW-invariants of symplectic manifolds. We will construct the
Euler class of $[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$ in details using the weakly smooth
approximations constructed in \cite{LT2}. In section three, we will construct a
collection of holomorphic weakly smooth approximations for projective manifolds.
The proof of the Comparisom Theorem will occupy the last section of this paper.
\section{Symplectic construction of GW invariants}
The goal of this section is to review the symplectic construction of
the GW-invariants of algebraic varieties. We will emphasize on those
parts that are relevant to our proof of the Comparison Theorem.
In this section,we will work mainly with real manifolds and will use the
standard notation in real differential geometry.
We begin with the symplectic construction of GW-invariants.
Let $X$ be a smooth complex projective variety, and let
$A\in H_2(X,{\mathbb Z})$ and let $g,\, n\in{\mathbb Z}$ be fixed once and for all.
We recall the notion of stable $C^l$-maps \cite[Definition 2.1]{LT2}.
\begin{defi}
\label{1.1}
An $n$-pointed stable map is a
collection $(f;\Sigma;x_1\ldots,x_n)$ satisfying the following property:
First,
$(\Sigma;x_1,\ldots,x_n)$ is an $n$-pointed connected prestable
complex curve with normal crossing
singularity; Secondly,
$f\!:\!\Sigma\to X$ is continuous, and the composite $f\circ\pi$ is
smooth, where $\pi\!:\!\tilde\Sigma\to\Sigma$ is
the normalization of $\Sigma$; And thirdly,
if we let $S\subset\Sigma$ be the union of singular locus of $\Sigma$
with its
marked points, then any rational component $R\subset\tilde{\Sigma}$
satisfying $(f\circ \pi)_{\ast}([R])=0\in H_2(X,{\mathbb Z})$ must
contains at least three points in $\pi^{-1}(S)$.
\end{defi}
For convenience, we will abbreviate $(f;\Sigma;x_1,\ldots,x_n)$ to
$(f;\Sigma;\{x_i\})$. Later, we will use ${\mathcal C}$ to denote an arbitrary stable
map and use $f_{{\mathcal C}}$ and $\Sigma_{{\mathcal C}}$ to denote its corresponding
mapping and domain.
Two stable maps $(f;\Sigma;\{x_i\})$ and $(f^{\prime};\Sigma^{\prime};\{x^{\prime}_i\})$
are said to be equivalent if there is an isomorphism
$\rho\!:\!\Sigma\to\Sigma^{\prime}$ such that $f^{\prime}\circ\rho=f$
and $x_i^{\prime}=\rho(x_i)$. When
$(f;\Sigma;\{x_i\})\equiv(f^{\prime};\Sigma^{\prime};\{x^{\prime}_i\})$,
such a $\rho$ is called an automorphism of $(f;\Sigma;\{x_i\})$.
We let ${\mathbf B}$ be the space of equivalence classes $[{\mathcal C}]$ of
$C^l$-stable maps ${\mathcal C}$ such that the arithmetic
genus of $\Sigma_{{\mathcal C}}$ is $g$ and
$f_{{\mathcal C}\ast}([\Sigma])=A\in H_2(X;{\mathbb Z})$.
Note that ${\mathbf B}$ was denoted by $\bar{{\mathcal F}}^l_A(X,g,n)$
in~\cite{LT2}.
Over ${\mathbf B}$ there is a generalized bundle ${\mathbf E}$ defined as follows.
Let ${\mathcal C}$ be any stable map and let
$\tilde f_{{\mathcal C}}\!:\!\tilde \Sigma_{{\mathcal C}}\to X$ be the composite of $f_{{\mathcal C}}$ with
$\pi\!:\!\tilde\Sigma_{{\mathcal C}}\to\Sigma_{{\mathcal C}}$. We define $\Lambda^{0,1}_{{\mathcal C}}$
to be the space of all $C^{l-1}$-smooth sections of $(0,1)$-forms of
$\tilde\Sigma$ with values in ${\tilde f}^{\ast} TX$. Assume ${\mathcal C}$ and ${\mathcal C}^{\prime}$
are two equivalent stable maps with $\rho\!:\!\Sigma_{{\mathcal C}}
\to\Sigma_{{\mathcal C}^{\prime}}$ the associated isomorphism, then there is a canonical
isomorphism $\Lambda^{0,1}_{{\mathcal C}^{\prime}}\cong\Lambda^{0,1}_{{\mathcal C}}$. We
let $\Lambda^{0,1}_{[{\mathcal C}]}$ be $\Lambda^{0,1}_{{\mathcal C}}/\Aut({\mathcal C})$. Then the union
$${\mathbf E}=\bigcup_{[{\mathcal C}]\in{\mathbf B}} \Lambda^{0,1}_{[{\mathcal C}]}
$$
is a fibration over ${\mathbf B}$ whose fibers are finite quotients of
infinite dimensional linear spaces.
There is a natural section
$$\Phi: {\mathbf B}\lra{\mathbf E}
$$
defined as follows. For any stable map ${\mathcal C}$, we define $\Phi({\mathcal C})$ to
be the image of $\bar\partial f_{[{\mathcal C}]}\in\Lambda^{0,1}_{{\mathcal C}}$ in
$\Lambda^{0,1}_{[{\mathcal C}]}$. Obviously, for ${\mathcal C}\sim{\mathcal C}^{\prime}$ we have
$\Phi({\mathcal C})=\Phi({\mathcal C}^{\prime})$. Thus $\Phi$ descends to a map
${\mathbf B}\to{\mathbf E}$, which we still denote by $\Phi$.
>From now on, we will denote by $\M_{g,n}(X,A)$ the moduli scheme of stable
moprhisms $f\!:\! C\to X$ with $n$-marked points such that
$C$ is (possibly with nodal singularities) has arithmetic
genus $g$ with $f_{\ast}([C])=A$.
\begin{lemm}
\label{1.2}
The vanishing locus of $\Phi$ is canonically homeomorphic to the
underlying topological space of $\M_{g,n}(X,A)$.
\end{lemm}
\begin{proof}
A stable $C^l$-stable map ${{\mathcal C}}$ in ${\mathbf B}$
belongs to the vanishing
locus of $\Phi$ if and only if $f_{{\mathcal C}}$ is holomorphic.
Since $\Sigma_{{\mathcal C}}$ is compact,
${\mathcal C}$ is the underlying analytic map of a stable morphism. Hence there
is a canonical map $\Phi^{-1}(0)\to \M_{g,n}(X,A)^{\rm top}$,
which is one-to-one and
onto. This proves the lemma.
\end{proof}
To discuss the smoothness of $\Phi$, we need the local uniformizing
charts of $\Phi\!:\! {\mathbf B}\to{\mathbf E}$ near $\Phi^{-1}(0)$.
Let $w\in{\mathbf B}$ be any point represented by
the stable map $(f_0;\Sigma_0;\{x_i\})$
with automorphism group $G_{w}$.
We pick integers $r_1, r_2>0$ and smooth ample divisors
$H_1,\ldots,H_{r_2}$ with $[H_i]\cdot[A]=r_1$ such that
all $f_0^{-1}(H_i)$ are contained in the
smooth locus of $\Sigma_0$ and that
for any $x\in f_0^{-1}(H_i)$ we have
\begin{equation}
\image(df_0(x))+T_{f(x)}H_i= T_{f(x)}X.
\label{eq:1.1}
\end{equation}
Now let $U\subset {\mathbf B}$ be a sufficiently small neighborhood of
$w\in{\mathbf B}$ and let
$\tilde U$ be the collection of all $({\mathcal C}; z_{n+1},\ldots,z_{n+r_1r_2})$
such that ${\mathcal C}\in U$ and the $z_i$'s
is a collection of smooth points of $\Sigma_{{\mathcal C}}$ such that
for each $1\leq j\leq r_2$ the subcollection
$(z_{n+(j-1)r_1+1},\ldots,z_{n+jr_1})$ contains distinct points
and is exactly $f_{{\mathcal C}}^{-1}(H_j)$.
Note that we do not require $(z_{n+1},\ldots,z_{n+r_1r_2})$ to be distinct.
\footnote{In case $X$ is a symplectic manifold, then we should use locally
closed real codimension 2 submanifold instead of $H_i$, as
did in \cite{LT2}.
Here we use this construction of uniformizing charts
because it is compatible to the construction of atlas of the
stack $\M_{g,n}(X,A)$ in algebraic geometry.}
Let $\pi_U\!:\!\tilde U\to U$ be the projection
that sends $({\mathcal C};z_{n+1},\ldots,z_{n+r_1r_2})$ to ${\mathcal C}$.
Clearly, $G_w$ acts on $\pi_U^{-1}(w)$ canonically by permuting
their $(n+r_1r_2)$-marked points. Namely, for any
$\sigma\in G_w$ and ${\mathcal C}\in\pi_U^{-1}(w)$ with
marked points $z_1,\ldots,z_{n+r_1r_2}$, $\sigma({\mathcal C})$ is the
same map with the marked points $\sigma(z_1),\ldots,\sigma(z_{n+r_1r_2})$.
In particular, we can view $G_w$ as a subgroup
of the permutation group $S_{n+r_1r_2}$.
Hence $G_w$ acts on $\tilde U$ by permuting the marked
points of ${\mathcal C}\in\tilde U$ according to the inclusion $G_w\subset S_{n+r_1r_2}$.
Note that if $H_i$'s are
in general position then elements in $\tilde U$ has no automorphisms
and have distinct marked points.
Let $G_{\tilde U}=G_w$. Since fibers of $\pi_U$ are invariant
under $G_{\tilde U}$, $\pi_U$ induces a map $\tilde U/G_{\tilde U}\to U$,
which is obviously a covering\!
\footnote{In this paper we call $p\!:\! A\to B$ a covering if $p$ is a
covering projection~\cite{Sp}
and $\#(p^{-1}(x))$ is independent of $x\in B$.
We call $p\!:\! A\to B$ a local covering if $p(A)$ is open in $B$ and
$p\!:\! A\to p(A)$ is a covering.} if $U$ is
sufficiently small. Further, if we let
$${\mathbf E}_{\tilde U}=\bigcup_{{\mathcal C}\in\tilde U} \Lambda^{0,1}_{{\mathcal C}}
$$
and let $\Phi_{\tilde U}\!:\!\tilde U\to{\mathbf E}_{\tilde U}$ be the section that sends
${\mathcal C}$ to $\bar\partial f_{{\mathcal C}}$, then $\Phi_{\tilde U}$ is
$G_{\tilde U}$-equivariant and $\Phi|_U\!:\! U\to {\mathbf E}|_U$ is the descent of
$\Phi_{\tilde U}/G_{\tilde U}\!:\!\tilde U/G_{\tilde U}\to{\mathbf E}_{\tilde U}/G_{\tilde U}$.
Note that fibers of ${\mathbf E}_{\tilde U}$ over $\tilde U$ are linear spaces. Following
the convention, we will call
$\Lambda=(\tilde U,\bE_{\tilde U},\Phi\ltilu,G_{\tilde U})$ a uniformizing chart of
$({\mathbf B},{\mathbf E},\Phi)$ over $U$. Let $V\subset U$ be an open subset
and let $\tilde V=\pi_U^{-1}(V)$, let $G_{\tilde V}=G_{\tilde U}$,
let ${\mathbf E}_{\tilde V}={\mathbf E}_{\tilde U}|_{_{\tilde V}}$ and let $\Phi_{\tilde V}=\Phi_{\tilde U}|_{\tilde V}$.
We will call $\Lambda^{\prime}=(\tilde V,{\mathbf E}_{\tilde V},\Phi_{\tilde V}, G_{\tilde V})$ a
uniformizing chart of $({\mathbf B},{\mathbf E},\Phi)$ that is
the {\it restriction} of the original chart to $V$, and denoted by $\Lambda|_V$.
We can also construct uniformizing charts by pull back. Let
$G_{\tilde V}$ be a finite group acting effectively on
a topological space $\tilde V$, let $G_{\tilde V}\to G_{\tilde U}$
be a homomorphism and $\varphi\!:\! \tilde V\to\tilde U$ be a
$G_{\tilde V}$-equivariant map so that $\tilde V/G_{\tilde V}\to \tilde U/G_{\tilde U}$
is a local covering map. Then we set ${\mathbf E}_{\tilde V}=\varphi^{\ast}{\mathbf E}_{\tilde U}$ and
$\Phi_{\tilde V}=\varphi^{\ast}\Phi_{\tilde U}$. The data
$\Lambda^{\prime}=(\tilde V,{\mathbf E}_{\tilde V},\Phi_{\tilde V}, G_{\tilde V})$
is also a uniformizing chart. We will call $\Lambda^{\prime}$ the
{\sl pull back} of $\Lambda$, and denoted by $\varphi^{\ast}\Lambda$.
In the following, we will denote the collection of all uniformizing charts of
$({\mathbf B},{\mathbf E},\Phi)$ by ${\mathfrak C}$.
The collection ${\mathfrak C}$ has the following compatibility
property. Let
$$\Lambda_i=(\tilde U_i,{\mathbf E}_{\tilde U_i},\Phi_{\tilde U_i}, G_{\tilde U_i}),
$$
where $i=1,\ldots,k$, be a collection of uniformizing charts in ${\mathfrak C}$
over $U_i\subset{\mathbf B}$ respectively. Let $p\in\cap_{i=1}^k U_i$
be any point. Then there is a uniformizing chart
$\Lambda=(\tilde V,{\mathbf E}_{\tilde V},\Phi_{\tilde V},G_{\tilde V})$ over $V\subset\cap^k U_i$
with $p\in V$
such that there are homomorphisms $G_{\tilde V}\to G_{\tilde U_i}$
and equivariant local covering maps $\varphi_i\!:\!\tilde V\to\pi_{U_i}^{-1}(V)
\subset\tilde U_i$ compatible with $\tilde V\to V$ and
$\pi_{U_i}^{-1}(V)\to V\subset U_i$, such that
$\varphi_i^{\ast}({\mathbf E}_{\tilde U_i},\Phi_{\tilde U_i})
\cong ({\mathbf E}_{\tilde V},\Phi_{\tilde V})$.
In this case, we say $\Lambda$ is
{\it finer than}
$\Lambda_i|_{V}$.
The main difficulty in constructing the GW invariants in this setting is that
the smoothness of $(\tilde U,\bE_{\tilde U},\Phi\ltilu)$ is unclear when $U$
contains maps whose domains are singular.
To overcome this difficulty, the authors introduced the notion of generalized
Fredholm bundles in~\cite{LT2}. The main result of~\cite{LT2}
is the following theorems,
which enable them to construct the GW invariants for all symplectic manifolds.
\begin{theo}
\label{1.3}
The data $[\Phi\!:\! {\mathbf B}\to{\mathbf E}]$ is a generalized oriented Fredholm
V-bundle of relative index $2r_{\rm exp }$, where $r_{\rm exp }=
c_1(X)\cdot A+n+(n-3)(1-g)$ is half of the virtual (real) dimension of
$\Phi^{-1}(0)$.
\end{theo}
\begin{theo}
\label{1.4}
For any generalized oriented Fredholm V-bundle
$[ \Phi\!:\! {\mathbf B}\to{\mathbf E}]$ of relative index $r$, we can assign
to it an Euler class $e([\Phi\!:\! {\mathbf B}\to{\mathbf E}])$ in $H_r({\mathbf B};{\mathbb Q})$
that satisfies all the expected properties of the Euler classes.
\end{theo}
As explained in the introduction, the pairing of the Euler class of
$[\Phi\!:\! {\mathbf B}\to{\mathbf E}]$ with the tautological topological class
will give rise to the symplectic version of the GW invariants of $X$.
Further, the Comparison Theorem we set out to prove amounts to compare
this Euler class with the image of the virtual moduli cycle $[\M_{g,n}(X,A)]^{{\rm vir}}$
in $H_r({\mathbf B};{\mathbb Q})$ via the inclusion $\M_{g,n}(X,A)^{\rm top}\subset{\mathbf B}$.
In the remainder part of this section, we will list all properties of
$ [\Phi\!:\! {\mathbf B}\to{\mathbf E}] $ that are relevant to the construction of its
Euler class. This list is essentially equivalent to saying
that $ [\Phi\!:\! {\mathbf B}\to{\mathbf E}] $ is a generalized oriented Fredholm V-bundle.
After that, we will construct the Euler class of $ [\Phi\!:\! {\mathbf B}\to{\mathbf E}] $
in details.
We begin with the notion of weakly smooth structure. A local smooth
approximation of $[\Phi\!:\! {\mathbf B}\to{\mathbf E}]$ over $U\subset {\mathbf B}$
is a pair $(\Lambda,V)$, where $\Lambda=(\tilde U,\bE_{\tilde U},\Phi\ltilu,G_{\tilde U})$
is a uniformizing chart over $U$ and $V$ is a finite equi-rank
$G_{\tilde U}$-vector bundle over $\tilde U$
that is a $G_{\tilde U}$-equivariant subbundle of $\bE_{\tilde U}$ such that
$R_V: = \Phi\ltilu^{-1}(V)\subset\tilde U$ is an equi-dimensional
smooth manifold, $V|_{R_V}$ is a smooth vector bundle and the lifting
$\phi_V\!:\! R_V\to V|_{R_V}$ of $\Phi\ltilu|_{R_V}$ is a smooth section.
An orientation of $(\Lambda,V)$ is a $G_{\tilde V}$-invariant
orientation of the real line bundle $\wedge^{\rm top}(TR_V)\otimes\wedge^{\rm top}
(V|_{R_V})^{-1}$ over $R_V$.
We call $\rank V-\dim R_V$ the
index of $(\Lambda,V)$ (We remind that all ranks and dimensions
in this section are over reals). Now
assume that $(\Lambda^{\prime},V^{\prime})$ is another weakly smooth structure of
identical index over $W\subset{\mathbf B}$. We say that $(\Lambda^{\prime},V^{\prime})$
is finer than $(\Lambda,V)$ if the following holds.
First, the restriction $\Lambda^{\prime}|_{W\cap U}$ is finer than
$\Lambda|_{W\cap U}$; Secondly, if we let $\varphi\!:\!
\pi_W^{-1}(W\cap U)\to\pi_U^{-1}(W\cap U)$
be the covering map then
$\varphi^{\ast} V\subset \varphi^{\ast}\bE_{\tilde U}\equiv
{\mathbf E}_{\tilde W}|_{\pi_W^{-1}(W\cap U)}$
is a subbundle of $V^{\prime}|_{\pi_W^{-1}(W\cap U)}$;
Thirdly, for any $w\in\tilde W$ the homomorphism $
T_wR_{V^{\prime}}\to \bigl( V^{\prime}/\varphi^{\ast} V\bigr) |_w$ induced by
$d\phi_{V^{\prime}}(w)\!:\! T_wR_V\to V^{\prime}|_w$ is surjective, and
the map $\phi_{V^{\prime}}^{-1}(\varphi^{\ast} V)\to R_V$
induced by $\varphi$ is
a local diffeomorphism between smooth manifolds.
Note that the last condition implies that if we identify
$T_{\varphi(w)}R_V$ with
$T_w\phi_{V^{\prime}}^{-1}(\varphi^{\ast} V)
\subset T_w R_{V^{\prime}}$, then the induced homomorphism
\begin{equation}
\label{eq:1.2}
T_wR_{V^{\prime}}
/T_{\varphi(w)}R_V\lra \bigl( V^{\prime}/\varphi^{\ast} V)|_w
\end{equation}
is an isomorphism. In case both $(\Lambda,V)$ and $(\Lambda^{\prime},V^{\prime})$
are oriented, then we require that the orientation of
$(\Lambda,V)$ coincides with that of $(\Lambda^{\prime},V^{\prime})$
based on the isomorphism
\begin{equation}
\label{eq:1.3}
\wedge^{\rm top}(T_w R_{V^{\prime}})\otimes\wedge^{\rm top}(V^{\prime}|_w)^{-1}\cong
\wedge^{\rm top}(T_{\varphi(w)}R_V)\otimes\wedge^{\rm top}(V|_{\varphi(w)})^{-1}
\end{equation}
induced by~\eqref{eq:1.2}.
Now let ${\mathfrak A}=\{(\Lambda_i,V_i)\}_{i\in{\mathcal K}}$ be a collection of oriented
smooth approximations of $({\mathbf B},{\mathbf E},\Phi)$. In the following, we will denote
by $U_i$ the open subsets of ${\mathbf B}$ such that $\wedge_i$ is a
smooth chart over $U_i$. We say ${\mathfrak A}$ covers $\phi^{-1}(0)$ if
$\phi^{-1}(0)$ is contained in the union of the images of $U_i$ in ${\mathbf B}$.
\begin{defi}\label{1.5}
An index $r$ oriented weakly smooth structure of $({\mathbf B},{\mathbf E},\Phi)$ is a
collection ${\mathfrak A}=\{(\Lambda_i,V_i)\}_{i\in{\mathcal K}}$ of index $r$ oriented smooth
approximations such that ${\mathfrak A}$ covers $\Phi^{-1}(0)$ and that
for any $(\Lambda_i,V_i)$ and $(\Lambda_j,V_j)$ in ${\mathfrak A}$ with
$p\in U_i\cap U_j$, there is a $(\Lambda_k,V_k)\in{\mathfrak A}$ such that
$p\in U_k$ and $(\Lambda_k,V_k)$ is finer than $(\Lambda_i,V_i)$ and
$(\Lambda_j,V_j)$.
\end{defi}
Let ${\mathfrak A}^{\prime}$ be another index $r$ oriented weakly smooth structure of
$({\mathbf B},{\mathbf E},\Phi)$. We say ${\mathfrak A}^{\prime}$ is finer than ${\mathfrak A}$ if for any
$(\Lambda,V)\in{\mathfrak A}$ over $U\subset{\mathbf B}$ and $p\in U\cap\Phi^{-1}(0)$,
there is a $(\Lambda^{\prime},V^{\prime})\in{\mathfrak A}^{\prime}$ over $U^{\prime}$ such that
$p\in U^{\prime}$ and $(\Lambda^{\prime},V^{\prime})$ is finer than
$(\Lambda,V)$.
We say that two weakly smooth structures ${\mathfrak A}_1$ and ${\mathfrak A}_2$
are equivalent if there is a third weakly smooth structure that is
finer than both ${\mathfrak A}_1$ and ${\mathfrak A}_2$.
\begin{prop}[\cite{LT2}]\label{1.6}
The tuple $({\mathbf B},{\mathbf E},\Phi)$ constructed at the beginning of this section
admits a canonical oriented weakly smooth structure of index $2r_{\rm exp }$.
\end{prop}
We remark that the construction of such a weakly smooth structure is
the core of the analytic part of [LT2].
In the following, we will use the weakly smooth structure
of $[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$ to construct its Euler class.
The idea of the construction is as
follows. Given a local smooth approximation $(\Lambda,V)$ over
$U\subset{\mathbf B}$, we obtain a smooth manifold $R_V$, a vector bundle $V|_{R_V}$
and a smooth section $\phi_V\!:\! R_V\to V|_{R_V}$. Following the
topological construction of the Euler classes, we shall perturb
$\phi_V$ to a new section $\tilde\phi_V\!:\! R_V\to V|_{R_V}$ so that
$\tilde\phi_V$ is transversal to the zero section of $V|_{R_V}$. Here
by a section transversal to the zero section, we mean that the
graph of this section is transversal to the zero section in the
total space of the vector bundle. Hence the
Euler class will be the cycle represented by ${\tilde\phi_V}^{-1}(0)$
near $U$. Since the weakly smooth structure of $\Phi\!:\! {\mathbf B}\to{\mathbf E}$
is given by a collection of compatible by not necessary matching local
smooth approximations, we need to work out this perturbation scheme
with special care so that $\{\tilde\phi_V^{-1}(0)\}$ patch together
to form a well-defined cycle.
Let ${\mathfrak A}=\{(\Lambda_{\alpha},V_{\alpha})\}_{\alpha\in{\mathcal K}}$ be the weakly smooth
structure provided by Proposition~\ref{1.6}. For convenience, for any
$\alpha\in{\mathcal K}$ we will denote the corresponding uniformizing chart
$\Lambda_{\alpha}$ by $(\tilde U_{\alpha},\tilde{\bB}_{\alpha},\tilde{\Phi}_{\alpha},G_{\alpha})$ and
will denote its descent by $(U_{\alpha},E_{\alpha},\Phi_{\alpha})$.
Accordingly, we will denote the projection $\pi_{U_{\alpha}}\!:\!\tilde U_{\alpha}\to U_{\alpha}$
by $\pi_{\alpha}$, denote ${\tilde\Phi_{\alpha}}^{-1}(V_{\alpha})$ by $R_{\alpha}$, denote
$V_{\alpha}|_{R_{\alpha}}$ by $W_{\alpha}$ and denote the lifting of $\tilde\Phi_{\alpha}|_{R_{\alpha}}
\!:\! R_{\alpha}\to\tilde{{\mathbf E}}_{\alpha}|_{R_{\alpha}}$ by $\phi_{\alpha}\!:\! R_{\alpha}\to W_{\alpha}$.
Without loss of generality, we can assume that for any approximation
$(\Lambda_{\alpha},V_{\alpha})\in{\mathfrak A}$ over $U_{\alpha}$ and any
$U^{\prime}\subset U_{\alpha}$, the restriction $(\Lambda_{\alpha},V_{\alpha})|_{U^{\prime}}$
is also a member in ${\mathfrak A}$.
In the following, we call $S\subset R_{\alpha}$ {\it symmetric} if
$S=\pi_{\alpha}^{-1}(\pi_{\alpha}(S))$.
Next, we pick a covering data for $\Phi^{-1}(0)\subset{\mathbf B}$
provided by the following covering lemma.
\begin{lemm}[\cite{LT2}]\label{1.20}
There is a finite collection ${\mathcal L}\subset{\mathcal K}$ and a total ordering of
${\mathcal L}$ of which the following holds.
the set $\Phi^{-1}(0)$ is contained in the union of
$\{R_{\alpha}\}_{\alpha\in{\mathcal L}}$ and
for any $\alpha$ and $\beta\in{\mathcal L}$ such that $\alpha<\beta$
then approximation $(\Lambda_{\beta},V_{\beta})$ is finer than the
approximation $(\Lambda_{\alpha},V_{\alpha})$.
\end{lemm}
\begin{proof}
The lemma is part of Proposition 2.2 in \cite{LT2}. It is proved there
by using the stratified structures of $({\mathbf B},{\mathbf E},\Phi)$.
Here we will give a direct proof of this by using the definition of
smooth approximations, when $\Phi^{-1}(0)$ is triangulable,
which is true when $X$ is projective.
Let $k$ be the real dimension of
$\Phi^{-1}(0)$. To prove the lemma, we will show that
there are $k+1$ subsets ${\mathcal L}_k,\ldots,{\mathcal L}_0\subset{\mathcal K}$ and that
for each $\alpha\in\cup_{i=0}^k{\mathcal L}_i$
there is an open symmetric subset $U_{\alpha}^{\prime}\Subset U_{\alpha}$
such that $R_{\alpha}^{\prime}=R_{\alpha}\cap\pi_{\alpha}^{-1}(U_{\alpha}^{\prime})\Subset R_{\alpha}$
of which the following holds: first, for each $i\leq k$ the set
$Z_i=\Phi^{-1}(0)-\cup_{j\geq i}\cup_{\alpha\in{\mathcal L}_j} U_{\alpha}^{\prime}$
is a triangulable space whose dimension is at most
$i-1$, and secondly, for any pair of distinct
$(\alpha,\beta)\in{\mathcal L}_i\times{\mathcal L}_j$
with $i\leq j$, the restriction
$(\Lambda_{\alpha},V_{\alpha})|_{U_{\alpha}^{\prime}\cap U_{\beta}^{\prime}}$
is finer than $(\Lambda_{\beta},V_{\beta})|_{U_{\alpha}^{\prime}\cap U_{\beta}^{\prime}}$.
We will construct ${\mathcal L}_i$ inductively, starting from ${\mathcal L}_k$. We first pick
a finite ${\mathcal L}_k\subset{\mathcal K}$ so that $\cup_{\alpha\in{\mathcal L}_k} U_{\alpha}
\supset \Phi^{-1}(0)$. This is possible since
$\Phi^{-1}(0)$ is compact. Since it is also triangulable, we can
find a symmetric $U_{\alpha}^{\prime}\subset U_{\alpha}$ for each $\alpha\in{\mathcal L}_k$
so that $\{U^{\prime}_{\alpha}\}_{\alpha\in{\mathcal L}_k}$ is disjoint,
$R_{\alpha}^{\prime}\Subset R_{\alpha}$ and
$Z_k$
is trangulable with dimension at most $k-1$.
Now we assume that we have found ${\mathcal L}_k,\ldots,{\mathcal L}_i$ as desired.
Then for each $x\in Z_i$ we can find a neighborhood $O$ of
$x\in{\mathbf B}$ such that for any $\alpha\in\cup_{j\geq i}{\mathcal L}_j$ either
$x\in U_{\alpha}$ or $O\cap U_{\alpha}^{\prime}=\emptyset$. Let ${\mathcal I}_x$ be those
$\alpha$ in $\cup_{j\geq i}{\mathcal L}_{\alpha}$ such that $x\in U_{\alpha}$. Then by the
property of ${\mathfrak A}$ there is a $\beta\in{\mathcal K}$ so that $(\Lambda_{\beta},V_{\beta})$ is
finer than $(\Lambda_{\alpha},V_{\alpha})|_{U_{\alpha}^{\prime}}$ for all $\alpha\in{\mathcal I}_x$.
Without loss of generality, we can assume that $U_{\beta}\subset O$. Then
$(\Lambda_{\beta},V_{\beta})$ is finer than $(\Lambda_{\alpha},V_{\alpha})$ for
all $\alpha\in\cup_{j\geq i}{\mathcal L}_j$.
Since $Z_i$ is compact, we can cover it by finitely many
such $(\Lambda_{\beta},V_{\beta})$'s, say indexed by
${\mathcal L}_{i-1}\subset{\mathcal K}$.
On the other hand, since
$Z_i$ is triangulable with dimension at most $i-1$, we can find symmetric
$U_{\alpha}^{\prime}\subset U_{\alpha}$ for each $\alpha\in{\mathcal L}_{i-1}$ so that
$R^{\prime}_{\alpha}\Subset R_{\alpha}$ for $\alpha\in{\mathcal L}_{i-1}$ and
$Z_i-\cup_{\alpha\in{\mathcal L}_{i-1}} U_{\alpha}^{\prime}$ is trianglable
with dimension at most $i-2$. This way, we can find the set ${\mathcal L}_k,\ldots,
{\mathcal L}_0$ as desired. In the end, we simply put ${\mathcal L}=\cup_{i=0}^k{\mathcal L}_i$.
We give it a total ordering so that whenever $\alpha\in{\mathcal L}_i$, $i\geq j$
and $\beta\in{\mathcal L}_j$ then $\alpha\leq\beta$. This proves the Lemma.
\end{proof}
We now fix such a collection ${\mathcal L}$ once and for all. Since
${\mathcal L}$ is totally ordered, in the following we will replace the
index by integers that range from 1 to $\#({\mathcal L})$ and use
$k$ to denote an arbitrary member of ${\mathcal L}$.
We first build the comparison data into the collection
$\{R_k\}_{k\in{\mathcal L}}$ and $\{W_k\}_{k\in{\mathcal L}}$. To distinguish the
projection $\pi_k\!:\!\tilde U_k\to U_k$ from the composite
$\tilde U_k\to U_k\to{\mathbf B}$, we will denote the later by $\iota_k$.
For any pair $k\geq l$, we set $R_{k,l}=\iota_k^{-1}(\iota_l(R_l))$.
Then there is a canonical map and a
canonical vector bundle inclusion
\begin{equation}
\label{eq:1.4}
f^l_k\!:\! R_{k,l}\to R_l
\quad{\rm and}\quad
(f^l_k)^{\ast}(W_l)\xrightarrow{\subset} W_k|_{R_{k,l}},
\end{equation}
that is part of the data making $(\Lambda_k,V_k)$
finer than $(\Lambda_l,V_l)$.
Note that $R_{k,l}\subset R_k$ is a
locally closed submanifold,
$f^l_k(R_{k,l})$ is open in $R_l$ and $f^l_k\!:\! R_{k,l}\to f^l_k(R_{k,l})$
is a covering map. Because of the compatibility condition, for any
$k>l>m$ if $R_{k,l}\cap R_{k,m}\ne\emptyset$ then
$f^l_k(R_{k,l}\cap R_{k,m})\subset R_{l,m}$ and
\begin{equation}
f^m_l\circ f^l_k=f^m_k:
R_{k,l}\cap R_{k,m}\lra R_m.
\label{eq:1.5}
\end{equation}
Further, restricting to $R_{k,l}\cap R_{k,m}$,
the pull backs
\begin{equation}
\label{eq:1.6}
(f_k^m)^{\ast}(W_m)|_{R_{k,l}\cap R_{k,m}}
=(f^l_k)^{\ast}(f^m_l)^{\ast}(W_m)|_{R_{k,l}\cap R_{k,m}}
\subset W_k|_{R_{k,l}\cap R_{k,m}}.
\end{equation}
In the following, we will use ${\mathfrak R}$ to denote the collection
of data $\{(R_{k,l},f^l_k)\}$ and use ${\mathfrak W}$ to denote
the data $\{(W_k,(f^l_k)^{\ast})\}$. We will call the pair $({\mathfrak R},{\mathfrak W})$
a good atlas
of the weakly smooth structure ${\mathfrak A}$ of $[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$.
For technical reason, later we need to shrink each $R_k$ slightly.
More precisely, let $\{ S_k\}_{k\in{\mathcal L}}$ be a collection of
symmetric open subsets $S_k\Subset R_k$ such that $\{ S_k\}$
still covers $\Phi^{-1}(0)$. We then let $S_{k,l}=(f^l_k)^{-1}(S_l)\cap
S_k$, let $W_k^{\prime}=W_k|_{S_k}$ and let $g^l_k$ and $(g^l_k)^{\ast}$ be the
restriction to $S_{k,l}$ of $f^l_k$ and $(f^l_k)^{\ast}$ respectively.
Then $({\mathfrak S},{\mathfrak W}^{\prime})$, where ${\mathfrak S}=\{(S_{k,l},g^l_k)\}$ and ${\mathfrak W}^{\prime}=
\{(W_k^{\prime},(g^l_k)^{\ast})\}$, is also a good atlas of $[\Phi\!:\! {\mathbf B}\to{\mathbf E}]$.
We call it a precompact sub-atlas of $({\mathfrak R},{\mathfrak W})$, and denote it
in short by ${\mathfrak S}\Subset{\mathfrak R}$.
To describe the collection $\{\phi_k\}$, we need to introduce the notion of
regular extension. Let $M$ be a manifold and $M_0\subset M$
be a locally closed submanifold. Let $V\to M$ be a smooth
vector bundle and $V_0\to M_0$ a subbundle of $V|_{M_0}$.
We assume that both $(M,V)$ and $(M_0,V_0)$ are
oriented. We say that a section $h\!:\! M\to V$ is a smooth extension of
$h_0\!:\! M_0\to V_0$ if both $h_0$ and $h$ are smooth and if
the induced section $M_0\xrightarrow{h_0} V_0\to V|_{M_0}$
is identical to the restriction $h|_{X_0}\!:\! X_0\to V_0$.
We say $h$ is a regular extension of $h_0$ if in addition to $h$
being a smooth extension of $h_0$ we have that for any $x\in X_0$ the
homomorphism
\begin{equation}
dh(x): T_xM/T_xM_0\lra (V/V_0)|_x
\label{eq:1.7}
\end{equation}
is an isomorphism and the orientation of $(M,V)$ and $(M_0,V_0)$ are
compatible over $M_0$ based on the isomorphism~\eqref{eq:1.7}.
\begin{defi}
A collection $\{h_k\}_{k\in{\mathcal L}}$ is called a smooth section of ${\mathfrak W}$ if
$h_k$ is a smooth section of $W_k$ for each $k\in{\mathcal L}$ and $h_k$
is a smooth extension of $h_l$ for any pair $k\geq l$ in ${\mathcal L}$.
If in addition that $h_k$ is a regular extension of
$h_l$ for all $k\geq l$, then we call $\{h_k\}$ a regular section of
${\mathfrak W}$.
\end{defi}
In the following, we will use ${\mathfrak h}\!:\!{\mathfrak R}\to{\mathfrak W}$ to denote a
smooth section with ${\mathfrak h}$ understood to be $\{h_k\}_{k\in{\mathcal L}}$.
We set ${\mathfrak h}^{-1}(0)$ to be the collection $\{h_k^{-1}(0)\}$
and set $\iota({\mathfrak h}^{-1}(0))$ to be the union of $\iota_k(h_k^{-1}(0))$
in ${\mathbf B}$.
We say ${\mathfrak h}^{-1}(0)$ is proper if
$\iota({\mathfrak h}^{-1}(0))$ is compact.
Without loss of generality, we can assume that $\dim R_k>0$ for all
$k\in{\mathcal L}$.
We say that ${\mathfrak h}$ is transversal to the zero section $\mathbf 0\!:\! {\mathfrak R}\to{\mathfrak W}$
if ${\mathfrak h}$ is a regular section and
if for any $k\in{\mathcal L}$ the graph $\Gamma_{h_k}$ of $h_k$ is
transversal to the 0 section of $W_k$ in the total space of $W_k$.
\begin{lemm}
\label{1.9}
Let the notation be as before. Then
${\mathfrak h}^{-1}(0)$ is proper if and only if there is a symmetric open
subsets $R_k^{\prime}\Subset R_k$ for each $k\in{\mathcal L}$ such that
$\cup_{k\in{\mathcal L}}\iota_k(h_k^{-1}(0))\subset
\cup_{k\in{\mathcal L}}\iota_k(R^{\prime}_k)$
and such that for each $k\in{\mathcal L}$,
\begin{equation}
\label{eq:1.8}
h_k^{-1}(0)\cap (R_k- R_k^{\prime})
\subset\bigl(\bigcup_{l<k}(f^l_k)^{-1}(R_l^{\prime})\bigr)
\cup\bigl( \bigcup_{l>k}f^l_k(R_{k,l}^{\prime})\bigr).
\end{equation}
\end{lemm}
\begin{proof}
We first assume that $Z=\cup_{k\in{\mathcal L}} \iota_k(h_k^{-1}(0))$ is compact.
Then since $\{R_k\}_{k\in{\mathcal L}}$ covers $Z$ and since $\dim R_k>0$,
for each $k\in{\mathcal L}$ we can find symmetric $R_k^{\prime}\Subset R_k$ so that
$\{R_k^{\prime}\}_{k\in{\mathcal L}}$
still covers $Z$. Obviously, this implies~\eqref{eq:1.8}.
Conversely, if we have found $R_k^{\prime}\subset R_k$
as stated in the lemma, then $\{\mathop{\rm cl\hspace{1pt}}(\iota_k(R_k^{\prime}))\cap Z\}$ will
cover $Z$, where $\mathop{\rm cl\hspace{1pt}}(A)$ is the closure of $A$.
Since $\mathop{\rm cl\hspace{1pt}}(\iota_k(R_k^{\prime}))$ are compact and since
$Z\cap \mathop{\rm cl\hspace{1pt}}(\iota_k(R_k^{\prime}))$ is closed in $\mathop{\rm cl\hspace{1pt}}(\iota_k(R_k^{\prime}))$, $Z$ is
compact as well. This proves the lemma.
\end{proof}
\begin{lemm}
\label{1.10}
Let ${\boldsymbol \phi}\!:\! {\mathfrak R}\to{\mathfrak W}$ be the collection $\{\phi_k\}$
induced by $\{\tilde\Phi_k\}_{k\in{\mathcal L}}$. Then ${\boldsymbol \phi}$ is
a regular section with proper vanishing locus.
\end{lemm}
\begin{proof}
This is equivalent to the fact that $[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$ is a
weakly Fredholm V-bundle, which was introduced and proved in
\cite{LT2}.
\end{proof}
Now let ${\mathfrak h}\!:\!{\mathfrak R}\to{\mathfrak W}$ be a regular section such that ${\mathfrak h}$ is
transversal to the zero section and ${\mathfrak h}^{-1}(0)$ is proper.
We claim that the data $\{h_k^{-1}(0)\}$ descends to an oriented current
in ${\mathbf B}$
with rational coefficients supported on a stratified set whose
boundary is empty. In particular, it defines a singular homology
class in $H_{\ast}({\mathbf B},{\mathbb Q})$.
Recall that for each $k\in{\mathcal L}$ the associated group $G_k$ acts on
$R_k$ such that $R_k/G_k$ is a covering of $\iota_k(R_k)$.
We let $m_k$ be the product of the order of $G_k$
with the number of the sheets
of the covering $R_k/G_k\to\iota_k(R_k)$. Note that
then the covering $R_{k,l}\to f^l_k(R_{k,l})$ is an $m_k/m_l$-fold
covering. Because $h_k$ is a regular extension of
$(f^l_k)^{\ast}(h_l)$, $(f^l_k)^{\ast}(h_l)^{-1}(0)$
is an open submanifold of $h_k^{-1}(0)$ with identical orientations.
Hence $\iota_k(h^{-1}_k(0))$ and $\iota_l(h_l^{-1}(0))$
patch together to form a stratified subset, and consequently
the collection $\{\iota_k(h_k^{-1}(0))\}_{k\in{\mathcal L}}$
patch together to form a stratified subset, say $Z$, in ${\mathbf B}$.
Now we assign multiplicities to open strata of $Z$.
Let $O_k=\iota_k(h_k^{-1}(0))$. Since $O_k\subset Z$ is
an open subset, we can assign multiplicities to $O_k$
so that as oriented current $[O_k]=\iota_{\ast}(\frac{1}{m_k}[
h_k^{-1}(0)])$, where $[h_k^{-1}(0)]$ is the current of
the oriented manifold $h_k^{-1}(0)$ with multiplicity one.
Here the orientation of $h_k^{-1}(0)$ is the one induced
by the orientation of $(R_k,W_k)$.
Using the fact that $R_{k,l}\to f^l_k(R_{k,l})$ is a covering
with $m_k/m_l$ sheets, the assignments of the multiplicities
of $O_k$ and $O_l$ over $\iota_k(R_k)\cap\iota_l(R_l)$ coincide.
Therefore $Z$ is an oriented stratified set of pure dimension
with rational multiplicities. We let $[Z]$ be the corresponding current.
It remains to check that $\partial[Z]=0$ as current.
Clearly, $\partial[Z\cap O_k]\subset\mathop{\rm cl\hspace{1pt}}(O_k)-O_k$. Since
$\{O_k\cap Z\}$ is an open covering of $Z$, $\partial[Z]=0$ if
$Z$ is compact. But this is what we have assumed in the first place.
Later, we will denote the so constructed cycle by
$$[{\mathfrak h}^{-1}(0)]\in H_{\ast}({\mathbf B},{\mathbb Q}).
$$
In the remainder of this section, we will perturb the section
${\boldsymbol \phi}\!:\!{\mathfrak R}\to{\mathfrak W}$ to a new section so that it is transversal to the zero section
and so that its vanishing locus is compact.
The current defined by the vanishing locus of the perturbed section
will define the Euler class of $[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$.
We begin with a collection ${\mathcal S}=\{S_l\}_{l\in{\mathcal L}}$
of symmetric open $S_k\Subset R_k$ such that $\{\iota_l(S_l)\}$
cover $\iota({\boldsymbol \phi}^{-1}(0))$.
For technical reason, we assume that for each $k\in{\mathcal L}$ the
boundary $\partial S_k$, which is defined to be $\mathop{\rm cl\hspace{1pt}}(S_k)-S_k$
in $R_k$, is a
smooth manifold of dimension $\dim S_k-1$. By slightly altering
$S_k$ if necessarily, we can and do assume that $\partial S_k$ is
transversal to $R_{k,l}$ along $\partial
S_k\cap (f^l_k)^{-1}(\mathop{\rm cl\hspace{1pt}}(S_l))$ for all $l<k$. (We will call
such ${\mathcal S}$ satisfying the transversality condition on
its boundary.) Following the convention,
we set $S_{k,l}=(f^l_k)^{-1}(S_l)\cap S_k$. We now construct a collection of
(closed) tubular neighborhoods of $S_{k,l}$ in $R_k$. We fix the
index $k$ and consider the closed submanifold
(with boundary) $\Sigma_l:=\mathop{\rm cl\hspace{1pt}}(S_{k,l})\subset R_k$. Because of the
transversality condition on $\partial S_l$ and on $\partial S_k$,
we can find a $D^h$-bundle $p_l\!:\! T_l\to \Sigma_l$,
where $D^h$ is the closed unit ball in ${\mathbb R}^h$ and
$h=\dim R_k-\dim R_{k,l}$, and a smooth embedding
$\eta_l\!:\! T_l\to R_k$ of which the following two conditions
holds. First, the restriction
of $\eta_l$ to the zero section $\Sigma_l\subset T_l$ is the original
embedding $\Sigma_l\subset R_k$, and secondly
\begin{equation}
\label{eq:1.9}
\eta_l(p_l^{-1}(\Sigma_l\cap \partial S_k ))\subset \partial S_k
\quad{\rm and}\quad
\eta_l(p_l^{-1}(S_{k,l} ))\subset S_k.
\end{equation}
For any $0<\epsilon<1$, we let $T_l^{\epsilon}\subset T_l$ be the closed
$\epsilon$-ball subbundle of $T_l$. By abuse of notation, in
the following we will not distinguish $T^{\epsilon}_l$
from its image $\eta_l(T_l^{\epsilon})$
in $R_k$. We will call $T^{\epsilon}_l$ the
$\epsilon$-tubular neighborhood of $\Sigma_l$ in $R_k$.
One property we will use later is that if $R_{k,l}\cap R_{k,l^{\prime}}\ne\emptyset$
for $l^{\prime}<l<k$, then $R_{k,l}\cap R_{k,l^{\prime}}$ is an open
subset of $R_{k,l^{\prime}}$, and hence for $0<\epsilon\ll 1$ we have
$\Sigma_{l^{\prime}}\cap T^{\epsilon}_l\subset \Sigma_{l^{\prime}}\cap \Sigma_l$.
Now consider $\Sigma_l\subset R_k$. Since $T_l$ is a disk bundle
over $\Sigma_l$, it follows that we can extend
the subbundle $(f^l_k)^{\ast}(W_l)|_{\Sigma_l}\subset W_k|_{\Sigma_l}$
to a smooth subbundle of $W_k|_{T_l}$, denoted by
$F_l\subset W_k|_{T_l}$.
We then fix an isomorphism and the inclusion
\begin{equation}
p_l^{\ast}\bigl( (f_k^l)^{\ast}(W_l)|_{\Sigma_l}\bigr)
\cong F_l.
\label{eq:1.10}
\end{equation}
In this way, we can extend any section $\zeta$ of
$(f_k^l)^{\ast}(W_l)|_{\Sigma_l}$ to a section of
$W_k|_{T_l}$ as follows. We first let $\zeta^{\prime}\!:\! T_l\to F_l$
be the obvious extension
using the isomorphism \eqref{1.10}. We then let
$\zeta_{{\rm ex}}\!:\! T_l\to W_k|_{T_l}$
be the induced section using the inclusion
$F_l\subset W_k|_{T_l}$.
We will call $\zeta_{{\rm ex}}$ the standard extension of $\zeta$ to $T_l$.
We
fix a Riemannian metric on $R_k$ and a metric on $W_k$. For any
section $\zeta$ as before, we say $\zeta$ is sufficiently small if its
$C^2$-norm is sufficiently small.
We now state a simple but important observation.
\begin{lemm}
\label{1.11}
Let the notation be as before. Then
there is an $\epsilon>0$ such that for any section
$g\!:\! T_l\to F_l\subset W_k|_{T_l}$
such that $\parallel\! g\!\parallel_{C^2}<\epsilon$, the section $h_k|_{T_l}+g$
is non-zero over $T^{\epsilon}_l-\Sigma_l$.
\end{lemm}
\begin{proof}
This follows immediately from the fact that
$\Sigma_l$ is compact and that for any $x\in R_{k,l}$ the
differential
$$dh_k: T_xR_k/T_x R_{k,l}\lra \bigl( W_k / (f^l_k)^{\ast}(W_l)\bigr)|_x
$$
is an isomorphism.
\end{proof}
We now state and prove the main proposition of this section.
\begin{prop}
\label{1.12}
Let ${\mathfrak h}\!:\! {\mathfrak R}\to{\mathfrak W}$ be a regular section with ${\mathfrak h}^{-1}(0)$
proper, let ${\mathfrak R}^{\prime}\Subset{\mathfrak R}$ be a good sub-atlas and
let ${\mathfrak h}^{\prime}$ be the the restriction of ${\mathfrak h}$ to ${\mathfrak R}^{\prime}$.
We assume that the vanishing locus of ${\mathfrak h}^{\prime}$ is still proper.
Then there is a smooth family of
regular sections $\mathfrak g(t)\!:\! {\mathfrak R}^{\prime}\to{\mathfrak W}^{\prime}$,
where ${\mathfrak W}^{\prime}$ be the restriction of ${\mathfrak W}$ to ${\mathfrak R}^{\prime}$,
parameterized by $t\in [0,1]$ such that
\begin{equation}
\bigcup_{t\in[0,1]}\iota\bigl(\mathfrak g(t)^{-1}(0)\bigr)\times\{t\}
\subset {\mathbf B}\times [0,1]
\label{eq:1.11}
\end{equation}
is compact, that $\mathfrak g(0)={\mathfrak h}^{\prime}$
and that $\mathfrak g(1)$ is transversal to the zero section of ${\mathfrak W}^{\prime}$.
\end{prop}
\begin{proof}
We will construct the perturbation over $R_1^{\prime}$ and then
successively extends it to the remainder of $\{R^{\prime}_k\}$.
We first fix
a collection of symmetric open subsets
$\{S_k\}_{k\in{\mathcal L}}$ such that $R_k^{\prime}\Subset S_k\Subset R_k$ and
that $S_k$ satisfies the transversality condition on its boundary.
Let $k$ be any positive
integer no bigger than $\#({\mathcal L})+1$. The induction hypothesis ${\mathcal H}_k$
states that for each integer $l<k$ we have constructed a symmetric
open $S_l^{\prime}$ satisfying
$R_l^{\prime}\Subset S_l^{\prime}\Subset S_l$ and a smooth family of small
enough sections $e_l(t)\!:\! R_l\to W_l$ such that
$e_l(0)\equiv0$ of which the following holds.
First, let ${\mathbf h}_l(t)=h_l+e_l(t)$, then for any $l<m<k$ the section
${\mathbf h}_{m}(t)|_{S_m^{\prime}}$ is a regular extension of
$(f^l_{m})^{\ast}({\mathbf h}_l(t))|_{S_{m,l}^{\prime}}$;
Secondly, for any $l<k$, the section ${\mathbf h}_l(1)$ is
transversal to the zero section of $W_l$
over $S_l^{\prime}$, and finally, for any $l<k$ and $t\in[0,1]$,
\begin{equation}
{\mathbf h}_l(t)^{-1}(0)\cap\bigl( S_l^{\prime}-R_l^{\prime}\bigr)\subset
\bigl(\bigcup_{i\leq l}(f_l^i)^{-1}(R_i^{\prime})\bigr)\bigcup
\bigl(\bigcup_{m\geq l}f_m^l(R_{m,l}^{\prime})\bigr).
\label{eq:1.12}
\end{equation}
Clearly, the condition ${\mathcal H}_1$ is automatically satisfied.
Now assume that we have found $\{ S_l^{\prime}\}_{l<k}$ and
$\{e_l\}_{l<k}$ required by the condition ${\mathcal H}_k$. We will demonstrate how
to find $e_k$ and a new sequence of open subsets
$\{S_l^{\prime}\}_{l\leq k}$ so that the condition ${\mathcal H}_{k+1}$ will hold for
$\{e_l\}_{l\leq k}$ and $\{ S^{\prime}_l\}_{l\leq k}$.
We continue to use the notation developed earlier. In particular, we
let $\Sigma_l$ be the closure of $S_{k,l}$,
let $T_l$ be the (closed) tubular neighborhood of $\Sigma_l\subset R_k$ with
the projection $p_l\!:\! T_l\to \Sigma_l$ and let $F_l$
be the subbundle of $W_k|_{T_l}$ with the isomorphism \eqref{1.10}.
Let $\zeta_l(t)$ be the standard extension of
$(f_k^l)^{\ast}(e_l(t))|_{\Sigma_l}$
to $T_l$. Note that $h_k|_{T_l}+\zeta_l(t)$ is a regular extension of
$(f_k^l)^{\ast}({\mathbf h}_l(t))|_{\Sigma_l}$. Because $\{{\mathbf h}_l\}_{l<k}$
satisfies condition ${\mathcal H}_k$, for $l<m<k$ and $x\in \Sigma_l\cap
\Sigma_{m}$ we have
$(f_k^l)^{\ast}({\mathbf h}_l(t))(x)=(f^{m}_k)^{\ast}({\mathbf h}_{m}(t))(x)$.
Now let
\begin{equation*}
A_l=p_l^{-1} \bigl((f_k^l)^{-1}(S^{\prime}_l)\bigr)-
\bigcup_{l<m<k} p_{m}^{-1}\bigl((f_k^{m})^{-1}(\mathop{\rm cl\hspace{1pt}}(R^{\prime}_{m}))\bigr)
\end{equation*}
and let
\begin{equation*}
B_l=\mathop{\rm cl\hspace{1pt}} (R_{k,l}^{\prime})-
\bigcup_{k>m>l}(f_k^{m})^{-1}(S_{m}^{\prime}).
\end{equation*}
Note that $\{A_l\}_{l<k}$ covers $\Int\bigl(\cup_{l<k}
T_l\bigr)$, that $B_l\Subset A_l$ and that $\{B_l\}_{l<k}$ \
is a collection of compact subsets of $R_k$.
Now let $\epsilon>0$ be sufficiently small. We choose a collection of
non-negative smooth functions $\{\rho_l\}_{l<k}$ that obeys the
requirement that $\supp(\rho_l)\Subset\Int(A_l
\cap T_l^{\epsilon})$, that $\rho_l\equiv1$ in a neighborhood of $B_l$
and that $\sum_{l<k}\rho_l\equiv1$ in a neighborhood of
$\cup_{l<k}\mathop{\rm cl\hspace{1pt}}(R_{k,l}^{\prime})$. This is possible because
the last set is compact and is contained in $\Int(
\cup_{l<k}A_l)$. We set
$$\zeta(t)=\sum_{l<k}\rho_l\cdot \zeta_l(t).
$$
Now we check that for each $l<k$ the section
$h_k+\zeta(t)$ is a regular extension of $(f_k^l)^{\ast}({\mathbf h}_l(t))$ in
a neighborhood of $\mathop{\rm cl\hspace{1pt}}(R_{k,l}^{\prime})$.
Let $x$ be any point in $\mathop{\rm cl\hspace{1pt}}(R_{k,l}^{\prime})$. We first consider the case
where $x$ is contained in $B_{m}$ for some $m\geq l$.
Let $y=f_k^{m}(x)$. Note that $y\in S_{m}^{\prime}$.
Then restricting to a sufficiently small neighborhood
of $x$ the section $h_k+\zeta(t)$ is equal to $h_k+\zeta_{m}(t)$.
Since $h_k+\zeta(t)$ is a regular extension of
$(f_k^{m})^{\ast}({\mathbf h}_{m}(t))$ near $x$ and
since ${\mathbf h}_{m}(t)$ is a regular extension of $(f^l_{m})^{\ast}
({\mathbf h}_l(t))$ in a neighborhood of $y\in S_{m}^{\prime}$, $h_k+\zeta(t)$ is
a regular extension of $(f^l_k)^{\ast}({\mathbf h}_l(t))$ near $x$.
We next consider the case where $x$ is not contained in any of the
$B_{m}$'s. Let $\Lambda$ be the set of all $m>l$
such that $x\in (f_k^{m})^{-1}(S_{m}^{\prime})$. Then for any
$m<k$ that is not in $\Lambda$, $\rho_{m}\equiv 0$
in a neighborhood of $x$. Here we have used the fact
that $\Sigma_m\cap T_l^{\epsilon}\subset\Sigma_m\cap\Sigma_l$ for
$0<\epsilon\ll 1$.
On the other hand, by induction hypothesis for each
$m\in\Lambda$ the section $h_k+\zeta_{m}(t)$ is a regular
extension of $(f_k^l)^{\ast}({\mathbf h}_l(t))$ near $x$. Therefore
since $\sum_{m\in\Lambda}\rho_{m}\equiv1$ near $x$, in a small
neighborhood of $x$
$$h_k+\zeta(t)=\sum_{m\in\Lambda}\rho_{m}\cdot (h_k+\zeta_{m}(t))
$$
is also a regular extension of $(f_k^l)^{\ast}({\mathbf h}_l(t))$.
Our last step is to extend $\zeta(t)$ to $R_k$.
We let $e_k(t)$ be a smooth family of sufficiently small sections
of $W_k$ such that $e_k(0)\equiv0$, that the restriction
of $e_k(t)$ to a neighborhood of $\cup_{l<k}\mathop{\rm cl\hspace{1pt}}\bigl(
(f_k^l)^{-1}(R_l^{\prime})\bigr)$ is $\zeta(t)$ and such that the section
${\mathbf h}_k(1)$ is transversal to the zero section in a
neighborhood of $\mathop{\rm cl\hspace{1pt}}(R_k^{\prime})$ in $S_k$. The last condition is
possible because $h_k+\zeta(1)$ is transversal to the zero section
in a neighborhood of
$\cup_{l<k}\mathop{\rm cl\hspace{1pt}}(R_{k,l}^{\prime})$.
Therefore, by possibly shrinking $S_l^{\prime}$ while still keeping
$R_l^{\prime}\Subset S_l^{\prime}$ for $l<k$ if necessary, we can find an
$S_k^{\prime}\Subset S_k$ satisfying $R_k^{\prime}\Subset S_k^{\prime}$
such that the induction hypothesis ${\mathcal H}_k$ holds
for $\{e_l\}_{l\leq k}$ and $\{S_l^{\prime}\}_{l\leq k}$, except possibly the
third condition.
We now show that the third condition of ${\mathcal H}_k$ holds as well.
We only need to check the inclusion \eqref{eq:1.12} for $l=k$.
First, by Lemma \ref{1.9} we can find an open $S\Subset S_k$
such that $R_k^{\prime}\Subset S$ and that
\begin{equation}
\label{eq:1.13}
{\mathbf h}_k^{-1}(0)\cap (\mathop{\rm cl\hspace{1pt}}(S)-R^{\prime}_k)\subset
\bigl(\bigcup_{i< k}(f^i_k)^{-1}(R_i^{\prime})\bigr)
\bigcup\bigl(\bigcup_{i> k}f^k_i(R_{i,k}^{\prime})\bigr).
\end{equation}
Now let
$$D_1={\mathbf h}_k^{-1}(0)\bigcap (\mathop{\rm cl\hspace{1pt}}(S)-R_k^{\prime})\bigcap
\bigl(\bigcup_{i< k}(f^i_k)^{-1}(R_i^{\prime})\bigr)
$$
and let
$$D_2={\mathbf h}_k^{-1}(0)\bigcap \bigl(\mathop{\rm cl\hspace{1pt}}(S)-R_k^{\prime}\bigr)\bigcap
\bigl(\bigcup_{i> k}f^k_i(R_{i,k} ^{\prime})\bigr).
$$
Since ${\mathbf h}_{k}(t)$ are small perturbations of $h_k$,
we can assume that ${\mathbf h}_k(t)$ are chosen so that
for any $t\in[0,1]$ the left hand side of \eqref{eq:1.13}
is contained in the union of neighborhood $V_1$ of $D_1$ and
a neighborhood
$V_2$ of $D_2$. We remark that if we choose $\{e_l\}_{l\leq k}$ so that
their $C^2$-norms are sufficiently small, then we can make $V_1$ and $V_2$
arbitrary small.
Then by Lemma \ref{1.11} the vanishing locus of ${\mathbf h}_k(t)$
inside $V_1$ is contained in $\cup_{i\leq k}(f^i_k)^{-1}(S_i^{\prime})$. On the
other hand, since $\cup_{i\geq k}f_i^k(R_{i,k}^{\prime})$ is
open, it contains $V_2$ since $D_2$ is compact and $V_2\supset D_2$
is sufficiently small. This proves the inclusion \eqref{eq:1.12}.
Therefore, by induction we have found $\{S_k^{\prime}\}_{k\in{\mathcal L}}$ and
$\{e_k(t)\}_{k\in{\mathcal L}}$ that satisfy the condition ${\mathcal H}_k$
for $k=\#({\mathcal L})+1$. Now let ${\mathbf g}_l(t)={\mathbf h}_l(t)|_{R_l^{\prime}}$.
Then $\mathfrak g(t)=\{{\mathbf g}_l(t)\}_{l\in{\mathcal L}}$ satisfies the condition of the
proposition. Note that the left hand side of \eqref{eq:1.11}
is compact because it is contained in the union of compact
sets $\{\iota_k(\mathop{\rm cl\hspace{1pt}}(R_k^{\prime}))\}_{k\in{\mathcal L}}$. This proves the
proposition.
\end{proof}
Let $\mathfrak g(t)$ be the perturbation constructed by Proposition \ref{1.12}
with ${\mathfrak h}={\boldsymbol \phi}$. We define the Euler class of
$[\Phi\!:\! {\mathbf B}\to{\mathbf E}]$ to be the homology class in
$H_{\ast}({\mathbf B};{\mathbb Q})$ represented by the current $[\mathfrak g(1)^{-1}(0)]$.
In the remainder of this section, we will sketch the argument that
shows that this class is independent of the choice of the chart
${\mathfrak R}$ and the perturbation $\mathfrak g$.
\begin{prop}
Let the notation be as before. Then the homology class $[\mathfrak g(1)^{-1}(0)]
\in H_{\ast}({\mathbf B};{\mathbb Q})$ so constructed is independent of the choice of
perturbations.
\end{prop}
\begin{proof}
First, we show that if we choose two perturbations $\mathfrak g_1(t)$ and
$\mathfrak g_2(t)$ based on identical sub-atlas ${\mathfrak R}^{\prime}\Subset{\mathfrak R}$ as stated in
Proposition \ref{1.12}, then we have $[\mathfrak g_1(1)^{-1}(0)]
=[\mathfrak g_2(1)^{-1}(0)]$.
To prove this, all we need is to construct a family of perturbations
$\mathfrak g_s(t)$, where $s\in [0,1]$, that satisfies conditions similar to that
of the perturbations constructed in Proposition \ref{1.12}. Since then we
obtain a current
$$\bigcup_{s\in [0,1]}\iota(\mathfrak g_s(1)^{-1}(0))\times\{s\}\subset{\mathbf B}\times [0,1]
$$
is a homotopy between the currents $\mathfrak g_0(1)^{-1}(0)_{\text{cur}}$
and $\mathfrak g_1(1)^{-1}(0)_{\text{cur}}$. The construction of
$\mathfrak g_s(t)$ is parallel to the construction of $\mathfrak g(t)$ in Proposition
\ref{1.12} by considering the data over $\{ R_k\times [0,1]\}_{k\in{\mathcal L}}$.
Next, we show that the cycle $[\mathfrak g(1)^{-1}(0)]$ does not depend on the
choice of ${\mathfrak R}_1\Subset {\mathfrak R}$. Let ${\mathfrak R}_1\Subset{\mathfrak R}$ and ${\mathfrak R}_2\Subset{\mathfrak R}$ be two
good sub-atlas and let $\mathfrak g_1(t)$ and $\mathfrak g_2(t)$ are two perturbations
subordinate to ${\mathfrak R}_1$ and ${\mathfrak R}_2$ respectively. Clearly,
we can choose a sub-atlas ${\mathfrak R}_0\Subset{\mathfrak R}$ such that ${\mathfrak R}_1\subset{\mathfrak R}_0$ and
${\mathfrak R}_2\subset{\mathfrak R}_0$. Let $\mathfrak g_0(t)$ be a perturbation given by Proposition
\ref{1.12} subordinate to ${\mathfrak R}_0$. Then $\mathfrak g_0(t)$ is also subordinate to
${\mathfrak R}_1$ and ${\mathfrak R}_2$. Hence by the previous argument
$$[\mathfrak g_1(1)^{-1}(0)]=[\mathfrak g_0(1)^{-1}(0)]=[\mathfrak g_2(1)^{-1}(0)].
$$
It remains to show that the class $[\mathfrak g(1)^{-1}(0)]$ does not depend on the
choice of the good atlas ${\mathfrak R}$. For this, it suffices to show that
for any two good atlas ${\mathfrak R}$ and ${\mathfrak R}^{\prime}$ so that ${\mathfrak R}$ is finer
than ${\mathfrak R}^{\prime}$, the respective perturbations $\mathfrak g(t)$ and $\mathfrak g^{\prime}(t)$
gives rise to identical homology classes
$[\mathfrak g(1)^{-1}(0)]=[\mathfrak g^{\prime}(1)^{-1}(0)]$. Let ${\mathfrak R}=\{R_k\}_{k\in{\mathcal K}}$ and
${\mathfrak R}^{\prime}=\{ R_k\}_{k\in{\mathcal L}}$, and let $U_k\subset{\mathbf B}$ (resp. $U_l\subset{\mathbf B}$)
be the open subsets so that $(R_k,W_k,\phi_k)$
(resp. $(R_l,W_l,\phi_l)$) are the smooth approximations of
$[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$ over $U_k$ (resp. $U_l$) for $k\in{\mathcal K}$
(resp. $l\in{\mathcal L}$). As before, we denote
$\iota_k\!:\! R_k\to U_k$ be the tautological map with $k\in{\mathcal K}$ or
$k\in{\mathcal L}$. Let $U_{kl}=U_k\cap U_l$.
We consider the good atlas ${\mathfrak R}_0$ with charts
$R_{kl}=\iota_k^{-1}(U_{kl})$, where $(k,l)\in{\mathcal K}\times{\mathcal L}$,
with bundles $W_{kl}=W_k|_{R_{kl}}$ and
$\phi_{kl}$ the restriction of $\phi_k$,
where $R_{kl}$ is considered to be an open subset of $R_k$.
Using the extension technique in the proof of Proposition \ref{1.12},
we can construct a perturbation $\mathfrak g_0(t)$ that is a regular
extension of $\mathfrak g^{\prime}(1)^{-1}(0)$ under the obvious
$\iota_l^{-1}(U_{kl})\to R_{kl}$ and $W_l|_{\iota_l^{-1}(U_{kl})}
\subset W_{kl}$. Therefore, $[\mathfrak g_0(1)^{-1}(0)]=[\mathfrak g^{\prime}(1)^{-1}(0)]$.
On the other hand, since $W_{kl}=W_k|_{R_{kl}}$, $\mathfrak g(t)$ induces a
perturbation $\mathfrak g_0^{\prime}(t)$ subordinate to ${\mathfrak R}_0$. Hence,
$[\mathfrak g_0(1)^{-1}(0)]=[\mathfrak g_0^{\prime}(1)^{-1}(0)]=[\mathfrak g(1)^{-1}(0)]$.
This proves the proposition.
\end{proof}
\section{Analytic charts}
The goal of this section is to construct a collection of local
smooth approximations $(\Lambda,V)$ so that the data
$\phi_V\!:\! R_V\to V|_{R_V}$ are analytic. Namely, $R_V$ are
complex manifolds, $V|_{R_V}$ are holomorphic vector bundles and
$\phi_V$ are holomorphic
sections. In the next section
we will show that such $\phi_V$'s are Kuranishi
maps, and hence the cones $\lim_{t\to\infty}\Gamma_{t\phi}$ are
the virtual cones constructed in \cite{LT1}.
We will use the standard notation in complex geometry in this section.
For instance, if $M$ is a complex manifold, we will denote by $T_xM$
the complex tangent space of $M$ at $x$ unless otherwise is mentioned.
We will use complex dimension throughout this section, unless otherwise is
mentioned. Accordingly
the complex dimension of a set is half of its real dimension.
We will use the words analytic and holomorphic interchangably in this
section as well.
We begin with the construction of such local smooth approximations.
Let $w\in {\mathbf B}$ be any point representing a holomorphic
stable map $f\!:\!\Sigma\to X$ with $n$-marked points.
We pick a uniformizing chart
$\Lambda=(\tilde U,{\mathbf E}_{\tilde U},\Phi_{\tilde U},G_{\tilde U})$
of $w$ over $U\subset{\mathbf B}$ such that the elements of
$\tilde U$ are stable maps $f_1\!:\!\Sigma_1\to X$ with (distinct)
$(n+k)$-marked points $\{x_i\}$ so that $\{f_1(x_m)\}_{m=n+1}^{n+k}$
are the $k$-distinct points of $f_1^{-1}(H)$, where $H$ is a smooth complex
hypersurface of $X$ in general position of degree $k=[H]\cdot
[A]$ and $A=f_{\ast}([\Sigma])$,
and that the stable maps resulting from discarding the
last $k$ marked points of $f_1$ are in $U$. Here as
usual we assume that $U$ is sufficiently small
so that all stable maps in $U$ intersect $H$ transversally and
positively. Note
that the later correspondence is the projection $\pi_U\!:\!
\tilde U\to U$.
Let ${\mathcal Y}$ over $\tilde U$ be the universal (continuous)
family of curves with $(n+k)$ marked
sections and let ${\mathcal F}\!:\!{\mathcal Y}\to X$ be the
universal map.
We let $\pi\!:\! \tilde U\to\M_{g,n+k}$ be the tautological map induced by
the family ${\mathcal Y}$ with its marked sections. Here
$\M_{g,n+k}$ is the moduli space of $(n+k)$-pointed stable curves of
genus $g$.
Without loss of generality, we can assume that no fibers of
${\mathcal Y}$ with the marked points have non-trivial
automorphisms. It follows that $\M_{g,n+k}$ is smooth near
$\pi(\tilde U)$. As in section 1, we view $G_{\tilde U}$ as a subgroup
of $S_{n+k}$. Then $G_{\tilde U}$ acts on $\M_{g,n+k}$ by permuting the
$(n+k)$-marked points of the curves in $\M_{g,n+k}$, and the map
$\pi\!:\!\tilde U\to\M_{g,n+k}$ is $G_{\tilde U}$-equivariant. Now let
$O\subset\M_{g,n+k}$ be a smooth $G_{\tilde U}$-invariant open neighborhood
of $\pi(\tilde U)\subset\M_{g,n+k}$ and let $p\!:\!{\mathcal X}\to O$ be the universal family of
stable curves over $O$ with $(n+k)$ marked sections
(In this section we will work with the analytic category unless otherwise
is mentioned). It follows that the $G_{\tilde U}$-action on $O$
lifts to ${\mathcal X}$ that permutes its marked sections.
For convenience, we
let ${\mathcal X}\times_O\tilde U$ be the topological subspace of
${\mathcal X}\times\tilde U$
that is the preimage of $\Gamma_{\pi}\subset O\times\tilde U$ under
${\mathcal X}\times\tilde U\to O\times\tilde U$, where
$\Gamma_{\pi}\subset O\times\tilde U$
is the graph of $\pi\!:\!\tilde U\to O$. Since no fibers of ${\mathcal Y}$
(with marked points) have non-trivial automorphisms, there is a
canonical $G_{\tilde U}$-equivariant isomorphism
${\mathcal Y}\cong {\mathcal X}\times_O\tilde U$ as family of pointed curves.
Let $\pi_{{\mathcal X}}$
and
$\pi_{\tilde U}$ be the first and the second projection of ${\mathcal X}\times_O\tilde U$.
Next, we let $({\mathcal X}_n,O_n;\Sigma,p_n,\varphi_n)$ be a semi-universal
family of the $n$-pointed curve $\Sigma$. Namely, ${\mathcal X}_n$ is a
(holomorphic) family of pointed prestable curves over
the pointed smooth complex manifold $p_n\in O_n$ whose
dimension is equal to $\dim_{{\mathbb C}}\Ext^1(\Omega_{\Sigma}(D),{\mathcal O}_{\Sigma})$,
where $D\subset\Sigma$ is the divisor of the $n$-marked points of
$\Sigma$, $\varphi_n\!:\! \Sigma\to{\mathcal X}_n|_{p_n}$ is an
isomorphism of $\Sigma$ with the fiber of ${\mathcal X}_n$ over $p_n$ as
$n$-pointed curve, and the Kuranishi map
$T_{p_n}O_n\to \Ext^1(\Omega_{\Sigma}(D),{\mathcal O}_{\Sigma})$ of the family ${\mathcal X}_n$
is an isomorphism. Note that $G_{\tilde U}$ acts canonically on $\Sigma$.
For convenience, we let $\Pi_n({\mathcal X})$ be the family of curves over $O$
that is derived from ${\mathcal X}$ by discarding its last $k$-marked sections.
We now let $B=\pi_U^{-1}(w)$ and
fix a $G_{\tilde U}$-equivariant isomorphism
\begin{equation}
\coprod_{z\in B}\Pi_n({\mathcal X})|_z\lra B\times \Sigma
\label{eq:2.50}
\end{equation}
over $B$.
Let $\aut_{p_n}({\mathcal X}_n)$ be
the group of those biholomorphisms of ${\mathcal X}_n$ that keep the
fiber ${\mathcal X}_n|_{p_n}$ invariant, that send fibers of
${\mathcal X}_n$ to fibers of ${\mathcal X}_n$ and that fix the $n$-sections
of ${\mathcal X}_n$. Possibly after shrinking $O_n$ if necessary,
we can assume that there is a homomorphism $\rho\!:\! G_{\tilde U}\to
\aut_{p_n}({\mathcal X}_n)$ such that for any $\sigma\in G_{\tilde U}$ the
$\rho(\sigma)$ action on ${\mathcal X}_n|_{p_n}$ is exactly the $\sigma$
action on $\Sigma$ via the isomorphism $\varphi_n$. Finally,
possibly after shrinking $U$ and $O$, we can
pick a $G_{\tilde U}$-equivariant holomorphic
map $\varphi\!:\! O\to O_n$ such that $\varphi(B)=p_n$ and that
there is a $G_{\tilde U}$-equivariant isomorphism of $n$-pointed curves
$\tilde{\varphi}\!:\!{\mathcal X}\to O\times_{O_n}{\mathcal X}_n$ that
extends the isomorphism \eqref{eq:2.50}.
We remark that the reason for doing this is to ensure that the
smooth approximation we are about to construct is $G_{\tilde U}$-equivariant.
Next, we let $l$ be an integer to be specified later
and let $U_i\subset\Sigma$, $i=1,\ldots,l$,
be $l$ disjoint open disks away from the marked points and the nodal points
of $\Sigma$. We assume that $\cup_{i=1}^l U_i$ is
$G_{\tilde U}$-invariant and that for any $\sigma\in G_{\tilde U}$
whenever $\sigma(U_i)=U_i$ then $\sigma|_{U_i}={\mathbf 1}_{U_i}$.
By shrinking $U$, $O$ and $O_n$
if necessary, we can find disjoint open subsets
${\mathcal U}_{n,i}\subset{\mathcal X}_n$ such that $\cup_{i=1}^l{\mathcal U}_{n,i}$ is
$G_{\tilde U}$-invariant, that $\cup_{i=1}^l{\mathcal U}_{n,i}$ is $G_{\tilde U}$-equivariantly
biholomorphic to $O\times\cup_{i=1}^l U_i$, that
${\mathcal U}_{n,i}\cap\Sigma=U_i$ and that the projections ${\mathcal U}_{n,i}\to O$ induced
by the projection
${\mathcal X}\to O$ is the first projection of $O\times U_i$ ($={\mathcal U}_{n,i}$).
For convenience, for each $i$
we will fix a biholomorphism between $U_i$ and the
unit disk in ${\mathbb C}$, and will denote by $U_i^{\hf}$ the open disk in
$U_i$ of radius $1/2$.
We let ${\mathcal U}_i$ be the
disjoint open subsets of ${\mathcal Y}$ defined by
$${\mathcal U}_i={\mathcal U}_{n,i}\times_{O_n}\tilde U\subset{\mathcal X}_n\times_{O_n}\tilde U
\cong{\mathcal X}\times_O\tilde U\cong{\mathcal Y}.
$$
We will call $U_i$ and ${\mathcal U}_i$ the distinguished open subsets
of $\Sigma$ and ${\mathcal Y}$ respectively.
Without loss of generality, we can assume that $\cup_{i=1}^l{\mathcal U}_i$ is
disjoint from the $(n+k)$-sections of ${\mathcal Y}$. We also assume that there
are holomorphic coordinate charts $V_i\subset X$ so that
${\mathcal F}({\mathcal U}_i)\subset V_i$.
We let $(w_{i,1},\ldots,w_{i,m})$, where $m=\dim
X$, be the coordinate variable of $V_i$
and let ${\mathbf v}_i=\partial/\partial w_{i,1}$.
For each $i$ we pick a nontrivial $(0,1)$-form $\gamma_i$ on $U_i$
with $\supp(\gamma_i)\Subset U_i^{\hf}$. We demand further that
if there is a $\sigma\in G_{\tilde U}$ so that $\sigma(U_i)=U_j$ then
$V_i=V_j$ as coordinate chart and $\sigma^{\ast}(\gamma_i)=\gamma_j$.
We then let $\sigma_i$ be the $(0,1)$-form over ${\mathcal U}_i$ with
values in ${\mathcal F}^{\ast}(T_X)|_{{\mathcal U}_i}$ that
is the product of the pull back of $\gamma_i$ via
${\mathcal U}_i\times_O\tilde U\to U_i$ with ${\mathcal F}^{\ast}({\mathbf v}_i)|_{{\mathcal U}_i}$,
and let $\tilde{\sigma}_i$ be the section over ${\mathcal Y}$ that is
the extension of ${\sigma}_i$ by zero. Obviously, $\tilde{\sigma}_i$ is
a section of
${\mathbf E}_{\tilde U}$, and $(\tilde{\sigma}_1,\ldots,\tilde{\sigma}_l)$ is
linearly independent fiberwise. Hence it spans a complex
subbundle of ${\mathbf E}_{\tilde U}$,
denoted by $V$. It follows from the construction that $V$ is
$G_{\tilde U}$-equivariant.
As in the previous section, we let $R=\Phi_{\tilde U}^{-1}(V)$,
let $W=V|_R$ and let $\phi\!:\! R\to W$ be the lifting of
$\Phi_{\tilde U}|_R\!:\! R\to {\mathbf E}_{\tilde U}|R$. The main task of this
section is to show that
we can choose $U_i$, $\gamma_i$ and $V_i$ so that $R$ admits a
canonical complex structure and that the section $\phi$ is holomorphic when
$W$ is endowed with the holomorphic structure so that the
basis $\tilde{\sigma}_1|_R,\ldots,\tilde{\sigma}_l|_R$ is holomorphic.
To specify our choice of $U_i$, $\gamma_i$ and $V_i$, we need first to
define the Dolbeault cohomology of holomorphic vector bundles over
singular curves. Let ${\mathcal E}$ be a locally free sheaf of ${\mathcal O}_{\Sigma}$-modules
and let $E$ be the associated vector bundle, namely, ${\mathcal O}_{\Sigma}(E)={\mathcal E}$.
We let $\Omega^0_{{\rm cpt}}(E)$ be the sheaf of smooth sections of $E$ that
are holomorphic in a neighborhood of $\sing(\Sigma)$ and let
$\Omega_{{\rm cpt}}^{0,1}(E)$ be the sheaf of smooth sections of $(0,1)$-forms
with values in $E$ that vanish in a neighborhood of $\sing(\Sigma)$.
Let
$$\bar\partial:
\Gamma(\Omega^0_{{\rm cpt}}(E))\lra \Gamma(
\Omega_{{\rm cpt}}^{0,1}(E))
$$
be the complex that send $\varphi\in \Omega^0_{{\rm cpt}}(E))$ to
$\bar\partial(\varphi)$. Since $\varphi$ is holomorphic near nodes of $\Sigma$,
$\bar\partial(\varphi)$ vanishes near nodes of $\Sigma$ as well, and hence
the above complex is well defined.
We define the Dolbeault cohomology
$H^0_{\bar\partial}(E)$ and $H^1_{\bar\partial}(E)$
to be the kernel and the cokernel of
$\bar\partial$.
\begin{lemm}
\label{2.0}
Let $H^i({\mathcal E})$ be the C\v{e}ch cohomology of the sheaf ${\mathcal E}$.
Then there are canonical isomorphisms
$H^0_{\bar\partial}(E)\cong H^0({\mathcal E})$ and $\Psi\!:\!
H^{0,1}_{\bar\partial}(E)\cong H^1({\mathcal E})$.
\end{lemm}
\begin{proof}
The proof is identical to the proof of the classical result that the
Dolbeault cohomology is isomorphic to the C\v{e}ch cohomology
for smooth complex manifolds. Obviously, $H^0_{\bar\partial}(E)$
is canonically isomorphic to $H^0({\mathcal E})$. We now construct $\Psi$.
We first cover $\Sigma$ by open subsets $\{W_i\}$ so that the
intersection of any of its subcollection is contractible. Now let
$\varphi$ be any global section in $\Omega^{0,1}_{{\rm cpt}}(E)$.
Then over each $W_i$ we can find a smooth function
$\eta_i\in\Gamma_{W_i}(\Omega_{{\rm cpt}}^0(E))$ such that
$\bar\partial\eta_i=\varphi|_{W_i}$. Clearly, the class in $H^1({\mathcal E})$
represented by the cocycle $[\eta_{ij}]$, where
$\eta_{ij}=\eta_i|_{W_i\cap W_j}-
\eta_j|_{W_i\cap W_j}$, is independent of the choice of $\eta_i$,
and thus defines a homomorphism
$\Gamma(\Omega_{{\rm cpt}}^{0,1}(E))\to
H^1({\mathcal E})$. It is routine to check that it is surjctive and its
kernel is exactly $\image(\overline{{\mathbf\partial}})$.
Therefore, we have $H^{0,1}_{\bar\partial}(E)\cong H^1({\mathcal E})$. Also, it
is direct to check that this isomorphism does not depend on the choice
of the covering $\{W_i\}$. This proves the lemma.
\end{proof}
For any $z\in\tilde U$, we denote by $\tilde{\sigma}_i(z)$ the restriction
of $\tilde{\sigma}_i$ to the fiber of ${\mathcal Y}$ over $z$.
We now choose the $l$ open disks $U_i\subset
\Sigma$, the $(0,1)$-forms $\gamma_i$ on
$U_i$ and the coordinate charts $V_i\subset X$ such that
for any $\tilde w\in\pi_U^{-1}(w)$ the collection
$\tilde{\sigma}_1(\tilde w),\ldots,
\tilde{\sigma}_l(\tilde w)$
spans $H^{0,1}_{\bar\partial}(f^{\ast} T_X)$.
This is certainly
possible if we choose $l$ large
because the locus of $U_i$ are arbitrary as long as
they are away from the nodal points of $\Sigma$ and the marked
points, and the charts $V_i$ can also be chosen with a lot of
choice.
We fix once and for all such choices of $U_i$, $V_i$ and $\gamma_i$.
We then let ${\mathcal U}_i\subset{\mathcal Y}$, $V\to{\mathbf E}_{\tilde U}$ and $R=\Phi_{\tilde U}^{-1}(V)$
be the objects constructed before according to this choice of
$U_i$, $\gamma_i$ and $V_i$. Let ${\mathcal Y}_R\to R$ be the restriction
to $R\subset\tilde U$ of the family ${\mathcal Y}\to\tilde U$
with the marked sections and let $F\!:\! {\mathcal Y}_R\to X$ be the
associated map.
We also fix a smooth function $\eta_i$ over $U_i$ so that
$\bar\partial\eta_i=\gamma_i$. We next extend the collection $\{ U_i\}_{i=1}^l$
to an open covering $\{U_i\}_{i=1}^L$ so that
the intersection of any subcollection of $\{U_i\}$ are
contractible, and that for any $i\leq l$ and $j\geq l+1$ the sets
$U_i^{\hf}$ and $U_j$ are disjoint. For convenience, we agree
that $\eta_j=0$ for $j>l$
From now on, we will fix an $\tilde w\in R$ over $w$.
\begin{lemm}\label{2.1}
There is a constant $A$ such that
for any C\v{e}ch 1-cocycle $[\tau_{ij}]$, where
$\tau_{ij}\in\Gamma_{U_i\cap U_j}(f^{\ast} {\mathcal T}_X)$,
there are constants $a_i$ and holomorphic sections
$\zeta_i\in\Gamma_{U_i}(f^{\ast} {\mathcal T}_X)$ for $i=1,\ldots,L$ such that
$$(\zeta_j+a_j\eta_j)|_{U_j\cap U_i}-(\zeta_i+a_i\eta_i)|_{U_j\cap U_i}
=\tau_{ji}
$$
and
$$\sum_{i=1}^L\bigl(\parallel\!\zeta_i\!\parallel_{L_2}+|a_i|\bigr)\leq
A\bigl(\sum_{i,j}^L\parallel\!\tau_{ij}\!\parallel_{L_2}\bigr).
$$
\end{lemm}
\begin{proof}
The existence of $\{a_i\}$ and $\{\zeta_i\}$ follows from
the fact that the images of $\tilde\sigma_1(\tilde w),
\ldots,\tilde\sigma_l(\tilde w)$ spans $H_{\bar\partial}^{0,1}(f^{\ast} T_X)$
and that $H_{\bar\partial}^{0,1}(f^{\ast} T_X)$ is isomorphic to $H^1(f^{\ast}{\mathcal T}_X)$.
The elliptic estimate is routine, using the harmonic theory on the
normalization of $\Sigma$. We will leave the details to the readers.
\end{proof}
We let $\Gamma(\Omega_{{\rm cpt}}^{0,1}(f^{\ast} T_X))^{\dag}$ be the quotient of
$\Gamma(\Omega_{{\rm cpt}}^{0,1}(f^{\ast} T_X))$ by the linear span of
$\tilde{\sigma}_1(\tilde w),\ldots,\tilde{\sigma}_l(\tilde w)$.
Because $\{\tilde\sigma_i(\tilde w)\}_{i=1}^L$
is invariant under the automorphism group of the stable
map $f$, $\Gamma(\Omega_{{\rm cpt}}^{0,1}(f^{\ast} T_X))^{\dag}$
is independent of the choice of $\tilde w\in\pi_U^{-1}(w)$. We let
$${\bar\partial}^{\dag}\!:\! \Gamma(\Omega_{{\rm cpt}}^0(f^{\ast} T_X))\to
\Gamma(\Omega_{{\rm cpt}}^{0,1}(f^{\ast} T_X))^{\dag}
$$
be the induced complex. We define $H^0_{\bar\partial}(f^{\ast})^{\dag}$
and $H^{0,1}_{\bar\partial}(f^{\ast} T_X)^{\dag}$ be the kernel and the
cokernel of the above complex.
\begin{coro}\label{2.2}
Let the notation be as before. Then
$H^{0,1}_{\bar\partial}(f^{\ast} T_X)^{\dag}=0$. Further,
the complex dimension of $H^0_{\bar\partial}(f^{\ast} T_X)^{\dag}$
is $\deg(f^{\ast} T_X)+m(1-g)+l$.
\end{coro}
\begin{proof}
The vanishing of $H_{\bar\partial}^{0,1}(f^{\ast} T_X)^{\dag}$ follows from the
surjectivity of ${\bar\partial}^{\dag}$. The second part follows from
$$\begin{array}{ll}
&\dim H^0_{\bar\partial}(f^{\ast} T_X)^{\dag}-\dim H^{0,1}_{\bar\partial}(f^{\ast}
T_X)^{\dag}\\
=&\dim H^0_{\bar\partial}(f^{\ast} T_X)-\dim H^{0,1}_{\bar\partial}(f^{\ast} T_X)+l
=\chi(f^{\ast} T_X)+l
\end{array}
$$
and the Riemann-Roch theorem.
\end{proof}
Next, we will describe the tangent space of $R$ at $\tilde w$.
By the smoothness result of \cite{LT2}, we know that $R$ is a
smooth manifold of (complex) dimension $r_{\rm exp }$.
As before, we let $D\subset \Sigma$ be the divisor of the first
$n$-marked points of $\tilde w$.
Since $f$ is holomorphic, $df^{\vee}$ is a homomorphism
of sheaves $f^{\ast}\Omega_X\to\Omega_{\Sigma}$.
We let
$${\mathcal D}_{\tilde w}^{\bullet}=[f^{\ast}\Omega_X
\xrightarrow{\alpha}\Omega_{\Sigma}(D)]
$$
be the induced complex indexed at $-1$ and $0$.
We will first define the extension space
$\Ext^1(\ddotw,\cO_{\Sigma})^{\dag}$ and then show that it is canonically isomorphic to
$T_{\tilde w}R$.
We begin with some more notations. Let ${\mathcal E}$ be a
sheaf of ${\mathcal O}_{\Sigma}$-modules that is locally free
away from the nodal points of $\Sigma$. Then there is a
holomorphic vector bundle $E$ over $\Sigma^0$, where $\Sigma^0$
is the smooth locus of $\Sigma$, such that ${\mathcal O}_{\Sigma^0}(E)
={\mathcal E}|_{\Sigma^0}$.
We define ${\mathcal E}^{\cA}$ to be the sheaf so that the germs of
${\mathcal E}^{\cA}$ at nodal points $p\in\Sigma$ (resp. smooth points
$p\in\Sigma^0$) are isomorphic to the
germs of ${\mathcal E}$ at $p$ (resp. germs of $\Omega_{\Sigma^0}^0(E)$ at $p$).
The set $\Ext^1(\ddotw,\cO_{\Sigma})^{\dag}$ is the set of equivalence classes
of pairs $(v_1,v_2)$ as follows. The data $v_1$ is an element in
$\Ext^1(\Omega_{\Sigma}(D),{\mathcal O}_{\Sigma})$, which defines an exact sequence
\begin{equation}
\begin{CD}
0@>>>{\mathcal O}_{\Sigma}@>{\varphi_1}>>
{\mathcal B}@>{\varphi_2}>>\Omega_{\Sigma}(D)@>>> 0,
\end{CD}
\label{eq:2.2}
\end{equation}
and equivalently a family of $n$-pointed nodal curves over
$T=\spec
{\mathbb C}[t]/(t^2)$, say ${\mathcal C}_T$ with $n$-marked sections $\tilde{x}_i$
(See \cite[section 1]{LT1}).
Note that ${\mathcal B}$ is locally free over $\Sigma^0$.
The data $v_2$ is a homomorphism $f^{\ast}\Omega_X\to {\mathcal B}^{\cA}$
such that, first of all, the diagram
\begin{equation}
\begin{CD}
@. @. f^{\ast}\Omega_X @= f^{\ast}\Omega_X\\
@. @. @Vv_2VV @V{df^{\vee}}VV\\
0 @>>> {\mathcal O}_{\Sigma}^{\cA} @>{\varphi_1}>> {\mathcal B}^{\cA} @>{\varphi_2}>>
\Omega_{\Sigma}(D)^{\cA} @>>> 0
\end{CD}
\label{eq:2.3}
\end{equation}
is commutative, where the lower sequence
is induced by \eqref{eq:2.2}. Secondly, since $v_2$ is
holomorphic near nodes of $\Sigma$,
the differential $\bar\partial v_2$ vanishes
near nodes of $\Sigma$, and since $df^{\vee}$ is holomorphic, $\bar\partial v_2$
lifts to a global section $\beta$ of $\Omega_{\rm cpt}^{0,1}(f^{\ast} T_X)$.
We require that there are constants $a_1,\ldots,a_l$ such that
$\beta=\sum a_i\tilde\sigma_i(\tilde w)$.
The equivalence relation of such pairs
are the usual equivalence relation of the diagrams \eqref{eq:2.3}.
Namely, two pairs
$(v_1,v_2)$ and $(v^{\prime}_1,v_2^{\prime})$ with the associated data
$\{{\mathcal B},\varphi_i\}$ and $\{{\mathcal B}^{\prime},\varphi_i^{\prime}\}$ are equivalent if
there is an isomorphism $\eta\!:\!{\mathcal B}\to{\mathcal B}^{\prime}$ so that
$\eta\circ\varphi_1=\varphi_1^{\prime}$, $\varphi_2=\varphi_2^{\prime}\circ\eta$
and $\eta\circ v_2=v_2^{\prime}$.
\begin{lemm}\label{2.3}
Let the notation be as before. Then
$\Ext^1_{\Sigma}({\mathcal D}_{\tilde w}^{\bullet},{\mathcal O}_{\Sigma}^{\cA})^{\dag}$ is canonically
a complex vector space of
complex dimension $r_{\rm exp }$.
\end{lemm}
\begin{proof}
The fact that $\Ext^1_{\Sigma}({\mathcal D}_{\tilde w}^{\bullet},{\mathcal O}_{\Sigma}^{\cA})^{\dag}$ forms a complex
vector space can be established using
the usual technique in homological algebra. For instance, if
$r\in \Ext^1_{\Sigma}({\mathcal D}_{\tilde w}^{\bullet},{\mathcal O}_{\Sigma}^{\cA})^{\dag}$
is represented by $\{{\mathcal B},\varphi_i,v_2\}$
shown in the diagram \eqref{eq:2.3}, then for any complex number
$a$ the element $ar$ is
represented by the same diagram with $\varphi_1$ replaced by
$a\varphi_1$. We now prove that
\begin{equation}
\dim\Ext^1_{\Sigma}({\mathcal D}_{\tilde w}^{\bullet},{\mathcal O}_{\Sigma}^{\cA})^{\dag}=r_{\rm exp }.
\label{eq:2.4}
\end{equation}
Clearly, the following familiar sequence is still exact
in this case:
$$
\Ext^0(\mathcal D^{\bullet}_f,{\mathcal O}_{\Sigma}^{\cA})\lra \Ext^0(\Omega_{\Sigma}(D),{\mathcal O}_{\Sigma})
\lra H^0_{\bar\partial}(f^{\ast} T_X)^{\dag}\qquad\qquad\qquad
$$
$$
\qquad
\lra \Ext^1(\Omega_{\Sigma}(D),{\mathcal O}_{\Sigma})\lra \Ext^1(\Omega_{\Sigma}(D),{\mathcal O}_{\Sigma})^{\dag}
\lra H^{0,1}_{\bar\partial}(f^{\ast} T_X)^{\dag}.
$$
Since $f$ is stable, $\Ext^0(\mathcal D^{\bullet}_f,{\mathcal O}_{\Sigma}^{\cA})=0$.
Hence \eqref{eq:2.4} follows from Corollary \ref{2.2} and the Riemann-Roch
theorem. This proves the lemma.
\end{proof}
Recall that
should $R$ be a scheme then the Zariski tangent space of $T_{\tilde w}R$
would be the space of morphisms $\spec {\mathbb C}[t]/(t^2)\to R$
that send their only closed points to $\tilde w$ modulo certain equivalence
relation. In the following,
we will imitate this construction and construct the space
of pre-${\mathbb C}$-tangents of $R$ at $\tilde w$.
We still denote by $U_1,\ldots, U_l$ the $l$-distinguished open subsets of
$\Sigma$ and let
$\{U_i\}_{i=1}^L$ be an extension of $\{U_i\}_{i=1}^l$
to an open covering of $\Sigma$ such that the intersection of any
of its subcollection are contractible.
Without loss of generality, we assume $U_j\cap U_i^{\hf}=\emptyset$ for
$j>l$ and $i\leq l$. We also assume that there are coordinate charts $V_i$ of
$X$ such that $f(U_i)\subset V_i$.
By abuse of notation, we will fix the embedding $V_i\subset {\mathbb C}^m$
and view any map to $ V_i$ as a map to ${\mathbb C}^m$.
We let $\iota_i\!:\! V_i\to X$ be the
tautological inclusion and let
$$g_{ij}\!:\! \iota_j^{-1}(\iota_i(V_i))\to \iota_i^{-1}(\iota_j(V_j))
\subset V_i\subset {\mathbb C}^m
$$
be the transition functions of $X$.
We define a pre-${\mathbb C}$-tangent $\xi$ of $R$ at $\tilde w$
to be a collection of data as follows: First, there is a flat
analytic family of $n$-pointed pre-stable curves $C_T$ over an open
neighborhood $T$ of $0\in {\mathbb C}$ such that
the fiber of $C_T$ over $0$, denoted by $C_0$, is isomorphic to $\Sigma$
as $n$-pointed curve; Secondly, there is
an open covering $\{\tilde U_i\}_{i=1}^L$ of
$C_T$ such that $\tilde U_i\cap C_0=U_i$, and that
for each $i\leq l$, there is a biholomorphism $\tilde U_i\cong U_i\times T$
such that its restriction to $U_i=\tilde U_i\cap
C_0$ is compatible to the identity map of $U_i$;
Thirdly, there is a collection of smooth maps
$\tilde f_i\!:\! \tilde U_i\to V_i$ such that for $i>l$,
all $\tilde f_i$ are
holomorphic and that for each
$i\leq l$ we have
${\bar\partial}_0(\tilde f_i)=0$ and
\begin{equation}
{\bar\partial}_i(\tilde f_i)=
\pi_T^{\ast}\varphi_i \cdot\pi_{U_i}^{\ast}(\gamma_i\cdot f^{\ast}({\mathbf v}_i)),
\label{eq:2.22}
\end{equation}
where $\pi_{U_i}$ and $\pi_T$ are the first and the second projection of
$U_i\times T$, $\varphi_i$ are holomorphic functions over $T$
and $\bar\partial_0$ (resp. $\bar\partial_i$) is the $\bar\partial$-differential with respect to
the holomorphic variable of $T$ (resp. $U_i$) using
$\tilde U_i\cong U_i\times T$
and the $\gamma_i$ and ${\mathbf v}_i$ are
the $(0,1)$-form and the vector field chosen before;
Forthly, if we let $z_0$ be the holomorphic variable of ${\mathbb C}\supset T$, then
we require that
\begin{equation}
\label{eq:2.27}
\tilde f_{ji}=\tilde f_i- g_{ij}\circ \tilde f_j:
\tilde U_{ij}\lra {\mathbb C}^m,
\end{equation}
where $\tilde U_{ij}$ is a neighborhood of $U_i\cap U_j$ in
$\tilde U_i\cap \tilde U_j$ over which $\tilde f_{ji}$
is well-defined, is divisible by $z^2_0$
(Namely, $\tilde f_{ji}$ has the form
$\pi_T^{\ast}(z^2_0)\cdot h_{ji}$ for some smooth function
$h_{ji}\!:\! \tilde U_{ij}\to {\mathbb C}^m$).
Intuitively, a pre-${\mathbb C}$-tangent is
a scheme analogue of a morphism $\spec {\mathbb C}[t]/(t^2)\to R$
should $R$ be a scheme. We denote the set of all pre-${\mathbb C}$-tangents by
$T_{\tilde w}^{{\rm pre}} R$. Note that $T_{\tilde w}^{{\rm pre}} R$ is merely a
collection of all pre-${\mathbb C}$-tangents.
We next define a canonical map
\begin{equation}
T_{\tilde w}^{{\rm pre}} R\lra \Ext^1({\mathcal D}_{\tilde w}^{\bullet},{\mathcal O}_{\Sigma})^{\dag}.
\label{eq:2.26}
\end{equation}
Let $\xi$ be any pre-${\mathbb C}$-tangent given by the data above. By
the theory of deformation of $n$-pointed curves
\cite[section 1]{LT1}, the analytic family $C_T$
defines canonically an exact sequence
\begin{equation}
0\lra{\mathcal O}_{\Sigma}\lra {\mathcal B}\lra \Omega_{\Sigma}(D)\lra 0,
\label{eq:2.23}
\end{equation}
associated to an extension class $v_1(\xi)\in\Ext^1(\Omega_{\Sigma}(D),{\mathcal O}_{\Sigma})$,
where away from the nodes of $\Sigma$ and the suport of $D$ the sheaf
${\mathcal B}$ is canonically isomorphic to $\Omega_{C_T}\otimes_{{\mathcal O}_{C_T}}{\mathcal O}_{C_0}$.
Because $\tilde f_i\!:\! \tilde U_i\to V_i$ are
holomorphic for $i>l$, it follows from \cite{LT1} that there is
a canonical homomorphism of sheaves $u_i\!:\! f^{\ast}\Omega_X|_{U_i}
\to {\mathcal B}|_{U_i}$ such that
\begin{equation}
\begin{CD}
@. @. f^{\ast}\Omega_X|_{U_i} @= f^{\ast}\Omega_X|_{U_i}\\
@. @. @VV{u_i}V @VV{df^{\vee}|_{U_i}}V\\
0 @>>> {\mathcal O}_{U_i}^{\cA} @>>> {\mathcal B}^{\cA}|_{U_i} @>>> \Omega_{U_i}(D)^{\cA}
@>>> 0\\
\end{CD}
\end{equation}
is commutative, where the lower sequence is induced by \eqref{eq:2.23}.
Indeed, at smooth point $p\in U_i$ away from the support of $D$ the dual of
$u_i\otimes k(p)$ is the differential
$$d\tilde f_i(p) : T_p C_T={\mathcal B}^{\vee}\otimes k(p)\lra f^{\ast} T_X|_p.
$$
Note that by our choice of $U_i$, for $i\leq l$ the distinguished
open subsets $U_i$ are disjoint from the support of the
$(n+k)$-marked points of $\tilde w$. Hence ${\mathcal B}^{\cA}|_{U_i}$ are canonically
isomorphic to $\Omega_{U_i}^0(TC_T|_{U_i})$, and
the dual of $d\tilde f_i$ define canonical homomorphisms $u_i\!:\!
f^{\ast}\Omega_X|_{U_i}\to{\mathcal B}|_{U_i}$ that make the above diagrams commutative.
Because of the condition \eqref{eq:2.22}, the lift
of $\bar\partial u_i$ is a constant multiple of $\tilde\sigma_i(\tilde w)|_{U_i}$.
Further, because of the condition that $\tilde f_{ji}$ is divisible by $z_0^2$,
the collection $\{ u_i\}_{i=1}^L$
patch together to form a homomorphism $v_2(\xi)\!:\! f^{\ast}\Omega_X\to{\mathcal B}^{\cA}$
that makes the diagram \eqref{eq:2.3} commutative. Hence $(v_1(\xi),v_2(\xi))$
defines an element in $\Ext^1(\ddotw,\cO_{\Sigma})^{\dag}$, which is defined to be the image of $\xi$.
We remark that in this construction we have only used the fact
that the stable map associated to $\tilde w$ is holomorphic, that
the domain $\Sigma$ of $\tilde w$ has $l$ distinguished open
subsets $U_i$ with $(0,1)$-forms $\tilde\sigma_i(\tilde w)$.
Because for any $z\in R$ its domain $\Sigma_z$ also has $l$ distinguished
open subsets, namely ${\mathcal U}_i\cap \Sigma_z\cong U_i$, and the forms
$\tilde\sigma(z)$, we can define the extension group
$\Ext^1({\mathcal D}^{\bullet}_z,{\mathcal O}_{\Sigma_z})^{\dag}$, the space of pre-${\mathbb C}$-tangents
of $R$ at $z$ and the analogut canonical map as in \eqref{eq:2.26}
if the map $f_z$ of $z$ is holomorphic.
To justify our choice of $\Ext^1(\ddotw,\cO_{\Sigma})^{\dag}$, we will construct, to each $v\in\Ext^1(\ddotw,\cO_{\Sigma})^{\dag}$,
a pre-${\mathbb C}$-tanget $\xi^v\in T_{\tilde w}^{{\rm pre}} R$ so that the
image of $\xi^v$ under \eqref{eq:2.26} is $v$.
Let $v=(v_1,v_2)$ be any element in $\Ext^1(\ddotw,\cO_{\Sigma})^{\dag}$ defined by the diagram
\eqref{eq:2.3}. Let $T\subset {\mathbb C}$ be a neighborhood of $0$ and
let $C_T$ be an analytic family of $n$-pointed curves so that $C_0\cong \Sigma$
and the Kuranishi map $T_0{\mathbb C}\to\Ext^1(\Omega_{\Sigma}(D),{\mathcal O}_{\Sigma})$ send
$1$ to $v_1$. For instance, we can take $C_T$ be the
pull back of ${\mathcal X}_n$ via an analytic map $(T,0)\to (O_n,p_n)$. We let
$\{U_i\}_{i=1}^L$ be a covering of
$\Sigma$ as before and let $\{\tilde U_i\}_{i=1}^L$ be a covering of $C_T$
that are the pull back of ${\mathcal U}_{n,i}$. Note that for $i\leq l$, they
come with biholomorphisms $\tilde U_i\cong U_i\times T$. Let
$V_i$ be open charts of $X$ as before with $f(U_i)\Subset V_i$. For $i>l$,
since the restriction of \eqref{eq:2.3} to $U_i$ is analytic, we can
find analytic $\tilde f_i\!:\! \tilde U_i\to V_i$, possibly after
shrinking $T$ if necessary, such that $\tilde f_i$ are related to
$v_2|_{U_i}$ as to how $u_i$ are related to $v_2(\xi)|_{U_i}$ before.
By analytic analogue of deformation theory (see \cite{LT1})
such $\tilde f_i$ do exist. For $i\leq l$, since $U_i$ are smooth
and ${\mathcal B}^{\cA}|_{U_i}$ are the sheaves $\Omega_{U_i}^0(T^{\ast} C_T|_{U_i})$,
we simply let $\tilde f_i$ be smooth so that in addition to $\tilde
f_i$ satisfying the condition on pre-${\mathbb C}$-tangents we require that
$v_2|_{U_i}$ coincide with the dual of $d\tilde f_1|_{U_i}$. Note that
$(C_T,\{\tilde f_i\})$ will be a pre-${\mathbb C}$-tangent if $\tilde f_{ji}$
in \eqref{eq:2.27} is divisible by $\pi_T^{\ast}(z_0^2)$.
But this is true because for any $p\in U_i\cap U_j$, the differential
$d\tilde f_i(p)$ and $d\tilde f_j(p)$ from $T_p C_T$ to $T_{f(p)}X$
are identical. We let the so constructed pre-${\mathbb C}$-tangent be $\xi^v$.
Of course $\xi^v$ are not unique. It is obvious from the construction that
the image of $\xi^v$ under \eqref{eq:2.26} is $v$.
We remark that it follows from the construction that for any complex
number $c\ne 0$ the pull back of $(C_T,\{\tilde f_i\})$ under $L_c
\!:\! {\mathbb C}\to {\mathbb C}$ defined by $L_c(z_0)=cz_0$ is a pre-${\mathbb C}$-tangent,
say $\xi^{cv}$, whose image under \eqref{eq:2.26} is $cv$.
We next construct a holomorphic coordinate chart of $R$ at $\tilde w$.
Let $r=\dim R$, which is $r_{\rm exp }+l=\dim\Ext^1(\ddotw,\cO_{\Sigma})^{\dag}$.
We fix a ${\mathbb C}$-isomorphism $T_0{\mathbb C}^r\cong\Ext^1(\ddotw,\cO_{\Sigma})^{\dag}$. Composed with
the canonical
$$\Ext^1(\ddotw,\cO_{\Sigma})^{\dag}\to \Ext^1(\Omega_{\Sigma}(D),{\mathcal O}_{\Sigma}),
$$
we obtain
\begin{equation}
T_0{\mathbb C}^r\lra \Ext^1(\Omega_{\Sigma}(D),{\mathcal O}_{\Sigma}).
\label{eq:2.20}
\end{equation}
Let ${\mathcal X}_n$ over $O_n$ be the semi-universal family of
the $n$-pointed curve $\Sigma$ given before. We let $S$ be a
neighborhood of $0\in{\mathbb C}^r$ and let $\varphi\!:\! S\to O_n$
be a holomorphic map with $\varphi(0)=0$ such that
$$d\varphi(0): T_0S\equiv T_0{\mathbb C}^r\lra T_{p_n}O_n\cong
\Ext^1(\Omega_{\Sigma}(D),{\mathcal O}_{\Sigma})
$$
is the homomorphism \eqref{eq:2.20}. We let $\pi_S\!:\! C_S\to S$ be the family
of $n$-pointed curves over $S$ that is the pull back of ${\mathcal X}_n$. Note that
$C_S|_0$, denoted by $C_0$, is canonically isomorphic to $\Sigma$.
We keep the open covering $\{U_i\}^L_{i=1}$ of $\Sigma$ ($\cong C_0$)
chosen before. We let $\{W_i\}_{i=1}^L$ be an open covering
of a neighborhood of $C_0\subset C_S$ so that $W_i\cap C_0=U_i$.
For $i\leq l$, we let $W_i$ be the pull back of ${\mathcal U}_{n,i}\subset {\mathcal X}_n$. For
$i>l$ and $U_i$ smooth, we choose $W_i$ so that there is a
holomorphic map $\pi_i\!:\! W_i\to U_i$ so that the restriction of
$\pi_i$ to $U_i$ is the identity map. For $i>l$ and $U_i$ contains
a nodal point, we assume that $W_i$ is biholomorphic to
the unit ball in ${\mathbb C}^{r+1}$ so that $U_i\subset W_i$ is defined by
$w_1w_2=0$ and $w_i=0$ for $i\geq 3$,
where $(w_i)$ are the coordinate variables of ${\mathbb C}^{r+1}$,
and the restriction
of $\pi_S$ to $W_i$ is given by
$$z_1=w_1w_2, \ z_2=w_3,\ldots,z_r=w_{r+1},
$$
where $(z_i)$ are the coordinate variables of ${\mathbb C}^r$. The upshot
of this is that if $h$ is a holomorphic function on $U_i$,
then we can extend it canonically to $W_i$ as follows. In case
$U_i$ is smooth, then the extension of $h$ is the
composite of $W_i\to U_i$ with $h$; In case
$U_i$ is singular, then $\varphi$ has a unique expression
$$a+w_1h_1(w_1)+w_2h_2(w_2),
$$
where $a\in{\mathbb C}$ and $h_1, h_2$ are holomophic.
We then let its extension be the holomorphic function on
$W_i$ that has the same expression.
We fix the choice of $\{U_i\}$ and $\{W_i\}$. Without loss of generality,
we can assume that there are coordinate charts $V_i\subset X$
so that $f(U_i)\Subset V_i$. Of course, for $i\leq l$ the charts $V_i$ are
the charts we have chosen before.
Our construction of the local holomorphic chart of $R$ is
parallel to the original construction of Kodaira-Spencer of semi-universal
family of deformation of holomorphic structures without obstructions.
To begin with, possibly after shrinking $W_i$ if necessary we
can assume that the maps $f|_{U_i}\!:\! U_i\to V_i$ can be extended
to a holomorphic $F_{0,i}\!:\! W_i\to V_i$ (Recall $f$ is holomorphic).
We now let ${\mathcal A}(W_i,V_i)$ be the space of smooth maps
from $W_i$ to ${\mathbb C}^m$ defined as follows.
If $i>l$, then ${\mathcal A}(W_i,V_i)$ consists of holomorphic maps
from $W_i$ to ${\mathbb C}^m$; If $i\leq l$, then using the isomorphism
$W_i\cong U_i\times S$ and holomorphic coordinate $z=(z_i)$ of
$S$ and holomorphic coordinate $\xi$ of $U_i$, any smooth function
$\varphi\!:\! W_i\to {\mathbb C}^m$ can be expressed in terms of its
$m$ components $\varphi_j(z, \xi)$, $j=1,\ldots,m$. We
define ${\mathcal A}(W_i,V_i)$ to be the set of
those smooth maps $\varphi\!:\!
W_i\to{\mathbb C}^m$ so that
$$\left\{
\begin{array}{l}
\bar\partial_{z_k}\varphi_j=0\quad \text{for}\quad k=1,\ldots,r \quad{\rm and}\quad j=1,\ldots,m;\\
\bar\partial_{\xi}\varphi_j=0\quad \text{for}\quad j\geq 2
\quad{\rm and}\quad \bar\partial_{\xi}\varphi_1=c\sigma_i^{\prime}\quad \text{for some}\quad
c\in{\mathbb C},
\end{array}
\right.
$$
where $\sigma^{\prime}_i$ is a $(0,1)$-form taking values in $\varphi^{\ast}
{\mathbb C}^n$ corresponding to the form $\sigma_i$ using the canonical
embedding $V_i\subset{\mathbb C}^n$.
Note that ${\mathcal A}(W_i,V_i)$ are ${\mathcal O}_S$-modules. In
particular, if we let ${\mathcal I}\subset {\mathcal O}_S $ be the ideal sheaf of $0\in S$,
then we denote by ${\mathcal I}^q {\mathcal A}(W_i,V_i)$ the image of
${\mathcal I}^q\otimes_{{\mathcal O}_S }{\mathcal A}(W_i,V_i)$ in ${\mathcal A}(W_i,V_i)$.
In the following, we will construct
a sequence of maps $F_{s,i}\in{\mathcal A}(W_i, V_i)$ indexed by
$s\geq 1$ and $1\leq i\leq L$ of which the following holds:
\noindent
1. For each $i$, $F_{s+1,i}-F_{s,i}\in {\mathcal I}^{s}{\mathcal A}(W_i, V_i)$;
\noindent
2. The restrictions $F_{1,i}|_{U_i}\!:\! U_i\to{\mathbb C}^m$ factor through
$V_i\subset{\mathbb C}^m$ and $\iota_i\circ (F_{1,i}|_{U_i})\!:\! U_i\to X$
is identical to $f|_{U_i}\!:\! U_i\to X$;
\noindent
3. In a neighborhood $W_{ij}$ of $U_i\cap U_j$ in $W_i\cap W_j$
over which the map
\begin{equation}
F_{s,ij}=g_{ij}\circ F_{s,j}- F_{s,i}\!:\! W_{ij}\to {\mathbb C}^m
\label{eq:2.21}
\end{equation}
is well defined,
$F_{s,ij}\in {\mathcal I}^{s}{\mathcal H}(W_{ij},{\mathbb C}^m)$, where ${\mathcal H}(W_{ij},{\mathbb C}^m)$
is the ${\mathcal O}_S $-module of holomorphic maps from $W_{ij}$ to ${\mathbb C}^m$;
\noindent
4. For any vector $\eta\in{\mathbb C}^r$, we let $L_{\eta}\!:\!{\mathbb C}\to{\mathbb C}^r$ be the
unique ${\mathbb C}$-linear map so that $L_{\eta}(1)=\eta$, and let $\eta^{{\rm pre}}$ be
the pre-${\mathbb C}$-tangent associated to the pull back of $(C_S,\{ F_{2,i}\})$
under $L_{\eta}$. Using the standard isomorphism $T_0 S\equiv T_0{\mathbb C}^r\cong
{\mathbb C}^r$, we obtain a map
\begin{equation}
T_0S\lra \Ext^1(\ddotw,\cO_{\Sigma})^{\dag}
\end{equation}
that send $\eta\in T_0 S$ to the image of $\eta^{{\rm pre}}$ under \eqref{eq:2.26}.
We require that this map is the isomorphism \eqref{eq:2.20}.
For $s=1$ we simply let $F_{1,i}$ be the standard extension
of $f|_{U_i}\!:\! U_i\to V_i$ to $W_i\to {\mathbb C}^m$.
We now show that we can construct $\{ F_{2,i}\}$ as required.
We let $\pi_1$ and $\pi_2$ be the first and the
second projection of ${\mathbb C}^r\times \Sigma$, where
we view ${\mathbb C}^r$ as the total space of $\Ext^1(\ddotw,\cO_{\Sigma})^{\dag}$. It follows
from the definition of the extension group that there is a
universal diagram
\begin{equation}
\begin{CD}
@.@.\pi_2^{\ast} f^{\ast}\Omega_X @= \pi_2^{\ast} f^{\ast}\Omega_X\\
@.@. @VV{{\mathcal V}_2}V @VV{\pi_2^{\ast}(df^{\vee})}V \\
0 @>>> \pi_2^{\ast}{\mathcal O}_{\Sigma}^{\cA} @>>> {\tilde{\mathcal B}}^{\cA}
@>>> \pi_2^{\ast}\Omega_{\Sigma}(D)^{\cA} @>>> 0
\end{CD}
\label{eq:2.28}
\end{equation}
such that its restriction to fibers of ${\mathbb C}^r\times\Sigma$ over
$\xi\in{\mathbb C}^r$ are the diagrams \eqref{eq:2.3} associated to $\xi
\in\Ext^1(\ddotw,\cO_{\Sigma})^{\dag}$. By deformation theory of pointed curves, for any smooth
point $p\in\Sigma$ the vector space $\tilde{\mathcal B}\otimes k(p)$ is
canonically isomorphic to the cotangent space $T_p^{\ast} C_S$.
By applying the construction of $\xi^v\in T_{\tilde w}^{{\rm pre}}
R$ from $v\in\Ext^1(\ddotw,\cO_{\Sigma})^{\dag}$ to the family version, we can construct the
family $\{ F_{2,i}\}$ as required. We will leave the details to the readers.
Now we show that we can successively construct $F_{s,i}$
that satisfies the four conditions above.
Assume that for some $s\geq 2$ we have constructed
$\{F_{s,i}\}$ that satisfies the four conditions above.
Let $W_{ij}$ be the neighborhood of $U_{ij}=U_i\cap U_j\subset
W_i\cap W_j$ so that \eqref{eq:2.21} is well-defined. Then by the
condition 3 above, $F_{s,ij}\in {\mathcal I}^{s} {\mathcal H}(W_{ij},{\mathbb C}^m)$.
Let $I=(i_1,\ldots,i_r)$ be any length $s$ mulptiple index,
namely, $i_j\geq 0$ and $\sum i_j=s$.
As usual, we will denote by $\partial^I$ the
symbol $\partial^{i_1}/\partial z_1^{i_1}\cdots\partial^{i_r}/\partial
z_r^{i_r}$
and by $z^I$ the term $z_1^{i_1}\cdots z_r^{i_r}$.
Then because of the
condition 3 above,
$\varphi_{I,ij}={\partial^I} F_{s,ij}|_{U_{ij}}$
is a holomorphic section of $f^{\ast} T_X|_{U_{ij}}$
using the standard isomorphism
$$TX|_{V_i}\cong
TV_i\cong V_i\times{\mathbb C}^m,
$$
and the collection
$[\varphi_{I,ij}]$ defines a C\v{e}ch 1-cocycle of
$f^{\ast}{\mathcal T}_X$. We let $\{\phi_{I,i}\}$, where $\phi_{I,i}=\zeta_i+a_i\eta_i$,
be the collection provided by Lemma \ref{2.1}.
Using the standard isomorphism $TX|_{V_i}\cong V_i\times{\mathbb C}^m$,
we can view $\phi_{I,i}$ as a map $V_i\to {\mathbb C}^m$. We let $\tilde\phi_{I,i}\!:\!
W_i\to{\mathbb C}^m$ be the standard extension of $\phi_{I,i}$ and let $G_{I,i}=
\pi_S^{\ast}(z^I)\tilde\phi_{I,i}$. Clearly, $\partial^I G_{I,i}=\phi_{I,i}$.
Now we let
$$F_{s+1,i}=F_{s,i}+\sum_{\ell(I)=s} G_{I,i}.
$$
It is direct to check that the collection $\{F_{s+1,i}\}$
satisfies the condition 1-4 before.
Finally, by the estimate in Lemma \ref{2.1}, there is a
neighborhood of $U_i\subset W_i$, say $W_i^0$, such that $\lim_s F_{s,i}$
converges over $W_i^0$. Let
$F_{\infty,i}$ be its limit.
Because $f(U_i)\Subset V_i$, there is a neighborhood
$\tilde{W}_i$ of $U_i\subset W_i^0$ such that $F_{\infty,i}(\tilde{W}_i)
\subset V_i\subset{\mathbb C}^m$. It follows that we can find a neighborhood
$S^0\subset S$ of $0\in S$ such that $\pi_S^{-1}(S^0)\subset \cup \tilde{W}_i$.
Finally, because $F_{\infty,i}$ is analytic near $U_i$ for $i>l$
and is analytic in $S$ direction using $W_i\cong U_i\times S$ otherwise,
the condition 3 implies that
the collection $F_{\infty,i}|_{{W}_i\cap\pi_S^{-1}(S^0)}$
defines a map
$$F_S: \pi_S^{-1}(S^0)\lra X.
$$
Clearly, $F_S$ is holomorphic away from the union of $W_1,\ldots, W_l$.
Further, for each $i\leq l$ if we let $\xi_i$ be a holomorphic
variable of $U_i$ and let $\pi_{U_i}$ and $\pi_{S^0}$ be the
first and the second projection of
$W_i\cap \pi_S^{-1}(S^0)\cong U_i\times S^0$, then
\begin{equation}
\frac{\partial}{\partial\bar\xi_i} F_S|_{W_i\cap\pi_S^{-1}(S^0)}d\bar\xi_i=
\pi_{S^0}^{\ast}(\varphi_i)\pi_{U_i}^{\ast}(\gamma_i) F_S^{\ast}({\mathbf v}_i)|_{W_i}
\label{eq:2.30}
\end{equation}
where $\varphi_i$ is a holomorphic
function over $S^0$.
Finally, we let $Z$ be the subset of
$$\pi_S^{-1}(S^0)\times_S\cdots\times_S\pi^{-1}_S(S^0)\qquad
(\text{$k$ times})
$$
consisting of $(s;x_{n+1},\ldots,x_{n+k})$ such that $s\in S^0$
and that
$ x_{n+1},\ldots,x_{n+k}$ are distinct points in $\pi_S^{-1}(s)$
that lie in $F_s^{-1}(H)$. Note that if we choose $U$ to be small
enough, then $F_s^{-1}(H)$ has exactly $k$ points.
Let $C_Z$ be the family of $(n+k)$-pointed curves over $Z$ so that
its domain is the pull back of $C_S$ via $Z\to S$, its first
$n$-marked sections is the pull back of the $n$-marked sections of $C_S$
and its last $k$-sections of the fiber of $C_Z$ over $(s;x_{n+1},\ldots,
x_{n+k})$ is $x_{n+1},\ldots,x_{n+k}$.
Coupled with the pull back of $F_S$, say $F_Z\!:\! C_Z\to X$,
we obtain a family of stable (continuous) maps from
$(n+k)$-pointed curves to $X$. Let $\eta\!:\! Z\to U$
be the tautological map.
We claim that $\eta(Z)\subset R$. Indeed, let $z\in Z$ be any
point and let $C_z$ be the domain of $z$. It follows from
our construction that $C_z$ has $l$ distinguished open subsets,
denoted by $U_1,\ldots, U_l$, such that $f_z=F_Z|_{C_z}$
is holomorphic away from $\cup_{i=1}^l U_i$ and
$\bar\partial f_z|_{U_i}$ is a constant multiple of $\gamma_i\cdot
f_z^{\ast}({\mathbf v}_i)$. Hence the value of the section $\Phi_{\tilde U}\!:\!
\tilde U\to {\mathbf E}_{\tilde U}$ at $\eta(z)$ is contained in the
subspace $V|_{\eta(z)}\subset {\mathbf E}_{\tilde U}|_{\eta(z)}$.
This shows that $\eta(z)\in R$.
\begin{prop}
The induced map $\eta\!:\! Z\to R$ is a local diffeomorphism near
those $z\in R$ whose associated map $f_z\!:\! C_z\to X$ are
holomorphic.
\end{prop}
\begin{proof}
This follows immediately from the proof of the basic Lemma in
\cite{LT2}. We will omit the details here.
\end{proof}
By shrinking $S^0$ if necesary, we can assume that
$\eta\!:\! Z\to R$ is a local diffeomorphism. We can further assume that
there is an open subset $Z^{\prime}\subset Z$ containing $\tilde w$
such that $\eta^{\prime}=\eta|_{Z^{\prime}}\!:\! Z^{\prime}\to R$ is one-to-one
and the image $\eta(Z^{\prime})\subset R$ is invariant under $G_{\tilde U}$.
$\eta^{\prime}\!:\! Z^{\prime}\to R$ is the analytic coordinate of $\tilde w\in R$
we want. For convenience,
we will view $Z^{\prime}$ as an open subset of $R$.
\begin{prop}
Let $V^{\prime}$ be the restriction of $W$ to $Z^{\prime}$ endowed with the
holomorphic structure so that $\tilde\sigma_1|_{Z^{\prime}},\ldots,
\tilde\sigma_l|_{Z^{\prime}}$ is a holomorphic frame. Then $\phi^{\prime}
\equiv \phi_V|_{Z^{\prime}}
\!:\! Z^{\prime}\to V^{\prime}$ is holomorphic.
\end{prop}
\begin{proof}
This follows immediately from \eqref{eq:2.30}.
\end{proof}
Let ${\phi^{\prime}}^{-1}(0)$ be any point and let $f_z\!:\! C_z\to X$ be the
associated (analytic) stable map with $D_z$ the divisor of its
first $n$-marked points. Then there is a caonical exact sequence of
vector spaces
$$
\Ext^1(\Omega_{C_z}(D_z),{\mathcal O}_{C_z})\lra
H^1(f_z^{\ast}{\mathcal T}_X)\lra \Ext^2({\mathcal D}^{\bullet}_z,{\mathcal O}_{C_z})\lra 0
$$
induced by the short exact sequence of complexes
$$
0\lra [0\to\Omega_{C_z}(D_z)]\lra
[f_z^{\ast}\Omega_{X}\to\Omega_{C_z}(D_z)]
\lra[f_z^{\ast}\Omega_X\to 0]\lra 0.
$$
Similarly, the differentil $d\phi_V(z)\!:\! T_z R\to W_z$ induces
an exact sequence of vector spaces
$$
\Ext^1({\mathcal D}^{\bullet}_z,{\mathcal O}_{C_z}^{\cA})^{!}\xrightarrow{d\phi_V(z)} W|_z
\lra \coker(d\phi_V(z))\lra 0.
$$
Note that there are canonical homomorphisms $\Ext^1({\mathcal D}^{\bullet}_z,{\mathcal O}_{C_z}
^{\cA})^{!}\to\Ext^1({\mathcal D}^{\bullet}_z,{\mathcal O}_{C_z})$ and $W|_z\to H^{0,1}_{\bar\partial}(
f_z^{\ast} T_X)\cong H^1(f_z^{\ast}{\mathcal T}_X)$.
\begin{lemm}
\label{2.22}
There is a canonical isomorphism $\xi$ (as shown below)
that fits into the diagram
\begin{equation*}
\begin{CD}
\Ext^1({\mathcal D}^{\bullet}_z,{\mathcal O}_{C_z})^{!} @>{d\phi_V(z)}>> W|_z
@>>> \coker(d\phi_V(z)) @>>> 0\\
@VVV @VVV @VV{\xi}V\\
\Ext^1(\Omega_{C_z}(D_z),{\mathcal O}_{C_z}) @>>> H^1(f_z^{\ast}{\mathcal T}_X) @>>>
\Ext^2({\mathcal D}^{\bullet}_z,{\mathcal O}_{C_z}) @>>> 0.
\end{CD}
\end{equation*}
\end{lemm}
\begin{proof}
This is obvious and will be left to the readers.
\end{proof}
\section{The proof of the comparison theorem}
In this section, we will prove that the algebraic and the symplectic
construction of GW-invariants yield identical invariants.
We will work with the category of algebraic schemes
as well as the category of
analytic schemes. Specifically, we will use the words schemes,
morphisms and \`etale neighborhoods
to mean the corresponding objects in algebraic category and use the
word analytic maps and open subsets to mean the corresponding objects in
analytic category. As before, the words analytic and holomorphic are
interchangable. Also, we will use ${\mathcal O}_S$ to mean the
sheaf of algebraic sections or the sheaf of analytic sections depending on
whether $S$ is an algebraic scheme or an analytic scheme.
We will continue to use the complex dimension through out this section.
We now clarify our usage of the notions of cycles and currents. Let
$W$ be a scheme. We denote by $Z_k^{\text{alg}} W$ the group of
formal sums of finitely many
$k$-dimensional irreducible subvarieties of $W$ with rational
coefficients. We call elements of $Z_k^{\text{alg}} W$ $k$-cycles of $W$.
Now let $W$ be any stratified topological space with stratification ${\mathcal S}$.
We say that a (complex) $k$-dimensional current
$C$ is stratifiable if there is a refinement of ${\mathcal S}$, say ${\mathcal S}^{\prime}$, such that
there are finitely many $k$-dimensional strata $S_i$
and rationals $a_i\in{\mathbb Q}$ such that
$C=\sum a_i[S_i]$ (All currents in this paper are oriented).
Here we assume that each stratum of
${\mathcal S}^{\prime}$ was given an orientation a priori and $[S_i]$ is the oriented
current defined by $S_i$. We identify two currents if they define identical
measures in the sense of rectifiable currents.
We denote the set of all stratifiable
$k$-dimensional currents modulo the equivalence relation by $Z_k W$.
Clearly, if $W$
is a scheme then any $k$-cycle has an associated current in
$Z_k W$, which defines a map $Z_k^{\text{alg}} W\to Z_k W$.
In the following, we will not
distinguish a cycle from its associated current.
Hence for $C\in Z_k^{\text{alg}}
W$ we will view it as an element of $Z_{k} W$.
Note that if
$C\in Z_k W$ has zero boundary in the sense of current and $C$ has
compact support, then $C$ defines canonically an element in $H_{2k}(W,{\mathbb Q})$.
Finally, if $C=\sum a_i[S_i]\in Z_k W$ and $F\subset W$ is a stratifiable
subset, we say that $C$ intersects $F$ transversally if $F$ intersects each
$S_i$ transversally as stratified sets (See
\cite{GM} for topics on stratifications).
In such cases, we can define the intersection current $C\cap F$ if the
orientation of the intersection can be defined according to the geometry of
$W$ and $F$.
We begin with a quick review of the algebraic construction of GW-invariants.
Let $X$ be any smooth projective variety and let $A\in H_2(X,{\mathbb Z})$ and
$g,n\in{\mathbb Z}$ as before be fixed once and for all. We let $\M_{g,n}(X,A)$ be the
moduli scheme of stable morphisms defined before. $\M_{g,n}(X,A)$ is
projective. The GW-invariants of $X$ is defined using the
virtual moduli cycle
$$[\M_{g,n}(X,A)]^{{\rm vir}}\in A_{\ast}\M_{g,n}(X,A).
$$
To review such a construction, a few
words on the obstruction theory of deformations of morphisms
are in order. Let $w\in\M_{g,n}(X,A)$ be any point associated to the
stable morphism ${\mathcal X}$. Let $(B,I,{\mathcal X}_{B/I})$ be any collection where $B$ is an
Artin ring, $I\subset B$ is an ideal annihilated by the maximal ideal
${\mathfrak m}_B$ of $B$ and ${\mathcal X}_{B/I}$ is a flat family of stable morphisms
over $\spec B/I$ whose restriction to the closed fiber of ${\mathcal X}_{B/I}$
is isomorphic to ${\mathcal X}$. An obstruction theory to
deformation of ${\mathcal X}$ consists of a ${\mathbb C}$-vector space $V$, called the
obstruction space, and an
assignment that assigns any data $(B,I,{\mathcal X}_{B/I})$ as before to an obstruction
class
$$\ob(B,B/I,{\mathcal X}_{B/I})\in I\otimes_{{\mathbb C}} V
$$
to extending ${\mathcal X}_{B/I}$ to $\spec B$. Here by an obstruction class,
we mean that its vanishing is the necessary and sufficient condition for
${\mathcal X}_{B/I}$ to be extendable to a family over $\spec B$.
We also require that such an assignment satisfies the obvious base change
property (For reference on obstruction theory please consult
\cite{Ob}). In case ${\mathcal X}$ is the map
$f\!:\! C\to X$ with
$D\subset C$ the divisor of its $n$ marked points, the space of
the first order deformations of ${\mathcal X}$ is parameterized by
$\Ext^1({\mathcal D}^{\bullet}_{{\mathcal X}},{\mathcal O}_C)$, where ${\mathcal D}^{\bullet}_{{\mathcal X}}=
[f^{\ast}\Omega_X\to\Omega_C(D)]$ is the complex as before,
and the standard obstruction theory to deformation of ${\mathcal X}$
takes values in $\Ext^2({\mathcal D}^{\bullet}_{{\mathcal X}},{\mathcal O}_C)$.
An example of obstruction theories is the following.
Let $R$ be the ring of formal power series in $m$ variables
and let ${\mathfrak m}_R\subset R$ be its maximal ideal.
Let $F$ be a vector space and let $f\in {\mathfrak m}_R\otimes_{{\mathbb C}}F$.
We let $(f)\subset R$ be the ideal generated by components of $f$. Then
there is a standard obstruction theory to deformations of
$0$ in $\spec R/(f)$ taking values in $V$, where $V$ is the
cokernel of $df\!:\! ({\mathfrak m}_R/{\mathfrak m}_R^2)^{\vee}\to F$,
defined as follows. Let $I\subset B$ be an ideal of an Artin ring as before and
let $\varphi_0\!:\!\spec B/I\to\spec R/(f)$
be any morphism. To extend $\varphi_0$ to $\spec B$, we first pick a
homomorphism $\sigma\!:\! R\to B$ extending the induced $R\to B/I$,
and hence a morphism $\varphi_{\text{pre}}\!:\! \spec B \to \spec R$.
The image $\sigma(f)\in B\otimes F$ is in $I\otimes F$,
and is the obstruction to
$\varphi_{\text{pre}}$ factor through $\spec R/(f)\subset\spec R$.
Let $\ob(B,B/I,\varphi_0)$ be the image of $\sigma(f)$ in
$I\otimes V$ via $F\to V$. It is direct to check that
$\ob(B,B/I,\varphi_0)=0$ if and only if there is an extension
$\varphi\!:\!\spec B\to\spec R/(f)$ of $\varphi_0$.
This assignment
\begin{equation}
\label{eq:3.0}
(B,B/I,\varphi_0)\mapsto \ob(B,B/I,\varphi_0)\in
I\otimes V
\end{equation}
is the induced obstruction theory of $\spec R/(f)$.
\begin{defi}
\label{3.1}
A Kuranishi family of the standard obstruction theory of ${\mathcal X}$ consists of
a vector space $F$, a ring of formal power series $R$
with ${\mathfrak m}_R$ its maximal ideal,
an $f\in{\mathfrak m}_R\otimes F$, a family ${\mathcal X}_{R/(f)}$
of stable morphisms over $\spec R/(f)$ whose closed fiber over $0\in
\spec R/(f)$ is isomorphic to ${\mathcal X}$ and an exact sequence
\begin{equation}
0\lra \Ext^1({\mathcal D}^{\bullet}_{{\mathcal X}},{\mathcal O}_C)\xrightarrow{\alpha} ({\mathfrak m}_R/{\mathfrak m}_R^2)^{\vee}
\xrightarrow{df} F\lra \Ext^2({\mathcal D}^{\bullet}_{{\mathcal X}},{\mathcal O}_C)
\lra 0
\label{eq:3.1}
\end{equation}
of which the following holds:
First, the composite
$$\Ext^1({\mathcal D}^{\bullet}_{{\mathcal X}},{\mathcal O}_C)\xrightarrow{\alpha} \ker(df)\equiv T_0\spec
R/(f) \xrightarrow{} \Ext^1({\mathcal D}^{\bullet}_{{\mathcal X}},{\mathcal O}_C),
$$
where the second arrow is the Kodaira-Spencer map of the family ${\mathcal X}_{R/(f)}$,
is the identity homomorphism;
Secondly, let $I\subset B$ and $\varphi_0\!:\!\spec B/I\to
\spec R/(f)$ be as before and let
$$\ob(B,B/I,\varphi_0^{\ast}{\mathcal X}_{R/(f)})\in I\otimes\Ext^2({\mathcal D}^{\bullet}_{{\mathcal X}},{\mathcal O}_C)
$$
be the obstruction to extending $\varphi_0^{\ast}{\mathcal X}_{B/I}$ to $\spec B$.
Then it is identical to $\ob(B,B/I,\varphi_0)$
under the isomorphism
$$\coker(df)\cong \Ext^2({\mathcal D}^{\bullet}_{{\mathcal X}},{\mathcal O}_C),
$$
where $\ob(B,B/I,\varphi_0)$ is the obstruction class in \eqref{eq:3.0}.
\end{defi}
We now sketch how the virtual moduli cycle $[\M_{g,n}(X,A)]^{{\rm vir}}$ was constructed.
Similar to the situation of the moduli of stable smooth maps,
we need to treat $\M_{g,n}(X,A)$ either as a ${\mathbb Q}$-scheme or as a
Deligne-Mumford stack. The key ingredient here is the notion of atlas,
which is a collection of charts of $\M_{g,n}(X,A)$. A chart of $\M_{g,n}(X,A)$
is a tuple $(S, G, {\mathcal X}_S)$, where $G$ is a finite group,
$S$ is a $G$-scheme
(with effective $G$-action) and ${\mathcal X}_S$ is a $G$-equivariant family of stable
morphisms so that the tautological morphism $\iota \!:\! S/G\to\M_{g,n}(X,A)$ induced by
the family ${\mathcal X}_S$ is an \'etale neighborhood.
For details of such an notion, please consult \cite{DM, Vi, LT1}.
We now let $f\!:\! C\to X$ be the representative of
${\mathcal X}_S$ with $D\subset C$ the divisor of the $n$-marked sections of ${\mathcal X}_S$.
Let $\pi\!:\! C\to S$ be the projection. We consider the relative extension sheaves
$\mathop{{\mathcal E} xt\hspace{1pt}}^i_{\pi}({\mathcal D}^{\bullet}_{{\mathcal X}_S},{\mathcal O}_C)$, where
${\mathcal D}^{\bullet}_{{\mathcal X}_S}=[f^{\ast}\Omega_X\to\Omega_{C/S}(D)]$ as
before. For short, we denote the sheaves
$\mathop{{\mathcal E} xt\hspace{1pt}}^i_{\pi}({\mathcal D}^{\bullet}_{{\mathcal X}_S},{\mathcal O}_C)$ by ${\mathcal T}^i_S$.
Because they vanish for $i=0$ and $i> 2$, for any $w\in S$, the
Zariski-tangent space $T_wS$ is ${\mathcal T}_S^1\otimes_{{\mathcal O}_S}k(w)$
and the
obstruction space to deformations of $w$ in $S$ is $V_w={\mathcal T}_S^2\otimes
_{{\mathcal O}_S}k(w)$. Now we choose a complex
of locally free sheaves of ${\mathcal O}_S$-modules $\cE^{\bullet}=[{\mathcal E}_1\to{\mathcal E}_2]$ so that
it fits into the exact sequence
\begin{equation}
0\lra {\mathcal T}_S^1\lra{\mathcal E}_1\lra{\mathcal E}_2\lra
{\mathcal T}_S^2\lra 0.
\label{eq:3.2}
\end{equation}
We let $F_i(w)={\mathcal E}_i\otimes_{{\mathcal O}_S}k(w)$.
Then we have the exact sequence of vector spaces
\begin{equation}
0\lra T_wS\lra F_1(w)\lra F_2(w) \lra V_w\lra 0.
\label{3.31}
\end{equation}
We let $K_w\in R(w)$ be a Kuranishi map of the
obstruction theory to deformations of $w$, where $R(w)=
\varprojlim\oplus _{k=0}^N \text{Sym}^k(F_1(w)^{\vee})$,
so that \eqref{3.31} is part of the data of the Kuranishi family specified
in Definition \ref{3.1}.
Let $(K_w)\subset R(w)$ be the ideal generated by the
components of $K_w$ and let $\spec R_w/(K_w)\subset\spec R_w$
be the corresponding subscheme. It follows that $\spec R_w/(K_w)$ is
isomorphic to the formal completion of $S$ along $w$, denoted $\hat w$.
We let $N_w$
be the normal cone to $\spec R_w/(K_w)$ in $\spec R_w$. Then $N_w$ is
canonically a subcone of $F_2(w)\times\hat w$. Here, by abuse of notation
we will use $F_2(w)$ to denote the total space of the vector
space $F_2(w)$.
Note that $N_w$ is the infinitesimal normal cone to $S$ in its obstruction theory at $w$.
To obtain a global cone over $S$, we need the following existence
and uniqueness theorem, which is the main result of [LT1].
In this paper, we will call a vector bundle $E$ the associated vector
bundle of a locally free sheaf ${\mathcal E}$ if ${\mathcal O}(E)\cong {\mathcal E}$. For
notational simplicity, we will not distinguish a vector bundle from
the total space (scheme)
of this vector bundle.
\begin{theo}[\cite{LT1}]
\label{3.15}
Let $E$ be the associated vector bundle of ${\mathcal E}_2$.
Then there is a cone scheme $N_S\subset E$ such that
for each $w\in S$ there is an isomorphism
\begin{equation}
F_2(w)\times\hat w\cong E \times_S\hat w
\label{eq:3.3}
\end{equation}
of cones over $\hat w$
extending the canonical isomorphism
$F_2(w) \cong E\times_S w$ such that under
the above isomorphism $N_w$ is
isomorphic to $N_S\times_S\hat w$.
In particular, the cycle defined by the scheme $N_S$ is uniquely
characterized by this condition.
\end{theo}
In the previous discussion, if we replace $F_1(w)$ and $F_2(w)$ by
$T_wS$ and $V_w$ respectively, we obtain a Kuranishi map and
correspondingly a cone scheme in $V_w\times \hat w$, denoted
by $N_w^0$.
\begin{theo}[\cite{LT1}]
\label{3.16}
Let the notation be as before. Then there is a vector bundle homomorphism
$r\!:\! E\times_S\hat w\to V_w\times \hat w$ extending the canonical
homomorphism $E|_w\to V_w$ induced by \eqref{3.31}
such that
$$N_w^0\times_{V_w\times\hat w} E\times_S\hat w= N_S\times_S \hat w.
$$
\end{theo}
To construct the virtual cycle $[\M_{g,n}(X,A)]^{{\rm vir}}$, we need to find a
global complex over $\M_{g,n}(X,A)$ analogous to $\cE^{\bullet}$.
For the purpose of comparing with the analytic construction of
the virtual cycles, we will use atlas of analytic charts.
We let $\{(R_i,W_i,\phi_i)\}_{i\in\Lambda}$ be the good
atlas of the smooth approximation of $[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$ chosen
in section 2.
Then the collection $Z_i=\phi_i^{-1}(0)$ with the tautological family of
stable analytic maps (with the last $k$-marked points discarded)
form an atlas of the underlying analytic scheme of $\M_{g,n}(X,A)\cong\Phi^{-1}(0)$.
Since we are only interested in constructing and working
with cone cycles in ${\mathbb Q}$-bundles (known as V-bundles) over
$\M_{g,n}(X,A)$, there is no loss of generality that we work with $\M_{g,n}(X,A)$ with the
reduced scheme structure. Hence, for simplicity we will endow $Z_i=\phi_i^{-1}(0)$ with the reduced analytic scheme structure.
We let ${\mathcal X}_i$ be the tautological
family of the $n$-pointed stable analytic maps over $Z_i$
that is derived by discarding the last $k$ marked points of
the restriction to $Z_i$ of the tautological family over $\tilde U_i$.
We let $G_i$ be the finite group associated to the
chart $(R_i,W_i,\phi_i)$, and let ${\mathcal X}_i$ be represented by
$f_i\!:\! C_i\to X$ with $D_i\subset C_i$ be the divisor of the $n$-marked sections
of $C_i$ and let $\pi_i\!:\! C_i\to Z_i$ be the projection.
In \cite{LT1}, to each $i$,
we have constructed a $G_i$-equivariant complex
of locally free sheaves of ${\mathcal O}_{Z_i}$-modules $\cE^{\bullet}_i=[{\mathcal E}_{i,1}
\to{\mathcal E}_{i,2}]$ such that $\mathop{{\mathcal E} xt\hspace{1pt}}_{\pi_i}^{\bullet}(
{\mathcal D}^{\bullet}_{{\mathcal X}_i},{\mathcal O}_{C_i})$ is the sheaf cohomology of $\cE^{\bullet}_i$.
It follows from the algebraic and the analytic constructions of charts that
each $(Z_i,{\mathcal X}_i)$ can be realized as an analytic open subset of
an algebraic chart, say $(S, G, {\mathcal X}_S)$, and the complex $\cE^{\bullet}_i$ is the
restriction to this open subset of an algebraic complex $\cE^{\bullet}$, as in
\eqref{3.2}. Therefore we can apply Theorem \ref{3.15} to obtain a unique
analytic cone cycle $M_i^{\text{alg}}\in Z_{\ast} E_i$, where $E_i$ is the associated
vector bundle of ${\mathcal E}_{i,2}$.
Let $\iota_i\!:\! Z_i/G_i\to\M_{g,n}(X,A) $ be the tautological map induced by
the family ${\mathcal X}_i$. One property that follows from the construction of the
complexes $\cE^{\bullet}_i$ which we did not mention is that to each $i$, the
cone bundle $E_i/G_i$ over $Z_i/G_i$ descends to a
cone bundle over $\iota_i(Z_i/G_i)$, denoted by $\tilde E_i$,
and
$\{\tilde E_i\}_{i\in\Lambda}$ patch together to form a global cone
bundle over $\M_{g,n}(X,A)$, denoted by $\tilde E$. Further, by the uniqueness
of the cone cycles $M_i^{\text{alg}}\in Z_{\ast} E_i$ in
Theorem 3.2 and 3.3, to each $i$ the cone cycle $M_i^{\text{alg}}/G_i$ in $E_i
/G_i$ descends to a cone cycle ${\mathcal M}_i^{\text{alg}}\in Z_{\ast} \tilde E_i$, and
$\{{\mathcal M}_i^{\text{alg}}\}_{i\in\Lambda}$ patch together to form a cone cycle in $Z_{\ast}\tilde E$,
denoted by ${\mathcal M}^{\text{alg}}$.
It follows from \cite{LT1} that $\tilde E$ is an algebraic cone
over $\M_{g,n}(X,A)$ and ${\mathcal M}^{\text{alg}}$ is an algebraic cone cycle in $\tilde E$.
In the end, we let $\eta_E\!:\! \M_{g,n}(X,A)\to \tilde E$ be the zero section
and let
$$\eta_E^{\ast}\!:\! \{\text{algebraic cycles in}\ Z_{\ast}\tilde E\}\lra H_{\ast}(\M_{g,n}(X,A);{\mathbb Q})
$$
be the Gysin homomorphism. Then the virtual moduli cycle is
$$[\M_{g,n}(X,A)]^{{\rm vir}}=\eta_E^{\ast}[{\mathcal M}^{\text{alg}}]\in A_{\ast}\M_{g,n}(X,A).
$$
There is an analogous way to construct the $GW$-invariants of algebraic
varieties using analytic method. We continue to use the notion developed
in section 1. Let $(R,W,\phi)$ be a smooth approximation of
$[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$ constructed in Lemma \ref{1.20}. Then we can
construct a cone current in the total space of $W$ as follows.
Let $\Gamma_{t\phi}$ be the graph of $t\phi$ in $W$ and
let $N_{0/\phi}$ be the limit current $\lim_{t\to\infty}\Gamma_{t\phi}$,
when it exists. Clearly, if such a limit does exist, then it is
contained in $W|_{\phi^{-1}(0)}$.
In general, though $\phi$ is smooth there is no guarantee
that such a limit will exist. However, if the approximation is analytic,
then we will show that such limit does exist as an
stratifiable current. Indeed, assume $(R,W,\phi)$ is an analytic smooth
approximation. Since the existence of $\lim\Gamma_{t\phi}$
is a local problem, we can
assume that there is a holomorphic basis of $W$, say $e_1,\ldots,e_r$.
Then $\phi$ can be expressed
in terms of $r$ holomorphic functions $\phi_1,\ldots,\phi_r$.
Now let ${\mathbf C}$ be the complex line with complex variable $t$,
let $w_i$ be the dual of $e_i$ and let $\Theta\subset W\times {\mathbf C}$ be the
analytic subscheme defined by the vanishing
of $tw_i-\phi_i$, $i=1,\ldots,r$.
We let $\Theta_0$ be the smallest closed analytic subscheme
of $\Theta$ that contains $\Theta\cap (W\times{\mathbf C}^{\ast})$, where ${\mathbf C}^{\ast}
={\mathbf C}-\{0\}$. By the Weierstrass preparation theorem, such $\Theta_0$ does
exist. Then we define $N_{0/\phi}$ to be the associated cycle of
the intersection of the
scheme $\Theta_0$ with $W\times\{0\}$. By \cite{Fu},
$N_{0/\phi}$ is the limit of $\Gamma_{t\phi}$.
Obviously, $N_{0/\phi}$ is stratifiable.
This shows that for any analytic approximation $(R,W,\phi)$ the
limit $\lim\Gamma_{t\phi}$ does exist.
We now state a simple lemma which implies that if $(R^{\prime},W^{\prime},
\phi^{\prime})$ is a smooth approximation that is finer than the
analytic approximation $(R,W,\phi)$, then $\lim\Gamma_{t\phi}$ exists as
well. We begin with the following situation. Let $V$ be a smooth
oriented vector bundle over a smooth oriented manifold $M$
and let $\varphi\!:\! M\to V$ be a smooth section. Let $V^{\prime}\subset
V$ be a smooth submanifold such that for any $x\in\varphi^{-1}(0)$
we have $\image(d\varphi(x))+V^{\prime}_x=V_x$.
Then $M_0=\varphi^{-1}(V^{\prime})$ is a smooth submanifold of
$M$ near $\varphi^{-1}(0)$. Let $V_0$ be the restriction of
$V^{\prime}$ to $M_0$ and let $\varphi_0\!:\! M_0\to V_0$ be
the induced section. We next let $N\subset TM|_{\varphi^{-1}(0)}$ be a
subbundle complement to $TM_0|_{\varphi^{-1}(0)}$
in $TM|_{\varphi^{-1}(0)}$.
Then the union of $d\varphi(x)(N_x)$ for all $x\in\varphi^{-1}(0)$
forms a subbundle of $V|_{\varphi^{-1}(0)}$. We denote this bundle
by $d\varphi(N)$.
Since $V|_{\varphi^{-1}(0)}\equiv V_0|_{\varphi^{-1}(0)}
\oplus d\varphi(N)$, there is a unique projection
$P\!:\! V|_{\varphi^{-1}(0)}\to V_0|_{\varphi^{-1}(0)}$ such that
whose kernel is $d\varphi(N)$ and the composite
of the inclusion $V_0|_{\varphi^{-1}(0)}\to V|_{\varphi^{-1}(0)}$
with $P$ is the identity map.
\begin{lemm}
\label{3.2}
Let the notation be as before and let $l=\dim M$ and
$l_0=\dim M_0$. Then $\lim\Gamma_{t\varphi}$
exists as an $l$-dimensional current in $V|_{\varphi^{-1}(0)}$ if and only if
$\lim\Gamma_{t\varphi_0}$ exists as an $l_0$-dimensional
oriented current in
$V_0|_{\varphi^{-1}(0)}$. Further, if they do exist then
$$\lim\Gamma_{t\varphi}=P^{\ast}(\lim\Gamma_{t\varphi_0}).
$$
Hence $\lim\Gamma_{t\phi}$ is stratifiable if $\lim\Gamma_{
t\phi_0}$ is stratifiable.
\end{lemm}
\begin{proof}
This is obvious and will be left to the readers.
\end{proof}
Now let $\{(R_{\alpha},W_{\alpha},\phi_{\alpha})\}_{\alpha\in\Xi}$ be a collection of
analytic smooth approximations of $[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$ such
that the images of $Z_{\alpha}=\phi_{\alpha}^{-1}(0)$
(in $\Phi^{-1}(0)$) covers $\Phi^{-1}(0)$.
It follows that we can choose a good atlas
$\{(R_i,W_i,\phi_i)\}_{k\in\Lambda}$
constructed in Lemma \ref{1.20} so that all
approximations in $\Lambda$ are finer than approximations in $\Xi$.
Now let $i\in\Lambda$ and let $x\in Z_i=\phi_i^{-1}(0)\subset R_i$ be any
point. Because charts in $\Xi$ cover $\Phi^{-1}(0)$, there is an
$\alpha\in\Xi$ such that the image of $R_{\alpha}$ in ${\mathbf B}$ contains
the image of $x$ in ${\mathbf B}$. Then because $(R_i,W_i,\phi_i)$ is finer than
$(R_{\alpha},W_{\alpha},\phi_{\alpha})$, by definition, there is a locally closed
submanifold $R_{i,\alpha}\subset R_i$, a local diffeomorphism $f_i^{\alpha}\!:\!
R_{i,\alpha}\to R_{\alpha}$ and a vector bundle inclusion $(f_i^{\alpha})^{\ast} W_{\alpha}
\subset W_i|_{R_{i,\alpha}}$ such that $(f_i^{\alpha})^{\ast}(\phi_{\alpha})=\phi_i$,
as in \eqref{eq:1.4}.
This is exactly the situation studied in Lemma \ref{3.2}. Hence
$\lim\Gamma_{t\phi_i}$ exists near fibers of $W$ over $x$.
Because $\{Z_{\alpha}\}$ covers $\Phi^{-1}(0)$,
$\lim\Gamma_{t\phi_i}$ exists and is a pure dimensional stratifiable
current of dimension $\dim R_i$. We denote this current by $N_{0/\phi_i}$.
Now it is clear how to construct the GW-invariants of algebraic
varieties using these analytically constructed cones.
By the property of good coverings,
for $j\leq i\in\Lambda$ the approximation $(R_i,W_i,\phi_i)$ is finer than
$(R_j,W_j,\phi_j)$.
We let $Z_i=\phi_i^{-1}(0)$ be as before and let $Z_{i,j}=
Z_i\cap R_{i,j}\subset Z_i$, where $R_{i,j}$ is defined before \eqref{eq:1.4}.
Let $\rho^j_i\!:\! Z_{i,j}\to Z_j$ be the restriction of $f^j_i$ to $Z_{i,j}$.
Note that $Z_{i,j}$ is an open subset of $Z_i$ and $\rho^j_i\!:\! Z_{i,j}\to
\rho^j_i(Z_{i,j})$ is a covering.
Let $F_i$ be the restriction of $W_i$ to $Z_i$ and let $p_i\!:\! F_i\to Z_i$
be the projection. Hence, $(\rho_i^j)^{\ast}(F_j)$ is canonically
a subvector bundle of $F_i|_{Z_{i,j}}$.
By Lemma \ref{3.2}, $(\rho^j_i)^{\ast}(F_j)$ intersects
transversally with $N_{0/\phi_i}\cap p_i^{-1}(Z_{i,j})$ and as currents,
$N_{0/\phi_i}\cap (\rho_i^j)^{\ast}(F_j)=(p_i^j)^{\ast}(N_{0/\phi_j})$.
For convenience, in the following we will call the collection
$\{F_i\}$ with transition functions $f^j_i$ a semi-${\mathbb Q}$-bundle and
denote it by ${\mathcal F}$, and will denote $\{ N_{0/\phi_i}\}$ by ${\mathcal N}^{\text{an}}$.
As in section two, we call a collection
${\mathbf s}=\{s_i\}_{i\in\Lambda}$ of smooth sections $s_i\!:\! Z_i\to F_i$
a global section of
${\mathcal F}$ if for $j\leq i\in\Lambda$ the restriction $s_i|_{Z_{i,j}}
\!:\! Z_{i,j}\to F_i|_{Z_{i,j}}$ coincides with the pull back section
$(\rho_i^j)^{\ast} s_j\!:\! Z_{i,j}\to (\rho_i^j)^{\ast} F_j$ under the canonical
inclusion $(\rho_i^j)^{\ast} F_j\subset F_i|_{Z_{i,j}}$. We say that the section
${\mathbf s}$ is transversal to ${\mathcal N}^{\text{an}}$ if for each $i\in\Lambda$,
the graph of the section $s_i$ is transversal to $N_{0/\phi_i}$ in $F_i$.
Obviously, if ${\mathbf s}$ is a global section of $F$
that is transversal to ${\mathcal N}^{\text{an}}$, then following the argument
after Lemma \ref{1.10}, currents
$$\frac{1}{m_i}\iota^{\prime}_{i\ast}\pi_{i\ast}(N_{0/\phi_i}\cap
\Gamma_{s_i}),\quad {i\in\Lambda},
$$
where $\iota^{\prime}_i\!:\! Z_i\to{\mathbf B}$ is the restriction of
$\iota_i\!:\! R_i\to{\mathbf B}$ to $Z_i\subset R_i$ and $m_i$ is the number of sheets
of the branched covering $\iota^{\prime}_i\!:\! Z_i\to\iota_i^{\prime}(Z_i)$, patch
together to form an oriented current in ${\mathbf B}$ without boundary.
We denote this current by ${\mathbf s}^{\ast}({\mathcal N}^{\text{an}})$.
It has pure dimension $r_{\rm exp }$ since the currents
$N_{0/\phi_i}$ has dimension $\dim R_i=\rank F_i+r_{\rm exp }$.
Hence it defines a homology class $[{\mathbf s}^{\ast}({\mathcal N}^{\text{an}})]$
in $H_{2r_{\rm exp }}({\mathbf B};{\mathbb Q})$.
\begin{prop}
$[{\mathbf s}^{\ast}({\mathcal N}^{\text{an}})]$ is the Euler class $e[\Phi\!:\! {\mathbf B}\to{\mathbf E}]$
constructed in section one.
\end{prop}
\begin{proof}
Recall that the class $e[\Phi\!:\!{\mathbf B}\to{\mathbf E}]$ was constructed
by first selecting a collection of perturbations $h_i(s)\!:\! R_i\to W_i$
of $\phi_i$ parameterized by $s\in [0,1]$
satisfying certain property and then form the current that is the
patch together of the currents $\frac{1}{ m_i}\iota_{k\ast}(\Gamma_{h_i(1)}
\cap\Gamma_0)$, where $\Gamma_{h_i(1)}$ and $\Gamma_0$
are the graph of $h_i(1)$ and $0\!:\! R_i\to W_i$.
Alternatively, we can perturb the $0$-section instead of
$\{\phi_i\}$ to obtain the same cycle. Namely, we let $h_i^{\prime}(s)\!:\!
R_i\to W_i$ be a collection of perturbations of the zero section
$0\!:\! R_i\to W_i$, such that it satisfies the obvious
compatibility and properness property similar to that of
$h_i(s)$ in section two.
Moreover, we require that the graph $\Gamma_{h^{\prime}_i(1)}$
is transversal to $N_{0/\phi_i}$ and transversal to
the graph $\Gamma_{t\phi_i}$ for sufficiently large $t$.
Of course such perturbations do exist following the proof of Proposition
\ref{1.12}. Let $C_t$ be the current in ${\mathbf B}$ that is the result of
patching together the currents
$\frac{1}{ m_i}\iota_{i\ast}p_{i\ast}(\Gamma_{h_i^{\prime}(1)}
\cap\Gamma_{t\phi_i})$, where $p_i$ is the projection $W_i\to R_i$.
Clearly, for $t\gg 0$, we have $\partial C_t=0$ and
$\supp(C_t)$ is compact. Hence $C_t$ defines a homology
class in $H_{2r_{\rm exp }}({\mathbf B};{\mathbb Q})$, denoted by $[C_t]$.
It follows from the uniqueness argument in the end of section two
that for sufficiently large $t$, the homology class
$[C_t]$ in
$H_{2r_{\rm exp }}({\mathbf B};{\mathbb Q})$ is exactly the Euler class.
On the other hand, we let $C_{\infty}$ be the current in
${\mathbf B}$ that is the patch together of the currents
$\frac{1}{ m_i}\iota_{i\ast}p_{i\ast}(\Gamma_{h_i^{\prime}(1)}\cap N_{0/\phi_0})$.
Because $N_{0/\phi_i}$ is the limit of
$\Gamma_{t\phi_i}$, and because $\Gamma_{h_i^{\prime}(1)}$ intersects
transversally with $\Gamma_{t\phi_i}$ for $t\gg 0$ and with $N_{0/\phi_i}$,
the union
$$\bigcup_{t\in [0,\epsilon]} \{t\}\times C_{1/t}\subset [0,\epsilon]\times{\mathbf B},
$$
where $1\gg\epsilon>0$, is a current whose boundary is $C_{1/\epsilon}- C_{\infty}$.
This implies that
$$[C_{\infty}]=[C_t]\in H_{2r_{\rm exp }}({\mathbf B};{\mathbb Q})\qquad \text{for}\ t\gg 0.
$$
Further, because the currents $N_{0/\phi_i}$
are contained in $F_i=W_i|_{Z_i}$,
$p_{i\ast}(N_{0/\phi_i}\cap
\Gamma_{h_i^{\prime}(1)})$ as current is identical to
$\pi_{i\ast}(N_{0/\phi_i}\cap \Gamma_{r_i})$, where $r_i\!:\!
Z_i\to F_i$ is the restriction of $h_i^{\prime}(1)$ to $Z_i$.
Hence $C_{\infty}={\mathbf r}^{\ast}({\mathcal N}^{\text{an}})$ with ${\mathbf r}=\{r_i\}$.
Finally, it is direct to check that the homology classes
$[{\mathbf s}^{\ast}({\mathcal N}^{\text{an}})]$ do not depend on the choices of the
sections ${\mathbf s}$ of ${\mathcal F}=\{F_i\}$ so long as they
satisfy the obvious transversality conditions.
Therefore,
$$[{\mathbf s}^{\ast}({\mathcal N}^{\text{an}})]=[{\mathbf r}^{\ast}({\mathcal N}^{\text{an}})]=
[C_{1/\epsilon}]=e[\Phi:{\mathbf B}\to{\mathbf E}].
$$
This proves the Proposition.
\end{proof}
In the end, we will compare the algebraic normal cones with the analytic
normal cones to demonstrate that the algebraic and analytic construction
of the GW-invariants give rise to the identical invariants.
Here is our strategy. Taking the good atlas
$\{(Z_i,{\mathcal X}_i)\}_{i\in\Lambda}$ of $\M_{g,n}(X,A)$ as before, we have two collections of
semi-${\mathbb Q}$-vector bundles, namely ${\mathcal E}=\{E_i\}$ and ${\mathcal F}=\{F_i\}$, and two
collections of cone currents ${\mathcal M}^{\text{alg}}=\{M_i^{\text{alg}}\}$ and
${\mathcal N}^{\text{an}}=\{N_{0/\phi_i}\}$ such that
$[\eta_E^{\ast}({\mathcal M}^{\text{alg}})]$ and $[\eta_F^{\ast}({\mathcal N}^{\text{an}})]$ are the algebraic and
the symplectic virtual moduli cycles of $\M_{g,n}(X,A)$ respectively.
Here $\eta_E$ and $\eta_F$ are generic sections of ${\mathcal E}$ and ${\mathcal F}$
respectively. To
compare these two classes, we will form a new
semi-${\mathbb Q}$-vector bundle ${\mathcal V}=\{V_i\}$, where
$V_i=E_i\oplus F_i$, and construct
a stratifiable cone current ${\mathcal P}$ in ${\mathcal V}$
such that the cycle ${\mathcal P}$ intersect ${\mathcal E}\subset{\mathcal V}$ and
${\mathcal F}\subset{\mathcal V}$ transversally and the intersection ${\mathcal P}\cap {\mathcal E}$ and ${\mathcal P}\cap
{\mathcal F}$ are ${\mathcal M}^{\text{alg}}$ and ${\mathcal N}^{\text{an}}$ respectively. Therefore, if we let
$\eta_V$ be a generic section of ${\mathcal V}$, then
$$[\eta_E^{\ast}({\mathcal M}^{\text{alg}})]=[\eta^{\ast}_V({\mathcal P})]=[\eta^{\ast}_F({\mathcal N}^{\text{an}})]
\in H_{\ast}(\M_{g,n}(X,A);{\mathbb Q}).
$$
This will prove the Comparison Theorem.
We now provide the details of this argument. We begin with any index
$i\in\Lambda$ and an open subset $S\subset Z_i$.
Let $f\!:\! C\to X$ be the restriction to $S$ of the
tautological family ${\mathcal X}_i$ of stable maps over $Z_i$, with
$D\subset C$ the divisor of its $n$-marked sections
and $\pi\!:\! C\to S$ the projection.
Note that $f$ is the restriction of a family of stable
morphisms over a scheme to an analytic open subset of the base scheme.
Following the construction in \cite[section 3]{LT1}, after fixing a
sufficiently ample line bundle over $X$, we canonically construct
a locally free sheaf of ${\mathcal O}_{C}$-modules ${\mathcal K}$ so that
$f^{\ast}\Omega_X$ is canonically a quotient sheaf of ${\mathcal K}$. Let
${\mathcal L}$ be the kernel of ${\mathcal K}\to f^{\ast}\Omega_X$. Then
the restriction to $S$ of the sheaf
${\mathcal E}_{i,1}$ (resp. ${\mathcal E}_{i,2}$)
mentioned before is the the relative extension
sheaf $\mathop{{\mathcal E} xt\hspace{1pt}}_{\pi}^1([{\mathcal K}\to\Omega_{C/S}(D)],{\mathcal O}_{C})$
(resp. $R\pi_{\ast}({\mathcal L}^{\vee})$).
We denote them by ${\mathcal E}_{S,1}$ and ${\mathcal E}_{S,2}$ respectively.
As usual, we let $E_{S,1}$ and $E_{S,2}$ be the
associated vector bundle of ${\mathcal E}_{S,1}$ and ${\mathcal E}_{S,2}$
respectively.
Following the notation in \cite{LT1}, the
tangent-obstruction complex $[{\mathcal T}_S^1\to{\mathcal T}_S^2]$ of ${\mathcal X}_i|_S$ is
$$\bigl[\sideset{}{^1_{\pi}}\mathop{{\mathcal E} xt\hspace{1pt}}([f^{\ast}\Omega_X\to\Omega_{C/S}(D)],{\mathcal O}_{C})
\xrightarrow{\times0}
\sideset{}{^2_{\pi}}\mathop{{\mathcal E} xt\hspace{1pt}}([f^{\ast}\Omega_X\to\Omega_{C/S}(D)],{\mathcal O}_{C})\bigr],
$$
and that there is a canonical homomorphism
$\epsilon: {\mathcal E}_{S,1}\lra {\mathcal E}_{S,2}$
so that the kernel and the cokernel of $\epsilon$ are
${\mathcal T}_S^1$ and ${\mathcal T}_S^2$ respectively.
The homomorphism $\epsilon$ is the middle arrow in the sequence
\eqref{eq:3.2}.
We now assume that there is an analytic approximation
$\alpha\in\Xi$ so that $(R_i,W_i,\phi_i)$ is finer than $\alpha$
and $\iota_i(S)\subset{\mathbf B}$
is contained in $\iota_{\alpha}(Z_{\alpha})$.
Let $\rho_{\alpha}\!:\! Z_i\to Z_{\alpha}$
be induced by $f_i^{\alpha}\!:\! R_{i,\alpha}\to R_{\alpha}$ (see \eqref{eq:1.4}).
Let $F_{S,\alpha}$
be the vector bundle over $Z_i$ that is the pull back of $F_{\alpha}$.
Note that $F_{S,\alpha}$ is a smooth vector bundle. Let
$G_{S,\alpha,2}= E_{S,2}\oplus F_{S,\alpha}$.
In the following, we will construct a holomorphic
vector bundle $G_{S,\alpha,1}$ and a
possibly degenerate
vector bundle homomorphism $\beta$ and non-degenerate
vector bundle inclusions $\tau_{\alpha,j}$ as shown below so that
\begin{equation}
\begin{CD}
E_{S,1} @>{\epsilon}>> E_{S,2}\\
@VV{\tau_{\alpha,1}}V @VV{\tau_{\alpha,2}}V\\
G_{S,\alpha,1} @>{\beta}>>G_{S,\alpha,2}
\end{CD}
\label{eq:3.5}
\end{equation}
is commutative.
Let $w$ be any point in $S$. We denote by $C_w$
the fiber of $C$ over $w$ and let $f_w$ (resp. ${\mathcal K}_w$, resp.
${\mathcal L}_w$) be the restriction of the respective objects to $C_w$.
As before, for any locally free
sheaf of ${\mathcal O}_{C_w}$-modules ${\mathcal W}$ that is locally free
away from the nodal points of $C_w$, we denote by ${\mathcal W}^{\cA}$ the sheaf
whose stalk at nodal points $z$ of $C_w$ are ${\mathcal W}_z$ and its stalks
at smooth points $z$ of $C_w$ are germs of smooth sections of the
associated vector bundle of ${\mathcal W}$ at $z$. We let $G_{S,\alpha,1}|_w$
be the vector space of the equivalence classes of
commutative diagrams
\begin{equation}
\begin{CD}
@. @. {\mathcal K}_w @>>> f_w^{\ast}\Omega_X\\
@.@. @VV{h}V @VV{df_w^{\vee}}V\\
0@>>> {\mathcal O}_{C_w}^{\cA} @>>> {\mathcal B}_w^{\cA} @>>> \Omega_{C_w}(D_w)^{\cA}
@>>> 0
\end{CD}
\label{eq:3.6}
\end{equation}
such that the lower exact sequences are induced by the exact sequences
of sheaves of ${\mathcal O}_{C_w}$-modules
\begin{equation*}
\begin{CD}
0 @>>> {\mathcal O}_{C_w} @>>> {\mathcal B}_w @>>> \Omega_{C_w}(D_w)
@>>> 0
\end{CD}
\end{equation*}
and that $h$ satisfies the following two requirements.
First, let
$c\!:\! {\mathcal L}_w\to {\mathcal B}_w^{\cA}$ be the composite of ${\mathcal L}_w\to{\mathcal K}_w$ with
$h$. Since ${\mathcal L}_w$ is the kernel of ${\mathcal K}_w\to f_w
^{\ast}\Omega_X$, $c$ automatically lifts to $h_{E}\!:\! {\mathcal L}_w
\to {\mathcal O}_{C_w}^{\cA}$. The first requirement is that $h_{E}$ is
holomorphic. Secondly, since both ${\mathcal K}_w$ and ${\mathcal L}_w$ are sheaves of
${\mathcal O}_{C_w}$-modules and since $h$ is analytic near nodal points of $C_w$,
$\bar\partial h$ is a $(0,1)$-form with compact support
\footnote{By which we mean that $\bar\partial h$ vanishes in
a neighborhood of the nodal points of $C_w$.}
taking values
in the associated vector bundle of ${\mathcal K}_w^{\vee}\otimes_{{\mathcal O}
_{C_w}}{\mathcal B}_w$. Because of the first requirement, it factors through
a section $h_{F}$ of $\Omega^{0,1}_{\rm cpt}(f_w^{\ast} T_X)$.
We require that $h_{ F}$ is an element in $\rho_{\alpha}^{\ast} W_{\alpha}|_w$.
Using Lemma \ref{2.1} and Corollary \ref{2.2} and
the fact that ${\mathcal K}^{\vee}$ is sufficiently ample which
was the precondition on our choice of ${\mathcal K}$,
it is direct to check that the collection
$\{G_{S,\alpha,1}|_w\mid w\in S\}$ forms a
smooth vector bundle, denoted $G_{S,\alpha,1}$, and the correspondence
that sends \eqref{eq:3.6} to $h_{E}- h_{F}$
form a possibly degenerate
vector bundle homomorphism
$\beta \!:\! G_{S,\alpha,1}\to G_{S,\alpha,2}$.
We next define the
homomorphisms $\tau_{\alpha,j}$. The
homomorphism $\tau_{\alpha,2}\!:\! E_{S,2}\to G_{S,\alpha,2}$
is the obvious homomorphism based on the definition
$G_{S,\alpha,2}=E_{S,2}\oplus F_{S,\alpha}$. For
$\tau_{\alpha,1}$, we recall that for any $w\in S$ the vector space
$E_{S,1}|_w$ is the set of equivalence classes of the diagrams
\eqref{eq:3.6} of which the $h$ are holomorphic. Namely,
$h$ are induced by homomorphisms
$f_w^{\ast}\Omega_X\to{\mathcal B}$.
Hence $E_{S,1}$ is canonically a subbundle of $G_{S,\alpha,1}$.
This shows that both
$\tau_{\alpha,1}$ and $\tau_{\alpha,2}$ are
inclusions of vector bundles.
Finally, let $\xi\in E_{S,1}|_w$ be any element associated to the diagram
\eqref{eq:3.6}, then
$\epsilon(\xi)$ is the section of ${\mathcal L}_w^{\vee}$
that is the lift of ${\mathcal L}_w\to{\mathcal K}_w\xrightarrow{h}{\mathcal B}_w$ to ${\mathcal L}_w\to{\mathcal O}_{C_w}$. It
follows that the square of \eqref{eq:3.5} is commutative.
We now show that $\coker(\tau_{\alpha,1})=
\coker(\tau_{\alpha,2})$. It suffices to show that the sequence
\begin{equation}
\begin{CD}
0 @>>> E_{S,1}@>{\tau_{\alpha,1}}>> G_{S,\alpha,1} @>{c}>> F_{S,\alpha} @>>> 0,
\end{CD}
\label{eq:3.9}
\end{equation}
where $c$ is the composite of $\beta$ with $G_{S,\alpha,2}\to F_{S,\alpha}$,
is an
exact sequence. But this follows directly from the definition of
$G_{S,\alpha,1}$ and Lemma \ref{2.1} and Corollary \ref{2.2}.
This proves that $\coker(\tau_{\alpha,1})=
\coker(\tau_{\alpha,2})$, and consequently
\begin{equation}
\coker(\beta|_w)=\coker(\epsilon|_w)={\mathcal T}^2_S|_w
\label{eq:3.31}
\end{equation}
for any $w\in S$.
In the following, we will construct the cone current $Q_{S,\alpha}\in Z_{\ast}
V_{S,\alpha,2}$. We first pick a subbundle $H_{\alpha}\subset G_{S,\alpha,2}$ such that
$H_{\alpha}\to G_{S,\alpha,2}\to\coker(\tau_{\alpha,1})$ is an isomorphism. We let
$P_{\alpha}\!:\! G_{S,\alpha,2}\to E_{S,2}$ be the projection so that
$\ker(P_{\alpha})=\beta(H_{\alpha})$ and
$P_{\alpha}\circ \tau_{\alpha,2}={\mathbf
1}_{E_{S,2}}$. We then take
$Q_{S,\alpha}$ to be the flat pull back current $P_{\alpha}^{\ast}(M_i^{\text{alg}})\in
Z_{\ast} G_{S,\alpha,2}$. It follows that $Q_{S,\alpha}$
intersects the subbundle $E_{S,2}
\subset G_{S,\alpha,2}$ transversally and the intersection
$Q_{S,\alpha}\cap E_{S,2}$ is exactly $M^{\text{alg}}_S=M_i^{\text{alg}}|_S$.
In the following, we will demonstrate that $Q_{S,\alpha}$ intersects
the subbundle $F_{S,\alpha}\subset G_{S,\alpha,2}$ transversally as well
and that the
intersection $Q_{S,\alpha}\cap F_{S,\alpha}$ is the current
$\rho_{\alpha}^{\ast} (N_{\alpha}^{\text{an}})\in Z_{\ast} F_{S,\alpha}$.
Let $w\in S$ ($\subset Z_i$) be any point. Since $T_{w^{\prime}}R_{\alpha}$, where
$w^{\prime}=\rho_{\alpha}(w)$, is the vector space
$\Ext^1({\mathcal D}^{\bullet}_w,{\mathcal O}_{C_w}^{\cA})^{\dag}$,
there is a canonical injective homomorphism
$\sigma_w \!:\! T_{w^{\prime}}R_{\alpha}\to G_{S,\alpha,1}|_w$ of vector spaces
that send the diagram \eqref{eq:2.3} to \eqref{eq:3.6} with
${\mathcal K}_w\to {\mathcal B}_w^{\cA}$ the composite of
${\mathcal K}|_w\to f_w^{\ast}\Omega_X$ with the $v_2$ in \eqref{eq:2.3}.
It is easy to see that the collection $\{\sigma_w \}_{w\in S}$
forms a smooth
non-degenerate vector bundle homomorphism
$\sigma \!:\! \rho_{\alpha}^{\ast}(TR_{\alpha})\to G_{S,\alpha,1}$. If follows
from the description of
$$\rho_{\alpha}^{\ast}(d \phi_{\alpha}):
\rho_{\alpha}^{\ast}(TR_{\alpha})\lra F_{S,\alpha}
$$
that the diagram of vector bundle homomorphisms
\begin{equation}
\begin{CD}
G_{S,\alpha,1} @>{\beta}>> G_{S,\alpha,2}\\
@AAA @AAA\\
\rho_{\alpha}^{\ast}(TR_{\alpha})
@>{\rho_{\alpha}^{\ast}(d\phi_{\alpha})}>> F_{S,\alpha}
\end{CD}
\end{equation}
is commutative,
where the second vertical arrow is the obvious
inclusion.
To compare $Q_{S,\alpha}$ with $\rho_{\alpha}^{\ast}(N_{\alpha}^{\text{an}})$, we need
the following two lemmas.
\begin{lemm}
\label{3.10}
Let $w\in S$ be any point and let $w^{\prime}=\rho_{\alpha}(w)$.
Let $d_2\!:\! G_{S,\alpha,2}|_w\to {\mathcal T}_S^2|_w$
be the homomorphism induced by \eqref{eq:3.31} and let
$F_{S,\alpha}|_w\to
{\mathcal T}_S^2|_w$ be the canonical homomorphism given in
Lemma \ref{2.22}.
Then the following squares are commutative:
\begin{equation}
\begin{CD}
F_{S,\alpha}|_w @>{\subset}>> G_{S,\alpha,2}|_w @<{\tau_{\alpha,2}}<<
E_{S,2}\\
@V{d_3}VV @V{d_2}VV @V{d_1}VV\\
{\mathcal T}_S^2|_w @= {\mathcal T}_S^2|_w @={\mathcal T}_S^2|_w
\end{CD}
\label{eq:3.40}
\end{equation}
\end{lemm}
\begin{lemm}
\label{3.11}
For any point $w\in Z_{\alpha}$, the germ of $\phi_{\alpha}\!:\! R_{\alpha}\to
W_{\alpha}$ at $w$ is a Kuranishi map of the standard obstruction theory
of the deformation of stable morphisms associated to the exact sequence
$$
0\lra {\mathcal T}_{\alpha}^1|_w\lra T_wR_{\alpha} \lra F_{\alpha}|_w\lra
{\mathcal T}_{\alpha}^2|_w\lra 0.
$$
\end{lemm}
\begin{proof}
We first prove Lemma \ref{3.10}.
Since $G_{S,\alpha,2}\equiv E_{S,2}\oplus F_{S,\alpha}$, $d_1$ and $d_3$ induces
a homomorphism $G_{S,\alpha,2}|_w\to {\mathcal T}^2_S|_w$. To prove
the lemma, it suffices to show that $d_2=d_1\oplus d_3$.
To accomplish this, we only need to show that for any
$\xi\in G_{S,\alpha,1}|_w$ with
$\xi_E$ and $-\xi_F$ its two components of
$\beta(\xi)$ according to the direct sum
decomposition $G_{S,\alpha,2}|_w=E_{S,2}|_w\oplus F_{S,\alpha}|_w$,
then
$d_1(\xi_E)=d_3(\xi_F)$.
To prove this, we first pick an $h_0\!:\! f_w^{\ast}\Omega_X\to {\mathcal B}^{\cA}_w$ such that
\begin{equation}
\begin{CD}
f_w^{\ast}\Omega_X @= f_w^{\ast}\Omega_X\\
@VV{h_0}V @VV{df_w^{\vee}}V\\\
{\mathcal B}^{\cA}_w @>>> \Omega_{C_w}(D_w)^{\cA}
\end{CD}
\end{equation}
is commutative. Let $h_0^{\prime}$ be the composite of ${\mathcal K}_w\to f_w^{\ast}\Omega_X$
with $h_0$.
Then $h^{\prime}-h_0$ factor through ${\mathcal O}_{C_w}^{\cA}\to{\mathcal B}_w^{\cA}$,
say $\tilde h\!:\! {\mathcal K}_w\to{\mathcal O}_{C_w}^{\cA}$. Clearly,
$\tilde h$ composed with ${\mathcal L}_w\to{\mathcal K}_w$ is the section
$\xi_E\in H^0({\mathcal L}_w^{\vee})$.
On the other hand, the lift of $\bar\partial \tilde h$ to
$\Omega^{0,1}_{\rm cpt}(f_w^{\ast} T_X)$
is $\xi_F-(\bar\partial h_0)^{\text{lift}}$. By the definition of
the connecting homomorphism $\delta\!:\! H^0({\mathcal L}_w^{\vee})\to
H^1(f_w^{\ast}\Omega_X^{\vee})$,
$$\delta(\xi_E)=\text{the image of}\ (\xi_F
-(\bar\partial h_0)^{\text{lift}})\ \text{in}\
H^{0,1}_{\bar\partial}(f_w^{\ast} T_X)\cong H^1(f_w^{\ast}\Omega_X^{\vee}).
$$
However, the image of $(\bar\partial h_0)^{\text{lift}}$ is contained in the image of
the connecting homomorphism
$$ \Ext^1(\Omega_{C_w}(D_w),{\mathcal O}_{C_w})\lra
\Ext^2([f_w^{\ast}\Omega_X\to 0],{\mathcal O}_{C_w})\equiv H^1(f_w^{\ast}\Omega_X^{\vee}).
$$
Hence $d_1(\xi_E)=d_3(\xi_F)$. This proves Lemma \ref{3.10}.
\end{proof}
\begin{proof}
We now prove Lemma \ref{3.11}.
Let $I\subset B$ be an ideal of an Artin ring annihilated by the
maximal ideal ${\mathfrak m}_B$ and let $\varphi\!:\!\spec B/I
\to R_{\alpha}$ be a morphism that sends the closed point of
$\spec B/I$ to $w$ and
such that $\varphi^{\ast}(\phi_{\alpha})=0$. By the description of
the tautological family ${\mathcal X}_{\alpha}$ over $R_{\alpha}$, the
pull back $\varphi^{\ast}({\mathcal X}_{\alpha})$ forms an algebraic family
of stable morphisms over
$\spec B/I$. We continue to use the open covering of the domain
${\mathcal X}_{\alpha}$ used before. Since $R_{\alpha}$ is smooth, we can extend
$\varphi$ to $\tilde\varphi\!:\!\spec B\to R_{\alpha}$. Let $C_B$
over $\spec B$ be the domain of the pull back of the domain of
${\mathcal X}_{\alpha}$ via $\tilde\varphi$ and
let $C_{B/I}$ be the domain of $C_B$ over $\spec B/I$. We let
$\{U_i\}$ (resp. $\{\tilde U_i\}$)
be the induced open covering of $C_{B/I}$ (resp. $C_B$)
and let $f_i\!:\! U_i\to X$
be the restriction to $U_i$ of the pull back of the stable maps in
${\mathcal X}_{\alpha}$. Because $\varphi^{\ast}(\phi_{\alpha})=0$, $f_i$ are
holomorphic. Hence they define a morphism $f\!:\! C_{B/I}\to X$.
Now we describe the obstruction to extending $f$ to $\spec B$. Let
$C_0$ be the closed fiber of $C_B$ and let $f_0\!:\! C_0\to X$
be the restriction of $f$. For each $i$, we pick a holomorphic
extension $\tilde f_i\!:\! \tilde U_i\to X$ of $f_i$. Then over
$\tilde U_{ij}=\tilde U_i\cap \tilde U_j$, $\tilde f_j-\tilde f_i$
is canonically an element in $\Gamma(f_0^{\ast}{\mathcal T}_X|_{U_i\cap U_j})\otimes I$,
denoted by
$f_{ij}$. Further, the collection $\{f_{ij}\}$
is a cocycle and hence defines an
element $[f_{ij}]\in H^1(f_0^{\ast}{\mathcal T}_X)\otimes I$.
The obstruction to extending $f$ to $\spec B$ is the image of
$[f_{ij}]$ in $\Ext^2(\mathcal D^{\bullet}_w,{\mathcal O}_{C_0})\otimes I$
under the homomorphism in the statement in Lemma \ref{2.22}
with $z$ replaced by $w$.
We denote the image by $\text{ob}^{\text{alg}}$.
The obstruction to extending $\varphi$ to
$\tilde\varphi\!:\!\spec B\to
R_{\alpha}$ so that $\tilde\varphi^{\ast}(\phi_{\alpha})=0$ can be constructed
as follows. Let $g_i\!:\! \tilde U_i\to X$ be the pull back of
the maps in ${\mathcal X}_{\alpha}$.
Note that $g_i$ are well defined since maps in ${\mathcal X}_{\alpha}$ depend
analytically on the base manifold $R_{\alpha}$. By the construction of
$R_{\alpha}$, for each $i>l$ the map $g_i$ is
holomorphic. For $i<l$, we have canonical biholomorphism
$\tilde U_i\cong \spec B\times (U_i\cap C_0)$.
Because $\varphi^{\ast}(\phi_{\alpha})\equiv0$, if we
let $\xi_i$ be a holomorphic variable of $U_i\cap C_0$, then
$\frac{\partial}{\partial\bar\xi_i}g_i\cdot d\bar\xi_i$,
denoted in short $\bar\partial g_i$, vanishes
over $U_i\subset \tilde U_i$. Hence $\bar\partial h$ is a section of
$\Gamma(\Omega^{0,1}_{\rm cpt}(f_0^{\ast} T_X)|_{U_i\cap C_0})\otimes I$.
Clearly they patch together to form a global section
$\gamma$ of $\Omega_{\rm cpt}^{0,1}(f_0^{\ast} T_X)\otimes I$.
The element $\gamma$ can be also defined as follows.
Let ${\tilde\varphi}^{\ast} \!:\! {\mathcal O}_{R_{\alpha}}\to B$ be the induced homomorphism
on rings. Then since the image of ${\tilde\varphi_{\alpha}}^{\ast}(\phi_{\alpha})\in
B\otimes_{{\mathcal O}_{R_{\alpha}}}\!{\mathcal O}_{R_{\alpha}}(W_{\alpha})$ in
$B/I\otimes_{{\mathcal O}_{R_{\alpha}}}\!{\mathcal O}_{R_{\alpha}}(W_{\alpha})$ vanishes,
it induces an element
$\gamma^{\prime}\in I\otimes W_{\alpha}|_w$.
By our construction of $R_{\alpha}$ and $\phi_{\alpha}$, $\gamma$
coincides with $\gamma^{\prime}$
under the inclusion $W_{\alpha}|_w\subset
\Gamma_{C_0}(\Omega_{\rm cpt}^{0,1}(f_0^{\ast} T_X))$.
Let $\text{ob}^{\text{an}}$ be the
image of $\gamma$ in the cokernel of
$d\phi_{\alpha}(w)\!:\! T_w R_{\alpha}\to W_{\alpha}|_w$. By definition,
$\text{ob}^{\text{an}}$ is the obstruction to extending
$\varphi$ to $\tilde\varphi\!:\! \spec B\to \{\phi_{\alpha}=0\}$.
To finish the proof of the lemma, we need to show that
$\text{ab}^{\text{alg}}=\text{ob}^{\text{an}}$ under the isomorphism
$$\coker\{d\phi_{\alpha}(w)\}\cong \Ext^1({\mathcal D}^{\bullet}_w,{\mathcal O}_{C_0})
$$
given in Lemma \ref{2.22}. For this, it suffices to show that the Dolbeault
cohomology class of $\gamma$, denoted $[\gamma]\in H^{0,1}_{\bar\partial}(f_0^{\ast}
T_X)\otimes I$, coincides with the C\v{e}ch cohomology class $[f_{ij}]\in
H^1(f^{\ast}_0{\mathcal T}_X)
\otimes I$ under the canonical isomorphism $H^{0,1}_{\bar\partial}(f_0^{\ast}
T_X)\cong H^1(f^{\ast}_0{\mathcal T}_X)$. But this is obvious since $\varphi_i
=\tilde f_i-g_i$ is in $\Gamma_{U_i\cap C_0}(
\Omega_{\rm cpt}^0(f_0^{\ast} T_X))\otimes I$ such that
$\varphi_j-\varphi_i=f_{ij}$
and $\bar\partial \varphi_i=-\bar\partial g_i$.
Hence, $[f_{ij}]=[\gamma]$ under the given
isomorphism. This proves the lemma.
\end{proof}
Now we come back to $Q_{S,\alpha}\in Z_{\ast} G_{S,\alpha,2}$. Let $w\in S$
be any point, let $\hat w$ be the formal completion of $S$ along $w$, let $V_w$
be the total space of ${\mathcal T}^2_S|_w$ and let $N_w^0\subset V_w\times \hat w$ be the
the cone in Theorem \ref{3.16}. We let $M_S^{\text{alg}}$,
$N_{S,\alpha}^{\text{an}}=\rho_{\alpha}^{\ast}(N^{\text{an}}_i)$ and $Q_{S,\alpha}$ be the cone
currents in $E_{S,2}$, $F_{S,\alpha}$ and $G_{S,\alpha,2}$ respectively as before.
Note that they are supported on union of closed subsets each diffeomorphic to
analytic variety. By Theorem \ref{3.15}, we have
vector bundle homomorphisms
$$e_1: E_{S,2}\times_S\hat w\lra V_w\times \hat w
\quad{\rm and}\quad
e_3: F_{S,\alpha}\times_S\hat w\lra V_w\times \hat w
$$
extending $E_{S,2}|_w\to{\mathcal T}^2_S|_w$ and $F_{S,\alpha}|_w\to{\mathcal T}^2_S|_w$ such that
$e_1^{\ast}(N_w^0)$ and $e_3^{\ast}(N_w^0)$ are the restrictions of $M_S^{\text{alg}}$
and $N_{S,2}^{\text{an}}$ to fibers over $\hat w$ in $S$ respectively. Let
$e_2\!:\! G_{S,\alpha,2}\times_S\hat w
\to V_w\times\hat w$ be induced by $P_{\alpha} \!:\! G_{S,\alpha,2}\to E_{S,2}$ and $e_1$.
Then $e_2^{\ast}(N_w^0)$ is the restriction of $Q_{S,\alpha}$ to $G_{S,\alpha,2}\times_S\hat w$.
Because the squares in \eqref{eq:3.40} are commutative,
\begin{equation*}
\begin{CD}
e_2\!:\! F_{S,\alpha}\times_S\hat w
@>{\subset}>> G_{S,\alpha,2}\times_S\hat
w @>{e_2|_{\hat w}}>> V_w\times\hat w
\end{CD}
\end{equation*}
is surjective. Hence $F_{S,\alpha}\times_S\hat w$
intersects $Q_{S,\alpha}$ transversally along fiber over $w$.
Let $e_3^{\prime}\!:\! F_{S,\alpha}\times_S\hat w\to V_w\times\hat w$ be induced by
$F_{S,\alpha}\to G_{S,\alpha,2}$ and $e_2$, then the intersection of $Q_{S,\alpha}$ with
$F_{S,\alpha}\times_S\hat w$ is $(e_3^{\prime})^{\ast}(N_w^0)$.
However, by the choice of $P_{\alpha}$, we have
$e_3^{\prime}\equiv e_3|_w$, therefore the support
of $Q_{S,\alpha}\cap F_{S,\alpha}|_w$ is identical to the support of $N_{S,\alpha}^{\text{an}}|_w$.
Because $w\in S$ is arbitrary, the support of $Q_{S,\alpha}\cap F_{S,\alpha}$ is
identical to the support of $N_{S,\alpha}^{\text{an}}$.
Further, for the same reason, for general point $p$ in $N_{S,\alpha}^{\text{an}}$
the multiplicity of $N_{S,\alpha}^{\text{an}}$ at $p$ is identical to the
multiplicity of the corresponding point in $Q_{S,\alpha}\cap F_{S,\alpha}$.
This proves that
the cycles (or currents) $Q_{S,\alpha}$ intersect $F_{S,\alpha}\subset G_{S,\alpha,2}$ transversally
and $Q_{S,\alpha}\cap F_{S,\alpha}=N_{S,\alpha}^{\text{an}}$. We remark that for the same reason, the
current $Q_{S,\alpha}$ is independent of the choice of the subbundles $H_{\alpha}\subset
G_{S,\alpha,2}$.
We now let $F_S=F_i|_S$ and let $G_{S,2}=E_{S,2}\oplus F_S$. Note that
$G_{S,\alpha,2}\subset G_{S,2}$.
Because $R_i$ is finer than $R_{\alpha}$, $\rho_{\alpha}^{\ast} TR_{\alpha}$ is
a subbundle of $TR_i|_S$. Let $K_{\alpha}\subset TR_i|_S$ be a complement of
$\rho_{\alpha}^{\ast} TR_{\alpha}\subset TR_i|_S$ and let $d\phi_i(K_{\alpha})\subset F_S$
be the image of this subbundle. Let $P_{S,\alpha}\!:\! F_S\to F_{S,\alpha}$ be the
projection so that $\ker P_{S,\alpha}=d\phi_i(K_{\alpha})$ and
the composite of $F_{S,\alpha}\subset F_S$ with $P_{S,\alpha}$ is
${\mathbf 1}_{F_{S,\alpha}}$. By Lemma \ref{3.2},
$N_i^{\text{an}}|_S=P_{S,\alpha}^{\ast}(N_{S,\alpha}^{\text{an}})$.
Now let $P_S$ be the projection
$$P_S= P_{\alpha}\circ ({\mathbf 1}_{E_{S,2}}\oplus P_{S,\alpha}):
G_{S,2}\lra G_{S,\alpha,2}\lra E_{S,2}
$$
and let $Q_S=P_S^{\ast}(M_i^{\text{alg}})$ be the pull back cone.
Let $\tilde d_3$ be
\begin{equation*}
\begin{CD}
\tilde d_3: F_S|_w @>{P_{S,\alpha}|_w}>> F_{S,\alpha}|_w @>{d_3}>> {\mathcal T}_S^2|_w,
\end{CD}
\end{equation*}
then clearly we have a commutative diagram of vector spaces
\begin{equation}
\begin{CD}
F_S|_w @>>> G_{S,2}|_w @<<< E_{S,2}|_w\\
@VV{\tilde d_3}V @VV{P_S|_w}V @VV{d_1}V\\
{\mathcal T}^2_S|_w @= {\mathcal T}^2_S|_w @= {\mathcal T}^2_S|_w.
\end{CD}
\end{equation}
Because $w$ is arbitrary, similar to the previous case, we have that $F_S$
intersects $Q_S$ transversally and $F_S\cap Q_S=N_i^{\text{an}}|_S$, as
stratifiable currents.
To enable us to patch $Q_S$, where $S\subset Z_i$, to form a current in $G_{i,2}=
E_{i,2}\oplus F_i$, we need to show that $Q_S$ is independent of the
choice of analytic chart $\alpha$. Namely if we let $\beta\in\Xi$ be another
analytic chart so that $\iota_i(S)\subset\iota_{\beta}(Z_{\beta})$, then the cone current
$Q_S^{\prime}\subset G_{S,2}$ constructed using $F_{\beta}$, etc., is identical to $Q_S$.
Again, following the same argument before, it suffices to show that
the homomorphism $\tilde d_3\!:\! F_S|_w\to{\mathcal T}^2_S|_w$ does
not depend on the choice of $\alpha$. Note that $\tilde d_3$ also fits into
the commutative diagram of exact sequences
\begin{equation}
\begin{CD}
T_{\rho_{\alpha}(w)}R_{\alpha} @>{d\phi_{\alpha}(\rho_{\alpha}(w))}>> F_{\alpha}|_{\rho_{\alpha}(w)}
@>>> {\mathcal T}_{\alpha}^2|_{\rho_{\alpha}(w)}@>>> 0\\
@VVV @VVV @| \\
T_w R_i @>{d\phi_i(w)}>> F_S|_w @>>> {\mathcal T}_S^2|_w @>>> 0.
\end{CD}
\label{eq:3.33}
\end{equation}
Now assume $\beta\in\Xi$ as before. Without loss of generality,
we can assume that
near $w$, the vector subbundles $\rho_{\alpha}^{\ast} F_{\alpha}$ and
$\rho_{\beta}^{\ast} F_{\beta}$ span
a $2l$-dimensional subvector bundle of $F_i$.
Now let $V_{\alpha}\to \tilde U_{\alpha}$ and $V_{\beta}\to\tilde U_{\beta}$
be the vector bundles that define
$R_{\alpha}$ and $R_{\beta}$ as in section 2 and let $V_{\alpha\beta}\to \tilde U_i$ be the
direct sum of the pull back of $V_{\alpha}$ and $V_{\beta}$ via the tautological map
$\tilde U_i\to \tilde U_{\alpha}$ and $\tilde U_i\to\tilde U_{\beta}$. Then near
a neighborhood of $w\in\tilde U_i$, the set
$\tilde\Phi^{-1}(V_{\alpha\beta})$ will form a base
of a smooth approximation containing $w$. We denote
$R_{\alpha\beta}=\tilde\Phi_i^{-1}(V_{\alpha\beta})$ and let
$\phi_{\alpha\beta}\!:\! R_{\alpha\beta}\to V_{\alpha\beta}|_{R_{\alpha\beta}}$ be the lift of
$\tilde\Phi_i$.
Clearly, $R_i$ is still finer than $R_{\alpha\beta}$. Hence we have commutative
diagrams
\begin{equation}
\begin{CD}
T_{\rho_{\alpha}(w)}R_{\alpha} @>{d\phi_{\alpha}(\rho_{\alpha}(w))}>> V_{\alpha}|_{w}
@>>> {\mathcal T}^2_{\alpha}|_{\rho_{\alpha}(w)} @>>> 0\\
@VVV @VVV @|\\
T_wR_{\alpha\beta} @>{d\phi_{\alpha\beta}(w)}>> V_{\alpha\beta}|_w @>>> {\mathcal T}^2_i|_{w}
@>>> 0\\
@VVV@VVV@|\\
T_w R_i @>{d\phi_i(w)}>> F_i|_w @>>> {\mathcal T}^2_i|_w @>>> 0\\
\end{CD}
\label{eq:3.34}
\end{equation}
with exact rows.
Note that $V_{\alpha\beta}|_w\to {\mathcal T}^2_i|_w$ is equal to
$$V_{\alpha}|_{\rho_{\alpha}(w)}\oplus V_{\beta}|_{\rho_{\beta}(w)}\lra
\Gamma(\Omega_{\rm cpt}^{0,1}(f_w^{\ast} T_X))\lra H^{0,1}_{\bar\partial}(f_w^{\ast} T_X)
\lra {\mathcal T}^2_i|_w.
$$
(Here that $V_{\alpha\beta}|_w\to {\mathcal T}_i^2|_w$ is defined apriori but
not $F_i|_w\to{\mathcal T}^2_i|_w$ because elements of $V_{\alpha}|_w$
and $V_{\beta}|_w$ are $(0,1)$-forms with compact support.)
Therefore, the homomorphism $\tilde d_3$ defined earlier is
independent of the choice of $\alpha$.
Now we are ready to prove the theorem. Let $i\in\Lambda$ be any
approximation and let $\{S_a\}$ be an open covering
of $Z_i$ so that to each $a$ there is an
$\alpha_a\in\Xi$ so that
$\iota_i(S_a)\subset\iota_{\alpha_a}(Z_{\alpha_a})$. We let
$G_{i,2}=E_{i,2}\oplus F_i$ and let
$Q_{S_a}$ be the cone in
$G_{i,2}|_{S_a}$ constructed before
using the analytic chart $\alpha$. We know that over
$G_{i,2}|_{S_a\cap
S_b}$, the currents $Q_{S_a}$ and $Q_{S_b}$ coincide.
Hence $\{Q_{S_a}\}$ patchs
together to form a stratifiable current, denoted $Q_i$. Assume that
$j<i\in\Lambda$ be any two indices. Let $Z_{i,j}\subset Z_i$ be the open
subset $\iota_i^{-1}(\iota_j(Z_j))$ and let $f^j_i\!:\! Z_{i,j}\to Z_j$
be the map induced by $Z_i$ being finer than $Z_j$. Then
$(f_i^j)^{\ast}(F_j)$ is canonically a subbundle of $F_i|_{Z_{i,j}}$,
and $(f_i^j)^{\ast}(E_{j,2})$ is
canonically isomorphic to $E_{i,2}|_{Z_{i,j}}$. Let
$(f_i^j)^{\ast}(G_{j,2})\to
G_{i,2}|_{Z_{i,j}}$ be the induced homomorphism. It follows from the
previous argument that $Q_i$
intersects $(f_i^j)^{\ast}(G_{j,2})$ transversally and the intersection
$Q_i\cap(f_i^j)^{\ast}(G_{j,2})$ is $(f_i^j)^{\ast}(Q_j)$. Finally, by our
construction, $Q_i$ intersects transversally with $E_{i,2}$ and
$F_i\subset G_{i,2}$, and $E_{i,2}\cap Q_i=M_i^{\text{alg}}$ and $F_i\cap
G_i=N_i^{\text{an}}$. Let ${\mathcal G}$ be the semi-${\mathbb Q}$-vector bundle $\{G_{i,2}\}$,
which is ${\mathcal E}\oplus{\mathcal F}$, and let ${\mathcal Q}$ be the cone $\{Q_i\}$.
It follows from the
perturbation argument in section two that for generic sections
$\eta_E$, $\eta_F$ and $\eta_G$ of ${\mathcal E}$, ${\mathcal F}$ and ${\mathcal G}$
respectively, we have
$$[\M_{g,n}(X,A)]^{{\rm vir}}=[\eta_E^{\ast}{\mathcal M}^{\text{alg}}]=[\eta_G^{\ast}{\mathcal Q}]
=[\eta_F^{\ast}{\mathcal N}^{\text{an}}]=e[\Phi\!:\!{\mathbf B}\to{\mathbf E}].
$$
This proves the comparison theorem.
|
1997-12-19T04:51:34 | 9712 | alg-geom/9712021 | en | https://arxiv.org/abs/alg-geom/9712021 | [
"alg-geom",
"math.AG"
] | alg-geom/9712021 | Alexander Polishchuk | Alexander Polishchuk | Analogue of Weil representation for abelian schemes | 39 pages, AMSLatex | null | null | null | null | In this paper we construct a projective action of certain arithmetic group on
the derived category of coherent sheaves on an abelian scheme $A$, which is
analogous to Weil representation of the symplectic group. More precisely, the
arithmetic group in question is a congruence subgroup in the group of
"symplectic" automorphisms of $A\times\hat{A}$ where $\hat{A}$ is the dual
abelian scheme. The "projectivity" of this action refers to shifts in the
derived category and tensorings with line bundles pulled from the base. In
particular, if $A$ is an abelian scheme over $S$ equipped with an ample line
bundle $L$ of degree 1 then we construct an action of a central extension of
$Sp_{2n}(\Bbb Z)$ by $\Bbb Z\times Pic(S)$ on the derived category of coherent
sheaves on $A^n$ (the $n$-th fibered power of $A$ over $S$). We describe the
corresponding central extension explicitly using the the canonical torsion line
bundle on $S$ associated with $L$. As a main technical result we prove the
existence of a representation of rank $d$ for a symmetric finite Heisenberg
group scheme of odd order $d^2$.
| [
{
"version": "v1",
"created": "Fri, 19 Dec 1997 03:51:33 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Polishchuk",
"Alexander",
""
]
] | alg-geom | \section{Heisenberg group schemes}
Let $K$ be a finite flat group scheme over a base scheme $S$.
A {\it finite Heisenberg group scheme} is a central
extension of group schemes
\begin{equation}\label{ext}
0\rightarrow{\Bbb G}_m\rightarrow G\stackrel{p}{\rightarrow} K\rightarrow 0
\end{equation}
such that the corresponding commutator form $e:K\times
K\rightarrow{\Bbb G}_m$ is a perfect pairing.
Let $A$ be an abelian scheme over $S$, $L$ be a line bundle
on $A$ trivialized along the zero section. Then the group
scheme $K(L)=\{x\in A\ |\ t_x^*L\simeq L\}$ has a canonical
central extension $G(L)$ by ${\Bbb G}_m$ (see \cite{Mum}).
When $K(L)$ is finite, $G(L)$ is a finite Heisenberg
group scheme.
A {\it symmetric} Heisenberg group scheme is an
extension $0\rightarrow{\Bbb G}_m\rightarrow G\rightarrow K\ra0$ as above together
with an isomorphism of central extensions $G\widetilde{\rightarrow} [-1]^*G$
(identical on ${\Bbb G}_m$), where $[-1]^*G$ is the pull-back of
$G$ with respect to the inversion morphism $[-1]:K\rightarrow K$.
For example, if $L$ is a symmetric line bundle on an
abelian scheme $A$ (i.e. $[-1]^*L\simeq L$) with a
symmetric trivialization along the zero section
then $G(L)$ is a symmetric Heisenberg group scheme.
For any integer $n$ we denote by
$G^n$ the push-forward of $G$ with respect to the morphism
$[n]:{\Bbb G}_m\rightarrow{\Bbb G}_m$. For any pair of central extensions
$(G_1,G_2)$ of the same group $K$ we denote by $G_1\otimes G_2$
their sum (given by the sum of the corresponding
${\Bbb G}_m$-torsors). Thus, $G^n\simeq G^{\otimes n}$.
Note that we have a canonical isomorphism of central
extensions
\begin{equation}\label{inver}
G^{-1}\simeq [-1]^*G^{op}
\end{equation}
where $[-1]^*G^{op}$ is the
pull-back of the opposite group to $G$ by the inversion
morphism $[-1]:K\rightarrow K$. In particular, a symmetric extension
$G$ is commutative if and only if $G^2$ is trivial.
\begin{lem}\label{mult}
For any integer $n$ there is a canonical
isomorphism of central extensions
$$[n]^*G\simeq G^{\frac{n(n+1)}{2}}\otimes [-1]^*G^{\frac{n(n-1)}{2}}$$
where $[n]^*G$ is the pull-back of $G$ with respect to the
multiplication by $n$ morphism $[n]:K\rightarrow K$. In particular,
if $G$ is symmetric then $[n]^*G\simeq G^{n^2}$.
\end{lem}
\noindent {\it Proof} . The structure of the central extension $G$ of $K$ by
${\Bbb G}_m$ is equivalent to the following data (see e.g.
\cite{Breen}): a cube structure on ${\Bbb G}_m$-torsor $G$ over
$K$ and a trivialization of the corresponding biextension
$\Lambda(G)=(p_1+p_2)^*G\otimes p_1^*G^{-1}\otimes p_2^*G^{-1}$ of $K^2$.
Now for any cube structure there is a canonical isomorphism
(see \cite{Breen})
$$[n]^*G\simeq G^{\frac{n(n+1)}{2}}\otimes [-1]^*G^{\frac{n(n-1)}{2}}$$
which is compatible with the natural isomorphism of biextensions
$$([n]\times [n])^*\Lambda(G)\simeq\Lambda(G)^{n^2}\simeq
\Lambda(G)^{\frac{n(n+1)}{2}}\otimes
([-1]\times [-1])^*\Lambda(G)^{\frac{n(n-1)}{2}}.$$
The latter isomorphism is compatible with the trivializations
of both sides when $G$ arises from a central extension.
\qed\vspace{3mm}
\begin{rem} Locally one can choose a splitting $K\rightarrow G$ so
that the central extension is given by a 2-cocycle
$f:K\times K\rightarrow{\Bbb G}_m$. The previous lemma says that for any
2-cocycle $f$ the functions $f(nk,nk')$ and
$f(k,k')^{\frac{n(n+1)}{2}}f(-k,-k')^{\frac{n(n-1)}{2}}$
differ by a canonical coboundary. In fact this coboundary can
be written explicitly in terms of the functions $f(mk,k)$ for various
$m\in{\Bbb Z}$.
\end{rem}
\begin{prop}\label{order}
Assume that $K$ is annihilated by an integer $N$.
If $N$ is odd then for any Heisenberg group $G\rightarrow K$
the central extension $G^N$ is canonically trivial,
otherwise $G^{2N}$ is trivial.
If $G$ is symmetric and $N$ is odd
then $G^N$ (resp. $G^{2N}$ if $N$ is even)
is trivial as a symmetric extension.
\end{prop}
\noindent {\it Proof} . Combining the previous lemma with (\ref{inver})
we get the following isomorphism:
$$[n]^*G\simeq
G^{\frac{n(n+1)}{2}}\otimes (G^{op})^{-\frac{n(n-1)}{2}}
\simeq G^n\otimes (G\otimes G^{op -1})^{\frac{n(n-1)}{2}}.$$
Now we remark that $G\otimes G^{op -1}$ is given by a
trivial ${\Bbb G}_m$-torsor over $K$ with the group law induced
by the commutator form $e:K\times K\rightarrow{\Bbb G}_m$ considered
as 2-cocycle. It remains to note that
$e^{\frac{n(n-1)}{2}}=1$ for $n=2N$ (resp. for $n=N$ if
$N$ is odd). Hence, the triviality of $G^n$ in these cases.
\qed\vspace{3mm}
\begin{cor} Let $G\rightarrow K$ be a symmetric Heisenberg group such
that the order of $K$ over $S$ is odd. Then the
${\Bbb G}_m$-torsor over $K$ underlying $G$ is trivial.
\end{cor}
\noindent {\it Proof} . The isomorphism (\ref{inver}) implies that the
${\Bbb G}_m$-torsor over $K$ underlying $G^2$ is trivial. Together
with the previous proposition this gives the result.
\qed\vspace{3mm}
If $G\rightarrow K$ is a (symmetric) Heisenberg group scheme, such
that $K$ is annihilated by an integer $N$,
$n$ is an integer prime to $N$ then $G^n$
is also a (symmetric) Heisenberg group.
When $N$ is odd this group depends only on the residue of $n$ modulo $N$
(due to the triviality of $G^N$).
We call a flat subgroup scheme $I\subset K$ $G$-{\it isotropic} if
the central extension (\ref{ext}) splits over $I$
(in particular, $e|_{I\times I}=1$).
If $\sigma:I\rightarrow G$ is the corresponding lifting, then
we have the reduced Heisenberg group scheme
$$0\rightarrow{\Bbb G}_m\rightarrow p^{-1}(I^\perp)/\sigma(I)\rightarrow I^\perp/I\rightarrow 0$$
where $I^\perp\subset K$ is the orthogonal complement to $I$
with respect to $e$.
If $G$ is a symmetric Heisneberg group, then $I\subset K$ is
called {\it symmetrically} $G$-isotropic if the restriction
of the central extension (\ref{ext}) to $I$ can be
trivialized as a symmetric extension. If $\sigma:I\rightarrow G$ is the
corresponding symmetric lifting them the reduced
Heisenberg group $p^{-1}(I^\perp)/\sigma(I)$ is also symmetric.
Let us define the Witt group $\operatorname{WH}_{\operatorname{sym}}(S)$
as the group of isomorphism classes of finite symmetric
Heisenberg groups over $S$ modulo the equivalence relation
generated by $[G]\sim [p^{-1}(I^\perp)/\sigma(I)]$ for a
symmetrically $G$-isotropic subgroup scheme $I\subset K$.
The (commutative) addition in
$\operatorname{WH}_{\operatorname{sym}}(S)$ is defined as follows: if $G_i\rightarrow K_i$ ($i=1,2$) are
Heisenberg groups with commutator forms $e_i$
then their sum is the central extension
$$0\rightarrow {\Bbb G}_m\rightarrow G_1\times_{{\Bbb G}_m} G_2\rightarrow K_1\times K_2\rightarrow 0$$
so that the corresponding commutator form on $K_1\times K_2$
is $e_1\oplus e_2$. The neutral element is the class of ${\Bbb G}_m$
considered as an extension of the trivial group.
The inverse element to $[G]$ is $[G^{-1}]$.
Indeed, there is a canonical splitting of $G\times_{{\Bbb G}_m} G^{-1}\rightarrow
K\times K$ over the diagonal $K\subset K\times K$, hence
the triviality of $[G]+[G^{-1}]$.
We define the order of a finite Heisenberg group scheme $G\rightarrow K$ over $S$
to be the order of $K$ over $S$ (specializing to a geometric
point of $S$ one can see easily that this number has form
$d^2$).
Let us denote by $\operatorname{WH}'_{\operatorname{sym}}(S)$ the analogous Witt group of
finite Heisenberg group schemes $G$ over $S$ of odd order.
Let also $\operatorname{WH}(S)$ and $\operatorname{WH}'(S)$ be the analogous groups
defined for all (not necessarily symmetric) finite
Heisenberg groups over $S$ (with equivalence relation given
by $G$-isotropic subgroups).
\begin{rem} Let us denote by $\operatorname{W}(S)$ the Witt group of finite
flat group schemes over $S$ with non-degenerate symplectic
${\Bbb G}_m$-valued forms (modulo the equivalence relation given
by global isotropic flat subgroup schemes).
Let also $\operatorname{W}'(S)$ be the analogous group for group schemes of
odd order. Then we have a natural homomorphism
$\operatorname{WH}(S)\rightarrow \operatorname{W}(S)$ and one
can show that the induced map $\operatorname{WH}'_{\operatorname{sym}}\rightarrow \operatorname{W}'(S)$ is an
isomorphism. This follows essentially from the fact
that a finite symmetric Heisenberg group of odd order is determined
up to an isomorphism by the corresponding commutator form,
also if $G\rightarrow K$ is a symmetric finite Heisenberg
group with the commutator form $e$,
$I\subset K$ is an isotropic flat subgroup scheme of odd order,
then there is a unique symmetric lifting $I\rightarrow G$.
\end{rem}
\begin{thm}\label{annih}
The group $\operatorname{WH}_{\operatorname{sym}}(S)$ (resp. $\operatorname{WH}'_{\operatorname{sym}}(S)$) is
annihilated by $8$ (resp. $4$).
\end{thm}
\noindent {\it Proof} . Let $G\rightarrow K$ be a symmetric finite Heisenberg group.
Assume first that the order $N$ of $G$ is odd. Then
we can find integers $m$ and $n$ such that
$m^2+n^2\equiv -1\mod(N)$. Let $\a$ be an automorphism of
$K\times K$ given by a matrix
$\left( \matrix m & -n\\ n & m \endmatrix \right)$. Let
$G_1=G\times_{{\Bbb G}_m} G$ be a Heisenberg extension of $K\times
K$ representing the class $2[G]\in \operatorname{WH}'_{\operatorname{sym}}(S)$. Then
from Lemma \ref{mult} and Proposition \ref{order} we get
$\a^*G_1\simeq G_1^{-1}$, hence $2[G]=-2[G]$, i.e. $4[G]=0$
in $\operatorname{WH}'(S)$.
If $N$ is even we can apply the similar argument to the
4-th cartesian power of $G$ and the automorphism of $K^4$
given by an integer $4\times 4$-matrix $Z$ such that
$Z^t Z=(2N-1)\operatorname{id}$. Such a matrix can be found by considering the
left multiplication by a quaternion $a+bi+cj+dk$ where
$a^2+b^2+c^2+d^2=2N-1$.
\qed\vspace{3mm}
\section{Schr\"odinger representations}\label{Schr}
Let $G$ be a finite Heisenberg group scheme of order $d^2$
over $S$. A representation of $G$ of weight 1 is a locally
free $\O_S$-module together with the action of $G$ such that
${\Bbb G}_m\subset G$ acts by the identity character.
We refer to chapter V of \cite{MB} for basic facts about such
representations. In this section we study the problem of
existence of a {\it Schr\"odinger representation} for $G$,
i.~e. a weight-1 representation of $G$ of
rank $d$ (the minimal possible rank).
It is well known that such a representation exists if $S$ is the
spectrum of an algebraically closed field (see e.g.
\cite{MB}, V, 2.5.5). Another example is the following.
As we already mentioned one can associate a
finite Heisenberg group scheme $G(L)$
(called the Mumford group) to a line bundle $L$ on an
abelian scheme $\pi:A\rightarrow S$ such that $K(L)$ is finite.
Assume that the base scheme $S$ is connected.
Then $R^i\pi_*(L)=0$ for $i\neq i(L)$ for some integer $i(L)$
(called the {\it index} of $L$) and $R^{i(L)}\pi_*(L)$ is a
Schr\"odinger representation for $G(L)$ (this follows from
\cite{Mum} III, 16 and \cite{Muk2}, prop.1.7).
In general, L. Moret-Bailly showed in \cite{MB} that a
Schr\"odinger representation
exists after some smooth base change. The main result of this
section is that for symmetric Heisenberg group schemes of odd
order a Schr\"odinger representation always exists.
Let $G$ be a symmetric finite Heisenberg group scheme of
order $d^2$ over
$S$. Then locally (in {\it fppf} topology) we can choose a
Schr\"odinger representation $V$ of $G$.
According to Theorem V, 2.4.2 of \cite{MB}
for any weight-1 representation $W$ of $G$ there is
a canonical isomorphism $V\otimes\underline{\operatorname{Hom}}_G(V,W)\widetilde{\rightarrow}
W$. In particular, locally $V$ is unique up to an isomorphism and
$\underline{\operatorname{Hom}}_G(V,V)\simeq\O$. Choose an open covering
$U_i$ such that there exist Schr\"odinger representations
$V_i$ for $G$ over $U_i$. For a sufficently fine covering
we have $G$-isomorphisms $\phi_{ij}:V_i\rightarrow V_j$ on
the intersections $U_i\cap U_j$, and
$\phi_{jk}\phi_{ij}=\a_{ijk}\phi_{ik}$ on the triple
intersections $U_i\cap U_j\cap U_k$ for some functions
$\a_{ijk}\in\O^*(U_i\cap U_j\cap U_k)$. Then $(\a_{ijk})$ is a
Cech 2-cocycle with values in ${\Bbb G}_m$ whose cohomology class
$e(G)\in H^2(S,{\Bbb G}_m)$ doesn't depend on the choices made.
Furthermore, by definition $e(G)$ is trivial if and only if
there exists a global weight-1 representation we are looking
for.
Using the language of gerbs (see e.g. \cite{Gir})
we can rephrase the construction above without fixing an open
covering. Namely, to each finite Heisenberg group $G$ we can
associate the ${\Bbb G}_m$-gerb $\operatorname{Schr}_G$ on $S$ such that
$\operatorname{Schr}_G(U)$ for an open set $U\subset S$ is the category of
Schr\"odinger representations for $G$ over $U$. Then $\operatorname{Schr}_G$
represents the cohomology class $e(G)\in H^2(S,{\Bbb G}_m)$.
Notice that the class $e(G)$ is actually represented by
an Azumaya algebra $\AA(G)$ which is defined as follows.
Locally, we can choose a Schr\'odinger representation $V$
for $G$ and put $\AA(G)=\underline{\operatorname{End}}(V)$. Now for two
such representations $V$ and $V'$ there is a canonical
isomorphism of algebras
$\underline{\operatorname{End}}(V)\simeq\underline{\operatorname{End}}(V')$ induced
by any $G$-isomorphism $f:V\rightarrow V'$ (since any other
$G$-isomorphism differs from $f$ by a scalar), hence these
local algebras glue together into a global Azumaya algebra
$\AA(G)$ of rank $d^2$. In particular, $d\cdot e(G)=0$ (see e.g.
\cite{Groth1}, prop. 1.4).
Now let $W$ be a {\it global}
weight-1 representation of $G$ which is locally free of rank
$l\cdot d$ over $S$. Then we claim that $\underline{\operatorname{End}}_G(W)$
is an Azumaya algebra with the class $-e(G)$. Indeed,
locally we can choose a representation $V$ of rank $d$ as
above and a $G$-isomorphism $W\simeq V^l$ which induces a
local isomorphism $\underline{\operatorname{End}}_G(W)\simeq\operatorname{Mat}_l(\O)$.
Now we claim that there is a global algebra isomorphism
$$\AA(G)\otimes\underline{\operatorname{End}}_G(W)\simeq\underline{\operatorname{End}}(W).$$
Indeed, we have
canonical isomorphism of $G$-modules of weight 1
(resp. $-1$)
$V\otimes\underline{\operatorname{Hom}}_G(V,W)\widetilde{\rightarrow}W$ (resp.
$V^*\otimes\underline{\operatorname{Hom}}_G(V^*,W^*)\widetilde{\rightarrow}W^*$). Hence,
we have a sequence of natural morphisms
\begin{align*}
&\underline{\operatorname{End}}(W)\simeq W^*\otimes W\simeq
V^*\otimes V\otimes\underline{\operatorname{Hom}}_G(V^*,W^*)\otimes\underline{\operatorname{Hom}}_G(V,W)\rightarrow\\
&\rightarrow\underline{\operatorname{End}}(V)\otimes\operatorname{Hom}_{G\times G}(V^*\otimes V,W^*\otimes W)\rightarrow
\underline{\operatorname{End}}(V)\otimes\underline{\operatorname{End}}_G(W)
\end{align*}
--- the latter map is obtained by taking the image of the
identity section $\operatorname{id}\in V^*\otimes V$ under a $G\times
G$-morphism $V^*\otimes V\rightarrow W^*\otimes W$. It is easy to see that
the composition morphism gives the required isomorphism.
This leads to the following statement.
\begin{prop}\label{obst}
For any finite Heisenberg group scheme $G$ over $S$
a canonical element $e(G)\in\operatorname{Br}(S)$ is defined such that
$e(G)$ is trivial if and only if a Schr\"odinger
representation for $G$ exists. Furthermore, $d\cdot e(G)=0$
where the order of $G$ is $d^2$, and if there exists
a weight-1 $G$ representation which is locally free of rank
$l\cdot d$ over $S$ then $l\cdot e(G)=0$.
\end{prop}
\begin{prop}\label{hom}
The map $[G]\mapsto e(G)$ defines a homomorphism
$\operatorname{WH}(S)\rightarrow\operatorname{Br}(S)$.
\end{prop}
\noindent {\it Proof} . First we have to check that if $I\subset K$ is a
$G$-isotropic subgroup, $\widetilde{I}\subset G$ its lifting, and
$\overline{G}=p^{-1}(I^{\perp})/\widetilde{I}$ then $e(\overline{G})=e(G)$.
Indeed, there is a canonical equivalence of ${\Bbb G}_m$-gerbs
$\operatorname{Schr}_G\rightarrow\operatorname{Schr}_{\overline{G}}$ given by the functor
$V\mapsto V^{\widetilde{I}}$ where $V$ is a (local) Schr\"odinger
representation of $G$. Next if $G=G_1\times_{{\Bbb G}_m} G_2$,
then for every pair $(V_1,V_2)$ of weight-1
representations of $G_1$ and $G_2$ there is a natural
structure of weight-1 $G$-representation on $V_1\otimes V_2$,
hence we get an equivalence of ${\Bbb G}_m$-gerbs
$\operatorname{Schr}_{G_1}+\operatorname{Schr}_{G_2}\rightarrow\operatorname{Schr}_G$ which implies the equality
$e(G)=e(G_1)+e(G_2)$. At last, the map
$V\rightarrow V^*$ induces an equivalence
$\operatorname{Schr}_{G}^{op}\rightarrow\operatorname{Schr}_{G^{-1}}$ so that $e(G^{-1})=-e(G)$.
\qed\vspace{3mm}
\begin{thm}\label{odd}
Let $G$ be a symmetric finite Heisenberg group scheme
of odd order. Then $e(G)=0$, that is there exists a global
Schr\"odinger representation for $G$.
\end{thm}
\noindent {\it Proof} . Let $[G]\in\operatorname{WH}'_{\operatorname{sym}}(S)$ be a class of $G$ in the Witt group.
Then $4[G]=0$ by Theorem \ref{annih}, hence $4e(G)=0$ by
Proposition \ref{hom}. On the other hand, $d\cdot e(G)=0$ by
Proposition \ref{obst} where $d$ is odd, therefore, $e(G)=0$.
\qed\vspace{3mm}
Let us give an example of a symmetric finite Heisenberg group
scheme of {\it even} order without a Schr\"odinger
representation. First let us recall the construction from
\cite{sympl} which associates to a group
scheme $G$ over $S$ which is a central extension of a finite commutative
group scheme $K$ by ${\Bbb G}_m$, and a $K$-torsor $E$ over $S$ a
class $e(G,E)\in H^2(S,{\Bbb G}_m)$. Morally, the map
$$H^1(S,K)\rightarrow H^2(E,{\Bbb G}_m): E\mapsto e(G,E)$$
is the boundary homomorphism corresponding
to the exact sequence
$$0\rightarrow {\Bbb G}_m\rightarrow G\rightarrow K\rightarrow 0.$$
To define it consider the category ${\cal C}$ of liftings of $E$ to
to a $G$-torsor. Locally such a lifting always exists and any
two such liftings differ by a ${\Bbb G}_m$-torsor. Thus, ${\cal C}$
is a ${\Bbb G}_2$-gerb over $S$, and by definition
$e(G,E)$ is the class of ${\cal C}$ in $H^2(S,{\Bbb G}_m)$
Note that $e(G,E)=0$ if and only if there exists a $G$-equivariant
line bundle $L$ over $E$, such that ${\Bbb G}_m\subset G$ acts on $L$
via the identity character.
A $K$-torsor $E$
defines a commutative group extension $G_E$ of $K$ by ${\Bbb G}_m$
as follows. Choose local trivializations of $E$ over some
covering $(U_i)$ and let $\a_{ij}\in K(U_i\cap U_j)$ be the
corresponding 1-cocycle with values in $K$. Now we glue $G_E$
from the trivial extensions ${\Bbb G}_m\times K$ over $U_i$ by the
following transition isomorphisms over $U_i\cap U_j$:
$$f_{ij}:{\Bbb G}_m\times K\rightarrow{\Bbb G}_m\times K:(\lambda,x)\mapsto
(\lambda e(x,\a_{ij}),x)$$
where $e:K\times K\rightarrow{\Bbb G}_m$ is the commutator form
corresponding to $G$. It is easy to see that $G_E$ doesn't
depend on a choice of trivializations. Now we claim that
if $G$ is a Heisenberg group then
\begin{equation}\label{diff}
e(G,E)=e(G\otimes G_E)-e(G).
\end{equation}
This is checked by a direct computation with Cech cocycles.
Notice that if $E^2$ is a trivial $K$-torsor
then $G_E^2$ is a trivial central extension of $K$, hence
$G_E$ is a symmetric extension. Thus, if $G$ is a symmetric
Heisenberg group, then $G\otimes G_E$ is also symmetric.
As was shown in \cite{sympl} the left hand side of (\ref{diff})
can be non-trivial.
Namely, consider the case when $S=A$ is a principally
polarized abelian variety over an algebraically closed field
$k$ of characteristic $\neq2$. Let $K=A_2\times A$ considered
as a (constant) finite group scheme over $A$. Then we can
consider $E=A$ as a $K$-torsor over $A$ via the morphism
$[2]:A\rightarrow A$. Now if $G\rightarrow A_2$ is a Heisenberg extension of
$A_2$ (defined over $k$) then we can consider $G$ as
a constant group scheme over $A$ and the class $e(G,E)$
is trivial if and only if $G$ embeds into the Mumford
group $G(L)$ of some line bundle $L$ over $A$ (this
embedding should be the identity on ${\Bbb G}_m$).
When $\operatorname{NS}(A)={\Bbb Z}$ this means, in particular, that
the commutator form $A_2\times A_2\rightarrow{\Bbb G}_m$
induced by $G$ is proportional to the symplectic
form given by the principal polarization.
When $\dim A\ge 2$ there is a plenty of other
symplectic forms on $A_2$, hence, $e(G,E)$ can be non-trivial.
Now we are going to show that one can replace $A$ by its
general point in this example. In other words, we
consider the base $S=\operatorname{Spec}(k(A))$ where $k(A)$ is the field
of rational functions on $A$. Then $E$ gets replaced by
$\operatorname{Spec}(k(A))$ considered as a $A_2$-torsor over itself
corresponding to the Galois extension
$$[2]^*:k(A)\rightarrow k(A): f\mapsto f(2\cdot)$$
with the Galois group $A_2$. Note that the class $e(G,E)$
for any Heisenberg extension $G$ of $A_2$ by $k^*$
is annihilated by the pull-back to $E$, hence, it is
represented by the class of Galois cohomology
$H^2(A_2,k(A)^*)\subset\operatorname{Br}(k(A))$ where $A_2$ acts
on $k(A)$ by translation of argument. It is easy
to see that this class is the image of the class
$e_G\in H^2(A_2,k^*)$ of the central extension $G$
under the natural homomorphism
$H^2(A_2,k^*)\rightarrow H^2(A_2,k(A)^*)$.
From the exact sequence of groups
$$0\rightarrow k^*\rightarrow k(A)^*\rightarrow k(A)^*/k^*\rightarrow 0$$
we get the exact sequence of cohomologies
$$0\rightarrow H^1(A_2,k(A)^*/k^*)\rightarrow H^2(A_2,k^*)\rightarrow
H^2(A_2,k(A)^*)$$
(note that $H^1(A_2,k(A)^*)=0$ by Hilbert theorem 90).
It follows that central extensions $G$ of $A_2$ by
$k^*$ with trivial $e(G,E)$ are classified by elements
of $H^1(A_2,k(A)^*/k^*)$.
\begin{lem} Let $A$ be a principally polarized abelian variety
over an algebraically closed field $k$ of characteristic
$\neq 2$. Assume that $\operatorname{NS}(A)={\Bbb Z}$. Then
$H^1(A_2,k(A)^*/k^*)={\Bbb Z}/2{\Bbb Z}$.
\end{lem}
\noindent {\it Proof} . Interpreting $k(A)^*/k^*$ as the group of divisors
linearly equivalent to zero we obtain the exact sequence
$$0\rightarrow k(A)^*/k^*\rightarrow\operatorname{Div}(A)\rightarrow\operatorname{Pic}(A)\rightarrow 0,$$
where $\operatorname{Div}(A)$ is the group of all divisors on $A$.
Note that as $A_2$-module $\operatorname{Div}(A)$ is decomposed into
a direct sum of modules of the form ${\Bbb Z}^{A_2/H}$ where
$H\subset A_2$ is a subgroup. Now by Shapiro lemma we have
$H^1(A_2,{\Bbb Z}^{A_2/H})\simeq H^1(H,{\Bbb Z})$, and the latter group
is zero since $H$ is a torsion group. Hence,
$H^1(A_2,\operatorname{Div}(A))=0$. Thus, from the above exact sequence
we get the identification
$$H^1(A_2,k(A)^*/k^*)\simeq
\operatorname{coker}(\operatorname{Div}(A)^{A_2}\rightarrow\operatorname{Pic}(A)^{A_2}).$$
Now we use the exact sequence
$$0\rightarrow\operatorname{Pic}^0(A)\rightarrow\operatorname{Pic}(A)\rightarrow\operatorname{NS}(A)\rightarrow 0,$$
where $\operatorname{Pic}^0(A)=\hat{A}(k)$.
Since the actions of $A_2$ on $\operatorname{Pic}^0(A)$ and $\operatorname{NS}(A)$ are trivial
we have the induced exact sequence
$$0\rightarrow\operatorname{Pic}^0(A)\rightarrow\operatorname{Pic}(A)^{A_2}\rightarrow\operatorname{NS}(A).$$
The image of the right arrow is the subgroup $2\operatorname{NS}(A)\subset\operatorname{NS}(A)$.
Note that $\operatorname{Pic}^0(A)=[2]^*\operatorname{Pic}^0(A)$, hence this subgroup
belongs to the image of $[2]^*\operatorname{Div}(A)\subset\operatorname{Div}(A)^{A_2}$.
Thus, we deduce that
$$H^1(A_2,k(A)^*/k^*)\simeq\operatorname{coker}(\operatorname{Div}(A)^{A_2}\rightarrow 2\operatorname{NS}(A)).$$
Let $[L]\subset\operatorname{NS}(A)$ be the generator corresponding to
a line bundle $L$ of degree 1 on $A$.
Then $L^4=[2]^*L$, hence $4\cdot [L]=[L^4]$ belongs to the image
of $\operatorname{Div}(A)^{A_2}$. On the other hand, it is easy to see
that there is no $A_2$-invariant divisor representing
$[L^2]$, hence
$$H^1(A_2,k(A)^*/k^*)\simeq{\Bbb Z}/2{\Bbb Z}.$$
\qed\vspace{3mm}
It follows that under the assumptions of this lemma there
is a unique Heisenberg extensions $G$ of $A_2$ by $k^*$
with the trivial class $e(G,E)$ (the Mumford
extension corresponding to $L^2$,
where $L$ is a line bundle of degree 1 on $A$). Hence,
for $g\ge 2$ there exists a Heisenberg extension with
a non-trivial class $e(G,E)\in\operatorname{Br}(k(A))$.
\section{Representations of the Heisenberg groupoid}
Recall that the Heisenberg group $H(W)$ associated with
a symplectic vector space $W$ is a central extension
$$0\rightarrow T\rightarrow H(W)\rightarrow W\rightarrow 0$$
of $W$ by the 1-dimensional torus $T$ with the commutator form
$\exp(B(\cdot,\cdot))$ where $B$ is the symplectic form.
In this section we consider an analogue of this
extension in the context of abelian schemes (see
\cite{Weilrep} , sect. 7, \cite{sympl}). Namely, we
replace a vector space $W$ by an abelian scheme $X/S$. Bilinear
forms on $W$ get replaced by biextensions of $X^2$.
Recall that
a {\it biextension} of $X^2$ is a line bundle ${\cal L}$ on $X^2$
together with isomorphisms
\begin{align*}
&{\cal L}_{x+x',y}\simeq {\cal L}_{x,y}\otimes {\cal L}_{x',y},\\
&{\cal L}_{x,y+y'}\simeq {\cal L}_{x,y}\otimes {\cal L}_{x,y'}
\end{align*}
--- this is a symbolic notation for isomorphisms
$(p_1+p_2,p_3)^*{\cal L}\simeq p_{13}^*{\cal L}\otimes p_{23}^*{\cal L}$ and
$(p_1,p_2+p_3)^*{\cal L}\simeq p_{12}^*{\cal L}\otimes p_{13}^*{\cal L}$ on $X^3$,
satisfying some natural cocycle conditions (see e.g. \cite{Breen}).
The parallel notion to the skew-symmetric form on $W$ is that of
a {\it skew-symmetric biextension} of $X^2$ which is a
biextension ${\cal L}$ of $X^2$ together with
an isomorphism of biextensions
$\phi:\sigma^*{\cal L}\widetilde{\rightarrow} {\cal L}^{-1}$, where $\sigma:X^2\rightarrow X^2$ is the
permutation of factors, and a trivialization
$\Delta^*{\cal L}\simeq\O_X$ of ${\cal L}$ over the diagonal
$\Delta:X\rightarrow X^2$ compatible with $\phi$.
A skew-symmetric biextension ${\cal L}$ is called {\it symplectic}
if the corresponding homomorphism $\psi_{{\cal L}}:X\rightarrow\hat{X}$ (where
$\hat{X}$ is the dual abelian scheme) is an isomorphism.
An {\it isotropic} subscheme (with respect to ${\cal L}$) is an abelian
subscheme $Y\subset X$ such that there is an isomorphism of skew-symmetric
biextensions ${\cal L}|_{Y\times Y}\simeq\O_{Y\times Y}$.
This is equivalent to the condition that the composition
$Y\stackrel{i}{\rightarrow} X\stackrel{\psi_{{\cal L}}}{\rightarrow}\hat{X}
\stackrel{\hat{i}}{\rightarrow} \hat{Y}$ is zero.
An isotropic subscheme $Y\subset X$ is called {\it lagrangian}
if the morphism $Y\rightarrow \ker(\hat{i})$ induced by $\psi_{{\cal L}}$ is an
isomorphism. In particular, for such a subscheme the quotient
$X/Y$ exists and is isomorphic to $\hat{Y}$.
Note that to define the Heisenberg group extension it is not sufficient
to have a symplectic form $B$ on $W$: one needs a bilinear form $B_1$
such that $B(x,y)=B_1(x,y)-B_1(y,x)$. In the case of the real symplectic
space one can just take $B_1=B/2$, however in our situation we have
to simply add necessary data.
An {\it enhanced} symplectic biextension $(X,{\cal B})$ is a
biextension ${\cal B}$ of $X^2$ such that
${\cal L}:={\cal B}\otimes\sigma^*{\cal B}^{-1}$ is a symplectic biextension.
The standard enhanced symplectic biextension
for $X=\hat{A}\times A$, where $A$ is any abelian scheme, is
obtained by setting
$${\cal B}=p_{14}^*\cal P\in\operatorname{Pic}(\hat{A}\times A\times \hat{A}\times A),$$
where $\cal P$ is the normalized Poincar\'e line bundle on $A\times\hat{A}$.
Given an enhanced symplectic
biextension $(X,{\cal B})$ one defines the {\it Heisenberg
groupoid} $H(X)=H(X,{\cal B})$ as
the stack of monoidal groupoids
such that $H(X)(S')$ for an $S$-scheme $S'$
is the monoidal groupoid generated by
the central subgroupoid ${\cal P}ic(S')$ of ${\Bbb G}_m$-torsors on $S'$ and the symbols
$T_x$, $x\in X(S')$ with the composition law
$$T_x\circ T_{x'}= {\cal B}_{x,x'} T_{x+x'}.$$
The Heisenberg groupoid is a central extension of $X$ by the
stack of line bundles on $S$ in the sense of Deligne \cite{Des}.
In \cite{Weilrep} we considered the action of $H(\hat{A}\times A)$
on ${\cal D}^b(A)$ which is similar to the standard representation of
the Heisenberg group $H(W)$ on functions on a lagrangian subspace of
$W$. Below we construct similar representations of the Heisenberg
groupoid $H(X)$ associated with lagrangian subschemes in $X$.
Further, we construct intertwining functors for two such
representations
corresponding to a pair of lagrangian subschemes, and consider
the analogue of Maslov index for a triple of lagrangian subschemes
that arises when composing these intertwining functors.
To define an action of $H(X)$ associated with a lagrangian subscheme one
needs some auxilary data described as follows.
An {\it enhanced} lagrangian subscheme (with respect to
${\cal B}$) is a pair $(Y,\a)$ where $Y\subset X$ is a lagrangian
subscheme with respect to $X$, $\a$ is a line bundle on $Y$
with a rigidification along the zero section such that an
isomorphism of symmetric
biextensions $\Lambda(\a)\simeq {\cal B}|_{Y\times Y}$ is given, where
$\Lambda(\a)=(p_1+p_2)^*\a\otimes p_1^*\a^{-1}\otimes p_2^*\a^{-1}$.
Note that an enhanced lagrangian subscheme is a particular
case of an {\it isotropic pair} as defined in
\cite{Weilrep} II, 7.3.
With every enhanced lagrangian subscheme $(Y,\a)$
one can associate a representation of $H(X)(S)$ as follows
(see \cite{Weilrep},\cite{sympl}). Let
${\cal D}(Y,\a)$ be the category of pairs $({\cal F},a)$ where
${\cal F}\in{\cal D}^b(X)$, $a$ is an isomorphism in ${\cal D}^b(Y\times X)$:
\begin{equation}\label{Schrsp}
a:(i_Yp_1+p_2)^*{\cal F}\widetilde{\rightarrow} {\cal B}^{-1}|_{Y\times X}
\otimes p_1^*\a^{-1}\otimes p_2^*{\cal F}
\end{equation}
where $i_Y:Y\hookrightarrow X$ is the embedding, such that $(e\times\operatorname{id})^*a=\operatorname{id}$.
These data should satisfy
the following cocycle condition:
$$(p_1+p_2,p_3)^*a=(p_2,p_3)^*a\circ (p_1,i_Yp_2+p_3)^*a$$
in ${\cal D}^b(Y\times Y\times X)$.
Then there is a natural action of the Heisenberg groupoid
$H(X)(S)$
on the category ${\cal D}(Y,\a)$ such that a line bundle $M$ on $S$
acts by tensoring with $p^*M$ and a generator $T_x$ acts
by the functor
\begin{equation}\label{act}
{\cal F}\mapsto {\cal B}|_{X\times x}\otimes t_x^*({\cal F}).
\end{equation}
If $S'$ is an $S$-scheme then this action is compatible
with the action of $H(X)(S')$ on ${\cal D}(Y_{S'},\a_{S'})$ via
pull-back functors.
Let $\delta_{Y,\a}\in{\cal D}(Y,\a)$ be the following object
(delta-function at $(Y,\a)$):
\begin{equation}\label{delta}
\delta_{Y,\a}=i_{Y*}(\a^{-1})
\end{equation}
where $i_Y:Y\rightarrow X$ is the embedding. It is easy to see that
$\delta_{Y,\a}$ has a canonical structure of an object of ${\cal D}(Y,\a)$
and for $y\in Y$ one has
$T_y(\delta_{Y,\a})\simeq\a_y^{-1}\delta_{Y,\a}$.
Let $(Y,\a)$, $(Z,\b)$ be a pair of enhanced lagrangian
subschemes in $X$, such that $Y\cap Z$ is finite over $S$. Then
the natural morphism $Y\rightarrow X/Z\simeq\hat{Z}$ is an isogeny,
hence, $Y\cap Z$ is flat over $S$. Note that we have isomorphisms of
biextensions $\Lambda(\a|_{Y\cap Z})\simeq\Lambda(\b|_{Y\cap Z})\simeq
{\cal B}|_{(Y\cap Z)^2}$, hence the trivialization of
$\Lambda(\b|_{Y\cap Z}\otimes \a^{-1}|_{Y\cap Z})$. Thus, the
${\Bbb G}_m$-torsor $G_{Y,Z}=\b|_{Y\cap Z}\otimes \a^{-1}|_{Y\cap Z}$
has a natural structure of a central extension of
$Y\cap Z$ by ${\Bbb G}_m$. Furthermore, the corresponding commutator
form $(Y\cap Z)^2\rightarrow{\Bbb G}_m$ is non-degenerate since it
corresponds to the canonical duality between
$Y\cap Z=\ker(Y\rightarrow\hat{Z})$ and $Y\cap Z=\ker(Z\rightarrow\hat{Y})$
(see \cite{sympl}, remark after Prop. 3.1).
Thus, $G_{Y,Z}$ is a finite Heisenberg group scheme over $S$.
If the line bundles $\a$ and $\b$
are symmetric then so is $G_{Y,Z}$.
Let $V$ be a Schr\"odinger representation of $G_{Y,Z}$.
Generalizing the construction of \cite{sympl}
we define the $H(X)(S)$-intertwining operator
$$R(V):{\cal D}(Y,\a)\rightarrow{\cal D}(Z,\b):
{\cal F}\mapsto
\underline{\operatorname{Hom}}_{G_{Y,Z}}(V,p_{2*}({\cal B}|_{Z\times X}\otimes p_1^*\b\otimes
(i_Zp_1+p_2)^*{\cal F})).$$
Here $p_1$ and $p_2$ are the projections of the product
$Z\times_S X$ onto its factors. The $G_{Y,Z}$-module structure
on $p_{2*}({\cal B}|_{Z\times X}\otimes p_1^*\b\otimes(i_Zp_1+p_2)^*{\cal F})$
comes from the natural $G_{Y,Z}$-action on
$I({\cal F})={\cal B}|_{Z\times X}\otimes p_1^*\b\otimes(i_Zp_1+p_2)^*{\cal F}$ which is
compatible with the action of $Y\cap Z$ on $Z\times X$ by
the translation of the first argument and arises from the
canonical isomorphism
\begin{equation}\label{integrand}
I({\cal F})_{(z+u,x)}\simeq \b_u\a_u^{-1} I({\cal F})_{(z,x)}
\end{equation}
where $z\in Z$, $x\in X$, $u\in Y\cap Z$ (one should
consider this as an isomorphism in ${\cal D}^b((Y\cap Z)\times
Z\times X)$). When $V$ is the representation
associated with a lagrangian subgroup scheme $H\subset G_{Y,Z}$
this functor coincides with the one defined in \cite{sympl}.
Let us call an enhanced lagrangian subscheme $(Y,\a)$
{\it admissible} if the projection $X\rightarrow X/Y$ splits.
For such a subscheme we have an equivalence
${\cal D}(Y,\a)\simeq{\cal D}^b(X/Y)$. Namely, let $s_{X/Y}:X/Y\rightarrow X$ be a
splitting of the canonical projection $q_{X/Y}:X\rightarrow X/Y$. Let
$q_Y=\operatorname{id}-s_{X/Y}q_{X/Y}:X\rightarrow Y$ be the corresponding projection to $Y$.
Then the functors ${\cal F}\mapsto s_{X/Y}^*{\cal F}$ and ${\cal G}\mapsto
(q_Y,s_{X/Y}q_{X/Y})^*{\cal B}^{-1}\otimes q_Y^*\a^{-1}\otimes q_{X/Y}^*{\cal G}$ where
${\cal F}\in{\cal D}(Y,\a)$, ${\cal G}\in{\cal D}^b(X/Y)$ give the required
equivalence.
When $(Y,\a)$ and $(Z,\b)$ are both admissible we can
represent the above functor $R(V):{\cal D}^b(X/Y)\rightarrow{\cal D}^b(X/Z)$
in the standard "integral" form.
\begin{lem}\label{compker}
Assume that $(Y,\a)$ and $(Z,\b)$ are
admissible, $Y\cap Z$ is finite. Then
$R(V)({\cal G})\simeq p_{2*}(p_1^*{\cal G}\otimes {\cal K}(V))$ where
$p_i$ are the projections of $X/Y\times X/Z$ on its factors,
${\cal K}(V)$ is the following vector bundle on $X/Y\times X/Z$:
\begin{eqnarray}\label{kernel}
{\cal K}(V)=(p_1-q_{X/Y}s_{X/Z}p_2)^*E(V)\otimes
(s_{X/Y}p_1-s_{X/Z}p_2,s_{X/Y}p_2)^*{\cal B}\otimes \nonumber\\
(s_{X/Y}(p_1-q_{X/Y}s_{X/Z}p_2),q_Ys_{X/Z}p_2)^*{\cal B}\otimes
(q_Ys_{X/Z}p_2)^*\a^{-1}
\end{eqnarray}
where $s_{X/Y}:X/Y\rightarrow X$ (resp. $s_{X/Z}:X/Z\rightarrow X$) is the
splitting of the projection $q_{X/Y}:X\rightarrow X/Y$ (resp.
$q_{X/Z}:X\rightarrow X/Z$), $q_Y=\operatorname{id}-s_{X/Y}q_{X/Y}$, $E(V)$ is the
following bundle on $X/Y$:
\begin{equation}\label{kernelaux}
E(V)=\underline{\operatorname{Hom}}_{G_{Y,Z}}(V,(q_{X/Y}i_Z)_*(\b\otimes (q_Yi_Z)^*\a^{-1}\otimes
(i_Z,s_{X/Y}q_{X/Y}i_Z)^*{\cal B}^{-1}))
\end{equation}
where $i_Y:Y\rightarrow X$, $i_Z:Z\rightarrow X$ are the embeddings.
\end{lem}
\noindent {\it Proof} . By definition we have
\begin{align*}
&R(V)({\cal G})_{\bar{x}}\simeq\\
&\underline{\operatorname{Hom}}(V,\int_Z \b_z {\cal B}_{z,s_{X/Z}(\bar{x})}
{\cal B}^{-1}_{q_Y(z+s_{X/Z}(\bar{x})),s_{X/Y}q_{X/Y}(z+s_{X/Z}
(\bar{x}))}\a^{-1}_{q_Y(z+s_{X/Z}(\bar{x}))}{\cal G}_{q_{X/Y}(z+s_{X/Z}
(\bar{x}))} dz)
\end{align*}
where $\bar{x}\in X/Z$, $z\in Z$. Using the isomorphism
$\a_{q_Y(z+s_{X/Z}(\bar{x}))}\simeq
\a_{q_Y(z)}\a_{q_Ys_{X/Z}(\bar{x})}
{\cal B}_{q_Y(z),q_Ys_{X/Z}(\bar{x})}$ and collecting together terms
depending only on $\bar{z}=q_{X/Y}(z)$ we get
\begin{align*}
&R(V)({\cal G})_{\bar{x}}\simeq
\underline{\operatorname{Hom}}(V,\int_Z \b_z \a_{q_Y(z)}^{-1}
{\cal B}^{-1}_{z, s_{X/Y}\bar{z}} \\
&\Bigl(
{\cal B}_{s_{X/Y}(\bar{z}+q_{X/Y}s_{X/Z}(\bar{x}))-s_{X/Z}(\bar{x}),
s_{X/Y}(\bar{z}+q_{X/Y}s_{X/Z}(\bar{x}))}
{\cal B}_{s_{X/Y}(\bar{z}),q_Ys_{X/Z}(\bar{x})}
\a^{-1}_{q_Ys_{X/Z}(\bar{x})}
{\cal G}_{\bar{z}+q_{X/Y}s_{X/Z}(\bar{x})}\Bigr) dz)\simeq\\
&\int_{X/Y} E(V)_{\bar{z}}
{\cal B}_{s_{X/Y}(\bar{z}+q_{X/Y}s_{X/Z}(\bar{x}))-s_{X/Z}(\bar{x}),
s_{X/Y}(\bar{z}+q_{X/Y}s_{X/Z}(\bar{x}))}
{\cal B}_{s_{X/Y}(\bar{z}),q_Ys_{X/Z}(\bar{x})}\\
&\a^{-1}_{q_Ys_{X/Z}(\bar{x})}
{\cal G}_{\bar{z}+q_{X/Y}s_{X/Z}(\bar{x})} d\bar{z}
\end{align*}
where $\bar{z}$ is now considered as a variable on $X/Y$.
Making the change of variables $\bar{z}\mapsto
\bar{z}-q_{X/Y}s_{X/Z}(\bar{x})$ we arrive to the formula
(\ref{kernel}).
\qed\vspace{3mm}
\begin{thm}\label{eq}
Assume that $(Y,\a)$ and $(Z,\b)$ are admissible
and $Y\cap Z$ is finite, then $R(V)$ is an equivalence of categories.
Let $(T,\gamma)$ be an admissible enhanced lagrangian
subscheme such that $Y\cap T$ and $Z\cap T$ are finite,
$W$ (resp. $U$) be a Schr\"odinger representation for
$G_{Y,T}$ (resp. $G_{Z,T}$). Then
$$R(U)\circ R(V)\simeq R(W)\otimes M[n]$$
for some line bundle $M$ on $S$ and some integer $n$.
\end{thm}
\noindent {\it Proof} . The direct computation shows that the
kernel ${\cal K}(V)\in{\cal D}^b(X/Y\times X/Z)$ constructed above
satisfies the "uniform" intertwining property
(with respect to $H(X)$-action) defined
in \cite{Weilrep}. Hence, the
analogue of Schur lemma for the action of $H(X)$ on
${\cal D}^b(X/Y)$ where $Y$ is an admissible lagrangian subscheme
(see \cite{Weilrep} Thm 7.9) implies that
$$p_{13*}(p_{12}^*{\cal K}(V)\otimes p_{23}^*{\cal K}(V^*))\simeq\Delta_*(F)$$
where ${\cal K}(V^*)$ is the similar kernel on $X/Z\times X/Y$
giving rise to the functor $R(V^*):{\cal D}^b(X/Z)\rightarrow{\cal D}^b(X/Y)$,
$p_{ij}$ are the projections of $X/Y\times X/Z\times X/Y$ on
the pairwise products, $\Delta:X/Y\rightarrow (X/Y)^2$ is the diagonal
embedding.
In the case when $S$ is the spectrum
of a field we know that $F\simeq N[n]$ for some line bundle
$N$ on $S$ and some integer $n$ (see \cite{Weilrep}). By
Prop.1.7 of \cite{Muk2} this implies that the same is true
when $S$ is connected. Therefore, in this case the composition
$R(V^*)\circ R(V)$ is isomorphic to the tensoring with $N[n]$.
Repeating this for the composition $R(V)\circ
R(V^*)$ we conclude that $R(V)$ is an equivalence.
Similar argument works for the proof of the second assertion.
\qed\vspace{3mm}
\begin{rems} 1. Most probably, one can extend this theorem
to the case of arbitrary enhanced lagrangian subschemes.
However, it seems that the definition of ${\cal D}(Y,\a)$ should be
modified in this case (one should start with appropriate
category of complexes and then localize it).
\noindent 2. An integer $n$ and a line bundle $M$ on $S$ appearing
in the above theorem should be considered as analogues of
the Maslov index (see \cite{LV}) for a triple $(Y,Z,T)$.
Note that different choices of Schr\"odinger representations $V$,
$W$, and $U$
above affect $M$ but not $n$, hence the function $n(Y,Z,T)$
behaves very much like the classical Maslov index
(cf. \cite{Orlov}).
\end{rems}
Let $(Y,\a)$, $(Z,\b)$ and $(T,\gamma)$ be a triple of enhanced
lagrangian subschemes in $X$. Let us denote by $K=K(Y,Z,T)$
the kernel of the homomorphism $Y\times Z\times T\rightarrow
X:(y,z,t)\mapsto y+z+t$. Let $p_Y:K\rightarrow Y$, $p_Z:K\rightarrow Z$ and
$p_T:K\rightarrow T$ be the restrictions to $K$ of the natural projections
from $Y\times Z\times T$ to its factors. Consider the
following line bundle on $K$:
\begin{equation}\label{MYZT}
M(Y,Z,T)=(-p_Y)^*\a^{-1}\otimes p_Z^*\b\otimes p_T^*\gamma\otimes
(p_Z,p_T)^*{\cal B}|_{Z\times T}.
\end{equation}
Then $M(Y,Z,T)$ has a canonical cube structure induced by
that of $\a$, $\b$, $\gamma$ and ${\cal B}$.
\begin{lem} There are canonical isomorphisms of line bundles
with cube structures on $K$
$$M(Y,Z,T)\simeq M(Z,T,Y)\simeq M(T,Y,Z).$$
There is a canonical isomorphism of biextensions of $K\times
K$:
$$\Lambda(M(Y,Z,T))\simeq (p_Zp_2,p_Tp_1)^*{\cal L}$$
where $p_i$ are the projections of $K\times K$ on its factors.
\end{lem}
\noindent {\it Proof} . We have
$$(M(Z,T,Y)\otimes M(Y,Z,T)^{-1})_{y,z,t}=
\a_y\a_{-y}\b_z^{-1}\b_{-z}^{-1}{\cal B}_{t,y}{\cal B}_{z,t}^{-1}\simeq
{\cal B}_{y,y}{\cal B}_{z,z}^{-1}{\cal B}_{t,y}{\cal B}_{z,-t}$$
where $y+z+t=0$ (here we used the isomorphism
$\a_y\a_{-y}\simeq {\cal B}_{y,-y}\simeq {\cal B}_{y,y}^{-1}$ and the
similar isomorphism for $\b$).
It is easy to see that when we substitute $t=-y-z$ the right
hand side becomes trivial.
The second isomorphism is obtained as follows:
$$\Lambda(M(Y,Z,T))_{(y,z,t),(y',z',t')}\simeq
{\cal B}_{-y,-y'}^{-1}{\cal B}_{z,z'}{\cal B}_{t,t'}{\cal B}_{z,t'}{\cal B}_{z',t}.$$
If we substitute $-y=z+t$, $-y'=z'+t'$ the right hand side
becomes ${\cal B}_{t,z'}^{-1}{\cal B}_{z',t}\simeq {\cal L}_{z',t}$.
\qed\vspace{3mm}
Consider the embedding $Z\cap T\hookrightarrow K:u\mapsto
(0,-u,u)$. Then the previous lemma implies that
$\Lambda(M(Y,Z,T))$ is trivial over $(Z\cap T)\times K$. Hence,
$M(Y,Z,T)|_{Z\cap T}$ has a structure of central extension
and the action of $Z\cap T$ on $K$ by translations lifts to
an action of this central extension on $M(Y,Z,T)$.
Moreover, we have a canonical isomorphism of central
extensions
$$M(Y,Z,T)_{(0,-u,u)}\simeq\gamma_u\b_{-u}{\cal B}_{-u,u}\simeq
\gamma_u\b_u^{-1}=(G_{Z,T})_u.$$
Hence, there is an action of $G_{Z,T}$ on $M(Y,Z,T)$
compatible with the action of $Z\cap T$ on $K$ by
translations. Using cyclic permutation we get embeddings of
$Y\cap T$ and $Y\cap Z$ into $K$ and it is easy to see that
the images of the three embeddings are independent so that
we get an embedding
$(Y\cap Z)\times_S (Z\cap T)\times_S (Y\cap T)\hookrightarrow K$
and the compatible
action of $G_{Y,Z}\times_{{\Bbb G}_m}G_{Z,T}\times_{{\Bbb G}_m} G_{T,Y}$
on $M(Y,Z,T)$.
\begin{thm}\label{intert}
With the notation and assumptions of Theorem
\ref{eq} we have
$$M[n]\simeq\underline{\operatorname{Hom}}_{G_{Y,Z,T}}(V_{Y,Z,T},p_*M(Y,Z,T))$$
where $G_{Y,Z,T}=G_{Y,Z}\times_{{\Bbb G}_m} G_{Z,T}\times_{{\Bbb G}_m}
\times G_{T,Y}$, $V_{Y,Z,T}=V\otimes U\otimes W^*$, $p:K\rightarrow S$ is the
projection.
\end{thm}
\noindent {\it Proof} . Let us compare the restrictions of $R(U)\circ
R(V)(\delta)$ and $R(W)(\delta)$ to the zero section, where
$\delta=\delta_{Y,\a}\in{\cal D}(Y,\a)$ is the delta-function at $Y$
defined by (\ref{delta}). On the one hand, we have
\begin{align*}
&R(U)\circ R(V)(\delta)_0\simeq\underline{\operatorname{Hom}}_{G_{Y,Z}\times_{{\Bbb G}_m} G_{Z,T}}
(V\otimes U,\int_{Z\times T} \gamma_t {\cal B}_{z,t}\b_z\delta_{z+t} dz dt)\simeq\\
&\simeq\underline{\operatorname{Hom}}_{G_{Y,Z}\times_{{\Bbb G}_m} G_{Z,T}}(V\otimes U,
\int_{K}M(Y,Z,T)).
\end{align*}
On the other hand,
$$R(W)(\delta)_0\simeq\underline{\operatorname{Hom}}_{G_{Y,T}}(W,
\int_{Y\cap T} \gamma_u\a_u^{-1} du)\simeq W^*$$
since $\int_{Y\cap T}G_{Y,T}\simeq W^*\otimes W$
by \cite{MB} V 2.4.2. Therefore,
$$\int_{K} M(Y,Z,T)\simeq V\otimes U\otimes W^*\otimes M[n]$$
as a representation of $G_{Y,Z,T}$.
\qed\vspace{3mm}
Consider the following example.
Let $X=\hat{A}\times A$, ${\cal B}=p_{14}^*\cal P$,
$(Y,\a)=(A,\O_A)$, $(T,\gamma)=(\hat{A},\O_{\hat{A}})$, and
$(Z,\b)=(Z_{\phi,m},\b)$ where $\phi=\phi_L:A\rightarrow\hat{A}$ is
the symmetric isogeny associated with a rigidified line bundle
$L$ on $A$, $Z_{\phi,m}=(\phi,m\operatorname{id}_A)(A)\simeq A/\ker(\phi_m)$
where $\phi_m=\phi|_{A_m}$, $\b$ is obtained from $L^m$ by
descent (such $\b$ always exists if $m$ is odd,
since $\ker(\phi_m)$ is isotropic
with respect to $e^{L^m}$). Then
$K(Y,Z,T)\simeq Z$, $M(Y,Z,T)\simeq\b$, $Y\cap T=0$,
$Y\cap Z\simeq \ker(\phi)/\ker(\phi_n)$, and
$Z\cap T\simeq A_n/\ker(\phi_n)$. Hence,
if we take $W=\O_S$ we get
$$M[n]\simeq\underline{\operatorname{Hom}}_{G_{Y,Z}\times_{{\Bbb G}_m}G_{Z,T}}(V\otimes U,p_*\b).$$
In particular, when $m=1$ we have $Z\simeq A$, $\b=L$,
$Z\cap T=0$, and $G_{Y,Z}=G(L)$. Thus, if we take $U=W=\O_S$
we obtain $M[n]\simeq\underline{\operatorname{Hom}}_{G(L)}(V,p_*L)$.
Note that if one of the pairwise intersections of $Y$, $Z$
and $T$ is trivial
then $K(Y,Z,T)$ is an abelian scheme over $S$. More precisely,
if say $Y\cap Z=0$ then $K(Y,Z,T)\simeq T$ and it is easy to
see from the above considerations that
in this case we have an isomorphism of Heisenberg groups
\begin{equation}\label{GYZT}
G(M(Y,Z,T))\simeq G_{Y,Z}\times_{{\Bbb G}_m}G_{Z,T}\times_{{\Bbb G}_m}G_{T,Y}.
\end{equation}
\section{Weil representation on the derived category of an abelian scheme}
In this section the base scheme $S$ is always assumed to
noetherian, normal and connected.
Let $K$ denotes the field of rational
functions on $S$.
\begin{lem}\label{extend}
Let $A$ and $A'$ be abelian schemes over $S$,
$A_K$ and $A'_K$ be their general fibers which are abelian
varieties over $K$. Then the restriction map
$$\operatorname{Hom}_S(A,A')\rightarrow\operatorname{Hom}_K(A_K,A'_K):f\mapsto f|_K$$
is an isomorphism. The morphism $f$ is an isogeny if and
only if $f|_K$ is an isogeny.
\end{lem}
\noindent {\it Proof} . The proof of the first assertion
is similar to the proof of the fact that an
abelian scheme over a Dedekind scheme is a N\'eron model of
its generic fiber (see \cite{Neron} 1.2.8).
We have to check that any homomorphism $f_K:A_K\rightarrow A'_{K}$
extends to a homomorphism $f:A\rightarrow A'$. Let $\phi:A\rightarrow A'$ be
the rational map defined by $f_K$. Since $A$ is normal, by a
valuative criterion of properness $\phi$ is defined in
codimension $\leq 1$. Let $V\subset A$ be a non-empty subscheme
over which $\phi$ is defined. Then since the projection
$p:A\rightarrow S$ is flat and of finite presentation it is open.
Thus, $U=p(V)$ is open, and $\phi_U:A_U\rightarrow A'_U$ is a
$U$-rational map in the terminology of \cite{Neron}.
By Weil's theorem (see \cite{Neron} 4.4.1) $\phi_U$ is
defined everywhere, hence we get a homomorphism $f_U:A_U\rightarrow
A'_U$ extending $f_K$. It remains to invoke Prop. I 2.9 of
\cite{FC} to finish the proof.
The part concerning isogenies can
be proven by exactly the same argument as in \cite{Neron}
7.3 prop. 6: starting with an isogeny $f|_K:A_K\rightarrow A'_K$ we
can find another isogeny $g|_K:A'_K\rightarrow A_K$ such that the
composition $g|_Kf|_K$ is the multiplication by an integer
$l$ on $A_K$. By the first part we can extend $g_K$ to a
homomorphism $g:A'\rightarrow A$. This implies that $gf=l_A$ --- the
multiplication by $l$ morphism on $A$. It follows that the
restriction of $f$ to each fiber is an isogeny, hence $f$ is
an isogeny itself.
\qed\vspace{3mm}
For an abelian scheme $A$ the group $\operatorname{SL}_2(A)$ is
defined as the subgroup of automorphisms of $\hat{A}\times
A$ preserving the line bundle $p_{14}^*\cal P\otimes
p_{23}^*\cal P^{-1}$ on $(\hat{A}\times A)^2$. More explicitly,
if we write an automorphism of $\hat{A}\times A$ as a matrix
$g=\left( \matrix a_{11} & a_{12}\\ a_{21} & a_{22}
\endmatrix \right)$ where $a_{11}\in\operatorname{Hom}(\hat{A},\hat{A})$,
$a_{12}\in\operatorname{Hom}(A,\hat{A})$ etc., then
$g\in\operatorname{SL}_2(A)$ if and only if the inverse automorphism
$g^{-1}$ is given by the matrix
$\left(\matrix \hat{a}_{22} & -\hat{a}_{12}\\
-\hat{a}_{21} & \hat{a}_{11} \endmatrix \right)$.
It follows from Lemma \ref{extend} that when the base $S$ is
normal we have $\operatorname{SL}_2(A)\simeq\operatorname{SL}_2(A_K)$.
Now similarly to the classical picture one has to consider
the group of automorphisms of the Heisenberg extension $H(X)$
corresponding to $X=\hat{A}\times A$ with the structure
of enhanced symplectic biextension given by ${\cal B}=p_{14}^*\cal P$.
Namely, we define $\widetilde{\operatorname{SL}}_2(A)$ as the group of
triples $g=(\bar{g},L^g,M^g)$ where
$\overline{g}=
\left( \matrix a_{11} & a_{12}\\ a_{21} & a_{22} \endmatrix \right)
\in\operatorname{SL}_2(A)$,
$L^g$ (resp. $M^g$) is a line bundle on $\hat{A}$ (resp. $A$)
rigidified along the zero section, such that
\begin{equation}
\phi_{L^g}=\hat{a}_{11}a_{21},\ \phi_{M^g}=\hat{a}_{22}a_{12},
\end{equation}
where for a line bundle $L$ on an abelian scheme $B$ we
denote by $\phi_L:B\rightarrow\hat{B}$ the symmetric homomorphism
corresponding to the symmetric biextension $\Lambda(L)$.
The group law on $\widetilde{\operatorname{SL}}_2(A)$
is defined uniquely from the condition that the there is
an action of $\widetilde{\operatorname{SL}}_2(A)$
on the stack of Picard groupoids ${\cal H}(A)$ such that
an element $g=(\bar{g},L^g,M^g)$ acts by the functor which is
identical on ${\cal P}ic$ and
sends the generator $T_{(x,y)}$ (where
$(x,y)\in\hat{A}\times A$) to
$L_x\otimes M_y\otimes \cal P_{(a_{12}y,a_{21}x)}\otimes T_{\bar{g}(x,y)}$.
We refer to \cite{Weilrep} for explicit formulas for the
group law in $\widetilde{\operatorname{SL}}_2(A)$. It is easy to see that the
natural projection $\widetilde{\operatorname{SL}}_2(A)\rightarrow\operatorname{SL}_2(A)$ is a
homomorphism with the kernel isomorphic to
$A(S)\times\hat{A}(S)$.
Consider the subgroup $\widehat{\operatorname{SL}}_2(A)\subset\widetilde{\operatorname{SL}}_2(A)$
consisting of triples with symmetric $L^g$ and $M^g$. Then we have
an isomorphism $\widehat{\operatorname{SL}}_2(A)\simeq\widehat{\operatorname{SL}}_2(A_K)$
since any symmetric line bundle on $A_K$ extends to a
symmetric line bundle on $A$ (see \cite{MB}, II.3.3).
Let $\Gamma(A)=\Gamma(A_K)$ be the image of the projection
$\widehat{\operatorname{SL}}_2(A_K)\rightarrow\operatorname{SL}_2(A_K)$.
Then $\Gamma(A)$ has finite index in $\operatorname{SL}_2(A_K)$
since it contains the subgroup $\Gamma(A,2)=\Gamma(A_K,2)\subset\operatorname{SL}_2(A_K)$
consisting of matrices with $a_{12}$ and $a_{21}$ divisible by 2.
In the case when $S$ is the spectrum of an algebraically closed field
it was shown in \cite{Weilrep} that there exist
intertwining functors between representations of Heisenberg
groupoid corresponding to the natural action
of $\widetilde{\operatorname{SL}}_2(A)$ on the Heisenberg groupoid
$H(\hat{A}\times A)$ which are analogous to the operators
of Weil-Shale representation. We are going
to extend this construction to the case of a normal base
scheme.
Recall (see \cite{Weilrep}, sect. 10) that there is a natural action of
$\widetilde{\operatorname{SL}}_2(A)$ on the set of enhanced lagrangian subvarieties
in $X=\hat{A}\times A$ such that a triple
$g=(\bar{g},L^g,M^g)\in\widetilde{\operatorname{SL}}_2(A)$
maps $(\hat{A},\O_{\hat{A}})$ to $\bar{g}(\hat{A})=
(a_{11},a_{21})(\hat{A})$
with the line bundle corresponding to $L^g\in\operatorname{Pic}(\hat{A})$.
Furthermore, there is a natural equivalence of categories
$$\overline{g}_*:{\cal D}(\hat{A},\O_{\hat{A}})\rightarrow
{\cal D}(\overline{g}(\hat{A}),L^g):{\cal F}\mapsto \overline{g}_*{\cal F}$$
such that the standard $H(X)$-action
on ${\cal D}(\hat{A},O_{\hat{A}})$ corresponds to the $g$-twisted
$H(X)$-action on ${\cal D}(\overline{g}(\hat{A}),L^g)$. On the other hand,
if $\hat{A}\cap \overline{g}(\hat{A})=\ker(a_{21})$ is finite
(hence, flat) over $S$ and there exists a
Schr\"odinger representation $V$ for the corresponding
Heisenberg extension
$G_g:=G_{\hat{A},\overline{g}(\hat{A})}$ of $\ker(a_{21})$ then
the construction of the previous section gives
another equivalence
$${\cal D}(\overline{g}(\hat{A}),L^g))\rightarrow{\cal D}(\hat{A},\O_{\hat{A}})$$
compatible with the standard $H(X)$-actions. Composing it
with the previous equivalence we get an equivalence
$\rho\widetilde{\rightarrow}\rho^g$ where $\rho$ is the representation
of $H(X)$ on ${\cal D}(\hat{A},\O_{\hat{A}})\simeq{\cal D}(A)$
given by (\ref{act}),
$\rho^g=\rho\circ g$ is the same representation twisted by
$g$. Using (\ref{kernel})
it is easy to compute that the kernel on $A\times_S A$
corresponding to this equivalence has form
\begin{equation}\label{kernelnew}
{\cal K}(g,V)=(p_2-a_{22}p_1)^*E\otimes (a_{12}\times\operatorname{id})^*\cal P^{-1}\otimes
(-p_1)^*M^g
\end{equation}
where
$$E=\underline{\operatorname{Hom}}_{G_g^{-1}}(V^*,a_{21*}((L^g)^{-1})).$$
Note that here $G_g^{-1}$ is the restriction of the
Mumford's extension $G((L^g)^{-1})\rightarrow\ker(\hat{a}_{11}a_{21})$
to $\ker(a_{21})$.
Hence,
if $\ker(a_{21})$ is finite over $S$ we get a functor from the
gerb of Schr\"odinger representations for $G_g$ to the stack of
$H(X)$-equivalences $\operatorname{Isom}_{H(X)}(\rho,\rho^g)$,
More precisely, the category of intertwining operators between
$\rho$ and $\rho^g$ is defined in terms of kernels in
${\cal D}^b(A\times_S A)$ (see \cite{Weilrep}) and the glueing property
is satisfied because the kernels corresponding to
equivalences are actually vector bundles (perhaps, shifted).
Indeed, the latter property is local with respect to the
{\it fppf} topology on the base $S$ and locally a Schr\"odinger
representation for $G_g$ exists and give rise to
the kernel (\ref{kernelnew}) in $\operatorname{Isom}_{H(X)}(\rho,\rho^g)$ which is a
vector bundle up to shift. Now any other object of
$\operatorname{Isom}_{H(X)}(\rho,\rho^g)$ is obtained from a given one
by tensoring with a line bundle on $S$ and a shift.
Thus, when $\ker(a_{21})$ is finite the
obstacle for the existence of a global equivalence
between $\rho$ and $\rho^g$ is given by the class
$e(G_g)\in\operatorname{Br}(S)$.
Let $U\subset\operatorname{SL}_2(A)$ be the subset of matrices such that
$a_{21}$ is an isogeny. It turns out that similarly to the
case of real groups one can deal with $U$ instead of
the entire group when defining representation of $\operatorname{SL}_2(A)$.
This observation can be formalized as follows.
Let us call a subset $B$ of a group
$G$ {\it big} if for any triple of elements $g_1, g_2,
g_3\in G$ the intersection $B^{-1}\cap Bg_1\cap Bg_2\cap Bg_3$
is non-empty. This condition first appeared
in \cite{Weil} IV. 42, while the term is due to D.~Kazhdan.
The reason for introducing this notion is the following lemma.
\begin{lem} Let $B\subset G$ be a big subset.
Then $G$ is isomorphic to
the abstract group generated by elements $[b]$ for $b\in B$
modulo the relations $[b_1][b_2]=[b]$ when $b,b_1,b_2\in B$
and $b=b_1b_2$.
If $c:G\times G\rightarrow C$
is a 2-cocycle (where $C$ is an abelian group with the trivial
$G$-action) such that $c(b_1,b_2)=0$ whenever
$b_1, b_2, b_1b_2\in B$ then $c$ is a coboundary.
\end{lem}
\noindent {\it Proof} . For the proof of the first statement we refer to
\cite{Weil}, IV, 42, Lem. 6.
Let $c:G\times G\rightarrow C$ be a 2-cocycle,
$H$ be the corresponding central extension of $G$ by $C$.
Consider the group $\widetilde{H}$ generated by the central
subgroup $C$ and generators $[b]$ for $b\in B$
subject to relations $[b_1][b_2]=c(b_1,b_2)[b_1b_2]$,
where $b_1, b_2, b_1b_2\in B$. Then
$\widetilde{H}/C\simeq G$, hence the natural homomorphism
$\widetilde{H}\rightarrow H$ is an isomorphism. If $c(b_1,b_2)=0$
whenever $b_1, b_2, b_1b_2\in B$, then the
extension $\widetilde{H}\simeq H\rightarrow G$ splits, hence
$c$ is a coboundary.
\qed\vspace{3mm}
At this point we need to recall
some results from \cite{Weilrep}, sect. 9 concerning
the group $\operatorname{SL}_2(A)$.
Since $\operatorname{SL}_2(A)=\operatorname{SL}_2(A_K)$ we can work with abelian varieties
over a field. First note that this group
can be considered as a group of ${\Bbb Z}$-points of an
group scheme over ${\Bbb Z}$. It turns out that the corresponding
algebraic group $\operatorname{SL}_{2,A,{\Bbb Q}}$ over ${\Bbb Q}$ is very close to be
semi-simple. Namely, if we fix a polarization on $A$ then the
latter group is completely determined by the algebra
$\operatorname{End}(A)\otimes{\Bbb Q}$ and the Rosati involution on it. Decomposing
(up to isogeny) $A$ into a product $A_1^{n_1}\times\ldots
A_l^{n_l}$ where $A_i$ are different simple abelian varieties
and choosing a polarization compatible with this decomposition
it is easy to see that
$$\operatorname{SL}_{2,A,{\Bbb Q}}\simeq\prod_i R_{K_{i,0}/{\Bbb Q}}\operatorname{U}^*_{2n_i,F_i}$$
where $F_i=\operatorname{End}(A_i)\otimes{\Bbb Q}$, $K_i$ is the center of $F_i$,
$K_{i,0}\subset K_i$ is the subfield of elements stable under
the Rosati involution, $\operatorname{U}^*_{2n_i,F_i}$ is the group of
$F_i$-automorphisms of $F_i^{2n_i}$
preserving the standard skew-hermitian form,
$R_{K_{i,0}/{\Bbb Q}}$ denotes the restriction of scalars from
$K_{i,0}$ to ${\Bbb Q}$. Thus, the only case
when the group $\operatorname{U}^*_{2n_i,F_i}$ is not semi-simple is when
the Rosati involution on $F_i$ is of the second kind, i.~e.
$K_{i,0}\neq K_i$. In the latter case, $\operatorname{U}^*_{2n_i,F_i}$
is a product of the semi-simple subgroup $\operatorname{SU}^*_{2n_i,F_i}$
(defined using the determinant with values in $K_i$)
and the central subgroup
$K^1_i=\{ x\in K_i\ |\ N_{K_i/K_{i,0}}=1\}$ consisting
of diagonal matrices. Furthermore, the intersections of these
two subgroup is finite. It follows that the group
$\operatorname{SL}_{2,A,{\Bbb Q}}$ always has an almost direct decomposition into
a product of the semi-simple subgroup
$H=\prod_i R_{K_{i,0}/{\Bbb Q}}\operatorname{SU}^*_{2n_i,F_i}$
and a central subgroup $Z$ consisting of diagonal matrices.
Now we can prove the following result.
\begin{lem}\label{big}
Let $\Gamma\subset\operatorname{SL}_2(A)$ be a subgroup of finite
index. Then the subset $\Gamma\cap U$ is big.
\end{lem}
\noindent {\it Proof} . By definition $U$ is an intersection of a Zariski open
subset $\underline{U}$ in the irreducible algebraic group
$\operatorname{SL}_{2,A_K,{\Bbb Q}}$ with $\Gamma$. If $\Gamma$
were Zariski dense in $\operatorname{SL}_{2,A_K,{\Bbb Q}}$ the proof would be
finished. This is not always true, however, we claim that
$\Gamma\cap H$ is dense in $H$ where $H$ is the subgroup of
$\operatorname{SL}_{2,A_K,{\Bbb Q}}$ introduced above. Indeed, this follows from
the fact that $H$ is semi-simple and the corresponding
real groups have no compact factors (see \cite{Weilrep} 9.4).
On the other hand, the set $\underline{U}$ is $Z$-invariant (since
$Z$ consists of diagonal matrices).
It follows that for any $g\in\Gamma$
the intersection $\underline{U}g\cap H$ is a non-empty Zariski
open subset of $H$ (as a preimage of a non-empty Zariski set
under the isogeny $H\rightarrow\operatorname{SL}_{2,A_K,{\Bbb Q}}/Z$). Therefore, for
any triple of elements $g_1,g_2,g_3\in\Gamma$ the intersection
$\underline{U}\cap\underline{U}g_1\cap\underline{U}g_2\cap\underline{U}g_3\cap H$
is a non-empty open subset in $H$, hence it contains an
element of $\Gamma\cap H$. As $U=U^{-1}$ this shows that $U$ is
big.
\qed\vspace{3mm}
\begin{prop}\label{obshom}
There exists a homomorphism
$\delta_A:\Gamma(A)\rightarrow\operatorname{Br}(S)$ such that for any $g\in\widehat{\operatorname{SL}}_2(A)$
lying over $\overline{g}\in\Gamma(A)$ there exists a global object in
$\operatorname{Isom}_{H(X)}(\rho,\rho^g)$ if and only if $\delta_A(\overline{g})=0$.
If the $a_{21}$-entry of $\overline{g}$ is an isogeny then
$\delta_A(\overline{g})=e(G_g)$.
\end{prop}
\noindent {\it Proof} . For any $g\in\widehat{\operatorname{SL}}_2(A)$ let us denote
by $\operatorname{Isom}_{H(X)}^0(\rho,\rho^g)\subset\operatorname{Isom}_{H(X)}(\rho,\rho^g)$
the full subcategory of kernels that belong to the core of
the standard $t$-structure on ${\cal D}^b(A\times_S A)$. We claim
that $\operatorname{Isom}_{H(X)}^0(\rho,\rho^g)$ is a gerb.
Indeed, we know this when $\overline{g}\in\Gamma(A)\cap U$. Now by
Lemma \ref{big} any element
in $\widehat{\operatorname{SL}}_2(A)$ can be written as a product
$gg'$ where $g$ and $g'$ lie over $\Gamma(A)\cap U$.
Locally over $S$
there exist Schr\"odinger representations for $G_g$ and
$G_{g'}$, hence (\ref{kernelnew}) defines the corresponding
kernels ${\cal K}(g)\in\operatorname{Isom}_{H(X)}(\rho,\rho^g)$ and
${\cal K}(g')\in\operatorname{Isom}_{H(X)}(\rho,\rho^{g'})$. Now their composition
${\cal K}(gg')=p_{13*}(p_{12}^*{\cal K}(g')\otimes p_{23}^*{\cal K}(g))$ is an element of
$\operatorname{Isom}_{H(X)}(\rho,\rho^{gg'})$ and Prop. 1.7 of \cite{Muk2}
implies that ${\cal K}(gg')$ has only one non-zero sheaf cohomology
which is flat over $S$
(since this is so in the case of an algebraically closed field
considered in \cite{Weilrep}). It follows that any object of
$\operatorname{Isom}_{H(X)}(\rho,\rho^{gg'})$ over any open subset of $U$
is a pure $S$-flat sheaf (perhaps shifted), hence, the gluing axiom is
satisfied.
Now we can define $\delta_A(\overline{g})\in H^2(S,{\Bbb G}_m)$ for an
element $\overline{g}\in\Gamma(A)$ as the class of the gerb
$\operatorname{Isom}_{H(X)}^0(\rho,\rho^g)$ where $g\in\widehat{\operatorname{SL}}_2(A)$ is
any element lying over $\overline{g}$.
When $\overline{g}\in U$ this class is equal to $e(G_g)\in\operatorname{Br}(S)$.
Clearly, $\delta_A$ is a homomorphism $\Gamma(A)\rightarrow H^2(S,{\Bbb G}_m)$.
Since $\Gamma(A)$ is generated by $\Gamma(A)\cap U$ we have
$\delta_A(\overline{g})\in\operatorname{Br}(S)$ for any $\overline{g}\in\Gamma(A)$.
\qed\vspace{3mm}
\begin{lem}\label{neutrcomp}
Let $g=(\overline{g},L^g,M^g)$ be an element of
$\widehat{\operatorname{SL}}_2(A)$ such that $\ker({a_{21}})$ is flat over
$S$, its neutral component $\ker(a_{21})^0$ is an abelian
subscheme of $\hat{A}$, and $L^g|_{\ker(a_{21})^0}$ is trivial.
Then $\delta_A(\overline{g})=0$ if and only if there exists a Schr\"odinger
representation for the Heisenberg extension
$G_g$ of
$\pi_0(\ker(a_{21}))=\ker(a_{21})/(\ker(a_{21}))^0$ induced by
$G(L^g)|_{\ker(a_{21})}$.
\end{lem}
\noindent {\it Proof} . Let $V$ be a Schr\"odinger representation for
$G_g$. Then we can define ${\cal K}(g,V)$ by the formula (\ref{kernelnew})
where
$E$ is defined as follows. First we descend $L^g$ to a
line bundle $\overline{L}$ on $\hat{A}/\ker(a_{21})^0$, then
we set
$$E=\underline{\operatorname{Hom}}_{G_g^{-1}}(V^*,\overline{a}_{21*}(\overline{L}^{-1}))$$
where $\overline{a}_{21}:\hat{A}/\ker(a_{21})^0\rightarrow A$ is the
finite map induced by $a_{21}$. Note that when $S$ is the
spectrum of an algebraically closed field this kernel
${\cal K}(g,V)$ coincides with the one defined in \cite{Weilrep},
12.3. By definition ${\cal K}(g,V)$
is the direct image of a bundle on an abelian subscheme
$$\operatorname{supp} {\cal K}(g,V)=\{(x_1,x_2)\in A^2\ | x_2-a_{22}x_1\in
a_{21}(\hat{A})\}$$
(note that $a_{21}(\hat{A})\subset A$ is an abelian subscheme
since $\ker(a_{21})$ is flat). Applying this to $g^{-1}$ we
get
$$\operatorname{supp} {\cal K}(g^{-1},V^*)=\{(x_1,x_2)\in A^2\ |\
x_2-\hat{a}_{11}x_1\in
\hat{a}_{21}(\hat{A})\}.$$
Hence, the sheaf $p_{12}^*{\cal K}(g^{-1},V^*)\otimes p_{23}^*{\cal K}(g,V)$ on
$A^3$ is supported on the abelian subscheme
$$X(g)=\{(x_1,x_2,x_3)\in A^3\ |\ x_3-a_{22}x_2\in
a_{21}(\hat{A}),
x_2-\hat{a}_{11}x_1\in\hat{a}_{21}(\hat{A})\}.$$
Note that for $(x_1,x_2,x_3)\in X(g)$ we have
$$x_3-a_{22}\hat{a}_{11}x_1\in a_{21}(\hat{A})+
a_{22}\hat{a}_{21}(\hat{A})=a_{21}(\hat{A})$$
since $a_{22}\hat{a}_{21}=a_{21}\hat{a}_{22}$. Now
$a_{22}\hat{a}_{11}x_1\equiv x_1\mod(a_{21}(\hat{A}))$,
hence $x_3\equiv x_1\mod(a_{21}(\hat{A}))$.
Thus, we have an isomorphism
\begin{align*}
&X(g)\rightarrow\{(x,x')\ |\ x-x'\in a_{21}(\hat{A})\}
\times\hat{a}_{21}(\hat{A}):\\
&(x_1,x_2,x_3)\mapsto ((x_1,x_3), x_2-\hat{a}_{11}x_1).
\end{align*}
It follows that the restriction of
the projection $p_{13}$ to $X(g)$ is flat and surjective.
Therefore, applying
Prop. 1.7 of \cite{Muk2} we conclude as in the proof of Theorem
\ref{eq} that $\overline{R}(V)$ is an equivalence.
Thus, the map $V\mapsto {\cal K}(g,V)$ gives a functor from
$\operatorname{Schr}_{G_g}$ to $\operatorname{Isom}_{H(X)}(\rho,\rho^g)$.
\qed\vspace{3mm}
\begin{prop}\label{compat}
Let $A$ and $B$ be abelian schemes over the
same base $S$, then there is a natural embedding
$i_A:\Gamma(A)\rightarrow\Gamma(A\times B)$ such that
$$\delta_A=\delta_{A\times B}\circ i_A.$$
\end{prop}
\noindent {\it Proof} . It is sufficient to check this identity on elements
of $\Gamma(A)\cap U$, in which case this follows immediately
from Lemma \ref{neutrcomp}.
\qed\vspace{3mm}
Consider the subgroup of finite index
$\Gamma_0(A)\subset\Gamma(A)$ consisting of matrices
$\left( \matrix a_{11} & a_{12}\\ a_{21} & a_{22} \endmatrix \right)$
in $\Gamma(A)$ for which $a_{21}$ is divisible by 2.
Note that $\Gamma_0(A)$ contains $\Gamma(A,2)$.
\begin{lem}\label{oddisog}
Let $A$ be an abelian variety over a field such
that there exists a symmetric line bundle $L$ on $A$
which induces an isogeny $f:A\rightarrow\hat{A}$ of odd degree.
Then the subgroup of $\Gamma(A)$ generated by its elements
$\left( \matrix a_{11} & a_{12}\\ a_{21} & a_{22} \endmatrix \right)$,
for which $a_{21}$ is an isogeny of odd degree, contains
$\Gamma_0(A)$.
\end{lem}
\noindent {\it Proof}. Recall that $U\subset\operatorname{SL}_2(A)$ is the subset defined by the
condition that $a_{21}$ is an isogeny. Consider the matrix
$\gamma_f=\left( \matrix \operatorname{id} & 0 \\ f & \operatorname{id} \endmatrix
\right)\in\Gamma(A)$. Let $U_1=U\cap U\gamma_f^{-1}$.
Then the argument similar to that of Lemma \ref{big} shows
that $\Gamma_0(A)\cap U_1$ is a big subset in $\Gamma_0(A)$, in particular,
$\Gamma_0(A)$ is generated by
$\Gamma_0(A)\cap U_1$. Now let $\gamma\in\Gamma_0(A)\cap U_1$, then
$\gamma \gamma_f\in U$ and its $a_{21}$-entry is an isogeny of odd
degree which implies the statement.
\qed\vspace{3mm}
\begin{prop}\label{triv1}
The restriction of the homomorphism
$\delta_A:\Gamma(A)\rightarrow\operatorname{Br}(S)$ to $\Gamma_0(A)$ is trivial.
\end{prop}
\noindent {\it Proof} . Recall that if $a_{21}$ is an isogeny then
$\delta_A(g)$ is defined as an obstacle for
the existence of a Schr\"odinger representation of the
(symmetric) Heisenberg extension $G_g$ of $\ker(a_{21})$ attached to $g$,
where $g$ projects to a matrix
$\left( \matrix a_{11} & a_{12}\\ a_{21} & a_{22} \endmatrix
\right)\in\operatorname{SL}_2(A)$. Hence, by Theorem \ref{odd} $\delta_A(g)=0$
if $a_{21}$ is an isogeny of odd degree. In particular,
if $A$ is principally polarized then Lemma \ref{oddisog}
implies that the restriction of $\delta_A$ to $\Gamma_0(A)$
is trivial.
By Zarhin's trick (see \cite{Zarh}) for any abelian scheme $A$ over $S$
there exists an abelian scheme $B$ over $S$ such that
$A\times B$ admits a principal polarization. Now by
Proposition \ref{compat} we have
$\delta_A=\delta_{A\times B}\circ i_A$. Therefore,
the restriction of $\delta_A$ to $\Gamma_0(A)$ is trivial.
\qed\vspace{3mm}
\begin{rem} It is easy to see that the kernel of $\delta_A$ is in
general bigger than $\Gamma_0(A)$. Namely, it contains also matrices
for which $a_{21}$ (or $a_{12}$) is an isogeny of odd degree,
those for which $a_{12}$ is divisible by 2, and those for which
$a_{11},a_{22}\in{\Bbb Z}$. Sometimes,
these elements together with $\Gamma_0(A)$ generate the entire
group $\Gamma(A)$,
however, it is not clear whether $\delta_A$ is always trivial.
In the section \ref{real} we will prove it in some special cases.
\end{rem}
Let $\widehat{\Gamma}_0(A)$ be the preimage of $\Gamma_0(A)$ in
$\widehat{\operatorname{SL}}_2(A)$. In other words, this is the subgroup
of elements $g\in\widehat{\operatorname{SL}}_2(A)$ such that for the
corresponding matrix in $\operatorname{SL}_2(A)$ the $a_{21}$-entry
is divisible by 2.
We say that there is a faithful action of a group
$G$ on a category ${\cal C}$ if there is an embedding of $G$ into
a group of autoequivalences of ${\cal C}$ (considered up to
isomorphism).
\begin{thm} For any abelian scheme $A$ over a normal connected
noetherian base $S$ there is a faithful action of a central extension of
the group $\widehat{\Gamma}_0(A)$ by ${\Bbb Z}\times\operatorname{Pic}(S)$ on ${\cal D}^b(A)$.
\end{thm}
\noindent {\it Proof} . According to Proposition \ref{triv1}
for every $g\in\widehat{\Gamma}_0(A)$ there
exists a global object in $\operatorname{Isom}_{H(X)}(\rho,\rho^g)$.
It is defined uniquely up to a shift and tensoring with a
line bundle on $S$. Hence, the required action of a central
extension. The fact that this action is faithful is
clear in the case when the base is a field: for example,
one can use explicit formulas for these functors from
Lemma \ref{neutrcomp}. Since the action of $\operatorname{Pic}(S)$ on
${\cal D}^b(A)$ is obviously faithful the general case follows.
\qed\vspace{3mm}
\begin{cor} With the assumptions of the above theorem, there
is a faithful action of a central extension of $\Gamma(A,2)$
by ${\Bbb Z}\times\operatorname{Pic}(S)$ on ${\cal D}^b(A)$.
\end{cor}
\noindent {\it Proof} . This action is obtained from the canonical homomorphism
$\Gamma(A,2)\rightarrow\widehat{\operatorname{SL}}_2(A)$ splitting
the projection $\widehat{\operatorname{SL}}_2(A)\rightarrow\operatorname{SL}_2(A)$. Namely,
under this splitting the matrix
$\overline{g}=\left( \matrix a_{11} & a_{12}\\ a_{21} & a_{22} \endmatrix
\right)\in\Gamma(A,2)$ maps to the element
$(\overline{g},(\hat{a}_{11}a_{21}/2,\operatorname{id})^*\cal P,
(\operatorname{id},\hat{a}_{22}a_{12}/2)^*\cal P)$ of $\widehat{\operatorname{SL}}_2(A)$,
where $\cal P$ is the Poincar\'e line bundle on $A\times\hat{A}$.
\qed\vspace{3mm}
\section{The induced action on a Chow motive}
In this section we will construct a projection action of
the algebraic group $\operatorname{SL}_{2,A,{\Bbb Q}}$ on the relative Chow motive
of an abelian scheme $\pi:A\rightarrow S$ with rational coefficients.
Let us denote by $\operatorname{Cor}(A)$ the Chow group
$\operatorname{CH}^*(A\times_S A)\otimes{\Bbb Q}$
considered as a
${\Bbb Q}$-algebra with multiplication given by the composition of
correspondences:
\begin{equation}\label{compcorr}
\beta\circ\a=p_{13*}(p_{12}^*(\a)\cdot p_{23}^*(\beta))
\end{equation}
where $\a,\beta\in\operatorname{CH}^*(A\times_S A)\otimes {\Bbb Q}$,
$p_{ij}$ are the projections from $A^3$ to $A^2$.
The unit of this algebra is
$[\Delta]\in\operatorname{CH}^g(A\times_S A)$ where $\Delta\subset A\times_S A$ is the
relative diagonal, $g=\dim A$.
Using the Riemann-Roch theorem
it is easy to see that the multiplication (\ref{compcorr})
is compatible with
the composition law on $K^0(A\times_S A)$ arising from the
interpretation of ${\cal D}^b(A\times_S A)$ as the category of
functors from ${\cal D}^b(A)$ to itself considered above,
via the map
\begin{equation}\label{ch}
K^0(A\times_S A)\otimes {\Bbb Q}\widetilde{\rightarrow}\operatorname{CH}^*(A\times_S A)\otimes{\Bbb Q}:
x\mapsto \operatorname{ch}(x)\cdot\pi^*\operatorname{Td}(e^*T_{A/S})
\end{equation}
where $\operatorname{ch}$ is the Chern character.
Let us consider
the embedding of algebras
$$\operatorname{CH}(S)_{{\Bbb Q}}\rightarrow\operatorname{Cor}(A):x\mapsto \pi^*(x)\cdot [\Delta]$$
where $\operatorname{CH}(S)_{{\Bbb Q}}=\operatorname{CH}(S)\otimes{\Bbb Q}$ is equipped with the
usual multiplication. In particular,
we have an embedding of groups of invertible elements
$\operatorname{CH}(S)_{{\Bbb Q}}^*\subset\operatorname{Cor}(A)^*$.
Applying the map (\ref{ch}) to the kernels giving
projective action of $\Gamma(A,2)$ on ${\cal D}^b(A)$ we obtain
a homomorphism
$$\widetilde{\phi}:\Gamma(A,2)\rightarrow(\operatorname{Cor}(A))^*/\pm\operatorname{Pic}(S),$$
where $\operatorname{Pic}(S)$ is embedded into $\operatorname{CH}(S)_{{\Bbb Q}}^*$
by Chern character (multiplication by $\pm$
arises from shifts in derived category).
Our aim is to approximate this homomorphism by
a morphism of algebraic groups over ${\Bbb Q}$.
More precisely, we have to replace $\widetilde{\phi}$ by the induced
homomorphism
$$\phi:\Gamma(A,2)\rightarrow(\operatorname{Cor}(A))^*/\operatorname{CH}(S)_{{\Bbb Q}}^*.$$
Now we claim that one can replace here source and target
by some algebraic groups over ${\Bbb Q}$ such that $\phi$ will be
induced by an algebraic homomorphism.
Naturally, the source should be replaced by
$\operatorname{SL}_{2,A,{\Bbb Q}}$ (see the previous section). To
approximate the target we have to replace algebras $\operatorname{Cor}(A)$ and
$\operatorname{CH}(S)_{{\Bbb Q}}$ by their finite-dimensional subalgebras.
\begin{thm}\label{actmotmain}
There exists a finite-dimensional ${\Bbb Q}$-subalgebra
$D\subset\operatorname{Cor}(A)$ and a morphism of algebraic
${\Bbb Q}$-groups $\rho:\operatorname{SL}_{2,A,{\Bbb Q}}\rightarrow D^*/(D\cap\operatorname{CH}(S)_{{\Bbb Q}})^*$
inducing $\phi$ on $\Gamma(A,2)$.
\end{thm}
\noindent {\it Proof} . For a pair of abelian schemes $A$ and $B$ over $S$
let us consider the map
$$\gamma_{A,B}:\operatorname{Hom}(A,B)\rightarrow\operatorname{CH}(A\times_S B)$$
that sends an $S$-morphism $f:A\rightarrow B$ to
the class $[\Gamma_f]$ of the (relative) graph of $f$.
One can extend naturally $\gamma$ to a
map
$$\gamma:\operatorname{Hom}(A)\otimes{\Bbb Q}\rightarrow\operatorname{CH}(A\times_S B)\otimes{\Bbb Q}$$
by sending
$f/n$ to $\gamma([n]_A)^{-1}\circ\gamma(f)$, where
$f\in\operatorname{End}(A)$, $n\neq0$ --- here we take the inverse
to $\gamma([n]_A)$ in the algebra $\operatorname{Cor}(A)$ and
use its natural action on $\operatorname{CH}(A\times_S B)\otimes{\Bbb Q}$.
It is easy to check that $\gamma$ is a polynomial map
(see \cite{Weilrep}, Lemma 13.3, for the case $A=B$).
Now let
$\overline{g}=\left( \matrix a_{11} & a_{12}\\ a_{21} & a_{22} \endmatrix
\right)$ be an element of $\Gamma(A,2)\cap U$ (recall that $U$
is defined by the condition that $a_{21}$ is an isogeny).
From the formula (\ref{kernelnew}) we
get the following expression for $\phi(\overline{g})$:
\begin{equation}\label{chker}
\phi(\overline{g})=(p_2-a_{22}p_1)^*(a_{21*}\operatorname{ch}(L^g)^{-1})\cdot
(a_{12}\times\operatorname{id})^*(\operatorname{ch}(\cal P))\cdot
p_1^*(\operatorname{ch}(M^g))\mod\operatorname{CH}(S)_{{\Bbb Q}}^*.
\end{equation}
Note that $\operatorname{ch}(L^g)$ (resp. $\operatorname{ch}(M^g)$) is a polynomial function of
$\hat{a}_{11}a_{21}$ (resp. $\hat{a}_{22}a_{12}$).
Also the functors $f^*$ and $f_*$ can be expressed as
compositions with the correspondence given by the graph of $f$.
It follows from the above remarks that the right hand side of (\ref{chker})
is obtained by evaluating at $\overline{g}$ of a polynomial map
$\psi:\operatorname{End}(\hat{A}\times A)\otimes{\Bbb Q}\rightarrow\operatorname{Cor}(A)$.
In particular, the image of $\psi$ belongs to a
finite-dimensional ${\Bbb Q}$-subspace of $\operatorname{Cor}(A)$.
Let $\underline{U}\subset\operatorname{SL}_{2,A,{\Bbb Q}}$ be the Zariski open
subset defined by $\deg(a_{21})\neq0$. Note that
$\underline{U}$ is stable under the inversion morphism $g\mapsto
g^{-1}$. Let us also denote
$\underline{U}^{(2)}=
\mu^{-1}(\underline{U})\cap(\underline{U}\times\underline{U})\subset\underline{U}\times
\underline{U}$ where $\mu$ is the group law.
Consider two polynomial maps
\begin{align*}
&a_1:\underline{U}\rightarrow\operatorname{Cor}(A):u\mapsto a_1(u)=\psi(u^{-1})\circ\psi(u),\\
&a_2:\underline{U}^{(2)}\rightarrow\operatorname{Cor}(A):
(u_1,u_2)\mapsto a_2(u_1,u_2)=\psi(u_1u_2)\circ\psi(u_2^{-1})\circ
\psi(u_1^{-1}).
\end{align*}
It is easy to see that the images of both maps belong to the
subalgebra $\operatorname{CH}(S)_{{\Bbb Q}}\subset\operatorname{Cor}(A)$. This can be done either
by direct computation using (\ref{chker}) or using the density
of $\Gamma$ in $H\subset\operatorname{SL}_{2,A,{\Bbb Q}}$ (see the previous section).
Also an easy direct computation shows that
$a_1(u)$ is invertible in the algebra $\operatorname{CH}(S)_{{\Bbb Q}}$ for all
$u\in\underline{U}$. This immediately implies that $a_2(u_1,u_2)$
is invertible for any $(u_1,u_2)\in\underline{U}^{(2)}$: indeed,
$a_2(u_1,u_2)$ is a divisor of
$a_1(u_1)a_1(u_2)a_1(u_2^{-1}u_1^{-1})$. Note that the
components of images of $a_1$ and $a_2$ span finite-dimensional subspaces
in $\operatorname{CH}(S)^i_{{\Bbb Q}}$ for any $i$. It follows that there exists a
finite dimensional subalgebra $D_S\subset\operatorname{CH}(S)_{{\Bbb Q}}$ such that
the images of $a_1$ and $a_2$ belong to $D_S^*$. Now we have
\begin{equation}\label{projmot}
\psi(u_1)\circ\psi(u_2)=a_2(u_1,u_2)^{-1}a_1(u_1)a_1(u_2)
\psi(u_1u_2)
\end{equation}
for $(u_1,u_2)\in\underline{U}^{(2)}$. Let $D\subset\operatorname{Cor}(A)$ be the
$D_S$-submodule generated by $\psi(u)$ with $u\in\underline{U}$.
Then $D$ is finite-dimensional as a ${\Bbb Q}$-vector space
and (\ref{projmot}) shows that
$\psi(u_1)\circ\psi(u_2)\in D$ for any
$(u_1,u_2)\in\underline{U}^{(2)}$. Since $\underline{U}^{(2)}$ is dense
in $\underline{U}\times\underline{U}$ it follows that $D$ is a subalgebra.
Now (\ref{projmot}) implies that $\psi$ uniquely extends to
a homomorphism $\operatorname{SL}_{2,A,{\Bbb Q}}\rightarrow D^*/D_S^*$.
\qed\vspace{3mm}
\section{Splittings of the extension
$\widetilde{SL}_2(A)\rightarrow SL_2(A)$}
Let $A/S$ be an abelian scheme with a principal
polarization $\phi:A\rightarrow\hat{A}$. Then we have the
Rosati involution
$$\varepsilon_{\phi}:\operatorname{End}(A)\rightarrow\operatorname{End}(A):
f\mapsto \phi^{-1}\circ \hat{f}\circ \phi.$$
The group $\operatorname{SL}_2(A)$ is completely determined by
the algebra $\operatorname{End}(A)$ with involution $\varepsilon_{\phi}$.
The definition of the group $\widetilde{\operatorname{SL}}_2(A)$
requires in addition the knowledge of the extension
\begin{equation}\label{extA}
0\rightarrow \hat{A}(S)\rightarrow\operatorname{Pic}(A)\rightarrow\operatorname{Hom}^{\operatorname{sym}}(A,\hat{A})
\end{equation}
together with the action of the multiplicative monoid of $\operatorname{End}(A)$ on it.
Thus, splittings of the homomorphism
$\widetilde{\operatorname{SL}}_2(A)\rightarrow\operatorname{SL}_2(A)$ should be related to
splittings of (\ref{extA}).
More precisely, it's natural to consider
splittings compatible with the $\operatorname{End}(A)$-action. We'll show that such
splittings of (\ref{extA}) correspond to simultaneous splittings
of homomorphisms $\widetilde{\operatorname{SL}}_2(A^n)\rightarrow\operatorname{SL}_2(A^n)$ for all $n$,
where $A^n/S$ is the $n$-th relative cartesian power of $A/S$.
More generally, we start with arbitrary subring $R\subset\operatorname{End}(A)$
stable under the Rosati involution. Let us denote by
$\varepsilon:R\rightarrow R$ the restriction of the Rosati involution to $R$,
let $R^+\subset R$ be the subring of elements stable under $\varepsilon$.
Then for any $n\ge 1$ we can consider
the subgroup $\operatorname{SL}_2(A^n,R)\subset\operatorname{SL}_2(A^n)$ consisting of
$2n\times 2n$ matrices
with all entries belonging to $R$ (we identify $\hat{A}$ with
$A$ via $\phi$). Let $\widetilde{\operatorname{SL}}_2(A^n,R)\subset\widetilde{\operatorname{SL}}_2(A^n)$
be the preimage of $\operatorname{SL}_2(A^n,R)$. We are interested in splittings
of the natural homomorphisms
\begin{equation}\label{homSL}
\widetilde{\operatorname{SL}}_2(A^n,R)\rightarrow\operatorname{SL}_2(A^n,R)
\end{equation}
It turns out that the following structure on $A$ is relevant for
this.
\begin{defi} A $\Sigma_{R,\varepsilon}$-structure for $\phi$ is a
homomorphism $R^+\rightarrow\operatorname{Pic}(A):r_0\mapsto L(r_0)$
such that
\begin{equation}\label{CM1}
\phi_{L(r_0)}=\phi\circ [r_0]_A
\end{equation}
for any $r_0\in R^+$ and
\begin{equation}\label{CM2}
[r]^*L(r_0)\simeq L(\varepsilon(r)r_0r)
\end{equation}
for any $r\in R$, $r_0\in R^+$.
\end{defi}
Note that (\ref{CM2}) for $r=-1$ implies that all line bundles
$L(r_0)$ are symmetric.
In \cite{thetaid} we studied the question of existence of
$\Sigma_{R,\varepsilon}$-structure for an
abelian variety.
For example, in the case of a complex elliptic curve $E$ with complex
multiplication such a structure for $R=\operatorname{End}(E)$ exists
if and only if $R$ is unramified at $2$. Another example
is the case when $R$ is a ring of integers in a
totally real number field unramified at 2 and $\varepsilon=\operatorname{id}$
(see next section).
\begin{thm}\label{Sig}
A $\Sigma_{R,\varepsilon}$-structure on $A$
induces canonical splittings of the homomorphisms
(\ref{homSL}) for all $n$.
\end{thm}
\noindent {\it Proof} . It is easy to see that a $\Sigma_{R,\varepsilon}$-structure
on $A$ induces a similar structure on $A^n$ with $R$
replaced by the matrix algebra $\operatorname{Mat}_n(R)$ and
$\varepsilon$ replaced by the corresponding
involution $(a_{ij})\mapsto (\varepsilon(a_{ji}))$ of $\operatorname{Mat}_n(R)$.
Hence, it is sufficient to consider the case $n=1$.
In this case we define the splitting
$$\operatorname{SL}_{2}(A,R)\rightarrow\widehat{\operatorname{SL}}_2(A,R):
g=\left( \matrix a_{11} & a_{12}\\ a_{21} & a_{22} \endmatrix
\right)\mapsto (g,
(\phi^{-1})^*L(\varepsilon(a_{11})a_{21}),
L(\varepsilon(a_{22})a_{12}))$$
\qed\vspace{3mm}
Now we are going to prove that conversely the existence
of splitting of (\ref{homSL}) for $n=2$ implies the
existence of $\Sigma_{R,\varepsilon}$-structure. We use the following
observation. For any abelian scheme $A$ there is a natural embedding of
the semi-direct product
$\operatorname{Aut}(A)\ltimes\operatorname{Hom}^{\operatorname{sym}}(A,\hat{A})$ into $\operatorname{SL}_2(A)$ as the
subgroup of matrices with $a_{21}=0$. Thus, a splitting of
the homomorphism $\widetilde{\operatorname{SL}}_2(A)\rightarrow\operatorname{SL}_2(A)$ restricts to
a splitting of the homomorphism of $\operatorname{Aut}(A)$-modules
$\operatorname{Pic}(A)\rightarrow\operatorname{Hom}^{\operatorname{sym}}(A,\hat{A})$. In the situation
considered above we have similar subgroups in
$\operatorname{SL}_n(A^n,R)$. Note that the subgroup of
$\operatorname{Hom}^{\operatorname{sym}}(A^n,\hat{A}^n)$ consisting of matrices with
entries in $R$ can be identified with the group of
hermitian matrices $\operatorname{Mat}^{\operatorname{herm}}_n(R)$ and the natural right
action of $\operatorname{GL}_n(R)$ on it induced by its action on $A^n$
is given by the formula
$$B\mapsto \overline{C}BC$$
where $C=(c_{ij})\in\operatorname{GL}_n(R)$, $B\in\operatorname{Mat}^{\operatorname{herm}}_n(R)$,
$\overline{C}=(\varepsilon(c_{ji}))$.
Thus, a splitting of the homomorphism
(\ref{homSL}) induces a splitting of the homomorphism of
$\operatorname{GL}_n(R)$-modules
$$\operatorname{Pic}(A^n,R)\rightarrow\operatorname{Mat}^{\operatorname{herm}}_n(R)$$
where $\operatorname{Pic}(A^n,R)\subset\operatorname{Pic}(A^n)$ is the subgroup
of line bundles $L$ such that
$\phi_L\in\operatorname{Hom}^{\operatorname{sym}}(A^n,\hat{A}^n)$ has entries in $R$.
Now we claim that such a splitting for $n=2$ leads to
a $\Sigma_{R,\varepsilon}$-structure on $A$.
\begin{thm} Any splitting of the homomorphism
of $\operatorname{GL}_2(R)$-modules
$\operatorname{Pic}(A^2,R)\rightarrow\operatorname{Mat}^{\operatorname{herm}}_2(R)$ is
induced by a unique $\Sigma_{R,\varepsilon}$-structure.
\end{thm}
\noindent {\it Proof} . Let
$$s:\operatorname{Mat}^{\operatorname{herm}}_2(R)\rightarrow\operatorname{Pic}(A^2,R)$$
be such a splitting. Then for $r_0\in R^+$ one has
\begin{equation}\label{s1}
s\left( \matrix r_0 & 0 \\ 0 & 0 \endmatrix
\right)=p_1^*L(r_0)\otimes p_2^*\eta(r_0)
\end{equation}
for some line bundle $L(r_0)$ and $\eta(r_0)$ on $A$ such that
$\phi_{L(r_0)}=\phi\circ [r_0]$, $\eta(r_0)\in\operatorname{Pic}^0(A)$.
The compatibility of $s$ with the action of $\operatorname{GL}_2(R)$
means that
$$[C]^*s(B)=s(\overline{C}BC)$$
where $B\in\operatorname{Mat}^{\operatorname{herm}}_2(R)$, $C=(c_{ij})\in\operatorname{GL}_2(R)$,
$\overline{C}=(\varepsilon(c_{ji}))$. Applying this to
$C=\left( \matrix 0 & 1 \\ 1 & 0 \endmatrix
\right)$ we deduce from (\ref{s1}) the equality
\begin{equation}\label{s2}
s\left( \matrix 0 & 0 \\ 0 & r_0 \endmatrix
\right)=p_2^*L(r_0)\otimes p_1^*\eta(r_0).
\end{equation}
Also using the identity
$$\left( \matrix 1 & 0 \\ \varepsilon(r) & 1 \endmatrix
\right)\cdot
\left( \matrix 1 & 0 \\ 0 & 0 \endmatrix
\right)\cdot
\left( \matrix 1 & r \\ 0 & 1 \endmatrix
\right)=
\left( \matrix 1 & r \\ \varepsilon(r) & \varepsilon(r)r \endmatrix
\right)$$
for any $r\in R$ we deduce that
$$s\left( \matrix 1 & r \\ \varepsilon(r) & \varepsilon(r)r \endmatrix
\right)=
\left( \matrix \operatorname{id} & [r] \\ 0 & \operatorname{id} \endmatrix
\right)^*(p_1^*L(1)\otimes p_2^*\eta(1))=
(p_1+[r]p_2)^*L(1)\otimes p_2^*\eta(1).$$
Combining this with (\ref{s1}) and (\ref{s2})
one can easily compute that
\begin{equation}\label{s3}
s\left( \matrix 0 & r \\ \varepsilon(r) & 0 \endmatrix
\right)=(\phi\times [r])^*\cal P\otimes p_1^*\eta(-\varepsilon(r)r)\otimes
p_2^*([r]^*L(1)\otimes L(-\varepsilon(r)r)).
\end{equation}
Note that the formulas (\ref{s1}), (\ref{s2}), and (\ref{s3})
completely determine $s$. Now the identity
$$\left( \matrix 1 & \varepsilon(r) \\ 0 & 1 \endmatrix
\right)\cdot
\left( \matrix 0 & 0 \\ 0 & r_0 \endmatrix
\right)\cdot
\left( \matrix 1 & 0 \\ r & 1 \endmatrix
\right)=
\left( \matrix \varepsilon(r)r_0r & \varepsilon(r)r_0 \\ r_0r & r_0 \endmatrix
\right)$$
which holds for any $r\in R$, $r_0\in R^+$ implies
the equality
\begin{equation}\label{s4}
\left( \matrix {\operatorname{id}} & 0 \\ {[r]} & {\operatorname{id}} \endmatrix
\right)^*(p_2^*L(r_0)\otimes p_1^*\eta(r_0))=
s\left( \matrix {\varepsilon(r)r_0r} & {\varepsilon(r)r_0} \\ {r_0r} & {r_0} \endmatrix
\right).
\end{equation}
Computing the right hand side using (\ref{s1})--(\ref{s3})
and restricting to $A\times 0$ we obtain the
identity
\begin{equation}\label{s5}
[r]^*L(r_0)=L(\varepsilon(r)r_0r)\otimes\eta(-r_0r\varepsilon(r)r_0).
\end{equation}
On the other hand, setting $r=1$ and restricting (\ref{s4})
to $0\times A$ we get
\begin{equation}\label{s6}
L(r_0^2)=[r_0]^*L(1)\otimes\eta(r_0).
\end{equation}
Setting $r=1$ in (\ref{s5}) we obtain the triviality
of $\eta(r_0^2)$. Then taking $r_0=1$ and $r\in R^+$
in (\ref{s5}) we obtain that
$$[r]^*L(1)=L(r^2)$$
for $r\in R^+$. Comparing this with (\ref{s6}) we deduce
the triviality of $\eta(r_0)$ for all $r_0\in R^+$.
Now (\ref{s5}) implies that $L(\cdot)$ gives a
$\Sigma_{R,\varepsilon}$-structure.
\qed\vspace{3mm}
\begin{cor} A $\Sigma_{R,\varepsilon}$-structure for $\phi$
exists if and only if a splitting of the homomorphism
(\ref{homSL}) for $n=2$ exists.
\end{cor}
\begin{ex} Let $E={\Bbb C}/{\Bbb Z}[i]$ be an elliptic curve with complex
multiplication by the ring of Gaussian numbers $R={\Bbb Z}[i]$,
so that the corresponding Rosati involution $\varepsilon$ is just
the complex conjugation. In this situation there is no
$\Sigma_{R,\varepsilon}$-structure corresponding to the standard
polarization of $E$. Indeed, the corresponding line
bundle $L(1)$ should be of the form $\O(x)$ where $x$ is
a point of order 2 on $E$. Now the identity
$[1+i]^*L(1)=L(2)$ leads to a contradiction (see
\cite{thetaid} for details).
\end{ex}
\section{Abelian schemes with real multiplication}\label{real}
Let $F$ be a totally real number field,
$R$ be its ring of integers.
Let $A\rightarrow S$ be an abelian scheme with real
multiplication by $R$, i.e. a ring homomorphism
$R\rightarrow\operatorname{End}_S(A):r\mapsto [r]_A$ is given. Then the dual
abelian scheme $\hat{A}$ also has a natural real
multiplication by $R$.
Let $J, \hat{J}\subset F$ be fractional ideals for $R$ (=
non-zero finitely generated $R$-submodules of $F$)
such that $J\hat{J}\subset R$.
\begin{defi} An $(J, \hat{J})$-polarization on $A$ is a pair
of $R$-module homomorphisms
\begin{align*}
&\lambda_J:J\rightarrow\operatorname{Hom}_{R}^{\operatorname{sym}}(A,\hat{A}),\\
&\lambda_{\hat{J}}:\hat{J}\rightarrow\operatorname{Hom}_{R}^{\operatorname{sym}}(\hat{A},A)
\end{align*}
where $\operatorname{Hom}_{R}^{\operatorname{sym}}(A,\hat{A})$ is an $R$-module of
symmetric $R$-linear homomorphisms $f:A\rightarrow\hat{A}$
(i.e. $\hat{f}=f$ and $f\circ [r]_A=[r]_{\hat{A}}\circ f$ for any $r\in
R$), such that $\lambda_{\hat{J}}(m)\circ\lambda_J(l)=[lm]_A$ and
$\lambda_J(l)\lambda_{\hat{J}}(m)=[lm]_{\hat{A}}$ for any $l\in J$, $m\in \hat{J}$.
\end{defi}
\begin{rem} Usually one also imposes some positivity
condition on a polarization. In the case of $(J,\hat{J})$-polarizations
one can fix an ordering on $J$: this means that for each
embedding $\sigma:F\rightarrow{\Bbb R}$ an orientation of the line
$J\otimes_{R,\sigma}{\Bbb R}$ is chosen. Then
one should require that if an element $l\in J$ is totally
positive then the homomorphism $\lambda_J(l):A\rightarrow\hat{A}$ is positive
(i.e. $\lambda_J(l)$ is a polarization in the classical sense).
\end{rem}
Note that the notion of $(J,\hat{J})$-polarization is equivalent to
that of $(Jx, \hat{J}x^{-1})$-polarization for any $x\in F^*$.
Also an $(J,\hat{J})$-polarization of $A$ is the same as
an $(\hat{J},J)$-polarization of $\hat{A}$.
When $\hat{J}=J^{-1}$ we recover the notion of
$J$-polarization in the sense of P.~Deligne and G.~Pappas \cite{DePa}
(except for the positivity condition).
Recall that they define
an $J$-polarization of an abelian scheme $A$ with real
multiplication by $R$ as an $R$-linear homomorphism
$\lambda:J\rightarrow\operatorname{Hom}_{R}^{\operatorname{sym}}(A,\hat{A})$ such that
the image of a totally positive element of $J$ under $\lambda$
is positive, and the induced morphism $A\otimes_{R} J\rightarrow\hat{A}$
is an isomorphism.
In this case we have also an isomorphism
$\hat{A}\otimes_{R} J^{-1}\widetilde{\rightarrow} A$ which induces an
$R$-linear homomorphism
$\lambda':J^{-1}\rightarrow\operatorname{Hom}_{R}^{\operatorname{sym}}(\hat{A},A)$,
hence we get an $(J,J^{-1})$-polarization in our sense.
Conversely, given an $(J,J^{-1})$-polarization as above
then the morphism
$\mu:A\otimes_R J\rightarrow\hat{A}$ induced by $\lambda_J$
and the morphism $\mu':\hat{A}\rightarrow A\otimes_R J$ induced by
$\lambda_{J^{-1}}$ are inverse to each other, so that $\lambda_J$
gives an $J$-polarization of $A$ (except for the positivity
condition).
For a pair $(J,\hat{J})$ as above we define the subgroup
$\Gamma(J,\hat{J})\subset\operatorname{SL}_2(F)$ as follows:
$$\Gamma(J,\hat{J})=\{
\left( \matrix a_{11} & a_{12}\\ a_{21} & a_{22} \endmatrix
\right)\in\operatorname{SL}_2(F): a_{11}, a_{22}\in R, a_{12}\in J,
a_{21}\in \hat{J} \}.$$
Note that for $x\in F^*$ the homomorphisms
$\rho_{J,\hat{J}}$ and $\rho_{Jx,\hat{J}x^{-1}}$ are compatible with
the natural isomorphism $\Gamma(J,\hat{J})\simeq \Gamma(Jx,\hat{J}x^{-1})$
(induced by the conjugation by
$\left( \matrix x^{\frac{1}{2}} & 0\\ 0 & x^{-\frac{1}{2}} \endmatrix
\right)$).
In particular, in the case $R={\Bbb Z}$ the group
$\Gamma(J,\hat{J})$ is always isomorphic to the principal congruenz-subgroup
$\Gamma_0(N)=\Gamma({\Bbb Z},N{\Bbb Z})\subset \operatorname{SL}_2({\Bbb Z})$ (for some $N>0$).
If $A$ is $(J,\hat{J})$-polarized then
using $\lambda_J$ and $\lambda_{\hat{J}}$ we can define a homomorphism
$$\rho_{J,\hat{J}}:\Gamma(J,\hat{J})\rightarrow\operatorname{SL}_2(A):
\left( \matrix a_{11} & a_{12}\\ a_{21} & a_{22} \endmatrix
\right)\mapsto
\left( \matrix [a_{11}]_{\hat{A}} & \lambda_J(a_{12})\\
\lambda_{\hat{J}}(a_{21}) & [a_{22}]_A \endmatrix
\right)$$
\begin{lem} For any non-zero element $r\in R$ (resp. $l\in
J$, $m\in \hat{J}$) the corresponding morphism $[r]_A$ (resp.
$\lambda_J(l)$, $\lambda_{\hat{J}}(m)$) is an isogeny.
\end{lem}
\noindent {\it Proof} . There exists a non-zero integer $N$ such that $r'=N/r\in R$, so
that $[r']_A\circ [r]_A=[N]_A$. Hence, $[r]_A$ is an isogeny on each
fiber, therefore, it is an isogeny. Similar argument works for
$\lambda_J(l)$ and $\lambda_{\hat{J}}(m)$.
\qed\vspace{3mm}
Recall that we denote by $\Gamma(A)$ the image of the
homomorphism $\widehat{\operatorname{SL}}_2(A)\rightarrow\operatorname{SL}_2(A)$.
Let assume for simplicity that the image $\rho_{J,\hat{J}}$ is
contained in $\Gamma(A)$. Otherwise, we can consider a
finite flat base change $S'\rightarrow S$ such that the image of
$\Gamma(J,\hat{J})$ is contained in $\Gamma(A_{S'})$ where $A_{S'}$ is the induced
abelian scheme over $S'$. Indeed, recall that one has an exact
sequence
$$0\rightarrow\hat{A}_2\rightarrow\operatorname{Pic}^{\operatorname{sym}}(A)\rightarrow\underline{\operatorname{Hom}}^{\operatorname{sym}}(A,\hat{A})\ra0$$
of sheaves in fppf topology, hence the boundary homomorphism
$J\stackrel{\lambda_J}{\rightarrow}\operatorname{Hom}^{\operatorname{sym}}(A,\hat{A})\rightarrow H^1(S,\hat{A}_2)$ which can
be considered as a $J^*\otimes\hat{A}_2$-torsor over $S$ where
$J^*=\operatorname{Hom}_{\Bbb Z}(J,{\Bbb Z})$ (similarly, for $\lambda_{\hat{J}}$ we get an
$\hat{J}^*\otimes A_2$-torsor). Now we can take $S'$ to be the
corresponding $J^*\otimes\hat{A}_2\times \hat{J}^*\otimes A_2$-torsor
over $S$.
By the above lemma we can define an obstruction homomorphism
$\delta_{J,\hat{J}}:\Gamma(J,\hat{J})\rightarrow\operatorname{Br}(S)$ as follows:
if $a_{21}$-entry of the matrix $h\in\Gamma(J,\hat{J})$
is non-zero then the same entry of $\overline{g}=\rho_{J,\hat{J}}(h)$ is an
isogeny and we can put $\delta_{J,\hat{J}}(h)=e(G_g)$ where
$g\in\widetilde{\operatorname{SL}}_2(A)$ lies above $\overline{g}$. Otherwise,
$\delta_{J,\hat{J}}(h)=0$. As in proposition \ref{obshom} one can check that
$\delta_{J,\hat{J}}$ is a homomorphism.
Let $I\subset R$ be a non-zero ideal.
Let us denote by $\overline{\Gamma}_I(J,\hat{J})$ the group of matrices
$\left( \matrix \bar{a}_{11} & \bar{a}_{12}\\ \bar{a}_{21} &
\bar{a}_{22} \endmatrix\right)$ where
$\bar{a}_{11},\bar{a}_{22}\in R/I$, $\bar{a}_{12}\in J/IJ$,
$\bar{a}_{21}\in \hat{J}/I\hat{J}$, such that
$\bar{a}_{11}\bar{a}_{22}-\bar{a}_{12}\bar{a}_{21}=1$.
Here we use the natural
homomorphism of $R/I$-modules $J/IJ\otimes \hat{J}/I\hat{J}\rightarrow R/IR$ induced
by $J\otimes \hat{J}\rightarrow R$.
\begin{lem}\label{surj} Assume that $I\subset J\hat{J}$. Then
the natural reduction homomorphism
$\pi_I:\Gamma(J,\hat{J})\rightarrow\overline{\Gamma}_I(J,\hat{J})$ is surjective.
\end{lem}
\noindent {\it Proof} . Let
$\left( \matrix \bar{a}_{11} & \bar{a}_{12}\\ \bar{a}_{21} &
\bar{a}_{22} \endmatrix\right)\in \overline{\Gamma}_I(J,\hat{J})$ be any element.
Choose any non-zero liftings
$a_{12}\in J$, $a'_{21}\in \hat{J}$ and $a'_{22}\in R$ of
$\bar{a}_{12}$, $\bar{a}_{21}$ and $\bar{a}_{22}$.
For an element $r\in R$ and a finite $R$-module $Q$ we say
that $r$ is relatively prime to $Q$ if $Q=rQ$, or
equivalently, $r\not\in{\goth p}$ for any prime ideal ${\goth p}$
associated with $Q$. By the Chinese
remainder theorem we can lift $\bar{a}_{11}$ to an element
$a_{11}\in R$ which is relatively prime to $J/Ra_{12}$.
On the other hand, $a_{11}$ is relatively prime to $R/J\hat{J}$ since
$a_{11}a'_{22}\equiv 1\mod(I)$ and $I\subset J\hat{J}$. Therefore, $a_{11}$ is
relatively prime to $J/J\hat{J}a_{12}$ (since
$\operatorname{supp}(J/J\hat{J}a_{12})\subset\operatorname{supp}(R/J\hat{J})\cup\operatorname{supp}(J/Ra_{12})$).
It follows that $J=a_{11}J+J\hat{J}a_{12}=(Ra_{11}+\hat{J}a_{12})J$ which
implies the equality $R=Ra_{11}+\hat{J}a_{12}$. Thus,
we can write $1=ra_{11}+ma_{12}$ where $r\in R$, $m\in \hat{J}$.
Let $x=a_{11}a'_{22}-a_{12}a'_{21}-1\in I$.
Then replacing $a'_{22}$ and $a'_{21}$ by
$a_{22}=a'_{22}-xr$, $a_{21}=a'_{21}-xm$
we achieve $a_{11}a_{22}-a_{12}a_{21}=1$.
\qed\vspace{3mm}
\begin{prop}\label{triv2}
The homomorphism $\delta_{J,\hat{J}}:\Gamma(J,\hat{J})\rightarrow\operatorname{Br}(S)$ is trivial.
\end{prop}
\noindent {\it Proof} .
Consider the reduction homomorphism
$\pi_I:\Gamma(J,\hat{J})\rightarrow\overline{\Gamma}_I(J,\hat{J})$
where $I\subset 2J\hat{J}$.
Then by Lemma \ref{surj} $\pi_I$ is surjective.
On the other hand, the kernel of $\pi_I$ is contained in the subgroup
$\Gamma(J,2\hat{J})\subset\Gamma(J,\hat{J})$.
By Proposition \ref{triv1} the restriction of $\delta_{J,\hat{J}}$ to
the subgroup $\Gamma(J,2\hat{J})$ is trivial. Hence,
$\delta_{J,\hat{J}}(\ker(\pi_I))=0$, so that
$\delta_{J,\hat{J}}=\overline{\delta}\circ\pi_I$ for some homomorphism
$\overline{\delta}:\overline{\Gamma}_I(J,\hat{J})\rightarrow\operatorname{Br}(S)$. Moreover,
since by Lemma \ref{surj} the homomorphism
$\Gamma(J,2\hat{J})\rightarrow\overline{\Gamma}_I(J,2\hat{J})$ is surjective,
it follows that $\overline{\delta}$ is trivial on
matrices with $\bar{a}_{21}\in 2\hat{J}/I\hat{J}$.
In particular, $\overline{\delta}$ vanishes on any diagonal matrix.
Let
$h=\left(\matrix \bar{a}_{11} & \bar{a}_{12}\\ \bar{a}_{21} &
\bar{a}_{22} \endmatrix\right)$ be any element of
$\overline{\Gamma}_I(J,\hat{J})$. Then $\bar{a}_{11}\bar{a}_{22}\equiv
1\mod (J\hat{J})$, hence $\bar{a}_{11}\mod (J\hat{J})$ is a unit in
$R/J\hat{J}$. Let $u\in (R/I)^*$ be any unit such that
$u\equiv a_{11}\mod (J\hat{J})$ (such a unit always exists since
$R/I$ is an artinian ring). Then replacing $h$ by
$h\cdot\left(\matrix u & 0\\ 0 & u^{-1} \endmatrix\right)$
we reduce the problem of showing that $\overline{\delta}(h)=0$
to the case when $\bar{a}_{11}\equiv 1\mod(J\hat{J})$.
Now we use
the result of L.~Vaserstein \cite{Vas} which asserts
that if $F\neq {\Bbb Q}$ then
the subgroup of $\Gamma(J,\hat{J})$ consisting of matrices with
$a_{11}\equiv 1\mod(J\hat{J})$ is generated by elementary
matrices, i.e. matrices of the form
$\left( \matrix 1 & l \\ 0 & 1 \endmatrix\right)$ and
$\left( \matrix 1 & 0 \\ m & 1 \endmatrix\right)$, where
$l\in L$, $m\in M$. In the case $F={\Bbb Q}$ we may assume that
$J={\Bbb Z}$, $\hat{J}=N{\Bbb Z}$ for some $N\in{\Bbb Z}$ and the corresponding
assertion for $\Gamma({\Bbb Z},N{\Bbb Z})=\Gamma_0(N)$ is trivial.
Note that $\delta_{J,\hat{J}}$ vanishes on
elementary matrices (the corresponding Heisenberg groups are
either Mumford groups of line bundles or trivial, so they
admit Shr\"odinger representations), hence it vanishes on any
matrix with $a_{11}\equiv 1\mod(J\hat{J})$ and we are done.
\qed\vspace{3mm}
Let $\widehat{\Gamma}(J,\hat{J})$ be the preimage of $\Gamma(J,\hat{J})$ under the
homomorphism $\widehat{\operatorname{SL}}_2(A)\rightarrow\operatorname{SL}_2(A)$.
\begin{thm} For every $(J,\hat{J})$-polarized abelian scheme $A$
over $S$ with
$R$-multiplication such that the image of $\rho_{J,\hat{J}}$
is contained in $\Gamma(A)$ there exists a faithful action of
a central extension of the group $\widehat{\Gamma}(J,\hat{J})$
by ${\Bbb Z}\times\operatorname{Pic}(S)$ on ${\cal D}^b(A)$.
Without this assumption we always have compatible faithful projective
actions of $\Gamma(2J,2\hat{J})$ on ${\cal D}^b(A)$ and of
$\widehat{\Gamma}(J,\hat{J})$
on $A_{S'}$ for some finite flat base change $S'\rightarrow S$.
\end{thm}
\begin{cor} Let $A$ be an abelian scheme over a normal
noetherian connected base $S$. Assume that the projection
$\widehat{\operatorname{SL}}_2(A)\rightarrow\operatorname{SL}_2(A)$ is surjective and
$\operatorname{End}_S(A)\simeq R$ is a totally real field. Then there is a
faithful
action of a central extension of $\widehat{\operatorname{SL}}_2(A)$ by
${\Bbb Z}\times\operatorname{Pic}(S)$ on ${\cal D}^b(A)$.
\end{cor}
\noindent {\it Proof} . By Prop. X 1.5 of \cite{Geer} the general fiber $A_K$ admits an
$R$-linear polarization $\lambda:A_K\rightarrow\hat{A}_K$. Hence,
$J=\operatorname{Hom}_K(A_K,\hat{A}_K)$ and $\hat{J}=\operatorname{Hom}_K(\hat{A}_K,A_K)$ can
be considered as fractional ideals for $R$, such that
$J\hat{J}\subset R$. By definition $\operatorname{SL}_2(A_K)=\Gamma(J,\hat{J})$ and
by Lemma \ref{extend} we have an $(J,\hat{J})$-polarization
on $A$.
\qed\vspace{3mm}
Now let us consider the case of abelian scheme $A$ with
$R$-linear principal polarization $\phi:A\widetilde{\rightarrow}\hat{A}$.
In this case we have a natural inclusion
$$i_{\phi}:\Sp_{2n}(R)\rightarrow\operatorname{SL}_2(A^n):
\left( \matrix M_{11} & M_{12}\\ M_{21} & M_{22} \endmatrix
\right)\mapsto
\left( \matrix [M_{11}]_{\hat{A}} & \phi_{(n)}[M_{12}]_A\\
{[M_{21}]_{\hat{A}}\phi_{(n)}^{-1}} & [M_{22}]_A \endmatrix
\right)$$
where $A^n$ is the relative $n$-th cartesian power of $A$
with the induced polarization $\phi_{(n)}$,
$M_{ij}\in\operatorname{Mat}_n(R)$, for every abelian scheme $A$ with
multiplication by $R$ we denote the natural map
$\operatorname{Mat}_n(R)\rightarrow\operatorname{End}(A^n)$ by $M\mapsto [M]_A$.
Now we claim that if $R$ is unramified at $2$ then one can split
the extension $\widehat{\operatorname{SL}}_2(A^n)\rightarrow\operatorname{SL}_2(A)$ over
$\Sp_{2n}(R)$ provided that a symmetric line bundle $L(1)$ on
$A$ is given such that $\phi_{L(1)}=\phi$.
According to Theorem \ref{Sig} it is sufficient to construct
a $\Sigma_{R,\operatorname{id}}$-structure for $\phi$.
Note that since $R$ is unramified at $2$
every element $r\in R$ can be represented in the form
$r=a^2+2b$ with $a,b\in R$. Now we define
$L(r)=[a]^*L(1)\otimes (\phi,[b]_A)^*\cal P$ where
$\cal P$ is the Poincar\'e line bundle. It is easy to see
that $L(r)$ doesn't depend on a choice of $a$ and $b$,
and satisfies (\ref{CM1}) and (\ref{CM2}) with $\varepsilon=\operatorname{id}$.
The induced structure for $A^n$ and $\operatorname{Mat}_n(R)$ is given by
the homomorphism
$$\operatorname{Mat}^{\operatorname{sym}}_n(R)\rightarrow\operatorname{Pic}^{\operatorname{sym}}(A^n):
B=(b_{ij})\mapsto L(B)=
\bigotimes_{i<j}(\phi p_i, [b_{ij}]_A p_j)^*\cal P\otimes
\bigotimes_i p_i^*L(b_{ii})$$
where $\operatorname{Mat}^{\operatorname{sym}}_n(R)$ denotes symmetric matrices with
entries in $R$, $p_i:A^n\rightarrow A$ is the
projection on the $i$-th factor. It is easy to see that
$\phi_{L(B)}=[B]_A$ and that
$[C]_A^*L(B)\simeq L(\sideset{^t}{}{C} B C)$ for any $C\in\operatorname{Mat}_n(R)$.
Now we can write the required splitting
$$\Sp_{2n}(R)\rightarrow\widehat{\operatorname{SL}}_2(A^n):
M=\left( \matrix M_{11} & M_{12}\\ M_{21} & M_{22} \endmatrix
\right)\mapsto (i_{\phi}(M),
(\phi^{-1}_{(n)})^*L(\sideset{^t}{_{11}}{M}M_{21}),
L(\sideset{^t}{_{22}}{M}M_{12})).$$
Using the above splitting we can construct a projective
action of $\Sp_{2n}(R)$ on $D^b(A^n)$.
The vanishing of the obstacle
follows in this case from the fact that $\Sp_{2n}(R)$ is
generated by elementary matrices established in \cite{BMS}.
\begin{thm} Let $A$ be an abelian scheme with real
multiplication by $R$ over $S$, $L(1)$ be a
symmetric line bundle on $A$ rigidified along the zero
section such that $\phi_{L(1)}:A\rightarrow\hat{A}$ is an $R$-linear
isomorphism.
Assume that $R$ is unramified at $2$. Then there is
a canonical faithful action of a central
extension of $\Sp_{2n}(R)$ by ${\Bbb Z}\times\operatorname{Pic}(S)$ on
${\cal D}^b(A^n)$ where $A^n$ is the relative cartesian power of
$A$, $n\ge 1$. These actions are compatible via the natural
embeddings ${\cal D}^b(A^n)\rightarrow{\cal D}^b(A^{n+1})$ and
$\Sp_{2n}(R)\rightarrow\Sp_{2n+2}(R)$.
\end{thm}
\noindent {\it Proof} . The same argument as in Proposition \ref{obshom} allows
to define an obstacle homomorphism $\delta:\Sp_{2n}(R)\rightarrow\operatorname{Br}(S)$
such that $\delta(h)=0$ if and only if there exists a global
object in $\operatorname{Isom}_{H(X)}(\rho,\rho^h)$ where
$X=\hat{A}^n\times_S A^n$, $\rho$ is the representation of the
Heisenberg groupoid on ${\cal D}^b(A^n)$.
It is easy to check that $\delta$ vanishes on elementary matrices,
hence it is zero.
\qed\vspace{3mm}
\section{The central extension}
In this section we describe
explicitly the central extension of $\Sp_{2n}({\Bbb Z})$ by
${\Bbb Z}\times\operatorname{Pic}(S)$ corresponding to the projective action
defined in the previous section.
We are going to use a presentation of $\Sp_{2n}({\Bbb Z})$ by
generators and relations borrowed from \cite{Cl}.
We always use the
standard symplectic basis $e_1,\ldots,e_n,f_1,\ldots f_n$ in
${\Bbb Z}^{2n}$ such that $(e_i,f_j)=\delta_{i,j}$).
First of all let us introduce
the relevant elementary matrices following the notation of \cite{Cl} 5.3.1.
Let $\S_{2n}$ be the set of pairs $(i,j)$ where
$1\le i,j\le 2n$ which
are not of the form $(2k-1,2k)$ or $(2k,2k-1)$. Then for for every
$(i,j)\in \S_{2n}$ we define an elementary
matrix $E_{ij}$ as follows:
$$E_{2k,2l}=
\left(\matrix 1 & 0 \\ \gamma_{k,l} & 1\endmatrix\right),$$
$$E_{2k-1,2l-1}=
\left(\matrix 1 & -\gamma_{k,l} \\ 0 & 1\endmatrix\right),$$
$$E_{2k-1,2l}=
\left(\matrix e_{kl} & 0 \\ 0 & e_{lk}^{-1}\endmatrix\right),$$
$$E_{2l,2k-1}=E_{2k-1,2l}$$
where $\gamma_{kl}$ has zero $(\a,\b)$-entry unless $(\a,\b)=(k,l)$
or $(\a,\b)=(l,k)$, in the latter case $(\a,\b)$-entry is 1;
$e_{kl}$ for $k\neq l$ is the usual elementary matrix with units on
the diagonal and at $(k,l)$-entry and zeros elsewhere.
Now theorem 9.2.13 of \cite{Cl} asserts that for $n\ge 3$
the group $\Sp_{2n}({\Bbb Z})$ has a presentation consisting of
the generators $E_{ij}=E_{ji}$ (where $(i,j)\in\S_{2n}$)
subject to the relations
\begin{enumerate}
\item $[E_{ij},E_{kl}]=1$, if
$(i,k), (i,l), (j,k), (j,l)$ are in $\S_{2n}$
\item $[E_{ij},E_{kl}]=E_{il}$, if $(j,k)\not\in\S_{2n}$, $j$
is even, and $i$, $j$, $k$, and $l$ are distinct
\item $[E_{ij},E_{ki}]=E_{ii}^2$, if $(j,k)\not\in\S_{2n}$, $j$
is even, and $i$, $j$, and $k$ are distinct
\item $[E_{ii},E_{kl}]=E_{il}E_{ll}^{-1}$ if
$(i,k)\not\in\S_{2n}$, $i$ is even, and $i$, $k$, and $l$ are
distinct
\item $[E_{ii},E_{kl}]=E_{il}^{-1}E_{ll}^{-1}$ if
$(i,k)\not\in\S_{2n}$, $i$ is odd, and $i$, $k$, and $l$ are
distinct
\item $(E_{11}E_{22}E_{11})^4=1$.
\end{enumerate}
It is convenient to introduce also the symplectic matrix
$$\varphi=
\left(\matrix 0 & -1 \\ 1 & 0\endmatrix\right).$$
Then one has the following relations
\begin{equation}\label{conj1}
\varphi^{-1}E_{2k-1,2l-1}\varphi=E_{2k,2l}
\end{equation}
for all $1\le k,l\le n$,
\begin{equation}\label{conj2}
\varphi^{-1}E_{2k-1,2l}\varphi=E_{2l-1,2k}^{-1}
\end{equation}
for all $k\neq l$.
In particular, the group $\Sp_{2n}({\Bbb Z})$ is generated by
$\varphi$ and $E_{ij}$ with $i$ odd. The latter set of
generators is more convenient from the point of view of
our projective
representation on ${\cal D}^b(A^n)$ since the functors corresponding
to $E_{ij}$ with $i$ odd are very easy to describe
(see the proof of the theorem below).
Let us denote by $\widetilde{\Sp}_{2n}({\Bbb Z})$ the group with generators
$E_{ij}=E_{ji}$ ($(i,j)\in\S_{2n}$) and one more generator
$\epsilon$ subject to relations (1)--(5) above, the
commutativity relation $[\epsilon,E_{ij}]=1$ for all
$(i,j)\in\S_{2n}$, and the modified relation (6)
$$(E_{11}E_{22}E_{11})^4=\epsilon.$$
Let $\pi:A\rightarrow S$ be an abelian scheme
with a symmetric
line bundle $L$ (rigidified along the zero section)
which induces a principal polarization
$\phi:A\widetilde{\rightarrow}\hat{A}$.
Let us also denote $\Delta(L)=2\pi_*L+e^*\omega_{A/S}\in\operatorname{Pic}(S)$
where $\omega_{A/S}$ is the relative canonical bundle.
It is known that
$4\cdot\Delta(L)=0$ (see e.~g. \cite{FC}, I, 5.1).
\begin{thm}\label{centrext}
Let $n\ge 3$.
The group $\widetilde{\Sp}_{2n}({\Bbb Z})$ is a central extension of
$\Sp_{2n}({\Bbb Z})$ by ${\Bbb Z}$.
The central extension of $\Sp_{2n}({\Bbb Z})$ by ${\Bbb Z}\times\operatorname{Pic}(S)$
corresponding to the projective action on ${\cal D}^b(A^n)$
is obtained from the $\widetilde{\Sp}_{2n}({\Bbb Z})$ by the
push-forward with respect to the homomorphism
${\Bbb Z}\rightarrow{\Bbb Z}\times\operatorname{Pic}(S):
1\mapsto (2g,2\Delta(L))$
\end{thm}
\noindent {\it Proof} . Let us choose the intertwining functors corresponding
to the generators $\varphi$ and $E_{ij}$ (with $i$ odd)
in the following way. The functor corresponding to
$\varphi$ is the composition $\phi_{(n)}^{-1}\circ F_{A^n}$
where $F_{A^n}:{\cal D}^b(A^n)\rightarrow{\cal D}^b(\hat{A}^n)$ is the Fourier-Mukai
transform. The functor corresponding to $E_{2k-1,2l-1}$
is simply tensor multiplication with the line bundle
$L(\gamma_{k,l})$. Note that $L(\gamma_{k,l})=(\phi p_k,p_l)^*\cal P$
if $k\neq l$ while $L(\gamma_{kk})=p_k^*L(1)$. At last the
functor corresponding to $E_{2k-1,2l}$ is $[e_{lk}]_A^*$.
We claim that these functors satisfy all the relations
(1)-(5).
Let $P\subset\Sp_{2n}({\Bbb Z})$ be the subgroup of matrices of
the form
$\left(\matrix * & * \\ 0 & *\endmatrix\right)$.
Then there is an obvious action
of $P$ on ${\cal D}^b(A^n)$ such that the element
$\left(\matrix\sideset{^t}{}C^{-1} & 0 \\ 0 & C
\endmatrix\right)
\left(\matrix 1 & B \\ 0 & 1\endmatrix\right)$
where $C\in\operatorname{GL}_n({\Bbb Z})$, $B\in\operatorname{Mat}^{\operatorname{sym}}(n,{\Bbb Z})$,
acts by the functor $[C^{-1}]_A^*\circ(\cdot \otimes L(-B))$.
It is easy to see that our definition of the functors
corresponding to the generators $E_{ij}$ for $i$ odd is
compatible with this action. This means that all the
relations out of (1)--(5) which contain only these generators
are satisfied by the corresponding functors. Furthermore,
using the relations (\ref{conj1}), (\ref{conj2}) one can see
that all the relations out of (1)--(5) containing a generator
$E_{ij}$ with $i$ and $j$ of opposite parity in the left
hand side, are satisfied by our functors. Similarly, the
relation (5) follows from (4) using the relations
(\ref{conj1}) and (\ref{conj2}).
It remains to check the relation (1) for $i$ and $j$ even, and
$k$ and $l$ odd, the relations (2), (3) for $i$ even and $l$
odd, and the relation (4) for $l$ odd. This can be done
directly applying the both sides of a relation to the object
$e_*\O_S\in{\cal D}^b(A^n)$. Thus, the relations (1)--(5) hold for
our functors. Now using that
$F_A^2\simeq [-1]^*(\cdot)\otimes\omega_{A/S}^{-1}[-g]$, where
$g=\dim A/S$, and that $F_A(L)\simeq L^{-1}\otimes\pi^*\pi_*L$
one can easily compute that the functor corresponding
to $(E_{11}E_{22}E_{11})^4$ is
$(\cdot)\otimes\Delta(L)^{\otimes 2}[2g]$.
\qed\vspace{3mm}
The ${\Bbb Z}$-part of the central extension of $\Sp_{2n}({\Bbb Z})$
acting on ${\cal D}^b(A^n)$ was computed in \cite{Orlov}.
Namely, for $g=1$ the corresponding class
in $H^2(\Sp_{2n}({\Bbb Z}),{\Bbb Z})$ is a half of the class of the
cocycle given by the Malsov index.
On the other hand, it is easy to see that
the class of the central extension
$\widetilde{\Sp}_{2n}({\Bbb Z})$ is a generator of $H^2(\Sp_{2n}({\Bbb Z}),{\Bbb Z})$
for sufficiently large $n$.
Indeed, it is known that $H^2(\Sp_{2n}({\Bbb Z}),{\Bbb Z})={\Bbb Z}$ for large $n$
while $\Sp_{2n}({\Bbb Z})=[\Sp_{2n}({\Bbb Z}),\Sp_{2n}({\Bbb Z})]$ for $n\ge 3$.
Moreover, the relations (2) and (4) easily imply that the element
$\epsilon$ belongs to $[\widetilde{\Sp}_{2n}({\Bbb Z}),\widetilde{\Sp}_{2n}({\Bbb Z})]$, hence
$\widetilde{\Sp}_{2n}({\Bbb Z})=[\widetilde{\Sp}_{2n}({\Bbb Z}),\widetilde{\Sp}_{2n}({\Bbb Z})]$.
It follows, that the central extension of $\Sp_{2n}({\Bbb Z})$ by
${\Bbb Z}/p{\Bbb Z}$ obtained from $\widetilde{\Sp}_{2n}({\Bbb Z})$ is non-trivial for every
prime $p$, so our claim follows.
Let $\Gamma_{1,2}\subset\Sp_{2n}({\Bbb Z})$ be the subgroup of matrices
$\left(\matrix M_{11} & M_{12} \\ M_{21} & M_{22}\endmatrix\right)$
such that $\sideset{^t}{_{11}}{M}M_{12}$ and
$\sideset{^t}{_{22}}{M}M_{21}$ have even diagonal entries.
Let $A$ be a principally polarized abelian scheme over $S$.
Then we have a
canonical splitting of the projection
$\widehat{\operatorname{SL}}_2(A^n)\rightarrow\operatorname{SL}_2(A^n)$
over $\Gamma_{1,2}$ which is constructed as in the previous
section using line bundles
$$L(B)=\bigotimes_{i<j}(\phi p_i, [b_{ij}]_A p_j)^*\cal P\otimes
\bigotimes_i (\phi p_i, [b_{ii}/2] p_i)^*\cal P$$
associated with symmetric integer even-diagonal matrices
$B=(b_{ij})$ (note that this time we don't need any additional
data on $A$). It is known (see \cite{ThetaII} A.4)
that $\Gamma_{1,2}$ is generated by
elements
$$\varphi,
\left(\matrix \sideset{^t}{^{-1}}{C} & 0 \\ 0 & C
\endmatrix\right),
\left(\matrix 1 & B \\ 0 & 1 \endmatrix\right)$$
where $C\in\operatorname{GL}_n({\Bbb Z})$, $B$ is symmetric integer with even
diagonal. Obviously, this implies
vanishing of the obstruction for the projective action of
$\Gamma_{1,2}$ on ${\cal D}^b(A^n)$ by intertwining operators,
hence this leads to a central extension of $\Gamma_{1,2}$ by
${\Bbb Z}\times\operatorname{Pic}(S)$.
\begin{prop} Let $A/S$ be a principally polarized abelian
scheme of dimension $g\ge 3$. Then the central extension of
$\Gamma_{1,2}$ by $\operatorname{Pic}(S)$ acting on ${\cal D}^b(A^n)$ up to shifts is
trivial.
\end{prop}
\noindent {\it Proof} . Considering a finite flat covering of $S$ corresponding
to a choice of a symmetric line bundle inducing
a principal polarization and using Theorem \ref{centrext}
one can see that the central extension in question is induced
by a central extension of $\Gamma_{1,2}$ by
the torsion subgroup $\operatorname{Pic}(S)^{\operatorname{tors}}\subset\operatorname{Pic}(S)$.
Note that it is sufficient to prove our assertion in the case
when $A$ is the universal abelian scheme over the moduli
stack $\AA_g$ of principally polarized abelian schemes.
It remains to notice that $\operatorname{Pic}(\AA_g)^{\operatorname{tors}}=0$ since
$\Sp_{2g}({\Bbb Z})$ has no abelian quotients for $g\ge 3$
(this is deduced using the Kummer exact sequence --- see \cite{Mu}).
\qed\vspace{3mm}
\begin{cor} The central extension of $\Sp_{2n}({\Bbb Z})$ by
${\Bbb Z}/2{\Bbb Z}$ obtained by push-forward from $\widetilde{\Sp}_{2n}({\Bbb Z})$
has a splitting over $\Gamma_{1,2}$.
\end{cor}
|
1997-12-03T21:03:27 | 9712 | alg-geom/9712006 | en | https://arxiv.org/abs/alg-geom/9712006 | [
"alg-geom",
"math.AG",
"math.DG",
"q-alg"
] | alg-geom/9712006 | Toshiyuki Akita | Toshiyuki Akita | Homological infiniteness of Torelli groups | AMSLaTeX v.1.2, 6 pages | null | null | Fukuoka University, Department of Applied Mathematics, preprint
1997/12/01 | null | We prove that rational homology of the Torelli group of genus g is infinite
dimensional, provided g>6. This means that rational homology of the Torelli
space of genus g>6 is infinite dimensional. The Torelli groups with marked
points are also considered. In addition, we prove that rational homology of the
subgroup of the Torelli group of genus g generated by all the Dehn twists along
separating simple closed curves is infinite dimensional for g>2.
| [
{
"version": "v1",
"created": "Wed, 3 Dec 1997 20:03:26 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Akita",
"Toshiyuki",
""
]
] | alg-geom | \section{Introduction}
Let $\Sigma_g$ be a closed orientable surface of genus $g\geq 2$.
Let $\mathcal{M}_{g,r}^n$ be the mapping class group of $\Sigma_g$ relative to
$n$ distinguished points and $r$ fixed embedded disks.
The action of $\mathcal{M}_{g,r}^n$ on the homology of $\Sigma_g$
induces a surjective homomorphism
\[
\mathcal{M}_{g,r}^n\rightarrow Sp(2g,\mathbb{Z}),
\]
where $Sp(2g,\mathbb{Z})$ is the Siegel modular group.
The Torelli group $\mathcal{I}_{g,r}^n$ is defined to be its kernel so that we
have an extension
\[
1\rightarrow\mathcal{I}_{g,r}^n\rightarrow\mathcal{M}_{g,r}^n\rightarrow
Sp(2g,\mathbb{Z})\rightarrow 1.
\]
We omit the decorations $n$ and $r$ when they are zero.
In a series of papers
\cite{johnson0,johnson1,johnson2,johnson3},
D. Johnson obtained several fundamental results
concerning the structure of $\mathcal{I}_g$ and $\mathcal{I}_{g,1}$ (see also
\cite{johnson-survey,hain-survey}).
In particular, he proved that
$\mathcal{I}_g$ and $\mathcal{I}_{g,1}$ are finitely generated for all $g\geq 3$.
On the contrary, A. Miller and D. McCullough \cite{miller} showed that
$\mathcal{I}_2$
(and hence $\mathcal{I}_{2,r}^n$ for all $n,r\geq0$)
is not finitely generated.
G. Mess \cite{mess} showed that $\mathcal{I}_2$ is a free group on
infinitely many generators.
Johnson and J. Millson showed that $H_3(\mathcal{I}_3,\mathbb{Z})$ contains a free
abelian group of infinite rank (cf. \cite{mess}).
It is not known whether $\mathcal{I}_{g,r}^n$ is finitely presented for $g\geq
3$.
In this paper, we will prove:
\begin{thm}\label{thm-torelli}
For all $n\geq 0$, the rational homology $H_*(\mathcal{I}_g^n,\mathbb{Q})$ of the
Torelli group $\mathcal{I}_g^n$ is infinite dimensional over $\mathbb{Q}$ if $g$ is
sufficiently large compared with $n$.
In particular, $H_*(\mathcal{I}_g,\mathbb{Q})$ and $H_*(\mathcal{I}_{g}^1,\mathbb{Q})$ are infinite
dimensional for $g\geq 7$.
\end{thm}\noindent
This theorem yields the negative answer to the question posed
by Johnson \cite{johnson-survey} which asks whether the Torelli
space $\mathbf{T}_{g}^1$ (see \S 2 for the definition) is homotopy
equivalent to a finite complex, provided $g\geq 7$.
For $n+r\leq 1$, let $\mathcal{K}_{g,r}^n$ be the subgroup of $\mathcal{M}_{g,r}^n$
generated by all the Dehn twists along separating simple closed
curves.
The groups $\mathcal{K}_{g,1}$ and $\mathcal{K}_g$
are related to the Casson invariants of homologly 3-spheres
through the work of S. Morita \cite{morita1989,morita1991}.
For $g=2$, $\mathcal{K}_2$ is equal to $\mathcal{I}_2$ so that it is a free
group on infinitely many generators.
In contrast, the group $\mathcal{K}_g$ is not free for $g\geq 3$ and
almost nothing is known about the structure of this group, however, we
can prove:
\begin{thm}\label{thm-kg}
For all $g\geq 3$,
$H_*(\mathcal{K}_g,\mathbb{Q})$ and $H_*(\mathcal{K}_{g}^1,\mathbb{Q})$ are infinite dimensional over
$\mathbb{Q}$.
\end{thm}\noindent
\begin{remark}
There is no general agreement on the definition of $\mathcal{I}_{g,r}^n$ when
$r+n>1$. We employ the one which was used in \cite{hain-survey}.
It differs from that which was given in \cite{johnson2}.
\end{remark}
\section{Preliminaries}
In this section, we recall relevant definitions and facts concerning
of Torelli groups which will be used later.
The reader should refer to \cite{hain-survey,harer} for further
detail.
Let $\mathcal{T}_g^n$ be the Teichm\"uller space with $n$ marked points.
$\mathcal{M}_g^n$ acts on $\mathcal{T}_g^n$ properly discontinuously and the quotient
space $\mathbf{M}_g^n=\mathcal{T}_g^n/\mathcal{M}_g^n$ is, by definition, the moduli
space of curves of genus $g$ with $n$ marked points.
The Torelli group $\mathcal{I}_g^n$ is torsion-free and hence it acts on
$\mathcal{T}_g^n$ freely so that the quotient space
$\mathbf{T}_g^n=\mathcal{T}_g^n/\mathcal{I}_g^n$ is a complex manifold.
Moreover, since $\mathcal{T}_g^n$ is contractible,
$\mathbf{T}_g^n$ is the classifying space of $\mathcal{I}_g^n$ so that
there is a canonical isomorphism
\[
H_*(\mathcal{I}_{g}^n,\mathbb{Z})\cong H_*(\mathbf{T}_{g}^n,\mathbb{Z}).
\]
$\mathbf{T}_g^n$ is called the {\em Torelli space} and is important in
algebraic geometry (cf. \cite{hain-survey}).
The action of $\mathcal{M}_g^n$ on $\mathcal{T}_g^n$ induces the properly
discontinuous action of $Sp(2g,\mathbb{Z})=\mathcal{M}_g^n/\mathcal{I}_g^n$ on $\mathbf{T}_g^n$ so
that the quotient space $\mathbf{T}_g^n/Sp(2g,\mathbb{Z})$ coincides with
$\mathbf{M}_g^n$.
Let $\mathfrak{S}_g$ be the Siegel upper half space of degree $g$.
The group $Sp(2g,\mathbb{Z})$ acts on $\mathfrak{S}_g$ properly discontinuously and
the quotient space $Sp(2g,\mathbb{Z})\backslash\mathfrak{S}_g$ is identified with
the moduli space of principally polarized abelian varieties of
dimension $g$.
For an integer $L\geq 3$,
let $\Gamma(L)$ be the principal congruence subgroup of $Sp(2g,\mathbb{Z})$ of
level $L$ defined to be the kernel of the canonical homomorphism
\[
Sp(2g,\mathbb{Z})\rightarrow Sp(2g,\mathbb{Z}/L\mathbb{Z}).
\]
$\Gamma(L)$ is a torsion-free subgroup of finite index of $Sp(2g,\mathbb{Z})$
and hence acts on $\mathfrak{S}_g$ freely.
The quotient space $\Gamma(L)\backslash\mathfrak{S}_g$ is identified with
the moduli space of principally polarized abelian varieties of
dimension $g$ with level $L$ structure.
It is known that $\Gamma(L)\backslash\mathfrak{S}_g$ is homotopy
equivalent to a finite complex (see
\cite{borel-serre,serre-arithmetic} for instance).
Let $\mathcal{M}_g^n(L)\subset\mathcal{M}_g^n$ be the full inverse image of $\Gamma(L)$
under the homomorphism $\mathcal{M}_g^n\rightarrow Sp(2g,\mathbb{Z})$ so that it fits
into the extension
\begin{equation}\label{mg-level-l}
1\rightarrow \mathcal{I}_g^n\rightarrow \mathcal{M}_g^n(L)\rightarrow
\Gamma(L)\rightarrow 1.
\end{equation}
$\mathcal{M}_g^n(L)$ is a torsion-free subgroup of finite index of $\mathcal{M}_g^n$ and
hence acts on $\mathcal{T}_g^n$ freely.
The quotient space $\mathbf{M}_g^n(L)=\mathcal{T}_g^n/\mathcal{M}_g^n(L)$ is, by
definition, the moduli space of curves of genus $g$ with $n$ marked
points and level $L$ structure.
The action of $\mathcal{M}_g^n(L)$ on $\mathcal{T}_g^n$ induces the free action
of $\Gamma(L)=\mathcal{M}_g^n(L)/\mathcal{I}_g^n$ on $\mathbf{T}_g^n$ so that the
quotient space $\mathbf{T}_g^n/\Gamma(L)$ coincides with
$\mathbf{M}_g^n(L)$.
According to the work of W. Harvey \cite{harvey}, $\mathbf{M}_g^n(L)$
is homotopy equivalent to a finite complex.
\section{Proof of Theorem \ref{thm-torelli}}
Fix an integer $L\geq 3$.
Since $\mathfrak{S}_g$ is contractible, the projection
$\mathfrak{S}_g\rightarrow\Gamma(L)\backslash\mathfrak{S}_g$ is the
universal principal $\Gamma(L)$-bundle.
The associated bundle
\begin{equation}\label{borel-const}
\mathbf{T}_g^n\rightarrow\mathfrak{S}_g\times_{\Gamma(L)}\mathbf{T}_g^n
\rightarrow \Gamma(L)\backslash\mathfrak{S}_g.
\end{equation}
is nothing but the Borel construction of the $\Gamma(L)$-space
$\mathbf{T}_g^n$.
Since $\Gamma(L)$ acts freely on $\mathbf{T}_g^n$, the total space
$\mathfrak{S}_g\times_{\Gamma(L)}\mathbf{T}_g^n$ is homotopy equivalent to
$\mathbf{T}_g^n/\Gamma(L)=\mathbf{M}_g^n(L)$.
Note that the associated bundle (\ref{borel-const}) is identified, up
to homotopy, with the fibration $B\mathcal{I}_g^n\rightarrow
B\mathcal{M}_g^n(L)\rightarrow B\Gamma(L)$ of classifying spaces induced from
the exact sequence (\ref{mg-level-l}).
Now suppose that $H_*(\mathcal{I}_g^n,\mathbb{Q})\cong H_*(\mathbf{T}_g^n,\mathbb{Q})$ is
finite dimensional.
As $\Gamma(L)\backslash\mathfrak{S}_g$ is homotopy equivalent to a finite
complex, we may apply the following lemma to the associated bundle
(\ref{borel-const}):
\begin{lem}\label{lem-q-euler-char}
Let $F\rightarrow E\rightarrow B$ be a fibration such that $B$ is a
finite complex and $\dim_{\mathbb{Q}}H_*(F,\mathbb{Q})<\infty$.
Then $\dim_{\mathbb{Q}}H_*(E,\mathbb{Q})<\infty$ and
\[
\chi_{\mathbb{Q}}(E)=\chi_{\mathbb{Q}}(F)\cdot\chi(B),
\]
where $\chi_{\mathbb{Q}}$ is defined by
$\chi_{\mathbb{Q}}(-)=\sum_i(-1)^i\dim_{\mathbb{Q}}H_i(-,\mathbb{Q})$.
\end{lem}\noindent
This lemma is a direct consequence of the Serre spectral sequence
applied to the fibration $F\rightarrow E\rightarrow B$.
As a result, one has
\[
\chi(\mathbf{M}_g^n(L))
=\chi_{\mathbb{Q}}(\mathbf{T}_g^n)\cdot\chi(\Gamma(L)\backslash\mathfrak{S}_g).
\]
Since both of the projections
$\Gamma(L)\backslash\mathfrak{S}_g\rightarrow Sp(2g,\mathbb{Z})\backslash\mathfrak{S}_g$
and $\mathbf{M}_g^n(L)\rightarrow\mathbf{M}_g^n$ are
$|Sp(2g,\mathbb{Z}/L\mathbb{Z})|$-fold branched coverings, one has
\begin{align*}\label{prod-formula}
\chi(\Gamma(L)\backslash\mathfrak{S}_g) &=|Sp(2g,\mathbb{Z}/L\mathbb{Z})|\cdot
e(Sp(2g,\mathbb{Z})\backslash\mathfrak{S}_g) \\
\chi(\mathbf{M}_g^n(L)) &=|Sp(2g,\mathbb{Z}/L\mathbb{Z})|\cdot
e(\mathbf{M}_g^n),
\end{align*}
and hence
\begin{equation}\label{prod-formula}
e(\mathbf{M}_g^n)=\chi_{\mathbb{Q}}(\mathbf{T}_g^n)\cdot
e(Sp(2g,\mathbb{Z})\backslash\mathfrak{S}_g),
\end{equation}
where $e$ denotes the orbifold Euler characteristics.
According to G. Harder \cite{harder}, one has
\[
e(Sp(2g,\mathbb{Z})\backslash\mathfrak{S}_g)=\prod_{k=1}^g\zeta(1-2k),
\]
while according to J. Harer and D. Zagier \cite{harer-zagier}, one has
\[
e(\mathbf{M}_g^n)=
\begin{cases}{\displaystyle\frac{1}{2-2g}\zeta(1-2g)}&\mbox{ if }n=0\\
{\displaystyle (-1)^{n-1}\frac{(2g+n-3)!}{(2g-2)!}\zeta(1-2g)}
&\mbox{ if }n>0,
\end{cases}
\]
where $\zeta$ is the Riemman $\zeta$-function.
See also \cite{penner,kont}.
Hence the equality (\ref{prod-formula}) leads to
\[
\chi_{\mathbb{Q}}(\mathbf{T}_g^n)=
\begin{cases}
{\displaystyle\frac{1}{2-2g}\prod_{k=1}^{g-1}
\frac{1}{\zeta(1-2k)}}&\mbox{ if }n=0\\
{\displaystyle (-1)^{n-1}\frac{(2g+n-3)!}{(2g-2)!}
\prod_{k=1}^{g-1}\frac{1}{\zeta(1-2k)}}&\mbox{ if }n>0
\end{cases}
\]
By definition, $\chi_{\mathbb{Q}}(\mathbf{T}_g^n)$ must be an integer and
the proof of Theorem \ref{thm-torelli} is then completed by
virtue of the following lemma which will be proven in the next
section.
\begin{lem}\label{lem-zeta}
For positive integers $m,n$, set
\[
e(m,n)=
\frac{(2m+n-1)!}{(2m)!}
\prod_{k=1}^{m}\frac{1}{|\zeta(1-2k)|}.
\]
Then, for each $n\geq 1$, $e(m,n)$ is not an integer for sufficiently large
$m$ compared with $n$.
In particular, $e(m,1)$ is not an integer for all $m\geq 6$.
\end{lem}\noindent
\section{The proof of Lemma \ref{lem-zeta}}
Recall that the Riemann $\zeta$-function is defined for
${\operatorname{Re}}\ s>1$ by
\[
\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s},
\]
On the other hand, the equality
\[
\zeta(1-2k)=(-1)^k\frac{2\cdot (2k-1)!}{(2\pi)^{2k}}\zeta(2k).
\]
holds for any integer $k\geq 1$.
It follows that
\[
|\zeta(1-2k)|
=\frac{2\cdot (2k-1)!}{(2\pi)^{2k}}
\sum_{n=1}^{\infty}\frac{1}{n^{2k}}
>\frac{2\cdot (2k-1)!}{(2\pi)^{2k}}
\]
for any integer $k\geq 1$, and hence
\[
e(m,n)<
\frac{(2m+n-1)!}{(2m)!}\prod_{k=1}^{m}
\frac{(2\pi)^{2k}}{2\cdot (2k-1)!}
\]
for any integer $m\geq 1$.
We claim that the right hand side of the inequality converges
to $0$ as $m\rightarrow\infty$.
Indeed, regarding the right hand side as a numerical sequence with
respect to $m$, the ratio of the $(m+1)$-th term to the $m$-th term is
given by
\[
\frac{(2m+n+1)(2m+n)}{(2m+2)(2m+1)}\cdot
\frac{(2\pi)^{2m+2}}{2\cdot (2m+1)!}.
\]
This converges to $0$ as $m\rightarrow\infty$, hence verifying the
claim.
We conclude that $e(m,n)<1$ and hence $e(m,n)$ is not an integer for
$m$ sufficiently large compared with $n$.
The first assertion is proved.
To prove the second assertion, observe that
${(2\pi)^{2k}}/({2\cdot (2k-1)!})<1$ for $k\geq 9$.
It follows that $e(m,1)$ is strictly decreasing with respect to $m$
for $m \geq 9$.
On the other hand, with the help of a computer, one has
\[
\prod_{k=1}^{14}\zeta(1-2k)=-297203.11\cdots .
\]
We see that, for all $m\geq 14$, $e(m,1)<1$ and hence $e(m,1)$ is
not an integer.
It remains to be proven that $e(m,1)$ is not an integer for $6\leq
m\leq 13$.
However, this can be verified by direct calculations and we omit
the detail.
\begin{remark}
Actually, $e(m,n)$ is not an integer for $n<678$ and $m\geq 6$ and
hence $H_*(\mathcal{I}_g^n,\mathbb{Q})$ is infinite dimensional for $n<678$ and $g\geq
7$.
We describe briefly how this can be proven.
Recall that $\zeta(1-2k)$ for positive integer $k$ is given by
\[
\zeta(1-2k)=-\frac{B_{2k}}{2k}\in\mathbb{Q},
\]
where $B_{2k}$ is the $2k$-th Bernoulli number defined as the
coefficient of $z^{2k}/(2k)!$ in the power series expansion of
$z/(e^z-1)$.
On the other hand, von Staudt's theorem asserts
that the denominator of $B_{2k}$ is not divisible by a prime $p$ if
$2k<p-1$.
By applying von Staudt's theorem to the primes 691 and 3617,
the numerators of $B_{12}$ and $B_{16}$ respectively,
we see that $e(m,n)$ is not an integer for $n<678$ and $6\leq
m\leq1470$.
Now the assertion follows from the inequality
\[
e(m,n)<\frac{(2m+677)!}{(2m)!}\cdot\prod_{k=1}^m
\frac{(2\pi)^{2m+2}}{2\cdot (2m+1)!}<1
\]
which holds for $n<678$ and $m\geq 37$.
\end{remark}
\section{Proof of Theorem \ref{thm-kg}}
To prove Theorem \ref{thm-kg}, we first
recall some of Johnson's results concerning the Torelli groups.
Suppose $g\geq 3$ and $[\Sigma_g]\in\wedge^2 H_1(\Sigma_g,\mathbb{Z})$
corresponds to the fundamental class of $\Sigma_g$.
Under these conditions, Johnson constructed in \cite{johnson0}
natural $Sp(2g,\mathbb{Z})$-equivariant surjective homomorphisms
\[
\tau_{2,1}:\mathcal{I}_{g,1}\rightarrow \wedge^3 H_1(\Sigma_g,\mathbb{Z})
\]
and
\[
\tau_{2}:\mathcal{I}_g\rightarrow \wedge^3 H_1(\Sigma_g,\mathbb{Z})/([\Sigma_g]\wedge
H_1(\Sigma_g,\mathbb{Z}))
\]
and proved in \cite{johnson2} that $\operatorname{ker}\tau_{2,1}=\mathcal{K}_{g,1}$ and
$\operatorname{ker}\tau_{2}=\mathcal{K}_g$.
The homomorphisms $\tau_{2,1}$ and $\tau_2$ are called {\em Johnson
homomorphisms}.
For simplicity, we abbreviate
$\wedge^3 H_1(\Sigma_g,\mathbb{Z})/([\Sigma_g]\wedge H_1(\Sigma_g,\mathbb{Z}))$
by $\wedge^3 H/H$ and
$\wedge^3 H_1(\Sigma_g,\mathbb{Z})$ by
$\wedge^3 H$.
As a consequence, $\mathcal{K}_g$ fits into the extension
\[
1\rightarrow\mathcal{K}_g\rightarrow\mathcal{I}_g\stackrel{\tau_2}{\rightarrow}
\wedge^3 H/H\rightarrow 1
\]
Take the classifying space of each group in the extension to yield a
fibration
\begin{equation}\label{fibre-kg}
B\mathcal{K}_{g}\rightarrow B\mathcal{I}_{g}\rightarrow B(\wedge^3 H/H).
\end{equation}
Observe that $B\mathcal{I}_{g}$ is homotopy equivalent to $\mathbf{T}_{g}$ and
$B(\wedge^3 H/H)$ is homotopy equivalent to the
$\binom{2g}{3}-2g$-dimensional torus since $\wedge^3
H/H$ is a free abelian group of rank $\binom{2g}{3}-2g$.
Now suppose $H_*(\mathcal{K}_g,\mathbb{Q})\cong H_*(B\mathcal{K}_g,\mathbb{Q})$ is finite dimensional.
It follows from Lemma \ref{lem-q-euler-char} that
$\dim_{\mathbb{Q}}H_*(\mathbf{T}_g,\mathbb{Q})<\infty$.
If $\dim_{\mathbb{Q}}H_*(\mathbf{T}_g,\mathbb{Q})<\infty$ (and hence $g\leq 6$), then,
as in the proof of Theorem \ref{thm-torelli},
$\chi_{\mathbb{Q}}(\mathbf{T}_g)$ is defined and satisfies
\[
\chi_{\mathbb{Q}}(\mathbf{T}_g)=\frac{1}{2-2g}\prod_{k=1}^{g-1}
\frac{1}{\zeta(1-2k)}\not=0.
\]
On the other hand,
by applying Lemma \ref{lem-q-euler-char} to the fibration
(\ref{fibre-kg}), one has
$\chi_{\mathbb{Q}}(\mathbf{T}_g)=\chi_{\mathbb{Q}}(B\mathcal{K}_g)\cdot \chi(B(\wedge^3
H/H))=0$ since $\chi(B(\wedge^3 H/H))=0$.
A contradiction.
To prove Theorem \ref{thm-kg} for the group $\mathcal{K}_g^1$, we will identify
$\mathcal{K}_g^1$ with the kernel of a variant of the Johnson homomorphism.
Recall that Torelli groups $\mathcal{I}_g^1$ and $\mathcal{I}_{g,1}$ fit into the
central extension
\[
1\rightarrow\mathbb{Z}\rightarrow\mathcal{I}_{g,1}\rightarrow\mathcal{I}_{g}^1\rightarrow 1,
\]
where the center $\mathbb{Z}$ is generated by the Dehn twist $\xi$ along a
simple closed curve parallel to the boundary of a fixed embedded disk
$D\subset\Sigma_g$.
Now the Dehn twist $\xi$ is contained in
$\mathcal{K}_{g,1}$ and hence $\tau_{2,1}$ induces a homomorphism
$\tau_2^1:\mathcal{I}_g^1\rightarrow \wedge^3 H$.
We claim that $\operatorname{ker}\tau_2^1=\mathcal{K}_{g}^1$.
Indeed, $\operatorname{ker}\tau_2^1$ coincides with the image of $\mathcal{K}_{g,1}$ under
the homomorphism $\mathcal{I}_{g,1}\rightarrow\mathcal{I}_g^1$.
But the image of $\mathcal{K}_{g,1}$ is nothing but $\mathcal{K}_g^1$ since any Dehn
twist along separating simple closed curve is isotopic to that
which fixes the embedded disk pointwise.
In summary, we have the extension
\[
1\rightarrow\mathcal{K}_{g}^1\rightarrow\mathcal{I}_{g}^1
\stackrel{\tau_2^1}{\rightarrow}\wedge^3 H\rightarrow 1.
\]
Take the classifying space for each group in the extension to yield a
fibration
\[
B\mathcal{K}_{g}^1\rightarrow B\mathcal{I}_{g}^1\rightarrow B(\wedge^3 H).
\]
Now $B\mathcal{I}_{g}^1$ is homotopy equivalent to $\mathbf{T}_{g}^1$ and
$B(\wedge^3 H)$ is homotopy equivalent to the
$\binom{2g}{3}$-dimensional torus since
$\wedge^3 H$ is a free abelian group of rank $\binom{2g}{3}$.
The rest of the proof is similar to that of $\mathcal{K}_g$.
\vspace{3mm}\\
{\em Acknowledgement.}
The author thanks to Professor Shigeyuki Morita for calling author's
attention to the groups $\mathcal{K}_{g,r}^n$.
\ifx\undefined\leavevmode\hbox to3em{\hrulefill}\,
\newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
|
1997-12-30T17:24:30 | 9712 | alg-geom/9712034 | en | https://arxiv.org/abs/alg-geom/9712034 | [
"alg-geom",
"hep-th",
"math.AG"
] | alg-geom/9712034 | null | Victor V. Batyrev | Toric Degenerations of Fano Varieties and Constructing Mirror Manifolds | 13 pages, AMS-Latex. This is an extended version of the author's talk
given during the Summer Symposium on Algebra at University of Niigata, July
22-25, 1997 | null | null | null | null | For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the
notion of a small toric degeneration. Using small toric degenerations of Fano
n-folds $X$, we propose a general method for constructing mirrors of Calabi-Yau
complete intersections in $X$. Our mirror construction is based on a
generalized monomial-divisor mirror correspondence which can be used for
computing Gromov-Witten invariants of rational curves via specializations of
GKZ-hypergeometric series.
| [
{
"version": "v1",
"created": "Tue, 30 Dec 1997 16:24:30 GMT"
}
] | 2007-05-23T00:00:00 | [
[
"Batyrev",
"Victor V.",
""
]
] | alg-geom | \section{Introduction}
Recent progress in understanding the mirror symmetry phenomenon
using explicit mirror constructions for Calabi-Yau hypersurfaces
and complete intersections in toric varieties \cite{BA,BS,BB0,Bo}
leads to the following natural question:
\medskip
{\em Is it possible to extend the mirror constructions for Calabi-Yau
complete intersections in toric Fano varieties to
the case of Calabi-Yau complete intersections in nontoric Fano varieties? }
\medskip
The first progress in this direction
has been obtained for Grassmannians \cite{BCKS1} and, more generally,
for partial flag manifolds \cite{BCKS2}. The key idea in both
examples is based on a degeneration of Grassmannians (resp.
partial flag manifolds) to some singular Gorenstein
toric Fano varieties. These
degenerations have been introduced and investigated by Sturmfels,
Gonciulea and Lakshmibai in \cite{GL1,GL2,L,S1,S2}.
The present paper is aimed to give a short systematic overview of our
method for constructing mirror manifolds and to formulate
some naturally arising questions and open problems.
In Section 2 we start with a review of a method for
constructing degenerations of unirational varieties $X$ to toric
varieties $Y$ using canonical subalgebra bases. This method has been
discovered by Kapur \& Madlener \cite{KM} and independently
by Robbiano \& Sweedler \cite{RS}. Further results on this topic have
been obtained in \cite{O,M,S1} (see also \cite{S2} for more details).
In Section 3 we introduce the notion of a
{\em small toric degeneration} of a Fano manifold and
discuss some examples. Finally, in Section 4 we explain our generalized
mirror construction which uses small toric degenerations.
\section{Canonical subalgebra bases}
Let $A$ be a finitely generated subalgebra of the polynomial
ring
$$K[{\bf u}]:=K[u_1, \ldots, u_n],$$
i.e., $X= Spec\, A$ is an unirational affine algebraic variety
together with a dominant morhism ${\Bbb A}^n \to X$.
We choose a weight vector
$\omega = (\omega_1, \ldots, \omega_d) \in {\Bbb R}^n$ and
set $$wt({\bf u}^{\bf a}) = wt(u_1^{a_1} \cdots u_n^{a_n}) :=
\sum_{i=1}^n a_i \omega_i.$$
The number $wt({\bf u}^{\bf a})$ will be called the {\bf weight} of the
monomial ${\bf u}^{\bf a}$.
We define
a partial order on the set of all monomials in $K[{\bf u}]$ as follows:
\[ {\bf u}^{\bf a} \prec {\bf u}^{\bf a'} \Leftrightarrow
wt({\bf u}^{\bf a}) \leq wt({\bf u}^{\bf a'}). \]
If $f \in K[{\bf u}]$ is a polynomial, then $in_{\prec}(f)$
denotes the {\bf initial part of $f$}, i.e., the sum of
those monomials in $f$ whose weight is maximal.
By definition, one has $in_{\prec}(fg) = in_{\prec}(f) in_{\prec}(g)$.
For suficiently general choice of the weight vector $\omega \in {\Bbb R}^n$
the initial part of a polynomial $f \in K[{\bf u}]$ is a single
monomial.
\begin{dfn}
{\rm The $K$-vector space spanned by initial terms of elements $f \in A$
is called the {\bf initial algebra} and is denoted by
\[ in_{\prec}(A) : = \{ in_{\prec}(f)\; : \; f \in A \}. \]}
\end{dfn}
\begin{dfn}
{\rm A subset ${\cal F} \subset A $ is called a {\bf canonical basis of
the subalgebra} $A \subset K[{\bf u}]$, if the initial
subalgebra $in_{\prec}(A)$ is generated
by the elements
$$\{ in_{\prec}(f)\; : \; f \in {\cal F} \}.$$ }
\end{dfn}
Fix a set of
polynomials ${\cal F} = \{f_1, \ldots, f_m\} \subset A$.
We set $K[{\bf v}]:=K[v_1, \ldots, v_m]$.
Let $I$ be the kernel of the
canonical epimorphism
$$\varphi \; : \; K[{\bf v}] \to A$$
$$v_i \mapsto f_i$$ and $I_{\prec}$
the kernel of the
canonical epimorphism
$$\varphi_0 \; : \; K[{\bf v}] \to in_{\prec}(A)$$
$$v_i \mapsto in_{\prec}(f_i)$$
\begin{rem}
{\rm It is easy to show that the ideal $I_{\prec}$ is generated by
binomials (see \cite{ES} for
general theory of binomial ideals). Hence, the spectrum
of $in_{\prec}(A)$ is
an affine toric variety (possibly not normal).}
\end{rem}
Now we assume that $\omega = (\omega_1, \ldots, \omega_d) \in {\Bbb Z}^n$ an integral
weight vector. If the set of polynomials
${\cal F} = \{f_1, \ldots, f_m\} \subset A$ form a canonical
basis of the subalgebra $A \subset K[{\bf u}]$ with respect to the
partial order defined by $\omega$, then we can define a $1$-parameter
family of subalgebras
\[ A_t := \{ f(t^{-\omega_1}u_1, \ldots, t^{-\omega_n} u_n) \;\; |\; \;
f(u_1, \ldots, u_n) \in A \},\; \;\; t \in K \setminus \{ 0 \} \}. \]
Setting $A_0: = in_{\prec}(A)$, we obtain a flat
family of subalgebras $A_t \subset K[{\bf u}]$ such that
$A_t \cong A$ for $ t\neq 0$ and $A_0 \cong K[{\bf v}]/I_{\prec}$.
This allows us to consider the affine toric
variety $Spec\, A_0$ as a flat degeneration of
$Spec\, A$.
\begin{rem}
{\rm It is important to remark that the above method for constructing
toric degenerations strongly depends on the choice of the coordinates
$u_1, \ldots, u_n$ on ${\Bbb A}^n$ and on the choice of a weight vector $\omega$.}
\end{rem}
\begin{exam}
{\rm
Let $A(r,s) \subset K[{\bf X}]: = K[X_{ij}]$ $(1 \leq i \leq r, \;
1 \leq j \leq s)$ be the subalgebra of the polynomial algebra
$K[{\bf X}]$ generated by all $r \times r$ minors of a generic
$r \times s$ matrix $( r \leq s)$, i.e., $A(r,s)$ is the homogeneous
cooordinate ring of the Pl\"ucker embedded Grassmannian
$G(r,s) \subset {\Bbb P}^{ { s \choose r } -1}$. Define the weights of monomials
as follows
\[ wt(X_{ij}) := (j-1) s^{i-1}, \;\; i,j \geq 1.\]
In particular, one has
\[ wt(X_{1,i_1} \cdots X_{r,i_r}) = (i_1-1) + (i_2-1)s + \cdots +
(i_r-1)s^{r-1} \]
and therefore the initial term of each
$(i_1, \ldots, i_r)$-minor $(1 \leq i_1 < \cdots < i_r \leq s)$ is
exactly the product of terms on the main diagonal:
\[X_{1,i_1} \cdots X_{r,i_r}. \]
The following result is due to Sturmfels \cite{S1,S2}:
\begin{theo}
The set of all $s \times s$-minors form a canonical base of the
subalgebra $A(r,s) \subset K[{\bf X}]$ with respect to the
partial order defined by the above weight vector. In particuar, one
obtains a natural toric degeneration of the Grassmanninan $G(r,s)$.
\end{theo}
\label{grass}
}
\end{exam}
\section{Small toric degenerations of Fano varieties}
\begin{dfn}
{\rm Let $X \subset {\Bbb P}^m$ be a smooth Fano variety of dimension $n$. A normal
Gorenstein
toric Fano variety $Y \subset {\Bbb P}^m$ is called a
{\bf small toric degeneration} of $X$, if
there exists a Zariski open neighbourhood $U$ of $0 \in {\Bbb A}^1$ and
an irreducible subvariety ${\frak X} \subset {\Bbb P}^m \times U$ such that
the morphism $\pi\; : \; {\frak X} \to U$ is flat and
the following conditions hold:
{(i)} the fiber $X_t := \pi^{-1}(t) \subset {\Bbb P}^m$ is smooth
for all $t \in U \setminus \{ 0 \}$;
{(ii)} the special fiber $X_0 := \pi^{-1}(0) \subset {\Bbb P}^m$ has
at worst Gorenstein terminal singularities (see \cite{KMM})
and $X_0$ is isomorphic to $Y \subset {\Bbb P}^m$;
{(iii)} the canonical homomorphism
\[ Pic({\frak X}/U) \to Pic(X_t) \]
is an isomorphism for all $t \in U$. }
\label{def-small}
\end{dfn}
\begin{rem}
{\rm It is weill-known that if $Y$ has at worst terminal singularities, then
the codimension of the singular locus of $Y$ is at least $3$.
On the other hand, it is easy to show that the only possible toric
Gorenstein terminal singularities in dimension $3$
are ordinary double points (or nodes): $x_1x_2 - x_3x_4=0$. So, if $Y$
is a small toric degeneration of $X$, then the singular locus of $Y$ in
codimension $3$ must consist of nodes.}
\label{codim3}
\end{rem}
\begin{exam}
{\rm Let $Y:= P(r,s) \subset {\Bbb P}^{ { s \choose r } -1}$ be
the toric degeneration of the Grassmannian $X:= Gr(r,s) \subset {\Bbb P}^{ { s \choose r } -1}$ (see Example \ref{grass}). Then
$Y$ is a small toric degeneration of $X$ \cite{BCKS1}. }
\end{exam}
\begin{exam}
{\rm Let $X:= F(n_1, \ldots,n_k ,n) \subset {\Bbb P}^{m}$ be the partial flag
manifold it is Pl\"ucker embedding. It is proved in
\cite{BCKS2} that the toric degenerations
introduced and investigated by
Gonciulea and Lakshmibai in \cite{GL1,GL2,L} are small toric degenerations
of $X$.
}
\end{exam}
\begin{exam}
{\rm Let $V_{d,n} \subset {\Bbb P}^{n+1}$
be a Gorenstein toric Fano hypersurface of degree $d$
$(d \geq 2)$
in projective space of dimension $n \geq 2d -2$ defined by
the homogeneous equation
\[ z_1 \cdots z_d = z_{d+1} \cdots z_{2d}. \]
It is easy to check that
irreducible components of the singular locus of $V_{d,n}$
are
\[ \frac{d^2(d-1)^2}{4} \]
codimension-3 linear subspaces
\[ z_i=z_j=z_k =z_l =0, \]
\[ \; \; \{i,j \} \subset \{1, \ldots, d\},\;
\{k,l \} \subset \{d+1, \ldots, 2d \}, \;i \neq j, \; k \neq l. \]
consisting of nodes.}
\end{exam}
\begin{theo}
$V_{d,n} \subset {\Bbb P}^{n+1}$ is a small toric degeneration of a smooth
Fano hypersurface $X_{d,n} \subset {\Bbb P}^n$ of degree $d$.
\label{sm-hyp}
\end{theo}
\noindent
{\em Proof.} Let us first consider the case $n = 2d-2$. In this case
the $2(d-1)$-dimensional fan $\Sigma_d$ defining the toric
variety $V_{d,2(d-1)}$ can be constructed as follows:
Let $e_1, \ldots, e_{d-1}, f_1, \ldots, f_{d-1}$ be a ${\Bbb Z}$-basis of the
lattice ${\Bbb Z}^{2(d-1)}$. We set $e_{d} := -e_1 - \cdots - e_{d-1}$
and $f_{d} := -e_1 - \cdots - f_{d-1}$. We denote by $h_{i,j}$ the
sum $e_i + f_j$ ($i, j \in \{1, \ldots, n\})$. If $\Delta_d^*$ denotes
the convex hull of $d^2$ points $h_{i,j}$, then the fan $\Sigma_d \subset
N_{{\Bbb R}}$ consists of cones over faces of the reflexive polyhedron
$\Delta_d^*$, where the integral lattice $N \subset {\Bbb Z}^{2(d-1)}$
is generated by all $d^2$ lattice vectors $h_{i,j}$ (the sublattice $N
\subset {\Bbb Z}^{2(d-1)}$ coincides with $ {\Bbb Z}^{2(d-1)}$ unless $d =2$).
Using the combinatorial characterisations of terminal toric
singularities \cite{KMM}, one immediately obtains
that all singularities of $V_{d,2(d-1)}$ are terminal, since
the only $N$-lattice points on the faces of $\Delta_d^*$ are their vertices.
If $d \geq 3$, then the Picard group of
$V_{d,2(d-1)}$ is generated by the class of the hyperplane section, i.e.,
$Pic(V_{d,2(d-1)})\cong {\Bbb Z}$ and the anticanonical class of
$V_{d,2(d-1)}$ is $d$-th multiple of the
generator of $Pic(V_{d,2(d-1)})$. The latter can be show as follows:
Consider a $(2d-3)$-dimensional face of $\Delta_d^*$ having vertices
\[ h_{i,j}, \;\; i \in \{1, \ldots, d-1\}, \; j \in \{1, \ldots, d\}. \]
Then every $\Sigma_d$-piecewise linear function $\varphi\, : \, N_{{\Bbb R}} \to
{\Bbb R}$, up to summing a linear function, can be normalized by
the condition
\[ \varphi(h_{i,j}) = 0, \;\; \forall i \in \{1, \ldots, d-1\}, \;
\forall j \in \{1, \ldots, d\}. \]
On the other hand, for any $j \neq j'$, $j, j' \in \{1, \ldots, d\}$
four lattice points
\[ h_{d,j}, h_{1,j}, h_{d,j'}, h_{1,j'} \]
generate a $3$-dimensional cone in $\Sigma_d$. Hence
\[ \varphi(h_{d,j}) = \varphi(h_{d,j'}) \;\; \forall
j, j' \in \{1, \ldots, d\}. \]
This means that the space of all $\Sigma_d$-piecewise linear functions
modulo linear functions is $1$-dimensional. The anticanonical
class is represented by the $\Sigma_d$-piecewise linear function $\varphi_1$
taking values $1$ on each vector $h_{i,j}$ $i,j \in \{1, \ldots, d\}$.
Considering the difference
\[ \varphi'_1 := \varphi_1 - \lambda, \]
where $\lambda$ is a linear function on $N_{{\Bbb R}}$ satisfying the conditions
\[ \lambda(e_1) = \cdots =\lambda(e_{d-1}) =1, \; \lambda(e_d) = -(d-1), \;
\lambda(f_1) = \cdots = \lambda(f_d) = 0, \]
we obtain a $\Sigma_d$-piecewise linear function having the properties
\[ \varphi_1'(h_{i,j}) = 0, \;\; \forall i \in \{1, \ldots, d-1\}, \;
\forall j \in \{1, \ldots, d\} \]
and
\[ \varphi(h_{d,j}) = d\;\; \forall
j \in \{1, \ldots, d\}. \]
So the class of $\varphi_1$ modulo linear functions is a $d$-th multiple
of a generator of $Pic(V_{d,2(d-1)})$.
The general case $n > 2(d-1)$ can be obtained by similar arguments using
the fact that $V_{d,n}$ is a projective cone over $V_{d,2(d-1)}$.
In order to construct the required
flat $1$-parameter family
${\frak X}$ (cf. \ref{def-small}), it suffices to consider a pencil of
hypersurfaces of degree $d$ in ${\Bbb P}^{n+1}$ joining $X_{d,n}$ and
$V_{d,n}$.
\hfill $\Box$
\begin{theo}
Let $X_d \subset {\Bbb P}^{n+1}$ be a smooth Fano
hypersurface of degree $d$. Then $X_d$ admits a small toric degeneration
if and only if $n \geq 2d -2$.
\label{hypersur}
\end{theo}
\noindent
{\em Proof.}
By \ref{sm-hyp}, it suffices to show that $X_d$ does not
admit a small toric degeneration if $n< 2d-2$.
Assume that $X_d$ admits a small toric degeneration $Y_d$. Then $Y_d$
is a toric hypersurface defined by a binomial equation $M_1 =M_2$ where
$M_1$ and $M_2$ are monomials in $z_0, \ldots, z_{n+1}$ of degree $d$
$(z_0, \ldots, z_{n+1}$ are homogeneous coordinates on ${\Bbb P}^{n+1}$).
If $n < 2d-2$, then at least one of the monomials $M_1$ and $M_2$
must be divisible by $z_i^2$ for some $i \in \{0, \ldots, n+1\}$.
We can assume that for instance $z_0^2$ divides $M_1$. If $z_k$ and
$z_l$ are two variables appearing in $M_2$, then $n-2$-dimensional
linear subspace
\[ z_0 = z_k = z_l = 0 \]
is contained in $Sing(Y_d)$. This contradicts the fact that
terminal singularities on $Y_d$ could appear only in codimension $\geq 3$
(see \ref{codim3}).
\hfill $\Box$
Using \ref{codim3}, one
immediately obtains:
\begin{prop}
If $X$ is a smooth Del Pezzo surface, then $X$ admits a small toric
degeneration if and only if $X$ is itself a toric variety (i.e.
$K_X^2 \geq 6$).
\end{prop}
As we have seen from \ref{sm-hyp}, a smooth quadric $3$-fold in
${\Bbb P}^4$ is an example of nontoric smooth Fano variety which
admits a small toric degeneration. By \ref{hypersur}, cubic and
quartic $3$-folds do not admit
small toric degenerations.
The compltete classification of smooth Fano $3$-folds has
been obtained in \cite{C,I,MM1,MM2,MU}. It is natural to ask the following:
\begin{ques}
Which $3$-dimensional nontoric smooth Fano varieties do admit small toric
degenerations?
\end{ques}
\section{The mirror construction}
For our convenience, we assume $K= {\Bbb C}$.
Let $X$ be a smooth Fano $n$-fold over ${\Bbb C}$
and $Y$ is its small toric degeneration.
The toric variety $Y$ is defined by some complete rational polyhedral fan
$\Sigma \subset N_{{\Bbb R}}$, where $N_{{\Bbb R}} = N \otimes {\Bbb R}$ is the
real scalar extension of a $N \cong {\Bbb Z}^n$.
We denote by $Cl(Y)$ (resp. by $Pic(Y)$) the group of Weil (resp.
Cartier) divisors on $Y$ modulo the rational equivalence.
One has a canonical embedding
$$\alpha\; : \; Pic(Y) \hookrightarrow Cl(Y).$$
If $\{ e_1, \ldots, e_k \}
\subset N$ is the set of integral generators of $1$-dimensional cones
in $\Sigma$, then $Cl(Y)$ is a finitely generated abelian
group of rank $k-n$ and the convex hull of $
e_1, \ldots, e_k$ is a reflexive polyhedron $\Delta^*$ (for definition
of reflexive polyhedra see \cite{BA}).
Assume that there exists a partition of the set
$I = \{ e_1, \ldots, e_k \}$ into $r$ disjoint subsets
$J_1, \ldots, J_r$ such that the union $D_i$
of toric strata in $Y$ corresponding
to elements of $J_i$ is a semiample Cartier divisor on $Y$ for each
$i \in \{1, \ldots, r\}$. Denote by $Z \subset Y$ a Calabi-Yau complete
intersection of $r$ hypersurfaces $Z_i \subset Y$ defined by vanishing of
generic global sections of ${\cal O}_Y(D_i)$.
By \cite{BS} (see also \cite{Bo}), the mirrors $Z^*$
of Calabi-Yau complete intersections $Z \subset Y$ are birationally
isomorphic to affine complete intersections in $({\Bbb C}^*)^n =
Spec\, {\Bbb C}[t_1^{\pm 1}, \ldots,
t_n^{\pm 1}]$ defined by $r$ equations
\[ 1 = \sum_{e_j \in J_i}^k a_j{\bf t}^{e_j}, \;\; i \in \{1, \ldots, r\},\]
where $(a_1, \ldots, a_k) \in {\Bbb C}^k$ is a general complex vector
and ${\bf t}^{e_1}, \ldots, {\bf t}^{e_k}$ are Laurent monomials in
variables $t_1, \ldots, t_n$ with the exponents
$e_1, \ldots, e_k$.
\begin{dfn}
{\rm A complex vector $(a_1, \ldots, a_k ) \in {\Bbb C}^k$ is called
$\Sigma$-{\bf admissible}, if there exists a $\Sigma$-piecewise linear
function
\[ \varphi\; : \; N_{{\Bbb R}} \to {\Bbb R}, \]
(i.e., a continuous function such that $\varphi|_{\sigma}$ is linear
for every $\sigma \in \Sigma$) having the property
\[ \varphi(e_i) = \log|a_i|,\;\;\forall i \in \{1, \ldots, k\}. \]
The set of all $\Sigma$-admissible vectors will be denoted
by $A(\Sigma)$.
}
\end{dfn}
\begin{rem}
{\rm It is easy to show that $A(\Sigma) \subset {\Bbb C}^k$ is
an irreducible closed subvariety which is isomorphic to an
affine toric variety of dimension $rk\, Pic(Y) + n \leq k$.}
\end{rem}
Now our generalization of the mirror construction
from \cite{BS} to the case of Calabi-Yau complete intersections
in a nontoric Fano variety $X$ can be formulated as follows:
\medskip
\noindent
{\bf Generalized mirror construction:}
{\em Mirrors $W^*$ of generic Calabi-Yau hypersurfaces $W \subset X$
are birationally isomorphic to the affine complete intersections
\[ 1 = \sum_{i=1}^k a_i {\bf t}^{e_i}, \]
where ${\bf a}:= (a_1, \ldots, a_k)$ is a general
point of $A(\Sigma)$.}
\medskip
\noindent
{\bf Monomial-divisor correspondence:}
Let us explain the monomial-divisor mirror correspondence for this
mirror construction (cf. \cite{AGM}). By \ref{def-small}(iii), the group
$Pic(Y)$ can be canonically identified with $Pic(X)$. The image of the
restriction homomorphism $Pic(X) \to Pic(W)$
defines a subgroup $G \subset Pic(W)$, whose elements correspond
to monomial deformations of the complex structure on mirrors:
{\em if $\psi$ is an integral $\Sigma$-piecewise linear function representing
an element $\gamma \in G$, then
the $1$-parameter family of hypersurfaces
\[ 1 = \sum_{i=1}^k t_0^{\varphi(e_i)} {\bf t}^{e_i},\;\; t_0 \in {\Bbb C} \]
defines the corresponding $1$-parameter deformation of the complex
structure on $W^*$ via the deformation of the coefficients $a_i =
t_0^{\varphi(e_i)}$.}
\medskip
\noindent
{\bf The main period:}
Let $R(\Sigma)$ the group of all vectors $(l_1, \ldots, l_k) \in
{{\Bbb Z}}^k$ satisfying the condition
$\sum_{i =1}^k l_i e_i = 0$
and
$L(\Sigma) \subset R(\Sigma)$ be the semigroup consisiting of
vectors $(l_1, \ldots, l_k) \in R(\Sigma)$
with nonnegative coordinates $l_i$ $(i =1, \ldots, k)$.
There exists a canonical
pairing
$ \langle *, * \rangle \; : \; R(\Sigma) \times Pic(Y) \to {\Bbb Z} $
which is the intersection pairing between $1$-dimensional cycles
and Cartier divisors on $Y$.
According to \cite{BS}, we can compute the main period in the
family of mirrors $W^*$ in our generalized mirror construction as
follows
\[ \Phi_0({\bf a}) = \sum_{ {\bf l}= (l_1, \ldots, l_k) \in L(\Sigma)}
\frac{ \langle l, D_1 + \cdots + D_r \rangle!}{ \langle l,
D_1\rangle! \cdots
\langle l, D_r\rangle !} \prod_{i=1}^k a_i^{l_i}, \; \; {\bf a} \in
A(\Sigma). \]
The condition ${\bf a} \in A(\Sigma)$ can be interpreted as
a specialization of $GKZ$-hypergeometric series from \cite{BS}.
\medskip
Some evidences in favor of our generalized
mirror construction were presented
in \cite{BCKS1,BCKS2}. For our next examples confirming
the proposed generalized
mirror construction we use the following simple combinatorial statement:
\begin{prop}
Let $S_d(m)$ be the set of all $d \times d$-matrices $K = (k_{ij})$
with nonnegative integral coefficients $k_{ij}$ satisfying
the equations
\[ \begin{pmatrix} 1 & \cdots & 1 \end{pmatrix}
\begin{pmatrix} k_{11} & \cdots & k_{1d} \\
\cdot & \cdots & \cdot \\ \cdot & \cdots & \cdot \\
\cdot & \cdots & \cdot \\ k_{d1} & \cdots & k_{dd} \end{pmatrix} =
\begin{pmatrix} m & \cdots & m \end{pmatrix}
\]
and
\[
\begin{pmatrix} k_{11} & \cdots & k_{1d} \\
\cdot & \cdots & \cdot \\ \cdot & \cdots & \cdot \\
\cdot & \cdots & \cdot \\ k_{d1} & \cdots & k_{dd} \end{pmatrix}
\begin{pmatrix} 1 \\ \cdot \\ \cdot \\ \cdot \\ 1 \end{pmatrix}=
\begin{pmatrix} m \\ \cdot \\ \cdot
\\ \cdot \\ m \end{pmatrix}.
\]
Then
\[ \sum_{K \in S_d(m)} \frac{(m!)^d}{\prod_{i,j =1}^{d} (k_{ij})!} =
\frac{(dm)!}{(m!)^d}. \]
\label{comb-f}
\end{prop}
\noindent
{\em Proof.} Let $A$ be the set $\{1, 2, \ldots, dm \}$ of first $dm$
natural numbers. We fix a
splitting $A$ into the disjoint union of $d$ subsets
\[ A_i := \{ (i-1)m +1, (i-1)m +2, \ldots, im \}, \;\; i =1, \ldots, d \]
consising of $m$ elements.
Let $\beta :
A = B_1 \cup \cdots \cup B_d$ be an arbitrary representation of $A$
as a disjoint union of the subsets $B_1, \ldots, B_d$ with the property
$|B_1| = \cdots = |B_d| =m$. Then every such a representation defines a
matrix $K(\beta) =(k_{ij}(\beta)) \in S_d(m)$ as
follows:
\[ k_{ij}(\beta) := |A_i \cap B_j|, \;\; i, j \in \{1, \ldots, d\}. \]
For a fixed matrix $K \in S_d(m)$ there exist exactly
\[ \prod_{j=1}^d \frac{(m!)}{\prod_{i =1}^{d} (k_{ij})!} \]
ways to construct a representation $\beta$ of $A$ as a dusjoint union of
$m$-element subsets $B_1, \ldots, B_d$ such that $K = K(\beta)$.
Therefore,
\[ \sum_{K \in S_d(m)} \frac{(m!)^d}{\prod_{i,j =1}^{d} (k_{ij})!} \]
is the total number of ways to split $A$ into a disjoint union of
$m$-element subsets $B_1, \ldots, B_d$. On the other hand, this number
is equal to the multinomial
\[ \frac{(dm)!}{(m!)^d} .\]
\hfill $\Box$
\begin{exam}
{\em Let $W$ be a generic Calabi-Yau complete intersection of
two hypersurfaces $V_d, V_d'$ in ${\Bbb P}^{2d-1}$.
By \ref{sm-hyp}, we can construct a small toric degeneration
of one smooth hypersurface $V_d'$ to the
$2(d-1)$-dimensional toric variety $Y_d \subset {\Bbb P}^{2d-1}$
\[ z_0z_1 \cdots z_{d-1} = z_d z_{d+1} \cdots z_{2d-1}. \]
Using an explicit description of the Picard group $Pic(Y_d)$ from
the proof of \ref{sm-hyp},
our generalized mirror construction suggests that mirrors $W^*$
for $W$ are birationally isomorphic to the
affine hypersurfaces $Z_F$ in the algebraic
torus
$$Spec\, {\Bbb C}[ t_1^{\pm1},\ldots, t_{d-1}^{\pm1},
u_1^{\pm1},\ldots, u_{d-1}^{\pm1}]$$
defined by the $1$-parameter family of the equations
\[ 1 = F(t_1,\ldots, t_{d-1},u_1, \ldots, u_{d-1},z) =
\sum_{i=1}^{d-1} \sum_{j=1}^{d-1}t_iu_j + (u_1 \cdots u_{d-1})^{-1}
\left( \sum_{i=1}^{d-1} t_i \right) + \]
\[ + z(t_1 \cdots t_{d-1})^{-1}
\left( (u_1 \cdots u_{d-1})^{-1} +
\sum_{i=1}^{d-1} u_j \right) ,\; \;\;\; z \in {\Bbb C} \]
On the other hand, it is known via a toric mirror construction
for Calabi-Yau complete intersection $W = V_d \cap V_d'$
(see \cite{BS}) that the power series
\[ \Phi_0(z) = \sum_{m \geq 0} \frac{(dm!)^2}{(m!)^{2d}} z^m \]
generates the Picard-Fuchs $D$-module discribing the
quantum differential system.
Now we compare our generalized mirror construction with the
known one from \cite{BS} computing the main period of the family $Z_F$
by the Cauchy residue formula:
\[ \Psi_F(z) := \frac{1}{(2\pi\sqrt{-1})^{2(d-1)}} \int_{\Gamma} \frac{1}{1 -
F({\bf t}, {\bf u}, z)} \frac{{\bf dt}}{{\bf t}}
\wedge \frac{{\bf du}}{\bf u} =
1 + a_1z + a_2z^2 + \cdots, \]
\[ \frac{{\bf dt}}{{\bf t}}:= \frac{dt_1}{t_1} \wedge \cdots
\wedge \frac{dt_{d-1}}{t_{d-1}}, \; \; \;
\frac{{\bf du}}{{\bf u}}:= \frac{du_1}{u_1} \wedge \cdots
\wedge \frac{du_{d-1}}{u_{d-1}}, \]
where the coefficients $a_m$ of the power series $\Psi_F(z)$
can be computed by the formula
\[ a_m = \sum_{K \in S_d(m)} \frac{(dm)!}{\prod_{i,j =1}^{d} (k_{ij})!}. \]
Using \ref{comb-f}, we obtain
that
\[ a_m = \frac{(dm!)^2}{(m!)^{2d}}, \]
i.e., the power series $\Psi_F(z)$ coincides with $\Phi_0(z)$ and therefore
our generalized mirror construction agrees with the already known one
from \cite{BS}.
For the special case $d=3$, we obtain a description
for mirrors $W^*$ of complete intersections $W$ of two cubics in ${\Bbb P}^5$
as smooth compactifications
of hypersurfaces in the $4$-dimensional algebraic
torus $$Spec\, {\Bbb C}[ t_1^{\pm1},t_2^{\pm1}, u_1^{\pm1},u_2^{\pm1}]$$
defined by the $1$-parameter family of the equations
\[ 1 = F(t_1,t_2,u_1,u_2, \lambda) = t_1u_1 + t_1u_2 + t_1(u_1u_2)^{-1} +
t_2u_1 + t_2u_2 + t_2(u_1u_2)^{-1} + \]
\[ + z(t_1t_2)^{-1}(u_1 + u_2 +
(u_1u_2)^{-1}),\; \;\;\; z \in {\Bbb C}. \]
This discription of mirrors is different from the one proposed by Libgober and
Teilelbaum in \cite{LT}, but it seems that both constructions are equivalent
to each other.}
\end{exam}
Now we want to suggest some problem which naturally arise from
the proposed generalized mirror construction.
\begin{prob}
Check the topological mirror duality test
\[ E_{\rm st}(W^*; u,v) =
(-u)^n E_{\rm st}(W; u^{-1},v) \]
for the above generalized mirror construction. Here $E_{\rm st}$ is
the stringy $E$-function introduced in \cite{B1}.
\end{prob}
\begin{rem}
{\rm The main difficulty of this checking arises from the fact that
the affine complete intersections in the above mirror construction
are not {\em generic}. For $\Delta^*$-regular affine hypersurfaces
there exists explicit combinatorial formula for their $E$-polynomials
(see \cite{BB1}). However, the affine hypersurfaces in our mirror
construction are not $\Delta^*$-regular and no explicit formula for their
$E$-polynomials (or Hodge-Deligne numbers) is known so far.
}
\end{rem}
\begin{prob}
Generalize the method of Givental \cite{G1,G2,G3} for computing
Gromov-Witten invariants of complete intersections in
smooth Fano varieties $X$ admitting small toric degenerations.
\end{prob}
\begin{rem}
{\rm If $X$ is a smooth Fano $n$-fold admitting a small toric degeneration
$Y$, then one can not expect that there exists a ${\Bbb C}^*$-action on $X$. So
the equivariant arguments from \cite{G1} can not be applied directly
to $X$. However,
one could try to use equivariant Gromov-Witten theory for the
ambient projective space ${\Bbb P}^m$ containing both $X$ and $Y$ and to
show that the virtual fundamental classes corresponding to
$Y$ and $X$ are the same. It seems that small quantum cohomology of $Y$
carry complete information about the subring in the small quantum cohomology
ring $QH^*(X)$ generated by the classes of divisors. This would give
an explicit description of such a subring (see \cite{ST}) as well as
of its gravitational version via Lax operators (see \cite{EHX}).
}
\end{rem}
|
1997-12-19T13:34:51 | 9712 | alg-geom/9712024 | en | https://arxiv.org/abs/alg-geom/9712024 | [
"alg-geom",
"dg-ga",
"math.AG",
"math.DG"
] | alg-geom/9712024 | Maxim Braverman | Maxim Braverman (Hebrew University) | Symplectic cutting of Kaehler manifolds | 11 pages, LaTeX 2e | null | null | Warwick preprint | null | We obtain estimates on the character of the cohomology of an
$S^1$-equivariant holomorphic vector bundle over a Kaehler manifold $M$ in
terms of the cohomology of the Lerman symplectic cuts and the symplectic
reduction of $M$. In particular, we prove and extend inequalities conjectured
by Wu and Zhang.
The proof is based on constructing a flat family of complex spaces $M_t$ such
that $M_t$ is isomorphic to $M$ for $t\not=0$, while $M_0$ is a singular
reducible complex space, whose irreducible components are the Lerman symplectic
cuts.
| [
{
"version": "v1",
"created": "Fri, 19 Dec 1997 12:34:50 GMT"
}
] | 2016-08-30T00:00:00 | [
[
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\begin{document}
\title{Symplectic cutting of K\"ahler manifolds}
\author{Maxim Braverman}
\address{Institute of Mathematics\\
The Hebrew University \\
Jerusalem 91904 \\
Israel
}
\email{[email protected]}
\thanks{This research was partially supported by grant No. 96-00210/1 from
the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel}
\begin{abstract}
We obtain estimates on the character of the cohomology of an $S^1$-equivariant
holomorphic vector bundle over a K\"ahler manifold $M$ in terms of the cohomology of the
Lerman symplectic cuts and the symplectic reduction of $M$. In particular, we prove and
extend inequalities conjectured by Wu and Zhang \cite{WuZhang}.
The proof is based on constructing a flat family of complex spaces $M_t \ (t\in\CC)$
such that $M_t$ is isomorphic to $M$ for $t\not=0$, while $M_0$ is a singular reducible
complex space, whose irreducible components are the Lerman symplectic cuts.
\end{abstract}
\maketitle
\sec{introd}{Introduction}
Let $M$ be a smooth K\"ahler manifold of complex dimension $n$ endowed with a holomorphic
Hamiltonian action of the circle group $S^1$. Let $\mu:M\to \RR$ denote the moment map
for this action. Assume that 0 is a regular value of $\mu$ and that $S^1$ acts freely
on $\mu^{-1}(0)$. Using a construction of E.~Lerman, \cite{Lerman-cut}, one can "cut"
$M$ into two smooth K\"ahler manifolds $M_+$ and $M_-$ endowed with a holomorphic circle
action. The symplectic reduction $M_{red}=\mu^{-1}(0)/S^1$ of $M$ is embedded into
$M_\pm$ as a connected component of the fixed point set.
Let $E$ be an equivariant holomorphic vector bundle over $M$. Then $E$ induces a
holomorphic vector bundle $E_{red}$ over $M_{red}$ and equivariant holomorphic bundles
$E_\pm$ over $M_\pm$. In this paper we show that there are Morse-type inequalities
which estimate the character of the $S^1$-action on the cohomology $\hm{}{*}$ of the
sheaf of holomorphic sections of $E$ in terms of the cohomology \/ $\hm{\pm}{*}, \
\hm{red}{*}$ \/ of the sheaves of holomorphic sections of the bundles $E_\pm$ and $E_{red}$
respectively. These inequalities were conjectured by Wu and Zhang \cite{WuZhang} for
the case when $E$ is a pre-quantum line bundle.
As a consequence, we obtain a new geometric proof of the "gluing formula" for the index
of $E$ (\cite{DGMW,Meinr-GS}, see \refe{glue}) for the case when the manifold $M$ is
K\"ahler.
Our proof is based on the following geometric construction which, we believe, is
interesting by itself. We consider the union $M_{cut}$ of $M_\pm$ along $M_{red}$. Thus
$M_{cut}$ is a singular reducible complex space, whose smooth irreducible components
$M_\pm$ intersect by the symplectic reduction $M_{red}$. We show that $M_{cut}$ may be
considered as a deformation of $M$. More precisely, we construct a family $M_t$ of
complex spaces parameterized by a complex parameter $t$, such that $M_t$ is complex
isomorphic to $M$ for any $t\not=0$ while $M_0$ is complex isomorphic to $M_{cut}$. It
turns out that $M_t$ is {\em a flat family of complex spaces}. That implies that the
dimension (and also the character) of the cohomology of $M_t$ is an upper
semi-continuous function of $t$. In particular, the character of $\hm{cut}{*}$ is
greater than the character of $\hm{}{*}$ (a partial order on the ring of characters is
introduced in \refd{polyn}). Moreover, there are Morse-type inequalities (cf.
\reft{Mcut>M}) which estimate the character of $\hm{}{*}$ in terms of the character of
$\hm{cut}{*}$.
The cohomology $\hm{cut}{*}$ of the space $M_{cut}$ can be, in turn, calculated by
means of a Mayer-Vietoris-type long exact sequence via the cohomology of $M_\pm$ and
$M_{red}$ (cf. \refss{Mcut}). That leads to estimates for the cohomology of $M$ in
terms of the cohomology of $M_\pm$ and $M_{red}$.
The paper is organized as follows. In \refs{main}, we formulate our main results. In
\refs{family}, we present our geometric construction of the family of complex spaces
and prove some important properties of this family. Finally, in \refs{proof}, we
present the proof of \reft{Mcut>M}.
\subsection*{Acknowledgments}
I would like to thank I.~Zakharevich for very useful and inspiring discussions. It was
I.~Zakharevich who suggested to consider the union of the Lerman symplectic cuts
$M_\pm$ as a singular complex space.
I would like to thank the University of Warwick, where this work was completed, for
hospitality.
\sec{main}{Main results}
In this section we formulate the main results of the paper. All these results are
consequences of \reft{Mcut>M}, which will be proved in \refs{proof}.
\ssec{char}{Weights and formal characters} Irreducible representation
of the circle group $S^1=\{e^{i\tet}:\, \tet\in\RR\}$ are classified by integer {\em
weights} (here we use the identification of the Lie algebra of $S^1$ with $\RR$ which
takes the {\em negative} primitive lattice element, $-2\pi i\in i\RR= Lie(S^1)$, to
$1$). A representation of weight $k\in \ZZ$ is isomorphic to the complex line $\CC$ on
which the element $e^{i\tet}\in S^1$ acts by multiplications by $e^{-ik\tet}$.
If $W$ is a finite dimensional representation of \/ $S^1$ \/ we denote by $\mult_k(W)$
the multiplicity of the weight $k\in\ZZ$ in $W$.
The {\em formal character} of $W$ is the formal sum
$$
\ch(W) \ = \ \sum_{k\in\ZZ}\mult_k(W)e^{-ik\tet}.
$$
It lies in the ring \/ $\calL=\ZZ[e^{i\tet},e^{-i\tet}]$ \/ of Laurent polynomials in
$e^{i\tet}$ with integer coefficients. This ring is called the {\em ring of formal
characters} of the circle group.
\ssec{mom-red}{Momentum map and symplectic reduction}
Let $V$ denote the vector field on $M$ that generates the $S^1$-action and let $\omega$
denote the K\"ahler form on $M$. We will assume that $S^1$-action is {\em Hamiltonian}, i.e.
there is a moment map $\mu: M\to\RR$ such that $\iot_V\ome=d\mu$. Note (\cite{Frankel})
that it is always the case if the fixed-point set of $S^1$ on $M$ is non-empty.
Assume that $0\in \RR$ is a regular value of the moment map $\mu$. Then
$\mu^{-1}(0)\subset M$ is a smooth submanifold endowed with a locally free action of
$S^1$. We will assume that this action is free. Then the quotient space
$M_{red}=\mu^{-1}(0)/S^1$ is a smooth symplectic manifold called the {\em symplectic
reduction of $M$ at level $0$}. The symplectic form $\omega_{red}$ on $M_{red}$ is defined
by the condition that its lift on $\mu^{-1}(0)$ coincides with the restriction of $\omega$
on $\mu^{-1}(0)$.
Recall now that our manifold $M$ is K\"ahler and that the K\"ahler structure on $M$ is preserved
by the circle action. In this case, {\em the $S^1$ action can be canonically extended
to a holomorphic action of the group of nonzero complex numbers $\CC^*$} (cf.
\cite[Lemma~3.3]{GuiSter82}). The set
$$
M_s \ = \ \big\{z\cdot x: \ z\in \CC^*, x\in \mu^{-1}(0)\subset M \big\},
$$
called {\em the set of stable points} for the $\CC^*$ action, is an open submanifold of
$M$, \cite[Lemma~4.5]{GuiSter82}, and the $\CC^*$ action on $M_s$ is free. Obviously,
the quotient of $M_s$ by this action is diffeomorphic to the reduced space:
\eq{M/C}
M_{red} \ \cong \ M_s/\CC^*.
\end{equation}
The equation \refe{M/C}, defines {\em a canonical complex structure on $M_{red}$}.
This structure is, in fact, K\"ahler, and the corresponding K\"ahler form coincides with the
form $\omega_{red}$ defined above.
Let now $E$ be a holomorphic vector bundle over $M$ which is equivariant for the $S^1$
action. Then the $\CC^*$ action on $M$ can be also lifted on $E$. There is a unique
holomorphic vector bundle $E_{red}$ over $M_{red}$ such that its pullback under the
projection \/ $M_s\to M_{red}$ \/ is isomorphic to the restriction of $E$ on $M_s$.
\ssec{cut}{Symplecting cuttings}
We now recall the Lerman symplectic cutting construction, \cite{Lerman-cut}. Let
$\CC_{\pm}$ denote the complex one-dimensional representations of the circle group of
weights $\pm 1$ respectively. We endow both $\CC_+$ and $\CC_-$ with the standard K\"ahler
form $\ome=\frac{i}{2}d z\wedge d\oz$. The diagonal actions of $S^1$ on
$M\times\CC_\pm$ are Hamiltonian and the corresponding moment maps are
$\mu\mp\frac12|z|^2$. One checks easily that 0 is a regular value for each one of these
moment maps. Let us denote by $M_\pm$ the symplectic quotients of $M\times\CC_\pm$ at
level $0$. The action of $S^1$ on the first factor of $M\times \CC_\pm$ reduces to a
Hamiltonian action on $M_\pm$. Thus, $(M_\pm,\ome_\pm)$ are smooth symplectic manifolds
with Hamiltonian $S^1$-actions. The reduced space $M_{red}$ is embedded into $M_\pm$
as one of the connected components (still denoted by $M_{red}$) of the fixed points
set; the compliments $M_\pm\backslash M_{red}$ are $S^1$-equivariantly symplectomorphic to
$\mu^{-1}(\RR^\pm)\subset M$, respectively. We refer to $M_\pm$ as {\em symplectic cuts
of $M$}.
The pull-back of the bundle $E$ under the natural projection $M\times \CC_\pm\to M$ is
an equivariant vector bundle over $M\times\CC_\pm$. Hence (cf. \refss{mom-red}), it
induces holomorphic vector bundles $E_\pm$ over $M_\pm$. One of the most important
facts about the cohomology of the symplectic cuts is the {\em gluing formula} (cf.
\cite{DGMW,Meinr-GS})
\begin{multline}\label{E:glue}
\sum_{p=0}^n (-1)^p\ch\hm{}{p} \ = \ \sum_{p=0}^n (-1)^p\ch\hm{+}{p} \\
\ + \ \sum_{p=0}^n (-1)^p\ch\hm{-}{p}
\ - \ \sum_{p=0}^{n-1} (-1)^p\dim_{\CC} \hm{red}{p}.
\end{multline}
\rem{symplectic}
Though the individual cohomology $\hm{}{p}$ has sense only for complex manifold $M$,
the alternating sums which appear in \refe{glue} may be defined in the case when $M$ is
an almost complex manifold. The formula \refe{glue} remains true for this more general
case \cite{DGMW,Meinr-GS} (see also \cite{SiKaTo} were the gluing formula is obtained
in a still more general situation).
\end{remark}
Let us return to the situation when $M$ is K\"ahler. The aim of this paper is to
strengthen the gluing formula \refe{glue} in order to obtain an information about
individual cohomology $\hm{}{p}$ of $M$ in terms of the cohomology of $M_\pm$ and
$M_{red}$.
\ssec{sing}{Symplectig cutting as a singular space}
Both manifolds $M_\pm$ contain the symplectic reduction $M_{red}$ as a submanifold.
Consider the union
$$
M_{cut}= M_+\cup_{M_{red}} M_-
$$
along $M_{red}$. Then $M_{cut}$ is a singular reducible complex space whose irreducible
components are $M_\pm$ and whose only singularities are the "double points" in
$M_{red}$. Let $E_{cut}$ denote the vector bundle over $M_{cut}$ whose restriction onto
$M_\pm$ is equal to $E_\pm$. The advantage of considering the singular space $M_{cut}$
rather then two disconnected manifolds $M_\pm$ is that $M_{cut}$ may be considered as a
deformation of $M$ (cf. \refs{family}). This implies (cf. \reft{Mcut>M}) estimates on
\/ $\ch H^*(M,\O(E))$ \/ in terms of the character \/ $\ch H^*(M_{cut},\O(E_{cut}))$ \/
of the cohomology of the sheaf of holomorphic sections of $E_{cut}$. The cohomology \/
$H^*(M_{cut},\O(E_{cut}))$ \/ my be, in turn, calculated in terms of the cohomology of
the sheaves \/ $E_\pm$ \/ and \/ $E_{red}$ \/ (cf. \refss{Mcut} bellow). That gives an
estimate on
\/ $\ch H^*(M,\O(E))$ \/ in terms of the spaces \/ $M_{\pm},M_{red}$.
To formulate the result we need the following
\defe{polyn} Let $q(\tet)= \sum_{k\in\ZZ}q_ke^{-ik\tet}\in \calL$ \/ be a formal
character of $S^1$, we say $q(\tet)\ge 0$ if $q_k\ge 0$ for all
$k\in\ZZ$. For two characters $p,q\in \calL$, we say that $p\ge q$
if $p-q\ge 0$.
Let $Q(\tet,t)= \sum_{m=0}^n q_m(\tet)t^m\in \ \calL[t]$ be a
polynomial of degree $n$ with coefficients in $\calL$, we say
$Q(\tet,t)\ge0$ if $q_m(\tet)\ge0$ for all $m$.
\end{defeni}
Our first result is the following Morse-type inequalities between the cohomology of $M$
and $M_{cut}$.
\th{Mcut>M} There exists a polynomial $Q(\tet,t)\in \calL[t]$, such that $Q\ge 0$ and
\eq{Mcut>M}
\sum_{p=0}^n t^p\ch\hm{cut}{p} \ = \ \sum_{p=0}^n t^p\ch\hm{}{p} \ + \
(1+t)Q(\tet,t).
\end{equation}
\eth
\reft{Mcut>M} is proven in \refs{proof}.
\rem{Mcut>M} \ 1. \ The Morse-type inequalities \refe{Mcut>M} imply
$$
\ch\hm{cut}{p} \ \ge \ \ch\hm{}{p} \quad \mbox{for any}\quad p=0\nek n.
$$
2. \ Substituting $t=-1$ into \refe{Mcut>M} we obtain the following index formula
\eq{Mcut=M}
\sum_{p=0}^n (-1)^p\ch\hm{}{p} \ = \ \sum_{p=0}^n (-1)^p\ch\hm{cut}{p}.
\end{equation}
\end{remark}
\ssec{Mcut}{Cohomology of $M_{cut}$} To calculate the cohomology of $M_{cut}$
with coefficients in $\O(E_{cut})$ consider the equivariant short exact sequence of
sheaves
$$\begin{CD}
0\to \O(E_{cut}) \ @>\alp>> \ \O(E_+)\oplus \O(E_-) \
@>\bet>> \O(M_{red}) \ \to 0,
\end{CD}
$$
Here the map \/ $\alp$ \/ sends the section $s$ of \/ $\O(E_{cut})$ \/ to the pair \/
$(s|_{M_+},s|_{M_-})$ \/ and the map \/ $\bet$ \/ sends the pair of sections \/
$(s_+,s_-)\in \O(E_+)\oplus \O(E_-)$ \/ to the section \/
$s_+|_{M_{red}}-s_-|_{M_{red}}\in \O(M_{red})$.
By standard arguments, the above short sequence leads to an equivariant long exact
sequence in cohomology
\begin{multline}\label{E:HMcut}
\cdots\to H^p(M_{cut},\O(E_{cut}))\to H^p(M_+,\O(E_+))\oplus H^p(M_-,\O(E_-)) \\
\to H^p(M_{red},\O(E_{red}))\to H^{p+1}(M_{cut},\O(E_{cut}))\to\cdots
\end{multline}
We think about $M_{cut}$ as being glued from $M_\pm$ along $M_{red}$. So we refer to
\refe{HMcut} as Mayer-Vietoris-type sequence.
\rem{WuZhang}
Wu and Zhang \cite[Remark~4.10]{WuZhang} conjectured that, if $E$ is a {\em pre-quantum
line bundle}, then (for a proper choice of the moment map) the cohomology $\hm{}{p}$ may
be calculated by a Mayer-Vietoris-type exact sequence of type \refe{HMcut}. In our terms,
that would mean that, in this case, $\hm{}{p}$ is isomorphic to $\hm{cut}{p}$.
\end{remark}
The long exact sequence \refe{HMcut} leads to the following Morse-type inequalities
\begin{multline}\label{E:MorseMcut}
\sum_{p=0}^n t^p\ch\hm{+}{p} \\
+ \sum_{p=0}^n t^p\ch\hm{-}{p} \ + \
\sum_{p=0}^{n-1} t^{p+1}\dim\hm{red}{p} \\
= \ \sum_{p=0}^n t^p\ch\hm{cut}{p} \ + \ (1+t)Q(\tet,t)
\end{multline}
for some $Q(\tet,t)\ge0$. Combining with \reft{Mcut>M} we obtain the following estimate
on $\hm{}{*}$ in terms of the cohomology of $M_\pm$ and $M_{red}$
\th{morse} There exists a polynomial $Q'(\tet,t)\in \calL[t]$, such that $Q\ge 0$ and
\begin{multline}\label{E:MorseM}
\sum_{p=0}^n t^p\ch\hm{+}{p} \ + \ \sum_{p=0}^n t^p\ch\hm{-}{p} \\
\ + \
\sum_{p=0}^{n-1} t^{p+1}\dim\hm{red}{p}
= \ \sum_{p=0}^n t^p\ch\hm{}{p} \ + \ (1+t)Q'(\tet,t)
\end{multline}
\eth
In the case when $E$ is a pre-quantum line bundle \reft{morse} was conjectured by Wu
and Zhang \cite[Remark~4.10]{WuZhang}.
\rem{gluing} \
1. \ The inequalities \refe{MorseMcut} a far from being exact. Hence, \reft{Mcut>M}
together with the Mayer-Vietoris sequence \refe{HMcut} give much more information about
the cohomology $\hm{}{*}$ than \reft{morse}.
2. \ The simplest consequence of \reft{morse} is the inequalities
\eq{simple}
\begin{aligned}
\ch\hm{+}{0} \ &+ \ \ch\hm{-}{0} \ \ge \ \ch\hm{}{0}; \\
\ch\hm{+}{p} \ &+ \ \ch\hm{-}{p} \ + \ \dim\hm{red}{p-1} \\
&\ge \ \ch\hm{}{p}, \quad\ \mbox{for any} \ \quad p=1\nek n.
\end{aligned}
\end{equation}
3. \ Substituting $t=-1$ into \refe{MorseM} and using \refe{Mcut=M}, we get the gluing
formula \refe{glue}. So we obtain a new proof of the gluing formula for
K\"ahler manifolds, which is based on the geometric construction described in \refs{family}.
Note that the standard proof of the gluing formula, \cite{DGMW,Meinr-GS,SiKaTo}, uses
the Atiyah-Segal-Singer equivariant index theorem. Dietmar Salamon pointed out that the
gluing formula for general symplectic manifold can also be proved using our geometric
construction by a method similar to \cite[Appendix~A]{McDSal2}.
\end{remark}
\rem{combine}
In the situation considered in this paper many other Morse-type inequalities may be
obtained (cf. \cite{WuZhang,TianZhang1,Br-HM}). It would be very interesting to compare
those inequalities.
\end{remark}
\ssec{example}{Example}
We finish this section with a very simple but typical example illustrating
\reft{morse}.
Let $M=\CC P^1$. We identify $M$ with the 2-dimensional sphere $S^2\subset\RR^3$ and we
let $S^1$ act on $M$ by rotations around the $z$-axis. This action has two fixed points
$P$ and $Q$ (the poles of the sphere). We normalize the K\"ahler structure on $M$ and the
moment map $\mu$ so that $\mu(P)=1, \ \mu(Q)=-1$. Then the image of $\mu$ is the
interval $[-1,1]$ and all the internal points of this interval are regular values of
$\mu$.
Let $E$ be an equivariant line bundle over $M$. Then $S^1$ acts on the fibers of this
bundle over the fixed points $P$ and $Q$. Denote by $r_Q, r_P$ the weights of these
actions. It is well known that $E$ is defined up to an equivariant isomorphism by these
weights. In particular (cf., for example, \cite[p.~330]{Witten84}) the character of
the representation of $S^1$ on the cohomology $\hm{}{p}$ is given by
\footnote{Note that our signs in the definition of weights and characters (cf.
\refss{char}) are different from \cite{Witten84} but agree with
\cite{WuZhang,SiKaTo,Br-HM}.}
\eq{cohom}
\begin{aligned}
\ch \hm{}{0} &=
\begin{cases}
\sum_{m=r_Q}^{r_P}e^{-im\tet}, \ \ \quad&\mbox{if}\quad r_Q\le r_P;\\
0, \ \ \quad&\text{if}\quad r_Q> r_P;
\end{cases} \\
%
\ch \hm{}{1} &=
\begin{cases}
0, \quad&\text{if}\quad r_Q\le r_P;\\
\sum_{m=r_P-1}^{r_Q-1}e^{-im\tet}, \quad&\mbox{if}\quad r_Q> r_P.
\end{cases}
\end{aligned}
\end{equation}
These formulas allow us to calculate the right hand side of \refe{MorseM}. Let us
calculate the left hand side of \refe{MorseM}.
Since, $M_{red}$ is a point, \/ $\dim\hm{red}{0}=1$ \/ and \/ $\dim\hm{red}{1}=0$.
Both manifolds $M_+$ and $M_-$ are isomorphic to $\CC P^1$. The weight of the fiber of
the bundle $E_+$ over $P$ is still equal to $r_P$ while the weight of the fiber over
$M_{red}$ is equal to zero. Similarly, the weight of the fiber of the bundle $E_-$ over
$Q$ is equal to $r_Q$ while the weight of the fiber over $M_{red}$ is equal to zero.
Using \refe{cohom}, one can now verify \reft{morse} in this simple case. For example,
if $r_Q=r_P=r>0$ (this corresponds to the trivial bundle $E=M\times\CC$ with a
nontrivial circle action), then the left hand side of \refe{MorseM} is equal to
$$
\sum_{m=0}^re^{-im\tet} + t\sum_{m=0}^{r-1}e^{-im\tet}
$$
while
$$
\sum_{p=0}^n t^p \ch\hm{}{p} = e^{-ir\tet}.
$$
It follows that in this case the polynomial $Q'$ of \reft{morse} does not depend on
$t$ and equals to \/ $\sum_{m=0}^{r-1}e^{-im\tet}$.
Note also, that {\em the inequalities \refe{MorseM}, \refe{simple} become equalities if
and only if $r_Q\le0$ and $r_P\ge0$}. This verifies the conjecture of Wu and Zhang
\cite{WuZhang} (cf. \refr{WuZhang}) in the case $M=\CC P^1$.
\sec{family}{The geometric construction}
In this section we present a geometric construction of a complex manifolds $\Phi$ and
a holomorphic map $p:\Phi\to \CC$ such that $p^{-1}(t)$ is isomorphic to $M$ for
$t\not=0$ while $p^{-1}(0)=M_{cut}$.
\ssec{family}{}
The idea of the construction is the same as in \refss{cut}. In fact, we just combine
the constructions of $M_+$ and $M_-$ together and consider the diagonal action of
$S^1$ on $M\times\CC_+\times\CC_-$. The moment map for this action is given by
$$
\tilmu(x,z_+,z_-) \ = \ \mu(x) \ - \ \frac12|z_+|^2+\frac12|z_-|^2, \qquad
x\in M, \ z_\pm\in\CC_{\pm}.
$$
Zero is a regular value of $\tilmu$ and we define $\Phi= \tilmu^{-1}(0)/S^1$ to be the
symplectic reduction of $M\times\CC_+\times\CC_-$ at zero level.
Clearly, the map $\tilp: \, M\times\CC_+\times\CC_-\to \CC$ defined by the formula
$$
\tilp: \ (x,z_+,z_-) \ \mapsto \ z_+z_-, \qquad x\in M, \ z_\pm\in\CC_{\pm}.
$$
is $S^1$ invariant and, hence, descends to a map $p:\Phi\to \CC$. We think about $\Phi$
as family of complex manifolds $M_t=p^{-1}(t)$ parameterized by a complex parameter
$t\in\CC$.
We endow $\Phi$ with an $S^1$-action induced by the action of the circle on the first
factor of $M\times\CC_+\times\CC_-$. Note that this action preserves the fibers
$M_t=p^{-1}(t)$ of $p$. Thus $M_t \ (t\in\CC)$ are also endowed with a holomorphic
circle action.
As in \refss{cut} the bundle $E$ induces a bundle $\tilE$ over $\Phi$. We denote the
restriction of $\tilE$ on $M_t$ by $E_t$.
The following lemmas describe the fibers of the projection $p$.
\lem{MtS}
For any $t\not=0$, the fiber $M_t=p^{-1}(t)$ is a smooth manifold which is
equivariantly symplectomorphic to $M$.
\end{lemma}
\begin{proof}
Fix a nonzero number $t\in\CC$. For any $x\in M$ set
$$
r(x) \ = \ \sqrt{\mu(x)+\sqrt{\mu(x)^2+|t|^2}}
$$
and define an embedding
\eq{it}
i_t: \, M\to \tilmu^{-1}(0)\cap\tilp^{-1}(t) \ \subset M\times\CC_+\times\CC_-,
\qquad
i_t: \ x \ \mapsto \Big(x,r(x),\frac{t}{r(x)}\Big).
\end{equation}
Clearly, the composition $q\circ i_t$ of the above embedding with the natural projection
$q: \, \tilmu^{-1}(0)\to \Phi$ is an equivariant diffeomorphism $M\to M_t=p^{-1}(t)$. Since
the map $i_t:M\to M\times\CC_+\times\CC_-$ is symplectic, so is the composition $q\circ i_t$.
\end{proof}
\lem{MtC}
For any $t\not=0$, the fiber $M_t=p^{-1}(t)$ is a smooth K\"ahler manifold which is
equivariantly complex isomorphic to $M$. In other words, there exists an
equivariant biholomorphic map $\phi_t:M\overset{~}{\to} M_t$.
The pullback
$\phi^*_tE_t$ of the bundle $E_t=\tilE|_{M_t}$ is equivariantly isomorphic to $E$.
\end{lemma}
\rem{isom}
The manifolds $M$ and $M_t$ are not isomorphic as K\"ahler manifolds. In particular, the map
$\phi_t$ of \refl{MtC} is different from the symplectomorphism of \refl{MtS}.
\end{remark}
\begin{proof}
Recall from \refss{mom-red} that the action of the circle group on $M$ extends
canonically to a holomorphic action of $\CC^*$ and consider the diagonal action of
$\CC^*$ on $M\times\CC_+\times\CC_-$ (here $z\in \CC^*$ acts on the second factor by
multiplication by $z$ and on the third factor by multiplication by $1/z$).
Let $U\subset M\times\CC_+\times\CC_-$ denote the set of stable points for this action
(cf. \refss{mom-red}). Then $\tilp^{-1}(t)\subset U$ for any $t\not=0$.
Indeed, if $v\in \tilp^{-1}(t)$ and if the absolute value of the number $z\in\CC^*$
is large enough, then $\tilmu(z\cdot v) < 0$, while $\tilmu(\frac1z\cdot v) > 0$.
Hence, one can find $z'\in\CC^*$ such that $z'\in \tilmu^{-1}(0)$.
Fix $t\not=0$ and consider the complex map
$$
j_t:M\to M\times\CC_+\times\CC_-, \qquad j_t: x \ \mapsto \big(x,1,t\big).
$$
Clearly, the image of $j_t$ belongs to $\tilp^{-1}(t)$ and, by the previous paragraph,
it belongs also to the set $U$ of stable points for $\CC^*$ action. Hence, the
composition of $j_t$ with the quotient map $q:U\to U/\CC^*$ defines an equivariant
holomorphic map $\phi_t:M\to M_t$. Clearly, $\phi_t$ is injective. We claim that
$\phi_t$ is an isomorphism. By \refl{MtS}, it
suffice to show that the image of $j_t$ contains the image of the map \refe{it}. But
this follows from the obvious inclusion \/ $\frac1r\cdot(x,r,\frac{t}r)\in \IM(j_t)$.
The first statement of the lemma is proven.
Consider the commutative diagram
\eq{diagr}
\begin{CD}
M\times\CC_+\times\CC_- &&\ &\ \hookleftarrow&\quad &U\cap \tilp^{-1}(t)\\
@V{\pi}VV \ &{\nearrow}&\quad &@VV{q}V\\
M &\ & @>\phi_t>>\quad &M_t
\end{CD}
\end{equation}
By definition of the bundle $E_t$ we have $q^*E_t=\pi^*E|_{U\cap \tilp^{-1}(t)}$. Hence,
using $\phi_t=q\circ j_t$ and $\pi\circ j_t=id$, we obtain
$$
\phi_t^*E_t \ = \ j_t^*q^*E_t \ = \ \phi_t^*\pi^*E \ = \ E.
$$
\end{proof}
\lem{M0}
The fiber $p^{-1}(0)$ is equivariantly complex isomorphic to the space $M_{cut}$. If we
identify $p^{-1}(0)$ with $M_{cut}$ using this isomorphism then the restriction $E_0$
of $\tilE$ to $p^{-1}(0)$ is isomorphic to $E_{cut}$.
\end{lemma}
\begin{proof}
The lemma is an obvious consequence of the equality
$$
\tilp^{-1}(0) \ = \ (M\times\CC_+\times\{0\}) \, \cup \,
(M\times\{0\}\times\CC_-).
$$
\end{proof}
For us the most important is the following consequence of the above lemmas
\cor{Phi}
The cohomology $H^*(M_t,\O(E_t))$ of the sheaf of holomorphic
sections of the bundle $E_t$ is equivariantly isomorphic to $\hm{}{*}$ if $t\not=0$ and
is equivariantly isomorphic to $\hm{cut}{*}$ if $t=0$.
\end{corol}
\sec{proof}{Flat morphisms. Proof of \reft{Mcut>M}}
We are in a position now to prove \reft{Mcut>M}. The proof is based on the properties
of flat morphisms in complex analysis.
\ssec{flat}{Flat morphisms}
First, we recall some basic facts about flat
morphisms. For the details we refer the reader to
\cite[Sections~II.2,III.4]{GraPetRem}.
If \/ $X$ \/ is a complex space and \/ $x\in X$ \/ we denote by \/ $\O(X)$ \/ the
sheaf of holomorphic functions on \/ $X$ \/ and by \/ $\O_x(X)$ \/ the ring of
germs of holomorphic functions at \/ $x$. Let \/ $f:X\to Y$ \/ be a holomorphic map
of complex spaces. For any \/ $y\in Y$, we denote by \/ $X_y=f^{-1}(y)$ \/ the fiber
of
\/ $f$ \/ over \/ $y$.
A holomorphic map \/ $f:X\to Y$ \/ is called {\em flat at a point \/ $x\in X$} \/ if
\/ $\O_x(X)$ \/ is a flat module over \/ $\O_{f(x)}(Y)$. Here \/ $\O_x(X)$ \/ is considered as
\/ $\O_{f(x)}(Y)$-module via the canonical map \/ $f^*:\O_{f(x)}(Y)\to
\O_x(X)$. The map \/ $f$ \/ is called {\em flat} if it is flat at any point \/ $x\in X$.
Let \/ $f:X\to Y$ \/ be a morphism of complex spaces and suppose \/ $V$ \/ is a
holomorphic vector bundle over \/ $X$. For any \/ $y\in Y$, let \/ $V_y=V|_{X_y}$ \/
denote the restriction of \/ $V$ \/ on the fiber \/ $X_y$ \/ and let \/ $\O(V_y)$
\/ denote the locally free sheaf of holomorphic sections of \/ $V_y$. If \/ $f$ \/
is a flat morphism, then, for any \/ $y\in Y$ \/ and for any pint \/ $\eta\in Y$ \/
closed to $y$, there exists a polynomial \/ $Q(t)$ \/ with nonnegative integer
coefficients, such that
\eq{family}
\sum_p t^p\dim H^p(X_y,\O(V_y)) \ = \
\sum_p t^p\dim H^p(X_\eta,\O(V_\eta))
\ + \ (1+t)Q(t).
\end{equation}
The equation \refe{family} implies, in particular, that the function \/ $y\mapsto
\dim H^p(X_y,\O(V_y))$ \/ is upper semi-continuous, while the {\em holomorphic index} of
the fibers
$$
\ind(V_y) \ = \ \sum_p (-1)^p\dim H^p(X_y,\O(V_y))
$$
is locally constant on \/ $Y$.
Suppose that in the situation described above a compact Lie group \/ $G$ \/ acts
holomorphically on \/ $X$ \/ and \/ $Y$ \/ and that this action commutes with \/
$f$. If the vector bundle \/ $E$ \/ is equivariant with respect to this action then
\/ $G$ \/ acts on the cohomology of the fibers. Let \/ $\ch H^p(X_y,\O(V_y)), \ y\in Y$ \/
denote the character of this action. Then, for any point \/ $\eta\in Y$ \/ closed
enough to \/ $y$, there exists a polynomial \/ $Q(t,\tet)\in \calL[t]$ \/ (cf.
\refd{polyn}) such that
\/ $Q\ge 0$ \/ and
\eq{eq-family}
\sum_p t^p\ch H^p(X_y,\O(V_y)) \ = \
\sum_p t^p\ch H^p(X_\eta,\O(V_\eta))
\ + \ (1+t)Q(t,\tet).
\end{equation}
\ssec{proof}{Proof of \reft{Mcut>M}}
It follows now from \refe{eq-family} and \refc{Phi}, that in order to prove
\reft{Mcut>M} it suffices to show that the projection \/ $p:\Phi\to \CC$ is flat. But,
by a theorem of Kaup and Kerner, \cite[Ch.~II, Theorem~2.13]{GraPetRem}, {\em any open
holomorphic map of smooth complex manifolds is flat}. Since, \/ $p$ \/ is open the
theorem is proven.
\hfill $\square$
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
|
1996-07-04T16:35:05 | 9607 | alg-geom/9607007 | en | https://arxiv.org/abs/alg-geom/9607007 | [
"alg-geom",
"dg-ga",
"math.AG",
"math.DG"
] | alg-geom/9607007 | Bernd Kreussler | Bernd Kreussler | On the algebraic dimension of twistor spaces over the connected sum of
four complex projective planes | 23 pages LaTeX 2e | null | null | null | null | We study the algebraic dimension of twistor spaces of positive type over
$4\bbfP^2$. We show that such a twistor space is Moishezon if and only if its
anticanonical class is not nef. More precisely, we show the equivalence of
being Moishezon with the existence of a smooth rational curve having negative
intersection number with the anticanonical class. Furthermore, we give precise
information on the dimension and base locus of the fundamental linear system
$|{-1/2}K|$. This implies, for example, $\dim|{-1/2}K|\leq a(Z)$. We
characterize those twistor spaces over $4\bbfP^2$, which contain a pencil of
divisors of degree one by the property $\dim|{-1/2}K| = 3$.
| [
{
"version": "v1",
"created": "Thu, 4 Jul 1996 14:24:33 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Kreussler",
"Bernd",
""
]
] | alg-geom | \section{Introduction}
\label{intro}
Twistor spaces usually arise in four--dimensional conformal geometry. Their
construction reflects the impossibility to equip in general a four--dimensional
conformal manifold $M$ with a compatible complex structure. It was shown in
\cite{AHS} that the conformal metric on $M$ is self--dual if and only if the
twistor space $Z$ associated to $M$ carries, in a natural way, the structure of
a complex manifold. Therefore, the conformal geometry of $M$ is closely related
to the holomorphic geometry of $Z$. Since we shall only work with methods of
complex geometry, we can use the following definition:
\\ A twistor space $Z$ is a complex three--manifold with the following
additional structure:
\begin{itemize}
\item
a proper differentiable submersion $\pi:Z\rightarrow M$ onto a real
differentiable four--manifold $M$. The fibres of $\pi$ are holomorphic curves
in $Z$ being isomorphic to ${\Bbb C}{\Bbb P}^1$ and having normal bundle in $Z$
isomorphic to ${\mathcal O}(1)\oplus{\mathcal O}(1)$;
\item
an anti--holomorphic fixed point free involution $\sigma:Z\rightarrow Z$
with $\pi\sigma=\pi$.
\end{itemize}
The fibres of $\pi$ are called ``real twistor lines'' and the involution
$\sigma$ is called the ``real structure''. A geometric object will be called
``real'' if it is $\sigma$--invariant. For example, a line bundle ${\mathcal L}$ on
$Z$ is real if $\sigma^\ast\bar{{\mathcal L}}\cong {\mathcal L}$, and a complex
subvariety $D\subset Z$ is real if $\sigma(D) = D$. Instead of $\sigma(D)$ we
shall often write $\bar{D}$.
We only consider compact and simply connected twistor spaces.
At the beginning of the 80's, the first classification result emerged in
\cite{FK}, \cite{Hit2}:\\
There exist exactly two compact K{\"a}hlerian twistor spaces. They are
automatically projective algebraic. The corresponding Riemannian
four--manifolds
are the $4$--sphere $S^4$ and the complex projective plane $\bbfC\bbfP^2$ (with
Fubini--Study metric).
This was generalized in \cite{Camsc} to the result that a twistor space which
is bimeromorphic to a compact K{\"a}hler manifold must be Moishezon and simply
connected. This implies (see \cite{Don}, \cite{F}) that $M$ is homeomorphic to
the connected sum $n\bbfC\bbfP^2$ for some $n\geq 0$.
New examples of Moishezon twistor spaces were constructed by Y.S.\ Poon
\cite{Po1} (case $n = 2$) and C.\ LeBrun \cite{LeB}, H.\ Kurke \cite{Ku} (case
$n\geq 3$).
Nowadays the situation is well understood for $n\leq 3$ (cf. \cite{Hit2},
\cite{FK}, \cite{Po1}, \cite{KK}, \cite{Po2}). To become more precise, we have
to introduce the notion of the ``type'' of a twistor space. By a result of
R.~Schoen \cite{Sch}, every conformal class of a compact Riemannian
four--manifold contains a metric of constant scalar curvature. Its sign will be
called the {\sl type\ } of the twistor space. This is an invariant of the
conformal class, hence of the twistor space.
It was shown in \cite{Po3} that a Moishezon twistor space is always of positive
type. If $n \leq 3$, the converse is also true.
In this paper we focus on the positive type case, for two reasons. One reason
is that we can then apply Hitchin's vanishing theorem (\ref{Hit}). The
other reason is the following: a result of P.\ Gauduchon \cite{Gau} implies
that any twistor space of negative type has algebraic dimension zero. From the
results of M.\ Pontecorvo \cite{Pon} we easily derive that a twistor space of
type zero over $n\bbfC\bbfP^2$ must also have algebraic dimension zero. It is not clear
whether there exist twistor spaces of non--positive type over $n\bbfC\bbfP^2$.
Computation of algebraic dimension is, therefore, interesting only in the case
of positive type. A very important tool to compute the algebraic dimension of
twistor spaces is the result of Y.S.\ Poon \cite{Po3} (see also \cite{Pon})
stating that the algebraic dimension is equal to the Iitaka dimension of the
anticanonical bundle (cf. Section \ref{prelim}).
From \cite{DonF} and \cite{Cam}, \cite{LeBP} it is known that the generic
twistor space over $n\bbfC\bbfP^2$ has algebraic dimension one (if $n = 4$),
respectively zero (if $n\geq 5$).
For the case $n=4$, the characterizing property $c_1^3 = 0$ is of central
importance.
In this paper we study the following
\begin{main}{Problem:}
Compute the algebraic dimension $a(Z)$ of a twistor space $Z$ over $4\bbfC\bbfP^2$ in
terms of geometric or numeric properties of certain divisors on $Z$.
\end{main}
A first attempt to tackle this problem was made by Y.S.\ Poon \cite[Section
7]{Po2}. He assumes, additionally, the existence of a divisor $D$ of degree one
on $Z$. He studies a birational map $D\rightarrow{\Bbb P}^2$, which is the
blow--up of four points. He seems to assume that these four points are
actually in ${\Bbb P}^2$ (no infinitesimally near blown--up points). If these four
points are in a special position he obtains $a(Z) = 3$. In the case of general
position he can only show: $a(Z) \leq 2$.
We shall, in general, not assume the existence of divisors of degree
one. Because in case $n=4$ there exists at least a pencil of so--called
fundamental divisors,
we shall study their geometry to obtain our results. If $S\subset Z$ is a real
fundamental divisor, we have a birational map $S\rightarrow\bbfP^1\times\bbfP^1$ which is the
blow--up of eight points. We shall study in detail the possible positions for
these points. We take into account that some of these points can be
infinitesimally near each other. We are able to derive the algebraic dimension
$a(Z)$ from the knowledge of the positions of these eight points.
Similar considerations were made for general $n\geq 4$ in the paper \cite{PP}.
But the authors of that paper are intersted in a study of small
deformations of well--known Moishezon twistor spaces, and so they investigate
only the case without infinitely near blown--up points.
As a consequence of our results, we give a new characterization of the
twistor spaces over $4\bbfC\bbfP^2$ which are first described by C.\ LeBrun \cite{LeB}
(with methods from differential geometry). From the point of
view of complex geometry the twistor space structure on these complex manifolds
was found by H.\ Kurke \cite{Ku}. Following the literature, we call them {\em
LeBrun twistor spaces}.
These twistor spaces are characterized in \cite{Ku} and \cite{Po2} by the
property to contain a pencil of divisors of degree one. In the case
$n=4$ we show (Theorem (\ref{cb})) that they can also be characterized by the
property $h^0(\fdb) = 4$ or by the structure of the base locus of $|\fund|$.
Besides this, our main results are a precise description of the set of
irreducible curves intersecting $\fdb$ negatively (Theorem \ref{ncurves}) and
the following theorems, where $Z$ denotes
always a simply connected compact twistor space of positive type over $4\bbfC\bbfP^2$:
\setcounter{section}{6}
\setcounter{thm}{1}
\begin{thm}
$a(Z) = 3 \iff \fdb$ is not nef;\\[-0.8mm]
$a(Z) = 2 \iff \fdb$ is nef and $\exists m\geq 1: h^1(\fb{m}) \ne 0$;%
\\[-0.8mm]
$a(Z) = 1 \iff \forall m\geq 1: h^1(\fb{m}) = 0$.
\end{thm}
\begin{thm}
The following conditions are equivalent:
\begin{enumerate}
\item $a(Z) = 3$;\vspace*{-1mm}
\item $\fdb$ is not nef;\vspace*{-1mm}
\item there exists a smooth rational curve $C\subset Z$ with $C.(\fund) < 0$.
\end{enumerate}
\end{thm}
\setcounter{thm}{5}
\begin{thm}
$a(Z) \geq \dim|\fund|$.
\end{thm}
\begin{thm}
If $\dim|\fund|\geq 2$, then:\\
$a(Z) = 2 \iff \fdb$ is nef $\iff |\fund|$ does not have base points.
\end{thm}
\setcounter{section}{1}
\setcounter{thm}{0}
This paper is organized as follows:\\
In Section \ref{prelim} well--known but necessary facts about simply connected
compact twistor spaces of positive type are collected.
Also Section \ref{eins} has preparatory character. We study there the structure
of fundamental divisors for general $n$, using results and techniques contained
in \cite{PP}. Technically important for the following sections will be
Proposition \ref{types} where the structure of effective anticanonical curves
on real fundamental divisors is described in detail.
In the remaining three sections we assume $n=4$.
In Section \ref{zwei} we study the case where the anti--canonical bundle
$K_Z^{-1}$ is nef (in the sense of Mori theory). We shall prove that the
algebraic dimension is, in this case, at most two. We also see how to
distinguish between algebraic dimension one and two. This generalizes results
of \cite{CK}.
In Section \ref{drei} we assume $K_Z^{-1}$ to be not nef. We
collect detailed information on the fundamental linear
system $|\fund|$ and on the set of curves which intersect $\fdb$ negatively. In
this cases the algebraic dimension is three.
The final Section \ref{vier} combines the results of the previous part to
prove the main theorems stated above.
\begin{main}{Acknowledgement}
I thank Fr\'ed\'eric Campana for encouragement and stimulating discussions.
\end{main}
\section{Preliminaries}
\label{prelim}
We briefly collect well--known facts which will be frequently used later. We
refer the reader to \cite{AHS}, \cite{ES}, \cite{Hit2}, \cite{Kr}, \cite{Ku}
and \cite{Po1}. For brevity, we assume, throughout this section, $Z$ to be a
simply connected compact twistor space of positive type. As mentioned in
Section \ref{intro} the corresponding Riemannian four--manifold $M$ is
homeomorphic
to $n\bbfC\bbfP^2$.
\subsubsection*{Cohomology ring of $Z$}
$H^i(Z,{\Bbb Z})$ is a free ${\Bbb Z}$--module.\\
$H^1(Z,{\Bbb Z}) = H^3(Z,{\Bbb Z}) = H^5(Z,{\Bbb Z}) = 0$ and $H^0(Z,{\Bbb Z}) \cong
H^6(Z,{\Bbb Z}) \cong {\Bbb Z}$. \\
$H^2(Z,{\Bbb Z})$ and $H^4(Z,{\Bbb Z})$ are free modules of rank $n+1$. There exists
a basis $x_1,\dots, x_n, w$ of $H^2(Z,{\Bbb Z})$ such that the pull--back
$H^2(Z,{\Bbb Z}) \rightarrow H^2(F,{\Bbb Z})\cong {\Bbb Z}$ (for any real twistor line
$F \subset Z$) sends $x_i$ to $0$ and $w$ to the positive generator.
The cohomology ring $H^\ast(Z,{\Bbb Z})$ is isomorphic to the graded ring ${\Bbb Z}
[x_1,\dots, x_n, w]/R$ where $R$ is the ideal generated by
\[ x_i^2 - x_j^2,\quad x_ix_j\; (i\ne j),\quad w^2 + w\sum_{i=1}^n x_i +
x_1^2.\]
The grading is given by $\deg x_i = \deg w = 2$.\\
$H^4(Z,{\Bbb Z})$ is a free ${\Bbb Z}$--module with generators $wx_1,\dots, wx_n,
w^2$. The dual class of a real twistor fibre $F\subset Z$ is $-x_i^2 \in
H^4(Z,{\Bbb Z})$.
$c_1(Z) = 4w + 2\sum_{i=1}^n x_i, \quad c_2(Z) = -6x_1^2 = 6F, \quad c_3(Z) =
2(n + 2)$. This yields the following Chern numbers:
$c_1^3 = 16(4 - n), \quad c_1c_2 = 24, \quad c_3 = 2(n + 2)$.
\subsubsection*{Cohomology of sheaves}
The main reason to assume $Z$ to be of positive type is Hitchin's vanishing
theorem. We shall only use the following special case:
\begin{thm}[Hitchin \cite{Hit}]\label{Hit}
If $Z$ is of positive type then we have for any ${\mathcal L}\in \Pic(Z)$
\begin{eqnarray*}
\deg({\mathcal L})\le -2 &\Rightarrow& H^1(Z,{\mathcal L})=0.
\end{eqnarray*}
\end{thm}
On the other hand, since the twistor lines cover $Z$, we obtain:
\begin{eqnarray*}
\deg({\mathcal L})\le -1 &\Rightarrow& H^0(Z,{\mathcal L})=0.
\end{eqnarray*}
By Serre duality this gives the following important vanishing results:
\begin{eqnarray}
\deg({\mathcal L})\ge -2 &\Rightarrow& H^2(Z,{\mathcal L})=0,\\
\deg({\mathcal L})\ge -3 &\Rightarrow& H^3(Z,{\mathcal L})=0.
\end{eqnarray}
In particular, we obtain $h^2({\mathcal O}_Z) = h^3({\mathcal O}_Z) = 0$. Because $Z$ is simply
connected, we also have $h^1({\mathcal O}_Z) = 0$. Hence, we obtain an isomorphism of
abelian groups, given by the first Chern class:
\[\Pic(Z)\stackrel{\sim}{\rightarrow}H^2(Z,{\Bbb Z}).\]
There exists a unique line bundle whose first Chern class is $\frac{1}{2}c_1$.
We shall denote it by $\fdb$. Following Poon, we call it the {\sl fundamental}
line bundle. The divisors in the linear system $|\fund|$ will be called {\sl
fundamental divisors}. The description of the cohomology ring gives
$(\fund)^3 = 2(4 - n)$. If $S\in|\fund|$ is a smooth fundamental divisor, we
obtain by the adjunction formula $\canS\cong\fdb\otimes{\mathcal O}_S$. If $n\leq 4$,
there exist smooth real fundamental divisors (cf.\ \cite[Lemma 3.1]{CK}).
The degree of a line bundle ${\mathcal L}\in \Pic(Z)$ will be by definition the
degree of its restriction to a real twistor line. For example,
$\deg(\fdb)=2$. We obtain in this way a {\sl surjective\/} degree map
\[\deg :\Pic(Z)\twoheadrightarrow {\Bbb Z}.\]
From the above equations on Chern numbers we obtain, by applying the
Riemann--Roch theorem,
\begin{equation}\label{RR}
\chi(Z,\fb{m})=m+1+2(4-n){\binom{m+2}{3}}.
\end{equation}
\subsubsection*{Algebraic dimension}
We denote by $a(Z)$ the algebraic dimension of $Z$, which is by definition the
transcendence degree of the field of meromorphic functions of $Z$ over ${\Bbb C}$.
If $\dim Z = a(Z)$, then $Z$ is called Moishezon. To compute the algebraic
dimension of twistor spaces we shall frequently use, without further reference,
the following theorem of Y.S.\ Poon:
\begin{thm}{\cite{Po3}, \cite[Prop.\ 3.1]{Pon}}\\
$ \begin{array}[t]{lcl}
\kappa(Z, K^{-1}) \geq 0& \Rightarrow& a(Z) = \kappa(Z, K^{-1})\\
\kappa(Z, K^{-1}) = -\infty& \Rightarrow& a(Z) = 0.
\end{array}$
\end{thm}
The number $\kappa(Z, K^{-1})$ is usually called the Iitaka dimension (or
L--dimension = line bundle dimension) of the line bundle $K^{-1}$. Its
definition generalizes the well--known notion of Kodaira dimension. For
details, including the following facts, we refer the reader to \cite{U}. \\ For
any line bundle ${\mathcal L}\in\Pic(Z)$ there holds: $\dim Z\geq a(Z)\geq
\kappa(Z, {\mathcal L}) $.\\
If $f:Z\rightarrow Y$ is a dominant morphism, then $a(Z) \geq
a(Y)$. Particularly, if $f:Z\rightarrow {\Bbb P}^N$ is a meromorphic map, then
$a(Z) \geq \dim f(Z)$, because any projective variety is Moishezon.\\
If we define
$g:=\gcd\{m\in {\Bbb Z}\; |\; m > 0, h^0(Z, L^m) \ne 0\}$ and denote by
$\Phi_{|L^m|}$ the meromorphic map given by the linear system $|L^m|$, then
$\kappa(Z, L) = \max\{\dim \Phi_{|L^m|}(Z)\;|\;m \in g{\Bbb Z}, \; m>0\}$. If
there exists a polynomial $P(X)$ such that for all large positive $m\in {\Bbb Z}$
we have $h^0(Z, L^{mg}) \leq P(m)$, then $\kappa(Z, L)\leq \deg P$.
We apply these basic facts to obtain our first result on the algebraic
dimension in case $n = 4$. The following proposition is a generalization of a
result contained in \cite{CK}. For convenience we introduce the following
\begin{main}{Definition:}
If there exists an integer $m\ge 1$ with $h^1(\fb{m})\ne 0$ then we define
$\tau := \min\{m | m\ge 1, h^1(\fb{m})\ne 0\}$. Otherwise we set $\tau :=
\infty$.
\end{main}
\begin{prop}
\label{tau}
Let $Z$ be a simply connected compact twistor space of positive type with
$c_1^3 = 0$. Then:\\
\begin{tabular}[t]{rl}
(i) & $a(Z)\ge 1$\\
(ii) & $a(Z) = 1 \iff \forall m\ge 1\quad h^1(\fb{m})=0$.
\end{tabular}
\end{prop}
{\sc Proof:\quad}
From Riemann--Roch and Hitchin's vanishing theorem we know:
$h^0(\fb{m})=m+1+h^1(\fb{m})$. Therefore, $a(Z)=\kappa(Z,\fdb)\ge 1$ and if
$\tau=\infty$ we have $\kappa(Z,\fdb)=1$.
Assume $\tau<\infty$ and $a(Z)=1$. Let $S\in|\fund|$ be smooth and real. (Such
a divisor exists, because we assume $n=4$, cf.\ \cite{CK}.) Since
$h^1(\fb{\tau-1})=0$ the exact sequence
\[0 \rightarrow\fb{\tau-1} \rightarrow \fb{\tau} \rightarrow \can{\tau}
\rightarrow 0\]
gives an exact sequence
\[
0 \rightarrow H^0(\fb{\tau-1}) \rightarrow H^0(\fb{\tau}) \rightarrow
H^0(\can{\tau}) \rightarrow 0.
\]
Since $h^1(\fb{\tau})\ge1$ we have, furthermore,
$h^0(\fb{\tau})=\tau+1+h^1(\fb{\tau})\ge\tau+2$. The linear system
$|\fun{\tau}|$ cannot have a fixed component since $\tau S\in |\fun{\tau}|$ and
$\dim |S| = \dim |\fund| \ge 1$. If necessary blow up $Z$ to obtain a morphism
$\Phi_\tau:\tilde{Z}\rightarrow {\Bbb P}^d$ defined by $|\fun{\tau}|$. Here
$d:=\dim |\fun{\tau}| \ge \tau+1\ge 2$. By assumption $\dim
\Phi_\tau(\tilde{Z})=1$. Since the curve $\Phi_\tau(\tilde{Z})$ is not
contained in a linear subspace of ${\Bbb P}^d$, its degree must be at least $d$.
Hence, a generic member of the linear system $|\fun{\tau}|$ is the sum of
$\lambda$ algebraically equivalent divisors and so it is linearly equivalent to
$\lambda S_0$ with $\lambda\ge d\ge\tau+1$. This gives
$2\tau=\deg(\fun{\tau})=\lambda\deg(S_0),$ which is only possible if $\lambda =
2\tau$ and $\deg(S_0)=1$. But then we have infinitely many divisors of degree
one in $Z$. This implies $a(Z)=3$ by the Theorem of Kurke--Poon (see \cite{Ku},
\cite{Po2}). This contradiction proves the proposition.\qed
\begin{rem}
If $|-K_S|$ contains a smooth curve $C$, then we computed in \cite{CK} that
$\tau$ is the order of $N:=\canS\otimes{\mathcal O}_C$ in the Picard group $\Pic{C}$
of the elliptic curve $C$. Under this additional assumption Proposition
\ref{tau} was shown in \cite{CK}.
\end{rem}
\section{The structure of fundamental divisors}
\label{eins}
In this section $Z$ always denotes a simply connected compact twistor space of
positive type.
\begin{lemma}\label{pic}
Let $S\in|\fund|$ be a smooth surface. Then the restriction map $\Pic{Z}
\rightarrow \Pic{S}$ is injective.
\end{lemma}
{\sc Proof:\quad}
By assumption we have $h^1({\mathcal O}_Z)=h^2({\mathcal O}_Z)=0$. Since $S$ is a rational
surface \cite{Po1}, we also have $h^1({\mathcal O}_S)=h^2({\mathcal O}_S)=0$. Therefore, taking
the first Chern class defines isomorphisms $\Pic{Z}
\stackrel{\sim}{\rightarrow} H^2(Z,{\Bbb Z})$
and $\Pic{S} \stackrel{\sim}{\rightarrow} H^2(S,{\Bbb Z})$. Let us denote the
inclusion of $S$ into $Z$ by $i$. The above isomorphisms transform then the
restriction morphism $\Pic{Z} \rightarrow \Pic{S}$ into the map $i^\ast$ on
cohomology groups.
We shall apply standard facts from algebraic topology to verify the injectivity
of $i^\ast$. Let $\mbox{\rm o}_S \in H_4(Z,{\Bbb Z})$ and $\mbox{\rm o}_Z \in H_6(Z,{\Bbb Z})$ be the
fundamental classes of $S$ and $Z$ respectively. By $d_Z(S) \in H^2(Z,{\Bbb Z})$
we denote the Poincar\'e dual of $i_\ast(\mbox{\rm o}_S) \in H_4(Z,{\Bbb Z})$, this means
$i_\ast(\mbox{\rm o}_S) = d_Z(S) \smallfrown \mbox{\rm o}_Z$ (cap--product).
For any cohomology class $\alpha \in H^2(Z,{\Bbb Z})$ we obtain by the
associativity of cap--product $\alpha \smallfrown i_\ast(\mbox{\rm o}_S) = \alpha
\smallfrown (d_Z(S) \smallfrown \mbox{\rm o}_Z) = (\alpha \smallsmile d_Z(S))
\smallfrown \mbox{\rm o}_Z$. The naturalness of cap--product implies $\alpha \smallfrown
i_\ast(\mbox{\rm o}_S) = i_\ast(i^\ast(\alpha) \smallfrown \mbox{\rm o}_S)$. Therefore, we obtain
a commutative diagram:
\centerline{
$\begin{array}[t]{ccc}
H^2(Z,{\Bbb Z}) & \stackrel{{\scriptstyle i^\ast}}{\longrightarrow} &
H^2(S,{\Bbb Z})\\
& & \downarrow {\scriptstyle\smallfrown \mbox{\rm o}_S}\\
\Bigg\downarrow {\scriptstyle \smallsmile d_Z(S)} & & H_2(S,{\Bbb Z})\\
& & \downarrow {\scriptstyle i_\ast}\\
H^4(Z,{\Bbb Z}) & \stackrel{{\scriptstyle \smallfrown \mbox{\rm o}_Z}}{\longrightarrow} &
H_2(Z,{\Bbb Z})
\end{array}$}
Since, by Poincar\'e duality, the cap--product with $\mbox{\rm o}_Z$ is an isomorphism,
we obtain $\ker(i^\ast) \subset \ker( \smallsmile d_Z(S))$.
The description of the cohomology ring given above allows us to compute the
kernel of the cup--product with the dual class $d_Z(S)$ of S. With the
notation of Section \ref{prelim} the elements $x_1,\dots, x_n,\omega$ form a
basis of the free ${\Bbb Z}$--module $H^2(Z,{\Bbb Z})$. The dual class of $S$ is
$d_Z(S) = c_1(\fdb) = 2\omega + x_1 + \dots + x_n$. If we use $\omega
x_1,\dots,\omega x_n,x_1^2$ as basis of $H^4(Z,{\Bbb Z})$ then the cup--product
with $d_Z(S)$ is described by the $(n+1)\times(n+1)$--matrix:
\[ \left(
\begin{array}{rrrrr}
2&0&\dots&0&1\\ 0&2&&0&1\\ \vdots& & &\vdots &\vdots\\ 0& &2 &0&1\\
0&\dots&0&2&1\\ -1&\dots&-1&-1&-2
\end{array}
\right)\]
whose determinant is equal to $2^{n-1}(n-4)$.
If $n\ne 4$ we obtain the injectivity of the map $\alpha \mapsto \alpha
\smallsmile d_Z(S)$ and thus of the restriction map $\Pic{Z} \hookrightarrow
\Pic{S}$. If $n = 4$ it is easy to see that $\alpha \smallsmile d_Z(S) = 0$ if
and only if $\alpha \in {\Bbb Z}\cdot d_Z(S) \subset H^2(Z,{\Bbb Z})$. To prove the
injectivity of $\Pic{Z} \rightarrow \Pic{S}$ it remains, therefore, to show
that $\fb{m}\otimes{\mathcal O}_S \cong {\mathcal O}_S$ implies $m = 0$.
By adjunction we have $\fb{m}\otimes{\mathcal O}_S \cong \can{m}$. But $S$ is rational,
hence $\Pic{S}$ is torsion free and $K_S\ncong {\mathcal O}_S$. Thus,
$\can{m}\cong{\mathcal O}_S$ if and only if $m=0$. This proves the Lemma. \qed
\begin{lemma}\label{conn}
Let $Z$ be a simply connected compact twistor space of positive type and
$D\subset Z$ a divisor of degree one. If $S\in|\fund|$ is a smooth surface,
then $C:=D\cap S$ is connected.
\end{lemma}
{\sc Proof:\quad}
We shall show $h^0({\mathcal O}_C)=1$, which implies connectedness of $C$.
Consider first the exact sequence
\begin{equation}
\label{seqelm}
0 \rightarrow {\mathcal O}_Z(-\bar{D}) \rightarrow {\mathcal O}_Z \rightarrow {\mathcal O}_{\bar{D}}
\rightarrow 0.
\end{equation}
From $h^1({\mathcal O}_Z)=0$ we obtain
$h^1({\mathcal O}(-\bar{D}))=h^0({\mathcal O}_{\bar{D}}) - h^0({\mathcal O}_Z)=0$ since $Z$ and
$\bar{D}$ are connected.
As $D+\bar{D}\in |\fund|$ we obtain an exact sequence
\[ 0 \rightarrow K(\bar{D}) \rightarrow \fdbd \rightarrow {\mathcal O}_D(-C)
\rightarrow 0. \] But the degree of $\fdbd$ is $-2$ and, therefore, Hitchin's
vanishing theorem gives $h^i(\fdbd)=0$ for all $i$. Therefore, using
Serre duality, $h^1({\mathcal O}_D(-C))= h^2(K(\bar{D})) = h^1({\mathcal O}_Z(-\bar{D})) = 0$.
Consider finally the exact sequence
\[ 0 \rightarrow {\mathcal O}_D(-C) \rightarrow {\mathcal O}_D \rightarrow {\mathcal O}_C \rightarrow
0. \] We have $h^0({\mathcal O}_D)=1$ since any divisor of degree one is connected
\cite{Po1}. Because $C$ is effective, $h^0({\mathcal O}_D(-C))$ must vanish. Hence,
$h^0({\mathcal O}_C)=h^0({\mathcal O}_D)=1$. \qed
\begin{lemma}[cf.\ \cite{PP}, p.\ 693]\label{noreal}
Let $Z$ be as above and $S\in|\fund|$ an irreducible real divisor. Then $S$
is smooth and contains a real twistor fibre $F\subset S$. The linear system
$|F|$ is one--dimensional and its real elements are precisely the real
twistor fibres contained in $S$.
\end{lemma}
{\sc Proof:\quad}
The smoothness of $S$ was shown in \cite[Lemma 2.1]{PP1}. If $S$ does not
contain a real twistor fibre, the restriction of the twistor fibration to $S$
would give an unramified double cover over a simply connected manifold, since
$Z$ does not contain real points. But $S$ is connected and must, therefore,
contain a real twistor fibre $F$. From the adjunction formula we obtain $F^2 =
0$ on $S$. Hence, we have an exact sequence
$0 \rightarrow {\mathcal O}_S \rightarrow {\mathcal O}_S(F) \rightarrow {\mathcal O}_F \rightarrow 0$.
From $h^1({\mathcal O}_S) = 0$ we infer, therefore, $\dim |F| = 1$.
Since the linear system $|F|$ defines a flat family of curves in $S$, its
elements form a curve in the Douady space ${\mathcal D}$ of curves on $Z$ (cf.\
\cite{Dou}). Since $h^0({\Bbb P}^1,{\mathcal O}(1)\oplus{\mathcal O}(1))=4$ and
$h^1({\Bbb P}^1,{\mathcal O}(1)\oplus{\mathcal O}(1))=0,\;\;{\mathcal D}$ is a four--dimensional
complex manifold near points which correspond to smooth rational curves on $Z$
with normal bundle ${\mathcal O}(1)\oplus{\mathcal O}(1)$. The real structure of $Z$ induces
one on ${\mathcal D}$. If the set of real points ${\mathcal D}({\Bbb R})$ is non--empty,
then it is a four--dimensional real manifold near points as before. Since the
real twistor lines are smooth rational curves with the above normal bundle, the
real manifold $M=4\bbfC\bbfP^2$ is a submanifold of ${\mathcal D}({\Bbb R})$. Since $M$ is
compact and has the same dimension as ${\mathcal D}({\Bbb R})$, it must be a connected
component.
The set $U$ of members of $|F|$ which are smooth rational curves with normal
bundle ${\mathcal O}(1)\oplus{\mathcal O}(1)$ is open and dense in ${\Bbb P}^1\cong|F|$ with
respect to the Zariski topology. Therefore, the set $U({\Bbb R})$ of real points
in $U$ is open and dense in the one--sphere of real members of $|F|$. Since
${\mathcal D}({\Bbb R})$ is smooth near $M$ and $M$ is a component of ${\mathcal
D}({\Bbb R})$, we have $U({\Bbb R})\subset M$. But $M$ is compact and must,
therefore, contain the closure of $U({\Bbb R})$ in ${\mathcal D}({\Bbb R})$, which is the
set of all real members of $|F|$. Therefore, any real member of $|F|$ is a real
twistor fibre and, in particular, smooth and ireducible. This proves the
claim.\qed
To obtain more information on the structure of real irreducible fundamental
divisors $S\in|\fund|$ one can study the morphism $S\rightarrow {\Bbb P}^1$ given
by $|F|$ (cf.\ \cite[p.\ 693]{PP}).
Since the general fibre of this morphism is a smooth rational curve it factors
through a rational ruled surface. Since $(-K_S)^2 = (\fund)^3 = 8-2n$, the
surface $S$ is a blow--up of a ruled surface at $2n$ points. The exceptional
curves of these blow--ups are contained in fibres of the morphism $S\rightarrow
{\Bbb P}^1$. By Lemma \ref{noreal} none of the exceptional curves is real and none
of the blown--up points lie on a real fibre of the ruled surface. Using this,
in \cite[Lemma 3.5]{PP} it has been shown that the ruled surface is isomorphic
to $\bbfP^1\times\bbfP^1$. Therefore, we obtain a morphism $\sigma:S\rightarrow\bbfP^1\times\bbfP^1$ which is a
succession of blow--ups. Let us equip $\bbfP^1\times\bbfP^1$ with the real structure given by
the antipodal map on the first factor and the usual real structure on the
second. Then $\sigma$ is equivariant (or ``real''). Since we can always
contract a conjugate pair of disjoint $(-1)$--curves, $\sigma$ is the
succession of $n$ blow--ups. At each step a conjugate pair of points is
blown--up to give a surface without real points.
We should bear in mind that it is possible to have
infinitesimally near blown--up points.
As in \cite{PP} we shall call curves of type $(1,0)$ on $\bbfP^1\times\bbfP^1$ ``lines'' and
curves of type $(0,1)$ ``fibres''. Then there do not exist real lines. But
the images of real twistor fibres in $|F|$ are exactly the real ``fibres''.
\begin{lemma}\label{smocomp}
Equip $\bbfP^1\times\bbfP^1$ with the real structure $((a_0:a_1),(b_0:b_1))\mapsto
((\bar{a}_1:-\bar{a}_0),(\bar{b}_0:\bar{b}_1))$ as described above. Then the
reduced components of any \underline{real} member of $|{\mathcal O}(2,2)| =
|-K_{\bbfP^1\times\bbfP^1}|$ are smooth. A non--reduced component of a real member of
$|{\mathcal O}(2,2)|$ can only be of the form $2F$ with a real curve
$F\in|{\mathcal O}(0,1)|$.
\end{lemma}
{\sc Proof:\quad}
As usual, ${\mathcal O}(k,l)$ denotes the locally free sheaf
$p_1^\ast{\mathcal O}_{{\Bbb P}^1}(k) \otimes p_2^\ast{\mathcal O}_{{\Bbb P}^1}(l)$ on
the smooth rational surface $\bbfP^1\times\bbfP^1$, where $p_i : \bbfP^1\times\bbfP^1 \rightarrow {\Bbb P}^1$ ($i
=1,2$) are the projections and $k, l$ are integers.
The Picard group $\Pic(\bbfP^1\times\bbfP^1)$ is free abelian of rank two with generators
${\mathcal O}(1,0)$ and ${\mathcal O}(0,1)$. In the proof we shall use the well--known fact
that, if $k<0$ or $l<0$, then the linear system $|{\mathcal O}(k,l)|$ is empty.
Let $C\in|{\mathcal O}(2,2)|$ be a real curve and $C_0\in |{\mathcal O}(a,b)|$ an {\sl
irreducible\/} component (with {\sl reduced} scheme structure) of $C$. Let
$\lambda\geq 1$ be the multiplicity of $C_0$ in $C$, that is the largest
integer with $\lambda C_0\subset C$. Then we must have $0\leq \lambda a\leq 2$
and $0\leq \lambda b\leq 2$. The case $\lambda a = \lambda b = 2$ can only
occur if $\lambda C_0 = C$.
Assume first $\lambda C_0\ne C$, hence $\lambda a \leq 1$ or $\lambda b \leq
1$. If $\lambda\ge 2$ or $C_0$ singular, there exists a point $y\in C_0$ such
that any curve not contained in $C_0$ but containing $y$ has intersection
number at least two with $\lambda C_0$. If $F\in |{\mathcal O}(0,1)|$ and
$G\in|{\mathcal O}(1,0)|$ are the unique curves in these linear systems containing the
point $y$, we obtain (as $C_0$ is irreducible) $F.(\lambda C_0) = \lambda a\ge
2$ or $G.(\lambda C_0) = \lambda b\ge 2$. By the above inequalities, this means
$\lambda a =2$ and $0\leq\lambda b\leq 1$ or $0\leq\lambda a\leq 1$ and
$\lambda b =2$. If $\lambda = 1$, the curve $C_0$ is, by assumption,
irreducible, reduced, singular and a member of $|{\mathcal O}(2,0)|$, $|{\mathcal O}(2,1)|$,
$|{\mathcal O}(0,2)|$, or $|{\mathcal O}(1,2)|$. But these linear systems do not contain such a
curve. Hence, we must have $\lambda = 2$ and, therefore, $C_0\in|{\mathcal O}(0,1)|$
or $C_0\in|{\mathcal O}(1,0)|$. In particular, $C_0$ is smooth.
If $C_0\in|{\mathcal O}(1,0)|$, this curve is not real, since by definition of the real
structure on $\bbfP^1\times\bbfP^1$ this linear system does not contain real members. Hence, the
component $2C_0$ of $C$ is not real, which implies $2C_0 + 2\bar C_0 \subset
C$, since $C$ is real. But $\bar C_0 \in|{\mathcal O}(1,0)|$ and so $2C_0 + 2\bar C_0
\in |{\mathcal O}(4,0)|$. Such a curve can never be contained in $C\in|{\mathcal O}(2,2)|$. So
we obtain $\lambda C_0 = 2F$ with some $F\in|{\mathcal O}(0,1)|$.
Again, since a curve of type $(0,4)$ can never be a component of $C$, the fibre
$F$ is necessarily real. This proves the lemma in the case $\lambda C_0 \ne C$.
Assume now $\lambda C_0=C$. Then $\lambda = 1$ or $\lambda = 2$. If
$\lambda=2$, we have, by reality of $C = 2C_0$, that $C_0\in|{\mathcal O}(1,1)|$ is a
real curve. Because $C_0$ is irreducible and $C_0.F =1$ for any
$F\in|{\mathcal O}(0,1)|$, the curve $C_0$ would intersect each real fibre
$F\in|{\mathcal O}(0,1)|$ at a real point. But on $\bbfP^1\times\bbfP^1$ real points do not exist. Hence
$\lambda=1$, which means $C = C_0$ is irreducible and real.
It remains to see that $C$ must be reducible if it is not smooth. Let $x\in C$
be a singular point of $C$. Since $C$ is real and $\bbfP^1\times\bbfP^1$ does not contain real
points, $\bar{x}\ne x$ is also a singular point on $C$. If we embed $\bbfP^1\times\bbfP^1$ by
$|{\mathcal O}(1,1)|$ as a smooth quadric into ${\Bbb P}^3,$ we easily see that the linear
system of curves of type $(1,1)$ on $\bbfP^1\times\bbfP^1$ containing $x$ and $\bar{x}$ is
one--dimensional. It is cut out by the pencil of planes in ${\Bbb P}^3$ containing
the line connecting $x$ and $\bar{x}$. Therefore, any point of $\bbfP^1\times\bbfP^1$ is
contained in such a curve. The intersection number of $C$ with a curve of type
$(1,1)$ is four. Since $x$ and $\bar{x}$ are singular points on $C$, any curve
of type $(1,1)$ containing $x$ and $\bar{x}$ and a third point of $C$ must have
a common component with $C$. Therefore, $C$ cannot be irreducible and
reduced.\qed
\begin{defi}
A reduced curve $C$ on a compact complex surface $S$ will be called a ``cycle
of rational curves'', if the irreducible components $C_1,\dots, C_m$ of $C$
are smooth rational curves with the following properties: (We use the
convention $C_{m+1} = C_1$.)\\
$m = 2$ and $C_1$ intersects $C_2$ transversally at two distinct points,
$C_1.C_2 = 2$, or\\
$m\geq 3$, $C_i.C_{i+1} = 1$ and $C_i\cap C_j \ne \emptyset$ implies
$j\in\{i-1, i, i+1\}$.
\end{defi}
\begin{prop}\label{types}
Assume $\dim |\fund|\geq 1$ and let $S\in|\fund|$ be smooth and real. Then
there exists a blow-down $S\rightarrow \bbfP^1\times\bbfP^1$ and a connected real member
$C\in|-K_S|$, such that:
\begin{itemize}
\item $\sigma$ is compatible with real structures, where we use the real
structure of Lemma \ref{smocomp} on $\bbfP^1\times\bbfP^1$,
\item the composition $\mbox{pr}_2\circ \sigma$ of $\sigma$ with the second
projection $\bbfP^1\times\bbfP^1\longrightarrow {\Bbb P}^1$ is the morphism given by the linear
system $|\widetilde{F}|$, where $\widetilde{F}\subset S$ is a real twistor
fibre,
\item the curve $C$ is reduced and
\item if $C$ is not smooth, it is a ``cycle of rational curves'' and its image
$C'$ in $\bbfP^1\times\bbfP^1$ has one of the following structures:
\end{itemize}
\renewcommand{\labelenumi}{(\Roman{enumi})}
\begin{enumerate}
\item
\mbox{
\begin{minipage}[t]{10.5cm}
$C'$ has four components $C'=F+\bar{F}+G+\bar{G}$ where $F\in|{\mathcal O}(0,1)|$ is a
non--real fibre and $G\in|{\mathcal O}(1,0)|$ is a line.
\end{minipage}
\begin{picture}(60,-60)(-15,-5)
\put(0,-40){\line(1,0){60}} \put(30,-42){\makebox(0,0)[t]{$\bar{G}$}}
\put(0,0){\line(1,0){60}} \put(30,2){\makebox(0,0)[b]{$G$}}
\put(10,10){\line(0,-1){60}} \put(12,-20){\makebox(0,0)[l]{$F$}}
\put(50,10){\line(0,-1){60}} \put(52,-20){\makebox(0,0)[l]{$\bar{F}$}}
\end{picture}\rule[-60pt]{0pt}{60pt}
}
\item
\begin{minipage}[t]{10.5cm}
$C'$ has two components $C'=F+C_0$ where $F\in|{\mathcal O}(0,1)|$ is a real fibre and
$C_0\in|{\mathcal O}(2,1)|$ is real, smooth and rational.
\end{minipage}
\begin{picture}(60,-60)(-15,-5)
\put(45,10){\line(0,-1){60}} \put(47,-20){\makebox(0,0)[l]{$F$}}
\put(50,-20){\oval(55,40)[l]} \put(18,-20){\makebox(0,0)[r]{$C_0$}}
\end{picture}\rule[-60pt]{0pt}{60pt}
\item
\begin{minipage}[t]{13.8cm}
$C'$ has two distinct components $C'=A'+\bar{A}'$ where $A',\bar{A}'\in
|{\mathcal O}(1,1)|.$\\
\mbox{\rm [By Corollary \ref{omit} this item can be omitted if
$n=4$ and $\fdb$ is not nef!]}
\end{minipage}
\end{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\end{prop}
{\sc Proof:\quad}
As $\dim|-K_S| = \dim|\fund| - 1$, we have by assumption $|-K_S|\ne\emptyset$.
As seen above, we can choose a real blow--down map $\sigma:S\rightarrow\bbfP^1\times\bbfP^1$
such that $(\mbox{pr}_2\circ\sigma)^\ast{\mathcal O}_{{\Bbb P}^1}(1)
={\mathcal O}_S(\widetilde{F})$.
If $|-K_S|$ contains a smooth member, we are done. Otherwise, take a reducible
real $C\in|-K_S|$ and let $C'\subset\bbfP^1\times\bbfP^1$ be the image of $C$. Since
$\sigma$ is a blow--up, $C'$ is a real member of $|{\mathcal O}(2,2)|$. By Lemma
\ref{smocomp} the components of $C'$ are smooth and a multiple component can
only be the multiple $2F$ of a real fibre $F\in|{\mathcal O}(0,1)|$. But Lemma
\ref{noreal} shows
that no point on such a real fibre is blown--up. Therefore, any other member of
$|2F|$ missing the $2n$ blown--up points, defines a divisor in $|-K_S|$.
Choosing, for example, a conjugate pair of appropriate fibres, we obtain a real
member in $|-K_S|$ whose image in $\bbfP^1\times\bbfP^1$ has only reduced components.
We assume for the rest of the proof that $C$ is chosen in this way. If $C'$
would be irreducible, it would be smooth by Lemma \ref{smocomp}. In this case
$C$ is smooth, too.
Assume $C'$ is reducible. A component of type $(1,a)$ with $a\in\{0,1,2\}$
cannot be real, since, otherwise, it would intersect real fibres at real
points. Therefore, such components appear in conjugate pairs, hence $a\le
1$. If $a=0$ we are in case (I). If $a\ne 0$ then $a=1$ and $C'=A'+\bar{A}'$
with two distinct curves $A'$, $\bar{A}'$ of type $(1,1)$. This is case (III).
Assume now that there is no component of type $(1,a)$. Then we must have a
component $C_0$ of $C'$ which has type $(2,a)$. If $a=2$ it must be smooth and
we are done. Therefore, $a=1$, because $|{\mathcal O}(2,0)|$ does not contain
irreducible reduced elements. Then we have $C'=C_0+F$ with $F\in|{\mathcal O}(0,1)|$
and $C_0\in|{\mathcal O}(2,1)|$. $F$ and $C_0$ must be real, since they have different
types. This is case (II) of our statement.
It remains to show that $C$ is a ``cycle of rational curves''. We have seen
this for the image $C'$ in $\bbfP^1\times\bbfP^1$. Exceptional components of $C$ are always
rational. Furthermore, $C'$ has at most ordinary nodes as singularities. To
obtain $C$ from $C'$, at every step of blowing--up, we have to subtract the
exceptional locus from the total transform of $C'$. At every step we blow up
either a conjugate pair of singular points or of smooth points. We obtain a
curve which has again, at most, singularities of multiplicity two and is a
``cycle of rational curves''. So we obtain this property for $C$, too.\qed
Using this structure result and assuming that the fundamental linear system
$|\fund|$ is a pencil, we can show that the structure of its base locus is
closely related to the effective divisors of degree one on $Z$.
\begin{prop}\label{basel}
Assume $\dim|\fund| = 1$ and denote by $C$ the base locus of the fundamental
linear system.\\
If $C$ is smooth, then $Z$ does not contain effective divisors of degree
one.\\
If $C$ is not smooth, then the number of effective divisors of degree one on
$Z$ is equal to the number of components of $C$.
\end{prop}
{\sc Proof:\quad}
Let $S\in|\fund|$ be a smooth real member. Then $|-K_S| = \{C\}$ and by
Proposition \ref{types} $C$ is smooth or a ``cycle'' of smooth rational curves.
If $C$ is smooth, there does not exist an effective divisor $D$ of degree one,
because $D + \bar{D}\in|\fund|$ would produce a reducible member in
$|-K_S|$. Let now $C$ be singular, hence reducible.
The rest of the proof is an adaption of an idea of Pedersen and Poon
\cite[p.\ 700]{PP}.
Now let $\{P,\bar{P}\}$ be any pair of singular points on $C$. The image $C'$
of $C$ in $\bbfP^1\times\bbfP^1$ does not contain a real fibre, because, otherwise, by
Proposition \ref{types} and Lemma \ref{noreal} the linear system $|-K_S|$ would
be at least one--dimensional. Hence, the real twistor line $L_P$ containing $P$
and $\bar{P}$ is not contained in $S$. Hence, $L_P$ meets $S$ transversally at
$P$ and $\bar{P}$. If $Q$ is a point on $L_P$ distinct from $P$ and$\bar{P}$,
then there exists a divisor $S_0\in |\fund|$ containing $Q$. Since $S_0$
contains also $C$ it contains three points of $L_P$. Hence, $L_P\subset S_0$.
Therefore, the real linear system of fundamental divisors containing $L_P$ is
non--empty. This implies that we can choose a real $S_0 \in |\fund|$ containing
$L_P$.
Since $S_0$ contains also $C$ and $P$ is a singular point of $C$, the surface
$S_0$ contains three curves meeting at $P$, namely $L_P$ and two components
(call them $A$ and $B$) of $C$. On the other hand, $L_P$ intersects $S$
precisely at $P$ and $\bar{P}$ as we have seen above. From $L_P.S=2$ we infer
that this is a transversal intersection. But $A$ and $B$ are contained in $S$
and are transversally there. We can conclude that the tangent space of $Z$ at
$P$ is generated by the tangent directions of $A, B$ and $L_P$ at $P$. Hence,
the real surface $S_0$ is singular at $P$. This implies that $S_0$ is singular
along $L_P$ (cf. \cite[p.141]{Hit2}) and by \cite[Lemma 2.1]{PP1} such a
divisor splits into the sum of two divisors of degree one. Therefore, we have
at least as many pairs of conjugate divisors of degree one as we have pairs of
conjugate singular points on $C$. In other words, the number of distinct
divisors of degree one is at least equal to the number of components of $C$.
Let $D$ and $\bar{D}$ be a conjugate pair of divisors of degree one on $Z$.
Then $C\subset D\cup\bar{D}$. $D\cap \bar{D}$ is a real twistor line (cf.\
\cite[Prop. 2.1]{Ku}), and no component of $C$ is a real twistor line.
Hence, every component of $C$ lies on exactly one of the surfaces $D$ and
$\bar{D}$. By Lemma \ref{conn} $C\cap D$ is connected. The same is true for the
conjugate curve $C\cap \bar{D}$. Since $C$ is a cycle of rational curves,
$(C\cap D)\cap(C\cap\bar{D})$ consists of a conjugate pair $\{P,\bar{P}\}$ of
singularities of $C$. Since $D$ and $\bar{D}$ are of degree one, the real
twistor line $L_P$ containing $P$ and $\bar{P}$ must be contained in $D$ and in
$\bar{D}$. Therefore, $D\cap\bar{D}=L_P$.
Let $D'$ be an arbitrary divisor of degree one containing $L_P$. Then
$D'\cap\bar{D}'=L_P$ and without loss of generality we may assume $D'\cap
C=D\cap C$, since the decomposition of $C$ into two conjugate connected curves
is determined by $\{P, \bar{P}\}=L_P\cap C$. By Lemma \ref{pic} the restriction
map $\Pic{Z}\rightarrow\Pic{S}$ is injective. Since $(D'+\bar{D}')\cap S=C$ we
have $D'\cap S=D'\cap C$ and $D\cap S=D\cap C$. Hence, we have ${\mathcal O}_Z(D)\cong
{\mathcal O}_Z(D')$, which means that $D$ and $D'$ are linearly equivalent. If $D\ne
D'$, then $\dim|D|\ge 1$ and $Z$ would contain infinitely many divisors of
degree one and by Kurke \cite{Ku} and Poon \cite{Po2} it must be a
conic--bundle twistor space. But then we should have $\dim|\fund|=3$ in
contradiction to our assumption. Hence, $D=D'$ and we have exactly as many
divisors of degree one on $Z$ as $C$ has components.\qed
For technical reasons we state here the following lemma needed in Section
\ref{drei}.
\begin{lemma}\label{normal}
Let $S$ be a smooth complex surface and $C\subset S$ a reduced curve. Assume
$C =\sum_{i=1}^{m} C_i$ is a ``cycle of rational curves'' as defined above.
If $L\in\Pic(S)$ is a line bundle, we define $l_i := L.C_i$. Let $I_\pm :=
\{i | \pm l_i>0\}$ and $C_\pm := \sum_{i\in I_\pm} C_i$. Let $\gamma$ denote
the number of connected components of $C\setminus C_-$. Assume $|I_-|\geq 2$
and each connected component of $C\setminus C_-$ contains a component of
$C_+$. Then we have:
\[h^0(C,L) = \sum_{i\in I_+} l_i - \gamma.\]
\end{lemma}
{\sc Proof:\quad}
Let $\eta:\tilde C = \sqcup_i C_i \rightarrow C$ be the normalization of $C$.
By $P_i$ we denote the intersection point of $C_i$ with $C_{i+1}$ $(1\leq i
\leq m)$. By assumption $m\geq 3$. Tensoring the exact sequence $0\rightarrow
{\mathcal O}_C \rightarrow \eta_\ast {\mathcal O}_{\tilde C} \rightarrow \oplus_i {\Bbb C}_{P_i}
\rightarrow 0$ with $L$ yields the exact sequence $0 \rightarrow L\otimes
{\mathcal O}_C \rightarrow \eta_\ast\eta^\ast(L\otimes{\mathcal O}_C) \rightarrow \oplus_i
{\Bbb C}_{P_i} \rightarrow 0$. Hence, $H^0(C, L\otimes{\mathcal O}_C) \cong \ker(\oplus_i
H^0(C_i, L_i) \stackrel{\rho}{\rightarrow} \oplus_i {\Bbb C}_{P_i})$. Here we
denote $L_i:= L\otimes{\mathcal O}_{C_i} \cong {\mathcal O}_{C_i}(l_i)$. Let $P'_i\in C_i$ and
$P''_i\in C_{i+1}$ be the two points on $\tilde C$ lying over $P_i$.
To describe $\rho$ we observe that the map $\eta$ gives isomorphisms $L_i(P'_i)
\stackrel{\sim}{\rightarrow} {\Bbb C}_{P_i}$ and $L_{i+1}(P''_i)
\stackrel{\sim}{\rightarrow} {\Bbb C}_{P_i}$. If $s_i\in H^0(C_i, L_i)$ is a
section, we denote by $s_i(P_i)$ the image of $s_i$ under the map $H^0(C_i,
L_i) \rightarrow L_i(P'_i) \stackrel{\sim}{\rightarrow} {\Bbb C}_{P_i}$.
Similarly, $s_i(P_{i-1})$ is the image of $s_i$ under $H^0(C_i, L_i)
\rightarrow L_i(P''_{i-1}) \stackrel{\sim}{\rightarrow} {\Bbb C}_{P_{i-1}}$. With
this notation we have:
\[ \rho(s_1,\dots, s_m) = (s_1(P_1) - s_2(P_1), s_2(P_2) - s_3(P_2),\dots,
s_m(P_m) - s_1(P_m)).\] Since $P'_i \ne P''_{i-1}$ on $C_i \cong {\Bbb P}^1$, the
restriction of $\rho$ $H^0(C_i, L_i) \rightarrow
{\Bbb C}_{P_i}\oplus{\Bbb C}_{P_{i-1}}$ is surjective if and only if $l_i > 0$. If
$C_i + \cdots + C_{i+r}$ is a connected component of $C\setminus C_-$, then we
obtain by induction on $r$ that the restriction of $\rho$ $\oplus_{\mu = 0}^r
H^0(C_{i+\mu}, L_{i+\mu}) \rightarrow \oplus_{\mu = -1}^r {\Bbb C}_{P_{i+\mu}}$ is
surjective. Because $H^0(C_i, L_i) = 0$ if and only if $l_i < 0$, we obtain
$\im(\rho) = \sum_{P_\mu\in C_0+C_+} {\Bbb C}_{P_\mu}$. (Here, we denote $I_0 := I
\setminus (I_- \cup I_+)$ and $C_0 := \sum_{\nu\in I_0} C_\nu$.)
The number of points
$P_\mu\in C_0+C_+$ is equal to $|I_0| + |I_+| + \gamma$. Therefore, we obtain
\[ \dim\ker(\rho) = \sum_i h^0(C_i, L_i) - (|I_0| + |I_+| +\gamma) =
\sum_{l_i\geq 0} (l_i + 1) - |I_0| - |I_+| -\gamma = \sum_{l_i > 0} l_i -
\gamma.\]
\qed
\section{The nef case}
\label{zwei}
For the rest of the paper we assume $n = 4$. Remember that $(\fund)^3 = 0$,
$\chi(\fb{m}) = m+1$ and $h^0(\fdb) \geq 2$ in this case. Remember from
Mori's theory that a line bundle $L\in\Pic(Z)$ is called {\em nef}, if for each
irreducible curve $C\subset Z$ there holds $L.C\geq 0$.
\begin{thm}\label{nef}
The following properties are equivalent:
\begin{enumerate}
\item
$\fdb$ is nef;
\item
for all smooth and real $S\in|\fund|$ and all $C\in|-K_S|$, every component
$C_0$ of $C$ has the property $C_0.(-K_S)=0$;
\item
there exists a smooth and real $S\in|\fund|$ and a divisor $C\in|-K_S|$, such
that all components $C_0$ of $C$ have the property $C_0.(-K_S)=0$.
\end{enumerate}
If $\fdb$ is nef, then $a(Z)\le 2$ and $\dim|\fund|\leq 2$.\\
If $\fdb$ is nef and $\dim|\fund| = 2$, then $a(Z) = 2$, $|\fund|$ does not
have base points and for any smooth real $S\in|\fund|$ the pencil $|-K_S|$
contains a smooth real member.
\end{thm}
{\sc Proof:\quad}
(i)$\Rightarrow$(ii):\\ Take any smooth real $S\in|\fund|$ and an arbitrary
curve $C\in|-K_S|$. Since $C.(\fund)=0$ and $\fdb$ is nef we obtain (ii).
(ii)$\Rightarrow$(iii) is obvious.
(iii)$\Rightarrow$(i):\\ If $\fdb$ were not nef, then there would exist an
irreducible curve $C_0\subset Z$ with $C_0.(\fund)<0$. If $S\in|\fund|$ is
smooth and real, then $C_0\subset S$ and $C_0.(-K_S)=C_0.(\fund)<0$.
Therefore, $C_0$ is a component of any element of $|-K_S|$ in contradiction to
(iii).
Assume for the rest of the proof that $\fdb$ is nef. Let $S\in|\fund|$ be
smooth and real and $C\in|-K_S|$ a real member. If $C$ is smooth, we have shown
in \cite{CK} that $a(Z)\leq 2$ and $\dim|\fund|\leq2$. Assume $C$ is not
smooth. To compute the algebraic dimension consider the exact sequences
\[0 \rightarrow \fb{m-1} \rightarrow \fb{m} \rightarrow \can{m} \rightarrow 0\]
and
\[0 \rightarrow \can{(m-1)} \rightarrow \can{m} \rightarrow N^{\otimes m}
\rightarrow 0\]
with $N:=\canS\otimes{\mathcal O}_C$. Since $\fdb$ is nef, $(-K_S).C_i = 0$ for any
component $C_i$ of $C$. But $C_i \cong {\Bbb P}^1$ and so $N\otimes{\mathcal O}_{C_i}
\cong {\mathcal O}_{C_i}$. This does not imply in general $N \cong {\mathcal O}_C$, because $C$
is a ``cycle'' of rational curves. But we obtain $h^0(N^{\otimes m}) = 1$ if
$N^{\otimes m} \cong {\mathcal O}_C$ and $h^0(N^{\otimes m}) = 0$ if $N^{\otimes m}
\ncong {\mathcal O}_C$. As in \cite{CK} this implies $a(Z)\leq 2$ and $h^0(\fdb) =
h^0({\mathcal O}_Z) + h^0(\canS) = 1 + h^0({\mathcal O}_S) + h^0(N) \leq 3$.
Assume now $\dim|\fund| = 2$. Hence, using the Riemann--Roch formula we have
$h^1(\fdb) = 1$. By Proposition \ref{tau} this implies $a(Z)\geq 2$. From the
above considerations we obtain $h^0(N) = 1$, hence $\canS\otimes{\mathcal O}_C \cong N
\cong {\mathcal O}_C$. (The same is true if $C$ is smooth, cf.\ \cite{CK}.) The exact
sequence $0 \rightarrow {\mathcal O}_S \rightarrow \canS \rightarrow N \rightarrow 0$
and $h^1({\mathcal O}_S) = 0$ give a surjective restriction map $H^0(\canS)
\twoheadrightarrow H^0({\mathcal O}_C) \cong {\Bbb C}$. Because $C\in|-K_S|$, this shows
that $|-K_S|$ does not have base points. Since $\dim|-K_S| = \dim|\fund| -1 =
1$, Bertini's Theorem \cite[I \S 1]{GH} states the existence of a smooth member
in $|-K_S|$. Hence, the generic divisor in $|-K_S|$ is smooth and so the
generic real member, too. On the other hand, we know from $h^1({\mathcal O}_Z) = 0$
that the restriction map $|\fund| \twoheadrightarrow |-K_S|$ is
surjective. From the freeness of $|-K_S|$ we conclude that $|\fund|$ does
not have base points.\qed
\begin{rem}\label{smnef}
If there exists a smooth real $S\in|\fund|$ and a smooth curve $C\in|-K_S|$,
then $\fdb$ is nef. This is clear from the theorem, because $C.(\fund) =
(\fund)^3 = 0$.
\end{rem}
\begin{cor}\label{omit}
In Proposition \ref{types} we can omit case (III) if $\fdb$ is not nef.
\end{cor}
{\sc Proof:\quad}
Assume $C'=A'+\bar{A}'$ as in the proof of Proposition \ref{types}.
$A'$ and $\bar{A}'$ intersect at a pair of conjugate points, say
$P$ and $\bar{P}$.
If $\sigma$ does not blow up $P$ and $\bar{P}$, then, by reality of the
blown--up set, on (the strict transforms of) $A'$ and $\bar{A}'$ exactly four
points are blown--up. If we denote by $A$ and $\bar{A}$ the strict transforms
of $A'$ and $\bar{A}'$ in $S$, then we have $C=A+\bar{A}$ and
$A^2=\bar{A}^2=-2$. Since $A$ and $\bar{A}$ are rational we obtain, by the
adjunction formula, $A.(-K_S)=\bar{A}.(-K_S)=0$. By Theorem \ref{nef} this
implies that $\fdb$ is nef.
If $\sigma$ blows up $P$ and $\bar{P}$, then we perform an elementary transform
to arrive at case (I) as follows. Let $\sigma_1:S^{(1)}\rightarrow\bbfP^1\times\bbfP^1$ be the
blow--up of
$P$ and $\bar{P}$, then we have an induced real structure on $S^{(1)}$. Since
$A'$ intersects any fibre at exactly one point, $P$ and $\bar{P}$ lie on a
conjugate pair of fibres. The strict transforms in $S^{(1)}$ of these fibres
form a conjugate pair of disjoint $(-1)$--curves. Contracting them we obtain a
blow--down map $\sigma_1':S^{(1)}\rightarrow\bbfP^1\times\bbfP^1$ which is again compatible with
real structures. If we denote by $E$ and $\bar{E}$ the exceptional curves of
$\sigma_1$, then the image of $C$ in $S^{(1)}$ is
$\sigma_1^\ast(A'+\bar{A}')-E-\bar{E} = A^{(1)}+\bar{A}^{(1)}+E+\bar{E}$. Here
$A^{(1)}$ and $\bar{A}^{(1)}$ are the strict transforms of $A'$ and $\bar{A}'$.
The morphism $\sigma_1'$ maps this curve onto a curve of type (I).\qed
\section{The non--nef case}
\label{drei}
Throughout this section we assume $\fdb$ to be not nef and $n=4$.\\
By Theorem \ref{nef}, Remark \ref{smnef} we know that in any smooth real
$S\in|\fund|$ the anticanonical system $|-K_S|$ contains only reducible
elements. By Proposition \ref{types} and Corollary \ref{omit} we can,
therefore, choose a real blow--down $S\rightarrow\bbfP^1\times\bbfP^1$ and a real reduced curve
$C\in|-K_S|$, whose image $C'$ in $\bbfP^1\times\bbfP^1$ is of type (I) or (II) as described
there. Observe that $C'$ has type (I) if and only if $C$ contains a real
irreducible component.
\begin{prop}\label{notnefi}
If there exists a real irreducible curve intersecting $|\fdb|$ negatively,
then $h^0(\fdb)=3$ and $a(Z) = 3$.\\
There exists a unique irreducible curve $C_0$ with $C_0.(\fund) < 0$. This
curve is real, smooth and rational and $C_0.(\fund)=-2$. The base locus of
$|\fund|$ is exactly $C_0$. $Z$ does not contain divisors of degree one.
\end{prop}
{\sc Proof:\quad}
Let $S\in|\fund|$ be smooth and real and choose $C\in|-K_S|$ and $\sigma:
S\rightarrow \bbfP^1\times\bbfP^1$ with the properties of Proposition \ref{types}. Because any
irreducible curve intersecting $\fdb$ negatively is contained in $C$, this
curve has a real component. Therefore, the image $C'$ of $C$ in $\bbfP^1\times\bbfP^1$ is of
type (II).
Let $C'=C_0'+F'$ be the decomposition of $C'$. By Lemma \ref{noreal} none of
the blown--up points lie on the real fibre $F'$. In particular, only smooth
points of $C_0'$ are blown--up. Hence, $C=C_0+F$ where $C_0$ and $F$ are the
strict transforms of $C_0'$ and $F'$ respectively. Therefore, the eight
blown--up points lie on $C_0'$ which implies $C_0^2 = {C'}_0^2-8=-4$.
By adjunction formula we obtain $-2=C_0.(-K_S)=C_0.(\fund)$. Hence,
$|-K_S|=C_0+|F|$ and we obtain: $\dim|-K_S|=1$ and $C_0$ is the base locus of
$|-K_S|$. Since $h^1({\mathcal O}_Z)=0$ the restriction map
$H^0(\fdb)\twoheadrightarrow H^0(\canS)$ is surjective. Hence, the linear
system $|\fund|$ has dimension two and its base locus is precisely $C_0$ (with
multiplicity one). $C_0$ is the unique irreducible curve in $Z$ having negative
intersection number with $\fund$, since any other such curve should be
contained in the base locus of $|\fund|$.
If $Z$ contains a divisor $D$ of degree one, then $D+\bar{D}\in|\fund|$. If
$D_0$ ($\bar{D}_0$ respectively) denotes the restriction of $D$ to $S$, then
$D_0+\bar{D}_0\in|-K_S|=C_0+F$. In the proof of Lemma \ref{noreal} we have
seen that the real elements of $|F|$ are irreducible. Therefore, any real
element of $|-K_S|$ consists of two distinct real irreducible curves with
multiplicity one. This shows that $|-K_S|$ cannot contain a member of the form
$D_0+\bar{D}_0$.
It remains to show that the {\bf algebraic dimension} of Z must be three in
this case.
Let $\sigma:\tilde{Z}\rightarrow Z$ be the blow--up of the smooth rational
curve $C_0$. By $E\subset \tilde{Z}$ we denote the exceptional divisor. Then we
obtain a morphism $\pi:\tilde{Z}\rightarrow{\Bbb P}^2$ defined by the linear
system $|\fund|$ such that $\pi^\ast{\mathcal O}(1)\cong
\sigma^\ast\fdb\otimes{\mathcal O}_{\tilde{Z}}(-E)$.
Since the restriction map $|\fund|\rightarrow|-K_S|$ is surjective, the
restriction $\pi_{|S}$ is given by the linear system $|-K_S|=C_0+|F|$. This
means that $\pi$ exibits $S$ as the blow--up of a ruled surface and $\pi(S)$ is
a line in ${\Bbb P}^2$. Since $\pi(\tilde{Z})$ is not contained in a linear
subspace, $\pi$ must be surjective. If we equip ${\Bbb P}^2$ with the usual real
structure, $\pi$ becomes compatible with real structures since the linear
system $|\fund|$ and the blown--up curve $C_0$ are real.
Since $Z$ does not contain divisors of degree one, any real fundamental divisor
$S$ is irreducible and, therefore, smooth. By $\tilde{S}\subset\tilde{Z}$ we
denote the strict transform of $S\in |\fund|$. Since $C_0$ is a smooth curve in
a smooth surface, $\sigma:\tilde{S}\rightarrow S$ is an isomorphism.
Furthermore, $E\cap\tilde{S}$ will be mapped isomorphically onto $C_0\subset
S$. Since $F.C_0=2$ and the restriction of $\pi$ onto $\tilde{S}$ is the map
defined by the linear system $|F|$, the restriction of $\pi$ exibits
$E\cap\tilde{S}$ as a double covering over $\pi(S)\cong{\Bbb P}^1$. Since real
lines cover ${\Bbb P}^2$ the morphism $\pi:E\rightarrow{\Bbb P}^2$ does not contract
curves and is of degree two.
Since generic fibres of $\pi$ are smooth rational curves, the line bundle
${\mathcal O}_{\tilde{Z}}(E)$ restricts to ${\mathcal O}_{{\Bbb P}^1}(2)$ on such fibres. Hence,
after replacing (if necessary) ${\Bbb P}^2$ by the open dense set $U$ of points
having smooth fibre, the adjunction morphism $\pi^\ast\pi_\ast
{\mathcal O}_{\tilde{Z}}(E) \rightarrow {\mathcal O}_{\tilde{Z}}(E)$ is surjective. This
defines a $U$--morphism $\Phi:\tilde{Z} \rightarrow
{\Bbb P}(\pi_\ast{\mathcal O}_{\tilde{Z}}(E))$, where $\pi_\ast{\mathcal O}_{\tilde{Z}}(E)$ is a
locally free sheaf of rank three. The restriction of $\Phi$ to smooth fibres
coincides with the Veronese embedding ${\Bbb P}^1\hookrightarrow{\Bbb P}^2$ of degree
two. Therefore, the image of $\Phi$ is a three--dimensional subvariety of the
${\Bbb P}^2$--bundle ${\Bbb P}(\pi_\ast{\mathcal O}_{\tilde{Z}}(E))\rightarrow U$. Hence,
$\tilde{Z}$ is bimeromorphically equivalent to a quasiprojective variety and
has, therefore, algebraic dimension three.\qed
For the rest of this section we assume that there does not exist a {\em real}
irreducible curve contained in the base locus of $|\fund|$. We keep the
assumptions $n=4$ and $\fdb$ is not nef. In this situation, we obtain:
\begin{lemma}\label{sub}
\begin{enumerate}
\item[(a)] If $A\subset Z$ is an irreducible curve, then $A.(\fund) \geq
-2$.
\item[(b)] If $A\subset Z$ is an irreducible curve with $A.(\fund) <0$, then
there
exists at least a one--parameter family of real smooth divisors
$S\in|\fund|$, containing a curve $C\in|-K_S|$ and possessing a
birational morphism $\sigma: S\longrightarrow \bbfP^1\times\bbfP^1$ as in Proposition
\ref{types}, such that moreover:\\
$A$ and $\overline{A}$ are components of $C$ and for twistor fibres
$F\subset S$ we have $F.A = F.\overline{A} = 1$.\\
In particular, the image $A'$ of $A$ in $\bbfP^1\times\bbfP^1$ is a ``line'', that means
$A'\in|{\mathcal O}(1,0)|$.
\end{enumerate}
\end{lemma}
{\sc Proof:\quad}
Our assumptions imply that $C'$ is a curve of type (I) in Proposition
\ref{types}. The components of $C'$ are curves in $\bbfP^1\times\bbfP^1$ with self--intersection
number zero. They are not real. Hence, after a succession of four blow--ups of
a
conjugate pair of points, each component $A$ of $C$ fulfills $A^2\geq-4$ in
$S$. The adjunction formula, together with the rationality of $A$, implies
$A.(\fund) = A.(-K_S) = A^2 + 2 \geq -2$. Because a curve $A$ with $A.(\fund) <
0$ must be a component of $C$, the assertion (a) is shown.
Let now $A\subset Z$ be an irreducible curve with $A.(\fund)<0$. Then we have
$A\subset C$. Let $x\in A\subset C$ be a smooth point of $C$ and $x\in F
\subset Z$ a twistor fibre. Since $|\fund|$ is at least a pencil, there exists
a divisor $S\in|\fund|$ containing a given point $y\in
F\setminus\{x,\overline{x}\}$. Because $F.S = 2$ and $S\cap F \supset \{y, x,
\overline{x}\}$ the twistor fibre $F$ is contained in $S$. So the real
subsystem $|\fund|_F\subset |\fund|$ of divisors containing $F$ is not
empty. Hence, we can choose a real smooth $S\in|\fund|$ containing $F$. By
construction, we have $F.A = F.\overline{A} \geq 1$. But $F.B\geq 0$ for any
curve $B\subset S$ together with $F.(-K_S)=2$ implies $F.A = F.\overline{A} =
1$. Because $S$ contains only a real one--parameter family of real twistor
lines, the intersection points with real twistor fibres form only a real
one--dimensional subset of points $z$ on $A$. Therefore, we obtain at least a
one--parameter family of such surfaces $S$. Proposition \ref{types} implies now
the claim.
\qed
\begin{prop}\label{notnefii}
Assume the existence of an irreducible (non--real) curve $A\subset Z$ with
$A.(\fund) = -2$. Then: $h^0(\fdb)=4$ and $a(Z) = 3$.\\
The curves $A$ and $\bar{A}$ are disjoint smooth and rational. $A$ and
$\bar{A}$ are the unique irreducible reduced curves having negative
intersection number with $\fund$. The base locus of $|\fund|$ is exactly the
union of $A$ and $\bar{A}$. $Z$ contains infinitely many divisors of degree
one and is one of the twistor spaces studied by LeBrun
\cite{LeB} and Kurke \cite{Ku}.
\end{prop}
{\sc Proof:\quad}
We choose $S\in|\fund|$ as in Lemma \ref{sub}(b). Then we have $C\in|-K_S|$
containing $A$ and $\overline{A}$ as smooth rational components which are
mapped to ``lines'' $A'$ and $\overline{A}'\in|{\mathcal O}(1,0)|$ in $\bbfP^1\times\bbfP^1$. We have by
the adjunction formula $A^2 = A.(-K_S) -2 = -4$, which implies that the eight
blown--up points lie on $A'+\overline{A}'$ (or the strict transforms after
partial blow--ups). Hence, any member of $A'+\overline{A}'+|{\mathcal O}(0,2)|$
contains the eight blown--up points and defines, therefore, a divisor in
$|-K_S|$. On the other hand, any curve in $|-K_S|$ is mapped onto a curve of
type $(2,2)$ on $\bbfP^1\times\bbfP^1$ containing the blown--up points. Such a curve must
contain $A'$ and $\overline{A}'$ since the intersection number with $A$ is two,
but four of the blown--up points lie on $A$. Hence, the image of $|-K_S|$ is
precisely the two--dimensional system $A'+\overline{A}'+|{\mathcal O}(0,2)|$. So we
have $\dim|-K_S|=2$. Using the exact sequence $0 \rightarrow {\mathcal O}_Z \rightarrow
\fdb \rightarrow \canS \rightarrow 0$, this implies $h^0(\fdb)=4$. Furthermore,
we see that $A+\bar{A}$ is in the base locus of $|-K_S|$ which coincides with
the base locus of $|\fund|$.
To see that $Z$ contains infinitely many divisors of degree one we modify an
idea of Pedersen, Poon \cite[p.\ 700]{PP}:
The above description of the image of $|-K_S|$ in $\bbfP^1\times\bbfP^1$ shows that there exist
infinitely many real curves $C\in|-K_S|$ whose image $C'$ in $\bbfP^1\times\bbfP^1$ is of type
(I) and the singular points of $C'$ are not blown--up. Then $C$ is the strict
transform of such a curve and $C'$ consists of four irreducible components.
Let $P$ and $\bar{P}$ denote a conjugate pair of singular points of $C$. Since
$A$ and $\bar{A}$ are disjoint, both points $P$ and $\bar{P}$ are contained in
$A+\bar{A}$, hence in the base locus of $|\fund|$. On the twistor space $Z$
there exists exactly one real twistor line $L_P$ connecting $P$ with $\bar{P}$.
Let $Q\in L_P$ be a point different from $P$ and $\bar{P}$. The linear system
$|\fund|_Q$ of all fundamental divisors containing $Q$ has at least dimension
$\dim|\fund|-1=\dim|-K_S|=2$. Since fundamental divisors have degree two and
any member of $|\fund|_Q$ contains $P,\,\bar{P}$ and $Q$ it also contains
$L_P$. Hence, $|\fund|_Q$ coincides with the {\sl real} linear subsystem of
divisors in $|\fund|$ containing $L_P$.
Choose now a point $R\in C$ which is not on $A+\bar{A}$. Then the real linear
system $|\fund|_{L_P,R,\bar{R}}$ is non--empty.
Let $C$ be decomposed as $A+\bar{A}+B+\bar{B}$. Then, by our choice of $C$,
$B.(\fund)=2$ and $A.(\fund)=-2$. These four curves intersect as indicated in
the following picture:
\begin{picture}(90,-90)(-30,0)
\put(0,-40){\line(1,0){60}} \put(30,-42){\makebox(0,0)[t]{$\bar{A}$}}
\put(0,0){\line(1,0){60}} \put(30,2){\makebox(0,0)[b]{$A$}}
\put(10,10){\line(0,-1){60}} \put(12,-20){\makebox(0,0)[l]{$B$}}
\put(50,10){\line(0,-1){60}} \put(52,-20){\makebox(0,0)[l]{$\bar{B}$}}
\end{picture}
\rule[-50pt]{0pt}{60pt}
\vspace*{12pt}
Any real member $S_0$ of $|\fund|_{L_P,R,\bar{R}}$ contains three distinct
points of $B$, namely $R$ and the intersection of $B$ with $A+\bar{A}$. Hence,
$B\subset S_0$. But this means that $S_0$ contains three curves, say $A, B$ and
$L_P$, which meet at $P\in S_0$. As in the proof of Proposition \ref{basel} we
obtain that $S_0$ is reducible, hence splits into the sum of two divisors of
degree one. Since the intersection of a conjugate pair of divisors of degree
one is a twistor line, we obtain in this way infinitely many divisors of degree
one on $Z$.
By the result of Kurke--Poon \cite{Ku}, \cite{Po2} we obtain that $Z$ is a
LeBrun twistor space and $A+\bar{A}$ is precisely the base locus of
$|\fund|$. Hence, no other curve can have negative intersection number with
$(\fund)$. This proves that really all properties of the Propositon are
fulfilled.\qed
\begin{prop}\label{notnefiii}
Assume $A.(\fund)\geq-1$ for all irreducible curves $A\subset Z$. Then:
$h^0(\fdb)=2$ and $a(Z) = 3$.\\
There exists a conjugate pair of irreducible curves $A$ and $\bar{A}$ which
are smooth and rational and $A.(\fund)=\bar{A}.(\fund)=-1$. The base locus
$C$ of $|\fund|$ consists of a cycle of an even number of rational curves.
The number of distinct divisors of degree one on $Z$ is equal to the number
of components of $C$.
\end{prop}
{\sc Proof:\quad}
By our assumption we obtain the existence of an irreducible curve $A\subset Z$
with $A.(\fund) = -1$. Now we choose $S\in|\fund|$ real and smooth and
$C\in|-K_S|$ as in Lemma \ref{sub}(b). The curve $C\in|-K_S|$ has $A$ and
$\overline{A}$ as components and the images $A'$ and $\overline{A}'$ of $A$ and
$\overline{A}$ in $\bbfP^1\times\bbfP^1$ are members of $|{\mathcal O}(1,0)|$. But $A^2 = A.(-K_S) - 2 =
-3$ implies that exactly one pair of blown--up points does not lie on $A' +
\overline{A}'$. This implies that the two components of $C'$, which are members
of $|{\mathcal O}(0,1)|$, are not movable. Hence, $|-K_S| = \{C\}$ and, as above,
$h^0(\fdb)=2$. Because $C'$ is of type (I), the curve $C$ consists of 2, 3, 4,
5 or 6 pairs of conjugate rational curves.
By Proposition \ref{basel} we have: the number of components of $C$ is equal
to the number of effective divisors of degree one.
It remains to be seen that the {\bf algebraic dimension} is three.
Since $|\fund|$ is a pencil we study in more detail the linear system $|-K|$ on
$Z$.
We need to investigate the structure of $C$ before we can collect more
information on the linear system $|-2K_S|$.
We know that the blow--up $\sigma:S\rightarrow S^{(0)} := \bbfP^1\times\bbfP^1$ factors through
a
succession of four blow--ups $S=S^{(4)} \rightarrow S^{(3)} \rightarrow S^{(2)}
\rightarrow S^{(1)} \rightarrow S^{(0)}$ such that at each step a conjugate
pair of points is blown--up. The image of $C$ in $S^{(i)}$ will be denoted by
$C^{(i)}$. The blown--up points in $S^{(i)}$ should lie on $C^{(i)}$. If they
are smooth points of $C^{(i)}$ then $C^{(i+1)} \stackrel{\sim}{\rightarrow}
C^{(i)}$. If we blow up a conjugate pair of singular points of $C^{(i)}$, the
curve $C^{(i+1)}$ has two components more than $C^{(i)}$. By assumption,
$C^{(0)} = C' \subset \bbfP^1\times\bbfP^1$ is of type (I). Each $C^{(i)}$ is a ``cycle of
rational curves''. We can choose the factorization of $\sigma$ in such a way
that at the first $k$ steps, only singular points of $C^{(i)}$ are blown--up
and at the last $4-k$ steps, only smooth points of $C^{(i)}$ are blown--up.
Then $C$ will have $2(2+k)$ components, where $0\leq k \leq 4$. If we would
have a component $A$ of $C$ with $A^2 =0$, then the image
$A^{(0)}$ of $A$ in $S^{(0)}$ would be a component of $C^{(0)}$ and none of the
blown--up points would lie on $A^{(0)}$. But then four of the blown--up points
must lie on a line or on a fibre in $S^{(0)}$, which implies that $C$ has a
component $B$ with $B^2 = -4$. This was excluded by assumption. Therefore, for
any component $A$ of $C$ we have $-1 \geq A^2 \geq -3$.
Since $A$ is a smooth rational curve, this means $A.(-K_S) \in \{-1, 0, +1\}$.
By assumption $C$ is reduced. Let $C = \sum\nolimits_{\nu = 1}^m C_\nu$ be the
decomposition of $C$ into irreducible components. By Proposition \ref{types} we
have $C_\nu \cong {\Bbb P}^1$, $C_\nu.C_{\nu+1} = 1$ and $C_\nu$ intersects only
$C_{\nu-1}$ and $C_{\nu+1}$. (This means ``$C$ is a cycle of rational curves''.
For convenience, we use cyclic subscripts, that is $C_\nu = C_{\nu+m}$.) For
$\varepsilon\in\{-1, 0, +1\}$ we define $I_\varepsilon := \{\nu | C_\nu.(-K_S)
= \varepsilon\}$ and $C_\varepsilon := \sum\nolimits_{\nu\in I_\varepsilon}
C_\nu$. In this way we split $C$ into three parts $C = C_- + C_0 + C_+$. As
$C.(-K_S) = 0$ we have $|I_-| = |I_+|$. The assumption that $\fdb$ is not nef
implies $I_- \ne \emptyset$. The curve $C$ has no real component, hence $|I_-|
= |I_+|\geq 2$.
{\bf Claim:} Any two components of $C_+$ are disjoint.\\
Let $C_\alpha$ and $C_\beta$ be two
distinct components of $C_+$. Then $C_\alpha^2 = C_\beta^2 = -1$. If $C_\alpha$
and $C_\beta$ are both
contracted to a point on $S^{(0)}$, they are obviously disjoint.
If $C_\alpha$ and $C_\beta$ are mapped to curves in $S^{(0)}$, by our choice of
$S$ both must be members of $|{\mathcal O}(0,1)|$, because $A$ and $\overline{A}$ are
components of $C_-$. Finally, we have to exclude the case where $C_\alpha$ is
mapped to a curve $C'_\alpha\in|{\mathcal O}(0,1)|$ in $\bbfP^1\times\bbfP^1$ and $C_\beta$ is
contracted to a point $P\in C'_\alpha$. This implies $C_\alpha \ne
\overline{C}_\beta$. Since $C_\beta$ is a component of the anticanonical
divisor
$C\subset S$, the point $P$ must be singular on $C'$. Thus, we can take
$S^{(1)} \longrightarrow S^{(0)} = \bbfP^1\times\bbfP^1$ to be the blow--up of $P$ and
$\overline{P}$. The curve $C^{(1)}$ consists then of six $(-1)$--curves among
which we find the images of $C_\alpha, C_\beta, \overline{C}_\alpha$ and
$\overline{C}_\beta$. Because those are $(-1)$--curves on $S$, the remaining
six blown--up points must lie on one pair of $(-1)$--curves giving
rise to $(-4)$--curves in $C$ contradicting our assumption.
Thus, the claim is proved and $C_+$ is the disjoint union of an even number of
smooth rational $(-1)$--curves.
Since $C$ is a cycle of rational curves, the curve $C\setminus C_+ = C_- + C_0$
has the same number of connected components as $C_+$.
We claim that each connected component of $C_- + C_0$ contains exactly one
component of $C_-$. Since $C_-$ and $C_+$ have the same number of components,
this is equivalent to the statement that each connected component of $C_- +
C_0$ contains at most one component of $C_-$.\\ Assume the contary, that is
there is a connected component of $C_- + C_0$ containing two irreducible
components of $C_-$. Then these two components of $C_-$ are not conjugate to
each other. (Two conjugate components of $C$ are ``opposite'' in the cycle
$C$.) Therefore, $C_-$ has at least four components and so $C_+$. Thus $C_- +
C_0$ has at least four connected components, hence at least six irreducible
ones. Therefore, $C$ contains at least ten irreducible components, that is
$k\geq 3$. Therefore, the image $C^{(3)}$ of $C$ in $S^{(3)}$ consists of a
cycle of ten rational curves with self--intersection numbers $-2, -1, -3, -1,
-2, -2, -1, -3, -1, -2$ (in this order). By assumption, at the last step of
blow--up, no point on a $(-3)$--curve is blown--up. To obtain a second pair of
$(-3)$--curves, we have to blow up points on a conjugate pair of
$(-2)$--curves.
Such curves are not neighbours in the cycle $C^{(3)}$, they are opposite to
each other. Thus, one easily sees that after the last step of blow--up, between
two $(-3)$--curves on the cycle $C$ we always have a $(-1)$--curve. This means
that no connected component of $C\setminus C_+$ contains two irreducible
components of $C_-$, as claimed.
We can now compute the dimension of $|-K|$ and $|-2K_S|$.\\ The Riemann--Roch
formula and $h^0(\fdb) = 2$ imply $h^1(\fdb) = 0$. Hence, the exact sequence $0
\rightarrow \fdb \rightarrow K^{-1} \rightarrow \can{2} \rightarrow 0$ gives
$h^0(K^{-1}) = h^0(\fdb) + h^0(\can{2}) = 2 + h^0(\can{2})$ and a surjective
restriction map $|-K| \twoheadrightarrow |-2K_S|$. With $N:=\canS \otimes
{\mathcal O}_C$ we obtain an exact sequence $0 \rightarrow \can{1} \rightarrow \can{2}
\rightarrow N^{\otimes 2} \rightarrow 0$. Since $h^1(\canS) = h^1(\fdb) = 0$,
this sequence yields $h^0(\can{2}) = h^0(\canS) + h^0(N^{\otimes 2}) = 1 +
h^0(N^{\otimes 2})$. We can apply Lemma \ref{normal} with $L=\can{2}$, because
we have seen that the components of $C_+$ are disjoint to each other and that
each connected component of $C\setminus C_+$ contains exactly one irreducible
component of $C_-$. We obtain $h^0(N^{\otimes 2}) = h^0(C,L) = \sum_{\nu\in
I_+} C_\nu.(-2K_S) - |I_-| = 2|I_+| - |I_-| = |I_+|$. Hence, $\dim |-2K_S| =
|I_+| \geq 2$ and $\dim |-K| = 2 + \dim |-2K_S| \geq 4$.
Next we study the base locus of $|-2K_S|$.\\ Since any component $A$ of $C_-$
fulfills $A.(-K_S) = -1$, $C_-$ is in the base locus of $|-2K_S|$. Now let $A$
be an irreducible component of $C_0$. Since, as we have shown above, any
connected component of $C_0 + C_-$ contains a curve $B\subset C_-$ there exists
a finite chain of components $A_1,\dots,A_r = A$ of $C_0$ with $B\cap A_1 \ne
\emptyset$ and $A_i \cap A_{i+1} \ne \emptyset\quad (1\leq i < r)$. But for a
component $A_i$ of $C_0$ we have $A_i.(-2K_S) = 0$ which implies: if $A_i$
intersects the base locus of $|-2K_S|$, it must be contained in this base
locus. Hence, by induction on $i$, we obtain that $A_i$ is contained in the
base locus of $|-2K_S|$ for all $1\leq i \leq r$. This shows that $C_0 + C_-$
is contained in the base locus of the linear system $|-2K_S|$. Therefore, we
have $|-2K_S| = C_0 + C_- + |-K_S + C_+|$ and the map defined by $|-2K_S|$
coincides with the map given by $|-K_S + C_+|$.
We have seen above that $C_+$ is a disjoint union of smooth rational
$(-1)$--curves. We can, therefore, contract $C_+$ to obtain a smooth surface
$S'$. If $\sigma':S\rightarrow S'$ is this contraction, we have
$\canS\otimes{\mathcal O}_S(C_+) \cong \sigma^{'\ast}(K^{-1}_{S'})$. Hence, $-K_{S'}$
is nef if and only if $-K_S + C_+$ is nef.
First we deal with the case where $C_+ - K_S$ is nef. In this case, the generic
member of the moving part of $|-2K_S|$ is irreducible. This can be seen as
follows:
Observe first that $S'$ can be blown down to ${\Bbb P}^2$.
This follows from \cite[Prop. 3, p. 48]{Dem} because $-K_{S'}$ is nef and
$K^2_{S'} = |I_+| > 0$.
Because $C_+$ has 2, 4 or 6 components, $S'\rightarrow{\Bbb P}^2$ is a blow--up of
7, 5 or 3 points. Therefore, we can apply a theorem of Demazure \cite[p.\ 39
and p.\ 55]{Dem} stating that $|-K_{S'}|$ contains a smooth irreducible member
and is base point free if $|-K_{S'}|$ is nef. Hence, there exists a smooth
irreducible curve in $|-K_{S'}|$ avoiding the blown--up points. Its preimage in
$S$ is a smooth irreducible member of $|\sigma^\ast K^{-1}_{S'}| = |C_+ - K_S|
= |-2K_S - C_0 -C_-|$ which is, therefore, the moving part of $|-2K_S|$.
Since $\dim |-2K_S| \geq 2$ and the generic member of the moving part of
$|-2K_S|$ is a smooth irreducible curve, the image of the map defined by
$|-2K_S|$ has dimension two. We have seen before that the restriction
$|-K|\twoheadrightarrow |-2K_S|$ is surjective which implies that the map
$\Phi_{|-K|}$ given by $|-K|$ on $Z$ coincides, after restriction to $S$, with
the map given by $|-2K_S|$. Since $2S\in|-K|$, the $\Phi_{|-K|}$--image of $S$
is contained in a hyperplane. But the $\Phi_{|-K|}$--image of $Z$ cannot be
contained in a hyperplane. Therefore, the image of $\Phi_{|-K|}$ has dimension
three. This implies $a(Z)=3$.
We are left with the case where $C_+ - K_S$ is not nef.\\
In this case we shall see that $\Phi_{|-K|}$ has only two--dimensional image
but equips $Z$ with a conic--bundle structure. Under the assumption that $C_+ -
K_S$ is not nef we study the structure of $C$.
Let $A$ be an irreducible curve in $S$ with $A.(C_+ - K_S) < 0$. If
$A\nsubseteq C$, then $A.C_+\geq 0$. But the base locus of $|-K_S|$ is
contained in $C$ and this implies $A.(-K_S)\geq 0$. Hence, we have necessarily
$A\subseteq C$. If $A\subseteq C_+$, then $A.(-K_S) = 1$ and $A.C_+ \geq
A^2$, hence, $A.(C_+ - K_S) \geq A^2 + 1 = 0$. If
$A\subseteq C_0$, then $A.(-K_S) = 0$ and $A.C_+\geq 0$, hence, $A.(C_+ - K_S)
\geq 0$. If, finally, $A\subseteq C_-$, then $A.(C_+ - K_S) = A.C_+ - 1 \geq
-1$. So, we obtain: the irreducible curves $A\subset S$ with $A.(C_+ - K_S) <
0$ are exactly those components of $C_-$ which are disjoint to $C_+$. They
fulfill $A.(C_+ - K_S) = -1$.
As we have seen above, each connected component of $C\setminus C_+$ contains
exactly one irreducible component of $C_-$. Hence, if $A$ is a component of
$C_-$ which does not meet $C_+$, then its connected component should contain
at least two curves from $C_0$. Thus, using reality, we see that $C$ has at
least eight components. This means, using the convention introduced above, the
image $C^{(2)}$ of $C$ in $S^{(2)}$ (the surface
obtained after two steps of blow--up) consists of eight curves whose
self--intersection numbers are alternately $-1$ and $-2$:\vspace{2mm}
\centerline{
\begin{picture}(90,90)(0,-90)
\put(2,-32){\line(1,1){30}} \put(15,-17){\makebox(0,0)[br]{$\scriptstyle -2$}}
\put(4,-70){\line(0,1){44}}
\put(2,-48){\makebox(0,0)[r]{$\scriptstyle -1$}}
\put(2,-64){\line(1,-1){30}} \put(15,-79){\makebox(0,0)[tr]{$\scriptstyle -2$}}
\put(27,-92){\line(1,0){43}}
\put(49,-94){\makebox(0,0)[t]{$\scriptstyle -1$}}
\put(64,-94){\line(1,1){30}} \put(80,-79){\makebox(0,0)[tl]{$\scriptstyle -2$}}
\put(92,-70){\line(0,1){44}}
\put(94,-48){\makebox(0,0)[l]{$\scriptstyle -1$}}
\put(94,-32){\line(-1,1){30}}\put(80,-17){\makebox(0,0)[bl]{$\scriptstyle -2$}}
\put(27,-4){\line(1,0){43}}
\put(49,-2){\makebox(0,0)[b]{$\scriptstyle -1$}}
\end{picture}
}
\vspace{1pt}
One now easily sees: if we were to blow up a pair of singular points on
$C^{(2)}$, in the resulting curve $C^{(3)}$ any $(-2)$--curve would meet a
$(-1)$--curve and any $(-3)$--curve would meet two $(-1)$--curves. Therfore,
after the last step of blow--up, no $(-3)$--curve is disjoint to all
$(-1)$--curves on $C$. Thus, we can only blow up smooth points in the last two
steps. If we were to blow up a conjugate pair of points on $(-2)$--curves, the
resulting $(-3)$--curve would intersect two $(-1)$--curves. Then, again, in $C$
there would be no $(-3)$--curve disjoint to all $(-1)$--curves. So we conclude
that the last two pairs
of conjugate blown--up points cannot lie on $(-2)$--components of $C^{(2)}$. If
each of the four $(-1)$--curves in $C^{(2)}$ contains one of the blown--up
points, then any component of $C$ has zero intersection number with $-K_S$.
But, then by Theorem \ref{nef}, $\fdb$ would be nef which contradicts our
general assumption. Thus, the four blown--up points lie on a pair of conjugate
$(-1)$--curves. The structure of $C$ is, therefore, the following:
\centerline{
\begin{picture}(90,90)(0,-90)
\put(2,-32){\line(1,1){30}} \put(15,-17){\makebox(0,0)[br]{$\scriptstyle -2$}}
\put(4,-70){\line(0,1){44}}
\put(2,-48){\makebox(0,0)[r]{$\scriptstyle -3$}}
\put(2,-64){\line(1,-1){30}} \put(15,-79){\makebox(0,0)[tr]{$\scriptstyle -2$}}
\put(27,-92){\line(1,0){43}}
\put(49,-94){\makebox(0,0)[t]{$\scriptstyle -1$}}
\put(49,-89){\makebox(0,0)[b]{$\scriptstyle C_+$}}
\put(64,-94){\line(1,1){30}} \put(80,-79){\makebox(0,0)[tl]{$\scriptstyle -2$}}
\put(92,-70){\line(0,1){44}}
\put(94,-48){\makebox(0,0)[l]{$\scriptstyle -3$}}
\put(94,-32){\line(-1,1){30}}\put(80,-17){\makebox(0,0)[bl]{$\scriptstyle -2$}}
\put(27,-4){\line(1,0){43}}
\put(49,-2){\makebox(0,0)[b]{$\scriptstyle -1$}}
\put(49,-6){\makebox(0,0)[t]{$\scriptstyle C_+$}}
\end{picture}
}
\vspace{11pt}
In particular, we obtain: $\dim |-2K_S| = |I_+| = 2$ and $\dim |-K| = 2 +
\dim|-2K_S| = 4$.
Furthermore, since both components of $C_-$ have negative intersection number
with $C_+ - K_S$, the curve $C_-$ is contained in the base locus of $|C_+ -
K_S|$. This means $|-2K_S| = |2C_+ + C_0| + C_0 + 2C_-$.
By our choice of $S$, the two components $A$ and $\overline{A}$ of $C_-$ are
mapped onto lines $A'$ and $\overline{A}'$ on $\bbfP^1\times\bbfP^1$. The above analysis of $C$
shows that we can decompose $\sigma : S \longrightarrow \bbfP^1\times\bbfP^1$ into the
following steps:
First we blow up a conjugate pair of singular points on the
curve $C'$ (which is of type (I)). This produces precisely two singular fibres
of the ruling (whose general fibre is the image of a twistor fibre). In the
second step we blow up the two singular points of these singular fibres. The
exceptional curves of this blow--up form the components of $C_+$. Because we
blow up points of multiplicity two on the fibres, the total transform of the
two singular fibres contains $2C_+$. In the remaining two steps we have to
blow up smooth points on $A' + \overline{A}'$. Hence, we obtain $C_0 + 2C_+ \in
|2F|$. So, we can write $|-2K_S| = C_0 + 2C_- + |2F|$. Since we have
$\dim|\fund| = 1$, this is true for the generic real surface $S\in|\fund|$ by
Lemma \ref{sub} (b).
Let us denote by $\Phi = \Phi_{|-K|}$ the meromorphic map
$Z\dashrightarrow{\Bbb P}^4$ defined by $|-K|$. If $\varphi:S\rightarrow{\Bbb P}^2$
is the restriction of $\Phi$ to a generic smooth real $S\in|\fund|$, then the
image of $\varphi$ is a conic in ${\Bbb P}^2$. The general fibre of $\varphi$ is a
twistor fibre, hence a smooth rational curve intersecting $C$ transversally at
two points lying on $A$ and $\overline{A}$ respectively.
Let $\tilde{Z}\rightarrow Z$ be a modification such that $\Phi$ becomes a
morphism $\tilde{\Phi}:\tilde{Z}\rightarrow{\Bbb P}^4$. Because the smooth real
fundamental divisors $S$ sweep out a Zariski dense subset of $Z$, the image of
this set is also Zariski dense in $\tilde{\Phi}(\tilde{Z})\subset{\Bbb P}^4$. As
the general fibre of $\Phi$, restricted to such surfaces $S$, is
one--dimensional, we obtain $\dim \tilde{\Phi}(\tilde{Z}) = 2$. Since
$\tilde{Z}\rightarrow Z$ is a modification, there exists an
open Zariski dense subset $U\subset\tilde{\Phi}(\tilde{Z})$ such that the
fibres of $\tilde{\Phi}$ are irreducible curves. Moreover, we can choose $U$
such that the fibres of $\tilde{\Phi}$ over $U$ are isomorphic to
${\Bbb P}^1$,because this is true over a Zariski dense subset of
$\tilde{\Phi}(\tilde{Z})$. Let $\tilde{\Phi}:\tilde{Z}_U\rightarrow U$ denote
the restriction of $\Phi$ over $U$. Then the preimage in $\tilde{Z}$ of the two
components of $C_-$ defines a pair of divisors $\Sigma$ and $\bar{\Sigma}$ in
$\tilde{Z}_U$ which are sections of $\tilde{Z}_U\rightarrow U$. Therefore,
${\mathcal E}:=\tilde{\Phi}_\ast{\mathcal O}_{\tilde{Z}_U}(\Sigma + \bar{\Sigma})$ is a
vector bundle of rank three on $U$ and the canonical morphism
$\tilde{\Phi}^\ast{\mathcal E}\rightarrow {\mathcal O}(\Sigma + \bar{\Sigma})$ is
surjective. This means that we obtain a morphism
$\tilde{Z}_U\rightarrow{\Bbb P}({\mathcal E})$ which is compatible with the
projections to $U$. Restricted to each fibre this morphism is the Veronese
embedding of degree two ${\Bbb P}^1\hookrightarrow{\Bbb P}^2$. Hence, the image of
$\tilde{Z}_U$ in the quasi--projective variety ${\Bbb P}({\mathcal E})$ is
three--dimensional. This implies $a(Z) = a(\tilde{Z}_U) = 3$, which completes
the proof.\qed
\section{Conclusions}
\label{vier}
In this section we collect the results of this paper to obtain a clear
picture of the situation considered.
By $Z$ we always denote a simply connected compact twistor space of positive
type over $4\bbfC\bbfP^2$.
We call ${\mathcal N}:= \{C\subset Z\mid C$ irreducible curve, $C.(\fund)<0\}$
the set of negative curves. By definition, $\fdb$ is nef if and only if
${\mathcal N}\ne \emptyset$. The structure of ${\mathcal N}$ is described by
the following
\begin{thm}\label{ncurves}
If ${\mathcal N}\ne \emptyset$ this set consists of a finite number of smooth
rational curves. More precisely, only the following cases are possible:
\begin{itemize}
\item[(a)] ${\mathcal N}$ contains a real member $C_0$. Then: ${\mathcal N} =
\{C_0\}$ and $C_0(\fund) = -2$, $\dim|\fund| = 2$ and $a(Z) = 3$.
\item[(b)] ${\mathcal N}$ contains a non--real member $A$ with $A.(\fund) =
-2$. Then, ${\mathcal N} = \{ A, \overline{A}\}$, $\dim|\fund| = 3, a(Z) = 3$
and $Z$ is a LeBrun twistor space.
\item[(c)] Each member $A\in {\mathcal N}$ fulfills $A.(\fund) = -1$. Then
$|{\mathcal N}| \in \{2, 4, 6\}$, $\dim|\fund| = 1$ and $a(Z) =3$.
\end{itemize}
\end{thm}
{\sc Proof:\quad} We have only to collect the results of Section \ref{drei}.
\qed
We can compute the algebraic dimension in the following way:
\begin{thm}\label{main}
$a(Z) = 3 \iff \fdb$ is not nef;\\
$a(Z) = 2 \iff \fdb$ is nef and $\exists m\geq 1: h^1(\fb{m}) \ne 0$;\\
$a(Z) = 1 \iff \forall m\geq 1: h^1(\fb{m}) = 0$.
\end{thm}
{\sc Proof:\quad}
This results from Proposition \ref{tau} and Theorems \ref{nef} and
\ref{ncurves}.\qed
We can characterize Moishezon twistor spaces as follows:
\begin{thm}
The following conditions are equivalent:
\begin{enumerate}
\item $a(Z) = 3$;
\item $\fdb$ is not nef;
\item there exists a smooth rational curve $C\subset Z$ with $C.(\fund) < 0$.
\end{enumerate}
\end{thm}
{\sc Proof:\quad}
Apply Theorems \ref{ncurves} and \ref{main}.\qed
\begin{rem}
Remembering that, by Poon's theorem, $\fdb$ is big if and only if $Z$ is
Moishezon, we obtain from the preceding theorem: the line bundle $\fdb$ is
never nef and big (under our special assumptions).
\end{rem}
LeBrun twistor spaces are characterized (see \cite{Ku}, \cite{Po2}) by
the property to contain a pencil of divisors of degree one. We can give (for
the case $n=4$) two further characterizations:
\begin{thm}\label{cb}
The following properties are equivalent:
\begin{enumerate}
\item $Z$ contains a pencil of divisors of degree one;
\item $\dim|\fund| = 3$;
\item there exists a smooth rational curve $A\subset Z$ with $A \ne \bar{A}$
and $A.(\fund) = -2$.
\end{enumerate}
\end{thm}
{\sc Proof:\quad}
The implications (i)$\Rightarrow$(ii) and (i)$\Rightarrow$(iii) follow from the
Kurke--Poon theorem. The reverse implications follow from Theorem
\ref{ncurves}.\qed
\begin{thm}
$a(Z) \geq \dim|\fund|$.
\end{thm}
{\sc Proof:\quad}
This follows directly from Proposition \ref{tau} and Theorems \ref{nef} and
\ref{ncurves}. \qed
If $|\fund|$ is not a pencil, we obtain the following nice result:
\begin{thm}
If $\dim|\fund|\geq 2$, then:\\
$a(Z) = 2 \iff \fdb$ is nef $\iff |\fund|$ does not have base points.
\end{thm}
{\sc Proof:\quad}
The first equivalence results from the previous theorem and Theorem \ref{main}.
If $|\fund|$ does not have base points, $\fdb$ is necessarily nef. If $\fdb$ is
nef and $\dim|\fund|\geq 2$ we have seen in Theorem \ref{nef} that $|\fund|$ is
base point free.\qed
\begin{cor}
$|\fund|$ is base point free $\Rightarrow a(Z) = 2$.
\end{cor}
{\sc Proof:\quad}
This is immediate from the previous theorem, because a pencil $|\fund|$ has
always base points. \qed
\begin{rem}
The reverse implication is not true, which follows from the existence theorem
in \cite{CK}. There, twistor spaces with $a(Z) = 2$ and
$\dim|\fund| = 1$ over $4\bbfC\bbfP^2$ were constructed.
\end{rem}
|
1997-01-17T12:54:38 | 9607 | alg-geom/9607015 | en | https://arxiv.org/abs/alg-geom/9607015 | [
"alg-geom",
"dg-ga",
"hep-th",
"math.AG",
"math.DG"
] | alg-geom/9607015 | Teleman | Christian Okonek, Alexander Schmitt and Andrei Teleman | Master Spaces for stable pairs | 26 pages. New introduction and applications LaTeX2e | null | null | null | null | We construct master spaces for oriented torsion free sheaves coupled with
morphisms into a fixed reference sheaf. These spaces are projective varieties
endowed with a natural $\C^*$-action. The fixed point set of this action
contains the moduli space of semistable oriented torsion free sheaves and the
quot scheme associated with the given data.
In the case of curves with trivial reference sheaf, our master spaces
compactify the moduli spaces constructed by Bertram, Daskalopoulos and
Wentworth. In the 2-dimensional case with trivial rank 1 reference sheaf,
master spaces provide algebraic analoga of compactified moduli spaces of
twisted quaternionic monopoles.
| [
{
"version": "v1",
"created": "Wed, 17 Jul 1996 09:10:39 GMT"
},
{
"version": "v2",
"created": "Fri, 17 Jan 1997 11:20:20 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Okonek",
"Christian",
""
],
[
"Schmitt",
"Alexander",
""
],
[
"Teleman",
"Andrei",
""
]
] | alg-geom | \section*{Introduction}
In this paper we construct master spaces for certain coupled vector
bundle problems over a fixed projective variety $X$.
From a technical point of view, master spaces classify oriented pairs $({\cal
E},\varepsilon,\varphi)$ consisting of a torsion free coherent sheaf
${\cal E}$ with
fixed Hilbert polynomial, an orientation $\varepsilon$ of the determinant
of ${\cal E}$,
and a framing $\varphi:{\cal E}\longrightarrow {\cal E}_0$ with values in a fixed
reference sheaf
${\cal E}_0$, satisfying certain semistability conditions. The relevant
stability concept
is new and does not involve the choice of a parameter, but it can easily be
compared to
the older parameter-dependent stability concepts for (unoriented) pairs.
The corresponding moduli spaces ${\cal M}$ have the structure of polarized
projective
varieties endowed with a natural ${\Bbb C}^*$-action which can be exploited in
two interesting
ways:
1. The fixed point set ${\cal M}^{{\Bbb C}^*}$ of the ${\Bbb C}^*$-action is a union
$${\cal
M}^{{\Bbb C}^*}={\cal M}_{source}\cup{\cal M}_{sink}\cup {\cal M}_R\ ,$$
where ${\cal
M}_{source}$ is a Gieseker moduli space of semistable oriented sheaves,
${\cal M}_{sink}$
is a certain (possible empty) Grothendieck Quot-scheme, and the third term
${\cal
M}_R:={\cal M}^{{\Bbb C}^*}\setminus({\cal M}_{source}\cup{\cal M}_{sink})$ is
the so-called
"variety of reductions", which consists essentially of lower rank objects. The structure as
a ${\Bbb C}^*$-space can be used to relate "correlation functions" associated
with the different
parts of ${\cal M}^{{\Bbb C}^*}$ to each other [OT2].
2. Master spaces are also useful for the investigation of the birational
geometry of
the moduli spaces ${\cal M}_\delta$ of $\delta$-semistable pairs in the
sense of [HL2].
Indeed, each of the ${\cal M}_\delta$'s can be obtained as a suitable
${\Bbb C}^*$-quotient of
the master space ${\cal M}$, and it can be shown that every two quotients
${\cal M}_\delta$,
${\cal M}_{\delta'}$ are related by a chain of generalized flips in the
sense of [Th].
When $X$ is a projective curve with trivial reference sheaf ${\cal E}_0={\cal
O}_X^{\oplus k}$, our master space can be considered as a natural
compactification of the
one described in [BDW]. Their space becomes an open subset of ours whose
complement is
the Quot-scheme ${\cal M}_{sink}$ alluded to above (${\cal M}_{sink}$ is
empty iff
$k<\mathop{\rm rk}({\cal E})$). Applying the ideas of 1.\ in this situation leads to
formulas for
volumina and characteristic numbers and to a new proof of the Verlinde
formula when $k=1$,
and allows to relate Gromov-Witten invariants for Grassmannians to simpler
vector bundle
data when
$k>\mathop{\rm rk}({\cal E})$.
In the case of an algebraic surface $X$, master spaces can be viewed as
algebraic analoga
of certain gauge theoretic moduli spaces of monopoles which can be used to
relate
Seiberg-Witten invariants and Donaldson polynomials [OT1], [T1]. The latter
application
was actually our original motivation for the construction of master spaces.
The study of non-abelian monopoles on K\"ahler surfaces leads naturally to the
investigation of a certain moment map on an infinite dimensional K\"ahler space.
The associated stability concept, which is expected to exist on general
grounds [MFK], is
precisely the one which gave rise to the stability definition for oriented
pairs [OT2].
Since the moduli space of non-abelian monopoles admit an Uhlenbeck type
compactification
[T1], it was natural to look for a corresponding Gieseker type
compactification of their
algebro-geometric analoga. These compactifications, the master spaces for
stable pairs,
provide very useful models for understanding the ends of monopole moduli
spaces in the more difficult gauge theoretical context [T2]. Understanding
these ends is the
essential final step in our program for relating Donaldson polynomials and
Seiberg-Witten
invariants [OT1], [T1]. Let us now briefly describe the main ideas and
results of this
paper.
The construction
of master spaces requires the study of GIT-quotients for direct sums of
representations, i.e.\ the construction of quotients $\P(A\oplus
B)^{ss}/\hskip-3pt/ G$, where $G$ is a reductive group acting linearly on vector
spaces $A$ and $B$. Since the Hilbert criterion is difficult to apply in
this situation, we have chosen another approach instead. The idea is to use
the natural ${\Bbb C}^*$-action $z\cdot\langle a,b\rangle:=\langle a,zb\rangle$ on
$\P(A\oplus B)$ which commutes with the given action of $G$. Our first main
result characterizes $G$-semistable points in $\P(A\oplus B)$ in terms of
$G$-semistability of their images in all possible
${\Bbb C}^*$-quotients of $\P(A\oplus B)$. The proof is based on a commuting
principle for actions of products of groups.
These results, which we prove in the first section, explain in particular
why chains of flips occur in GIT-problems for $G\times{\Bbb C}^*$-actions [DH], [Th].
In the second section of our paper, after defining stability for oriented pairs
$({\cal E},\varepsilon,\varphi)$, we prove a crucial boundedness result and
construct the corresponding parameter space ${\frak B}$. This space admits
a morphism
$\iota:{\frak B}\longrightarrow \P({\frak Z})$ into a certain Gieseker space
$\P({\frak Z})$ which is equivariant w.r.t.\ a natural action of a product
$\mathop{\rm SL}\times{\Bbb C}^*$ of ${\Bbb C}^*$ with a special linear group. The $\mathop{\rm SL}$-action on
$\P({\frak Z})$ possesses a linearization in a suitable line bundle, and the
preimage of the subset $\P({\frak Z})^{ss}$ of $\mathop{\rm SL}$-semistable points is
precisely the open subspace ${\frak B}^{ss}\subset {\frak B}$ of points
representing semistable oriented pairs. In order to prove this, we apply
our GIT-Theorem from the first section to the $\mathop{\rm SL}\times{\Bbb C}^*$-action on
$\P({\frak Z})$, and thereby reduce the proof to results in [G] and [HL1].
Then we show that the induced map $\iota|_{{\frak B}^{ss}}:{\frak B}^{ss}\longrightarrow
\P({\frak Z})^{ss}$ is finite and hence descends to a finite map
$\bar\iota:{\frak B}^{ss}/\hskip-3pt/ \mathop{\rm SL}\longrightarrow \P({\frak Z})^{ss}/\hskip-3pt/ \mathop{\rm SL}$. The
quotient ${\frak B}^{ss}/\hskip-3pt/ \mathop{\rm SL}$, which is therefore a projective
variety, is our master space.
The ideas and techniques of this paper can also be applied
to construct master spaces in other interesting situations, e.g.\ by
coupling with
sections in twisted endomorphism bundles. When $X$ is a curve and the twisting
line bundle
is the canonical bundle, one obtains a natural compactification of the moduli
spaces of
Higgs bundles [H], [S].
Similar ideas should also apply to coupling with singular
objects like parabolic structures. We refer to [OT2] for a general
description of
the underlying coupling principle and its application to computations of
correlation
functions.
\subsection*{Conventions}
Our ground field is ${\Bbb C}$. A \it polarization \rm on a quasi-projective
variety $X$ is
an equivalence class $[L]$ of ample line bundles,
where two line bundles $L_1$ and $L_2$ are \it equivalent\rm , if there
exist positive integers $n_1$ and $n_2$ such that
$L_1^{\otimes n_1}\cong {L_2}^{\otimes n_2}$.
\par
If $W$ is a finite dimensional vector space, we denote by $\P(W)$ its
projectivization in the sense of Grothendieck, i.e., the closed
points of $\P(W)$
correspond to
lines in the dual space $W^\vee$.
We do not distinguish notationally between a vector space $W$ and its
associated scheme.
\section{A theorem from Geometric Invariant Theory}
\subsection{Background material from GIT}
\label{BackGIT}
Let $G$ be a reductive algebraic group and let
$\gamma\colon G\longrightarrow \mathop{\rm GL}(W)$
be a rational representation in the finite dimensional vector space $W$.
The map $\gamma$ defines an action of $G$ on the dual space $W^\vee$
given by
$$g\cdot w:= w\circ \gamma(g^{-1})\qquad \forall g\in G; w\in W^\vee,$$
an action $\overline{\gamma}$ on the projective space $\P(W)$,
and a linearization of this action in $\O_{\P(W)}(1)$.
In the following we identify $H^0(\P(W),\O_{\P(W)}(k))$ with
$S^kW$.\par
Recall that a point $x\in \P(W)$ is \it $\gamma$-semistable \rm
if and only if the orbit closure $\overline{G\cdot w}$
of any lift $w\in W^\vee\backslash\{0\}$ does not contain $0$.
Denote by $\P(W)_{\gamma}^{ss}\subset\P(W)$ the open set of semistable
points and by $\P(W)_\gamma^{ps}$ the set of \it $\gamma$-polystable \rm
points, i.e.\ the semistable points whose orbit is closed in
$\P(W)_{\gamma}^{ss}$.
Equivalently, a point $x\in \P(W)$ is polystable if and only if the
orbit $G\cdot w$ of any lift $w\in W^\vee\backslash\{0\}$
is closed in $W^\vee$.
With this terminology, $x\in \P(W)$ is \it $\gamma$-stable \rm if and only if it
is polystable and its stabilizer $G_x$ is finite.
Let $\pi_\gamma\colon \P(W)_\gamma^{ss}\longrightarrow Q_\gamma:= \P(W)/\hskip-3pt/_\gamma G$
be the categorical quotient.
For sufficiently large $n$, $Q_\gamma$ admits a projective
embedding
$j_n\colon Q_\gamma\hookrightarrow \P({S^nW}^G)$
such that the following diagram commutes:
\begin{equation}
\label{eqemb}
\begin{array}{c}
\unitlength=1mm
\begin{picture}(70,24)(0,9)
\put(0,29){$\P(W)^{ss}_\gamma\subset \P(W)$}
\put(34,30){\vector(1,0){20}}
\put(34,31){\oval(3,1.8)[l]}
\put(40,32){${\scriptstyle v_n}$}
\put(56,29){$\P(S^n W)\ $}
\put(7,25){\vector(0,-1){10}}
\multiput(64,24)(0,-2){4}{\line(0,1){1}}
\put(64,16){\vector(0,-1){1}}
\put(5,9){$Q_\gamma$}
\put(14,11){\oval(3,1.8)[l]}
\put(14,10){\vector(1,0){40}}
\put(56,9){$\P(S^nW^G)$}
\put(2,20){$\scriptstyle\pi_\gamma$}
\put(38,12){$\scriptstyle j_n$}
\put(66,20){${\scriptstyle p_G}$}
\end{picture}
\end{array}
\end{equation}
In this diagram, $v_n$ stands for the $n$-th Veronese embedding and
$p_G$ is the projection induced by the inclusion ${S^nW}^G\subset
S^nW$.
The space $Q_\gamma$ comes with a natural polarization represented by
$L_n:= j_n^*\O_{\P({S^nW}^G)}(1)$.
Indeed, by (\ref{eqemb})
we have $\pi_\gamma^*L_n\cong\O_{\P(W)_\gamma^{ss}}(n)$,
and from the commutative diagram
\begin{equation}
\label{eqemb}
\begin{array}{c}
\unitlength=1mm
\begin{picture}(120,24)(0,9)
\put(0,29){$\P(W)^{ss}_\gamma\subset \P(W)$}
\put(34,30){\vector(1,0){10}}
\put(34,31){\oval(3,1.8)[l]}
\put(36,32){${\scriptstyle v_{n_1}}$}
\put(46,29){$\P(S^{n_1} W)$}
\put(7,25){\vector(0,-1){10}}
\put(5,9){$Q_\gamma$}
\put(14,11){\oval(3,1.8)[l]}
\put(14,10){\vector(1,0){15}}
\put(31,9){$\P(S^{n_1}W^G)$}
\put(2,20){$\scriptstyle\pi_\gamma$}
\put(18,12){$\scriptstyle j_{n_1}$}
\put(110,20){${\scriptstyle p_G}$}
\put(53,10){\vector(1,0){10}}
\put(53,11){\oval(3,1.8)[l]}
\put(55,12){$\scriptstyle v_{n_2}$}
\put(66,9){$\P(S^{n_2}(S^{n_1}W^G))$}
\multiput(116,24)(0,-2){4}{\line(0,1){1}}
\put(116,16){\vector(0,-1){1}}
\multiput(97,10)(2,0){3}{\line(1,0){1}}
\put(104,10){\vector(1,0){1}}
\put(107,9){$\P(S^{n_1n_2} W^G)$}
\put(67,30){\vector(1,0){36}}
\put(67,31){\oval(3,1.8)[l]}
\put(80,32){${\scriptstyle v_{n_2}}$}
\put(106,29){$\P(S^{n_1n_2} W)$}
\end{picture}
\end{array}
\end{equation}
we infer $L_{n_1}^{\otimes n_2}\cong L_{n_1n_2}$, hence
\begin{equation}
\label{polind}
L_{n_1}^{\otimes n_2}\cong L_{n_2}^{\otimes n_1},\quad \forall n_1,n_2\quad
\hbox{large enough}.
\end{equation}
\begin{Rem}
\label{MoreGeneral}
In the following, we will mainly consider actions on projective spaces.
However, if $X$ is a quasi-projective variety with an action of an
algebraic group $G$ which is
linearized in an ample line bundle $L$, then $L^{\otimes n}$ induces,
for $n$ large enough, a $G$-invariant embedding of $X$
into $\P:=\P(H^0(L^{\otimes n}))$ such that the semistable,
polystable, and stable points of $X$ are mapped to the semistable, polystable,
and stable points of $\P$.
Hence all the results which we will prove hold also in this more general
setting,
and will be used in this generality in Section 2.
\end{Rem}
\subsection{Polarized ${\Bbb C}^*$-quotients}
\label{PolC*Quot}
Let $\lambda\colon{\Bbb C}^*\rightarrow \mathop{\rm GL}(W)$ be a rational representation of ${\Bbb C}^*$ in the
finite dimensional vector space $W$ and let
$\overline{\lambda}\colon {\Bbb C}^*\times\P(W)\longrightarrow \P(W)$ be the induced action.
The space $W^\vee$ splits as a direct sum
$$W^\vee=\bigoplus_{i=1}^m W^\vee_i,$$
where $W^\vee_i$ is the eigenspace of the character
$\chi_{d_i}\colon{\Bbb C}^*\longrightarrow{\Bbb C}^*, z\longmapsto z^{d_i}$.
We assume $d_1<d_2<\cdots<d_m$.
Let $x\in \P(W)$ and choose a lift $w\in W^\vee\backslash\{0\}$ of $x$.
Define
\begin{eqnarray*}
d^\lambda_{\min}(x)&:=&\min\bigl\{\, d_i\ \vert\ w\hbox{ has a non-trivial
component in } W_i^\vee\,\bigr\}\\
d^\lambda_{\max}(x)&:=&\max\bigl\{\, d_i\ \vert\ w\hbox{ has a non-trivial
component in } W_i^\vee\,\bigr\} .
\end{eqnarray*}
\begin{Prop}
\label{C^*-ss}\hfill{\break}
{\rm i)} A point $x\in\P(W)$ is $\lambda$-semistable if and only if
$d^\lambda_{\min}(x)\le 0\le d^\lambda_{\max}(x)$.\\
{\rm ii)} A point $x\in\P(W)$ is $\lambda$-polystable if and only if
either $d^\lambda_{\min}(x)=0=d^\lambda_{\max}(x)$ or
$d^\lambda_{\min}(x)<0<d^\lambda_{\max}(x)$.
\end{Prop}
\begin{pf}
Let $w=(w_1,...,w_n)\in W^\vee\backslash\{0\}$ be a lift of $x$, where we take
coordinates with respect to a basis of eigenvectors.
For $z\in {\Bbb C}^*$, we get
$$z\cdot w=(0,...,0,z^{d^\lambda_{\min}(x)}\cdot
w_{i_0},...,z^{d^\lambda_{\max}(x)}\cdot
w_{i_r},0,...,0).$$ Using this description, the assertion becomes obvious.
\end{pf}
As remarked above, we can view $\lambda$ as a linearization
of the action $\overline{\lambda}$.
There are two natural ways of changing this linearization:
\begin{enumerate}
\item Multiplying $\lambda$ by a character: Let $d$ be an integer, and denote by
$\lambda_d$ the representation
$z\longmapsto z^d\cdot \lambda(z)$ of ${\Bbb C}^*$ in $\mathop{\rm GL}(W)$. This means that we change the
$\O_{\P(W)}(1)$-linearization
of $\overline{\lambda}$ by multiplying it with the character $\chi_{d}\colon
{\Bbb C}^*\longrightarrow{\Bbb C}^*, z\longmapsto z^{d}$.
\item Replacing $\lambda$ by a symmetric power:
Let $\lambda^k\colon{\Bbb C}^*\longrightarrow \mathop{\rm GL}(S^kW)$ be the $k$-th symmetric power of $\lambda$.
This induces an $\O_{\P(W)}(k)$-linearization of $\overline{\lambda}$.
\end{enumerate}
Now we combine both methods, i.e.,
we change $\lambda^k$ to the representation $\lambda^k_d$ of ${\Bbb C}^*$
in $\mathop{\rm GL}(S^kW)$. As above, this defines an $\O_{\P(W)}(k)$-linearization of
$\overline{\lambda}$.
Altogether, we have a family $\lambda_d^k$, $k\in{\Bbb Z}_{>0}$, $d\in{\Bbb Z}$,
of linearizations of $\overline{\lambda}$.
Since two $\O_{\P(W)}(k)$-linearizations of $\overline{\lambda}$ differ
by a character of ${\Bbb C}^*$, these are indeed all possible
linearizations.\par
Every linearization $\lambda_d^k$ yields a polarized
GIT-quotient $\bigl(Q_d^k:=\P(W)/\hskip-3pt/_{\lambda_d^k}{\Bbb C}^*, [L^k_d]\bigr)$, and
$(Q_d^k,[L^k_d])$ and $(Q_{d^\prime}^{k^\prime},[L^{k^\prime}_{d^\prime}])$ are isomorphic
as polarized varieties
when the ratios $d/ k$ and $d^\prime/ k^\prime$ coincide.
To see this, one just has to observe that, for any positve integer $t$, the
linearization $\lambda_{t\cdot d}^{t\cdot k}$ is the $t$-th
symmetric power of the linearization $\lambda_d^k$.
\par
Since for a point $x\in \P(W)$ we have
$$d^{\lambda^k_d}_{\min}(x)=k\cdot d^\lambda_{\min}-d,\quad
d^{\lambda_d^k}_{\max}(x)=k\cdot d^{\lambda}_{\max}-d,$$
we obtain the following corollary to Proposition~\ref{C^*-ss}:
\begin{Prop}
\label{C^*-ss2} \hfill{\break}
{\rm i)} The point $x$ is $\lambda_d^k$-semistable if and only if
$d^{\lambda}_{\min}(x)\le d/ k\le d^\lambda_{\max}(x)$.
\\
{\rm ii)} The point $x$ is
$\lambda_d^k$-polystable if and only if either
$d^{\lambda}_{\min}(x)=d/ k= d^\lambda_{\max}(x)$ or
$d^{\lambda}_{\min}(x)< d/ k< d^\lambda_{\max}(x)$.
In particular, every point $x\in\P(W)$ is $\lambda_d^k$-polystable
for suitable numbers
$k\in{\Bbb Z}_{>0}$, $d\in{\Bbb Z}$.
\end{Prop}
For integers $i$ with $1\le i\le 2m$ we define the following intervals in
$\P^1_{\Bbb Q}$:
$$I_i:=\cases \P_{\Bbb Q}^1\setminus[d_m,d_1] &\hbox{if $i=2m$}\\
\{d_{{i+1\over 2}}\} & \hbox{if $i$ is odd}\\
(d_{{i\over 2}},d_{{i\over 2}+1}) & \hbox{if $i$ is even.}
\endcases
$$
\begin{Cor}
$\P(W)^{ss}_{\lambda^k_d}=\P(W)^{ss}_{\lambda^{k^\prime}_{d^\prime}}$
if and only if there is an $i$ with $1\le i\le 2m$,
such that $I_i$ contains both $d/ k$ and $d^\prime/ k^\prime$.
\end{Cor}
We see that for the given action $\overline{\lambda}$
there are exactly $2m$ notions of stability. Denote
by $Q_i$, $i=1,...,2m$, the corresponding unpolarized GIT-quotients,
where $Q_{2m}=\emptyset$.
Then, for any $i=1,...,2m$, there is a $k$ with $Q_i=Q^k_2$.\par
\begin{Rem}
Bia\l ynicki-Birula and Sommese \cite{BS}
investigated ${\Bbb C}^*$-actions in a more
general context.
Specialized to our situation, their main result is the following:
Let $\lambda$ be a ${\Bbb C}^*$-action on $W$ with
a decomposition of the dual space $W^\vee=\bigoplus_{i=1}^m W_i^\vee$
as above. The fixed point set of the induced ${\Bbb C}^*$-action on $\P(W)$ is
given by
$\bigcup_{i=1}^m \P(W_i)$. Set $F_i:=\P(W_i)$, and
define for each index $i$:
\begin{eqnarray*}
X_i^+&:=&\bigl\{\,x\in \P(W)\ \vert\ \mathop{\rm lim}_{z\longrightarrow 0} z\cdot x\in F_i
\,\bigr\}=\P(W_i\oplus\cdots\oplus W_m)\\
X_i^-&:=&\bigl\{\,x\in \P(W)\ \vert\ \lim_{z\longrightarrow \infty} z\cdot x\in
F_i\,\bigr\}=
\P(W_1\oplus\cdots \oplus W_i),
\end{eqnarray*}
and for $i\neq j$ set
$C_{ij}:=(X_i^+\backslash F_i)\cap (X_j^-\setminus F_j)$. This means
$C_{ij}$ is empty for $i\ge j$ and equal to
$\P(W_i\oplus\cdots\oplus W_j)\setminus (\P(W_i)\cup\P(W_j))$ for $i<j$.
We write $F_i<F_j$ when $C_{ij}\neq\emptyset$, i.e.
$$F_1<F_2<\cdots<F_m.$$
In the terminology of \cite{BS}, $F_1$ is the \sl source \rm and $F_m$ is
the \sl sink. \rm
For each $i$ with $1\le i\le m-1$,
one has a partition of $A:=\{1,...,m\}$:
$$A=A_i^-\cup A_i^+,\quad \hbox{with } A_i^-:=\{1,...,i\} \hbox{ and } A_i^+=\{
i+1,...,m\},$$
and an associated
open set
$$U_i:=\bigcup_{\mu\in A_i^-, \nu\in A_i^+} C_{\mu\nu}.$$
The main theorem of \cite{BS} asserts that the $U_i$ are the only Zariski-open
${\Bbb C}^*$-invariant subsets of $\P(W)$ not intersecting the fixed point set
whose quotients by the ${\Bbb C}^*$-action are compact.
One checks directly that $U_i$ is the set of
$\lambda^k_d$-semistable points for any pair $k,d$ with
$d/ k\in(d_i,d_{i+1}).$
\end{Rem}
\begin{Ex}
\label{C^*-ex}
Consider an action $\lambda$ of ${\Bbb C}^*$ on a finite dimensional vector space
$W$
such that the dual space decomposes as
$W^\vee=W_1^\vee\oplus W_2^\vee$ with weights $d_1<d_2$.
If $d\in{\Bbb Z}$ and $k\in{\Bbb Z}_{>0}$ are such that $d_1<d/k <d_2$, then
the set of $\lambda^k_d$-semistable points is
$\P(W_1\oplus W_2)\backslash
\left(\P(W_1)\cup\P(W_2)\right)$ and the quotient $Q_{\lambda^k_d}$ is
naturally isomorphic to $\P(W_1)\times\P(W_2)$. The quotient map
$$\pi\colon\P(W_1\oplus W_2)\backslash
\left(\P(W_1)\cup\P(W_2)\right)
\subset \P(W_1\oplus W_2)\dasharrow \P(W_1)\times\P(W_2)$$
is the obvious one.
\end{Ex}
\begin{Claim}
The polarization induced by $\lambda^k_d$ on $\P(W_1)\times\P(W_2)$
is the equivalence class of the bundle
$\O_{\P(W_1)\times\P(W_2)}(kd_2-d,-kd_1+d)$.
In particular, for every $m,n\in {\Bbb Z}_{>0}$, the class
$[\O_{\P(W_1)\times\P(W_2)}(m,n)]$ occurs as an
induced polarization.
\end{Claim}
{\it Proof}.
Let $L:=\O_{\P(W_1)\times\P(W_2)}(m,n)$ represent the induced polarization.
From the description of $\pi$ it follows that
$H^0(\pi^*L)^{{\lambda_d^k}}=\pi^*H^0(L)=S^mW_1\otimes S^nW_2$ is
the set of bihomogenous polynomials of bidegree $(m,n)$, for some $m,n$.
If $S^mW_1\otimes S^nW_2$
occurs as an eigenspace of the induced ${\Bbb C}^*$-action on the space
$H^0(\O_{\P(W_1\oplus W_2)}(m\cdot n))$, then it must obviously be an
eigenspace for
the character $\chi_{-(md_1+nd_2)+((m+n)/ k) d}$.
Now invariance implies $md_1+nd_2-((m+n)/ k) d=0$,
which can be written as $m (kd_1-d) + n (kd_2-d)=0$.
This yields the first assertion.\par
To prove the second part of the claim one has to find
positive integers $k$, $r$ and an integer $d$ such that the following
equations hold
\begin{eqnarray*}
kd_2-d &=& r m\\
-kd_1+d &=& r n;
\end{eqnarray*}
this results from a straightforward computation.
The other quotients are $\P(W_1)$, $\P(W_2)$ with
the obvious polarizations, and $\emptyset$.
\subsection{Stability for actions of products of groups}
Consider now two reductive groups $G$, $H$ and a rational representation
$\rho\colon G\times H\longrightarrow \mathop{\rm GL}(W)$ in the finite dimensional space $W$.
We denote by $\gamma$ and $\lambda$ the induced representations of
$G$ and $H$, respectively.
Choose $n$ large enough in order to obtain an embedding
$j_n\colon Q_\gamma\hookrightarrow \P({S^nW}^G)$.
Since the actions of $G$ and $H$ commute, $\lambda$ induces actions of $H$
on $Q_\gamma$, on ${S^nW}^G$, and on
$\P({S^nW}^G)$; for these
actions $j_n$ is $H$-equivariant.
The action of $H$ on $Q_\gamma$ possesses a natural linearization
in
$j_n^*\O_{\P({S^nW}^G)}(1)$.
By \ref{BackGIT}(\ref{polind}), the corresponding concept of stability does
not depend
on the choice of $n$.
Let us denote the set of semistable points by $Q_\gamma^{ss}$ and the set
of polystable points by $Q_\gamma^{ps}$.
\begin{Prop}
\label{prodss}
The set of $\rho$-semistable points in the projective space $\P(W)$ is given by
$\P(W)^{ss}_\rho=\P(W)^{ss}_\gamma\cap \pi_\gamma^{-1}(Q^{ss}_\gamma)$,
and there exists a natural isomorphism
$Q_\gamma/\hskip-3pt/_\lambda H\cong Q_\rho$.
\end{Prop}
\begin{pf}
Suppose $x\in\P(W)$ is $\gamma$-semistable and its image
$\pi_\gamma(x)$ is $\lambda$-semistable in $Q_\gamma$.
If $n$ is large, $j_n(\pi_\gamma(x))$ is semistable in
$\P({S^nW}^G)$, so that there exists an integer $k\ge 1$
and a section $\overline{s}\in
H^0(\P({S^nW}^G),\O_{\P({S^nW}^G)}(k))^H$ not vanishing at \
$j_n(\pi_\gamma(x))$.
Identifying $\overline{s}\in {S^k({S^nW}^G)}^H$ with an element of
${S^{kn}W}^{G\times H}$, we obtain a $G\times H$-invariant section
in $\O_{\P(W)}(kn)$ not vanishing at $x$, hence $x$ is $\rho$-semistable.
\par
Conversely, suppose $x\in \P(W)_\rho^{ss}$. Then there exists, for some
$m\ge 1$, a section $s\in H^0(\P(W),\O_{\P(W)}(m))^{G\times H}$ with $s(x)\neq 0$.
Viewing $s\in {S^mW}^{G\times H}$ as an $H$-invariant element of ${S^mW}^G$,
we see that $x\in \P(W)^{ss}_\gamma\cap \pi_\gamma^{-1}(Q_\gamma^{ss}).$
This proves the first assertion.
\par
The second assertion follows immediately from the first one
and the universal property of the categorical quotient.
\end{pf}
The corresponding result for the polystable points is
\begin{Prop} The set of $\rho$-polystable points is
$\P(W)^{ps}_\rho=\P(W)^{ps}_\gamma\cap \pi_\gamma^{-1}(Q^{ps}_\gamma)$.
\end{Prop}
\begin{pf}
Let $x\in \P(W)$ be a $\gamma$-polystable point with $\pi_\gamma(x)\in
Q_\gamma^{ps}$.
By \ref{prodss}, $x$ is $\rho$-semistable. Choose
a $\rho$-polystable point
$y\in \overline{(G\times H)\cdot x}\cap \P(W)_\rho^{ss}$.
Projecting onto $Q_\gamma$, it follows that $\pi_\gamma(y)$ is contained
in $\overline{H\cdot \pi_\gamma(x)}$ and hence in $H\cdot \pi_\gamma(x)$,
because $\pi_\gamma(x)$ is polystable by assumption.
Therefore, there exists an $h\in H$ with
$\pi_\gamma(x)=h\cdot \pi_\gamma(y)=\pi_\gamma(h\cdot y)$.
But this means that the closures of the $G$-orbits of $x$ and $h\cdot y$
intersect, so that $G\cdot x\subset \overline{G\cdot (h\cdot y)}\cap
\P(W)^{ss}_\gamma$,
since $x$ is $\gamma$-polystable.
In particular, $x\in \overline{(G\times H)\cdot y}\cap\P(W)_\rho^{ss}=
(G\times H)\cdot y.$ Hence $x$ is also $\rho$-polystable.
\par
To prove the converse, suppose $x$ is a $\rho$-polystable point.
We first show that $x$ is $\gamma$-polystable, too.
Let $y\in \overline{G\cdot x}\cap\P(W)_\gamma^{ss}$ be a $\gamma$-polystable
point. Since $\pi_\gamma(y)=\pi_\gamma(x)$, it follows from \ref{prodss} that
$\pi_\gamma(y)\in Q_\gamma^{ss}$.
Applying \ref{prodss} again, we see that $y\in\P(W)_\rho^{ss}$.
The orbit $(G\times H)\cdot x$ being closed in $\P(W)_\rho^{ss}$,
there exist $g\in G$ and $h\in H$ with $y=g\cdot h\cdot x$,
i.e.\ $x= h^{-1}\cdot g^{-1}\cdot y$.
Now $g^{-1}\cdot y$ is $\gamma$-polystable, hence $x$ is $\gamma$-polystable
too, because $\gamma$ and $\lambda$ commute.
Finally, we must show that $\pi_\gamma(x)\in Q_\gamma^{ps}$.
Choose $y$ such that $\pi_\gamma(y)\in\overline{H\cdot\pi_\gamma(x)}\cap
Q_\gamma^{ps}$.
We may assume that $y$ is $\gamma$-polystable.
By what we have already proved, $y$ is $\rho$-polystable.
Now $\pi_\gamma(y)$ and $\pi_\gamma(x)$ are mapped to the
same point in $Q_\gamma/\hskip-3pt/_\lambda G=Q_\rho$.
But the projection $\pi_\rho\colon\P(W)_\rho^{ss}\longrightarrow Q_\rho$
separates closed $\rho$-orbits, thus $(G\times H)\cdot x=(G\times H)\cdot y$,
and therefore $H\cdot \pi_\gamma(x)=H\cdot \pi_\gamma(y)$ is closed
in $Q_\gamma^{ss}$.
\end{pf}
\subsection{Applications to $G\times{\Bbb C}^*$-actions}
Let $G$ be a reductive algebraic group possessing only the trivial
character, so that for any action of $G$ on a projective
variety $V$ and any
line bundle $L$ on $V$ there is at most one $L$-linearization
of the given action.
Consider a rational representation
$\rho$ of $G\times{\Bbb C}^*$ in the finite dimensional vector space $W$.
As above we denote by $\gamma$ and $\lambda$ the induced representations
of $G$ and ${\Bbb C}^*$, respectively, and by $\overline{\rho}$,
$\overline{\gamma}$, and $\overline{\lambda}$ the induced action
of $G\times{\Bbb C}^*$, $G$, and ${\Bbb C}^*$ on $\P(W)$.
Let $\P(W)_i^s\subset\P(W)_i^{ps}\subset\P(W)_i^{ss}$
be the set stable, polystable, or semistable points w.r.t.\ the
$i$-th stability concept for the action $\overline{\lambda}$,
and let $I_i$, $i=1,...,2m$, be the associated intervals of rational
numbers.
The representation $\rho$ induces an action of $G$ on $Q^k_d$
which is equipped with a natural linearization in the ample line
bundle $L^k_d$, and there is no natural way to alter this linearization,
because $G$ does not possess a non-trivial character.
The corresponding concept of $G$-stability depends only on the
rational parameter $d/ k$.\par
Now fix a rational parameter $\eta:=d/ k\in I_i$ for some
index $i$.
A point $y\in Q_i$ is called \it
$\eta$-stable ($\eta$-polystable, $\eta$-semistable) \rm if it is
$G$-stable ($G$-polystable, $G$-semistable) w.r.t.\ the
$G$-linearized line bundle $L^k_d$ on $Q_i=Q_d^k$.
\par
Recall that every point $x\in\P(W)$ lies in $\P(W)_i^{ps}$
for a suitable index $i$; let $\pi_i(x)\in Q_i$ be its
image under $\pi_i\colon \P(W)_i^{ps}\longrightarrow Q_i$.
\begin{Thm}
\label{GITThm}
Fix a point $x\in\P(W)$. Then the following conditions are equivalent:
\par
{\rm i)} The point $x$ is $G$-semistable ($G$-polystable).\par
{\rm ii)} There exists an index $i$ and a parameter $\eta\in I_i$
such that $x\in \P(W)_i^{ss}$ ($x\in \P(W)_i^{ps}$) and
$\pi_i(x)$ is $\eta$-semistable ($\eta$-polystable).
\end{Thm}
\begin{pf} We explain the semistable case; the arguments in the
polystable case are similar.
Suppose first that $x\in \P(W)$ is $G$-semistable.
Choose $n$ large enough (cf.\ Section~\ref{BackGIT}) in order to obtain
a commutative diagram as in \ref{BackGIT}(\ref{eqemb}).
Since $\gamma$ and $\lambda$ commute, the representation
$\lambda^n\colon {\Bbb C}^*\longrightarrow \mathop{\rm GL}(S^nW)$
induces a representation
$\lambda^\prime\colon {\Bbb C}^*\longrightarrow \mathop{\rm GL}({S^nW}^G).$
By \ref{C^*-ss2}, we find $k\in {\Bbb Z}_{>0}$ and $d\in{\Bbb Z}$
such that $\pi_\gamma(x)$ is semistable w.r.t.\ the stability
concept induced by $(\lambda^\prime)^k_d$ on $Q_\gamma$.
Since $(\lambda^\prime)^k_d$ is induced by the representation
$$\lambda^{nk}_d\colon {\Bbb C}^*\longrightarrow \mathop{\rm GL}\left(S^k(S^nW)\right),$$
we may replace $n$ by $kn$ and, therefore, assume that
$\pi_\gamma(x)$ is semistable w.r.t.\ the stability
concept induced by $(\lambda^\prime)_d$ on $Q_\gamma$, for some integer $d$.
We now apply Proposition~\ref{prodss} to the representation
$$(\gamma^n\times \lambda^n_d)\colon G\times{\Bbb C}^*\longrightarrow \mathop{\rm GL}(S^nW).$$
(Note that this representation induces the action $\overline{\rho}$ on $\P(W)$.)
Since $x\in\P(W)$ is $\gamma$-semistable, it is also $\gamma^n$-semistable.
By construction, $\pi_\gamma(x)$ is semistable w.r.t.\ the induced
${\Bbb C}^*$-action on $Q_\gamma$, and hence $x$ is $\gamma^n\times\lambda_d^n$-semistable
by \ref{prodss}.
Applying \ref{prodss} the other way round, setting $\eta:=d/ k$
and choosing $i$ with $\eta\in I_i$,
it follows that $x\in\P(W)_i^{ss}$ and that $\pi_i(x)$ is $\eta$-semistable.
This settles the implication i)$\Rightarrow$ii).
\par
To prove the other implication suppose $x\in \P(W)$ fulfills
the assumptions of ii).
By definition and by Proposition~\ref{prodss}, this means that
there are $k\in{\Bbb Z}_{>0}$ and
$d\in{\Bbb Z}$ with $\eta=d/ k$
such that $x\in\P(W)$ is $\gamma^k\times\lambda^k_d$-semistable.
This implies that $x$ is $\gamma^k$- and hence $\gamma$-semistable.
This concludes the proof.
\end{pf}
\begin{Rem}
\label{ChainsofFlips}
At this point it becomes clear why chains of flips appear:
Let $G$, $\rho$, $\gamma$, and $\lambda$ be as above.
We have constructed a family
$(\gamma^k\times\lambda^k_d)$ of linearizations of the action $\overline{\rho}$
on $\P(W)$.
Each of these linearizations yields a GIT-quotient of $\P(W)$ by the action
$\overline{\rho}$.
This family of quotients can be constructed in another manner:
First take the $G$-quotient in order to obtain
a polarized variety $(\tilde{Q}:=\P(W)/\hskip-3pt/_{\gamma} G,[L])$.
The resulting ${\Bbb C}^*$-action on this variety yields a
family of quotients $Q_i$, $i=1,...,2n$, where $2n$ is usually (much) larger
than $2m$, the number of \sl unpolarized \rm ${\Bbb C}^*$-quotients of $\P(W)$
(see~\ref{FlipsEx}). But the family $Q_i$, $i=1,...,2n$,
coincides with the family $\P(W)/\hskip-3pt/_{\gamma^k\times \lambda_d^k}G\times{\Bbb C}^*$,
$k\in {\Bbb Z}_{>0}, d\in{\Bbb Z}$.
This phenomenon is responsible for the occurence of chains of flips
in these situations. It explains the question which was left open in
\cite{R}, 2.4 Remark (2), 2.5.
\end{Rem}
\begin{Ex}
\label{FlipsEx}
Let $W^\vee:=S^3{{\Bbb C}^2}^\vee\oplus {{\Bbb C}^2}^\vee$ and let $\mathop{\rm SL}_2({\Bbb C})$ act on
$W^\vee$ in the following way: Given $(f,p)\in W^\vee$ and $m\in \mathop{\rm SL}_2({\Bbb C})$,
we interpret $f$ and $p$ as functions on ${\Bbb C}^2$ and set
$(m\cdot f)(v):= f(m^t\cdot v)$ and $(m\cdot p)(v):=p(m^t\cdot v)$;
then we define $m\cdot (f,p) := (m\cdot f, m\cdot p).$
Let ${\Bbb C}^*$ act on $W^\vee$ by multiplication with $z^{d_1}$ on the first
factor and by multiplication with $z^{d_2}$ on the second one.
The quotient $V:=W^\vee/\hskip-3pt/\mathop{\rm SL}_2({\Bbb C})$ is of the form
$\mathop{\rm Spec}{\Bbb C}[I,J,D,R]$, where $I$, $J$, $D$, and $R$ are certain
bihomogenous polynomials of bidegrees $(2,2)$, $(3,3)$, $(4,0)$, and
$(1,3)$ in the coordinates of $S^3{{\Bbb C}^2}^\vee$
and ${{\Bbb C}^2}^\vee$.
Furthermore, $I$, $D$, and $R$ are algebraically independent,
and there is a relation
$$27 J^2={1\over 256}DR^2+I^3.$$
We examine the $\mathop{\rm SL}_2({\Bbb C})\times{\Bbb C}^*$-action on $\P(W)$.
The quotient $Q:=\P(W)/\hskip-3pt/\mathop{\rm SL}_2({\Bbb C})$ is given by $\mathop{\rm Proj}{\Bbb C}[I,J,D,R]$
where $I$, $J$, $D$, and $R$ have weights 4, 6, 4, and 4, respectively.
The ring ${\Bbb C}[I,J,D,R]_{(12)}$ is generated by its elements in degree 1, i.e.
by $I^3, I^2D, I^2R, ID^2, IR^2, IDR, J^2, D^3, D^2R, DR^2, R^3$;
hence there is an embedding $Q\hookrightarrow \P(S^{12}W^{\mathop{\rm SL}_2({\Bbb C})})$.
The ${\Bbb C}^*$-action on $Q$ can be extended to $\P(S^{12}W^{\mathop{\rm SL}_2({\Bbb C})})$ such
that the weights of the corresponding action on ${S^{12}W^{\mathop{\rm SL}_2({\Bbb C})}}^\vee$
are
$$6d_1+6d_2, 8d_1+4d_2, 5d_1+7d_2, 10 d_1+2d_2, 4d_1+8d_2,
7d_1+5d_2, 12 d_1, 9d_1+3d_2, 3d_1+9d_2.$$
For a point in $p\in Q$, $d_{\min}(p)$ and $d_{\max}(p)$ can take the
values $6d_1+6d_2$, $12d_1$, and $3d_1+9d_2$.
Hence, for $d_1\neq d_2$, there are
6 different notions of semistability on $Q$, hence 6 different
notions of
$\mathop{\rm SL}_2\times{\Bbb C}^*$-semistability on $\P(W)$, whereas there are only 4
different notions of
${\Bbb C}^*$-semistability on $\P(W)$.
\end{Ex}
\section{Oriented pairs and their moduli}
Let $X$ be a smooth projective variety over the field of complex numbers
and fix an ample divisor $H$ on $X$.
All degrees will be taken with respect to $H$ and
the corresponding line bundle will
be denoted by $\O_X(1)$.
Fix a torsion free
coherent sheaf ${\cal E}_0$ and a Hilbert polynomial $P$.
Finally, let $\mathop{\rm Pic}(X)$ be the Picard scheme of $X$
and choose a Poincar\' e line bundle $\L$ over $\mathop{\rm Pic}(X)\times X$.
If $S$ is a scheme and ${\frak E}_S$ a flat family of coherent sheaves
over $S\times X$, then there is a morphism $\det_S\colon S\longrightarrow \mathop{\rm Pic}(X)$
mapping a closed point $s$ to $[\det({\frak E}_{S\vert \{s\}\times X})]$.
We set $\L[{\frak E}_S]:=(\det_S\times\mathop{\rm id})^*(\L)$; this line bundle depends only
on the isomorphism class of the family ${\frak E}_S$.
The Hilbert polynomial of a sheaf ${\cal F}$ will be denoted by $P_{\cal F}$.
For any non-trivial
torsion free coherent sheaf ${\cal F}$ there is a unique subsheaf
${\cal F}_{\max }$ for which $P_{\cal F}/\mathop{\rm rk}{\cal F}$ is maximal and whose rank is maximal among
the subsheaves ${\cal F}^\prime$ with $P_{{\cal F}^\prime}/\mathop{\rm rk}{\cal F}^\prime$ maximal.
Set $\mu_{\max }({\cal F}):=\mu({\cal F}_{\max })$.
\subsection{Oriented pairs}
An \it oriented pair of type $(P,\L,{\cal E}_0)$ \rm is a triple $({\cal E},\varepsilon,\phi)$
consisting of a torsion free coherent sheaf ${\cal E}$ with
Hilbert polynomial $P_{\cal E}=P$, a homomorphism $\varepsilon\colon\det{\cal E}\longrightarrow\L[{\cal E}]$,
and a homomorphism $\phi\colon {\cal E}\longrightarrow {\cal E}_0$.
The homomorphisms $\varepsilon$ and $\phi$ will be called
the \it orientation \rm and the \it framing \rm of the pair $({\cal E},\varepsilon,\phi)$.
Two oriented pairs $({\cal E}_1,\varepsilon_1,\phi_1)$ and $({\cal E}_2,\varepsilon_2,\phi_2)$ are
said to be \it equivalent\rm , if there is an isomorphism
$\Psi\colon{\cal E}_1\longrightarrow {\cal E}_2$ with $\varepsilon_1
=\varepsilon_2\circ\det\Psi$ and $\phi_1=\phi_2\circ\Psi$. When $\ker(\phi)\neq
0$, we set
$$\delta_{{\cal E},\phi}:=P_{\cal E}-{\mathop{\rm rk}{\cal E}\over\mathop{\rm rk}\ker(\phi)_{\max}}P_{\ker(\phi)_{\max
}}.$$
An oriented pair $({\cal E},\varepsilon,\phi)$ of type $(P,\L,{\cal E}_0)$ is \it semistable\rm ,
if either $\phi$ is injective, or $\varepsilon$ is an isomorphism, $\ker(\phi)\neq 0$,
$\delta_{{\cal E},\phi}\ge 0$, and for all non-trivial subsheaves ${\cal F}\subset{\cal E}$
$${P_{\cal F}\over \mathop{\rm rk}{\cal F}}-{\delta_{{\cal E},\phi}\over\mathop{\rm rk}{\cal F}}\quad\le\quad {P_{\cal E}\over\mathop{\rm rk}{\cal E}}-
{\delta_{{\cal E},\phi}\over\mathop{\rm rk}{\cal E}}.$$
\\
The corresponding stability concept is slightly more
complicated: An oriented pair $({\cal E},\varepsilon,\phi)$ of type $(P,\L,{\cal E}_0)$ is \it
stable\rm ,
if either $\phi$ is injective, or $\varepsilon$ is an isomorphism, $\ker(\phi)\neq 0$,
$\delta_{{\cal E},\phi}> 0$, and one of the following conditions holds:
\begin{enumerate}
\item For all non-trivial proper subsheaves ${\cal F}\subset {\cal E}$:
$${P_{\cal F}\over \mathop{\rm rk}{\cal F}}-{\delta_{{\cal E},\phi}\over\mathop{\rm rk}{\cal F}}\quad
<\quad{P_{\cal E}\over\mathop{\rm rk}{\cal E}}-{\delta_{{\cal E},\phi}\over\mathop{\rm rk}{\cal E}}.$$
\item $\phi\neq 0$, $\ker(\phi)_{\max}$ is stable, and
${\cal E}\cong\ker(\phi)_{\max}\oplus{\cal E}^\prime$, where the pair $({\cal E}^\prime,\phi)$ satisfies
\begin{eqnarray*}
{P_{\cal F}\over\mathop{\rm rk}{\cal F}}-{\delta_{{\cal E},\phi}\over\mathop{\rm rk}{\cal F}}\quad <\quad{P_{{\cal E}^\prime}\over\mathop{\rm rk}{\cal E}^\prime}
-{\delta_{{\cal E},\phi}\over\mathop{\rm rk}{\cal E}^\prime} &\ & \hbox{$\forall$ proper subsheaves
$0\neq{\cal F}\subset{\cal E}^\prime\ ,$}
\\
{P_{\cal F}\over\mathop{\rm rk}{\cal F}}\quad <\quad{P_{{\cal E}^\prime}\over\mathop{\rm rk}{\cal E}^\prime}-{\delta_{{\cal E},\phi}
\over\mathop{\rm rk}{\cal E}^\prime} &\ &
\hbox{$\forall$ proper subsheaves $0\neq {\cal F}\subset{\cal E}^\prime\cap\ker(\phi)$.}
\end{eqnarray*}
\end{enumerate}
Our (semi)stability concept is related to the \sl parameter dependent \rm
(semi)stability concept of \cite{HL1} and \cite{HL2}
in the following way:
Let $\delta$ be a polynomial over the rationals with positive leading
coefficient.
Recall that a pair $({\cal E},\phi)$ consisting of a torsion free coherent sheaf
${\cal E}$ with $P_{{\cal E}}=P$ and a non-zero homomorphism $\phi\colon {\cal E}\longrightarrow{\cal E}_0$
is called \it (semi)stable w.r.t.\ $\delta$\rm , if for any non-trivial
proper subsheaf
${\cal F}\subset{\cal E}$ the following conditions hold:
\begin{eqnarray*}
{P_{\cal F}\over\mathop{\rm rk}{\cal F}}-{\delta\over\mathop{\rm rk}{\cal F}}&(\le)&{P_{\cal E}\over\mathop{\rm rk}{\cal E}}-{\delta\over\mathop{\rm rk}{\cal E}
}\ ,\\
{P_{\cal F}\over\mathop{\rm rk}{\cal F}}&(\le)&{P_{\cal E}\over\mathop{\rm rk}{\cal E}}-{\delta\over\mathop{\rm rk}{\cal E}}
,\qquad\hbox{when ${\cal F}\subset\ker(\phi)$}.
\end{eqnarray*}
In this terminology, (semi)stable oriented pairs can be characterized as
follows:
\begin{Lem}
\label{HLcharac}
{\rm i)} An oriented pair $({\cal E},\varepsilon,\phi)$ is semistable if and only if it
satisfies
one of the following three conditions:
\begin{enumerate}
\item $\phi$ is injective.
\item ${\cal E}$ is semistable and $\varepsilon$ is an isomorphism.
\item $\phi\neq 0$, $\varepsilon$ is an isomorphism, and $({\cal E},\phi)$ is semistable
w.r.t.\ some $\delta>0$.
\end{enumerate}\par
{\rm ii)} An oriented pair $({\cal E},\varepsilon,\phi)$ is stable if and only if it
satisfies
one of the following four conditions:
\begin{enumerate}
\item $\phi$ is injective.
\item ${\cal E}$ is stable and $\varepsilon$ is an isomorphism.
\item $\phi\neq 0$, $\varepsilon$ is an isomorphism, and $({\cal E},\phi)$ is stable
w.r.t.\ some $\delta>0$.
\item $\phi\neq 0$, $\delta_{{\cal E},\phi}>0$, $\varepsilon$ is an isomorphism, and
${\cal E}$ splits as $\ker(\phi)_{\max}\oplus{\cal E}^\prime$, where $\ker(\phi)_{\max}$ is
stable and
$({\cal E}^\prime,\phi)$ is stable
w.r.t.\ $\delta_{{\cal E},\phi}$.
\end{enumerate}
\end{Lem}
We note that the stable oriented pairs appearing in
Lemma~\ref{HLcharac}.ii)4.
are precisely those pairs $({\cal E},\varepsilon,\phi)$, for which
$\varepsilon$ is isomorphic, $\phi\neq 0$, $\delta_{{\cal E},\phi}>0$,
the pair $({\cal E},\phi)$ is polystable w.r.t.\ $\delta_{{\cal E},\phi}$,
and which have \sl only finitely many automorphisms\rm .
To see this, recall from
\cite{HL2} that for a given $\delta\in{\Bbb Q}[x]$, $\delta>0$, the polystable
pairs
$({\cal E},\phi)$
are those for which ${\cal E}$ splits in the form
$${\cal E}\cong {\cal E}_1\oplus\cdots\oplus{\cal E}_{s-1}\oplus{\cal E}_s\ ,$$
where the sheaves ${\cal E}_1,...,{\cal E}_{s-1}$ are stable subsheaves of $\ker(\varphi)$,
$({\cal E}_s,\phi)$ is a
stable pair w.r.t.\ $\delta$, and
$P_{{\cal E}_1}/\mathop{\rm rk}{\cal E}_1=\cdots=
P_{{\cal E}_{s-1}}/\mathop{\rm rk}{\cal E}_{s-1}=
P_{{\cal E}_s}/\mathop{\rm rk}{\cal E}_s-\delta/\mathop{\rm rk}{\cal E}_s$.
This makes our assertion obvious.
\begin{Rem}
\label{properlysemistablepairs}
Let $({\cal E},\varepsilon,\phi)$ be a stable oriented pair of type 4.
(see~\ref{HLcharac}.ii)).
Then $\delta_{{\cal E},\phi}$
is the only rational polynomial with positive
leading coefficient w.r.t.\ which the pair $({\cal E},\phi)$ is semistable.
This follows from
the equalities
\begin{eqnarray*}
{P_{{\cal E}^\prime}\over\mathop{\rm rk}{\cal E}^\prime}-{\delta_{{\cal E},\phi}\over\mathop{\rm rk}{\cal E}^\prime}&=
&{P_{{\cal E}}\over\mathop{\rm rk}{\cal E}}-{\delta_{{\cal E},\phi}\over\mathop{\rm rk}{\cal E}}\ ,
\\
{P_{\ker(\phi)_{\max}}\over\mathop{\rm rk}\ker(\phi)_{\max}}&=
&{P_{{\cal E}}\over\mathop{\rm rk}{\cal E}}-{{\delta_{{\cal E},\phi}}\over{\mathop{\rm rk}{\cal E}}}\stackrel{.}{}
\end{eqnarray*}
\end{Rem}
For all stability concepts introduced so far, there are analogous notions of
\it slope-(semi)stability\rm . As usual, slope-stability implies stability and
semistability implies slope-semistability.
\par
Let $S$ be a noetherian scheme. \it A family of oriented pairs
parametrized by $S$ \rm is a quadruple $({\frak E}_S,\varepsilon_S,
\widehat{\phi}_S,{\frak M}_S)$
consisting of a flat family ${\frak E}_S$ of torsion free
coherent sheaves
over the product $S\times X$, an invertible sheaf ${\frak M}_S$ on $S$,
a morphism
$\varepsilon_S\colon \det{\frak E}_S\rightarrow \L[{\frak E}_S]\otimes
\pi_S^*{\frak M}_S$,
and a morphism
$\widehat{\phi}_S\colon S^r{\frak E}_S\rightarrow \pi_X^*S^r{\cal E}_0\otimes
\pi_S^*{\frak M}_S$
with $\widehat{\phi}_{S|\{s\}\times X}=S^r\phi_s$ for any
closed point $s\in S$ and a
suitable
$\phi_s\in\mathop{\rm Hom}({\frak E}_{S|\{s\}\times X},{\cal E}_0)$, so that
the pair
$(\varepsilon_{S|\{s\}\times X},
\widehat{\phi}_{S|\{s\}\times X})$ is non-zero.
Two families $({\frak E}^i_S,\varepsilon^i_S,\widehat{\phi}_S^i,{\frak M}_S^i)$,
$i=1,2$, are
called
\it equivalent\rm , if there exist an isomorphism
$\Psi_S\colon {\frak E}_S^1\longrightarrow {\frak E}_S^2$ and an isomorphism
${\frak m}\colon {\frak M}_S^1\longrightarrow {\frak M}_S^2$ such that
$(\mathop{\rm id}_{\L[{\frak E}_S^1]}\otimes \pi_S^*{\frak m})\circ
\varepsilon_S^1=\varepsilon_S^2\circ \det\Psi$ and
$(\mathop{\rm id}_{\pi_X^*S^r{\cal E}_0}\otimes \pi_S^*{\frak m})\circ
\widehat{\phi}_S^1=\widehat{\phi}_S^2\circ S^r\Psi$.
\par
With these notions, we define the functors
$M^{ss}_{(P,\L,{\cal E}_0)}$ and $M^s_{(P,\L,{\cal E}_0)}$
of equivalence classes of families of semistable and stable oriented pairs
of type $(P,\L,{\cal E}_0)$.
\begin{Rem}
Though the definition of a family may appear a little odd at first sight,
it will become clear that families must be defined in this way
for technical reasons. Families of the above type are precisely those
which are locally induced by the universal family on the
parameter space which we will construct in Section~\ref{ParSpace}.
The functors defined above do depend on the
choice of the Poincar\' e bundle and there is no natural way
to compare functors associated to different Poincar\' e bundles.
\end{Rem}
\subsection{A boundedness result}
\label{Bound}
Here we show that the family of isomorphism classes of torsion free
coherent sheaves occuring in oriented slope-semistable pairs of type
$(P,\L,{\cal E}_0)$ is
bounded. We use Maruyama's boundedness criterion:
\begin{Thm}\cite{Ma}
Let $C$ be some constant.
The set of isomorphism classes of torsion free coherent sheaves with
Hilbert polynomial $P$ and $\mu_{\max }\le C$ is bounded.
\end{Thm}
\begin{Prop}
The set of isomorphism classes of torsion free sheaves occuring in a
slope-semistable oriented pair of type $(P,\L,{\cal E}_0)$ is bounded.
\end{Prop}
\begin{pf}
Set $C:=\max\{\,\mu_{\max}({\cal E}_0), \mu({\cal E})\,\}$.
Let $({\cal E},\varepsilon,\phi)$ be a slope-semistable oriented pair of type
$(P,\L,{\cal E}_0)$.
We claim that $\mu_{\max}({\cal E})\le C;$ in view of Maruyama's theorem, this
assertion proves
the proposition.
\par
Write a given non-trivial subsheaf ${\cal F}$ of ${\cal E}$ as an extension
$$0\longrightarrow {\cal F}\cap\ker(\phi)\longrightarrow {\cal F}\longrightarrow\phi({\cal F})\longrightarrow 0.$$
If ${\cal F}$ is entirely contained in the kernel of $\phi$, the definition of
slope-semistability
implies $\mu({\cal F})\le \mu({\cal E})\le C$.
If ${\cal F}$ is isomorphic to $\phi({\cal F})$, then obviously
$\mu({\cal F})\le\mu_{\max}({\cal E}_0)\le C$.
In the remaining cases
\begin{eqnarray*}
\mu({\cal F}) &=& {\mu({\cal F}\cap\ker(\phi))\mathop{\rm rk}({\cal F}\cap\ker(\phi))+
\mu(\phi({\cal F}))\mathop{\rm rk}\phi({\cal F})\over\mathop{\rm rk}{\cal F}}\\
&\le& {\mathop{\rm rk}({\cal F}\cap \ker(\phi))\over\mathop{\rm rk}{\cal F}}\mu({\cal E})+
{\mathop{\rm rk}\phi({\cal F})\over\mathop{\rm rk}{\cal F}}\mu_{\max}({\cal E}_0) \le C.
\end{eqnarray*}
\end{pf}
\subsection{The parameter space for semistable oriented pairs}
\label{ParSpace}
By the boundedness result of the previous paragraph, there is a natural
number $m_0$ such that for all torsion free coherent sheaves
${\cal E}$ occuring in a
semistable oriented pair, and for all $m\ge m_0$ the following
properties hold true: ${\cal E}(m)$ is globally generated and $H^i(X,{\cal E}(m))=0$
for $i>0$.
Let $V$ be a complex vector space of dimension $p:=P(m)$.
There exists a quasi-projective scheme ${\frak Q}$, the $\mathop{\rm Quot}$-scheme of
torsion free coherent quotient
sheaves of
$V\otimes\O_X(-m)$
with Hilbert polynomial $P$, and a universal
quotient on ${\frak Q}\times X$:
$$q_{\frak Q}\colon V\otimes\pi_X^*\O_X(-m) \longrightarrow {\frak E}_{\frak Q}.$$
Let ${\cal N} $ be the sheaf $\pi_{\frak Q *}(\det({\frak E}_{\frak Q})^\vee
\otimes \L[{\frak
E}_{\frak Q}])$. By the universal property of the Picard scheme, there is a line bundle
${\frak M}$ on ${\frak Q}$ such that
$$\det({\frak E}_{\frak Q})^\vee\otimes \L[{\frak E}_{\frak Q}]\cong \pi_{\frak
Q}^*\frak M.$$
This implies that ${\cal N}$ is invertible and
$${\cal N}\langle [q]\rangle\cong H^0(X,\det({\frak E}_{{\frak Q}\vert\{[q]\}\times
X}^\vee)\otimes\L[{\frak E}_{{\frak Q}\vert\{[q]\}\times X}])\ .$$
Let ${\frak N}\buildrel
\over\longrightarrow {\frak Q}$ be the associated geometric line bundle.
The space ${\frak N}$ is a parameter space for equivalence classes
$[q\colon V\otimes\O_X(-m)\longrightarrow {\cal E},\varepsilon]$
consisting of a quotient \linebreak
$q\colon V\otimes\O_X(-m)\longrightarrow {\cal E}$
and an orientation
$\varepsilon\colon \det({\cal E})\longrightarrow \L[{\cal E}]$.
Here two objects $(q_i\colon V\otimes \O_X(-m)\longrightarrow{\cal E}_i,\varepsilon_i)$,
$i=1,2$, are
\it equivalent\rm , if there is an isomorphism $\Psi\colon {\cal E}_1\longrightarrow{\cal E}_2$
with $\Psi\circ q_1=q_2$ and $\varepsilon_1=\varepsilon_2\circ \det(\Psi)$.
\par
Next we have to construct a parameter space for all oriented
pairs.
We choose $m\ge m_0$ so large that ${\cal E}_0(m)$ is also globally generated.
Every oriented pair yields an element
in $K:=\mathop{\rm Hom}(V,H^0({\cal E}_0(m)))$ and hence an element in $S^rK$.
On the projective bundle ${\frak P}:=\P(({\frak N}\times S^rK)^\vee)
\stackrel{\frak p}
{\longrightarrow}{\frak Q}$ there is a (nowhere vanishing) tautological section
$${\frak s}\colon
\O_{\frak P}\longrightarrow {\frak p}^*({\cal N}\oplus (S^rK\otimes{\cal O}_{\frak
Q}))\otimes\O_{\frak P}(1).$$
Let
$$q_{\frak P}\colon V\otimes \pi_X^*\O_X(-m)\longrightarrow {\frak E}_{\frak P}$$
be the pullback of the universal quotient on ${\frak Q}\times X$
to ${\frak P}\times X$.
We view the pullback $\pi_{\frak P}^*{\frak s}$ of ${\frak s}$ to
${\frak P}\times X$
as a pair consisting of a homomorphism
$$\varepsilon_{\frak P}\colon \det({\frak E}_{\frak P})\longrightarrow \L[{\frak E}_{\frak
P}]\otimes
\pi_{\frak P}^*\O_{\frak P}(1)$$ and a homomorphism
$$\kappa_{\frak P}\colon S^rV\otimes\O_{{\frak P}\times X}\longrightarrow
S^rH^0({\cal E}_0(m))\otimes
\pi_{\frak P}^*\O_{\frak P}(1).$$
\begin{Rem}
\label{MorphtoN}
For a scheme $S$, giving a morphism
$f\colon S\longrightarrow {\frak P}$ is equivalent to giving a map
$\overline{f}\colon S\longrightarrow
{\frak Q}$ - which yields the family ${\frak
E}_S:=(\overline{f}\times\mathop{\rm id}_X)^*{\frak
E}_{\frak Q}$ - , a line bundle ${\frak M}_S$ on $S$, and
homomorphisms
$$\varepsilon_S\colon \det({\frak E}_S)\longrightarrow \L[{\frak E}_S]\otimes
\pi_S^*{\frak
M}_S\ ,$$
$$\kappa_S\colon S^rV\otimes\O_{S\times X}\longrightarrow
S^rH^0({\cal E}_0(m))\otimes\pi_S^*{\frak
M}_S\ $$
on $S\times X$ such that the pair $(\varepsilon_{S|\{s\}\times
X},\kappa_{S|\{s\}\times X})$ is
non-zero for every closed point $s\in S$. Of course, for the morphism $f$
determined by
$\overline{f}$ and $(\varepsilon_S,\kappa_S,{\frak M}_S)$, we have
$\overline{f}={\frak p}\circ
f$, and there is an isomorphism
${\frak m}\colon {\frak M}_S\longrightarrow \overline{f}^*\O_{\frak P}(1)$ such that
$$({\rm id}_{\L[{\frak E}_S]}\otimes \pi_S^*{\frak
m})\circ\varepsilon_S=(f\times{\rm id}_X)^*(\varepsilon_{{\frak P}})\ ,$$
$$ ({\rm id}_{\pi_X^*S^rH^0({\cal E}_0(m))}\otimes\pi_S^*{\frak m})\circ
\kappa_S=(f\times{\rm id}_X)^*(\kappa_{\frak P})\ .$$
\end{Rem}
Our parameter space ${\frak B}$ will be a closed subscheme of
${\frak P}$ whose closed points are of the form
$[[q\colon V\otimes\O_X(-m)\longrightarrow {\cal E},\varepsilon], S^rk]$, with
$[q,\varepsilon]\in {\frak N}$ and $k\in K$, such that there is a map
$\phi\colon {\cal E}\longrightarrow {\cal E}_0$ making the following diagramm
commutative:
\begin{center}
\unitlength=1mm
\begin{picture}(70,24)(0,8)
\put(0,29){$V\otimes{\cal O}_X(-m)$}
\put(28,30){\vector(1,0){30}}
\put(42,32){${\scriptstyle q}$}
\put(62,29){${\cal E}$}
\put(7,25){\vector(0,-1){10}}
\put(63,25){\vector(0,-1){10}}
\put(-15,9){$H^0({\cal E}_0(m))\otimes{\cal O}_X(-m)$}
\put(28,10){\vector(1,0){30}}
\put(62,9){${\cal E}_0$}
\put(2,20){$\scriptstyle k$}
\put(42,12){$\scriptstyle ev$}
\put(66,20){${\scriptstyle \varphi}$}
\end{picture}
\end{center}
Scheme-theoretically, ${\frak B}$ is constructed as follows:
On ${\frak P}\times X$, there is a homomorphism
$$\overline{\phi}_{\frak P}\colon S^rV\otimes\pi_X^*\O_X(-rm)\longrightarrow
\pi_X^*S^r{\cal E}_0\otimes\pi_{\frak P}^*\O_{\frak P}(1).$$
Set $\widehat{\cal G}:=\ker(S^rq_{\frak P})$, choose $n\ge m$
large enough so that
$\widehat{\cal G}_{\vert \{b\}\times X}(n)$ is globally generated
and without higher cohomology for any closed point $b\in {\frak P}$,
and let
$$\widehat{\gamma}\colon {\cal G}:=\widehat{\cal G}\otimes\pi_X^*
\O_X(n)\longrightarrow \pi_X^*
S^r{\cal E}_0(n)\otimes\pi_{\frak P}^*\O_{\frak P}(1)$$ be the induced
homomorphism.
We first define a scheme $\widehat{\frak B}$ whose closed points
are those elements $b\in {\frak P}$ for which
$\widehat{\gamma}_{|\{b\}\times X}$
is the zero map.
Since ${\cal G}_{|\{b\}\times X}$ and $S^r{\cal E}_0(n)$ are globally generated
for any closed point $b\in {\frak P}$, the scheme $\widehat{\frak B}$
is the zero locus of the following homomorphism between locally free
sheaves:
$$\gamma:=\pi_{{\frak P}*}(\widehat{\gamma})\colon
\pi_{{\frak P}*}{\cal G}\longrightarrow \pi_{{\frak P}*}(\pi_X^*S^r{\cal E}_0(n)\otimes
\pi_{\frak
P}^*\O_{\frak P}(1))=H^0(S^r{\cal E}_0(n))\otimes \O_{\frak P}(1).$$ The
scheme ${\frak B}$ we
are looking for is the scheme-theoretic intersection of
$\widehat{\frak B}$ with the
image in ${\frak P}$ of the weighted projective bundle associated with
the vector
bundle ${\frak N}\times K$ over ${\frak Q}$. There exists a universal family
$({\frak
E}_{\frak B},\varepsilon_{\frak B},\widehat{\phi}_{\frak B},{\frak M}_{\frak B})$:
${\frak
M}_{\frak B}$ is the restriction of
$\O_{\frak P}(1)$ to ${\frak B}$,
$q_{\frak B}$ and $\varepsilon_{\frak B}$ are the restrictions of $q_{\frak P}$ and
$\varepsilon_{\frak P}$, and $\widehat{\phi}_{\frak B}$ is induced by the restriction
of $\widehat{\phi}_{\frak P}$ which factorizes through
$S^r{\frak E}_{\frak B}$ by definition.
In the following, a closed point $b=[[q\colon V\otimes\longrightarrow \O_X(-m),\varepsilon],
S^rk]\in
{\frak B}$ will be denoted by
$[q,\varepsilon,\phi]$; here $\phi$ is the unique framing on ${\cal E}$ induced by $k$.
\begin{Rem}
By construction, a morphism $\widehat{f}\colon S\longrightarrow {\frak P}$ factorizes
through ${\frak B}$ if and only if it factorizes through the image
of the associated weighted projective bundle of ${\frak N}\times K$, and
$(\widehat{f}\times\mathop{\rm id}_X)^*(\widehat{\phi}_{\frak P})$ is identically
zero on the kernel
of the map $(\widehat{f}\times \mathop{\rm id}_X)^*(S^rq_{\frak P})$.
\end{Rem}
On the parameter space ${\frak B}$, there is a natural action (from the
right) of
the group $\mathop{\rm SL}(V)$.
To define this action, it suffices to construct a $\mathop{\rm SL}(V)$-action on
${\frak P}$ which leaves the scheme ${\frak B}$ invariant.
The standard representation of $\mathop{\rm SL}(V)$ on $V$ gives us
the homomorphism
$$\Gamma\colon V\otimes \O_{{\frak Q}\times\mathop{\rm SL}(V)\times X}\longrightarrow V\otimes
\O_{{\frak
Q}\times\mathop{\rm SL}(V)\times X}.$$ Moreover, on ${\frak Q}\times\mathop{\rm SL}(V)\times X$
there is the
pullback of the universal quotient
$$\pi_{{\frak Q}\times X}^*(q_{\frak Q})\colon V\otimes
\pi_X^*\O_X(-m)\longrightarrow
\pi_{{\frak Q}\times X}^*{\frak E}_{\frak Q}.$$
By the universal property of the $\mathop{\rm Quot}$-scheme,
$\pi_{{\frak Q}\times X}^* (q_{\frak
Q})\circ \bigl(\Gamma\otimes\mathop{\rm id}_{\pi_X^*\O_X(-m)}\bigr)$ yields a morphism
$\overline{f}\colon {\frak Q}\times\mathop{\rm SL}(V)\longrightarrow {\frak Q}$ such that there is
a well-defined isomorphism
$$\Psi_{{\frak Q}\times \mathop{\rm SL}(V)}\colon (\overline{f}
\times\mathop{\rm id}{}_X)^* {\frak E}_{\frak
Q}\longrightarrow \pi^*_{{\frak Q}\times X}{\frak E}_{\frak Q}$$ with $\Psi_{{\frak
Q}\times\mathop{\rm SL}(V)}\circ (\overline{f}\times\mathop{\rm id}_X)^*(q_{\frak Q})=
\pi_{{\frak Q}\times
X}^*(q_{\frak Q})\circ
\bigl(\Gamma\otimes\mathop{\rm id}_{\pi_X^*\O_X(-m)}\bigr)$. Let
$\Psi_{{\frak P}\times \mathop{\rm SL}(V)}$ be
the pullback of
$\Psi_{{\frak Q}\times\mathop{\rm SL}(V)}$ to
${\frak P}\times \mathop{\rm SL}(V)\times X$, and set ${\frak M}_{{\frak P}\times\mathop{\rm SL}(V)}:=
\pi_{\frak P}^*\O_{\frak P}(1)$,
\begin{eqnarray*}\varepsilon_{{\frak P}\times\mathop{\rm SL}(V)}&:= &\pi_{{\frak P}\times X}^*
(\varepsilon_{{\frak
P}})\circ \det\Psi_{{\frak P}\times\mathop{\rm SL}(V)}\ ,\\
\kappa_{{\frak P}\times\mathop{\rm SL}(V)} &:=& \pi^*_{{\frak P}\times X}
(\kappa_{\frak P})\circ
S^r\left(({\frak p}\times\mathop{\rm id}{}_{\mathop{\rm SL}(V)\times X})^*\Gamma\right)\ .
\end{eqnarray*}
By Remark~\ref{MorphtoN}, the data $\overline{f}$ and ($\varepsilon_{{\frak
P}\times\mathop{\rm SL}(V)},
\kappa_{{\frak P}\times \mathop{\rm SL}(V)}, {\frak M}_{{\frak P}\times\mathop{\rm SL}(V)})$ define
an action
$$f\colon {\frak P}\times \mathop{\rm SL}(V)\longrightarrow {\frak P}.$$
\begin{Prop}
\label{LocUnivProp}
Let $S$ be a noetherian scheme and let $({\frak E}_S,\varepsilon_S,\widehat{\phi}_S,
{\frak
M}_S)$ be a family of semistable oriented pairs parametrized by $S$.
Then $S$ can be covered by open subschemes $S_i$ for which there exist
morphisms $\beta_i\colon S_i\longrightarrow {\frak B}$
such that the restricted families
$({\frak E}_{S|S_i},\varepsilon_{S|S_i},\widehat{\phi}_{S|S_i},{\frak M}_{S|S_i})$
are equivalent
to the
pullbacks of $({\frak E}_{\frak B}, \varepsilon_{\frak B}, \widehat{\phi}_{\frak B},
{\frak
M}_{\frak B})$ via the maps $\beta_i\times\mathop{\rm id}_X$.
\end{Prop}
\begin{pf}
The scheme $S$ can be covered by open subschemes $S_i$ such that
the family ${\frak E}_{S\vert S_i}$ over $S_i\times X$ can be written as a
family
of quotients:
$$q_{S_i}\colon V\otimes \pi_X^*\O_X(-m)\longrightarrow {\frak E}_{S|S_i}.$$
Each $q_{S_i}$ defines a morphism $\overline{f}_i\colon S_i\longrightarrow {\frak Q}$
such that there is a well defined isomorphism
$\Psi_{S_i}\colon {\frak E}_{S_i}:=(\overline{f}_i\times\mathop{\rm id}_X)^*
{\frak E}_{\frak Q}\longrightarrow {\frak E}_{S|S_i}$.
Define ${\frak M}_{S_i}:={\frak M}_{S|S_i}$,
$$\begin{array}{cl}
\varepsilon_{S_i}\colon & \det({\frak E}_{S_i})\textmap{\det\Psi_{S_i}}
\det{\frak E}_{S|S_i}\textmap{\varepsilon_{S|S_i}} \L[{\frak E}_{S\vert S_i}]
\otimes
\pi_{S_i}^*{\frak M}_{S_i}\ \hbox{,}\\
\widehat{\phi}_{S_i}\colon & S^r{\frak E}_{S_i}\textmap{S^r\Psi_{S_i}}
S^r{\frak
E}_{S|S_i}\textmap{\widehat{\phi}_{S|S_i}}\pi_X^*S^r{\cal E}_0\otimes\pi_{S_i}^*
{\frak
M}_{S_i}.
\end{array}$$
The homomorphism $\widehat{\phi}_{S_i}$
yields a homomorphism
$$\overline{\kappa}_{S_i}\colon S^rV\otimes\O_{S_i\times X}\longrightarrow
\pi_X^*S^r{\cal E}_0(m)\otimes\pi_{S_i}^*{\frak M}_{S_i}$$
and hence a homomorphism
$$\kappa_{S_i}:=\pi_{S_i}^*\pi_{S_i*}(\overline{\kappa}_{S_i})\colon S^rV
\otimes\O_{S_i\times X}
\longrightarrow S^rH^0({\cal E}_0(m))\otimes \pi^*_{S_i}{\frak M}_{S_i};$$
here we have used the fact that our definition of a family implies that the map
$$\pi_{S_i*}(\overline{\kappa}_{S_i})\colon S^rV\otimes\O_{S_i}\longrightarrow
H^0(S^r{\cal E}_0(m))\otimes{\frak M}_{S_i}$$
factorizes through $S^rH^0({\cal E}_0(m))\otimes {\frak
M}_{S_i}$.
By Remark~\ref{MorphtoN}, the quadruple
$(\overline{f}_i,\varepsilon_{S_i},\kappa_{S_i},{\frak M}_{S_i})$ determines a morphism
$\beta_i\colon S_i\longrightarrow {\frak P}$.
It is clear that the morphism
$\beta_i$ factorizes through ${\frak B}$ and that the family
$({\frak E}_{S_i},\varepsilon_{S_i}, \widehat{\phi}_{S_i}, {\frak M}_{S_i})$ is the
pullback of
the universal family by $\beta_i\times \mathop{\rm id}_X$.
The family $({\frak E}_{S_i},\varepsilon_{S_i}, \widehat{\phi}_{S_i},{\frak M}_{S_i})$
is equivalent to $({\frak E}_{S|S_i},\varepsilon_{S|S_i},\widehat{\phi}_{S|S_i},
{\frak M}_{S|S_i})$ by construction.
\end{pf}
Let ${\frak B}^{\mathop{\rm iso}}$ be the open subscheme of oriented pairs
$[q,\varepsilon,\phi]$ for which
$H^0(q(m))$ is an isomorphism. The maps constructed in
the above proof factorize through ${\frak B}^{\mathop{\rm iso}}$.
\begin{Prop}
\label{GlueTog}
Let $S$ be a noetherian scheme and let $\beta_i\colon S\longrightarrow {\frak
B}^{\mathop{\rm iso}}$,
$i=1,2$,
be two morphisms such that the pullbacks of $({\frak E}_{\frak B},
\varepsilon_{\frak B},
\widehat{\phi}_{\frak B},{\frak M}_{\frak B})$ via the maps
$(\beta_i\times\mathop{\rm id}_X)$
are equivalent families. Then there exists an \'etale cover $\eta\colon
T\longrightarrow S$ and a
morphism $g\colon T\longrightarrow \mathop{\rm SL}(V)$ such that
$\beta_1\circ\eta=(\beta_2\circ\eta)\cdot g.$
\end{Prop}
\begin{pf}
Denote the two families by $({\frak E}_S^i,\varepsilon_S^i,\widehat{\phi}_S^i,
{\frak M}_S^i)$,
and let $\Psi_S\colon {\frak E}_S^1\longrightarrow {\frak E}_S^2$ be the corresponding
isomorphism.
The bundles ${\frak E}_S^i$ can be written as quotients
$q_S^i\colon V\otimes \pi_X^*\O_X(-m)\longrightarrow {\frak E}_S^i$, and there is a
morphism $g_S\colon S\longrightarrow \mathop{\rm GL}(V)$ making the following diagramm commutative:
\begin{center}
\unitlength=1mm
\begin{picture}(70,24)(6,9)
\put(0,29){$V\otimes\pi_X^*{\cal O}_X(-m)$}
\put(32,30){\vector(1,0){20}}
\put(37,32){${\scriptstyle g_S\otimes{\rm id}}$}
\put(56,29){$V\otimes\pi_X^*{\cal O}_X(-m)$}
\put(7,25){\vector(0,-1){10}}
\put(64,25){\vector(0,-1){10}}
\put(5,9){${\frak E}^1_S$}
\put(16,10){\vector(1,0){42}}
\put(62,9){${\frak E}^2_S$}
\put(2,20){$\scriptstyle q^1_S$}
\put(35,12){$\scriptstyle \Psi_S$}
\put(66,20){${\scriptstyle q^2_S}$}
\end{picture}
\end{center}
As in the proof of \cite{HL1}, Lemma 1.15, one constructs an
\'etale cover $\eta\colon
T\longrightarrow S$ such that there is a morphism ${\frak d}\colon T\longrightarrow {\Bbb C}^*$ with
$({\frak
d}(t))^p=\det(g_S(\eta(t)))$ for any closed point $t\in T$. Now
define $g:={\frak d}\cdot
(g_S\circ \eta)$. In view of the description of the $\mathop{\rm SL}(V)$-action at the
beginning of
this section, the assertion is obvious.
\end{pf}
\subsection{The GIT-construction}
\label{GITconst}
Let ${\frak A}$ be the union of the finitely many components of $\mathop{\rm Pic}(X)$
meeting the image of $\det_{\frak B}\colon {\frak B}\longrightarrow \mathop{\rm Pic}(X)$.
We may choose $m$ so large that the restriction of the line bundle
$\L_{\vert{\frak
A}\times X}\otimes\pi_X^*\O_X(rm)$ to $\{a\}\times X$ is globally generated
and without
higher cohomology for any closed point $a\in {\frak A}$. The direct image sheaf
$\pi_{{\frak A}_*}(\L_{\vert{\frak A}\times X}\otimes\pi_X^*\O_X(rm))$ is then
locally
free and commutes with base change. The same holds for ${\cal H}om(\bigwedge^r
V\otimes\O_{\frak A},\pi_{{\frak A}*}(\L_{\vert{\frak A}\times X}
\otimes\pi_X^*\O_X(rm)))$; let ${\frak H}$ be the geometric
vector bundle associated to this locally
free sheaf.
Consider the homomorphism
$$\sigma_{\frak N}\colon \bigwedge^rV\otimes
\O_{{\frak N}\times X}\longrightarrow
\det{\frak E}_{\frak N}\otimes
\pi_X^*\O_X(rm)\stackrel{\varepsilon_{\frak N}}{\longrightarrow} \L[{\frak E}_{\frak N}]
\otimes\pi_X^*\O_X(rm).$$
By the
universal property of the scheme ${\frak H}$, the pushforward
$\pi_{{\frak N}*}
(\sigma_{\frak N})$ determines a morphism of schemes
${\frak N}\longrightarrow {\frak H}$ and hence a
morphism
${\frak N}\times S^rK\longrightarrow {\frak H}\times S^rK$. Let ${\frak Z}$ be the
vector bundle
$({\frak H}\times S^rK)^\vee$ over
${\frak A}$, and denote by $\P({\frak Z})$ the associated projective bundle.
$\P({\frak
Z})$ can be polarized by tensorizing ${\cal O}_{\P({\frak Z})}(1)$ with the
pull back
of a very ample line bundle from ${\frak A}$.
On $\P({\frak Z})$ there is a natural action of the group $\mathop{\rm SL}(V)$ from
the right,
which is trivial on the base ${\frak A}$
and admits a canonical linearization in the polarizing line bundle.
We have a natural morphism
$$\iota\colon {\frak B}\hookrightarrow {\frak P}\longrightarrow
\P({\frak Z}) $$
which is equivariant w.r.t.\ the given actions.
Let us describe the effect of $\iota$
on closed points:
Given $b\in {\frak B}$, let $({\cal E}_b,\varepsilon_b,\phi_b)$ be the oriented pair
induced by the restriction of $({\frak E}_{\frak B},\varepsilon_{\frak B},
\widehat{\phi}_{\frak B})$ to $\{b\}\times X$, i.e., ${\cal E}_b$ and $\varepsilon_b$ are the
restrictions of ${\frak E}_{\frak B}$ and $\varepsilon_{\frak B}$ and
$\phi_b$ is
a framing with
$S^r\phi_b=\widehat{\phi}_{{\frak B}\vert \{b\}\times X}$ ($\phi_b$ is
unique up to an
$r$-th root of unity). The point $b$ is
mapped to $[\L[{\cal E}_b], h,S^rk]$ with
$$h\colon \bigwedge^rV\longrightarrow H^0(\det({\cal E}_b)(rm))\textmap{H^0(\varepsilon_b(rm))}
H^0(\L[{\cal E}_b](rm))$$
and $k=H^0((\phi_b\circ q)(m))$.
A point in $\P({\frak Z})$ is $\mathop{\rm SL}(V)$-\it (semi)stable \rm if
it is semistable in the projective space
$\P((\mathop{\rm Hom}(\bigwedge^rV,H^0(\L_{\vert\{a\}\times X}(rm)))\oplus
S^rK)^\vee),$ where
$a$ is its image in ${\frak A}$.
\par
Let ${\frak B}^{ss}$ (${\frak B}^s$)
be the open subscheme of points
$[q,\varepsilon,\phi]$ such that the triple
$({\cal E},\varepsilon,\phi)$ is a semistable (stable)
oriented pair and such that the homomorphism $H^0(q(m))\colon V\longrightarrow
H^0({\cal E}(m))$ is
an isomorphism.
\par
\begin{Thm}
\label{MyStabCrit}
For $m$ large enough,
${\frak B}^{ss}=\iota^{-1}(\P({\frak Z})^{ss})$,
and ${\frak B}^s=\iota^{-1}(\P({\frak Z})^{s})$.
\end{Thm}
Before we can start with the proof, we have to recall some definitions and
results from \cite{HL1} and \cite{HL2}.
Let $({\cal E},\phi)$ be a pair consisting of a torsion free coherent sheaf ${\cal E}$
with $P_{\cal E}=P$
and
a non-trivial framing $\phi$.
\par
Let $\overline{\delta}$ be any positive rational number.
The pair $({\cal E},\phi)$ is called \it sectional (semi)stable w.r.t.\ \rm
$\overline{\delta}$, if there is a
subspace $V\subset H^0({\cal E})$ of dimension $\chi({\cal E})=P(0)$ such that the
following conditions are satisfied:
\begin{enumerate}
\item For all non-trivial submodules ${\cal F}$ of $\ker(\phi)$:
$$(\mathop{\rm rk}{\cal E})\dim \left(H^0({\cal F})\cap V\right)(\le)
\mathop{\rm rk}{\cal F}(\chi({\cal E})-\overline{\delta}).$$
\item For all non-trivial submodules ${\cal F}\neq {\cal E}$:
$$(\mathop{\rm rk}{\cal E})\dim \left(H^0({\cal F})\cap V\right)(\le) \mathop{\rm rk}{\cal F}(\chi({\cal E})-\overline{\delta})+
(\mathop{\rm rk}{\cal E})\overline{\delta}.$$
\end{enumerate}
\par
Then we have the following result \cite{HL2}{, Th.\ 2.1}:
\begin{Thm}
\label{SecStab}
For any polynomial $\delta$, there exists a natural number $m_1$
such that for all $m\ge m_1$
the following conditions are equivalent for a pair $({\cal E},\phi)$:
\par
{\rm i)} $({\cal E},\phi)$ is (semi)stable w.r.t.\ the polynomial $\delta$.\par
{\rm ii)} $({\cal E},\phi)(m)$ is sectional (semi)stable w.r.t.\ $\delta(m)$.
\end{Thm}
Let $(q\colon V\otimes\O_X(-m)\longrightarrow{\cal E},\phi)$ be a pair consisting of
a \sl generically \rm surjective map $q$ of $V\otimes \O_X(-m)$ to a
torsion free sheaf ${\cal E}$ with $P_{\cal E}=P$ and a non-zero homomorphism
$\phi\colon{\cal E}\longrightarrow{\cal E}_0$.
We can associate to this pair an element
$([h], [k])\in\P(H^\vee)\times\P(K^\vee)$, where
$H:=\mathop{\rm Hom}(\bigwedge^rV,H^0(\L[{\cal E}](rm)))$.
There is a natural $\mathop{\rm SL}(V)$-action on
$\P(H^\vee)\times\P(K^\vee)$ which
can be linearized in every sheaf $\O(a_1,a_2)$,
where $a_1$ and $a_2$ are positive integers.
Define $\nu:=a_2/a_1$ and $\overline{\delta}:= p\nu/(\mathop{\rm rk}{\cal E} +\nu)$.
The proof of \cite{HL1}, Proposition 1.18 is valid in any dimension
and yields
the following
\begin{Thm}
\label{StabCrit}
Let $(q\colon V\otimes \O_X(-m)\longrightarrow{\cal E},\phi)$ be as above.
The associated element $([h], [k])$ is (semi)stable
w.r.t.\ the linearization in $\O(a_1,a_2)$ if and only if the
following two conditions are satisfied:
\par
{\rm i)} The homomorphism $H^0(q(m))$ is injective.\par
{\rm ii)} The pair $({\cal E},\phi)(m)$ is sectional (semi)stable w.r.t.\
$\overline{\delta}$.
\end{Thm}
We also need the following obvious observation:
\begin{Lem}
\label{EasyStabCrit}
Let $(q\colon V\otimes\O_X(-m)\longrightarrow {\cal E},\phi)$ be as above.
The following conditions are
equivalent:
\par
{\rm i)} The homomorphism $k=H^0((\phi\circ q)(m))$ is injective.\par
{\rm ii)} The associated element $[k]\in \P(K^\vee)$ is stable.
\end{Lem}
After these preparations, we return to our situation.
Let ${\frak B}_0\subset {\frak B}$ be
the open set of all oriented pairs $[q,\varepsilon,\phi]$
for which ${\cal E}$ is semistable, and define for each polynomial $\delta$
the set ${\frak B}_\delta$ as the open set of oriented pairs $({\cal E},\varepsilon,\phi)$
with $\phi\neq 0$ such that $({\cal E},\phi)$ is semistable w.r.t.\ $\delta$.
The union ${\frak B}^\prime:={\frak B}_0\cup\bigcup {\frak B}_\delta$
is quasi-projective,
hence quasi-compact, so that there exist finitely many polynomials, say,
$\delta_1$,...,$\delta_s$ with
${\frak B}^\prime={\frak B}_0\cup {\frak B}_{\delta_1}\cup\cdots\cup
{\frak B}_{\delta_s}$.
Let $M$ be some constant.
By \cite{Ma}{, Theorem 1.7}, the set of points $b \in {\frak B}$
such that $\mu_{\max}({\frak E}_{{\frak B}\vert\{b\}\times X})\le M$
is open. Since ${\frak B}$ is quasi-compact,
there is a constant $\mu_0$ such that
$\mu_{\max}({\frak E}_{{\frak B}\vert\{b\}\times X})\le \mu_0$ for all
$b\in {\frak B}$.
We also know that the family ${\frak Ker}$ of kernels of framings of semistable
oriented pairs
is bounded. It follows that $\mu_{\max}(\ker(\phi))$, for $\ker(\phi)\in
{\frak Ker}$,
can only take finitely many values.
As in \cite{HL2}, Lemma 2.7, this implies that there are only finitely many
polynomials of the form $P_{\ker(\phi)_{\max}}$.
In particular, there are only finitely many polynomials of the form
$$P_{\cal E}-(\mathop{\rm rk}{\cal E}/\mathop{\rm rk}\ker(\phi)_{\max})P_{\ker(\phi)_{\max}}.$$
We assume in the following that these polynomials are among
$\delta_1,...,\delta_s$, and
that the chosen $m$ is large enough, so that Theorem~\ref{SecStab} holds
for all
$\delta_i$ and
set $\overline{\delta}_i:=\delta_i(m)$.
\begin{Thm}
\label{TheProp}
Suppose $m$ is sufficiently large. Let $[q,\varepsilon,\phi]\in {\frak B}$
be a pair with $\phi\neq 0$
which is not (semi)stable.
Then there is no positive rational number
$\overline{\delta}$ such that $({\cal E},\phi)(m)$ is sectional (semi)stable
w.r.t.\ $\overline{\delta}$.
\end{Thm}
\begin{pf}
Denote by ${\frak S}$ the bounded set of equivalence classes of pairs
$({\cal E},\phi)$
for which there is an element $[q,\varepsilon,\phi]\in {\frak B}$.
\par
By the above, any pair $({\cal E},\phi)\in {\frak S}$ satisfies
$\mu_{\max}({\cal E})\le\mu_0$.
Let $\tilde\delta$ be a rational polynomial of degree $\dim X-1$
whose leading coefficient $\tilde{\delta}_0$ satisfies
$\mu({\cal E})+\tilde{\delta}_0\ge \max\{\,0,\mu_0\,\}$.
One can now copy the proof of \cite{HL2}{, page 305}, to show that
there is a constant $C$ such that for any submodule $(\tilde{{\cal E}},\tilde{\phi})$
of a pair $({\cal E}, \phi)\in {\frak S}$ either
$\vert \deg(\tilde{{\cal E}})-\mathop{\rm rk}\tilde{{\cal E}}\mu({\cal E})\vert <C$,
or for all $m$ large enough
\begin{eqnarray*}
{h^0(\tilde{{\cal E}}(m))\over \mathop{\rm rk}\tilde{{\cal E}}}-
{\tilde{\delta}(m)\over \mathop{\rm rk}\tilde{{\cal E}}}&<&
{P_{\cal E}(m)\over \mathop{\rm rk}{\cal E}}-
{\tilde{\delta}(m)\over \mathop{\rm rk}{\cal E}}\qquad
\hbox{if $\tilde{{\cal E}}\not\subset \ker(\phi)\ ,$}\\
\\
{h^0(\tilde{{\cal E}}(m))\over \mathop{\rm rk}\tilde{{\cal E}}} &<&{P_{\cal E}(m)\over \mathop{\rm rk}{\cal E}}-
{\tilde{\delta}(m)\over
\mathop{\rm rk}{\cal E}}\qquad\hbox{otherwise.}\\
\end{eqnarray*}
Recall that a submodule $\tilde{{\cal E}}\subset {\cal E}$ is called \it saturated\rm ,
if the quotient ${\cal E}/\tilde{{\cal E}}$ is torsion free.
The family of saturated submodules $\tilde{{\cal E}}$ of modules ${\cal E}$
with $({\cal E},\phi)\in {\frak S}$
satisfying $\vert \deg(\tilde{{\cal E}})-\mathop{\rm rk}\tilde{{\cal E}}\mu({\cal E})\vert <C$ is bounded
(\cite{HL2}{, Lemma 2.7}).
Denote this family by $\tilde{\frak S}$.
There are only finitely many
possibilities for the Hilbert polynomials of those submodules.
Let
$\delta_j^\prime$ be the finite family of polynomials of the form
$P_{\cal E}-(\mathop{\rm rk}{\cal E}/ \mathop{\rm rk}{\cal E}^\prime)P_{{\cal E}^\prime}$ where ${\cal E}^\prime$ is a saturated submodule
of $\ker(\phi)$ for some $({\cal E},\phi)\in \tilde{\frak S}$, and
$\delta_k^{\prime\p}$ be the finite family of polynomials of the form
$(\mathop{\rm rk}{\cal E}^{\prime\p} P_{\cal E}- \mathop{\rm rk}{\cal E} P_{{\cal E}^{\prime\p}})/(\mathop{\rm rk}{\cal E}-\mathop{\rm rk}{\cal E}^{\prime\p})$
where ${\cal E}^{\prime\p}$ is a saturated submodule of a pair $({\cal E},\phi)\in
\tilde{\frak S}$
not contained in the kernel of $\phi$.
We may assume that $\tilde{\delta}$, the $\delta_j^\prime$'s and the
$\delta_k^{\prime\p}$'s
with positive leading coefficients are among $\delta_1,...,\delta_s$.
Next, we choose $m$ large enough, so that
$\tilde{{\cal E}}(m)$ is globally generated and has no higher cohomology for all
$\tilde{{\cal E}}\in \tilde{\frak S}$.
Let $({\cal E},\phi)$ be a pair which is not semistable
w.r.t.\ any of the polynomials $\delta_1,...,\delta_s$.
This is equivalent to $({\cal E},\phi)(m)$ not being sectional semistable w.r.t.\
any of the numbers
$\overline{\delta}_1,...,\overline{\delta}_s$.
Since $({\cal E},\phi)$ is not semistable w.r.t.\ $\tilde{\delta}$, there
is either a saturated submodule ${\cal E}_0^\prime\subset\ker(\phi)$ with
$\delta_{{\cal E}_0^\prime}:=P_{\cal E}-(\mathop{\rm rk}{\cal E}/ \mathop{\rm rk}{\cal E}_0^\prime)P_{{\cal E}_0^\prime}<\tilde{\delta}$, or there
exists a saturated submodule ${\cal E}_0^{\prime\p}\not\subset\ker(\phi)$ such that
$$\delta_{{\cal E}_0^{\prime\p}}:=(\mathop{\rm rk}{\cal E}_0^{\prime\p} P_{\cal E}- \mathop{\rm rk}{\cal E}
P_{{\cal E}_0^{\prime\p}})/(\mathop{\rm rk}{\cal E}-\mathop{\rm rk}{\cal E}_0^{\prime\p})>\tilde{\delta}\ .$$
In the first case suppose that
$\delta_{{\cal E}_0^\prime}$ is minimal and in the second that
$\delta_{{\cal E}_0^{\prime\p}}$ is maximal.
We consider only the first case, since the second can be treated similarly.
If $\delta_{{\cal E}_0^\prime}\le 0$, then we are done.
Otherwise, set $\delta^\prime_{i_0}:=\delta_{{\cal E}_0^\prime}$.
By the minimality of $\delta_{i_0}^\prime$, any submodule ${\cal E}^\prime$ of
$\ker(\phi)$ satisfies
$$(\mathop{\rm rk}{\cal E})\dim H^0({\cal E}^\prime(m)) \le \mathop{\rm rk}{\cal E}^\prime (p-\overline{\delta^\prime}_{i_0}),$$
and for ${\cal E}^\prime={\cal E}_0^\prime$ we have equality.
Since ${\cal E}$ is not sectional semistable w.r.t.\ $\overline{\delta^\prime}_{i_0}$,
there must exist a submodule ${\cal E}^{\prime\p}\not\subset\ker(\phi)$ with
$$(\mathop{\rm rk}{\cal E})\dim H^0({\cal E}^{\prime\p}(m))> \mathop{\rm rk}{\cal E}^{\prime\p}
(p-\overline{\delta^\prime}_{i_0})+(\mathop{\rm rk}{\cal E})\overline{\delta^\prime}_{i_0}.$$
This makes it obvious that $({\cal E},\phi)$ cannot be sectional semistable w.r.t.\
to any positive rational number.\par
We still have to prove the ``stable'' version of the proposition.
For this we enlarge the constant $C$ such that $-C\le -\delta_i^0$, $i=1,...,s$,
where $\delta_i^0$ is the leading coefficient of $\delta_i$.
If $({\cal E},\phi)$ is a pair which is semistable w.r.t.\ the polynomial, say,
$\delta_{i_0}$ but not stable
w.r.t.\ any other polynomial $\delta$, then there must exist submodules
${\cal E}^\prime\subset\ker(\phi)$
and ${\cal E}^{\prime\p}$ belonging to $\tilde{\frak S}$ with
$${P_{{\cal E}^\prime}\over\mathop{\rm rk}{\cal E}^\prime}={P_{{\cal E}}-
\delta_{i_0}\over\mathop{\rm rk}{\cal E}}\quad\hbox{and}\quad {P_{{\cal E}^{\prime\p}}-
\delta_{i_0}\over
\mathop{\rm rk}{\cal E}^{\prime\p}}={P_{{\cal E}^\prime}-\delta_{i_0}\over\mathop{\rm rk}{\cal E}}.$$
Since $m$ was so large that all modules in $\tilde{\frak S}$ are globally
generated and
without higher cohomology, this gives
\begin{eqnarray*}
(\mathop{\rm rk}{\cal E})\dim H^0({\cal E}^\prime(m)) &=& (\mathop{\rm rk}{\cal E}^\prime)(p-\overline{\delta}_{i_0})\\
(\mathop{\rm rk}{\cal E})\dim H^0({\cal E}^{\prime\p}(m)) &=& (\mathop{\rm rk}{\cal E}^{\prime\p})(p-\overline{\delta}_{i_0})+
(\mathop{\rm rk}{\cal E})\overline{\delta}_{i_0},\\
\end{eqnarray*}
and hence the assertion.
\end{pf}
\subsection{Proof of Theorem~\ref{MyStabCrit}}
\label{PfMyStabCrit}
For $b\in {\frak B}$, put $H_b:=\mathop{\rm Hom}(\bigwedge^rV,H^0(\L[{\cal E}_b](rm)))$ and
$\P_b:=\P((H_b\oplus S^rK)^\vee)$. The space
$\P_b$
admits the following natural ${\Bbb C}^*$-action:
$$z\cdot [h,\widehat{k}]:= [h, z\widehat{k}]=[z^{-1} h,\widehat{k}].$$
By \ref{C^*-ex}, this ${\Bbb C}^*$-action can be linearized
in such a way that the quotient is either $\P(H_b^\vee)$, $\P((S^rK)^\vee)$,
or
$\P(H_b^\vee)\times\P((S^rK)^\vee)$
equipped with the
polarization $[\O(a_1,a_2)]$ for any prescribed ratio $a_2/a_1$.
We are now able to apply our GIT-Theorem~\ref{GITThm}
to reduce Theorem~\ref{MyStabCrit}
to Theorem~\ref{StabCrit}.
\par
First we explain the assertion about semistability:
Suppose that $b=[q,\varepsilon,\phi]$ lies in ${\frak B}^{ss}$.
Then either $\phi$ is injective, or ${\cal E}$ is semistable, or $\phi\neq 0$ and
the pair
$({\cal E},\phi)$ is semistable w.r.t.\ some $\delta_i$.
If $\phi$ is injective, we linearize in such a way that we obtain
$\P((S^rK)^\vee)$ as the quotient. By \ref{EasyStabCrit}, the point $[k]$ is
semistable in $\P(K^\vee)$ and hence $[S^rk]$ is semistable in
$\P((S^rK)^\vee)$.
This implies by~\ref{GITThm} that $[h, S^rk]$ is semistable in $\P_b$.
If ${\cal E}$ is semistable, we linearize the ${\Bbb C}^*$-action in such a way that
the quotient $\P_b/\hskip-3pt/{\Bbb C}^*$ is given by $\P(H_b^\vee)$.
By \cite{Gi}, Theorem 0.7 (which does not depend on dimension 2),
the point $[h]$ is then semistable in $\P(H_b^\vee)$, and
hence $[h,S^rk]$ is $\mathop{\rm SL}(V)$-semistable in $\P_b$ by \ref{GITThm}.
If $\phi\neq 0$, $\varepsilon\neq 0$ and $({\cal E},\phi)$ is semistable w.r.t.\
$\delta_i$, we choose the linearization of the ${\Bbb C}^*$-action in
such a way that the quotient is $\P(H_b^\vee)\times \P((S^rK)^\vee)$,
equipped with
a polarization $[\O(ra_1,a_2)]$ satisfying
$(a_2/a_1)= \mathop{\rm rk}{\cal E} \overline{\delta}_i/(p-\overline{\delta}_i)$.
By Theorem~\ref{StabCrit}, $([h], [S^rk])$ is semistable and thus $[h,S^rk]$
is semistable.
\par
Conversely, suppose $[h,S^rk]$ is $\mathop{\rm SL}(V)$-semistable.
By \ref{GITThm}
there is a linearization of the ${\Bbb C}^*$-action such that the image
of $[h,S^rk]$ is $\mathop{\rm SL}(V)$-semistable in the quotient
$\P_b/\hskip-3pt/{\Bbb C}^*$. There are three possible quotients:
If the quotient is $\P((S^rK)^\vee)$, then semistability implies that $[k]$
is semistable in $\P(K^\vee)$ and hence that $k$
is injective. It follows that ${\cal E}$ is a subsheaf of ${\cal E}_0$,
since we may assume that $m$ is so large that $\ker(\phi(m))$ is globally
generated.
If the quotient is $\P(H_b^\vee)$, then
${\cal E}$ is semistable by \cite{Gi}, loc.\ cit..
If the quotient is $\P(H_b^\vee)\times\P((S^rK)^\vee)$ with polarization
$[\O(a_1,a_2)]$, then $({\cal E},\phi)$ is sectional semistable w.r.t.\
$$\overline{\delta}:=p(ra_2/a_1)/(\mathop{\rm rk}{\cal E} +(ra_2/a_1))\ .$$
In view of \ref{SecStab} and \ref{TheProp}, $({\cal E},\phi)$ is semistable
w.r.t.\ some $\delta$,
hence $[q,\varepsilon,\phi]$ lies in ${\frak B}^{ss}$.
\par
We still have to identify the stable points.
As the proof of \ref{GlueTog} shows, the oriented pair $({\cal E},\varepsilon,\phi)$ given by
a point $b=[q,\varepsilon,\phi]\in {\frak B}$ has only finitely many automorphisms if
and only if the associated point $[h,S^rk]\in\P_b$ has a finite
$\mathop{\rm SL}(V)$-stabilizer.
Let $b=[q,\varepsilon,\phi]$ be a point whose associated element $[h,S^rk]$ in
$\P_b$ is stable.
If $h=0$ or $k=0$, then it is easy to see that the corresponding element
$[S^rk]\in\P((S^rK)^\vee)$
or $[h]\in\P(H_b^\vee)$ is stable.
Hence $H^0(q(m))$ is an isomorphism and either $\phi$ is injective or ${\cal E}$
is a stable sheaf.
In both cases, the oriented pair $({\cal E},\varepsilon,\phi)$ is stable and $H^0(q(m))$
is an isomorphism,
in other words $b\in {\frak B}^s$.
If both $h\neq 0$ and $k\neq 0$, then by \ref{GITThm} $([h],[S^rk])\in
\P(H_b^\vee)\times\P((S^rK)^\vee)$
is a polystable point w.r.t.\ the polarization, say, $\O(a_1,a_2)$.
By what we have already proved, $({\cal E},\phi)$ is a semistable pair.
Remark~\ref{properlysemistablepairs} shows that either $({\cal E},\phi)$ is a
stable pair
or there is an $i\in\{\, 1,...,s\,\}$ such that $({\cal E},\phi)$ is polystable
w.r.t.\ $\delta_i$.
In the first case, we are done.
In the second case, the finiteness of the stabilizer of $[h,S^rk]$ implies
that
the oriented pair $({\cal E},\varepsilon,\phi)$ has only finitely many automorphisms, hence
it is a stable oriented pair.
\par
Suppose now that $b\in {\frak B}^s$.
If $\phi=0$, then ${\cal E}$ must be a stable coherent sheaf and thus
$[h]\in\P(H_b^\vee)$
is a stable point.
It follows that $[h,0]$ is a polystable point.
But as $[h,0]$ is a fixed point of the ${\Bbb C}^*$-action, the $\mathop{\rm SL}(V)$-stabilizer
of $[h,0]\in\P_b$
can be identified
with the $\mathop{\rm SL}(V)$-stabilizer of $[h]\in \P(H_b^\vee)$,
so that $[h,0]$ is indeed
a stable point.
If $\varepsilon=0$, then $\phi$ must be injective and we may argue in the same manner.
If both $\varepsilon\neq 0$ and $\phi\neq 0$, it suffices to show that $[h,S^rk]$ is
a polystable point, since its stabilizer is finite by definition.
By the stability of $({\cal E},\phi)$, by the ``stable'' version of \ref{TheProp},
and by the choice
of the $\delta_i$, there exists an index $i\in \{\, 1,...,s\,\}$
such that $({\cal E},\phi)$ is polystable w.r.t.\ $\delta_i$.
This in turn shows that $([h],[S^rk])\in \P(H_b^\vee)\times\P((S^rK)^\vee)$
is polystable w.r.t.\
the linearization in $\O(ra_1,a_2)$ satisfying
$\overline{\delta}_i=p(a_2/a_1)/(\mathop{\rm rk}{\cal E} +(a_2/a_1))$.
\subsection{Moduli spaces of stable oriented pairs}
We need the following proposition
\begin{Prop}
\label{Proper}
The map ${\iota}_{\vert {\frak B}^{ss}}\colon {\frak B}^{ss}\longrightarrow \P({\frak
Z})^{ss}$ is
finite.
\end{Prop}
\begin{pf}
We claim that ${\iota}_{\vert {\frak B}^{ss}}$ is proper and injective.
Injectivity follows by standard arguments.
For the proof of properness, we will make use of the discrete valuative
criterion.
Let $C=\mathop{\rm Spec} R$ be the spectrum of a discrete valuation ring,
$c_0\in C$ the closed point,
and $C_0:=C\setminus \{c_0\}$.
Suppose there is a commutative diagram:
\begin{center}
\unitlength=0.08mm
\begin{picture}(400,400)(0,60)
\put(0,390){$C_0$}
\put(75,400){\vector(1,0){220}}
\put(170,410){${\scriptstyle u}$}
\put(320,390){${\frak B}^{ss}$}
\put(20,350){\vector(0,-1){200}}
\put(340,350){\vector(0,-1){200}}
\put(0,90){$C$}
\put(70,100){\vector(1,0){220}}
\put(320,90){$\P({\frak Z})^{ss}$}
\put(170,120){$\scriptstyle \bar u$}
\put(380,240){$\scriptstyle \iota|_{{\frak B}^{ss}}$}
\multiput(70,140)(80,80){3}{\line(1,1){45}}
\put(250,320){\vector(1,1){36}}
\put(160,280){$\scriptstyle\tilde u$}
\end{picture}
\end{center}
We have to construct a lifting $\tilde{u}$ of the map $\overline{u}$.
By assumption, we are given a family
$({\frak E}_{C_0}, \varepsilon_{C_0},\widehat{\phi}_{C_0},\O_{C_0})$
of semistable oriented pairs over
$C_0\times X$. Note that ${\frak E}_{C_0}$ is
torsion free.
We claim that we can extend the quotient map
$$q_{C_0}\colon V\otimes \pi_X^*\O_X(-m)\longrightarrow
{\frak E}_{C_0}$$
to a homomorphism $q_C\colon V\otimes \pi_X^*\O_X(-m)\longrightarrow {\frak E}_C$
over $C\times X$,
where ${\frak E}_C$ is a flat family of torsion free coherent sheaves
extending ${\frak E}_{C_0}$,
$q_{C}$ extends $q_{C_0}$, and its restriction to $\{c_0\}\times X$ is
generically
surjective.
In order to prove this claim, we first extend the family ${\frak E}_{C_0}$
to a flat family
of quotients
$$\tilde{q}_C\colon V\otimes \pi_X^*\O_X(-m)\longrightarrow \tilde{\frak E}_C.$$
There is a locally free sheaf ${\cal H}$ on $X$ and an epimorphism
$\pi_X^*{\cal H}\longrightarrow \tilde{\frak E}^\vee_C$.
This yields a homomorphism
$$\lambda\colon V\otimes\pi_X^*\O_X(-m)\longrightarrow \tilde{\frak E}_C\longrightarrow \tilde {\frak
E}_C^{\vee\vee}\longrightarrow \pi_X^*{\cal H}^\vee.$$ Let ${\frak E}_C$ be the maximal
subsheaf of
$\pi_X^*{\cal H}^\vee$ with the following properties
$${\frak E}_{C\vert C_0\times X}={\frak E}_{C_0};\qquad \Im\lambda\subset
{\frak
E}_C;\qquad
\dim(\mathop{\rm supp}({\frak E}_C/\Im\lambda))<\dim X.$$
Note that the set of subsheaves of $\pi_X^*{\cal H}^\vee$
having the above properties contains $\Im\lambda$.
One checks that ${\frak E}_{C\vert \{c_0\}\times X}$ is torsion free,
using arguments as in \cite{HL1}, p.85.
Let $t\in R$ be a generator of the maximal ideal.
There is a well defined integer $\alpha$ such that $(t^{\alpha}\varepsilon_{C_0},
t^\alpha\widehat{\phi}_{C_0})$ extends to the family ${\frak E}_C$.
The classifying map to $\P({\frak Z})^{ss}$ induced by the resulting family
$$(q_C\colon V\otimes\pi_X^*\O_X(-m)\longrightarrow {\frak
E}_C,\tilde{\varepsilon}_C,\tilde{\widehat{\phi}}_C,\O_C)$$
is the same as the one induced by
$\overline{u}$. By the various stability criteria we have encountered so far,
it follows
that $H^0(q_{C\vert \{c_0\}\times X}(m))$ is injective and that the triple
$({\frak E}_{C\vert \{c_0\}\times X}, \tilde{\varepsilon}_{C\vert \{c_0\}\times X},
\tilde{\phi}_{c_0})$, where $\phi_{c_0}$ is a framing induced by
$\tilde{\widehat{\phi}}_{C|\{c_0\}\times
X}$, is a semistable oriented pair.
Thus, ${\frak E}_{C\vert \{c_0\}\times X}(m)$ is globally generated
and without higher cohomology, the map $q_{C\vert \{c_0\}\times X}$ is
surjective,
and hence
$q_C\colon V\otimes \pi_X^*\O_X(-m)\longrightarrow {\frak E}_C$ is a
flat family of torsion free quotients.
The family $(q_C\colon V\otimes\pi_X^*\O_X(-m)\longrightarrow {\frak E}_C,\tilde{\varepsilon}_C,
\tilde{\widehat{\phi}}_C,\O_C)$ defines by \ref{LocUnivProp} a morphism
$$\tilde{u}\colon C\longrightarrow {\frak B}^{ss}$$
which extends $u$ by construction.
\end{pf}
By Proposition 2.6.1. and \cite{Gi}, Lemma 4.6, the
quotient ${\frak B}^{ss}/\hskip-3pt/\mathop{\rm SL}(V)$ exists as a projective scheme. We set
$${\cal M}_{(P,\L,{\cal E}_0)}^{ss}:={\frak B}^{ss}/\hskip-3pt/\mathop{\rm SL}(V)\ ,$$
$${\cal
M}_{(P,\L,{\cal E}_0)}^{s}:={\frak B}^s/\hskip-3pt/\mathop{\rm SL}(V)\ .$$
\begin{Thm}
\label{ModuliSpaces}
{\rm i)} There is a natural transformation of functors
$$ \tau\colon M_{(P,\L,{\cal E}_0)}^{ss}\longrightarrow h_{{\cal M}_{(P,\L,{\cal E}_0)}^{ss}},$$
\hspace*{0.5cm} such that for any scheme $\tilde{\cal M}$ and any natural
transformation
of functors
$$\tau^\prime\colon M_{(P,\L,{\cal E}_0)}^{ss}\longrightarrow h_{\tilde{\cal M}}$$
\hspace*{0.5cm}there is a unique morphism $\vartheta\colon {\cal
M}_{(P,\L,{\cal E}_0)}^{ss}\longrightarrow
\tilde{\cal M}$
such that
$\tau^\prime=h(\vartheta)\circ \tau$.\\
\hspace*{0.5cm}{\rm ii)} The space ${\cal M}_{(P,\L,{\cal E}_0)}^s$ is a coarse moduli
space for stable
oriented pairs.
\end{Thm}
\begin{pf}
The existence of the natural transformation is a direct consequence
of Proposition~\ref{LocUnivProp} and \ref{GlueTog}.
The minimality property of ${\cal M}_{(P,\L,{\cal E}_0)}^{ss}$ follows from
the universal property of the categorical quotient.\par
Since ${\frak B}^s$ is contained in the set of $\mathop{\rm SL}(V)$-stable points,
the set of closed points of ${\cal M}_{(P,\L,{\cal E}_0)}^s$ is the set of
equivalence classes of stable oriented pairs which means that
${\cal M}_{(P,\L,{\cal E}_0)}^s$ is a coarse moduli space.
\end{pf}
\vspace{0.3cm}
In our applications [OT2] we shall also need a slightly modified version of the
constructions and results above. We fix a line bundle ${\cal
L}_0\in\mathop{\rm Pic} (X)$ and consider only torsion free sheaves of determinant
isomorphic to
${\cal L}_0$.
More precisely, an ${\cal L}_0$-{\it oriented pair of type
$(P,{\cal E}_0)$} is a triple $({\cal E},\varepsilon,\varphi)$ consisting of
a torsion free coherent sheaf ${\cal E}$ with Hilbert polynomial $P$ and
with $\det{\cal
E}$ isomorphic to ${\cal L}_0$, a homomorphism
$\varepsilon:\det{\cal E}\longrightarrow {\cal L}_0$,
and a homomorphism $\varphi:{\cal E}\longrightarrow {\cal E}_0$.
Equivalence classes of such ${\cal L}_0$-oriented pairs, families,
equivalence classes
of families, (semi)stability and the corresponding functors
$M^{ss}_{(P,{\cal L}_0,{\cal
E}_0)}$ are defined as in 2.1. The same methods as above yield the
following result:
\begin{Thm}
\label{ModuliSpaces}
There exist moduli spaces ${\cal M}^{ss}_{(P,\L_0,{\cal E}_0)}$ and ${\cal
M}^{s}_{(P,\L_0,{\cal E}_0)}$ with the following properties:\\
\hspace*{0.5cm} {\rm i)} There is a natural transformation of
functors
$$ \tau\colon M_{(P,\L_0,{\cal E}_0)}^{ss}\longrightarrow h_{{\cal M}_{(P,\L_0,{\cal E}_0)}^{ss}},$$
\hspace*{0.5cm} such that for any scheme $\tilde{\cal M}$ and any natural
transformation
of functors
$$\tau^\prime\colon M_{(P,\L_0,{\cal E}_0)}^{ss}\longrightarrow h_{\tilde{\cal M}}$$
\hspace*{0.5cm} there is a unique morphism $\vartheta\colon {\cal
M}_{(P,\L_0,{\cal E}_0)}^{ss}\longrightarrow
\tilde{\cal
M}$ such that
$\tau^\prime=h(\vartheta)\circ \tau$.\\
\hspace*{0.5cm} {\rm ii)} The space ${\cal M}_{(P,\L_0,{\cal E}_0)}^s$ is a
coarse moduli space
for
stable
${\cal L}_0$-oriented pairs.
\end{Thm}
\subsection{The closed points of ${\cal M}^{ss}_{(P,\L,{\cal E}_0)}$}
Let $({\cal E},\varepsilon,\phi)$ be a semistable oriented pair of type $(P,\L,{\cal E}_0)$.
If $({\cal E},\varepsilon,\phi)$ is stable, then it defines a closed point in ${\cal
M}_{(P,\L,{\cal E}_0)}^{ss}$. If $({\cal E},\varepsilon,\phi)$ is not stable, then either ${\cal E}$ is a
semistable but not stable coherent sheaf, or $\phi\neq 0$ and there exists a
$\delta\in{\Bbb Q}[x]$, $\delta>0$, such that
$({\cal E},\phi)$ is semistable but not stable w.r.t.\ $\delta$.
In both cases, there is a Harder-Narasimhan filtration
$$0= {\cal E}_0\subset {\cal E}_1\subset\cdots\subset{\cal E}_s={\cal E}$$
of ${\cal E}$, whose associated graded sheaf $\mathop{\rm gr}({\cal E}):=\bigoplus_{i=1}^s
{\cal E}_i/{\cal E}_{i-1}$ inherits a well-defined orientation $\varepsilon_{\mathop{\rm gr}}$ and
a well-defined framing $\phi_{\mathop{\rm gr}}$
from $({\cal E},\varepsilon,\phi)$.
As usual, the resulting object $({\mathop{\rm gr}}({\cal E}),\varepsilon_{\mathop{\rm gr}},
\phi_{\mathop{\rm gr}})$ is determined up to equivalence.
We call it the \it graded object associated to $({\cal E},\varepsilon,\phi)$\rm .
Using the techniques of Section~\ref{PfMyStabCrit}, i.e., applying
\ref{GITThm} in the ``polystable'' version, we reduce the polystability of
$({\mathop{\rm gr}}({\cal E}),\varepsilon_{\mathop{\rm gr}},\phi_{\mathop{\rm gr}})$ to the
respective
results of \cite{HL1}, \cite{HL2}, \cite{Gi}, and \cite{Ma2}.
Finally, one easily adapts the proof in \cite{HL2}, p.312, to show that a
semistable
oriented pair
$({\cal E},\varepsilon,\phi)$ can be deformed into its graded object.
If we call two semistable oriented pairs $({\cal E}_i,\varepsilon_i,\phi_i)$, $i=1,2$,
\it gr-equivalent \rm if
their associated graded objects are equivalent, then
we see that the closed points of ${\cal M}^{ss}_{(P,\L,{\cal E}_0)}$
correspond to gr-equivalence classes of semistable oriented pairs of
type $(P,\L,{\cal E}_0)$.
\subsection{The ${\Bbb C}^*$-action on ${\cal M}^{ss}_{(P,\L,{\cal E}_0)}$}
\label{Flips}
The moduli space possesses a natural ${\Bbb C}^*$-action, given by
$$z\cdot [{\cal E},\varepsilon,\phi]:=[{\cal E},\varepsilon, z\phi]=[{\cal E},z^{-r}\varepsilon,\phi].$$
The set of fixed points of this action can easily be described:
It consists of classes $[{\cal E},0,\phi]$, $[{\cal E},\varepsilon,0]$,
and of classes $[\ker(\phi)_{\max}\oplus{\cal E}^\prime, \varepsilon, \phi]$ with
$0\neq \ker(\phi)_{\max}$.
\par
The ${\Bbb C}^*$-action is naturally linearized in an ample line bundle
coming from the description of ${\cal M}^{ss}_{(P,\L,{\cal E}_0)}$
as GIT-quotient.
This line bundle and the polarization which it represents
may, however, depend on an integer $m$ chosen
in the course of the construction.
Nevertheless, we can state the following result which clarifies
the birational geometry of the moduli spaces ${\cal
M}^{ss}_{\delta}(X;{\cal E}_0,P)$ constructed in \cite{HL2}:
\begin{Thm}
Let $\delta_i\in{\Bbb Q}[x]$, $i=1,2$,
be polynomials with positive leading coefficients.
For a suitable choice of the polarization on ${\cal M}^{ss}_{(P,\L,{\cal E}_0)}$
the following properties hold true:
\par
{\rm i)}
${\cal M}^{ss}_{\delta_i}(X;{\cal E}_0,P)$, $i=1,2$, are ${\Bbb C}^*$-quotients of
the master space ${\cal M}^{ss}_{(P,\L,{\cal E}_0)}$.
\par
{\rm ii)}
${\cal M}^{ss}_{\delta_1}(X;{\cal E}_0,P)$ and ${\cal M}^{ss}_{\delta_2}(X;{\cal E}_0,P)$
are related by a chain of generalized flips.
\end{Thm}
\begin{pf}
Let $m$ be so large that a pair $({\cal E},\phi)$ is
semistable w.r.t.\ $\delta_i$ if and only the pair $({\cal E}(m),\phi(m))$
is sectional semistable w.r.t.\ $\delta_i(m)$, $i=1,2$, and that
all the other requirements needed in the constructions are met.
Then our proof of Theorem~2.4.1 together with the results
of Section~1 easily yields the assertions of the theorem.
\end{pf}
We note that the $\delta_i$ for which the corresponding
set of ${\Bbb C}^*$-stable points meets the fixed point set of the
${\Bbb C}^*$-action, i.e.,
for which the corresponding set of ${\Bbb C}^*$-stable points
contains stable oriented pairs of the type
$[\ker(\phi)_{\max}\oplus{\cal E}^\prime, \varepsilon, \phi]$ with
$0\neq \ker(\phi)_{\max}$ are uniquely determined.
The corresponding polynomial is
$\mathop{\rm rk}{\cal E}^\prime(P_{{\cal E}^\prime}-P_{\ker(\phi)_{\max}}/\mathop{\rm rk}\ker(\phi)_{\max})$.
The associated moduli spaces ${\cal M}_{\delta_i}$ are
those which show up ``at the top'' of the flips.
\newpage
%
\vspace*{1.5cm}
|
1996-07-03T21:01:01 | 9607 | alg-geom/9607003 | en | https://arxiv.org/abs/alg-geom/9607003 | [
"alg-geom",
"math.AG"
] | alg-geom/9607003 | Indranil Biswas | Indranil Biswas | A remark on the jet bundles over the projective line | AMS-Latex file, to appear in Mathematical Research Letters | null | null | null | null | This is a footnote of a recent interesting work of Cohen, Manin and Zagier,
where they, among other things, produce a natural isomorphism between the sheaf
of (n-1)-th order jets of the n-th tensor power of the tangent bundle of a
Riemann surface equipped with a projective structure and the sheaf of
differential operators of order n (on the trivial bundle) with vanishing 0-th
order part. We give a different proof of this result without using the
coordinates, and following the idea of this proof we prove:
Take a line bundle L with $L^2 = T$ on a Riemann surface equipped with a
projective structure. Then the jet bundle $J^n(L^n)$ has a natural flat
connection with $J^n(L^n) = S^n(J^1(L))$. For any $m >n$ the obvious surjection
$J^m(L^n) \rightarrow J^n(L^n)$ has a canonical splitting. In particular,
taking $m = n+1$, one gets a natural differential operator of order $n+1$ from
$L^n$ to $L^{-n-2}$.
| [
{
"version": "v1",
"created": "Wed, 3 Jul 1996 18:56:39 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Biswas",
"Indranil",
""
]
] | alg-geom | \section{Introduction}
Let $X$ be a Riemann surface equipped with a projective
structure (i.e., a covering by coordinate charts such that
the transition functions are of the form $z \longmapsto
(az +b)/(cz+d)$). Let ${\cal L}$ be a line bundle on $X$ such that
${{\cal L}}^2 = T_X$. Let $J^m({{\cal L}}^{\otimes n}) \longrightarrow X$
denote the jet bundle of
order $m$ for the line bundle ${{\cal L}}^{\otimes n}$. For $i \geq j$,
there is a natural restriction homomorphism from
$J^i({{\cal L}}^{n})$ onto $J^j({{\cal L}}^{n})$. We prove that
for any $m\geq n$, the surjective homomorphism
$$
J^m({{\cal L}}^{n}) \, \longrightarrow \, J^n({{\cal L}}^{n})
$$
admits a canonical splitting [Theorem 4.1]. As a consequence,
for each $n\geq 0$ we
construct a differential operator of order
$n$ from ${{\cal L}}^{n -1}$ to ${{\cal L}}^{-n-1}$ whose symbol is the
constant function $1$. Theorem 4.1 follows from the results on the
jet bundles over the projective line established in Section 2.
In \cite{CMZ} certain
differential operators on a Riemann surface
equipped with a projective structure are explicitly constructed
(see (3.1)). As an application
of the set-up we use to prove Theorem 4.1, in Section 3 we derive the
differential operators
constructed in \cite{CMZ}. The present work was inspired by
\cite{CMZ}; in fact, it grew out of attempts to
reconstruct the differential operators there without using
the coordinates.
\section{Constructions on the projective line}
Let $V$ be a two dimensional vector space over ${\Bbb C}$. Let ${\Bbb P}(V)$
denote the projective space given by the space of all one
dimensional quotients of $V$. Define the line bundle
$$
L \, := \, {{\cal O}}_{{\Bbb P}(V)}(1)
$$
on ${\Bbb P}(V)$, whose fiber over the quotient line $[q]$ is
the line $[q]$ itself.
We will recall the definition of the jet bundles for a line bundle.
For a line bundle $\xi$ on a Riemann surface $X$, the $n$-th
jet bundle, denoted by $J^n(\xi)$, is the rank $n+1$ vector
bundle on $X$ whose fiber over $x \in X$ is
$$
{\xi}_x {\otimes}_{{\Bbb C}} ({{\cal O}}_{X,x}/{\bf m}^{n+1}_x)
$$
where ${\xi}_x$ is the fiber of $\xi$ over $x$, and ${{\cal O}}_{X,x}$ is the
ring of functions defined around $x$ with ${\bf m}_x$ being the
maximal ideal consisting of functions vanishing at $x$.
The inclusion of ${\bf m}^{n+1}_x$ in
${\bf m}^n_x$ induces the following short exact
sequence of vector bundle on $X$:
$$
0 \, \longrightarrow \, K^{\otimes n}_X\otimes \xi \,
\longrightarrow \, J^n(\xi) \, \longrightarrow \, J^{n-1}(\xi) \,
\longrightarrow \, 0 \leqno{(2.1)}
$$
where $K_X$ is the canonical bundle of $X$.
Let ${\cal V}$ denote the rank two trivial vector bundle on ${\Bbb P}(V)$ with
$V$ as the fiber. For any $n \geq 0$, $S^n({\cal V})$ will denote the $n$-th
symmetric power of ${\cal V}$, with $S^0({\cal V})$ being the trivial line
bundle.
\medskip
\noindent {\bf Lemma 2.2.}\, {\it For any integer $n \geq 0$, the
vector bundle $J^n(L^n)$ on
${\Bbb P}(V)$ is canonically isomorphic to the symmetric power $S^n({\cal V})$.
For any $m \geq n$, the surjection
$$
J^m(L^n) \, \longrightarrow \, J^n(L^n) \, \longrightarrow \, 0
$$
given by (2.1), admits a canonical splitting (i.e., a homomorphism
from $J^n(L^n)$ to $J^m(L^n)$ such that the composition is identity
on $J^n(L^n)$).}
\medskip
\noindent {\bf Proof.}\, Take any integer $n \geq 0$. Since $S^n(V) =
H^0({\Bbb P}(V), L^n)$, for any $x \in {\Bbb P}(V)$ there is a natural homomorphism
of $S^n(V)$ into the
fiber $J^n(L^n)_x$ given by the restriction of sections to the $n$-th
order infinitesimal neighborhood of $x$. Since for any integer $j$ with
$j\leq n$,
$$
\dim H^0({\Bbb P}(V), L^n \otimes {{\cal O}}_{{\Bbb P}(V)}(-jx)) \, - \,
\dim H^0({\Bbb P}(V), L^n\otimes {{\cal O}}_{{\Bbb P}(V)}(-(j+1)x)) \, = \, 1
$$
the above obtained homomorphism must be an isomorphism.
This proves the first part of the lemma.
Take any integer $m$ such that $m\geq n$. We may restrict a section
of $L^n$ to the $m$-th order infinitesimal neighborhood
of $x$ to get a homomorphism from the vector space $S^n(V)$ (=
$H^0({\Bbb P}(V), L^n)$) to the fiber $J^m(L^n)_x$. Now
using the previous identification of $S^n(V)$ with $J^n(L^n)_x$ we
get the required splitting. $\hfill{\Box}$
\medskip
Setting $m = n+1$ in Lemma 2.2 we obtain the following:
\medskip
\noindent {\bf Corollary 2.3.}\, {\it For any integer
$n \geq 0$, the exact sequence
$$
0\, \longrightarrow \, K^{n+1}_{{\Bbb P}(V)}\otimes L^n \, \longrightarrow
\, J^{n+1}(L^n) \, \longrightarrow \, J^n(L^n)\, \longrightarrow \,0
$$
admits a canonical splitting.}
\medskip
Choose and fix a trivialization of ${\stackrel{2}{\wedge}V}$; this is
equivalent to fixing a nonzero vector $\theta $ in ${\stackrel{2}{\wedge}}V$.
The canonical bundle $K_{{\Bbb P}(V)} = L^{-2}\otimes {\rm det}\, {\cal V}$. Using
the trivialization of ${\stackrel{2}{\wedge}}V$ we have,
$K_{{\Bbb P}(V)} = L^{-2}$.
The sheaf of differential operators of order $k$ from the sections
of a line bundle $\xi$ to the sections of a line bundle $\eta$
is precisely the sheaf ${\rm Hom}(J^k(\xi), \eta)$. Consider
the projection of $J^{n+1}(L^n)$ onto
$K^{n+1}_{{\Bbb P}(V)}\otimes L^n = L^{-n-2}$ defining the splitting in
Corollary 2.3. This gives a global differential operator of order $n+1$,
$$
{{\cal D}}({n+1}) \, \in \, H^0({\Bbb P}(V), {\rm Diff}^{n+1}(L^n,
L^{-n-2})) \leqno{(2.4)}
$$
The symbol of a differential operator in ${\rm Diff}^{n+1}(L^n, L^{-n-2})$
is a section of of the line bundle $T^{n+1}_{{\Bbb P}(V)}\otimes L^{-2n-2} =
{\cal O}_{{\Bbb P}(V)}$.
Since the differential operator ${{\cal D}}(n+1)$ in (2.4) gives a splitting of
the jet sequence -- it's symbol, which is a constant function, must be
the constant function $1$.
Let $SL(V)$ denote the subgroup of $GL(V) = {\rm Aut}(V)$ that acts
trivially
on ${\stackrel{2}{\wedge}}V$. The group $SL(V)$ has a natural action
on ${\Bbb P}(V)$, and ${\rm Aut}({\Bbb P}(V)) = SL(V)/{{\Bbb Z}}_2$. There is a
natural induced
action of $SL(V)$ on any sheaf $J^m(L^n)$ that lifts the action
on ${\Bbb P}(V)$. The isomorphism between $S^n({\cal V})$
and $J^n(L^n)$, and the splitting in Lemma 2.2, are both equivariant for
this action. Indeed, this follows from the
canonical nature of the construction in Lemma 2.2. So, in particular,
the differential operator ${{\cal D}}(n)$ in (2.4) is an invariant for the
action of $SL(V)$ on the space of all global sections of ${\rm
Diff}^n(L^{n-1}, L^{-n-1})$.
Note that ${{\cal D}}(n)$ is not an invariant for the action $GL(V)$ since
the trivialization of ${\stackrel{2}{\wedge}}V$ was used in its
construction. The identification between $K_{{\Bbb P}(V)}$ and $L^{-2}$
is not equivariant for the action of the center of $GL(V)$.
Setting $n=2$ in Lemma 2.2 we get that $J^2(L^2) = S^2({\cal V})$. This
implies that the homomorphism
$$
\rho \, : \, H^0({\Bbb P}(V), J^2(L^2)) \, \longrightarrow \, H^0({\Bbb P}(V), L^2)
$$
induced by the obvious projection, namely $J^2(L^2) \longrightarrow L^2$,
is actually an isomorphism. Moreover, $\rho $ is the identity map of
$S^2(V)$. Thus, after identifying the
tangent bundle $T_{{\Bbb P}(V)}$ with $L^2$ using the trivialization
of ${\stackrel{2}{\wedge}}V$, the Lemma 2.2
implies that
$$
H^0({\Bbb P}(V) , T_{{\Bbb P}(V)}) \, = \, H^0({\Bbb P}(V), J^2(L^2))
\, = \, S^2(V) \leqno{(2.5)}
$$
The Lie-bracket operation equips the vector space
$H^0({\Bbb P}(V), T_{{\Bbb P}(V)})$ with the structure of a Lie algebra. The
action of $SL(V)$ on ${\Bbb P}(V)$ gives a Lie algebra homomorphism from
its Lie algebra, $sl(V)$, into $H^0({\Bbb P}(V), T_{{\Bbb P}(V)})$. This
homomorphism is actually an isomorphism. The Lie
algebra structure on $S^2(V)$ induced by the equality (2.5) can be
seen directly as follows: using contraction, $S^2(V)$ maps
$V^*$ into $V$; on the other hand, $\theta $ identifies $V^*$ with $V$ --
combining these, the resulting homomorphism from $S^2(V)$ into $sl(V)$
is an isomorphism.
Let $C \in S^2(H^0({\Bbb P}(V), T_{{\Bbb P}(V)}))$ be the Casimir of the Lie
algebra $H^0({\Bbb P}(V), T_{{\Bbb P}(V)})$. The section $C$ is evidently
an invariant for the obvious action of $SL(V)$ on the vector
space $S^2(H^0({\Bbb P}(V), T_{{\Bbb P}(V)}))$. For a section $s$ of
$T_{{\Bbb P}(V)}$, let $L_{s}$ denote
the Lie derivative with respect to $s$. The (second order) Lie
derivative with respect to $s{\otimes} s$ is defined to be
$L_{s}\circ L_{s}$. Thus $C$ acts as a differential operator,
denoted by $L_C$, on all vector bundles associated to ${\Bbb P}(V)$. This
differential operator is actually of order zero (i.e., a constant
scalar multiplication).
Using (2.5) and Lemma 2.2 we get that
$$
S^2(H^0({\Bbb P}(V), T_{{\Bbb P}(V)})) \, = \, H^0({\Bbb P}(V) ,S^2(J^2(T_{{\Bbb P}(V)})))
$$
Let ${\bar C} \in H^0({\Bbb P}(V), S^2(J^2(T_{{\Bbb P}(V)})))$ be the element
corresponding to the Casimir $C$; ${\bar C}$ is actually the Casimir for
the Lie algebra $S^2(V)$ (which is the fiber of $J^2(T_{{\Bbb P}(V)})$).
Let $p : J^2(T_{{\Bbb P}(V)}) \longrightarrow T_{{\Bbb P}(V)}$ be the obvious
projection. If (locally)
$$
{\bar C} \, = \, \sum_{i} A_i\otimes A_i \leqno{(2.6)}
$$
where $A_i$ are local sections of $J^2(T_{{\Bbb P}(V)})$,
consider the operator
$$
L_{{\bar C}} \, = \, \sum L_{p(A_i)}\circ L_{p(A_i)}
$$
with $L_{\phi (A_i)}$ being the Lie derivative with respect to the
vector field $p(A_i)$. It is easy to
check that the operator $L_{{\bar C}}$ does not depend upon the choice of the
decomposition of ${\bar C}$, and that $L_{C} = L_{{\bar C}}$.
\section{Jets of the trivial line bundle on the projective line}
Let ${\rm Diff}^n({\cal O} , {\cal O}) = J^n({\cal O})^*$ be the sheaf of differential
operators on the trivial line bundle over ${\Bbb P}(V)$. The symbol map, which
is the dual of the injection in (2.1), gives a surjective homomorphism
$$
\sigma \, : \, {\rm Diff}^n({\cal O}, {\cal O}) \, \longrightarrow \, T^n_{{\Bbb P}(V)}
$$
Let $\gamma $ denote the obvious projection of $J^{n-1}(T^n_{{\Bbb P}(V)})$ onto
$T^n_{{\Bbb P}(V)}$.
For any $n \geq 1$, let $J^n_0({\cal O}) \subset J^n({\cal O})$ be the kernel of
the obvious homomorphism from $J^n({\cal O})$ onto $J^0({\cal O}) = {\cal O}$. This subsheaf
has a canonical splitting given by the constant functions. Define the
subsheaf, ${\rm Diff}^n_0({\cal O}, {\cal O}) := J^n_0({\cal O})^*$,
of ${\rm Diff}^n({\cal O},{\cal O})$.
For a function $f$ on ${\Bbb C}$ and any integer $n \geq 1$, in Proposition 1
(page 4) of \cite{CMZ} (where it is called ${{\cal L}}_{-n}(f)$) the following
differential operator of order $n$ is constructed:
$$
D_{n}(f) \, := \, \sum_ {i=0}^{n-1} {{(2n-i)!}\over {i!(n-i)!(n-i-1)!}}
f^{(i)}{\partial}^{n-i} \leqno{(3.1)}
$$
with $f^{(i)} = {\partial}^{i}f$ being the $i$-th derivative of $f$.
The operator
${D}_{n}$ has the property that for any M\"obius transformation,
$M(z) = (az+b)/(cz+d)$, of ${\Bbb C}{\Bbb P}^1$, the following equality holds:
$$
{D}_{n}(f)\circ M \, = \, {D}_{n}(M_n.(f \circ M)) \leqno{(3.2)}
$$
where $M_n(z) =(cz+d)^{2n}$.
Thus ${D}_{n}$ is a $SL(V)$
equivariant ${{\cal O}}_{{\Bbb P}(V)}$ linear isomorphism (in other words, a
canonical isomorphism)
$$
\phi \, : \, J^{n-1}(T^n_{{\Bbb P}(V)}) \, \longrightarrow \,
{\rm Diff}^n_0({\cal O},{\cal O}) \leqno{(3.3)}
$$
The operator ${D}_{n}$ has the further property that $\sigma \circ \phi =
\gamma $. It is shown in \cite{CMZ} that this splitting condition together
with the automorphic property (3.2) actually determine the operator
${D}_{n}$. In this section we
want to deduce the above result of \cite{CMZ} in the set-up of Section 2.
Take a point $x \in {\Bbb P}(V)$. The long exact sequence of cohomology
for the exact sequence of sheaves on ${\Bbb P}(V)$
$$
0 \, \longrightarrow \, {{\cal O}}(-(n+1).x) \, \longrightarrow \, {\cal O}
\, \longrightarrow \, J^n({\cal O})_x \, \longrightarrow \, 0
$$
gives the equality
$$
J^n_0({\cal O})_x \, = \, H^1({\Bbb P}(V) , {{\cal O}}(-(n+1).x))
$$
where $J^n_0({\cal O})_x$ is the fiber of $J^n_0({\cal O})$ over $x$.
Choose and fix an isomorphism between the two line bundles ${{\cal O}}(x)$
and $L$. Since the fiber ${{\cal O}}(x)_x = T_{{\Bbb P}(V), x} = L^2_x$, fixing
such an isomorphism is equivalent
to fixing a nonzero vector $\omega $ in $L_x$.
Using Serre duality for ${{\cal O}}(-(n+1).x)$, and then
identifying $K_{{\Bbb P}(V)}$ with ${{\cal O}}(-2x)$ using $\omega $, we have
$$
{\rm Diff}^n_0({\cal O}, {\cal O})_x \, = \, H^0({\Bbb P}(V), {\cal O}((n-1).x))
\, = \, H^0({\Bbb P}(V), L^{n-1}) \, = \, S^{n-1}(V) \leqno{(3.4)}
$$
Consider the restriction of sections of $T^n_{{\Bbb P}(V)}$ to the
$(n-1)$-th order infinitesimal neighborhood of $x$, namely
$$
\beta \, : \, S^{2n}(V) \, = \, H^0({\Bbb P}(V), T^n_{{\Bbb P}(V)})
\, \longrightarrow \, J^{n-1}(T^n_{{\Bbb P}(V)})_x \leqno{(3.5)}
$$
which is clearly a surjective homomorphism. Indeed, in the proof
of Lemma 2.2 we saw that $S^{2n}(V)$ surjects onto $J^{2n}(L^{2n})_x
= J^{2n}(T^n_{{\Bbb P}(V)})_x$. We want to identify the kernel of the
homomorphism $\beta $.
The symplectic form on $V$ given by the trivialization of
${\stackrel{2}{\wedge}}V$ identifies $V$ with $V^*$.
Let $v$ be the vector in the kernel of the quotient homomorphism
$V \longrightarrow L_x$ which corresponds to $\omega $ using the symplectic
form on $V$. (This vector $v \in V$ corresponds to the section
of the sheaf ${{\cal O}}(x)$ given by the constant function $1$.)
Consider the homomorphism, $m_v : S^n(V) \longrightarrow S^{2n}(V)$,
defined by multiplication with $v^{\otimes n}$. The inclusion
$m_v$ corresponds to the natural inclusion of $H^0({\Bbb P}(V), {{\cal O}}(n.x))$
into $H^0({\Bbb P}(V), {\cal O}(2n.x))$. The image of $m_v$ is precisely the kernel
of $\beta $ in (3.5).
Consider the homomorphism $i_{\omega } : S^{2n}(V) \longrightarrow S^{n-1}(V)$
given by the contraction with ${\omega }^{\otimes (n+1)}$. (The vector $\omega $
is considered as an element of $V^*$.) This homomorphism
vanishes on the image $m_v(S^n(V))$. Indeed, this follows from the
fact that $\omega (v) = 0$.
Thus using the equality (3.4) and $i_{\omega }$ we have the homomorphism
$$
{\phi }_x \, : \, J^{n-1}(T^n_{{\Bbb P}(V)})_x \, \longrightarrow \,
{\rm Diff}^n_0({\cal O}, {\cal O})_x
$$
It is easy to check that the homomorphism ${\phi }_x$ does not depend
upon the choice of the nonzero vector $\omega \in L_x$. The resulting
homomorphism
$\phi $ from $J^{n-1}(T^n_{{\Bbb P}(V)})$ to ${\rm Diff}^n_0({\cal O},{\cal O})$ satisfies
the condition that $\sigma \circ \phi = \gamma $. The canonical nature of the
construction of $\phi $ ensures that it is equivariant for the action
of $SL(V)$. Since ${\phi }_x$ is an isomorphism, $\phi $ is an isomorphism.
Let $U \subset SL(V)$ be the unipotent subgroup which fixes the
vector $v$. Let
${\frak n}$ be the nilpotent part of the Lie algebra of $U$. Let $N$
denote the unique element in $\frak n$ which maps a preimage of $\omega $
(in $V$) to $v$. For any $0\leq i \leq 2n$, the image of $S^i(V)$ in
$S^{2n}(V)$, for the homomorphism given by the multiplication with
$v^{2n-i}$, is denoted by
$S^{2n}_i(V)$. In this notation, $N$ maps $S^{2n}_{i+1}(V)$ onto
$S^{2n}_i(V)$;
the resulting homomorphism from $S^{i+1}(V)$ onto $S^i(V)$ is
the contraction by $\omega $. From this
it is easy to deduce that any homomorphism from $S^{2n}(V)/m_v(S^n(V))$
to $S^{n-1}(V)$, which is equivariant
for the actions of $N$, must be a scalar multiple of $i_{\omega }$.
Now the condition, $\sigma \circ \phi = \gamma $, uniquely determines
the homomorphism $\phi $.
\section{Jets on a Riemann surface with a projective structure}
Let $X$ be a Riemann surface, not necessarily compact.
A {\it projective structure}
on $X$ is a maximal atlas of holomorphic coordinate charts, $\{U_{\alpha },
f_{\alpha }\}_{\alpha \in I}$, covering $X$, such that any $f_{\alpha }$ maps $U_{\alpha }$
biholomorphically onto some analytic open set in ${\Bbb P}(V)$ and the
transition function $f_{\alpha }\circ f^{-1}_{\beta }$, for any $\alpha , \beta \in I$,
is a restriction of an automorphism of ${\Bbb P}(V)$, \cite{G}, \cite{D},
\cite{T}. We note that any Riemann surface admits a
projective structure, since, from the uniformization theorem, the
universal cover has a natural projective structure. It is know that for
any projective structure, it is possible
to choose a sub-cover such that the transition functions have a compatible
lift to $SL(V)$ (from ${\rm Aut}({\Bbb P}(V))$). Actually, more than
one inequivalent lifts are possible. For a compact Riemann surface,
the set of equivalence classes of lifts correspond to the set of square
roots of the canonical bundle (called
{\it theta characteristics}) \cite{G},
\cite{T}. Henceforth, by a projective structure we will always mean
a lift of the structure group to $SL(V)$.
Let $X$ be a Riemann surface equipped with
a projective structure in the above sense.
Since the natural action of $SL(V)$ on ${\Bbb P}(V)$ lifts to the bundle $L$,
the projective structure gives a line bundle on $X$ associated to $L$.
Let ${\cal L}$ denote this line bundle on $X$. Since the isomorphism
between $L^2$ and $T_{{\Bbb P}(V)}$ is $SL(V)$ equivariant, we have
${{\cal L}}^{\otimes 2} = T_X$.
Since $J^n(L^n)$ on ${\Bbb P}(V)$ is a trivial bundle (Lemma 2.2), it has
a natural flat connection, which is equivariant under the action of
$SL(V)$. We now
have the following consequence of Lemma 2.2, Corollary 2.3 and (2.4):
\medskip
\noindent {\bf Theorem 4.1.}\, {\it For any $n \geq 0$, the jet bundle
$J^n({{\cal L}}^n)$ on
$X$ has a natural flat connection, and $S^n(J^1({\cal L})) = J^n({{\cal L}}^n)$,
with the identification being compatible with the flat connections.
For any $m\geq n$, the natural surjection
$$
J^m({{\cal L}}^n) \, \longrightarrow \, J^n({{\cal L}}^n) \, \longrightarrow \, 0
$$
has a canonical splitting. Setting $m=n+1$, a global differential
operator of order $n+1$
$$
{{\cal D}}_X (n+1) \, \in \, H^0(X, {\rm Diff}^n_X({{\cal L}}^{n}, {{\cal L}}^{-n-2}))
$$
is obtained. The symbol of ${{\cal D}}_X(n)$ is the constant function $1$.
The fibers of $J^2(T_X)$ have the structure of a Lie algebra compatible
with the flat connection on $J^2(T_X)$. The Lie derivative action
of the Casimir
$$
C_X \, \in \, H^0(X, S^2(J^2(T_X)))
$$
on any tensor power of ${\cal L}$ is a multiplication by a constant scalar.}
\medskip
Similarly, since the isomorphism $\phi $ in (3.3) is $SL(V)$ equivariant,
we have an isomorphism of vector bundles on $X$
$$
{\phi }_X \, : \, J^{n-1}(T^n_X) \, \longrightarrow \, {\rm Diff}^n_0({\cal O}, {\cal O})
$$
(${\rm Diff}^n_0 ({\cal O},{\cal O}) \subset {\rm Diff}^n_X({\cal O},{\cal O})$ is the canonical
complement of ${\rm Diff}^0_X({\cal O},{\cal O})$)
such that the composition of the symbol map on ${\rm Diff}^n_X({\cal O}, {\cal O})$
with the isomorphism ${\phi }_X$ is the natural projection
of $J^{n-1}(T^n_X)$ onto $T^n_X$.
|
1996-07-03T13:29:54 | 9607 | alg-geom/9607002 | en | https://arxiv.org/abs/alg-geom/9607002 | [
"alg-geom",
"math.AG"
] | alg-geom/9607002 | Carlos Simpson | Carlos Simpson | A relative notion of algebraic Lie group and applications to $n$-stacks | 83 pages Latex | null | null | null | null | If $S$ is a scheme of finite type over $k=\cc $, let $\Xx /S$ denote the big
etale site of schemes over $S$. We introduce {\em presentable group sheaves}, a
full subcategory of the category of sheaves of groups on $\Xx /S$ which is
closed under kernel, quotient, and extension. Group sheaves which are
representable by group schemes of finite type over $S$ are presentable;
pullback and finite direct image preserve the notions of presentable group
sheaves; over $S=Spec (k)$ then presentable group sheaves are just group
schemes of finite type over $Spec(k)$; there is a notion of connectedness
extending the usual notion over $Spec(k)$; and a presentable group sheaf $G$
has a Lie algebra object $Lie(G $. If $G$ is a connected presentable group
sheaf then $G/Z(G)$ is determined up to isomorphism by the Lie algebra sheaf
$Lie (G)$.
We envision the category of presentable group sheaves as a generalisation
relative to an arbitrary base scheme $S$, of the category of algebraic Lie
groups over $Spec (k)$.
The notion of presentable group sheaf is used in order to define {\em
presentable $n$-stacks} over $\Xx$. Roughly, an $n$-stack is presentable if
there is a surjection from a scheme of finite type to its $\pi_0$ (the actual
condition on $\pi_0$ is slightly more subtle), and if its $\pi_i$ (which are
sheaves on various $\Xx /S$) are presentable group sheaves. The notion of
presentable $n$-stack is closed under homotopy fiber product and truncation.
We propose the notion of presentable $n$-stack as an answer in characteristic
zero for A. Grothendieck's search for what he called ``schematization of
homotopy types''.
| [
{
"version": "v1",
"created": "Wed, 3 Jul 1996 11:33:02 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Simpson",
"Carlos",
""
]
] | alg-geom | \section*{A relative notion of algebraic Lie group and applications to
$n$-stacks}
Carlos Simpson\newline
{\small Laboratoire Emile Picard
(UMR 5580 CNRS) \newline
Universit\'e Paul Sabatier\newline
31062 Toulouse CEDEX, France}
\bigskip
Let ${\cal X}$ be the big etale site of schemes over $k={\bf C}$.
If $S$ is a scheme of finite type over $k$, let ${\cal X} /S$ denote the big
etale site of schemes over $S$.
The goal of this paper is to introduce a full subcategory of the category
of sheaves of groups on ${\cal X} /S$, which we will call {\em the category
of presentable group sheaves} (\S 2), with the following properties.
\newline
1. \, The category of presentable group sheaves contains those group sheaves
which are representable by group schemes of finite type over $S$ (Corollary
\ref{uvw}).
\newline
2. \, The category of presentable group sheaves is closed under kernel,
quotient (by a normal subgroup sheaf which is presentable), and extension
(Theorem \ref{I.1.e}).
\newline
3. \, If $S'\rightarrow S$ is a morphism then pullback takes presentable group
sheaves on $S$ to presentable group sheaves on $S'$ (Lemma \ref{I.1.h}).
\newline
4. \, If $S'\rightarrow S$ is a finite morphism then direct image takes
presentable group sheaves on $S'$ to presentable group sheaves on $S$ (Lemma
\ref{I.1.i}).
\newline
5. \, If $S=Spec (k)$ then presentable group sheaves are just group schemes of
finite type over $Spec (k)$ (Theorem \ref{I.1.m}). In particular if ${\cal G}$ is
a
presentable group sheaf over any $S$ then the pullback to each point $Spec (k
)\rightarrow S$ is an algebraic group.
\newline
6. \, There is a notion of connectedness extending the usual
notion over $Spec(k )$ and compatible with quotients, extensions, pullbacks
and
finite direct images; and a presentable group sheaf ${\cal G}$ has a
largest connected presentable subsheaf ${\cal G} ^0\subset {\cal G}$ which we call the
{\em connected component} (Theorem \ref{I.1.o}). \newline
7. \, A presentable group sheaf ${\cal G}$ has a Lie algebra
object $Lie({\cal G} )$ (Theorem \ref{lmn}) which is a vector sheaf with bracket
operation (see below for a
discussion of the notion of vector sheaf---in the case $S=Spec (k)$ it is
the same thing as a finite dimensional $k$-vector space).
\newline
8. \, If ${\cal G}$ is a connected presentable group sheaf then ${\cal G} /Z({\cal G} )$ is
determined up to isomorphism by the Lie algebra sheaf $Lie ({\cal G} )$ (where
$Z({\cal G}
)$ denotes the center of ${\cal G}$). This is Theorem \ref{abc} below.
\bigskip
We envision the category of presentable group sheaves as a generalisation
relative to an arbitrary base scheme $S$, of the category of algebraic Lie
groups
over $Spec ({\bf C} )$.
We mention here a few questions related to the analogy with classical
algebraic
groups. Property 8 poses an obvious existence problem: given a Lie
algebra object
in the category of vector sheaves, does it come from a presentable
group sheaf with
vector sheaf center? I don't know the answer to this question.
We do know,
however, that $Aut(L)$ is a presentable group sheaf (Lemma \ref{AutLie}).
Another
question is the existence of a ``universal covering'', i.e. a morphism
$\tilde{{\cal G}
}\rightarrow {\cal G}$ surjective with finite kernel such that for any other such
morphism ${\cal F} \rightarrow {\cal G}$ there is a factorization $\tilde{{\cal G} }
\rightarrow
{\cal F} \rightarrow {\cal G}$. There are obvious questions about the generalisation of
the theory of representations to the case of presentable group sheaves.
The first
among these is whether there always exists a faithful representation into
$Aut(V)$
for $V$ a vector sheaf. I suspect that the answer is no, but don't have a
counterexample. For connected group sheaves this problem concerns only the
center, because we always have the adjoint representation of ${\cal G}$ on
$Lie ({\cal G}
)$. Beyond the question of the description of the representations, there
is also
the question of whether a suitable tannakian theory exists, namely given
a group
${\cal G} \subset Aut (V)$, is ${\cal G}$ defined as the stabilizer of some
${\cal G}$-invariant
sub-vector-sheaf $U$ in a tensor power of $V$?
The motivation for introducing presentable group sheaves comes from the
theory of
homotopy types over $Spec ({\bf C} )$, or what Grothendieck
called ``schematization of
homotopy types'' in \cite{Grothendieck}. We will discuss the application to
this theory at the end of the paper---note also that it is explained in
essentially the same way in \cite{kobe} where some applications to
nonabelian de
Rham cohomology are also announced. Briefly, the considerations are as
follows. A
homotopy type over ${\cal X}$ (which we call an ``$n$-stack'') is a presheaf of
topological spaces on ${\cal X}$ satisfying a homotopic descent condition
(``fibrant''
in the terminology of Jardine \cite{Jardine1}, cf \cite{kobe}). This
condition is
the generalisation of the descent condition that goes into the definition of
$1$-stack. An $n$-stack or fibrant presheaf $T$ has homotopy sheaves
as follows.
First, $\pi _0(T)$ is a sheaf of sets on ${\cal X}$. Then for $i\geq 1$ if
$S\in {\cal X}$
and $t\in T(S)$, $\pi _i (T|_{{\cal X} /S},t)$ is a sheaf of groups on ${\cal X} /S$
(abelian if $i\geq 2$). In the fibrant presheaf point of view, these
homotopy
sheaves are the sheafifications of the presheaves which one defines in
the obvious
way. These things satisfy the same sorts of properties as in the homotopy
theory of
spaces. In particular there are notions of homotopy fiber products and (as
special cases) homotopy fibers and loop or path spaces. The homotopy
groups of the
homotopy fiber of a morphism fit into the usual long exact sequence
(and there is
a similar exact sequence for homotopy fiber products in general).
There are also
notions of morphism spaces $Hom (T,T')$ which are spaces or $n$-groupoids
(depending on the point of view) and internal morphism objects
$\underline{Hom}(T,T')$ which are $n$-stacks whose global sections
are the morphism
spaces.
The main particularity of this situation is that $\pi _0(T)$ can be nontrivial
and not just the union a set of points. Because of this, one must consider
basepoints not only in $T(Spec (k ))$ but in $T(S)$ for any scheme $S$
(say, of
finite type) in order to get the full picture of $T$. One is thus lead to
consider sheaves of groups on ${\cal X} /S$.
We would like to define a restricted class of $n$-stacks or
fibrant presheaves of
spaces which we
will call {\em presentable}. We would like this category to be closed under
homotopy fiber products and also under the truncation (or coskeleton)
operations of
eliminating the homotopy groups above a certain level. From these
requirements it
follows that the condition for inclusion in the class of presentable presheaves
of spaces should be expressed solely in terms of the homotopy group sheaves.
From the exact sequences for homotopy fibers or more generally fiber products,
one can see that the category of group sheaves allowable as homotopy
group sheaves
of presentable spaces must be closed under kernel, cokernel and extension.
We would like our allowable group sheaves to be the algebraic Lie
groups when the
base space is $Spec (k )$, and of course for doing anything useful we need
notions of connectedness and an infinitesimal (Lie algebra) picture. These are
the reasons which lead us to look for a notion of sheaf of groups on
${\cal X} /S$ with
the properties listed above.
I should add a note of caution about the terminology, for we propose the
terminology {\em presentable group sheaf} and also {\em presentable $n$-stack}.
If ${\cal G}$ is a sheaf of sets on ${\cal X} /S$ (i.e. a $0$-stack) which happens to have
a group structure, then the condition that ${\cal G}$ be presentable as a $0$-stack
is {\em not} the same as the condition that ${\cal G}$ be a presentable group sheaf
on ${\cal X} /S$. The right way to think of a sheaf of groups is as
corresponding to a $1$-stack which we can denote $K({\cal G} , 1)$ or $B{\cal G}$ (with a
morphism to $S$). From this point of view the terminologies are compatible:
${\cal G}$ is a presentable group sheaf over $S$ if and only if $K({\cal G} , 1)$ is
a presentable $1$-stack.
Let's look more carefully at the reasoning that leads to our definition of
presentable $n$-stack.
What are we going to do with a presentable $n$-stack
$T$? If $W$ is (the $n$-truncation
of) a finite CW complex considered as a constant $n$-stack on ${\cal X}$ then we can
look at the $n$-stack $Hom (W, T)$. This is the {\em nonabelian cohomology of
$W$ with coefficients in $T$}. If $T= K({\cal O} , n)$ is the Eilenberg-MacLane
presheaf with homotopy group sheaf equal to the structure sheaf of rings ${\cal O}$
on ${\cal X}$ in degree $n$, then $\pi _0Hom (W, T)$ is just the cohomology $H^n(W,
{\bf C} )$---or rather the sheaf on ${\cal X}$ represented by this vector space.
Similarly, if $G$ is a group scheme over ${\bf C}$ then for $T=K(G, 1)=BG$ we get
that $Hom (W, G)$ is the moduli stack for flat principal
$G$-bundles or equivalently representations $\pi _1(W)\rightarrow G$.
We hope to
obtain an appropriate mixture of these cases by considering a more general
class
of $n$-stacks $T$. In particular we would like to have a {\em Kunneth formula}
for two CW complexes $V$ and $W$,
$$
\underline{Hom} (U, \underline{Hom}(V,T))=\underline{Hom}(U\times V, T).
$$
One can imagine for example the problem of trying to compute the moduli stack of
flat principal $G$-bundles on $U\times V$ in terms of a Kunneth formula as
above. One is forced to consider the cohomology of $U$ with coefficients in the
moduli stack $T'=\underline{Hom}(V,BG)$, and this stack is not necessarily
connected ($\pi _0(T')$ is roughly speaking the moduli space of principal
$G$-bundles).
The Kunneth formula is not an end in itself, as it is rare for a
space to decompose into a product. It points the way to a
``Leray-Serre theory''
which could be more generally useful. If $W\rightarrow U$ is a morphism we would
be led to consider a relative morphism stack $T'=\underline{Hom}(W/U, T)
\rightarrow
U$ and then try to take the $n$-stack of sections $U\rightarrow T'$, a sort of
{\em nonabelian cohomology with twisted coefficients}. I haven't fully
thought
about this yet (and in particular not about the de Rham
theory---see below---which
seems to be significantly more complicated than that which is needed in the
constant coefficient case, for example to make sense of the Kunneth formula).
The motivation for all of this is to be able to do geometric versions of the
nonabelian cohomology in the case where $W$ is, say, a smooth projective
variety. It is announced with some sketches of proofs in \cite{kobe}, how to
get a de Rham version of the morphism space $\underline{Hom}(W_{DR}, T)$ when
$T$ is a presentable $n$-stack. We want of course to have the (analytic)
isomorphism between de Rham and Betti cohomology. Needless to say, this will
not work for an arbitrary $n$-stack $T$ on $X$ (for example if one takes $T=W$
to a constant stack associated to a CW complex which is an algebraic variety
then
there will probably be nothing in $Hom (W_{DR}, W)$ corresponding to the
identity in $Hom (W, W)$). We need to impose conditions on $T$ which
guarantee
that it is reasonably close to the examples $K({\cal O} , n)$ or $K(G, 1)$
given above
(in these cases, the de Rham-Betti isomorphism works as is already well
known).
As a first approach, the condition we want seems to be that the homotopy group
sheaves should be representable by group schemes over the base $S$. In the case
where $T$ is the moduli stack of flat principal $G$-bundles on a space $V$,
encountered above when looking at the Kunneth formula, the $\pi _1$ sheaves are
indeed representable (the moduli stack is an algebraic stack).
Unfortunately the
condition of being representable is not stable under cokernels, but as explained
above this is important if we want our notion of good $n$-stack to be stable
under homotopy fiber products.
Before going directly to the conclusion that we need a category stable under
kernels, cokernels and extensions, we can analyze a bit more precisely just what
is needed. Notice first of all that the algebraic de Rham theory is not
going to
work well in the case of higher cohomology with coefficients in the
multiplicative
group scheme, i.e. when $T= K({\bf G}_m, n)$ for $n\geq 2$. I won't go into the
explanation of that here! Thus, at least for the algebraic de Rham theory we
would like to have an appropriate notion of unipotent abelian group sheaf. Not
yet having come up with a reasonable general theory of this, we can replace this
notion by the (possibly more restrictive) notion of {\em vector sheaf}.
The notion of vector sheaf is explained in \S 4 below.
The reader may actually wish to start by reading this section, since the
theory of
vector sheaves
is in some sense a paradigm, applicable only for abelian group sheaves, of
what we are trying to do in general. The notion of vector sheaf was
introduced by
A. Hirschowitz \cite{Hirschowitz} who called it ``U-coherent sheaf''.
He defined
the category of U-coherent sheaves as the smallest abelian category
of sheaves of
abelian groups containing the coherent sheaves (note that the category of
coherent sheaves is not abelian on the big etale site or any big site).
We take a
more constructive approach, defining the notion of vector sheaf in terms of
presentations, although in the end the two notions are equivalent.
The notion of vector sheaf doesn't work too nicely in characteristic $p>0$,
basically because the Frobenius automorphism of the sheaf ${\cal O}$ is not
linear, so
the linear structure is no longer encoded in the sheaf structure. As
we try in the
beginning of the paper to put off the hypothesis of characteristic zero as long
as possible, and as the notion of vector sheaf comes into the analysis
at a later
stage (the infinitesima study related to properties 7 and 8 listed above), I
have decided not to put the section on vector sheaves at the beginning.
Still, it
is essentially self-contained for the reader who wishes to start there.
In considering the algebraic
de Rham theory we will only be looking at $n$-stacks $T$ with $\pi _i(T,t)$ a
vector sheaf on $S$ for $t\in T(S)$ and $i\geq 2$. What does this mean for
our restriction on $\pi _1(T, t)$? Going back to the question of stability
under fiber products, we see from looking at the long exact homotopy sheaf
sequence that the minimum that is absolutely necessary is that our class of
group
sheaves $G$ be stable under central extension by a vector sheaf. On the other
hand it also must be stable under taking kernels.
One could thus hope to make
good on a {\em minimalist approach} saying that we should look at the
category of
group sheaves generated by representable group sheaves (affine, say---this again
would be needed to make the de Rham theory work), under the operations
of kernel
and central extension by a vector sheaf. A vector sheaf always has a
presentation
as the cokernel of a morphism of {\em vector schemes}, i.e. representable
vector-space objects over the base $S$ (these are sometimes called {\em linear
spaces} in the complex analytic category \cite{Grauert} \cite{Fischer}
\cite{Axelsson-Magnusson}). The most natural approach then is to say,
suppose a
group sheaf $G$ has a presentation as the cokernel of a morphism $F_2
\rightarrow
F_1$ of representable group sheaves, and suppose $E$ is a central
extension of $G$
by a vector sheaf $U$ which is itself the cokernel of a morphism $V_2
\rightarrow
V_1$ of vector schemes. Then try to combine these into a presentation of $E$
with, for example, a surjection $V_1\times F_1\rightarrow E$. The
problem (which
I was not able to solve although I don't claim that it is impossible)
is then to
lift the multiplication of $E$ to an associative multiplication on $V_1\times
F_1$.
As I didn't see how to do this, a slightly more general approach was needed,
wherein we consider groups which have presentations by objects where the
multiplication lifts but not necessarily to a multiplication satisfying the
associativity property. This is the reasoning that leads to the definition of
{\em $S$-vertical morphism:} \, a morphism where one can lift things such as
multiplications in a nice way cf \S 2. We finally come to the definition of
{\em presentable group sheaf} as a group sheaf $G$ which admits a vertical
surjection $X\rightarrow G$ from a scheme of finite type over $S$, and such
that there is a vertical surjection $R\rightarrow X\times _{G}X$ again from
a scheme of finite type over $S$.
One could, on the other hand, take a {\em maximalist approach} and try to
include anything that seems vaguely algebraic. This would mean, for example,
looking at sheaves $G$ such that there are surjections (in the etale sense,
although not necessarily etale morphisms) $X\rightarrow G$ and $R\rightarrow
X\times _GX$ with $X$ and $R$ schemes of finite type over the base $S$. We
call this condition P2. This might also work (in fact it might even be the
case that a P2 group sheaf is automatically presentable). However, I was not
able to obtain a reasonable infinitesimal analysis which could lead, for
example, to the notion of connected component---though again, I don't claim
that
this could never work.
In a similar vein, one might point out that there is a fairly limited range of
situations in which we use the lifting properties going into the definition of
verticality.
I have chosen to state the
condition of verticality in what seems to be the most natural setting,
but this leads to requiring that many more lifting properties be satisfied than
what we actually use. One could rewrite the definition of verticality
to include only those lifting properties that we use afterward. It might be
interesting to see if this change makes any difference in which group sheaves
are allowed as presentable.
All in all, the definitions we give here of presentable group sheaf and of
presentable $n$-stack are first attempts at giving useful and reasonable
notions,
but is is quite possible that they would need to be altered
in the future in view
of applications.
A word about the characteristic of the ground field (or base scheme). While our
aim is to work over a field of characteristic zero, there are certain parts of
our discussion valid over any base scheme, namely
those concerning the abstract method for defining conditions of presentability.
When it comes down to finding conditions which result in a nice theory (and in
particular which result in a theory having the required local structure) we must
restrict ourselves to characteristic zero. It is possible that a variant could
work nicely in positive characteristic, so we will present the first part of the
argument concerning the definition of presentability (which is valid over
any base
scheme), in full generality (\S 1) before specifying in characteristic zero
which
morphisms we want to use in the presentations (\S 2). Actually the definition
given in \S 2 works in any characteristic but we can only prove anything about
local properties in characteristic zero (\S\S 4-9), so it is probably the
``right'' definition only in characteristic zero. With an appropriate
different
definition of verticality (certainly incorporating divided powers) what we do in
these later sections might be made to work in any characteristic.
\subnumero{Notations}
We fix a noetherian ground
ring $k$, for sections 1-3. From section 4 on, we assume that $k$ is an
uncountable field of characteristic zero, and we may when necessary assume that
the ground field is $k={\bf C}$.
Let ${\cal X}$ denote the site of noetherian schemes over $k$ with the etale topology
(this is known as the ``big etale site'').
If $S\in {\cal X}$ then we denote by ${\cal X} /S$ the site of schemes over $S$ (again
with the etale topology).
A {\em sheaf} on ${\cal X}$ means (unless otherwise specified) a sheaf of sets. For
a sheaf of groups, we sometimes use the terminology {\em group sheaf}.
We will confuse notations between an object of ${\cal X}$ and the sheaf it
represents.
Denote by $\ast$ the sheaf on ${\cal X}$ with values equal to the one-point set; it
is represented by $Spec (k)$ (and we can interchange these notations at will).
If $S$ is a sheaf on ${\cal X}$ (most often represented by an object) then we have
the site ${\cal X} /S$ of objects of ${\cal X}$ together with morphisms to $S$.
There is an
equivalence between the notions of sheaf on ${\cal X} /S$ and sheaf on ${\cal X}$ with
morphism to $S$. Since we will sometimes need to distinguish these, we
introduce
the following notations.
If ${\cal F}$ is a sheaf on ${\cal X}$ then its {\em restriction up} to ${\cal X} /S$ is
denoted
by ${\cal F} |_{{\cal X} /S}$, with the formula
$$
{\cal F} |_{{\cal X} /S}(Y\rightarrow S)= {\cal F} (Y).
$$
If ${\cal F}$ is a sheaf on ${\cal X} /S$ then we denote by $Res_{S/\ast}{\cal F}$ the
corresponding sheaf on ${\cal X}$ together with its morphism
$$
Res_{S/\ast}{\cal F} \rightarrow S.
$$
It is defined by the statement that $Res_{S/\ast}{\cal F} (Y)$ is equal to the set of
pairs $(a, f)$ where $a: Y\rightarrow S$ and $f\in {\cal F} (
Y\stackrel{a}{\rightarrow}
S)$. We call this the {\em restriction of ${\cal F}$ from $S$ down to $\ast$}.
More
generally if $S'\rightarrow S$ is a morphism and if ${\cal F}$ is a sheaf on ${\cal X}
/S'$ then we obtain a sheaf $Res _{S'/S}{\cal F}$ on ${\cal X}/S$ called the
{\em restriction of ${\cal F}$ from $S'$ down to $S$}.
The operations of restriction up and restriction down are not inverses: we have
the formula, for a sheaf ${\cal F} $ on ${\cal X} /S$,
$$
Res _{S'/S}({\cal F} |_{{\cal X} /S'}) = {\cal F} \times _SS' .
$$
On the other hand, suppose $p:{\cal F} \rightarrow S'$ is a morphism of sheaves on
${\cal X}/S$. Then we denote by ${\cal F} /S'$ the corresponding sheaf on ${\cal X} /S'$ (the
data of the morphism is implicit in the notation). It is defined by the
statement
that ${\cal F} /S' ( Y\rightarrow S')$ is equal to the set of $u \in {\cal F}
(Y\rightarrow
S)$ such that $p(u)\in S'(Y\rightarrow S)$ is equal to the given morphism
$Y\rightarrow S'$. For another point of view note that there is a tautological
section of $(S'/S)|_{ {\cal X} /S'}$, and ${\cal F} /S'$
is the preimage of this section in ${\cal F} |_{{\cal X} /S'}$.
As a special case we get that if ${\cal F}$ is a sheaf on ${\cal X} = {\cal X} /\ast$ with a
morphism ${\cal F} \rightarrow S$ then we obtain a sheaf ${\cal F} /S$ on ${\cal X} /S$.
The operations
$$
({\cal F} \rightarrow S')\mapsto {\cal F} /S'
$$
from sheaves on ${\cal X} /S$ with morphisms to $S'$ to sheaves on ${\cal X} /S'$,
and
$$
{\cal F} ' \mapsto (Res _{S'/S}{\cal F} ' \rightarrow S'
$$
from sheaves on ${\cal X} /S'$ to sheaves on ${\cal X} /S$ with morphisms to $S'$, are
inverses. For this reason it is often tempting to ignore the strict notational
convention and simply use the same notations for the two objects. This is not
too dangerous except in the last section of the paper where we will try to be
careful.
If a sheaf ${\cal F}$ on ${\cal X}$ is representable by an object $F\in {\cal X}$ and
if $F\rightarrow S$ is a morphism then ${\cal F} /S$ is representable by the same
object $F$ together with its morphism, considered as an object of ${\cal X} /S$.
For this reason we will sometimes drop the notation ${\cal F} /S$ and just denote
this as ${\cal F}$ when there is no risk of confusion (and in fact the attentive
reader will notice that even in the definition two paragraphs ago we have
written
$S'$ when we should have written $S'/S$ in the first sentence...but the second
version would have been impossible because not yet defined...!)
Finally there is another natural operation: suppose $\pi : S'\rightarrow S$ is a
morphism and ${\cal F}$ is a sheaf on ${\cal X} /S'$. Its {\em direct image} is the sheaf
$\pi _{\ast}({\cal F} )$ defined by the statement that
$$
\pi _{\ast}({\cal F}
)(Y\rightarrow S):= {\cal F} (Y\times _SS' \rightarrow S').
$$
This is {\em not} the same thing as the restriction down from $S'$ to $S$.
Think of the case where $S$ is one point and $S'$ is a collection of several
points. The value of $Res_{S'/S}({\cal F} )$ at $S$ is the {\em union} of the values
of ${\cal F}$ over the points in $S'$ whereas the value of $\pi _{\ast}({\cal F} )$ at $S$
is the {\em product} of the values of ${\cal F}$ at the points in $S'$.
\numero{Presentability conditions for sheaves}
We will define several conditions, numbered $P1$, $P2$, $P4({\cal M} )$,
$P5({\cal M} )$ (whereas two other conditions $P3$ and $P3\frac{1}{2}$ will be
defined
later, in \S 2). The last two depend on a choice of a class ${\cal M}$ of morphisms
in ${\cal X}$ subject to certain properties set out below. In the upcoming section
we then specify which class ${\cal M}$ we are interested in (at least in
characteristic zero), the class of {\em vertical morphisms}. Since the
preliminary results depend only on the formal properties of ${\cal M}$ we thought it
might be useful to state them in general rather than just for the class of
vertical morphisms, this is why we have the seeming complication of introducing
${\cal M}$ into the notations for our properties.
We also introduce {\em boundedness conditions} denoted $B1$ and $B2$. These
conditions sum up what is necessary in order to be able to apply Artin
approximation.
Fix a base scheme $S\in {\cal X}$.
In what follows, we work in the category of sheaves on ${\cal X} /S$. Thus a sheaf
is supposed to be on ${\cal X} /S$ unless otherwise specified.
\noindent
{\bf P1.}\,\, We say that ${\cal F}$ is {\em P1} if there is a surjective morphism of
sheaves $X\rightarrow {\cal F}$ where $X$ is represented by a scheme of finite type
over $S$. We may assume that $X$ is affine.
\noindent
{\bf P2.}\,\, We say that ${\cal F}$ is {\em P2} if there are surjective morphisms of
sheaves $X\rightarrow {\cal F}$ and $Y\rightarrow X\times _{{\cal F}}X$ where $X$ and $Y$
are represented
by schemes of finite type over $S$. We may assume that $X$ and $Y$ are affine.
\begin{lemma}
\mylabel{I.t}
If $G$ is a sheaf of groups which is P1, and $G$ acts on a sheaf $F$ which is
P2, then the quotient sheaf $F/G$ is again P2.
\end{lemma}
{\em Proof:}
Choose surjections $\varphi :X\rightarrow F$ and $(p_1,p_2):Y\rightarrow X\times
_FX$. The action is a map $G\times F\rightarrow F$, and
we can choose a surjection $(q_1,q_2):W\rightarrow G\times X$ (with $W$ an
affine scheme, by condition P1 for $G$), such that the action lifts to a map
$m:W\rightarrow X$. There is obviously a surjection $X\rightarrow F/G$. We
have a map
$$
W\times _X Y\rightarrow X\times X
$$
(where the maps used in the fiber product are $m:W\rightarrow X$ and
$p_1:Y\rightarrow X$), defined by
$$
(w,y)\mapsto (q_2(w), p_2(y)).
$$
This map surjects onto the fiber product $X\times _{F/G}X$. It clearly maps
into this fiber product. The map is surjective because if $(x,x')\in X\times
X$ with $g\varphi (x)=\varphi (x')$ then for a point $w$ of $W$ lying above
$(g,x)$ we have $\varphi (m(w))= g\varphi (x)=\varphi (x')$; in particular there
is a point $y$ of $Y$ with $p_1(y)=m(w)$ and $p_2(y)=x'$, so the point
$(w,y)$ maps to $(x,x')$.
Our surjection
$$
W\times _X Y\rightarrow X\times _{F/G}X
$$
now shows that $F/G$ is P2.
\hfill $\Box$\vspace{.1in}
{\em Remark:}
These conditions are independent of base scheme $S$ for finite-type morphisms.
More precisely if $S'\rightarrow S$ is a morphism of finite type and if
${\cal F} '$ is
a sheaf on ${\cal X} /S'$ then denoting by ${\cal F} = Res _{S'/S}{\cal F} '$ its restriction
down to $S$ we have that ${\cal F} $ is $P1$ (resp. $P2$) if and only
if ${\cal F} '$ is $P1$ (resp. $P2$).
\subnumero{Boundedness conditions}
We consider the following boundedness conditions for a sheaves on ${\cal X}$. These
two conditions are designed to contain exactly the information needed to apply
the Artin approximation theorem \cite{Artin}.
\newline {\bf B1.} \,\,
We say that a sheaf ${\cal F}$ is {\em B1} if, for
any $k$-algebra $B$, we have that
$$
\lim _{\rightarrow}{\cal F} (Spec (B')) \rightarrow {\cal F} (Spec (B))
$$
is an isomorphism, where the limit is taken over the subalgebras $B'\subset
B$ which are of finite type over $k$. This is equivalent to the local finite
type
condition of Artin \cite{Artin}.
\noindent
{\bf B2.} \,\, We say that a sheaf ${\cal F}$ is {\em B2} if, for any complete local
ring $A$, we have that the morphism
$$
{\cal F} (Spec (A)) \rightarrow \lim _{\leftarrow} {\cal F} (Spec (A/{\bf m} ^i)
$$
is an isomorphism.
The {\bf Artin approximation theorem} (\cite{Artin}) can now be stated as
follows.
{\em
Suppose ${\cal F}$ is a sheaf of sets which is B1 and B2. If $S=Spec (A)$ is an
affine scheme with point $P\in S$ corresponding to a maximal ideal ${\bf m}
\subset
A$ then for any
$$ \eta \in
\lim _{\leftarrow} {\cal F} (Spec (A/{\bf m} ^i))
$$
and for $i_0\geq 0$ there exists an etale neighborhood $P\in
S' \rightarrow S$ and
an element $\eta ' \in {\cal F} (S')$ agreeing with $\eta$ over $Spec (A/{\bf m}
^{i_0})$.
}
\begin{lemma}
\mylabel{I.t.1}
1.\,\, If ${\cal F}$ and ${\cal G}$ are B1 (resp. B2) and $f,g$ are two morphisms from
${\cal F}$ to ${\cal G}$ then the equalizer is again B1 (resp. B2).
\newline
2.\,\, Suppose ${\cal F}\rightarrow
{\cal G}$ is a surjective morphism of sheaves. If ${\cal F}$ and ${\cal F} \times _{{\cal G}}{\cal F}$
are B1 then ${\cal G}$ is B1.
\end{lemma}
{\em Proof:}
Fix $S=Spec (A)$ and $\{ B_i\}$ our directed system of $A$-algebras. Let $B=
\lim _{\rightarrow}B_i$.
Suppose $\eta \in {\cal G} (B)$. There is a natural morphism
$$
\lim _{\rightarrow} {\cal G} (B_i)\rightarrow {\cal G} (B).
$$
First we prove injectivity. Suppose $\varphi , \psi \in {\cal G} (B_i)$ map to the
same element of ${\cal G} (B)$. We may choose an etale surjection of finite type
$Spec (B'_i)\rightarrow Spec (B_i)$ such that the restrictions $\varphi '$ and
$\psi '$ lift to elements $u,v\in {\cal F} (B_i)$. Their images in ${\cal F} (B')$
give a
point $(u,v)_{B'}$ in ${\cal F} \times _{{\cal G}} {\cal F} (B')$ (here $B':= B\otimes
_{B_i}B'_i$). By the condition B1 for the fiber product, there is a $j\geq i$
such that this point comes from a point $\eta \in {\cal F} \times _{{\cal G}}{\cal F} (B'_j)$.
On the other hand, note that the product ${\cal F} \times {\cal F}$ is B1. The image of
$\eta$ in ${\cal F} \times {\cal F} (B')$ is the same as that of $(u,v)$; and
by the B1 condition for the product, there is $k\geq j$ such that the image of
$\eta$ in ${\cal F} \times {\cal F} (B'_k)$ is equal to the image of $(u,v)$.
In particular, $(u|_{Spec (B'_k)},v|_{Spec (B'_k)})$ lies in the fiber product
${\cal F} \times _{{\cal G}}{\cal F} (B'_k)$. In other words, $u$ and $v$ have the same images
in ${\cal G} (B'_k)$. These images are the restrictions of the original $\varphi ,
\psi$. Since $Spec (B'_k)\rightarrow Spec (B_k)$ is an etale surjection, the
images of $\varphi$ and $\psi $ in ${\cal G} (B_k)$ are the same. This proves the
injectivity.
Now we prove surjectivity.
Then there exists an etale surjection of
finite type
$$
Spec (B')\rightarrow Spec (B)
$$
such that $\eta |_{Spec (B')}$ comes
from an element $\rho \in {\cal F} (B')$. The functor ``etale surjections of
finite type'' is itself B1, so there is an etale $Spec (B'_i)\rightarrow Spec
(B_i)$ inducing $B'$. Then $B'=\lim _{\rightarrow} B'_j$ where $B'_j=
B_j\otimes _{B_i}B'_i$ for $j\geq i$. By the property B1 for ${\cal F}$ there is
some $j$ such that $\rho$ comes from $\rho _j\in {\cal F} ( B'_j)$. We obtain an
element $\eta '_j\in {\cal G} (B'_j)$ mapping to $\eta ':=\eta |_{Spec (B')}$. The
two
pullbacks of $ \eta '$ to $Spec (B'\otimes _BB')$ are equal. Note that
$$
B'\otimes _BB' = \lim _{\rightarrow} B'_k\otimes _{B_k}B'_k,
$$
so by the above injectivity, there is some $k$ such that the two pullbacks of
$\eta _j|_{Spec (B'_k)}$ to $Spec (B'_k\otimes _{B_k}B'_k)$ are equal. Now
the sheaf condition for ${\cal G}$ means that $\eta _j|_{Spec (B'_k)}$ descends to an
element $\eta _k\in {\cal G} (B_k)$. The restriction of $\eta _k$ to $B'$ is equal
to the restriction of $\eta$, so the sheaf condition for ${\cal G}$ implies that
the restriction of $\eta _k$ to $Spec (B)$ is $\eta$.
\hfill $\Box$\vspace{.1in}
{\em Remark:} The direct product of a finite number of B1 (resp. B2) sheaves is
again B1 (resp. B2) so part 1 of the lemma implies that the properties B1 and B2
are maintained under fiber products.
\begin{theorem}
\mylabel{I.t.2}
Suppose ${\cal F}$ is a sheaf which is P2. Then ${\cal F}$ is B1. If the ground field
is uncountable, then ${\cal F}$ is B2.
\end{theorem}
{\em Proof:}
The condition B1 follows from the previous lemma. Indeed, let $X\rightarrow
{\cal F}$ and $R\rightarrow X\times _{{\cal F}}X$ be the morphisms given by the property
P2, with $X$ and $R$ of finite type (in particular, B1). Note that $R\times
_{X\times _{{\cal F}}X}R= R\times _{X\times X}R$ is a scheme of finite type, so the
lemma implies that $X\times _{{\cal F}}X$ is B1; another application of the lemma
then shows that ${\cal F}$ is B1.
For B2, let $S=Spec (A)$ with $A$ a complete local ring, and let $S_n:= Spec
(A/{\bf m}^{n+1})$.
Let $X\rightarrow
{\cal F}$ and $R\rightarrow X\times _{{\cal F}}X$ be the morphisms given by the property
P2, with $X$ and $R$ of finite type over $S$. Schemes of finite type are B2.
We show surjectivity of the map
$$
{\cal F} (S)\rightarrow \lim _{\leftarrow} {\cal F} (S_n).
$$
Suppose $(\varphi _n )$ is a compatible system of elements of ${\cal F} (S_n)$. Let
$$
E_n := \{ x_n \in X(S_n):\;\;\; x_n \mapsto \varphi _n \} .
$$
Note that $E_n$ is a nonempty closed subset of the scheme $X(S_n )$ (that is,
the
scheme whose $Spec (k)$-valued points are $X(S_n)$). Let
$$
E'_n:= \bigcap _{m\geq n} {\rm im}(E_m \rightarrow E_n);
$$
this is an intersection of a decreasing family of nonempty constructible subsets
of $E_n$. Since $k$ is uncountable, this intersection is nonempty. Indeed,
the closures of the images form a decreasing family of closed sets, which
stabilizes by the noetherian property of $E_n$; then within this closed subset,
the dense constructible subsets contain open sets which are complements of
proper closed subsets. The union of countably many proper closed subsets is a
proper subset, so the intersection of the open complements is nonempty. (Note
however that $E'_n$ is not necessarily constructible).
The morphism $E'_{n+1} \rightarrow E' _n$ is surjective. To see this, suppose
$u\in E' _n$. We can consider the subsets
$$
D_m := \{ v\in E_m, \;\; v\mapsto u\} .
$$
These are closed subsets of $E_m$, nonempty by the condition $u\in
E'_n$. Let $D' _{n+1}= \bigcap _{m\geq n+1} {\rm im} (D_m \rightarrow D_{n+1})$.
By the same proof as above, $D'_{n+1}$ is nonempty. But it is contained in
$E'_{n+1}$ and maps to $u\in E'_n$.
The surjectivity of the maps implies that the inverse limit $\lim _{\leftarrow}
E'_n $ is nonempty. It is a subset of $\lim _{\leftarrow} X(S_n)=X(S)$,
consisting of elements mapping to $(\varphi _n)$ in $\lim _{\leftarrow}{\cal F}
(S_n)$. (In fact, this subset is equal to the inverse image of $(\varphi _n)$.)
We obtain an element of $X(S)$, hence an element of ${\cal F} (S)$, mapping to
$(\varphi _n)$. This proves surjectivity. Note that this part of the proof
only used property P1 for ${\cal F}$.
We now prove injectivity. Note that $X\times _{{\cal F}}X$ is P1, so by the proof
above, we obtain surjectivity of the morphism
$$
X\times _{{\cal F}}X(S)\rightarrow \lim _{\leftarrow}X\times _{{\cal F}}X(S_n).
$$
Suppose two elements $u,v\in G(S)$ go to the same element of $G(S_n)$ for all
$n$ (we write this $u_n=v_n$). We can lift them to elements $x,y\in X(S)$, and
we obtain a compatible sequence of elements $(x_n, y_n)\in X\times_{{\cal F}}X
(S_n)$.
By the surjectivity of the above morphism, there is an element $(x',y')\in
X\times _{{\cal F}}X(S)$ with $x'_n=x_n$ and $y'_n=y_n$. The images $u'$ and $v'$
of $x'$ and $y'$ in ${\cal F} (S)$ are equal. By
the B2 property for $X$, this implies that $x'=x$ and $y'=y$, which shows that
$u=v$.
\hfill $\Box$\vspace{.1in}
We have the following Krull-type property.
\begin{lemma}
\mylabel{Krull}
Suppose ${\cal F}$ is a sheaf which is B1 and B2. Then for any scheme $S$ of
finite type the natural morphism is an injection
$$
{\cal F} (S ) \hookrightarrow \prod _{{\rm Art.} \, S'\rightarrow S} {\cal F} (S' )
$$
where the product is taken over $S'\rightarrow S$ which are artinian and of
finite type.
\end{lemma}
{\em Proof:}
Suppose $f,f'\in {\cal F} (S)$ agree over all artinian subschemes. Let ${\cal G} = S
\times _{{\cal F} } S$ be the fiber product where $f$ and $f'$ provide the two
morphisms from $S$ to ${\cal F}$. Then ${\cal G}$ is B1 and B2 (by the remark following
Lemma \ref{I.t.1}). But ${\cal G}$ has a (unique) section over any artinian
$S'\rightarrow S$ and applying B2, B1 and Artin approximation we obtain
sections of ${\cal G}$ over an etale covering of $S$.
\hfill $\Box$\vspace{.1in}
\subnumero{Choice of a class of morphisms ${\cal M}$}
Fix a base scheme $S\in {\cal X}$.
We assume fixed for the rest of this section a subset ${\cal M}\subset
Mor ({\cal X} /S)$ of
morphisms in ${\cal X} /S$, containing the identities and closed under composition
(i.e.
${\cal M}$ is the set of morphism of a subcategory of ${\cal X} /S$) subject to the
following
axioms: \newline
{\bf M1}\,\, If $a$ and $b$ are composable morphisms such that $a$ and $ba$
are in
${\cal M}$, and $a$ is surjective, then $b$ is in ${\cal M}$.
\newline
{\bf M2}\,\, Compatibility with base change: if ${\cal F} \rightarrow {\cal G}$ is an
${\cal M}$-morphism and ${\cal H} \rightarrow {\cal G}$ any morphism, then ${\cal F}
\times _{{\cal G}}{\cal H} \rightarrow {\cal H}$ is an ${\cal M}$-morphism; and conversely if
$a:{\cal F}\rightarrow {\cal G}$ is a morphism such that ${\cal F} \times _{\cal G}
Y\rightarrow Y$ is
in ${\cal M}$ for every $S$-scheme and morphism $Y\rightarrow {\cal G}$, then $a$ is in
${\cal M}$.
\newline
{\bf M3}\,\, Etale morphisms between schemes are in ${\cal M}$.
{\em Remark:}
It follows from these axioms that the direct product of morphisms in ${\cal M}$ is
again a morphism in ${\cal M}$.
In the next section we will specify a certain such subcategory ${\cal M}$, the class
of {\em vertical morphisms}, and show that it satisfies these axioms. But there
may be other interesting examples of such a class of morphisms ${\cal M}$ to
which the
following definitions and lemmas could be applied.
We can now extend our list of presentability properties which refer to the
class ${\cal M}$. We use the notation ${\cal M}$-morphism for morphism lying in ${\cal M}$.
The gap in the numbering is to leave a place for the property $P3$ later.
This property (which is absolute rather than relative to a base scheme $S$)
will come up only at the end of the paper, but it turns out to be more
logical to
number it in between $P2$ and $P4$ (this is the numbering used in \cite{kobe}).
\noindent
${\bf P4({\cal M} )}$\,\, We say that a sheaf ${\cal F}$ is $P4({\cal M} )$ if there exist
surjective ${\cal M}$-morphisms
$$
X\rightarrow {\cal F}
$$
and
$$
R\rightarrow X\times _{{\cal F}}X
$$
with $X$ and $R$ represented by affine schemes of finite type over $S$.
\noindent
${\bf P5({\cal M} )}$\,\,
We say that ${\cal F}$ is $P5({\cal M} )$ if it is $P5({\cal M} )$ and if, in addition, the
structural morphism ${\cal F} \rightarrow S$ is in ${\cal M}$.
\begin{lemma}
\mylabel{I.z.1}
If ${\cal F}$ and ${\cal G}$ are $P4({\cal M} )$ (resp. $P5({\cal M} )$) then so is
${\cal F} \times _S{\cal G}$.
\end{lemma}
{\em Proof:} The presentation is just the product of the presentations for
${\cal F}$ and ${\cal G}$.
\hfill $\Box$\vspace{.1in}
\subnumero{Kernels and extensions}
\begin{lemma}
\mylabel{I.1.a}
If $f,g:{\cal G} \rightarrow {\cal H}$ are two morphisms, and if ${\cal G}$ and ${\cal H}$ are
$P4({\cal M} )$,
then the equalizer ${\cal F}$ is $P4({\cal M} )$.
\end{lemma}
{\em Proof:}
Let $X\rightarrow {\cal H}$, $R\rightarrow X\times _{{\cal H}}X$, $Z\rightarrow {\cal G}$ and
$T\rightarrow Z\times _{{\cal G}}Z$ be ${\cal M}$-morphisms with $X$, $R$, $Z$ and $T$
schemes of finite type over $S$. Assume that we have liftings
$f',g': Z\rightarrow X$ of $f$ and $g$. Set
$$
W:= Z\times _{X\times _SX}R.
$$
It is a scheme of finite type over $S$. Note that the composed map
$Z\times _{{\cal G}}
{\cal F} \rightarrow Z\rightarrow X\times _SX$ factors through $X\times _{{\cal H}}X$, and
we have
$$
W= (Z\times _{{\cal G}} {\cal F} )\times _{X\times _{{\cal H}}X}R.
$$
From this and property $M2$, it is clear that the morphism $W\rightarrow {\cal F}$ is
surjective and in ${\cal M}$. Now set
$$
V:= (W\times _SW)\times _{Z\times _SZ}T.
$$
Again, this is of finite type over $S$. We have
$$
W\times _{{\cal F}}W= W\times _{{\cal G}}W = (W\times _SW)\times _{Z\times _SZ}(Z\times
_{{\cal G}}Z).
$$
Therefore
$$
V= (W\times _{{\cal F}} W)\times _{Z\times _{{\cal G}}Z}T.
$$
From this and property $M2$ it is clear that the morphism
$V\rightarrow W\times _{{\cal F}}W$ is surjective and in ${\cal M}$.
We obtain the property $P4({\cal M} )$ for ${\cal F}$.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{I.1.a.1}
If ${\cal F}\rightarrow {\cal H}$ and ${\cal G} \rightarrow {\cal H}$ are two morphisms between
$P4({\cal M} )$
sheaves, then the fiber product ${\cal F} \times _{{\cal H}}{\cal G}$ is $P4({\cal M} )$.
\end{corollary}
{\em Proof:}
The fiber product is the equalizer of the two morphisms ${\cal F} \times _S{\cal G}
\rightarrow {\cal H}$.
\hfill $\Box$\vspace{.1in}
\begin{lemma}
\mylabel{I.1.b}
Suppose ${\cal H}$ is a group sheaf which is $P5({\cal M} )$. If ${\cal H}$ acts freely on a
sheaf ${\cal G}$ with quotient ${\cal F} = {\cal G} /{\cal H}$, then the morphism ${\cal G} \rightarrow
{\cal F}$ is in ${\cal M}$.
\end{lemma}
{\em Proof:}
Make a
base change by a scheme $Y\rightarrow {\cal F}$. Let ${\cal G} '':= {\cal G} \times
_{{\cal F}}Y$. Then ${\cal H}$ acts freely on ${\cal G} ''$ with quotient $Y$. Since the
morphism ${\cal G} '' \rightarrow Y$ is surjective in the etale topology, we may
find an etale morphism (of finite type and surjective) $Y'\rightarrow Y$ such
that the base change ${\cal G} ^3$ of ${\cal G} ''$ to $Y'$ admits a section. Then ${\cal G}
^3= Y'\times _S{\cal H}$. In particular, the morphism ${\cal G} ^3\rightarrow Y'$ is
in ${\cal M}$, hence also the morphism ${\cal G} ^3 \rightarrow Y$. Finally, the morphism
${\cal G} ^3 \rightarrow {\cal G} ''$ is surjective, since $Y'\rightarrow Y$ is
surjective,
and is an ${\cal M}$-morphism because it becomes an etale morphism after
base change to
any scheme. By property $M1$, the morphism ${\cal G} '' \rightarrow Y$ is
in ${\cal M}$; then by $M2$ the morphism ${\cal G} \rightarrow {\cal F}$ is
in ${\cal M}$.
\hfill $\Box$\vspace{.1in}
\begin{lemma}
\mylabel{I.1.c}
Suppose ${\cal G}$ is a $P4({\cal M} )$ sheaf, and suppose $X\rightarrow {\cal G}$ is a
morphism with $X$ a scheme of finite type over $S$.
Then there exists a surjective ${\cal M}$-morphism $R\rightarrow X\times _{{\cal G}}X$
with $R$ a scheme of finite type over $S$.
\end{lemma}
{\em Proof:}
Let $Y\rightarrow {\cal G}$ and $Q\rightarrow Y\times _{{\cal G}}Y$ be the surjective
${\cal M}$-morphisms.
There is an etale surjection $X'\rightarrow X$ such that the lifting
$X'\rightarrow Y$ exists. Note that
$$
X'\times _{{\cal G}} X' = (X' \times _S X')\times _{Y\times _SY}(Y\times _{{\cal G}}Y).
$$
We get that
$$
R:= (X'\times _{{\cal G}}X')\times _{Y\times _{{\cal G}}Y}Q=
(X' \times _S X')\times _{Y\times _SY}Q
$$
is a scheme of finite type. But also the morphism
$$
R= (X'\times _{{\cal G}}X')\times _{Y\times _{{\cal G}}Y}Q\rightarrow
X'\times _{{\cal G}}X'
$$
is in ${\cal M}$, by property $M2$. Finally,
$$
X'\times _{{\cal G}} X' = (X\times _{{\cal G}}X) \times _{X\times _SX} X'\times _SX'
$$
and $X'\times _SX'\rightarrow X\times _SX$ is an ${\cal M}$-morphism by $M3$ and the
remark following the properties $M$. Thus $X'\times _{{\cal G}} X'\rightarrow
X\times
_{{\cal G}} X$ is in ${\cal M}$ (it is also surjective), so the surjection $R\rightarrow
X\times _{{\cal G}}X$ is in ${\cal M}$. \hfill $\Box$\vspace{.1in}
\begin{theorem}
\mylabel{I.1.d}
Suppose ${\cal H}$ is a group sheaf which is $P5({\cal M} )$, and suppose that ${\cal H}$
acts freely
on a sheaf ${\cal G}$ with quotient ${\cal F} = {\cal G} /{\cal H} $.
Then ${\cal F}$ is $P4({\cal M} )$ (resp. $P5({\cal M} )$) if and only if ${\cal G}$ is $P4({\cal M} )$
(resp. $P5({\cal M} )$).
\end{theorem}
{\em Proof:}
By the lemma, the morphism ${\cal G} \rightarrow {\cal F}$ is in ${\cal M}$. If ${\cal G}$ is
$P4({\cal M} )$
then there is a surjective ${\cal M}$-morphism $X\rightarrow {\cal G}$
with $X$ a scheme of finite type over $S$. The morphism $X\rightarrow
{\cal F}$ is then surjective and in ${\cal M}$.
Let $Y\rightarrow {\cal H}$ be a surjective ${\cal M}$-morphism.
Now we have a surjective ${\cal M}$-morphism
$$
X\times _SY\rightarrow X\times _S{\cal H} = X\times _{{\cal F}}{\cal G} ,
$$
and another surjective ${\cal M}$-morphism
$$
X\times _{{\cal F}}{\cal G} \rightarrow {\cal F} \times _{{\cal F}}{\cal G} ={\cal G} .
$$
Apply the previous lemma to the composition of these two morphisms, using the
property $P4({\cal M} )$ of ${\cal G}$. We obtain the existence of a surjective
${\cal M}$-morphism
$$
T\rightarrow (X\times _SY)\times _{{\cal G}} (X\times _SY)
$$
with $T$ a scheme of finite type over $S$.
On the other hand, note that we have a surjective ${\cal M}$-morphism
$$
X\times _{{\cal F}}X\times _S{\cal H}=(X\times _{{\cal F}}{\cal G} )\times _{{\cal G}}
(X\times _{{\cal F}}{\cal G} )\rightarrow X\times _{{\cal F}} X,
$$
and a surjective ${\cal M}$-morphism
$$
(X\times _SY )\times _{{\cal G}}
(X\times _SY )\rightarrow
(X\times _{{\cal F}}{\cal G} )\times _{{\cal G}}
(X\times _{{\cal F}}{\cal G} ).
$$
Composing these three morphisms we obtain a surjective ${\cal M}$-morphism
$$
T\rightarrow X\times _{{\cal F}}X.
$$
This proves that ${\cal F}$ is $P4({\cal M} )$.
Suppose now that ${\cal F}$ is $P4({\cal M} )$. Let
$$
X\rightarrow {\cal F} , \;\;\; R\rightarrow X\times _{{\cal F}} X
$$
be the presentation given by the property $P4({\cal M} )$. We may choose $X$ in
such a way
that there exists a lifting $X\rightarrow {\cal G}$ (the freedom to
replace $X$ by an etale cover comes from Property $M3$ and Lemma
\ref{I.1.c}). This gives an isomorphism $X\times _{{\cal F}}{\cal G}\cong X\times _S{\cal H}$.
Let $$
Y\rightarrow {\cal H} , \;\;\; W\rightarrow Y\times _{{\cal H}}Y
$$
be the presentation given by the property $P4({\cal M} )$ of ${\cal H}$. We obtain
surjective ${\cal M}$-morphisms
$$
X\times _SY\rightarrow X\times _S{\cal H}
$$
and (defining $U:= X\times _SW$)
$$
U:=X\times _SW \rightarrow (X\times _SY)\times _{X\times _S{\cal H} }(X\times _SY).
$$
Put $Z:= X\times _SY$. Then we have surjections in ${\cal M}$
$$
Z\rightarrow X\times _{{\cal F}}{\cal G} \rightarrow {\cal G}
$$
(giving the first part of property $P4({\cal M} )$), and
$$
U\rightarrow Z\times _{X\times _{{\cal F}}{\cal G}}Z.
$$
Now,
$$
(X\times _{{\cal F}}{\cal G} )\times _{{\cal G}} (X\times _{{\cal F}}{\cal G} )=
$$
$$
X\times _{{\cal F}}(X\times _{{\cal F}}{\cal G} )=(X\times _{{\cal F}}X)\times _{{\cal F}}{\cal G} ,
$$
and we have an ${\cal M}$-surjection
$$
R\times _{{\cal F}}{\cal G} \rightarrow (X\times _{{\cal F}}X)\times _{{\cal F}} {\cal G} .
$$
Since $R\rightarrow {\cal F}$ lifts to $R\rightarrow {\cal G}$ we have $R\times
_{{\cal F}}{\cal G}=R\times _S{\cal H}$ and letting $V\rightarrow R\times _SY$ be an etale
surjection (needed for a certain step below), we obtain ${\cal M}$-surjections
$$
V\rightarrow R\times _SY \rightarrow R\times _S{\cal H} \rightarrow (X\times
_{{\cal F}}{\cal G}
)\times _{{\cal G}} (X\times _{{\cal F}}{\cal G} ).
$$
On the other hand,
$$
Z\times _{{\cal G}} Z= Z\times _{X\times _{{\cal F}}{\cal G} }((X\times _{{\cal F}}{\cal G} )
\times _{{\cal G}}
(X\times _{{\cal F}}{\cal G} ))\times _{X\times _{{\cal F}}{\cal G} }Z
$$
so we obtain a surjection in ${\cal M}$
$$
Z\times _{X\times _{{\cal F}}{\cal G} }V\times _{X\times _{{\cal F}}{\cal G} }Z\rightarrow Z\times
_{{\cal G}}Z.
$$
We can assume (by choosing $V$ appropriately) that the morphism
$$
V\rightarrow (X\times _{{\cal F}}{\cal G} )\times
_{{\cal F}} (X\times _{{\cal F}}{\cal G} )
$$
lifts to a morphism
$$
V\rightarrow Z\times_{{\cal F}} Z.
$$
We then have an ${\cal M}$-surjection
$$
U\times _ZV\times _ZU
\rightarrow
(Z\times _{X\times _{{\cal F}}{\cal G} }Z)\times _ZV\times _Z
(Z\times _{X\times _{{\cal F}}{\cal G} }Z)
$$
(where the two maps from $V$ to $Z$ used in the fiber product are the two
projections composed with $V\rightarrow Z\times _{{\cal F}}Z$). Note that the right
hand side is equal to
$$
Z\times _{X\times _{{\cal F}}{\cal G} }V\times _{X\times _{{\cal F}}{\cal G} }Z,
$$
which admits, as we have seen above, an ${\cal M}$-surjection to $Z\times _{{\cal G}}Z$.
Since $U\times _ZV\times _ZU$ is a scheme of finite type over $S$, this
completes the verification of the property $P4({\cal M} )$ for ${\cal G}$.
We have now shown the equivalence of the conditions $P4({\cal M} )$ for ${\cal F}$ and
${\cal G}$.
By the lemma, the morphism ${\cal G} \rightarrow {\cal F}$ is in ${\cal M}$.
By Property $M1$, the structural morphism ${\cal F} \rightarrow S$ is in ${\cal M}$
if and only if the structural morphism ${\cal G} \rightarrow S$ is. Given the
equivalence of the conditions $P4({\cal M} )$, this gives equivalence of the
conditions $P5({\cal M} )$.
\hfill $\Box$\vspace{.1in}
Finally we give a lemma which allows us some flexibility in specifying
resolutions.
\begin{lemma}
\mylabel{I.1.j}
Suppose that $F$ is a sheaf on $S$ with surjective ${\cal M}$-morphisms
$X\rightarrow F$ and $R \rightarrow X\times _F X$ such that $X$ and $R$ are
$P4({\cal M} )$.
Then $F$ is $P4({\cal M} )$.
\end{lemma}
{\em Proof:}
Let $X'\rightarrow X$ and $Q\rightarrow X'\times _XX'$, and $R'\rightarrow R$
be the ${\cal M}$-surjections given by the hypotheses.
We obtain a surjection $X'\rightarrow F$ in ${\cal M}$. On the other hand,
$R'\rightarrow X\times _FX$ is in ${\cal M}$ and surjective, so
$$
X'\times _XR'\times _XX' = R'\times _{X\times _FX}(X'\times _FX')\rightarrow
X'\times _FX'
$$
is an ${\cal M}$-surjection. But the left side is equal to
$$
(X'\times _XX')\times _{X'}R' \times _{X'}(X'\times _XX')
$$
if we choose (as we may assume is possible) a lifting $R'\rightarrow X'\times
_FX'$ over $X\times _FX$. There is thus a surjection in ${\cal M}$
$$
Q\times _{X'}R'\times _{X'}Q\rightarrow X'\times _XR'\times _XX'.
$$
Composing we get the required
$$
Q\times _{X'}R'\times _{X'}Q\rightarrow X'\times _FX'.
$$
\hfill $\Box$\vspace{.1in}
\subnumero{Stability of the condition $P5({\cal M} )$}
In the following corollary and theorem we will make use of a supplementary
condition on the class ${\cal M}$:
\newline
{\bf M4}\,\, If $f: {\cal F} \rightarrow {\cal G}$
is a surjective morphism of sheaves of groups, then $f$ is in ${\cal M}$.
\begin{corollary}
\mylabel{I.z}
Suppose ${\cal M}$ satisfies condition M4 in addition to the conditions M1-3.
If ${\cal G}$ is a $P4({\cal M} )$ group sheaf then it is also $P5({\cal M} )$.
\end{corollary}
Indeed, M4 applied with ${\cal G} = \{ 1\}=S$ gives that the
structural morphism ${\cal F} \rightarrow S$ for any sheaf of groups, is in ${\cal M}$.
\hfill $\Box$\vspace{.1in}
\begin{theorem}
\mylabel{I.1.e}
Suppose ${\cal M}$ satisfies condition M4 in addition to the conditions M1-3.
Then if
$$
1\rightarrow {\cal F} \rightarrow {\cal E} \rightarrow {\cal G} \rightarrow 1
$$
is an extension of group sheaves and if any two of the elements are
$P5({\cal M} )$, the
third one is too.
\end{theorem}
{\em Proof:} Suppose that ${\cal F}$ is $P5({\cal M} )$. Then ${\cal E}$ is $P5({\cal M} )$ if and
only if ${\cal G}$
is $P5({\cal M} )$ (by applying the previous theorem in view of the fact that ${\cal F}$
acts
freely on ${\cal E}$ with quotient ${\cal G}$). The remaining case is if ${\cal E}$ and ${\cal G}$
are $P5({\cal M} )$. Then by Lemma \ref{I.1.a}, the kernel ${\cal F}$ (which is an
equalizer of
two maps ${\cal E} \rightarrow {\cal G}$) is $P4({\cal M} )$. By the above corollary,
${\cal F}$ is
$P5({\cal M} )$.
\hfill $\Box$\vspace{.1in}
\numero{Lifting properties and verticality}
We now fill in what class of morphisms ${\cal M}$ we would like to use in the theory
sketched above.
We could, of course, take ${\cal M} = {\cal X}$ to be the full set of morphisms of ${\cal X}$.
This might well be a reasonable choice, but I don't see how to get a good
infinitesimal theory in characteristic zero out of this choice. We could
also try,
for example, to take ${\cal M}$ as the class of flat (or maybe smooth) morphism
s. But
then any non-flat group scheme over $S$ would be a counterexample to
property M4,
and as we have seen this property is essential to be able to specify a class of
presentable groups closed under kernels, cokernels and extensions. Thus we have
to work a little harder to find an appropriate class of morphisms.
We say that a morphism of sheaves $a:{\cal F} \rightarrow {\cal G}$, is {\em vertical} (or
{\em $S$-vertical}, if the base needs to be specified), if it satisfies the
following lifting properties for all $n\geq 1$:
Suppose $Y$
is a scheme with $n$ closed subschemes $Y_i\subset Y$, with retractions
$r_i:Y\rightarrow Y_i$---commuting pairwise ($r_ir_j=r_jr_i$)---such that for
$j\leq i$, $r_i$ retracts $Y_i$ to $Y_j\cap Y_i$. Suppose given a morphism
$Y\rightarrow {\cal G}$, and liftings $\lambda _i:Y_i\rightarrow {\cal F}$ such
that $\lambda
_i|_{Y_i\cap Y_j}= \lambda _j|_{Y_i\cap Y_j}$. Then for any $P\in Y$
lying on at
least one of the $Y_i$ there exists an etale neighborhood $P\in Y' \rightarrow
Y$ and a lifting $\lambda : Y' \rightarrow {\cal F}$ which agrees with the given
liftings $\lambda _i|_{Y_i\times _YY'}$ on $Y_i\times _YY'$.
For future reference we call this lifting property $Lift _n(Y; Y_i)$.
\begin{lemma}
\mylabel{I.u.1}
Suppose $f:{\cal F} \rightarrow {\cal G}$ is a morphism of sheaves which are P2.
Then $f$ is vertical if and only if $Lift _n(Y; Y_i)$ holds for all systems
$(Y; Y_i)$ with $Y$ (and hence $Y_i$) artinian.
\end{lemma}
{\em Proof:} Suppose given a system $(Y, Y_i)$ which is not artinian. Choose a
point $y_0$ (in one of the $Y_i$) and try to find a lifting in an etale
neighborhood of $y_0$. We can find liftings on $Y^{(n)}$ (the infinitesimal
neighborhoods of $y_0$) by hypothesis. Using the P2 property of ${\cal F}$ and an
argument similar to that of Theorem \ref{I.t.2}, we can choose a compatible
sequence of liftings. Since ${\cal F}$ is B2 we obtain a lifting over the spectrum
of the complete local ring, then by Artin approximation (using B1) we obtain a
lifting on an etale neighborhood of $y_0$.
\hfill $\Box$\vspace{.1in}
\begin{theorem}
\mylabel{I.u}
We have the following statements: \newline
1. \,\, If ${\cal F} \rightarrow {\cal G}$ is vertical and if ${\cal H} \rightarrow
{\cal G}$ is any morphism of sheaves, then ${\cal F} \times _{{\cal G}}{\cal H} \rightarrow {\cal H}$ is
vertical.
\newline
2. \,\, If $a:{\cal F} \rightarrow {\cal G}$ is a morphism of sheaves such that for any
$S$-scheme $Y$ and morphism $Y\rightarrow {\cal G}$, we have that ${\cal F} \times
_{{\cal G}}Y\rightarrow Y$ is vertical, then $a$ is vertical;
\newline
3. \,\, If $a:{\cal F} \rightarrow {\cal G}$ and $b:{\cal G} \rightarrow {\cal H}$ are two morphisms
which are vertical, then $ba$ is vertical (also the identity is vertical); and
\newline
4. \,\, If $a:{\cal F} \rightarrow {\cal G}$ and $b:{\cal G} \rightarrow {\cal H}$ are two
morphisms such that $a$ and $ba$ are vertical, and $a$ is surjective, then
$b$ is vertical.
\newline
5.\,\, The etale
surjections between schemes are vertical. \newline
6.\,\, Any injective morphism ${\cal F} \hookrightarrow S$ is vertical.
\newline
7.\,\, If $f: {\cal F} \rightarrow {\cal G}$
is a surjective morphism of sheaves of groups, then $f$ is vertical.
\end{theorem}
{\em Proof:}
The lifting property concerns only maps from
schemes to ${\cal G}$, so it obviously satisfies parts 1 and 2. For part 3, just
lift two times successively (for the identity the lifting property is
tautological). For part 4, the proof is by induction on $n$. Keep the
notations
$a$, $b$, ${\cal F}$, ${\cal G}$ and ${\cal H}$ of part 4. Suppose $n=1$. Then we
just have to
note that if we have a lifting $Y_1 \rightarrow {\cal G}$ for $b$, then since $a$ is
surjective, we can lift further to $Y_1\rightarrow {\cal F}$ (locally in the etale
topology). The lifting for $ba$ gives $Y\rightarrow {\cal F}$ and we just
project back
to ${\cal G}$ to get the lifting for ${\cal G}$. This gives the case $n=1$. We may
assume that the present lemma is known when there are strictly fewer than $n$
subschemes. Suppose we have liftings $\lambda _i:
Y_i\rightarrow {\cal G}$; in order to get a lifting $\lambda$, and using the lifting
property for the morphism $ba$, it suffices to choose liftings $\mu _i :
Y_i\rightarrow {\cal F}$ with $\mu _i|_{Y_i\cap Y_j}= \mu _j|_{Y_i\cap Y_j}$. We can
do this by induction. Suppose we have chosen $\mu _1,\ldots , \mu _{k-1}$.
Since $k-1<n$, we know the lemma when there are $k-1$ subschemes; apply the
lifting property for the morphism $a$ with respect to the morphism
$Y_k\rightarrow {\cal G}$, with respect to the subschemes $Y_k\cap Y_i$,
$i=1,\ldots ,
k-1$, and with respect to the liftings $\mu _j|_{Y_k\cap Y_j}$. We obtain a
lifting $\mu _k:Y_k\rightarrow {\cal F}$ such that
$\mu _k|_{Y_k\cap Y_j}=\mu _j|_{Y_k\cap Y_j}$. By induction now we obtain all
of the liftings $\mu _1,\ldots , \mu _n$. The lifting property for $ba$ gives
a lifting $\mu$ and we can set $\lambda := a\mu$. This completes the
verification of part 4.
For the etale surjections (part 5), use the previous lemma. Suppose $i:{\cal F}
\rightarrow S$ is injective (part 6). To verify the lifting property for
$Y\rightarrow S$ we just have to verify that this morphism factors through
$Y\rightarrow {\cal F}$. For this, use the facts that $Y$ retracts onto $Y_1$ (over
$S$) and that the morphism $Y_1\rightarrow S$ factors through $Y_1\rightarrow
{\cal F}$.
Finally we verify $Lift _n(Y, Y_i)$ for the morphism $f:{\cal F} \rightarrow
{\cal G}$ in part 7.
Let $r_i: Y\rightarrow Y_i$ denote the retractions. Suppose given $\mu :
Y\rightarrow {\cal G}$ and $\lambda _i : Y_i \rightarrow {\cal F}$ satisfying the
necessary compatibility conditions. Since $f$ is surjective, we may suppose
that there is a lifting $\sigma : Y\rightarrow {\cal F}$ of $\mu$ (by
restricting to an etale neighborhood in $Y$). We construct inductively $\phi
_i : Y\rightarrow {\cal F}$ lifting $\mu$, with $\phi _i |_{Y_j}=\lambda _j$ for
$j\leq i$. Denote the multiplication operations in ${\cal F}$ or ${\cal G}$ by $\cdot$.
Let $$
h_1:= (\lambda _1\cdot (\sigma |_{Y_1})^{-1})\circ r_1: Y\rightarrow \ker (f).
$$
Put $\phi _1 := h_1\cdot \sigma $.
Then $\phi _1$ restricts to $\lambda _1$ on $Y_1$, and lifts $\mu$. Suppose we
have chosen $\phi _i$. Let
$$
h_{i+1}:= (\lambda _{i+1}\cdot (\phi _i |_{Y_{i+1}})^{-1})\circ r_{i+1}
:Y\rightarrow \ker (f),
$$
and put $\phi _{i+1}:= \phi _i \cdot h_{i+1}$. This lifts $\mu$ because
$h_{i+1}$ is a section of $\ker (f)$. For $j\leq i$, $r_{i+1}$ maps
$Y_j$ to $Y_j\cap Y_{i+1}$, and there $\lambda _{i+1}=\lambda _j$ agrees with
$\phi _i$ so $h_{i+1}|_{Y_j}=1$. We don't destroy the required property for
$j\leq i$. On the other hand, we gain the required property for $j=i+1$, by
construction. This completes the inductive step to construct $\phi _i$.
Finally, the $\phi _n$ is the lifting required for property $Lift _n(Y,Y_i)$.
This completes the proof of part 7.
\hfill $\Box$\vspace{.1in}
From the above results, the class ${\cal M}$ of vertical morphisms satisfies the
axioms M1, M2, M3, {\em and} M4 of the previous section.
This is the principal class ${\cal M}$ to which we will refer, in view of which we
drop ${\cal M}$ from the notation when ${\cal M}$ is the class of vertical morphisms.
Thus
the conditions $P4$ and $P5$ refer respectively to
$P4({\cal M} )$ and $P5({\cal M} )$ with ${\cal M}$ the class of vertical morphisms.
In particular we obtain the results \ref{I.z.1}, \ref{I.1.a}, \ref{I.1.a.1},
\ref{I.1.b}, \ref{I.1.c}, \ref{I.1.d}, \ref{I.z}, and \ref{I.1.e} for
the properties P4 and P5.
We have some further results about $P4$ and $P5$.
\begin{lemma}
\mylabel{I.x}
Suppose that ${\cal F}$ is $P4$.
In the situation of the lifting property $Lift_n(Y; Y_i)$ for the morphism
$X\rightarrow {\cal F}$ given by property $P4$, suppose that $Y$ is the
scheme-theoretic union of $Y_1,\ldots , Y_n$. Then the lifting is unique.
\end{lemma}
{\em Proof:}
In effect, for morphisms $Y\rightarrow X$ with $X$ a scheme, if $Y$ is the
scheme theoretic union of the $Y_i$ then the morphism is determined by its
restrictions to the $Y_i$.
\hfill $\Box$\vspace{.1in}
\begin{proposition}
\mylabel{aaa}
The property of being vertical is stable under base change
of $S$: suppose $p:S'\rightarrow S$ is a morphism of schemes. If $f:{\cal F}
\rightarrow {\cal G}$ is vertical then $p^{\ast}(f):p^{\ast}({\cal F})\rightarrow
p^{\ast}({\cal G} )$ is vertical. Furthermore if ${\cal H} \rightarrow {\cal K}$ is an
$S'$-vertical morphism of sheaves on ${\cal X} /S'$ then the restriction down to $S$,
$$
Res _{S'/S}({\cal H} )\rightarrow Res _{S'/S}({\cal K} )
$$
is $S$-vertical.
\end{proposition}
{\em Proof:}
This follows from the form of the lifting
properties.
\hfill $\Box$\vspace{.1in}
{\em Remark:} We often ignore the notation of ``restriction down'', then the
first part of the proposition states that if ${\cal F} \rightarrow {\cal G} \rightarrow
S$ with the first morphism being $S$-vertical, then
${\cal F} \times _SS'\rightarrow {\cal G} \times _SS'$ is $S'$-vertical.
The last part of
the proposition states that if ${\cal H} \rightarrow {\cal K} \rightarrow S'$ with the
first morphism being $S'$-vertical, then it is also $S$-vertical.
\begin{corollary}
\mylabel{I.1.j.1}
Suppose ${\cal F}$ is a sheaf over $S$, and suppose $S'\rightarrow S$ is a
surjective etale morphism such that ${\cal F} |_{S'}$ is $P4$ over $S'$.
Then ${\cal F}$
is $P4$ over $S$.
\end{corollary}
{\em Proof:}
If $(Y,Y_i)$ is a system for the lifting property over $S$, then their
pullbacks $(Y',Y_i')$ form such a system over $S'$. If a morphism of sheaves
over $S'$ satisfies the lifting property, then we can lift for the system
$(Y', Y'_i)$. This gives a lifting over $Y'$ for the system $(Y,Y_i)$, that
is a lifting etale locally, thus satisfying the lifting property over $S$.
Thus a morphism which is $S'$-vertical is also $S$-vertical. It follows that
${\cal F} |_{S'}$ is $P4({\cal M} )$ over $S$. Now
$$
({\cal F} |_{S'})\times _{{\cal F}} ({\cal F} |_{S'})= {\cal F} |_{S'}\times _{S'} (S'\times
_SS'),
$$
and $S'\times _SS'$ is $P4({\cal M} )$ over $S$. Thus
$({\cal F} |_{S'})\times _{{\cal F}} ({\cal F}
|_{S'})$ is $P4({\cal M} )$ over $S$. We can now apply Lemma \ref{I.1.j} with
$X={\cal F} |_{S'}$ and $R=({\cal F} |_{S'})\times _{{\cal F}} ({\cal F}
|_{S'})$.
\hfill $\Box$\vspace{.1in}
\subnumero{Presentable group sheaves}
In view of the nice properties of $P5$ group sheaves, we make the following
change
of notation. A sheaf of groups ${\cal G}$ over ${\cal X} /S$ is a {\em presentable group
sheaf} if it is $P5$.
Note that we use this terminology only for sheaves of groups.
\begin{corollary}
\mylabel{uvw}
A sheaf of groups which is representable by
a scheme of finite type $G$ over $S$, is presentable.
\end{corollary}
{\em Proof:}
This is because we can
take $X=G$ and $R$ equal to the diagonal $G$ in the definition of property
$P4$; and property $P5$ is then Corollary \ref{I.z}.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{vwx}
The category of presentable group sheaves contains the
category generated by representable group sheaves under the operations of
extensions, kernels, and division by normal subgroups.
\end{corollary}
{\em Proof:}
Theorem \ref{I.1.e}.
\hfill $\Box$\vspace{.1in}
In particular, the category of presentable group sheaves is
much bigger than the category of representable group sheaves. I believe that
the category of presentable group sheaves is strictly larger than the category
generated generated by representable group sheaves under the operations of
kernel, cokernel and extension. For example the group sheaves $Aut (V)$ for a
vector sheaf $V$, which are presentable as shown below, are probably not
generated
from representable group sheaves by kernels, extensions and quotients
(although I
don't have a counterexample). In an intuitive sense, however, the two
categories
are about the same.
The two previous corollaries would also hold for the category of $P5({\cal M} )$
group
sheaves for any class ${\cal M}$ satisfying $M1$ through $M4$.
We now give the main argument where we use the lifting properties and the notion
of verticality, i.e. the special definition of ${\cal M}$.
\begin{lemma}
If ${\cal G}$ is a sheaf of groups and $X\rightarrow {\cal G}$ is a vertical
surjection, with identity section $e:S\rightarrow X$ then (choosing a point
$P$ on $e(S)$) there is a lifting of the multiplication to a map of etale germs
$$
\mu : (X,P)\times _S(X,P) \rightarrow (X,P)
$$
such that $\mu (x,e)=\mu (e,x)=x$.
\end{lemma}
{\em Proof:}
Let $Y=X\times _SX$ and $Y_1 = X\times _Se(S)\cong X$ and $Y_2 = e(S)\times _SX
\cong X$.
We have retractions $Y\rightarrow Y_1$ and $Y\rightarrow Y_2$ as in the lifting
property. The multiplication map ${\cal G} \times _S{\cal G} \rightarrow {\cal G}$ composes to
give a map $Y=X\times _SX\rightarrow {\cal G}$. The identity gives liftings
$Y_1\rightarrow X$ and $Y_2\rightarrow X$ agreeing on $Y_1\cap Y_2 = e(S)\times
_S e(S)$. By the definition of verticality of the morphism $X\rightarrow {\cal G}$,
there is an etale neighborhood $P\in Y'\rightarrow Y$ and a lifting to a map
$Y'\rightarrow X$ agreeing with our given lifts on $Y'_1$ and $Y'_2$.
This gives
the desired map (note that when we have written the product of two etale germs,
this means the germ of the product rather than the product of the two spectra of
henselian local rings).
\hfill $\Box$\vspace{.1in}
We use this result in the following way.
A map $\mu : X\times X\rightarrow X$ (defined on germs at a point $P$) such that
$\mu (e,x)=\mu (x,e) = x$, gives rise to an exponential map
$T(X)_e^{\wedge}\rightarrow X$ where $T(X)_e$ is the tangent vector scheme (see
\S\S 5-8 below) to $X$ along the identity section $e$ and $T(X)_e^{\wedge}$
denotes the formal completion at the zero section. To define this exponential
map note that the multiplication takes tangent vectors at $e$ to tangent vector
fields on $X$ which we can then exponentiate in the classical way. The formal
exponential map is an isomorphism between $T(X)_e^{\wedge}$ and the completion
$X^{\wedge}$ along $e$. This is a fairly strong condition on $X$ which we will
exploit below, notably to get $Lie ({\cal G} )$ and to develop a theory of
connectedness.
In particular this technique allows us to prove directly (in \S 6 below) that
when $k$ is a field of characteristic zero, presentable group sheaves over $Spec
(k)$ are just algebraic Lie groups over $k$.
It is possible that in characteristic $p$ there would be an appropriate
notion of
verticality taking into account divided powers, which would have the same effect
of enabling a good infinitesimal theory. This is why we have left the class
${\cal M}$
as an indeterminate in the first part of our discussion above.
\subnumero{The conditions $P3$ and $P3\frac{1}{2}$}
We now add the following two conditions, which will be used as conditions on
$\pi _0$ in the last section (in contrast to the condition $P5$ which is to be
used on $\pi _1$ and even $\pi _i$, $i\geq 2$). These conditions depend on a
functorial choice of class ${\cal M} (Y)$ of morphisms of sheaves over $Y$ for each
$Y\in {\cal X}$. We will leave to the reader the (easy) job of stating these
properties in this generality, and instead we will state them directly when ${\cal M}
(Y)$ is taken as the class of $Y$-vertical morphisms. Note that the
properties we
are about to state are {\em absolute} properties of sheaves on ${\cal X}$ rather than
relative properties of sheaves over some base $S$.
\noindent
{\bf P3.} \,\, A sheaf ${\cal F}$ on ${\cal X}$ is $P3$ if there is a surjection
$X\rightarrow {\cal F}$ from a scheme $X$ of finite type over $Spec (k)$, and
if there
is a surjection $\varphi : R\rightarrow X\times _{{\cal F}}X$ from a scheme $R$ of
finite type over $Spec (k)$ such that $\varphi$ is an $X\times X$-vertical
morphism.
\noindent
${\bf P3\frac{1}{2}.}$ \,\, A sheaf ${\cal F}$ on ${\cal X}$ is $P3\frac{1}{2}$ if
there is a surjection
$X\rightarrow {\cal F}$ from a scheme $X$ of finite type over $Spec (k)$, and
if there
is a surjection $\varphi : R\rightarrow X\times _{{\cal F}}X$ from a scheme $R$ of
finite type over $Spec (k)$ such that $\varphi$ is an $X$-vertical
morphism, where the map to $X$ is the first projection of $X\times _{{\cal F}}X$.
{\em Remark:} These properties seem almost identical. The first was refered to
in \cite{kobe} (already as property $P3$). However it will turn out that the
second version (which I hadn't yet thought of at the time of writing
\cite{kobe})
seems more useful---cf \S 10 below. The author apologizes for this complication
of the notation!
{\em Remark:}
$$
P5 \Rightarrow P4\Rightarrow P3\frac{1}{2}\Rightarrow P3 \Rightarrow P2
\Rightarrow P1.
$$
These properties will not come into our study of group sheaves over a base $S$.
Rather, they come in as conditions on $\pi _0$ of $n$-stacks on ${\cal X}$, in our
brief discussion at the end of the paper. In fact we could have put off stating
these properties until \S 10, but the reader had probably been wondering for
some time already why we are skipping number $3$ in our list of properties.
We quickly give the analogues, for $P3\frac{1}{2}$, of some of the basic
facts about our other properties. We leave to the reader the task of elicudating
the corresponding properties for $P3$.
\begin{lemma}
\mylabel{P3a}
Suppose ${\cal G}$ is $P3\frac{1}{2}$ and suppose $X$ is a scheme of finite type with
a morphism $X\rightarrow {\cal G}$. Then there is a surjection from a scheme of
finite type $R\rightarrow X\times _{{\cal G}} X$ which is vertical with respect to
the first factor $X$.
\end{lemma}
{\em Proof:}
Let $Y\rightarrow {\cal G}$ and $W\rightarrow Y\times _{{\cal G}}Y$ be the surjections
with the second one being vertical with respect to the first factor $Y$.
There is an etale covering $X' \rightarrow X$ and a lifting of our morphism to
$X'\rightarrow Y$. Then
$$
X'\times _{{\cal G}}X'=(X'\times X') \times _{Y\times Y} (Y\times _{{\cal G}}Y)
$$
so
$$
R:=(X' \times X' )\times _{Y\times Y} W \rightarrow X'\times _{{\cal G}}X'
$$
is surjective. It is vertical with respect to the first factor $Y$ and hence
vertical with respect to the first factor $X'$. Since $X'\rightarrow X$
is etale,
this morphism is also vertical with respect to $X$ (via the first factor).
The surjection
$$
X'\times _{{\cal G}}X' \rightarrow X\times _{{\cal G}}X
$$
is the pullback of the etale morphism $X'\times X'\rightarrow X\times X$ so it
is also vertical with respect to the first factor $X$. Composing we obtain
$$
R\rightarrow X\times _{{\cal G}}X
$$
vertical with respect to the first factor.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{P3b}
Suppose ${\cal G}$ is $P3\frac{1}{2}$ and suppose ${\cal F} \subset {\cal G}$. If ${\cal F}$ is $P1$
then it is $P3\frac{1}{2}$.
\end{corollary}
{\em Proof:}
Let $X\rightarrow {\cal F}$ be a surjection from a scheme of finite type $Y$.
From the above lemma we get a surjection $R\rightarrow X\times _{{\cal G}}X$
which is vertical with respect to the first factor, but since ${\cal F}\rightarrow
{\cal G}$ is injective $X\times _{{\cal G}} X=X\times _{{\cal F}}X$ and we're done.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{P3c}
Suppose ${\cal G}$ is $P3\frac{1}{2}$ and ${\cal H}$ is $P2$, then the equalizer
${\cal F}$ of any two morphisms $f,g: {\cal G} \rightarrow {\cal H}$ is again
$P3\frac{1}{2}$.
\end{corollary}
{\em Proof:}
By Lemma \ref{I.1.a} with ${\cal M}$ being the class of all morphisms,
we obtain that ${\cal F}$ is $P2$ and in particular $P1$. Since it is a subsheaf of
${\cal G}$, the previous corollary applies to show that ${\cal F}$ is
$P3\frac{1}{2}$.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{P3d}
Suppose ${\cal F} \rightarrow {\cal H}$ and ${\cal G} \rightarrow {\cal H}$ are two morphisms such
that ${\cal H}$ is $P2$ and ${\cal F}$ and ${\cal G}$ are $P3\frac{1}{2}$. Then the fiber
product ${\cal F} \times _{{\cal H}} {\cal G}$ is $P3\frac{1}{2}$.
\end{corollary}
{\em Proof:}
The fiber product is the equalizer of two morphisms ${\cal F} \times {\cal G}
\rightarrow {\cal H}$. Note that the product of two $P3\frac{1}{2}$ sheaves is
again $P3\frac{1}{2}$---this comes from the general statement that if ${\cal A}
\rightarrow {\cal B}$ is $S$-vertical and if ${\cal A} '\rightarrow
{\cal B}'$ is $S'$-vertical then ${\cal A} \times {\cal A}'\rightarrow {\cal B} \times {\cal B} '$
is $S\times S'$-vertical (a direct consequence of the form of the lifting
properties).
\hfill $\Box$\vspace{.1in}
Finally we have the analogue of one half of Theorem \ref{I.1.d}. I didn't quite
see how to do the other half.
\begin{proposition}
\mylabel{P3e}
Suppose $S$ is a scheme of finite type, and suppose ${\cal H}$ is a group sheaf over
$S$ which is $P5$. Suppose that ${\cal G}\rightarrow S$ is a sheaf and that ${\cal H}$
acts freely on ${\cal G}$ over $S$, with quotient ${\cal F} = {\cal G} /{\cal H}$. If ${\cal F}$ is
$P3\frac{1}{2}$ then ${\cal G}$ is $P3\frac{1}{2}$ (here ${\cal F}$ and ${\cal G}$ are being
considered as the restrictions down to $Spec (k)$ of the corresponding
sheaves over $S$).
\end{proposition}
{\em Proof:}
We follow the proof of the second half of Theorem \ref{I.1.d}.
Let
$$
X\rightarrow {\cal F} , \;\;\; R\rightarrow X\times _{{\cal F}} X
$$
be the presentation given by the property $P3\frac{1}{2}$. We may choose
$X$ in
such a way
that there exists a lifting $X\rightarrow {\cal G}$,
giving an isomorphism
$X\times _{{\cal F}}{\cal G}\cong X\times _S{\cal H}$. Let
$$
Y\rightarrow {\cal H} , \;\;\; W\rightarrow Y\times _{{\cal H}}Y
$$
be the presentation given by the property $P4({\cal M} )$ of ${\cal H}$. We obtain
surjective $S$-vertical morphisms
$$
X\times _SY\rightarrow X\times _S{\cal H}
$$
and (defining $U:= X\times _SW$)
$$
U:=X\times _SW \rightarrow (X\times _SY)\times _{X\times _S{\cal H} }(X\times _SY).
$$
Put $Z:= X\times _SY$. Then we have a surjection
$$
Z\rightarrow X\times _{{\cal F}}{\cal G} \rightarrow {\cal G}
$$
and an $S$-vertical surjection
$$
U\rightarrow Z\times _{X\times _{{\cal F}}{\cal G}}Z.
$$
Now,
$$
(X\times _{{\cal F}}{\cal G} )\times _{{\cal G}} (X\times _{{\cal F}}{\cal G} )=
$$
$$
X\times _{{\cal F}}(X\times _{{\cal F}}{\cal G} )=(X\times _{{\cal F}}X)\times _{{\cal F}}{\cal G} ,
$$
and we have a surjection vertical with respect to the first factor $X$,
$$
R\times _{{\cal F}}{\cal G} \rightarrow (X\times _{{\cal F}}X)\times _{{\cal F}} {\cal G} .
$$
Since $R\rightarrow {\cal F}$ lifts to $R\rightarrow {\cal G}$ we have $R\times
_{{\cal F}}{\cal G}=R\times _S{\cal H}$ and letting $V\rightarrow R\times _SY$ be an etale
surjection, we obtain surjections
$$
V\rightarrow R\times _SY \rightarrow R\times _S{\cal H} \rightarrow
(X\times _{{\cal F}}{\cal G}
)\times _{{\cal G}} (X\times _{{\cal F}}{\cal G} ).
$$
The first is etale, the second is $S$-vertical, and the third is $X$-vertical
for the first factor, so the composition is $X$-vertical.
As before
$$
Z\times _{{\cal G}} Z= Z\times _{X\times _{{\cal F}}{\cal G} }((X\times _{{\cal F}}{\cal G} )
\times _{{\cal G}}
(X\times _{{\cal F}}{\cal G} ))\times _{X\times _{{\cal F}}{\cal G} }Z
$$
so we obtain an $X$-vertical surjection
$$
Z\times _{X\times _{{\cal F}}{\cal G} }V\times _{X\times _{{\cal F}}{\cal G} }Z\rightarrow Z\times
_{{\cal G}}Z.
$$
We can assume by choosing $V$ appropriately that the morphism
$$
V\rightarrow (X\times _{{\cal F}}{\cal G} )\times
_{{\cal F}} (X\times _{{\cal F}}{\cal G} )
$$
lifts to a morphism
$$
V\rightarrow Z\times_{{\cal F}} Z.
$$
We then have an $S$-vertical surjection
$$
U\times _ZV\times _ZU
\rightarrow
(Z\times _{X\times _{{\cal F}}{\cal G} }Z)\times _ZV\times _Z
(Z\times _{X\times _{{\cal F}}{\cal G} }Z).
$$
The right
hand side is equal to
$$
Z\times _{X\times _{{\cal F}}{\cal G} }V\times _{X\times _{{\cal F}}{\cal G} }Z,
$$
which admits, as we have seen above, an $X$-vertical surjection to $Z\times
_{{\cal G}}Z$. By composing we obtain an $X$-vertical, and hence $Z$-vertical
surjection
$$
U\times _ZV\times _ZU\rightarrow Z\times _{{\cal G}}Z.
$$
This completes the proof.
\hfill $\Box$\vspace{.1in}
\numero{Functoriality}
Suppose $F$ is a sheaf over $S$, and suppose $\pi : S'\rightarrow S$ is a
morphism. We denote by $\pi ^{\ast}(F)$ the restriction $F|_{{\cal X} /S'}$, which
is the sheaf associated to the presheaf $Y\rightarrow S' \mapsto F(Y\rightarrow
S)$. If $F$ is representable then $\pi ^{\ast}F$ is also representable by the
fiber product $F\times _SS'$. In general, we allow ourselves to use the
notations $\pi ^{\ast}F$, $F\times _SS'$ and $F|_{S'}$ interchangeably.
We have defined, for a sheaf $G$ on $S'$, the {\em restriction down $Res
_{S'/S}(G )$}.
Suppose $G$ is a
sheaf on $S'$. We defined the direct image by
$$
\pi _{\ast}G(Y\rightarrow S):= G(Y\times _SS' \rightarrow S').
$$
The morphism $F(Y\rightarrow S) \rightarrow F(Y\times _SS' \rightarrow S)$
gives a natural morphism
$$
F\rightarrow \pi _{\ast} \pi ^{\ast}(F),
$$
and the morphism $G(Y\times _S S'\rightarrow S) \rightarrow G (Y\rightarrow
S')$ (coming from the graph morphism $Y\rightarrow Y\times _S S'$) gives a
natural morphism
$$
\pi ^{\ast}\pi _{\ast} (G)\rightarrow G.
$$
These functors are adjoints and the above are the adjunction morphisms. More
precisely, we have a natural isomorphism
$$
Hom (F, \pi _{\ast}G)\cong Hom (\pi ^{\ast}F,G).
$$
This may be verified directly.
{\em Remark:} If $f:A\rightarrow B$ is a vertical morphism over
$S'$ then $\pi _{\ast}(f):\pi _{\ast}A\rightarrow \pi _{\ast}B$ is vertical
over $S$. To see this, note that if $Y, Y^{(n)}$ is a collection of
$S$-schemes with retractions etc. as in the definition of verticality, then
$Y\times _S S', Y^{(n)} \times _SS'$ is a collection with retractions over
$S'$. The verticality of $\pi _{\ast}(f)$ for the case of $Y, Y^{(n)}$
follows from the verticality of $f$ for the case of
$Y\times _S S', Y^{(n)} \times _SS'$.
{\em Remark:} Direct and inverse images are compatible with fiber products.
For inverse images this is easy. For direct images, suppose we have morphisms
$A\rightarrow C$ and $B\rightarrow C$ on $S'$. We obtain morphisms $A\times
_CB\rightarrow A$ and $A\times _CB$ satisfying a universal property. These
give morphisms
$$
\pi _{\ast}(A\times _CB)\rightarrow \pi _{\ast} A \;\; (resp. \;\; \pi _{\ast}B
\, ).
$$
We show the universal property: suppose
$$
(u,v)\in (\pi _{\ast}A\times _{\pi _{\ast}C}\pi _{\ast}B )(Y),
$$
that is $u\in A(Y\times _SS') $ and $v\in B(Y\times _SS')$ with the same image
in $C(Y\times _SS')$. We obtain a unique element of $(A\times _CB)(Y\times
_SS')$ mapping to $(u,v)$. This gives the claim.
\begin{lemma}
\mylabel{I.1.g.2}
if ${\cal F}$ is a coherent sheaf on $S'$ and $\pi :S'\rightarrow S$ is a finite
morphism then $\pi _{\ast}({\cal F} )$ is a coherent sheaf on $S$.
\end{lemma}
{\em Proof:}
We may assume $S$ and $S'$ affine, so that $S=Spec (A)$ and $S'=Spec (A')$ with
$A'$ a finite $A$-algebra. The coherent sheaf ${\cal F}$ corresponds to an
$A'$-module $M$. This implies that
$$
\pi _{\ast}({\cal F} ) (Spec (B)\rightarrow Spec (A)) = {\cal F} ( Spec (B\otimes _AA'))
$$
$$
= M\otimes _{A'}(B\otimes _AA') = M\otimes _AB.
$$
This formula means that $\pi _{\ast}({\cal F} )$ corresponds to the same
module $M$ considered as an $A$-module; in particular it is coherent.
\hfill $\Box$\vspace{.1in}
\begin{lemma}
\mylabel{I.1.h}
If $F$ is $P4$ (resp. $P5$) on $S$ then $\pi ^{\ast}F$ is $P4$ (resp. $P5$)
on $S'$.
\end{lemma}
{\em Proof:}
Note first of all that $S$-verticality of a morphism of sheaves over $S'$
implies $S'$-verticality.
Now if $F$ is $P4$, let $X\rightarrow F$ and $R\rightarrow X\times _FX$ be the
corresponding vertical surjections. We get $\pi ^{\ast}(X) \rightarrow \pi
^{\ast}(F)$ and
$$
\pi ^{\ast}(R)\rightarrow \pi ^{\ast}(X\times _FX)=
\pi ^{\ast}(X)\times _{\pi ^{\ast}(F)}\pi ^{\ast}(X),
$$
surjective and $S$-vertical (hence $S'$-vertical) morphisms. Note that $\pi
^{\ast}(X)$ and $\pi ^{\ast}(R)$ are schemes of finite type over $S'$, so we
obtain the proof for P3. For $P5$ note that $\pi ^{\ast}(F)=F\times _SS'$,
so by Theorem \ref{I.u}, $\pi ^{\ast}(F)\rightarrow S'$ is $S$-vertical; hence
it is $S'$-vertical as required.
\hfill $\Box$\vspace{.1in}
\begin{lemma}
\mylabel{I.1.i}
Suppose $\pi :S'\rightarrow S$ is a finite morphism and suppose $G'$ is a $P4$
sheaf on $S'$. Then $\pi _{\ast}(G')$ is a $P4$ sheaf on $S'$.
\end{lemma}
{\em Proof:}
Let $X'\rightarrow G'$ and $R'\rightarrow X'\times_{G'}X'$ be the surjective
vertical morphisms with $X'$ and $R'$ schemes of finite type over $S'$.
Let $G:= \pi _{\ast}(G')$ and similarly for $X$ and $R$.
By the above remark, we obtain vertical morphisms $X\rightarrow G$ and
$R\rightarrow X\times _GX$. (Note that $X\times _GX= \pi _{\ast}(X'\times
_{G'}X'$ by above.)
In the case of a finite morphism $\pi : S'\rightarrow S$, note that if
$f:A\rightarrow B$ is surjective over $S'$ then $\pi _{\ast}(f):\pi
_{\ast}A\rightarrow \pi _{\ast}B$ is surjective. This is a general property of
sheaves on the etale topology, for which we sketch the proof (an application
of Artin approximation). If $\eta \in \pi _{\ast}(B)(Y)$, this means $\eta :
Y\times _SS'\rightarrow B$. For $y'\in Y\times _SS'$ there is an etale
neighborhood $U\rightarrow Y\times _SS'$ and a lifting $U\rightarrow A$. We
need to find an etale neighborhood $V$ of the image $y\in Y$ and a lifting
$V\times _SS' \rightarrow U$. Define a functor $L(V/Y)$ to be the set of
liftings $V\times _SS' \rightarrow U$ over $Y\times _SS'$. It is B1, and a
lifting exists on $\hat{V}= Spec ({\cal O} _{Y,y}^{\wedge})$, so by Artin
approximation there is an etale neighborhood $V$ with a lifting.
Applying this to our case, the morphisms $X\rightarrow G$ and
$R\rightarrow X\times _GX$ are surjective. By Lemma \ref{I.1.j}, it suffices to
prove that $X$ and $R$ are $P4$. Thus it suffices in general to show:
if $Z$ is a
scheme of finite type over $S'$ then $\pi _{\ast}(Z)$ is $P4$. We make a
further
reduction: a scheme of finite type can be presented as the kernel of a morphism
${\bf A}^n \rightarrow {\bf A}^m$; the direct image is then the kernel of $\pi
_{\ast}{\bf A}^n \rightarrow \pi _{\ast}{\bf A}^m$. The kernel of a morphism
of $P4$ sheaves is again $P4$ (Lemma \ref{I.1.a}) so it suffices to
treat the case
$Z={\bf A}^n$. But in this case, $Z$ is a coherent sheaf and its direct image
is also a coherent sheaf. One can see directly that a coherent sheaf ${\cal F}$ is
$P4$ by using the fact that it has a resolution of the form
$$
{\cal O} ^a \rightarrow {\cal O} ^b \rightarrow {\cal F} \rightarrow 0
$$
(exact even on the big site ${\cal X} /S$), or by looking at Lemmas
\ref{I.1.g.1} (below) and \ref{I.1.g.2}. This completes the proof.
\hfill $\Box$\vspace{.1in}
\begin{lemma}
\mylabel{restrictionPreserves?}
Suppose $S'\rightarrow S$ is a morphism of schemes of finite type. Suppose
${\cal F}$ is a sheaf on ${\cal X} /S'$. Then $Res _{S'/S}{\cal F}$ is $P3\frac{1}{2}$ if and
only if ${\cal F}$ is $P3\frac{1}{2}$.
\end{lemma}
{\em Proof:}
This is only a matter of terminology since, $P3\frac{1}{2}$ being a global
property, the statement that ${\cal F}$ is $P3\frac{1}{2}$ really means that the
restriction of ${\cal F}$ down to $Spec (k)$ is $P3\frac{1}{2}$. It is obviously
equivalent to say this after first restricting down to $S$.
\hfill $\Box$\vspace{.1in}
\numero{Vector sheaves}
With this section we begin the part of our study which requires working over a
ground field $k$ of characteristic zero. From now on ${\cal X}$ denotes the big
etale
site of schemes over $Spec (k)$. Before returning to the definition of
presentability and its infinitesimal study, we make a detour to discuss {\em
vector sheaves}. These are objects which will be the linearizations of
presentable group sheaves---we are also interested in vector sheaves as
candidates for the $\pi _i (T,t)$ with $t\in T(S)$, for an $n$-stack $T$ on ${\cal X}
/S$ for $i\geq 2$.
To be slightly more precise,
suppose $S\in {\cal X}$ is a scheme over $k$, and let ${\cal X} /S$ denote the
category of
schemes over $S$. We will define a notion of {\em vector sheaf} on ${\cal X} /S$.
This notion is what was called ``U-coherent sheaf'' by Hirschowitz in
\cite{Hirschowitz}.
The particular case which we call ``vector scheme'' below has already been well
known for some time as the ``linear spaces'' of Grauert \cite{Grauert},
appearing
notably in Whitney's tangent cones \cite{Whitney}.
We feel that the terminology ``vector sheaf'' is more
suggestive. Many of the results below seem to be due to Hirschowitz
\cite{Hirschowitz} (in particular, the observation that duality is involutive)
although some parts of the theory are certainly due to
\cite{Fischer}, \cite{Grauert}, \cite{Whitney}. We
have integrated these results into our treatment for the reader's convenience.
Essentially the only thing new in our treatment is the first lemma (and the
analogous statement about extensions).
Before starting in on the definition, I would like to make one note of caution.
The category of vector sheaves will not satisfy any nice (ascending or
descending) chain condition. This is one of the principal differences with
vector spaces or modules over a noetherian ring, and could in the long run pose
a major problem if one wants to consider an ``infinite dimensional'' version of
the theory such as by looking at $ind$- or $pro$- objects.
We have a sheaf of rings ${\cal O}$ on ${\cal X}$, defined by ${\cal O} (X):= \Gamma (X, {\cal O}
_X)$. Note that it is represented by the affine line.
\begin{lemma}
\mylabel{I.a}
Suppose $F$ is a sheaf of abelian groups on ${\cal X} /S$, representable by a scheme
which is affine and of finite type over $S$. If there exists a structure of
${\cal O}$-module for $F$, then this structure is unique. If $F$ and $G$ are two
such sheaves, and if $a:F\rightarrow G$ is a morphism of
sheaves of groups, then $a$ is a morphism of ${\cal O}$-modules.
\end{lemma}
{\em Proof:}
The first statement of the lemma follows from the second. For the second
statement, suppose $u\in F_X$. Consider the element $tu\in F_{X\times {\bf
A}^1}$. For any positive integer $n$ we have $tu|_{X\times \{ n\}}=u+\ldots +u$
($n$ times). The same is true for the image $a(u)$. Therefore
$$
a(tu)|_{X\times \{ n\}} =ta(u)|_{X\times \{ n\}} .
$$
We obtain two morphisms $X\times {\bf A}^1\rightarrow G$ which are equal on
the subschemes $X\times \{ n\}$; this implies that they are equal.
(Here is a proof of this: we may suppose that $X$ and the base $S$ are
affine, so $X=Spec (A)$ and $G=Spec (B)$ and a morphism $X\times {\bf
A}^1\rightarrow G$ corresponds to a morphism $\phi : B\rightarrow
A[t]$.
Pick any $b\in B$ and write
$$
\phi (b)= \sum _{j=1}^m p_{j}t^j;
$$
but the matrix $a_{nj}= n^j$ for $n,j=1,\ldots , m$ is invertible as a matrix
with coefficients in $k$, so there is a matrix $c_{nj}$ with
$$
p_j= \sum _{n=1}^m c_{nj}\phi (b)(n) .
$$
Thus $\phi (b)$ is determined by the values at positive integers $\phi (b)(n)$.)
\hfill $\Box$\vspace{.1in}
A {\em vector scheme over $S$} is a sheaf $V$ of abelian groups on ${\cal X} /S$
which
is a sheaf of ${\cal O}$-modules and such that there exists an etale covering $\{
S_{\alpha}\rightarrow S\}$ such that each $V|_{S_{\alpha}}$ is representable
by a scheme $F_{\alpha}$ which is affine of finite type over $S_{\alpha}$.
The
above lemma shows that the category of vector
schemes is a full subcategory of the category of sheaves of abelian groups on
${\cal X}$. In the complex analytic category these objects were called ``linear
spaces'' by Grauert and were studied in \cite{Grauert}, \cite{Fischer}.
The first remark is that, in fact, the locality in the definition
of vector scheme was
extraneous. In effect, since the representing schemes $F_{\alpha}$ are unique
up to unique isomorphism, they glue together to give a scheme $F$, affine and
locally of finite type over $S$.
\begin{lemma}
\mylabel{I.b}
Suppose $V$ is a vector scheme on ${\cal X} /S$, and suppose $S$ is affine. Then
there
is an exact sequence
$$
0\rightarrow V\rightarrow {\cal O} ^m \rightarrow {\cal O} ^n
$$
of abelian sheaves on ${\cal X}$.
\end{lemma}
{\em Proof:}
Write $S=Spec (A)$ and $V=Spec
(B)$. The action of ${\bf G} _m$ gives a decomposition
$$
B = \bigoplus B^{\lambda}
$$
where $B^{\lambda}$ consists of functions $b$ such that $b(tv)=
t^{\lambda}b(v)$.
The sum is over $\lambda \geq 0$ (integers), since the action extends to an
action of the multiplicative monoid ${\bf A}^1$. Furthermore, if $b\in B^0$
then $b(tv)=b(v)$ for all $t$ (including $t=0$), in particular $b(v)=b(0)$.
Thus $B^0= A$. If $b\in B^{\lambda }$ for $\lambda >0$ then $b(0)=b(0\cdot
O)= 0$. Thus the zero section corresponds to the projection onto $B^0=A$.
The decomposition is compatible with multiplication in $B$. It is also
compatible with the comultiplication $B\rightarrow B\otimes _A B$
corresponding to the addition law on $V$. The comultiplication is
$$
B^{\lambda } \rightarrow \bigoplus _{\mu + \nu = \lambda} B^{\mu} \otimes _A
B^{\nu},
$$
and furthermore the coefficients $B^{\lambda} \rightarrow B^{\lambda}\otimes
_A B^0= B^{\lambda}$ and $B^{\lambda} \rightarrow B^0\otimes
_A B^{\lambda}= B^{\lambda}$ are the identity (corresponding to the formula
$v+0=v=0+v$). On the other hand, the composition $B\rightarrow B\otimes _A B
\rightarrow B$ corresponds to the map $v\mapsto v+v=2v$, which is also scalar
multiplication by $t=2$. Thus the composition
$$
B^{\lambda } \rightarrow \bigoplus _{\mu + \nu = \lambda} B^{\mu} \otimes _A
B^{\nu}\rightarrow B^{\lambda}
$$
is equal to multiplication by $2^{\lambda}$. The first and last terms in the
sum give a contribution of $b\mapsto 2b$ (by the observation $v+0=v=0+v$), so
for $\lambda \geq 2$, the composition
$$
B^{\lambda } \rightarrow \bigoplus _{\mu + \nu = \lambda , 0< \mu , \nu <
\lambda } B^{\mu} \otimes _A B^{\nu}\rightarrow B^{\lambda}
$$
is multiplication by $2^{\lambda}-2$, invertible. Hence every element of
$B^{\lambda}$ is expressed as a sum of products of elements of $B^{\mu}$ and
$B^{\nu}$ for $\mu , \nu < \lambda$. This proves that $B^1$ generates $B$ as
an $A$-algebra. Since $B$ is of finite type over $A$ (a consequence of the
fact that we have supposed all of our schemes noetherian), we can choose a
finite
number of elements of $x_1,\ldots , x_m \in B^1$ which generate $B$ as an
$A$-algebra, and these elements give an embedding $V\subset {\cal O} ^m$. This
embedding is linear, since the elements are elements of $B^1$ (from the above
discussion one sees that for $b\in B^1$ we have $b(u+v)=b(u)+ b(v)$). Write $$
B= A[x_1,\ldots , x_m]/I
$$
for a homogeneous ideal $I=\bigoplus I^{\lambda}$. We claim that $I$ is
generated as an ideal by $I^1$. To see this, let $I'$ be the ideal generated
by $I^1$ and put $B'= A[x]/I'$. Under the comultiplication of $A[x]$ we have
$$
I^1 \rightarrow I^1 \otimes _A A \oplus A \otimes _A I^1 ,
$$
so $I^1$ maps to zero in $B'\otimes _A B'$. Thus so does $I'$. We obtain a
comultiplication
$$
B'= A[x]/I'\rightarrow B'\otimes _A B',
$$
so $Spec (B')$ is a vector scheme too. But the map $B'\rightarrow B$ is
surjective and an isomorphism on the pieces of degree $1$. It is compatible
with the comultiplication. We claim that it is injective, showing this on the
part of degree $\lambda$ by induction on $\lambda$ (starting at $\lambda =2$).
If an element $b\in (B')^{\lambda}$ maps to zero in $B$, then by applying the
process given above (in the algebra $B'$) we can write $b= \sum b_{\mu} b_{\nu}$
for $\mu , \nu < \lambda$. But $b_{\mu}$ and $b_{\nu}$ map to the elements in
$B$
given by applying the same process to the image of $b$; as this image is $0$, so
are the images of $b_{\mu}$ and $b_{\nu}$. By the induction hypothesis, the
map
is injective on the pieces of degrees $\mu , \nu$, so $b_{\mu}=b_{\nu}=0$,
giving $b=0$. This induction shows that $B'\cong B$, so $I'=I$ is generated by
$I^1$. Since $B$ is of finite type over $A$ (which is noetherian), $I$ is
generated by a finite number of elements. This implies that it is generated by
a finite number of elements $y_1, \ldots , y_n$ of $I^1$. These elements give
a linear map ${\cal O} ^n \rightarrow {\cal O} ^m$, and $V$ is the kernel.
\hfill $\Box$\vspace{.1in}
We come now to the main definition of this section.
A {\em vector sheaf on $S$} is a sheaf of abelian groups $F$ on ${\cal X} /S$ such
thatthere exists an etale covering $\{
S_{\alpha}\rightarrow S\}$ such that for each $\alpha$ there exists an exact
sequence
$$
U_{\alpha}\rightarrow V_{\alpha}\rightarrow F|_{S_{\alpha}}\rightarrow 0
$$
of sheaves of abelian groups, with $U_{\alpha}$ and $V_{\alpha}$ vector schemes
over $S_{\alpha}$.
Denote by ${\cal V} (S)$ the category of vector sheaves over $S$.
If $X\rightarrow S$ is an element of
${\cal X} /S$, we denote by $F|_X$ the restriction of $F$ to the category ${\cal X} /X$.
It is a vector sheaf over $X$ (this is easy to see from the definitions).
If $F$ is a vector sheaf and $f\in F(Y)$ and $a:X\rightarrow Y$ is a morphism,
we denote the restriction of $f$ to $X$ by $a^{\ast}(f)$ or just $f|_X$.
\begin{lemma}
\mylabel{I.c}
If $F$ is a vector sheaf, and $S$ is an affine variety, then the cohomology
groups $H^i (S, F)$ vanish for $i>0$. If
$$
F_1\rightarrow F_2\rightarrow F_3
$$
is an exact sequence of vector sheaves (that is, an exact sequence in the
category of abelian sheaves on ${\cal X}$, where the elements are vector sheaves)
then for any $X$ over $S$ which is itself an affine scheme, the sequence
$$
F_1(X)\rightarrow F_2(X)\rightarrow F_3(X)
$$
is exact.
\end{lemma}
{\em Proof:}
Treat first the case where $F$ is a vector scheme. We have an exact sequence
$$
0\rightarrow F\rightarrow {\cal O} ^a \rightarrow {\cal O} ^b
$$
by Lemma \ref{I.b}. Let $G$ be the kernel of the morphism ${\cal O} ^a \rightarrow
{\cal O} ^b$ on the small etale site over $S$. It is a coherent sheaf. Let $F'$ be
the sheaf on ${\cal X}$ whose value on $Y\rightarrow S$ is the space of sections of
the pullback (of coherent sheaves) of $G$ to $Y$. There is a surjective
morphism $F'\rightarrow F$, which induces $F'(U)\stackrel{\cong}{\rightarrow}
F(U)$ for any $U$ etale over $S$ (or even any $U$ which is flat over $S$).
Let $K$ denote the kernel of $F'\rightarrow F$. We claim that if $Y$ is any
scheme etale over $S$, then $H^i(Y, K)=0$. Prove this by ascending induction
on $i$. If the cohomology in degrees $<i$ of all fiber products of elements in
all etale covering families of $Y$ vanishes, then the degree $i$ sheaf
cohomology is equal to the degree $i$ \v{C}ech cohomology. But the \v{C}ech
cohomology is calculated only in terms of the values of the sheaf on the fiber
products, and here the values of $K$ are zero. Thus $H^i(Y,K)=\check{H}^i(Y,
K)=0$, completing the induction. We obtain $H^i(S, F)= H^i(S, F')$. But the
higher cohomology of a coherent sheaf on an affine scheme $S$ vanishes (even in
the big etale site).
We obtain the desired vanishing.
For the second part, suppose that $X=S$ is affine. The
restriction of the exact sequence to the small etale site (over $X$) remains
exact. It can be completed to a $5$-term exact sequence where the first and
last terms are also coherent sheaves; then broken down into short exact
sequences. The vanishing of $H^1$ of coherent sheaves on the small etale site
yields the desired exactness of all the short exact sequences of global
sections, and hence the exactness of the sequence in question.
\hfill $\Box$\vspace{.1in}
{\em Remark:} One can show that a vector sheaf $V$ over an affine $S$ has a
resolution by vector schemes, over $S$ rather than over an etale covering of
$S$ \cite{Hirschowitz}.
\begin{lemma}
\mylabel{I.d}
Suppose $F$ is a vector sheaf over $S$. Then for any $X\in {\cal X} /S$ and $Y$ a
scheme of finite type over $k$, we have
$$
F(X\times _{Spec (k)}Y )=
F(X)\otimes _k {\cal O} (Y).
$$
The isomorphism is given by the pullback $F(X)\rightarrow F(X\times _kY)$
and the
scalar multiplication by the pullback of functions on $Y$.
\end{lemma}
{\em Proof:}
We first prove this when $F$
is a vector scheme. There is an exact sequence
$$
0\rightarrow F\rightarrow {\cal O} ^a \stackrel{M}{\rightarrow} {\cal O} ^b .
$$
We have $F(X)= \ker (M(X))$ and $F(X\times _kY)= \ker
(M(X\times _kY))$. But ${\cal O} (X\times _kY)={\cal O} (X)\otimes _k{\cal O} (Y)$,
and $M(X\times _kY)=M(X)\otimes 1$. Since tensoring over $k$ is exact,
$$
\ker (M(X)\otimes 1) = \ker (M(X)) \otimes _k {\cal O} (Y)
$$
as desired.
Now suppose $F$ is a vector sheaf. There is an
exact sequence
$$
U\rightarrow V\rightarrow F\rightarrow 0.
$$
If $Z$ is affine then the sequence
$$
U(Z)\rightarrow V(Z)\rightarrow F(Z)\rightarrow 0
$$
remains exact. To see this, replace $F$ by a coherent
sheaf $F'$ on the small etale site over $Z$. The restriction of $F$ to the
small etale site over $Z$ is the quotient $F'$ of the restriction of
$U\rightarrow V$ to the small etale site over $Z$, that is to say the sections
of $F$ and $F'$ are the same on schemes etale over $Z$ (and in particular over
$Y$). But if $Z$ is affine, then taking global sections preserves surjectivity
of a morphism of coherent sheaves. This gives the desired exact sequence
(proceed in a similar way for exactness at $V(Z)$). Suppose now that $X$ and
$Y$ are affine. Then applying the above to $Z=X$ and $Z=X\times _k Y$ we get
$$
U(X)\otimes _k {\cal O} (Y) \rightarrow V(X)\otimes _k{\cal O} (Y) \rightarrow
F(X\times _kY)\rightarrow 0 .
$$
The first morphism is the same as in the tensor product of
$$
U(X)\rightarrow
V(X)\rightarrow F(X)\rightarrow 0
$$
with ${\cal O} (Y)$, so the two quotients are isomorphic: $F(X\times _kY)\cong
F(X)\otimes _k {\cal O} (Y)$. This completes the case where $X$ and $Y$ are
affine. But both sides of the equation have the property that they are
sheaves in each variable $X$ and $Y$ separately; thus we may first localize
on $X$ and then localize on $Y$, to reduce to the case where $X$ and $Y$ are
affine.
Finally, suppose $F$ is a
vector sheaf, and write $F= \bigcup _{i\in I} F_i$ as a directed union of vector
sheaves. The tensor product of the union is equal to the union of the tensor
products:
$$
F(X)\otimes _k{\cal O} (Y) = \bigcup _{i\in I} F_i (X)\otimes _k{\cal O} (Y) =
\bigcup _{i\in I} F(X\otimes _kY) = F(X\otimes _kY).
$$
Note that
the inclusion maps in the two directed unions are the same (since the
isomorphisms established above are uniquely determined by compatibility with the
morphisms $F_i (X)\rightarrow F_i (X\otimes _kY)$ and with scalar
multiplication by elements of ${\cal O} (Y)$). This completes the proof.
\hfill $\Box$\vspace{.1in}
{\em Remark:} We will mostly use this lemma in the following two cases.
Suppose $F$ is a vector sheaf over $S$. Then for any $X\in {\cal X} /S$ we have
$F(X\times {\bf A}^1)= F(X)\otimes _k k[t]$. The isomorphism is given by the
pullback $F(X)\rightarrow F(X\times {\bf A}^1)$ and the scalar multiplication
by the pullback of the coordinate function $t$ on ${\bf A}^1$.
Similarly, $F(X\times {\bf G} _m) = F(X)\otimes _k k[t,t^{-1}]$, with
the isomorphism uniquely determined by compatibility with the previous one
under
the inclusion ${\bf G} _m \subset {\bf A}^1$.
\begin{lemma}
\mylabel{I.e}
A vector sheaf has a unique structure of ${\cal O}$-module, and any morphism of
vector sheaves is automatically compatible with the ${\cal O}$-module structure.
\end{lemma}
{\em Proof:}
Suppose that $\phi :F\rightarrow G$ is a morphism of vector sheaves. Suppose
$X\in {\cal X} /S$. Suppose $f\in F(X\otimes {\bf A}^1)$. The difference $g=\phi
(tf)-t\phi (f )$ is an element of $G(X\times {\bf A}^1)$ which restricts to zero
on $X\otimes \{ n\}$ for any integer $n$. We can write
$$
g=\sum _{i=1}^pg_i t^i
$$
with $g^i\in G(X)$ (by the previous lemma). We know that
$$
g(n)=\sum _{i=1}^pg_i n^i = 0
$$
for any integer $n$. But in $k$ the matrix $(n^i)_{1\leq n, i\leq p}$
has an inverse $(c_{ni})$, and we have
$$
g_i = \sum _{n=1}^p c_{ni}g(n) = 0.
$$
Therefore $\phi
(tf)-t\phi (f )=g=0$, for any $f$. Thus $\phi $ is compatible with
multiplication by $t$. Now suppose $\lambda \in {\cal O} (X)$. This gives a
morphism $\gamma : X\rightarrow X\times {\bf A} ^1$ such that $\gamma ^{\ast}
(tp_1^{\ast}(f))= \lambda f$ for any $f\in F(X)$ or $G(X)$ (here $p_1:X\times
{\bf A}^1\rightarrow X$ is the projection). The fact that $\phi$ is a
morphism of sheaves means that it is compatible with $\gamma ^{\ast}$ and
$p_1^{\ast}$, so we have
$$
\phi (\lambda f)= \phi (\gamma ^{\ast} (tp_1^{\ast}(f)))=
\gamma ^{\ast} (\phi (tp_1^{\ast}(f)))
$$
$$
= \gamma
^{\ast}(t\phi (p_1^{\ast} (f)))= \gamma ^{\ast}(tp_1^{\ast}(\phi (f)))=
\lambda \phi (f).
$$
Thus $\phi$ is compatible with scalar multiplication. This fact, applied to
the identity of $F$, implies that the scalar multiplication is unique if it
exists.
For existence, note that any morphism of vector schemes is automatically a
morphism of ${\cal O}$-modules, so the quotient has a structure of ${\cal O}$-module.
Thus any vector sheaf has a structure of ${\cal O}$-module. If $F$ is a
vector sheaf expressed as a directed union $F= \bigcup _{i\in I} F_i$ of finite
vector sheaves, then the inclusions in the directed union are compatible
with the ${\cal O}$-module structures; thus the union has an ${\cal O}$-module
structure.
\hfill $\Box$\vspace{.1in}
The conclusion of this lemma is that the category of vector sheaves, with
morphisms equal to those morphisms of abelian sheaves compatible with the
${\cal O}$-module structure, is a full subcategory of the category of sheaves of
abelian groups on ${\cal X} /S$.
Next we establish a Krull-type property.
\begin{lemma}
\mylabel{I.f}
Suppose that $F$ is a vector sheaf over $S$, with $f\in F(Y)$, and suppose that
for every $X\rightarrow Y$ where $X$ is an artinian scheme, $f|_X=0$. Then
$f=0$. Suppose $\phi : F\rightarrow G$ is a morphism of vector sheaves such
that for every $X\rightarrow S$ with $X$ artinian, $\phi |_X=0$. Then $\phi
=0$.
\end{lemma}
{\em Proof:}
We work with vector schemes over base schemes which are not necessarily of
finite type over $k$ (the definition is the same, but we require
additionally that the vector scheme be of finite type over the base). If
$U\rightarrow V$ is a morphism of vector schemes over a henselian local ring
$A$, and if $v$ is a section of $V$ over $A$ such that for each $n$ there exists
$u_n\in U(Spec (A/{\bf m}^n))$ with $u_n$ mapping to the restriction of $v$,
then
there exists a section $u$ of $U$ over $A$ which maps to $v$. This follows
from the strong Artin approximation theorem at maximal ideals, applied to
finding sections of the morphism $U\times _VSpec (A)\rightarrow Spec (A)$.
Now onto the proof of the lemma.
For the first statement, any section $f$ is contained in a vector
subsheaf of $F$, so we may suppose that $F$ is a vector sheaf. Choose a
presentation
$$
U\rightarrow V \rightarrow F \rightarrow 0
$$
by vector schemes. We may replace $X$ by a covering, so we may suppose that our
section $f$ comes from a section $v$ of $V$. From the previous paragraph, for
every henselized local ring $A$ of $X$, there exists a section $u$ of $U (Spec
(A)$ mapping to $v$. But any such $A$---henselization at a point $P$---is the
direct limit of algebras $A_i$ etale of finite type over $X$ (which give etale
neighborhoods of $P$), and the space of sections is the direct limit:
$$
U(Spec (A))= \lim _{\rightarrow } U( Spec (A_i )).
$$
Thus there is a section $u_i$ over some $Spec (A_i)$ mapping to $v$. Thus
every point $P$ of $X$ has an etale neighborhood on which there is a lifting
of $v$ to a section of $U$. This implies that the image of $v$ in the cokernel
$F$ in the etale topology, is zero. This gives the first statement, and
the second statement follows easily from this.
\hfill $\Box$\vspace{.1in}
{\em Remark:} An alternative to the above proof is to use Lemma
\ref{Krull}.
The utility of this property comes from the following fact.
\begin{corollary}
\mylabel{I.g}
If $F$ is a vector scheme, and if $Y\rightarrow S$ is an element of
${\cal X} /S$ with $Y$ artinian, then the functor $F_Y: Z\mapsto
F(Y\times _{Spec (k)}Z)$
from schemes over $Spec (k)$ to sets, is represented by an additive group
scheme (that is, a finite dimensional vector space) over $k$. This vector space
is the $k$-module $F(Y)$.
\end{corollary}
{\em Proof:}
By Lemma \ref{I.d}, we have $F_Y(Z)= F(Y)\otimes _k {\cal O} (Z)$ which is the space
of morphisms of schemes from $Z$ to the vector space $F(Y)$. Thus $F_Y$ is
represented by the vector space $F(Y)$. Note that from the exact sequences
used in the proof of Lemma \ref{I.d}, $F(Y)$ is a finite-dimensional $k$-vector
space.
\hfill $\Box$\vspace{.1in}
The group scheme ${\bf G} _m$ acts on every vector sheaf, by scalar
multiplication. This action may be thought of as an action of the functor
${\bf G}
_m (X)$ on $F(X)$, or as an automorphism of $F(X\otimes {\bf G} _m )$
(multiplication by $t$) which is natural in $X$. We have seen above that if
$F\rightarrow G$ is a morphism of sheaves of abelian groups between two vector
sheaves, then it is compatible with the ${\bf G} _m$ action.
Suppose $A$ is a vector scheme, and $F$ is a vector sheaf.
We look at $F(A)$, the space of sections over the scheme $A$. Let
$F(A)^{\lambda}$ denote the subgroup of elements $f\in F(A)$ such that
$f(ta)=t^{\lambda} f(a)$. Here $a\mapsto ta$ is considered as a morphism
$A\times {\bf G} _m\rightarrow A$ over $S$, and $f(a)\mapsto
t^{\lambda}f(a)$ is the automorphism of $F(A\times {\bf G} _m)$ given by scalar
multiplication by $t^{\lambda} \in k[t,t^{-1}]$; the notation $f$ in the
second half of the formula actually denotes the pullback of $f$ to $A\times
{\bf G}
_m$.
\begin{lemma}
\mylabel{I.h}
With the above notations, $F(A)$ decomposes as a direct sum
$$
F(A) = \bigoplus _{\lambda \in {\bf Z} ,\lambda \geq 0} F(A)^{\lambda} .
$$
This direct sum decomposition is natural with respect to morphisms
$F\rightarrow G$, and the linear piece $F(A)^1$ is exactly the space of
morphisms of vector sheaves $A\rightarrow F$.
\end{lemma}
{\em Proof:}
Recall that $F(A\times {\bf A}^1)= F(A)\otimes _kk[t]$, which we will just
write as $F(A)[t]$. The morphism of scalar multiplication $A\times{\bf
A}^1\rightarrow A$ gives $\Psi _t: F(A)\rightarrow F(A)[t]$ defined
by $(\Psi _tf)(a):= f(ta)$ (to be accurate, this should be defined in terms of
restriction maps for the morphisms involved, but we keep this notation for
simplicity). Then $F(A)^{\lambda}$ is the set of $f$ such that $\Psi _tf =
t^{\lambda}f$ in $F(A)[t]$. Let $\Psi _s[t]: F(A)[t]\rightarrow F(A)[s,t]$
denote the extension of $\Psi _s: F(A)\rightarrow F(A)[s]$ to the polynomials in
$t$. We have
$$
(\Psi _s[t]\Psi _tf)(a)= f(tsa)= (\Psi _{st}f)(a).
$$
Write
$$
\Psi _t(f)= \sum _{i=0}^{\infty} \psi _i(f)t^i,
$$
where $\psi _i(f)\in F(A)$ and for any $f$, there are only a finite number of
nonzero $\psi _i (f)$. Our previous formula
becomes
$$
\sum _{i,j} \psi _i (\psi _j (f))s^it^j = \sum _k \psi _k (f)(st)^k.
$$
Comparing terms we see that $\psi _i(\psi _j (f))=0$ for $i\neq j$ and $\psi
_i(\psi _i(f))=\psi _i (f)$. But in general $f\in F(A)^{\lambda}$ if and only
if
$\psi _i(f)=0$ for $i\neq \lambda$ and $\psi _{\lambda}(f)=f$. Therefore $\psi
_i (f)\in F(A)^i$. Restrict to $t=1$, and note that the composed morphism
$a\mapsto (a,1)\mapsto a$ is the identity so $\Psi _1(f)=f$. We get
$$
f= \sum _{i=0}^{\infty} \psi _i (f) ,
$$
and this sum is actually finite. Thus every element of $F(A)$ can be expressed
as a finite sum of elements of the $F(A)^{\lambda}$. On the other hand, this
expression is unique: if $f= \sum f_i$ with $f_i\in F(A)^i$ then
$$
\sum \psi _i(f)t^i=\Psi _t(f)=\sum \Psi _t(f_i)= \sum \psi _i (f_i)t^i = \sum
f_i t^i,
$$
and comparing coefficients of $t^i$ we get $f_i = \psi _i (f)$.
This completes the proof of the decomposition (note that in working with ${\bf
A}^1$ instead of ${\bf G} _m$ we obtain automatically that the exponents are
positive).
We have to show that $F(A)^1$ is equal to the space of linear morphisms from $A$
to $F$. A linear morphism gives an element of $F(A)^1$ (since it is compatible
with the action of ${\cal O}$ by Lemma \ref{I.e}), and the resulting map from the
space of morphisms to $F(A)^1$ is injective, since $F(A)$ is the space of
morphisms of functors $A\rightarrow F$.
Finally, we show surjectivity. For this, suppose
given an element $\phi \in F(A)^1$.
Suppose $Y$ is artinian, and $F$ is a vector sheaf over $S$. Then the
functor $Z\mapsto F(Y\times Z )$ is represented by a vector space $F_Y$ over
$k$ (Lemma \ref{I.g}). Our element of $F(A)$ now gives a morphism of schemes
$\phi _Y: A_Y \rightarrow F_Y$ between these two vector spaces. It is compatible
with scalar multiplication, so it is linear. In particular, if $u,v\in A_Y(Spec
(k))= A(Y)$ then $\phi (u+v)= \phi (u)+\phi (v)$ in $F_Y(Spec (k))= F(Y)$.
Now suppose $X$ is any element of ${\cal X} /S$. We show that $\phi : A(X)
\rightarrow
F(X)$ is a morphism of abelian groups. Suppose $u,v\in A(X)$. Let $f=\phi
(u+v)-\phi (u)-\phi (v)\in F(X)$. By the previous paragraph, for any
$Y\rightarrow X$ with $Y$ artinian, we have $f|_Y=0$. But the Krull property
of Lemma \ref{I.f} then implies that $f=0$. This shows that $\phi$ is a
morphism
of sheaves of abelian groups.
\hfill $\Box$\vspace{.1in}
If $F$ and $G$ are sheaves of abelian groups, we denote by $Hom (F,G)$ the
internal $Hom$, that is the sheaf of homomorphisms of sheaves of abelian
groups from $F$ to $G$. The value $Hom (F,G)(X)$ is the space of morphisms of
sheaves of abelian groups from $F|_{{\cal X} /X}$ to $G|_{{\cal X} /X}$ (this is already
a sheaf).
\begin{corollary}
\mylabel{I.i}
If $F\rightarrow G $ is a surjection of vector sheaves, and if $A$
is a vector scheme, then the morphism sheaves
$$
Hom (A, F) \rightarrow Hom (A, G)
$$
is surjective. If $X$ is affine then
$$
Hom (A,F)(X)\rightarrow Hom (A, G)(X)
$$
is surjective.
\end{corollary}
{\em Proof:}
It suffices to prove the second statement. We may assume that $X=S$. We have
$$
Hom (A, G)(S)= G(A)^1
$$
by the last statement of the lemma. Since $A$ is
affine, the morphism $F(A)\rightarrow G(A)$ is surjective, and by the previous
lemma this implies that $Hom (A,F)(S)=F(A)^1\rightarrow G(A)^1$ is surjective,
giving the corollary.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{I.j}
If $\phi : F\rightarrow G$ is a morphism of vector sheaves, then ${\rm
coker}(\phi )$ and ${\rm ker} (\phi )$ are vector sheaves.
\end{corollary}
{\em Proof:}
We may suppose that $S$ is affine and small enough. Choose presentations
by vector schemes (cf the remark before Lemma
\ref{I.d})
$$
U\rightarrow V\rightarrow F\rightarrow 0
$$
and
$$
0\rightarrow P \rightarrow R\rightarrow T \rightarrow G \rightarrow 0
$$
(note that the kernel $P$ is automatically a vector scheme).
The morphism $V\rightarrow G$ lifts to a morphism $V\rightarrow T$, by the
previous corollary, and we obtain a presentation
$$
R \oplus V \rightarrow T \rightarrow {\rm coker} (\phi ) \rightarrow 0.
$$
The fiber
products $V\times _T R$ and $V\times _T R$ are vector schemes,
and we have a presentation
$$
U\times _T R\rightarrow V\times _T R \rightarrow {\rm ker}(\phi )\rightarrow
0.
$$
\hfill $\Box$\vspace{.1in}
Now we have shown that the category of vector sheaves is an abelian subcategory
of the category of sheaves of abelian groups on ${\cal X} /S$.
Suppose $A$ is a vector scheme and $F$ is a vector sheaf. Let ${\bf 3}$ denote
the automorphism of $A$ obtained by multiplication by the scalar $3$ (any
integer $\neq 0, \pm 1$ will do). We have
$$
F(A)=\bigoplus F(A)^{\lambda }
$$
(the decomposition given by Lemma \ref{I.h}) where $F(A)^{\lambda}$ may be
characterized as the subspace of elements $f$ such that ${\bf 3}^{\ast}(f)=
3^{\lambda}f$. In particular, the linear subspace $Hom (A, F)= F(A)^1$ is
characterized as the subspace of elements $f$ such that
${\bf 3}^{\ast}(f)=
3f$.
\begin{theorem}
\mylabel{I.k}
Suppose $E$ and $G$ are vector sheaves, and
$$
0\rightarrow E \rightarrow F\rightarrow G \rightarrow 0
$$
is an extension in the category of sheaves of abelian groups on ${\cal X} /S$. Then
$F$ is a vector sheaf.
\end{theorem}
{\em Proof:}
We proceed in several steps. We may assume that $S$ is affine and small
enough. Let
$$
\begin{array}{ccccccc}
& V & & & & B & \\
& \downarrow & & & & \downarrow & \\
& U & & & & A & \\
& \downarrow & & & & \downarrow & \\
0 \rightarrow & E &\rightarrow & F & \rightarrow & G & \rightarrow 0 \\
& \downarrow & & & & \downarrow & \\
& 0 & & & & 0 &
\end{array}
$$
be presentations for $E$ and $G$.
{\em Step 1.} {\em There exists a lifting of the morphism $A\rightarrow G$ to
an element $\phi \in F(A)$ with $({\bf 3}^{\ast} - 3)^2\phi =0$.}
The cohomology of $E$ over the affine $S$ is zero, so
$$
0\rightarrow E(A)\rightarrow F(A)\rightarrow G(A)\rightarrow 0
$$
is exact. Let $\alpha : A\rightarrow G$ denote the morphism in the
presentation above, and choose $f\in F(A)$ mapping to $\alpha$. Then write
$$
({\bf 3}^{\ast} - 3)f = \sum e_i
$$
with $e_{\lambda} \in E(A)^{\lambda}$ (thus ${\bf 3}^{\ast} e_{\lambda} =
3^{\lambda}e_{\lambda}$). Let
$$
\phi = f-\sum c_{\lambda} e_{\lambda}
$$
for $c_{\lambda} = (3^{\lambda }-3)^{-1}$ when $\lambda \neq 1$ (and $c_1=0$).
We then have
\begin{eqnarray*}
({\bf 3}^{\ast} - 3)\phi &=& ({\bf 3}^{\ast} - 3)f -\sum c_{\lambda}({\bf
3}^{\ast} - 3) e_{\lambda} \\
&=&\sum e_{\lambda}-\sum c_{\lambda}(3^{\lambda }-3)e_{\lambda} \\
&=& e_1.
\end{eqnarray*}
On the other hand, $({\bf 3}^{\ast} - 3)e_1=0$, so we get
$$
({\bf 3}^{\ast} - 3)^2\phi =0.
$$
In other words, $\phi$ is in the generalized eigenspace for the eigenvalue $3$
of the transformation ${\bf 3}^{\ast}$.
{\em Step 2.} {\em The extension $F$ satisfies the Krull property of Lemma
\ref{I.f}: if $f\in F(X)$ such that for any artinian $Y\rightarrow X$,
$f|_Y=0$, then $f=0$.}
Under these hypotheses, $f$ maps to an element $g\in G(X)$ satisfying the same
vanishing, so by Lemma \ref{I.f} we have $g=0$; thus $f$ comes from an element
$e\in E(X)$. This element again satisfies the same vanishing, so by Lemma
\ref{I.f}, $e=0$.
{\em Step 3.} {\em If $A$ is a vector scheme and $F$ is an extension of two
vector sheaves, then any element $\phi \in F(A)$ with $({\bf 3}^{\ast} -
3)^2\phi =0$ is a morphism of sheaves of abelian groups from $A$ to $F$.}
Suppose $Y$ is artinian, and $G$ is a vector sheaf over $S$. Then the
functor $Z\mapsto G(Y\times Z )$ is represented by a vector space $G_Y$ over
$k$. If $F$ is an extension of two finite vetor sheaves $E$ and
$G$, then let $F_Y$ denote the functor $Z\mapsto F(Y\times Z)$. We obtain an
extension $$
0\rightarrow E_Y \rightarrow F_Y \rightarrow G_Y \rightarrow 0
$$
in the category of sheaves of abelian groups over $Spec (k)$. But since the
cohomology of the affine space $G_Y$ with coefficients in the additive group
$E_Y$ vanishes, there is a lifting of the identity to a section $u\in F_Y(G_Y)$.
Using $u$ we obtain an isomorphism of functors $F_Y \cong E_Y \times G_Y$, so
$F_Y$ is a scheme. Since $F_Y$ is a sheaf of abelian groups, $F_Y$ is an
abelian group-scheme over $k$. Since it is an extension of two additive
groups, it is additive. Our element of $F(A)$ now gives a morphism of schemes
$\phi _Y: A_Y \rightarrow F_Y$ between these two vector spaces.
We still have $({\bf 3}^{\ast}-3)^2)\phi _Y =0$. But $F_Y(A_Y)$ decomposes
into eigenspaces
$$
F_Y(A_Y)=\bigoplus F_Y(A_Y)^{\lambda}
$$
where $f\in F_Y(A_Y)^{\lambda} \Leftrightarrow f(ta)= t^{\lambda}f(a)$.
In particular, $F_Y(A_Y)$ is the $3^{\lambda}$-eigenspace for ${\bf
3}^{\ast}$. But since the space $F_Y(A_Y)$ is the direct sum of eigenspaces,
the generalized eigenspaces are equal to the eigenspaces, so $\phi _Y \in
F_Y(A_Y)^1= Hom (A_Y, F_Y)$. In particular, if $u,v\in A_Y(Spec (k))= A(Y)$
then $\phi (u+v)= \phi (u)+\phi (v)$ in $F_Y(Spec (k))= F(Y)$.
Now suppose $X$ is any element of ${\cal X} /S$. We show that $\phi : A(X)
\rightarrow
F(X)$ is a morphism of abelian groups. Suppose $u,v\in A(X)$. Let $f=\phi
(u+v)-\phi (u)-\phi (v)\in F(X)$. By the previous paragraph, for any
$Y\rightarrow X$ with $Y$ artinian, we have $f|_Y=0$. But the Krull property
of Step 2 then implies that $f=0$. This shows that $\phi$ is a morphism of
sheaves of abelian groups.
{\em Step 4.} {\em There is a surjection from a vector scheme to $F$.}
The direct sum of the morphism $U\rightarrow F$ with our lifting $\phi :
A\rightarrow F$ gives a surjection $U\oplus A \rightarrow F \rightarrow 0$.
In fact, this fits into a diagram
$$
\begin{array}{ccccccc}
0 \rightarrow &U& \rightarrow & U\oplus A& \rightarrow & A&\rightarrow 0\\
& \downarrow & & \downarrow & & \downarrow & \\
0 \rightarrow & E &\rightarrow & F & \rightarrow & G & \rightarrow 0\\
& \downarrow & & \downarrow & & \downarrow & \\
& 0 & & 0 & & 0 . &
\end{array}
$$
{\em Step 5.} {\em There is a surjection from a vector scheme to the
kernel of $U\oplus A \rightarrow F$ (proving the theorem).}
Taking the kernels along the top row of the above diagram gives
$$
\begin{array}{ccccccc}
0 \rightarrow & K & \rightarrow & L & \rightarrow & M & \rightarrow 0\\
& \downarrow & & \downarrow & & \downarrow & \\
0 \rightarrow &U& \rightarrow & U\oplus A& \rightarrow & A&\rightarrow 0\\
& \downarrow & & \downarrow & & \downarrow & \\
0 \rightarrow & E &\rightarrow & F & \rightarrow & G & \rightarrow 0\\
& \downarrow & & \downarrow & & \downarrow & \\
& 0 & & 0 & & 0 . &
\end{array}
$$
But $K$ and $M$ are vector sheaves, and we have surjections
$V\rightarrow K \rightarrow 0$ and $B\rightarrow M \rightarrow 0$. By
repeating the above argument in this case, we obtain a surjection
$$
V\oplus B \rightarrow L \rightarrow 0,
$$
finally giving our presentation
$$
V\oplus B \rightarrow U\oplus A \rightarrow F \rightarrow 0.
$$
Thus $F$ is a vector sheaf.
\hfill $\Box$\vspace{.1in}
Our abelian category ${\cal V}$ of vector sheaves is therefore closed
under extensions of sheaves of abelian groups.
\subnumero{Duality}
Suppose $F, G$ are vector sheaves. We have defined $Hom (F,G)$ which
is for now a sheaf of abelian groups. Put
$$
F^{\ast} := Hom (F, {\cal O} ).
$$
If $\phi :F\rightarrow G$ is a morphism of vector schemes, then we obtain
a morphism $\phi ^t:G^{\ast} \rightarrow F^{\ast}$, and the construction $\phi
\mapsto \phi ^t$ preserves composition (reversing the order, of course).
\begin{lemma}
\mylabel{I.l}
{\rm (Hirschowitz \cite{Hirschowitz})}
Suppose
$$
0\rightarrow U\rightarrow V\rightarrow W \rightarrow F \rightarrow 0
$$
is an exact sequence with $U$, $V$ and $W$ vector schemes. Then taking the dual
gives an exact sequence
$$
0\rightarrow F^{\ast}\rightarrow W^{\ast}\rightarrow
V^{\ast} \rightarrow U^{\ast} \rightarrow
0.
$$
\end{lemma}
{\em Proof:}
Note first that the compositions are zero, since taking the dual is
compatible with compositions (and the dual of the zero map is zero!).
The map $F^{\ast}\rightarrow W^{\ast}$ is injective
because $W\rightarrow F$ is surjective (so any morphism $F\rightarrow {\cal O}$
restricting to $0$ on $W$, must be zero).
The morphism $V^{\ast }\rightarrow U^{\ast}$ is surjective: if
$a:U\rightarrow {\cal O}$ is a morphism, it can be interpreted as a section of ${\cal O}
(U)^1$; but since $U\subset V$ is a closed subscheme, we can extend this to a
section $a'\in {\cal O} (V)$, then let $a''$ be the component of $a'$ in ${\cal O}
(V)^1$; restriction from ${\cal O} (V)$ to ${\cal O} (U)$ is compatible with the ${\bf G}
_m$ action, hence with the decomposition of Lemma \ref{I.h}, so $a''$ restricts
to $a$.
Suppose $b : W\rightarrow {\cal O}$ restricts to zero on $V$; then it factors
through the quotient sheaf $F=W/V$, so it comes from $F^{\ast}$. Thus the
sequence is exact at $W^{\ast}$.
We still have to prove exactness at $V^{\ast}$. Choose embeddings
$U\hookrightarrow {\cal O} ^m$ and $W\hookrightarrow {\cal O} ^n$. Then extend the first
to a function $V\rightarrow {\cal O} ^m$; combining with the second we obtain
$V\hookrightarrow {\cal O} ^{m+n}$, fitting into a diagram
$$
\begin{array}{ccccccc}
0\rightarrow & {\cal O} ^m& \rightarrow & {\cal O} ^{m+n} & \rightarrow & {\cal O} ^n &
\rightarrow 0 \\
&\downarrow &&\downarrow && \downarrow & \\
0\rightarrow & U& \rightarrow & V & \rightarrow & W & .
\end{array}
$$
Furthermore, $U= {\cal O} ^m \cap V$ as subschemes of ${\cal O} ^{m+n}$ (by the
injectivity of $W\rightarrow {\cal O} ^n$). Given a linear map $\lambda :
V\rightarrow {\cal O}$ such that $\lambda |_{U}=0$, extend it to $\varphi : {\cal O}
^{m+n}\rightarrow {\cal O}$ such that $\varphi |_{{\cal O} ^{m}}=0$. Replace $\varphi$
by its linear part under the decomposition of Lemma \ref{I.h} (this will
conserve the property $\varphi |_{{\cal O} ^{m}}=0$ as well as the property of
restricting to $\lambda$). Our $\varphi$ now descends to a map ${\cal O}
^n\rightarrow {\cal O}$, restricting to $\varphi |_W$ which extends $\lambda$.
Note in the previous paragraph, we have used the following general fact: if
$X,Y\subset Z$ are closed subschemes of an affine scheme, and $\lambda \in {\cal O}
(X)$ such that $\lambda |_{X\cap Y}=0$, then there exists $\varphi \in {\cal O} (Z)$
such that $\varphi |_X=\lambda$ and $\varphi |_Y=0$. To prove this, let $I_X$,
$I_Y$ and $I_{X\cap Y}$ denote the ideals of $X$, $Y$ and $X\cap Y$ in the
coordinate ring ${\cal O} (Z)$. The definition of the scheme-theoretic intersection
$X\cap Y$ is that $I_{X\cap Y}= I_X+I_Y$, and our statement follows from the
translation that
$$
I_Y \rightarrow I_{X\cap Y}/I_X \subset {\cal O} (Z)/I_X
$$
is surjective.
We have completed the proof of the lemma.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{I.m}
The functor $F\mapsto F^{\ast}$ is an exact functor from the category of finite
vector sheaves, to the category of sheaves of abelian groups.
\end{corollary}
{\em Proof:}
Suppose
$$
0\rightarrow F' \rightarrow F \rightarrow F'' \rightarrow 0
$$
is an exact sequence of vector schemes. Choose presentations
$$
0\rightarrow U'\rightarrow V'\rightarrow W'\rightarrow F' \rightarrow 0
$$
and
$$
0\rightarrow U''\rightarrow V''\rightarrow W''\rightarrow F \rightarrow 0,
$$
and combine these into a presentation
$$
0\rightarrow U\rightarrow V\rightarrow W\rightarrow F \rightarrow 0
$$
with $U=U'\oplus U''$, $V=V'\oplus V''$ and $W=W'\oplus W''$ (using the method
of Theorem \ref{I.k}, which is easier since we now have the required lifts
automatically). These fit together into a diagram
$$
\begin{array}{ccccccc}
&0&&0&&0 & \\
& \downarrow & & \downarrow && \downarrow \\
0\rightarrow &U'& \rightarrow &U& \rightarrow &U''& \rightarrow 0 \\
& \downarrow & & \downarrow && \downarrow \\
0\rightarrow &V'& \rightarrow &V& \rightarrow &V''& \rightarrow 0 \\
& \downarrow & & \downarrow && \downarrow \\
0\rightarrow &W'& \rightarrow &W& \rightarrow &W''& \rightarrow 0 \\
& \downarrow & & \downarrow && \downarrow \\
0\rightarrow &F'& \rightarrow &F& \rightarrow &F''& \rightarrow 0 \\
& \downarrow & & \downarrow && \downarrow \\
&0&&0&&0 &
\end{array}
$$
where all the rows and columns are exact. Apply duality to this diagram; we
obtain a diagram with the arrows reversed, with the columns exact, by the lemma.
Furthermore, the same lemma shows that the upper three rows are exact (in fact,
this is easier because the rows in the original diagram are split, by
construction). This implies that the bottom row is exact, as desired.
\hfill $\Box$\vspace{.1in}
A {\em coherent sheaf} is a sheaf which (locally) has a presentation of the
form
$$
{\cal O} ^n\rightarrow {\cal O} ^m\rightarrow F\rightarrow 0.
$$
In particular, note that it is a vector sheaf. This coincides with the
usual definition: if $S$ is affine and $X\rightarrow S$ is a morphism, then
$F(X)=F(S)\otimes _{{\cal O} (S)}{\cal O} (X)$ (this is because the same is true for
${\cal O}$, and the presentation remains exact on the right after tensoring).
As usual, we can assume that a presentation as above exists globally over any
affine base.
\begin{corollary}
\mylabel{I.n}
The dual of a coherent sheaf is a vector scheme and vice-versa.
\end{corollary}
{\em Proof:}
Note that ${\cal O} ^{\ast}={\cal O}$. Taking the dual of a presentation of a coherent
sheaf gives
$$
0\rightarrow F^{\ast} \rightarrow {\cal O} ^m \rightarrow {\cal O} ^n,
$$
so $F^{\ast}$ is a vector scheme (the kernel here is a closed subscheme of ${\cal O}
^n$). Conversely, if $V$ is a vector scheme, take an exact sequence such as
given in Lemma \ref{I.b}, and apply the dual. We obtain a presentation for
$V^{\ast}$ as a coherent sheaf.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{I.o}
The dual of a vector sheaf is again a vector sheaf.
\end{corollary}
{\em Proof:}
If $F$ is a vector sheaf, choose a presentation
$$
U\rightarrow V\rightarrow F\rightarrow 0
$$
by vector schemes. Taking the dual gives
$$
0\rightarrow F^{\ast}\rightarrow U^{\ast}\rightarrow V^{\ast}.
$$
By the previous corollary, $U^{\ast}$ and $V^{\ast}$ are coherent sheaves, in
particular vector schemes. Thus $F^{\ast}$ is the kernel of a morphism
of vector sheaves, so $F^{\ast}$ is a vector sheaf.
\hfill $\Box$\vspace{.1in}
\begin{lemma}
\mylabel{I.p}
If $F$ is a vector sheaf, then $F^{\ast\ast}=F$ (via the natural
morphism).
\end{lemma}
{\em Proof:}
If $F$ is a vector scheme, this follows from the construction given in
Corollary \ref{I.n}: write $F=\ker (M)$ as the kernel of a matrix
$M:{\cal O} ^m\rightarrow {\cal O} ^n$; then $F^{\ast} = {\rm coker} (M^t)$ is
the cokernel
of the transpose matrix (and this $M^t$ is really just the transpose,
keeping the
same coefficients as in $M$). Finally, $F^{\ast\ast}=\ker (M^{tt})$, but the
transpose of the transpose is the same matrix $M=M^{tt}$, so $F=F^{\ast\ast}$.
(The same argument works for coherent sheaves, of course). If $F$ is any
vector scheme, choose a presentation
$$
U\rightarrow V\rightarrow F\rightarrow 0
$$
and take the double dual. Since $U^{\ast\ast}=U$ and $V^{\ast\ast}=V$ we get
$$
U\rightarrow V\rightarrow F^{\ast\ast}\rightarrow 0,
$$
so $F^{\ast\ast}=F$.
\hfill $\Box$\vspace{.1in}
We have now shown that duality is an exact contravariant involution on the
category ${\cal V} $ of vector sheaves, interchanging vector schemes and
coherent sheaves.
\begin{lemma}
\mylabel{I.q}
The vector schemes are projective objects in ${\cal V} $, and the coherent
sheaves are injective objects. There exist enough projectives and injectives
(assuming that $S$ is affine).
\end{lemma}
{\em Proof:}
The argument given above shows that a vector scheme $A$ is a projective object:
if $F\rightarrow G$ is a surjection of vector sheaves then, since $A$
is affine, $F(A)^1\rightarrow G(A)^1$ is surjective. By definition, every
vector sheaf admits a surjection from a vector scheme, so there are
enough projectives. By duality, the coherent sheaves are injective and there
are enough injectives.
\hfill $\Box$\vspace{.1in}
Taking the dual of the three step resolution by vector schemes shows that every
vector sheaf $F$ admits a resolution
$$
0\rightarrow F \rightarrow U\rightarrow V\rightarrow W\rightarrow 0,
$$
with $U$, $V$ and $W$ coherent sheaves (in particular, injective).
\subnumero{Internal $Hom$ and tensor products}
We begin with a corollary to the last lemma.
\begin{corollary}
\mylabel{I.r}
If $A$ is a vector scheme, then the functor $V\mapsto Hom (A,V)$ from ${\cal V} $
to the category of abelian sheaves, is exact. If $F$ is a coherent sheaf, then
the functor $V\mapsto Hom (V,F)$ is exact.
\end{corollary}
{\em Proof:}
If $S$ is affine, the functors $V\mapsto Hom (A,V)(S)$ and $V\mapsto Hom
(V,F)(S)$ are exact, by the lemma. But the restriction of a vector scheme or a
coherent sheaf, to any object $X\in {\cal X} /S$ is again a vector scheme or coherent
sheaf over $X$, so we obtain exactness over every affine object; and since
exactness is a local condition, we get exactness.
\hfill $\Box$\vspace{.1in}
\begin{lemma}
\mylabel{I.s}
If $F$ and $G$ are vector sheaves, then $Hom (F,G)$ is a vector
sheaf.
\end{lemma}
{\em Proof:}
Suppose $F$ and $G$ are vector schemes. Then the exact sequence
$$
0\rightarrow G\rightarrow {\cal O} ^a \rightarrow {\cal O} ^b
$$
yields an exact sequence
$$
0\rightarrow Hom (F,G)\rightarrow Hom (F, {\cal O} ^a) \rightarrow Hom (F,{\cal O} ^b);
$$
but the middle and right terms are direct sums of the dual $F^{\ast}$ which is
a vector sheaf, so the kernel $Hom (F,G)$ is a vector sheaf.
Now suppose $F$ is a vector scheme and $G$ is a vector sheaf; resolving
$G$ by vector schemes we obtain a resolution of $Hom (F,G)$ by vector
sheaves, from the previous sentence. Thus $Hom (F,G)$ is a vector sheaf
in this case too. Now suppose $F$ is a vector scheme, and choose a
resolution
$$
U\rightarrow V\rightarrow F\rightarrow 0
$$
by vector schemes. The functor $W\mapsto Hom (W,G)$ is contravariant and left
exact for any $G$, so we obtain an exact sequence
$$
0\rightarrow Hom (F,G)\rightarrow Hom (V,G)\rightarrow Hom (U,G).
$$
The middle and right terms are vector sheaves by the previous arguments,
so the kernel is also. This completes the proof in general.
\hfill $\Box$\vspace{.1in}
We now define the {\em tensor product} $F\otimes ^{{\cal V}} G$ of two vector
sheaves to be
$$
F\otimes _{{\cal O}}G:= (Hom (F, G^{\ast}))^{\ast}.
$$
Beware that this is not just the tensor product of sheaves of ${\cal O}$-modules
(although this will be the case if $F$ and $G$ are coherent sheaves).
We can also define the {\em cotensor product}
$$
F\otimes ^{{\cal O}} G:= Hom (F^{\ast} , G).
$$
Again, beware here that this is not equal to the tensor product. The
difference is seen in noting that the tensor product is right exact as usual,
whereas the cotensor product is left exact. (These exactness statements hold
in both variables since the tensor and cotensor products are commutative, as we
see below). Duality permutes the tensor and cotensor products:
$$
(F\otimes _{{\cal O}}G)^{\ast}= F^{\ast}\otimes ^{{\cal O}}G^{\ast}
$$
and
$$
(F\otimes ^{{\cal O}}G)^{\ast}= F^{\ast}\otimes _{{\cal O}}G^{\ast}.
$$
Define recursively
$$
V_1\otimes \ldots \otimes V_n := V_1\otimes (V_2\otimes \ldots \otimes V_{n})
$$
for either one of the tensor products.
By {\em multilinear form} $V_1\times \ldots V_n\rightarrow W$ we mean simply a
multilinear morphism of sheaves of groups. In the same way as above for the
linear morphisms, we obtain a vector sheaf $Mult(V_1\times \ldots \times V_n ,
W)$ of multilinear forms (denoted $Bil (\;\;\; )$ when $n=2$).
\begin{proposition}
\mylabel{I.s.1}
1. \,\, There is a natural isomorphism
$\alpha _{U,V}:Hom (U^{\ast}, V)\cong Hom (V^{\ast}, U)$ and $\alpha
_{U,V}\alpha _{V,U}$ is the identity.
\newline
2. \,\, There is a natural isomorphism
$$
Multi (V_1\times \ldots \times V_n, W)\cong
Hom (W^{\ast}, Multi (V_1\times \ldots \times V_n, {\cal O} ).
$$
3. \,\, There is a natural isomorphism
$$
Multi (V_1\times \ldots \times V_n , W)\cong Hom (V_1, Multi (V_2\times\ldots
\times V_n, W)).
$$
\end{proposition}
{\em Proof:}
In each case one defines natural maps in both directions and checks that the
two compositions are the identity.
\hfill $\Box$\vspace{.1in}
\begin{theorem}
\mylabel{I.s.2}
Suppose $V_i$ are vector sheaves, $i=1,\ldots , n$.
There is a multilinear form
$$
\mu : V_1\times \ldots V_n \rightarrow V_1 \otimes _{{\cal O}} \ldots \otimes _{{\cal O}}
V_n
$$
which is universal in the sense that if
$$
\phi: V_1\times \ldots \times V_n\rightarrow W
$$
is a
multilinear form then there is a unique morphism
$$
\psi : V_1 \otimes _{{\cal O}} \ldots \otimes _{{\cal O}}
V_n \rightarrow W
$$
such that $\phi = \psi \circ \mu $.
\end{theorem}
{\em Proof:}
Note first that for $n=2$ there is a natural bilinear map $U\times V
\rightarrow Hom (U, V^{\ast})^{\ast}= U\otimes _{{\cal O}}V$. Inductively this
gives the multilinear map for any $n$. The universal property says that the
induced map
$$
Hom (V_1\otimes _{{\cal O}}\ldots \otimes _{{\cal O}}V_n,W)\rightarrow Multi (V_1,\ldots
, V_n , W)
$$
should be an isomorphism. We prove this by induction on $n$, so we may suppose
it is true for $n-1$. By the definition of the multiple tensor product, the
quantity on the left is
$$
Hom (Hom (V_1, (V_2\otimes _{{\cal O}}\ldots \otimes _{{\cal O}}V_n)^{\ast})^{\ast},W).
$$
By part 1 of the proposition, this is equal to
$$
Hom (W^{\ast}, Hom (V_1, (V_2\otimes _{{\cal O}}\ldots \otimes
_{{\cal O}}V_n)^{\ast})).
$$
By induction, $(V_2\otimes _{{\cal O}}\ldots \otimes
_{{\cal O}}V_n)^{\ast}=Multi (V_2\times \ldots \times V_n,{\cal O} )$. Coupled with part
3 of the proposition we get
$$
Hom (W^{\ast}, Multi (V_1\times \ldots \times V_n,{\cal O} )
$$
which then is equal to the right hand side above, by part 2 of the proposition.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{I.s.3}
The tensor and cotensor products have natural commutativity and associativity
isomorphisms satisfying the usual constraints.
\end{corollary}
{\em Proof:}
For the tensor product this follows from the universal property and the fact
that the notion of multilinear form is independent of the order of the
variables. For the cotensor product this follows because it is the dual of the
tensor product.
\hfill $\Box$\vspace{.1in}
We can define symmetric and exterior powers, either with respect to the tensor
product or with respect to the cotensor product. Let $S_n$ denote the
symmetric group on $n$ objects. Let $V^{\otimes _{{\cal O}}n}$ (resp.
$V^{\otimes ^{{\cal O}}n}$) denote the tensor product (resp. cotensor product) of $n$
copies of a vector sheaf $V$. Then $S_n$ acts on $V^{\otimes _{{\cal O}}n}$
(resp. $V^{\otimes ^{{\cal O}}n}$) because of the commutativity and associativity.
This representation is completely reducible (this can be see object-by-object).
The components are vector sheaves; this can be seen by noting that the fixed
part is the kernel of a morphism of vector sheaves (the direct sum of $1-\gamma
$ for $\gamma \in S_n$); to get the other components, apply that to tensor
products with the irreducible representations of $S_n$. The trivial component
of
$V^{\otimes _{{\cal O}}n}$ (resp. $V^{\otimes ^{{\cal O}}n}$) is denoted by $Sym
_{{\cal O}}^n(V)$, the symmetric power (resp. $Sym ^{{\cal O}}_n(V)$, the symmetric
copower). The component corresponding to the sign representation is denoted by
$\bigwedge _{{\cal O}}^n(V)$, the exterior power (resp. $\bigwedge ^{{\cal O}}_n(V)$, the
exterior copower).
{\em Remark:}
There is a natural morphism $U\otimes _{{\cal O}}V\rightarrow U\otimes ^{{\cal O}}V$,
that is to say
$$
Hom (U,V^{\ast})^{\ast}\rightarrow Hom (U^{\ast}, V).
$$
However, this is not an isomorphism. A counterexample can be constructed by
looking for a case where the cotensor product is left exact but not right exact
(and noting that the tensor product is right exact), or vice-versa.
We have the expression $U\otimes ^{{\cal O}}V= Bil (U^{\ast}\times V^{\ast},{\cal O})$.
The inequality mentionned above implies in particular that there are
bilinear functions on $U^{\ast}\times V^{\ast}$ which are not sums of tensors
$u\otimes v$. This is a big difference from the case of schemes (for example
if $U$ and $V$ are coherent so that $U^{\ast}$ and $V^{\ast}$ are vector
schemes, then the bilinear functions {\em are} sums of tensor products.
\subnumero{Automorphisms of vector sheaves}
We end our discussion of vector sheaves by showing how they give examples of
presentable group sheaves.
\begin{lemma}
\mylabel{I.1.g.1}
If $V$ is a vector sheaf over $S$, then $V$ is presentable.
\end{lemma}
{\em Proof:}
Suppose $V$ is a vector scheme. Then taking $X=V$ and $R= V\times _VV=V$
we obtain the required presentation (note that the identity morphisms are
vertical)---so $V$ is $P4$, and then $P5$ by Corollary
\ref{I.z}. It follows from Theorem \ref{I.1.d} that the quotient of one vector
scheme by another is again $P5$; and finally that the quotient of a
vector scheme by such a quotient is $P5$. In view of the 3-stage resolution of
any vector sheaf by vector schemes, we obtain the lemma.
\hfill $\Box$\vspace{.1in}
One of the main examples of presentable group sheaves is given by the following
theorem.
\begin{theorem}
\mylabel{I.1.g}
Suppose $V$ is a vector sheaf over $S$. Then the group sheaf $Aut (V)$ is a
presentable.
\end{theorem}
{\em Proof:}
By the previous lemma and Lemma \ref{I.s}, $Hom (V,V)$ is $P5$. We can express
$$
Aut (V)\subset Hom (V,V)\times Hom (V,V)
$$
as the equalizer of the two morphisms
$$
\begin{array}{ccc}
Hom (V,V)\times Hom (V,V)&\rightarrow &Hom (V,V)\times Hom (V,V)\\
(a,b) & \mapsto & (ab,ba) \\
(a,b)&\mapsto & (1,1).
\end{array}
$$
Apply Lemma \ref{I.1.a} to obtain that $Aut (V)$ is $P4$, and then Corollary
\ref{I.z} to obtain that it is $P5$.
\hfill $\Box$\vspace{.1in}
A particular case of this construction is when $V$ is a coherent sheaf which we
denote by ${\cal F}$. There is a presentation
$$
U_2 \stackrel{\phi}{\rightarrow} U_1 \rightarrow {\cal F} \rightarrow 0
$$
where $U_i = {\cal O} ^{a_i}$. Let $Aut (U_2, U_1, \phi )$ denote the group sheaf of
automorphisms of the morphism $U_2 \rightarrow U_1$. Any such automorphism
gives
an automorphism of ${\cal F}$ so we have a morphism
$$
Aut (U_2, U_1, \phi )\rightarrow Aut ({\cal F} ).
$$
\begin{lemma}
\mylabel{surjection}
This morphism is a surjection onto $Aut ({\cal F} )$, and $Aut (U_2, U_1, \phi )$
is represesentable by a group scheme over $S$.
\end{lemma}
{\em Proof:}
The representability by a group scheme is clear, since $Aut (U_i)$ are group
schemes (isomorphic to $GL(a_i)$) and the condition of compatibility with $\phi$
is a closed condition so $Aut (U_2, U_1, \phi )$ is a closed subscheme of $Aut
(U_1)\times Aut (U_2)$.
Suppose $S' \rightarrow S$ is a scheme and $P\in S'$ is a point. Suppose
$\eta :
{\cal F} |_{S'}\rightarrow {\cal F} |_{S'}$ is an automorphism. Let
$$
U'_2 \stackrel{\phi '}{\rightarrow }U'_1 \rightarrow {\cal F} |_{S'}\rightarrow 0
$$
be a minimal resolution of ${\cal F} |_{S'}$ at the point $P$ (that is to say that
the
value $\phi '(P)$ is identically zero and the rank of $U'_2$ is minimal). Then
there are locally free $W_i\cong {\cal O} ^{b_i}$ on $S'$ such that $U_i|_{S'} \cong
U'_i \oplus W$ and such that the map $\phi |_{S'}$ can be written in block
form with respect to this decomposition, with a morphism
$\psi '$ in the block of the $W_i$ and
the map $\phi '$ in the block of the $U'_i$, such that $\psi '$ is surjective.
Our
morphism $\eta$ extends to a morphism of resolutions
$U'_{\cdot} \rightarrow U'_{\cdot}$ which is an isomorphism near $P$ by the
minimality of the resolution (in fact the values $U'_i(P)$ are the $Tor
^i_{{\cal O} _{S'}}({\cal F} |_{S'}, k_P)$ and an isomorphism of ${\cal F} |_{S'}$ induces an
isomorphism on the $Tor ^i$). We can complete this with the identity in the
block of the $W_i$ to get an isomorphism of resolutions $U_i |_{S'}$ inducing
$\eta$. This gives the desired surjectivity.
\hfill $\Box$\vspace{.1in}
{\em Question:} Does a similar result hold for the automorphisms of any vector
sheaf?
\numero{Tangent sheaves of presentable sheaves}
Suppose $S'\rightarrow S$ is an $S$-scheme. Put
$$
Y:= S' \times Spec (k[\epsilon _1
,\epsilon _2, \epsilon _3]/(\epsilon _i^2, \epsilon _i\epsilon _j ))
$$
with the subschemes
$$
Y_i:= S' \times Spec (k[\epsilon _i ]/(\epsilon
_i^2))
$$
and
$$
Y_{ij}:= S' \times Spec (k[\epsilon _i ,\epsilon _j]/(\epsilon
_i^2, \epsilon _j^2, \epsilon _i\epsilon _j )).
$$
Note that $Y=Y_1\cup Y_2\cup Y_3$, and $Y_{ij}=Y_i\cup Y_j$, as well as
$Y_i\cap Y_k =
S'\subset Y$ and
$Y_{ij}\cap
Y_{jk}=Y_j$ (for $i\neq k$). It
should be stated explicitly that $Y_{ij}$ is the closed subscheme defined by the
ideal $(\epsilon _k)$, $k\neq i,j$; and $Y_i$ is the closed subscheme defined
by
the ideal $(\epsilon _j,\epsilon _k)$, $j,k\neq i$.
We need a weaker version of the notion of verticality.
We say that a morphism ${\cal F} \rightarrow {\cal G}$ of sheaves is {\em $T$-vertical}
if it satisfies the lifting property $Lift_2(Y_{ij};Y_i ,Y_j)$
and $Lift _3(Y; Y_{12},Y_{23}, Y_{13})$ (for any $S'$). Note that these systems
satisfy the retraction hypotheses in the lifting property, so the property of
$T$-verticality is weaker than the property of verticality.
The result of Theorem \ref{I.u} holds also for $T$-verticality, so the class
${\cal T}$ of $T$-vertical morphisms satisfies the axioms M1-M4. In particular
the properties $P4$ and $P5$ imply $P4({\cal T} )$ and $P5({\cal T})$ respectively.
The advantage of th weaker property of $T$-verticality is that if
$X\rightarrow Z$
is a morphism of schemes over $S$, then it is $T$-vertical. To prove this, note
that the properties $Y=Y_1\cup Y_2\cup Y_3$, $Y_{ij}=Y_i\cup Y_j$,
$Y_i\cap Y_k =
S'\subset Y$ and $Y_{ij}\cap
Y_{jk}=Y_j$ mean that for defining morphisms from $Y$ to a scheme (or from
$Y_{ij}$ to a scheme) it suffices to have compatible morphisms on the $Y_{ij}$
or on the $Y_i$.
({\em Caution:} We did not include the lifting condition $Lift _1(Y_1; S')$
in the notion of $T$-verticality; morphisms of schemes do not necessarily
satisfy this lifting property!)
The conclusion of the previous paragraph and property $M1$ for
$T$-verticality is that if ${\cal F}$ is a $P4({\cal T} )$ sheaf then the structural
morphism $p:{\cal F} \rightarrow S$ is $T$-vertical; thus $P4({\cal T} )\Leftrightarrow
P5({\cal T})$.
\begin{lemma}
\mylabel{I.1.e.1}
Suppose $f:{\cal F}\rightarrow {\cal G} $ is a morphism of $P4$ sheaves.
Then $f$ is $T$-vertical.
Furthermore, the
liftings in the lifting properties for $f$, for the systems
$(Y_{ij};Y_i ,Y_j)$
and $(Y; Y_{12},Y_{23}, Y_{13})$, are unique.
\end{lemma}
{\em Proof:}
For $T$-verticality, we can choose vertical surjections $X\rightarrow {\cal F}$ and
$Y\rightarrow {\cal G}$ so that there is a lifting $X\rightarrow Y$. This lifting
is $T$-vertical since it is a morphism between schemes (cf the above remark).
Hence the composition $X\rightarrow {\cal G}$ is $T$-vertical. By Theorem \ref{I.u},
part 4 for $T$-verticality, applied to the composition $X\rightarrow {\cal F}
\rightarrow {\cal G}$, we obtain $T$-verticality of the morphism $f$.
To prove the uniqueness, note that liftings to schemes are unique
since $Y=Y_1\cup Y_2\cup Y_3$ and $Y_{ij}=Y_i\cup Y_j$. Then descend the
uniqueness down from $X$ to ${\cal F}$ where $X\rightarrow {\cal F}$ is the vertical
(hence
$T$-vertical) morphism provided by the property $P4$. This descent of the
uniqueness property is immediate from the lifting property for $X\rightarrow
{\cal F}$.
\hfill $\Box$\vspace{.1in}
In the statement of the following theorem, the condition is $P4$ and not
$P4({\cal T} )$ (i.e. that isn't a misprint).
\begin{theorem}
\mylabel{I.1.f}
Suppose ${\cal F}\rightarrow {\cal G} $ is a morphism of $P4$ sheaves on $S$. Suppose $u
:S\rightarrow {\cal F}$ is a section. Then the relative tangent sheaf $T(f )_{u}$
over $S$, defined by
$$
T({\cal F} )_{u} (b:S'\rightarrow S):= \{ \eta :S' \times Spec (k[\epsilon
]/(\epsilon ^2))\rightarrow {\cal F}\;\; :\;\;\;\; f \eta = fubp_1 \;\; \mbox{and}
\;\; \eta |_{S'}= ub \} ,
$$
has a natural structure of sheaf of abelian groups making it a vector sheaf.
\end{theorem}
{\em Proof:}
We first define the natural abelian group structure on this sheaf. Suppose
$$
\eta _i: S'\times Spec(k[\epsilon _i]/(\epsilon _i^2))\rightarrow {\cal F}
$$
are sections of $T(f )_u$ over $S'$ ($i=1,,\ldots , 3$). ({\em Nota:} for the
definition of the group law we only need $i=1,2$; we need $i=1,2,3$ only to
check that it is associative.) Here (and below) we attach various subscripts to
the variables $\epsilon$. Use the notations established above:
$$
Y:= S' \times Spec (k[\epsilon _1
,\epsilon _2, \epsilon _3]/(\epsilon _i^2, \epsilon _i\epsilon _j ))
$$
with the subschemes
$$
Y_i:= S' \times Spec (k[\epsilon _i ]/(\epsilon
_i^2))
$$
and
$$
Y_{ij}:= S' \times Spec (k[\epsilon _i ,\epsilon _j]/(\epsilon
_i^2, \epsilon _j^2, \epsilon _i\epsilon _j )).
$$
Note that $Y=Y_1\cup Y_2\cup Y_3$, and $Y_{ij}=Y_i\cup Y_j$. Again $Y_{ij}$ is
the closed subscheme defined by the ideal $(\epsilon _k)$, $k\neq i,j$; and
$Y_i$ is the closed subscheme defined by the ideal $(\epsilon _j,\epsilon _k)$,
$j,k\neq i$. The systems $(Y_{ij};Y_i ,Y_j)$ and $(Y; Y_{12},Y_{23}, Y_{13})$
satisfy a unique lifting property for the morphism $f$ (Lemma \ref{I.1.e.1}).
Note that $Y_{ij}\cap Y_{jk}=Y_j$ (for $i\neq k$). We apply this first to the
system $(Y_{ij}; Y_i,Y_j)$. There is a unique morphism
$$
\eta _{ij}: Y_{ij}\rightarrow {\cal F}
$$
over the base morphism $Y_{ij}\rightarrow S\rightarrow {\cal G}$ and
agreeing with $\eta _i$ (resp. $\eta _j$) on $Y_i$ (resp. $Y_j$). Let
$$
\delta _{ij}: S' \times Spec (k[\epsilon ]/(\epsilon ^2))\rightarrow
Y_{ij}
$$
be the diagonal and---for future use---let
$$
\delta _{123}: S' \times Spec (k[\epsilon ]/(\epsilon ^2))\rightarrow
Y
$$
be the triple
diagonal.
Then we put
$$
\eta _i+\eta _j:= \eta _{ij} \circ \delta _{ij} .
$$
This gives a composition which is obviously commutative (the definition is
symmetric in the two variables). To check that it is associative, apply unique
lifting for $(Y,Y_{ij})$ to get a unique $\eta _{123}: Y\rightarrow {\cal F}$
restricting to the $\eta _{ij}$ on $Y_{ij}$. Next, note that the triple
diagonal
is equal to the composition of $1\times \delta _{23}$ with the diagonal
$$
S' \times Spec (k[\epsilon _0]/(\epsilon _0^2))\rightarrow
Spec (k[\epsilon _1,\epsilon ]/(\epsilon _1^2, \epsilon ^2, \epsilon
_1\epsilon )).
$$
Using this, we get
$$
\epsilon _1+(\eta _2 +\eta _3)= \eta _{123}\circ \delta _{123}.
$$
Similarly, we have
$$
(\epsilon _1+\eta _2) +\eta _3= \eta _{123}\circ \delta _{123},
$$
giving associativity.
The identity element (which we denote by $0$) is the composition
$$
S'\times Spec (k[\epsilon ]/(\epsilon ^2))\rightarrow S \rightarrow {\cal F} .
$$
This construction is natural: if
$$
\begin{array}{ccc}
{\cal F} & \rightarrow & {\cal F} ' \\
\downarrow &&\downarrow \\
{\cal G} & \rightarrow & {\cal G} '
\end{array}
$$
is a diagram with vertical arrows vertical, and if $u:S\rightarrow {\cal F}$
is a section projecting to $u':S\rightarrow {\cal F} '$, then composition with the
morphism ${\cal F} \rightarrow {\cal F} '$ respects the conditions in the definition of
the tangent sheaves, and so it gives a morphism $T(f)_u\rightarrow
T(f')_{u'}$. The addition we have defined is natural, so this morphism
of tangent sheaves respects the addition (it also respects the identity).
The inverse is obtained by applying the
automorphism $\epsilon \mapsto -\epsilon$.
This completes the construction of the natural structure of sheaf of abelian
groups.
Next, we show that if
$$
{\cal F} \stackrel{a}{\rightarrow }{\cal G} \stackrel{b}{\rightarrow }{\cal H}
$$
is a sequence of morphisms of $P4$ sheaves, and if $u:S\rightarrow {\cal F}$ is
a section, then we have an exact sequence
$$
0\rightarrow T(a)_u\rightarrow T(ba)_u \rightarrow T(b)_{au}
$$
We certainly get such a sequence with the
composition being zero. Furthermore, $T(a)_u$ is the subsheaf of $T(ba)_u$
consisting of those elements projecting to zero in $T(b)_{au}$ (this follows
immediately from the definition).
Furthermore, if $a$ is vertical, then the sequence is exact on the right. This
follows from the lifting property in the definition of vertical, in view of the
fact that $S'$ is a retraction of $S'\times Spec (k[\epsilon ]/(\epsilon ^2))$.
(Note that we have not required this lifting property in the definition of
$T$-verticality.)
Let $p: {\cal F} \rightarrow S$ denote the structural morphism for a $P4$ sheaf
${\cal F}$,
and define the tangent sheaf $T({\cal F} )_u:= T(p)_u$. If $f:{\cal F} \rightarrow
{\cal G}$ is
a morphism of $P4$ sheaves, the exact sequence of the previous paragraph becomes
$$
0\rightarrow T(f)_u\rightarrow T({\cal F})_u \rightarrow T({\cal G} )_{fu}.
$$
Again, if $f$ is vertical then this sequence is exact on the right also.
Finally, we show that if ${\cal F}$ is $P4$ then $T({\cal F} )_u$ is a vector sheaf. The
above exact sequence implies that if $f$ is a morphism of $P4$ sheaves then
$T(f)_u$ is a vector sheaf. Let $f: X\rightarrow {\cal F}$ be the vertical morphism
given by the property $P4$. Since the question is etale local on $S$, we may
assume that our section $u: S\rightarrow {\cal F}$ lifts to a section $v:
S\rightarrow X$. We have an exact sequence
$$
0\rightarrow T(f)_v \rightarrow T(X)_v\rightarrow T({\cal F} )_u\rightarrow 0.
$$
Note that $T(X)_v$ is a vector scheme (an easy thing to see---it is given by the
linear parts of the equations of $X$ at the section $v$).
Let $g:R\rightarrow X\times _{{\cal F}} X$ be the other vertical morphism given by
the property $P4$.
We claim that we have an exact sequence
$$
0\rightarrow T(X\times _{{\cal F}}X)_{(v,v)} \rightarrow T(X)_v \oplus T(X)_v
\rightarrow T({\cal F} )_u \rightarrow 0.
$$
To see this, note that an element of $T(X\times _{{\cal F}}X)_{(v,v)}$ consists of
an element of $T(X\times _SX)_{(v,v)}$ mapping to $T({\cal F} )_u\subset T({\cal F}
\times _S{\cal F} )_{(u,u)}$. Note that
$$
T(X\times _SX)_{(v,v)}=T(X)_v \oplus T(X)_v,
$$
and
$$
T({\cal F} \times _S{\cal F} )_{(u,u)}=T({\cal F} )_u\oplus T({\cal F} )_u
$$
with the map from $T({\cal F} )_u$ being the diagonal. The quotient of
$T({\cal F} \times _S{\cal F} )_{(u,u)}$ by the diagonal $T({\cal F} )_u$ is thus isomorphic
to $T({\cal F} )_u$ and we obtain the exact sequence in question. The surjectivity
on
the right is from surjectivity of $T(X)_v\rightarrow T({\cal F} )_u$.
Lift $(v,v)$ to a section $w:S\rightarrow R$.
The exact sequence for $g$ gives a surjection
$$
T(R)_w \rightarrow T(X\times _{{\cal F}}X)_{(v,v)}\rightarrow 0.
$$
Combining this with the above exact sequence, we obtain the right exact sequence
$$
T(R)_w \rightarrow T(X)_v \oplus T(X)_v
\rightarrow T({\cal F} )_u \rightarrow 0.
$$
Since $T(R)_w$ and $T(X)_v$ are vector schemes, this shows that $T({\cal F} )_u$
is a vector sheaf.
\hfill $\Box$\vspace{.1in}
\numero{The case $S=Spec (k)$}
We now analyse the definitions of the previous sections in the case where the
base scheme is $S=Spec (k)$ (a hypothesis we suppose for the rest of this
section).
{\em Caution:} We will use throughout this section certain properties of
vertical morphisms etc. which hold only in the context $S=Spec (k)$. The
reader should not extrapolate these properties to other cases.
Our first lemma is a preliminary version of the next lemma which we include
because the argument may be easier to understand in a simpler context.
\begin{lemma}
\mylabel{I.1.k}
Suppose $f: X\rightarrow Spec (k)$ is morphism of finite type.
Then $f$ is vertical if and only if $f$ is a smooth morphism.
\end{lemma}
{\em Proof:}
Suppose $X$ is smooth. Then the required lifting properties hold. Indeed, $X$
is etale locally a vector space, and Theorem \ref{I.u} (part 7) implies that
vector
spaces are vertical over $Spec(k)$.
Conversely, suppose $f$ is vertical, and suppose $x\in X$. The first claim is
that for any $v\in T(X)_x$ there is a smooth germ of curve $(C,0)$ mapping to
$(X,x)$ with tangent vector $v$ at the origin. Since $X$ is of finite type,
and by Artin approximation, it suffices to construct a compatible family of
morphisms
$$
\gamma _n:Spec (k[t]/t^n)\rightarrow X
$$
sending $Spec(k)$ to $x$ and with tangent vector $v$ (that is, the map
$\gamma _2$ represents $v$). Before starting the construction, choose a
morphism
$$
\mu : X\times X\rightarrow X
$$
with $\mu (x,y)=\mu (y,x)=y$ for any $y$ (the possibility of finding $\mu$
follows from the definition of verticality). We now construct $\gamma _n$ by
induction, starting with $\gamma _2$ given by $v$. Suppose we
have constructed $\gamma _{n}$ by the inductive procedure. Let $Y(n):= Spec
(k[r]/r^{n})\times Spec (k[s]/s^2)$. The composition gives a morphism
$$
\phi _n:= \mu \circ (\gamma _n , \gamma _2): Y(n)\rightarrow X.
$$
We will show that $\phi _n$ factors through the morphism
$$
d: Y(n)\rightarrow Spec (k[t]/t^{n+1})
$$
which is dual to the morphism
$$
k[t]/t^{n+1} \rightarrow k[r,s]/(r^n,s^2)
$$
$$
t\mapsto r+s.
$$
We will then choose $\gamma _{n+1}$ equal to the resulting morphism
$Spec (k[t]/t^{n+1})\rightarrow X$, that is with $\phi _n =\gamma _{n+1}d$.
Since $\gamma _n$ restricts to $\gamma _{n-1}$, and since we have chosen
$\gamma _n$ by the inductive procedure, we have that
$$
\phi _n|_{Y(n-1)}=\phi _{n-1} = \gamma _n d.
$$
Writing $X=Spec (A)$ (in a neighborhood of $x$) the morphism $\phi _n$
corresponds to
$$
\phi _n^{\ast}: A\rightarrow k[r,s]/(r^n,s^2).
$$
We have that $\phi _n^{\ast} (a)$ reduces modulo $r^{n-1}$ to
$d^{\ast}\gamma _n^{\ast} (a)$. Writing
$$
\gamma _n^{\ast}(a)= \sum _{j=0}^{n-1}b_jt^j
$$
we
have
$$
\phi _n^{\ast} (a)= \sum _{j=0}^{n-1}b_j (r+s)^j + \alpha r^{n-1} +
\beta r^{n-1}s.
$$
Write, on the other hand, the equation $\phi _n |_{Spec (k[r]/r^n)} = \gamma
_n$. We get that
$$
\phi _n^{\ast} (a) \sim \sum _{j=0}^{n-1}b_j r^j \;\; \mbox{mod} (s).
$$
This gives $\alpha = 0$ in the above equation. Finally, note that $(r+s)^n=
nr^{n-1}s$ modulo $(r^n, s^2)$. Thus we may set $b_n:= \beta / n$ and
get
$$
\phi _n^{\ast} (a)= \sum _{j=0}^{n}b_j (r+s)^j .
$$
Put
$$
\gamma _{n+1}^{\ast} (a):= \sum _{j=0}^{n}b_j t^j ,
$$
and we get the desired factorization $\phi _n = \gamma _{n+1}d$. This
completes the inductive step for the construction of the $\gamma _n$. We
obtain the desired formal curve and hence a curve $(C,0)$ as claimed.
{\em Remark:} Intuitively what we have done above is to integrate the vector
field on $X$ given by the tangent vector $v$ and the multiplication $\mu$.
Of course, the curve $C$ is an approximation to the integral curve, which
might only exist formally.
The next step in the proof of the lemma is to choose a collection of vectors
$v_1,\ldots , v_m$ generating $T(X)_x$, and to choose resulting curves $C_1,
\ldots , C_m$. Using the map $\mu$ in succession (or applying directly the
definition of verticality) we obtain a map
$$
\Phi : (U,0):=(C_1\times \ldots \times C_m , 0)\rightarrow (X,x),
$$
inducing the given morphisms on the factors $C_i$ (considered as subspaces of
the product by putting the origin in the other places). By construction the
differential $d\Phi _0$ is given by the vectors $v_1,\ldots , v_m$, in
particular it gives a surjection
$$
d\Phi _0: T(U)_0 \rightarrow T(X)_x \rightarrow 0.
$$
Note that $U$ is smooth of dimension $m$.
We claim that this implies $dim _x (X) \geq dim T(X)_x$. To see this, let $d:=
dim _x(X)$ and $n:= dim T(X)_x$. By semicontinuity, the dimension of the fiber
$\Phi ^{-1}(x)$ at the origin is at least equal to $m-d$. In particular, the
tangent space to the fiber has dimension at least $m-d$; but this gives a
subspace of dimension $m-d$ of $T(U)_0$ which maps to zero in $T(X)_x$; by the
surjectivity of $d\Phi _0$ we get $n \leq m-(m-d)=d$, the desired inequality.
Finally, it follows from this inequality that $X$ is regular at $x$ and hence
smooth at $x$ (and, of course, the inequality is an equality!). This proves
the lemma.
\hfill $\Box$\vspace{.1in}
\begin{lemma}
\mylabel{smooth}
Suppose $X$ and $Y$ are schemes of finite type over $k$ and $f: X\rightarrow Y$
is a morphism. Then $f$ is $Spec (k)$-vertical if and only if $f$ is smooth.
\end{lemma}
{\em Proof:}
Note first that if $f$ is smooth then it is etale-locally a product with affine
space so we get all of the lifting properties.
Suppose now that $f$ is vertical. If $Q\in Y$ and $P\in f^{-1}(Q)$
then $Lift _1(Y, Q)$ implies that, after replacing $Y$ by an etale neighborhood
of $Q$ we may suppose that there is a section $\sigma : Y\rightarrow X$
with $\sigma (Q)=P$. Let $T(X/Y)_{\sigma}$ denote the relative tangent vector
scheme along the section $\sigma$.
It is easy to see that the morphism $T(X/Y)_{\sigma}\rightarrow Y$ is
$Spec(k)$-vertical. We then obtain that the morphism
$$
\Gamma (Y, T(X/Y)_{\sigma})\rightarrow (T(X/Y)_{\sigma})_Q=T(f^{-1}(Q))_P
$$
is surjective, and this then implies that $T(X/Y)_{\sigma}$ is a vector bundle
over $Y$. The same argument as in the previous lemma allows us to
``exponentiate'' in a formal neighborhood of $P$, to get a map $\varphi$ from
$T(X/Y)_{\sigma}^{\wedge}$ (the formal completion in a neighborhood of
$0(Q)$) to $X$, which sends the zero section $0$ to $\sigma$ and whose tangent
map is the identity along $\sigma$.
We claim that if $S'$ is artinian local with a morphism
$S''\rightarrow X$ sending the origin to $P$, then the morphism factors via
$\varphi$ through a map $S'\rightarrow T(X/Y)_{\sigma}^{\wedge}$ sending the
origin to $0(Q)$. Prove this claim using the standard deformation theory
argument by induction on the length of $S'$: suppose $S''\subset S'$ is defined
by an ideal $I$ annihilated by the maximal ideal, and suppose we know the claim
for $S''$. Then there exists a map $S'\rightarrow T(X/Y)_{\sigma}^{\wedge}$
extending the known map on $S''$ since $T(X/Y)_{\sigma}^{\wedge}$ is a vector
bundle over $Y$. The space of such extensions is a principal homogeneous space
over $I\otimes _k (T(X/Y)_{\sigma})_Q$ whereas the space of extensions of
$S''\rightarrow X$ to morphisms $S'\rightarrow X$ is a principal homogeneous
space over $I\otimes _kT(f^{-1}(Q))_P$. The map $\varphi$ induces an isomorphism
$$
(T(X/Y)_{\sigma})_Q\cong T(f^{-1}(Q))_P
$$
so there is an extension to a map $S' \rightarrow T(X/Y)_{\sigma}^{\wedge}$
which projects to our given map $S'\rightarrow X$. This proves the claim.
Now we can prove that $X\rightarrow Y$ is formally smooth at $P$. If $S''\subset
S'$ are artinian local and if $a:S'\rightarrow Y$ is a map lifting over $S''$ to
a map $b:S'' \rightarrow X$ sending the origin to $P$, then we get
(from the previous claim) that
the map $b$ factors through a map $S'' \rightarrow
T(X/Y)_{\sigma}^{\wedge}$. Since $T(X/Y)_{\sigma}^{\wedge}$ is a vector bundle
and in particular smooth over $Y$, this extends to a map
$S' \rightarrow
T(X/Y)_{\sigma}^{\wedge}$. This extension projects into $X$ to an extension
$S' \rightarrow X$ of the map $b$. This shows formal smoothness. Since $X$
and $Y$ are of finite type, $f$ is smooth.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{I.1.l}
Suppose $G$ is a presentable sheaf of groups on ${\cal X} /Spec (k)$ (which is
equal to
${\cal X}$ in this case), and suppose $f:X\rightarrow G$ is a vertical morphism.
Then
$X$ is smooth over $Spec (k)$.
\end{corollary}
{\em Proof:}
The morphism $G\rightarrow Spec (k)$ is vertical by Theorem \ref{I.u} (7).
The composed morphism $X\rightarrow Spec (k)$ is vertical hence smooth
by Lemma \ref{I.1.k}.
\hfill $\Box$\vspace{.1in}
\begin{theorem}
\mylabel{I.1.m}
If $G$ is a presentable group sheaf on ${\cal X} /Spec (k)$ then it is represented
by a
smooth separated scheme of finite type over $k$ (in other words it is an
algebraic Lie group over $k$).
\end{theorem}
{\em Proof:}
We assume $k={\bf C}$ for this proof.
Choose vertical surjections
$f:X\rightarrow G$ and $R\rightarrow X\times _GX$. Note that $R\rightarrow G$
is
vertical, so $X$ and $R$ are smooth schemes of finite type. By adding some
factors of affine spaces we can assume that the components of $X$ and $R$ all
have the same dimension.
By the previous
section, the morphism $df:T(X)\rightarrow f^{\ast}T(G)$ is a morphism of vector
sheaves on $X$, hence it is a morphism of vector bundles. It is surjective, so
the kernel is a strict sub-vector bundle ${\cal F} \subset T(X)$.
For each $x\in X$ we have
$$
{\cal F} _x:= \ker (T(X)_x \rightarrow T(G)_{f(x)}).
$$
The morphism $p_1: R\rightarrow X$ is vertical (since $X\times _GX\rightarrow
X$ is the pullback of the vertical $X\rightarrow G$ by the morphism
$X\rightarrow G$, and $p_1$ is the composition of the vertical $R\rightarrow
X\times _GX$ with this projection). Therefore, by Lemma \ref{smooth} $p_1$
is smooth. Suppose $r\in R$ maps to $(x,y)\in X\times X$. Let $g\in G$ denote
the common image of $x$ and $y$. We have an exact sequence
$$
T(R)_r \rightarrow T(X)_x \oplus T(X)_y \rightarrow T(G)_g \rightarrow 0.
$$
From this we get that the image of the map on the left always has the same
dimension; in particular this shows that the map $T(R)\rightarrow (p_1,
p_2)^{\ast}T(X\times X)$ is strict. For any point $g$ in $G$ we can identify
$T(G)_g\cong T(G)_1$ by left multiplication. The morphism on the right in the
exact sequence then comes from a morphism of the form $p_1^{\ast}(\alpha
)-p_2^{\ast}(\alpha )$ where $\alpha : T(X) \rightarrow T(G)_1$ is obtained
from the differential of $f$ by the left-multiplication trivialization. This
morphism is a morphism of vector bundles from the tangent bundle of $X\times X$
to the constant bundle $T(G)_1$,
so its kernel is a distribution in the tangent
bundle of $X\times X$. The image of $R$ is an integral leaf of this
distribution. In particular, the image of $R$ is a smooth complex submanifold
of $X\times X$ (note that the map from $R$ to the leaf is smooth since, by the
above exact sequence, the differential is surjective at any point---this implies
that the image is open in the leaf).
Choose a subvariety $X'\subset X$ which is everywhere transverse to the
distribution ${\cal F}$, and which meets every subvariety of $X$ of the form
$p_2(p_1^{-1}(x))$ for $p_i$ denoting the projections $R\rightarrow X$. We may
assume that $X'$ is of finite type. Let $R'$ be the intersection of $X'\times
X'$ with the image of $R$ in $X\times X$. We claim that the morphism
$X'\rightarrow G$ is surjective and vertical, and that $R'= X'\times _GX'$. To
see this, note that by hypothesis $X'\times _X R\rightarrow X$ is surjective on
closed points. By our transversality assumptions this morphism is also smooth.
Thus any point in $X$ is equivalent via $R$ (etale-locally) to a point in
$X'$.
For verticality, it suffices to prove that $X'\times _GX \rightarrow X$ is
vertical (Theorem \ref{I.u}, parts 3 and 4). And for this it suffices to note
that $X'\times _X R \rightarrow X' \times _GX$ is surjective and vertical
(being the pullback of $X\times _XR\rightarrow X\times _GX$ by $X'\times
_GX\rightarrow X\times _GX$), that $X'\times _XR\rightarrow X$ is smooth and
hence vertical, and to apply Theorem \ref{I.u}, part 4. We get $X'\rightarrow
G$ surjective and vertical. If we put $R'' $ equal to the pullback of $R$ to
$X'\times X'$ then $R'' \rightarrow X'\times _G X'$ is surjective and vertical
(it being also the pullback of $R$ via $X'\times _GX' \rightarrow X\times
_GX$). The previous proof applied to this case shows that $R''$ is smooth over
its image $R'$, and that $R'$ is a smooth subvariety of $X'\times X'$. But
now, by our previous transversality assumptions, the projections $R'\rightarrow
X'$ are etale.
We can now conclude that $G$, which is the quotient of $X'$ by the equivalence
relation $R'$, is a smooth algebraic space. We will find an open subset
$U\subset G$ which is a smooth variety over $k$. In order to do this,
let $d$ be the maximum number of points in the fibers of $X'\rightarrow G$.
The fiber through a point $x$ is equal to $p_2(p_1^{-1}(x))$ where $p_i: R'
\rightarrow X'$ here denote the projections. Let $W\subset X$ be the set of
points $x$ where the maximum number $d$ of points in the fiber $p_1^{-1}(x)$ is
achieved. Since the morphism $p_1: R'\rightarrow X$ is etale, it is easy to see
that $W$ is an open subset, and that if we let $R'_W $ denote $p_1^{-1}(W)$
then $R'_W\rightarrow W$ is a finite etale morphism of degree $d$. On the
other hand, if $x\in W$ and $y$ is in the fiber through $x$ then $y$ is also in
$W$. This means that $p_2(R'_W)\subset W$. The correspondence
$$
x\mapsto p_2(p_1^{-1}(x))
$$
gives a morphism $\chi$ from $W$ to the symmetric product $W^{(d)}$ having image
in the complement of the singular locus. Then $W\times _{W^{(d)}}W= R'_W$. In
particular, the quotient of $W$ by the equivalence relation $R'_W$ is the
image of $\chi$. Note that $\chi $ is etale over its image, which is thus a
locally closed subscheme of $W^{(d)}$. This shows that the quotient of $W$ by
the equivalence relation is a scheme $U$ of finite type. It is also smooth.
The morphism $W\rightarrow G$ factors through $U\rightarrow G$.
We claim
that the morphism $U\rightarrow G$ is an open subfunctor, that is for any
$Y\rightarrow G$ the fiber product $U\times _GY$ is an open subset of $Y$.
The fiber product is the quotient of $W\times _GY$ by the
induced equivalence relation; and the quotient of $X'\times _GY$ by the
equivalence relation is equal to $Y$. Choosing local liftings
$Y\rightarrow X'$ we find that $X'\times _GY$ is the image of $R'\times
_{X'\times X'}(X'\times Y)\rightarrow X'\times Y$, that is it is the
pullback of $R'$. In particular it is a subscheme of $X'\times Y$. This
subscheme surjects to $Y$ by a vertical morphism, a morphism which is hence
smooth. The image of the open subset $W\times _GY$ (which is the
intersection of $X'\times _GY$ with $W\times Y$) is therefore an open set in
$Y$. This shows that $U\subset G$ is an open subfunctor.
We can choose a finite number of elements $g_i \in G(S)$ such that $g_i\cdot U$
cover $G$. For the finiteness use the surjection $X\rightarrow G$ with $X$ of
finite type (in particular, quasi-compact).
We now apply Grothendieck's theorem about representability which says
that if a
functor $G$ is a sheaf (in the Zariski topology, which is the case here since
Zariski is coarser than etale), and if it is covered by a finite number of open
subfunctors $G_i$ which are representable by schemes, then the functor $G$ is
representable by a scheme (the union of the schemes $G_i$). In our case the
$G_i$ are the $g_i\cdot U$, representable by $U$. Since $U$ is of finite type,
the union of a finite number of copies is again of finite type.
We obtain that $G$ is a scheme of finite type. Note that $U$ is smooth so $G$
is smooth (alternatively, use that any group scheme is smooth). To complete
the proof we just have to show that $G$ is separated. Note first that all
connected components of $G$ must have the same dimension, so we can speak of
the dimension of $G$ without problem.
Let $\Delta \subset G\times G$ denote the diagonal. It is preserved by the
diagonal left action of $G(k)$ on $G\times G$ (that is, the action $g(a,b)=(ga,
gb)$). The
complement $K:=\overline{\Delta}-\Delta$ is a closed subset of $G\times G$, of
dimension strictly smaller than the dimension of $G$. But $K$ is invariant
under the diagonal left action of $G(k)$, so its image $pr_1(K)\subset G$ is
invariant by the left action of $G(k)$. Since $dim (K)< dim (G)$ the image $pr
_1(K)$ (which is a constructible subset of dimension $\leq dim (K)$) is not
dense in $G$. On the other hand, if $K$ were nonempty then this image, being
left invariant, would contain a right translate of $G(k)$ which is Zariski
dense. This contradiction implies that $K$ is empty, in other words $G$ is
separated. This completes the proof of the theorem.
\hfill $\Box$\vspace{.1in}
{\em Application:}
Suppose $S$ is any base scheme of finite type over $Spec (k)$ now, and
suppose $S'\rightarrow S$ is an artinian scheme of finite type. Let $\pi :
S' \rightarrow Spec (k)$ denote the structural morphism. If $G$ is a
presentable
group sheaf over $S$ the pullback $G|_{S'}$ is presentable (Lemma
\ref{I.1.h}) and the direct image $\pi _{\ast} (G|_{S'})$ is presentable over
$Spec (k)$ (Lemma \ref{I.1.i}). By Theorem \ref{I.1.m}, $\pi _{\ast}(G|_{S'})$
is represented by a group scheme of finite type which we denote $G_{S'}$ over
$k$. We have
$$
G(S')= G_{S'}(Spec (k)).
$$
Furthermore, if $X\rightarrow G$ is a vertical surjection then we obtain a
scheme of finite type $X_{S'}= \pi _{\ast}(X|_{S'})$ with a morphism
$X_{S'}\rightarrow G_{S'}$. This morphism is smooth.
\numero{Local study of presentable group sheaves}
In this section we return to the case of general base scheme $S$ (in
particular, the hypothesis $S=Spec (k)$ is no longer in effect).
First we establish some notations for formal completions.
Suppose $G$ is a presentable group sheaf. Let $\widehat{G}$ denote the sheaf
which associates to $Y\in {\cal X}$ the set of values in $G(Y)$ which restrict to
the identity on $Y^{\rm red}$. More generally, use the same notation
$\widehat{{\cal F}}$ whenever ${\cal F}$ is a sheaf with a given section playing the role
of the identity section (usually the section in question is understood from the
context).
\subnumero{Local structure}
\begin{lemma}
\mylabel{I.1.n}
Suppose $G$ is a presentable group over a base $S$. Suppose $Z\rightarrow G$
is a vertical surjection with $Z$ an affine scheme of finite type over $S$. Let
$T(Z)_e\rightarrow S$ be the tangent vector scheme at a lift $e$ of the identity
section. For any $s\in S$ there is an etale neighborhood
$$
e(s)\in W \stackrel{p}{\rightarrow} Z
$$
and an etale $S$-morphism $q:U\rightarrow TZ$, such that $q=p$ over the
section $e$ (which maps to the zero section of $TZ$).
\end{lemma}
{\em Proof:}
Verticality of $Z\rightarrow G$ means that
we can choose a lifting of the multiplication of $G$ to $m: Z\times Z
\rightarrow Z$ such that $m(x,e)=x$ and $m(e,y)=y$.
Let $Q: Z\rightarrow Z$ be the automorphism $Q(x):= m(x,x)$. It has the
effect of multiplication by $2$ on the tangent scheme $TZ$ at the identity
section, because
$$
\frac{\partial }{\partial x}m(x,x)(e)= \frac{\partial }{\partial x}m(x,e)+
\frac{\partial }{\partial x}m(e,x)(e)= 2 \frac{\partial x}{\partial x} =2.
$$
If we embedd $Z\subset {\bf A}^N_S$ as
a closed subscheme with the identity section going to the origin-section,
then we
may extend $Q$ to a morphism $Q': {\bf A}^N_S\rightarrow {\bf A}^N_S$ such
that $Q'$ acts by multiplication by two on the tangent space at the
origin. Let $\widehat{{\bf A}^N_S}$ denote the formal completion of the
affine space along the origin-section. Then $Q'$ induces an
automorphism of $\widehat{{\bf A}^N_S}$, and it is well known---and easy
to see using power series---that such an automorphism is conjugate to its linear
part (since the eigenvalues are different
from $1$).
We obtain an automorphism $F:
\widehat{{\bf A}^N_S}\rightarrow \widehat{{\bf A}^N_S}$ such that
$F^{-1}\circ Q'\circ F = 2$. Let $\widehat{Z}\subset \widehat{{\bf A}^N_S}$
be the closed formal subscheme obtained by completing $Z$ at the identity
section. Note that $\widehat{Z}$ is preserved by $Q'$. Thus the image
$F(\widehat{Z})$ is a formal subscheme which is preserved by
multiplication by $2$. It follows that it is a cone, and in particular
that the linear parts of the equations defining $F(\widehat{Z})$ vanish
on $F(\widehat{Z})$. This means that $F(\widehat{Z})$ is included in its
tangent scheme $T(F(\widehat{Z}))$ along the identity section.
Translating back by $F$ we obtain an immersion
$$
\widehat{Z}\hookrightarrow TZ
$$
which is the identity on the tangent space at the identity section.
The image is a closed formal subscheme preserved by scalar multiplication.
For any artinian scheme $S'$ over $S$, $Z(S')$ is a
smooth scheme over $Spec (k)$ and $\widehat{Z(S')}\subset TZ(S')$ is
a closed formal subscheme at the origin, with the same Zariski tangent space,
and which is formally preserved by scalar multiplication. Therefore
$\widehat{Z(S')}\cong \widehat{TZ(S')}$. Now $\widehat{Z}(S')$ is the
inverse image of $e\in \widehat{Z(Spec (k))}$ via the map
$$
\widehat{Z(S')}\rightarrow \widehat{Z(Spec (k))}.
$$
The same is true for the tangent scheme $TZ$. From these properties we get
that $\widehat{Z}(S')\rightarrow \widehat{TZ}(S')$ is an isomorphism for any
$S'$.
As that holds true for all artinian schemes
$S'$ over $S$ we get that the morphism $\widehat{Z} \rightarrow \widehat{TZ}$
is an isomorphism.
Artin approximation now
gives the existence of such an isomorphism (inducing the same map on tangent
schemes along the identity section) over an etale neighborhood in $Z$, as
required for the lemma.
\hfill $\Box$\vspace{.1in}
\subnumero{Theory of the connected component}
We need to develop a suitable theory of the connected component of a
presentable group sheaf $G$.
\begin{theorem}
\mylabel{I.1.o}
If $G$ is a presentable group sheaf over $S$, then there is a unique subsheaf of
groups $G^0\subset G$ such that $G^0$ is presentable and such that for any
artinian $S$-scheme $S'$, we have $G^0(S')$ equal to the connected component of
$G(S')$ (when these are considered as algebraic groups over the ground field of
$S'$---cf the application at the end of the section on the situation over
$Spec (k)$).
\end{theorem}
{\em Proof:}
We first show existence. Let $Z\rightarrow G$ be a vertical surjection with $Z$
a scheme of finite type. Let $\sigma : S\hookrightarrow Z$ be the identity
section. We claim that there is an open neighborhood $U\subset Z$ of $\sigma
(S)$ such that for any artinian $S$-scheme $S'$, $U(S')$ is connected. By
Lemma \ref{I.1.n}, there is an etale neighborhood of the zero section $W
\rightarrow TZ$ and another etale morphism $W\rightarrow Z$ giving an etale
neighborhood of the section $\sigma$. We claim that (possibly throwing out a
closed subset of $W$ not meeting the section) we can assume that the $W(S')$
are connected. In what follows we refer to the lifting of the zero section of
$TZ$ as the section $\sigma$ of $W$.
For any given $S'$, artinian located at $s\in S$, there is a
surjection of vector spaces
$$
(TZ)(S')\rightarrow V_i \subset (TZ)(s),
$$
for some subspace $V_i$ which depends on $S'$.
If $W\rightarrow TZ$ is our etale morphism, then we have
$$
W(S')=W(s)\times _{TZ(s)}(TZ)(S')
= W(s)\times _{TZ(s)}V_i \times_{V_i}(TZ)(S'),
$$
since a point $S'\rightarrow TZ$ has a unique lifting to $W$ once the lifting
is specified on the closed point. Thus $W(S')$ is connected if and only if,
for all subspaces $V_i \subset (TZ)(s)$ we have that $W(s)\times _{TZ(s)}V_i$
is connected.
Let $Gr (TZ)\rightarrow S$ be the disjoint union of the grassmanian schemes of
subspaces of different dimensions. It is proper over $S$. We have a universal
subscheme
$$
{\cal V} \subset Gr (TZ)\times _S TZ.
$$
Note that the map ${\cal V} \rightarrow TZ$ is proper.
Let $\tilde{W}:= W\times _{TZ} {\cal V} $; this is an etale covering of ${\cal V}$, and
is proper over $W$. Let $\tilde{W}^N\subset \tilde{W}$ be the union of the
connected components in fibers which do not pass through the section $\sigma$
(relative to $Gr (TZ)$). Note that $\tilde{W}^N$ is a constructible subset of
$\tilde{W}$ (one can see this by noetherian induction). Let $W^N\subset W$
be the image of $\tilde{W}^N$. It is again a constructible subset. A point
$w\in W$ is in $W^N$ if and only if there exists a vector subspace $V_i \subset
(TZ)(s)$ such that $w$ is in a different connected component of $V_i \times
_{TZ}W$ from $\sigma (s)$. In particular, if we choose an analytic neighborhood
of the section $\sigma$ which is isomorphic to a tubular neighborhood of the
zero-section of $TZ$, then this analytic neighborhood doesn't meet $W^N$.
Thus there is a Zariski open neighborhood of $\sigma$ not meeting $W^N$.
Since taking a Zariski open subset doesn't affect connectivity (the schemes
$W_{S'}$ in question being smooth), we may replace $W$ by this open subset
and hence assume that $W^N$ is empty. From the discussion of the previous
paragraph, this implies that the $W_{S'}$ are connected, proving the first
claim.
Let $U$ be the image of $W$ in $Z$. Note that the set-theoretic image is an
open set and is equal to the image of the functor, since $W\rightarrow Z$ is
etale.
Let $\eta : Z\times _SZ \rightarrow Z$ be a lifting of the multiplication map
$(g,h)\mapsto gh$ such that $\eta (z, 1)= z$ and $\eta (1,z)=z$.
We claim that the composition law $Z\times _SZ \rightarrow G$ is a vertical
morphism. Note that $Z\times _SZ\rightarrow G\times _SG$ is vertical, so it
suffices to prove that the composition $G\times _SG\rightarrow G$ is
vertical. For this, notice that there is an isomorphism $G\times _SG\cong
G\times _S G$ sending $(a,b)$ to $(ab,b)$, and which interchanges the
multiplication and the first projection. Since the first projection is
vertical (this comes from the fact that $G\rightarrow S$ is vertical), we
obtain that the composition law is vertical, yielding the claim.
By Lemma \ref{I.1.c}, there exists a vertical surjection
$$
R\rightarrow (Z\times _SZ )\times _G (Z\times _S Z)
$$
with $R$ a scheme of finite type.
Let $G^0\subset G$ be the image of the morphism $U\times _SU\rightarrow G$.
Then the morphism $U\times _S U\rightarrow G^0$ is a vertical surjection, and
we have a vertical surjection
$$
R'\rightarrow (U\times _SU )\times _{G^0} (U\times _S U)
$$
obtained by letting $R'$ be the inverse image of
$(U\times _SU )\times _{G^0} (U\times _S U)$ in $R$.
Note that $R'$ is also equal to the fiber product
$$
U\times _SU\times _SU\times _SU\times _{Z\times _SZ\times _S Z\times _S Z}R,
$$
so $R'$ is a scheme of finite type over $S$.
We
claim that for any artinian $S'$, the $G^0(S')$ is equal to the connected
component of $G(S')$. To see this, note first of all that $G^0(S')$ is
connected (since it is the image of $U(S')\times U(S')$ which is connected).
And secondly, note that the morphism
$$
Z(S')\rightarrow G(S')
$$
is an open map (this is a map of smooth varieties---cf the section on what
happens over a field and in particular the application at the end). Therefore
the image of $U(S')$ is an open subset $V\subset G(S')$. It is connected
since $U(S')$ is connected.
The image of $(U\times _SU)(S')$ is equal to the
image of the multiplication map $V\times V\rightarrow G(S')$. It is easy to
see that if $V$ is a connected Zariski open subset of an algebraic group over a
field (containing the identity), then the image of the multiplication map is a
subgroup. Thus $G^0(S')$ is a subgroup of $G(S')$. It contains an open
neighborhood of the identity and it is connected, so it is equal to the
connected component. We claim now that $G^0$ is a sheaf of subgroups of $G$.
If $g,h\in G^0(S')$ then the product $gh$ restricts into $G^0(S'')$ for any
artinian ring $S''$ over $S'$. The sheaf $G^0$ is P2, hence it is B1 and B2
(Theorem \ref{I.t.2}). The inverse image of the section $gh$ by the morphism
$G^0\rightarrow G$ is again B1 and B2. This inverse image is nonempty
artinian $S''$. By Artin approximation, the inverse image has a section
locally over $S'$, and since this section is unique if it exists, it gives
a section $gh\in G^0(S')$.
We have now shown existence of $G^0$ as required by the theorem. For
uniqueness, suppose that $G^1$ were another candidate. Then $G^0$ and $G^1$
are both B1 and B2 subsheaves of $G$ having the same points over artinian
$S'$. Artin approximation implies that they are equal.
\hfill $\Box$\vspace{.1in}
We say that a presentable group sheaf $G$ is {\em connected} if $G= G^0$.
The above theorem immediately gives the characterization that $G$ is connected
if and only if $G(S')$ is connected for all artinian $S'$.
\begin{corollary}
\mylabel{connex} We have the following properties.
\newline
1. \, If $G$ is connected then any quotient group of $G$ is
connected;
\newline
2.\, Of $G$ and $H$ are connected then any extension of $G$ by $H$ is connected;
\newline
3. \, If $G$ is a connected group sheaf over a base $S$ and if $Y\rightarrow S$
is any morphism of schemes then $G|_{{\cal X} /Y}$ is a connected group sheaf over
$Y$; and
\newline
4. \, If $f:Y\rightarrow S$ is a finite morphism and if $G$ is a connected group
sheaf over $Y$ then $f_{\ast}(G)$ is a connected group sheaf over $S$.
\newline
5.\, If $G$ is any presentable group sheaf then the connected component $G^0$ is
the largest connected presentable subgroup.
\end{corollary}
{\em Proof:}
Items 1-3 are immediate from the characterization. To prove 4 note that if
$S'\rightarrow S$ then $f_{\ast}(G)(S')= G(Y\times _SS')$ and $Y\times _SS'$ is
artinian, so this latter group is connected, thus by the above characterization
$f_{\ast}(G)$ is connected. To prove 5 note that if $H$ is any connected
subgroup of $G$ then $H(S') \subset G^0(S')$ for all artinian $ S'$, hence
$H\subset G^0$.
\hfill $\Box$\vspace{.1in}
\subnumero{Finite presentable group sheaves}
We say that a presentable group sheaf $G$ is {\em finite} if $G^0=\{ 1\}$.
If $G$ is any presentable group sheaf, then the connected component $G^0$ is a
normal subgroup sheaf, and the quotient $C:=G/G^0$ is again presentable. Over
artinian $S'$, this quotient is just the group of connected components, in
particular the connected component is trivial. Thus $C$ is finite.
\begin{lemma}
\mylabel{I.1.p}
If $G$ is a finite presentable group sheaf, then there is an integer $N$
such that for any henselian local $S$-scheme $S'$ (with algebraically closed
residue field), the number of elements in $G(S')$ is less than or equal to $N$.
\end{lemma}
{\em Proof:}
We first treat the case where $S'$ is artinian local with algebraically closed
residue field. Let $Z\rightarrow G$ and $R\rightarrow Z\times _GZ$ be the
vertical surjections given by the fact that $G$ is $P4$. There is an etale
neighborhood $U\rightarrow Z\times _SZ$ of the diagonal such that $U$ is
isomorphic to an etale neighborhood of the zero section in the total scheme $TZ$
(and this isomorphism is compatible with the first projection to $Z$). This is
seen as in the argument above. Furthermore, as above we may assume that the
fibers of the first projection $U\rightarrow Z$ are connected (over any artinian
scheme). Then for any artinian scheme $S'\rightarrow U$, the two elements of
$G(S')$ obtained from the two projections $U\rightarrow Z$ are the same, by the
hypothesis that $G$ is finite. (To see this, compare $(a,b): S'\rightarrow U$
with $(a,a): S\rightarrow U$; they are in the same fiber over $a$, and this
fiber is connected, so they have to have the same image in $G(S')$.) Thus, any
artinian subscheme of $U$ lifts into $R$. This implies that there is (locally
in the etale topology) a lifting $U\rightarrow R$. Let $V\subset Z\times _SZ$
be the image of $U$. It is a Zariski neighborhood of the diagonal, and locally
there is a lifting from $V$ into $R$. Let $F\subset Z\times _SZ$ be the
reduced closed subscheme corresponding to the closed subset which is the
complement of $V$. Suppose $Y\rightarrow S$ is an artinian local scheme (with
acrf). If $(\alpha _1,\ldots , \alpha _n)$ is an $n$-tuple of distinct
points of $G(Y)$, then there is a lifting $(a_1,\ldots , a_n) \in Z\times _S
\ldots \times _S Z(Y)$ such that for any $i,j$ we have that $(a_i, a_j):
Y\rightarrow Z\times _SZ$ is not contained in $V$. In particular, the
reduced point $(a_i,a_j)^{\rm red}$ is contained in $F$. Thus the reduced
point
$(a_1,\ldots , a_n)^{\rm red}$ is contained in the closed subscheme
$$
F^{(n)}:= \bigcap _{i,j} pr_{ij}^{-1}(F)\subset X\times _S\ldots \times _SX.
$$
We claim that there is an $n$ such that $F^{(n)}$ is empty. For any
$(x_1,\ldots , x_k)\in F^{(k)}$, let
$$
\Phi (x_1,\ldots , x_k):= \{ y\in X, \;\; \pi (y)= \pi (x_i)\in S,\;\;
(y,x_1,\ldots , x_k)\in F^{(k+1)}\} .
$$
Note that these are closed subschemes of $X$ with strict inclusions
$$
\Phi (x_1,\ldots , x_k) \subset \Phi (x_1,\ldots , x_{k-1}).
$$
Furthermore, $\Phi (x_1,\ldots , x_k)$ varies algebraically with $(x_1,\ldots ,
x_k)$.
Let $d=dim (X)$ and let $\Lambda = {\bf N} ^d$ with the lexicographic ordering
giving the most importance to the $d$th coordinate. For any algebraic set
$Y$ of dimension $\leq d$, let $\lambda (Y)= (\lambda _1, \ldots , \lambda _d)$
be defined by setting $\lambda _d$ equal to the number of irreducible components
of dimension $d$. Note that if $Y'\subset Y$ is a strict inclusion of a
closed subset then $\lambda (Y')< \lambda (Y)$. Let $\Lambda ^{(k)}$ be the
finite set of all $\lambda (\Phi (x_1,\ldots , x_k))$ for $(x_1,\ldots ,
x_k)\in F^{(k)}$ (it is finite because
$\Phi (x_1,\ldots , x_k)$ varies algebraically with $(x_1,\ldots , x_k)$).
Introduce an order relation on subsets $\Sigma \subset \Lambda$ by saying
$$
\Sigma < \Sigma \; \Leftrightarrow \forall \sigma \in \Sigma ,\, \exists \sigma
' \in \Sigma ',\;\; \sigma < \sigma ' .
$$
Then the sequence $\Lambda ^{(k)}$ is a sequence of finite subsets which is
strictly decreasing for this order relation. We claim that this implies (by
combinatorics) that one of the $\Lambda ^{(k)}$ is empty. To see this, assume
that the combinatorial claim is true for $d-1$. We will show that the set of
upper bounds for $\lambda _d$ on $\Lambda ^{(k)}$ doesn't stabilize. If it
were to stabilize after $k_0$ at a certain $y$, then for $k\geq k_0$ we could
let
$A^k\subset {\bf N} ^{(d-1)}$ be the subset of elements $(a_1,\ldots, a_{d-1})$
such that $(a_1,\ldots , a_{d-1},y)\in \Lambda ^{(k)}$. We obtain a strictly
decreasing sequence of subsets for the case of $d-1$, so it is eventually
empty, meaning that in fact the upper bound for $\lambda _d$ didn't stabilize.
A decreasing sequence which doesn't stabilize can't exist, so eventually there
is
no upper bound, in other words $\Lambda ^{(k)}$ becomes empty. This gives the
claim.
Since one of the $\Lambda ^{(k)}$ is empty, one of the $F^{(k)}$ is empty. Let
$N$ be chosen so that $F^{(N)}$ is empty (and consequently $F^{(k)}$ is empty
for
$k\geq N$). Then
by the above argument, if $(\alpha _1,\ldots , \alpha _n)$ is an $n$-tuple of
distinct points of $G(Y)$, we must have $n<N$. This gives the theorem in the
case of an artinian local $Y$.
Now suppose $A$ is a henselian local ring and $S'= Spec (A)$. Let $S'_n:=Spec
(A/{\bf m}_A^n)$. In the inverse system $\lim _{\leftarrow} G(S'_n)$ we have
that all of the $G(S'_n)$ have cardinality bounded by $N$. In particular, the
cardinality of the inverse limit is bounded by $N$. Now suppose that there are
$N+1$ distinct points $y_i$ in $G(S')$. Two of the points go to the same point
in $\lim _{\leftarrow} G(S'_n)$, which means that for two of the points, the
liftings $z_i,z_j\in Z(S')$ give a point $(z_i,z_j)$ in $Z\times _SZ$ which
lifts, over any $S'_n$, into $R$. By strong artin approximation (check here
!!!), the point $(z_i,z_j)$ must lift into $R$ so the two points in
$G(S')$ are equal, a contradiction.
This completes the proof of the lemma.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{I.1.q}
If $G$ is a presentable group sheaf, then $G$ is finite if and only if $G(S')$
is finite for any henselian (resp. artinian) $S'$.
\end{corollary}
{\em Proof:}
The lemma provides one direction. For the other, note that if $G(S)$ is finite
for artinian $S'$ then $G^0(S')=\{ 1\}$. By unicity in the characterization of
$G^0$ we get $G^0= \{ 1\}$.
\hfill $\Box$\vspace{.1in}
\numero{Local study of presentable subgroups}
In this section we show that if $H\subset G$ is a presentable subgroup of a
presentable group $G$ then locally at the identity, in an appropriate sense,
$H$ is defined by the vanishing of a section of a vector sheaf.
This is a generalisation of the basic result that a subgroup of an algebraic
group is smooth, and hence a local complete intersection---cut out by a
section of its normal bundle. We obtain this result only in a ``neighborhood of
the identity'', or more precisely upon pullback by a vertical morphism
$X\rightarrow G$ such that $X$ admits a lift of the identity. If $Y$ is a
scheme with morphism $Y\rightarrow G$ such that $P\in Y$ maps to the identity
section in $G$, then there will be an etale neighborhood of $P\in Y$ lifting
to $X$ (which is why we can think of $X$ as a neighborhood of the identity).
This result will be used in a future study of de Rham cohomology (results
announced in \cite{kobe}). There, it will be important to have a structure
theory for presentable subgroups because of the general principle that if $G$
is a presentable group sheaf then $G/Z(G)\subset Aut ({\cal L} )$ where ${\cal L} = Lie
(G)$ is the Lie algebra vector sheaf of $G$ (see \S 9 below). A good
understanding of the structure of presentable subgroups will allow us to
reduce to looking at de Rham cohomology with coefficients in $Aut ({\cal L} )$ for
${\cal L}$ a vector sheaf, and here we have a more concrete hold on what happens.
\begin{theorem}
\mylabel{D.1}
Suppose $G$ is a connected presentable group sheaf over $S$, and suppose
$H\subset G$ is a presentable subgroup sheaf. Suppose that $X_1\rightarrow
G$ is a vertical morphism with lift of the identity
section $e:S\rightarrow X_1$. Suppose $P\in S$. Then there is an etale
neighborhood $X\rightarrow X_1$ of $e(P)$ with a lift of the identity $e:
S\rightarrow X$ (possibly after localizing in the etale topology of $S$ here)
and an etale morphism $\rho : X\rightarrow TX_e$ sending $e$ to the zero
section, such that
$$
X\times _GH = \rho ^{-1}(TX_e \times _{TG_e} TH_e).
$$
In particular, there is a vector sheaf $V$ over $S$ and a section $\sigma :
X\rightarrow V$ such that $X\times _GH= \sigma ^{-1}(0)$.
\end{theorem}
{\em Proof:}
Let $X_1\rightarrow G$ be a surjective vertical morphism with $X_1(S')$
connected for all artinian $S'$ (with $X_1$ a scheme of finite type).
Put $Y_1:= X_1\times _GH$. It is a subsheaf of $X_1$.
We can choose a vertical surjection $Z_1\rightarrow Y_1$ (with $Z_1$ a
scheme of finite type over $S$) together with a lift of $(e,e)$ also denoted by
$e$. Note that the morphism $Z_1\rightarrow H$
is also vertical (using the composition property of vertical morphisms and
the fact that the morphism $Y_1\rightarrow H$ is vertical by the pullback
property).
There is an etale neighborhood of $(e,e)\in X_1\times _SX_1$ denoted by
$U_1\rightarrow X_1\times _SX_1$ together with a lifting $\psi : U_1\rightarrow
X_1$ of the multiplication in $G$, such that $\psi $ restricted to the inverse
images of $\{ e\} \times _SX_1$ or $X_1\times _S\{ e\} $ are the identity.
We obtain a morphism
$$
U_1\times _{X_1\times _SX_1}(Y_1\times _SY_1)\rightarrow Y_1
$$
compatible with the multiplication in $H$ and again having the property that
the restrictions to the inverse images of the two ``coordinate axes'' are the
identity. Now pull back our multiplication to
$$
U_1\times _{X_1\times _SX_1}(Z_1\times _SZ_1)
$$
and note that $Z_1\rightarrow Y_1$ being vertical, there is an etale
neighborhood of the identity section (all of this is local on $S$!)
$$
V_1\rightarrow U_1\times _{X_1\times _SX_1}(Z_1\times _SZ_1)
$$
(which we can consider just as an etale neighborhood $V_1\rightarrow Z_1\times
_SZ_1$) and a good lift of our multiplication
$$
V_1\rightarrow Z_1
$$
restricting to the identity on the inverse images of the ``coordinate
axes''.
We obtain in this way morphisms on the etale germs
$$
{\bf 2}_{Z_1}: (Z_1,e)\rightarrow (Z_1,e)
$$
and
$$
{\bf 2}_{X_1}: (X_1,e)\rightarrow (X_1,e)
$$
compatible with the morphism $Z_1\rightarrow X_1$.
These morphisms induce multiplication by $2$ on the tangent vector schemes.
There are unique analytic isomorphisms of complex analytic germs
$$
(X_1,e)^{\rm an}\cong (T(X_1)_e,0)^{\rm an}
$$
and
$$
(Z_1,e)^{\rm an}\cong (T(Z_1)_e,0)^{\rm an}
$$
transforming the automorphisms ${\bf 2}$ into multiplication by $2$ and
inducing the identity on tangent spaces at the identity section. (To see
uniqueness, note that over artinian bases these are germs of vector spaces, and
any germ of automorphism $f$ of a vector space, such that $f(2x)=2f(x)$, is
linear; hence fixing it at the identity fixes it.)
By uniqueness, these isomorphisms are compatible with the morphism
$Z_1\rightarrow X_1$.
On the formal level, we have an etale morphism of formal germs
$$
\hat{\varphi}:\widehat{T(X_1)_e}\rightarrow X_1
$$
such that $\widehat{T(Z_1)_e}$ maps into $Y_1$. The {\em first claim} is that,
in fact, this gives a map
$$
Spec (\widehat{{\cal O}} _{T(X_1)_e,e(S)}) \rightarrow X_1
$$
such that
$$
T(Z_1)_e \times _{T(X_1)_e}Spec (\widehat{{\cal O}} _{T(X_1)_e,e(S)})
$$
maps into $Y_1$.
We can now apply Artin approximation to find an etale neighborhood
$W_1\rightarrow T(X_1)_e$ of the identity section (of course locally on $S$)
together with a morphism $W_1\rightarrow X_1$ inducing the identity on tangent
vector schemes at the identity section, and sending
$$
T(Z_1)_e\times _{T(X_1)_e}W_1\rightarrow Y_1.
$$
We can suppose that the morphism $W_1\rightarrow X_1$ is etale. In particular
the morphism $W_1\rightarrow G$ is vertical. We obtain two subsheaves
$$
im (T(Z_1)_e\times _{T(X_1)_e}W_1\stackrel{pr_2}{\rightarrow} W_1)
\subset
W_1\times _{X_1} Y_1 \subset W_1.
$$
They have the same tangent subsheaves at the identity.
Our {\em main claim} is that by taking an open subset of $W_1$ (still a
neighborhood of $e(P)$ for a given basepoint $P\in S$) we can assume that these
two subsheaves are equal.
The first subsheaf is given by the vanishing of the morphism
$$
W_1\rightarrow T(X_1)_e /T(Z_1)_e = T(G)/T(H),
$$
while the second subsheaf is equal to $W_1\times _GH$. Setting $X=W_1$ we
obtain the result of the theorem.
We just have to prove the {\em first claim} and the {\em main claim}.
{\em Proof of the first claim:}
By the sheaf condition and the finite type condition B1 and B2 for $Y_1$, it
suffices to prove that for any artinian $S'$, we have
$$
T(Z_1)_e \times _{T(X_1)_e}Spec (\widehat{{\cal O}} _{T(X_1)_e,e(S)})(S')
$$
mapping into $Y_1(S')$. That is to say, we have to prove that for any point
$S'\rightarrow T(Z_1)_e$ mapping to a point of $T(X_1)_e$ located near the
origin (that is to say factoring through
$Spec (\widehat{{\cal O}} _{T(X_1)_e,e(S)})$), this point maps into $Y_1(S')$.
We change to an algebraic notation. We can suppose that $S=Spec(A)$,
$T(X_1)_e=Spec(B)$ and $T(Z_1)_e=Spec(C)$. Further we can suppose that
$S'=Spec
(K)$ with $K$ artinian (although not necessarily of finite type). We have
$C\rightarrow K$. Since $T(Z_1)_e$ is a vector scheme we have a map
$C\rightarrow C[t]$ corresponding to multiplication by $t$ (and compatible with
the same map on $B$). Let $\hat{B}$ denote the completion of $B$ around the
zero section (which corresponds to an ideal ${\bf b}\subset B$). We are
provided with a factorisation $B\rightarrow \hat{B}\rightarrow K$.
We can assume that $K$ is of finite type over $\hat{B}$, and in particular
that $K$ is the total fraction ring of a subring $R\subset K$ such that $R$
is finite over $\hat{B}$. Let ${\bf
r}\subset R$ denote the ideal corresponding to ${\bf b}\subset B$ (note that
$R$ is complete with respect to ${\bf r}$). Let $K\{ t\} \subset K[[ t]]$ denote
the set of formal series of the form $\sum a_it^i$ such that there exists $\eta
\in R$ such that $\eta a_i \in {\bf r}^i$. With the same notations for $B$,
multiplication by $t$ provides a map $\hat{B}\rightarrow B\{ t\}$ compatible
with the map $B\rightarrow B[t]$, hence we get a map $\hat{B} \rightarrow K\{
t\}$. On the other hand we get a map $C\rightarrow C[t]\rightarrow
K[t]\rightarrow K\{ t\}$. Putting these together we get a map
$$
\hat{B}\otimes _BC \rightarrow K\{ t\}
$$
corresponding to multiplication by $t$. There is an evaluation at $t=1$ which
is a map $K\{ t\} \rightarrow K$ (this summability of the formal series comes
from the definition of $K\{ t\}$ and the completeness of $R$),
and the above map
is compatible with this and with the map $\hat{B}\otimes _BC\rightarrow K$ given
at the start. All in all we obtain a map
$$
Spec (K\{ t\}) \rightarrow
T(Z_1)_e \times _{T(X_1)_e}Spec (\widehat{{\cal O}} _{T(X_1)_e,e(S)})
$$
which induces on the subscheme $Spec (K)\rightarrow Spec (K\{ t\})$
(evaluation at $t=1$) the original inclusion. Now compose with the projection
into $G$. We obtain a morphism
$$
Spec (K\{ t\} )\rightarrow G
$$
which sends $Spec (K[[t ]])$ into $H$ (this comes from the condition that
$\widehat{T(Z_1)}$ maps into $Y_1$ together with B1 and B2 for $Y_1$ or $H$)
and we would like to show that it sends $Spec (K)$ (at $t=1$) into $H$.
It suffices to show that $Spec (K\{ t\} )\rightarrow H$.
By Noether normalization there is a morphism $R'\rightarrow R$ such that
$R'$ is integral and $R$ is finite over $R'$. Let $K'$ be the total fraction
ring of $R'$: it is a field, and $K$ is finite over $K'$. There is an ideal
${\bf r'}\subset R'$ which induces ${\bf r}$, and $K\{ t\}$ is finite over the
ring $K'\{ t\}$ defined in the same way as above with respect to this ideal.
Let $G'$ and $H'$ denote the direct images to $Spec (K')$ of the groups $G$
and $H$ pulled back to $K$. We have that $H'$ is a presentable subgroup of
the presentable group $G'$ (Lemma \ref{I.1.i}),
but since $K'$ is a field, $H'\subset G'$ is a closed subgroup of the
algebraic group $G'$ over $K'$. Since $K$ is finite over $K'$ we have
$$
K\{ t\} = K'\{ t\} \otimes _{K'}K,
$$
whence our point $Spec (K\{ t\} )\rightarrow G$ gives a point
$Spec (K'\{ t\} )\rightarrow G'$ sending $Spec (K'[[ t]])$ into $H'$. Now
since $H'$ is a closed subgroup of $G'$ both of which are algebraic groups (of
finite type) over $K'$, we get that $Spec (K' \{ t\} )\rightarrow H'$, meaning
that $Spec (K\{ t\} )\rightarrow H$.
This completes the proof of the first claim.
\hfill $\Box$\vspace{.1in}
{\em Proof of the main claim:}
Suppose that the main claim is not true. Note that there is a scheme of finite
type surjecting to $W_1\times _{X_1}Y_1$. The falsity of the main claim means
that the morphism from this scheme to $T(X_1)_e/T(Z_1)_e$ is nonzero on any
subset of the form pullback of an open subset of $W_1$ containing $P$. In
particular we can find a (possibly nonreduced) curve inside this scheme, such
that the section pulls back to something nonzero on the generic (artinian)
point, but such that the image of the curve in $W_1$ contains $P$ in its
closure. We get an $S$-scheme $S'$ with reduced scheme equal to a curve, and a
morphism $\psi :S'\rightarrow W_1\times _GH$ such that the projection into
$T(X_1)_e/T(Z_1)_e$ is nontrivial at the generic point of $S'$, such that $P$
is in the closure of the image of $S'$.
Let $\overline{S}'$ be a closure of $S'$ relative to $W_1$ obtained by adding
one point over $P$. Call this point $P'$. Then for any $n$ there is an etale
neighborhood of $P\in W_1$ on which the squaring map $n$-times is defined.
We obtain an etale $\overline{S}'_n\rightarrow \overline{S}'$ on which the
squaring map $n$-times is defined. We may assume that $\overline{S}'_n$
consists of an etale morphism $S'_n\rightarrow S'$, union one point $P'_n$ over
$P'$. Denote by $\psi _n:
\overline{S}'_n\rightarrow W_1$ the result of the squaring operation
iterated $n$
times. There is an analytic isomorphism of a neighborhood of
$P'_n$ in $\overline{S}'_n$ with a neighborhood of $P'$ in $S'$, and an
analytic trivialization of a neighborhood of $P$ in $W_1$ (isomorphism with the
tangent vector scheme) such that $\psi _n= 2^n\psi$ as analytic germs
around the
point $P'_n$.
{\em Step 1.} There is an $n_0$ such that for any $n\geq n_0$, the projection of
$S'_n$ into $T(X_1)_e/T(Z_1)_e$ is nontrivial at the generic point of $S'_n$.
In particular for any $m$ the projection of
$S'_{mn_0}$ into $T(X_1)_e/T(Z_1)_e$ is nontrivial at the generic point of
$S'_{mn_0}$.
Let $v: W_1\rightarrow T(X_1)_e/T(Z_1)_e$ denote our section. With respect
to our analytic trivialization of $W_1$ where the squaring map becomes
multiplication by $2$, can take a Taylor expansion for $v$ around the identity
section of $W_1$,
$$
v= v_1 + v_2+ v_3 + \ldots + v_{i-1} + w_i,
$$
with $v_j(2x)= 2^jv(x)$ and $w_i$ vanishes to order $i$ along $e$;
this notion can be defined by considering $w_i$ as a section of a coherent
sheaf ${\cal F}$ which contains $T(X_1)_e/T(Z_1)_e$. By hypothesis the restriction
of $v$ to $S'$ is nonzero at the generic point of $S'$. Let ${\cal G} _{S'}$ be the
quotient of ${\cal F} |_{S'}$ by the ``torsion'' subsheaf (i.e. the subsheaf
of sections supported in dimension zero). That a
section is nonzero at the generic point means that its projection into ${\cal G}
_{S'}$ is nonzero. We may choose $i$ big enough so that $v$ is nonzero in ${\cal G}
_{S'}$ modulo the image of sections which vanish to order $i$ along $e$.
Let
$\overline{v}_j$ denote the projection of $v_j$ into the space of sections of
${\cal G}_{S'}$ modulo the image of the sections vanishing to order $i$.
At least one of the $\overline{v}_j$ is nonzero.
Now notice that the projection of $v(2^nx)$ is equal to
$$
\overline{v(2^nx)} = 2^n\overline{v}_1(x) + 2^{2n}\overline{v}_2(x)
+ \ldots + 2^{(i-1)n}\overline{v}_{i-1}(x).
$$
A little $2$-adic argument shows that there is $n_0$ such that for $n\geq n_0$
this quantity must be nonzero. We obtain that
$\overline{v(2^nx)}\neq 0$ and hence that $v(2^nx)=v(\psi_nx)$ is nonzero at
the generic point of $S'_n$, as claimed for Step 1.
{\em Step 2.} The Zariski closure of the union of the images of the $\psi
_{mn_0}$ contains the zero-section. To prove this, note that in the formal
completion at $P$, the union of the closures of the $S'_{mn_0}$ is a subset
stable
under multiplication by $2^{n_0}$, hence its Zariski closure is stable under
(fiberwise) multiplication by $2$, hence it is fiberwise homogeneous and thus
contains the zero-section. The completion of the Zariski closure contains the
Zariski closure of the intersection with the completion, so the zero-section is
in the closure.
{\em Step 3.} Over the generic point of $S$, the zero section is in the
Zariski closure of the $S'_{mn_0}$. Otherwise we would obtain a function
nonvanishing on the zero section and vanishing on the $S'_{mn_0}$; clearing
denominators this function can be assumed defined over $S$ rather than the
generic point of $S$, and since (we may assume) the $S'_{mn_0}$ are all schemes
of
pure dimension $1$ dominating $S$, this function defined over $S$ which
vanishes generically on the $S'_{mn_0}$, must vanish identically on the
$S'_{mn_0}$.
This would contradict the fact that the zero section is in the Zariski closure
globally over $S$.
{\em End of proof of claim:} Now we work over the generic geometric artinian
point of $S$. Change notations now to suppose that $S$ is artinian and $S'=S$;
we note the schemes $S'_{mn_0}$ by $S_{mn_0}$ (they are all isomorphic to $S$)
with $S'=:S_1$. We have points $S_{mn_0}\rightarrow W_1\times _GH$ all mapping
to something nonzero in $T(X_1)_e/T(Z_1)_e$.
Note, as a bit of a detour, that the connected component of the identity in
$W_1\times _GH (S)$, must map to zero in $T(X_1)_e/T(Z_1)_e(S)$. This is
because $T(X_1)_e/T(Z_1)_e(S)= T(X_1)_e(S)/T(Y_1)_e(S) = T(G)_e(S)/T(H)_e(S)$,
whereas verticality of $X_1\rightarrow G$ implies that
$X_1(S)\rightarrow G(S)$ is smooth. In particular
$W_1\times _GH (S)$ is a smooth local complete intersection so a morphism from
$W_1(S)$ to the normal space $T(G)_e(S)/T(H)_e(S)$ of $W_1\times _GH (S)$,
with zero set contained in the complete intersection, must have
zero set which is a union of connected components of $W_1\times _GH (S)$.
Containing the identity, it contains the connected component of
the identity.
In particular, our points $S_{mn_0}\rightarrow W_1\times _GH$ from before are
never in the connected component of $W_1\times _GH (S)$ which contains the
identity. On the other hand, these points all lift to $Z_2\rightarrow W_1$
(a scheme of finite type surjecting vertically to $W_1\times _GH$). Let
$Z_2(S)'$ denote the union of components of $Z_2(S)$ which contain liftings of
our points $S_{mn_0}\rightarrow W_1$. We have a morphism $Z_2(S)'\rightarrow
W_1(Spec (k))$ whose image is a constructible set. But the image contains all
of the points where the $S_{mn_0}$ are located, so the image must contain a
generic point of any irreducible component of the Zariski closure of the
$S_{mn_0}$. In particular, there is a component of $Z_2(S)'$ which maps to
something in $W_1(Spec (k))$ containing the identity in its closure.
Let $W_1(S)_e$ denote the inverse image of $e\in W_1(Spec (k))$ in $W_1(S)$.
Let $N\subset G(S)$ denote the image of $W_1(S)_e\rightarrow G(S)$. We claim:
that $N$ is a unipotent subgroup of $G(S)$, and that the morphism
$W_1(S)_e\rightarrow N$ is a fibration with connected fibers.
Assume this claim for the moment.
The image of $W_1(S)\rightarrow W_1(Spec (k))$ is a closed subvariety
$R\subset W_1(Spec (k))$ (this can be seen since $W_1$ is etale over the vector
scheme $TX_1$). We have a morphism $R\rightarrow G(S)/N$. On the other hand,
the above morphism $Z_1(S)' \rightarrow W_1(Spec (k))$ factors through a
morphism $Z_1(S)'\rightarrow R$, and the image of this map contains $1\in R$
in its closure.
The morphism $W_1(S)\rightarrow R$ is a fibration with fiber $W_1(S)_e$ in the
etale topology.
It suffices to prove that $TX_1 (S) \rightarrow TX_1(Spec (k))$ is a fibration
over its image, since locally in the etale topology $W_1$ is isomorphic to
$TX_1$.
In fact if $V$ is any vector scheme then $V(S)$ and $V(Spec (k))$ are vector
spaces so the morphism $V(S)\rightarrow V(Spec (k))$ is a fibration over its
image, with fiber the inverse image of the origin.
We now show that the morphism
$$
W_1\times _GH(S) \rightarrow R \times _{G(S)/N} H(S)
$$
is a fibration in the etale topology with fiber the kernel of
$W_1(S)_e\rightarrow N$. Locally on $R$ we can choose a lifting $\lambda : R
\rightarrow W_1(S)$ and then we have a morphism
$$
R\times _{G(S)/N}H(S)\rightarrow N
$$
given by $(r,h)\mapsto h^{-1}im(\lambda (r))$. We claim that (locally over
$R$)
$$
W_1(S)\times _{G(S)}H(S) = W_1(S)_e \times _N (R\times _{G(S)/N}H(S)).
$$
The morphism from right to left associates to the point
$(a,r,h)$ the point $(i(a)\ast \lambda (r) , h)$ where $i: W_1\rightarrow
W_1$ is
an etale-locally defined morphism covering the inverse. This shows
that the morphism at the start of the paragraph is a fibration.
Suppose $A$ is an algebraic group with connected algebraic subgroups $B\subset
A$ and $N\subset A$. Then the morphism
$$
B / (B\cap N) \rightarrow A/N
$$
is proper over an open neighborhood of the class of the identity in $A/N$.
To prove this, proceed as follows.
Let $I\subset A/N$ denote the image. Let $Z\subset A/N$ denote the subset of
points over which the map in question is not proper. This can be constructed
as follows. Let $X:= B/(B\cap N)$, and let $\overline{X}$ be a
relative completion with proper morphism $\overline{X}\rightarrow A/N$; and
suppose that $X\subset \overline{X}$ is open and dense. Then $Z$ is the image
of $\overline{X}-X$. Since the map $X\rightarrow A/N$ is injective, we have
that the dimension of the image $Z$ is strictly less than the dimension of
the image $I$ of $X$. In particular, there is a point $y\in I$ such that
the morphism in question is proper over a neighborhood $U$ of $y$. But since
$B$ acts on $X$ and compatibly on $A$ (by left
multiplication) the morphism in question is proper over any translate of the
form $bU$. Setting $b\in B$ equal to the inverse of a representative in for
$y$ we obtain a neighborhood $bU$ of the identity over which the map is proper.
Note that by the above claim that we are accepting for now, the fiber of
the fibration $W_1(S)_e\rightarrow N$ is connected. On the other hand, by
the previous paragraph the map $H(S)\rightarrow G(S)/N$ induces a map
$H(S)/(H(S)\cap N)\rightarrow G(S)/N$ which is proper over a neighborhood of
the class of the identity. Let $I\subset G(S)/N$ be the image of this map. It is
a locally closed subset and the subset topology coincides with the topology of
the base of the fibration, at least near the identity. Note that $H(S)$ is
fibered over $I$ with fibers $H(S)\cap N$ which are connected because $N$ and
hence $H(S)\cap N$ are unipotent groups (unipotent groups are always connected).
Finally we have the following situation:
$$
W_1\times _GH(S)\rightarrow R\times _{G(S)/N} I
$$
is a fibration with connected fiber, whereas $I\subset G(S)/N$ is a locally
closed subset. Since an etale fibration is an open map, the image of the
connected component of the identity in $W_1\times _GH(S)$ is an open
neighborhood of the identity in $R\times _{G(S)/N}I$. Since the fibers of the
fibration are connected, the image of the complement of the identity component
is the complement of the image of the identity component. In particular,
there is an open neighborhood of the identity in $R\times _{G(S)/N}I\subset R$
(and hence an open neighborhood of the identity in $R$) whose inverse image
doesn't meet any other connected component of $W_1\times _GH(S)$. Finally,
since $R$ is a closed subset of $W_1(Spec(k))$, we obtain an open neighborhood
of the identity in $W_1(Spec (k))$ whose inverse image in $W_1\times _GH(S)$
is contained in the connected component of the identity. This is a
contradiction to our earlier situation where $Z_2(S)' \rightarrow W_1(Spec
(k))$ has image a constructible set with the identity in its closure. This
completes the proof modulo the following part.
We have to show that $N$ is a unipotent subgroup of $G(S)$ and that the
morphism $W_1(S)_e\rightarrow N$ is a fibration with connected fibers.
Write $S=Spec (A)$ with $A$ artinian, and choose a sequence of ideals
$I_j\subset A$ for example $I_j= {\bf m}^j$. Let
$$
W_1(S)_j
$$
be the set of points of $W_1(S)$ which restrict to the identity on $Spec
(A/I_j)$. In particular $W_1(S)_1=W_1(S)_e$. Choose a good lift of the
multiplication
$$
W_1\times W_1 \rightarrow W_1
$$
in a formal neighborhood of our point $P$. We obtain
$$
\ast : W_1(S)_e\times W_1 (S)_e\rightarrow W_1(S)_e.
$$
This operation is not a group, however we have the following property.
$$
\ast : W_1(S)_j\times W_1 (S)_j\rightarrow W_1(S)_j.
$$
Next, note that the fact that $W_1$ is etale over a vector scheme gives
another operation which we denote
$$
+ : W_1(S)_e\times W_1 (S)_e\rightarrow W_1(S)_e
$$
which is an abelian group structure.
We can write
$$
a*b = a+b + F(a,b)
$$
where
$$
F: W_1(S)_j\times W_1 (S)_j\rightarrow W_1(S)_{j+1}.
$$
This is because there is a unique good operation on the set of
elements of $W_1(Spec (A/I_{j+1})$ which restrict to the identity in $W_1(Spec
(A/I_j)$. (Also note that the $+$-quotient $W_1(S)_j/W_1(S)_{j+1}$ injects into
this subset of $W_1(Spec(A/I_{j+1})$).
Because of this formula, we can define the quotient $W_1(S)_j/W_1(S)_{j+1}$
with respect to the operation $\ast$ and it is the same as the quotient with
respect to $+$. In particular note that the morphism
$$
W_1(S)_j\rightarrow W_1(S)_j/W_1(S)_{j+1}
$$
is a fibration with fiber a vector space.
The morphism $W_1(S)_e\rightarrow G(S)$ is compatible with the operation
$\ast$. Let $N_j$ denote the image of $W_1(S)_j$ in $G(S)$ (in particular
$N_1=N$). The $N_j$ are constructible sets and subgroups so they are
algebraic subgroups of $G(S)$. We obtain a surjective morphism
$$
W_1(S)_j /W_1(S)_{j+1} \rightarrow N_j/N_{j+1},
$$
but from the previous formula the operation on the left is a unipotent
algebraic group, this shows that $N_j/N_{j+1}$ is a unipotent group, and since
extensions of unipotent groups are unipotent (and $N_j=\{ 1\}$ for $j$ large),
$N=N_1$ is unipotent.
We claim that
$W_1(S)_e\rightarrow N$ is smooth. Suppose $R'\subset R$ is an inclusion
of artinian schemes over $Spec(k)$. Look at the map
$$
W_1(S)_e(R)\rightarrow W_1(S)_e(R')\times _{N(R')}N(R).
$$
Suppose that we have a map $S\times R\rightarrow
G$ and a lifting over $S\times R'$ to $W_1$, sending $Spec(k)\times R$ to $e$.
We would like to find a lifting over $S\times R$ sending $Spec(k)\times R$ to
$e$. We can do this whenever $R'$ is a union of $R_i$ and we have commuting
retracts from $R'$ to $R_i$, just apply the verticality property to $R\times
S$ with retracts to $R_i\times S$ and $R\times Spec( k)$. This proves that
the morphism $W_1(S)_e\rightarrow G(S)$ is vertical with respect to
$Spec(k)$. It is then immediate that the morphism $W_1(S)_e\rightarrow N$
is surjective (since $N$ is the image of the previous map). Note that $N$ is
presentable, so by Lemma \ref{smooth}, the morphism
$W_1(S)_e\rightarrow N$ is smooth.
Let $K\subset W_1(S)_e$ be the inverse image of $1\in N$.
Then for any two points $a,b$ with $f(a)=f(b)$ there is a unique element
$k\in K$ such that $b=k\ast a$ (the existence and uniqueness of such an
element $k\in W_1(S)_e$ can be seen using the above grading and the expression
for $\ast$, and then it is immediate that $k\in K$ from compatibility of
$f$ with $\ast$). Any point $n\in N$ has an etale neighborhood $n\in
U\stackrel{p}{\rightarrow} N$ with a section $\sigma :U\rightarrow W_1(S)_e$.
Then we obtain a morphism
$$
K\times U \rightarrow W_1(S)_e\times _N U
$$
obtained by sending $(k,u)$ to $(k\ast \sigma (u), p(u))$. This is an
isomorphism on the level of points by the above property for $K$, and both
sides are smooth, so it is an isomorphism. This proves that
$W_1(S)_e\rightarrow N$ is a fibration in the etale topology.
Finally, to
show that the fibers are connected it suffices to show that $K$ is connected.
But since $W_1(S)_e$ and $N$ are vector spaces and the morphism $f$ is a
fibration in the etale topology, the associated analytic morphism is a
fibration in the usual topology, so the fiber is contractible.
\hfill $\Box$\vspace{.1in}
\numero{The Lie algebra sheaf}
\begin{theorem}
\mylabel{lmn}
If ${\cal G}$ is a presentable group sheaf, and if
we set $Lie ({\cal G}) := T({\cal G} )_1$, then there is a unique bilinear form
(Lie bracket)
$$
[\cdot , \cdot
] : Lie ({\cal G} )\times Lie ({\cal G} )\rightarrow Lie ({\cal G} )
$$
which, over artinian base schemes, reduces to the usual Lie bracket.
\end{theorem}
{\em Proof:}
A section of $Lie ({\cal G} )$ over $S'= Spec (A)$ is a morphism $Spec (A[\epsilon
])\rightarrow {\cal G}$ sending $Spec (A)$ to the identity section (in our notation
here $\epsilon$ denotes an element with $\epsilon ^2=0$).
Given two
such morphisms which we denote $\alpha$ and $\beta$ we obtain
$$
\alpha p_1, \, \beta p_2 : Spec (A[\epsilon , \epsilon '])\rightarrow {\cal G}
$$
(where also $(\epsilon ')^2=0$) and we can form the morphism
$$
\gamma := \alpha p_1 \cdot \beta p_2 \cdot (\alpha m p_1) \cdot (\beta m
p_2)
: Spec (A[\epsilon , \epsilon '])\rightarrow {\cal G}
$$
where the $g\cdot h$ denotes composition in ${\cal G}$, and where $m= A[\epsilon
]\rightarrow A[\epsilon ]$ is the involution sending $\epsilon$ to $-\epsilon$.
The morphism $\gamma$ restricts to the identity on $Spec (A[\epsilon , \epsilon
']/(\epsilon \epsilon '))$. Let $$
q: Spec (A[\epsilon , \epsilon '])\rightarrow Spec (A[\delta])
$$
denote the morphism sending $\delta$ to $\epsilon \epsilon '$ (here again
$\delta
^2=0$).
Our first claim is that if the morphism $\gamma$ factors as $\gamma = \varphi
\circ q$ then $\varphi $ is unique. To see this suppose that $\phi$ and
$\varphi$ were two morphisms $Spec (A[\delta])\rightarrow {\cal G}$ with $\phi \circ
q = \varphi \circ q$. Let $X\rightarrow {\cal G}$ and $R\rightarrow X\times _{{\cal G}}X$
be the morphisms in a presentation of ${\cal G}$, with a chosen lift of the identity
section into $X$. Choose liftings $\tilde{\varphi}$ and $\tilde{\phi}$
from $Spec
(A[\delta ])$ into $X$ sending $Spec (A)$ to the identity section of $X$
(here we
may have to localize on $S'=Spec (A)$ in the etale topology---but henceforth
ignore this point, much as we have already ignored it in lifting the identity
section into $X$\ldots ). The fact that the compositions with $q$ are the same
means that the pair $(\tilde{\varphi}\circ q, \tilde{\phi}\circ q)$
defines a point which we denote
$$
\eta :Spec (A[\epsilon , \epsilon '])\rightarrow
X\times _{{\cal G}}X.
$$
Note that $Spec (A[\epsilon , \epsilon ']/(\epsilon \epsilon
'))$ projects by $q$ to $Spec (A)\subset Spec (A[\delta ])$ and both
$\tilde{\varphi}$ and $\tilde{\phi}$ send $Spec (A)$ to the identity section
(by
hypothesis on our liftings) so in particular $\eta$ sends $Spec (A[\epsilon ,
\epsilon ']/(\epsilon \epsilon '))$ to the identity pair $(e,e)$ in
$X\times _{{\cal G}
}X$. On the other hand we can take $Y=Spec (A[\epsilon , \epsilon '])$ and
$Y_1=Spec (A[\epsilon ])$ and $Y_2 = Spec (A[\epsilon '])$ and then apply the
lifting property $Lift _2(Y, Y_i)$ which holds for the morphism $R\rightarrow
X\times _{{\cal G}}X$ because (from the hypothesis in the property $P4$) this
morphism
is vertical.
Fix a lifting $e_R: S\rightarrow R$ of the identity pair in $X\times
_{{\cal G}}X$ and fix the values of the morphisms (denoted $\lambda _i$ in the
definition of the lifting property) as being $e_R$ on $Y_1$ and $Y_2$.
These are
indeed liftings of our given morphisms $Y_i \rightarrow X\times _{{\cal G}}X$
since, as
we have seen above, both $Y_1$ and $Y_2$ map to the identity pair (the subscheme
defined by $(\epsilon \epsilon ')$ is the union of $Y_1$ and $Y_2$).
We obtain by
the lifting property a lifting $Y\rightarrow R$ which agrees with $e_R$ on $Y_1$
and $Y_2$. If we write (locally) $R=Spec (B)$ then this morphism corresponds to
a morphism $a:B\rightarrow A[\epsilon , \epsilon ']$ such that the projection of
$B$ modulo $\epsilon$ or modulo $\epsilon '$ is a constant morphism
$B\rightarrow
A$. It now follows that $a$ factors through $B\rightarrow A[\delta ]$. We
obtain a morphism $Spec (A[\delta ])\rightarrow R$ whose projection into
$X\times X$ is the pair $(\tilde{\varphi}, \tilde{\phi} )$ (that this is the
case is easy to check directly again by supposing that $X$ is affine). This
implies that $(\tilde{\varphi}, \tilde{\phi} )$ has image in $X\times _{{\cal G}}X$,
in other words that the morphisms $\tilde{\varphi}$ and $\tilde{\phi}$ from
$Spec
(A[\delta ])$ into $X$ project to the same morphism into ${\cal G}$. Thus $\varphi =
\phi$, completing the proof of uniqueness.
Now we show existence of the factorization $\gamma = \varphi \circ q$. The
preceding uniqueness result implies that it is sufficient to construct $\varphi$
after etale localization on $S'$. Thus we may assume that $\alpha$ and $\beta$
lift to points $\tilde{\alpha}, \tilde{\beta}: Spec (A[\epsilon ])\rightarrow X$
sending $Spec (A)$ to the identity section. There is a good lifting of the
multiplication in ${\cal G}$ to a multiplication $X\times X \rightarrow X$ which we
still denote $x\cdot y$, where goodness means the property $x\cdot e = e
\cdot x =
x$.
We can
now put
$$
\tilde{\gamma }:= \tilde{\alpha }p_1 \cdot \tilde{\beta }p_2 \cdot
(\tilde{\alpha }m p_1)
\cdot (\tilde{\beta }m p_2)
: Spec (A[\epsilon , \epsilon '])\rightarrow X.
$$
We still have the formula that
$$
\alpha \cdot (\alpha m) = e
$$
(this is because the first order term of the composition is just addition of
vectors) and from this formula it follows that $\tilde{\gamma }$ sends the
subschemes $Spec (A[\epsilon ])$ and $Spec (A[\epsilon '])$ to $e$ (through
their
projections to $Spec (A)$). Since now $X$ is a scheme, this implies directly
the
existence of $\tilde{\varphi } : Spec (A[\delta ])\rightarrow X$ such that
$\tilde{\gamma } = \tilde{\varphi } \circ q$. Projecting from $X$ to ${\cal G}$ we
get the factorization $\varphi$ desired.
Finally, we set $[\alpha , \beta ] := \varphi$ from the above construction.
It is of course completely clear from the construction that if $S'$ is artinian,
this gives the usual Lie bracket on the algebraic group ${\cal G} (S' )$.
it remains to be seen that this morphism is bilinear and satisfies the Jacobi
identity (i.e. that a certain deduced trilinear form vanishes). But these
properties can be checked on values over artinian schemes $S'$, and there since
the bracket we have defined coincides with the usual one, we get bilinearity and
the Jacobi identity.
\hfill $\Box$\vspace{.1in}
{\em Remark:} The subtlety in our whole situation
being essentially that the factorization, while immediate and obviously unique
in the case where the target of the map is a scheme, does not necessarily exist
and may not be unique even if it does exist, when the target of the
map is just a
sheaf. One can give examples of $P2$ sheaves ${\cal H}$ on ${\cal X} /S$ together with
morphisms $Spec (A[\epsilon , \epsilon '])\rightarrow {\cal H}$ restricting to a
given
section $S\rightarrow {\cal H}$ over the subscheme defined by
$(\epsilon \epsilon ')$,
and where the morphism either doesn't factor through $Spec (A[\delta ])$ or else
such that the factorization isn't unique.
We indicate here a simpler example which shows the way toward the examples
refered to in the above paragraph. Let $Y\rightarrow X$ be a degree $2$
morphism
of smooth curves completely ramified above a point $x\in X$. Let ${\cal F}$ be the
image of this morphism (considered as a sheaf on ${\cal X}$). Let $y$ be the point
lying over $x$ and suppose $f: Spec (k[\epsilon]/\epsilon ^3)\rightarrow
Y$ is a nonzero tangent vector located at $y$. Then the associated element
of ${\cal F}( Spec (k[\tau ]/\tau ^3))$ is constant (equal to the constant
point $y$) on the subscheme
$$
Spec (k[\tau ]/\tau ^2)\subset Spec (k[\tau ]/\tau ^3).
$$
Nevertheless there exists no factorization of the form
$$
Spec (k[\tau ]/\tau ^3)\subset Spec (k[\tau ]/\tau ^2)\rightarrow
{\cal F}
$$
(this factorization would have existed had ${\cal F}$ been a scheme).
We can obtain
an example where a factorization of the type needed in the above theorem doesn't
exist, simply by composing this example with the morphism $Spec (k[\epsilon ,
\epsilon ']) \rightarrow Spec(k[\tau ]/\tau ^3$ sending $\tau$ to $\epsilon +
\epsilon '$.
The sheaf ${\cal F}$ in this example is not $P4$ with respect to $Spec (k)$.
Of course ${\cal F}$ is not a group sheaf. As stated elsewhere, I am not
sure about whether a $P2$ group sheaf might not automatically have to be $P4$
for example (or at least satisfy some of the properties we use here).
For example we have seen that an algebraic space (of finite type) which is a
group
is automatically a scheme.
\subnumero{The adjoint representation}
Suppose ${\cal G}$ is a presentable group sheaf. Then ${\cal G}$ acts on itself by
conjugation, by the formula
$$
Int (g)(h):= ghg^{-1}.
$$
More precisely this action is a morphism ${\cal G} \times {\cal G} \rightarrow {\cal G}$
and if we put in the identity map on the first projection we obtain a morphism
${\cal G} \times{\cal G} \rightarrow {\cal G} \times {\cal G}$ which is a morphism of group objects
(the second variable) over the first variable ${\cal G}$. From this and from the
invariance of the above definition of the Lie algebra object, this action
induces
an action (the {\em adjoint action})
$$
{\cal G} \times Lie ({\cal G} )\rightarrow Lie ({\cal G} )
$$
which preserves the bracket.
If $({\cal L} , [,])$ is a Lie algebra sheaf (that is to say, ${\cal L}$ is a vector sheaf
with bilinear morphism $[,]: {\cal L} \times {\cal L} \rightarrow {\cal L}$ satisfying the
Jacobi identity) then we obtain a group sheaf $Aut ({\cal L} , [,])$.
\begin{lemma}
\mylabel{AutLie}
If $({\cal L} , [,])$ is a Lie algebra sheaf then $Aut ({\cal L} , [,])$
is a presentable group sheaf.
\end{lemma}
{\em Proof:}
The group sheaf $Aut ({\cal L} )$ of automorphisms of the vector sheaf ${\cal L}$ is
presentable and in particular $P4$ by Theorem \ref{I.1.g}. The Lie bracket can
be considered as a morphism
$$
{\cal L} \otimes _{{\cal O}} {\cal L} \rightarrow {\cal L} .
$$
The subgroup $Aut ({\cal L} , [,])\subset Aut ({\cal L} )$ may thus be represented as the
equalizer of two morphisms
$$
Aut ({\cal L} ) \rightarrow Hom ({\cal L} \otimes _{{\cal O}} {\cal L} , {\cal L} ).
$$
Note that $Hom ({\cal L} \otimes _{{\cal O}}{\cal L} , {\cal L} )$ is a vector sheaf by
Lemma \ref{I.s} and the definition of tensor product following that lemma; and
presentable by Theorem \ref{I.1.g}. In particular $Aut ({\cal L} , [,])$ is $P4$ by
Lemma \ref{I.1.a} and presentable by Corollary \ref{I.z}. \hfill $\Box$\vspace{.1in}
The adjoint action may be interpreted as a morphism of presentable group sheaves
$$
Ad : {\cal G} \rightarrow Aut (Lie ({\cal G} ), [,] ).
$$
We can of course forget about the bracket and compose this with the morphism
into
$Aut(Lie ({\cal G} ))$ which is just the automorphism sheaf of a vector sheaf.
\begin{proposition}
\mylabel{Adjoint}
Suppose ${\cal G}$ is a connected presentable group sheaf. Then the kernel of the
morphism $Ad$ is the center $Z({\cal G} )$ (that is to say the sheaf whose values
are the centers of the values of ${\cal G}$).
\end{proposition}
{\em Proof:}
The statement amounts to saying that a section $g$ of ${\cal G}$ acts trivially on
${\cal G}$ if and only if it acts trivially on $Lie ({\cal G} )$. This statement is true,
in fact, of any automorphism (defined over any base scheme $S'\rightarrow S$).
It
suffices to prove this last statement for the values over artinian base schemes
(if an automorphism agrees with the identity on the values over all artinian
base schemes then it must be equal to the identity). In the case of values over
artinian base schemes it is just the statement that an automorphism which acts
trivially on the Lie algebra of a connected algebraic group must act trivially
on
the whole group. \hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{centerPres}
If ${\cal G}$ is a connected presentable group sheaf, then the center
$Z({\cal G} )$ is again presentable.
\end{corollary}
{\em Proof:} By Proposition \ref{Adjoint} the center is the kernel of a morphism
of presentable group sheaves. By Theorem \ref{I.1.e}, this kernel is
presentable.
\hfill $\Box$\vspace{.1in}
{\em Question}
Suppose ${\cal G}$ is a presentable group sheaf, not necessarily connected. Is the
center $Z({\cal G} )$ presentable?
This is related to the following question.
{\em Question}
Suppose $H$ is a finite presentable group sheaf. Is $Aut (H)$ presentable?
A positive response here would allow us to prove that the center $Z({\cal G} )$ is
connected, because it is the kernel of the action of
$Z({\cal G} ^o)$ on the group of connected components $H={\cal G} /{\cal G} ^o$.
\subnumero{Determination of presentable group sheaves by their Lie algebras}
The object of this section is to prove the following theorem, which is a
generalization of the well known principle that a Lie group is determined by its
Lie algebra, up to finite coverings, if the center is unipotent.
\begin{lemma}
\mylabel{123}
Suppose $F, G \subset H$ are two presentable group subsheaves of a presentable
group sheaf $H$, and suppose $F$ and $G$ are connected. If $Lie (F)=Lie (G)$ as
subsheaves of $Lie (H)$ then $F=G$.
\end{lemma}
{\em Proof:}
By the properties B1 and B2 and artin approximation, it suffices to show that
for
any artinian $S'$ we have $F(S')=G(S')$. But these two are connected Lie
subgroups of $H(S')$ which by hypothesis have the same Lie algebras; thus they
are equal.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{abc}
Suppose $F$ and $G$ are connected presentable group sheaves on ${\cal X}$.
Suppose $Lie (F)\rightarrow Lie(G)$ is an isomorphism of Lie algebras. Then
this isomorphism lifts to a unique isomorphism
$F/Z(F)\cong G/Z(G)$ where $Z()$ denotes the center.
\end{corollary}
{\em Proof:}
Note that the center of a
connected presentable group sheaf is presentable by \ref{centerPres}, so
$F/Z(F)$ and $G/Z(G)$ are presentable.
Let $L=Lie (F)=Lie (G)$ and let $A=Aut (L)$ (automorphisms of the vector
sheaf or of the Lie algebra sheaf, we don't care). We get maps $F\rightarrow A$
and $G\rightarrow A$. Let $F_1$ and $G_1$ denote the images. We have
$$
Lie (F_1) = im (L\rightarrow Lie (A)) = Lie (G_1)
$$
as subsheaves of $Lie (A)$, so by Lemma \ref{123} we have $F_1=G_1$. On the
other
hand, note that $Z(F)$ is the kernel of the map $F\rightarrow A$ because if an
element of $F$ acts trivially on $Lie (F)$ then by exponentiation and the fact
that $F$ is connected, it acts trivially on all $F(S')$ for $S'$ artinian
hence in
fact it acts trivially on $F$. Thus $F_1 = F/Z(F)$ and similarly $G_1=G/Z(G)$.
\hfill $\Box$\vspace{.1in}
We have now finished verifying that the class of presentable group sheaves
satisfies the properties set out in the introduction. In effect:
\newline
Property 1 is Corollary \ref{uvw};
\newline
Property 2 is Theorem \ref{I.1.e};
\newline
Property 3 is Lemma \ref{I.1.h};
\newline
Property 4 is Lemma \ref{I.1.i};
\newline
Property 5 is Theorem \ref{I.1.m};
\newline
Property 6 is Theorem \ref{I.1.o} and Corollary \ref{connex};
\newline
Property 7 is Theorem \ref{lmn}; and
\newline
Property 8 is Theorem \ref{abc}.
\subnumero{Questions}
We present in further detail some other questions analogous to well known
properties of algebraic Lie groups, which seem to be more difficult here.
{\bf 1.} \,
(Existence) {\em If $({\cal L} , [,])$ is a
Lie algebra sheaf (i.e. a vector sheaf with bilinear operation satisfying the
Jacobi identity) then does there exist a presentable group sheaf ${\cal G}$ with $Lie
({\cal G} )= ({\cal L} , [,])$?} One has the following idea for a proof of existence in
a formal sense. Take a resolution of ${\cal L}$ by vector schemes, and lift the
bracket to a bracket (not necessarily satisfying the Jacobi identity) on the
vector scheme $X$ surjecting to ${\cal L}$. Then use an explicit version of
Baker-Campbell-Hausdorff to define a composition law on the formal completion of
$X$ along the zero section. This composition law will not be associative, but
one should be able to use the second part of the resolution of ${\cal L}$ to define a
relation scheme $R$ (formally), such that when we set ${\cal G}$ to be the quotient
of $X$ by $R$ we get a group sheaf. One would have to check that the maps are
vertical. Of course this idea for a proof skirts the main question of how to
integrate the formal structure out into an actual presentable group sheaf.
{\bf 2.} \, {\em Does every (connected, say) presentable group sheaf have a
faithful representation on a vector sheaf?} I guess that the answer is probably
no, but I don't have a specific example in mind.
{\bf 3.} \, {\em Suppose $Lie ({\cal F} )\rightarrow Lie ({\cal G} )$ is a morphism of
vector Lie algebras. Under what conditions does this lift to a morphism ${\cal F}
'\rightarrow {\cal G}$ where ${\cal F}' \rightarrow {\cal F}$ is a finite covering?}
{\bf 4.} \, {\em What happens in Theorem \ref{abc} if we don't divide out by the
centers?}
{\bf 5.} \, {\em Suppose $G\subset Aut (V)$ is a presentable subgroup of the
automorphisms of a vector sheaf. Is there a vector subsheaf
$U\subset T^{a,b}(V)$ of a tensor power of $V$ (or possibly a cotensor power
or a
mixture\ldots ) such that $U$ is preserved by the action of $G$ and such that
$G$
is characterized as the subgroup of $Aut (V)$ preserving $U$?} One of the main
problems in trying to
prove such a statement is that the vector sheaves (and similarly $P4$ or $P5$
sheaves) don't satisfy any nice chain condition.
Note that in the situation of question 4, for any sub-vector sheaf $U$ of a
tensor and cotensor combination of $V$, the subgroup of $Aut (V)$ of elements
preserving $U$ is a presentable subgroup, so at least we obtain a way of
constructing examples, even if we don't know whether we get everything this way.
\numero{Presentable $n$-stacks}
Recall that an $n$-groupoid in the sense of \cite{Tamsamani} is essentially the
same thing as an $n$-truncated homotopy type \cite{Tamsamani2}. In
view of this,
we can approach the theory of $n$-stacks (we assume from here on that this
means $n$-stack of
$n$-groupoids and drop the word ``groupoid'' from the notation) via the
theory of
presheaves of topological spaces or equivalently simplicial
presheaves\cite{Jardine1}. We adopt a working convention that by {\em
$n$-stack}
we mean the presheaf of $n$-groupoids associated to a fibrant presheaf of spaces
\cite{Jardine1} \cite{kobe} or, a bit more generally, any presheaf of
$n$-groupoids
such that the associated simplicial presheaf (taking the diagonal of
the nerve) is
fibrant in the sense of \cite{kobe} which means that it satisfies the
global part
of the fibrant condition of Jardine \cite{Jardine1}.
Some special cases are worth mentioning. A $0$-stack is simply a sheaf of
sets.
A $1$-stack is what is usually called a stack---it is a sort of sheaf of
groupoids. The notions of $2$-stack and $3$-stack were explored heuristically
from the category-theoretic point of view in \cite{Breen23}.
We suppose given an adequate theory of morphism $n$-stacks $Hom (R,T)$; and of
homotopy fiber products $T\times _RT'$ for $n$-stacks. These can be had, for
example, within the realm of presheaves of spaces \cite{Jardine1} \cite{kobe}
\cite{flexible}.
The path-stack
$P^{t_1,t_2}T$ on ${\cal X} /S$ between two basepoints (i.e. objects)
$t_1,t_2\in T(S)$
is then well defined. We denote by $\pi _0(T)$ the truncation
down to a sheaf of
sets, and from this and the path space construction we obtain the homotopy group
sheaves $\pi _i(T,t)$ over ${\cal X} /S$ for an $n$-stack $T$ and object $t\in T(S)$.
In terms of the easier-to-understand version version of the theory involving
presheaves of spaces, the homotopy group sheaves are defined as follows. If
$t\in T(S)$ then for any $Y\rightarrow S$ we get a basepoint $t|_Y\in T(Y)$.
The
functor $$
Y/S\mapsto \pi _i (T(Y), t|_Y)
$$
is a presheaf on ${\cal X} /S$ which we denote by $\pi _i^{\rm pre}(T,t)$. Then $\pi
_i(T,t)$ is sheaf associated to this presheaf.
There is probably a good extension of the theory to $\infty$-stacks which would
correspond to presheaves of spaces which are not necessarily truncated (and I
suppose that it again becomes equivalent to Jardine's theory but there may be a
few subtleties hidden here). Generally below when we speak of $n$-stacks,
$n$ will
be indeterminate. There is probably not too much difference between the
theory of
$\infty$-stacks and the projective limit of the theories of $n$-stacks, so we
will stick to the notation $n$-stack.
For $t_1, t_2\in T(S)$ use the notation $\varpi _1(T,t_1,t_2)$ for the sheaf on
${\cal X} /S$ of paths in $T|_{{\cal X} /S}$ from $t_1$ to $t_2$ up to homotopy. Thus
$$
\varpi _1(T,t_1,t_2)= \pi _0 (P^{t_1,t_2}T).
$$
We make the
following definition.
\newline
---We say that an $n$-stack $T$ on ${\cal X}$ is {\em presentable} if it satisfies
the
following conditions:
\begin{enumerate}
\item The sheaf $\pi _0(T)$ is P1 over $k$.
\item For any finite type morphism of schemes $Z\rightarrow Y$ and any
two sections
$\eta : Y\rightarrow T$ and $\eta ': Z\rightarrow T$ the sheaf
$\varpi _1(T|_{{\cal Z}
/Z}, \eta |_Z, \eta ' )$, when restricted down from $Z$ to $Y$, is $P4$
over $Y$.
\item For any scheme $Y$ and section $\eta : Y\rightarrow T$, the higher
homotopy group sheaves $\pi _i( (T|_{{\cal Z} /Y}), \eta )$, for $i\geq 1$, are
presentable group sheaves ($P5$) over $Y$.
\end{enumerate}
(Recall that if ${\cal H}$ is a sheaf on ${\cal X} / Z$ then it can also be considered
as a
sheaf on ${\cal X}$ with a map to $Z$; the restriction down to $Y$ is the same sheaf
taken
with the composed map to $Y$, then considered as a sheaf on ${\cal X} /Y$. This
shouldn't be confused with the direct image from $Z$ to $Y$. In
heuristic topological terms the fiber over $y\in Y$ of the restriction is
obtained by taking the direct union of the fibers of ${\cal H}$ over the points $z$
lying over $y$, whereas the fiber of the direct image is obtained by taking the
direct product of the fibers of ${\cal H}$ over points $z$ lying over $y$.)
{\bf Caution:} This definition of presentability is very slightly different
from
the definition given in \cite{kobe}. The older version of presentability for
$T$ as defined in \cite{kobe}
corresponds to the property $P3$ for $\pi _0$ (see Theorem
\ref{I.1.q.1kobe} below); whereas the
present definition corresponds to the property $P3\frac{1}{2}$ (see
Theorem
\ref{I.1.q.1} below). I
hope that the present version corresponding to $P3\frac{1}{2}$ will be the most
useful. The reason for changing the definition was to be able to state
Theorem \ref{stability} in a nice way, i.e. to have a reasonable definition of
{\em presentable morphism} of $n$-stacks.
{\em Caution:} If $T$ is $0$-truncated, that is a sheaf of sets, and happens
to have a group structure, then this notion is not the same as the notion
that $T$ be a presentable group sheaf. The presentability in $T$ as
defined here refers to the higher homotopy groups. In fact, presentability in
this case corresponds to the property $P3\frac{1}{2}$ rather than $P4$
(see below).
We can also reasonably use the notations {\em presentable homotopy sheaf}; {\em
presentable space over ${\cal X}$} or just {\em presentable space}; or {\em
presentable fibrant presheaf of spaces}, for the notion of presentable
$n$-stack.
Property $1$ implies the seemingly stronger statement that
there is a section $f: Z\rightarrow T$ over a
scheme $Z$ of finite type over $k$, such that the associated morphism
$Z\rightarrow \pi _0(T)$ is surjective.
The second condition reduces, in the case $\eta = \eta '$, to the statement
that for any scheme $Y$ and section $\eta : Y\rightarrow T$, the fundamental
group sheaf $\pi _1( (T|_{{\cal Z} /Y}), \eta )$ is a presentable group sheaf over
$Y$.
We can give an alternative characterization, from which it follows
that any truncation $\tau _{\leq n}T$ of a presentable space is again
presentable.
Recall that we have defined a
condition $P3\frac{1}{2}$ which is intermediate between $P2$ and $P4$.
\begin{theorem}
\mylabel{I.1.q.1}
Suppose $T$ is an $n$-stack over $X$. Then $T$ is presentable
if and only if the sheaf $\pi _0$ is $P3\frac{1}{2}$, and for any $Y\in {\cal X}$ and
$t\in T(Y)$, the sheaves $\pi _i (T|_{{\cal X} /Y}, t)$ are presentable group sheaves
($P5$) over $Y$.
\end{theorem}
{\em Proof:}
Suppose $T$ is presentable. Then we just have to show that $\pi _0$ is
$P3\frac{1}{2}$. We know that it is P1, so there is a surjection
$Y\rightarrow \pi _0$. By replacing $Y$ by an etale cover, we may assume that
this comes from a point $t\in T(Y)$. The path space $P^{p_1^{\ast}t,
p_2^{\ast}t}T$ maps to $Y\times Y$, and
$$
\varpi _1(T, p_1^{\ast}t,
p_2^{\ast}t )=\pi _0(P^{p_1^{\ast}t,
p_2^{\ast}t}T)\rightarrow Y\times _{\pi _0} Y
$$
is surjective. Let $G\rightarrow Y$ be the sheaf of groups $\pi _1(T|_Y,t)$.
It is presentable by hypothesis, and $G$ acts freely on
(the restriction from $Y\times Y$ down to $Y$ of) $\varpi _1(T, p_1^{\ast}t,
p_2^{\ast}t )$ with quotient $Y\times _{\pi _0}Y$. Finally, we know
that (the restriction from $Y\times Y$ down to $Y$ of)
$\varpi _1(T, p_1^{\ast}t,
p_2^{\ast}t )$ is $P4$ over $Y$; thus the quotient
$Y\times _{\pi _0}Y$ is $P4$ over $Y$ by Theorem
\ref{I.1.d}. Now by definition there exists a surjective morphism
$Q\rightarrow Y\times _{\pi _0}Y$ which is $Y$-vertical. This is what
is required to show that $\pi _0$ is $P3\frac{1}{2}$.
Now suppose that $\pi _0$ is $P3\frac{1}{2}$ and that the other homotopy
group sheaves are presentable. We obtain immediately that $\pi _0$ is P1. Let
$X\rightarrow \pi _0$ be the surjection given by the property $P3\frac{1}{2}$.
Then we have
an $X$-vertical surjection $Q\rightarrow X\times _{\pi _0}X$ (where the morphism
to $X$ is the first projection). Suppose $X'\rightarrow X$ is an etale
surjection
chosen so that the map $X\rightarrow \pi _0$ lifts to $t\in T(X')$. Let $Q'$ be
the pullback of $X' \times X'$ to $Q$. Then $Q'= (X'\times _{\pi _0}X')\times
_{X\times _{\pi _0}X}Q$ so $Q'\rightarrow X'\times _{\pi _0}X'$ is
$X$-vertical, and hence $X'$-vertical. This implies that $X'\times _{\pi
_0}X'$ is $P4$ over $X'$, because we can take as the relation scheme $$
Q'\times
_{X'\times _{\pi _0}X'}Q'= Q'\times _{X'\times X'}Q'
$$
which is already a
scheme of finite type (and the identity is vertical). Now we have a sheaf of
groups $G= \pi _1(T|_{X'}, t)$ over $X'$ which is by hypothesis presentable, and
$G$ acts freely on $\varpi _1(T, p_1^{\ast}t,
p_2^{\ast}t )$ with
quotient $X'\times _{\pi _0}X'$. By Theorem \ref{I.1.d},
$\varpi _1(T, p_1^{\ast}t,
p_2^{\ast}t )$ is $P4$ over $X'$.
Now suppose that we have a finite type morphism $q:Z\rightarrow Y$ and two
points $\eta _1 \in T(Y)$ and $\eta _2 \in T(Z)$, and we show that the
restriction
from $Z$ to $Y$ of the path space $\varpi _1(T, \eta _1 |_Z,\eta _2)$
is $P4$ over
$Y$. There are etale surjections $ Y'\rightarrow Y$ and $Z'\rightarrow Z$ (of
finite type) with $Z'\rightarrow Y'$ and there are morphisms
$f_1:Y'\rightarrow X'$ and $f_2: Z'\rightarrow X'$ such that $f_1^{\ast} (t)$ is
homotopic to $\eta _1|_{Y'}$ and
$f_2^{\ast} (t)$ is
homotopic to $\eta _2|_{Z'}$. Let $(f_1|_{Z'},f_2): Z'\rightarrow X'\times
X'$ denote the resulting morphism (the first projection of which factors through
$Y'$).
Then
$$
\varpi _1(T,\eta _1|_{Z}, \eta _2)|_{Z'}=
\varpi _1(T,\eta _1|_{Z'}, \eta _2|_{Z'})= (f_1|_{Z'},f_2)^{\ast}
\varpi _1(T,p_1^{\ast}t, p_2^{\ast}t)
$$
$$
= (q, f_2)^{\ast}[\varpi _1(T,p_1^{\ast}t, p_2^{\ast}t)|_{Y'\times X'}].
$$
Note that $\varpi _1(T,p_1^{\ast}t, p_2^{\ast}t)|_{Y'\times X'}$ is
$P4$ with respect to $Y'$, so by the appendix to the proof below,
one gets that the restriction down to $Y'$ of $\varpi _1(T,\eta _1|_{Z}, \eta
_2)|_{Z'}$ is $P4$ with respect to $Y'$. By Corollary \ref{I.1.j.1}, the
restriction down to $Y$ of $\varpi _1(T,\eta _1|_{Z}, \eta _2)$ is $P4$
over $Y$.
\hfill $\Box$\vspace{.1in}
{\em Appendix to the proof:}
Suppose $Z\rightarrow Y$ is a finite type morphism, and suppose ${\cal F}$ is a
sheaf on
$Y$. Then the restriction from $Z$ down to $Y$ of the pullback ${\cal F} |_Z$ is
equal to the fiber product $Z\times _Y{\cal F}$. Note also that $Z$ is $P4$ over
$Y$. Thus if ${\cal F}$ is $P4$ over $Y$ then the restriction of the pullback is
again $P4$.
\begin{corollary}
\mylabel{truncation}
If $T$ is a presentable $n$-stack and if $m<n$ then $\tau _{\leq m}T$ is a
presentable $m$-stack.
\end{corollary}
{\em Proof:}
Indeed the truncation operation preserves the homotopy group sheaves (and the
homotopy sheaf $\pi _0$). By the theorem, presentability is expressed solely in
terms of these sheaves so it is preserved by truncation.
\hfill $\Box$\vspace{.1in}
We have a similar theorem for the old version of presentability of $T$ \cite
{kobe}.
\begin{theorem}
\mylabel{I.1.q.1kobe}
Suppose $T$ is an $n$-stack over $X$. Then $T$ is presentable in the sense of
\cite{kobe} if and only if the sheaf $\pi _0$ is $P3$, and for any
$Y\in
{\cal X}$ and $t\in T(Y)$, the sheaves $\pi _i (T|_{{\cal X} /Y}, t)$ are presentable
group
sheaves ($P5$) over $Y$.
\end{theorem}
{\em Proof:} The proof is the same as above only very slightly easier.
The details
are left to the reader.
\hfill $\Box$\vspace{.1in}
\subnumero{Very presentable $n$-stacks}
We make the following more restrictive definition. Say that a presentable
group sheaf $G$ on ${\cal X} /S$ is {\em affine} if, for any artinian $S$-scheme
$S'$, the group scheme $G(S')$ over $Spec (k)$ is affine. A truncated homotopy
sheaf $T$ is {\em very presentable} if $T$ is presentable and if for any $\eta
\in T_Y$ we have that $\pi _1(T/Y,\eta )$ is affine, and $\pi _i(T/Y, \eta )$
are vector sheaves for $i\geq 2$.
The idea behind the definition of ``very presentable'' is that we want to
require
the higher homotopy groups to be unipotent.
Note that
if we don't require $\pi _1$ to be affine, or $\pi _i$ to be unipotent $(i\geq
2$), then the comparison between algebraic and analytic de Rham cohomology
(announced in \cite{kobe}) is no longer true, even over the base $S=Spec (k)$
when all of the groups are representable. This is the reason for making the
definition of ``very presentable''.
I make the following conjecture:
\begin{conjecture}
\mylabel{I.1.r}
If $G$ is an abelian affine presentable group sheaf on ${\cal X} /S$ such that
for any
artinian $S'\rightarrow S$ the group scheme $G(S')$ over $k$ is a direct sum
of additive groups, then $G$ is a vector sheaf.
\end{conjecture}
If we knew this conjecture, we could replace the condition of being a vector
sheaf by the condition that the $G(S')$ are unipotent (hence additive) for
$G=\pi _i$, $i\geq 2$; this would then be along the same lines as the affineness
condition for $\pi _1$. As it is, we need to require the condition of $\pi
_i$ being vector schemes ($i\geq 2$) for many of the arguments concerning
de Rham
cohomology sketched in \cite{kobe} to work.
{\em Remark:} The categories of presentable and very presentable $n$-stacks are
closed under weak equivalences and fiber products but not under cofiber products
(push-outs); thus they are not closed model categories.
{\em Remark:} We have the same statement as Corollary \ref{truncation} for very
presentable stacks (if $T$ is very presentable then $\tau _{\leq m} T$ is very
presentable).
\subnumero{Other presentability conditions}
Recall from \cite{kobe} that we used the notation $P6$ for affine presentable
group sheaves and $P7$ for vector sheaves. An $n$-stack $T$ on ${\cal X}$ is
{\em $(a_0,\ldots , a_n)$-presentable} (with $a_i \in \{ 0,1, 2 ,3,
3\frac{1}{2},
4, 5,6, 7\}$) if $\pi _0(T)$ is $Pa_0$ and if for any scheme $Y$ and
$t\in T(Y)$,
$\pi _i (T, t)$ is $Pa_i$ over $Y$. Here by convention $P0$ means no
condition at
all. Thus a presentable $n$-stack in our previous notation becomes a
$(3\frac{1}{2},5,5, \ldots )$-presentable $n$-stack in this notation. A very
presentable $n$-stack is a $( 3\frac{1}{2},
6, 7, 7, \ldots )$-presentable $n$-stack. The old notions of presentability
and very presentability as defined in \cite{kobe} are respectively
$(3,5,5,\ldots
)$-presentability and $(3,6,7,7, \ldots )$ presentability. There may be some
interest in considering, for example, the $(2,2,2,\ldots )$-presentable
$n$-stacks, or the $(0,0, 7,7,7,\ldots )$-presentable $n$-stacks.
Some other useful versions might be $(4, 5, 5, \ldots
)$-presentable $n$-stacks, or $(4, 6, 7, 7, \ldots )$-presentable $n$-stacks
for example. Here the condition $P4$ on $\pi _0$ would be with
respect to $S=Spec
(k)$. For example an algebraic stack with smooth morphisms from the morphism
scheme to the object scheme (or even more strongly a Deligne-Mumford
stack where
these morphisms are etale) would be a $(4, 5)$-presentable stack. The converse
is not true since in the condition of $(4,5)$-presentability, the morphism
sheaves are not necessarily representable. In fact we will never see the
condition of representability of the morphism sheaves in our context, since this
is unnatural from the point of view of higher-order stacks (and even in the
context of algebraic stacks, one may wonder why the morphism object itself was
never allowed
to be an algebraic space?).
{\em Remark:}
Again we have the statement of Corollary \ref{truncation}: if $T$ is an
$(a_0,\ldots , a_n)$-presentable $n$-stack then $\tau _{\leq m}T$ is an
$(a_0,\ldots , a_m)$-presentable $m$-stack.
{\em Remark:} A good convention for using all of these different notions would
be to chose some variables $A$, $B$, etc. and set them to be specific $(a_0,
a_1, \ldots )$ at the start of a discussion, then to use the notation
``$A$-presentable'' or ``$B$-presentable'' throughout the discussion.
\subnumero{A relative version of presentability}
We can make a relative definition. In general, say that a morphism
$T\rightarrow
R$ of $n$-stacks is {\em $(a_0,\ldots , a_n)$-presentable} if for any scheme
$Y\in {\cal X}$ and any morphism $Y\rightarrow R$, the fiber $T\times _RY$ is
$(a_0,\ldots , a_n)$-presentable. In particular we obtain the notions of
presentable and very presentable morphisms by taking
$(3\frac{1}{2},5,5, \ldots )$ and $(3\frac{1}{2},6,7,7, \ldots )$ respectively.
It is clear that if $T\rightarrow R$ is an $(a_0,\ldots , a_n)$-presentable
morphism and if $R'\rightarrow R$ is any morphism of $n$-stacks then the
morphism
$T':= T\times _RR'\rightarrow R'$ is $(a_0,\ldots , a_n)$-presentable.
\begin{lemma}
\mylabel{structural}
Suppose that $a_0 \leq 5$.
An $n$-stack $T$ on ${\cal X}$ is
$(a_0,\ldots , a_n)$-presentable if and only if the structural morphism
$T\rightarrow \ast$ is $(a_0,\ldots , a_n)$-presentable.
\end{lemma}
{\em Proof:}
Since $\ast$ is itself a scheme of finite type (it is $Spec (k)$) the structural
morphism being $(a_0,\ldots , a_n)$-presentable implies that $T$ is
$(a_0,\ldots , a_n)$-presentable.
For the other implication, suppose $T$ is
$(a_0,\ldots , a_n)$-presentable, then for any scheme of finite type $Y$ we have
that $T\times Y = T\times _{\ast}Y$ is $(a_0,\ldots , a_n)$-presentable (since
a scheme $Y$ is $a_0$-presentable for any $a_0 \leq 5$).
\hfill $\Box$\vspace{.1in}
{\em Remark:} If ${\cal G}$ is a sheaf of groups on ${\cal X} /S$ then ${\cal G}$ is a
presentable group sheaf if and only if $K({\cal G} , 1)\rightarrow S$ is a
presentable morphism of $1$-stacks. This is the correct point of view relating
our terminologies ``presentable group sheaf'' and ``presentable morphism'' or
``presentable $n$-stack'', i.e. the answer to the terminological problem posed
by the caution at the start of this section.
\begin{theorem}
\mylabel{stability}
Suppose $R$ is a presentable (resp. very presentable) $n$-stack. Then
a morphism
$T\rightarrow R$ is presentable (resp. very presentable) if and only if $T$
itself is presentable (resp. very presentable).
\end{theorem}
The proof of this theorem will be given in the next subsection below.
We first state a few corollaries.
\begin{corollary}
\mylabel{fiberprod}
Suppose $T\rightarrow R$ and $S\rightarrow R$ are morphisms between presentable
(resp. very presentable) $n$-stacks. Then the fiber product $T\times _RS$ is
presentable (resp. very presentable).
\end{corollary}
{\em Proof:}
From the theorem, the morphism $T \rightarrow R$ is presentable, hence the
morphism $T\times _RS$ is presentable and since $S$ is presentable, again from
the theorem we conclude that $T\times_RS$ is presentable. The same goes for very
presentable. \hfill $\Box$\vspace{.1in}
\begin{lemma}
\mylabel{basechange}
Suppose $R'\rightarrow R$ is a morphism inducing a surjection on $\pi _0$.
Then a morphism $T\rightarrow R$ is presentable (resp. very presentable) if and
only if the morphism $T':= T\times _RR'\rightarrow R'$ is presentable (resp.
very
presentable). \end{lemma}
{\em Proof:}
One direction follows directly from the first remark after the definition above.
For the other direction, suppose that $T'\rightarrow R'$ is presentable
(resp. very
presentable). Then
for any scheme $Y\rightarrow R$ there is an etale covering $Y' \rightarrow Y$
and a lifting $Y'\rightarrow R'$, and we have
$$
(T\times _RY)\times _YY'=T\times _RY'=T' \times _{R'}Y',
$$
which is presentable (resp. very
presentable) by hypothesis. The conditions on homotopy sheaves for
being presentable (resp. very
presentable) are etale-local, so $T\times _RY$ is presentable (resp. very
presentable). \hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{composition}
Suppose $R\rightarrow S$ and $S\rightarrow T$ are presentable (resp. very
presentable) morphisms of $n$-stacks. Then the composition $R\rightarrow T$ is a
presentable (resp. very presentable) morphism.
\end{corollary}
{\em Proof:}
Suppose $X$ is a scheme of finite type with a morphism $X\rightarrow T$.
Then
$$
X\times _TR = (X\times _TS) \times _SR.
$$
By hypothesis, $(X\times _TS)$ is presentable (resp. very presentable),
and by the other hypothesis and the base change property given at the start of
the subsection, the morphism $(X\times _TS) \times _SR\rightarrow
(X\times _TS)$ is presentable (resp. very presentable). Theorem \ref{stability}
now implies that $X\times _TR$ is presentable (resp. very presentable). By
definition then, the morphism $R\rightarrow T$ is presentable (resp. very
presentable).
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{check}
Suppose $f:T\rightarrow R$ is a morphism such that $R$ is presentable (resp.
very presentable), and suppose $X\rightarrow
R$ is a morphism from a scheme of finite type $X$ which is surjective on
$\pi _0$.
Then $T$ and the morphism $f$ are presentable (resp. very presentable)
if and only if $T\times _RX$ is presentable (resp. very presentable).
\end{corollary}
{\em Proof:}
By Lemma \ref{basechange} the morphism $f$ is presentable if and only if the
morphism $p_2: T\times _RX\rightarrow X$ is presentable. On the other
hand, $T$ is
presentable if and only if $f$ is presentable, from Theorem \ref{stability}.
Similarly $T\times _RX$ is presentable if and only if $p_2$ is presentable
again by \ref{stability}. This gives the desired statement (the
same proof holds for
very presentable).
\hfill $\Box$\vspace{.1in}
We now give some results that will be used in the proof of Theorem
\ref{stability}.
\begin{lemma}
\mylabel{vector?}
Suppose $V$ is a vector sheaf and $G$ is a presentable group sheaf on
${\cal X} /S$. If
$f: V\rightarrow G$ is a morphism of group sheaves then the kernel of $f$ is a
vector sheaf.
\end{lemma}
{\em Proof:}
There is a natural isomorphism of vector sheaves $\varphi : V \cong Lie (V)$,
such that $\varphi$ reduces to the exponential on the values over artinian $S'$.
To construct $\varphi$ note that a section of $V$ may be interpreted as a map
${\cal O} \rightarrow V$. We have a tautological section of $Lie ({\cal O} )$ so for
every section of $V$ the image of this tautological section is a section of
$Lie (V)$. This map is an isomorphism on values over artinian schemes,
so it is an isomorphism.
Let $U \subset Lie (V)$ be the kernel of
$$
Lie (f) : Lie
(V)\rightarrow Lie (G).
$$
Since $Lie (f)$ is a morphism of vector sheaves, its
kernel $U$ is a vector sheaf. We claim that $\varphi ^{-1}(U)$ is the kernel of
$f$. In order to prove this claim it suffices to prove it for the values over
artinian $S'$ (since both are presentable and contained in $V$, and using
\ref{Krull}). Here it reduces to the following statement about Lie groups: the
kernel of an algebraic morphism from a vector space to a Lie group is the
exponential of the kernel of the corresponding morphism of Lie algebras. To
prove this notice first that this exponential is a subvector subspace; we can
take the quotient and then we are reduced to the case where the map is injective
on Lie algebras. The kernel is thus a finite subgroup, but a vector space
contains no finite subgroups so we are done.
\hfill $\Box$\vspace{.1in}
\begin{proposition}
\mylabel{I.1.s.3}
Suppose $R$, $S$ and $T$ are $n$-stacks over ${\cal X}$, with morphisms
$R\rightarrow T$ and $S\rightarrow T$. Suppose $Z\in {\cal X}$ and $(r,s)\in
R\times _TS(Z)$. Let $t\in T(Z)$ be the common image of $r$ and $s$. Then we
have the following long exact sequence of homotopy group sheaves on ${\cal X} /Z$:
$$
\ldots \rightarrow \pi _i (R\times _TS|_{{\cal X} /Z},(r,s))\rightarrow
\pi _i(R|_{{\cal X} /Z},r)\times \pi _i (S|_{{\cal X} /Z})\rightarrow
$$
$$
\pi _i (T|_{{\cal X} /Z},t)\rightarrow \pi _{i-1}(R\times _TS|_{{\cal X} /Z},(r,s))
\rightarrow \ldots ,
$$
terminating with the sequence
$$
\pi _2(R|_{{\cal X} /Z},r)\times \pi _2(S|_{{\cal X} /Z},s)\rightarrow
\pi _2(T|_{{\cal X} /Z},t)\rightarrow \pi _1(R\times
_TS|_{{\cal X} /Z}, (r,s))\rightarrow
$$
$$
\pi _1(R|_{{\cal X} /Z},r)\times
\pi _1(S|_{{\cal X} /Z},s) \stackrel{\displaystyle
\rightarrow }{\rightarrow } \pi _1(T|_{{\cal X} /Z},t)
$$
(the last part meaning that the image is equal to the equalizer of the two
arrows). Furthermore, we have a similar sequence for the path spaces. Suppose
$(r_1,s_1)$ and $(r_2,s_2)$ are two points, with images $t_1$ and $t_2$.
We have the exact sequence
$$
\pi _2(R|_{{\cal X} /Z},r_1)\times \pi _2(S|_{{\cal X} /Z},s_1)\rightarrow
\pi _2(T|_{{\cal X} /Z},t_1)\stackrel{acts\; on}{\rightarrow} \varpi _1(R\times
_TS|_{{\cal X} /Z}, (r_1,s_1),(r_2,s_2))
$$
$$
\mbox{with quotient the equalizer of} \varpi
_1(R|_{{\cal X} /Z},r_1,r_2)\times \varpi _1(S|_{{\cal X} /Z},s_1,s_2)
\stackrel{\displaystyle \rightarrow }{\rightarrow } \varpi _1(T|_{{\cal X}
/Z},t_1,t_2). $$
\end{proposition}
{\em Proof:}
We show that we have similar exact sequences at the homotopy presheaf level;
then the sequences for the homotopy sheaves follow by sheafification. To
define the exact sequences at the presheaf level, we can work object by
object, so it suffices to give functorial exact sequences for fibrations of
topological spaces $R\rightarrow T$ and $S\rightarrow T$ with basepoints
$(r,s)$ mapping to $t$. The morphisms are defined as follows. The morphism
from $\pi _i (R\times _TS,(r,s))$ to $\pi _i(R,r)\times \pi _i (S,s)= \pi _i
(R\times S, (r,s))$ comes from the inclusion $R\times _TS\rightarrow R\times
S$. The morphism from the product to $\pi _i (T,t)$ is the difference of the
two projection maps. The morphism from $\pi _i (T,t)$ to $\pi _{i-1}(R\times
_TS,(r,s))$ is obtained as a composition
$$
\pi _i (T,t)\stackrel{\delta}{\rightarrow} \pi _{i-1} (R_t,r)
\stackrel{(1, 0_s)}{\rightarrow }\pi _{i-1}(R_t\times S_t,(r,s))
\stackrel{i}{\rightarrow}\pi _{i-1}(R\times _TS,(r,s))
$$
where $\delta$ is the connecting homomorphism for the fibration $R\rightarrow
T$, $0_s$ is the constant class at the basepoint $s$, and $i$ is the inclusion
of the fiber $i: R_t\times S_t\rightarrow R\times _TS$. If we took $(1,1)$
instead of $(1,0_s)$ we would get the connecting morphism for the fibration
$R\times _TS\rightarrow T$, which goes to zero in the homotopy of the total
space $R\times _TS$. Thus, our map is the same as the map which would be
obtained by putting in $-(0_r,1)$ instead. From the equality of these two
maps, one obtains that the composition of this map with the difference of
projections, is equal to zero. That the other compositions are zero is easy to
see. Exactness follows by making a diagram with this sequence on one
horizontal row, with the sequence
$$
\pi _i(R_t\times S_t, (r,s))= \pi _i(R_t,r)\times \pi _i (S_t,s)\rightarrow
0\rightarrow \ldots
$$
on the row above, and the sequence
$$
\pi _i (T,t)\rightarrow \pi _i (T,t)\times \pi _i (T,t)\rightarrow \pi _i
(T,t) \stackrel{0}{\rightarrow} \pi _{i-1}(T,t)
$$
on the row below. The vertical rows then have the exact fibration sequences
going downwards. One obtains the exactness of the sequence of homotopy
groups in question (this works at the end by using the extension of the
homotopy sequence for a fibration, to the action of $\pi _1$ of the base on
$\pi _0$ of the fiber, with the $\pi _1$ of the total space being the
stabilizer of the component of $\pi _0$ of the fiber containing the
basepoint.
Finally, we treat the case of the path spaces. What is written on the left
means, more precisely, that the cokernel of the first map acts freely on the
middle sheaf, with quotient equal to the equalizer. The action in question is
by the map to $\pi _1(R\times _TS, (r_1,s_1))$ which itself acts on the path
space. Now if $\varpi _1(R\times _TS, (r_1,s_1), (r_2,s_2))$ is empty then
the equalizer in question is also empty (any element of the equalizer can be
realized as a pair of paths mapping to exactly the same path in $T$, giving a
path in the fiber product). Note that we count an action on the empty set as
free. So we may assume that $\varpi _1(R\times _TS, (r_1,s_1), (r_2,s_2))$ is
nonempty, and choose an element. This choice gives compatible choices in all
the other path spaces, so composing with the inverse of this path we reduce to
the exact sequence for fundamental groups. \hfill $\Box$\vspace{.1in}
{\em Remark:} We can extend this sequence to a statement involving $\pi _0$,
specially in the case of a fibration sequence. This will be done as we need it
below.
\begin{lemma}
\mylabel{kernel}
Suppose $S$ is a base scheme and suppose ${\cal F}$ is a sheaf on ${\cal X} /S$ whose
restriction down to ${\cal X}$ is $P3\frac{1}{2}$. Suppose that ${\cal G}$ is a
$P4$ sheaf on ${\cal X} /S$ with morphism ${\cal G} \rightarrow {\cal F}$, and finally suppose
that $\eta : S\rightarrow {\cal F}$ is a section. Then the inverse image ${\cal H}
\subset
{\cal G}$ of the section $\eta$ is a $P4$ sheaf.
\end{lemma}
{\em Proof:}
Let $X\rightarrow {\cal G}$ and $W\rightarrow X\times _{{\cal G}}X$ be the $S$-vertical
surjections for ${\cal G}$. Fix a surjection $Z\rightarrow {\cal F}$ and a surjection
$W\rightarrow Z\times _{{\cal F}}Z$ which is vertical with respect to the first
projection to $Z$. Fix a lifting $\eta '$ of the section to $Z$ (note that we
are allowed to etale-localize on the base $S$). Let $U:= S\times _{Z}W$
where the
morphism in the fiber product is the first projection from $W$ to $Z$ (note that
$U$ is a scheme of finite type over $S$). The surjective morphism
$$
U\rightarrow S\times _{Z}(Z\times _{{\cal F}}Z) = S\times _{{\cal F}} Z
$$
is $S$-vertical since the morphism $W\rightarrow Z\times _{{\cal F}}Z$
was $Z$-vertical.
We can choose a lifting $X\rightarrow Z$ of the morphism ${\cal G} \rightarrow {\cal F}$.
Then
$$
S\times _{{\cal F}} X= (S\times _{{\cal F}}Z)\times _ZX
$$
so there is an $S$-vertical morphism
$$
U\times _Z X \rightarrow S\times _{{\cal F}} X.
$$
On the other hand the $S$-vertical morphism $X\rightarrow {\cal G}$ gives
an $S$-vertical morphism
$$
S\times _{{\cal F}} X\rightarrow S\times _{{\cal F}} {\cal G} = {\cal H} .
$$
Note that $Y:= U\times _Z X$ is a scheme of finite type with a surjective
vertical
morphism to ${\cal H}$. Since ${\cal G}$ is $P4$ there exists a scheme of finite type $V$
and an $S$-vertical morphism
$$
V\rightarrow Y\times _{{\cal G}}Y = Y\times _{{\cal H}} Y.
$$
This gives the condition $P4$ for ${\cal H}$.
\hfill $\Box$\vspace{.1in}
The following lemma is a consequence of Corollary
\ref{fiberprod}, but we need
it in the proof of Theorem \ref{stability}.
\begin{lemma}
\mylabel{I.1.s.?}
If $R$ and $S$ are presentable (resp. very presentable) $n$-stacks over
${\cal X}$ and $X$ a scheme of finite type, with morphisms $R\rightarrow S$ and
$X\rightarrow S$, then the homotopy fiber product $X\times _SR$ is presentable
(resp. very presentable).
\end{lemma}
{\em Proof:}
Suppose $f:Y\rightarrow X\times _SR$ is a morphism. Let $r: Y\rightarrow R$ and
$s: Y\rightarrow S$ be the composed morphisms. Then (since $X$ is
zero-truncated)
for $i\geq 1$ we have
$$
\pi _i(X\times _SR |_{{\cal X} /Y}, f)= \pi _i (Y\times _SR/Y, r).
$$
The latter fits into a homotopy exact sequence, which we can therefore write
$$
\ldots \pi _{i+1}(S|_{{\cal X} /Y}, s)\rightarrow
\pi _i(X\times _SR |_{{\cal X} /Y}, f)\rightarrow \pi _i(R|_{{\cal X} /Y}, r)\rightarrow
\ldots .
$$
In the presentable case we obtain immediately from Theorem \ref{I.1.e} that
$\pi _i(X\times _SR |_{{\cal X} /Y}, f)$ is a presentable group sheaf over $Y$.
In the very presentable case,
for $i\geq 3$ we obtain immediately (from Corollary \ref{I.j} and Theorem
\ref{I.k}) that $\pi _i(X\times _SR |_{{\cal X} /Y}, f)$ is a vector sheaf. For
$i=2$ we obtain the same conclusion but must also use Lemma \ref{vector?}.
For $i=1$ we obtain that $\pi _1(X\times _SR |_{{\cal X} /Y}, f)$ is $P5$. In fact it
is an extension of the kernel of a morphism of $P6$ group sheaves, by a vector
sheaf. Therefore it is also affine (since kernels and extensions by vector
sheaves at least, preserve the affineness property). Thus it is $P6$.
We just have to prove (in both the presentable and very presentable case)
that $\pi _0(X\times _SR)$ is $P3\frac{1}{2}$. Let $a: X\rightarrow S$ denote
the given morphism. Recall that $\pi _0(X\times _SR)/X$ denotes this sheaf
considered as a sheaf on ${\cal X} /X$.
We have an action of $\pi _0(S|_{{\cal X} /X}, a)$ (which is a $P5$ group sheaf over
$X$) on $\pi _0(X\times _SR)/X$, and the quotient is the fiber product $X\times
_{\pi _0(S)}\pi _0(R)/X$ (i.e. again considered as a sheaf over ${\cal X} /X$). This
is the same thing as the inverse image of the given section $a$ via the map $\pi
_0(R|_{{\cal X} /X})\rightarrow \pi _0(S|_{{\cal X} /X})$. By Corollary \ref{P3c} or
\ref{P3d} the quotient by the action is $P3\frac{1}{2}$. Finally by Proposition
\ref{P3e}, the sheaf $\pi _0(X\times _SR)$ is $P3\frac{1}{2}$.
\hfill $\Box$\vspace{.1in}
{\em Remark:}
A similar technique allows one to directly prove Corollary \ref{fiberprod}, that
if $R$, $S$ and $T$ are presentable (resp. very presentable) $n$-stacks with
morphisms $R\rightarrow S$ and $T\rightarrow S$ then the fiber product $R\times
_ST$ is presentable (resp. very presentable). This is left to the reader. Our
technique is to use only the above special case to get Theorem \ref{stability},
and then to deduce Corollary \ref{fiberprod} as a consequence.
\subnumero{The proof of Theorem \ref{stability}}
Lemma \ref{I.1.s.?} immediately implies one direction in Theorem
\ref{stability},
namely that if $R$ and $S$ are presentable then the morphism $f$ is presentable.
We have to show the other direction: suppose $S$ is a presentable
$n$-stack, $R$ is an $n$-stack, and $f:R\rightarrow S$ is a presentable
morphism. Choose a scheme of finite type $X$ with a morphism $X\rightarrow S$
inducing a surjection on $\pi _0$. We will show that if $X\times _SR$ is
presentable then $R$ is presentable.
First of all the morphism $\pi _0(X\times _SR)\rightarrow \pi _0(R)$ is
surjective so if $\pi _0(X\times _SR)$ is $P1$ then so is $\pi _0(R)$.
For the higher homotopy groups, suppose that $s:Z\rightarrow R$ is a morphism.
Lift the projection into $S$ (denoted by $s$) to a morphism $Z\rightarrow X$.
This gives a point $f: Z\rightarrow X\times _SR$ and by composition
$f_Z: Z\rightarrow Z\times _SR= Z\times _X(X\times _SR)$.
Then we have the exact sequence
$$
\ldots \rightarrow \pi _i(Z\times _SR |_{{\cal X} /Z}, f_Z)
\rightarrow \pi _i(R|_{{\cal X} /Z}, r)\rightarrow \pi _i(S|_{{\cal X} /Z}, s)\rightarrow
\ldots .
$$
But since $Z$ and $X|_{{\cal X} /Z}$ are zero-truncated, and we have that
$Z\times _SR = Z\times _X(X\times _SR)$, the higher
homotopy groups $\pi _i(Z\times _SR |_{{\cal X} /Z}, f_Z)$ are the same as the
$\pi _i(X\times _SR |_{{\cal X} /Z}, f)$.
Thus we can write the exact sequence as
$$
\ldots \rightarrow \pi _i(X\times _SR |_{{\cal X} /Z}, f)
\rightarrow \pi _i(R|_{{\cal X} /Z}, r)\rightarrow \pi _i(S|_{{\cal X} /Z}, s)\rightarrow
\ldots .
$$
Note that (in the very
presentable case) the kernel of the morphism
$$
\pi _2(S|_{{\cal X} /Z}, s)\rightarrow \pi _1(X\times _SR |_{{\cal X} /Z}, f)
$$
is a vector sheaf by Lemma \ref{vector?}. In the other cases the kernel (and
the
cokernel on the other end) are automatically vector sheaves by Corollary
\ref{I.j}. Since the property of being a vector sheaf is preserved under
extension we get the condition that the $\pi _i(R|_{{\cal X} /Z}, r)$ are vector
sheaves ($i\geq 2$). In the presentable case the exact sequence immediately
gives the property $P5$ for the $\pi _i(R|_{{\cal X} /Z}, r)$ for ($i\geq 2$).
We have to treat the case of $\varpi _1$. Suppose $Z\rightarrow Y$ is a
morphism
of finite type and suppose $r, r': Z\rightarrow R$ are points such that $r$
factors through $Y$. Let $s,s'$ denote the images in $S$ and assume that they
lift to points $f, f'$ and $f_Z, f'_Z$ as above (with $f$ or $f_Z$ factoring
through $Y$).
We first study everything on the level of sheaves on ${\cal X} /Z$. Note first that
$$
Z \times _{S|_{{\cal X} /Z}}(R|_{{\cal X} /Z}) \rightarrow R|_{{\cal X} /Z}\rightarrow
S|_{{\cal X} /Z}
$$
is a fibration sequence (this should actually have been pointed out above in the
treatment of the $\pi _i$, $i\geq 2$), over the basepoint $s\in S(Z)$.
On the other hand note that $r': Z\rightarrow R$ is a point lying over $s'$.
Consider the map
$$
\varpi _1(S|_{{\cal X} /Z}, s, s')\rightarrow
\pi _0(Z \times _{S|_{{\cal X} /Z}}(R|_{{\cal X} /Z}))
$$
which sends a path to the point obtained by transporting $f'_Z$ along the path
from $s'$ back to $s$.
The fibration sequence gives the following statement:
{\em The group $\pi _1(Z \times _{S|_{{\cal X} /Z}}(R|_{{\cal X} /Z}, f_Z)$ acts on
$\varpi _1(R|_{{\cal X} /Z}, r, r')$ with quotient the inverse image in
$\varpi _1(S|_{{\cal X} /Z}, s, s')$ of the section $f_Z: Z \rightarrow \pi _0(
Z \times _{S|_{{\cal X} /Z}}(R|_{{\cal X} /Z})$.
}
Now we note that
$$
\pi _1(Z \times _{S|_{{\cal X} /Z}}(R|_{{\cal X} /Z}, f_Z) =
\pi _1(X\times _SR|_{{\cal X} /Z}, f),
$$
and
$$
\pi _0(Z \times _{S|_{{\cal X} /Z}}(R|_{{\cal X} /Z})\subset \pi _0(X\times _S
R|_{{\cal X} /Z}).
$$
The transport of $f'$ along the path from $s'$ to $s$ again gives a map
$$
\varpi _1(S|_{{\cal X} /Z}, s, s')\rightarrow
\pi _0(X\times _SR|_{{\cal X} /Z})
$$
and we obtain the following statement.
{\em The group $\pi _1(X\times _SR|_{{\cal X} /Z}, f)$ acts on
$\varpi _1(R|_{{\cal X} /Z}, r, r')$ with quotient the inverse image in
\linebreak
$\varpi _1(S|_{{\cal X} /Z}, s, s')$ of the section $f: Z \rightarrow \pi
_0( X\times _SR|_{{\cal X} /Z})$.
}
Now we look at everything in terms of sheaves on ${\cal X} /Y$.
Let $Res _{Z/Y}$ denote the restriction from $Z$ down to $Y$, and let
$\tilde{f}$ denote the $Y$-valued point corresponding to $f$.
Note that
$$
Res _{Z/Y} \pi _0(X\times _SR|_{{\cal X} /Z}) = \pi _0(X\times _SR|_{{\cal X} /Y})\times
_YZ.
$$
In general if ${\cal A}$ is a sheaf over $Z$ and ${\cal B}$ a sheaf over $Y$ with a
section $Y\rightarrow {\cal B}$ then
$$
Res _{Z/Y}({\cal A} \times _{{\cal B} |_{{\cal X} /Z}}Z) = (Res _{Z/Y}{\cal A} )\times _{{\cal B}}Y.
$$
In particular the inverse image in $\varpi _1(S|_{{\cal X} /Z}, s, s')$ of the
section $f: Z \rightarrow \pi _0(
X\times _SR|_{{\cal X} /Z})$ restricts down to $Y$ to the inverse image in
$Res _{Z/Y}\varpi _1(S|_{{\cal X} /Z}, s, s')$ of the section
$\tilde{f}: Y \rightarrow \pi _0(X\times
_SR|_{{\cal X} /Y})$.
Another general principal is that if ${\cal G}$ is a group sheaf on $Y$ such that
${\cal G} |_{{\cal X} /Z}$ acts on a sheaf ${\cal H}$ then ${\cal G}$ acts on $Res _{Z/Y}{\cal H}$ with
quotient equal to $Res _{Z/Y}({\cal H} /({\cal G} |_{{\cal X} /Z}))$.
With these things in mind,
our above statement becomes:
{\em The group $\pi _1((X\times _SR |_{{\cal X} /Y}, \tilde{f})$ acts on
$Res _{Z/Y}\varpi _1(R|_{{\cal X} /Z}, r, r')$ with quotient the inverse image in
$Res _{Z/Y}\varpi _1(S|_{{\cal X} /Z}, s, s')$ of the section
$\tilde{f}: Y \rightarrow \pi _0(X\times
_SR|_{{\cal X} /Y})$.
}
Now the facts that $\pi _0(X\times
_SR|_{{\cal X} /Y})$ is $P3\frac{1}{2}$ and that
$Res _{Z/Y}\varpi _1(S|_{{\cal X} /Z}, s, s')$ is $P4$ (which comes by
hypothesis)
imply that the inverse image
in question is $P4$ (Lemma \ref{kernel}); then the theorem on quotients (Theorem
\ref{I.1.d}) and the fact that the group $\pi _1((X\times _SR |_{{\cal X} /Y},
\tilde{f})$ is $P5$ over $Y$ gives the condition that $Res _{Z/Y}\varpi
_1(R|_{{\cal X} /Z}, r, r')$ is $P4$ over $Y$.
This is the condition on $\varpi _1$ needed to insure that $R$ is presentable.
This completes the proof of Theorem \ref{stability}.
\hfill $\Box$\vspace{.1in}
We have the following characterization of presentable morphisms via the relative
homotopy group sheaves.
\begin{proposition}
\mylabel{characterization}
Suppose $f: R\rightarrow S$ is a morphism of $n$-stacks. Then
$f$ is presentable (resp. very presentable) if and only if the following
conditions are satisfied for any scheme $X$ of finite type:
\newline
---for any morphism $X\rightarrow S$,
the sheaf $\pi
_0(X\times _SR)$ is $P3\frac{1}{2}$; and
\newline
---for any morphism $r: X\rightarrow R$ the sheaves $\pi _i(X\times _SR/X, r)$
on ${\cal X} /X$ are presentable group sheaves over $X$ (resp. $\pi _1$ is affine
presentable and $\pi _i$ are vector sheaves for $i\geq 2$).
\end{proposition}
{\em Proof:}
This falls out of the proof of \ref{stability}.
\hfill $\Box$\vspace{.1in}
{\em Exercise:} For which values of $(a_0,a_1,\ldots )$ does Theorem
\ref{I.1.s.?} hold for $(a_0,a_1,\ldots )$-presentable spaces? Place these
conditions in Corollary \ref{I.1.u} below.
\subnumero{Going to the base of a fibration}
It is an interesting question to ask, if $R\rightarrow S$ is a morphism
of $n$-stacks such that $R$ is presentable and such that for every scheme-valued
point $X\rightarrow S$ the fiber product $X\times _SR$ is presentable, then
is $S$ presentable? The answer is surely no in this generality. We need to make
additional hypotheses. Directly from the fibration exact sequences, one can see
that if $\pi _0(S)$ is assumed to be $P3\frac{1}{2}$ (a hypothesis which seems
unavoidable) and if we suppose that for any point $a:X\rightarrow S$, the action
of $\pi _1(S|_{{\cal X} /X}, a)$ on $\pi _0(X\times _SR)$ factors through a
presentable group sheaf over $X$, then $S$ will be presentable.
As a particular case, if the morphism $R\rightarrow S$ is relatively
$0$-connected (i.e. the fibers are connected) and surjective on $\pi _0$, then
presentability of $R$ implies presentability of $S$.
One might look for other weaker conditions, for example that the fibers satisfy
some sort of artinian condition (e.g. there is a surjection from a scheme finite
over $X$, to $\pi _0(X\times _SR)$). I don't know if this can be made to work.
\subnumero{Presentable shapes}
We have a notion of internal $Hom$ for $n$-stacks. In the topological presheaf
interpretation (\cite{kobe} \S 2), recall that $\underline{Hom}(R,T)$ is
defined to be the presheaf $X\mapsto Hom (R'_X,T|_{{\cal X} /X})$ where $R'_X$ is a
functorial replacement of $R|_{{\cal X} /X}$ by a cofibrant presheaf.
\begin{corollary}
\label{I.1.u}
Suppose $W$ is a finite CW complex, and let $W_{{\cal X}}$ denote the constant
$n$-stack with values $\Pi _n(W)$ (or in terms of presheaves of spaces,
it is the fibrant presheaf associated to the constant presheaf with values
$\tau _{\leq n}W$). If $T$ is a presentable (resp. very presentable)
$n$-stack over $X$
then the $n$-stack $\underline{Hom}(W_{{\cal X}}, T)$ is presentable (resp.
very presentable).
\end{corollary}
{\em Proof:}
We first show this for $W=S^m$, the $m$-sphere. Do this by induction on $m$.
It is clear for $m=0$ because then $W$ consists of two points and
$\underline{Hom}(W_{{\cal X}}, T)=T\times T$. For any $m$, write $S^m$ as the
union of two copies of $B^m$ joined along $S^{m-1}$. We get
$$
\underline{Hom}(S^m_{{\cal X}}, T)=T\times _{
\underline{Hom}(S^{m-1}_{{\cal X}}, T)}T,
$$
since $\underline{Hom}(B^m_{{\cal X}}, T)=T$.
By Theorem \ref{I.1.s.?}, $\underline{Hom}(S^m_{{\cal X}}, T)$ is presentable
(resp. very presentable). This shows the corollary for the spheres.
We now treat the case of general $W$, by induction on the number of cells. We
may thus write $W=W'\cup B^m$ with the cell $B^m$ attached over an attaching
map $S^{m-1}\rightarrow W'$, and where we know the result for $W'$. Then
$$
\underline{Hom}(W_{{\cal X}}, T)=\underline{Hom}(W'_{{\cal X}}, T)\times _{
\underline{Hom}(S^{m-1}_{{\cal X}}, T)}T.
$$
Again by Theorem \ref{I.1.s.?}, we obtain the result for $W$.
\hfill $\Box$\vspace{.1in}
Let $Pres ^n/{\cal X}$ denote the $n+1$-category of presentable
$n$-stacks. We define the {\em presentable shape} of $W$ to be the
$n+1$-functor
$$
Shape (W):T\mapsto \underline{Hom}(\underline{W}, T)
$$
from $Pres ^n/{\cal X}$ to $Pres ^n/{\cal X}$.
One can show (using the calculations of \cite{kobe} Corollary 3.9 over $S=Spec
(k)$) that if $W$ is connected and simply connected then this functor is
homotopy-representable by an object $Hull (W)\in Pres /{\cal X} $. On the other hand,
in most cases where $W$ is not simply connected, the presentable shape is not
representable. We could try to interpret the hull of $W$ as the inverse limit
of $Shape (W)$, but this is not a standard kind of inverse limit. It is a
question for further study, just what information is contained in $Shape (W)$.
{\em Example:} Take $G=GL(n)$ and $T= K(G, 1)$.
Fix a finite CW complex $U$.
Then $M:=\underline{Hom}(U, T)$
is the moduli stack for flat principal $G$-bundles (i.e. flat vector bundles of
rank $n$) on $U$.
More generally it should be interesting to look at presentable or
very presentable {\em connected} $T$, these are objects whose homotopy group
sheaves are algebraic Lie groups over $Spec (k)$. Note that if $k$ is
algebraically closed then there is an essentially unique choice of basepoint
$t\in T(Spec (k))$. If $G= \pi _1(T, t)$ then we have a fibration
$T\rightarrow K(G,1)$ and we get a morphism
$$
\underline{Hom} (U, T) \rightarrow \underline{Hom}(U, K(G, 1)).
$$
This expresses $\underline{Hom} (U, T)$ as a presentable $n$-stack over the
moduli stack $M$ of flat principal $G$-bundles over $U$.
One can see from this example that we should consider the notion of vector
sheaf as a candidate for the higher homotopy group sheaves.
\subnumero{Leray theory}
We develop here a nonabelian Leray theory and K\"unneth formula. This is in
some sense one of the principal reasons for going to nonconnected $n$-stacks, as
they can intervene as intermediate steps even when the original coefficient
stacks
were connected.
We give some notation for the stack of sections. If $T\rightarrow S$ is a
morphism of $n$-stacks on ${\cal X}$ (or on any site) then we denote by
$\underline{\Gamma}(S, T)$ the $n$-stack of sections, i.e. of diagrams
$$
\begin{array}{ccc}
S&\rightarrow &T\\
& {\displaystyle =}\searrow&\downarrow \\
& & S
\end{array}
$$
(with homotopy making the diagram commutative).
We also have a notion of relative morphism stack. Suppose that $T\rightarrow S$
and $T' \rightarrow S$ are two morphisms of $n$-stacks. Then we obtain an
$n$-stack together with morphism to $S$
$$
\underline{Hom}(T/S, T'/S) \rightarrow S.
$$
In topological language this corresponds to the space whose fiber over $s$
is the
space of morphisms from $T_s$ to $T'_s$. This should not be confused with
another useful construction in the same situation, the space
$$
\underline{Hom}_S(T, T')
$$
which is the $n$-stack of diagrams
$$
\begin{array}{ccc}
T&\rightarrow &T'\\
& \searrow&\downarrow \\
& & S
\end{array}
$$
(again with homotopy making the diagram commutative).
These things can be constructed using the point of view of simplicial
presheaves or presheaves of spaces---cf for example \cite{flexible}.
It remains to
be seen how to give constructions of these things purely within the
realm of stacks
(and consequently to extend the same constructions to stacks of $n$-categories
which are not necessarily $n$-groupoids).
We have the following relationships among the above constructions.
First of all,
$\underline{\Gamma}(S, T) = \underline{Hom}_S(S, T)$. Then,
\begin{lemma}
Suppose $T\rightarrow S$ and $T' \rightarrow S$ are morphisms of $n$-stacks.
There is a natural equivalence
$$
\underline{\Gamma} (S, \underline{Hom}(T/S, T'/S)) \cong
\underline{Hom}_S(T,T').
$$
\end{lemma}
{\em Proof:} From the point of view of presheaves of spaces, see
\cite{flexible}.
\hfill $\Box$\vspace{.1in}
Finally note that if $T$ is an $n$-stack and $R\rightarrow S$ is a morphism of
$n$-stacks then
$$
\underline{Hom}_S(R/S, T\times S/S) \cong \underline{Hom}(R, T).
$$
From the above lemma we obtain a method of ``devissage'':
\begin{corollary}
Suppose $T$ is an $n$-stack and $R\rightarrow S$ is a morphism of
$n$-stacks, then
$$
\underline{Hom}(R, T) \cong \underline{\Gamma} (S, \underline{Hom}(R/S, T\times
S/S)).
$$
\end{corollary}
\vspace*{-.5cm}
\hfill $\Box$\vspace{.1in}
In words this says that to calculate the stack of morphisms from $R$ to $T$ we
first look at the fiberwise morphisms from $R/S$ to $T$, and then we take the
sections over $S$.
Rather than taking the internal morphism and section spaces we can take the
external ones, removing the underline in the notation which means taking the
sections over $\ast$ (which is $Spec (k)$ in our case). We get
the statement
$$
Hom(R, T) \cong \Gamma (S, \underline{Hom}(R/S, T\times
S/S)).
$$
Note that it is still essential to look at the internal $\underline{Hom}$ inside
the space of sections.
It might be worthwhile looking at how this works in the case
of usual cohomology. Suppose ${\cal A}$ is a sheaf of abelian groups on ${\cal X}$.
Let $T= K({\cal A} , n)$, so that $Hom(R, T)$ is an $n$-groupoid with
homotopy groups
$$
\pi _i = H^{n-i}(R, {\cal A} ).
$$
Similarly $\underline{Hom}(R/S, T)$ is an $n$-stack over $S$ whose relative
homotopy group sheaves over $S$ are the higher direct images
$$
\pi _i = R^{n-i}f_{\ast} ({\cal A} |_R).
$$
There is a spectral sequence for the $n$-stack of sections going from the
cohomology of $S$ with coefficients in the relative homotopy sheaves to the
homotopy groups of the space of sections, which turns out to be the Leray
spectral
sequence in this case.
This version of Leray theory is due to
Thomason \cite{Thomason}, who developed it mostly
in the context of presheaves of spectra.
We finally introduce one more bit of notation combining the previous notations,
that is the {\em relative section stack}. Suppose $R\rightarrow S\rightarrow T$
are morphisms of $n$-stacks. Then we obtain the $n$-stack
$$
\underline{\Gamma}(S/T, R/T)\rightarrow T
$$
which is geometrically the ``fiberwise space of sections of the morphism
$R\rightarrow S$ along the fibers of $S\rightarrow T$''.
The above Leray theory can itself be presented in a relative context:
\begin{lemma}
\mylabel{RelativeLeray}
Suppose $R\rightarrow S\rightarrow T\rightarrow U$ are morphisms of $n$-stacks.
Then
$$
\underline{\Gamma}(T/U, \underline{\Gamma}(S/T, R/T)/U) \cong
\underline{\Gamma}(T/U, R/U).
$$
\end{lemma}
\hfill $\Box$\vspace{.1in}
Of course, given four morphisms there should be a diagram expressing
compatibility
of these Leray equivalences (and further diagrams of homotopy between
the homotopies).
\subnumero{Leray theory for presentable and very presentable $n$-stacks}
Now we get back to presentable and very presentable $n$-stacks. Our goal is to
show that in certain cases the Leray theory stays within the world of
presentable
$n$-stacks.
The first task is to generalize Corollary
\ref{I.1.u} to the case of a local coefficient system, i.e. a presentable
morphism of $n$-stacks to our given finite CW complex.
\begin{lemma}
\mylabel{Leray2}
Suppose $U$ is a constant $n$-stack associated to the $n$-groupoids
associated to a finite CW complex.
Suppose $T\rightarrow U$ is a presentable (resp. very presentable) morphism of
$n$-stacks. Then the $n$-groupoid of sections
$\underline{\Gamma}(U, T)$ is a presentable (resp. very presentable) $n$-stack.
\end{lemma}
{\em Proof:}
The proof is identical to that of Corollary \ref{I.1.u} but we repeat it here
for the reader's convenience.
As before, we first treat the case $U=S^m$ by induction on $m$. It is clear for
$m=0$ because then $W$ consists of two points $a,b$ and $\underline{\Gamma
}(W_{{\cal X}}, T)=T_a\times T_b$, with the fibers $T_a$ and $T_b$ being presentable
(resp. very presentable). Now for any $m$, write $S^m$ as the union of two
copies of $B^m$ joined along $S^{m-1}$ and let $T_a$ be the fiber of $T$ over a
basepoint. This fiber is presentable (resp. very presentable). We get
$$
\underline{\Gamma }(S^m_{{\cal X}}, T)=T_a\times _{
\underline{\Gamma }(S^{m-1}_{{\cal X}}, T)}T_a,
$$
since $\underline{\Gamma }(B^m_{{\cal X}}, T)\cong T_a$.
By the induction hypothesis and Theorem \ref{I.1.s.?}, $\underline{\Gamma
}(S^m_{{\cal X}}, T)$ is presentable (resp. very presentable). This shows the
lemma for the spheres.
We now treat the case of general $U$, by induction on the number of cells. We
may thus write $U=U'\cup B^m$ with the cell $B^m$ attached over an attaching
map $S^{m-1}\rightarrow U'$, and where we know the result for $U'$. Again let
$T_a$ be the fiber over a basepoint in the attached cell. Then
$$
\underline{\Gamma }(U_{{\cal X}}, T)=\underline{\Gamma }(U'_{{\cal X}}, T)\times _{
\underline{\Gamma }(S^{m-1}_{{\cal X}}, T)}T_a.
$$
By Theorem \ref{I.1.s.?} and the above result for spheres, we obtain the result
for $U$.
\hfill $\Box$\vspace{.1in}
Say that a morphism $U\rightarrow V$ of $n$-stacks is {\em of finite CW type}
if for any scheme of finite type $X$ with morphism $X\rightarrow V$ there
is a covering family $\{ Y_{\alpha} \rightarrow X\}$ and finite CW complexes
$W^{\alpha}$ such that $Y_{\alpha} \times _V U \cong Y_{\alpha} \times
W^{\alpha}_{{\cal X}}$ (with $W^{\alpha}_{{\cal X}}$ being the constant $n$-stack
associated to $\Pi _n(W^{\alpha})$ as defined previously).
\begin{theorem}
\mylabel{Leray}
Suppose $U\rightarrow V$ is a morphism of $n$-stacks of finite CW type,
and suppose $T\rightarrow U$ is a presentable (resp. very presentable) morphism
of $n$-stacks. Then $\underline{\Gamma} (U/V, T/V)\rightarrow V$ is a
presentable (resp. very presentable) morphism.
\end{theorem}
{\em Proof:}
Suppose $X$ is a scheme of finite type with a morphism $X\rightarrow V$.
Let $\{ Y^{\alpha} \rightarrow X\}$ be the covering family and $\{ W^{\alpha}\}$
the collection of finite CW complexes
with isomorphisms $U\times _VY^{\alpha}\cong W^{\alpha}_{{\cal X}}$
given by the fact
that $U\rightarrow V$ is a morphism of finite CW type. It suffices to
prove that
$$
\underline{\Gamma} (U/V, T/V)\times _V Y^{\alpha}=
\underline{\Gamma} (U\times _VY^{\alpha}/Y^{\alpha}, T\times
_VY^{\alpha}/Y^{\alpha})
$$
is presentable (resp. very presentable). Thus it suffices to prove the theorem
in the case where $V$ is a scheme of finite type and $U=V\times W_{{\cal X}}$ for
a finite CW complex $W$. With these hypotheses we return to the notations of the
theorem. If $W$ is a finite union of components then the section space in
question will be the product of the section spaces of each of the components.
Thus we may assume that $W$ is connected. The $n$-stack of sections from
$W_{{\cal X}}$ to $V\times W_{{\cal X}}$ is isomorphic to $V$. Thus the $n$-stack of
sections of the morphism $T\rightarrow W_{{\cal X}}$ maps to $V$, and this $n$-stack
of sections is the same as the relative section stack $\underline{\Gamma}(U/V,
T/V)$.
It suffices to prove that $\underline{\Gamma}(W_{{\cal X}}, T)$ is presentable (resp.
very presentable).
But the morphism $V\times W_{{\cal X}}\rightarrow W_{{\cal X}}$ is very
presentable, so by Corollary \ref{composition} the morphism $T\rightarrow
W_{{\cal X}}$ is presentable (resp. very presentable), and Lemma \ref{Leray2}
applies to give that $\underline{\Gamma}(W_{{\cal X}}, T)$ is presentable (resp.
very presentable) as needed.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{Leray1}
Suppose $U\rightarrow V$ is a morphism of $n$-stacks of finite CW type
and
Suppose $T\rightarrow V$ is a presentable morphism of
$n$-stacks. Then the morphism
$$
\underline{Hom}(U/V, T/V)\rightarrow V
$$
is a presentable morphism.
\end{corollary}
{\em Proof:}
We have
$$
\underline{Hom}(U/V, T/V) = \underline{\Gamma}(U/V, T\times _VU/V)
$$
and $T\times _VU\rightarrow U$ is presentable by \ref{fiberprod}, so Theorem
\ref{Leray} applies.
\hfill $\Box$\vspace{.1in}
\begin{corollary}
\mylabel{Leray1a}
Suppose $T$ is a presentable $n$-stack, and suppose $V\rightarrow U$ is a
morphism whose fibers are finite CW complexes in the sense of the above
theorem.
Then
$$
\underline{Hom}(V/U, T\times U/U)\rightarrow U
$$
is a presentable morphism.
\end{corollary}
{\em Proof:}
Indeed, the morphism $T\times U\rightarrow U$ is presentable.
\hfill $\Box$\vspace{.1in}
We look at the case of a morphism of $n$-groupoids
$f:U\rightarrow V$ such that $U$ and $V$ are the $n$-groupoids associated to
finite
CW complexes. Suppose that the fibers of $f$ are
the $n$-groupoids associated to finite CW complexes. This is the case for
example
if $f$ comes from a smooth morphism of manifolds.
For a presentable
$n$-stack $T$ we can calculate
$$
\underline{Hom}(V, T) = \underline{\Gamma}(U, \underline{Hom}(V/U,
T\times U/U)).
$$
Corollary \ref{Leray1a} states that
$\underline{Hom}(V/U, T\times U/U)\rightarrow U$ is a presentable morphism, and
Lemma \ref{Leray2} (which is also a corollary of Theorem \ref{Leray})
states that
for any presentable morphism $R\rightarrow U$ the space of sections is
presentable. We obtain in particular the presentability of $\underline{Hom}(V,
T)$ (which we already knew beforehand). The Leray devissage process thus stays
within the realm of presentable $n$-stacks.
{\em The K\"unneth formula:} We can apply the above discussion to the
particular
case where $V=U\times U'$ is a product. In this case the formula is
simplified:
$$
\underline{Hom}(U\times U', T) = \underline{Hom}(U, \underline{Hom}(U', T))
$$
and again (this time using only Corollary \ref{I.1.u})
this process of first taking $\underline{Hom}(U', T)$ and then
$\underline{Hom}(U, -)$ stays within the realm of presentable $n$-stacks.
Of course the entire discussion above works equally well if we replace
``presentable'' by ``very presentable''.
{\em Example:} Take $G=GL(n)$ and $T= K(G, 1)$.
Then $M':=\underline{Hom}(U', T)$
is the moduli stack for flat principal $G$-bundles (i.e. flat vector bundles of
rank $n$) on $U'$. After that, assuming that $U$ is connected,
$\underline{Hom}(U,
M')$ is the moduli stack of flat $G$-bundles on $U\times U'$.
More generally it should be interesting to look at presentable or
very presentable {\em connected} $T$, these are objects whose homotopy group
sheaves are algebraic Lie groups over $Spec (k)$. Note that if $k$ is
algebraically closed then there is an essentially unique choice of basepoint
$t\in T(Spec (k))$. If $G= \pi _1(T, t)$ then we have a fibration
$T\rightarrow K(G,1)$ and we get a morphism
$$
\underline{Hom} (U, T) \rightarrow \underline{Hom}(U, K(G, 1)).
$$
This expresses $\underline{Hom} (U, T)$ as a presentable $n$-stack over the
moduli stack $M$ of flat principal $G$-bundles over $U$.
|
1996-07-18T21:42:53 | 9607 | alg-geom/9607018 | en | https://arxiv.org/abs/alg-geom/9607018 | [
"alg-geom",
"math.AG"
] | alg-geom/9607018 | Pablo Ares Gastesi | Pablo Ares Gastesi (Tata Institute, Bombay) | Torelli groups and Jacobian varieties of non-orientable compact Klein
surfaces | AMSLaTeX, 18 pages, xypic, available from ftp://ftp.math.tifr.res.in/
with dvi file at http://www.math.tifr.res.in/~pablo/ | null | null | null | null | The Torelli group of a compact non-orientable Klein surface is the subgroup
of the modular group consisting of the mapping classes that act trivially on
the first homology group of the surface. We prove that if a surface has genus
at least $3$, then the Torelli group acts fixed points free on the
Teichm\"{u}ller space of the surface. That gives an embedding of the Torelli
space of a Klein surface in the Torelli space of its complex double. We also
construct real tori associated to Klein surfaces, which we call the Jacobian of
the surface. We prove that this Jacobian is isomorphic to a component of the
real part of the Jacobian of the complex double.
| [
{
"version": "v1",
"created": "Thu, 18 Jul 1996 19:42:55 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Gastesi",
"Pablo Ares",
"",
"Tata Institute, Bombay"
]
] | alg-geom | \section{Statement of results}
Klein surfaces are the natural generalization of Riemann surfaces to the
non-orientable situation: one considers holomorphic and anti-holomorphic
changes of coordinates. One of the points of interest in the
study of Klein surfaces is to determine which results of the theory of
deformation of Riemann surfaces hold for the non-orientable
case. A common approach to this problem is to consider a Klein
surface $\Sigma$, as a Riemann surface $\Sigma^c$, with an
anti-holomorphic involution
$\sigma$, and thus one wants to find $\sigma$-invariant objects. In this
paper we follow these two points of view to
show two related results in the theory of Klein surfaces,
that is, we will do some constructions on Klein surfaces, and then find
the corresponding invariant objects related to the Riemann surface
$\Sigma^c$. More precisely, we
construct the {\bf Torelli space} $Tor(\Sigma)$, and prove that it can
be identified with the set of fixed points of an involution on
$Tor(\Sigma^c)$. We also construct the {\bf Jacobian variety} $J(\Sigma)$
of $\Sigma$ by integrating a basis of the space of real
harmonic forms over the free part of
${\mathrm H}_1(\Sigma,{\Bbb Z})$. We prove that $J(\Sigma)$ is
isomorphic to a component of the real part of the Jacobian $J(\Sigma^c)$
of the complex double $\Sigma^c$ of $\Sigma$.
Given a compact smooth non-orientable surface $\Sigma$, the {Teichm\"{u}ller space}
$T(\Sigma)$ of $\Sigma$ is defined as $T(\Sigma)={\cal
M}(\Sigma)/Diff_0(\Sigma)$, where ${\cal M}(\Sigma)$ is the set of Klein
surface structures on $\Sigma$ that agree with the given smooth
structure, and $Diff_0(\Sigma)$ is the group of diffeomorphisms of
$\Sigma$ homotopic to the identity \cite[pg. 145]{sep:book}.
We will use $\Sigma$ for a Klein
surface, if it is clear from the context what the structure is, or we
will write $(\Sigma,X)$ if we need to specify more.
The {\bf modular} or {\bf mapping
class group}, $Mod(\Sigma)=Diff(\Sigma)/Diff_0(\Sigma)$, acts on
$T(\Sigma)$ by pull-back of dianalytic structures (see \S $2$).
The {\bf Torelli group} $U(\Sigma)$ is the subgroup of
$Mod(\Sigma)$ consisting of the mapping classes that act trivially on
${\mathrm H}_1(\Sigma,{\Bbb Z})$. The parallel result to the following theorem
is a classical fact on Riemann surfaces.
\setcounter{section}{3}
\setcounter{thm}{0}
\begin{thm}Let $\Sigma$ be a compact non-orientable surface of genus $g
\geq 3$. Let $[f]\in Mod(\Sigma)$, and suppose that there exists a Klein
surface structure $X$ on $\Sigma$ such that $f:(\Sigma,X)\rightarrow
(\Sigma,X)$ is dianalytic. Then $[f]=[id]$. Therefore, $U(\Sigma)$ acts
fixed-points free on $T(\Sigma)$, and the Torelli space
$Tor(\Sigma)=T(\Sigma)/U(\Sigma)$ is a smooth real manifold of dimension
$3g-6$.
\end{thm}
Assume now that $\Sigma$ has a fixed Klein surface structure.
Then there exists an
unramified double covering of $\Sigma$ by a Riemann surface $\Sigma^c$,
known as the {\bf complex double}. Moreover, $\Sigma$ is isomorphic to
$\Sigma^c/<\!\sigma\!>$, where $\sigma$ is an anti-holomorphic
involution. The mapping $\sigma$ induces involutions
$\sigma^*$ and $\tilde\sigma$ on $T(\Sigma)$ and $Tor(\Sigma)$,
respectively. It is a well known fact that $T(\Sigma)$ can be identified
with the set of fixed points of $\sigma^*$. A similar result holds for
Torelli spaces, as the next proposition shows.
\setcounter{thm}{2}
\begin{prop}The Torelli space $Tor(\Sigma)$ can be identified with the
set of fixed points of $\tilde\sigma$ on $Tor(\Sigma^c)$.
\end{prop}
Torelli spaces are intimately related to the Jacobian
variety of a compact Riemann surface.
Recall that this variety $J(\Sigma^c)$, is a $g$-dimensional complex
torus ($g$ is the genus of $\Sigma^c$) given by ${\Bbb C}^g/\Gamma$, where
$\Gamma$ is the lattice generated by integration of a basis of holomorphic
forms on $\Sigma^c$ over a basis of ${\mathrm H}_1(\Sigma^c,{\Bbb Z})$.
We can also construct the Jacobian by
considering the lattice $\Gamma'$ generated by integration of harmonic
forms and then taking the quotient
${\Bbb R}^{2g}/\Gamma'$, which is a real torus. The Hodge-$*$ operator gives
a complex structure to this real torus in such a way that it becomes
$J(\Sigma^c)$. This point of view can be generalized to construct a
Jacobian variety $J(\Sigma)$ of a non-orientable Klein surface.
\setcounter{section}{4}
\setcounter{thm}{0}
\begin{thm}
Let $\Sigma$ be a compact non-orientable surface of
genus $g\geq3$. Then we can associate to $\Sigma$ a real torus of
dimension $g-1$, the {\bf Jacobian variety} $J(\Sigma)$ of $\Sigma$,
such that $J(\Sigma)$ is isomorphic to any component of the real part of
the Jacobian $J(\Sigma^c)$ of the complex double. This last set is
defined as the set of fixed points of the symmetry $\sigma_1$ of
$J(\Sigma^c)$ induced by $\sigma$.
\end{thm}
{\bf Acknowledgments}: the idea of using harmonic forms to construct
$J(\Sigma)$ was suggested by S. Nag; I would like to thank him for
useful conversations regarding this topic.
I would like also to express my gratitude to D. S.
Nagaraj and R. R. Simha for many helpful conversations while this paper
was being written.
\setcounter{section}{1}
\setcounter{thm}{0}
\section{Some general facts about Klein surfaces}
A {\bf Klein surface} (or {\bf dianalytic}) structure $X$ on a surface
without boundary $\Sigma$ is a covering by open sets $U_i$, and a
collection of homeomorphisms $z_i:U_i\rightarrow V_i$, where
$V_i\subset{\Bbb C}$ are open sets, such that $z_i\circ z_j^{-1}$ is
holomorphic or anti-holomorphic, whenever $U_i\cap U_j\neq\emptyset$
\cite{all:klein}. Observe that a
Klein surface structure on an orientable surface is just a pair of
conjugate Riemann surface structures \cite{natan:klein}.
A compact non-orientable surface $\Sigma$ is homeomorphic to
the connected sum of $g\geq 1$ real projective planes \cite{blackett:topo}.
The number $g$ is
called the {\bf genus} of $\Sigma$. If $g=2n+1$, then the fundamental
group of $\Sigma$ has a presentation given by generators $c$,
$a_1,\ldots,a_n$, $b_1,\ldots,b_n$, satisfying
$c^2\prod_{j=1}^n[a_j,b_j]=1$, where $[a,b]=aba^{-1}b^{-1}$. If the
genus is even, $g=2n+2$, then we can choose generators $c$, $d,$
$a_1,\ldots,a_n$, $b_1,\ldots,b_n$, satisfying the relation
$c^2d^2\prod_{j=1}^n[a_j,b_j]=1$. An alternative presentation for this
latter case is given by generators $\gamma$, $\delta$, $a_1,\ldots,a_n,$
$b_1,\ldots,b_n,$ and the relation
$\gamma\delta\gamma^{-1}\delta\prod_{j=1}^n[a_j,b_j]=1$.
For the rest of this paper,
we will assume that all surfaces are compact without boundary. We will
further assume that non-orientable surfaces have genus $g\geq 3$, while
orientable surfaces satisfy $g\geq 2$.
The {\bf complex double} \cite{all:klein} of a Klein surface $\Sigma$
of genus $g$ is a triple $(\Sigma^c,\pi,\sigma)$, where:\\
\noindent (1) $\Sigma^c$ is a Riemann surface of genus $g-1$;\\
\noindent (2) $\pi:\Sigma^c\rightarrow\Sigma$ is an unramified double
covering;\\
\noindent (3) there exist local coordinates $z$ and $w$ on $\Sigma^c$
and $\Sigma$, respectively, such that the function $w\circ\pi\circ
z^{-1}$ is either holomorphic or anti-holomorphic (i.e. $\pi$ is a
morphism of Klein surfaces);\\
\noindent (4) $\sigma:\Sigma^c\rightarrow\Sigma^c$ is a
symmetry such that $\pi\circ\sigma=\pi$.
Let $S$ be a compact orientable surface, with a fixed orientation and a
smooth structure. The {\bf Teichm\"{u}ller space} $T(S)$ of $S$ is $T(S)={\cal
M}(S)/Diff_0(S)$, where ${\cal M}(S)$ is the set of Riemann surface
structures on $S$ that agree with the given orientation and smooth
structure \cite{sep:book}.
The classical definition of $T(S)$ involves quasiconformal
mappings; to see that it is equivalent to the above definition, it
suffices to observe that on a compact surface, any homeomorphism is
homotopic to a smooth one, and diffeomorphisms are quasiconformal.
The {\bf modular} or {\bf mapping class} group $Mod(S)$
is the group of homotopic classes of orientation preserving
diffeomorphisms of $S$, that is $Mod(S)=Diff^+(S)/Diff_0(S)$. This group
acts on $T(S)$ by pull-back of complex structures: if $[f]\in Mod(S)$,
and $[X]\in T(S)$, then $[f]^*([X])=[f^*(X)]$, where $f^*(X)$ is the
Riemann surface structure on $S$ that makes
$f:(\Sigma,f^*(X))\rightarrow (\Sigma,X)$ biholomorphic. However, this
action has fixed points; it is therefore interesting to find subgroups of
$Mod(\Sigma)$ that act without fixed points on $T(S)$. A subgroup $G$ of
$Mod(S)$ has the {\bf Hurwitz-Serre} property \cite{nag:teic}
if $G$ satisfies that for
any element $[g]\in G$ such that there exists an $[X]\in{\cal M}(S)$
with $g:(S,X)\rightarrow (S,X)$ biholomorphic, one has that $[g]=[id]$.
A group with this property will act fixed-points free on Teichm\"{u}ller space and,
therefore, the quotient $T(S)/G$ will be a smooth finite dimensional
complex manifold. The {\bf Torelli group} $U(S)=\{[f]\in Mod(S);~
f~\mathrm{acts~trivially~on~H}_1(S,{\Bbb Z})\}$, is known to satisfy the
Hurwitz-Serre property \cite{fk:book}.
The quotient space $Tor(S)=T(S)/U(S)$ is called
the {\bf Torelli space} of $S$.
The Jacobian variety $J(S)$ of a compact Riemann surface is an abelian
variety constructed as follows: let ${\cal
B}^c=\{\alpha_1,\ldots,\alpha_g,\beta_1,\ldots,\beta_g\}$ be a symplectic
basis of ${\mathrm H}_1(S,{\Bbb Z})$. Then we can find a dual basis for
${\mathrm H}^0(S,\Omega_S^1)$, the space of holomorphic forms on $S$,
consisting of forms $\{\omega_1,\ldots,\omega_g\}$, satisfying
$$\int_{\alpha_j}\omega_k=\begin{cases}
1 & \text{if } j=k,\\
0 & \text{otherwise}.\end{cases}$$
Let $\Gamma^c$ be the lattice
on ${\Bbb C}^g$ generated by the vectors $(\int_c\omega_1,\ldots,\int_c\omega_g)$,
$c\in{\cal B}^c$; then we define $J(S)={\Bbb C}^g/\Gamma^c$.
\section{Torelli groups of non-orientable compact surfaces}
In this section we will show that the Torelli group of a compact
non-orientable Klein surface $\Sigma$ has the Hurwitz-Serre property.
The main idea of the proof is to see that, if a diffeomorphism of a
Klein surface $\Sigma$ acts trivially on ${\mathrm H}_1(\Sigma,{\Bbb Z})$,
its orientation preserving lift to the complex double will act
trivially on ${\mathrm H}_1(\Sigma^c,{\Bbb Z})$; then we use the fact that the
Hurwitz-Serre property is satisfied for Riemann surfaces.
The quotient space $Tor(\Sigma)=T(\Sigma)/U(\Sigma)$ is a
smooth real manifold. We will show that $Tor(\Sigma)$ can be
identified with the set of fixed points of a symmetry $\tilde\sigma$ on
$Tor(\Sigma^c)$.
Let us start with a smooth non-orientable surface $\Sigma$, of genus
$g=2n+1$, and a diffeomorphism $f:\Sigma\rightarrow\Sigma$ such that $f$
acts trivially on ${\mathrm H}_1(\Sigma,{\Bbb Z})$. We can find a unique
orientation preserving diffeomorphism $\tilde f$ of $\Sigma^c$ such that
the following diagram commutes \cite{sep:spaces}:
$$\diagram
\Sigma^c\rto^{\tilde f}\dto_\pi & \Sigma^c\dto^\pi \\
\Sigma\rto^f & \Sigma.\enddiagram$$
We want to show that the mapping ${\tilde f}_\#$ induced by $\tilde f$
on ${\mathrm H}_1(\Sigma^c,{\Bbb Z})$ is trivial. For that purpose we need
to recall the way $\Sigma^c$ is constructed, from the
topological viewpoint. The reader can find more details in
\cite{blackett:topo}.
By the presentation of the fundamental group of $\Sigma$, we can identify
this surface with a $(4n+2)$-polygon, whose sides are labeled
to satisfy the relation of the fundamental group. Then $\Sigma^c$
is given by two polygons with boundary relations:
$$c_1c_2\prod_{j=1}^n[a_{j,1},b_{j,1}]=1\hspace{5mm}
\mathrm{and}\hspace{5mm}
c_2c_1\prod_{j=1}^n[a_{j,2},b_{j,2}]=1.$$
To obtain a single relation, we find the value of $c_2$ on the right
hand side equation and substitute it on the left hand one
(equivalently, we glue the two polygons by the $c_2$ sides):
$$c_2=(\prod_{j=1}^n[b_{n+1-j,2},a_{n+1-j,2}])c_1^{-1};$$
therefore
$$c_1\big(\prod_{j=1}^n[b_{n+1-j,2},a_{n+1-j,2}]\big)c_1^{-1}
\big(\prod_{j=1}^n[a_{j,1},b_{j,1}]\big)=$$
$$\big(\prod_{j=1}^n[c_1b_{n+1-j,2}c_1^{-1},c_1a_{n+1-j,2}c_1^{-1}]\big)
\big(\prod_{j=1}^n[a_{j,1},b_{j,1}]\big)=1.$$
{}From this formula we see that $\Sigma^c$ is a compact surface of genus
$g-1=2n$; we can choose the following paths as generators of the
fundamental group of $\Sigma$:
$$\alpha_1=c_1b_{n,2}c_1^{-1},\ldots,\alpha_n=c_1b_{n,2}c_1^{-1},
\alpha_{n+1}=a_{1,1}\ldots,\alpha_{2n}=a_{n,1},$$
$$\beta=c_1a_{n,2}c_1^{-1},\ldots,\beta_n=c_1a_{n,2}c_1^{-1},
\beta_{n+1}=b_{1,1}\ldots,\beta_{2n}=b_{n,1}.$$
These loops satisfy $\prod_{j=1}^{2n}[a_j,b_j]=1$. Let $\cal B$ and
${\cal B}^c$ denote the basis of ${\mathrm H}_1(\Sigma,{\Bbb Z})$ and
${\mathrm H}_1(\Sigma^c,{\Bbb Z})$ induced by the two given sets of generators of the
corresponding fundamental groups. By an abuse of notation, we will use
the same letters for the elements of the fundamental group and their
classes in homology. We can see that ${\cal B}^c$ is a symplectic basis
of ${\mathrm H}_1(\Sigma^c,{\Bbb Z})$; that is, the intersection matrix is
given by
$J=\left(\begin{matrix}
0 & {\mathrm I} \\
-{\mathrm I} & 0 \end{matrix}\right) ,$ where ${\mathrm I}$ is the identity
matrix. The covering map $\pi:\Sigma^c\rightarrow\Sigma$ induces a
mapping on homology, with associated matrix
$$\pi_\#=\left(\begin{matrix}
0 & 0 & 0 & 0 \\
0 & {\mathrm I} & K & 0 \\
K & 0 & 0 & {\mathrm I} \end{matrix}\right),$$ with respect to ${\cal
B}^c$ and $\cal B$. The matrix $K$ is given by
$$K=\left(\begin{matrix}
0 & \cdots & \cdots & 1 \\
0 & \cdots & 1 & 0 \\
\vdots & & & \vdots \\
1 & 0 & \cdots & 0 \end{matrix}\right).$$
The symmetry $\sigma$ maps $a_{j1}$ (resp. $b_{j1}$) to $a_{j2}$ (resp.
$b_{j2}$); it is not difficult to see that the map $\sigma_\#$
induced on ${\mathrm H}_1(\Sigma^c,{\Bbb Z})$ is given by $\sigma_\#=K$.
Let ${\tilde f}_\#=\left(A_{jk}\right)_{j,k=1}^4$;
since $f_\#$ acts trivially on ${\mathrm
H}_1(\Sigma,{\Bbb Z})$, we have $\pi_\#{\tilde f}_\#=\pi_\#$. By the
uniqueness of ${\tilde f}_\#$ we get
${\tilde f}_\#\sigma_\#=\sigma_\#{\tilde f}_\#$.
Finally, ${\tilde f}^t_\#J{\tilde
f}_\#=J$, where ${\tilde f}_\#^t$ is the transpose of ${\tilde f}_\#$,
since $\tilde f$ preserves the intersection matrix
(\cite[theorem N13, pg. 178]{mag:comb}).
The condition $\pi_\#{\tilde f}_\# =
\pi_\#$ is equivalent to the following set of equations:
$$\left\{\begin{array}{lclccclcl}
A_{21} + KA_{31} & = & 0 & & & & KA_{11} + A_{41} & = & K \\
A_{22} + KA_{32} & = & {\mathrm I} & & & & KA_{12} + A_{42} & = & 0 \\
A_{23} + KA_{33} & = & K & & & & KA_{13} + A_{43} & = & 0 \\
A_{24} + KA_{34} & = & 0 & & & & KA_{14} + A_{44} & = & {\mathrm I}.
\end{array} \right .$$
Therefore, the matrix ${\tilde f}_\#$ can be written as
$${\tilde f}_\# = \left(\begin{matrix}
A_{11} & A_{12} & A_{13} & A_{14} \\
A_{21} & A_{22} & A_{23} & A_{24} \\
-KA_{21} & K-KA_{22} & {\mathrm I}-KA_{23} & -KA_{24} \\
K-KA_{11} & -KA_{12} & -KA_{13} & {\mathrm I}-KA_{14} \end{matrix}\right).$$
We now use the fact that $\tilde f$ and $\sigma$ commute, to obtain the
following relations among the entries of the matrix ${\tilde f}_\#$:
\begin{equation}\left\{ \begin{array}{lclccclcl}
A_{14}K & = & K(K-KA_{11}) & & & & A_{24}K & = & K(-KA_{21}) \\
A_{13}K & = & K(-KA_{12}) & & & & A_{23}K & = & K(K-KA_{22}) \\
A_{12}K & = & K(KA_{13}) & & & & A_{22}K & = & K({\mathrm I}-KA_{23}) \\
A_{11}K & = & K({\mathrm I}-KA_{14})& & & & A_{21}K & = & K(-KA_{24}).
\end{array}\right .\label{eq:f}\end{equation}
Consider the equation ${\tilde f}_\#^t J {\tilde f}_\# = J$; looking
at the first row of the matrices on both sides of the equality, we get,
after using \eqref{eq:f} to simplify the result,
$$\left\{\begin{array}{rcr}
A_{21}^tK-KA_{21} & = & 0 \\
A_{11}^tK-KA_{22} & = & 0 \\
-2{\mathrm I}+A_{11}^t+KA_{22}K & = & 0 \\
A_{21}^t+KA_{21}K & = & 0
\end{array}\right .$$
Solving these equations, we get $A_{21}=0$ and $A_{11}={\mathrm I}$, which
imply that $A_{24}=0$ and $A_{14}=0$.
Using this, we now consider the equality between the second rows of the
matrices ${\tilde f}_\#^t J {\tilde f}_\#$ and $J$, to obtain
$A_{22}={\mathrm I}$, and $A_{12}=0$. By \eqref{eq:f}, we get $A_{23}=0$
and $A_{13}=0$. Therefore, we have that ${\tilde f}_\#$ is the identity
matrix.
{\bf Remark:} if we would have chosen the orientation reversing lift of $f$,
say ${\tilde f}_1$, then ${\tilde f}_1={\tilde f}\sigma$, so $({\tilde
f}_1)_\#={\tilde f}_\#\sigma_\#=\sigma_\#$.
If $\Sigma$ has even genus $g=2n+2$, we use the first of the two presentations
of its fundamental group given in \S $2$.
We have that $\Sigma^c$ is given by two polygons with boundary relations:
$$c_1c_2d_1d_2\prod_{j=1}^n[a_{j,1},b_{j,1}] = 1\hspace{5mm}
\mathrm{and}\hspace{5mm}
c_2c_1d_2d_1\prod_{j=1}^n[a_{j,2},b_{j,2}] = 1.$$
{}From the second equation we get
$$d_2=c_1^{-1}c_2^{-1}\big(\prod_{j=1}^n[b_{n+1-j,2},a_{n+1-j,2}]\big)
d_1^{-1},$$
which reduces the first equation to
$$c_1c_2d_1c_1^{-1}c_2^{-1}\big(\prod_{j=1}^n[b_{n+1-j,2},a_{n+1-j,2}]\big)
d_1^{-1}\big(\prod_{j=1}^n[a_{j,1},b_{j,1}]\big) = $$
$$c_1c_2d_1c_1^{-1}c_2^{-1}d_1^{-1}
\big(\prod_{j=1}^n[d_1b_{n+1-j,2}d_1^{-1},d_1a_{n+1-j,2}d_1^{-1}]\big)
\big(\prod_{j=1}^n[a_{j,1},b_{j,1}]\big) = 1.$$
We therefore obtain that the fundamental group of $\Sigma^c$ is
generated by the loops
$$\alpha_1=c_1d_1^{-1},~\alpha_2=d_1b_{n,2}d_1^{-1},\ldots,
\alpha_{n+1}=d_1b_{1,2}d_1^{-1},
\alpha_{n+2}=a_{1,1},\ldots,\alpha_{2n+1}=a_{n,1},$$
$$\beta_1=d_1c_2,~\beta_2=d_1a_{n,2}d_1^{-1},\ldots,
\beta_{n+1}=d_1\a_{1,2}d_1^{-1},
\beta{n+2}=a_{1,1},\ldots,\beta_{2n+1}=a_{n,1},$$
satisfying the relation
$\prod_{j=1}^{2n+1}[\alpha_j,\beta_j]=1.$
The basis $\{\alpha_1, \alpha_2, \ldots, \alpha_{2n+1}, \beta_1, \beta_2,
\ldots,$\newline
\noindent $\beta_{2n+1} \}$
is symplectic, but computations are easier if we
rearrange the basis as
${\cal B}^c=\{ \alpha_1, \beta_1, \alpha_2, \ldots,\alpha_{2n+1},
\beta_2, \ldots, \beta_{2n+1} \}$,
whose intersection matrix is given by:
$$J=\left(\begin{matrix}
N & 0 & 0 & 0 & 0 & \\
0 & 0 & 0 & {\mathrm I} & 0 & \\
0 & 0 & 0 & 0 & {\mathrm I} & \\
-{\mathrm I} & 0 & 0 & 0 & 0 & \\
0 & -{\mathrm I} & 0 & 0 & 0 & \end{matrix}\right) ,$$
where $N=\left(\begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix}\right) .$
With respect to $\cal B^c$, the symmetry $\sigma$ has the following
matrix representation, for its action on ${\mathrm H}_1(\Sigma^c,{\Bbb Z})$:
$$\sigma_\#=\left(\begin{matrix}
M & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & K & 0 \\
0 & 0 & K & 0 & 0 \\
0 & K & 0 & 0 & 0 \\
K & 0 & 0 & 0 & 0 \end{matrix}\right) ,$$
where the matrix
$M=\left(\begin{matrix} 1& 0 \\
2 & -1 \end{matrix}\right) .$
To obtain the matrix
$M$, observe that in homology one has $\alpha=c_1+d_1$ and
$\beta=d_1+c_2$, so $\sigma(\alpha)=c_2+d_2$,
$\sigma(\beta)=d_2+c_1.$ Substituting the value of $d_2$ obtained
previously, we get $M$.
Similarly we have that the expression for the action of the covering
map $\pi$ on homology is given by:
$$\pi_\#=\left(\begin{matrix}
{\mathrm I} & 0 & 0 & 0 & 0 \\
0 & 0 & {\mathrm I} & K & 0 \\
0 & K & 0 & 0 & {\mathrm I} \end{matrix}\right) .$$
Computing in a way similar to the odd genus case, we get that if
$f:\Sigma\rightarrow\Sigma$ is a diffeomorphism of $\Sigma$ that acts
trivially on ${\mathrm H}_1(\Sigma,{\Bbb Z})$, and $\tilde f$ is a lift of $f$
to $\Sigma^c$, then the action of this last mapping on ${\mathrm
H}_1(\Sigma^c,{\Bbb Z})$ is given by either the identity or $\sigma_\#$,
depending on whether $\tilde f$ is orientation preserving or reversing.
Recall that a map $f:\Sigma\rightarrow\Sigma$ on a Klein
surface is {\bf dianalytic} if, when expressed in local coordinates $(U,z)$,
$f\circ z^{-1}$ is either holomorphic or anti-holomorphic. The above
computations show that $U(\Sigma)$ satisfies the equivalent of the
Hurwitz-Serre property.
\begin{thm}Let $\Sigma$ be a compact non-orientable surface of genus
$g\geq 2$. Let $[f]\in Mod(\Sigma)$, and suppose that there exists a
Klein surface structure $X$ on $\Sigma$ such that
$f:(\Sigma,X)\rightarrow (\Sigma,X)$ is dianalytic. Then $f$ is
homotopic to the identity.
\end{thm}
\begin{cor}The Torelli space $Tor(\Sigma)=T(\Sigma)/U(\Sigma)$
is a smooth manifold of real dimension $3g-6$.
\end{cor}
\begin{proof}{of the Proposition} Since $f$ is dianalytic on the Klein surface
$(\Sigma,X)$, the orientation preserving lift $\tilde f$ is
biholomorphic on the Riemann surface $\Sigma^c$. But
then, since the genus of $\Sigma^c$ is at least $2$, we have that
$\tilde f$ is the identity, which proves the result.
\end{proof}
The involution $\sigma$ of $\Sigma^c$ induces a symmetry $\sigma^*$ on
$T(\sigma^c)$. The Teichm\"{u}ller space $T(\Sigma)$ can be identified with the set
of fixed points of $\sigma^*$, which proves the corollary.
It is clear that $\sigma^*$ descends to a symmetry
$\tilde\sigma$ on $Tor(\Sigma^c)$.
\begin{prop}The Torelli space $Tor(\Sigma)=T(\Sigma)/U(\Sigma)$ can be
identified with the set of fixed points of $\tilde\sigma$ in
$Tor(\Sigma^c)$.\end{prop}
\begin{pf}The proof follows immediately from the definition of Torelli
spaces. In fact, we have that two elements $[X_1]$ and $[X_2]$ of ${\cal
M}(\Sigma)$ project to the same point in $Tor(\Sigma)$ {\it if and only if}
there exists a diffeomorphism $h\in Diff(\Sigma)$ such that
$h_\#:{\mathrm H}_1(\Sigma,{\Bbb Z})\rightarrow {\mathrm H}_1(\Sigma,{\Bbb Z})$ is
the identity, and $h:(\Sigma,X_2)\rightarrow (\Sigma,X_1)$ is
dianalytic. The rest of the proof is similar to the proof that
$T(\Sigma)$ can be identified with the set of fixed points of
$\sigma^*$; see \cite{sep:book} for more details.\end{pf}
\section{Jacobi varieties of Klein surfaces}
Throughout this section, $\Sigma$ will denote a fixed compact
non-orientable Klein surface of genus $g\geq 3$, and $\Sigma^c$ its
complex double. We can take $\Sigma^c$ to be defined by a
polynomial, $p(z,w)=0$, with real coefficients (\cite{all:klein} and
\cite{natan:gordon}). Then the involution
$\sigma$ is given by $\sigma(z,w)=(\overline z,\overline w)$, and
conjugation $z\mapsto\overline z$ in ${\Bbb C}^{g-1}$ induces an involution
$\sigma_1$ on the Jacobian $J(\Sigma^c)$. The set of fixed points of
$\sigma_1$, that is, the {\bf real part} of $J(\Sigma^c)$, is a real manifold
of dimension $g-1$; the pair $J(\Sigma^c,\sigma_1)$ is usually
considered as the Jacobian of $\Sigma$.
On a Klein surface the concept of harmonic forms makes
sense; it is not difficult to see that
the space ${\cal H}_{\Bbb R}^1(\Sigma)$ of such forms has dimension precisely $g-1$. One
can choose a basis of ${\mathrm H}_1(\Sigma,{\Bbb Z})_f$, the free part of
${\mathrm H}_1(\Sigma,{\Bbb Z})\cong{\Bbb Z}^{g-1}\oplus{\Bbb Z}/2Z$,
and a dual basis for ${\cal H}_{\Bbb R}^1(\Sigma)$;
these two basis generate a lattice $\Gamma$ in
${\Bbb R}^{g-1}$. We will call the real torus ${\Bbb R}^{g-1}/\Gamma$
the {\bf Jacobian variety} of $\Sigma$, and denote it by $J(\Sigma)$. On the
other hand, the real part of a holomorphic form (on $\Sigma^c$) is a
harmonic form, so one can expect some relationship between $J(\Sigma)$
and the real part of $J(\Sigma^c)$. We prove that, in fact, $J(\Sigma)$
is isomorphic to a component of the set of fixed points of $\sigma_1$ in
$J(\Sigma^c)$.
A continuous function $f:W\rightarrow{\Bbb R}$ defined on an open set of a
Klein or Riemann surface is called {\bf harmonic} if for any local
coordinate $(U,z)$, with $U\cap W\neq\emptyset$, the function $f\circ
z^{-1}$ is harmonic. Since precomposition with holomorphic and
anti-holomorphic functions preserves harmonicity, the above definition
makes sense. Actually, a Klein surface is the most general surface in
which the notion of harmonic function is well defined \cite{all:klein}.
Similarly, a (real) form $\psi$ is {\bf harmonic} if it
can be written locally as $\psi=df$, where $f$ is harmonic. We will
denote by ${\cal H}_{\Bbb R}^1(\Sigma)$ the space of harmonic forms on $\Sigma$. Let $\sigma^*$
be the pull-back map induced by $\sigma$ on forms on $\Sigma^c$.
If $\omega=gdz$, with $g$ holomorphic, we have that $\sigma^*(\omega) =
g(\sigma)\sigma_{\overline z}d\overline z$, so $\sigma^*$ is
anti-holomorphic. We also have that, for any holomorphic form $\omega\in
{\mathrm H}^0(\Sigma^c,\Omega^1)$, and for any cycle $c$ on $\Sigma^c$,
$\int_c\sigma^*(\omega)=\int_{\sigma(c)}\omega$. Observe that this last
equality agrees with \cite{natan:gordon}, while it differs
of \cite{silhol:comess} and \cite{all:klein}, since these two authors
define $\sigma^*$ as the conjugate of our definition (in order
to have that $\sigma^*$ preserves holomorphic forms).
To compute the dimension of ${\cal H}_{\Bbb R}^1(\Sigma)$, it suffices to observe that
$\sigma^*$ takes harmonic forms to harmonic forms; therefore, ${\cal H}_{\Bbb R}^1(\Sigma)$ will
be isomorphic to the set of fixed points of $\sigma^*$ in ${\cal H}_{\Bbb R}^1(\Sigma^c)$. By
Hodge theory, $\sigma^*$ acts like $\sigma_\#=K$; so $dim\,{\cal H}_{\Bbb R}^1(\Sigma)=g-1$.
This result agrees with \cite{all:obst}.
In order to justify later computations, we need the following lemma.
\begin{lemma}[Duality Lemma]On a non-orientable compact Klein surface
$\Sigma$, of genus $g \geq 3$, the space of harmonic forms, ${\cal H}_{\Bbb R}^1(\Sigma)$,
and the dual space to the homology with real coefficients,
${\mathrm H}_1(\Sigma,{\Bbb R})^*$, are isomorphic.\end{lemma}
\begin{pf}From Differential Topology \cite[Theorem 15.8]{bott:forms}
we know that the \v{C}ech cohomology with coefficients in the constant
presheaf $\Bbb Z$, ${\mathrm H}^1_{\Bbb Z}(\Sigma)^*,$ is isomorphic to the
singular cohomology ${\mathrm H}^1(\Sigma)$. Furthermore, by the Universal
Coefficients Theorem
\cite[Corollary 15.14.1]{bott:forms},
we have that the space ${\mathrm H}^1(\Sigma)$ is isomorphic to the free
part of ${\mathrm H}_1(\Sigma)$, which is just ${\Bbb Z}^{g-1}$.
Tensoring with ${\Bbb R}$ we
have that ${\mathrm H}_{\Bbb R}^1(\Sigma)$ is isomorphic to ${\Bbb R}^{g-1}$.
On the other hand, ${\mathrm H}_{\Bbb R}^1(\Sigma)$ is isomorphic to the de Rham
cohomology ${\mathrm H}_{DR}^1(\Sigma)$ \cite[8.9, 9.8 and 14.28]{bott:forms}.
By the compactness of $\Sigma$, on each de Rham class there exists at most a
harmonic form. A counting of dimensions shows that there exists
exactly one harmonic form, which completes the proof. Therefore, ${\cal H}_{\Bbb R}^1(\Sigma)$
and ${\mathrm H}_{DR}(\Sigma)$ are isomorphic, and since this last space is
isomorphic (by integration) to ${\mathrm H}_1(\Sigma,{\Bbb R})$, we are done.
\end{pf}
To make matters more clear, we will construct $J(\Sigma)$ on the cases
of genus $3$ and $4$. The general case will follow easily from our
examples. Let us start with a Klein surface $\Sigma$ of
genus $g=3$.
Then from \S $2$
we have that ${\cal B}=\{a,b\}$ is a basis of ${\mathrm
H}_1(\Sigma,{\Bbb Z})_f$. The loops $\alpha_1$, $\alpha_2$, $\beta_1$,
$\beta_2$ of \S $3$ give a basis of ${\mathrm H}_1(\Sigma^c,{\Bbb Z})$; but
for computational purposes, it is better to choose the basis
$${\cal B}^c=\{\gamma_1 = -(\alpha_2+\beta_1), \gamma_2 =
-(\alpha_1+\beta_2), \delta_1 = \alpha_1+\alpha_2+\beta_1, \delta_2 =
\alpha_1+\alpha_2+\beta_2\}.$$
Observe that the change of basis in ${\mathrm H}_1(\Sigma^c,{\Bbb Z})$ is
given by the matrix
$$C=\left(\begin{matrix}
0 & -1 & 1 & 1 \\
-1 & 0 & 1 & 1 \\
-1 & 0 & 1 & 0 \\
0 & -1 & 0 & 1 \end{matrix}\right) ,$$
which has determinant equal to $1$, and satisfies $C^tJC=J$.
Therefore ${\cal B}^c$ is a symplectic basis. The mapping $\pi_\#$ gives
$\pi_\#(\gamma_1) = -2b,$ $\pi_\#(\gamma_2) = -2a.$ This suggests that
we should take a basis ${\cal B}_h=\{\phi_1, \phi_2\}$ of ${\cal H}_{\Bbb R}^1(\Sigma)$ normalized
by $\int_a\phi_1=\int_b\phi_2=0$, and $\int_b\phi_1=\int_a\phi_2=-1/2.$
Observe that this normalization is possible because of the Duality
Lemma, and the fact that $a$ and $b$ are not torsion classes in
${\mathrm H}_1(\Sigma,{\Bbb Z})$.
We can use the pull-back mapping $\pi^*$ induced by $\pi$ to get forms
$\psi_j=\pi^*(\phi_j)$ on $\Sigma^c$. These forms are real harmonic, so
$\omega_j=\psi_j+i*\phi_j$ are holomorphic forms (where $*$ stands for
the Hodge-$*$ operator). By the expression of
$\sigma$ and the formula relating $\sigma^*$ with integrals, we have
that
$$\int_{\gamma_j}\overline{\omega}_k = \int_{\gamma_j}\sigma^*(\omega_k)
= \int_{\sigma(\gamma_j)}\omega_k = \int_{\gamma_j}\omega_k .$$
In particular, we see that $\int_{\gamma_j}\omega_k$ is real. But since
by \cite[Theorem 1.0.7, pg. 74]{all:klein}
${\mathrm Re}\int_{\gamma_j}\omega_k = \int_{\gamma_j}\psi_k =
\int_{\pi(\gamma_j)}\phi_k,$ we have that ${\cal B}^*=\{\omega_1,
\omega_2\}$ is normalized with respect to ${\cal B}^c$. Let $P$ denote
the corresponding period matrix, that is the entries of this matrix are
given by $p_{jk} = \int_{\delta_k}\omega_j$. The mapping
$\sigma_\#:{\mathrm H}_1(\Sigma^c,{\Bbb Z})\rightarrow {\mathrm
H}_1(\Sigma^c,{\Bbb Z})$ has the following expression with respect to the
basis ${\cal B}^c$:
$$\sigma_\#=\left(\begin{matrix}
1 & 0 & -2 & -1 \\
0 & 1 & -1 & -2 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \end{matrix}\right) .$$
This results agrees with the one obtained by Natanzon in
\cite{natan:gordon} except in that he gets all sign positive.
Nevertheless, the computation of the real part of $J(\Sigma^c)$ yields
the same result.
We have that $P$ satisfies $\overline{P}=A-P$, where
$A=\left(\begin{matrix}
-2 & -1 \\
-1 & -2 \end{matrix}\right) .$ The Jacobian variety of $\Sigma^c$ is
then given by $J(\Sigma^c) = {\Bbb C}^2/\Gamma^c$, where $\Gamma^c={\Bbb Z}^2+P{\Bbb Z}^2$.
To compute the real part of $J(\Sigma^c)$, we write, for any $z\in{\Bbb C}^2$,
$z=P\alpha+\beta$, where $\alpha,~\beta\in{\Bbb R}^2$. Then
$\sigma_1(z)=\overline z\equiv z$ is equivalent to $P\alpha+\beta =
\overline{P}\alpha+\beta+Pn+m,$ for some $n,~m\in{\Bbb Z}^2$. The imaginary
part of this equation gives $({\mathrm Im} P)\alpha = -({\mathrm Im}
P)\alpha + ({\mathrm Im} P)n$. By Riemann bilinear relations
(\cite{fk:book}; see also \cite{simha:rs} for a nice introduction to
Jacobians and of Riemann surfaces from an algebro-geometric approach)
we have
that $({\mathrm Im} P)$ is invertible, so we obtain
$\alpha=n/2$. On the other hand, taking real parts in the above
equation we get $({\mathrm Re} P)\alpha+\beta = (A-{\mathrm Re}
P)\alpha+\beta+({\mathrm Re} P)n+m$. Since $\overline P = A-P$, we have
that $({\mathrm Re} P)=A-({\mathrm Re} P)$, so this equation reduces to
$0=\frac{1}{2}An+m$, or equivalently
$$\left\{\begin{array}{ccc}
n_1+\frac{n_2}{2}+m_1 & = & 0 \\
\frac{n_1}{2}+n_2+m_2 & = & 0, \end{array}\right .$$ where notation the
should be clear. This implies that $n_j\in 2{\Bbb Z}$, so $\alpha\in{\Bbb Z}^2$.
Therefore, the set of fixed points of $\sigma_1$ is given by the real
torus ${\mathrm Re}(J(\Sigma^c))=\{P{\Bbb Z}^2+\beta;~\beta\in{\Bbb R}^2\}/\Gamma^c$.
which agrees with the results obtained by Silhol
\cite[pgs. $349$ and 359]{silhol:comess} and Natanzon \cite{natan:klein}.
By the form of the lattice $\Gamma^c$,
it is clear that ${\mathrm Re}(J(\Sigma^c))\cong ({\Bbb R}/{\Bbb Z})^2$.
In a similar way to the construction of $J(\Sigma^c)$, we can form a
lattice in ${\Bbb R}^2$ using the basis ${\cal B}_h=\{\phi_1,\phi_2\}$ and
${\cal B}$ of ${\cal H}_{\Bbb R}^1(\Sigma)$ and
${\mathrm H}_1(\Sigma,{\Bbb Z})_f$, respectively.
Let us denote this lattice by $\Gamma$. We define the {\bf
Jacobian variety} of $\Sigma$ as the quotient $J(\Sigma)={\Bbb R}^2/\Gamma$.
It is clear that $[z]\mapsto[-\frac{1}{2}z]$ induces an isomorphism
between ${\mathrm Re}(J(\Sigma^c))$ and $J(\Sigma)$, which proves our
result for $g=3$. The general case of a surface of odd genus is done in
a similar way.
To see the even genus case, we take a surface with $g=4$,
and we choose the second
of the two presentations of the fundamental group of $\Sigma$ given in
\S $2$; i.e. the generators are the loops
$c$, $d$, $a$ and $b$, and the relation is
$cdc^{-1}d[ab] = 1$. To construct the
complex double we proceed as in \S $3$; we do not include the
computations here, since they are done as in \S $3$. We get that
the fundamental group of $\Sigma^c$ is generated by the loops
$$\alpha_1=c_2c_1, ~\alpha_2=(d_1^{-1}c_2)b_1(d_1^{-1}b_1c_2)^{-1},
{}~\alpha_3=a_2,$$
$$\beta_1=d_1^{-1}, ~\beta_2=(d_1^{-1}c_2)a_1(d_1^{-1}b_1c_2)^{-1},
{}~\beta_3=b_2,$$
We again change our basis of ${\mathrm H}_1(\Sigma^c,{\Bbb Z})$ to
$${\cal B}^c=\{\gamma_1=\alpha_1,~ \gamma_2=-(\alpha_3+\beta_2),~
\gamma_3=-(\alpha_2+\beta_3),~ \delta_1=\alpha_1+\beta_1,~
\delta_2=\alpha_2+\alpha_3+\beta_2,~$$
$$\delta_3=\alpha_2+\alpha_3+\beta_3\}.$$
It is not hard to see that ${\cal B}^c$ is symplectic; one simply has
to chech that the matrix
$$C=\left(\begin{matrix}
1 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 & 1 & 1 \\
0 & -1 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 & 1 & 0 \\
0 & -1 & 0 & 0 & 0 & 1 \end{matrix}\right) ,$$
satisfies $C^tJC=J$. The projection $\pi$ acts on these loops by
$\pi_\#(\gamma_1)=2c$, $\pi_\#(\gamma_2)=-2a$, $\pi_\#(\gamma_1)=-2b\,$;
so we take a basis $\{\phi_1, \phi_2, \phi_3\}$ of ${\cal H}_{\Bbb R}^1(\Sigma)$, and normalize
it by requiring
$$\int_c\phi_1 = -\int_a\phi_2 = -\int_b\phi_3 = \frac{1}{2}, $$
and the other integrals equal to $0$.
As in the previous
situation, we have that if $\psi_j=\pi^*(\phi_j)$, then ${\cal B}^* = \{
\omega_j=\psi_j+i*\psi_j;~ j=1,2,3\}$ is a basis of holomorphic
$1$-forms dual to ${\cal B}^c$. It is not hard to see that $\sigma_\#$
is given by the following matrix, when computed with respect to ${\cal
B}^c$:
$$\sigma_\#=\left(\begin{matrix}
1 & 0 & 0 & 2 & 0 & 0 \\
0 & 1 & 0 & 0 & -2 & -1 \\
0 & 0 & 1 & 0 & -1 & -2 \\
0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 \end{matrix}\right) ,$$
and the period matrix $P$ satisfies $\overline P=A-P$, where
$$A=\left(\begin{matrix}
2 & 0 & 0 \\
0 & -2 & -1 \\
0 & -1 & -2 \end{matrix}\right).$$
The above matrix of $\sigma_\#$ is different from the one given in
\cite{natan:gordon}; we have not been able to obtain the matrix of that
reference, but nevertheless, we obtain similar result in the computation
of the real part of $J(\Sigma^c)$. In this case we have that the real part
of $J(\Sigma^c)$ (which is found in the same way that the $g=3$ case)
has two components, namely
$T_1=\{{\Bbb Z}^3n+\beta;~\beta\in{\Bbb R}^3,~ n=(n_1,n_2,n_3),~
n_1,n_2,n_3\in{\Bbb Z}\}$ and
$T_2=T_1+{\Bbb Z}^3(\frac{1}{2},0,0)^t$. We again obtained the results of
\cite{silhol:comess} and \cite{natan:klein}.
An isomorphism similar to the
previous case holds in this situation, except that we have $J(\Sigma)$
is isomorphic to any of the two sets $T_1$ or $T_2$.
{\bf Remark}: the results about the fixed points of $\sigma_1$ on
$J(\Sigma^c)$ can also be obtained from the expression of the period
matrix given in \cite[Proposition $4$, pg. 351]{silhol:comess}.
The above results can be put together in the following theorem:
\begin{thm}
Let $\Sigma$ be a compact non-orientable surface of
genus $g\geq3$. Then we can associate to $\Sigma$ a real torus of
dimension $g-1$, the {\bf Jacobian variety} $J(\Sigma)$ of $\Sigma$,
such that $J(\Sigma)$ is isomorphic to any component of the real part of
the Jacobian $J(\Sigma^c)$ of the complex double. This last set is
defined as the set of fixed points of the symmetry $\sigma_1$ of
$J(\Sigma^c)$ induced by $\sigma$.
\end{thm}
|
1996-07-23T14:52:20 | 9607 | alg-geom/9607023 | en | https://arxiv.org/abs/alg-geom/9607023 | [
"alg-geom",
"math.AG"
] | alg-geom/9607023 | Stefan Mueller-Stach | Pedro Luis del Angel and Stefan M\"uller-Stach | Motives of uniruled 3-folds | Latex with amsfonts, 12 pages | null | null | null | null | We construct projectors in the ring of correspondences of a complex uniruled
3-fold $X$ which lift the Kuenneth components of the diagonal in singular
cohomology and have other properties which were conjectured by J. Murre. Such
Murre decompositions have been already obtained for curves, surfaces, abelian
varieties and varieties with cell decompositions by the work of Manin,
Shermenev, Beauville, Murre et.al.. In particular they define a natural
filtration on the Chow groups of $X$ which was conjectured by Bloch and
Beilinson. To do this we use Mori theory and construct projectors in the
situation of a fiber space over a surface. These projectors may also be used in
more general situations.
| [
{
"version": "v1",
"created": "Tue, 23 Jul 1996 12:39:46 GMT"
}
] | 2014-10-24T00:00:00 | [
[
"del Angel",
"Pedro Luis",
""
],
[
"Müller-Stach",
"Stefan",
""
]
] | alg-geom | \section{Introduction}
Let $F$ be a subfield of ${\Bbb C}$. We denote by $V(F)$ the category of smooth,
projective varieties over $F$ with the usual morphisms. Let $CV(F)$ be the
category with the same underlying object, but where the morphisms are
replaced by correspondences of degree zero, i.e. for two irreducible varieties $X,Y$ we have
$Mor(X,Y):=CH^{\dim(X)}(X \times Y)$. If $f \in Mor(X,Y)$ we view it as a
homomorphism $f_\ast: CH^*(X) \to CH^*(Y)$, by defining
$f_*(W)=(pr_2)_* ( (W \times X) \cap f) $.
Given $X_1,X_2,X_3 \in V(k)$ the composition of correspondences
$f \in Mor(X_1,X_2)$ and $g \in Mor(X_2,X_3)$ is defined by
$$ g \circ f = (pr_{13})_*\{( pr_{12})^* f \cap (pr_{23})^* g \} $$
An element $p \in Mor(X,X)$ is called a {\bf projector} if $p \circ p = p$.
A special example is the diagonal, denoted by $\Delta$.
Finally denote by $M(F)$ the category of {\bf effective Chow motives}, where
objects are pairs $(X,p)$ with $X \in V(F)$ and $p \in Mor(X,X)$ a projector.
The morphisms are described by $Mor((X,p),(Y,q)):=q \circ Mor(X,Y) \circ p$.\\
\begin{definition} Let $M=(X,p) \in M(F)$. Define
$$ CH^i(M):= p_*CH^i(X) \otimes {\Bbb Q} $$
\end{definition}
\begin{definition}
Let $X \in V(F)$ be a smooth projective variety of dimension $d$.
We say that $X$ has a {\bf Murre decomposition}, if there exist projectors
$p_0,p_1,...,p_{2d}$ in $CH^d(X \times X) \otimes {\Bbb Q}$ such that the
following properties hold (modulo rational equivalence for (1) and (2)):\\
(1) $p_j \circ p_i = \delta_{i,j} \cdot p_i $\\
(2) $\Delta = \sum p_i $\\
(3) In cohomology the $p_i$ induce the $(2d-i,i)-$th K\"unneth
component of the diagonal.\\
(4) $p_0,...,p_{j-1}$ and $p_{2j+1},..,p_{2d}$ act trivially on $CH^j(X)
\otimes {\Bbb Q}$.\\
(5) If we put $F^0 CH^j(X) \otimes {\Bbb Q} = CH^j(X) \otimes {\Bbb Q}$ and inductively
$F^k CH^j(X) \otimes {\Bbb Q} := Ker(p_{2j+1-k} \mid_{F^{k-1}}) $, then this
descending filtration does not depend on the particular choice of the $p_i$.\\
(6) Always $F^1 CH^j(X) \otimes {\Bbb Q} = CH^j_{hom}(X) \otimes {\Bbb Q}$.
\end{definition}
The motives $(X,p_i)$ are traditionally denoted by
$h^i(X)$ and we write $h(X)=h^0(X)+...+h^{2d}(X)$.
In (6) one also wants to have that $F^2 CH^j(X) \otimes {\Bbb Q}$ is
the kernel of the cycle class map in rational Deligne cohomology, but this is
very hard to verify in general. \\
(1) - (6) have been proved for curves,
surfaces (\cite{6}), abelian varieties (\cite{7})
and certain varieties close to projective
varieties. Recently B. Gordon and
J. Murre \cite{GM} computed the Chow motive of elliptic modular varieties
using work of A. Scholl \cite{Tony}.\\
S. Saito has proposed a filtration in \cite{8}
which has property (6).
Manin (\cite{3}) and Murre (\cite{6})
have quite generally defined
$p_0,p_1,p_{2d-1},p_{2d}$ for every $X$. A. Scholl has refined this
in \cite{Tony} to have also the property
that $p_1=p_{2d-1}^t$, where $p^t$ denotes a transpose of a projector $p$.
Murre has formulated the following \\
\ \\
{\bf Conjecture:} {\sl Every smooth projective F-variety $X$ admits
a Murre decomposition.}\\
J. Murre has also studied the case of a product of a curve with a surface
where one in fact has a Murre decomposition. Inspired by this, we have
tried to construct projectors in the following situation: Let $f:Y \to S$
be a morphism from a smooth 3-fold $Y$
to a smooth surface $S$ with connected fibers.
Choose a smooth hyperplane section
$i: Z \hookrightarrow Y$ and let $h=f|_Z$.
Look the following cycles
$$\pi_{i0}:={1 \over m} (i \times 1)_* (h\times f)^*\pi _i(S),$$
$$\pi_{i2}:={1 \over m} (1 \times i)_* (f\times h)^*\pi _i(S),$$
in $CH^3(Y\times Y)\otimes{\Bbb Q}$. Here the $\pi_i(S)$ are
orthogonal projectors of a
Murre decomposition of $S$ as constructed by Murre (\cite{6})
and $m$ is the number of points on a general fiber of $h$. These cycles are
not orthogonal in general but we are able to construct orthogonal projectors
$\pi_0,\cdots, \pi_6$ in the following way:
$$\pi_0:=\pi_{00},\quad \pi_1:=\pi_{10}, \quad
\pi_2:=\pi_{20}+\pi_{02}-\pi_{20}\cdot\pi_{02}$$
$$\pi_4:=\pi_{40}+\pi_{22}-\pi_{40}\cdot\pi_{22}, \quad
\pi_5:=\pi_{32}, \quad \pi_6:=\pi_{42}, \quad
\pi_3:=\Delta-\sum_{i\ne 3}\pi_i
$$
The $\pi_j$ do not operate in the right way on cohomology, but if
all higher direct images sheaves $R^if_*{\cal O}_Y$ vanish for $i \ge 1$,
they can be modified to form a Murre decomposition.
In particular a suitable blow up $Y$ of any smooth {\bf uniruled} 3-fold $X$
over a subfield of the complex numbers has this property.
Recall that a 3-fold $X$ is called uniruled, if there exists
a dominant rational map $\varphi: S \times {{\Bbb P}}^1 - - - \to X$ for some
smooth projective surface $S$. By a theorem of Mori and
Miyaoka (\cite{3}), this is equivalent to saying that $X$
has Kodaira dimension $-\infty$.
There is no structure theorem for these varieties which is as simple as in
the case of ruled surfaces, but there is a version in the category of
3-folds with ${\Bbb Q}$-factorial and terminal singularities (\cite{4})
stating that $X$ is birationally equivalent
to a 3-fold $Y$ which has a fiber structure with rationally connected fibers
over a base variety which can be a point, a smooth curve or a
normal surface. Using this and suitable modifications of the
projectors above we can therefore prove:
\begin{thm} Let $X$ be a smooth uniruled complex projective 3-fold.
Then $X$ admits a Murre decomposition.
\end{thm}
In the proof of this theorem, which makes heavy use of Fulton's
machinery of intersection theory, the Murre decomposition provides the
following description of the {\bf Chow motive } of a complex
uniruled 3-fold $X$ (ignoring torsion):\\
\[
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|} \hline
Motive $M$ & $h^0(X)$ & \ $h^1(X)$ \ & \ $h^2(X)$ \ & \ $h^3(X)$ \ & \
$h^4(X)$ \ & \ $h^5(X)$ \ & $h^6(X)$ \ \\ \hline
$CH^0(M)$ & ${\Bbb Z}$ & \ 0 \ & \ 0 \ & \ 0 \ & \ 0 \ & \ 0
\ & $0$ \ \\ \hline
$CH^1(M)$ & $0$ & ${\rm Pic}^0(X)$ & \ ${\rm NS}(X)$ \ & \ $0$ \
& \ $0$ \ & \ $0$ \ & $0$ \ \\ \hline
$CH^2(M)$ & $0$ & \ $0$ \ & \ ${\rm Ker}(\psi)$ \ & \ ${\rm Im}(\psi)$ \
& $H^{2,2}(X,{\Bbb Z})$ & \ $0$ \ & $0$ \ \\ \hline
$CH^3(M)$ & $0$ & \ $0$ \ & \ $0$ \ & \ $0$ \ &
${\rm Ker}(alb_X)$
& \ $Alb(X)$ \ & ${\Bbb Z}$ \ \\ \hline
\end{tabular}
\]\\
\ \\
Here $\psi: CH^2_{\rm hom}(X) \to J^2(X)$ is the Abel-Jacobi map.
We hope that our approach may also be used to construct projectors in
other situations. \\
{\bf Acknowledgements}: It is a pleasure to thank H. Esnault,
B. Gordon, J. Murre and E. Viehweg for several discussions.
The DFG and the universities of Essen and Leiden
have supported the authors during this project.
\newpage
\section{\bf Projectors for special varieties}
\bigskip
The easiest case in which one has a Murre decomposition is the
case of projective space,
because there $H^{2k+1}(X,{\Bbb C})=0$ for all $k\ge 0$ and the other groups
admit a basis represented by algebraic cycles. One has a more general theorem:\\
\begin{thm} \label{thm1}
Let $X$ be a smooth variety of dimension $n$ and assume that
for certain $1\le q\le n-1$ there is a basis $\{E_1,\cdots ,E_t\}$ of
$H^{2q}(X,{\Bbb Q})$ and a basis $\{\ell_1, \cdots ,\ell_t\}$
of $H^{2(n-q)}(X,{\Bbb Q})$ represented by classes of algebraic
cycles. Then: \\
a) There exists a matrix $B=(b_{ij})\in {\bf GL_n}({\Bbb Q})$ such that the cycle
$p=\sum b_{ij}(\ell_i\times E_j)\in CH^n(X\times X)\otimes{\Bbb Q}$ operates as the
identity on $H^{2q}(X,{\Bbb Q})$.\\
b) For the same choice of $b_{ij}$,
$p^t=\sum b_{ij}(E_j\times \ell_i)
\in CH^n(X\times X)\otimes{\Bbb Q}$ operates as the
identity on $H^{2(n-q)}(X,{\Bbb Q})$.\\
c) Both cycles, $p$ and $p^t$ are idempotent and therefore projectors.
\end{thm}
{\noindent{\bf Proof. }} Let $A=(E_i\cdot \ell_j)$ be the intersection matrix, then take
$B=A^{-1}$. {\hfill $\square$\\}
Moreover, one can explicitely say how these projectors operate on
cycles, namely:\\
\begin{prop} Let $p$ be as before and let $k\ne q$. Then, for all
$Z\in CH^k(X)\otimes{\Bbb Q}$ one has $p(Z)=0$ as an element of
$CH^k(X) \otimes {\Bbb Q}$.
\end{prop}
{\noindent{\bf Proof. }} By dimension reasons, as $p(Z)\in <E_i>\subset CH^q(X)\otimes{\Bbb Q}$. {\hfill $\square$\\}
\begin{lemma} Let $p$ be as before and $Z\in CH^q(X)\otimes{\Bbb Q}$. If $[Z]$
denotes the homology class of $Z$ on $H^{2q}(X,{\Bbb Q})$, then
$[p(Z)]=p([Z])=[Z]$.
\end{lemma}
{\noindent{\bf Proof. }} $p$ operates as the identity on $H^{2q}(X,{\Bbb Q})$ and
$p(Z)=\sum b_{ij}(\ell_i\cdot Z)E_j$. {\hfill $\square$\\}
\begin{cor} Let $p$ be as before, then
$({\rm Ker}\; p)\cap CH^q(X)\otimes{\Bbb Q} =CH_{\rm
hom}^q(X)\otimes{\Bbb Q}$.
\end{cor}
{\bf Examples:} Smooth Fano 3-folds and Calabi-Yau 3-folds have the property
that the Hodge numbers $h^{i,0}$ are always zero for $i=1,2$ and therefore
theorem \ref{thm1} applies. Another example is a del Pezzo fibration
$f: X \to B$:
to illustrate this, let $\ell$ be the extremal
rational curve, $F$ a general fiber,
$Y$ be a section of $|-mK_X|$, $C$ a twofold intersection in
the linear system $|Y|$ and hence a
multisection of $f$ over $B$, such that $C$ is a smooth
curve dominating $B$. $H^2(X,{\Bbb Q})$ is free of rank two. Then
theorem \ref{thm1} produces the following projector
$$p_2:= {1 \over r}(C \times F)+{1 \over m }({\ell} \times Y)-
{d \over {m \cdot r}}({\ell} \times F) $$
where $d=Y^3$ and $r:=(C.F)$. Note that $(-K_X.\ell)=1$.
$p_2$ is unique as a cycle up to the choices of $Y,C,F$ and
$\ell$.
\section{\bf Murre decompositions of birational conic bundles}
\bigskip
Let $f:Y\longrightarrow S$ be a morphism from a smooth projective 3-fold $Y$
to a smooth projective surface $S$, such that every fiber of $f$ is
rationally
connected and the general fiber of $f$ is isomorphic to ${{\Bbb P}}^1$.
Choose a smooth hyperplane section $i:Z\hookrightarrow Y$ such that
$h:=f_{|Z}:Z\longrightarrow S$
is surjective and generically finite. Then define cycles
$$\pi_{i0}:={1 \over m} (i \times 1)_* (h\times f)^*\pi _i(S),$$
$$\pi_{i2}:={1 \over m}(1 \times i)_* (f\times h)^*\pi _i(S),$$
in $CH^3(Y\times Y)\otimes{\Bbb Q}$ for $0\le i\le 4$.
Here the $\pi_i(S)$ are the orthogonal projectors of a
Murre decomposition of $S$ as constructed by Murre (\cite{6})
(and improved by A. Scholl in \cite{Tony} to have also the property
that $\pi_1=\pi_3^t$)
and $m$ is the number of points on a general fiber of $h$. The following is
our {\bf key result} in some sense:
\begin{lemma}\quad \\
a) $\pi_{i0}\circ \pi_{j0} = \delta_{ij}\pi_{i0}$ \\
b) $\pi_{i2}\circ \pi_{j2} = \delta_{ij}\pi_{i2}$ \\
c) $\pi_{j2}\circ \pi_{i0} = 0$
\end{lemma}
{\noindent{\bf Proof. }} a) Using the projection formula and the theory of Gysin maps for
l.c.i. morphisms from \cite[prop.6.6 (c)]{Fu} in the
following diagram
$$
\matrix{Y \times Y \times Y & \to & Y \times Y \cr
\downarrow && \downarrow \cr
Z \times Y \times Y & \to& Z \times Y \cr
\downarrow && \downarrow \cr
Z\times S \times Y &\to& Z \times Y \cr
\downarrow && \downarrow \cr
S \times S \times S & \to & S \times S }
$$
where the vertical maps are canonical l.c.i. morphisms, one obtains: \\
\begin{small}
$\pi_{i0}\circ \pi_{j0}$\\
$ ={1 \over m^2}(pr_{13}^{Y\times Y\times Y})_*
((i\times 1)_*((h\times f)^*(\pi_j(S))\times Y \cap
Y\times (i\times 1)_*((h\times f)^*(\pi_i(S))))$\\
$={1 \over m^2}(pr_{13}^{Y\times Y\times Y})_*
((i\times 1\times 1)_*(h\times f\times f)^*(\pi_j(S)\times S) \cap
(1\times i\times 1)_*(f\times h\times f)^*(S\times \pi_i(S)))$\\
$={1 \over m^2}(pr_{13}^{Y\times Y\times Y})_*
(i\times 1\times 1)_*[(h\times f\times f)^*(\pi_j(S)\times S) \cap
(i\times 1\times 1)^*(1\times i\times 1)_*(f\times h\times f)^*
(S\times \pi_i(S))]$\\
$={1 \over m^2}(i\times 1)_*(pr_{13}^{Z\times Y\times Y})_*
[(h\times f\times f)^*(\pi_j(S)\times S) \cap
(1\times i\times 1)_*(i\times 1\times 1)^*(f\times h\times f)^*
(S\times \pi_i(S))]$\\
$={1 \over m^2}(i\times 1)_*(pr_{13}^{Z\times Y\times Y})_*
[(h\times f\times f)^*(\pi_j(S)\times S) \cap
(1\times i\times 1)_*(h\times h\times f)^*(S\times \pi_i(S))]$\\
$={1 \over m^2}(i\times 1)_*(pr_{13}^{Z\times Y\times Y})_*(1\times i\times 1)_*
[(1\times i\times 1)^*(h\times f\times f)^*(\pi_j(S)\times S) \cap
(h\times h\times f)^*(S\times \pi_i(S))]$\\
$={1 \over m^2}(i\times 1)_*(pr_{13}^{Z\times S\times Y})_*(1\times h\times 1)_*
(h\times h\times f)^*[(\pi_j(S)\times S) \cap (S\times \pi_i(S))]$\\
$={1 \over m}(i\times 1)_*(pr_{13}^{Z\times S\times Y})_*
(h\times 1\times f)^*[(\pi_j(S)\times S) \cap (S\times \pi_i(S))]$
\hfill (\cite[prop. 6.6 (c)]{Fu}) \\
$={1 \over m}(i\times 1)_*(h\times f)^*(pr_{13}^{S\times S\times S})_*
(\pi_j(S)\times S) \cap (S\times \pi_i(S))$\\
$={1 \over m}(i\times 1)_*(h\times f)^*
(\pi_i(S) \circ \pi_j(S))=\delta_{ij}\pi_{i0}$.\\
\end{small}
Similarly one proves b).\\
c) As before, one finds that\\
\begin{small}
$\pi_{j2}\cdot \pi_{i0}$\\
$={1 \over m^2}(i\times i)_*(pr_{13}^{Z\times S\times Z})_*
(1\times f\times 1)_*
[(1\times f\times 1)^*(h\times 1\times h)^*
(\pi_i(S)\times S\cap S\times \pi_j(S))\cap (Z\times Y\times Z)]$\\
$={1 \over m^2}(i\times i)_*(pr_{13}^{Z\times S\times Z})_*
[(h\times 1\times h)^*(\pi_i(S)\times S\cap S\times \pi_j(S))\cap
(1\times f\times 1)_*(Z\times Y\times Z)]= 0$\\
\end{small}
because $(1\times f\times 1)_*(Z\times Y\times Z)= 0$ due to
dimension reasons. {\hfill $\square$\\}
\ \\
Define now a set of cycles $\pi_0,\cdots,\pi_6$ in the following way:\\
\begin{small}
$\pi_0:=\pi_{00}$\\
$\pi_1:=\pi_{10}$\\
$\pi_2:=\pi_{20}+\pi_{02}-\pi_{20} \circ \pi_{02}$\\
$\pi_4:=\pi_{40}+\pi_{22}-\pi_{40} \circ \pi_{22}$\\
$\pi_5:=\pi_{32}$\\
$\pi_6:=\pi_{42}$\\
$\pi_3:=\Delta-\sum_{i\ne 3}\pi_i$
\end{small}
\begin{cor}
The $\pi_j$ defined above form a set of orthogonal projectors
such that $\pi_k=\pi_{6-k}^t$ .
\end{cor}
\begin{thm} \label{thm2} $\pi_i=\delta_{ij}$ on
$\cases{f^*H^j(S,{\Bbb Q}) \quad if \quad j = 0,1 \cr
f^*H^j(S,{\Bbb Q}) \oplus {\Bbb Q}\cdot [Z] \quad if \quad j=2 \cr
f^*H^j(S,{\Bbb Q}) \oplus [Z]\cdot f^*H^2(S,{\Bbb Q}) \quad if \quad j=4 \cr
[Z]\cdot f^*H^3(S,{\Bbb Q}) \quad if \quad j=5 \cr
[Z]\cdot f^*H^4(S,{\Bbb Q}) \quad if \quad j=6}$
\end{thm}
{\noindent{\bf Proof. }} First note that one has the equation:
$\pi_{i0}(f^*\alpha)=
{1 \over m}(i \times 1)_* (h\times f)^*\pi _i(S)(f^*\alpha)$\\
$={1 \over m}(pr^{Y \times Y}_2)_*[(i \times 1)_*
(h\times f)^*\pi _i(S)\cap (f^*\alpha\times Y)]$\\
$ ={1 \over m}(pr^{Y \times Y}_2)_*(i \times 1)_*
[(h\times f)^*\pi _i(S)\cap (i\times 1)^*(f^*\alpha\times Y)]$\\
$ ={1 \over m}(pr^{Y \times Y}_2)_*(i \times 1)_*
(h\times f)^*[\pi _i(S)\cap \alpha\times S]$\\
$ ={1 \over m}(pr^{Z \times Y}_2)_* (h\times f)^*[\pi _i(S)
\cap \alpha\times S]$\\
$ ={1 \over m}(pr^{S \times Y}_2)_* (h\times 1)_*(h\times f)^*
[\pi _i(S) \cap \alpha \times S)]$\\
$ =(pr^{S \times Y}_2)_*(1\times f)^*[\pi _i(S)\cap \alpha\times S)]$\\
$ =f^*(pr^{S \times S}_2)_*[\pi _i(S)\cap \alpha\times S]
=f^*\pi_i(S)(\alpha)$.\\
Therefore $\pi_{i0}$ operates as $\delta_{ij}$ on $f^*H^j(S)$,
proving the assertion for $\pi_0$ and $\pi_1$.\\
On the other hand, using projection formula, one gets\\
\noindent
$\pi_{i2}(f^*\alpha)={1 \over m}(pr^{Y\times Y}_2)_*
[ (1 \times i)_* (f\times h)^*\pi _i(S)\cap (f^* \alpha\times Y)$ \\
$\quad\quad\quad\quad = {1 \over m} i_* (pr^{S\times Z}_2)_*
(f\times 1)_*[(f\times 1)^*(1\times h)^*(\pi_i(S)\cap (\alpha\times S))
\cap (Y\times Z)]$\\
$\quad\quad\quad\quad = {1 \over m} i_* (pr^{S\times Z}_2)_*
[(1\times h)^*(\pi_i(S)\cap (\alpha\times S))
\cap (f\times 1)_*(Y\times Z)]=0$,\\
since $(f\times 1)_*(Y\times Z)=0.$\\
Take any $D\in H^k(S,{\Bbb Q})$ with $k=0,2,3,4$ and consider
$C:=i_*h^*(D)$. Observe that
$[C]=f^*(D)\cdot [Z]$. The same computation as above in
cohomology shows that
$$\pi_{i2}([C])=:{1 \over m}(pr_2^{Y\times Y})_*
[(1\times i)_*(f\times h)^*\pi_i(S)\cap [C]\times [Y]]=
i_*h^*(\pi_i(S)(D))$$
As the $\pi_i(S)$ induce the K\"unneth decomposition of
$\Delta_S$ on cohomology,
it follows that $\pi_i(S)([D])=\delta_{ik}([D])$ and therefore one gets
$\pi_{i2}([C])=\delta_{ik}[C]$.\\
Moreover, a similar argument together with Chow's moving lemma shows that\\
$\pi_{i0}([C])={1 \over m}(pr_2^{Y\times Y})_*
[(i \times 1)_*(h \times f)^*\pi_i(S)\cap [C] \times [Y]]$\\
$={1 \over m}(pr_2^{Y\times Y})_*
(i \times 1)_*[(h \times f)^*\pi_i(S)\cap (i \times 1)^* [C] \times [Y]]$\\
$={1 \over m}(pr_2^{Z \times Y})_*
[(h \times f)^*\pi_i(S)\cap [C \cap Z] \times [Y]]$\\
$={1 \over m}(pr_2^{S \times Y})_*(h \times 1)_*
[(h \times 1)^*(1 \times f)^*
\pi_i(S)\cap [C \cap Z] \times [Y]]$\\
$={1 \over m}(pr_2^{S \times Y})_*[(1 \times f)^*
\pi_i(S)\cap h_* [C \cap Z] \times [Y]]$\\
$={1 \over m} f^* (pr_2^{S \times S})_*[ \pi_i(S) \cap h_*
[C \cap Z] \times [S]]$\\
$={1 \over m} f^* \pi_i(S)(h_* [C \cap Z])=0$,\\
if $i \ne k+2$. As a consequence one also gets
$\pi_{i0} \circ \pi_{j2}([C])=\delta_{jk}\pi_{i0}([C])$, which proves the
assertion for $\pi_2,\pi_4,\pi_5$ and $\pi_6$ and the theorem. {\hfill $\square$\\}
\ \\
Now assume additionally that $f: Y \to S$ is a desingularization of a conic
bundle morphism $f': X' \to S' $ in the sense of \cite{4},
i.e. there is a commutative diagram
$$\matrix{ Y & {\buildrel f \over \longrightarrow} & S \cr
\quad \downarrow \sigma && \quad \downarrow \tau \cr
X' & {\buildrel f' \over {\longrightarrow }} & S' }
$$
with blow-up morphisms $\sigma, \tau$. Also we assume $Z \subset Y$ is
a sufficiently general smooth hyperplane section
of $Y$ that dominates $S$.
Then we can choose irreducible
divisors $H_1,...,H_r$ in $Y$ such that $H_1=Z$ and
$$ H^{1,1}(Y,{\Bbb Q})= \bigoplus_{i=1}^r {\Bbb Q}[H_i]
$$
form a basis of $H^{1,1}(Y,{\Bbb Q})$ and such that
$f_*H_i=0$ in $CH^0(S)$ for $i \ge 2$, i.e. $H_i$ is exceptional with
respect to $f$ for $i \ge 2$.
\begin{lemma} \label{le1}
For every cycle $W$ one has $\pi_{20}(W)={1 \over m}
f^* \pi_2(S) (h_* (W \cap Z)) \in f^* CH^*(S) \otimes {\Bbb Q}$.
Let $W$ be a cycle with $f_*(W) =0$.
Then $\pi_{02}(W)=0$ already in the Chow group of $Y$.
\end{lemma}
{\noindent{\bf Proof. }}
$\pi_{02}(W)={1 \over m}(pr_2^{Y\times Y})_*
[(1\times i)_*(f\times h)^*\pi_0(S)\cap (W \times Y) ]$\\
$={1 \over m} i_* (pr_2^{S \times Z})_*[(1 \times h )^*
\pi_0(S) \cap (f \times 1)_*(W \times Z)] =0 $ \\
by \cite[prop.6.6 (c)]{Fu} and
since $f_*(W)=0 \in CH^*(S)$. \\
On the other hand\\
$\pi_{20}(W)={1 \over m}(pr_2^{Y\times Y})_*
[(i \times 1)_* (h \times f)^*\pi_2(S) \cap (W \times Y) ]$\\
$={1 \over m}(pr_2^{Z \times Y})_*
[(h \times f)^*\pi_2(S) \cap ((W \cap Z) \times Y) ]$\\
$={1 \over m} (pr_2^{S \times Y})_*[(1 \times f)^*
\pi_2(S) \cap (h \times 1)_*((W \cap Z) \times Y)] $\\
$={1 \over m} (pr_2^{S \times Y})_*(1 \times f)^*[
\pi_2(S) \cap h_*(W \cap Z) \times S)] $\\
$={1 \over m} f^* (pr_2^{S \times S})_*
[\pi_2(S) \cap h_*(W \cap Z) \times S ] $\\
$={1 \over m} f^* \pi_2(S)(h_*(W \cap Z)) \in f^* CH^*(S) \otimes {\Bbb Q}$.
{\hfill $\square$\\}
\begin{cor} $\pi_2(Y)(H_i)={1 \over m} f^*(h_* (H_i \cap Z))
\in f^* CH^1(S) \otimes {\Bbb Q} $ for $i \ge 2$.
\end{cor}
By theorem \ref{thm2} $\pi_2(Y)$ operates as zero on ${\rm Pic}^0(Y)$,
therefore the image of $\pi_2(Y)$ in $CH^1(Y) \otimes {\Bbb Q}$ is a finite
dimensional vector space.
By changing our generators $H_i$ above modulo classes in ${\rm Pic}^0(Y)=
f^*{\rm Pic}^0(S)$, we may assume that they
generate ${\rm Im}(\pi_2) \subset CH^1(Y) \otimes {\Bbb Q}$.
Then we write uniquely
$$\pi_2(Y)(H_i)=\sum_k a_{i,k} H_k \in CH^1(Y) \otimes {\Bbb Q}$$
with a matrix $A=(a_{i,k}) \in {\rm Mat}(r \times r,{\Bbb Q})$.
$\pi_2(Y)$ being a projector implies that $A^2=A$.
Choose algebraic cycles $\ell_1,...,\ell_r $ such that $\ell_1=F$, a general
fiber of $f$, and such that
their cohomology classes form a basis of $H^{2,2}(Y,{\Bbb Q})$.
By Poincar\'e duality the intersection matrix
$M=(m_{i,j}):=
(\ell_1,...,\ell_r)^{T}(H_1,...,H_r)$
has nonzero determinant.\\
We define
$$ q_2:= \pi_2(Y) + \sum b_{i,j}(\ell_i \times H_j) -
\sum b_{i,j}(\ell_i \times H_j) \circ \pi_2
$$
with some matrix $B=(b_{i,j}) \in {\rm Mat}(r \times r,{\Bbb Q})$.\\
\begin{lemma}{\label{le2}} If $B=M^{-1}({\bf 1} -A)$, then $q_2$
is a projector and operates as the
identity on $H^2(Y,{\Bbb Q})$.
\end{lemma}
{\noindent{\bf Proof. }}
$\pi_2$ acts as the identity on $f^*H^2(S,{\Bbb Q})$ by
theorem \ref{thm2}. The higher direct images $R^if_*{\cal O}_Y$
vanish for $i \ge 1$ by \cite{4}. Therefore by the Leray spectral sequence
$H^2(Y,{\cal O}_Y)=f^*H^2(S,{\cal O}_S)$ and it is
enough to show that $q_2$ operates as the identity on $H^{1,1}(Y,{\Bbb Q})$ too.
But $q_2$ acts via the matrix $MB + A +BA$
on $H^{1,1}(Y,{\Bbb Q})$ with respect to
the basis $\{H_i\}$. Now $\pi_2^2=\pi_2$ and we get
$A^2=A$ and therefore $BA=0$. By definition of $B$, we obtain
that $MB + A +BA= M(M^{-1}({\bf 1}-A)) +A={\bf 1}$. \\
To show that $q_2$ is a projector, let us write
$q_2=\pi_2+ \beta - \beta \pi_2$. Note that $\beta \beta = \beta$,
since $BMB=B$. From $BA=0$ we deduce that $\pi_2 \beta=0$.
Therefore\\
$q_2 \circ q_2= \pi_2^2 +\beta^2 + \beta \pi_2 \beta \pi_2 + \pi_2 \beta
- \pi_2 \beta \pi_2 + \beta \pi_2 - \beta \beta \pi_2 - \beta \pi_2 \pi_2
-\beta \pi_2 \beta = \pi_2 +\beta -\beta \pi_2 = q_2$ is a projector.
{\hfill $\square$\\}
\begin{thm} \label{thm3} The following cycles
$ p_0(Y):=\pi_0(Y), \ p_1(Y):=\pi_1(Y) $, \\
$ p_2(Y):= q_2 -\pi_1(Y) \circ \sum b_{i,j}(\ell_i \times H_j)
-\pi_1(Y) \circ \sum b_{i,j}(\ell_i \times H_j) \circ \pi_2(Y) $\\
$ p_4:= p_2^{tr}(Y), \quad p_5(Y):= \pi_5(Y),
\quad p_6(Y):=\pi_6(Y), \quad p_3(Y):=\Delta - \sum_{i \ne 3} p_i $\\
define orthogonal projectors, which satisfy properties (1)-(6) of a
Murre decomposition.
\end{thm}
{\noindent{\bf Proof. }} By lemma \ref{le2} above, (1),(2) and (3) are straightforward. \\
To prove (4),(5) and (6)
for $j=1$, note that ${\rm Pic}(Y) \otimes {\Bbb Q} =f^*{\rm Pic}^0(S) \otimes {\Bbb Q}
\oplus \bigoplus_i {\Bbb Q} \cdot H_i$. By theorem \ref{thm2} above,
$p_1$ operates on $Pic^0(Y) \otimes {\Bbb Q}=f^*Pic^0(S) \otimes {\Bbb Q}$
as the identity and trivially on
$\bigoplus_i {\Bbb Q} \cdot H_i$. Vice versa $p_2$ is the identity on
$\bigoplus_i {\Bbb Q} \cdot H_i$ and zero on $f^*Pic^0(S) \otimes {\Bbb Q}$, because it
acts trivially on $f^*H^1(S,{\Bbb Q})$.
All the other projectors are zero on $CH^1(Y) \otimes {\Bbb Q}$.
Therefore we get (4)-(6) for $j=1$ with $F^2CH^1(Y) \otimes {\Bbb Q}=0$.\\
For $j=2$, property (4) follows from the analogous assertion for $S$. By
construction $F^1 CH^2(Y) \otimes {\Bbb Q}={\rm Ker}(p_4)=
CH^2_{\rm hom}(Y) \otimes {\Bbb Q}$. Then
$F^2CH^2(Y) \otimes {\Bbb Q}={\rm Ker}(p_3)\cap{\rm Ker}(p_4)={\rm Im}(p_2)=
{\rm Im}(\pi_2(Y))$.\\
Now we show that $F^2CH^2(Y) \otimes {\Bbb Q} \cong f^* F^2 CH^2(S) \otimes {\Bbb Q}$:
$\pi_{02}$ operates as zero on $CH^2(Y)$
by Chow's moving lemma and if $C$ is any
curve homologous to zero on $Y$, then by Lemma \ref{le1},
$\pi_{20}(C)= f^* h_*(C \cap Z) \in
f^* F^2 CH^2(S) \otimes {\Bbb Q}$.\\
This proves that
$F^2CH^2(Y) \otimes {\Bbb Q} \subset f^* F^2CH^2(S) \otimes {\Bbb Q}$, but
since $\pi_2(Y)$ operates as the identity on every fiber of $f$, we get
equality. This is then independent of all choices, because this is the case
for $F^2 CH^2(S)$ by \cite{6}.
Finally $F^3 CH^2(Y) \otimes {\Bbb Q}=0$, since $p_2$ acts as the
identity on $F^2CH^2(Y) \otimes {\Bbb Q} ={\rm Im}(p_2)$.
Hence we get (5) and (6) for $j=2$.\\
Finally consider $CH^3(Y)$: Clearly $F^1CH^3(Y) \otimes {\Bbb Q}=Ker(\pi_6)=
CH^3_{\rm hom}(Y) \otimes {\Bbb Q}$. Further
$F^2CH^3(Y) \otimes {\Bbb Q} = Ker (\pi_5|_{F^1CH^3(Y) \otimes {\Bbb Q}})$ and we
claim that $F^2 CH^3(Y) \otimes {\Bbb Q} \cong Ker(alb_Y) \otimes {\Bbb Q}$, where
$alb_Y: CH^3(Y)_{\rm hom} \to Alb(Y)$ is the Albanese map. But
there is a commutative diagram
$$\matrix{ CH^3(Y)_{\rm hom} & \to & Alb(Y) \cr
f_* \downarrow && \downarrow f_* \cr
CH^2(S)_{\rm hom} & \to & Alb(S) }
$$
Both vertical maps are isomorphisms. To compute
$F^2CH^3(Y) \otimes {\Bbb Q} $ we take any closed point $P$ in $Y$ and compute that
$f_* \pi_5(P)= f_* {1 \over m} i_* h^* (\pi_3(S)(P))= \pi_3(S)(f_*(P))$.\\
This shows that $f_* F^2CH^3(Y) \otimes {\Bbb Q} \cong
F^2 CH^2(S) \otimes {\Bbb Q} \cong Ker(alb_S) \otimes {\Bbb Q} $ by \cite{6}. Therefore
$F^2CH^3(Y) \otimes {\Bbb Q} \cong Ker(alb_Y) \otimes {\Bbb Q}$, which is
independent of all choices again by \cite{6}. Finally
$F^3CH^3(Y) \otimes {\Bbb Q} = 0$, since if $P=\sum a_i P_i$ is a zero
cycle on $Y$ with $\sum a_i=0$, then
$f_* \pi_4(P)=f_* \pi_{20}^{t}(P) + f_* \pi_{02}^{t}(P) =
f_*{1 \over m} (1 \times i)_*(f \times h)^*\pi_2(S) (P) +
f_*{1 \over m} (i \times 1)_* (h \times f)^* \pi_4(S) (P)$.
But $\pi_4(S)=S \times e$ , hence the last term is zero and the first
term becomes $\pi_2(S)(f_*P)$. But $\pi_2(S)$ acts as the identity
on $F^2 CH^2(S) \otimes {\Bbb Q}$. Thus $f_* F^3 CH^3(Y) \otimes {\Bbb Q}
\subset F^3 CH^2(S) \otimes {\Bbb Q} = 0$. \\
This finishes the proof of the theorem. {\hfill $\square$\\}
\newpage
\section{Murre decompositions of uniruled 3-folds}
Let $k={\Bbb C}$. By a 3-fold we just mean a normal 3-dimensional complex
variety.\\
\begin{definition} A 3-fold $X$ is called {\bf uniruled}, if there exists
a dominant rational map $\varphi: S \times {{\Bbb P}}^1 - - - \to X$ for some
surface $S$.
\end{definition}
\begin{thm} (\cite{3}): A smooth projective 3-fold $X$ is uniruled
if and only if it has Kodaira dimension $- \infty$, i.e. no multiple of
$K_X$ has sections.
\end{thm}
\begin{thm} (\cite{4}): Let $X$ be a uniruled 3-fold
with only ${\Bbb Q}$-factorial terminal singularities. Then there exists a
birational mapping $r : X ---\to Y$ which is a composition of flips and
divisorial contractions, such that $Y$ has an extremal ray $R$ whose
extremal contraction map $f: Y \to Z$ satisfies one of the following cases:\\
(a) dim(Z)=0, $Y$ is a ${\Bbb Q}$-Fano 3-fold with $\rho(Y)=1$, i.e. $-mK_Y$ is
an ample Cartier divisor for some $m \ge 1$ and the divisor class group is
free with one generator.\\
(b) $Z$ is a smooth curve and $Y$ is a del Pezzo fibration over $Z$, i.e.
the general fibre of $f$ is a del Pezzo surface.\\
(c) $Z$ is a surface with at most quotient singularities and $Y$ is a
conic bundle over $Z$.\\
In cases (b) and (c) the reduced preimage of any irreducible divisor is again
irreducible.
\end{thm}
\begin{thm} Let $X$ be a smooth complex
uniruled 3-fold. Then $X$ admits a Murre decomposition.
\end{thm}
\bigskip
{\noindent{\bf Proof. }} Since $X$ is uniruled, it is birational to one of the following
varieties:\\
(a) A ${\Bbb Q}$-Fano 3-fold $Y$ with $\rho(Y)=1$, i.e. $-mK_Y$ is
an ample Cartier divisor for some $m \ge 1$ and the divisor class group is
free with one generator.\\
(b) A del Pezzo fibration over a smooth curve. \\
(c) A conic bundle over a normal
surface with at most quotient singularities.\\
In cases (a), (b) $H^2(X,{\Bbb Q})$ and $H^4(X,{\Bbb Q})$ are
generated by classes of
algebraic cycles. Thus we define $p_0(X)=\{e\}\times X$
and $p_6(X)=X \times\{e\}$ for some rational point $e\in X$, $p_1(X)$ and
$p_5(X)$ as in \cite{6} and $p_2(X)$ and $p_4(X)=p_2(X)^{tr}$ as in
theorem \ref{thm1}. Then it is immediate to verify all properties
(2)-(6) similar to the proof of \ref{thm3} while property (1) can be achieved
like in \cite[remark 6.5.]{6}, by the non-commutative Gram-Schmidt process.\\
In case (c) we may assume that after blowing up $X$ along several
smooth subvarieties, there is a situation as in the previous section:\\
Let $\varphi: Y \to X$ be the blow-up and assume that $f:Y \to S$
is a morphism to a smooth surface $S$ with rationally connected fibers.
Take the projectors $p_0(Y),...,p_6(Y)$ as defined in the last section.\\
To define the projectors for $X$,
consider the graph $\Gamma_\varphi \subset
Y \times X$ of $\varphi$. Define
$$p_i(X):=\Gamma_\varphi \circ p_i(Y) \circ \Gamma_\varphi^{tr}=
(\varphi\times\varphi)_*(p_i(Y))$$
(by Liebermann's lemma \cite{2})
for $0 \le i \le 2$. We claim that all $p_i(X)$ are orthogonal projectors.\\
By induction on the number of blow-ups we may assume that there is
just one blow-up along a smooth subvariety $W\subset X$.\\
Consider the canonical diagram
$$\matrix{Y \times Y \times Y & {\buildrel pr_{13} \over \to}
& Y \times Y \cr
\downarrow &&
\downarrow \cr
X \times Y \times X & {\buildrel pr_{13} \over \to} & X \times X}
$$
where the vertical maps are $\varphi \times 1 \times \varphi$ and
$\varphi \times \varphi$.
Let $E$ be the exceptional divisor. Then we compute for $0 \le i,j \le 2$:\\
$p_i(X)\circ p_j(X)= (pr_{13})_*((\varphi\times id)_*p_j(Y)\times X\cap
X\times (id\times \varphi)_*p_i(Y))=$\\
$= (\varphi\times\varphi)_*
(pr_{13})_*(p_j(Y)\times Y\cap
Y\times (id\times \varphi)^*(id\times \varphi)_*p_i(Y))$\\
$= (\varphi\times\varphi)_*
(pr_{13})_*(p_j(Y)\times Y\cap
Y\times (p_i(Y)+(id\times j)_*Q_{i,j}))$ \\
$=(\varphi\times\varphi)_* (pr_{13})_*(p_j(Y)\times Y\cap Y\times p_i(Y))+
(\varphi\times\varphi)_*
(pr_{13})_*(p_j(Y)\times Y\cap Y \times (id\times j)_*Q_{i,j})$\\
$=(\varphi\times\varphi)_* (p_i(Y) \circ p_j(Y)+
(pr_{13})_*(p_j(Y)\times Y \cap Y \times (id\times j)_*Q_{i,j})$\\
where $Q_{i,j}\in CH_3(Y\times E)$ and $j: E \hookrightarrow Y$ is
the inclusion. Hence \\
${\cal C}_i:=p_i(X)\circ p_i(X)-p_i(X)= (\varphi\times id)_*
(pr_{13})_*(p_i(Y)\times X \cap
Y \times (id\times i)_*(id\times \varphi^E)_*Q_{i,i}))$.\\
$p_i(Y)={1 \over m} (i \times 1)_*(h \times f)^*\pi_i(S) + T_i$
with $T_0,T_1=0$ and
$T_2=\sum c_{ij}(\ell_i \times H_j)-
\sum b_{i,j}(\ell_i \times H_j) \circ \pi_2(Y)$ for some integers $c_{i,j},
b_{i,j}$ which
is supported on $(Z \times Y) \cup (\ell_i \times Y)$. Therefore
${\cal C}_i$ is supported on $\varphi(Z) \times W$.
Here $i:W\to X$ is the inclusion and $\varphi^E: E \to W$ is the restriction
of $\varphi$ to $E$.\\
If $W$ is a point, ${\cal C}_i=0$ by dimension reasons. If $W$ is a curve,
${\cal C}_i=a(\varphi(Z)\times W)$ with $a \in {\Bbb Z}$. But
${\cal C}_i=p_i(X)\circ p_i(X)-p_i(X)$
operates as zero on the cohomology classes of every curve $T \in CH^2(X)$,
since by Chow's moving lemma we
can choose $T$ to be disjoint from $W$ and use that
$p_i(Y)(T)=0$ in cohomology for $i=0,1,2$.
Therefore $a=0$ and $p_i(X)$ is a projector.\\
For $ i \neq j$,
$p_i(X)\circ p_j(X)=
(\varphi\times\varphi)_*
(pr_{13})_*(p_j(Y)\times Y \cap Y \times (id\times j)_*Q_{i,j})$\\
since $p_i(Y)$ and $p_j(Y)$ are orthogonal. As above this implies that
$ p_i(X) \circ p_j(X)$ is supported on
$\varphi(Z) \times W$ for all $j$. By the same argument
with Chow's moving lemma for $CH^2(X)$ as before, $p_i(X)\circ p_j(X)=0$. \\
Now define
$$p_4(X)=p_2(X)^{tr}, p_5(X)=p_1(X)^{tr}, p_6(X)=p_0^{tr} \quad {\rm and}
\quad p_3(X)=\Delta-\sum_{i \ne 3} p_i(X)$$
Properties (3)-(6) follow from theorem
\ref{thm3} together with the split exact sequences (\cite[prop. 6.7]{Fu})
$$ 0 \to CH_k(W) \to CH_k(E) \oplus CH_k(X) \to CH_k(Y) \to 0 $$
(1) and (2) can be obtained again via the
Gram-Schmidt process. {\hfill $\square$\\}
\bigskip
|
1997-03-21T17:15:49 | 9607 | alg-geom/9607021 | en | https://arxiv.org/abs/alg-geom/9607021 | [
"alg-geom",
"math.AG"
] | alg-geom/9607021 | Miles Reid | F. Catanese, M. Franciosi, K. Hulek and M. Reid | Embeddings of curves and surfaces | LaTeX2e with packages: amstex, amssymb, theorem. 32 pp | null | null | To be issued as Univ. of Pisa preprint. Requests for copies to the
second author at Pisa | null | We prove a general embedding theorem for Cohen--Macaulay curves (possibly
nonreduced), and deduce a cheap proof of the standard results on pluricanonical
embeddings of surfaces, assuming vanishing H^1(2K_X)=0.
| [
{
"version": "v1",
"created": "Mon, 22 Jul 1996 10:30:14 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Catanese",
"F.",
""
],
[
"Franciosi",
"M.",
""
],
[
"Hulek",
"K.",
""
],
[
"Reid",
"M.",
""
]
] | alg-geom | \section{Introduction}
Let $C$ be a curve over a field $k$ of characteristic $p\ge0$, and $H$ a
Cartier divisor on $C$. We assume that $C$ is projective and Cohen--Macaulay
(but possibly reducible or nonreduced). Write $HC=\deg\Oh_C(H)$ for the degree
of $H$, $p_aC=1-\chi(\Oh_C)$ for the arithmetic genus of $C$, and $\om_C$ for
the dualising sheaf (see \cite{Ha}, Chap.~III, \S7).
Our first result is the following. (A {\em cluster} $Z$ of {\em degree}
$\deg Z=r$ is simply a \hbox{$0$-dimensional} subscheme with
$\length\Oh_Z=\dim_k\Oh_Z=r$; a curve $B$ is {\em generically Gorenstein} if,
outside a finite set, $\om_B$ is locally isomorphic to $\Oh_B$. The remaining
definitions and notation are explained below.)
\begin{TEO}[Curve embedding theorem]\label{th:curve} $H$ is very ample on $C$
if for every generically Gorenstein subcurve $B\subset C$, either
\begin{enumerate}
\item $HB\ge2p_aB+1$, or
\item $HB\ge2p_aB$, and there does not exist a cluster $Z\subset B$ of
degree $2$ such that $\sI_Z\Oh_B(H)\iso\om_B$.
\end{enumerate}
More generally, suppose that $Z\subset C$ is a cluster (of any degree) such
that the restriction
\begin{equation}
H^{0}(C,\Oh_C(H))\to\Oh_Z(H)=\Oh_C(H)\otimes\Oh_Z
\label{eq:rest}
\end{equation}
is not onto. Then there exists a generically Gorenstein subcurve $B$ of $C$
and an inclusion $\fie\colon\sI_Z\Oh_B(H)\into\om_B$ not induced by a map
$\Oh_B(H)\to\om_B$. In particular, (\ref{eq:rest}) is onto if
\begin{equation}
HB>2p_aB-2+\deg(Z\cap B)
\nonumber
\end{equation}
for every generically Gorenstein subcurve $B\subset C$.
\end{TEO}
Theorem~\ref{th:curve} is well known for nonsingular curves $C$. Although
particular cases were proved in \cite{Ca1}, \cite{Ba2}, \cite{C-F}, \cite{C-H},
it was clear that the result was more general. In discussion after a lecture
on the Gorenstein case by the first author at the May 1994 Lisboa AGE meeting,
the fourth author pointed out the above result, where $C$ is only assumed to
be a pure 1-dimensional scheme. For divisors on a nonsingular surface, Mendes
Lopes \cite{ML} has obtained results analogous to Theorem~\ref{th:curve} and to
Theorem~\ref{th:hh}. We apply these ideas to the canonical map of a Gorenstein
curve in \S\ref{sec:cc}.
The proof of Theorem~\ref{th:curve} is based on two ideas from Serre and
Grothendieck duality:
\begin{enumerate}
\renewcommand\labelenumi{(\alph{enumi})}
\item we use Serre duality in its ``raw'' form
\begin{equation}
H^1(C,\sF)\dual\Hom(\sF,\om_C)\quad\text{for $\sF$ a coherent sheaf,}
\nonumber
\end{equation}
where $\dual$ denotes duality of vector spaces.
\item If $\Oh_C$ has nilpotents, a nonzero map $\fie\colon\sF\to\om_C$ is
not necessarily generically onto; however (because we are $\Hom$'ming into
$\om_C$), duality gives an automatic factorisation of $\fie$ of the form
\begin{equation}
\sF\onto\sF_{|B}\to\om_B\into\om_C,
\nonumber
\end{equation}
via a purely 1-dimensional subscheme $B\subset C$, where $\sF_{|B}\to\om_B$ is
generically onto. See Lemma~\ref{lem:adj} for details.
\end{enumerate}
Since our main result might otherwise seem somewhat abstract and useless, we
motivate it by giving a short proof in \S\ref{sec:pluri}, following the methods
of \cite{C-F}, of the following result essentially due to Bombieri (when
$\chara k=0$) and to Ekedahl and Shepherd-Barron in general. Recall that a
{\em canonical surface} (or canonical model of a surface of general type) is a
surface with at worst Du Val singularities and $K_X$ ample. The remaining
notation and definitions are explained below.
\begin{TEO}[Canonical embeddings of surfaces]\label{th:surf} $X$ is a
canonical surface. Assume that $H^1(2K_X)=0$. Then $mK_X$ is very ample if
$m\ge5$, or if $m=4$ and $K_X^2\ge2$, or if $m=3$, $p_g\ge2$ and $K_X^2\ge3$.
\end{TEO}
Here $H^1(2K_X)=0$ follows at once in characteristic 0 from Kodaira vanishing
or Mumford's vanishing theorem. One can also get around the assumption
$H^1(2K_X)=0$ in characteristic $p>0$ (see \cite{Ek} or \cite{S-B}). In fact
Ekedahl's analysis (see \cite{Ek}, Theorem~II.1.7) shows that $H^1(2K_S)\ne0$
is only possible in a very special case, when $p=2,\chi(\Oh_S)=1$ and $S$ is
(birationally) an inseparable double cover of a K3 surface or a rational
surface.
In \S\ref{sec:tri} and \S\ref{sec:bi} we apply these ideas to prove the following theorems on
tricanonical and bicanonical linear systems of a surface of general type.
\begin{TEO}[Tricanonical embeddings]\label{th:tri} Suppose that $X$ is a
canonical surface with $K_X^2\ge3$. Then $3K_X$ is very ample if either
\begin{enumerate}
\renewcommand\labelenumi{(\alph{enumi})}
\item $q=h^1(\Oh_X)=0$; or
\item $\chi(\Oh_X)\ge1$, $\dim\Pico X>0$ and $H^1(2K_X-L)=0$ for all
$L\in\Pico X$.
\end{enumerate}
Note that (a) or (b) cover all cases with $\chara k=0$. Thus the cases {\em
not} covered by our argument are in $\chara k=p>0$, with either $p_g<q$ or
$\dim\Pico X=0$. \end{TEO}
Theorem~\ref{th:tri} in characteristic 0 is a result of Reider \cite{Rei}, but
see also \cite{Ca2}. Without the condition $K_X^2\ge3$, the double plane with
branch curve of degree 8 (that is, $X_8\subset\proj(1,1,1,4)$) is a
counter\-example. It follows from a result of Ekedahl (\cite{Ek},
Theorem~II.1.7) that if $\chi(\Oh_X)\ge1$ then $H^1(2K_X-L)=0$ for all $L\ne0$.
The remaining assumption in Theorem~\ref{th:tri} is that $H^1(2K_X)=0$, and
this can also be got around, as shown by Shepherd-Barron \cite{S-B}.
\begin{TEO}[Bicanonical embeddings]\label{th:bi} We now assume that $q=0$
and $p_g\ge4$.
\begin{enumerate}
\renewcommand\labelenumi{(\alph{enumi})}
\item $2K_X$ is very ample if every $C\in|K_X|$ is numerically $3$-connected
(in the sense of Definition~\ref{def:m-conn}, see also Lemma~\ref{lem:n-conn}).
More precisely, $|2K_X|$ separates a cluster $Z$ of degree $2$ provided that
every curve $C\in|K_X|$ through $Z$ is $3$-connected.
\item Assume in addition that $K_X^2\ge10$, and let $Z$ be a cluster of
degree $2$ contained in $X$. Then $Z$ is contracted by $|2K_X|$ if and only if
$Z$ is contained in a curve $B\subset X$ with
\begin{equation}
K_XB=p_aB=1\text{ or }2
\nonumber
\end{equation}
(a {\em Francia curve}, compare Definition~\ref{def:Frc}), and
$\sI_Z\Oh_B(2K_X)\iso\om_B$.
\item In particular, $|2K_X|$ defines a birational morphism unless $X$ has a
pencil of curves of genus $2$.
\end{enumerate}
\end{TEO}
\begin{REMS}
(1) A cluster $Z$ of degree 2 is automatically contracted by $|2K_X|$ if it
is contained in a curve $C\subset X$ for which $\sI_Z\Oh_C(2K_X)\iso\om_C$ (for
a non\-singular curve, this reads ${2K_X}_{|C}=K_C+P+Q$). Thus (b) says in
particular that if this happens for some $C$ then it also happens for a
Francia curve.
(2) The assumptions $q=0$ and $p_g\ge4$ are needed for the simple minded
``restriction method'' of this paper, but we conjecture that (b) holds without
them (at least in characteristic zero, or assuming $q=0$); the case
$Z=\{x,y\}$ with $x\ne y$ (that is, ``separating points'') follows in
characteristic zero by Reider's method. We believe that the conjecture can be
proved quite generally by a different argument based on Ramanujam--Francia
vanishing, or by Reider's method applied to reflexive sheaves on $X$. Stay
tuned!
(3) In characteristic 0, Theorem~\ref{th:bi} (without the assumption $q=0$) is
due essentially to Francia (unpublished, but see \cite{Fr1}--\cite{Fr2}) and
Reider \cite{Rei}. Theorem~\ref{th:bi}, (a) is a consequence of
Theorem~\ref{th:hh} on canonical embeddings of curves and the generalisation
of hyper\-elliptic curves. The results in Theorem~\ref{th:bi} are only a
modest novelty, in that there is no restriction on the characteristic of the
ground field (see \cite{S-B}, Theorems~25, 26 and~27 for $\chara k\ge11$).
Further results on the bicanonical map $\fie_{2K}$ for smaller values of
$p_g$, $K_X^2$ (in characteristic 0) require a more intricate analysis, and we
refer to recent or forthcoming articles (\cite{C-F-M}, \cite{C-C-M}). Other
applications of our methods can be found in \cite{F}.
\end{REMS}
\subsection*{Acknowledgment} It is a pleasure to thank Ingrid Bauer for
interesting discussions on linear systems on surfaces, out of which this paper
originated.
\subsection*{Conventions}
This paper deals systematically with reducible and nonreduced curves and their
subschemes $B\subset C$. A coherent sheaf $\sF$ on a curve $C$ is {\em torsion
free} if there are no sections $s\in\sF$ supported at points; on a
1-dimensional scheme, this is obviously equivalent to $\sF$ {\em
Cohen--Macaulay}. We say that $C$ is {\em purely \hbox{$1$-dimensional}\/} or
{\em Cohen--Macaulay} if $\Oh_C$ is torsion free.
A map $\fie\colon\sF\to\sG$ between coherent sheaves on $B$ is {\em
generically injective} if it is injective at every generic point of $B$; if
$\sF$ is torsion free then $\fie$ is automatically an inclusion $\sF\into\sG$.
If we know that the generic stalks of $\sF$ and $\sG$ have the same length at
every generic point of $C$ then a generically injective map
$\fie\colon\sF\to\sG$ is an isomorphism at each generic point, and therefore
$\ker\fie$ and $\coker\fie$ have finite length. Indeed, they are both coherent
sheaves supported at a finite set, and by the Nullstellensatz, each stalk is
killed by a power of the maximal ideal. This applies, for example, to the map
$\fie\colon\sI_Z\Oh_B(H)\into\om_B$ of Theorem~\ref{th:curve}, see
Lemma~\ref{lem:gg} below.
A scheme $B$ is {\em Gorenstein in codimension $0$} or {\em generically
Gorenstein} if $\om_B$ is locally isomorphic to $\Oh_B$ at every generic point
of $B$.
A {\em cluster} of degree $r$ is a 0-dimensional subscheme $Z\subset X$
supported at finitely many points, with ideal sheaf $\sI_Z$, structure sheaf
$\Oh_Z=\Oh_X/\sI_Z$, and having $\deg Z=h^0(\Oh_Z)=\length\Oh_Z=r$.
We sometimes write $Z=(x,y)$ for a cluster of degree 2, where $x,y$ are
either 2 distinct points of $X$, or a point $x$ plus a tangent vector $y$ at
$x$. We say that a linear system $|H|$ on $X$ {\em separates} $Z$ (or
separates $x$ and $y$) if $H^0(X,\Oh_X(H))\to\Oh_Z(H)$ is onto, or {\em
contracts} $Z$ if $Z$ does not meet the base locus $\Bs|H|$, and
$\rank\{H^0(X,\Oh_X(H))\to\Oh_Z(H)\}=1$.
\subsection*{Notation}
\begin{enumerate}
\item[$X$] A projective scheme over an arbitrary field $k$. We sometimes (not
always consistently) write $k\subset\overline k$ for the algebraic closure,
and $X_{\overline k}=X\otimes_k\overline k$.
\item[$\om_X$] Dualising sheaf of $X$ (see \cite{Ha}, Chap.~III, \S7).
\item[$|H|$] Linear system defined by a Cartier divisor $H$ on $X$.
\item[$C$] A curve, that is, a projective scheme over $k$ which is purely
1-dimensional, in the sense that $\Oh_C$ is Cohen--Macaulay (torsion free).
\item[$p_aC$] The arithmetic genus of $C$, $p_aC=1-\chi(\Oh_C)$.
\item[$K_C$] A canonical divisor of a Gorenstein curve $C$, that is, a Cartier
divisor such that $\Oh_C(K_C)\iso\om_C$ (only defined if $C$ is Gorenstein).
\item[$\deg\sL$] The degree of a torsion free sheaf of rank 1 on $C$; it can be
defined by
\begin{equation}
\deg\sL=\chi(\sL)-\chi(\Oh_C).
\nonumber
\end{equation}
If $H$ is a Cartier divisor on $C$, we set $HC=\deg\Oh_C(H)$.
\item[$S$] A nonsingular projective surface.
\item[$DD'$] Intersection number of divisors $D,D'$ on a nonsingular
projective surface.
\item[$K_S$] A canonical divisor on $S$.
\item[$K_X^2$] If $X$ is a Gorenstein surface, $K_X$ is a Cartier divisor with
$\om_X=\Oh_X(K_X)$, and $K_X^2$ is the selfintersection number of the Cartier
divisor $K_X$. If $X$ has only Du Val singularities and $\pi\colon S\to X$
is the minimal nonsingular model then $K_S=\pi^*K_X$ and $K_X^2=K_S^2$.
\item[$p_g,q$] The geometric genus $p_g=h^0(S,K_S)=h^0(X,K_X)$ of $S$ or $X$
(respectively the irregularity $q=h^1(S,\Oh_S)=h^1(X,\Oh_X)$).
\item[$P_n$] The $n$th plurigenus $P_n=h^0(S,nK_S)$ of $S$.
\end{enumerate}
\section{Embedding curves} We start with a useful remark.
\begin{REM}\label{rem:1} Let $H$ be a Cartier divisor on a scheme $X$. Then
$H$ is very ample if and only if the restriction map
\begin{equation}
H^0(\Oh_X(H))\to\Oh_Z(H)
\label{eq1}
\end{equation}
is onto for every cluster $Z\subset X$ (more precisely,
for every $Z\subset X_{\overline k}$) of degree $\le2$.
\end{REM}
\begin{pf} By the standard embedding criterion of \cite{Ha}, Chap.~II,
Prop.~7.3, we have to prove that (\ref{eq1}) is onto for all the ideals
$\sI_Z=m_x$ or $m_xm_y$ with $x,y\in X$. For $x\ne y$, we are done.
By assumption $H^0(\Oh_C(H))\to\Oh_C/m_x$ is onto for every $x\in X$. Now if
the image of $H^0(m_x\Oh_C(H))\to m_x/m_x^2$ is contained in a hyperplane
$V\subset m_x/m_x^2$, then the inverse image of $V$ in $\Oh_{C,x}$ generates an
ideal $\sI\subset\Oh_{X,x}$ defining a cluster $Z$ of degree 2 supported at
$x$ such that $H^0(\Oh_C(H))\to\Oh_Z$ is not onto. \QED \end{pf}
\begin{REM}\label{rem:2} The chain of reasoning we use below is that, by
Remark~\ref{rem:1} and cohomology, $H$ is very ample if and only if
$H^1(\sI_Z\Oh_X(H))\to H^1(\Oh_X(H))$ is injective for each cluster $Z$ of
degree $2$, or dually (if $X=C$ is a curve),
$\Hom(\Oh_C(H),\om_C)\to\Hom(\sI_Z(H),\om_C)$ is onto.
\end{REM}
\begin{LEM}\label{lem:gg} Let $C$ be a curve. Assume that there is a Cartier
divisor $H$ on $C$ and a cluster $Z\subset C$ for which the sheaf
$\sL=\sI_Z\Oh_C(H)$ has an inclusion $\sL\into\om_C$. Then $C$ is
generically Gorenstein.
\end{LEM}
\begin{pf} By assumption, $\sL\iso\Oh_C$ at every generic point of $C$. We must
prove that an inclusion $\sL\into\om_C$ maps onto every generic stalk
$\om_{C,\eta}$, or equivalently, that the cokernel $\sN=\om_C/\sL$ has finite
length. We give two slightly different proofs, one based on RR, and one using
properties of dualising modules.
Let $\Oh_C(1)$ be an ample line bundle on $C$. Then by Serre vanishing (see
\cite{Se1}, n$^\circ$~66, Theorem~2 or \cite{Ha}, Chap.~III, Theorem~5.2), for
$n\gg0$, the exact sequence
\begin{equation}
0\to\sL(n)\to\om_C(n)\to\sN(n)\to0
\nonumber
\end{equation}
is exact on global sections, and all the $H^1$ vanish. Now by RR and duality,
\begin{equation}
h^0(\om_C(n))=h^1(\Oh_C(-n))=-\chi(\Oh_C)+n\deg\Oh_C(1)\quad\text{for $n\gg0$.}
\nonumber
\end{equation}
On the other hand, RR also gives
$h^0(\sL(n))=\chi(\Oh_C)+\deg\sL+n\deg\Oh_C(1)$ for $n\gg0$, since
$\sL\iso\Oh_C$ at every generic point. Thus
\begin{equation}
h^0(\sN(n))=-2\chi(\Oh_C)+\deg\sL\quad\text{for all $n\gg0$,}
\nonumber
\end{equation}
and therefore $\sN$ has finite length.
The alternative proof of the lemma uses the ``well-known fact'' (see below)
that the generic stalk $\om_{C,\eta}$ of the dualising sheaf at a generic
point $\eta\in C$ is a dualising module for the Artinian local ring
$\Oh_{C,\eta}$, so that they have the same length, and therefore an inclusion
$\sL\into\om_C$ is generically an isomorphism. The above proof in effect
deduces $\length\om_{C,\eta}=\length\Oh_{C,\eta}$ from RR together with Serre
duality, the defining property of $\om_C$.
\paragraph{Proof of the ``well-known fact''} This is proof {\em by
incomprehensible reference}. First, if $\eta\in X$ is a generic point of a
scheme, more-or-less by definition, a dualising module of the Artinian ring
$\Oh_{X,\eta}$ is an injective hull of the residue field
$\Oh_{X,\eta}/m_{X,\eta}=k(\eta)$ (see \cite{Gr-Ha}, Proposition~4.10); in
simple-minded terms, $\Oh_{X,\eta}$ clearly contains a field $K_0$ such that
$K_0\subset k(\eta)$ is a finite field extension, and the vector space dual
$\Hom_{K_0}(\Oh_{X,\eta},K_0)$ is a dualising module. Next, if $\eta\in X$ is
a generic point of a subscheme $X\subset\proj=\proj^N$ of pure codimension $r$,
then by \cite{Ha}, Chap.~III, Prop.~7.5, the dualising sheaf of $X$ is
$\om_X=\sExt^r_{\Oh_\proj}(\Oh_X,\om_\proj)$. On the other hand, the local
ring $\Oh_{\proj,\eta}$ of projective space along $\eta$ is an
\hbox{$r$-dimensional} regular local ring, and therefore Gorenstein, so that
by \cite{Gr-Ha}, Prop.~4.13,
$\Ext^r_{\Oh_{\proj,\eta}}(\Oh_{X,\eta},\om_{\proj,\eta})$ is a dualising
module of $\Oh_{X,\eta}$ (an injective hull of the residue field
$\Oh_{X,\eta}/m_{X,\eta}=k(\eta)$). \QED \end{pf}
\begin{LEM}[Automatic adjunction]\label{lem:adj} Let $\sF$ be a coherent
sheaf on $C$, and $\fie\colon\sF\to\om_C$ a map of $\Oh_C$-modules. Set
$\sJ=\Ann\fie\subset\Oh_C$, and write $B\subset C$ for the subscheme defined by
$\sJ$. Then $\fie$ has a canonical factorisation of the form
\begin{equation}
\sF\onto\sF_{|B}\to\om_B=\sHom_{\Oh_C}(\Oh_B,\om_C)\subset\om_C,
\label{eq:adj}
\end{equation}
where $\sF_{|B}\to\om_B$ is generically onto.
\end{LEM}
\begin{pf} By construction of $\sJ$, the image of $\fie$ is contained in the
submodule
\begin{equation}
\bigl\{s\in\om_C\bigm|\sJ s=0\bigr\}\subset\om_C
\nonumber
\end{equation}
But this clearly coincides with $\sHom(\Oh_B,\om_C)$. Now the inclusion
morphism $B\into C$ is finite, and $\om_B=\sHom_{\Oh_C}(\Oh_B,\om_C)$ is just
the adjunction formula for a finite morphism (see, for example, \cite{Ha},
Chap.~III, \S7, Ex.~7.2, or \cite{Re}, Prop.~2.11).
The factorisation (\ref{eq:adj}) goes like this: $\fie$ is killed by $\sJ$, so
it factors via the quotient module $\sF/\sJ\sF=\sF_{|B}$. As just observed, it
maps into $\om_B\subset\om_C$. Finally, it maps onto every generic stalk of
$\om_B$, again by definition of $\sJ$: a submodule of the sum of generic stalks
$\bigoplus\om_{B,\eta}$ is the dual to the generic stalk
$\bigoplus\Oh_{B',\eta}$ of a purely 1-dimensional subscheme $B'\subset B$,
and $\fie$ is not killed by the corresponding ideal sheaf $\sJ'$. \QED
\end{pf}
\begin{REM} We define $B$ to be the {\em scheme theoretic support} of $\fie$.
Note that if $C=\sum n_i\Ga_i$ is a Weil divisor on a normal surface and $\sF$
a line bundle, the curve $B\subset C$ defines a splitting $C=A+B$ where $A$ is
the {\em divisor of zeros} of $\fie$: at the generic point of $\Ga_i$, the
map $\fie$ then looks like $y_i^{a_i}$, where $y_i$ is the local equation
of $\Ga_i$, and $A=\sum a_i\Ga_i$. In the general case however, $A$ does not
make sense. \end{REM}
\begin{pfof}{Theorem~\ref{th:curve}} Let $H$ be a Cartier divisor, and $\sI$
the ideal sheaf of a cluster for which $H^1(\sI\Oh_C(H))\ne0$. Then
$\Hom(\sI\Oh_C(H),\om_C)\ne0$ by Serre duality. First pick any nonzero map
$\fie\colon\sI\Oh_C(H)\to\om_C$. By Lemma~\ref{lem:adj}, $\fie$ comes from an
inclusion $\sI\Oh_B(H)\into\om_B$ for a subscheme $B\subset C$, and $B$ is
generically Gorenstein by Lemma~\ref{lem:gg}.
Finally, if $H^0(\Oh_C(H))\to\Oh_Z(H)$ is not onto, then the next arrow in
the cohomology sequence
\begin{equation}
H^1(\sI\Oh_C(H))\to H^1(\Oh_C(H))
\nonumber
\end{equation}
is not injective, and dually, the restriction map
\begin{equation}
\Hom(\Oh_C(H),\om_C)\to\Hom(\sI\Oh_C(H),\om_C)
\nonumber
\end{equation}
is not onto. Thus we can pick $\fie\colon\sI\Oh_C(H)\to\om_C$ which is not the
restriction of a map $\Oh_C(H)\to\om_C$. Then also the map
$\sI\Oh_B(H)\into\om_B$ given by Lemma~\ref{lem:adj} is not the restriction of
a map $\Oh_B(H)\into\om_B$.
For the final part, an inclusion $\sI\Oh_B(H)\into\om_B$ has cokernel of
finite length, so that $\chi(\sI\Oh_B(H))\le\chi(\om_B)$. Plugging in the
definition of degree gives
\begin{equation}
1-p_aB+HB-\deg(Z\cap B)\le p_aB-1,
\nonumber
\end{equation}
that is,
\begin{equation}
HB\le2p_aB-2+\deg(Z\cap B).
\nonumber
\end{equation}
Thus, assuming the inequality (2) of Theorem~\ref{th:curve}, no such inclusion
$\sI\Oh_B(H)\into\om_B$ can exist, so that $H^0(\Oh_C(H))\to\Oh_Z(H)$ is
onto. \QED \end{pfof}
\section{The canonical map of a Gorenstein curve}\label{sec:cc} We now discuss
the canonical map $\fie_{K_C}$ of a Gorenstein curve, writing $K_C$ for a
canonical divisor of $C$, that is, a Cartier divisor for which
$\om_C\iso\Oh_C(K_C)$. Our approach is motivated in part by the examples and
results in the reduced case treated in \cite{Ca1}.
\begin{DEF}\label{def:m-conn} A Gorenstein curve $C$ over an algebraically
closed field
$k$ is {\em numerically $m$-connected} if
\begin{equation}
\deg\Oh_B(K_C)-\deg\om_B=\deg(\om_C\otimes\Oh_B)-(2p_aB-2)\ge m
\nonumber
\end{equation}
for every generically Gorenstein strict subcurve $B\subset C$. For $C$ over
any field, we say that $C$ is numerically $m$-connected if
$C\otimes\overline k$ is numerically $m$-connected.
\end{DEF}
\begin{REM}\label{rem:m-conn} Note that for divisors on a nonsingular
surface,
\begin{equation}
\deg\Oh_B(K_C)-\deg\om_B=(K_S+C)B-(K_S+B)B=(C-B)B.
\nonumber
\end{equation}
In this context, Franchetta and Ramanujam define numerically connected in
terms of the intersection numbers $AB=(C-B)B$ for all effective decompositions
$C=A+B$. The point of our definition is to use the numbers
$\deg\Oh_B(K_C)-\deg\om_B$ in the more general case as a substitute for
$(C-B)B$. In effect, we think of the adjunction formula as defining the
``degree'' of the ``normal bundle'' to $B$ in $C$, in terms of the difference
between $K_C{}_{|B}$ and $\om_B$. \end{REM}
\begin{TEO}\label{th:free} Let $C$ be a Gorenstein curve over a field $k$.
\begin{enumerate}
\renewcommand\labelenumi{(\alph{enumi})}
\item If $C$ is numerically $1$-connected then
$H^0(\Oh_C)=k$ (the constant functions).
\item If $C$ is numerically $2$-connected then either $|K_C|$ is free or
$C\iso\proj^1$ (over the algebraic closure $\overline k$, of course). In
particular, in this case $p_aC=0$ implies $C\iso\proj^1$.
\end{enumerate}
\end{TEO}
\begin{pfof}{(a)} First, if $f\in H^0(\Oh_C)$ is a nonzero section vanishing
along some reduced component of $C$, then applying Lemma~\ref{lem:adj} to the
multiplication map $\mu_f\colon\Oh_C(K_C)\to\om_C$ gives an inclusion
$\Oh_B(K_C)\into\om_B$, which is forbidden by numerically 1-connected (because
$\deg\Oh_B(K_C)>\deg\om_B$). Now if $H^0(\Oh_C)\ne k$, there exists a nonzero
section $f\in H^0(\Oh_{C\otimes\overline k})$ vanishing at any given point
$x\in C\otimes\overline k$. An inclusion $\Oh_C\into m_x$ contradicts at once
$0=\deg\Oh_C>\deg m_x=-1$, so that $f$ must vanish along some component of
$C$, and we have seen that this is impossible. \QED \end{pfof}
\begin{pfof}{(b)} As discussed in Remark~\ref{rem:2}, the standard chain of
reasoning is as follows:
\begin{enumerate}
\item $x\in C$ is a base point of $|K_C|$ if and only if
$H^0(\Oh_C(K_C))\to\Oh_x(K_C)$ is not onto, and then
\item $H^1(m_x\Oh_C(K_C))\to H^1(\Oh_C(K_C))$ is not injective,
\item dually, $\Hom(\Oh_C(K_C),\om_C)\to\Hom(m_x\Oh_C(K_C),\om_C)$ is not
onto,
\item therefore there exists a map $s\colon m_x\Oh_C(K_C)\to\om_C$ linearly
independent of the identity inclusion.
\end{enumerate}
Now by Lemma~\ref{lem:adj}, the map $s$ factors via an inclusion
$m_x\Oh_B(K_C)\into\om_B$ on a generically Gorenstein curve $B$. But then
$B\subsetneq C$ is forbidden by the numerically 2-connected assumption
$\deg m_x\Oh_B(K_C)-\deg\om_B\ge1$.
Therefore $B=C$, that is, $s\colon m_x\Oh_C(K_C)\into\om_C$ is an inclusion.
After tensoring down with $-K_C$, this gives an inclusion
$i\colon m_x\into\Oh_C$ linearly independent of the identity. Write
$\sF=i(m_x)\subset\Oh_C$. Then $\deg\sF=-1$, and therefore $\sF=m_z$ for some
$z\in C$.
Now for any point $y\in C\setminus\{x\}$, there exists a linear combination
$s'=s+\la\id$ vanishing at $y$, which therefore defines an isomorphism
$m_x\iso m_y$. This implies that every point $y\in C$ is a Cartier divisor,
hence a nonsingular point. Since $C$ is clearly connected, and
$\Oh_C(x-y)\iso\Oh_C$ for every $x,y\in C$, it follows that $C\iso\proj^1$.
For the final statement, if $p_aC=0$ then $1=h^0(\Oh_C)=h^1(\om_C)$ by
(a) and duality, hence $h^0(\om_C)=0$ by RR, so that $H^0(\Oh_C(K_C))\to\Oh_x$
is not onto for any $x\in C$. \QED \end{pfof}
\begin{DEF} We say that a Gorenstein curve $C$ is {\em honestly hyperelliptic
(\cite{Ca1}, Definition~3.18)} if there exists a finite morphism $\psi\colon
C\to\proj^1$ of degree $2$ (that is, $\psi$ is finite and $\psi_*\Oh_C$ is
locally free of rank $2$ on $\proj^1$). The linear system $\psi^*|\Oh_C(1)|$
defining $\psi$ is called an {\em honest $g^1_2$.} \end{DEF}
We note the immediate consequences of the definition.
\begin{LEM}\label{lem:hh} An honestly hyperelliptic curve $C$ of genus
$p_aC=g\ge0$ is isomorphic to a divisor $C_{2g+2}$ in the weighted projective
space $\proj(1,1,g+1)$, not passing through the vertex $(0,0,1)$, defined by an
equation
\begin{equation}
w^2+a_{g+1}(x_1,x_2)w+b_{2g+2}(x_1,x_2)=0.
\nonumber
\end{equation}
It follows that every point of $C$ is either nonsingular or a planar double
point, and that $C$ is either irreducible, or of the form $C=D_1+D_2$ with
$D_1D_2=g+1$.
The projection $\fie\colon C\to\proj^1$ is a finite double cover, and the
inverse image of any $x\in\proj^1$ is a Cartier divisor which is a cluster
$Z\subset C$ of degree $2$. In other words, $Z$ is either $2$ distinct
nonsingular points of $C$, a nonsingular point with multiplicity $2$, or a
section through a planar double point of $C$. \qed\end{LEM}
\begin{TEO}\label{th:hh} Let $C$ be a numerically $3$-connected Gorenstein
curve. Then either $|K_C|$ is very ample or $C$ is honestly hyperelliptic.
In particular, in this case if\/ $p_aC\ge2$ then $K_C$ is ample, and if\/
$p_aC=1$ then $C$ is honestly hyperelliptic (over the algebraic closure
$\overline k$, of course).
\end{TEO}
\begin{pf} Let $Z$ be a cluster of degree 2 for which
$H^0(\Oh_C(K_C))\to\Oh_Z(K_C)$ is not onto. The previous chain of reasoning
gives a map $\sI_Z\Oh_C(K_C)\to\om_C$ linearly independent of the identity
inclusion. An inclusion $\sI_Z\Oh_B(K_C)\into\om_B$ with $B\subsetneq C$ is
forbidden as before by $C$ numerically 3-connected. Therefore we get an
inclusion $s\colon\sI_Z\Oh_C(K_C)\into\om_C$ linearly independent of the
identity inclusion. Note that any linear combination $s'=s+\la\id$ of the two
sections is again generically injective, since an inclusion
$\sI_Z\Oh_B(K_C)\into\om_B$ with $B\subsetneq C$ is forbidden by numerically
3-connected.
The image $\sF=s(\sI_Z\Oh_C(K_C))\subset\om_C$ is a submodule of colength 2,
therefore of the form $\sF=\sI_{Z'}\om_C$ for some cluster $Z'\subset C$.
Tensoring down the iso\-morphism $s\colon\sI_Z\Oh_C(K_C)\to\sI_{Z'}\om_C$
gives an isomorphism $s\colon\sI_Z\iso\sI_{Z'}$, still linearly independent
of the identity inclusion $\sI_Z\into\Oh_C$.
Logically, there are 3 cases for $Z$ and $Z'$. The first of these
corresponds to an honest $g^1_2$ on $C$; the other two, corresponding to a
$g^1_2$ with one or two base points, lead either to $p_aC\le1$ or to a
contradiction. The case division is as follows:
\paragraph{Case $Z\cap Z'=\emptyset$} Then the isomorphism
$\sI_Z\iso\sI_{Z'}$ implies that both $Z$ and $Z'$ are Cartier
divisors, and the two linearly independent inclusions $\sI_Z\into\Oh_C$
define an honest $g^1_2$ on $C$. In more detail: $\Oh_C(Z)$ has 2 linearly
independent sections with no common zeroes, and no linear combination of these
vanishes on any component of $C$. Therefore $|Z|$ defines a finite 2-to-1
morphism $C\to\proj^1$.
\paragraph{Case $Z=Z'$} This case leads to an immediate contradiction.
Indeed, take any point $x\notin\Supp Z$; then some linear combination of the
two isomorphisms $s,\id\colon\sI_Z\to\sI_Z$ vanishes at $x$, and
therefore vanishes along any reduced component of $C$ containing $x$. But we
have just said that this is forbidden.
\paragraph{Case $Z\cap Z'=x$} Here the case assumption can be rewritten
$\sI_Z+\sI_{Z'}=m_x$. This case is substantial, and it really happens in
two examples:
\begin{enumerate}
\item if $C$ is an irreducible plane cubic with a node or cusp $P$, and
$Q,Q'\in C\setminus P$ then $m_Pm_Q\iso m_Pm_{Q'}$;
\item $\proj^1$ has an incomplete $g^1_2$ with a fixed point, of the form
$P+|Q|$.
\end{enumerate}
We prove that we are in one of these cases. In either example, the curve $C$
has an honest $g^1_2$ (not directly given by our sections $s,\id$), so the
theorem is correct.
\begin{CLA}\label{cla:mov_y} For any point $y\in C\setminus\{x\}$, there
exists a linear combination $s'=s+\la\id$ defining an isomorphism $\sI_Z\iso
m_xm_y$.
\end{CLA}
\begin{pfof}{Claim}
Since $\sI_Z,\sI_{Z'}\subset m_x$, we have two linearly independent maps
$s,\id\colon\sI_Z\into m_x$, and some linear combination $s'=s+\la\id$
vanishes at $y$. Also, no map $\sI_Z\to m_x$ vanishes along a component of
$C$. Thus $s'(\sI_Z)=m_xm_y$. \QED \end{pfof}
It follows from the claim that $m_xm_y\iso m_xm_{y'}$ for any two points
$y,y'\ne x$, so that $y,y'$ are nonsingular, and $C$ is reduced and
irreducible. Now let $\si\colon C_1\to C$ be the blow up of $m_x$. Then,
essentially by definition of the blow up, $m_x\Oh_{C_1}\iso\Oh_{C_1}(-E)$
where $E$ is a Cartier divisor on $C_1$. Then $m_{C_1,y}\iso m_{C_1,y'}$
for general points $y,y'\in C_1$, hence as usual $C_1\iso\proj^1$. If
$C_1\iso C$ there is nothing more to prove.
If $C_1\not\iso C$, the conductor ideal
$\sC=\sHom_{\Oh_C}(\si_*\Oh_{C_1},\Oh_C)$ of $\si_*\Oh_{C_1}$ in $\Oh_C$ is
$m_x$. Indeed, let $f\in k(C)$ be the rational function such that
multiplication by $f$ gives $m_xm_y\iso m_xm_{y'}$; then $f$ is an affine
parameter on $C_1=\proj^1$ outside $y$, so that all regular functions on $C_1$
are regular functions of $f$, and $fm_x=m_x$ implies
$\si_*(m_x\Oh_{C_1})=m_x\subset\Oh_C$. Now it is known that the only
Gorenstein curve singularity $x\in C$ with conductor ideal $m_x$ is a node or
cusp (see \cite{Se2}, Chap.~IV, \S11 or \cite{Re}, Theorem~3.2): indeed,
$m_x\subset\Oh_C\subset\si_*\Oh_{C_1}$, and the Gorenstein assumption $n=2\de$
gives $\length(\si_*\Oh_{C_1}/\Oh_C)=\length(\Oh_C/m_x)=1$. Therefore
$p_aC=1$.
For the final statement, if $p_aC=1$ then $1=h^0(\Oh_C)=h^1(\om_C)$ by
Theorem~\ref{th:free}, (a) and duality, hence $h^0(\om_C)=1$ by RR, so that
$H^0(\Oh_C(K_C))\to\Oh_Z$ is not onto for any cluster $Z\in C$ of degree
2. \QED \end{pf}
\begin{REM} If $C$ is a numerically $3$-connected Gorenstein curve with
$p_aC\ge2$, then Theorem~\ref{th:hh} says that $K_C$ is automatically ample,
and the usual dichotomy holds: either $K_C$ is very ample, or $C$ is honestly
hyperelliptic.
Now assume instead that the dualising sheaf $\om_C=\Oh_C(K_C)$ is ample and
generated by its $H^0$. Equivalently, that $|K_C|$ is a free linear system,
defining a finite morphism (the {\em canonical morphism})
$\fie=\fie_{K_C}\colon C\to\proj^{p_a-1}$. In \cite{Ca1}, Definition~3.9, $C$
was defined to be {\em hyperelliptic} if $\fie_{K_C}$ is not birational on
some component of $C$. Thus by Theorem~\ref{th:hh}, in the $3$-connected case,
hyperelliptic and honestly hyperelliptic coincide.
\end{REM}
\section{Canonical maps of surfaces of general type}\label{sec:pluri} We give
a slight refinement of a useful lemma due independently to J.~Alexander and
I.~Bauer.
\begin{LEM}[Alexander--Bauer]\label{lem:ab} Suppose that $H$ is a Cartier
divisor on an irreducible projective scheme $X$. Assume given effective Cartier
divisors $D_1,D_2$, $D_3$ such that
\begin{enumerate}
\renewcommand\labelenumi{(\roman{enumi})}
\item $H^0(\Oh_X(H))\to H^0(\Oh_{D_i}(H))$ is onto.
\item $H$ is very ample on every $\De\in|D_i|$ for $i=1,2,3$.
\end{enumerate}
Then $H$ is very ample on $X$ if either
\begin{enumerate}
\renewcommand\labelenumi{(\alph{enumi})}
\item $H\lineq D_1+D_2$ and $\dim|D_2|\ge1$, or
\item $H\lineq D_1+D_2+D_3$ and $\dim|D_i|\ge1$ for $i=1,2,3$.
\end{enumerate}
\end{LEM}
\begin{pf} (a) is proved in \cite{Ba1}, Claim~2.19 and \cite{Ra}, Lemma~3.1,
and also in \cite{C-F}, Prop.~5.1.
We prove (b). By Remark~\ref{rem:1}, we need to prove that if $x$ is any point
of $X$, and $y$ is either another point of $X$ or a tangent vector at $x$,
then $|H|$ separates $x$ from $y$. If some $\De_i\in|D_i|$ contains both $x$
and $y$, we are done by the assumptions (i) and (ii). In particular, since
$\dim|D_i|\ge1$, such a $\De_i$ exists if $x$ or $y$ belong to the base
locus of $|D_i|$.
Finally, if none of the above possibilities occurs, we can find $\De_1$
containing $x$ but not $y$, and $\De_2,\De_3$ containing neither $x$ nor $y$.
Then $\De_1+\De_2+\De_3$ separates $x$ from $y$. \QED\end{pf}
\begin{pfof}{Theorem~\ref{th:surf}} Let $\pi\colon S\to X$ be the natural
birational morphism from a minimal surface of general type $S$ to its
canonical model $X$; write $K_S$ and $K_X$ for the canonical divisors of $S$
and $X$. Then $\om_X$ is invertible and $\pi^*(\om_X)\iso\om_S$; in particular
$H^0(X,mK_X)\iso H^0(X,mK_S)$ and$K_X^2=K_S^2$.
\subparagraph{Step I} If $C\in|(m-2)K_X)|$, then $H^0(\Oh_X(mK_X))\to
H^0(\Oh_C(mK_X))$ is onto. This follows from our assumption
$H^1(\Oh_X(2K_X))=0$.
\subparagraph{Step II} If $C\in|(m-2)K_X|$, then $\Oh_C(mK_X)$ is very ample.
\begin{pf} By the curve embedding theorem Theorem~\ref{th:curve}, it is enough
to prove that $mK_XB\ge2p_aB+1$ for every subcurve $B\subset C$. Note that by
adjunction $K_C=(m-1){K_X}_{|C}$, so that we can write
$m{K_X}_{|B}={K_X}_{|B}+{K_C}_{|B}$. Since $K_X$ is ample,
$K_XB\ge1$, and therefore we need only prove that $K_XC\ge3$ and
\begin{equation}
\deg\Oh_B(K_C)-\deg\om_B\ge2\quad\text{for every strict subcurve $B\subset C$,}
\nonumber
\end{equation}
that is, that $C$ is numerically 2-connected.
The corresponding fact for the minimal nonsingular model $S\to X$ is easy and
well known.\footnote{{\bf Tutorial}\enspace This is an easy consequence of the
Hodge algebraic index theorem. If $D$ is nef and big and $D=A+B$ then
$A^2+AB\ge0$, $AB+B^2\ge0$. The index theorem says that $A^2B^2\le(AB)^2$, with
equality only if $A,B$ are numerically equivalent to rational multiples of one
another. The reader should carry out the easy exercise of seeing that $AB\le0$
gives a contradiction, and proving all the connected assertions we need. Or
see \cite{Bo}, \S4, Lemma~2 for details (the exceptional case $n=2$,
$2K_S=A+B$, with $A\numeq B\numeq K_S$ and $K_S^2=1$ is
excluded by the assumption $K_S^2\ge2$ if $m=4$ of Theorem~\ref{th:surf}).}
Therefore $C$ numerically 2-connected follows from the next result, whose
proof we relegate to an appendix.
\begin{LEM}\label{lem:n-conn} Let $X$ be a surface with only Du Val
singularities, and $\pi\colon S\to X$ the minimal resolution of singularities.
Let $C\subset X$ be an effective Cartier divisor, and $C^*=\pi^*C$ the total
transform of $C$ on $S$. Then
\begin{equation}
\text{$C^*$ numerically $k$-connected} \implies \text{so is $C$.}
\nonumber
\end{equation}
Moreover, if $C^*$ is numerically $2$-connected, and is only $3$-disconnected
by expressions $C^*=A+B$ where $A$ or $B$ is a $-2$-cycle exceptional for
$\pi$ then $C$ is numerically $3$-connected.
\end{LEM}\unskip\end{pf}
\subparagraph{Step III} $h^0((m-2)K_X)\ge3$ if $m\ge5$, and $\ge2$ if $m=3$ or
4.
\begin{pf} For $m=3$ this is just the assumption $p_g\ge2$.
For $m\ge4$, if $p_g\ge2$, then clearly $h^0((m-2)K_X)\ge3$. Otherwise, in
the case $p_g\le1$, we use the traditional numerical game of \cite{B-M},
based on Noether's formula $12\chi(\Oh_X)=(c_1^2+c_2)(X)$. It consists of
writing out Noether's formula using Betti numbers for the etale cohomology, in
the form
\begin{equation}
10+12p_g=8h^1(\Oh_X)+2\De+b_2+K_X^2.
\label{eq:No}
\end{equation}
Here the nonclassical term $\De=2h^1(\Oh_X)-b_1$ satisfies $\De\ge0$, and
$\De=0$ if $\chara k=0$. Since all the terms on the right hand side of
(\ref{eq:No}) are $\ge0$, it follows immediately that
\begin{equation}
\begin{aligned}
&p_g\le1\implies h^1(\Oh_X)\le2\\
&p_g\le0\implies h^1(\Oh_X)\le1.
\end{aligned}
\nonumber
\end{equation}
Therefore, $p_g\le1$ implies $\chi(\Oh_X)\ge0$; hence, for $m\ge4$, by
RR
\begin{equation}
h^0((m-2)K_X)\ge\chi(\Oh_X)+\binom{m-2}2K_X^2\quad
\begin{cases}
\ge3&\text{if $m\ge5$,}\\
\ge2&\text{if $m=4$.}
\end{cases}
\nonumber
\end{equation}
\end{pf}
\unskip
\subparagraph{Step IV} For $m=3$, we simply apply Lemma~\ref{lem:ab}, (b) to
$3K\lineq K+K+K$. For $m=4$ we apply Lemma~\ref{lem:ab}, (a) to $4K\lineq 2K+2K$:
the assumptions (i) and (ii) of the lemma hold by Steps~I, II and~III.
For $m\ge5$, we want to show that $H^0(\Oh_X(mK_X))\to\Oh_Z$ is onto for any
cluster $Z\subset X$ of degree 2. But by Step~III, there exists
$C\in|(m-2)K_X|$ containing $Z$. The result then follows by Steps~I and~II.
\QED \end{pfof}
\subsection*{Appendix: Proof of Lemma~\ref{lem:n-conn}}
Suppose that $B\subset C$ is a strict subcurve. Write $B'$ for the birational
(=strict or proper) transform of $B$ in $S$ and $C^*=\pi^*C$ for the total
transform of $C$. For the proof, we find a divisor $\Bh$ (the {\em hat
transform}) with the properties
\begin{enumerate}
\renewcommand\labelenumi{(\roman{enumi})}
\item $B'\le\Bh\le C^*$ and $\Bh-B'$ contains only exceptional curves;
\item $p_a\Bh=p_aB$.
\end{enumerate}
Suppose first that we know $\Bh$ satisfying these conditions. Then
\begin{equation}
(C^*-\Bh)\Bh\ge k
\nonumber
\end{equation}
by the assumption on $C^*$, which we write
\begin{equation}
(K_S+C^*)\Bh-(K_S+\Bh)\Bh\ge k.
\nonumber
\end{equation}
Here the first term equals $(K_X+C)B=\deg\Oh_B(K_C)$, and the second
$2p_a\Bh-2=2p_aB-2$. Thus
\begin{equation}
\deg\Oh_B(K_C)-(2p_aB-2)=(K_S+C^*)\Bh-(2p_a\Bh-2)\ge k.
\nonumber
\end{equation}
So it is enough to find $\Bh$. For this, following the methods of
\cite{Ar1}--\cite{Ar2}, let $\bigl\{\Ga_i\bigr\}$ be all the exceptional
$-2$-curves. Define $\Bh=B'+\sum e_i\Ga_i$ with $e_i\in\Z$, $e_i\ge0$ minimal
with respect to the property $\Bh\Ga_i\le0$; this exists, because $C^*-A'$ has
the stated property (where $A'$ is the birational transform of the residual
Weil divisor $C-B$).
\begin{CLA}\label{cla:bhat} The curve $\Bh$ has the following properties:
\begin{enumerate}
\renewcommand\labelenumi{(\roman{enumi})}
\setcounter{enumi}2
\item $\om_B=\pi_*\om_{\Bh}$;
\item $R^1\pi_*\om_{\Bh}=0$.
\end{enumerate}
Therefore $p_a\Bh=p_aB$.
\end{CLA}
\begin{pfof}{Claim} Taking $\pi_*$ of the short exact sequence
\begin{equation}
0\to\Oh_S(K_S)\to\Oh_S(K_S+\Bh)\to\om_{\Bh}\to0
\nonumber
\end{equation}
gives $0\to\Oh_X(K_X)\to\Oh_X(K_X+B)\to\pi_*\om_{\Bh}\to0$ and
$R^1\pi_*\Oh_S(K_S+\Bh)=R^1\pi_*\om_{\Bh}$. The first of these implies that
$\om_B=\pi_*\om_{\Bh}$. Indeed, if $B\subset X$ is an effective Weil divisor
on any Cohen--Macaulay variety then the adjunction formula
$\om_B=\sExt^1_{\Oh_X}(\Oh_B,\om_X)$ (see, for example, \cite{Re},
Theorem~2.12, (1)) boils down to an exact sequence
$0\to\Oh_X(K_X)\to\Oh_X(K_X+B)\to\om_B\to0$. This proves (iii).
By the method of \cite{Ar1}--\cite{Ar2},
\begin{equation}
R^1\pi_*\Oh_S(K_S+\Bh)=\varprojlim H^1(D,\Oh_D(K_S+\Bh)),
\nonumber
\end{equation}
where the inverse limit is taken over effective divisors $D=\sum a_j\Ga_j$. If
all the $H^1=0$, the limit is zero, as required.
Suppose then by contradiction that $D=\sum a_j\Ga_j$ has
$H^1(\Oh_D(K_S+\Bh))\ne0$. Then dually, $\Hom(\Oh_D(K_S+\Bh),\om_D)\ne0$, and
Lemma~\ref{lem:adj} gives an inclusion $\Oh_D(K_S+\Bh)\into\om_D$ (for a
possibly smaller $D$). Writing out the adjunction formula for $\om_D$ and
tensoring down by $K_S+\Bh$ gives $\Oh_D\into\Oh_D(D-\Bh)$. Therefore
$(\Bh-D)\Ga_i\le0$ for every $\Ga_i\subset D$, and by construction of $\Bh$
for the other $\Ga_i$. Now $\Bh-D=B'+\sum e'_j\Ga_j$ contradicts the
minimality of $\Bh$, provided we show that the $e_j'\ge0$. For this, note that
\begin{equation}
\bigl(\sum e'_j\Ga_j\bigr)\Ga_i=(\Bh-D)\Ga_i-B'\Ga_i\le0
\quad\text{for every $i$}
\nonumber
\end{equation}
and the intersection form on the $\Ga_i$ is negative definite, so that the
standard argument implies $\sum e'_j\Ga_j\ge0$ (write it as $A-B$ where
$A,B\ge0$ have no common divisor, and calculate $B^2$). \QED \end{pfof}
\section{The tricanonical map}\label{sec:tri}
We state the following three points as independent lemmas in order to tidy up
our proofs, and because they might be useful elsewhere. The first is a
particular case of the numerical criterion for flatness, see \cite{Ha},
Chap.~III, Theorem~9.9.
\begin{LEM}[Flat double covers]\label{lem:flat} If $\fie\colon X\to Y$ is a
generically $2$-to-$1$ morphism (say with $Y$ integral), then for any $y\in Y$,
the condition $\length\fie\1(y)=2$ implies that $\fie$ is flat over a
neighbourhood of $y$. \qed \end{LEM}
\begin{LEM}[Push-down of invariant linear systems]\label{lem:push} Let
$\fie\colon X\to Y$ be a finite morphism of degree $2$, where $X$ and $Y$ are
normal. Suppose that $L$ is a linear system of Cartier divisors on $X$ with
the property that $\fie_{|D}\colon D\to\Ga_D=\fie(D)$ has degree $2$ for every
$D\in L$. Then the $\Ga_D$ are linearly equivalent Weil divisors, that is, they
are all members of one linear system.
\end{LEM}
\begin{pf} For any $D,D'\in L$, note that $2\Ga_D=\pi_*D$ is a Cartier
divisor on $Y$, and $2\Ga_D\lineq 2\Ga_{D'}$, because if $D$ is locally
defined by $f\in k(X)$ (or $D-D'=\div f$) then $2\Ga_D$ is locally defined by
$\Norm(f)$, where $\Norm=\Norm_{k(X)/k(Y)}$.
Thus the Weil divisor class $\Ga_D-\Ga_{D'}$ is a 2-torsion element of the
Weil divisor class group $\WCl Y$ (modulo linear equivalence). The group of
Weil divisors numerically equivalent to zero is an algebraic group of finite
type, so that its 2-torsion subgroup is a finite algebraic group scheme $G$.
Now for fixed $D_0\in L$, taking $D\mapsto\Ga_D-\Ga_{D_0}$ defines a morphism
from the parameter space of the linear system $L$ to $G$, which must be the
constant morphism to 0. This proves what we need.
Assuming that $\fie$ is separable make this argument more intuitive, since
then it is Galois, and $\fie_*\Oh_X$ splits into invariant and antiinvariant
parts: $\fie_*\Oh_X=\Oh_Y\oplus\sL$, with $\sL$ a divisorial sheaf. Then
$\Ga_D$ is locally either a Cartier divisor or in the local Weil divisor class
of $\sL$, and $\Ga_D-\Ga_{D'}$ is in the kernel of $\fie^*$, which is a finite
algebraic group scheme, etc. \QED \end{pf}
\begin{LEM}\label{lem:sing} Let $\La$ be a linear system of Weil divisors
through a point $P$ on a normal surface $Y$. Then the curves in $\La$ singular
at $P$ form a projective linear subspace of codimension $\le2$.
\end{LEM}
\begin{pf} Easy exercise involving the resolution and birational transform.
\qed \end{pf}
\begin{pfof}{Theorem~\ref{th:tri}, Case~(a)} Since $q=0$, we have
$\chi(\Oh_X)\ge1$, and $K_X^2\ge3$ gives $P_2=h^0(2K_X)\ge4$. Let $Z$ be a
cluster of degree 2 on $X$. Since $P_2\ge4$, the linear subsystem
$|2K_X-Z|$ consisting of curves $D\in|2K_X|$ through $Z$ has dimension
$\ge1$, and any $D\in|2K_X|$ is 3-connected by the final part of
Lemma~\ref{lem:n-conn} (whose assumptions are easily verified as in \cite{Bo},
\S4, Lemma~2). By $H^1(K_X)=0$, the sequence
\begin{equation}
0\to H^0(X,\Oh_X(K_X))\to H^0(X,\Oh_X(3K_X))\to H^0(D,\om_D)\to0
\nonumber
\end{equation}
is exact. Since $|\om_D|$ is free by Theorem~\ref{th:free}, it follows that
$\fie=\fie_{3K_X}$ is a finite morphism $\fie\colon X\to Y\subset\proj^N$,
where $N=P_3-1$. Assume that $|3K_X|$ does not separate $Z$. Then, by
Theorem~\ref{th:hh}, $D$ is honestly hyperelliptic. Since the same argument
applies to any $D\in|2K_X-Z|$, it follows that $\deg\fie\ge2$.
On the other hand, for any point $y\in Y$, if the scheme theoretic fibre
$\fie\1(y)$ is a cluster of degree $\ge3$, then there is a curve $D'\in|2K_X|$
containing $\fie\1(y)$, and $\fie\1(y)$ is contained in a fibre of
$\fie_{\om_{D'}}\colon D'\to\proj^1$, which contradicts Lemma~\ref{lem:hh}.
Hence $\fie\colon X\to Y$ is of degree 2 (possibly inseparable if $\chara
k=2$). In particular $2\mid9K^2$, so that $K^2$ is even and $K^2\ge4$; thus
$P_2\ge5$, and $\dim|2K_X-Z|\ge2$ for any cluster $Z$ of degree 2. By
changing $Z$ if necessary, we can assume that $\fie(Z)=y\in Y$ is a general
point, and is thus nonsingular. We have just shown that every fibre
$\fie\1(y)$ has degree exactly 2, so that $\fie$ is flat by
Lemma~\ref{lem:flat}; it is easy to see that this implies that $Y$ is normal.
Now for any $D\in|2K_X-Z|$, the image $\fie(D)=\Ga_D\subset Y$ is a curve
through $y=\fie(Z)$ isomorphic to $\proj^1$, and $\deg\fie_{|D}=\deg\fie=2$.
By Lemma~\ref{lem:push} the $\Ga_D\subset Y$ are linearly equivalent, so that
they are all contained in a linear system. This contradicts
Lemma~\ref{lem:sing}: in any linear system of curves through $y$, curves
singular at $y$ form a linear subsystem of codimension $\le2$, whereas the
$\Ga_D$ for $D\in|2K_X-Z|$ form an algebraic subfamily of nonsingular curves
depending with a complete parameter space of dimension $\ge2$ made up of
curves isomorphic to $\proj^1$. \QED \end{pfof}
\begin{REM} Here we have assumed that $\fie(Z)=y\in Y$ is a general
point only for simplicity (see Lemma~\ref{lem:sing}).
\end{REM}
\begin{pfof}{Theorem~\ref{th:tri}, Case~(b)} Let $Z$ be a cluster of degree
2 on $X$ and $x\in Z$ a reduced point; that is, $Z$ is either a pair
$(x,y)$ of distinct points, or a point $x$ plus a tangent vector $y$ at $x$.
We assume that $|3K_X|$ does not separate $Z$, and gather together a number
of deductions concerning the curves
\begin{equation}
C_L\in|K_X+L|\quad\text{and}\quad
D_L\in|2K_X-L|\quad\text{for all}\quad L\in\Pico X,
\nonumber
\end{equation}
arriving eventually at a contradiction.
\subparagraph{Step A} $h^0(K_X+L)\ge1$ for all $L\in\Pico X$. In fact if
$L\ne0$ then $h^2(K_X+L)=0$, and hence $h^0(K_X+L)\ge\chi(K_X)\ge1$.
\subparagraph{Step B} $Z\not\subset C_L$ for all $L\in\Pico X$ and all
$C_L\in|K_X+L|$. Indeed
\begin{equation}
H^0(X,\Oh_X(3K))\to H^0(C_L,\Oh_{C_L}(3K_X))
\nonumber
\end{equation}
is onto by the assumption $H^1(\Oh_X(2K_X-L))=0$, and $\Oh_{C_L}(3K_X)$ very
ample follows from Theorem~\ref{th:curve} exactly as in \S\ref{sec:pluri},
Step~II. Therefore if $Z\subset C_L$ then $|3K_X|$ separates $Z$, which we
are assuming is not the case.
\subparagraph{Step C} For general $L\in\Pico X$ and all $C_L\in|K_X+L|$ we
have $x\in C_L$.
First of all, since $\dim\Pico X\ge1$, there is an $L\in\Pico X$ and a curve
$C_L\in|K_X+L|$ containing $x$, and $C_L$ does not contain $Z$ by Step~B.
Now if $L_1,L_2\in\Pico X$ is a general solution of $L+L_1+L_2=0$, and $x\notin
C_{L_1}$, $x\notin C_{L_2}$, then $C_L+C_{L_1}+C_{L_2}$ separates $x$ and
$Z$, a contradiction.
\subparagraph{Step D} $h^0(K_X+L)=1$ and $H^1(K_X+L)=0$ for general $L\in\Pico
X$.
By Step~C, every $s\in H^0(K_X+L)$ vanishes at $x$. If $h^0(K_X+L)\ge2$ then
some nonzero section would vanish also at $y$. The statement about $H^1$
follows from RR:
\begin{equation}
1=h^0(\Oh_X(K_X+L))\ge\chi(\Oh_X(K_X+L))=\chi(\Oh_X)\ge1.
\nonumber
\end{equation}
\subparagraph{Step E} $x\in\Bs|2K_X-L|$ for general $L\in\Pico X$. For if
$D_L\in|2K_X-L|$ does not contain $x$ then $D_L+C_L$ separates $x$ from
$Z$ (since by Step~B already $C_L$ separates them).
\subparagraph{Step F} For general $L,L_1\in\Pico X$, the point $x$ is a base
point of the linear system $\bigl|(2K_X-L_1)_{\textstyle{|C_L}}\bigr|$ on
$C_L$, and hence
\begin{equation}
H^1(m_x\Oh_{C_L}(2K_X-L_1))\ne0.
\nonumber
\end{equation}
This follows from $x\in\Bs|2K_X-L_1|$ because by Step~D, restriction from $X$
maps onto $H^0(\Oh_{C_L}(2K_X-L_1))$.
\subparagraph{Step G} We now observe that Step~B implies that $x$ is a
singular point of $C_L$. If $x\in\Sing X$ then it is automatically singular on
$C_L$. On the other hand, if $x$ is nonsingular on $X$ and on $C_L$, consider
the blowup $\si\colon X_1\to X$ of $x$ and the algebraic system
$C'_L=\si^*C_L-E$, where $E$ is the exceptional divisor. Let $y\in X_1$ be the
point corresponding either to the other point or to the tangent vector of the
cluster $Z$. Since the curves $C'_L$ move in a positive dimensional system,
there is a curve $C'_L$ through $y$, and therefore a curve $C_L$ containing
$Z$, contradicting Step~B.
\subparagraph{Step H} For general elements $L,L_2\in\Pico X$, there is an
isomorphism $m_x\Oh_{C_L}(L_2)\iso m_x$.
This follows as usual by automatic adjunction (Lemma~\ref{lem:adj}) applied to
the conclusion $H^1(m_x\Oh_{C_L}(2K_X-L_1))\ne0$ of Step~F, where
$L_1=-L-L_2$. We first get a nonzero homomorphism
\begin{equation}
m_x\Oh_{C_L}(2K_X-L_1)\to\om_{C_L}=\Oh_{C_L}(2K_X+L),
\nonumber
\end{equation}
that is, a map $m_x\Oh_{C_L}(L_2)\to\Oh_{C_L}$; since $C_L$ is 2-connected
this must be an inclusion, and the image is the ideal of a point $m_z$. But
$x$ is a singular point of $C_L$ (by Step~G), and thus $x=z$.
\subparagraph{Step I} Let $\si\colon C'\to C=C_L$ be the blowup at $x$.
Step~H implies that $L_2'=\si^*L_2$ is trivial on $C'$ for every general $L_2$,
and hence for every $L_2\in\Pico X$ (by the group law). We derive a
contradiction from this. Consider the diagram
\begin{equation}
\renewcommand\arraystretch{1.5}
\begin{array}{rcl}
\Pico X @>\res_C>> & \Pico C & @>\si^*>> \Pico(C')\\
& \uparrow \\
& G
\end{array}
\nonumber
\end{equation}
where $G$ is the kernel of $\si^*$. Now the key point (exactly as in Ramanujam
and Francia vanishing) is that $G$ is an affine group scheme. Since the
composite $\si^*\circ\res_C$ is zero, $\Pico X$ maps to $G$. Since $\Pico
X$ is complete $\res_C$ is the constant morphism to zero. But this is
obviously nonsense: for example, since $H^1(\Oh_X(2K_X+L))=0$ for all
$L\in\Pico X$, the exact sequence
\begin{equation}
0\to\Oh_X(-K_X-L+N)\to\Oh_X(N)\to\Oh_C(N)\to0
\nonumber
\end{equation}
is exact on global sections if $L\ne N$. Thus $H^0(\Oh_C(N))=0$ and
the restriction of $N$ to $C$ is nontrivial. \QED \end{pfof}
\section{The bicanonical map}\label{sec:bi}
\subsection*{Preliminaries and the proof of Theorem~\ref{th:bi}, (a) and (c)}
This section proves Theorem~\ref{th:bi}. We start by remarking that $|2K_X|$ is
free. Indeed, for any $C\in|K_X|$, the restriction $\Oh_X(2K_X)\to\Oh_C(K_C)$
is surjective on $H^0$, and $|K_C|$ is free by Theorem~\ref{th:free}. For a
cluster $Z$ of degree 2 in $X$, note the following obvious facts:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item If $Z$ is contracted by $|2K_X|$ then $|K_X|$ does not separate
$Z$; thus
\begin{equation}
h^0(\sI_Z\Oh_X(K_X))\ge p_g-1\quad\text{or}\quad \dim|K_X-Z|\ge p_g-2.
\nonumber
\end{equation}
\item If $|2K_X|$ contracts $Z$ then so does $|K_C|$ for any curve
$C\in|K_X-Z|$.
\end{enumerate}
\begin{pfof}{Theorem~\ref{th:bi}, (a)} We suppose that every curve
$C\in|K_X-Z|$ is \hbox{3-connected}, and derive a contradiction from the
assumption that $|2K_X|$ contracts $Z$. By Theorem~\ref{th:hh}, every
$C\in|K_X-Z|$ is honestly hyperelliptic. As in the proof of
Theorem~\ref{th:tri}, Case~(a), it follows that $\fie_{2K}\colon X\to Y$ has
degree 2, and maps every $C\in|K_X-Z|$ as a double cover of a curve
$\Ga_C\subset Y$ isomorphic to $\proj^1$. Then $\Ga_C$ for $C\in|K_X-Z|$
form an algebraic subfamily of a linear system of curves through
$y=\fie_{2K}(Z)$, with a complete parameter space of dimension $\ge2$. As
before, this contradicts Lemma~\ref{lem:sing} (but $y\in Y$ may now be
singular). \QED
\end{pfof}
\begin{DEF}\label{def:Frc} Let $X$ be a projective surface with at worst Du
Val singularities and with $K_X$ nef. A {\em Francia curve} or {\em Francia
cycle} is an effective Weil divisor $B$ on $X$ satisfying
\begin{equation}
K_XB=p_aB=1\text{ or }2.
\nonumber
\end{equation}
If $K_X$ is ample and $B$ is Gorenstein (for example if $B$ is a Cartier
divisor), it is clearly either an irreducible curve of genus 1, or a
numerically 2-connected curve of arithmetic genus $p_a=2$. It would be
interesting to know if $B$ is necessarily Gorenstein.
\end{DEF}
\begin{pfof}{Theorem~\ref{th:bi}, (b) $\implies$ (c)} The argument is standard
and we omit some details. Suppose that the 2-canonical map
$\fie=\fie_{2K}\colon X\to Y$ is not birational. Every point $x\in X$ is
contained in a cluster $Z$ of degree 2 contracted by $\fie$; we choose
$x\in\NonSing X$. Theorem~\ref{th:bi}, (b) gives a Francia curve $B_0\subset X$
through $Z$. Write $S\to X$ for the minimal nonsingular model of $X$ and
$B=\Bh_0$ for the hat transform of $B_0$ (as in the proof of
Lemma~\ref{lem:n-conn}). Then by Claim~\ref{cla:bhat}, $B$ is also a Francia
cycle on $S$, that is, $1\le K_SB=p_aB\le2$. An easy argument in quadratic
forms shows that there are at most finitely many effective divisors $B\subset
S$ with $K_SB=1$ and $B^2=-1$ (compare \cite{Bo}, pp.~191--192 or \cite{BPV},
p.~224). Therefore every general point of $S$ is contained in a curve $B$ with
$K_SB=p_aB=2$, and hence $B^2=0$. Now the same argument in quadratic forms
shows that divisors with $K_SB=2$ and $B^2=0$ belong to finitely many
numerical equivalence classes, so one class must contain an algebraic family
of curves. This gives a genus 2 pencil on $S$, and therefore also on $X$. \QED
\end{pfof}
We use the following obvious lemma at several points in what follows.
\begin{LEM}[Dimension lemma]\label{lem:dim} Let $\eta\subset X$ be a cluster
of degree $d$ which is contracted by $|2K_X|$, and $C\in|K_X|$ a curve
containing $\eta$. Then
\begin{equation}
h^1(\sI_\eta\Oh_C(K_C))=\dim\Hom(\sI_\eta,\Oh_C)=d.
\nonumber
\end{equation}
In particular, for any $x\in C$, we have
\begin{equation}
h^1(m_x^2\Oh_C(K_C))=\dim\Hom(m_x^2\Oh_C,\Oh_C)=1+\dim T_{\fie,x}\le4,
\nonumber
\end{equation}
where $T_{\fie,x}$ is the Zariski tangent space to the scheme theoretic fibre
of $\fie_{2K_X}$ through $x$. \end{LEM}
\begin{pf} Since $|K_C|$ is free and contracts $\eta$, the evaluation map
$H^0(\Oh_C(K_C))\to\Oh_\eta(K_C)=k^d$ has rank 1, so that
$h^1(\sI_\eta\Oh_C(K_C))=d$ comes from the exact sequence
\begin{equation}
\renewcommand\arraystretch{1.3}
\begin{array}{l}
0\to H^0(\sI_\eta\Oh_C(K_C))\to H^0(\Oh_C(K_C))\to k^d \\
\hphantom{0}\to H^1(\sI_\eta\Oh_C(K_C))\to H^1(\Oh_C(K_C))=k.
\end{array}
\nonumber
\end{equation}
As usual, Serre duality gives
\begin{equation}
\Hom(\sI_\eta,\Oh_C)=\Hom(\sI_\eta\Oh_C(K_C),\om_C)\dual H^1(\sI_\eta\Oh_C(K_C)).
\nonumber
\end{equation}
We obtain the last part by taking $\eta$ to be the intersection of the scheme
theoretic fibre $\fie\1(\fie(x))$ with the subscheme $V(m_x^2)\subset C$
corresponding to the tangent space. \QED \end{pf}
\subsection*{Case division and plan of proof of~(b)}
Throughout this section, $Z$ is a cluster of degree 2, and we argue by
restricting to a curve $C\in|K_X-Z|$, usually imposing singularities on $C$ at
a point $x\in Z$. As usual, the assumption that $Z$ is contracted by $K_C$
gives a homomorphism $\sI_Z\to\Oh_C$ linearly independent of the identity
inclusion. By passing to a suitable linear combination $s'=s+\la\id$ if
necessary, we assume that $s\in\Hom(\sI_Z,\Oh_C)$ is injective, and hence
$s(\sI_Z)=\sI_{Z'}$ for some cluster $Z'$ of degree 2; the family of clusters
$Z'$ as $s$ runs through injective elements $s\in\Hom(\sI_Z,\Oh_C)$ is an
analog of a $g^1_2$ on $C$.
The argument is modelled on the proof of Theorem~\ref{th:hh}. As there, we use
different arguments depending on how $Z$ and $Z'$ intersect, or, to put it
another way, how $Z'$ moves as $s$ runs through injective elements
$s\in\Hom(\sI_Z,\Oh_C)$. (In other words, how the $g^1_2$ corresponding to
$\Hom(\sI_Z,\Oh_C)$ breaks up into a ``base locus'' plus a ``moving part''.)
Let $s\in\Hom(\sI_Z,\Oh_C)$ be a general element, and $\sI_{Z'}=s(\sI_Z)$.
Logically, there are 4 cases for $Z$ and $Z'$.
\begin{enumerate}
\item $\Supp Z\cap\Supp Z'=\emptyset$.
\item $\Supp Z\cap\Supp Z'\ne\emptyset$, but $\Supp Z\ne\Supp Z'$.
\item $Z=Z'$.
\item $Z\ne Z'$ are nonreduced clusters supported at the same point $x\in
X$.
\end{enumerate}
In Case~2, $|Z|$ has a fixed point plus a moving point; as we see in
Lemma~\ref{lem:case2}, this contradicts $K_X$ ample. In Case~1, $|Z|$ is a
free $g^1_2$, and the isomorphism $\sI_Z\iso\sI_{Z'}$ with $\Supp Z\cap\Supp
Z'=\emptyset$ implies that $\sI_Z$ is locally free, so that $Z$ is a Cartier
divisor on $C$. If $p_g\ge4$, it turns out that we can choose $C$ to be
``sufficiently singular'' at a point $x\in Z$ so that $Z\subset C$ is not
Cartier, and Case~1 is excluded for such $C$ (see Lemma~\ref{lem:notCt}).
In Cases~3--4, when the support of $Z$ does not move, we must find a map
$s'\colon\sI_Z\to\Oh_C$ vanishing on a ``fairly large'' portion of $C$, so that
its scheme theoretic support $B\subset C$ is ``fairly small''. The key idea is
to look for $s'$ as a nilpotent or idempotent (see Lemma~\ref{lem:pot} and
Corollary~\ref{cor:Art}). The assumption of Case~3 is
$\Hom(\sI_Z,\Oh_C)=\End(\sI_Z)$, which is a 2-dimensional Artinian algebra;
this makes it is rather easy to find a nilpotent or idempotent element, and to
prove Theorem~\ref{th:bi}, (b).
In Case~4, $Z'$ is $x$ plus a tangent vector $y$ which moves in $T_{C,x}$ as
$s\in\Hom(\sI_Z,\Oh_C)$ runs through injective elements; this is an {\em
infinitesimal} $g^1_2$, an interesting geometric phenomenon in its own right
(see Remark~\ref{rem:g23} and the proof of Proposition~\ref{pro:m2}, Step~6 for
more details). The key point in this case is to prove that the extra
homomorphism $s\colon\sI_Z\to\Oh_C$ takes $m_x^2$ to itself, so that
$\End(m_x^2)$ is a nontrivial Artinian algebra; see Proposition~\ref{pro:m2}.
\begin{REM}\label{rem:g23} In Case~4, reversing the usual argument proves
that $\fie_{K_C}$ also contracts $Z'$, and so it contracts a cluster $\eta$ of
degree $\ge3$ contained in the first order tangent scheme $V(m_x^2)\subset C$.
If $C$ is numerically 3-connected, this is of course impossible by
Theorem~\ref{th:hh}. In this case, $\Hom(\sI_\eta,\Oh_C)$ is a certain analog
of a $g^2_3$ or $g^3_4$ on $C$.
Case~4 certainly happens on abstract numerically 2-connected Gorenstein
curves, and more generally, the analog of a $g^{m-1}_m$. Example: let $C_i$ for
$i=1,\dots,m$ be nonhyperelliptic curves of genus $g_i\ge3$ with marked points
$x_i\in C_i$, and assemble the $C_i$ into a curve $C=\bigcup C_i$ by glueing
together all the $x_i$ to one point $x$, at which the tangent directions are
subject to a single nondegenerate linear relation, so that the singularity
$x\in C$ is analytically equivalent to the cone over a frame of reference
$\{P_1,P_2,\dots,P_m\}$ in $\proj^{m-2}$. Then $C$ is Gorenstein and
$K_C$ restricted to each $C_i$ is $K_{C_i}+2x_i$ (see \cite{Ca1},
Proposition~1.18, (b), p.~64, or \cite{Re}, Theorem~3.7), so that $|K_C|$
contracts the whole $(m-1)$-dimensional tangent space $T_{C,x}$ to a point.
A cluster $Z$ of degree 2 supported at $x$ corresponds to a point
$Q\in\proj^{m-2}=\proj(T_{C,x})$. Since $Z$ is contracted by $K_C$ (together
with the whole tangent space), by our usual argument, the group
$\Hom(\sI_Z,\Oh_C)$ is 2-dimensional and a general $s\colon\sI_Z\to\Oh_C$
has image $\sI_{Z'}$ where $Z'$ is a moving cluster of degree 2 at $x$,
corresponding to a moving point $Q'\in\proj^{m-2}$. It is an amusing exercise
to see that if $Q$ is linearly in general position with respect to the frame
of reference $\{P_1,P_2,\dots,P_m\}$ then $Q'$ moves around the unique
rational normal curve of degree $m-2$ passing through
$\{P_1,P_2,\dots,P_m,Q\}$. On the other hand, if $Z$ is in the tangent cone
to $C$ (say, tangent to the branch $C_1$), then $\sI_Z$ is not isomorphic to
any other cluster of degree 2, so that $\Hom(\sI_Z,\Oh_C)=\End(\sI_Z)$;
this has 2 idempotents vanishing on $C_1$ and on $C_2+\cdots+C_m$.
The following easy exercises may help to clarify things for the reader:
\begin{enumerate}
\item Let $x\in C$ be an ordinary triple point of a plane curve, say defined
by an equation $f(u,v)=u^3+v^3+$ higher order terms; then for general $\la$,
the ideals $(u+\la v,v^2)$ in $\Oh_{C,x}$ are all locally isomorphic. [Hint:
Multiply by the rational function $(u+\mu v)/(u+\la v)$.]
\item If $C$ is the planar curve defined by $vw=v^3+w^3$ then $m_x=(v,w)$ is
locally isomorphic to $\sI_Z=(v,w^2)$ and to $\sI_{Z'}=(v^2,w)$.
\item If $C$ is the planar curve locally defined by $v^2=w^3$ then $m_x=(v,w)$
is locally isomorphic to $\sI_Z=(v,w^2)$.
\end{enumerate}
(Compare the proof of Proposition~\ref{pro:m2}, Step~6.)
\end{REM}
\begin{LEM}\label{lem:case2} Case~2 is impossible. \end{LEM}
\begin{pf} Since $x\in Z\cap Z'$ and $\Supp Z\ne\Supp Z'$, we can interchange
$Z$ and $Z'$ if necessary and assume that $Z'=\{x,y\}$ with $x\ne y$. Consider
the inclusion $s\colon\sI_Z\into\Oh_C$ with image $s(\sI_Z)=\sI_{Z'}=m_xm_y$
and the identity inclusion. One of these vanishes at $y$ and the other
doesn't, so their restrictions to a component $\Ga$ containing $y$ are linearly
independent on $\Ga$, and, as in Claim~\ref{cla:mov_y}, for any general point
$y'\in\Ga$, some linear combination $s'=s+\la\id$ defines an isomorphism
$s'\colon\sI_Z\iso m_xm_{y'}$. Reversing our usual argument shows that $x$ and $y'$ are
contracted to the same point by $|K_C|$ or $|2K_X|$, so that the free linear
system $|2K_X|$ contracts $\Ga$ to a point. This contradicts $K_X$ ample. \QED
\end{pf}
\subsection*{Clusters on singular curves}
Our immediate aim is to exclude Case~1, but at the same time we introduce some
ideas and notation used throughout the rest of this section. Choose a point
$x\in Z$. Since $X$ has at worst hypersurface singularities and $C$ is a
Cartier divisor in $X$, it is a local complete intersection, that is, locally
defined by $F=G=0$. (Of course, $X$ may be nonsingular.) We think of $x\in
Z\subset C\subset X\subset\aff^3$ as local, and write $\Oh_{\aff^3}$, $\Oh_C$,
etc.\ for the local rings at $x$. We take local coordinates $u,v,w$ in $\aff^3$
so that $Z$ is defined by $u=v=w=0$ in the reduced case, or $u=v=w^2=0$
otherwise.
\begin{LEM}\label{lem:notCt}
\begin{enumerate}
\renewcommand{\labelenumi}{{\rm(\arabic{enumi})}}
\item The quotient $\sI_{\aff^3,Z}/m_{\aff^3,x}\sI_{\aff^3,Z}$ is a
$3$-dimensional vector space, and $Z\subset C$ is a Cartier divisor at $x$ if
and only if $F,G$ map to linearly independent elements of it.
\item Suppose that $p_g\ge4$ and $Z$ is contracted by $|2K_X|$. Then the curve
$C\in|K_X-Z|$ can be chosen such that $Z$ is not a Cartier divisor. For this
$C$, Case~1 is excluded.
\end{enumerate}
\end{LEM}
\begin{pf} (1) says that a minimal set of generators of the ideal
$\sI_{\aff^3,Z}$ consists of 3 elements, which is obvious because
$\sI_{\aff^3,Z}$ is locally generated at $x\in Z$ by the regular sequence
$(u,v,w)$ or $(u,v,w^2)$. Now $Z$ is a Cartier divisor on $C$ if and only if
$\sI_{C,Z}$ is generated by 1 element, that is, $F$ and $G$ provide two of the
minimal generators of $\sI_{\aff^3,Z}$. This proves (1).
For (2), suppose that $F=0$ is the local equation of $X\subset\aff^3$. If $F\in
m_{\aff^3,x}\sI_{\aff^3,Z}$ then by (1), $Z$ is not a Cartier divisor on any
curve $C\in|K_X-Z|$. Suppose then that $F\notin m_{\aff^3,x}\sI_{\aff^3,Z}$, so
that $F$ provides one of the minimal generators of $\sI_{\aff^3,Z}$. Then the
ideal $\sI_{X,Z}$ of $Z\subset X$ is generated by 2 elements, in other words,
$\dim_k\sI_{X,Z}/m_{X,x}\sI_{X,Z}=2$. Therefore
\begin{equation}
h^0(m_x\sI_Z\Oh_X(K_X))\ge h^0(\sI_Z\Oh_X(K_X))-2\ge p_g-3\ge1
\nonumber
\end{equation}
(by remark (i) at the beginning of this section). Thus we can find a curve
$C\in|K_X-Z|$ whose local equation at $x$ is $g\in m_{X,x}\sI_{X,Z}$. Then
$g$ has a local lift $G\in m_{\aff^3,x}\sI_{\aff^3,Z}$, so that (1) applies to
$C$. \QED \end{pf}
\begin{REM}\label{rem:geom} The same argument can be expressed more
geometrically. If $Z$ contains $x$ as a reduced point, that is,
$\sI_{\aff^3,Z}=m_x$, then $x\in C$ is Cartier if and only if $C$ defined by
$(F,G)$ is nonsingular at $x$, that is, $F,G$ map to linearly independent
elements of $m_x/m_x^2$.
To interpret the nonreduced case $\sI_{\aff^3,Z}=(u,v,w^2)$, note that
\begin{equation}
F\notin m_{\aff^3,x}\sI_{\aff^3,Z} \iff F=Pu+Qv+Rw^2
\quad\text{with one of $P,Q,R\notin m_x$.}
\nonumber
\end{equation}
In other words, the surface $Y$ locally defined by $F=0$ is either nonsingular
at $x$, or has a double point with $Z$ not in the tangent cone. In the opposite
case $F\in m_{\aff^3,x}\sI_{\aff^3,Z}$, it is easy to see that $x\in C$ is
either a complete intersection defined by two singular hypersurfaces, so has
3-dimensional tangent space $T_{C,x}$, or is a planar curve, which is either a
double point with $Z$ in the tangent cone, or a point of multiplicity $\ge3$.
\end{REM}
\subsection*{The nilpotent--idempotent lemma}
Our proof of Theorem~\ref{th:bi}, (b) in Cases~3--4 is based on the following
result. Note first that $\Hom(\sI_Z,\Oh_C)\subset H^0(C\setminus\Supp
Z,\Oh_C)$, and the latter is a ring. (We usually write $\sI_Z$ for
$\sI_{C,Z}$ in what follows.) In other words, maps $\sI_Z\to\Oh_C$ can be
viewed as rational sections of $\Oh_C$ that are regular outside $\Supp Z$, so
that it is meaningful to multiply them (the product is again a rational
section of $\Oh_C$ that is regular away from $Z$).
\begin{LEM}\label{lem:pot} Assume that $K_X^2\ge10$, and let $C\in|K_X-Z|$.
Suppose that $s\colon\sI_Z\to\Oh_C$ is a nonzero homomorphism which is either
nilpotent with $s^4=0$, or a nontrivial idempotent with $s(1-s)=0$. Then the
scheme theoretic support of $s$ (respectively, in the idempotent case, either
$s$ or $1-s$) is a Francia curve $B$, and $\sI_Z\Oh_B(2K_X)\iso\om_B$.
More generally, suppose that $s_i\colon\sI_Z\to\Oh_C$ for $i=1,\dots,4$ are
nonzero homomorphisms such that $s_1s_2s_3s_4=0$. Then one of the $s_i$ has
scheme theoretic support a Francia curve $B_i$ with
$\sI_Z\Oh_{B_i}(2K_X)\iso\om_{B_i}$.
\end{LEM}
The final part is more general, because we allow some $s_i=\id$, or some of
the $s_i$ to coincide. Notice that $\Oh_C$ has no sections supported at
finitely many points, so we need only check the conditions $s^4=0$ etc.\
in each generic stalk of $\Oh_C$, that is, as rational functions on $C$.
\begin{pf} If $s\colon\sI_Z\Oh_C(K_C)\to\om_C$ is not generically injective,
the factorisation provided by automatic adjunction (Lemma~\ref{lem:adj}) gives
a subcurve $B\subset C$ satisfying $\sI_Z\Oh_B(K_C)\iso\om_B$; we are in the
limiting case of numerically 2-connected. Write $C=A+B$ for the decomposition
of Weil divisors, so that $A$ is the divisor of zeros of $s$. Passing to the
minimal nonsingular model $S$ and taking the hat transform $\Bh$ as in
Lemma~\ref{lem:n-conn} and Claim~\ref{cla:bhat} gives a decomposition
$K_S\lineq f^*C=A_1+\Bh$ such that $A_1\Bh=2$.
Therefore by the Hodge algebraic index theorem, $A_1^2\Bh^2\le(A_1\Bh)^2=4$. If
both $A_1^2$, $\Bh^2\ge1$, it follows that $K_S^2\le9$, a contradiction, so
that either $A_1^2\le0$ or $\Bh^2\le0$. Then (because $K_S=A_1+\Bh$ and
$A_1\Bh=2$), either $K_XA=K_SA_1\le2$ or $K_XB=K_S\Bh\le2$. Suppose for the
moment that $K_S\Bh\le2$. Since $2p_a\Bh-2=\Bh^2+K_S\Bh$, it follows at once
that we are in one of the two cases
\begin{equation}
\Bh^2=-1,K_S\Bh=1,p_a\Bh=1\quad\text{or}\quad
\Bh^2=0,K_S\Bh=2,p_a\Bh=2.
\nonumber
\end{equation}
But by Lemma~\ref{lem:n-conn} and Claim~\ref{cla:bhat} we have $K_S\Bh=K_XB$
and $p_a\Bh=p_aB$, so that $B$ is the required Francia curve.
It remains to get rid of the possibility that $K_XA=K_SA_1\le2$ in the
different cases. If $s$ is a nontrivial idempotent, we can swap $A\bij B$ by
$s\bij1-s$ if necessary, so that $K_XB\le2$. In the nilpotent case, since
$A$ equals the Weil divisor of zeros of $s$ and $s^4=0$, it follows that
$C\le4A$. Then $K_XA\le2$ would imply $K_X^2\le8$, a contradiction.
The last part is exactly the same: each $s_i$ (for $i=1,2,3,4$) is either
injective or has scheme theoretic support a subcurve $B_i\subset C$ with
$\sI_Z\Oh_{B_i}(K_C)\iso\om_{B_i}$, and divisor of zeros $A_i=C-B_i$. Since
$\prod s_i=0$ it follows that $C\le\sum A_i$. Now arguing as above gives that
one of $K_XA_i$ or $K_XB_i\le2$; if the first alternative holds for all $i$
then $K_X^2=K_XC\le \sum K_XA_i\le8$, a contradiction. This proves the lemma.
\QED \end{pf}
We apply Lemma~\ref{lem:pot} via a simple algebraic trick.
\begin{COR}\label{cor:Art} If $A=\End_{\Oh_C}(\sI_{C,Z})$ is an Artinian
algebra of length $\ge2$ then it has a nontrivial idempotent or a nonzero
nilpotent with $s^2=0$. More generally, if\/ $\Hom(\sI_{C,Z},\Oh_C)$ is a
$2$-dimensional vector space contained in an Artinian algebra $A\subset
H^0(C\setminus\Supp Z,\Oh_C)$ of dimension $\le4$ then there exist nonzero
elements $s_1,\dots,s_4\in\Hom(\sI_{C,Z},\Oh_C)$ with zero product. Under
either assumption, Lemma~\ref{lem:pot} gives a Francia curve $B\subset C$
containing $Z$. \end{COR}
This completes the proof of Theorem~\ref{th:bi}, (b) in Case~3, since the case
assumption is that $s\colon\sI_Z\to\sI_Z\subset\Oh_C$, so that
$\Hom(\sI_Z,\Oh_C)=\End(\sI_Z)$ is a 2-dimensional Artinian algebra.
\begin{pf} In the main case $\dim A=2$, this is completely trivial: if
$k\subset A$ is the constant subfield, any $s\in A\setminus k$ satisfies a
quadratic equation over $k$ of the form
\begin{equation}
0=s^2+as+b=(s-\al)(s-\be).
\nonumber
\end{equation}
If $\al=\be$ then $s'=s-\al$ is nilpotent with $s'{}^2=0$; otherwise,
$s'=(s-\al)/(\al-\be)$ and $1-s'=(s-\be)/(\be-\al)$ are orthogonal idempotents.
More generally, an Artinian algebra is a product $A=A_1\times\cdots\times A_l$
with local Artinian rings $(A_i,n_i)$ as factors; the maximal ideals of $A$
are codimension 1 vector subspaces $m_i\subset A_i$ given by $n_1\times
A_2\times\cdots\times A_l$ (say). The projection to the factors (if $l\ge2$)
give nontrivial idempotents; if $l=1$ then $A$ is local, with nilpotent maximal
ideal. This proves the first part.
We now prove the more general statement: a 2-dimensional vector subspace
$V\subset A$ in an Artinian algebra has nonzero intersection with every
maximal ideal, say $s_i\in V\cap m_i$. If the local factors $(A_i,n_i)$ have
dimension $d_i$ then $n_i^{d_i}=0$, and the product $\prod s_i^{d_i}$ maps to
zero in each factor, so is zero in $A$. Taking $\sum d_i=\dim A\le4$ gives the
final part of the claim. \QED \end{pf}
\subsection*{Proof in Case~4}
In the following proposition, $x\in C\subset\aff^3$ is a {\em local} curve
which is a local complete intersection at $x$. We choose local coordinates
$u,v,w$ on $\aff^3$ so that $\sI_{\aff^3,Z}\subset\Oh_{\aff^3}$ is generated at
$x$ by the regular sequence $u,v,w^2$. As before, we write $\Oh_C$ for the
local ring $\Oh_{C,x}$ and $\sI_Z=\sI_{C,Z}$ for the $\Oh_C$ module obtained
as the stalk at $x$ of the corresponding ideal sheaf. (Thus the statement of
the proposition only concerns homomorphisms $s\colon\sI_Z\to\Oh_C$ of
modules over the local ring $\Oh_C$.)
\begin{PROP}\label{pro:m2} Let $Z\subset C$ be a cluster of degree $2$
supported at $x$. We assume
\begin{enumerate}
\renewcommand{\labelenumi}{\rm(\roman{enumi})}
\item $Z$ is not a Cartier divisor on $C$;
\item there exists a homomorphism $s_0\colon\sI_Z\to\Oh_C$ such that for
general $\la\in k$, $s_0+\la\id$ defines an isomorphism $\sI_Z\iso\sI_{Z_\la}$
with $Z_\la$ a cluster of degree $2$ supported at $x$, and $Z_0\ne Z$.
\end{enumerate}
Then any homomorphism $s\colon\sI_Z\to\Oh_C$ takes $m_{C,x}^2$ to
$m_{C,x}^2$, that is,
\begin{equation}
\renewcommand\arraystretch{1.5}
\begin{matrix}
\sI_Z & @>{\quad s\quad}>> & \Oh_C\\
\bigcup && \bigcup\\
m_x^2 & @>{\hphantom{\quad s\quad}}>> & m_x^2
\end{matrix}
\nonumber
\end{equation}
\end{PROP}
\begin{pfof}{Theorem~\ref{th:bi}, (b) in Case~4} We apply the proposition to
the {\em global} homomorphism $s\colon\sI_Z\to\Oh_C$, using the assumption
of Case~4. We get
\begin{equation}
\Hom(\sI_Z,\Oh_C) \subset \End(m_x^2) \subset \Hom(m_x^2,\Oh_C).
\nonumber
\end{equation}
Now Lemma~\ref{lem:dim} gives $\dim\Hom(\sI_Z,\Oh_C)=2$ and
$\dim\Hom(m_x^2,\Oh_C)\le4$; but $A=\End(m_x^2)$ is a subring of
$H^0(C\setminus\Supp Z,\Oh_C)$, so that Corollary~\ref{cor:Art} gives the
result. \QED \end{pfof}
\begin{pfof}{Proposition~\ref{pro:m2}, Step 1} If $s\in\Hom(\sI_Z,\Oh_C)$
is any element then $s(\sI_Z)\subset m_x$; for otherwise $s$ would be an
isomorphism
$\sI_Z\iso\Oh_C$ near $x$, contradicting the assumption that $Z\subset C$ is
not Cartier.
\subparagraph{Step 2} Note that $m_x^2\subset\sI_Z$, so that we can
restrict $s\colon\sI_Z\to\Oh_C$ to $m_x^2$. Also, $m_x\sI_Z\subset m_x^2$,
and obviously $s(\sI_Z)\subset m_x$ implies that $s(m_x\sI_Z)\subset m_x^2$.
\subparagraph{Step 3} It is enough to prove that $s(w^2)\in m_x^2$. Indeed,
\begin{equation}
m_x\sI_Z=(u,v,w)\cdot(u,v,w^2)=(u^2,uv,v^2,uw,vw,w^3),
\nonumber
\end{equation}
so that
\begin{equation}
m_x^2=(u,v,w)^2=(u^2,uv,v^2,uw,vw,w^2)=m_x\sI_Z+\Oh_Cw^2\subset\Oh_C.
\nonumber
\end{equation}
\subparagraph{Step 4} Since $C$ is a local complete intersection,
$\sI_{\aff^3,C}=(F,G)$, where $F,G\in\Oh_{\aff^3}$ is a regular sequence. Now
$Z\subset C$ gives $F,G\in\sI_{\aff^3,Z}$, so that
\begin{equation}
\begin{aligned}
F&=Pu+Qv+Rw^2,\\ G&=P'u+Q'v+R'w^2,
\end{aligned}
\quad\text{with}\quad P,Q,R,P',Q',R'\in\Oh_{\aff^3}.
\end{equation}
The set of local homomorphisms $\sI_Z\to\Oh_C$ is a module over $\Oh_C$;
this is the stalk at $x$ of the sheaf $\sHom$. For the moment, we take on
trust the following general fact (see Appendix to \S\ref{sec:bi} for a
discussion and a detailed proof.)
\begin{CLA}\label{cla:PQ-PQ} The $\Oh_C$ module
$\sHom_{\Oh_C}(\sI_Z,\Oh_C)$ is generated by two elements, the identity
inclusion $\id\colon\sI_{C,Z}\into\Oh_C$ and the map
$t\colon\sI_{C,Z}\to\Oh_C$ determined by the minors of the $2\times3$ matrix
of coefficients of $F,G$:
\begin{equation}
t(u)=QR'-RQ',\quad t(v)=-PR'+RP',\quad t(w^2)=PQ'-QP'.
\end{equation}
\end{CLA}
\subparagraph{Step 5} According to Steps~3--4, to prove
Proposition~\ref{pro:m2}, we need only prove that $PQ'-QP'\in
m_{\aff^3,x}^2$. We are home if all four of $P,Q,P',Q'\in m_x$. Thus in what
follows, we assume (say) that $P'\notin m_x$. Then $P'$ is a unit, and $G=0$
defines a nonsingular surface $Y$ containing $C$. Dividing by $P'$, we can
rewrite $G$ in the form $u=-(Q'/P')v-(R'/P')w^2$. Then subtracting a multiple
of this relation from $F$ gives $f=qv+rw^2$ as the local equation of $C\subset
Y$ (where $q=Q-PQ'/P'$ and $r=R-PR'/P'$).
Therefore it only remains to prove that if $C$ is the planar curve defined by
$f=qv+rw^2$, the two assumptions of Proposition~\ref{pro:m2} imply that
$q\in m_{Y,x}^2$. As in Lemma~\ref{lem:notCt}, assumption (i) implies that
$q,r\in m_{Y,x}$, so that $q\in m_{Y,x}^2$ is equivalent to saying that $x\in
C\subset Y$ has multiplicity $\ge3$
\subparagraph{Step 6} Consider the linear terms of the given isomorphism
$s_0\colon\sI_Z\to\sI_{Z_0}$:
\begin{equation}
s_0(v)=av+bw\mod m_{Y,x}^2,\quad s_0(w^2)=cv+dw\mod m_{Y,x}^2.
\nonumber
\end{equation}
Because $Z_0\ne Z$, it follows that $(b,d)\ne(0,0)$. However, if $b=0$ and
$d\ne0$, then for general $\la$, the two generators of
$\sI_{Z_\la}=(s_0(v)+\la v,s(w^2)+\la w^2)$ would have linearly independent
linear terms, so that $\sI_{Z_\la}=m_{C,x}$. This contradicts assumption (ii).
Therefore $b\ne0$, and $\sI_{Z_\la}$ has a generator with the {\em variable}
linear term $(a+\la)v+bw$. It follows that $Z_\la$ runs linearly around the
tangent space to $x$ in $C$.
Now we claim that $x\in C\subset Y$ is a planar curve singularity of
multiplicity $\ge3$. Indeed, the isomorphism $\sI_Z\iso\sI_{Z_\la}$ implies
that $Z_\la\subset C$ cannot be a Cartier divisor; but if $x\in C\subset Y$
were a double point, this would restrict $Z_\la$ to be in the tangent cone,
contradicting what we have just proved. This completes the proof of
Proposition~\ref{pro:m2}. \QED \end{pfof}
\subsection*{Appendix: Proof of Claim~\ref{cla:PQ-PQ}}
We start by slightly generalising the set-up: let $\Oh_{\aff}$ be a local
ring, assumed to be regular (for simplicity only), and $x,y,z$ a regular
sequence generating a codimension 3 complete intersection ideal
$\sI_Z=(x,y,z)$. Consider a regular sequence $F,G\in\sI_Z$. Note that
\begin{equation}
F=Px+Qy+Rz\quad\text{and}\quad G=P'x+Q'y+R'z
\nonumber
\end{equation}
for some $P,\dots,R'\in\Oh_{\aff}$. Write $\Oh_C=\Oh_{\aff}/(F,G)$ and
$\sI_{C,Z}=\sI_Z\Oh_C=(x,y,z)\subset\Oh_C$. (In the application,
$Z\subset\aff=\aff^3$ was a nonreduced cluster defined by $(x,y,z)=(u,v,w^2)$
and $C\subset\aff^3$ a complete intersection curve through $Z$.)
\begin{LEM}
\begin{enumerate}
\renewcommand{\labelenumi}{\rm(\arabic{enumi})}
\item A presentation of $\sI_{C,Z}$ over $\Oh_C$ is given by
\begin{equation}
\Oh_C^{\oplus5}@>M>>\Oh_C^{\oplus3}
@>\left(\begin{matrix} x\\y\\z\end{matrix}\right)>>\sI_{C,Z}\to0,
\quad\text{where}\quad
\renewcommand\arraystretch{1.2}
M=\left(\matrix
P&Q&R\\
P'&Q'&R'\\
0&z&-y\\
-z&0&x\\
y&-x&0
\endmatrix\right).
\nonumber
\end{equation}
\item $\sHom(\sI_{C,Z},\Oh_C)$ is generated over $\Oh_C$ by the two
elements $\id$ and $t$, where
\begin{equation}
t\colon\left(\matrix x\\y\\z\endmatrix\right)\mapsto
\left(\matrix QR'-RQ'\\-PR'+RP'\\PQ'-QP'\endmatrix\right).
\label{eq:t}
\end{equation}
\end{enumerate}
\end{LEM}
\begin{pf} (1) An almost obvious calculation: because $\sI_{C,Z}=(x,y,z)$,
there is a surjective map $\fie\colon\Oh_C^{\oplus3}\to\sI_{C,Z}$, such that
$(h_1,h_2,h_3)\in\ker\fie$ if and only if $h_1x+h_2y+h_3z=0\in\Oh_C$. Write
$H_1,H_2,H_3\in\Oh_\aff$ for lifts of the $h_i$. Then
$H_1x+H_2y+H_3z\in\sI_{\aff^3,C}=(F,G)$. Subtracting off multiples of $F$ and
$G$ means exactly subtracting multiples of the first two rows of $M$ from
$(H_1,H_2,H_3)$, to give identities $H_1'x+H_2'y+H_3'z=0\in\Oh_\aff$. Now
$x,y,z\in\Oh_C$ is a regular sequence, so it follows that $(H'_1,H'_2,H'_3)$ is
in the image of the Koszul matrix given by the bottom 3 rows of $M$. This
proves (1).
(2) A homomorphism $s\colon\sI_{C,Z}\to\Oh_C$ is determined by
$(x,y,z)\mapsto(a,b,c)$ where $a,b,c\in\Oh_C$ satisfy $M(a,b,c)^\mathrm{tr}=0$
(we write $(a,b,c)^\mathrm{tr}$ for the column vector). It is easy to check
that (\ref{eq:t}) gives a map $t$ in this way.
The condition $M(a,b,c)^\mathrm{tr}=0$ consists of 5 equalities in
$\Oh_C=\Oh_{\aff}/(F,G)$. We choose lifts $A,B,C$ to $\Oh_{\aff}$, and write
out the last 3 of these as identities in $\Oh_{\aff}$:
\begin{equation}
\begin{array}{cccl}
&-zB&+yC&=\al F -\al'G \\
zA&&-xC&=\be F -\be'G \\
-yA&+xB&&=\ga F -\ga'G
\end{array}
\quad\text{for some $\al,\dots,\ga'\in\Oh_{\aff}$.}
\label{eq:al}
\end{equation}
Taking $x$ times the first plus $y$ times the second plus $z$ times the
third, the left-hand sides cancel, giving the identity
\begin{equation}
(\al x+\be y+\ga z)F=(\al'x+\be'y+\ga'z)G\in\Oh_{\aff}.
\nonumber
\end{equation}
Now since $F,G$ is a regular sequence in $\Oh_{\aff}$, this implies that
\begin{equation}
\begin{aligned}
\al x+\be y+\ga z&=DG=D(P'x+Q'y+R'z)\\
\al'x+\be'y+\ga'z&=DF=D(Px+Qy+Rz)
\end{aligned}
\label{eq:D}
\end{equation}
for some $D\in\Oh_{\aff}$.
Now subtracting $D$ times the given generator $t$ changes
\begin{equation}
\left(\begin{matrix}
A\\B\\C
\end{matrix}\right)
\mapsto
\left(\begin{matrix}
A\\B\\C
\end{matrix}\right)
-\left(\begin{matrix}
QR'-Q'R\\-PR'+P'R\\PQ'-P'Q\\
\end{matrix}\right)D
\nonumber
\end{equation}
and has the following effect on the quantities $\al,\dots,\ga'$ introduced in
(\ref{eq:al}):
\begin{align*}
(\al,\be,\ga)&\mapsto(\al+DP',\be+DQ',\ga+DR'),\\
(\al',\be',\ga')&\mapsto(\al'+DP,\be'+DQ,\ga'+DR).
\end{align*}
To see this, note that the first equation of (\ref{eq:al}) is
\begin{equation}
-zB+yC=\al F -\al'G=\al(Px+Qy+Rz)-\al'(P'x+Q'y+R'z),
\nonumber
\end{equation}
so that the effect of the two substitutions $\al\mapsto\al+DP'$ and
$\al'\mapsto\al'+DP$ on the right exactly cancels out $B\mapsto B+D(PR'-P'R)$
and $C\mapsto C-D(PQ'-P'Q)$ on the left. The upshot is that we can assume
$D=0$ in (\ref{eq:D}).
But then since $(x,y,z)$ is a regular sequence, (\ref{eq:D}) with $D=0$ gives
\begin{equation}
\begin{array}{rccc}
\al=&&ly&-mz\\
\be=&-lx&&+nz\\
\ga=&mx&-ny
\end{array}
\quad\text{and}\quad
\begin{array}{rccc}
\al'=&&l'y&-m'z\\
\be'=&-l'x&&+n'z\\
\ga'=&m'x&-n'y
\end{array}
\nonumber
\end{equation}
for some $l,\dots,n'\in\Oh_\aff$.
Finally (\ref{eq:al}) can now be rearranged as
\begin{equation}
\begin{aligned}
(C-lF+l'G)y&=(B-mG+m'G)z\\
(C-lF+l'G)x&=(A-nF+n'G)z\\
(B-mF+m'G)x&=(A-nF+n'G)y
\end{aligned}
\quad\text{therefore}\quad
\begin{aligned}
A-nF+n'G&=Ex\\
B-mF+m'G&=Ey\\
C-lF+l'G&=Ez
\end{aligned}
\nonumber
\end{equation}
for some $E\in\Oh_\aff$. This means that the map $s$ given by $(a,b,c)$ is a
linear combination of $t$ and the identity, as required. \QED \end{pf}
A less pedestrian method of arguing is to say that all three of $\Oh_\aff$,
$\Oh_C$ and $\Oh_Z$ are Gorenstein, so that adjunction gives
\begin{equation}
0\to\om_C\to\sHom(\sI_Z,\om_C)\to\om_Z=\Ext^1(\Oh_Z,\om_C)\to0.
\nonumber
\end{equation}
The two generators $\id$ and $t$ correspond naturally to the generators of
$\om_C$ and $\om_Z$.
|
1996-07-04T13:19:50 | 9607 | alg-geom/9607005 | en | https://arxiv.org/abs/alg-geom/9607005 | [
"alg-geom",
"math.AG"
] | alg-geom/9607005 | Sandro Manfredini | Fabrizio Catanese, Sandro Manfredini | The orbifold fundamental group of Persson-Noether-Horikawa surfaces | LaTeX file, 19 pages with 1 figure | null | null | null | null | The Noether-Horikawa surfaces are the minimal surfaces S with K^2=2p_g-4. For
8 | K^2 they belong to two families of respective type C and N (connected,
resp. non connected branch locus for the canonical map). For 16 | K^2 the two
types are homeomorphic. Ulf Persson constructed surfaces of type N with a
maximally singular canonical model X, whose topology encodes information on the
differentiable structure of S. A similar analysis was done by the first author
for type C. In this paper we study the genus 2 fibration on X and, in
particular, our main result is (X^# being the nonsingular locus of X)
\pi_1(X^#)= Z_4 x Z_4 if 8 | K^2 but 16 does not | K^2 \pi_1(X^#)= Z_4 x Z_2 if
16 | K^2.
| [
{
"version": "v1",
"created": "Thu, 4 Jul 1996 11:18:05 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Catanese",
"Fabrizio",
""
],
[
"Manfredini",
"Sandro",
""
]
] | alg-geom | \section{Introduction.}
Among the minimal surfaces of general type, the
Noether surfaces are
those for which the Noether inequality $K^2 \geq
2p_g -4$ is an equality
($K^2$ is the self intersection of a canonical
divisor, $p_g$ is the
dimension of the space of holomorphic 2-forms).\\
These surfaces were described by Noether (\cite{No})
and more recently by
Horikawa (\cite{Ho}) who proved that if $8\ |\ K^2$
then there are two
distinct deformation types, namely the
Noether-Horikawa surfaces of
connected type (for short, N-H surfaces of type C),
and those of non
connected type (for short, of type N). This notation
refers to the fact
that, the canonical map being a double covering of a
rational ruled
surface, for type C the branch locus is connected,
whereas for type N it
is not connected.\\
In particular Horikawa proved that the intersection
forms are both of the same
parity (in fact, both odd) if and only if $16\ |\
K^2$.\\
From M. Freedman's theorem (\cite{Fr}) follows that
if $16\ |\ K^2$ type N
and type C provide two orientedly homeomorphic
compact 4-manifolds.\\
Horikawa posed the question whether type N and type
C provide
two orientedly diffeomorphic compact 4-manifolds.\\
It looked like a natural problem to try to see
whether the two differentiable
structures could be distinguished by means of the
invariants introduced by
S. Donaldson in \cite{Do}.\\
In the case of type C we have been able (\cite{Ca})
to calculate the constant
Do\-nald\-son invariants (corresponding to
zero-dimensional moduli spaces)
using some singular canonical models of these
surfaces with very many
singularities, and an approach introduced by P.
Kronheimer (\cite{Kr}) for
the case of the Kummer surfaces.
The number we obtained, namely $ 2^{2k}$ when $K^2=8k$,
is the leading term of the Donaldson series (see
\cite{K-M}), which was
later fully calculated by Fintushel and Stern in the
case of N-H surfaces
of type C via the technique of rational blow-downs
(\cite{F-S}).\\
The Donaldson series for N-H surfaces of type N has
not yet, to our
knowledge, been calculated;
although, after the Seiberg-Witten theory (\cite{W})
has been introduced,
and after Pidstrigach and Tyurin (\cite{P-T}) have
announced the equality
between Kronheimer-Mrowka and Seiberg-Witten
classes, the two series
should be equal.\\
Our original aim was to extend the application of
the Kronheimer theory to the
case of N-H surfaces of type N using a very singular
model constructed by
Ulf Persson (\cite{Per}), describing its orbifold
fundamental group, its
representations into $SO(3)$, and then trying to see
which of those have
virtual dimension zero.\\
In this article we consider the singular N-H
surfaces of type N with maximal
Picard number constructed by Persson, henceforth called
Persson-Noether-Horikawa surfaces (P-N-H for short),
and we determine
their orbifold fundamental group.\\
This is our main result:\\
{\bf Theorem.} {\em The orbifold fundamental group
of the P-N-H surfaces is
$$\zeta_4\oplus\zeta_2$$ if $16\ |\ K^2,$
$$\zeta_4\oplus\zeta_4$$\nopagebreak
in the other case where $8\ |\ K^2$ but $16$ does
not divide
$K^2$.}\\
It follows immediately that we have, for $16\ |\
K^2$, only six nontrivial
classes of orbifold $SO(3)$-representations, and a
result which we do not
prove here is that we do not get anyone of virtual
dimension zero.\\
This is not surprising in view of (\cite{P-T}),
since if Kronheimer's
approach would have worked, we would have had only a
finite number of
constant Donaldson invariants.\\
On the other hand, the algebro-geometric technique
of studying canonical
mo\-dels with many rational double points produces
on the smooth model
configurations of (-2)-projective lines (spheres)
whose tubular neighborhood
has a unique holomorphic structure and, in
particular, a unique compatible
$C^{\infty}$ structure. In this way one produces a
decomposition of the
4-manifold in geometric pieces, one of which is the
nonsingular part of the
singular canonical model.\\
From this point of view, the calculation of the
orbifold fundamental group
leads to a better understanding of the
differentiable structures of the
smooth model.\\
Since our proof is rather involved technically we
would like to give a brief
geometrical "explanation" of our result.\\
Persson's construction starts with a plane nodal
cubic $C$ meeting a conic
$Q$ at only one point $P$. Moreover, $C$ and $Q$
have two common tangents
$L_{-1}$ and $L_1$ which meet in a point $O$
collinear with $P$ and the node
of $C$.\\
Blowing up $O$ we get a $\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1$-bundle
$f':\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_1\smash{\mathop{\longrightarrow}}\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1$ with a
section $\Sigma_{\infty}$, a bisection $Q'$ and a
3-section $C'$ ($'$
denoting the proper transform under the blow up).\\
A cyclic cover of order $2k\piu2$ branched on
$L'_{-1}$ and $L'_1$ yields a
new $\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1$-bundle $f'':\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_{2k+2}\smash{\mathop{\longrightarrow}}\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1$
with a section
$\Sigma''_{\infty}$ disjoint from a 3-section $C''$
and two sections
$Q_1''$, $Q_2''$ (the inverse image $Q''$ of $Q'$
splits into two
components).\\
The curve $B=C''\cup Q_1''\cup
Q_2''\cup\Sigma''_{\infty}$ has many singular
points, and our canonical model $X_{2k+2}$ is the
double cover of
$\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_{2k+2}$ branched on $B$. By construction
$X_{2k+2}$ has a genus 2
fibration onto $\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1$, whence the orbifold
fundamental group
$\pi_1(X_{2k+2}^{\#})$, $X_{2k+2}^{\#}$ being the
nonsingular part of
$X_{2k+2}$, is a quotient of $\pi_1(F)$, where $F$
is a fixed genus 2
fibre.\\
$F$ being a double cover of $\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1$ branched in six
points
$P_0\hbox{\mat \char61}\hskip1pt\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1\cap\Sigma''_{\infty}$,
$P_1\hbox{\mat \char61}\hskip1pt\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1\cap Q''_1$, $P_2\hbox{\mat \char61}\hskip1pt\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1\cap Q''_2$,
$\{P_3,P_4,P_5\}=\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1\cap C''$, $\pi_1(F)$ is the
subgroup of a free
product $\hbox{$\cal F\!$}_5(2)$ of five copies of $\zeta_2$, given
by words of even
length.\\
$\hbox{$\cal F\!$}_5(2)$ is generated by elements
$\unoenne{\hbox{$\varepsilon$}}{6}$ such that
$\cunoenne{\hbox{$\varepsilon$}}{6}\ugu1$ ($\hbox{$\varepsilon$}_i$ corresponds to a
loop in $\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1$ around
the point $P_{i-1}$).\\
The first main point (we must be rather vague here,
else we must give the
full proof) is that, since curve $C''$ is
irreducible, when the fibre $F$
moves around, $\hbox{$\varepsilon$}_4,\hbox{$\varepsilon$}_5,\hbox{$\varepsilon$}_6$ become
identified.\\
Thus we only have $\unoenne{\hbox{$\varepsilon$}}{4}$ with
$\cunoenne{\hbox{$\varepsilon$}}{4}\ugu1$, and
therefore we have "proved" that our group is abelian
, being a quotient of
the fundamental group $\Gamma$ of a curve of genus 1
obtained as the double
cover of $\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1$ branched in four points. More
precisely, $\Gamma$ is an
abelian group with generators $\hbox{$\varepsilon$}_1\hbox{$\varepsilon$}_2$,
$\hbox{$\varepsilon$}_1\hbox{$\varepsilon$}_3$.\\
We must still take into account the fact that, when
the fibre $F$ moves
towards a singular point (corresponding to points of
intersection $C''\cap
Q_1''$, $C''\cap Q_2'', Q_1''\cap Q_2''$), further
relations are introduced.
These relations are hard to control globally but if
we look locally
around these points of intersection, and accordingly
take a new basis
$\unoenne{\hbox{$\varepsilon$}'}{4}$, the situation becomes simpler.\\
In fact, the local equation of the double cover is
$z^2\hbox{\mat \char61}\hskip1pt y^2\hbox{\rmp \char123}\hskip1pt x^{2c}$,
where $c\ugu6$ or $c\hbox{\mat \char61}\hskip1pt k+1$, and $x$ is the
pullback of a local coordinate
on $\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1$, so that the corresponding local braid
yields the relation
$(\hbox{$\varepsilon$}'_j\hbox{$\varepsilon$}'_i)^c=(\hbox{$\varepsilon$}'_i\hbox{$\varepsilon$}'_j)^c$. In turn,
using $(\hbox{$\varepsilon$}'_i)^2\ugu1$,
we obtain the relation $(\hbox{$\varepsilon$}'_j\hbox{$\varepsilon$}'_i)^{2c}\ugu1$.\\
That's how one shows that the two generators of the
abelian group have
period 2 or 4.\\
The paper is organized as follows:\\
In section two we take up Persson's construction
using explicit
equations showing that the surface is defined over a
real quadratic field.\\
In the third section we describe the five steps
leading to a presentation
of our fundamental group in terms of the braid
monodromy of the plane
curve $D=C\cup Q$.\\
Finally, in section four we apply combinatorial
group theory arguments in
order to give the main result concerning the
orbifold fundamental group.\\
Acknowledgements : Both authors acknowledge support
from the AGE Project
H.C.M. contract ERBCHRXCT 940557 and from 40\%
M.U.R.S.T..\\
The first author would like to express his gratitude
to the Max-Planck
Institut in Bonn where this research was initiated
(in 1993), and to the
Accademia dei Lincei where he is currently
Professore Distaccato.
\section{Persson's configuration.}
In this section we will provide explicit equations
for the configuration
constructed by Ulf Persson in \cite{Per}.\\This is
the configuration formed
by a smooth conic $Q$ and a nodal cubic $C$
intersecting in only one point
$P$ which is smooth for $C$. Moreover $Q$ and $C$
have two common tangents
$L_{1}$ and $L_{-1}$ meeting in a point $O$ lying on
the line joining $P$
and the node of $C$.\\ Let $Q\subset\ci\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^2$
be the conic $\{(x,y,z)\app\ci\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^2\,|\,x^2\raise 1pt\hbox{\mat \char43}\hskip1pt
2zy\raise 1pt\hbox{\mat \char43}\hskip1pt z^2\hbox{\mat \char61}\hskip1pt 0\}$.\\
Since $$x^2\raise 1pt\hbox{\mat \char43}\hskip1pt 2zy\raise 1pt\hbox{\mat \char43}\hskip1pt z^2\hbox{\mat \char61}\hskip1pt(x\raise 1pt\hbox{\mat \char43}\hskip1pt z)^2\raise 1pt\hbox{\mat \char43}\hskip1pt
2z(y\hbox{\rmp \char123}\hskip1pt x)\hbox{\mat \char61}\hskip1pt (x\hbox{\rmp \char123}\hskip1pt
z)^2\raise 1pt\hbox{\mat \char43}\hskip1pt 2z(y\raise 1pt\hbox{\mat \char43}\hskip1pt x)$$
$Q$ is tangent to the lines $L_1=\{x\hbox{\rmp \char123}\hskip1pt y\hbox{\mat \char61}\hskip1pt 0\}$ and
$L_{-1}=\{x\raise 1pt\hbox{\mat \char43}\hskip1pt y\hbox{\mat \char61}\hskip1pt 0\}$.\\
The tangency points are:
$$ x\hbox{\rmp \char123}\hskip1pt y\hbox{\mat \char61}\hskip1pt x\raise 1pt\hbox{\mat \char43}\hskip1pt z\hbox{\mat \char61}\hskip1pt 0 \hbox{$\,\Rightarrow\,$} (1,1,\men1)$$
$$x\raise 1pt\hbox{\mat \char43}\hskip1pt y\hbox{\mat \char61}\hskip1pt x\hbox{\rmp \char123}\hskip1pt z\hbox{\mat \char61}\hskip1pt 0\hbox{$\,\Rightarrow\,$} (1,\men1,1).$$
Note that $Q$ is also tangent to the line $z\hbox{\mat \char61}\hskip1pt 0$
at the point
$(0,1,0)=P$.\\
We want to find an irreducible nodal cubic $C$ such
that $C\cdot Q=6P$ and
such that $C$ is tangent to the lines $x\hbox{\mat \char61}\hskip1pt\pm y$
in points different from
those of $Q$.\\
Let $C$ be a cubic s.t. $P\app C$ and $C\cdot Q=6P$.
Note that if $C$ were
reducible, then the previous condition would imply
that $z\ugu0$ is a
component of $C$.\\
We then have ${\rm div}(C)={\rm div}(z^3)\ ({\rm
mod}Q)$, so $C=z^3\raise 1pt\hbox{\mat \char43}\hskip1pt QL$
with $L$ a linear form, and thus
$$C=z^3\raise 1pt\hbox{\mat \char43}\hskip1pt (x^2\raise 1pt\hbox{\mat \char43}\hskip1pt 2zy\raise 1pt\hbox{\mat \char43}\hskip1pt z^2)(ax\raise 1pt\hbox{\mat \char43}\hskip1pt by\raise 1pt\hbox{\mat \char43}\hskip1pt cz).$$
Since we want $C$ to be tangent to the two lines
$L_{1}$ and $L_{-1}$
we obtain that the following homogeneous polynomials
in $(x,z)$
\begin{equaz}z^3\raise 1pt\hbox{\mat \char43}\hskip1pt (x\raise 1pt\hbox{\mat \char43}\hskip1pt z)^2((a\raise 1pt\hbox{\mat \char43}\hskip1pt
b)(x\raise 1pt\hbox{\mat \char43}\hskip1pt z)\raise 1pt\hbox{\mat \char43}\hskip1pt z(c\hbox{\rmp \char123}\hskip1pt a\hbox{\rmp \char123}\hskip1pt b))\label{1}\end{equaz}
\begin{equaz}z^3\raise 1pt\hbox{\mat \char43}\hskip1pt (x\hbox{\rmp \char123}\hskip1pt
z)^2((a\hbox{\rmp \char123}\hskip1pt b)(x\hbox{\rmp \char123}\hskip1pt z)\raise 1pt\hbox{\mat \char43}\hskip1pt z(c\raise 1pt\hbox{\mat \char43}\hskip1pt a\hbox{\rmp \char123}\hskip1pt
b))\label{21}\end{equaz} must have
a double root.\\
Set $\ze\hbox{\mat \char61}\hskip1pt (\frac{z}{x+z})$ and $\hat\ze\hbox{\mat \char61}\hskip1pt
(\frac{z}{x-z})$ and rewrite
\ref{1}, \ref{21} as:
$$\ze^3\raise 1pt\hbox{\mat \char43}\hskip1pt \ze (c\hbox{\rmp \char123}\hskip1pt a\hbox{\rmp \char123}\hskip1pt b)\raise 1pt\hbox{\mat \char43}\hskip1pt (a\raise 1pt\hbox{\mat \char43}\hskip1pt b)=0\ \ \
\ \hat\ze^3\raise 1pt\hbox{\mat \char43}\hskip1pt\hat\ze
(c\raise 1pt\hbox{\mat \char43}\hskip1pt a\hbox{\rmp \char123}\hskip1pt b)\raise 1pt\hbox{\mat \char43}\hskip1pt (a\hbox{\rmp \char123}\hskip1pt b)=0.$$
We recall that if $\ze$ is a double root of $z^3\raise 1pt\hbox{\mat \char43}\hskip1pt
pz\raise 1pt\hbox{\mat \char43}\hskip1pt q\hbox{\mat \char61}\hskip1pt 0$ then
$$3\ze^2+p=0{ \rm\ \ whence\ \ } \frac23\ze p\raise 1pt\hbox{\mat \char43}\hskip1pt
q\hbox{\mat \char61}\hskip1pt 0$$ and this implies
that $$\ze\hbox{\mat \char61}\hskip1pt\hbox{\rmp \char123}\hskip1pt \frac32\frac pq{\rm\ \ thus\ \
}27q^2\raise 1pt\hbox{\mat \char43}\hskip1pt 4p^3\hbox{\mat \char61}\hskip1pt 0.$$
Therefore we have a double root of \ref{1} if and
only if $$\exists\, A\ :\
\ze\hbox{\mat \char61}\hskip1pt
A,\ q\hbox{\mat \char61}\hskip1pt 2A^3,\ p\hbox{\mat \char61}\hskip1pt\hbox{\rmp \char123}\hskip1pt 3A^2,\ {\rm i.e.}\
\left\{\begin{array}{l} a\raise 1pt\hbox{\mat \char43}\hskip1pt
b\hbox{\mat \char61}\hskip1pt 2A^3 \\ c\hbox{\rmp \char123}\hskip1pt (a\raise 1pt\hbox{\mat \char43}\hskip1pt b)\hbox{\mat \char61}\hskip1pt\hbox{\rmp \char123}\hskip1pt
3A^2.\end{array}\right.$$
Similarly if we set $\hat\ze\hbox{\mat \char61}\hskip1pt\hbox{\rmp \char123}\hskip1pt B$
we have $$\left\{\begin{array}{l} a\hbox{\rmp \char123}\hskip1pt b\hbox{\mat \char61}\hskip1pt\hbox{\rmp \char123}\hskip1pt 2B^3\\
c\hbox{\rmp \char123}\hskip1pt (b\hbox{\rmp \char123}\hskip1pt a)\hbox{\mat \char61}\hskip1pt\men3B^2\end{array}\right.$$ and so
$$\left\{\begin{array}{l}a\hbox{\mat \char61}\hskip1pt A^3\hbox{\rmp \char123}\hskip1pt B^3 \\ b\hbox{\mat \char61}\hskip1pt
A^3\raise 1pt\hbox{\mat \char43}\hskip1pt B^3 \\
c\hbox{\mat \char61}\hskip1pt 2A^3\hbox{\rmp \char123}\hskip1pt 3A^2\hbox{\mat \char61}\hskip1pt 2B^3\hbox{\rmp \char123}\hskip1pt 3B^2.
\end{array}\right.$$
Then $A$ and $B$ must satisfy $2(A^3\hbox{\rmp \char123}\hskip1pt B^3)\hbox{\mat \char61}\hskip1pt
3(A^2\hbox{\rmp \char123}\hskip1pt B^2)$.\\
Recall that (we make no distinction between a curve
and its equation)
$$C=z^3\raise 1pt\hbox{\mat \char43}\hskip1pt (x^2\raise 1pt\hbox{\mat \char43}\hskip1pt 2zy\raise 1pt\hbox{\mat \char43}\hskip1pt z^2)((A^3\hbox{\rmp \char123}\hskip1pt
B^3)x\raise 1pt\hbox{\mat \char43}\hskip1pt (A^3\raise 1pt\hbox{\mat \char43}\hskip1pt B^3)y\raise 1pt\hbox{\mat \char43}\hskip1pt (2A^3\hbox{\rmp \char123}\hskip1pt 3A^2)z)$$
while $x\hbox{\rmp \char123}\hskip1pt y\hbox{\mat \char61}\hskip1pt 0$ is tangent to $C$ at the point
where
$$\ze\hbox{\mat \char61}\hskip1pt\frac{z}{x+z}\hbox{\mat \char61}\hskip1pt A.$$ Therefore the
tangency point is
$(1\hbox{\rmp \char123}\hskip1pt A,1\hbox{\rmp \char123}\hskip1pt A,A)$.\\ Similarly $x\raise 1pt\hbox{\mat \char43}\hskip1pt y\hbox{\mat \char61}\hskip1pt 0$
is tangent to $C$ at the
point $(B\hbox{\rmp \char123}\hskip1pt 1,1\hbox{\rmp \char123}\hskip1pt B,B)$.\\
Let us now search for a cubic $C$ with a singular
point on the line
$x\ugu0$, as in Persson's construction.\\
Since $\frac{\partial C}{\partial x}$ on the line
$x\ugu0$ equals $aQ$ and
the singular point is different from $P$ it follows that
$a\ugu0$. Whence $A^3\hbox{\rmp \char123}\hskip1pt B^3\hbox{\mat \char61}\hskip1pt A^2\hbox{\rmp \char123}\hskip1pt B^2\hbox{\mat \char61}\hskip1pt 0$
and so $A\hbox{\mat \char61}\hskip1pt B$.\\
If $A\hbox{\mat \char61}\hskip1pt B$ then $C$ contains only the monomial
$x^2$ as a polynomial in
$x$, so the involution $x{\longmapsto} \hbox{\rmp \char123}\hskip1pt x$
leaves the curve $C$ invariant. From this we deduce
that a singular point of
$C$ must have its $x$ coordinate equal to $0$ and
$C$ has then a
singularity on the line $x\hbox{\mat \char61}\hskip1pt 0$ if and only if
$$z^3\raise 1pt\hbox{\mat \char43}\hskip1pt (z^2\raise 1pt\hbox{\mat \char43}\hskip1pt 2zy)(2A^3y\raise 1pt\hbox{\mat \char43}\hskip1pt (2A^3\hbox{\rmp \char123}\hskip1pt 3A^2)z)\
\ {\rm has\ a\ double\
root.}$$ Remembering that it can't be $A\hbox{\mat \char61}\hskip1pt B\hbox{\mat \char61}\hskip1pt
0$, the double root
cannot be $z\hbox{\mat \char61}\hskip1pt 0$ and we can write the above as
$$z(z^2\raise 1pt\hbox{\mat \char43}\hskip1pt (z\raise 1pt\hbox{\mat \char43}\hskip1pt
2y)(2A^3y\raise 1pt\hbox{\mat \char43}\hskip1pt (2A^3\hbox{\rmp \char123}\hskip1pt 3A^2)z)).$$ So we must check
that
$$z^2(1\raise 1pt\hbox{\mat \char43}\hskip1pt 2A^3\hbox{\rmp \char123}\hskip1pt 3A^2)\raise 1pt\hbox{\mat \char43}\hskip1pt 2zy(A^3\raise 1pt\hbox{\mat \char43}\hskip1pt 2A^3\hbox{\rmp \char123}\hskip1pt
3A^2)\raise 1pt\hbox{\mat \char43}\hskip1pt 4A^3y^2=$$
$$=z^2(1\raise 1pt\hbox{\mat \char43}\hskip1pt 2A^3\hbox{\rmp \char123}\hskip1pt 3A^2)\raise 1pt\hbox{\mat \char43}\hskip1pt 2zy3A^2(A\hbox{\rmp \char123}\hskip1pt 1)\raise 1pt\hbox{\mat \char43}\hskip1pt
4A^3y^2$$
has a double root.\\ This is the case when
$$9A^4(A\hbox{\rmp \char123}\hskip1pt 1)^2\hbox{\mat \char61}\hskip1pt 4A^3(1\raise 1pt\hbox{\mat \char43}\hskip1pt 2A^3\hbox{\rmp \char123}\hskip1pt 3A^2)\ \
{\rm\ i.e.}$$
$$9A^6\hbox{\rmp \char123}\hskip1pt 18A^5\raise 1pt\hbox{\mat \char43}\hskip1pt 9A^4\hbox{\mat \char61}\hskip1pt 4A^3\raise 1pt\hbox{\mat \char43}\hskip1pt 8A^6\hbox{\rmp \char123}\hskip1pt 12A^5.$$
Upon dividing by $A^3\ugu\hskip -7pt / \kern 2pt 0$ we get
$$A^3\hbox{\rmp \char123}\hskip1pt 6A^2\raise 1pt\hbox{\mat \char43}\hskip1pt 9A\hbox{\rmp \char123}\hskip1pt 4\hbox{\mat \char61}\hskip1pt 0.$$
Observe that $1$ is a root of this equation, but if
$A\hbox{\mat \char61}\hskip1pt 1$ then the
singular point is $(0,0,1)$ and coincides with the
point of tangency of
$x\raise 1pt\hbox{\mat \char43}\hskip1pt y\hbox{\mat \char61}\hskip1pt 0$ so this root has to be discarded. Since
$$A^3\hbox{\rmp \char123}\hskip1pt 6A^2\raise 1pt\hbox{\mat \char43}\hskip1pt 9A\hbox{\rmp \char123}\hskip1pt 4\hbox{\mat \char61}\hskip1pt (A\hbox{\rmp \char123}\hskip1pt 1)(A^2\hbox{\rmp \char123}\hskip1pt
5A\raise 1pt\hbox{\mat \char43}\hskip1pt 4)\hbox{\mat \char61}\hskip1pt (A\hbox{\rmp \char123}\hskip1pt
1)^2(A\hbox{\rmp \char123}\hskip1pt 4)$$
the other possible root is then $A\hbox{\mat \char61}\hskip1pt 4$, and in
this case we have $B\hbox{\mat \char61}\hskip1pt
A\hbox{\mat \char61}\hskip1pt 4$, $a\hbox{\mat \char61}\hskip1pt 0$, $b\hbox{\mat \char61}\hskip1pt 8\cdot4^2$, $c\hbox{\mat \char61}\hskip1pt
5\cdot4^2$.\\
Then
$$C=z^3\raise 1pt\hbox{\mat \char43}\hskip1pt 4^2(x^2\raise 1pt\hbox{\mat \char43}\hskip1pt 2yz\raise 1pt\hbox{\mat \char43}\hskip1pt z^2)(8y\raise 1pt\hbox{\mat \char43}\hskip1pt 5z)$$
The tangency points are $(\hbox{\rmp \char123}\hskip1pt 3,\hbox{\rmp \char123}\hskip1pt 3,4)$ and
$(3,\hbox{\rmp \char123}\hskip1pt 3,4)$, while for
the singular point we have $x\hbox{\mat \char61}\hskip1pt 0$ and a double
root of
$$z^2\raise 1pt\hbox{\mat \char43}\hskip1pt 4^2(2y\raise 1pt\hbox{\mat \char43}\hskip1pt z)(8y\raise 1pt\hbox{\mat \char43}\hskip1pt 5z)\hbox{\mat \char61}\hskip1pt 0\iff
81z^2\raise 1pt\hbox{\mat \char43}\hskip1pt 4^218zy\raise 1pt\hbox{\mat \char43}\hskip1pt
4^4y\hbox{\mat \char61}\hskip1pt 0\iff 9z\raise 1pt\hbox{\mat \char43}\hskip1pt 4^2y\hbox{\mat \char61}\hskip1pt 0$$
so the singular point is $(0,9,\hbox{\rmp \char123}\hskip1pt 16)$.\\
With this choice of $A$ and $B$, $C$ is irreducible
(since $z\hbox{\mat \char61}\hskip1pt 0$ is
not a component of $C$).\\
We want to find the lines through $(0,0,1)$ and
tangent to $C$.\\
Let $A\hbox{\mat \char61}\hskip1pt B\hbox{\mat \char61}\hskip1pt\hbox{$\lambda$}$ and consider more generally the
1-parameter family
of curves: $$C_{\hbox{$\lambda$}}=z^3\raise 1pt\hbox{\mat \char43}\hskip1pt (x^2\raise 1pt\hbox{\mat \char43}\hskip1pt 2yz\raise 1pt\hbox{\mat \char43}\hskip1pt
z^2)(2\hbox{$\lambda$}^3y\raise 1pt\hbox{\mat \char43}\hskip1pt (2\hbox{$\lambda$}^3\hbox{\rmp \char123}\hskip1pt
3\hbox{$\lambda$}^2)z)\hbox{\mat \char61}\hskip1pt 0.$$
The tangency points on the two fixed lines $x\raise 1pt\hbox{\mat \char43}\hskip1pt
y=0$, $x\hbox{\rmp \char123}\hskip1pt y=0$ are, as
we know, $(1\hbox{\rmp \char123}\hskip1pt\hbox{$\lambda$},1\hbox{\rmp \char123}\hskip1pt\hbox{$\lambda$},\hbox{$\lambda$})$ and $(\hbox{$\lambda$}\hbox{\rmp \char123}\hskip1pt
1,1\hbox{\rmp \char123}\hskip1pt\hbox{$\lambda$},\hbox{$\lambda$})$.\\
Rewriting the last equation in powers of $z$ we obtain:
$$z^3(1\raise 1pt\hbox{\mat \char43}\hskip1pt 2\hbox{$\lambda$}^3\hbox{\rmp \char123}\hskip1pt 3\hbox{$\lambda$}^2)\raise 1pt\hbox{\mat \char43}\hskip1pt
z^26y\hbox{$\lambda$}^2(\hbox{$\lambda$}\hbox{\rmp \char123}\hskip1pt 1)\raise 1pt\hbox{\mat \char43}\hskip1pt z\hbox{$\lambda$}^2(4\hbox{$\lambda$}
y^2\raise 1pt\hbox{\mat \char43}\hskip1pt (2\hbox{$\lambda$}\hbox{\rmp \char123}\hskip1pt 3)x^2)\raise 1pt\hbox{\mat \char43}\hskip1pt 2\hbox{$\lambda$}^3x^2y\hbox{\mat \char61}\hskip1pt 0.$$
Since we know what happens for $\hbox{$\lambda$}\hbox{\mat \char61}\hskip1pt 0$, we can
divide by $\hbox{$\lambda$}^3$, set
$w\hbox{\mat \char61}\hskip1pt \frac z{\lambda}$ and obtain:
$$w^3(1\raise 1pt\hbox{\mat \char43}\hskip1pt 2\hbox{$\lambda$}^3\hbox{\rmp \char123}\hskip1pt 3\hbox{$\lambda$}^2)\raise 1pt\hbox{\mat \char43}\hskip1pt w^26y\hbox{$\lambda$}(\hbox{$\lambda$}\hbox{\rmp \char123}\hskip1pt
1)\raise 1pt\hbox{\mat \char43}\hskip1pt w(4\hbox{$\lambda$}
y^2\raise 1pt\hbox{\mat \char43}\hskip1pt (2\hbox{$\lambda$}\hbox{\rmp \char123}\hskip1pt 3)x^2)\raise 1pt\hbox{\mat \char43}\hskip1pt 2x^2y\hbox{\mat \char61}\hskip1pt 0.$$
We let now $\hbox{$\Delta$}$ be the discriminant of $C_{\hbox{$\lambda$}}$
with respect to the
variable $w$, and using a
standard formula for $\hbox{$\Delta$}$, we find a degree 6
equation in $x$ and $y$ which
is divisible by $x^2(x^2\hbox{\rmp \char123}\hskip1pt y^2)$.\\
Remembering that the discriminant of $a_0x^3\raise 1pt\hbox{\mat \char43}\hskip1pt
a_1x^2\raise 1pt\hbox{\mat \char43}\hskip1pt a_2x\raise 1pt\hbox{\mat \char43}\hskip1pt a_3$
is: $$\hbox{$\Delta$}\hbox{\mat \char61}\hskip1pt a_1^2a_2^2\hbox{\rmp \char123}\hskip1pt 4a_0a_2^3\hbox{\rmp \char123}\hskip1pt
4a_1^3a_3\hbox{\rmp \char123}\hskip1pt 27a_0^2a_3^2\raise 1pt\hbox{\mat \char43}\hskip1pt 18a_0a_1a_2a_3$$
and applying this formula for simplicity when
$\hbox{$\lambda$}\hbox{\mat \char61}\hskip1pt 4$, we obtain:
$$y^22^63^4(16y^2\raise 1pt\hbox{\mat \char43}\hskip1pt 5x^2)^2\hbox{\rmp \char123}\hskip1pt 2^23^4(16y^2\raise 1pt\hbox{\mat \char43}\hskip1pt
5x^2)^3\hbox{\rmp \char123}\hskip1pt$$
$$\men2^{12}3^6x^2y^4 \hbox{\rmp \char123}\hskip1pt 2^23^{11}x^4y^2\raise 1pt\hbox{\mat \char43}\hskip1pt
2^53^8(16y^2\raise 1pt\hbox{\mat \char43}\hskip1pt
5x^2)x^2y^2$$
and factoring this binary form we get:
$$x^2(x^2\hbox{\rmp \char123}\hskip1pt y^2)2^23^4(2^7y^2\hbox{\rmp \char123}\hskip1pt 5^3x^2).$$
So we have that the tangent lines to $C$ passing
through $(0,0,1)$ are
$x\hbox{\mat \char61}\hskip1pt\pm y$, $x\hbox{\mat \char61}\hskip1pt\pm\sqrt{\frac{128}{125}}y$
while $x\hbox{\mat \char61}\hskip1pt 0$ passes
through the node of $C$. We denote by $L_0$ the line
$x\ugu0$ and by
$L_+,L_-$ the two lines $x\hbox{\mat \char61}\hskip1pt\sqrt{\frac{128}{125}}y$,
$x\hbox{\mat \char61}\hskip1pt-\sqrt{\frac{128}{125}}y$ respectively.\\
In order to find the tangency point on the lines
$L_+,L_-$
we by symmetry may restrict to the line $L_+$.\\
Writing $x\hbox{\mat \char61}\hskip1pt 2^3\sqrt2\,a$,
$y\hbox{\mat \char61}\hskip1pt 5\sqrt5\,a$ we have that
\begin{equaz}\label{pol}z^3\raise 1pt\hbox{\mat \char43}\hskip1pt 2^4(2^7a^2\raise 1pt\hbox{\mat \char43}\hskip1pt
10\sqrt5\,az\raise 1pt\hbox{\mat \char43}\hskip1pt
z^2)(40\sqrt5\,a\raise 1pt\hbox{\mat \char43}\hskip1pt
5z)\hbox{\mat \char61}\hskip1pt 0\end{equaz} has a double root. Since for
its derivative we have
$$3z^2\raise 1pt\hbox{\mat \char43}\hskip1pt 2^4(10\sqrt5\,a\raise 1pt\hbox{\mat \char43}\hskip1pt 2z)(40\sqrt5\,a\raise 1pt\hbox{\mat \char43}\hskip1pt
5z)\raise 1pt\hbox{\mat \char43}\hskip1pt 2^45(2^7a^2\raise 1pt\hbox{\mat \char43}\hskip1pt
10\sqrt5\,az\raise 1pt\hbox{\mat \char43}\hskip1pt z^2)\hbox{\mat \char61}\hskip1pt 0$$
$$(15\raise 1pt\hbox{\mat \char43}\hskip1pt {3\over16})z^2\raise 1pt\hbox{\mat \char43}\hskip1pt 180\sqrt5\,az\raise 1pt\hbox{\mat \char43}\hskip1pt
2640a^2\ugu0$$
$${a\over z}\hbox{\mat \char61}\hskip1pt{-90\sqrt5\pm\sqrt{90^25-
2640(15+{3\over16})}\over2640}\hbox{\mat \char61}\hskip1pt\sqrt5{-30\pm3\over880}.$$
Thus $\frac{y}{z}={-25(30\pm3)\over880}$,
$\frac{x}{z}={-8\sqrt{10}(30\pm3)\over880}$ and the
point of tangency is
one of the points $(\men33\cdot8\sqrt{10},\hbox{\rmp \char123}\hskip1pt
25\cdot 33,880)$,
$(\men27\cdot8\sqrt{10},\hbox{\rmp \char123}\hskip1pt 25\cdot 27,880)$.\\
Upon substituting these values in the polynomial
\ref{pol} we find that the
correct choice is $(\men24\sqrt{10},\hbox{\rmp \char123}\hskip1pt 75,80)$.\\
By symmetry the point
$(24\sqrt{10},\hbox{\rmp \char123}\hskip1pt 75,80)$ is the tangency point of
the line $L_-$.\\
Let us write
$$C=4^2(8y\raise 1pt\hbox{\mat \char43}\hskip1pt 5z)x^2\raise 1pt\hbox{\mat \char43}\hskip1pt z(16y\raise 1pt\hbox{\mat \char43}\hskip1pt 9z)^2\hbox{\mat \char61}\hskip1pt 0$$
and let us set $u\hbox{\mat \char61}\hskip1pt 16y\raise 1pt\hbox{\mat \char43}\hskip1pt 9z$. We have:
$$C= zu^2\raise 1pt\hbox{\mat \char43}\hskip1pt 8x^2(u\raise 1pt\hbox{\mat \char43}\hskip1pt z)\hbox{\mat \char61}\hskip1pt 0$$
In these coordinates the singular point of $C$ is
$(0,0,1)$, so the
tangents at the singular point are given by:
$$8x^2\raise 1pt\hbox{\mat \char43}\hskip1pt u^2\hbox{\mat \char61}\hskip1pt 0$$
whence they are complex and we have an isolated point.\\
In order to draw $C$, let's compute its flexes.
Using the
coordinates $x$, $u$, and $z$ the Hessian matrix is:
$$\left(\begin{array}{ccc}16(u\raise 1pt\hbox{\mat \char43}\hskip1pt
z)&16x&16x\\16x&2z&2u\\16x&2u&0\end{array} \right)$$
The Hessian curve is then given by the determinant of
$$\left(\begin{array}{ccc}(u\raise 1pt\hbox{\mat \char43}\hskip1pt z)&0&x\\0&z\hbox{\rmp \char123}\hskip1pt
2u&u\\8x&u&0\end{array}
\right)$$
which equals
$$\hbox{\rmp \char123}\hskip1pt (u\raise 1pt\hbox{\mat \char43}\hskip1pt z)u^2\hbox{\rmp \char123}\hskip1pt 8x^2(z\hbox{\rmp \char123}\hskip1pt 2u)\ugu0.$$\\
Eliminating $8x^2$ from the two equations we get
$$(u\raise 1pt\hbox{\mat \char43}\hskip1pt z)^2u^2\hbox{\rmp \char123}\hskip1pt zu^2(z\hbox{\rmp \char123}\hskip1pt 2u)\ugu0$$
so either $u=0$, and this implies either $x\hbox{\mat \char61}\hskip1pt 0$
(the singular point) or
$z\hbox{\mat \char61}\hskip1pt 0$ that gives the point $(1,0,0)$, or
$$(u\raise 1pt\hbox{\mat \char43}\hskip1pt z)^2\hbox{\rmp \char123}\hskip1pt z(z\hbox{\rmp \char123}\hskip1pt 2u)\hbox{\mat \char61}\hskip1pt u^2\raise 1pt\hbox{\mat \char43}\hskip1pt 4uz\hbox{\mat \char61}\hskip1pt 0$$
that gives ($u\diverso0$) $u\hbox{\mat \char61}\hskip1pt \hbox{\rmp \char123}\hskip1pt 4z$, that is
$z\hbox{\mat \char61}\hskip1pt\hbox{\rmp \char123}\hskip1pt 1$, $u\hbox{\mat \char61}\hskip1pt 4$,
$y\hbox{\mat \char61}\hskip1pt{13\over16}$, $x\hbox{\mat \char61}\hskip1pt\pm\sqrt{2/3}$.\\
For these points ${x\over
y}=\pm\sqrt{2/3}{16\over13}$.\pagebreak
\section{Fundamental groups.}
In this section we are going to describe the five
steps leading to the
determination of the orbifold fundamental group of
the Persson's surfaces.\\
{\bf Step 1.}\\
Let $\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_1$ be the blow up of $\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^2$ at the point
$(0,0,1)$ and let
$\Sigma_{\infty}$ be the exceptional divisor.\\
We consider the fibre bundle
$\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_1\mapright{f'}\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1$ and its
restriction $f$ $$\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_1\meno( C\cup
Q\cup\Sigma_{\infty}\cup L_1\cup L_{-1}
\cup L_+\cup L_-\cup L_0)=\tilde{\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}}_1$$
$$f\big\downarrow$$
$$\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1\meno\{P_1,P_{-1},P_+,P_-,P_0\}=\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1\meno\{5\ {\rm
pts.}\}.$$ $f$ is again a fibre bundle and we have a
corresponding homotopy
exact sequence of fundamental groups $$1\smash{\mathop{\longrightarrow}}
\hbox{$\cal F\!$}_5\smash{\mathop{\longrightarrow}}\tilde{\PI}\smash{\mathop{\longrightarrow}}\hbox{$\cal F\!$}_4\mapdestra1$$
\puntif
where $\hbox{$\cal F\!$}_k$ denotes the free group with $k$
generators and
$\tilde\PI=\pi_1(\tilde{\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}}_1)$.\\
Here we choose a small positive real number
$\hbox{$\varepsilon$}\mag0$ and $x\hbox{\mat \char61}\hskip1pt\hbox{$\varepsilon$}$,
$y\ugu1$ as base point on $\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1\meno\{5\ {\rm
pts.}\}$ and $x\hbox{\mat \char61}\hskip1pt\hbox{$\varepsilon$}$,
$y\ugu1$, $z\hbox{\mat \char61}\hskip1pt\men4\sqrt{-1}$ as base point on
$\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_1$.\\
We let $\unoenne{\delta}{5}$ be a natural geometric
basis of the free
group $$\hbox{$\cal F\!$}_5=\pi_1(f^{-1}({\rm base\
pt.}))=\pi_1(L_{\hbox{$\varepsilon$}}\meno(C\cup
Q\cup\Sigma_{\infty}))$$ where the five points
$L_{\hbox{$\varepsilon$}}\cap C$,
$L_{\hbox{$\varepsilon$}}\cap
Q$ are ordered by lexicografic order on $\rm Re(\frac
zy)$, $\rm Im(\frac zy)$.\\
$\hbox{$\cal F\!$}_4$ is generated by the five geometric paths
$\gamma_i'$ around the five
critical values described in figure \ref{figura} and
whose product is the
identity.\\
For these elements we choose lifts to $\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_1$ using
a $C^{\infty}$ section
of a tubular neighborhood of $\Sigma_{\infty}$
meeting $\Sigma_{\infty}$
just in the point $\infty$ $(y\ugu0)$ with
intersection number equal to
$-1$.\\
Therefore such lifts give paths $\gamma_i$ such that
$$\prod\gamma_i=\prod\delta_i$$ and more specifically
$$\gamma_+\gamma_1\gamma_0\gamma_-\gamma_{-1}=\delta_1\cdots\delta_5=
\gamma_{-1}\gamma_+\gamma_1\gamma_0\gamma_-.$$
We have that, indeed, $\tilde{\PI}$ occurs as a
semidirect
product described by the relations
$$\gamma_j^{-1}\delta_i\gamma_j=(\delta_i)\beta_j$$
where the $\beta_j$'s are suitable braids in
$$\hbox{\tengt B}_5
=\hbox{\matem \char60}\sigma_1,\ldots,\sigma_{4}|\:\:\:\,
\sigma_i\sigma_j \hbox{\mat \char61}\hskip1pt\sigma_j\sigma_i\ \;\forall\:
1\hbox{\mpic \char20}\hskip1pt
i\hbox{\mate \char60}\hskip1pt j\piu1\hbox{\mpic \char20}\hskip1pt 5
$$
\hspace*{15.4em}\immediate\vspace*{-1ex}
$\sigma_i\sigma_{i+1}\sigma_i\hbox{\mat \char61}\hskip1pt
\sigma_{i+1}\sigma_i\sigma_{i+1}\
\forall\: 1\hbox{\mpic \char20}\hskip1pt i\hbox{\mate \char60}\hskip1pt 4\ \hbox{\matem \char62}$
\vspace{\baselineskip}\\
the braid group on 5 strings which acts on the right
on the free group
$\hbox{$\cal F\!$}_5$
by the formulae
\begin{eqnarray*}(\delta_h)\sigma_k&=&\delta_h\ \ \
\ {\rm
if}\ h\ugu\hskip -7pt / \kern 2pt k,k\piu1\\
(\delta_k)\sigma_k&=&\delta_{k+1}\\
(\delta_{k+1})\sigma_k&=&\delta_{k+1}^{-1}\delta_k\delta_{k+1}.
\end{eqnarray*}
The braids $\beta_j$ are constructed by following
the motion of the five
points of the intersection of $f'^{-1}(P)$ with
$C\cup Q$ while $P$ goes
along $\gamma_j'$.\\
With our choice of the $\gamma'$'s we have, as the
reader can easily verify,
\begin{eqnarray*}
\beta_0&=&\sigma_4^{12}\sigma_2^2\\
\beta_1&=&\sigma_1^{-1}\sigma_2\sigma_3\sigma_1\sigma_2^{-1}\sigma_1\\
\beta_{-1}&=&\sigma_4^{-6}\sigma_2^{-1}\beta_1\sigma_2\sigma_4^6\\
\beta_+&=&\sigma_1^{-2}\sigma_2\sigma_3\sigma_4\sigma_3^{-1}\sigma_2^{-1}
\sigma_1^2\\
\beta_-&=&\sigma_4^{-6}\sigma_2^{-1}\beta_+\sigma_2\sigma_4^6.
\end{eqnarray*}
{\bf Step 2.}\\
By taking $\sqrt{\frac{x-y}{x+y}}$ we have a new
fibre bundle
$\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_2\mapright{g'}\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1$ obtained by base change.
Under this base change
the inverse image $Q'$ of the conic $Q$ splits into
two sections of $g'$
which we will denote by $Q_1'$ and $Q_2'$. Again, by
restriction we have a
fibre bundle $g$
$$\hat{\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}}_2=\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_2\meno (C'\cup Q_1'\cup Q_2'\cup
\Sigma_{\infty}'
\cup \{8\ {\rm fibres}\})\mapright{g}\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1\meno \{8\
{\rm
pts.}\}.$$ Correspondingly we get an exact sequence
$$1\smash{\mathop{\longrightarrow}}\hbox{$\cal F\!$}_5=\hbox{\matem \char60}\unoenne{\delta}{5}\hbox{\matem \char62}\smash{\mathop{\longrightarrow}}\hat{\PI}
\smash{\mathop{\longrightarrow}}\hbox{$\cal F\!$}_7=
\hbox{\matem \char60}\gamma_0,\gamma_-,\gamma_+,\bar{\gamma}_0,\bar{\gamma}_-,
\bar{\gamma}_+,\gamma_1^2\hbox{\matem \char62}\mapdestra1$$
where
$\bar\gamma_i\hbox{\mat \char61}\hskip1pt\gamma_i^{\gamma_1}\hbox{\mat \char61}\hskip1pt\gamma_1\gamma_i\gamma_1^{-1}$
and $\hat\PI=\pi_1(\hat{\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}}_2)$.\\
The fact that $\hbox{$\cal F\!$}_7$ has seven generators as above
follows since the double
cover of $\hbox{{\uyu I}\hskip-.05truecm{\uyu P}}^1\meno\{5\ {\rm pts.}\}$
corresponds to the homomorphism
$\hbox{$\cal F\!$}_4\rightarrow\zeta_{\!2}$ sending
$\gamma_1',\gamma_{-1}'\mapsto\bar1$, and
$\gamma_0',\gamma_+',
\gamma_-'\mapsto\bar0$.\\
If we want to keep track of the eight critical
values, we can also use
$(\gamma_{-1}^2)^{\gamma_1}$ as a generator. In fact
$$(\delta_1\cdots\delta_5)^2=(\gamma_+\gamma_1\gamma_0\gamma_-\gamma_{-1})
(\gamma_{-1}\gamma_+\gamma_1\gamma_0\gamma_-)$$
thus
$$\gamma_+\gamma_0^{\gamma_1}\gamma_-^{\gamma_1}(\gamma_{-1}^2)^{\gamma_1}
\gamma_+^{\gamma_1}\gamma_1^2\gamma_0\gamma_-=(\delta_1\cdots\delta_5)^2.$$
The geometric meaning of the above formula is
related to the fact that
$(\Sigma_{\infty}')^2=-2,$ and more precisely to the
fact that the new
generators of $\hbox{$\cal F\!$}_7$ lie in a $C^{\infty}$ section
meeting
$\Sigma_{\infty}'$ in one point with intersection
number $(-2)$, and not
meeting the other curves.\\
A presentation of $\hat{\PI}$ is thus given by
$$\hbox{\matem \char60}\unoenne{\delta}{5},\gamma_0,\gamma_-,\gamma_+,
\bar{\gamma}_0,\bar{\gamma}_-,\bar{\gamma}_+,\Gamma\hbox{\mat \char61}\hskip1pt\gamma_1^{-2}\ |\
\gamma_0^{-1}\delta_i\gamma_0=(\delta_i)\beta_0\ \ \
\ \ \ \ \ \ \ $$
$$\hspace*{16.5em}\vdots$$
$$\hspace*{18.5em}\bar{\gamma}_0^{-1}\delta_i\bar{\gamma}_0=(\delta_i)
\beta_1\beta_0\beta_1^{-1}$$ $$\hspace*{16.5em}\vdots$$
$$\hspace*{17em}\Gamma\delta_i\Gamma^{-1}=(\delta_i)\beta_1^2\hbox{\matem \char62}$$
{\bf Step 3.}\\
The fundamental group
$$\PI'=\pi_1(\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_2\meno (C'\cup Q_1'\cup Q_2'\cup
\Sigma_{\infty}'
\cup L_1'\cup L_{-1}'))$$ is a quotient of $\hat\PI$.
The presentation of $\PI'$ is readily accomplished
simply by introducing in
the above presentation the further relations
$$\gamma_0=\gamma_-=\gamma_+=\bar{\gamma}_0=\bar{\gamma}_-=
\bar{\gamma}_+=1.$$
Then $\PI'$ is presented as
$$\hbox{\matem \char60}\unoenne{\delta}{5},\Gamma\ |\
\delta_i=(\delta_i)\beta_0\ \ \
\delta_i=(\delta_i)\beta_-\ \ \
\delta_i=(\delta_i)\beta_+$$
$$\hspace*{7em}\delta_i=(\delta_i)\beta_1\beta_0\beta_1^{-1}\ \ \
\delta_i=(\delta_i)\beta_1\beta_-\beta_1^{-1}$$
$$\hspace*{8em}\delta_i=(\delta_i)\beta_1\beta_+\beta_1^{-1}\ \ \
\Gamma\delta_i\Gamma^{-1}=(\delta_i)\beta_1^2\hbox{\matem \char62}$$
{\bf Remark:} with the new relations we get, setting
$\Gamma_{-1}=(\gamma_{-1}^2)^{\gamma_1},$
$$\Gamma_{-1}\Gamma=(\delta_1\cdots\delta_5)^2$$
{\bf Step 4'}.\\
We denote by $X_2^{\#}$ the non singular
part of the double cover $X_2$ of $\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_2$ (branched
over $C'\cup Q_1'\cup
Q_2' \cup\Sigma_{\infty}'$) and by $Z_2^{\#}$ the
complement
in $X_2^{\#}$ of $L_1'',L_{-1}''$, the respective
inverse images of
$L_1',L_{-1}'$.\\ We finally let $Y_2^{\#}$ be the
double cover of
$\hbox{{\uyu I}\hskip-.05truecm{\uyu F}$\!$}_2\meno (C'\cup Q_1'\cup Q_2'\cup
\Sigma_{\infty}' \cup L_1'\cup
L_{-1}')$.\\ Thus $Y_2^{\#}\subset Z_2^{\#}\subset
X_2^{\#}$.\\
Clearly $\pi_1(Y_2^{\#})=\ker(\PI'\smash{\mathop{\longrightarrow}}\zeta_2)$,
where $\delta_i{\longmapsto}\bar1$ and $\Gamma{\longmapsto}\bar0$, is
generated by
$\Gamma$, $\sigma=\delta_1\Gamma\delta_1^{-1}$,
$A_i=\delta_1\delta_i$
$(i\ugu1,\ldots , 5)$ and $B_j=\delta_j\delta_1^{-1}$
$(j\ugu2,\ldots , 5)$.\\
To find the relations we apply the
Reidemeister-Shreier rewriting process
to the relations $R_{\alpha}$ of $\PI'$ and to the
relations
$\delta_1R_{\alpha}\delta_1^{-1}$.\\
{\bf Step 4''}.\\
Clearly, $\pi_1(Y_2^{\#})$ maps onto
$\pi_1(Z_2^{\#})$ surjectively with
kernel normally generated by $\delta_1^2$,
$\delta_i^2=B_iA_i$ $(i\ugu2,\ldots ,
5)$ and $(\delta_1\cdots\delta_5)^2$, thus
$\pi_1(Z_2^{\#})$ is generated by $A_2,\ldots ,
A_5,\Gamma$ and has for
relations the relations coming from the rewriting of
$R_{\alpha}$,
$\delta_1R_{\alpha}\delta_1^{-1}$, and the rewriting of
$(\delta_1\cdots\delta_5)^2=1\!$, i.e.
$A_2A_3^{-1}A_4A_5^{-1}
A_2^{-1}A_3A_4^{-1}A_5=1$.\\
{\bf Remark:} This relation says that the four
generators $A_2,\ldots , A_5$
are the generators of $\pi_1({\rm fibre})=\pi_1({\rm
genus\ 2\ curve})$.\\
{\bf Step 5.}\\ Let $m\hbox{\mat \char61}\hskip1pt k\piu1$ and
consider $X_{2m}^{\#}$, the non singular part of the
m-fold cyclic cover of
$X_2$ totally branched over $L_1''$ and $L_2''$.\\
To find a presentation of $X_{2m}^{\#}$ we first
need a presentation
of the kernel of the map
$\pi_1(Z_{2}^{\#})\smash{\mathop{\longrightarrow}} \zeta_m$ such that
$A_i{\longmapsto}\bar0$ and $\Gamma,\sigma{\longmapsto}\bar 1$, and
then we add the relations
$\Gamma^m=\sigma^m=1$.\\
Applying the Reidemeister-Shreier method, we find
that the kernel
is generated by $\Gamma^m$, $\Gamma^iA_j\Gamma^{-i}$
for $i\ugu1,\ldots , m\men1$
and $j\ugu2,\ldots , 5$, by $\Gamma^i\sigma\Gamma^{-i-1}$
for $i\ugu1,\ldots , m\men2$
and $\Gamma^{m-1}\sigma$; it has for relations the
rewriting in the
new generators of the relations $R_{\alpha}''$ of
$\pi_1(Z_2^{\#})$
and the rewriting of
$\Gamma^iR_{\alpha}''\Gamma^{-i}$ for
$i\ugu1,\ldots , m\men1$.
\section{Calculations.}
{\bf Step 3.}\\
We have \begin{eqnarray*}
\beta_0&=&\sigma_4^{12}\sigma_2^2\\
\beta_1&=&\sigma_1^{-1}\sigma_2\sigma_3\sigma_1\sigma_2^{-1}\sigma_1\\
\beta_{-1}&=&\sigma_4^{-6}\sigma_2^{-1}\beta_1\sigma_2\sigma_4^6\\
\beta_+&=&\sigma_1^{-2}\sigma_2\sigma_3\sigma_4\sigma_3^{-1}\sigma_2^{-1}
\sigma_1^2\\
\beta_-&=&\sigma_4^{-6}\sigma_2^{-1}\beta_+\sigma_2\sigma_4^6
\end{eqnarray*}
The relations $\delta_i\hbox{\mat \char61}\hskip1pt(\delta_i)\beta_0$ are
equivalent to the two
relations \begin{equaz}\label{fofififo}
(\delta_4\delta_5)^6=(\delta_5\delta_4)^6\end{equaz}
\begin{equaz}\label{tttt}
\delta_2\delta_3=\delta_3\delta_2.\end{equaz}
The relations $\delta_i\hbox{\mat \char61}\hskip1pt(\delta_i)\beta_+$ amount to
\begin{equaz}\label{totot}
\delta_5=\delta_2^{-1}\delta_1^{-1}\delta_2\delta_1\delta_2.\end{equaz}
In fact, here and in the sequel, we use the
following argument:
$\beta_+$ is a conjugate $\sigma\sigma_4\sigma^{-1}$
of the braid $\sigma_4$
and the braid $\sigma_4$ yields the relation
$\delta_4\hbox{\mat \char61}\hskip1pt\delta_5$.
Therefore, if we set $\delta_4'\hbox{\mat \char61}\hskip1pt
(\delta_4)\sigma^{-1}$, $\delta_5'\hbox{\mat \char61}\hskip1pt
(\delta_5)\sigma^{-1}$, we get the relation
$\delta_4'\hbox{\mat \char61}\hskip1pt\delta_5'$.
By our particular choice of $\sigma$ \begin{eqnarray*}
\delta_5'&\hbox{\mat \char61}\hskip1pt&\delta_5\\
\delta_4'&\hbox{\mat \char61}\hskip1pt&(\delta_4)\sigma_3^{-1}\sigma_2^{-1}\sigma_1^2\\
&\hbox{\mat \char61}\hskip1pt&(\delta_3)\sigma_2^{-1}\sigma_1^2\\
&\hbox{\mat \char61}\hskip1pt&(\delta_2)\sigma_1^2\\&\hbox{\mat \char61}\hskip1pt&(\delta_2^{-1}\delta_1\delta_2)\sigma_1\\
&\hbox{\mat \char61}\hskip1pt&\delta_2^{-1}\delta_1^{-1}\delta_2\delta_1\delta_2\end{eqnarray*}
Similarly, the relations
$\delta_i\hbox{\mat \char61}\hskip1pt(\delta_i)\beta_-$ are equivalent to
the relation \begin{equaz}\label{thothoth}
\delta_3^{-1}\delta_1^{-1}\delta_3\delta_1\delta_3=
(\delta_4\delta_5)^{-3}\delta_5(\delta_4\delta_5)^3.
\end{equaz}
We write down, for convenience of the reader, the
action of the braid
$\beta_1^{-1}$, since the new relations $\delta_i
\hbox{\mat \char61}\hskip1pt(\delta_i)\beta_1\beta_j\beta_1^{-1}$ will be
obtained from the
relations equivalent to
$\delta_i\hbox{\mat \char61}\hskip1pt(\delta_i)\beta_j$ simply by applying
the automorphism $\beta_1^{-1}$.\begin{eqnarray*}
(\delta_1)\beta_1^{-1}&\hbox{\mat \char61}\hskip1pt&
\delta_1\delta_2\delta_3\delta_2^{-1}\delta_1^{-1}
\delta_2^{-1}\delta_1\delta_2\delta_4\delta_2^{-1}\delta_1^{-1}
\delta_2\delta_1\delta_2\delta_3^{-1}\delta_2^{-1}\delta_1^{-1}\\
(\delta_2)\beta_1^{-1}&\hbox{\mat \char61}\hskip1pt&
\delta_1\delta_2\delta_3\delta_2^{-1}\delta_1^{-1}\\
(\delta_3)\beta_1^{-1}&\hbox{\mat \char61}\hskip1pt&\delta_2^{-1}\delta_1\delta_2\delta_4^{-1}
\delta_2^{-1}\delta_1^{-1}\delta_2\delta_1\delta_2\delta_4\delta_2^{-1}
\delta_1^{-1}\delta_2\\
(\delta_4)\beta_1^{-1}&\hbox{\mat \char61}\hskip1pt&\delta_2^{-1}\delta_1\delta_2\\
(\delta_5)\beta_1^{-1}&\hbox{\mat \char61}\hskip1pt&\delta_5
\end{eqnarray*}
Thus, the relations
$\delta_i\hbox{\mat \char61}\hskip1pt(\delta_i)\beta_1\beta_0\beta_1^{-1}$
are equivalent to the relations
\begin{equaz}\label{otwtwo}
(\delta_1\delta_2)^6=(\delta_2\delta_1)^6\end{equaz}
\begin{equaz}\label{ftffff}
\delta_5\delta_3\delta_5^{-1}\delta_4^{-1}\delta_5\delta_4=
\delta_4^{-1}\delta_5\delta_4\delta_5\delta_3\delta_5^{-1}
\end{equaz} where we have used \ref{totot}.\\
The relations
$\delta_i\hbox{\mat \char61}\hskip1pt(\delta_i)\beta_1\beta_+\beta_1^{-1}$ are
equivalent to the relation
\begin{equaz}\label{ftotfftfftot}
\delta_5=\delta_2^{-1}\delta_1\delta_2\delta_4^{-1}\delta_5
\delta_3\delta_5^{-1}\delta_4\delta_2^{-1}\delta_1^{-1}\delta_2\end{equaz}
where we have used \ref{totot}.\\
The relations
$\delta_i\hbox{\mat \char61}\hskip1pt(\delta_i)\beta_1\beta_-\beta_1^{-1}$
pop up to
the relation \begin{equaz}\label{thototh}
\delta_3^{-1}\delta_1\delta_2\delta_1^{-1}\delta_3=
(\delta_4\delta_5)^{-2}\delta_5(\delta_4\delta_5)^2.
\end{equaz} In fact, using \ref{totot} we have
$(\delta_4\delta_5)\beta_1^{-1}=\delta_1\delta_2$
and so
$$((\delta_4\delta_5)^{-3}\delta_5(\delta_4\delta_5)^3)
\beta_1^{-1}=(\delta_1\delta_2)^{-4}\delta_2(\delta_1\delta_2)^4$$
On the other side,
$$(\delta_1\delta_3)\beta_1^{-1}=\delta_1\delta_2
\delta_3\delta_5^{-1}\delta_4\delta_5\delta_3^{-1}\delta_5^{-1}\delta_4^{-1}
\delta_5\delta_4\delta_2^{-1}\delta_1^{-1}\delta_2$$
and using \ref{ftffff} and again \ref{totot}
$$(\delta_1\delta_3)\beta_1^{-1}=\delta_1\delta_2
\delta_3\delta_4\delta_5\delta_3^{-1}\delta_5^{-1}
\delta_2^{-1}\delta_1^{-1}\delta_2=\delta_1\delta_2
\delta_3\delta_4\delta_5\delta_3^{-1}
\delta_2^{-1}\delta_1^{-1}.$$
With the same method we have \begin{eqnarray*}
((\delta_1\delta_3)^{-1}\delta_3\delta_1\delta_3)\beta_1^{-1}&=&
\delta_1\delta_2\delta_3\delta_5^{-1}\delta_4^{-1}(\delta_3^{-1}\delta_5^{-1}
\delta_4^{-1}\delta_5\delta_4\delta_5\delta_3)\delta_4\delta_5\delta_3^{-1}
\delta_2^{-1}\delta_1^{-1}\\
&=&\delta_1\delta_2\delta_3\delta_5^{-1}\delta_4^{-1}
(\delta_5^{-1}\delta_4^{-1}
\delta_5\delta_4\delta_5)\delta_4\delta_5\delta_3^{-1}
\delta_2^{-1}\delta_1^{-1}\\
&=&\delta_1\delta_2\delta_3(\delta_4\delta_5)^{-2}
\delta_5(\delta_4\delta_5)^2\delta_3^{-1}
\delta_2^{-1}\delta_1^{-1}\end{eqnarray*}
So the relation is
$$\delta_3(\delta_4\delta_5)^{-2}\delta_5(\delta_4\delta_5)^2\delta_3^{-1}=
(\delta_1\delta_2)^{-5}\delta_2(\delta_1\delta_2)^5=
\delta_1\delta_2\delta_1^{-1}$$ where we have used
\ref{otwtwo}.\\
Finally we have to write the relations
$\Gamma\delta_i\Gamma^{-1}=
(\delta_i)\beta_1^2$, i.e.
\begin{eqnarray*}
\Gamma\delta_1\Gamma^{-1}&\!\!\!=\!\!\!&\Gamma\delta_2\Gamma^{-1}\delta_4^{-1}
\delta_2^{-1}\delta_1\delta_2\delta_4\Gamma\delta_2^{-1}\Gamma^{-1}
\hbox to
0pt{\hspace*{13.6em}\begin{numera}\label{GoG}\end{numera}\hss}\\
\Gamma\delta_2\Gamma^{-1}&\!\!\!=\!\!\!&\delta_2^{-1}\delta_1\delta_2
\delta_3^{-1}\delta_5\delta_3\delta_2^{-1}\delta_1^{-1}\delta_2
\hbox to
0pt{\hspace*{15.4em}\begin{numera}\label{GtwG}\end{numera}\hss}\\
\Gamma\delta_3\Gamma^{-1}&\!\!\!=\!\!\!&
\delta_4^{-1}\delta_2^{-1}\delta_1^{-1}\delta_2\delta_4\delta_2^{-1}
\delta_1\delta_2
\delta_3^{-1}\delta_5^{-1}\delta_3\delta_5\delta_3
\delta_2^{-1}\delta_1^{-1}\delta_2\delta_4^{-1}\delta_2^{-1}\delta_1\delta_2
\delta_4
\ \ \begin{numera}\label{GthG}\end{numera}\\
\Gamma\delta_4\Gamma^{-1}&\!\!\!=\!\!\!&
\delta_4^{-1}\delta_2^{-1}\delta_1^{-1}\delta_2
\delta_4\delta_2^{-1}\delta_1\delta_2\delta_4
\hbox to
0pt{\hspace*{15em}\begin{numera}\label{GfoG}
\end{numera}\hss}\\
\Gamma\delta_5\Gamma^{-1}&\!\!\!=\!\!\!&\delta_5
\hbox to
0pt{\hspace*{24.8em}
\begin{numera}\label{GfiG} \end{numera}\hss}
\end{eqnarray*}
{\bf Step 4.}\\
We take as Shreier set for the left cosets of the
kernel the set
$\{S_0=1,S_1=\delta_1\}$,
so applying the Reidemeister-Shreier method we get
the generators
$\Delta=\delta_1^2$, $\Gamma$,
$\sigma=\delta_1\Gamma\delta_1^{-1}$,
$A_i=\delta_1\delta_i$ and
$B_i=\delta_i\delta_1^{-1}$ for $i\ugu2,3,4,5$.
For the relations we must rewrite the relations
\ref{fofififo},...,\ref{GfiG} and their conjugate by
$\delta_1$ in terms of
the new generators. The rewriting process goes as
follows (cf.
\cite{makaso}, pages 86-98): \begin{eqnarray*}
S_0\delta_1=S_1&&
S_1\delta_1=\Delta S_0\\
S_0\delta_i=B_iS_1&&
S_1\delta_i=A_iS_0\ \ \ \ {\rm for}\ i\ugu2,3,4,5\\
S_0\Gamma=\Gamma S_0&&
S_1\Gamma=\sigma S_1\end{eqnarray*}
We want to show that it suffices to rewrite only the
relations
\ref{fofififo},...,\ref{GfiG}.\\
Observe that all our relations can be written in the
form
$W\delta_iW^{-1}=\delta_k$ for a suitable word $W$.
Assume that $\Gamma$
doesn't appear in the relation and do the rewriting
after moding out by
the relations
\begin{equaz}\hbox{$\Delta$}=B_iA_i=1.\label{canc1}\end{equaz}
Since $S_0\delta_i\hbox{\mat \char61}\hskip1pt A_i^{-1}S_1$ and also
$S_0\delta_i^{-1}\hbox{\mat \char61}\hskip1pt
A_i^{-1}S_1$, if we write
$\displaystyle
W=\prod_{\lambda=1}^h\delta_{j_{\lambda}}^{\pm1}$,
the rewriting of
$W\delta_iW^{-1}\delta_k^{-1}$ is given by
$$A_{j_1}^{-1}A_{j_2}\cdots A_i^{\pm1}\cdots
A_{j_1}^{-1}A_k$$
(note that $A_1\ugu1$). The rewriting of the same
relation conjugated by
$\delta_1$ yields instead
$$A_{j_1}A_{j_2}^{-1}\cdots A_i^{\mp1}\cdots
A_{j_1}A_k^{-1}.$$
We get thus two relations of respective form
$UA_k=1$, $U^{-1}A_k^{-1}=1$,
which are obviously equivalent.\\
If instead $\Gamma$ appears in the relation, we have
one of the
\ref{GoG},...,\ref{GfiG} which are of the form
$\Gamma\delta_i\Gamma^{-1}=W\delta_iW^{-1}$ where we
can in fact assume that
$\Gamma$ doesn't appear in the word $W$.\\ The
rewriting of
$\Gamma\delta_i\Gamma^{-1}W\delta_i^{-1}W^{-1}$
yields, again a relation of
the form
$$\Gamma A_i^{-1}\sigma^{-1}U^{-1}=1,$$ whereas the
rewriting of the
conjugate by $\delta_1$ gives a relation $$\sigma
A_i\Gamma^{-1}U=1,$$ which
is an equivalent relation.\\
For convenience of notation we shall keep the
generators $B_i=A_i^{-1}$.\\
To calculate $\pi_1(Z_2^{\#})$ we must add the
rewriting of
$(\prod_{i=1}^5\delta_i)^2=1$ which gives
$$A_2B_3A_4B_5B_2A_3B_4A_5=1.$$
We have thus that $\pi_1(Z_2^{\#})$ is generated by
$A_2$, $A_3$, $A_4$,
$A_5$, $\Gamma$ and $\sigma$ and has the
following set of relations
\begin{equaz}\label{3)'} (B_4A_5)^6=(B_5A_4)^3
\end{equaz}
\begin{equaz}\label{1)'} B_3A_2=B_2A_3\end{equaz}
\begin{equaz}\label{4)'} B_5=B_2^3 \end{equaz}
\begin{equaz}\label{2)'} B_3^3=(B_5A_4)^6B_5 \end{equaz}
\begin{equaz}\label{5)'} A_2^{12}=1 \end{equaz}
\begin{equaz}\label{6)'}
B_5A_3B_5A_4B_5A_4=B_4A_5B_4A_5B_3A_5 \end{equaz}
\begin{equaz}\label{7)'}
B_5=B_2^2A_4B_5A_3B_5A_4B_2^2 \end{equaz}
\begin{equaz}\label{8)'} B_3B_2B_3=(B_5A_4)^4B_5
\end{equaz}
\begin{equaz}\label{12)'} \sigma
A_2^2\Gamma^{-1}=A_4B_2^2A_4 \end{equaz}
\begin{equaz}\label{10)'}\Gamma
B_2\sigma^{-1}=B_2^2A_3B_5A_3B_2^2\end{equaz}
\begin{equaz}\label{13)'}\Gamma B_3\sigma^{-1}=
B_4A_2^2B_4A_2^2B_3A_5B_3A_5B_3A_2^2B_4A_2^2B_4\end{equaz}
\begin{equaz}\label{11)'}\Gamma
B_4\sigma^{-1}=B_4A_2^2B_4A_2^2B_4\end{equaz}
\begin{equaz}\label{9)'} \Gamma
B_5\sigma^{-1}=B_5\end{equaz}
\begin{equaz}\label{14)'}
A_2B_3A_4B_5B_2A_3B_4A_5=1\end{equaz}
where $B_i=A_i^{-1}$.\\
Let's reduce this presentation. Using \ref{4)'}
relation \ref{7)'}
becomes
\begin{equaz}B_4A_2B_4=B_5A_3B_5\label{7)''}\end{equaz} and with
this \ref{6)'}
becomes $$B_2^4=1$$ which implies \ref{5)'}, and
changes \ref{4)'} into
$$B_5=A_2.$$
Moreover, using \ref{8)'} and the last equation,
relation
\ref{2)'} gives $$(A_2A_4)^2=A_3A_2B_3^2$$ and with
this, using also
\ref{1)'}, \ref{8)'} becomes
\begin{equaz}\label{8)''}B_3B_2=A_2A_3\end{equaz}
thus transforming \ref{7)''} into
$$B_3=B_4A_2B_4$$ which allows us to delete the
generator $A_3$.
Upon substituting the expressions of $A_5$ and $A_3$
into \ref{14)'} and
\ref{8)''} we have $$A_2A_4=A_4A_2$$ $$A_4^4=1.$$
We can then see that the relations
\ref{1)'},...,\ref{14)'}
are equivalent to the following
$$A_5=A_2^{-1}\ \ \ \ \ \ \
A_3=A_2^{-1}A_4^2$$
$$A_2^4=A_4^4=1\ \ \ \ \ \ \
A_2A_4=A_4A_2$$
$$\sigma A_2^2\Gamma^{-1}=A_2^2A_4^2$$
$$\Gamma A_2^{-1}=A_2^{-1}\sigma$$
$$\Gamma A_2A_4^2=A_2A_4^2\sigma$$
$$\Gamma A_4^{-1}=A_4\sigma$$
$$\Gamma A_2=A_2\sigma.$$
{\bf Step 5.}\\
We take as Shreier set for the left cosets of the
kernel the set
$$\{R_i\hbox{\mat \char61}\hskip1pt\Gamma^i\ |\ i\ugu0,1,\ldots , m\men1\}$$ and
we apply the
Reidemeister-Shreier method.\\
The generators are $\hat\Gamma=\Gamma^m$,
$A_{2,i}=\Gamma^iA_2\Gamma^{-i}$,
$A_{4,i}=\Gamma^iA_4\Gamma^{-i}$ for
$i\ugu0,\ldots , m\men1$
$\sigma_i=\Gamma^i\sigma\Gamma^{-(i+1)}$ for
$i\ugu0,\ldots , m\men2$ and
$\sigma_{m-1}=\Gamma^{m-1}\sigma$.\\
For the rewriting process we have
\begin{eqnarray*}R_iA_j&=&A_{j,i}R_i\ \ \ \ {\rm
for}\ j\ugu2,4\ i\ugu0,\ldots ,
m-1\\
R_i\Gamma&=&R_{i+1}\ \ \ \ \ {\rm for}\ i\ugu0,\ldots , m-2\\
R_{m-1}\Gamma&=&\hat\Gamma R_0\\
R_i\sigma&=&\sigma_i R_{i+1}\ \ \ \ {\rm for}\
i\ugu0,\ldots , m-2\\
R_{m-1}\sigma&=&\sigma_{m-1}R_0.
\end{eqnarray*}
Thus, taking indices $i$ (mod$m$) and adding (as we
must) the relation
$\hat\Gamma=1$, we obtain the relations
$$A_{2,i}^4=A_{4,i}^4=1$$
$$A_{2,i}A_{4,i}=A_{4,i}A_{2,i}$$
$$\sigma_iA_{2,i+1}^2=A_{2,i}^2A_{4,i}^2$$
$$A_{2,i+1}^{-1}=A_{2,i}^{-1}\sigma_i$$
$$A_{2,i+1}A_{4,i+1}^2=A_{2,i}A_{4,i}^2\sigma_i$$
$$A_{4,i+1}^{-1}=A_{4,i}\sigma_i$$
$$A_{2,i+1}=A_{2,i}\sigma_i.$$
To simplify this presentation we write
\begin{eqnarray*}\sigma_i&=&A_{2,i}^2A_{4,i}^2A_{2,i+1}^2\\
&=&A_{2,i}A_{2,i+1}^{-1}\\
&=&A_{4,i}^2A_{2,i}^{-1}A_{2,i+1}A_{4,i+1}^2=
A_{2,i}^{-1}A_{4,i}^2A_{4,i+1}^2A_{2,i+1}\\
&=&A_{4,i}^{-1}A_{4,i+1}^{-1}\\
&=&A_{2,i}^{-1}A_{2,i+1}\end{eqnarray*}
From the last and the second equations we get
$$A_{2,i}^2=A_{2,i+1}^2=A_{2,0}^2$$ and from the
first one,
remembering that $A_{2,i}$ commutes with $A_{4,i}$
and that $A_{2,0}^4=1$,
$$\sigma_i=A_{4,i}^2.$$
The fourth equation then
gives $$A_{4,i}=A_{4,i+1}=A_{4,0}$$ which makes the
last and the
third relations equivalent. These two cancellation
relations
enable us to delete all the generators
$\sigma_j$ and $A_{4,i}$ for $i\ugu1,\ldots , m-1$.\\
We may rewrite the five relations above as
\begin{eqnarray*}\sigma_i&=&A_{4,0}^2\\
A_{4,i}&=&A_{4,0}\\
A_{4,0}^2&=&A_{2,i}^{-1}A_{2,i+1}\\
A_{4,0}^2&=&A_{2,i}A_{2,i+1}^{-1}.
\end{eqnarray*}
Clearly the last two equations are equivalent and give
\begin{equaz}A_{2,2i}=A_{2,0}\ \ \ \ \ \
A_{2,2i+1}=A_{2,0}A_{4,0}^2\label{abv}.\end{equaz}
Moreover, if we add the relation $\sigma^m=1$, that
in the generators of
$\pi_1(X_{2m}^{\#})$ reads out as
$\sigma_0\sigma_1\cdots\sigma_{m-1}=1$, we
get $A_{4,0}^{2m}=1,$ i.e., if $m$ is odd,
$A_{4,0}^2=1$, while if $m$ is
even we have no new relations. Observe that this is
in accordance with the
fact that in \ref{abv} the index is cyclic mod$(m)$.\\
Summing up, we have a commutative group with only
two generators, namely
$a=A_{2,0}$ and $b=A_{4,0}$, such that $a^4=1$ and
$b^4=1$ if $m$ is even,
$b^2=1$ if $m$ is odd, i.e.
$$\pi_1(X^{\#}_{2k+2})=\zeta_4\times\zeta_4$$ if $k$
is odd and
$$\pi_1(X^{\#}_{2k+2})=\zeta_4\times\zeta_2$$ if $k$
is even.
|
1996-07-23T20:36:16 | 9607 | alg-geom/9607024 | en | https://arxiv.org/abs/alg-geom/9607024 | [
"alg-geom",
"math.AG"
] | alg-geom/9607024 | Rahul Pandharipande | R. Pandharipande | The Equivariant Chow Ring of SO(4) | 10 pages, Latex2e | null | null | null | null | The integral equivariant Chow ring of S0(4) is computed via the geometry of
ruled quadric surfaces in P^3.
| [
{
"version": "v1",
"created": "Tue, 23 Jul 1996 18:32:10 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Pandharipande",
"R.",
""
]
] | alg-geom | \section{\bf{Introduction}}
Let $\mathbf{G}$ be a reductive algebraic group.
The algebraic analogue of $E\mathbf{G}$ is attained by
approximation. Let $V$ be a $\mathbb{C}$-vector space. Let
$\mathbf{G}\times V \rightarrow V$ be an
algebraic representation of $\mathbf{G}$.
Let $W\subset V$ be a $\mathbf{G}$-invariant open set satisfying:
\begin{enumerate}
\item[(i)] The complement of $W$ in $V$ is of codimension greater than
$q$.
\item[(ii)] $\mathbf{G}$ acts freely on $W$.
\item[(iii)] There exists a geometric quotient $W\rightarrow W/\mathbf{G}$.
\end{enumerate}
$W$ is an approximation of $E\mathbf{G}$ up to codimension $q$.
Let $e=dim( W/{\mathbf{G}})$.
The equivariant Chow groups of $\mathbf{G}$ (acting on a point)
are defined by:
\begin{equation}
\label{defff}
A^{\mathbf{G}}_{-j}(\text{point})= A_{e-j}(W/\mathbf{G})
\end{equation}
for $0\leq j \leq q.$
An argument is required to check the
Chow groups are well-defined (see [EG1]).
The basic properties of equivariant
Chow groups are
established in [EG1]. In particular,
there is a natural intersection
ring structure on $A_i^{\mathbf{G}}(\text{point})$.
For notational convenience, a superscript will
denote the Chow group codimension:
$$A^{\mathbf{G}}_{-j}(\text{point}) = A^j_\mathbf{G}(\text{point}).$$
Equation (\ref{defff}) becomes:
$$\forall\ 0\leq j \leq q, \ \
A_{\mathbf{G}}^{j}(\text{point})= A^j(W/\mathbf{G}).$$
$W/\mathbf{G}$ is an approximation of $B \mathbf{G}$.
$A_{\mathbf{G}}^*(\text{point})$ is called the (equivariant) Chow ring of
$\mathbf{G}$.
$A_{\mathbf{G}}^*(\text{point})$ is naturally isomorphic to the
ring of algebraic characteristic classes of (\'etale locally
trivial) principal $\mathbf{G}$-bundles.
The equivariant Chow ring of $\mathbf{G}$ was
first defined by B. Totaro in [T].
Consider now the
orthogonal and special orthogonal algebraic groups
(over $\mathbb{C}$).
The Chow ring of $\mathbf{O} (n)$ is
generated by the Chern classes
of the standard representation. The odd classes
are 2-torsion:
$$A^*_{\mathbf{O} (n)}(\text{point})=
\mathbb{Z}[c_1, \ldots, c_n]/(2c_1, 2c_3, 2c_5, \ldots).$$
The Chow ring of $\mathbf{SO} (n=2k+1)$
is also generated by the Chern classes
of the standard representation. The odd classes
are 2-torsion and $c_1=0$:
$$A^*_{\mathbf{SO} (n=2k+1)}(\text{point})=
\mathbb{Z}[c_1, \ldots, c_n]/(c_1, 2c_3, 2c_5, \ldots).$$
$A^*_{\mathbf{O} (n)}(\text{point})$ was first computed
by B. Totaro. Algebraic computations of
$A^*_{\mathbf{O} (n)}(\text{point})$ and $A^*_{\mathbf{SO} (2k+1)}(\text{point})$
can be found in [P2]. The Chow ring of
$\mathbf{SO}(n)$ has been computed with $\mathbb{Q}$-coefficients
in [EG2]. The integral Chow ring of
$\mathbf{SO}(n=2k)$ is not known in general.
Since
$B \mathbf{O} (n)$ is approximated by the set of
non-degenerate quadratic forms in $Sym^2 S^*$ (where
$S \rightarrow \mathbf G(n, \infty)$ is the
tautological sub-bundle), the Chow ring
can be analyzed by degeneracy calculations ([P2]).
Algebraic
$B \mathbf{SO} (n)$ double covers $B \mathbf{O}(n)$.
If $n=2k+1$, there
is a product decomposition $\mathbf{O}(n)\stackrel{\sim}{=} \mathbb{Z}/2\mathbb{Z} \times \mathbf{SO}(n)$.
As a result,
the double cover geometry for $n=2k+1$ is tractable and
a computation of $A_{\mathbf{SO}(2k+1)}^*(\text{point})$ can be made.
In case $n=2k$, the double cover geometry is more
complicated.
Since
$\mathbf{SO}(2)\stackrel{\sim}{=} \mathbb{C}^*$, the first non-trivial even case is
$\mathbf{SO}(4)$.
In this paper,
the ring $A^*_{\mathbf{SO}(4)}(\text{point})$ is determined.
Let $c_1,c_2,c_3,c_4$ be Chern classes of the
standard representation of $\mathbf{SO}(4)$. Let $F$ be one of the
two distinct irreducible $3$-dimensional representations of $\mathbf{SO}(4)$, and
let $f_2$ be the second Chern class $F$.
\begin{tm}
$A_{\mathbf{SO}(4)}^*(\text{\em point})$ is generated by the Chern classes
$c_1, c_2, c_3,$ $c_4,$ and $f_2$. Define $x\in A_{\mathbf{SO}(4)}^{2}$ by
$x=c_2-f_2$.
$$A_{\mathbf{SO}(4)}^*(\text{\em point})= \mathbb{Z}[c_1,c_2,c_3,c_4,x]/
(c_1, 2c_3, xc_3, x^2-4c_4)$$
\end{tm}
\noindent Let $\tilde{F}$ be the other irreducible $3$-dimensional
representation of $\mathbf{SO}(4)$. Let $\tilde{f}_2$ be the
second Chern class of $\tilde{F}$. Since (see [FH])
$$F \oplus \tilde{F} \stackrel{\sim}{=} \wedge^2 V,$$
the relation $\tilde{f}_2= 2c_2-f_2$ is obtained.
Hence, $c_2-\tilde{f}_2= -x$. The presentation in
Theorem 1 does not depend upon the choice of $3$-dimensional
representation.
Thanks are due to D. Edidin, W. Fulton, W. Graham, and B. Totaro for
conversations about $B \mathbf{SO}(n)$. The $\mathbf{SO} (4)$ calculation
presented here is similar in spirit to the $\mathbf{SO}(2k+1)$
calculations of [P2]. In [P1] and [P2], equivariant Chow
rings are used to compute ordinary Chow rings of certain
moduli spaces of maps and Hilbert schemes of rational curves.
\section{\bf{Ruled Quadric Surfaces}}
\label{rrr}
Let $V\stackrel{\sim}{=} \mathbb{C}^4$ be equipped with
a non-degenerate quadratic form $Q$. A {\em ruled quadric
surface} in $\mathbf P(V)$ is a pair $(X,r)$ where $X\subset \mathbf P(V)$
is a nonsingular quadric surface and $r$ is a choice of ruling.
Let
$\mathcal{X}\subset \mathbf P(Sym^2 V^*)$ be the parameter space
of nonsingular quadrics.
The parameter space of ruled quadrics,
$\mathcal{X}_{ruled}$, is an \'etale double
cover of $\mathcal{X}$ via the natural map:
$\mathcal{X}_{ruled} \rightarrow \mathcal{X}$.
There are natural maps $\mathbf{SO}(V) \rightarrow \mathbf P\mathbf{SO}(V)\subset \mathbf{PGL}(V)$
and
$\mathbf{O}(V) \rightarrow \mathbf P\mathbf{O}(V) \subset \mathbf{PGL}(V)$.
Let $\mathbf{SO}(V)$ and $\mathbf{O}(V)$ act on $\mathbf{PGL}(V)$ on the
right via these maps.
There exist geometric
quotients (see [P2]):
$$\mathbf{PGL}(V)/\mathbf{SO}(V) \rightarrow \mathbf{PGL}(V)/\mathbf{O}(V).$$
Consider the quadric surface $(Q)\subset \mathbf P(V)$
obtained from the quadratic form.
The standard left action $\mathbf{PGL}(V)\times \mathbf P(V) \rightarrow
\mathbf P(V)$ yields a transitive $\mathbf{PGL}(V)$-action on the
space of nonsingular quadric surfaces. The
stabilizer of $(Q)$ for this action is exactly
$\mathbf P\mathbf{O}(V) \subset \mathbf{PGL}(V)$. Hence, there is a
canonical isomorphism
$$\mathbf{PGL}(V)/\mathbf{O}(V) \stackrel{\sim}{=} \mathcal{X}.$$
For the entire paper, fix a ruling $r$ of $(Q)$.
Since $\mathbf{PGL}(V)$ acts transitively on the
space of ruled quadrics and the stabilizer of
$((Q),r)$ is exactly $\mathbf P\mathbf{SO}(V) \subset \mathbf{PGL}(V)$,
there is a canonical isomorphism determined by $((Q),r)$:
$$\mathbf{PGL}(V)/\mathbf{SO}(V) \stackrel{\sim}{=} \mathcal{X}_{ruled}.$$
There is a canonical Pl\"ucker embedding
$\mathbf G(2, V) \hookrightarrow \mathbf P(\wedge^2 V)$.
Let $Z\subset \mathbf G(3, \wedge^2 V)$ be the
open locus of $2$-planes in $\mathbf P(\wedge^2 V)$
which intersect $\mathbf G(2,V)$ transversely in a nonsingular
conic curve.
\begin{lm}
\label{trick}
There is a canonical isomorphism $Z \stackrel{\sim}{=} \mathcal{X}_{ruled}$.
\end{lm}
\noindent {\em Proof.}
The family of lines determined by
a nonsingular
plane conic $C \subset \mathbf G(2,V) \subset \mathbf P(\wedge^2 V)$
sweeps out an irreducible degree $2$ surface in $\mathbf P(V)$.
There are three possibilities for this degree $2$ surface:
a double plane, a quadric cone, or a nonsingular
quadric surface. If the conic $C$ sweeps out a a double
plane $H\subset \mathbf P(V)$,
then $C \subset P \subset \mathbf G(2,V)$ where
$P$ is the plane of all lines contained in $H$.
If $C$ sweeps out a quadric cone, then $C\subset P \subset
\mathbf G(2,V)$ where $P$ is the plane of all lines
passing through the vertex of the cone. Hence, if
$C$ is the transverse intersection $P\cap \mathbf G(2,V)$
of a plane, then $C$ must correspond to a ruling
of a unique nonsingular quadric surface.
Conversely, a ruling
of a nonsingular quadric surface yields a conic curve
in $\mathbf G(2,V)$ which is the transverse intersection of
a unique $2$-plane in $\mathbf P(V)$. These maps
are easily seen to be algebraic.
\qed \vspace{+10pt}
By Lemma 1,
the ruled quadric $((Q),r)$ corresponds to
a $3$-dimensional subspace $F\subset \wedge^2 V$.
$F$ is $\mathbf{SO}(V)$-invariant since $((Q),r)$ is stabilized
by $\mathbf{SO}(V)$. $F$ is therefore a $3$-dimensional representation
of $\mathbf{SO}(V)$. Let $s$ be the {\em other} ruling of $(Q)$.
$((Q),s)$ similarly corresponds to an invariant
$3$-dimensional subspace
$\tilde{F} \subset \wedge^2 V$. The $\mathbf{SO}(V)$ representation
$\wedge^2 V$ decomposes as $\wedge^2 V \stackrel{\sim}{=} F \oplus \tilde{F}$.
\section{ $B\mathbf{SO}(V)$}
\label{bsofour}
Let $V\stackrel{\sim}{=} \mathbb{C}^4$ be equipped with a non-degenerate
quadratic form as before. Approximations to
$E\mathbf{SO}(V)$ and $B\mathbf{SO}(V)$ are obtained via direct sums of the
representation $V^*$.
Let $m>>0$ and let
$$W_m \subset \oplus_{1}^{m} V^*$$
denote the spanning locus. $W_m$ is the
locus of $m$-tuples of vectors of $ V^*$
which span $V^*$.
The natural action of $\mathbf{SO} (V)$ on
$W_m$ is free and has a geometric quotient
(see [P2]).
The codimension of the complement of
$W_m$ in $\oplus_{1}^{m} V^*$ is $m-3$.
$W_m$ is an approximation of
$E\mathbf{SO} (V)$ up to codimension $m-4$.
Therefore
$$B\mathbf{SO} (V)=\ \stackrel{Lim}{m \rightarrow \infty} \ W_m/\mathbf{SO} (V),$$
$$A^*_{\mathbf{SO} (V)}(\text{point})=\ \stackrel{Lim}{m \rightarrow \infty} \
A^*(W_m/\mathbf{SO} (V)).$$
There is a scalar $\mathbb{C}^*$-action on $W_m$.
Let $\mathbf P(W_m)=W_m/\mathbb{C}^*$. Since this $\mathbb{C}^*$-action
commutes with the $\mathbf{SO}(V)$-action, there is diagram of
quotients:
\begin{equation}
\label{diax}
\begin{CD}
W_m @>{\tau_1}>> W_m/ \mathbf{SO}(V) \\
@V{i_1}VV @V{i_2}VV \\
\mathbf P(W_m)= W_m/\mathbb{C}^* @>{\tau_2}>>\mathbf P( W_m)/ \mathbf{SO} (V)\\
\end{CD}
\end{equation}
All the maps in (\ref{diax}) are quotient maps:
\begin{enumerate}
\item[(i)] $i_1$ is a free $\mathbb{C}^*$-quotient.
\item[(ii)] $i_2$ is a free $\mathbb{C}^*/(\pm)$-quotient.
\item[(iii)] $\tau_1$ is a free $\mathbf{SO}(V)$-quotient.
\item[(iv)] $\tau_2$ is a free $\mathbf P\mathbf{SO}(V)$-quotient.
\end{enumerate}
The existence of these quotients is easily deduced (see [P2]).
First consider the space $\mathbf P(W_m)/ \mathbf{SO}(V)$.
Let $Q$ be the quadratic form on $V$ and let
$r$ be the ruling of the quadric surface $(Q)\subset \mathbf P(V)$
fixed in section \ref{rrr}.
An element $f \in \mathbf P(W_m)$ yields a canonical embedding
$$\mu_f: \mathbf P(V) \hookrightarrow \mathbf P(\mathbb{C}^m).$$
The image under $\mu_f$ of $((Q),r)$ is
a ruled quadric surface in $\mathbf P(\mathbb{C}^m)$
associated canonically to $f \in \mathbf P(W_m)$.
Since $\mathbf P\mathbf{SO}(V)\subset \mathbf{PGL}(V)$
is exactly the stabilizer of the ruled quadric $((Q),r)$,
it follows that $\mathbf P(W_m)/ \mathbf{SO}(V)$ is isomorphic
to the parameter space of ruled quadric surfaces in $\mathbf P(\mathbb{C}^m)$.
Since a ruled quadric surface in $\mathbf P(\mathbb{C}^m)$
spans a unique $3$-plane in
$\mathbf P(\mathbb{C}^m)$, the parameter space is fibered over $\mathbf G(4,m)$.
By Lemma \ref{trick}, the parameter space of ruled
quadric surface in $\mathbf P(\mathbb{C}^m)$ is canonically
isomorphic to an
open set $$Z \subset \mathbf G(3, \wedge^2 S)$$
where $S\rightarrow \mathbf G(4,m)$
is the tautological sub-bundle.
The Chow computations in section \ref{calcc} will require two
results about line bundles. We have seen
$\mathbf P(W_m)/\mathbf{SO}(V)$ is canonically fibered over
$\mathbf G(4,m)$. Let $c_1$ be the first Chern
class of the tautological bundle $S$ on $\mathbf G(4,m)$.
Let $c_1$ also denote the pull-back of this class
to $\mathbf P(W_m)/ \mathbf{SO}(V)$.
For $m>4$, $A^1(\mathbf P(W_m)) \stackrel{\sim}{=} \mathbb{Z}$ with generator
$c_1({\mathcal{O}}_{\mathbf P}(-1))$ (which is the Chern class of the
line bundle associated
to the $\mathbb{C}^*$-bundle $i_1$.)
\begin{lm}
\label{lastl} $\tau_2^*(c_1)= c_1({\mathcal{O}}_{\mathbf P}(-4))$.
\end{lm}
\noindent {\em Proof.} Elements of $\mathbf P(W_m)$
correspond to embeddings of $\mathbf P(V)$ in $\mathbf P(\mathbb{C}^m)$.
The class $\tau_2^*(-c_1)$ is the divisor class
of embeddings that meet a fixed $(m-5)$-plane in $\mathbf P(\mathbb{C}^m)=
\mathbf P^{m-1}$. This divisor class is determined by a $4\times 4$
determinant. Hence, $\tau_2^*(-c_1)=c_1({\mathcal{O}}_{\mathbf P}(4))$.
\qed \vspace{+10pt}
The map $i_2: W_m/ \mathbf{SO}(V) \rightarrow \mathbf P(W_m)/ \mathbf{SO}(V)$
is a $\mathbb{C}^*/(\pm)$-bundle.
Since there is an abstract isomorphism $\mathbb{C}^*/(\pm)\stackrel{\sim}{=} \mathbb{C}^*$,
$i_2$ is also a $\mathbb{C}^*$-bundle. Let $N$ be the
line bundle on $\mathbf P(W_m)/\mathbf{SO}(V)$
canonically associated to $i_2$.
\begin{lm}
\label{twot}
$\tau_2^*(N) \stackrel{\sim}{=} {\mathcal{O}}_{\mathbf P}(-2)$.
\end{lm}
\noindent {\em Proof.}
Let $i_1/(\pm): W_m/(\pm) \rightarrow \mathbf P(W_m)$. The map
$i_1/(\pm)$ is a free $\mathbb{C}^*/(\pm)$-quotient.
The line bundle associated to the $\mathbb{C}^*/(\pm)$-bundle
$i_1/(\pm)$ is ${\mathcal{O}}_{\mathbf P}(-2)$.
The map $\tau_1/(\pm): W_m/(\pm) \rightarrow W_m/ \mathbf{SO}(V)$ is
$\mathbb{C}^*/(\pm)$-equivariant. Hence, $\tau_2^*(N)\stackrel{\sim}{=}
{\mathcal{O}}_{\mathbf P}(-2)$.
\qed \vspace{+10pt}
\section{\bf{ Chow Calculations}}
\label{calcc}
In this section,
the Chow ring of $W_m/ \mathbf{SO}(V)$ is determined
(up to codimension $m-4$).
Consider the parameter space of ruled quadrics in
$\mathbf P(\mathbb{C}^m)$:
$$Z \subset \mathbf G(3,\wedge^2 S) \rightarrow \mathbf G(4,m).$$
Let $D$ be the complement of $Z$ in $\mathbf G(3, \wedge^2 S)$.
Following the notation of
section \ref{bsofour}, $W_m/\mathbf{SO}(V)$ is the
$\mathbb{C}^*$-bundle associated to a line bundle $N\rightarrow Z$. Therefore,
$$A^*(W_m/\mathbf{SO}(V)) \stackrel{\sim}{=} A^*(Z)/ (c_1(N)) \stackrel{\sim}{=}
A^*(\mathbf G(3,\wedge^2 S))/(I_D, c_1(\overline{N}))$$
where $I_D \subset A^*(\mathbf G(3,\wedge^2 S))$ is the ideal
generated by cycles supported on $D$ and
$\overline{N}$ is any extension of $N$ to $\mathbf G(3,\wedge^2 S)$.
The ideal $I_D$ is
determined by constructing a well-behaved variety which surjects
onto $D$.
Let
$\mathbf G(2,S) \hookrightarrow \mathbf P(\wedge^2 S)$
be the canonical relative Pl\"ucker embedding.
$D$ is exactly the locus of $2$-planes in the
the fibers of $\mathbf P(\wedge^2 S)$ which do not
meet $\mathbf G(2,S)$ transversely in a nonsingular
conic curve. Equivalently, $D$ is the locus
of $2$-planes $P$ in the fibers of $\mathbf P(\wedge^2 S)$ which
satisfy one of the following conditions:
\begin{enumerate}
\item[(i)] $P\cap \mathbf G(2,S)$ is a pair of distinct lines in $P$.
\item[(ii)] $P \cap \mathbf G(2,S)$ is a double line in $P$.
\item[(iii)] $P \cap \mathbf G(2,S)= P$.
\end{enumerate}
$D$ is dominated by a canonical Grassmannian bundle over
$\mathbf G(2,S)$.
Let $B\rightarrow \mathbf G(2,S)$ be the tautological sub-bundle.
By wedging, there is canonical surjective
bundle map on $\mathbf G(2,S)$:
$$\wedge^2 S \otimes \wedge^2 B \rightarrow \wedge^4 S$$
which induces a
canonical sequence on $\mathbf G(2,S)$:
\begin{equation}
\label{aaa}
0 \rightarrow K \rightarrow \wedge^2 S \rightarrow \wedge^4 S
\otimes (\wedge^2 B)^*\rightarrow
0.
\end{equation}
There is a canonical inclusion $\wedge^2 B \subset K$ and
a quotient sequence
\begin{equation}
\label{bbb}
0 \rightarrow \wedge^2 B \rightarrow K \rightarrow E \rightarrow 0
\end{equation}
on $\mathbf G(2,S)$.
The geometric interpretation of these sequences is as
follows. Let $\xi\in \mathbf G(2,S)$. $\mathbf P(K_\xi)\subset\mathbf P(
\wedge^2 S_\xi)$ is the projective tangent space to
$\mathbf G(2, S_\xi)$ at $\xi$. $\mathbf P(\wedge^2 B_\xi)$ in
$\mathbf P(\wedge^2 S_\xi)$ is the Pl\"ucker image of the point
$\xi$. The fiber of the Grassmannian bundle
$$\mathbf G(2,E) \rightarrow \mathbf G(2,S)$$ over $\xi$
corresponds to the $2$-planes $P$ of $\mathbf P(\wedge^2 S_{\xi})$
that are tangent to $\mathbf G(2,S)$ at $\xi$.
There is a canonical map
$$\rho: \mathbf G(2,E) \rightarrow D$$
which is a surjection of algebraic varieties.
Let $[P]\in D$. The fiber of $\rho$ over $[P]$
is simply the set of points of
$P \cap \mathbf G(2,S)$ where $P$ is tangent to
$\mathbf G(2,S)$. In case (i) above, the fiber is a point.
In case (ii), the fiber is a straight line in $\mathbf P(\wedge^2 S)$.
In case (iii), the fiber is $2$-plane in $\mathbf P(\wedge^2 S)$.
Hence, there is stratification of $D$ by intersection type (i-iii)
where $\rho$ is a projective bundle over each stratum.
The Chow groups of $\mathbf G(2,E)$ therefore surject upon
the Chow groups of $D$.
The ideal $I_D$ is determined by calculating the
push-forwards of the Chow classes of $\mathbf G(2,E)$ to
$\mathbf G(3, \wedge^2 S)$. Consider the projection
$$\pi:\mathcal{G}= \mathbf G(3, \wedge^2 S) \times
_{\mathbf G(4,m)} \mathbf G(2,S) \rightarrow \mathbf G(3,\wedge^2 S).$$
The sequences (\ref{aaa}) and (\ref{bbb}) pull-back to
$\mathcal{G}$.
Let $F\rightarrow \mathbf G(3, \wedge^2 S)$ denote the
tautological sub-bundle (and also let $F$ denote the
pull-back to $\mathcal{G}$ of this bundle).
There is a canonical inclusion
$$\iota: \mathbf G(2,E) \hookrightarrow \mathcal{G}$$
determined by the sequences (\ref{aaa}) and (\ref{bbb}).
$\mathbf G(2,E)\subset \mathcal{G}$ is the closed
subvariety of points $g \in \mathcal{G}$ where
$$\wedge^2 B_g \subset F_g \subset K_g.$$
The class $[\mathbf G(2,E)]$ in Chow ring
of $A^*(\mathcal{G})$ is easily found by degeneracy calculations.
Let $c_1,c_2, c_3, c_4$ be the Chern classes of $S\rightarrow\mathbf G(4,m)$.
Let $b_1, b_2$ be the Chern classes of $B \rightarrow \mathbf G(2,S)$.
Let $f_1,f_2,f_3$ be the Chern classes of
$F\rightarrow \mathbf G(3,\wedge^2 S)$. Since $\mathcal{G}$ is
a tower of Grassmannian bundles, these Chern classes
$c_i, b_j, f_k$ generate $A^*(\mathcal{G})$.
Let $Y$ be
the locus of points $g\in \mathcal{G}$ such that $F_g \subset K_g$.
$Y$
is the nonsingular degeneracy locus of the canonical
bundle map on $\mathcal{G}$,
$$ F \rightarrow \wedge^4 S \otimes (\wedge^2 B)^*,$$
obtained from the inclusion $F \subset \wedge^2 S$ and
sequence (\ref{aaa}).
By the Thom-Porteous formula on $\mathcal{G}$ (see [F]),
\begin{eqnarray*}
A^*(\mathcal{G})\ni
[Y] &= &c_3(F^*\otimes \wedge^4 S \otimes (\wedge^2 B)^*) \\
&=& -f_3 +(c_1-b_1)f_2- (c_1-b_1)^2 f_1+(c_1-b_1)^3.
\end{eqnarray*}
$Y$ is canonically isomorphic to
the Grassmannian bundle $\mathbf G(3, K) \rightarrow \mathbf G(2,S)$.
There is natural bundle quotient sequence on $Y$:
\begin{equation}
\label{ccc}
0 \rightarrow F \rightarrow K \rightarrow K/F \rightarrow 0.
\end{equation}
The locus $\mathbf G(2,E)\subset Y$ is the set
of points $y\in Y$ such that
$\wedge^2 B_y \subset F_y$. $\mathbf G(2,E)\subset Y$
is the nonsingular degeneracy
locus of the canonical bundle map on $Y$,
$$\wedge^2 B \rightarrow K/F,$$
obtained from the sequences (\ref{bbb}) and (\ref{ccc}).
By the Thom-Porteous formula on $Y$,
\begin{eqnarray*}
A^*(Y)\ni
[\mathbf G(2,E)] & = &c_2( (K/F)\otimes (\wedge^2 B)^*) \\
& =& b_1^2-c_1b_1+ c_1^2-2c_1 f_1+ f_1^2-f_2+2c_2.
\end{eqnarray*}
The class $[\mathbf G(2,E)] \in A^*(\mathcal{G})$ is there
expressed by
\begin{eqnarray*}
A^*(\mathcal{G}) \ni [\mathbf G(2,E)] & = &
\big(-f_3 +(c_1-b_1)f_2- (c_1-b_1)^2 f_1+(c_1-b_1)^3\big) \\
& &
\cdot \big(b_1^2-c_1b_1+ c_1^2-2c_1 f_1+ f_1^2-f_2+2c_2\big).
\end{eqnarray*}
Since $\mathbf G(2,E)$ is a Grassmannian bundle over $\mathbf G(2,S)$,
the Chow ring of $\mathbf G(2,E)$ is generated over $A^*(\mathbf G(2,S))$
by the Chern classes $h_1, h_2$ of the tautological
sub-bundle $H\rightarrow \mathbf G(2,E)$. Via
the embedding $\iota:\mathbf G(2,E) \hookrightarrow
\mathcal{G}$, $H$ is isomorphic to $\iota^*(F)/
\iota^*(\wedge^2 B)$.
The Chern classes
$h_1$ and $h_2$ can be expressed via
$\iota$ in terms of the
classes $b_j$ and $f_k$.
Therefore, the classes
\begin{equation*}
\mathbf G(2,E) \cap M(c_1,c_2,c_3,c_4,b_1,b_2,f_1,f_2,f_3)
\end{equation*}
(where $M$ is monomial in the Chern classes) span the
Chow ring of $\mathbf G(2,E)$. The ideal $I_D \subset A^*(\mathbf G(3,
\wedge^2 S))$ is generated by the $\pi$
push-forwards of the classes (\ref{clazz}):
\begin{equation}
\label{clazz}
[\mathbf G(2,E)] \cdot M(c_1,c_2,c_3,c_4,b_1,b_2,f_1,f_2,f_3) \in
A^*(\mathcal{G})
\end{equation}
Since the classes $c_i, f_k$ in $A^*(\mathcal{G})$ are
pull-backs from $\mathbf G(3,\wedge^2 S)$, $I_D$ is generated
by the elements
$$\pi_*([\mathbf G(2,E)] \cdot M(b_1,b_2)).$$
By the standard relations satisfied by the classes
$b_1$ and $b_2$ over $A^*(\mathbf G(4,m))$, it follows that
$I_D$ is generated by:
$$\pi_*([\mathbf G(2,E)])$$
$$\pi_*([\mathbf G(2,E)]\cdot b_1)$$
$$\pi_*([\mathbf G(2,E)]\cdot b_1^2) \ \ \ \pi_*([\mathbf G(2,E)]\cdot b_2)$$
$$\pi_*([\mathbf G(2,E)]\cdot b_1b_2)$$
$$\pi_*([\mathbf G(2,E)]\cdot b^2_1b_2).$$
Since the class $[\mathbf G(2,E)]$ is determined explicitly in
$A^*(\mathcal{G})$,
these six push-forwards can be easily computed by hand or by the
the Maple package Schubert ([KS]).
The first push-forward is:
$$\pi_*([\mathbf G(2,E)])=13c_1-2f_1.$$
\begin{lm}
\label{killl}
The pair $(13c_1-2f_1, c_1(\overline{N}))$ generates
$A^1(\mathbf G(3,\wedge^2 S))$.
\end{lm}
\noindent {\em Proof.}
Recall the notation of diagram (\ref{diax}):
$$\tau_2: \mathbf P(W_m) \rightarrow Z \subset \mathbf G(3,\wedge^2 S).$$
Let $L=c_1({\mathcal{O}}_{\mathbf P}(-1))$ be a generator of
$A^1(\mathbf P(W_m))$.
By Lemma 2, $\tau_2^*(c_1)\stackrel{\sim}{=} 4L$.
Since $\tau_2^*([D])=\tau_2^*(13c_1-2f_1)=0$,
$\tau_2^*(f_1)=26 L$. Therefore the
image of $\tau_2^*$ is the subgroup $\mathbb{Z}(2L)$.
Since $[D]=13c_1-2f_1$ is not divisible in
$A^1(\mathbf G(3,\wedge^2 S))$, $A^1(Z)\stackrel{\sim}{=} \mathbb{Z}$ and
$\tau_2^*$ is an isomorphism:
$$\tau_2^*: A^1(Z) \stackrel{\sim}{\rightarrow} \mathbb{Z}(2L).$$
By Lemma 3, $\tau_2^*(c_1(N))=2L$. Therefore,
$c_1(N)$ generates $A^1(Z)$. It now follows that the pair
$(13c_1-2f_1, c_1(\overline{N}))$ generates the group
$A^1(\mathbf G(3,\wedge^2 S))$.
\qed \vspace{+10pt}
\noindent
Therefore, $(I_D, c_1(\overline{N}))=(I_D,c_1, f_1).$
By Lemma \ref{killl}, it suffices to compute
the five remaining push-forwards
modulo the ideal $J=(c_1,f_1)$. The
results are (modulo $J$):
\begin{enumerate}
\item[{}] $\pi_*([\mathbf G(2,E)]\cdot b_1)=0.$
\item[{}] $\pi_*([\mathbf G(2,E)]\cdot b_1^2)=-2f_3.$
\item[{}] $\pi_*([\mathbf G(2,E)]\cdot b_2)=c_3-f_3.$
\item[{}] $\pi_*([\mathbf G(2,E)]\cdot b_1b_2)=(c_2-f_2)^2-4c_4.$
\item[{}] $\pi_*([\mathbf G(2,E)]\cdot b_1^2 b_2)=c_2f_3+f_2c_3.$
\end{enumerate}
Hence $(I_D, c_1(\overline{N}))=(c_1,f_1,2c_3, c_3-f_3,
(c_2-f_2)^2-4c_4, (c_2-f_2)c_3)$.
The ring $A^*(\mathbf G(4,m))$ is freely generated
(up to codimension $m-4$) by $c_1,c_2,c_3,c_4$.
The ring $A^*(\mathbf G(3,\wedge^2 S))$ has the
following presentation (up to codimension $m-4$):
$$A^*(\mathbf G(3,\wedge^2 S))\stackrel{\sim}{=} \mathbb{Z}[c_1,c_2,c_3,c_4, f_1,f_2,f_3]/
(t_4,t_5,t_6)$$
where the $t_4,t_5, t_6$ are the Chern classes of the tautological
quotient bundle $T$: $0 \rightarrow F \rightarrow \wedge^2 S \rightarrow T \rightarrow 0.$
We find (modulo $J$):
\begin{enumerate}
\item[{}] $t_4= (c_2-f_2)^2-4c_4$.
\item[{}] $t_5= 2f_2f_3-2c_2f_3$.
\item[{}] $t_6= f_2(-(c_2-f_2)^2+4c4)+f_3^2-c_3^2$.
\end{enumerate}
There is a presentation:
$A^*(\mathbf G(3,\wedge^2 S))/(I_D,c_1(\overline{N})) \stackrel{\sim}{=}$
\begin{equation}\label{prezz}
\mathbb{Z}[c_1,c_2,c_3,c_4,f_1,f_2,f_3]/(I_D, c_1, f_1, t_4,t_5,t_6)
\end{equation}
(up to codimension $m-4$).
Surprisingly, the relations $t_4,t_5,t_6$
are contained in the ideal $(I_D, c_1,f_1)$. By the limit
procedure,
$$A_{\mathbf{SO}(4)}^*(\text{point})\stackrel{\sim}{=}
\mathbb{Z}[c_1,c_2,c_3,c_4, f_2]/(c_1,2c_3,(c_2-f_2)c_3, (c_2-f_2)^2-4c_4).$$
The vector bundles $S$, $F \subset \wedge^2 S$ on the
approximation $W_m/\mathbf{SO}(V)$ are easily seen to be obtained
from the principal $\mathbf{SO}(V)$-bundle
$$W_m \rightarrow W_m/ \mathbf{SO}(V)$$
and the representations $V$, $F \subset \wedge^2 V$ defined
in section \ref{rrr}. Define $$x=c_2-f_2.$$ Theorem 1 is
proved.
|
1996-07-22T15:31:03 | 9607 | alg-geom/9607022 | en | https://arxiv.org/abs/alg-geom/9607022 | [
"alg-geom",
"math.AG"
] | alg-geom/9607022 | Flavio. Angelini | Flavio Angelini | Ample divisors on the blow up of P^3 at points | AMS-LaTeX, 9 pages | null | null | null | null | We give a condition for certain divisors on the blow up of P^3 at points in
general position to be ample. The result extends a theorem of G. Xu on the blow
up of the projective plane.
| [
{
"version": "v1",
"created": "Mon, 22 Jul 1996 14:26:14 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Angelini",
"Flavio",
""
]
] | alg-geom | \section{Introduction}
In this note we will prove a theorem on divisors on the blow up of
$\Bbb{P}^3$ at points which extends a theorem of G. Xu \cite{Xu} on the
blow up of $\Bbb{P}^2$. The central idea of the proof works for any dimension
and therefore opens the doors to a generalization of the theorem to higher
dimension, once one overcomes certain technical difficulties that arise.
Basically we will give a new proof of Xu's theorem
that works also for $\Bbb{P}^3$ and so in most of this note $n$ will be
either 2 or 3.
\noindent
{\bf Theorem.} Let $n=2$ or $3$. Let $\Bbb{P}^n$ be the projective space over
the field of
complex numbers. Let $p_1,\dots,p_k$ be k points in $\Bbb{P}^n$ in general
position and let $\pi: X \longrightarrow \Bbb{P}^n$ be the blow up of $\Bbb{P}^n$
at $p_1,\dots,p_k$ with exceptional divisors $E_1,\dots,E_k$. Let
$H=\pi^* \cal{O}_{\Bbb{P}^n}(1)$. Then, if $d\geq d_0(n)$, the divisor
$L=dH- \displaystyle{\sum_{i=1}^k} E_i$ is ample if and only if $L^n >0$, i.e. $d^n>k$,
where $d_0(2)=3$, $d_0(3)=5$.
\noindent
{\bf Remark 1.1.} In the case of $\Bbb{P}^2$ we obtain the same bound as in Xu's,
which is sharp. For $\Bbb{P}^3$ we believe, following a conjecture, that
the theorem holds for $d\geq3$.
Xu's proof is based on an estimate for the self-intersection of moving
singular curves in $\Bbb{P}^2$. The same estimate was obtained also
by Ein and Lazarsfeld in the context of Seshadri constants on smooth
surfaces \cite{EL} and used by K\"uchle to prove the above theorem in
the case of a smooth surface \cite{Ku}.
The basic idea of the proof comes from an example of R. Miranda regarding
ample divisors on a smooth surface with arbitrarily small Seshadri constant
\cite[\S5]{L}.
The strategy is as follows:
we will use the fact that ampleness is an open condition to reduce to
the case when the points are part of the base locus of a $(n-1)$-dimensional
general linear system of hypersurfaces of degree $d$. The corresponding
line bundle
$L$ on the blow up at the entire base locus is
nef and this will imply
that the divisor obtained by blowing down some of the exceptional loci
is ample. For this last step we will use in a determinant way the fact that the
fibres of the morphism to $\Bbb{P}^{n-1}$ determined by $L$ are irreducible, under
the stated conditions on $d$.
We will prove this fact in Lemma 2.1 which concerns curves in $\Bbb{P}^3$ which are
complete intersection. The problem is to give a lower bound for the
codimension, in the space of such curves, of the space of reducible ones.
We emphasize that it is this technical lemma that
gives, apart from the lower bound on $d$, that may be not sharp, but
reasonable, the restriction for the dimension of the projective space for which
the theorem holds.
It is actually possible to prove Lemma 2.1, and therefore the theorem,
also in the cases
$n=4$ and $5$,
in a very similar way and we will spend a couple of words about it at
the end of the proof of the
lemma.
The rest of the proof works for any dimension and therefore we will present it
as much as possible in its generality in section 3.
This note is part of my Ph. D. thesis at UCLA, and I would like to
thank Rob Lazarsfeld for his guidance and encouragement.
\section{Preliminary material and lemmas}
Let $\Bbb{P}^N$ be the projective space parametrizing hypersurfaces in
$\Bbb{P}^n$ of degree $d$. So $N=N(d,n)=\binom{n+d}{n} -1$. If $F$ is such
a hypersurface, we will denote by $\left[ F \right]$ the
corresponding point in $\Bbb{P}^N$.
Recall that there is an action
of $PGL(N)$ on $\Bbb{P}^N$. If $\sigma$ is an element of $PGL(N)$ and $\Lambda$ is
a linear subspace of $\Bbb{P}^N$, we will denote by $\Lambda^{\sigma}$ the linear
subspace obtained by letting $\sigma$ act on $\Lambda$. We will say that a
property holds for a general linear subspace $\Lambda$ of $\Bbb{P}^N$ if it
holds for $\Lambda^{\sigma}$ for $\sigma$ outside a union (possibly countable)
of proper subvarieties of $PGL(N)$.
We will denote
by $\Sigma_{d}$ the locus of singular hypersurfaces which
is an irreducible
subvariety of $\Bbb{P}^N$ of codimension one.
We will also be dealing with curves which are
complete intersection
of hypersurfaces of same degree $d$ in $\Bbb{P}^3$. These are
parametrized by an open subset of
the Grassmannian $Gr(\Bbb{P}^1,\Bbb{P}^N)$
(of course for $n=2$ this is just $\Bbb{P}^{N(d,2)}$).
We will denote by
$\left[l_C\right] \in Gr(\Bbb{P}^1,\Bbb{P}^N)$
the point corresponding to a curve $C$.
Also we have
$dimGr(\Bbb{P}^k,\Bbb{P}^r)=(k+1)(r-k)$.
Let now
$$\begin{array}{rl}
NL_{d,3}=&\lbrace \text{smooth surfaces}\phantom{.} S\subset\Bbb{P}^3
\phantom{.}\text{of degree}\phantom{.} d \\
&\phantom{.}: Pic(S)\phantom{.} \text{is not
generated by the hyperplane class} \rbrace.
\end{array}$$
$NL_{d,3}$ is called the
{\em Noether-Lefschetz
locus} and may be viewed as a subset of $\Bbb{P}^{N(d,3)}$. This locus is
pretty well understood, at least as far as we are concerned here.
$NL_{d,3}$ is a countable union of
quasi-projective algebraic varieties.
The Noether-Lefschetz theorem asserts that, for $d\geq 4$, the general
surface of degree $d$ in $\Bbb{P}^3$ has Picard group generated by the
hyperplane section, i.e. is not contained in $NL_{d,3}$. In other words
the theorem says that the codimension of all the irreducible components
$N_i$ of
$NL_{d,3}$
is at least one.
What we will need
is an
explicit Noether-Lefschetz theorem (see \cite{Gr} for a nice proof of it)
giving a precise bound for the codimension of any of the $N_i$.
\noindent
{\bf Theorem (Green).} For $d\geq 4$, the codimension of any irreducible
component of $NL_{d,3}$ in $\Bbb{P}^N$ is at least $(d-3)$.
The main technical lemma we need is:
\noindent
{\bf Lemma 2.1.} Let $n=2,3$. Let $\Bbb{P}^N$ be the projective space parametrizing
hypersurfaces
of degree $d$ in $\Bbb{P}^n$ and assume $d\geq d_0(n)$, with $d_0(2)=3$ and
$d_0(3)=5$. Let $\Lambda$ be a general linear subspace
of $\Bbb{P}^N$ of dimension $(n-1)$ whose base locus consists of $d^n$ points.
Then, for every $(n-1)$ linearly independent elements
$\left[ F_1 \right] , \dots, \left[ F_{n-1}
\right]$ of $\Lambda$, the intersection $F_1 \cap \dots \cap F_{n-1}$ is a
curve and is irreducible.
In the case $n=2$ this is just saying that every element of a general pencil
of curves of degree $d$ is irreducible for $d\geq3$.
\noindent
{\bf Remark 2.1.} Lemma 2.1, and therefore the theorem, would be proven
for any dimension if one found
a good bound for the codimension, in the space of curves in $\Bbb{P}^n$ which are
a complete intersection of hypersurfaces of degree $d$, of the space of
reducible ones. It is easy to conjecture that this codimension should be at
least $(n-1)(d-1)$, being this the codimension of such curves which have a
line as a component. This conjecture would imply Lemma 2.1 with $d_0(n)=3$
and for any $n$.
\noindent
{\it Proof.} The proof in the case $n=2$ is rather easy. The set $\Sigma_d$
of singular curves of degree $d$ in $\Bbb{P}^2$ is an irreducible subvariety
of codimension one. The reducible curves are union of closed subvarieties
all lying in $\Sigma$. It is then enough to notice that, for $d\geq3$,
there are irreducible singular curves to establish that the codimension
of the reducible ones is at least two and therefore conclude that a
general pencil does not contain reducible curves.
For $n=3$ we take $d\geq5$ and we need to prove Lemma 2.1 for
a general plane $\Lambda\subset \Bbb{P}^N$ with $N=\binom{d+3}{3}-1$. A curve $C$
which is a complete intersection of two elements of $\Lambda$ corresponds to
a line
$l_C \subset\Lambda$, i.e. to an element
$\left[l_C\right] \in Gr(\Bbb{P}^1,\Lambda) \subset Gr(\Bbb{P}^1,\Bbb{P}^N)$. First
we have the following:
\noindent
{\bf Claim 2.1.} For any $l_C$ in $\Lambda$, we can find a {\em smooth}
surface $S$ with $\left[ S\right] \in l_C$ such
that $\left[ S \right]$ is not in $NL_{d,3}$.
For this we need the following sublemma:
\noindent
{\bf Lemma 2.2.} For $d\geq5$, the intersection of $\Lambda \subset\Bbb{P}^N$
with the Noether-Lefschetz locus consists of at most countably many points.
This is to say that there are at most countably many smooth surfaces $S$ with
$\left[ S\right] \in \Lambda$ and with $Pic(S)$ not generated by the hyperplane
section.
\noindent
{\it Proof.} We use here Kleiman's Transversality Theorem \cite[Thm. III.10.8]
{H} for the action of $PGL(N)$ on $\Bbb{P}^N$. We will apply the Theorem to
a plane $\Lambda \subset \Bbb{P}^N$ and the closure $\overline{N_i}$ of one irreducible
component $N_i$
of $NL_{d,3}$.
The Theorem says that $\Lambda^{\sigma} \cap \overline{N_i}$ is either empty or
of dimension
$$dim\Lambda - codim\overline{N_i}$$
for $\sigma$ in a non-empty open subset $V_i \subset PGL(N)$.
So, by the explicit Noether-Lefschetz theorem and for $d\geq5$, we have
$$dim(\Lambda^{\sigma} \cap \overline{N_i}) \leq 2-(d-3) \leq 0$$
for $\sigma \in V_i$.
This means that the intersection of a general plane with a component of
$NL_{d,3}$ consists at most of a finite number of points. Therefore, for $\sigma$
outside a possibly countable union of closed proper subvarieties, the
intersection of $\Lambda^{\sigma}$ with $NL_{d,3}$ consists of at most countably many
points. $\hspace{1cm}\Box$
Also observe that the intersection of $\Lambda$ with $\Sigma_{d}$ does not
contain any line $l_C \subset \Lambda$.
In other words, for any curve $C$ which is a complete intersection of
elements of $\Lambda$ there exist at most finitely many surfaces $S$ with
$\left[ S\right] \in l_C$ that are singular.
This is because $\Sigma_{d}$ is an
irreducible subvariety of codimension one and high degree of $\Bbb{P}^N$, and hence,
for a general $\Lambda$,
$\Sigma_d \cap \Lambda$ is an irreducible curve of degree strictly greater than one.
\noindent
{\it Proof of Claim 2.1.} For any $l_C$ in $\Lambda$, by Lemma 2.2
there are at most countably many surfaces $S$ with $\left[ S\right] \in
l_C$ and $Pic(S)$ not isomorphic to $\Bbb{Z}$ and by the observation above there
are at most finitely many singular ones. It is then possible to find
a smooth $S$ with $Pic(S)$ isomorphic to $\Bbb{Z}$. $\hspace{1cm}\Box$
Now we can prove Lemma 2.1 for $n=3$.
\noindent
{\it Proof of Lemma 2.1.} Let $C$ be any curve which is complete intersection
of elements of $\Lambda$. We need to prove that $C$ is irreducible.
Choose a surface $S$ as in Claim 2.1.
Pick another surface $T$ with $\left[ T \right]$ in $l_C$ so that
$C=S\cap T$.
If $C$ were reducible, say $C=\cup C_i$, then any $C_i \subset S$
would be a complete intersection $S \cap T_i$, with $T_i$ a hypersurface
in $\Bbb{P}^3$ of degree less than $d$, since $Pic(S)$ is generated
by the hyperplane section. But then
$$C=S \cap T=\cup(S \cap T_i)=S \cap (\cup T_i)$$
This means that $\left[ \cup T_i \right] \in l_C \subset \Lambda$.
But $\Lambda$ misses
reducible surfaces, so $C$ has to be irreducible. $\hspace{1cm}\Box$
\noindent
{\bf Remark 2.2.} As mentioned in the introduction, it is possible to prove
Lemma 2.1 in the case $n=4$ and $5$ along the same line as for $n=3$, using
a generalized explicit Noether-
Lefschetz theorem, due to S. Kim \cite{Kim}, regarding smooth surfaces
which are complete intersection of hypersurfaces \cite{A}.
The proof does not work in higher dimension due to the fact that the Noether-
Lefschetz theorem does not give any information about singular surfaces and
it is not possible anymore to ensure the existence of a smooth surface
containing the curve.
Another tool for the proof of the theorem is the fact that
ampleness is an open condition. We will state this very well known fact
in the form of:
\noindent
{\bf Proposition 2.1.} Let $\cal{L} \longrightarrow T$ be a flat family of line
bundles
over a flat family $\cal{X} \stackrel{f}\longrightarrow T$ of
projective varieties of dimension
$n$.
Then the set
$$\lbrace t\in T\phantom{.}:\phantom{.}
\cal{L}_t=\cal{L}_{|_{X_t}}\text{is ample on}\phantom{.}X_t \rbrace$$
is open in $T$.
(See \cite{A} for a proof of it).
\section{Proof of Theorem}
The central idea of the proof works for any dimension and we will
present the argument as much as possible in its generality. The restriction
on the
dimension comes
uniquely from
Lemma 2.1, to ensure the irreducibility of the fibers of the morphism $\mu$
below.
We are given $k$ points in general position in $\Bbb{P}^n$ and we need to
prove that the divisor $dH-\displaystyle{\sum_{i=1}^k} E_i$ is ample when $d^n>k$ for $d\geq d_0(n)$.
Clearly this condition is necessary.
We consider a general linear system $\Lambda$ of hypersurfaces of degree $d$
and dimension $(n-1)$ and we let $p_1^\prime ,\dots , p_{d^n} ^\prime$ be
the base locus of $\Lambda$.
Let $\pi ^\prime : X^\prime \longrightarrow \Bbb{P}^n$ be the blow up at $k$ of
these points $p_1 ^\prime , \dots , p_k ^\prime$,
$H^\prime ={\pi ^\prime}^* \cal{O}_{\Bbb{P}^n}(1)$ and $E_i ^\prime, \dots ,E_k ^\prime$
the exceptional
divisors. We will prove that $dH^\prime- \displaystyle{\sum_{i=1}^k} E_i^\prime$ is ample and
the theorem
will follow from Proposition 2.1.
To do so we consider the blow up $\nu : Z \longrightarrow \Bbb{P}^n$ at the whole
base locus $p_1 ^\prime, \dots , p_{d^n} ^\prime$ and set
$H=\nu^* \cal{O}_{\Bbb{P}^n}(1)$,
$E_i$ for $i=1, \dots ,k$, $F_i$ for $i=k+1, \dots , d^n$ the ecxeptional
divisors of $\nu$ and $L=dH-\displaystyle{\sum_{i=1}^k} E_i -\displaystyle{\sum_{i=k+1}^{d^n}} F_i$.
We have the following diagram:
$$\begin{array}{rcl}
Z\phantom{.} &\stackrel{\delta}\longrightarrow&X^\prime \\
&\stackrel{\nu}\searrow\phantom{.}\stackrel{\pi ^\prime}\swarrow& \\
&\phantom{.}\Bbb{P}^n&
\end{array}$$
where $\delta$ is the blow down of $F_{k+1}, \dots ,F_{d^n}$.
Now $L$ is globally generated and therefore is nef. We will show that,
for $d\geq d_0(n)$, this implies that $dH^\prime-\displaystyle{\sum_{i=1}^k} E_i^\prime$ is ample.
Now we have a morphism $\mu :Z \longrightarrow \Bbb{P}^{n-1}$
whose fibers are curves which are complete intersection of elements of $\Lambda$.
Moreover, if $C$ is a curve,
$$L\cdot C=0 \Longleftrightarrow \mu(C)\phantom{..}
\mbox{\text{is a set of points}}.$$
By Lemma 2.1, for $n=2,3$ and $d\geq d_0(n)$, by taking $\Lambda$
sufficiently general,
we may arrange for these fibers to be all irreducible.
What we need is the following:
\begin{align}
\tag{3.1} &(L+\displaystyle{\sum_{i=k+1}^{d^n}} F_i)^m \cdot Y_m \geq0 \\
\tag{3.2} &(L+\displaystyle{\sum_{i=k+1}^{d^n}} F_i)^m \cdot Y_m =0 \Longleftrightarrow Y_m \subset F_i
\phantom{...} \mbox{\text{for one}}\phantom{.}i=k+1, \dots ,d^n,
\end{align}
where $Y_m$ is any irreducible subvariety of dimension $m$ of $Z$, for any
$0<m<n$.
Notice that:
$$L+\displaystyle{\sum_{i=k+1}^{d^n}} F_i=dH-\displaystyle{\sum_{i=1}^k} E_i=\delta^*(dH^\prime-\displaystyle{\sum_{i=1}^k} E_i^\prime).$$
(3.1) and (3.2) prove, by Nakai's criterion and the projection formula, that
$dH^\prime-\displaystyle{\sum_{i=1}^k} E_i^\prime$ is ample on $X^\prime$.
To simplify notation let $L^\prime=L+\displaystyle{\sum_{i=k+1}^{d^n}} F_i$. From here on we restrict to
$n=2$ and 3 because we will use Lemma 2.1. If Lemma 2.1 were proven
for any $n$ we would proceed by induction on $m$, using the same
arguments.
Here we just have to check (3.1) and (3.2) for $m=1$ in
the case $n=2$ and for $m=1$ and 2 in the case $n=3$.
Let $m=1$ so that $Y_1$ is a curve. If $Y_1 \subseteq F_i$ for one $i$, then,
being $L^\prime$ trivial on $F_i$, $L^\prime \cdot Y_1 =0$ and, if
$Y_1\not\subseteq F_i$
for all $i$, then, being $L \cdot Y_1 \geq 0$ ($L$ is nef)
and $\displaystyle{\sum_{i=k+1}^{d^n}} F_i \cdot Y_1\geq 0$,
$L^\prime \cdot Y_1 \geq 0$.
This proves (3.1) and the easy direction of (3.2). For the other
direction suppose
$L^\prime \cdot Y_1=0$ but $Y_1 \not\subseteq F_i$ for all $i$. But then, being
$\displaystyle{\sum_{i=k+1}^{d^n}} F_i \cdot Y_1 \geq 0$, $L \cdot Y_1$ has to be zero and therefore
$Y_1$ is a component of a fiber of $\mu$. Since
every fiber is irreducible,
$Y_1$ is exactly a fiber and so it meets all the exceptional divisors of $\nu$.
Therefore $\displaystyle{\sum_{i=k+1}^{d^n}} F_i \cdot Y_1$ is strictly positive which is a contradiction.
In the case $n=3$ we need also to consider
any irreducible subvariety $Y_m$ of $Z$ of dimension $m=2$. As
before, if $Y_m \subseteq F_i$ for some $i$, then $L^\prime
\cdot Y_m =0$.
If $Y_m \not\subseteq F_i$ for all $i$, then we write:
$${L^\prime}^m \cdot Y_m=L \cdot {L^\prime}^{m-1}\cdot Y_m +\displaystyle{\sum_{i=k+1}^{d^n}} F_i \cdot
{L^\prime}^{m-1}
\cdot Y_m.$$
Now $\displaystyle{\sum_{i=k+1}^{d^n}} F_i \cdot Y_m$ has dimension strictly less than $m$ (or is empty)
and $L \cdot Y_m$ can be represented by a cycle of dimension strictly
less than $m$ (or empty) because $L$ is globally generated. So
both $L \cdot {L^\prime}^{m-1}\cdot Y_m$ and $\displaystyle{\sum_{i=k+1}^{d^n}} F_i \cdot {L^\prime}^{m-1}
\cdot Y_m$
are greater or equal than zero by the previous step and so is
${L^\prime}^m \cdot Y_m$.
For the remaining direction of (3.2) suppose ${L^\prime}^m \cdot Y_m=0$ while
$Y_m \not\subseteq F_i$ all $i$. Then, since as before $\displaystyle{\sum_{i=k+1}^{d^n}} F_i \cdot
{L^\prime}^{m-1} \cdot Y_m \geq 0$, we have $L\cdot {L^\prime}^{m-1}
\cdot Y_m=0$. By the previous step, this implies that
$L \cdot Y_m \subseteq F_i$ for
some $i$. We claim that this is a contradiction, i.e. we claim that, if
$Y_m \not\subseteq F_i$ for any $i$, then $L \cdot Y_m \not\subseteq F_i$
for any $i$. Indeed, if $Y_m \not\subseteq F_i$ for any $i$, then $\nu_*(Y_m)
\subseteq \Bbb{P}^3$ has dimension 2. Consider a divisor $D \in | \Lambda|$.
By Bezout's Theorem $dim(\nu_*(Y_m) \cdot D) \geq 1$. Also
$$dim(\nu_*(Y_m) \cdot D)=dim(\nu_*(Y_m \cdot \nu^*D))=
dim(\nu_*(Y_m \cdot L)).$$
So $dim(\nu_*(Y_m\cdot L))\geq 1$ and therefore $Y_m \cdot L\not\subseteq F_i$
for any $i$. $\hspace{1cm}\Box$
|
1997-09-22T16:38:52 | 9607 | alg-geom/9607020 | en | https://arxiv.org/abs/alg-geom/9607020 | [
"alg-geom",
"math.AG"
] | alg-geom/9607020 | Paul Bressler | P.Bressler, M.Saito, B.Youssin | Filtered Perverse Complexes | AMSLaTeX v 1.1. This version is a major revision. With the new
co-author (M.Saito) it contains substantially new results, improvements and
corrections | null | null | null | null | We introduce the notion of filtered perversity of a filtered differential
complex on a complex analytic manifold $X$, without any assumptions of
coherence, with the purpose of studying the connection between the pure Hodge
modules and the \lt-complexes. We show that if a filtered differential complex
$(\cM^\bullet,F_\bullet)$ is filtered perverse then
$\aDR(\cM^\bullet,F_\bullet)$ is isomorphic to a filtered $\cD$-module; a
coherence assumption on the cohomology of $(\cM^\bullet,F_\bullet)$ implies
that, in addition, this $\cD$-module is holonomic. We show the converse: the de
Rham complex of a holonomic Cohen-Macaulay filtered $\cD$-module is filtered
perverse.
| [
{
"version": "v1",
"created": "Fri, 19 Jul 1996 15:38:35 GMT"
},
{
"version": "v2",
"created": "Sun, 22 Sep 1996 18:48:19 GMT"
},
{
"version": "v3",
"created": "Wed, 9 Oct 1996 14:36:13 GMT"
},
{
"version": "v4",
"created": "Mon, 22 Sep 1997 14:38:51 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Bressler",
"P.",
""
],
[
"Saito",
"M.",
""
],
[
"Youssin",
"B.",
""
]
] | alg-geom | \section{Introduction}
\subsection{Cheeger---Goresky---MacPherson conjectures}
J.~Cheeger, M.~Goresky and R.~MacPherson \cite{CGM} conjectured some fifteen
years ago that
the intersection cohomology of a singular complex projective algebraic variety
is naturally isomorphic to its $L^2$ cohomology and the K\"ahler
package holds for them.
Their motivation was as follows.
The intersection cohomology was discovered by M.~Goresky and R.~MacPherson
\cite{GM1}, \cite{GM2} as an invariant of stratified spaces which for
complex algebraic varieties might serve as a replacement of the usual
cohomology: it had some properties that
the usual cohomology of smooth projective varieties possessed but
the usual cohomology of singular projective varieties did not.
One of such properties was Poincar\'e duality which is a part of the ``K\"ahler
package'' of properties that hold in the smooth case.
At the same time, J.~Cheeger discovered that the $L^2$ cohomology
groups of varieties with conical singularities have properties similar
to those of intersection cohomology, and he proved in this case
the Hodge---de Rham isomorphism between the $L^2$ cohomology that he defined
and studied, and the intersection cohomology \cite{C}.
The hope that underlied these conjectures was that it would be possible to
use the $L^2$ K\"ahler methods to prove the K\"ahler package for intersection
cohomology similarly to the way the K\"ahler package was proved for the
usual cohomology of complex projective manifolds.
The most important part of the K\"ahler package is the $(p,q)$-decomposition
in the cohomology groups (the ``Hodge structure'').
The definition of $L^2$ cohomology involves a metric
(Riemannian or K\"ahler) defined almost everywhere on the variety (e.g.\ on
its nonsingular part). The most important metric comes from a projective
embedding of the variety and is induced by the Fubini---Studi metric on
the projective space. (The $L^2$ cohomology is independent of the choice of
the imbedding.)
The isomorphism with intersection cohomology is known in case of surfaces
\cite{HP}, \cite{Nag1} and in case of isolated singularities of any dimension
both for Fubini---Studi metric \cite{O2}, \cite{O2a} and for a different,
complete metric, introduced by L.~Saper,
which is defined on the nonsingular part of the variety
and blows up near the singularities \cite{Sap}.
The $(p,q)$-decomposition is known for the case of Fubini---Studi metrics
only in cases of dimension two \cite{Nag2} (except for the middle degree
cohomology groups) while a classical
result of Andreotti---Vesentini implies the $(p,q)$-decomposition
for {\em any\/} complete metric.
The general case is still open, despite the announcement of T.~Ohsawa
\cite{O3}.
In the meantime the second author \cite{S1}, \cite{S2} developed a theory of
polarizable Hodge modules which implied the K\"ahler package
for the intersection cohomology.
His main tool was the theory of ${\cal D}$-modules and his methods were essentially
algebraic, reducing the intersection cohomology to the intersection cohomology
of a curve with coefficients
in a polarised variation of Hodge structure \cite{Z1}.
\subsection{The comparison between the Hodge structures}
Assuming that the Cheeger---Goresky---MacPherson
conjectures are true, one is faced with the question of comparison between the
two Hodge structures on the intersection cohomology: one induced by the
isomorphism with {$L^2$} cohomology, the other coming from the theory of
polarised Hodge modules.
In fact, different metrics give different $L^2$ cohomology theories and hence,
pose different comparison problems.
In case of isolated singularities, S.~Zucker \cite{Z} proved the coincidence
between the Hodge structures coming from polarized Hodge modules and from
$L^2$ cohomology with respect to the
Saper metric (or arithmetic quotient metrics similar to it).
Some partial results are also known in case of Fubini---Studi metric, see
\cite{Z} and \cite{Nag2}.
It is interesting to note that the original purpose of the conjectures was
to construct the Hodge structure on the intersection cohomology.
The $L^2$ methods, however, turned out to be so difficult
that the Hodge structure was constructed by different,
algebraic methods and now we
are faced with the problem of comparison between the two Hodge structures.
\subsection{The local comparison problem}
A major component of a polarizable Hodge module is a regular
holonomic ${\cal D}$-module $M$ with a good filtration $F_\bullet$.
Suppose $M$ corresponds to the intersection cohomology
complex of a complex projective subvariety $Z$;
the correspondence is given by taking the de Rham complex $\operatorname{DR}(M)$ of $M$
(so that $\operatorname{DR}(M)$ is isomorphic to the intersection cohomology complex of $Z$).
Then the filtration $F_\bullet$ induces a filtration on $\operatorname{DR}(M)$
which yields the Hodge structure on the intersection cohomology.
The complex $\operatorname{DR}(M,F_\bullet)$ is a filtered differential complex~\cite{S1}:
a complex of sheaves which are modules over the sheaf of analytic functions
and the differentials are differential operators.
This filtered differential complex completely determines $(M,F_\bullet)$ as
there is an inverse functor $\operatorname{DR}^{-1}$ \cite{S1}.
If the metric used in the construction of the {$L^2$} complex
(it is a K\"ahler metric on the nonsingular
part of $Z$) is bounded below with respect to Fubini---Studi metric,
the {$L^2$} complex is a filtered differential complex (see, e.g.,~\ref{subsec:l2}
below).
The {\em local comparison problem\/} is as follows: is it true that
the de Rham complex of the Hodge module is isomorphic to the {$L^2$} complex
in the derived category of filtered differential complexes?
The intersection cohomology can be taken with coefficients in a local system
defined on the non-singular part of $Z$ or a Zariski-open subset of it.
If this local system underlies a polarized variation of Hodge
structures then a corresponding polarized Hodge module can be constructed
and the intersection cohomology with coefficients in this local system has
a Hodge structure.
On the other hand, the $L^2$ cohomology can be taken with coefficients in
the same polarized variation, and we can ask the same local comparison
question in this situation.
\subsection{Weak filtered perversity}
A way to approach this problem is to try to identify the properties of
a filtered differential complex $({\cal M}^\bullet,F_\bullet)$ which
would imply that $\operatorname{DR}^{-1}({\cal M}^\bullet,F_\bullet)$ is isomorphic to the filtered
${\cal D}$-module which underlies a polarized Hodge module.
In general, $\operatorname{DR}^{-1}({\cal M}^\bullet,F_\bullet)$ is a
{\em complex\/} of filtered ${\cal D}$-modules; in this paper we study
properties of $({\cal M}^\bullet,F_\bullet)$ which imply that this complex
is isomorphic to one filtered ${\cal D}$-module in the filtered derived
category.
We call these properties {\em weak filtered perversity\/} (see
Definition~\ref{cond:wfp}); it means that, first, the complex
$({\cal M}^\bullet,F_\bullet)$ is locally trivial along the strata --- in a certain
filtered sense --- with respect to some analytic stratification, and second,
it satisfies certain local filtered cohomology vanishing which is similar
to the local cohomology vanishing of perverse sheaves.
No coherence assumption is being made on $({\cal M}^\bullet,F_\bullet)$.
In case $({\cal M}^\bullet,F_\bullet)$ is the $L^2$\ complex, the cohomology that must
vanish, turn out to be a version of the $L^2$-$\overline{\partial}$-cohomology,
see~\S\ref{rem:bound-cond} below for the discussion.
\subsection{The main results}
We show (see Theorem~\ref{thm:main})
that if $({\cal M}^\bullet,F_\bullet)$ is weakly filtered perverse then,
indeed, $\operatorname{DR}^{-1}({\cal M}^\bullet,F_\bullet)$ is isomorphic to a filtered ${\cal D}$-module.
We show the converse (see Theorem~\ref{thm:converse}):
if $(M^\bullet,F_\bullet)$ is a coherent filtered ${\cal D}$-module
which is Cohen-Macaulay (i.e., its dual in the filtered sense is also a
complex of filtered ${\cal D}$-modules isomorphic to one filtered ${\cal D}$-module)
then $\operatorname{DR}(M,F_\bullet)$ is weakly filtered perverse.
We show (see Proposition~\ref{cor:holon}) that a coherence assumption
together with filtered perversity of $({\cal M}^\bullet,F_\bullet)$ implies
that the filtered ${\cal D}$-module $\operatorname{DR}^{-1}({\cal M}^\bullet,F_\bullet)$ is holonomic.
\subsection{Plan of the paper}
In Section~\ref{sec:fdm-dc} we review the necessary background material
from~\cite{S1}.
In Section~\ref{sec:fil.perv} we introduce the notion of weak filtered
perversity.
In Sections \ref{sec:fpc-to-dm} and~\ref{sec:coh.case} we prove the results
listed above.
In Section~\ref{sec:appl} we give a modest application: we strengthen the
results of \cite{KK2} and \cite{S1} and show (in the situation of \cite{KK2})
that filtered perversity of the $L^2$ complex implies
the local filtered isomorphism
(in the sense of derived category) between the $L^2$ complex
and the de Rham complex of
the ${\cal D}$-module that underlies the corresponding pure Hodge module.
\subsection{Acknowledgements}
It is our pleasant duty to express our heartful thanks to all people who
helped us with their advice and helpful discussions: Daniel Barlet,
Alexander Beilinson, Joseph Bernstein, Jean-Luc Brylinski,
Michael Kapranov, Masaki Kashiwara, David Kazhdan,
Takeo Ohsawa, Claude Sabbah.
\section{Filtered ${\cal D}$-modules and differential complexes}
\label{sec:fdm-dc}
In this section we
make a brief survey of the necessary parts of \cite{S1}.
\subsection{General notation}
Throughout this paper $X$ will denote a complex manifold, ${\cal O}_X$ the sheaf of
holomorphic functions, $\omega_X$ the canonical sheaf of $X$,
${\cal D}_X$ the sheaf of differential operators.
Unless specified otherwise, a ${\cal D}_X$-module will always refer to a
sheaf of {\em right} modules over ${\cal D}_X$.
For a complex of sheaves ${\cal F}^\bullet$, we shall denote by
$H^j{\cal F}^\bullet$ its {\em sheaf\/} cohomology, and by
$H^j_{\{x\}}{\cal F}^\bullet$ its (hyper)cohomology with supports in a one-point
set $\{x\}$.
\subsection{Filtered ${\cal D}$-modules}
Recall that ${\cal D}_X$ is a filtered ring when equipped with the filtration
$F_\bullet{\cal D}_X$, where $F_p{\cal D}_X$ is the ${\cal O}_X$-module of ${\cal D}_X$ of
operators of order at most $p$.
A filtered ${\cal D}_X$-module is a pair $(M,F_\bullet)$ consisting of a
${\cal D}_X$-module $M$ and a filtration $F_\bullet M$ of $M$ by ${\cal O}_X$
submodules compatible with the action of ${\cal D}_X$ and the filtration
on the latter.
We refer the reader to~\cite{S1}, \S2.1 for the precise definition of the
category of filtered ${\cal D}_X$-modules and its derived category, in various
flavors; what is important for us now, is that the derived category
of filtered ${\cal D}_X$-modules is
isomorphic to the derived category of the category whose objects are filtered
${\cal O}_X$-modules and whose morphisms are differential operators
that agree with the filtration in a certain way.
This equivalence is given by the two functors, $\operatorname{DR}_X$ and $\operatorname{DR}^{-1}_X$ (loc.~cit.)
which act as follows.
\subsection{The de Rham functor $\protect\operatorname{DR}_X$}
For a filtered ${\cal D}_X$-module $(M,F_\bullet)$, the filtered differential
complex $\operatorname{DR}_X(M,F_\bullet)$ is the usual de Rham complex of $M$, given by
\[
\operatorname{DR}_XM=
M\otimes_{{\cal D}_X}\left({\cal D}_X\otimes_{{\cal O}_X}
\textstyle\bigwedge^{-\bullet}\Theta_X\right)
=M\otimes_{{\cal O}_X}\textstyle\bigwedge^{-\bullet}\Theta_X
\]
where $\Theta_X$ is the tangent sheaf to $X$,
$\bigwedge^{-\bullet}\Theta_X$ its exterior algebra
with $p$-th exterior power placed in degree $-p$,
the differential is given by
\begin{equation*}
\begin{split}
{d}(P\otimes\xi_1\wedge\ldots\wedge\xi_p) & =\sum_{i=1}^{p}
(-1)^{i-1}P\xi_i\otimes\xi_1\wedge\ldots\wedge\widehat{\xi_i}
\wedge\ldots\wedge\xi_p \\
& +\sum_{1\leq i < j\leq p}(-1)^{i+j}P\otimes
\lbrack\xi_i,\xi_j\rbrack\wedge\xi_1\wedge\ldots\wedge\widehat{\xi_i}
\wedge\ldots\wedge\widehat{\xi_j}\wedge\ldots\wedge\xi_p
\end{split}
\end{equation*}
(it corresponds to the differential in
${\cal D}_X\otimes_{{\cal O}_X}\bigwedge^{-\bullet}\Theta_X$
which makes it the standard
Koszul resolution of ${\cal O}_X$ as a ${\cal D}_X$-module),
and the filtration on $\operatorname{DR}_XM$ is given by
\[
F_p\left(M\otimes_{{\cal O}_X}\textstyle\bigwedge^{-i}\Theta_X\right)=
F_{p+i}M\otimes_{{\cal O}_X}\textstyle\bigwedge^{-i}\Theta_X\ .
\]
For a complex of filtered ${\cal D}_X$-modules $(M^\bullet,F_\bullet)$, the
filtered differential complex $\operatorname{DR}_X(M^\bullet,F_\bullet)$ is the total
complex of $\operatorname{DR}_X(M^q,F_\bullet)$ for all $q$.
(Note that what we denote by $\operatorname{DR}_X$, was denoted by
$\widetilde\operatorname{DR}$ in~\cite{S1}.)
\subsection{The inverse de Rham functor $\protect\operatorname{DR}^{-1}_X$}
For a filtered differential complex $({\cal M}^\bullet,F_\bullet)$,
the complex of filtered ${\cal D}_X$-modules $\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet)$
is described as the complex of differential operators from ${\cal O}_X$
into $({\cal M}^\bullet,F_\bullet)$ with the obvious differential and filtration.
The action of ${\cal D}_X$, i.~e., differential operators ${\cal O}_X\to{\cal O}_X$,
is by composition.
The individual terms of this complex are simply ${\cal M}^j\otimes_{{\cal O}_X}{\cal D}_X$.
The two functors $\operatorname{DR}_X$ and $\operatorname{DR}^{-1}_X$ are inverse to each other in
the derived categories.
\subsection{Duality}
\label{subsec:duality}
For a bounded complex of filtered ${\cal D}_X$-modules
$(M^\bullet,F_\bullet)$, its dual
${\Bbb D}(M,F_\bullet)$ is is another complex of filtered ${\cal D}_X$-modules
defined (\cite{S1}, 2.4.3)
in such way that it agrees with various other duality functors, as follows.
There is a duality functor (also denoted by ${\Bbb D}$) on filtered differential
complexes which on an individual ${\cal O}_X$-module $L$ is defined as
${\Bbb D}(L)=\operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal O}_X}(L,\omega_X[n])$ where $n=\dim X$
(\cite{S1}, 2.4.11),
and such that in the appropriate derived categories the functors
${\Bbb D}\circ\operatorname{DR}_X$ and $\operatorname{DR}_X\circ{\Bbb D}$ are isomorphic.
For a filtered differential complex $({\cal M}^\bullet,F_\bullet)$ we have
\begin{equation}
\label{eqn:D.vs.Gr}
\operatorname{Gr}^F_\bullet{\Bbb D}({\cal M}^\bullet,F_\bullet)\overset{\sim}{=}
\operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal O}_X}(\operatorname{Gr}^F_\bullet{\cal M}^\bullet,\omega_X[n])\ .
\end{equation}
In case $(M^\bullet,F_\bullet)$ is a complex of filtered coherent
${\cal D}_X$-modules (i.e., coherent ${\cal D}_X$-modules
with good filtrations), the complex of ${\cal D}_X$-modules which underlies
${\Bbb D}(M^\bullet,F_\bullet)$, is the usual dual of $M^\bullet$.
In addition, under the same assumptions
${\Bbb D}\bbD(M^\bullet,F_\bullet)\overset{\sim}{=}(M^\bullet,F_\bullet)$.
\subsection{Restriction to a noncharacteristic submanifold}
Let $Y$ be a smooth submanifold of codimension $d$ in $X$; denote
the embedding $i:Y\hookrightarrow X$ and
$\omega_{Y/X}=\omega_Y\otimes_{{\cal O}_X}\omega_X^{-1}$.
We say that
a bounded complex of filtered ${\cal D}_X$-modules $(M^\bullet,F_\bullet)$
is {\em weakly noncharacteristic\/} with respect to $Y$ (or $Y$ with
respect to $(M^\bullet,F_\bullet)$) if it satisfies the property
\begin{equation}
\label{eqn:tor-nonchar}
{\underline{\operatorname{Tor}}}^{{\cal O}_X}_k(H^j(\operatorname{Gr}^F_p M^\bullet),{\cal O}_Y) = 0
\text{ \ \ for all $k\ne0$, $j$ and $p$.}
\end{equation}
Under this assumption, the noncharacteristic restriction
$(M^\bullet,F_\bullet)_Y$ is defined as follows:
\[
(M^\bullet,F_\bullet)_Y=
(M^\bullet,F_\bullet)\otimes^{\Bbb L}_{{\cal D}_X}({\cal D}_{X\leftarrow Y},F_\bullet)
\]
where
${\cal D}_{X\leftarrow Y}={\cal D}_X\otimes_{{\cal O}_X}\omega_{Y/X}$
is the usual $({\cal D}_X,{\cal D}_Y)$-bimodule with the filtration
$F_p{\cal D}_{X\leftarrow Y}=
F_{p-d}{\cal D}_X\otimes_{{\cal O}_X}\omega_{Y/X}$;
the restriction $(M^\bullet,F_\bullet)_Y$ thus defined,
is a complex of right ${\cal D}_Y$-modules.
As a complex of ${\cal O}_Y$-modules, it can be described as
$(M^\bullet)_Y=M^\bullet\otimes^{\Bbb L}_{{\cal O}_X}\omega_{Y/X}$ with
the filtration
$F_p(M^\bullet)_Y=F_{p-d}M^\bullet\otimes^{\Bbb L}_{{\cal O}_X}\omega_{Y/X}$.
We have
\begin{equation}
\label{eqn:HGr(restr)}
H^j\operatorname{Gr}^F_p\left((M^\bullet,F_\bullet)_Y\right)\overset{\sim}{=}
H^j(\operatorname{Gr}^F_{p-d} M^\bullet)\otimes_{{\cal O}_X}\omega_{Y/X}\ .
\end{equation}
Suppose that $(M^\bullet,F_\bullet)$ has the property that the complex
$\operatorname{Gr}^F_\bullet M^\bullet$ has bounded $\operatorname{Gr}^F_\bullet{\cal D}_X$-coherent
cohomology; in such case we say that $(M^\bullet,F_\bullet)$
is {\em noncharacteristic\/} with respect to $Y$ if, first,
\eqref{eqn:tor-nonchar} is satisfied, and second,
$\operatorname{Gr}^F_\bullet (M^\bullet)_Y$ also has bounded $\operatorname{Gr}^F_\bullet{\cal D}_Y$-coherent
cohomology.
In the particular case when the complex $(M^\bullet,F_\bullet)$
is actually a filtered coherent ${\cal D}_X$-module $(M,F_\bullet)$,
this definition is equivalent to the definition in \cite{S1}, 3.5.1 because
the condition of coherence of $(M^\bullet,F_\bullet)_Y$ is is equivalent
to the finiteness of the projection
$(Y\times_X T^\ast X)\cap\operatorname{Ch}(M)\to T^\ast Y$ where $\operatorname{Ch}(M)$ denotes the
characteristic variety of $M$.
In such case if $Y$ is noncharacteristic, we have
$i^\ast(M,F_\bullet)=(M^\bullet,F_\bullet)_Y[d]$ and
$i^!(M,F_\bullet)$ is isomorphic to $(M^\bullet,F_\bullet)_Y[-d]$ up to a
shift of filtration.
\begin{defn}
A filtered coherent ${\cal D}_X$-module $(M,F_\bullet)$ is
{\em Cohen---Macaulay\/} if $\operatorname{Gr}^F_\bullet M$ is a Cohen---Macaulay module over
$\operatorname{Gr}^F_\bullet{\cal D}$.
\end{defn}
A Cohen---Macaulay ${\cal D}_X$-module $(M,F_\bullet)$ is holonomic iff
the dimension of $\operatorname{Gr}^F_\bullet M$ over $\operatorname{Gr}^F_\bullet{\cal D}$ is equal
to $\dim X$.
\begin{lemma}
\label{lem:dual-to-restriction}
Suppose that $(M,F_\bullet)$ is a coherent holonomic filtered ${\cal D}_X$-module
noncharacteristic with respect to $Y$.
Then $(M,F_\bullet)$ is holonomic Cohen---Macaulay at a point $y\in Y$
if and only if $(M,F_\bullet)_Y$ is.
\end{lemma}
\begin{pf}
We shall denote $(M,F_\bullet)_Y$ by $(M_Y,F)$.
Let $\dim X=n$, $\operatorname{codim}_X Y=d$.
Let $R=\operatorname{Gr}^F{\cal D}_{X,y}$ and $R'=\operatorname{Gr}^F{\cal D}_{Y,y}$, and
let ${\frak m}$ (respectively, ${\frak m}'$) denote the maximal ideal in $R$ (respectively,
$R'$) corresponding to the origin of $T^\ast_y X$ (respectively, $T^\ast_y Y$).
Both $R$ and $R'$ are graded rings, $\operatorname{Gr}^F M_y$ and $\operatorname{Gr}^F M_{Y,y}$ are
graded modules over them, and hence, the support of $\operatorname{Gr}^F M_y$ in $\operatorname{Spec} R$
corresponds to a homogeneous closed analytic subspace of $T^\ast U$ where
$U$ is a sufficiently small open neighborhood of $y$ in $X$, and
similarly for the support of $\operatorname{Gr}^F M_{Y,y}$ in $\operatorname{Spec} R'$.
We need to show that $\operatorname{Gr}^F M_y$ is Cohen---Macaulay of dimension $n$ over $R$
iff $\operatorname{Gr}^F M_{Y,y}$ is Cohen---Macaulay of dimension $n-d$ over $R'$.
The Cohen---Macaulay property of $\operatorname{Gr}^F M_y$
is equivalent to vanishing of $\operatorname{Ext}^j_R(\operatorname{Gr}^F M_y,R)$ for $j\ne n$;
as the support of $\operatorname{Ext}^j_R(\operatorname{Gr}^F M_y,R)$
is homogeneous in $\operatorname{Spec} R$, this property holds at all points of
$\operatorname{Spec} R$ iff it holds at the origin of $T^\ast_y X$, i.e., at the
maximal ideal ${\frak m}$.
In other words, $\operatorname{Gr}^F M_y$ is Cohen---Macaulay of dimension $n$ over $R$ iff
$(\operatorname{Gr}^F M_y)_{\frak m}$ is Cohen---Macaulay of dimension $n$ over $R_{\frak m}$.
Similarly, $\operatorname{Gr}^F M_{Y,y}$ is a Cohen---Macaulay $R'$-module of
dimension $n-d$ iff
$(\operatorname{Gr}^F M_{Y,y})_{{\frak m}'}$ is a Cohen---Macaulay $R'_{{\frak m}'}$-module of
dimension $n-d$.
It follows that we need to show that $(\operatorname{Gr}^F M_y)_{\frak m}$
is Cohen---Macaulay of dimension $n$ over $R_{\frak m}$
iff $(\operatorname{Gr}^F M_{Y,y})_{{\frak m}'}$ is Cohen---Macaulay of dimension $n-d$
over $R'_{{\frak m}'}$.
Let $N=\operatorname{Gr}^F M_{Y,y}$.
Let $x_1,\dots,x_n$ be a local coordinate system in $X$ at $y$
such that $x_1,\dots,x_d$ is a system of local equations of $Y$ in $X$.
Since $Y$ is noncharacteristic, $x_1,\dots,x_d$ is a regular
$\operatorname{Gr}^F M_y$-sequence in $R$, and
$N\overset{\sim}{=}\operatorname{Gr}^F M_y/(\sum_{l=1}^d x_l\operatorname{Gr}^F M_y)$.
Hence, $N$ is a module over $R/(\sum_{l=1}^d x_l R)$; its structure of an
$R'$-module comes from the embedding $R'\hookrightarrow R/(\sum_{l=1}^d x_l R)$.
Let $A$ and $A'$ be the quotients of $R/(\sum_{l=1}^d x_l R)$ and $R'$,
respectively, by the annihilators of $N$. Then $A'\hookrightarrow A$;
since $\operatorname{Gr}^F M_Y$ is $\operatorname{Gr}^F{\cal D}_Y$-coherent, $N$ is finite over $R'$ and
hence, $A$ is a finite $A'$-module.
Denote by $\tilde{\frak m}$ and $\tilde{\frak m}'$, respectively, the maximal ideals of
$A$ and $A'$ that correspond to the maximal ideals ${\frak m}$ and ${\frak m}'$ of $R$
and $R'$, respectively.
As $A$ is a finite $A'$-module, there are only finitely many ideals in $A$
lying over $\tilde{\frak m}'$, and clearly, $\tilde{\frak m}$ is one of them.
By a homogeneity argument, $\tilde{\frak m}$ is the only
ideal of $A$ lying over $\tilde{\frak m}'$.
Hence, $A_{\tilde{\frak m}}$ is a finite $A'_{\tilde{\frak m}'}$-module.
The localization $N_{{\frak m}'}=(\operatorname{Gr}^F M_{Y,y})_{{\frak m}'}$ of $N$ at ${\frak m}'$ as an
$R'$-module is the same as the localization $N_{\tilde{\frak m}'}$
of $N$ at $\tilde{\frak m}'$ as an
$A'$-module, and is isomorphic to the localization $N_{\tilde{\frak m}}$
of $N$ at $\tilde{\frak m}$ as an $A$-module (since $\tilde{\frak m}$ is the only
ideal of $A$ lying over $\tilde{\frak m}'$).
By~\cite{Se}, Ch.~IV, Proposition~12,
$\operatorname{depth}_{A'_{\tilde{\frak m}'}}N_{\tilde{\frak m}'}=\operatorname{depth}_{A_{\tilde{\frak m}}}N_{\tilde{\frak m}}$ and
$\dim_{A'_{\tilde{\frak m}'}}N_{\tilde{\frak m}'}=\dim_{A_{\tilde{\frak m}}}N_{\tilde{\frak m}}$.
Clearly, $\operatorname{depth}_{R'_{{\frak m}'}}N_{{\frak m}'}=\operatorname{depth}_{A'_{\tilde{\frak m}'}}N_{\tilde{\frak m}'}$
and $\dim_{R'_{{\frak m}'}}N_{{\frak m}'}=\dim_{A'_{\tilde{\frak m}'}}N_{\tilde{\frak m}'}$.
Since $x_1,\dots,x_d$ is a regular $\operatorname{Gr}^F M_y$-sequence in $R$, it is
a regular $(\operatorname{Gr}^F M_y)_{\frak m}$-sequence in $R_{\frak m}$.
We have $N_{\tilde{\frak m}}\overset{\sim}{=}(\operatorname{Gr}^F M_y)_{\frak m}/(\sum_{l=1}^d x_l(\operatorname{Gr}^F M_y)_{\frak m})$,
and hence,
$\operatorname{depth}_{A_{\tilde{\frak m}}}N_{\tilde{\frak m}}=\operatorname{depth}_{R_{\frak m}}(\operatorname{Gr}^F M_y)_{\frak m}-d$ and
$\dim_{A_{\tilde{\frak m}}}N_{\tilde{\frak m}}=\dim_{R_{\frak m}}(\operatorname{Gr}^F M_y)_{\frak m}-d$.
Altogether, we see that
$\operatorname{depth}_{R'_{{\frak m}'}}N_{{\frak m}'}=\operatorname{depth}_{R_{\frak m}}(\operatorname{Gr}^F M_y)_{\frak m}-d$ and
$\dim_{R'_{{\frak m}'}}N_{{\frak m}'}=\dim_{R_{\frak m}}(\operatorname{Gr}^F M_y)_{\frak m}-d$.
It follows that $\operatorname{depth}_{R_{\frak m}}(\operatorname{Gr}^F M_y)_{\frak m}=\dim_{R_{\frak m}}(\operatorname{Gr}^F M_y)_{\frak m}=n$
iff $\operatorname{depth}_{R'_{{\frak m}'}}N_{{\frak m}'}=\dim_{R'_{{\frak m}'}}N_{{\frak m}'}=n-d$, i.e.,
$(\operatorname{Gr}^F M_y)_{\frak m}$ is Cohen---Macaulay of dimension $n$ over $R_{\frak m}$ iff $N_{{\frak m}'}$
is Cohen---Macaulay of dimension $n-d$ over $R'_{{\frak m}'}$.
\end{pf}
\begin{remark}
\label{rem:right-exact}
It is not hard to see from the definitions that
the functors $\operatorname{DR}_X$ and $\operatorname{DR}^{-1}_X$ are right exact in the filtered sense:
if a filtered complex $(M^\bullet,F_\bullet)$ has the property that
$H^j\operatorname{Gr}^F_\bullet M^\bullet=0$ for $j>j_0$ then
$H^j\operatorname{Gr}^F_\bullet\operatorname{DR}_X(M^\bullet,F_\bullet)=0$ for $j>j_0$, and vice versa,
and the same holds for the functor of noncharacteristic restriction
$(\bullet)_Y$.
The functor $\operatorname{DR}^{-1}_X$ is also left exact in the similar sense.
\end{remark}
\section{Filtered perversity}
\label{sec:fil.perv}
In this section we introduce the notion of a {\em weakly filtered perverse}
differential complex; its meaning is that the complex is ``locally trivial''
in a certain filtered sense made precise below, and satisfies filtered
cohomology vanishing conditions which are similar to the cohomology vanishing
conditions for perverse sheaves.
The stratifications need to be defined only locally, which is made precise
by the notion of {\em stratified chart.}
This notion of {\em weak filtered perversity} is precisely
the assumption that we need to use; we call it {\em weak} because
we suspect that some stronger
property of ``local triviality'' along the strata will appear eventually.
We introduce also the notion of {\em coherent filtered perversity\/}
which is somewhat stronger than coherence together with weak
filtered perversity; we shall show in Proposition~\ref{cor:holon}
that it implies holonomicity of the corresponding ${\cal D}_X$-module.
\subsection{Stratified charts}
\begin{defn}
A {\em stratified chart} ${\cal U}$ on $X$ is the following
collection of data:
\begin{enumerate}
\item an open subset $U$ of $X$;
\item an analytic stratification of $U$;
\item for every point $x$ of any stratum $S$ of this stratification,
an open neighborhood $U_x$ of $x$ in $U$ and
an analytic submersion $\pi_x : U_x\to U_x\cap S$
which restricts
to the identity on $U_x\cap S$.
\end{enumerate}
\end{defn}
\subsection{The definition of filtered perversity}
In what follows we denote by $Y$ the fiber $\pi_x^{-1}(x)$.
Given a filtered differential complex $({\cal M}^\bullet,F_\bullet)$ on $X$, we use
the notation ${\cal F}^j_p=H^j\operatorname{Gr}^F_p\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet)$;
this is a sheaf of ${\cal O}_X$-modules.
\begin{defn}
\label{cond:wfp}
A filtered differential complex $({\cal M}^\bullet,F_\bullet)$ on $X$ is called
{\em weakly filtered perverse} if $X$ can be covered by stratified charts
${\cal U}$ which satisfy
the following properties for every point $x$ of any stratum $S$ of ${\cal U}$:
\begin{itemize}
\item[(i)]
for all $j$ and $p$, the sheaf ${\cal F}^j_p$ has the property that
for all $i>0$, and for all
$y\in Y$, we have
${\operatorname{Tor}}^{{\cal O}_{X,y}}_i({\cal F}^j_{p,y},{\cal O}_{Y,y}) = 0$;
\item[(ii)]
for all $j$, $p$, if
${\cal F}^j_{p,x}\otimes_{{\cal O}_{X,x}}{\cal O}_{Y,x} = 0$ then
${\cal F}^j_{p,x}=0$;
\item[(iii)] for all $p$, all $j<0$, we have
$H^j_{\{x\}}\operatorname{Gr}^F_p\operatorname{DR}_Y\left((\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet))_Y\right) = 0$;
\item[(iv)] for all $p$, all $j>0$, we have $H^j\operatorname{Gr}^F_p{\cal M}^\bullet = 0$.
\end{itemize}
We say that $({\cal M}^\bullet,F_\bullet)$ is {\em coherent filtered perverse\/}
if it is weakly filtered perverse, the complex
$\operatorname{Gr}^F_p\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet)$ has bounded $\operatorname{Gr}^F_\bullet{\cal D}_Y$-coherent
cohomology, and for any point $x$, $({\cal M}^\bullet,F_\bullet)$ is
noncharacteristic with respect to $Y$.
\end{defn}
Note that
property (i) means that the complex $\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet)$
satisfies the condition \eqref{eqn:tor-nonchar}, and hence, the
noncharacteristic restriction $(\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet))_Y$
which appears in (iii), is defined.
(We shall actually see that if $({\cal M}^\bullet,F_\bullet)$ is weakly filtered
perverse then $\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet)$ is isomorphic to one filtered
${\cal D}_X$-module $(M,F_\bullet)$, and the condition of coherent filtered
perversity is equivalent to the condition that $(M,F_\bullet)$ coherent
holonomic Cohen-Macaulay.)
\subsection{Construction of stratified charts in the coherent case}
Suppose that $({\cal M}^\bullet,F_\bullet)$ is a filtered differential complex
such that $\operatorname{Gr}^F_\bullet{\cal M}^\bullet$ has bounded coherent cohomology.
Then the condition (ii) is always satisfied.
We shall see here that under this assumption, there always exist
stratified charts satisfying also (i).
\begin{prop}
\label{prop:hom.shf.flat}
Suppose that $p:E\to X$ is a holomorphic vector bundle on $X$
and $\left\{{\cal F}_i\right\}_I$ is a finite collection of homogeneous coherent
sheaves on $E$. Then at any point $x_0$ of
$X$ there exists a stratified chart ${\cal U}$ such that for every
$x\in U$ and $i\in I$ the sheaf ${\cal F}_i\vert_{p^{-1}U_x}$ is
$(\pi_xp)$-flat over $U_x\cap S$.
\end{prop}
\begin{pf}
Let $n=\dim X$.
We shall construct inductively a stratified chart ${\cal U}^k$ containing
$x_0$ such that it satisfies the required flatness property at all
points $x$ of all the strata $S$ of codimension smaller than $k$.
We shall show how to construct ${\cal U}^{k+1}$ once ${\cal U}^k$ has been constructed.
Let $U^k$ be the open set containing $x_0$ which underlies ${\cal U}^k$, and
let $X^k$ be the union of the closures of
all the strata of ${\cal U}^k$ of codimension $k$.
Then $X^k$ is a closed analytic subset of $U^k$ of pure dimension $n-k$.
We choose an open polydisc $\Delta^n$ embedded in $U^k$ which
contains $x_0$ and such that there is a projection $q:\Delta^n\to\Delta^{n-k}$
with the property that the map
$q|_{X^k\cap \Delta^n}:X^k\cap \Delta^n\to\Delta^{n-k}$ is finite.
Consider the composite projection $qp:p^{-1}\Delta^n\to\Delta^{n-k}$.
Let $Z$ be the set of points in $p^{-1}\Delta^n\subset E$
where one of the sheaves ${\cal F}_i$ is not $qp$-flat.
By Frisch's theorem on the openness of the flat locus
(\cite{F}, Theorem (IV,9) or \cite{BS}, Theorem V.4.5), $Z$ is a closed analytic
subset in $p^{-1}\Delta^n$ such that its image in $\Delta^{n-k}$ is negligible.
All the sheaves ${\cal F}_i$ are homogeneous, and hence, $Z$ is homogeneous;
it follows that $p(Z)$ is a closed analytic subset of $\Delta^n$.
The intersection $p(Z)\cap X^k$ is negligible in $X^k\cap \Delta^n$
since its image
under $q$ is negligible and $q$ is finite on $X^k\cap \Delta^n$; it follows that
$p(Z)\cap X^k$ is a proper closed analytic subset of $X^k\cap \Delta^n$.
Construct ${\cal U}^{k+1}$ as follows.
Take $U^{k+1}=\Delta^n$.
All the strata of ${\cal U}^{k+1}$
of codimension less than $k$ are the intersections with
$U^{k+1}$ of the strata of ${\cal U}^k$.
Any $(n-k)$-stratum $S'$ is obtained from a $(n-k)$-stratum $S$ of ${\cal U}^k$
by intersecting with $U^{k+1}$ and then removing, first, all points where
$q|_{S\cap U^{k+1}}:S\cap U^{k+1}\to\Delta^{n-k}$ is ramified,
and second, the intersection with $p(Z)$.
The complement of these strata in $U^{k+1}$ has codimension at least
$k+1$; stratifying it, we complete the stratification of ${\cal U}^{k+1}$.
Clearly, the stata of codimension less than $k$ satisfy the required
flatness condition.
Let $S'$ be any $(n-k)$-stratum constructed as above.
At any point $x\in S'$ we take a neighborhood $U_x\subset U^{k+1}$
in such way that $q(U_x)=q(U_x\cap S')$ and
$q$ is an isomorphism on $U_x\cap S'$.
Take the projection $\pi_x:U_x\to U_x\cap S'$ such that $q\pi_x=q$, i.e.,
$\pi_x=(q|_{U_x\cap S'})^{-1}q$.
Then all the sheaves ${\cal F}_i$ are $qp$-flat on $p^{-1}(U_x\cap S')$
since $S'$ does not intersect $p(Z)$, and hence, they are
$\pi_xp$-flat.
\end{pf}
\begin{cor}
\label{cor:strat}
Suppose that $({\cal M}_i^\bullet,F_\bullet)$ is a finite collection of
filtered differential complexes on $X$
such that $\operatorname{Gr}^F_\bullet{\cal M}_i^\bullet$ have bounded coherent cohomology.
Then, locally
at any point of $X$ there exists a stratified chart such that
the properties (i) and (ii) of Definition~\ref{cond:wfp} hold for each
$({\cal M}_i^\bullet,F_\bullet)$.
\end{cor}
\begin{pf}
Consider the homogeneous coherent sheaves
$\left(H^j\operatorname{Gr}^F_\bullet\operatorname{DR}^{-1}({\cal M}_i^\bullet,F_\bullet)\right)^\sim$
on $T^*X$
obtained by localizing the corresponding $\operatorname{Gr}^F_\bullet{\cal D}_X$-modules.
Proposition~\ref{prop:hom.shf.flat} yields a stratified chart satifying
the condition (i) for each
$({\cal M}_i^\bullet,F_\bullet)$; the condition (ii) is satisfied by coherence.
\end{pf}
\section{Filtered perverse complexes correspond to filtered
$\protect{\cal D}$-modules}
\label{sec:fpc-to-dm}
\subsection{The main theorem}
Given a filtered complex $(M^\bullet,F_\bullet)$, the property that
$H^j\operatorname{Gr}^F_pM^\bullet =0$ for all $p$ and all $j\neq 0$, means that
$(M^\bullet,F_\bullet)$ is strict and $H^jM^\bullet = 0$ for $j\neq 0$.
Another formulation of the same property is that
$(M^\bullet,F_\bullet)$ is isomorphic to
$H^0(M^\bullet,F_\bullet)$ in the filtered derived category,
where $H^0(M^\bullet,F_\bullet)$ denotes $H^0M^\bullet$ equipped with the
induced filtration.
\begin{thm}
\label{thm:main}
Suppose that $({\cal M}^\bullet,F_\bullet)$ is weakly filtered perverse.
Then, for all $p$, all $j\neq 0$,
\begin{equation*}
H^j\operatorname{Gr}^F_p\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet) = 0
\end{equation*}
Consequently the filtered complex
$\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet)$ is
strict and isomorphic in the filtered derived category
to $H^0\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet)$ equipped with the induced filtration.
\end{thm}
\begin{pf}
The statement is local so we may assume that $X=U$ in the definition of
weak filtered perversity, $x$ lies in the stratum $S$, $\pi_x: U_x\to S$ is
an analytic submersion which restricts to the identity on $S$, and
$Y=\pi_x^{-1}(x)$.
We are going to show by induction on $\operatorname{codim} S$
that the conclusion holds for the stalk of
$\operatorname{Gr}^F_p\operatorname{DR}^{-1}({\cal M}^\bullet,F_\bullet)$ at $x$. Thus we may assume that
the conclusion holds on the complement of the stratum $S$.
Condition (iv) of Definition \ref{cond:wfp} implies that, for all $p$,
\begin{equation}\label{van:above}
\text{$H^j\operatorname{Gr}^F_p\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet) = 0$ for $j>0$}\ .
\end{equation}
By~\eqref{eqn:HGr(restr)} we have
\begin{equation}\label{iso:restr}
H^j\operatorname{Gr}^F_p\left((\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet))_Y\right)\overset{\sim}{=}
H^j\operatorname{Gr}^F_p\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet)\otimes_{{\cal O}_X}\omega_{Y/X}\ .
\end{equation}
The induction hypothesis and \eqref{iso:restr} imply that
\begin{equation}\label{van:on-Y-S}
\text{
$H^j\operatorname{Gr}^F_p\left((\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet))_Y\right)
\vert_{Y\setminus\{x\}} = 0$
for $j\neq 0$}\ .
\end{equation}
Let $i:\{x\}\hookrightarrow Y$ be the embedding map.
Condition (iii) of Definition \ref{cond:wfp} implies that
$\operatorname{\bold R} i^!\operatorname{Gr}^F_p\operatorname{DR}_Y\left((\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet))_Y\right)$
is acyclic in negative degrees.
Here $\operatorname{\bold R} i^!$ is the derived functor of the functor $i^!$ which assigns
to a sheaf ${\cal F}$ its sections supported in $x$; if ${\cal F}$ is an ${\cal O}_Y$-module
or a ${\cal D}_Y$-module then $\operatorname{\bold R} i^!{\cal F}$ is an an ${\cal O}_{Y,x}$-module
or a ${\cal D}_{Y,x}$-module, and $\operatorname{\bold R} i^!$ commutes with the functors
$\operatorname{Gr}^F_p$ and $\operatorname{DR}_Y$. Hence,
$\operatorname{Gr}^F_p\operatorname{DR}_Y\operatorname{\bold R} i^!\left((\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet))_Y\right)$
is acyclic in negative degrees; as $\operatorname{DR}^{-1}_Y$ is left exact
(Remark~\ref{rem:right-exact}), the complex
$\operatorname{Gr}^F_p\operatorname{\bold R} i^!\left((\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet))_Y\right)$
is acyclic in negative degrees, so that
\begin{equation}\label{van:loc-coh}
H^j_{\{x\}}\operatorname{Gr}^F_p\left((\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet))_Y\right)= 0
\text{ for $j<0$ .}
\end{equation}
Examination of the long exact sequence in cohomology associated to the
inclusion $Y\setminus\{x\}\subset Y$ in the light of \eqref{van:on-Y-S} and
\eqref{van:loc-coh} shows that
\[
\text{$H^j\operatorname{Gr}^F_p\left((\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet))_Y\right)= 0$
for $j<0$}\ .
\]
Together with \eqref{iso:restr} and the condition (ii) of Definition
\ref{cond:wfp} this shows that
\begin{equation}
\label{eqn:van:j<0}
\text{$H^j\operatorname{Gr}^F_p\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet)_x = 0$ for $j<0$}\ .
\end{equation}
The statement of the Theorem is the combination of \eqref{van:above}
and \eqref{eqn:van:j<0}.
\end{pf}
\section{The coherent case}
\label{sec:coh.case}
In this section we study the property of filtered perversity of
a filtered differential complex $({\cal M}^\bullet,F_\bullet)$ under the
assumption of coherence of $H^\bullet\operatorname{Gr}^F_\bullet{\cal M}^\bullet$; in particular,
this implies that the cohomology of $\operatorname{Gr}^F_\bullet{\cal M}^\bullet$ is bounded.
By~\cite{S1}, (2.2.10.5), this is equivalent to the
$\operatorname{Gr}^F_\bullet{\cal D}_X$-coherence of
$H^\bullet\operatorname{Gr}^F_\bullet\operatorname{DR}^{-1}({\cal M}^\bullet,F_\bullet)$; in case
$\operatorname{DR}^{-1}({\cal M}^\bullet,F_\bullet)$ is isomorphic to a single filtered
${\cal D}_X$-module, this property means that the module is ${\cal D}_X$-coherent and
its filtration is good.
\subsection{Duality for coherent complexes}
The following technical lemma is a standard application of duality theory.
\begin{lemma}\label{lemma:duality}
Suppose that $X$ is a complex manifold of dimension $n$,
$L^\bullet$ is a bounded complex of coherent ${\cal O}_X$-modules, and $x\in X$.
Then for each $j$, there is a nondegenerate pairing between
the spaces $H^{-j}_{\{x\}}L^\bullet$ and
$\left(H^j\operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal O}_X}(L^\bullet,\omega_X[n])\right)_x$;
the same is true for the spaces
$H^jL^\bullet_x$ and
$H^{-j}_{\{x\}}\operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal O}_X}(L^\bullet,\omega_X[n])$.
\end{lemma}
A nondegenerate pairing between two vector spaces
is a pairing that induces a monomorphism from each of them into
the (algebraic) dual of the other. (Actually, each of the vector spaces
can be given a topology so that they become topologically dual.
More precisely, the pairs of
spaces indicated in the Lemma, are strong dual to each other with respect to
certain natural FS and DFS topologies.
In case the complex $L^\bullet$ is zero except in one degree,
this statement
is a particular case of a theorem of Harvey: take $K=\{x\}$ in Theorem~5.12
of~\cite{ST}.
However, we do not need the topological duality;
all we need is that these vector spaces are either both zero or both nonzero.)
\begin{pf*}{Proof of Lemma \protect\ref{lemma:duality}}
We shall establish the duality between the spaces
$H^{-j}_{\{x\}}L^\bullet$ and\break
$\left(H^j\operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal O}_X}(L^\bullet,\omega_X[n])\right)_x$;
the other duality would follow by substituting the dual complex
$\operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal O}_X}(L^\bullet,\omega_X[n])$ in place of $L^\bullet$.
Replacing $L^\bullet$ by its bounded free resolution in a neighborhood of $x$,
we may assume that all the sheaves $L^k$ are free.
Then $\operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal O}_X}(L^\bullet,\omega_X[n])$ is represented
by $\underline{\operatorname{Hom}}^\bullet_{{\cal O}_X}(L^\bullet,\omega_X[n])$, and
$\left(H^j\operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal O}_X}(L^\bullet,\omega_X[n])\right)_x\overset{\sim}{=}
H^j\operatorname{Hom}^\bullet_{{\cal O}_{X,x}}(L^\bullet_x,\omega_{X,x}[n])$.
The complex $\operatorname{Hom}^\bullet_{{\cal O}_{X,x}}(L^\bullet_x,\omega_{X,x}[n])$
is a complex of free finitely generated ${\cal O}_{X,x}$-modules.
Each of them has a canonical DFS topology and the differential is
continuous with respect to it; moreover, the image of the differential
is closed since it is closed with respect to the weaker topology of
coefficientwise convergence of formal power series (Theorem~6.3.5 of~\cite{H}).
Since each $L^k$ is free,
by a theorem of Martineau (\cite{ST}, Theorem 5.9) we have that $H^j_{\{x\}}L^k$
is zero for $j\ne n$, and $H^n_{\{x\}} L^k$ can be given a natural
Hausdorff FS topology in which it is a strong dual to
$\operatorname{Hom}_{{\cal O}_{X,x}}(L_x^k,\omega_{X,x})$; in particular, it follows that
$H^{-j}_{\{x\}} L^\bullet\overset{\sim}{=} H^{-j-n}(H^n_{\{x\}} L)^\bullet$ where
we denote
$(H^n_{\{x\}} L)^\bullet=
\{\dots\to H^n_{\{x\}} L^k\to H^n_{\{x\}} L^{k+1}\to\dots\}$.
The pairing between $H^n_{\{x\}} L^k$ and
$\operatorname{Hom}_{{\cal O}_{X,x}}(L_x^k,\omega_{X,x})$, is given by the composition
of the multiplication
$H^n_{\{x\}}L^k\otimes\operatorname{Hom}_{{\cal O}_{X,x}}(L_x^k,\omega_{X,x})\to
H^n_{\{x\}}\omega_X$
and the residue map $H^n_{\{x\}}\omega_X\to{\Bbb C}$, and hence, the complex
$(H^n_{\{x\}} L)^\bullet$ is the strong dual to the complex
$\operatorname{Hom}^\bullet_{{\cal O}_{X,x}}(L_x^\bullet,\omega_{X,x}[n])$.
As the latter is a complex of DFS spaces with Hausdorff cohomology,
the former is a complex of FS spaces with Hausdorff cohomology
$H^{-j-n}(H^n_{\{x\}} L)^\bullet$ strong dual to
$\left(H^j\operatorname{Hom}^\bullet_{{\cal O}_X}(L^\bullet,\omega_X[n])\right)_x$.
This yields a nondegenerate pairing between $H^{-j}_{\{x\}} L^\bullet$
and $\left(H^j\operatorname{Hom}^\bullet_{{\cal O}_X}(L^\bullet,\omega_X[n])\right)_x$.
\end{pf*}
\subsection{Holonomicity}
\begin{prop}
\label{cor:holon}
Suppose that $({\cal M}^\bullet,F_\bullet)$ is a coherent filtered perverse
complex on a complex manifold $X$. Then
$\operatorname{DR}^{-1}({\cal M}^\bullet,F_\bullet)$ is isomorphic in the filtered derived category
to a filtered holonomic Cohen-Macauley ${\cal D}_X$-module.
\end{prop}
\begin{pf}
The question is local and we need to prove it in a neighborhood of
any point $x\in X$.
The point $x$ is covered by a stratified chart ${\cal U}$ satisfying properties
(i)--(iv) of Definition~\ref{cond:wfp}; we keep the notation introduced there.
Our assumptions imply that $\operatorname{DR}^{-1}_X({\cal M}^\bullet,F_\bullet)$ is isomorphic to a
coherent filtered ${\cal D}_X$-module; we shall denote this module by
$(M,F_\bullet)$.
We argue by induction by the codimension of the stratum $S$ containing
$x$; by the inductive assumption, we may assume that $(M,F_\bullet)$
is holonomic Cohen-Macauley in the complement to $S$.
By Lemma~\ref{lem:dual-to-restriction},
this implies that $(M,F_\bullet)_Y$ is holonomic
Cohen-Macauley everywhere on $Y$ except possibly at $x$; hence, it
is holonomic.
Property (iii) implies that $H^j_{\{x\}}\operatorname{Gr}^F_p\operatorname{DR}_Y((M,F_\bullet)_Y)=0$
if $j<0$.
By Lemma~\ref{lemma:duality} this yields
$\left(H^j\operatorname{\bold R}\underline{\operatorname{Hom}}^\bullet_{{\cal O}_Y}
(\operatorname{Gr}^F_p\operatorname{DR}_Y((M,F_\bullet)_Y),\omega_Y[\dim Y])\right)_x=0$
if $j>0$.
By \S\ref{subsec:duality}
this implies that $H^j\operatorname{Gr}^F_p\operatorname{DR}_Y{\Bbb D}((M,F_\bullet)_Y)=0$ if $j>0$,
and by right exactness of $\operatorname{DR}^{-1}_Y$ we get
$H^j\operatorname{Gr}^F_p{\Bbb D}((M,F_\bullet)_Y)=0$ if $j>0$.
Since $(M,F_\bullet)_Y$ is a filtered ${\cal D}_Y$-module, we have
$H^j\operatorname{Gr}^F_p\operatorname{DR}_Y((M,F_\bullet)_Y)=0$ for $j>0$.
By Lemma~\ref{lemma:duality} and \S\ref{subsec:duality}, this implies
$H^j_{\{x\}}\operatorname{Gr}^F_p\operatorname{DR}_Y{\Bbb D}((M,F_\bullet)_Y)=0$ for $j<0$.
So we get $H^j_{\{x\}}\operatorname{Gr}^F_p{\Bbb D}((M,F_\bullet)_Y)=0$ for $j<0$
by the left exactness of $\operatorname{DR}^{-1}_Y$.
The long exact sequence of the inclusion $Y\setminus\{x\}\subset Y$
(cf.\ the proof of Theorem~\ref{thm:main}) yields
$H^j\operatorname{Gr}^F_p{\Bbb D}((M,F_\bullet)_Y)=0$ if $j<0$.
(Actually, this vanishing also follows from the vanishing --- see, for
example,~\cite{Borel}, V.2.2.2 --- of
$\operatorname{Ext}^i_{\operatorname{Gr}^F{\cal D}_{Y,x}}(\operatorname{Gr}^F M_{Y,x},\operatorname{Gr}^F{\cal D}_{Y,x})$ for $i<d$ where
$d$ is the codimension of the support of $\operatorname{Gr}^F M_{Y,x}$ in
$\operatorname{Spec}\operatorname{Gr}^F{\cal D}_{Y,x}$; in our case $d=\dim Y$.)
Altogether, we see that $H^j\operatorname{Gr}^F_p{\Bbb D}((M,F_\bullet)_Y)=0$ if $j\ne 0$,
i.e., the filtered complex ${\Bbb D}((M,F_\bullet)_Y)$ is isomorphic to one
filtered module. Hence, $(M,F_\bullet)_Y$ is Cohen-Macaulay at $x$.
It follows by Lemma~\ref{lem:dual-to-restriction}
that $(M,F_\bullet)$ is also Cohen-Macaulay at $x$.
\end{pf}
\subsection{The converse to the Main Theorem}
\begin{thm}
\label{thm:converse}
If a coherent filtered ${\cal D}_X$-module $(M,F_\bullet)$ is holonomic
and Cohen-Macauley, then $\operatorname{DR}_X(M,F_\bullet)$ is coherent filtered perverse.
\end{thm}
\begin{pf}
Consider a point $x\in X$.
The coherence of $(M,F_\bullet)$ implies that
$H^\bullet\operatorname{Gr}^F_\bullet\operatorname{DR}(M,F_\bullet)$ is ${\cal O}_X$-coherent.
Consequently, Corollary~\ref{cor:strat}
implies that in a neighborhood of $x$ there
exists a stratified chart ${\cal U}$ such that $\operatorname{DR}(M,F_\bullet)$
satisfies properties (i) and (ii) with respect to it.
We shall keep the notation of Definition~\ref{cond:wfp}.
As $M$ is holonomic, we may assume that that the characterisic
variety $\operatorname{Ch}(M)$ is contained in the union of the conormal bundles
to the strata of the Whitney stratification that underlies ${\cal U}$.
It follows that the projection $(Y\times_X T^\ast X)\cap\operatorname{Ch}(M)\to T^\ast Y$
is finite (it is even an embedding), and hence, $(M,F_\bullet)_Y$
is a coherent filtered ${\cal D}_Y$-module.
This implies that $Y$ is noncharacteristic with respect to $(M,F_\bullet)$.
The property (iv) of Definition~\ref{cond:wfp}
at $x$ is satisfied by $\operatorname{DR}(M,F_\bullet)$
since it is satisfied by the de Rham complex of any filtered ${\cal D}_X$-module.
Let us show the property (iii) at $x$ with respect to this stratified chart.
By Lemma~\ref{lem:dual-to-restriction},
$(M,F_\bullet)_Y$ is holonomic Cohen-Macaulay at $x$.
Hence, the complex
${\Bbb D}((M,F_\bullet)_Y)$ is isomorphic to a filtered ${\cal D}_Y$-module,
and consequently, the complex $\operatorname{DR}_Y{\Bbb D}((M,F_\bullet)_Y)$ satisfies (iv)
at $x$:
\[
H^j\operatorname{Gr}^F_\bullet\operatorname{DR}_Y{\Bbb D}((M,F_\bullet)_Y)=0\text{ \ at $x$ for all $j>0$.}
\]
By \S\ref{subsec:duality} and Lemma~\ref{lemma:duality} we get
\[
H^j_{\{x\}}\operatorname{Gr}^F_\bullet\operatorname{DR}_Y((M,F_\bullet)_Y)=0\text{ \ for all $j<0$.}
\]
This is the property (iii) at $x$ for the filtered complex $\operatorname{DR}_X(M,F_\bullet)$.
\end{pf}
\section{An application to {\protect$L^2$} cohomology}
\label{sec:appl}
\renewcommand{F^\bullet}{{F_\bullet}}
In this section we give an application to our results and show that
in the situation of \cite{KK2} (and under the assumption of filtered
perversity of the $L^2$\ complex) there is a local filtered isomorphism
(in the sense of derived category) between the $L^2$ complex
and the de Rham complex of
the ${\cal D}$-module that underlies the corresponding pure Hodge module.
\subsection{}
\label{subsec:l2}
Let $X$ denote a K\"ahler manifold of dimension $n$, let
$j:{X^\circ}\hookrightarrow X$ be the inclusion map of the complement of
a divisor with normal crossings, and
${\Bbb E} = ({\Bbb E}_{{\Bbb Q}},({\cal O}_{X^\circ}\otimes_{{\Bbb Q}}{\Bbb E}_{{\Bbb Q}},F^\bullet))$
a quasiunipotent polarised variation of pure Hodge structure of weight
$w$ on $X^\circ$.
Let $(N,F_\bullet)$ denote the filtered ${\cal D}_X$-module underlying the
polarizable Hodge module~\cite{S1} which restricts to
$(\omega_{X^\circ}\otimes_{{\Bbb Q}}{\Bbb E}_{{\Bbb Q}},F^\bullet)$ on $X^\circ$.
Let $({\cal A}^{\bullet}_{(2)}({\Bbb E}),F^\bullet)$ denote
the {$L^2$}-complex with coefficients in ${\Bbb E}$ constructed using the Hodge
inner product in the fibers of ${\Bbb E}$ and a certain complete metric $\eta$ on
$X^\circ$ as in~\cite{KK2}, \cite{CKS}; to keep up with our degree conventions,
we shall assume that the grading of $({\cal A}^{\bullet}_{(2)}({\Bbb E}),F^\bullet)$
is chosen in such way that ${\cal A}^i_{(2)}({\Bbb E})$ contains forms of
degree $i+n$.
As the metric $\eta$ satisfies $\eta>C\eta_X$ locally in a neighborhood
of any point of $X$, where $\eta_X$ is
the metric on $X$ and $C$ a suitable positive constant,
the holomorphic forms on $X$ are bounded in the pointwise
norm with respect to $\eta$ and the {$L^2$}-complex is an ${\cal O}_X$-module:
if $\omega$ is a section of ${\cal A}^{\bullet}_{(2)}({\Bbb E})$ and $f$ is a holomorphic function
then $f\omega$ is also a section of ${\cal A}^{\bullet}_{(2)}({\Bbb E})$ since both $f\omega$
and $d(f\omega)=df\wedge\omega+fd\omega$ are $L^2$.
We shall assume here that $({\cal A}^{\bullet}_{(2)}({\Bbb E}),F^\bullet)$ is weakly filtered
perverse, and so is every direct summand of it
(in the sense of derived category).
By Theorem~\ref{thm:main} this implies that the complex
$\operatorname{DR}^{-1}({\cal A}^{\bullet}_{(2)}({\Bbb E}),F^\bullet)$ is strict and isomorphic in the filtered derived category
to its zeroeth cohomology with the induced filtration.
By~\cite{KK2} and \cite{CKS}, ${\cal A}^{\bullet}_{(2)}({\Bbb E})$ is isomorphic in the derived category of
complexes
of sheaves on $X$ to the intersection complex with coefficients in ${\Bbb E}$.
We shall assume, moreover, that for any cross-section $Y$
appearing in the definition of weak filtered perversity, the
complex
$\operatorname{DR}_Y\left(\left(\operatorname{DR}^{-1}_X({\cal A}^{\bullet}_{(2)}(X,{\Bbb E}))\right)_Y\right)$
is isomorphic
to the intersection cohomology complex on $Y$ with the coefficients in
${\Bbb E}|_{Y\capX^\circ}$
in the derived category of complexes (without filtration).
(One would even expect that the filtered complex
$\operatorname{DR}_Y\left(\left(\operatorname{DR}^{-1}_X({\cal A}^{\bullet}_{(2)}(X,{\Bbb E}),F_\bullet)
\right)_Y\right)$
is isomorphic in the filtered derived category to
$({\cal A}^{\bullet}_{(2)}(Y,{\Bbb E}|_{Y\capX^\circ}),F_\bullet)$.)
In case $X$ is compact, both complexes of global sections
$\Gamma(X,{\cal A}^{\bullet}_{(2)}({\Bbb E}))$ and $\Gamma(X,\operatorname{DR}(N))$ are strict (\cite{KK2}, \cite{S1}),
and their cohomology have pure Hodge structures.
Their cohomology groups are isomorphic (\cite{KK2}) together with the Hodge
filtrations (\cite{S2}, p.~294). Here we strengthen these results and
show the isomorphism at the level of sheaves (without the assumption of
compactness), in the filtered derived categories:
\begin{prop}
\label{thm:applic}
Assume that
\begin{enumerate}
\item any direct summand of
$({\cal A}^{\bullet}_{(2)}(X,{\Bbb E}),F_\bullet)$
in the filtered derived category of filtered differential complexes
(in particular,
$({\cal A}^{\bullet}_{(2)}(X,{\Bbb E}),F_\bullet)$ itself)
is weakly filtered perverse;
\item for any cross-section $Y$
as in Definition \ref{cond:wfp} (weak filtered perversity), the
complex
$\operatorname{DR}_Y\left(\left(\operatorname{DR}^{-1}_X({\cal A}^{\bullet}_{(2)}(X,{\Bbb E}))\right)_Y\right)$
is isomorphic
to the intersection cohomology complex on $Y$ with the coefficients in
${\Bbb E}|_{Y\capX^\circ}$.
\end{enumerate}
Then the filtered differential complexes $\operatorname{DR}_X(M,F_\bullet)$ and
$({\cal A}^{\bullet}_{(2)}(X,{\Bbb E}),F_\bullet)$ are isomorphic
in the filtered derived category.
Equivalently, the filtered ${\cal D}_X$-modules
$(M,F_\bullet)$ and $H^0\operatorname{DR}^{-1}({\cal A}^{\bullet}_{(2)}({\Bbb E}),F_\bullet)$ are isomorphic.
\end{prop}
\begin{pf}
By Remark 3.15 of~\cite{S2} (the idea actually going back
to~\cite{KK}), there is a direct sum decompostion in the
derived category of filtered differential complexes
\begin{equation*}
({\cal A}^{\bullet}_{(2)}({\Bbb E}),F^\bullet)\overset{\sim}{=} \operatorname{DR}_X(N,F_\bullet)\oplus ({\cal M}^\bullet,F_\bullet)
\end{equation*}
and we need to show that the second summand is trivial.
Our assumptions imply that
\begin{enumerate}
\item
$({\cal M}^\bullet,F_\bullet)$ is weakly filtered
perverse, therefore, by Theorem~\ref{thm:main}, isomorphic to
$\operatorname{DR}_X(M,F_\bullet)$ where $(M,F_\bullet)$ is a filtered ${\cal D}_X$-module;
\item
for any cross-section $Y$ as in Definition \ref{cond:wfp}, the complex
$\operatorname{DR}_Y((\operatorname{DR}^{-1}_X{\cal M}^\bullet)_Y)\overset{\sim}{=}\operatorname{DR}_Y(M_Y)$ is acyclic.
\end{enumerate}
The weak filtered perversity of $({\cal M}^\bullet,F_\bullet)$
implies that $X$ is covered by stratified charts satisfying properties
(i)--(iv) of Definition \ref{cond:wfp}. It is sufficient to show that the
intersection of the support of $H^\bullet\operatorname{Gr}^F_\bullet({\cal M}^\bullet,F_\bullet)$
with any of the charts is empty.
Assume to the contrary and consider a stratified chart and a point
$x\in\operatorname{Supp} H^\bullet\operatorname{Gr}^F_\bullet({\cal M}^\bullet,F_\bullet)$ which lies
on a stratum which is maximal among those which have a nonempty
intersection with $\operatorname{Supp} H^\bullet\operatorname{Gr}^F_\bullet({\cal M}^\bullet,F_\bullet)$.
Let $Y$ denote the cross-section at $x$. Then $\operatorname{Supp} M_Y\subseteq\{x\}$.
In addition we have
$H^j\operatorname{Gr}^F_\bullet\operatorname{DR}_Y((M,F_\bullet)_Y)=0$ for $j>0$ by the right exactness
of $\operatorname{DR}_Y$, and for $j<0$ by property (iii) observing that
$H^j\operatorname{Gr}^F_\bullet\operatorname{DR}_Y((M,F_\bullet)_Y)\overset{\sim}{=}
H^j_{\{x\}}\operatorname{Gr}^F_\bullet\operatorname{DR}_Y((M,F_\bullet)_Y)$
since $\operatorname{Supp}\operatorname{Gr}^F_\bullet\operatorname{DR}_Y((M,F_\bullet)_Y)\subseteq\{x\}$.
It follows that the filtered complex $\operatorname{DR}_Y((M,F_\bullet)_Y)$ is strict.
Since the complex $\operatorname{DR}_Y(M_Y)$ is acyclic it follows that
the complex $\operatorname{DR}_Y((M,F_\bullet)_Y)$ is filtered acyclic, and hence,
$M_Y$ is trivial.
The property (ii) of the weak filtered perversity shows that $M$
is trivial at $x$ (cf.\ the proof of Theorem~\ref{thm:main}),
which contradicts our assumption. Hence, $M$ is trivial.
\end{pf}
\begin{cor}
In the assumptions of Proposition~\ref{thm:applic} the sheaves
$H^\bullet\operatorname{Gr}^F_\bullet{\cal A}^{\bullet}_{(2)}({\Bbb E})$ (the $L^2$-$\overline{\partial}$-cohomology, see
below) are coherent.
\end{cor}
\subsection{Remarks on the $L^2$-$\overline\partial$-cohomology of
a singular variety}
\label{rem:bound-cond}
Let $({\cal M}^\bullet,F_\bullet)$ be the $L^2$-complex of a singular subvariety
$Z$ (the complex of sheaves of forms with locally summable coefficients
on the nonsingular part $Z^\circ$ of $Z$ which
are $L^2$\ together with their differentials near all points of $Z$,
both smooth and singular).
This is the sheafification of the presheaf assigning to each open set
$U\subset Z$ the domain of the maximal closed extension of the
differential $d$ on the Hilbert space of the $L^2$-forms on $U\cap Z^\circ$;
there is another flavor of the $L^2$-complex constructed in a similar
way by sheafification of the {\em minimal\/} closed extension, see details
in~\cite{Y}, \S2.3.
The complex $\operatorname{Gr}^F_{-p}{\cal M}^\bullet$ consists of $(p,q)$-forms
$\omega^{pq}$ on $Z^\circ$ (for any $q$)
which have the following properties:
$\omega^{pq}$ and $\overline{\partial}\omega^{pq}$ are $L^2$\ and moreover,
there exists a form $\omega=\omega^{pq}+\omega^{p+1,q-1}+\dots$ such
that both $\omega$ and $d\omega$ are $L^2$.
The
differential in the complex $\operatorname{Gr}^F_p{\cal M}^\bullet$ is the operator $\overline{\partial}$.
Let us {\em assume\/} that this is a closed extension of $\overline{\partial}$.
(To be precise, this means that the sections of this sheaf over an
open set $U\subset Z$ form a closed extension of $\overline{\partial}$ in the
Fr\'echet space of forms on $U\cap Z^\circ$ which are $L^2$\ locally
in a neighborhood of any point of $U$; the topology on this Fr\'echet
space is given by the seminorms $\|\bullet\|_K$ where $K$ is a relatively
compact open subset
of $U$; for a form $\omega$, the value $\|\omega\|_K$ is the $L^2$\ norm
of $\omega$ on $K$.)
In such case the complex $\operatorname{Gr}^F_p{\cal M}^\bullet$ can be viewed as an
``ideal boundary condition'' (the notion due to J. Cheeger)
for the operator $\overline{\partial}$ at the singularities of $Z$;
this complex contains the minimal closed extension of $\overline{\partial}$
and is contained in the maximal one (their sheafifications
can be defined in a way similar to those of the operator $d$).
If the operator $d$ on $L^2$\ forms has the property that its minimal
extension coincides with the maximal one (this is called the {\em $L^2$
Stokes property\/}~\cite{C}, \cite{Y}; it is known for conical
singularities~\cite{C} and seems from~\cite{O3}
to be a reasonable conjecture in general) then, under our assumptions,
it is not hard to see that the boundary condition
of $\operatorname{Gr}^F_{-p}{\cal M}^\bullet$ is dual to the boundary condition
of $\operatorname{Gr}^F_{-p'}{\cal M}^\bullet$ if $p+p'=\dim Z$. (More precisely, this
means the following.
For any open $U\subset Z$, the dual to
the Fr\'echet space of forms which are locally
L2 on $U$ --- with the topology described above --- can be identified
by the pairing
$<\omega,\phi>=\int_{U\cap Z^\circ}\omega\wedge\phi$ with the DF space
of forms $\phi$ on $U\cap Z^\circ$ such that the closure of $\operatorname{Supp}\phi$
in $U$ is compact.
The $L^2$ Stokes property means that the adjoint of maximal extension of
$d$ in the first space is, up to sign,
the maximal extension of $d$ in the second one.
The differential $\overline{\partial}$ on the sections of $\operatorname{Gr}^F_{-p}{\cal M}^\bullet$ on $U$
is an unbounded operator on the subspace of $(p,q)$-forms of the above
Fr\'echet space; similarly, the differential $\overline{\partial}$ on the sections of
$\operatorname{Gr}^F_{-p'}{\cal M}^\bullet$ with compact support is an unbounded operator
on the subspace of $(p',q)$-forms of the above DF space.
The duality between the boundary conditions means that these two operators
are adjoint up to sign.)
It is easy to see that under our assumptions,
the differential in $\operatorname{Gr}^F_{-\dim Z}{\cal M}^\bullet$ is the maximal closed
extension of $\overline{\partial}$; it follows by duality that the differential in
$\operatorname{Gr}^F_0{\cal M}^\bullet$ is the minimal closed extension of $\overline{\partial}$.
In case the metric on $Z^\circ$ is complete (e.g., Saper metric),
the minimal closed extension of $\overline{\partial}$ is known to coincide with the
maximal one, and hence, $\operatorname{Gr}^F_{-p}{\cal M}^\bullet$ is just the domain
of $\overline{\partial}$ with any of the boundary conditions.
In case the metric is incomplete (e.g., the restriction of the Fubini---Studi
metric on the projective space to $Z^\circ$), it is known that the
minimal closed extension of $\overline{\partial}$ may be different from the
maximal one~\cite{P}. The results of~\cite{PS} and~\cite{FH}
suggest that in case
$p=0$ the ``correct'' boundary condition for $\overline{\partial}$ is the minimal
(Dirichlet) one, and in case
$p=\dim Z$ the ``correct'' boundary condition for $\overline{\partial}$ is the maximal
(Neumann) one.
This suggests that under our assumptions,
the complex $\operatorname{Gr}^F_{-p}{\cal M}^\bullet$ is the most natural
boundary condition for the operator $\overline{\partial}$, and its cohomology
$H^j\operatorname{Gr}^F_{-p}{\cal M}^\bullet$ is the most natural notion of the
$L^2$-$\overline{\partial}$-cohomology sheaves.
|
1996-07-23T20:38:11 | 9607 | alg-geom/9607025 | en | https://arxiv.org/abs/alg-geom/9607025 | [
"alg-geom",
"math.AG"
] | alg-geom/9607025 | Rahul Pandharipande | R. Pandharipande | The Chow Ring of the Hilbert Scheme of Rational Normal Curves | 24 pages, Latex2e | null | null | null | null | Let H(d) be the (open) Hilbert scheme of rational normal curves of degree d
in P^d. A presentation of the integral Chow ring of H(d) is given via
equivariant Chow ring computations. Included also in the paper are algebraic
computations of the integral equivariant Chow rings of the algebraic groups
O(n), SO(2k+1). The results for S0(3)=PGL(2) are needed for the Hilbert scheme
calculation.
| [
{
"version": "v1",
"created": "Tue, 23 Jul 1996 18:30:38 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Pandharipande",
"R.",
""
]
] | alg-geom | \section{\bf{Introduction}}
\subsection{Summary}
Let $\mathbb{C}$ be the ground field of complex numbers.
A rational normal curve in $\mathbf P^d$ is an irreducible,
nonsingular, non-degenerate, degree $d$ rational curve.
For $d\geq 1$,
let $H(d)$ be the open Hilbert scheme of rational
normal curves of degree $d$ in $\mathbf P^d$.
$H(d)$ is a nonsingular, irreducible, quasi-projective, algebraic
variety. Let $A^*(d)$ be the integral Chow ring of $H(d)$.
In case $d=1$, there is a unique rational normal curve in $\mathbf P^1$.
Hence, $H(1)$ is a point. $H(2)$ is the space
of nonsingular plane conics.
The dimension of $H(d)$ is $d^2+2d-3$.
In this paper, a presentation
of $A^*(d)$ is computed via the theory of equivariant Chow
groups. The idea is to exhibit $H(d)$ as a quotient of
an appropriate variety $X$
by a free $\mathbf{G}$-action. For free actions, the
equivariant Chow ring $A^*_{\mathbf{G}}(X)$ is isomorphic
to the ordinary Chow ring $A^*(d)$ of the quotient $X/\mathbf{G}\stackrel{\sim}{=} H(d)$.
The equivariant Chow ring $A^*_{\mathbf{G}}(X)$ is then computed
in the required cases
via Chow rings of projective bundles and
Chow ideals of degeneracy loci.
The geometry of
$H(d)$ depends significantly on the parity of $d$.
The quotient approaches and the presentations
of $A^*(d)$ differ for $d$ even and odd.
In the even case, a $\mathbf{PGL}(2)$-quotient
approach is taken. The geometry of
algebraic $B\mathbf{PGL}(2)$ is studied as a necessary
first step. There is an isomorphism of linear
algebraic groups:
$\mathbf{PGL}(2) \stackrel{\sim}{=} \mathbf{SO}(3)$. The space $B \mathbf{SO}(3)$
is analyzed via conic geometry in projective space.
It is no more difficult to study
algebraic $B \mathbf{O}(n)$ and $B \mathbf{SO} (n=2k+1)$ via
higher dimensional quadrics. The equivariant
Chow rings of these two series are computed.
The $d$ odd case
is simpler.
In this case, the quotient
group is taken to be a central extension of
$\mathbf{SL}(2)$ in $\mathbf{GL}(2)$.
Presentations of $A^*(d)$
in case $d$ is even and odd are determined
in Theorems \ref{heven} and \ref{hodd} respectively.
The equivariant Chow rings of the groups
$\mathbf{O}(n)$ an $\mathbf{SO}(n=2k+1)$ are computed
in
Theorem \ref{chor}.
The equivariant Chow ring
of $\mathbf{O}(n)$ was first determined by
B. Totaro using
complex cobordism theory and topology.
After
the intial algebraic calculation of the ring of $\mathbf{SO}(3)$
presented here, it was realized
both Totaro's methods and
the $\mathbf{SO}(3)$ computation generalize to $\mathbf{SO}(2k+1)$.
An algebraic approach to
$B \mathbf{SO} (3)$ is required for the application to
Theorem 1.
In [P], Chow rings (with $\mathbb{Q}$-coefficients)
of certain moduli spaces of maps are computed via
equivariant Chow groups. The integral
computations presented here were motivated by the
calculations in [P]. These arguments show the
equivariant constructions in [T] and [EG] can
be used effectively to compute ordinary Chow rings
of quotients.
\subsection{Presentations of $A^*(d)$}
\label{prezz}
Equivariant Chow theory is reviewed in section \ref{chow}.
Let $\mathbf{G}$ be a reductive algebraic group. Let $\mathbf{G}\times X \rightarrow X$
be a linearized algebraic group action on a
nonsingular quasi-projective
variety $X$. An equivariant Chow ring $A^*_\mathbf{G}(X)$ is defined
via algebraic approximations to $E\mathbf{G}$ and $B \mathbf{G}$.
Let $V$ be a fixed $2$-dimensional $\mathbb{C}$-vector space.
Let $\mathbf P^1\stackrel{\sim}{=} \mathbf P(V)$. There is a canonical isomorphism
$H^0(\mathbf P^1, {\mathcal{O}}_{\mathbf P^1}(d)) \stackrel{\sim}{=} Sym^{d}(V^*)$.
Let $$U\subset \bigoplus_{0}^{d} Sym^d(V^*)$$
denote the non-degenerate locus (this is
the open set consisting of linearly independent
$(d+1)$-tuples of vectors of $Sym^d(V^*)$).
$U$ parameterizes bases of the linear series of ${\mathcal{O}}_{\mathbf P^1}(d)$
on $\mathbf P^1$.
There is a canonical
$\mathbf{GL} (V)$-action on $U$ with geometric quotient $H(d)$.
The required existence results for the
algebraic quotient problems encountered in this paper
are developed in the Appendix (section \ref{appx}).
$\mathbf{GL} (V)$ acts with finite stabilizers on $U$
(the stabilizer of a point $u\in U$ is the subgroup
of scalar $d^{th}$ roots of unity), By a theorem of
D. Edidin and W. Graham ([EG]), there is a canonical
isomorphism of graded
rings
$$A^*(d) \otimes_{\mathbb{Z}}{\mathbb{Q}} \stackrel{\sim}{=} A^*_{\mathbf{GL} (V)}(U) \otimes _{\mathbb{Z}}{\mathbb{Q}}.$$
The equivariant Chow ring $A^*_{\mathbf{GL} (V)}(U)$ is determined
in section \ref{abe}.
$A^*_{\mathbf{GL} (V)}(U)$ is generated (as a ring)
in codimensions $1$, $2$ by elements $c_1$, $c_2$ respectively.
There are $d+1$ relations given as follows. Let $S$ be a rank
$2$ bundle with Chern classes $c_1$ and $c_2$. The $d+1$
Chern classes of $Sym^d(S)$ are the relations.
It is not difficult to see $A^i_{\mathbf{GL} (V)}(U) \otimes_{\mathbb{Z}} \mathbb{Q}= 0$
for $i>0$.
\begin{pr}
\label{alltor}
$A^*(d)$ is torsion in codimension $i>0$.
\end{pr}
\noindent Note that $\mathbf{GL}(Sym^d (V^*))$ acts transitively
on $U$ and $H(d)$ is a homogeneous space for
$\mathbf{GL}(Sym^d(V^*))=\mathbf{GL}(d+1)$.
Let $\mathbf P(U) \subset \mathbf P(\bigoplus_{0}^{d} Sym^d(V^*))$
be the projective non-degenerate locus.
$\mathbf P(U)$ is exactly the space of parameterized rational
normal curves.
There is a canonical
$\mathbf{PGL} (V)$-action on $\mathbf P(U)$ with geometric quotient
$H(d)$. This is a free action. Hence, there is a canonical
isomorphism of graded rings (see [EG]):
$$A^*(d) \stackrel{\sim}{=} A^*_{\mathbf{PGL} (V)}(\mathbf P(U)).$$
Assume $d\geq 2$ is even. Let $d=2n$ (where $n\geq 1$).
$\mathbf P(U) \rightarrow H(d)$ is a principal $\mathbf{PGL}(V)$-bundle.
Let $S$ be the rank $3$ algebraic vector bundle on $H(d)$
obtained from the principal bundle $\mathbf P(U) \rightarrow H(d)$
and the representation $Sym^2 (V)$ of $\mathbf{PGL}(V)$.
A discussion of algebraic principal bundles
can be found in the Appendix.
For $1 \leq i \leq 3$, let $c_i\in A^*(d)$ be
the Chern classes of $S$.
Let $\mathcal{H}\in A^1(d)$ be the
divisor class of curves meeting a fixed codimension
$2$ linear space in $\mathbf P^d$.
Let $\mathcal{L}= n \mathcal{H}$.
In section \ref{evan},
the equivariant Chow ring $A^*_{\mathbf{PGL} (V)}(\mathbf P(U))$ is
evaluated in the even case.
\begin{tm}
\label{heven}
$A^*(d=2n)$ is generated by $c_1$, $c_2$, $c_3$, and $\mathcal{L}$.
The first relations are:
$$ c_1 =0$$
$$2c_3=0.$$
There are $d+1$
additional relations given by the first
$d+1$ Chern classes of the formal expansion:
$$ \frac{(1+\mathcal{L})^{d+1} \cdot c(Sym^{n-2} (S))}
{c(Sym^{n} (S))}.$$
(If $n=1$ or $2$, then $c(Sym^{n-2}(S))= 1$.)
\end{tm}
\noindent
It is easily seen from Theorem \ref{heven}
that $A^1(d=2n) \stackrel{\sim}{=} \mathbb{Z}/(d+1)\mathbb{Z}$ with
generator $\mathcal{L}$. The equation $\mathcal{L} = n\mathcal{H}$
can then be uniquely solved to obtain $\mathcal{H}=2d \mathcal{L}
=-2 \mathcal{L}$.
Now assume $d\geq 1$ is odd.
Let $d=2n-1$ (where $n\geq 1$).
Let $$det: \mathbf{GL} (V) \rightarrow \mathbb{C}^*$$ be the determinant homomorphism.
Let $\mathbb{Z}/n\mathbb{Z} \subset \mathbb{C}^*$ be the subgroup of the $n^{th}$ roots
of unity.
Let $\mathbf{SL} (V,n)= det^{-1}(\mathbb{Z}/n\mathbb{Z})$.
Consider again $\bigoplus_0^{2n-1} Sym^{2n-1}(V^*)$.
There is a canonical, $\mathbf{GL} (V)$-equivariant, multilinear map
$$\mu:\bigoplus_0^{2n-1}
Sym^{2n-1}(V^*) \rightarrow \bigwedge^{2n} Sym^{2n-1}(V^*)$$
given by the exterior product:
$$(\omega_0, \omega_1, \ldots, \omega_{2n-1}) \mapsto
\omega_0 \wedge \omega_1 \wedge \ldots \wedge
\omega_{2n-1}.$$
$\mathbf{SL} (V,n)$ acts trivially on the 1 dimensional space
$\bigwedge^{2n} Sym^{2n-1}(V^*)$. Let
$Y= \mu^{-1} (p)$ where $0 \neq p \in\bigwedge^{2n} Sym^{2n-1}(V^*)$.
There is an $\mathbf{SL} (V,n)$-action on $Y$.
In Lemma \ref{freeaq}, it is shown
this is a free action with
geometric quotient $H(d)$.
Hence, there is a canonical
isomorphism of graded rings
$$A^*(d=2n-1) \stackrel{\sim}{=} A^*_{\mathbf{SL} (V,n)}(Y).$$
Let $S$ now denote the rank $2$ algebraic
vector bundle obtained from the principal
$\mathbf{SL}(V,n)$-bundle $Y \rightarrow H(d)$ and the
standard representation $V$.
For $1\leq i \leq 2$, let $c_i\in A^*(d)$
be the Chern classes of $S$.
The equivariant Chow ring $A^*_{\mathbf{SL} (V,n)}(Y)$ is evaluated
in section \ref{ode}.
\begin{tm}
$A^*(d=2n-1)$ is generated by $c_1$ and $c_2$.
The first relation is
$$ nc_1 =0.$$
There are $d+1$
additional relations given by the first
$d+1$ Chern classes of $Sym^d(S)$.
\label{hodd}
\end{tm}
\noindent
It is easily seen that $A^1(d=2n-1) \stackrel{\sim}{=} \mathbb{Z}/n\mathbb{Z}$.
\subsection{Chow rings of the Orthogonal Groups}
The Chow ring of a reductive algebraic group $\mathbf{G}$ is, by
definition, the equivariant Chow ring $A^*_\mathbf{G}(
\text{point})$.
Let $\mathbf{O} (n)$ and $\mathbf{SO} (n)$ denote the orthogonal and
special orthogonal algebraic groups.
The equivariant calculations of Theorem \ref{heven}
require knowledge of $B\mathbf{PGL} (2)$. $\mathbf{PGL} (2)$ is isomorphic
to $\mathbf{SO} (3)$. The following Theorem will be established:
\begin{tm}
\label{chor}
The integral Chow ring of $\mathbf{O} (n)$ is
generated by the Chern classes $c_1, \ldots, c_n$
of the standard representation. The odd classes
are 2-torsion:
$$A^*_{\mathbf{O} (n)}(\text{point})=
\mathbb{Z}[c_1, \ldots, c_n]/(2c_1, 2c_3, 2c_5, \ldots).$$
\noindent The integral Chow ring of $\mathbf{SO} (n=2k+1)$
is generated by the Chern classes $c_1, \ldots, c_n$
of the standard representation. The odd classes
are 2-torsion and $c_1=0$:
$$A^*_{\mathbf{SO} (n)}(\text{point})=
\mathbb{Z}[c_1, \ldots, c_n]/(c_1, 2c_3, 2c_5, \ldots).$$
\end{tm}
\noindent
The Chow ring of $\mathbf{SO} (2k)$ is not generated by the
Chern classes of the standard representation. The
main difference in the odd and even cases is that
$\mathbf{SO} (2k+1) \stackrel{\sim}{=} \mathbb{Z}/2\mathbb{Z} \times \mathbf{O}(2k+1)$ while
such a product decomposition does not hold
for $\mathbf{SO} (2k)$.
The methods
of this paper do not yield a computation of $A^*_{\mathbf{SO} (2k)}(
\text{point})$. The Chow ring of
$\mathbf{SO}(n)$ has been computed with $\mathbb{Q}$-coefficients in [EG2].
\subsection{Acknowledgments}
Equivariant Chow groups were first defined in [T].
Thanks are due to D. Edidin, W. Graham, and B. Totaro
for conversations in which the theory of
equivariant Chow groups was explained.
The author particularly wishes to thank B. Totaro
for his insights on $\mathbf{O}(n)$ and $\mathbf{SO}(n)$.
Discussions with
W. Fulton on many related issues have also been helpful.
\section{\bf{Chow Ideals of Degeneracy Loci}}
\subsection{Presentations}
\label{idealz}
For the Chow computations in this paper,
presentations of four ideals associated to tautological
degeneracy loci are needed.
Let $E$ be a rank $e$ vector bundle on a
nonsingular algebraic variety $M$.
We will consider two affine and
two projective fibrations over $M$:
\begin{enumerate}
\item[(i)] $\oplus_{1}^e E \rightarrow M$,
\item[(ii)] $\mathbf P(\oplus_{1}^e E) \rightarrow M$,
\item[(iii)] $Sym^2 E^* \rightarrow M$,
\item[(iv)] $\mathbf P(Sym^2 E^*) \rightarrow M$.
\end{enumerate}
The
subspace projectivization is taken in (ii) and (iv).
Let $r=e^2$ denote the
rank of $\oplus_{1}^{e} E$.
Let $L$ in $A^1(\mathbf P(\oplus_{1}^{e} E))$ be the
class of ${\mathcal{O}}_{\mathbf P}(1)$ obtained from the
projectivization. The Chow ring of $\mathbf P(\oplus_{1}^{e} E)$
has a standard presentation:
$$A^*(M)[L]\ /\ \big(L^{r}+
c_1(\oplus_{1}^{e} E)\cdot
L^{r-1} + \ldots +c_{r}(\oplus_{1}^{e} E)\big).$$
Similarly, the Chow ring of $\mathbf P(Sym^2 E^*)$
has a presentation:
$$A^*(M)[L]\ /\ \big(L^{s}+
c_1(Sym^2 E^*)\cdot
L^{s-1} + \ldots +c_{s}(Sym^2 E^*)\big)$$
where $s=\frac{1}{2}(e^2+e)$ is the rank of
$Sym^2 E^*$ and $L$ is again the class of ${\mathcal{O}}_{\mathbf P}(1)$
obtained from the projectivization.
The Chow rings of the affine fibrations (i) and
(iii) are canonically isomorphic to $A^*(M)$.
There are intrinsic, fiberwise degeneracy loci
in these fibrations.
Let $D_1 \subset \oplus_{1}^{e} E$ and
$\mathbf P(D_1) \subset \mathbf P(\oplus_{1}^{e} E)$
be the
closed subvariety of linearly dependent $e$-tuples
of vectors in the fibers of $E$.
Let $D_2 \subset Sym^2 E^*$ and
$\mathbf P(D_2) \subset \mathbf P(Sym^2 E^*)$
be the closed subvariety of degenerate quadratic forms
on the fibers of $E$.
Let $$I_1\subset A^*(\oplus_{1}^{e} E)\stackrel{\sim}{=} A^*(M), \ \
J_1\subset A^*(\mathbf P(\oplus_{1}^{e} E)),$$
$$I_2 \subset A^*(Sym^2 E^*)\stackrel{\sim}{=} A^*(M), \ \
J_2 \subset A^*(\mathbf P(Sym^2 E^*))$$ be the
ideals generated by classes supported on the
degeneracy loci $D_1$, $\mathbf P(D_1)$, $D_2$,
and $\mathbf P(D_2)$ respectively.
In this section, simple sets of generators of the
ideals $I_1$, $J_1$, $I_2$, and $J_2$ are determined.
The results of this section are essentially
special cases of
Pragacz's presentations of the ideals of Chow classes supported
on degeneracy loci of bundle maps
([Pr]). Pragacz considers more general degeneracy
loci and obtains presentations of their
universal Chow ideals via Schur $S$-polynomials.
Actual (not universal) Chow ideal presentations
are needed here. Since the geometry of the cases
(i)-(iv) is particularly simple, the actual and
the universal presentations coincide.
A full proof will be given here.
For a rank $f$ bundle $F$, let
$c(F)=1+c_1(F)+ \ldots +c_f(F)$.
\begin{lm}
\label{petey}
$I_1\subset A^*(M)$ is generated by
$(\alpha_1, \ldots, \alpha_e)$ where
$$\frac{1}{c(E^*)}=
1+ \alpha_1 + \ldots + \alpha_e+ \ldots.$$
\end{lm}
\begin{lm}
\label{pete}
$J_1\subset A^*(\mathbf P(\oplus_{1}^{e} E))$ is generated by
$(\alpha'_1, \ldots, \alpha'_e)$ where
$$\frac{c(\oplus_{1}^{e} {\mathcal{O}}_{\mathbf P}(1))}{c(E^*)}=
1+ \alpha'_1 + \ldots + \alpha'_e+ \ldots.$$
\end{lm}
\begin{lm}
\label{paul}
$I_2\subset A^*(M)$ is generated by
$(\beta_1, \ldots, \beta_e)$ where
$$\frac{c(E^*)}{c(E)}=
1+ \beta_1 + \ldots + \beta_e+ \ldots.$$
\end{lm}
\begin{lm}
\label{pauly}
$J_2\subset A^*(\mathbf P(Sym^2 E^*))$ is generated by
$(\beta'_1, \ldots, \beta'_e)$ where
$$\frac{c(E^*\otimes {\mathcal{O}}_{\mathbf P}(1))}{c(E)}=
1+ \beta'_1 + \ldots + \beta'_e+ \ldots.$$
\end{lm}
\noindent
The proofs of
Lemmas \ref{petey} -- \ref{pauly}
are essentially the same.
The first step is to find a tower of bundles dominating
the degeneracy loci $D_1$, $\mathbf P(D_1)$, $D_2$,
and $\mathbf P(D_2)$.
First consider $D_1$ and $\mathbf P(D_1)$.
Let $\eta:\mathbf P(E^*) \rightarrow M$ be the projective bundle.
A point $\xi \in \mathbf P(E^*)$ is a pair $(m,h)$ where
$m\in M$ and $h \in \mathbf P(E^*_m)$.
Let $B$ be the vector bundle on $\mathbf P(E^*)$ determined
as follows. The fiber of $B$ at the point $(m,h)$
is the linear subspace of $\oplus_{1}^{e} E_m$
consisting of $e$-tuples of vectors annihilated by
$h$.
$B$ is a sub-bundle of $\eta^*(\oplus_{1}^{e} E)$.
There are canonical, proper, surjective projections:
$$\rho: B \rightarrow D_1 \subset \oplus_{1}^{e} E,$$
$$\mathbf P(\rho): \mathbf P(B) \rightarrow \mathbf P(D_1)
\subset \mathbf P(\oplus_{1}^{e} E).$$
There are stratifications of $D_1$ and $\mathbf P(D_1)$
by the
rank of the span of the $e$-tuple of vectors. Over these strata,
$\rho$ and $\mathbf P(\rho)$ are projective bundles.
Hence $\rho$ and $\mathbf P(\rho)$ induce {\em surjections}
on the integral Chow rings
via push-forward:
$$\rho_*: A^*(B) \rightarrow A^*(D_1),$$
$$\mathbf P(\rho)_*: A^*(\mathbf P(B)) \rightarrow A^*(\mathbf P(D_1)).$$
Lemmas \ref{petey} and \ref{pete}
are proven by computing the images of
the generators of $A^*(B)$ and $A^*(\mathbf P(B))$
respectively.
Consider the commuting diagrams:
\begin{equation}
\label{heyhey}
\begin{CD}
B @>{\rho}>> D_1 \subset \oplus_{1}^{e} E \\
@V{\pi}VV @VVV \\
\mathbf P(E^*) @>{\eta}>> M \\
\end{CD}
\end{equation}
\begin{equation}
\label{heyheyy}
\begin{CD}
\mathbf P(B) @>{\mathbf P(\rho)}>> \mathbf P( D_1)
\subset \mathbf P(\oplus_{1}^{e} E) \\
@V{\pi}VV @VVV \\
\mathbf P(E^*) @>{\eta}>> M \\
\end{CD}
\end{equation}
$A^*(B)$ is generated over $A^*(M)$ by
the class corresponding to ${\mathcal{O}}_{\mathbf P(E^*)}(1)$.
Let
this class be denoted by $\zeta$.
It follows that
$$I_1=(\rho_{*}(1), \rho_{*}(\zeta^1), \rho_{*}(\zeta^2),
\ldots, \rho_{*}(\zeta^{e-1})).$$
Similarly, there is a presentation of $J_1$:
$$J_1=(\mathbf P(\rho)_*(1), \mathbf P(\rho)_{*}(\zeta^1),
\mathbf P(\rho)_{*}(\zeta^2),
\ldots, \mathbf P(\rho)_{*}(\zeta^{e-1})).$$
To prove Lemmas \ref{petey} and \ref{pete},
it is sufficient to establish
the equalities
\begin{equation}
\label{eqql}
\rho_{*}(\zeta^{i-1})= \alpha_i, \ \
\mathbf P(\rho)_{*}(\zeta^{i-1})= \alpha'_i
\end{equation}
for $1 \leq i \leq e$.
First the equalities (\ref{eqql}) for Lemma
\ref{petey} are proven. By definition,
$B\subset \eta^*(\oplus_{1}^{e} E)$.
In fact, there is a natural exact squence on $\mathbf P(E^*)$:
\begin{equation}
\label{yess1}
0 \rightarrow B \rightarrow \eta^*(\oplus_{1}^{e} E) \rightarrow \oplus_{1}^{e}
{\mathcal{O}}_{\mathbf P(E^*)}(1) \rightarrow 0.
\end{equation}
As a first
step, the class of $[B] \in A^*(\eta^*(\oplus_{1}^{e} E))$
is computed. Since
$\eta^*(\oplus_{1}^{e} E)$
is a projective bundle over $\oplus_{1}^{e} E$,
$A^*(\eta^*(\oplus_{1}^{e} E))$
is generated over $A^*(M)$ by $\zeta$ (which
satisfies the Chern relation).
By sequence (\ref{yess1}) and Lemma \ref{fullt} below,
it follows that $[B] = \zeta ^{e} \in A^*(\eta^*(\oplus_{1}^{e} E))$.
Denote the natural projection $\eta^*(\oplus_{1}^{e} E))
\rightarrow \oplus_{1}^{e}
E$ by $\phi$.
There is a fundamental equality:
$$\rho_{*}(\zeta ^{i-1}) = \phi_{*} (\zeta^{i-1} \cap [B]) \in A^*(M).$$
The right side is easy to calculate.
$$\phi_{*}(\zeta^{i-1}\cap [B])= \phi_{*}(\zeta^{e-1+i}).$$
For $1\leq i \leq e$, the latter is simply the $i^{th}$ Segre
class of $E^*$. Lemma \ref{petey} is proved.
Lemma \ref{pete} is only slightly more complicated.
The class of $[\mathbf P(B)] \in A^*(\mathbf P(\eta^*(\oplus_{1}^{e} E)))$
is computed. Again
$A^*(\mathbf P(\eta^*(\oplus_{1}^{e} E)))$
is generated over $A^*(\mathbf P(\oplus_{1}^{e} E))$ by $\zeta$.
By sequence (\ref{yess1}) and Lemma \ref{fullt},
it follows that $[\mathbf P(B)]
= (L+\zeta) ^{e} \in A^*(\mathbf P(\eta^*(\oplus_{1}^{e} E)))$
where $L$ is the class of ${\mathcal{O}}_{\mathbf P(\oplus_{1}^{r} E)}(1)$.
Denote the natural projection $\mathbf P(\eta^*(\oplus_{1}^{e} E))
\rightarrow \mathbf P(\oplus_{1}^{e}
E)$ by $\mathbf P(\phi)$.
There are equalities:
$$\mathbf P(\rho)_{*}(\zeta ^{i-1})
= \mathbf P(\phi)_{*} (\zeta^{i-1} \cap [B])
=\mathbf P(\phi)_{*}
(\zeta^{i-1}\cap (L+\zeta)^e)$$
in $A^*(\mathbf P
(\oplus_{1}^{e} E))$.
Lemma \ref{ssegre} now yields Lemma \ref{pete}.
The degeneracy loci
$D_2$ and $\mathbf P(D_2)$ are considered next. The notation will
parallel the notation used in the proofs of Lemmas \ref{petey}
and \ref{pete}.
Let $\eta:\mathbf P(E) \rightarrow M$ be the projective bundle.
A point $\xi \in \mathbf P(E)$ is a pair $(m,p)$ where
$m\in M$ and $p \in \mathbf P(E_m)$.
Let $B$ be the vector bundle on $\mathbf P(E)$ determined
as follows. The fiber of $B$ at the point $(m,p)$
is the linear subspace of quadratic forms on $E_m$
singular at $p$.
$B$ is a sub-bundle of $\eta^*(Sym^2 E^*)$.
There are canonical, proper, surjective projections:
$$\rho: B \rightarrow D_2 \subset Sym^2 E^*,$$
$$\mathbf P(\rho): \mathbf P(B) \rightarrow \mathbf P(D_2) \subset Sym^2 E^*.$$
There are stratifications of $D_2$ and $\mathbf P(D_2)$
by the
rank of the quadratic form. Over these strata,
$\rho$ and $\mathbf P(\rho)$ are projective bundles.
Hence $\rho$ and $\mathbf P(\rho)$ induce {\em surjections}
on the integral Chow rings
via push-forward:
$$\rho_*: A^*(B) \rightarrow A^*(D_2),$$
$$\mathbf P(\rho)_*: A^*(\mathbf P(B)) \rightarrow A^*(\mathbf P(D_2)).$$
Lemmas \ref{paul} and \ref{pauly}
are proven by computing the images of
the generators of $A^*(B)$ and $A^*(\mathbf P(B))$
respectively.
As before,
$$I_2=(\rho_{*}(1), \rho_{*}(\zeta^1), \rho_{*}(\zeta^2),
\ldots, \rho_{*}(\zeta^{e-1})),$$
$$J_2= (\mathbf P(\rho)_{*}(1), \mathbf P(\rho)_{*}
(\zeta^1), \mathbf P(\rho)_{*}(\zeta^2),
\ldots, \mathbf P(\rho)_{*}(\zeta^{e-1})).$$
To prove Lemma \ref{paul} and \ref{pauly},
it is sufficient to establish
the equalities
\begin{equation}
\label{eqql2}
\rho_{*}(\zeta^{i-1})= \beta_i, \ \
\mathbf P(\rho)_{*}(\zeta^{i-1})= \beta'_i
\end{equation}
for $1 \leq i \leq e$.
First the equalities (\ref{eqql2}) for Lemma
\ref{paul} are proven. There
is an exact squence on $\mathbf P(E)$:
\begin{equation}
\label{yess2}
0 \rightarrow B \rightarrow \eta^*(Sym^2 E^*) \rightarrow E^*\otimes {\mathcal{O}}_{\mathbf P(E)}(1)
\rightarrow 0.
\end{equation}
The class of $[B] \in A^*(\eta^*(\oplus_{1}^{e} E))$
is computed.
$A^*(\eta^*(Sym^2 E^*))$
is generated over $A^*(M)$ by $\zeta$.
By sequence (\ref{yess2}) and Lemma \ref{fullt} below,
it follows that
$$[B] = c_e (E^*\otimes {\mathcal{O}}_{\mathbf P(E)}(1) )\in A^*(\eta^*(Sym^2 E)).$$
Denote the natural projection $\eta^*(Sym^2 E^*))
\rightarrow Sym^2
E^*$ by $\phi$.
There is an equality:
$$\rho_{*}(\zeta ^{i-1}) = \phi_{*} (\zeta^{i-1} \cap [B]) \in A^*(M).$$
Lemma \ref{ssegre} now yields Lemma \ref{paul}.
Lemma \ref{pauly} is established next.
By sequence (\ref{yess2}) and Lemma \ref{fullt} below,
it follows that
$$[\mathbf P(B)] = c_e \bigg(
\frac{c(E^*\otimes {\mathcal{O}}_{\mathbf P(E)}(1))}
{c({\mathcal{O}}_{\mathbf P(Sym^2 E^*)}(-1))} \bigg)\in
A^*(\mathbf P(\eta^*(Sym^2 E^*))).$$
There is an equality (since $E^*\otimes {\mathcal{O}}_{\mathbf P(E)}(1)$
is a rank $e$ bundle):
$$c_e \bigg(
\frac{c(E^*\otimes {\mathcal{O}}_{\mathbf P(E)}(1))}
{c({\mathcal{O}}_{\mathbf P(Sym^2 E^*)}(-1))} \bigg)=
c_e(E^*\otimes {\mathcal{O}}_{\mathbf P(E)}(1) \otimes {\mathcal{O}}_{\mathbf P(Sym^2 E^*)}(1)).$$
Denote the natural projection $\mathbf P(\eta^*(Sym^2 E^*))
\rightarrow \mathbf P(Sym^2
E^*)$ by $\mathbf P (\phi)$.
There is an equality:
$$\mathbf P(\rho)_{*}(\zeta ^{i-1}) =
\mathbf P(\phi)_{*} (\zeta^{i-1} \cap [B]) \in A^*(\mathbf P(Sym^2 E^*)).$$
Lemma \ref{ssegre} now yields Lemma \ref{pauly}.
\subsection{Lemmas}
The following Lemmas were used in the proofs
of Lemmas \ref{petey} -- \ref{pauly}.
Let $F \rightarrow N$ be a vector
bundle on a nonsingular algebraic
variety $N$.
\begin{lm}
\label{fullt}
Let
$\ 0 \rightarrow B \rightarrow F \rightarrow Q \rightarrow 0$
be an exact sequence of bundles on $N$.
Let $q$ be the rank of $Q$.
The class $[B]\in A^*(F)\stackrel{\sim}{=} A^*(N)$ is determined
by
$$[B]= c_q(Q).$$
The class $[\mathbf P(B)] \in A^*(\mathbf P(F))$ is determined
by
$$[\mathbf P(B)]= c_q \bigg(\frac {c(Q)}{c({\mathcal{O}}_{\mathbf P(F)}(-1))}
\bigg).$$
\end{lm}
\noindent {\em Proof.}
This is an application of the Thom-Porteous formulas
for degeneracy loci of bundle maps
(see [F]).
\qed \vspace{+10pt}
\noindent
Let $f$ be the rank $F$.
Let $\phi: \mathbf P(F) \rightarrow N$ be the projection.
\begin{lm}
\label{ssegre}
Let $G$ be a bundle of rank $g=f$ on $N$.
Let $\zeta= c_1({\mathcal{O}}_{\mathbf P(F)}(1))$.
Let $\gamma_{i}$
be determined by
$$\frac{c(G)}{c(F)} = 1 + \gamma_{1} + \ldots+
\gamma_{f} + \ldots.$$
Then, for $1 \leq i \leq f$,
$\gamma_{i}= \phi_{*}\big( \zeta^{i-1} \cap
c_f(G \otimes
{\mathcal{O}}_{\mathbf P(F)}(1))\big).$
\end{lm}
\noindent {\em Proof.}
A simple Segre class argument yields the result.
\qed \vspace{+10pt}
\section{\bf{Equivariant Chow Groups}}
\label{chow}
Let $\mathbf{G}$ be a group.
Let $\mathbf{G}\times X \rightarrow X$ be a left group action.
In topology, the $\mathbf{G}$-equivariant cohomology of
$X$ is defined as follows. Let $E\mathbf{G}$ be a contractible
topological space equipped with a free left $\mathbf{G}$-action and
quotient $E\mathbf{G}/\mathbf{G}=B\mathbf{G}$.
Consider the left action of $\mathbf{G}$ on $X\times E\mathbf{G}$ defined
by:
$$g(x,b)= (g(x), g(b)).$$
$\mathbf{G}$ acts freely on $X\times E\mathbf{G}$. Let
$X\times^{\mathbf{G}} E\mathbf{G}$ be the (topological) quotient.
The $\mathbf{G}$-equivariant cohomology of
of $X$, $H_\mathbf{G}^*(X)$, is defined by:
$$H_\mathbf{G}^*(X) = H^*_{sing}(X\times^{\mathbf{G}} E\mathbf{G}).$$
If $X$ is a locally trivial principal $\mathbf{G}$-bundle,
then $X\times^{\mathbf{G}} E\mathbf{G}$ is a locally trivial
fibration of $E\mathbf{G}$ over the quotient $X/\mathbf{G}$.
In this case, $X\times^{\mathbf{G}} E\mathbf{G}$ is homotopy equivalent
to $X/\mathbf{G}$ and
$$H_\mathbf{G}^*(X) = H^*_{sing}(X\times^{\mathbf{G}} E\mathbf{G}) \stackrel{\sim}{=}
H^*_{sing}(X/\mathbf{G}).$$
For principal bundles, computing the
equivariant cohomology ring is equivalent
to computing the cohomology of the quotient.
There is an analogous equivariant theory of Chow groups
developed by B. Totaro in case $X$ is a point and
generalized by D. Edidin and W. Graham to arbitrary $X$
([T], [EG]). Let $\mathbf{G}$ be a reductive algebraic group.
Let $\mathbf{G}\times X \rightarrow X$ be a linearized algebraic $\mathbf{G}$-action.
The algebraic analogue of $E\mathbf{G}$ is attained by
approximation. Let $V$ be a $\mathbb{C}$-vector space. Let
$\mathbf{G}\times V \rightarrow V$ be an
algebraic representation of $\mathbf{G}$.
Let $W\subset V$ be a $\mathbf{G}$-invariant open set satisfying:
\begin{enumerate}
\item[(i)] The complement of $W$ in $V$ is of codimension greater than
$q$.
\item[(ii)] $\mathbf{G}$ acts freely on $W$ (see the Appendix for
the definition).
\item[(iii)] There exists a geometric quotient $W\rightarrow W/\mathbf{G}$.
\end{enumerate}
$W$ is an approximation of $E\mathbf{G}$ up to codimension $q$.
By (iii) and the
assumption of linearization,
a geometric quotient $X\times ^{\mathbf{G}} W$ exists as
an algebraic variety. Let $d=dim(X)$, $e=dim( X\times ^{\mathbf{G}} W)$.
The equivariant Chow groups are defined by:
\begin{equation}
\label{defff}
A^{\mathbf{G}}_{d-j}(X)= A_{e-j}(X\times ^{\mathbf{G}} W)
\end{equation}
for $0\leq j \leq q.$
An argument is required to check these equivariant
Chow groups are well-defined (see [EG]).
The basic functorial properties of equivariant
Chow groups are
established in [EG]. In particular, if $X$ is
nonsingular, there is a natural intersection
ring structure on $A_i^{\mathbf{G}}(X)$.
Let $Z$ be a variety of dimension $z$.
For notational convenience, a superscript will
denote the Chow group codimension:
$$A^{\mathbf{G}}_{z-j}(Z) = A^j_\mathbf{G}(Z), \ A_{z-j}(Z)=A^j(Z).$$
In particular, equation (\ref{defff}) becomes:
$$\forall\ 0\leq j \leq q, \ \
A_{\mathbf{G}}^{j}(X)= A^j(X\times ^{\mathbf{G}} W).$$
The following result of [EG]
will be used.
\begin{pr}
\label{dane}
Let $\mathbb{C}$ be the ground field of complex numbers.
Let $X$ be a quasi-projective variety. Let $\mathbf{G}$ be a reductive
group.
Let $\mathbf{G}\times X \rightarrow X$ be a linearized proper $\mathbf{G}$-action.
Let $X\rightarrow X/\mathbf{G}$ be a
quasi-projective
geometric quotient.
\begin{enumerate}
\item[(i)] If the action is free, then there is a canonical
isomorphism of graded rings:
$$A^*_\mathbf{G}(X) \stackrel{\sim}{=} A^*(X/\mathbf{G}).$$
\item[(ii)] If $\mathbf{G}$ acts with finite stabilizers on $X$, then
there is a canonical isomorphism of graded rings:
$$ A^*_\mathbf{G}(X) \otimes \mathbb{Q} \stackrel{\sim}{=} A^*(X/\mathbf{G}) \otimes \mathbb{Q}.$$
\end{enumerate}
\end{pr}
\noindent Proposition \ref{dane} is a characteristic 0
specialization of Theorem 2 of [EG].
\section{\bf The Chow Rings of $\mathbf{O} (k)$ and $\mathbf{SO} (2k+1)$}
\subsection{$B\mathbf{O} (V)$ and $B\mathbf{SO} (V)$}
\label{orthoo}
Let $V$ be a complex vector space equipped with
a non-degenerate quadratic form. Let $\mathbf{O} (V)$, $\mathbf{SO} (V)$
be the orthogonal and special orthogonal groups
respectively.
Approximations to $E\mathbf{O} (V)$ and $E\mathbf{SO} (V)$ are
obtained via direct sums of the
representation $V^*$.
Let $m>>0$ and let
$$W_m \subset \oplus_{1}^{m} V^*$$
denote the spanning locus. $W_m$ is the
locus of $m$-tuples of vectors of $ V^*$
which span $V^*$.
The natural actions of $\mathbf{O} (V)$ and $\mathbf{SO} (V)$ on
$W_m$ are free and have a geometric quotients
(see section \ref{appx}).
The codimension of the complement of
$W_m$ in $\oplus_{1}^{m} V^*$ is $m-dim(V^*)+1$.
$W_m$ is an approximation of $E\mathbf{O} (V)$ and
$E\mathbf{SO} (V)$ up to codimension $m-dim(V^*)$.
By the general theory of equivariant Chow groups
(section \ref{chow}), we have approximations:
$$B\mathbf{O} (V)= \stackrel{Lim}{m \rightarrow \infty} \ W_m/\mathbf{O} (V),$$
$$B\mathbf{SO} (V)= \stackrel{Lim}{m \rightarrow \infty} \ W_m/\mathbf{SO} (V).$$
In this section, equivariant Chow rings of $\mathbf{O} (k)$ and
$\mathbf{SO} (2k+1)$ are computed via the approximations
$$A^*_{\mathbf{O} (V)}(\text{point})= \stackrel{Lim}{m \rightarrow \infty} \
A^*(W_m/\mathbf{O} (V)),$$
$$A^*_{\mathbf{SO} (V)}(\text{point})= \stackrel{Lim}{m \rightarrow \infty} \
A^*(W_m/\mathbf{SO} (V)),$$
and the degeneracy loci results of section \ref{idealz}.
\subsection{The Chow Ring of $\mathbf{O} (k)$}
Let $k\geq 1$.
Let $V\stackrel{\sim}{=} \mathbb{C}^k$ be equipped with a non-degenerate
quadratic form $Q$ preserved by $\mathbf{O} (k)$. The quotient
$W_m/ \mathbf{O} (k)$ can be explicitly realized as follows.
Let $\mathbf G(k,m)$ be the Grassmannian of linear $k$-spaces
in $\mathbb{C}^m$.
Let $S \rightarrow \mathbf G(k,m)$ be the tautological
sub-bundle. Let $Y_m \subset Sym^2 S^*$ be the
open locus of non-degenerate quadratic forms
on the fibers of $S$.
\begin{lm}
\label{fbb}
There is canonical $\mathbf{O} (k)$-invariant
map $\tau: W_m \rightarrow Y_m$ which induces
an isomorphism $W_m/\mathbf{O} (k) \stackrel{\sim}{=} Y_m$.
\end{lm}
\noindent {\em Proof.}
Let $w\in W_m$. By the definitions,
$w$ naturally induces an injection $\iota_{w}:V \rightarrow \mathbb{C}^m$.
The quadratic form $Q$ then induces a non-degenerate
quadratic form $\iota_{w}(Q)$ on $\iota_{w}(V)$.
Let
$$\tau(w) = \iota_{w}(Q) \in Y_m.$$
It is easily checked that $\tau$ is an
algebraic morphism.
Let $g\in \mathbf{O} (k)$. Then,
$$\iota_{g(w)}= \iota_{w} \circ g : V \rightarrow \mathbb{C}^m.$$
Hence, $\tau$ is $\mathbf{O} (k)$-invariant.
Since the fibers of $\tau$ are exactly the
$\mathbf{O} (k)$ orbits, the induced map
$$W_m/ \mathbf{O} (k) \rightarrow Y_m$$
is a bijective morphism of nonsingular
complex algebraic varieties and thus an algebraic
isomorphism.
\qed \vspace{+10pt}
$Y_m$ is an approximation to $B\mathbf{O} (k)$
up to codimension $m-k$.
$W_m\rightarrow Y_m$ is a principal $\mathbf{O}(k)$-bundle
(see the Appendix).
The pull-back of the
tautological sub-bundle $S \rightarrow \mathbf G(k,m)$
to $Y_m$ is the vector bundle on $Y_m$
induced by the
principal $\mathbf{O}(k)$-bundle $W_m\rightarrow Y_m$
and the representation $V$.
The Chow ring of the Grassmannian $\mathbf G(k,m)$
is freely generated by the Chern classes
$c_1, \ldots, c_k$ of $S$ up to codimension $m-k$
(the relations start in codimension $m-k+1$).
By Lemma \ref{paul}, the Chow ring of
$Y_m$ is isomorphic to
$$\mathbb{Z}[c_1, \ldots, c_k] \ / \ (\beta_1, \ldots, \beta_k)$$
up to codimension $m-k$ where
\begin{equation}
\label{bbbt}
\frac{c(S^*)}{c(S)}=
1+ \beta_1 + \ldots + \beta_k+ \ldots.
\end{equation}
Induction and simple algebra establishes:
$$(\beta_1, \ldots, \beta_k)=(2c_1, 2c_3, 2c_5, \ldots).$$
The Chow ring limit $m\rightarrow \infty$ of $A^*(Y_m)$
is now easily seen to yield:
$$A^*_{\mathbf{O} (k)}(\text{point})=
\mathbb{Z}[c_1, \ldots, c_k] \ / \ (2c_1, 2c_3, 2c_5, \ldots).
$$
Theorem \ref{chor} is proven for $\mathbf{O} (k)$.
\subsection{The Chow ring of $\mathbf{SO} (2k+1)$}
\label{ort}
Let $k\geq 0$.
Let $V\stackrel{\sim}{=} \mathbb{C}^{2k+1}$ be equipped with a non-degenerate
quadratic form preserved by $$\mathbf{SO} (2k+1)\subset \mathbf{GL} (V).$$
Let $\mathbb{C}^* \subset \mathbf{GL} (V)$ be the scalars.
Since $V$ is odd dimensional $$\mathbb{C}^* \cap \mathbf{SO} (2k+1)= \{1\},
\ \ \mathbb{C}^* \times \mathbf{SO}(2k+1) \subset \mathbf{GL}(V).$$
The approximations $W_m$ to $E\mathbf{SO} (2k+1)$ are
used.
There is a natural free $\mathbf{GL} (V)$-action on $W_m$
which induces a free
$\mathbf{SO} (2k+1)$-action and a free
scalar $\mathbb{C}^*$-action on $W_m$.
The $\mathbf{SO} (2k+1)$-action and the $\mathbb{C}^*$-action
commute.
There is a commutative diagram:
\begin{equation}
\label{ffibb}
\begin{CD}
W_m @>>> W_m/ \mathbf{SO}(2k+1) \\
@VVV @VVV \\
W_m/\mathbb{C}^* @>{\tau}>> W_m/\ \mathbb{C}^* \times \mathbf{SO} (2k+1)\\
\end{CD}
\end{equation}
All morphisms are group quotients:
the horizontal maps are free $\mathbf{SO} (2k+1)$-quotients,
the vertical maps are free $\mathbb{C}^*$-quotients.
See the Appendix for a discussion of these algebraic
quotient problems.
The quotients in diagram (\ref{ffibb}) are
analyzed. Let $S \rightarrow \mathbf G(2k+1, m)$
be the tautological sub-bundle over the Grassmannian.
By an argument identical to Lemma \ref{fbb}, it
is seen that
$$W_m/ \ \mathbb{C}^* \times \mathbf{SO} (2k+1) \stackrel{\sim}{=} Z_m$$
where $Z_m \subset \mathbf P(Sym^2 S^*)$
is the locus of non-degenerate quadratic forms
on the fibers of $S$.
Hence, $W_m/ \mathbf{SO} (2k+1) \rightarrow Z_m$
is a $\mathbb{C}^*$-bundle. Let $N \rightarrow Z_m$
be the line bundle associated to this $\mathbb{C}^*$-bundle.
On the left side of the diagram,
$$W_m/\mathbb{C}^* \subset \mathbf P (\oplus_{1}^m V^*)$$
is the projective spanning locus.
$A^1( W_m/\mathbb{C}^*)= \mathbb{Z}$ and $W_m \rightarrow W_m/\mathbb{C}^*$
is the $\mathbb{C}^*$-bundle associated to the
generator ${\mathcal{O}}_{\mathbf P}(-1)$ of $A^1(W_m/\mathbb{C}^*)$.
\begin{lm}
\label{ddd}
$A^1(Z_m) \stackrel{\sim}{=} \mathbb{Z}$ and $c_1(N)$ is a generator.
\end{lm}
\noindent {\em Proof.}
Consider the inclusion $Z_m \subset \mathbf P(Sym^2 S^*)$.
Let
$$\tau: W_m/\mathbb{C}^* \rightarrow Z_m \subset \mathbf P(Sym^2 S^*)$$
be the natural map. Let $\overline{N}$ denote
an extension of $N$ to $A^1(\mathbf P(Sym^2 S^*))$.
Since $\tau^*(\overline{N})= {\mathcal{O}}_{\mathbf P}(-1)$
generates $A^1(W_m/\mathbb{C}^*)\stackrel{\sim}{=} \mathbb{Z}$,
the kernel $K$ of
$$ \tau^*: A^1(\mathbf P(Sym^2 S^*)) \rightarrow A^1(W_m/ \mathbb{C}^*)$$
is isomorphic to $\mathbb{Z}$.
The class $[D]$ of the locus of degenerate
quadratic forms is in $K$.
$A^1(\mathbf P(Sym^2 S^*))\stackrel{\sim}{=} \mathbb{Z} c_1 \oplus \mathbb{Z} L$
where $c_1=c_1(S)$ and $L$ is
the canonical class ${\mathcal{O}}_{\mathbf P}(1)$. The class of $[D]$ is
$-2c_1+ (2k+1) L$ which is not divisible in
$A^1(\mathbf P(Sym^2 S^*))$.
Hence $K$ is generated by $[D]$. Therefore
$\tau^*: A^1(Z_m) \rightarrow A^1(W_m/ \mathbb{C}^*)$ is
an isomorphism and $c_1(N)$ is a generator of $A^1(Z_m)$.
\qed \vspace{+10pt}
There is now enough information to compute the
Chow ring of the approximation $W_m/\mathbf{SO} (2k+1)$
to $B\mathbf{SO} (2k+1)$.
As before, $W_m/\mathbf{SO} (2k+1)$ is an approximation up to codimension
$m-(2k+1)$.
The Chow ring of $\mathbf G(2k+1, m)$
is freely generated by the Chern classes
$c_1, \ldots, c_{2k+1}$ of the tautological sub-bundle
$S$ up to codimension $m-(2k+1)$.
By Lemma \ref{pauly}, the
Chow ring of $Z_m$ (up to codimension $m-(2k+1)$) has
a presentation:
$$\mathbb{Z}[c_1, \ldots, c_{2k+1}, L]/
(p(L),\beta_1', \ldots, \beta_{2k+1}')$$
where $L$ is the class of ${\mathcal{O}}_{\mathbf P}(1)$, $p(L)$ is the
Chern polynomial satisfied by $L$, and
$$\frac{c(S^*\otimes {\mathcal{O}}_{\mathbf P}(1))}{c(S)}=
1+ \beta'_1 + \ldots + \beta'_{2k+1}+ \ldots.$$
Finally, since $W_m/\mathbf{SO} (2k+1)$ is the
total space of the $\mathbb{C}^*$-bundle associated
to the line bundle $N\rightarrow Z_m$,
\begin{equation}
\label{ttoott}
\mathbb{Z}[c_1, \ldots, c_{2k+1}, L]/
(c_1(N),p(L),\beta_1', \ldots, \beta_{2k+1}')
\end{equation}
is a presentation of the Chow ring of $W_m/\mathbf{SO} (2k+1)$
(up to codimension $m-(2k+1)$).
Since $c_1(N)$ generates $A^1(Z_m)$ and the pair $\{c_1, L\}$
also generate $A^1(Z_m)$, (\ref{ttoott}) is
equivalent to:
$$\mathbb{Z}[c_1, \ldots, c_{2k+1}, L]/
(c_1, L ,p(L),\beta_1', \ldots, \beta_{2k+1}').$$
By the defintions of
$p(L)$ and the elements $\beta'_i$, there
is an equality of ideals
$$(c_1,L, p(L), \beta'_1,\ldots, \beta_{2k+1}')
=
(c_1,L, c_s(Sym^2 S^*), \beta_1, \ldots, \beta_{2k+1})$$
where $s= rank(Sym^2 S^*)$ and the $\beta_i$ are determined
by (\ref{bbbt}).
\begin{lm}
$c_s(Sym^2 S^*) \in (\beta_1, \ldots, \beta_{2k+1})$.
\end{lm}
\noindent {\em Proof.}
Consider the total space $Sym^2 S^*$.
There is an isomorphism $A^*(Sym^2 S^*) \stackrel{\sim}{=}
A^*(\mathbf G(2k+1,m))$.
The pull-back of the bundle $$Sym^2 S^* \rightarrow \mathbf G(2k+1,m)$$
to the total space $Sym^2 S^*$ has a canonical section $\tau$.
The zero scheme of $\tau$ is contained in the
locus of degenerate quadratic forms $D\subset Sym^2 S^*$.
Also, the zero scheme of $\tau$ represents the class
$$c_s(Sym^2 S^*)\in A^*(Sym^2 S^*).$$
Therefore, $c_s(Sym^2 S^*) \in I_2$. The proof is
complete by Lemma \ref{paul}. \qed \vspace{+10pt}
\noindent As before,
$(\beta_1, \ldots, \beta_{2k+1})= (2c_1, 2c_3,2c_5, \ldots, 2c_{2k+1}).$
Hence, the Chow ring of $W_m/ \mathbf{SO} (2k+1)$ up to
codimension $m-(2k+1)$ has a presentation:
$$\mathbb{Z}[c_1, \ldots, c_{2k+1}]/
(c_1,
2c_3, 2c_5,\ldots, 2c_{2k+1}).$$
The limit process yields Theorem \ref{chor} for $\mathbf{SO} (2k+1)$.
\section{\bf The Proof of Proposition \ref{alltor}}
\subsection{}
We follow the notation of section \ref{prezz}.
Let $V$ be a fixed $2$-dimensional $\mathbb{C}$-vector space.
Let $\mathbf P^1\stackrel{\sim}{=} \mathbf P(V)$.
Let $$U\subset \bigoplus_{0}^{d} Sym^d(V^*)$$
denote the non-degenerate locus parameterizing
bases of the linear series of ${\mathcal{O}}_{\mathbf P^1}(d)$
on $\mathbf P^1$. $\mathbf{GL}(V)$ acts on $U$ properly with finite
stabilizers and geometric quotient (see the Appendix)
isomorphic to $H(d)$.
By Proposition \ref{dane},
\begin{equation}
\label{xeq}
A^*(d) \otimes_{\mathbb{Z}}{\mathbb{Q}} \stackrel{\sim}{=} A^*_{\mathbf{GL} (V)}(U) \otimes _{\mathbb{Z}}{\mathbb{Q}}.
\end{equation}
By the definition of equivariant Chow groups,
$$A^*_{\mathbf{GL}(V)}(U)= A^*(U \times^{\mathbf{GL}(V)} E \mathbf{GL} (V)).$$
\subsection{The Chow Rings of $\mathbf{GL}(V)$ and $\mathbf{SL}(V,n)$}
\label{slvn}
Algebraic approximations to $E\mathbf{GL} (V)$ are easily found.
Since related results about the groups $\mathbf{SL}(V,n)$ are
need in section \ref{ode}, a unified development
is presented here. Recall $SL(V,n) \subset \mathbf{GL}(V)$
is defined to be $det^{-1}(\mathbb{Z}/n\mathbb{Z})$ where $det: \mathbf{GL} \rightarrow \mathbb{C}^*$
is the determinant homomorphism and
$\mathbb{Z}/ n\mathbb{Z}$ is th group of $n^{th}$ roots of unity.
As in the orthogonal cases, the easiest approach to
$E \mathbf{GL} (V)$ and $E \mathbf{SL} (V,n)$ is via sums of the
representation $V^*$.
As before, let $m>>0$ and let
$$W_m \subset \oplus_{1}^{m} V^*$$
be the spanning locus.
The induced $\mathbf{GL} (V)$ and $\mathbf{SL} (V,n)$-actions on
$W_m$ are free and have geometric quotients (see the
Appendix) which
approximate $B \mathbf{GL} (V)$ and $B \mathbf{SL} (V,n)$
up to codimension $m-2$.
It is easily seen that $W_m / \mathbf{GL}(V) \stackrel{\sim}{=} \mathbf G(2,m)$.
Since the Chow ring of this Grassmannian (up to
codimension $m-2$) is
freely generated by the Chern classes $c_1$ and $c_2$
of the tautological sub-bundle,
$$A^*(W_m/ \mathbf{GL} (V)) \stackrel{\sim}{=} \mathbb{Z}[c_1, c_2]$$
up to codimension $m-2$.
Taking the $m\rightarrow \infty $ limit,
$$A^*_{\mathbf{GL} (V)}(\text{point}) \stackrel{\sim}{=} \mathbb{Z}[c_1, c_2].$$
Similarly, $W_m/ \mathbf{SL} (V, n)$ is the total
space of the $n^{th}$ tensor power of the
line bundle $\bigwedge^2 S$ over $\mathbf G(2,m)$.
Hence up to codimension $m-2$,
$$A^*(W_m/ \mathbf{SL} (V,n)) \stackrel{\sim}{=} \mathbb{Z}[c_1, c_2]/ (nc_1).$$
Taking the $m \rightarrow \infty$ limit,
$$A^*_{\mathbf{SL} (V,n)}(\text{point}) \stackrel{\sim}{=} \mathbb{Z}[c_1, c_2]/(nc_1).$$
\subsection{Proposition \ref{alltor}}
\label{abe}
The quotient $U \times ^{\mathbf{GL} (V)} E\mathbf{GL} (V)$
is analyzed via approximation.
$V \times ^{\mathbf{GL}(V)} W_m$ is the tautological
sub-bundle $S$ over $\mathbf G(2,m)$.
$$U \times ^{\mathbf{GL}(V)} W_m \subset \oplus_{0}^{d} Sym^d(V^*)
\times ^{\mathbf{GL}(V)} E$$ is the non-degenerate
open locus in the total space of the bundle
$\oplus_{0}^{d} Sym^d (S^*)$ over $\mathbf G(2,m)$.
By Lemma \ref{petey}, there
is an isomorphism
$$A^*(U \times ^{\mathbf{GL}(V)} W_m) \stackrel{\sim}{=}
\mathbb{Z}[c_1, c_2]/ (\alpha_1, \ldots, \alpha_{d+1})$$
up to codimension $m-2$ where
$$\frac{1}{c(Sym^d (S))}=
1+ \alpha_1+ \ldots + \alpha_{d+1} + \ldots.$$
The
ideal generated by $(\alpha_1, \ldots, \alpha_{d+1})$
is equal to the ideal generated by the
first $d+1$ Chern classes of $Sym^d (S)$.
Taking the $m \rightarrow \infty$ limit, a presentation
of $A^*_{\mathbf{GL} (V)} (U)$ is obtained.
$A^*_{\mathbf{GL} (V)}(U)$ is generated (as a ring)
in codimensions $1$, $2$ by elements $c_1$, $c_2$ respectively.
There are $d+1$ relations given as follows. Let $S$ be a rank
$2$ bundle with Chern classes $c_1$ and $c_2$. The $d+1$
Chern classes of $Sym^d(S)$ are the relations.
\begin{lm}
\label{ater}
$A^*_{\mathbf{GL}(V)}(U) \otimes \mathbb{Q}$ is zero is positive codimension.
\end{lm}
\noindent {\em Proof.}
A standard calculation yields:
\begin{equation}
\label{see1}
c_1(Sym^d(S))=\frac{d(d+1)}{2} c_1,
\end{equation}
\begin{equation}
\label{see2}
c_2(Sym^d(S))= \frac{d(d-1)(d+1)(3d+2)}{24} c_1^2 +
\frac{d(d+1)(d+2)}{6} c_2.
\end{equation}
Since the coefficients of $c_1$ and $c_2$
never vanish for positive $d$ in equations
(\ref{see1}) and (\ref{see2}) respectively,
the first two Chern classes of $Sym^d(S)$ generate
the ideal $(c_1, c_2)$ in $\mathbb{Q}[c_1,c_2]$.
\qed \vspace{+10pt}
\noindent
Lemma \ref{ater} and the isomorphism (\ref{xeq}) establish
Proposition \ref{alltor}.
\section{{\bf $A^*(d)$, $d$ Even}}
\label{evan}
The notation of section \ref{prezz} is used.
Let $d=2n$ (where $n\geq 1$).
Let $V\stackrel{\sim}{=} \mathbb{C}^2$.
There is a free $\mathbf{PGL} (V)$-action
on $\mathbf P(U) \subset \mathbf P(\oplus_{0}^{d} Sym^d V^*)$
with geometric quotient (see the Appendix)
isomorphic to $H(d)$.
By Proposition \ref{dane},
$$A^*(d) \stackrel{\sim}{=} A^*_{\mathbf{PGL} (V)} (\mathbf P (U)).$$
By the definition of equivariant Chow groups,
$$A^*_{\mathbf{PGL} (V)} (\mathbf P (U)) \stackrel{\sim}{=} A^*( \mathbf P(U) \times
^{\mathbf{PGL} (V)} E \mathbf{PGL} (V)).$$
The Chow ring $A^*( \mathbf P(U) \times
^{\mathbf{PGL} (V)} E \mathbf{PGL} (V))$ is computed in this section
for $d=2n$.
Consider the $3$-dimensional representation
$Sym^2(V)$ of $\mathbf{PGL}(V)$. This respresentation
leaves invariant a unique
(up to $\mathbb{C}^*$) quadratic form $Q$ on $Sym^2(V)$.
A group isomorphism
$\mathbf{PGL} (V) \stackrel{\sim}{=} \mathbf{SO} (3)$ is induced by this quadratic
form.
The dual of the standard $3$-dimensional representation
of $\mathbf{SO}(3)$ corresponds to the representation
$Sym^2 (V^*)$ of $\mathbf{PGL} (V)$.
Let
$$A_m \subset \oplus_{1}^{m} Sym^2 (V^*)$$
be the spanning locus. The approximations
$A_m/ \mathbf{PGL} (V)$ to $B \mathbf{PGL} (V)$ correspond
exactly to the approximations
$W_m/ \mathbf{SO} (3)$ to $B \mathbf{SO} (3)$ defined in section \ref{orthoo}.
$A_m/ \mathbf{PGL}(V)$ is therefore the total space of
a $\mathbb{C}^*$-bundle $N \rightarrow Z_m$.
$Z_m$ is the open set of non-degenerate
quadratic forms in
$\mathbf P(Sym^2 (S^*))$ over the Grassmannian $\mathbf G(3,m)$.
Let $B_m$ denote this approximation to $B \mathbf{PGL} (V)$.
$Sym^d(V^*)$ is a $\mathbf{PGL}(V)$ representation
for $d$ even ({\em not} for $d$ odd). Hence,
$$ Sym^d(V^*) \times^{\mathbf{PGL}(V)} A_m$$
is a rank $d+1$ vector bundle $F_d \rightarrow B_m$.
The quotient
$$\mathbf P(U) \times
^{\mathbf{PGL} (V)} A_m \subset \mathbf P(\oplus_{0}^{d} Sym^d(V^*))
\times ^{\mathbf{PGL} (V)} A_m$$
is simply the projective non-degenerate locus
in $\mathbf P(\oplus_{0}^{d} F_d)$.
The first step is to identify the bundle $F_d \rightarrow B_m$.
There is a tautological sub-bundle $S \rightarrow B_m$
obtained from the Grassmannian. There is a
tautological equivalence
$S^* \stackrel{\sim}{=} F_2$. More generally, there
is a tautological sequence on $\mathbf P(Sym^2 (S^*))$:
\begin{equation}
\label{toto}
0 \rightarrow {\mathcal{O}}_{\mathbf P}(-1) \otimes Sym^{n-2} (S^*)
\rightarrow Sym^n (S^*) \rightarrow Q_n \rightarrow 0
\end{equation}
for all $n\geq 2$.
Let $([q], P)\in \mathbf P(Sym^2 (S^*))$ where
$P\subset \mathbb{C}^m$ is a linear $3$-space
and $0 \neq q \in Sym^2(P^*)$. The fiber
of ${\mathcal{O}}_{\mathbf P}(-1)$ over $([q],P)$ is
simply $\mathbb{C} \cdot q$.
The left inclusion in sequence (\ref{toto}) is
determined by the canonical multiplication map:
$$0 \rightarrow \mathbb{C} \cdot q \otimes Sym^{n-2} (P^*) \rightarrow
Sym^n(P^*).$$
Again, there is a tautological equivalence
$F_{2n} \stackrel{\sim}{=} Q_n$ on $B_m$.
Note $A^1(B_m)=0$ by Lemma \ref{ddd}.
The Chern polynomial of $F_d$ on $B_m$ is therefore:
$$c(F_d)= \frac{c(Sym^n (S^*))}{c(Sym^{n-2}(S^*))}.$$
Now, by Lemma \ref{pete}, a presentation
of $A^*(\mathbf P(U) \times^{\mathbf{PGL}(V)} B_m)$
up to codimension $m-3$ is obtained by
$$A^*(B_m)[\mathcal{L}]/
(p(\mathcal{L}), \alpha'_1, \ldots, \alpha'_{d+1})$$
where $\mathcal{L}$ is the class of
${\mathcal{O}}_{\mathbf P(\oplus_{0}^{d} F_d)}(1)$
and
$$\frac{ (1+\mathcal{L})^{d+1}}{c(F^*_d)}=
\frac{(1+\mathcal{L})^{d+1} \cdot c(Sym^{n-2} S)} {c(Sym^n S)} =
1+ \alpha'_1+ \ldots+ \alpha'_{d+1}
\ldots .$$
By the presentation of $A^*(B_m)$ in section \ref{ort},
it follows
$$A^*(\mathbf P(U) \times^{\mathbf{PGL}(V)} B_m) \stackrel{\sim}{=}
\mathbb{Z}[c_1, c_2, c_3, \mathcal{L}]/
(p(\mathcal{L}), c_1, 2c_3, \alpha'_1, \ldots, \alpha'_{d+1})$$
up to codimension $m-3$. Taking the $m\rightarrow \infty$
limit,
$$ A^*(d) \stackrel{\sim}{=} A^*_{\mathbf{PGL}(V)}(\mathbf P(U)) \stackrel{\sim}{=}
\mathbb{Z}[c_1, c_2, c_3, \mathcal{L}]/
(p(\mathcal{L}), c_1, 2c_3, \alpha'_1, \ldots, \alpha'_{d+1}).$$
The relation $p(\mathcal{L})$ is of codimension $(d+1)^2$.
Since the dimension of $H(d)$ is $d^2+ 2d-3=(d+1)^2-4$ and
the generators $c_1,c_2, c_3$, and $\mathcal{L}$ have
dimension at most 3, $p(\mathcal{L})$
is a relation among classes that are already zero.
Hence,
\begin{equation}
\label{freedy}
A^*(d) \stackrel{\sim}{=} A^*_{\mathbf{PGL}(V)}(\mathbf P(U)) \stackrel{\sim}{=}
\mathbb{Z}[c_1, c_2, c_3, \mathcal{L}]/
(c_1, 2c_3, \alpha'_1, \ldots, \alpha'_{d+1}).
\end{equation}
Following [EG], there is natural
map
$$\mathbf P(U)\times^{\mathbf{PGL}(V)} A_m \rightarrow \mathbf P(U)/ \mathbf{PGL}(V) \stackrel{\sim}{=} H(d)$$
which expresses $\mathbf P(U)\times^{\mathbf{PGL}(V)} A_m$ as an open set
of a vector bundle over $H(d)$. This fibration
induces an isomorphism on Chow rings (up to
codimension $m-3$). The classes $c_i\in A^*(d)$ are
easily identified via this isomorphism (up to
codimension $m-3$):
\begin{equation}
\label{ddod}
A^*(\mathbf P(U)\times^{\mathbf{PGL}(V)} A_m) \stackrel{\sim}{=} A^*(d).
\end{equation}
They are the Chern classes of the vector
bundle obtained from the
the principal $\mathbf{PGL}(V)$-bundle $\mathbf P(U) \rightarrow
H(d)$ and the representation $Sym^2(V)$.
Let $\mathcal{H}\in A^1(d)$ be the class of
curves meeting a fixed codimension 2 linear
space $P$ of $\mathbf P^d$. $\mathcal{H}$ corresponds via the isomorphism
(\ref{ddod}) to a resultant
class in $A^1(\mathbf P(U)\times^{\mathbf{PGL}(V)} A_m)$.
Routine calculations show $\mathcal{H}=2d \mathcal{L}$
where $2d$ is the degree of the resultant of
degree $d$ polynomials.
Since $(d+1)\mathcal{L}=0$ by the presentation (\ref{freedy}).
$$n \mathcal{H} = d^2 \mathcal{L} = (d^2-1+1)\mathcal{L} = (d-1)(d+1)\mathcal{L}
+\mathcal{L} = \mathcal{L}.$$
The proof of Theorem \ref{evan} is complete.
\section{{\bf $A^*(d)$, $d$ Odd}}
\label{ode}
Let $d=2n-1$ (where $n\geq 1$).
Let $V\stackrel{\sim}{=} \mathbb{C}^2$.
There is a canonical, $\mathbf{GL} (V)$-equivariant, multilinear map
$$\mu:\bigoplus_0^{2n-1}
Sym^{2n-1}(V^*) \rightarrow \bigwedge^{2n} Sym^{2n-1}(V^*)$$
given by the exterior product (see section \ref{prezz}).
\begin{lm} The
$\mathbf{SL} (V,n)$-action on
$\bigwedge^{2n} Sym^{2n-1}(V^*)$ is trivial.
\end{lm}
\noindent {\em Proof.}
Since the $1$-dimensional representations of
$\mathbf{SL}(V)$ are trivial, the action of $\mathbf{SL}(V)$
on
$\bigwedge^{2n} Sym^{2n-1}(V^*)$ is certainly trivial. Let
$H\subset\mathbf{SL}(V,n)$ be the subgroup of scalars.
$H$ is the multiplicative group of scalar $2n^{th}$
roots of unity,
Let $\xi \in H$ be a scalar. $\xi$ acts
on $\bigwedge^{2n} Sym^{2n-1}(V^*)$ by the
scalar $\xi^{(2n)(2n-1)}= 1$.
It is easily checked that $\mathbf{SL}(V,n)$
is generated (as a group) by $H$ and $\mathbf{SL}(V)$.
Hence, the $\mathbf{SL}(V,n)$-action is trivial.
\qed \vspace{+10pt}
\noindent
Let
$Y= \mu^{-1} (p)$ where $0 \neq p \in\bigwedge^{2n} Sym^{2n-1}(V^*)$.
There is an $\mathbf{SL} (V,n)$-action on $Y$.
\begin{lm}
\label{freeaq}
The $\mathbf{SL} (V,n)$-action on $Y$ is free with geometric
quotient $H(d)$.
\end{lm}
\noindent {\em Proof.}
Certainly $\mathbf{SL}(V,n)$ acts on $Y$ since the
$\mathbf{SL}(V,n)$-action on $\bigwedge^{2n} Sym^{2n-1}(V^*)$
is trivial.
Let
$$U \subset \oplus_0^{2n-1} Sym^{2n-1} (V^*)$$
be the non-degenerate locus.
First, it is shown that the $\mathbf{SL}(V,n)$-action
on $U$ is free. Since $Y\subset U$, $\mathbf{SL}(V,n)$
acts freely on $Y$.
Let $u\in U$. Suppose $g \in \mathbf{SL}(V,n)$ satisfies
$g\cdot u = u$. $\mathbf{PGL}(V)$ acts freely
on $\mathbf P(U)$. Let $\pi: \mathbf{SL}(V,n) \rightarrow \mathbf{PGL}(V)$.
Then,
$$\pi(g) \cdot \mathbf P(u)= \mathbf P(u).$$
Hence, $\pi(g) = 1 \in \mathbf{PGL} (V)$. The
element $g$ is therefore a scalar in $\mathbf{SL}(V,n)$
equal to a $2n^{th}$ root of unity $\xi$.
Then, $g$ acts on $u$ by
the scalar $\xi^{2n-1}$. Since
$g \cdot u=u$, $\xi^{2n-1} =1$.
Since $(2n, 2n-1)=1$, $\xi^{2n-1}=1$ implies
$\xi=1$. Therefore, $g=1 \in \mathbf{SL}(V,n)$.
The $\mathbf{SL}(V,n)$-action on $U$ is free.
It is now shown the quotient $Y/ \mathbf{SL}(V,n)$ is isomorphic
to $H(d)$. There are natural, equivariant, algebraic projection
maps:
$$Y \rightarrow \mathbf P(U),$$
$$\pi: \mathbf{SL}(V,n) \rightarrow \mathbf{PGL}(V).$$
These maps induce a natural surjective map on quotients:
$$\phi: Y/ \mathbf{SL}(V,n) \rightarrow \mathbf P(U)/ \mathbf{PGL}(V) \stackrel{\sim}{=} H(d).$$
It suffice to prove $\phi$ is injective. (A bijective
map of nonsingular complex algebraic varieties
is an algebraic isomorphism.)
Let $y_1, y_2 \in Y$ be points. Let
$[y_1], [y_2]\in \mathbf P(U)$ denote the corresponding
points. Suppose there exists an element $\gamma \in \mathbf{PGL}(V)$
satisfying $\gamma \cdot [y_1]=[y_2]$.
To prove $\phi$ is injective,
it must be shown that $y_1$ and $y_2$ are in the same
$\mathbf{SL}(V,n)$ orbit.
Let $g\in \mathbf{SL}(V,n)$ satisfy $\pi(g)=\gamma$.
Then,
$[g \cdot y_1] =[y_2]$.
Hence $g \cdot y_1= (\lambda y_2)$
where $\lambda \in \mathbb{C}^*$ is a scalar.
By the conditions $g \cdot y_1, y_2 \in Y$,
it follows $\lambda^{2n}=1$.
Since $(2n, 2n-1)=1$, a $2n^{th}$ root
of unity $\xi \in \mathbf{SL}(V,n)$ can be found satisfying
$\xi^{2n-1}=\lambda^{-1}$.
Let $h \in \mathbf{SL}(V,n)$ be determined by $h= \xi \cdot g$.
$$h\cdot y_1= \xi \cdot g \cdot y_1= \xi \cdot (\lambda y_2)
= \xi^{2n-1}(\lambda y_2)= y_2.$$
Therefore $y_1$ and $y_2$ are in the same $\mathbf{SL}(V,n)$ orbit.
\qed \vspace{+10pt}
\noindent
There is a canonical
isomorphism of graded rings
$$A^*(d=2n-1) \stackrel{\sim}{=} A^*_{\mathbf{SL} (V,n)}(Y).$$
The equivariant Chow ring $A^*_{\mathbf{SL} (V,n)}(Y)$
is computed in this section.
The approximations $W_m$ and $W_m/ \mathbf{SL}(V,n)$ to
$E \mathbf{SL}(V,n)$ and $B \mathbf{SL}(V,n)$ determined in
section \ref{slvn} are used here.
Recall $W_m/ \mathbf{SL}(V,n) \rightarrow \mathbf G(2,m)$
is the $\mathbb{C}^*$-bundle associated to the $n^{th}$
tensor power of $\wedge^2 S$ (where $S$ is
the tautological sub-bundle over $\mathbf G(2,m)$).
Since $$Y \subset U \subset \oplus_{0}^{2n-1} Sym^{2n-1} (V^*),$$
there are inclusions:
$$ Y \times^{\mathbf{SL}(V,n)} W_m \ \subset \
U \times ^{\mathbf{SL}(V,n)} W_m \ \subset \
\oplus_{0}^{2n-1} Sym^{2n-1}(V^*) \times ^{\mathbf{SL}(V,n)} W_m.$$
Let $F_n=V^* \times^{\mathbf{SL}(V,n)} W_m$. $F_n$ is an algebraic
vector bundle over $W_m/ \mathbf{SL}(V,n)$. $F_n$ is
easily identified as the pull-back of $S^*$ to
$W_m/ \mathbf{SL}(V,n)$.
$U \times ^{\mathbf{SL}(V,n)} W_m$ is the
affine non-degenerate locus (i) of section \ref{idealz}
associated to the bundle $Sym^{2n-1} F_n\stackrel{\sim}{=} Sym^{2n-1} S^*$.
\begin{lm}
\label{qwq}
There is an isomorphism
$$ \epsilon: \mathbb{C}^* \times (Y \times^{\mathbf{SL}(V,n)} W_m)
\stackrel{\sim}{=} U \times ^{\mathbf{SL}(V,n)} W_m.$$
\end{lm}
\noindent {\em Proof.}
Let $\mathbf{SL}(V,n)$ act trivially on $\mathbb{C}^*$.
Define a $\mathbf{SL}(V,n)$ equivariant
isomorphism
$$\delta: \mathbb{C}^* \times Y \rightarrow U$$
by the following:
$$\delta\big( \lambda, (\omega_0, \omega_1\ldots, \omega_{2n-1})\big)
= (\lambda \omega_0, \omega_1, \ldots, \omega_{2n-1}).$$
The isomorphism $\delta$ induces
isomorphisms:
$$\mathbb{C}^* \times (Y \times^{\mathbf{SL}(V,n)} W_m) \stackrel{\sim}{=}
(\mathbb{C}^* \times Y) \times^{\mathbf{SL}(V,n)} W_m \stackrel{\sim}{=}
U \times^{\mathbf{SL}(V,n)} W_m.$$
Let $\epsilon$ be the composition.
\qed \vspace{+10pt}
$W_m/ \mathbf{SL}(V,n)$ approximates $B \mathbf{SL}(V,n)$
up to codimension $m-2$.
The Chow ring of $Y \times ^{\mathbf{SL}(V,n)} W_m$
is now computed (up to codimension $m-2$).
By Lemma \ref{qwq}, there is an isomorphism:
$$A^*(Y \times ^{\mathbf{SL}(V,n)} W_m) \stackrel{\sim}{=}
A^*(U \times ^{\mathbf{SL}(V,n)} W_m).$$
Since $U$ is the affine non-degenerate
locus associated to the bundle
$$Sym^{2n-1} S^* \rightarrow (W_m / \mathbf{SL}(V,n)),$$
Lemma \ref{pete} can be applied.
Recall the Chow ring of $W_m / \mathbf{SL}(V,n)$ (up to
codimension $m-2$)
has a presentation $\mathbb{Z}[c_1, c_2]/ (nc_1)$.
Hence, there is an isomorphism (up to
codimension $m-2$):
$$A^*(Y \times ^{\mathbf{SL}(V,n)} W_m) \stackrel{\sim}{=}
\mathbb{Z}[c_1, c_2]/ (nc_1, \alpha_1, \ldots, \alpha_{d+1})$$
where
$$\frac{1}{c(Sym^d S)}=
1+ \alpha_1 + \ldots + \alpha_{d+1}+ \ldots.$$
The proof of Theorem \ref{hodd} is complete.
\section {\bf Examples}
Since $H(1)$ is a point, $A^*(1)$ is the trivial
$\mathbb{Z}$-algebra (which agrees with the presentation
of Theorem 2).
$H(2)$ is the space of nonsingular plane conics.
By Theorem 1, $A^*(2)$ is generated
by $c_2$, $c_3$, and $\mathcal{L}=\mathcal{H}$ subject to
$4$ relations:
$$2c_3=0,$$
$$3\mathcal{H}=0, \ -c_2+3\mathcal{H}^2=0, \ -c_3+\mathcal{H}^3=0.$$
Since $\mathcal{H}$ is 3-torsion, $c_2=0$.
Since $c_3$ is two torsion, the last
equation can be reduced to $c_3=\mathcal{H}^3=0$.
Therefore $A^*(2)$ is given by
$$\mathbb{Z}[\mathcal{H}]/ (3\mathcal{H}, \mathcal{H}^3).$$
Since $H(2)$ is an open set of the projective
space of plane conics, another approach to $A^*(2)$
is possible.
The class $\mathcal{H}$ is simply the restriction
of the hyperplane class which necessarily generates $H(2)$.
The relation $3\mathcal{H}$
can be obtained from the degree $3$ degeneracy locus
of singular plane conics.
The relation $\mathcal{H}^3$ is a
consequence of the fact that the locus of
conics singular at a {\em fixed} point in $\mathbf P^2$
is a linear $\mathbf P^2$ in the $\mathbf P^5$ of conics.
Let $d=3$, $n=2$, $d=2n-1$. By Theorem 2,
$A^*(3)$ is generated by $c_1$ and $c_2$
with relations:
$$ 2c_1=0,$$
$$6c_1=0, \ 11c_1^2+ 10c_2=0, \ 6c_1^3+30c_1c_2=0,
\ 18c_1^2c_2+9c_2^2=0.$$
These relations simplify to yield the presentation:
$$A^*(3)= \mathbb{Z}[c_1,c_2]/ (2c_1, c_1^2+10c_2, c_1^3, c_1^2c_2,c_2^2).$$
In particular, $A^i(3)=0$ for $i\geq 4$.
\section {\bf Appendix On Algebraic Quotients}
\label{appx}
Let $\mathbb{C}$ be the ground field of complex numbers.
The geometric invariant theory terminology of [MFK]
is used here.
Let $\mathbf{G}$ be a reductive linear algebraic group.
A group action $\mathbf{G} \times X \rightarrow X$ is {\em proper}
if the natural map
$$\Psi: \mathbf{G} \times X \rightarrow X \times X$$
(given by the action and projection onto the second
factor) is a proper morphism.
The main result needed is the following:
\begin{pr}
Let $X$ be a quasi-projective variety with
a linearized $\mathbf{G}$-action satisfying $X^{stable}_{(0)}
=X$.
\label{qquot}
Then, the
$\mathbf{G}$-action on $X$ is proper and
there is a quasi-projective
geometric quotient $X \rightarrow X/\mathbf{G}$.
\end{pr}
\noindent {\em Proof.}
Properness of the action is exactly Corollary 2.5
of [MFK]. The geometric quotient
is the main construction in geometric invariant
theory (Theorem 1.10 of [MFK]).
\qed \vspace{+10pt}
\noindent
The stable locus $X_{(0)}^{stable}$ is detected
by the Numerical Criterion.
Let $V\stackrel{\sim}{=} \mathbb{C}^k$ be a vector space equipped
with a quadratic form.
All of the linear algebraic groups
considered in this paper are reductive:
$\mathbf{GL}(V)$, $\mathbf{SL}(V)$, $\mathbf{PGL}(V)$, $\mathbf{SL}(V,n)$, $\mathbf{O}(V)$, $\mathbf{SO} (V)$,
$\mathbb{C}^*$, $\mathbb{C}^* \times \mathbf{SO}(V)$.
Let
$$Span_m(V,d) \subset \oplus _{1}^{m} Sym^d(V*)$$
be the spanning locus (the locus of $m$-tuples
of vector of $Sym^d(V^*)$ which span $Sym^d(V^*)$).
The spanning loci
$U \subset \oplus_{0}^{d} Sym^d(\mathbb{C}^{2*})$
and
$W_m \subset \oplus_{1}^{m} V^*$
are special cases of $Span_m(V,d)$.
The group actions considered in the paper are
of three forms:
\begin{enumerate}
\item[(i)] The natural $\mathbf{G}$-action on $X=Span_m(V,d)$
where $\mathbf{G} \subset \mathbf{GL}(V)$ is
a reductive subgroup.
\item[(ii)] The $\mathbf{G}$-action on a $\mathbf{G}$-invariant
subvariety $Y\subset Span_m(V,d)$ where
$\mathbf{G} \subset \mathbf{GL}(V)$ is a reductive
subgroup.
\item[(iii)] The natural $\mathbf{PGL}(V)$-action on $X=\mathbf P(Span_m(V,d))$.
\end{enumerate}
For example, the $SL(V,n)$-action
on $Y$ considered in section \ref{ode} is of form (ii).
Consider first (i) and (ii).
A linearization of the $\mathbf{GL}(V)$-action
can be found on $X$ satisfying $X^{stable}_{(0)}=X$.
Such a linearization is found in section 1 of
[P]. Since the stable locus
is detected by the Numerical Criterion, the
result for $\mathbf{GL}(V)$ implies $X^{stable}_{(0)}=X$
for the induced action of any reductive
subgroup $\mathbf{G} \subset \mathbf{GL}(V)$.
It is similarly simple to find a linearization
in case (iii) satisfying $X^{stable}_{(0)}=X$.
Therefore, Proposition \ref{qquot} applies
to all the quotient problems in the paper.
In [MFK], the $\mathbf{G}$-action $\mathbf{G} \times X \rightarrow X$
is defined to be {\em free} if
the natural map $\Psi$
is a closed embedding.
An action is {\em set-theoretically free}
if the stabilizers are trivial.
For the set-theoretically free
actions considered in this paper,
the following Lemma is utilized.
\begin{lm}
Let $X$ be nonsingular.
Let $\mathbf{G}\times X \rightarrow X$ be a proper action.
In this case, set-theoretically free implies free.
\end{lm}
\noindent {\em Proof.}
Let $I \subset X\times X$ be the image of $\Psi$.
$I$ is a closed subvariety since $\Psi$ is proper.
It must be shown that $\mathbf{G} \times X \rightarrow I$
is an isomorphism.
First it is shown that $I$ is nonsingular.
For this, it suffices to prove the
differential of $\Psi$ is injective at each point
of $\mathbf{G} \times X$. It is well known (over $\mathbb{C}$) that
set-theoretically trivial stabilizers are
also scheme-theoretically trivial. From this,
the injectivity of the differential $d\Psi$ is easily deduced.
Now $\Psi: \mathbf{G} \times X \rightarrow I$ is
a bijective map of nonsingular complex
algebraic varieties and thus an algebraic
isomorphism.
\qed \vspace{+10pt}
\noindent A result of [MFK] (Proposition 0.9)
relates free quotients to principal $\mathbf{G}$-bundles.
\begin{pr}
Let $\mathbf{G} \times X \rightarrow X$ be an algebraic
group action with geometric quotient $X \rightarrow Y$.
If the action is free, then $X \rightarrow Y$ is
a (\'etale locally trivial) principal $\mathbf{G}$-bundle.
\end{pr}
\noindent
A principal $\mathbf{G}$-bundle $X \rightarrow Y$ and a representation
$\mathbf{G}\rightarrow\mathbf{GL}$ together yield a principal $\mathbf{GL}$-bundle
$X \times^{\mathbf{G}} \mathbf{GL} \rightarrow Y$.
Since every principal $\mathbf{GL}$-bundle is
Zariski locally trivial ($\mathbf{GL}$ is a {\em special}
group in the sense of Grothendieck (see
{\em Anneau de Chow et Applications}, Seminaire
Chevalley 1958)), an algebraic vector bundle
over $Y$ is obtained.
|
1996-11-04T11:53:12 | 9607 | alg-geom/9607004 | en | https://arxiv.org/abs/alg-geom/9607004 | [
"alg-geom",
"math.AG"
] | alg-geom/9607004 | E. Looijenga | Richard Hain and Eduard Looijenga | Mapping Class Groups and Moduli Spaces of Curves | We expanded section 7 and rewrote parts of section 10. We also did
some editing and made some minor corrections. latex2e, 46 pages | null | null | null | null | This is a survey paper that also contains some new results. It will appear in
the proceedings of the AMS summer research institute on Algebraic Geometry at
Santa Cruz.
| [
{
"version": "v1",
"created": "Thu, 4 Jul 1996 08:49:56 GMT"
},
{
"version": "v2",
"created": "Mon, 4 Nov 1996 10:48:45 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Hain",
"Richard",
""
],
[
"Looijenga",
"Eduard",
""
]
] | alg-geom | \section{Introduction}
\label{sec:intro}
It is classical that there is a very strong relation between
the topology of ${\mathcal M}_g$, the moduli space of smooth projective curves
of genus $g$, and the structure of the mapping class group $\Gamma_g$,
the group of homotopy classes of orientation preserving diffeomorphisms
of a compact orientable surface of genus $g$. The geometry of ${\mathcal M}_g$,
the topology of ${\mathcal M}_g$, and the structure of $\Gamma_g$ are all intimately
related. Until recently, the principal tools for studying these topics
were Teichm\"uller theory (complex analysis and hyperbolic geometry),
algebraic geometry, and geometric topology. Recently, a fourth
cornerstone has been added, and that is physics which
enters through the theories of quantum gravity and conformal field theory.
Already these new ideas have had a remarkable impact on the subject through the
ideas of Witten and the work of Kontsevich. In this article, we survey some
recent developments in the understanding of moduli spaces. Some of these
are classical (do not use physical ideas), while others are modern. One
message we would like to convey is that algebraic geometers, topologists,
and physicists who work on moduli spaces of curves may have a lot to learn
from each other.
Having said this, we should immediately point out that,
partly due to our own limitations, there are important developments
that we have not included in this survey. Our most notable omission is
the arithmetic aspect of the theory, much of which originates in
Grothendieck's fundamental works \cite{groth:marche}, \cite{groth:esq}.
We direct readers to the volume \cite{groth:dessins} and to the recent
papers of Ihara, Nakamura and Oda for other recent developments (see
Nakamura's survey \cite{nakamura:survey} for references). Other
topics we have
not covered include conformal field theory and recent work
of Ivanov \cite{ivanov:rigid}
and Ivanov and McCarthy \cite{ivanov-mccarthy} on homomorphisms
from mapping class groups and arithmetic groups to mapping class groups.
Of particular importance is Ivanov's version of Margulis rigidity
for mapping class groups \cite{ivanov:rigid} which he obtains using some
recent fundamental work of Kaimanovich and Masur \cite{masur} on the
ergodic theory of Teichm\"uller space.
We shall denote the moduli space of $n$ pointed smooth projective
curves of genus $g$ by ${\mathcal M}_g^n$. Knudsen, Mumford and Deligne
constructed a
canonical compactification $\overline{\M}_g^n$ of it. It is the moduli space
of stable $n$ pointed projective curves of genus $g$. It is a projective
variety with only finite quotient singularities. Perhaps the most important
developments of the decade concern the Chow rings%
\footnote{All Chow rings and cohomology groups in this paper are with
${\mathbb Q}$ coefficients except when explicit coefficients are used.}
of ${\mathcal M}_g^n$ and $\overline{\M}_g^n$.
The first Chern class of the relative cotangent bundle of the universal
curve associated to the $i$th point is a class $\overline{\tau}_i$ in
$\CH^1(\overline{\M}_g^n)$. One can consider monomials in the $\overline{\tau}_i$'s of
polynomial degree equal to the
dimension of some $\overline{\M}_g^n$. For such a monomial, one can take the
degree of the monomial as a zero cycle on $\overline{\M}_g^n$ to obtain a
rational number. These can be assembled into a generating function. Witten
conjectured that this formal power series satisfies a system of
partial differential operators. Kontsevich proved this using topological
arguments, and thereby provided inductive formulas for these intersection
numbers. These developments are surveyed in Section~\ref{sec:ribbon}.
For each positive integer $i$, Mumford defined a {\it tautological class}
$\overline{\kappa} _i$
in $\CH^i(\overline{\M}_g)$. The restrictions $\kappa _i$ of these classes to
$\CH^{\bullet}({\mathcal M}_g)$ generate a subalgebra of $\CH^{\bullet}({\mathcal M}_g)$ which is called
the {\it tautological algebra} of ${\mathcal M}_g$. Faber has conjectured that this
ring has the structure of the $(p,p)$ part of the cohomology ring of a
smooth complex projective variety of complex dimension $g-2$. That is, it
satisfies
Poincar\'e
duality and has the ``Hard Lefschetz Property'' with respect to $\kappa _1$.
Considerable evidence now exists for this conjecture, much of which is
presented in Section~\ref{sec:chow}.
Other developments on the Chow ring, such as explicit computations in
low genus, are also surveyed there.
In the early 80s, Harer proved that the cohomology in a given degree
of ${\mathcal M}_g$ is independent of the genus once the genus
is sufficiently large relative to the degree. These stable cohomology
groups form a graded commutative algebra which is known to be free.
The tautological classes $\kappa_i$ freely generate a polynomial algebra
inside the stable cohomology ring. Mumford and others have conjectured
that the stable cohomology of ${\mathcal M}_g$ is generated by the $\kappa_i$'s.
Some progress has been made towards this conjecture which we survey
throughout the paper. In Section~\ref{sec:agstability}
we consider the stabilization maps from an algebro-geometric point of view,
and in Section~\ref{sec:algebras} we survey Kontsevich's methods for
constructing classes in the cohomology of the ${\mathcal M}_g^n$.
We have also tried to advertise the fecund work of Dennis Johnson
on the Torelli groups. The Torelli group $T_g$ is the subgroup of
the mapping class group $\Gamma_g$ consisting of those diffeomorphism
classes that act trivially on the homology of the reference
surface. This mysterious group, in some sense, measures the
difference between curves and abelian varieties and appears to play
a subtle role in the geometry of ${\mathcal M}_g$. Johnson proved
that $T_g$ is finitely generated when $g\ge 3$ and computed its
first integral homology group. These computations have
direct geometric applications, especially when combined with M.~Saito's
work in Hodge theory --- for example, they restrict the
normal functions defined over ${\mathcal M}_g$ and its standard level covers.
{}From this, one can give a computation of the Picard group of the
generic curve with a level $l$ structure. Johnson's work and its
applications is surveyed in Section~\ref{sec:torelli}.
Since $\Gamma_g$ is the orbifold fundamental group of ${\mathcal M}_g$, an algebraic
variety, one should be able to apply Hodge theory and Galois theory to study
its structure. In Section~\ref{sec:hodgemap} we survey recent work on
applications of Hodge theory to understanding the structure of the Torelli
groups, mainly via Malcev completion. In Section~\ref{sec:algebras}
we combine this Hodge theory with recent results of Kawazumi and Morita
to show that the cohomology of ${\mathcal M}_g$ constructed
by Kontsevich using graph cohomology are, after stabilization, polynomials
in the $\kappa_i$'s. Thus Hodge theory provides some evidence for Mumford's
conjecture that the stable cohomology of the
mapping class group is generated by the $\kappa_i$'s.
Some of the results we discuss have not yet appeared in the literature,
at least not in the form in which we present them. Rather than mention
all such results, we simply mention a few instances where we believe our
presentation to be novel: the correspondences in Section~\ref{subsec:correspondences},
the r\^ole of the fundamental normal function for orbifold fundamental groups in
Section~\ref{fundgroup}, Theorem~\ref{tautbound} and the contents of
Section~\ref{subsec:relation}.
\medskip
\noindent{\it Notation and Conventions.}
All varieties will be defined over ${\mathbb C}$ unless explicitly stated
to the contrary. Unless explicit coefficients are used, all (co)homology
groups are with rational coefficients. We will often abbreviate {\it
mixed Hodge structure} by MHS. The sub- or superscript {\it pr}
on a (co)homology group will denote the primitive part in both the
context of the Hard Lefschetz Theorem and in the context of Hopf algebras.
\medskip
\noindent{\it Acknowledgements.} We would like to thank Carel Faber for his
comments on part of an earlier version of this paper and Shigeyuki Morita for explaining to us some of his recent work. We also appreciate the useful
comments by a referee.
We gratefully acknowledge support by the AMS that enabled us to attend this conference.
\section{Mapping Class Groups}
\label{sec:groups}
Fix a compact connected oriented reference surface $S_g$ of genus $g$, and
a sequence of distinct points $(x_0,x_1,x_2,\dots )$ in $S_g$. Let us write
$S_g^n$ for the open surface $S-\{x_1,\dots ,x_n\}$ and
$\pi_g^n$ for its fundamental group $\pi_1(S_g^n ,x_0)$. This group admits a
presentation with generators $\alpha _{\pm 1},\dots ,\alpha_{\pm g},\beta_1,\dots
,\beta_n$ and relation
$$
(\alpha_1,\alpha_{-1})\cdots (\alpha_g,\alpha_{-g})=\beta_1\cdots\beta_n,
$$
where $(x,y)$ denotes the commutator of $x$ and $y$.\footnote{For $n=0$ the
righthand side is to be interpreted as the unit element.} The generators are
represented by loops that do not meet outside the base point;
$\beta_i$ is represented by a loop that follows an arc to a point close to
$x_i$, makes a simple loop around $x_i$, and returns to the base point along
the same arc.
Let $\Diff^+(S)^n_r$
denote the group of orientation preserving diffeomorphisms of $S$ that fix
the $x_i$ for $i=1,\dots ,n+r$, and are the identity on $T_{x_i}S$ for
$i=n+1,\dots ,n+r$. Although not really necessary at this stage, it is
convenient to assume that $2g-2+n+2r>0$. In other words, we do not consider
the cases where $(g,n,r)$ is $(0,0,0)$, $(0,1,0)$, $(0,0,1)$,
$(0,2,0)$ or $(1,0,0)$. We will keep this assumption throughout the paper.
The {\it mapping class group} $\Gamma_{g,r}^n$
is defined to be the group of connected components of this group:
$$
\Gamma_{g,r}^n = \pi_0 \Diff^+(S)^n_r.
$$
We omit the decorations $n$ and $r$ when they are zero.
The mapping class group $\Gamma_g^n$ acts on $\pi_g^n$ by outer
automorphisms. A theorem that goes back to Baer (1928) and Nielsen (1927)
\cite{nielsen} identifies $\Gamma_g$, via this representation, with the
subgroup of $\Out (\pi_g)$ (of index two) that acts trivially on
$H_2(\pi_g)\cong H_2(S_g)$.
When $n\ge 1$ we can consider the diagonal action of $\Aut (\pi_g^n)$ on
$(\pi_g^n)^n$. Clearly, $\Out (\pi_g^n)$ acts on the set
of orbits of $\pi_g^n$ (which acts by inner automorphisms on each component) in
$(\pi_g^n)^n$.
Now $\Gamma_g^n$ can be identified with the group of outer automorphisms of
$\pi_g^n$ that preserve the image of $(\beta_1,\dots
,\beta_n)$ in $\pi_g^n\backslash (\pi_g^n)^n$.
If we choose $x_n$ as a base point, then a corresponding
assertion holds: $\Gamma_g^n$ can be identified with a subgroup of
$\Aut (\pi_1(S_g^{n-1} ,x_n))$ that is characterized in a similar way.
The evident homomorphism $\Gamma_g^n\to \Gamma_g^{n-1}$ is surjective and
its kernel can be identified with $\pi_1(S_g^{n-1} ,x_n)$
(acting by inner automorphisms).
Ivanov and McCarthy \cite{ivanov-mccarthy}
recently showed that the resulting exact sequence cannot be split.
\subsection{Generators and basic properties}\label{subsec:basic}
Although a lot is known about these groups they are still poorly
understood. Let us quickly review some of their basic
properties. Dehn proved in \cite{dehn} that the mapping class groups are
generated by the `twists' that are now named after him:
if $\alpha$ is a simple (unoriented) loop on
$S_g^{n+r}$, then parameterize a regular
neighborhood of $\alpha $ in $S_g^{n+r}$ by the cylinder
$[0,1]\times S^1$ (preserving orientations) and define an
automorphism of $S_g$ that on this neighborhood is given by
$(t,z)\mapsto (t ,e^{2\pi it}z)$ and is the identity elsewhere. The isotopy
class of this automorphism only depends on the isotopy class of
$\alpha $ and is called the {\it Dehn twist} along $\alpha$.
(Perhaps we should add that $\alpha$ is, in turn, already determined by
its free homotopy class, in other words, by the associated conjugacy
class in $\pi_g^n$.) The corresponding element of
$\Gamma_{g,r}^n$ is the identity precisely when $\alpha$ bounds a disk in
$S_g-\{ x_{n+1},\dots ,x_{n+r}\}$ which meets $\{ x_1,\dots ,x_{n}\}$ in at
most one point. Several people have found a finite presentation for the
mapping class groups. One with few generators was given by Waynryb
\cite{waynryb}. From this presentation one sees that the mapping class
groups considered here are perfect when $g \ge 3$ (a result due to Powell
\cite{powell} in the undecorated case).
There is an obvious homomorphism
$\Gamma_{g,n}\to \Gamma_g^n$. It is easy to see that it is surjective and that the
kernel is generated by the Dehn twists around the points $x_1,\dots ,x_n$.
These Dehn twists generate a free abelian central subgroup of $\Gamma_{g,n}$ of
rank $n$. Now recall that a central extension of a discrete group
$G$ by ${\mathbb Z}$ determines an extension class in $H^2(G;{\mathbb Z} )$; it has a
geometric interpretation
as a first Chern class. In the present case we have $n$ such
classes $\tau _i\in H^2(\Gamma_g^n;{\mathbb Z} )$, $i=1,\dots ,n$.
Conversely, each subgroup of $H^2(G;{\mathbb Z} )$ determines a central
extension of $G$ by that subgroup. Harer proved that $H^2(\Gamma _{g,r};{\mathbb Z} )$
is infinite cyclic if $g\ge 3$ \cite{harer:h2}, so that
there is a corresponding central extension
$$
0\to{\mathbb Z}\to\widetilde\Gamma _{g,r}\to\Gamma _{g,r}\to 1.
$$
Since $H_1(\Gamma_{g,r};{\mathbb Z} )$ vanishes, this central extension is perfect (and
universal).
A nice presentation of it was recently given by Gervais
\cite{gervais}. The (imperfect) central extension by $\frac{1}{12}{\mathbb Z}$
containing this extension appears in the theory of conformal blocks; it
has a simple geometric description which we will give in
Section~\ref{sec:moduli}.
\subsection{Stable cohomology}\label{subsec:stable}
The mapping class groups $\Gamma_{g,r}^n$ turn up in a connected sum construction
that we describe next. It is convenient to do this in a somewhat abstract
setting. Suppose we are given a closed, oriented (but not necessarily
connected) surface $S$, a finite subset $Y\subset S$, and a fixed point free
involution $\iota $ of $Y$. Assume that $\iota $ has been lifted to an
orientation reversing linear involution $\tilde\iota$ on the spaces of rays
$\Ray (TS|Y)$. The {\it real oriented blow up} $S_Y\to S$ is a surface
with boundary
canonically isomorphic to $\Ray (TS|Y)$. So $\tilde\iota$ defines an
orientation reversing involution of this boundary. Welding the boundary
components of $S_Y$ by means of this involution produces a closed surface
$S({\tilde\iota})$. Some care is needed to give it a differentiable structure
inducing the given one on $S_Y$. Although there is no unique way to do
this, all natural choices lie in the same isotopy class. If $S$ happens to
have a complex structure, then each choice of a real ray $L$ in
$T_pS\otimes_{{\mathbb C}} T_{\iota (p)}S$ determines a lift of $\iota$ over the
pair $\{p, \iota (p)\}$: if $l$ is a ray in $T_pS$, then $\tilde\iota (l)$
is determined uniquely by the condition $l\otimes_{{\mathbb C}}\tilde\iota (l)=L$.
If $S({\tilde\iota})$ is connected, then each finite subset
$X$ of $S-Y$ determines a natural homomorphism from the
mapping class group which is perhaps best denoted by $\Gamma (S)_Y^X$
(a product of groups of the type $\Gamma_{g,r}^n$) to the mapping class group
$\Gamma (S(\tilde\iota))^X$. The image of this homomorphism is simply the
stabilizer of the simple loops indexed by $Y/\iota $ that are images of
boundary components of $S_Y$. Its kernel is a free abelian group whose
generators can be labeled by a system of representatives $R$ of $\iota$
orbits in $Y$. Indeed, for each element $y$ of $R$, take the composite of the
Dehn twist around $y$ and the inverse of the Dehn twist around $\iota (y)$.
These maps appear in the stability theorems and are at the root of the
recent operad theoretic approaches to the study of the cohomology of
mapping class groups.
\begin{theorem}[Stability theorem, Harer \cite{harer:stab}]
There exists a positive constant $c$ with the following property.
If $S(\tilde\iota )$ is connected and $S'$ is a connected component of $S$
and $X$ a finite subset of $S'\setminus Y$, then the homomorphism
$$
\Gamma (S')_{Y\cap S'}^X\to \Gamma (S(\tilde\iota ))^X
$$
induces an isomorphism on integral cohomology in degree $\le
c.\text{genus}(S')$.
\end{theorem}
The constant $c$ appearing in this theorem was $1/3$ in Harer's
original paper. It was later improved to $1/2$ by Ivanov in
\cite{ivanov:teichm}. Most recently, Harer \cite{harer:imp_stab} has showed
that we can take $c$ to be about $2/3$ and that this is the minimal
possible value. There is also a version for twisted coefficients, due to
Ivanov \cite{ivanov}.
Harer's theorem says essentially that the $k$th cohomology group of
$\Gamma_{g,r}^n$ depends only on $n$, provided that $g$ is large
enough. These stable cohomology groups are the cohomology of a single group,
namely the group $\Gamma_{\infty}^n$ of compactly supported mapping classes of a
surface $S_{\infty}$ of infinite genus (with one end, say) that fix a given
set of $n$ distinct points.
Among the homomorphisms defined above are maps
$\Gamma_{g,1}\times \Gamma_{g',1}\to \Gamma_{g+g'}$. These stabilize and define
homomorphisms of
${\mathbb Q}$ algebras
$$
\mu : H^{\bullet}(\Gamma_{\infty})\to H^{\bullet}(\Gamma_{\infty})\otimes
H^{\bullet}(\Gamma_{\infty}).
$$
This defines a coproduct on $H^{\bullet}(\Gamma_{\infty})$. Together
with the cup product, this gives $H^{\bullet}(\Gamma_{\infty})$ the
structure of a connected graded-bicommutative Hopf algebra.
The classification of such Hopf algebras implies that
$H^{\bullet}(\Gamma_{\infty})$ is free as a graded algebra and is
generated by its set of primitive elements
$$
H^{\bullet}_{{\rm pr}}(\Gamma_{\infty}) := \{x\in H^+(\Gamma_{\infty}):
\mu (x)=x\otimes 1+1\otimes x\}.
$$
For each $i>0$, Mumford \cite{mumford} and Morita \cite{morita:classes}
independently found a class $\kappa_i$ in $H^{2i}_{{\rm pr}}(\Gamma_{\infty})$
(we shall recall the definition in Section~\ref{sec:agstability}) and
Miller \cite{miller} and Morita \cite{morita:classes} independently showed
that each $\kappa_i$ is nonzero. So the $\kappa_i$'s generate
a polynomial subalgebra of the stable cohomology.
Mumford conjectured that they span all of
$H^{\bullet}_{{\rm pr}}(\Gamma_{\infty})$. This has been verified by Harer in a
series of papers \cite{harer:h2}, \cite{harer:h3}, \cite{harer:h4} in
degrees $\le 4$.\footnote{He also tells us that he has checked that there are
no stable primitive classes in degree 5.}
The first Chern class $\tau _i\in H^2(\Gamma_g^n;{\mathbb Z} )$ stabilizes also and we may
think of it as an element of $H^{\bullet}(\Gamma_{\infty}^n;{\mathbb Z} )$ ($i=1,\dots
,n$).
The forgetful map $\Gamma_{\infty}^n\to \Gamma_{\infty}$ gives
$H^{\bullet}(\Gamma_{\infty}^n)$ the structure of a module over this Hopf
algebra. From the stability theorem one can deduce:
\begin{theorem}[Looijenga \cite{looijenga}]
The algebra
$H^{\bullet}(\Gamma_{\infty}^n;{\mathbb Z} )$ is freely generated by the classes
$\tau _1,\dots ,\tau _n$ as a graded-commutative $H^{\bullet}(\Gamma_{\infty};{\mathbb Z}
)$ algebra.
\end{theorem}
\section{Moduli Spaces}
\label{sec:moduli}
A conformal structure and an orientation on $S_g$ determine a complex
structure on $S_g$. The {\it Teichm\"uller space} ${\mathcal X}_{g,r}^n$ is
the space of conformal structures on $S_g$ (with some reasonable
topology) up to isotopies that
fix $\{ x_1,\dots ,x_{n+r}\}$ pointwise and act trivially on the tangent
spaces $T_{x_i}S$ for $i=n+1,\dots, n+r$. It is, in a natural way, a complex
manifold of dimension $3g-3+n+2r$.
As a real manifold it is diffeomorphic to a cell. The group $\Gamma_{g,r}^n$
acts naturally on it. This action is properly discontinuous and a subgroup of
finite index acts freely.
If $\Gamma$ is any subgroup of $\Gamma_{g,r}^n$ that
acts freely, then the orbit space $\Gamma\backslash {\mathcal X}_{g,r}^n$ is a classifying
space for $\Gamma$ and so its singular integral cohomology coincides with
$H^{{\bullet}}(\Gamma ;{\mathbb Z} )$. This is even true with twisted coefficients: if $V$
is a $\Gamma$ module,
then the trivial sheaf over ${\mathcal X}_{g,r}^n$ with fiber $V$ comes with an obvious
(diagonal) action of $\Gamma$. Passing to $\Gamma$ orbits yields a locally constant
sheaf ${\mathbb V}$ on $\Gamma\backslash {\mathcal X}_{g,r}^n$. The cohomology of this sheaf equals
$H^{{\bullet}}(\Gamma ;V)$. For an arbitrary subgroup $\Gamma$ of $\Gamma_{g,r}^n$ these
statements still hold as long as we take our coefficients to be ${\mathbb Q}$ vector
spaces (but ${\mathbb V}$ need no longer be locally constant). For $\Gamma =\Gamma_{g,r}^n$,
we denote the orbit space by ${\mathcal M}_{g,r}^n$.
The space ${\mathcal M}_{g,r}^n$ is, in a natural way, a normal analytic
space and the obvious forgetful maps such as ${\mathcal M}_{g,r}^n\to {\mathcal M}_g^n$ are
analytic. An interpretation as a coarse moduli space makes it possible
to lift this analytic structure to the algebraic category. To see this,
we first choose a nonzero
vector in each tangent space $T_{x_i}S_g$. Each triple $(C;x,v)$, where
$C$ is a connected nonsingular complex projective curve $C$ of genus $g$,
$x$ an
injective map $x:\{ 1,\dots ,n+r\} \to C$, and $v$ a nowhere zero section of
$TC$ over $\{n+1,\dots ,n+r\}$, determines an element of ${\mathcal M}_{g,r}^n$.
This point depends only on the isomorphism class of $(C,x,v)$ with
respect to the obvious notion of isomorphism. Since each conformal structure
on $S$ gives $S$ the structure of a
nonsingular complex projective curve, ${\mathcal M}_{g,r}^n$ can be identified can be
identified
with the space of isomorphism classes of such triples. From the work of
Knudsen, Mumford and Deligne, we know that ${\mathcal M}_g^n$ is, in a natural way, a
quasi-projective orbifold. Recall that they also constructed a projective
completion $\overline{\M}_g^n$ of ${\mathcal M}_g^n$, the {\it Deligne-Mumford completion}
\cite{del_mum}, that also admits the interpretation of a coarse moduli
space. Its points parameterize the connected stable $n$ pointed curves
$(C,x)$ of arithmetic genus $g$, where we now allow $C$ to have ordinary
double points, but still require $x$ to map to the smooth part of $C$ and
the automorphism group of $(C,x)$ to be finite. The {\it Deligne-Mumford
boundary} $\overline{\M}_g^n-{\mathcal M}_g^n$ is a normal crossing divisor in the orbifold
sense. There is a projective morphism $\overline{\M}_g^{n+1}\to\overline{\M}_g^n$, defined
by forgetting the last point. It comes with $n$ sections
$x_1,\dots ,x_n$. The fibers of this morphism are stable $n$ pointed curves
(modulo finite automorphism groups) and the morphism can be regarded as the
universal stable $n$ pointed curve (in an
orbifold sense). Let $\omega$ denote the relative dualizing sheaf of
this morphism, considered as a line bundle in the orbifold sense.
We can then think of ${\mathcal M}_{g,r}^n$ as the set of $(v_{n+1},\dots ,v_{n+r})$
in the total space of $x_{n+1}^*\omega \oplus\cdots \oplus x_{n+r}^*\omega$
restricted to ${\mathcal M}_g^{n+r}$
that have each component nonzero. So ${\mathcal M}_{g,r}^n$ is also quasi-projective.
Each finite quotient group $G$ of $\Gamma_g^n$ determines, in an obvious way,
a Galois cover ${\mathcal M}_g^n[G]\to {\mathcal M}_g^n$. The Deligne-Mumford completion
$\overline{\M}_g^n[G]$ of this cover is, by definition, the normalization of
$\overline{\M}_g^n$ in ${\mathcal M}_g^n[G]$.
\begin{theorem}[Looijenga \cite{looijenga:cover}]
There exists a finite group $G$ such that
$\overline{\M}_g[G]$ is smooth with a normal crossing divisor as
Deligne-Mumford boundary.
\end{theorem}
This has been extended by De Jong and Pikaart \cite{jong}
to arbitrary characteristic,
and by Boggi and Pikaart (independently) to the $n$-pointed case.
(They show that it also can be arranged that each irreducible component of
the Deligne-Mumford boundary of $\overline{\M}_g^n[G]$ is smooth.)
This makes it relatively easy to define the Chow algebra of
$\overline{\M}_g^n$: if $\overline{\M}_g^n[G]$ is smooth, then define
$\CH^{{\bullet}} (\overline{\M} _g^n)$ to be the $G$ invariant part of
$\CH^{{\bullet}} (\overline{\M} _g^n[G])$ (we take algebraic cycles modulo rational
equivalence
with coefficients in ${\mathbb Q}$). It is easy to see that this is
independent of the choice of $G$.
The central extension of $\Gamma _g$ by ${\mathbb Z}$ ($g\ge 3$) discussed in
Section~\ref{subsec:basic} takes the geometric form of a complex line
bundle over Teichm\"uller space with $\Gamma _g$ action and hence yields an
orbifold line bundle over ${\mathcal M}_g$.
Its twelfth tensor power has a concrete description: it is the determinant
bundle of the direct image of the relative dualizing sheaf of
${\mathcal M}_g^1\to{\mathcal M}_g$ (this is a rank $g$ vector bundle). The orbifold fundamental
group of the associated ${\mathbb C}^{\times}$ bundle is just the central extension
of $\Gamma_g$ by $\frac{1}{12}{\mathbb Z}$ mentioned in Section~\ref{subsec:basic}.
Since $\Gamma_g$ is perfect when $g\ge 3$, we have $H^1(\Gamma _g)=0$. Ivanov has
asked the following question:
\begin{question}[Ivanov]
Is it true that $H^1(\Gamma)$ vanishes for all finite index subgroups $\Gamma$
of $\Gamma_g$, at least when $g$ is sufficiently large?
\end{question}
This would imply that the Picard group of each finite unramified cover
of ${\mathcal M}_g$ (in the orbifold sense) is finitely generated. The answer
to Ivanov's questions is affirmative, for example, for subgroups of finite
index of $\Gamma _g$, $g\ge 3$, that contain the Torelli group --- see
(\ref{van_h1}).
\section{Algebro-Geometric Stability}
\label{sec:agstability}
The Deligne-Mumford completion $\overline{\M}_g^n$ comes with a natural
stratification into orbifolds, with each stratum parameterizing stable
$n$ pointed curves of a fixed topological type $T$. Denote this stratum
by ${\mathcal M} (T)$. It has codimension equal to the number of singular points of $T$.
The
normalization of the topological type $T$ is an oriented closed surface $S$
that comes with $n$ distinct numbered points $X=\{ x_1,\dots ,x_n\}$ and a
finite subset $Y$ of $S-X$ with a fixed point free involution $\iota$,
so that $T$ is recovered by identifying the points of $Y$ according to
$\iota$.
These topological data define a moduli space ${\mathcal M} (S)^{X\cup Y}$
of the same type (we hope that the notation is self-explanatory) and
there is a natural morphism ${\mathcal M} (S)^{X\cup Y}\to {\mathcal M}(S/\iota )^X$ that is a
Galois cover of orbifolds. This morphism extends to a finite surjective
morphism from the Deligne-Mumford completion $\overline{\M} (S)^{X\cup Y}$ to the
closure of ${\mathcal M} (T)$ in $\overline{\M}_g^n$. The resulting morphism
$\overline{\M} (S)^{X\cup Y}\to\overline{\M}_g^n$ has
only self-intersections of normal crossing type and so carries a normal
bundle in the orbifold sense. This normal bundle is a direct sum of line
bundles with one summand for each $\iota$ orbit $\{ p,p'\}$, namely
$p^*\omega^{-1}\otimes p'{}^*\omega^{-1}$. (To see this, notice that
the restriction of the universal curve to $\overline{\M} (T)$ has a quadratic
singularity along the locus defined by the pair $\{ p,p'\}$. Associating
to a local defining equation its hessian determines a natural isomorphism
between $p^*\omega^{-1}\otimes p'{}^*\omega^{-1}$ and the normal bundle
of a divisor in the Deligne-Mumford boundary passing through $\overline{\M}(T)$.)
We now see before us an algebro-geometric
incarnation of the map that appears in the stability theorem:
the set of normal vectors that point towards the interior ${\mathcal M}_g^n$
is the restriction to ${\mathcal M}(S)$ of the total space of the direct sum of
${\mathbb C}^{\times}$ bundles in this normal bundle. So ${\mathcal M} (S)_Y^X$ maps to the
latter space, and although we do not have a morphism ${\mathcal M} (S)_Y^X\to {\mathcal M}_g^n$,
the map on cohomology behaves as if there were. In particular, the map
$H^{{\bullet}}({\mathcal M}_g^n)\to H^{{\bullet}}({\mathcal M} (S)_Y^X)$ is a MHS morphism. So the stability
theorem implies:
\begin{theorem}
[Algebro-geometric stability]
Suppose that the finite set $X$ is contained in a connected component $S'$ of
$S$ of genus $g'$, so that ${\mathcal M} (S')^X_{Y\cap S'}$ appears as
a factor of ${\mathcal M}(S)_Y^X$. Choose points in the remaining factors
so that we have an inclusion
of ${\mathcal M} (S')^X_{Y\cap S'}$ in ${\mathcal M} (S)_Y^X$. Then for
$k\le cg'$ the composite map
$$
H^k({\mathcal M}_g^n)\to H^k({\mathcal M} (S)_Y^X)\to H^k({\mathcal M} (S')^X_{Y\cap S'})
$$
is an isomorphism and so is the map
$$
H^k({\mathcal M}_{g'}^n)\to H^k({\mathcal M} (S')^X_{Y\cap S'})
$$
induced by the forgetful morphism ${\mathcal M} (S')^X_{Y\cap S'}\to {\mathcal M} (S')^X\cong {\mathcal M}
_{g'}^n$.
These maps are also MHS morphisms.
\end{theorem}
So the stable rational cohomology $H^{{\bullet}}(\Gamma_{\infty}^n)$
comes with a natural MHS.
A geometric consequence of this result is that each stable rational
cohomology class of ${\mathcal M}_g^n$ (that is, a class whose degree is in
the stability
range) extends across the open part of the blow up of $\overline{\M} (T)$
parameterizing
the normal directions pointing towards the interior. Pikaart showed that
these partial extensions can be made to come from a single extension
to $\overline{\M}_g^n$, at least if $g$ is large compared with $k$.
But then it is not hard to show that if this is possible for large $g$,
then it is possible in the stable range and so the conclusion is:
\begin{theorem}[Pikaart \cite{pikaart}]\label{pikaart_purity}
The restriction map $H^k(\overline{\M}_g^n)\to H^k({\mathcal M}_g^n)$ is surjective
in the stable range. Consequently, the MHS on
$H^k(\Gamma_{\infty}^n)$ is pure of weight $k$.
\end{theorem}
Mumford's Conjecture, if known, would imply this result, and so Pikaart's
Theorem is evidence for the truth of this conjecture.
We illustrate this theorem with the known stable classes.
We have seen in the previous section that $\overline{\M}_g^n$ comes with $n$
orbifold line bundles $x_i^*\omega$, $i=1,\dots ,n$. Let $\overline{\tau}_{n,i}$ denote
the first Chern class of this line bundle, regarded as an element of
$\CH^1(\overline{\M}_g^n)$. The restriction of this class to
$\CH^1({\mathcal M}_g^n)$ is a pull-back of the restriction of $\overline{\tau} _{n-1,i}$ to
${\mathcal M}_g^{n-1}$ (when $n\ge 1$) and so we denote that restriction simply by
$\tau_i$. The underlying cohomology class of $\tau_i$ in $H^{2i}({\mathcal M}_g^n)\cong
H^{2i}(\Gamma_g^n)$ is what we denoted earlier by that symbol, in particular, it is stable.
For the definition of the tautological classes of ${\mathcal M} _g^n$, we shall not use
Mumford's original definition, but a modification proposed by Arbarello-Cornalba. This might begin with the observation that
the ``functor'' which associates to an $(n+1)$-pointed stable
genus $g$ curve $(C;x_1,\dots ,x_n,x)$ the cotangent space $T_x^*C$
defines an orbifold line bundle over $\overline{\M}_g^{n+1}$. It is not quite the
same as the relative dualizing sheaf $\omega$ of the forgetful map
$\overline{\M}_g^{n+1}\to \overline{\M}_g^n$: a little computation shows that it is
in fact $\omega (\sum_{i=1}^n (x_i))$. This is perhaps a more natural
bundle to consider than $\omega$. In any case, we denote the direct image of
$c_1(\omega (\sum_{i=1}^n (x_i))^{i+1}\in\CH ^{i+1}(\overline{\M}_g^{n+1})$ under the
projection $\overline{\M}_g^{n+1}\to \overline{\M}_g^n$ by $\overline{\kappa}_{n,i}\in \CH^i({\mathcal M}_g^n)$
and its restriction to ${\mathcal M}_g^n$ by $\kappa_{n,i}$. The cohomology class underlying $\kappa_{n,i}$ can be regarded as an element of
$H^{2i}(\Gamma^n_g)$ (of Hodge bidegree $(i,i)$). These cohomology classes
stabilize and, for $n=0$, they define the nonzero primitive elements of degree
$2i$ alluded to in \ref{subsec:stable}.
We regard (for $k=0,1,\dots ,n$) $\CH ^{{\bullet}}(\overline{\M} _g^n)$ as a
$\CH ^{{\bullet}}(\overline{\M}_g^k)$-algebra via the obvious forgetful morphism, and
view the classes $\overline{\kappa}_{k,i}$ as elements of $\CH ^{{\bullet}}(\overline{\M}_g^n)$
when appropriate. The class $\overline{\kappa}_{n,i}$ is then not equal to
$\overline{\kappa}_{n-1,i}$, but according to formula (1.10) of \cite{arb_cor}
we have:
$$
\overline{\kappa}_{n,i}=\overline{\kappa}_{n-1,i} +(\overline{\tau}_{n,n})^i.
$$
As Arbarello-Cornalba explain, the classes $\overline{\kappa}_{n,i}$ possess
a nice property not enjoyed by Mumford's classes. First recall that
every stratum of $\overline{\M}_g^n$ is the image of a finite map
$\overline{\M} (S)^{X\cup Y}\to\overline{\M}_g^n$ and that $\overline{\M} (S)^{X\cup Y}$ is a
product of varieties of the type $\overline{\M}_{g_\alpha}^{n_\alpha}$. The
pull-back of $\overline{\kappa}_{n,i}$ along this map is the
sum of the classes $\overline{\kappa}_{n_\alpha ,i}$ (pulled back along the
projection $\overline{\M} (S)^{X\cup Y} \to \overline{\M}_{g_\alpha }^{n_\alpha}$).
Carel Faber pointed out to us that a similar property is enjoyed by
the divisor class of the Deligne-Mumford boundary, but we know of no other
examples. Since this behaviour is reminiscent of that of a primitive
element in a Hopf algebra under the coproduct, we ask:
\begin{question}
What other collections $\left\{\mu_{g,n}\in \CH ^k({\mathcal M}_g^n)\right\}_{g,n}$
have this property?
\end{question}
\subsection{Correspondences between moduli
spaces}\label{subsec:correspondences}
There is an altogether different way to relate the cohomology of the
moduli spaces ${\mathcal M}_g^n$ for different values of $g$. This involves certain
Hecke type correspondences. For simplicity we shall restrict ourselves
to the undecorated case $n=0$. We return
to the reference surface $S_g$ and suppose that we are given a subgroup
$\pi$ of $\pi_g$ of finite index $d$, say.
(For what follows only its conjugacy class will matter.)
This subgroup determines an unramified
finite covering $\tilde S\to S$ of closed oriented surfaces. The genus
$\tilde g$ of $\tilde S$ is then equal to $d(g-1)+1$.
Consider the group of pairs $(\tilde h,h)\in \Diff^+(\tilde S)\times
\Diff^+(S)$ such that $\tilde h$ is a lift of $h$. Let $\Gamma_g(\pi )$ be
its group of connected components. The projection $\Gamma_g(\pi )\to \Gamma_g$
has as kernel the group of covering transformations of $\tilde S \to S$
(so is finite) and its image consists of the outer automorphisms of $\pi_g$
that come from an automorphism which preserves the subgroup $\pi$
(so is of finite index, $e$, say). There is a corresponding finite covering
of moduli spaces $p_1:{\mathcal M}_g(\pi)\to {\mathcal M}_g$, where ${\mathcal M}_g(\pi )$ is simply the
coarse moduli space of finite unramified coverings of nonsingular complex
projective curves $\tilde C\to C$ topologically equivalent to $\tilde S\to S$.
There is also a finite map $p_2:{\mathcal M}_g(\pi)\to {\mathcal M}_{\tilde g}$. Together they
define a one-to-finite correspondence $p_2p_1^{-1}$ from ${\mathcal M}_g$ to
${\mathcal M}_{\tilde g}$. This extends over
the Deligne-Mumford compactifications: if $p_1:\overline{\M}_g(\pi )\to \overline{\M}_g$
denotes the normalization of $\overline{\M}_g$ in ${\mathcal M}_g(\pi)$, then $p_2$ extends
to a finite morphism $p_2 :\overline{\M}_g(\pi)\to\overline{\M}_{\tilde g}$. We have
an induced map
$$
T_{\pi}:=e^{-1}p_{1*}p_2^*: \CH^{{\bullet}}(\overline{\M}_{\tilde g})\to
\CH^{{\bullet}}(\overline{\M}_g)
$$
and likewise on cohomology. A computation shows that any monomial in the
tautological classes is an ``eigen class'' for such correspondences:
\begin{proposition}
The map $T_{\pi}$ sends $\overline{\kappa}_{i_1}\overline{\kappa}_{i_2}\cdots
\overline{\kappa}_{i_r}$ to
$d^r\overline{\kappa}_{i_1}\overline{\kappa}_{i_2}\cdots \overline{\kappa}_{i_r}$.
\end{proposition}
This proposition suggests the consideration, for given positive integers
$r$ and $s$, of sequences of classes $(x_g\in\CH^s(\overline{\M}_g))_{g\ge 2}$ of
fixed degree that have the property that $T_{\pi}(x_g)=d^r x_{(d-1)g+1}$
for each index $d$ subgroup $\pi$ of $\pi_g$.
\begin{question}
Is for such a system the image of $x_g$ in $H^{{\bullet}}({\mathcal M}_g)$ stable?
Is it in fact a polynomial of degree $r$ in primitive stable classes?
\end{question}
An affirmative answer would give us a notion of stability for the Chow groups
of the moduli spaces ${\mathcal M}_g$.
\section{Chow Algebras and the Tautological Classes}
\label{sec:chow}
We have already encountered some of the basic classes on $\overline{\M}_g^n$:
the first Chern classes $\overline{\tau}_i\in \CH^1(\overline{\M}_g^n)$
($i=1,\dots ,n$) and the tautological classes $\overline{\kappa}_i$ ($i=1,2,\dots $).
More such classes come from the boundary: if $\prod _i\overline{\M} _{g_i}^{n_i}\to
\overline{\M}_g$ is a Galois covering of a stratum of the boundary as in
Section~\ref{sec:agstability},
then we can add to these the push-forwards along this map of the exterior
products of the corresponding
classes on the factors. Let us call the subalgebra of
$\CH^{{\bullet}}(\overline{\M}_g^n)$ generated by all these classes the
{\it tautological subalgebra} and denote it by ${\mathcal R}^{{\bullet}}(\overline{\M}_g^n)$.
The image of this algebra in $\CH^{{\bullet}}({\mathcal M}_g^n)$ is denoted by
${\mathcal R}^{{\bullet}}({\mathcal M}_g^n)$; it is generated by the classes
$\kappa_{n,i}$ ($i=1,2,\dots $) and $\tau _i$ ($i=1,\dots ,n$). It is possible
that these classes generate the rational Chow ring of $\overline{\M}_g^n$ modulo
homological equivalence, but this is of course unknown.
In any case, these subalgebras are preserved under pull-back and
push-forward along the natural maps that we have met so far.
The first computations
were done by Mumford \cite{mumford} who found a presentation of
$\CH^{{\bullet}}(\overline{\M}_2)$.
Subsequently Faber \cite{faber} calculated $\CH^{{\bullet}}(\overline{\M}_2^1)$,
$\CH^{{\bullet}}(\overline{\M}_3)$ and obtained partial results
on $\CH^{{\bullet}}(\overline{\M}_4)$. In all these cases the tautological algebra
is the whole Chow algebra.
This is also the case for $\overline{\M}_0^n$, whose Chow algebra was computed
by Keel. This is a very remarkable algebra which appears in other
contexts. Because of this, we describe it explicitly. We first introduce
notation for the divisor classes on $\overline{\M}_0^n$.
The boundary divisor $\overline{\M}_0^n - {\mathcal M}_0^n$ parameterizes
all singular stable $n$ pointed rational curves. Its components
correspond to the topological types of $n$ pointed stable rational
curves with exactly one singular point.
Such curves have exactly two irreducible components.
By collecting the points $x_i$ lying on the same
component, we obtain a partition $P$ of $\{1,\dots ,n\}$ into two subsets.
The stability property implies that both members of $P$ have at least
two elements. We denote the corresponding class in $\CH^1(\overline{\M}_0^n)$ by
$D(P)$.
\begin{theorem}[Keel \cite{keel}] The Chow algebra $\CH^{\bullet}(\overline{\M}_0^n)$
coincides with $H^{\bullet}(\overline{\M}_0^n)$ and, as a ${\mathbb Q}$ algebra, is generated
by the $D(P)$'s subject to the following relations:
\begin{enumerate}
\item[(i)] If $\{i,j,k\}$ are distinct integers in $\{1,\dots ,n\}$,
then the sum of the $D(P)$'s for which $P$ separates $i$ from
$\{ j,k\}$ is independent of $j$ and $k$ (and equals $\overline{\tau}_i$).
\item[(ii)] $D(P)\cdot D(P')=0$ if $P$ and $P'$ are independent in the sense
that the partition they generate has four nonempty members.
\end{enumerate}
\end{theorem}
The relations (ii) are geometrically obvious since the divisors
$D(P)$ and $D(P')$ do not meet if $P$ and $P'$ are independent.
The additive relations (i) are not difficult to see either:
if $C$ is a stable $n$ pointed rational curve, then a moment of thought
shows that there is a unique morphism $z:C\to {\mathbb P}^1$ that is an isomorphism on
one irreducible component, constant on the
other irreducible components, and is such that $z(x_i)=1$, $z(x_j)=0$ and
$z(x_k)=\infty$.
The differential $z^{-1}dz$ restricted to $x_i$ defines a section
of $x^*_i\omega $. The image of $z^{-1}dz$ in $T^*_{x_i}C$ vanishes
precisely when $z$ collapses the irreducible component containing $x_i$.
In \cite{manin:trees}, Manin derives a formula for the
Poincar\'e polynomial of $\overline{\M}_0^n$. Such a formula was independently
found by Getzler \cite{getzler} who also obtained the
$\mathcal{S}_n$ equivariant Poincar\'e polynomial of $H^{\bullet}(\overline{\M}_0^n)$.
That is, he determined the character
of the $\mathcal{S}_n$ representations $H^k(\overline{\M}_0^n)$,
$k\ge 0$. Kaufmann \cite{k_m:product} recently gave a formula for
the intersection number of classes of strata of complementary dimension.
We now turn to the Chow and cohomology algebras of the moduli spaces
${\mathcal M}_g$. First we list some results about the Chow algebras.
\begin{enumerate}
\item[]$\CH^{{\bullet}}({\mathcal M}_1^n)={\mathbb Q}$ for $n=1,2$ (folklore)\par
\item[]$\CH^{{\bullet}}({\mathcal M}_2 )={\mathbb Q}$ (folklore),\par
\item[]$\CH^{{\bullet}}({\mathcal M}_2^1)={\mathbb Q} [\tau]/(\tau^2)$ (Mumford \cite{mumford})\par
\item[]$\CH^{{\bullet}}({\mathcal M}_3 )={\mathbb Q} [\kappa_1]/(\kappa_1^2)$ (Faber
\cite{faber}),\par
\item[]$\CH^{{\bullet}}({\mathcal M}^1_3)={\mathbb Q} [\kappa_1,\tau ]/(\kappa_1^2,
4\tau ^2-\tau\kappa_1)$ (Faber\cite{faber}),\par
\item[]$\CH^{{\bullet}}({\mathcal M}_4 )={\mathbb Q} [\kappa_1]/(\kappa_1^3)$ (Faber
\cite{faber}),\par
\item[]$\CH^{{\bullet}}({\mathcal M}_5 )={\mathbb Q} [\kappa_1]/(\kappa_1^4)$ (Izadi \cite{izadi}
combined with Faber \cite{faber:hyp}).
\end{enumerate}
The reason that such computations can be made is that, when $g$ and
$n$ are both small, the moduli space ${\mathcal M}_g^n$ has a concrete
description. For example, when $g=2$, each curve is hyperelliptic
and therefore given by configuration of $6$ points
on the projective line. In the case $g=3$ a nonhyperelliptic curve is
realized by its canonical system as a quartic curve in ${\mathbb P}^2$. The
double cover of the projective plane along this curve is a Del Pezzo
surface of degree $2$, i.e., is obtained by blowing up $7$ points in
the plane in general position. General curves of genus 4 and 5 can be
described as complete intersections of multidegrees $(2,3)$ (in ${\mathbb P}^3$)
and $(2,2,2)$ (in ${\mathbb P}^4$), respectively.
\subsection{The tautological algebra of ${\mathcal M}_g$ and Faber's Conjecture}
\label{subsec:tautalg}
On the basis of numerous calculations, Faber, around 1993, made the
following conjecture.
\begin{conjecture}[Faber\cite{seminar}]
The tautological algebra ${\mathcal R}^{{\bullet}}({\mathcal M}_g)$ is a graded Frobenius algebra
with socle in degree $g-2$. That is, $\dim {\mathcal R} ^{g-2}({\mathcal M}_g)=1$, and the
intersection product defines a
nondegenerate bilinear form ${\mathcal R} ^i({\mathcal M}_g)\times {\mathcal R} ^{g-2-i}({\mathcal M}_g)\to
{\mathcal R} ^{g-2}({\mathcal M}_g)$ $(i=0,\dots ,g-2)$. Moreover, $\kappa_1$ has the
Lefschetz property in ${\mathcal R}^{{\bullet}}({\mathcal M}_g)$ in the
sense that multiplication by $(\kappa_1)^{g-2-2i}$ maps
${\mathcal R}^i({\mathcal M}_g)$ isomorphically onto ${\mathcal R}^{g-2-i}({\mathcal M}_g)$ for $0\le i\le
(g-2)/2$.
\end{conjecture}
Since the conjecture was made, evidence for it has been growing. For example:
\begin{theorem}[Looijenga \cite{looijenga:taut}] The algebra
${\mathcal R}^{{\bullet}}({\mathcal M}_g)$ is trivial in degree
$>g-2$ and ${\mathcal R}^{g-2}({\mathcal M}_g)$ is generated by the class of the
hyperelliptic locus (a closed irreducible variety of codimension
$g-2$).
\end{theorem}
In particular $\kappa_1^{g-1}=0$. Since $\kappa_1$ is ample on ${\mathcal M}_g$,
we recover a theorem of Diaz \cite{diaz} which asserts that every
complete subvariety of ${\mathcal M}_g$ must be of $\dim \le g-2$.
Actually, in \cite{looijenga:taut} a stronger result is proven, which, among
other
things, implies that
${\mathcal R}^k({\mathcal M}_g^n)=0$ for $k>g-2+n$. An induction argument then shows that
${\mathcal R}^{3g-3+n}(\overline{\M}_g^n)$ is spanned by the classes of the zero dimensional
strata. But zero dimensional strata can be connected by one dimensional strata
and the one dimensional strata are all rational. This shows that
${\mathcal R}^{3g-3+n}(\overline{\M}_g^n)\cong{\mathbb Q}$.
Faber recently proved that the tautological class $\kappa_{g-2}$ is nonzero.
To describe his result, we find it convenient to
introduce a compactly supported version
of the tautological algebra: let ${\mathcal R}^{{\bullet}}_c({\mathcal M}_g^n)$ be defined as the
set of elements in ${\mathcal R}^{{\bullet}}(\overline{\M}_g^n)$ that restrict trivially to the
Deligne-Mumford boundary. This is a graded ideal in
${\mathcal R}^{{\bullet}}(\overline{\M}_g^n)$ and the intersection product defines a map
$$
{\mathcal R} ^{{\bullet}}({\mathcal M}_g^n)\times{\mathcal R} ^{{\bullet}}_c({\mathcal M}_g^n)\to{\mathcal R} _c^{{\bullet}}({\mathcal M}_g^n)
$$
that makes ${\mathcal R} _c^{{\bullet}}({\mathcal M}_g^n)$ a ${\mathcal R} ^{{\bullet}}({\mathcal M}_g^n)$-module.
Notice that every complete subvariety of ${\mathcal M}_g$ of
codimension $d$ whose class is in ${\mathcal R} ^{{\bullet}}(\overline{\M}_g^n)$
defines a nonzero element of ${\mathcal R}^d_c({\mathcal M}_g^n)$ (but it is
by no means clear that such elements span ${\mathcal R}^{{\bullet}}_c({\mathcal M}_g^n)$).
A somewhat stronger form of the first part of Faber's Conjecture is:
\begin{conjecture}\label{strongfaber}
The intersection pairings
$$
{\mathcal R}^k({\mathcal M}_g)\times {\mathcal R}^{3g-3-k}_c({\mathcal M}_g)\to {\mathcal R}^{3g-3}_c({\mathcal M}_g)\cong{\mathbb Q} ,
\quad k=0,1,2,\dots
$$
are perfect (Poincar\'e duality) and ${\mathcal R} ^{{\bullet}}_c({\mathcal M}_g)$
is a free ${\mathcal R}^{{\bullet}}({\mathcal M}_g)$ module of rank one.
\end{conjecture}
Faber \cite{faber:hyp} finds a compactly supported class
$I_g$ in ${\mathcal R}_c^{2g-1}({\mathcal M}_g)$ with $\kappa_{g-2}\cdot I_g\neq 0$.
So ${\mathcal R}^{{\bullet}}_c({\mathcal M}_g)$ should be the ideal generated by this element.
Faber verified his conjecture for genera $\le 15$ by writing down many
relations in ${\mathcal R} ^{{\bullet}}({\mathcal M} _g)$ (this evidently gives an upper bound)
and using the nonvanishing of $\kappa_{g-2}$ (this gives a surprisingly strong
lower bound).
A refined form of Conjecture~\ref{strongfaber} (which we shall not state here) also takes care of the Lefschetz property.
\begin{question}
Does the tautological ring of $\overline{\M} _g^n$ satisfy Poincar\'e duality?
Does it have the Lefschetz property with respect to $\overline{\kappa}_{n,1}$?
(It is known that $\overline{\kappa}_{n,1}$ is ample \cite{cor}.)
\end{question}
\subsection{Cohomology of some moduli spaces}\label{subsec:somecohom}
As may be expected, even less is known about the cohomology algebras.
Here is an incomplete list of special results. In genus 0 we have that
the Chow algebra of $\overline{\M} _0^n$ maps isomorphically onto its rational
cohomology algebra. The cohomology of ${\mathcal M}_0^n$ is easily computed if we
start out from the observation that this space is the projective
arrangement of type $A_{n-2}$. It then follows for instance, that its
cohomology in degree $p$ is of type $(p,p)$. There are similar
descriptions of the moduli spaces of $n$-pointed hyperelliptic curves of genus
$g$ when $n=0,1,2$ that involve arrangements of type $A$ or $D$. These again
should enable us to determine their rational cohomology ring, but it seems that
this hasn't been done yet. In the same spirit arrangements of various types
(among them $E_6$ and $E_7$) were used in \cite{looijenga:mthree} to prove that
$$
H^{{\bullet}}({\mathcal M}_3 )=\CH^{{\bullet}}({\mathcal M}_3)+{\mathbb Q} u,
$$
where $u$ is a class of degree $6$ of Hodge bidegree $(6,6)$ and
$$
H^{{\bullet}}({\mathcal M}^1_3)=\CH^{{\bullet}}({\mathcal M}^1_3)+{\mathbb Q} u +{\mathbb Q} u\tau + {\mathbb Q} u\kappa _1+{\mathbb Q} v,
$$
where $v$ is a class of degree $7$ and of Hodge bidegree $(6,6)$.
\begin{question}
The image of the tautological algebra
in $H^{2p}({\mathcal M}_g^n)$ consists of classes of type $(p,p)$. Are all such classes
of this form?
\end{question}
A version of the Hodge conjecture asserts that the rational classes in degree
$2p$ of type $(p,p)$ are in the image of the Chow algebra, so modulo this conjecture we are asking whether
every Chow class on ${\mathcal M}_g^n$ is homologically equivalent to a tautological class.
\section{The Ribbon Graph Picture}
\label{sec:ribbon}
Around 1981 Thurston, Mumford and Harer observed that partial completions
of the Teichm\"uller spaces ${\mathcal X} _g^n$ with $n>0$ possess two natural
$\Gamma _g^n$ equivariant triangulations. One is based on the hyperbolic
geometry of $S_g^n$ (Thurston) and the other based on the singular
euclidean geometry of $S_g^n$ (Mumford, Harer).
The last approach was actually a direct, but very powerful application of
work that Jenkins and Strebel had done 10--20 years earlier.
It is this approach that we shall explain.
The basic notion is that of a {\it ribbon graph}. This is a finite
graph\footnote{For us a graph is a cell complex of pure dimension one;
its zero cells are called {\it vertices} and its one cells {\it edges}.
So it has no isolated vertices.} $G$ together with a cyclic order on the
set of oriented edges\footnote{An oriented edge of a graph
is an edge together with an orientation of it.} emanating from each vertex. As we
shall see, there is
a canonical construction of a surface that contains $G$ and of which
$G$ is a deformation retract. This construction should explain the
name. We first give a somewhat more abstract
characterization of ribbon graphs which is very useful in some applications.
Let $X(G)$ be the set of oriented edges of $G$. Let $\sigma _1$ be
the involution of $X(G)$ that reverses the orientation of each edge. The
set $X_1(G)$ of $\sigma _1$ orbits can be identified with the set of edges
of $G$.
The cyclic orderings define another permutation $\sigma _0$ of $X(G)$
as follows. Each oriented edge $e$ has an initial vertex $\initial (e)$ and a
terminal vertex $\term (e)$. Define $\sigma _0(e)$ to be the successor of
$e$ with respect to the given cyclic order on the set of oriented edges that
have $\initial (e)$ as their initial vertex. The set of orbits
$X_0(G)$ of $\sigma _0$ can be identified with the set of vertices of $G$.
Put $\sigma _{\infty}:= (\sigma _1\sigma _0)^{-1}=\sigma _0^{-1}\sigma _1$.
Call an orbit of this permutation a {\it boundary cycle}. (Draw a
picture to see why.) The set of boundary cycles wil be denoted
$X_{\infty}(G)$. These data form a complete invariant of $G$, for
we can reverse the construction
and associate to a finite nonempty set $X$ endowed with a fixed point free
involution $\sigma _1$ and a permutation $\sigma _0$ of $X$,
a ribbon graph $G(X,\sigma _0,\sigma _1)$ whose oriented edges are indexed
by $X$ and such that $\sigma_0$ and $\sigma_1$ are the permutations defined
above.
For every oriented edge $e$ of $G$ we form the one point compactification
$\Delta _e$ of the half strip $e\times [0,\infty )$; this is just a
$2$-simplex, parameterized in an unusual way. We make identifications
along the boundaries of these simplices with the help of $\sigma _0$
and $\sigma _{\infty}$:
$e\times \{0\}$ is identified with $\sigma _1e\times \{0\}$ and
and $\{\term (e)\}\times [0,\infty )$ with
$\{\initial (\sigma _{\infty}e)\}\times [0,\infty )$ (in either case, the
identification map is essentially the identity). This is easily seen to be
a compact, triangulated surface $S(G)$ that contains $G$ as a
subcomplex. Its vertex set can be identified with the disjoint union of
the vertex set of $G$ (so $X_0(G)$) and $X_{\infty}(G)$. We call vertices
of the latter type {\it cusps}. Notice that $G$ is a deformation retract
of $S(G)-X_{\infty}(G)$ and that the surface is canonically oriented
if we insist that the cyclic orderings of the edges emanating from each
vertex are induced by the orientation.
Let us say that the ribbon graph $G$ is {\it $n$-pointed} if we are given a injection $y:\{ 1,\dots ,n\}\hookrightarrow X_{\infty}(G)\cup X_0(G)$ whose image contains
$X_{\infty}(G)$ and the vertices of valency $\le 2$.
Suppose that we are given a {\it metric} $l$ on $G$. That is, a function
that assigns to every (unoriented) edge of $X$ a positive real number.
Give $[0,\infty )$ the standard metric and every half strip
$e\times [0,\infty )$ the product metric. This defines (at least locally) a
metric on $S(G)$. This metric is euclidean except possibly at the vertices.
However, it is not difficult to show that the underlying conformal structure
extends across all the vertices of $S(G)$ so that we end up with a compact
Riemann surface $C(G,l)$. Notice that each cusp has a ``circumference'' ---
this is the length of the associated boundary cycle. It is clear that we get
the same complex structure if $l$ is replaced by a positive multiple of it
and so we may just as well assume that the total length of $G$ is $1$. With
this convention, the sum of the circumferences of the cusps is $2$.
The work of Jenkins and Strebel shows that all compact Riemann
surfaces arise in this way:
\begin{theorem}[Strebel \cite{strebel}]
Let $(C;x:\{ 1,\dots,n\}\hookrightarrow C)$ be an $n$-pointed connected Riemann surface (so that the complement of the image of $x$ has negative Euler
characteristic as usual) and let $c_1,\dots,c_n$ be nonnegative real numbers, not all zero. Then there exists an $n$-pointed metrized ribbon graph $(G,y,l)$, with $y(i)$ a cusp of $G$ of circumference $c_i$ when $c_i>0$ and a vertex of $G$ otherwise, such that
$(C(G,l),y)$ and $(C,x)$ are isomorphic as $n$-pointed Riemann surfaces.
Moreover, $(G,y,l)$ is unique up to the obvious notion of isomorphism.
\end{theorem}
The results of Strebel also include a continuity property:
a continuous variation
of the complex structure on $C$ corresponds to a continuous variation of
$(G,y,l)$ in a sense that we make precise. Denote by ${\mathcal R}{\mathcal G} _g^n$ the set
of isomorphism classes of $n$-pointed ribbon graphs $(G,y)$ that are marked in the sense that we are given an isotopy class of homeomorphisms
$h: S_g\to S(G)$ with $h(x_i)=y(i)$, $i=1,\dots ,n$. On this set $\Gamma _g^n$ acts, and it is easy to see that the number of orbits of markings is finite.
Suppose that $(G,y,[h])$ represents an element of ${\mathcal R}{\mathcal G} _g^n$. Denote the
geometric realization of the
abstract simplex on the set $X_1(G)$ by $\Delta (G)$. Notice that the
metrics $l$ on $G$ that give $G$ unit length are parameterized by the
interior of $\Delta (G)$. The circumferences of the cusps add up to two, so
half the cicumferences are the barycentric coordinates of
a simplicial projection $\lambda :\Delta (G)\to\Delta ^{n-1}$.
Let $s$ be an edge of $G$ that is not a loop and does not connect two
vertices in the image of $y$. Then
collapsing that edge yields a member $(G/s ,y/s, [h]/s)$ of ${\mathcal R}{\mathcal G} _g^n$.
We can regard $\Delta (G/s)$ as a face of $\Delta (G)$. Making these
identifications produces a simplicial complex which we will denote by
$\widehat{{\mathbb X}} _g^n$. It comes with a simplicial map
$\lambda:\widehat{{\mathbb X}} _g^n\to \Delta ^{n-1}$. We have a simplicial action of $\Gamma _g^n$ on $\widehat{{\mathbb X}} _g^n$ which preserves the fibers of $\lambda$. The union of relative interiors of
simplices of $\widehat{{\mathbb X}} _g^n$ indexed by the elements of ${\mathcal R}{\mathcal G} _g^n$ is
an open subset ${\mathbb X} _g^n$ of $\widehat{{\mathbb X}} _g^n$. The results of Strebel can
be strengthened to:
\begin{proposition}[cf.\ \cite{looijenga:cell}]
The above construction defines a $\Gamma _g^n$ equivariant homeomorphism of
${\mathbb X} _g^n$ onto ${\mathcal X} _g^n\times\Delta ^{n-1}$.
\end{proposition}
Now consider the quotient space
$$
\widehat{{\mathbb M}} _g^n:=\Gamma _g^n\backslash\widehat{{\mathbb X}} _g^n.
$$
This is a finite simplicial orbicomplex that is equipped with a simplicial
map $\lambda :\widehat{{\mathbb M}} _g^n\to\Delta ^{n-1}$. We regard this complex as a
compactification of its open subset ${\mathbb M} _g^n:=
\Gamma _g^n\backslash{\mathbb X} _g^n$. According to the above theorem,
the latter is canonically homeomorphic with ${\mathcal M} _g^n\times\Delta ^{n-1}$.
This raises the question of how this compactification compares to that
of Deligne-Mumford. The answer is essentially due to Kontsevich:
\begin{theorem}[Kontsevich \cite{kontsevich:airy}, see also
\cite{looijenga:cell}]
The simplicial orbicomplex $\widehat{{\mathbb M}} _g^n$ is a
quotient space of $\overline{\M} _g^n\times\Delta ^{n-1}$. Moreover,
the part of $\widehat{{\mathbb M}} _g^n$ where $\lambda _i>0$ carries an
oriented piecewise linear circle bundle
(in the orbifold sense) whose pull-back to
$\overline{\M} _g^n\times \{\lambda\in\Delta ^{n-1} | \lambda _i>0\}$
is the oriented circle bundle coming from the standard line bundle
$\overline{\tau}_i$.
In particular, the part of $\widehat{{\mathbb M}} _g^n$ lying over the interior of $\Delta
^{n-1}$
carries the tautological cohomology classes underlying $\overline{\tau} _{n,i}$, $i=1,\dots ,n$.
\end{theorem}
The defining equivalence relation on
$\overline{\M} _g^n\times\Delta ^{n-1}$ is a little subtle and we refer to
\cite{looijenga:cell} for details regarding both statement and
proof.\footnote{The space used by Kontsevich is not quite
$\widehat{{\mathbb M}} _g^n$, but basically the part lying over
the interior of $\Delta ^{n-1}$ times a half line. In this case the
circumference map $\lambda$ has image $(0,\infty)^n$.}
This compactification of ${\mathcal M} _g^n\times\Delta ^{n-1}$ plays a crucial r\^ole
in Kontsevich's proof of the Witten conjectures. There are however
earlier applications. These include Harer's stability theorem we met before,
the computation of the Euler characteristic of ${\mathcal M} _g^n$, and the proof
that $\Gamma _g^n$ is a virtual duality group of dimension
$4g-4+n$. We shall not explain the relation with stability here,
but we will briefly touch on the other applications.
There is also a remarkable arithmetic aspect of ribbon graphs that
is presently under intense investigation, but which
we merely mention in passing. This
is the observation, made by Grothendieck in a research proposal
\cite{groth:esq}, that for a metrized ribbon graph $(G,l)$ all of whose
edges have equal length, the corresponding Riemann surface $C(G,l)$ is,
in a canonical way, a ramified covering of the Riemann sphere ${\mathbb P} ^1$ with
ramification
locus contained in $\{ 0,1,\infty\}$. The graph $G$ appears here as the
preimage of the interval $[0,1]$,
its vertex set as the preimage of $0$ and the set of cusps as the preimage
of $\infty$. The preimage of $1$ consists of the midpoints of the edges.
At these points we have simple ramification. A covering of this type is
naturally an algebraic curve defined over some number field.
Conversely, every connected covering of
the Riemann sphere of this type arises in this manner. The absolute Galois
group of ${\mathbb Q}$ acts on the collection of isomorphism types of such coverings,
and thus also on each finite set ${\mathcal R}{\mathcal G} _g^n$. It is very difficult to come
to grips with this action. For more information we refer to the collection
\cite{groth:dessins} and to Grothendieck's manuscripts \cite{groth:marche}
and \cite{groth:esq}.
Since these metrized ribbon graphs represent the barycenters of the simplices
of ${\mathbb M} _g^n$, one can also think of this as an action of the absolute Galois
group on the simplices of ${\mathbb M} _g^n$, but the significance of this is not
clear to us.
\subsection{Virtual duality and virtual Euler characteristic}
\label{subsec:virtual}
We first make some observations about simplicial complexes. Let $K$ be a
simplicial complex, $L$ a subcomplex. Set $U:=K-L$. Then $U$ admits a
canonical deformation retraction onto the union of the closed
simplices of the barycentric subdivision of $K$ that lie in $U$.
This is a subcomplex, called the {\it spine} of $U$, whose $k$-simplices
correspond to strictly increasing chains
$\sigma _0\subset\sigma _1\subset\cdots\subset\sigma _k$
of simplices of $K$ not in $L$.
Further, if $\Gamma$ is a group of automorphisms of $K$ that preserves $L$,
and if
\begin{enumerate}
\item[(i)] $U$ is contractible,
\item[(ii)] a subgroup of $\Gamma$ of finite index acts freely on $U$,
\end{enumerate}
then $\Gamma\backslash U$ is a simplicial `orbicomplex' that is also a virtual
classifying space for $\Gamma$.\footnote{This means there is a normal subgroup
$\Gamma_1\subset\Gamma$ of finite index such that a $\Gamma /\Gamma_1$-cover of this space classifies $\Gamma_1$.} It has $\Gamma\backslash \spine (U)$ as deformation
retract, and so the dimension of this spine is an upper
bound for the virtual homological dimension of $\Gamma$.
We apply this in the situation where $K$ is the preimage of the first vertex of
$\Delta ^{n-1}$ in $\widehat{{\mathbb X}} _g^n$ under $\lambda$ and $U=K\cap {\mathbb X} _g^n$. Notice
that $U\cong {\mathcal X} _g^n$. The simplices meeting $U$ are indexed by the elements of
${\mathcal R}{\mathcal G} _g^n$ with a single boundary cycle. A simple calculation shows that
when $g\ge 1$, the number of edges of such a graph is at most $6g-5+2n$ and at
least $2g-1+n$. For $g=0$ these numbers are $2n-5$, resp.\ $n-2$.
So the spine of $U$ has dimension $\le 4g-4+n$, resp. $n-3$. One can verify
that this is, in fact, an equality. It follows that $U$ admits a subcomplex of
this dimension as an equivariant deformation retract. Hence:
\begin{theorem}[Harer \cite{harer:virt}] If $n\ge 1$, then for every level
structure, the moduli space ${\mathcal M} _g^n[G]$ contains a subcomplex of dimension
$4g-4+n$ (when $g>0$) or $n-3$ (when $g=0$) as a deformation retract.
\end{theorem}
From this he deduces a similar result for the case when $n=0$: ${\mathcal M} _g[G]$
has the homotopy type of complex of dimension $4g-5$.
\begin{problem}
Is there a Lefschetz type of proof of this fact? For instance, the Lefschetz
property would follow if one can find an orbifold stratification of ${\mathcal M}_g^n$ with all strata affine subvarieties of codimension $\le g$ ($n\ge 1$)
or $\le g-1$ ($n=0$). That
would also show that the cohomological dimension of ${\mathcal M}_g^n$ for quasicoherent sheaves is $\le g-1$ ($n\ge 1$) or $\le g-2$ ($n=0$).
\end{problem}
Let us return to the general situation considered earlier and
suppose, in addition, that
\begin{enumerate}
\item[(iii)] $\Gamma\backslash K$ is a finite complex, and
\item[(iv)] $U$ is a simplicial manifold of dimension $d$, say.
\end{enumerate}
These conditions are satisfied in the case at hand.
It is then natural (and standard) to assign to each simplex of $K$
the weight that is the reciprocal of the order of its $\Gamma$ stabilizer.
This weighting is constant on orbits. Wall's Euler characteristic of $\Gamma$
is simply the usual alternating sum of the number of $\Gamma$ orbits of
simplices not in $L$, except that each is counted with its weight.
Equivalently, it is the orbifold Euler characteristic of the quotient
$\Gamma\backslash K$.
In the present case, a ribbon graph $G$ defining a member of ${\mathcal R}{\mathcal G} _g^n$
gives a contribution $|\Aut (G)|^{-1}(-1)^{|X_1(G)|}$ to the virtual Euler
characteristic. The computation of the resulting sum is a combinatorial
problem that was first solved by Harer and Zagier. Kontsevich
\cite{kontsevich:airy} later gave a shorter proof. The answer is:
\begin{theorem}[Harer-Zagier \cite{harer_zagier}] The orbifold Euler
characteristic of ${\mathcal M}_g^n$ equals
$$
(-1)^{n-1}\,\frac{(2g+n-3)!}{(2g-2)!}\zeta (1-2g).
$$
Here $\zeta$ denotes the Riemann zeta function.
\end{theorem}
Harer and Zagier also find formulae for the actual Euler characteristics of
${\mathcal M} _g^1$ and ${\mathcal M} _g$. These are often negative so that there must be lot
of cohomology in odd degrees.
For the discussion of virtual duality we go back to the general situation
and assume that beyond the four conditions already imposed we have:
\begin{enumerate}
\item[(v)] $L$ has the homotopy type of a bouquet of $r$-spheres.
\end{enumerate}
Then the theory of Bieri-Eckmann can be invoked in a virtual setting:
if we set $D:=\tilde H_r(L;{\mathbb Z} )$ and regard $D$ as a
$\Gamma$ module in an obvious way, then $H_{d-r-1}(\Gamma ;D)$ is of rank one
and for any $\Gamma$-module $V$ with rational coefficients
the cap products
$$
\cap :H^k(\Gamma ;V )\otimes H_{d-r-1}(\Gamma ;D )\to
H_{d-r-1-k}(\Gamma ;V\otimes D ),\quad k=0,1,2,\dots
$$
are isomorphisms. One calls $D$ the {\it Steinberg module} of $\Gamma$.
Harer \cite{harer:virt} proves that in the present case hypothesis (v) is
satisfied: $L$ is a subcomplex of dimension $2g-3-n$, resp.\ $n-4$ which is
$(2g-4-n)$-connected, resp.\ $(n-5)$-connected when $g>0$, resp.\ $g=0$.
We shall call the corresponding orbifold local system ${\mathbb D}$ over ${\mathcal M}_g^n$
the {\it Steinberg sheaf}. The homology group $H_{4g-4+n}({\mathcal M}_g^n ;{\mathbb D} )$
is of rank one. For every orbifold local system ${\mathbb V}$ of rational
vector spaces on ${\mathcal M} _g^n$, cap product with a generator of this homology group
defines isomorphisms
$$
H^k({\mathcal M}_g^n ;{\mathbb V} )\stackrel{\sim}{\to}
H_{4g-4+n-k}({\mathcal M} _g^n;{\mathbb V}\otimes{\mathbb D} ),\quad k=0,1,2,\dots
$$
when $g>0$ and $n>0$ (and similar isomorphisms in the remaining cases).
In particular, taking ${\mathbb V}$ to be ${\mathbb Q}$, we see that $H_{\bullet}({\mathcal M}_g^n;{\mathbb D})$
has a canonical MHS. This suggests that ${\mathbb D}$ has some Hodge theoretic
significance. Unfortunately it is not of finite rank, yet we wonder:
\begin{question}
Is the Steinberg sheaf motivic? In particular, does it have natural
completions that carry (compatible) Hodge and \'etale structures?
\end{question}
\subsection{Intersection numbers on the Deligne-Mumford completion}
\label{subsec:intersection}
The intersection numbers in question are those defined by monomials in
the $\overline{\tau} _i$'s. To be precise, define for
every such monomial $\overline{\tau} _1^{d_1}\dots \overline{\tau} _n^{d_n}$
(with all $d_i \ge 0$) the intersection number
$\int _{\overline{\M}_g^n} \overline{\tau} _1^{d_1}\dots \overline{\tau} _n^{d_n}$
where $g$ is chosen in such a way that this has a possibility of being
nonzero:
$3g-3+n=d_1+\cdots +d_n$. A physics interpretation suggests that we
should combine these numbers into the generating function
\begin{equation*}
\sum _{n=1}^{\infty} \frac{1}{n!}\sum _{g>1-{\frac{1}{2}} n}\,
\sum_{d_1 + \cdots +d_n = 3g-3+n} t_{d_1}\dots t_{d_n}
\int _{\overline{\M}_g^n} \overline{\tau} _1^{d_1}\cdots \overline{\tau} _n^{d_n}.
\end{equation*}
Now pass to a new set of variables $T_1,T_3,T_5,\dots$ by
setting
$$
t_i=1.3.5.\cdots(2i+1)T_{2i+1}.
$$
The resulting expansion $F(T_1,T_3,T_5,\dots )$ encodes all these
intersection numbers.
Witten \cite{witten}
conjectured two other characterizations of this function, both of which allow
computation of its coefficients. These were proved by Kontsevich in his
celebrated paper \cite{kontsevich:airy}. Perhaps the most useful
characterization is the one which says that $F$ is killed by a Lie algebra
of differential operators isomorphic to the Lie algebra of polynomial vector
fields in one variable. This Lie algebra comes with a basis $(L_k)_{k\ge -1}$
corresponding to the vector fields $(z^k\partial /\partial z)_{k\ge -1}$ and
Witten verified the identities $L_k(F)=0$ for $k=-1,0$ within the realm of
algebraic geometry. However no such proof is known for $k\ge 1$. Kontsevich's
strategy is to represent the classes
$\overline{\tau} _1^{d_1}\cdots \overline{\tau} _n^{d_n}$ by piecewise differential forms
on the ribbon graph model that can actually be integrated.
This allows him to convert the intersection numbers into weighted sums over
ribbon graphs. This leads to a new characterization of the generating function
that is more manageable. Still a great deal of ingenuity is needed to
complete the proof of Witten's Conjecture.
\section{Torelli Groups and Moduli}
\label{sec:torelli}
In the early 80s, Dennis Johnson published a series of
pioneering papers \cite{johnson:fg,johnson:ker,johnson:h1} on the
Torelli groups. Although this work is in geometric topology, it has
several interesting applications to algebraic geometry. Here
we review some of his work.
First a remark on notation.
In the remainder of the paper we will write
$V_g$ for the symplectic vector space $H_1(S_g)$ and
$Sp_g({\mathbb Z})$ for the
group $\Aut (H_1(S_g,{\mathbb Z}),\langle \phantom{x},\phantom{x} \rangle)$;
this does not really clash with standard notation, since a choice
of a symplectic basis of $H_1(S_g;{\mathbb Z})$ identifies this with the standard
integral symplectic group of genus $g$. Likewise, $Sp_g$ will stand for
the algebraic ${\mathbb Q}$-group defined by the symplectic transformations of
$V_g$; so its group of ${\mathbb Q}$-points, $Sp_g({\mathbb Q} )$, is just
the group of symplectic automorphisms of $V_g$.
The mapping class group $\Gamma_{g,r}^n$ acts on the homology of
the reference surface $S_g$. Since each of its elements preserves
the orientation of $S_g$, we have a homomorphism
\begin{equation}\label{homom}
\Gamma_{g,r}^n \to Sp_g({\mathbb Z}).
\end{equation}
which is surjective. The {\it Torelli group} $T_{g,r}^n$ is defined to be
its kernel\footnote{Note that there is no general agreement on the
definition of $T_{g,r}^n$ when $r + n > 1$.} so that we have an extension
$$
1 \to T_{g,r}^n \to \Gamma_{g,r}^n \to Sp_g({\mathbb Z}) \to 1.
$$
The homology groups of $T_{g,r}^n$ are therefore $Sp_g({\mathbb Z})$ modules.
The simplest kind of element of $T_{g,r}^n$ is a Dehn
twist along a simple loop in $S_g^{n+r}$ that separates
$S$ into two connected components. We call such a loop a {\it separating
simple loop}. Another type of element of $T_{g,r}^n$ is determined by a
{\it separating pair of simple loops}. This is a pair of two disjoint
nonisotopic loops $\alpha _1,\alpha _2$ on $S_g^{n+r}$
that together separate $S$ into two connected components. The Dehn
twist along
$\alpha _1$ composed with the inverse of the Dehn twist along $\alpha _2$
is in $T_{g,r}^n$. The first of Johnson's results is:
\begin{theorem}[Johnson \cite{johnson:fg,johnson:ker,johnson:h1}]
\label{johnson}
When $g\ge 3$, $T_{g,r}^n$ is generated by elements
associated to a finite number of separating simple loops and
a finite number of separating pairs of simple loops.
If $[S_g]\in \wedge ^2 H_1(S_g ;{\mathbb Z})$ corresponds to the fundamental
class of $S_g$, then there are natural $Sp_g({\mathbb Z})$ equivariant surjective
homomorphisms
$$
\tau_g^1 : T_g^1 \to \wedge ^3 H_1(S_g;{\mathbb Z} ) \text{ and }
\tau_g : T_g \to \wedge^3 H_1(S_g;{\mathbb Z} )/([S_g]\wedge H_1(S_g;{\mathbb Z} )).
$$
In both cases, the kernel of $\tau$ is the subgroup generated by the
elements associated to simple separating loops. Finally, the kernels
of the induced homomorphisms
$$
H_1(T_g^1;{\mathbb Z} ) \to\wedge ^3 H_1(S_g ;{\mathbb Z})\text{ and }
H_1(T_g^1;{\mathbb Z} ) \to\wedge ^3 H_1(S_g ;{\mathbb Z} )/([S_g]\wedge H_1(S_g ;{\mathbb Z} ))
$$
are both 2-torsion.
\end{theorem}
Johnson also finds an explicit description of this 2-torsion. We will
give it in a moment, but first we want to point out an algebro-geometric
consequence of this theorem. Let $\widetilde{{\mathcal M}}_g\subset\overline{\M} _g$ be the
complement of the irreducible divisor whose
generic point parametrizes irreducible singular stable curves, and let
$\widetilde{{\mathcal M}} _g^1$ be its preimage in $\overline{\M} _g^1$.
\begin{corollary}\label{cor:pioftildem}
When $g\ge 3$, the orbifold fundamental group of $\widetilde{{\mathcal M}}_g$
(resp.\ $\widetilde{{\mathcal M}}_g^1$) is isomorphic to an extension of $Sp_g({\mathbb Z} )$
by $\wedge ^3H_1(S_g;{\mathbb Z})/([S_g]\wedge H_1(S_g;{\mathbb Z} ))$ (resp.\
$\wedge ^3H_1(S_g;{\mathbb Z} )$).
\end{corollary}
Johnson's theorem shows that the $Sp_g({\mathbb Z})$ action on $H_1(T_{g,r}^n)$
is the restriction of a representation of the algebraic
group $Sp_g$. We shall see shortly the importance of this property.
Let $\lambda_1, \lambda_2,\dots ,\lambda_g$ be a
fundamental set of weights of $Sp_g$ so that $\lambda_j$ corresponds to
the $j$th fundamental representation of $Sp_g$. This last representation
can be realized as the natural $Sp_g$ action on the primitive part of
$\wedge^j V_g$.
The next result follows from Johnson's Theorem by standard arguments.
\begin{corollary}
For each $g\ge 3$, there is a natural $Sp_g({\mathbb Z})$ equivariant
isomorphism
$$
\tau_{g,r}^n : H^1(T_{g,r}^n) \stackrel{\sim}{\to}
V(\lambda_3) \oplus V(\lambda_1)^{\oplus(r+n)}.
$$
\end{corollary}
A theorem of Ragunathan \cite{ragunathan} implies that when $g\ge 2$,
the first cohomology of each finite index subgroup of $Sp_g({\mathbb Z})$ with
coefficients in a rational representation of $Sp_g({\mathbb Q})$ vanishes.
So Johnson's computation also gives:
\begin{corollary}\label{van_h1}
If $g\ge 3$, then every finite index subgroup of $\Gamma_{g,r}^n$ that
contains $T_{g,r}^n$ has zero first Betti number.
\end{corollary}
The situation is very different when $g < 3$. The Torelli groups $T_1$
and $T_1^1$ are trivial, while Geoff Mess \cite{mess} proved that when
$g=2$, $T_2$ is a countably generated free group. He also computed
$H_1(T_2;{\mathbb Z})$. It is the $Sp_2({\mathbb Z})$ module obtained by inducing the
trivial representation up to $Sp_2({\mathbb Z})$ from the stabilizer
$({\mathbb Z}/2)\ltimes (SL_2({\mathbb Z})\times SL_2({\mathbb Z}))$ of a decomposition of
$H_1(S_2;{\mathbb Z})$ into two symplectic modules each of rank 2. (We shall
sketch a proof in the next subsection.) It is still unknown whether,
for any $g\ge 3$, $T_g$ is finitely presented.
\begin{problem}
Determine whether $T_g$ is finitely presented when $g$ is sufficiently large.
\end{problem}
Next, we describe Johnson's computation of the torsion in $H_1(T_g;{\mathbb Z})$.
Denote the field of two elements by ${\mathbb F}_2$. Recall that an ${\mathbb F}_2$
quadratic form on $H_1(S_g;{\mathbb F}_2)$ associated to the mod two symplectic form
$\langle\phantom{x} ,\phantom{x}\rangle$ on $H_1(S_g;{\mathbb F}_2)$
is a function $\omega : H_1(S_g;{\mathbb F}_2) \to {\mathbb F}_2$ satisfying
$$
\omega(a+b) = \omega(a) + \omega(b) + \langle a,b\rangle .
$$
The difference between any two such is an element of $H^1(S_g;{\mathbb F}_2)$.
This makes the set $\Omega_g$ of such quadratic forms an affine space
over the ${\mathbb F} _2$ vector space $H^1(S_g;{\mathbb F}_2)$. Denote the algebra of
${\mathbb F} _2$ valued functions on $\Omega_g$ by $S\,\Omega_g$. All such functions
are given by polynomials and so we have a filtration
$$
{\mathbb F}_2 = S_0\Omega_g \subset S_1\Omega_g \subset S_2\Omega_g \subset
\dots \subset S\,\Omega_g,
$$
where $S_d\Omega_g$ denotes the space of polynomial functions of degree
$\le d$.
Since $f=f^2$ for each $f\in S\,\Omega_g$, the associated graded algebra
is naturally isomorphic to the exterior algebra $\wedge^{\bullet} H_1(S_g;{\mathbb F}_2)$.
The algebra $S\,\Omega_g$ has as a distinguished element which is called
the {\it Arf invariant}, denoted here by ${\rm arf}$. If
$a_1,\dots, a_g, b_1,\dots,b_g$
is a symplectic basis of $H^1(S;{\mathbb F}_2)$, then ${\rm arf}$ is defined by
$$
{\rm arf} : \omega \mapsto \sum_i \omega(a_i)\omega(b_i).
$$
It is an element of $S_2\Omega_g$, and its zero set $\Psi _g$ is an
affine quadric in $\Omega_g$. Let $S_d\Psi _g$ denote the image of
$S_d\Omega_g$ in the set of ${\mathbb F}_2$ valued functions on $\Psi_g$.
\begin{theorem}[Johnson \cite{johnson:h1}]
There are natural isomorphisms
\begin{gather*}
\sigma_{g,1} : H_1(T_{g,1};{\mathbb F}_2) \stackrel{\sim}{\to} S_3 \Omega_g,\quad
\sigma_g^1 : H_1(T_g^1;{\mathbb F}_2)\stackrel{\sim}{\to}
S_3\Omega_g/{\mathbb F}_2\,{\rm arf} ,\\
\sigma_g : H_1(T_g;{\mathbb F}_2)\stackrel{\sim}{\to} S_3\Psi_g
\end{gather*}
which are equivariant with respect to the $Sp_g({\mathbb F}_2)$-action.
These induce natural isomorphisms
$$
H_1(T_{g,1};{\mathbb Z})_{\rm tor}\cong S_2\Omega_g,\quad
H_1(T_g^1;{\mathbb Z})_{\rm tor} \cong H_1(T_g;{\mathbb Z})_{\rm tor} \cong S_2\Psi_g.
$$
Moreover, the natural isomorphisms
$$
\phi_{g,r}^n : H_1(T_{g,r}^n;{\mathbb F}_2)/H_1(T_{g,r}^n;{\mathbb Z})_{\rm tor}
\stackrel{\sim}{\to}
\left[H_1(T_{g,r}^n;{\mathbb Z})/\text{\rm torsion}\right]\otimes{\mathbb F}_2
$$
correspond, under the isomorphisms $\sigma_{g,r}^n$ and $\tau_{g,r}^n$,
to the obvious isomorphisms
\begin{gather*}
\phi_{g,1} : S_3\Omega_g/S_2\Omega_g\stackrel{\sim}{\to}
\wedge^3H_1(S_g;{\mathbb F}_2),\quad
\phi_g^1 : S_3\Omega_g/({\mathbb F}_2\,{\rm arf} + S_2\Omega_g) \stackrel{\sim}{\to}
\wedge^3H_1(S_g;{\mathbb F}_2),\\
\phi_g : S_3\Psi_g/S_2\Psi_g \stackrel{\sim}{\to}
\wedge^3H_1(S_g;{\mathbb F}_2)/([S_g]\wedge H_1(S_g;{\mathbb F}_2)).
\end{gather*}
\end{theorem}
The homomorphisms $\tau _g^1$ and $\tau _g$ admit direct conceptual
definitions that we will give later. Here we give a formula for the
image of the standard generators of $T_{g,1}$ in $\wedge^3H_1(S_g;{\mathbb Z})$
and in $S_3\Omega _g$ under $\tau_{g,1}$ and $\sigma_{g,1}$, respectively.
Let $(\alpha _1,\alpha _2)$ be a separating pair of simple loops.
Let $t$ be the corresponding element of $T_{g,1}$ --- recall that this is
the product of the Dehn twist about $\alpha_1$ and the {\em inverse} of
the Dehn twist about
$\alpha_2$. The two loops decompose $S_g$ into two pieces $S'$ and $S''$, say,
where we suppose that $S'$ contains the point $x_1$.
We orient $\alpha _1$ and $\alpha _2$ as boundary components of $S''$.
The resulting cycles are opposite in $H_1(S'';{\mathbb Z})$: $[\alpha_2]=-[\alpha_1]$,
and each spans the radical of the intersection pairing on
this group. So there is a well-defined element in
$\wedge ^2H_1(S'';{\mathbb Z})/[\alpha_1]\wedge H_1(S'';{\mathbb Z})$ representing the
intersection pairing on $H_1(S'';{\mathbb Z})$. Its wedge with $[\alpha_1]$ can be
regarded as an element of $\wedge ^3H_1(S'';{\mathbb Z})$. Since the inclusion
$S''\subset S_g$ induces an injection on first homology, we can also
view the latter as an element of $\wedge ^3H_1(S_g;{\mathbb Z})$. This is the
element $\tau _{g,1}(t)$; it is clear that it only depends on the
image of $t$ in $T_g^1$.
Next we associate to $t$ a function
$\sigma_t:\Omega_g\to {\mathbb F}_2$ as follows. If $\omega\in\Omega_g$ takes the
value 1 on $[\alpha]$, then we put $\sigma_t(\omega )=0$; if it takes
the value 0 on $[\alpha]$, then the restriction of $\omega$ to
$H_1(S'';{\mathbb F}_2)$ factors through a nondegenerate quadratic function
on $H_1(S'';{\mathbb F}_2)/{\mathbb F}_2[\alpha]$. Then $\sigma_{g,1}(t)(\omega)$ is its
Arf invariant. It can be shown that $\sigma_{g,1}(t)$ lies in $S_3\Omega$.
Now suppose that $t$ is the element of $T_{g,1}$ associated to a
separating simple loop $\alpha$. Denote the pieces $S'$ and $S''$ as before.
In this case, $\tau_{g,1}(t)$ is trivial and $\sigma_{g,1}(t)$ is the element
of $S\Omega_g$ that assigns to $\omega$ the Arf invariant of its restriction to
$H_1(S'';{\mathbb F}_2)$. Notice that if $\alpha$ is a simple loop around $x_1$, then
$\sigma_{g,1}(t)$ is just the function ${\rm arf}$. (This explains why we mod out by
this function when passing from $T_{g,1}$ to $T_g^1$.)
Without a base point there is no way of telling $S'$ and $S''$ apart. It is
because of this ambiguity that we have to restrict functions to $\Psi_g$ in
order to obtain a well defined function.
A diffeomorphism of $S_g$ onto a smooth projective curve $C$ determines
a natural isomorphism between $\Omega_g$ and the space of
theta characteristics of $C$ (i.e., square roots of the canonical bundle
$K_C$; see for instance Appendix B of \cite{acgh}). This suggests that
Johnson's computation should have
an algebro-geometric interpretation, if not interesting applications
to the geometry of curves.
\begin{problem}
Give an algebro-geometric construction of the epimorphism $T_g\to S_3\Omega$.
\end{problem}
Van Geemen has suggested such a construction (unpublished).
\subsection{Torelli space and period space}\label{subsec:torelli}
The group $T_g$ acts freely on ${\mathcal X}_g$. The quotient ${\mathcal T}_g$ is
therefore a complex manifold. It is called {\it Torelli space}. According
to the discussion at the beginning of Section~\ref{sec:moduli}, ${\mathcal T}_g$
is then a classifying space for $T_g$ so that there is a canonical isomorphism
$H_{\bullet}(T_g;{\mathbb Z} ) \cong H_{\bullet}({\mathcal T}_g;{\mathbb Z} )$.
Torelli space has a moduli
interpretation; it is the moduli space of smooth projective curves
$C$ of genus $g$ together with a symplectic isomorphism
$$
{\mathbf \gamma} : H_1(S_g;{\mathbb Z}) \to H_1(C;{\mathbb Z}).
$$
There are also decorated versions ${\mathcal T}_{g,r}^n$ of Torelli space. Their
points are points of ${\mathcal M}_{g,r}^n$ together with a symplectic
isomorphism ${\mathbf \gamma}$ of $H_1(S;{\mathbb Z})$ with the first homology of the curve
corresponding to the point of ${\mathcal M}_g$. It is clear that
the map ${\mathcal T}_{g,r}^n \to {\mathcal M}_{g,r}^n$ is Galois with Galois group
$Sp_g({\mathbb Z})$.
Denote the Siegel space associated to $V_g$ by ${\mathfrak h}_g$. To be precise,
${\mathfrak h}_g$ is the set of pure Hodge structures on $V_g$
with Hodge numbers $(-1,0)$ and $(0,-1)$, polarized by the intersection form.
This is a contractible complex manifold of dimension $g(g+1)/2$ on
which the group $Sp_g({\mathbb R})$ acts properly and transitively.
We can also regard ${\mathfrak h}_g$ as the moduli space of pairs consisting of a
$g$ dimensional principally polarized abelian
variety $A$ plus a symplectic isomorphism
$$
{\mathbf \gamma} : H_1(S;{\mathbb Z}) \to H_1(A;{\mathbb Z}).
$$
This interprets the $Sp_g({\mathbb Z})$ orbit space of ${\mathfrak h}_g$ as the moduli space
of principally polarized abelian varieties of dimension $g$, ${\mathcal A} _g$. We
regard ${\mathcal A} _g$ as an orbifold with orbifold fundamental group $Sp_g({\mathbb Z})$,
although $Sp_g({\mathbb Z})$ does not act faithfully on ${\mathfrak h}_g$. The kernel of this
action is $\{\pm 1\}$.
Assigning to a smooth projective curve the Hodge structure on
its first homology group defines
a map ${\mathcal T}_g \to {\mathfrak h}_g$, the {\it period map} for $T_g$. It is an isomorphism
in genus 1, an open imbedding when $g=2$, and 2:1 with ramification along
the hyperelliptic locus when $g\ge 3$.%
\footnote{It is stated incorrectly in \cite{hain:normal} that ${\mathcal T}_2 \to
{\mathfrak h}_2$ is an unramified 2:1 map onto its image.}
The reason for this is that
for all abelian varieties we have the equality
$$
[A;{\mathbf \gamma}] = [A;-{\mathbf \gamma}]
$$
of points of ${\mathfrak h}_g$ as $-\id$ is an automorphism of each abelian variety.
On the other hand, we have the equality
$$
[C;{\mathbf \gamma}] = [C;-{\mathbf \gamma}]
$$
of points of ${\mathcal T}_g$ if and only if $C$ is hyperelliptic.
Mess's result (mentioned at the beginning of the section)
can now be deduced from this: ${\mathcal T}_2$ is the complement
in ${\mathfrak h}_2$ of the locus of principally polarized abelian varieties
that are products of two elliptic curves. The locus of such reducible
abelian varieties is a countable disjoint union of copies of
${\mathfrak h}_1 \times {\mathfrak h}_1$. The group $Sp_2({\mathbb Z})$ permutes them transitively,
and each is stabilized by a product of two copies of $SL_2({\mathbb Z})$ and
an involution that switches the two copies of the upper half plane.
Mess's result follows easily using the stratified Morse theory of
Goresky and MacPherson --- use distance from a generic point of
${\mathfrak h}_2$ as the Morse function. Since each component of ${\mathfrak h}_2 -{\mathcal T}_2$ is
a totally geodesic divisor, the distance function has a unique critical
point (necessarily a minimum) on each stratum. It follows that ${\mathcal T}_2$
has the homotopy type of a wedge of circles, one for each component of
${\mathfrak h}_2 -{\mathcal T}_2$.
The period map gives, after passage to $Sp _g({\mathbb Z} )$ orbit spaces, a morphism
${\mathcal M}_g\to{\mathcal A}_g$, the period mapping for ${\mathcal M} _g$. This period mapping extends
to the partial completion
$\widetilde{{\mathcal M}}_g$ of ${\mathcal M}_g$ and the resulting map
$\widetilde{{\mathcal M}}_g\to {\mathcal A}_g$ is proper.
Now assume $g\ge 3$ and
denote the image of the period map ${\mathcal T}_g \to {\mathfrak h}_g$ by ${\mathcal S}_g$. This
space is the quotient of ${\mathcal T} _g$ by the subgroup $\{\pm 1\}$ of $Sp_g({\mathbb Z})$.
Consequently
$$
H^{\bullet}({\mathcal S}_g) \cong H^{\bullet}(T_g)^{\{\pm 1\}}.
$$
Observe that ${\mathcal S}_g$ is a locally closed analytic subvariety of
${\mathfrak h}_g$, but not closed. The $\{\pm 1\}$ cover ${\mathcal T}_g\to{\mathcal S}_g$ extends as
a $\{\pm 1\}$ cover $\overline{{\mathcal T}}_g\to\overline{{\mathcal S}}_g$ over the closure
of ${\mathcal S}_g$ in ${\mathfrak h} _g$, and the $\{\pm 1\}$ action on the total space
is the restriction of an $Sp_g({\mathbb Z})$ action. Both $\overline{{\mathcal T}}_g$ and
$\overline{{\mathcal S}}_g$ are rather singular along the added locus
(which is of codimension $3$). If we pass to $Sp_g({\mathbb Z})$ orbit spaces, then
the natural map
$$
\widetilde{{\mathcal M}}_g\to Sp_g({\mathbb Z})\backslash \overline{{\mathcal T}}_g\cong
Sp_g({\mathbb Z})\backslash \overline{{\mathcal S}}_g
$$
resolves these singularities in an orbifold sense. A resolution of a normal analytic variety always induces a surjection on fundamental groups and so it follows from (\ref{johnson}) that the fundamental group of $\overline{{\mathcal T}}_g$ is abelian and is $Sp_g({\mathbb Z})$ equivariantly
a quotient of $\wedge ^3 H_1(S_g ;{\mathbb Z} )/([S_g]\wedge H_1(S_g ;{\mathbb Z} ))$.
\begin{problem}
Understand the topology of ${\mathcal S}_g$ and its closure $\overline{{\mathcal S}}_g$ in
${\mathfrak h}_g$. In particular, how close is ${\mathcal S}_g$ to being a finite complex? (Observe
that if it has a finite 2-skeleton, then $T_g$ is finitely presented.)
\end{problem}
Related, but formally independent of this problem, is the question of
whether the cohomology of $T_g$ stabilizes in a suitable sense:
\begin{question}
Is $H^k(T_g)$ expressible as an $Sp_g({\mathbb Z} )$ module in a manner that is
independent of $g$ if $g$ is large enough? For example, from Johnson's
Theorem, we know that $H^1(T_g)$ is the third fundamental representation
of $Sp_g$ for all $g\ge 3$.
\end{question}
\subsection{The Johnson homomorphism}\label{subsec:johnson}
The proof of Johnson's Theorem is non-trivial and uses geometric
topology, but the homomorphism $\tau_g^1$ is easily described.
Since $T_g$ is torsion free, the projection ${\mathcal T} _g^1\to {\mathcal T}_g$ defines the
universal curve over ${\mathcal T}_g$. Denote the corresponding bundle of jacobians by
${\mathcal J}_g \to {\mathcal T}_g$. Since the local system of first homology groups
associated to the universal curve is canonically framed, this jacobian
bundle ${\mathcal J}_g \to {\mathcal T}_g$ is analytically trivial as a bundle of Lie groups: we
have a natural trivializing projection $p:{\mathcal J} _g \to \Jac S_g$, where
$\Jac S_g:= H_1(S_g; {\mathbb R} /{\mathbb Z} )$ is the ``jacobian'' of the reference surface.
The usual Abel-Jacobi map, which assigns to an ordered pair of points $(x,y)$
on
a smooth curve $C$ the divisor class of $(x)-(y)$, induces a morphism
$$
{\mathcal T}_g^1\times _{{\mathcal T} _g} {\mathcal T}_g^1\to {\mathcal J} _g.
$$
over ${\mathcal T} _g$. This provides a correspondence
$$
\begin{CD}
{\mathcal T}_g^1\times _{{\mathcal T} _g} {\mathcal T}_g^1 @>>> {\mathcal J} _g @>p>> \Jac S_g \cr
@VV{pr_2}V \cr
{\mathcal T}_g^1 \cr
\end{CD}
$$
from ${\mathcal T}_g^1$ to $\Jac S_g $. It induces homomorphisms
$$
H_k(T_g^1) \cong H_k({\mathcal T}_g^1) \to H_{k+2}(\Jac S_g ).
$$
The first of these is the Johnson homomorphism
$$
\tau_g^1 : H_1(T_g^1) \to H_3(\Jac S_g )
$$
for $T_g^1$.
Since ${\mathcal T} _g^1\to {\mathcal T}_g$ is a fibration of Eilenberg-MacLane spaces, we have
an exact sequence of fundamental groups:
$$
1 \to \pi_g \to T_g^1 \to T_g \to 1.
$$
This induces an exact sequence
$$
H_1(S_g;{\mathbb Z} ) \to H_1(T_g^1;{\mathbb Z} ) \to H_1(T_g;{\mathbb Z} ) \to 0
$$
on homology. Since $\Jac S_g $ is a topological group with torsion free
homology, its integral homology has a product --- the {\it Pontrjagin
product}. It is not difficult to check that the composite
$$
H_1(S_g;{\mathbb Z} ) \to H_1(T_g^1;{\mathbb Z} ) \to H_3(\Jac S_g;{\mathbb Z} )
$$
is the map given by Pontrjagin product with the class $[S_g]$.
It follows that there is a natural homomorphism
$$
H_1(T_g;{\mathbb Z} ) \to H_3(\Jac S_g;{\mathbb Z} )/\left([S_g ]\times H_1(S_g;{\mathbb Z} )\right).
$$
This is the Johnson homomorphism $\tau_g$ for $T_g$.
Johnson's Theorem, alone and in concert with Saito's theory of
Hodge modules, has several interesting applications to the
geometry of moduli spaces of curves as we shall see in subsequent
sections.
\subsection{Monodromy of roots of the canonical bundle}
In this subsection we assume that $g\ge 2$. Suppose that $C$ is a smooth
projective curve of genus $g$. Since its canonical bundle $K_C$ is of degree
$2g-2$ and since $\Pic^0 C$ is a divisible group, $K_C$ has $n$th roots
whenever $n$ divides $2g-2$. Any two such $n$th roots will differ by an
$n$ torsion point of $\Pic^0 C$. Because of this, $n$th roots of $K_C$
are rigid under deformation. It follows that they form a locally constant
sheaf (in the orbifold sense) $\Rt^n$ over ${\mathcal M}_g$. The fiber over
$C$, denoted $\Rt^n C$, is a principal homogenous space over
$H_1(C;{\mathbb Z}/n)$, the group of $n$ torsion points of $\Pic^0 C$.
Choose a conformal structure on $S_g$. Denote the corresponding
algebraic curve by $C$.
Sipe \cite{sipe} determined the monodromy representation
$$
\rho^n : \Gamma_g \to \Aut \Rt^n(C).
$$
of this sheaf. Before giving it, we make some remarks. Since the Torelli
group acts trivially on the $n$ torsion of $\Pic^0 C$, it follows that
the restriction of $\rho^n$ to $T_g$ factors through a representation
$T_g \to H_1(S;{\mathbb Z}/(2g-2)) \to H_1(S;{\mathbb Z}/n)$. However, the action of $\Gamma_g$ on
the set $\Rt^2 C$ of square roots of $K_C$ (the set of theta characteristics
of $C$) factorizes through $Sp_g({\mathbb Z})$ also (even through $Sp_g({\mathbb F}_2)$) --- this
is because there is a canonical correspondence between square roots of $K_C$
and ${\mathbb F}_2$ quadratic forms on $H_1(S;{\mathbb F}_2)$ associated to the intersection form.
It follows that the image of the monodromy representation $\rho^n$ will
be contained in an extension of $Sp_g({\mathbb Z}/n)$ by a subgroup of
$2\cdot H^1(S_g;{\mathbb Z}/n)$. In fact, it is all of this group.
\begin{theorem}[Sipe \cite{sipe}]\label{sipe}
The monodromy group of $\Rt^n$ is an extension of
$Sp_g({\mathbb Z}/n)$ by the subgroup $2\cdot H^1(S_g;{\mathbb Z}/n)$ of $H_1(S_g;{\mathbb Z}/n)$.
\end{theorem}
The subgroup $2\cdot H^1(S_g;{\mathbb Z}/n)$ appears as a quotient of the Torelli group
$T_g$. In \cite{hain:normal} it is shown that the restriction of the monodromy
representation to $T_g$ is the composite of the Johnson homomorphism with a natural surjection
$$
\wedge^3H_1(S_g;{\mathbb Z})/([S_g]\wedge H_1(S_g;{\mathbb Z})\to H_1(S_g;{\mathbb Z}/(g-1)) \to 2\cdot H_1(S_g;{\mathbb Z}/n).
$$
\subsection{Picard groups of level covers}\label{subsec:picard}
Denote the moduli space of smooth projective genus $g$
curves with a level $l$ structure by ${\mathcal M}_g[l]$. This is convenient
shorthand for the notation ${\mathcal M}_g[Sp_g({\mathbb Z}/l)]$ introduced in
Section~\ref{sec:moduli}. Denote the kernel of the reduction mod $l$ map
$$
Sp_g({\mathbb Z}) \to Sp_g({\mathbb Z}/l)
$$
by $Sp_g({\mathbb Z})[l]$, and its full inverse image in $\Gamma_g$ by
$\Gamma_g[l]$. Then ${\mathcal M}_g[l]$ is the quotient of Teichm\"uller
space ${\mathcal X}_g$ by $\Gamma_g[l]$. As in the case of ${\mathcal M}_g$, there is
a canonical isomorphism
$$
H^{\bullet}({\mathcal M}_g[l]) \cong H^{\bullet}(\Gamma_g[l]).
$$
This holds with rational coefficients for all $l$, and arbitrary
coefficients whenever $\Gamma_g[l]$ is torsion free, which holds
whenever $Sp_g({\mathbb Z})[l]$ is torsion free --- $l \ge 3$.
We know from (\ref{van_h1}) that
$$
H^1(\Gamma_g[l]) \cong H^1({\mathcal M}_g[l]) = 0
$$
when $g\ge 3$. By standard arguments (cf.\ \cite[\S 5]{hain:normal}),
this implies that
$$
c_1 : \Pic {\mathcal M}_g[l]\otimes{\mathbb Q} \to H^2({\mathcal M}_g[l])
$$
is injective, and therefore that $\Pic {\mathcal M}_g[l]$ is finitely
generated when $g\ge 3$.
The stable cohomology of an arithmetic group depends only on
the ambient real algebraic group \cite{borel:triv}. Based on
this, one might expect that the natural map
$$
H^k(\Gamma_g) \to H^k(\Gamma_g[l])
$$
is an isomorphism for all $l\ge 0$, once the genus $g$ is sufficiently
large compared to the degree $k$. It follows from Johnson's work
that this is true when $k=1$ (cf.\ \cite{hain:normal}), but the only
evidence for it when $k>1$ is Harer's computation of the second homology
of the spin mapping class groups \cite{harer:spin}, and Foisy's theorem
from which Harer's computation now follows:
\begin{theorem}[Foisy \cite{foisy}]
For all $g\ge 3$, the natural map $H^2(\Gamma_g) \to H^2(\Gamma_g[2])$ is
an isomorphism. Consequently, $\Pic {\mathcal M}_g[2]$ is finitely generated
of rank 1.
\end{theorem}
\begin{question}
Is $\Pic {\mathcal M}_g[l]$ rank 1 for all $g\ge 3$ and all $l\ge 1$?
\end{question}
This would be the case if we knew that the $Sp_g({\mathbb Z})$ action
on $H^2(T_g)$ extended to an algebraic action of $Sp_g$,
for we could then invoke Borel's computation of the
stable cohomology of arithmetic groups \cite{borel:triv}.
\subsection{Normal Functions}
\label{subsec:normal} Each rational representation $V$ of
$Sp_g$ gives rise to an orbifold local system ${\mathbb V}$ over ${\mathcal M}_g[l]$. Such
a local system underlies an admissible variation of Hodge structure. First,
if $V$ is irreducible, then $V$ underlies a variation of Hodge
structure unique up to Tate twist (\cite[(9.1)]{hain:normal}).
Every polarized ${\mathbb Q}$ variation of Hodge structure whose monodromy
representation comes from a rational representation of $Sp_g$
has the property that each of its
isotypical components is an admissible variation of Hodge structure
of the form $A_\lambda\otimes {\mathbb V}(\lambda)$, where $A_\lambda$ is a
Hodge structure and ${\mathbb V}(\lambda)$ is a variation of Hodge structure
corresponding to the $Sp_g$ module with highest weight $\lambda$ ---
cf.\ \cite[(9.2)]{hain:normal}.
For a Hodge structure $V$ of weight $-1$ one
defines
the corresponding {\it intermediate jacobian} $JV$ by
$$
JV = V_{\mathbb C}/(F^0 V + V_{\mathbb Z}).
$$
Its interest comes from the fact that it parametrizes the extensions
of ${\mathbb Z}$ by $V$ in the MHS category:
if $E$ is an extension of the ${\mathbb Z}$ (with its trivial Hodge structure of
weight zero) by $V$, then choose an integral lift $e\in E$
of $1$ and consider the image of $e$ in
$$
E_{\mathbb C}/(F^0 E + V_{\mathbb Z})\cong V_{\mathbb C}/(F^0 V + V_{\mathbb Z}).
$$
This is independent of the lift and yields a complete invariant of the
extension. There is an inverse construction that makes $JV$ support
a variation of mixed Hodge structure $\E$ that is universal as an
extension of the trivial Hodge structure ${\mathbb Z}$ by the constant
Hodge structure $V$:
$$
0 \to {\mathbb V} _{JV}\to \E \to {\mathbb Z} _{JV}\to 0
$$
(see \cite{carlson}). This immediately generalizes to a relative setting:
if ${\mathbb V}$ is an admissible variation of ${\mathbb Z}$ Hodge structure
of weight $-1$ over a smooth variety $X$, then we have a
corresponding bundle $\pi :{\mathcal J}{\mathbb V}\to X$ of intermediate jacobians over $X$
supporting a universal extension
$$
0 \to \pi ^*{\mathbb V} \to \E \to \pi ^*{\mathbb Z} _X\to 0.
$$
A section $\sigma$ of ${\mathcal J}{\mathbb V}$ over $X$ determines an extension of
Hodge structures:
$$
0 \to {\mathbb V} \to \sigma ^*\E \to {\mathbb Z} _X\to 0.
$$
A {\it normal function} is a section of ${\mathcal J}{\mathbb V}$ such that the corresponding
extension $\E$ is an admissible variation of mixed Hodge structure. The normal
functions arising from algebraic cycles are normal functions in this
sense --- cf.\ \cite[\S6]{hain:normal}.
We briefly recall Griffiths' construction of a normal function
associated to a family of homologically trivial algebraic
cycles. First we consider the case where the base is a point.
Suppose that $X$ is a smooth projective variety. A homologically trivial
algebraic $d$-cycle $Z$ in $X$ canonically determines an extension of
${\mathbb Z}$ by $H_{2d+1}(X;{\mathbb Z}(-d))$ by pulling back the exact sequence
$$
0 \to H_{2d+1}(X;{\mathbb Z}(-d)) \to H_{2d+1}(X,|Z|;{\mathbb Z}(-d)) \to H_{2d}(|Z|;{\mathbb Z}(-d))
\to \cdots
$$
of MHSs along the inclusion
$$
{\mathbb Z} \to H_{2d}(|Z|;{\mathbb Z}(-d))
$$
that takes 1 to the class of $Z$. So an integral lift of $1$ is
given by an integral singular $2d+1$ chain $W$ in $X$ whose boundary is $Z$.
Integration identifies $JH_{2d+1}(X;{\mathbb Z} (-d))$ with the {\it Griffiths
intermediate
Jacobian}
$$
J_d(X):=\Hom_{\mathbb C} (F^dH^{2d+1}(X);{\mathbb C} (-d))/H^{2d+1}(X;{\mathbb Z} (-d)),
$$
and under this isomorphism
the extension class in question is just given by integration over $W$.
Families of homologically trivial cycles give rise to normal functions:
Suppose that ${\mathcal X} \to T$ is a family of smooth projective varieties
over a smooth base $T$ and that ${\mathcal Z}$ is an algebraic cycle in
${\mathcal X}$ which is proper over $T$ of relative dimension $d$. Then the local
system whose fiber over $t\in T$ is $H_{2d+1}(X_t;{\mathbb Z}(-d))$ naturally
underlies a variation of Hodge structure ${\mathbb V}$ over $T$ of weight $-1$
so that we can form the $d$th {\it relative intermediate jacobian}
${\mathcal J}_d ({\mathcal X} /T)\to T$,
whose fiber over $t\in T$ is $J_d(X_t)$. The family of
cycles ${\mathcal Z}$ defines a section of this bundle which is a normal function.
\begin{theorem}[Hain \cite{hain:normal}]\label{norm_classn}
Suppose that ${\mathbb V}$ is an admissible variation of Hodge structure
of weight $-1$ over ${\mathcal M}_g[l]$ whose monodromy representation factors
through a rational representation of $Sp_g$. If $g \ge 3$,
then the space of normal functions associated to ${\mathbb V}$ is finitely
generated of rank equal to the number of copies of the variation
${\mathbb V}(\lambda_3)$ of weight $-1$ that occur in ${\mathbb V}$.
\end{theorem}
The theorem implies that, up to torsion and multiples, there is only
one normal function over ${\mathcal M}_g$ associated to a variation of Hodge
structure whose monodromy factors through a rational representation of
$Sp_g$. So what is the generator of these normal functions?
To answer this question, recall that if $C$ is a smooth projective curve of
genus $g$ and
$x\in C$, we have the Abel-Jacobi morphism
$$
C \to \Jac C, \quad y\mapsto (y)-(x).
$$
Denote the image 1-cycle in $\Jac C$ by $C_x$
and the cycle $i_\ast C_x$ by $C_x^-$, where $i : \Jac C \to \Jac C$
takes $u$ to $-u$. The cycle $C_x - C_x^-$ is homologous to zero, and
therefore defines a point $\nu ^1(C,x)$ in $J_1(\Jac C)$.
Pontrjagin product with the class of $C$ induces a homomorphism
$$
A:\Jac C \to J_1(\Jac C).
$$
We call the cokernel of $A$ the {\it primitive first intermediate Jacobian}
$J_1^{{\rm pr}}(\Jac C)$ of $\Jac C$. The family of such primitive intermediate
jacobians over ${\mathcal M}_g$ is the unique one (up to isogeny) associated to
the variation of Hodge structure of weight $-1$ over ${\mathcal M}_g$ whose associated
$\Gamma_g$ module is $V(\lambda_3)$. It is not difficult to show that
$$
\nu ^1(C,x) - \nu ^1(C,y) = 2A(x-y).
$$
It follows that the image of $\nu ^1(C,x)$ in $J_1^{{\rm pr}}(\Jac C)$ is
independent of $x$. This is the value of the normal function associated with
$C-C^-$ over $[C]$.
We can do better and realize half of this generator by a generalized
normal function as follows.
Let $A$ be a principally polarized abelian variety of dimension $g\ge 3$.
The polarization determines a distinguished element $\omega$ of $H_2(A;{\mathbb Z})$.
If $Z$ and $Z'$ are two piecewise smooth cycles representing
$\omega$, then their difference is the
boundary of a piecewise smooth $3$-chain $W$ on $A$.
Represent the dual of $H_3(A;{\mathbb R})$ by translation invariant $3$-forms on $A$.
Then integrating these forms over $A$ determines an element
of $H_3(A;{\mathbb R} )$. Another choice of $W$ gives a class
that differs from this one by an element of $H_3(A;{\mathbb Z} )$, and so we have
a well-defined element $[Z-Z']$ of $H_3(A;{\mathbb R}/{\mathbb Z} )$. Notice that the latter torus
is naturally identified with the first intermediate jacobian $J_1(A)$ of $A$.
We declare $Z$ and $Z'$ to be equivalent if $[Z-Z']=0$ and denote the space of
piecewise smooth cycles representing $\omega$ modulo this equivalence relation
by $D(A)$. This is clearly a torsor of $J_1(A)$ and so it has a natural
complex structure. In view of its connection with Deligne cohomology, we call
it the {\it Deligne torsor} of $A$.
This torsor contains naturally a subtorsor $D(A)[2]$ of the $2$-torsion in $J_1(A)$,
$J_1(A)[2]\cong H_3(A;\frac{1}{2}{\mathbb Z}/{\mathbb Z} )$: Let $a=(a_1,a_{-1},\dots ,a_g,a_{-g})$ be a symplectic basis of $H_1(A;{\mathbb Z} )$. Each basis element $a_i$ is uniquely represented by a homomorphism $\alpha_i:S^1\to A$ and so $\omega$ is represented by the $2$-cycle $\sum_{i=1}^g \alpha_i\times\alpha_{-i}$. This cycle defines an element $z(a)\in D(A)$. It is easily verified that $z(a)$ only depends on the mod two reduction of $a$ and that if $a$ runs over all symplectic bases, $z(a)$ runs over an entire orbit $D(A)[2]$ of $J_1(A)[2]$. (So $J_1(A)[2]\backslash D(A)$ has a canonical point which identifies it with $J_1(A)$.) The group $Sp(H_1(A;{\mathbb Z}))$ acts on $D(A)$ as an affine transformation group in a way that is easily made explicit. The lifts of these
transformations to a universal covering of $D(A)$ form a group of affine symplectic transformations. It is an extension of $Sp(H_1(A;{\mathbb Z}))$ by $H_3(A;{\mathbb Z} )$ which splits if we enlarge the extension to $H_3(A;\frac{1}{2}{\mathbb Z} )$.
The Pontrjagin product with $\omega$ defines a homomorphism $A\to J_1(A)$ which gives rise to corresponding primitive notions:
the {\it primitive Deligne torsor} $D^{{\rm pr}}(A):=A\backslash D(A)$ is a torsor
of the primitive intermediate Jacobian $J^{\rm pr} _1(A):=A\backslash J_1(A)$.
We have corresponding universal Deligne torsors over ${\mathcal A}_g$ which we denote
${\mathcal D} _g\to{\mathcal A}_g$ and ${\mathcal D} _g^{{\rm pr}}\to{\mathcal A}_g$. By the above argument, these torsors become trivial on the Galois cover of ${\mathcal A}_g$ representing principally polarized abelian varieties with a level $2$ structure. The torsors themselves are nontrivial, for it can be shown that the orbifold fundamental groups of these torsors are nonsplit extensions of the integral symplectic group of genus $g$
For $C$ a nonsingular projective curve of genus $g\ge 3$ and $x\in C$,
the Abel-Jacobi morphism $C \to \Jac C$ defined by $y\mapsto (y)-(x)$
defines a cycle
in the homology class of the natural polarization of $\Jac C$ and
so we get an element $[(C,x)]$ of $D(\Jac C)$. Its image in
$D^{{\rm pr}}(\Jac C)$ is independent of $x$ and so can be denoted by $[C]$.
Universally this produces holomorphic lifts of the period map:
$$
\nu_g^1: {\mathcal M}_g^1\to {\mathcal D}_g\text{ and } \nu_g: {\mathcal M}_g\to {\mathcal D}_g^{{\rm pr}}.
$$
We call $\nu_g$ the {\it fundamental normal function} on ${\mathcal M}_g$.
\subsection{Picard group of the generic curve with a level $l$
structure} \label{subsec:genericpicard}
The classification of normal functions (\ref{norm_classn}) implies that there
are no sections of $\Pic^0$ of infinite order defined over ${\mathcal M}_g[l]$ when $g\ge 3$.
This, combined with Sipe's computation (\ref{sipe}) of the monodromy of
roots of the canonical bundle allows one
to determine the Picard group of the generic point of ${\mathcal M}_g[l]$.
The case $l=1$ was the subject of the Franchetta Conjecture which
was deduced from Harer's computation of $\Gamma_g$ by Beauville
(unpublished) and by Arbarello and Cornalba \cite{arb_cor:pic}.
\begin{theorem}[Hain \cite{hain:derham}]
The Picard group of the generic curve of genus $g\ge 3$ with a level
$l$ structure is of rank 1, has torsion subgroup isomorphic to the
$l$ torsion points $H_1(\Jac S_g;{\mathbb Z}/l)$, and, modulo torsion, is
generated by the canonical bundle if $l$ is odd, and a theta
characteristic if $l$ is even.
\end{theorem}
\section{Relative Malcev Completion}
\label{sec:malcev}
Fundamental groups of smooth algebraic varieties are quite special
as we know from the work of Morgan \cite{morgan} and others.
The least trivial restrictions on these groups come from Hodge theory
and Galois theory. Since $\Gamma_g$ is the (orbifold) fundamental group
of ${\mathcal M}_g$, a smooth orbifold, Hodge theory and Galois theory should
have something interesting to say about its structure. To put a MHS on a group
one needs to linearize it. One way to do this is to replace the
group by some kind of algebraic envelope and put a MHS on the
coordinate ring of this (pro)algebraic group.
In this section we introduce these linearizations and use them
to establish a relation between the fundamental normal function and a
remarkable central extension that is hidden in a
quotient of the mapping class group. Here the impact of mixed Hodge theory
is not yet felt, but we are setting the stage for
Section~\ref{sec:hodgemap} where it is omnipresent.
\subsection{Classical Malcev completion}
\label{subsec:malcev} Suppose that $\pi$ is a
finitely generated group. The classical Malcev (or unipotent) completion of
$\pi$
consists of a prounipotent group ${\mathcal U}(\pi)$ (over ${\mathbb Q}$) and a homomorphism
$\pi \to {\mathcal U}(\pi)$. It is characterized by the following universal mapping
property: if $U$ is a unipotent group, and $\phi : \pi \to U$ is a
homomorphism, there is a unique homomorphism of prounipotent groups
${\mathcal U}(\pi) \to U$ through which $\phi$ factors. There are several well known
constructions of the unipotent completion, which can be found in
\cite{hain:comp}, for example. Each (pro)unipotent group $U$ is isomorphic
to its Lie algebra ${\mathfrak u}$, a (pro)nilpotent Lie algebra via the exponential
map. Thus, to give the Malcev group ${\mathcal U}(\pi)$ associated to $\pi$ it suffices
to give its associated pronilpotent Lie algebra ${\mathfrak u}(\pi)$. This Lie algebra is
called the {\it Malcev Lie algebra associated to $\pi$}. It comes with a
natural descending filtration whose $k$th term ${\mathfrak u} ^{(k)}(\pi)$
is the closed ideal of ${\mathfrak u} (\pi )$ generated by its $k$-fold commutators
$(k=1,2,\dots )$ and it is complete with respect to this filtration.
We will refer to this filtration as the {\it Malcev filtration}.
When $\pi$ is the fundamental group $\pi_1(X,x)$ of a smooth complex
algebraic variety, ${\mathfrak u}(\pi)$ has a
canonical MHS which was first constructed by Morgan \cite{morgan}.
If $X$ is also complete, or more generally, when $H_1(X)$ has a pure Hodge
structure of weight $-1$, then the weight filtration is the Malcev
filtration:
$$
W_{-k}{\mathfrak u} (\pi _1(X,x))={\mathfrak u} ^{(k)}(\pi _1(X,x)).
$$
Alternatively, this MHS determines and is determined by a MHS
on the coordinate ring ${\mathcal O}({\mathcal U}(\pi))$ of the associated Malcev group.
We shall denote the Malcev completion of $\pi_g^n=\pi_1(S_g^n,x_0)$ by
${\mathfrak p}_g^n$.
\subsection{Relative Malcev completion}
\label{subsec:relmalcev} The Malcev completion
of a group $\pi$ is trivial when $H_1(\pi)$ vanishes, for
then $\pi$ has no non-trivial unipotent quotients. Since the
first homology of $\Gamma_g$ vanishes for all $g$, its Malcev
completion will be trivial. Deligne has defined the notion
of Malcev completion of a group $\pi$ relative to a Zariski dense
homomorphism $\rho:\pi \to S$, where $S$ is a reductive algebraic
group defined over a base field $F$ (that we assume to be of
characteristic zero).
The {\it Malcev completion of $\pi$ relative to $\rho : \pi \to S$}
is a a proalgebraic $F$-group ${\mathcal G}(\pi,\rho)$, which is an extension
$$
1 \to {\mathcal U} \to {\mathcal G}(\pi,\rho) \to S \to 1
$$
of $S$ by a prounipotent group, together with a lift
$\tilde{\rho} : \pi \to {\mathcal G}(\pi,\rho)$ of $\rho$.\footnote{In many cases
the completion of $\pi$ over an algebraic closure $\bar F$ of $F$ is the
set of $\bar F$ points of the completion of $\pi$ over $F$. This is the
case for the mapping class groups when $g\ge 3$, but we do not know
whether this is true in general, except when $S$ is trivial.} It is
characterized by the following universal mapping property: if
$G$ is an $F$-group which is an extension of $S$ by a unipotent
group $U$, and if $\phi : \pi \to G$ is a homomorphism, then there
is a unique homomorphism ${\mathcal G}(\pi,\rho) \to G$ through which $\phi$ factors:
$$
\phi : \pi \stackrel{\tilde{\rho}}{\to} {\mathcal G}(\pi,\rho) \to G.
$$
Since $S$ is reductive, we should think of ${\mathcal U}$ as the prounipotent radical
of ${\mathcal G} (\pi ,\rho )$. One can show, for instance, that ${\mathcal U}$ has a Levi
supplement so that ${\mathcal G} (\pi ,\rho )$ is a semidirect product of $S$ and
${\mathcal U}$. The Lie algebra ${\mathfrak g} (\pi ,\rho )$ of ${\mathcal G} (\pi ,\rho )$ also comes
with a Malcev filtration with respect to which it is complete:
${\mathfrak g} (\pi ,\rho )^{(0)}={\mathfrak g} (\pi ,\rho )$, and for $k\ge 1$,
${\mathfrak g} (\pi ,\rho )^{(k)}$ is the closed ideal generated by $k$-fold
commutators in the Lie algebra of ${\mathcal U}$.
We will often write ${\mathcal G}(\pi)$ instead of ${\mathcal G}(\pi,\rho)$ when the
representation
$\rho$ is clear from the context. We shall denote the completion of
the (orbifold) fundamental group of a pointed orbifold $(X,x)$ with
respect to a Zariski dense reductive representation $\rho:\pi_1(X,x) \to S$
by ${\mathcal G}(X,x;\rho)$, or simply ${\mathcal G}(X,x)$ when $\rho$ is clear from the
context.
When $S$ is trivial, we recover the classical Malcev completion.
The universal property of the Malcev completion of $\ker \rho$ yields a
natural homomorphism of proalgebraic $F$-groups
${\mathcal U} (\ker \rho)(F)\to {\mathcal G} (\pi)$. In general, it is neither
surjective nor injective as the following two examples show.
\begin{example}
The fundamental group of the
symplectic Lie group $Sp_g({\mathbb R})$ is infinite cyclic and hence so is its
universal cover $\widehat{Sp}_g({\mathbb R})\to Sp_g({\mathbb R} )$. This universal cover
is not an algebraic group (which follows for instance from the fact that
the complexification of $Sp_g({\mathbb R} )$, $Sp_g({\mathbb C} )$, is simply connected).
The preimage $\widehat{Sp}_g({\mathbb Z})$ of $Sp_g({\mathbb Z})$ in this covering contains
the universal central extension of $Sp_g({\mathbb Z})$ by ${\mathbb Z}$. Now take for $\pi$
this central extension and for $\rho$ its natural homomorphism to
$Sp _g({\mathbb C} )$. The corresponding relative Malcev completion is then reduced
to $Sp _g({\mathbb C} )$ itself, so that the homomorphism from ${\mathcal U} ({\mathbb Z})({\mathbb C})$ (which
is just the abelian group ${\mathbb C}$) to ${\mathcal G} (\widehat{Sp}_g({\mathbb Z} ))$ is
trivial. We will see that this example is realized inside a quotient of
the mapping class
group.
\end{example}
\begin{example}
In this example, $\ker \rho$ is trivial, but ${\mathcal U}(\pi)$ is not.
The basic fact we need (see \cite[(10.3)]{hain:derham}) is that there is
always a natural $S$ equivariant isomorphism
$$
H_1({\mathfrak u}(\pi)) \cong
\prod_{\alpha \in \Check{S}} H_1(\pi;V_\alpha)\otimes V_\alpha^\ast,
$$
where $V_\alpha$ denotes a representation with highest weight $\alpha$.
For $\pi$ we take $\Gamma$, a finite index subgroup of $SL_2({\mathbb Z})$, for
$S$ we take $SL_2({\mathbb Q})$, and for $\rho$ we take the natural inclusion.
Denote the $n$th power of the fundamental representation of $SL_2$ by
$S^nV$.
For all such $\Gamma$, there is an infinite number of integers $n\ge 0$ such
that $H^1(\Gamma;S^nV)$ is non-trivial.\footnote{This is easily seen when
$\Gamma$ is free, for example. In general it is related to the theory of
modular forms.} It follows that ${\mathcal U}(\Gamma)$
has an infinite dimensional $H_1$, even though $\ker \rho$ is trivial.
\end{example}
This example suggests the following problem:
\begin{problem}
Investigate the relationship between the theory of modular forms
associated to a finite index subgroup $\Gamma$ of $SL_2({\mathbb Z})$ and the
completion of $\Gamma$ relative to the inclusion $\Gamma \hookrightarrow
SL_2({\mathbb Q})$.
\end{problem}
\subsection{The relative Malcev completion of $\Gamma_g$}
\label{malcevofgamma} The natural
homomorphism $\rho:\Gamma_{g,r}^n \to Sp_g$ has Zariski dense image.
Denote the completion of $\Gamma_{g,r}^n$ relative to $\rho$ by
${\mathcal G}_{g,r}^n$, its prounipotent radical by ${\mathcal U}_{g,r}^n$ and their Lie algebras
by ${\mathfrak g}_{g,r}^n$ and ${\mathfrak u}_{g,r}^n$.
The following theorem indicates the presence
of essentially one copy of the universal central extension of $Sp _g({\mathbb Z})$ in
quotients of each mapping class group of genus $g$ when $g\ge 3$.
\begin{theorem}[Hain \cite{hain:comp}] When $g\ge 2$, the homomorphism
\begin{equation}\label{nat_map}
{\mathcal U}(T_{g,r}^n) \to {\mathcal U}_{g,r}^n
\end{equation}
is surjective. When $g \ge 3$, its kernel
is a central subgroup isomorphic to the additive group.
\end{theorem}
This phenomenon is intimately related to the cycle $C-C^-$ and
its normal function as we shall now explain.
\subsection{The central extension}
\label{subsec:cent_extn} The existence of the central extension
has both a group theoretic and a geometric explanation. It is also
related to the Casson invariant through the work of Morita
\cite{morita:casson,morita:cocycles}. We begin with the group
theoretic one.
The group analogue of the Malcev filtration for the Torelli group $T_g$ is
the most rapidly descending central series of $T_g$ with
torsion free quotients:
$$
T_g = T_g^{(1)}\supset T_g^{(2)}\supset T_g^{(3)} \supset \cdots
$$
Note that $T_g^{(1)}/T_g^{(2)}$ is the maximal torsion free abelian
quotient of $T_g$, which is
$$
V(\lambda _3)_g{\mathbb Z} := \wedge^3 H_1(S_g;{\mathbb Z})/\left([S_g]\times H_1(S_g;{\mathbb Z} )\right)
$$
by Johnson's Theorem (\ref{johnson}). The group $\Gamma_g/T_g^{(3)}$
can be written as an extension
\begin{equation}
\label{ext1}
1 \to T_g^{(2)}/T_g^{(3)} \to \Gamma_g/T_g^{(3)}\to \Gamma_g/T_g^{(2)} \to 1.
\end{equation}
It turns out that this sequence contains a multiple of the universal
central extension of $Sp_g({\mathbb Z} )$ by ${\mathbb Z}$.
Since $V_g(\lambda _3)$ is a rational representation of
$Sp_g$, and since the surjection
$$
\wedge^2 V_g(\lambda _3)_{\mathbb Z} \to T_g^{(2)}/T_g^{(3)}
$$
induced by the commutator
is $Sp_g({\mathbb Z})$ equivariant, it follows that $T_g^{(2)}/T_g^{(3)}\otimes {\mathbb Q}$
is also a rational representation of $Sp_g$. Because $V_g(\lambda _3)$ is
an irreducible symplectic representation,
there is exactly one copy of the trivial representation in
$\wedge^2 V_g(\lambda _3)$. This copy of the trivial representation
survives in $T_g^{(2)}/T_g^{(3)}\otimes {\mathbb Q}$ \cite{hain:comp} so that there
is an $Sp_g({\mathbb Z})$ equivariant projection $T_g^{(2)}/T_g^{(3)}\to {\mathbb Z}$.
Pushing the extension (\ref{ext1}) out along this map gives an extension
\begin{equation}
\label{ext2}
0 \to {\mathbb Z} \to E \to \Gamma_g/T_g^{(2)}\to 1
\end{equation}
Note that $E$ is a quotient of $\Gamma_g$. We will manufacture a multiple
of the universal central extension of $Sp_g({\mathbb Z})$ from this group that
turns out to be the obstruction to the map ${\mathcal U}(T_g) \to {\mathcal U}_g$
being injective. (Full details can be found in \cite{hain:comp}.)
The group $\Gamma_g/T_g^{(2)}$ can be written as an extension
\begin{equation}
\label{ext3}
0 \to V_g(\lambda _3)_{\mathbb Z} \to \Gamma_g/T_g^{(2)}\to Sp_g({\mathbb Z}) \to 1.
\end{equation}
Morita \cite{morita:conj} showed that this
extension is {\it semisplit}, that is, if we replace $V_g(\lambda _3)_{\mathbb Z}$ by
$\frac{1}{2}V_g(\lambda _3)_{\mathbb Z}$, it splits.
(This can also be seen using the normal function of $C-C^-$.)
\begin{theorem}[Morita \cite{morita:cocycles}, Hain \cite{hain:comp}]
\label{nosplit}
The extension of $Sp_g({\mathbb Z})$ by ${\mathbb Z}$ obtained by pulling back the extension
(\ref{ext2}) along a semisplitting of (\ref{ext3}) contains
the universal central extension of $Sp_g({\mathbb Z})$.
\end{theorem}
The geometric picture uses the fundamental normal function $\nu_g$.
The lifted period maps $\nu_g$ and $\nu_g^1$ to the Deligne torsors
are easily seen to extend over the partial completions $\widetilde{{\mathcal M}}_g$ resp.\ $\widetilde{{\mathcal M}}^1_g$:
$$
\tilde\nu_g: \widetilde{{\mathcal M}}_g\to {\mathcal D}_g^{{\rm pr}}, \quad
\tilde\nu_g^1: \widetilde{{\mathcal M}}_g^1\to {\mathcal D}_g.
$$
The orbifold fundamental group of ${\mathcal D}_g^{{\rm pr}}$ resp.\ ${\mathcal D}_{g,1}$
is an extension of $Sp_g({\mathbb Z} )$ by $V_g(\lambda _3)_{\mathbb Z}$ resp.\ $\wedge ^3V_{g,{\mathbb Z}}$
as both the base and fiber are
Eilenberg MacLane spaces with these groups as orbifold fundamental groups. But by \ref{cor:pioftildem} the orbifold fundamental group of
$\widetilde{{\mathcal M}}_g$, resp.\ $\widetilde{{\mathcal M}}^1_g$, also has such a structure.
Indeed:
\begin{theorem}\label{fundgroup}
For $g\ge 3$, the normal functions $\tilde\nu_g: \widetilde{{\mathcal M}}_g\to {\mathcal D}_g^{{\rm pr}}$ and $\tilde\nu_g^1: \widetilde{{\mathcal M}}_g^1\to {\mathcal D}_g$
induce an isomorphism on orbifold fundamental groups. (The former can be identified with $\Gamma_g/T_g\!{}^{(2)}$ and the latter with
$\Gamma_g^1/(T_g^1)^{(2)}$.)
\end{theorem}
{}From this theorem we recover the fact that (\ref{ext3}) is semisplit, not split. But we get more, since it should also lead to a description of that extension.
The extension (\ref{ext2}) can also be realized geometrically.
\begin{proposition}[Hain \cite{hain:comp}]
There is a canonical (locally homogeneous) line bundle ${\mathcal B}_g$ over
the bundle ${\mathcal D}_{g,1}^{{\rm pr}}\to{\mathcal A}_g$
that realizes the central extension (\ref{ext2}) via the isomorphism of the previous proposition
as an extension of orbifold fundamental groups.
In particular, both $\tilde\nu^\ast {\mathcal B}_g$ and
$\nu^\ast {\mathcal B}_g$ have nonzero rational first Chern class.
The bundle $\nu^\ast {\mathcal B}_g$ is canonically metrized and
its square is isomorphic (as a metrized line bundle) to the metrized line bundle
associated to the archimedean height of the cycle $C-C^-$.
\end{proposition}
\section{Hodge Theory of the Mapping Class Group}\label{sec:hodgemap}
One reason that mixed Hodge theory is so powerful is that the MHS category is
abelian. In many situations this turns out to have
topological implications for algebraic varieties that are difficult, if
not impossible, to obtain directly. A somewhat
related (but less exploited) property is that a MHS
is canonically split over ${\mathbb C}$. This implies that the weight filtration
(which often has a topological interpretation) splits in a way that is
compatible
with all the algebraic structure naturally present. So, for many purposes,
there is no loss of information regarding this algebraic structure if we pass
to the corresponding weight graded object. For example, the Malcev
filtration on the Malcev Lie algebra of a smooth projective variety is minus
the weight filtration, and it therefore splits over ${\mathbb C}$ in a natural way.
This splitting is natural in the sense that it respects the Lie algebra
structure
and is preserved under all base point preserving morphisms. But if
we vary the complex structure on $X$ or the base point $x$, then
the splitting will, in general, vary with it.
A basic example is the Malcev Lie algebra ${\mathfrak p}_g^1$ of $\pi_g^1=\pi
_1(S_g^1,x_0)$.
The group $\pi _g^1$ is free on $2g$ generators and it is a classical
fact that the graded of ${\mathfrak p}_g^1$ with respect to the
Malcev filtration is just the free Lie algebra generated by $V_g$. If
$S_g$ is given a conformal structure, then $V_g$ has a pure Hodge
structure of weight $-1$ and the weight filtration of ${\mathfrak p}_g^1$
is minus the Malcev filtration. The splitting allows us to identify
${\mathfrak p}_g^1\otimes{\mathbb C}$ with the completion of $\Lie (V_g)\otimes{\mathbb C}$.
We shall come back to this example in Section~\ref{subsec:relation}. But
for now we will focus on the relative Malcev completions introduced in the
previous section.
\subsection{Hodge theory of ${\mathcal G}_{g,r}^n$}
\label{subsec:hodge} A choice of a conformal structure on $S_g$
and nonzero tangent vectors at $x_{n+1},\dots ,x_{n+r}$,
determines a point $x_o$ of the moduli space
${\mathcal M}_{g,r}^n$. We can thus identify $\Gamma_{g,r}^n$ with the
orbifold fundamental group of $({\mathcal M}_{g,r}^n,x_o)$.
This induces an isomorphism of ${\mathcal G}_{g,r}^n$ with ${\mathcal G}({\mathcal M}_{g,r}^n,x_o)$,
the completion of $\pi_1({\mathcal M}_{g,r}^n,x_o)$ with respect to the standard
symplectic representation. We shall write ${\mathcal G}_{g,r}^n(x_o)$ for
${\mathcal G}({\mathcal M}_{g,r}^n,x_o)$ and denote its prounipotent radical by ${\mathcal U}_{g,r}^n(x_o)$.
There is a general Hodge de~Rham theory of relative Malcev completion \cite{hain:derham}. Applying it to $({\mathcal M}_{g,r}^n,x_o)$, one obtains the following
result:
\begin{theorem}[Hain \cite{hain:torelli}]\label{mhs}
For each choice of a base point $x_o$ of ${\mathcal M}_{g,r}^n$, there is a
canonical MHS on the coordinate ring ${\mathcal O}({\mathcal G}_{g,r}^n(x_o))$ which
is compatible with its Hopf algebra structure. Consequently, the Lie
algebra ${\mathfrak g}_{g,r}^n(x_o)$ of ${\mathcal G}_{g,r}^n(x_o)$ and
the Lie algebra ${\mathfrak u}_{g,r}^n(x_o)$ of its prounipotent radical
both have a natural MHS.
\end{theorem}
Denote the Malcev Lie algebra of the subgroup of $\pi_1({\mathcal M}_{g,r}^n,x_o)$
corresponding to the Torelli group $T_{g,r}^n$ by ${\mathfrak t}_{g,r}^n(x_o)$.
The normal function of $C-C^-$ can be used to lift the MHS
from ${\mathfrak u}_{g,r}^n(x_o)$ to ${\mathfrak t}_{g,r}^n(x_o)$.
\begin{theorem}[Hain \cite{hain:torelli}]\label{cent_extn}
For each $g\ge 3$ and for each choice of a base point $x_o$ of
${\mathcal M}_{g,r}^n$, there is a canonical MHS on ${\mathfrak t}_{g,r}^n(x_o)$ which
is compatible with its bracket. Moreover, the canonical central
extension
$$
0 \to {\mathbb Q}(1) \to {\mathfrak t}_{g,r}^n(x_o) \to {\mathfrak u}_{g,r}^n(x_o) \to 0
$$
is an extension of MHSs, and the weight filtration equals the
Malcev filtration.
\end{theorem}
\subsection{A presentation of ${\mathfrak t}_g$}
\label{subsec:torelli_presentn}
We denote the Malcev Lie algebra of $T_{g,r}^n$ by ${\mathfrak t}_{g,r}^n$. The
existence of a MHS on ${\mathfrak t}_{g,r}^n(x_o)$ implies that, after tensoring
with ${\mathbb C}$, there is a canonical isomorphism
$$
{\mathfrak t}_{g,r}^n(x_o)\otimes {\mathbb C} \cong
\prod_m \Gr^W_{-m} {\mathfrak t}_{g,r}^n(x_o)\otimes {\mathbb C}.
$$
Since the left hand side is (noncanonically) isomorphic to
${\mathfrak t}_{g,r}^n\otimes {\mathbb C}$, to give a presentation of ${\mathfrak t}_{g,r}^n\otimes{\mathbb C}$,
it suffices to give a presentation of its associated graded.
It follows from Johnson's Theorem (\ref{johnson}) that each graded
quotient of the lower central series of ${\mathfrak t}_g$ is a representation
of the algebraic group $Sp_g$. We will give a presentation of
$\Gr^W_{\bullet} {\mathfrak t}_g$ in the category of representations of $Sp_g$. Recall
that $\lambda_1,\dots, \lambda_g$ is a set of fundamental weights of
$Sp_g$. For a nonnegative integral linear combination of the fundamental
weights $\lambda =\sum _{i=1}^gn_i\lambda _i$ we denote by
$V_g(\lambda)$ the representation of $Sp_g$ with highest weight $\lambda$.
For all $g \ge 3$, the representation $\wedge^2 V_g(\lambda_3)$
contains a unique copy of $V_g(2\lambda_2) + V_g(0)$. Denote the
$Sp_g$ invariant complement of this by $R_g$. Since the quadratic
part of the free Lie algebra $\Lie (V_g)$ is $\wedge^2 V_g$, we can
view $R_g$ as being a subspace of the quadratic elements of
$\Lie (V_g(\lambda_3))$.
As mentioned earlier, it is unknown whether any $T_g$
is finitely presented when $g\ge 3$. But the following theorem says
that its de~Rham incarnation is:
\begin{theorem}[Hain \cite{hain:torelli}]\label{presentation}
For all $g \ge 3$, ${\mathfrak t}_g$ is isomorphic to the completion of its
associated graded $\Gr^W_{\bullet}{\mathfrak t}_g$. When $g\ge 6$, this has presentation
$$
\Gr^W_{\bullet} {\mathfrak t}_g = \Lie (V_g(\lambda_3))/(R_g),
$$
where $R_g$ is the set of quadratic relations defined above.
When $3 \le g < 6$, the relations in $\Gr^W_{\bullet}{\mathfrak t}_g$ are generated by
the quadratic relations $R_g$, and possibly some cubic relations.
In particular, ${\mathfrak t}_{g,r}^n$ is finitely presented whenever $g \ge 3$.
\end{theorem}
Note that this, combined with (\ref{cent_extn}) gives a presentation
of $\Gr^W_{\bullet}{\mathfrak u}_g$ when $g\ge 6$:
\begin{corollary}
For all $g \ge 3$, ${\mathfrak u}_g$ is isomorphic to the completion of its
associated graded $\Gr^W_{\bullet}{\mathfrak u}_g$. When $g\ge 6$, this has quadratic
presentation
$$
\Gr^W_{\bullet} {\mathfrak u}_g = \Lie (V_g(\lambda_3))/(R_g + V_g(0)),
$$
where $R_g$ is the set of quadratic relations defined above and where
$V_g(0)$ is the unique copy of the trivial representation in
$\wedge^2 V_g(\lambda_3)$.
When $3 \le g < 6$, the relations in $\Gr^W_{\bullet}{\mathfrak u}_g$ are generated by
the quadratic relations $R_g+V_g(0)$, and possibly some cubic relations.
In particular, ${\mathfrak u}_{g,r}^n$ is finitely presented whenever $g \ge 3$.
\end{corollary}
The proof that the relations in the presentation of ${\mathfrak t}_g$ are
generated by quadratic relations when $g\ge 6$ and quadratic
and cubic ones when $g \ge 3$ is not topological, but uses deep
Hodge theory and, surprisingly, intersection homology. The
key ingredients are a result of Kabanov, which we state below,
and M.~Saito's theory of Hodge modules.
We define the {\it Satake compactification} $\overline{\M}^{\rm sat}_g$ of
${\mathcal M}_g$ as the closure of ${\mathcal M}_g$ inside the (Baily-Borel-)Satake
compactification of ${\mathcal A}_g$.
\begin{theorem}[Kabanov \cite{kabanov}]
For each irreducible representation $V$ of $Sp_g$, the
natural map
$$
IH^2(\overline{\M}^{\rm sat}_g;{\mathbb V}) \to H^2({\mathcal M}_g;{\mathbb V})
$$
is an isomorphism when $g\ge 6$. Here ${\mathbb V}$ denotes the generically
defined local system corresponding to $V$.
\end{theorem}
Such a local system ${\mathbb V}$ is, up to a Tate twist, canonically
a variation of Hodge structure. Saito's purity theorem then implies that
$H^2({\mathcal M}_g;{\mathbb V})$ is pure of weight $2+$ the weight of ${\mathbb V}$ when $g\ge 6$.
It is this purity result that forces $H^2({\mathfrak t}_g)$ to be of weight 2, and
implies that no higher order relations are needed.
\subsection{Understanding ${\mathfrak t}_g$}\label{subsec:understanding}
Even though we have a presentation of ${\mathfrak t}_g$, we still do not have a
good understanding of its graded quotients, either as vector spaces
or as $Sp_g$ modules. There is an exact sequence
$$
0 \to {\mathfrak p}_g \to {\mathfrak t}_g^1 \to {\mathfrak t}_g \to 0
$$
of Lie algebras (recall that ${\mathfrak p}_g$ stands for the Malcev Lie algebra
of $\pi _g=\pi_1(S,x_0)$).
It is the de~Rham incarnation of the exact sequence
of fundamental groups associated to the universal curve. Fix a conformal
structure on $(S,x_0)$.
Then this sequence is an exact sequence of MHSs.
Since $\Gr^W$ is an exact functor, and since
$\Gr^W_{\bullet} {\mathfrak p}_g$ is well understood, it suffices to
understand $\Gr^W_{\bullet} {\mathfrak t}_g^1$.
There is a natural representation
\begin{equation}\label{rep}
{\mathfrak t}_g^1 \to \Der {\mathfrak p}_g
\end{equation}
It is a morphism of MHS, and therefore determined by the graded
Lie algebra homomorphism
$$
\Gr^W_{\bullet}{\mathfrak t}_g^1 \to \Der \Gr^W_{\bullet} {\mathfrak p}_g.
$$
One can ask how close it is to being an isomorphism. Since this
map is induced by the natural homomorphism
\begin{equation}\label{can_hom}
\Gamma_g^1 \to \varprojlim \Aut {\mathbb C}\pi_g/I^m,
\end{equation}
the homomorphism (\ref{rep}) factors through the projection
${\mathfrak t}_g^1 \to {\mathfrak u}_g^1$,
and therefore cannot be injective. On the other hand, we have
the following (reformulated) result of Morita:
\begin{theorem}[Morita \cite{morita:trace}]
There is a natural Lie algebra surjection
$$
Tr_M : W_{-1}\Der \Gr^W_{\bullet} {\mathfrak p}_g \to \oplus_{k \ge 1} S^{2k+1}H_1(S)
$$
onto an abelian Lie algebra whose composition with (\ref{rep}) is
trivial. Here $S^m$ denotes the $m^{\text{th}}$ symmetric power.
\end{theorem}
One may then hope that the sequence
$$
0 \to {\mathbb C} \to \Gr^W_{\bullet}{\mathfrak t}_g^1 \to W_{-1}\Der \Gr^W_{\bullet} {\mathfrak p}_g \to
\oplus_{k \ge 1} S^{2k+1}H_1(S) \to 0
$$
is exact. However, there are further obstructions to exactness
at $W_{-1}\Der \Gr^W_{\bullet} {\mathfrak p}_g $ which were discovered by Nakamura
\cite{nakamura:obstn}.
They come from Galois theory and use the fact that ${\mathcal M}_g^1$ is defined
over ${\mathbb Q}$.\footnote{Actually, Nakamura proves his result for a
corresponding sequence for ${\mathfrak t}_{g,1}$, but his obstructions most likely
appear in this case too.} On the other hand, one can ask:
\begin{question}
Is the map ${\mathfrak u}_g^1 \to \Der {\mathfrak p}_g$ injective? Equivalently,
is ${\mathcal G}_g^1$ the Zariski closure of the image of the representation
(\ref{can_hom})?
\end{question}
A good understanding of ${\mathfrak t}_g$ may help in understanding the
stable cohomology of $\Gamma_g$ as we shall explain in the next
subsection.
\subsection{Torelli Lie algebras and the cohomology of $\Gamma_g$}
\label{subsec:torelli&coho}
Each Malcev Lie algebra ${\mathfrak g}$ can be viewed as a complete topological
Lie algebra. A basis for the neighbourhoods of 0 being the
terms ${\mathfrak g}^{(k)}$ of the Malcev filtration. One can define the
continuous cohomology of such a ${\mathfrak g}$ to be
$$
H^{\bullet}({\mathfrak g}) := \varinjlim H^{\bullet}({\mathfrak g}/{\mathfrak g}^{(k)}).
$$
If ${\mathfrak g}$ has a MHS, then so will $H^{\bullet}({\mathfrak g})$.
The continuous cohomology of ${\mathfrak t}_{g,r}^n$, ${\mathfrak u}_{g,r}^n$, etc. each has
an action of $Sp_g$. The general theory of relative Malcev completion
\cite{hain:derham} gives a canonical homomorphism
\begin{equation}\label{nat_homom}
H^{\bullet}({\mathfrak u}_{g,r}^n)^{Sp_g} \to H^{\bullet}({\mathcal M}_{g,r}^n)
\end{equation}
One can ask how much of the cohomology of ${\mathcal M}_{g,r}^n$ is captured by
this map.
Fix a base point $x_o$ of ${\mathcal M}_{g,r}^n$. Then ${\mathfrak t}_{g,r}^n$, etc.\ all
have compatible MHSs,
and these induce MHSs on their continuous cohomology
groups. These groups have the property that the weights on $H^k$ are
$\ge k$.
\begin{theorem}[Hain \cite{hain:torelli}]\label{morphism}
The map~(\ref{nat_homom}) is a morphism of MHS.
\end{theorem}
Since $H_1(T_g)$ is a quotient of ${\mathfrak u}_g$, there is an induced map
\begin{equation}\label{ext_map}
H^{\bullet}(H_1(T_g)) \to H^{\bullet}({\mathfrak u}_g).
\end{equation}
This is also a morphism of MHS. The following result follows
directly from \cite[(9.2)]{hain:cycles}, the presentation
(\ref{presentation}) of ${\mathfrak t}_g$, and the existence of the
MHS on ${\mathfrak u}_g$.
\begin{proposition}
If $g\ge 3$, the map~(\ref{ext_map}) surjects onto the lowest weight
subring
$$
\oplus_{k\ge 0} W_kH^k({\mathfrak u}_g)
$$
of $H^{\bullet}({\mathfrak u}_g)$, and the kernel is generated by the ideal generated
by the unique copy of $V_g(2\lambda_2)$ in $H^2(H_1(T_g))$.
\end{proposition}
Similar results hold when $r+n>0$ --- cf.\ \cite[\S 14.6]{hain:torelli}.
The following result of Kawazumi and Morita tells us that the
image of the lowest weight subring of $H^{\bullet}({\mathfrak u}_g)^{Sp_g}$ contains
no new cohomology classes.
\begin{theorem}[Kawazumi-Morita \cite{kawazumi-morita}]
\label{kawazumi-morita}
The image of the natural map
$$
H^{\bullet}(H_1(T_g))^{Sp_g} \to H^{\bullet}({\mathcal M}_g)
$$
is precisely the subring generated by the $\kappa_i$'s.
\end{theorem}
If we combine this with the previous two results and Pikaart's Purity
Theorem (\ref{pikaart_purity}), we obtain the following strengthening
of the theorem of Kawazumi and Morita (and obtained independently by Morita,
building on our work):
\begin{theorem}\label{tautbound}
When $k\le g/2$, the image of $H^k({\mathfrak u}_g)^{Sp_g} \to H^k({\mathcal M}_g)$
is the degree $k$ part of the subring generated by the $\kappa_i$'s.
\end{theorem}
To continue the discussion further, it seems useful to consider
cohomology with symplectic coefficients.
\subsection{Cohomology with symplectic coefficients}
The irreducible representations of $Sp_g$ are parametrized by Young
diagrams with $\le g$ rows (and no indexing of the boxes), in other
words, by nonincreasing sequences of nonnegative integers whose
terms with index $>g$ are zero.
So any such sequence $\alpha =(\alpha_1,\alpha_2,\dots)$ defines an
irreducible representation of $Sp_h$ for all $h\ge g$. We will
denote the representation of $Sp_g$ corresponding to $\alpha$
by $V_{g,\alpha}$, and the corresponding (orbifold) local system over
${\mathcal M}_g$ by ${\mathbb V}_{g,\alpha}$.
A theorem of Ivanov \cite{ivanov} (that in fact pertains to more
general local systems) implies that, when $r\ge 1$, the group
$H^k(\Gamma_{g,r}^n;V_{g,\alpha})$ is independent of $g$ once $g$ is
large enough. In the case at hand we have a more explicit result that we
state here for the undecorated case (a case that Ivanov actually excludes).
\begin{theorem}[Looijenga \cite{looijenga}]
Let $\alpha =(\alpha_1,\alpha_2,\dots)$ be a nonincreasing sequence of
nonnegative integers that is eventually zero, and let $c_1,c_2,\dots$
be weighted variables with
$\deg (c_i)=2i$. Put $|\alpha |:=\sum _{i\ge 1} \alpha_i$.
Then there exists a finitely generated, evenly graded
${\mathbb Q} [c_1,\dots ,c_{|\alpha |}]$-module $A^{{\bullet}}_{\alpha}$ (that can be
described explicitly) and a
graded homomorphism of $H^{\bullet} (\Gamma _{\infty})$ modules
$$
A^{\bullet} _{\alpha}[-|\alpha |]\otimes H^{\bullet}(\Gamma _{\infty})
\to H^{\bullet} (\Gamma_g;V_{g,\alpha })
$$
that is an isomorphism in degree $\le cg-|\alpha |$. It is also
a MHS morphism if we take
$A^{2k}_{\alpha}$ to be pure of type $(k,k)$. In particular, we have
$$
A^{\bullet}_{\alpha}[-|\alpha |]\otimes H^{\bullet}(\Gamma _{\infty})\cong
H^{\bullet}(\Gamma_\infty;V_\alpha )
$$
both as MHSs and as graded $H^{\bullet}(\Gamma _{\infty})$
modules. So, by (\ref{pikaart_purity}), $H^k(\Gamma_\infty;V_\alpha)$ is
pure of weight $k+|\alpha|$.
\end{theorem}
It is useful to try to understand all cohomology groups with
symplectic coefficients at the same time. To do this we take a leaf
out of the physicist's book and consider the ``generating function''
\begin{equation}\label{gen_fn}
\oplus_\alpha H^{\bullet}(\Gamma_g;V_\alpha^\ast)\otimes V_\alpha
\end{equation}
where $\alpha$ ranges over all partitions with $\le g$ rows,
and $\ast$ denotes dual.
This is actually a graded commutative ring as the Peter-Weyl Theorem
implies that the coordinate ring ${\mathcal O}_g$ of $Sp_g$ is
$$
{\mathcal O}_g = \oplus_\alpha \left(\End V_\alpha\right)^\ast \cong
\oplus_\alpha V_\alpha^\ast\otimes V_\alpha.
$$
The mapping class group acts on ${\mathcal O}_g$ by composing the
right translation action of $Sp_g$ on ${\mathcal O}_g$ with the canonical
representation $\Gamma_g \to Sp_g$. The corresponding cohomology group
$H^{\bullet}(\Gamma_g;{\mathcal O})$ is then the ``generating function''~(\ref{gen_fn}).
Note that ${\mathcal O}_g$ is a variation of Hodge structure of weight 0, so
the group $H^k(\Gamma_g;{\mathcal O}_g)$ is stably of weight $k$ by the above theorem.
There is a canonical algebra homomorphism
$$
H^{\bullet}({\mathfrak u}_g) \to H^{\bullet}(\Gamma_g;{\mathcal O}_g)
$$
whose existence follows from the de~Rham theory of relative completion
suggested by Deligne --- cf.\ \cite{hain:derham}. The map~(\ref{nat_homom})
of the previous subsection is just its invariant part.
This map is a MHS morphism for each choice of complex
structure on $S$. The $\alpha$ isotypical
part of both sides stabilizes as $g$ increases. It is natural
to ask:
\begin{question}
Is this map stably an isomorphism?
\end{question}
This has been verified by Hain and Kabanov (unpublished) in degrees
$\le 2$ for all weights, and in degree 3 and weight 3.
If the answer is yes, or even if one has surjectivity, then it will
follow from the theorem of Kawazumi and Morita (\ref{kawazumi-morita})
that the stable cohomology of ${\mathcal M}_g$ is generated by the $\kappa_i$'s.
A consequence of injectivity
and Pikaart's Purity Theorem would be that for each $k$, $H^k({\mathfrak u}_g)$ is
pure of weight $k$ once the genus is sufficiently large. This
is equivalent to the answer to the following question being affirmative.
\begin{question}
Are $H^{\bullet}({\mathfrak u}_g)$ and $U\Gr^W_{\bullet} {\mathfrak u}_g$ stably Koszul dual?
\end{question}
Note that $U\Gr^W_{\bullet} {\mathfrak u}_g$ and the lowest weight subalgebra of
$H^{\bullet}({\mathfrak u}_g)$ have dual quadratic presentations.
\section{Algebras Related to the Cohomology of Moduli Spaces of Curves}
\label{sec:algebras}
The ribbon graph description is the root of a number of ways of
constructing (co)homology classes on moduli spaces of curves from certain
algebraic structures. These constructions have in common that they actually
produce cellular (co)chains on ${\mathbb M} _g^n$, and so they
are recipes that assign numbers to `oriented' ribbon graphs. The typical
construction, due to Kontsevich \cite{kontsevich:feynman}, goes like this:
assume that we are given a complex vector space $V$,
a symmetric tensor $p\in V\otimes V$, and linear forms
$T_k:V^{\otimes k}\to {\mathbb C}$ that are cyclically invariant. If $\Gamma$ is a
ribbon graph, then the decomposition of $X(G)$ into $\sigma _1$ and
$\sigma _0$-orbits gives isomorphisms
$$
\otimes _{s\in X_1(\Gamma)}V^{\otimes\ori (s)}\cong V^{\otimes X(G)}\cong
\otimes _{v\in X_0(\Gamma)}V^{\otimes\out (v)},
$$
where $\ori (s)$ stands for the two-element set of orientations of the edge
$s$ and $\out (v)$ for the set of oriented edges that have $v$ as initial
vertex. Now
$p^{\otimes X_1(\Gamma)}$ defines a vector of the lefthand side
and a tensor product of certain $T_k$'s defines a linear form on the
righthand side. Evaluation of the linear form on the vector gives a number,
which is clearly an invariant of the ribbon graph.
Since this invariant does not depend on an orientation on the set of
edges of $G$, it cannot be used directly to define a cochain on the
combinatorial moduli spaces. To this end we need some sign rules so
that, for instance, the displayed isomorphisms acquire a sign.
The tensors $p$ and $T_k$ are sometimes referred to as the {\it propagator}
and the {\it interactions}, respectively,
to remind us of their physical origin.
If $p$ is nondegenerate, then we may use it to identify $V$ with its dual.
In this case $T_k$ defines a linear map $V^{\otimes (k-1)}\to V$.
The properties one needs to impose on propagator and interactions in order
that the above recipe produce a cocycle on ${\mathbb M} _g^n$ is that they define a
${\mathbb Z} /2$ graded $A_{\infty}$ algebra with inner product. A similar recipe
assigns cycles on ${\mathbb M}_g^n$ to certain ${\mathbb Z} /2$ graded differential algebras.
The cocycles can be evaluated on the cycles and this, in principle, gives a
method of showing that some of the classes thus obtained are nonzero.
We shall not be more precise,
but instead refer to \cite{kontsevich:feynman} or \cite{seminar}
for an overview. A simple example is to take $V={\mathbb C}$, $p:=1\otimes1$ and
$T_k(z^k)$ arbitrary for $k\ge 3$ odd, and zero otherwise. Kontsevich
asserts that the classes thus obtained are all tautological.
\subsection{Outer space}\label{subsec:outer}
In Section~\ref{sec:ribbon} we encountered a beautiful combinatorial
model for a virtual classifying space of the mapping class group $\Gamma _g^n$.
There is a similar, but simpler, combinatorial model that does the same job
for the outer automorphism group of a free group.
We fix an integer $r\ge 2$ and consider connected graphs $G$ with first
Betti number equal to $r$ and where each vertex has degree
$\ge 3$. Let us call these graphs {\it $r$-circular graphs}.
The maximal number of edges (resp.\ vertices) such a graph can
have is $3r-3$ (resp.\ $2r-2$). These bounds are realized by all
trivalent graphs of this type. Notice that an $r$-circular graph $G$
has fundamental group isomorphic to the free group on $r$ generators, $F_r$.
We say that $G$ is {\it marked} if we are given an isomorphism
$\phi :F_r\to\pi _1(G,\text{base point})$ up
to inner automorphism. The group $\Out (F_r)$ permutes these markings
simply transitively. There is an obvious notion of isomorphism for marked
$r$-circular graphs. We shall denote the collection of isomorphism classes
by
${\mathcal G} _r$.
Let $(G,[\phi ])$ represent an element of ${\mathcal G} _r$. The metrics on $G$ that
give $G$ total length $1$ are parameterized by the interior of a simplex
$\Delta (G)$. We fit these simplices together in a way analogous to the
ribbon graph case: if $s$ is an edge of $G$ that is not a loop, then
collapsing it defines another element $(G/s,[\phi ]/s)$ of ${\mathcal G} _r$. We
may then identify $\Delta (G/s)$ with a face of $\Delta (G)$. After we have
made these identifications we end up with a simplicial complex $\widehat{{\mathbb O}} _r$.
The union of the interiors of the simplices $\Delta (G)$ (indexed by ${\mathcal G} _r$)
will be denoted by ${\mathbb O} _r$; it is the complement of a closed subcomplex
of $\widehat{{\mathbb O}} _r$. This construction is due to Culler-Vogtmann
\cite{culler}. We call ${\mathbb O} _r$ the {\it outer space
of order $r$} for reasons that will become apparent in a moment.
Observe that $\widehat{{\mathbb O}} _r$ comes with a simplicial action of
$\Out (F_r)$. We denote the quotient of $\widehat{{\mathbb O}} _r$ (resp.\ ${\mathbb O} _r$)
by $\Out (F_r)$ by $\widehat{{\mathbb G}}_r$ (resp.\ ${\mathbb G} _r$). It is easy to see that
$\widehat{{\mathbb G}} _r$ is a finite orbicomplex. The open subset ${\mathbb G} _r$ is the moduli
space of metrized $r$-circular graphs. It has a spine of dimension $2r-3$.
\begin{theorem}[Culler-Vogtmann \cite{culler}, Gersten]
The outer space of order $r$ is contractible and a subgroup of finite
index of
$\Out
(F_r)$ acts freely on it. Hence ${\mathbb G} _r$ is a virtual classifying space for
$\Out (F_r)$ and $\Out (F_r)$ has virtual homological dimension $\le 2r-3$.
\end{theorem}
In contrast to the ribbon graph case, ${\mathbb O} _r$ is not piecewise smooth. If we
choose $2g-1+n$ free generators for the fundamental group of our reference
surface $S_g^n$, then each ribbon graph without vertices of degree $\le 2$
determines an element of ${\mathbb G} _{2g-1+n}$: simply forget the ribbon structure.
The ribbon data is finite and it is therefore not surprising that forgetting
the ribbons defines a finite map
$$
\widehat{f} : {\mathcal S} _n\backslash\widehat{{\mathbb M}} _g^n\to \widehat{{\mathbb G}} _{2g-1+n}
$$
of orbicomplexes. Here ${\mathcal S} _n$ stands for the symmetric group, which acts
in the obvious way on $\widehat{{\mathbb M}} _g^n$. Following Strebel's theorem, the
preimage of
${\mathbb G} _{2g-1+n}$ can be identified with ${\mathcal S} _n\backslash ({\mathcal M} _g^n\times
\inn \Delta ^{n-1})$. We denote the resulting map by
$$
f: {\mathcal S} _n\backslash ({\mathcal M} _g^n\times \inn\Delta ^{n-1})\to {\mathbb G} _{2g-1+n}.
$$
It induces the evident map
$$
f_*: H_k(\Gamma _g^n)_{{\mathcal S} _n}\to H_k(\Out F_{2g-1+n})
$$
on rational homology. It is unclear whether there is such an
interpretation for the induced map on cohomology with compact supports.
We remark that ${\mathcal M} _g^n\times \inn\Delta ^{n-1}$ is canonically oriented,
but that its ${\mathcal S} _n$-orbit space is not (since transpositions reverse this
orientation). Poincar\'e duality therefore takes the form
$$
H_k({\mathcal M} _g^n)_{{\mathcal S} _n}\cong
H_k({\mathcal S} _n\backslash ({\mathcal M} _g^n\times \inn\Delta ^{n-1});\epsilon)\cong
H_c^{6g-7+3n-k}({\mathcal S} _n\backslash ({\mathcal M} _g^n\times \inn\Delta ^{n-1});
\epsilon ),
$$
where $\epsilon$ is the signum representation of ${\mathcal S} _n$.
If $\delta$ denotes the (signum) character of $\Out (F_r)$ on
$\wedge ^rH_1(F_r)$, then the adjoint of $f_*$ is a map
$$
H_c^{6g-7+3n-k}({\mathbb G} _{2g-1+n};\delta )\to
H_c^{6g-7+3n-k}({\mathcal S} _n\backslash ({\mathcal M} _g^n\times \inn\Delta ^{n-1});
\epsilon)\cong H_k({\mathcal M} _g^n)_{{\mathcal S} _n}.
$$
So, when $m\ge 1$, we have maps
$$
H_c^{i+m-1}({\mathbb G} _{m+1};\delta )\to\oplus _{2g-2+n=m}H_{2m-i}
({\mathcal M} _g^n)_{{\mathcal S} _n}\to H_{2m-i}({\mathbb G} _{m+1}),\quad i=0,1,\dots .
$$
There is a remarkable interpretation of this sequence that we will discuss
next.
\subsection{Three Lie algebras}\label{subsec:threeLie}
We describe Kontsevich's three functors from the category of symplectic
vector spaces to the category of Lie algebras and their relation with
the cohomology of the moduli spaces ${\mathcal M} _g^n$. The basic
references are \cite{kontsevich:symp} and \cite{kontsevich:feynman}.
We start out with a finite dimensional ${\mathbb Q}$ vector space $V$ endowed with a
nondegenerate antisymmetric tensor $\omega _V\in V\otimes V$.
Let $\Ass (V)$ be the tensor algebra (i.e., the free associative algebra)
generated by $V$. We grade it by giving $V$ degree $-1$. The Lie
subalgebra generated by $V$ is free and so we denote it by $\Lie (V)$.
It is well-known that $\Ass (V)$ may be identified with the universal
enveloping algebra of $\Lie (V)$. If we mod out $\Ass (V)$ by the two-sided
ideal generated by the degree $\le -2$ part of $\Lie (V)$,
we obtain the symmetric algebra $\Com (V)$ of $V$. Define
${\mathfrak g} _{{\rm ass}} (V)$ (resp.\ ${\mathfrak g} _{{\rm lie}} (V)$) to be the Lie algebra of
derivations of $\Ass (V)$ (resp.\ $\Lie (V)$) of degree $\le 0$ that kill
$\omega _V$. Since each derivation of $\Lie (V)$ extends canonically to its
universal enveloping algebra, we have an inclusion
${\mathfrak g} _{{\rm lie}} (V)\subset{\mathfrak g} _{{\rm ass}} (V)$. There is also a corresponding Lie
algebra ${\mathfrak g} _{{\rm com}} (V)$ of derivations of degree $\le 0$ of $\Com (V)$
that kill $\omega _V$. Here we regard the latter as a two-form on the
affine space $\spec \Com (V)$. This Lie algebra is a quotient of ${\mathfrak g} _{{\rm ass}}
(V)$. All three Lie algebras are graded and have as degree zero
summand the Lie algebra $\sp (V)$ of the group $\Symp (V)$ of symplectic
transformations of $V$. A simple verification shows that the degree $-1$ summands
have as $\sp (V)$ representations the following natural descriptions:
$$
{\mathfrak g}_{{\rm com}}(V)_{-1}\cong S^3(V),\quad
{\mathfrak g}_{{\rm ass}}(V)_{-1}\cong S^3(V)\oplus \wedge ^3V,\quad
{\mathfrak g}_{{\rm lie}}(V)_{-1}\cong \wedge^3V.
$$
These Lie algebras are functorial with respect to symplectic injections
$(V,\omega _V)\hookrightarrow (W,\omega _W)$. Note that $\Symp (V)$ acts
trivially on this cohomology of the Lie algebra in question because
$\sp (V)\subset{\mathfrak g} _*(V)$. This implies that $H^k({\mathfrak g} _*(V))$,
$\ast \in \{{\rm lie},{\rm ass},{\rm com}\}$, depends
only on $\dim V$. We form the inverse limit:
$$
H^k({\mathfrak g} _*):=\varprojlim_{V}H^k({\mathfrak g} _*(V)).
$$
The sum over $k$, $H^{\bullet}({\mathfrak g} _*)$, has the structure of a connected graded bicommutative Hopf
algebra; the coproduct comes from the direct sum operation on symplectic
vector spaces.
It is actually bigraded: apart from the cohomological grading
there is another coming from the grading of the Lie algebras. Notice that the
latter grading has all its degrees $\ge 0$. The primitive part
$H^{{\bullet}}_{{\rm pr}}({\mathfrak g} _*)$ inherits this bigrading. Furthermore, the
natural maps
$$
H^{{\bullet}}({\mathfrak g} _{{\rm com}})\to H^{{\bullet}}({\mathfrak g} _{{\rm ass}})\to H^{{\bullet}}({\mathfrak g} _{{\rm lie}})
$$
are homomorphisms of bigraded Hopf algebras. Consequently, we have induced
maps between the bigraded pieces of their primitive parts.
\begin{theorem}[Kontsevich \cite{kontsevich:symp}, \cite{kontsevich:feynman}]
\label{thm:lie}
For $\ast \in \{{\rm lie},{\rm ass},{\rm com}\}$ we have
$$
H^k_{{\rm pr}}({\mathfrak g} _*)_0=H^k_{{\rm pr}}(\sp_\infty )\cong
\begin{cases}
{\mathbb Q} &\text{for }k=3,7,11,\dots\, ;\\
0 &\text{otherwise}.
\end{cases}
$$
Furthermore,
$H^k_{{\rm pr}}({\mathfrak g} _*)_l=0$ when $l$ is odd and, when $m>0$, we have a natural
diagram
$$
\begin{CD}
H^k_{{\rm pr}}({\mathfrak g} _{{\rm com}})_{2m} @>>>
H^k_{{\rm pr}}({\mathfrak g} _{{\rm ass}})_{2m} @>>>
H^k_{{\rm pr}}({\mathfrak g} _{{\rm lie}})_{2m} \cr
@V{\cong }VV @V{\cong }VV @V{\cong }VV \cr
H^{k+m-1}_c({\mathbb G} _{m+1},\delta ) @>{f_c^*}>>
\oplus _{2g-2+n=m}H_{2m-k}({\mathcal M} _g^n)_{{\mathcal S} _n} @>{f_*}>>
H_{2m-k}({\mathbb G}_{m+1})\cr
\end{CD}
$$
which commutes up to sign and whose rows are complexes.
The maps in the top row are the natural maps and the bottom row is the
sequence defined in Section~\ref{subsec:outer}.
\end{theorem}
The proof is an intelligent application of classical invariant
theory. For each of the three Lie algebras one writes down the standard
complex. The subcomplex of invariants with respect to the symplectic
group is quasi-isomorphic to the full complex. Weyl's invariant theory
furnishes a natural basis for this subcomplex. Kontsevich then
observes that this makes the subcomplex naturally isomorphic to a cellular
chain (or cochain) complex of one of the cell complexes ${\mathbb G} _*$ and
${\mathbb M} _*^*$ whose (co)homology appears in the bottom row.
The diagram in this theorem suggests that the
sequence of natural transformations $\Lie\to\Ass\to\Com$
is self dual in some sense. This
can actually be pinned down by looking at the corresponding operads:
Ginzburg and Kapranov \cite{ginz} observed that these operads have
``quadratic relations'' and they proved the self duality of the operad
sequence in a Koszul sense.
However, our main reason for displaying this diagram is that it pertains
to the cohomology of the moduli spaces of curves in two apparently
unrelated ways. The first one is evident. The ${\mathcal S} _n$ coinvariants of
the homology of ${\mathcal M} _g^n$ features in the middle column, but the righthand
column has something to do with the cohomology of a
`linearization' of $\Gamma _{\infty}$: we will see that ${\mathfrak g} _{{\rm lie}}$
is intimately related to the Lie algebra of the relative Malcev completions
discussed in Section~\ref{sec:malcev}. We explain this in the next subsection
after a giving a restatement of Kontsevich's Theorem.
In this restatement the Lie algebra cohomology of ${\mathfrak g}_\ast (V)$ is replaced
by the relative Lie algebra cohomology
of the pair $({\mathfrak g}_\ast (V),{\mathfrak k}(V))$, where ${\mathfrak k}(V)$ is a maximal compact Lie
subalgebra of ${\mathfrak g}_\ast (V)_0$, and therefore of ${\mathfrak g}_\ast (V)$ (${\mathfrak k}(V)$ is a
unitary Lie algebra of rank $\dim V/2$). As above, the Lie algebra cohomology
$H^k({\mathfrak g}_\ast (V),{\mathfrak k}(V))$ depends only on the dimension of $V$ and stabilizes
once $\dim V$ is sufficiently large. We denote the inverse limit of these groups
by $H^{\bullet}({\mathfrak g}_\ast,{\mathfrak k}_\infty)$. Likewise, we denote the stable cohomology of the
pair $(\sp_g,{\mathfrak k}_g)$ by $H^{\bullet}(\sp_\infty,{\mathfrak k}_\infty)$. By a theorem of Borel
\cite{borel:triv}, this is naturally isomorphic to the stable cohomology of ${\mathcal A}_g$
and is a polynomial algebra generated by classes $c_1, c_3, c_5, \dots$, where
$c_k$ has degree $2k$.
Combining Kontsevich's Theorem~\ref{thm:lie} with Borel's computation
and an elementary spectral sequence argument, we obtain the following
result. (Use the fact that $(\sp_g,{\mathfrak k}_g)$
is both a sub and a quotient of $({\mathfrak g}_\ast(V),{\mathfrak k}(V))$.)
\begin{corollary}\label{rel_lie}
We have
$$
H^k_{{\rm pr}}({\mathfrak g}_\ast,{\mathfrak k}_\infty)_0=H^k_{{\rm pr}}(\sp_\infty,{\mathfrak k}_\infty )\cong
\begin{cases}
{\mathbb Q} &\text{for }k=2,6,10,\dots\, ;\\
0 &\text{otherwise}.
\end{cases}
$$
Furthermore, when $m > 0$, the natural maps $H^{\bullet}({\mathfrak g}_\ast,{\mathfrak k}_\infty)_m
\to H^{\bullet}({\mathfrak g}_\ast)_m$ are isomorphisms.
\end{corollary}
\subsection{Relation with the relative Malcev completion}
\label{subsec:relation}
We begin with an observation. For a symplectic vector space $V$ and
$\ast \in \{{\rm lie},{\rm ass},{\rm com}\}$, denote by ${\mathfrak g} _*^{\flat}(V)$ the subalgebra of
${\mathfrak g} _*(V)$ generated by its summands of weight $0$ and $-1$.
Kontsevich's computation shows:
\begin{proposition} The graded cohomology groups
$H^k ({\mathfrak g} _*^{\flat} (V),{\mathfrak k}(V))_l$ stabilize and the sum of the stable terms is a bigraded
bicommutative Hopf algebra $H^{\bullet}({\mathfrak g}_*^{\flat},{\mathfrak k}_\infty)_{{\bullet}}$. In addition,
the restriction map
$H^k_{{\rm pr}}({\mathfrak g}_*,{\mathfrak k}_\infty)_l\to H^k_{{\rm pr}}({\mathfrak g}_*^{\flat}(V),{\mathfrak k}_{\rm lie}(V))_l$
is an isomorphism when $l\le k$.
\end{proposition}
The case of interest here is that of lie where
${\mathfrak g} _{\rm lie}^{\flat}(V)_0=\sp (V)$ and ${\mathfrak g} _{\rm lie}^{\flat}(V)_{-1}\cong \wedge ^3V$.
Denote by $z_m$ the element of $H^{2m}_{{\rm pr}}({\mathfrak g}_{\rm lie} )_{2m}$ that corresponds,
via Theorem~\ref{thm:lie}, to $1\in H_0({\mathbb G} _{m+1})$. The preceding proposition
yields:
\begin{corollary}\label{cor:weightcontrol}
We have a natural isomorphism of bigraded Hopf algebras
$$
\sum_{l\le k} H^k({\mathfrak g}_{{\rm lie}}^{\flat},{\mathfrak k}_\infty)_l\cong
H^{\bullet}(\sp_\infty,{\mathfrak k}_\infty)[z_1,z_2,\dots ]
\cong {\mathbb C}[c_1,c_3,c_5,\dots,z_1,z_2,z_3,\dots]
$$
where each $z_i$ and $c_j$ is primitive.
\end{corollary}
The graded Lie algebra ${\mathfrak g}^\flat_{\rm lie}(V)$ is the semi-direct product of
$\sp(V)$ and its elements of positive weight, which we shall denote by
${\mathfrak u}^\flat_{\rm lie}$. Consequently, there are natural inclusions
$$
H^{\bullet}({\mathfrak u}^\flat_{\rm lie}(V))^{Sp} \hookrightarrow H^{\bullet}({\mathfrak g}^\flat_{\rm lie}(V),{\mathfrak k}(V))
\text{ and } H^{\bullet}(\sp(V),{\mathfrak k}(V)) \hookrightarrow
H^{\bullet}({\mathfrak g}^\flat_{\rm lie}(V),{\mathfrak k}(V)).
$$
Together these induce an algebra homomorphism
$$
H^{\bullet}(\sp(V),{\mathfrak k}(V))\otimes H^{\bullet}({\mathfrak u}^\flat_{\rm lie}(V))^{Sp}
\to H^{\bullet}({\mathfrak g}^\flat_{\rm lie}(V),{\mathfrak k}(V))
$$
which is compatible with stabilization.
\begin{proposition}\label{stab_prod}
Upon stabilization, these maps induce an isomorphism
$$
H^{\bullet}(\sp_\infty,{\mathfrak k}_\infty)\otimes H^{\bullet}({\mathfrak u}^\flat_{\rm lie})^{Sp}
\to H^{\bullet}({\mathfrak g}^\flat_{\rm lie},{\mathfrak k}_\infty).
$$
\end{proposition}
Next we relate the graded Lie algebra ${\mathfrak g}^\flat_{\rm lie}$ to the filtered
Lie algebra ${\mathfrak g}_{g,1}$ of the relative Malcev completion ${\mathcal G} _{g,1}$ of
$\Gamma_{g,1}\to Sp_g$.
Recall from Section~\ref{sec:groups} that $\pi_g^1$ is freely generated by $2g$
generators
named $\alpha _{\pm},\dots ,\alpha_{\pm g}$ so that
the commutator $\beta:=(\alpha_1,\alpha_{-1})\cdots (\alpha_g,\alpha_{-g})$
represents a simple loop around $x_1$. Using Latin letters for the
logarithms of the images of elements of $\pi _g^1$ in its Malcev completion,
we find that
$$
b\equiv [a_1,a_{-1}]+\cdots +[a_g,a_{-g}]\mod{({\mathfrak p} _g^1)^{(3)}}.
$$
So the image of $b$ in $\Gr^2{\mathfrak p}_g^1\cong \wedge^2 V_g$ is the symplectic
form $\omega_S$.
The obvious homomorphism $\Gamma_{g,1} \to \Aut (\pi_g^1)$ induces a Lie algebra
homomorphism
\begin{equation}\label{map}
{\mathfrak g} _{g,1} \to \Der {\mathfrak p}_g^1.
\end{equation}
whose image we denote by $\overline{\g} _{g,1}$.\footnote{It is possible that
this map is injective so that ${\mathfrak g} _{g,1}\cong\overline{\g} _{g,1}$. Note that it
is not surjective --- see \cite{morita:trace} and \cite{nakamura:obstn},
and Section~\ref{subsec:understanding}.}
Notice that $\overline{\g}_{g,1}$ is contained in the subalgebra $\Der({\mathfrak p}_{g,1},b)$
consisting of those derivations that kill $b$. Since (\ref{map}) is (Malcev)
filtration preserving, it induces Lie
algebra homomorphisms
$$
\Gr^{\bullet} {\mathfrak g}_{g,1}\to \Gr^{\bullet}\overline{\g}_{g,1}\to \Der^{\bullet}(\Gr{\mathfrak p}_g^1,\omega_S).
$$
Notice that the last term is just ${\mathfrak g} _{{\rm lie}}(V_g)$.
In view of (\ref{stab_prod}), to construct an algebra homomorphism
$$
H^{\bullet}({\mathfrak g}^\flat_{\rm lie},{\mathfrak k}_\infty) \to H^{\bullet}(\Gamma_\infty)
$$
it suffices to construct an algebra homomorphism
$H^{\bullet}({\mathfrak u}^\flat_{\rm lie})^{Sp_g} \to H^{\bullet}(\Gamma_\infty)$.
At this stage we need Hodge theory. Choose a conformal structure
on $S_g$. Then, by~(\ref{mhs}), there are
natural MHSs on ${\mathfrak g}_{g,1}$ and ${\mathfrak p}_{g,1}$ whose weight filtrations are the Malcev filtrations and such that (\ref{map}) is a MHS morphism. Hence
the image $\overline{\g}_{g,1}$ has a natural MHS.
Since $\Gr^{\bullet} {\mathfrak g}_{g,1}$ is generated by
its summands in degree $0$ and $1$, the same is true for $\Gr^{\bullet}\overline{\g}_{g,1}$.
On the other hand, the summands in degree $0$ and $1$ of
$\Gr^{\bullet}\overline{\g}_{g,1}$ are equal to the summands of weight $0$ and $-1$ of ${\mathfrak g}
_{{\rm lie}}(V_g)$, and so the graded Lie algebra $\Gr^{\bullet}\overline{\g}_{g,1}$
may be identified with ${\mathfrak g}_{\rm lie}^\flat (V_g)_{{\bullet}}$ (except that the indexing
of the summands differs by sign).
Denote the pronilpotent radical $W_{-1}\overline{\g}_{g,1}$ of $\overline{\g}_{g,1}$ by
$\overline{\u}_{g,1}$. We know from Section~\ref{sec:hodgemap} that the homomorphisms
\begin{equation}\label{sequence}
H^{\bullet} (\overline{\u}_{g,1})^{Sp_g}\to H^{\bullet}({\mathfrak u}_{g,1})^{Sp_g}\to H^{\bullet}(\Gamma_{g,1})
\end{equation}
are morphisms of MHS. After weight grading these become bigraded algebra
homomorphisms
$$
H^{\bullet}({\mathfrak u}_{\rm lie}^\flat (V_g)_{{\bullet}})^{Sp_g}\to H^{\bullet}(\Gr^W_{{\bullet}}{\mathfrak u}_{g,1})^{Sp_g}
\to \Gr^W_{\bullet} H^{\bullet}(\Gamma_{g,1}).
$$
The sequence (\ref{sequence}) stabilizes with $g$ to a sequence of Hopf
algebras in the MHS category. The corresponding weight graded sequence is
$$
H^{\bullet}(({\mathfrak u}_{\rm lie}^\flat )_{{\bullet}})^{Sp_g}\to
H^{\bullet}(\Gr^W_{{\bullet}}{\mathfrak u}_{\infty,1})^{Sp_g}
\to \Gr^W_{\bullet} H^{\bullet}(\Gamma_{\infty}).
$$
Each term in this sequence is a Hopf algebra and each map a Hopf algebra
homomorphism. But by Pikaart's Purity Theorem we know that the last term
is pure of weight $k$ in degree $k$, so that we can replace it by
$H^{\bullet}(\Gamma_{\infty})$ and obtain a map $H^{\bullet}({\mathfrak u}^\flat_{\rm lie})^{Sp_g}
\to H^{\bullet}(\Gamma_\infty)$. We therefore have a Hopf algebra homomorphism
$$
H^{\bullet}({\mathfrak g}^\flat_{\rm lie},{\mathfrak k}_\infty) \cong
H^{\bullet}(\sp_\infty,{\mathfrak k}_\infty)\otimes H^{\bullet}({\mathfrak u}^\flat_{\rm lie})^{Sp_g}
\to H^{\bullet}(\Gamma_\infty).
$$
If we compose the natural restriction map
$H^{\bullet}({\mathfrak g}_{\rm lie}, {\mathfrak k}_\infty)\to H^{\bullet}({\mathfrak g}_{\rm lie}^\flat,{\mathfrak k}_\infty)$ with the
above maps we get a homomorphism
$$
H^{\bullet}({\mathfrak g}_{\rm lie},{\mathfrak k}_\infty)\to H^{\bullet}(\Gamma_{\infty}).
$$
Kontsevich asked (at the end of \cite{kontsevich:feynman}) about the meaning
of this map.\footnote{Actually, Kontsevich asks this for with $H^{\bullet}({\mathfrak g}_{\rm lie})$
in place of the relative Lie algebra cohomology. However, there
does not seem to be a natural homomorphism to $H^{\bullet}(\Gamma_\infty)$ in this
case.} This can now be answered by invoking the theorem of Kawazumi and Morita
(\ref{kawazumi-morita}), or rather a weaker form, which says that $z_i$ is
mapped to a nonzero multiple of $\kappa_i$. This result was obtained
with Kawazumi and Morita.
\begin{theorem}
There is a natural Hopf algebra homomorphism
$$
H^{\bullet}({\mathfrak g}_{\rm lie},{\mathfrak k}_\infty)\to H^{\bullet}(\Gamma_{\infty}).
$$
The left hand side is a polynomial algebra generated by primitive
elements $z_1, z_2,\dots$ and $c_1, c_3, \dots $ where $z_i$ has degree
$2i$ and $c_j$ has degree $2j$. The image of this homomorphism is
precisely the subalgebra generated by the $\kappa_i$s. The kernel is
generated by elements of the form
$$
c_{2k+1} - a_k z_{2k+1} - P_{2k+1}, \quad k \in \{0,1,2,\dots \}
$$
where $P_{2k+1}$ is a polynomial in the $z_i$ and $c_j$ with no linear
terms , and $a_k$ is a non zero rational number.
\end{theorem}
Here we have used the fact, due to Mumford \cite{mumford}, that the image
of $c_{2k+1}$ in $H^{\bullet} ({\mathcal M}_{g,1})$ is a polynomial in the odd $\kappa_i$'s.
The theorem indicates that no new stable classes are to be expected from
Hodge theory --- that is, a de~Rham version of the Mumford conjecture
holds. Recent work of Kawazumi and Morita attempts to explain the kernel
of the homomorphism $H^{{\bullet}}(\overline{\g}_{\infty})\to H^{{\bullet}}(\Gamma_{\infty })$
in terms of secondary characteristic classes of surface bundles. The first
element of the kernel is the difference $c_1 - 12\, z_1$. Its restriction
to the Torelli group can be interpreted as the Casson Invariant (cf.\ \cite{morita:casson}.)
|
1996-07-17T01:43:24 | 9607 | alg-geom/9607014 | en | https://arxiv.org/abs/alg-geom/9607014 | [
"alg-geom",
"math.AG"
] | alg-geom/9607014 | null | Eriko Hironaka | Torsion Points on an Algebraic Subset of an Affine Torus | LaTeX, 25 pages, 4 figures. email: [email protected] | null | null | null | null | Work of Laurent and Sarnak, following a conjecture of Lang, shows that the
number of torsion points of order n on an algebraic subset of an affine complex
torus is polynomial periodic. In this paper, we find bounds on the degree and
period of this number as a function of n. Some examples, including the number
of n torsion points on Fermat curves, are computed to illustrate the methods.
| [
{
"version": "v1",
"created": "Tue, 16 Jul 1996 23:33:34 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Hironaka",
"Eriko",
""
]
] | alg-geom | \section{Introduction.}
Let $V \subset ({\Bbb C}^*)^r$ be an algebraic subset.
Laurent has shown that the torsion points on $V$ lie on a finite
union of translates of affine subtori contained in $V$. It follows
that the number, $p_V(n)$, of
torsion points of order $n$ on $V$ is polynomial periodic.
In this paper, we find formulas and bounds for the degree and period
of $p_V(n)$ in terms of defining equations for $V$.
Counting torsion points on algebraic subsets of an the affine torus
is useful for studying abelian representations of a finitely
presented group $\Gamma$. For example, the Alexander invariants
define a stratification of the character variety for $\Gamma$,
which naturally embeds in $({\Bbb C}^*)^r$,
and the first Betti number of unbranched coverings can be computed
from the torsion points on these strata (cf. \cite{A-S:Betti},
\cite{Hiro:Alex}).
In \cite{Laur:Equ}, Laurent gives the following description
of the set of torsion points ${\mathrm{Tor}}(V)$ on $V$.
\begin{theorem} (Laurent)
For any algebraic subset $V$ of $({\Bbb C}^*)^r$ there is a finite set
of rational planes $Q_1,..,Q_\ell \subset V$ such that
$$
{\mathrm{Tor}}(V) = \bigcup_{i=1}^\ell {\mathrm{Tor}}(Q_i).
$$
\end{theorem}
Here, a {\it rational plane} $Q \subset ({\Bbb C}^*)^r$ is a subset of the form
$\eta P$, where $P \subset ({\Bbb C}^*)^r$
is an {\it affine subtorus}, or connected algebraic subgroup,
and $\eta \in ({\Bbb C}^*)^r$ is an element of finite order.
Laurent's result extends further than the statement we give here and
settles a more general conjecture of Lang (\cite{Lang:Conj}, p.220).
It follows from Theorem 1 (cf. \cite{A-S:Betti})
that $p_V(n)$ is a {\it polynomial periodic} function in $n$,
that is, there exist periodic functions $a_0(n), \dots, a_d(n)$ such
that
$$
p_V(n) = a_0(n) + a_1(n)n + \dots + a_d(n)n^d.
$$
For, by Theorem 1, the Zariski closure of $V$ has a decomposition
into a finite union of rational planes
$$
\Zar{{\mathrm{Tor}}(V)} = Q_1 \cup \dots \cup Q_\ell
$$
and hence $p_V(n)$ is given by the following formula
\begin{eqnarray}
p_V(n) = \sum_{k=1}^\ell \quad
\sum_{1\leq i_1 < \dots <
i_k \leq\ell}
(-1)^{\ell-k}
p_{Q_{i_1} \cap \dots \cap Q_{i_k}}(n).
\end{eqnarray}
It is not hard to see that a finite intersection of rational
planes is a finite union of disjoint rational planes (see Prop. 3.6
for a more precise description). Furthermore, the number of
$n$-torsion points on a rational plane $Q$ is given by
\begin{eqnarray*}
p_Q(n) &=& \left\{\begin{array}{ll}
n^{\dim (Q)} &\quad\mbox{if ${\mathrm{ord}}(Q)\ |\ n$,}\\
0 &\quad\mbox{otherwise,}
\end{array}\right .
\end{eqnarray*}
where ${\mathrm{ord}}(Q)$ is the least integer $n$ such that $Q^n$ contains the identity
element of
$({\Bbb C}^*)^r$.
We will concentrate on finding the degree and period of $p_V$.
The {\it degree} of $p_V$, written $\deg (p_V)$, is
the largest $d$ such
that $a_d(n)$ is not constantly
zero and
the {\it period} of $p_V$, written ${\mathrm{per}} (p_V)$,
is the least common multiple of the periods
of $a_0(n),\dots,a_d(n)$. Thus, for example, if $Q \subset ({\Bbb C}^*)^r$ is a
rational plane, then ${\mathrm{ord}} (Q) = {\mathrm{per}} (p_Q)$.
As with ordinary polynomials the degree of $p_V$ determines the order of
growth of $p_V(n)$:
if $\deg(p_V) = d$, then
$$
p_V(n) = O(n^d)
$$
and for some fixed integer $c$
$$
p_V(n) \asymp n^d
$$
for all $n \equiv c\ ({\mathrm{mod}}\ {\mathrm{per}}(p_V))$.
If $\Zar{{\mathrm{Tor}}(V)} = Q_1 \cup \dots \cup Q_\ell$, where $Q_1,\dots,Q_\ell$
are rational planes, then (1) implies that
\begin{description}
\item{(i)} $\deg (p_V) = \max\ \{ \dim(Q_i) : i=1,\dots,\ell \} $,
\item{(ii)} ${\mathrm{per}} (p_V)$ divides the least common multiple of
${\mathrm{ord}}(Q)$ where $Q$ ranges among connected components of
$$
Q_{i_1} \cap \dots \cap Q_{i_k}, \quad
1 \leq i_1 < \dots < i_k \leq \ell.
$$
\end{description}
In section 2, Theorem 2, we give bounds for $\deg(p_V)$
and ${\mathrm{per}}(p_V)$, when $V$ is defined over ${\Bbb Q}$.
More precise formulas for $\deg(p_V)$ and ${\mathrm{per}}(p_V)$, for general $V$,
are obtained later in section 5 after developing notation and
theory in sections 3 and 4.
The main ingredients of Laurent's proof of Theorem 1
can be stated as follows. Given an algebraic subset
$V \subset ({\Bbb C}^*)^r$, one defines a finite set $\Pi$
of mappings,
$$
\phi: ({\Bbb C}^*)^r \rightarrow \Gamma_{\phi}, \qquad \phi \in \Pi
$$
to algebraic groups $\Gamma_{\phi}$. These mappings have the
following properties.
\begin{description}
\item{(A)} All the fibers of $\phi$ are finite
unions of rational planes; and
\item{(B)}
$$
{\mathrm{Tor}}(V) = \bigcup_{\phi\in \Pi} \phi^{-1}({\cal S}_{\phi}),
$$
where ${\cal S}_{\phi} \subset \Gamma_{\phi}$ are finite subsets.
\end{description}
We will go further in this paper by finding the fibers in
(A) and the sets $S_\phi$ in (B)
explicitly and relating them to the degree and period of $p_V$.
Techniques for finding the fibers in (A) are given in section 3,
where we develop some tools for studying rational planes and give
properties of monomial mappings. Our approach
is to focus on the relationship between algebraic subgroups of
$({\Bbb C}^*)^r$ and subgroups of ${\Bbb Z}^r$. We use this correspondence
to describe the rational planes in fibers of
monomial mappings (see Lemma 3.4 and Cor. 3.5) and, as
a consequence, in the intersection
of a finite set of rational planes (see Prop. 3.6).
In many natural applications, for example, the Alexander strata
mentioned above, the algebraic subsets $V$ are defined over ${\Bbb Q}$, so we
will concentrate on this case.
Then, the sets $S_\phi$ in (B)
are related to formal ${\Bbb Q}$-linear combinations of roots of
unity.
In section 4, we review some of the theory of ${\Bbb Q}$-linear relations
among roots of unity using ideas of Schoenberg. The main idea is to
view formal ${\Bbb Q}$-linear relations as convex polygons with rational sides
and angles. Schoenberg shows in \cite{Sch:Cyc}
that all the convex polygons can be obtained from regular $p$-gons,
where $p$ ranges over prime numbers. We describe these ideas in section
4 and include proofs
of Mann's bound on the order of roots of unity satisying a linear equation
of a given length (see Prop. 4.3)
and of Schoenberg's result (see Cor. 4.4).
The results in sections 3 and 4 are used in section 5
to find the rational planes in an algebraic subset
$V \subset ({\Bbb C}^*)^r$ (see Prop. 5.2)
and to give formulas for $\deg(p_V)$
and bounds on ${\mathrm{per}}(p_V)$ (see Prop. 5.3 and Theorem 3) in terms of
defining equations for $V$.
In section 6 we conclude with some illustrative examples.
For instance, we find $p_V$ when $V$ is the Fermat curve
$$
x^m + y^m = 1, \qquad m \ge 1;
$$
and show (see Example 3)
that, in this case, the bounds for the degree and period
of $p_V$ given
in Theorem 2 are attained.
\heading{Acknowledgement.} I would like to thank E. Jahangard for useful
discussions during the research for this paper.
\section{Notation and main result.}
In this section we will set up some notation to state our main result:
Theorem 2. This theorem gives bounds on the degree and period of
$p_V$ rather than exact formulas for them and applies to varieties $V$
defined over ${\Bbb Q}$. A more general result, Theorem 3, is
stated and proved in section 5, but the bounds in Theorem 2 are
easier to state and compute.
\heading{Notation.}
Let $\Lambda_r = {\Bbb C}[t_1,\dots,t_r]$ denote the
ring of Laurent polynomials. A monomial $t_1^{\lambda_1}\dots t_r^{\lambda_r}
\in \Lambda_r$ will be written
as $t^\lambda$ where $\lambda = (\lambda_1,\dots,\lambda_r)$. Thus,
coordinate functions $t_1,\dots,t_r$ and the usual basis for ${\Bbb Z}^r$
determine a canonical isomorphism between $\Lambda_r$ and ${\Bbb C}[{\Bbb Z}^r]$.
A Laurent polynomial $f \in \Lambda_r = {\Bbb C}[t_1^{\pm 1},\dots,t_r^{\pm 1}]$
will be written as
$$
f(t) = \sum_{\lambda \in {\cal L}(f)} a_\lambda t^{\lambda},
$$
where ${\cal L}(f) \subset {\Bbb Z}^r$ is a finite subset and $a_\lambda \in {\Bbb C}^*$.
For a finite subset ${\cal F} \subset \Lambda_r$,
our main theorem will describe
the rational planes in the zero set $V({\cal F})$ and bounds for
the degree and period
of $p_{V({\cal F})}$ in terms of the functions in ${\cal F}$.
Two main ingredients will be numbers $N[R({\cal F})]$ and $D({\cal F})$
which can be computed from the number of coefficients of functions
in ${\cal F}$ and certain subgroups of ${\Bbb Z}^r$ associated to the
exponents of functions in ${\cal F}$.
Define
$$
R({\cal F}) = \max\ \{|{\cal L}(f)| : f \in {\cal F}\},
$$
where $|{\cal L}(f)|$ denotes the number of elements in ${\cal L}(f)$.
For any positive integer $R$, let $N[R]$ denote the product of primes
less than or equal to $R$.
The number $D({\cal F})$ takes longer to define.
For any $f \in \Lambda_r$,
let $\Pi_f$ be the set of all partitions ${\cal P}$ of ${\cal L}(f)$
such that each $\nu \in {\cal P}$ has at least two elements
and for any finite subset ${\cal F} = \{f_1,\dots,f_\ell\} \in \Lambda_r$, let
$$
\Pi_{{\cal F}} = \Pi_{f_1} \times \dots \times \Pi_{f_\ell}.
$$
For any partition ${\cal P} \in \Pi_f$,
let $\varepsilon({\cal P},f) \subset {\Bbb Z}^r$ be the subgroup generated by
$$
\{\lambda-\mu : \ \exists\nu \in {\cal P},\ \lambda,\mu \in \nu\};
$$
for any $\pi = ({\cal P}_1,\dots,{\cal P}_k) \in \Pi_{{\cal F}}$, let
$\varepsilon(\pi,{\cal F}) \subset {\Bbb Z}^r$ be the sum
$$
\varepsilon({\cal P}_1,f_1) + \dots + \varepsilon({\cal P}_k,f_k);
$$
and for any subset ${\cal U} \subset \Pi_{{\cal F}}$, let
$\varepsilon({\cal U},{\cal F}) \subset {\Bbb Z}^r$ be the subgroup generated by
$$
\bigcup_{\pi \in {\cal U}}\varepsilon(\pi,{\cal F}).
$$
For any finite abelian group $G$, let $D(G)$
be the largest order of any element in $G$.
Define
$$
D({\cal U}, {\cal F}) = D(\overline{\varepsilon({\cal U},{\cal F})}/{\varepsilon({\cal U},{\cal F})})
$$
and
$$
D({\cal F}) = \mathop{{\mathrm{lcm}}}\ \{D({\cal U},{\cal F}) : {\cal U} \subset \Pi_{{\cal F}}\}.
$$
\smallskip
Our main result is the following.
\begin{theorem} Let $V = V({\cal F}) \subset ({\Bbb C}^*)^r$ be any algebraic subset
defined over ${\Bbb Q}$.
\begin{description}
\item{(i)} For each maximal
rational plane $Q$ in $V$, there is a $\pi \in \Pi_{{\cal F}}$
so that $Q$ is a translate of the affine subtorus of $({\Bbb C}^*)^r$ defined by
the set of binomials
$$
\{t^\lambda - 1 : \lambda \in \overline{\varepsilon(\pi,{\cal F})} \};
$$
\item{(ii)}
$$\deg(p_V) \leq r - \min_{\pi\in \Pi_{{\cal F}}} {\mathrm{rank}}(\overline{\varepsilon(\pi,{\cal F})});
$$
\item{(iii)}
$$
{\mathrm{per}}(p_V) \quad | \quad N[R({\cal F})]\ D({\cal F}).
$$
\end{description}
\end{theorem}
This theorem together with finer versions will be proved in section 5.
\section{Character groups and rational planes.}
The main tool we use in this paper is a natural correspondence between
between affine subtori of $({\Bbb C}^*)^r$ and subgroups of ${\Bbb Z}^r$.
The relation is explained in section 3.1 and applied in section 3.2
to give properties of rational planes.
\subsection{Subgroups of ${\Bbb Z}^r$ and algebraic subgroups of $({\Bbb C}^*)^r$.}
For any subgroup $\varepsilon \subset {\Bbb Z}^r$, let $V(\varepsilon)$ be the
closed points of ${\mathrm{Spec}} ({\Bbb C}[{\Bbb Z}^r/\varepsilon])$. Then, there is a natural
identification of $V(\varepsilon)$ with the group of characters of
${\Bbb Z}^r/\varepsilon$, since any closed point corresponds to a ring
epimorphism ${\Bbb C}[{\Bbb Z}^r/\varepsilon] \rightarrow {\Bbb C}$.
Define $I_\varepsilon \subset \Lambda_r$ to be the ideal generated by
$$
\{ t^\lambda - 1 : \lambda \in \varepsilon\}.
$$
Then the kernel of the map ${\Bbb C}[{\Bbb Z}^r] \rightarrow {\Bbb C}[{\Bbb Z}^r/\varepsilon]$
is the image of $I_\varepsilon$ in ${\Bbb C}[{\Bbb Z}^r]$ under the identification
$\Lambda_r \cong {\Bbb C}[{\Bbb Z}^r]$. Thus, there is a natural embedding of $V(\varepsilon)$
into $({\Bbb C}^*)^r$ with $V(\varepsilon) = V(I_\varepsilon)$ and
$I_\varepsilon = I(V(\varepsilon))$.
This embedding preserves group structure as well as the algebraic structure
of $V(\varepsilon)$.
For any algebraic subset $V \subset ({\Bbb C}^*)^r$, define
$$
\varepsilon(V) = \{\lambda \in {\Bbb Z}^r : t^\lambda - 1\ \mbox{vanishes on}\ V\}.
$$
\begin{lemma} If $\varepsilon \subset {\Bbb Z}^r$ is any
subgroup, then $\varepsilon(V(\varepsilon)) = \varepsilon$.
\end{lemma}
\heading{Proof.}
Observe that for
any subgroup $\varepsilon \subset {\Bbb Z}^r$, if $t^\lambda-1 \in I_\varepsilon$,
then $\lambda$ must be a sum of elements in $\varepsilon$.
This gives the inclusion $\varepsilon(V(\varepsilon)) \subset \varepsilon$.
The other inclusion is clear.
\qed
\begin{lemma} Any subtorus $P$ of $({\Bbb C}^*)^r$ is of the form $V(\varepsilon)$
for some $\varepsilon \subset {\Bbb Z}^r$.
\end{lemma}
\heading{Proof.}
Since $P$ is itself isomorphic to $({\Bbb C}^*)^s$ for some $0 \leq s \leq r$,
the embedding $\psi : P \rightarrow ({\Bbb C}^*)^r$
defines a ring epimorphism on coordinate rings
$$
\psi^*: {\Bbb C}[{\Bbb Z}^r] \rightarrow {\Bbb C}[{\Bbb Z}^s].
$$
We will show that $\psi^*$ restricted to
${\Bbb Z}^r$ induces an epimorphism ${\Bbb Z}^r \rightarrow {\Bbb Z}^s$.
This is the same as saying that the components of
$\psi$ are monomials. Since $\psi$ preserves multiplication
it also preserves the multiplication of $({\Bbb C}^*)$. Thus,
the components of $\psi$ must be homogeneous. Furthermore,
since the components of $\psi$ can have no zeros or poles, other than
the origin, they must be monomials.
Let $\varepsilon \subset {\Bbb Z}^r$ be the kernel of the epimorphism
${\Bbb Z}^r \rightarrow {\Bbb Z}^s$ induced
by $\psi$. Then, the kernel of $\psi^*$ is $I_\varepsilon$, so
$P = V(\varepsilon)$.\qed.
\heading{Remark.} Another way to say the above is that
there is a naturally duality between algebraic subgroups of $({\Bbb C}^*)^r$
and quotient groups of ${\Bbb Z}^r$ given by the contravariant functors
${\mathrm{Hom}}_{{\cal A}}(-,{\Bbb C}^*)$ and ${\mathrm{Hom}}(-,{\Bbb C}^*)$, where ${\mathrm{Hom}}_{{\cal A}}(-,-)$ are
morphisms of algebraic subgroups of $({\Bbb C}^*)^r$ which preserve both
the algebraic and multiplicative structure. For brevity and because
they are not necessary for the results of this paper, we omit the details.
\subsection{Rational planes and monomial mappings.}
Given a subgroup $\varepsilon \subset {\Bbb Z}^r$, let
$\overline{\varepsilon}$ be the subgroup of ${\Bbb Z}^r$ defined by
$$
\overline{\varepsilon} = \{\lambda \in {\Bbb Z}^r : \exists n \in {\Bbb N},\
n\lambda \in \varepsilon \}.
$$
For any subgroup $\varepsilon \subset {\Bbb Z}^r$
and $m \in {\Bbb N}$, let
$$
\varepsilon_{m} = \{\lambda \in {\Bbb Z}^r : m\lambda \in \varepsilon \}.
$$
\smallskip
\begin{lemma} For any subgroup $\varepsilon \in {\Bbb Z}^r$,
$V(\overline\varepsilon)$ is an affine subtorus of $({\Bbb C}^*)^r$.
Furthermore, there is an injective (non-canonical) endomorphism
$$
T : \chargp{\overline{\Epsilon}/\Epsilon} \hookrightarrow ({\Bbb C}^*)^r,
$$
from the character group $\chargp{\overline{\Epsilon}/\Epsilon}$ into $({\Bbb C}^*)^r$,
such that $V(\varepsilon)$ decomposes as a disjoint union
$$
V(\varepsilon) = \bigcup_{\eta \in T(\chargp{\overline{\Epsilon}/\Epsilon})} \eta V(\overline\varepsilon).
$$
\end{lemma}
\heading{Proof.}
The epimorphisms
$$
{\Bbb Z}^r \rightarrow {\Bbb Z}^r/\varepsilon \mapright{} {\Bbb Z}^r/{\overline\varepsilon}
$$
induce inclusions
$$
V(\overline{\varepsilon}) \hookrightarrow V(\varepsilon) \hookrightarrow
({\Bbb C}^*)^r.
$$
Since ${\Bbb Z}^r/{\overline{\varepsilon}}$ is a free abelian group,
$V(\overline{\varepsilon})$ is an affine subtorus of $({\Bbb C}^*)^r$.
The inclusion $\overline{\Epsilon}/\Epsilon \hookrightarrow {\Bbb Z}^r/\varepsilon$
induces a surjective map on character groups
$$
\psi : V(\varepsilon) \rightarrow \chargp{\overline{\Epsilon}/\Epsilon}
$$
whose identity fiber $F_1$ equals $V(\overline\varepsilon)$.
We need to find a splitting for $\psi$.
Write ${{\Bbb Z}^r}/\varepsilon$ as the product
$$
{{\Bbb Z}^r}/\varepsilon = {\Bbb Z}^s \times G,
$$
where $G$ is finite. Then $G \cong \overline{\Epsilon}/\Epsilon$ and
${\Bbb Z}^s \cong {\Bbb Z}^r/\overline{\varepsilon}$.
Thus, there is a surjection ${\Bbb Z}^r/\varepsilon \rightarrow \overline{\Epsilon}/\Epsilon$
whose restriction to $\overline{\Epsilon}/\Epsilon$ is the identity.
Let $T : \chargp{\overline{\Epsilon}/\Epsilon}\rightarrow V(\varepsilon)$ be the
induced endomorphism of character groups.
Then $T$ defines a splitting for $\psi$ and we are done.
\qed
\smallskip
\heading{Notation.}
A map
$$
\psi : \chargp{H} \rightarrow \chargp{G}
$$
between character groups
which is induced by a homomorphism $\psi^* : G \rightarrow H$
is called a {\it monomial mapping}, since the induced
map on coordinate rings is just the linear extension of $\psi^*$
to a mapping ${\Bbb C}[G] \rightarrow {\Bbb C}[H]$.
Given a monomial mapping $\psi : ({\Bbb C}^*)^r \rightarrow \chargp{G}$,
let $\varepsilon(\psi) \subset {\Bbb Z}^r$ be the image of the induced map
$$
\psi^* : G \rightarrow {\Bbb Z}^r.
$$
\smallskip
\begin{lemma} Let $\psi : ({\Bbb C}^*)^r \rightarrow \chargp{G}$
be a monomial map and set
$\varepsilon = \varepsilon(\psi)$. For each $\mu \in {\mathrm{Tor}}({\mathrm{im}}(\psi))$,
there is a $\tau \in \overline{\Epsilon}/\Epsilon$
such that, for some $\eta \in ({\Bbb C}^*)^r$ with ${\mathrm{ord}}(\eta) = {\mathrm{ord}}(\mu){\mathrm{ord}}(\tau)$,
$$
\psi^{-1}(\mu) = \eta V(\varepsilon).
$$
\end{lemma}
\heading{Proof.} If $m = {\mathrm{ord}}(\mu)$, then the fiber
$\psi^{-1}(\mu)^m$ lies in the identity fiber $V(\varepsilon)$.
Furthermore, if $\eta \in \psi^{-1}(\mu)$ has finite order, then
$m$ divides ${\mathrm{ord}} (\eta)$.
We have $V(\varepsilon)^m = V(\varepsilon_{m})$, so
by Lemma 3.3,
$$
V(\varepsilon_{m}) = \bigcup_{\gamma \in T(\overline{\varepsilon}/
{\varepsilon_{m}})}
\gamma V(\overline{\varepsilon}).
$$
Thus, for any $\eta \in \psi^{-1}(\mu)$,
$$
\psi^{-1}(\mu)^m = \eta^m V(\varepsilon)^m = \eta^m V(\varepsilon_{m})
$$
and $\psi^{-1}(\mu)^m$ is a translate of $V(\varepsilon_{m})$
in $V(\varepsilon)$.
Thus, $\psi^{-1}(\mu)^m$ must contain an element of $T(V(\overline{\Epsilon}/\Epsilon))$.
This means
we could have chosen $\eta \in \psi^{-1}(\mu)$ such that
$\eta^m \in T(V(\overline{\Epsilon}/\Epsilon))$.
If $\varepsilon_{m} = \varepsilon$, then $V(\varepsilon)^m = V(\varepsilon)$
and we could have chosen $\eta$ so that $\eta^m = 1$,
which would imply ${\mathrm{ord}}(\eta) = m$. Otherwise, set
$\tau = \eta^m$. We will show
that ${\mathrm{ord}}(\eta) = {\mathrm{ord}}(\tau) m$.
We've seen that $m$ divides ${\mathrm{ord}}(\eta)$. If $k = {\mathrm{ord}}(\eta)/m$,
then $\tau^k = \eta^{mk} = 1$, so ${\mathrm{ord}}(\tau)$ divides $k$.
Since $\eta^{{{\mathrm{ord}}(\tau)}m} = \tau^{{\mathrm{ord}}(\tau)} = 1$, we have
${\mathrm{ord}}(\eta) = {\mathrm{ord}}(\tau)m$.\qed
\begin{corollary} Let $\psi : ({\Bbb C}^*)^r \rightarrow \chargp{G}$ be a
monomial map and let $\varepsilon = \varepsilon(\psi)$. Then, for
any $\mu \in {\mathrm{im}} (\psi)$ and connected component
$Q \subset \psi^{-1}(\mu)$, we have
\begin{description}
\item{(i)} $Q$ is a translate of $V(\overline{\varepsilon})$,
\item{(ii)} $\dim(Q) = r - {\mathrm{rank}}(\overline{\varepsilon})$, and
\item{(iii)} if $\mu$ has finite order, then
$$
{\mathrm{ord}}(\mu)\ | \ {\mathrm{ord}} (Q) \ |
\ {\mathrm{ord}}(\mu)D({\overline{\varepsilon}}/\varepsilon),
$$
where $D(\overline{\Epsilon}/\Epsilon)$ is the largest order of any element of
$\overline{\Epsilon}/\Epsilon$.
\end{description}
\end{corollary}
We will
now describe the dimensions and orders of intersections of rational
planes.
\begin{proposition} Let $\varepsilon_1,\dots,\varepsilon_k \subset {\Bbb Z}^r$ be
any subgroups of ${\Bbb Z}^r$ and let $\eta_1,\dots,\eta_k \in {\mathrm{Tor}}(({\Bbb C}^*)^r)$.
Let $Q_1,\dots,Q_k \subset ({\Bbb C}^*)^r$ be defined by
$$
Q_i = \eta_i V(\varepsilon_i), \qquad i=1,\dots,k.
$$
Let $\varepsilon = \varepsilon_1 + \dots + \varepsilon_k$ and
$\eta = (\eta_1,\dots,\eta_k)$. Let
$$
\begin{array}{rcl}
\rho: \varepsilon_1 \times \dots \times \varepsilon_k &\rightarrow& \varepsilon\\
(\lambda_1,\dots,\lambda_k) &\mapsto& \lambda_1+\dots+\lambda_k
\end{array}
$$
and let $\gamma : \varepsilon_1 \times \dots \times \varepsilon_k
\rightarrow ({\Bbb Z}^r )^k$ be the inclusion map.
Then $Q_1 \cap \dots \cap Q_k$ is nonempty if and only if
$\gamma^*(\eta) \in {\mathrm{im}}(\rho^*)$ and
for any connected component $Q \subset Q_1 \cap \dots \cap Q_k$,
we have
\begin{description}
\item{(i)} $Q$ is a translate of $V(\overline{\varepsilon})$,
\item{(ii)} $\dim (Q) = r - {\mathrm{rank}} (\overline\varepsilon)$, and
\item{(iii)} if $\eta$ has finite order, then
$$
{\mathrm{ord}}(\eta)\ |\ {\mathrm{ord}}(Q) \ | \ {\mathrm{ord}}(\eta)D(\overline{\Epsilon}/\Epsilon).
$$
\end{description}
\end{proposition}
\heading{Proof.}
Consider the commutative diagram
$$
\cd
{
&{\Bbb Z}^r/{\varepsilon} &\leftarrow &{{\Bbb Z}^r}/{\varepsilon_1} \times\dots\times {{\Bbb Z}^r}/{\varepsilon_k}\cr
&\uparrow{} &&\uparrow{}\cr
&{\Bbb Z}^r &\leftarrow &{\Bbb Z}^r \times \dots \times {\Bbb Z}^r \cr
&\uparrow{}&&\uparrow{}\cr
&\varepsilon &\leftarrow
&\varepsilon_1 \times \dots \times \varepsilon_r
}
$$
where the horizontal arrows are given by adding coordinates,
the bottom vertical arrows are inclusions and the top vertical
arrow are quotient maps. For $i=1,\dots,k$, let $P_i = V(\varepsilon_i)$.
Then we
have a commutative diagram of induced maps on the associated algebraic sets:
$$
\cd
{
&V(\varepsilon) &\mapright{} &P_1 \times \dots \times P_k \cr
&\downarrow{} &&\downarrow{}\cr
&({\Bbb C}^*)^r &\mapright{\alpha} &({\Bbb C}^*)^r\times\dots\times ({\Bbb C}^*)^r\cr
&\downarrow{\beta} &&\downarrow{\gamma^*}\cr
&\chargp{\varepsilon} &\mapright{\rho^*}&\chargp{\varepsilon_1}
\times \dots \times \chargp{\varepsilon_k}.}
$$
The preimage $\alpha^{-1}(Q_1\times\dots \times Q_k)$
equals the intersection $Q_1 \cap \dots \cap Q_k$.
If $Q \subset Q_1 \cap \dots \cap Q_k$ is a connected component, then
it is a connected component of a fiber of $\beta$. The intersection
is nonempty if and only if
$$
\gamma^*(Q_1\times\dots\times Q_k) \cap {\mathrm{im}}(\rho^*) \neq \emptyset.
$$
The claim now follows
from Cor. 3.5.\qed
\smallskip
\section{Formal ${\Bbb Q}$-linear relations.}
In this section, we describe ${\Bbb Q}$-linear
relations between roots of unity and polar rational polygons
following the work of Schoenberg \cite{Sch:Cyc}
and Mann \cite{Mann:LinRels}. The aim is to give a constructive
method for producing all ${\Bbb Q}$-linear relations and give a
bound on the orders of roots of unity satisfying a linear equation
in terms of the number of coefficients of the equation.
Other work in this area can be found in {\cite{Len:Van}} and \cite{C-J:Trig}.
For any $n \in N$, let $\zeta_n = \exp(2 \pi \sqrt{-1}/n)$. For
$z\in {\Bbb C}^*$, let $\theta(z) = \arg(z)/{2 \pi}$. Thus,
$$
z = |z|\exp(2 \pi \sqrt{-1}\ \theta(z)).
$$
A {\it polar rational polygon}, or prp, is an oriented polygon
in the complex plane with no parallel edges, such that each edge is a vector
with rational length and with angle equal to a rational multiple of $2 \pi$.
Then prp's can be put into one-to-one correspondence
with formal
${\Bbb Q}$-linear relations of the form
\begin{eqnarray}
\sum_{i=1}^k a_i \epsilon_i = 0.
\end{eqnarray}
where $a_i \in {\Bbb Q}^*$ and
$$
0 \leq \theta({\mathrm{sign}}(a_1)\epsilon_1) < \dots
< \theta({\mathrm{sign}}(a_k)\epsilon_k) < 1.
$$
Given a prp $T$ defined by (2), we will call $a_i\epsilon_i$ the
sides of $T$, $|a_i|$ the {\it side lengths} and $\epsilon_i$
the {\it side angles}.
The {\it order} of $T$, written
${\mathrm{ord}}(T)$, is the
least $n$ such that, for some root of unity $\eta$,
we have
$$
\eta^n \epsilon^n = 1,
$$
for all side angles $\epsilon$ of $T$.
The length of $T$, written ${\mathrm len} (T)$ is $k$, the number
of sides.
Any formal ${\Bbb Q}$-linear equation
$$
\sum_{j=1}^k b_j \eta_j
$$
can be put
in the form (2) by the following:
\begin{description}
\item{(i)} if $\theta(\eta_j)\ ({\mathrm{mod}}\ 1) \ge 1/2$,
then replace $a_j$ by $-a_j$ and $\eta_j$ by $\zeta_2 \eta_j$;
\item{(ii)} if $\theta(\eta_j) = \theta(\eta_\ell)$, then replace
$b_j \eta_j + b_\ell \eta_\ell$ by $(b_j + b_\ell)\eta_j$;
\item{(iii)} remove any summand whose coefficient is zero; and
\item{(iv)} reorder the summands.
\end{description}
It is easy to see that to any formal ${\Bbb Q}$-linear equation there is
a unique corresponding equation of the form (2), which can be obtained
using the above steps, and hence a unique
prp.
It follows that the set of prp's forms a commutative
${\Bbb Q}[{\mathrm{Tor}}({\Bbb C}^*)]$-algebra, where ${\mathrm{Tor}}({\Bbb C}^*)$ is the set of
roots of unity. One just notes that formal ${\Bbb Q}$-linear equations
are closed under addition, multiplication and scalar multiplication
by elements of ${\mathrm{Tor}}({\Bbb C}^*)$.
The algebra operations on a prp will be the same as those for
the corresponding formal ${\Bbb Q}$-linear equation. The resulting
${\Bbb Q}$-linear equation may not be of the form (2), but there is
a unique prp associated to it which we take as the result of
the operation.
The algebra operations have the following geometric interpretations.
Multiplication by a positive rational
$a \in {\Bbb Q}_+$
scales the corresponding polygon by $a$. Multiplication by any root of
unity $\eta \in {\mathrm{Tor}}({\Bbb C}^*)$ rotates the polygon by
the argument of $\eta$. For example, multiplying the polygon in
figure 1 by
$\eta = -1$ yields the polygon rotated by 180 degrees shown in figure 2.
$$
\epsffile{neg}
$$
Summing is like taking the union except that one needs to reorder the
sides and get rid of redundancies.
Two prp's will be said to be {\it disjoint} if they do not share any sides
angles.
A prp $T$ is {\it primitive} if there are no disjoint prp's $S$ and $U$ such
that $T = S + U$.
Geometrically, a prp is primitive if there is no
way to rearrange the edges of the polygon to get a union of polygons
joined at vertices.
While any prp has a decomposition into a sum of disjoint primitives, this
decomposition is not necessarily unique.
For example, consider the ${\Bbb Q}$-linear equation given by expanding out
\begin{eqnarray}
(1 + \zeta_3 + \zeta_3^2)(1 + \zeta_5 + \zeta_5^2 + \zeta_5^3 + \zeta_5^4) = 0.
\end{eqnarray}
The associated prp is given in figure 3.
$$
\epsffile{tot}
$$
The prp defined by (3) can be decomposed into primitive
prp's as in figure 4 and figure 5.
$$
\epsffile{decomp}
$$
The decomposition in figure 4 comes from writing (3) as
$$
\sum_{i=1}^2 \zeta_3^i\ [1 + \zeta_5 + \zeta_5^2 + \zeta_5^3 + \zeta_5^4] = 0
$$
and figure 5 comes from writing (3) as
$$
\sum_{i=1}^4 \zeta_5^i\ [1 + \zeta_3 + \zeta_3^2] = 0.
$$
\smallskip
The sum of nondisjoint prp's could have smaller length than the
sum of the total as we see in the next example.
Let $A$ be the prp defined by
$$
\zeta_6 + \zeta_6^2 + (-1) = 0
$$
Then $\zeta_5\ A + T_5$ is the prp (see figure 6)
given by the following ${\Bbb Q}$-linear
equation
$$
1 + (\zeta_6 + \zeta_6^5)\zeta_5 + \zeta_5^2 + \zeta_5^3 + \zeta_5^4 = 0
$$
or
$$
1 + \zeta_{30} + \zeta_{30}^{11} + \zeta_5^2 + \zeta_5^3 + \zeta_5^4 = 0.
$$
$$
\epsffile{pent}
$$
In general we have the following.
\begin{lemma} The lengths of sums satisfies the following
inequalities
$$
\max\ \{{\mathrm len}(T_1),{\mathrm len}(T_2)\} \leq
{\mathrm len}(T_1 + T_2) \leq {\mathrm len} (T_1) + {\mathrm len} (T_2),
$$
with equality on the right hand side if $T_1$ and $T_2$
are disjoint and equality on the left hand side if the set of
side angles of one prp is contained in the other's.
\end{lemma}
We will make use of the following types of prp's.
For any prime $p$, let $\sigma_p(x)$ be the cyclotomic
polynomial
$$
\sigma_p(x) = 1 + x + \dots + x^{p-1}.
$$
Let $T_p$ be the prp defined by $\sigma_p(\zeta_p) = 0$.
Let $n = mp$, where $p$ is a prime not dividing $m$.
(For example, if $n$ is prime then $m = 1$.)
Then the minimal polynomial $\sigma_{n,p}$
for $\zeta_n$ over ${\Bbb Q}[\zeta_m]$ is given by
$$
\sigma_{n,p}(x) = \zeta_m^{a(p-1)} + \zeta_m^{a(p-2)}x
+ \dots + x^{p-1},
$$
where $a$ is the integer $0 \leq a < m$, such that
$ap \equiv 1\ ({\mathrm{mod}}\ m)$.
Let $T_{n,p}$ be the prp defined by $\sigma_{n,p}(\zeta_n) = 0$.
That is, the prp corresponding to the formal ${\Bbb Q}$-linear relation
$$
\sum_{i=0}^{p-1} \zeta_n^{ap(p-1-i) + i} = 0.
$$
\begin{lemma} Let $n$ be an integer, $p$ a prime dividing $n$, such
that $p^2$ doesn't divide $n$. The prp $T_{n,p}$ is a multiple of $T_p$
by some element of ${\Bbb Q}[{\mathrm{Tor}}({\Bbb C}^*)]$.
\end{lemma}
\heading{Proof.} Let $m = n/p$ and let $a$ be such that $ap \equiv 1
\ ({\mathrm{mod}}\ m)$. Let $r$ be any integer such that $ap = 1 + mr$.
Then, since $\zeta_n^m = \zeta_p$, we have
\begin{eqnarray*}
\zeta_n^{ap(p-1-i) + i} &=& \zeta_n^{mr(p-1-i) + (p-1)}\\
&=& \zeta_p^{r(p-1-i)} \zeta_n^{p-1}.
\end{eqnarray*}
Thus, $T_{n,p}$ is defined by the formal linear relation
$$
\zeta_n^{p-1}\sum_{i=0}^{p-1} \zeta_p^{r(p-1-i)} = 0.
$$
Since $p$ doesn't divide $r$, $r(p-1-i)$ ranges in $0,\dots,p-1$
as $i$ ranges in $0,\dots,p-1$. Therefore,
$$
T_{n,p} = \zeta_n^{p-1} T_p.
$$
\qed
Our proof of the following result uses essentially the same ideas
as Mann uses in (\cite{Mann:LinRels}, Theorem 1), except that we
take more advantage of the structure of prp's as a
${\Bbb Q}[{\mathrm{Tor}}({\Bbb C}^*)]$-algebra.
\begin{proposition} Let $T$ be a primitive prp of length $r$
and order $n$. Then we have
\begin{description}
\item{(i)} $n$ is square free;
\item{(ii)} for any prime $p$ dividing $n$, $p \leq r$;
\item{(iii)} for any prime $p$ dividing $n$,
$T$ is a ${\Bbb Q}[{\mathrm{Tor}}({\Bbb C}^*)]$-linear combination
of $T_{n,p}$ and prp's of order $n/p$.
\end{description}
\end{proposition}
\heading{Proof.}
Let
\begin{eqnarray}
\sum_{i=1}^r a_i \epsilon_i = 0
\end{eqnarray}
be the ${\Bbb Q}$-linear relation corresponding to $T$ of the form (2).
If $T$ has order $n$, then by
multiplying $T$ by an element of ${\mathrm{Tor}}({\Bbb C}^*)$, if necessary, we
can assume that all the $\epsilon_i$ in (4) have order $n$.
We will show that $n$ is square free and for any prime $p$
dividing $n$, $p \leq r$.
Let $p$ be a prime dividing $n$ and let $m = n/p$. Since
$\zeta_n^p = \zeta_m$, we can write
$$
\epsilon_i = \eta_i\zeta_{n}^{\alpha_i},
$$
for $i=1,\dots,r$, where $\eta_i \in {\Bbb Q}[\zeta_{m}]$ and
$0 \leq \alpha_i \leq p-1$. Define
$$
q_{T,n,p}(x) = \sum_{i=1}^r a_i \eta_i x^{\alpha_i}.
$$
This is a polynomial with coefficients in ${\Bbb Q}[\zeta_{m}]$
satisfied by $\zeta_{n}$ and hence the minimal polynomial
$\sigma_{n,p}(x)$ for $\zeta_{n}$ over ${\Bbb Q}[\zeta_{m}]$
divides $q_{T,n,p}(x)$.
If $p^2$ divides $n$ then $\deg(\sigma_{n,p}) = p$ which is
strictly greater than the degree of $q_{T,n,p}(x)$, and hence
$q_{T,n,p}(x)$ is identically zero. Since $T$ is
primitive, this means that all the $\alpha_i$ are the
same, but then we can multiply $T$ by $\zeta_p^{-1}$ to get
a prp of order $n/p$ which is a contradiction, since the order
of a prp is preserved under multiplication by roots of unity.
Thus, $n$ is square free (proving (i)) and $\sigma_{n,p}(x)$ divides
$q_{T,n,p}(x)$.
Set
$$
A_\alpha = \sum_{\alpha_i = \alpha} a_i \eta_i
$$
and write $q_{T,n,p}(x)$ as
$$
q_{T,n,p}(x) = \sum_{\alpha = 0}^{p-1} A_\alpha x^\alpha.
$$
Since $q_{T,n,p}(x)$ is not identically
zero and has degree less than or equal to $p-1$,
we have
\begin{eqnarray}
q_{T,n,p}(x) = B \sigma_{n,p}(x)
\end{eqnarray}
for some invertible $B \in {\Bbb Q}[\zeta_m]$.
By (5), evaluating $B^{-1}\ A_\alpha$ as a
complex number gives the $\ell$th
coefficient of $\sigma_{n,p}(x)$.
Thus, the formal ${\Bbb Q}$-linear equation
$$
B^{-1}\ A_\alpha - \zeta_m^{a(p-1-\ell)} = 0
$$
defines a prp, which we'll call $T_\alpha$.
Then $T$ has the following decomposition into a sum of prp's
$$
T = B\ \sum_{\alpha = 0}^{p-1} T_\alpha + B\ T_{n,p}.
$$
Since ${\mathrm{ord}}(T_\alpha) = m$, we have proved (iii).
By Lemma 4.1, ${\mathrm len} (T) \ge {\mathrm len} (T_{n,p}) = p$, so $p \leq r$
which proves (ii).
\qed
Proposition 4.3 leads to a geometric method for generating all prps as we see
in the following result, originally proved by Schoenberg in \cite{Sch:Cyc}.
\begin{corollary} The set of prp's, considered as a ${\Bbb Q}[{\mathrm{Tor}}({\Bbb C}^*)]$-module,
is generated by the set of $T_p$, where $p$ ranges over all primes.
\end{corollary}
\heading{Proof.} By Lemma 4.2,
$T_{n,p}$ is a multiple of $T_p$ by an element of ${\Bbb Q}[\zeta_{n/p}]$.
The rest follows from Prop. 4.3 and induction on the number
of primes dividing $n$.
\qed
It follows that geometrically one can construct all prp's by starting
with regular $p$-gons, for primes $p$, and doing the operations of
rotation, stretching and ``adding", where ``adding" means taking the
union of sides, grouping like side angles, getting rid of sides of
zero length and reordering the sides, as described above.
\section{Behavior of torsion points.}
We start in this section with a review of the proof of Theorem 1.
Then, using the
relation between finitely generated groups and rational planes
developed in section 3, we study the degree and period of $p_V$.
Finally, we give a proof of Theorem 2.
\subsection{Proof of Theorem 1.}
Let $V = V(f_1,\dots,f_r) \subset ({\Bbb C}^*)^r$ be any algebraic subset.
We want to find a finite number of rational planes $Q_1,\dots,Q_\ell$
contained in $V$ so that
$$
\Zar{Tor(V)} = \bigcup_{i=1}^\ell Q_i.
$$
\heading{Step 1.} We can reduce to the case of hypersurfaces, since
$V = V(f_1) \cap \dots \cap V(f_r)$ and, by Prop. 3.6, a
finite intersection of rational planes is a finite union of rational
planes.
\smallskip
\heading{Step 2.} Let $V = V(f)$, where $f \in \Lambda_r$ defined by
$$
f = \sum_{\lambda \in {\cal L}(f)} a_\lambda t^\lambda, \quad a_\lambda \in {\Bbb C}^*,
$$
for ${\cal L}(f) \subset {\Bbb Z}^r$ is a finite subset. We'll show that
$$
\Zar{{\mathrm{Tor}}(V)} = \bigcup_{i=1}^k \psi_i^{-1}(S_i),
$$
for some finite collection of monomial maps $\psi_i: ({\Bbb C}^*)^r \rightarrow
\chargp{G_i}$, some character groups $\chargp{G_i}$, and some finite sets
of torsion elements $S_i \subset {\mathrm{Tor}}(\chargp{G_i})$.
It then follows from Cor. 3.5 that $\Zar{{\mathrm{Tor}}(V)}$ is a finite union of
rational planes.
Let ${\cal L} = {\cal L}(f)$ and
let $\ell(f)$ be the linear polynomial
$$
\ell(f) = \sum_{\lambda \in {\cal L}} a_\lambda x_\lambda
$$
where $x_\lambda$
are independent variables, for $\lambda$ ranging in
${\cal L}$.
Then
$$
V(f) = \alpha_f^{-1}(V(\ell(f))),
$$
where $\alpha_f : ({\Bbb C}^*)^r \rightarrow ({\Bbb C}^*)^{{\cal L}}$
is the monomial map induced by
\begin{eqnarray*}
{\Bbb Z}^{{\cal L}} &\rightarrow& {\Bbb Z}^r\\
e_\lambda &\mapsto& \lambda.
\end{eqnarray*}
\smallskip
\heading{Step 3.} Let $\Pi_f$ be the set of
partitions of ${\cal L}$, where for all $\nu \in {\cal P} \in \Pi_f$,
$\nu$ has at least 2 elements.
Fix ${\cal P} \in \Pi_f$.
Let $L_{{\cal P}}$ be the system of linear equations
$$
\ell_\nu(f) = \sum_{\lambda \in \nu}a_{\lambda}t_{\lambda},
$$
for each $\nu \in {\cal P}$.
This defines a system of linear equations
defined on
$$
W({\cal P}) = \prod_{\nu \in {\cal P}} ({\Bbb C}^*)^{\nu}.
$$
Let ${\cal T}({\cal P},f)$ be the set of
elements $(\epsilon_\lambda)_{\lambda \in \nu, \nu \in {\cal P}} \in W({\cal P})$
satisfying the system $L_{{\cal P}}$:
$$
\sum_{\lambda \in \nu} a_\lambda \varepsilon_\lambda = 0,
\qquad\mathrm{for all}\ \nu \in {\cal P}.
$$
Then
$$
V(\ell(f)) = \bigcup_{{\cal P} \in \Pi_f} {\cal T}({\cal P},f).
$$
For each $\nu \in {\cal P}$, choose $\lambda_\nu \in \nu$
and let $\nu^* = \nu \setminus \{\lambda_\nu\}$.
Define
$$
W({\cal P})^* = \prod_{\nu \in {\cal P}} ({\Bbb C}^*)^{\nu^*}.
$$
Let
$$
\beta_\nu : ({\Bbb C}^*)^\nu \rightarrow ({\Bbb C}^*)^{\nu^*}
$$
be the map defined by
\begin{eqnarray*}
{\Bbb Z}^{\nu^*} &\rightarrow& {\Bbb Z}^\nu\\
e_\lambda &\mapsto& e_\lambda - e_{\lambda_\nu}.
\end{eqnarray*}
Let
$$
\beta_{\cal P} : W({\cal P}) \rightarrow W({\cal P})^*
$$
be the product of the maps $\beta_{\nu}$.
Let ${\cal V}({\cal P},f)$ be the set of solutions
$(\varepsilon_\lambda)_{\lambda \in \nu^*, \nu \in {\cal P}} \in W({\cal P})^*$
to the inhomogeneous equations
$$
\sum_{\lambda \in \nu^*} (-a_{\lambda}/{a_{\lambda_\nu}})
t_\lambda = 1, \qquad \nu \in {\cal P}.
$$
Then ${\cal T}({\cal P},f) = \beta_{{\cal P}}^{-1}({\cal V}({\cal P},f))$, so we have
\begin{eqnarray}
V(f) = \bigcup_{{\cal P}\in \Pi_f} \alpha_f^{-1} \beta_{{\cal P}}^{-1}({\cal V}({\cal P},f)).
\end{eqnarray}
\smallskip
\heading{Step 4.} Let
$$
\psi_{{\cal P},f} : ({\Bbb C}^*)^r \rightarrow W({\cal P})^*
$$
be the composition $\psi_{{\cal P},f} = \beta_{{\cal P}}\circ \alpha_f$.
Then $\psi_{{\cal P},f}$ is induced by the maps
\begin{eqnarray*}
\bigoplus_{\nu \in {\cal P}} {\Bbb Z}^{\nu^*} &\rightarrow& {\Bbb Z}^r\\
e_\lambda &\mapsto& \lambda - \lambda_\nu \quad \mbox{if $\lambda \in \nu$.}
\end{eqnarray*}
Equation (6) now becomes
\begin{eqnarray}
V(f) = \bigcup_{{\cal P}\in \Pi_f}\psi_{{\cal P},f}^{-1}({\cal V}({\cal P},f)).
\end{eqnarray}
Since monomial maps take torsion points to torsion points, (7) implies
$$
\Zar{{\mathrm{Tor}}(V(f))} = \bigcup_{{\cal P}\in \Pi_f} \psi_{{\cal P},f}^{-1}({\mathrm{Tor}}({\cal V}({\cal P},f)))
$$
If ${\cal P}$ and ${\cal P}'$ are two elements in $\Pi_f$,
${\cal P}'$ is a {\it refinement} of ${\cal P}$ if
for any $\nu' \in {\cal P}'$,
there is a $\nu \in {\cal P}$ such that $\nu' \subset \nu$.
A refinement ${\cal P}'$ of ${\cal P}$ is {\it proper} if ${\cal P}'$ is not equal to ${\cal P}$.
If ${\cal P}'$ is a refinement of ${\cal P}$, then
$$
{\cal V}({\cal P}',f) \subset {\cal V}({\cal P},f).
$$
Let ${\cal V}_m({\cal P},f)$ be the set of elements of ${\cal V}({\cal P},f)$ which are
not in ${\cal V}({\cal P}',f)$ for any proper refinement ${\cal P}'$ of ${\cal P}$.
Let $S({\cal P},f) = {\mathrm{Tor}}({\cal V}_m({\cal P},f))$.
Then we have
\begin{eqnarray}
\Zar{{\mathrm{Tor}}(V(f))}
= \bigcup_{{\cal P} \in \Pi_f} \psi_{{\cal P},f}^{-1}(S({\cal P},f)).
\end{eqnarray}
Thus, the proof of Theorem 1 is completed by the following lemma.
\begin{lemma} (Laurent, Sarnak) The set $S({\cal P},f)$ is
finite.
\end{lemma}
This follows from (\cite{Laur:Equ}, Theorem 1) and (\cite{A-S:Betti},
Lemma 3.1). \qed
The rational planes inside a given algebraic subset of $({\Bbb C}^*)^r$ can now be
described in the following way.
\begin{proposition}
Let $V\subset ({\Bbb C}^*)^r$ be any algebraic subset defined
by Laurent polynomials ${\cal F} = \{f_1,\dots,f_k\} \subset \Lambda_r$.
Let $\Pi_{\cal F} =\Pi_{f_1} \times \dots \times \Pi_{f_k}$.
Then
\begin{eqnarray}
\Zar{{\mathrm{Tor}}(V)}
= \bigcup_{{\cal P}_1 \times \dots \times {\cal P}_k \in \Pi_{{\cal F}}}\ \bigcap_{i=1}^k
\psi_{{\cal P}_i,f_i}^{-1}(S({\cal P}_i,f)).
\end{eqnarray}
\end{proposition}
\heading{Proof.}
We have
$$
\Zar{{\mathrm{Tor}}(V)} = \bigcap_{i=1}^k \Zar{{\mathrm{Tor}}(V(f_i))}
$$
so the statement follows from (8).
\qed
\smallskip
\subsection{Degree and Periodicity.}
We will now give some results concerning the degree and period of
$p_V$.
First let $f \in \Lambda_r$ be a Laurent polynomial. We will begin
by studying the hypersurface $V(f) \subset ({\Bbb C}^*)^r$ defined by $f$.
For a partition ${\cal P} \in \Pi_f$, define
$$
\delta_{{\cal P},f} = \left \{ \begin{array}{ll}
1 &\mbox{if $S({\cal P},f) \neq \emptyset$,}\\
0 &\mbox{otherwise.}
\end{array}\right .
$$
For ${\cal P} \in \Pi_f$, let
$\varepsilon({\cal P},f) \subset {\Bbb Z}^r$ be the subset generated
by
$$
\{ \lambda - \mu : \exists \nu \in {\cal P}, \lambda,\mu \in \nu\}
$$
Then $\varepsilon({\cal P},f) = \varepsilon(\psi_{{\cal P},f})$, where $\psi_{{\cal P},f}$
is the map defined in the previous section,
so by Cor. 3.5, any connected component of a fiber of $\psi_{{\cal P},f}$
is a translate of $V(\overline{\varepsilon({\cal P},f)})$.
For any subset ${\cal U} \subset \Pi_f$, let
$\varepsilon({\cal U},f) \subset {\Bbb Z}^r$ be the subgroup generated by
$$
\bigcup_{{\cal P} \in {\cal U}} \varepsilon({\cal P},f).
$$
Then by Prop. 3.6 any rational plane $Q$ in the intersection
$$
\bigcap_{{\cal P} \in {\cal U}} F_{{\cal P}}
$$
where each $F_{{\cal P}}$ is a fiber of $\psi_{{\cal P},f}$, is a translate
of $V(\overline{\varepsilon({\cal U},f)})$.
Let
$$
D({\cal U},f) = D(\overline{\varepsilon({\cal U},f)}/{\varepsilon({\cal U},f)})
$$
and let
$$
D(f) = \mathop{{\mathrm{lcm}}} \ \{D({\cal U},f) : {\cal U} \subset \Pi_f\}.
$$
Let
$$
M({\cal P},f) = \max \ (\{{\mathrm{ord}}(x) : x \in S({\cal P},f)\} \cup \{1\})
$$
and let
$$
M(f) = \mathop{{\mathrm{lcm}}}_{{\cal P} \in \Pi_f}\ M({\cal P},f).
$$
\begin{proposition} Let $f \in \Lambda_r$ any Laurent polynomial
and $V = V(f)$. Then
any rational plane $Q \subset V(f)$
is a translate of $V(\overline{\varepsilon({\cal P},f)})$, for some
partition ${\cal P} \in \Pi_f$.
Furthermore,
$$
\deg(p_{V(f)}) = \max_{{\cal P} \in \Pi_f}\ (r - {\mathrm{rank}}(\overline{\varepsilon({\cal P},f)})
\delta_{{\cal P},f}
$$
and
$$
{\mathrm{per}}(p_{V(f)}) \quad | \quad M(f)D(f).
$$
\end{proposition}
\heading{Proof.} Recall that
$$
\Zar{{\mathrm{Tor}}(V(f))} = \bigcup_{{\cal P}\in \Pi_f} \psi_{{\cal P},f}^{-1}(S({\cal P},f)).
$$
Thus, the degree of $p_{V(f)}$ is the maximum dimension of rational
planes in $\psi_{{\cal P},f}^{-1}(S({\cal P},f))$.
Since $\varepsilon(\psi_{{\cal P},f}) = \varepsilon({\cal P},f)$,
by Cor. 3.5, if $S({\cal P},f)$ is not empty, then
for any rational plane $Q \subset \psi_{{\cal P},f}^{-1}(S({\cal P},f))$,
we have
$$
\dim(Q) = r - {\mathrm{rank}}(\overline{\varepsilon({\cal P},f)}).
$$
This gives the formula for $\deg(p_{V(f)})$.
The period of $p_V$ depends on the orders of rational planes
in intersections of fibers of $\psi_{{\cal P},f}^{-1}$ over
$S({\cal P},f)$. Any such intersection is empty if it involves
more than one fiber of $\psi_{{\cal P},f}$, for some ${\cal P}$.
Let ${\cal U} \subset \Pi_f$ be any subset such that, for any
${\cal P} \in {\cal U}$, $\delta_{{\cal P},f} = 1$. Choose a connected
component $Q_{\cal P} \subset \psi_{{\cal P},f}^{-1}(S({\cal P},f))$
for each ${\cal P} \in{\cal U}$.
Let
$$
M = \mathop{{\mathrm{lcm}}}_{{\cal P} \in {\cal U}} \ {\mathrm{ord}}(\psi_{{\cal P}}(Q_{{\cal P}})).
$$
Then, by Prop. 3.6, for any rational plane $Q$ in the intersection
$$
\bigcap_{{\cal P} \in {\cal U}}Q_{{\cal P}},
$$
we have
$$
{\mathrm{ord}}(Q) \ | \ M\ D({\cal U},f).
$$
Taking the least common multiple of both sides, the claim follows.
\qed
\smallskip
While the various groups denoted by $\varepsilon(-)$ can be computed
routinely, the sets $S({\cal P},f)$ need to be studied in a case by
case manner. When $f$ is defined over ${\Bbb Q}$ the results described in
section 4 are aids to computation.
In particular, we have the following result of Mann \cite{Mann:LinRels}
(cf. Prop. 4.3).
\begin{proposition} (Mann)
Let $\epsilon = (\epsilon_1,\dots,\epsilon_r)$ be any
finite order maximal solution to a linear equation
\begin{eqnarray*}
\sum_{i=1}^r a_i \epsilon_i = 1,
\end{eqnarray*}
with $a_i \in {\Bbb Q}$ for all $i=1,\dots,r$. Let $n$ be the order
of $\epsilon$ as an element of $({\Bbb C}^*)^r$. Then $n$ is square
free and, if $n=p_1\dots p_k$ is a factorization, then
$p_i \leq r+1$ for $i=1,\dots,k$.
\end{proposition}
As a consequence of Prop. 5.4, we have the following bound on $M({\cal P},f)$.
For any partition ${\cal P}$, let
$$
R({\cal P}) = \max\ \{|\nu | : \nu \in {\cal P}\}.
$$
For any positive integer $R$, let $N[R]$ be the product of distinct
primes less than or equal to $R$.
\begin{corollary} If $f \in \Lambda_r$ is defined over ${\Bbb Q}$ and has
$R$ coefficients, then
$$
M({\cal P},f) \ | \ N[R({\cal P})]
$$
for any partition ${\cal P} \in \Pi_f$.
\end{corollary}
We will now give bounds on $p_{V({\cal F})}$, where ${\cal F} = \{f_1,\dots,f_k\}
\subset \Lambda_r$ is
any finite subset.
For any $\pi= ({\cal P}_1,\dots,{\cal P}_k) \in \Pi_{{\cal F}}$, let
\begin{eqnarray*}
\varepsilon(\pi,{\cal F}) &=& \varepsilon({\cal P}_1,f_1) + \dots + \varepsilon({\cal P}_k,f_k),\\
S(\pi,{\cal F}) &=& S({\cal P}_1,f_1) \times \dots \times S({\cal P}_k,f_k),\\
R(\pi) &=& \max\ \{R({\cal P}_i) : i=1,\dots,k\}, \\
M(\pi,{\cal F}) &=& \mathop{{\mathrm{lcm}}} \ \{{\mathrm{ord}}(x) : x \in S(\pi,{\cal F})\}, \ \mbox{and}\\
M({\cal F}) &=& \mathop{{\mathrm{lcm}}}\ \{M(\pi,{\cal F}) : \pi \in \Pi_{{\cal F}}\}.
\end{eqnarray*}
For any subset ${\cal U} \subset \Pi_{\cal F}$, define
$\varepsilon({\cal U},{\cal F}) \subset {\Bbb Z}^r$ to be the subgroup generated by
$$
\bigcup_{\pi \in {\cal U}}\varepsilon(\pi,{\cal F}).
$$
Let
\begin{eqnarray*}
D({\cal U}, {\cal F}) &=& D(\overline{\varepsilon({\cal U},{\cal F})}/{\varepsilon({\cal U},{\cal F})})\\
D({\cal F}) &=& \mathop{{\mathrm{lcm}}}\ \{D({\cal U},{\cal F}) : {\cal U} \subset \Pi_{{\cal F}}\}.
\end{eqnarray*}
Thus, for any $\lambda \in \overline{\varepsilon({\cal U},{\cal F})}/{\varepsilon({\cal U},{\cal F})}$,
we have ${\mathrm{ord}}(\lambda)\ | \ D({\cal F})$.
Let $\rho$ be defined by
\begin{eqnarray*}
\rho: \varepsilon({\cal P}_1,f_1) \times \dots \times \varepsilon({\cal P}_k,f_k)
&\rightarrow &
\varepsilon({\cal P}_1,f_1) + \dots + \varepsilon({\cal P}_k,f_k)\\
(\lambda_1,\dots,\lambda_k)&\mapsto& \lambda_1 + \dots + \lambda_k
\end{eqnarray*}
and let
$$
\gamma: \varepsilon({\cal P}_1,f_1) \times \dots \times \varepsilon({\cal P}_k,f_k)
\hookrightarrow
Z^r \times \dots \times {\Bbb Z}^r
$$
be the inclusion map. Let
$$
\delta_{\pi,{\cal F}} = \left\{\begin{array}{ll} 1 &\quad\mbox{if
$\gamma^*(S(\pi,{\cal F})) \cap {\mathrm{im}} (\rho^*) \neq \emptyset$,} \\
0 &\quad\mbox{otherwise.}
\end{array}\right .
$$
Prop. 5.3 extends to arbitrary algebraic subsets as follows.
\begin{theorem} Let
${\cal F} \subset \Lambda_r$ be a finite subset let and
$V = V({\cal F})$. Then any rational plane
$Q \subset V$ is a translate of $V(\overline{\varepsilon(\pi,{\cal F})})$ for
some partition $\pi \in \Pi_{{\cal F}}$.
Furthermore,
$$
\deg(p_V) = \max_{\pi \in \Pi_{{\cal F}}}
\ (r - {\mathrm{rank}}(\overline{\varepsilon(\pi,{\cal F})})) \delta_{{\cal P},{\cal F}}
$$
and
$$
{\mathrm{per}}(p_V) \ | \ M({\cal F})\ D({\cal F}).
$$
\end{theorem}
\heading{Proof.} From (9), we know that $\Zar{{\mathrm{Tor}}(V({\cal F}))}$
is the union over $\pi \in \Pi_{{\cal F}}$ of
$$
Q(\pi,{\cal F}) = \bigcap_{i=1}^k \psi_{{\cal P}_i,f}^{-1}(S({\cal P}_i,f)).
$$
Hence $\deg(p_V)$ is the maximum dimension of any rational plane
$Q \subset Q(\pi,{\cal F})$.
By Prop. 3.6, $Q(\pi,{\cal F})$ is nonempty if and only if
$\delta_{\pi,{\cal F}} = 1$.
Also by Prop. 3.6, any rational plane $Q \subset Q(\pi,{\cal F})$
is a translate of $V(\overline{\varepsilon(\pi,{\cal F})})$ and its dimension
equals $r-{\mathrm{rank}}(\overline{\varepsilon(\pi,{\cal F})})$. This gives the formula
for the degree of $p_V$.
The period ${\mathrm{per}}(p_V)$ is the $\mathop{{\mathrm{lcm}}}$ of ${\mathrm{ord}}(Q)$, where $Q$ is
a rational plane in the intersection $\bigcap_{\pi \in {\cal U}} Q_{\pi}$,
where $Q_{\pi} \subset Q(\pi,{\cal F})$,
is some choice of rational planes
and $\pi$ ranges
in a subset ${\cal U} \subset \Pi_{\cal F}$.
Note that
$$
M({\cal U},{\cal F}) = \mathop{{\mathrm{lcm}}}\ \{ M(\pi,{\cal F}) : \pi\in {\cal U}\}
$$
divides $M({\cal F})$.
By Prop. 3.6, we have
$$
{\mathrm{ord}}(Q) \ | \ M({\cal U},{\cal F})\ D({\cal U},{\cal F})
$$
and taking the least common multiple of both sides, we get
the bound for the period of $p_{V(f)}$.
\qed
\subsection{Proof of Theorem 2.} Theorem 2 follows easily from
Theorem 3 and Corollary 5.5. To prove (i), we need to find
find polynomials defining the rational planes in $V$. Since by
Theorem 3, any rational plane $Q$ in $V$ is of the form $Q = \eta
V(\varepsilon)$,
where $\varepsilon = \overline{\varepsilon(\pi,{\cal F})}$ and $\eta$ is a finite
order element of $({\Bbb C}^*)^r$. By Lemma 3.1, $V(\varepsilon)$ is
defined by $I_\varepsilon$ and (i) follows.
The bound for $\deg(p_V)$ in (ii)
comes from ignoring the $\delta_{{\cal P},V}$ in Theorem 3.
By Corollary 5.5, we have the inequality
$$
M({\cal F}) \ | \ N[R({\cal F})],
$$
where $R({\cal F})$ is the maximum number of coefficients of a Laurent polynomial
$f \in {\cal F}$. This implies the bound for ${\mathrm{per}}(p_V)$ in (iii).
\qed
\section{Examples.}
In this section we consider some algebraic subsets $V \subset ({\Bbb C}^*)^r$
defined by a finite set of Laurent polynomials ${\cal F} \subset \Lambda_r$
and study the torsion points on $V$ in terms of ${\cal F}$.
We start with some notation.
For any $m \in {\Bbb N}$, let ${\cal C}_m : {\Bbb N} \rightarrow {\Bbb N}$
be the periodic function defined by
$$
{\cal C}_m(n) = \left\{\begin{array}{ll}1 &\quad\mbox{if $m$ divides $n$,}\\
0 &\quad\mbox{otherwise}\end{array}\right .
$$
For any root of unity $\epsilon$, let $\theta(\epsilon) \in {\Bbb Q}/{\Bbb Z}$
be such that
$$
\epsilon = \exp(2\pi\ \sqrt {-1} \ \theta(\epsilon)).
$$
For any positive integer $n$, denote by
$\zeta_{n}$ the primitive $n$th root of unity
such that $\theta(\zeta_n) = 1/n$.
For any $\theta \in ({\Bbb Q}/{\Bbb Z})^r$, let ${\mathrm{ord}}(\theta)$ be the
least positive integer $n$, such that $n\theta = 0 ({\mathrm{mod}}\ 1)$.
For any torsion point $\epsilon = (\epsilon_1,\dots,\epsilon_r)$
in $({\Bbb C}^*)^r$ define
$$
\theta({\epsilon}) = (\theta(\epsilon_1),\dots,\theta(\epsilon_r))
\ \in \ ({\Bbb Q}/{\Bbb Z})^r.
$$
Then
$$
{\mathrm{ord}}(\epsilon) = {\mathrm{ord}}(\theta(\epsilon)).
$$
A binomial equation
\begin{eqnarray}
t^\lambda - \zeta = 0
\end{eqnarray}
where
$\lambda=(\lambda_1,\dots,\lambda_r) \in {\Bbb Z}^r$ and $\zeta$ is a root
of unity $\zeta = \zeta_{n}^k$, $0 \leq k \leq n-1$,
corresponds to a linear equation
\begin{eqnarray}
\lambda_1 \theta_1 + \dots + \lambda_r \theta_r = k/n\ ({\mathrm{mod}}\ 1),
\end{eqnarray}
in the sense that $\epsilon$ satisfies (10) if and only if $\theta(\epsilon)$ is
a solution to (11).
We'll call equation (11) the {\it exponential form} of equation (10).
We'll use the following simplification of Theorem 3
in Examples 2 and 3.
\begin{proposition} Suppose $\Pi_{{\cal F}}$ contains only one element
$\pi$.
Let
$$
M({\cal F}) = \mathop{{\mathrm{lcm}}} \ \{{\mathrm{ord}}(x) : x \in S(\pi,{\cal F})\} \cup \{1\}
$$
and let
$$
D({\cal F}) = D(\overline{\varepsilon(\pi,{\cal F})}/{\varepsilon(\pi,{\cal F})}).
$$
If $S(\pi,{\cal F}) = 0$, then $p_{V({\cal F})} = 0$.
Otherwise,
$$
\deg (p_{V({\cal F})}) = r - {\mathrm{rank}}(\overline{\varepsilon(\pi,{\cal F})}).
$$
and
$$
M({\cal F}) \ | \ {\mathrm{ord}}(p_{V({\cal F})}) \ |\ M({\cal F})\ D({\cal F}).
$$
\end{proposition}
\heading{Proof.} We have
$$
\Zar{{\mathrm{Tor}}(V)} = \psi_{\pi,{\cal F}}^{-1}(S(\pi,{\cal F})),
$$
which is a union of disjoint rational planes
contained in fibers over the points in $S(\pi,{\cal F})$.
Since $\varepsilon(\pi,{\cal F}) = \varepsilon(\psi_{\pi,{\cal F}})$,
the rest follows from Cor. 3.5.
\qed
\heading{Example 1.} We begin with examples
where the degree bound
in Theorem 2 is attained. Let $r$ be an even number $r = 2k$.
Let
$$
f = \sum_{i=1}^r (-1)^i t_i \in \Lambda_r
$$
and $V = V(f)$.
\smallskip
We can see immediately that $V(f)$ contains the affine subtorus
$$
P = V(\{t_it_{i+1}^{-1} - 1 : i=2j-1, j=1,\dots,k\}).
$$
The dimension of $P$ equals the dimension of solutions
$\theta=(\theta_1,\dots,\theta_r) \in ({\Bbb Q}/{\Bbb Z})^r$ such that
$$
\theta_{2j-1} - \theta_{2j} = 0\ ({\mathrm{mod}}\ 1), \qquad \mbox{for all
$j = 1,\dots,k$}.
$$
The dimension is clearly $r - k = k$, so $\deg (p_{V(f)}) \ge k$.
Now take any partition ${\cal P} \in \Pi_f$. Then
$\varepsilon({\cal P},f) = \overline{\varepsilon({\cal P},f)}$ and
$$
{\mathrm{rank}}(\varepsilon({\cal P},f)) = \sum_{\nu \in {\cal P}} (|\nu|-1) \ge k.
$$
Therefore,
$$
\deg (p_{V(f)}) \leq n - k = k
$$
and we have equality.
\smallskip
\heading{Example 2.} Let $V(f) \in ({\Bbb C}^*)^r$ be defined by
$$
f = a_1 t^{\lambda_1} + a_2 t^{\lambda_2}
+ a_3 t^{\lambda_3}, \qquad a_i \in {\Bbb Q}^*
$$
Since there are only three coefficients there is only one parition
in $\Pi_f$, namely ${\cal P} = \{1,2,3\}$.
We have
\begin{eqnarray*}
&&\varepsilon = \varepsilon({\cal P},f)
= \{\lambda_1 - \lambda_3, \lambda_2 - \lambda_3\}\\
&& L_{{\cal P},f} : \frac{-a_1}{a_3}x + \frac{-a_1}{a_2}y - 1\\
\end{eqnarray*}
The following is a table of all possible
values for $a=-a_1/{a_3}$ and $b=-a_2/a_3$ which give nonempty
$S({\cal P},f)$.
\bigskip
\begin{center}
\begin{tabular}{|l|l|}\hline
$(a,b)$ & $S({\cal P},f)$\\
\hline
$(1,1)$&$(\zeta_{6},\zeta_{6}^{-1}),(\zeta_{6}^{-1},\zeta_{6})$\\
$(1,-1)$&$(\zeta_{6},\zeta_{3}),(\zeta_{6}^{-1},\zeta_{3}^{-1})$\\
$(-1,1)$&$(\zeta_{3},\zeta_{6}),(\zeta_{3}^{-1},\zeta_{6}^{-1})$\\
$(-1,-1)$&$(\zeta_{3},\zeta_{3}^{-1}),(\zeta_{3}^{-1},\zeta_{3})$\\
$(1/2,1/2)$&$(1,1)$\\
$(1/2,-1/2)$&$(1,-1)$\\
$(-1/2,1/2)$&$(-1,1)$\\
$(-1/2,-1/2)$&$(-1,-1)$\\
\hline
\end{tabular}
\end{center}
\bigskip
Thus, for example, if $f = t_1 + t_2 + t_3$, then
$$
p_{V(f)}(n) = 2n \ {\cal C}_3(n).
$$
If ${\cal F} \subset \Lambda_r$ is any finite collection
of Laurent polynomials with three coefficients, then $\Pi_{{\cal F}}$ contains
a single element $\pi$ so we can use Prop. 6.1.
Let $V = V({\cal F})$, ${\cal F} = \{f_1,f_2\} \subset
\Lambda_r$, where
\begin{eqnarray*}
f_1 &=& t_1t_3 + t_4 + \alpha\\
f_2 &=& t_1 + t_2 + \beta t_3
\end{eqnarray*}
where $\alpha,\beta \in \{ \pm 1\}$.
Then $\Pi_{{\cal F}}$ has a single element $\pi = ({\cal P},{\cal P})$, where ${\cal P}
= \{1,2,3\}$. Since
\begin{eqnarray*}
&&\varepsilon = \varepsilon(\pi,{\cal F}) = \varepsilon({\cal P},f_1) + \varepsilon({\cal P},f_2)\\
&&\quad = \{(1,0,1,0),(0,0,0,1), (1,0,-1,0),(0,1,-1,0)\}\\
&&\overline{\varepsilon} = {\Bbb Z}^4,\\
&&D({\cal F}) = D(\overline{\varepsilon}/\varepsilon) = D({\Bbb Z}/{2{\Bbb Z}}) = 2,
\end{eqnarray*}
by Prop. 6.1,
the bounds on the degree and period of $p_V$ are
$$
0 \leq \deg(p_V) \leq r - {\mathrm{rank}}(\overline{\varepsilon})=4 - 4 = 0\\
$$
which implies $\deg(p_V) = 0$ and
$$
M({\cal F}) \ | \ {\mathrm{per}}(p_V) \ | \ 2M({\cal F}).
$$
For any $\alpha$ and $\beta$, the exponential form of equations
defining
$$
\Zar{{\mathrm{Tor}}(V)} = \psi_{\pi,{\cal F}}^{-1}(S(\pi,{\cal F})),
$$
are the linear equations
\begin{eqnarray*}
\theta_1 + \theta_3 &=& c\ ({\mathrm{mod}}\ 1)\\
\theta_4 &=& -c\ ({\mathrm{mod}}\ 1)\\
\theta_1 - \theta_3 &=& d\ ({\mathrm{mod}}\ 1)\\
\theta_2 - \theta_3 &=& -d\ ({\mathrm{mod}}\ 1).\\
\end{eqnarray*}
where $c$ and $d$ range in $A \times B$, and depend on $\alpha$ and
$\beta$. The following table shows the $p_V$ corresponding to the
different choices of $\alpha$ and $\beta$.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}\hline
$(\alpha,\beta)$ & $A$ & $B$ & $M({\cal F})$ & $p_V$ & ${\mathrm{per}}(p_V)$\\
\hline
$(1,1)$ & $\{1/3,2/3\}$ & $\{1/3,2/3\}$ & $3$ & $4\ {\cal C}_3 + 4\ {\cal C}_6$
&$6$\\
$(1,-1)$ & $\{1/3,2/3\}$ & $\{1/6,5/6\}$ & $6$ & $8\ {\cal C}_{12}$
&$12$\\
$(-1,1)$ & $\{1/6,5/6\}$ & $\{1/3,2/3\}$ & $6$ & $8\ {\cal C}_{12}$
&$12$\\
$(-1,-1)$ & $\{1/6,5/6\}$ & $\{1/6,5/6\}$ & $6$ & $8\ {\cal C}_6$&
$6$\\
\hline
\end{tabular}
\end{center}
Note that only in the last example where $(\alpha,\beta) = (-1,-1)$
is the period of $p_V$ strictly less than $M({\cal F})\ D({\cal F})$.
We will now justify the entries in the column under $p_V$ in the table.
If $\alpha = \beta =1$, we have solutions
\bigskip
\begin{center}
\begin{tabular}{|l|l|}\hline
$(c,d)$ & solutions $(\theta_1,\theta_2,\theta_3,\theta_4)$\\
\hline
$(1/3,1/3)$&$(1/3,2/3,0,2/3), (5/6,1/6,1/2,2/3)$\\
$(1/3,2/3)$&$(0,2/3,1/3,2/3), (1/2,1/6,5/6,2/3)$\\
$(2/3,1/3)$&$(0,1/3,2/3,1/3), (1/2,5/6,1/6,1/3)$\\
$(2/3,2/3)$&$(2/3,1/3,0,1/3), (1/6,5/6,1/2,1/3)$\\
\hline
\end{tabular}
\end{center}
There are four solutions with order 3 and four with order 6,
thus,
$$
p_V(n) = \left\{\begin{array}{ll} 8 &\mbox{if $6 \ |\ n$,}\\
4 &\mbox{if $3 \ |\ n$ and $6 \ \not | \ n$}\\
0 &\mbox{otherwise}
\end{array}\right .
$$
and hence $p_V(n) = 4 \ {\cal C}_6(n) + 4 \ {\cal C}_3(n)$.
\smallskip
If $\alpha = 1$ and $\beta = -1$, we have the solutions:
\bigskip
\begin{center}
\begin{tabular}{|l|l|}\hline
$(c,d)$& solutions $(\theta_1,\theta_2,\theta_3,\theta_4)$\\
\hline
$(1/3,1/6)$&$(1/4,11/12,1/12,2/3), (3/4,5/12,7/12,2/3)$\\
$(1/3,5/6)$&$(1/12,5/12,1/4,2/3), (7/12,11/12,3/4,2/3)$\\
$(2/3,1/6)$&$(11/12,7/12,3/4,1/3), (5/12,1/12,1/4,1/3)$\\
$(2/3,5/6)$&$(3/4,1/12,11/12,1/3), (1/4,7/12,5/12,1/3)$\\
\hline
\end{tabular}
\end{center}
All solutions have order $12$ so we have
$p_V(n) = 8 \ {\cal C}_{12}(n)$.
\smallskip
If $\alpha = -1$ and $\beta = 1$, we have the solutions:
\bigskip
\begin{center}
\begin{tabular}{|l|l|}\hline
$(c,d)$& solutions $(\theta_1,\theta_2,\theta_3,\theta_4)$\\
\hline
$(1/6,1/3)$&$(1/4,7/12,11/12,5/6), (3/4,1/12,5/12,5/6)$\\
$(1/6,2/3)$&$(5/12,1/12,3/4,5/6), (11/12,7/12,1/4,5/6)$\\
$(5/6,1/3)$&$(7/12,11/12,1/4,1/6), (1/12,5/12,3/4,1/6)$\\
$(5/6,2/3)$&$(3/4,5/12,1/12,1/6), (1/4,11/12,7/12,1/6)$\\
\hline
\end{tabular}
\end{center}
All solutions have order $12$ so
$p_V(n) = 8\ {\cal C}_{12}(n)$.
If $a = b = -1$, we have solutions
\bigskip
\begin{center}
\begin{tabular}{|l|l|}\hline
$(c,d)$ & solutions $(\theta_1,\theta_2,\theta_3,\theta_4)$\\
\hline
$(1/6,1/6)$&$(1/6,5/6,0,5/6), (2/3,1/3,1/2,5/6)$\\
$(1/6,5/6)$&$(0,1/3,1/6,5/6), (1/2,5/6,2/3,5/6)$\\
$(5/6,1/6)$&$(0,2/3,5/6,1/6), (1/2,1/6,1/3,1/6)$\\
$(5/6,5/6)$&$(5/6,1/6,0,1/6), (1/3,2/3,1/2,1/6)$\\
\hline
\end{tabular}
\end{center}
All solutions have order 6,
so $p_V(n) = 8 \ {\cal C}_6(n)$.
\bigskip
\heading{Example 3.}
We now study the Fermat curve $V(f)$, defined by
$$
f(t_1,t_2) = t_1^m + t_2^m -1.
$$
Then
\begin{eqnarray*}
&&\Zar{{\mathrm{Tor}}(V(f))} = \phi_{{\cal P},f}^{-1}(S({\cal P},f))\\
&&S({\cal P},f) = \{\mu_1,\mu_2\}
\qquad \mu_1 = (\zeta_{6},\zeta_{6}^{-1}),
\quad \mu_2 = (\zeta_{6}^{-1},\zeta_{6}),\\
&&M(f) = 6,\\
&&\varepsilon = \varepsilon({\cal P},f) = \{(m,0),(0,m)\},\\
&&\overline{\varepsilon} = \{(1,0),(0,1)\} = {\Bbb Z}^2,\\
&&\overline{\varepsilon}/\varepsilon = ({\Bbb Z}/{m{\Bbb Z}})^2,\\
&&D(f) = D(({\Bbb Z}/{m{\Bbb Z}})^2) = m\\
\end{eqnarray*}
We thus have
$$
\deg(p_{V(f)})= 0
$$
and
$$
6= M(f) \ | \ {{\mathrm{per}}}(p_{V(f)})\ |\ M(f)\ D(f)= 6m.
$$
For integers $a,b \in {\Bbb Z}$, let $(a,b)$ denote their greatest common divisor.
We'll show that
$$
p_V(f) = \left\{\begin{array}{ll}
2(m,n)^2&\mbox{if $6 \ |\ \frac{n}{(n,m)}$,}\\
0 & \mbox{otherwise.}
\end{array}\right .
$$
and hence $p_{V(f)}$ has period $6m$.
\smallskip
For instance, if $m = 3$, $p_V(n) = 18\ C_{18}(n)$.
\smallskip
The exponential linear equations associated to
$\phi_{{\cal P},f}$ are
\begin{eqnarray}
\left\{\begin{array}{rcl}
m\theta_1 &=& 1/6\ ({\mathrm{mod}}\ 1)\\
m\theta_2 &=& 5/6\ ({\mathrm{mod}}\ 1)
\end{array} \right .
\qquad
\left\{\begin{array}{rcl}
m\theta_1 &=& 5/6\ ({\mathrm{mod}}\ 1)\\
m\theta_2 &=& 1/6\ ({\mathrm{mod}}\ 1)
\end{array} \right .
\end{eqnarray}
Consider the first equation and suppose there is a solution
$$
(\theta_1, \theta_2) = (a/n,b/n) \in ({\Bbb Q}/{\Bbb Z})^2.
$$
We will show that $6$ divides $n/{(m,n)}$.
The equations in (12) imply that
$$
ma/n - 1/6 \in {\Bbb Z}
$$
so $6n$ divides $6ma - n$ and hence $6$ divides $n$.
Setting $n_1 = n/6$, we have $n$ divides $ma - n_1$.
Set $m_1 = m/(m,n)$.
Then
$$
m_1(m,n)a = n_1\ ({\mathrm{mod}}\ n).
$$
Since $(m_1,n) = 1$, $m_1$ is invertible modulo $n$ and there is some
$m_2 \in {\Bbb Z}$ such that $m_2m_1 = 1 ({\mathrm{mod}}\ n)$.
Thus,
$$
(m,n)a = m_2n_1\ ({\mathrm{mod}}\ n)
$$
and hence
$$
(m,n)a = m_2n_1 + nr,
$$
for some integer $r$.
Since $n$ is relatively prime
to $m_1$ and $m_2$, so is $(m,n)$. Thus,
$(m,n)$ divides $n_1$ and hence $6$ divides $n/(m,n)$.
Conversely, suppose $6$ divides $n/{(m,n)}$. Let $m_1$ be a representative
for the multiplicative inverse of $m/{(m,n)}$ modulo $n$. Then
$$
a = \frac{n}{6(m,n)}m_1 \qquad b = \frac{-n}{6(m,n)}m_1
$$
is a solution to the first system and $(b,a)$ is a solution to the second.
To find the number of solutions of order $n$. We need to look
at the homogeneous equations
$$
m(r/n) = 0\ ({\mathrm{mod}}\ 1) \qquad m (s/n) = 0\ ({\mathrm{mod}}\ 1).
$$
These have solutions $r,s\in {\Bbb Q}/{\Bbb Z}$ given by
$$
r = kn/{(m,n)},\quad s = \ell n/{(m,n)}
$$
for $k,\ell=0,\dots,(m,n) -1$.
Thus, in total, there are $2(m,n)^2$ possible solutions to (12) of order
$n$ when $6$ divides $n/{(m,n)}$.
\qed
\bibliographystyle{math}
|
1996-07-24T15:36:56 | 9607 | alg-geom/9607026 | en | https://arxiv.org/abs/alg-geom/9607026 | [
"alg-geom",
"hep-th",
"math.AG",
"math.QA",
"q-alg"
] | alg-geom/9607026 | Ashok Raina | Indranil Biswas, A.K. Raina | Projective structures on a Riemann surface | Plain LATEX file, to appear in Int. Math. Res. Not | Int.Math.Res.Not. 15 (1996) 753 | null | TIFR/TH/96-17 | null | For a compact Riemann surface $X$ of any genus $g$, let $L$denote the line
bundle $K_{X\times X}\otimes {\cal O}_{X\times X}(2\Delta)$ on $X\times X$,
where $K_{X\times X}$ is the canonical bundle of $X\times X$ and $\Delta$ is
the diagonal divisor. We show that $L$ has a canonical trivialisation over the
nonreduced divisor $2\Delta$. Our main result is that the space of projective
structures on $X$ is canonically identified with the space of all
trivialisations of $L$ over $3\Delta$ which restrict to the canonical
trivialisation of $L$ over $2\Delta$ mentioned above. We give a direct
identification of this definition of a projective structure with a definition
of Deligne.We also describe briefly the origin of this work in the study of the
so-called "Sugawara form" of the energy-momentum tensor in a conformal quantum
field theory.
| [
{
"version": "v1",
"created": "Wed, 24 Jul 1996 15:59:17 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Biswas",
"Indranil",
""
],
[
"Raina",
"A. K.",
""
]
] | alg-geom | \section{Introduction}
A {\it projective structure} (also called a {\it projective
connection}) on a Riemann surface is an equivalence class of
coverings by holomorphic coordinate charts such
that the transition functions are all M\"obius transformations.
There are several equivalent notions of a projective
structure \cite{D}, \cite{G}.
For a compact Riemann surface $X$ of any genus $g$, let $L$
denote the line bundle $K_{X\times X}\otimes {{\cal O}}_{X\times
X}(2\Delta)$ on $X\times X$, where $K_{X\times X}$ is the
canonical bundle of $X\times X$ and $\Delta$ is the diagonal
divisor. This line bundle $L$ is trivialisable over a Zariski
open neighborhood of $\Delta$ and has a {\it canonical
trivialisation} over the nonreduced divisor $2\Delta$.
Our main result [Theorem 3.2] is that the space
of projective structures on $X$ is canonically identified with
the space of all trivialisations of $L$ over $3\Delta$ which
restrict to the canonical trivialisation
of $L$ over $2\Delta$ mentioned above.
In (\cite{D}, page 31, Definition 5.6 bis) Deligne gave another
definition of a projective structure (what he calls
``forme infinit\'esimale''). We give a direct
identification of this definition with our definition of a
projective structure [Theorem 4.2].
In Section 5, which is independent of the rest of the
paper, we describe briefly the
origin of this work in the study of the
so-called ``Sugawara form" of the energy-momentum tensor in a
conformal quantum field theory.
\section{Trivialisability of the line bundle $L$}
Let $X$ be a compact connected Riemann surface, equivalently,
a smooth connected projective curve over ${\Bbb C}$, of genus $g$.
We denote by $S$ the complex surface $X\times X$, by $\Delta$
the diagonal divisor of $S$, and by
$K_S$ the canonical bundle of $S$. Thus
$K_S = p^*_1K_X\otimes p^*_2K_X$, where $p_i$ ($i=1,2$)
is the projection of $X\times X$ onto the $i$-th factor
and $K_X$ is the canonical bundle of $X$.
Let $\sigma $ be the involution of $S$ defined by $(x,y) \longmapsto
(y,x)$, of which $\Delta$ is the fixed point set. We note that $\sigma $
has a canonical lift ${\tilde \sigma}$ to $L=K_S\otimes {{\cal O}}_S(2\Delta)$; in
other words, ${\tilde \sigma}$ is an
isomorphism between $L$ and $\sigma ^*(L)$ with ${\tilde \sigma}\circ {\tilde \sigma}$ being
the identity isomorphism.
The aim of this section is to establish the
following theorem:
\noindent {\bf Theorem 2.1.}~ {\it The line bundle
$L := K_S\otimes {{\cal O}}_S(2\Delta)$
on $S$ is (a), trivialisable on every infinitesimal neighborhood
$n\Delta$ of $\Delta$ in $S$ and (b), has a canonical trivialisation
on the first infinitesimal neighborhood $2\Delta$, which is the
unique trivialisation of $L$ on $2\Delta$ invariant under the
action of ${\tilde \sigma}$ and coinciding with the canonical trivialisation
of $L$ on $\Delta$.}
We denote by $J^d$ the component of the Picard group of $X$
consisting of all line bundles of degree $d$ and by ${{\cal O}}(\Theta )$
the line bundle on $J^{g-1}$ corresponding to the theta divisor,
viz. the reduced theta divisor on $J^{g-1}$ defined by the
subset $\{\xi \in J^{g-1}\vert H^0(X , \xi) \neq 0\}$ (when $g
=0$, it is the zero divisor). Let $\theta $ denote the natural
section of the line bundle ${{\cal O}}(\Theta )$ on $J^{g-1}$ given by the
constant function $1$; it vanishes precisely on the theta
divisor. For $\xi\in J^{g-1}$ let $\xi^*\Theta $ denote the divisor
$\{\zeta\otimes
\xi^{-1}\mid \zeta\in\Theta \}\subset J^0$. We now recall \cite{NR} that
the linear equivalence class of $\xi^*\Theta +
(K\otimes\xi^{-1})^*\Theta $ on $J^0$ is independent of $\xi\in
J^{g-1}$ and defines {\it canonically} a line bundle on $J^0$,
which we denote by ${{\cal O}}(2\Theta _0)$.
We require the following property of the line bundle $L$ in the
proof of Theorem 2.1:
\noindent {\bf Lemma 2.2.}~ {\it Let $\phi~: ~~S~\longrightarrow ~J^0$ be
the morphism defined by $(x,y)\longmapsto {\cal O}_X(x-y)$. Then
$$
L~~=~~\phi^*{{\cal O}}(2\Theta_0)\leqno{(2.3)}
$$}
\noindent {\bf Proof.} Clearly we can write
$$
L ~~ =~~ {\cal M}_{\alpha }\otimes {\sigma }^*{\cal M}_{\alpha } \leqno{(2.4)}
$$
where, for $\alpha \in J^{g-1}$, the line bundle ${\cal M}_{\alpha }$ on $S$
is defined as follows:
$$
{\cal M}_{\alpha } ~ := ~ p^*_1(K_X\otimes {\alpha }^{-1})
\otimes p^*_2({\alpha })\otimes {{\cal O}}_S(\Delta) \leqno{(2.5)}
$$
As shown in \cite{R1}, ${\cal M}_\alpha $ is
isomorphic to $\phi _\alpha ^*{{\cal O}}(\Theta )$, where
$$
{\phi }_{\alpha } ~ : ~~ S ~ \longrightarrow ~ J^{g-1} \leqno{(2.6)}
$$
is the morphism defined by $(x,y) \longmapsto \alpha \otimes {{\cal O}}_X(x-y)$.
Theorem 2.2 of \cite{BR} is, however, preferable, since it
gives, in this special case, a {\it natural} isomorphism between
${\phi }^*_{\alpha }{{\cal O}}(\Theta )$ and ${\cal M}_{\alpha }\otimes {\zeta }_{\alpha }$, where
${\zeta }_{\alpha }$ denotes the trivial line bundle on $S$
with fiber ${\Theta }_{\alpha }$, the fiber of ${{\cal O}}(\Theta )$ at the point $\alpha $.
(For any $\alpha $ outside the theta divisor, the nonzero vector
${\theta }(\alpha ) \in {\Theta }_{\alpha }$ identifies ${\zeta }_{\alpha }$ with
the trivial line bundle.)
Using this in (2.4), we see that Lemma 2.2 follows immediately
from the definition of ${{\cal O}}(2\Theta _0)$.$\hfill{\Box}$
\noindent {\it Proof of part (a) of Theorem 2.1.}~ Simply observe
that the image ${\phi }(\Delta) = 0\in J^0$. In fact, since
${{\cal O}}(2\Theta _0)$ has no base points, $L$ has a global section
which is nowhere zero on $\Delta$.$\hfill{\Box}$
\medskip
\noindent {\bf Corollary 2.7.}~ {\it Let $\alpha \in J^{g-1}\setminus\Theta $.
Then the section
$$
{\omega }_{\alpha } ~~ = ~~ {\phi }^*_{\alpha }\theta \otimes
({\phi }_{\alpha }\circ \sigma )^*\theta ~~ \in ~~ H^0(S,L)
$$
is 1 at any point of the diagonal $\Delta$. In particular,
this section gives a trivialisation of $L$ over some Zariski open
neighborhood of $\Delta$.
The existence of ${\omega }_{\alpha }$ implies that
$$
\dim H^0(S,L)\, \geq \, \dim H^0(S, K_S) +1 \, = \,
\dim H^0(X,K_X)^{\otimes 2} +1 \, = \, g^2+1\leqno{(2.8)}
$$}
\noindent {\bf Proof.}~ Using the natural trivialisation of ${{\cal O}}(\Theta )$
outside the theta divisor given by the section $\theta $ and the above
identification of ${\cal M}_{\alpha }$ with
the pullback of ${{\cal O}}(\Theta )$, we have a
trivialisation of ${\cal M}_{\alpha }$ over
some Zariski open neighborhood of
$\Delta$. This gives a trivialisation of ${\sigma }^*{\cal M}_{\alpha }$ over
some Zariski open neighborhood of $\Delta$. Now the equality (2.4)
completes the proof.$\hfill{\Box}$
\medskip
\noindent {\bf Notation:}~ For $n\geq 1$, we shall denote the
restriction of $L$ to the divisor $n\Delta$ (the $(n-1)$-th
order infinitesimal neighborhood of $\Delta$) by $L\mid
n\Delta$.
\medskip
\noindent {\it Proof of part (b) of Theorem 2.1.}~ Now $L\mid
\Delta={{\cal O}}_{\Delta}$ and ${{\cal O}}_{\Delta}$ has a one-dimensional
space of sections invariant under the action induced by the
involution $\sigma $ on $S$. Hence $L$ has a
canonical trivialisation on $\Delta$ defined by the section ``1".
The situation on $2\Delta$ is more complicated. We know
that $\omega _{\alpha }$ in Corollary 2.7, which defines a trivialisation
of $L$ on $2\Delta$, is symmetric under $\sigma $. The claim that
$L$ has a {\it canonical}
trivialisation on $2\Delta$ will then follow from the following lemma:
\noindent {\bf Lemma 2.9.}~ {\it The restriction of $L$ to $ 2\Delta$ has
a one-dimensional space of sections invariant under the action induced by
the involution $\sigma : (x,y)\mapsto (y,x)$ on $S$.}
\noindent {\bf Proof.}~ Consider the exact sequence
$$
0~\longrightarrow K_{\Delta}~\longrightarrow ~L\mid 2\Delta~\longrightarrow ~{{\cal O}}_{\Delta}~\longrightarrow 0
$$
where we have made use of the canonical trivialisation of $L$ on $\Delta$.
Now note that the global sections form a short exact sequence. Observe
that the natural invariant section ``1" of ${{\cal O}}_{\Delta}$ lifts
(by averaging over ${\tilde \sigma}$) to an invariant section of $L\mid
2\Delta$, so that the dimension of the space of invariant
sections of the latter is at least one. On the other hand,
${\tilde \sigma}$ operates on $H^0(K_{\Delta})$ as {\it -Id}. Indeed, the
tangent space at $(x,x)\in\Delta$ is $T_xX \oplus T_xX$, and it
is the direct sum of the subspace spanned by $(v_x,v_x)$ with
the subspace spanned by $(v_x,-v_x)$, where $v_x$ is a nonzero
vector in $T_xX$. The former are invariant under ${\tilde \sigma}$ and
belong to the tangent bundle of $\Delta$, while the latter are
anti-invariant under ${\tilde \sigma}$ and belong to the normal bundle of
$\Delta$. Now, since $K_{\Delta}$ is the conormal bundle to
$\Delta$, the involution ${\tilde \sigma}$ operates as {\it -Id} on
$H^0(K_{\Delta})$. Thus we conclude that $K_{\Delta}$ has no
nonzero section which is invariant under ${\tilde \sigma}$. This proves the
lemma and also completes the proof of Theorem 2.1.
What is happening is that, under the quotient map $q:S\rightarrow
S/\sigma $, the line bundle $L$ descends to ${\tilde L}$ on $S/\sigma $, since
${\tilde \sigma}$ acts trivially on the fibers of $L$ at each point of the
fixed point set $\Delta$ of $\sigma $. The trivialisation of $L$
over $\Delta$ induces a trivialisation of ${\tilde L}$ over
$\Delta/\sigma $. Since the scheme-theoretic inverse image
$q^{-1}(\Delta /\sigma )$ is $2\Delta$ and $q^*{\tilde L}=L$, the
trivialisation of ${\tilde L}$ over $\Delta/\sigma $ induces a
trivialisation of $L$ over $2\Delta$.$\hfill{\Box}$
It is useful to have an alternative view of the canonical
trivialisation of $L$ on $2\Delta$:
{\bf Proposition 2.10.}~{\it The canonical trivialisation of
$L$ on $2\Delta$ is given by the unique section of $L\mid 2\Delta$
which restricts to the canonical trivialisation on $\Delta$
and lifts to a global section of $L$.}
The proof rests on the following lemma, which shows that the
inequality (2.8) is actually an equality.
\medskip
\noindent {\bf Lemma 2.11.}~~~~~ $\dim H^0(S,L) ~ = ~ g^2+1$.
\medskip
\noindent {\bf Proof.}~ In view of (2.8), we merely have to establish
the upper bound. Indeed, tensoring the following exact
sequence of sheaves on $S$
$$
0 ~ \longrightarrow ~ {{\cal O}}_S(-\Delta) ~ \longrightarrow ~{{\cal O}}_S ~
\longrightarrow ~ {{\cal O}}_{\Delta}
~ \longrightarrow ~ 0 \leqno{(2.12)}
$$
by $L$ and passing to cohomology, this follows from the observation
that $K_S(\Delta)$ has $g^2$ sections. To establish the latter, tensor
$(2.12)$ by $K_S(\Delta)$
and pass to cohomology; it then suffices to show that the injection
$H^0(S,K_S)\rightarrow H^0(S,K_S(\Delta))$ is an isomorphism.
Taking the direct image of this short exact sequence by the
projection, $p_1$, to the first factor of $S$, gives the long
exact sequence
$$
0 ~ \longrightarrow ~K_X\otimes {\Bbb C}^g ~\stackrel{\iota}{\longrightarrow} ~
p_{1*}(K_S(\Delta))~\longrightarrow ~K_X~ \longrightarrow ~\cdots
$$
of which the first three terms are locally free sheaves.
The first two terms are rank $g$ vector bundles and
hence $\iota$ must be an isomorphism, which completes the proof.
$\hfill{\Box}$
\medskip
\noindent {\it Proof of Proposition 2.10.}~ From the exact sequence
$$
0~\longrightarrow ~K_S~\longrightarrow ~L~\longrightarrow ~ L\mid 2\Delta ~\longrightarrow~0
$$
and the fact that $L$ has only $g^2 + 1$ sections, we conclude
that the space of sections of $L$ has a one-dimensional image
in $L\mid 2\Delta$.$\hfill {\Box}$
\section{Projective structures and the line bundle $L$}
We will recall the definition of
a {\it projective structure on a Riemann surface subordinate to the
complex structure}. This is defined (see \cite{G} page 167) to
be a holomorphic
coordinate covering, $\{U_i,z_i\}_{i \in I}$, of $X$
such that for any pair $i, j \in I$, the holomorphic transition
function $f_{i,j}$ (defined by $z_i = f_{i,j}(z_j)$)
is a M\"obius transformation, i.e., a function of the form
$$
z ~~ \longmapsto ~~ {{az+b}\over {cz+d}} \leqno{(3.1)}
$$
where $a,b,c,d \in {\Bbb C}$ with $ad-bc = 1$. The space ${\mbox{\Large $\wp$}}$ of all
projective structures on $X$ subordinate to the
complex structure is an affine space for the complex vector space
$H^0(X, K^2_X)$ (\cite{G} page 172).
The main result of this section is the following theorem:
\noindent {\bf Theorem 3.2.}~{\it Let ${\cal Q}$ denote the
space of all trivialisations of $L\mid 3\Delta$, which, on
restriction to $2\Delta$, give the canonical trivialisation of
$L\mid 2\Delta$. Then ${\cal Q}$ is an affine space for the
vector space $H^0(X, K_X^2)$, which is canonically isomorphic to
the affine space $\mbox{\Large $\wp$}$ of projective structures on $X$.}
{\bf Proof.} The obvious exact sequence
$$
0 ~ \longrightarrow ~ K^2_X ~ \longrightarrow ~ L\mid 3\Delta ~
\longrightarrow~ L\mid 2\Delta ~ \longrightarrow ~ 0 \leqno{(3.3)}
$$
shows that ${\cal Q}$ is an affine space for the vector space
$H^0(X,K^2_X)$.
We shall now construct a map from ${\mbox{\Large $\wp$}}$
to ${\cal Q}$.
Let $M = CP^1\times CP^1$, and consider the {\it trivial} line bundle
$L_M ~ := ~ K_M \otimes {{\cal O}}_M(2{\Delta}_M)$
on $M$, where ${\Delta}_M$ is the diagonal on $M$. Let
$$
s ~\in ~ H^0(M,L_M) \leqno{(3.4)}
$$
be the trivialisation of $L_M$ whose restriction to ${\Delta}_M$
coincides with the canonical trivialisation given by Theorem
2.1(b). The group of all automorphisms of $CP^1$, namely
${\rm Aut}(CP^1)$,
acts naturally on $M$ by the diagonal action; this action lifts
to $L_M$. The section $s$ in (3.4) is evidently invariant
under the induced action of ${\rm Aut}(CP^1)$ on $H^0(M, L_M)$.
This
invariance property of the section $s$ immediately implies that
if we have a projective structure on $X$, the section $s$ induces a
trivialisation of $L$ on some analytic open neighborhood of the
diagonal $\Delta$. Now, restricting this trivialisation of $L$ to
$3\Delta$ we get an element in $\cal Q$. This gives the required
map
$$
F~: ~ \mbox{\Large $\wp$} ~ \longrightarrow ~ {\cal Q} \leqno{(3.5)}
$$
The above construction of the map $F$ has been motivated by
\cite{Bi}.
The proof of Theorem 3.2 is now completed by the following lemma
which describes how the map $F$ relates the affine structures on
$\mbox{\Large $\wp$}$ and ${\cal Q}$.
\medskip
\noindent {\bf Lemma 3.6.}~ {\it For any ${\cal I}\in {\mbox{\Large $\wp$}}$ and
$\gamma \in H^0(X, K^2_X)$, the following equality holds.
$$
F({\cal I} +{\gamma }) ~~ =~~ F({\cal I} ) ~+ ~{\gamma \over 6}
$$}
\medskip
\noindent {\bf Proof.}~ Let us first recall
how the affine $H^0(X, K^2_X)$ structure of ${\mbox{\Large $\wp$}}$ is defined
(\cite{G} page 170, Theorem 19). We start by recalling
the definition of the {\it Schwarzian derivative},
denoted by $\mbox{${\cal{S}}$}$, which is the differential operator:
$$
\mbox{${\cal{S}}$} (f)(z) ~:= ~ {{2f'(z)f'''(z) - 3(f''(z))^2}
\over {2(f'(z))^2}}
$$
defined over ${\Bbb C}$.
Take any ${\cal I} =
\{U_i,z_i\}_{i\in I} \in {\mbox{\Large $\wp$}}$ and $\gamma \in H^0(X,K^2_X)$.
On each $U_i$ there is a
holomorphic function $h_i$ such that $\gamma = h_i dz_i\otimes dz_i$.
For $i \in I$, let $w_i$ be a holomorphic function on $z_i(U_i)$
satisfying the equation
$$
h_i ~= ~ \mbox{${\cal{S}}$} (w_i)(z_i) \leqno{(3.7)}
$$
Another function $w'_i$ satisfies the equation (3.7) if and only if
$w'_i (z_i) = \rho \circ w_i(z_i)$, where $\rho $ is a M\"obius
transformation. The element ${\cal I} + \gamma \in {\mbox{\Large $\wp$}}$ is given by $\{U_i,
w_i\circ z_i\}_{i \in I}$. (Actually we may have to shrink each
$U_i$ a bit so that $w_i\circ z_i$ is a coordinate function.)
We require an explicit description
of the section $s$ defined in (3.4) in terms of local coordinates.
Identify
$CP^1$ with ${\Bbb C} \cup \{\infty\}$ and let $(z_1,z_2)$ be the natural
coordinates on $M$. In these coordinates the section $s$ can
be written as:
$$
s_z ~~:= ~~{{dz_1\wedge dz_2}\over {(z_1-z_2)^2}} \leqno{(3.8)}
$$
Let $ {\cal I} ~ := ~ \{ U_i,z_i\}_{i\in I}$ be a projective
structure on $X$, as before. Take a coordinate chart $(U , z)$ in ${\cal I}$.
On the open set $U\times U \subset S$ there is a
natural coordinate function $(z_1,z_2)$ obtained from $z$. Now
$s_z$ in (3.8) gives a trivialisation of $L$ over $U\times U$.
Let $(V,y)$ be another coordinate chart in ${\cal I}$ with
$y = (az+b)/(cz+d)$ as in (3.1). This
implies that the following identity holds:
$$
s_z ~:= ~ {{dz_1\wedge dz_2}\over {(z_1-z_2)^2}} ~~ = ~~
{{dy_1\wedge dy_2}\over {(y_1-y_2)^2}} ~ =: ~ s_y
$$
where $(y_1,y_2)$ is the coordinate function on $V\times V$.
This equality implies that the two local sections of $L$,
viz. $s_z$ and $s_y$, coincide on the intersection
$(U\cap V) \times (U\cap V) \subset S$.
Thus various local trivialisations of $L$ of the
form $s_z$ patch together to give a trivialisation of $L$ on
some analytic open neighborhood of $\Delta$. In particular,
we get a trivialisation of $L\mid n\Delta$ for any $n$. Since
the section $s_z$ takes the value $1$ on $U\times U$ and is
invariant under the involution
${\tilde \sigma}$, the trivialisation of $L\mid 2\Delta$ obtained this way
is the canonical trivialisation. Thus the trivialisation
of $L\mid 3\Delta$ is actually an element of ${\cal Q}$.
Evidently, the element of $\cal Q$ obtained in this way coincides
with $F({\cal I})$, where $F$ is the map defined in (3.5).
Let $(U_i,z_i)$, $i\in I$, be a coordinate chart around $x \in X$,
with $z_i(x) = 0$.
Assume that
$$w_i ~= ~ z_i + \sum_{j=2}^{\infty}a_jz_i^j \leqno{(3.9)}
$$
is a solution of (3.7) (we may assume that $w_i$ is of this form
since we may
compose $w_i$ with any M\"obius transformation).
Then equation (3.7)
gives
$$
h(0) ~=~ 6a_3 -6a^2_2 \leqno{(3.10)}
$$
Set $y = w_i\circ z_i$, and define $s_y$ as in (3.8). For
${\tilde x} := (x,x) \in S$, using (3.9) we find that
$$
s_y({\tilde x} ) ~=~ s_{z_i}({\tilde x} ) + (a_3 -a^2_2)dz_i\otimes dz_i
$$
Comparing this with (3.10) we get that $s_y({\tilde x} ) = s_{z_i}({\tilde x} )
+ \gamma (x)/6$. This completes the proof of the lemma and
also of Theorem 3.2.$\hfill{\Box}$ \medskip
\noindent {\bf Remark 3.11.}~ A consequence of Theorem 3.2
is the following alternative
definition of the Schwarzian derivative.
Let $f$ be a holomorphic function around $z_0 \in {\Bbb C}$ such that
$f'(z_0) \neq 0$. Then the function ${\bar f} := (f,f)$ is a
biholomorphism defined on some neighborhood, $U$, of
$(z_0,z_0) \in {\Bbb C}\times{\Bbb C}$. Consider the section $s$ defined in
(3.4). The restriction of
$$
{\hat s} ~ := ~ {\bar f}^*s ~ - ~ s
$$
to the (nonreduced) divisor $3{\Delta}_U$ is actually a local
section of $K^{2}_{{\Bbb C}}$ around
$z_0$. From the computation in the proof of Lemma 3.6 it follows
that $\hat s$ is actually ${\mbox{${\cal{S}}$}}(f)(dz)^{\otimes 2}/6$.
\medskip
\noindent {\bf Remark 3.12.} An interesting question, arising
naturally from Theorem 3.2, is whether an element of ${\cal Q}$
comes necessarily from a global section of $L$. Thus
let $\Lambda \subset H^0(S,L)$ denote the affine subspace consisting
of those sections of $L$ which restrict to the canonical
trivialisation on $2\Delta$. Then $\Lambda $ is an affine space
for the subspace $H^0(S,K_S)=H^0(X,K_X)^{\otimes 2}$.
Associating to any $s \in \Lambda $, the corresponding trivialisation
of $L$ over $3\Delta$, we get a map from
$\Lambda $ to ${\cal Q}$. Then from Theorem 3.2 we have a
(holomorphic) map $\lambda $ from $\Lambda $ to $\mbox{\Large $\wp$}$, the space of all
projective structures on $X$. Our question is now whether $\lambda $
is surjective. Let
$$
R ~: ~ H^0(X, K_X)^{\otimes 2} ~= ~
H^0(S,K_S) ~\longrightarrow ~ H^0(\Delta , K_S{\vert}_{\Delta}) ~ = ~
H^0(X, K^2_X) \leqno{(3.13)}
$$
denote the obvious restriction
map. From Lemma 3.6 it follows that for any $s \in \Lambda $ and $\beta
\in H^0(X,K_X)^{\otimes 2}$, the equality $$
\lambda (s + \beta ) ~ = ~ \lambda (s) + R(6\beta ) ~\in ~ \mbox{\Large $\wp$} \leqno{(3.14)}
$$
holds. This equality implies that $\lambda $ is surjective if and only if
the homomorphism $R$ in (3.13) is surjective. From M. Noether's theorem
(\cite{ACGH}, page 117) we know that if $X$ is non-hyperelliptic then
$R$ is surjective. Moreover, for elements $s,t\in \Lambda $ to have
the same image under $\lambda $, we must have $s-t\in H^0(S,K_S(-\Delta))$.
A similar argument shows that in the non-hyperelliptic case $\lambda $
remains surjective when restricted to the subspace of $\Lambda $ consisting
of sections symmetric under the map ${\tilde \sigma}$ induced from $\sigma :S\rightarrow S
~((x,y)\mapsto (y,x))$. Related observations have been made by
Tyurin \cite{T}.
\medskip
\noindent{\bf Remark 3.15.}~ Let $(X_T, {\Gamma}_T) \longrightarrow T$
be a family of Riemann surfaces
with theta characteristic. This means that $\Gamma_T$ is a holomorphic
line bundle on $X_T$, the total space of the family of Riemann
surfaces, and for any $t\in T$, the restriction
${\Gamma}_t$ to the Riemann surface $X_t$ satisfies the condition that
${\Gamma}^{\otimes 2}_t = K_{X_t}$. Consider the line bundle
$$
{\cal M}_T ~ := ~ p^*_1(K_{\rm rel}\otimes {\Gamma}^*_T) \otimes
p^*_2({\Gamma}_T)\otimes {{\cal O}}(\Delta_T)
$$
on the fiber product $X_T\times_TX_T$, where $p_i$ denote the projection
to the $i$-th factor, ${\Delta}_T$ is the diagonal divisor in
the fiber product, and $K_{\rm rel}$ is the relative canonical
bundle on $X_T$. Let ${\cal M}_t$ be the line bundle on
$X_t \times X_t$ obtained by setting $\alpha = \Gamma_t$ in the proof of Lemma
2.2. Clearly the restriction of ${\cal M}_T$ to $X_t\times X_t$ is
${\cal M}_t$. The natural isomorphism between
${\phi}^*_{\alpha }{{\cal O}}(\Theta)$ and ${\cal M}_{\alpha }\otimes {\zeta }_{\alpha }$
mentioned in the proof of Lemma 2.2 shows that the restriction
of ${\cal M}_T$ to the diagonal $\Delta_T$ is the trivial line bundle.
We may extend this trivialisation to some analytic neighborhood
of $\Delta_T$.
Now using the equality (2.4) for the given family of Riemann surfaces
we get a holomorphic family of trivialisations of
the restriction of $L$ to some neighborhood of the diagonal.
Using Theorem 3.2 this family of trivialisations
equips the family $X_T$ with a holomorphic family of
projective structures.
Given a family of Riemann surfaces, $X_{T'} \longrightarrow T'$, consider
the corresponding family of Riemann surfaces with theta characteristic
$$
(X_T ,{\Gamma}_T) ~\longrightarrow ~T
$$
where $p : T\longrightarrow T'$ is the finite \'etale Galois cover with
the fiber of $p$ over $t \in T'$ being the set of all theta
characteristics on the corresponding Riemann
surface $X_t$. We earlier saw that there is a holomorphic family of
projective structures for the family $X_T \longrightarrow T$. For $x \in
T$ let ${\mbox{\Large $\wp$}}_x$ denote the projective structure on the Riemann surface
over $x$. For any $t \in T'$ consider the projective structure
on $X_t$ given by the average
$$
{1\over {\# p^{-1}(t)}}\sum_{x \in p^{-1}(t)} {\mbox{\Large $\wp$}}_x
$$
which is defined using the affine space structure on the space of all
projective structures on $X_t$. Using this construction we conclude that
the family of Riemann surfaces, $X_{T'}$, admits a holomorphic
family of projective structures.
\section{Relation with Deligne's definition}
We shall now recall another definition of a projective structure
given in \cite{D} (Definition 5.6 bis).
The fibers of the natural
projection, $\nu$, of the second order infinitesimal neighborhood
of the diagonal $\Delta$ (in $S$) onto $\Delta$ are isomorphic to
${\rm Spec}(R)$, where $R$ is the algebra ${{\Bbb C}}[\epsilon
]/{\epsilon}^3$. Let $P$ denote the
principal ${\rm Aut}({\rm Spec}(R))$ bundle on $X$ whose fiber over
$x \in X$ is the space of all isomorphisms between ${\rm Spec}(R)$
and the fiber of $\nu$ over $x$. On the other hand, ${\rm Aut}({\rm
Spec}(R))$ is same as the group of all
automorphisms of $CP^1$ that fix the point $0 \in {\Bbb C} \cup
\{\infty\} = CP^1$. Let $P_{tg}$ denote the
projective bundle on $X$ associated to $P$. Since ${\rm
Aut}({\rm Spec}(R))$
fixes a point in $CP^1$, the bundle $P_{tg}$ has a natural
section which we shall denote by $\tau$. There is a natural
isomorphism between the second order infinitesimal neighborhood of
$\Delta$ and
the second order neighborhood of the image of $\tau$ (in $P_{tg}$).
\medskip
\noindent {\bf Definition 4.1} ``Definition 5.6 bis of
\cite{D}''. A projective structure on $X$ is
an isomorphism between the third order infinitesimal neighborhood
of the diagonal $\Delta$ (in $S$) with the third order infinitesimal
neighborhood of $\tau$
(in $P_{tg}$) such that
the restriction of this isomorphism to the second order
infinitesimal neighborhood
of $\Delta$ is the canonical isomorphism with the second order
infinitesimal neighborhood of the image of $\tau$ mentioned above.
\medskip
If $X = CP^1$, the projective line, then $P_{tg} = CP^1\times CP^1$.
Thus there is a canonical projective structure on $CP^1$
in the sense of (\cite{D}, Definition 5.6 bis) given by the
identity map of of the third order neighborhood of the diagonal
in $CP^1 \times CP^1$.
Let $\cal H$ denote the sheaf on $X$ which to any
open set, $U \subset X$, associates the space of all embeddings
of the third order infinitesimal neighborhood of the diagonal
of $U\times U$
into the restriction of $P_{tg}$ to $U$ which lift the canonical
embedding of the second order neighborhood of the diagonal of
$U\times U$.
Let us recall \cite{D} that a
$K^2_X$-{\it torsor} is a holomorphic fiber bundle over $X$ such that
its fiber over any $x\in X$ is equipped with a free, transitive
holomorphic action of the fiber $K^2_x$. In other words, the result of
the action of a local holomorphic section of $K^2_X$ on a local
holomorphic section of the torsor is again a local holomorphic section.
Proposition 5.8 (page 32) of \cite{D} says that the
sheaf $\cal H$ defined above is a $K^2_X$-torsor.
We shall denote the restriction of $L$ to $n\Delta$ by $L(n)$.
For an analytic open set $U$ of $X$ let ${\Delta}_U$ be the
diagonal divisor on $U\times U$. Let $\cal G$ denote the sheaf
on $X$ which to any open set
$U \subset X$ associates the space of all
trivialisations of the restriction of $L(3)$ to $3\Delta_U$
giving the canonical trivialisation on $2\Delta_U$. From (3.3)
it follows that the restriction ${\cal G} (U)$ is an affine
space for $H^0(U,K^2_U)$, where $K_U$ is the canonical bundle of
$U$. In other words, $\cal G$ is a torsor for the sheaf $K^2_X$.
Our aim in this section is to prove the following theorem:
\medskip
\noindent {\bf Theorem 4.2.}~ {\it The two
$K^2_X$-torsors on $X$, namely $\cal G$ and $\cal H$,
are canonically isomorphic.}
\medskip
Theorem 4.2 gives a natural identification of the
space ${\cal Q}$, the space of global
sections of $\cal G$, with the space of global sections of $\cal H$,
which is the space of all projective structures on $X$ in
the sense of (\cite{D}, Definition 5.6 bis).
\medskip
\noindent {\it Proof of Theorem 4.2.}~ We shall prove the
theorem by constructing a third $K^2_X$-torsor, $\cal T$, on $X$
and identifying both $\cal G$ and $\cal H$ with $\cal T$.
A reason for introducing $\cal T$ as the intermediate step is that
its construction might be of some interest.
For $n\geq 0$, let $J^n(X)$ denote the sheaf of jets of
order $n$ on $X$, which is a vector bundle on $X$ of rank $n+1$.
Define ${J}^n_0(X)$ to be the kernel of the obvious projection
of $J^n(X)$ onto $J^0(X)$.
Note that there is a canonical splitting of
the inclusion of ${J}^n_0(X)$ into $J^n(X)$ given by the constant
functions. Let $\mbox{${\cal{P}}$} (X)$ denote the subset of the total
space of ${J}^3_0(X)$ given by the
inverse image of $\{K_X -0\}$ (the
set of all nonzero vectors in the total space of $K_X$) under the
projection of ${J}^3_0(X)$ onto ${J}^1_0(X) = K_X$. The
space $\mbox{${\cal{P}}$} (X)$ admits a natural action (by composition of
functions) of the group $M(0)$, the
isotropy group of $0 \in {\Bbb C}$ for the M\"obius group action on
$CP^1$. The action
of $M(0)$ on $\mbox{${\cal{P}}$}(X)$ is free, since the only M\"obius
transformation of $CP^1$, which acts as the
identity map on the second order neighborhood of a point, is actually
the identity transformation (\cite{D}, page 29).
Let $\cal T$
denote the quotient of $\mbox{${\cal{P}}$} (X)$ by $M(0)$. A projective
structure on $X$ gives maps from neighborhoods of points of $X$
into $CP^1$ which differ only by a M\"obius transformation, and
hence gives a section of the obvious projection
of $\cal T$ onto $X$. An identification between the
space of all sections of $\cal T$ and $\mbox{\Large $\wp$}$, the space of projective
structures on $X$, is obtained in this way.
We shall now give a $K^2_X$-torsor structure on $\cal T$.
Let $f \in {\mbox{${\cal{P}}$}}(X)$ be an element over $x \in X$, and let $v \in
(K^2_X)_x$ be an element of the fiber of $K^2_X$ over $x$. Let
${\bar f}$ be a function defined around $x$ which represents $f$.
Since $d{\bar f}(x) \neq 0$ (by the definition of ${\mbox{${\cal{P}}$}}(X)$),
there is a number $\lambda \in {\Bbb C}$ such that $v = \lambda .d{\bar f}(x)\otimes
d{\bar f}(x)$. Consider the function
$$
{\bar f}_{\lambda }~ := ~ {\bar f} ~+ ~ \lambda . {\bar f}^3 \leqno{(4.3)}
$$
defined around $x$. The element in ${\mbox{${\cal{P}}$}}(X)$ over $x$
represented by the function ${\bar f}_{\lambda }$ clearly does not
depend upon the choice of the representative $\bar f$ of $f$.
An action of $K^2_X$ on ${\mbox{${\cal{P}}$}}(X)$ is obtained by
mapping the pair $(v ,f)$ to the element of ${\mbox{${\cal{P}}$}}(X)_x$
represented by ${\bar f}_{\lambda }$. This is a free (but not transitive)
action of the
abelian group scheme $K^2_X$ over $X$. This action of $K^2_X$ on
${\mbox{${\cal{P}}$}} (X)$ induces a $K^2_X$-torsor structure on the
quotient space $\cal T$ of ${\mbox{${\cal{P}}$}}(X)$. Indeed, this is a consequence
of the following fact: let $J^n_0(0)$ be the jets of order $n$ of
functions vanishing at $0\in {\Bbb C}$; in this notation, the group
$M(0)$ acts freely and transitively on
the subset of $J^2_0(0)$ consisting of all elements
whose image in $J^1_0(0)$ is nonzero. (If $i$ denotes the isomorphism
from the space of all sections of $\cal T$ to $\mbox{\Large $\wp$}$, then
$i(A +\gamma ) = i(A)+ 6\gamma $ for any $\gamma \in H^0(X,K^2_X)$.)
Theorem 4.2 is a consequence of the assertion that both $\cal G$
and $\cal H$ coincide with this $K^2_X$-torsor $\cal T$. We shall first
show that $\cal G$ coincides with $\cal T$.
Take any $x \in X$, and let $f \in {\cal T}_x$ be an element of the
fiber over $x$. Let $z$ be a function defined in a
neighborhood, $U$, of $x$, that represents $f$.
Since $d{z}(x) \neq 0$, we may
assume that $z$ is a biholomorphism onto its image. Let ${\bar
z} = (z,z)$ be the biholomorphism defined on $U\times U$.
Pull back the section $s$ (defined in (3.4)) to $U\times U$ using
this map $\bar z$. Let $\hat f$ denote the local
section of $\cal G$ obtained by restricting this section to the second
order infinitesimal neighborhood of the diagonal. The
evaluation at $x$, namely ${\hat f}(x)$, depends
only on $f$ and not on the representing function $z$. Thus we
have a map from $\cal T$ to $\cal G$ which is evidently an
isomorphism. We want to check that this isomorphism preserves
the $K^2_X$-torsor structures of $\cal T$ and $\cal G$.
Take an element $v = \lambda (dz)^{\otimes 2} \in K^2_x$, where $\lambda
\in {\Bbb C}$. From the
definition of the $K^2_X$-torsor structure on $\cal T$ in (4.3)
it follows that the local function $z +\lambda z^3$ represents the
result of the action of $v$ on $f$. Remark 3.11 says that the two
sections of $\cal G$,
represented by $z$ and $z+\lambda z^3$ respectively, differ by $\mbox{${\cal{S}}$}
(z + \lambda z^3)(0)/6$. Since
$$
\mbox{${\cal{S}}$} (z+\lambda z^3)(0) ~ = ~ 6\lambda
$$
the preservation of the $K^2_X$-torsor structures of $\cal T$
and $\cal G$ is established.
Next we want to show that $\cal H$ coincides with $\cal T$.
Take $x$, $f$ and $z$ as above. We noted earlier that $CP^1$ has
a canonical projective structure (in the sense of \cite{D},
Definition 5.6 bis) given by the identity map of the third order
neighborhood of the diagonal. This projective structure induces
a projective structure on $U$ by the biholomorphism $z$. The
evaluation of the section (over $U$) of $\cal H$, thus obtained,
at the point $x$, does not depend upon the choice of the
representative $z$ of $f$. This gives the required
$K^2_X$-torsor structure preserving isomorphism
between $\cal T$ and $\cal H$.
As an alternative proof of Theorem 4.2 we shall give a
direct identification between $\cal G$ and $\cal H$ using
coordinate charts.
Let $(U,z)$ be a coordinate chart around $x \in X$ with
$z(x) = 0$. Using (3.8) we get a section of $\cal G$ over $U$.
We shall denote this section as $f_z$.
Since the only M\"obius transformation of $CP^1$, which acts as the
identity map on the second order neighborhood of a point, is actually
the identity map, and the group of M\"obius transformations
acts transitively on $CP^1$, there is a natural projective
structure on $CP^1$ in the sense of (\cite{D}, Definition
5.6 bis).
Since the function $z$ identifies $U$ with an open set in
$CP^1$, we get a local section of $\cal H$, which we shall
denote by $g_z$.
By mapping the section $f_z$ to $g_z$ we get an identification of
the restriction ${\cal G} (U)$ with ${\cal H} (U)$ which preserves
the torsor structures. We shall show that this identification does
not depend upon the choice of the coordinate function $z$.
Let $(V,w)$ be another coordinate chart around $x$. Thus
$$
w ~ = ~ \sum_{i=0}^{\infty}a_i z^i \leqno{(4.4)}
$$
with $a_1 \neq 0$. Let $f_w$ (resp. $g_w$) denote the local
section of $\cal G$ (resp. $\cal H$) for $(V,w)$.
It is a simple calculation using (4.4) to check that
$$
f_w(x)\, - \, f_z(x)~= ~ {{a_1a_3 - a^2_2} \over
{a^2_1}} dz\otimes dz
~ = ~ g_w(x) \, - \, g_z(x)
$$
This completes the proof of the theorem.$\hfill{\Box}$
\section{Genesis in conformal field theory}
In this section we explain how the above definition of projective
connection in terms of trivialisations of $L$ on $3\Delta$ came out
of some investigations on a model quantum field theory on a curve
(see \cite{R1}-\cite{R3}), which give it the intuitive picture of a
{\it generalized cross ratio} on a compact Riemann surface, in the
limit when all of its arguments are made to coalesce.
The application of algebraic geometry to quantum field theory in
\cite{R1}-\cite{R3} rests on replacing the study of ``quantum
fields", which are not geometric objects, by their so-called
``$n$-point functions" which are hypothesised to be so. Thus in
\cite{R1} and
\cite{R2} we identified the ``$n$-point functions" of the defining
``quantum fields" of the model with meromorphic sections of
certain line bundles on the $n$-fold Cartesian product of the
curve $X$. In \cite{R3} we showed how the $n$-point functions
of the ``current" $j$, which is a ``regularised product" of the
defining fields, could be computed by the use of {\it schemes}
having {\it nilpotent elements} to give a precise meaning to the
coalescing of arguments involved in the definition of the
current.
A similar regularised product of currents gives the ``energy-momentum
tensor" $T$ of the system, a fact usually expressed by
saying that $T$ is in ``Sugawara form". This is a feature
of many conformal quantum field theories and plays an important role
in the theory of the Virasoro algebra \cite{KR}. The heuristic
expectation in (conformal) quantum field theory \cite{BPZ} is
that its ``one point function" $<T(z)>$ is a {\it projective
connection}.
Our study of $<T(z)>$ proceeds from the
calculation of the two point function of currents $<j(z)j(w)>$ in
\cite{R3}. The salient point is the introduction of the
remarkable line bundle ${\cal A}:={\cal O}(D_{12}+D_{34}-D_{14}-D_{23})$
on $X^4:=X_1\times X_2\times X_3\times X_4$, the product of 4 copies of X,
where $D_{ij}$ denotes the divisor of $X^4$ defined by the diagonal
of $X_i$ and $X_j$. It was pointed out in \cite{R3} that the
canonical meromorphic section $1_{\cal A}$, associated with the divisor
defining ${\cal A}$, is a natural generalisation
to an arbitrary compact, connected Riemann surface of the
{\it cross ratio} of 4 points in the complex plane. It was shown in
\cite{R3} that the calculation of
$<j(z)j(w)>$ requires the trivialisability of ${\cal A}$ on
the product scheme $Z:=2\Delta_{13}\times 2\Delta_{24}$, where
$\Delta_{ij}$ is the diagonal of $X_i\times X_j$ and $2\Delta_{ij}$
denotes its first infinitesimal neighborhood.
\noindent {\bf Proposition 5.1}.~ {\it The line bundle
${\cal A}:={\cal O}(D_{12}+D_{34}-D_{14}-D_{23}) $
is trivialisable on $Z~ := 2\Delta_{13}\times 2\Delta_{24}$
and, moreover, if $\rho\in H^0(Z,{\cal A}\mid Z)$ denotes such
a trivialisation, then
$$
1_{\cal A}\mid Z~-~ \rho = \omega_B \leqno{(5.2)}
$$
where $\omega_B$ denotes a symmetric meromorphic section of
$K_{X\times X}$
with double pole on the diagonal, defined by a holomorphic section of
$K_{X\times X}(2\Delta)$ which restricts to $1$ on the diagonal.}
As pointed out in \cite{R3}, equation (5.2) can be regarded as the
precise algebro-geometric formulation of the following
well known formula
expressing the meromorphic bidifferential $\omega_B$ in terms
of the ``prime form" $E(x,y)$ (see Fay\cite{F}, eqn.(28) p.20):
$$
\omega_B(x,y)~=~\frac{\partial^2\ln E(x,y)}{\partial x\partial
y}\leqno{(5.3)}
$$
In this way it was shown in \cite{R3} that the two point
function of currents $<j(z)j(w)>$ is a {\it symmetric
meromorphic bidifferential with a double pole on the diagonal}.
The computation of the ``one point function" $<T(z)>$ from
$<j(z)j(w)>$ now requires that the line bundle $K_{X\times
X}(2\Delta)$ should be trivialisable on the {\it second}
infinitesimal neighborhood $3\Delta$ of $\Delta$ in $X\times X$
(see \cite{R4} for further details), the validity of which
follows from results in \cite{R1}. In this way we arrive at our
proposed definition of projective connection, having started
with the generalised cross ratio and ended with the coalescing
of all of its arguments.
The fact
that a symmetric meromorphic bidifferential gives rise to a projective
connection appears to have been first observed in \cite{HS} (see also
\cite{F} p.20, following eqn.(28) cited above). The techniques
used in these references, however, do not give a {\it characterisation}
of a projective structure, as is provided by Theorem 3.2, nor an
understanding of when all projective structures arise in this way,
as is provided by Remark 3.12. Moreover, their approach cannot be
adapted to the study of $<T(z)>$, for which we require an
algebro-geometric approach, which will make possible the study of the
{\it higher} point functions as well as other related problems. It also
appears to be a fact that \cite{HS} and \cite{F} are
inaccessible to most geometers and so we hope that the present treatment
clarifies some of these results.
The present formulation of the concept of projective
connection was announced in several conferences and also in \cite{R4},
where the interested reader will find, in addition, a survey for
mathematicians of the papers \cite{R1}, \cite{R2} and \cite{R3}.
\medskip
\noindent {\bf Acknowledgments:} The authors are very grateful
to Prof. M. S. Narasimhan for his useful comments. The
first named author is thankful to the Institut Fourier and the
Acad\'emie des Sciences, Paris, for their hospitality and support.
The second named author thanks the International
Centre for Theoretical Physics, Trieste, for its hospitality.
|
1996-07-19T01:44:05 | 9607 | alg-geom/9607019 | en | https://arxiv.org/abs/alg-geom/9607019 | [
"alg-geom",
"math.AG"
] | alg-geom/9607019 | Richard Hain | Richard Hain | The Hodge de Rham theory of relative Malcev completion | 36 pages. Author supplied dvi available at
http://www.math.duke.edu/faculty/hain/ | null | null | null | null | The Hodge de Rham theory of relative Malcev completion is developed in this
paper. In the special case where one takes the corresponding reductive group to
be trivial, one recovers Chen's de Rham theory of the fundamental group and the
corresponding Hodge theory due to Morgan and the author. This work is a
principal technical tool in the author's work on the mapping class groups.
| [
{
"version": "v1",
"created": "Thu, 18 Jul 1996 23:41:08 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Hain",
"Richard",
""
]
] | alg-geom | \section{Introduction}
Suppose that $\pi$ is an abstract group, that $S$ is a reductive
algebraic group defined over a field $F$ of characteristic zero,
and that $\rho : \pi \to S(F)$ is a homomorphism with Zariski dense
image. The completion of $\pi$ relative to $\rho$ is a proalgebraic
group ${\mathcal G}$ which is an extension
$$
1 \to {\mathcal U} \to {\mathcal G} \stackrel{p}{\to} S \to 1
$$
where ${\mathcal U}$ is prounipotent, and a homomorphism $\tilde{\rho} : \pi \to {\mathcal G}(F)$
which lifts $\rho$:
$$
\begin{CD}
\pi @>{\rho}>> S \cr
@V{\tilde{\rho}}VV @| \cr
{\mathcal G} @>p>> S
\end{CD}
$$
It is characterized by the following universal mapping property. If
$\phi$ is a homomorphism of $\pi$ to a (pro)algebraic group $G$
over $F$ which is an extension
$$
1 \to U \to G \to S \to 1
$$
of $S$ by a unipotent group $U$, and if the the composite
$$
\pi \to G \to S
$$
is $\rho$, then there is a unique homomorphism ${\mathcal G} \to G$ of
$F$-proalgebraic groups which commutes with the projections to $S$ and
through which $\phi$ factors.
When $S$ is the trivial group, ${\mathcal G}$ is simply the classical Malcev
(or unipotent) completion of $\pi$. In this case, with $F={\mathbb R}$ or ${\mathbb C}$,
and $\pi$ the fundamental group of a smooth manifold, there is a de~Rham
theorem for ${\mathcal O}({\mathcal G})$ which was proved by K.-T.~Chen \cite{chen}.
In these notes we generalize Chen's de~Rham Theorem from the unipotent
case to the general case. Our approach is based on the notes
\cite{deligne:letter} of Deligne where an approach to computing the
Lie algebra of the prounipotent radical of ${\mathcal G}$ via Sullivan's minimal
models is sketched. Before explaining our result in general, we recall
Chen's de~Rham Theorem in the unipotent case.
If $M$ is a smooth manifold and $w_1,\dots, w_r$ are smooth
1-forms on $M$, then Chen defined
$$
\int_\gamma w_1\dots w_r =
\idotsint\limits_{0\le t_1\le \cdots \le t_r \le 1} f_1(t_1)\dots
f_r(t_r)\, dt_1 \dots dt_r
$$
where $\gamma : [0,1] \to M$ is a piecewise smooth path and
$\gamma^\ast w_j = f_j(t)\, dt$. These are viewed as functions on the
path space of $M$. An iterated integral is a linear combination of
such functions and the constant function. Fix a base point $x\in M$.
Set $\pi = \pi_1(M,x)$. Denote the iterated integrals on the space
of loops in $M$ based at $x$ by ${\mathcal I}_x$. Denote by $H^0({\mathcal I}_x)$ those
elements of ${\mathcal I}_x$ whose value on a loop depends only on its homotopy
class. Then Chen's $\pi_1$ de~Rham Theorem asserts that integration
induces a Hopf algebra isomorphism
$$
{\mathcal O}({\mathcal U}) \cong H^0({\mathcal I}_x)
$$
where ${\mathcal U}$ denotes the real points of the unipotent completion of $\pi$
and ${\mathcal O}({\mathcal U})$ its coordinate ring.
Another important ingredient of Chen's theorem is that it gives an
algebraic description of ${\mathcal I}_x$ and $H^0({\mathcal I}_x)$ in terms of the (reduced)
bar construction on the de~Rham complex of $M$ and the augmentation induced
by the base point.
In this paper we generalize the definition of iterated integrals and
prove a more general de~Rham theorem in which the Hopf algebra ${\mathcal O}({\mathcal G})$
of functions on the completion of $\pi_1(M,x)$ relative to a homomorphism
$\rho : \pi_1(M,x) \to S$ is isomorphic to a Hopf algebra of ``locally
constant iterated integrals,'' defined algebraically in terms of a suitable
(2-sided) bar construction on a complex $\Efin^{\bullet}(M,{\mathcal O}(P))$. This complex
of forms plays a central role in all our constructions and was introduced by
Deligne in his notes \cite{deligne:letter}, the main result of which is
that the pronilpotent Lie algebra associated to the 1-minimal model of
$\Efin^{\bullet}(M,{\mathcal O}(P))$ is the Lie algebra of the prounipotent radical
${\mathcal U}$ of ${\mathcal G}$.
In Section~\ref{groupoid} we define the completion of the fundamental
groupoid of a manifold $M$ with respect to the representation $\rho$.
This is a category (in fact, a groupoid) whose objects are the points
of $X$ and where the Hom sets are proalgebraic varieties; the
automorphism of the object $x\in M$ is the completion of $\pi_1(M,x)$
relative to $\rho$. There is a canonical functor of the fundamental
groupoid of $M$ to this category. We give a de~Rham description of the
coordinate ring of each Hom variety in terms of a suitable 2-sided bar
construction on $\Efin^{\bullet}(M,{\mathcal O}(P))$ and of the functor from the
fundamental groupoid to its relative completion using iterated integrals.
One of the main applications of Chen's $\pi_1$ de~Rham Theorem is to
give a direct construction of Morgan's mixed Hodge structure \cite{morgan}
on the unipotent completion of the fundamental group of a pointed complex
algebraic variety as is explained in \cite{hain:geom}. In this paper we
prove that if $X$ is a smooth complex algebraic variety (or the complement
of a normal crossings divisor in a compact K\"ahler manifold) and
${\mathbb V} \to X$ is an admissible variation of Hodge structure with polarization
$\langle\phantom{x},\phantom{x}\rangle$ whose monodromy representation
$$
\rho : \pi_1(X,x) \to S := \Aut(V_x,\langle\phantom{x},\phantom{x}\rangle)
$$
has Zariski dense image\footnote{The assumption that the monodromy
have Zariski dense monodromy can probably be removed. What one needs
to know is that the Zariski closure of the image of $\rho$ is reductive
and that its coordinate ring has a natural real Hodge structure --- see
Remark~\ref{extended}. This should follow from the work of Simpson and Corlette
as each of them has pointed out.}, then the coordinate ring ${\mathcal O}({\mathcal G})$ of the
completion of $\pi_1(X,x)$ relative to $\rho$ has a natural mixed
Hodge structure. More generally, we show that the coordinate rings
of the Hom sets of the relative completion of the fundamental groupoid
of $X$ with respect to $\rho$ have canonical mixed Hodge structures.
Our principal application of the Hodge theorem for relative completion
appears in \cite{hain:torelli} where we use it to prove that the unipotent
completion of each Torelli group (genus $\neq 2$) has a canonical mixed
Hodge structure given the choice of a smooth projective curve of genus $g$.
Another application suggested by Ludmil Kartzarkov, and proved in
Section~\ref{hodge_str}, is a generalization
of the theorem of Deligne-Griffiths-Morgan-Sullivan (DGMS) on fundamental
groups of compact K\"ahler manifolds: If $X$ is a compact K\"ahler
manifold and ${\mathbb V}\to X$ is a polarized variation of Hodge structure
with Zariski dense monodromy, then the prounipotent radical of the
completion of $\pi_1(X,x)$ relative to the monodromy representation
has a presentation with only quadratic relations. The theorem of
DGMS is recovered by taking ${\mathbb V}$ to be the trivial variation ${\mathbb Q}_X$.
In Section~\ref{connection} we show that if $X$ is a smooth variety
and ${\mathbb V}$ is an admissible variation of Hodge structure over $X$ with
Zariski dense monodromy representation $\rho$, then there is a canonical
integrable 1-form
$$
\omega \in E^1(X')\comptensor \Gr^W_{\bullet} {\mathfrak u}
$$
where $X'$ is the Galois covering of $X$ with Galois group $\im\rho$,
and ${\mathfrak u}$ the Lie algebra of the prounipotent radical ${\mathcal U}$ of the
completion ${\mathcal G}$ of $\pi_1(X,x)$ with respect to $\rho$. This
form is $\im \rho$ invariant under the natural actions of $\im \rho$
on $X'$ and ${\mathfrak u}$. It can be integrated to the canonical representation
$$
\tilde{\rho} : \pi_1(X,x) \to S\ltimes {\mathcal U} \cong {\mathcal G}.
$$
In the particular case where $X$ is the complement of the discriminant
locus in ${\mathbb C}^n$, where $\pi_1(X,x)$ is the braid group$B_n$ and
$S$ the symmetric group, this connection is the standard one
$$
\omega = \sum_{i<j} d\log(x_i-x_j)\, X_{ij}
$$
on $X'$, the complement in ${\mathbb C}^n$ of the hyperplanes $x_i=x_j$. Kohno
\cite{kohno} used the $\Sigma_n$ invariant form $\omega$ and
finite dimensional representations of $\Gr^W_{\bullet} {\mathfrak u}$ to
construct Jones's representations of $B_n$.
Our construction is used in \cite{hain:torelli} to construct an
analogous ``universal projectively flat connection'' for the mapping
class groups in genus $\ge 3$.
I am very grateful to Professor Deligne for sharing his notes on
the de~Rham theory of relative completion with me and for his interest
in this work. I would also like to thank M.~Saito for explaining some
of his work to me, and Hiroaki Nakamura for his careful reading the
manuscript and his many useful comments. I'd also like to thank Kevin
Corlette and Carlos Simpson for freely sharing their ideas on
(\ref{extended}). The bulk of this
paper was written when I was visiting Paris in spring 1995. I would
like to thank the Institute Henri Poincar\'e and the Institute des
Hautes \'Etudes Scientifiques for their generous hospitality and support.
\section{Conventions}
Here, to avoid confusion later on, we make explicit our basic conventions
and review some basic constructions that depend, so some extent, on these
conventions.
Throughout these notes, $X$ will be a connected smooth manifold. By a path
in $X$ from $x\in X$ to $y\in Y$, we shall mean a piecewise smooth map
$\alpha : [0,1] \to X$ with $\alpha(0)=x$ and $\alpha(1) =y$. The set
of all paths in $X$ will be denoted by $PX$. There is a natural
projection $PX \to X\times X$; it takes $\alpha$ to its endpoints
$(\alpha(0),\alpha(1))$.
The fiber of this map over $(x,y)$ will be denoted by $P_{x,y}X$, and the
inverse image of $\{x\}\times X$ will be denoted by $P_{x,-}$. The sets
$PX$, $P_{x,y}X$, $P_{x,-}X$, each endowed with the compact-open
topology, are topological spaces.
We shall multiply paths in their natural order, as distinct from the
functional order. That is, if $\alpha$ and $\beta$ are two paths in $X$
with $\alpha(1) = \beta(0)$, then the path $\alpha\beta$ is defined and
is the path obtained by first traversing $\alpha$, and then $\beta$.
Suppose that $(\widetilde{X},\tilde{x}_o) \to (X,x_o)$ is a pointed universal
covering of $X$. With our path multiplication convention, $\pi_1(X,x_o)$
acts on the {\em left} of $\widetilde{X}$. One way to see this clearly is to
note that there is a natural bijection
$$
\coprod_{y\in X}\pi_0(P_{x_o,y}X) \to \widetilde{X}.
$$
This bijection is constructed by taking the homotopy class of the path
$\alpha$ in $X$ that starts at $x_o$ to the endpoint $\tilde{\alpha}(1)$ of
the unique lift $\tilde{\alpha}$ of $\alpha$ to $\widetilde{X}$ that starts at
$\tilde{x}_o$. With respect to this identification, the action of
$\pi_1(X,x_o)$ is by left multiplication.
Another consequence of our path multiplication convention is that
$\pi_1(X,x_o)$ naturally acts on the {\em right} of the fiber over $x_o$
of a flat bundle over $X$, as can be seen from an elementary computation.
Conversely, if
$$
\rho : F\times \pi_1(X,x_o) \to F
$$
is a right action of $\pi_1(X,x_o)$ on $F$, then one can define
$F\times_\rho \widetilde{X}$ to be the quotient space $F\times \widetilde{X}/\sim$,
where the equivalence relation is defined by
$$
(f,gx) \sim (fg,x)
$$
for all $g \in \pi_1(X,x_o)$. This bundle has a natural flat structure
--- namely the one induced by the trivial flat structure on the bundle
$F \times \widetilde{X} \to \widetilde{X}$.
The composite
$$
F \cong F \times \{\tilde{x}_o\} \hookrightarrow
F \times \widetilde{X} \to F\times_\rho\widetilde{X}
$$
gives a natural identification of the fiber over $x_o$ with $F$. With
respect to this identification, the monodromy representation of the flat
bundle $F\times_\rho \widetilde{X} \to X$ is $\rho$.
Of course, left actions can be converted into right actions by using
inverses. Presented with a natural left action of $\pi_1(X,x_o)$ on a
space, we will convert it, in this manner, into a right action in order
to form the associated flat bundle.
The flat bundle over $X$ corresponding to the right $\pi_1(X,x_o)$-module
$V$ will be denoted by ${\mathbb V}$.
For a flat vector bundle ${\mathbb V}$ over $X$, we shall denote the complex of
smooth forms with coefficients in the corresponding $C^\infty$ vector
bundle by $E^{\bullet}(X,{\mathbb V})$. This is a complex whose cohomology
is naturally isomorphic to $H^{\bullet}(X,{\mathbb V})$. In particular, the $C^\infty$
de~Rham complex of $X$ will be denoted by $E^{\bullet}(X)$.
By definition, mixed Hodge structures (MHSs) are usually finite
dimensional. When studying MHSs on completions of fundamental
groups, one encounters two kinds of infinite dimensional MHSs
$$
((V_{\mathbb R},W_{\bullet}),(V_{\mathbb C},W_{\bullet},F^{\bullet})).
$$
In both cases, the weight graded quotients are finite dimensional.
In one, the weight filtration is bounded below (i.e. $W_lV=0$, for some
$l$) so that each $W_mV$ is finite dimensional. In this case we require
that each $W_mV$ with the induced filtrations be a finite dimensional
MHS in the usual sense. The other case is dual. Here the
weight filtration is bounded above (i.e., $V=W_lV$ for some $l$). In
this case, each $V/W_mV$ is finite dimensional. We require that $V$
be complete in the topology defined by the weight filtration (i.e.,
$V$ is the inverse limit of the $V/W_mV$), that each part of the
Hodge filtration be closed in $V$, and that each $V/W_mV$ with the
induced filtrations be a finite dimensional MHS in the usual sense.
Such mixed Hodge structures form an abelian category, as is easily
verified.
Finally, if $V^{\bullet}$ is a graded module and $r$ is a integer, $V[r]^{\bullet}$
denotes the graded module with
$$
V[r]^n = V^{r+n}.
$$
\section{The Coordinate Ring of a Reductive Linear Algebraic Group}
\label{coord}
Suppose that $S$ is a reductive linear
algebraic group over a field $F$ of characteristic zero. The right and
left actions of $S$ on itself induce commuting left and right actions of
$S$ on its coordinate ring ${\mathcal O}(S)$.
If $V$ is a right $S$ module, its dual $V^\ast := \Hom_F(V,F)$ is a left
$S$ module via the action
$$
(s \cdot \phi) (v) := \phi(v\cdot s),
$$
where $s\in S$, $\phi \in \Hom_F(V,F)$ and $v\in V$.
The following result generalizes to reductive groups a well known fact
about the group ring of a finite group.
\begin{proposition}\label{decomp} If $\left(V_\alpha\right)_\alpha$ is a
set of representatives of the isomorphism classes of irreducible
right $S$-modules, then, as an $(S,S)$ bimodule, ${\mathcal O}(S)$ is canonically
isomorphic to
$$
\bigoplus_\alpha V_\alpha^\ast\boxtimes V_\alpha.
$$
\end{proposition}
\begin{proof} This follows from the following facts:
\begin{enumerate}
\item If $V$ is an $S$ module, then the set of matrix entries of
$V$ is the dual $(\End V)^\ast$ of $\End V$. It has commuting right
and left $S$ actions. The right action is induced by left multiplication
of $S$ on itself by left translation, and the left action by the right
action of $S$ on itself.
\item As a vector space, $(\End_F V)^\ast$ is naturally isomorphic to
$V^\ast\otimes V$. The isomorphism takes
$\phi\otimes v\in V^\ast \otimes V$ to the matrix entry
$$
\{f:V \to V\} \mapsto \left\{F \stackrel{v}{\to} V \stackrel{f}{\to}
V \stackrel{\phi}{\to} F \right\}.
$$
It is easily checked that this isomorphism gives an isomorphism
$(\End V)^\ast \cong V^\ast \boxtimes V$ of $(S,S)$-bimodules.
\item By standard arguments (cf.\ \cite{cartier}), the fact that
$S$ is reductive implies that the subspace of ${\mathcal O}(S)$ spanned by
the matrix entries of all irreducible linear representations
is a subalgebra of ${\mathcal O}(S)$. That is, the image of the linear
map
$$
\Phi : \sum_\alpha V_\alpha^\ast \boxtimes V_\alpha \to {\mathcal O}(S)
$$
is a subalgebra of ${\mathcal O}(S)$. Since $\Phi$ is $S\times S$ equivariant, and
since the $V_\alpha^\ast \boxtimes V_\alpha$ are pairwise non-isomorphic
irreducible representations of $S\times S$, $\Phi$ is injective.
\item Since $S$ is linear, it has a faithful linear representation $V_0$,
say and ${\mathcal O}(S)$ is generated by the matrix entries of $V_0$. It follows
that $\Phi$ is surjective, and therefore an algebra isomorphism.
\end{enumerate}
\hfill \end{proof}
Recall that if $G$ is an affine algebraic group, then the Lie
algebra ${\mathfrak g}$ of $G$ can be recovered from ${\mathcal O}(G)$ as follows:
Denote the maximal ideal in ${\mathcal O}(G)$ of functions that vanish at
the identity by ${\mathfrak m}$. Then, as a vector space, ${\mathfrak g}$ is isomorphic to
the dual ${\mathfrak m}/{\mathfrak m}^2$ of the Zariski tangent space of $G$ at the identity.
The bracket is induced by the comultiplication
$$
\Delta : {\mathcal O}(G) \to {\mathcal O}(G)\otimes {\mathcal O}(G)
$$
as we shall now explain.
Evaluation at the identity and inclusion of scalars give linear
maps ${\mathcal O}(G) \to k$ and $k\to {\mathcal O}(G)$. There is therefore a canonical
isomorphism
$$
{\mathcal O}(G) \cong k \oplus {\mathfrak m}.
$$
Using this decomposition, we see that the diagonal induces a
diagonal map
$$
\overline{\Delta} : {\mathfrak m} \to {\mathfrak m}\otimes {\mathfrak m}.
$$
Denote the involution $f\otimes g \mapsto g\otimes f$ of ${\mathfrak m} \otimes {\mathfrak m}$
by $\tau$. The map
$$
\overline{\Delta} - \tau \circ \overline{\Delta} : {\mathfrak m} \to {\mathfrak m}\otimes {\mathfrak m}
$$
induces the map
$$
\Delta^c : {\mathfrak m}/{\mathfrak m}^2 \to {\mathfrak m}/{\mathfrak m}^2\otimes {\mathfrak m}/{\mathfrak m}^2
$$
dual to the bracket.
\section{A Basic Construction}
{}From this point on $S$ will be a linear algebraic group defined over
${\mathbb R}$. We will abuse notation and also denote its group of real points by
$S$. We will assume now that we have a representation
$$
\rho : \pi_1(X,x_o) \to S
$$
whose image is Zariski dense.
We will fix a set of representatives $\left(V_\alpha\right)_\alpha$ of
the isomorphism classes of rational representations of $S$.
Composing $\rho$ with the action of $S$ on itself by {\it right}
multiplication, we obtain a right action of $\pi_1(X,x_o)$ on $S$.
Denote the corresponding flat bundle by
$$
p: P \to X.
$$
This is a left principal $S$ bundle whose fiber $p^{-1}(x_o)$ over
$x_o$ comes with an identification with $S$ ; the $S$ action and the
marking of $p^{-1}(x_o)$ are induced by the obvious left action of
$S$ on $S \times \widetilde{X}$ and by the composite
$$
S \cong S \times \{x_o\} \hookrightarrow S \times \widetilde{X} \to P.
$$
The point $\tilde{x}_o$ of $p^{-1}(x_o)$ corresponding to $1 \in S$
will be used as a basepoint of $P$.
Each rational representation of $S$ gives rise to a representation of
$\pi_1(X,x_o)$, and therefore to a local system over $X$. We shall call
such a local system a {\it rational local system}.
The action of $\pi_1(X,x_o)$ on $S$ by right multiplication induces a
left action of $S$ on ${\mathcal O}(S)$, the coordinate ring of $S$. Convert this
to a right action using inverses:
$$
(f\gamma)(s) = f(s\gamma^{-1}),
$$
where $f\in {\mathcal O}(S)$, $\gamma\in \pi_1(X,x_o)$, and $s\in S$. Denote the
associated flat bundle by
$$
{\mathcal O}(P) \to X.
$$
This is naturally a {\em right} flat principal $S$ bundle over $X$. It
follows from (\ref{decomp}) that it is the direct sum of its rational
sub-local systems:
\begin{equation}\label{decomp2}
{\mathcal O}(P) = \bigoplus_\alpha {\mathbb V}_\alpha^\ast \otimes V_\alpha.
\end{equation}
In particular, it is the direct limit of its rational sub-local systems.
Define
$$
\Efin^{\bullet}(X,{\mathcal O}(P)) = \lim_\to E^{\bullet}(X,{\mathbb M}),
$$
where ${\mathbb M}$ ranges over the rational sub-local systems of ${\mathcal O}(P)$.
Denote the cohomology
$$
\lim_\to H^{\bullet}(X,{\mathbb M})
$$
of this complex by $H^{\bullet}(X,{\mathcal O}(P))$.
The right action of $S$ on ${\mathcal O}(P)$ induces a right action of $S$ on
$$
H^{\bullet}(X,{\mathcal O}(P)).
$$
{}From (\ref{decomp2}), it follows that there is a natural isomorphism
$$
\Efin^{\bullet}(X,{\mathcal O}(P)) \cong
\bigoplus_\alpha E^{\bullet}(X,{\mathbb V}_\alpha^\ast)\otimes V_\alpha
$$
of right $S$ modules. The following result is an immediate consequence.
\begin{proposition}\label{isom}
For each irreducible representation $V$ of $S$, there is a natural
isomorphism
$$
\left[ H^k(X,{\mathcal O}(P))\otimes V\right]^S \cong H^k(X,{\mathbb V}). \qed
$$
\end{proposition}
The bundle $P\to X$ is foliated by its locally flat sections. Denote
this foliation by ${\mathcal F}$. We view it as a sub-bundle of $TP$, the tangent
bundle of $P$. Denote by $E^k(P,{\mathcal F})$ the vector space consisting
of $C^\infty$ sections of the dual of the bundle
$$
\Lambda^k {\mathcal F} \to P.
$$
One can differentiate sections along the leaves to obtain an
exterior derivative map
$$
d : E^k(P,{\mathcal F}) \to E^{k+1}(P,{\mathcal F}).
$$
With this differential, $E^{\bullet}(P,{\mathcal F})$ is a differential graded algebra.
Moreover, the left action of $S$ on $P$ induces a natural right action
of $S$ on it, and the natural restriction map
\begin{equation}\label{res}
E^{\bullet}(P) \to E^{\bullet}(P,{\mathcal F})
\end{equation}
is an $S$-equivariant homomorphism of differential graded algebras.
The base point $\tilde{x}_o\in P$ induces augmentations
$$
\Efin^{\bullet}(X,{\mathcal O}(P)) \to {\mathbb R}\text{ and } E^{\bullet}(P,{\mathcal F}) \to {\mathbb R}.
$$
\begin{proposition}\label{dga_homom}
There is a natural, augmentation preserving d.g.\ algebra
homomorphism
$$
\Efin^{\bullet}(X,{\mathcal O}(P)) \to E^{\bullet}(P,{\mathcal F})
$$
which is injective and $S$-equivariant with respect to the natural
right $S$ actions.\hfill \qed
\end{proposition}
\section{Iterated Integrals and Monodromy of Flat Bundles}
Consider the category ${\mathcal B}(X,S)$ whose objects are flat vector
bundles ${\mathbb V}$ over $X$ that admit a finite filtration
$$
{\mathbb V} = {\mathbb V}^0 \supset {\mathbb V}^1 \supset {\mathbb V}^2 \supset \cdots
$$
by sub-local systems with the properties:
\begin{enumerate}
\item the intersection of the ${\mathbb V}^i$ is trivial;
\item each graded quotient ${\mathbb V}^i/{\mathbb V}^{i+1}$ is the local system
associated with a rational representation of $S$.
\end{enumerate}
Denote the fiber over the base point $x_o$ by $V_o$. It has
a filtration corresponding to the filtration ${\mathbb V}^{\bullet}$ of
${\mathbb V}$:
$$
V_o = V_o^0 \supset V_o^1 \supset V_o^2 \supset \cdots
$$
The second condition above implies that there are rational
representations $\tau_i : S \to \Aut \Gr^i V_o$ such that
the representation of $\pi_1(X,x_o)$ on $\Gr^i V_o$ is the
composite
$$
\pi_1(X,x_o) \stackrel{\rho}{\to} S
\stackrel{\tau_i}{\to} \Aut \Gr^i V_o.
$$
Let $\tau : S \to \prod \Aut \Gr^i V_o$ be the product of the
representations $\tau_i$.
Let
$$
G = \left\{\phi \in \Aut V_o :
\phi\text{ preserves } V_o^{\bullet}\text{ and }
\Gr^{\bullet}\phi\in \im \tau \right\}.
$$
This is a linear algebraic group which is an extension of $\im \tau$
by the unipotent group
$$
U = \left\{\phi \in \Aut V_o : \phi\text{ preserves }
V_o^{\bullet}\text{ and acts trivially on } \Gr^{\bullet} V_o \right\}
$$
whose Lie algebra we shall denote by ${\mathfrak u}$.
We shall denote the monodromy representation at $x_o$ of ${\mathbb V}$ by
$$
\tilde{\rho} : \pi_1(X,x_o) \to G.
$$
Denote the $C^\infty$ vector bundles associated to the flat bundles
${\mathbb V}$ and ${\mathbb V}^i$ by ${\mathcal V}$ and ${\mathcal V}^i$, respectively.
We would like to trivialize ${\mathcal V}$. In order to do this, we pull it
back to $P$ along the projection $p:P \to X$.
\begin{proposition}\label{triv}
There is a trivialization
$$
p^\ast {\mathcal V} \stackrel{\cong}{\to} P\times V_o
$$
and a splitting of the natural map $G \to \im \tau$ which satisfy
\begin{enumerate}
\item the corresponding connection form%
\footnote{Our convention is that the connection form associated to the
trivialized bundle $V\times X \to X$ with connection $\nabla$ is the
1-form $\omega$ on $X$ with values in $\End V$ which is characterized
by the property that for all sections $f:X \to V$
$$
\nabla f = df - f\omega \in E^1(X)\otimes V.
$$}
$\widetilde{\omega}$ satisfies
$$
\widetilde{\omega} \in E^1(P)\otimes {\mathfrak u};
$$
\item\label{cond} the isomorphism $V_o \to V_o$, induced by the
trivialization of $p^\ast {\mathcal V}$ between the fiber over the points $\tilde{x}_o$
and $s\cdot\tilde{x}_o$ of $p^{-1}(x_o)$, is $\tau(s)^{-1}$.
\end{enumerate}
\end{proposition}
Note that the second condition implies that the isomorphism $V_o \to V_o$,
induced by the trivialization of $p^\ast {\mathcal V}$ between the fiber over the
points $a\cdot \tilde{x}_o$
and $sa\cdot\tilde{x}_o$ of $p^{-1}(x_o)$, is $\tau(s)^{-1}$.
The first step in the proof is the following elementary result. It
can be proved by induction on the length of the filtration. It gives
the splitting of $G \to \im \tau$.
\begin{lemma}\label{splitting}
There is an isomorphism
$$
{\mathcal V} \cong \bigoplus_{i\ge 0} \Gr^i {\mathcal V}
$$
of $C^\infty$ vector bundles that splits the filtration ${\mathcal V}^{\bullet}$.
That is,
\begin{enumerate}
\item the sub-bundle ${\mathcal V}^i$ corresponds to $\oplus_{j\ge i} \Gr^j {\mathcal V}$;
\item the isomorphism
$$
\Gr^i {\mathcal V} \to {\mathcal V}^i/{\mathcal V}^{i+1}
$$
induced by the trivialization is the identity. \hfill \qed
\end{enumerate}
\end{lemma}
\begin{proof}[Proof of (\ref{triv})]
Pulling back the splitting given by (\ref{splitting}) of the
filtration ${\mathcal V}^i$ to $P$, we obtain a splitting
$$
p^\ast{\mathcal V} \cong \bigoplus_i p^\ast \Gr^i {\mathcal V}
$$
of $p^\ast {\mathcal V}$. So it suffices to trivialize each $p^\ast \Gr^i {\mathcal V}$.
To do this, we first do it on a single leaf ${\mathcal L}$ of $P$. The
restriction of the monodromy representation $\tau$ to ${\mathcal L}$
is clearly trivial. Consequently, the restriction of $p^\ast {\mathcal V}$
to ${\mathcal L}$ is trivial as a flat bundle. Observe that if this leaf contains
$\tilde{x}_o$, then this trivialization satisfies condition (\ref{cond})
in the statement of (\ref{triv}).
Next, change the trivialization of $p^\ast\Gr^i{\mathcal V}$ on $p^{-1}(x_o)$
so that it satisfies condition (\ref{cond}) in the statement of
(\ref{triv}). Extend this to a trivialization of $p^{-1}\Gr^i{\mathcal V}$
on all of $P$ by parallel transport along the leaves of $P$. This
gives a well defined local trivialization which is a global
trivialization by the argument in the previous paragraph.
We thus obtain a trivialization of $p^\ast{\mathcal V}$ which is
compatible with the filtration ${\mathcal V}^{\bullet}$ and which is flat on
each $\Gr^i{\mathcal V}$. It follows that the connection form $\widetilde{\omega}$
associated to this trivialization satisfies
$\widetilde{\omega} \in E^1(P)\otimes {\mathfrak u}$.
\end{proof}
If $S$ is not finite, this connection is not flat as it is
not flat in the vertical direction. We can make it flat by
restricting it to the leaves of the foliation ${\mathcal F}$ of $P$.
Denote the image of $\widetilde{\omega}$ under the restriction
homomorphism
$$
E^1(P)\otimes {\mathfrak u} \to E^1(P,{\mathcal F}) \otimes {\mathfrak u}
$$
by $\omega$. It defines the connection in the leaf direction.
This connection is clearly flat, and it follows that $\omega$ is
integrable.
The following assertion is a consequence of the
properties (1) and (2) in the statement of Proposition \ref{triv}
and (\ref{dga_homom}). Note that we view $S$ as acting on the
left of ${\mathfrak u}$ via the adjoint action --- that is, via the
composite $S \to \im \tau \hookrightarrow G \to \Aut {\mathfrak u}$.
\begin{proposition}
The connection form $\omega$ is integrable and lies in the subspace
$\Efin^1(X,{\mathcal O}(P))\otimes {\mathfrak u}$ of $E^1(P,{\mathcal F})\otimes {\mathfrak u}$. Moreover,
if $s\in S$, then
$s^\ast \omega = Ad(s)\omega$. \qed
\end{proposition}
\begin{remark}\label{converse}
There is a converse to this result. Suppose that ${\mathfrak u}$ is a nilpotent
Lie algebra in the category of rational representations of $S$. Then
we can form the semi-direct product $G=S\ltimes U$, where $U$ is the
corresponding unipotent group. If $V$ is a $G$ module, and if
$$
\omega \in \Efin^1(X,{\mathcal O}(P))\otimes {\mathfrak u}
$$
satisfies the conditions
\begin{enumerate}
\item $d\omega + \omega \wedge \omega = 0$;
\item $s^\ast \omega = Ad(s)\omega$;
\end{enumerate}
then we can construct an object of ${\mathcal B}(X,S)$ with fiber $V$ over $x_o$
whose pullback to $P$ has connection form $\omega$ with respect to an
appropriate trivialization.
\end{remark}
We are now ready to express the monodromy representation of
${\mathbb V}$ in terms of iterated integrals of $\omega$. Recall that
K.-T.~Chen \cite{chen} defined, for 1-forms $w_i$ on a manifold $M$
taking values in an associative algebra $A$,
$$
\int_\gamma w_1 w_2 \dots w_r
$$
to be the element
$$
\idotsint\limits_{0 \le t_1 \le \dots \le t_r \le 1}
f_1(t_1)f_2(t_2) \dots f_r(t_r)\, dt_1dt_2 \dots dt_r
$$
of $A$. This is regarded as an $A$-valued function $PM \to A$ on the
path space of $M$. An $A$-valued iterated integral is a function
$PM \to A$ which is a linear combination of functions
of this form together with a constant function.
Suppose that $V\times M \to M$ is a trivial bundle with a connection
given by the connection form
$$
\omega \in E^1(M) \otimes \End(V).
$$
In this case we can define the parallel transport map
$$
T : PM \to \Aut(V)
$$
where $PM$ denotes the space of piecewise smooth paths in $M$. A
path goes to the linear transformation of $V$ obtained by parallel
transporting the identity along it. Chen \cite{chen} obtained the
following expression for $T$ in terms of $\omega$.
\begin{proposition}\label{transp}
With notation as above, we have
$$
T(\gamma) = 1 + \int_\gamma \omega + \int_\gamma \omega\omega
+ \int_\gamma \omega\omega\omega + \cdots \qed
$$
\end{proposition}
Note that since ${\mathfrak u}$ is nilpotent, this is a finite sum.
Armed with this formula, we can express the monodromy
of ${\mathbb V} \to X$ in terms of $\omega \in \Efin^1(X,{\mathcal O}(P))$. Suppose
that $\gamma \in P_{x_o,x_o}X$. Denote the unique lift of
$\gamma$ to $P$ which is tangent to ${\mathcal F}$ and begins at
$\tilde{x}_o\in p^{-1}(x_o)$, by ${\tilde{\gamma}}$.
\begin{proposition}\label{monod}
The monodromy of ${\mathbb V} \to X$ takes $\gamma \in P_{x_o,x_o}X$
to
$$
\tilde{\rho}(\gamma) =
\left(1 + \int_{{\tilde{\gamma}}} \omega + \int_{{\tilde{\gamma}}}
\omega\omega + \int_{{\tilde{\gamma}}} \omega\omega\omega + \cdots
\right)\tau(\rho(\gamma)) \in G.
$$
\end{proposition}
The proof is a straightforward consequence of Chen's formula
(\ref{transp}) and condition (\ref{cond}) of (\ref{triv}).
This formula motivates the following generalization of Chen's iterated
integrals.
\begin{definition}\label{defn}
For $\phi\in {\mathcal O}(S)$ and $w_1,\dots,w_r$ elements of
$\Efin^1(X,{\mathcal O}(P))$, we define
$$
\int \left(w_1 \dots w_r | \phi\right) : P_{x_o,x_o}X \to {\mathbb R}
$$
by
$$
\int_\gamma \left(w_1 \dots w_r | \phi\right)
= \phi(\rho(\gamma))\int_{{\tilde{\gamma}}}w_1\dots w_r.
$$
\end{definition}
We will call linear combinations of such functions {\it iterated
integrals with coefficients in ${\mathcal O}(S)$}. They will be regarded
as functions $P_{x_o,x_o}X \to {\mathbb R}$. We will denote the set of them
by $I(X,{\mathcal O}(S))_{x_o}$. Such an iterated integral will
be said to be {\it locally constant} if it is constant on each
connected component of $P_{x_o,x_o}X$. We shall denote the set of
locally constant iterated integrals on $P_{x_o,x_o}X$ by
$H^0(I(X,{\mathcal O}(S))_{x_o})$. Evidently, each such locally
constant iterated iterated integral defines a function
$$
\pi_1(X,x_o) \to {\mathbb R}.
$$
By taking matrix entries in (\ref{monod}), we obtain the following
result.
\begin{corollary}\label{matrix_entries}
Each matrix entry of the monodromy representation
$$
\tilde{\rho}: \pi_1(X,x_o) \to G
$$
of an object of ${\mathcal B}(X,S)$ can be expressed as a locally constant
iterated integral on $X$ with coefficients in ${\mathcal O}(S)$. \qed
\end{corollary}
The following results imply that $H^0(I(X,{\mathcal O}(S))_{x_o})$
is a Hopf algebra with coproduct dual to the multiplication of
paths, and antipode dual to the involution of $P_{x_o,x_o}X$ that
takes each path to its inverse.
\begin{proposition}\label{props}
Suppose that $\gamma$ and $\mu$ are in $P_{x_o,x_o}X$, that
$\phi,\psi\in {\mathcal O}(S)$ and that $w_1,w_2,\dots \in \Efin^1(X,{\mathcal O}(P))$.
Then we have:
\begin{equation}
\int_\gamma \left(w_1 \dots w_p | \phi\right)
\int_\gamma \left(w_{p+1} \dots w_{p+q} | \phi\right)
= \sum_{\sigma \in Sh(p,q)}
\int_\gamma \left( w_{\sigma(1)}\dots w_{\sigma(p+q)}|
\phi\psi \right)
\end{equation}
where $Sh(p,q)$ denotes the set of shuffles of type $(p,q)$;
\begin{equation}
\int_{\gamma^{-1}} \left(w_1 \dots w_r | \phi\right) = (-1)^r
\int_\gamma
\left(\rho(\gamma^{-1})^\ast w_r \dots \rho(\gamma^{-1})^\ast w_1|
i_S^\ast\phi\right)
\end{equation}
where $i_S^\ast : {\mathcal O}(S) \to {\mathcal O}(S)$ is the antipode of ${\mathcal O}(S)$;
\begin{equation}
\int_{\gamma\mu}\left(w_1 \dots w_r | \phi\right) =
\sum_{i=0}^r \sum_j \int_\gamma\left( w_1\dots w_i| \phi_j'\right)
\int_\mu \left(\rho(\gamma)^\ast w_{i+1}\dots \rho(\gamma)^\ast w_r |
\phi_j''\right)
\end{equation}
where $\Delta_S : {\mathcal O}(S) \to {\mathcal O}(S)\otimes {\mathcal O}(S)$ is the coproduct of
${\mathcal O}(S)$, and
$$
\Delta_S \phi = \sum_j \phi_j'\otimes \phi_j''.
$$
\end{proposition}
\begin{proof}
This proof is a straightforward using the definition (\ref{defn})
and basic properties of classical iterated integrals due to Chen
\cite{chen}.
\end{proof}
\begin{corollary}
The set of iterated integrals $I(X,{\mathcal O}(S))_{x_o}$ is a commutative Hopf
algebra.
\end{corollary}
\begin{remark}
Let $\pi_1(X,x_o) \to {\mathcal G}$ be the completion of $\pi_1(X,x_o)$ relative
to $\rho : \pi_1(X,x_o) \to S$. Since the coordinate ring ${\mathcal O}({\mathcal G})$ of
${\mathcal G}$ is the ring of matrix entries of representations of $G$, it follows
from (\ref{matrix_entries}) that there is a Hopf algebra inclusion
$$
{\mathcal O}({\mathcal G}) \hookrightarrow H^0(I(X,{\mathcal O}(S))_{x_o}).
$$
To prove this assertion, it would suffice to show that
$$
H^0(I(X,{\mathcal O}(P)))\otimes_{{\mathcal O}(S)}{\mathbb R}
$$
is the direct limit of coordinate rings of a directed system of unipotent
groups, each with an $S$ action. This is surely true, but we seek a more
algebraic de~Rham theorem for ${\mathcal O}({\mathcal G})$ which is more convenient for Hodge
theory.
\end{remark}
\section{Higher Iterated Integrals}
As a preliminary step to defining the algebraic analogue of
$I(X,{\mathcal O}(S))_{x_o}$, we generalize the the definition of iterated
integrals with values in ${\mathcal O}(S)$ to higher dimensional forms.
Denote by $E^n(P_{x_o,x_o}X)$ the differential forms of degree $n$ on
the loop space $P_{x_o,x_o}X$. One can surely use any reasonable
definition of differential forms on $P_{x_o,x_o}X$, but we will use
Chen's definition from \cite{chen} where, to specify a differential form
on $P_{x_o,x_o}X$, it is enough to specify its pullback along each
``smooth map'' $\alpha : U\to P_{x_o,x_o}X$ from an open subset $U$
of some finite dimensional euclidean space. By a smooth map,
we mean a map $\alpha : U\to P_{x_o,x_o}X$ whose ``suspension''
$$
\widehat{\alpha} : [0,1] \times U \to X; \quad (t,u) \mapsto \alpha(u)(t)
$$
is continuous and smooth on each $[t_{j-1},t_j]\times U$ for some
partition
$$
0=t_0 \le t_1 \le \dots \le t_m = 1
$$
of $[0,1]$.
\begin{definition}\label{higherdef}
Suppose that $\phi \in {\mathcal O}(S)$, and that $w_j\in \Efin^{n_j}(X,{\mathcal O}(P))$ with
each $n_j>0$. Set $n=-r + \sum_j n_j$. Define
$$
\int \left(w_1 \dots w_r | \phi\right) \in E^n(P_{x_o,x_o}X)
$$
by specifying that for each smooth map $\alpha : U \to P_{x,x}X$,
$$
\alpha^\ast\int \left(w_1 \dots w_r | \phi\right)
$$
is the element
$$
\idotsint\limits_{0\le t_1 \le \cdots \le t_r \le 1}
\widehat{w}_1(t_1)\wedge \dots \wedge \widehat{w}_r(t_r)\,
dt_1\! dt_2\dots dt_r\, \phi(\rho(\alpha(u)))
$$
of $E^n(U)$, where
$$
\widehat{w}_j : (\partial/\partial t) \lrcorner \tilde{\alpha}^\ast w_j
$$
and $\tilde{\alpha} : [0,1]\times U \to P$ is the smooth map with the
property that for each $x\in U$, the map $t\mapsto \tilde{\alpha}(t,x)$
is the unique lift of $t\mapsto \widehat{\alpha}(t,x)$ that begins at
$\tilde{x}_o$ and is tangent to ${\mathcal F}$.
\end{definition}
These iterated integrals form a subspace $I^{\bullet}(X,{\mathcal O}(S))_{x_o}$
of $E^{\bullet}(P_{x_o,x_o}X)$. Chen's arguments \cite{chen} can be adapted
easily to show that this is, in fact, a sub d.g.\ Hopf algebra of
$E^{\bullet}(P_{x_o,x_o}X)$. In particular, we have:
\begin{proposition}
The space of locally constant iterated integrals on $X$ with
coefficients in ${\mathcal O}(S)$ is $H^0(I^{\bullet}(X,{\mathcal O}(S))_{x_o})$. \qed
\end{proposition}
\section{The Reduced Bar Construction}
In this section we review Chen's definition of the reduced bar
construction which he described in \cite{chen:bar}.
Suppose that $A^{\bullet}$ is a commutative differential graded algebra
(hereafter denoted d.g.a.) and that $M^{\bullet}$ and $N^{\bullet}$ are
complexes which are modules over $A^{\bullet}$. That is, the
structure maps
$$
A^{\bullet} \otimes M^{\bullet} \to M^{\bullet} \text{ and }
A^{\bullet}\otimes N^{\bullet} \to N^{\bullet}
$$
are chain maps. We shall suppose that $A^{\bullet}$, $M^{\bullet}$ and $N^{\bullet}$
are all positively graded. Denote the subcomplex of $A^{\bullet}$ consisting
of elements of positive degree by $A^+$.
The {\it (reduced) bar construction} $B(M^{\bullet},A^{\bullet},N^{\bullet})$ is
defined as follows. We first describe the underlying graded vector
space. It is a quotient of the graded vector space
$$
T(M^{\bullet},A^{\bullet},N^{\bullet}) :=
\bigoplus_s M^{\bullet} \otimes\left(A^+[1]^{\otimes r}\right) \otimes N^{\bullet}.
$$
We will use the customary notation $m[a_1|\dots|a_r]n$ for
$$
m\otimes a_1\otimes \dots \otimes a_r \otimes n
\in T(M^{\bullet},A^{\bullet},N^{\bullet}).
$$
To obtain the vector space underlying the bar construction, we mod out
by the relations
$$
m[dg|a_1|\dots|a_r]n = m[ga_1|\dots|a_r]n - m\cdot g[a_1|\dots|a_r]n;
$$
\begin{multline*}
m[a_1|\dots|a_i|dg|a_{i+1}|\dots|a_r]n = \hfill \cr
m[a_1|\dots|a_i|g\,a_{i+1}|\dots|a_r]n
- m[a_1|\dots|a_i\,g|a_{i+1}|\dots|a_r]n \quad 1\le i < s;
\end{multline*}
$$
m[a_1|\dots|a_r|dg]n = m[a_1|\dots|a_r]g\cdot n - m[a_1|\dots|a_r\,g]n;
$$
$$
m[dg]n = 1 \otimes g\cdot n - m\cdot g \otimes 1
$$
Here each $a_i \in A^+$, $g\in A^0$, $m\in M^{\bullet}$, $n\in N^{\bullet}$,
and $r$ is a positive integer.
Before defining the differential, it is convenient to define an
endomorphism $J$ of each graded vector space by
$J: v\mapsto (-1)^{\deg v}v$. The differential is defined as
$$
d = d_M\otimes 1_T \otimes 1_N + J\otimes d_B \otimes 1 +
J_M \otimes J_T \otimes d_N + d_C.
$$
Here $T$ denotes the tensor algebra on $A^+[1]$, $d_B$ is defined by
\begin{multline*}
d_B[a_1|\dots|a_r] =
\sum_{1\le i \le r} (-1)^i [Ja_1|\dots|Ja_{i-1}|da_i|a_{i+1}|\dots|a_r]
\cr \hfill + \sum_{1 \le i < r}
(-1)^{i+1}[Ja_1|\dots|Ja_{i-1}|Ja_i\wedge a_{i+1}|a_{i+2}|\dots|a_r]
\end{multline*}
and $d_C$ is defined by
$$
d_C m[a_1|\dots|a_r]n = (-1)^s
Jm[Ja_1|\dots|Ja_{r-1}]a_r \cdot n - Jm\cdot a_1 [a_2|\dots|a_r]n.
$$
One can check that these differentials are well defined.
If both $M^{\bullet}$ and $N^{\bullet}$ are themselves d.g.a.s over $A^{\bullet}$,
then $B(M^{\bullet},A^{\bullet},N^{\bullet})$ is also a differential graded algebra.
The product is defined by
\begin{multline}\label{prod}
m'[a_1|\dots|a_p]n' \otimes m''[a_{p+1}|\dots |a_{p+q}]n'' \mapsto
\hfill \cr \hfill \sum_{\sigma\in \Sigma(p,q)} \pm m'\wedge m''
[a_{\sigma(1)}| a_{\sigma(2)}|\dots |a_{\sigma(p+q)}]n'\wedge n''.
\end{multline}
Here $\Sigma(p,q)$ denotes the set of shuffles of type $(p,q)$. The
sign in front of each term on the right hand side is determined by
the usual sign conventions that apply when moving a symbol of degree
$k$ past one of degree $l$ --- one considers each $a_j$ to be of
degree $-1 + \deg a_j$.
The reduced bar construction $B(M^{\bullet},A^{\bullet},N^{\bullet})$ has a standard
filtration
$$
{\mathbb R} = B_0(M^{\bullet},A^{\bullet},N^{\bullet}) \subseteq B_1(M^{\bullet},A^{\bullet},N^{\bullet})
\subseteq B_2(M^{\bullet},A^{\bullet},N^{\bullet}) \subseteq \cdots
$$
which is often called the {\it bar filtration}. The subspace
$$
B_s(M^{\bullet},A^{\bullet},N^{\bullet})
$$
is defined to be the span of those $m[a_1|\dots|a_r]n$ with $r\le s$.
When $A^{\bullet}$ has connected homology (i.e., $H^0(A^{\bullet}) = {\mathbb R}$), the
corresponding spectral sequence, which is called the {\it Eilenberg-Moore
spectral sequence}, has $E_1$ term
$$
E_1^{-s,t} =
\left[M^{\bullet}\otimes H^+(A^{\bullet})^{\otimes s}\otimes N^{\bullet}\right]^t.
$$
A proof of this can be found in \cite{chen:bar}.
The following basic property of the reduced bar construction is
a special case of a result proved in \cite{chen:bar}. It is easily
proved using the Eilenberg-Moore spectral sequence. Suppose that
$\psi : A_1^{\bullet} \to A_2^{\bullet}$ is a d.g.a.\ homomorphism, and that
$M^{\bullet}$ is a right $A_2^{\bullet}$ module and $N^{\bullet}$ a right $A_2^{\bullet}$
module. Then $M^{\bullet}$ and $N^{\bullet}$ can be regarded as $A_1^{\bullet}$ modules
via $\psi$. We therefore have a chain map
\begin{equation}\label{map}
B(M^{\bullet},A_1^{\bullet},N^{\bullet}) \to B(M^{\bullet},A_2^{\bullet},N^{\bullet}).
\end{equation}
\begin{proposition}\label{qism}
If $\psi$ is a quasi-isomorphism, then so is (\ref{map}). \qed
\end{proposition}
\section{The Construction of $\Gdr$}
\label{gdr}
In this section we construct a proalgebraic group $\Gdr$ which
is an extension
$$
1 \to \Udr \to \Gdr \stackrel{p}{\to} S \to 1,
$$
where $\Udr$ is prounipotent, and a homomorphism
$$
\tilde{\rho} : \pi_1(X,x_o) \to \Gdr
$$
whose composition with $p : \Gdr \to S$ is $\rho$. We do this
by constructing the coordinate ring of $\Gdr$ using the
bar construction. In the two subsequent sections, we will show
that $\tilde{\rho} : \pi_1(X,x_o) \to \Gdr$
is the Malcev completion of $\pi_1(X,x_o)$ relative to $\rho$.
The fixed choice of a base point $\tilde{x}_o \in p^{-1}(x_o)$
determines augmentations
$$
\epsilon_{\tilde{x}_o} : \Efin^{\bullet}(X,{\mathcal O}(P)) \to {\mathbb R}
$$
and
$$
\delta_{x_o} : \Efin^{\bullet}(X,{\mathcal O}(P)) \to {\mathcal O}(S)
$$
as we shall now explain.
Since these augmentations are compatible with restriction, it suffices
to give them in a neighbourhood of $x_o$. Over a contractible
neighbourhood $U$ of $x_o$, the local system $P$ is trivial and
may therefore by identified with the trivial flat bundle
$S\times U \to S$ in such a way that $\tilde{x}_o$ corresponds to
$(1,x_o) \in S\times U$. The restriction of an element of
$\Efin^k(X,{\mathcal O}(P))$ to $U$ is then of the form
$$
\sum_i \phi_i \otimes w_i
$$
where $\phi_i \in {\mathcal O}(S)$, and $w_i \in E^k(U)$. Denote the augmentation
$E^{\bullet}(U) \to {\mathbb R}$ induced by $x_o$ by $\mu_{x_o}$. Then the augmentations
$\delta_{x_o}$ and $\epsilon_{\tilde{x}_o}$ are defined by
$$
\delta_{x_o} : \sum_i \phi_i \otimes w_i
\mapsto \sum_i \mu_{x_o}(w_i)\, \phi_i
$$
and
$$
\epsilon_{\tilde{x}_o} : \sum_i \phi_i \otimes w_i \mapsto
\sum_i \mu_{x_o}(w_i)\, \phi_i(1).
$$
One can regard ${\mathbb R}$ and ${\mathcal O}(S)$ as algebras over $\Efin^{\bullet}(X,{\mathcal O}(P))$
where the actions of $\Efin^{\bullet}(X,{\mathcal O}(P))$ on these is defined using
these two augmentations. We can therefore form
the bar construction
$$
B({\mathbb R},\Efin^{\bullet}(X,{\mathcal O}(P)),{\mathcal O}(S))
$$
which we shall denote by $B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)})$.
It is a commutative d.g.a.\ when endowed with the product (\ref{prod}).
It is, in fact, a d.g.~Hopf algebra, with coproduct defined as follows:
\begin{multline*}
\Delta : [w_1|\dots|w_r]\phi \mapsto\\
\sum_i [w_1|\dots |w_i]
\left(\sum \psi_i^{(k_i)}\dots\psi_r^{(k_r)}\phi'\right)
\otimes [w_{i+1}^{(k_i)}|\dots|w_r^{(k_r)}]\left(\sum \phi''\right)
\end{multline*}
where
$$
\Delta_S(\phi) = \sum \phi'\otimes \phi''
$$
is the diagonal of ${\mathcal O}(S)$, and the map
$$
\Efin^{\bullet}(X,{\mathcal O}(P)) \to {\mathcal O}(S) \otimes \Efin^1(X,{\mathcal O}(P))
$$
which gives the $S$ action takes $w_j$ to
$$
\sum \psi_j^{(k_j)} \otimes w_j^{(k_j)}.
$$
The following proposition is a direct consequence of the definition
(\ref{higherdef}) and the basic properties of iterated integrals
which may be found in \cite{chen}.
\begin{proposition}\label{pro}
The map
$$
B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)}) \to I^{\bullet}(X,{\mathcal O}(S))_{x_o}
$$
defined by
$$
[w_1|w_2|\dots |w_r]\phi \mapsto \int\left(w_1 w_2 \dots w_r|\phi\right)
$$
is a well defined d.g.~Hopf algebra homomorphism. \qed
\end{proposition}
\begin{proposition}\label{alg_gp}
If $\pi_1(X,x_o)$ is finitely generated, then
$$
H^0(B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)}))
$$
is the coordinate ring of a linear proalgebraic group which is an
extension of $S$ by a prounipotent group.
\end{proposition}
In the proof, we shall need the following technical result, the proof of
which is a straightforward modification of Sullivan's proof of the
existence of minimal models (cf.\ \cite{sullivan}.)
\begin{proposition}\label{sub}
There is a d.g.~subalgebra $A^{\bullet}$ of $\Efin^{\bullet}(X,{\mathcal O}(P))$ with
$A^0={\mathbb R}$, which is also an $S$ submodule, with the properties that the
inclusion is a quasi-isomorphism. \qed
\end{proposition}
\begin{proof}[Proof of (\ref{alg_gp})]
Choose a d.g.\ subalgebra $A^{\bullet}$ of $\Efin^{\bullet}(X,{\mathcal O}(P))$ as given by
(\ref{sub}). It follows from (\ref{qism}) that the natural map
$$
H^0(B({\mathbb R},A^{\bullet},{\mathcal O}(S))) \to H^0(B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)}))
$$
is an isomorphism. Since $A^0={\mathbb R}$, we have that
$$
H^0(B({\mathbb R},A^{\bullet},{\mathcal O}(S))) = H^0(B({\mathbb R},A^{\bullet},{\mathbb R}))\otimes {\mathcal O}(S).
$$
It is not difficult to check that ${\mathcal O}(S)$ is a sub Hopf algebra, and
that this is a tensor product of algebras, but where the coproduct is
twisted by the action of $S$ on $H^0(B({\mathbb R},A^{\bullet},{\mathbb R}))$. So, if we can
show that $H^0(B({\mathbb R},A^{\bullet},{\mathbb R}))$ is the limit of the coordinate rings
of an inverse system of unipotent groups, each with an $S$ action, then
we will have shown that
$$
H^0(B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)}))
$$
is the coordinate ring of
$$
S\ltimes \spec H^0(B({\mathbb R},A^{\bullet},{\mathbb R}))
$$
and therefore proved the proposition. From \cite{hain:bar}, we
know that there is a canonical splitting (in particular, it is
$S$ equivariant) of the projection
$$
H^0(B({\mathbb R},A^{\bullet},{\mathbb R})) \to QH^0(B({\mathbb R},A^{\bullet},{\mathbb R})) =: Q
$$
onto the indecomposable elements $Q$. This splitting induces an
$S$-equivariant algebra isomorphism
$$
{\mathbb R}[Q] \to H^0(B({\mathbb R},A^{\bullet},{\mathbb R})).
$$
The bar filtration induces a filtration
$$
Q_1\subseteq Q_2 \subseteq Q_3 \subseteq \cdots \subseteq Q
$$
of the indecomposables such that $Q= \cup Q_r$. Each $Q_r$ is a Lie
coalgebra, and the cobracket $\Delta^c$ satisfies
$$
\Delta^c : Q_r \to \sum_{i+j = r} Q_i \otimes Q_j
$$
and is injective when $r>1$. Since $\pi_1(X,x_o)$ is finitely generated,
each of the cohomology groups
$H^1(X,{\mathbb V})$ is finite dimensional for each rational local system ${\mathbb V}$
over $X$. It follows from (\ref{isom}) that each isotypical component of
$H^1(X,{\mathcal O}(P))$ is finite dimensional. Since $Q_r/Q_{r-1}$ is a
subquotient of
$$
H^1(X,{\mathcal O}(P))^{\otimes r},
$$
it follows that each $S$-isotypical component of each $Q_r$ is finite
dimensional. One can now prove by induction on $r$ using the nilpotence,
that as an $S$-module, each $Q_r$ is the direct limit of duals of
nilpotent Lie algebras, each of which has an $S$ action. This completes
the proof.
\end{proof}
\begin{definition}\label{def}
Define proalgebraic groups $\Gdr$ and $\Udr$ by
$$
\Gdr = \spec H^0(B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)}))
$$
and
$$
\Udr = \spec H^0(B({\mathbb R},\Efin^{\bullet}(X,{\mathcal O}(P)),{\mathbb R})).
$$
\end{definition}
Evidently, we have an extension
$$
1 \to \Udr \to \Gdr \to S \to 1
$$
of proalgebraic groups, where $\Udr$ is prounipotent.
When we want to emphasize the dependence of $\Gdr$ and $\Udr$
on $(X,x)$, we will write them as $\Gdr(X,x)$ and $\Udr(X,x)$,
respectively.
\begin{proposition}\label{homom}
There is a natural homomorphism $\tilde{\rho} : \pi_1(X,x_o) \to \Gdr$ whose
composition with $\Gdr \to S$ is $\rho$.
\end{proposition}
\begin{proof}
Define a map from $P_{x_o,x_o}X$ to the linear functionals on
$$
B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)})
$$
by
$$
\gamma : [w_1|\dots|w_r]\phi
\mapsto \int_\gamma\left(w_1\dots w_r|\phi\right).
$$
This induces a function
$$
\Phi : \pi_1(X,x_o) \to \Hom_{\mathbb R}({\mathcal O}(\Gdr),{\mathbb R}).
$$
Define $\tilde{\rho}$ by taking the class of $\gamma$ in $\pi_1(X,x_o)$
to the maximal ideal of
$$
H^0(B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)}))
$$
consisting of those elements on which $\gamma$ vanishes. (Note
that $\gamma$ acts via integration.) That this
is a group homomorphism follows from (\ref{props}).
\end{proof}
\section{Construction of Homomorphisms from $\Gdr$}
Suppose that $G$ is a linear algebraic group which can be
expressed as an extension
$$
1 \to U \to G \to S \to 1
$$
where $U$ is unipotent. Choose an isomorphism of $G$ with $S\ltimes U$.
Denote the Lie algebra of $U$ by ${\mathfrak u}$.
\begin{proposition}
Each one form $\omega \in \Efin^1(X,{\mathcal O}(P))\otimes {\mathfrak u}$ that satisfies
\begin{enumerate}
\item $d\omega + \omega\wedge \omega = 0$;
\item for all $s \in S$, $s^\ast \omega = Ad(s)\omega$;
\end{enumerate}
determines a homomorphism $\Gdr \to G$ that commutes with projection to
$S$.
\end{proposition}
\begin{proof}
First note that since the exponential map ${\mathfrak u} \to U$ is a polynomial
isomorphism, ${\mathcal O}(U)$ is isomorphic to the polynomials ${\mathbb R}[{\mathfrak u}]$ on the
vector space ${\mathfrak u}$. Further, there is a natural isomorphism
\begin{equation}\label{iso}
{\mathcal O}(U) \cong {\mathbb R}[{\mathfrak u}] \to \lim_\to \Hom(U{\mathfrak u}/I^n,{\mathbb R})
\end{equation}
which is defined by noting that $U{\mathfrak u}$ is, by the PBW Theorem, the
symmetric coalgebra $S^c{\mathfrak u}$ on ${\mathfrak u}$. The isomorphism (\ref{iso})
is an isomorphism of Hopf algebras.
Set
$$
T = 1 + [\omega] + [\omega|\omega] + [\omega|\omega|\omega] + \cdots
$$
which we view as an element of
$$
B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)}) \comptensor \widehat{U}{\mathfrak u}
$$
of degree zero, where $\widehat{U}{\mathfrak u}$ denotes the completion
$$
\lim_\leftarrow U{\mathfrak u}/I^n
$$
of $U{\mathfrak u}$ with respect to the powers of its augmentation ideal,
and where $\comptensor$ denotes the completed tensor product
$$
\lim_\leftarrow
B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)})\otimes \widehat{U}{\mathfrak u}/I^n
$$
The coordinate ring of $G$ is isomorphic to ${\mathcal O}(U)\otimes{\mathcal O}(S)$.
Define a linear map
$$
\Theta : {\mathcal O}(G) \to B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)})^0
$$
by
$$
f\otimes \phi \mapsto \langle T,f\rangle \cdot \phi.
$$
It is not difficult to check that $\Theta$ is a well defined Hopf algebra
homomorphism. This uses the fact that $s^\ast \omega = Ad(s)\omega$. That
$\omega$ satisfies the integrability condition
$$
d \omega + \omega \wedge \omega = 0
$$
implies that $\im \Theta$ is contained in
$H^0(B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)}))$. It follows that $\Theta$
induces a Hopf algebra homomorphism
$$
{\mathcal O}(G) \to H^0(B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,(x_o)}))
$$
and therefore a group homomorphism
$$
\theta : \Gdr \to G
$$
which commutes with the projections to $S$.
Finally, it follows from (\ref{converse}), (\ref{transp}) and (\ref{homom})
that the composite
$$
\pi_1(X,x_o) \to \Gdr \to G
$$
is the homomorphism induced by $\omega$.
\end{proof}
\begin{corollary}\label{monod_rep}
If ${\mathbb V}$ is a local system in ${\mathcal B}(X,S)$, then the monodromy representation
$$
\tau : \pi_1(X,x_o) \to \Aut V_o
$$
factors through $\tilde{\rho} : \pi_1(X,x_o) \to \Gdr$. \qed
\end{corollary}
\begin{corollary}\label{factor}
If $\tau: \pi_1(X,x_o) \to G$ is a homomorphism into a linear algebraic
group which is an extension of $S$ by a unipotent group, and whose
composite with the projection to $S$ is $\rho$, then there is a
homomorphism $\Gdr \to G$ whose composite with
$$
\tilde{\rho} : \pi_1(X,x_o) \to \Gdr
$$
is $\tau$.
\end{corollary}
\begin{proof}
Denote the kernel of $G \to S$ by $U$. One can construct a faithful, finite
dimensional representation $V$ of $G$ which has a filtration
$$
V = V^0 \supset V^1 \supset V^2 \supset \cdots
$$
by $G$-submodules whose intersection is zero and where each $V^j/V^{j+1}$
is a trivial $U$-module. The corresponding local system over $X$ lies in
${\mathcal B}(X,S)$. The result now follows from (\ref{monod_rep}).
\end{proof}
\section{Isomorphism with the Relative Completion}
Denote $\pi_1(X,x_o)$ by $\pi$. In Section~\ref{gdr} we constructed
a homomorphism $\pi \to \Gdr$. In this section, we prove:
\begin{theorem}\label{main}
If $\pi$ is finitely generated, then the homomorphism $\pi \to \Gdr$
is the completion of $\pi$ relative to $\rho$.
\end{theorem}
To prove the theorem, we first fix a completion
$\pi \to {\mathcal G}$ of $\pi$ relative to $\rho$.
The universal mapping property of the relative completion gives a
homomorphism ${\mathcal G} \to \Gdr$ of proalgebraic groups that commutes with
the canonical projections to $S$. It follows from (\ref{factor}), that
there is a natural homomorphism
\begin{equation}\label{univ}
\Gdr \to {\mathcal G}
\end{equation}
that also commutes with the projections to $S$. It follows from the
universal mapping property of the relative completion that the composite
$$
{\mathcal G} \to \Gdr \to {\mathcal G}
$$
is the identity.
Denote the prounipotent radical of ${\mathcal G}$ by ${\mathcal U}$. Since $\pi$ is finitely
generated, each of the groups $H^1(\pi,V)$ is finite dimensional for
each rational representation $V$ of $S$.
In view of the following proposition and the assumption that $\pi$
is finitely generated, all we need do to show that the natural
homomorphism ${\mathcal G} \to \Gdr$ is an isomorphism is to show that either
of the induced maps
$$
\Hom_S(H_1({\mathcal U}),V) \to \Hom_S(H_1(\Udr),V) \to \Hom_S(H_1({\mathcal U}),V)
$$
is an isomorphism for all $S$-modules $V$.
\begin{proposition}
Suppose that $G_1$ and $G_2$ are extensions of the reductive group $S$ by
unipotent groups $U_1$, $U_2$, respectively:
$$
1 \to U_j \to G_j \to S \to 1.
$$
Suppose that $\theta : G_1 \to G_2$ is a split surjective homomorphism of
algebraic groups that commutes with the projections to $S$. If either of
the induced maps
$$
\Hom_S(H_1(U_1),V) \to \Hom_S(H_1(U_2),V) \to \Hom_S(H_1(U_1),V)
$$
is an isomorphism for all $S$ modules $V$, then both are, and $\theta$
is an isomorphism.
\end{proposition}
\begin{proof}
The proof reduces to basic fact that a split surjective homomorphism
between nilpotent Lie algebras is an isomorphism if and only if it
induces an isomorphism on $H_1$. The details are left to the reader.
\end{proof}
Our first task in the proof of Theorem~\ref{main} is to
compute $\Hom_S(H_1({\mathcal U}),V)$.
\begin{proposition}\label{h1_comp}
For all $S$-modules $V$, there is a canonical isomorphism
$$
H^1(\pi,V) \cong \Hom_S(H_1({\mathcal U}),V).
$$
\end{proposition}
\begin{proof}
We introduce an auxiliary group for the proof. Let
$$
\Hom_\rho(\pi,S\ltimes V)
$$
be the set of group homomorphisms $\pi \to S\ltimes V$ whose composite
with the projection $S\ltimes V \to S$ is $\rho$. Then there is a natural
bijection between $\Hom_\rho(\pi,S\ltimes V)$ and the set of splittings
$\pi \to \pi \ltimes V$ of the projection $\pi \ltimes V \to \pi$: the
splitting $\sigma$ corresponds to $\tilde{\rho} : \pi \to S\ltimes V$ if and
only if the diagram
$$
\begin{CD}
\pi @>\sigma>> \pi\ltimes V \cr
@| @VV{\rho\ltimes id}V\cr
\pi @>\tilde{\rho}>> S\ltimes V \cr
\end{CD}
$$
commutes.
The kernel $V$ acts on both $\Hom_\rho(\pi,S\ltimes V)$ and on the set of
splittings, in both cases by inner automorphisms. The action commutes
with the bijection. Since $H^1(\pi,V)$ is naturally isomorphic to the set
of splittings of $\pi\ltimes V \to \pi$ modulo conjugation by $V$
\cite[p.~106]{maclane}, the bijection induces a natural isomorphism
$$
H^1(\pi,V) \cong \Hom_\rho(\pi,S\ltimes V)/\sim.
$$
On the other hand, by the universal mapping property of the relative
completion, each element of $\Hom_\rho(\pi,S\ltimes V)$ induces a
homomorphism ${\mathcal G} \to S\ltimes V$ which commutes with the projections to
$S$. Such a homomorphism induces a homomorphism ${\mathcal U} \to V$, and therefore an
$S$-equivariant homomorphism $H_1({\mathcal U}) \to V$. Since $V$ is central, this
induces a homomorphism
$$
\Hom_\rho(\pi,S\ltimes V) \to \Hom_S(H_1({\mathcal U}),V).
$$
To complete the proof, we show that this is an isomorphism. Denote the
commutator subgroup of ${\mathcal U}$ by ${\mathcal U}'$. Then the quotient ${\mathcal G}/{\mathcal U}'$ is an
extension of $S$ by $H_1({\mathcal U})$; the latter being a possibly infinite
product of representations of $S$ in which each isotypical factor is
finite dimensional. Using the fact that every extension of $S$ by a
rational representation in the category of algebraic groups splits and
that any two such splittings are conjugate by an element of the kernel,
we see that the extension
\begin{equation}\label{seqce}
0 \to H_1({\mathcal U}) \to {\mathcal G}/{\mathcal U}' \to S \to 1
\end{equation}
is split and that any two splittings are conjugate by an element of
$H_1({\mathcal U})$. Choose a splitting of this sequence. This gives an isomorphism
$$
{\mathcal G}/{\mathcal U}' \cong S\ltimes H_1({\mathcal U}).
$$
An $S$-equivariant homomorphism $H_1({\mathcal U}) \to V$ induces a homomorphism
$$
{\mathcal G}/{\mathcal U}' \cong S\ltimes H_1({\mathcal U}) \to S\ltimes V
$$
of proalgebraic groups. Composing this with the homomorphism
$$
\pi \to {\mathcal G} \to {\mathcal G}/{\mathcal U}',
$$
we obtain an element of $\Hom_\rho(\pi,S\ltimes V)/\sim$. Since all
splittings of (\ref{seqce}) differ by an inner automorphism by an element
of $H_1({\mathcal U})$, we have constructed a well defined map
$$
\Hom_S(H_1({\mathcal U}),V) \to \Hom_\rho(\pi,S\ltimes V).
$$
This is easily seen to be the inverse of the map constructed above. This
completes the proof.
\end{proof}
The following result completes the proof of Theorem~\ref{main}.
\begin{proposition}
The map
$$
H_1({\mathcal U}) \to H_1(\Udr)
$$
induced by (\ref{univ}) is an isomorphism.
\end{proposition}
\begin{proof}
It suffices to show that for all rational representations $V$ of
$S$, the map
$$
\left[H^1(\Udr)\otimes V\right]^S \to \left[H^1({\mathcal U})\otimes V\right]^S
$$
is an isomorphism. Both groups are isomorphic to $H^1(X,{\mathbb V})$.
Choose a de~Rham representative $w \in E^1(X,{\mathbb V})$ of a class in
$H^1(X,{\mathbb V})$. Let $\delta \in V^\ast\otimes V$ be the element
corresponding to the identity $V\to V$. Set
$$
\omega := w\otimes \delta \in E^1(X,{\mathbb V})\otimes V^\ast \otimes V.
$$
Regard $V$ as an abelian Lie algebra. Then
$$
\omega \in \Efin^1(X,{\mathcal O}(P))\otimes V.
$$
It is closed, and therefore satisfies the integrability condition
$d\omega + \omega\wedge\omega = 0$. Since the identity $V\to V$ is
$S$ equivariant,
$$
s^\ast \omega = Ad(s) \omega
$$
for all $s\in S$.
Set $W = V\oplus {\mathbb R}$. Filter this by
$$
W = W^0 \supset W^1 \supset W^2 =0
$$
where $W^1 = V$. Then $V\subset \End W$. It follows from (\ref{converse})
that $\omega$ defines a connection on $P\times V$ which is flat along the
leaves of the foliation ${\mathcal F}$ and descends to a flat bundle over $X$. The
monodromy representation of this bundle is a homomorphism
$$
\tau:\pi_1(X,x_o) \to S\ltimes V \subset \Aut W.
$$
It follows from the monodromy formula (\ref{transp}) that $\tau$ takes the
class of the loop $\gamma$ to
$$
\left(\rho(\gamma),\int_{\tilde{\gamma}} w\right) \in S\ltimes V.
$$
The result follows.
\end{proof}
\section{Naturality}
Suppose that $\pi_X$ and $\pi_Y$ are two groups, and that
$\rho_X : \pi_X \to S_X$ and $\rho_Y : \pi_Y \to S_Y$ are homomorphisms
into the $F$-points of reductive algebraic groups, each with Zariski
dense image. We have the two corresponding relative completions
$$
\tilde{\rho}_X : \pi_X \to {\mathcal G}_X \text{ and } \tilde{\rho}_Y : \pi_Y \to {\mathcal G}_Y.
$$
Fix an algebraic group homomorphism $\Psi : S_X \to S_Y$.
\begin{proposition}\label{induced}
If $\psi : \pi_X \to \pi_Y$ is a homomorphism such that the diagram
$$
\begin{CD}
\pi_X @>{\rho_X}>> S_X \cr
@V{\psi}VV @VV{\Psi}V \cr
\pi_Y @>{\rho_Y}>> S_Y
\end{CD}
$$
commutes, then there is a canonical homomorphism
$\widehat{\Psi} : {\mathcal G}_X \to {\mathcal G}_Y$ such that the diagram
$$
\begin{CD}
\pi_X @>{\tilde{\rho}_X}>> {\mathcal G}_X \cr
@V{\psi}VV @VV{\widehat{\Psi}}V \cr
\pi_Y @>{\tilde{\rho}_Y}>> {\mathcal G}_Y
\end{CD}
$$
\end{proposition}
\begin{proof}
Let $\Psi^\ast {\mathcal G}_Y$ be the pullback of $G_Y$ along $\Psi$:
$$
\begin{CD}
\Psi^\ast {\mathcal G}_Y @>>> S_X \cr
@VVV @VV{\Psi}V \cr
{\mathcal G}_Y @>>> S_Y.
\end{CD}
$$
This group is an extension of $S_X$ by the prounipotent radical
of ${\mathcal G}_Y$.
The homomorphisms $\pi_X \to S_X$ and $\pi_X \to \pi_Y \to {\mathcal G}_Y$
induce a homomorphism $\pi_X \to \Psi^\ast {\mathcal G}_Y$. By the universal
mapping property of $\tilde{\rho}_X : \pi_X \to {\mathcal G}_X$, there is a
homomorphism ${\mathcal G}_X \to \Psi^\ast{\mathcal G}_Y$ which extends the homomorphism
$\pi_X \to \Psi^\ast{\mathcal G}_Y$. The sought after homomorphism $\widehat{\Psi}$
is the composite ${\mathcal G}_X \to \Psi^\ast{\mathcal G}_Y \to {\mathcal G}_Y$.
\end{proof}
Next we explain how to realize $\widehat{\Psi}$ using the bar construction.
Suppose that $(X,x)$ and $(Y,y)$ are two pointed manifolds. Denote
$\pi_1(X,x)$ and $\pi_1(Y,y)$ by $\pi_X$ and $\pi_Y$, respectively.
Suppose that $f:(X,x) \to (Y,y)$ is a smooth map which induces the
homomorphism $\psi : \pi_X \to \pi_Y$ on fundamental groups. Denote
the principal bundles associated to $\rho_X$ and $\rho_Y$ by $P_X \to X$
and $P_Y \to Y$. We have the d.g.a.s
$$
\Efin^{\bullet}(X,{\mathcal O}(P_X)) \text{ and } \Efin^{\bullet}(Y,{\mathcal O}(P_Y)).
$$
Since the diagram in Proposition~\ref{induced} commutes, $f$ and $\psi$
induce a d.g.a.\ homomorphism
$$
(f,\phi)^\ast : \Efin^{\bullet}(Y,{\mathcal O}(P_Y)) \to \Efin^{\bullet}(X,{\mathcal O}(P_X)).
$$
This homomorphism respects the augmentations
induced by $x \in X$ and $y \in Y$, and therefore induces
a d.g.~Hopf algebra homomorphism
$$
B(\Efin^{\bullet}(Y,{\mathcal O}(P_Y))_{\tilde{y},(y)}) \to
B(\Efin^{\bullet}(X,{\mathcal O}(P_X))_{\tilde{x},(x)}).
$$
This induces a homomorphism
\begin{equation}\label{induced_homom}
\Gdr(X,x) \to \Gdr(Y,y)
\end{equation}
after taking $H^0$ and then $\spec$.
\begin{proposition}
Under the canonical identifications of $\Gdr(X,x)$ with ${\mathcal G}(X,x)$ and
$\Gdr(Y,y)$ with ${\mathcal G}(Y,y)$, the homomorphism
$\widehat{\Psi} : {\mathcal G}(X,x) \to {\mathcal G}(Y,y)$
corresponds to the homomorphism
(\ref{induced_homom}).
\end{proposition}
\begin{proof}
If $\gamma$ is a loop in $X$ based at $x$,
$w_1,\dots,w_r \in \Efin^{\bullet}(Y,{\mathcal O}(P_Y))$, and $\phi \in {\mathcal O}(S_Y)$, then
$$
\int_{f\circ \gamma}\left(w_1 \dots w_r | \phi\right)
\int_{\gamma}\left((f,\phi)^\ast w_1 \dots (f,\phi)^\ast w_r |
\Psi^\ast\phi\right).
$$
It follows that (\ref{induced_homom}) is the homomorphism $\widehat{\Psi}$
induced by $f$ and $\psi$.
\end{proof}
\section{Relative Completion of the Fundamental Groupoid}
In this section we explain how the fundamental groupoid of $X$
can be completed with respect to $\rho : \pi_1(X,x) \to S$ and we
give a de~Rham construction of it. In the unipotent case, the de~Rham
theorem is implicit in Chen's work \cite{chen}, and is described
explicitly in \cite{hain-zucker}.
Recall that the fundamental groupoid $\pi(X)$ of a topological space $X$
is the category whose objects are the points of $X$ and whose morphisms
from $a \in X$ to $b \in X$ are homotopy classes $\pi(X;a,b)$ of paths
$[0,1] \to X$ from $a$ to $b$. We can think of $\pi(X)$ as a torsor over
$X\times X$; the fiber over $(a,b)$ being $\pi(X;a,b)$. Observe that
there is a canonical isomorphism between the fiber over $(a,a)$ and
$\pi_1(X,a)$. The torsor is the one over $X\times X$ corresponding to
the representation
\begin{equation}\label{action}
\pi_1(X\times X,(a,a)) \cong \pi_1(X,a)\times \pi_1(X,a) \to \Aut \pi_1(X,a)
\end{equation}
where
$$
(\gamma, \mu) \to \left\{g \mapsto \gamma^{-1}g \mu\right\}.
$$
As in previous sections, $X$ will be a connected smooth manifold and
$x_o$ a distinguished base point. Suppose, as before, that
$\rho : \pi_1(X,x_o) \to S$ is a Zariski dense homomorphism to a reductive
real algebraic group. Denote the completion of $\pi_1(X,x_o)$ relative to
$\rho$ by $\pi_1(X,x_o) \to {\mathcal G}$. The representation (\ref{action})
extends to a representation
$$
\pi_1(X\times X,(x_o,x_o)) \cong \pi_1(X,x_o)\times \pi_1(X,x_o) \to \Aut {\mathcal G}
$$
Denote the corresponding torsor over $X\times X$ by $\boldsymbol{\G}$. This is easily
seen to be a torsor of real proalgebraic varieties. Denote the fiber
of $\boldsymbol{\G}$ over $(a,b)$ by ${\mathcal G}_{a,b}$. There is a canonical map
$$
\pi(X;a,b) \to {\mathcal G}_{a,b}
$$
which induces a map of torsors. It follows from standard arguments that,
for all $a$, $b$ and $c$ in $X$, there is a morphism of proalgebraic
varieties
\begin{equation}\label{mult_map}
{\mathcal G}_{a,b} \times {\mathcal G}_{b,c} \to {\mathcal G}_{a,c}
\end{equation}
which is compatible with the multiplication map
$$
\pi(X;a,b) \times \pi(X;b,c) \to \pi(X;a,c).
$$
An efficient way to summarize the properties of $\boldsymbol{\G}$ and the
multiplication maps is to say that they form a category (in fact, a
groupoid) whose objects are the elements of $X$ and where $\Hom(a,b)$
is ${\mathcal G}_{a,b}$ with composition defined by (\ref{mult_map}). In addition,
the natural map $\pi(X;a,b) \rightsquigarrow {\mathcal G}_{a,b}$ from the fundamental
groupoid of $X$ to this category is a functor. We shall call this functor
the {\it relative completion of the fundamental groupoid of $X$ with respect
to $\rho$}.\footnote{We shall see that the torsor $\boldsymbol{\G}$ is independent
of the choice of base point $x_o$, so it may have been better to call
$\boldsymbol{\G}$ the completion of the fundamental groupoid relative to the principal
bundle $P$.}
Our goal is to give a description of it in terms of differential
forms.
We also have the torsor $\boldsymbol{\cP}$ over $X\times X$ associated to the
representation
$$
\pi_1(X\times X,x_o) \cong \pi_1(X,x_o)\times \pi_1(X,x_o) \to \Aut S
$$
where
$$
(\gamma, \mu)\to\left\{g \mapsto\rho(\gamma)^{-1}g\rho(\mu)\right\}.
$$
given by $\rho$. Denote the fiber of $\boldsymbol{\cP}$ over $(a,b)$ by ${\mathcal P}_{a,b}$.
As above, we have a category whose objects are the points of $X$ and
where $\Hom(a,b)$ is ${\mathcal P}_{a,b}$. There is also a functor from the
fundamental groupoid of $X$ to this category which is the identity on
objects. Denote the restriction of this torsor to $\{a\}\times X$
by $\boldsymbol{\cP}_{a,\underline{\blank}}$. We shall view this as a torsor over $(X,a)$. The
image $\id_a$ of the identity in $\pi_1(X,a)$ in ${\mathcal P}_{a,a}$ gives a
canonical lift of the base point $a$ of $X$ to $\boldsymbol{\cP}_{a,\underline{\blank}}$.
Observe that $\boldsymbol{\cP}_{x_o,\underline{\blank}}$ is the principal $S$ bundle $P$ used in
the construction of ${\mathcal G}$.
For each $a \in X$, we can from the corresponding local system
${\mathcal O}(P_{a,\underline{\blank}})$ whose fiber over $b\in X$ is the coordinate ring
of ${\mathcal P}_{a,b}$. We can form the complex
$$
\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}})) = \varinjlim E^{\bullet}(X,{\mathbb M})
$$
where ${\mathbb M}$ ranges over all finte dimensional sub-local systems of
${\mathcal O}({\mathcal P}_{a,\underline{\blank}})$. This has
augmentations
$$
\epsilon_{a} : \Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}})) \to {\mathbb R}
$$
and
$$
\delta_{a,b} : \Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}})) \to {\mathcal O}({\mathcal P}_{a,b})
$$
given by evaluation at $\id_a$ and on the fiber over $b$, respectively.
We view ${\mathbb R}$ as a right $\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}}))$ module via
$\epsilon_a$ and ${\mathcal O}({\mathcal P}_{a,b})$ as a left
$\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}}))$ module via $\delta_{a,b}$.
We can therefore form the two sided bar construction
$$
B(\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}}))_{\id_a,(b)}) :=
B({\mathbb R},\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}})),{\mathcal O}({\mathcal P}_{a,b}))
$$
Define
\begin{multline}\label{comult}
B(\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}}))_{\id_a,(c)}) \\ \to
B(\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}}))_{\id_a,(b)}) \otimes
B(\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{b,\underline{\blank}}))_{\id_b,(c)})
\end{multline}
by
\begin{multline*}
\Delta : [w_1|\dots|w_r]\phi \mapsto \cr
\sum_i [w_1|\dots |w_i]
\left(\sum \psi_i^{(k_i)}\dots\psi_r^{(k_r)}\phi'\right)
\otimes [w_{i+1}^{(k_i)}|\dots|w_r^{(k_r)}]\left(\sum \phi''\right)
\end{multline*}
where
$$
\Delta_S(\phi) = \sum \phi'\otimes \phi''
$$
is the map ${\mathcal O}(P_{a,c})\to {\mathcal O}(P_{a,b})\otimes{\mathcal O}(P_{b,c})$ dual to
the multiplication map $P_{a,b}\times P_{b,c}\to P_{a,c}$; and the map
$$
\Efin^{\bullet}(X,{\mathcal O}(P_{a,\underline{\blank}})) \to
{\mathcal O}(P_{a,b}) \otimes \Efin^1(X,{\mathcal O}(P_{b,\underline{\blank}}))
$$
induced by multiplication $P_{a,b} \times \boldsymbol{\cP}_{b,\underline{\blank}} \to \boldsymbol{\cP}_{a,\underline{\blank}}$
takes $w_j$ to
$$
\sum \psi_j^{(k_j)} \otimes w_j^{(k_j)}.
$$
Definition~\ref{defn} generalizes:
\begin{definition}
For $\gamma$ a path in $X$ from $a$ to $b$, $\phi\in {\mathcal O}(P_{a,b})$ and
$w_1,\dots,w_r$ elements of $\Efin^1(X,{\mathcal O}(P_{a,\underline{\blank}}))$, we define
$$
\int_\gamma \left(w_1 \dots w_r | \phi\right)
= \phi({\tilde{\gamma}}(1))\int_{{\tilde{\gamma}}}w_1\dots w_r
$$
where ${\tilde{\gamma}}$ is the unique lift of $\gamma$ to a horizontal
section of $\boldsymbol{\cP}_{a,\underline{\blank}}$ which begins at $\id_a \in P_{a,a}$.
\end{definition}
There is an analogous extension of the definition of higher
iterated integrals (\ref{higherdef}) to this situation. As in
that case, one has a d.g.\ algebra homomorphism
$$
B(\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}}))_{\id_a,(b)}) \to E^{\bullet}(P_{a,b}X)
$$
to the de~Rham complex of $P_{a,b}X$, the space of paths in $X$
from $a$ to $b$. It is defined by
$$
[w_1|\dots|w_r]\phi \mapsto \int(w_1\dots w_r|\phi).
$$
By taking a homotopy class $\gamma \in \pi(X;a,b)$ to the ideal of
functions that vanish on it, we obtain a function
$$
\pi(X;a,b) \to
\spec H^0(B(\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}}))_{\id_a,(b)})).
$$
\begin{theorem}\label{gpoid_dr}
This function gives a natural algebra isomorphism
$$
{\mathcal O}({\mathcal G}_{a,b}) \cong
H^0(B(\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}}))_{a,(b)})).
$$
Moreover, the map
$$
{\mathcal O}({\mathcal G}_{a,c}) \to {\mathcal O}({\mathcal G}_{a,b})\otimes {\mathcal O}({\mathcal G}_{b,c})
$$
induced by (\ref{mult_map}) corresponds to (\ref{comult}) under this
isomorphism.
\end{theorem}
\begin{proof}[Sketch of Proof]
Define $\Gdr_{a,b}$ by
$$
\Gdr_{a,b} = \spec H^0(B(\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}}))_{a,(b)})).
$$
The coproduct above induces morphisms
$$
\Gdr_{a,b} \times \Gdr_{b,c} \to \Gdr_{a,c}.
$$
We therefore have a groupoid whose objects are the points of $X$
and where $\Hom(a,b)$ is $\Gdr_{a,b}$ and a function
$$
\pi(X;a,b) \to \Gdr_{a,b}.
$$
This map is easily seen to be compatible with path multiplication
(use the generalization of the last property of (\ref{props})),
and therefore a functor of groupoids. Since $X$ is connected, it
suffices to prove that
$\Gdr_{a,b}$ is isomorphic to ${\mathcal G}_{a,b}$ for just one pair of points
$a,b$ of $X$. But these are isomorphic in the case $a=b=x_o$ by
Theorem~\ref{main}.
\end{proof}
\section{Hodge Theory}
\label{hodge_str}
Now suppose that $X$ is a smooth complex algebraic variety (or the
complement of a normal crossings divisor in a compact K\"ahler manifold)
and that ${\mathbb V}$ is an admissible variation of Hodge structure over $X$.
Denote the semisimple group associated to the fiber $V_o$ over
the base point $x_o\in X$ by $S$. This is the ``orthogonal'' group
$$
S = \Aut(V_o,\langle\phantom{x},\phantom{x}\rangle)
$$
associated to the polarization $\langle\phantom{x},\phantom{x}\rangle$. It is
semi-simple. Suppose that the image of the monodromy representation
$$
\rho : \pi_1(X,x_o) \to S
$$
is Zariski dense. Denote the completion of $\pi_1(X,x_o)$ relative to
$\rho$ by
$$
\tilde{\rho} : \pi_1(X,x_o) \to {\mathcal G}(X,x_o).
$$
\begin{theorem}\label{hodge}
Under these assumptions, the coordinate ring ${\mathcal O}({\mathcal G}(X,x_o))$ of the
completion of $\pi_1(X,x_o)$ with respect to $\rho$ has a canonical
real mixed Hodge structure with weights $\ge 0$ for which the product,
coproduct, antipode and the natural inclusion
$$
{\mathcal O}(S) \hookrightarrow {\mathcal O}({\mathcal G}(X,x_o))
$$
are all morphisms of mixed Hodge structure. Moreover the canonical
homomorphism ${\mathcal G}(X,x_o) \to S$ induces an isomorphism
$\Gr^W_0{\mathcal O}({\mathcal G}(X,x_o)) \cong {\mathcal O}(S)$.
\end{theorem}
Denote the Lie algebra of $S$ by ${\mathfrak s}$. This has a canonical Hodge
structure of weight 0.
The following result is an important corollary of the proof of Theorem~%
\ref{hodge}. It follows immediately from the theorem and the standard
description of the Lie algebra of an affine algebraic group given at the
end of Section~\ref{coord}.
\begin{corollary}\label{hodge-lie}
Under the assumptions of the theorem, the Lie algebra ${\mathfrak g}(X,x_o)$ of
${\mathcal G}(X,x_o)$ has a canonical MHS with weights $\le 0$, and the homomorphism
${\mathfrak g}(X,x_o) \to {\mathfrak s}$ is a morphism of MHS which induces an isomorphism
$$
\Gr^W_0 {\mathfrak g} \cong {\mathfrak s}.
$$
In particular, there is a canonical
MHS with weights $< 0$ on ${\mathfrak u}(X,x_o)$, the Lie algebra of the prounipotent
radical of ${\mathcal G}(X,x_o)$. \qed
\end{corollary}
The principal assertion of Theorem~\ref{hodge} is a special case of the
following result when $a=b=c=x_o$.
\begin{theorem}\label{groupoid}
With $X$, ${\mathbb V}$ and $S$ as above, if $a,\, b\in X$, then the coordinate
ring ${\mathcal O}({\mathcal G}_{a,b})$ of the completion of $\pi(X;a,b)$ relative to $\rho$
has a canonical mixed Hodge structure with weight $\ge 0$ and whose
multiplication is a morphism of MHS. If $a$, $b$ and $c$ are three points
of $X$, then the map
$$
{\mathcal O}({\mathcal G}_{a,c}) \to {\mathcal O}({\mathcal G}_{a,b}) \otimes {\mathcal O}({\mathcal G}_{b,c})
$$
induced by path multiplication is a morphism of MHS. Moreover, the
mixed Hodge structure on ${\mathcal O}({\mathcal G}_{a,b})$ depends only on the variation
${\mathbb V}$ and not on the choice of the base point $x_o$.
\end{theorem}
Because of the last assertion, it may be more appropriate to say
that {\it ${\mathcal G}_{a,b}$ is the completion of $\pi(X;a,b)$ with respect to
the variation ${\mathbb V}$.}
The reader is assumed to be familiar with the basic methods for
constructing mixed Hodge structures on the cohomology of bar
constructions as described in \cite[\S3]{hain:dht}.
In the previous section we showed how to express ${\mathcal O}({\mathcal G}_{a,b})$ as the
0th cohomology group of a suitable reduced bar construction. So in
order to show that it has a canonical MHS we need only find a suitable
augmented, multiplicative mixed Hodge complex $\mathbf{A}^{\bullet}$ which is
quasi-isomorphic to $\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}}))$. To do this, we
shall use the work of M.~Saito on Hodge modules.
First, some notation: Assume that $X = {\overline{X}} - D$, where
${\overline{X}}$ is a compact K\"ahler manifold and $D$ is a normal
crossings divisor. Denote the inclusion $X \hookrightarrow {\overline{X}}$ by
$j$. Denote Deligne's canonical extension of
${\mathbb V}\otimes{\mathcal O}_X$ to ${\overline{X}}$ by $\overline{\mathcal{V}}$. Saito proves that there is a
Hodge module over ${\overline{X}}$ canonically associated to ${\mathbb V}$, whose
complex part is a bifiltered $D$-module $(M,W_{\bullet},F^{\bullet})$, and whose
real part is $Rj_\ast{\mathbb V}_{\mathbb R}$ endowed with a suitable weight filtration.
There is a canonical inclusion
$$
\Omega^{\bullet}_{\overline{X}}({\overline{X}}\log D)\otimes_{\mathcal O} \overline{\mathcal{V}}
\hookrightarrow M\otimes_{\mathcal O} \Omega_{\overline{X}}^{\bullet}.
$$
Saito defines Hodge and weight filtrations on
$\Omega^{\bullet}_{\overline{X}}({\overline{X}}\log D)\otimes_{\mathcal O} \overline{\mathcal{V}}$
by restricting those of $M$. The Hodge filtration is simply the
tensor product of those of $\Omega^{\bullet}_{\overline{X}}({\overline{X}}\log D)$
and $\overline{\mathcal{V}}$. The weight filtration is more difficult to describe.
\begin{theorem}[Saito \protect{\cite[(3.3)]{saito}}]\label{saito:mhc}
The pair
\begin{equation}\label{mhc}
M^{\bullet}(X,{\mathbb V}) :=
((Rj_\ast{\mathbb V}_{\mathbb R}, W_{\bullet}),
(\Omega^{\bullet}_{\overline{X}}({\overline{X}}\log D)\otimes_{\mathcal O} \overline{\mathcal{V}}, F^{\bullet}, W_{\bullet}))
\end{equation}
is a cohomological mixed Hodge complex whose cohomology is canonically
isomorphic to $H^{\bullet}(X,{\mathbb V})$. \qed
\end{theorem}
We can therefore obtain a mixed Hodge complex (MHC) which computes
$H^{\bullet}(X,{\mathbb V})$ by taking the standard fine resolution of these sheaves
by $C^\infty$ forms. (So the complex part of this will be the $C^\infty$
log complex $E^{\bullet}({\overline{X}}\log D,\overline{\mathcal{V}})$ with suitable Hodge and weight
filtrations.)
To apply Saito's machinery to the current situation, we will need
to know that ${\mathcal O}({\mathcal P}_{x_o,\underline{\blank}})$ is a direct limit of admissible
variations over $X$.
\begin{lemma}\label{loc_sys}
The local system associated with an irreducible representation
of $S({\mathbb R})$ underlies an admissible variation of Hodge structure
over $X$. These structures are compatible with the decomposition
of tensor products. Moreover, these variations are unique up to Tate
twist.
\end{lemma}
\begin{proof}
The connected component of the identity of $S({\mathbb R})$ is a real form
of $Sp_n({\mathbb C})$ when ${\mathbb V}$ has odd
weight, and $SO_n({\mathbb C})$ when ${\mathbb V}$ has even weight. In both cases
each irreducible representations of the complex group can be
constructed by applying a suitable Schur functor the the
fundamental representation and then taking the intersection of the
kernels of all contractions with the polarization.
(This is Weyl's construction of the irreducible representations; it
is explained, for example, in \cite[\S17.3,\S19.2]{fulton-harris}.)
Since these operations preserve variations of Hodge structure, it
follows that a local system corresponding to an irreducible
representation of $S$ underlies a variation of Hodge structure.
Since the monodromy representation of ${\mathbb V}$ is Zariski dense, the
structure of a polarized variation of Hodge structure on this local
system is unique up to Tate twist. (Cf.\ the proof of
\cite[(9.1)]{hain:normal}.)
\end{proof}
This, combined with (\ref{decomp}) yields:
\begin{corollary}
With our assumptions, ${\mathcal O}({\mathcal P}_{x_o,\underline{\blank}})$ is a direct limit of
admissible variations of Hodge structure over $X$ of weight 0,
and the multiplication map is a morphism. \qed
\end{corollary}
\begin{corollary}
For each $b\in X$, there is a canonical Hodge structure on
${\mathcal O}(P_{x_o,b})$. \qed
\end{corollary}
Combining (\ref{loc_sys}) with (\ref{h1_comp}), we obtain:
\begin{corollary}\label{varmhs}
The local system over $X$ whose fiber over $x\in X$ is
$H^1({\mathcal U}(X,x))$ is an admissible variation of mixed Hodge structure
whose weights are positive.
\end{corollary}
Using Saito's machine \cite{saito}, we see that there is a MHC $\mathbf{A}^{\bullet}$
which is quasi-isomorphic to $\Efin^{\bullet}(X,{\mathcal O}({\mathcal P}_{a,\underline{\blank}}))$. The
complex part of this MHC is simply the complex of $C^\infty$ forms
on ${\overline{X}}$ with logarithmic singularities along $D$ and which have
coefficients in the canonical extension $\overline{\O}$ of ${\mathcal O}({\mathcal P}_{a,\underline{\blank}})$
to ${\overline{X}}$.
The Hodge filtration is the obvious one inducedd by the Hodge filtration
of $\overline{\O}$ and the Hodge filtration of forms on ${\overline{X}}$. The weight
filtration is not so easily described, and we refer to Saito's paper
for that.
We need to know that the multiplication is compatible with the
Hodge and weight filtrations. This follows from the next
result.
\begin{proposition}
If ${\mathbb V}_1\otimes{\mathbb V}_2 \to {\mathbb W}$ is a pairing of admissible
variations of Hodge structure over $X$, then the multiplication
map
$$
M^{\bullet}(X,{\mathbb V}_1) \otimes M^{\bullet}(X,{\mathbb V}_2) \to M^{\bullet}(X,{\mathbb W})
$$
is a morphism of cohomological mixed Hodge complexes.
\end{proposition}
\begin{proof}
This follows immediately from the naturality of Saito's construction,
its compatibility with exterior tensor products,
and the fact that admissible variations of Hodge structure are closed
under exterior products --- use restriction to the diagonal.
\end{proof}
There are two augmentations
$$
{\mathbb R} \leftarrow \mathbf{A}^{\bullet} \to {\mathcal O}(P_{a,b})
$$
corresponding to the inclusions $P_{a,b}\hookrightarrow {\mathcal P}_{a,\underline{\blank}}$
and $\id_a \in P_{a,a}$, and these are compatible with
all filtrations. It follows from \cite[(3.2.1)]{hain:dht}, (\ref{qism})
and (\ref{gpoid_dr}) that
$$
B({\mathbb R},\mathbf{A}^{\bullet},{\mathcal O}(P_{a,b}))
$$
is a MHC whose $H^0$ is isomorphic to ${\mathcal O}({\mathcal G}_{a,b})$. Moreover, the
multiplication is compatible
with the Hodge and weight filtrations. Consequently,
$$
{\mathcal O}({\mathcal G}_{a,b}) \cong H^0(B({\mathbb R},\mathbf{A}^{\bullet},{\mathcal O}(P_{a,b})))
$$
has a canonical MHS and its multiplication is a morphism
of MHS. Since the MHS on $P_{a,b}$ depends only on ${\mathbb V}$ and
not on $x_o$, the same is true of the MHS on ${\mathcal O}({\mathcal G}_{a,b})$.
If $a$, $b$, and $c$ are three points of $X$, then it follows directly
from the definitions that the map
$$
B({\mathbb R},\mathbf{A}^{\bullet},{\mathcal O}(P_{a,c})) \to
B({\mathbb R},\mathbf{A}^{\bullet},{\mathcal O}(P_{a,b}))\otimes B({\mathbb R},\mathbf{A}^{\bullet},{\mathcal O}(P_{b,c}))
$$
corresponding to path multiplication is a morphism of MHCs. It
follows that the induced map
$$
{\mathcal O}({\mathcal G}_{a,c}) \to {\mathcal O}({\mathcal G}_{a,b}) \otimes {\mathcal O}({\mathcal G}_{b,c})
$$
is a morphism of MHS. This completes the proof of Theorem~\ref{groupoid};
Theorem~\ref{hodge} follows by taking $a=b=x_o$ except for the assertion
that ${\mathcal O}(S) \hookrightarrow {\mathcal O}({\mathcal G})$ is a morphism of MHS. This follows
as this is induced by the natural inclusion
$$
{\mathcal O}(S) \hookrightarrow B({\mathbb R},\mathbf{A}^{\bullet},{\mathcal O}(S)),
$$
that takes $\phi$ to $[\phantom{x}]\phi$. It is a morphism of MHCs. This
completes the proofs of Theorems~\ref{hodge} and \ref{groupoid}. \qed
We now turn our attention to the variation of the Hodge filtration.
Suppose that $X$ and $S$ are as above. Consider the real local
system over $X\times X$ whose fiber over $(a,b)$ is ${\mathcal O}({\mathcal G}_{a,b})$.
Denote it by $\boldsymbol{\O}$.
Next we establish that this underlies a ``pre-variation of MHS.''
Denote by ${\mathcal F}^p\boldsymbol{\O}$the subset of the associated complex local system
with fiber $F^p{\mathcal O}({\mathcal G}_{a,b}({\mathbb C}))$ over $(a,b)$. Denote by $W_m\boldsymbol{\O}$
the subset of $\boldsymbol{\O}$ with fiber $W_m{\mathcal O}({\mathcal G}_{a,b})$ over $b$.
\begin{theorem}\label{pre_var}
The subset $W_m\boldsymbol{\O}$ is a flat sub-bundle of $\boldsymbol{\O}$, and $F^p\boldsymbol{\O}$ is a
holomorphic sub-bundle of $\boldsymbol{\O}_{\mathbb C}$ whose corresponding sheaf of
sections ${\mathcal F}^p$ satisfies Griffiths transversality:
$$
\nabla : {\mathcal F}^p \to {\mathcal F}^{p-1}\otimes \Omega^1_X.
$$
\end{theorem}
\begin{proof}[Sketch of Proof]
We will prove the result for the restriction $\boldsymbol{\O}_a$ of $\boldsymbol{\O}$ to
$\{a\}\times X$. The result for the restriction of $\boldsymbol{\O}$ to
$X\times \{b\}$ is proved similarly. The general result then
follows as the tangent spaces of $\{a\}\times X$ and $X\times \{b\}$
span the tangent space of $X\times X$ at $(a,b)$.
First we need a formula for the connection on $\boldsymbol{\O}_a$ at the point
$b$ in $X$. Fix a path $\gamma$ in $X$ from $a$ to $b$.
Suppose that $\mu : [-\epsilon, \epsilon] \to X$ is a smooth path
with $\mu(0) = b$. For $s\in [-\epsilon, \epsilon]$ let
$\gamma_s : [0,1] \to X$ be the piecewise smooth path obtained by
following $\gamma$ and then $\mu$ from $t=0$ to $t=s$. Suppose that
$w_1,\dots, w_r$ are in $\Efin^{\bullet}(X,{\mathcal O}(P_{a,b}))$. Suppose that
$U$ is a contractible neighbourhood of $b$ in $X$. With respect to
a flat trivialization of the restriction of ${\mathcal P}_{a,\underline{\blank}}$ to $U$,
we have
$$
w_r|_U = \sum_j w_r^j\otimes \psi_j
$$
where $w_r^j\in E^1(U)$ and $\psi_j \in {\mathcal O}(P_{a,b})$.
It follows from the analogue of (\ref{props}) in this situation that
$$
\frac{d}{ds}\bigg\vert_{s=0} \int_{\gamma_s}(w_1\dots w_r|\phi)
= \sum_j \int_\gamma (w_1\dots w_{r-1}|\phi \psi_j)
\langle w_r^j,\dot{\mu}(0)\rangle.
$$
The restriction of the connection on $\boldsymbol{\O}_a$ to the stalk at $b$ is
therefore induced by the map
$$
[w_1|\dots |w_r]\phi \mapsto
\sum_j [w_1|\dots |w_{r-1}]\phi \psi_j \otimes w_r^j
$$
on the bar construction.
The flatness of the weight filtration follows immediately from
the definition of the weight filtration on the bar construction.
Further, if $(z_1,\dots, z_n)$ is a holomorphic coordinate in $X$
centered at $b$, then it follows immediately from the definition of
the Hodge and weight filtrations on ${\mathcal O}({\mathcal G}_{a,b}({\mathbb C}))$ and the formula
for the connection that
$$
\nabla_{\partial/\partial {\overline{z}}_k} : {\mathcal F}^p \to {\mathcal F}^p
$$
for each $k$, so that the Hodge filtration varies holomorphically
at $b$. Similarly, on the stalk of $F^p$ at $b$ we have
$$
\nabla_{\partial/\partial z_k} : {\mathcal F}^p \to {\mathcal F}^{p-1}
$$
as each $w_r^j$ contributes at most 1 to the Hodge filtration
of ${\mathcal O}({\mathcal G}_{a,b})$.
\end{proof}
When $X$ is compact, we have:
\begin{corollary}\label{good_varn}
If $X$ is a compact K\"ahler manifold, then $\boldsymbol{\O}$ is an admissible
variation of MHS over $X\times X$. \qed
\end{corollary}
In order to prove the corresponding result in the non-compact case,
it is necessary to study the asymptotic behaviour of $\boldsymbol{\O}$. I plan
to consider this in a future paper.
We now consider naturality. Suppose that $X$ and ${\mathbb V}$ are as
above, and that $Y$ is a smooth variety and that ${\mathbb W}$ is an
admissible variations of Hodge structure over $Y$. We will now
denote $S$ by $S_X$:
$$
S_X = \Aut(V_o,\langle\phantom{x},\phantom{x}\rangle).
$$
Set
$$
S_Y = \Aut(W_o,\langle\phantom{x},\phantom{x}\rangle)
$$
where $W_o$ denotes the fiber of ${\mathbb W}$ over $y_o$. Suppose that
the monodromy representation
$$
\rho_Y : \pi_1(Y,y_o) \to S_Y
$$
has Zariski dense image. Denote the completion of $\pi_1(X,x_o)$
with respect to $\rho_X:\pi_1(X,x_o) \to S_X$ by $\pi_1(X,x_o)\to {\mathcal G}_X$,
and the completion of $\pi_1(Y,y_o)$ with respect to $\rho_Y$ by
$\pi_1(Y,y_o) \to {\mathcal G}_Y$.
Suppose that $f:(Y,y_o) \to (X,x_o)$ is a morphism of varieties, and
that we have fixed an inclusion
$$
\End {\mathbb V} \hookrightarrow \End f^\ast {\mathbb W}
$$
of variations of Hodge structure. This fixes a group homomorphism
$$
\Psi : S_X \hookrightarrow S_Y
$$
that is compatible with the Hodge theory.
By (\ref{induced}), there is a homomorphism $\widehat{\Psi} : {\mathcal G}_X \to {\mathcal G}_Y$
such that the diagram
$$
\begin{CD}
\pi_1(X,x_o) @>>> {\mathcal G}_X @>>> S_X \cr
@V{f_\ast}VV @VV{\widehat{\Psi}}V @VV{\Psi}V \cr
\pi_1(Y,y_o) @>>> {\mathcal G}_Y @>>> S_Y \cr
\end{CD}
$$
commutes.
\begin{theorem}\label{hodge_nat}
Under these hypotheses, the induced map
$$
\widehat{\Psi}^\ast : {\mathcal O}({\mathcal G}_Y) \to {\mathcal O}({\mathcal G}_X)
$$
is a morphism of MHS.
\end{theorem}
\begin{proof}
First, choose smooth compactifications ${\overline{X}}$ of $X$ and ${\overline{Y}}$
of $Y$ such that $X={\overline{X}} - D$ and $Y = {\overline{Y}} - E$, where $D$ and
$E$ are normal crossings divisors. We may choose these such that $f$
extends to a morphism ${\overline{X}} \to {\overline{Y}}$, which we shall also denote
by $f$.
Denote by $P_X$ the flat left $S_X$ principal bundle over $X$
associated the the
representation of $\pi_1(X,x_o)$ on $S_X$ via $\rho_X$. Denote the
analogous principal $S_Y$ principal bundle over $Y$ by $P_Y$.
Associated to these we have the local systems ${\mathcal O}(P_X)$ over $X$
and ${\mathcal O}(P_Y)$ over $Y$.
The construction above gives multiplicative mixed Hodge complexes
$$
\mathbf{A}^{\bullet}(X,{\mathcal O}(P_X)),\quad \mathbf{A}^{\bullet}(X,f^\ast{\mathcal O}(P_Y)),
\text{ and } \mathbf{A}^{\bullet}(Y,{\mathcal O}(P_Y))
$$
which compute the canonical mixed Hodge structures on
$$
H^{\bullet}(X,{\mathcal O}(P_X)),\quad H^{\bullet}(X,f^\ast{\mathcal O}(P_Y)),\text{ and }
H^{\bullet}(X,f^\ast{\mathcal O}(P_Y))
$$
respectively. The map $f$ induces a morphism
$$
\mathbf{A}^{\bullet}(Y,{\mathcal O}(P_Y)) \to \mathbf{A}^{\bullet}(X,f^\ast{\mathcal O}(P_Y))
$$
of MHCs, while the inclusion $S_X \hookrightarrow S_Y$ induces a
morphism
$$
\mathbf{A}^{\bullet}(X,f^\ast{\mathcal O}(P_Y)) \to \mathbf{A}^{\bullet}(X,{\mathcal O}(P_X))
$$
of MHCs. The composition of these corresponds to the induced map
$$
\Efin^{\bullet}(Y,{\mathcal O}(P_Y)) \to \Efin^{\bullet}(X,{\mathcal O}(P_X))
$$
induced by $f$ under the canonical quasi-isomorphisms. It follows that
the induced map
$$
B({\mathbb R},\mathbf{A}^{\bullet}(Y,{\mathcal O}(P_Y)),{\mathcal O}(P_Y)) \to B({\mathbb R},\mathbf{A}^{\bullet}(Y,{\mathcal O}(P_X)),{\mathcal O}(P_X))
$$
is a morphism of MHCs and that the induced the map
$$
f^\ast : {\mathcal O}({\mathcal G}_Y) \to {\mathcal O}({\mathcal G}_X)
$$
on $H^0$ is the ring homomorphism induced by $f$. The result follows.
\end{proof}
\begin{remark}\label{extended}
Suppose that ${\mathbb V}$ is an admissible variation of Hodge structure over
the complement $X$ of a normal crossings divisor in a compact K\"ahler
manifold. We will say that the pair $(X,{\mathbb V})$ is {\it neat} if the Zariski
closure $S$ of the image of the monodromy map
$$
\rho : \pi_1(X,x_o) \to \Aut(V_o,\langle\phantom{x},\phantom{x}\rangle)
$$
is semi-simple, and that the canonical MHS on the coordinate ring of
$$
\Aut(V_o,\langle\phantom{x},\phantom{x}\rangle)
$$
induces one on $S$. For example, every variation where $S$ is finite
is neat. I believe that every admissible $(X,{\mathbb V})$ is neat, but have not
yet found a proof.
The results (\ref{hodge}), (\ref{hodge-lie}), (\ref{groupoid}),
(\ref{varmhs}), (\ref{pre_var}), (\ref{good_varn}), (\ref{hodge_nat})
and their proofs are valid with the assumption that $\im \rho$ be
Zariski dense in $\Aut(V_o,\langle\phantom{x},\phantom{x}\rangle)$ replaced
by the assumption that the pairs $(X,{\mathbb V})$ and $(Y,{\mathbb W})$ be neat.
\end{remark}
The following is an application suggested by Ludmil Kartzarkov.
\begin{theorem}
Suppose that $X$ is a compact K\"ahler manifold and that ${\mathbb V}$ is
a polarized variation of Hodge structure over $X$ whose monodromy
representation $\rho$ has Zariski dense image. Then the prounipotent
radical of the completion of $\pi_1(X,x_o)$ relative to $\rho$ has
a quadratic presentation.
\end{theorem}
\begin{proof}
It is well known that if $X$ is compact K\"ahler and ${\mathbb V}$ is a
polarized variation of Hodge structure over $X$ of weight $m$,
then $H^k(X,{\mathbb V})$ has a pure Hodge structure of weight $k+m$.
In particular, as the variation ${\mathcal O}(P)$ over $X$ has weight
zero, $H^k(X,{\mathcal O}(P))$ is pure of weight weight $k$ for all $k$.
Denote the Lie algebra of the prounipotent radical of the relative
completion of $\pi_1(X,x_o)$ by ${\mathfrak u}$.
It follows from (\ref{def}) and (\ref{main}) that ${\mathfrak u}$ is the Lie
algebra canonically associated to the d.g.a.\ $\Efin^{\bullet}(X,{\mathcal O}(P))$
by rational homotopy theory (either via Sullivan's theory of minimal
models, or via Chen's theory as the dual of the indecomposables
of the bar construction on $\Efin^{\bullet}(X,{\mathcal O}(P))$.) There are canonical
maps
$$
H^1({\mathfrak u}) \cong H^1(X,{\mathcal O}(P))\text{ and }
H^2({\mathfrak u}) \hookrightarrow H^2(X,{\mathcal O}(P)).
$$
These are morphisms of MHS \cite[(7.2)]{hain:torelli}. It follows
that $H^1({\mathfrak u})$ is pure of weight 1 and $H^2({\mathfrak u})$ is pure of weight 2.
The result now follows from \cite[(5.2),(5.7)]{hain:torelli}.
\end{proof}
\begin{remark}
It is not necessarily true that ${\mathfrak u}$ is a quotient of the unipotent
completion of $\ker \rho$. A criterion for surjectivity is given in
\cite[(4.6)]{hain:comp}. For this reason it may not be easy to apply this
result in general situations without artificially restrictive hypotheses.
\end{remark}
\section{A Canonical Connection}
\label{connection}
For the time being, let $X$, $x_o$, ${\mathbb V}$, $\rho$, etc.\ be as in the
previous section. However, all groups and Lie algebras in this section
will be complex, and ${\mathcal G}$, ${\mathcal U}$, ${\mathfrak u}$, etc.\
denote the {\em complex} points of the relative completion of
$\pi_1(X,x_o)$, its prounipotent radical, its Lie algebra, etc.
Denote the image of $\rho$ by $\Gamma$,
and the Galois covering of $X$ with Galois group $\Gamma$ by $X'$.
In this section, we show how the Hodge theory of ${\mathcal G}$ gives a
canonical (given the choice of $x_o$), $\Gamma$ invariant integrable
1-form
$$
\omega \in E^1(X')\comptensor \Gr^W_{\bullet}{\mathfrak u}
$$
on $X'$ which can be integrated to the canonical representation
$$
\tilde{\rho} : \pi_1(X,x_o) \to S \ltimes {\mathcal U} \cong {\mathcal G}.
$$
Here $\comptensor$ denotes the completed tensor product, which
is defined below.
At the end of the section, we shall explain what this means when
$X$ is the complement of the discriminant locus in ${\mathbb C}^n$ and $S$ is
the symmetric group $\Sigma_n$. In this case, $X'$ is the
complement of the hyperplanes $x_i = x_j$ in ${\mathbb C}^n$, $\pi_1(X,x_o)$
is the classical braid group $B_n$, and
the form is
$$
\omega = \sum_{i<j} d\log(x_i-x_j).
$$
First, we shall define the completed tensor product $\comptensor$.
Suppose that ${\mathfrak u}$ is a topological Lie algebra and that
$$
{\mathfrak u} = {\mathfrak u}^1 \supseteq {\mathfrak u}^2 \supseteq {\mathfrak u}^3 \supseteq \cdots
$$
is a base of neighbourhoods of 0. Suppose that
$E$ is a vector space. Define
$$
E\comptensor {\mathfrak u} = \lim_{\leftarrow} E\otimes {\mathfrak u}/{\mathfrak u}^m.
$$
We can regard a graded Lie algebra ${\mathfrak u} = \oplus_{m < 0} {\mathfrak u}_m$ as a
topological Lie algebra where the basic neighbourhoods of 0 are
$$
\bigoplus_{l\le m} {\mathfrak u}_l, \quad m < 0.
$$
The definition of completed tensor product therefore extends to
the case where ${\mathfrak u}$ is graded. Finally, if ${\mathfrak u}$ is a Lie algebra
in the category of mixed Hodge structures where ${\mathfrak u} = W_{-1}{\mathfrak u}$
which is complete with respect to the weight filtration, and if $E$
is a complex vector space, then there is a canonical isomorphism
$$
E\comptensor {\mathfrak u}_{\mathbb C} \cong E \comptensor \Gr^W_{\bullet} {\mathfrak u}_{\mathbb C}
$$
as ${\mathfrak u}_{\mathbb C}$ is canonically isomorphic to
$\prod \Gr^W_m{\mathfrak u}_{\mathbb C}$. (Cf.\ \cite[(5.2)]{hain:torelli}.)
We view a principal bundles with structure group a
proalgebraic group to be the inverse limit of the principal
bundles whose structure groups are the finite dimensional
quotients of the proalgebraic group. A connection on a
principal bundle with proalgebraic structure group is
the inverse limit of compatible connections on the
corresponding bundles with finite dimensional structure
group.
\subsection{The unipotent case.}
We begin with the unipotent case, $S=1$. Suppose that $X$ is a
smooth manifold with distinguished base point $x_o$. Denote the complex
form of the unipotent completion of $\pi_1(X,x_o)$ by ${\mathcal G}$ and the
complex points of the completion of $\pi(X;x_o,x)$ by ${\mathcal G}_{x_o,x}$. The
family
$$
\left({\mathcal G}_{x_o,x}\right)_{x\in X}
$$
forms a flat principal left ${\mathcal G}$ bundle over $X$ that we shall
denote by ${\mathcal G}_{x_o,\underline{\blank}}$. Since the structure group is contractible,
(more precisely, an inverse limit of contractible groups), this bundle
has a section. Pulling back the canonical connection form, we obtain
an integrable connection form
$$
\omega \in E^{\bullet}(X)\comptensor {\mathfrak g}
$$
where ${\mathfrak g}$ denotes the Lie algebra of ${\mathcal G}$. The monodromy
representation of this form is the monodromy of ${\mathcal G}_{x_o,\underline{\blank}}$,
which is the canonical homomorphism
$$
\pi_1(X,x_o) \to {\mathcal G}.
$$
When $X$ is an algebraic manifold, there is a canonical choice of
section and therefore a canonical connection form. To see this,
note that for each $a \in X$, the weights on ${\mathcal O}({\mathcal G}_{x_o,a})$ are
$\ge 0$ and that
$$
\Gr^W_0 {\mathcal O}({\mathcal G}_{x_o,a}) \cong {\mathbb C}.
$$
Since there is a canonical ring isomorphism
$$
{\mathcal O}({\mathcal G}_{x_o,a}) \cong
\bigoplus_{l\ge 0} \Gr^W_l {\mathcal O}({\mathcal G}_{x_o,a})
$$
there is a canonical augmentation
$$
{\mathcal O}({\mathcal G}_{x_o,a}) \to {\mathbb C}
$$
whose kernel is
$$
\bigoplus_{l > 0} \Gr^W_l {\mathcal O}({\mathcal G}_{x_o,a}).
$$
This determines a canonical point in ${\mathcal G}_{x_o,a}$.
Since the family $\left\{{\mathcal O}({\mathcal G}_{x_o,a})\right\}_{a\in X}$ is a variation
of MHS over $X$ (see \cite{hain-zucker}), these distinguished points
vary smoothly as $a$ varies. They therefore determine a smooth section
of ${\mathcal G}_{x_o,\underline{\blank}}$. We therefore have a canonical integrable
1-form
$$
\omega \in E^1(X)\comptensor{\mathfrak g} \cong E^1(X)\comptensor\Gr^W_{\bullet}{\mathfrak g}.
$$
\subsection{The general case}
The first step in doing this in general is to explain the necessary
constructions in the $C^\infty$ case. So suppose for the time being that
$X$ is a smooth manifold; $\rho$, $S$, ${\mathcal G}$ and $P\to X$ are as before.
We also have the torsor
$$
{\mathcal G}_{x_o,\underline{\blank}} \to X.
$$
It is a flat principal left ${\mathcal G}$ bundle over $X$. There is a map
$$
\begin{CD}
{\mathcal G}_{x_o,\underline{\blank}} @>\pi>> P \cr
@VVV @VVV \cr
X @= X
\end{CD}
$$
of flat bundles. It is compatible with the canonical homomorphism
${\mathcal G} \to S$. Denote by $X'$ the leaf of $P$ containing the distinguished
lift $\tilde{x}_o$ to $P$ of $x_o$. The projection $P\to X$ induces a
covering map $X' \to X$. It is the Galois covering corresponding to
$\ker \rho$. Define ${\mathcal U}_{x_o,\underline{\blank}}$ to be the subset $\pi^{-1}X'$
of ${\mathcal G}_{x_o,\underline{\blank}}$.
There is a natural projection ${\mathcal U}_{x_o,\underline{\blank}}\to X'$ indued by $\pi$.
Note that the fiber of this over $\tilde{x}_o$ is ${\mathcal U}$, the prounipotent
radical of ${\mathcal G}$. Denote the fiber of ${\mathcal U}_{x_o,\underline{\blank}}$ over $a\in X'$ by
${\mathcal U}_{x_o,a}$.
Each point $a$ of $P$ determines an augmentation
$$
\epsilon_a : \Efin^{\bullet}(X,{\mathcal O}(P)) \to {\mathbb C}.
$$
Given two points $a$ and $b$ of $P$, we may form the two sided
bar construction
\begin{equation}\label{bar}
B({\mathbb C},\Efin^{\bullet}(X,{\mathcal O}(P)),{\mathbb C})
\end{equation}
where the left hand ${\mathbb C}$ is viewed as a module over $\Efin^{\bullet}(X,{\mathcal O}(P))$
via $\epsilon_a$, and the right hand ${\mathbb C}$ via $\epsilon_b$. We shall
denote the d.g.a.\ (ref{bar}) by $B(\Efin^{\bullet}(X,{\mathcal O}(P))_{a,b})$
\begin{proposition}
Each ${\mathcal U}_{x_o,a}$ is a proalgebraic variety with coordinate ring
$$
{\mathcal O}({\mathcal U}_{x_o,a}) \cong H^0(B(\Efin^{\bullet}(X,{\mathcal O}(P))_{\tilde{x}_o,a})).
$$
Moreover, ${\mathcal U}_{x_o,\underline{\blank}}\to X'$ is a principal ${\mathcal U}$ bundle with respect
to the natural ${\mathcal U}$ action on ${\mathcal G}_{x_o,\underline{\blank}}$.
\end{proposition}
Choose a splitting $S\to {\mathcal G}$ of the natural homomorphism ${\mathcal G} \to S$.
This induces an isomorphism ${\mathcal G}\cong S\ltimes {\mathcal U}$. The splitting enables
us to lift the action of $S$ to ${\mathcal G}_{x_o,\underline{\blank}}$ in such a way that
the projection ${\mathcal G}_{x_o,\underline{\blank}}\to P$ is $S$ equivariant. Since $\Gamma$
is a subgroup of $S$, and since it preserves $X'\subset P$, it follows
that there is a natural left action of $\Gamma$ on ${\mathcal U}_{x_o,\underline{\blank}}$ and
that, with respect to this action, the projection ${\mathcal U}_{x_o,\underline{\blank}}$
is $\Gamma$ equivariant.
Denote the pullback of the extension
$$
1 \to {\mathcal U} \to {\mathcal G} \to S \to 1
$$
along $\Gamma \hookrightarrow S$ by ${\mathcal G}_\Gamma$. This is an extension
$$
1 \to {\mathcal U} \to {\mathcal G}_\Gamma \to \Gamma \to 1.
$$
The splitting $S\to {\mathcal G}$ induces a splitting $\Gamma \to {\mathcal G}_\Gamma$,
and therefore a semi-direct productu decomposition
${\mathcal G}_\Gamma \cong \Gamma \ltimes {\mathcal U}$.
The image of the canonical homomorphism $\pi_1(X,x_o) \to {\mathcal G}$ lies in
${\mathcal G}_\Gamma$. The composite ${\mathcal U}_{x_o,\underline{\blank}} \to X' \to X$ is a
flat principal left ${\mathcal G}_\Gamma$ bundle over $X$. The associated monodromy
representation is the canonical homomorphism $\pi_1(X,x_o) \to {\mathcal G}_\Gamma$.
The monodromy therefore induces the canonical homomorphism
$$
\pi_1(X,x_o) \to {\mathcal G} \cong S \ltimes {\mathcal U}.
$$
Next, we explain that the pullback of this bundle to $X'$ is trivial,
and therefore given by an integrable 1-form.
\begin{proposition}
There is a $\Gamma$ equivariant section of ${\mathcal U}_{x_o,\underline{\blank}}\to X'$.
\end{proposition}
\begin{proof}
The action of $\Gamma$ on $X'$ is free. It follows that the action
of $\Gamma$ on ${\mathcal U}_{x_o,\underline{\blank}}$ is also free. Consequently, the
square
$$
\begin{CD}
{\mathcal U}_{x_o,\underline{\blank}} @>>> \Gamma\backslash {\mathcal U}_{x_o,\underline{\blank}} \cr
@VVV @VVV \cr
X' @>>p> X
\end{CD}
$$
is a pullback square. Since the fibers of
$\Gamma\backslash {\mathcal U}_{x_o,\underline{\blank}} \to X$
are connected, it has a $C^\infty$ section. This section pulls back
to a $\Gamma$ invariant section of ${\mathcal U}_{x_o,\underline{\blank}} \to X'$.
\end{proof}
Let $\Gamma$ act on ${\mathcal U}$ on the left via the adjoint action:
$$
Ad(\gamma) : u \mapsto \gamma u \gamma^{-1}.
$$
Then $\Gamma$ acts on $X'\times {\mathcal U}$ on the left via the diagonal
action. It follows from the previous result that the flat principal
bundle ${\mathcal U}_{x_o,\underline{\blank}}\to X'$ has a $\Gamma$ invariant trivialization.
We therefore have a connection form
$$
\omega \in E^1(X')\comptensor {\mathfrak u}.
$$
\begin{proposition}
This connection form satisfies $\gamma^\ast \omega = Ad(\gamma)\omega$
for all $\gamma \in \Gamma$.
\end{proposition}
\begin{proof}
Since the bundle is trivial, its (locally defined) sections can be
identified with (locally defined) functions $X' \to {\mathcal U}$. Since $\Gamma$
preserves the connection, we see that for each $\gamma\in \Gamma$ the
local section $u$ is flat if and only if the local section
$(\gamma^{-1})^\ast Ad(\gamma)(u)$ is flat. That is, $Ad(\gamma)(u)$
is flat if and only if $\gamma^\ast u$ is flat. The result now follows
from a standard and straight forward computation.
\end{proof}
Now suppose that $X$ is an algebraic manifold. We have to show
that this construction can be made canonical. Note that, given the
choice of the base point $x_o$, the only
choices made in the construction of $\omega$ were the choice of a
splitting of the homomorphism ${\mathcal G} \to S$, and the choice of a $\Gamma$
invariant section of ${\mathcal U}_{x_o,\underline{\blank}} \to X'$. We will now explain
how Hodge theory provides canonical choices of both.
It follows from (\ref{hodge-lie}) that $\Gr^W_0{\mathfrak g} \cong {\mathfrak s}$.
Consequently, there is a canonical splitting of the homomorphism
${\mathfrak g} \to {\mathfrak s}$. This induces a canonical
splitting of the homomorphism ${\mathcal G} \to S$, and therefore a
canonical action of $\Gamma$ on ${\mathcal U}_{x_o,\underline{\blank}}$ and a canonical
identification ${\mathcal G} \cong S \ltimes {\mathcal U}$.
It remains to explain why there is a $\Gamma$ equivariant
section of ${\mathcal U}_{x_o,\underline{\blank}}$. This is an elaboration of the
argument in the unipotent case.
For each $b\in X$, Hodge theory provides canonical ring isomorphisms
$$
{\mathcal O}(G_{x_o,b}) \cong \bigoplus_{m\ge 0} Gr^W_m {\mathcal O}({\mathcal G}_{x_o,b})
$$
and
$$
{\mathcal O}({\mathcal P}_{x_o,b}) \cong \Gr^W_0 {\mathcal O}({\mathcal G}_{x_o,b}).
$$
Moreover, it follows from (\ref{pre_var}) that these identifications
depend smoothly on $b$. Consequently, there is a smooth section $\sigma$
of the canonical projection ${\mathcal G}_{x_o,\underline{\blank}} \to {\mathcal P}_{x_o,\underline{\blank}}$.
Restricting to $X'$, we obtain a canonical smooth section of the
projection ${\mathcal U}_{x_o,\underline{\blank}} \to X'$.
\begin{proposition}
This section is $\Gamma$ equivariant.
\end{proposition}
\begin{proof}
For each $x\in X$ we have the action ${\mathcal G}\times {\mathcal G}_{x_o,x}\to{\mathcal G}_{x_o,x}$.
By (\ref{gpoid_dr}) the corresponding map of coordinate rings is a
morphism of MHS. By the choice of splitting of ${\mathcal G} \to S$, the action
of $S$ given by the splitting preserves the canonical isomorphism
$$
{\mathcal O}({\mathcal G}_{x_o,x}) \cong \bigoplus_{l\ge 0}\Gr^W_l {\mathcal O}({\mathcal G}_{x_o,x}).
$$
It follows that the section of ${\mathcal G}_{x_o,\underline{\blank}} \to {\mathcal P}_{x_o,\underline{\blank}}$
defined above is equivariant with respect to the left $S$ actions.
It follows that the restriction of this section to $X'$ is $\Gamma$
equivariant.
\end{proof}
\begin{example}
In this example, we take $X$ to be the complement in ${\mathbb C}^n$ of
the universal discriminant locus. (View ${\mathbb C}^n$ as the space of
monic polynomials of degree $n$.) Pick a base point $x_o$. The
fundamental group of this
space is the classical braid group. Denote the symmetric
group on $n$ letters by $\Sigma_n$. There is a natural
homomorphism $\rho : B_n \to \Sigma_n$. Denote the
corresponding covering of $X$ by $\pi : X' \to X$. Its fundamental
group is the pure braid group $P_n$. As is
well known, $X'$ is the complement of the hyperplanes
$x_i = x_j$ in ${\mathbb C}^n$ where $i\neq j$. The projection takes
$(x_1,\dots,x_n)$ to the monic polynomial $\prod(T-x_j)$. The
natural left action of $\Sigma_n$ on $X'$ is given by
$$
\sigma : (x_1,\dots,x_n) \mapsto
(x_{\sigma^{-1}(1)},\dots,x_{\sigma^{-1}(n)}).
$$
The local system $\pi_\ast{\mathbb Q}_{X'}$ is an admissible variation of
Hodge structure over $X$ of weight 0, rank $n$, and type $(0,0)$.
The closure of the image of the monodromy is $\Sigma_n$, a
semi-simple group. So we can
apply Theorem~\ref{extended} to deduce the existence of a MHS
on the relative completion, and the existence of a universal
connection. In this case, the canonical connection is well known
by the work \cite{kohno} of Kohno.
Denote the free Lie algebra over ${\mathbb C}$ generated by the $Y_j$ by
${\mathbb L}(Y_1,\dots,Y_m)$. Denote the unipotent completion of $P_n$
by ${\mathbb P}_n$ and its Lie algebra by ${\mathfrak p}_n$.
The associated graded of ${\mathfrak p}_n$ of is the graded Lie algebra
$$
{\mathbb L}(X_{ij} : ij\text{ is a two element subset of }\{1,\dots,n\})/R
$$
where $R$ is the ideal generated by the quadratic relations
\begin{align*}\label{braid_relns}
[X_{ij},X_{kl}]&\text{ when $i,j,k$ and $l$ are distinct;}\cr
[X_{ij},X_{ik} + X_{jk}]& \text{ when $i,j$ and $k$ are distinct}.
\end{align*}
The natural (left) action of the symmetric group on it is defined by
$$
Ad(\sigma): X_{ij} \mapsto X_{\sigma(ij)}.
$$
The canonical invariant form
$$
\omega \in E^1(X')\otimes \Gr^W_{\bullet} {\mathfrak p}_n
$$
is
$$
\omega = \sum_{ij} d\log(x_i - x_j) X_{ij}.
$$
It is invariant because
$$
\sigma^\ast \omega =
\sum_{ij} d\log(x_{\sigma^{-1}(i)} - x_{\sigma^{-1}(j)}) X_{ij}
= \sum_{ij} d\log(x_i - x_j) X_{\sigma(ij)}
= Ad(\sigma)\omega.
$$
We therefore obtain a homomorphism
$$
B_n \to \Sigma_n\ltimes {\mathcal P}_n
$$
where ${\mathcal P}_n$ denotes the complex form of the Malcev completion of
$P_n$. This is the completion of $B_n$ relative to
$\rho : B_n \to \Sigma_n$.
\end{example}
|
1997-01-17T19:28:33 | 9607 | alg-geom/9607006 | en | https://arxiv.org/abs/alg-geom/9607006 | [
"alg-geom",
"math.AG"
] | alg-geom/9607006 | Mikhail Zaidenberg | G. Dethloff, S. Orevkov, M. Zaidenberg | Plane curves with a big fundamental group of the complement | 23 pages LaTeX. A revised version. The unnecessary restriction $d \ge
2g - 1$ of the previous version has been removed, and the main result has
taken its final form | Amer. Math. Soc. Transl. (2) 184, 63-84 (1998) | null | Duke preprint DUKE-M-95-00 | null | Let $C \s \pr^2$ be an irreducible plane curve whose dual $C^* \s \pr^{2*}$
is an immersed curve which is neither a conic nor a nodal cubic. The main
result states that the Poincar\'e group $\pi_1(\pr^2 \se C)$ contains a free
group with two generators. If the geometric genus $g$ of $C$ is at least 2,
then a subgroup of $G$ can be mapped epimorphically onto the fundamental group
of the normalization of $C$, and the result follows. To handle the cases
$g=0,1$, we construct universal families of immersed plane curves and their
Picard bundles. This allows us to reduce the consideration to the case of
Pl\"ucker curves. Such a curve $C$ can be regarded as a plane section of the
corresponding discriminant hypersurface (cf. [Zar, DoLib]). Applying
Zariski--Lefschetz type arguments we deduce the result from `the bigness' of
the $d$-th braid group $B_{d,g}$ of the Riemann surface of $C$.
| [
{
"version": "v1",
"created": "Thu, 4 Jul 1996 12:40:41 GMT"
},
{
"version": "v2",
"created": "Fri, 17 Jan 1997 18:28:10 GMT"
}
] | 2014-12-01T00:00:00 | [
[
"Dethloff",
"G.",
""
],
[
"Orevkov",
"S.",
""
],
[
"Zaidenberg",
"M.",
""
]
] | alg-geom | \section*{Introduction}
\bigskip
\noindent The fundamental groups of the plane curve complements are of permanent
interest (see e.g. [Di, DoLib, Lib, MoTe, No, O, Zar] and the literature
therein). Here we look for the most coarse properties of these groups (cf. e.g.
[MoTe]). Namely, we distinguish between {\it big} and {\it small} groups.
\bigskip
\noindent {\bf 0.1. Definition.} We say that a group $G$ is {\it big} if it
contains a non--abelian free subgroup. We call $G$ {\it small} if it
is {\it almost solvable}, i.e. it has a solvable subgroup of finite index.
\bigskip
Recall the Tits alternative [Ti]: {\it any subgroup $G$ of a general linear
group $GL(n,\,k)$ over a field $k$ of characteristic zero is either
big or small.} This alternative holds true, even in a stronger form, for some
classes of discrete groups, such as hyperbolic groups in
sense of Gromov and the mapping class groups (see sect.1 below for references).
\smallskip
In [MoTe] classes of plane Pl\"ucker curves were indicated with
infinite almost solvable (i.e. small) non--abelian fundamental groups
of the complement. An example
is the branching divisor of a generic projection of the Veronese surface
$V_3$ of order $3$ onto $ I \!\! P^2$ [MoTe].
The well known Deligne--Fulton Theorem asserts that the complement of a nodal
plane curve has abelian fundamental group. Here we show (and this is the main
purpose of the paper) that
the fundamental group of
the complement of the dual of a nodal plane curve is big
(with two evident exceptions). More precisely, we have
\bigskip
\noindent {\bf 0.2. Theorem.} {\it Let $C \subset I \!\! P^2$ be an irreducible immersed
curve\footnote{i.e. all the analytic branches at the singular points of $C$
are smooth.}
which is neither a line nor a conic nor a nodal cubic.
Let $C^* \subset I \!\! P^{2*}$ be the dual curve.
Then the group $\pi_1( I \!\! P^{2*} \setminus C^*)$ is big.}
\bigskip
\noindent Obviously, the statement does not hold for
a line, nor for a conic.
If $C$ is a nodal cubic then
$C^*$ is a three--cuspidal quartic and $\pi_1 ( I \!\! P^{2*} \setminus C^*)$ is the
metacyclic group of order $12$ [Zar, p.143--145].
Let $d$ be the degree and $g$ be the geometric genus of $C$.
For $g\ge2$ the proof of Theorem 0.2 is easy.
Indeed,
denote by $\nu: C_{\rm norm} \to C$ a normalization of $C$.
Set
$R = \{(p,l)\in C_{\rm norm} \times I \!\! P^{2*}\,|\,\nu(p)\in l\}$.
It is a smooth surface. Let $\mu_1 : R \to C_{\rm norm}$
and $\mu_2 : R \to I \!\! P^{2*}$ be the canonical
projections. Since $C$ is immersed, it is easily seen that
$\mu_2$ is ramified exactly over $C^*$. Denote by
$R_0$ the part of $R$ over $ I \!\! P^{2*} \setminus C^*$.
Then $\mu_1 : R_0 \to C_{\rm norm}$ is a holomorphic surjection
with connected fibres. It follows that
$(\mu_1)_* : \pi_1(R_0) \to \pi_1(C_{\rm norm})$ is an epimorphism,
and hence the group $\pi_1(R_0)$ is big as soon as
$g(C_{\rm norm}) \ge 2$ (see e.g. 1.2(a) below).
Since $\mu_2 : R_0 \to I \!\! P^{2*} \setminus C^*$
is a finite unramified covering, we have that
$(\mu_2)_* (\pi_1(R_0))$ is a finite index subgroup of
$\pi_1( I \!\! P^{2*} \setminus C^*)$.
Therefore, the group $\pi_1( I \!\! P^{2*} \setminus C^*)$ is also big.
Thus, the only non-trivial cases are $g=0$ and $g=1$.
However, the proofs of most of the intermediate results
needed for these two cases
are valid for any $g$, some of them under the additional assumption that
$d\ge 2g-1$ (which is automatically fulfilled for $g = 0,\,1$).
Therefore, we formulate everything for an arbitrary genus.
This provides another proof of Theorem 0.2
for the case $g\ge2$, $d\ge 2g-1$.
\smallskip
The paper is organized as follows.
In Section 1 we provide some (mostly well known)
examples of big groups.
Besides, by several examples we illustrate a conjectural
relation between bigness of the fundamental group and C-hyperbolicity.
These include, in particular,
the quasi--projective
quotients of bounded symmetric domains and the complements
of certain reducible plane curves.
The proof of Theorem 0.2 is done in Sect. 4. The
results in Sect. 2 and 3 (which we believe to be of some independent interest)
reduce the proof to the
case of a nodal Pl\"ucker curve. In Theorem 2.1 we show that the part
$Imm_{d,\,g}$ of
the Hilbert scheme of degree $d$ genus $g$ plane curves, which corresponds to
the immersed curves, is smooth,
and the universal family of curves admits a simultaneous normalization
over $Imm_{d,\,g}$ (see [AC, Ha] for related results, especially concerning
($a$) and
($b$) of Theorem 2.1). We show also that the nodal and
(for $d \ge 2g-1$) the Pl\"ucker curves form Zariski open subsets of
$Imm_{d,\,g}$.
\medskip
A preliminary version of Theorem 0.2
(under the additional restriction $d\ge 2g-1$)
was announced in [DeZa1, Sect. 7] (see also [DeZa2]).
After the preprint [DeOrZa] had been
distributed the authors have received the preprint\footnote{We are thankful to
I. Shimada for sending us this preprint.} [KuShi] where a presentation
of the group $\pi_1( I \!\! P^{2*} \setminus C^*)$ for a generic nodal curve
$C \subset I \!\! P^2$ of genus $g$ and degree $d \ge 2g+1$ has been computed
(see Remark 4.6 below).
It is our pleasure to thank D. Akhiezer, E. Artal, H. Flenner, V. Guba, S.
Kosarev, V. Lin, V. Sergiescu for their friendly assistance and contributions
to the paper.
\section{Big groups and C-hyperbolicity}
\noindent {\bf 1.1. Generalities on big groups}
\smallskip
\noindent By a theorem of von Neumann, a big group is non--amenable.
The converse is not true, in general; the corresponding
examples are due to A. Ol'shanskij, S. I. Adian and M. Gromov
(see [OSh]).
Note that the group in all these examples
is not finitely presented. For a finitely presented
group the
equivalence of bigness and non--amenability is unknown\footnote
{We are grateful to V. Sergiescu and V. Guba for this information.}.
Being non--amenable, a
big group can not be almost nilpotent or even almost
solvable. As follows from the Nielsen--Schreier Theorem, a subgroup of
finite index of a big group is big, as well
as a normal subgroup with a solvable quotient. Clearly, a group with
a big quotient is big.
\medskip
We remind several classical examples of big groups. First of all, for
$g \ge 1$ the Siegel modular
group $Sp_{2g}( Z \!\!\! Z)$ is big. In addition, it has no infinite
normal solvable subgroup (see (1.3)-(1.4) below).
\smallskip
Another examples are: the Artin group $B_{d, \,g}$ of the $d$--string
braids of a
genus $g$ compact Riemann surface $R_g$, and the mapping class group
Mod$_{g,\,n}$, i.e. the group of classes of isotopy of
orientation preserving diffeomorphisms of a genus $g$ Riemann surface with
$n$ punctures (see e.g. [Bi]). Namely, we have the following
\bigskip
\noindent {\bf 1.2. Lemma.}
\noindent {\it ($a$) If $g\ge2$ then $\pi_1(R_g)$ is big.
\noindent ($b$) The braid group $B_{d, \,g}\,\,(d \ge 1)$ is big iff $(d,\,g)
\neq (1,\,0),\,(2,\,0),\,(3,\,0),\,(1,\,1)$.
\noindent ($c$) The mapping class group Mod$_{g,\,n}$ is big iff $g \ge 1$, or
$g = 0$ and $n \ge 4$.}
\bigskip
\noindent {\it Proof.} ($a$)
By a theorem of Magnus [CoZi, (2.5.1)], after removing any of the standard
generators $a_1,\,b_1,\dots,\,a_g,\,b_g$ of
$\pi_1(R_g)$, the subgroup generated by the remaining ones is the free
group {\bf F}$_{2g - 1}$.
($b$)
By definition, $B_{d, \,g} =
\pi_1 (S^dR_g \setminus {\Delta}_{d, \,g})$, where $S^dR_g$ denotes the $d$-th
symmetric power of $R_g$ and ${\Delta}_{d, \,g} \subset S^dR_g$ denotes the
discriminant hypersurface consisting of the $d$-tuples of points with
coincidences. The pure braid group $P_{d,\,g} := \pi_1((R_g)^{\,d} \setminus
D_{d,\,g})$, where $D_{d,\,g} \subset (R_g)^{\,d}$ is the union of diagonal
hypersurfaces, is the normal subgroup of
$B_{d, \,g}$ of index $d$! which corresponds to the Vieta covering
$(R_g)^{\,d} \setminus D_{d,\,g} \to S^dR_g \setminus {\Delta}_{d, \,g}$.
The fibration $(R_g)^{\,d+1} \setminus D_{d+1,\,g} \to (R_g)^{\,d} \setminus
D_{d,\,g}$
with the fibre $R_g \setminus \{d\,\,\,{\rm points}\}$ yields the short
exact sequence [Bi, sect. 1.3]
$${\bf 1} \to \pi_1(R_g \setminus \{d\,\,\,{\rm points}\})
\to P_{d+1,\,g} \to P_{d,\,g} \to {\bf 1}\,.$$
For $d > 0$ the group
$\pi_1(R_g \setminus \{d\,\,\,{\rm points}\})$ is a free group
{\bf F}$_k$ with $k = 2g + d - 1$ generators. For $d = 0$
see (a).
Hence, under
the above restrictions the pure braid group $P_{d,\,g}$, and therefore also the
braid group $B_{d, \,g}$,
contains a subgroup isomorphic to
a non-abelian free group. In the exceptional cases when $(d,\,g) = (1,\,1)$ or
$g = 0,\,1 \le d \le 3$ the same exact sequence
shows that the corresponding group $B_{d, \,g}$ is not big. This proves ($b$).
($c$) There is a natural
surjection $j\,:\,$ Mod$_{g,\,n} \to $ Mod$_g :=$ Mod$_{g,\,0}$, where the
kernel Ker$\,j$ is
the braid group $B_{n, \,g}$ if $g \ge 2$ and its
quotient by the center if $g = 1,\,n\ge 2$ or $g = 0,\, n\ge 3$
[Bi, Theorem 4.3]. Therefore, the group Mod$_{g,\,n}$ is big as soon as
the corresponding braid group $B_{n, \,g}$ is so.
For $g \ge 1$
the induced representation of Mod$_{g}$ into the first homology group of $R_g$
yields a surjection Mod$_{g}\to$ Sp$_{2g}( Z \!\!\! Z)$ (actually, Mod$_{1} \cong
GL(2,\, Z \!\!\! Z)$).
This shows that Mod$_{g},\,g \ge 1$, is a big group.
For $g = 0$ we have that Mod$_{0,\,3} = B_{3, \,0}/$(center) is a finite
group, the groups Mod$_{0,\,0}$ and Mod$_{0,\,1}$ are trivial, whereas
Mod$_{0,\,2} = Z \!\!\! Z/2 Z \!\!\! Z$ [Bi, Theorem 4.5]. This completes the proof. \hfill $\Box$
\bigskip
\noindent {\it Remark.} In fact, the Tits alternative holds in
Mod$_{g},\,g \ge 1$ [Iv, MC] (note that for
$g \ge 2$ the latter group is not isomorphic to any arithmetic linear group
[Iv]). Furthermore, for $g\ge 2$ any almost solvable subgroup of Mod$_{g}$ is
almost abelian [BiLuMC].
\bigskip
Let us make certain remarks concerning a conjectural relation of bigness
of the fundamental group of a complex space $X$ and its C-hyperbolicity.
Recall that $X$ is said to be {\it (almost) C-hyperbolic} if it has an
{\it (almost) Carath\'eodory hyperbolic} covering $Y \to X$, i.e. such that
the bounded holomorphic functions on $Y$ separate points of $Y$
(up to finite subsets). As follows from Lin's Theorem
[Lin, Theorem B], {\it the fundamental group of an almost C-hyperbolic
algebraic variety can not be almost nilpotent.} Note that for $g \ge 1$ the
complement $ I \!\! P^{2*} \setminus C^*$ is C--hyperbolic, and it is almost C-hyperbolic
if $C$ is a generic rational curve of degree $d \ge 5$
[DeZa1, Thm. 1.1]. Thus, by Lin's Theorem, in all these cases
the group $\pi_1( I \!\! P^{2*} \setminus C^*)$ is not almost nilpotent. Actually,
by Theorem 0.2 above this group is big. This leads us to
the following
\medskip
\noindent {\bf Question.} {\it Let $X$ be an almost C--hyperbolic
algebraic variety. Is then necessarily $\pi_1(X)$ a big group?}
\medskip
By another theorem of Lin [Lin, Thm. B($b$)], $\pi_1(X)$
cannot be an amenable group with a non--trivial center assuming
that the universal covering space $\tilde X$ is Carath\'eodory hyperbolic.
An easy
observation is that the answer is `yes' for dim$\,X=1$. Indeed,
an algebraic curve $C$ is C--hyperbolic iff it is hyperbolic, or, in turn,
iff its normalization $C_{\rm norm}$ has a
non-abelian fundamental group. In the latter case the group
$\pi_1(C_{\rm norm})$ is big (see 1.2($a$)).
Note, however, that by a result of [LySu], any compact Riemann
surface $R$ of
genus $g \ge 2$ admits a Galois covering $\tilde R$ with a metabelian (i.e.
two-step
solvable) Galois group such that $\tilde R$ carries a non--constant bounded
holomorphic function. Modifying this result, one may even assume $\tilde R$
being Carath\'eodory hyperbolic [LinZa, Sect. 3].
\medskip
More generally, we have the following fact. Its proof given below was
communicated to us by D. Akhiezer\footnote{and it is placed here with his
kind permission.}.
\bigskip
\noindent {\bf 1.3. Theorem.} {\it Let $D \subset I \!\!\!\! C^n$ be a bounded
symmetric domain, and let $\Gamma \subset
{\rm Aut}\,D$ be a discrete subgroup. If the Bergman volume of a
fundamental domain of $\Gamma$ is finite, then $\Gamma$ is a
big group and it has no infinite solvable normal subgroup.}
\bigskip
\noindent {\it Proof.} According to a result of A. Borel and J.--L. Koszul
[Bo, Kos], a homogeneous domain $D$ is symmetric iff
the identity component $G$ of the automorphism group Aut$\,D$ is semisimple.
Recall that $G$ has trivial center, and therefore it is a connected linear
group [He, Ch. VIII.6]. Being semisimple
$G$ is not solvable. Moreover, since $G$ is connected, it is not small. We
have $D \cong G/K$, where $K \subset G$ is a maximal compact subgroup
[ibid, Ch. VIII. 7]. The automorphism group Aut$\,D$ has finitely many
connected components, i.e. [Aut$\,D : G] < \infty$ (indeed, being a compact Lie
group the stabilizer
Stab$_z \subset$ Aut$\,D$ of a point $z \in D$ has
a finite number of connected components, which is the same as those of
Aut$\,D$, because the quotient $D \simeq$ Aut$\,D/$Stab$_z$ is connected).
Hence, $\Gamma \cap G$ has finite index in $\Gamma$, and the Bergman volume
of $(\Gamma \cap G) \setminus D$ is finite, too. Therefore, the invariant volume
Vol$\,((\Gamma\cap G) \setminus G)$ is finite, and so $\Gamma\cap G$ is a lattice of $G$.
Fix a faithful linear representation $G \hookrightarrow GL(n,\, I \!\!\!\! C)$. Let
$G_{ I \!\!\!\! C}$ be the Zariski closure of $G$ in $GL(n,\, I \!\!\!\! C)$. By Borel's Density
Theorem (see e.g. [Ra, 5.16]), the conditions
"$G$ is semisimple and Vol$\,((\Gamma\cap G) \setminus G) < \infty$" imply that the
subgroup
$\Gamma\cap G$ is Zariski dense in $G_{ I \!\!\!\! C}$. Hence, if $\Gamma$ is almost
solvable,
$G_{ I \!\!\!\! C}$ should be also almost solvable, which is not the case. By the
Tits alternative, $\Gamma$ must be big.
The last assertion follows from a theorem of V. Gorbatsevich [GoShVi,
Proposition 3.7]. According to this theorem, the lattice $\Gamma\cap G$ in a
connected Lie group $G$
possesses no infinite solvable normal subgroup iff $G$ is reductive and
its
semisimple part has a finite center. It is easily seen that in our case
both conditions are fulfilled. \hfill $\Box$
\bigskip
\noindent {\bf 1.4.} {\it Remark.} In fact, it would be enough in the
above theorem that $\Gamma$ was a Zariski dense subgroup
of a semisimple linear algebraic group $G$ with a finite center, which acts
holomorphically in $D$. This may be illustrated by the following
example 1.5($a$).
\medskip
\noindent {\bf 1.5.} {\bf Examples.}
\smallskip
\noindent ($a$) Let $D = {\cal Z}_g$ be the Siegel upper half--plane
and $\Gamma = $Sp$_{2g}( Z \!\!\! Z),\,\,G =$ Sp$_{2g}( I \!\! R),\,g \ge 1$, are resp.
the Siegel modular
group and the simplectic group. Then $\Gamma \setminus D$ is a coarse moduli space
of principally polarized abelian varieties of dimension $g$, which is
a quasiprojective variety. Here $\Gamma$
is Zariski dense in $G$. Actually, by a theorem of A. Borel and
Harish--Chandra [BoHC, Thm. 7.8], the arithmetic subgroup $G_{ Z \!\!\! Z}$ of a
semisimple real algebraic group $G_{ I \!\! R}$ defined over $\bf Q$ is a lattice
in $G_{ I \!\! R}$,
and so by Borel's Density Theorem, it is Zariski dense in $G_{ I \!\!\!\! C}$.
(By the way, these arguments show that $\Gamma = $Sp$_{2g}( Z \!\!\! Z)$ is a big group
without infinite normal solvable subgroups.)
\medskip
\noindent ($b$) Let, furthermore, $D := T_{g,\,n} \subset\s I \!\!\!\! C^{3g-3+n}$ be
the Teichm\"uller space of the $n$--punctured genus $g$ marked
Riemann surfaces under the Bers realization, where $2-2g-n <0$. By
Royden's Theorem, $\Gamma:=$ Aut$\,D$ is the Teichm\"uller modular
group, which coincides with
the mapping class group Mod$_{g,\,n}$. The quotient $\Gamma \setminus D$ is
a coarse moduli space ${\cal M}_{g,\,n}$ of genus $g$ $\,\,n$--punctured
Riemann surfaces, which is a quasiprojective variety. By Lemma 1.2 above,
except the case when $(g,\,n) = (0,\,3)$ the group Mod$_{g,\,n}$ is big.
\medskip
\noindent (c) (see e.g. [Sh1, 2]).
A smooth projective surface
$S$ is called {\it a Kodaira surface} if there is a smooth fibration
$\pi\,:\,S \to B$ over a curve $B$, where both $B$ and a generic fibre $F$
of $\pi$ are of genus $\ge 2$ (usually $\pi$ is supposed being a non-trivial
deformation of $F$, but we don't need this assumption here).
It is well known that
the universal covering $\tilde S$ of $S$ can be realized as a bounded
pseudo--convex Bergman domain in $ I \!\!\!\! C^2$. Thus, the projective surface
$S = \Gamma \setminus D$ is C--hyperbolic; clearly, $\Gamma \simeq \pi_1(S)$ is a big
group.
More generally, the same is true when both $B$ and $F$ are quasiprojective
hyperbolic curves.
\bigskip
Next we pass to the simplest examples of reducible plane projective curves
with a big fundamental group of the complement.
\medskip
\noindent {\bf 1.6.} {\bf Examples.}
\smallskip
\noindent ($a$) Let $C \subset I \!\! P^2$ be a finite line
arrangement. If these lines are in general position, then by the
Deligne--Fulton Theorem, $\pi_1( I \!\! P^2 \setminus C)$ is abelian. Otherwise,
this group is big. Indeed, let $C$ has a point $A$ of
multiplicity at least $3$. The union $L$ of lines in $C$ passing through $A$
contains at least three members of the associated linear pencil. The linear
projection $ I \!\! P^2 \setminus C \to I \!\! P^1 \setminus \{3\,\,{\rm points}\}$ with
center at $A$ yields an epimorphism of the fundamental groups. Thus,
$\pi_1( I \!\! P^2 \setminus C)$ dominates the free group {\bf F}$_2 = \pi_1( I \!\! P^1
\setminus \{3\,\,{\rm points}\})$, and therefore, it is big.
In particular, if $C$ is an arrangement of six lines with four triple points,
then $\pi_1( I \!\! P^2 \setminus C)$ is a finite index subgroup of
the mapping class group Mod$_{0,\,5}$ (see [DeZa1, 6.1($a$)];
cf. also 1.5($b$) above).
\medskip
\noindent ($b$) Consider further a configuration $C \subset I \!\! P^2$ of a plane conic
together with two of its tangent lines (cf. [DeZa1, 6.1($b$)]).
The Zariski--van Kampen method yields a presentation
$$G := \pi_1( I \!\! P^2 \setminus C) = \langle\,a,\,b\,|\,abab = baba \,\rangle\,.$$
The following proof of the bigness of $G$ was communicated to us by V.
Lin\footnote{We are grateful to
V. Lin for a kind permission to place here this proof.}.
Remind that the Coxeter group ${\rm {\bf B}}_k$ is the group generated by
the orthogonal reflections
in $ I \!\! R^k$ with respect to the coordinate planes and the diagonals $x_i - x_j
= 0,\,i,j=1,\dots,k$. The corresponding Artin--Brieskorn braid group is
the fundamental group $\pi_1(G_k({\rm {\bf B}}_k))$ of the domain $$G_k({\rm
{\bf B}}_k) := \{z = (z_1,\dots,z_k) \in I \!\!\!\! C^k\,|\,d_k(z)\cdot z_k \neq 0\}\,,
$$ where $d_k(z)$ is the discriminant of the universal polynomial $p_k(t) =
p_k(t,\,z):= t^k + z_1t^{k-1} +\dots +z_k$
of degree $k$. Put $G_k := \{z \in I \!\!\!\! C^k\,|\,d_k(z) \neq 0\}$, and let $E^1_k
\to G_k$ be the standard $k$--sheeted covering over $G_k$, where
$$E^1_k := \{(z,\,\lambda) = (z_1,\dots,z_k,\,\lambda) \in I \!\!\!\! C^{k+1}\,|\, p_k(\lambda,\,z)
= 0\}\,.$$ Define a mapping $\,\varphi\,:\,E^1_{k+1} \to G_k({\rm {\bf B}}_k)
\times \, I \!\!\!\! C$ as follows:
$$\varphi (z_1,\dots,z_{k+1},\,\lambda) = (q_k,\,\lambda) = (\xi_1,\dots,\xi_k,\,\lambda)\,,$$
where
$$q_k = q_k(t,\,\xi) = t^k + \xi_1t^{k-1} +\dots+\xi_k:= p_{k+1}(t+\lambda,\,z)/t
\in G_k({\rm {\bf B}}_k) \,.$$
Note that $t\,|\,p_{k+1}(t+\lambda,\,z)$, because
$p_{k+1}(\lambda,\,z) \equiv 0$ for $(z,\,\lambda) \in E^1_{k+1}$. Since
$p_{k+1}(t+\lambda)$ is a polynomial with simple roots, the same is true for
$q_k(t)$. Moreover,
$q_k(0) \neq 0$; thus, indeed, $q_k \in G_k({\rm {\bf B}}_k)$. It is easily
seen
that $\varphi$ is a biregular isomorphism. Hence, the isomorphism
$$\pi_1(G_k({\rm {\bf B}}_k)) \cong \pi_1(E^1_{k+1}) \hookrightarrow
\pi_1(G_{k+1})$$
represents the Artin--Brieskorn braid group $\pi_1(G_k({\rm {\bf B}}_k))$ as a
subgroup of finite
index (equal to $k+1$) of the standard
Artin braid group\footnote{From now on we denote $B_m = \pi_1(G_m)$ the
standard Artin braid group with $n$ strings; don't confuse with the Coxeter
group ${\rm {\bf B}}_k$.} $B_{k+1} := \pi_1(G_{k+1})$. Therefore, the former
group is
big as soon as the latter one is so. Both of them are big starting with $k = 2$
(for the Artin group $B_{k+1}$ this can be checked in the same way as
it was done in the proof of Lemma 1.2 for the braid groups $B_{k,\,g}$). It
remains to note that $ I \!\! P^2 \setminus C \cong G_2({\rm {\bf B}}_2)$, and
therefore $G = \pi_1( I \!\! P^2 \setminus C)$ is isomorphic to the braid group
$\pi_1(G_2({\rm {\bf B}}_2))$ which is big.
\section{Nodal approximation of immersed curves}
Due to Theorem 2.1 below, Theorem 0.2 can be reduced
to the case where $C$ is a generic nodal Pl\"ucker curve. We also believe that
Theorem 2.1 has an independent interest.
We use the following notation and terminology.
Let $ I \!\! P^N,\,N = N(d) = {d+2 \choose 2} - 1,\,d \ge 1$, be the
Hilbert scheme of degree $d$ plane curves. Denote $Imm_{d,\,g}$ the locus of
points of $ I \!\! P^N$ which correspond to reduced irreducible immersed curves of
geometric genus $g,\,\, 0 \le g \le {d-1 \choose 2}$, and by $Nod_{d,\,g}$
resp. $PlNod_{d,\,g}$ the subset of points of $Imm_{d,\,g}$
which correspond to the nodal resp. to the Pl\"ucker nodal curves. Remind that
an irreducible curve $C \subset I \!\! P^2$ is called {\it Pl\"ucker} if the only
singular points of $C$ and the dual curve $C^*$ are ordinary nodes and cusps.
Let $Pl\ddot uNod_{d,\,g} \subset PlNod_{d,\,g}$ be the subset of curves
which have no flex at a node.
Denote ${\cal S}_d \to I \!\! P^N$ the universal family of curves over the
Hilbert scheme $ I \!\! P^N$, and let ${\cal S}_{d,\,g} \to Imm_{d,\,g}$ be its
restriction to $Imm_{d,\,g}$. By {\it a family of curves} we mean a proper
morphism
$\varphi\,:X \to Y$ of relative dimension one of quasiprojective varieties. If
$X,\,Y$ are smooth and $\varphi$ is a submersion, then the family $\varphi$ is called
{\it smooth}. We say that $\varphi$ admits {\it a simultaneous normalization} if
$Y$ is smooth and there exists a smooth family
of curves $\varphi'\,:\,X' \to Y$ and a morphism $f\,:\,X' \to X$ commuting with
the projections onto $Y$ such that for every point $y \in Y$ the restriction
$f\,|\,X'_y\,:\,X'_y \to X_y$ onto the fibre over $y$ is a normalization map.
\bigskip
\noindent {\bf 2.1. Theorem.} {\it
\noindent a) $Imm_{d,\,g} \subset I \!\! P^N$ is a smooth locally closed subvariety of
pure dimension $3d + g - 1$.
\smallskip
\noindent b) The universal family of curves ${\cal S}_{d,\,g} \to Imm_{d,\,g}$ admits
a simultaneous normalization $f\,:\, {\cal M}_{d,\,g} \to {\cal S}_{d,\,g}$.
\smallskip
\noindent c) $Nod_{d,\,g}$ and, for $n \ge 2g-1,\,\,\,PlNod_{d,\,g}$ are
Zariski open subsets of $Imm_{d,\,g}$.}
\bigskip
\noindent {\it Remark.} The first statement of (c) and the dimension count in ($a$)
can be found
in [Ha, Sect. 2], while the proofs are quite different. Note that, by Harris
[Ha], the variety $Imm_{d,\,g}$ is irreducible; it is non--empty for any
$(d,\,g)$ with $0 \le g \le {d - 1 \choose 2}$ [Se, sect.11, p.347; Ha; O].
\medskip
In this section we prove ($a$), ($b$) and the first part of ($c$) of Theorem
2.1; the proof of ($c$) is completed in sect. 3.
First we study $Imm_{d,\,g}$ locally,
in a neighborhood of a given curve $C \in Imm_{d,\,g}$.
This needs certain preparation, including a portion of plane curve
singularities.
\bigskip
\noindent {\bf 2.2. The Gorenstein--Rosenlicht invariant, the boundary braid and
its algebraic length}
\medskip
\noindent Recall that the Gorenstein--Rosenlicht invariant
${\delta}_P$ of a singular
analytic plane curve germ $(A,\,P)$ can be expressed as ${\delta}_P = {1 \over 2}(\mu
+ r - 1)$, where $\mu$ is the Milnor number and $r$ is the number of
local branches of $A$ at $P$ [Mi, sect. 10]. For a reduced curve $F$ on
a smooth
surface $W$ we set ${\delta} (F) = \sum_{P \in {\rm Sing} F} {\delta}_P$.
If $F$ is a complete irreducible curve, then by the genus formula and the
adjunction formula [BPVV, II.11] we have
\begin{equation}
\pi_a (F) = g(F) + {\delta} (F) = 1/2\, F (K_W + F) + 1\,,
\end{equation}
where $\pi_a$ resp. $g$ denotes arithmetic resp. geometric genus,
$K_W$ is the canonical divisor of
$W$, and where for a non--compact surface $W$ we put $F K_W = {\rm
deg}\,(K_W\,|\,F)$.
\smallskip
Let $U \subset I \!\!\!\! C$ be the unit disc, ${\Sigma} = U \times I \!\!\!\! C \subset I \!\!\!\! C^2$ be the solid
cylinder ${\Sigma} = \{(u,\,v) \in I \!\!\!\! C^2\,|\,|u| < 1\}$, and $p\,:\, I \!\!\!\! C^2 \to I \!\!\!\! C$
be the first projection. Let $A \subset {\Sigma}$ be an
analytic curve extendible transversally through the boundary $\partial {\Sigma}$, so
that {\it the link} $\partial A = {\bar A} \cap \partial {\Sigma}$ is smooth.
Suppose also that the projection $p\,:\,A \to U$ is proper, i.e. it is a
(ramified)
covering over the unit disc $U$ of degree, say, $m$. The link $\partial A$
carries a (closed) braid with $m$ strings $b_A \in B_m$ defined uniquely up
to conjugation, where $B_m$ is the Artin braid group (see (1.6($b$) above)
\footnote{
To define the braid $b_A$, cut the cylinder
$\partial U = S^1 \times I \!\!\!\! C$ along
its generator $1 \times I \!\!\!\! C$ and then identify it with
$[0,\,1] \times I \!\! R^2 \subset I \!\! R^3$. Fix a numbering of the points of
the fibre of $\partial A$ over $1 \in \partial U$. Passing once along the circle
$S^1 = \partial U$ counterclockwise, we obtain the braid $b_A $.}.
Let
$\sigma_1,\dots, \sigma_{m-1}$ be the standard generators of $B_m$.
For a braid $b = \sigma_{i_1}^{\alpha_1}\dots
\sigma_{i_n}^{\alpha_n} \in B_m$ its {\it algebraic length}
is defined as $l($b$) := \sum_{k=1}^n \alpha_k$.
\bigskip
\noindent {\bf 2.3. Lemma.} {\it Let $A \subset {\Sigma}$ as above be a nodal curve with ${\delta}$
nodes. Suppose that all the ramification points of the covering $p\,:\,A \to U$
are simple (i.e. with ramification indices $2$) and no two of them are at the
same
fibre. If the branching divisor $D \subset U$ consists of ${\delta} + \tau$ points,
then $$l(b_A) = 2{\delta} + \tau\,.$$}
{\it Proof.} Choose small disjoint discs $\omega_i$ in $U$,
$i=1,\dots,{\delta} + \tau$, centered at the points of
$D$. Fix a point at the boundary of the disc $\omega_i$ and join it by
a path $\gamma_i$ with the point $1 \in \partial U$, where $\gamma_i,\,
i=1,\dots,{\delta} + \tau$, are disjoint. The complement $U \setminus \bigcup_{i=1}^{{\delta} +
\tau} ({\bar \omega}_i \cup \gamma_i)$ being simply connected, the braid $b_A$
is the product of the local braids $b_{A_i}$ which correspond to
the curves $A_i := A \cap p^{-1}(\omega_i)$. It is easily seen that the local
braid which corresponds to a node of $A$ is conjugate in the braid group $B_m$
with the square of a generator, and those at an irreducible ramification point
is conjugate with a generator. Now the lemma easily follows. \hfill $\Box$
\bigskip
With each plane curve singularity $(A,\,{\bar 0}) \subset ( I \!\!\!\! C^2,\,{\bar 0})$
we associate its {\it braid} $b_{A,\,{\bar 0}}$ defined as follows. Fix
a generic linear projection $p\,:\,( I \!\!\!\! C^2,\,{\bar 0}) \to ( I \!\!\!\! C,\,{\bar 0})$,
so that the direction
of $p$ is different from the tangent directions of the branches of $A$ at
$\bar 0$,
and proceed in the same way as above.
\bigskip
\noindent {\bf 2.4. Lemma.} {\it Suppose that
$(A,\,{\bar 0}) \subset ( I \!\!\!\! C^2,\,{\bar 0})$ is an
immersed singularity (i.e. a singular point of a reduced curve having only
smooth local branches $A_1,\dots, A_r$) with
the Gorenstein--Rosenlicht invariant ${\delta} = {\delta}(A, \,{\bar 0})$.
Then
\noindent a) $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\delta} = {1 \over 2} \,
l(b_{A,\,{\bar 0}})\,.$
\smallskip
\noindent b) Let $\tilde A$ be a small nodal deformation
of $A$ defined in a fixed small ball $B_{\epsilon}$ centered at the origin.
Denote by $r$ resp. $\tilde r$ the number of irreducible components of
$A$ resp. of $\tilde A$ in $B_{\epsilon}$.
Then ${\delta} ({\tilde A}) \le {\delta} (A)$, and ${\delta} ({\tilde A}) = {\delta} (A)$ iff $r =
{\tilde r}$. In the latter case the irreducible components ${\tilde A}_1,\dots,
{\tilde A}_r$ of ${\tilde A}$ in $B_{\epsilon}$ approximate the corresponding
irreducible components $A_1,\dots, A_r$ of $A \cap B_{\epsilon}$. }
\bigskip
\noindent {\it Proof.} ($a$) We have ${\delta} = \sum_{1\le k < l \le r}
(A_k\,\cdot\,A_l)_{\bar 0}$ [Mil, (10.20)]. Let $A_i' \subset B_{\epsilon}$ be
a small
generic deformation of the branch $A_i,\,\,i=1,\dots,r$. Set
$A' = \bigcup_{i=1}^r A_i'$. Then $A'$ is a nodal curve with
$${\delta} = \sum_{1\le k < l \le r} A'_k \cdot A'_l =
\sum_{1\le k < l \le r} (A_k\,\cdot\,A_l)_{\bar 0}$$ nodes, and
clearly, $b_{A,\,{\bar 0}} = b_{A',\,{\bar 0}}$. Since for all $i=1,\dots,r$
the generic linear projection $p\,:\,A_i \to U_{\epsilon'}$ is
non--ramified, the same is true for the branches
$A'_i,\,i=1,\dots,r$. Thus, $p\,:\,A' \to U_{\epsilon'}$ is ramified only
at ${\delta}$ nodes, and therefore, in the notation of Lemma 2.3, $\tau = \tau (A')
=0$. By this lemma, we have ${\delta} = 1/2\, l(b_{A',\,{\bar 0}})
= 1/2\, l(b_{A,\,{\bar 0}})$. This proves ($a$).
\smallskip
\noindent ($b$) Once again here
$b_{A,\,{\bar 0}} = b_{{\tilde A},\,{\bar 0}}$.
Due to ($a$) and to Lemma 2.3, we have
$$2{\delta}(A) = l(b_{A,\,{\bar 0}}) =
l(b_{{\tilde A},\,{\bar 0}}) = 2{\delta} ({\tilde A}) + \tau ({\tilde A})\,,$$
and the inequality of ($b$) follows. The equality holds iff $ \tau ({\tilde A})
=0$, which means that the projection $p \,:\,{\tilde A} \to U_{{\tilde
\epsilon}}$ is ramified only at nodes of ${\tilde A}$. Therefore, for
any irreducible component ${\tilde A}_i$ of ${\tilde A} \cap B_{\epsilon}$
the composition of the normalization map
$({\tilde A}_i)_{\rm norm} \to {\tilde A}_i$ with the projection
$p \,:\,{\tilde A}_i \to U_{{\tilde \epsilon}}$ is non--ramified and hence,
one--sheeted. It follows that both of these mappings are biholomorphic,
so that the irreducible components ${\tilde A}_i$ of ${\tilde A} \cap
B_{\epsilon}$ are smooth. The degree of the
branched covering $p\,:\,{\tilde A} \to U_{{\tilde \epsilon}}$ being equal to
$r,\,\,{\tilde A} \cap B_{\epsilon}$ consists of $r$ smooth irreducible
components close to those of $A$. \hfill $\Box$
\bigskip
Let $X$ be a smooth projective surface, $C \subset X$ be an irreducible immersed
curve with a normalization $\varphi_0\,:\,M_0 \cong C_{\rm norm} \to C$. By
[No, (1.8)-(1.12)], there exists a smooth open complex surface $V$ which
contains $M_0$ as a closed subvariety, and a holomorphic immersion
$\varphi\,:\, V \to X$ that extends $\varphi_0$; it is called {\it a tubular
neighborhood of} $\varphi_0$. To obtain $V$ one simply normalizes
$C$ together with a tubular neighborhood of $C$ in $X$.
\bigskip
\noindent {\bf 2.5. Lemma.} {\it Let $C \subset X$, $M_0$ and $V$ be as above,
and let $N \to M_0$ be the normal bundle of
$M_0$ in $V$. Then
\begin{equation} {\rm deg}\, N = M_0^2 = C^2 - 2{\delta} (C)\,.\end{equation}
If $X = I \!\! P^2$ and $C \in Imm_{d,\,g}$, then
\begin{equation}{\rm deg}\, N = 3d +2(g-1)\,.\end{equation}}
\noindent {\it Proof.} By the adjunction formula, we have
\begin{equation} 2g-2 = C^2 + CK_X - 2 {\delta} (C) = M_0^2 + M_0 K_V \,.\end{equation}
Since $K_V = \varphi^* K_X$, by the projection formula we have $M_0 K_V =
CK_X$, and
so (2) follows. (3) is a corollary of (2) and the genus formula (1). \hfill $\Box$
\bigskip
\noindent {\bf 2.6. Corollary.} {\it a) $N$ is very ample iff $C^2 - 2{\delta} (C)
\ge 2g+1$. For $X = I \!\! P^2$ and $C \in Imm_{d,\,g}$ this is always the case,
and furthermore, $h^1 (M_0,\,{\cal O}(N)) = 0$
and $h^0 (M_0,\,{\cal O}(N)) = 3d + g - 1$.
\smallskip
\noindent b) For any pair of points $p_1,\,p_2 \in M_0$ the line bundle
$N_{p_1,\,p_2} = N - [p_1] - [p_2]$ on $M_0$ is spanned\footnote{i.e. the
linear system $|N_{p_1,\,p_2}|$ has no base point.} if $C^2 - 2{\delta} (C)
\ge 2g+2$. In particular, this is so if $X = I \!\! P^2$ and $C \in Imm_{d,\,g}$,
where $d \ge 2$.}
\bigskip
\noindent {\it Proof.} The first statement of (a) and (b) follow from Lemma 2.5
by the well known criteria of ampleness or spannedness
of a line bundle over a curve (see e.g. [Hart,IV.3.2] or [Na, 5.1.12]). By
the Kodaira Vanishing Theorem, we obtain that $h^1 (M_0,\,{\cal O}(N)) = 0$,
and hence, by the Riemann--Roch Formula,
we have $h^0 (M_0,\,{\cal O}(N)) = {\rm deg}\, N + 1 - g = 3d + g - 1$. \hfill $\Box$
\bigskip
The Kodaira Theorem on embedded deformations [Ko] implies such a
\bigskip
\noindent {\bf 2.7. Corollary.} {\it There exists a maximal smooth family
$\pi_{loc}\,:\,{\cal M}_{loc} \to T_{loc}$ of embedded deformations
of the curve $M_0 \cong
\pi^{-1} (t_0)$ in $V$ over a smooth base $T_{loc}$ such that the
Kodaira--Spencer
map $T_{s_0}T_{loc} \to H^0 (M_0,\,{\cal O} (N))$ is an isomorphism. In
particular, if $X = I \!\! P^2$ and $C \in Imm_{d,\,g}$, then\footnote{cf. [GH,
sect.2.4; Ha].} ${\rm dim}\,T_{loc} = 3d + g - 1$.}
\bigskip
\noindent {\bf 2.8. Definition.} We say that a curve $C \in Imm_{d,\,g}$ is {\it
strongly approximated} by curves $C' \subset Imm_{d,\,g}$ if $C'$ approximate $C$
in the Hausdorff topology, and for any
singular point $P$ of $C$ of multiplicity $r(C,\,P)$ and for a fixed small
neighborhood $B_{\epsilon, \,P}$ of $P$, the number $r(C',\,P)$ of irreducible
components of $C' \cap B_{\epsilon, \,P}$ is equal to $r(C,\,P)$, and the
irreducible components of $C' \cap B_{\epsilon, \,P}$ approximate those of $C
\cap B_{\epsilon, \,P}$. Or, which is equivalent, if for a
given tubular neighborhood $\varphi\,:\,V \to I \!\! P^2$ of a normalization
$\varphi_0\,:\,M_0 \to C$, the curves $C'$ have normalizations
$\varphi\,|\,M'\,:\,M' \to C'$, where $M' \subset V$
are obtained from $M_0$ by a small deformation.
\bigskip
We use below the following simple observation: a curve $C \in Nod_{d,\,g}$ is
Pl\"ucker iff $C$ has only ordinary flexes,
no multitangent line, i.e. a line tangent to $C$ in at least three points,
and no bitangent line which is an inflexional tangent. One says that
a curve $C \subset I \!\! P^2$ {\it has
only ordinary singularities} iff all the local branches of $C'$ at any of its
singular point are smooth and pairwise transversal. Denote by $Ord_{d,\,g}$ the
set of all such curves of degree $d$ and genus $g$; clearly, $Ord_{d,\,g} \subset
Imm_{d,\,g}$.
\smallskip
The next proposition should be known at least partially; in view of the lack of
references, we give its proof.
\bigskip
\noindent {\bf 2.9. Proposition.} {\it The subspaces $Ord_{d,\,g},\,
Nod_{d,\,g},\,PlNod_{d,\,g}$ and
$Pl\ddot uNod_{d,\,g}$ are dense in $Imm_{d,\,g}$ in the topology of strong
approximation, and hence also in the Hausdorff topology of $ I \!\! P^N$.}
\bigskip
\noindent {\it Proof.} Fix an arbitrary curve $C \in Imm_{d,\,g}$. We may assume
that $d \ge 3$. First we show that $C$ can be strongly approximated by curves
$C' \in Ord_{d,\,g}$. For a curve $C' \in Imm_{d,\,g}$ denote by
${\delta}_1(C')$ the number of all non--ordered pairs $(A'_i,\,A'_j)$ of local
analytic branches of $C'$ which meet normally at their common center $P \in
C'$, so that $(A'_i,\,A'_j)_P = 1$. Clearly,
$C' \in Ord_{d,\,g}$ iff ${\delta}(C') = {\delta}_1(C')$.
Suppose that $C$ as above has a non--ordinary singular point $P$ of
multiplicity $m$. Consider the blow up $\sigma\,:\,X \to I \!\! P^2$ of
$ I \!\! P^2$ at $P$, and let ${\hat C} \subset X$ be the proper transform of $C$.
It is easily seen that ${\delta}({\hat C}) = {\delta}(C) - {m \choose 2}$. Since
${\hat C}^2 = C^2 - m^2$ we have
$${\hat C}^2 - 2{\delta}({\hat C}) = C^2 - 2 {\delta}(C) - m = 3d + 2(g-1) - m \ge 2g
+2\,.$$ Let $\varphi \,:\,V \to X$ be a tubular neighborhood of a normalization
$\varphi_0\,:\, M_0 \to {\hat C}$ of ${\hat C}$. For a pair $(A_i,\,A_j)$
of local branches of $C$ at $P$ with $(A_i,\,A_j)_P > 1$ let $\hat P \in X$
be the common center of their proper preimages ${\hat A_i},\,{\hat A_j}$
in $X$,
and let $P_i,\,P_j \in M_0$ be resp. the centers of the branches
$\varphi^{-1}({\hat A_i}),\,\varphi^{-1}({\hat A_j})$ of the curve $M_0 \subset V$. By Lemma
2.5, for the normal bundle $N$ of $M_0$ in $V$ we have
$${\rm deg}\,(N - [P_i]) = M_0^2 - 1 = {\hat C}^2 - 2{\delta}({\hat C}) - 1
\ge 2g + 1\,.$$ Therefore, being spanned, the line bundle $N - [P_i]$ possesses
a section which does not vanish at $P_j,\,j \neq i$. It follows that $N$ has a
section
that vanishes at $P_i$, but not at $P_j$. This yields a deformation
$M'_0$ of $M_0$ in $V$ which passes through $P_i$, but not through $P_j$. Thus,
for the curve $C' := \sigma\varphi (M_0') \in Imm_{d,\,g}$ close enough to $C$ we have
${\delta}_1(C') > {\delta}_1(C)$. By induction on ${\delta}_1(C')$, we get a strong
approximation $C' \in Ord_{d,\,g}$ of $C$.
Suppose further that $C \in Ord_{d,\,g}$
is not nodal, i.e. it has a point $P$ of multiplicity $m \ge 3$. Applying
the same procedure as above to a triple of points $P_i,\,P_j,\,P_k \in M_0$
which lie over $P$, and using the inequality $${\rm deg}\,(N - [P_i] - [P_j])
\ge 2g\,,$$ by the spannedness of the line bundle $N - [P_i] - [P_j]$,
we obtain a section of $N$ which vanishes at the points $P_i$ and $P_j$, but
not at $P_k$. This leads to a curve $C' \in Ord_{d,\,g}$ which strongly
approximates $C$
and is simpler than $C$ in the following sense: $m(C') < m(C)$, where
$$m(C) := \sum_{P_i \in {\rm sing}\,(C)} ({\rm mult}\,(P_i) - 1)\,.$$
Induction on $m(C')$ now shows that $C$ can be strongly approximated by curves
$C' \in Nod_{d,\,g}$.
\smallskip
Next we show that a curve $C \in Nod_{d,\,g}$ can be strongly approximated by
curves
$C' \in Nod_{d,\,g}$ with only ordinary flexes. We proceed by induction on the
number $ofl(C')$ of ordinary flexes of $C'$. Since such a flex is a normal
intersection point of
the Hesse curve of $C'$ with a smooth local branch of $C'$, clearly,
the bounded function $ofl(C')$ is lower semi--continuous on $Nod_{d,\,g}$ with
respect to the Hausdorff topology.
Suppose that $C$ has
a non--ordinary flex at a local branch $A$ of $C$ centered at $P \in C$, so
that $(A,\,L)_P \ge 4$, where $L$ is the tangent line to $A$ at $P$. In the
notation as above, let $\sigma\,:\,X \to I \!\! P^2$ be the composition of three
successive blow ups over $P$ with centers at the proper preimages of $A$.
Let ${\hat L} \subset X$ be the proper transform of $L$ and $\hat P$ be the center
of the proper transform ${\hat A} \subset X$ of $A$. We have
$({\hat A},\, {\hat L})_{\hat P} \ge 1$. If $P$ is a smooth point of $C$ then
${\hat C}^2 = C^2 - 3$ and ${\delta}({\hat C}) = {\delta} (C)$. If $P$ is a node of $C$
then ${\hat C}^2 = C^2 - 6$ and ${\delta}({\hat C}) = {\delta} (C) - 1$. In any case,
$${\rm deg}\,N = {\hat C}^2 - 2{\delta}({\hat C}) \ge C^2 - 2{\delta}(C) - 4 \ge 2g\,.$$
Therefore, the normal bundle $N$ of $M_0$ in $V$ is spanned, and hence it has a
section which does not vanish
at the point $\hat P$. The corresponding Kodaira-Spencer deformation yields a
curve $M_0'$ on $X$ close enough to $M_0$ which does not pass through $\hat P$.
It is easily seen that the projection $C' := \sigma\varphi(M_0') \subset I \!\! P^2$ is a nodal
curve with an ordinary flex at $P$ and such that $ofl(C') > ofl(C)$. After a
finite
number of steps we obtain a strong approximation $C' \in Nod_{d,\,g}$ of $C$
with only ordinary flexes.
Suppose further that $C \in Nod_{d,\,g}$ has only ordinary flexes
(note that this is an open condition). We will
find a strong approximation $C'$ of $C$ without multiple tangents.
Denote by $b(C')$ the total number of distinct intersection points with $C$ of
all the bitangent lines of $C'$. Clearly, the bounded function $b(C')$
is lower semi--continuous on $Nod_{d,\,g}$.
Let $C$ have a multitangent line $L$ which is tangent to $C$ at points
$P,\,Q,\,R \in C$ and, perhaps, at some other points.
Let $\sigma\,:\,X \to I \!\! P^2$ be the composition of the blow-ups of $ I \!\! P^2$ at the
points $P,\,Q$ and $R$, and let $\hat C$ be the proper transform of $C$ at $X$.
Note that $d = $ deg$\,C = L\cdot C \ge 6$. As above, the blow up at a smooth
point (resp. at a node) of $C$ decreases the difference $C^2 - 2{\delta} (C)$ by $1$
(resp. by $2$). Thus, we have
$${\hat C}^2 - 2{\delta} ({\hat C}) \ge C^2 - 2{\delta} (C) - 6 = 2g + 3d - 8 \,.$$
Let $\varphi\,:\,V \to X$ be a tubular neighborhood of a normalization
$\varphi_0\,:\,M_0 \to {\hat C}$ of $\hat C$, and let $N$ be the normal bundle of
$M_0$ in $V$.
The line bundle $N - [{\hat P}] - [{\hat Q}]$ on $M_0$ of degree
$\ge 2g + 3d - 10 > 2g + 2$ is spanned (cf. Corollary 2.6). This yields a
deformation $C' := \sigma \varphi(M_0') \subset Nod_{d,\,g}$ of $C$ such that $L$
is still tangent to $C'$ at the points $P$ and $Q$, and meets $C'$ normally
at $R$, so that $b(C') > b(C)$. Maximizing $b(C')$ we get a
strong approximation $ C' \in Nod_{d,\,g}$ of $C$ with only ordinary flexes
and without multiple tangents.
Suppose now that $C \in Nod_{d,\,g}$ has only ordinary flexes and no multiple
tangent line, which is an open condition. To find a strong Pl\"ucker
approximation
$C'$ of $C$, we will proceed by induction on the total number $inf(C')$ of
distinct intersection points of $C'$ with all of its inflexional tangent lines.
We have to ensure that no inflexional tangent line of $C'$ is a bitangent line.
Let a bitangent line $L$ of $C$ be an inflexional tangent of $C$ at a point
$P \in C$ and tangent to $C$ at a point $Q \in C$. Then
$d =$ deg$\,C = C\cdot L \ge 5$. Blowing up $ I \!\! P^2$ at $Q$
we get a surface $X = \sigma_Q( I \!\! P^2)$. In the notation as above, we have
$${\rm deg}\,(N - 3[{\hat P}]) \ge 2g + 3d - 7 > 2g + 2\,.$$ Therefore,
there exists a deformation $ C' = \sigma \varphi(M_0') \in Nod_{d,\,g}$ of $C$ such
that $L$ is still an inflexional tangent of $C$ at $P$, but it meets $C$
normally at $Q$. Thus, $inf(C') > inf(C)$. By induction, we obtain
a strong approximation $C'$ of $C$ which belongs to $PlNod_{d,\,g}$.
Suppose finally that $C \in PlNod_{d,\,g} \setminus Pl\ddot uNod_{d,\,g}$,
so that, although all the flexes of $C$ are
ordinary, one of them, say $(A,\,P)$, is located at a node of $C$ with the
second branch, say, $B$. This time we proceed by induction on
the number $sfl(C')$ of flexes of $C'$ which are smooth
points. Evidently, $sfl(C')$ is a bounded lower
semi--continuous function on $Nod_{d,\,g}$.
Performing two successive blow ups, the first one at $P \in C$ and the second
one at the center of the proper transform of the branch $A$, we obtain a
surface $X$. Denote by
${\hat Q}$ the center of the proper preimage $\hat B$ of the branch $B$
in $X$. We have
$${\rm deg}\,N = {\hat C}^2 - 2{\delta}({\hat C}) = C^2 - 2{\delta}(C) - 3
\ge 2g + 1\,,$$
so that the line bundle $N - [{\hat Q}]$ on $V$ is spanned.
Hence, we can find a
section of $N$ which vanishes at $\hat Q$ and does not vanish at $\hat P$.
This yields a small deformation $M_0'$ of $M_0$ on $V$ which passes through
$\hat Q$ but not through $\hat P$. The curve $C' := \sigma\varphi(M_0') \subset I \!\! P^2$
is close enough to $C$, still has a node at $P$ which is not any more
a flex, while $L$ is still a tangent line of $C'$ at $P$. Note that a small
deformation of $C$ yields a small deformation of the Hesse curve $H_C$ of $C$,
so that the flexes of $C$ which are the (normal) intersection points of $C$
and $H_C$ are also perturbed a little. Thus, $C'$ has a flex at a smooth point
close to $P$. It follows that $sfl(C') > sfl(C)$. In a finite number of steps
we obtain a desired strong approximation $C' \in Pl\ddot uNod_{d,\,g}$ of $C$.
This completes the proof. \hfill $\Box$
\bigskip
The next lemma shows that the strong approximation of immersed
curves coincides with the usual one.
\bigskip
\noindent {\bf 2.10. Lemma.} {\it Let $C \in Imm_{d,\,g}$, and let
$\varphi\,:\, V \to I \!\! P^2$ be a tubular neighborhood of its normalization
$\varphi_0\,:\,M_0 \to C$.
Then any curve $C' \in Imm_{d,\,g}$ close enough to $C$ in
the Hausdorff topology of $ I \!\! P^N$ (or, which is the
same, coefficientwise) is the image of a unique smooth curve
$M \cong C'_{\rm norm} \subset V$ under the holomorphic mapping
$\varphi\,:\,V \to I \!\! P^2$.}
\bigskip
\noindent {\it Proof.} Let $P$ be a singular point of $C$, and let $B_{\epsilon,
\,P}$ be a fixed small neighborhood of $P$. Denote by $r(C,\,P)$ the
multiplicity of $C$ at $P$, and by $r(C',\,P)$ the number of irreducible
components in $B_{\epsilon, \,P}$ of a curve $C'$ close enough to $C$
(cf. Definition 2.8).
Once we show that $\,r(C,\,P) = r(C',\,P)$ for any singular point
$P$ of $C$, then the irreducible components
of $C' \cap B_{\epsilon, \,P}$ approximate those of $C \cap B_{\epsilon, \,P}$,
i.e. $C'$ is a strong approximation of $C$, and the statement follows.
Actually, it is sufficient to prove the equality $r(C,\,P) = r(C',\,P)$ under
the additional assumption that the approximating curve $C'$ is nodal.
Indeed, by Proposition 2.9, the curve
$C'\subset Imm_{d,\,g}$ can be, in turn, strongly approximated by a curve $C'' \in
Nod_{d,\,g}$. Since $C''$ approximates both $C$ and $C'$ in the Hausdorff
topology, from the
equalities $r(C'',\,P) = r(C,\,P)$ and $r(C'',\,P) = r(C',\,P)$ it follows
that $r(C',\,P) = r(C,\,P)$.
Assuming further that $C'$ is nodal, by (1)
and Lemma 2.4($b$), we obtain
$${n-1 \choose 2} - g = {\delta} (C') = \sum_{P\in {\rm Sing}\,C'} {\delta} (C'\cap
B_{\epsilon, \,P}) \le \sum_{P\in {\rm Sing}\,C} {\delta}(C,\,P) =
{n-1 \choose 2} - g\,.$$ Henceforth,
${\delta} (C'\cap B_{\epsilon, \,P}) = {\delta} (C, \,P)$ for all
$P\in {\rm Sing}\,C$.
Applying Lemma 2.4($b$) once again, we get that $r(C',\,P) = r(C,\,P)$
for all $P\in {\rm Sing}\,C$, as desired. \hfill $\Box$
\bigskip
\noindent {\bf 2.11. Lemma.} {\it ($a$) $Imm_{d,\,g}$ is a locally closed complex
analytic submanifold of $ I \!\! P^N$ of dimension $3d + g - 1$.
\smallskip
\noindent ($b$) The universal family of curves ${\cal S}_{d,\,g} \to Imm_{d,\,g}$
over $Imm_{d,\,g}$ admits a complex analytic simultaneous normalization
$f = f_{d,\,g}\,:\,{\cal M}_{d,\,g} \to {\cal S}_{d,\,g}$.}
\bigskip
\noindent {\it Proof.} Fix a curve $C \in Imm_{d,\,g}$, and consider a tubular
neighborhood $\varphi\,:\,V \to I \!\! P^2$ of a normalization $\varphi_0$ of $C$. By
Corollary 2.7 and Lemma 2.10, the projection $\varphi$ yields a local
analytic chart $U_C$ of dimension $3d + g - 1$ on $Imm_{d,\,g}$ centered at
$C$ which covers the whole intersection of $Imm_{d,\,g}$ with a sufficiently
small ball in $ I \!\! P^N$ around $C$. This proves ($a$).
\smallskip
To prove ($b$) denote by ${\cal S}_C$ the restriction of the family ${\cal S}_{d,\,g}$ onto
the chart $U_C$. Note that the same projection $\varphi$ yields an
analytic simultaneous normalization $f_C\,:\,{\cal M}_C \to {\cal S}_C$ of ${\cal S}_C$. Any two
such normalizations $f_C\,:\,{\cal M}_C \to {\cal S}_C$ and $f'_C\,:\,{\cal M}'_C \to {\cal S}_C$ over
the
same chart $U_C$ which arise from two different tubular neighborhoods
$\varphi,\,\varphi'$, can be naturally biholomorphically identified via their
projections. Hence, the equivalence class of these normalizations over the
same chart $U_C$ in $Imm_{d,\,g}$ can be regarded as an equivalence
class of charts on a new complex manifold ${\cal M}_{d,\,g}$ of dimension $3d + g$.
Indeed, suppose that two charts $U_C$ and $U_{C'}$ on $Imm_{d,\,g}$ have a
non--empty intersection $U_{C,\,C'} := U_C \cap U_{C'}$. Consider a fibrewise
bimeromorphic mapping of smooth manifolds
$f_{C,\,C'} := f_{C'}^{-1} \circ f_C\,:\, {\cal M}_C\,|\,U_{C,\,C'} \to
{\cal M}_{C'}\,|\,U_{C,\,C'}$. It is biholomorphic at the complement of the
`multiple point locus'
$D_{C,\,C'} := f_C^{-1}($sing$\,S_{C,\,C'})$, where $S_{C,\,C'}:= S_C\,|\,
U_{C,\,C'}$, and by Riemann's extension Theorem, it has a holomorphic extension
through $D_{C,\,C'}$. Clearly, the projection
$f_{d,\,g}\,:\,{\cal M}_{d,\,g} \to {\cal S}_{d,\,g}$ induced by the local mappings
$f_C\,:\,{\cal M}_C \to {\cal S}_C$ is a holomorphic simultaneous normalization, which
proves ($b$). \hfill $\Box$
\bigskip
Next we show that all the above subvarieties of the Hilbert scheme $ I \!\! P^N$
are algebraic. Although
the following statement holds in much bigger generality\footnote{We are
grateful to H. Flenner who introduced to us this circle of ideas.} (cf. e.g.
[BinFl, Theorem 2.2]), it will be enough for us this
restricted version which has a rather easy proof.
\bigskip
\noindent {\bf 2.12. Lemma.} {\it Let $f\,:\,X \to Y$ be a family of curves over an
irreducible base $Y$. Then there exists a Zariski open subset $U \subset Y$ such
that the restriction $f\,|\,f^{-1}(U)$ of $f$ over $U$ admits a simultaneous
normalization.}
\bigskip
\noindent {\it Proof.} Without loss of generality we may suppose $Y$ being smooth.
Let $\nu\,:\,X_{\rm norm} \to X$ be a normalization. Consider the induced
family of curves $f' := f \circ \nu$. Since the singular locus $S$ of
the normal variety $X_{\rm norm}$ has codimension at least $2$, its image
$f'(S) \subset Y$ has codimension at least $1$. Restricting $f$ and $f'$ onto
the complement of the Zariski closure $\overline {f'(S)}$ of the constructible
subset $f'(S)$ in $Y$, we may suppose $X_{\rm norm}$ being smooth. By the
Bertini--Sard Theorem [Hart, III.10.7], $f'$ is an immersion over a Zariski
open
subset $U \subset Y$. Therefore, each fibre $(f')^{-1}(y),\,y \in U$, is smooth,
and the restriction $\nu\,|\,(f')^{-1}(y)$ yields a normalization of the curve
$X_y :=f^{-1}(y)$. Thus, we have obtained the desired simultaneous
normalization of the original family $f$ over $U$. \hfill $\Box$
\bigskip
We use below the following notation. Given a family of curves $f\,:\,X \to Y$,
for any $g \ge 0$ denote by $Curv_g(f)$ the subset of points $y \in Y$ such
that the fibre $X_y$ over $y$ is a reduced irreducible curve of geometric
genus $g$. For the universal family $f_d\,:\,{\cal S}_d \to I \!\! P^N$ of degree $d$
curves in $ I \!\! P^2$, set $Curv_{d,\,g} = Curv_g(f_d)$.
We say that an abstract reduced irreducible curve $C$ is {\it of immersed type}
if its normalization map $\nu\,:\,C_{\rm norm} \to C$ has a nowhere vanishing
differential. Let $Imm_g(f)$ be the subset of points $y \in Curv_g(f)$ which
correspond to the curves of immersed type, so that, in particular, $Imm_g(f_d)=
Imm_{d,\,g}$.
\bigskip
\noindent {\bf 2.13. Corollary.} {\it ($a$) Given a family of curves $f\,:\,X \to
Y$, the base $Y$ can be represented as a disjoint union of smooth irreducible
quasi--projective subvarieties $Y_i \subset Y,\,\,i=1,\dots,n = n(f)$, such that
for
each $i=1,\dots,n $ the restriction of $f$ onto $Y_i$ admits a simultaneous
normalization.
\smallskip
\noindent ($b$) For any $g \ge 0$ the subsets $Curv_g(f) \subset Y$ and $Imm_g(f) \subset Y$
are constructible. In particular, $Curv_{d,\,g}$ and $Imm_{d,\,g}$ are
constructible subsets of the Hilbert scheme $ I \!\! P^N$.}
\bigskip
\noindent {\it Proof.} ($a$) Assuming for simplicity that $Y$ is irreducible
we start with $Y_1 := U$, where $U \subset Y$ is as in Lemma 2.12 above. Next we
apply Lemma 2.12 to the restriction of $f$ onto each of the irreducible
components of the regular part of the Zariski closed subvariety $Y^{(1)}:=Y
\setminus Y_1$ of $Y$. Following this way, in a finite number of
steps we obtain the desired partition of $Y$. \hfill $\Box$
\smallskip
\noindent ($b$) Since $f\,|\,Y_i$ admits a simultaneous normalization, for any
$i=1,\dots,n$ the number and the geometric genera of the irreducible components
of a fibre $X_y = f^{-1}(y)$ do not depend on $y \in Y_i$. Thus, $Curv_g(f)$ is
a union of some of the $Y_i$, and hence it is constructible.
Set $X_i = f^{-1}(Y_i)$ and $f_i = f\,|\,X_i$, where $Y_i \subset Curv_g(f)$ is a
stratum of the above stratification. Let
$$
\begin{picture}(800,60)
\unitlength0.2em
\thicklines
\put(88,1){$Y_i$}
\put(64,23){$X_i'$}
\put(108,23){$X_i$}
\put(83,24){$\vector(1,0){15}$}
\put(105,19){$\vector(-1,-1){12}$}
\put(72,19){$\vector(1,-1){12}$}
\put(70,10){$p_i$}
\put(89,29){$\nu_i$}
\end{picture}
$$
be a simultaneous normalization. Denote by $T_{Y_i} X_i' = $ Ker$\,dp_i$ the
relative tangent bundle of $p_i$; $p_i$ being a smooth family of curves,
$T_{Y_i} X_i'$ is a smooth line bundle on $X_i'$. Let $D_i \subset X_i'$ be the
locus of points where the restriction $d\nu_i\,|\,T_{Y_i} X_i'$ vanishes.
Since $D_i$ is Zariski closed its image $p_i(D_i) \subset Y_i$ is a constructible
subset of $Y_i$. Clearly, the complement $Y_i \setminus p_i(D_i)$ coincides
with $Imm_g(f) \cap Y_i$. Thus, the latter subset is constructible for all
$i= 1,\dots,n$. Hence, $Imm_g(f)$ is constructible, too. \hfill $\Box$
\bigskip
\noindent {\bf 2.14.} {\it Starting the proof of Theorem 2.1.} ($a$)
directly follows from Lemma 2.11($a$) and Corollary 2.13($b$). From
Corollary 2.13($b$) it also follows
that the total space ${\cal S}_{d,\,g}$ of the universal
family of curves ${\cal S}_{d,\,g} \to Imm_{d,\,g}$ is a quasi--projective variety.
The holomorphic mapping $f = f_{d,\,g}\,:\,{\cal M}_{d,\,g} \to {\cal S}_{d,\,g}$ which
realizes an analytic simultaneous normalization is finite and proper
(see Lemma 2.11($b$)). Therefore, by the Grauert--Remmert Theorem [Ha, B 3.2],
${\cal M}_{d,\,g}$ possesses a structure
of a quasi--projective variety, so that $f$ is a finite morphism of
quasi--projective varieties. Thus, $f$ yields an algebraic simultaneous
normalization of the universal family of curves over $Imm_{d,\,g}$. This
proves ($b$).
To prove the first part of (c) denote $T = Imm_{d,\,g},\,\,{\cal S}_T =
{\cal S}_{d,\,g},\,\,{\cal M}_T = {\cal M}_{d,\,g}$ and $ I \!\! P^2_T = I \!\! P^2 \times T$. There is
a natural embedding $\,i\,:\,{\cal S}_T \hookrightarrow I \!\! P^2_T$. Consider the
composition $\varphi:= i \circ f\,:\,{\cal M}_T \to I \!\! P^2_T$ and its relative square
$\varphi^{(2)} := \varphi^2_T \,:\,{\cal M}_T^2 \to ( I \!\! P^2_T)^2$, where ${\cal M}_T^2 := {\cal M}_T
\times_T {\cal M}_T$ and $( I \!\! P^2_T)^2 := I \!\! P^2_T\times_T I \!\! P^2_T$. Let ${\cal D}_T \subset
{\cal M}_T^2$ resp. $D_T \subset ( I \!\! P^2_T)^2$ be the diagonals. Clearly, $E:=
(\varphi^{(2)})^{-1}(D_T) \setminus
{\cal D}_T$ is a closed subvariety of ${\cal M}_T^2$, and the restriction
$\pi^{(2)}\,|\,E\,:\,E \to T$ of the projection $\pi^{(2)}\,:\,{\cal M}_T^2 \to T$
has finite fibres. Its fibre over a point $t \in T = Imm_{d,\,g}$ corresponds
to the multiple point divisor on the normalization $M_t$ of the immersed curve
$S_t \subset I \!\! P^2$. The restriction $\varphi^{(2)}\,|\,E\,:\,E \to D_T$ is a finite
morphism. The image ${\tilde E} := \varphi^{(2)}(E)$ is proper over $T$. Moreover,
the fibre $\tau^{-1}(t) \subset {\tilde E}$ over a point $t \in T$ under the
restriction to ${\tilde E}$ of the projection $\tau\,:\,D_T \to T$ corresponds
to the set of singular points of the curve $S_t$. Therefore, it
consists of ${\delta} = {d-1 \choose 2} - g$ points iff $S_t$
is a nodal curve. By Proposition 2.9, any irreducible component of $T$ contains
points which correspond to nodal curves. Thus, the finite morphism
$\tau\,:\,{\tilde E} \to T$
has degree ${\delta}$ over every such component, and so, the complement
$Imm_{d,\,g} \setminus Nod_{d,\,g} \subset T = Imm_{d,\,g}$ coincides with the
ramification divisor $R_{\tau}$ of $\tau$. Hence, $Nod_{d,\,g} \subset Imm_{d,\,g}$
is, indeed, a Zariski open subvariety. \hfill $\Box$
\bigskip
\noindent {\it Remark.} If $S_t$ is a nodal curve with
${\delta} = {d-1 \choose 2} - g$ nodes, then the fibre $p^{-1}(t)$ of the
above projection $p := \pi^{(2)}\,|\,E\,:\,E \to T$ consists of $2{\delta}$ points.
The latter holds true if $S_t$ has only ordinary singularities. Hence, the
subset $Imm_{d,\,g} \setminus Ord_{d,\,g}$ is contained in the ramification
divisor $R_p \subset T$ of $p$.
\section{Pl\"ucker conditions}
It is known [Au] that in general, the subset of
the rational Pl\"ucker curves is not Zariski open
in the space ${\cal R}_d$ of all the rational plane curves of a given
degree $d$, although it always contains a Zariski open subset of
${\cal R}_d$. Nevertheless, we will show that $PlNod_{d,\,g}$ is a Zariski
open subset of $Imm_{d,\,g}$ for $d \ge 2g-1$, which proves Theorem 2.1(c).
\bigskip
\noindent {\bf 3.1. Lemma.} {\it Let $C \subset I \!\! P^2$ be an irreducible nodal curve
of
degree $d$ with the normalization $M$, and let $g^2_d$ be
the linear system on $M$
of all line cuts of $C$. Then $C$ is a Pl\"ucker curve iff $g^2_d$ contains
no divisor $D$ of the form
$$(i) \,\,\,D = 4p_1 +\dots\,;\,\,\,\,\,\,or\,\,\,\,\,\,(ii)\,\,\,D = 3p_1 +
2p_2 +\dots\,;\,\,\,\,\,\,or\,\,\,\,\,\,(iii)
\,\,\,D = 2p_1 + 2p_2 + 2p_3+\dots\,,$$ where $p_i \in M$ are not necessarily
distinct.}
\bigskip
\noindent {\it Proof.} The system $g^2_d$ contains no divisor of type (i)
iff all the flexes of $C$
are ordinary, i.e. all the singular branches of $C^*$ are ordinary cusps.
Under this condition, at most two of the local branches of $C^*$ meet at
a point
iff $g^2_d$ does not contain any divisor of type (iii).
Furthermore, two branches of $C^*$
meet at a point and one of them is singular iff $g^2_d$ contains a
divisor $D$ as in (ii). Since
$C$ being nodal has no tacnode, $C^*$ has no one, too. Therefore, $C$ is a
Pl\"ucker curve iff $g^2_d$ does not contain any divisor $D$ as in (i)--(iii).
\hfill $\Box$
\bigskip
Recall the following notion (see e.g. [ACGH]).
\medskip
\noindent {\bf 3.2. Picard bundles}
\smallskip
Let $M$ be a smooth projective curve of genus $g$. The $d$--th symmetric
power $S^dM$ (which is a smooth manifold) might be regarded as the space of
degree
$d$ effective divisors on $M$. Let $J_d(M) = $Pic$\,^d(M)$ be the component
of the Picard group Pic$\,(M)$ which parametrizes the degree $d$ line
bundles on
$M$, and let $\phi_d \,:\,S^dM \to J_d(M)$ be the morphism sending a
degree $d$ effective divisor on $M$ into its linear equivalence class.
Chosing a base point $p_0 \in M$ we may identify $ J_d(M)$ with the
Jacobian variety
$J_0(M)$ and $\phi_d$ with the $d$-th Abel--Jacobi mapping.
By a theorem of Mattuck [Ma] (see also [ACGH, Ch.IV]) for $d \ge 2g -1$
the morphism $\phi_d$ is a submersion and moreover, it defines a projective
bundle (i.e. a projectivization of an algebraic vector bundle) with the
standard fibre $ I \!\! P^{d-g}$. This bundle is called {\it the $d$-th Picard
bundle of $M$}.
Given a smooth family $\pi\,:\,{\cal M} \to T$ of complete genus $g$
curves and given $d \ge 2g -1$,
there is the associated Picard bundle $\Phi_d \,:\,S^d{\cal M} \to {\cal J}_d({\cal M})$
of relative smooth schemes over $T$. Consider also the associated
grassmanian bundle
$Grass_{2,\,d-g}({\cal M}) \to {\cal J}_d({\cal M})$ which parametrizes the two--dimensional
linear series $g^2_d$ of degree $d$ on the fibres $M_t =
\pi^{-1}(t),\,t\in T$.
Let $\pi\,:\,{\cal M} \to T,\,\,T = Imm_{d,\,g}$, be the family constructed in 2.14
above. Then for each $t \in T$ there is the linear series
$g^2_d = g^2_d(t)$ on $M_t$ of the line
cuts of the plane curve $C_t = f(M_t) \subset I \!\! P^2$. This defines a regular
section $\sigma\,:\,T \to Grass_{2,\,d-g}({\cal M})$.
\bigskip
\noindent {\bf 3.3.} {\it Finishing up the proof of Theorem 2.1(c).} Let
$\pi\,:\,{\cal M} := {\cal M}_T \to T$ be the family as in 2.14, and let
$\Phi_d \,:\,S^d{\cal M} \to {\cal J}_d({\cal M})$
be the associated Picard bundle. Denote ${\cal D}^{(i)}$ resp.
${\cal D}^{(ii)},\,\,{\cal D}^{(iii)}$ the subvariety of $S^d{\cal M}$ which consists of the
degree $d$ effective divisors on the fibres $M_t$ of $\pi$ of the form
(i) resp. (ii), (iii) of Lemma 3.1. Set ${\cal D} = {\cal D}^{(i)} \cup {\cal D}^{(ii)}
\cup {\cal D}^{(iii)}$. Note that ${\cal D}$ is a closed subvariety of $S^d{\cal M}$ of
codim$\,_{S^d{\cal M}}{\cal D} \ge 3$ (and moreover,
codim$\,_{S^dM_t}{\cal D}_t \ge 3$ for each $t \in T$). Indeed, to be in ${\cal D}_t$
a divisor on $M_t$ must satisfy a system of three independent equations.
Let ${\cal Z} \subset Grass_{2,\,d-g}({\cal M}) \times S^d{\cal M}$ be the incidence relation.
Its fibre ${\cal Z}_t$ over a point $t \in T$ consists of all pairs $(L,\,v)$, where
$L$ is a two--plane in $ I \!\! P^{d-g}_j := \phi_d^{-1}(j),\,\,j \in J_d(M_t)$, and
$v \in I \!\! P^{d-g}_j$ is a point of $L$. Let $pr_1\,:\,{\cal Z} \to
Grass_{2,\,d-g}({\cal M}),\,\,pr_2\,:\,{\cal Z} \to S^d{\cal M}$ be the canonical projections,
and let $\sigma\,:\,
T \to Grass_{2,\,d-g}({\cal M})$ be the regular section as in (3.2) above.
Put
${\cal Z}_{{\cal D}} := pr_2^{-1}({\cal D}) \subset {\cal Z}$, ${\hat {\cal D}} := pr_1({\cal Z}_{{\cal D}}) \subset
Grass_{2,\,d-g}({\cal M})$ and $T' := \sigma^{-1}({\hat {\cal D}}) \subset T$.
Since the projection
$pr_1$ is proper, ${\hat {\cal D}} \subset Grass_{2,\,d-g}({\cal M})$, and therefore also
$T' \subset T$ are closed subvarieties of the corresponding varieties. Clearly,
$t \in T'$ iff
the linear series $g^2_d(t) = \sigma(t)$ on $M_t$ contains a divisor from ${\cal D}_t$.
Recall that $Nod_{d,\,g} = T \setminus R_{\tau}$, where $R_{\tau}$ is the
ramification divisor as in (2.14). By Lemma 3.1, we have that
$PlNod_{d,\,g} = T \setminus (R_{\tau} \cup T')$. By Proposition 2.9, any irreducible
component $I$ of $T = Imm_{d,\,g}$ contains a Pl\"ucker curve. Thus,
$T' \cap I$ is a proper subvariety of $I$; in particular,
codim$\,_{T}T' \ge 1$. Hence, $PlNod_{d,\,g}$ is, indeed, a Zariski open
subset
of $T = Imm_{d,\,g}$. This completes the proof of Theorem 2.1. \hfill $\Box$
\bigskip
Theorem 2.1 implies
\bigskip
\noindent {\bf 3.4. Corollary.} {\it Any irreducible plane curve $C^*$ of genus
$g$
and degree $n = 2(d + g - 1)$, where $d \ge 2g-1$, whose dual $C$ is an
immersed curve, is a specialization
of generic maximal cuspidal Pl\"ucker curves $C'^*$ of the same degree and
genus\footnote{i.e. $C'^*$ has the maximal number of cusps allowed by
Pl\"ucker's formulas.}. Hence, there is an epimorphism
$\pi_1( I \!\! P^{2*} \setminus C^*) \to \pi_1( I \!\! P^{2*} \setminus C'^*)$.
In particular, the former group is big (resp. non--amenable, non--almost
solvable, non--almost nilpotent) if the latter one is so.}
\bigskip
\noindent {\it Proof.} By the class formula [Na, 1.5.4], the dual of an
irreducible immersed plane
curve of degree $d$ and genus $g$ has degree $d^* = 2(g + d - 1)$. By
Theorem 2.1($a$) and ($b$),
there is the diagram
$$
\begin{picture}(800,60)
\unitlength0.2em
\thicklines
\put(88,23){${\cal M}_T$}
\put(64,1){$ I \!\! P^2_T$}
\put(108,1){$ I \!\! P^{2^*}_T$}
\put(86,2){$\longleftrightarrow$}
\put(87,18){$\vector(-1,-1){12}$}
\put(93,18){$\vector(1,-1){12}$}
\put(70,14){$f$}
\put(110,14){$f^*$}
\end{picture}
$$
where the morphism $f^*$ yields a simultaneous normalization of the dual
family, so that for each $t \in T = Imm_{d,\,g}$
the image $f^*(M_t) =S_t^*$ is the dual curve of the curve $S_t = f(M_t)$
(see e.g. [Na, 1.5.1]).
By ($c$), the subset $PlNod_{d,\,g}\subset T$ is Zariski open. The dual $S_t^*$,
where $t \in PlNod_{d,\,g}$, is a
maximal cuspidal curve of degree $d$ and genus $g$. Vice versa, any such curve
is the dual $S_t^*$ of a nodal Pl\"ucker curve $S_t,\,t \in PlNod_{d,\,g}$.
This yields the first assertion. The second one follows from a well known
theorem of Zariski (see [Zar, p.131, Thm.5] or [Di, 4.3.2]). As for the third
one, see (1.1) above. \hfill $\Box$
\section{Proof of Theorem 0.2}
\bigskip
The following lemma is a particular case of the Varchenko Equisingularity
Theorem [Va, Theorem 5.3].
\bigskip
\noindent {\bf 4.1. Lemma.} {\it Let $p\,:\,E \to B$ be a surjective morphism,
where $E$, $B$ are smooth connected quasi--projective varieties. Then
there exist a proper subvariety $A \subset B$ such that the restriction $p\,|\,(E
\setminus H)$, where
$H = p^{-1}(A)$, determines
a smooth locally trivial fibre bundle $p \,:\,E \setminus H \to B \setminus A$.}
\bigskip
Let ${\Delta}$ be a hypersurface in a complex manifold $E$, $e \in$
reg$\,{\Delta}$ be a smooth point of ${\Delta}$, and ${\omega}$ be a small
disc in $E$ centered at $e$ and transversal to ${\Delta}$.
By {\it a vanishing loop} of ${\Delta}$ at $e$
we mean a loop ${\delta}$ in $E \setminus {\Delta}$ consisting of a path $\alpha$ which
joins a base point $e_0 \in E \setminus {\Delta}$ with a point $e' \in {\omega} \setminus
{\Delta}$ and a loop $\beta$ in ${\omega} \setminus {\Delta}$ with the base point $e'$
(i.e. $e$ is in the interior of $\beta$ in $\omega$).
The next simple lemma is well known; for the sake of completeness
we give its proof.
\bigskip
\noindent {\bf 4.2. Lemma.} {\it Let, as before, ${\Delta}$ be a hypersurface in a
complex manifold $E$, and let ${\gamma}_0,\,{\gamma}_1\,:\,S^1 \to E \setminus {\Delta}$ be two loops
with
the base point $e_0 \in E \setminus {\Delta}$ joined in $E$ by a smooth homotopy ${\gamma}\,:\,
S^1 \times [0,\,1] \to E$ transversal to ${\Delta}$, such that the image $S =$
Im$\,{\gamma}$ meets ${\Delta}$ at the points $e_1,\dots,e_k \in {\rm reg}\,{\Delta}$. Then
${\gamma}_0$ is homotopic in $E \setminus {\Delta}$ to a product
${\gamma}_1{\delta}_{i_1}\dots {\delta}_{i_k}$, where $(i_1,\dots,i_k)$
is a permutation of $(1,\dots,k)$ and ${\delta}_i$ is a vanishing loop of ${\Delta}$ at
the point $e_i,\,\,i=1,\dots,k$. }
\bigskip
\noindent {\it Proof.} Slightly modifying the original homotopy and changing the
numeration of the intersection points $e_1,\,\dots,\,e_n \in {\rm
reg}\,{\Delta}$
we may assume that $e_i \in \gamma_{t_i} \cap {\Delta},\,\,i=1,\dots,k,$
correspond to different
values $0 < t_1 < \dots < t_n < 1$ of the parameter of homotopy
$t \in [0, 1]$.
If $s_i \in [0,\,1],\,\,0 < s_1 < t_1 < \dots < t_n < s_{n+1} < 1,$ and ${\bar
\gamma}_i =
\gamma_{s_i}\,:\,S^1 \to E \setminus {\Delta} , \, i=1,\dots,n+1$, then clearly
${\bar \gamma}_{i+1}^{-1}\cdot {\bar \gamma}_i \approx {\delta}_i$, i.e.
${\bar \gamma}_i \approx
{\bar \gamma}_{i+1}\cdot{\delta}_i$ in $E \setminus {\Delta}$,
where ${\delta}_i$ is a
vanishing
loop of ${\Delta}$ at the point $e_i$, and ${\bar\gamma}_1 \approx
{\gamma}_0,\,\,{\bar\gamma}_{n+1} \approx {\gamma}_1$. Thus, $\gamma_0 \approx
{\gamma}_1
{\delta}_n\cdot \dots \cdot {\delta}_1$ in
$E \setminus {\Delta}$, and the lemma follows. \hfill $\Box$
\bigskip
In the proof of Theorem 0.2 below we use the following proposition.
Actually, it follows from Lemma 1.5(C) in [No]. However, we give a
proof which is different from that in [No].
\bigskip
\noindent {\bf 4.3. Proposition.} {\it Let a morphism $p\,:\,E \to B$ of smooth
quasiprojective varieties be a smooth fibration over $B$ with
a connected generic fibre $F$ of positive dimension.
Let ${\Delta} \subset E$ be a Zariski closed
hypersurface which contains no entire fibre of $p$, i.e.
$p^{-1}(b) \not\subset {\Delta}$ for each $b \in B$. Then we have
the following exact sequence}:
$$\pi_1(F \setminus {\Delta}) \stackrel{i_*}{\rightarrow} \pi_1(E \setminus {\Delta})
\stackrel{p_*}{\rightarrow} \pi_1(B) \to {\bf 1}\,.$$
\noindent {\it Proof.} By Lemma 4.1,
there exist hypersurfaces $A \subset B$ and $D = H \cup {\Delta} \subset E$, where
$H := f^{-1}(A)$, such that $p\,|\,(E \setminus D) \,:\, E \setminus D \to B \setminus A$
is a
smooth fibration. In particular, $p\,|\,(E \setminus D)$ induces an epimorphism of
the fundamental groups. Since the same is also true for
the embedding $i\,:\,B \setminus A \hookrightarrow B$, and since
$p_* = i_* \circ (p\,|\,E \setminus D)_*$, the exactness at the third term follows.
It remains to prove that the homomorphism
$$i_*\,:\,\pi_1(F \setminus {\Delta}) \to {\rm Ker}\,p_* \subset \pi_1(E \setminus {\Delta})$$
is surjective.
Fix a generic fibre $F \not\subset D$ and base points
$e_0 \in F \setminus D$ and $b_0 = p(e_0) \in B \setminus A$. Let a
class $[{\gamma}_0] \in {\rm Ker}\,p_*$ be
represented by a loop ${\gamma}_0\,:\,S^1 \to E \setminus {\Delta}$ with the base point
$e_0$. We will show that ${\gamma}_0$ is homotopic in $E \setminus {\Delta}$ to a loop
${\gamma}'_0\,:\,S^1 \to F \setminus {\Delta}$ with the same base point.
The loop ${\bar {\gamma}}_0 := p \circ {\gamma}_0\,:\,S^1 \to B$ with the base point
$b_0 \in B$ is contractible. Let ${\bar {\gamma}} \,:\,S^1 \times [0;\,1] \to B$ be
a contraction to the constant loop ${\bar {\gamma}}_1 \equiv b_0$. Since
$p\,:\,E \to B$ is a fibration, there exists a covering homotopy
${\gamma}\,:\,S^1 \times [0;\,1] \to E$. Thus, we have
${\bar {\gamma}} = p \circ {\gamma}$ and
${\gamma}_1\,:\,S^1 \to F$.
Fix a stratification of $D = {\Delta} \cup H$ which satisfies the
Whitney condition A and
contains the regular part ${\rm reg}\,D$ of $D$ as an open stratum.
By Thom's Transversality Theorem, the homotopy ${\gamma}$ can be chosen
being transversal to the strata of this stratification, and therefore
such that its image
meets the divisor $D$ only in a finite number of its regular points. Let it
meet ${\Delta}$ at the points $e_1,\dots,e_k \in$ reg$\,({\Delta} \setminus H)$. We may also
assume that the loop ${\gamma}_1\,:\,S^1 \to F$ does not meet $D$; in
particular, $[{\gamma}_1] \in \pi_1 (F \setminus {\Delta};\,e_0)$. By Lemma 4.2, ${\gamma}_0$ is
homotopic in
$E \setminus {\Delta}$ to the product ${\gamma}_1{\delta}_{i_1}\dots {\delta}_{i_k}$, where
${\delta}_{i}$ is a vanishing loop of ${\Delta}$ at the point $e_i,\,\, i=1,\dots,k$.
Note that all the transversal discs to ${\Delta}$ in $E$ centered at $e_i$ are
homotopic (via the family of such discs). Hence, all the simple
positive local vanishing loops of ${\Delta}$ at $e_i$ are freely homotopic in
$E \setminus {\Delta}$. Therefore, performing further deformation of the vanishing
loops ${\delta}_{i},\,\, i=1,\dots,k$, and taking into account our assumptions
that dim$\,F > 0$ and ${\Delta}$ does not contain entirely a fibre of $p$,
we may suppose that
\smallskip
\noindent (i) for each $ \,\, i=1,\dots,k$ the fibre of $p$ through the point
$e_i$ is transversal to ${\Delta}$;
\smallskip
\noindent (ii) the loops ${\delta}_{i},\,\, i=1,\dots,k$, do not meet $H$, and
the corresponding local loops $\beta_i,\,\,i=1, \dots, k$, are contained
in the fibres of $p$.
\smallskip
\noindent Since $p^{-1}(A) \subset D$, we have that for each
$i = 1,\dots,k$ the projection ${\bar {\delta}}_i := p \circ {\delta}_i$ of the
loop ${\delta}_i$ does not
meet the hypersurface $A \subset B$. By the construction, the loops ${\bar
{\delta}}_i,\,\,i = 1,\dots,k,$ are
contractible in $B \setminus A$. Applying the covering homotopy theorem
to the smooth fibration $p\,:\,E \setminus D \to B \setminus A$ we may conclude that for
each $i = 1,\dots,k,$ the loop ${\delta}_i$ is homotopic in
$E \setminus D \subset E \setminus {\Delta}$ to a loop ${\delta}_i'\,:\,S^1 \to F \setminus D \subset F \setminus {\Delta}$.
Hence, ${\gamma}_0$ is homotopic in $E \setminus {\Delta}$ to the product ${\gamma}'_0 :=
{\gamma}_1{\delta}'_{i_1}\dots {\delta}'_{i_k}\,,\,\,{\gamma}'_0\,:\,S^1 \to F \setminus {\Delta}$, and we are
done. \hfill $\Box$
\bigskip
\noindent {\bf 4.4. Duality, discriminants and the Zariski embedding}
\medskip
The following construction was used, for instance, in [Zar, pp.307, 326]
and in [DoLib, sect.1, 3]. Let $M$ be an irreducible smooth projective
variety, and let $L \subset H^0(M,\,{\cal L})$ be a linear system of effective
divisors on $M$, where $\cal L$ is a linear bundle on $M$. It
defines a rational mapping $\Phi_L\,:\,M \to I \!\! P (L^*)$.
If $K \subset L$ is a linear subsystem, then the mapping
$\Phi_K\,:\,M \to I \!\! P (K^*)$ is composed of the mapping $\Phi_L$ followed by
the linear projection $\pi_{L,\,K}\,:\, I \!\! P (L^*) \to I \!\! P (K^*)$ which is dual
to the tautological embedding $\rho_{K,\,L}\,:\, I \!\! P (K) \hookrightarrow I \!\! P
(L)$.
Set $C_L := \Phi_L (M) \subset I \!\! P (L^*)$ and $C_K := \Phi_K (M)
\subset I \!\! P (K^*)$, so that $C_K = \pi_{L,\,K} (C_L)$. The dual variety
${\Delta}_L \subset I \!\! P(L)$
of $C_L \subset I \!\! P (L^*)$ is usually a hypersurface, which is called {\it the
discriminant hypersurface of the linear system $L$}. The embedding
$\rho_{K,\,L}$ yields the embedding of the discriminants
${\Delta}_K = I \!\! P (K) \cap {\Delta}_L \hookrightarrow {\Delta}_L$.
\smallskip
In particular, starting with a degree $d$ irreducible plane curve $C \subset I \!\! P^2$
with a normalization $M \to C$, denote by $K = g^2_d$ the linear system on $M$
of line cuts of $C$ and by $L = |g^2_d|$ the corresponding complete linear
system. Since $g^2_d$ and therefore, also $L$ are base point free, they define
morphisms $\Phi_K\,:\,M \to C \subset I \!\! P^2 = I \!\! P (K^*)$ resp.
$\Phi_L\,:\,M \to C_L := \Phi_L(M) \hookrightarrow I \!\! P (L^*)$, and
$C \subset I \!\! P^2$ is a projection of the curve $C_L \subset I \!\! P (L^*)$. Set $ I \!\! P^2_C :=
I \!\! P(K) \hookrightarrow I \!\! P(L)$. The discriminant
${\Delta}_L = C_L^*$ is, indeed, a projective hypersurface, and the dual curve
$C^* \subset I \!\! P^2_C$ is an irreducible component of the plane cut ${\Delta}_K = I \!\! P(K)
\cap {\Delta}_L$. The other irreducible components of ${\Delta}_K$ are special tangent
lines of $C^*$ dual to the cusps of $C$ (by {\it a cusp} we mean here a
singular point of a local irreducible analytic branch of $C$). We call these
tangent lines {\it artifacts} [DeZa1]. Thus, the plane cut $ I \!\! P(K) \cap {\Delta}_L$
of the discriminant hypersurface ${\Delta}_L$ is irreducible iff $C$ is an
immersed curve.
The embedding $ I \!\! P^{2*} \cong I \!\! P^2_C \hookrightarrow I \!\! P(L)$ which represents
$C^*$ as a plane cut of the discriminant hypersurface ${\Delta}_L$ is called
{\it the Zariski embedding} (see [Zar, pp.307, 326; DeZa1]).
By definition, the dual variety $C_L^* = {\Delta}_L$ consists of the points
$x \in I \!\! P(L)$ such that the dual hyperplane $x^* \subset I \!\! P_L$ cuts out of $C_L$
a non-reduced divisor on the normalization $M$ of $C_L$. If $x \in C_L$ is
a cusp, then, clearly, the dual hyperplane $x^*$ is an irreducible component
of ${\Delta}_L$. Thus, the discriminant ${\Delta}_L$ is irreducible iff $C_L$ was an
immersed curve. In particular, this is the case if $C =
\pi_{L,\,K} (C_L)$ is an immersed curve. Vice versa, if $C_L$ is an immersed
curve, then the same is true for its generic projection onto the plane. Or,
what is the same, if the discriminant ${\Delta}_L$ is irreducible, then its generic
plane section is irreducible, too.
The projectivization $ I \!\! P(L)$ of the complete linear system $L$ of degree $d$
divisors on $M$ coincides with a fibre of the Abel--Jacobi map
$\phi_d\,:\,S^dM \to J_d(M)$ (see (3.2)), so that $ I \!\! P^2_C$ is a plane in this
fibre. We still call the morphism $ I \!\! P^2_C \hookrightarrow S^dM$
{\it the Zariski embedding}. The hypersurface ${\Delta}_d \subset S^dM$ which consists of
the non-reduced degree $d$ effective divisors on $M$ is also called
{\it the discriminant hypersurface}. It is the image of the diagonal
hypersurfaces of the direct product $M^d$ via the Vieta map $M^d \to S^dM$.
Thus, ${\Delta}_L = I \!\! P(L) \cap {\Delta}_d$, where $ I \!\! P(L)$ has been identified with a
fibre $F_j := \phi_d^{-1}(j),\,j \in J_d(M)$, of $\phi_d$.
\bigskip
\noindent {\bf 4.5.} {\it Proof of Theorem 0.2.} By Corollary 3.4, we may
suppose that $C$ is a generic nodal Pl\"ucker curve of degree $d$ and
geometric genus $g$, where $d \ge 2g-1$. By Mattuck's Theorem (see (3.2)),
the $d$-th Picard bundle
$\phi_d\,:\,S^dM \to J_d(M)$, where $M$ is a normalization of $C$, is a
projective bundle with a generic fibre
$F \cong I \!\! P^{d - g}$. By (4.4), the dual curve $C^*$ can be identified
with the plane cut of the discriminant hypersurface ${\Delta}_d \subset S^dM$
by the plane $ I \!\! P^2_C$ via its Zariski embedding $ I \!\! P^2_C \hookrightarrow F_0
:= I \!\! P(L) \subset S^dM$, where $L = |g^2_d|$ and $g^2_d$ is the linear system
on $M$ of line cuts of $C$.
If the group $\pi_1 ( I \!\! P^2_C \setminus \Delta_d)$ is big for a generic plane
$ I \!\! P^2_C \subset F_0 \cong I \!\! P^{d - g}$, then by Zariski's Lefschetz type Theorem
[Zar, p.279; Di, 4.1.17], it is big for any such plane, so that
$ I \!\! P^2_C \subset F_0$ might be assumed being generic. Indeed, a section $S$ of the
discriminant hypersurface ${\Delta}_L = F_0 \cap {\Delta}_d$ by a generic plane $ I \!\! P \subset
F_0$ is an irreducible curve with the same normalization $M$ and with the dual
$S^* \subset I \!\! P^2$ an immersed curve of degree $d$ and genus $g$. Thus, we may
start with $C = S^*$ and obtain $C^* = S = I \!\! P^2_C \cap {\Delta}_d$. Note that
such a generic linear system $K = g^2_d \subset L$, where $ I \!\! P = I \!\! P(K)$, defines
a morphism $M \to I \!\! P^2$ such that its image coincides with $C = S^*$.
Since by Theorem 2.1(c), $PlNod_{d,\,g}$ is a Zariski open subset of
$Imm_{d,\,g}$, the curve $C=S^*$ obtained in this way is a nodal Pl\"ucker one.
Applying the Zariski Lefschetz type Theorem we get an isomorphism
$$\pi_1 (F_0 \setminus \Delta_d) \cong \pi_1 ( I \!\! P^2_C \setminus \Delta_d) \cong
\pi_1 ( I \!\! P^2 \setminus C^*)\,.$$
By Proposition 4.3, we have
the exact sequence $$\pi_1 (F_0 \setminus \Delta_d) \to \pi_1 (S^d M \setminus \Delta_d)
\to \pi_1 (J_d(M)) \cong Z \!\!\! Z^{2g} \to {\bf 1}\,.$$
It follows that $ \pi_1 ( I \!\! P^2 \setminus C^*)$ is a big group if $\pi_1 (S^d M \setminus
\Delta_d)$ is big (cf. (1.1)). But $\pi_1 (S^d M \setminus \Delta_d)$
is the braid group $B_{d,\,g}$ of $M$ with $d$ strings which is big (see
Lemma 1.2($b$)). This completes the proof. \hfill $\Box$
\bigskip
\noindent {\it Remark.} A presentation of the group $\pi_1( I \!\! P^2 \setminus C)$
for a generic maximal cuspidal curve $C \subset I \!\! P^2$ of genus 0 or 1
was found by Zariski [Zar, p. 307]; see also [Ka] for $g \le {d-1 \over 2}$,
where $d = {\rm deg}\,C^*$. The result of [Ka] is based on the statement in
[DoLib] that for $d \ge 2g-1$ the $d$--th
Abel--Jacobi mapping $\phi_d\,:\,S^dM \to J(M)$ restricted to the
complement of the discriminant hypersurface ${\Delta}_d \subset S^dM$ is a Serre
fibration, so that the long exact homotopy sequence is available. But the
indication given in [DoLib] does not seem to be sufficient
for the proof. Another proof of the exactness of the above sequence of
fundamental groups extended to the left by the term $\bf 1$ has been recently
obtained in [KuShi]. Once again, this leads to a presentation of
the group $\pi_1( I \!\! P^2 \setminus C)$.
\bigskip
\begin{center} {\LARGE References} \end{center}
{\footnotesize
\noindent [ACGH] E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris, {\sl
Geometry of algebraic curves. I}, N.Y. e.a.: Springer, 1985
\noindent [AC] E. Arbarello, M. Cornalba. {\sl Su una conjettura di Petri},
Comment. Math. Helv. 56 (1981), 1-38
\noindent [Au] A. B. Aure. {\sl Pl\"ucker conditions on plane rational curves},
Math. Scand. 55 (1984), 47--58, with {\sl Appendix} by S. A. Str\"omme,
ibid. 59--61
\noindent [BPVV] W. Barth, C. Peters, A. Van de Ven. {\sl Compact complex
surfaces},
N.Y. e.a.: Springer, 1984
\noindent [Be] D. Bennequin. {\sl Entrelacements et \'equations de Pfaff},
Ast\'erisque, 107--108 (1983), 87--161
\noindent [BinFl] J. Bingener, H. Flenner. {\sl On the fibres of analytic
mappings}, in: Complex Analysis and Geometry, V. Ancona and A. Silva, eds.,
N.Y.: Plenum Press, 1993, 45--101
\noindent [Bi] J.S. Birman, {\sl Braids, links, and mapping class groups},
Princeton Univ. Press, Princeton, NJ, 1974
\noindent [Bo] A. Borel. {\sl K\"ahlerian coset spaces of semisimple Lie groups},
Proc. Nat. Acad. Sci., USA, 40, No. 12 (1954), 1147--1154
\noindent [BoHC] A. Borel, Harish--Chandra. {\sl Arithmetic subgroups of
algebraic groups}, Ann. of Math. 75 (1962), 485--535
\noindent [CoZi] D. J. Collins, H. Zieschang. {\sl Combinatorial group theory and
fundamental groups}, In: Algebra VII, Encyclopaedia of Math. Sci. Vol. 58,
Berlin e.a.: Springer, 1993, 3--166
\noindent [DeZa1] G. Dethloff, M. Zaidenberg. {\sl Plane curves
with hyperbolic and C--hyperbolic complements},
Ann. Sci. Ecole Norm. Super. Pisa (to appear); Pr\'epublication de l'Institut
Fourier de Math\'ematiques, 299, Grenoble 1995, 44p.
\noindent [DeZa2] G. Dethloff, M. Zaidenberg. {\sl Examples of plane curves of low
degrees with hyperbolic and C--hyperbolic complements.} In:
Geometric Complex Analysis, J. Noguchi e.a. eds. World Scientific Publ. Co.,
Singapore 1996., 176--193
\noindent [DeOrZa] G. Dethloff, S. Orevkov, M. Zaidenberg.
{\sl Plane curves with a
big fundamental group of the complement.} Pr\'epublication de l'Institut
Fourier de Math\'ematiques, 354, Grenoble 1996, 26p.; E-print alg-geom/9607006
\noindent [Di] A. Dimca, {\sl Singularities and topology of hypersurfaces},
Berlin e.a.: Springer, 1992
\noindent [DoLib] I. Dolgachev, A. Libgober. {\sl On the fundamental group of the
complement to a discriminant variety}, In: Algebraic Geometry,
Lecture Notes in Math. 862, 1--25, N.Y. e.a.: Springer, 1981
\noindent [GoShVi] V. V. Gorbatsevich, O. V. Shvartsman, E. B. Vinberg.
{\sl Discrete subgroups of Lie groups.} In: Lie Groups and Lie Algebras II,
Encyclopaedia of Math. Sci. Vol. 21, Springer : NY e.a., 1995
\noindent [Ha] J. Harris. {\sl On the Severi problem}, Invent. Math. 84 (1986),
445--461
\noindent [Hart] R. Hartshorn. {\sl Algebraic geometry}, NY e.a.: Springer, 1977
\noindent [He] S. Helgason. {\sl Differential geometry, Lie groups and
symmetric spaces}, N.Y. e.a.: Academic Press, 1978
\noindent [Iv] N. V. Ivanov. {\sl Algebraic properties of the Teichm\"uller
modular group}, Soviet Math. Dokl. 29 (1984), No.2, 288--291
\noindent [Ka] J. Kaneko. {\sl On the fundamental group of the complement to
a maximal cuspidal plane curve}, Mem. Fac. Sci. Kyushu Univ. Ser. A. 39
(1985), 133-146
\noindent [Ko] K. Kodaira. {\sl A theorem of completeness of characteristic
systems for analytic families of compact submanifolds of complex manifolds},
Ann. Math. 75 (1962), 146--162
\noindent [Kos] J.-L. Koszul. {\sl Sur la forme hermitienne canonique des espaces
homog\`enes complexes}, Can. J. Math. 7 (1955), 562--576
\noindent [KuShi] Vik. S. Kulikov, I. Shimada. {\sl On the fundamental group of
complements to the dual hypersurfaces of projective curves.} Preprint
Max-Planck-Institute f\"ur Mathematik, MPI 96-32, Bonn, 1996, 1--15
\noindent [Lib] A. Libgober. {\sl Fundamental groups of the complements to plane
singular curves}, Proc. Sympos. in Pure Mathem. 46 (1987), 29--45
\noindent [Lin] V. Ja. Lin. {\sl Liouville coverings of complex spaces, and
amenable groups}, Math. USSR Sbornik, 60 (1988), 197--216
\noindent [LinZa] V. Ja. Lin, M. Zaidenberg. {\sl Liouville and Carath\'eodory
coverings in Riemannian and complex geometry.} Preprint Max-Planck-Institute
f\"ur Mathematik, MPI 96-110, Bonn, 1996, 1--20 (see in this volume)
\noindent [LySu] T. Lyons, D. Sullivan. {\sl Function theory, random paths,
and covering spaces}, J. Diff. Geom. 19 (1984), 299--323
\noindent [Ma] A. Mattuck. {\sl Picard bundles}, Illinois J. Math. 5 (1961),
550--564
\noindent [MC] J. McCarthy. {\sl A "Tits-alternative" for subgroups of surface
mapping class groups}, Trans. Amer. Math. Soc. 291 (1985), 583--612
\noindent [Mi] J. Milnor. {\sl Singular points of complex hypersurfaces},
Princeton, New Jersey : Princeton University Press, 1968
\noindent [MoTe] B. Moishezon, M. Teicher. {\sl Fundamental groups of complements
of branch curves as solvable groups}, Duke E-print alg--geom/9502015, 1995,
17p.
\noindent [Na] M. Namba. {\sl Geometry of projective algebraic curves},
N.Y. a.e.: Marcel Dekker, 1984
\noindent [No] M.V. Nori. {\sl Zariski's conjecture and related problems},
Ann. scient. Ec. Norm. Sup. 16 (1983), 305--344
\noindent [O] M. Oka. {\sl Symmetric plane curves with nodes and cusps}, J.
Math. Soc. Japan, 44, No. 3 (1992), 375--414
\noindent [OSh] A. Yu. Ol'shanskij, A. L. Shmel'kin. {\sl Infinite groups},
In: Algebra IV, Enciclopaedia of Math. Sci. 37, Berlin e.a.: Springer, 1993,
3--95
\noindent [Ra] M. S. Raghunathan. {\sl Discrete subgroups of Lie groups},
Berlin e.a.: Springer, 1972
\noindent [Se] F. Severi. {\sl Vorlesungen \"{u}ber algebraische Geometrie},
Leipzig: Teubner, 1921
\noindent [Sh1] G. B. Shabat. {\sl The complex structure of domains covering
algebraic surfaces}, Functional Analysis Appl. 11 (1977), 135--142
\noindent [Sh2] G. B. Shabat. {\sl On families of curves covered by bounded
symmetric domains}, Serdica Bulg. Math. Public. 11 (1985), 185--188
(in Russian)
\noindent [Ti] J. Tits. {\sl Free subgroups in linear groups}, J. Algebra, 20
(1972), 250--270
\noindent [Va] A.N. Varchenko. {\sl Theorems on the topological equisingularity
of families of algebraic varieties and families of polynomial mappings},
Math. USSR Izvestija, Vol. 6 (1972), No. 5, 957--1019
\noindent [Zar] O. Zariski. {\sl Collected Papers}. Vol III : {\sl Topology of
curves and surfaces, and special topics in the theory of algebraic
varieties}, Cambridge, Massachusets e. a.: The MIT Press, 1978
\bigskip
\noindent Gerd Dethloff,
Mathematisches Institut der Universit\"at G\"ot\-tin\-gen,
Bunsenstrasse 3-5,
37073 G\"ot\-tin\-gen,
Germany.
e-mail: [email protected]
\bigskip
\noindent Stepan Orevkov,
System Research Institute RAN,
Moscow, Avtozavodskaja 23, Russia.
e-mail: [email protected]
\bigskip
\noindent Mikhail Zaidenberg,
Universit\'{e} Grenoble I,
Institut Fourier et Laboratoire de Math\'ematiques
associ\'e au CNRS,
BP 74,
38402 St. Martin d'H\`{e}res--c\'edex,
France.
e-mail: [email protected]}
\end{document}
|
1995-01-18T06:20:53 | 9501 | alg-geom/9501009 | en | https://arxiv.org/abs/alg-geom/9501009 | [
"alg-geom",
"math.AG"
] | alg-geom/9501009 | Fantechi Barbara | Barbara Fantechi and Rita Pardini | On the Hilbert scheme of curves in higher-dimensional projective space | latex, 12 pages, no figures | null | null | UTM 448 | null | In this paper we prove that, for any $n\ge 3$, there exist infinitely many
$r\in \N$ and for each of them a smooth, connected curve $C_r$ in $\P^r$ such
that $C_r$ lies on exactly $n$ irreducible components of the Hilbert scheme
$\hilb(\P^r)$. This is proven by reducing the problem to an analogous statement
for the moduli of surfaces of general type.
| [
{
"version": "v1",
"created": "Tue, 17 Jan 1995 15:27:37 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Fantechi",
"Barbara",
""
],
[
"Pardini",
"Rita",
""
]
] | alg-geom | \section{Introduction}
It is well-known that the Hilbert scheme parametrizing
subschemes of
$\P^r$ can be singular at points corresponding to smooth curves as soon as
$r\ge 3$; actually Mumford \cite{Mu} gave an example of an everywhere singular
irreducible component. If $r=3$, it has been proven in \cite{EHM} that the
open subset of the Hilbert scheme parametrizing smooth curves in $\P^3$ with
given genus and degree can have arbitrarily many components when the genus and
the degree grow (in fact, they prove that no polynomial estimate on the number
of such components holds).
Our main result is the following:
\smallskip
\noindent
{\bf Theorem \ref{mainthm2}.}{\em\ Let $n\ge 3$ be an integer. Then there
exist infinitely many integers $r$, and for each of them a smooth, irreducible
curve
$C_r\subset \P^r$ such that $C_r$ lies exactly on $n$ components of the
Hilbert scheme of $\P^r$.}
\smallskip
The idea of the proof is very simple. Firstly, we modify a construction of
\cite{FP} to obtain a regular surface $S$ of general type which lies on $n$
components of the moduli space; secondly, we
consider a suitable pluricanonical embedding of this surface and intersect its
image with a high-degree hypersurface $F$ to construct the curve $C$ we are
interested in. Finally, we prove that all embedded deformations of $C$
are induced by embedded deformations of $F$ and $S$.
\par\noindent
{\em Acknowledgements}. We are grateful to Ciro
Ciliberto, who told us about this problem and suggested that we might apply
to it the results of \cite{FP}.
\section{Notation and preliminaries}
All varieties will be assumed smooth and projective over the complex numbers
unless the contrary is explicitly stated. A variety $Y$ will be called
regular if $H^1(Y,\O_Y)=0$. If ${\cal F}$ is a sheaf on $Y$, let $h^i(Y,{\cal F})=\dim
H^i(Y,{\cal F})$. If $t$ is a real number, we denote its integral part by $[t]$.
Let $\zeta_3=\exp(2\pi i/3)$.
In this paper we will be concerned with abelian covers of a very special
type; we collect here the necessary notational set-up.
Let $n$ be an integer $\ge 2$, and let $G={\bf Z}_3^n$, $G^*$ its dual; let
$e_1,\ldots,e_n$ be the canonical basis of $G$, and $\chi_1,\ldots,\chi_n$
the dual basis of $G^*$ (i.e., $\chi_j(e_i)=1$ if $i\ne j$ and
$\chi_j(e_j)=\zeta_3$). Let $e_0=-(e_1+\ldots+e_n)$. Let
$I=\{0,\ldots,n\}$,
and to each
$i\in I$ associate the pair
$(H_i,\psi_i)$ where
$H_i$ is the cyclic subgroup of $G$ generated by $e_i$, and $\psi_i\in
H_i^*$ is the character such that $\psi_i(e_i)=\zeta_3$.
Let
$Y$ be a smooth projective variety, and $(G,I)$ as above: a $(G,I)$-cover of
$Y$ is a normal variety $X$ and a Galois cover $f:X\to Y$ with
Galois group $G$ and (nonempty) branch divisors $D_i$ (for $i\in I$) having
$(H_i,\psi_i)$ as inertia group and induced character (see \cite{Pa} for
details).
\begin{lem}
To give a smooth $(G,I)$-cover of $Y$ is equivalent to giving
line bundles $L$ and $F_j$, for $j=1,\ldots,n$, together with smooth
nonempty divisors $D_i\in |M_i|$ {\rm(}where $M_0=L$ and, for $i\ge 1$,
$M_i=L-3F_i${\rm)} such that the union of the $D_i$'s has normal crossings.
\end{lem}
\begin{Pf}
{}From \cite{Pa} we know that the cover is determined by its reduced building
data, divisors $D_i$ for $i\in I$ and line bundles $L_j$ for $j=1,\ldots,n$
satisfying the relation $3L_j\equiv D_j+2D_0$. Letting $M_i=\O(D_i)$, and
putting $M_0=L$, $F_j=L-L_j$, the equations become precisely $M_j=L-3F_j$.
\end{Pf}
As the natural map
$\bigoplus_{i\in I}H_i\to G$ is
surjective, the covers we consider will be totally ramified. For $\chi\in
G^*$, let as usual $L_\chi^{-1}$ be the corresponding eigensheaf in the
direct sum decomposition of $f_*\O_X$; in the above notation, we will have
(for $\chi=\chi_1^{\alpha_1}\cdots\chi_n^{\alpha_n}$):
\setcounter{equation}{0}
\begin{equation}\label{lchi}
L_\chi=n_\chi L-\sum_{j=1}^n \alpha_jF_j,
\end{equation}
where $n_\chi=-[(-\alpha_1-\ldots-\alpha_n)/3]$. In particular note
that $n_\chi\ge 1$ when $\chi\ne 1$, and $n_\chi=1$ if and only if $1\le\sum
\alpha_j\le 3$. We will write $L_j$ instead of $L_{\chi_j}$.
Recall from \cite{Pa}, proof of proposition 4.2 on page 208, that
\begin{equation}\label{canonico}
3K_X=\pi^*(3K_Y+2(n+1)L-6\sum F_j).
\end{equation}
\smallskip
We now recall some results from \cite{FP} in a simplified form (fit for our
situation). For details and proofs see \cite{FP}, \S 5.
\begin{rem}\label{rema}{\rm (1) Let ${\cal Y}\to B$ be a
smooth projective morphism (with $B$ a smooth, connected quasiprojective
variety) together with an isomorphism between ${\cal Y}_o$ and $Y$ for some $o\in
B$, and assume that $Y$ is regular and that the morphism ${\cal Y}\to B$ has a
section $\sigma$. Let
$L$ be a line bundle on
$Y$; assume that $c_1(L)$ is kept fixed by the
monodromy action of
$\pi_1(B,o)$ on
$H^2(Y,{\bf Z})$. Then for each
$b\in B$ there is a canonical induced class
$c_1(L_b)$ on ${\cal Y}_b$. If, for all $b\in B$, the class $c_1(L_b)$ is of type
$(1,1)$, then
$L$ can be extended to a line bundle $\L$ over ${\cal Y}$, flat over $B$; this
extension is unique if we require that its restriction to $\sigma(B)$ be
trivial. This follows by applying the results on p.~20 of \cite{mumford}, and
by noting that the relative Picard scheme of ${\cal Y}$ over $B$
is \'etale over $B$ since all fibres are smooth and regular (it is surjective
as $c_1(L_b)$ is always of type $(1,1)$); the condition on the monodromy action
implies then that the component of the relative Picard scheme containing $[L]$
is in fact isomorphic to $B$. Let
$L_b$ be the restriction of
$\L$ to
${\cal Y}_b$.
\noindent (2) If $h^0({\cal Y}_b,L_b)$ is either constant in $b$, or if it only
assumes the values $1$ (for $b\in Z$) and $0$, then there is a
(nonunique) quasiprojective variety $W^L\to B$ such that $W^L_b$ is canonically
isomorphic to $H^0({\cal Y}_b,L_b)$; $W^L$ is smooth and irreducible in the former
case, while in the latter it is the union of one component isomorphic to $B$
and another being the total space of a line bundle over $Z$ (compare with
\cite{FP}, theorem 5.8 and remark 5.11). }\end{rem}
\begin{assu}\label{ass}{\rm
Let $S=\{({i,\chi})\in I\times G^*|\chi_{|H_i}\ne\psi_i^{-1}\}$.
Let $X\to Y$ be a smooth $(G,I)$-cover as in lemma 2.1, and
${\cal Y}\to B$ be a smooth projective morphism (with $(B,o)$ a pointed space, and
${\cal Y}_o$ isomorphic to $Y$), such that remark
\ref{rema}, (1) applies to ${\cal Y}\to B$, for the line bundle $L$ and for each of
the $F_j$'s. Assume moreover that remark \ref{rema}, (2) applies for the line
bundles
$M_i-L_\chi$ for $({i,\chi})\in S$, yielding varieties $W^{i,\chi}$: let $W$ be the
fibred product of the $W^{i,\chi}$ over $B$. Finally, assume that the germ of $B$
at $o$ maps smoothly to the base of the Kuranishi family of $Y$, and that the
cohomology groups $H^1(Y,L_\chi^{-1})$ and $H^1(Y,T_Y\otimes L_\chi^{-1})$
vanish
for each $\chi\in G^*\setminus 1$.
}\end{assu}
\begin{thm}\label{fromFP}
Assume that assumption {\rm\ref{ass}} holds, and let
$w\in W$ be a point over $o\in B$ corresponding to sections $s_{i,\chi}$ such that
$s_{i,\chi}=0$ if $\chi\ne 1$, and $s_{i,1}$ defines $D_i$ for $i=0,\ldots,n$.
Assume also that $X$ has ample canonical class. One can construct a family of
natural deformations of
$(G,I)$-covers
${\cal X}\to W$; the induced map from the germ of $w$ in $W$ to the Kuranishi family
of
$X$ is smooth {\rm(}and, in particular, surjective{\rm)}. Moreover, the flat,
projective morphism
${\cal X}\to W$ defines a rational map from $W$ to the moduli of surfaces with
ample canonical class, regular at $w$; this map is
dominant on each irreducible component of the moduli containing $X$.
\end{thm}
\begin{Pf}
Let $\L$ (resp.~${\cal F}_j$) be the line bundle induced by $L$
(resp.~$F_j$) on $W$; as we {\em define} ${\cal M}_0$ to be $\L$, $\L_j$ to be
$\L-{\cal F}_j$ and ${\cal M}_j$ to be $\L-3{\cal F}_j$ for $j=1,\ldots,n$, there are global,
canonical isomorphisms $\phi_j:3\L_j\to {\cal M}_j+2{\cal M}_0$.
By \cite{FP}, theorem 5.12, the germ of $W$ at $w$ maps smoothly to the
base of the Kuranishi family of $X$.
If $M$ is
an irreducible component of the moduli containing $[X]$, by the previous result
the image of $W$ contains an open set in $M$ (in the strong topology), hence it
cannot be contained in a closed subset (in the Zariski topology) and is
therefore dominant.
\end{Pf}
\section{Moduli of surfaces of general type}
The aim of this section is the proof of theorem \ref{mainthm1}, i.e., the
explicit construction of regular surfaces with ample canonical class lying on
arbitrarily many components of the moduli. This construction can be carried
out in a much more general setting (see remark \ref{+gen}); we consider only
the case needed for our applications, since it is easier to describe.
For $S$ a smooth projective surface and
$\xto.n$ pairwise distinct points of $S$, we let $B\ell(S;\xto.n)$ denote the
surface obtained by blowing up $S$ at $\xto.n$.
\begin{constr}{\rm Let $S$ be a regular surface, $x_0\in S$, $n$ a positive
integer; let
$B=B(S,n)$ be the variety parametrizing data $(\xto.n,\yto.n)$ where
the $x_i$'s are pairwise distinct points in $S$ (for $i=0,\ldots,n$), the
$y_i$'s
are pairwise distinct points in $B\ell(S;\xto.n)$, such that $y_i$ is not
infinitely near to $x_j$ for $i\ne j\ge 1$ and none of the $y_i$'s lies
over $x_0$.
$B$ is a smooth quasiprojective variety, which is naturally isomorphic to an
open subset of the product of $n$ copies of
$S\times S$ blown up along the diagonal. Let ${\cal Y}\to B$ be the smooth
projective family such that ${\cal Y}_b$, the fibre of ${\cal Y}$ over the point $b$, is
isomorphic to
$B\ell(B\ell(S;\xto.n);\yto.n)$ for $b=(\xto.n,\yto.n)$.}
\end{constr}
Note that the morphism ${\cal Y}\to B$ has a section, given by mapping $b\in B$ to
the inverse image of $x_0$ in ${\cal Y}_b$.
\begin{lem} Assume that
$S$ is rigid. Let $B^0$ be the open set in $B$ where
$Aut(S)$ acts freely (the action being the natural one). Then if $b\in B^0$,
the natural map from the germ of $B$ in $b$ to the Kuranishi family of ${\cal Y}_b$
is smooth of relative dimension $h^0(S,T_S)$.
\end{lem}
\begin{Pf} The proof is easy and left to the reader.
\end{Pf}
\begin{rem}{\rm For any $b\in B$, $b=(\xto.n,\yto.n)$, there is a canonical
isomorphism
$$N\!S({\cal Y}_b)=N\!S(S)\oplus{\bf Z} e'_1\oplus\ldots\oplus{\bf Z} e_n'\oplus{\bf Z} e''_1\oplus
\ldots\oplus{\bf Z} e_n'',$$
where $e_i'$ is the pullback from $B\ell(S;\xto.n)$ of the class of the
exceptional divisor over $x_i$, and $e_i''$ is the class of the exceptional
divisor over $y_i$. We will consider this isomorphism fixed, and denote this
group by $N\!S$. We also let $f_i$ denote $e_i'-e_i''$.
Since $S$ is regular, so are all the ${\cal Y}_b$'s and
we will not need to distinguish between line bundles and their Chern
classes.
}\end{rem}
\begin{defn}{\rm Let $L\in N\!S$, $G={\bf Z}_3^n$ as in \S
2; for $\chi\in G^*$, let $L_\chi\in N\!S$ be defined by equation (\ref{lchi}),
with $F_i=f_i$. Let $B_{L}$ be the open subset
of $B$ consisting of the $b$'s such that
\begin{enumerate} \item the cohomology groups $H^1({\cal Y}_b,L_\chi^{-1})$,
$H^1({\cal Y}_b,T_{{\cal Y}_b}\otimes L_\chi^{-1})$ are zero for each $\chi\in G^*\setminus
1$;
\item the line bundles $L$ and $L-3F_j$ are very ample on ${\cal Y}_b$,
for $j=1,\ldots,n$;
\item the line bundles $L-K_{{\cal Y}_b}$ and $L-3F_j-K_{{\cal Y}_b}$ are ample on ${\cal Y}_b$,
for $j=1,\ldots,n$;
\item the line bundle $3K_Y+2(n+1)L-6\sum F_j$ is ample on ${\cal Y}_b$.
\end{enumerate}}
\end{defn}
Note that the first condition is needed to ensure that assumption \ref{ass}
is satisfied; the second allows one to choose smooth divisors in the linear
systems $|L|$ and $|L-3F_j|$ meeting transversally; the third implies that
these linear systems have constant dimension when $b$ varies; and
the fourth ensures, in view of equation (\ref{canonico}), that the cover so
obtained has ample canonical class (recall that the pullback of an ample line
bundle via a finite map is again ample).
\begin{lem}\label{basic}
Let $Y$ be a smooth surface containing $m$
disjoint irreducible curves $C_1,\ldots,C_m$, such that $C_i^2<0$. Then:
\begin{enumerate}
\item for any choice of nonnegative integers $a_1,\ldots,a_m$,
the linear system $|a_1C_1+\ldots+a_mC_m|$ contains only the divisor
$a_1C_1+\ldots+a_mC_m$;
\item for any choice of nonnegative integers $a_1,\ldots,a_{m-1}$, and for
any $b>0$, the linear system $|a_1C_1+\ldots+a_{m-1}C_{m-1}-bC_m|$ is empty.
\end{enumerate}\end{lem}
\begin{Pf} (1) We prove the theorem by induction on $a_1+\ldots+a_m$, the
case where this sum is zero being trivial. Assume without loss of generality
that $a_1\ge 1$, and let $C\in |a_1C_1+\ldots+a_mC_m|$; then $C\cdot
C_1=a_1C_1^2<0$, hence $C$ must have a common component with $C_1$; therefore
$C=C_1+C'$, $C'\in |(a_1-1)C_1+\ldots+a_mC_m|$, and by induction the proof is
complete.\par
\noindent
(2) Assume that there exists $C\in |a_1C_1+\ldots+a_{m-1}C_{m-1}-bC_m|$. Then
$C+bC_m\in |a_1C_1+\ldots+a_{m-1}C_{m-1}|$, contradicting (1).
\end{Pf}
\begin{cor} Let $b\in B$, $b=(\xto.n,\yto.n)$ and let $a_1,\ldots,a_m$ be
nonnegative integers. Then the line bundle
$a_1f_1+\ldots+a_mf_m$ on ${\cal Y}_b$ is effective if and only if $y_i$ is
infinitely near to
$x_i$ for every $i$ such that $a_i>0$, and in this case it has only one
section.
\end{cor}
\begin{Pf} If $y_i$ is infinitely near to $x_i$, then $f_i$ is a $(-2)$
curve; otherwise it is the difference of two disjoint $(-1)$ curves. In the
former case lemma \ref{basic}, (1) applies and in the latter case
\ref{basic}, (2) applies.
\end{Pf}
\begin{nota}{\rm We will denote by $E$ the closed subset of $B$
consisting of the points $b$ such that $f_i$ is effective on ${\cal Y}_b$ for
$i=1,\ldots,n$.}
\end{nota}
\begin{lem} Let $L\in NS$ be a line bundle and assume that $E\cap
B_L\ne\emptyset$. Then assumption {\rm\ref{ass}} holds for the restriction of
${\cal Y}\to B$ to $B_L$; applying theorem {\rm\ref{fromFP}} yields a quasiprojective
variety $W$. In this case
$W$ is the union of
$2^n$ smooth irreducible components
$W_A$, indexed by subsets
$A\subset
\{1,\ldots,n\}$. The dimension of $W_A$ and $W_{A'}$ are equal if $\#A=\#A'$,
and in particular one has:
$$\dim W_A-\dim W_{\emptyset}=\frac{1}{6}(\#A^3+6\#A^2-\#A).$$
The $W_A$'s have a nonempty intersection.
\end{lem}
\begin{Pf}
The verification that assumption \ref{ass} holds is easy and we leave it to
the reader. For $A\subset \{1,\ldots,n\}$, let $E_A=\{b\in B_L|f_i\ \mbox{is
effective for}\ i\in A\}$ and let $W_A\subset W$ be defined by
$$W_A=\{(b,s_{{i,\chi}})|b\in E_A\ \mbox{and}\ s_{{i,\chi}}=0\ \mbox{for}\ \chi \ne
1\ \mbox{and}\ i\notin A\}.$$ It is easy to check that $W_A$ is smooth over
$E_A$ of dimension $1/6(\#A^3+6\#A^2+5\#A)$; on the other hand $E_A$ is
smooth of codimension $\#A$ in $B_L$. Finally, $W$ is the union of the
$W_A$'s, which are easily seen to be irreducible components.
Moreover, the intersection of the $W_A$'s is clearly equal to
$W_\emptyset$ intersected with the inverse image of $E\cap B_L$.
\end{Pf}
\begin{thm}\label{mainthm1}
Let $L\in N\!S$ and $b\in E\cap B_L\cap B^0$. Let
$f:X\to Y={\cal Y}_b$ be a smooth
$(G,I)$-cover with building data $(D_i,L_\chi)$; let $w\in W_b$ be a point
corresponding to a choice of equations $s_i\in \O_Y(D_i)$ defining $D_i$,
with $s_{{i,\chi}}=0$ for all $\chi\ne 1$. Then the natural map from the germ of
$W$ in $w$ to the Kuranishi family of $X$ is smooth. In particular, the
Kuranishi family of $X$ is the union of $2^n$ irreducible components,
$n+1$ of which have pairwise different dimension.
Moreover, the surface $X$ lies on exactly $n+1$ components of the
moduli space, having pairwise different dimensions.
\end{thm}
\begin{Pf}
The first statement is a straightforward application of theorem \ref{fromFP},
in view of the previous lemma.
To prove the second, note that the map from $W$ to the moduli space factors
through the action of the symmetric group on $n$ letters, $\Sigma_n$. The
quotient $W/\Sigma_n$ has exactly $n+1$ irreducible components of pairwise
different dimensions. By the previous result, each of these components
dominates a component of the moduli; the $n+1$ components so obtained must
all be distinct, as they have different dimensions.
\end{Pf}
\begin{rem}\label{suffample}
{\rm
If $L\in NS$ is sufficiently ample, then the intersection $E\cap B_L\cap B^0$
is nonempty, hence the theorem applies yielding infinitely many surfaces with
different invariants. }
\end{rem}
\begin{cor}\label{HilbX}
Given integers $n\ge 2$ and $m\ge 5$, for infinitely
many values of $r$ there exists a smooth, regular surface $X$ in $\P^r$
such that $\O_X(1)=mK_X$ and $X$ lies on exactly $n+1$ irreducible components
of the Hilbert scheme.
\end{cor}
\begin{Pf}
Let $X$ be a regular surface,
with ample $K_X$, lying on exactly $n+1$ irreducible components of the moduli;
let $M$ be the union of the irreducible components of the moduli space of
surfaces with ample canonical class which contain $[X]$. Let
$r=h^0(X,mK_X)-1$; infinitely many such $X$'s (with distinct values of $r$) can
be constructed by applying theorem \ref{mainthm1}, in view of remark
\ref{suffample}.
Fix an $m$-canonical embedding of $X$ in $\P^r$. Every small embedded
deformation of $X$ in $\P^r$ is again a smooth surface, $m$-canonically
embedded
as
$X$ is regular. Let $H$ be the union of the irreducible components of the
Hilbert
scheme of $\P^r$ containing $[X]$, and $H^0$ be the open dense subset of $H$
parametrizing smooth, $m$-canonically embedded surfaces.
The natural map $H^0\to M$ is dominant, and each fibre is irreducible of
dimension $(r+1)^2-1$; in fact, the fibre over $[X']$ is the set of
bases of $H^0(X',mK_{X'})$ modulo the action of the finite group $Aut(X')$ (and
modulo the obvious ${\bf C}^*$-action).
In particular there is an induced bijection between irreducible
components of $M$ and of $H$, which increases the dimension by $(r+1)^2-1$.
\end{Pf}
\begin{rem}\label{+gen}
{\rm The constructions in this section generalize easily to the case where $Y$
is neither regular nor rigid and $G$ is any abelian group. In fact, they also
work if the dimension of $Y$ is bigger than $2$ (using a suitable, modified
form of lemma \ref{basic}).}
\end{rem}
\section{The Hilbert scheme of curves in $\P^r$}
In this section we apply the results on the Hilbert scheme of surfaces to the
Hilbert scheme of curves. We first introduce some notation. If $Z\subset
\P^r$ is a subscheme, we will denote by $\hbox{\rm Hilb}(Z)$ the union of the
irreducible components of the Hilbert scheme of $\P^r$ containing $[Z]$; we
let $H(Z)$ be the germ of $\hbox{\rm Hilb}(Z)$ at $[Z]$.
Note that if $F$ is a hypersurface of degree $l$, $\hbox{\rm Hilb}(F)$ is naturally
isomorphic to $\P(H^0(\P^r,\O(l)))$.
\begin{lem} \label{splitting}
Let $X\subset \P^r=\P$ be a smooth surface,
$F\subset
\P$ be a smooth hypersurface of degree $l$ transversal to $X$, and let
$C=X\cap
F$. Then there is a natural isomorphism
$N_{C/\P}|_C\cong N_{X/\P}|_C\oplus \O_C(l)$.
\end{lem}
\def\phantom{.}\\{\phantom{.}\\}
\begin{Pf} We have a natural diagram: $$
\begin{array}{ccccccccc}
&&&& 0 &&&&\\
&&&& \downarrow &&&&\\
\phantom{.}\\
&\phantom{\sum_I^J}&&& N_{C/F} &&&&\\
\phantom{.}\\
&\phantom{\sum_I^J}&&& \downarrow &\searrow&&&\\
\phantom{.}\\
0&\longrightarrow
&N_{C/X}&\longrightarrow&\phantom{\sum_I^J}N_{C/\P}\phantom{\sum_I^J}&\longrightarrow&N_{X/\P}|_{C}&\longrightarrow&0\\
\phantom{.}\\
&\phantom{\sum_I^J}&&\searrow& \downarrow &&&&\\
\phantom{.}\\
&\phantom{\sum_I^J}&&& N_{F/\P}|_{C} &&&&\\
\phantom{.}\\
&&&& \downarrow &&&&\\
&&&& 0 &&&&
\end{array}$$
However $N_{F/\P}|_{C}=N_{C/X}=\O_C(l)$, and it is easy
to check that the morphism $N_{C/X}\to N_{F/\P}|_{C}$ in the diagram is
the identity. This implies that the natural map $N_{C/F}\to N_{X/\P}|_{C}$ is
also an isomorphism, hence the claimed splitting.
\end{Pf}
\begin{prop}\label{smooth}
Let $X\subset \P^r=\P$ be a smooth,
regular, projectively normal surface. Let $F$ be a
smooth hypersurface of degree $l$ in $\P$ cutting
$X$ transversally along a curve $C$, and let $U\subset \hbox{\rm Hilb}(X)\times
\hbox{\rm Hilb}(F)$ be the open set of pairs $({X'},{F'})$ such that ${X'}$ and
${F'}$ are smooth and transversal and ${X'}$ is projectively normal.
If $l>>0$, then for every $({X'},{F'})\in U$ the map
$H({X'})\times H({F'})\to H({C'})$ induced by intersection is smooth, where
${C'}={X'}\cap{F'}$.
\end{prop}
\begin{Pf} The germ of the Hilbert scheme $H(Z)$ represents the functor of
embedded deformations of $Z$ in $\P$; when $Z$ is smooth, this functor has
tangent (resp.~obstruction) space $H^0(Z,N_{Z/\P})$ (resp.\
$H^1(Z,N_{Z/\P})$). Let $({X'},{F'})\in U$, and ${C'}={X'}\cap {F'}$. The
map $H({X'})\times H({F'})\to H({C'})$
induces natural maps on tangent and obstruction spaces; to prove the
required smoothness it is enough to prove that the induced maps are
surjective on tangent spaces and injective on obstruction spaces.
Note that $H^i({C'},N_{{C'}/\P})=
H^i({C'},N_{{X'}/\P}|_{{C'}})\oplus H^i({C'},N_{{C'}/\P}|_{{C'}})$ by
lemma \ref{splitting}. Via this isomorphism, the maps we are interested in are
induced by the long exact sequences associated to:$$
\begin{array}{ccccccccc}
0&\to &N_{{X'}/\P}\otimes{\cal I}_{{C'}\subset
{X'}}&\longrightarrow&N_{{X'}/\P}& \longrightarrow
N_{{X'}/\P}|_{{C'}}&\to&0\\ \phantom{1}\\
0&\to &N_{{F'}/\P}\otimes{\cal I}_{{C'}\subset
{F'}}&\longrightarrow&N_{{F'}/\P}& \longrightarrow
N_{{F'}/\P}|_{{C'}}&\to&0. \end{array}$$
Therefore it is enough to prove that
$$H^1({X'},N_{{X'}/\P}\otimes{\cal I}_{{C'}\subset {X'}})=0$$ and that
$H^0({F'},N_{{F'}/\P})\to H^0(C,N_{{F'}/\P}|_{{C'}})$ is surjective
(remark that $N_{{F'}/\P}=\O_{F'}(l)$, hence $H^1({F'},N_{{F'}/\P})=0$ by
Kodaira vanishing).
For the claimed surjectivity, note that there is a commutative diagram $$
\begin{array}{ccc}
H^0(\P,\O_{\P}(l))&\longrightarrow &H^0({X'},\O_{{X'}}(l))\cr
\downarrow&\phantom{{2 choose 3}}&\downarrow\cr
H^0({F'},\O_{{F'}}(l))&\longrightarrow &H^0({C'},\O_{{C'}}(l))
\end{array}
$$
As ${X'}$ is projectively normal, the upper horizontal arrow is onto, and as
${X'}$ is regular, the right vertical arrow is onto. Hence the lower
horizontal arrow is also onto.
To prove the vanishing, as ${\cal I}_{{C'}\subset {X'}}=\O_{X'}(-l)$, it is enough
to prove that
$H^1({X'},N_{{X'}/\P}(-l))=0$ if $l$ is sufficiently large. For any given
${X'}$, this follows immediately from the definition of ampleness; on the
other hand it is easy to prove (by a standard semicontinuity argument) that in
fact an $l_0$ can be found such that the claimed vanishing holds for all $l\ge
l_0$ and for all ${X'}\in \hbox{\rm Hilb}(X)$.
\end{Pf}
\begin{prop}\label{inj}
Let $X\subset \P^r=\P$ be a smooth surface
and let $F$ be a smooth hypersurface of degree $l$ meeting $X$ transversally
in a smooth curve $C$. Let $U\subset \hbox{\rm Hilb}(X) \times \hbox{\rm Hilb}(F)$ be the open set
of pairs $({X'},{F'})$ such that ${X'} \cap {F'}$ is a smooth curve. If
$l>>0$, then each fibre of the map $U\to \hbox{\rm Hilb}(C)$ given
by $({X'},{F'}) \mapsto {X'}\cap {F'}$ is contained in a fibre of the
projection $U\to\hbox{\rm Hilb}(X)$. In other words, each curve contained in the image
of $U$ in
$\hbox{\rm Hilb}(C)$ lies on exactly one surface in $\hbox{\rm Hilb}(X)$.
\end{prop}
\begin{Pf}
Let ${\cal X}\to
\hbox{\rm Hilb}(X)$ be the universal family. Inside the product ${\cal X} \times {\cal X}$,
consider the diagonal subvariety ${\cal I}$ consisting of the pairs $(x,x)$.
Let $W$ be the locus of $\hbox{\rm Hilb}(X)\times \hbox{\rm Hilb}(X)$ over which the map ${\cal
I}\to \hbox{\rm Hilb}(X)\times \hbox{\rm Hilb}(X)$ has one-dimensional fibres. One may choose a
stratification $\{W_j\}$ of $W$ such that each of the restricted
families ${\cal I}_j\to W_j$ is flat. Thus, one has induced maps from $W_j$ to the
Hilbert scheme of one-dimensional subschemes of $\P$. Since the union of the
images of the $W_j$'s is contained in a finite number of components of the
Hibert scheme, the degree of the curves contained in the intersection of two
distinct surfaces of $\hbox{\rm Hilb}(X)$ is bounded by an integer $l_0$. Therefore it
is enough to choose $l>l_0$.
\end{Pf}
\begin{thm}\label{mainthm2}
Let $n\ge 3$ be an integer. Then there exist
infinitely many integers $r$, and for each of them a smooth, irreducible curve
$C_r\subset \P^r$ such that $C_r$ lies exactly on $n$ components of the
Hilbert scheme of $\P^r$.
\end{thm}
\begin{Pf}
By corollary \ref{HilbX}, for infinitely many values of $r$ one can construct
a regular surface $X$ of general type, embedded in $\P^r$ by a complete
$m$-canonical system, such that $X$ lies
on exactly
$n$ components of the Hilbert scheme of $\P^r$, having pairwise different
dimensions. By \cite{And}, $X$ is projectively normal in $\P^r$ if
$m>>0$: in fact, by the theorem on page 362 together with the fact that if
$K_X$ is ample then $5K_X$ is very ample, it is enough to assume
$m\ge 11$. Choose an integer
$l$ with
$l>>0$, such that propositions
\ref{smooth} and \ref{inj} hold for $l$.
Let $F$ be a smooth hypersurface of degree $l$ meeting $X$ transversally.
Let $U\subset \hbox{\rm Hilb}(X)\times \hbox{\rm Hilb}(F)$ be the locus of pairs $({X'},{F'})$
where both are smooth and meeting transversally, and ${X'}$ is projectively
normal. $U$ is the union of
$n$ irreducible components of pairwise different dimensions, each of them being
the inverse image of an irreducible component of $\hbox{\rm Hilb}(X)$.
Let now $C=C_r$ be the intersection of $X$ and $F$, and consider the natural
map $U\to \hbox{\rm Hilb}(C)$ given by $({X'},{F'})\mapsto {X'}\cap {F'}$. By
proposition
\ref{smooth} this morphism is dominant and smooth on its image $V$. By
\ref{inj} there is an induced morphism $V\to \hbox{\rm Hilb}(X)$, which is also
dominant and smooth on its image. Therefore there is a natural
bijection between the irreducible components of $\hbox{\rm Hilb}(X)$ and those of
$\hbox{\rm Hilb}(C)$.
\end{Pf}
|
1995-09-15T05:57:21 | 9501 | alg-geom/9501001 | en | https://arxiv.org/abs/alg-geom/9501001 | [
"alg-geom",
"dg-ga",
"math.AG",
"math.DG"
] | alg-geom/9501001 | Misha Verbitsky | Misha Verbitsky | Cohomology of compact hyperkaehler manifolds | 87 pages LaTeX 2.09 | null | null | null | null | Let M be a compact simply connected hyperk\"ahler (or holomorphically
symplectic) manifold, \dim H^2(M)=n. Assume that M is not a product of
hyperkaehler manifolds. We prove that the Lie algebra so(n-3,3) acts by
automorphisms on the cohomology ring H^*(M). Under this action, the space
H^2(M) is isomorphic to the fundamental representation of so(n-3,3). Let A^r be
the subring of H^*(M) generated by H^2(M). We construct an action of the Lie
algebra so(n-2,4) on the space A, which preserves A^r. The space A^r is an
irreducible representation of so(n-2,4). This makes it possible to compute the
ring A^r explicitely.
| [
{
"version": "v1",
"created": "Tue, 3 Jan 1995 02:00:13 GMT"
},
{
"version": "v2",
"created": "Sat, 13 May 1995 21:49:21 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Verbitsky",
"Misha",
""
]
] | alg-geom | \section{Introduction.}\label{_introduction_to_so(b_2,...)_Section_}
Here we give a brief introduction to results of this
paper. The subsequent sections are independent from the
introduction.
The main object of this paper is theory of compact hyperk\"ahler manifolds.
A hyperk\"ahler manifold (see \ref{_hyperkaehler_manifold_Definition_}
for more precise wording) is a Riemannian manifold $M$ equipped
with three complex structures $I$,
$J$ and $K$, such that $I\circ J=-J\circ I=K$ and $M$
is K\"ahler with respect to the complex structures $I$,
$J$ and $K$.
Let $M$ be a complex manifold which admits a hyperk\"ahler
structure. A simple linear-algebraic argument implies that
$M$ is equipped with a holomorphic symplectic form. Calabi-Yau theorem
shows that, conversely, every compact holomorphically symplectic
K\"ahler manifold admits a hyperk\"ahler structure, which is uniquely
defined by these data.
Further on, we do not discriminate between compact holomorphic
symplectic manifolds of K\"ahler type and compact hyperk\"ahler
manifolds.
Let $(M,I)$ be a compact K\"ahler manifold which admits a holomorphic
symplectic form $\Omega$. For simplicity of statements, we assume
in this introduction that $\Omega$ is unique up to a constant.
Denote by $M$ the $C^\infty$-manifold underlying $(M,I)$.
Let $(M,I')$ be another compact holomorphically symplectic
manifold which lies in the same deformation class as $(M,I)$.
Fixing a diffeomorphism of underlying $C^\infty$-manifolds,
we may identify the smooth manifold which underlies $(M,I)$
with that underlying $(M,I')$.
Let $X$ be an arbitrary compact K\"ahler manifold,
$\dim_{\Bbb C} X=n$. We associate with a K\"ahler structure on $X$
so-called Riemann-Hodge pairing
\[
(\cdot,\cdot): \;\; H^2(X,{\Bbb R})\times H^2(X,{\Bbb R})\longrightarrow {\Bbb R}
\]
which is a map associating a number
\begin{equation}\label{_Hodge-Riemann-correct-Equation_}
\int_X \omega^{n-2}\wedge \eta_1\wedge\eta_2 -
\frac{n-2}{(n-1)^2}\cdot
\frac{\int_X \omega^{n-1}\eta_1 \cdot \int_X\omega^{n-1}\eta_2}
{\int_X \omega^n}
\end{equation}
to classes $\eta_1,\eta_2 \in H^2(X,{\Bbb R})$, where
$\omega\in H^2(X,{\Bbb R})$ is the K\"ahler class
(see also \ref{_Hodge_Riema_general_Claim_}).
Consider the positively defined scalar product
$(\cdot,\cdot)_{Metr}$ induced by Riemannian metric
on the space of harmonic 2-forms, identified by Hodge
with $H^2(X,{\Bbb R})$.
The pairing \eqref{_Hodge-Riemann-correct-Equation_}
is defined in such a way that on primitive $(1,1)$-forms
with coefficients in ${\Bbb R}$, $(\cdot,\cdot)$ is equal
to $-(\cdot,\cdot)_{Metr}$. Similarly, on the space
\[ \bigg({\Bbb R}\omega \oplus H^{2,0}(X)\oplus H^{0,2}(X)\bigg) \cap H^2(X,{\Bbb R}), \]
the form $(\cdot,\cdot)$ is equal to $(\cdot,\cdot)_{Metr}$.
\hfill
Let $(\cdot,\cdot)$,
$(\cdot,\cdot)': \; H^2(M,{\Bbb R})\times H^2(M,{\Bbb R})\longrightarrow {\Bbb R}$
be the Hodge-Riemann forms associated with $(M,I)$ and $(M,I')$
respectively. The most surprising result of this paper is
following (see \ref{_Hodge_Riemann_independent_Theorem_}
for a different wording of the same statement):
\hfill
\theorem \label{_Riemann_Hodge_unique_in_intro_Theorem_}
The forms $(\cdot,\cdot)$ and $(\cdot,\cdot)'$
are {\bf proportional}.
\hfill
Taking a K\"ahler class $\omega$ such that
$Vol(M)= 1$, where $Vol(M):= \int_M \omega^{n}$,
we get rid of the ambiguity in the choice of a constant.
If $(M,I)$ and $(M,I')$ both satisfy $Vol(M)= 1$,
then the Hodge-Riemann forms $(\cdot,\cdot)$ and $(\cdot,\cdot)'$
are {\bf equal}.
We call this form {\bf the normalized Hodge-Riemann pairing},
denoted as $(\cdot,\cdot)_{\c H}$. This form is an
invariant of a deformation class of complex manifolds.
One may regard $(\cdot,\cdot)_{\c H}$ as {\bf topological} invariant.
\footnote{In fact, the form
$(\cdot,\cdot):\;H^2(M,{\Bbb R})\times H^2(M,{\Bbb R})\longrightarrow {\Bbb R}$ of
\eqref{_Hodge-Riemann-correct-Equation_} is a
{\it topological} invariant associated
with every cohomology class $\omega\in H^2(M,{\Bbb R})$.
Our proof immediately implies that
for a dense set of $\omega,\omega'\in H^2(M,{\Bbb R})$,
\ref{_Riemann_Hodge_unique_in_intro_Theorem_} holds.}
We use the form $(\cdot,\cdot)_{\c H}$ to
compute the cohomology algebra $H^*(M)$. We explicitely
compute the subalgebra of $H^*(M)$ generated by $H^2(M)$.
These computations also give the relations between elements
generated by $H^2(M)$ and the rest of $H^*(M)$.
The main ideas of this computation are due to the observations which
involve Torelli theorem and deformation spaces. As follows
from results of Bogomolov
(\cite{_Bogomolov_}, \cite{_Besse:Einst_Manifo_}, \cite{_Todorov_}),
the deformation space of a holomorphically symplectic manifold
is a smooth complex manifold, which is a quotient of a
Stein space by an arithmetic group. From this description
we use only the calculation of dimension of this moduli space.
The map of Kodaira and Spencer
\[ KS:\; T_{[M]}\c M\longrightarrow H^1(TM)
\]
is a linear homomorphism
from the Zariski tangent space $T_{[M]}\c M$ of a moduli space $\c M$ of
deformations of $M$ to the first cohomology of the sheaf of holomorphic
vector fields on $M$. It is proven by Kodaira and Spencer
\cite{_Kodaira_Spencer_} that this
map is an embedding. Results of Bogomolov et al
imply that $\c M$ is smooth and
$KS$ is an isomorphism. This is often called ``Torelli
theorem''. This statement could be translated
to the language of period maps.
Together with \ref{_Riemann_Hodge_unique_in_intro_Theorem_},
this observation implies a nice description of the period map.
By ``period map'' associated with the holomorphic symplectic manifolds
(see Section \ref{_perio_and_forge_Section_}
for details) we mean the following geometrical object.
We define the (coarse, marked) moduli of holomorphic symplectic
manifolds as the space of different complex holomorphic symplectic
structures on a given $C^\infty$-manifold $M$ up to
diffeomorphisms which act trivially on
$H^*(M)$. Let $Comp$ be a connected component of this moduli space.
For more accurate definition of $Comp$, we refer the reader to
\ref{_Comp_Hyp_Definition_}. For all points $I\in Comp$, we
denote the corresponding complex manifold by $(M,I)$.
{}From the definition of $Comp$ we obtain a canonical
identification of cohomology spaces $H^*(M,I)$ for all $I\in Comp$.
For simplicity of statements, we deal with the simply connected
holomorphically symplectic manifolds $M$ such that $H^{2,0}(M)={\Bbb C}$
(such manifold are called {\bf simple}).
The period map (Section \ref{_perio_and_forge_Section_})
in this context is a map $P_c:\; Comp \longrightarrow \Bbb P(H^2(M,{\Bbb C}))$
which relies a line $H^{2,0}(M,I)\subset H^2(M,{\Bbb C})$ to every $I\in Comp$.
Classical results of Bogomolov et al imply that
$P_c$ is an immersion.
Hodge-Riemann relations together with
\ref{_Riemann_Hodge_unique_in_intro_Theorem_}
let one to describe the image of $P_c$ in following
terms. An immediate consequence of Hodge-Riemann relations
is that for all lines $x\in Im(P_c)\subset \Bbb P(H^2(M,{\Bbb C}))$,
and all vectors $l\in x\subset H^2(M,{\Bbb C})$, we have
$(l,l)_{\c H} =0$. Therefore, $P_c$ maps $Comp$ to a
quadric $C\subset \Bbb P(H^2(M,{\Bbb C}))$, which is defined
by the quadratic form associated with the Riemann-Hodge pairing.
Dimension of $Comp$ is computed from Torelli theorem.
As one can easily check, it is equal to the dimension
of $C$. Since $P_c$ is an immersion, we obtain the following theorem:
\hfill
\theorem
The period map $P_c:\; Comp \longrightarrow C$ is etale.
\hfill
This is the main
observation used in computations of the cohomology algebra
$H^*(M)$.
\hfill
Let $I\in Comp$. Let $ad I\in End(H^*(M))$ be a linear endomorphism
which maps a $(p,q)$-form $\eta\in H^{p,q}(M)$ to $(p-q)\sqrt{-1}\: \eta$.
Let $\goth M\subset End(H^*(M))$ be a Lie algebra generated
by the endomorphisms $ad I$ for all $I\in Comp$.
Hodge-Riemann relations imply that the action
of $\goth M$ on $V= H^2(M)$ preserves the scalar
product $(\cdot,\cdot)_{\c H}$. This defines a
Lie algebra homomorphism
\begin{equation} \label{_from_Mum-Tate_to_SO(V)_homom_Equation_}
\rho:\; \goth M\longrightarrow \goth{so}(V).
\end{equation}
Using period maps and estimation on dimensions of moduli spaces, we
prove that $\rho$ is an {\bf isomorphism}
(\ref{_g_0_computed_Theorem_}, \ref{_g_0_is_Mumf_Tate_Theorem_}).
This statement is an ingredient in the computation of the
algebra $H^*(M)$.
The algebraic structure on $H^*(M)$ is studied using the
general theory of Lefschetz-Frobenius algebras, introduced
in \cite{_Lunts-Loo_}. Lefschetz-Frobenius
algebras are associative graded commutative algebras whose
properties approximate that of cohomology of compact
manifolds which admit K\"ahler structure. We give an
exact definition of this term in Section \ref{_Lefshe_Frob_Section_}
(\ref{_Lefschetz_Frob_alge_Definition_}).
With no loss of generality, reader may think of
these algebras as of cohomology algebras.
With the Lefschetz-Frobenius algebra $A$ we associate
so-called {\bf structure Lie algebra} ${\goth g}(A)\subset End(A)$
which acts on $A$ by linear endomorphism. The action of structure
Lie algebra is often sufficient to reconstruct multiplication
on $A$. This algebra is defined using the algebraic version
of the strong Lefschetz theorem. Let $A= \bigoplus\limits^{2d}_{i=0} A_i$
be a graded commutative associative algebra over a field of
characteristic zero. Let $H\in End(A)$ be a linear endomorphism
of $A$ such that for all $\eta \in A_i$, $H(\eta)= (i-d) \eta$.
This endomorphism is usually considered in Hodge theory.
For all $a\in A_2$, denote by $L_a:\;\; A\longrightarrow A$ the linear map which
associates with $x\in A$ the element $ax\in A$. Again, this
operator is a counterpart of the operator $L$ considered in Hodge theory.
The triple $(L_a, H, \Lambda_a)\in End(A)$ is called {\bf a Lefschetz triple}
if
\[ [ L_a, \Lambda_a] = H,\;\; [ H, L_a ] = 2 L_a, \;\;
[ H, \Lambda_a] = -2 \Lambda_a.
\]
A Lefschetz triple establishes a representation of the
Lie algebra $\goth{sl}(2)$ in the space $A$. For cohomology
algebras, this representation
arises as a part of Lefschetz theory. V. Lunts noticed that
in a Lefschetz triple, the endomorphism $\Lambda_a$ is
uniquely defined by the element $a\in A_2$
(\ref{_Lefshe_tri_unique_Proposition_}). For arbitrary
$a\in A_2$, $a$ is called {\bf of Lefschetz type} if the
Lefschetz triple $(L_a, H, \Lambda_a)$ exists. If $A= H^*(X)$
where $X$ is a compact complex manifold of K\"ahler type,
then all K\"ahler classes $\omega\in H^2(M)$ are elements
of Lefschetz type. On the other hand, not all elements of Lefschetz
type are K\"ahler classes. For instance, if $\omega$ is
of Lefschetz type, then $-\omega$ is also of Lefschetz type,
but $\omega$ and $-\omega$ cannot be K\"ahler classes
simultaneously. As one can easily check (see \cite{_Lunts-Loo_}),
the set $S\subset A_2$ of all elements of Lefschetz type
is Zariski open in $A_2$.
Now we can define the structure Lie algebra ${\goth g}(A)$ of $A$.
By definition, ${\goth g}(A)\subset End(A)$ is a Lie subalgebra
of $End(A)$ generated by $L_a$, $\Lambda_a$, for all
elements of Lefschetz type $a\in S$.
This Lie algebra is often sufficient to reconstruct the
multiplicative structure on $A$.
Returning to the hyperk\"ahler manifolds, we consider the structure Lie
algebra ${\goth g}(A)$ of the ring $A= H^*(M)$, where $M$ is a compact
hyperk\"ahler manifold. It turns out that the structure Lie algebra
${\goth g}(A)$ can be explicitely computed. In particular,
\ref{_g(A)_for_hyperkae_Theorem_} gives us the following theorem:
\hfill
\theorem
Let $M$ be a compact holomorphically symplectic manifold. Assume for
simplicity\footnote{The structure Lie algebra can be computed
without this trivial assumption, but the statement is
less starightforward. See the discussion after
\ref{_simple_hyperkaehler_mfolds_Definition_} for details.}
that $M$ admits a unique up to a constant holomorphic symplectic form.
Let $n= \dim(H^2(M))$.
Then ${\goth g}(A)= \goth{so}(4, n-2)$.
\hfill
This computation, which
also involves the computation of the Lie algebra $\goth M$ of
\eqref{_from_Mum-Tate_to_SO(V)_homom_Equation_},
takes up 4 sections of this paper. However, the main idea of
this computation is easy.
\hfill
Let $M$ be a compact hyperk\"ahler manifold with the complex
structures $I$, $J$, $K$. Consider the K\"ahler forms
$\omega_I$, $\omega_J$, $\omega_K$ associated with these
complex structures. Let $\rho_I:\; \goth{sl}(2)\longrightarrow End(H^*(M))$,
$\rho_J:\; \goth{sl}(2)\longrightarrow End(H^*(M))$,
$\rho_K:\; \goth{sl}(2)\longrightarrow End(H^*(M))$
be the corresponding Lefschetz homomorphisms.
Let $\goth a\subset End(H^*(M))$ be the minimal
Lie subalgebra which contains images of $\rho_I$,
$\rho_J$, $\rho_K$. The algebra $\goth a$ was computed
explicitely in \cite{_so5_on_cohomo_}.
\hfill
\theorem \label{_so_5_Theorem_}
(\cite{_so5_on_cohomo_})
The Lie algebra $\goth a$ is isomorphic to $\goth{so}(4,1)$.
\hfill
This statement can be regarded as a ``hyperk\"ahler Lefschetz theorem''.
Indeed, its proof parallels the proof of Lefschetz theorem.
One can check that the cohomology classes
$\omega_I$, $\omega_J$, $\omega_K\in H^2(M,{\Bbb R})$ are orthogonal
with respect to Riemann-Hodge. Let $Hyp$ be the classifying
space of the hyperk\"ahler structures on $M$ (see Section
\ref{_moduli_Section_}). Let
$P_{hyp}:Hyp\longrightarrow H^2(M)\times H^2(M)\times H^2(M)$
be the map which associates with the hyperk\"ahler structure
$\c H= (I, J, K, (\cdot,\cdot))$ the triple
$(\omega_I,\omega_J,\omega_K)$. Then the image of $P_{hyp}$ in
$H^2(M)\times H^2(M)\times H^2(M)$ satisfies
\begin{equation}\label{_image_of_P_hyp_Equation_}
\forall (x,y,z)\in im P_{hyp}\;\; \bigg |\;\;
\begin{array}{l}
(x,y)_{\c H}=(x,z)_{\c H}=(y,z)_{\c H}=0,\\[3mm]
(x,x)_{\c H}=(y,y)_{\c H}=(z,z)_{\c H},
\end{array}
\end{equation}
where $(\cdot,\cdot)_{\c H}$ is the Hodge-Riemann pairing
of \eqref{_Hodge-Riemann-correct-Equation_}.
Let $D\subset H^2(M)\times H^2(M)\times H^2(M)$ be the set
defined by the equations \eqref{_image_of_P_hyp_Equation_}.
Using Torelli theorem and Calabi-Yau, we prove the following
statement:
\hfill
\theorem\label{_image_of_P_hyp_Theorem_}
The image of $P_{hyp}$ is Zariski dense in $D$.
\hfill
\ref{_image_of_P_hyp_Theorem_} shows that all algebraic relations
which are true for \[ (x,y,z)\in P_{hyp}(Hyp) \] are true
for all $(x,y,z)\in D$. Computing the Lie algebra $\goth a$
as in \ref{_so_5_Theorem_}, we obtain a number of relations
between $x,y,z\in H^2(M)$ which hold for all $(x,y,z)\in Im(P_{hyp})$.
Using the density argument, we obtain that these relations
are universally true. This idea leads to the following theorem.
\hfill
\theorem \label{_structure_alge_for_coho_hyperkahe_Theorem_}
Let $A= H^*(M)$ be a cohomology algebra of a compact
simply connected holomorphically symplectic manifold $M$
with $H^{2,0}(M)\cong {\Bbb C}$. Let $n = \dim H^2(M)$. Then
the structure Lie algebra ${\goth g}(A)$ of $A$ is isomorphic
to $\goth{so}(4,n-2)$.
\hfill
It remains to recover the multiplication on $H^*(M)$ from
the structure Lie algebra. This is done as follows. Let
$A = \oplus A_i$ be a Lefschetz-Frobenius algebra,
${\goth g} = {\goth g}(A)$ be its structure Lie algebra. Clearly,
${\goth g}$ is graded: ${\goth g}=\bigoplus\limits_i{\goth g}_{2i}$,
${\goth g}_{2i}(A_j)\subset A_{j+2i}$. Let $k$ denote the
one-dimensional commutative Lie algebra. In the case
of \ref{_structure_alge_for_coho_hyperkahe_Theorem_},
${\goth g}\cong \goth{so}(4,n-2)$, ${\goth g}_0 \cong \goth{so}(3,n-3)\oplus k$,
$\dim {\goth g}_2=\dim{\goth g}_{-2}=n$ and $\dim {\goth g}_{2i}=0$ for $|i|>1$
(see \ref{_calculation_of_g(A)_for_minim_Theorem_}).
We say that the Lefschetz-Frobenius algebra $A$ is
{\bf of Jordan type} if ${\goth g}_{2i}(A)=0$ for $|i|>1$.
For such algebras, the subspaces ${\goth g}_2(A)$,
${\goth g}_{-2}(A)\subset A$ are commutative Lie
subalgebras of ${\goth g}(A)$. Let $U_{\goth g}$ be the universal
enveloping algebra of ${\goth g}= {\goth g}(A)$, and $U_{{\goth g}_{2}}\subset U_{\goth g}$
be the enveloping algebra of the subalgebra ${\goth g}_2={\goth g}_2(A)\subset {\goth g}$.
Consider the space $A$ as $A$-module. Then, for all
$v\in A$ we have a map $t_v:\; U_{{\goth g}}\longrightarrow A$ which
associates $P(v)$ with $P\in U_{{\goth g}}$. The Lie algebra
${\goth g}_2$ is commutative, and therefore $U_{{\goth g}_2}\cong S^*({\goth g}_2)$.
According to \cite{_Lunts-Loo_}, the natural map
$A_2\longrightarrow {\goth g}_2$, $a \longrightarrow L_a$,
is an isomorphism (see \ref{_g_2_is_A_2_Corollary_}
for details). Let $v\in H^0(M)\subset A$ be a unit element of the
ring $A$. Consider the restriction $t$ of $t_v:\; U_{\goth g}\longrightarrow A$
to $U_{{\goth g}_2}\subset U_{\goth g}$. Then $t$ is a map from
$S^* {\goth g}_2\cong S^* A_2$ to $A$. Clearly, this
map coinsides with the map $S^* A_2\longrightarrow A$ defined
by the multiplication. This implies that multiplication
by elements from $H^2(M)$ can be recovered from the action
of the structure Lie algebra ${\goth g}$.
Using the calculations of
\ref{_structure_alge_for_coho_hyperkahe_Theorem_},
we obtain, in particular, the following theorem
(see Section \ref{_cohomolo_compu_Section_}):
\hfill
\theorem \label{_S^*H^2_is_H^*M_intro-Theorem_}
Let $M$ be a compact hyperk\"ahler manifold. Let
\[ \bar H^* (M)\subset H^*(M)
\]
be the subalgebra in $H^*(M)$
generated by $H^2(M)$. Let $\dim_{\Bbb C} M=2n$.
Then
\[ \bar H^{2i}(M)\cong S^i H^2(M) \]
for $i\leq n$, and
\[ \bar H^{2i}(M)\cong S^{2n-i} H^2(M) \]
for $i\geq n$.
\hfill
\hfill
\centerline{\Large \bf Contents:}
\hfill
\begin{description}
\item [Section \ref{_introduction_to_so(b_2,...)_Section_}:]\hspace{2mm}
{Introduction.}
\item [Section \ref{hyperk_manif_Section_}:]\hspace{2mm}
{Hyperk\"ahler manifolds.}
\item [Section \ref{_moduli_Section_}:]\hspace{2mm}
{Moduli spaces for hyperk\"ahler and
holomorphically symplectic manifolds.}
\item [Section \ref{_perio_and_forge_Section_}:]\hspace{2mm}
{Periods and forgetful maps.}
\item [Section \ref{_Period_and_Hodge_Riemann_Section_}:]\hspace{2mm}
{Hodge-Riemann relations for the
hyperk\"ahler manifolds and period map.}
\item [Section \ref{_Hodge-Rie_independent_Section_}:]\hspace{2mm}
{The Hodge-Riemann metric on $H^2(M)$ does not depend
on complex structure.}
\item [Section \ref{_Q_c_defini_Section_}:]\hspace{2mm}
{Period map and the space of 2-dimensional
planes in \\ $H^2(M,{\Bbb R})$.}
\item [Section \ref{_Lefshe_Frob_Section_}:]\hspace{2mm}
{Lefschetz-Frobenius algebras.}
\item [Section \ref{_minimal_Fro_Section_}:]\hspace{2mm}
{The minimal Frobenius algebras and cohomology of compact
K\"ahler surfaces.}
\item [Section \ref{_^dA(V)_Section_}:]\hspace{2mm}
{Representations of $SO(V,+)$ leading to Frobenius algebras.}
\item [Section \ref{_computing_g_for_hyperk_pt-I_Section_}:]\hspace{2mm}
{Computing the structure Lie algebra for the
cohomology of a hyperk\"ahler manifold, part I.}
\item [Section \ref{_compu_g_0_part_1_Section_}:]\hspace{2mm}
{Calculation of a zero graded part of the structure Lie
algebra of the cohomology of a hyperk\"ahler manifold, part I.}
\item [Section \ref{_compu_g_0_part_2_Section_}:]\hspace{2mm}
{Calculation of a zero graded part of the structure Lie
algebra of the cohomology of a hyperk\"ahler manifold, part II.}
\item [Section \ref{_computing_g_for_hyperk_pt-2_Section_}:]\hspace{2mm}
{Computing the structure Lie algebra for the
cohomology of a hyperk\"ahler manifold, part II.}
\item [Section \ref{_cohomolo_compu_Section_}:]\hspace{2mm}
{The structure of the cohomology ring for
compact hyperkaehler manifolds.}
\item [Section \ref{_calcu_dimensi_Section_}:]\hspace{2mm}
{Calculations of dimensions.}
\end{description}
\hfill
\hfill
\begin{itemize}
\item Section \ref{_introduction_to_so(b_2,...)_Section_}
tries to supply motivations and heuristics for the further
study. In the body of this article, we never refer to Section
\ref{_introduction_to_so(b_2,...)_Section_}. Reading
Section \ref{_introduction_to_so(b_2,...)_Section_}
is not necessarily for understanding of this paper.
\item In Section \ref{hyperk_manif_Section_}, we give the definition of a
hyperk\"ahler manifold. We explain the geopetry of quaternionic action
on $H^*(M)$. This section ends with the statement of Calabi-Yau
for compact holomorphically symplectic manifolds. Results
and definitions of this section are well known.
\item Section \ref{_moduli_Section_} begins with a definition
of simple hyperk\"ahler manifolds. This notion stems from the theory
of holonomy groups. Let $M$ be a compact hyperk\"ahler manifold,
$\dim_{\Bbb R} M=4n$. Then the holonomy group of $M$ is naturally
a subgroup of $Sp(n)$. The simple hyperk\"ahler manifold
is a hyperk\"ahler manifold such that its
restricted holonomy group in exactly $Sp(n)$ and
not a proper subgroup of $Sp(n)$. In
\ref{_simple_hyperkaehler_mfolds_Definition_} we give a different,
but equivalent treatment of this notion
(see also \cite{_Beauville_}).Using a formalism
by de Rham and Berger, Bogomolov and Beauville proved that a
hyperk\"ahler manifold $M$ is simple if and only if
there is no finite covering $\tilde M$ of $M$ such that
$\tilde M$ can be represented as a product of two (or more)
non-trivial hyperk\"ahler manifolds. Therefore, it is
usually harmless to assume that a given compact hyperk\"ahler
manifold is simple. For simple hyperk\"ahler
manifolds, $\dim H^{2,0}(M)=1$.
\item After we define simple hyperk\"ahler manifolds, we go for
the marked moduli spaces. We define moduli spaces for complex,
hyperk\"ahler and holomorphically symplectic structures. We
do this in topological, rather than in algebro-geometrical setting.
The most obvious reason for this lopsided treatment is that
moduli of hyperk\"ahler structures don't
have any structures in addition to topology. Results
and definitions of this section are well known.
\item In Section \ref{_perio_and_forge_Section_},
we define several kinds of period maps. Let $Comp$ be
the marked moduli of complex structures on a simple holomorphically
symplectic manifold. For all complex structures $I\in Comp$,
the space $H^{2,0}(M,I)$ of $(2,0)$-forms is one-dimensional.
With every complex structure $I\in Comp$,
the period maps $P_c:\; Comp \longrightarrow {\Bbb P}(H^2(M, {\Bbb C}))$
associates a line $H^{2,0}(M,I)\subset H^2(M)$
in ${\Bbb P}(H^2(M, {\Bbb C}))$. With every holomorphic symplectic
structure, period map associates a class in $H^2(M,{\Bbb C})$
represented by a holomorphic symplectic form. Finally, with
every hyperk\"ahler structure, period map $P_{hyp}$ associates
a triple of K\"ahler classes which correspond to the complex
structures $I$, $J$, $K$. We define a number of forgetful maps
(from hyperk\"ahler moduli to complex moduli etc.)
and compare these maps against period mappings.
\item Sections \ref{hyperk_manif_Section_} -
\ref{_perio_and_forge_Section_} contain no new results.
We establish setting for the further study of hyperk\"ahler manifolds.
\item Section \ref{_Period_and_Hodge_Riemann_Section_} answers the
following query. Hodge-Riemann pairing on cohomology satisfies a certain
type of positivity conditions, called {\bf Hodge-Riemann relations}.
What happens with these relations on a hyperk\"ahler manifold?
It turns that for a Hodge-Riemann form $(\cdot,\cdot)$ defined
on $H^2(M)$ and an action of $SU(2)$ on $H^2(M)$ which
comes from quaternions, the following conditions are satisfied:
\begin{description}
\item [{\rm (i)}] Let $H^{inv}\subset H^2(M)$ be the space
of $SU(2)$-invariants, and $H^\bot$ be its orthogonal complement.
Then the Riemann-Hodge pairing is negatively defined on
$H^{inv}$ and positively defined on $H^\bot$.
\item [{\rm (ii)}] The space $H^\bot$ is three-dimensional.
\item [{\rm (iii)}] The space $H^\bot$ is generated
by the K\"ahler forms associated with the complex
structures $I$, $J$ and $K$.
\end{description}
\item Section \ref{_Hodge-Rie_independent_Section_}
is the crux of the first part of this paper. We prove that
the Hodge-Riemann form \eqref{_Hodge-Riemann-correct-Equation_}
on $H^2(M)$ is independent
(up to a constant) from the K\"ahler structure.
The idea of the proof is the following.
\item The group $SO(3)$ acts on the
space of the hyperk\"ahler structures, replacing the triple $(I, J, K)$
by another orthogonal triple of imaginary quaternions.
As Section \ref{_Period_and_Hodge_Riemann_Section_}
shows, this action does not change the Hodge-Riemann form.
Let $\c H \in Hyp$, and let
$(\omega_I, \omega_J,\omega_K)\in
H^2(M,{\Bbb R})\times H^2(M,{\Bbb R})\times H^2(M,{\Bbb R})$ be the image
of $\c H$ under the action of period map $P_{hyp}$.
By definition, the Hodge-Riemann form depends only on $\omega_I$.
Therefore, we may replace $\c H$ by $\c H'\in Hyp$ such that
$P_{hyp}(\c H')= (\omega_I, \omega_J',\omega_K')$, and
such replacement does not change the Hodge-Riemann
pairing. We show that by iterating such replacements
and action of $SO(3)$, we can connect any two hyperk\"ahler
structure $\c H$ and $\c H'$ satisfying
$Vol_{\c H} (M)= Vol_{\c H'}(M)$, where by $Vol_{\c H}(M)$
we undertstand the volume of $M$ computed with respect to
the Riemannian structure associated with $\c H$.
Since these operations don't change the Riemann-Hodge form,
this form is equal for all hyperk\"ahler structure
$\c H$ of given volume. Section
\ref{_Hodge-Rie_independent_Section_} depends
on Sections
\ref{hyperk_manif_Section_} - \ref{_Period_and_Hodge_Riemann_Section_}.
\item Section \ref{_Q_c_defini_Section_} gives a description of the period
map \[ P_c:\; Comp\longrightarrow \Bbb B(H^2(M, {\Bbb C}))\] in terms of the
manifold $Pl$ of 2-dimensional planes in $H^2(M, {\Bbb R})$. It turns
out that there exist an etale mapping $Q_c:\; Comp\longrightarrow Pl$.
This is a standard material, covered also in \cite{_Todorov_}.
Our exposition adds a twist to \cite{_Todorov_}, because we use
the normalized Hodge-Riemann form, which was unknown before.
Otherwise, this section depends only on Sections
\ref{hyperk_manif_Section_} - \ref{_perio_and_forge_Section_}.
\item Sections \ref{_Lefshe_Frob_Section_} - \ref{_^dA(V)_Section_}
are completely independed on the preceding sections.
\item In Section \ref{_Lefshe_Frob_Section_}, we give a number
of algebraic definitions. We give an exposition of the theory
of Lefschetz-Frobenius algebras, following \cite{_Lunts-Loo_}.
The aim of this section is a purely algebraic version
of strong Lefschetz theorem.
\item Roughly speaking, Frobenius algebra is a graded algebra
for which an algebraic version of Poincare duality holds.
A typical example of such algebra is an algebra of cohomology
of a compact manifold. Similarly, the Lefschetz-Frobenius algebra
is a Frobenius algebra for which the strong Lefschetz theorem holds -
typically, an algebra of cohomology of a K\"ahler manifold.
In Section \ref{_Lefshe_Frob_Section_} we explain these notions
and define a structure Lie algebra ${\goth g}$ of a Lefschetz-Frobenius algebra.
Here we also give a definition of Lefschetz-Frobenius algebras
of Jordan type.
\item Section \ref{_minimal_Fro_Section_} is dedicated to
explicit examples of Lefschetz-Frobenius algebras, called
{\bf minimal Lefhsetz-Frobenius algebras}.
By definition, a minimal Lefschetz-Frobenius algebra
is a Lefschetz-Frobenius algebra $A= A_0 \oplus A_2\oplus A_4$.
The ``Poincare'' form on $A$ defines a bilinear symmetric pairing on
$A_2$. It turns out that this pairing uniquely determines $A$.
Conversely, with every linear space equipped with
non-degenerate symmetric sclalar product
we associate a minimal Lefschetz-Frobenius algebra.
We prove that every minimal Lefschetz-Frobenius algebra
is of Jordan type, and explicitely compute its structure
Lie algebra. For a minimal Lefschetz-Frobenius algebra
$A(V)$ associated with a space $V$, we denote the
corresponding structure Lie algebra by $\goth{so}(V,+)$.
If $V$ is a linear space over ${\Bbb R}$ equipped with a scalar product
of signature $(p,q)$, then $\goth{so}(V,+)\cong \goth{so}(p+1,q+1)$.
\item In Section \ref{_^dA(V)_Section_} we find all
reduced Lefschetz-Frobenius algebras
$A= A_0\oplus A_2 \oplus ... \oplus A_n$ with the structure Lie
algebra $\goth{so}(V,+)$, where $\dim V\geq 3$. By ``reduced'' we
understand the Lefschetz-Frobenius algebras generated by $A_2$.
It turns out that for $n$ even, such algebra
is unique (we denote it by ${}^{\frac{n}{2}}A(V)$),
and for $n$ odd, there is no such algebras.
\item Sections \ref{_Lefshe_Frob_Section_} - \ref{_^dA(V)_Section_}
are purely algebraic, and Sections \ref{hyperk_manif_Section_} -
\ref{_Q_c_defini_Section_} are dealing with geometry. These
sections are mutually independent, and Sections
\ref{_computing_g_for_hyperk_pt-I_Section_} -
\ref{_computing_g_for_hyperk_pt-2_Section_} draw heavily on
both parts, geometrical and algebraic.
In these sections, we compute the structure Lie algebra
of an algebra of cohomology of a simple hyperk\"ahler manifold.
The basic result is that this algebra is isomorphic
to $\goth{so}(V,+)$, where $V$ is the linear space $H^2(M, {\Bbb R})$
equipped with the normalized Hodge-Riemann pairing.
\item In Section \ref{_computing_g_for_hyperk_pt-I_Section_},
we prove that the Lefschetz-Frobenius algebra $A=H^*(M)$ is of
Jordan type: ${\goth g}(A)={\goth g}_{-2}(A)\oplus {\goth g}_0(A)\oplus {\goth g}_2(A)$.
To prove this we introduce the standard ``density and periods''
argument, which is also used in Sections
\ref{_compu_g_0_part_2_Section_} -
\ref{_computing_g_for_hyperk_pt-2_Section_}.
\item In Section \ref{_compu_g_0_part_1_Section_}, we
construct a map ${\goth g}_0(A)\longrightarrow \goth{so}(V)\oplus k$, where
$k$ is a one-dimensional commutative Lie algebra.
We prove that this map is an isomorphism. We also
prove that the Lie subalgebra $\goth M\subset End(V)$
generated by $ad I$ for all $I\in Comp$ is isomorphic
to $\goth{so}(V)$. By definition, $ad I:\; H^i(M)\longrightarrow H^i(M)$
is an endomorphism which maps $\eta\in H^{p,q}_I(M)$
to $(p-q)\sqrt{-1}\: \eta$. We use the computations related to the
$\goth{so}(5)$-action on $H^*(M)$ (see \cite{_so5_on_cohomo_}).
\item In Section \ref{_compu_g_0_part_2_Section_}, we prove
that the map $u:\; {\goth g}_0\longrightarrow \goth{so}(V)\oplus k$, constructed
in Section \ref{_compu_g_0_part_1_Section_}, is an isomorphism.
The proof is computational.
\item In Section \ref{_computing_g_for_hyperk_pt-2_Section_},
we use the results of Sections \ref{_computing_g_for_hyperk_pt-I_Section_}
- \ref{_compu_g_0_part_2_Section_} to finish the computation of
the structure Lie algebra ${\goth g}(A)$. This is done by writing down
a linear isomorphism ${\goth g}(A)\longrightarrow \goth{so}(4,n-2)$ explicitely.
By computations, we check that this isomorphism is in fact
an isomorphism of Lie algebras.
\item Section \ref{_cohomolo_compu_Section_} is, again, algebraic.
It depends only on Sections
\ref{_Lefshe_Frob_Section_} - \ref{_^dA(V)_Section_}.
In this section, we explicitely compute the graded commutative
algebra ${}^dA(V)$ of Section \ref{_^dA(V)_Section_}.
This computation has the following
geometrical interpretation. Let $A=H^*(M)$ be the algebra of
cohomology of a simple hyperk\"ahler manifold $M$, $\dim_{\Bbb R} M=4d$,
and $A^r\subset A$ be its subalgebra generated by $V= H^2(M)$. The
main theorem of \ref{_^dA(V)_Section_}, together with an isomorphism
${\goth g}(A)\cong\goth{so}(V,+)$ immediately imply
that $A^r\cong {}^dA(V)$. Therefore, by computing
${}^dA(V)$, we compute a big part of the cohomology
algebra of $M$. This way, we obtain a proof of
\ref{_S^*H^2_is_H^*M_intro-Theorem_}. The computation
of ${}^dA(V)$ is based on the classical theory of representations
of $\goth{so}(V)$ and their tensor invariants (\cite{_Weyl_}).
\item The final section applies the result of
\ref{_S^*H^2_is_H^*M_intro-Theorem_} to obtain numerical
lower bounds on Betti and Hodge numbers of a hyperk\"ahler manifold.
\end{itemize}
\section{Hyperk\"ahler manifolds.}\label{hyperk_manif_Section_}
\definition \label{_hyperkaehler_manifold_Definition_}
(\cite{_Beauville_},
\cite{_Besse:Einst_Manifo_}) A {\bf hyperk\"ahler manifold} is a
Riemannian manifold $M$ endowed with three complex structures $I$, $J$
and $K$, such that the following holds.
\hspace{5mm} (i) $M$ is K\"ahler with respect to these structures and
\hspace{5mm} (ii) $I$, $J$ and $K$, considered as endomorphisms
of a real tangent bundle, satisfy the relation
$I\circ J=-J\circ I = K$.
\hfill
This means that the hyperk\"ahler manifold has the natural action of
quaternions ${\Bbb H}$ in its real tangent bundle.
Therefore its complex dimension is even.
Let $\mbox{ad}I$, $\mbox{ad}J$ and $\mbox{ad}K$ be the endomorphisms of
the bundles of differential forms over a hyperk\"ahler manifold
$M$ which are defined as follows. Define $\mbox{ad}I$.
Let this operator act as a complex structure operator
$I$ on the bundle of differential 1-forms. We
extend it on $i$-forms for arbitrary $i$ using Leibnitz
formula: $\mbox{ad}I(\alpha\wedge\beta)=\mbox{ad}I(\alpha)\wedge\beta+
\alpha\wedge \mbox{ad}I(\beta)$. Since Leibnitz
formula is true for a commutator in a Lie algebras, one can immediately
obtain the following identities, which follow from the same
identities in ${\Bbb H}$:
\[
[\mbox{ad}I,\mbox{ad}J]=2\mbox{ad}K;\;
[\mbox{ad}J,\mbox{ad}K]=2\mbox{ad}I;\;
\]
\[
[\mbox{ad}K,\mbox{ad}I]=2\mbox{ad}J
\]
Therefore, the operators $\mbox{ad}I,\mbox{ad}J,\mbox{ad}K$
generate a Lie algebra $\goth g_M\cong \goth{su}(2)$ acting on the
bundle of differential forms. We can integrate this
Lie algebra action to the action of a Lie group
$G_M=SU(2)$. In particular, operators $I$, $J$
and $K$, which act on differential forms by the formula
$I(\alpha\wedge\beta)=I(\alpha)\wedge I(\beta)$,
belong to this group.
\hfill
\proposition \label{_there_is_action_of_G_M_Proposition_}
There is an action of the Lie group $SU(2)$
and Lie algebra $\goth{su}(2)$ on the bundle of differential
forms over a hyperk\"ahler manifold. This action is
parallel, and therefore it commutes with Laplace operator.
{\bf Proof:} Clear.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
If $M$ is compact, this implies that there is
a canonical $SU(2)$-action on $H^i(M,{\Bbb R})$ (see
\cite{_so5_on_cohomo_}).
\hfill
Let $M$ be a hyperk\"ahler manifold with a Riemannian form
$\inangles{\cdot,\cdot}$.
Let the form $\omega_I := \inangles{I(\cdot),\cdot}$ be the usual K\"ahler
form which is closed and parallel
(with respect to the connection). Analogously defined
forms $\omega_J$ and $\omega_K$ are
also closed and parallel.
The simple linear algebraic
consideration (\cite{_Besse:Einst_Manifo_}) shows that \hfill
$\omega_J+\sqrt{-1}\omega_K$ is of
type $(2,0)$ and, being closed, this form is also holomorphic.
\hfill
\definition \label{_canon_holo_symple_form_Definition_
Let $\Omega:= \omega_J+\sqrt{-1}\omega_K$. This form is called
{\bf the canonical holomorphic symplectic form
of a manifold M}.
\hfill
Let $M$ be a complex manifold which admits a
holomorphic symplectic form $\Omega$. Take the Riemannian
metric $(\cdot,\cdot)$ on $M$, and the corresponding Levi-Civitta
connection. Assume that $\Omega$ is parallel
with respect to the Levi-Civitta connection.
Then the metric $(\cdot,\cdot)$ is hyperkaehler%
\footnote%
{This means that the $(\cdot,\cdot)$ is induced
by a hyperk\"ahler structure on $M$.} (\cite{_Besse:Einst_Manifo_}).
If some {\bf it compact} K\"ahler manifold $M$ admits non-degenerate
holomorphic symplectic form $\Omega$, the Calabi-Yau
(\cite{_Yau:Calabi-Yau_}) theorem
implies that $M$ is hyperk\"ahler
(\ref{_symplectic_=>_hyperkaehler_Proposition_})
This follows from the existence of a K\"ahler
metric on $M$ such that $\Omega$ is parallel under the
Levi-Civitta connection associated with this metric.
\hfill
Let $M$ be a hyperk\"ahler manifold with complex structures
$I$, $J$ and $K$. For any real numbers $a$, $b$, $c$
such that $a^2+b^2+c^2=1$ the operator $L:=aI+bJ+cK$ is also
an almost complex structure: $L^2=-1$.
Clearly, $L$ is parallel with respect to a connection.
This implies that $L$ is a complex structure, and
that $M$ is K\"ahler with respect to $L$.
\hfill
\definition \label{_induced_structures_Definion_}
If $M$ is a hyperk\"ahler manifold,
the complex structure $L$ is called {\bf induced
by a hyperk\"ahler structure}, if $L=aI+bJ+cK$ for some
real numbers $a,b,c\:|\:a^2+b^2+c^2=1$.
\hfill
\hfill
If $M$ is a hyperk\"ahler manifold and $L$ is induced complex structure,
we will denote $M$, considered as a complex manifold with respect to
$L$, by $(M,L)$ or, sometimes, by $M_L$.
\hfill
Consider the Lie algebra $\goth{g}_M$ generated by ${ad}L$ for all $L$
induced by a hyperk\"ahler structure on $M$. One can easily see
that $\goth{g}_M=\goth{su}(2)$.
The Lie algebra $\goth{g}_M$ is called {\bf isotropy algebra} of $M$, and
corresponding Lie group $G_M$ is called an {\bf isotropy group}
of $M$. By Proposition 1.1, the action of the group is parallel,
and therefore it commutes with the action of Laplace operator on differential
forms. In particular, this implies that the action of the isotropy
group $G_M$ preserves harmonic forms, and therefore this
group canonically acts on cohomology of $M$.
\hfill
\proposition \label{_G_M_invariant_forms_Proposition_}
Let $\omega$ be a differential form over
a hyperk\"ahler manifold $M$. The form $\omega$ is $G_M$-invariant
if and only if it is of Hodge type $(p,p)$ with respect to all
induced complex structures on $M$.
{\bf Proof:} Assume that $\omega$ is $G_M$-invariant.
This implies that all elements of ${\goth g}_M$ act trivially on
$\omega$ and, in particular, that $\mbox{ad}L(\omega)=0$
for any induced complex structure $L$. On the other hand,
$\mbox{ad}L(\omega)=(p-q)\sqrt{-1}\:$ if $\omega$ is of Hodge type $(p,q)$.
Therefore $\omega$ is of Hodge type $(p,p)$ with respect to any
induced complex structure $L$.
Conversely, assume that $\omega$ is of type $(p,p)$ with respect
to all induced $L$. Then $\mbox{ad}L(\omega)=0$ for any induced $L$.
By definition, ${\goth g}_M$ is generated by such $\mbox{ad}L$,
and therefore ${\goth g}_M$ and $G_M$ act trivially on $\omega$. $\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\definition \label{_degree_Definition_
Let $M$ be a Kaehler manifold and
$\omega\in H^2(M,{\Bbb R})$ be the Kaehler class of $M$.
Let $dim_{\Bbb C}(M)=n$. Let $\eta\in H^{2i}(M)$.
We define the {\bf degree} of the cohomology class $\eta$
by the formula
\[ deg(\eta):=
\frac{\int\limits_M \eta\wedge\omega^{n-i}}
{{\mbox Vol}(M)}.
\]
Clearly, if $\eta$ is pure of Hodge type $(p,q)$ and
$deg(\eta)\neq 0$, then $p=q$.
Let $M$ be a hyperkaehler manifold, and $I$ be an induced complex
structure. Then $(M,I)$ is equipped with the canonical Kaehler metric.
Consider $(M,I)$ as a Kehler manifold. We define {\bf the degree
associated with the induced complex structure $I$}
as the linear homomorphism $deg_I:\; H^{2i}(M, {\Bbb R})\longrightarrow {\Bbb R}$
which is equal to degree map
\[
deg:\; H^{2i}((M,I), {\Bbb R})\longrightarrow {\Bbb R}
\]
defined on the cohomology of the Kaehler manifold $(M,I)$.
\hfill
The following statement follows from a trivial local
computation. The more general form of this claim is proven in
\cite{Verbitsky:Symplectic_I_}.
\hfill
\claim \label{_inv_2-forms_have_zero_degree_Claim_}
Let $M$ be a hyperkaehler manifold and $\eta\in H^2(M)$
be a $G_M$-invariant hohomology class. Then $deg_I(\eta)=0$
for all induced complex structures $I$.
{\bf Proof:} See Theorem 2.1 of
\cite{Verbitsky:Symplectic_I_} $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Calabi-Yau theorem provides an elegant way to think of
hyperkaehler manifolds in holomorphic terms. Heuristically speaking,
compact hyperkaehler manifolds are holomorphic manifolds which admit
a holomorphic symplectic form.
\hfill
\definition \label{_holomorphi_symple_Definition_}
The compact complex manifold $M$ is called
holomorphically symplectic if there is a holomorphic 2-form $\Omega$
over $M$ such that $\Omega^n=\Omega\wedge\Omega\wedge...$ is
a nowhere degenerate section of a canonical class of $M$.
There, $2n=dim_{\Bbb C}(M)$.
Note that we assumed compactness of $M$.%
\footnote{If one wants to define a holomorphic symplectic
structure in a situation when $M$ is not compact,
one should require also the equation $\nabla'\Omega$ to held.
The operator $\nabla':\;\Lambda^{p,0}(M)\longrightarrow\Lambda^{p+1,0}(M)$
is a holomorphic differential defined on differential $(p,0)$-forms
(\cite{_Griffiths_Harris_}).}
One observes that the holomorphically symplectic
manifold has a trivial canonical bundle.
A hyperk\"ahler manifold is holomorphically symplectic
(see Section \ref{hyperk_manif_Section_}). There is a converse proposition:
\hfill
\theorem \label{_symplectic_=>_hyperkaehler_Proposition_
(\cite{_Beauville_}, \cite{_Besse:Einst_Manifo_})
Let $M$ be a holomorphically
symplectic K\"ahler manifold with the holomorphic symplectic form
$\Omega$, a K\"ahler class
$[\omega]\in H^{1,1}(M)$ and a complex structure $I$. Assume
that
\[ deg ([\Omega]\wedge \bar [\Omega])
=2 deg([\omega]\wedge[\omega])
\]
Then there exists a {\it unique} hyperk\"ahler structure
$(I,J,K,(\cdot,\cdot))$ over $M$ such that the cohomology
class of the symplectic form $\omega_I=(\cdot,I\cdot)$ is
equal to $[\omega]$ and the canonical symplectic form
$\omega_J+\sqrt{-1}\:\omega_K$ is equal to $\Omega$.
\hfill
\ref{_symplectic_=>_hyperkaehler_Proposition_} immediately
follows from the Calabi-Yau theorem (\cite{_Yau:Calabi-Yau_}).
$\:\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\section{Moduli spaces for hyperkaehler and holomorphically symplectic
manifolds.}
\label{_moduli_Section_}
\definition \label{_simple_hyperkaehler_mfolds_Definition_}
(\cite{_Beauville_}) The connected simply connected compact hyperkaehler
manifold
$M$ is called {\bf simple} if $M$ cannot be represented
as a Cartesian product of two (non-trivial) hyperkaehler manifolds:
\[ M\neq M_1\times M_2, \] where $M_1$, $M_2$ are
hyperkaehler manifolds such that $dim\; M_1>0$, $dim\; M_2>0$.
\hfill
Bogomolov proved that every compact hyperkaehler manifold has a finite
covering which is a Cartesian product of a compact torus
and simple hyperkaehler manifolds. Even if our results
could be easily carried over for all compact hyperkaehler manifolds,
we restrict ourselves to the case of non-decomposable manifold
to simplify the argument.
\hfill
Let $M$ be a simple hyperkaehler manifold.
According to Bogomolov's theorem (\cite{_Beauville_}),
for every induced complex structure $I$,
\[dim_{\Bbb C} \bigg(H^{2,0}((M,I))\bigg)=1.\]
This means that the space of holomorphic
symplectic forms on $(M,I)$ is one-dimensional.
{}From now on, we assume that $M$ is a simple compact hyperkaehler
manifold, which is not a torus.
\hfill
The moduli spaces of the hyperkaehler and holomorphically
symplectic manifolds were first studied by Bogomolov
(\cite{_Bogomolov_}). The studies were continued by
Todorov ([Tod]).
\hfill
Let $M_{C^\infty}$ be $M$ considered as a
differential manifold. Let $\mbox{\it Diff}$ be the group of diffeomorphisms
of $M$. Recall that the {\bf hyperkaehler structure} on
$M_{C^\infty}$ was defined as a quadruple
$(I, J, K, (\cdot, \cdot ))$ where
\[ I, J, K \in End(TM_{C^\infty}), \;\; I^2=J^2=K^2=-1 \]
are operators on the tangent bundle $TM_{C^\infty}$
and $(\cdot, \cdot )$ is a Riemannian form. This
quadruple must satisfy certain relations
(\ref{_hyperkaehler_manifold_Definition_}).
Let $\widetilde{\mbox{\it Hyp}}$ be the set of all hyperkaehler
structures on $M_{C^\infty}$. Clearly, the group $\mbox{\it Diff}$ acts
on $\widetilde{\mbox{\it Hyp}}$. The set of all non-isomorphic
hyperkaehler structures on $M_{C^\infty}$ is in bijective
correspondence with the set of orbits of $\mbox{\it Diff}$ on
$\widetilde{\mbox{\it Hyp}}$. However, the geometrical properties of
$\widetilde{\mbox{\it Hyp}}/\mbox{\it Diff}$ are not satisfactory: as a rule,
the natural topology on $\widetilde{\mbox{\it Hyp}}/\mbox{\it Diff}$
is not separable etc. To produce a more geometrical
moduli space of the hyperkaehler structures,
we will refine the space $\widetilde{\mbox{\it Hyp}}/\mbox{\it Diff}$ in
accordance with the general algebro-geometrical formalism
of marked coarse moduli spaces.
\hfill
Let $\underline{\widetilde{Comp}}$ be the set of all integrable complex
structures on $M_{C^\infty}$. In other words,
$\underline{\widetilde{Comp}}$ is the set of all operators
\[
I\in End(TM_{C^\infty}),\;\; I^2=-1
\]
such that the almost complex structure defined by $I$ is integrable.
The set of all non-isomorphic complex structures on $M_{C^\infty}$
is in one-to-one correspondence with
$\underline{\widetilde{Comp}}/\mbox{\it Diff}$.
For every $I\in \underline{\widetilde{Comp}}$,
we say that $I$ {\bf admits a hyperkaehler
structure} when there exist a hyperkaehler structure
$(I, J, K, (\cdot,\cdot))$ on $M$. We say that $I$
is {\bf holomorphically symplectic} when the manifold $(M, I)$
admits a non-degenerate holomorphic symplectic form.
The set $\underline{\widetilde{Comp}}$ is endowed with
a natural topology.
\hfill
Let $M$ be a compact hyperkaehler manifold.
Clearly, each of induced complex
structures on $M$ is contained in the same
connected component of $\underline{\widetilde{Comp}}$.
Denote this component by $\underline{\widetilde{Comp}}^\circ $.
\ref{_symplectic_=>_hyperkaehler_Proposition_}
immediately implies the following statement:
\hfill
\corollary \label{_defo_simple_if_it's_Kaeh_Corollary_}
Let $I\in \underline{\widetilde{Comp}}^\circ$. Then
$(M,I)$ admits the hyperkaehler structure if and only if
$(M,I)$ admits a holomorphic symplectic structure and is
of Kaehler type%
\footnote{We say that a complex manifold $X$ is {\bf of Kaehler type}
if $X$ admits a Kaehler metric.}.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Denote the set of all $I\in \underline{\widetilde{Comp}}^\circ$
which admit the hyperkaehler structure by $\widetilde{Comp}^\circ$.
\hfill
The pair $(I,\Omega)$ is called {\bf the holomorphic symplectic
structure on the differential manifold
$M_{C^\infty}$} if $I$ is a complex structure
on $M_{C^\infty}$ and $\Omega$ is a holomorphic symplectic form
over $(M,I)$. Let $\widetilde{Symp}^\circ$ denote the set of
all holomorphic symplectic structures $(I,\Omega)$ on $M_{C^\infty}$
such that $I \in {\widetilde{Comp}}^\circ$.
Let $\mbox{\it Diff}\,^\circ $ be the set of all
$x\in\mbox{\it Diff}$ which act trivially on the cohomology
$H^*(M,{\Bbb R})$.
Denote by $\widetilde{\mbox{\it Hyp}}^\circ $ the connected component
of $\widetilde{\mbox{\it Hyp}}$ which contains the
initial hyperkaehler structure on $M$.
Let $\mbox{\it Hyp}:=
\widetilde{\mbox{\it Hyp}}^\circ /\mbox{\it Diff}\,^\circ$,
$Symp:=\widetilde{Symp}^\circ/\mbox{\it Diff}\,^\circ$
and $Comp:= \widetilde{Comp}^\circ /\mbox{\it Diff}\,^\circ $.
These spaces are endowed with the natural topology.
Their points can be considered as the classes of
hyperkaehler (respectively, holomorphically symplectic and complex)
structures on $M_{C^\infty}$ up to the action of
$\mbox{\it Diff}\,^\circ $. Slightly abusing the
language, we will refer to these points as to hyperkaehler (resp.,
holomorphically symplectic and
complex) structures. For each $I\in Comp$,
we denote $M$, considered as a complex manifold
with the complex structure $I$, by $(M,I)$.
It is clear that $(M,I)$ is holomorphically symplectic
and admits a hyperkaehler structure for all $I\in Comp$.
\hfill
\definition \label{_Comp_Hyp_Definition_}
The spaces $\mbox{\it Hyp}$, $Symp$, $Comp$
are called {\bf the coarse moduli spaces
of deformations of the hyperkaehler (respectively,
holomorphically symplectic and complex) structure
on the marked compact manifold of
hyperkaehler type.}
\hfill
The word {\bf marked} refers to considering
the factorization by $\mbox{\it Diff}\,^\circ $
instead of $\mbox{\it Diff}$. This is roughly equivalent
to fixing the basis in the cohomology $H^*(M)$, hence
``marking''.
\section {Periods and forgetful maps.}\label{_perio_and_forge_Section_}
In assumptions of Section \ref{_moduli_Section_},
let $\mbox{\it Hyp}$, $Symp$, $Comp$ be the moduli spaces
of \ref{_Comp_Hyp_Definition_}.
We define the {\bf period map}
\[
\tilde P_{hyp}: \;\widetilde{\mbox{\it Hyp}}\longrightarrow H^2(M,{\Bbb R})\otimes {\Bbb R}^3
\]
as a rule which associates with every hyperkaehler
structure $(I,J,K,(\cdot,\cdot))$ on $M_{C^\infty}$
the triple
\[ ([\omega_I],\: [\omega_J],\: [\omega_K])\in
H^2(M_{C^\infty},{\Bbb R})\times H^2(M_{C^\infty},{\Bbb R})
\times H^2(M_{C^\infty},{\Bbb R})
\]
of Kaehler classes corresponding to $I$, $J$ and
$K$ respectively.
By definition, the group $\mbox{\it Diff}\,^\circ $
acts trivially on $H^2(M)$. Therefore, $\tilde P_{hyp}$
descends to a map
\[
P_{hyp}: \;\mbox{\it Hyp}\longrightarrow H^2(M,{\Bbb R})\otimes {\Bbb R}^3
\]
Similarly, define the Griffiths'
period map
\[
P_c:\; Comp\longrightarrow {\Bbb P}^1(H^2(M_{C^\infty},{\Bbb C}))
\]
as a rule which relies the 1-dimensional complex subspace
\[
H^{2,0}((M_{C^\infty},I))\subset H^2(M_{C^\infty},{\Bbb C})
\]
to the complex structure $I$ on $M_{C^\infty}$. Using Dolbeault
spectral sequence, one can easily see that
the subspace $H^{2,0}((M_{C^\infty},I))\subset H^2(M_{C^\infty},{\Bbb C})$
is defined independently on the Kaehler metric.
Let $P_s:\; Symp\longrightarrow H^2(M,{\Bbb C})$
map a pair $(I,\Omega)\in Symp$ to the class
$[\Omega]\in H^2(M,{\Bbb C})$ which is represented by the
closed 2-form $\Omega$.
There exist a number of natural ``forgetful maps'' between
the spaces $\mbox{\it Hyp}$, $Symp$ and $Comp$.
Here we define some of these maps and find how these
maps relate to period maps.
Let $\c H= (I,J,K, (\cdot,\cdot))\in \mbox{\it Hyp}$ be a hyperkaehler
structure. As in \ref{_canon_holo_symple_form_Definition_},
consider the canonical holomorphic symplectic form
$\Omega:= \omega_J+\sqrt{-1}\: \omega_K$ associated with $\c H$.
Let $\Phi^{hyp}_s:\; \mbox{\it Hyp}\longrightarrow Symp$ map
$\c H$ to the pair $(I, \Omega)\in Symp$. Let
$\Phi^{hyp}_s:\; \mbox{\it Hyp}\longrightarrow Comp$ map
$\c H$ to $I\in Comp$. Let $\Phi^s_c:\; Symp\longrightarrow Comp$
map $\c S=(I, \Omega)\in Symp$ to $I\in Comp$.
For $h=(x_1,x_2,x_3)\in H^2(M,{\Bbb R})\times H^2(M,{\Bbb R})\times H^2(M,{\Bbb R})$,
let $\pi_i(h)=x_i$.
\hfill
\claim \label{_forgetting-n-periods_Claim_}
Let $\c H\in Hyp$, $\c S\in Symp$.
Then
(i) $P_s(\Phi^{hyp}_s(\c H))= \pi_2(P_{hyp}(\c H))+
\sqrt{-1}\:\pi_3(P_{hyp}(\c H))$
(ii) The point $P_c(\Phi^{s}_c(\c S))\in {\Bbb P} H^2(M,{\Bbb C})$
corresponds to a line generated by $P_s(\c S)$.
{\bf Proof:} Clear. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
According to the general formalism of Kodaira and Kuranishi, $Comp$ is
endowed with a canonical structure of a complex variety.
Using the complex structure on $Comp$, we describe the map
$P_c^s:\; Symp\longrightarrow Comp$ in terms of algebraic geometry.
Let $L$ be a holomorphic vector bundle over a complex variety $X$.
Let $Tot(L)$ be the total space of $L$.
By definition, $Tot(L)$ is a complex variety which is
smoothly fibered over $X$. Every holomorphic section
$f\in \Gamma_X(L)$ gives a standard holomorphic map
$s_f:\; X\longrightarrow Tot(L)$. Consider the map $s_0:\; X\longrightarrow Tot(L)$
corresponding to the zero section of $L$. This map identifies
$X$ with the closed analytic subspace of $Tot(L)$.
Let $Tot^*(L):= Tot(L)\backslash X$ be the completion
of $Tot(L)$ to $X$.
\hfill
\proposition \label{_Symp_as_a_total_space_Proposition_}
Let $M$ be a hyperkaehler manifold, $Comp$ and $Symp$ be the
moduli spaces associated with $M$ as in \ref{_Comp_Hyp_Definition_}.
Then there exist a natural holomorphic linear bundle $\tilde \Omega$
on $Comp$ such that the following conditions hold.
(i) There exist a natural homeomorphism
$i:\; Tot^*(\tilde \Omega)\longrightarrow Symp$.
(ii) Let $\pi:\; Tot^*(\tilde \Omega)\longrightarrow Comp$ be the
standard projection. Then the diagram
\[
\begin{array}{ccccc}
Tot^*(\tilde \Omega)&\!\!\!\stackrel {i}\longrightarrow \!\!\!& Symp\\[3mm]
\;\;\;\;\;\searrow\!\!{}^\pi\!\!\!\!\!
&& \!\!\!\swarrow\!\!{}_{\Phi^s_c}\;\;\;\;\; \\[3mm]
& \!\!\!\!\!Comp &\\
\end{array}
\]
is commutative.
\hfill
The homeomorphism $i$ defines a complex analytic structure on $Symp$.
Further on, we consider both $Comp$ and $Symp$ as complex analytic
varieties.
\hfill
{\bf Proof of \ref{_Symp_as_a_total_space_Proposition_}:}
Let ${A}:= Comp\times H^2(M, {\Bbb C})$.
The holomorphic symplectic structure $(I,\Omega)\in Symp$
is uniquely defined by $I\in Comp$ and the cohomology class
$[\Omega]\in H^2(M, {\Bbb C})$. This defines an injection
\[ j:\; Symp\hookrightarrow {A}, \;\;
j(I,\Omega)= (I, [\Omega]).
\]
Consider ${A}$ as the total space of a trivial holomorphic
bundle ${A}_b$ with the fiber $H^2(M,{\Bbb C})$.
We construct $\tilde \Omega$ as a linear subbundle of
${A}_b$, such that its total space coinsides with
$j(Symp)$.
\hfill
Let $U\subset Comp$ be an open set. We say that
{\bf there exists a universal fibration over $U$}
if there exist a smooth complex analytic fibration
$\pi:\;\goth M\longrightarrow U$ such that for all $J\in U$,
the fiber $\pi^{-1}(J)$ is isomorphic to $(M,J)$.
\hfill
\claim \label{_unive_fibra_exi_loca_Claim_}
For all $I\in Comp$, there exist an open set $U\subset Comp$,
$I\in U$, which admits universal fibration.
{\bf Proof:} This is a consequence of Kodaira-Spencer theory
(see \cite{_Kodaira_Spencer_}). $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Let $U\subset Comp$ be an open subset which admits a universal fibration
$\goth M\stackrel \pi \longrightarrow U$. Let ${\Bbb C}_{\goth M}$ be the
constant sheaf over $\goth M$, and $\pi_\bullet$ be the sheaf-theoretic
direct image. Let $H^2:=R^2\pi_\bullet {\Bbb C}_{\goth M}$ be the second derived
functor of $\pi_\bullet$ applied to ${\Bbb C}_{\goth M}$. Since ${\Bbb C}_{\goth M}$
is a constructible sheaf, and $\pi$ is a proper morphism,
the sheaf $H^2$ is also constructible.
For every point $I\in U$, the restriction $H^2\restrict{I}$
is isomorphic to $H^2(M,{\Bbb C})$. Hence, $H^2\restrict{I}$ is a locally
constant sheaf. This sheaf is equipped with a natural flat connection,
known as Gauss-Manin connection. Since $U$ is a
subset in the space of {\it marked} deformations of $M$,
the monodromy of Gauss-Manin connection is trivial. Hence,
the bundle $H^2$ is naturally isomorphic to ${A}_b\restrict{U}$.
Let $F^0\subset F^1\subset F^2={A}_b\restrict{U}$ be the variation of
Hodge structures associated with $\pi$. By definition,
$F^i$ are holomorphic sub-bundles of ${A}_b\restrict{U}$.
For every $I\in U$, we have $F^0\cong H^{2,0}((M,I))$.
Therefore, $F^0$ is a linear sub-bundle of ${A}_b\restrict{U}$.
\hfill
\lemma \label{_hol_bundle_from_Hodge_str_and_j_Lemma_}
Let $U\subset Comp$ be an open set which admits an universal
fibration $\goth M\longrightarrow U$. Let $F^0\subset {A}_b\restrict{U}$
be the holomorphic linear bundle defined as above.
Let
\[
Symp(U):= \{ (I,\Omega)\in Symp\;\;|\;\; I\in U\}.
\]
Let $Tot(F^0)\subset U\times H^2(M,{\Bbb R})$ be the total space
of $F^0$ considered as a subspace in a total space of
${A}_b\restrict{U}$. Then $Tot^*(F^0)$ coinsides with $j(Symp(U))$
{\bf Proof:} Clear. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
For every open set which admits an universal
fibration $\goth M\longrightarrow U$, we defined the
holomorphic linear bundle $F^0\subset {A}_b\restrict{U}$.
\ref{_hol_bundle_from_Hodge_str_and_j_Lemma_} implies that
$F^0\subset {A}_b\restrict{U}$ is independent from the choice of the
universal fibration, and that the locally defined
sub-bundles $F^0$ can be glued to a globally defined
holomorphic linear sub-bundle in ${A}_b$. Denote this
linear sub-bundle by $\tilde \Omega$. It is clear that
$Tot^*(\tilde \Omega)$ coinsides with $j(Symp)$.
Since $j$ is injective, there exist an inverse homomorphism
$i:\; Tot^*(\tilde \Omega)\longrightarrow Simp$. The condition
(ii) of \ref{_Symp_as_a_total_space_Proposition_} is
obvious. It remains to show that the bijective maps $i$ and
$j= i^{-1}$ are continuous. This is left to the reader
as an exercise. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\section[Hodge-Riemann relations for the
hyperkaehler manifolds and period map.]
{Hodge-Riemann relations for the \\
hyperkaehler manifolds and period map.}
\label{_Period_and_Hodge_Riemann_Section_}
\subsection{What do we do in this section:}
With every hyperkaehler manifold $M$, we associate
the action of the group $G_M\cong SU(2)$ on the cohomology
of $M$ (see \ref{_there_is_action_of_G_M_Proposition_}).
Let $P_{hyp}(M):= (\omega_I,\omega_J,\omega_K)$ be the periods
of $M$. We show that the action of $G_M$ may be reconstructed
from the periods. This follows from \ref{_Lambda_dual_to_L_Proposition_}
and \ref{_g_m_from_L_Lambda_Claim_}.
The action of $G_M\cong SU(2)$ on $H^2(M)$ induces a weight
decomposition of $H^2(M)$. Using this decomposition, we obtain
an interesting version of Hodge-Riemann relations
(\ref{_restrictions_of_pairings_to_H^2_Lemma_}). In
particular, we obtain that for every hyperkaehler structure
$(I,J,K,(\cdot,\cdot))$ on $M$, the Hodge-Riemann
pairings associated with the complex structures $I$, $J$ and
$K$ are equal (\ref{_pairings_on_H^2_are_equal_Proposition_}).
\subsection{Hodge-Riemannian pairing.}
\vspace{3mm}
\hspace{5.5mm}In this subsection we follow \cite{_Weil_}.
\hfill
Let $X$ be a compact Kaehler manifold, and
$\Lambda^*(X)=\oplus \Lambda^{p,q}(X)$
be a space of differential forms equipped with Hodge decomposition.
The Riemannian structure on $X$ equips $\Lambda^*(X)$ with a positively
defined Hermitian metric (see \cite{_Weil_} for correct normalization of
this metric). Integrating the scalar product of two forms over $X$,
we obtain a Hermitian positively defined pairing on the space
of global sections of $\Lambda^*(X)$. Let us identify the cohomology
space of $X$ with the space of harmonic differential forms. This gives
a positively defined Hermitian product on the space of cohomology
$H^*(X)$. We denote it by
\[ (\cdot,\cdot)_{Her}:\; H^i(X,{\Bbb C})\times H^i(X,{\Bbb C}) \longrightarrow {\Bbb C}. \]
Let $I:\; H^i(X,{\Bbb C})\times H^i(X,{\Bbb C}) \longrightarrow {\Bbb C}$ map $(x,y)$ to
$(x,\bar y)_{Her}$. Clearly, $I$ is a complex-linear non-degenerate
2-form on $H^i(X,{\Bbb C})$, which is defined over reals.
Let $A:\; H^i(X,{\Bbb C})\times H^i(X,{\Bbb C}) \longrightarrow {\Bbb C}$ map
$x,y \in H^i(X,{\Bbb C})$ to
\[ \int_X x\wedge y\wedge \omega^{n-i},
\]
where $n=\dim_{\Bbb C} X$, and $i\leq n$. Let $C:\; H^*(X,{\Bbb C})\longrightarrow H^*(X,{\Bbb C})$
be the Weil operator, which maps a cohomology class
$\omega\in H^{p,q}(X)\subset H^{p+q}(X)$ to $\sqrt{-1}\:^{p-q} \omega$.
Let $L:\; H^i(X)\longrightarrow H^{i+2}(X)$,
$\Lambda:\; H^i(X)\longrightarrow H^{i-2}(X)$
be the Hodge operators, and $P^i(X)\subset H^i(X)$
be the space of primitive cohomology classes:
\[ P^i(X) = \{ \alpha \in H^i(X) \;\; |\;\; \Lambda(\alpha) \} =0 \]
By Lefshetz theorem,
\[
H^i(X) = \oplus L^r P^{i-r}(X).
\]
Let $p_r:\; H^i(X) \longrightarrow P^{i-r}(X)$ be a map corresponding to
this decomposition, such that for all $a\in H^*(M)$,
\[ a = \sum_r L^r p_r(a). \]
The forms $A$ and $I$ are related by the so-called
Hodge-Riemann equation:
\begin{equation}\label{_Hodge_Riemann_general_Equation_}
(-1)^\frac{(n-i)(n-i-1)}{2} A(a,C b) =
\sum_r \mu_r\frac{(n-p+r)!}{r!} I(L^r p_r(a), L^r p_r(b)),
\end{equation}
where $\mu_r$ are positive real constants which depend only on
$r$ and dimension of $X$.
Let $\omega\in H^2(X, {\Bbb R})$ be a Kaehler class of $X$.
We call the form
\begin{equation}\label{_Hodge_Riemann_form_general_Equation_}
(-1)^\frac{(n-i)(n-i-1)}{2} A(a,C b):
\; H^i(X,{\Bbb C})\times H^i(X,{\Bbb C}) \longrightarrow {\Bbb C}
\end{equation}
{\bf the Hodge-Riemann pairing associated with a Kaehler class
$\omega$} and
denote this form by $(\cdot,\cdot)_{\omega}$.
\hfill
\claim \label{_Hodge_Riema_general_Claim_}
(i) The form $(\cdot,\cdot)_{\omega}$
depends only on the Kaehler class $\omega\in H^2(X,{\Bbb R})$ of $X$.
In other words, $(\cdot,\cdot)_{\omega}$
would not change if we modify the complex structure or Kaehler
metric, provided that the Kaehler class stays the same.
(ii) The form $(\cdot,\cdot)_{\omega}$ is defined over reals.
(iii) If $X$ is a surface, then the restriction of $(\cdot,\cdot)_{\omega}$
to the primitive cohomology $P^2(X)\subset H^2(X)$
coincides with the intersection form.
(iv) Restriction of $(\cdot,\cdot)_{\omega}$ to $H^2(X)$ can be written
as follows:
\begin{equation} \label{_H_R_to_H^2_formula_Equation_}
(\eta_1,\eta_2)_{\omega}=\int_X \omega^{n-2}\wedge \eta_1\wedge\eta_2
- \frac{n-2}{(n-1)^2} \cdot \frac{
\int_X \omega^{n-1}\eta_1 \cdot \int_X\omega^{n-1}\eta_2}
{\int_X \omega^n}.
\end{equation}
\hfill
{\bf Proof:} Follows from \eqref{_Hodge_Riemann_general_Equation_}
(see also \cite{_Weil_}). $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\subsection{Riemann-Hodge relations in hyperkaehler case.}
Let $M$ be a compact hyperkaehler manifold and $\c H\in Hyp$ be
a hyperkaehler structure on $M$. Let $P_{hyp}(\c H)$ be
denoted by $(x_1, x_2, x_3)$, $x_i\in H^2(M,{\Bbb R})$.
The hyperkaehler structure $\c H\in Hyp$ defines a Riemannian
metric on $M$. This metric establishes a positively
defined Hermitian scalar product on the space of ${\Bbb C}$-valued
differential forms over $M$. Realizing the cohomology classes
as harmonic forms, we obtain a Hermitian pairing
$(\cdot,\cdot)_{Her}$ on $H^i(M,{\Bbb C})$.
Let $\Lambda_{x_i}\!:\; H^i(M)\longrightarrow H^{i-2}(M)$ be
the operator adjoint to $L_{x_i}$ with respect to the
pairing $(\cdot,\cdot)_{Her}$. Clearly,
$\Lambda_{x_i}$ is the Hodge operator associated with the Kaehler
structure on $M$ which is defined by $x_i$ and ${\cal H}$.
Let $\inangles{\cdot,\cdot}_{x_i}$ be the Hodge-Riemann form
\eqref{_Hodge_Riemann_form_general_Equation_} associated
with the Kaehler form $x_i$. Let $L\check{\;}_{x_i}$ be an
operator adjoint to $L_{x_i}$ with respect to $\inangles{\cdot,\cdot}_{x_i}$.
\hfill
\proposition \label{_Lambda_dual_to_L_Proposition_}
\ \ $L\check{\;}_{x_j}=\Lambda_{x_j}$ \ \ \ for $j=1,2,3$.
{\bf Proof:} Fix a choice of $j\in\{1,2,3\}$. For simplicity,
assume that $j=1$. We abbreviate $L_{x_1}$ by $L$, $\Lambda_{x_1}$
by $\Lambda$.
Let $I$ be the complex structure
induced by ${\cal H}$, such that $x_1\in H^2(M,{\Bbb R})$
is the Kaehler class of $I$. Take the Lefschetz decomposition
(\cite{_Griffiths_Harris_})
\[ H^k(M)=\bigoplus L^i P^{k-2i}(M),\;\;\;
P^{k-2i}\subset H^{k-2i}(M),
\]
where the space $P^i(M)$ is a space of all primitive
classes:
\[ P^i(M)=
ker\bigg(\Lambda:\; H^i(M)\longrightarrow H^{i+2}(M)\bigg).
\]
Let $P^{p,q}(M):= P^{p+q}(M)\cap H^{p,q}(M)$.
It is well known that
\[ P^{i}(M)=\bigoplus\limits_{p+q=i}P^{p,q}(M). \]
(see \cite{_Griffiths_Harris_}, \cite{_Weil_}).
Hodge-Riemann relations (\eqref{_Hodge_Riemann_general_Equation_};
see also \cite{_Weil_})
describe $(\zeta_1,\zeta_2)_{Her}$ in terms of
$\inangles{\zeta_1,\bar\zeta_2}_{x_1}$ and Lefschetz
decomposition. Let $n=dim_{\Bbb C}(M)$.
When
\[ \zeta_1\in L^i P^{p,q}(M), \;\; \zeta_2\in L^{i'} P^{p',q'}(M)
\;\;\;\mbox{\it and} \;\;\;(i,p,q)\neq (i',p',q'),
\]
both scalar products vanish:
\begin{equation}\label{_Hodge_Riemann_relations_vanishing_Equation}
(\zeta_1,\zeta_2)_{Her}=
\inangles{\zeta_1,\bar\zeta_2}_{x_1}=0.
\end{equation}
\hfill
When $\zeta_1\in L^i P^{p,q}(M)$ and
$\zeta_2\in L^i P^{p,q}(M)$, we have
\begin{equation}\label{_Hodge_Riemann_relations_Equation_}
(\zeta_1,\zeta_2)_{Her}=
\sqrt{-1}\:^{p-q}(-1)^{\frac{(n-p-q)(n-p-q-1)}{2}}
\inangles{\zeta_1,\bar\zeta_2}_{x_1}.
\end{equation}
\hfill
The operator $L\check{\;}_{x_1}$ is adjoint to $L_{x_1}$ with respect
to $\inangles{\cdot,\cdot}_{x_1}$ and $\Lambda_{x_1}$
is adjoint to $L_{x_1}$ with respect to $(\cdot,\cdot)_{Her}$.
Let $\zeta\in L^i P^{p,q}(M)$. By definition,
$t=L\check{\;}_{x_1}(\zeta)$ is the element of
$H^{2i-2+p+q}(M)$ such that $\forall \xi \in H^{2i-2+p+q}(M)$
we have
\begin{equation} \label{_L_galochka_definition_Equation_}
\inangles{t,\xi}_{x_1} =
\inangles{\zeta,L_{x_1}(\xi)}_{x_1}.
\end{equation}
Using \eqref{_Hodge_Riemann_relations_vanishing_Equation},
we see that $t\in L^{i-1} P^{p,q}(M)$. For
$t\in L^{i-1} P^{p,q}(M)$,
\eqref{_Hodge_Riemann_relations_vanishing_Equation}
shows that if \eqref{_L_galochka_definition_Equation_}
holds for all $\xi\in L^{i-1} P^{p,q}(M)$,
this equation holds for all $\xi \in H^{2i-2+p+q}(M)$.
On the other hand, Hodge-Riemann relations
\eqref{_Hodge_Riemann_relations_Equation_}
imply that for $\xi,t\in L^{i-1} P^{p,q}(M)$,
\[ (t,\bar \xi)_{Her}=
\sqrt{-1}\:^{p-q}(-1)^{\frac{(n-p-q)(n-p-q-1)}{2}}
\inangles{t,\xi}_{x_1} =
\]
\[
= \sqrt{-1}\:^{p-q}(-1)^{\frac{(n-p-q)(n-p-q-1)}{2}}
\inangles{\zeta,L_{x_1}(\xi)}_{x_1}=
(\zeta,L_{x_1}\bar\xi)_{Her}.
\]
Therefore $L\check{\;}_{x_1}$ is adjoint to
$L_{x_1}$ with respect to $(\cdot,\cdot)_{Her}$.
\ref{_Lambda_dual_to_L_Proposition_} is proven.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
In Section \ref{hyperk_manif_Section_},
we defined the action of $G_M\cong SU(2)$
on the cohomology of a hyperkaehler manifold.
For every hyperkaehler structure ${\cal H}\in \mbox{\it Hyp}$,
there is an action of $SU(2)$ on $H^*(M_{C^\infty})$
which is determined by ${\cal H}$. We proceed to describe this
$SU(2)$-action in terms of the triple
\[ (x_1,x_2,x_3)=P_{hyp}({\cal H})\in
H^2(M)\oplus H^2(M)\oplus H^2(M).
\]
\ref{_Lambda_dual_to_L_Proposition_} expresses
the Hodge operators $\Lambda_{x_i}$ via the multiplicative
structure on $H^*(M_{C^\infty})$. Let
$\goth a_{\cal H}\subset End(H^*(M,{\Bbb R})$ be the
Lie algebra generated by $L_{x_i}$, $\Lambda_{x_i}$, $i=1,2,3$.
According to \cite{_so5_on_cohomo_},
$\goth a_{\cal H}\cong \goth{so}(4,1)$.
\hfill
Let ${\goth g}_{\cal H}\subset \goth a_{\cal H}$ be the subalgebra of
$\goth a_{\cal H}$ consisting of all elements which
respect the grading on $H^*(M)$ induced by the degree:
\[ {\goth g}_{\cal H}:=
\{ x\in \goth a_{\cal H}\;\; | \;\; x(H^i(M))\subset H^i(M),
\;\;i=0,1, ... \; 2n.\}
\]
\hfill
\claim \label{_g_m_from_L_Lambda_Claim_}
The Lie algebra ${\goth g}_{\cal H}$ is isomorphic to
$\goth{su}(2)$. Its action coincides with that
of ${\goth g}_M$ defined in Section \ref{hyperk_manif_Section_}.
{\bf Proof:} This is Theorem 2 of
\cite{_so5_on_cohomo_}. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
The forms $(\cdot,\cdot)_{x_i}$, $i=1,2,3$ depend only on the
value of $x_i\in H^2(M,{\Bbb R})$. Therefore,
\ref{_Lambda_dual_to_L_Proposition_} has the following
interesting consequence:
\hfill
\corollary \label{_so(5)_inde_from_H_Corollary_} %
Let $\c H\in Hyp$ be a hyperkaehler structure
on $M$, and ${\goth g}_{\cal H}\cong \goth{su}(2)$,
$\goth a_{\c H}\cong \goth{so}(4,1)$ be the corresponding
Lie algebras defined as above. Then the action of
${\goth g}_{\cal H}$, $\goth a_{\c H}$ on $H^*(M,{\Bbb R})$ depends only
on hyperkaehler periods of $\c H$.
In other words, if $\c H_1$,
$\c H_2$ are hyperkaehler structures such that
\[P_{hyp}(\c H_1)= P_{hyp}(\c H_2),\] then action
of $\goth a_{\c H_1}$, ${\goth g}_{\c H_1}$ on $H^*(M)$ coinsides
with action of $\goth a_{\c H_2}$, ${\goth g}_{\c H_2}$.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\proposition \label{_pairings_on_H^2_are_equal_Proposition_}
In assumptions of \ref{_Lambda_dual_to_L_Proposition_}, let
$\inangles{\cdot,\cdot}_i$ be the restriction of the
pairing $\inangles{\cdot,\cdot}_{x_i}$ to $H^2(M)$.
Then $\inangles{\cdot,\cdot}_1 =
\inangles{\cdot,\cdot}_2=\inangles{\cdot,\cdot}_3$.
{\bf Proof:} Let $(\cdot,\cdot)$ be the restriction
of $(\cdot,\cdot)_{\cal H}$ to $H^2(M)$.
Let $V$ be the subspace of $H^2(M)$ spanned by
$(x_1,x_2,x_3)$.
Earlier, we defined the action of the Lie algebra
${\goth g}_{\cal H}\cong\goth{su}(2)$ on $H^2(M)$.
Let $H_{inv}$ be the space of all ${\goth g}_{\cal H}$-invariant
elements in $H^2(M)$. According to \cite{_so5_on_cohomo_},
the action of ${\goth g}_{\cal H}$ on $H^*(M)$ induces the Hodge
decomposition. Namely, for every induced complex structure
$I$ there exist a Cartan subalgebra $\goth h\in {\goth g}_{\cal H}$
such that the weight decomposition on $H^*(M)$ induced by
$\goth h$ coincides with the Hodge decomposition
\[ H^i(M)=\bigoplus\limits_{p+q=i}H^{p,q}(M). \]
The space $H^{2,0}(M)$ is one-dimensional
for every induced complex structure. Using the theory
of representations of $\goth{sl}(2)$, one can check that
this implies that $H^2(M)/H_{inv}$ is a simple
3-dimensional representation of ${\goth g}_{\cal H}$.
Another trivial calculation shows that
$V$ is a ${\goth g}_{\cal H}$-invariant subspace of
$H^2(M)$, and ${\goth g}_{\cal H}$ acts on $V$ non-trivially.
Therefore $H^2(M)=H_{inv}\oplus V$.
\ref{_pairings_on_H^2_are_equal_Proposition_}
is implied by the following lemma.
\hfill
\lemma \label{_restrictions_of_pairings_to_H^2_Lemma_}
Consider the restrictions of $\inangles{\cdot,\cdot}_i$
and $(\cdot,\cdot)$ to $V$ and $H_{inv}$. Then
\begin{equation} \label{_restriction_to_V_Equation_}
\inangles{x,\bar y} = (x,\bar y)
\;\;\;\mbox{for} \;\;x,y\in V,
\end{equation}
\begin{equation} \label{_restriction_to_H_inv_Equation_}
\inangles{x,\bar y}_i =
-(x,\bar y)\;\;\;\mbox{for}\;\; x,y\in H_{inv}, \;\:
\mbox{\it\ and finally,}
\end{equation}
\begin{equation} \label{_restriction_to_H_and_V_Equation_}
\inangles{x,\bar y}_i =
(x,\bar y)=0\;\;\;\mbox{for} \;\;x\in V,\; y\in H_{inv}.
\end{equation}
\hfill
{\bf Proof:} Let $I$ be an induced complex structure.
Then $V=H^{2,0}\oplus L_I (H^{0,0})\oplus H^{0,2}(M)$.
where the Hodge decomposition is taken with respect
to $I$ and $L_I$ is the Hodge operator of exterrior
multiplication by the Kaehler class of $I$. Then
\eqref{_restriction_to_V_Equation_} immediately follows
from Hodge-Riemann relations
\eqref{_Hodge_Riemann_relations_Equation_}.
By \ref{_inv_2-forms_have_zero_degree_Claim_}
all elements of $H_{inv}$ are primitive.
On the other hand,
$H_{inv}\subset H^{1,1}$ by
\ref{_G_M_invariant_forms_Proposition_}. Therefore
\eqref{_restriction_to_H_inv_Equation_}
follows from
\eqref{_Hodge_Riemann_relations_Equation_},
and \eqref{_restriction_to_H_and_V_Equation_}
follows from \eqref{_Hodge_Riemann_relations_vanishing_Equation}.
\ref{_restrictions_of_pairings_to_H^2_Lemma_}
and consequently \ref{_pairings_on_H^2_are_equal_Proposition_}
are proven.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\corollary \label{_indu_comple_same_HR_Corollary_}
Let $\c H= (I, J, K, (\cdot,\cdot))$ be a hyperkaehler
structure on $M$, and $L=a I + bJ + cK$ be an induced
complex structure, $a^2+b^2+c^2=1$. Let $\omega_1\in H^2(M,{\Bbb R})$
be the Kaehler class associated with the Kaehler manifold
$(M, I)$, and $\omega\in H^2(M,{\Bbb R})$ be the Kaehler
class associated with the Kaehler manifold $(M, L)$.
Let $(\cdot,\cdot)_\omega$,
$(\cdot,\cdot)_{\omega_1}:\; H^2(M)\times H^2(M)\longrightarrow {\Bbb C}$
be the Hodge-Riemann forms associated with $\omega$, $\omega_1$.
Then $(\cdot,\cdot)_{\omega_1}=(\cdot,\cdot)_\omega$
{\bf Proof:} Follows from \ref{_restrictions_of_pairings_to_H^2_Lemma_}
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\section{The Hodge-Riemann metric on $H^2(M)$ does not depend
on complex structure.}
\label{_Hodge-Rie_independent_Section_}
Let $M_{C^\infty}$ be a compact manifold which
admits a hyperkaehler structure. Let $Hyp$, $Symp$, $Comp$ be the moduli
spaces constructed in Section \ref{_moduli_Section_}.
\hfill
\definition \label{_Hodge_Riemann_asso_w_hyperkeahler_Definition_}
Let $M$ be a hyperkaehler manifold, $\c H =(I, J, K, (\cdot, \cdot))$
be its hyperkaehler structure and $\omega_1$, $\omega_2$,
$\omega_3\in H^2(M, {\Bbb R})$ be Kaehler
classes associated with induced complex structures $I$, $J$, $K$.
Consider the Riemann-Hodge pairing
$(\cdot,\cdot)_{\omega_i}:\; H^2(M,{\Bbb R})\times H^2(M, {\Bbb R})\longrightarrow {\Bbb R}$, $i=1,2,3$
defined as in \eqref{_Hodge_Riemann_form_general_Equation_}
(see also \eqref{_H_R_to_H^2_formula_Equation_}).
According to \ref{_pairings_on_H^2_are_equal_Proposition_},
\[ (\cdot,\cdot)_{\omega_1}= (\cdot,\cdot)_{\omega_2} =
(\cdot,\cdot)_{\omega_3}.
\]
Let
\[ \inangles{x,y}_{\c H} := (x,y)_{(vol(M))^{-1/n}\cdot x_i}
\]
where the volume $vol(M)= \int_M x_i^n$ is volume
calculated with respect to the Riemannian metric
$(\cdot,\cdot)$, and $n=\frac{\dim_{\Bbb R}(M)}{2}$.
This pairing is called {\bf the normalized Hodge-Riemann pairing
associated with the hyperkaehler structure $\c H$}.
According to \eqref{_H_R_to_H^2_formula_Equation_},
the normalized Hodge-Riemann pairing
$\inangles{\cdot,\cdot}_{\c H}$ can be expressed by
\begin{equation} \label{_normalized_HR_Equation_}
(\eta_1,\eta_2)_{\omega}=
\lambda^{n-2}\int_X \omega^{n-2}\wedge \eta_1\wedge\eta_2
- \frac{n-2}{(n-1)^2}
\lambda^{2n-2}\int_X \omega^{n-1}\eta_1 \cdot \int_X\omega^{n-1}\eta_2
\end{equation}
where $\lambda = (vol(M))^{-1/n}$.
\hfill
The main result of this section is the following theorem:
\hfill
\theorem \label{_Hodge_Riemann_independent_Theorem_}
Let $\c H_1$, $\c H_2\in \mbox{\it Hyp}$ be hyperkaehler structures on
$M_{C^\infty}$. Then
$\inangles{\cdot, \cdot}_{\c H_1} =
\inangles{\cdot, \cdot}_{\c H_2}$
In other words, the normalized Hodge-Riemann pairing
associated with the point $\c H\in \mbox{\it Hyp}$ does not depend
on the choice of $\c H$ in $Hyp$.
\hfill
{\bf Proof:} The space $\mbox{\it Hyp}$ is endowed with the homogenous
action of the group $SO(3)$ as follows. Let
$(I, J, K, (\cdot, \cdot))\in \mbox{\it Hyp}$ be a hyperkaehler structure.
Consider the complex structures $I$, $J$, $K$ as endomorphisms
of the tangent bundle $TM$. We express this by
$I, J, K\in \Gamma_M(End(TM))$. Consider the
three-dimensional subspace $V\subset \Gamma_M(End (TM))$ generated
by $I$, $J$, $K$. By definition of a hyperkaehler structure,
$V$ is a three-dimensional vector space equipped with a canonical
isomorphism with the space of anti-self-adjoint quaternions.
The space $\c V$ of anti-self-adjoint quaternions is a Lie subalgebra
of the quaternion algebra. The space of sections
$\Gamma_M(End (TM))$ has a canonical algebra structure. By definition,
$V$ is a Lie subalgebra of $\Gamma_M(End(TM))$, and the Lie
algebra structure on $V$ coincides with that on $\c V\subset \Bbb H$.
The Lie algebra $\c V\subset \Bbb H$ is isomorphic to $\goth{so}(3)$.
Consider the adjoint action of $SO(3)$ on $V\cong \c V\cong \goth {so}(3)$.
Let $A\in SO(3)$. By definition of adjoint action,
\begin{equation} \label{_A_of_quate_quate_Equation_}
\begin{array}{l}
A(I)^2=A(J)^2=A(K)^2, \mbox{ \ and \ } \\
A(I)\circ A(J)=A(K)=-A(J)\circ A(I)
\end{array}
\end{equation}
The operators $I$, $J$, $K$ are parallel with respect
to the Levi-Civita connection. The operators $A(I)$, $A(J)$, $A(K)$
are linear combinations of $I$, $J$, $K$ with constant coefficients.
Hence, these operators are also parallel. They are orthogonal
by trivial reasons.
\hfill
\claim \label{_Newla_Niere_for_para_Claim_}
Let $X$ be a Riemannian
manifold equipped with Levi-Civita connection. Let $\c I$ be an
orthogonal almost complex structure which is parallel with respect
to the connection. Then $\c I$ is an integrable (i. e., defines a
complex structure).
Moreover, the Riemannian metric on $X$ is Kaehler.
{\bf Proof:} This follows from Newlander-Nierenberg theorem.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\ref{_Newla_Niere_for_para_Claim_} implies that
$A(I)$, $A(J)$, $A(K)$ are operators of complex structure.
Now, \eqref{_A_of_quate_quate_Equation_} implies that
$(A(I), A(J), A(K), (\cdot, \cdot))$ is a hyperkaehler structure.
We obtain an action of $SO(3)$ on $Hyp$.
\hfill
\definition \label{_action_SO(3)_on_Hyp_Definition_}
This action of $SO(3)$ on $\mbox{\it Hyp}$ is called {\bf a standard
action of $SO(3)$ on the space of hyperkaehler structures}.
Two hyperkaehler structures are called {\bf equivalent}
if one can be obtained from another by the standard action
of $SO(3)$.
\hfill
It is easy to describe the action of $SO(3)$ on $\mbox{\it Hyp}$
in terms of the period map:
\hfill
\claim\label{_action_SO(3)_on_Hyp_via_periods_Lemma_}
Let $\c H \in \mbox{\it Hyp}$ and $A\in SO(3)$. Consider
$P_{hyp}(\c H)$ and $P_{hyp}(A(\c H))$ as elements
of the space
\[ W:= H^2(M,{\Bbb R})\otimes{\Bbb R}^3
\cong H^2(M,{\Bbb R})\oplus H^2(M,{\Bbb R})\oplus H^2(M,{\Bbb R}).
\]
Then $P_{hyp}(A(\c H))$ is obtained from $P_{hyp}(\c H)$
by applying $Id\otimes A$ to $P_{hyp}(\c H)\in H^2(M,{\Bbb R})\otimes{\Bbb R}^3$.
{\bf Proof:} Clear. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\lemma \label{_Hodge_Riemann_independe_for_equiva_hyperkae_Lemma_}
In assumptions of \ref{_Hodge_Riemann_independent_Theorem_},
Let $\c H_1$, $\c H_2\in \mbox{\it Hyp}$ be
equivalent hyperkaehler structures on
$M_{C^\infty}$. Then
\[ \inangles{\cdot, \cdot}_{\c H_1} =
\inangles{\cdot, \cdot}_{\c H_2}
\]
{\bf Proof:} Follows from
\ref{_indu_comple_same_HR_Corollary_}. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\definition \label{_admissible_substi_Definition_}
Let $\c H_1$, $\c H_2\in \mbox{\it Hyp}$. We say that $\c H_2$
{\bf is obtained from $\c H_1$ by an admissible substitution}
if either of the following two conditions hold:
(i) There exists $\lambda\in {\Bbb R}$ such that
$P_3(\c H_1)= \lambda P_3(\c H_2)$.
(ii) $\c H_1$ is equivalent to $\c H_2$.
We say that $\c H_1$ and $\c H_2$ are {\bf well connected}
if $\c H_2$ can be obtained from $\c H_1$ by a sequence of
admissible substitutions. Obviously, this relation is an equivalence relation.
\hfill
\lemma \label{_Hodge_Riemann_independe_for_well_connected_Lemma_}
In assumptions of \ref{_Hodge_Riemann_independent_Theorem_},
Let $\c H_1$, $\c H_2\in \mbox{\it Hyp}$ be the hyperkaehler structures on
$M_{C^\infty}$. Assume that $\c H_1$ and $\c H_2$ are well connected.
Then
\[ \inangles{\cdot, \cdot}_{\c H_1} =
\inangles{\cdot, \cdot}_{\c H_2}.
\]
{\bf Proof:} It is sufficient to prove
\ref{_Hodge_Riemann_independe_for_well_connected_Lemma_}
assuming that $\c H_2$ is obtained from $\c H_1$ by admissible
substitution. In other words, we may assume that one
of conditions (i) and (ii) of \ref{_admissible_substi_Definition_}
holds. When (i) holds,
\ref{_Hodge_Riemann_independe_for_well_connected_Lemma_}
follows from
\ref{_Hodge_Riemann_independe_for_equiva_hyperkae_Lemma_}. When (ii)
holds, \ref{_Hodge_Riemann_independe_for_well_connected_Lemma_}
is a direct consequence of \eqref{_normalized_HR_Equation_}
(see also \ref{_Hodge_Riema_general_Claim_} (iv)).
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
We obtain that \ref{_Hodge_Riemann_independent_Theorem_}
is a consequence of
\ref{_Hodge_Riemann_independe_for_well_connected_Lemma_}
and the following statement:
\hfill
\proposition \label{_hyperk_are_well_connected_Proposition_}
Let $\c H_1$, $\c H_2\in \mbox{\it Hyp}$ be the hyperkaehler structures on
$M_{C^\infty}$. Then $\c H_1$ and $\c H_2$ are well connected.
{\bf Proof:}
\hfill
\lemma \label{_hyp.st._w/the_same_I_w/conne_Lemma_}
Let $\c H=(I, J, K, (\cdot,\cdot))$
and $\c H'=(I', J', K', (\cdot,\cdot)')$ be two hyperkaehler
structures with $I=I'$. Then $\c H$ and $\c H'$ are well
connected.
{\bf Proof:} Since $I=I'$, we have
\[
P_c(\Phi^{hyp}_c(\c H))=P_c(\Phi^{hyp}_c(\c H')).
\]
By \ref{_forgetting-n-periods_Claim_}, the spaces spanned by
$\inangles{P_2(\c H), P_3(\c H)}$ and $\inangles{P_2(\c H'), P_3(\c H')}$
coinside. Denote $\inangles{P_2(\c H), P_3(\c H)}$ by $W$.
Let $U$ be the group of linear automorphisms
of $W$ which preserve the Hodge-Riemann pairing
$(\cdot,\cdot)_{\c H}$ associated with $\c H$. Using the
basis $W=\inangles{P_2(\c H), P_3(\c H)}$, we may identify
$U$ with $U(1)$. Let $u\in U\cong U(1)$ be represented by
the matrix
\[ u = \bigg(\begin{array}{rr}
\cos(\alpha) & \sin(\alpha) \\
-\sin(\alpha) & \cos(\alpha)
\end{array}
\bigg).
\]
Let $u(J)= \cos(\alpha) J+ \sin(\alpha) K$ and
$u(K)= \cos(\alpha) K- \sin(\alpha) J$. Checking the
definition of the hyperkaehler structure,
one obtains that $u(\c H):= (I, u(J), u(K), (\cdot, \cdot))$
is a hyperkaehler structure which is equivalent to
$\c H$. By \ref{_action_SO(3)_on_Hyp_via_periods_Lemma_},
$P_2(u(\c H))= u(P_2(\c H))$ and $P_3(u(\c H))= u(P_3(\c H))$.
Choosing a suitable $u$, we can make $P_3(u(\c H))$
proportional to $P_3(\c H')$. For such $u$, $u(\c H)$
is well connected to $\c H'$. Since $u(\c H)$ is equivalent
to $\c H$, we obtain that $\c H$ is well connected to $\c H'$.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\lemma\label{_hyper_w/same_ind_comp_str_well_connect_Corollary_
Let $\c H_1$, $\c H_2\in \mbox{\it Hyp}$ be the hyperkaehler structures, and
$I\in Comp$ be the complex structure. Assume that $I$ is induced by
$\c H_1$ and $\c H_2$. Then $\c H_1$ is well connected with $\c H_2$.
{\bf Proof:} Clearly,
\ref{_hyper_w/same_ind_comp_str_well_connect_Corollary_}
follows from \ref{_hyp.st._w/the_same_I_w/conne_Lemma_}
and the following statement:
\hfill
\sublemma \label{_induced_compl_str_turn_to_I_Sublemma_}
Let $(I, J, K, (\cdot,\cdot))=\c H\in \mbox{\it Hyp}$,
$I'\in Comp$ be a complex structure which
is induced by $\c H$. Then $\c H$ is equivalent to a hyperkaehler
structure $\c H' = (I', J', K', (\cdot,\cdot)')$ for some
$J', K', (\cdot,\cdot')$.
{\bf Proof:} Consider the action of $SO(3)$ on the space
$V:=\inangles{I,J,K}\subset \Gamma(End(TM))$ (see
\ref{_action_SO(3)_on_Hyp_Definition_}). Clearly,
$I'$, considered as a section of $\Gamma(End(TM))$, belongs
to $V$. Take a matrix $A\in SO(3)$ which maps
$I\in V$ to $I'$. Then $\c H':=A(\c H)$ satisfies conditions
of \ref{_induced_compl_str_turn_to_I_Sublemma_}.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\definition \label{_well_conne_comple_str_Definition_
Let $I_1$, $I_2\in Comp$. The complex structures $I_1$, $I_2$
are called {\bf well connected} if there exist well connected
hyperkaehler structures $\c H_1$, $\c H_2$ such that
$\c H_1$ induces $I_1$ and $\c H_2$ induces $I_2$.
By \ref{_hyper_w/same_ind_comp_str_well_connect_Corollary_},
this is an equivalence relation.
\hfill
Let $\omega\in H^2(M,{\Bbb R})$. Let $Comp^\omega$ be the set
of all $I\in Comp$ such that $\omega$ belongs to the Kaehler
cone of $I$.
\hfill
\claim \label{_Comp^omega_well_conne_Claim_}
Let $\omega\in H^2(M,{\Bbb R})$, $I, I'\in Comp^\omega$. Then
the complex structures $I$ and $I'$ are well connected.
{\bf Proof:} Clear. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Let $\mbox{\it Kah}$ be the set of all $\omega\in H^2(M,{\Bbb R})$ such that
$\omega$ is a Kaehler class for some complex structure $I\in Comp$.
\hfill
\definition \label{_well_conne_kah_str_Definition_
Let $\omega$, $\omega'\in \mbox{\it Kah}$. The classes $\omega$ and $\omega'$
are called {\bf well connected} if there exist well connected
hyperkaehler structures $\c H$, $\c H'$ such that
$P_1(\c H)= \omega$ and $P_1(\c H')= \omega'$.
\hfill
\lemma \label{_well_conne_Kah_classe_indu_w/c_hype_Lemma_}
Let $\omega$, $\omega'$ be two well connected classes from $\mbox{\it Kah}$.
Let $\c H$, $\c H'$ be the hyperkaehler structures such that
$P_1(\c H)= \omega$ and $P_1(\c H')= \omega'$. Then
$\c H$, $\c H'$ are well connected.
{\bf Proof:} Consider the spaces $Comp^\omega$ and $Comp^{\omega'}$.
Since $\omega$ is well connected to $\omega'$, there exist
well connected hyperkaehler structures
\[ \c F=\bigg(A, B, C, (\cdot,\cdot)\bigg),\;\; \;
\c F'=\bigg(A', B', C', (\cdot,\cdot)'\bigg)
\]
such that
$P_1(\c F)= \omega$ and $P_1(\c F')= \omega'$. Therefore,
$A\in Comp^\omega$ is well connected to $A'\in Comp^{\omega'}$.
Let $\c H = (I, J, K, (\cdot,\cdot))$,
$\c H'=(I', J', K', (\cdot,\cdot)')$. By definition,
$I\in Comp^\omega$ and $I'\in Comp^{\omega'}$. By
\ref{_Comp^omega_well_conne_Claim_}, $I$ is well
connected to $A$ and $I'$ is well connected to $A'$.
Therefore, $I$ is well connected to $I'$. By definition
of well connected complex structures, there exist
well connected hyperkaehler structures $\c G$ and $\c G'$
such that $\c G$ induces $I$ and $\c G'$ induces $I'$
By \ref{_hyper_w/same_ind_comp_str_well_connect_Corollary_},
$\c G$ is well connected to $\c H$ and $\c G'$ is well
connected to $\c H'$. Since the relation of being well connected
is transitive, $\c H$ is well connected to $\c H'$.
\ref{_well_conne_Kah_classe_indu_w/c_hype_Lemma_}
is proven. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
To finish the proof of \ref{_hyperk_are_well_connected_Proposition_},
it is sufficient to show that for all $x,y \in \mbox{\it Kah}$, the cohomology
classes $x$ and $y$ are well connected. By
\ref{_symplectic_=>_hyperkaehler_Proposition_},
$\mbox{\it Kah}=P_1(Hyp)$. Since $\mbox{\it Hyp}$ is connected, $\mbox{\it
Kah}$ is also connected.
Therefore \ref{_hyperk_are_well_connected_Proposition_} is
implied by the following lemma:
\hfill
\lemma \label{_C(omega)_open_in_Kah_Lemma_}
Let $\omega\in \mbox{\it Kah}$, $C(\omega)$ be the set of all
classes $\omega'\in \mbox{\it Kah}$ which are well connected to $\omega$.
Then $C(\omega)$ is open in $\mbox{\it Kah}$.
{\bf Proof:} Since $\mbox{\it Kah}\subset H^2(M,{\Bbb R})$,
it is sufficient to show that $C(\omega)$ is open in $H^2(M,{\Bbb R})$.
Since the relation of being well connected is transitive,
it is sufficient to show that $C(\omega)\subset \mbox{\it Kah}$
contains an open neighbourhood of $\omega$ for all $\omega\in Kah$.
Let $I\in Comp^\omega$. Let $\c H$ be a hyperkaehler
structure associated with $I$ and $\omega$
as in \ref{_symplectic_=>_hyperkaehler_Proposition_}.
Let $K(\c H)\subset Kah$ be the set of all $\eta\in H^2(M,{\Bbb R})$
such that the following condition holds. The hyperkaehler structure
$\c H$ induces a complex structure $L$ such that $\eta\in K(L)$.
As usually, $K(L)$ is the Kaehler cone of $L$.
As the following lemma implies, $K(\c H)\subset C(\omega)$.
\hfill
\sublemma \label{_K(H)_well_conne_to_omega_Sublemma_}
Let $\omega'\in K(\c H)$. Then $\omega'$ is well connected to $\omega$.
{\bf Proof:} Let $\c H\in Hyp$, $\omega=P_1(\c H)$.
Let $L$ be an induced complex structure such that
$\omega'\in K(L)$. By \ref{_induced_compl_str_turn_to_I_Sublemma_},
there exist a hyperkaehler structure $\c H'=(L, J, K, (\cdot,\cdot))$
which is equivalent to $\c H$. Then, $\c H$ is well connected to $\c H'$.
Let $\c H''$ be the hyperkaehler structure associated with $L$ and $\omega'$
as in \ref{_symplectic_=>_hyperkaehler_Proposition_}. By
\ref{_hyper_w/same_ind_comp_str_well_connect_Corollary_},
$\c H'$ and $\c H''$ are well connected. By definition,
$P_1(\c H'')=\omega'$. Since $\c H$ is well connected to $\c H''$,
$\omega$ is well connected to $\omega'$.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Let $\c H \in Hyp$ and $L$ be an induced complex structure.
As usually, we denote the intersection $H^{1,1}(M,L)\cap H^2(M,{\Bbb R})$
by $H^{1,1}_L(M,{\Bbb R})$. Let $\omega\in H^{1,1}_L(M,{\Bbb R})$.
The hyperkaehler structure $\c H$ induces a Riemannian metric on $M$.
Let $\tilde \omega\in \Lambda^2(M,{\Bbb R})$ be the harmonic
form which represents the cohomology
class $\omega$. Hodge theory implies that $\tilde\omega$ is
a form of Hodge type (1,1) with respect to the complex structure
$L$. Under these assumptions, we introduce the following
definition.
\hfill
\definition \label{_positive_classes_Definition_}
We say that the cohomology class $\omega$ is {\bf positive}
with respect to $(\c H,L)$ if the corresponding harmonic
(1,1)-form $\tilde \omega$ is everywhere positively defined.
In other words, $\omega\in H^{1,1}_L(M,{\Bbb R})$ is {\bf positive}
if the symmetric form
\[ S_p: \; T_pM\times T_pM \longrightarrow {\Bbb R},\;\;
S_p(x,y):= \tilde \omega(x, L(y))
\]
is positively defined in every point of $p\in M$.
We denote by $K_{\c H}(L)$
the set of all $\omega\in H^{1,1}_L(M,{\Bbb R})$
such that $\omega$ is positive with respect to $(\c H,L)$.
\hfill
\claim \label{_positive_form_is_Kaehler_Claim_}
In assumptions of \ref{_positive_classes_Definition_},
let $\omega\in H^{1,1}_L(M,{\Bbb R})$ be the two-form which is positive
with respect to $(\c H,L)$. Then $\omega$ is a Kaehler class:
$\omega\in K(L)$.
{\bf Proof:} Clear. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Let $\c H\in Hyp$. Denote by $K_c(\c H)$ the set of all
$\omega\in H^2(M,{\Bbb R})$ such that there exists an induced
complex structure $L$ and $\omega\in K_{\c H}(L)$.
By \ref{_positive_form_is_Kaehler_Claim_},
$K_c(\c H)\subset K(\c H)$. Let $\omega\in C(\omega)$.
This means that $\omega=P_1(\c H)$ for some $\c H\in Hyp$.
Since $K_c(\c H)\subset K(\c H)\subset C(\omega)$, to prove that
$C(\omega)$ is open in $H^2(M,{\Bbb R})$ it is sufficient to show that
$K_c(\c H)$ contains an open neighbourhood of $\omega$
(we use here the transitiveness of well-connectedness).
Therefor, \ref{_C(omega)_open_in_Kah_Lemma_} is
a consequence of the following statement:
\hfill
\proposition \label{_K(H)_open_in_H^2(M,R)_Sublemma_}
Let $M$ be a hyperkaehler manifold with the hyperkaehler
structure $\c H$. Then the set $K_c(\c H)\subset H^2(M,{\Bbb R})$
contains an open neighbourhood of $P_1(\c H)$.
{\bf Proof:} Consider the action of the group of unit quaternions
$G_M\cong SU(2)$ defined as in Section
\ref{hyperk_manif_Section_}. The action of $G_M$ is defined on the
tangent bundle $T(M)$. We naturally extend this action to the tensor
powers of $T(M)$, including $End(T(M))\cong T(M)\otimes T^*(M)$.
Consider the set $R$ of induced complex structures as subset of
the space of sections $\Gamma_M(End(TM))$. An easy local computation
shows that $G_M$ acts transitively on $R\cong S^2$ (see
also \ref{_induced_compl_str_turn_to_I_Sublemma_}). Let $L$ be an
induced complex structure, $\omega\in K_{\c H}(L)$. Let $g\in G_M$,
$L':=g(L)$. Consider the Kaehler form $\omega$ as the section of
$\Lambda^2(TM)\subset T^*(M)\otimes T^*(M)$. Obviously, the 2-form
\[ \inbfpare{\cdot,\cdot}:= g(\omega)(L'(\cdot),\cdot)=
\omega(g \circ g^{-1}\circ L\circ g (\cdot),g(\cdot))
= \omega (L(g(\cdot)),g(\cdot))
\]
is symmetric and positively defined. To show that
the Riemannian form $\inbfpare{\cdot,\cdot}$
is Kaehler, we have to prove that the form
\[ \inbfpare{L'(\cdot),\cdot} = - g(\omega)(\cdot,\cdot) \]
is symplectic. Since $G_M$ commutes with Laplacian,
it maps harmonic forms to harmonic ones. Hence, $g(\omega)$ is
a symplectic form. Therefore $\inbfpare{\cdot,\cdot}$ is a
Kaehler metric. This implies that $g(\omega)\in K(L')$.
We proved the following statement:
\hfill
\claim \label{_G_M_acts_transi_on_K(H)_Claim_}
Let $M$ be a hyperkaehler manifold with the hyperkaehler
structure $\c H$. Consider the action of the group of unit
quaternions $G_M\cong SU(2)$ on $H^2(M,{\Bbb R})$
(see \ref{_there_is_action_of_G_M_Proposition_}).
Let $g\in G_M$, $L, L'\in R$, $L'=g(L)$. Then
$g:\; H^2(M, {\Bbb R})\longrightarrow H^2(M,{\Bbb R})$ induces
an isomorphism from $K_{\c H}(L)\subset H^2(M,{\Bbb R})$ to
$K_{\c H}(L')\subset H^2(M,{\Bbb R})$.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
The following statement is clear:
\hfill
\claim \label{_K_c(L)_is_open_Claim_}
In assumptions of \ref{_positive_classes_Definition_},
the set $K_{\c H}(L)$ is open in $H^{1,1}_L(M,{\Bbb R})$.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\ref{_G_M_acts_transi_on_K(H)_Claim_}
establishes a smooth map $\delta:\; G_M\times K_{\c H}(I)\longrightarrow K_c(\c H)$.
Let $b_2:=dim(H^2(M))$. By \ref{_K_c(L)_is_open_Claim_},
$dim_{\Bbb R} K_{\c H}(I)= dim_{\Bbb C} (H^{11}((M, I))$.
Since $M$ is a simple hyperkaehler manifold,
$dim_{\Bbb C} (H^{11}((M, I)) = b_2-2$. According to Section
\ref{hyperk_manif_Section_}, $R\cong S^2$. Therefore,
$dim_{\Bbb R}(R\times K_{\c H}(I))= b_2$. By definition, $G_M$ is
identified with the group of unit quaternions, and $R$
is identified with the set $x\in G_M \ \ | \ \ x^2=-1$.
This identification defines a canonical embedding $R\hookrightarrow G_M$.
Let $\phi:\; R\times K_{\c H}(I)\longrightarrow K_c(\c H)$ be the restriction
of $\delta$ to $R\times K_{\c H}(I)\subset G_M\times K_{\c H}(I)$.
The dimension of $R\times K_{\c H}(I)$ is equal to $b_2$.
Therefore, to prove that $K_c(\c H)$ is open
in $H^2(M,{\Bbb R})$ it is sufficient to prove the following:
\hfill
\sublemma \label{_K(H)cong_S^2_times_K(I)_Sublemma_}
The map $\phi:\; R\times K_{\c H}(I)\longrightarrow K_c(\c H)$ is
a diffeomorphism.
{\bf Proof:}
As in \ref{_restrictions_of_pairings_to_H^2_Lemma_},
consider the decomposition $H^2(M,{\Bbb R})= H_{inv}\oplus V$.
As we have established previously, $V$ is generated
by $P_i(\c H)$, $i=1,2,3$ and $(H_{inv},V)_{\c H}=0$.
For $x\in H^2(M, {\Bbb R})$, let $\pi_i(x)$ be the orthogonal projection
of $x$ to $H_{inv}$ and $\pi_v(x)$ be the orthogonal projection
of $x$ to $V$. The bilinear form $(\cdot,\cdot)_{\c H}$ is $G_M$-invariant by
\ref{_Hodge_Riemann_independe_for_equiva_hyperkae_Lemma_}.
Therefore, $\pi_i(g(x))=g(\pi_i(x))$.
For every induced complex structure $L$, $\c H$
defines a Kaehler structure on the complex manifold $(M, L)$.
Hence, for every induced complex structure $L$, the hyperkaehler
structure $\c H$ defines a Kaehler form $\omega_L$ and a
degree map $deg_L:\; H^{2i}(M,{\Bbb R})\longrightarrow {\Bbb R}$. According to
\ref{_inv_2-forms_have_zero_degree_Claim_}, for all
$x\in H_{inv}$ and all induced complex structures $L$,
$deg_L(x)=0$. Therefore for all $x\in K_c(\c H)$, we have
$\pi_v(x)\neq 0$.
Let $y\in K_c(\c H)$. Let $l(y)$ be the line
in the three-dimensional space $V$ generated by $\pi_v(y)$.
The space $V$ is generated by the set of induced complex structures,
which constitute a unit sphere in $V$. Hence, the space
of lines in $V$ is canonically identified with the set of
complex structures up to a sign.
Let $R^{\pm}\cong {\Bbb R} P^2$ be the quotient of $R$ by $\pm 1$.
Let $\theta: \; K_c(\c H)\longrightarrow R^{\pm}$ map $y$ to the
point of $R^{\pm}$ which corresponds to $l(y)$.
Denote the induced complex structures which correspond
to $\theta(y)$ by $L_1$, $L_2$, where $L_2=-L_1$.
Denote the Hodge decomposition
associated with an arbitrary complex structure $L\in Comp$ by $H^{pq}_L$.
According to \ref{_G_M_invariant_forms_Proposition_},
$x\in H^{pp}_L$ if and only if $L(x)=x$.
Obviously,
$L(y)=\pi_i(y)+ L(\pi_v(y))$. Realizing $L$ and $\pi_v(y)$
as quaternions in a usual way, we may check that
$L(\pi_v(y))= L\pi_v(y)L^{-1}$. Since the centralizator
of all elements in $SU(2)$ is one-dimensional, $L\in l$
whenever $L(y)=y$. Therefore for $y\in H^{11}_L$, we
have $L=\pm L_1$. Since $\pi_v$ is an orthogonal projection,
\begin{equation} \label{_pi_v(y)_Equation_}
\pi_v(y)=\frac{deg_L(y)}{deg_L(\omega_L)}\omega_L,
\end{equation}
where $\omega_L$ is the Kaehler form of $(M,L)$ considered
as an element of $V$, where
\[ deg_L(y):= \int_M \omega_L^{n-1}\wedge y. \]
By definition, $deg_{L_1}(x)=-deg_{L_2}(x)$.
On the other hand, for $x\in K(L)$, we have $deg_L(x)>0$.
Therefore for all $y\in K_c(\c H)$, there exist only
one induced complex structure $L$ such that $y\in K(L)$.
This implies that $\phi$ is a monomorphism. We need to construct the inverse
of $\phi$ and prove that it is smooth. Let
$\Theta^{\pm}: K_c(\c H) \longrightarrow R^{\pm}\times (H_{inv}\oplus {\Bbb R})$
map $y\in K_c(\c H)$ to the pair
\[ \bigg(\theta (y),\; \pi_i(y)\oplus |deg_{\pi_i(y)}(y)|\bigg) \]
where $deg_{\pi_i(y)}$ is well defined up to a sign. According
to \eqref{_pi_v(y)_Equation_}, up to a sign, one can reconstruct
$\pi_v(y)$ by $\Theta^{\pm}(y)$. Therefore,
$\Theta^{\pm}$ is a double covering. Let
$\rho:\;R^{\pm}\times (H_{inv}\oplus {\Bbb R})\longrightarrow H^2(M,{\Bbb R})$ map
$(s,h+t)\in R^{\pm}\times(H_{inv}\oplus {\Bbb R})$ to
\[ \frac{t}{deg_I(\omega_I)}\omega_I +h. \]
The $I$-degree of $\omega\in K_{\c H}(I)$ is positive.
Therefore \eqref{_pi_v(y)_Equation_} implies that
the map
\[ \phi\circ \Theta^{\pm} \circ \rho:\;
R\times K_c(\c H)\longrightarrow R^{\pm}\times K_c(\c H)
\]
acts as identity on $K_c(\c H)$ and acts as a double covering on
$R$. Since $\Theta^{\pm}$ is a double covering, $\phi$ is an
open embedding. \ref{_K(H)cong_S^2_times_K(I)_Sublemma_}
and \ref{_C(omega)_open_in_Kah_Lemma_} is proven. The proof of
\ref{_hyperk_are_well_connected_Proposition_} and
consequently \ref{_Hodge_Riemann_independent_Theorem_}
is finished. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\section{Period map and the space of 2-dimensional
planes in $H^2(M,\protect {\Bbb R})$.} \label{_Q_c_defini_Section_}
There is an alternative way of looking at Griffiths period map
$P_c:\; Comp\longrightarrow {\Bbb P}(H^2(M,{\Bbb C}))$. This enhanced version
of period map is a map from $Comp$ to an open subset in Grassmanian
of all 2-dimensional planes in $H^2(M, {\Bbb R})$. To define this
map, we remind the reader certain well-known
results from linear algebra.
Let $V_{\Bbb R}$ be an ${\Bbb R}$-linear space endowed with the
non-degenerate symmetric bilinear form $(\cdot,\cdot)_{\Bbb R}$.
Let $V_{\Bbb C}:= V_{\Bbb R}\otimes{\Bbb C}$ be the complexification of
$V_{\Bbb R}$, and $(\cdot,\cdot)_{\Bbb C}$ be the ${\Bbb C}$-linear form on
$V_{\Bbb C}$ obtained as a complexification of $(\cdot,\cdot)_{\Bbb R}$.
In applications, $V_{\Bbb R}= H^2(M,{\Bbb R})$, $V_{\Bbb C}= H^2(M,{\Bbb C})$,
and $(\cdot,\cdot)_{{\Bbb R}}$ is the normalized
Hodge-Riemann pairing $(\cdot,\cdot)_{\c H}$.
\hfill
Consider the projectivization ${\Bbb P} V_{\Bbb C}$ as a space
of lines in $V_{\Bbb C}$. For all $x\in V_{\Bbb C}$, let $\bar x$ denote
the complex conjugate to $x$. Let
\[ C:= \{ t\in V_{\Bbb C} \;\; |\;\; \forall x\in t,\, (x,x)_{\Bbb C}=0,
(x,\bar x)_{\Bbb C}> 0 \}.
\]
Let $\tilde Pl$ be the space of all oriented 2-dimensional
linear subspaces in $V_{\Bbb R}$. Let $Pl\subset \tilde Pl$
be the set of all $L\in \tilde Pl$ such that the restriction
of $(\cdot,\cdot)_{\Bbb R}$ to the 2-dimensional space $L\subset V_R$
is positively defined. Clearly, $Pl$ is open in $\tilde Pl$.
For $t\in {\Bbb P} V_{\Bbb C}$, take $x\in t$, $x\neq 0$.
Let $i_x(t)\subset V_{\Bbb R}$ be the linear span of
$Re(x)$, $Im(x)\in V_{\Bbb R}$. If $i_x(t)$ is two-dimensional,
we consider $i_x(t)$ as the oriented space with the
orientation defined by the basis $(Re(x), Im(x))$.
\hfill
\proposition \label{_from_PV_C_to_Pl_linear-alg_Proposition_}
Let $t\in C\subset {\Bbb P} V_{\Bbb C}$. Then the space $i_x(t)$
is 2-dimensional and independent on the choice of $x\in t$.
Let $i:\; C\longrightarrow \tilde Pl$ map $t\in C$ to $i_x(t)$.
The image of $i:\; C\longrightarrow \tilde Pl$
coinsides with $Pl\subset \tilde Pl$. Established this
way map $i:\; C\longrightarrow Pl$ is bijective.
{\bf Proof:} Let $t\in C$, $x\in t\subset V_{\Bbb C}$. Let
$y=Re(x), \;z= Im(x)$. Then
\[
(x,x)_{\Bbb C}= (y,y)_{\Bbb R}-(z,z)_{\Bbb R} + 2\sqrt{-1}\: (y,z)_{\Bbb R}= 0.
\]
Therefore $(y,y)_{\Bbb R}=(z,z)_{\Bbb R}$ and $(y,z)_{\Bbb R}=0$. On the other hand,
\[
(x,\bar x)_{\Bbb C} = (y,y)_{\Bbb R}+ (z,z)_{\Bbb R}> 0.
\]
We obtain that
\begin{equation} \label{_y^2=z^2>0_Equation_}
(y,y)_{\Bbb R} = (z,z)_{\Bbb R} > 0 \;\; \mbox{and}\;\; (y,z)_{\Bbb R}=0
\end{equation}
This implies that the vectors $y$ and $z$ are linearly independent.
For all $c=\lambda e^{\sqrt{-1}\: \alpha}\in {\Bbb C}$, where $\lambda,\alpha\in {\Bbb R}$,
we have
\[ Re(cx)=\lambda \cos (\alpha) y +\lambda \sin(\alpha) z, \;
Im(cx)=-\lambda \sin (\alpha) y +\lambda \cos(\alpha) z.
\]
Therefore, $i_x(t)=i_{cx}(t)$. This implies that the map
$i:\; C\longrightarrow \tilde Pl$ is well defined. According to
\eqref{_y^2=z^2>0_Equation_}, $i(C)\subset Pl$.
\hfill
Let us construct the inverse map $j:\; Pl\longrightarrow C$. For
$L\in Pl$, take an oriented orthonormal basis $(y,z)$ in the
Euclidean space $L$. For another orthonormal basis
$(y',z')$ in $L$, we have
\begin{equation} \label{_rotation_on_y,z_Equation_}
\begin{array}{rrrrr}
y'& =& \cos (\alpha) y &+&\sin(\alpha) z \\[3mm]
z'& = & -\sin (\alpha) y &+&\cos(\alpha) z
\end{array}
\end{equation}
for some $\alpha\in {\Bbb R}$. Let $j(L)\in {\Bbb P}V_{\Bbb C}$
be the line generated by $y+ \sqrt{-1}\: z\in V_{\Bbb C}$. The equations
\eqref{_rotation_on_y,z_Equation_} imply that $j(L)$ is independent
of the choice of the oriented orthonormal basis $(y,z)$.
Since $(y,z)$ is orthonormal basis, \eqref{_y^2=z^2>0_Equation_}
holds. This equation immediately implies that
$j(L)\in C\subset {\Bbb P}V_{\Bbb C}$.
Finally, it is clear from defintions
that $j\circ i= Id$ and $i\circ j= Id$.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Returning to the hyperkaehler manifolds, consider the space
\[
\goth C\subset {\Bbb P}(H^2(M,{\Bbb C}))
\]
consisting of lines
$l\in H^2(M,{\Bbb C})$ such that for all $x\in l$, $(x,x)_{\c H}=0$
and $(x,\bar x)_{\c H}>0$.
Here, as elsewhere,
\[
(\cdot,\cdot)_{\c H}:\; H^2(M,{\Bbb C})\times H^2(M,{\Bbb C})\longrightarrow {\Bbb C}
\]
is a complexification of the normalized Hodge-Riemann pairing.
Hodge-Riemann relations imply that $P_c(Comp)\subset \goth C$.
\ref{_from_PV_C_to_Pl_linear-alg_Proposition_}
establishes a diffeomorphism $\goth i:\; \goth C\longrightarrow Pl$, where
$Pl$ is the space of 2-dimensional subspaces $L\subset H^2(M,{\Bbb R})$
such that $(\cdot,\cdot)_{\c H}$ is positively defined on $L$.
Let $Q_c:\; Comp\longrightarrow Pl$ be the composition of $P_c$ and $\goth i$.
As results of Bogomolov and Todorov imply (see
\cite{_Bogomolov_}, \cite{_Beauville_}, \cite{_Todorov_}),
the map $Q_c$ is an immersion. Since
$\dim Comp = \dim \goth C= \dim H^2(M) -2$, this map is
etale. It can be described in more straightforward terms
as follows. Let $I\in Comp$. Let $\tilde \Omega$ be a holomorphic
symplectic form over $(M, I)$. Let $\Omega\in H^2(M, {\Bbb C})$ be the cohomology
class represented by the closed differential form $\tilde \Omega$.
Let $\omega_2:= Re(\Omega)$, $\omega_3:= Im(\Omega)$. Then one can
define $Q_c(I)$ as the linear span of $\omega_2$, $\omega_3$.
\section {Lefschetz-Frobenius algebras.}
\label{_Lefshe_Frob_Section_}
In this section, we give a number of preliminary definitions,
which eventually
lead to a calculation of the cohomology of a compact hyperkaehler
manifold. Some of these definitions are due to V. Lunts (see
\cite{_Lunts-Loo_}).
Further on, by ``algebra'' we understand
an associative algebra with unit.
\hfill
\definition
Let $A$ be an algebra over a field $k$ and $(\cdot,\cdot)$
be a $k$-valued bilinear form on $A$. The form $(\cdot,\cdot)$
is called {\bf invariant} if for all $a,b,c\in A$,
$(ab,c)=(a,bc)$. A $k$-algebra equipped with an invariant
non-degenerate bilinear form is called {\bf Frobenius algebra}.
\hfill
\definition
Let $A=\oplus A_i, i=0,...,d$ be a graded supercommutative algebra
over a field $k$ of characteristic zero. Assume that $A$ is equipped
with an invariant bilinear form, such that for all $a\in A_n, b\in A_m$,
the following holds:
\begin{equation} \label{_graded_scalar_pro_Equation_}
\begin{array}{l}
(a,b) = (-1)^{nm}(b,a), \mbox{\ and} \ \ \ \ \ \ \ \ \ \ \ \ \\[2mm]
(a,b)=0 \mbox{\ for\ } n+m\neq d. \ \ \ \ \ \ \ \ \ \ \ \ \\[2mm]
\end{array}
\end{equation}
Then $A$ is called {\bf a graded Frobenius algebra of degree $d$}.
The prototypical example of graded Frobenius algebras is the cohomology
algebra of a compact manifold.
\hfill
Further on, we consider only the graded Frobenius algebras.
For brevity, we sometimes omit the word ``graded''.
Let $A=\oplus A_i, i=0,...,d$ be a graded Frobenius algebra of
degree $d$ over the field $k$. Let $End_k(A)$ be the space of all
$k$-linear endomorphisms of $A$. Let $H\in End_k(A)$ be the endomorphism
which maps $a\in A_i$ to $(2d-i)\cdot a$. This endomorphism is
introduced by Hodge in his study of harmonic forms and Lefschetz
isomorphism. One can check that $H$ is a derivative of an algebra $A$:
\[ H(ab)= H(a) b + a H(b). \]
For all $A\in A$, let $L_a:\; A\longrightarrow A$ map $b\in A$ to $ab$.
\hfill
\definition \label{_Lefschetz_triple_Definition_}
Let $A=\oplus A_i$ be a graded Frobenius algebra, $a\in A_2$.
Let $L_a$, $H\in End_k(A)$ be as above. The triple of endomorphisms
$L_a$, $H$, $\Lambda_a\in End_k(A)$ is called {\bf a Lefshets
triple} if
\[ [H, L_a] = 2L_a, [H,\Lambda_a] = -2 \Lambda_a,
[L_a,\Lambda_a] =H.
\]
Clearly, Lefschetz triples correspond to some representations
of the Lie algebra $\goth{sl}(2)$ in $End_k(A)$.
Lefschetz theorem (\cite{_Griffiths_Harris_}) gives examples
of Lefschetz triples for $A=H^*(M)$ and $M$ is a Kaehler manifold.
\hfill
\proposition \label{_Lefshe_tri_unique_Proposition_}
Let $A$ be a graded Frobenius algebra, $a\in A_2$.
Let $(L_a, H, \Lambda_a)$, $(L_a, H, \Lambda'_a)$
be two Lefschetz triples. Then $\Lambda_a=\Lambda'_a$.
In other words, $\Lambda_a$ is uniquely determined by $a$.
{\bf Proof:} (V. Lunts) Consider the representations $\rho$, $\rho'$
of the Lie algebra $\goth{sl}(2)$ associated with these triples.
Take a basis $(x,y,h)$ in $\goth{sl}(2)$,
\[
[h,x]=2x, [h,y] =-2y, [x,y]=h,
\]
such that $\rho(x) =\rho'(x) = L_a$, $\rho(h)= \rho'(h)= H$,
$\rho(y)=\Lambda_a$, $\rho'(y)=\Lambda'_a$.
Let $T:= \Lambda_a-\Lambda'_a$. Consider the adjoint
action of $\goth{sl}(2)$ on the space $End(A)$
obtained from $\rho$:
\[
ad (\rho):\; \goth{sl}(2)\longrightarrow End(End(A))
\]
Clearly, $ad (\rho)(x)(T)=[L_a,\Lambda_a]-[L_a,\Lambda'_a]=0$.
Therefore, $T$ is a highest vector is an $\goth{sl}(2)$-submodule
of $End(A)$ generated by $T$ and $ad (\rho)$. On the other hand,
$ad (\rho)(h)(T)=-2T$, and therefore, the weight of $T$ is $-2$.
This is impossible because $ad (\rho)$ is a finite dimensional
representation of $\goth {sl}(2)$. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\definition
Let $A=\oplus A_i$ be a graded Frobenius algebra, $a\in A_2$.
Then $a$ is called {\bf of Lefschetz type} if a Lefschetz triple
$(L_a, H, \Lambda_a)$ exists.
\hfill
\lemma \label{_a_Lefshe_if_a^i_iso_Lemma_}
Let $A=A_0\oplus A_1\oplus ... A_{2d}$ be a
graded space. Let $L\in End(A)$ be an endomorphism of
grading 2: $L:\; A_i\longrightarrow A_{i+2}$. Let $H$ act on $A_i$ as the
multiplication by $d-i$, $i=0,1, ... , 2d$.
Then the following conditions are equivalent:
\hfill
(i) There exist an endomoprhism $\Lambda\in End(A)$ of grading
-2 such that the relations
\[ [H,\Lambda] =-2 \Lambda, [H,L] =-2 L,
[L,\Lambda] = H
\]
hold\footnote{The first two of these relations hold trivially
because of the grading.}.
\hfill
(ii) For all $i=0,1,...,d-1$, the map
\[ L^{d-i}:\; A^i\longrightarrow A^2d-i,
\]
is an isomorphism.
\hfill
{\bf Proof:} Clear (see \cite{_Lunts-Loo_} for details).
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
When $A$ is a cohomology algebra of a compact
Kaehler manifold $M$, all Kaehler classes
are obviously of Lefschetz type. On the other hand, a class
of Lefschetz type is not necessarily a Kaehler class. For example,
for a Kaehler class $\omega$, the class $-\omega$ is of Lefschetz
type, but $-\omega$ cannot be a Kaehler class by trivial reasons.
\hfill
\definition \label{_Lefschetz_Frob_alge_Definition_}
Let $A$ be a graded Frobenius algebra. Let $S\subset A_2$
be the set of all elements of Lefschetz type. The algebra
$A$ is called {\bf a Lefschetz-Frobenius algebra} if the following
conditions hold:
(i) The space $A_0$ is one-dimensional over $k$.
(ii) The set $S$ is Zariski dense in $A_2$.
\hfill
{\bf Example}:
Let $M$ be a compact Kaehler manifold. Then the algebras
$H^*(M)$ and $\oplus H^{p,p}(M)$ are Lefschetz-Frobenius, as
\ref{_Lefshe_Frob_if_a_Lefshe_ele_exists_Proposition_}
implies.
\hfill
By \ref{_a_Lefshe_if_a^i_iso_Lemma_},
the set $S$ of all elements of Lefschetz type
is given by an open condition. Therefore $S$ is
open in $A_2$. Therefore, $A$ is
a Frobenius-Lefschetz algebra if and only if
$S$ is non-empty in $A_2$. We obtained the following
statement:
\hfill
\proposition \label{_Lefshe_Frob_if_a_Lefshe_ele_exists_Proposition_}
Let $A$ be a graded Frobenius algebra. Assume that $A_2$ contains at
least one element of Lefschetz type. Then $A$ is Lefschetz-Frobenius.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
For every Lefschetz triple $T$, we define a Lie algebra homomorphism
\[
\rho_T:\; \goth{sl}(2) \longrightarrow End_k(A)
\]
in an obvious way. For a Lefschetz-Frobenius algebra $A$, let ${\goth g}(A)$
be the Lie subalgebra of $End_k(A)$ generated by the images
of $\rho_T$ for all Lefschetz triples $T$. This algebra is our main object
of study. The algebra ${\goth g}(A)$ is graded: ${\goth g}(A)= \oplus{\goth g}_{2i}(A)$,
$g(A_n)\subset A_{n+2i}$ for all $g\in {\goth g}_{2i}(A)$. This
algebra is called {\bf the structure Lie algebra of $A$}.
\hfill
\definition \label{_Lefshe_Fro_Definition_}
Let $A$ be a Lefschetz-Frobenius algebra. Assume that
${\goth g}_{2i}(A)=0$ for $i\neq -1,0,1$:
\[
{\goth g}(A)={\goth g}_{-2}(A)\oplus {\goth g}_0(A)\oplus {\goth g}_2(A).
\]
Then $A$ is called {\bf a Lefschetz-Frobenius algebra of Jordan type}.
Such $A$ are closely related with Jordan algebras (\cite{_Springer_}).
\hfill
If $A$ is generated by $A_2$ and
$A_0\cong k$, $A$ is called {\bf reduced}. The subalgebra $A^r\subset A$
generated by $A_2$ and $A_0$ is called {\bf reduction of $A$}.
We use the following result.
\hfill
\proposition \label{_Lunts_about_FLJ_Proposition_}
(\cite{_Lunts-Loo_})
Let $A$ be a Frobenius-Lefschetz algebra, ${\goth g}=\oplus {\goth g}_{2i}$ be its
structure Lie algebra. Then the following conditions are equivalent:
(i) ${\goth g}_2$ is spanned by $L_a$ for all Lefschetz elements $a$,
(ii) $A$ is of Jordan type,
(iii) $[\Lambda_a,\Lambda_b]=0$ for all Lefschetz elements $a,b\in A_2$.
{\bf Proof:} See Proposition 2.6 and Claim 2.6.1 of \cite{_Lunts-Loo_}.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\ref{_Lunts_about_FLJ_Proposition_} immediately implies the following
statement:
\hfill
\corollary \label{_g_2_is_A_2_Corollary_}
Let $A=\oplus A_i$ be a Lefschetz-Frobenius algebra of Jordan type,
${\goth g}={\goth g}_{-2}\oplus {\goth g}_0\oplus {\goth g}_2$ be its sturtcure Lie algebra.
Then ${\goth g}_2=A_2$.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
In the future, we often assume that $A$ is reduced.
In this case, the multiplicative structure on $A$ can be
recovered from the ${\goth g}(A)$-action. This is done as follows.
Since $[{\goth g}_2(A),{\goth g}_2(A)]\subset {\goth g}_4(A) =0$, the space
${\goth g}_2(A)\subset {\goth g}(A)$ is a commutative
subalgebra of ${\goth g}(A)$. Consider the corresponding embedding of enveloping
algebras:
\[
U_{{\goth g}_2(A)}\cong S^*({\goth g}_2(A))\hookrightarrow U_{{\goth g}(A)}.
\]
Let ${\Bbb I}\in A_0$ be the unit. The representation ${\goth g}(A)\longrightarrow End(A)$
induces the canonical map
\[
U_{{\goth g}(A)} \stackrel{\tilde p}\longrightarrow A, \; P\longrightarrow P({\Bbb I}),
\]
where $P\in U_{{\goth g}(A)}$ is an ``polynomial''
over ${\goth g}(A)$. Consider the restriction of $\tilde p$ to
$U_{{\goth g}_2(A)}\subset U_{{\goth g}(A)}$:
\begin{equation}\label{_p_from_U_g_2_to_A_Equation_}
p:\; U_{{\goth g}_2(A)} \longrightarrow A.
\end{equation}
Clearly, for all $a\in A_2$, $a$ of Lefschetz type, $L_a\in {\goth g}_2(A)$.
Since the set of elements of Lefschetz type is Zariski dense in $A_2$,
we have $L_a\in {\goth g}(A)\subset End(A)$ for all $a\in A_2$.
One can easily check that the corresponding map $i:\; A_2\longrightarrow {\goth g}_2(A)$
is an isomorphism (see \ref{_Lunts_about_FLJ_Proposition_}).
Therefore, $S^*({\goth g}_2(A))\cong S^*(A_2)$. Applying the isomorphism
$U_{{\goth g}_2(A)}\cong S^*({\goth g}_2(A))\cong S^*(A_2)$ to the map
\eqref{_p_from_U_g_2_to_A_Equation_}, we obtain the map
\[ p':\; S^* A_2\longrightarrow A. \]
\hfill
\claim \label{_p_is_multiplication_Claim_}
The map $p'$ coinsides with the map induced by multiplication
in $A$.
{\bf Proof:} Clear. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\ref{_p_is_multiplication_Claim_} implies that the kernel of the map
$p:\; U_{{\goth g}_2(A)} \longrightarrow A$ is an ideal in $U_{{\goth g}_2(A)}$. This leads
to a more general construction.
\hfill
\definition \label{_multi_associ_with_representa_Definition_}
Let ${\goth g}={\goth g}_{-2}\oplus {\goth g}_0\oplus {\goth g}_2$ be a graded Lie algebra.
Let $V$ be a representation of ${\goth g}$ and $v\in V$ be a vector.
Assume that by applying ${\goth g}_2$ to $v$ repeatedly we obtain
the whole space $V$ (i. e., the vector $v$ generates $V$ as a
representation of ${\goth g}_2$). Assume that ${\goth g}_{-2}(v)=0$,
and that for all $g\in {\goth g}_0$, $g(v)$ is proportional to $v$. Let
$p:\; U_{{\goth g}_2}\longrightarrow V$ be the map which associates with the polynomial
$P\in U_{{\goth g}_2}$ the vector $P(v)\in V$. Clearly, $\ker(p)$ is a left
ideal in $U_{{\goth g}_2}$. Since ${\goth g}_2$ is commutative, this ideal is two-sided.
Therefore, $V\cong \bigg(U_{{\goth g}_2}/ \ker (p)\bigg)$ is equipped with a structure
of commutative algebra. We denote this algebra by $V_{{\goth g}, v}$.
\hfill
\claim \label{_g_structure_defines_algebr_Claim_
Let $A$ be a Lefschetz-Frobenius algebra of Jordan type. Assume that
$A$ is reduced (generated by $A_2$). Consider $A$ as a representation of
${\goth g}(A)\cong {\goth g}_{-2}(A)\oplus {\goth g}_0(A)\oplus {\goth g}_2(A)$. Take the
unity vector ${\Bbb I}\in A_0$. Then the algebra $A_{{\goth g}(A),{\Bbb I}}$
coinsides with $A$.
{\bf Proof:} Follows from definitions. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\hfill
{\large\bf Appendix. Reduction and the structure Lie algebra.}
\hfill
Let $A$ be a Lefschetz-Frobenius algebra, and $A^r$ be its reduction.
Assume that the restriction $(\cdot,\cdot)_r$ of $(\cdot,\cdot)$ to $A^r$
is non-degenerate. Then $(\cdot,\cdot)_r$
establishes a structure of Frobenius algebra on $A^r$.
We are going to show that $A^r$ is Lefschetz-Frobenius, and relate ${\goth g}(A)$
to ${\goth g}(A^r)$.
\hfill
\proposition \label {_A^r_Lef-Frob,_pres_by_g(A)_Proposition_}
Let $A$ be a Lefschetz-Frobenius algebra.
Assume that the restriction of $(\cdot,\cdot)$ to $A^r$
is non-degenerate. Then $A^r$ is also Lefschetz-Frobenius.
Moreover, the action of ${\goth g}(A)$ on $A$ preserves the subspace
$A^r\subset A$.
{\bf Proof:} Let $A^r_\bot$ be the orthogonal complement to $A^r$
in $A$. Since $(\cdot,\cdot)\restrict{A^r}$ is nondegenerate,
$A=A^r\oplus A^r_\bot$.
\hfill
\lemma \label{_A^r^bot_preserved_by_mult_by_A^r_Lemma_}
Let $a\in A^r$, $b\in A^r_\bot$. Then $ab\in A^r_\bot$.
{\bf Proof:} It is sufficient to show that for all $c\in A^r$,
$(ab,c)=0$. Since $(\cdot,\cdot)$ is invariant, for all $c\in A$ we have
$(ab,c)=(b, ac)$. Since $ac\in A^r$ and $b\in A^r_\bot$,
$(ab,c)=0$. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
{}From \ref{_A^r^bot_preserved_by_mult_by_A^r_Lemma_}, we obtain that
the operators $L_a$, $a\in A_2$ preserve the decomposition
$A=A^r\oplus A^r_\bot$. By \ref{_a_Lefshe_if_a^i_iso_Lemma_},
the map $L_a^{d-i}:\; A_i\longrightarrow A_{2d-i}$ is an isomorphism.
Therefore, the restriction of $L_a^{d-i}$ to the $i$-th grading component
$(A^r)_i$ of $A^r$ is an embedding to $(A^r)_{2d-i}$. Since $A^r$
is Frobenius, $dim_k(A^r)_{2d-i}=dim_k(A^r)_i$. Therefore,
the restriction of $L_a^{d-i}$ to $(A^r)_i$
is an isomporphism. Applying \ref{_a_Lefshe_if_a^i_iso_Lemma_}
again, we obtain that for all Lefschetz-type elements
$a\in A_2$, these elements are of Lefschetz type
in $A^r$. Therefore, $A^r$ is a Lefschetz-Frobenius algebra.
It remains to prove that $A^r$ is preserved by ${\goth g}(A)$.
Clearly, the generators $L_a$ and $H$ of ${\goth g}(A)$ preserve
$A^r$. Therefore, to show that ${\goth g}(A)$ preserves $A^r$
it is sufficient to prove that $\Lambda_a$ preserve $A^r$
for all Lefschetz-type elements $a\in A_2$. Let $L=L_a$.
Let $L_r$, $H_r$ be the restrictions of $L$, $H$ to $A^r$ and
$L_{\bot}$ be the restrictions of $L$, $H$ to $A^r_\bot$.
By \ref{_a_Lefshe_if_a^i_iso_Lemma_}, there exists an
endomorphism $\Lambda_\bot:\; A^r_\bot\longrightarrow A^r_\bot$
of grading $-2$ such that $[L_\bot,\Lambda_\bot]= H_\bot$.
Let $\Lambda_r:\; A^r\longrightarrow A^r$ be the endomorphism
of $A^r$ such that $(L_r, H_r, \Lambda_r)$
is a Lefschetz triple. Let $\Lambda_r+\Lambda_\bot$ be an
endomorphism of $A$ such that for all $a=b+c$, $b\in A^r, c\in A_\bot^r$,
\[
\Lambda_r+\Lambda_\bot(a)= \Lambda_r(b)+\Lambda_\bot(c).
\]
Checking relations, we obtain that $(L, H, \Lambda_r+\Lambda_\bot)$ is
a Lefschetz triple. By \ref{_Lefshe_tri_unique_Proposition_},
$\Lambda_a=\Lambda_r+\Lambda_\bot$. This implies that
$\Lambda_a$ preserves $A^r$. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\ref{_A^r_Lef-Frob,_pres_by_g(A)_Proposition_} immediately
implies the following useful statement:
\hfill
\corollary
Let $A$ be a Lefschetz-Frobenius algebra such that
its reduction $A^r$ is also Frobenius. Then $A^r$
is Lefschetz-Frobenius, and there exists a natural
Lie algebra epimorphism ${\goth g}(A)\longrightarrow {\goth g}(A^r)$.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\section {The minimal Frobenius algebras and cohomology of compact
Kaehler surfaces.} \label{_minimal_Fro_Section_}
In this section we concentrate on the simplest case of
Frobenius algebras related to Lefschetz theory. Namely,
we analyze the graded Frobenius algebras $A= A_0\oplus A_2\oplus A_4$,
where $dim_k A_0=1$. Such algebras are called minimal.
These algebras are naturally related to the complex surfaces.
\hfill
\definition
Let $A= \oplus A_i$ be a graded Frobenius algebra.
Assume that $A= A_0\oplus A_2\oplus A_4$, $dim_k A_0=dim_k A_4=1$.
Then $A$ is called {\bf a minimal graded Frobenius algebra}.
\hfill
\proposition \label{_minimal_is_Lefschetz_Proposition_}
Let $A= A_0\oplus A_2\oplus A_4$ be a minimal graded
Frobenius algebra. Then $A$ is Lefschetz-Frobenius.
{\bf Proof:}
Let $(\cdot,\cdot):\; A\times A\longrightarrow k$ denote the
invariant scalar product on $A$. The restriction
of $(\cdot,\cdot)$ to $A_2$ is a non-degenerate
bilinear symmetric form (it is non-degenerate
because of grading conditions \eqref{_graded_scalar_pro_Equation_}).
The following statement immediately implies
\ref{_minimal_is_Lefschetz_Proposition_}:
\hfill
\lemma \label{_el-t_with_non_zero_square_Lefschetz_Lemma_}
Let $A= A_0\oplus A_2\oplus A_4$ be a minimal graded
Frobenius algebra. Let $a\in A_2$ be a vector such that
$(a,a)\neq 0$. Then $a$ is a Lefschetz element.
{\bf Proof:} Let $a^\bot$ be the orthogonal complement of $a$ in $A_2$:
\[
a^\bot:= \{b\in A_2\; |\; (a,b)=0 \}.
\]
Let ${\Bbb I}\in A_0$ be the unit in $A$.
For all $b\in a^\bot$, $(ab,{\Bbb I})=(a,b)=0$.
Since $A_4$ is one-dimensional and its generator
has non-zero scalar product with ${\Bbb I}$, we have
\begin{equation} \label{_ab=0_for_all_b_in_a^bot_Equation_}
\forall b\in a^\bot,\; \;\; ab=0.
\end{equation}
Let $k_a\subset A_2$ be the one-dimensional space generated
by $a$. Let $A_a:= A_0\oplus ka \oplus A_4$.
Clearly, $A_a$ is a subalgebra of $A$. By
\eqref{_ab=0_for_all_b_in_a^bot_Equation_}, the operator
$L_a$ vanishes on $a^\bot$. Since $H(A_2) =0$, the operator
$H$ also vanishes on $a^\bot\subset A_2$. Therefore
it is sufficient to show that $a$ is a Lefschetz element in the algebra
$A_a$. Since $A_a\cong k[x]/(x^3=0)\cong H^*(\Bbb P^2, k)$, this follows
from Lefschetz theory. \ref{_el-t_with_non_zero_square_Lefschetz_Lemma_}
and consequently, \ref{_minimal_is_Lefschetz_Proposition_},
is proven. We also obtained the following result:
\hfill
\corollary \label{_Lambda_vanish_Corollary_}
Let $(L_a, H, \Lambda_a)$ be the Lefschetz triple on $A$, where
$A$ is a minimal graded Frobenius algebra. Then
$(L_a, H, \Lambda_a)$ all vanish on $a^\bot$.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
There is an easy way to construct the minimal graded Frobenius
algebras using spaces with non-degenerate symmetric bilinear forms.
Namely, let $V$ be a linear space over $k$, equipped witha bilinear form
$(\cdot,\cdot)_V$. Consider the linear space
\[
A(V):= k{{\Bbb I}}\oplus V\oplus k\Omega,
\]
where $k{\Bbb I}$ and $k\Omega$ are one-dimensional spaces
generated, respectively, by ${\Bbb I}$ and $\Omega$. We introduce a graded
Frobenius algebra structure on $A(V)$ in the following way. The grading
of $V$ is 2, the grading of ${\Bbb I}$ is 0, the grading of $\Omega$ is 4.
The product on $A(V)$ is defined as follows:
\hfill
(i) ${\Bbb I}$ is a unit.
(ii) for $v_1, v_2\in A_2(V)\cong V$, $v_1 v_2= (v_1,v_2)_V \Omega$.
\hfill
It remains to establish the invariant bilinear symmetric form
$(\cdot,\cdot)$ on $A(V)$.
\hfill
(iii) On $A_2(V)\cong V$, $(\cdot,\cdot)$
is equal to $(\cdot,\cdot)_V$.
(iv) The product of ${\Bbb I}$ and $\Omega$ is 1.
\hfill
Together with \eqref{_graded_scalar_pro_Equation_}, relations
(iii) and (iv) define the form $(\cdot,\cdot)$ is a unique way.
One can trivially check that this construction results in a
graded Frobenius algebra. Exactly this algebra appears as the
even cohomology of the compact Kaehler surface $M$, where
$V=H^2(M)$ and $(\cdot,\cdot)_V$ is the intersection form.
In fact, every minimal graded Frobenius algebra can be obtained
this way (\ref{_all_minim_alge_are_associ_Claim_}).
\hfill
\definition \label{_associ_gra_Fro_Definition_
The graded Frobenius algebra $A(V)$ is called {\bf the minimal graded
Frobenius algebra associated with $V$, $(\cdot,\cdot)_V$}.
\hfill
\claim\label{_all_minim_alge_are_associ_Claim_}
Let $A= A_0\oplus A_2\oplus A_4$ be the minimal graded
Frobenius algebra. Denote the restriction of
the invariant scalar product to $A_2$ by
$(\cdot,\cdot):\; A_2\times A_2\longrightarrow k$.
Then $A$ is is isomorphic to the minimal graded
Frobenius algebra associated with $A_2$, $(\cdot,\cdot)$.
{\bf Proof:} Clear. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\proposition\label{_minimal_alge_are_Jordan_Proposition_
Let $A= A_0\oplus A_2\oplus A_4$ be a minimal graded Frobenius
algebra. Then $A$ is of Jordan type.
{\bf Proof:} By \ref{_Lunts_about_FLJ_Proposition_}, it is sufficient to
show that for every two elements $a_1, a_2\in A_2$ of Lefschetz
type, $[ \Lambda_{a_1},\Lambda_{a_2}]=0$. Denote the generators of
$A_0$, $A_4$, by ${\Bbb I}$, $\Omega$, as in
\ref{_associ_gra_Fro_Definition_}. The endomorphism
$[ \Lambda_{a_1},\Lambda_{a_2}]$ has a grading $-4$. Therefore it is
a map from $A_4$ to $A_0$. To prove that
$[ \Lambda_{a_1},\Lambda_{a_2}]=0$ it is sufficient to show
that
\[
[ \Lambda_{a_1},\Lambda_{a_2}]\:\bigg(\Omega\bigg)=0
\]
\ref{_Lambda_vanish_Corollary_} implies that $\Lambda_{a_i}(\Omega)$
is proportional to $a_i$. An easy calculation in $\goth{sl}(2)$
implies that $\Lambda_{a_i}(\Omega)=-a_i$. Similarly,
$\Lambda_{a_i}(a_j)=(a_i,a_j) {\Bbb I}$. Therefore
\[ [ \Lambda_{a_1},\Lambda_{a_2}]\:\bigg(\Omega\bigg) = (a_1,a_2)\cdot
{\Bbb I} - (a_2,a_1) \cdot{\Bbb I} =0
\]
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
We proceed to compute the Lie algebra ${\goth g}(A)$ associated with
the minimal Frobenius algebra $A=A(V)$, where $V$ is a linear
space equipped with a scalar product. Denote by $\goth{so}(V)$
the Lie algebra of skew-symmetric endomorphisms of $V$. Let
$\goth H$ be the 2-dimensional space over $k$ with the hyperbolic
scalar product. In other words, $\goth H$ has a basis $x, y$
such that $(x,y)=1$, $(x,x)=0$, $(y,y)=0$.
By $\goth{so}(V)\oplus k$ we understand a direct sum of $\goth{so}(V)$
and a trivial Lie algebra of dimension 1.
\hfill
\theorem\label{_calculation_of_g(A)_for_minim_Theorem_}
Let $V$ be a $k$-linear space equipped with a non-degenerate
scalar product. Let $A=A_0\oplus A_2\oplus A_4$
be the minimal graded Frobenius algebra $A(V)$
constructed by $V$. Take the graded Lie algebra
${\goth g}(A)={\goth g}_{-2}(A)\oplus {\goth g}_0(A)\oplus {\goth g}_2(A)$
(\ref{_Lefshe_Fro_Definition_}).
Then
\hfill
(i) ${\goth g}_0(A)\cong \goth{so}(V)\oplus k$,
(ii) ${\goth g}_2(A)\cong {\goth g}_{-2}(A)\cong V$,
(ii) ${\goth g}(A)\cong \goth{so}(V\oplus \goth H)$.
\hfill
{\bf Proof:} Denote the invariant bilinear form on $A(V)$ by
$(\cdot,\cdot)$. Let $(\cdot,\cdot)'$ another bilinear symmetric form,
defined by
\hfill
$(a,b)'=(a,b)$ if $a,b\in A_2$,
$(a,b)'=-(a,b)$ if $a,b\in A_0\oplus A_4$,
$(a,b)'=0$ if $a\in A_2$ and $b\in A_0\oplus A_4$.
\hfill
Let $A'$ be $A$ equipped with the scalar product $(\cdot,\cdot)'$.
Obviously,
\[ A'\cong V\oplus \goth H. \]
{\bf Step 1:} We are going to show
that ${\goth g}(A)\subset \goth{so}(A')$.
By trivial reasons,
$L_a$ and $H$ belong to $\goth{so}(A')$ for all $a\in A_2$.
Let $a\in A_2$ be an element of the Lefschetz type.
To prove that $\Lambda_a\in \goth{so}(A')$, we consider
the decomposition $A' = a^\bot \oplus A_a$ (see the proof
of \ref{_el-t_with_non_zero_square_Lefschetz_Lemma_}).
By \ref{_Lambda_vanish_Corollary_}, $\Lambda_a$ acts trivially
on $a^\bot$. Since the decomposition $A' = a^\bot \oplus A_a$
is orthogonal, it is sufficient to prove that the restriction
of $\Lambda_a$ to $A_a$ is skew-symmetric.
Three-dimensional representation $\rho:\; \goth{sl}(2)\longrightarrow A_a$
obtained from the Lefschetz triple
$(L_a, \Lambda_a, H)$ is naturally isomorphic to the
adjoint representation of $\goth{sl}(2)$. Using this isomorphism,
we obtain that $(\cdot,\cdot)$ is the Killing form of the
Lie algebra ${\goth g}(A_a)\cong \goth{sl}(2)$. Therefore,
${\goth g}(A_a)\subset End(A_a)$ consists of skew-symmetric matrices.
This finishes Step 1.
\hfill
{\bf Step 2:} We prove \ref{_calculation_of_g(A)_for_minim_Theorem_} (i).
Since ${\goth g}_0(A)$ is the grade-preserving part of ${\goth g}(A)$, we have a
homomorphism
\[
{\goth g}_0(A)\stackrel {\mu}\longrightarrow \goth{so}(A_2) \cong \goth{so}(V)
\]
which maps an endomorphism $h\in End(A)$ to its restriction
$h\restrict{A_2}\in End(A_2)$. The kernel of this homomorphism
is the space of all $h\in g_0(A)$ such that $h$ vanishes on $A_2$.
Therefore, $\ker\mu\in \goth{so}(A_0\oplus A_4)$. The algebra
$\goth{so}(A_0\oplus A_4)\cong \goth{so}(\goth H)\cong \goth{so}(1,1)$
is commutative and one-dimensional. Combining $\mu$ and the embedding
\[ \ker\mu\stackrel i\hookrightarrow
\goth{so}(A_0\oplus A_4)\cong k,
\]
we obtain an embedding
\[
{\goth g}_0(A)\stackrel m\hookrightarrow \goth{so}(V) \oplus k.
\]
It remains to show that $m$ is a surjection. Consider the Hodge
endomorphism $H\in {\goth g}_0(A)\subset End(A)$ introduced a few sentences
before \ref{_Lefschetz_triple_Definition_}. By obvious reasons,
$H\in \ker\mu$.
The map $i:\; \ker \mu \longrightarrow \goth{so}(A_0\oplus A_4)$ is surjective
because $i(H)$ is non-zero, and $\goth{so}(A_0\oplus A_4)$
is one-dimensional. Therefore, to prove that $m$ is surjective,
it is sufficient to show that $\mu:\;{\goth g}_0(A)\longrightarrow \goth{so}(A_2)$
is surjective.
Let
\begin{equation} \label{_condi_on_a,b_Equation_}
a,b\in A_2,\;\; (a,a)\neq, \;(b,b)\neq 0,\;(a,b)=0.
\end{equation}
Let $\inangles{a,b}\subset V$ be the plane generated
by $a$ and $b$, and $\inangles{a,b}^\bot\subset V$ be its orthogonal
completion.
Let $T_{ab}\subset \goth{so}(A_2)$ be the set of all
skew-symmetric endomorphisms which vanish on $\inangles{a,b}^\bot$.
Since $\goth{so}(2)$ is one-dimensional,
for given $a,b\in V$, all elements of $T_{ab}$
are proportional. We notice that the union of all
$T_{ab}$ generates the lie algebra $\goth{so}(A_2)$.
Therefore, to prove that $\mu$ is surjective
it is sufficient to show that $T_{ab}\subset \mu({\goth g}_0(A))$ for all
$a,b$ satisfying conditions \eqref{_condi_on_a,b_Equation_}.
\ref{_Lambda_vanish_Corollary_} implies that
$[L_a,\Lambda_b]\in T_{ab}$. It is easy to check that
this element is non-zero. The proof of
\ref{_calculation_of_g(A)_for_minim_Theorem_} (i) is finished.
\ref{_calculation_of_g(A)_for_minim_Theorem_} (ii) immediately
follows from \ref{_Lunts_about_FLJ_Proposition_}.
\ref{_calculation_of_g(A)_for_minim_Theorem_} (iii) follows fom
the inclusion ${\goth g}(A)\subset \goth{so}(A')$ and comparing
dimensions, where dimension of ${\goth g}(A)$ is computed via
\ref{_calculation_of_g(A)_for_minim_Theorem_} (i) and
\ref{_calculation_of_g(A)_for_minim_Theorem_} (ii).
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
We denote the graded Lie algebra ${\goth g}(A)$ constructed by $V$ as in
\ref{_calculation_of_g(A)_for_minim_Theorem_} by $\goth{so}(V,+)$.
Clearly, over ${\Bbb R}$, when the symmetric form on $V$ has a signature
$(a,b)$%
\footnote{Of course, this means that
$\goth{so}(V)=\goth{so}(a,b)$}%
,
\[ \goth{so}(V,+)\cong \goth{so}(a+1,b+1). \]
\section[Representations of $SO(V,+)$
leading to Frobenius algebras.]
{Representations of $SO(V,+)$
leading to \\ Frobenius algebras.}
\label{_^dA(V)_Section_}
In this section, we describe all reduced Lefschetz-Frobenius
algebras $A= A_0\oplus A_2 \oplus ... \oplus A_{2d}$ with
${\goth g}(A)\cong \goth{so}(V,+)$. It turns out that such algebras
are uniquely defined by the number $d$, which is {\it even},
whenever $dim V>2$.
Let $V$ be a linear space supplied with a non-degenerate symmetric
bilinear form $(\cdot,\cdot)$. Let $A=A(V)$ be the minimal
Frobenius algebra constructed in Section \ref{_minimal_Fro_Section_}.
Take the tensor product
\[ A^{\otimes d}:=
\underbrace{A\otimes_k A\otimes_k ... A}_{d\mbox{\ \ times}}.
\]
There is a natural action of $\goth{so}(V,+)$ on $A^{\otimes d}$.
Let ${}^{(d)}A$ be the irreducible $\goth{so}(V,+)$-module generated
by ${}^d{\Bbb I}:= {\Bbb I} \otimes {\Bbb I}\otimes ... {\Bbb I}$,
where ${\Bbb I}\in A$ is the unit. The space ${}^{(d)}A$ is not necessarily a
subalgebra in $A^{\otimes d}$. We introduce a new
algebra structure on ${}^{(d)}A$ which does
not necessarily come from the algebra structure on
$A^{\otimes d}$, but instead comes from $\goth{so}(V,+)$-action
as in \ref{_multi_associ_with_representa_Definition_}.
Denote the graded Lie algebra $\goth{so}(V,+)$ by
${\goth g}= {\goth g}_{-2}\oplus {\goth g}_0\oplus {\goth g}_2$. The algebra
${\goth g}_0$ has an additional decomposition:
${\goth g}_0= \goth{so}(V)\oplus kH$
(\ref{_calculation_of_g(A)_for_minim_Theorem_}).
Clearly, all elements of ${\goth g}_{-2}$ vanish on ${}^d{\Bbb I}$,
$\goth{so}(V)\subset {\goth g}_0$ vanish on ${}^d{\Bbb I}$ and
$H$ acts on ${}^d{\Bbb I}\in {}^{(d)}A$ as multiplication by $-2d$.
Therefore we may apply
\ref{_multi_associ_with_representa_Definition_}
to the representation ${}^{(d)}A$ and the vector ${}^d{\Bbb I}$.
Denote the resulting algebra ${}^{(d)}A_{{\goth g}, {\Bbb I}}$
by ${}^dA(V)$.
\hfill
\theorem \label{_all_alg_with_so_are_^dA_Theorem_}
Let $\tilde A=A_0\oplus A_2\oplus ... \oplus A_{2n}$ be a
Lefschetz-Frobenius algebra of Jordan type. Assume that the graded
Lie algebra ${\goth g}(A)$ is isomorphic to $\goth{so}(V,+)$ and $dim_k V> 2$.
Let $A$ be the subalgebra of $\tilde A$ generated by $A_0$ and $A_2$
(also known as {\bf reduction} of $\tilde A$).
Then
(i) $n$ is even
(ii) $A$ is isomorphic to ${}^{n/2} A(V)$ as a graded algebra%
\footnote{The invariant Frobenius pairing is unique up to a scalar,
which is easy to see.}%
{}.
\hfill
{\bf Remark:} In particlural, this theorem implies that the reduction
$A$ of $\tilde A$ is Frobenius, which is not immediately clear
in general case.
\hfill
{\bf Proof:} Let ${\goth g}={\goth g}_{-2}\oplus {\goth g}_0\oplus {\goth g}_2$ be the graded
Lie algebra $\goth{so}(V,+)$. Denote the unit in $A$ by ${\Bbb I}$.
By definition, $A_0$ is ${\goth g}_0$-invariant one-dimensional space.
We have shown that
${\goth g}_0= \goth{so}(V)\oplus kH$. Since $\goth{so}(V)$ is a simple
Lie algebra, $\goth{so}(V)({\Bbb I})=0$. Therefore, to prove
\ref{_all_alg_with_so_are_^dA_Theorem_} (i), it is sufficient
to prove the following lemma.
\hfill
\lemma \label{_repres_so(V,+)_even-weight_Lemma_}
Let $\rho:\;{\goth g}\longrightarrow End(M)$ be a simple representation
of ${\goth g}\cong \goth{so}(V,+)$, where $dim_k(V)>2$. Let
${\Bbb I}\in M$ be a vector such that ${\goth g}_{-2}({\Bbb I})=0$,
$\goth{so}(V)({\Bbb I})=0$,%
\footnote{As elsewhere, we use the decomposition
${\goth g}_0\cong \goth{so}(V)\oplus kH$ provided by
\ref{_calculation_of_g(A)_for_minim_Theorem_}.}
and $H({\Bbb I})=-2n\cdot{\Bbb I}$. Then $n$ is even.
{\bf Proof:}
{\bf Step 1:} We show that $k\neq 1$.
Assume the contrary.
Consider the decomposition $M\cong M_{-1}\oplus M_1$, given
by the weights of $H$. Since ${\Bbb I}$ is a highest
weight vector for some root system in ${\goth g}$
(see \ref{_Cartan_exists_in_so(V,+)_Claim_}),
the corresponding weight space is one-dimensional.
This implies that $dim_k(M_{-1})=1$. There exists an automorphism
of $\goth{so}(V,+)$ which maps $H$ to $-H$ (an easy check).
Therefore, $dim_k(M_{-1})=dim_k(M_1)=1$. In other words,
the representation $M$ is two-dimensional. Consider $M$
as a representation of $\goth{so}(V)\subset \goth{so}(V,+)$.
Since ${\Bbb I}$ is invariant with respect to $\goth{so}(V)$,
the space $M$ is decomposed into a sum of two
one-dimensional $\goth{so}(V)$-invariant subspaces.
For $dim_k(V)>2$, the Lie algebra $\goth{so}(V)$
is simple. Therefore its one-dimensional representations
are trivial. We obtain that $\goth{so}(V)(M)=0$.
Since ${\goth g}=\goth{so}(V,+)$ is a simple Lie algebra,
the homomorphism $\rho:\; {\goth g}\longrightarrow End(M)$
cannot have proper non-zero kernel. Therefore,
$\rho({\goth g})=0$.
{\bf Step 2:} We use the following statement, which is easy to check.
\hfill
\claim \label{_Cartan_exists_in_so(V,+)_Claim_}
Let $\goth{so}(V)\oplus kH \stackrel i \hookrightarrow \goth{so}(V,+)$
be the embedding provided by \ref{_calculation_of_g(A)_for_minim_Theorem_}
(i). Then there exists a Cartan subalgebra $\goth h'\subset \goth{so}(V)$
such that $i(\goth h'\oplus kH)$ is a Cartan subalgebra in $\goth{so}(V,+)$.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Take a Cartan subalgebra $\goth h:= i(\goth h'\oplus kH)\subset {\goth g}$
provided by \ref{_Cartan_exists_in_so(V,+)_Claim_}. Clearly,
the linear map
\[ -H\check{\;}:= -(H,\cdot), \;\;-H\check{\;}:\; \goth h\longrightarrow k\]
is a root. Taking a root system $\alpha_1, ... ,\alpha_m$ in
$\goth h'\subset \goth{so}(V)$, we obtain that
$\alpha_1, ... ,\alpha_m, -H\check{\;}$ is a root system in
${\goth g}$. In this root system, ${\Bbb I}\in M$ is a highest weight
vector of the representation $M$. It is known that the
set\footnote{Here, $\goth h\check{\;}$ means $\goth h$ dual}
$W\subset \goth h\check{\;}$ of possible
weights of the highest weight vector coinsides with the intersection
of a weight lattice $L$ in $\goth h\check{\;}$ and a Weyl chamber.
In particular, $W\subset\goth h\check{\;}$ is an abelian semi-group
with group structure induced from $\goth h\check{\;}$.
The weight of ${\Bbb I}$ corresponding to the root system
$\alpha_1, ... ,\alpha_m, -H\check{\;}$ is $(0,0 ,..., 2n)$.
Let $W_0\subset W$ be the set of all
$(0,0 ,..., 2n)\subset W\subset \goth h\check{\;}$ which correspond to
representations $M$ satisfying conditions of
\ref{_repres_so(V,+)_even-weight_Lemma_}.
Clearly,
\[ W_0= \bigg \{ (x_1, .... , x_m)\in
W \;\;|\;\; x_1=...=x_{m-1}=0\bigg \}
\]
By Step 1, $n\neq 1$. Since Weyl chamber is invariant with
respect to homotheties, the semigroup $W_0$ is isomorphic
to ${\Bbb Z}_{\geq 0}$. Therefore, $n$ is never odd.
\ref{_repres_so(V,+)_even-weight_Lemma_} is proven.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
To prove \ref{_all_alg_with_so_are_^dA_Theorem_},
we notice that the simple representation of a reductive Lie algebra
is uniquely determined by its highest weight.
The weights of representations of $\goth{so}(V,+)$
corresponding to Lefschetz-Frobenius algebras are computed
a few lines above. In particular, we obtained that for
some root system in ${\goth g}$, the highest weight of $A$
is $(0,0,0... , 2n)$. Simple representations with a given
highest weight are isomorphic. Therefore,
as a representation of ${\goth g}$, $A\cong {}^dA(V)$.
By \ref{_g_structure_defines_algebr_Claim_},
the action of ${\goth g}$ detrermines multiplication in $A$ uniquely.
This finishes the proof of \ref{_all_alg_with_so_are_^dA_Theorem_}.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\section[Computing the structure Lie algebra for the
cohomology of a hyperkaehler manifold,
part \ I.]{Computing the structure Lie algebra for the
cohomology of a hyperkaehler manifold, \\part \ I.}
\label{_computing_g_for_hyperk_pt-I_Section_}
Let $M$ be a compact Kaehler manifold. According to
\ref{_Lefshe_Frob_if_a_Lefshe_ele_exists_Proposition_},
the ring $A:=H^*(M)$ is Lefschetz-Frobenius. The aim of this
section is to compute ${\goth g}(A)$ in the case when
$M$ is a simple compact hyperkaehler manifold. The answer is
hinted at by the following statement, which is proven in
\cite{_so5_on_cohomo_}:
\hfill
\proposition \label{_subalg_in_g_genera_by_three_Kae_forms_Proposition_
Let $\c H\in Hyp$ be a hyperkaehler structure on $M$. Consider the
Kaehler classes
\[ \omega_I= P_1(\c H),\;\; \omega_J=P_2(\c H),\;\;
\omega_K= P_3(\c H), \;\;\omega_I, \omega_J,\omega_K\in H^2(M,{\Bbb R}).
\]
Cohomology classes $\omega_I, \omega_J,\omega_K$ are Lefschetz elements in
the graded Frobenius algebra $A:= H^*(M,{\Bbb R})$. Consider the graded
subalgebra ${\goth g}(\c H)$ in ${\goth g}(A)$ generated by
$L_{\omega_I},L_{\omega_J},L_{\omega_K}$,
$\Lambda_{\omega_I}, \Lambda_{\omega_J},\Lambda_{\omega_K}$ and
$H$. Then ${\goth g}(\c H)$ is isomorphic to ${\goth g}({\Bbb R}^3)$, where
${\goth g}({\Bbb R}^3)\cong \goth{so}(4,1)$ is the structure Lie algebra
of the minimal graded Lefschetz-Frobenius algebra corresponding
to the linear space ${\Bbb R}^3$ with positively defined scalar product.
In particular, the graded algebra
${\goth g}(\c H)$ is independent from $\c H$ and $M$.
{\bf Proof:} This statement is proven in \cite{_so5_on_cohomo_}.
It is based on the commutation relations in ${\goth g}(\c H)$ given as follows.
Denote $P_i(\c H)$ by $\omega_i$, $i=1,2,3$. Denote $L_{\omega_i}$ by
$L_i$, and $\Lambda_{\omega_i}$ by $\Lambda_i$. Let
$K_{ij}:= [L_i,\Lambda_j]$, $i\neq j$.
Then the following relations are true:
\begin{equation} \label{_so5_relations_Equation_}
\begin{array}{l}
{}[ L_i, L_j] = [\Lambda_i,\Lambda_j] =0;\;\; \\[2mm]
[L_i,\Lambda_i]= H;\;\; [H, L_i] = 2 L_i;
\;\; [H, \Lambda_i]=-2\Lambda_i \\[2mm]{}
K_{ij}=-K_{ji}, [K_{ij}, K_{jk}]=2 K_{ik}, [K_{ij}, H] =0 \\[2mm]{}
[K_{ij} L_j]=2 L_i;\;\; [K_{ij} \Lambda_j] = 2 \Lambda_i \\[2mm]{}
[K_{ij}, L_k] = [K_{ij}, \Lambda_k] =0\;\; (k\neq i,j)\\[2mm]
\end{array}
\end{equation}
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Let $M$ be a simple hyperkaehler manifold, $A=H^*(M,{\Bbb R})$ be its
cohomology ring, equipped with the invariant pairing provided
by the Poincare duality. We consider $A=\oplus A_i=\oplus H^i(M,{\Bbb R})$
as a graded Frobenius algebra over ${\Bbb R}$.
In Section \ref{_Hodge-Rie_independent_Section_},
we defined the normalized Hodge-Riemann pairing $(\cdot,\cdot)_{\c H}$
on $H^2(M,{\Bbb R})$. Let $V$ be a linear space
$H^2(M)$ equipped with this pairing.
In Section \ref{_minimal_Fro_Section_}, we defined a graded
Frobenius algebra ${\goth g}(V)$, also denoted by $\goth{so}(V,+)$.
By definition, over ${\Bbb R}$ the algebra $\goth{so}(V,+)$ is isomorphic to
$\goth{so}(m+1,n+1)$, where $(m,n)$ is the signature of $V$.
\hfill
\theorem\label{_g(A)_for_hyperkae_Theorem_}
In this notation, ${\goth g}(A)$ is isomorphic to ${\goth g}(V)$.
\hfill
\ref{_g(A)_for_hyperkae_Theorem_} is the main result of this paper.
It is proven in the subsequent sections. The present section is
dedicated to proving that the Lefschetz-Frobenius
algebra $A$ is of Jordan type. This
is a crucial step in proof of \ref{_g(A)_for_hyperkae_Theorem_}.
\hfill
{\bf Remarks:} For a hyperkaehler
manifold with $dim \; H^2(M) =3$ (minimal dimension which
is not obviously impossible), \ref{_g(A)_for_hyperkae_Theorem_}
is equivalent to \ref{_subalg_in_g_genera_by_three_Kae_forms_Proposition_}.
For a hyperkaehler manifold with $dim\; H^2(M) =n$,
the Riemann-Hodge metric on $H^2(M,{\Bbb R})$ has a signature $(3,n-3)$.
This means that ${\goth g}(V)$ is isomorphic to $\goth{so}(4,n-2)$.
\hfill
\proposition \label{_H^*_hyp_Jordan_type_Proposition_}
Let $A$ be the Lefschetz-Frobenius
algebra of cohomology of a simple compact hyperkaehler
manifold. Then $A$ is of Jordan type.
{\bf Proof:} According to \ref{_Lunts_about_FLJ_Proposition_},
it is sufficient to show that for all $a,b \in A_2$,
$a$, $b$ of Lefschetz type, $[\Lambda_a,\Lambda_b]=0$.
Let $a,b\in A_2$, $\c H\in Hyp$, $\c H = (I,J,K, (\cdot,\cdot))$.
We write $a\bullet_{\c H}b$ if there exist complex structures
$I_1, I_2$ which are induced by $\c H$ such that $a, b$ are
Kaehler classes corresponding to $I_1$, $I_2$ and the metric
$(\cdot,\cdot)$.\footnote{the metric $(\cdot,\cdot)$ is Kaehler
with respect to the complex structures $I_1$, $I_2$, as the
definition of induced complex structures implies.}
Clearly, if $a\bullet_{\c H} b$, then $a$ and $b$
are of Lefschetz type.
\hfill
\lemma \label{_a_bullet_H_b=>_Lambdas_commute_Lemma_}
Let $\alpha,\beta\in A_2$, $\c H\in Hyp$, $\alpha\bullet_{\c H}\beta$.
Then $[\Lambda_\alpha,\Lambda_\beta]=0$.
{\bf Proof:} Let ${\goth g}(\c H)$ be the graded Lie algebra defined
in \ref{_subalg_in_g_genera_by_three_Kae_forms_Proposition_}.
Since ${\goth g}(\c H)={\goth g}({\Bbb R}^3)$,
the $-4$-th component of ${\goth g}(\c H)$ vanishes: ${\goth g}_{-4}(\c H)=0$.
Therefore for all $\lambda,\mu\in {\goth g}_{-2}(\c H)$, we have
$[\lambda,\mu]=0$. Therefore, to prove
\ref{_a_bullet_H_b=>_Lambdas_commute_Lemma_} it is sufficient
to show that $\Lambda_\alpha,\Lambda_\beta\in {\goth g}_{-2}(\c H)$.
This is implied by the following lemma.
\hfill
\lemma\label{_Kaehle_cla_indu_by_H_in_g(H)_Lemma_}
Let $\c H\in Hyp$, $\c H=(I,J,K,(\cdot,\cdot))$
and $I'$ be a complex structure induced
by $\c H$. Let $\omega\in A_2$ be the Kaehler class corresponding
to $I'$ and the metric $(\cdot,\cdot)$. Then
$\Lambda_\omega \in {\goth g}_{-2}(\c H)$.
{\bf Proof:} Let $\omega_i:=P_i(\c H)$, $i=1,2,3$.
By definition of induced complex structures,
$I'= a I+b J+ c K$, where $a,b,c\in {\Bbb R}$, $a^2+b^2+c^2=1$.
Since $\omega(x,y)= (x, I'y)$, we have
$\omega=a\omega_1+b\omega_2+c\omega_3$. Let
\[ \Lambda:= a\Lambda_{\omega_1}+
b\Lambda_{\omega_2} + c\Lambda_{\omega_3}.
\]
Using \eqref{_so5_relations_Equation_}, we see that
$[L_\omega,\Lambda]=H$. The other relations defining Lefschetz
triples checked automatically, we obtain that
$(L_\omega, H, \Lambda)$ is a Lefschetz triple. Therefore,
$\Lambda=\Lambda_\omega\in{\goth g}_2(\c H)$. This proves
\ref{_a_bullet_H_b=>_Lambdas_commute_Lemma_}
and \ref{_Kaehle_cla_indu_by_H_in_g(H)_Lemma_}.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Let $a,b\in A_2$. We write $a\bullet b$ if there exist $\c H\in Hyp$,
$\lambda\in {\Bbb R}$, $\lambda\neq 0$ such that $a\bullet_{\c H}\lambda b$.
Clearly, $\Lambda_{\lambda b} = \lambda^{-1}\Lambda_b$.
Therefore, \ref{_a_bullet_H_b=>_Lambdas_commute_Lemma_}
implies the following statement.
\hfill
\claim \label{_a_bullet_b_Lambdas_commute_Claim_}
Let $a,b\in A_2$, $a\bullet b$. Then $[\Lambda_a,\Lambda_b]=0$.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
The relation $a\bullet b$ is prominent further on in this paper.
We could have given an alternative definition of $a\bullet b$ as follows.
\hfill
\lemma \label{_a_bullet_b_if_hyperk_exists_Lemma_}
Let $a,b \in A_2$. The following conditions are equivalent:
(i) $a \bullet b$
(ii) There exists $\c H\in Hyp$ such that $a$ and $b$
can be expressed as linear combinations of $P_i(\c H)$,
$i=1,2,3$.
{\bf Proof:} Clear from \ref{_action_SO(3)_on_Hyp_via_periods_Lemma_}.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
{\bf Remark:} Let $a\bullet b$, $a,b\neq 0$. Then both $a$ and $b$ are
elements of Lefschetz type.
\hfill
Let $S\in A_2$ be a set of all elements of Lefschetz type.
By \ref{_a_Lefshe_if_a^i_iso_Lemma_}, $S$ is Zariski open in $A_2$.
Let $\nu:\; S\longrightarrow End(A)$ map $a\in S$ to $\Lambda_a\in End(A)$.
An easy linear-algebraic check implies that $\nu$ is an algebraic
map. Therefore the map $\eta:\; S\times S\longrightarrow End(A)$,
$a,b \longrightarrow [\Lambda_a,\Lambda_b]$ is also an algebraic map.
To prove \ref{_H^*_hyp_Jordan_type_Proposition_}
it is sufficient to show that $\eta$ is identically zero,
as shown by \ref{_Lunts_about_FLJ_Proposition_}.
By \ref{_a_bullet_b_Lambdas_commute_Claim_},
for all $a,b\in A_2$, such that $a\bullet b$,
we have $\eta(a,b)=0$. Therefore,
\ref{_H^*_hyp_Jordan_type_Proposition_} is implied by
the following statement:
\hfill
\lemma \label{_a_bullet_b_dense_in_A_2_x_A_2_Lemma_}
Let $D\subset A_2\times A_2$ be the set of all pairs
$(a,b)\in A_2\times A_2$ such that $a\bullet b$. Then
$D$ is Zariski dense in $A_2$.
{\bf Proof:} We use the following trivial
statement:
\hfill
\sublemma \label{_a_bullet_b_linear_span_Sublemma_}
Let $(a,b)$ and $(a',b')$ be two pairs in $A_2$ such that
the linear span of $(a,b)$ is equal to the linear span
of $(a',b')$. Then
\[ a\bullet b\;\;\Leftrightarrow \;\; a'\bullet b'. \]
{\bf Proof:} See
\ref{_a_bullet_b_if_hyperk_exists_Lemma_}.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Denote the Grassmanian of all 2-dimensional planes in $H^2(M,{\Bbb R})$ by $Gr$.
Let $Gr^\bullet$ be the space of planes generated by $a, b$ with
$a\bullet b$. To prove \ref{_a_bullet_b_dense_in_A_2_x_A_2_Lemma_}
it is sufficient to show that $Gr^\bullet$ is Zariski dense in $Gr$.
\hfill
In Section \ref{_Q_c_defini_Section_}, we defined the period map
$Q_c:\; Comp\longrightarrow G_r$.
\hfill
\claim \label{_D_in_Q_c(Comp)_Claim_}
The space $Gr^\bullet$ coincides with $Q_c(Comp)$.
{\bf Proof:} Clear from definitions. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Since the map $Q_c$ is etale (see the end of Section
\ref{_Q_c_defini_Section_}), its image contains an open set
in $Gr$. Therefore $Q_c(Comp)$ is Zariski dense in $Gr$.
\ref{_a_bullet_b_dense_in_A_2_x_A_2_Lemma_}, and consequently,
\ref{_H^*_hyp_Jordan_type_Proposition_}, is proven.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\section{Calculation of a zero graded part of the structure Lie
algebra of the cohomology of a hyperkaehler manifold, part I.}
\label{_compu_g_0_part_1_Section_}
In this section, we make steps related to computation of
the grade zero part of the Lie algebra
${\goth g}(A)={\goth g}_{-2}(A)\oplus {\goth g}_0(A)\oplus {\goth g}_2(A)$.
As in the previous section, $V$ denotes the space $H^2(M,{\Bbb R})$,
considered as a linear space with the scalar product
$(\cdot, \cdot)_{\c H}$, and $A$ is the Frobenius algebra
$H^*(M,{\Bbb R})$. We construct an epimorphism
${\goth g}_0(A)\longrightarrow \goth{so}(V)\oplus kH$, where $kH$ is one-dimensional
Lie algebra.
\hfill
With every $I\in Comp$, we associate the semisimple
endomorphism $ad I$ of $A$, defined as follows (see also
Section \ref{hyperk_manif_Section_}). Consider the Hodge decomposition
$H^i(M,{\Bbb C})=\oplus_{p+q=i}H^{p,q}(M)$. Let
$ad^cI:\; H^i(M,{\Bbb C}) \longrightarrow H^i(M,{\Bbb C})$ multiply $(p,q)$-forms
by the scalar $(p-q)\sqrt{-1}\:$. One can check that $ad^c I$ commutes
with the standard real structure on $H^i(M,{\Bbb C})$. Therefore,
$ad^c I$ is a complexification of a (uniquely defined) endomorphism of
$H^i(M,{\Bbb R})$. Denote this endomorphism by $ad I$. This definition
coinsides with one given in Section \ref{hyperk_manif_Section_}.
Consider the action of $ad I$ on $V=H^2(M, {\Bbb R})$. Using
Hodge-Riemann relations, we immediately obtain that
$ad I\restrict{V}\in \goth{so}(V)$. Let $\c M_2\subset End(V)$
be the Lie algebra generated by endomorphisms $ad I\restrict{V}$,
for all $I\in Comp$. Clearly, $\c M_2\subset \goth{so}(V)$.
One can show that $\c M_2$ is a Mumford-Tate group of
$(M, I)$ for generic $I\in Comp$, although we never use this
observation.
Let $v:\; B\longrightarrow B$ be an endomorphism of a linear space $B$.
We call the endomorphism
\[ v^{\circ{}}\in End(B),
\;\;v^{\circ{}}:= v-\frac{1}{\dim B} Tr(B) Id_B
\]
{\bf the traceless part of $v$}. Clearly, the map
$Tl:\; \goth{gl}(V)\longrightarrow \goth{sl}(V)$,
$Tl(v)= v^{\circ{}}$ is a Lie algebra homomorphism.
For all $g\in {\goth g}_0(A)$, consider the restriction
$g\restrict{A_2}:\; A_2\longrightarrow A_2$. Let $g^\circ\in End(A_2)$ be the
traceless part of $g\restrict{A_2}$. This defines a Lie
algebra homomorphism\footnote{$V$ is $A_2$ is $H^2(M,{\Bbb R})$}
$t:\; {\goth g}_0(A)\longrightarrow \goth{sl}(V)$, $t(g)= g^\circ$.
Consider the one-dimensional Lie algebra in $End(A)$,
generated by the Hodge endomorphism $H\in End(A)$.
Denote this algebra by $k H$.
Let $s:\; {\goth g}_0(A)\longrightarrow k H$ map $g\in {\goth g}_0(A)$ to
\[
-\frac{Tr(g\restrict{A_0})}{\frac{1}{2}\dim_{\Bbb R} M} H\in kH.
\]
The map $s$ is defined in such a way that $s(H)= H$.
Clearly, $s$ is also a Lie algebra homomorphism.
\hfill
\proposition \label{_str_of_g_0_Proposition_}
The following statements are true:
\hfill
(i) $t({\goth g}_0(A))\subset \c M_2$
(ii) The inclusion $\c M_2\subset \goth{so}(V)$ is an equality:
$\c M_2= \goth{so}(V)$.
(iii) The map $t\oplus s:\; {\goth g}_0(A)\longrightarrow \goth{so}(V)\oplus k H$
is an epimorphism.
\hfill
{\bf Proof:} We use the following simple lemma.
\hfill
\lemma \label{_g_0_gener_by_[L,Lambda]_Lemma_}
(see also \cite{_Lunts-Loo_})
Let $\c A$ be a Lefschetz-Frobenius algebra of Jordan type,
${\goth g}(\c A)={\goth g}_{-2}(\c A)\oplus{\goth g}_0(\c A)\oplus{\goth g}_2(\c A)$ be its
structure Lie algebra. Then ${\goth g}_0(\c A)$ is a linear span
of the elements $[L_a,\Lambda_b]$, where $a,b$ are Lefschetz
elements in $\c A_2$.
{\bf Proof:} Clear.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Let $S\subset A_2$ be the set of all elements of Lefschetz type.
Let $\nu:\; S\times S \longrightarrow {\goth g}_0(A)$ be the map
$a,b\longrightarrow [L_a,\Lambda_b]$. \ref{_g_0_gener_by_[L,Lambda]_Lemma_}
implies that ${\goth g}_0(A)$ is a linear span of the set
$\nu(S\times S)$. Therefore, to prove \ref{_str_of_g_0_Proposition_}
(i) it is sufficient to show that for all $a,b\in S$,
$t(\nu(a,b))\in \c M_2$. As we have seen, $S$ is a
Zariski open subset in $A_2$. Consider $S$ as an algebraic
manifold with the algebraic structure induced from $A$.
With respect to this algebraic structure, both $t$ and
$\nu$ are algebraic maps. Therefore it is sufficient to
show that for a Zariski dense subset $D\subset S\times S$,
$t(\nu(D))\subset \c M_2$. By
\ref{_a_bullet_b_dense_in_A_2_x_A_2_Lemma_},
the set of all $a,b\in S$, $a\bullet b$ is
Zariski dense in $S$. Therefore, the inclusion
$t(\nu(S\times S))\subset \c M_2$ is implied by the following
statement:
\hfill
\lemma \label{_[L_a,Lambda_b]=I_for_(a,b)_in_Q_c(I)_Lemma_}
Let $a,b \in S$, $a\bullet b$, where $a$ is not proportional to $b$.
Let $\c L$ be a plane in
$H^2(M,{\Bbb R})$ spanned by $a$ and $b$. According to
\ref{_D_in_Q_c(Comp)_Claim_}, there exist $\c I\in Comp$ such
that $Q_c(I)= \c L$. Let
\[ \xi_1:=
\bigg([L_a,\Lambda_b]\bigg|_{{}_{H^2(M,{\Bbb R})}}\bigg)^\circ
\in End(H^2(M, {\Bbb R})
\]
be the traceless part of the restriction of the linear operator
$[L_a,\Lambda_b]\in End(A)$
to $H^2(M, {\Bbb R})=A_2\subset A$. Let $\xi_2$ be the restriction
of $ad I$ to $H^2(M, {\Bbb R})$. Then $\xi_1$ is proportional to $\xi_2$.
{\bf Proof:} The space $A_2\cong H^2(M, {\Bbb R})$ is equipped with the
normalized Hodge-Riemann pairing $(\cdot,\cdot)_{\c H}$.
Let $b,x$ be an orthogonal basis of $\c L$. Clearly,
$x=\lambda a +\mu b$, where $\lambda,\mu\in {\Bbb R}$. This implies that
\begin{equation} \label{_adding_L_in_[L,Lambda]_Equation_}
[L_x,\Lambda_b] = \lambda H +\mu [L_a,\Lambda_b].
\end{equation}
Since the endomorphism $\lambda H\restrict{A_2}\in End(A_2)$ is proportional
to identity, the traceless parts
\[ \bigg([L_a,\Lambda_b]\bigg)^\circ, \;\;
\bigg([L_x,\Lambda_b]\bigg)^\circ
\]
are proportional. Therefore, proving
\ref{_[L_a,Lambda_b]=I_for_(a,b)_in_Q_c(I)_Lemma_}, we may
assume that
\[ (a,b)_{\c H}=0, \;\;
(a,a)_{\c H}= (b,b)_{\c H}>0.
\]
Let $\c H\in Hyp$ be a hyperkaehler structure such that
$\Phi^{hyp}_c(\c H)= \c I$. The relation $\Phi^{hyp}_c(\c H)= \c I$
means that $\c H= (\c I, J, K, (\cdot,\cdot))$ for some $J, K, (\cdot,\cdot)$.
By definition of $Q_c$, the linear span of $P_2(\c H)$, $P_3(\c H)$
coinsides with $\c L$. Using
\ref{_action_SO(3)_on_Hyp_via_periods_Lemma_},
we can easily find a hyperkaehler structure $\c H'\in Hyp$,
$\c H'$ equivalent to $\c H$, such that $P_2(\c H)=\lambda a$,
$P_3(\c H)=\lambda b$ for some $\lambda \in {\Bbb R}$. Now,
\ref{_[L_a,Lambda_b]=I_for_(a,b)_in_Q_c(I)_Lemma_}
is a consequence of the following simple statement:
\hfill
\claim \label{_[L_2,Lambda_3]=adI_Claim_}
Let $\c H\in Hyp$, $\c H= (I, J, K, (\cdot,\cdot))$,
$a=P_2(\c H)$, $b=P_3(\c H)$.
Then the following endomorphisms of $A$
coinside:
\[ [L_a,\Lambda_b]=ad I \]
{\bf Proof:} See \cite{_so5_on_cohomo_}. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
This finishes the proof of \ref{_str_of_g_0_Proposition_} (i).
To prove \ref{_str_of_g_0_Proposition_} (ii), we recall the
following linear-algebraic construction. Let $V$ be a linear space
equipped with non-degenerate symmetric bilinear form $(\cdot,\cdot)$,
and $\c L\subset V$ be a 2-dimensional plane in $V$,
such that the restriction of $(\cdot,\cdot)$ to $\c L$
is non-degenerate. Let $\c L^\bot$ be the orthogonal complement
of $V$ to $\c L$.
Let $T_{\c L}$ be the set of all non-trivial skew-symmetric endomorphisms
of $V$ which vanish on $\c L^\bot$. As we have seen
previously, all elements of $T_{\c L}$ are proportional.
Let $Gr^\circ$ be the space of all 2-dimensional planes
$\c L\subset V$ such that the restriction
$(\cdot,\cdot)_{\c H}\restrict{\c L}$ is non-degenerate.
\hfill
\claim \label{_T_L_generate_SO_Claim_}
The linear span of the union
\[ \bigcup\limits_{\c L\in Gr^\circ} T_{\c L} \]
coinsides with $\goth{so}(V)$.
{\bf Proof:} Clear. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
By \ref{_T_L_generate_SO_Claim_}, to prove that $\c M_2=\goth{so}(V)$ it
is sufficient to show that for a Zariski dense set $\c E\subset Gr^\circ$,
we have
\[
\forall \c L\in \c E, \;\; T_{\c L}\subset \c M_2.
\]
For $\c E$, we take the set $Gr^\bullet$ of
\ref{_D_in_Q_c(Comp)_Claim_}. Since $Gr^\bullet = Q_c(Comp)$
is Zariski dense in $Gr$, it is sufficient to show that for all
$\c L\in Q_c(Comp)$, we have $T_{\c L}\subset \c M_2$. This is
implied by \ref{_[L_2,Lambda_3]=adI_Claim_} and the following
statement.
\hfill
\claim \label{_T_Q(I)_is_[L,Lambda]_Claim_}
In assumptions of \ref{_[L_2,Lambda_3]=adI_Claim_},
the following two sets coinside:
\[ \bigg\{\lambda [L_a,\Lambda_b],\;\lambda\in{\Bbb R}\backslash 0\bigg\} =
T_{Q_c(I)}.
\]
{\bf Proof:} Since all elements of $T_{Q_c(I)}$ are proportional
and $[L_a,\Lambda_b]\neq 0$, it is sufficient to show
that $[L_a,\Lambda_b]\in T_{Q_c(I)}$. Let $\c L\subset V$ be the linear
span of $a$ and $b$. Obviously from definition, $Q_c(I)=\c L$.
Let $\c L^\bot$ be the orthogonal complement to $\c L$ in $V$.
We need to show that the restriction of $[L_a,\Lambda_b]$
to $V=H^2(M, {\Bbb R})$ vanishes on $\c L^\bot$. Consider the
Hodge decomposition on $H^2(M)$ associated with the complex
structure $I$. By definition,
\[ \c L =
\bigg( H^{2,0}_I(M)\oplus H^{0,2}_I(M)\bigg) \cap H^2(M, {\Bbb R}).
\]
Hodge-Riemann relations imply that
\[
\c L^\bot= H^{1,1}_I(M)\cap H^2(M, {\Bbb R})
\]
(see Section
\ref{_Period_and_Hodge_Riemann_Section_} for details). By
defintion of $ad I$, we have
$ad I(H^{1,1}_I(M))=0$. Since $[L_a,\Lambda_b] = ad I$, we
obtain that $[L_a,\Lambda_b]\in T_{Q_c(I)}$. This proves
\ref{_T_Q(I)_is_[L,Lambda]_Claim_}, and consequently,
proves \ref{_str_of_g_0_Proposition_} (ii). $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Using the fact that
\[ \{ a,b\in A_2\times A_2 \;\; | \; \; a\bullet b \} \]
is Zariski dense in $A_2\times A_2$, we can derive
from \ref{_T_Q(I)_is_[L,Lambda]_Claim_} the following
corollary:
\hfill
\corollary \label{_[L,Lambda]_in_T_a,b_Corollary_}
Let $a,b\in A_2$ be elements of Lefschetz type. Let
$\c L\subset A_2$ be a plane generated by $a$ and $b$
and $\c L^\bot$ be its orthogonal complement in $A_2=V$.
Let $[L_a,\Lambda_b]\bigg|_{{}_{V}}^\circ$ be a traceless part of
$[L_a,\Lambda_b]\bigg|_{{}_{V}}\in End(V)$. Then
$[L_a,\Lambda_b]\bigg|_{{}_{V}}^\circ$ vanish on $\c L^\bot$.
{\bf Proof:} Using the argument with Zariski dense sets,
we see that it is sufficient to check
\ref{_[L,Lambda]_in_T_a,b_Corollary_}
for $a,b$ such that $a\bullet b$, and $(b,b)_{\c H}\neq 0$.
When $(a,b)_{\c H} =0$,
\ref{_[L,Lambda]_in_T_a,b_Corollary_}
is a direct consequence of \ref{_T_Q(I)_is_[L,Lambda]_Claim_}.
If $(a,b)_{\c H} \neq 0$, take
\[ x:= a - \frac{(a,b)_{\c H}}{(b,b)_{\c H}} b. \]
Clearly, $(x,b)_{\c H} =0$.
The traceless part of $[L_b, \Lambda_b]\bigg|_{{}_{V}}= H\restrict{V}$ is zero.
Therefore,
\[ [L_a,\Lambda_b]\bigg|_{{}_{V}}^\circ =
[L_x,\Lambda_b]\bigg|_{{}_{V}}^\circ
\]
This reduces \ref{_[L,Lambda]_in_T_a,b_Corollary_}
to the case $(a,b)_{\c H} =0$.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
To prove \ref{_str_of_g_0_Proposition_} (iii), we notice that from
the proof of \ref{_str_of_g_0_Proposition_} (ii) it follows also that
$t:\; {\goth g}_0(A)\longrightarrow \goth{so}(V)$ is an epimorphism. Therefore, it is
sufficient to find an element $e\in {\goth g}_0(A)$ such that $t(e)=0$,
$s(e)\neq 0$. This element is a Hodge endomorphism
$H\in {\goth g}_0(A)\subset End(A)$. We proved
\ref{_str_of_g_0_Proposition_} (iii). $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\section{Calculation of a zero graded part of the structure Lie
algebra of the cohomology of a hyperkaehler manifold, part II.}
\label{_compu_g_0_part_2_Section_}
We work in assumptions of the previous section.
\hfill
\theorem \label{_g_0_computed_Theorem_}
In assumptions of \ref{_str_of_g_0_Proposition_},
consider the homomorphism
\[
u:\; {\goth g}_0(A)\longrightarrow \goth{so}(V) \oplus k H,
\]
$u=t\oplus s$. Then $u$ is an isomorphism.
{\bf Proof:} We use the following technical result:
\hfill
\proposition \label{_[L_a,Lambda_b]-[Lambda_a,L_b]_Proposition_}
Let $A$ be a Lefschetz-Frobenius algebra of cohomology of a compact
simple hyperkaehler manifold. Let $a, b\in A_2$ be elements of Lefschetz
type. Then
\[ {{(b,b)_{\c H}}} [L_a, \Lambda_b] -
{{(a,a)_{\c H}}} [\Lambda_a,L_b] =2(a,b)_{\c H} H,
\]
where $(\cdot,\cdot)_{\c H}$ is a normalized Hodge-Riemann pairing.
{\bf Proof:} First of all, we prove the following lemma.
\hfill
\lemma \label{_Lambda_additive_for_a_bullet_b_Lemma_
In assumptions of
\ref{_[L_a,Lambda_b]-[Lambda_a,L_b]_Proposition_},
let $a,b\in A_2$, $a\bullet b$.
Then $a+b$ is also of Lefschetz type, and
\[ {(a+b,a+b)_{\c H}}\Lambda_{a+b} =
{(a,a)_{\c H}}\Lambda_a + {(b,b)_{\c H}}\Lambda_b.
\]
{\bf Proof:} Let $\c H\in Hyp$ be a hyperkaehler structure,
such that there exist $x_i, y_i\in {\Bbb R}$, $i=1,2,3$, such that
\[
a =\sum x_i\omega_i, \;\; b=\sum y_i\omega_i,
\]
where $\omega_i=P_i(\c H)$, $i=1,2,3$. Such $\c H$ exists
by definition of the relation $a \bullet b$.
Clearly,
\[ (\omega_1,\omega_1)_{\c H}= (\omega_2,\omega_2)_{\c H} =
(\omega_3,\omega_3)_{\c H}.
\]
Let $c:= {(\omega_1,\omega_1)_{\c H}}$.
We are going to show that
\begin{equation}\label{_Lambda_a_as_lin_combi_Lambda_omega_Equation_}
\Lambda_a = c\frac{\sum x_i\Lambda_{\omega_i}}{{(a,a)_{\c H}}.}
\end{equation}
First of all, we notice that for all
$a =\sum x_i\omega_i, \;\; a\neq 0,$ the triple
\[
L_a,\;\; H,\;\; \Lambda :=
\frac{\sum x_i\Lambda_{\omega_i}}{{\sum x_i^2}}
\]
is a Lefschetz triple.
This can be shown by an easy calculation which uses
\eqref{_so5_relations_Equation_}. On the other hand,
$\omega_i$ are orthogonal with respect to $(\cdot,\cdot)_{\c H}$,
and therefore $(a,a)_{\c H}= c\sum x_i^2$. Therefore,
\[ c\frac{\sum x_i\omega_i}{{(a,a)_{\c H}}} =
\frac{\sum x_i\omega_i}{{\sum x_i^2}}
\]
Now, \eqref{_Lambda_a_as_lin_combi_Lambda_omega_Equation_}
immediately implies \ref{_Lambda_additive_for_a_bullet_b_Lemma_},
as an easy calculation shows:
\[ {(a+b,a+b)_{\c H}}\Lambda_{a+b} =^*
c\sum (x_i+y_i)\Lambda_{\omega_i} =
\]
\[ = c\sum x_i\Lambda_{\omega_i}+c\sum y_i\Lambda_{\omega_i}
=^* {(a,a)_{\c H}}\Lambda_{a} +{(b,b)_{\c H}}\Lambda_{b},
\]
where $=^*$ marks an application of
\eqref{_Lambda_a_as_lin_combi_Lambda_omega_Equation_}.
This proves \ref{_Lambda_additive_for_a_bullet_b_Lemma_}.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Since
\[ \{ (a,b)\in A_a \;\; |\;\; a\bullet b \} \]
is Zariski dense in $A_2$,
\ref{_Lambda_additive_for_a_bullet_b_Lemma_}
has the following corollary:
\hfill
\corollary \label{_Lambda_additive_Corollary_}
Let $a,b\in A_2$, be the elements of Lefschetz type, such that
$a+b$ is also of Lefschetz type. Then
\[ {(a+b,a+b)_{\c H}}\Lambda_{a+b} =
{(a,a)_{\c H}}\Lambda_a + {(b,b)_{\c H}}\Lambda_b.
\]
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\ref{_Lambda_additive_Corollary_} implies the following interesting
result. The set $S\subset A_2$ of Lefschetz elements
is preserved by homotheties.
Therefore we may speak of homogeneous functions from
$S$ to some linear space. Consider the function
$\Lambda:\; S\longrightarrow {\goth g}_{-2}$, $x\longrightarrow \Lambda_2$.
Clearly, this map is homogeneous of degree $-1$.
Therefore, the map $\tilde r:\;S\longrightarrow {\goth g}_{-2}$,
$x\longrightarrow (x,x)_{\c H} \Lambda_x$ is homogeneous of degree 1.
By \ref{_Lambda_additive_Corollary_}, $\tilde r$ is linear
on $S$. Therefore, $\tilde r$ is a restriction of a linear
map $r:\; A_2 \longrightarrow {\goth g}_{-2}$
\hfill
\corollary \label{_g_-2_is_quotie_of_A_2_Corollary_}
The map $r:\; A_2\longrightarrow {\goth g}_{-2}$ is a surjection
of linear spaces.
{\bf Proof:} For all Lefschetz-Frobenius
algebras $C$, ${\goth g}_{-2}(C)$ is spanned by $\Lambda_x$
for $x$ of Lefschetz type (\cite{_Lunts-Loo_}).
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
To prove \ref{_[L_a,Lambda_b]-[Lambda_a,L_b]_Proposition_},
we need only the formula
\eqref{_Lambda_a_as_lin_combi_Lambda_omega_Equation_}.
Since
\[ \{ (a,b)\in A_a \;\; |\;\; a\bullet b \} \]
is Zariski dense in $A_2$, it is sufficient to prove
\ref{_[L_a,Lambda_b]-[Lambda_a,L_b]_Proposition_}, for $a,b$
such that $a\bullet b$. Consider notation introduced
in the proof of \ref{_Lambda_additive_for_a_bullet_b_Lemma_}.
The formula \eqref{_Lambda_a_as_lin_combi_Lambda_omega_Equation_}
together with relations \eqref{_so5_relations_Equation_}
implies that
\begin{equation}\label{_[Lambda_a,L_b]_via_x_y_Equation_}
[ \Lambda_a, L_b ] =
\bigg[ \frac{\sum x_i \Lambda_{\omega_i}}{\sum x_i^2},
\sum y_i L_{\omega_i} \bigg]
= \frac{1}{\sum x_i^2} \bigg(-\sum x_i y_i H +
\sum\limits_{i\neq j} x_i y_j [\Lambda_{\omega_i}, L_{\omega_j}]\bigg),
\end{equation}
and
\begin{equation}\label{_[L_a,Lambda_b]_via_x_y_Equation_}
[ L_a, \Lambda_b ] =
-\bigg[ \frac{\sum y_i \Lambda_{\omega_i}}{\sum y_i^2},
\sum x_i L_{\omega_i} \bigg]
= \frac{1}{\sum y_i^2} \bigg(\sum x_i y_i H -
\sum\limits_{i\neq j} x_i y_j [L_{\omega_i}, \Lambda_{\omega_j}]\bigg).
\end{equation}
Let $c\in {\Bbb R}$ be the constant defined in the proof
of \ref{_Lambda_additive_for_a_bullet_b_Lemma_}.
We have
\[ (a,a)_{\c H}= c \sum x_i^2,\;\; (b,b)_{\c H}= c \sum y_i^2,\;\;
(a,b)_{\c H} = c \sum x_i y_i.
\]
Making the corresponding substitutions in
\eqref{_[Lambda_a,L_b]_via_x_y_Equation_},
\eqref{_[L_a,Lambda_b]_via_x_y_Equation_}, we obtain
\begin{equation}\label{_[Lambda_a,L_b]_via_()_H_and_x_i_Equation_}
[ \Lambda_a, L_b ]
= \frac{1}{(a,a)_{\c H}} \bigg(-(a,b)_{\c H} H +
\frac{1}{c}
\sum\limits_{i\neq j} x_i y_j [\Lambda_{\omega_i}, L_{\omega_j}]\bigg)
\end{equation}
and
\begin{equation}\label{_[L_a,Lambda_b]_via_()_H_and_x_i_Equation_}
[ L_a, \Lambda_b ]
= \frac{1}{(b,b)_{\c H}} \bigg((a,b)_{\c H} H +
\frac{1}{c}
\sum\limits_{i\neq j} x_i y_j [L_{\omega_i}, \Lambda_{\omega_j}]\bigg).
\end{equation}
A linear combination of these equations yields
\[ {(b,b)_{\c H}}[L_a, \Lambda_b]-{(a,a)_{\c H}}[\Lambda_a, L_b] =
(a,b)_{\c H} H +
\]
\[ +\frac{1}{c}
\sum\limits_{i\neq j} x_i y_j
\bigg([\Lambda_{\omega_i}, L_{\omega_j}] -
[L_{\omega_i}, \Lambda_{\omega_j}] \bigg)
\]
To prove \ref{_[L_a,Lambda_b]-[Lambda_a,L_b]_Proposition_}
it remains to show that
\[ \sum\limits_{i\neq j} x_i y_j
\bigg([\Lambda_{\omega_i}, L_{\omega_j}] -
[L_{\omega_i}, \Lambda_{\omega_j}] \bigg) =0,
\]
which follows from the relation
\[ [\Lambda_{\omega_i}, L_{\omega_j}] =
[L_{\omega_i}, \Lambda_{\omega_j}], \;\;i \neq j
\]
from \eqref{_so5_relations_Equation_}.
\ref{_[L_a,Lambda_b]-[Lambda_a,L_b]_Proposition_}
is proven. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Let $s:\; {\goth g}_0\longrightarrow kH$ be a Lie algebra homomorphism
of \ref{_str_of_g_0_Proposition_}.
Either of equations \eqref{_[L_a,Lambda_b]_via_()_H_and_x_i_Equation_}
and \eqref{_[Lambda_a,L_b]_via_()_H_and_x_i_Equation_} implies the
following useful corollary:
\hfill
\corollary \label{_s_of_[L_a,Lambda_b]_Corollary_}
Let $a,b \in S$. Then
\[ s([L_a,\Lambda_b]) = \frac{(a,b)_{\c H}}{(b,b)_{\c H}} H. \]
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Now we can easily prove \ref{_g_0_computed_Theorem_}.
Let $n=\dim V$. Clearly, $\dim \goth{so}(V)=\frac{n(n-1)}{2}$.
Since $u:\; {\goth g}_0\longrightarrow \goth{so}(V)\oplus k$ is an epimorphism by
\ref{_str_of_g_0_Proposition_}, it is sufficient to show that
$\dim {\goth g}_0\leq \frac{n(n-1)}{2}+1$. The element $H$ belongs to the center
of ${\goth g}_0$. Let $\bar {\goth g}_0:= {\goth g}_0/k H$ be the quotient Lie algebra, and
$q:\; {\goth g}_0\longrightarrow \bar{\goth g}_0$ be the quotient map. Let $S\subset A_2$ be the
set of all elements of Lefschetz type. By
\ref{_g_0_gener_by_[L,Lambda]_Lemma_}, the space
$\bar {\goth g}_0$ is spanned by all vectors
$q([L_a,\Lambda_b])$, where $a,b\in S$.
Denote the map $S\times S\longrightarrow \bar{\goth g}_0$,
$a,b \longrightarrow q([L_a,\Lambda_b])$ by
$\tilde\nu:\; S\times S \longrightarrow \bar{\goth g}_0$. Let
$\nu:\; S\times S\longrightarrow \bar{\goth g}_0$,
\[
\nu(a,b):= \frac{q([L_a,\Lambda_b])}{(b,b)_{\c H}}.
\]
Consider the Zariski open set $S\subset A_2$ as a space with an
associative commutative group structure, which is defined by rational
maps. In other words, $S$ is equipped with an addition, which
is defined not everywhere, but in a Zariski open subset of
$S$. This addition is induced from $A_2\supset S$, which is a linear space.
By \ref{_Lambda_additive_Corollary_}, the map $\nu$ is bilinear with respect
to this operation. Consider $\nu$ as a rational map from $A_2\times A_2$
to $\bar {\goth g}_0$. This rational map is also bilinear. An easy check
shows that a linear rational map of linear spaces is defined everywhere.
Hence, $\nu$ can be uniquely lifted to a bilinear
map
\[
\nu:\; A_2\times A_2 \longrightarrow \bar{\goth g}_0,
\]
such that the square
\[ \begin{array}{ccc}
S\times S & \hookrightarrow & A_2\times A_2 \\[3mm]
\bigg\downarrow \nu &&\bigg\downarrow \nu \\[3mm]
\bar{\goth g}_0 &\stackrel{Id}{\longrightarrow} &\bar{\goth g}_0
\end{array}
\]
is commutative.
By \ref{_[L_a,Lambda_b]-[Lambda_a,L_b]_Proposition_},
the bilinear map $\nu:\; A_2\times A_2\longrightarrow \bar{\goth g}_0$ is
skew-symmetric. Let $\eta$ be the corresponding linear map from the
exterrior square of $A_2$ to $\bar {\goth g}_0$:
\[ \eta:\;
\bigwedge^2 A_2\longrightarrow \bar{\goth g}_0.
\]
Clearly, $\nu(A_2\times A_2) = \eta(\bigwedge^2 A_2)$.
As \ref{_g_0_gener_by_[L,Lambda]_Lemma_} implies,
$\bar{\goth g}_0$ is generated by the image of $\nu(A_2\times A_2)$.
Hence, $\eta$ is an epimorphism. Therefore,
$\dim \bar{\goth g}_0 \leq \dim \bigwedge^2 A_2 = \frac{n(n-1)}{2}$.
We obtained an upper bound on $\dim{\goth g}_0$:
\[
\dim {\goth g}_0\leq \frac{n(n-1)}{2} + 1.
\]
Since the Lie algebra $\goth{so}(V)\oplus k$ has the same dimension,
$\frac{n(n-1)}{2}+1$ and the map $u:\; {\goth g}_0\longrightarrow \goth{so}(V)\oplus k$
is an epimorphism as we have seen previously, the map $u$
is an isomorphism. \ref{_g_0_computed_Theorem_} is proven.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Consider the map $r:\; A_2\longrightarrow {\goth g}_{-2}$ constructed in
\ref{_g_-2_is_quotie_of_A_2_Corollary_}. Let $r':\; A_2\longrightarrow {\goth g}_2$
be a standard isomorphism. Then $\nu$ can be defined
as a composition of $r'\times r:\; A_2\times A_2 \longrightarrow {\goth g}_2\times {\goth g}_{-2}$
and a commutator map $[,]:\; {\goth g}_2\times {\goth g}_{-2}\longrightarrow {\goth g}_0$.
The map $\nu$ was described in terms of the standard bilinear
map $A_2\times A_2\longrightarrow \Lambda^2A_2$. This description
easily implies that for all $y\in A_2$ there exists
$x\in A_2$ such that $\nu(x,y)\neq 0$. Since
$\nu=r'\times r\circ [,]$, the map $r$ has no kernel.
It is epimorphic by \ref{_g_-2_is_quotie_of_A_2_Corollary_}.
We obtained the following statement:
\hfill
\corollary \label{_g_-2_is_A_2_Corollary_}
Consider the linear map $r:\; A_2\longrightarrow {\goth g}_{-2}$ constructed in
\ref{_g_-2_is_quotie_of_A_2_Corollary_}. Then $r$ is
an isomorphism of linear spaces.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\ref{_g_0_computed_Theorem_} gives a nice insight on the
Hodge structures on $H^*(M)$ corresponding to various
complex structures $I\in Comp$. As elsewhere,
for each $I\in Comp$ we define an endomorphism $ad I\in End(A)$,
$ad I(\omega)= (p-q)\sqrt{-1}\: \omega$ for all $\omega \in H^{p,q}(M)$.
Let $\goth M\subset End(A)$ be a Lie algebra generated by
$ad I$ for all $I\in Comp$.
\hfill
\lemma \label{_M_acts_by_deriva_Lemma_}
The Lie algebra $\goth M$ acts on $A$ by derivations:
for all $m\in \goth M$, $a,b \in A$, we have
\[
m(ab) = m(a) b+ am(b).
\]
{\bf Proof:} For all $I\in Comp$, the operator $ad I$ is a derivation,
as a calculation shows.
A commutator of derivations is a derivation by obvious reasons.
$\;\; \hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\theorem\label{_g_0_is_Mumf_Tate_Theorem_}
The following Lie subalgebras of $End(A)$ coinside:
\[
{\goth g}_0(A)\cong \goth M\oplus kH.
\]
{\bf Proof:} The inclusion $\goth M\subset {\goth g}_0(A)$ is implied by
\ref{_[L_2,Lambda_3]=adI_Claim_}. The inclusion
${\goth g}_0(A)\subset \goth M\oplus kH$ is proven as follows.
We have seen that ${\goth g}_0(A)$ is linearly spanned by $H$ and
endomorphisms $[L_a,\Lambda_b]$, where $a\bullet b$, $(a,b)_{\c H}=0$,
$(a,a)_{\c H}=(b,b)_{\c H}\neq 0$. Let $\c H'$ be a hyperkaehler
structure such that $a\bullet_{\c H'} b$. Applying
\ref{_action_SO(3)_on_Hyp_via_periods_Lemma_}, we
obtain a hyperkaehler structure $\c H$, $\c H$ equivalent to
$\c H'$, such that $a=P_2(\c H)$ and
$b=P_3(\c H)$. Then, \ref{_[L_2,Lambda_3]=adI_Claim_}
implies that $[L_a,\Lambda_b]\in \goth M$. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
By trivial reasons, $H$ acts on $A$ as a derivation. Therefore,
\ref{_g_0_is_Mumf_Tate_Theorem_} together with
\ref{_M_acts_by_deriva_Lemma_} implies the following:
\hfill
\corollary \label{_g_0_derivatives_Corollary_}
The Lie algebra ${\goth g}_0\subset End(A)$ acts on $A$ by derivations.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\section[Computing the structure Lie algebra for the
cohomology of a hyperkaehler manifold, part \ II.]
{Computing the structure Lie algebra for the
cohomology of a hyperkaehler manifold, \\part \ II.}
\label{_computing_g_for_hyperk_pt-2_Section_}
In this section, we prove the isomorphism ${\goth g}\cong \goth{so}(V, +)$.
This is done as follows. In previous sections, we
have computed dimensions of the components
of ${\goth g}\cong {\goth g}_{-2}\oplus {\goth g}_0\oplus{\goth g}_2$. Let $n:= \dim V$.
\ref{_g_-2_is_A_2_Corollary_} implies that $\dim {\goth g}_2=n$,
\ref{_g_-2_is_A_2_Corollary_} implies that $\dim{\goth g}_{-2}=n$,
and \ref{_g_0_computed_Theorem_} implies that
$\dim{\goth g}_0 = \frac{n(n-1)}{2}+1$. Therefore,
\[
\dim{\goth g}= \frac{n(n-1)}{2}+1 +2n = \frac{(n+2)(n+1)}{2}.
\]
A trivial computation yields
\[
\dim \goth{so}(V,+)= \frac{(n+2)(n+1)}{2}.
\]
We see that dimensions of ${\goth g}$ and $\goth{so}(V, +)$ are equal;
it is easy to see that ${\goth g}$ is isomorphic to $\goth{so}(V, +)$
as a graded linear space.
We construct an isomorphism of graded linear spaces
$U:\; {\goth g} \longrightarrow \goth{so}(V, +)$ and prove that
it commutes with the Lie algebra operation.
\hfill
Let $B= B_0\oplus B_2\oplus B_4$ be the minimal graded Frobenius
algebra associated with $V$, $(\cdot,\cdot)_{\c H}$.
By definition, ${\goth g}(B)= \goth{so}(V,+)$. We are going to construct
an isomorphism of linear spaces $U=U_{-2}\oplus U_0\oplus U_2$,
\[
U:\; {\goth g}(A)\longrightarrow {\goth g}(B),
\;\; U_{2i}:\; {\goth g}_{2i}(A)\longrightarrow {\goth g}_{2i}(B).
\]
We have canonical isomorphisms
\[ u_B:\; {\goth g}_0(B)\longrightarrow \goth{so}(V)\oplus kH,\;\;
u_A:\; {\goth g}_0(A)\longrightarrow \goth{so}(V)\oplus kH.
\]
These isomorphisms yield a natural Lie algebra isomorphism
$U_0:\; {\goth g}_0(A)\longrightarrow {\goth g}_0(B)$. The homomorphism
$U_2:\; {\goth g}_2(A)\longrightarrow {\goth g}_2(B)$ is provided by the natural
isomorphism ${\goth g}_2(A)\cong A_2$ (\ref{_g_2_is_A_2_Corollary_}),
which exists for every Lefschetz-Frobenius algebra of Jordan type.
To construct the isomorphism $U_{-2}:\; {\goth g}_{-2}(A)\longrightarrow {\goth g}_{-2}(B)$,
we use \ref{_g_-2_is_A_2_Corollary_}. According to this statement,
${\goth g}_{-2}(A)$ is naturally isomorphic to $V$.
The natural isomorphism ${\goth g}_{-2}(B)\cong V$ is constructed in
\ref{_calculation_of_g(A)_for_minim_Theorem_}. Composing
these isomorphisms, we obtain $U_{-2}$.
Now, the isomorphism ${\goth g}(A)\cong \goth{so}(V, +)$
is implied by the following proposition:
\hfill
\proposition \label{_U_is_Lie_homomo_Proposition_}
The map $U:\; {\goth g}(A)\longrightarrow {\goth g}(B)$
is a homomorphism of Lie algebras.
{\bf Proof:} By our construction, the restriction of $U$ to ${\goth g}_0(A)$
is a homomorphism of Lie algebras. Therefore, it suffices to
check that
\begin{equation}\label{_U_commu_with_commutator_Equation_}
U([X,Y])=[U(X),U(Y)]
\end{equation}
in the following cases:
(i) $X\in {\goth g}_2$, $Y\in {\goth g}_{-2}$
(ii) $X\in {\goth g}_0$, $Y\in {\goth g}_2$
(iii) $X\in {\goth g}_0$, $Y\in {\goth g}_{-2}$.
\hfill
We start the proof of \eqref{_U_commu_with_commutator_Equation_}
with (i). We represent $u_A:\; {\goth g}_0(A)\longrightarrow \goth{so}(V)\oplus k H$
as a sum $u_A=t_A\oplus s_A$, where $t_A:\; {\goth g}_0(A)\longrightarrow \goth{so}(V)$
and $s_A:\; {\goth g}_0(A)\longrightarrow k H$ are components of $u_A$. Consider
the similar decomposition of $u_B:\; {\goth g}_0(B)\longrightarrow \goth{so}(V)\oplus k H$,
$u_B=t_B\oplus s_B$.
Let $S_A\subset A_2=V$ be the set of all elements of Lefschetz type
in $A$, and $S_B\subset B_2=V$ be the set of all elements of Lefschetz type
in $B$. Let $S:= S_A\cap S_B$. For $x\in S_A$ ($S_B)$, we denote
by $L^A_x$, $\Lambda^A_x$ ($L^B_x$, $\Lambda^B_x$) the corresponding
elements in ${\goth g}(A)$ (${\goth g}(B)$). Clearly, $S$ is Zariski open in $V$.
Therefore, to prove \eqref{_U_commu_with_commutator_Equation_}
in case (i) it is sufficient to show that for all $x,y \in S$,
\begin{equation}\label{_U_[L,Lambda]=[U(L),U(Lambda)]_Equation_}
U([L^A_x, \Lambda^A_y]) =
[ U(L^A_x), U(\Lambda^A_y) ].
\end{equation}
Checking the definition of $U$, one can easily see that
$U(L_x^A)= L_x^B$ and $U(\Lambda_x^A)= \Lambda_x^B$.
Therefore, \eqref{_U_[L,Lambda]=[U(L),U(Lambda)]_Equation_}
is equivalent to
\begin{equation}\label{_U_[L,Lambda]=[L^B,Lambda^B]_Equation_}
U([L^A_x, \Lambda^A_y]) =
[L^B_x, \Lambda^B_y ].
\end{equation}
By definition of $U$, \eqref{_U_[L,Lambda]=[L^B,Lambda^B]_Equation_}
is equivalent to
\[
u_A([L^A_x, \Lambda^A_y]) = u_B([L^B_x, \Lambda^B_y ]).
\]
Using the decomposition of $u_A$, $u_B$, we obtain that
this equation is implied by the following two relations:
\begin{equation}\label{_t_commu_with_commutato_Equation_}
t_A([L^A_x, \Lambda^A_y]) = t_B([L^B_x, \Lambda^B_y ]),
\end{equation}
\begin{equation}\label{_t_commu_with_commutato_another_Equation_}
s_A([L^A_x, \Lambda^A_y]) = s_B([L^B_x, \Lambda^B_y ]).
\end{equation}
The second of these relations is implied by the equation
\[ s_A([L^A_x, \Lambda^A_y]) =
\frac{(x,y)_{\c H}}{(y,y)_{\c H}} H
\]
(\ref{_s_of_[L_a,Lambda_b]_Corollary_}).
We proceed to prove \eqref{_t_commu_with_commutato_Equation_}.
Consider the action of the operators
$t_A([L^A_x, \Lambda^A_y]),
t_B([L^B_x, \Lambda^B_y ]) \in \goth{so}(V)$
on $V$. Let $\c L$ be the two-dimensional plane
generated by $x,y\in V$. Let $\c L^\bot$ be its
orthogonal complement. By
\ref{_[L,Lambda]_in_T_a,b_Corollary_},
$t_A([L^A_x, \Lambda^A_y])$ acts as zero on $\c L^\bot$.
By \ref{_Lambda_vanish_Corollary_},
$t_B([L^B_x, \Lambda^B_y])$ also vanish on $\c L^\bot$.
The space of skew-symmetric endomorphisms of $V$ which
vanish on $\c L^\bot$ is one-dimensional. Hence,
the operators $t_A([L^A_x, \Lambda^A_y])$ and
$t_B([L^B_x, \Lambda^B_y])$ are proportional.
To prove that they are equal we have to compute
only the coefficient of proportionality.
\hfill
Denote the result of application of $\xi\in \goth{so}(V)$ to $x\in V$ by
$\xi x$. To prove \eqref{_t_commu_with_commutato_Equation_}
it is sufficient to show that
\begin{equation}\label{_L_x_Lambda_y_to_y_Equation_}
t_A([L_x^A,\Lambda_y^A])y=t_B([L_x^B,\Lambda_y^B])y\neq 0.
\end{equation}
In the case when $x,y$ are collinear,
\[
t_A([L_x^A,\Lambda_y^A])=t_B([L_x^B,\Lambda_y^B])=0.
\]
(see \eqref{_t(L_y_Lambda_y)=0_Equation_}).
Therefore, in this case \eqref{_L_x_Lambda_y_to_y_Equation_}
is not true. However, \eqref{_t_commu_with_commutato_Equation_}
is vacuously true in this case. Therefore, to prove
\eqref{_t_commu_with_commutato_Equation_} it is sufficient
to prove \eqref{_L_x_Lambda_y_to_y_Equation_} assuming
that $x$ and $y$ are not collinear.
\hfill
Let $x,y\in V$ be two vectors which are not collinear.
We prove \eqref{_L_x_Lambda_y_to_y_Equation_} as follows.
Denote the Hodge operators in $A$, $B$ by $H^A$, $H^B$
respectively. By definition of the Lefschetz triple,
\[
[L_y^A,\Lambda_y^A])= H^A, \;\;[L_y^B,\Lambda_y^B] = H^B.
\]
This implies that
\begin{equation}\label{_t(L_y_Lambda_y)=0_Equation_}
t_A([L_y^A,\Lambda_y^A])=t_B([L_y^B,\Lambda_y^B])=0
\end{equation}
Let $\lambda\in R$.
Since the expressions $t_A([L_x^A,\Lambda_y^A])$,
$t_B([L_x^B,\Lambda_y^B])$ are bilinear by $x$,
we have
\[
t_A([L_{x+\lambda y}^A,\Lambda_y^A])=t_A([L_x^A,\Lambda_y^A])
\]
and
\[
t_B([L_{x+\lambda y}^B,\Lambda_y^B])=t_B([L_x^B,\Lambda_y^B]).
\]
Therefore, \eqref{_L_x_Lambda_y_to_y_Equation_}
is equivalent to
\[
t_A([L_{x+\lambda y}^A,\Lambda_y^A])y=
t_B([L_{x+\lambda y}^B,\Lambda_y^B])y\neq 0.
\]
By \ref{_el-t_with_non_zero_square_Lefschetz_Lemma_},
$z\in S_B$ if and only if $(z,z)_{\c H}\neq 0$. Since
$S\subset S_B$, the number $(y,y)_{\c H}$ is non-zero.
Take $\lambda=\frac{(x,y)_{\c H}}{(y,y)_{\c H}}$.
Then $(x+\lambda y,y)_{\c H}=0$. Replacing $x$
by $x+\lambda y$, we see that it is sufficient
to prove \eqref{_L_x_Lambda_y_to_y_Equation_} in the
case when $(x,y)_{\c H}=0$.
Let $\mu\in {\Bbb R}$, $\mu\neq 0$.
If we replace $x$ by a vector $\mu x$,
both sides of \eqref{_L_x_Lambda_y_to_y_Equation_}
are multiplied by the number $\mu$. Choosing the
appropriate coefficient $\mu$, we may assume that
$(x,x)_{\c H}=(y,y)_{\c H}>0$. We obtained that we may prove
\eqref{_L_x_Lambda_y_to_y_Equation_} under the following set
of assumptions:
\[
(x,x)_{\c H}=(y,y)_{\c H}>0, \;\; (x,y)_{\c H}=0.
\]
This is implied by the following lemma.
\hfill
\lemma \label{_[L_x,Lambda_y]_to_y_is_-2x_Lemma_}
Let $x,y\in S$, $(x,x)_{\c H}=(y,y)_{\c H}>0$,
$(x,y)_{\c H}=0$. Then
\hfill
(i) $t_A([L_x^A,\Lambda_y^A])y =-2x$, and
(ii) $t_B([L_x^B,\Lambda_y^B])y =-2x$.
\hfill
{\bf Proof:} (i) Let
\[ T:= \bigg \{ (x,y)\in S \;\; | \;\; (x,x)_{\c H}=(y,y)_{\c H},\;\;
(x,y)_{\c H}=0 \bigg\}.
\]
Let
\[ T^\bullet:= \bigg \{ (x,y)\in S \;\; | \;\;
(x,x)_{\c H}=(y,y)_{\c H}, \;\;
(x,y)_{\c H}=0, \;\;x \bullet y \bigg\}.
\]
A standard argument with periods and comparing dimensions
implies that $T^\bullet$ is Zariski dense in $T$.
Therefore, we may prove (i) assuming that $x\bullet y$.
Let $\tilde {\c H}$ be a hyperkaehler structure such that
$x\bullet_{\tilde{\c H}} y$. Using
\ref{_action_SO(3)_on_Hyp_via_periods_Lemma_}, we replace
$\tilde {\c H}$ by an equivalent hyperkaehler structure
$\c H=(I,J, K, (\cdot,\cdot))$ such that $P_2(\c H)=x$,
$P_3(\c H) =y$. In this case
\[
[ L^A_x, L^A_y] = ad I
\]
(\ref{_[L_2,Lambda_3]=adI_Claim_}). Let $\Omega:= x+ \sqrt{-1}\: y$.
By definition of $ad I$, $adI (\Omega) = 2\sqrt{-1}\: \Omega$.
This immediately implies
\ref{_[L_x,Lambda_y]_to_y_is_-2x_Lemma_} (i).
\hfill
{\bf Proof of \ref{_[L_x,Lambda_y]_to_y_is_-2x_Lemma_} (ii):}
By definition, for all $a, b\in B_2$,
\begin{equation}\label{_ab_in_B_Equation_}
ab= (a,b)_{\c H}\Omega_B
\end{equation}
for a fixed vector $\Omega_B\in B_4$. Therefore, $L_x^B y=0$.
We obtain that
\begin{equation}\label{_commut_applied_to_y__in_B_Equation_}
[ L_x^B,\Lambda_y^B]y = L_x^B\Lambda_y^B y.
\end{equation}
Let ${\Bbb I}_B\in B_0$ be the unit in $B$. Then
$\Lambda_y^B y= \Lambda_y^B L_y^B {\Bbb I}_B$. Since
$[L_y^B,\Lambda_y^B] =H^B$ and $\Lambda_y^B {\Bbb I}_B=0$,
we have
\[ \Lambda_y^B L_y^B {\Bbb I}_B =
- H^B (\Bbb I) = -2 {\Bbb I}_B.
\]
Using \eqref{_ab_in_B_Equation_}, we can easily check that
\[
[L_x^B,\Lambda_y^B]{\Bbb I}_B=0.
\]
Therefore,
\[
s_B([L_x^B,\Lambda_y^B])=0.
\]
This implies that the action of $[ L_x^B,\Lambda_y^B]$ on
$B_2$ coinsides with the action of $t_B([ L_x^B,\Lambda_y^B])$ on $V$.
Hence, \eqref{_commut_applied_to_y__in_B_Equation_} implies
\ref{_[L_x,Lambda_y]_to_y_is_-2x_Lemma_} (ii). This finishes
the proof of \eqref{_U_commu_with_commutator_Equation_},
case (i). $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Relation \eqref{_U_commu_with_commutator_Equation_},
case (ii) is implied by the following explicit
description of the commutators between ${\goth g}_0(A)$
and ${\goth g}_2(A)$, ${\goth g}_{-2}(A)$, which is valid for
many Lefschetz-Frobenius algebras of Jordan type.
\hfill
\proposition \label{_commu_between_g_0_and_g_+-2_Proposition_}
Let $C=\oplus C_i$ be a Lefschetz-Frobenius algebra of Jordan type.
Let $\xi\in {\goth g}_0(C)$. Assume that ${\goth g}_0(C)$ acts on $C$ by derivations.
For any $x\in C_2$, denote by
$\xi(x)$ the image of $x$ under an action of $\xi:\; C_2\longrightarrow C_2$.
Then $[\xi, L_x]= L_{\xi(x)}$ for all $x\in A_2$, $\xi\in{\goth g}_0$.
{\bf Proof:} By definition of a derivation, $\xi(xa)= \xi(x) a +x\xi(a)$.
By definition, $\xi L_x(a)= \xi(xa)$ and $L_x \xi (a)=x\xi(a)$.
Substracting one from another, we obtain $[\xi, L_x](a)= L_{\xi(x)}(a)$.
$\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
According to \ref{_g_0_derivatives_Corollary_},
${\goth g}_0(A)$ acts on $A$ by derivations.
To show that ${\goth g}_0(B)$ acts on $B$ by derivations, we notice
the following. Let $C$ be an associative algebra over a field
and ${\goth g}$ be a Lie algebra, ${\goth g}\subset End(C)$. Consider the
corresponding Lie group $G\subset End(C)$. Then
${\goth g}$ acts on $C$ by derivations if and only if $G$ acts
on $C$ by algebra automorphisms. Now, ${\goth g}_0(B)\cong \goth{so}(V)\oplus kH$.
The algebra $kH$ acts on $B$ by derivations for obvious reasons.
On the other hand, the Lie group $SO(V)$ acts on $B$
by automorphisms, as follows from definition of
$B$. Therefore, \ref{_commu_between_g_0_and_g_+-2_Proposition_}
can be applied to $A$ and $B$. Let $g\in \goth{so}(V)\oplus kH$.
Let $g^A$, $g^B$ be the elements of ${\goth g}_0(A)$, ${\goth g}_0(B)$
which correspond to $g$. Then,
\eqref{_U_commu_with_commutator_Equation_},
case (ii) is equivalent to
\[
U^A([g^A, L_x^A]) = U^B([g^B, L_x^B]),\;\; \forall x\in V.
\]
By \ref{_commu_between_g_0_and_g_+-2_Proposition_},
this is equivalent to
\[ g^A(x)=g^B(x), \]
which is clear from definitions. We proved
\eqref{_U_commu_with_commutator_Equation_},
case (ii). $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
It remains to prove \eqref{_U_commu_with_commutator_Equation_}
in case (iii). Consider the action of ${\goth g}_0(A)$, ${\goth g}_0(B)$ on
${\goth g}_{-2}(A)$, ${\goth g}_{-2}(B)$ by commutators.
The action of ${\goth g}_0(B)$ on ${\goth g}_{-2}(B)$ consides
with that on $B_2\cong V$ (we use the standard isomorphism
${\goth g}_{-2}(B)\cong V$ which is apparent from the explicit
description of ${\goth g}(B)$). This means that $k H\subset {\goth g}_0(B)$
acts on ${\goth g}_{-2}(B)$ trivially, and $\goth{so}(V)\subset {\goth g}_0(B)$
acts on ${\goth g}_{-2}(B)\cong V$ in a standard fashion. Denote this
action by $\rho_1:\; \goth{so}(V)\longrightarrow End(V)$.
Similarly, $kH\subset {\goth g}_0(A)$ acts trivially on
${\goth g}_{-2}(A)$. It remains to compare the action
of $\goth{so}(V)\subset {\goth g}_0(A)$ on ${\goth g}_{-2}(A)$
to $\rho_1$. Consider the isomorphism $r:\; A_2\longrightarrow {\goth g}_{-2}(A)$
constructed in \ref{_g_-2_is_A_2_Corollary_}. The action
of ${\goth g}_0(A)$ on ${\goth g}_{-2}(A)$ defines an action
of $\goth{so}(V)\subset {\goth g}_0(A)$ on $A_2$. Using
the isomorphism $A_2\cong V$, we write this action
as a homomorphism $\rho_2:\;\goth{so}(V)\longrightarrow End(V)$.
In this notation, the equation
\eqref{_U_commu_with_commutator_Equation_},
case (iii) is equivalent to the following statement:
\hfill
\lemma \label{_rho_1_is_rho_2_Lemma_}
The representations $\rho_1$, $\rho_2$ coinside.
{\bf Proof:} Let $I\in Comp$.
Consider the endomorphism $ad I\in {\goth g}_0(A)$.
We identify ${\goth g}_0(A)$ and ${\goth g}_0(B)$ using the isomorphism
$U_0$. Let $t:\; {\goth g}_0(A)\longrightarrow \goth{so}(V)$ be a projection
on a summand.
Let $C\subset \goth{so}(V)$ be the union of $t(ad I)$
for all $I\in Comp$. As we have seen, $C$ is Zariski
dense in $\goth{so}(V)$. Therefore it is sufficient
to show that $\rho_1$, $\rho_2$ coinside on $C$.
Let $x\in A_2$. By definition, $\rho_1(t(ad I))(x)= ad I(x)$,
and $\rho_2(t(ad I))x = r^{-1}[ad I, r(x)]$, where
$r:\; V\longrightarrow {\goth g}_{-2}(A)$ is a map of \ref{_g_-2_is_A_2_Corollary_}.
To prove \ref{_rho_1_is_rho_2_Lemma_}
we have to show the following:
\begin{equation} \label{_r_commu_w_so(V)_Equation_}
r(ad I(x)) = [ad I, r(x)].
\end{equation}
Both sides of \eqref{_r_commu_w_so(V)_Equation_} are
linear by $x$. Therefore it is sufficient to check
\eqref{_r_commu_w_so(V)_Equation_} in two cases:
\hfill
(i) $x\in H^{1,1}_I(M)$
(ii) $x\in H^{2,0}_I(M)\oplus H^{0,2}_I(M)$,
\hfill
\hspace{-6mm}where $H^{p,q}_I(M)$ is Hodge decomposition
associated with the complex structure $I\in Comp$.
In case (i), $ad I(x)=0$, so \eqref{_r_commu_w_so(V)_Equation_}
is equivalent to
\begin{equation} \label{_adI_commu_w_r(x)_for_x_in_H^1,1_Equation_}
[ad I, r(x)] =0.
\end{equation}
Since elements of Lefschetz type are Zariski dense in $A_2$, it is
sufficient to prove \eqref{_adI_commu_w_r(x)_for_x_in_H^1,1_Equation_}
assuming that $x$ is of Lefschetz type. In this case,
$r(x)=(x,x)_{\c H}\Lambda_x$. Therefore,
\eqref{_adI_commu_w_r(x)_for_x_in_H^1,1_Equation_}
follows from the equation $[ad I, \Lambda_x] =0$. Since
$x\in H^{1,1}_I(M)$, the operator $L_x$ preserves weights
of the Hodge decomposition. An easy linear algebraic check insures that
in this case $\Lambda_x$ also preserves Hodge weights.
Therefore, by definition of $ad I$, we have $[ad I, \Lambda_x] =0$.
We proved \eqref{_r_commu_w_so(V)_Equation_}, case (i).
\hfill
Consider a non-zero holomorphic symplectic form $\tilde \Omega$
over the complex manifold $(M,I)$. Let $\Omega\in H^2(M)$ be
cohomology class represented by $\Omega$. Then $Im(\Omega)$,
$Re(\Omega)$ constitute basis in two-dimensional space
$H^{2,0}_I(M)\oplus H^{0,2}_I(M)$. Let
$\c H = (I, J, K, (\cdot,\cdot))$ be a hyperkaehler
structure such that $P_2(\c H)= Re(\Omega)$,
$P_3(\c H)= Im(\Omega)$. Such $\c H$ exists by
Calabi-Yau theorem. Since \ref{_r_commu_w_so(V)_Equation_}
is linear by $x$, we may check
\ref{_r_commu_w_so(V)_Equation_} case (ii) only for $x_2=P_2(\c H)$,
$x_3=P_3(\c H)$.
Clearly from definitions,
\[
ad I(x_2) = 2 x_3, \; ad I(x_3) = -2 x_2.
\]
Let $c=(x_2,x_2)_{\c H}=(x_3,x_3)_{\c H}$.
By definition, $r(x_i)=c\Lambda_{x_i}$, $i=2,3$.
Therefore, \eqref{_r_commu_w_so(V)_Equation_} case (ii)
is equivalent to the following pair of equations:
\[ 4c \Lambda_{2x_3} = [ad I, c\Lambda_{x_2}], \]
and
\[ -4 c\Lambda_{2x_2} = [ad I, c\Lambda_{x_3}] \]
Since $\Lambda_{2a}=1/2\Lambda_a$, these two equations can
be rewritten as
\begin{equation}\label{_adI_on_Lambdas_Equation_}
2\Lambda_{x_3} = [ad I, \Lambda_{x_2}], \;
-2\Lambda_{x_2} = [ad I, \Lambda_{x_3}]
\end{equation}
\ref{_[L_2,Lambda_3]=adI_Claim_} implies that in notation of
\eqref{_so5_relations_Equation_}, $ad I = K_{23}$.
Therefore \eqref {_adI_on_Lambdas_Equation_}
is a consequence of \eqref{_so5_relations_Equation_}.
This proves \ref{_rho_1_is_rho_2_Lemma_}, and consequently,
\eqref{_U_commu_with_commutator_Equation_}
case (iii). Proof of \ref{_U_is_Lie_homomo_Proposition_}
is finished. This also finishes the proof
of \ref{_g(A)_for_hyperkae_Theorem_}, which spanned
four sections of this paper. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\section{The structure of the cohomology ring for
compact hyperkaehler manifolds.}
\label{_cohomolo_compu_Section_}
Let $M$ be a compact simple hyperkaehler manifold and $A=H^*(M)$
be its cohomology ring. Let $\c V=H^2(M,{\Bbb R})$ considered as a linear
space with non-degenerate symmetric pairing $(\cdot,\cdot)_{\c H}$.
Applying \ref{_g(A)_for_hyperkae_Theorem_} and
\ref{_all_alg_with_so_are_^dA_Theorem_} to $A$, we immediately
obtain the following statement.
\hfill
\theorem\label{_cohomo_of_hyperk_are_^dA(V)_Theorem_}
Let $A^r\subset H^*(M)$ be the subalgebra of $A=H^*(M)$
generated by $A_0$, $A_2$. Then $A^r\cong {}^dA(\c V)$, where
${}^dA(\c V)$ is a Frobenius algebra considered in
\ref{_all_alg_with_so_are_^dA_Theorem_}
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
We see that $A^r\subset H^*(M)$ can be expressed purely in
linear-algebraic terms. In this section, we make this
description as explicit as possible. Fix a linear space
$V$ with a non-degenerate bilinear symmetric form $B:\; S^2V\longrightarrow k$,
where $k$ is a ground field. We express ${}^dA(V)$ in terms of
$V$ and $B$ as follows.
Let
\[
C = C_0\oplus C_2\oplus ... \oplus C_{4d}
\]
be a graded linear space, with
\[ C_{2i} = S^iV, \;\;i\leq 2d, \]
\[ C_{2i} = S^{2i-d}V, \;\;i\geq 2d. \]
We describe a multiplicative structure on $C$ using some
classical results of linear algebra. Let $V$ be a linear space
equipped with non-degenerate bilinear symmetric product
$B:\; V\otimes V\longrightarrow k$. Consider the Lie group $SO(V)$
associated with $V$ and $B$. This group naturally acts
on the symmetric powers $S^nV$ for all $n$. The representation
$SO(V)\longrightarrow End(S^nV)$ is not irreducible. Its irreducible
decomposition is a classical result of linear algebra.
We describe this decomposition explicitely, and
define $SO(V)$-invariant multiplicative structure on $C$
in terms of this decomposition.
Let $\goth V:= \{ x_1,...., x_n\}$ be a basis in $V$.
We represent the vectors from $S^nV$ by the polynomials
\[
\sum \alpha_{i_1,...,i_m} x_{i_1},..., x_{i_m},
\]
where $\alpha_{i_1,...,i_m}\in k$ and $x_{i_j}\in \goth V$, $k$
is a ground field. Consider an $SO(V)$-invariant vector $r\in S^2 V$
represented by the polynomial
\[
r:= \sum_{i,j} B(x_i,x_j) x_i x_j.
\]
Let $L_r:\; S^nV\longrightarrow S^{n+2}V$ be the map multiplying
the polynomial $P$ by $r$. Since the product in $S^*V$
commutes with the $SO(V)$-action and $r$ is $SO(V)$-invariant,
the map $L_r$ is a homomorphism of $SO(V)$-representations.
The scalar product $B$ on $V$ can be extended to an $SO(V)$-invariant
scalar product $(\cdot,\cdot)_{V^{\otimes_n}}$
on the space of $n$-tensors $\otimes^n V$ by the law
\[ (x_1\otimes x_2\otimes ... \otimes x_n,
y_1\otimes y_2\otimes ... \otimes y_n)_{V^{\otimes_n}}
= B(x_1,y_1) B(x_2,y_2) ... B(x_n,y_n).
\]
The space $S^nV\subset \otimes ^n V$ is $SO(V)$-invariant.
Using Schur's lemma, one can see that the restriction
of $(\cdot,\cdot)_{V^{\otimes_n}}$ to $S^nV$ is non-degenerate.
Denote this scalar product by $(\cdot,\cdot)_{S^n V}$.
For all $n>1$, the map $L_r:\; S^{n-2}V\longrightarrow S^n V$
is an embedding. Let $R^n V\subset S^n V$ be the orthogonal
complement to the image of $L_r\; S^{n-2}V\longrightarrow S^n V$.
Using the embedding $L_r:\; S^{i-2}V\longrightarrow S^i V$
for different $i$,we obtain a decomposition
\begin{equation}\label{_decompo_of_S^nV_Equation_}
S^n V\cong R^n V\oplus R^{n-2} V \oplus ... \oplus R^{n\;
\mbox{\tiny mod}\; 2} V,
\end{equation}
where $R^0V= k$ and $R^1 V = V$.
\proposition \label{_R^nV_is_irredu_Proposition_}
For all $n\in \Bbb N$, the $SO(V)$-representation $R^n V$ is irreducible.
{\bf Proof:} See \cite{_Weyl_}. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
We obtain that \eqref{_decompo_of_S^nV_Equation_}
is an irreducible decomposition of an $SO(V)$-representation
$S^n V$. By definition, the representation
\[ S^nV / R^n V \oplus R^{n-2}V \oplus...
\oplus R^{n-2i +2}V
\]
is canonically isomorphic to $S^{n-2i}V$. Denote the corresponding
quotient map by $B^n_{n-2i}:\; S^n V\longrightarrow S^{n-2i}V$. Using the
maps $B^n_{n-2i}$ we define the multiplicative structure on $C$
as follows. Let $S^* V= \oplus S^n V$ be the algebra
of symmetric tensors over $V$. Let $\phi:\; S^* V\longrightarrow C$,
$\phi=\oplus \phi_i$, where $\phi_i:\; S^iV \longrightarrow C_{2i}$ is the following
map. For $2i\leq 2d$, $C_{2i}\cong S^i V$. For such $i$,
$\phi_i$ is defined as identity map. For $2d<2i\leq 4d$,
we have $C_{2i}= S^{2d-i}V$. Let $\phi_i: S^i V\longrightarrow C_{2i}$
be equal to $B^i_{2d-i}:\; S^i V\longrightarrow S^{2d-i}V$. For
$2i>4d$, $C_{2i}=0$ and we take $\phi_i=0$.
Clearly, the map $\phi$ is onto.
\hfill
\lemma \label{_ker_phi_ideal_in_S^*V_Lemma_}
Let $I$ be a kernel of $\phi:\; S^*V\longrightarrow C$. Then
$I$ is an ideal in $S^*V$.
{\bf Proof:}
Let $x\in S^l V$, $x\in I$, and $y\in S^m V$. We have to show that
$xy\in I$. This relation is
vacuously true except in case when $d<l<l+m <2d$.
Let $\Lambda_r:\; S^n V\longrightarrow S^{n-2} V$
be equal to $B^n_{n-2}$ for all $n = 2,3,...$. Clearly,
$B^n_{n-2i}= \Lambda^i_r$, where $\Lambda^i_r$ is $\Lambda_r$
to the power of $i$.
Therefore $x\in I$ is equivalent to $\Lambda^{l-d}_r(x)=0$,
and $xy\in I$ is equivalent to $\Lambda^{l+m-d}_r(xy)=0$.
Therefore, \ref{_ker_phi_ideal_in_S^*V_Lemma_}
is a special case of the following statement.
\hfill
\lemma \label{_Lambda_r_to_multi_Lemma_}
Let $l,m,n$ be positive integer numbers, $x\in S^l V$,
$y\in S^m V$. Assume that $\Lambda^{n}_r(x) =0$.
Then $\Lambda^{n+m}_r(xy) =0$.
{\bf Proof:} Consider an element of $S^n V$ as polynomial function
on $V$ considered as an affine space. Let $\Delta:\; S^n V\longrightarrow S^{n-2}V$
be the Laplace operator associated with the metric structure
$B$ on $V$. By definition,
\[ \Delta(P) = \sum B(x_i,x_j)
\frac{\partial^2}{\partial x_i \partial x_j} P,
\]
where $x_1,..., x_n$ is a basis in $V$. This definition
is independent of the choice of basis in $V$. The operator
$\Delta$ commutes with an action of $SO(V)$.
Checking that $\Delta(S^n V)$ contains $y^{n-2}$ for all $y\in V$,
we obtain that the map $\Delta$ is onto. This imples that
$\ker(\Delta^i:\; S^n V \longrightarrow S^{n-2i} V)$ coinsides
with $R^n V\oplus R^{n-2} V\oplus ... \oplus R^{n-2i+2}V\subset S^n V$.
Therefore, $\ker(\Delta^i)=\ker(\Lambda_r^i)= \ker(B^n_{n-2i})$.
We obtain that \ref{_Lambda_r_to_multi_Lemma_} is equivalent
to the following statement:
\hfill
\lemma \label{_laplace_multi_Lemma_}
Let $x\in S^l V$, $y\in S^m V$. Assume that $\Delta^i x =0$.
Then $\Delta^{i+m}(xy)=0$.
{\bf Proof:} We prove \ref{_laplace_multi_Lemma_}
using induction by $l$, $m$. We denote by $\bf L(l_0, m_0)$
the statement of \ref{_laplace_multi_Lemma_} applied to $l=l_0$,
$m=m_0$. Clearly,
\begin{equation}\label{_Laplace_of_produ_Equation_}
\Delta(xy) = \Delta(x) y + x\Delta(y)
+ 2\sum \frac{\partial x}{\partial x_i}
\frac{\partial y}{\partial x_j} B(x_i,x_j).
\end{equation}
By $\bf L(l-1, m)$, we have
\[ \Delta^{i+m -1}(\Delta(x) y) =0. \]
By $\bf L(l, m-2)$,
\[ \Delta^{i+m -1}(\Delta(x) y) =0. \]
Laplacian commutes with partial derivatives, and therefore
$\Delta^{i}(\frac{\partial x}{\partial x_i})=0$. Hence, by the
virtue of $\bf L(l, m-1)$,
\[ \Delta^{i+m -1} \bigg(\sum \frac{\partial x}{\partial x_i}
\frac{\partial y}{\partial x_j} B(x_i,x_j)\bigg) =0
\]
Therefore, $\Delta^{i+m -1}$ applied to the right hand side
of \eqref{_Laplace_of_produ_Equation_} is zero.
This finishes the proof of \ref{_laplace_multi_Lemma_}.
We proved \ref{_Lambda_r_to_multi_Lemma_} and
\ref{_ker_phi_ideal_in_S^*V_Lemma_}. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
By definition, $C= S^*V / I$. Since $I$ is an ideal in $S^* V$,
the space $C$ inherits a canonical ring structure. Denote this
algebra by ${}^dC(V)$. The following
theorem characterizes the $H^2(M)$-generated subring of
cohomology ring of a simple compact hyperkaehler manifold
in terms of $C$.
\hfill
\theorem \label{_^dA(V)_is_C_Theorem_}
Let $V$ be a linear space equipped with bilinear symmetric pairing $B_V$.
Then the algebra ${}^dA(V)$ is naturally isomorphic to ${}^dC(V)$.
{\bf Proof:} We prove \ref{_^dA(V)_is_C_Theorem_} as follows.
We consider ${}^dC(V)$ and ${}^dA(V)$ as graded linear spaces
with an action of the group $SO(V)$. These spaces are isomorphic
as graded $SO(V)$-representations. We notice that ${}^dA(V)$
is by definition a quotient of $SO(V)$ by the $SO(V)$-invariant
ideal $J$. We show that there is a unique graded $SO(V)$-invartiant
ideal $J$ in $S^*V$ such that $S^*V/J$ is isomorphic as a graded
$SO(V)$-representation to ${}^dC(V)\cong S^*(V)/I$. This implies
that $I$ coinsides with $J$, which proves \ref{_^dA(V)_is_C_Theorem_}.
In Section \ref{_minimal_Fro_Section_} we considered the graded Lie algebra
$\goth{so}(V,+)$. By definition, $\goth{so}(V,+)$ is a Lie algebra
of skew-symmetric endomorphisms of the space $V\oplus \goth H$.
Denote $V\oplus \goth H$ by $W$. The minimal Frobenius algebra
$A(V)\cong {}^1A(V)$ is isomorphic to $W$ as
$\goth{so}(W)$-representation. Therefore ${}^dA(V)$ is an
irreducible $\goth{so}(W)$-subrepresentation of $S^d W$ generated
by ${\Bbb I}\otimes{\Bbb I}\otimes...\otimes{\Bbb I}$. Consider the action
of the group $SO(W)$ on ${}^dA(V)$ which corresponds to
this Lie algebra action. We immediately obtain the following:
\hfill
\claim \label{_^dA(V)_is_R^d(W)_Claim_}
Let $V$ be a linear space equipped with a non-degenerate bilinear
symmetric form, and ${}^dA(V)$ be a Frobenius
algebra defined in Section \ref{_^dA(V)_Section_}.
Let $\goth H$ be the two-dimensional
space with the hyperbolic metric, and $W:= V\oplus \goth H$.
As shown above, there is a natural action of $SO(W)$ on
the space ${}^dA(V)$. Earlier in this section, we defined the
$SO(W)$-representation $R^d W$. Then
${}^dA(V)$ is isomorphic to $R^d W$
as a representation of $SO(W)$.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
Consider a natural embedding $SO(V)\subset SO(W)$
which corresponds to the decomposition $W=V\oplus \goth H$.
Consider $R^d W$ as a graded space, with the grading inherited
from ${}^dA(V)$. We proceed to demonstrate that, as a graded
$SO(V)$-representation, $R^d W\cong {}^dC(V)$. This is done
as follows. Let $r_V\in S^2 V$, $r_W\in S^2 W$ be $SO(V)$-invariant
(respectively, $SO(W)$-invariant) polynomials of degree 2defined earlier in
this
section. Denote the scalar product in $W$ by $B_W(\cdot,\cdot)$.
Earlier we denoted the scalar product in $V$ by $B_V(\cdot,\cdot)$.
Let $x_1, ..., x_n$ be a basis in $V$. Consider the vectors
${\Bbb I}$, $\Omega\in W$ which were introduced when we gave definition
of $A(V)\cong W$. By definition, all vectors $x_i$ are orthogonal
to ${\Bbb I}$ and $\Omega$, and $B_W({\Bbb I}, \Omega)=1$,
while $B_W({\Bbb I}, {\Bbb I})=0$ and $B_W(\Omega, \Omega)=0$.
Clearly, the vectors ${\Bbb I}, \Omega, x_1,...,x_n$ form
a basis in $W$. By definition,
\[
r_W = {\Bbb I}\Omega - r_V.
\]
Consider a linear map $\gamma:\; S^d W\longrightarrow S^d W$ which maps a monomial
$P = {\Bbb I}^i \Omega^j T$
to
\[ \gamma(P) :=
\begin{array}{l}
r_V^i \Omega^{j-i},
\;\; j\geq i \\[5mm]
{\Bbb I}^{i-j} r_V^j,
\;\; j< i,
\end{array}
\]
where $T= x_1^{\alpha_1} x_2^{\alpha_2}... x_n^{\alpha_n}$,
$\sum \alpha_i= d-i-j$.
Clearly, $\ker \gamma = r_W S^{d-2}W$. Therefore, the image of
$\gamma$ is naturally isomorphic as a linear space to $R^dW$. Consider a
multiplicative grading on $S^d W$ defined as follows:
$gr(\Bbb I)=0$, $gr(x_i)=2$, $i=1,2,..., n$, $gr(\Omega)=4$.
Clearly, this grading induces the standard one on
the space ${}^dA(V)\cong R^dW\subset S^d V$. By definition, the map
$\gamma:\; S^d W\longrightarrow S^d W$ preserves this grading.
Since $\Bbb I$, $r_V$ and $\Omega$ are $SO(V)$-invariant,
the map $\gamma$ commutes with an action of $SO(V)$
on $S^d W$. Therefore, $\gamma(S^d W)$ is isomorphic to
$R^d W$ as a graded representation of $SO(V)$.
On the other hand, $\gamma(S^d W)$ is isomorphic
to ${}^d C(V)$ (again, as a graded representation of $SO(V)$
as the following argument shows.
For $2i\leq 2d$, the grading-$2i$ subspace
$\bigg( \gamma(S^d W)\bigg)_{2d}\subset \gamma(S^d W)$
is a linear span of monomials
\[ {\Bbb I}^{d-i} x_1^{\alpha_1} x_2^{\alpha_2} ... x_n^{\alpha_n},
\;\; \sum \alpha_i = i.
\]
Similarly, for $2i>2d$, the space $\bigg( \gamma(S^d W)\bigg)_{2d}$
is a linear span of monomials
\[ \Omega^{i-d} x_1^{\alpha_1} x_2^{\alpha_2} ... x_n^{\alpha_n},
\;\; \sum \alpha_i = 2d-i.
\]
Therefore, the grade $2i$ part of $R^dW\cong {}^d(W)$ is
$S^iV$ for $i<d$ and is $S^{2d-i}V$ for $i>d$. We proved the
following statement:
\hfill
\lemma \label{_C_iso_to_A(V)_as_SO(V)_repre_Lemma_}
The spaces ${}^d C(V)$ and ${}^dA(V)$ are isomorphic as graded
representations of $SO(V)$.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
By construction, ${}^d C(V)$ and ${}^dA(V)$ are quotient
algebras of $S^*V$. The canonical epimorphisms
$S^*V \longrightarrow {}^d C(V)$ and $S^*V \longrightarrow {}^d A(V)$
are $SO(V)$-invariant. Therefore, \ref{_^dA(V)_is_C_Theorem_}
is a consequence of \ref{_C_iso_to_A(V)_as_SO(V)_repre_Lemma_}
and the following lemma.
\hfill
\lemma \label{_quotie_alge_of_S^*V_iso_as_repre_iso_as_alge_Lemma_}
Let $V$ be a linear space equipped with a scalar product.
Let $D$, $E$ be quotients of $S^*V$ by graded ideals
$I$, $J\subset S^*V$. Consider $D$, $E$ as graded algebras,
with the grading inherited from $S^*V$. Assume that $I$, $J$ are
$SO(V)$-invariant subspaces in $S^*V$ and $D$ is
isomorphic to $E$ as a graded representation of $SO(V)$.
Then $D$ is isomorphic to $E$ as an algebra.
{\bf Proof:} Consider the irreducible decomposition of $S^i V$
given by \eqref{_decompo_of_S^nV_Equation_}.
The summands of this decomposition are pairwise non-isomorphic.
Therefore, by Schur's lemma every $SO(V)$-invariant subspace
of $S^nV$ is a direct sum of several components of the
decomposition \eqref{_decompo_of_S^nV_Equation_}.
Let $I_n$, $J_n\subset S^n V$
be the $n$-th grade components of $I$, $J$. Since $I_n$, $J_n$
are $SO(V)$-invariant, these subspaces are direct sum of
several components of \eqref{_decompo_of_S^nV_Equation_}.
The quotient spaces $D_n =S^n V/I_n$, $E_n =S^n V/J_n$ are isomorphic
as $SO(V)$-representations. These spaces can be identified with the direct
sums of those components of the decomposition
\eqref{_decompo_of_S^nV_Equation_}
which don't appear in the decomposition of $I_n$, $J_n$.
Since $E_n$ is isomorphic to $D_n$, the spaces $I_n$ and
$J_n$ are isomorphic (as representations of $SO(V)$).
Since the components of \eqref{_decompo_of_S^nV_Equation_}
are pairwise non-isomorphic, the spaces $I_n$ and $J_n$ coinside.
This proves \ref{_quotie_alge_of_S^*V_iso_as_repre_iso_as_alge_Lemma_}.
\ref{_^dA(V)_is_C_Theorem_} is proven. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\section{Calculations of dimensions.}
\label{_calcu_dimensi_Section_}
We obtain easy numerical lower bounds on the dimensions
\[ \dim H^{p,q}(M)\] of Hodge components of cohomology spaces
of compact hyperkaehler manifolds. We have computed the
part of cohomology generated by $H^2(M)$.
The dimension of $\dim H^{p,q}(M)$ cannot be
lower than the dimension of the space $\bar H^{p,q}(M)$ of all
$(p,q)$ cohomology classes which are generated
by $H^2(M)$. In this section, we
compute dimensions of $\bar H^{p,q}$ for all $p$, $q$.
Let $\bar H^*(M)= \oplus_{p,q}\bar H^{p,q}\subset H^*(M)$ be the subring of
$H^*(M)$ generated by $H^2(M)$. Clearly,
\[
\dim H^{p,q}(M) \geq \dim \bar H^{p,q}(M).
\]
By $p(n,m)$, we denote dimension of the space of
homogeneous polynomials of degree $m$ of $n$ variables.
This number is known from combinatorics as partition number.
It is given by the following formal serie, which was discovered
by Euler:
\[ \sum p(n,m) s^n t^m = \prod\limits_{i=1}^\infty
\bigg(\frac{1}{1-t^is}\bigg)
\]
Consider the ring $S_2$ of polynomials of $n+1$ variables,
where $n$ variables are assigned degree 1 and one variable is
assigned degree 2. Let $p_2(n,m)$ be the space of homogeneous
polynomials of degree $m$ in $S_2$. Clearly,
\[
p_2(n,m) = \sum_{i=0}^{i=[\frac{m}{2}]} p(n,m-i).
\]
\hfill
\theorem \label{_dimens_of_bar_H_^pq_in_terms_of_p_Theorem_}
Let $M$ be a simple compact hyperkaehler manifold, $\dim_{\Bbb R} M =4d$,
$b_2(M)=n$ (we denote by $b_2$ the second Betti number). Then
\hfill
(i) $\dim \bar H^*(M) = p(n,d) - p(n, d-2)$.
\hfill
(ii) $\dim \bar H^{2i}(M)= \begin{array}{l} p(n,i),\;\; i\leq d\\[2mm]
p(n,2d-i),\;\; i\geq d
\end{array}$
\hfill
(iii) $\dim\bar H^{p,q}(M)=0$ for $p+q$ odd.
\hfill
(iv) $\dim \bar H^{p,q}(M) = \dim \bar H^{p,2d-q}(M) =$ \\
\centerline{$=\dim \bar H^{2d-p,q}(M)=\dim \bar H^{2d-p,2d-q}(M)$.}
\hfill
(v) For $p+q\leq 2d$, $p\leq q$,
\[ \dim \bar H^{p,q}(M) = p_2(n-2,p) \]
\hfill
{\bf Proof:}
\hfill
(i) Follows from \ref{_^dA(V)_is_R^d(W)_Claim_}.
\hfill
(ii) \ref{_^dA(V)_is_C_Theorem_}
\hfill
(iii) Clear
\hfill
(iv) See \cite{_so5_on_cohomo_}
\hfill
(v) Let $V= H^2(M)$. \ref{_^dA(V)_is_C_Theorem_}
implies that $\bar H^{2m}(M) \cong S^m V$. Clearly, the Hodge decomposition
on $\bar H^{2m}(M)= S^mV$ is induced from that on
$V= H^{2,0}(M)\oplus H^{1,1}(M)\oplus H^{0,2}(M)$. The spaces
$H^{2,0}(M)$ and $H^{0,2}(M)$ are one-dimensional. Let $z\in H^{2,0}(M)$,
$\bar z \in H^{0,2}(M)$ be the non-zero vectors, and
$z,\bar z, x_1,...,x_{n-2}$ be the basis in $H^2(M)\cong V$.
Then the space $\bar H^{p,q}(M)$, $p+q\leq 2d$, $p\leq q$,
is a linear span of the monomials
\[ T_{a,b,\alpha_1,...\alpha_{n-2}} =
z^a \bar z^b x_1^{\alpha_1} x_2^{\alpha_2}... x_{n-2}^{\alpha_{n-2}}
\]
where $b-a=q-p$, $\sum \alpha_i = p+q-(b+a)$.
Let $\Theta=z \bar z\in S^2 V$. Then
\[ T_{a,b,\alpha_1,...\alpha_{n-2}} = \Theta^{a} \bar z^{b-a}
x_1^{\alpha_1} x_2^{\alpha_2}... x_{n-2}^{\alpha_{n-2}},
\]
where $a+b + \sum \alpha_i = p+q$. Since $b-a= q-p$,
\[ T_{a,b,\alpha_1,...\alpha_{n-2}} = \Theta^{a} \bar z^{p-q}
x_1^{\alpha_1} x_2^{\alpha_2}... x_{n-2}^{\alpha_{n-2}},
\]
where $2a + (q-p) +\sum \alpha_i = p+q$. Translating $q-p$ to the
right hand side, we obtain that $T_{a,b,\alpha_1,...\alpha_{n-2}}$
is numbered by the different combinations of $a, \alpha_1,...\alpha_{n-2}$,
which satisfy the condition $2a +\sum \alpha_i = 2p$.
This number is by definition $ p_2(n-2,p)$. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
{\bf Acknowledgements:}
I am very grateful to my advisor David Kazhdan for warm support
and encouragement. This paper owes its existence to F. A. Bogomolov
and A. Todorov, whose studies of hyperkaehler manifolds made this
work possible. The last parts of this paper were inspired by
joint work with Valery Lunts.
I am extremely grateful to
Pierre Deligne, who was most kind and helpful. I owe Deligne several
important corrections in the final version of this manuscript.
Thanks due to B. A. Dubrovin, P. Etingof,
R. Bezrukavnikov, L. Positsel'sky, A. Todorov,
A. Polishchuk, T. Pantev and M. Bershadsky for stimulating
discussions. Prof. Y.-T. Siu and Prof. J. Bernstein kindly answered the
questions vital for the development of this work.
I am also grateful to MIT math department
for allowing me the use of their computing facilities.
\hfill
|
1996-03-05T06:15:37 | 9501 | alg-geom/9501003 | en | https://arxiv.org/abs/alg-geom/9501003 | [
"alg-geom",
"math.AG"
] | alg-geom/9501003 | Martin Pikaart | Martin Pikaart and Johan de Jong | Moduli of curves with non-abelian level structure | 25 pages, latex, only hand-drawn figures. | null | null | null | null | Following Deligne and Mumford we construct a coarse moduli space of smooth
curves with non-abelian level structure, involving higher order commutators. We
prove that its Deligne-Mumford compactification is smooth over an open part of
Spec(${\msy Z}$).
| [
{
"version": "v1",
"created": "Mon, 9 Jan 1995 12:02:53 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Pikaart",
"Martin",
""
],
[
"de Jong",
"Johan",
""
]
] | alg-geom | \section{Introduction}
Deligne and Mumford introduced the moduli stack
${}_G{\cal M}_g} \def\GMg{{}_GM_g$ parametrizing smooth genus $g$ curves with Teichm{\"u}ller
structure of level $G$, a finite group. For example,
if $G=\{e\}$, resp.\ $G\cong (\msy Z/n\msy Z)^{2g}$, this moduli
stack is just the moduli stack of smooth curves ${\cal M}_g} \def\Mg{M_g$, resp.\
of smooth curves with abelian level structure (sometimes
denoted ${\cal M}_g} \def\Mg{M_g[n]$). They also defined $\overline{{\cal M}_g}} \def\bMg{\overline{M_g}$, the moduli
stack of stable curves and proved it is proper over
${\rm Spec}(\msy Z)$. Taking the normalization of $\overline{{\cal M}_g}} \def\bMg{\overline{M_g}\times
{\rm Spec}(\msy Z[1/\# G])$ in the function field
of ${}_G{\cal M}_g} \def\GMg{{}_GM_g$ defines $\overline{{}_G{\cal M}_g}} \def\bGMg{\overline{{}_GM_g}$, proper over ${\rm Spec}(\msy Z[1/\# G])$.
Let $\Pi$ denote the standard fundamental group of a compact
Riemann surface of genus $g$. The nth powers together with
the kth order commutators generate a normal subgroup $\Pi^{(k),n}$.
(We regard $[a,b]$ as a commutator of order 2.) Let $G$ be the
quotient $\Pi/\Pi^{(k+1),n}$. We show that if $k\geq 1$, $n\geq 3$
the coarse moduli scheme $\bGMg$ for $\overline{{}_G{\cal M}_g}} \def\bGMg{\overline{{}_GM_g}$ exists, and we actually
have $\bGMg\cong\overline{{}_G{\cal M}_g}} \def\bGMg{\overline{{}_GM_g}$. The following theorem is our main result.
\medskip
\noindent{\bf Theorem\enspace \ref{glad}} {\sl
Suppose $k\in \{1,2,3\}$ and $n\geq 3$. The structural morphism
$\bGMg\to {\rm Spec}(\msy Z[1/n])$ is smooth if and only if
\begin{itemize}
\item{$k=1$} and $g=2$,
\item{$k=2$} and $n$ is odd,
\item{$k=3$} and $n$ is odd or $n$ is divisible by $4$.
\end{itemize}
Furthermore, if $k \geq 4$, $n \geq 3$ and $n$ relatively prime to $6$,
then $\bGMg\to {\rm Spec}(\msy Z[1/n])$ is smooth.}
\medskip
Looijenga constructed a smooth and compact cover
$\overline{M_g[_2^n]}^{an},\ n\geq 3$ of $\bMgan$ using Prym level
structures. He proves that these coverings are universally ramified
along the boundary of ${\cal M}^{an}_g} \def\Mgan{M_g^{an}$, see \cite{Looijenga}.
This holds also for our coverings
as can be seen from \ref{mono}. These coverings can be applied
to the construction of the Chow rings of the stacks $\overline{{\cal M}_g}} \def\bMg{\overline{M_g}[n]$:
for example it is clear that the specialization maps are ring homomorphisms.
The paper is organized as follows. Section 2 deals with the definition
of the moduli problem. The formulation is in terms of the relative
fundamental group of a family of curves.
Its final proposition states that $\bGMg\to {\rm Spec}(\msy Z[1/n])$
is smooth if and only if the associated compact analytic space $\overline{{}_GM_g}^{an}$ is
is a manifold. Section 3 contains a precise description of the monodromy
along the boundary of $\overline{{}_GM_g}^{an}$ for general groups $G$ and the statement
of the main result. Sections 4 and 5 are of a topological nature.
We describe the monodromy on the relative fundamental group
of the universal curve along the boundary in terms of Dehn twists.
We have to describe the situation in some detail in order to
understand how this monodromy acts on the finite quotients $\Pi/\Pi^{(k),n}$.
These considerations prove the main theorem, using Section 6,
which contains the necessary computations for a free group on three
generators.
The authors were stimulated by the article \cite{Looijenga}.
We thank Prof.~Looijenga for numerous discussions explaining
his and other results. We thank Prof.~Oort, who remarked that
it should be possible to do everything algebraically and
drew our attention to the article \cite{Oda}.
\subsection{Notations and conventions}\label{notation}
\begin{enumerate}
\item Throughout the paper $g$ is a fixed natural number at least 2.
\item $G$ is a finite group.
\item $\msy L$ is a set of primes, $\msy L$ contains the primes dividing
$\# G$.
\item The stack of stable curves of genus $g$ is denoted $\overline{{\cal M}_g}} \def\bMg{\overline{M_g}$, the
open substack of smooth curves ${\cal M}_g} \def\Mg{M_g$. Stacks are denoted
by script letters. For definitions and results concerning stacks
we refer to \cite{DM}.
\item Suppose $\Gamma$ is a (pro-finite) group. A characteristic
subgroup of $\Gamma$ is a normal subgroup fixed by any automorphism
of $\Gamma$. A characteristic quotient is one whose kernel is a
characteristic subgroup. If $\Gamma$ is profinite and topologically
finitely generated then it is the direct limit of its finite
characteristic quotients.
\item Let $\Gamma$ denote a (finitely generated) group. For $a,b\in \Gamma$ we
put $[a,b]=a^{-1}b^{-1}ab$, so that $ab=ba[a,b]$. We define the lower
central series $\Gamma^{(k)}$ of $\Gamma$ by $\Gamma^{(1)}=\Gamma$ and
$\Gamma^{(k+1)}=[\Gamma^{(k)},\Gamma]$. The subgroup of $\Gamma$ generated
by nth-powers is denoted $\Gamma^n$. We write $\Gamma^{(k),n}$ to indicate
the subgroup generated by $\Gamma^{(k)}$ and nth-powers,
$\Gamma^{(k),n}=\Gamma^{(k)}\cdot \Gamma^n$. Any group homomorphism
maps commutators to commutators and nth-powers to nth-powers, hence
preserves these subgroups. In particular, the subgroups $\Gamma^{(k),n}$
are characteristic subgroups of $\Gamma$.
\item Let $\Pi=\Pi_g$ denote the standard fundamental group $\Pi=\pi_1(S)$
of a compact Riemann surface $S$ of genus $g$. \end{enumerate}
\section{Definition of the moduli problem}
In this section we recall the definition of the moduli problem of
Teichm\"uller level structures, see \cite[ Section 5]{DM}. Furthermore,
we prove that the Deligne-Mumford compactification of the
associated stack is smooth if and only if the corresponding
analytic orbifold is smooth.
\subsection{The relative fundamental group}
In this section we define the relative fundamental
group for a proper smooth morphism $f: X\to S$ with connected
fibres and endowed with a section $s: S\to X$. In order to
motivate the definition in the algebraic case (and since we
need it also) we first do the analytic case.
\subsubsection{The analytic case}
Here $f:X\to S$ is a proper smooth morphism of analytic spaces
with connected fibres. In addition we are given a section $s:S\to X$
of $f$. We define a locally constant sheaf of groups
$\pi_1(X/S,s)$ over $S$ such that for all points $p\in S$ we have an isomorphism
of groups
$$\pi_1(X/S,s)_p\cong \pi_1(X_p,s(p)),$$
of the fibre of the sheaf at $p$ with the topological fundamental group
of the fibre $X_p$ of $f$ at $p$ with base point $s(p)$.
To construct $\pi_1(X/S,s)$ we choose for any point $p$ of $S$ a connected open
neighbourhood $U_p\subset S$ and a topological isomorphism
$$\phi_p: f^{-1}(U_p)\cong X_p\times U_p.$$
Such can be found compatible with $f$ and the projection to $U_p$,
inducing the identity on $X_p$ and
such that $\phi_p\circ s$ equals $q\mapsto (s(p), q)$. Over $U_p$ we
take $\pi_1(X/S,s)$ constant with fibre $\pi_1(X_p,s(p))$.
To glue these we note that given two points $p_1, p_2\in S$ there is for
any $q\in U_{p_1}\cap U_{p_2}$ an identification
$$X_{p_1}=X_{p_1}\times \{q\}
\mapright{\phi_{p_1}^{-1}} X_q \mapright{\phi_{p_2}}
X_{p_2}\times \{q\}=X_{p_2}.$$
This identification is compatible with base points $s(p_i)$ and
depends continuously on $q\in U_{p_1}\cap U_{p_2}$. This means that
the induced isomorphism
$$ \pif{p_1}\cong \pif{p_2}$$
is constant on the connected components of $U_{p_1}\cap U_{p_2}$. Hence we
get the desired gluing. We leave to the reader the trivial verification
that these gluings satisfy the desired cocycle condition on
$U_{p_1}\cap U_{p_2}\cap U_{p_3}$.
If we choose other $\phi_p$, say $\phi'_p$, then for $q\in U_p$ the map
$X_q\to X_p\times \{q\}\to X_q$, using first $\phi_p^{-1}$ then $\phi'_p$,
is homotopic to the identity. Hence the resulting sheaves $\pi_1(X/S,s)$
are canonically isomorphic. A similar argument deals with the shrinking
of the neighbourhoods $U_p$.
\begin{proposition}The construction given above defines a locally constant
sheaf of groups $\pi_1(X/S,s)$ over $S$, characterized by the following
properties:
\begin{enumerate}
\item For any point $p\in S$
there is given an isomorphism $\pi_1(X/S,s)_p\cong \pi_1(X_p,s(p))$.
\item The monodromy action of $\pi_1(S,p)$ on the fibre of the locally
constant sheaf
$$\rho : \pi_1(S,p)\to {\rm Aut}\big(\pi_1(X/S,s)_p\big) $$
agrees, via the isomorphism of 1), with the action of $\pi_1(S,p)$
on $\pi_1(X_p,s(p))$ deduced from the split exact sequence
$$1\longrightarrow \pi_1(X_p,s(p)) \longrightarrow \pi_1(X, s(p))
\lower3pt\hbox{$\longrightarrow$}\llap{\raise3pt\hbox{$\mapleft{s_\ast}$}}
\pi_1(S,p)\longrightarrow 1.$$
\item The construction of $\pi_1(X/S,s)$ commutes with arbitrary base change
$S'\to S$.
\end{enumerate}\end{proposition}
\begin{proof}Suppose $\gamma\in \pi_1(S,p)$ and $\alpha\in \pi_1(X_p,s(p))$. It is
well known and easy to prove that $s_\ast(\gamma)\alpha s_\ast(\gamma^{-1})$
is equal to the horizontal transportation of the loop $\alpha$ over
$\gamma$. Clearly this describes the monodromy representation for the sheaf
$\pi_1(X/S,s)$. The proof of the other assertions is left to the reader.\end{proof}
\begin{remark}\label{innerconjugation}
{\rm Suppose $s':S\to X$ is a second section of $f$. In
general $\pi_1(X/S,s)$ is not isomorphic to $\pi_1(X/S,s')$. However, locally
on $S$, say over $U\subset S$, we can choose a homotopy $H$ between
$s$ and $s'$. This will induce an identification
$$ i_H: \pi_1(X/S,s)|_U\cong \pi_1(X/S,s')|_U.$$
This is unique up to an {\it inner automorphism} of $\pi_1(X/S,s)$. Indeed,
if $H'$ is another such homotopy, then combining $H$ and $H'$ gives a
familly of loops in $X$ over $U$, with base points $s(u)$, i.e., a section
of $\pi_1(X/S,s)$ over $U$. The map $(i_{H'})^{-1}\circ i_H$ is equal to
conjugation with this section.}\end{remark}
\subsubsection{The algebraic case}
Here we consider a proper smooth morphism of schemes
$f:X\to S$ with connected geometric fibres. As before we have a section
$s:S\to X$ of the morphism $f$. Further, we assume given a set of primes
$\msy L} \def\cC{{\cal C}$ such that all residue characteristics of $S$ are {\it not} in $\msy L} \def\cC{{\cal C}$.
We recall some general notations concerning algebraic fundamental groups.
We refer to \cite{SGA1} and \cite{Murre} for more details.
If $Y$ is a scheme, then ${\rm \acute Et}(Y)$ denotes the category of finite \'etale
coverings $Y'\to Y$. If ${\bar p}$ is a geometric point of $Y$ then we denote
by $F_{Y,{\bar p}}$ the fundamental functor (or fibre functor)
$ F_{Y,{\bar p}} : {\rm \acute Et}(Y)\longrightarrow Set $
which associates to $Y'$ the set of geometric points ${\bar p}\to Y'$ lying over
${\bar p}\to Y$. By definition we have $\pi_1(Y,{\bar p})={\rm Aut}(F_{Y,{\bar p}})$; this is
the fundamental group of $Y$ with base point ${\bar p}$. To compare the fundamental
groups with base points ${\bar p}$, resp.\ ${\bar q}$ we use a path from ${\bar p}$ to ${\bar q}$,
i.e., an isomorphism of fibre functors
$ \alpha: F_{Y,{\bar p}}\longrightarrow F_{Y,{\bar q}}.$
Obviously, $\alpha$ gives an isomorphism
$\alpha_\ast : \pi_1(Y,{\bar p})\longrightarrow \pi_1(Y,{\bar q}).$
We note that it is independent of the choice of $\alpha$ up to conjugation.
A morphism of schemes $h: Y\to Z$, defines a functor
$h^\ast : {\rm \acute Et}(Z)\to {\rm \acute Et}(Y)$, which satisfies $F_{Y,{\bar p}}=F_{Z,h({\bar p})}\circ
h^\ast$.
Therefore we get $h_\ast$ on loops and on paths.
A slight modification of the above gives $\pip Y{\bar p}$, the algebraic fundamental
group classifying Galois coverings of degree in $\msy L} \def\cC{{\cal C}$. Formally it can be
defined as
$$\pip Y{\bar p} = \lim_{{\longleftarrow}} G, $$
where the limit is taken over all surjections $\pi_1(Y,{\bar p})\to G$ onto finite
groups $G$ whose orders have only prime factors from $\msy L} \def\cC{{\cal C}$.
In the sequel we will use the following results from \cite{SGA1}: The
sequence
$$\pi_1(X_{\bar p},s({\bar p}))\longrightarrow \pi_1(X,s({\bar p})) \longrightarrow
\pi_1(S,{\bar p})\longrightarrow 1 $$
is exact. If we take the pushout of this sequence with the surjection
$\pif{\bar p}\to\pip {X_{\bar p}}{\bar p}$ then the sequence also becomes left exact
$$ 1\longrightarrow \pip {X_{\bar p}}{\bar p} \longrightarrow\pi'_1(X,s({\bar p}))
\lower3pt\hbox{$\longrightarrow$}\llap{\raise3pt\hbox{$\mapleft{s_\ast}$}}
\pi_1(S,{\bar p})\longrightarrow 1.\eqno{(*)}$$
See \cite[Expos\'e XII 4.3, 4.4]{SGA1}.
As before the section $s$ defines a splitting
$s_\ast : \pi_1(S,{\bar p})\to \pi_1(X,s({\bar p}))$.
\begin{proposition}(\cite[Expos\'e XII 4.5]{SGA1})\label{piprel}
There is a pro-object in the category of locally
constant sheaves of groups on $S_{\acute et}$, denoted $\pi_1^{\L}(X/S,s)$,
determined up to unique isomorphism by the following properties:
\begin{enumerate}
\item For any geometric point ${\bar p}$ of $S$ there is given an
isomorphism $$\pi_1^{\L}(X/S,s)_{\bar p}\longrightarrow \pip {X_{\bar p}}{s({\bar p})}.$$
\item The monodromy presentation
$$\rho : \pi_1(S,{\bar p})\longrightarrow {\rm Aut}\big(\pi_1^{\L}(X/S,s)_{\bar p}\big)$$
equals, via 1), the action of $\pi_1(S,{\bar p})$ on $\pip {X_{\bar p}}{s({\bar p})}$
deduced from {\rm (*)}.
\item The construction of $\pi_1^{\L}(X/S,s)$ commutes with arbitrary base change
$S'\to S$.\end{enumerate}\end{proposition}
\begin{proof}Of course we may assume that $S$ is connected. Take a
geometric point ${\bar p}$ of $S$. First we note that, since $\pi_1^{\L}(X_\p, s(\p))$ is
topologically finitely generated, it is the direct limit of its
characteristic finite quotients:
$$\pi_1^{\L}(X_\p, s(\p))=\lim_{{\scriptstyle \longleftarrow}\atop {\scriptstyle \omega}}
G_\omega$$
The action of $\pi_1(S,{\bar p})$ on $\pi_1^{\L}(X_\p, s(\p))$ deduced from (*)
gives an action $\rho_\omega$ on each $G_\omega$. This defines a finite locally
constant \'etale sheaf ${\cal F}_\omega$ on $S_{\acute et}$ whose fibre in ${\bar p}$
is given by $G_\omega$ and monodromy action equal to $\rho_\omega$.
We put
$$ \pi_1^{\L}(X/S,s)=\lim_{{\scriptstyle \longleftarrow}\atop {\scriptstyle \omega}}
{\cal F}_\omega.$$
This immediately gives 1) and 2) for our chosen point ${\bar p}$. Part 3) is also
clear if there exists a lift of ${\bar p}$ to a geometric point of $S'$, see
\cite{SGA1}.
Thus it suffices to prove 1) and 2) for a second geometric point ${\bar q}$ of $S$.
Note that by definition
$$\pi_1(X,s({\bar p}))={\rm Aut}\big(F_{S,{\bar p}}\circ s^\ast\big).$$
Hence if we choose an isomorphism $\alpha : F_{S,{\bar p}}\to F_{S,{\bar q}}$
then we get a commutative diagram
$$\matrix{\pi_1(X, s({\bar p}))&
\lower3pt\hbox{$\longrightarrow$}\llap{\raise3pt\hbox{$\mapleft{s_\ast}$}}&
\pi_1(S,{\bar p})\cr
\mapdown{\alpha_\ast}&&\mapdown{\alpha_\ast}\cr
\pi_1(X, s({\bar q}))&
\lower3pt\hbox{$\longrightarrow$}\llap{\raise3pt\hbox{$\mapleft{s_\ast}$}}&
\pi_1(S,{\bar q}).\cr}$$
By (*) this induces an isomorphism
$$ \alpha^{\msy L} \def\cC{{\cal C}} : \pi_1^{\L}(X_\p, s(\p)) \longrightarrow \pi_1^{\msy L} \def\cC{{\cal C}}(X_{\bar q}, s({\bar q})).$$
This is an isomorphism compatible with the actions of
$\pi_1(S,{\bar p})$ and $\pi_1(S,{\bar q})$, compared via $\alpha_\ast$.
The fibre of $\pi_1^{\L}(X/S,s)$ at ${\bar q}$ is by definition
$$F_{S,{\bar q}}\big(\pi_1^{\L}(X/S,s)\big)\mapright{\alpha^{-1}}
F_{S,{\bar p}}\big(\pi_1^{\L}(X/S,s)\big)= \pi_1^{\L}(X_\p, s(\p)).$$
If we use $\alpha^{\msy L} \def\cC{{\cal C}}$ to identify this with $\pi_1^{\msy L} \def\cC{{\cal C}}(X_{\bar q}, s({\bar q}))$ then
we see by the above that the monodromy action on this exactly corresponds
to the action deduced from (*) (with ${\bar q}$ in stead of ${\bar p}$).
We leave to the reader the verification that another choice of $\alpha$
gives the same identification
$$\pi_1^{\L}(X/S,s)_{\bar q}\longrightarrow \pi_1^{\msy L} \def\cC{{\cal C}}(X_{\bar q}, s({\bar q})).$$\end{proof}
Suppose $s': S\to X$ is a second section of $f$. Take a geometric point
${\bar p}$ of $S$. We write $i_{\bar p}$ for the morphism $X_{\bar p}\to X$.
We say that a path $\beta$ on $X$ connecting $s({\bar p})$ to
$s'({\bar p})$ lies in $X_{\bar p}$ if there exists a path $\tilde \beta$ in $X_{\bar p}$
such that $i_{{\bar p},\ast}(\tilde \beta)=\beta$. We remark that any path
$\beta$ connecting $s({\bar p})$ to $s'({\bar p})$ lies in $X_{\bar p}$ if and only if
$f_\ast(\beta)=1$ (in $\pi_1(S,{\bar p})$). This is easily seen using the
first exact sequence above.
Let us take such a $\beta$ lying in $X_{\bar p}$. It gives rise to a commutative
diagram
$$\matrix{1&\longrightarrow&
\pi_1^{\L}(X_\p, s(\p))&\longrightarrow&\pi'_1(X,s({\bar p}))&\longrightarrow&\pi_1(S,{\bar p})&
\longrightarrow&1\cr
&&\mapdown{\cong}&&\mapdown{\beta_\ast}&&\mapdown{{\rm id}}&&\cr
1&\longrightarrow&\pi_1^{\msy L} \def\cC{{\cal C}}(X_{\bar p}, s'({\bar p}))
&\longrightarrow&\pi'_1(X,s'({\bar p}))&\longrightarrow&\pi_1(S,{\bar p})&
\longrightarrow&1\cr}$$
The isomorphism $\pi_1^{\L}(X_\p, s(\p))\cong \pi_1^{\msy L} \def\cC{{\cal C}}(X_{\bar p}, s'({\bar p}))$ determined in this way
is unique up to inner conjugation. We constructed $\pi_1^{\L}(X/S,s)$, resp.\
${\pi_1^{\msy L} \def\cC{{\cal C}}(X/S,s')}$ as the limit of sheaves ${\cal F}_\omega$, resp.\
${\cal F}'_\omega$ corresponding to characteristic quotients
$\pi_1^{\L}(X_\p, s(\p))\to G_\omega$, resp.\ ${\pi_1^{\msy L} \def\cC{{\cal C}}(X/S,s')}\to G_\omega$. Note
that the isomorphisms $G_\omega\to G'_\omega$ induced from the above
are also unique up to inner conjugation.
Let $S_\omega\to S$ be the finite \'etale covering of $S$ trivializing
the action of $\pi_1(S,{\bar p})$ on both $G_\omega$ and $G'_\omega$. We can
use the above to get an isomorphism of (constant) sheaves of groups
$$ {\cal F}_\omega|_{S_\omega}\longrightarrow {\cal F}'_\omega|_{S_\omega}.$$
We claim this isomorphism is unique up to inner conjugation.
This is clear if we only change $\beta$, but what happens if we change ${\bar p}$
to ${\bar q}$? Take $\alpha : F_{S,{\bar p}}\to F_{S,{\bar q}}$ as in the proof
of Proposition \ref{piprel}. What we have to check is that
$s'_\ast(\alpha)\circ\beta\circ s_\ast(\alpha^{-1})$ is a path
connecting $s({\bar q})$ to $s'({\bar q})$ lying in $X_{\bar q}$. But this is clear
since $f_\ast\big(s'_\ast(\alpha)\circ\beta\circ s_\ast(\alpha^{-1})\big)
=\alpha\circ \alpha^{-1}=1.$
\begin{corollary}\label{invariance}The construction above defines
locally in the \'etale topology on $S$ identifications of the
finite quotients of the sheaves $\pi_1^{\L}(X/S,s)$ and ${\pi_1^{\msy L} \def\cC{{\cal C}}(X/S,s')}$.
These identifications are unique up to inner conjugation
and agree via 1) of Proposition \ref{piprel} with the usual
identifications of $\pip {X_{\bar p}}{s({\bar p})}$ and
$\pip {X_{\bar p}}{s'({\bar p})}$.\end{corollary}
\subsection{Exterior homomorphisms}
In this section we define the sheaf of exterior homomorphisms
of the relative fundamental group of $X$ over $S$ into a fixed finite
group $G$. As in \cite[5.5]{DM} this sheaf will be denoted ${\cal H}om^{ext}(\pi_1(X/S),G)$ and will
be a finite locally constant sheaf of sets on $S$ (or $S_{\acute et}$).
We note that $\pi_1(X/S)$ has not been defined.
\subsubsection{The analytic case}
Here $f:X\to S$ is a proper smooth morphism of analytic spaces
with connected fibres.
If $f$ has a section $s$ then we can look at the locally constant
sheaf
$$ {\cal F}={\cal H}om(\pi_1(X/S,s), G)$$
on $S$. It has finite fibres since $\pi_1(X_p,s(p))$ is finitely generated for all
$p$ in $S$. There is a natural action of the sheaf $\pi_1(X/S,s)$ on the
sheaf ${\cal F}$ given by conjugation. We define the sheaf of exterior
homomorphisms as the quotient of ${\cal F}$ by this action
$$ {\cal H}om^{ext}(\pi_1(X/S),G):={\cal F}\big/\pi_1(X/S,s) .$$
It is clear from Remark \ref{innerconjugation} that the right hand side
does not depend on the chosen section $s$.
In general, we choose an open covering $S=\bigcup U_i$ such that
$X\to S$ has a section over each $U_i$. The sheaf ${\cal H}om^{ext}(\pi_1(X/S),G)$ is then
defined by gluing the sheaves constructed above.
The fibres are described by the formula:
$${\cal H}om^{ext}(\pi_1(X/S),G)_p={\rm Hom}(\pi_1(X_p,q),G)\big/ \pi_1(X_p,q), $$
where $q$ is any point of the fibre $X_p$. Note that the monodromy
action of $\gamma\in \pi_1(S,p)$ on this is given by horizontal
transport of loops in $\pi_1(X_p,q)$.
\subsubsection{The algebraic case}
Here we assume that $S$ is a scheme over ${\rm Spec}(\msy Z[1/\#G])$.
The morphism $f:X\to S$ is still assumed proper smooth with
connected geometric fibres.
The construction of ${\cal H}om^{ext}(\pi_1(X/S),G)$ in this case is exactly the same as
for the analytic case. We use $\pi_1^{\L}(X/S,s)$, where $\msy L} \def\cC{{\cal C}$ is the set of
primes dividing $\# G$, and we use that sections of $f$ exist
locally in the \'etale topology on $S$. We use also Corollary
\ref{invariance}.
The geometric fibres of the resulting sheaf can be described as follows:
$${\cal H}om^{ext}(\pi_1(X/S),G)_{\bar p}={\rm Hom}(\pi^{\msy L} \def\cC{{\cal C}}_1(X_{\bar p},{\bar q}),G)\big/ \pi^{\msy L} \def\cC{{\cal C}}_1(X_{\bar p},{\bar q})=
{\rm Hom}(\pi_1(X_p,q),G)\big/ \pi_1(X_p,q), $$
the last equality holds in view of our definition of $\msy L} \def\cC{{\cal C}$.
This justifies dropping $\msy L} \def\cC{{\cal C}$ from the notations.
\subsubsection{Comparison}
We note that if $S$ is a scheme of finite type over $\msy C$
there is a canonical homomorphism
$\pi_1(X^{an}/S^{an},s)\to \pi_1^{\L}(X/S,s)^{an}$, identifying the relevant
finite quotient sheaves. Clearly this gives rise to an identification
of sheaves of exterior homomorphisms.
\subsection{Teichm\"uller level structures}
In this section the morphism $f:X\to S$ will be a familly of smooth
projective curves of genus $g$. The abstract finite group $G$ will
be fixed. In both the analytic case and the algebraic case we make the
following definition.
\begin{definition}{\rm \cite[5.6]{DM}} A Teichm\"uller
structure $\alpha$ of level $G$ on $X\to S$
is a surjective exterior homomorphism
$$\alpha\in \Gamma(S, {\cal H}om^{ext}(\pi_1(X/S),G)).$$
Thus locally on $S$ (resp.\ $S_{\acute et}$) $\alpha$ corresponds to a
surjective homomorphism $\pi_1(X/S,s)\to G$.\end{definition}
We want to consider the moduli spaces parametrizing
smooth stable curves of genus $g$ with a Teichm\"uller structure
of level $G$. However, as usual, it is more convenient to work
with stacks. Thus let ${}_G{\cal M}_g} \def\GMg{{}_GM_g$ denote the stack whose
category of sections over the scheme $S$
(lying over ${\rm Spec}(\msy Z[1/\#G])$) is the category of smooth stable
curves $X\to S$ of genus $g$ endowed with a Teichm\"uller structure
of level $G$, see \cite[Section 5]{DM}. The construction of ${\cal H}om^{ext}(\pi_1(X/S),G)$
shows that the stack ${}_G{\cal M}_g} \def\GMg{{}_GM_g$ is representable
finite \'etale over ${\cal M}_g} \def\Mg{M_g[1/\#G]$. Thus
${}_G{\cal M}_g} \def\GMg{{}_GM_g$ is a separated algebraic stack, smooth over
${\rm Spec}(\msy Z[1/\#G])$.
In a similar way we define the analytic stack
${}_G{\cal M}^{an}_g} \def\GMgan{{}_GM_g^{an}$ classifying complex curves with a Teichm\"uller structure of
level $G$. By comparing the stacks ${}_G{\cal M}_g} \def\GMg{{}_GM_g$ and ${}_G{\cal M}^{an}_g} \def\GMgan{{}_GM_g^{an}$ with the
stack ${\cal M}_g} \def\Mg{M_g$ and using the known result for $M_g$ one derives easily
the following result.
\begin{theorem}\label{existence}With notations as above.
\begin{enumerate}
\item A coarse moduli scheme ${}_GM_g$ for ${}_G{\cal M}_g} \def\GMg{{}_GM_g$ exists. It is separated
of finite type over ${\rm Spec}(\msy Z[1/\#G])$.
\item A coarse analytic moduli space ${}_GM_g^{an}$ for ${}_G{\cal M}^{an}_g} \def\GMgan{{}_GM_g^{an}$
exists. It is canonically isomorphic to the analytic space associated
to the complex variety ${}_GM_g\otimes \msy C$.
\end{enumerate}\end{theorem}
Suppose we have a surjection $G\to G'$. There is a natural forgetful
morphism ${}_G{\cal M}_g} \def\GMg{{}_GM_g\to {}_{G'}{\cal M}_g} \def\Mg{M_g$. This morphism is representable
finite \'etale. Hence, if ${}_{G'}{\cal M}_g} \def\Mg{M_g$ is a scheme (i.e., isomorphic
to ${}_{G'}M_g$), then so is ${}_G{\cal M}_g} \def\GMg{{}_GM_g$. In this case the morphism
${}_{G'}M_g\to \GMg$ is finite \'etale (perhaps of degree $0$).
Finally, we note that ${}_G{\cal M}_g} \def\GMg{{}_GM_g$ might be empty, this being the case
if $G$ is not isomorphic to a quotient of the fundamental group of
a Riemann surface of genus $g$.
\subsubsection{Abelian level structures}
In this section we treat the algebraic case. Let $f:X\to S$
be a smooth stable curve of genus $g$ over $S$. Fix a natural number
$m$. An abelian structure structure of level $m$ on $X$ over $S$ is
defined as an isomorphism of \'etale sheaves over $S$
$$ (\msy Z/m\msy Z)^{2g}_S\longrightarrow R^1f_\ast(\msy Z/m\msy Z).$$
Such a level structure can only exist if the base scheme $S$ lies
over ${\rm Spec}(\msy Z[1/m])$; let us assume this is the case.
Note that the sheaf ${\cal H}om^{ext}(\pi_1(X/S),\msy Z/m\msy Z)$
is isomorphic to the subsheaf of primitive elements
in $R^1f_\ast(\msy Z/m\msy Z)$.
Thus we see that the moduli stack of curves with an abelian level $m$
structure is isomorphic to the stack ${}_G{\cal M}_g} \def\GMg{{}_GM_g$
with $G=(\msy Z/m\msy Z)^{2g}$.
In particular, if $m\geq 3$, then $\GMg$ is a fine moduli
scheme smooth over ${\rm Spec}(\msy Z[1/m])$. The following is deduced
from the above.
\begin{proposition}
If the finite group $G$ allows a surjection onto $(\msy Z/m\msy Z)^{2g}$
for some $m\geq 3$ then the coarse moduli scheme ${}_GM_g$ is a
fine moduli scheme.\end{proposition}
\subsubsection{Compactifications}
In order to get compact moduli spaces we just take the normalization
with respect to the Deligne-Mumford compactification. In this subsection
the finite group $G$ will be fixed, of order $n=\#G$.
Consider the Deligne-Mumford compactification ${\cal M}_g} \def\Mg{M_g\subset \overline{{\cal M}_g}} \def\bMg{\overline{M_g}$.
We define $\overline{{}_G{\cal M}_g}} \def\bGMg{\overline{{}_GM_g}$ as the normalization of $\bMg[1/n]$ with
respect to ${}_G{\cal M}_g} \def\GMg{{}_GM_g$. Similarly we define $\bcGMgan$ as the
normalization of $\overline{{\cal M}_g}^{an}} \def\bMgan{\overline{M_g}^{an}$ with respect to ${}_G{\cal M}^{an}_g} \def\GMgan{{}_GM_g^{an}$.
As per convention we denote $\bGMg$ (resp.\ $\overline{{}_GM_g}^{an}$) the associated
coarse moduli scheme (resp.\ analytic moduli space).
The morphism
$$\bGMg\longrightarrow {\rm Spec}(\msy Z[1/n])$$
is proper, since $\bMg[1/n]$ is proper over ${\rm Spec}(\msy Z[1/n])$.
To see whether this morphism is smooth we have the following
criterion.
\begin{proposition}\label{criterium}
For finite groups $G$ which allow a surjection
$G\to (\msy Z/m\msy Z)^{2g}$ for some $m\geq 3$ the following
statements are equivalent:
\begin{enumerate}
\item The morphism $\bGMg\longrightarrow {\rm Spec}(\msy Z[1/n])$ is smooth.
\item The analytic space $\overline{{}_GM_g}^{an}$ is a (nonsingular) complex manifold.
\end{enumerate}\end{proposition}
\begin{proof}We use that $\overline{{}_GM_g}^{an}$ is isomorphic to the analytic space
associated to the variety $\bGMg\otimes \msy C$. Thus it suffices
to show that $\bGMg\otimes \msy C$ nonsingular implies
$\bGMg\otimes \bar {\msy F}_p$ nonsingular, where $p>0$ and $p$ does not
divide $n$. The argument will be based on the fact that the morphism
$\varphi : \bGMg\to \bMg[1/n]$ is tamely ramified along the boundary.
To see this we need a description of the complete local rings
of $\bGMg$ in points on the boundary. Suppose that $C$ is a stable curve
of genus $g$ over an algebraically closed field $k$ of characteristic $p$.
The singular points of $C$ are $P_1,\ldots, P_\ell$.
Let $\cC\to {\rm Spf}\big(W(k)[[t_1,\ldots,t_{3g-3}]]\big)$ be the universal
deformation of $\cC$. We choose the parameters $t_i$ such that $t_i=0$
defines the locus where $P_i$ survives as a singular point,
for $i=1,\ldots, \ell$.
Put $A=W(k)[[t_1,\ldots,t_{3g-3}]]$. Since $\cC$ is (uniquely) algebraizable
we have a morphism ${\rm Spec}(A)\to \overline{{\cal M}_g}} \def\bMg{\overline{M_g}$ and
$${\rm Spec}\big(A[1/t_1\ldots t_\ell]\big)\longrightarrow {\cal M}_g} \def\Mg{M_g.$$
We consider the fibre product
$${\rm Spec}\big(A[1/t_1\ldots t_\ell]\big)\times_{{\cal M}_g} \def\Mg{M_g} {}_G{\cal M}_g} \def\GMg{{}_GM_g .$$
This is finite \'etale over $A[1/t_1\ldots t_\ell]$ hence affine. The
normalization of this over ${\rm Spec}(A)$ is ${\rm Spec}(B)$; here $B$ is
a product of complete local rings finite over $A$, ramified only over
$t_1\ldots t_\ell=0$. Thus we have
$${\rm Spec}\big(B[1/t_1\ldots t_\ell]\big)=
{\rm Spec}\big(A[1/t_1\ldots t_\ell]\big)\times_{{\cal M}_g} \def\Mg{M_g} {}_G{\cal M}_g} \def\GMg{{}_GM_g .$$
We claim that the morphism ${\rm Spec}(B)\to \bGMg$ identifies complete
local rings at the points of $\bGMg$ lying over $[C]\in \bMg(k)$. By
general theory we know that the completion of $\bMg$ at $[C]$ is
${\rm Spf}(A)/{\rm Aut}(C)$. A formal argument gives that the completion
of $\bGMg$ along $\varphi^{-1}([C])$ is isomorphic to
${\rm Spf}(B)/{\rm Aut}(C)$. Thus it suffices to show that
the action of ${\rm Aut}(C)$ on $\varphi^{-1}([C])$ is free. By comparing
levels, it suffices to show this for $G=(\msy Z/m\msy Z)^{2g}$; this is
the content of \cite[Proposition 3.5]{De}. See also references in remark
below.
We use Abhyankar's lemma which asserts
that any normal local domain $A'$, finite generically
\'etale over $A$ and ramified only along $t_1\ldots t_\ell=0$ is contained
in $A[t_i^{1/n_i}]$ for some $n_1,\ldots,n_\ell$ relatively prime to $p$.
In addition, it is easily seen that $A'$ is formally smooth over $W(k)$
if and only if $A'$ is actually equal to $A[t_i^{1/n_i}]$
for some $n_1,\ldots,n_\ell$.
Let $C_K$ denote the lift of $C$ to $K=Q\big(W(k)\big)$ given by setting
$t_1=\ldots=t_{3g-3}=0$. The homomorphism
$A\to K[[t_1,\ldots,t_{3g-3}]]$ is such that
$$B\otimes_A K[[t_1,\ldots,t_{3g-3}]]$$
describes the complete local rings of $\bGMg$ along $\varphi^{-1}([C_K])$.
The result follows by comparing $B$ to this ring.\end{proof}
\begin{remark} {\rm The arguments above actually show that in the
situation of the proposition the stacks $\overline{{}_G{\cal M}_g}} \def\bGMg{\overline{{}_GM_g}$ are schemes, i.e.,
$\overline{{}_G{\cal M}_g}} \def\bGMg{\overline{{}_GM_g}\cong \bGMg$. Thus we get a stable curve over $\bGMg$ from
the morphism $\bGMg\cong \overline{{}_G{\cal M}_g}} \def\bGMg{\overline{{}_GM_g}\to \overline{{\cal M}_g}} \def\bMg{\overline{M_g}$; this also follows
from the existence of such a stable curve in the case
of abelian level structures, see \cite[Thm. 10.9]{Po},
\cite[page 12]{GO} and \cite[Bemerkung 1]{Mo}.}\end{remark}
\section{Monodromy along the boundary}
In this section we study the moduli spaces $\bGMg$ along the boundary.
To do this it suffices to understand the monodromy along the boundary
on the relative fundamental group of the universal curve over ${\cal M}^{an}_g} \def\Mgan{M_g^{an}$.
\subsection{The results}\label{results}
Let us formulate the main result. Let $\Pi$ denote the standard fundamental
group of a compact Riemann surface of genus $g$. We fix natural numbers
$k,n$ with $k\geq 1$ and $n\geq 3$. We will consider moduli of curves
of genus $g$ with a Teichm\"uller structure of level $G$, where
$$G=\Pi\big/ \Pi^{(k+1),n}.$$
(For notations see \ref{notation}.) By a result of Labute \cite{Labute},
the quotients $\Pi^{(k)}/\Pi^{(k+1)}$ are finitely generated free
abelian groups. Thus $G$ has a filtration whose successive quotients
are finite abelian groups of exponent $n$. Any prime dividing $\# G$
also divides $n$. Further, there is a surjection $G\to (\msy Z/n\msy Z)^{2g}$.
By Section 2 we get a moduli scheme
$$\bGMg\longrightarrow {\rm Spec}(\msy Z[1/n])$$
whose interior $\GMg$ classifies smooth genus $g$ curves $C$ with a surjection
$\pi_1(C)\to G$ given up to inner automorphisms.
\begin{theorem}\label{glad}
Suppose $k\in \{1,2,3\}$ and $n\geq 3$. The structural morphism
$\bGMg\to {\rm Spec}(\msy Z[1/n])$ is smooth if and only if
\begin{itemize}
\item{$k=1$} and $g=2$,
\item{$k=2$} and $n$ is odd,
\item{$k=3$} and $n$ is odd or $n$ is divisible by $4$.
\end{itemize}
Furthermore, if $k \geq 4$, $n \geq 3$ and $n$ relatively prime to $6$,
then $\bGMg\to {\rm Spec}(\msy Z[1/n])$ is smooth.
\end{theorem}
\begin{remark}{\rm The abelian case of this theorem, i.e., the case $k=1$,
has been proven by Mostafa \cite{Mo} and Van Geemen-Oort \cite{GO}.
A smooth and compact cover $\overline{M_g[_2^n]}^{an} \rightarrow \bMgan$
with $n$ even and $n\geq 6$
has been constructed by Looijenga using Prym level structures,
see \cite{Looijenga}. This has been extended to higher level structures
$\overline{ M_g[_q^n]}^{an}$ by one of us (Pikaart). Using an
analog of Proposition \ref{criterium} this result may be extended
to an open set of ${\rm Spec}(\msy Z)$, at least if $q\geq 3$.
We note that $\bGMg\to\bMg[1/n]$ is a Galois cover\label{galois}
with group ${\rm Aut}(G)/{\rm Inn}(G)$ if $g\geq 3$ (if $g=2$ one has to divide
out an additional $\{\pm 1\}$). This follows since for
any surjection $\Pi\to G$ the kernel is equal to $\Pi^{(k),n}$. However,
$\bGMg\otimes\msy C$ need not be connected, see \cite[5.13]{DM} for
a description of the set of connected components.}\end{remark}
To prove the theorem, it suffices to consider $\overline{{}_GM_g}^{an}$, see Proposition
\ref{criterium}. We already know that $\GMgan$ is smooth. As in the
proof of Proposition \ref{criterium} we describe analytic neighbourhoods of
points in the boundary.
Let $C$ be a complex stable curve of genus $g$ with singular points
$\{ P_1,\ldots,P_\ell\}$. Let $\Gamma=\Gamma(C)$ be its dual graph;
an edge for each point $P_j$, a vertex for an irreducible component of $C$.
Let $ \pi: (\cC,C) \rightarrow (B,0)$ be a local
universal deformation of $C$,
where $B\subset \msy C^{3g-s}$ is a polydisc neighbourhood of $0$.
The coordinates $z_i$ are chosen such that $z_j=0$, $1\leq j\leq \ell$
parametrizes curves where the singular point $P_j$ subsists.
The discriminant locus $\Delta\subset B$ of $\pi$ is thus given
by $z_1\ldots z_\ell=0$. Put $U=B\setminus \Delta$, let $x\in U$
and choose $y\in \cC_x=\pi^{-1}(x)$.
The fundamental group of $U$ is an abelian group, freely generated by
simple loops
around the divisors $z_j=0$, thus naturally isomorphic to the free
abelian group on the edges of $\Gamma$, i.e., $\pi_1(U,x)
\cong \bigoplus_{e \in {\rm Edges}( \Gamma )} {\msy Z} e$.
The map $\cC|_U\to U$ is a locally trivial fibration, hence we have
the exact sequence
$$1 \longrightarrow \pi_1(\cC_x,y) \longrightarrow \pi_1(\cC|_U,y)
\longrightarrow \pi_1(U,x) \longrightarrow 1.$$
(Use that $\pi_2(U)=(0)$.) This short exact sequence provides us
with the monodromy representation
$$\rho: \pi_1(U,x) \longrightarrow {\rm Out}\big(\pi_1(\cC_x,y)\big).$$
The points $P_j$ determine non-trivial distinct isotopy classes
of circles on $\cC_x$, which have pairwise disjoint
representatives $c_j$. The fundamental group of $U$ is also naturally
isomorphic to the free abelian group on these circles, $\pi_1(U,x)
\cong \bigoplus_{i=1}^l {\msy Z} c_i$. Under this identification
we have that
$$\rho(c_i)=D_{c_i},$$
where $D_{c_i}$ is the exterior automorphism of $\pi_1(C_x)$ given by a Dehn
twist (also written $D_{c_j}$) around the circle $c_i$ (see
\cite{Dehn},\cite{Lamotke}).
We will describe of a neighbourhood of a point in $\overline{{}_GM_g}^{an}$
lying above $[C]$. Let $Z$ be the fibre product
$$Z=U\times_{\Mgan}\GMgan.$$
The normalization of $B$ in the function field of $Z$ is denoted $\bar Z$.
Note that $Z\to U$ is a finite topological covering space given by the set
$$S=\hbox{Hom-surj}\big(\pi_1(\cC_x,y),G\big)\big/\pi_1(\cC_x,y)$$
with $\pi_1(U,x)$-action defined via $\rho$. As in the proof of Proposition
\ref{criterium} there is an action of ${\rm Aut}(C)$ on $\bar Z$ and
$\bar Z/{\rm Aut}(C)$ defines a neighbourhood of
$\varphi^{-1}([C])\subset \overline{{}_GM_g}^{an}$. As in that proof we get that
$\overline{{}_GM_g}^{an}$ is smooth along $\varphi^{-1}([C])$ if and only if $\bar Z$
is smooth. (Here we use again that ${\rm Aut}(C)$ acts freely
on $\varphi^{-1}([C])$.) Finally there is the following
criterion: $\bar Z$ is smooth
if and only if for all $s\in S$ we have
$${\rm Stab}(s)=\bigoplus_{e \in {\rm Edges}(\Gamma)}n_e {\msy Z} e $$
for certain $n_e\in \msy Z$. Notice that if $m \in \msy Z $, then $mF^0 +m F^1$
($F^i$ as below) is of this form, but $2mF^0 +m F^1$ is not.
We remark that the arguments above go through for arbitrary finite
groups $G$, with a surjection onto $(\msy Z/n\msy Z)^{2g}$. In order
to describe the stabilizers in our more special situation we introduce the
following notation:
\begin{eqnarray*}
\Akn &= &{\rm Ker}\left( {\rm Aut}(\Pi) \to {\rm
Aut}(\Pi/\Pi^{(k+1),n})\right),\\
\Ikn &= &{\rm Ker}\left( {\rm Inn}(\Pi) \to {\rm
Inn}(\Pi/\Pi^{(k+1),n})\right),\\
\Okn &= &{\rm Ker}\left( {\rm Out}(\Pi) \to {\rm
Out}(\Pi/\Pi^{(k+1),n})\right).
\end{eqnarray*}
Oda et al.\ consider a variant with $n=0$.
By choosing an isomorphism $\pi_1(\cC_x,y)\cong\Pi$ we may view $\rho$
as a map into ${\rm Out}(\Pi)$. It is clear that
${\rm Stab}(s)=\rho^{-1}(\Okn)$ for any $s\in S$ (use Remark \ref{galois}).
Therefore, Theorem \ref{glad} follows from Theorem \ref{mono} below.
We will describe a decreasing
filtration $F^i$ on $\bigoplus_{e \in {\rm Edges}( \Gamma)}
{\msy Z} e$.
An edge $e$ such that $\Gamma \backslash e$ is disconnected is called a
{\em bridge}. A bridge $b$ is said to {\em bound a genus one curve}
if one of the two
components of $C_x \backslash \{ $
the circle corresponding to $b \}$ has genus one.
A pair of distinct edges $\{e,f\}$ is called a {\em cut pair} if neither $e$
nor
$f$
is a bridge and $\Gamma \backslash \{e,f \}$ is disconnected.
A subset $E$ of the edges of $\Gamma$ is called a {\em maximal cut system}
if $E$ contains at least one cut pair, any two elements of $E$ form a cut pair
and no element of $E$ forms a cut pair with an element outside $E$.
Let $B$ be the set of bridges of $\Gamma$ and let $B_1$ be the subset of $B$
consisting of bridges which bound genus one curves. Let $\{E_i\}_{i \in I}$
denote the maximal cut systems.Set
$$D_i:=Ker(\bigoplus_{e \in E_i}
\msy Z e \stackrel{deg}{\rightarrow } \msy Z).$$
We define a decreasing filtration $F^i$ on $\bigoplus_{e \in {\rm
Edges}(\Gamma)}\msy Z e$, as follows:
\begin{eqnarray*}
F^0&= & \bigoplus\nolimits_{e \in {\rm Edges}(\Gamma)}{\msy Z} e,\\
F^1&= & \bigoplus\nolimits_{i\in I} D_i \bigoplus
\left(\bigoplus\nolimits_{b \in B} {\msy Z}b\right), \\
F^2&= & \bigoplus\nolimits_{b \in B} {\msy Z}b ,\\
F^3&= & (0).\end{eqnarray*}
Furthermore, set $F^2_1:=\bigoplus_{b \in B_1} {\msy Z}b$. This refines the
filtration into $F^0\supset F^1\supset F^2\supset F^2_1\supset F^3=(0)$.
Here is the main result of this article.
\begin{theorem}\label{mono} Notations as above. For $n,l \in \msy Z$, define
$n_l:=n/ {\rm gcd}(l,n)$.
\[ \begin{array}{cll}
1. & \mbox{If $k=1$, then} & \rho^{-1}(\Okn ) =nF^0 +F^1 \\
2. & \mbox{If $k=2$, then} & \rho^{-1}(\Okn )=nF^0 +n_2F^1+F^2 \\
3. & \mbox{If $k=3$ and $2 ||n$, then}
& \rho^{-1}(\Okn )=nF^0 + \frac{1}{2}nF^1+n_2F^2 +n_6F^2_1 \\
& \mbox{If $k=3$ and $n$ is odd or $4 | n$, then}
& \rho^{-1}(\Okn ) =nF^0 + nF^1+n_2F^2 +n_6F^2_1\\
4. & \mbox{If $k \geq 4$ and $(n,6)=1$, then} & \rho^{-1}(\Okn ) =nF^0
\end{array} \] \end{theorem}
\begin{remark} {\rm The case $k=1$ has been proven by Brylinski
(\cite{Brylinski}). The case $k \geq 4$ follows from 3, the easy inclusions
of Section \ref{zeer easy} and the inclusions $\Okn \subset O^{(l),n}$ if
$k \geq l$.
If we take $n=0$ then Theorem \ref{mono}
reduces to the non-pointed case of \cite[Main Theorem]{Oda}.} \end{remark}
\section{Description of Dehn twists and easy inclusions} \label{zeer easy}
In this section we will prove the inclusions ``$\supset$'' from Theorem
\ref{mono}.
Let $\Gamma$ be the graph of a stable curve and $(S,\{ c_i \})$ the smooth
model
with a set of circles as described in Subsection \ref{results}. We will
describe the Dehn twist associated to a bridge, cut pair or circle
separatedly.
\subsection{Bridges}\label{bridges}
Let $b$ be a bridge on $S$, let $g$ be the genus of $S$. Modulo a
homeomorphism the situation looks as follows:
\hfill \vspace{4 cm}
$$\hbox{\sl Fig.~1}$$
Cutting $S$ along $b$ yields the decomposition $S= S_1 \cup S_2$.
Let $g_i$ be the genus of $S_i$.
Choose a base point $p$ in $S_1$ and standard generators
$\alpha_{\pm i},$ $1 \leq i \leq g$, for $\pi :=\pi_1(S,p)$ such that
$\alpha_{\pm i}$ are in $S_1$ if $i \leq g_1$ and $\alpha_{\pm i}$
for $i\geq g_1+1$ hits $b$ exactly twice. We set
$v=[\alpha_1,\alpha_{-1}] \cdots [\alpha_{g_1},
\alpha_{-g_1}]$, it is freely homotopic to $b$ for a suitable orientation
of $b$.
We list the effect of the Dehn twist $D_b$ on the standard generators:
\[ \begin{array}{ll}
i \in \{\pm 1,\ldots, \pm g_1\} & {D}_{b}: \alpha_i \mapsto \alpha_i ,\\
i \in \{ \pm ( g_1 +1),\ldots,\pm g \} & {D}_{b}: \alpha_i \mapsto v^{-1}
\alpha_i v
\end{array} \]
This gives for the mth-powers:
\[ \begin{array}{ll}
i \in \{\pm 1,\ldots, \pm g_1\} & {D}^m_{b}: \alpha_i \mapsto \alpha_i ,\\
i \in \{ \pm ( g_1 +1),\ldots,\pm g \} & {D}^m_{b}: \alpha_i \mapsto v^{-m}
\alpha_i v^m
\end{array} \]
We see that $D_b(\alpha_{\pm i})\alpha_{\pm i}^{-1} \in \pi^{(3)}$
for all $i$. This proves that $\rho(b) \in O^{(2)}$ and thus
by linearity $\rho(F^2) \subset O^{(2)}$.
Suppose $b$ bounds a genus one curve, say $S_2$ has genus one.
In this case $v = [\alpha_{-g},\alpha_g]$ and we only have to consider
$$D_b^m(\alpha_{\pm g})\alpha_{\pm g}^{-1}=
[[\alpha_g,\alpha_{-g}]^m,\alpha_{\pm g}]\equiv
[[\alpha_g,\alpha_{-g}],\alpha_{\pm g}]^m \bmod \pi^{(4)}.$$
To prove this element lies in $\pi^{(4),n}$ we define $f:F(x,y,z) \ra \pi$ by
$x \mapsto \alpha_g$, $y \mapsto \alpha_{-g}$ and $z \mapsto 1$.
Then $f([[x,y],x])=[[\alpha_g,\alpha_{-g}],\alpha_{ g}]$
and $f([[x,y],y])=[[\alpha_g,\alpha_{-g}],\alpha_{ -g}]$.
{}From Lemma \ref{berekening} we see that $n_6|m$ implies
that the element above lies in $\pi^{(4),n}$.
Thus $\rho(n_6F^2_1) \subset O^{(3),n}$.
Suppose $b$ does not bound a genus one curve. In that case we have
\begin{eqnarray*}
D_b^m(\alpha_{\pm i})\alpha_{\pm i}^{-1}&=& [v^m,\alpha_{\pm i}^{-1}]\ \equiv\
[v,\alpha_{\pm i}]^{-m}\\
&\equiv & [[\alpha_{1},\alpha_{-1}],\alpha_{\pm i}]^{-m}\cdots
[[\alpha_{g_1},\alpha_{-g_1}],\alpha_{\pm i}]^{-m}
\end{eqnarray*}
for $i>g_1$ in $\pi/\pi^{(4)}$. Define $g_j:F(x,y,z) \ra \pi$ by $x \mapsto
\alpha_j,~y \mapsto \alpha_{-j}$
and $z \mapsto \alpha_{\pm i}$. Then $g_j([[x,y],z]^m)=
[[\alpha_j,\alpha_{-j}],\alpha_{\pm i}]^m$.
{}From Lemma \ref{berekening} we see that
$n_2 |m$ implies $\rho(mb) \in O^{(3),n}$, and
thus $\rho(n_2F^2) \subset O^{(3),n}$.
\subsection{Edges which are not bridges}\label{not bridges}
Let $c$ be a circle on $S$ which is not a bridge. Modulo a homeomorphism the
situation looks as follows:
\hfill \vspace{4 cm}
$$\hbox{\sl Fig.~2}$$
We choose a point $p$ in $S$ and standard generators $\alpha_{\pm i}$ for
$\pi =\pi_1(S,p)$ such that $\alpha_{-g}$ is the only one intersecting $c$
and $\alpha_g$ is freely homotopic to $c$.
The action of the Dehn twist becomes:
$D_c(\alpha_i)=\alpha_i$ if $i \neq -g$ and $D_c(\alpha_{-g}) =\alpha_g
\alpha_{-g}$.
Thus $D_c^m(\alpha_i)\alpha_i^{-1}=1$ or $\alpha_g^m$.
Evidently, if $n|m$ then $\rho(mc) \in O^{(k),n}$ for all $k$.
Together with the results of \ref{bridges} this gives
$\rho(nF^0)\subset O^{(k),n}$ for all $k$.
\subsection{Cut systems}\label{cut systems}
Let $e_1,e_2$ be a cut pair on $S$. Modulo a homeomorphism the situation is
as follows:
\hfill \vspace{4 cm}
$$\hbox{\sl Fig.~3}$$
Cutting $S$ along $e_1$ and $e_2$ yields the decomposition $S=S_1 \cup
S_2$. Let $g_1$ be the genus of $S_1$. Choose a base point $p$ in $S_1$.
We take standard generators $\alpha_{\pm i}$ for $\pi_1(S,p)$, such that
for $i \in \{ \pm 1,\ldots, \pm g_1, g_1+1\} $, $ \alpha_i $ is in $S_1$,
for $i \in \{ \pm (g_1+2),\ldots, \pm g \} $, $ \alpha_i $ enters $S_2$ via
$e_1$
and leaves $S_2$ also via $e_1$ and
$\alpha_{-(g_1+1)}$ enters $S_2$ via $e_1$ and leaves $S_2$ via $e_2$.
Furthermore, we want $\alpha_{g_1+1}$ to be freely homotopic to $e_2$, for
a suitable orientation of $e_2$. It follows that after suitable orientation
of $e_1$ the loop $[\alpha_1,\alpha_{-1}] \cdots
[\alpha_{g_1},\alpha_{-g_1}]\alpha_{g_1+1}^{-1}$ is freely homotopic to
${e_1}$. Let us write $v=[\alpha_1,\alpha_{-1}] \cdots
[\alpha_{g_1},\alpha_{-g_1}]$.
We list the effect of the Dehn twists ${D}_{e_i}$ on these generators:
\[ \begin{array}{ll}
\mbox{$i \in \{ \pm 1,\ldots, \pm g_1, g_1+1\} $}&
\left\{ \begin{array}{l} {D}_{e_1}: \alpha_i \mapsto \alpha_i \\
{D}_{e_2}: \alpha_i \mapsto \alpha_i
\end{array} \right. \\[12pt]
\mbox{$i = -(g_1+1)$} &
\left\{ \begin{array}{l} {D}_{e_1}: \alpha_i \mapsto
\alpha_i v \alpha_{g_1+1}^{-1}\\
{D}_{e_2}: \alpha_i \mapsto
\alpha_{g_1+1}^{-1}\alpha_i
\end{array} \right. \\[12pt]
\mbox{$i \in \{ \pm (g_1+2),\ldots, \pm g \}$} &
\left\{ \begin{array}{l} {D}_{e_1}: \alpha_i \mapsto \alpha_{g_1+1} v^{-1}
\alpha_i v \alpha_{g_1+1}^{-1}\\
{D}_{e_2}: \alpha_i \mapsto \alpha_i
\end{array} \right.
\end{array} \]
Thus we get the following formulae for $D_{e_2}D_{e_1}^{-1}$:
\[ \begin{array}{ll}
|i| \leq g_1 & D_{e_2}D_{e_1}^{-1}(\alpha_i)=\alpha_i, \\[2pt]
i=g_1+1 & D_{e_2}D_{e_1}^{-1}(\alpha_i)=\alpha_i, \\[2pt]
i = -(g_1+1)& D_{e_2}D_{e_1}^{-1}(\alpha_i)
=\alpha_{g_1+1}^{-1}\alpha_i\alpha_{g_1+1}v^{-1}\alpha_i^{-1}\alpha_i
=[\alpha_{g_1+1},\alpha_i^{-1}]v^{-1}[v^{-1},\alpha_i^{-1}]\alpha_i,\\[2pt]
|i| \geq g_1+2 & D_{e_2}D_{e_1}^{-1}(\alpha_i)
=v\alpha_{g_1+1}^{-1}\alpha_i\alpha_{g_1+1}v^{-1}\alpha_i^{-1}\alpha_i
=v[\alpha_{g_1+1},\alpha_i^{-1}]v^{-1}[v^{-1},\alpha^{-1}]\alpha_i
\end{array} \]
and for the mth powers:
\[ \begin{array}{ll}
i \in \{ \pm 1,\ldots, \pm g_1, g_1+1\}&
(D_{e_2}D_{e_1}^{-1})^m(\alpha_i)=\alpha_i,\\[2pt]
i = -(g_1+1)& (D_{e_2}D_{e_1}^{-1})^m(\alpha_i)
=\alpha_{g_1+1}^{-m}\alpha_i (\alpha_{g_1+1}v^{-1})^m,\\[2pt]
i \in \{ \pm (g_1+2),\ldots, \pm g \}&
(D_{e_2}D_{e_1}^{-1})^m(\alpha_i)
=(v\alpha_{g_1+1}^{-1})^m \alpha_i (\alpha_{g_1+1}v^{-1})^m.
\end{array} \]
This proves that for a cut pair $\{e_1,e_2\}$ we have
$\rho(e_2-e_1) \in O^{(1)}$. It follows from this and \ref{bridges} that
$\rho(F^1) \subset O^{(1)}$, because
$F^1$ is generated by elements of the form $e_2-e_1$ for a cut pair
$\{e_1,e_2\}$ and the elements $b,~b \in B$.
To finish the proof of the inclusions ``$\supset$''
we have to show that $\rho(me_2-me_1)\in O^{(2),n}$ if
$n_2|m$ and $\rho(me_2-me_1)\in O^{(3),n}$ if $n|m$, or
$\frac{1}{2}n|m$ in case $2||m$. We have to show that the
divisibility conditions imply
$(D_{e_2}D_{e_1}^{-1})^m(\alpha_i)\alpha_i^{-1}\in \pi^{(3),n}$,
respectively $\pi^{(4),n}$. We will make computations in $\pi$
modulo $\pi^{(4)}$. In the case that $i\in\{\pm1,\ldots,\pm g_1, g_1+1\}$
there is nothing to prove. We have
$(\alpha_{g_1+1}v^{-1})^m\equiv
\alpha_{g_1+1}^mv^{-m}[v^{-1},\alpha_{g_1+1}]^{\frac{1}{2}m(m-1)}
$ mod $ \pi^{(4)}$, as one can prove by induction.
{}From equality 5 of Subsection \ref{group structure} we have
$$\alpha_{g_1+1}^{-m}\alpha_i\alpha_{g_1+1}^m \alpha_i^{-1}\equiv
[\alpha_{g_1+1},\alpha_i^{-1}]^m
[[\alpha_{g_1+1},\alpha_i^{-1}],\alpha_{g_1+1}]^{\frac{1}{2}m(m-1)}.$$
Thus for $i=-(g_1+1)$ we get
\begin{eqnarray*}
&&(D_{e_2}D_{e_1}^{-1})^m(\alpha_i)\alpha_i^{-1}\\
&\equiv&\alpha_{g_1+1}^{-m} \alpha_i (\alpha_{g_1+1}v^{-1})^m \alpha_i^{-1}\\
&\equiv&\alpha_{g_1+1}^{-m} \alpha_i
\alpha_{g_1+1}^mv^{-m} [v^{-1},\alpha_{g_1+1}]^{\frac{1}{2}m(m-1)}
\alpha_i^{-1}\\
&\equiv&{[\alpha_{g_1+1},\alpha_i^{-1}]}^{m}
[[\alpha_{g_1+1},\alpha_i^{-1}],\alpha_{g_1+1}]^{\frac{1}{2}m(m-1)}
v^{-m} [v, \alpha_i]^m [v,\alpha_{g_1+1}]^{-\frac{1}{2}m(m-1)}\\
&\equiv&
{[\alpha_{g_1+1},\alpha_i^{-1}]}^{m}
[[\alpha_{g_1+1},\alpha_i^{-1}],\alpha_{g_1+1}]^{\frac{1}{2}m(m-1)}
[\alpha_1,\alpha_{-1}]^m \cdots [\alpha_{g_1},\alpha_{-g_1}]^m
[[\alpha_1,\alpha_{-1}],\alpha_i]^m \cdots \\
&&
{[[\alpha_{g_1},\alpha_{-g_1}],\alpha_i]}^m
{[[\alpha_1,\alpha_{-1}],\alpha_{g_1+1}]}^{-\frac{1}{2}m(m-1)} \cdots
[[\alpha_{g_1},\alpha_{-g_1}],\alpha_{g_1+1}]^{-\frac{1}{2}m(m-1)}.
\end{eqnarray*}
In this product there are, apart from factors in $\pi^{(3)}$,
a number of terms of the form
$[x,y]^m$, $x,y\in \pi$. By the divisibility conditions
all these are in $\pi^{(4),n}$, since this is true in the
case of a free group $\langle x, y, z\rangle$ by Lemma \ref{berekening}.
Thus it is clear that the whole product lies in $\pi^{(3),n}$ for
$n_2|m$.
The other terms in the product are of the form
$[[x,y],z]^{\frac{1}{2}m(m-1)}$, where $x,y,z\in \pi$.
It follows from Lemma \ref{berekening}, by mapping $G=\langle x, y, z\rangle$
into $\pi$ that these terms are in $\pi^{(4),n}$ as soon as
$n_2| \frac{1}{2}m(m-1)$. This is equivalent to the condition
$n|m$ or $\frac{1}{2}n|m$ in case $2||n$.
For $i \geq g_1+2$ we have:
\begin{eqnarray*}
(D_{e_2}D_{e_1}^{-1})^m(\alpha_i)\alpha_i^{-1}&=&
(v\alpha_{g_1+1}^{-1})^m \alpha_i (\alpha_{g_1+1}v^{-1})^m
\alpha_i^{-1} \\
&\equiv&
v^m \alpha_{g_1+1}^{-m} [\alpha_{g_1+1}^{-1},v]^{\frac{1}{2}m(m-1)}\alpha_i
\alpha_{g_1+1}^mv^{-m} [v^{-1},\alpha_{g_1+1}]^{\frac{1}{2}m(m-1)}
\alpha_i^{-1}\\
&\equiv &
{[\alpha_{g_1+1},\alpha_i^{-1}]}^m
[[\alpha_{g_1+1},\alpha_i^{-1}],\alpha_{g_1+1}]^{\frac{1}{2}m(m-1)}
\end{eqnarray*}
The same argument works to show that our divisibility conditions
imply this product lies in either $\pi^{(3),n}$ or $\pi^{(4),n}$.
\section{Completion of proof}
In this section we will use the following argument.
Suppose we are given a morphism $f:S \ra S'$ of compact connected oriented
surfaces and a circle $c$ on $S$. Assume that $f$ maps a tubular neighbourhood
of $c$ isomorphically into $S'$, denote by $c'$ the image of $c$.
Then we have: $f_\ast \circ D_c =D_{c'} \circ f_\ast$.
Let $\Gamma$ be the graph of a stable curve and $(S,\{ c_i \})$
the smooth model with a set of circles as described in Subsection
\ref{results}.
Let $H$ be the set of circles which are not bridges and are not
involved in any cut system.
Let $B$ be the set of bridges of $\Gamma$ and let $\{ E_i|i=1,\ldots,s \}$
be the set of maximal cut systems.
Choose a numbering $E_i= \{ e_{i,0},\ldots,e_{i,f_i} \}$ for each $i$.
Remark that ${\rm Edges}(\Gamma)=H\cup \bigcup E_i\cup B$.
\subsection{The case $k=1$}
We want to prove the inclusion ``$\subset$'' for $k=1$.
Suppose $\sigma \in \oplus {\msy Z}e$ is a counterexample which involves a
minimal number of edges. This means that $\rho(\sigma)$ lies in $O^{(1),n}$
but not all coefficients of $\sigma$ are divisible by $n$; the number
of nonzero coefficients is minimal. We write $\sigma =\sum n_c c$.
We may subtract elements of $nF_0+F_1$ from $\sigma$:
we already know that $nF_0+F_1$ maps into $O^{(1),n}$.
Thus, by minimality, we may suppose that:
(\romannumeral1) none of the nonzero coefficients $n_c$ is divisible by $n$,
(\romannumeral2) none of the circles $c$ is a bridge, and
(\romannumeral3) of each cutsystem $E_i$ at most one element occurs in
$\sigma$. Modulo a homeomorphism the situation looks as follows:
\hfill \vspace{4 cm}
$$\hbox{\sl Fig.~4}$$
Take a point $p$ near $c$. It is clear that we can find a loop $\alpha$
which intersects $c$ exactly once and none of the other edges involved in
$\sigma$. But now we have a contradiction, because by Section \ref{not bridges}
we know: $\rho(\sigma)(\alpha)\alpha^{-1}= c^{n_c} \notin \pi^n$
if $n$ does not divide $n_c$.
Thus we have proven $\rho^{-1}(O^{(1),n})\subset nF^0+F^1$, equality follows
from sections \ref{not bridges} and \ref{cut systems}.
\subsection{The case $k=2$}\label{k is 2}
We want to prove the inclusion ``$\subset$'' for $k=2$.
Let $\sigma=\sum_{b \in B}m_b b
+\sum_{i,j}m_{i,j}(e_{i,j}-e_{i,0})
+\sum_i m_i e_{i,0} + \sum_{c \in H} n_c c$ be such that $\rho(\sigma ) \in
O^{(2),n}$. By the above we know that $n|m_i$ and $n|n_c, c\in H$. We have
to show that $n_2|m_{i,j}$ for all possible $i,j$.
Suppose this does not hold and suppose furthermore that $\sigma$ is
a minimal counterexample with respect to the number of edges involved.
Arguing as above we may assume that
$\sigma =\sum_{i,j}m_{i,j}(e_{i,j}-e_{i,0})$, where no (nonzero)
$m_{i,j}$ is divisible by $n_2$.
\begin{proposition} \label{nice cut system}
There is a maximal cut system which is such that if we cut $S$ along
every element of the cut system, one connected component contains all other
cut systems. \end{proposition}
\begin{proof} (Cf.\ \cite{Oda}, Lemma 5.3.)
Take any cut system $E$. If it does not have the required
property, that means that there is more than one connected component of $S
\setminus E$, say $S_{E,1}, \dots , S_{E,l}$, which contain maximal
cut systems. Suppose $S_{E,j}$ contains $a_j$ maximal cut systems and
suppose $a_1= \mbox{ min}_j \{ a_j \}$. Take a maximal cut system $F$ in
$S_{E,1}$. Clearly there is a connected component of $S\setminus F$ which
contains at least $\sum_{j=2}^la_j+1$ components, namely the one containing
$E$. This component contains more maximal cut systems then the one we started
with.
Continuing in this way, we arrive at our result.
\end{proof}
Let $E_1=\{e_{1,0}, \dots, e_{1,f_1} \}$ be a
maximal cut system such that one of the two components bounded by $e_{1,0}$ and
$e_{1,f_1}$, contains all other cut systems involved in $\sigma$.
We suppose these edges are numbered cyclically: one of the connected
components of $S\setminus \{e_{1,i},e_{1,i+1}\}$ contains all other
$e_{1,j}$ (see figure 6). If $E_1$ contains more than two edges,
we proceed as follows. Replace
the component of $S \setminus \{ e_{1,1},~e_{1,f_1} \}$
which does not contain $e_{1,0}$
by a cylinder to get an oriented surface $S'$.
There is a continuous map $f: S\to S'$ such that for all elements
$e_{1,j}$, except $e_{1,0}$, we have that $f(e_{1,j})$ is homotopic
to $f(e_{1,1})$. As explained above, we get that
$f_\ast \circ \rho(\sigma)=\rho(f_\ast\sigma)\circ f_\ast$.
Thus $\rho(\sigma)\in O^{(2),n}$
implies $\rho(f_\ast \sigma)\in O^{(2),n}$.
It is also
clear that $f_\ast(\sigma)= m_{1,1}(f(e_{1,1})-f(e_{1,0}))+
\sum_{i>1} m_{i,j}(f(e_{i,j})-f(e_{i,0}))$. We are reduced to the case
that $E_1=\{e_{1,0},e_{1,1}\}$ consists of two elements.
The situation now is as in \ref{cut systems} (figure 3) where
all the cut systems $E_i$, $i>1$, lie in the component $S_2$.
The generator $\alpha_{-(g_1+1)}$ may be chosen such that it does not
intersect the circles $e_{i,j}$, $i>1$. Thus we get from the formulae
of Section \ref{cut systems} that
\begin{eqnarray*}
\rho(\sigma)(\alpha_{-(g_1+1)})\cdot \alpha_{-(g_1+1)}^{-1}&
=&(D_{e_{1,1}}D_{e_{1,0}}^{-1})^{m_{1,1}}(\alpha_{-(g_1+1)})
\cdot \alpha_{-(g_1+1)}^{-1}\\
&=&\alpha_{g_1+1}^{-m_{1,1}}\alpha_{-(g_1+1)}(\alpha_{g_1+1}v^{-1})^{m_{1,1}}
\alpha_{-(g_1+1)}^{-1}.\end{eqnarray*}
We define $f:\pi\to G=\langle x,y,z\rangle$ by
$\alpha_1, \alpha_{-g} \mapsto x$
, $\alpha_{-1}, \alpha_g \mapsto y$, $\alpha_{g_1+1} \mapsto z$,
other generators $ \mapsto 1$. (The defining relation for
$\pi$ is indeed mapped to 1.) The expression above is mapped
to $z^{-m_{1,1}}\cdot(z\cdot [y,x])^{m_{1,1}}$.
Modulo $G^{(3)}$ this equals $[y,x]^{m_{1,1}}$, hence
$\rho(\sigma)\in O^{(3),n}$ implies $n_2|m_{1,1}$ (see Corollary
\ref{free comp}). In this way we prove the desired contradiction;
the inclusion ``$\subset$'' for $k=2$ follows.
\subsection{The case $k=3$}
We want to prove the inclusion ``$\subset$'' for $k=3$.
Let $\sigma=\sum_{b \in B}m_b b
+\sum_{i,j}m_{i,j}(e_{i,j}-e_{i,0})
+\sum_i m_i e_{i,0} + \sum_{c \in H} n_c c$ be such that
$\rho(\sigma) \in O^{(3),n}$. We have to show that
$n_2|m_{i,j}$ or $n|m_{i,j}$, for all $i,j$,
depending on $n$ being exactly divisible by $2$ or not;
and $n_6$ respectively $n_2$ divides $m_b$ depending on $b \in B_1$ or not.
Suppose this does not hold and suppose furthermore that $\sigma$ is
a minimal counterexample with respect to the number of edges involved.
Thus $\sigma =\sum_{b \in B}m_b b +\sum_{i,j}m_{i,j}(e_{i,j}-e_{i,0})$
and none of the nonzero $m_b$, $m_{i,j}$ satisfy the divisibility conditions.
Case 1. For some $b$ involved in $\sigma$ one of the connected
components, say $S'$, of $S\setminus \{b\}$ contains no other
edges involved in $\sigma$. The situation looks as follows.
\hfill \vspace{4.8 cm}
$$\hbox{\sl Fig.~5}$$
We take a basepoint $p$ in $S'$ and standard generators
$\alpha_{\pm i}$ which are loops in $S'$ for $i \leq g(S')$
and which are not loops in $S'$ for $i > g(S')$.
We may choose $\alpha_{g(S')+1}$ such that it does not intersect any
circles involved in $\sigma$ but $b$. To see this let $S''$ denote
the connected component of $S\setminus \{c\in \sigma, c\not = b\}$
containing $S'$. If $g(S'')>g(S')$, then there is a `hole' between
$b$ and the boundary of $S''$, which we can take to be hole number
$g(S')+1$ and take $\alpha_{g(S')+1}$ accordingly. This is the
case for example if $S''=S$ (i.e., if $\sigma = m_b b$)
or if $S''$ is bounded by another bridge $b'$. If $g(S'')=g(S')$,
then $S''$ is bounded by a cut pair $\{e_1, e_2\}$ (belonging to
some maximal cut system $E_i$), the part $S''\setminus S'$
looks like a pair of pants. In this case we can choose $\alpha_{g(S')+1}$
to go around a pants leg of $S''\setminus S'$, for example freely homotopic
to $e_2$ as in Section \ref{cut systems}.
It is now clear that $\rho(\sigma)(\alpha_{g(S')+1})=
D_b^{m_b}(\alpha_{g(S')+1})$ and by Section \ref{bridges} we get
$$ \xi:= \rho(\sigma)(\alpha_{g(S')+1})\cdot \alpha_{g(S')+1}^{-1}=
v^{-m_b}\alpha_{g(S')+1}v^{m_b}\alpha_{g(S')+1}^{-1}$$
with $v=[\alpha_1,\alpha_{-1}]\cdot\ldots\cdot
[\alpha_{g(S')},\alpha_{-g(S')}]$.
Thus $\rho(\sigma)\in O^{(3),n}$ implies $\xi\in \pi^{(4),n}$.
If $b$ bounds a curve of genus $1$, say $g(S')=1$ (the case $g-g(S')=1$
is similar), then $v=[\alpha_1,\alpha_{-1}]$. We define $f :\pi \to
G=\langle x,y,z\rangle$ by $\alpha_1,\alpha_{-2}\mapsto x$,
$\alpha_{-1},\alpha_{2}\mapsto y$ and other generators $\mapsto 1$.
We see that $f(\xi)= [x,y]^{-m_b}y[x,y]^{m_b}y^{-1}\equiv [[x,y],y]^{-m_b}$
modulo $G^{(4)}$. Thus $\xi\in \pi^{(4),n}$ implies $n_6 | m_b$ by
Lemma \ref{berekening}.
If $b$ does not bound a curve of genus 1 (i.e., $g(S')\geq 2$ and
$g-g(S')\geq 2$) we define $g:\pi \to G=\langle x,y,z\rangle$ by
$\alpha_1, \alpha_{-g} \mapsto x$, $\alpha_{-1}, \alpha_g \mapsto y$
, $\alpha_{g(S')+1} \mapsto z^{-1}$ and other generators $\mapsto 1$.
We get $g(\xi)=[x,y]^{-m_b}z^{-1}[x,y]^{m_b}z\equiv [[x,y],z]^{m_b}$.
Application of Lemma \ref{berekening} gives $n_2|m_b$ as desired.
Case 2: case 1 does not occur. This means that every
connected component of $S$ cut out by one bridge involved in $\sigma$ contains
a maximal cut system. Choose a maximal cut system as in Proposition \ref{nice
cut system}. Let $E_1=\{e_{1,0}, \dots, e_{1,f_1} \}$
be this cut system. Suppose the edges are numbered cyclically and
suppose the connected component cut out by $e_{1,0}$ and $e_{1,1}$
which does not contain any edge of $E_1$ contains all other
maximal cut systems. Call this component $S'$. By the choice of $E_1$ and
the assumption, we get that $S \setminus S'$ does not contain any bridges
involved in $\sigma$. The situation thus looks as shown below:
\vspace{6.5 cm}
$$\hbox{\sl Fig.~6}$$
Arguing as in the proof for $k=2$,
we arrive at a curve with a cut pair $e_{1,0}$ and $e_{1,1}$ which is a maximal
cut system with the property that one of the two components cut out
by the cut pair contains all other edges involved in the counter example
$\sigma$. Considering $f : \pi \to G$ as in
Section \ref{k is 2}, we get, as in case 1, that the element
$z^{-m_{1,1}}\cdot (z\cdot [y,x])^{m_{1,1}}$ lies in $G^{(4),n}$.
Note that
$$z^{-m_{1,1}}\cdot (z\cdot [y,x])^{m_{1,1}}\equiv
[y,x]^{m_{1,1}}[[y,x],z]^{\frac{1}{2}m_{1,1}(m_{1,1}-1)}.$$
Thus we get our divisibility condition on $m_{1,1}$ by Lemma \ref{berekening}.
\section{Computations for a free group on three generators}
\label{group structure}
\label{berekeningen}
\begin{lemmatje} Let $G$ be a free group on three generators, $G=\langle
x,y,z\rangle$. Then
$$ G/G^{(4)} \cong \{(i_1,\ldots,i_{14})\in {\msy Z}^{14} \}$$
where multiplication on the right-hand side is given as follows:
$$(i_1,i_2,i_3,i_4,i_5,i_6,i_7,i_8,i_9,i_{10},i_{11},i_{12},i_{13},i_{14})
\cdot $$
$$(j_1,j_2,j_3,j_4,j_5,j_6,j_7,j_8,j_9,j_{10},j_{11},j_{12},j_{13},j_{14})=$$
\[ \left( \begin{array}{c}
i_1 +j_1 \\
i_2 +j_2 \\
i_3 +j_3 \\
i_4+j_4 -i_2 j_1 \\
i_5+j_5-i_3 j_2 \\
i_6+j_6-i_3j_1 \\
i_7+j_7 +i_4j_1- \frac{1}{2}(j_1-1)j_1i_2 \\
i_8+j_8 +i_4j_2- \frac{1}{2}(i_2-1)i_2j_1 -i_2j_1j_2 \\
i_9+j_9 +i_4j_3+i_6j_2 -i_2i_3j_1-i_2j_1j_3-i_3j_1j_2 \\
i_{10}+j_{10}+i_5j_1+i_6j_2 -i_3j_1j_2 \\
i_{11}+j_{11}+i_5j_2 -\frac{1}{2}(j_2-1)j_2i_3 \\
i_{12}+j_{12}+i_5j_3-\frac{1}{2}(i_3-1)i_3j_2-i_3j_2j_3 \\
i_{13}+j_{13}+i_6j_1-\frac{1}{2}(j_1-1)j_1i_3 \\
i_{14}+j_{14}+i_6j_3-\frac{1}{2}(i_3-1)i_3j_1-i_3j_1j_3
\end{array} \right) \]
\end{lemmatje}
\begin{proof} We have the exact sequences
$$1 \rightarrow \Gk/\Gkeen \rightarrow G/ \Gkeen
\rightarrow G/ \Gk \rightarrow 1 ,$$
where $\Gk/\Gkeen$ is a finitely generated free abelian group
(\cite[Th.5.12]{Magnus}) of rank $N_i:=\frac{1}{i} \sum_{d|i} \mu(d) 3^{i/d}$
(\cite[Th.5.11]{Magnus}), where $\mu$ is the M\"obius-function.
Explicitly, we have, with $r:=[x,y]$, $s:=[y,z]$ and $t:=[x,z]$:
\[ \begin{array}{l}
G^{(1)}/G^{(2)} = \langle x,y,z\rangle_{ab} ~,\\
G^{(2)}/G^{(3)} = \langle r,s,t\rangle_{ab}~, \\
G^{(3)}/G^{(4)} =
\langle [r,x],[r,y],[r,z],[s,x],[s,y],[s,z],[t,x],[t,z]\rangle_{ab}~,
\end{array} \]
by the Jacobi-relation $[t,y] \equiv [r,z][s,x]$ mod $G^{(4)}$.
Now recall that if $a \in \Gk$ and $b \in G^{(l)}$ then $ab=ba[a,b]$
with $[a,b] \in G^{(k+l)}$, so that
\begin{eqnarray} ab\equiv ba \mbox{ mod } G^{(k+l)}. \end{eqnarray}
Furthermore we have the following identities: (\cite[Th.5.1]{Magnus})
\begin{eqnarray}
\ [a,b] & = & [b,a]^{-1},\\
\ [a,bc] & = & [a,c]~[a,b]~[[a,b],c], \\
\ [ab,c] & = & [a,c]~[[a,c],b]~[b,c].
\end{eqnarray}
It is clear now that any element $a$ of $G/G^{(4)}$ can be written in the form
$a=x^{i_{1}}\cdots [t,z]^{i_{14}}$ in a unique manner. Note
that the last eleven factors commute by (1).
We define a map $\phi:G/G^{(4)} \rightarrow \{(i_1,\ldots,i_{14}) \in {\msy
Z}^{14} \}$
by
$$x^{i_1}y^{i_2}z^{i_3}r^{i_4}s^{i_5}t^{i_6}[r,x]^{i_7}[r,y]^{i_8}[r,z]^{i_9}
[s,x]^{i_{10}}[s,y]^{i_{11}}[s,z]^{i_{12}}[t,x]^{i_{13}}[t,z]^{i_{14}}
\mapsto (i_1,\ldots,i_{14}) .$$
Before we start the computation, notice that modulo $G^{(4)}$ we have
\begin{eqnarray}
[a^i,b^j]\equiv
[a,b]^{ij}[[a,b],a]^{\frac{1}{2}ij(i-1)}[[a,b],b]^{\frac{1}{2}ij(j-1)},
\end{eqnarray}
as one proves easily by induction. We set $(n,m):=\frac{1}{2}(n-1)nm$.
The following identities hold in the group $G/G^{(4)}$:
\begin{eqnarray*}
&&x^{i_1}y^{i_2}z^{i_3}x^{j_1}y^{j_2}z^{j_3}\\
&=&
x^{i_1+j_1}x^{-j_1}y^{i_2}z^{i_3}x^{j_1}z^{-i_3}y^{-i_2}y^{i_2}z^{i_3}
y^{j_2}z^{j_3}\\
&=&
x^{i_1+j_1}[x^{j_1},z^{-i_3}y^{-i_2}]y^{i_2+j_2}y^{-j_2}z^{i_3}
y^{j_2}z^{-i_3}z^{i_3+j_3}\\
&=&
x^{i_1+j_1}[x^{j_1},y^{-i_2}][x^{j_1},z^{-i_3}][[x^{j_1},z^{-i_3}],y^{-i_2}]
y^{i_2+j_2}[y^{j_2},z^{-i_3}]z^{i_3+j_3}\\
&=&
x^{i_1+j_1}r^{-i_2j_1}[r,x]^{(j_1,-i_2)}[r,y]^{(-i_2,j_1)}t^{-i_3j_1}
[t,x]^{(j_1,-i_3)}[t,z]^{(-i_3,j_1)} \\
&&\ \ \ y^{i_2+j_2}s^{-i_3j_2}[s,y]^{(j_2,-i_3)}
[s,z]^{(-i_3,j_2)}z^{i_3+j_3}[t,y]^{i_2i_3j_1}\\
&=&
x^{i_1+j_1}y^{i_2+j_2}z^{i_3+j_3}r^{-i_2j_1}s^{-i_3j_2}t^{-i_3j_1}
[r,y]^{-i_2j_1(i_2+j_2)}[t,y]^{-i_3j_1(i_2+j_2)}[r,z]^{-i_2j_1(i_3+j_3)} \\
&&\ \ \ [t,z]^{-i_3j_1(i_3+j_3)}[s,z]^{-i_3j_2(i_3+j_3)}
[r,x]^{(j_1,-i_2)}[r,y]^{(-i_2,j_1)}[t,x]^{(j_1,-i_3)}[t,z]^{(-i_3,j_1)} \\
&&\ \ \
[s,y]^{(j_2,-i_3)}[s,z]^{(-i_3,j_2)}[r,z]^{j_1i_2i_3}[s,x]^{j_1i_2i_3}\\
&=&x^{i_1+j_1}y^{i_2+j_2}z^{i_3+j_3}r^{-i_2j_1}s^{-i_3j_2}t^{-i_3j_1}
[r,x]^{(j_1,-i_2)}[r,y]^{-(i_2,j_1)-i_2j_1j_2}[r,z]^{-i_2j_1j_3-i_3j_1j_2
-i_2i_3j_1} \\
&&\ \ \ [s,x]^{-i_3j_1j_2}[s,y]^{(j_2,-i_3)}[s,z]^{-(i_3,j_2)-i_3j_2j_3}
[t,x]^{(j_1,-i_3)}[t,z]^{-(i_3,j_1)-i_3j_1j_3}.
\end{eqnarray*}
Also
\begin{eqnarray*}
r^{i_4}s^{i_5}t^{i_6}x^{j_1}y^{j_2}z^{j_3}r^{j_4}s^{j_5}t^{j_6}&=&
x^{j_1}y^{j_2}z^{j_3}r^{i_4}s^{i_5}t^{i_6}[r^{i_4}s^{i_5}t^{i_6},
x^{j_1}y^{j_2}z^{j_3}]r^{j_4}s^{j_5}t^{j_6}\\
&=&
x^{j_1}y^{j_2}z^{j_3}r^{i_4+j_4}s^{i_5+j_5}t^{i_6+j_6}[r,x]^{i_4j_1}
[r,y]^{i_4j_2}[r,z]^{i_4j_3+i_6j_2}\\
&&\ \ \
{[s,x]}^{i_5j_1+i_6j_2}[s,y]^{i_5j_2}
[s,z]^{i_5j_3}[t,x]^{i_6j_1}[t,z]^{i_6j_3}.
\end{eqnarray*}
Combining these proves the lemma. \end{proof}
\begin{lemmatje} $(i_1,\cdots,i_{14})^n=$
\[ \left( \begin{array}{c}
ni_1 \\ ni_2 \\ ni_3 \\ni_4-\frac{1}{2}(n-1)ni_2i_1\\
ni_5-\frac{1}{2}(n-1)ni_3i_2\\ ni_6-\frac{1}{2}(n-1)ni_3i_1\\
ni_7+\frac{1}{2}(n-1)ni_4i_1-\frac{1}{12}(n-1)n(2n-1)i_1^2i_2
+\frac{1}{4}(n-1)ni_1i_2 \\
ni_8 +\frac{1}{2}(n-1)ni_4i_2-\frac{1}{6}(n-1)n(2n-1)i_1i_2^2
+\frac{1}{4}(n-1)ni_1i_2-\frac{1}{4}(n-1)ni_1i_2^2\\
ni_9 +\frac{1}{2}(n-1)ni_4i_3 +\frac{1}{2}(n-1)ni_6i_2
-\frac{1}{3}(n-1)n(2n-1)i_3i_1i_2 -\frac{1}{2}(n-1)ni_1i_2i_3\\
ni_{10} +\frac{1}{2}(n-1)ni_5i_1 +\frac{1}{2}(n-1)ni_6i_2
-\frac{1}{6}(n-1)n(2n-1)i_3i_1i_2 \\
ni_{11}+\frac{1}{2}(n-1)ni_5i_2-\frac{1}{12}(n-1)n(2n-1)i_2^2i_3
+\frac{1}{4}(n-1)ni_2i_3 \\
ni_{12} +\frac{1}{2}(n-1)ni_5i_3-\frac{1}{6}(n-1)n(2n-1)i_2i_3^2
+\frac{1}{4}(n-1)ni_3i_2-\frac{1}{4}(n-1)ni_2i_3^2\\
ni_{13}+\frac{1}{2}(n-1)ni_6i_1-\frac{1}{12}(n-1)n(2n-1)i_1^2i_3
+\frac{1}{4}(n-1)ni_1i_3 \\
ni_{14} +\frac{1}{2}(n-1)ni_6i_3-\frac{1}{6}(n-1)n(2n-1)i_1i_3^2
+\frac{1}{4}(n-1)ni_3i_1-\frac{1}{4}(n-1)ni_1i_3^2\\
\end{array} \right) \] \end{lemmatje}
\begin{proof} By induction on $n$. \end{proof}
\begin{lemmatje} \label{berekening} Notations as above.
\[ \begin{array}{rcl}
G^{(4),n}/G^{(4)} & \cong & \{(i_1,\ldots,i_{14})|
i_1,i_2,i_3 \in n{\msy Z},~
i_4,i_5,i_6 \in n_2{\msy Z}, \\
& & i_7,\ldots,i_{14} \in
n_6{\msy Z},~i_9 + i_{10} \in n_2 {\msy Z} \}. \end{array} \]
(Recall that $n_i:= n / {\rm gcd} (n,i).) $\end{lemmatje}
\begin{proof}
The inclusion ``$\subset$'' follows at once from Lemma 4.2 and the following
two
observations:
$$-\frac{1}{12}(n-1)n(2n-1)i_1^2i_2+\frac{1}{4}(n-1)ni_1i_2=$$
$$\frac{1}{12}(n-1)ni_1i_2[-(2n-1)i_1+3]$$
is an element of $n_6 \msy Z$ (either $i_1$ is even or the two terms between
brackets have the same parity).
The same holds for
$$-\frac{1}{6}(n-1)n(2n-1)i_1i_2^2
+\frac{1}{4}(n-1)ni_1i_2-\frac{1}{4}(n-1)ni_1i_2^2=$$
$$\frac{1}{12}(n-1)ni_1i_2[-(4n+1)i_2+3].$$
The other inclusion follows from a direct computation:
\[ \begin{array}{rl}
a:= & (1,0,\ldots,0)^n(-1,1,0,\ldots,0)^n(0,-1,0,\ldots,0)^n \\
=&
(0,0,0,\frac{1}{2}(n-1)n,0,0,-\frac{1}{6}(n-1)n(n+1),-\frac{1}{6}(n-1)n(n+1),
0,\ldots,0),\\
b:= & (-1,0,\ldots,0)^n (1,1,0\ldots,0)^n (0,-1,\ldots,0)^n\\
=& (0,0,0,-\frac{1}{2}(n-1)n,0,0,-\frac{1}{6}(n-2)(n-1)n,
\frac{1}{6}(n-1)n(n+1),0,\ldots,0),\\
c:= & (1,0\ldots,0)^n(-1,-1,0,\ldots,0)^n(0,1,0,\ldots.,0)^n \\
=&
(0,0,0,-\frac{1}{2}(n-1)n,0,0,\frac{1}{6}(n-1)n(n+1),-\frac{1}{6}(n-2)(n-1)n,
0,\ldots,0).
\end{array} \]
So we have:
$$ab=(0,0,0,0,0,0,-\frac{1}{6}(n-1)n(2n-1),0,\ldots,0),$$
$$ac=(0,0,0,0,0,0,0,-\frac{1}{6}(n-1)n(2n-1),0,\ldots,0).$$
The same computation holds for the eleventh till the fourteenth coefficient,
so it is possible to get $n_6$ as one of the coefficients
$i_7,i_8,i_{11},..,i_{14}$ and all the other coefficients zero.
Thus, looking again at $a$,
we see that it is possible to get $n_2$ as the coefficient
$i_4$ and all the other coefficients zero. By symmetry the same holds for
the coefficients $i_5,i_6$.\\
Furthermore, if we put
$$d:= (0,0,-1,0,\ldots,0)^n(0,0,1,1,0,\ldots,0)^n(0,0,0,-1,0,\ldots.,)^n,$$
$$e:= (-1,0,\ldots,0)^n(1,0,0,0,1,0,\ldots,0)^n(0,0,0,0,-1,0,\ldots.,)^n,$$
then we have:
\[ \begin{array}{l}
d=(0,0,0,0,0,0,0,0,\frac{1}{2}(n-1)n,0,\ldots,0), \\
e=(0,0,0,0,0,0,0,0,0,\frac{1}{2}(n-1)n,0,\ldots,0).
\end{array} \]
Thus, we can get $n_2$ in the
ninth and tenth coefficient, independently.
The rest now follows from what we have proven already
and the computation of
$$(-1,0,\ldots,0)^n(0,-1,0,\ldots,0)^n(0,0,-1,0,\ldots,0)^n(1,1,1,0,\ldots,0)^n.$$
\end{proof}
\begin{cortje} \label{free comp} Notations as above. \hfill \break
1. $(G^{(2)} \cap G^{(3),n})/G^{(3)} \cong n_2G^{(2)}/G^{(3)}.$ \hfill\break
2. $(i_7,\ldots,i_{14}) \in (G^{(3)} \cap G^{(4),n})/G^{(4)} $ if and only if
$i_7,\ldots,i_{14} \in
n_6 {\msy Z},~i_9 +i_{10} \in n_2 {\msy Z}.$
\end{cortje}
|
1996-03-05T06:16:17 | 9501 | alg-geom/9501007 | en | https://arxiv.org/abs/alg-geom/9501007 | [
"alg-geom",
"math.AG"
] | alg-geom/9501007 | Gerd Dethloff | Gerd Dethloff and Mikhail Zaidenberg | Plane Curves with Hyperbolic and C-hyperbolic Complements | Final version, published in 1996, subsuming version 1 of this
preprint and another work by the same authors on "Examples of Plane Curves
with Hyperbolic and C-hyperbolic Complements" | Ann. Scuola Norm. Sup. Pisa 23, 749-778 (1996) | null | null | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ | The general problem which initiated this work is:
What are the quasiprojective varieties which can be uniformized by means of
bounded domains in $\cz^n$ ?
Such a variety should be, in particular, C--hyperbolic, i.e. it should have a
Carath\'{e}odory hyperbolic covering. We study here the plane projective curves
whose complements are C--hyperbolic. For instance, we show that most of the
curves whose duals are nodal or, more generally, immersed curves, belong to
this class. We also give explicit examples of irreducible such curves of any
even degree d greater or equal 6.
| [
{
"version": "v1",
"created": "Tue, 17 Jan 1995 11:49:50 GMT"
},
{
"version": "v2",
"created": "Sat, 29 Nov 2014 14:18:27 GMT"
}
] | 2014-12-02T00:00:00 | [
[
"Dethloff",
"Gerd",
""
],
[
"Zaidenberg",
"Mikhail",
""
]
] | alg-geom | \section{Introduction}
\noindent {\bf 1.1.} A complex space $X$ is called {\it C--hyperbolic} if it has
a (non--ramified) covering $\tilde{X}$ which is {\it Carath\'{e}odory hyperbolic},
i.e.
the points of $\tilde{X}$ can be separated by bounded holomorphic functions [Ko1, pp.
129--130] (see also [LiZa, 1.3]). In this paper we study C--hyperbolicity of
the complements of plane projective curves. In particular, we are interesting
in what the minimal degree of a plane curve with C--hyperbolic complement is.
It is well known that any C--hyperbolic space is Kobayashi hyperbolic [Ko1, p.
130]. The opposite property to Carath\'{e}odory hyperbolicity is {\it
liouvilleness}.
A complex space $X$ is called {\it Liouville} if it has no non-constant
bounded holomorphic functions.
For example, any quasi--projective variety $X$ is Liouville, and by the theorem
of V. Lin its
liouvilleness is preserved by passing to a nilpotent covering over $X$,
i.e. a Galois covering with nilpotent group of deck transformations
[Li, Theorem B] (see also [LiZa], Theorem 3 at p.119).
Thus, if $X$ is a
quasi--projective variety whose Poincar\'{e} group $\pi_1(X)$ is (almost)
nilpotent, then any covering over $X$ is Liouville and therefore
$X$ can not be C--hyperbolic.
In particular, this is the case for $X= I \!\! P^2 \setminus C$, where $C$ is a (not
necessarily irreducible) nodal curve, i.e. a
plane curve with normal crossing singularities only.
Indeed, in this case by the Deligne-Fulton theorem [Del, Fu]
the fundamental group $\pi_1 (X)$ is abelian, and thus by Lin's Theorem any
covering over $X= I \!\! P^2 \setminus C$ is a Liouville one.
The fundamental group $\pi_1 ( I \!\! P^2 \setminus C)$ for an irreducible plane curve $C$
of degree $d$, which is not necessarily nodal, is known to be abelian in a
number of other cases, and hence to be isomorphic to $ Z \!\!\! Z / d Z \!\!\! Z$ (see e.g. the
survey article [Lib] and the references therein). For instance, this is so for
any rational or elliptic Pl\"ucker curve except those of even degree with the
maximal number of cusps, and therefore also for the curves that can be
specialized to such ones [Zar, pp. 267, 327-330] (cf. [DL], [Kan]). This is
true as well for any irreducible curve of degree $d \le 4$ with the only
exception of the three--cuspidal quartic; in the latter case $\pi_1 ( I \!\! P^2 \setminus
C)$ is a finite non-abelian metacyclic group of order $12$ [Zar, pp. 135, 145],
and so it is almost abelian. Therefore, in all these cases any covering over
$ I \!\! P^2 \setminus C$ is a Liouville one. \\
[1ex]
\noindent {\bf 1.2.} At the same time, the complement of a nodal plane curve
can be Kobayashi complete hyperbolic and hyperbolically embedded
into $ I \!\! P^2$. The well known example is the
complement of five lines in $ I \!\! P^2$ in general position [Gr3; KiKo, Corollary 3
in section 4]; for further examples of reducible curves see e.g.
[DSW1,2] and the literature therein.
There exist even the irreducible smooth quintics with these
properties [Za3].
Moreover, Y.-T. Siu and S.-K. Yeung [SY] have announced recently a proof of a
long standing conjecture that generic (in Zariski sense) smooth curve in
$ I \!\! P^2$ of degree $d$ large enough ($d \ge 1,200,000$) has the complement which
is Kobayashi complete hyperbolic and hyperbolically embedded into $ I \!\! P^2$
(while all its coverings are Liouville).
This shows that for the complement of a curve in $ I \!\! P^2$ the property to be
C--hyperbolic is much stronger than those of Kobayashi hyperbolicity, and it
can occure only for the curves with singularities worse than the ordinary
double points.\\[1ex]
\noindent {\bf 1.3.} However, plane curves with C--hyperbolic complements do
exist. The simplest example is
a reducible quintic $C_5$ with the ordinary triple points as
singularities at worst. Namely, $C_5$ is the union of five lines which is
given in homogeneous coordinates $(x_0:x_1:x_2)$ in $ I \!\! P^2$ by the equation
$$x_0 x_1 x_2 (x_0 -x_1 )(x_0 -x_2) =0\,\,$$
\begin{center}
\begin{picture}(300,90)
\thicklines
\put(30,85){\line(0,-1){85}}
\put(20,85){\line(1,-1){85}}
\put(25,85){\line(1,-2){45}}
\put(20,10){\line(1,0){95}}
\put(20,47){\line(2,-1){90}}
\put(50,-15){$C_5$}
\put(200,85){\line(0,-1){85}}
\put(190,85){\line(1,-1){85}}
\put(195,85){\line(1,-2){45}}
\put(190,10){\line(1,0){95}}
\put(190,47){\line(2,-1){90}}
\put(190,0){\line(1,1){55}}
\put(218,-15){$C_6$}
\end{picture}
\end{center}
\begin{center}
Figure 1
\end{center}
\noindent Indeed, $ I \!\! P^2 \setminus C_5$ is biholomorphic to $( I \!\!\!\! C^{**})^2$, where
$ I \!\!\!\! C^{**} = I \!\! P^1 \setminus \{3\,{\rm points}\}$, and thus its universal covering is
the bidisk $\Delta^2$ (hereafter $\Delta=\{z\in I \!\!\!\! C\,|\,|z| < 1\}$ denotes the unit
disc).
Slightly modifying the previous example, consider further the reducible sextic
$C_6 \subset I \!\! P^2$ which is the line arrangement given by the equation $$x_0 x_1
x_2 (x_0 -x_1 )(x_0 -x_2)(x_1 - x_2) =0\,\,.$$
\noindent It is known [Kal] that the universal covering of the complement $M_2 :=
I \!\! P^2 \setminus C_6$ is biholomorphic to the Teichm\"uller space $T_{0,\, 5}$ of the
Riemann sphere with five punctures. Furthermore, via the Bers embedding $T_{0,
\,5} \hookrightarrow I \!\!\!\! C^2$ it is biholomorphic to a bounded Bergman domain of
holomorphy in $ I \!\!\!\! C^2$, which is contractible and Kobayashi complete
hyperbolic. The automorphism group of $T_{0,\,5}$ is discrete and isomorphic to
the mapping class group, or modular group, ${\rm Mod}(0,\,5)$ [Ro].
Note that $5$ is the minimal degree of a plane curve whose complement is
C--hyperbolic. Indeed, the complement of a quartic curve is not even Kobayashi
hyperbolic [Gr2, section 6]. While there do exist irreducible plane sextics
with C--hyperbolic complements (see Proposition 4.5 below), there does not
exist such an irreducible plane quintic (see (4.8)-(4.10)), and so the minimal
degree of an irreducible plane curve with C--hyperbolic complement is $6$
(Proposition 4.10). \\[1ex]
\noindent {\bf 1.4.} To obtain examples of irreducible plane curves whose
complements are C--hyperbolic we can apply the method that was used
by M. Green [Gr1] (see also [CaGr, GP]) for constructing curves with hyperbolic
complements.
In this paper we study systematically the class of curves with C--hyperbolic
complements which can be obtained by this method. Let us describe briefly its
main idea.
Let $S^n X$ denote the n--th symmetric power of a variety $X$ and $R_n \subset S^n
X$ be its discriminant variety, i.e. the ramification locus of the branched
Galois covering $s_n \,:\, X^n \to S^n X$.
For a plane curve $C \subset I \!\! P^2$ there is a natural embedding $\rho_C \,:\, I \!\! P^2
\hookrightarrow S^n C^*_{norm}$, where $C^* \subset I \!\! P^{2*}$ is the dual curve, $n
= {\rm deg}\, C^*$ and $C^*_{norm}$ is the normalization of $C^*$. It may
happen that this gives an embedding of the complement $ I \!\! P^2 \setminus C$ into the
n--th configuration space $S^n C^* \setminus R_n$, and that either this configuration
space, or some subspace of it containing the image $\rho_C ( I \!\! P^2 \setminus C )$ has
nice hyperbolic properties.
Here we give an example.
Suppose that the dual curve $C^* \subset I \!\! P^{2*}$ is smooth and
of degree $n \ge 4$. For $z = (a:b:c) \in I \!\! P^2$ denote by $\rho_C (z)$ the
non--ordered set of $n$ points of intersection $l_z \cap C^*$, where $l_z$ is
the dual line $ax_0^* + bx_1^* + cx_2^* = 0$ in $ I \!\! P^{2*}$; here an
intersection point of multiplicity $m$ is repeated $m$ times. In this way
we obtain a morphism $\rho_C : I \!\! P^2 \to S^nC^*$ into $n$--th symmetric
power of $C^*$, which is a smooth variety (see e.g. [Zar, p.253] or [Na,
(5.2.15)]).
The ramified covering $s_n : {C^*}^n \to S^nC^*$ has the ramification
divisors $D_n := \bigcup\limits_{1\le i< j\le n} D_{ij} \subset {C^*}^n$ resp. $R_n
= s_n
(D_n) \subset S^nC^*$, where $D_{ij} := \{ x = (x_1,\dots,x_n)
\in {C^*}^n\,|\,x_i=x_j\}$ is a diagonal hypersurface. Following Zariski [Zar,
p.266] we call $R_n$ {\it the discriminant hypersurface}. Since $C^*$ is
smooth,
the preimage $\rho_C^{-1} (R_n)$ coincides with $C$, and so we have the
commutative diagram
\begin{center}
\begin{picture}(1000,60)
\thicklines
\put(100,5){$ I \!\! P^2 \setminus C = X$}
\put(155,45){$Y$}
\put(157,38){\vector(0,-1){20}}
\put(130,25){${\tilde s}_n$}
\put(200,58){${\tilde \rho}_C$}
\put(200,20){$\rho_C$}
\put(200,45){$\hookrightarrow$}
\put(200,5){$\hookrightarrow$}
\put(245,5){$S^n C^* \setminus R_n \,\,\,\,$}
\put(250,45){${C^*}^n \setminus D_n$}
\put(275,38){\vector(0,-1){20}}
\put(255,25){$s_n$}
\put(400,25){(1)}
\put(320,45){$\hookrightarrow$}
\put(350,45){${C^*}^n$}
\end{picture}
\end{center}
\noindent
where ${\tilde s}_n \,:\,Y \to X$ is the induced covering. The genus $g(C^*)
\ge 3$, therefore ${C^*}^n$ has the polydisc
$\Delta ^n$ as the universal covering. Passing to the induced covering $Z \to Y$ we
can extend (1) to the diagram
\begin{center}
\begin{picture}(1000,80)
\thicklines
\put(155,75){$Z$}
\put(158,69){\vector(0,-1){18}}
\put(155,40){$Y$}
\put(158,36){\vector(0,-1){18}}
\put(100,5){$ I \!\! P^2 \setminus C = X$}
\put(133,23){${\tilde s}_n$}
\put(200,75){$\hookrightarrow$}
\put(200,40){$\hookrightarrow$}
\put(200,5){$\hookrightarrow$}
\put(200,55){${\tilde \rho}_C$}
\put(200,20){$\rho_C$}
\put(253,75){$\Delta^n$}
\put(258,69){\vector(0,-1){18}}
\put(252,40){${C^*}^n$}
\put(258,36){\vector(0,-1){18}}
\put(247,5){$S^n C^*$}
\put(264,25){$s_n$}
\put(400,40){(2)}
\end{picture}
\end{center}
\noindent Being a submanifold of the polydisc, $Z$ is Carath\'eodory hyperbolic,
and so $X$ is C--hyperbolic. Therefore, we have proved the following
assertion.\\[1ex]
\noindent {\bf 1.5. Proposition.} {\it Let $C \subset I \!\! P^2$ be an irreducible curve
whose dual curve $C^*$ is smooth and of degree at least $4$. Then $ I \!\! P^2 \setminus
C$ is C--hyperbolic.}\\[1ex]
\noindent {\bf 1.6.} Note that, furthermore, $ I \!\! P^2 \setminus C$ is Kobayashi complete
hyperbolic and hyperbolically embedded
into $ I \!\! P^2$. The latter is also true under the weaker assumptions that (a)
the geometric genus $G$ of $C$ is at least two; (b) each tangent line to $C^*$
intersects with $C^*$ in at least two points, and (c) the following inequality
is fulfilled: $2n < d$, where $d = {\rm deg}\,C$ and $n = {\rm deg}\,C^*$, or,
what is equivalent,
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sum \limits_{i=1}^l (m^*_i - 1) < 2g-2 \,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)$$
where $m^*_1,\dots,m^*_l$ are the multiplicities of the singular branches of
$C^*$ [Gr1, CaGr, GP]. Moreover, under these assumptions a stronger conclusion
is valid. Namely, there exists a continuous hermitian metric on $ I \!\! P^2 \setminus C$
whose holomorphic sectional curvature is bounded from above by a negative
constant and which dominates some positive multiple of the Fubini-Study metric
on $ I \!\! P^2$ [GP]. \\
[1ex]
\noindent {\bf 1.7.} It is clear that a subspace of a C--hyperbolic space is also
C--hyperbolic. In particular, if $D \subset I \!\! P^n$ is a hypersurface whose
complement $ I \!\! P^n \setminus D$ is C--hyperbolic, then any plane section of $D$ is a
plane curve with C--hyperbolic complement. In this way, considering curve
complements, one might at least obtain necessary conditions for $ I \!\! P^n \setminus D$
to be C--hyperbolic (cf. [Za2]).\\[1ex]
\noindent {\bf 1.8.} {\it Contents of the paper.} The main results are summarized
at Theorem 7.12 at the very end of the paper. Besides this Introduction, the
paper contains six sections. Sections 2 and 3 are
preliminary. The first of them deals with some necessary facts from hyperbolic
analysis, while in the second one certain generalities on plane curves are
given. In section 4 we prove C--hyperbolicity of the complements of
irreducible curves of genus at least $1$, whose duals are immersed curves (for
instance, nodal curves)
(see Theorem 4.1). We give examples of such curves of any even degree $d \ge
6$.
Furthermore, we study the general case when the dual curve $C^*$ may have
cusps. Then, under the morphism $\rho_C \,:\, I \!\! P^2 \to S^n C^*_{norm}$, the
discriminant divisor $R_n \subset S^n C^*_{norm}$, besides the curve $C$ itself,
cuts out a line configuration $L_C \subset I \!\! P^2$ which consists of the dual lines
of cusps of $C^* \subset I \!\! P^{2*}$; they are the inflexional tangents and some
cuspidal tangents of $C$. We call $L_C$ {\it the artifacts of $C$} (see (3.3)).
In Theorem 4.1 we prove C--hyperbolicity of $ I \!\! P^2 \setminus (C \cup L_C )$, where
$C$ is an irreducible curve of genus $g \ge 1$.
The case of rational curves is studied in sections 5 - 7. In section 5 we give
necessary preliminaries. If $C$ is rational, then $S^n C^*_{norm} \cong I \!\! P^n
\,,\,\,\,\rho_C \,:\, I \!\! P^2 \to I \!\! P^n \cong S^n C^*_{norm}$ is a linear
embedding and the discriminant hypersurface $R_n \subset I \!\! P^n$ is the projective
hypersurface defined by the usual discriminant of the universal polynomial of
degree $n$. Therefore, we have the following assertion.\\[1ex]
\noindent {\bf 1.9. Lemma} (cf. {\rm [Zar, p.266])}. {\it Any rational curve $C \subset
I \!\! P^2$ whose dual $C^*$ is of degree $n$, together with its artifacts $L_C$ is
a plane section of the discriminant hypersurface $R_n \subset I \!\! P^n$.} \\[1ex]
\noindent We call $\rho_C$ {\it the Zariski embedding} \footnote{In an unexplicit
way it is contained already in [Ve, Ch. IV, p.208]}.
Using this lemma as well as a duality between the Zariski embedding and a
projection of the rational normal curve (see 5.4-5.5), we establish an analog
of Theorem 4.1 for a rational curve whose dual has at least one cusp. This is
done in Theorem 6.5, where also all exceptions are listed. A classification of
the orbits of the natural $ I \!\!\!\! C^*$--actions is an important ingredient of the
proof. We give several concrete examples.
In section 7 we deal with the rational curves whose duals are nodal Pl\"ucker
curves, i.e. with the maximal cuspidal rational curves. For such a curve of
degree $d \ge 8$ we prove that its complement is almost C--hyperbolic
(Corollary 7.10; see 2.4 below for the terminology). The proof is based on
passing to the moduli space of the punctured Riemann sphere and on a study of
the orbits of the natural representation of the group $ I \!\! P GL (2;\, I \!\!\!\! C )$ on
the projectivized space of binary forms.
\\[1ex]
The second of the authors had fruitful discussions on the content of section 7
with D. Akhiezer, M. Brion, Sh. Kaliman and H. Kraft; its his pleasure to thank
all of them. He also is gratefull to the SFB-170 `Geometry and Analysis'
at G\"ottingen University for its hospitality and excellent
working conditions. \\[2ex]
\section {Preliminaries in hyperbolic complex analysis}
\noindent {\bf 2.1.} {\it Lin's Theorem.} Here we recall some definitions and facts
from [Li]. A complex space $X$ is called {\it ultra--Liouville} if any bounded
plurisubharmonic function on $X$ is constant. For instance, any
quasi--projective variety is ultra--Liouville. By Lin's Theorem [Li, Theorems B
and 3.5] any almost nilpotent (or even almost $\omega$--nilpotent) Galois
covering of an ultra--Liouville complex space $X$ is Liouville. A covering is
called {\it almost nilpotent} (resp. {\it almost $\omega$--nilpotent}) if its
group of deck transformations is so. Recall that a group $G$ is {\it almost
nilpotent} if it has a nilpotent subgroup of a finite index ($G$ is {\it almost
$\omega$--nilpotent} if the union of the members of its upper central series is
a subgroup of $G$ of finite index; for a finitely generated group $G$ the
almost $\omega$--nilpotency is equivalent to the almost nilpotency).
\\
[1ex]
\noindent {\bf 2.2.} {\it Super--liouvilleness}. Let us say that a complex space
$X$ is
{\it super-Liouville} if any covering over $X$ is Liouville.
Super-liouvilleness is a property which in a sense is opposite to
C-hyperbolicity. It is clear that $X$ is super-Liouville iff the universal
covering $U_X$ of $X$ is Liouville. By Lin's theorem any ultra-Liouville
complex space $X$ which has almost $\omega$--nilpotent fundamental group
$\pi_1(X)$
is a super-Liouville one. In particular, a smooth quasi-projective curve $C$ is
super-Liouville
iff the group $\pi_1(C)$ is abelian, i.e. iff $C$ is non-hyperbolic.
Note that if any two points of $X$ can be connected by a finite
chain of Liouville subspaces (which are assumed to be connected but
not necessarily closed), then $X$ itself is Liouville. More generally,
we have the following lemma. \\[1ex]
\noindent {\bf 2.3. Lemma.} {\it Let $X$ be a complex space (with countable
topology) such that any two points of $X$ can be connected by a finite
chain of super-Liouville subspaces of $X$. Then $X$ is super-Liouville. In
particular, if $X$ is a quasi-projective variety
such that each pair of points of $X$ can be connected by a finite chain of
non-hyperbolic curves, then $X$ is super-Liouville.} \\[1ex]
\noindent {\it Proof.} Let $\, \pi : U_X \rightarrow X \,$ be the universal covering.
Suppose that there exists a non-constant bounded holomorphic function
$f$ on $U_X$. Let ${\cal F}$ be the collection of all super-Liouville
subspaces of $X$, and let $\tilde{{\cal F}}$ be the collection of subspaces
consisting of all connected components of preimages $\, \pi^{-1}(A)$,
where $\, A \in {\cal F}$. We define an equivalence relation on $U_X$ as
follows:
\noindent {\it Two points in $U_X$ are equivalent iff they can be connected by a
finite
chain of members of $\tilde{{\cal F}}$.}
\noindent By the condition of the lemma it is
easily seen that the union of the equivalence classes of the points of a
given fibre of $\pi$ coincides with the whole space $U_X$. Since $\pi_1(X)$
is an at most countable group, the fibre of $\pi$ is at most countable, too,
and therefore there exists an at most countable set of equivalence classes.
If $M$ is any of them, then clearly $f|M \equiv const$. Therefore, $f$
takes at most countable set of values, which is impossible.
The second statement is an easy corollary of the first one. \qed
\noindent {\bf 2.4.} {\it Weak C--hyperbolicity}. We say that a complex space $X$
is {\it almost} resp. {\it weakly Carath\'eodory hyperbolic} if for any point
$p \in X$ there exist only finitely many resp. countably many points $q \in X$
which cannot be separated from $p$ by bounded holomorphic functions (i.e. if
the equivalence relation defined on $X$ by the functions from the algebra
$H^{\infty} (X)$ is finite resp. at most countable). It will be called {\it
almost} resp. {\it weakly C--hyperbolic} if $X$ has a covering $Y \to X$,
where $Y$ is almost resp. weakly Carath\'eodory hyperbolic.
These notions are meaningful due to the following reasons. It is unknown
whether the universal covering space $U_X$ of a C--hyperbolic complex space $X$
is Carath\'eodory hyperbolic, or more generally, whether there is a
Carath\'eodory hyperbolic covering $Y \to X$ which can be defined in a
functorial way. In contrary, one can make the following observation.\\
\noindent {\it A complex space $X$ is weakly C--hyperbolic iff the universal
covering space $U_X$ is weakly Carath\'eodory hyperbolic.}\\
\noindent Hereafter we assume $X$ to be reduced and with countable topology. In
particular, the universal covering $U_X$ of a C--hyperbolic space $X$ is weakly
Carath\'eodory hyperbolic. One may consider on $X$ the pseudo--distance which
is the quotient of the Carath\'eodory pseudo--distance $c_{U_X }$ on $U_X$
(resp. the quotient of the inner Carath\'eodory pseudo--distance $c'_{U_X }$
resp. of the differential Carath\'eodory--Reiffen pseudo--distance $C_{U_X }$;
see [Re]). All three of these quotient pseudo--distances are contracted by
holomorphic mappings. Furthermore, the deck transformations on $U_X$ being
isometries, the quotient of $C_{U_X }$ on $X$ is locally isometric to $C_{U_X
}$ itself, and thus it is non-degenerate iff $C_{U_X }$ is so (for a weakly
C--hyperbolic space $X$ it is at least non--trivial).
The proof of the following lemma is easy and can be omitted.\\
[1ex]
\noindent {\bf 2.5. Lemma.} {\it Let $f: Y \to X$ be a holomorphic mapping of
complex spaces. If f is injective (resp. f has finite resp. at most countable
fibres) and $X$ is C--hyperbolic (resp. almost resp. weakly C--hyperbolic),
then so is $Y$.} \\[1ex]
\noindent {\bf 2.6.} {\it Brody hyperbolicity}. Recall that a complex space $X$ is
{\it Brody hyperbolic} if it contains no entire curve, i.e. if every
holomorphic mapping $ I \!\!\!\! C \to X$ is constant. Note that sometimes by Brody
hyperbolicity one means absence of {\it Brody entire curves} in $X$, i.e.
entire curves whose derivatives are uniformly bounded with respect to a fixed
hermitian metric on $X$ (see e.g. [Za3]). Usually this is enough in
applications. But in this paper we do not need such a precision.
It is clear that any weakly C--hyperbolic complex space is Brody hyperbolic. \\
[1ex]
\noindent {\bf 2.7.} {\it Kobayashi hyperbolicity}.
For a curve $C \subset I \!\! P^2$ denote by ${\rm sing}\, C$ the set of all singular
points of $C$. Put ${\rm reg}\, C = C \setminus {\rm sing}\, C$.
The next statement follows from Theorem 2.5 in [Za1]. \\[1ex]
\noindent {\bf Proposition.} {\it Let the Riemann surface ${\rm reg}\, C$ be
hyperbolic and $ I \!\! P^2 \setminus C$ be Brody hyperbolic (the latter is true, in
particular, if $ I \!\! P^2 \setminus C$ is weakly C--hyperbolic). Then $ I \!\! P^2 \setminus C$ is
Kobayashi complete hyperbolic and hyperbolically embedded into $ I \!\! P^2$.}\\
[1ex]
Note that in Example 1.3 in the Introduction the first condition fails while
the second one is fulfilled. It is easily seen that in this example $ I \!\! P^2 \setminus
C$ is not hyperbolically embedded into $ I \!\! P^2$. In fact, the condition ` ${\rm
reg}\, C$ is hyperbolic' is necessary for $ I \!\! P^2 \setminus C$ being hyperbolically
embedded into $ I \!\! P^2$ [Za1, Corollary 1.3].\\[1ex]
\noindent {\bf 2.8.} {\it Relative hyperbolicities}. Let $X$ be a complex space
resp. a quasi--projective variety and $Z \subset X$ be a closed analytic subset
resp. a closed algebraic subvariety. We say that $X$ is {\it Brody hyperbolic
modulo $Z$} if any (non-constant) entire curve $\, I \!\!\!\! C \to X$ is contained in
$Z$.
For instance, this is the case if $X$ is Kobayashi hyperbolic modulo $Z$ (see
[KiKo]). (We mention that in [Za4] the above property of relative Brody
hyperbolicity was called {\it strong algebraic degeneracy}.)
We will say that $X$ is {\it C--hyperbolic modulo $Z$} ({resp. \it almost resp.
weakly C--hyperbolic modulo $Z$}) if there is a covering $\pi \,:\,Y \to X$
such that for each point $p \in Y$ any other point $q \in Y \setminus \pi^{-1} (Z)$
(resp. any other point $q \in Y \setminus \pi^{-1} (Z)$ besides only finitely many
resp. besides only countably many such points) can be separated from $p$ by
bounded holomorphic functions.
By the monodromy theorem weak C--hyperbolicity of $X$ modulo $Z$ implies Brody
hyperbolicity of $X$ modulo $Z$.
The next lemma, which is a generalization of Lemma 2.5, easily follows from the
definitions. \\[1ex]
\noindent {\bf 2.9. Lemma.} {\it Let $f: Y \to X$ be a holomorphic mapping of
complex spaces and let $Z$ be a closed complex subspace of $X$. If $f\,|\,(Y
\setminus f^{-1} (Z) )$ is injective (resp. has finite resp. at most countable
fibres) and $X$ is C--hyperbolic (resp. almost resp. weakly C--hyperbolic)
modulo $Z$, then $Y$ is C--hyperbolic (resp. almost resp. weakly C--hyperbolic)
modulo $f^{-1} (Z).$} \\
[2ex]
\section {Background on plane algebraic curves}
\noindent {\bf 3.1.} {\it Classical singularities. Immersed curves}. We say that a
curve $C$ in $ I \!\! P^2$ has {\it classical singularities} if its singular points
are nodes and ordinary cusps. It is called {\it a Pl\"ucker curve} if both $C$
and the dual curve $C^*$ have classical singularities only and no flex at a
node. If the normalization mapping $\nu : C^*_{norm} \to C \hookrightarrow
I \!\! P^2$ is an immersion, or, which is equivalent, if all irreducible local
analytic branches of $C$ are smooth, then we say that $C$ is {\it an immersed
curve}. An immersed curve $C$ is called {\it a curve with tidy} or {\it
ordinary singularities}, or simply {\it a tidy curve}, if at each point $p \in
C$ the local irreducible branches of $C$ have pairwise distinct tangents. By M.
Noether's Theorem [Co] any plane curve can be transformed into a tidy curve by
means of Cremona transformations.\\
\noindent {\bf 3.2.} {\it Cusps and flexes}. In the sequel by {\it a cusp} we mean
an irreducible plane curve singularity. In particular, an irreducible local
analytic branch $A$ of a plane curve $C$ with centrum $p \in C$ is a cusp iff
it is singular. The tangent line to a cusp $A$ at $p$ is called {\it a cuspidal
tangent of $C$}. Recall that {\it the multiplicity sequence} of a plane
analytic germ $A$ at $p_0 \in A$ is the sequence of multiplicities of $A$ at
$p_0$ and its infinitely nearby points. Following [Na, (1.5)] $A$ is called
{\it a simple cusp} if its first Puiseaux pair is $(m, m+1)$, where $m = {\rm
mult}_p A \ge 2$, or, what is the same, if the multiplicity sequence of $A$ is
$(m,\, 1, \,1, \dots)$. By Lemma 1.5.7 in [Na] a cusp $A$ is a simple cusp iff
the corresponding branch $A^*$ of the dual curve $C^*$ is smooth. In this case
$A^*$ is a flex of order $m-1$ (see the definition below), and vice versa, if
$A^*$ is a flex of order $m-1$, then $A$ is a simple cusp of multiplicity $m$.
A simple c
sp $A$ of multiplicity $2$ is called {\it an ordinary cusp} if $C$ is locally
irreducible at $p$.
A smooth irreducible local branch $A$ of the curve $C$ at a point $p \in C$ is
called {\it a flex of order} $k$ if $i(A, T_p A; p) = k+2 \ge 3$. The tangent
line $T_p A$ to a flex $A$ at $p$ is called {\it an inflexional tangent}. {\it
An ordinary flex} is a flex of order $1$ at a smooth point of $C$.
Thus, the dual $C^*$ is an immersed curve iff $C$ has no flex and all its cusps
are simple. \\[1ex]
\noindent {\bf 3.3.} {\it Artifacts}. If $C^*$ has cusps, denote by $L_C$ the union
of their dual lines in $ I \!\! P^2$. Clearly, $L_C$ consists of the inflexional
tangents of $C$ and the cuspidal tangents at those cusps of $C$ which are not
simple. Due to some analogy in tomography, we call $L_C$ {\it the artifacts} of
$C$.
Note that the dual curve $C^*$ of $C$ is an immersed curve iff $L_C =
\emptyset$. Such a curve $C$ may have complicated reducible singularities,
which correspond to multiple tangents of $C^*$; for instance, it may have
tacnodes, etc. \\[1ex]
\noindent {\bf 3.4.} {\it The Class Formula}. Let $C \subset I \!\! P^2$ be an irrducible
curve of degree $d\ge 2$, of geometric genus $g$ and of class $c$. Then $c=d^*
={\rm deg}\, C^*$ is defined by the Class Formula (see [Na, (1.5.4)])
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,c=d^* = 2(g+d-1) - \sum\limits_{p \in {\rm sing}\, C} (m_p - r_p) =\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,\,\,\,$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,\,\,\, = 2(g+d-1) - \sum\limits_{A = (A, p),\,p \in {\rm
sing}\, C} (m_A - 1) \,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,\,\,\,(4)$$
where $m_p = {\rm mult}_p C ,\,\,\,r_p$ is the number of irreducible analytic
branches of $C$ at $p$, $A = (A, p)$ is a local analytic branch of $C$ at $p$
and $m_A$ is its multiplicity at $p$. In particular, $C$ is an immersed curve
iff $d^* = 2(g+d-1)$ (indeed, this is the case iff the last sum in (4)
vanishes). Furthermore, for an immersed curve $C$ one has $\,\,\,d^* \ge 2d+2
\ge 10\,\,\,$ if $\,\,\,g\ge 2 ,\,\,\,$ $d^* = 2d \ge 6\,\,\,$ if $g=1 ,\,\,\,$
and $d^* = 2d-2 \ge 2\,\,\,$
if $g=0$.
For reader's convenience we recall here also the usual Pl\"ucker formulas: $$g
= 1/2 (d - 1)(d - 2) - \delta - \kappa = 1/2 (d^* - 1)(d^* - 2) - b - f$$
$$d^* = d(d - 1) - 2\delta - 3\kappa\,\,\,\,\,{\rm and}\,\,\,\,\,d = d^* (d^* -
1) - 2b - 3f$$ for a Pl\"ucker curve $C$ with $\delta$ nodes, $\kappa$ cusps,
$b$ bitangent lines and $f$ flexes. \\[2ex]
\noindent {\bf 3.5.} {\it The $n$--th Abel--Jacobi map}. Let $M$ be a compact
Riemann surface of genus $g$, and let $j\,:\,M \to J(M)$ be a fixed
Abel--Jacobi embedding of $M$ into its Jacobian variety $J(M) \cong {\rm
Pic^0}\, (M)$. The $n$--th symmetric power $S^n M$ may be identified with the
space of effective divisors of degree $n$ on $M$. Let $\phi_n \,:\, S^n M \to
J(M)$ be the $n$--th Abel--Jacobi map $D = p_1 + \dots + p_n \longmapsto \phi_n
(D) := j(p_1 ) + \dots + j(p_n )$, so that $j = \phi_1$. We recall the
following well known facts (see e.g. [GH, 2.2], [Zar, pp.352--353] or [Na,
(5.2), (5.3)] and references therein): \\
\noindent i) $\phi_n$ is holomorphic;
\noindent ii) (Abel's Theorem) $\phi_n^{-1} (\phi_n (D)) = |D| = I \!\! P H^0 (M,
O([D])) \cong I \!\! P^{{\rm dim}\,|D|}$ is the complete linear system of $D$, where
$D \in S^n M$ is an effective divisor of degree $n$ on $M$;
\noindent iii) the natural injection $|D| \hookrightarrow S^n M$ is a holomorphic
embedding, i.e. $|D| = \phi_n^{-1} (\phi_n (D))$ is a smooth subvariety in $S^n
M$;
\noindent iv) if $n \le g$, then $\phi_n \,:\,S^n M \to J(M)$ is generically
one--to--one; in particular, the image $\phi_{g-1} (S^{g-1} M) \subset J(M)$ is a
translation of the theta divisor $\Theta$ on $J(M)$;
\noindent v) (Jacobi Inversion) if $n \ge g$, then $\phi_n \,:\,S^n M \to J(M)$ is
surjective. For $n > 2g - 2$ it is an algebraic projective bundle (see [Ma]);
in particular, if $g = 1$, then it is a $ I \!\! P^{n-1}$--bundle over $J(M) \cong
M$. \\[1ex]
\noindent {\bf 3.6.} {\it The Zariski embedding}. Let $C \subset I \!\! P^2$ be an
irreducible curve of degree $d \ge 2$ and let $\nu : C^*_{norm} \to C^*$ be the
normalization of the dual curve. Following Zariski [Zar, p.307, p.326] and M.
Green [Gr1] (see also [DL]), as in (1.4) we consider the mapping $\rho_C \,:
\, I \!\! P^2 \to S^n C^*_{norm}$ of $ I \!\! P^2$ into the $n$-th symmetric power of
$C^*_{norm}$, where $n = {\rm deg}\, C^*$. We put $\rho_C (z) = \nu^* (l_z) \subset
S^n C^*_{norm}$, where $z \in I \!\! P^2$ and $l_z \subset I \!\! P^{2*}$ is the dual line.
Clearly, $ \rho_C: I \!\! P^2 \to S^n C^*_{norm}$ is holomorphic. We still denote by
$D_n$ the union of the diagonal divisors in $(C^*_{norm})^n$ and by $R_n = s_n
(D_n)$ the discriminant divisor, i.e. the ramification locus of the branched
covering $s_n: (C^*_{norm})^n \to S^n C^*_{norm}$ (cf. (1.4)).
It is clear that $\rho_C$ is a holomorphic embedding, which we call in the
sequel {\it the Zariski embedding}. More precisely, it is composed of two
embeddings $i_1 \,:\, I \!\! P^2 \hookrightarrow I \!\! P^{h(C)}$ and $i_2 \,:\,
I \!\! P^{h(C)} \hookrightarrow S^n C^*_{norm}$ which are described below.
Denote by $H$ a divisor of degree $n$ on $M:= C^*_{norm}$, which is the trace
of a line cut of $C^*$ in $ I \!\! P^{2*}$. The two dimensional linear system $g^2_n$
of all such line cuts is naturally identified with the original projective
plane $ I \!\! P^2 = ( I \!\! P^{2*})^*$. Let $h(C) := {\rm dim}\,|H|$; then $i_1
\,:\, I \!\! P^2 = g^2_n \hookrightarrow I \!\! P^{h(C)} = |H| $ is defined to be the
canonical linear embedding of $g^2_n $ into the complete linear system $|H|$.
(Let us mention that $g^2_n$ itself might be complete; for instance, this is
the case when $C^*$ is a nodal curve with $\delta$ nodes, where $\delta <
{n(n-2) \over 4}$ for $n$ even or $\delta < {(n-1)^2 \over 4}$ for $n$ odd; see
[Na, p.115].) The Abel embedding $i_2 \,:\, I \!\! P^{h(C)} = |H| \hookrightarrow
S^n C^*_{norm} = S^n M$ identifies $|H|$ with the fibre $ \phi_n^{-1} (\phi_n
(H))$ of the $n$--th Abel--Jacobi map $\phi_n \,:\,S^n M \to J(M)\,$ (see
(3.5)).
What we really need in section 4 is that the restriction $\rho_C\,|\, ( I \!\! P^2
\setminus C)$ is injective, which can also be shown as follows.
We have to show that any projective line $l$ in $ I \!\! P^{2*}$
which is not tangent to $C^*$ meets $C^*$ in at least two distinct points, and
so it is uniquely defined by its image $\rho (l) = l \cap C^*$. Suppose that
there exists a line $l_0 \subset I \!\! P^{2*}$ which has only one point $p_0$ in common
with $C^*$ and which is not tangent to $C^*$ at this point. Let $l_1$ be the
tangent to a local analytic branch of $C^*$ at $p_0$. Then we have
$$i(l_1, C^*; p_0) > i(l_0, C^*; p_0) = n = l_1 \cdot C^* \,\,\,,$$
which is impossible. In the same elementary way it can be shown that $\rho_C$
is a holomorphic injection. \\[1ex]
\noindent {\bf 3.7. Lemma.} {\it The preimage $\rho_C^{-1} (R_n) \subset I \!\! P^2$ is the
union of $C$ with its artifacts $L_C$. In particular, $\rho_C^{-1} (R_n) = C$
iff the dual curve $C^*$ is an immersed curve. }\\
\noindent {\it Proof.} Note that a point $z \in I \!\! P^2 \setminus C$ is contained in
$\rho_C^{-1} (R_n)$ iff its dual line $l_z$ passes through a cusp of $C^*$. The
lines in $L_C$ are just the dual lines to the cusps of $C^*$. \qed
\section {C-hyperbolicity of complements of plane curves of genus $g \ge 1$}
\noindent In this section we keep all the notation from 3.6. The main result here
is the following theorem. \\[1ex]
\noindent {\bf 4.1. Theorem.} {\it Let $C \subset I \!\! P^2$ be an irreducible curve of
genus $g \ge 1$ and $L_C$ be its artifacts. Then
\noindent a) $ I \!\! P^2 \setminus (C\cup L_C)$ is C--hyperbolic.
\noindent b) If the dual curve $C^* \subset I \!\! P^{2*}$ is an immersed curve, then $ I \!\! P^2
\setminus C$ is C--hyperbolic, Kobayashi complete hyperbolic and hyperbolically
embedded into $ I \!\! P^2$.} \\
[1ex]
\noindent {\it Proof.} a)
Consider first the case when $g\ge 2$. In this case
$S^n C^*_{norm}\setminus R_n$ is C--hyperbolic (cf. (1.4)). Indeed, its covering
$(C^*_{norm})^n \setminus D_n$ is a domain in $(C^*_{norm})^n$. Since the universal
covering of $(C^*_{norm})^n$ is the polydisc $\Delta^n$, it
is C--hyperbolic. Therefore, $ (C^*_{norm})^n \setminus D_n$, and hence also $S^n
C^*_{norm} \setminus R_n$ are C--hyperbolic, too. By Lemma 3.7 the image $\rho_C
( I \!\! P^2 \setminus (C \cup L_C )) \subset S^n C^*_{norm}$ does not meet the discriminant
variety $R_n$ and by 3.6 $\rho_C\,|\, ( I \!\! P^2 \setminus (C \cup L_C )) \,:\, I \!\! P^2 \setminus
(C \cup L_C ) \to S^n C^*_{norm} \setminus R_n$ is injective. Therefore, by Lemma 2.5
$ I \!\! P^2 \setminus (C \cup L_C )$ is C--hyperbolic.
Next we consider the case when $C$ is an elliptic curve. Denote $E =
C^*_{norm}$. Note that both $E^n \setminus D_n$ and $S^nE \setminus R_n$ are not
C--hyperbolic or even hyperbolic, and so we can not apply the same arguments as
above.
Represent $E$ as $E = J(E) = I \!\!\!\! C / \Lambda_{\omega}$, where $\Lambda_{\omega}$
is the lattice generated by $1$ and $\omega \in I \!\!\!\! C_{+}$ (here $ I \!\!\!\! C_{+} := \{z
\in I \!\!\!\! C\,|\,{\rm Im}z > 0\}$). By Abel's Theorem we may assume this
identification of $E$ with its jacobian $J(E)$ being chosen in such a way that
the image $\rho_C ( I \!\! P^2)$ is contained in the hypersurface $s_n (H_0) =
\phi_n^{-1} ({\bar 0}) \cong I \!\! P^{n-1} \subset S^n E$, where $$H_0 :=
\{z=(z_1,\dots, z_n) \in E^n\,|\,\sum\limits_{i=1}^n z_i = 0\}$$ is an abelian
subvariety in $E^n$ (see (3.5)). The universal covering $\tilde{H}_0$ of $H_0$
can be identified with the hyperplane $\sum\limits_{i=1}^n x_i = 0$ in $ I \!\!\!\! C^n =
\tilde{E}^n$.
Consider the countable families $\tilde{D}_{ij}$ of parallel affine hyperplanes
in $ I \!\!\!\! C^n$ given by the equations $x_i - x_j \in \Lambda_{\omega}\,,\,\,\,i,j =
1,\dots,n,\,\,i<j$.\\[1ex]
\noindent {\it Claim. The domain $\tilde{H}_0 \setminus \bigcup\limits_{i=1}^{n-1}
\tilde{D}_{i,i+1}$ is biholomorphic to $( I \!\!\!\! C \setminus \Lambda_{\omega})^{n-1}$.} \\
[1ex]
Indeed, put $y_k := (x_k - x_{k+1})\,|\,\tilde{H}_0\,,\,\,i = 1,\dots,n-1$. It
is easily seen that $(y_1,\dots,y_{n-1})\,:\, \tilde{H}_0 \to I \!\!\!\! C^{n-1}$ is a
linear isomorphism whose restriction yields a biholomorphism as in the claim.
The universal covering of $( I \!\!\!\! C \setminus \Lambda_{\omega})^{n-1}$ is the polydisc
$\Delta^n$, and so $( I \!\!\!\! C \setminus \Lambda_{\omega})^{n-1}$ is C--hyperbolic. Put
$\tilde{D}_n:= \bigcup\limits_{i,j=1,\dots,n} \tilde{D}_{ij}$. The open subset
$\tilde{H}_0 \setminus \tilde{D}_n$ of $\tilde{H}_0 \setminus \bigcup\limits_{i=1}^{n-1}
\tilde{D}_{i,i+1} \cong ( I \!\!\!\! C \setminus \Lambda_{\omega})^{n-1} $ is also
C--hyperbolic.
Denote by $p$ the universal covering map $ I \!\!\!\! C^n \to ( I \!\!\!\! C /\Lambda_{\omega})^n$.
The restriction $$p\,|\,\tilde{H}_0 \setminus \tilde{D}_n\,:\,\tilde{H}_0 \setminus
\tilde{D}_n \to H_0 \setminus D_n \subset E^n \setminus D_n$$ is also a covering mapping.
Therefore, $H_0 \setminus D_n$ is C--hyperbolic, and so $s_n (H_0) \setminus R_n$ is
C--hyperbolic, too. Since $\rho_C\,|\, ( I \!\! P^2 \setminus (C \cup L_C )) \,:\, I \!\! P^2
\setminus (C \cup L_C ) \to s_n (H_0) \setminus R_n$ is an injective holomorphic mapping,
by Lemma 2.5 $ I \!\! P^2 \setminus (C \cup L_C )$ is C--hyperbolic.
b) Assume further that $C^*$ is an immersed curve. Then $C$ can not be smooth.
Indeed, being smooth $C$ would have flexes at the points of intersection of $C$
with its Hesse curve (see [Wa]), and hence $C^*$ would have cusps. Thus, ${\rm
reg}\,C$ is hyperbolic, and by that what has been proven above $ I \!\! P^2 \setminus C$
is C--hyperbolic. Therefore, by Proposition 2.7 it is Kobayashi complete
hyperbolic and hyperbolically embedded into $ I \!\! P^2$.
This completes the proof. \qed
\noindent {\bf 4.2. Remark.} The complement to a ${\underline {\rm rational}}$
curve whose dual is
an immersed curve is not necessarily C--hyperbolic; it even may be not Brody or
Kobayashi hyperbolic. An example is a plane quartic $C$ with three cusps. Such
a quartic $C$ is projectively equivalent to the curve given by the equation
$$4x_1^2 (x_1 -2x_0 )(x_1 + x_2 ) - (2x_0 x_2 - x_1^2 )^2 =0$$
(see [Na, (2.2.5)]). Its dual curve $C^*$ is a nodal cubic with equation
$$x_0 x_1^2 + x_1^3 - x_0 x_2^2 =0\,\,\,.$$
Thus, $g(C) = 0$ and $C^*$ is an immersed curve. The complement $ I \!\! P^2 \setminus C$
is not Kobayashi hyperbolic, because its Kobayashi pseudo--distance vanishes
on any of three cuspidal tangent lines of $C$, on any of three lines passing
through two cusps of $C$ each one and on the only bitangent line
of $C$. Indeed, each of these seven lines meets $C$ in only two points; but
$k_{ I \!\!\!\! C^*} =0 $, where $ I \!\!\!\! C^* = I \!\! P^1 \setminus \{2 \,{\rm points}\}$. Therefore,
$ I \!\! P^2 \setminus C$ is not C--hyperbolic. Note that $\pi_1 ( I \!\! P^2 \setminus C)$ is a finite
group of order $12$ [Zar, p.145], and thus $ I \!\! P^2 \setminus C$ is super--Liouville
(see (2.2)).
Note, furthermore, that by an analogous reason the complement of any
quartic curve $C \subset I \!\! P^2$ is neither Kobayashi hyperbolic nor Brody
hyperbolic [Gr2]. The fundamental group $\pi_1 ( I \!\! P^2 \setminus C)$ for an
irreducible quartic $C$ being almost abelian (see (1.1)), by Lin's Theorem
$ I \!\! P^2 \setminus C$ is super--Liouville. \\[1ex]
Next we give some examples, or at least computations of numerical characters of
plane curves which satisfy the assumptions of Theorem 4.1. First we consider
examples of curves of genus $g \ge 2$ with the dual an immersed curve. \\[1ex]
\noindent {\bf 4.3. Example.} Let $C \subset I \!\! P^2$ be an irreducible curve whose dual
$C^*$ is a nodal curve of degree $n=d^* \ge 3$ with $\delta$ nodes. Assume that
the genus $g(C) = g(C^*) = {(n-1)(n-2)\over 2} - \delta$ is at least $2$. Such
a curve does exist for any given $\delta$ with $0\le\delta \le {(n-1)(n-2)\over
2} -2$ (see [Se, \S 11, p.347]; [O, Proposition 6.7]). By the Class Formula (4)
$C$ has degree $d=n(n-1) -2\delta $, which can be any even integer from the
interval $[2(n+1), n(n-1)]$. The least possible value of $n$ resp. $d$ is $n=4$
resp. $d=10$, which corresponds to the case when $C^*$ is a nodal quartic with
one node ($\delta =1$) (see e.g. [Na, p.130]). To be a Pl\"ucker curve (see
(3.1)) such $C$ should be a curve of degree $10$ with $16$ nodes and $18$
ordinary cusps (cf. [Zar, p.176]).
If $C$ is a curve of genus $g\ge 2$ whose dual is a nodal curve, then by
Theorem
4.1 the complement $ I \!\! P^2 \setminus C$ is C--hyperbolic, Kobayashi complete
hyperbolic and hyperbolically embedded
into $ I \!\! P^2$. This yields examples of such curves $C$ of any even degree $d\ge
10$. \\[1ex]
There are similar examples with elliptic curves.\\
\noindent {\bf 4.4. Example.} If the dual $C^*$ of $C$ is an immersed elliptic
curve, then by the Class Formula (4) $d= {\rm deg}\,C =2n \ge 6$, where $n =
{\rm deg}\,C^* \ge 3$ (see 3.4). Thus, the least possible value of the degree
$d$ of such a curve $C$ is $d=6$. Let $C$ be a sextic in $ I \!\! P^2$ with $9$
cusps. Then $C$ is an elliptic Pl\"ucker curve whose dual $C^*$ is a smooth
cubic; vice versa, the dual curve to a smooth cubic is a sextic with $9$
ordinary cusps (see e.g. [Wa]). By Theorem 4.1 the complement of such a curve
is C--hyperbolic, Kobayashi complete hyperbolic and hyperbolically embedded
into $ I \!\! P^2$. The same is true if $C$ is the dual curve to a nodal quartic
$C^*$ with two nodes; here $d= {\rm deg}\,C =8$ (see e.g. [Na, p.133]). To be
Pl\"ucker such a curve $C$ must have $8$ nodes and $12$ ordinary cusps. \\[1ex]
These examples lead to the following conclusion.\\[1ex]
\noindent {\bf 4.5. Proposition.} {\it For any even $d\ge6$ there are irreducible
plane curves of degree $d$ and of genus $g\ge 1$ whose duals are nodal curves,
and so which satisfy the assumptions of Theorem 4.1, b).} \\[1ex]
Next we pass to examples to part a) of Theorem 4.1. \\
\noindent {\bf 4.6. Examples.} Let $C^* \subset I \!\! P^{2*}$ be an irreducible curve of
genus $g\ge 1$ with classical singularities. If $C^*$ has $\delta$ nodes and
$\kappa$ cusps, then the dual curve $C \subset I \!\! P^2$ has $\kappa$ ordinary flexes
as the only flexes, and so $L_C$ is the union of inflexional tangents of $C$.
By the Class Formula (4) we have $d = {\rm deg}\,C = 2(n+g-1)- \kappa$. Since
all $\kappa$ inflexional tangents of $C$ are distinct, it follows that ${\rm
deg}\,(C\cup L_C) = 2(n+g-1) \ge 2n \ge 6$.. Assume that $\kappa>0$ to
exclude the case considered in Examples 4.3 and 4.4 above, when $C^*$ was an
immersed curve. Since $g\ge 1$, the case when $C$ is a singular cubic has also
been excluded. Thus, we have $n\ge 4$, and hence ${\rm deg}\,(C\cup L_C) \ge
8$.
The simplest example is a quartic $C^*$ with an ordinary cusp and a node as the
only singularities; such a quartic does exist (see [Na, p.133]). The dual curve
$C$ is an elliptic septic with the only inflexional tangent line $l=L_C$. To be
a Pl\"ucker curve, $C$ must have $4$ nodes and $10$ ordinary cusps.
Another example is a quartic $C^*$ with two ordinary cusps as the only
singular points; it also does exist [Na, p.133]. Here $C$ is an elliptic
sextic and $L_C$ is the union of two inflexional tangents of $C$. To be a
Pl\"ucker curve, $C$ should have one node and $8$ ordinary cusps.
In all these examples the assumptions of Theorem 4.1 are fulfilled. \\[1ex]
\noindent {\bf 4.7. Remark.} Of course, it may happen that the complement of the
artifacts $ I \!\! P^2 \setminus L_C$ is itself C--hyperbolic. For instance, this is so if
$L_C$ contains an arrangement of five lines with two triple points on one of
them, which is projectively equivalent to those $C_5$ of Example 1.3 in the
Introduction. But this is not the case if $L_C$ consists only of few lines like
in the examples 4.6, or if it consists of lines in general position (cf.
(1.2)). For instance, if $C = F_d$ is the Fermat curve of degree $n \ge 3$ in
$ I \!\! P^2$, then it is easily checked that the inflexional tangents are in general
position (note that here all the flexes are hyperflexes of high order). We
suppose that for a generic smooth plane curve $C$ of degree $d \ge 4$ its
inflexional tangents are in general position, and therefore the complement
$ I \!\! P^2 \setminus L_C$ is super--Liouville. \\[1ex]
Next we consider the problem of existence of an irreducible quintic with
C--hyperbolic complement. \\
\noindent {\bf 4.8.} We have already noted that there is no plane quartic $C$ with
Kobayashi hyperbolic complement $ I \!\! P^2 \setminus C$ [Gr2]. The obstructions are lines
in $ I \!\! P^2$ which intersect $C$ in at most two points; e.g. cuspidal tangents,
inflexional tangents, bitangents, etc. Recall that if $C \subset I \!\! P^2$ is a nodal
quintic, then $ I \!\! P^2 \setminus C$ is super--Liouville (see (1.1) and (2.2)). Although
there are irreducible quintics whose complements are Kobayashi hyperbolic
[Za3], there is no one with C--hyperbolic complement (cf. (1.3) and (6.1) for
examples of reducible quintics with C--hyperbolic complements). It is shown in
the next lemma that there is no one among the non-Pl\"ucker quintics; as for
the Pl\"ucker ones, see Proposition 4.10 below. \\[1ex]
\noindent {\bf 4.9. Lemma.} {\it Let $C \subset I \!\! P^2$ be an irreducible quintic.
Suppose that $C$ is not a Pl\"ucker curve. Then $ I \!\! P^2 \setminus C$ is not Brody
hyperbolic. Moreover, there exists a line $l_0 \subset I \!\! P^2$ which intersects with
$C$ in at most two points.} \\[1ex]
\noindent {\it Proof.} Assume that $C$ has a singular point $p_0$ which is not a
classical one. Let $l_0$ be the tangent line to an irreducible local analytic
branch of $C$ at $p_0$. If ${\rm mult}_{p_0}\,C \ge 3$, then $i(C,\,l_0 ;\,p_0
) \ge 4$, and so the line $l_0$ has at most one more intersection point with
$C$. If ${\rm mult}_{p_0}\,C = 2$, then either
\noindent 1) $C$ has two smooth branches at $p_0$ with the same tangent $l_0$
(e.g. $p_0 \in C$ is {\it a tacnode}),
\noindent or
\noindent 2) $C$ is locally irreducible in $p_0$ and has the multiplicity sequence
$(2,\,2,\,\dots)$ at $p_0$ (see (3.2)).
\noindent In both cases we still have $i(C,\,l_0 ;\,p_0 ) \ge 4$, and the same
conclusion as before holds. It holds also in the case when $l_0$ is the
inflexional tangent to $C$ at a point where $C$ has a flex of order at least
$2$ (see (3.2)).
Therefore, from now on we may suppose that $C$ has only classical singularities
and ordinary flexes.
Let $q_0$ be a singular point of $C^*$ which is not classical.
It can not be locally irreducible, since $C$ has only ordinary flexes.
If one of the irreducible local branches of $C^*$ at $q_0$ is singular,
then the dual line $l_0$ of $q_0$ is a multiple tangent line to $C$ which
is an inflexional tangent at some flex of $C$. Therefore, by the Bezout Theorem
$l_0$ is a bitangent line with intersection indices $2$ and $3$. The
remaining case to consider is the case when $C^*$ has only smooth local
branches at $q_0$. If two of them, say, $A^*_0$ and $A^*_1$, are tangent
to each other, then by duality the corresponding local branches $A_0$
and $A_1$ of $C$ should have common center and moreover, they should be
tangent to each other, too. But this is impossible since $C$ is assumed
to have only classical singularities. Thus, we are left with the case that
$q_0$ is a tidy singularity consisting of at least three disinct irreducible
local branches of $C^*$. But then the dual line $l_0$ of $q_0$ is tangent
in at least three different points of $C$. Since $C$ is of degree five, by
Bezout's Theorem this is also impossible. This completes the proof. \qed
From this lemma, Proposition 4.5 and the computations done by A. Degtjarjov
[Deg 1, 2] we obtain the following statement.
\\[1ex]
\noindent {\bf 4.10. Proposition.} {\it The minimal degree of an irreducible plane
curve with C--hyperbolic complement is 6}. \\
\noindent {\it Proof}. Indeed, it is shown in [Deg 1, 2] that two irreducible plane
quintics with the same type of singular points are isotopic in $ I \!\! P^2$, and
there are only two types of them such that the fundamental groups of the
complement are not abelian.. In both cases these quintics have non-classical
singularities. It follows that for an irreducible Pl\"ucker quintic $C \subset
I \!\! P^2$ the complement $ I \!\! P^2 \setminus C$ has cyclic fundamental group, and therefore
it is super-Liouville. \qed
\section {Rational plane curves and duality}
Here we precise the construction of (3.6) in the case of a rational curve. \\
\noindent {\bf 5.1.} {\it The Vieta covering}. The symmetric power $S^n I \!\! P^1$ can
be identified with $ I \!\! P^n$ in such a way that the canonical projection $s_n
\,:\, ( I \!\! P^1 )^n \to S^n I \!\! P^1$ becomes {\it the Vieta ramified covering},
which is given by
$$((u_1 : v_1),\dots,(u_n : v_n)) \longmapsto $$
$$\longmapsto (\prod\limits_{i=1}^n v_i)\,(1 : \sigma_1 (u_1 /v_1 ,\dots, u_n
/v_n )\,:\,\dots \,:\,\sigma_n (u_1 /v_1 ,\dots, u_n /v_n ))\,\,\,,$$
where $\sigma_i (x_1 ,\dots, x_n)\,,\,\,i=1,\dots,n$, are elementary symmetric
polynomials. This is a Galois covering with the Galois group being the n-th
symmetric group $S_n$. In the case when $z_i := u_i /v_i \in I \!\!\!\! C
,\,\,i=1,\dots,n$, we have $s_n (z_1 ,\dots, z_n) = (a_0 : \dots : a_n)$, where
the equation $a_0 z^n + \dots +a_n =0$ has the roots $z_1,\dots, z_n$ (see
[Zar, p.252] or [Na, (5.2.18)]). In general, $z_i \in I \!\! P^1 ,\,i= 1,\dots,
n,\,$ are the roots of the binary form $\, \sum\limits_{i=0}^n a_i u^{n - i}
v^i$ of degree $n$. \\
\noindent {\bf 5.2.} {\it Plane cuts of the discriminant hypersurface}. If $C \subset
I \!\! P^2$ is a rational curve of degree $d > 1$, then $C^*_{norm} \cong I \!\! P^1$,
and so the Zariski embedding is a morphism $\rho_C \,:\, I \!\! P^2 \to I \!\! P^n \cong
S^n I \!\! P^1$, where $n = {\rm deg}\,C^*$. The normalization map $ \nu \,:\, I \!\! P^1
\to C^* \subset I \!\! P^2$ can be given as $\nu = (g_0 : g_1 : g_2)\,$, where $g_i (z_0
, z_1) = \sum\limits_{j=0}^n b^{(i)}_j z_0^{n-j} z_1^j \,\,,\,\,i=0,1,2, $ are
homogeneous polynomials of degree $n$ without common factor.
If $x = (x_0 : x_1 : x_2) \in I \!\! P^2$ and $l_x \subset I \!\! P^{2*}$ is the dual line,
then $\rho_C (x) = \nu^* (l_x) \in S^n I \!\! P^1 = I \!\! P^n$ is defined by the
equation $\sum\limits_{i=0}^2 x_i g_i (z_0 : z_1 ) =0$. Thus, $\rho_C (x) =
(a_0 (x) : \dots : a_n (x) )$, where $a_j (x) = \sum\limits_{i=0}^2 x_i
b^{(i)}_j$.
Therefore, in the case of a plane rational curve $C$ the Zariski embedding
$\rho_C \,:\, I \!\! P^2 \to I \!\! P^n$ is the linear embedding given by the $3 \times
(n+1)$--matrix $B_C := (b^{(i)}_j )$, $i=0,1,2,\,j=0,\dots,n$. In what follows
we denote by $ I \!\! P^2_C$ the image $\rho_C ( I \!\! P^2)$, which is a plane in $ I \!\! P^n$.
By Lemma 3.7 the curve $C$ is an irreducible component of the plane cut of the
discriminant hypersurface $R_n \subset I \!\! P^n$, which has degree $2n-2$, by the plane
$\rho_C ( I \!\! P^2)$; the other irreducible components come from the artifacts
$L_C$ of $C$. This yields Lemma 1.9 in the Introduction:
$$ I \!\! P_C^2 \cap R_n = C \cup L_C \,\,.$$
The embedding $C^* \hookrightarrow I \!\! P^{2*}$ composed with the normalization
$\nu\,:\, I \!\! P^1 \to C^*$ is uniquely, up to projective equivalence, defined by
the corresponding linear series $g^2_n$ on $ I \!\! P^1$, and the embedding $\rho_C$
is uniquely defined by $C$ up to a choice of normalization of $C^*$. Thus,
$ I \!\! P^2_C$ is uniquely defined by $g^2_n$ up to the action on $ I \!\! P^n$ of the
group $ I \!\! P {\rm GL} (2,\, I \!\!\!\! C )
\times I \!\! P {\rm GL} (3,\, I \!\!\!\! C )$ via its natural representation in $ I \!\! P {\rm GL}
(n+1,\, I \!\!\!\! C )$, where the second factor leaves $ I \!\! P^2_C$ invariant. \\
\noindent {\bf 5.3.} {\it The rational normal curve.} The dual map ${\rho_C}^*
\,:\, I \!\! P^{n*} \to I \!\! P^{2*}$, given by the transposed $(n+1) \times 3$--matrix
$\,^t B_C = (b^{(j)}_i )$, $i=0,\dots,n,\, j=0,1,2$, defines a linear
projection with the center $N_C := {\rm Ker\,} \,^t B_C \subset I \!\! P^{n*}$ of
dimension $n - 3$. The curve $C^*$ is the image under this projection of the
rational normal curve $C_n^* = (z_0^n : z_0^{n-1} z_1 : \dots : z_1^n ) \subset
I \!\! P^{n*}$ (cf. [Ve, p.208]), i.e. $$\rho_C^* (C_n^*) = C^* \,\,..$$
Furthermore, $C_n^*$ is the image of $ I \!\! P^1 \cong C^*_{norm}$ under the
embedding $i \,:\, I \!\! P^1 \hookrightarrow I \!\! P^{n*}$ defined by the complete
linear system $|H| = |n(\infty)| \cong I \!\! P^n$. Therefore, $\nu = {\rho_C}^*
\circ i \,:\, I \!\! P^1 \to C^* \subset I \!\! P^{2*}$ is the normalization map. \\
\noindent {\bf 5.4.} {\it The duality picture}. It is easily seen that the rational
normal curve $C_n^* \subset I \!\! P^{n*}$ and the discriminant hypersurface
$R_n \subset I \!\! P^{n}$ are dual to each other. This yields the following duality
picture:
\begin{center}
\begin{picture}(500,70)
\put(150,55){$( I \!\! P^2 ,\,C \cup L_C)\,\,\,\,\,\,\, \hookrightarrow
\,\,\,\,\,\,\, ( I \!\! P^n ,\,R_n )$}
\put(155,5){$( I \!\! P^{2*} ,\,C^* ) \,\,\,\,\,\,\,\,\,\, \longleftarrow
\,\,\,\,\,\,\, ( I \!\! P^{n*} ,\,C_n^* )$}
\put(175,30){$\updownarrow$}
\put(291,30){$\updownarrow$}
\put(238,20){$\rho^*_C$}
\put(240,68){$\rho_C$}
\end{picture}
\end{center}
\noindent To describe this duality in more details, fix a point
$q=(z_0^n:z_0^{n-1}z_1:...:z_1^n) \in C_n^* \subset I \!\! P^{n*}$, and let $$F_q C^*_n =
\{ T^0_q C_n^* \subset T^1_q C_n^* \subset \dots \subset T^{n - 1}_q C_n^* \subset I \!\! P^{n*} \}$$ be
the flag of the osculating subspaces to $C_n^*$ at $q$, where ${\rm dim}\,T^k_q
C_n^* = k,\,\,T^0_q C_n^* = \{q\}$ and $T^1_q C_n^* = T_q C_n^*$ is the tangent
line to $C_n^*$ at $q$ (see [Na, p.110]). For instance, for $q = q_0 = (1 : 0 :
\dots : 0) \in C_n^*$ we have $T^k_q C_n^* = \{x_{k + 1} = \dots = x_n = 0 \}
\subset I \!\! P^{n*}$.
The dual curve $C_n \subset I \!\! P^n$ of $C_n^*$ is in turn projectively equivalent to
a rational normal curve; namely, $$C_n = \{ p \in I \!\! P^n \,|\, p = (z_1^n :
-nz_0 z_1^{n - 1} : \dots : (-1)^k { n \choose k } z_0^k z_1^{n - k} : \dots :
(-1)^n z_0^n )\}$$ Furthermore, the dual flag $F_q^{\perp} = \{ I \!\! P^n \supset
H^{n - 1}_q \supset \dots \supset H^0_q \}$, where $H^{n - k}_q := (T^{k - 1}_q
C_n^*)^{\perp}$, is nothing else but the flag of the osculating subspaces $F_p
C_n = \{T^{k - 1}_p C_n \}_{k = 1}^n$ of the dual rational normal curve $C_n \subset
I \!\! P^n$. An easy way to see this is to look at the flags at the dual points $q_0
= (1 : 0 : \dots : 0 ) \in C_n^*$ and $p_0 = (0 : \dots : 0 : 1 ) \in C_n$,
where all the flags consist of coordinate subspaces, and then to use the ${\rm
Aut}\, I \!\! P^1$-homogeneity (cf. 7.2 - 7.4 below).
The points of the osculating subspace $H^k_q = T^k_p C_n$ correspond to the
binary forms of degree $n$ for which $(z_0 : z_1) \in I \!\! P^1$ is a root of
multiplicity at least $n - k$. In particular, $H_q^{n - 2} = (T_q C^*_n
)^{\perp}$ consists of the binary forms which have $(z_0 : z_1)$ as a multiple
root. Therefore, the discriminant hypersurface $R_n$ is the union of these
linear subspaces $H_q^{n - 2} \cong I \!\! P^{n - 2}$ for all $q \in C_n^*$, and
thus it is the dual hypersurface of the rational normal curve $C_n^*$, i.e.
each of its points corresponds to a hyperplane in $ I \!\! P^{n*}$ which contains a
tangent line of $C_n^*$. At the same time, $R_n$ is the developable
hypersurface of the $(n-2)$--osculating subspaces $H_q^{n - 2} = T_p^{n - 2}
C_n$ of the dual rational normal curve $C_n \subset R_n$; here $T_p^{n - 2} C_n \cap
C_n = \{p\}$.
If $z_0 \neq 0$, then the subspace $H_q^{n - 2}$ in $ I \!\! P^n$ can be given as the
image of the linear embedding
$$ I \!\! P^{n-2} \ni (c_0:...:c_{n-2}) \longmapsto (a_0:a_1:a_2:...:a_n) =$$
$$= (\sum_{k=2}^n c_{k-2} (k-1) z_0^{n-k} z_1^k: -\sum_{k=2}^n c_{k-2} k
z_0^{n-k+1} z_1^{k-1} : c_0 z_0^n :...: c_{n-2} z_0^n) \in (T_qC_n^*)^* \subset
I \!\! P^n \,$$ (and symmetrically for $z_0 = 0$).
Consider the decomposition $D_{ij} = d_{ij} \times ( I \!\! P^1 )^{n-2}$ of the
diagonal hyperplane $D_{ij} \subset D_n$, where $d_{ij} \equiv I \!\! P^1$ is the
diagonal line in $ I \!\! P^1_i \times I \!\! P^1_j$, as the trivial fibre bundle $D_{ij}
\to I \!\! P^1$ with the fibre $( I \!\! P^1 )^{n-2}$. The subspaces $H_q^{n - 2} \subset R_n$
are just the images of the fibres under the Vieta map; moreover, the
restriction of the Vieta map $s_n\, :\,( I \!\! P^1)^n \to I \!\! P^n$ to a fibre yields
the Vieta map $s_{n-2}\, :\,( I \!\! P^1)^{n-2} \to I \!\! P^{n-2}$. The dual rational
normal curve $C_n \subset I \!\! P^n$ is the image $s_n (d_n )$ of the diagonal line $d_n
:= \bigcap_{i,\,j} D_{ij} = \{z_1 = \dots = z_n \} \subset ( I \!\! P^1 )^n$. \\[1ex]
\noindent {\bf 5.5.} {\it Artifacts as linear sections and the hyperplanes dual to
the cusps}.
By duality we have $N_C={\rm Ker}\,
{\rho_C}^* = ({\rm Im}\,\rho_C )^{\perp}$, i.e. $N_C = ( I \!\! P^2_C )^{\perp}$.
Therefore,
$$ I \!\! P^2_C = N_C^{\perp} = \bigcap\limits_{x^* \in N_C } {\rm Ker}\,x^* = \{ x
\in I \!\! P^n \,|\, <x, x^* > = 0 \,\,{\rm for\,\, all\,\,}x^* \in N_C \} \,\,\,.$$
A point $q$ on the rational normal curve $C_n^* \subset I \!\! P^{n*}$ corresponds to a
cusp of $C^*$ under the projection ${\rho_C}^*$ iff the center $N_C$ of the
projection meets the tangent developable $TC_n^*$, which is a smooth ruled
surface in $ I \!\! P^{n*}$, in some point $x_{q^*}$ of the tangent line $T_q C_n^*$
(see [Pi]). In this case it meets $T_q C_n^*$ at the only point $x_q^*$,
because otherwise $N_C$ would contain $T_q C_n^*$ and thus also the point $q$,
which is impossible since ${\deg}\,C^* = {\deg}\,C_n^* = n$.
Let $C^*$ have a cusp $B$ at the point $ q_0 = {\rho_C}^* (q)$, which
corresponds to a local branch of $C_n^*$ at the point $q \in C_n^*$ under the
normalizing projection ${\rho_C}^* \,:\,C_n^* \to C^*$. Define $L_{B, q_0} :=
{\rm Ker}\,x_q^* \subset I \!\! P^n$ to be the dual hyperplane of the point $x_q^*
\in N_C \cap T_q C_n^*$. Since $x_q^* \in N_C$, this hyperplane $L_{B, q_0}$
contains the image $ I \!\! P^2_C = \rho_C ( I \!\! P^2 )$. This yields a
correspondence between the cusps of $C^*$
and certain hyperplanes in $ I \!\! P^n$ containing the plane $ I \!\! P^2_C$. From the
definition it follows that $L_{B,q_0}$ contains also the dual linear space
$H_q^{n - 2} = (T_q C_n^*)^{\perp} \subset R_n$ of dimension $n - 2$. Since the
plane $ I \!\! P^2_C$ is not contained in $R_n$, we have $L_{B,q_0} = {\rm span}\,
( I \!\! P^2_C ,\,H_q^{n - 2} )$. It is easily seen that the intersection $ I \!\! P^2_C
\cap H_q^{n - 2}$ coincides with the tangent line $l_{q_0} \subset L_C$ of $C$,
which is dual to the cusp $q_0$ of $C^*$. Thus, the artifacts $L_C$ of $C$ are
the sections of $ I \!\! P^2_C$ by those osculating linear subspaces $H_q^{n - 2} \subset
R_n$ for which $q$ is a cusp of $C^*$; any other subspace $H_q^{n - 2}$ meets
the plane $ I \!\! P^2_C$ in one point of $C$ only. \\[1ex]
\noindent {\bf 5.6. Lemma.} {\it Let $C \subset I \!\! P^2$ be a rational curve whose
dual curve $C^* \subset I \!\! P^{2*}$ has degree $n$. Let $B$ be a cusp of $C^*$ with
center
$q_0 \in C^*$, and let $L_{B, q_0} \subset I \!\! P^n$ be the corresponding hyperplane
which contains the plane $ I \!\! P^2_C = \rho_C ( I \!\! P^2 )$ (see (5.5)). Then under
a suitable choice of a normalization of the dual curve $C^*$ we have $L_{B,
q_0} = {\bar A}_1$, where $${\bar A}_1 := \{(a_0 : \dots : a_n) \in I \!\! P^n
\,|\,a_1 = 0 \}\,\,.$$
\noindent The preimage ${\bar H}_0 := s_n^{-1}
({\bar A}_1) \subset ( I \!\! P^1)^n$ is the closure of the linear hyperplane in $ I \!\!\!\! C^n$
$${H}_0 := \{z = (z_1,\dots, z_n ) \in I \!\!\!\! C^n \,|\, \sum\limits_{i=1}^n z_i =0
\} \,\,\,.$$}
\noindent {\it Proof.} Up to a choice of coordinates in $ I \!\! P^{2*}$, which does not
affect the statement, we may assume that $C^*$ has a cusp $B$ at the point $q_0
= (0:0:1)$. Let $\infty = (1:0) \in I \!\! P^1$, and let $\nu \,:\, I \!\! P^1 \cong
C^*_{norm} \to C^* \hookrightarrow I \!\! P^2$ be composition of an isomorphism
$ I \!\! P^1 \cong C^*_{norm}$ with the normalization map. This isomorphism may be
chosen in such a way that the cusp $B$ corresponds to the local branch of
$ I \!\! P^1$ at $\infty$, and so
$\nu (\infty) = q_0$. Here as above $\nu = (g_0 : g_1 : g_2)\,$ is given by a
triple of homogeneous polynomials $g_i (z_0 , z_1) = \sum\limits_{j=0}^n
b^{(i)}_j z_0^{n-j} z_1^j \,\,,\,\,i=0,1,2, $ of degree $n$. Since $\nu
(\infty) = q_0$ we have
${\rm deg}_{z_0}\,g_0 < n\,,\,{\rm deg}_{z_0}\,g_1 < n\,,\,{\rm deg}_{z_0}\,g_2
= n$, i.e. $b^{(0)}_0 =b^{(1)}_0 =0\,,\,b^{(2)}_0 \neq 0$. Performing
Tschirnhausen transformation
$$ I \!\! P^1 \ni (z_0 \,:\,z_1) \longmapsto (z_0 - {b_1^{(2)} \over nb_0^{(2)}} z_1
\,: \, z_1) \in I \!\! P^1$$
we may assume, furthermore, that $b_1^{(2)} =0$.\\
\noindent {\it Claim 1. Under the above choice of parametrization the image
$ I \!\! P^2_C = \rho_C ( I \!\! P^2 )$ is contained in the hyperplane ${\bar A}_1 :=
\{(a_0 :\dots : a_n) \in I \!\! P^n\,|\,a_1 = 0\}$. } \\
Indeed, since $C^*$ has a
cusp at $q_0$ we have $(g_0 / g_2)'_{z_1} = (g_1 / g_2)'_{z_1} = 0$ at
the point $(1:0) \in I \!\! P^1$, i.e. $(g_0)'_{z_1} = (g_1)'_{z_1} = 0$ when $z_1
= 0$. This means that ${\rm deg}_{z_0}\,g_0 < n-1\,,\,{\rm deg}_{z_0}\,g_1 <
n-1$, i.e. $b^{(0)}_1 =b^{(1)}_1 =0$. And also $b_1^{(2)} =0$,
as it has been achieved above by making use of Tschirnhausen transformation.
Since $b_1^{(i)} =0\,,\,i=0,1,2$, we have $a_1 (x) \equiv 0$. Therefore,
$\rho_C (x) \in {\bar A}_1$ for any $x \in I \!\! P^2$, which proves the claim. \qed
\noindent {\it Claim 2. The dual space $H_q^{n - 2}$ to $T_qC_n^*$ is contained in
${\bar A}_1$.}\\
Indeed, since $\nu ( \infty) = q_0$ and $\nu = \rho_C^* \circ i$ with
$i: I \!\! P^1 \rightarrow C_n^* \subset I \!\! P^{n*}$ we get $q=(1:0:...:0)$. Thus, by (5.4) the
subspace $H_q^{n - 2} = (T_qC_n^*)^{\perp}$ is given by the equations
$\{a_0=a_1=0 \} $, and hence it is contained in ${\bar A}_1$. \qed
By (5.5) we have $L_{B,q_0} = {\rm span}\, ( I \!\! P^2_C ,\,H_q^{n - 2} )$.
Therefore, from these two claims we get $L_{B,q_0} = {\bar A}_1$. \\
To conclude the proof of the lemma it is enough to note that if $a_0 \neq 0$
and $a_1 = 0$, then the sum of the roots $z_1 + \dots + z_n$ of the equation
$a_0 z^n + a_1 z^{n-1} + \dots + a_n = 0$ is identically zero. Thus,
$${\bar H}_0 = s_n^{-1} ({\bar A}_1) = \{(\,(u_1 : v_1),\dots,(u_n : v_n)\,)
\in ( I \!\! P^1)^n \,|\, \sum\limits_{i=1}^n u_i / v_i = 0 \}\,,$$
which is the closure of the linear hyperplane $H_0 \subset I \!\!\!\! C^n$ as in the lemma.
\qed
\noindent {\bf 5.7.} {\it Monomial and quasi--monomial rational plane curves}. A
rational curve $C \subset I \!\! P^2$, which can be normalized (up to permutation) as
follows: $(x_0 (t) : x_1 (t) : x_2 (t)) = (at^k : bt^m : g(t))$, where $a,\,b
\in I \!\!\!\! C^* , \,k,\,m \in Z \!\!\! Z_{\ge 0}$ and $g \in I \!\!\!\! C [t]$, will be called {\it a
quasi--monomial curve}.
If here $g(t) = ct^l ,\, c\in I \!\!\!\! C^* ,\,l \in Z \!\!\! Z_{\ge 0}$, then $C$ is {\it a
monomial curve}; in this case we may assume that ${\rm min} \,\{k,\,l,\,m
\} =m =0$ and gcd$(k, \,l) =1\,,\,l > k$. Note that a linear pencil of monomial
curves $C_{\mu} = \{\alpha x_0^l + \beta x_1^{l-k} x_2^k =0\}$, where $\mu =
\alpha /\beta \in I \!\! P^1$, is self--dual, i.e. the dual curve of a monomial one
is again monomial and belongs to the same pencil. In contrast, the dual curve
to a quasi--monomial one is not necessarily projectively equivalent to a
quasi--monomial curve (recall that two plane curves $C,\,C'$ are {\it
projectively equivalent} if $C' = \alpha ( C)$ for some $\alpha \in I \!\! P {\rm
GL} (3;\, I \!\!\!\! C ) \cong {\rm Aut}\, I \!\! P^2$). The simplest example is the nodal
cubic $C = \{(x_0 : x_1 : x_2 ) = (t : t^3 : t^2 -1 )\}$. Indeed, its dual
curve is a quartic with three cusps (cf. Remark 4.2); but a quasi--monomial
curve may have at most two cusps. \\
The statement of the next lemma is easy to check, so the proof is omitted.
\\[1ex]
\noindent {\bf 5.8. Lemma.} {\it A quasi--monomial curve $C = (t^k : t^m : g(t))$,
where $k < m$ and $g(t) = \sum\limits_{j=0}^n b_j t^{n-j}$ is a polynomial of
degree $n \ge 3$, has no cusp iff it is one of the following curves:}
$$(t : t^{n \pm 1} : g(t)),\,\,b_n \neq 0$$
$$(t : t^n : g(t)),\,\,b_1 \neq 0, b_n \neq 0$$
$$(1 : t^{n \pm 1} : g(t)),\,\,b_{n-1} \neq 0$$
$$(1 : t^n : g(t)),\,\,b_1 \neq 0 \,\,{\rm and}\,\,b_{n-1} \neq 0 \,\,.$$
{\it In particular, a monomial curve $C = (t^k : t^l : 1)$, where $k < l $ and
gcd$(k,\,l) = 1$, has no cusp iff it is a smooth conic $C = (t : t^2 : 1)$.}
\\[1ex]
\noindent {\bf 5.9.} {\it Parametrized rational plane curves}. Note that, while the
action of the projective group $PGL(3, I \!\!\!\! C)$ on $ I \!\! P^2$ does not affect the
image $ I \!\! P^2_C = \rho_C ( I \!\! P^2 ) \subset I \!\! P^n = S^n I \!\! P^1$, the choice of the
normalization $ I \!\! P^1 \to C^*$, defined up to the action of the group $PGL(2,
I \!\!\!\! C ) = {\rm Aut} I \!\! P^1$, usually does (cf. (5.2)). This is why in the next
lemma we have to fix the normalization of a rational plane curve $C$. This
automatically fixes a normalization of its dual curve $C^*$, and vice versa.
Indeed, recall that if $C = (g_0 : g_1 : g_2 )$, where $g_i \in I \!\!\!\! C [t]
,\,\,i=0,1,2$, is a parametrized rational plane curve, then the dual curve
$C^*$ has, up to cancelling of the common factors, the parametrization $C^* =
(M_{12} : M_{02} : M_{01} )$, where $M_{ij}$ are the $2 \times 2$--minors of
the matrix
$$ \left( \begin{array}{ccc}
g_0 & g_1 & g_2 \\
g'_0 & g'_1 & g'_2
\end{array} \right)$$
\noindent Furthermore, the equation of $C$ can be written as $\frac{1}{x_2^d }
{\rm Res} \,(x_0 g_2 - x_2 g_0 , x_1 g_2 - x_2 g_1 ) = 0$, where $d = {\rm
deg}\,C$ and ${\rm Res}$ means resultant (see e.g. [Au, 3.2]).
We will use the following terminology. By {\it a parametrized rational plane
curve} we will mean a rational curve $C$ in $ I \!\! P^2$ with a fixed normalization
$ I \!\! P^1 \to C$ of it. {\it A parametrized monomial} resp. {\it a parametrized
quasi--monomial plane curve} is a parametrized rational plane curve such that
all resp. two of its coordinate functions are monomials.
Clearly, projective equivalence between parametrized curves is a stronger
relation than just projective equivalence between underlying projective curves
themselves. \\[1ex]
\noindent {\bf 5.10. Lemma.} {\it A parametrized rational plane curve $C^* \subset
I \!\! P^{2 *}$ of degree $n$ is projectively equivalent to a parametrized monomial
resp. quasi--monomial curve iff $ I \!\! P^2_C \subset I \!\! P^n$ is a coordinate plane resp.
containes a coordinate axis. This axis is unique iff $C^*$ is projectively
equivalent to a parametrized quasi--monomial curve, but not to a monomial one.}
\\[1ex]
\noindent {\it Proof.} Let $\nu \,:\,t \longmapsto (at^k : bt^m : g(t))$, where
$a,\,b \in I \!\!\!\! C^* ,\,\,g \in I \!\!\!\! C [t]$ and $t=z_0 / z_1 \in I \!\! P^1$, define a
parametrized quasi--monomial curve $C^* \subset I \!\! P^{2 *}$ of degree $n$. Denote
$e_k = (0 : \dots : 0 : 1_k : 0 : \dots : 0) \in I \!\! P^n$. Then $\rho_C$ is given
by the matrix $B_C = (b^{(0)},\,b^{(1)},\,b^{(2)}) = (ae_{n-k} ,\, be_{n-m}
,\,b^{(2)})$, and therefore $ I \!\! P^2_C = \rho_C ( I \!\! P^2 ) = {\rm
span}\,(b^{(0)},\,b^{(1)},\,b^{(2)})$ contains the coordinate axis
$l_{n-k,\,n-m}$, where $l_{i,j} := {\rm span}\,(e_i ,\,e_j ) \subset I \!\! P^n$.
If $C^*$ is a paramatrized monomial curve, i.e. if $g(t) = ct^r$, where $c \in
I \!\!\!\! C^*$, then clearly $ I \!\! P^2_C$ is the coordinate plane $ I \!\! P_{n-k,\,n-m,\,n-r}
:= {\rm span}\,(e_{n-k},\,e_{n-m},\,e_{n-r})$.
Since the projective equivalence of parametrized plane curves does not affect
the $ I \!\! P^2_C$, this yields the first statement of the lemma in one direction.
Vice versa, suppose that $ I \!\! P^2_C$ coincides with the coordinate plane
$ I \!\! P_{n-k,\,n-m,\,n-r}$. Performing a suitable linear coordinate change in
$ I \!\! P^{2 *}$ we may assume that $b^{(0)} = e_{n-k} , \,b^{(1)} = e_{n-m}
,\,b^{(2)} = e_{n-r}$, i.e. that $\nu (t) = (t^k : t^m : t^r)$. Therefore, in
this case the parametrized curve $C^*$ is projectively equivalent to a monomial
curve.
Suppose now that $ I \!\! P^2_C$ contains the coordinate axis $l_{n-k ,\,n-m}$.
Performing as above a suitable linear coordinate change in $ I \!\! P^{2 *}$ we may
assume that $b^{(0)} = e_{n-k} , \,b^{(1)} = e_{n-m}$, and so $\nu (t) = (t^k :
t^m : g(t))$. In this case $C^*$ is projectively equivalent to a parametrized
quasi--monomial curve. This proves the first assertion of the lemma.
Let $C^* = (at^{n-k} : bt^{n-m} : g(t))$ be a parametrized quasi--monomial
curve which is not projectively equivalent to a monomial one. Then as above
$ I \!\! P^2_C \supset l_{k,\,m}$, and this is the only coordinate axis contained in
$ I \!\! P^2_C$. Indeed, if $l_{i,\,j} \subset I \!\! P^2_C$, where $\{i,\,j\} \neq \{k,\,m\}$,
then $ I \!\! P^2_C$ would contain at least three distinct vertices $e_{\alpha}$,
where $\alpha \in \{i,\,j,\,k,\,m\}$, and so $ I \!\! P^2_C$ would be a coordinate
plane, what has been excluded by our assumption. The opposite statement is
evidently true. This concludes the proof. \qed
\noindent {\bf 5.11. Remarks.} {\it a}. Let $C^* = (at^k : bt^m : ct^r)$, where
$a,\, b,\, c \in I \!\!\!\! C^*$, be a parametrized monomial curve of degree $n$. To be
a normalization, this parametrization should be irreducible, i.e. up to
permutation there should be $0 = k < m < r = n$, where ${\rm gcd}\,(m,\,n) =
1$. Thus, $ I \!\! P^2_C = I \!\! P_{0,\,n-m,\,n}$ is a rather special coordinate plane.
\\[1ex]
{\it b}. Let $C^*$ be the parametrized quasi--monomial curve $C_{k, m, g} :=
(at^k : bt^m : g(t))$, which is not equivalent to a monomial one. Then the only
coordinate axis contained in $ I \!\! P^2_C$ is the axis $l_{n-k,\,n-m} := {\rm
span}\,(e_{n-k},\,e_{n-m}) = \rho_C (l_2 )$, where $l_2 := \{x_2 = 0 \} \subset
I \!\! P^2$. Furthermore, if $C^*$ is obtained from such a curve by a permutation of
the coordinates, then still the only coordinate axis contained in $ I \!\! P^2_C$ is
$l_{n-k,\,n-m}$. \\[1ex]
\noindent {\bf 5.12.} {\it An equivariant meaning of the Vieta map}. Consider the
following $\, I \!\!\!\! C^*$--actions on $( I \!\! P^1)^n$ resp. on $ I \!\! P^n = S^n I \!\! P^1$:
$${\tilde{G}} \,:\, I \!\!\!\! C^* \times ( I \!\! P^1)^n \ni (\lambda,\,((u_1 : v_1),\dots, (u_n
: v_n))) \longmapsto ((\lambda u_1 : v_1), \dots, (\lambda u_n : v_n)) \in
( I \!\! P^1)^n$$
\noindent resp.
$$G \,:\, I \!\!\!\! C^* \times I \!\! P^n \ni (\lambda,\, (a_0 : a_1 : \dots : a_n )) \longmapsto
(a_0 : \lambda a_1 : \lambda^2 a_2 : \dots : \lambda^n a_n ) \in I \!\! P^n$$
\noindent Note that the Vieta map $s_n : ( I \!\! P^1)^n \to I \!\! P^n$ (see (5.1)) is
equivariant with respect to these $\, I \!\!\!\! C^*$--actions and its branching
divisors $D_n$ resp. $R_n$ are invariant under ${\tilde{G}}$ resp. $G$.
Identifying
$ I \!\!\!\! C$ with $ I \!\! P^1 \setminus \{(1:0)\}$, we fix an embedding $ I \!\!\!\! C^n \hookrightarrow
( I \!\! P^1)^n$; denote its image by $ I \!\!\!\! C_z^n$. Both this Zariski open part of
$( I \!\! P^1)^n$ and its complementary divisor are ${\tilde{G}}$--invariant. In
turn, the hyperplane $ I \!\! P_0^{n-1} := \{a_0 = 0 \}$ in $ I \!\! P^n$, as well as any
other coordinate linear subspace of $ I \!\! P^n$, and its complement $ I \!\!\!\! C_a^n :=
I \!\! P^n \setminus I \!\! P_0^{n-1}$ are $G$--invariant. \\[1ex]
The next lemma is a usefull supplement to Lemma 5.10. \\
\noindent {\bf 5.13. Lemma.} {\it A parametrized rational plane curve $C^* \subset
I \!\! P^{2*}$ is projectively equivalent to a parametrized quasi--monomial curve
iff $ I \!\! P^2_C \subset I \!\! P^n$ contains a one--dimensional $G$--orbit. This orbit is
unique iff $C^*$ is projectively equivalent to a parametrized quasi--monomial
curve, but not to a monomial one.} \\[1ex]
\noindent {\it Proof.} Let $\lambda \longmapsto (a_0 : \lambda a_1 :\dots
:\lambda^n a_n )$, where $\lambda \in I \!\!\!\! C^*$, be a parametrization of the
$G$--orbit $O_p$ throuh the point $p = (a_0 : \dots : a_n ) \in I \!\! P^n$. Since
the non-zero coordinates here are linearly independent as functions of
$\lambda$, the orbit $O_p \subset I \!\! P^n$ is contained in a projective plane iff all
but at most three of coordinates of $p$ vanish. If $p$ has exactly three
non--zero coordinates, then the only plane that containes ${\bar O}_p$ is a
coordinate plane. If only two of the coordinates of $p$ are not zero, then the
closure ${\bar O}_p$ is a coordinate axis. Since we consider a one--dimensional
orbit, the case of one non--zero coordinate is excluded. Now the lemma easily
follows from Lemma 5.10. \qed
\section{C--hyperbolicity of complements of rational curves in presence of
artifacts}
Before proving an analog of Theorem 4.1 for the case of a rational curve (see
Theorem 6.5 below), let us consider simple examples which illustrate some ideas
used in the proof.
In (1.3) we gave an example of a quintic $C_5 \subset I \!\! P^2$ (union of five lines)
whose complement is C--hyperbolic. Here is another one.\\[1ex]
\noindent {\bf 6.1. Example.} Let $C \subset I \!\! P^2$ be a smooth conic and $L = l_1 \cup
l_2 \cup l_3$ be the union of three distinct tangents of $C$.
\begin{center}
\begin{picture}(500, 90)
\thicklines
\put(212,85){\line(1,-2){40}}
\put(222,85){\line(-1,-2){40}}
\put(174,16){\line(1,0){92}}
\put(217,34){\circle{37}}
\end{picture} \\
Figure 2
\end{center}
\noindent {\it Claim. a) $X_1 := I \!\! P^2 \setminus (C \cup l_1)$ is super--Liouville and
its Kobayashi pseudo--distance $k_{X_1}$ is identically zero. \\
\noindent b) Put $X_2 = I \!\! P^2 \setminus (C \cup l_1 \cup l_2 )$. Let $(C_{\alpha}),
\,\alpha \in I \!\! P^1 $, be the linear pencil of conics generated by $C$ and $l_1
+ l_2$, where $C = C_{(1 : 1)}$ and $l_1 + l_2$ = $C_{(1 : 0)}$. Then the image
of any entire curve $ I \!\!\!\! C \to X_2$ is contained in one of the conics
$C_{\alpha}$, and $k_{X_2} (p,\,q) = 0$ iff $p,\,q \in C_{\alpha}$ for some
$\alpha \in I \!\! P^1$. Furthermore, $X_2$ is neither C--hyperbolic, nor
super--Liouville (see (2.2)). \\
\noindent c) $X = X_3 := I \!\! P^2 \setminus (C \cup L)$ is C--hyperbolic, Kobayashi
complete hyperbolic and hyperbolically embedded into $ I \!\! P^2$.} \\
\noindent {\it Proof.} a) is easily checked by applying, for instance, Lemma 2.3.
An alternative way is to note that $X_1$ is isomorphic to the product $ I \!\!\!\! C
\times I \!\!\!\! C^*$. \qed
\noindent b) Consider the affine chart $ I \!\!\!\! C^2 \cong I \!\! P^2 \setminus l_2$ in $ I \!\! P^2$. We
have $X_2 \cong I \!\!\!\! C^2 \setminus \Gamma$, where the affine curve $\Gamma := (C \cup
l_1) \setminus l_2$ can be given in appropriate coordinates by the equation $y(x^2 -
y) = 0$. Let the double covering $\pi \,:\, I \!\!\!\! C^2 \to I \!\!\!\! C^2$ branched over the
axis $l_1$ be given as $(x, y) = \pi (x, z) := (x, z^2)$. It yields the
non--ramified double covering $Y \to X_2$, where $Y := I \!\!\!\! C^2 \setminus \pi^{-1}
(\Gamma )$. Here $ \pi^{-1} (\Gamma )$ is union of three affine lines $m_0 =
\{z = 0 \},\,\,m_1 = \{x = z \},\,\,m_{-1} = \{x = -z \}$, which are level sets
of the rational function $\phi (x,\,z) := z / x$. It defines a holomorphic
mapping $\phi \,|\,Y \,:\, Y \to I \!\! P^1 \setminus \{0,\,1,\,-1 \}$. Therefore, for any
entire curve $f\,:\, I \!\!\!\! C \to X_2$ its covering curve ${\tilde f} \,:\, I \!\!\!\! C \to
Y$ has the image contained in an affine line $l_{\beta_0}$ from the linear
pencil $l_{\beta} := \{x = \beta z\} ,\,\,\beta \in I \!\!\!\! C$. Thus, the image $f
( I \!\!\!\! C )$ is contained
the conic $C_{\alpha_0}$ from the linear pencil $C_{\alpha} = \{x^2 = \alpha y
\}$, where $\alpha = \beta^2$. This proves the first assertion in b). The
second one easily follows from the inequality $k_Y \ge \phi^* k_{ I \!\! P^1 \setminus \{3
\,{\rm points}\}}$ and the equality $k_{X_2} = \pi_* k_Y$. Finally, since the
tautological line bundle $\phi \,:\, I \!\!\!\! C^2 \setminus \{{\bar 0}\} \to I \!\! P^1$ is
trivial over $ I \!\! P^1 \setminus \{{\rm a\,\,point}\}$, there is an isomorphism $Y \cong
I \!\!\!\! C^* \times ( I \!\! P^1 \setminus \{3 \,{\rm points}\})$.. Therefore, the universal
covering $U_Y \cong U_{X_2}$ of $Y$ resp. of $X_2$ is biholomorphic to $ I \!\!\!\! C
\times \Delta$. Hence, $X_2$ is neither C--hyperbolic, nor super--Liouville.
\qed
\noindent c) We can treat the dual curve of $C \cup L$ as the dual conic $C^* \subset
I \!\! P^{2*}$ with three distinguished points $q_1 ,\, q_2 , \,q_3$ on it, whose
dual lines are, respectively, $l_1 , \,l_2 ,\, l_3$. Choose an isomorphism $C^*
\cong I \!\! P^1$ in such a way that $q_1 , \,q_2 ,\, q_3 \in C^*$ correspond,
respectively, to the points $(0 : 1),\, (1 : 0),\, (1 : 1) \in I \!\! P^1$. The
Vieta map $s_2 \,:\,( I \!\! P^1 )^2 \to I \!\! P^2 = S^2 I \!\! P^1$ is given by the formula
$$s_2 \,:\,((u_1 : v_1),\, (u_2 : v_2)) \longmapsto (v_1 v_2 : -(u_1 v_2 + u_2
v_1 ) : u_1 u_2 )\,\,.$$
To the distinguished points $(0 : 1),\, (1 : 0),\, (1 : 1) \in I \!\! P^1$ there
correspond six generators of the quadric $ I \!\! P^1 \times I \!\! P^1$, three vertical ones
and three horizontal ones. Their images under the Vieta map $s_2$ is the union
$L_0$ of three lines $x_0 =0 ,\, x_2 =0, \,x_0 + x_1 + x_2 =0\,$ in $ I \!\! P^2 =
S^2 I \!\! P^1$, which are tangent to the conic $C_0 := s_2 ({\bar D}_2 ) = \{x_1^2
- 4x_0 x_2 =0 \} \subset I \!\! P^2$, where ${\bar D}_2 ={\bar D}_{1,2} $ is the diagonal
of $( I \!\! P^1 )^2$. Thus, we have the commutative diagram: \\
\begin{picture}(270,150)
\unitlength0.2em
\thicklines
\put(45,65){$Y\hookrightarrow {\bar Y}$}
\put(66,67){{\vector(1,0){40}}}
\put(112,65){$( I \!\! P^1)^2$}
\put(40,49){${\tilde s}_2$}
\put(63,57){${\tilde\rho}_C$}
\put(50,62){{\vector(1,-1){10}}}
\put(47,60){\vector(0,-1){23}}
\put(120,60){\vector(0,-1){23}}
\put(60,44){$( I \!\!\!\! C^{**} )^2 \setminus D_2 \hookrightarrow ( I \!\!\!\! C^{**} )^2$}
\put(105,52){{\vector(1,1){10}}}
\put(76,41){\vector(0,-1){23}}
\put(8,29){${ I \!\! P^2} \setminus (C\cup L) = X \hookrightarrow I \!\! P^2$}
\put(102,29){$ I \!\! P^2 = S^2 I \!\! P^1 \hbox{\hskip 1cm} $}
\put(67,30){\vector(1,0){30}}
\put(45,20){$\rho_C$}
\put(48,27){{\vector(1,-1){10}}}
\put(60,11){${ I \!\! P^2} \setminus (C_0 \cup L_0)$}
\put(92,18){{\vector(1,1){10}}}
\put(160,35){(5)}
\put(122,49){$s_2$}
\end{picture}
\noindent where ${\tilde{s_2}} \,:\,Y \to X$ is the induced covering, $\, I \!\!\!\! C^{**} =
I \!\! P^1 \setminus \{3 \,{\rm points}\}$ and the horizontal arrows are isomorphisms. It
follows that $Y \cong ( I \!\!\!\! C^{**} )^2 \setminus D_2 \subset ( I \!\!\!\! C^{**} )^2$ is C--hyperbolic,
and therefore, $X$ is C--hyperbolic, too.
It is easily seen that ${\rm reg}\,(C\cup L) = (C\cup L) \setminus {\rm sing}\,(C\cup
L)$ is hyperbolic. Therefore, by Proposition 2.7 $X = I \!\! P^2 \setminus (C\cup L)$ is
Kobayashi complete hyperbolic and hyperbolically embedded into $ I \!\! P^2$. \qed
Here is one more example of a curve with properties as in Claim c) above.
\\[1ex]
\noindent {\bf 6.2. Example.} Let the things be as in the previous example.
Performing the Cremona transformation $\sigma$ of $ I \!\! P^2$ with center at the
points of intersections of the lines $l_1 , l_2 , l_3$, we obtain a 3-cuspidal
quartic $C' := \sigma (C)$ together with three new lines $m_1 , m_2 , m_3$,
passing each one through a pair of cusps of $C'$ (they are images of the
exceptional curves of the blow-ups by $\sigma$ at the above three points; see
Fig. 3).
\begin{center}
\begin{picture}(500, 90)
\thicklines
\put(212,85){\line(1,-2){40}}
\put(222,85){\line(-1,-2){40}}
\put(174,15){\line(1,0){92}}
\end{picture} \\
Figure 3
\end{center}
$$
$$
\noindent Put $L' = m_1 \cup m_2 \cup m_3$ and $X' = I \!\! P^2 \setminus (C' \cup L' )$.
Since $\sigma\,|\,X\,:\,X \to X'$ is an isomorphism and $X$ is C--hyperbolic,
we have that $X'$ is also C--hyperbolic. The same reasoning as above ensures
that $X'$
is Kobayashi complete hyperbolic and hyperbolically embedded into $ I \!\! P^2$.
\\[1ex]
The next lemma will be used in the proof of Theorem 6.5. From now on `bar' over
a letter will denote a projective object, in contrast with the affine ones.
\\[1ex]
\noindent {\bf 6.3. Lemma.} {\it Let ${\bar H}_0$ be the hyperplane in $ I \!\! P^{n-1}$
given by the equation $\sum\limits_{i=1}^n x_i = 0$, and let ${\bar D}_{n-1} =
\bigcup\limits_{1\le i<j\le n} {\bar D}_{ij}$ be the union of the diagonal
hyperplanes, where ${\bar D}_{ij} \subset I \!\! P^{n-1}$ is given by the equation $x_i
- x_j =0$. Then ${\bar H}_0 \setminus {\bar D}_{n-1}$ is C--hyperbolic, Kobayashi
complete hyperbolic and hyperbolically embedded
into ${\bar H}_0 \cong I \!\! P^{n-2}$.}\\
[1ex]
\noindent {\it Proof.} Put $y_i = x_1 - x_{i+1}\,\,,\,\,i=1,\dots,n-1$. Then $z_i =
y_i / y_{n-1}\,\,,\,\,i=1,\dots,n-2 $, are coordinates in the affine chart
${\bar H}_0 \setminus {\bar D}_{1, n} \cong I \!\!\!\! C^{n-2}$. In these coordinates ${\bar
D}_{1, i+1} \cap {\bar H}_0$ resp. ${\bar D}_{i+1, n} \cap {\bar H}_0$ is given
by the equation $z_i = 0$ resp. $z_i =1\,\,,\,\,i=1,\dots,n-2 $. Thus, ${\bar
H}_0 \setminus {\bar D}_{n-1} \hookrightarrow ( I \!\!\!\! C^{**} )^{n-2}$, where $ I \!\!\!\! C^{**} :=
I \!\! P^1 \setminus \{3\,\,points\}$. By Lemma 2.5 it follows that ${\bar H}_0 \setminus {\bar
D}_{n-1}$ is C--hyperbolic.
To prove Kobayashi complete hyperbolicity and hyperbolic embededdness we may
use the following criterion [Za1, Theorem 3.4] :\\
\noindent {\it The complement of a finite set of hyperplanes $L_1,\dots,L_N$ in
$ I \!\! P^n$ is hyperbolically embedded into $ I \!\! P^n$ iff (*) for any two distinct
points $p, q$ in $ I \!\! P^n$ there is a hyperplane $L_i\,\,,\,\,i \in
\{1,\dots,N\},$ which does not contain any of them. }\\
Note that the complement of a hypersurface is locally complete hyperbolic
[KiKo, Proposition 1], and therefore its hyperbolic embededdness implies the
complete hyperbolicity (see [Ki] or [KiKo, Theorem 4]). Therefore, it is enough
to check that the union of hyperplanes ${\bar H}_0 \cap {\bar D}_{n-1}$ in
${\bar H}_0 \cong I \!\! P^{n-1}$ satisfies the above condition (*).
Supposing the contrary we would have that there exists a pair of points
$p, q \in {\bar H}_0 \,\,,\,\,p \neq q\,,$ such that each of the diagonal
hyperplanes ${\bar D}_{ij}$ contains at least one of these points.
Put $p= (x'_1 :\dots :x'_n )$ and $q= (x''_1 :\dots :x''_n )$.
Since $(\bigcap\limits_{i, j} {\bar D}_{ij} ) \cap {\bar H}_0 = \emptyset$, we
may
assume that up to permutation $x'_1 =\dots =x'_k$ and
$x''_{k+1} =\dots =x''_n$, where $2\le k \le n-1$, and moreover,
that $x'_l \neq x'_i$ for each $i\le k < l \le n$. The latter means that $p
\notin {\bar D}_{il}$ for such $i\,,\,l$. Therefore, we must have $q \in {\bar
D}_{il}$ for $i\le k < l$. In particular, $q\in {\bar D}_{i, k+1}
\,\,,\,\,i=1,\dots,k$, and so $x''_1=\dots =x''_k = x''_{k+1} =\dots =x''_n$,
which is impossible, since $q \in {\bar H}_0$. \qed
\noindent {\bf 6.4. Remark.} If $n = 4$, so that
${\bar H}_0 \cong I \!\! P^2$, it is easily seen that ${\bar D}_3 \cap {\bar H}_0$
is a complete quadruple in $ I \!\! P^2$, i.e. the union of six lines defined by four
points in general position. \\[1ex]
Now we are ready to extend Theorem 4.1, under certain additional restrictions,
to the case of a rational curve. \\[1ex]
\noindent {\bf 6.5. Theorem.} {\it Let $C \subset I \!\! P^2$ be a rational curve whose dual
curve $C^*$ has at least one cusp, so that $C$ has the artifacts $L_C \neq
\emptyset$. Let $X := I \!\! P^2 \setminus (C \cup L_C)$. Then the following statements
hold. \\
a) If the dual curve $C^*$ is not projectively equivalent to a quasi--monomial
one, then $X$ is almost C--hyperbolic.\\
b) $X$ is still almost C--hyperbolic if $C^*$ is projectively equivalent to a
quasi--monomial curve $C_{k,\,m,\,g} :=\{(t^k : t^m : g(t))\}$ of degree $n$,
but not to a monomial one, except the cases when, up to a choice of
normalization, $C_{k,\,m,\,g}$ is one of the curves $\{(1 : t^n : g(t))\}$ or
$\{(t : t^n : g(t))\}$, where $g \in I \!\!\!\! C [t]$ and ${\rm deg}\,g \le n-2$. In
the latter cases $X$ is almost C--hyperbolic modulo the line $l_2 := \{x_2 = 0
\} \subset I \!\! P^2$ in the coordinates where $\,C^* = C_{k,\,m,\,g}$.\\
c) Let $C=C_{\mu_0}$ be a monomial curve \footnote {The case when $C^*$ is
projectively equivalent to a monomial curve is easily deduced to this one.}
from the linear pencil $C_{\mu} = \{\alpha x_0^n + \beta x_1^k x_2^{n-k} =0\}$,
where $\mu = (\alpha : \beta ) \in I \!\! P^1$. Then $k_X (p,\,q) = 0$ iff $p,\,q
\in C_{\mu}$ for some $\mu \in I \!\! P^1 \setminus \{\mu_0 \}$. In particular, any entire
curve $ I \!\!\!\! C \to X$ is contained in one of the curves of the linear pencil
$(C_{\mu})$.} \\[1ex]
\noindent {\it Proof.} The proof will be done in several steps. We will start with
the main construction used in the proof. \\
\noindent {\it Basic construction.} Fix a cusp $q_0$ of $C^*$, and let $q_0^* \subset
I \!\! P^2$
be the dual line of $q_0$. Clearly, $q_0^* \subset L_C$. Choosing an appropriate
isomorphism $ I \!\! P^1 \cong C^*_{norm}$ and coordinates in $ I \!\! P^2$ as in the proof
of Lemma 5.6, by this lemma we may assume that $\nu (\infty ) = q_0 =
(0 : 0 : 1) \in I \!\! P^{2*}$, $q_0^* = l_2 = \{x_2 = 0 \} \subset I \!\! P^2$ and
$ I \!\! P^2_C = \rho_C ( I \!\! P^2 ) \subset {\bar A}_1 \subset I \!\! P^n = S^n I \!\! P^1$, where $n =
{\rm deg}\,C^*$ and ${\bar A}_1 = \{(a_0 :\dots : a_n) \in I \!\! P^n\,|\,a_1 = 0\}
$. Let $ I \!\!\!\! C^n_z \subset ( I \!\! P_1)^n$ and $ I \!\!\!\! C^n_{(a)} \subset I \!\! P^n$ be as in (5.12). Then,
as it is easily seen, $\rho_C (X) \subset \rho_C ( I \!\! P^2 \setminus l_2 ) \subset s_n
( I \!\!\!\! C^n_{(z)} ) \cong I \!\!\!\! C^n_{(a)} \subset I \!\! P^n$, where $s_n \,:\, I \!\!\!\! C^n_{(z)} \to
I \!\!\!\! C^n_{(a)}$ is the restriction of the Vieta map (see (5.1)).
By (5.12) this affine Vieta map yields the non--ramified covering
$s_n\,:\,{H}_0 \setminus D_n \to A_1 \setminus R_n$, where as in Lemma 5.6 above ${H}_0 =
\{z = (z_1,\dots, z_n ) \in I \!\!\!\! C^n \,|\, \sum\limits_{i=1}^n z_i =0 \}$, $D_n$
is the union of the affine diagonal hyperplanes $D_{ij} = \{z \in I \!\!\!\! C^n \,|\,
z_i = z_j \}\,\,,\,1\le i < j \le n$,
$A_1 := \{a = (a_1,\dots, a_n ) \in I \!\!\!\! C^n_{(a)} \,|\,a_1 = 0 \} \cong I \!\!\!\! C^{n-1}
$ and $R_n \subset I \!\!\!\! C^n_{(a)}$ is the affine discriminant hypersurface.
The Zariski map gives the linear embedding $\rho_C \,|\,X \,:\,X \to A_1 \setminus
R_n$. Let $ {\tilde{s}}_n \,:\,Y \to X$ be the non--ramified covering induced
by the Vieta covering via this embedding.
Denote by $\pi$ the canonical projection $ I \!\!\!\! C^n_{(z)} \setminus \{{\bar 0}\} \to
I \!\! P^{n-1}$. Put ${\bar H}_0 := \pi ({H}_0 ) \cong I \!\! P^{n-2} \subset I \!\! P^{n-1}$ and
${\bar D}_{ij} := \pi (D_{ij} )\,,\,{\bar D}_{n-1} := \pi (D_n ) =
\bigcup\limits_{1\le i < j \le n} {\bar D}_{ij}$. By Lemma 6.3 ${\bar H}_0 \setminus
{\bar D}_{n-1}$ is C--hyperbolic, Kobayashi complete hyperbolic and
hyperbolically embedded into ${\bar H}_0 \cong I \!\! P^{n-2}$.
Thus, we have the following commutative diagram: \\
\begin{picture}(250,80)
\unitlength0.2em
\thicklines
\put(-4,5) {${ I \!\! P^2} \setminus (C\cup L_C) = X$}
\put(35,25){$Y$}
\put(37,22){\vector(0,-1){11}}
\put(27,16){${\tilde s}_n$}
\put(44,27){{\vector(1,0){14}}}
\put(47,30){${\tilde\rho}_C$}
\put(44,6){\vector(1,0){12}}
\put(47,10){$\rho_C$}
\put(62,5){$A_1 \setminus R_n$}
\put(62,25){${H}_0 \setminus D_n$}
\put(70,22){\vector(0,-1){11}}
\put(75,16){$s_n$}
\put(170,15){(6)}
\put(82,27){{\vector(1,0){14}}}
\put(87,30){$\pi$}
\put(100,25){${\bar H}_0 \setminus {\bar D}_{n-1} \hookrightarrow I \!\! P^{n-2}$}
\end{picture}
\noindent
where ${\tilde\rho}_C$ is an injective holomorphic mapping. Note that here the
Vieta map $s_n$ is equivariant with respect to the $\, I \!\!\!\! C^*$--actions
${\tilde{G}}$ on $ {H}_0 \setminus D_n$ and $G$ on $A_1 \setminus R_n$, respectively, and
all the fibres of the projection $\pi$ are one--dimensional
${\tilde{G}}$--orbits (see(5.12)). \\
\noindent {\it Proof of a)}. Under the assumption of a) $C^*$ is not projectively
equivalent to a quasi--monomial curve. Then we have the following assertion.\\
\noindent {\it Claim. The mapping $\pi \circ {\tilde \rho}_C \,:\,Y \to {\bar H}_0
\setminus {\bar D}_{n-1}$ has finite fibres.}\\
\noindent Indeed, since the fibres of $\pi$ are $\tilde G$--orbits, it is enough to
show that any $\tilde G$--orbit in $H_0 \subset I \!\!\!\! C^n_{(z)}$ has a finite
intersection with ${\tilde \rho}_C (Y)$. Or, what is equivalent, that any
$G$--orbit in $A_1 \subset I \!\!\!\! C^n_{(a)}$ has a finite intersection with $\rho_C (X)
\subset I \!\! P^2_C$. We have shown in Lemma 5.13 above that if the latter fails, i.e.
if $ I \!\! P^2_C$ contains a one-dimensional $G$--orbit, then $C^*$ (paramatrized as
above) is projectively equivalent to a (paramatrized) quasi--monomial curve,
which is assumed not to be the case. This yields the claim.
Since ${\bar H}_0 \setminus {\bar D}_{n-1}$ is C--hyperbolic, by Lemma 2.5 this
implies that $X$ is almost C--hyperbolic. \qed
\noindent {\it Proof of b)}. We still fix a parametrization of $C^*$ as in the
basic construction above, and so we fix the $ I \!\! P^2_C$ in $ I \!\! P^n$. If $ I \!\! P^2_C$
does not contain any coordinate line, we can finish up the proof like in a) and
conclude that $X$ is almost C--hyperbolic. So, assume further that $ I \!\! P^2_C$
does contain a coordinate line. By Lemma 5.10 this means that $C^*$ as a
parametrized curve is projectively equivalent to a quasi--monomial curve. Since
by our assumption it is not equivalent to a monomial one, the plane $ I \!\! P^2_C$
is not a coordinate one. After an appropriate change of coordinates in $ I \!\! P^2$
which does not affect $ I \!\! P^2_C$ we may assume that $C^* = C_{f,\,g,\,h} := (f :
g : h)$, where $f,\,g,\,h \in I \!\!\!\! C[t]$ and two of them are the monomials $t^k
,\,t^m$. We have that $l_{n-k,\,n-m} \subset I \!\! P^2_C$ is the only coordinate axis
contained in $ I \!\! P^2_C$ (see Lemma 5.10 and Remark 5.11, b)). By Lemma 5.13 it
is the closure of the only one-dimensional $G$--orbit $O_p$ contained in
$ I \!\! P^2_C$.
Now we have to distinguish between two cases: \\
\noindent i) $\rho_C^{-1} (l_{n-k,\,n-m}) \subset L_C$ {\hskip 0.5in} and {\hskip 0.5in}
ii) $\rho_C^{-1} (l_{n-k,\,n-m}) \not\subset L_C$. \\
\noindent In case i) we have, as in the Claim above, that $\pi \circ {\tilde
\rho}_C \,:\,Y \to {\bar H}_0 \setminus {\bar D}_{n-1}$ has finite fibres, and
therefore $X$ is almost C--hyperbolic. In case ii) we have $O_p \subset \rho_C (X)$;
the preimage ${\tilde s}_n^{-1} (O_p)$ is the union of $n!$ distinct $\tilde
G$--orbits, which are $\pi$--fibres, and all the others $\pi$--fibres in $Y$
are finite. Thus, by Lemma 2.9 it follows that $Y$ is almost C--hyperbolic
modulo ${\tilde s}_n^{-1} (O_p)$, and hence $X$ is almost C--hyperbolic modulo
$O_p$.
Next we show that ii) corresponds exactly to the two exceptional cases
mentioned in b), which proves b).
By the assumption of the theorem $C^*$ has a cusp, and we suppose as above this
cusp being at the point $q_0 = (0 : 0 : 1)$ and corresponding to the value $t =
\infty$. This means that ${\rm deg}\,f \le n-2,\, {\rm deg}\,g \le n-2$ and
${\rm deg}\,h = n$ (see the proof of Lemma 5.6). Thus, the dual line $l_2 =
q_0^* \subset I \!\! P^2$ belongs to $L_C$. If $f = t^k$ and $g = t^m$ are monomials,
then $\rho_C^{-1} (l_{n-k,\,n-m}) = l_2$ and we have case i). Therefore, up to
the transposition of $f$ and $g$ we may suppose further that $f = t^k$ and $h =
t^m$ are monomials, while $g(t)$ is not. In that case $k \le n-2, \,{\rm
deg}\,g \le n-2 ,\,m = n$ and $\rho_C^{-1} (l_{n-k,\,n-m}) = l_1 := \{x_1 = 0
\} \subset I \!\! P^2$. The dual point $q_1 = (0 : 1 : 0) = l_1^* \in I \!\! P^{2*}$ is a cusp
of $C^*$ iff $k \ge 2$. Hence, ii) occurs iff here $k \le 1$, i.e. iff $C^*$
was projectively equivalent to one of the curves $(1 : t^n : g(t))$ or $(t :
t^n : g(t))$, where
${\rm deg}\,g \le n-2$. If $C^*$ is one of these curves, then $X = I \!\! P^2 \setminus (C
\cup L_C)$ is C--hyperbolic modulo $l_2 = \rho_C^{-1} (l_{n-k,\,0})$. \qed
\noindent {\it Proof of c)}. Let $C = C_{\mu_0}$ be a monomial curve from the
linear pencil $C_{\mu} = \{\alpha x_0^n + \beta x_1^k x_2^{n - k} =0\}$, where
$\mu = (\alpha : \beta ) \in I \!\! P^1$. The pencil $(C_{\mu})$ is self--dual, i.e.
$C_{\mu}^* = C_{\mu^*}$, where $\mu^*$ depends on $\mu$ (see 5.7, 5.9), and so
without loss of generality we may assume that $C^* = C_{\mu}^* = C_{(1 : -1)}$.
Thus, $C^*$ has the parametrizations $C^* = (\tau^k : \tau^n : 1) = (t^{n - k}
: 1 : t^n )$, where $\tau = t^{-1}$. Since $C^*$ has a cusp, we have $n = {\rm
deg}\, C^* \ge 3$ and ${\rm max}\,(k, n-k) \ge 2$. By permuting coordinates, if
necessary, we may assume that $k \ge 2$. In this case the second
parametrization, which we denote by $\nu$, fits in with the basic construction,
i.e. $\nu (\infty ) = q_0 = (0 : 0 : 1)$ is a cusp of $C^*$ and $b^{(i)}_1 =
0,\,i = 0,\,1,\,2$.
The parametrization $\nu$ being fixed as above, the Zariski embedding $\rho_C$
is given by the matrix $B_C = (e_k ,\,e_n ,\,e_0 )$. Therefore, $\rho_C
\,:\, I \!\! P^2 \to I \!\! P^2_C = I \!\! P_{0, k, n} \subset {\bar A}_1 \subset I \!\! P^n$ is
coordinatewise (cf. Remark 5.11, a): $$\rho_C (x_0 : x_1 : x_2 ) = (a_0 : \dots
: a_n ) = (x_2 : 0 : \dots : \underbrace{x_0}_k : 0 : \dots : 0 : x_1)$$ The
$ I \!\!\!\! C^*$--action $G$ on $ I \!\! P^n$ induces the $ I \!\!\!\! C^*$--action $G'$ on $ I \!\! P^2$,
where $$G' \,:\,(\lambda, \,(x_0 : x_1 : x_2 )) \longmapsto (\lambda^k x_0 :
\lambda^n x_1 : x_2 ) = (x_0 /\lambda^{n - k} : x_1 : x_2 /\lambda^n )$$ It is
easily seen that the closure of a one-dimensional $G'$--orbit is an irreducible
component of a member of the linear pencil $(C_{\mu})$.
In what follows we identify $X$ resp. $Y$ with its image under $\rho_C$ resp.
$\tilde \rho_C$. Let $f\,:\, I \!\!\!\! C \to X$ be an entire curve and ${\tilde
f}\,:\, I \!\!\!\! C \to Y$ be its covering curve. From Lemma 6.3 it follows that the map
$\pi \circ {\tilde \rho}_C \circ {\tilde f}$ is constant. This means that
${\tilde f}( I \!\!\!\! C)$ is contained in an orbit of $\tilde G$, and so $f( I \!\!\!\! C)$ is
contained in a $G$--orbit, which in turn is contained in one of the curves
$C_{\mu}$, as it is stated in c).
Furthermore, ${\bar H}_0 \setminus {\bar D}_{n-1}$ being Kobayashi hyperbolic, the
$k_Y$--distance between any two distinct $\tilde G$--orbits in $Y$ is positive.
Over each $G$--orbit $O$ in $X$ there is $n! \,\,\tilde G$--orbits in $Y$, and
each of them is maped by ${\tilde s}_n$ isomorphically onto $O$. Therefore, the
$k_X$--distance between two different $G$--orbits in $X$, which is equal to the
$k_Y$--distance between their preimages in $Y$, is positive, too. This proves
c). Now the proof of Theorem 6.5 is complete. \qed
\noindent {\bf 6.6. Remark.} In general, b) is not true for a plane curve whose
dual is a quasi-monomial curve without cusps. Indeed, if $C$ is a
three--cuspidal plane quartic, then $C^*$ is a nodal cubic, which is
projectively equivalent to a quasi--monomial curve $t \longmapsto (t : t^3 :
t^2 - 1 )$, where the node corresponds to $t = \pm 1$. The Kobayashi
pseudo--distance of $ I \!\! P^2 \setminus C$ is degenerate on at least seven lines (see
Remark 4.2), and thus $ I \!\! P^2 \setminus C$ is not C--hyperbolic modulo a line.
\\[1ex]
The next examples illustrate Theorem 6.5. \\[1ex]
\noindent {\bf 6.7. Example.} Let $C \subset I \!\! P^2$ be the cuspidal cubic $4x_0^3 -
27x_1^2 x_2 =0$. Its dual curve $C^* \subset I \!\! P^{2*}$ is the cuspidal cubic with
the equation $y_0^3 + y_1^2 y_2 =0$. The cusp of $C^*$ at the point $q_0 = (0
:0 :1)$ corresponds to the only flex of $C$ at the point $p_0 = (0 : 1 : 0)$,
with the inflexional tangent $l_2 = \{x_2 = 0 \} \subset I \!\! P^2$, so that $L_C =
l_2$. Consider the curve $C \cup l_2$.
Its complement $X:= I \!\! P^2 \setminus (C \cup l_2)$ is neither C--hyperbolic nor
Kobayashi hyperbolic. Indeed, $C$ is a member of the linear pencil of cubics
$C_{\mu} = \{\alpha x_0^3 - \beta x_1^2 x_2 = 0 \}$, where $\mu = (\alpha :
\beta ) \in I \!\! P^1$ (here $C = C_{\mu_0}$, where $\mu_0 = (4 : 27)$). This
pencil is generated by its only non--reduced members $C_{(1:0)} = 3 l_0$ and
$C_{(0 : 1)} = 2 l_1 + l_2$, where $l_i = \{x_i =0 \}\,,\,i=0,1,2$. The
Kobayashi pseudo-distance $k_X$ is identically zero along any of the cubics
$C_{\mu}\,,\mu \neq \mu_0$, because $C_{\mu} \cap X = C_{\mu} \setminus (C \cup l_2 )
\cong I \!\!\!\! C^*$ and $k_{ I \!\!\!\! C^*} \equiv 0$.
Nevertheless, by Theorem 6.5, c) any entire curve $ I \!\!\!\! C \to X = I \!\! P^2 \setminus (C
\cup l_2)$ is contained in one of the cubics $C_{\mu}$, where $\mu \in I \!\! P^1
\setminus \{\mu_0\}$. Moreover, $k_X (p,\,q) = 0 $ iff $p,\,q \in C_{\mu}$ for some
$\mu \in I \!\! P^1 \setminus \{\mu_0 \}$. \\[1ex]
\noindent {\bf 6.8. Example.} Let $C \subset I \!\! P^2$ be the nodal cubic $x_1^2 x_2 =
x_0^3 + x_0^2 x_2$, and let $l_1 ,\,l_2 ,\,l_3$ be the three inflexional
tangents of $C$.
They correspond to the cusps of the dual curve $C^* \subset I \!\! P^{2*}$, which is the
3-cuspidal quartic $(2y_1 y_2 + y_0^2 )^2 = 4y_0^2 (y_0 - 2y_2 )(y_0 +y_2)$
(see Remark 4.2). Thus, $L_C = l_1 \cup l_2 \cup l_3 $. By Theorem 6.5, a) we
have that $X := I \!\! P^2 \setminus (C \cup L_C) $ is almost C--hyperbolic. Hence, it is
also Brody hyperbolic (see (2.6)).
By Bezout Theorem three cusps of $C^*$ are not at the same line in $ I \!\! P^{2*}$.
Therefore, their dual lines, which are inflexional tangents $l_1 ,\,l_2 ,\,l_3$
of $C$, are not passing through the same point. From this it easily follows
that ${\rm reg}\,(C \cup l_1 \cup l_2 \cup l_3 )$ is hyperbolic. Thus, by
Proposition 2.7 $X$ is Kobayashi complete hyperbolic and hyperbolically
embedded
into $ I \!\! P^2$. \\[1ex]
\noindent {\bf 6.9. Example.} Let $C \subset I \!\! P^2$ be the rational quintic $t
\longmapsto (2t^5 - t^2 : -(4t^3 + 1) : 2t)$ with a cusp at the only singular
point $(1 : 0 : 0)$. The dual curve $C^* \subset I \!\! P^{2*}$ is the quasi--monomial
quartic $t \longmapsto (1 : t^2 : t^4 + t)$ given by the equation $(y_0 y_2 -
y_1^2 )^2 = y_0^3 y_1$. It has the only singular point $q_0 = (0 : 0 : 1)$,
which is a ramphoid cusp, i.e. it has the multiplicity sequence
$(2,\,2,\,2,\,1,\,\dots)$ and $\delta = \mu /2 = 3$, where $\mu$ is the Milnor
number. Any rational quartic with a ramphoid cusp is projectively equivalent to
$C^*$ (see [Na, 2.2.5(a)]). The artifacts $L_C$ consist of the only cuspidal
tangent line $l_2 = \{x_2 = 0 \}$ of $C$. By Theorem 6.5, b) the complement
$ I \!\! P^2 \setminus (C \cup l_2)$ is almost C--hyperbolic. Note that $\Gamma := C \setminus
l_2$ is a smooth rational affine curve in $ I \!\!\!\! C^2 \cong I \!\! P^2 \setminus l_2$, which is
isomorphic to $ I \!\!\!\! C^* := I \!\!\!\! C \setminus \{0\}$. Thus, $X := I \!\!\!\! C^2 \setminus \Gamma$ is almost
C--hyperbolic. \\[1ex]
\noindent {\bf 6.10. Example.} Let $C' \subset I \!\! P^2$ be the rational quartic $t
\longmapsto (t^3 (2t + 1) : -t(4t + 3) : -2)$. It has two singular points, a
double cusp at the point $(0 : 0 : 1)$ (i.e. a cusp with the multiplicity
sequence $(2, \,2,\,1,\,\dots)$ and $\delta = 2$) and another one, which is an
ordinary cusp. The dual curve $C'^* \subset I \!\! P^{2*}$ is the quasi--monomial
quartic $t \longmapsto (1 : t^2 : t^4 + t^3)$ given by the equation $(y_0 y_2 -
y_1^2 )^2 = y_0 y_1^3$. It has the same type of singularities as $C'$, namely a
double cusp at the point $q_0 = (0 : 0 : 1)$ and an ordinary cusp at the point
$(1 : 0 : 0)$. Therefore, $L_{C'} = l_0 \cup l_2$, where $l_0 = \{x_0 = 0 \}$
and $l_2 = \{x_2 = 0 \}$. By Theorem 6.5, b) the complement $X:= I \!\! P^2 \setminus (C'
\cup L_{C'})$ is almost C--hyperbolic. \\[2ex]
\section{C--hyperbolicity of complements of maximal cuspidal rational curves}
In Corollary 7.10 below we show that the complement of a maximal cuspidal
rational curve of degree $d \ge 8$ in $ I \!\! P^2$ is almost C--hyperbolic. In a
sense, this completes the study on C--hyperbolicity of $ I \!\! P^2 \setminus (C \cup
L_C)$. The deep reason of this fact, which actually does not appear in the
proof, is that the Teichm\"uller space $T_{0,n}$ of the Riemann sphere with $n$
punctures is a bounded domain in $ I \!\!\!\! C^n$ (cf. [Kal]).\\
Let us start with necessary preliminaries. \\[2ex]
\noindent {\bf 7.1.} {\it Maximal cuspidal rational curves as generic plane
sections of the discriminant.} Let $C \subset I \!\! P^2$ be a rational curve of degree
$d > 1$. By the Class Formula (4) its dual curve $C^* \subset I \!\! P^{2*}$ is an
immersed curve (or, equivalently, $L_C = \emptyset$) iff $d = 2(n - 1)$, where
$n = {\rm deg}\,C^*$ (cf. 3.4). If in addition $C$ is a Pl\"ucker curve, then
it has the maximal possible number of ordinary cusps, which is equal to $3(n -
2)$, and besides this it has also $2(n - 2)(n - 3)$ nodes. Such a curve $C$ is
called {\it a maximal cuspidal rational curve} [Zar, p. 267]. Note that the
dual $C^*$ of such a curve $C$ is a rational nodal curve of degree $n$ in
$ I \!\! P^{2*}$. In particular, a generic maximal cuspidal rational curve $C$
naturally appears via the Zariski embedding $\rho_C \,:\, I \!\! P^2 \to I \!\! P^2_C
\hookrightarrow I \!\! P^n$ as a generic plane section of the discriminant
hypersurface $R_n \subset I \!\! P^n$ (see 3.6-3.7, 5.4). \\[2ex]
\noindent {\bf 7.2.} {\it The moduli space of the $n$--punctured sphere as an orbit
space}. Note, first of all, that the Vieta map $s_n\,:\,( I \!\! P^1)^n \to S^n I \!\! P^1
= I \!\! P^n$
is equivariant with respect to the natural actions of the group $ I \!\! P GL(2,\,
I \!\!\!\! C) = {\rm Aut}\, I \!\! P^1$ on $ ( I \!\! P^1)^n$ and on $ I \!\! P^n$, respectively. The
branching divisors $D_n$ (the union of the diagonals) resp. $R_n$ (the
discriminant divisor), as well as their complements are invariant under the
corresponding actions. It is easily seen that for $n \ge 3$ the orbit space of
the $ I \!\! P GL(2,\, I \!\!\!\! C)$--action on $ I \!\! P^n \setminus R_n$ is naturally isomorphic to
the moduli space $M_{0,\,n}$ of the Riemann sphere with $n$ punctures. Denote
by $\tilde M_{0,\,n}$ the quotient $(( I \!\! P^1)^n \setminus D_n)\, / I \!\! P GL(2,\, I \!\!\!\! C)$.
We have the following commutative diagram of equivariant morphisms
\begin{picture}(200,80)
\unitlength0.2em
\thicklines
\put(54,25){$( I \!\! P^1)^n \setminus D_n$}
\put(110,25){$\tilde M_{0,\,n}$}
\put(87,27){$\vector(1,0){15}$}
\put(60,5){$ I \!\! P^n \setminus R_n$}
\put(87,6){$\vector(1,0){15}$}
\put(110,5){$M_{0,\,n}$}
\put(69,22){$\vector(0,-1){11}$}
\put(114,22){$\vector(0,-1){11}$}
\put(90,10){$\pi_n$}
\put(90,30){$\tilde \pi_n$}
\put(60,16){$s_n$}
\put(170,15){(7)}
\end{picture}
\noindent {\bf 7.3.} {\it Description of $\tilde M_{0,\,n}$}. The cross--ratios
$\sigma_i (z) = (z_1,\,z_2;\,z_3 ,\,z_i)$, where $z = (z_1 ,\dots, z_n ) \in
( I \!\! P^1)^n$ and $4 \le i \le n$, define a morphism $$\sigma^{(n)} = (\sigma_4
,\,\dots, \,\sigma_n )\,:\,( I \!\! P^1)^n \setminus D_n \to ( I \!\!\!\! C^{**})^{n - 3} \setminus D_{n -
3}$$
(here as before $\, I \!\!\!\! C^{**} = I \!\! P^1 \setminus \{0,\,1,\,\infty\}$). By the invariance
of cross--ratio $\sigma^{(n)}$ is constant along the orbits of the action of
$ I \!\! P GL(2,\, I \!\!\!\! C)$ on $( I \!\! P^1 )^n \setminus D_n$. Therefore, it factorizes through a
mapping of the orbit space $\tilde M_{0,\,n} \to ( I \!\!\!\! C^{**})^{n - 3} \setminus D_{n -
3}$. On the other hand, for each point $z \in ( I \!\! P^1)^n \setminus D_n$ its $ I \!\! P
GL(2,\, I \!\!\!\! C)$--orbit $O_z$ contains the unique point $z'$ of the form $z' =
(0,\,1,\,\infty,\,z'_4 ,\,\dots,\,z'_n )$. This defines a regular section
$\tilde M_{0,\,n} \to ( I \!\! P^1)^n \setminus D_n$, and its image coincides with the
image of the biregular embedding $$( I \!\!\!\! C^{**})^{n - 3} \setminus D_{n - 3} \ni u =
(u_4 ,\,\dots ,\,u_n ) \longmapsto (0,\,1,\,\infty,\,u_4 ,\,\dots,\,u_n ) \in
( I \!\! P^1)^n \setminus D_n \,\,.$$
This shows that the above mapping $\tilde M_{0,\,n} \to ( I \!\!\!\! C^{**})^{n - 3} \setminus
D_{n - 3}$ is an isomorphism. \\[2ex]
\noindent {\bf 7.4.} {\it $ I \!\! P GL(2,\, I \!\!\!\! C)$--orbits.} Here as before we treat
$ I \!\! P^n$ as the projectivized space of the binary forms of degree $n$ in $u$ and
$v$. For instance, $e_k = (0:\dots :0:1_k :0:\dots :0) \in I \!\! P^n$ corresponds
to the forms $cu^{n - k}v^k$, where $c \in I \!\!\!\! C^*$. Denote by $O_q$ the $ I \!\! P
GL(2,\, I \!\!\!\! C)$--orbit of a point $q \in I \!\! P^n$. Clearly, $O_{e_i} = O_{e_{n -
i}},\,i=0,\dots, n$; $O_{e_0}$ is the only one--dimensional orbit and, at the
same time, the only closed orbit; $O_{e_i},\,i=1,\dots,[n/2]$, are the only
two-dimensional orbits, and any other orbit has dimension $3$. Note that
$O_{e_0} = C_n$ is the dual rational normal curve, and $S:= O_{e_0} \cup
O_{e_1}$ is its developable tangent surface (see 5.4).
If $O_q$ is an orbit of dimension $3$, then its closure $\bar {O_q}$ is the
union of the orbits $O_q, O_{e_0}$ and those of the orbits $O_{e_i}, i=1,\dots,
n-1$, for which the form $q$ has a root of multiplicity $i$ [AlFa, Proposition
2.1]. Furthermore, for any point $q \in I \!\! P^n \setminus R_n$, i.e. for any binary
form $q$ without multiple roots, its orbit $O_q$ is closed in $ I \!\! P^n \setminus R_n$
and $\bar {O_q} = O_q \cup S$, where $S = \bar {O_q} \cap R_n$. Therefore, any
Zariski closed subvariety $Z$ of $ I \!\! P^n$ such that ${\rm dim}\,(O_q \cap Z) >
0$ must meet the surface $S$. These observations yield the following lemma.
\footnote {We are gratefull to H. Kraft who pointed out to us an approach which
is based on the notion of the associated cone of an orbit [Kr] (here we have
used a simplified version of it), and to M. Brion for mentioning to us of the
paper [AlFa].} \\[2ex]
\noindent {\bf 7.5. Lemma.} {\it If a linear subspace $L$ in $ I \!\! P^n$ does not meet
the surface $S = {\bar O_{e_1}} \subset R_n$, then it has at most finite
intersection with any of the orbits $O_q$, where $q \in I \!\! P^n \setminus R_n$. In
particular, this is so for a generic linear subspace $L$ in $ I \!\! P^n$ of
codimension at least $3$.} \\[2ex]
\noindent {\bf 7.6. Remark.} Fix $k$ distinct points $z_1 ,\dots, z_k \in I \!\! P^1$,
where $3 \le k \le n$. Let $g_0$ be a binary form of degree $k$ with the roots
$z_1 ,\dots, z_k$. Consider the projectivized linear subspace $L_0 \subset I \!\! P^n$ of
codimension $k$ consisting of the binary forms of degree $n$ divisible by
$g_0$. It is easily seen that $L_0 \cap S = \emptyset$. This gives a concrete
example of such a subspace. \\
The next tautological lemma is used below in the proof of Theorem 7.9.\\[2ex]
\noindent {\bf 7.7. Lemma.} {\it Let $C \subset I \!\! P^2$ be a rational curve. Put $n =
{\rm deg}\,C^*$, and let as before $ I \!\! P^2_C = \rho_C ( I \!\! P^2 ) \hookrightarrow
I \!\! P^n$ be the image under the Zariski embedding. \\
\noindent a) The plane $ I \!\! P^2_C$ meets the surface $S = {\bar O_{e_1}}$ iff there
exists a local irreducible analytic branch $(A^*, p^* )$ of the dual curve
$C^*$
such that $\,i(T_{p^*} A^* , \,A^* ;\,p^*) \ge n - 1$. \\
\noindent b) Furthermore, if $C^*$ has a cusp $(A^*, p^* )$ of multiplicity $n -
1$, then $\rho_C (l_{p^*}) \subset I \!\! P^2_C \cap S$, where $l_{p^*} \subset L_C \subset I \!\! P^2$
is the dual line of the point $p^* \in I \!\! P^{2*}$. \\
\noindent c) If the dual curve $C^*$ has only ordinary cusps and flexes and $n =
{\rm deg}\,C^* \ge 5$, then $ I \!\! P^2_C \cap S = \emptyset$. } \\[2ex]
\noindent{\it Proof.} a) By the definition of the Zariski embedding $q \in I \!\! P^2_C
\cap S$ iff, after passing to the normalization $\nu :\, I \!\! P^1 \to C^*$ and
identifying $ I \!\! P^2$ with its image $ I \!\! P^2_C$ under the Zariski embedding
$\rho_C$, the dual line $l_q \subset I \!\! P^2$ cuts out on $C^*$ a divisor of the form
$(n - 1)a + b$, where $a, b \in I \!\! P^1$. Then $p^* := \nu (a) \in C^*$ is the
center of a local branch $A^*$ of $C^*$ which satisfies the condition in a).
The converse is evidently true. \qed
\noindent b) For any point $q \in l_{p^*}$ its dual line $l_q \subset I \!\! P^{2*}$ passes
through $p^*$, and hence by the above consideration we have $\rho_C (q) \in
I \!\! P^2_C \cap S$. \qed
\noindent c) By the condition we have that $i(T_{p^*} A^* , \,A^* ;\,p^*) \le 3 < n
- 1$ for any local analytic branch $(A^*, p^* )$ of $C^*$. Now the result
follows from a). \qed
\noindent {\bf 7.8. Lemma.} {\it Let $C^* \subset I \!\! P^{2*}$ be a rational curve of
degree $n$. Then the complement $X = I \!\! P^2 \setminus (C \cup L_C)$ is almost
C--hyperbolic, whenever $ I \!\! P^2_C \cap S = \emptyset$.} \\[1ex]
\noindent{\it Proof.} Consider the following commutative diagram of morphisms: \\
\begin{picture}(250,80)
\unitlength0.2em
\thicklines
\put(-5,5) {${ I \!\! P^2} \setminus (C \cup L_C) = X$}
\put(35,25){$Y$}
\put(36,22){\vector(0,-1){11}}
\put(27,16){${\tilde s}_n$}
\put(44,27){{\vector(1,0){14}}}
\put(49,30){${\tilde\rho}_C$}
\put(44,6){\vector(1,0){14}}
\put(49,10){$\rho_C$}
\put(66,5){$ I \!\! P^n \setminus R_n$}
\put(62,25){$( I \!\! P^1)^n \setminus D_n$}
\put(76,22){\vector(0,-1){11}}
\put(128,22){\vector(0,-1){11}}
\put(124,5){$M_{0,\,n}$}
\put(67,16){$s_n$}
\put(170,15){(8)}
\put(92,27){{\vector(1,0){14}}}
\put(94,6){{\vector(1,0){20}}}
\put(95,30){${\tilde \pi}_n$}
\put(100,10){$\pi_n$}
\put(110,25){$ ( I \!\!\!\! C^{**})^{n - 3} \setminus D_{n - 3} \hookrightarrow ( I \!\!\!\! C^{**})^{n -
3}$}
\end{picture}
\noindent where ${\tilde s}_n \,:\,Y \to X$ is the induced covering (cf. (7) and
7.2--7.3 above).
From Lemma 7.5 it follows that the mapping $\pi_n \circ \rho _C \,:\,X \to
M_{0,\,n}$ has finite fibres. Hence, the same is valid for the mapping ${\tilde
\pi}_n \circ {\tilde\rho}_C \,:\, Y \to ( I \!\!\!\! C^{**})^{n - 3} \setminus D_{n - 3}$. By
Lemma 2.5 $Y$, and thus also $X$, are almost C--hyperbolic. \qed
{}From this lemma and Lemma 7.7 we have the following theorem, which is a
useful supplement to Theorem 6.5. \\[1ex]
\noindent {\bf 7.9. Theorem.} {\it Let $C^* \subset I \!\! P^{2*}$ be a rational curve of
degree $n$ such that $\,i(T_{p^*} A^* , \,A^* ;\,p^*) \le n - 2$ for any local
analytic branch $(A^* , \,p^* )$ of $C^*$. Let $C = (C^*)^* \subset I \!\! P^2$ be the
dual curve. Then the complement $X = I \!\! P^2 \setminus (C \cup L_C)$ is almost
C--hyperbolic. In particular, this is so if $n \ge 5$ and $C^*$ has only
ordinary cusps and flexes.} \\[2ex]
The next corollary is an addition to Theorem 4.1, b). \\
\noindent {\bf 7.10. Corollary.} {\it Let $C \subset I \!\! P^2$ be a maximal cuspidal
rational curve of degree $d = 2(n - 1) \ge 8$. Then $X = I \!\! P^2 \setminus C$ is almost
C--hyperbolic, Kobayashi complete hyperbolic and hyperbolically embedded into
$ I \!\! P^2$. In particular, this is the case if the dual curve $C^*$ is a generic
rational nodal curve of degree $n \ge 5$ in $ I \!\! P^{2*}$. } \\
\noindent {\it Proof.} The first statement immediately follows from Theorem 7.9,
while the second one follows from Proposition 2.7. Indeed, under our
assumptions we have $n \ge 5$, and therefore the curve $C$ has at least $9$
cusps. Hence ${\rm reg}\,C$ is a hyperbolic curve. The last statement is
evident. \qed
The next example shows that our method is available not for all rational
curves whose dual curves are nodal. \\
\noindent {\bf 7.11. Example.} Let $C^* = (p(t) : q(t) : 1)$ be a parametrized
plane rational curve, where $p, q \in I \!\!\!\! C [t]$ are generic polynomials of
degrees $n$ and $n - 1$, respectively. Then $C^*$ is a nodal curve of degree
$n$ which is the projective closure of an affine plane polynomial curve with
one place at infinity, at the point $(1 : 0 : 0)$, and this is a smooth point
of $C^*$. Thus, the line $l_2 = \{x_2 = 0 \}$ is an inflexional tangent of
order $n - 2$ of $C^*$, and so by Lemma 7.7, a) $ I \!\! P^2_C \cap S \neq
\emptyset$. Therefore, we can not apply in this case the same approach as
above. \\[2ex]
At last, we can summarize the main results of the paper (cf. Theorems 4.1, 6.5
and 7.9). \\[1ex]
\noindent {\bf 7.12. Theorem.} {\it Let $C \subset I \!\! P^2$ be an irreducible curve of
genus $g$. Put $n = {\rm deg}\,C^*$ and $X = I \!\! P^2 \setminus (C \cup L_C )$. \\
\noindent a) If $g \ge 1$, then $X$ is C--hyperbolic. If $g = 0$, then $X$ is
almost C--hyperbolic if at least one of the following conditions is fulfilled:
\\
\noindent i) $\,i(T_{p^*} A^* , \,A^* ;\,p^*) \le n - 2$ for any local analytic
branch $(A^* , \,p^* )$ of $C^*$; \\
\noindent ii) $C^*$ has a cusp and it is not projectively equivalent to one of the
curves $(1 : g(t) : t^n ),\,\,(t : g(t) : t^n )$, where $g \in I \!\!\!\! C [t],\, {\rm
deg}\,g \le n - 2$. \footnote{The monomial curves correspond here to $g(t) =
t^k,\,k \le n - 2$. Note that the curves $(1 : t^{n - 1} : t^n )$ and $(1 : t :
t^n )$, being considered as non--parametrized ones, are projectively
equivalent, and therefore all monomial curves have been excluded.} \\
\noindent b) Let, furthermore, $C^*$ be an immersed curve. If $g \ge 1$, then
$ I \!\! P^2 \setminus C$ is C--hyperbolic. If $g = 0$ and i) is fulfilled, then $ I \!\! P^2 \setminus
C$ is almost C--hyperbolic; in particular, this is so if $C^*$ is a generic
rational nodal curve of degree $n \ge 5$. In both cases $ I \!\! P^2 \setminus C$ is
Kobayashi complete hyperbolic and hyperbolically embedded into $ I \!\! P^2$. }
\\[2ex]
\newpage
\begin{center} {\LARGE References} \end{center}
\noindent [AlFa] P. Aluffi, C. Faber. {\sl Linear orbits of $d$--tuples of points
in $ I \!\! P^1$}, J. reine angew. Math. 445 (1993), 205--220 \\
\noindent [Au] A. B. Aure. {\sl Pl\"ucker conditions on plane rational curves},
Math. Scand. 55 (1984), 47--58, with {\sl Appendix} by S. A. Str\"omme, ibid.
59--61 \\
\noindent [CaGr] J. A. Carlson, M. Green. {\sl Holomorphic curves in the plane},
Duke Math. J. 43 (1976), 1--9 \\
\noindent [Co] J. L. Coolidge. {\sl A Treatise on Algebraic Plane Curves}, N.Y.:
Dover, 1959 \\
\noindent [Deg 1] A. I. Degtjarjov. {\sl Isotopy classification of complex plane
projective curves of degree 5}, Preprint LOMI P--3-87, Leningrad, 1987, 1--17
\\
\noindent [Deg 2] A. I. Degtjarjov. {\sl Topology of plane projective algebraic
curves}, PhD Thesis, Leningrad State University, 1987 (in Russian) \\
\noindent [Del] P. Deligne. {\sl Le groupe fondamental du complement d'une courbe
plane n'ayant que des points doubles ordinaires est ab\'elien}, Sem. Bourbaki,
1979/1980, Lect. Notes in Math. vol. 842, Springer-Verlag (1981), 1--25 \\
\noindent [DSW1] G. Dethloff, G. Schumacher, P.-M. Wong. {\sl Hyperbolicity of the
complements of plane algebraic curves}, preprint, Math. G\"otting. 31
(1992), 38 p. (to appear in Amer. J. Math.) \\
\noindent [DSW2] G. Dethloff, G. Schumacher, P.-M. Wong. {\sl On the hyperbolicity
of the complement of curves in algebraic surfaces: The three component case},
preprint, Essen e.a., 1993, 25 p. (to appear in Duke Math. J.) \\
\noindent [DL] I. Dolgachev, A. Libgober. {\sl On the fundamenthal group of the
complement to a discriminant variety}, In: Algebraic Geometry, Lecture Notes in
Math. 862, 1--25, N.Y. e.a.: Springer, 1981 \\
\noindent [Fu] W. Fulton. {\sl On the fundamental group of the complement to a node
curve}, Ann. of Math. (2) 111 (1980), 407--409 \\
\noindent [GP] H. Grauert, U. Peternell. {\sl Hyperbolicity of the complement of
plane curves}, Manuscr. Math. 50 (1985), 429--441 \\
\noindent [Gr1] M. Green. {\sl The complement of the dual of a plane curve and some
new hyperbolic manifolds}, in: 'Value Distribution Theory`, Kujala and Vitter,
eds., N.Y.: Marcel Dekker, 1974, 119--131 \\
\noindent [Gr2] M. Green. {\sl Some examples and counterexamples in value
distribution theory}. Compos. Math. 30 (1975), 317-322 \\
\noindent [Gr3] M. Green. {\sl The hyperbolicity of the complement of $2n+1$
hyperplanes in general position in $ I \!\! P_n$ and related results}, Proc. Amer.
Math. Soc. 66 (1977), 109--113 \\
\noindent [GK] G. M. Greuel, U. Karras. {\sl Families of varieties with prescribed
singularities}, Compositio Math. 69 (1989), 83--110 \\
\noindent [GH] Ph. Griffiths, J. Harris. {\sl Principles of Algebraic Geometry}.
N.Y. e.a.: J. Wiley and Sons Inc., 1978 \\
\noindent [Kal] Sh. I. Kaliman. {\sl The holomorphic universal covers of spaces of
polynomials without multiple roots}, Selecta Mathem. form. Sovietica, 12 (1993)
No. 4, 395--405 \\
\noindent [Kan] J. Kaneko. {\sl On the fundamental group of the complement to a
maximal cuspidal plane curve}, Mem. Fac. Sci. Kyushu Univ. Ser. A. 39 (1985),
133--146 \\
\noindent [Ki] P. Kiernan. {\sl Hyperbolically imbedded spaces and the big Picard
theorem}, Math. Ann. 204 (1973), 203--209 \\
\noindent [KiKo] P. Kiernan, Sh. Kobayashi. {\sl Holomorphic mappings into
projective space with lacunary hyperplanes}, Nagoya Math. J. 50 (1973),
199-216 \\
\noindent [Ko1] Sh. Kobayashi. {\sl Hyperbolic manifolds and holomorphic mappings}.
N.Y. a.e.: Marcel Dekker, 1970 \\
\noindent [Ko2] Sh. Kobayashi, {\sl Intrinsic distances, measures and geometric
function theory}, Bull. Amer. Math. Soc 82 (1976), 357-416 \\
\noindent [Kr] H. Kraft. {\sl Geometrische Methoden in der Invariantentheorie}.
Braunschweig/Wiesbaden: Vieweg und Sohn, 1985 \\
\noindent [Lib] A. Libgober. {\sl Fundamental groups of the complements to plane
singular curves}, Proc. Sympos. in Pure Mathem. 46 (1987), 29--45 \\
\noindent [Li] V. Ja. Lin. {\sl Liouville coverings of complex spaces, and amenable
groups}, Math. USSR Sbornik, 60 (1988), 197--216\\
\noindent [LiZa] V. Ya. Lin, M.G. Zaidenberg, {\sl Finiteness theorems for
holomorphic mappings}, Encyclopaedia of Math. Sci. 9 (1986), 127-194 (in
Russian). English transl. in
Encyclopaedia of Math. Sci. Vol.9. Several Complex Variables III. N.Y. e.a.:
Springer Verlag, 1989, 113-172 \\
\noindent [Ma] A. Mattuck, {\sl Picard bundles}, Illinois J. Math. 5 (1961),
550--564 \\
\noindent [Na] M. Namba. {\sl Geometry of projective algebraic curves}, N.Y. a.e.:
Marcel Dekker, 1984 \\
\noindent [Pi] R. Piene. {\sl Cuspidal projections of space curves}, Math. Ann. 256
(1981), 95--119 \\
\noindent [O] M. Oka. {\sl Symmetric plane curves with nodes and cusps}, J. Math.
Soc. Japan, 44, No. 3 (1992), 375--414 \\
\noindent [Re] H. J. Reiffen. {\sl Die Carath\'eodorysche Distanz und ihre
zugeh\"orige Differentialmetrik}, Math. Ann. 161 (1965), 315--324 \\
\noindent [Ro] H. Royden. {\it Automorphisms and isometries of Teichm\"uller
space}, In: Advances in the Theory of Riemann Surfaces, 1969 Stony Brook Conf.
Ann. of Math. St. 66, Princeton, N.J.: Princeton Univ. Press, 1971, 369--383 \\
\noindent [Se] F. Severi. {\sl Vorlesungen \"uber algebraische Geometrie}, Leipzig:
Teubner, 1921 \\
\noindent [SY] Y.--T. Siu, S.--K. Yeung. {\sl Hyperbolicity of the complement of a
generic smooth curve of high degree in the complex projective plane}, preprint,
1994, 56 p. \\
\noindent [Ve] G. Veronese. {\sl Behandlung der projectivischen Verh\"altnisse der
R\"aume von verschiedenen Dimensionen durch das Princip des Projicirens und
Schneidens}, Math. Ann. 19 (1882), 193--234 \\
\noindent [Wa] R. J. Walker. {\sl Algebraic curves}. Princeton Meth. Series 13,
Princeton, N.J.: Princeton University Press, 1950 \\
\noindent [Za1] M. Zaidenberg. {\sl On hyperbolic embedding of complements of
divisors and the limiting behavior of the Kobayashi-Royden metric}, Math.
USSR Sbornik 55 (1986), 55--70 \\
\noindent [Za2] M. Zaidenberg. {\sl The complement of a generic hypersurface of
degree $2n$ in $ I \!\!\!\! C I \!\! P^n$ is not hyperbolic}. Siberian Math. J. 28
(1987), 425--432 \\
\noindent [Za3] M. Zaidenberg. {\sl Stability of hyperbolic imbeddedness and
construction of examples}, Math. USSR Sbornik 63 (1989), 351--361
\\
\noindent [Za4] M. Zaidenberg. {\sl Hyperbolicity in projective spaces}, Proc.
Conf. on Hyperbolic and Diophantine Analysis, RIMS, Kyoto, Oct. 26--30, 1992.
Tokyo, TIT, 1992, 136--156 \\
\noindent [Zar] O. Zariski. {\sl Collected Papers}. Vol III : {\sl Topology of
curves and surfaces, and special topics in the theory of algebraic varieties}.
Cambridge, Massachusets e. a.: The MIT Press, 1978 \\
\noindent Gerd Dethloff\\
Mathematisches Institut der Universit\"at G\"ottingen\\
Bunsenstra\3e 3-5\\
3400 G\"ottingen\\
Germany\\
\vspace{0.8cm}e-mail: [email protected]\\
\noindent Mikhail Zaidenberg\\
Universit\'{e} Grenoble I \\
Institut Fourier des Math\'ematiques\\
BP 74\\
38402 St. Martin d'H\`{e}res--c\'edex\\
France\\
\vspace{0.8cm}e-mail: [email protected]
\end{document}
|
1995-01-23T06:20:11 | 9501 | alg-geom/9501010 | en | https://arxiv.org/abs/alg-geom/9501010 | [
"alg-geom",
"math.AG"
] | alg-geom/9501010 | E. Looijenga | Eduard Looijenga | On the tautological ring of $\M _g$ | 6 pages, amstex 2.1 | null | 10.1007/BF01884306 | null | null | We prove among other things that any product of tautological classes of
$\M_g$ of degree $d$ (in the Chow ring of $\M _g$ with rational coefficients)
vanishes for $d>g-2$ and is proportional to the class of the hyperelliptic
locus in degree $g-2$.
| [
{
"version": "v1",
"created": "Fri, 20 Jan 1995 09:07:01 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Looijenga",
"Eduard",
""
]
] | alg-geom | \section{\global\advance\headnumber
by1\global\labelnumber=0{{\the\headnumber}.\ }}
\define\label{(\global\advance\labelnumber by1 \the\headnumber
.\the\labelnumber )\enspace}
\NoBlackBoxes
\topmatter
\title
On the tautological ring of $\M _g$
\endtitle
\rightheadtext{Tautological ring}
\author
Eduard Looijenga
\endauthor
\address Faculteit Wiskunde en Informatica,
Universiteit Utrecht,
P.O. Box 80.010, 3508 TA Utrecht,
The Netherlands\endaddress
\email looijeng\@math.ruu.nl\endemail
\abstract
We prove that any product of tautological classes of $\M _g$ of degree $d$
(in the Chow ring of $\M _g$ with rational coefficients) vanishes for $d>g-2$
and is proportional to the class of the hyperelliptic locus in degree $g-2$.
\endabstract
\endtopmatter
\document
\head
\section Results
\endhead
Fix an integer $g\ge 2$ and denote by $\CC _g^n$ the moduli space
of tuples $(C,x_1,\dots ,x_n)$, where $C$ is a smooth connected projective
curve
of genus $g$ and $x_1,\dots ,x_n$ are (not necessarily distinct) points of $C$;
we also write $\M _g$ for $\CC _g^0$.
Forgetting the $i$th point defines a morphism $\CC _g^n\to\CC _g^{n-1}$
whose relatively dualizing sheaf is denoted by $\omega _i$ ($i=1,\dots ,n)$.
We
write $K_i$ for the first Chern class of $\omega _i$, considered as an
element
of the Chow group $A^1(\CC _g^n)$ (with rational coefficients); for $n=1$ we
also write $K$. Our main result is:
\proclaim{\label Theorem}
Any monomial of degree $d$ in the classes $K_1,\dots ,K_n$ is a linear
combination of the classes of the irreducible components of the locus
parametrizing tuples $(C,x_1,\dots ,x_n)$ admitting a morphism $C\to\Pone$ of
degree $\le g+n$ such that the fiber over $0$ (resp.\ $\infty$) has at most
$g+n-d-1$ points (resp.\ is a singleton) and $\{ x_1,\dots ,x_n\}$ is
contained in one of these two fibers. (Hence such a class is zero when
$d>g+n-2$.) All monomials of degree $g+n-2$ are proportional to the class of
the locus parametrizing tuples $(C,x_1,\dots ,x_n)$ with $C$ hyperelliptic and
$x_1=\cdots =x_n$ a Weierstra\ss\ point.
\endproclaim
The direct image of $K^{d+1}$ in $A^d(\M _g)$ is the Mumford--Morita--Miller
{\it tautological class} $\kappa _d$. Mumford showed in his fundamental paper
\cite{4} that the subring of $A^{\bullet}(\M _g)$ generated by these classes
(the {\it tautological ring} of $\M _g$) is already generated by $\kappa
_1,\dots ,\kappa _{g-2}$. On the basis of many calculations Carel Faber has
made
the intriguing conjecture that this ring has the formal properties of the
even-dimensional cohomology ring of a projective manifold of dimension $g-2$,
i.e., satisfies Poincar\'e duality and a Lefschetz decomposition.
We offer the following support for this conjecture:
\proclaim{\label Theorem}
Any product of tautological classes that has degree $d$ is a linear combination
of the classes of the irreducible components of the locus parametrizing curves
$C$ admitting a morphism $C\to\Pone$ of degree $\le 2g-2$ totally ramified over
$\infty$ and with at most $g-1-d$ points over $0$ (hence is zero when
$d>g-2$).
All such classes of degree $g-2$ are proportional to the class of the
hyperelliptic locus.
\endproclaim
A finer analysis of our proof may well yield that $\kappa _1^{g-2}$ is a
nonzero multiple of the hyperelliptic class, but it is not known whether the
latter is actually nonzero.
The proof of the theorems uses the flag of subvarieties of $\M _g$ introduced
by
Arbarello \cite{1}, a variant of which was exploited by Diaz \cite{2} to prove
that $\M _g$ has no complete subvarieties of dimension $>g-2$. Our simple key
result
\refer{2.4} serves as a substitute for Diaz's lemma $2$ in \cite{2} and can be
used in that paper to eliminate the use of compactifications of Hurwitz schemes
(see \refer{2.8}).
The proof of the second assertion of each theorem involves an application of
the Fourier transform for abelian varieties, due to Mukai and Beauville.
\smallskip
In this paper we only consider Chow groups with respect to rational
equivalence, tensorized with $\Q$, and graded by codimension, notation:
$A^{\bullet}$. If $X$ is a variety that is smooth, or more generally, that
admits a smooth Galois covering, then there is an intersection product
$A^k(X)\otimes A^l(X)\to A^{k+l}(X)$.
\smallskip
I thank Johan de Jong for drawing my attention to the paper by Deninger--Murre
\cite{2} and for comments on a first draft.
\head
\section Proofs
\endhead
\label Let $C$ be a smooth projective curve of genus $g$ and let $D_0$ and
$D_{\infty}$ be positive divisors on $C$ that are linearly
equivalent, but whose supports are disjoint. Then there is a
finite morphism $\pi :C\to\Pone$ such that $\pi ^*(i)=D_i$ ($i=0,\infty $). If
$p\in C$ occurs in $D_i$ with multiplicity $m_p>0$, then $\pi$
determines an isomorphism of $\C\cong T^*_i\Pone$ onto $T^*_pC ^{\otimes
m_p}$.
However, $\pi$ is not unique for it is defined up to natural action of $\C
^{\times}$ on $\Pone$. That ambiguity can be eliminated as follows.
Let $R$ denote the part of the ramification divisor of $\pi$ that lies over
$\Pone -\{ 0,\infty \}$. If $c$ denotes the number of points of
$\supp (D_0+D_{\infty})$, then the Riemann-Hurwitz formula implies that the
degree $r$ of $R$ is equal to $2g-2+c$. If $\pi _*(R)= \sum _i n_i(z_i)$, then
$\pi$ can be
normalized in such a way that $\prod _i z_i ^{n_i}=1$. This normalization is
unique up to multiplication by an $r$th root of unity.
So for $p$ and $m_p$ as above, and $\pi$ normalized, the corresponding
generator of $T^*_pC ^{\otimes m_p}$ raised to the $r$th power
gives a {\it canonical} generator of $T^*_pC ^{\otimes m_pr}$.
This argument works just as well in families and so we obtain:
\proclaim{\label Proposition}
Let $f:\CC \to S$ be a projective family of smooth genus
$g$ curves with reduced base. Let $D_0$ and $D_{\infty}$ be positive relative
divisors on $\CC$ whose supports are disjoint and are \'etale over $S$. Suppose
that their difference is
linearly equivalent to the pull-back of a divisor on $S$. Then for every
section
$x:S\to \CC$ of $f$ with image in the support of $D_0+D_{\infty}$, a suitable
positive tensor power of $x^*\omega _{\CC /S}$ is trivial.
\endproclaim
We shall use the following simple fact:
\proclaim{\label Lemma}
Let $L_1,\dots ,L_d$ be line bundles on a variety $V$ and let $V=V^0\supset
V^1\supset\cdots\supset V^d$ be a chain of closed
subvarieties such that $L_k$ is trivial on $V^{k-1}-V^k$. Then $c_1(L_1)\cdots
c_1(L_d)$ has support in $V^d$.
\endproclaim
The key result we need is:
\proclaim{\label Lemma}
Let $d$ be a positive integer and let $\{(C_t,x_t,P_t)\} _{t\in\Delta}$ be
an analytic family of triples consisting of a smooth connected projective curve
$C_t$, a point $x_t\in C_t$, and a pencil $P_t$ on $C_t$ containing $d(x_t)$.
Assume that for $t\not=0$, $P_t$ has no base points and let $R_t$ be the part
of
the ramification divisor on $C_t-x_t$ of the associated morphism $C_t\to P_t$.
If $R_0$ is the limit of $R_t$ for $t\to 0$, then the multiplicity of $x_0$ in
$R_0$ is also the multiplicity of $x_0$ as a fixed point of $P_0$.
\endproclaim \demo{Proof} Represent the family by a smooth analytic morphism
$t:\CC\to\Delta$ with section $x:\Delta \to\CC$. Extend $t$ to a chart $(z,t)$
at $x_0$ such that $z=0$ is the image of $x$ at $x_0$. In terms of these
coordinates generators of $P_t$ can be represented by $z^d$ and a holomorphic
function $A(z,t)=\sum _{i\not= d} a_i(t)z^i$ which is divisible neither by $t$
nor by $z$. The first index $k$ for which $a_k(0)\not= 0$ is $<d$ and is equal
to the multiplicity of $x_0$ as fixed point of $P_0$. In the domain of the
chart, the divisor $R_t$ is given by locus where the $z$-derivatives of $A$ and
$z^d$ are proportional, i.e., by the divisor of $\sum _{i \not= d}
(i-d)a_i(t)z^i$ ($t\not= 0$). This expression is not divisible by $z$ or $t$ so
that $R_0$ is given by $\sum _{i\not= d} (i-d)a_i(0)z^i$. So $x_0$ occurs with
multiplicity $k$ in $R_0$. \enddemo
An immediate consequence is an amplification of a result due to Arbarello
\cite{1} and Diaz \cite{2}:
\proclaim{\label Corollary} Suppose that in the situation of \refer{2.4} there
exists an analytic section $\{ D_t\in P_t\} _{t\in\Delta}$ such that for
$t\not=
0$, $\supp (D_t)$ is disjoint with $x_t$ and has $d-r$ points, whereas
$D_0=d(x_0)$. Then $P_0$ can be written as $r(x_0) +P'$. \endproclaim
\medskip
\label If $d$ is positive integer, then we have moduli space $P(d)$ of triples
$(C,x,P)$ with $C$ a smooth projective curve of genus $g$, $x\in C$ and $P$ a
pencil on $C$ containing $(d)x$. The existence of this is clear if $d>2g-2$,
for then this is just a bundle of projective spaces of dimension $d-g-1$ over
$\CC _g$; the remaining cases $d\le 2g-2$ follows from this by simply viewing
$P(d)$ as the locus in $P(2g-1)$ parametrizing triples $(C,x,P)$ for which $x$
is a fixed point in $P$ of multiplicity $2g-1-d$. This implies that we also
have defined a moduli space $Z$ of tuples $(C,x_1,\dots ,x_n,x,D,P)$
with $C$ a smooth projective curve of genus $g$, $x_1,\dots ,x_n,x\in C$, $P$ a
pencil on $C$ containing $(n+g)x$, $D$ a degenerate member of $P$ and $\{
x_1,\cdots ,x_n\}\subset \supp (D)$.
Notice that $D$ and $x$ determine $P$ unless $D=(n+g)(x)$. The forgetful
morphism $f:Z\to\CC ^n_g$ is clearly proper.
The tuples for which $\supp (D)$ has at most $g+n-1-k$ points outside $x$
define a closed subvariety $Z^k$ of $Z$. It is clear that $Z^{n+g-1}$ can be
identified with the set of tuples $(C,x,\dots ,x,x,(n+g)x,P)$ with $P$ a pencil
through $(n+g)(x)$.
\proclaim{\label Lemma}
For $k<g+n-1$, $Z^k-Z^{k+1}$ is Zariski-open in an affine variety of pure
dimension $3g-3+n-k$ and $f^*K_i|Z^k-Z^{k+1}=0$ ($i=1,\dots ,n$).
\endproclaim
\demo{Proof}
Let $k<g+n-1$ and let $W$ be a connected component of $Z^k-Z^{k+1}$.
If $(C,x_1,\dots ,x_n,x,D,P)$ represents an element of $W$, then write
$D=m(x)+D'$ with $x\notin\supp (D')$ so that $\supp (D')$ has exactly $n+g-k$
points. There is a finite morphism $\pi :C\to\Pone$ with $\pi ^*(0)=D'$ and
$\pi ^*(\infty )=(g+n-m)(x)$. The part of the ramification divisor $R$ of $\pi$
over $\Pone -\{ 0,\infty \}$ has by Riemann-Hurwitz degree
$2g-2+(g+n-k)=3g-2+n-k$.
The multiplicity $m$, the multiplicity of $x_i$ in $D$, and the stratum of the
diagonal stratification of $\CC ^{n+1}_g$ containing $(C,x_1,\dots ,x_n,x)$
only
depend on $W$. So assigning to $(C,x_1,\dots ,x_n,x,P)$ the $\C
^{\times}$-orbit of $\pi _*R$ defines a flat, quasi-finite morphism from $W$ to
the quotient of a $(3g-3+n-k)$-dimensional torus by an action of the symmetric
group. So $W$ is pure of dimension $3g-3+n-k$. Proposition \refer{2.2} implies
that $f^*K_i|W$ is trivial. \enddemo
\demo{Proof of the first clause of \refer{1.1}} Let $X^k$ be the union of
irreducible components of $Z^k$ that are distinct from $Z^{n+g-1}$. (It can be
shown that $Z^{n+g-1}$ is actually an irreducible component of $Z$
and so $X^0\not= Z$.) The restriction $f: X^0\to\CC ^n_g$ is clearly proper.
It is also surjective, because for given $(C,x_1,\dots ,x_n)$, the morphism
$$
(y,y_1,\dots ,y_{g-1})\in C^g\mapsto [-(n+g)y + 2(x_1)+\sum _{i=2}^n(x_i)+\sum
_{j=1}^{g-1}(y_j)]\in J(C)
$$
is onto. Observe that $X^{n+g-1}=\emptyset$.
We claim that $f(X^k\cap Z^{n+g-1})\subset f(X^{k+1})$. For if
$(C,x,\dots ,x,x,(n+g)x,P)$ represents an element of $X^k\cap Z^{n+g-1}$,
then by \refer{2.5}, $P$ will be of the form $(k+1)x +P'$ with $P'$ a pencil of
degree $n+g-k-1$. So $P$ has a member $\not= (n+g)(x)$ with at most $n+g-k-2$
points.
It follows that the pre-image $U^k$ of
$f(X^k)-f(X^{k+1})$ in $X^k$ is contained in $Z^k-Z^{k+1}$. In particular,
$f^*K_i|U^k=0$ for $i=1,\dots ,n$. Since $f:U ^k\to f(X^k)-f(X^{k+1})$
is proper and onto, we also have $K_i|f(X^k)-f(X^{k+1})$ =0. So by \refer{2.3},
a monomial of degree $k$ in $K_1,\dots ,K_n$ is a linear combination of
irreducible components of $f(X^k)$ of codimension $k$. One easily checks that
these components are as described in the theorem.
\enddemo
\label Since $f(X_k)-f(X_{k+1})$ admits a finite covering that is Zariski-open
in an affine variety, it cannot contain a complete curve. From this we recover
Diaz's theorem which asserts that $\CC _g^n$ does not contain a complete
subvariety of dimension $>g+n-2$.
In order to complete the proof of \refer{1.1} we need two more results, one
algebraic, one topological.
\proclaim{\label Lemma}
Let $f:\A \to S$ be a family of abelian
varieties of dimension $g$ and let $d$ be a positive integer.
Then the class of the locus $\A \la d\ra $ of points of order $d$ is
a positive multiple of the class of the zero section in $A^g(\A )$. (The
coefficient is the number of elements in $(\Z /d)^{2g}$ of order $d$.)
\endproclaim
\demo{Proof} We use the Fourier transform for abelian varieties introduced by
Mukai, developed by Beauville and extended to abelian schemes by
Deninger--Murre \cite{2}. Mukai's transform gives an (in
general inhomogeneous) isomorphism $\FF : A(\A )\to A(\hat\A )$, where
$\hat\A\to S $ is the dual family. We shall compare the images of
the two classes in $A(\hat\A )$ under $\FF$.
Let $k$ be an positive integer relative prime to $d$.
Multiplication by $k$ in $A$ maps $\A \la d\ra $ isomorphically onto itself. So
the class of $\A \la d\ra $ in $A^g(\A )$ is fixed under $k_*$.
Lemma \refer{2.18} of \cite{2} implies that then $\FF ([\A \la d\ra ])\in
A^0(\hat\A )$. Since the projection induces an isomorphism
$A^0(S )\to A^0(\hat\A )$, the lemma follows.
\enddemo
\proclaim{\label Lemma} Let $\pi :C\to\Pone$ be a covering of degree $d$
by a smooth connected curve that is totally ramified over $0$ and
$\infty$ such that the part $D$ of the discriminant in $\Pone -\{ 0,\infty\}$
is
reduced. Then there exists a disk neighborhood $B$ of
$\supp (D)$ in $\Pone -\{0,\infty\}$ such that for $p\in\partial B$, the
monodromy group
of $\pi$ over $(B-\supp (D),p)$ is a single transposition $(a',a'')$. Moreover,
if $\sigma$ is the monodromy of a simple loop in $\Pone -\inw (B)$ around $0$
based at $p$, then
$a''=\sigma ^r(a')$ for some divisor $r$ of $d$ and $\pi$ factorizes through
the covering $z\in\Pone\to z^r\in\Pone$.
\endproclaim
\demo{Proof}
We choose a base point $p\in\Pone$ outside the discriminant and we put $F:=\pi
^{-1}(p)$. By a {\it simple arc} we shall mean an embedded interval connecting
$p$ with a point of the discriminant that does not meet the discriminant along
the way. A simple arc $\alpha$
determines up to isotopy (relative $p$ and the discriminant) a simple loop
based at $p$ around a point of the discriminant and hence a
monodromy transformation $\tau _{\alpha}\in\Aut (F)$. A collection of simple
arcs that do not meet outside $p$ shall be called an {\it arc system}. Notice
that the directions of departure of the members of such a collection determine
a cyclic ordering (our preference is clockwise) of these.
We begin by fixing a simple arc $\omega$ connecting $p$ with $0$. We write
$\sigma$ for $\tau _{\omega}$; this is a $d$-cycle in $\Aut (F)$.
Any transposition $\tau$ in $\Aut (F)$ can be written $(a,\sigma ^l(a))$ for
some $l\in\{ 0,1,\dots {1\over 2}d\}$; this means that $\sigma\tau$ is the
product of two disjoint cycles of length $l$ and $d-l$. Let us call $l$ the
{\it mesh} of $\tau$.
Let $\alpha _1$ be an simple arc to a point of $\supp (D)$ that forms with
$\omega$ an arc
system and is such that $\tau :=\tau _{\alpha _1}$ has minimal mesh $r$. Write
$\sigma\tau =\sigma '\sigma ''$ with
$\sigma '$ and $\sigma ''$ disjoint cycles of length $r$ resp.\ $d-r$ and
denote by $F'$ and $F''$ the corresponding parts of $F$. Notice that $\tau
_{\alpha _1}$ interchanges some $a'\in F'$ with some $a''\in F''$.
Let $\beta$ be another simple arc to a point of $\supp (D)$ such that $(\omega
,\alpha _1
,\beta )$ is a clockwise oriented arc system. Then $\tau _{\beta}$ cannot
commute with $\sigma ''$: if it did, then it would interchange two points of
$F'$ and would therefore have a mesh $<r$.
It may happen that $\tau _{\beta}$ commutes with $\sigma '$. But not every
choice for $\beta$ can be like this, for then $\sigma '$ would commute with the
monodromy around $\infty$ and this is impossible as the latter is a $d$-cycle.
So for some $\beta$, $\tau _{\beta}$
interchanges some $b'\in F'$ with some $b''\in F''$. If we modify $\beta$
by letting it first wind $k$ times
around the union of $\omega$ and $\alpha _1$, then its monodromy gets
conjugated by $(\sigma '\sigma '')^{\pm k}$. In this way we can arrange that
$b''=a''$. If $b'\not= a'$, then a straightforward verification shows that
$\tau _{\beta}$ would have a smaller mesh than $r$. So $b'=a'$ and hence $\tau
_{\beta}=\tau$. This argument proves more: the fact that for every integer $k$
the mesh of the $(\sigma '\sigma '')^k$-conjugate of $\tau _{\beta}$ is $\ge r$
implies that $r$ divides $d$. We put $\alpha _2:=\beta$.
We now prove with induction on $l$ that for $l\le \deg (D)$ there is an arc
system $(\alpha _1,\alpha _2,\dots ,\alpha _l)$ in clockwise cyclic order such
that $\tau _{\alpha _i}=\tau$ for $i=1,\dots ,l$.
The lemma then follows: we already showed that $r$ divides $d$, and it is easy
to see that the asserted factorization exists. So suppose we found such an arc
system $(\alpha _1,\alpha _2,\dots ,\alpha _l)$ for some $l\ge 2$.
First assume $l$ even. Then the monodromy around the union of these arcs is
equal to $\sigma$ and so the above argument yields simple arcs $\beta _1,\beta
_2$ such that $\tau _{\beta _1}=\tau _{\beta _2}$ and $(\omega ,\alpha _1,\dots
,\alpha _l,\beta _1,\beta _2)$ is an arc system in clockwise order.
Since $\tau _{\beta _i}$ does not commute with $\sigma ''$, we can modify
$\beta _1$ and $\beta _2$ by letting both go round the union of $(\omega
,\alpha _1,\dots ,\alpha _l)$ the same number of times first, to ensure that
$\tau _{\beta _1}=\tau _{\beta _2}$ moves $a''$.
If $\tau _{\beta _i}$ does not commute with $\sigma '$, then the argument above
shows that in fact $\tau _{\beta _i}=\tau$ and so we managed to take two
induction steps.
If $\tau _{\beta _i}$ does commute with $\sigma '$, then let $\beta '_i$ be
obtained from $\beta _i$ by going round $\alpha _l$ first. Then $(\omega
,\alpha _1,\dots ,\alpha _{l-1},\beta '_1,\beta ' _2,\alpha _l)$ is in
clockwise order and
$\tau _{\beta '_1}=\tau _{\beta '_2}$ interchanges an element of $F'$ with an
element of $F''$. Next modify the $\beta '_1$ and $\beta '_2$ by letting them
first encircle $(\omega ,\alpha _1,\dots ,\alpha _{l-1})$ the same number of
times as to arrange that $\tau _{\beta '_i}$ moves $a''$ (this might cause them
to meet $\alpha _l$ in a point $\not= p$). Then $\tau _{\beta '_i}=\tau$ and
hence we have made the induction step.
It remains to do the induction step for $l$ odd. That is handled in the same
way as the case $l=1$.
\enddemo
\demo{Proof of the second clause of \refer{1.1}}
Notice that $X^{n+g-2}$ parametrizes the triples $(C,x,y)$, where $C$
is smooth of genus $g$, $x,y\in C$ are distinct and $d(x)\equiv d(y)$ for some
$d\in \{ 2,\dots ,n+g\}$. By our previous discussion this defines a closed
subvariety $Y$ of $\CC ^2_g$ of pure codimension $g$. The assertion that is to
be proved will follow if we show that the classes of the irreducible components
of $Y$ are proportional in $A^g(\CC ^2_g)$. Our first business is therefore to
describe these irreducible components.
For $d\ge 2$, let $Y_d\subset \CC ^2_g$ be the locus parametrizing triples
$(C,x,y)$ for which $(x)-(y)$ has order $d$ in $J(C)$. For such
$(C,x,y)$ we have a morphism $\pi :C\to\Pone$ of degree $d$ such that $\pi
^*(0)=d(x)$, $\pi ^*(\infty )=d(y)$ and $\pi$ does not factor through a cover
$z\in\Pone\mapsto z^r\in\Pone$ for some $r>1$. The previous lemma shows that
all such covers are of the same topological type. This
implies that $Y_d$ is irreducible. So every irreducible component of $Y$ is
equal to some $Y_d$.
Let $\JJ_g\to\M _g$ be the universal Jacobian and let $q:\CC _g^2\to\JJ
_g$ be the Abel-Jacobi map $(C,x,y)\mapsto (x)-(y)\in J(C)$. Then
$Y_d=q^{-1}\JJ
_g\la d\ra $. Since $Y_d$ has the correct codimension $g$ in $\CC _g^2$, it
follows that $[Y_d]$ is a positive multiple of $q^*[\JJ _g\la d\ra ]$.
According
to \refer{2.9}, $[\JJ _g\la d\ra ]$ is a positive multiple of the class of the
zero section in $A^g(\JJ _g )$ and so the proof is complete.
\enddemo
\demo{Proof of \refer{1.2}} First observe that the direct image of
$K_1^{1+d_1}\cdots K_n^{1+d_n}$ under the forgetful morphism $\CC _g^n\to\M _g$
equals $\kappa _{d_1}\cdots\kappa _{d_n}$. Now the direct image of the class of
an irreducible component of $X^k$ of codimension $k$ under $\CC _g^n\to\M _g$
is
zero unless the image has the correct codimension $k-n$. In particular, a
nonzero image requires $k\ge n$. It follows that any product in the
tautological
classes of degree $d$ can be represented by a linear combination of the
irreducible components of the locus in $\M _g$ that parametrizes the curves $C$
that admit a covering $\pi :C\to \Pone$ of degree $\le 2g-2$ totally ramified
over $\infty$ and with at most $g-1-d$ points over $0$. The rest follows
immediately from \refer{1.1}.
\enddemo
\Refs
\ref\no 1
\paper Weierstrass points and moduli of curves
\by E.\ Arbarello
\jour Compositio Math.
\vol 29
\yr 1974
\pages 325--342
\endref
\ref\no 2
\paper Motivic decomposition of abelian schemes and the Fourier transform
\by C.\ Deninger \&\ J.\ Murre
\jour J. reine angew. Math.
\vol 422
\yr 1991
\pages 201--219
\endref
\ref\no 3
\paper A bound on the dimensions of complete subvarieties of $\M _g$
\by S.\ Diaz
\jour Duke Math.\ J.
\vol 51
\yr 1984
\pages 405--408
\endref
\ref\no 4
\by D. Mumford
\paper Towards an enumerative geometry of the moduli space of curves
\inbook Arithmetic and Geometry.~{\rm II}
\eds M. Artin and J. Tate
\publ Birkha\"user Verlag
\publaddr Boston--Basel--Berlin
\pages 271--328
\yr 1983
\endref
\endRefs
\enddocument
|
1995-10-19T05:20:10 | 9411 | alg-geom/9411006 | en | https://arxiv.org/abs/alg-geom/9411006 | [
"alg-geom",
"math.AG"
] | alg-geom/9411006 | Fumiharu Kato | Fumiharu Kato | Logarithmic Embeddings and Logarithmic Semistable Reductions | LaTeX | null | null | null | null | In this paper, we give a criterion for the existence of logarithmic
embeddings -- which was first introduced by Steenbrink -- for general normal
crossing varieties. Using this criterion, we also give a new proof of the
theorem of Kawamata--Namikawa which states a criterion for the existence of the
log structures of semistable type.
| [
{
"version": "v1",
"created": "Fri, 11 Nov 1994 05:04:56 GMT"
},
{
"version": "v2",
"created": "Wed, 18 Oct 1995 16:34:51 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Kato",
"Fumiharu",
""
]
] | alg-geom | \section{Introduction}
Let $X$ be a connected, geometrically reduced algebraic scheme over a
field $k$. Then $X$ is said to be a {\it normal crossing variety} of
dimension $n-1$ if there exists an isomorphism of $k$-algebras
$$
\widehat{\O}_{X,x}\stackrel{\sim}{\longrightarrow}
k(x)[[T_1,\ldots,T_n]]/(T_1\cdots T_{l_x})
$$
for each closed point $x\in X$, where $\widehat{\O}_{X,x}$ denotes the
completion of the local ring $\O_{X,x}$ along its maximal ideal
(Definition \ref{ncvdef}).
Normal crossing varieties usually appear in contexts of algebraic geometry
via degenerations and normal crossing divisors.
In the first case, they appear as a specialization of a family of smooth
varieties.
Normal crossing varieties are usually considered and expected to be limits
of smooth varieties, and --- as is well--known --- they
are important to the theory of moduli.
As for the second situation, a {\it normal crossing divisor} is a divisor
of a smooth variety which itself is a normal crossing variety.
Normal crossing divisors play important roles
in various fields of algebraic geometry.
For example, a pair of smooth variety and its normal crossing divisor is
usually called a log variety.
Considering log varieties instead of smooth varieties ---
or usually admitting some mild singularities --- alone,
some algebro geometric theories ({\it e.g.}, minimal model theory, etc.)
are well generalized.
Relating with a normal crossing variety $X$, there are two problems,
{\it smoothings} and {\it embeddings},
in light of degenerations and normal crossing divisors, respectively.
The {\it smoothing problem} is a problem to find a Cartesian diagram
$$
\begin{array}{ccc}
X&\longrightarrow&\hbox{\maxid X}\\
\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\
0&\longrightarrow&\Delta\rlap{,}
\end{array}
$$
for a normal crossing variety $X$ (in this situation, we should assume that
$X$ is proper over $k$), where $\Delta$ is a one-dimensional regular scheme,
$\hbox{\maxid X}$ is a regular scheme proper flat and generically smooth over $\Delta$,
and $0$ is a closed point of $\Delta$ whose residue field is $k$.
We usually take, as the base scheme $\Delta$, the spectrum of a discrete
valuation ring, {\it e.g.}, the
ring of formal power series over $k$ or --- in case $k$ is perfect ---
the ring of Witt vectors over $k$.
In the complex analytic situation, Friedmann \cite{Fri1}
studied the smoothing problem generally, and solve it
for degenerated K3 surfaces.
Recently, Kawamata--Namikawa \cite{K-N1} approached this problem
by introducing a new method; the {\it logarithmic}
method.
The Cartesian diagram as above with $\Delta$ a
spectrum of an Artinian local ring $A$ is
called an {\it infinitesimal smoothing},
if it is \'{e}tale locally isomorphic to the diagram
$$
\begin{array}{ccc}
\mathop{\mbox{\rm Spec}}\nolimits k[Z_1,\ldots,Z_n]/(Z_1\cdots Z_l)&\longrightarrow&
\mathop{\mbox{\rm Spec}}\nolimits A[Z_1,\ldots,Z_n]/(Z_1\cdots Z_l-\pi)\\
\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\
\mathop{\mbox{\rm Spec}}\nolimits k&\longrightarrow&\mathop{\mbox{\rm Spec}}\nolimits A\rlap{,}
\end{array}
$$
where $\pi$ is an element of the maximal ideal of $A$ and $\pi\neq 0$.
The central problem to find such an infinitesimal smoothing is to compute
the obstruction class of $X$ to have such a diagram and to show vanishing
or non--vanishing of it.
The {\it embedding problem} is a problem to find
a closed embedding $X\hookrightarrow V$ over $k$ of $X$ as a normal crossing
divisor, where $V$ is a smooth variety over $k$.
If $X$ is smoothable in the above sense
with $\Delta$ a smooth algebraic variety over $\mathop{\mbox{\rm Spec}}\nolimits k$, the smoothing family
$X\hookrightarrow\hbox{\maxid X}$ gives an embedding of this sense.
If $X$ is smooth, this problem becomes trivial, since we can take as $V$ the
product of $X$ and, for example, ${\mbox{\bf P}}^1$.
But for a general normal crossing variety, this problem seems
far from satisfactory solutions.
Similarly to the smoothing problem, we can consider
this problem in the infinitesimal sense.
In this paper, we consider the above problems in a {\it logarithmic} sense.
We consider logarithmic generalizations of smoothings and embeddings of
normal crossing varieties according to Kajiwara \cite{Kaj1},
Kawamata--Namikawa \cite{K-N1} and Steenbrink \cite{Ste1}, and we solve
their existence problems.
These generalizations are done in terms of logarithmic geometry of
Fontaine, Illusie and Kazuya Kato.
{\it Logarithmic geometry} --- or {\it log geometry} --- was first founded by
Fontaine and Illusie based on their idea of, so--called,
{\it log structures};
afterwards, it was established as a generally
organized theory and applied to various fields of algebraic and
arithmetic geometry by Fontaine, Illusie and Kazuya Kato (cf. \cite{Kat1},
\cite{Kat4}).
In various kinds of geometries including algebraic geometry,
we usually consider local ringed spaces, {\it i.e.}, the pairs of topological
spaces --- possibly in the sense of Grothendieck topologies ---
and sheaves of local rings over them.
The basic idea of Fontaine and Illusie is that, instead of
local ringed spaces alone, they consider local ringed spaces equipped with some
additional structure --- which they call the logarithmic structures ---
written in
terms of sheaves of commutative and unitary monoids (see \cite{Kat1} for the
precise definition). In algebro geometric situations,
these log structures usually
represent ``something'' of the underlying local ringed spaces, {\it e.g.},
divisors or the structure of torus embeddings, etc. Through these
foundations, they suggested to generalize the ``classical'' geometries by
considering ``log objects'' --- such as {\it log schemes} --- which are
the pairs of local ringed spaces and log structures on them.
In the present paper, we recall and generalize the
{\it logarithmic embedding}
(Definition \ref{logembdef}) introduced by Steenbrink \cite{Ste1}.
A logarithmic embedding --- which is regarded as a logarithmic generalization
of a log variety --- is a certain log scheme $(X,{\cal M}_X)$ with $X$ a
normal crossing variety.
Then we prove the following theorem which
gives a criterion for the existence of logarithmic embeddings:
\vspace{3mm}\noindent
{\bf Theorem}\ ({\it Theorem \ref{mainthm}})\ {\it
For a normal crossing variety $X$, a logarithmic embedding of $X$ exists if
and only if there exists a line bundle ${\cal L}$ on $X$ such that
${\cal L}\otimes_{\O_X}\O_D\stackrel{\sim}{\rightarrow}
{\cal T}^1_X$, where $D$ is the singular locus of $X$.}
\vspace{3mm}
Here, ${\cal T}^1_X$ is an invertible $\O_D$-module, called the {\it infinitesimal
normal bundle} (cf. \cite{Fri1}), which is naturally isomorphic to
${\cal E}xt^1_{\O_X}(\Omega^1_X,\O_X)$; we recall the construction of it in
\S 3.
A normal crossing variety $X$ is said to be
{\it $d$-semistable} if
${\cal T}^1_X$ is a trivial bundle on $D$ (cf. \cite{Fri1}).
By the above theorem, any $d$-semistable normal crossing variety $X$ has a
logarithmic embedding.
As for the smoothing problem, we recall and generalize the concept,
the {\it logarithmic semistable reduction} (Definition \ref{logsemidef})
introduced by Kajiwara \cite{Kaj1} (in one dimensional case) and
Kawamata--Namikawa \cite{K-N1} (by a different but essentially the same
method).
Using the above theorem, we get a criterion for the existence of
logarithmic semistable reductions, which was first proved by
Kawamata--Namikawa \cite{K-N1} in the complex analytic situation, as
follows:
\vspace{3mm}\noindent
{\bf Theorem}\ ({\it Theorem \ref{mainthm2}})
{\rm (cf. \cite{K-N1})}\
{\it For a normal crossing variety $X$,
the log structure of semistable type
on $X$ exists if and only if $X$ is $d$-semistable.}
\vspace{3mm}
The composition of this paper is as follows.
In \S 2, we study the geometry of normal crossing varieties in general.
In particular, we define good \'{e}tale local charts on normal crossing
varieties, and prove the existence of them.
In \S 3, we recall the basic construction of the tangent complex of a
normal crossing variety, and introduce the invertible sheaf
${\cal T}^1_X$ on $D$.
We introduce the logarithmic embedding in \S 4.
This section also contains the proof of our main theorem.
The logarithmic semistable reduction is studied in \S 5.
The author thanks T. Fujisawa for useful communications.
The author is also grateful to Professors K. Ueno, S. Usui and T. Yusa
for their helpful comments.
{\sc Conventions}:\
All sheaves are considered with respect to \'{e}tale topology.
By a monoid, we mean --- as usual in the contexts of log geometry ---
a set with a commutative and associative binary operation and the neutral
element.
For such a monoid $M$, we denote by $\gp{M}$ the Grothendieck group of $M$.
We denote by $\mbox{\bf N}$ the monoid of non--negative integers.
\section{Normal crossing varieties}
Throughout this paper, we always work over a fixed base field $k$.
As usual, an algebraic $k$-scheme is, by definition, a seperated
scheme of finite type over $k$.
Let $X$ be an algebraic $k$-scheme and $x\in X$ a point.
We denote the residue field at $x\in X$ by $k(x)$.
\begin{dfn}\label{ncvdef}
Let $X$ be a connected and geometrically reduced algebraic $k$-scheme.
Then $X$ is said to be a {\it normal crossing variety} over $k$ of
dimension $n-1$ if the following condition is satisfied: For any closed
point $x\in X$, there exists an isomorphism
\begin{equation}\label{ncvdefloc}
\widehat{\O}_{X,x}\stackrel{\sim}{\longrightarrow}
k(x)[[T_1,\ldots,T_n]]/(T_1\cdots T_{l_x})
\end{equation}
of $k$-algebras, where $l_x$ is an integer $(1\leq l_x\leq n)$
depending on $x$.
Here, we denote by $\widehat{\O}_{X,x}$ the completion of the local ring
$\O_{X,x}$ by its maximal ideal.
\end{dfn}
The integer $l_x$ is called the {\it multiplicity} at $x\in X$.
We sometimes denote it by $l^X_{x}$ if we want to emphasize the scheme $X$.
The Zariski closure of the set of closed points whose multiplicity is greater
than 1 is the singular locus of $X$, which we denote by $D$.
A standard example of normal crossing varieties is an affine scheme
\begin{equation}\label{ncvstandard}
\mathop{\mbox{\rm Spec}}\nolimits k[T_1,\ldots,T_n]/(T_1\cdots T_l)\ \ (1\leq l\leq n).
\end{equation}
This scheme consists of $l$ irreducible components which intersect
transversally along the singular locus
\begin{equation}
\mathop{\mbox{\rm Spec}}\nolimits k[T_1,\ldots,T_n]/(T_1\cdots\widehat{T_j}\cdots T_l\ :\ 1\leq j\leq l).
\end{equation}
Each irreducible component is isomorphic to the affine $(n-1)$-space
over $k$.
In general, a normal crossing variety $X$ is said to be {\it simple}
if each irreducible component of $X$ is smooth over $k$.
For example, a smooth $k$-variety is a simple normal crossing variety.
Let $V$ be a smooth $k$-variety of dimension $n$.
A reduced divisor $X$ on $V$ is called a {\it normal crossing divisor}
if $X$ itself is a normal crossing variety
of dimension $n-1$.
In this case, the closed embedding $X\hookrightarrow V$ is called a {\it NCD
embedding} of $X$.
For example, the affine normal crossing variety (\ref{ncvstandard}) is
a normal crossing divisor in the affine $n$-space over $k$.
The proof of the following proposition is straightforward and is left
to the reader.
\begin{pro}\label{ncvetale}
Let $Y$ be a connected scheme \'{e}tale over a connected algebraic
$k$-scheme $X$.
If $X$ is a normal crossing variety, then so is $Y$.
The converse is also true if the \'{e}tale morphism $Y\rightarrow X$
is surjective.
\end{pro}
It is clear that an \'{e}tale morphism leaves invariant the multiplicity at
every closed point, {\it i.e.}, if $\varphi\colon Y\rightarrow X$ is an
\'{e}tale
morphism of normal crossing varieties and $y\in Y$ is a closed point,
then we have $l^Y_y=l^X_{\varphi(y)}$.
In the following paragraphs of this section, we shall study the local nature
of normal crossing varieties for the later purpose.
In the subsequent sections, we need to take a good \'{e}tale
neighborhood around every closed point.
We require that these \'{e}tale neighborhoods have good coordinate systems
which serve for several explicit calculations.
To clarify the notion of ``good'' \'{e}tale neighborhoods, we define them
as follows:
\begin{dfn}\label{ncvchart}{\rm
Let $X$ be a normal crossing variety and $x\in X$ a closed point.
Let $\varphi\colon U\rightarrow X$ be an \'{e}tale morphism with $U$ a simple
normal crossing variety and
$z_1,\ldots,z_{l_x}\in\Gamma(U,\O_U)$, where $l_x$ is the multiplicity
at $x$.
Then $(\varphi\colon U\rightarrow X;\ z_1,\ldots,z_{l_x})$ is said to be a {\it
local chart} around $x$ if the following conditions are satisfied:
\begin{description}
\item[{\rm (a)}] There exists a unique point $y\in U$ such that
$\varphi(y)=x$.
\item[{\rm (b)}] There exists a closed immersion $\iota\colon U\hookrightarrow
V$, where $V$ is an affine smooth $k$-scheme.
\item[{\rm (c)}] There exist $Z_1,\ldots,Z_n\in\Gamma(V,\O_V)$ which form
a regular parameter system at $\iota(y)\in V$ such that $z_i=\iota^*Z_i$
for $1\leq i\leq l_x$, and $U$ is defined as a closed
subset in $V$ by the ideal $(Z_1\cdots Z_{l_x})$.
\item[{\rm (d)}] each ideal
$(z_i)$ is prime and the irreducible components of $U$ are precisely the
closed subsets of $U$ corresponding to the ideals $(z_1),\ldots,(z_{l_x})$.
\end{description}
}
\end{dfn}
Note that $\iota\colon U\hookrightarrow V$ is, due to (c), a NCD embedding.
Moreover, due to (d), all the irreducible components intersect and contain
the point $y$.
The following theorem assures the existence of local chart around every
closed point of normal crossing variety $X$.
We prove this theorem later in this section.
\begin{thm}\label{chartexist}
Let $X$ be a normal crossing variety and $x\in X$ a closed point.
Then there exists a local chart
$(\varphi\colon U\rightarrow X;\ z_1,\ldots,z_{l_x})$ around $x$.
\end{thm}
Since any \'{e}tale open set of $X$ is again a normal crossing variety,
we have the following:
\begin{cor}\label{specialcov}
Let $X$ be a normal crossing variety.
Then the set of all local charts forms an open basis with respect to the
\'{e}tale topology on $X$.
\end{cor}
\begin{rem}\label{ncvemb}{\rm
Theorem \ref{chartexist} implies that any normal crossing variety is
realized as a simple normal crossing divisor on some smooth $k$-variety
\'{e}tale locally.
But a normal crossing variety, in general, cannot be a normal crossing
divisor globally on a smooth $k$-variety.
In the next section, we will see a necessary condition for a normal
crossing variety to be a normal crossing divisor
(Proposition \ref{suffcondemb}).
}
\end{rem}
For the proof of Theorem \ref{chartexist}, we need one lemma:
\begin{lem}\label{htzero}
Let $\hbox{\maxid q}$ be a height zero prime ideal in $K[T_1,\ldots,T_n]/(T_1\cdots T_l)$
$(1\leq l\leq n)$, where $K$ is a field.
Then $\hbox{\maxid q}=(T_j)$ for some $j$ $(1\leq j\leq l)$.
\end{lem}
{\sc Proof.}\hspace{2mm}
By Krull's principal ideal theorem, any non--zero element in $\hbox{\maxid q}$ is a
zero factor. Hence any element in $\hbox{\maxid q}$ is a multiple of $T_j$'s $(1\leq
j\leq l)$. Since $\hbox{\maxid q}$ is a prime ideal, $\hbox{\maxid q}$ must contain $T_j$ for some
$j$ $(1\leq j\leq l)$, {\it i.e.}, $(T_j)\subseteq\hbox{\maxid q}$. But since the height of
$\hbox{\maxid q}$ is zero and $(T_j)$ is a prime ideal, we have $\hbox{\maxid q}=(T_j)$.
$\Box$
\vspace{3mm}
{\sc Proof of Theorem \ref{chartexist}.}\hspace{2mm}
The complete local ring $\widehat{\O}_{X,x}$ is isomorphic to the complete
local
ring $k(x)[[T_1,\ldots,T_n]]/(T_1\cdots T_{l_x})$ which is a completion of
the local ring $(k(x)[T_1,\ldots,T_n]/(T_1\cdots T_{l_x}))_{0}$.
Then due to \cite[Corollary (2.6)]{Art1}, there exist a scheme
$U$ and \'{e}tale morphisms $\varphi\colon U\rightarrow X$ and
$\phi\colon U\rightarrow\mathop{\mbox{\rm Spec}}\nolimits k(x)[T_1,\ldots,T_n]/(T_1\cdots T_{l_x})$ such
that
$\varphi(y)=x$ and $\phi(y)=0$ for some $y\in U$.
We fix this closed point $y\in U$.
Since $\varphi$ is \'{e}tale, we may assume --- replacing $U$ by its Zariski
open subset if necessary --- that $y$ is the only point which
is mapped to $x$ by $\varphi$.
Obviously we may assume that $U$ is connected and affine.
We can remove all the irreducible components which do not contain $y$.
Then we may assume that all the irreducible components of $U$ contain $y$.
We set $U=\mathop{\mbox{\rm Spec}}\nolimits A$ and $B\colon =k(x)[T_1,\ldots,T_n]/(T_1\cdots T_{l_x})$.
Since $U$ is \'{e}tale over a reduced $k$-scheme $\mathop{\mbox{\rm Spec}}\nolimits B$,
the $k$-algebra $A$ is reduced.
Take a minimal prime factorization
\begin{equation}\label{prmfac}
(0)=\hbox{\maxid p}_1\cap\cdots\cap\hbox{\maxid p}_{l}.
\end{equation}
of the ideal $(0)=\sqrt{(0)}$.
Since each $\hbox{\maxid p}_i$ is minimal in the set of all prime ideals, the height of
each $\hbox{\maxid p}_i$ is zero.
Obviously the prime decomposition (\ref{prmfac}) precisely corresponds to the
decomposition of $U$ into irreducible components.
Set $\hbox{\maxid q}_i\colon =\phi(\hbox{\maxid p}_i)$ which is a prime ideal of height zero in $B$ for
$1\leq i\leq l$.
Due to Lemma \ref{htzero}, we have $\hbox{\maxid q}_i=(T_{j_i})$ for some $j_i$
$(1\leq j_i\leq l_x)$, {\it i.e.}, any generic point of a irreducible component
of
$U$ is mapped by $\phi$ to a generic point of a irreducible component of
$\mathop{\mbox{\rm Spec}}\nolimits B$.
Let us suppose that the map $i\mapsto j_i$ is not injective,
{\it i.e.}, there exist $i$ and $j$
$(i\neq j)$ such that $\hbox{\maxid q}_i=\hbox{\maxid q}_j$.
Consider the Cartesian diagram
$$
\begin{array}{ccc}
\overline{\{\hbox{\maxid q}_i\}}\times_{\mathop{\mbox{\rm Spec}}\nolimits B}U&\lhook\joinrel\longrightarrow&U\\
\llap{$\phi_i$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$\phi$}\\
\overline{\{\hbox{\maxid q}_i\}}&\lhook\joinrel\longrightarrow&\mathop{\mbox{\rm Spec}}\nolimits B\rlap{,}
\end{array}
$$
where the horizontal arrows are closed immersions and the vertical ones are
\'{e}tale.
The scheme $U_i\colon =\overline{\{\hbox{\maxid q}_i\}}\times_{\mathop{\mbox{\rm Spec}}\nolimits B}U$ is also a normal
crossing variety. Since $\overline{\{\hbox{\maxid p}_i\}}\cap\overline{\{\hbox{\maxid p}_j\}}$ is a
closed subscheme (which contains $y$) of $U_i$,
the multiplicity $l^{U_i}_y$ at $y$ in $U_i$ is greater than 1.
But since the irreducible component $\overline{\{\hbox{\maxid q}_i\}}$ is smooth,
we have $l_{\phi_i(y)}=1$.
This is a contradiction since $l^{U_i}_y=l_{\phi_i(y)}$.
Thus, the map $i\mapsto j_i$ is injective, {\it i.e.}, there is at
most one component over each component of $\mathop{\mbox{\rm Spec}}\nolimits B$.
Moreover, in this case, we have $U_i=\overline{\{\hbox{\maxid p}_i\}}$.
Then the irreducible component $U_i$ is \'{e}tale
over a smooth scheme $\overline{\{\hbox{\maxid q}_i\}}$,
and hence the normal crossing
variety $U$ is simple. Moreover, the prime ideal $\hbox{\maxid p}_i$ is a principal ideal
$(z_i)$, where $z_i\colon =\phi^*T_{j_i}$ for $1\leq i\leq l$,
since $\overline{\{\hbox{\maxid p}_i\}}=\overline{\{\hbox{\maxid q}_i\}}\times_{\mathop{\mbox{\rm Spec}}\nolimits B}U$ implies that
$\hbox{\maxid p}_i=\hbox{\maxid q}_i\otimes_BA$.
Since the map $i\mapsto j_i$ is injective, we have $l\leq l_x$.
Note that the multiplicity $l_y$ at y in $U$ equals to $l_x$.
Since the simple normal crossing variety $U$ consists of $l$ irreducible
components, we have $l_y=l_x\leq l$.
Hence we have $l_x=l$.
The scheme $\mathop{\mbox{\rm Spec}}\nolimits B$ is a normal crossing divisor in the $n$ dimensional
affine space over $k(x)$.
Hence, due to \cite[Expos\'{e} 1. Proposition 8.1]{Gro1}, any point in $U$
has a Zariski open neighborhood which is embedded in a smooth
$k(x)$-variety as a normal crossing divisor.
This implies that, replacing $U$ by its Zariski open neighborhood of $y$,
we may assume that $U$ can be embedded in an affine smooth $k(x)$-scheme
$V=\mathop{\mbox{\rm Spec}}\nolimits R$ of dimension $n$ as a normal crossing divisor.
Let $\iota\colon U\hookrightarrow V$ be the closed immersion.
Finally, consider the Cartesian diagram
$$
\begin{array}{ccc}
U&\stackrel{\iota}{\lhook\joinrel\longrightarrow}&V\\
\llap{$\phi$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$\Phi$}\\
\mathop{\mbox{\rm Spec}}\nolimits B&\lhook\joinrel\longrightarrow&\mathop{\mbox{\rm Spec}}\nolimits k[T_1,\ldots,T_n]\rlap{,}
\end{array}
$$
where $\Phi$ is an \'{e}tale morphism.
Set $Z_i\colon =\Phi^*T_i\in\Gamma(V,\O_V)$ for $1\leq i\leq n$.
Then $Z_1,\ldots,Z_n$ form a regular parameter system at $\iota(y)\in V$.
We also have $z_i=\iota^* Z_i$ for $1\leq i\leq l_x$.
It is clear that the closed subscheme $U$ in $V$ is defined by an ideal
$(Z_1\cdots Z_{l_x})$.
Then the proof of the theorem is completed.
$\Box$
The following lemma will be needed in the later arguments.
\begin{lem}\label{inj-zero}
Let $(\varphi'\colon U'\rightarrow X;z'_1,\ldots,z'_{l'})$ be a local chart
on $X$ around some closed point and
$(\psi\colon U\rightarrow U';z_1,\ldots,z_{l})$ a local chart on $U'$
around some closed point.
Then $\psi\colon U\rightarrow U'$ is injective in codimension zero,
{\it i.e.}, it maps the generic points of irreducible components on $U$
injectively to those of $U'$.
\end{lem}
{\sc Proof.}\hspace{2mm}
Let $\eta\in U'$ be a codimension zero point.
Since $U'$ is simple, $\overline{\{\eta\}}$ is regular and so is
$\overline{\{\eta\}}\times_{U'}U$ whenever it is not empty.
Then each connected component of $\overline{\{\eta\}}\times_{U'}U$ is
irreducible and its generic point is of codimension zero.
Hence each connected component of $\overline{\{\eta\}}\times_{U'}U$ is
an irreducible component of $U$. Since any two of irreducible
components of $U$ intersect, $\overline{\{\eta\}}\times_{U'}U$ itself is an
irreducible component of $U$.
Hence, if $\xi\in U$ is a codimension zero point such that $\psi(\xi)=\eta$,
we have $\overline{\{\xi\}}=\overline{\{\eta\}}\times_{U'}U$.
In particular, there exists at most one such $\xi$.
$\Box$
For a normal crossing variety $X$, the {\it normalization}
$\nu\colon\widetilde{X}\rightarrow X$ of $X$ is defined as usual:
The scheme $\widetilde{X}$ is defined by the disjont union of the
normalizations of irreducible components of $X$ and $\nu\colon\widetilde{X}
\rightarrow X$ is the natural morphism. The normalization
$\widetilde{X}$ is a smooth $k$-scheme due to Theorem \ref{chartexist} and
the following lemma.
\begin{lem}\label{normalization}
Let $U\rightarrow Z$ be a \'{e}tale morphism of $k$-varieties.
Let $\widetilde{U}\rightarrow U$ and $\widetilde{Z}\rightarrow Z$ be
normalizations of $U$ and $Z$, respectively. Then there exists a natural
isomorphism
$\widetilde{U}\stackrel{\sim}{\rightarrow}U\times_{Z}\widetilde{Z}$.
In particular, the natural morphism $\widetilde{U}\rightarrow\widetilde{Z}$
is \'{e}tale.
\end{lem}
{\sc Proof.}\hspace{2mm}
Since $U\times_{Z}\widetilde{Z}\rightarrow\widetilde{Z}$ is \'{e}tale and
$\widetilde{Z}$ is normal, $U\times_{Z}\widetilde{Z}$ is a normal variety.
Hence there exists a unique morphism $\phi\colon
U\times_{Z}\widetilde{Z}\rightarrow
\widetilde{U}$ which factors the morphism $U\times_{Z}\widetilde{Z}\rightarrow
U$. Moreover $\phi$ also factors the morphism $U\times_{Z}\widetilde{Z}
\rightarrow\widetilde{Z}$ since the last morphism is the unique morphism
determined by the morphism $U\times_{Z}\widetilde{Z}\rightarrow Z$. Hence
the natural morphism $\varphi\colon \widetilde{U}\rightarrow
U\times_{Z}\widetilde{Z}$
is the inverse morphism of $\phi$.
$\Box$
For a local chart $(\varphi\colon U\rightarrow X; z_1,\ldots,z_l)$, the
normalization of $U$ is given by the disjoint union of all irreducible
components and the natural morphism, {\it i.e.},
$$
\nu_U\colon\widetilde{U}=\coprod^l_{i=1}U_i\longrightarrow U,
$$
where $U_i$ is the irreducible component of $U$ corresponding to the
ideal $(z_i)$.
Set $\overline{D}\colon =D\times_X\widetilde{X}$, which is a divisor of
$\widetilde{X}$.
\begin{lem}
$\overline{D}$ is a normal crossing divisor of $\widetilde{X}$.
\end{lem}
{\sc Proof.}\hspace{2mm}
Let $(\varphi\colon U\rightarrow X; z_1,\ldots,z_l)$ be a local chart on $X$.
Then $D_U\colon =D\times_{X}U$ is nothing but the singular locus of $U$ and is
\'{e}tale over $D$.
Consider the normalization $\nu_U\colon \widetilde{U}\rightarrow U$ as above.
Set
$$
\overline{D_U}\colon =D_U\times_{U}\widetilde{U}.
$$
Clearly, $\overline{D_U}$ is a normal crossing divisor of $\widetilde{U}$
defined by an ideal $(z_1\cdots\widehat{z_i}\cdots z_l)$ on $U_i$.
There exists a natural morphism $\overline{D_U}\rightarrow\overline{D}$.
Since one can easily see that there exists a natural isomorphism
$$
\overline{D_U}\stackrel{\sim}{\longrightarrow}
\overline{D}\times_{\widetilde{X}}\widetilde{U}
$$
and the morphism $\widetilde{U}\rightarrow\widetilde{X}$ is \'{e}tale
due to Lemma \ref{normalization}, the morphism $\overline{D_U}\rightarrow
\overline{D}$ is \'{e}tale.
Then, considering all the local charts on $X$, $\overline{D}$ is a normal
crossing divisor on $\widetilde{X}$ due to Proposition \ref{ncvetale}.
$\Box$
\section{Tangent complex on a normal crossing variety}
In this section, we recall the tangent complex and the infinitesimal normal
bundle ${\cal T}^1_X$ of a normal crossing variety $X$ which will play
important roles in the subsequent sections.
Let $X$ be a normal crossing variety over a field $k$.
For a local chart $(\varphi\colon U=\mathop{\mbox{\rm Spec}}\nolimits A\rightarrow X; z_1,\ldots,z_l)$ of
$X$ around some closed point, we use the folowing notation in this and
subsequent sections:
Let $V=\mathop{\mbox{\rm Spec}}\nolimits R$ and $Z_1,\ldots,Z_l$ be as in Definition \ref{ncvchart}.
Set $I_j\colon=(Z_j)$ and $J_j\colon=(Z_1\cdots\widehat{Z_j}\cdots Z_l)$
for $1\leq j\leq l$. (If $l=1$, we set $J_1=R$ for the convention.)
Then $A=R/I$ where $I\colon=I_1\cdots I_l$.
Moreover, the ideal $I_j/I\subset A$ is generated by $z_j=(Z_j\,\mbox{\rm
mod}\,I)$
and is prime of height zero.
Set $J\colon=J_1+\cdots+J_l$.
Then the singular locus $D_U\colon=D\times_XU$ of $U$ is the closed subscheme
defined by $J$. We set $Q\colon=R/J$.
Note that, for $1\leq j\leq l$, $I_j/II_j$ is a free $A$-module of rank
one and is generated by $\zeta_j\colon=(Z_j\,\mbox{\rm mod}\,II_j)$.
There exists a natural isomorphism $I_j/II_j\otimes_AQ\stackrel{\sim}
{\rightarrow}I_j/JI_j$ of $Q$-modules which maps $\zeta_j\otimes 1$ to
$\xi_j\colon=(Z_j\,\mbox{\rm mod}\,JI_j)$.
Moreover, there exists a natural isomorphism
\begin{equation}\label{conormal}
I/I^2\stackrel{\sim}{\rightarrow}
I_1/II_1\otimes_A\cdots\otimes_AI_l/II_l
\end{equation}
of $A$-modules, and hence, the $A$-module $I/I^2$ is free of rank one and is
generated by $\zeta_1\otimes\cdots\otimes\zeta_l$.
We denote by $\pi_j$ the natural projection $I_j/II_j\rightarrow I_j/I\subset
A$.
The cotangent complex of the morphism $k\rightarrow A$ is given by
$$
L^{\cdot}:0\longrightarrow R\otimes_{R}A\stackrel{\delta}{\longrightarrow}
\Omega^1_{R/k}\otimes_{R}A\longrightarrow 0,
$$
where $\delta$ is defined by $R\rightarrow F\cdot R
\stackrel{d}{\rightarrow}\Omega^1_{R/k}$ with $F\colon=Z_1\cdots Z_l$
(cf. \cite{L-S1}).
Then the tangent complex of $U$ is the complex
$$
\mathop{\mbox{\rm Hom}}\nolimits_A(L^{\cdot},A):0\longrightarrow\Theta_{R/k}\otimes_{R}A
\stackrel{\delta^*}{\longrightarrow}\mathop{\mbox{\rm Hom}}\nolimits_A(R\otimes_{R}A,A)
\longrightarrow 0,
$$
where $\Theta_{R/k}\colon =\mathop{\mbox{\rm Hom}}\nolimits_R(\Omega^1_{R/k},R)$.
We define
\begin{equation}\label{tangentdef}
T^1_A=\mathop{\mbox{\rm Hom}}\nolimits_A(R\otimes_{R}A,A)/\delta^*(\Theta_{R/k}\otimes_{R}A).
\end{equation}
\begin{lem}\label{tangentloc}
We have the natural isomorphism
\begin{equation}\label{tangentloc1}
T^1_A\stackrel{\sim}{\rightarrow}\mathop{\mbox{\rm Hom}}\nolimits_A(I/I^2,A)\otimes_{A}Q.
\end{equation}
\end{lem}
{\sc Proof.}\hspace{2mm}
Consider the exact sequence
$$
0\longrightarrow I/I^2\longrightarrow\Omega^1_{R/k}\otimes_{R}A
\longrightarrow\Omega^1_{A/k}\longrightarrow 0.
$$
By definition, we have
$T^1_A=\mathop{\mbox{\rm Coker}}\nolimits(\mathop{\mbox{\rm Hom}}\nolimits_A(\Omega^1_{R/k}\otimes_{R}A,A)\rightarrow
\mathop{\mbox{\rm Hom}}\nolimits_A(I/I^2,A))$.
Then one can show --- by direct calculations --- that
$\mathop{\mbox{\rm Hom}}\nolimits_A(I/I^2,A)\rightarrow T^1_A$ is nothing but the ``tensoring'' morphism
$\otimes_AQ$. Moreover, we have
$T^1_A\stackrel{\sim}{\rightarrow}\mathop{\mbox{\rm Ext}}\nolimits^1_A(\Omega^1_{A/k},A)$.
$\Box$
Considering all the local charts $U$ on $X$, these modules $T^1_A$ glue to an
invertible $\O_D$-module on $X$, which is denoted by ${\cal T}^1_X$;
this is well--known (cf. \cite{L-S1}) but, for the later purpose, we prove
it in the following.
Suppose we have two local charts $(\varphi\colon U\rightarrow X;
z_1,\ldots,z_{l})$ and $(\varphi'\colon U'\rightarrow X;
z'_1,\ldots,z'_{l'})$ and an \'{e}tale morphism $\psi\colon U\rightarrow U'$
such that $\varphi=\varphi'\circ\psi$.
(Because we are interested in the singular locus, we shall assume
$l>1$ and $l'>1$.)
For these local charts, we use all the notation as above. (For $U'$, we denote
them by $A'$, $I'$, $J'$, $\zeta'_j$, etc.)
Let $f\colon A'\rightarrow A$ be the ring homomorphism corresponding to
$\psi$.
We shall show that the morphism $\psi$ induces naturally an isomorphism
$T^1_{A'}\otimes_{Q'}Q\stackrel{\sim}{\rightarrow}T^1_A$ of $Q$-modules.
Let $U_j$ (resp. $U'_j$) be the irreducible component of $U$ (resp. $U'$)
corresponding to $I_j/I$ (resp. $I'_j/I'$) for $1\leq j\leq l$
(resp. $1\leq j\leq l'$).
Since $\psi$ is \'{e}tale and injective in codimension zero (Lemma
\ref{inj-zero}), we may assume that the generic point of $U_j$ is mapped to
that of $U'_j$ by $\psi$ for $1\leq j\leq l$.
In particular, we have $l\leq l'$.
Then one sees easily that $U\times_{U'}U'_j\cong U_j$ for $1\leq j\leq l$.
This implies that $A/(I_j/I)\cong (A'/(I'_j/I')\otimes_{A'}A)
(\cong A/((I'_j/I')\otimes_{A'}A))$, and hence,
\begin{equation}\label{ideals}
I_j/I=(I'_j/I')\otimes_{A'}A,\ (1\leq j\leq l)
\end{equation}
as ideals in $A$.
For $1\leq j\leq l$, we can set $f(z'_j)=u_jz_j$ for some $u_j\in A$.
Here, each $u_j$ is determined up to modulo $J_j/I$.
Due to (\ref{ideals}), $u_jz_j$ generates the ideal $I_j/I$,
and hence, $u_j$ is a unit in $A/(J_j/I)$ (and, of course, in $A/(J/I)$).
(Note that $u_j$ is not necessarily a unit in $A$, since $A$ is not an integral
domain for $l>1$.)
Then there exists an isomorphism (naturally induced by $f$) of $Q$-modules
\begin{equation}\label{def-tau}
\tau_j\colon I'_j/I'I'_j\otimes_{A'}Q\stackrel{\sim}{\longrightarrow}
I_j/II_j\otimes_AQ
\end{equation}
by $\xi'_j\mapsto(u_j\,\mbox{\rm mod}\,J/I)\xi_j$.
The natural projection $\pi'_j\colon I'_i/I'I'_i\rightarrow I'_i/I'
\subset A'$
$(1\leq i\leq l')$ and $f$ induce an $A$-module morphism
\begin{equation}\label{def-proj}
\widetilde{\rho}_i\colon I'_i/I'I'_i\otimes_{A'}A\longrightarrow A.
\end{equation}
For $1\leq j\leq l$, $\widetilde{\rho}_j$ maps $I'_i/I'I'_i\otimes_{A'}A$
surjectively onto
$I_j/I$, and for $i>l$, $\widetilde{\rho}_i$ is an isomorphism;
because, for $i>l$, one sees that
$\widetilde{\rho}_i(\zeta'_i\otimes 1)=f(z'_i)$
is an invertible element of $A$ as folows: Since $\psi$ is injective in
codimension zero, the point $I'_i/I'$ does not belong to $\psi(U)$;
hence $\psi$ maps $U=\mathop{\mbox{\rm Spec}}\nolimits A$ to $\mathop{\mbox{\rm Spec}}\nolimits A'_{(I'_i/I')}$, and this
implies the image of elements in $I'_i/I'$ under $f$ is invertible.
Set $\rho_i\colon=\widetilde{\rho}_i\otimes_AQ$.
Then these isomorphisms induce
\begin{equation}\label{def-tau2}
\tau\colon=\tau_1\otimes_Q\cdots\otimes_Q\tau_l\otimes_Q
\rho_{l+1}\otimes_Q\cdots\otimes_Q\rho_{l'}\colon
I'/I'^2\otimes_{A'}Q\stackrel{\sim}{\rightarrow}I/I^2\otimes_AQ.
\end{equation}
The $Q$-dual of $\tau$ is the desired isomorphism (cf. Lemma \ref{tangentloc}).
One can easily check that this isomorphism $\tau$ does not depends on
parameters $z'_j$, $z_j$; it is cannonically induced by
$f\colon A'\rightarrow A$.
Hence, for any sequence of \'{e}tale morphisms of local charts
$U\stackrel{\psi}{\rightarrow}U'\stackrel{\psi'}{\rightarrow}U''$, we obviously
have $\tau''=\tau\circ(\tau'\otimes_{Q'}Q)$, where
$\tau\colon I'/I'^2\otimes_{A'}Q\stackrel{\sim}{\rightarrow}I/I^2\otimes_AQ$,
$\tau'\colon I''/I''^2\otimes_{A''}Q'\stackrel{\sim}{\rightarrow}
I'/I'^2\otimes_{A'}Q'$ and
$\tau''\colon I''/I''^2\otimes_{A''}Q\stackrel{\sim}{\rightarrow}
I/I^2\otimes_AQ$ are the isomorphisms defined as above with respect to
$\psi$, $\psi'$ and $\psi'\circ\psi$, respectively.
Then one sees easily that there exists a unique $\O_D$-module whose restriction
to each $U$ is the $\O_{D_U}$-module corresponding to $T^1_A$; and it is
nothing but our desired $\O_D$-module ${\cal T}^1_X$.
Note that there exists a natural isomorphism
${\cal T}^1_X\stackrel{\sim}{\rightarrow}{\cal E}xt^1_{\O_X}(\Omega^1_{X/k},\O_X)$.
Suppose $X$ has a global NCD embedding $X\hookrightarrow V$.
Then by Lemma \ref{tangentloc}, the restriction of the normal bundle
${\cal N}_{X|V}$ to the singular locus $D$ is isomorphic to ${\cal T}^1_X$.
Hence we have the following:
\begin{pro}\label{suffcondemb}
If a normal crossing variety $X$ over $k$ is embedded into a smooth $k$-variety
as a normal crossing divisor, then there exists a line bundle $\L_X$ on $X$
such that $\L_X\otimes_{\O_X}\O_D\stackrel{\sim}{\rightarrow}{\cal T}^1_X$.
\end{pro}
Let $\hbox{\maxid X}\rightarrow\Delta$ be a semistable reduction of schemes,
{\it i.e.},
a flat and generically smooth morphism between regular schemes with
$\Delta$ one-dimensional and every closed fiber is a normal crossing
variety.
Suppose $X\rightarrow \mathop{\mbox{\rm Spec}}\nolimits k$ is isomorphic to a closed fiber of this
family.
Then one sees that the normal bundle ${\cal N}_{X|V}$ is trivial on $X$, and so
is ${\cal T}^1_X$.
\begin{dfn}{\rm (cf. \cite{Fri1})}\ \label{dsemistable}{\rm
A normal crossing variety $X$ is said to be {\it $d$-semistable}
if ${\cal T}^1_X$ is the trivial line bundle on $D$.}
\end{dfn}
Due to the above observation, we have the following:
\begin{pro}{\rm (cf. \cite{Fri1})}\label{suffcondsmoothing}
The $d$-semistablilty is a necessary condition for the existence of global
smoothings of $X$.
\end{pro}
\section{Logarithmic embeddings}
In this section, we define the logarithmic embedding of a
normal crossing varieties
(cf. \cite{Ste1}).
This concept is defined in terms of log geometry of Fontaine, Illusie,
and Kazuya Kato (cf. \cite{Kat1}).
Let $X$ be a normal crossing variety over a field $k$.
Suppose that $X$ has a NCD embedding $\iota\colon X\hookrightarrow V$.
We denote the open immersion $V\setminus X\hookrightarrow V$ by $j$.
We define a log structure on $X$ by
$$
\iota^*(\O_V\bigcap j_*{\cal O}^\times_{V\setminus X})\longrightarrow\O_X,
$$
where $\iota^*$ denotes the pull--back of log structures
(cf. \cite[(1.4)]{Kat1}).
We call this the log structure associated to the NCD embedding
$\iota\colon X\hookrightarrow V$.
For a general normal crossing variety $X$, we cannot define the log structure
of this type on $X$, because $X$ may not have a NCD embedding.
But, as we have seen in Remark \ref{ncvemb}, $X$ has \'{e}tale locally
a NCD embedding.
Then we can consider the log strcuture of this type for a general $X$
defined as follows:
\begin{dfn}{\rm (cf. \cite{Ste1})}\ \label{logembdef}{\rm
A log structure ${\cal M}_X\rightarrow\O_X$ is said to be of {\it embedding type},
if the following condition is satisfied:
There exists an \'{e}tale covering $\{\varphi_{\lambda}\colon U_{\lambda}\rightarrow
X\}_{\lambda\in\Lambda}$ by local charts --- with the NCD embeddings
$\iota_{\lambda}\colon U_{\lambda}\hookrightarrow V_{\lambda}$ as in
Definition \ref{ncvchart} --- such that, for each $\lambda\in\Lambda$, the
restriction
$$
{\cal M}_{U_{\lambda}}\colon =\varphi^*_{\lambda}{\cal M}_X\longrightarrow\O_{U_{\lambda}}
$$
is isomorphic to the log structure
associated to the NCD embedding $\iota_{\lambda}$.
If ${\cal M}_X\rightarrow\O_X$ is a log structure of embedding type of $X$, we call
the log scheme $(X,{\cal M}_X)$ the {\it logarithmic embedding}.
}
\end{dfn}
Let $(X,{\cal M}_X)$ be a logarithmic embedding.
We can explicitly write this log structure ${\cal M}_X$ \'{e}tale locally.
Let $\nu\colon \widetilde{X}\rightarrow X$ be a normalization of $X$.
Take a local chart $\varphi\colon U\rightarrow X$ with parameters
$z_1,\ldots,z_l$
such that ${\cal M}_U\colon =\varphi^*{\cal M}_X\rightarrow\O_U$ is the log structure
associated to the NCD embedding $\iota\colon U\hookrightarrow V$.
Let $U=\bigcup^l_{i=1}U_i$ be the decomposition into irreduclbile components,
where $U_i$ is the irreducible component corresponding to the ideal $(z_i)$.
The normalization $\widetilde{U}=\coprod^{l}_{i=1}U_i\rightarrow U$ is denoted
by $\nu_U$.
Note that, due to Lemma \ref{normalization}, we have $U\times_X\widetilde{X}
\cong\widetilde{U}$.
Define a homomorphism of monoids
\begin{equation}\label{logembchart}
\alpha\colon (\nu_U)_*\mbox{\bf N}_{\widetilde{U}}\rightarrow\O_U
\end{equation}
by
$\alpha(e_{U_i})=z_i$ for $i=1,\ldots,l$, where $(e_{U_i})$ is the standard
base of $(\nu_U)_*\mbox{\bf N}_{\widetilde{U}}=\bigoplus^l_{i=1}\mbox{\bf N}_{U_i}$.
Then $\alpha$ induces a log structure
\begin{equation}\label{logembloc}
{\cal O}^\times_U\bigoplus(\nu_U)_*\mbox{\bf N}_{\widetilde{U}}\rightarrow\O_U.
\end{equation}
\begin{pro}\label{logembloc-imp}
The log structure ${\cal M}_U\rightarrow\O_U$ is isomorphic to (\ref{logembloc}).
\end{pro}
{\sc Proof.}\hspace{2mm}
Let $Z_1,\ldots,Z_l\in\Gamma(V,\O_V)$ be as in Definition \ref{ncvchart}.
By definition of the log structure associated to the embedding
$\iota\colon U\hookrightarrow V$, these $Z_1,\ldots,Z_l$ are sections of the
sheaf
${\cal M}_U$.
Define a morphism
$$
\psi\colon (\nu_U)_*\mbox{\bf N}_{\widetilde{U}}\longrightarrow{\cal M}_U
$$
by $\psi(e_{U_i})\colon =Z_i$ for $1\leq i\leq l$.
Let
$$
\widetilde{\psi}\colon (\nu_U)_*\mbox{\bf N}_{\widetilde{U}}\rightarrow
{\cal M}_U/{\cal O}^\times_U
$$
be the composition of $\psi$ followed by the natural projection
${\cal M}_U\rightarrow{\cal M}_U/{\cal O}^\times_U$.
It is easy to see that $\widetilde{\psi}$ is injective.
Since sections of $\O_V\cap j_*{\cal O}^\times_{V\setminus U}$ are precisely those of
$\O_V$
which may take zeros along $\iota(U)$, these are written in the
form $uZ^{a_1}_1\cdots Z^{a_l}_l$ where $u\in{\cal O}^\times_V$ and $a_1,\ldots,a_l
\in\mbox{\bf N}$.
This implies that the morphism $\widetilde{\psi}$ is an isomorphism.
Then, consider the exact sequence of sheaves of monoids
$$
1\rightarrow{\cal O}^\times_U\rightarrow{\cal M}_U\rightarrow{\cal M}_U/{\cal O}^\times_U\rightarrow 1,
$$
where the second arrow is injective.
This exact sequence splits since $\widetilde{\psi}$ is an isomorphism and
$\psi$ defines a cross section ${\cal M}_U/{\cal O}^\times_U\rightarrow{\cal M}_U$.
By this, we can easily obtain the desired result.
$\Box$
Thus, a log structure of embedding type is determined by the morphism
$\alpha\colon (\nu_U)_*\mbox{\bf N}_{\widetilde{U}}\rightarrow\O_U$ such that
$\alpha(e_{U_i})$ is a local defining function of the component $U_i$ for
each $i=1,\ldots,l$.
Let ${\alpha}'$ be another such homomorphism.
Then --- replacing $U$ by sufficiently small Zariski open subset ---
we can take $u_i\in\Gamma(U,{\cal O}^\times_X)$ such that
${\alpha}'(e_{U_i})=u_i\alpha(e_{U_i})$ for each $i$.
Then the isomorphism of log structures of embedding type determined by
$\alpha$ and ${\alpha}'$ is described by the following commutative diagram
\begin{equation}\label{logemb-iso}
\begin{array}{ccccc}
{\cal O}^\times_U\oplus(\nu_U)_*\mbox{\bf N}_{\widetilde{U}}&&\stackrel{\phi}{\longrightarrow}&&
{\cal O}^\times_U\oplus(\nu_U)_*\mbox{\bf N}_{\widetilde{U}}\\
&\llap{$\mbox{\rm by}\ {\alpha}'$}\searrow&&\swarrow\rlap{$\mbox{\rm by}\
\alpha$}\\
&&\O_X\rlap{,}
\end{array}
\end{equation}
where $\phi$ is defined by $\phi(1,e_{U_i})=(u_i,e_{U_i})$ for each
$i=1,\ldots,l$.
In particular, the log structure of embedding type exists \'{e}tale locally,
and is unique up to isomorphisms.
\begin{cor}
For any logarithmic embedding $(X,{\cal M}_X)$,
we have an exact sequence of abelian sheaves
\begin{equation}\label{abelian}
1\longrightarrow{\cal O}^\times_X\longrightarrow\gp{{\cal M}_X}\longrightarrow
\nu_*\mbox{\bf Z}_{\widetilde{X}}\longrightarrow 0.
\end{equation}
\end{cor}
{\sc Proof.}\hspace{2mm}
Due to the local expression (\ref{logembloc}).
$\Box$
In the rest of this section, we prove the following theorem, which is the
main theorem of this paper.
\begin{thm}\label{mainthm}
For a normal crossing variety $X$, the logarithmic embedding of $X$ exists
if and only if there exists a line bundle ${\cal L}$ on $X$ such that
${\cal L}\otimes_{\O_X}\O_D\stackrel{\sim}{\rightarrow} {\cal T}^1_X$.
\end{thm}
For the proof of this theorem, we shall prove some lemmas as follows.
Let $(\varphi\colon U=\mathop{\mbox{\rm Spec}}\nolimits A\rightarrow X; z_1,\ldots,z_l)$ be a local chart
on $X$.
Let the NCD embedding $U\hookrightarrow V=\mathop{\mbox{\rm Spec}}\nolimits R$ and the ideals $I_j,I,
J_j,J$ of $R$ be as in the previous section.
\begin{lem}\label{lem-1}
The natural morphism
$$
\bigoplus^l_{j=1}J_j/I\longrightarrow J/I
$$
of $A$-modules, induced by $J_j\hookrightarrow J$, is an isomorphism.
\end{lem}
{\sc Proof.}\hspace{2mm}
The surjectivity is clear.
We are going to show the injectivity.
Take $a_jZ_1\cdots\widehat{Z_j}\cdots Z_l\in J_j$ --- where $Z_1,\ldots,Z_l$
are as in the previous section --- for $1\leq j\leq l$ such that
$$
\sum^l_{j=1}a_jZ_1\cdots\widehat{Z_j}\cdots Z_l=b\cdot Z_1\cdots Z_l,
$$
where $a_j,b\in R$.
Since $R$ is an integral domain, $a_j$ is divisible by $Z_j$, and hence, we
have $a_jZ_1\cdots\widehat{Z_j}\cdots Z_l\equiv 0\ (\mbox{\rm mod}\,I)$.
$\Box$
Let $\pi_j\colon I_j/II_j\rightarrow I_j/I$ and $q_j\colon I_j/I
\rightarrow I_j/JI_j(\cong I_j/II_j\otimes_AQ \ \mbox{\rm where}\ Q=R/J)$ be
the
natural projections and set $p_j\colon=q_j\circ\pi_j$.
Let $q\colon I/I^2\rightarrow I/JI(\cong I/I^2\otimes_AQ)$ be the natural
projection.
\begin{lem}\label{lem-2}
Let $M_1,\ldots,M_l$ be free $A$-modules of rank one and set
$M\colon=M_1\otimes_A\cdots\otimes_AM_l$.
Suppose we are given an $A$-module isomorphism
$\widetilde{g}\colon M\stackrel{\sim}{\rightarrow}I/I^2$ and $A$-module
homomorphisms
$g_j\colon M_j\rightarrow I_j/I$, for $1\leq j\leq l$, such that,
\begin{enumerate}
\item
for each $j$, there exists a free generator $\delta_j$ of $M_j$ such that
$g_j(\delta_j)=z_j$,
\item
$(q_1\circ g_1)\otimes_Q\cdots\otimes_Q(q_l\circ g_l)=q\circ\widetilde{g}$.
\end{enumerate}
Then there exists a unique collection $\{\widetilde{g}_j\colon M_j
\stackrel{\sim}{\rightarrow}I_j/II_j\}^{l}_{j=1}$ of $A$-isomorphisms
such that
$\pi_j\circ\widetilde{g}_j=g_j$ for each $j$ and
$\widetilde{g}_1\otimes_A\cdots\otimes_A\widetilde{g}_l=\widetilde{g}$.
\end{lem}
{\sc Proof.}\hspace{2mm}
We fix the free generators $\delta_j$ of $M_j$ as above.
Then $M$ is generated by $\delta_1\otimes\cdots\otimes\delta_l$.
Set $\widetilde{g}(\delta_1\otimes\cdots\otimes\delta_l)=
v\zeta_1\otimes\cdots\otimes\zeta_l$ where $v\in A^{\times}$.
By the second condition, we have
$v\equiv 1\ (\mbox{\rm mod}\,J/I)$, {\it i.e.},
$$
v=1+\sum^l_{j=1}a_jz_1\cdots\widehat{z_j}\cdots z_l
$$
for $a_j\in A$.
We set $u_j=1+a_jz_1\cdots\widehat{z_j}\cdots z_l$ and define
$\widetilde{g}_j$ by $\widetilde{g}_j(\delta_j)\colon=u_j\zeta_j$ for
$1\leq j\leq l$.
Then, since $v=u_1\cdots u_l$, each $u_j$ is a unit in $A$ and
$\widetilde{g}_j$ is an isomorphism. Moreover, we have
$\widetilde{g}_1\otimes\cdots\otimes\widetilde{g}_l=\widetilde{g}$
as desired.
The uniqueness follows from Lemma \ref{lem-1}.
$\Box$
\vspace{3mm}
{\sc Proof of Theorem \ref{mainthm}}.
We first prove the ``if'' part. This part is divided into four steps.
{\sc Step 1}:
Here, we shall describe the log structure of embedding type by another
\'{e}tale local expression.
Let $(\varphi\colon U=\mathop{\mbox{\rm Spec}}\nolimits A\rightarrow X; z_1,\ldots,z_l)$ be a local chart.
For $m=(m_1,\ldots,m_l)\in\mbox{\bf N}^l$, define an $A$-module $P_m$ by
$$
P_m\colon=(I_1/II_1)^{\otimes m_1}\otimes_A\cdots\otimes_A
(I_l/II_l)^{\otimes m_l}.
$$
Each $P_m$ is a free $A$-module of rank one and $P_{(1,\ldots,1)}\cong
I/I^2$.
The natural projections $\pi_j$ induce a natural $A$-homomorphism
$$
\sigma_m\colon P_m\longrightarrow A.
$$
Define a monoid
$$
M\colon =\left\{
\begin{array}{c|l}
(m,a)&m\in\mbox{\bf N}^l,\\
&a:\mbox{a generator of $P_m$}
\end{array}
\right\},
$$
and a homomorphism $M\rightarrow A$ of monoids by $(m,a)\mapsto\sigma_m(a)$.
Then the associated log structure $\alpha_U\colon{\cal M}_U\rightarrow\O_U$ of the
pre--log structure $M\rightarrow A$ is that of embedding type on $U$.
{\sc Step 2}:
Now, we assume that we are given a line bundle $\L$ on $X$ satisfying
$\L\otimes_{\O_X}\O_D\cong({\cal T}^1_X)^{\vee}$.
Suppose we have two local charts $(\varphi\colon U\rightarrow X;
z_1,\ldots,z_{l})$ and $(\varphi'\colon U'\rightarrow X;
z'_1,\ldots,z'_{l'})$ and an \'{e}tale morphism $\psi\colon U\rightarrow U'$
such that $\varphi=\varphi'\circ\psi$.
For these local charts, we use the notation as in the previous
section; such as $U=\mathop{\mbox{\rm Spec}}\nolimits A\hookrightarrow V=\mathop{\mbox{\rm Spec}}\nolimits R$,
$U'=\mathop{\mbox{\rm Spec}}\nolimits A'\hookrightarrow V'=\mathop{\mbox{\rm Spec}}\nolimits R'$, $f\colon A'\rightarrow A$,
$I$, $I'$, etc.
As in the previous section, we may assume
$(I'_j/I')\otimes_{A'}A=I_j/I$ as ideals in $A$ for
$1\leq j\leq l$, and set $f(z'_j)=u_jz_j$ (each $u_j$ is determined up to
modulo $J_j/I$).
To give the line bundle $\L$ as above is equivalent to give a compatible
system of isomorphisms
$$
\widetilde{\tau}\colon
I'/I'^2\otimes_{A'}A\stackrel{\sim}{\longrightarrow}I/I^2,
$$
for all such $U\rightarrow U'$, with $\widetilde{\tau}\otimes_AQ=\tau$,
where $\tau$ is defined as in (\ref{def-tau2}).
Then we shall show that $\widetilde{\tau}$ induces canonically an isomorphism
of log structures $\psi^*{\cal M}_{U'}\stackrel{\sim}{\rightarrow}{\cal M}_U$, and
prove that these
isomorphisms form so a compatible system that the log structures ${\cal M}_U$ glue to
a log structure of embedding type on $X$.
Moreover --- since local charts form an \'{e}tale open basis (Corollary
\ref{specialcov}) --- we can pass through this procedure replacing $U$ by
its Zariski open subset if necessary.
In particular, we may assume that each $u_j$ as above is a unit in $A$,
because $(u_j\,\mbox{\rm mod}\,J/I)$ is a unit in $A/(J/I)$ (in case $l>1$).
Fix a locally constant section $w\in\H^0(D,{\cal O}^\times_D)$.
(Actually, we can take $w$ as any
global section in $\H^0(D,{\cal O}^\times_D)$ but, if we do so, the following argument
have to be modified slightly.)
{\sc Step 3}:
(i) If $l=l'=1$, {\it i.e.}, $I_1=I$ and $I'_1=I'$, then we set
$\widetilde{\tau}_1\colon I'_1/I'I'_1\otimes_{A'}A\stackrel{\sim}{\rightarrow}
I_1/II_1$ by $\widetilde{\tau}_1\colon=\widetilde{\tau}$.
(ii) If $l=1$ and $l'>1$, we define
$\widetilde{\tau}_1\colon I'_1/I'I'_1\otimes_{A'}A\stackrel{\sim}{\rightarrow}
I_1/II_1$ as follows:
Suppose $\widetilde{\tau}$ maps $\zeta'_1\otimes\cdots\otimes\zeta'_{l'}\otimes
1$ to
$v\zeta_1$, where $v\in A^{\times}$.
Let $\widetilde{\rho}_j\colon I'_i/I'I'_i\otimes_{A'}A\rightarrow A$ be as
(\ref{def-proj}), for $1\leq i\leq l'$.
Suppose, moreover, each $\widetilde{\rho}_i$, for $i>1$, maps $\zeta'_i\otimes
1$ to $v_i\in A^{\times}$.
Then, define $\widetilde{\tau}_1$ by $\widetilde{\tau}_1(\zeta'_1\otimes 1)
\colon=w_Uvv^{-1}_2\cdots v^{-1}_{l'}\zeta_1$, where $w_U$ is a non--zero
scalar which coincides with $w$ restricted to $D_U$.
(iii) Suppose $l>1$ and $l'>1$.
We claim that, under the conditions
\begin{equation}\label{cond1}
\pi_j\circ\widetilde{\tau}_j=\widetilde{\rho}_j,\ (1\leq j\leq l)
\end{equation}
and
\begin{equation}\label{cond2}
\widetilde{\tau}_1\otimes_A\cdots\otimes_A\widetilde{\tau}_l\otimes_A
\widetilde{\rho}_{l+1}\otimes_A\cdots\otimes_A\widetilde{\rho}_{l'}=
\widetilde{\tau},
\end{equation}
the $A$-isomorphisms
$$
\widetilde{\tau}_j\colon I'_j/I'I'_j\otimes_{A'}A\stackrel{\sim}
{\longrightarrow}I_j/II_j
$$
exist uniquely for $1\leq j\leq l$.
Set $M_j\colon=I'_j/I'I'_j\otimes_{A'}A$ and
$g_j\colon=\widetilde{\rho}_j$ for $1\leq j\leq l$.
Define
$\widetilde{g}$ by $\widetilde{g}\otimes_A\widetilde{\rho}_{l+1}\otimes_A
\cdots\otimes_A\widetilde{\rho}_{l'}=\widetilde{\tau}$ (this is possible
since $\widetilde{\rho}_i(\zeta'_i\otimes 1)$ is a unit element in $A$ for
$i>l$), which is obviously an isomorphism.
Then --- since we assumed each $u_j$ to be a unit in $A$ --- $M_j\colon=
I'_j/I'_jI'\otimes_{A'}A$, $g$, and $g_j$ satisfy the conditions in Lemma
\ref{lem-2}.
Hence our claim follows from this lemma.
Note that, in any cases, we have the following commutative diagram:
\begin{equation}\label{com-mor}
\begin{array}{ccc}
I'_j/I'I'_j&\longrightarrow&I_j/II_j\\
\llap{$\pi'_j$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$\pi_j$}\\
A'&\underrel{\longrightarrow}{f}&A\rlap{,}
\end{array}
\end{equation}
for $1\leq j\leq l$; this follows from (\ref{cond1}) in case $l, l'>1$, and is
quite obvious in the other cases.
{\sc Step 4}:
These morphisms $\widetilde{\tau}_j$ induce the morphisms
$$
\gamma_{m'}\colon P'_{m'}\longrightarrow P_{m},
$$
where $m=(m_1,\ldots,m_l)$ for $m'=(m_1,\ldots,m_{l'})\in\mbox{\bf N}^{l'}$.
Then these $\gamma_{m'}$ induce naturally a morphism of monoids
$M'\rightarrow M$ compatible with $M'\rightarrow A'$, $M\rightarrow A$ and
$f$.
By the construction of these morphisms, the induced morphism of sheaves
of monoids
$\gamma\colon \psi^*{\cal M}_{U'}\stackrel{\sim}{\rightarrow}{\cal M}_U$ is an isomorphism.
By the commutative diagram (\ref{com-mor}), this isomorphism commutes the
following diagram:
$$
\begin{array}{ccc}
\psi^*{\cal M}_{U'}&\stackrel{\sim}{\longrightarrow}&{\cal M}_{U}\\
\llap{$\psi^*\alpha_{U'}$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$\alpha_U$}\\
\O_U&=&\O_U\rlap{;}
\end{array}
$$
hence $\gamma$ is an isomorphism of log structures.
Our construction of the isomorphism $\gamma$ is canonical in the following
sense: Suppose we are given a sequence of \'{e}tale morphisms
$U\stackrel{\psi}{\rightarrow}U'\stackrel{\psi'}{\rightarrow}U''$ of
local charts (with $U$ and $U'$ sufficiently small),
we have $\gamma''=\gamma\circ(\psi^*\gamma')$, where
$\gamma\colon \psi^*{\cal M}_{U'}\stackrel{\sim}{\rightarrow}{\cal M}_U$,
$\gamma'\colon \psi'^*{\cal M}_{U''}\stackrel{\sim}{\rightarrow}{\cal M}_{U'}$ and
$\gamma''\colon \psi^*\psi'^*{\cal M}_{U''}\stackrel{\sim}{\rightarrow}{\cal M}_U$
are the isomorphisms of log structures defined as above corresponding to
$\psi$, $\psi'$ and $\psi'\circ\psi$, respectively.
This follows from the naturality of $\pi_j$ and $\widetilde{\rho}_j$, and
the compatibility of $\widetilde{\tau}$'s. Then there exists a unique
log structure ${\cal M}_X$ on $X$ which is of embedding type.
Hence the ``if'' part is now proved.
Conversely, suppose we are given a log structure ${\cal M}_X$ of embedding type.
Then we have an exact sequence (\ref{abelian}) of abelian sheaves.
Considering the cohomology exact sequence, we obtain a morphism
$$
\delta\colon \H^0(X,\nu_*\mbox{\bf Z}_{\widetilde{X}})\longrightarrow\H^1(X,{\cal O}^\times_X)
(\cong\mathop{\mbox{\rm Pic}}\nolimits X).
$$
In $\H^0(X,\nu_*\mbox{\bf Z}_{\widetilde{X}})$, we consider the element $\hbox{\maxid{d}}$
which is defined by
the image of $1\in\mbox{\bf Z}_X$ under the diagonal morphism $\mbox{\bf Z}_X\rightarrow
\nu_*\mbox{\bf Z}_{\widetilde{X}}$.
Then $\delta(\hbox{\maxid{d}})$ defines a line bundle $\L=\L_{{\cal M}_X}$ on $X$.
We shall show that this line bundle satisfies $\L\otimes_{\O_X}\O_D
\stackrel{\sim}{\rightarrow}({\cal T}^1_X)^{\vee}$.
The line bundle $\L$ is constructed as follows: the inverse image of
$\hbox{\maxid{d}}$ under $\gp{{\cal M}_X}\rightarrow\nu_*\mbox{\bf Z}_{\widetilde{X}}$ defines a
principally homogeneous space over ${\cal O}^\times_X$ and hence defines a line bundle,
which is nothing but $\L$.
Let $U=\mathop{\mbox{\rm Spec}}\nolimits A$ be a local chart as above.
Then the inverse image of $\hbox{\maxid{d}}$ restricted to $U$ gives a generator
of an $A$-module $I/I^2$ which is --- due to Lemma \ref{tangentloc} ---
a local lifting of ${\cal T}^1_X$ restricted to
$U$. Hence $\L$ satisfies the desired condition.
$\Box$
\begin{rem}\label{important}{\rm
1.
As we have seen above, the log structure of embedding type exists locally and
is unique up to isomorphisms.
The sheaf of germs of automorphisms of such a log structure
is naturally isomorphic to ${\cal K}$, where ${\cal K}$ is defined by the exact sequence
\begin{equation}\label{important2}
1\longrightarrow{\cal K}\longrightarrow{\cal O}^\times_X\longrightarrow
{\cal O}^\times_D\longrightarrow 1.
\end{equation}
This can be shown by the following steps:
(i) any automorphism over a sufficiently small local chart $U$ is given by $
\phi$ in the diagram
(\ref{logemb-iso}) with $\alpha=\alpha'$; (ii) $\phi$ is determined by
$\{u_i\}$ with $u_i\in\Gamma(U,{\cal O}^\times_X)$ such that $z_i=u_i\cdot z_i$ for each
$i$; (iii) hence such $u_i$'s are written in the form of $u_i=
1+a_i\cdot z_1\cdots\widehat{z_i}\cdots z_l$; (iv) due to Lemma \ref{lem-1},
to give a system
$\{u_i\}$ is equivarent to give $u=u_1\cdots u_l$ which is a section of ${\cal K}$.
Hence the obstruction for the existence of log structures of embedding type
lies
in $\H^2(X,{\cal K})$.
The proof of Theorem \ref{mainthm} shows that this class coincides with the
obstruction class for a lifting of $(T^1_X)^{\vee}$ on $X$, {\it i.e.}, the
image
of $(T^1_X)^{\vee}$ under $\H^1(D,{\cal O}^\times_D)\rightarrow\H^2(X,{\cal K})$.
2.
One sees easily --- by the proof of Theorem \ref{mainthm} --- that there
exists a natural surjective map
\begin{equation}\label{important1}
\left\{
\begin{array}{ll}
\mbox{isom. class of log structures}\\
\mbox{of embedding type on $X$}
\end{array}
\right\}
{\longrightarrow}
\left\{
\begin{array}{c|c}
\L\in\mathop{\mbox{\rm Pic}}\nolimits X&\L\otimes_{\O_X}\O_D\stackrel{\sim}{\rightarrow}({\cal T}^1_X)^{\vee}
\end{array}
\right\}
\end{equation}
by ${\cal M}\mapsto\L_{{\cal M}}$, where $\L_{{\cal M}}$ is defined as in the proof of Theorem
\ref{mainthm}.
If ${\cal M}_X$ is associated to a global NCD embedding $X\hookrightarrow V$, then
$\L_{{\cal M}_X}$ is nothing but the conormal bundle of $X$ in $V$.
The set of isomorphism classes of log structures of embedding type on $X$, is
a principally homogeneous space over $\H^1(X,{\cal K})$.
Then one sees easily that the map (\ref{important1}) is equivariant to
$\H^1(X,{\cal K})\rightarrow
\mathop{\mbox{\rm Ker}}\nolimits(\H^1(X,{\cal O}^\times_X)\rightarrow\H^1(D,{\cal O}^\times_D))$ induced by the
cohomology exact sequence of (\ref{important2}). In particular, if $X$ is
proper and $D$ is connected, the map (\ref{important1}) is a bijection
since $\H^1(X,{\cal K})\stackrel{\sim}{\rightarrow}\mathop{\mbox{\rm Ker}}\nolimits(\H^1(X,{\cal O}^\times_X)\rightarrow
\H^1(D,{\cal O}^\times_D))$; in this case, the logarithmic embeddings are determined by
their ``normal bundles.''
3.
By the exact sequence (\ref{abelian}), a log structure of embedding type
${\cal M}_X$ on
$X$ defines an extension class in
$\mathop{\mbox{\rm Ext}}\nolimits^1_{\mbox{\bf Z}_X}(\nu_*\mbox{\bf Z}_{\widetilde{X}},{\cal O}^\times_X)$.
Under the morphism $\mathop{\mbox{\rm Ext}}\nolimits^1_{\mbox{\bf Z}_X}(\nu_*\mbox{\bf Z}_{\widetilde{X}},{\cal O}^\times_X)
\rightarrow\mathop{\mbox{\rm Ext}}\nolimits^1_{\mbox{\bf Z}_X}(\mbox{\bf Z}_X,{\cal O}^\times_X)$, induced by the diagonal morphism
$\mbox{\bf Z}_X\rightarrow\nu_*\mbox{\bf Z}_{\widetilde{X}}$, and the natural identification
$\mathop{\mbox{\rm Ext}}\nolimits^1_{\mbox{\bf Z}_X}(\mbox{\bf Z}_X,{\cal O}^\times_X)\stackrel{\sim}{\rightarrow}\mathop{\mbox{\rm Pic}}\nolimits X$, this class is
mapped
to the class corresponding to the line bundle $\L_{{\cal M}_X}$ defined as above.
(The proof is straightforward and left to the reader.)
}
\end{rem}
\section{Logarithmic semistable reductions}
\begin{dfn}{\rm (cf. \cite{Kaj1}, \cite{K-N1})}\label{logsemidef}\ {\rm
A log strcuture of embedding type ${\cal M}_X\rightarrow\O_X$ is said to be
of {\it semistable type}, if there exists a homomorphism
$\mbox{\bf Z}_X\rightarrow\gp{{\cal M}_X}$ of abelian sheaves on $X$ such that the diagram
$$
\begin{array}{ccccc}
\gp{{\cal M}_X}&&\longrightarrow&&\nu_*\mbox{\bf Z}_{\widetilde{X}}\\
&\nwarrow&&\nearrow\rlap{$\hbox{\maxid{d}}$}\\
&&\mbox{\bf Z}_X
\end{array}
$$
commutes, where $\hbox{\maxid{d}}\colon\mbox{\bf Z}_X{\rightarrow}\nu_*\mbox{\bf Z}_{\widetilde{X}}$ is
the diagonal homomorphism, and $\gp{{\cal M}_X}\rightarrow\nu_*\mbox{\bf Z}_{\widetilde{X}}$
is the projection in (\ref{abelian}).
}
\end{dfn}
If ${\cal M}_X$ is a log strcuture of semistable type, the homomorphism
$\mbox{\bf Z}_X\rightarrow\gp{{\cal M}_X}$ induces the homomorphism $\mbox{\bf N}_X\rightarrow{\cal M}_X$
of monoids by the following Cartesian diagram:
$$
\begin{array}{ccc}
\mbox{\bf N}_X&\longrightarrow&{\cal M}_X\\
\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\
\mbox{\bf Z}_X&\longrightarrow&\gp{{\cal M}_X}\rlap{;}
\end{array}
$$
this follows easily from the local expression (\ref{logembloc}).
This morphism defines a morphism of log schemes
$$
(X,{\cal M}_X)\longrightarrow(\mathop{\mbox{\rm Spec}}\nolimits k,\mbox{\bf N})
$$
Here, $(\mathop{\mbox{\rm Spec}}\nolimits k,\mbox{\bf N})$ is the {\it standard point} defined by $\mbox{\bf N}\rightarrow k$
which maps $m\in\mbox{\bf N}$ to $0^m$.
We call this morphism of log schemes the {\it logarithmic semistable
reduction}.
Logarithmic semistable reductions are {\it log smooth}
in the sense of \cite{Kat1}.
\begin{rem}\label{genuine-semistable}{\rm
Let $f\colon\hbox{\maxid X}\rightarrow\Delta$ be a semistable reduction of schemes;
{\it i.e.}, a proper flat generically smooth morphism $f$ with $\hbox{\maxid X}$ a
regular scheme and $\Delta$ a one-dimensional regular local scheme,
with the closed fiber $X\rightarrow 0=\mathop{\mbox{\rm Spec}}\nolimits k$ a normal crossing variety.
Then this morphism induces canonically a logarithmic semistable reduction
$(X,{\cal M}_X)\rightarrow(\mathop{\mbox{\rm Spec}}\nolimits k,\mbox{\bf N})$ on the closed fiber as follows:
We define a log structure ${\cal M}_{\hbox{\maxid X}}\rightarrow\O_{\hbox{\maxid X}}$ by
$$
{\cal M}_{\hbox{\maxid X}}\colon=\O_{\hbox{\maxid X}}\bigcap j_*{\cal O}^\times_{\hbox{\maxid X}\setminus X}
\lhook\joinrel\longrightarrow\O_{\hbox{\maxid X}}
$$
where $j\colon \hbox{\maxid X}\setminus X\hookrightarrow\hbox{\maxid X}$ is an open
immersion.
Take a local parameter $t\in\O_{\Delta}$ around $0=\mathop{\mbox{\rm Spec}}\nolimits k$.
Then $f^{-1}(t)$ belongs to ${\cal M}_{\hbox{\maxid X}}$.
We define a homomorphism of monoids $\mbox{\bf N}_{\hbox{\maxid X}}\rightarrow{\cal M}_{\hbox{\maxid X}}$ by
$1\mapsto f^{-1}(t)$. Then this homomorphism extends to a morphism of
log schemes
\begin{equation}\label{france}
(\hbox{\maxid X},{\cal M}_{\hbox{\maxid X}})\longrightarrow(\Delta,0),
\end{equation}
where the log structure on $\Delta$ is the associated log structure of
$$
\mbox{\bf N}\longrightarrow\O_{\Delta}\quad \mbox{\rm by}\quad
m\mapsto t^m.
$$
Taking the pull--back of (\ref{france}) to the closed fiber, we
get a logarithmic semistable reduction.
Note that the monoid morphism $\mbox{\bf N}_{\hbox{\maxid X}}\rightarrow{\cal M}_{\hbox{\maxid X}}$ induces
$\mbox{\bf Z}_X\rightarrow\gp{{\cal M}_X}$ which satisfies the condition in Definition
\ref{logsemidef}.
Hence, such a morphism $\mbox{\bf Z}_X\rightarrow\gp{{\cal M}_X}$ for a general log structure
of semistable type can be regarded as a ``parametrization.''
}
\end{rem}
\begin{rem}{\rm
The logarithmic semistable reduction induced by a semistable reduction
family, as in Remark \ref{genuine-semistable}, is regarded as the
``closed fiber'' of the morphism (\ref{france}) of log schemes.
Then, conversely, one can consider the theory of deformations which
deal with liftings of the logarithmic semistable reductions.
This is nothing but the {\it logarithmic deformation} of Kawamata--Namikawa
\cite{K-N1}, and also a part of the {\it log smooth deformation} developed
in \cite{Kat2}.
}
\end{rem}
Using Theorem \ref{mainthm} --- which is proved in the previous section ---
we get a new proof of the theorem of Kawamata--Namikawa as follows:
\begin{thm}\label{mainthm2}{\rm (cf. \cite{K-N1})}\
For a normal crossing variety $X$, the log structure of semistable type
on $X$ exists if and only if $X$ is $d$-semistable.
\end{thm}
To prove the theorem, we need the following lemma:
\begin{lem}\label{l1}
Let $(X,{\cal M}_X)$ be a logarithmic embedding.
Consider the exact sequence (\ref{abelian}) of abelian sheaves and the
induced morphism
$$
\mathop{\mbox{\rm Hom}}\nolimits_{\mbox{\bf Z}_X}(\mbox{\bf Z}_X,\nu_*\mbox{\bf Z}_{\widetilde{X}})\stackrel{\delta}{\longrightarrow}
\mathop{\mbox{\rm Ext}}\nolimits^1_{\mbox{\bf Z}_X}(\mbox{\bf Z}_X,{\cal O}^\times_X).
$$
Let $\hbox{\maxid{d}}\in\mathop{\mbox{\rm Hom}}\nolimits_{\mbox{\bf Z}_X}(\mbox{\bf Z}_X,\nu_*\mbox{\bf Z}_{\widetilde{X}})$ be the diagonal
morphism.
Then, under the natural identification $\mathop{\mbox{\rm Ext}}\nolimits^1_{\mbox{\bf Z}_X}(\mbox{\bf Z}_X,{\cal O}^\times_X)
\stackrel{\sim}{\rightarrow}\mathop{\mbox{\rm Pic}}\nolimits X$, we have
$$
\delta(\hbox{\maxid{d}})=[\L_{{\cal M}_X}],
$$
where $\L_{{\cal M}_X}$ is the line bundle defined in the previous section.
\end{lem}
{\sc Proof.}\hspace{2mm}
This lemma follows from the commutative diagram
$$
\begin{array}{ccc}
\mathop{\mbox{\rm Hom}}\nolimits_{\mbox{\bf Z}_X}(\mbox{\bf Z}_X,\nu_*\mbox{\bf Z}_{\widetilde{X}})&\stackrel{\delta}{\longrightarrow}&
\mathop{\mbox{\rm Ext}}\nolimits^1_{\mbox{\bf Z}_X}(\mbox{\bf Z}_X,{\cal O}^\times_X)\\
\llap{$\cong$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\rlap{$\cong$}\\
\H^0(X,\nu_*\mbox{\bf Z}_{\widetilde{X}})&\longrightarrow&\H^1(X,{\cal O}^\times_X)
\end{array}
$$
where the vertical morphisms are natural isomorphisms and the definition
of the line bundle $\L_{{\cal M}_X}$.
$\Box$
\vspace{3mm}
{\sc Proof of Theorem \ref{mainthm2}.}\hspace{2mm}
Suppose ${\cal M}_X$ is a log structure of semistable type.
Consider the exact sequence
\begin{equation}\label{extension}
\mathop{\mbox{\rm Hom}}\nolimits_{\mbox{\bf Z}_X}(\mbox{\bf Z}_X,\gp{{\cal M}_X})\stackrel{\pi}{\longrightarrow}
\mathop{\mbox{\rm Hom}}\nolimits_{\mbox{\bf Z}_X}(\mbox{\bf Z}_X,\nu_*\mbox{\bf Z}_{\widetilde{X}})\stackrel{\delta}{\longrightarrow}
\mathop{\mbox{\rm Ext}}\nolimits^1_{\mbox{\bf Z}_X}(\mbox{\bf Z}_X,{\cal O}^\times_X)
\end{equation}
induced by (\ref{abelian}).
The ``parametrization'' morphism $\mbox{\bf Z}_X\rightarrow\gp{{\cal M}_X}$ is mapped to
$\hbox{\maxid{d}}$ by $\pi$.
This implies that the line bundle $\L_{{\cal M}_X}$ is trivial.
Then so is $(T^1_X)^{\vee}$ because $\L_{{\cal M}_X}\otimes_{\O_X}\O_D$ is
isomorphic to $(T^1_X)^{\vee}$.
Conversely, if $X$ is $d$-semistable, there exists at least one
log structure of embedding type on $X$ due to Theorem \ref{mainthm}.
Since $\L_{{\cal M}_X}\otimes_{\O_X}\O_D$ is trivial, we can take the log structure
${\cal M}_X$ of embedding type such that the corresponding line bundle $\L_{{\cal M}_X}$
is trivial (due to the natural surjection (\ref{important1})).
Since the obstruction for the existence of a morphism
$\mbox{\bf Z}_X\rightarrow\gp{{\cal M}_X}$ which is mapped to $\hbox{\maxid{d}}$, is nothing but the
class $[\L_{{\cal M}_X}]$, we deduce that ${\cal M}_X$ is of semistable type.
$\Box$
As is shown in the above proof, the log structure of semistable type on $X$
is --- considering the natural surjection (\ref{important1}) --- the log
structure of embedding type which is mapped
to the trivial bundle on $X$. Hence we have the following:
\begin{cor}
Let $X$ be a proper, $d$-semistable normal crossing variety, and assume that
the singular locus $D$ is connected.
Then, the log structure of semistable type on $X$ exists uniquely.
\end{cor}
\begin{exa}{\rm
Let $X\colon =X_0\cup\cdots\cup X_N$ be a chain of surfaces defined as follows:
Each $X_i$ is the Hirzebruch surface of degree $a_i\leq 0$.
The surfaces $X_{i-1}$
and $X_i$ are connected by identifying the section $s'_{i-1}$ on $X_{i-1}$
and the one $s_i$ on $X_i$, where $(s'_{i-1})^2=a_{i-1}$ and $(s_i)^2=
-a_i$, for $1\leq i\leq N$. Then $X$ has a
log structure of embedding type if and only if $a_i|(a_{i-1}+a_{i+1})$
for $1\leq i\leq N-1$, while $X$ has a log structure of semistable type
if and only if $a_0=a_1=\cdots =a_N$.
}
\end{exa}
\begin{small}
|
2006-03-24T21:30:51 | 9411 | alg-geom/9411012 | en | https://arxiv.org/abs/alg-geom/9411012 | [
"alg-geom",
"math.AG"
] | alg-geom/9411012 | null | David B. Jaffe | On sextic surfaces having only nodes (preliminary report) | 14 pages, AMS-LaTeX | null | null | null | null | Let S be a surface in CP^3, having only nodes as singularities. Let pi: S~
--> S be a minimal resolution of singularities. A set N of nodes on S is EVEN
if there exists a divisor Q on S~ such that 2Q ~ pi^{-1}(N).
Suppose that S has degree 6. It is known (Basset) that S cannot have 67 or
more nodes. It is also known (Barth) that S can have 65 nodes. It is not known
if S can have 66 nodes. Likewise, it is not known exactly what sizes can occur
for an even set of nodes on S.
We show that an nonempty even set of nodes on S must have size 24, 32, 40,
56, or 64. We do not know if the sizes 56 and 64 can occur. We show that if S
has 66 nodes, then it must have an even set of 64 nodes, and it cannot have an
even set of 56 nodes. THUS IF ONE COULD RULE OUT THE CASE OF A 64 NODE EVEN
SET, IT WOULD FOLLOW THAT S CANNOT HAVE 66 NODES.
The existence or nonexistence of large even node sets is related to the
following vanishing problem. Let S be a normal surface of degree s in CP^3. Let
D be a Weil divisor on S such that D is Q-rationally equivalent to rH, for some
r \in \Q. Under what circumstances do we have H^1(O_S(D)) = 0? For instance,
this holds when r < 0. For s=4 and r=0, H^1 can be nonzero. For s=6 and r=0, if
a 56 or 64 node even set exists, then H^1 can be nonzero. The vanishing of H^1
is also related to linear normality, quadric normality, etc. of set-theoretic
complete intersections in P^3. Hard copy is available from the author. E-mail
to [email protected].
| [
{
"version": "v1",
"created": "Thu, 17 Nov 1994 21:44:35 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Jaffe",
"David B.",
""
]
] | alg-geom | \section{#1}}
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\vskip 0.15in
\par\noindent {\footnotesize Department of Mathematics and Statistics,
University of Nebraska}
\par\noindent {\footnotesize Lincoln, NE 68588-0323, USA\ \ (jaffe{\kern0.5pt}@{\kern0.5pt}cpthree.unl.edu)}}
\def \def\arabic{footnote}}\setcounter{footnote}{0}{\fnsymbol{footnote}{ \def\arabic{footnote}}\setcounter{footnote}{0}{\fnsymbol{footnote}}
\par\noindent David B. Jaffe\protect\footnote{Partially supported by
the National Science Foundation.}
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\begin{document}
\vskip 0.15in
\def2 \left[ 1 + {s-1 \choose 3} \right]{2 \left[ 1 + {s-1 \choose 3} \right]}
\begin{center}
\bf\Large On sextic surfaces having only nodes\\
(preliminary report)
\end{center}
\vskip 0.15in
\def\arabic{footnote}}\setcounter{footnote}{0}{\fnsymbol{footnote}
\vspace{0.25in}
\block{Introduction}
Over ${\Bbb C}\kern1pt$, let $S \subset \xmode{\Bbb P\kern1pt}^3$ be a surface having only nodes as singularities.
Let \mp[[ \pi || {\tilde{S}} || S ]] be a minimal resolution of singularities.
A set $N$ of nodes on $S$ is {\it even\/} if there exists a divisor $Q$ on
${\tilde{S}}$ such that $2Q \sim \pi^{-1}(N)$.
Suppose that $S$ has degree $6$. It is known (Basset) that $S$ cannot have
$67$ or more nodes. It is also known (Barth) that $S$ can have $65$ nodes.
It is not known if $S$ can have $66$ nodes. Likewise, it is not known exactly
what sizes can occur for an even set of nodes on $S$.
We show that an nonempty even set of nodes on $S$ must have size
$24$, $32$, $40$, $56$, or $64$. We do not know if the sizes $56$ and $64$
can occur. We show that if $S$ has $66$ nodes, then it must have an even
set of $64$ nodes%
,\footnote{J.\ Wahl informed me that he has independently obtained
these results, but I have not seen his methods.}
and it cannot have an even set of $56$ nodes. {\bf\it Thus if one could
rule out the case of a 64 node even set, it would follow that S cannot
have 66 nodes.}
The existence or nonexistence of large even node sets is related to the
following vanishing problem.
Let $S \subset \xmode{\Bbb P\kern1pt}^3$ be a normal surface of degree $s$. Let $D$ be a Weil
divisor on $S$ such that $D \sim_{\Bbb Q}\kern1pt rH$, for some $r \in {\Bbb Q}\kern1pt$. Under what
circumstances do we have $H^1({\cal O}_S(D)) = 0$? For instance, this holds when
$r < 0$. For $s=4$ and $r=0$, $H^1$ can be nonzero. For $s=6$ and $r=0$, if
a $56$ or $64$ node even set exists, then $H^1$ can be nonzero. The
vanishing of $H^1$ is also connected to linear normality, quadric normality,
etc.\ of set-theoretic complete intersections\ in $\xmode{\Bbb P\kern1pt}^3$. See
(\Lcitemark 5\Rcitemark \ 11.8--11.11). There is no further
discussion of these issues in this paper.
\block{Basic tools and notation}
Since some of the calculations in this report apply as well to surfaces of
arbitrary degree, we will for a while allow $S$ to be any surface of
degree $s$ in $\xmode{\Bbb P\kern1pt}^3$ having only nodes as singularities.
Many ideas used here are taken from Beauville\Lspace \Lcitemark 3\Rcitemark
\Rspace{},
who analyzes quintic surfaces having only nodes. The following result
is well-known:
\begin{lemma}\label{torsion-free}
$\mathop{\operatoratfont Pic}\nolimits({\tilde{S}})$ is torsion-free.
\end{lemma}
\begin{proof}
Let ${\cal{L}}$ be a line bundle on ${\tilde{S}}$ such that ${\cal{L}}^{\o* n} \cong {\cal O}_{\tilde{S}}$ for
some $n \in \xmode{\Bbb N}$. Let $S_0$ be the disjoint union of the spectra of the local
rings of the singular points of $S$, and let ${\tilde{S}}_0 = \pi^{-1}(S_0)$. Since
${\cal{L}}$ restricted to each exceptional curve is trivial, it follows
(see\Lspace \Lcitemark 7\Rcitemark \Rspace{}\ \S10, p.\ 157) that
${\cal{L}}|_{{\tilde{S}}_0}$ is trivial. Therefore there exist effective divisors
$D_1, D_2$ on ${\tilde{S}}$ which do not meet the exceptional curves and are such
that ${\cal{L}} \cong {\cal O}_{\tilde{S}}(D_1-D_2)$. Hence ${\cal{L}}$ is the pullback of a line
bundle from $S$. But $\mathop{\operatoratfont Pic}\nolimits(S)$ is torsion-free and $\mathop{\operatoratfont Pic}\nolimits(\pi)$ is
injective, so ${\cal{L}} \cong {\cal O}_{\tilde{S}}$. {\hfill$\square$}
\end{proof}
The even sets of nodes on $S$ comprise the codewords of a binary linear
code $C$. That is, if $C$ is the set of all even sets of nodes, then $C$
may be regarded as a sub-vector-space of ${\Bbb F}\kern1pt_2^n$, where $n$ is the number of
nodes on $S$. We use without much explanation some standard tools from
coding theory to investigate $C$; the standard reference is
\Lcitemark 6\Rcitemark \Rspace{}.
For $w \in C$, let $\abs{w}$ denote the size of the node set, which in
coding theory would be called the {\it weight\/} of $w$. It is the number
of $1$'s which appear in the expression for $w$ as an element of ${\Bbb F}\kern1pt_2^n$.
For $n \in \xmode{\Bbb N}$, let $C_n$ denote the number of words of weight $n$ in $C$.
Let $C^\perp$ denote the dual code to $C$.
If $J$ is a set of positive integers, by a $[n,k,J]$ code we mean a code
with lives in ${\Bbb F}\kern1pt_2^n$, has dimension $k$, and has nonzero weights in the
set $J$. We also underscore weights to indicate that they must occur.
Thus, for example, a $[66,13,\setof{24,32,40,\underline{64}}]$ code is a code which
lives in ${\Bbb F}\kern1pt_2^{66}$, has dimension $13$, has nonzero weights in the set
\setof{24,32,40,64}, and has a word of weight $64$.
For $w \in C$, we let $Q_w$ denote a divisor on ${\tilde{S}}$ with the property
that $2Q_w$ is rationally equivalent to the sum of the exceptional curves
corresponding to the points in $w$. By \pref{torsion-free}, $Q_w$ is
determined up to rational equivalence.
For $w \in C - \setof{0}$, let \mp[[ \tau_w || X_w || {\tilde{S}} ]] be the
double cover branched along the union of the exceptional curves corresponding
to the singular points in $w$, and determined by $Q_w$.
(See\Lspace \Lcitemark 2\Rcitemark \Rspace{}\ I\ \S17 for background on this.)
Since
anything in $\mathop{\operatoratfont Ker}\nolimits[\mathop{\operatoratfont Pic}\nolimits(\tau_w)]$ must be $2$-torsion
(\Lcitemark 2\Rcitemark \ I\ 16.2), it follows from \pref{torsion-free} that
$\mathop{\operatoratfont Pic}\nolimits(\tau_w)$ is injective.
For any divisor $D$ on ${\tilde{S}}$, we can push $D$ down to a Weil divisor $\overline{D}$
on $S$; the exceptional components of $D$ (if any) are ignored.
\block{Results on linear systems and curves}
If ${\cal{L}}$ is a globally generated line bundle on a smooth projective variety
$T$, we consider the complete linear system corresponding to
${\cal{L}}$, and let $\varphi_{\cal{L}}$ denote the induced morphism from $T$ to its
image, which is a projective variety. The following result is no
doubt well-known; we include it for lack of a reference.
\begin{lemma}\label{birational}
Let \mp[[ f || W || T ]] be a surjective morphism of smooth projective
varieties of dimension $\geq 2$, which is generically two-to-one, and such that
$\mathop{\operatoratfont Pic}\nolimits(f)$ is injective. Let ${\cal{L}}$ be a globally generated line bundle
on $T$. Assume that $h^0(f^*{\cal{L}}) > h^0({\cal{L}})$. Assume that
$\varphi_{\cal{L}}$ is birational. Then $\varphi_{f^*{\cal{L}}}$ is birational.
\end{lemma}
\begin{proof}
Let $D$ be an effective divisor in the complete linear system associated to
${\cal{L}}$.
Let $Q$ be a general member of $\abs{f^*D}$. By Bertini's theorem, $Q$ is
smooth and connected. Since $\dim \abs{f^*D} > \dim \abs{D}$, we see that
$Q \notin f^*\abs{D}$. Let $f_*(Q)$ denote $f(Q)$, thought of as a reduced
effective divisor. The divisor $f^*f_*(Q) - Q$ is effective. If
$f^*f_*(Q) = Q$, then $f^*({\cal{L}}) \cong f^*{\cal O}_T(f_* Q)$, so
${\cal{L}} \cong {\cal O}_T(f_*Q)$ [since $\mathop{\operatoratfont Pic}\nolimits(f)$ is injective], and hence
$Q \in f^*\abs{D}$: contradiction. Hence $f^*f_*(Q) - Q \not= 0$. Since $Q$
is general it follows that $Q \subsetneq f^{-1}(f(Q))$ as sets. Therefore if
$x \in Q$ is a general point,
$\setof{x} \subsetneq f^{-1}(f(x))$. Since $Q$ passes
through $x$ and there is exactly one other point $y \in f^{-1}(f(x))$, which
$Q$ does not pass through, we see that
the restriction of $\varphi_{f^*{\cal{L}}}$ to $f^{-1}(f(x))$ is
injective. Since $x$ ranges over general points on general members of
$\abs{f^*D}$, in fact $x$ ranges over a nonempty open subset of $W$. Hence
there is a nonempty open subset of $W$ on which $\varphi_{f^*{\cal{L}}}$ is
injective. {\hfill$\square$}
\end{proof}
We state Castelnuovo's bound for the genus of a curve, in a form which
follows immediately from (\Lcitemark 1\Rcitemark \ p.\ 116).
\begin{theorem}\label{castelnuovo}
Let $M$ be a smooth curve of genus $g$, that admits a birational mapping
onto a nondegenerate curve of degree $d$ in $\xmode{\Bbb P\kern1pt}^r$, $r \geq 2$. Let
$x = \floor{{d-1 \over r-1}}$, $y = {d-1 \over r-1} - x$.
Then:
$${g \over r-1} \leq {x \choose 2} + xy.$$
\end{theorem}
\block{Results valid for any degree {\it s}}
First we make an elementary calculation, and then we give a very useful
proposition, which is probably known in some sense.
\begin{lemma}\label{chi}
Let $w \in C - \setof{0}$, and write $X = X_w$ and $Q = Q_w$ for simplicity.
Let $p = \abs{w}$. Let $n \in \xmode{\Bbb Z}$. Then
$$\chi({\cal O}_X(n))\ =\ n(n+4-s)s + 2 \left[ 1 + {s-1 \choose 3} \right] - p/4.$$
\end{lemma}
\begin{proof}
\begin{eqnarray*}
\chi({\cal O}_X(n)) & = & \chi({\cal O}_{\tilde{S}}(n)) + \chi({\cal O}_{\tilde{S}}(-Q)(n))\\
& = & {1\over2}(nH) \cdot (nH-K_S) + {1\over2}(-Q + nH) \cdot (-Q + nH - K_S)
+ 2\chi({\cal O}_{\tilde{S}})\\
& = & (nH)\cdot(n+4-s)H + 2\chi({\cal O}_{\tilde{S}}) + {1\over2} Q^2\\
& = & n(n+4-s)s + 2 \left[ 1 + {s-1 \choose 3} \right] + {1\over8}(\svec E1p)^2.\kern1.15cm\square
\end{eqnarray*}
\end{proof}
Let $w \in C - \setof{0}$, and simplify notation by writing
$Q = Q_w$, $X = X_w$, $\tau = \tau_w$. Let $E$ be the union of the
exceptional curves corresponding to the singular points in $w$.
For any $n \in \xmode{\Bbb Z}$, consider the exact sequence
\sesdot{{\cal O}_{\tilde{S}}(nH-Q)}{{\cal O}_{\tilde{S}}(nH+Q)}{{\cal O}_E(nH+Q)%
}Since the restriction of $nH+Q$ to any component of $E$ has degree $-1$,
$h^j({\cal O}_E(nH+Q)) = 0$ for all $j$. Hence
$h^k({\cal O}_{\tilde{S}}(nH-Q)) = h^k({\cal O}_{\tilde{S}}(nH+Q))$
for all $k$. This fact will be used a number of times.
\begin{prop}\label{mmmmm}
Assume that $s \geq 5$. Assume that $h^0({\cal O}_X(1)) \geq 5$. Then
$h^0({\cal O}_X(1)) = 5$ and $\abs{w} \leq s(s-1)$.
\end{prop}
\begin{proof}
Let $p = \abs{w}$. Since $h^0({\cal O}_{{\tilde{S}}}(1)) = h^0({\cal O}_S(1)) = 4$, it follows by
\pref{birational} that $\varphi_{{\cal O}_X(1)}$ is birational. Let $H$ be a
general member of the complete linear system on ${\tilde{S}}$ corresponding to
${\cal O}_{{\tilde{S}}}(1)$. Let $M = \tau^{-1}(H)$. By Bertini's theorem, $M$ is a
smooth connected curve.
Let ${\cal O}_M(1)$ denote the pullback of ${\cal O}_X(1)$ to $M$.
Since $\varphi_{{\cal O}_X(1)}$ is birational, $\varphi_{{\cal O}_M(1)}$ is birational.
Let $r = h^0({\cal O}_M(1)) - 1$. The map \mapx[[ M || H ]] is an \'etale double
cover. Since $H$ has genus $(s-1)(s-2)/2$, $M$ has genus $(s-1)(s-2)-1$.
Since the image of $H$ in $\xmode{\Bbb P\kern1pt}^3$ has degree $s$, it follows that
$\varphi_{{\cal O}_M(1)}(M)$ is a nondegenerate curve of degree $2s$ in $\xmode{\Bbb P\kern1pt}^r$.
We have $r \geq 3$; we will show that in fact $r = 3$. Apply
\pref{castelnuovo}. We assume that $s \geq 8$, leaving the cases
$s = 5,6,7$ to the reader. We have
$x = \floor{{2s-1 \over r-1}}$. Since $y \leq (r-2)/(r-1)$, we have:
\begin{eqnarray*}
{g \over r-1} & = & {(s-1)(s-2)-1 \over r-1}\ \leq\ {x(x-1) \over 2} +
\left( {r-2 \over r-1} \right) x\\
& \leq & {(2s-1)(2s-r) \over 2(r-1)^2} + {(2s-1)(r-2) \over (r-1)^2},
\end{eqnarray*}
which implies that
$$2(r-1)(s^2 - 3s + 1)\ \leq\ (2s-1)(2s+r-4).$%
$In particular, if $r \geq 4$, then the above inequality holds even when
$r = 4$ is formally substituted in, and it follows that $s^2 - 8s + 3 \leq 0$.
Hence $s < 8$: contradiction. Hence $r = 3$.
{}From the exact sequence
\ses{{\cal O}_X}{{\cal O}_X(1)}{{\cal O}_M(1)%
}we get $h^0({\cal O}_X(1)) \leq 5$, so in fact $h^0({\cal O}_X(1)) = 5$. Since
$h^0({\cal O}_X(1)) = h^0({\cal O}_{{\tilde{S}}}(H)) + h^0({\cal O}_{{\tilde{S}}}(H-Q))$, we get
$h^0({\cal O}_{{\tilde{S}}}(H-Q)) = 1$.
Let $I \in \abs{H-Q}$. Since $I \cdot H = s$, $\pi(\red{I})$ is a curve
(possibly reducible) of degree $\leq s$ on $S$, which passes through at least
$p$ of its nodes. It follows that $p \leq s(s-1)$. {\hfill$\square$}
\end{proof}
\block{Words of weight < 24 in {\it C}}
{\bf From now on we will assume that {\it s} = 6.} We will want to get as
much information as possible about the code $C$. First, an argument of Reid
(\Lcitemark 4\Rcitemark \ 2.11) shows
that all the weights of words in $C$ are divisible by $8$.
Suppose that $w \in C$ is a nonzero word of weight $< 24$. Then
$\abs{w} \leq 16$. Let $X = X_w$. By \pref{chi}, $\chi({\cal O}_X(1)) \geq 12$.
By Serre duality (roughly), $h^0({\cal O}_X(1)) = h^2({\cal O}_X(1))$, so
$h^0({\cal O}_X(1)) \geq 6$, contradicting \pref{mmmmm}. Hence
$C$ has no nonzero words of weight $< 24$.
\block{Properties of words of weight 24 in {\it C}}\label{weight-24-section}
Let $w \in C_{24}$, and let $X = X_w$.
By \pref{chi}, $\chi({\cal O}_X(1)) = 10$, and since
$h^0({\cal O}_X(1)) = h^2({\cal O}_X(1))$ (as above), we have $h^0({\cal O}_X(1)) \geq 5$.
By \pref{mmmmm}, $h^0({\cal O}_X(1)) = 5$. Hence $h^0({\cal O}_{\tilde{S}}(H-Q_w)) = 1$.
In other words $\abs{H-Q_w}$ has a unique element, which we henceforth
denote by $D_w$.
We will show now that $\overline{D_w}$ is reduced. Since $\overline{D_w}$ passes through
$\geq 24$ singular points of $S$, $\deg(\red{(\overline{D_w})}) \geq 5$. Suppose
that $\overline{D_w}$ is nonreduced. Then $\deg(\red{(\overline{D_w})}) = 5$ so $\overline{D_w}$
contains a double line. Let $M$ be the remainder of $\overline{D_w}$, sans the
doubled line. Since $\overline{D_w}$ must pass with
{\it odd multiplicity\/} through exactly $24$ singular points of $S$, we
conclude that $M$ must pass with odd multiplicity through exactly $24$
singular points of $S$. Since $\deg(M) = 4$, this is impossible. Hence
$\overline{D_w}$ is reduced.
Now there is a ``quadric surface'' $V_w \subset \xmode{\Bbb P\kern1pt}^3$ such that
$2\overline{D_w} = V_w \cap S$ as Weil divisors on $S$. There are four
possibilities for $V_w$. The worst case, that $V_w$ might be a doubled plane,
can be ruled out, since then $\overline{D_w} = \red{(V_w)} \cap S$ as
Weil divisors on $S$, so $\overline{D_w}$ is Cartier on $S$, which yields a
contradiction. Unfortunately, we do not know if $V_w$ can be the union of
two distinct planes, a possibility which substantially complicates the
proofs. For starters, we must analyze this case. We could rule it out
if we could show that a cubic curve on $S$ cannot pass through $15$ singular
points of $S$.
\begin{prop}\label{two-planes}
Let $w \in C_{24}$. Suppose that $V_w$ is the union of two distinct planes
$H_1$, $H_2$. For each $i$, let $\overline{D_i} = \red{(H_i \cap S)}$. Then:
\begin{alphalist}
\item for each $i$, $\overline{D_i}$ passes through exactly $15$ singular points of
$S$, and it passes with multiplicity one through each of these;
\item of the $15$ singular points, exactly $12$ are in $w$, and the remaining
$3$ lie on the line $H_1 \cap H_2$. Both $\overline{D_1}$ and $\overline{D_2}$
pass with multiplicity one through these $3$ nodes.
\end{alphalist}
\end{prop}
\begin{proof}
Let $D_i \subset {\tilde{S}}$ be the strict transform of $\overline{D_i}$, for each $i$. Let
$L = H_1 \cap H_2$. Because $2|\deg(S)$ and $\overline{D_w}$ is reduced, we cannot
have $L \subset S$. Hence $D_1$ meets $D_2$ properly. Note that $\deg(D_i) = 3$.
Let $Q_i = {1\over2} \sum_E (D_i \cdot E) E$, as $E$ ranges over the exceptional
curves of ${\tilde{S}}$. Then $Q$ is a ${\Bbb Q}\kern1pt$-divisor, but (caution) it is not a
divisor. We have $D_i \sim_{\Bbb Q}\kern1pt {1\over2} H - Q_i$.
Suppose that $\overline{D_1}$ has only nodes as singularities. Then
$\chi({\cal O}_{D_1})$ is equal to the number (say $n$) of such singular points
which are resolved by $\pi$. Then $D_1^2 = -2n-6$, from which it
follows that $Q_1$ has the form
$${1\over2}\left(\svec E1{15} + \sum_{i=1}^n 2E_i'\right),$%
$where $\vec E1{15}, E_1',\ldots,E_n'$
are distinct exceptional curves. But $\overline{D_1}$
has degree $3$, and so can pass through at most $3(\deg(S)-1) = 15$ singular
points of $S$. Hence $n = 0$, so (a) holds, for $D_1$.
In the three cases where $\overline{D_1}$ has a singularity other than a node,
arguments similar to those of the last paragraph still yield the conclusion
that (a) holds, for $D_1$.
Of course, the results of the last two paragraphs apply equally to $\overline{D_2}$.
Since $\overline{D_w}$ cannot pass with multiplicity one through a node outside $w$,
it follows that the number of nodes of $w$ (say $r$) which $\overline{D_i}$ passes
through is independent of $i$.
Since $\overline{D_1}$ and $\overline{D_2}$ must together pass with multiplicity one
through all $24$ of the singular points in $w$, we must
have $r = 12$. Statement (b) follows. {\hfill$\square$}
\end{proof}
\begin{lemma}\label{line-through-four}
Let $L \subset S$ be a line. Let $w \in C$. Then $L$ can pass through
at most $4$ singular points of $S$ which are in $w$.
\end{lemma}
\begin{proof}
Let ${\tilde{L}} \subset {\tilde{S}}$ be the strict transform of $L$.
Certainly $L$ can pass through at most $5$ singular points of $S$.
But ${\tilde{L}} \cdot Q_w \in \xmode{\Bbb Z}$, so the lemma follows. {\hfill$\square$}
\end{proof}
\begin{prop}\label{empty-intersection}
Let $w_1, w_2 \in C$, and assume that
$\abs{w_1} = \abs{w_2} = \abs{w_1+w_2} = 24$. Then either (a)
$D_{w_1} \cap D_{w_2} = \varnothing$, or else (b) we can write
$V_{w_1} = H_1 \cup H_2$ and $V_{w_2} = H_2 \cup H_3$, where $H_1, H_2, H_3$
are planes.
\end{prop}
\begin{proof}
First note that $(H-Q_{w_1}) \cdot (H-Q_{w_2}) = 6 - {1\over2}(12) = 0$, so for
case (a) it suffices to show that $D_{w_1}$ and $D_{w_2}$ share no component.
Let $M$ be the shared part of $D_{w_1}$ and $D_{w_2}$.
We may assume that $V_{w_1}$ meets $V_{w_2}$ properly, since otherwise
conclusion (b) holds. Then $\deg(V_{w_1} \cap V_{w_2}) = 4$. We have
${\overline{M}} \subset V_{w_1} \cap V_{w_2}$, and since $V_{w_i}$ is tangent to $S$
along ${\overline{M}}$ for each $i$, in fact $V_{w_1}$ is tangent to $V_{w_2}$ along
${\overline{M}}$. Hence $2{\overline{M}} \subset V_{w_1} \cap V_{w_2}$. Hence $\deg(M) \leq 2$.
If $\deg(M) = 2$, then $V_{w_1} \cap V_{w_2} = 2{\overline{M}}$.
Hence ${\overline{M}}$ passes through
all $12$ singular points in $w_1 \cap w_2$. But $\deg({\overline{M}}) = 2$, so ${\overline{M}}$
can pass through at most $2[\deg(S)-1] = 10$ singular points of $S$:
contradiction. Hence $\deg(M) \leq 1$. We have
$$0\ \leq\ (H-Q_{w_1}-M) \cdot (H-Q_{w_2}-M) = M(Q_{w_1}+Q_{w_2}-2H)+M^2.
\eqno(*)$%
$We will complete the proof by showing that if $M \not= 0$, then the right hand side\ of
$(*)$ is negative. Write $M = R+E$, where $R$, $E$ are effective, $E$ is
exceptional, and $R$ has no exceptional components. Expanding out the
right hand side\ of $(*)$ yields
$$(R+E)\cdot(Q_{w_1}+Q_{w_2}) - 2\deg(R) + R^2 + 2(E \cdot R) + E^2.$%
$If $R=0$, then $E=0$, or else this quantity is negative. Hence $\deg(R) = 1$.
By the adjunction formula, $R^2 = -4$. Therefore
to get a contradiction, it is enough to show that
$$(R+E) \cdot (Q_{w_1} + Q_{w_2}) + 2(E \cdot R) + E^2 < 6.$%
$By \pref{line-through-four}, $R \cdot (Q_{w_1} + Q_{w_2}) \leq 4$, so it is
enough to show that
$$E \cdot (Q_{w_1} + Q_{w_2}) + 2(E \cdot R) + E^2 < 2.$%
$Thus it is sufficient to show that if $F$ is an exceptional curve, then for
all $n \in \xmode{\Bbb N}$,
$$nF \cdot (Q_{w_1} +Q_{w_2}) + 2(nF \cdot R) + (nF)^2 \leq 0.$%
$Clearly it is enough to do the case $n=1$, so we need
$$F \cdot(Q_{w_1} + Q_{w_2}) + 2(F \cdot R) \leq 2.$%
$Since $F \cdot R \leq 1$, this is clear. {\hfill$\square$}
\end{proof}
\block{Words of weight 48 in {\it C}}
We will show that $C$ has no words of weight $48$. So let
$v \in C_{48}$, heading for a contradiction. Write $Q$ for $Q_v$, $X$ for
$X_v$. The first step is to show that
$h^0({\cal O}_{\tilde{S}}(2H-Q)) = 1$. Let $N \subset {\Bbb F}\kern1pt_2^{48}$ be $\setof{w \in C: w \subset v}$.
We have $h^0({\cal O}_{\tilde{S}}(H-Q)) = 0$, since otherwise there is a curve of degree
$6$ on $S$ which passes through $\geq 48$ singular points of $S$. Hence
$h^0({\cal O}_X(1)) = 4$. By \pref{chi} we have $\chi({\cal O}_X) = 10$, so
$h^2({\cal O}_X) - h^1({\cal O}_X) = 9$. But $h^2({\cal O}_X) \geq h^2({\cal O}_{\tilde{S}}) = 10$, so
$h^1({\cal O}_X) \geq 1$. Hence $\dim_{{\Bbb F}\kern1pt_2}(\mathop{\operatoratfont Pic}\nolimits(X)_2) \geq 2$. Hence by
(\Lcitemark 3\Rcitemark \ Lemma 2), $\dim(N) \geq 3$. Since we know
that $w \in N-\setof{0}\ \Longrightarrow\ \abs{w} \geq 24$, we in fact have
$\setof{\abs{w}: w \in N} = \setof{0,24,48}$. Choose $w_1, w_2 \in N$ with
$\abs{w_1} = \abs{w_2} = 24$ and $w_1 \cap w_2 = \varnothing$. By the first
paragraph of \S\ref{weight-24-section},
$h^0({\cal O}_{\tilde{S}}(H-Q_{w_i})) \not= 0$. Since $v = w_1 + w_2$,
$h^0({\cal O}_{\tilde{S}}(2H-Q)) \not= 0$. By Serre duality,
$h^2({\cal O}_{\tilde{S}}(Q)) \not= 0$. Hence $h^2({\cal O}_{\tilde{S}}(-Q)) \not= 0$. Then
$h^2({\cal O}_X) = h^2({\cal O}_{\tilde{S}}) + h^2({\cal O}_{\tilde{S}}(-Q)) \geq 11$, so $h^1({\cal O}_X) \geq 2$.
Hence $\dim(N) \geq 5$.
Suppose that $h^0({\cal O}_{\tilde{S}}(2H-Q)) > 1$. Then (arguing as above)
$h^1({\cal O}_X) \geq 3$, so $\dim(N) \geq 7$. Hence there exists a
$[48,7,\setof{24,48}]$ code, which contradicts the Griesmer bound for
codes. (One also gets a contradiction by the linear programming method.)
Hence $h^0({\cal O}_{\tilde{S}}(2H-Q)) = 1$.
Let $w \in N_{24}$. Since $h^0({\cal O}_{\tilde{S}}(2H-Q)) = 1$, it follows that
$D_w + D_{v+w}$ is independent of $w$.
Since $\dim(N) \geq 3$, we can find $w_1, w_2 \in N$ with
$\abs{w_1} = \abs{w_2} = \abs{w_1+w_2} = 24$. We will show that
$V_{w_1}$ does not meet $V_{w_2}$ properly. Suppose otherwise.
We have $D_{w_1} + D_{v+w_1} = D_{w_2} + D_{v+w_2}$.
By \pref{empty-intersection}, $D_{w_1} \cap D_{w_2} = \varnothing$.
Hence $D_{w_1} \subset D_{v+w_2}$, so $\overline{D_{w_1}} = \overline{D_{v+w_2}}$, so
$w_1 = v+w_2$: contradiction. Hence $V_{w_1}$ does not meet
$V_{w_2}$ properly, i.e.\ each consists of two planes, with one in common.
The same holds for the pair $(V_{w_1}, V_{v+w_2})$, the pair
$(V_{v+w_1}, V_{w_2})$, and the pair $(V_{v+w_1}, V_{v+w_2})$.
We consider now the configuration $\cal C$ of planes which occur in $V_{w_1}$,
$V_{w_2}$, $V_{v+w_1}$, and $V_{v+w_2}$. In total, there are four planes.
First suppose that no three of these planes share a common line. Then
we may visualize the configuration by means of a tetrahedron $T$. (The faces
correspond to planes.) For any pair of faces, the intersection of the
corresponding two planes with $S$ gives rise to a word $w$ of weight $24$,
and so \pref{two-planes} applies. Therefore every edge of $T$ has exactly $3$
singular points of $S$ on it, and every face of $T$ has exactly $15$ singular
points of $S$ on it. Here is the generic picture, showing only the singular
points which appear on the edges:
\vspace{0.1in}
\par\noindent{\bf[There was a postscript picture of a tetrahedron here, with
three little spheres along each edge.]}
\vspace{0.1in}
\par\noindent
The picture is generic because some of the edge singular points could be
at the vertices. Restricting to the generic case for the moment, let us
compute the total number of singular points of $S$ which lie on $T$. Since
each face contains $15$, the interior of each face has $6$. Hence
$$\abs{\mathop{\operatoratfont Sing}\nolimits(S) \cap T}\ =\ (4 \cdot 6) + (6 \cdot 3)\ =\ 42.$%
$But $T$ must go through at least $\abs{v} = 48$ singular points of $S$:
contradiction. In the nongeneric cases, one similarly gets a
contradiction. (In the least generic case, where all $4$ vertices of $T$ are
in $\mathop{\operatoratfont Sing}\nolimits(S)$, one gets $\abs{\mathop{\operatoratfont Sing}\nolimits(S) \cap T} = 46$, which still gives a
contradiction.)
Hence at least $3$ of the planes in $\cal C$ must share
a common line.
Suppose now that the four planes in ${\cal{C}}$ do not all share a common line.
Taking a cross section by a suitable plane, we may represent each plane by
a line segment and each intersection of two planes by a vertex. (The single
point where all planes meet is not shown.) In the generic case, we can
label each vertex in the picture with a $3$.
\widepost{85}{25}{
newpath 0 -15 moveto -50 50 lineto stroke
newpath 0 -15 moveto 50 50 lineto stroke
newpath 0 -15 moveto 0 50 lineto stroke
newpath -50 50 moveto 50 50 lineto stroke
0 -15 PointAt -50 50 PointAt 50 50 PointAt 0 50 PointAt
{6 270 (3) AnglePrint} 0 -15 xyput
{6 90 (3) AnglePrint} 0 50 xyput
{6 135 (3) AnglePrint} -50 50 xyput
{6 45 (3) AnglePrint} 50 50 xyput}
\par\noindent One finds that the four planes together pass through at
most $45$ singular points of $S$: contradiction. In the nongeneric case,
one finds that ${\cal{C}}$ passes through $47$ points of $\mathop{\operatoratfont Sing}\nolimits(S)$, still shy
of $48$.
Now we know that the four planes in ${\cal{C}}$ must contain a common line.
It follows that there exists a line $L$ such that for any
$w \in N_{24}$, $L$ is contained in both planes of $V_w$.
Let ${\cal{C}}^+$ be the configuration
of all planes which appear in $V_w$ for some $w \in N_{24}$.
Since $\dim(N) \geq 5$, $\abs{N_{24}} \geq 30$. Since
${8 \choose 2} < 30$, ${\cal{C}}^+$ must have at least $9$ planes in it. Hence
${\cal{C}}^+$ must pass through at least $(9 \cdot 12) + 3 = 111$ singular points
of $S$, which is absurd. We conclude that $C$ has no word of weight $48$.
\block{Apply coding theory}
Suppose now that $S$ has $66$ nodes. Since $H^2(S,\xmode{\Bbb Z}) \cong \xmode{\Bbb Z}^{106}$, it
follows\Lspace \Lcitemark 3\Rcitemark \Rspace{} that $\dim(C) \geq 66 - 106/2 =
13$.
We know that the nonzero weights appearing in $C$ are all in the set
\setof{24,32,40,56,64}. Now we
show that weight $64$ must occur and that weight $56$ cannot occur.
\begin{theorem}\label{code-doesnt-exist}
There is no $[66,13,\setof{24,32,40,56}]$ code.
\end{theorem}
\begin{proof}
Let $R = \setof{24,32,40,56}$ and $T = \setof{4,8,12,16,20}$. By the linear
programming method (LP), there is no $[21,10,T]$ code. (See the appendix
for some {\sc Maple} code to do calculations like this.) Therefore by taking
the residual code with respect to\ a codeword of weight $40$ (hereafter referred to as
res!40), one sees that there is no $[61,11,\setof{24,32,\underline{40},56}]$ code.
Let $C$ be a $[58,11,\setof{24,32,56}]$ code. By LP, $C_{56} = 1$. By LP,
$C^\perp_1 = 2$. By LP, $C^\perp_2 = 6.5$: contradiction. Hence there is no
such code $C$. Now let $C$ be a $[59,11,\setof{24,32,56}]$ code. Then
$C^\perp_1 = 0$. Then LP yields a contradiction, so there is no such code $C$.
Similarly, one sees that there is no $[60,11,\setof{24,32,56}]$ code, and
finally in the same way that $(*)$ there is no $[61,11,\setof{24,32,56}]$ code.
By the preceeding paragraph, we conclude $(\dag)$ that there is no $[61,11,R]$
code.
By LP, there is no $[22,11,T]$ code. Therefore by res!40, there is no
$[62,12,\setof{24,32,\underline{40},56}]$ code. By LP, there is no
$[62,12,\setof{24,32,56}]$ code $C$ with $C^\perp_1 = 0$, and therefore by
$(*)$ of the preceeding paragraph, there is no $[62,12,\setof{24,32,56}]$ code
at all. Hence there is no $[62,12,R]$ code.
{}From $(\dag)$ we see that a $[63,13,R]$ code $C$ must have
$C^\perp_2 = C^\perp_3 = 0$. By LP, one concludes that there is no
$[63,13,R]$ code.
{}From the preceeding three paragraphs we see that a $[64,13,R]$ code $C$ must
have $C^\perp_1 = C^\perp_2 = C^\perp_3 = 0$. By LP, one concludes that
a $[64,13,R]$ code $C$ must have $C_{56} = 2.5$: contradiction. Hence there
is no such code $C$.
Let $C$ be a $[65,13,R]$ code. Then $C^\perp_1 = 0$. By LP,
$C^\perp_2 \geq 5$. Consider two distinct words of weight two in $C^\perp$.
If they are disjoint [resp.\ not disjoint], then there is a
$[61,11,R]$ code [resp.\ $[62,12,R]$ code], in either case contradicting
earlier results. Hence there is no $[65,13,R]$ code.
Let $C$ be a $[66,13,R]$ code. (This will lead to a contradiction.) Then
$C^\perp_1 = 0$. Form a graph $G$ whose vertices correspond to the coordinate
positions covered by the words of weight $2$ in $C^\perp$, and such that the
edges of $G$ correspond to the words of weight $2$ in $C^\perp$. Then $G$ is a
disjoint union of complete graphs. By LP, $C^\perp_2 \geq 7$, so $G$ has at
least $7$ edges. Now $G$ has no $K_4$ component, because then one would have a
$[62,12,R]$ code. Similar arguments show that $G$ cannot have a $K_n$ (for
$n \geq 5$), it cannot have $2$ $K_3$'s, and it cannot have a $K_3$ and a
$K_2$. Therefore, since $G$ has at least $7$ edges, it must consist entirely
of $K_2$'s. Otherwise said, the words of weight $2$ in $C^\perp$ are pairwise
disjoint.
If $C$ has a word $v$ of weight $56$, it occupies all but $10$ bits of $C$, and
so at most $5$ of the words of weight $2$ in $C^\perp$ are disjoint from $v$.
Thus we can choose some $w \in C^\perp_2$ with the property that if such
a word $v$ exists, then $w \subset v$.
Let $D$ be the subcode of $C$ consisting of words disjoint from $w$. Then $D$
is a $[64,12,\setof{24,32,40}]$ code and $D^\perp$ has at least $7-1=6$ words
of weight $2$. Adjoin the unique word of weight $64$ to $D$,
yielding a $[64,13,\setof{24,32,40,64}]$ code $E$ with $E^\perp_2 \not= 0$.
Hence there exists a $[62,12,\setof{24,32,40}]$ code: contradiction. {\hfill$\square$}
\end{proof}
We conclude that $C$ must be a
$[66,13,\setof{24,32,40,\underline{64}}]$ code. Unfortunately, such a code exists.
\vspace{0.1in}
\par\noindent{\bf Appendix}
\vspace{0.1in}
Here we give some {\sc Maple} code which can be used to carry out the
linear programming calculations which appear in the proof of
\pref{code-doesnt-exist}.
The first four lines can be adjusted to check
for particular types of codes. As written, the program will analyze
$[64,13,\setof{24,32,40,56}]$ codes $C$ with
$C^\perp_1 = C^\perp_2 = C^\perp_3 = 0$. If you
run it, you will find that such a code must have $C_{56} = 2.5$ (which is
absurd), as claimed in the proof of \pref{code-doesnt-exist}.
\vspace{0.2in}
\begin{verbatim}
n := 64;
k := 13;
w := {24,32,40,56};
extraconstraints := [mu1=0, mu2=0, mu3=0];
K := (x,i) -> sum( (-1)^j * binomial(i,j) * binomial(n-i,x-j), j=0..x ):
for m from 0 to n/2 do
mu.m := (K(m,0) + convert(map(i -> 'a.i' * K(m,i), w), `+`)) / 2^k:
od:
objective := 1 + convert(map(z -> a.z, w), `+`);
mus := {seq( mu.m >= 0, m=0..n/2 ), objective = 2^k, op(extraconstraints)}:
for i from 1 to 2 do
M := simplex[minimize](mu.i,mus,NONNEGATIVE);
`minimum value of mu`.i, evalf(subs(M,mu.i));
M := simplex[maximize](mu.i,mus,NONNEGATIVE);
`maximum value of mu`.i, evalf(subs(M,mu.i));
od;
for q in w do
M := simplex[minimize](a.q,mus,NONNEGATIVE);
q, `min`, evalf(subs(M,a.q));
M := simplex[maximize](a.q,mus,NONNEGATIVE);
q, `max`, evalf(subs(M,a.q));
od;
quit():
\end{verbatim}
\message {DOCUMENT TEXT}
\section*{References}
\addcontentsline{toc}{section}{References}
\ \par\noindent\vspace*{-0.25in}
\hfuzz 5pt
\message{REFERENCE LIST}
\bgroup\Resetstrings%
\def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{
}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
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\def\Ftest{ }\def\Fstr{1}%
\def\Atest{ }\def\Astr{Arbarello\Revcomma E\Initper %
\Acomma M\Initper Cornalba%
\Acomma P\Initper \Initgap A\Initper Griffiths%
\Aandd J\Initper Harris}%
\def\Ttest{ }\def\Tstr{Geometry of Algebraic Curves, Vol.\ I}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Dtest{ }\def\Dstr{1985}%
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\def\Ftest{ }\def\Fstr{2}%
\def\Ven{Van de Ven}{}%
\def\Atest{ }\def\Astr{Barth\Revcomma W\Initper %
\Acomma C\Initper Peters%
\Aandd A\Initper \Ven}%
\def\Ttest{ }\def\Tstr{Compact Complex Surfaces}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Dtest{ }\def\Dstr{1984}%
\def\Qtest{ }\def\Qstr{access via "barth peters ven"}%
\def\Xtest{ }\def\Xstr{local-triv: I 10.1 (theorem of Grauert-Fischer)}%
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\bgroup\Resetstrings%
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\def\Ftest{ }\def\Fstr{3}%
\def\Atest{ }\def\Astr{Beauville\Revcomma A\Initper }%
\def\Ttest{ }\def\Tstr{Sur le nombre maximum de points doubles d'une surface in
$\xmode{\Bbb P\kern1pt}^3$ $(\mu(5) = 31)$}%
\def\Btest{ }\def\Bstr{Algebraic Geometry, Angers, 1979}%
\def\Etest{ }\def\Estr{A\Initper Beauville}%
\def\Itest{ }\def\Istr{Sijthoff \& Noordhoff}%
\def\Dtest{ }\def\Dstr{1980}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{207--215}%
\def\Qtest{ }\def\Qstr{access via "beauville double points"}%
\Refformat\egroup%
\bgroup\Resetstrings%
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}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
\def\NoArev{1}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}%
\def\Ftest{ }\def\Fstr{4}%
\def\Atest{ }\def\Astr{Catanese\Revcomma F\Initper }%
\def\Ttest{ }\def\Tstr{Babbage's conjecture, contact of surfaces, symmetric
determinantal varieties and applications}%
\def\Jtest{ }\def\Jstr{Invent. Math.}%
\def\Vtest{ }\def\Vstr{63}%
\def\Dtest{ }\def\Dstr{1981}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{433--465}%
\def\Qtest{ }\def\Qstr{access via "catanese contact"}%
\Refformat\egroup%
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}\def\Edcapsmallcapstest{}\def\Underlinetest{ }%
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\def\Ftest{ }\def\Fstr{5}%
\def\Atest{ }\def\Astr{Jaffe\Revcomma D\Initper \Initgap B\Initper }%
\def\Ttest{ }\def\Tstr{Applications of iterated curve blowup to set theoretic
complete intersections in $\xmode{\Bbb P\kern1pt}^3$}%
\def\Rtest{ }\def\Rstr{preprint}%
\def\Qtest{ }\def\Qstr{access via "jaffe rational singularities" or preferably
"jaffe applications intersections"}%
\def\Htest{ }\def\Hstr{7}%
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\bgroup\Resetstrings%
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\def\Ftest{ }\def\Fstr{6}%
\def\Atest{ }\def\Astr{MacWilliams\Revcomma F\Initper \Initgap J\Initper %
\Aand N\Initper \Initgap J\Initper \Initgap A\Initper Sloane}%
\def\Ttest{ }\def\Tstr{The Theory of Error-Correcting Codes}%
\def\Itest{ }\def\Istr{North-Holland}%
\def\Ctest{ }\def\Cstr{Amsterdam}%
\def\Dtest{ }\def\Dstr{1977}%
\def\Qtest{ }\def\Qstr{access via "macwilliams sloane"}%
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\def\Ftest{ }\def\Fstr{7}%
\def\Atest{ }\def\Astr{Pinkham\Revcomma H\Initper }%
\def\Ttest{ }\def\Tstr{Singularites rationnelles de surfaces}%
\def\Btest{ }\def\Bstr{S\'eminaire sur les Singularit\'es des Surfaces}%
\def\Stest{ }\def\Sstr{Lecture Notes in Mathematics}%
\def\Itest{ }\def\Istr{Spring\-er-Ver\-lag}%
\def\Ctest{ }\def\Cstr{New York}%
\def\Vtest{ }\def\Vstr{777}%
\def\Dtest{ }\def\Dstr{1980}%
\def\Ptest{ }\def\Pcnt{ }\def\Pstr{147--178}%
\def\Qtest{ }\def\Qstr{access via "pinkham rational singularities"}%
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\end{document}
|
1996-03-31T05:41:55 | 9411 | alg-geom/9411010 | en | https://arxiv.org/abs/alg-geom/9411010 | [
"alg-geom",
"math.AG"
] | alg-geom/9411010 | Miles Reid | Yukari Ito and Miles Reid | The McKay correspondence for finite subgroups of SL(3,\C) | AMSTeX, amsppt and optional epsf.tex , This paper will appear in
Higher Dimensional Complex Varieties (Trento, Jun 1994), ed. M. Andreatta, De
Gruyter, Mar 1996. It has been circulated as a Univ. of Tokyo, Dept. of Math
Sciences preprint, UTMS 94--66, 19 pp | null | null | null | null | This is the final draft, containing very minor proof-reading corrections. Let
G in SL(n,\C) be a finite subgroup and \fie: Y -> X = \C^n/G any resolution of
singularities of the quotient space. We prove that crepant exceptional prime
divisors of Y correspond one-to-one with ``junior'' conjugacy classes of G.
When n = 2 this is a version of the McKay correspondence (with irreducible
representations of G replaced by conjugacy classes). In the case n = 3, a
resolution with K_Y = 0 is known to exist by work of Roan and others; we prove
the existence of a basis of H^*(Y, \Q) by algebraic cycles in one-to-one
correspondence with conjugacy classes of G. Our treatment leaves lots of open
problems.
| [
{
"version": "v1",
"created": "Wed, 16 Nov 1994 14:00:39 GMT"
},
{
"version": "v2",
"created": "Wed, 10 Jan 1996 09:13:15 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Ito",
"Yukari",
""
],
[
"Reid",
"Miles",
""
]
] | alg-geom | |
1994-11-16T06:20:22 | 9411 | alg-geom/9411009 | en | https://arxiv.org/abs/alg-geom/9411009 | [
"alg-geom",
"math.AG"
] | alg-geom/9411009 | V. Batyrev | Victor V. Batyrev and Yuri Tschinkel | Rational Points of Bounded Height on Compactifications of Anisotropic
Tori | 45 pages, Latex | null | null | null | null | We investigate the analytic properties of the zeta-function associated with
heights on equivariant compactifications of anisotropic tori over number
fields. This allows to verify conjectures about the distribution of rational
points of bounded height.
| [
{
"version": "v1",
"created": "Tue, 15 Nov 1994 17:59:52 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Batyrev",
"Victor V.",
""
],
[
"Tschinkel",
"Yuri",
""
]
] | alg-geom | \section{Toric varieties over arbitrary fields}
\subsection{Algebraic tori}
Let $K$ be an arbitrary field, $\overline{K}$ the algebraic closure of
$K$, ${\bf G}_m(\overline{K}) = \overline{K}^*$ the multiplicative
group of $\overline{K}$.
Let $X$ be an arbitrary algebraic
variety over $\overline{K}$. Let $E/K$ be a finite extension such that $X$ is
defined over $E$. To stress this fact we sometimes will denote $X$ also
by $X_E$. The set of $E$-rational points of $X_E$ will be
denoted by $X_E(E)$.
\begin{dfn}
{\rm A linear algebraic group $T$
over $K$ is
called a {\em $d$-dimen\-sio\-nal algebraic torus} if
its base extension $T_{\overline{K}} =
T \times_{{\rm Spec}(K)} {\rm Spec }(\overline{K})$
is isomorphic to
$({\bf G}_m(\overline{K}))^d$. }
\label{opr.tori}
\end{dfn}
We notice that an isomorphism
between ${T}$ and $({\bf G}_m(\overline{K}))^d$ is
always defined over a finite Galois extension $E$ of $K$.
\begin{dfn}
{\rm Let $T$ be an algebraic torus over $K$. A
finite Galois extension
$E$ of $K$ such that $T_E = T \times_{{\rm Spec}(K)}
{\rm Spec}(E)$ is isomorphic to $({\bf G}_m(E))^d$
is called a {\em splitting field } of $T$.}
\end{dfn}
\begin{dfn}
{\rm We denote by $\hat{T} = {\rm Hom}\,( T, \overline{K}^*)$
the group of regular $\overline{K}$-rational
characters of $T$. For any subfield $E \subset
\overline{K}$
containing $K$, we denote by $\hat{T}_E$ the group of characters of $T$
defined over $E$. }
\end{dfn}
There is well-known correspondence between
Galois representations by integral matrices and algebraic tori
\cite{ono1,vosk}:
\begin{theo}
Let $G = {\rm Gal }(E/K)$
be the Galois group of the splitting field $E$ of a
$d$-dimen\-sional torus $T$. Then ${\hat{T}}$ is
a free abelian group of rank $d$ with a structure of $G$-module
defined by the natural representation
\medskip
\[ \rho \; :\; G \rightarrow {\rm Aut}(\hat{T}) \cong
{\rm GL}(d, {\bf Z}).\]
Every d-dimensional integral representation
of $G$ defines a $d$-dimen\-sional algebraic torus over $K$ which splits
over $E$. One obtains a one-to-one correspondence
between $d$-dimensional
algebraic tori over
$K$ with the splitting field $E$ up to isomorphism, and $d$-dimensional
integral representations of $G$ up to equivalence.
\label{represent}
\end{theo}
\begin{rem}
{\rm The group $\hat{T}_K$ is a sublattice in
$\hat{T} \cong {\bf Z}^d$ consisting of all $G$-invariant elements.}
\label{prop.char}
\end{rem}
\begin{dfn}
{\rm An algebraic torus $T$ over $K$ is called
{\em anisotropic} if $\hat{T}_K$ has rank zero.}
\end{dfn}
\begin{exam}
{\rm
Let $f(z) \in K\lbrack z \rbrack$ be a separable
polynomial of degree $d$. Consider the $d$-dimensional $K$-algebra
\[ A(f) = K \lbrack z \rbrack / (f(z)). \]
Then the multiplicative group $A^*(f)$ is a $d$-dimensional algebraic
torus over $K$. This torus has the following properties:
(i) The rank of the group of characters of $A^*(f)$ is
equal to the number of irreducible components of ${\rm Spec}(A(f))$.
(ii) If $f(z)$ splits in linear factors over some finite Galois extension $E$
of $K$, then
$A(f) \otimes_K E \cong E^n$, and $A^*(f)\otimes_K E \cong (E^*)^n$.
Thus, $E$ is a splitting field of $A^*(f)$.
(iii) Since the classes of $1, z, \ldots , z^{d-1}$ in $A(f)$ give
rise to a $K$-basis of the $d$-dimensional algebra $A(f)$, we can consider
$A^*(f)$ as a commutative subgroup in ${\rm GL}(d,K)$. Thus, the
determinant of the matrix defines a regular $K$-character
\[{\cal N} \; : \; A^*(f) \rightarrow K^*. \]
We denote by $A^*_1(f)$ the $(d-1)$-dimensional algebraic torus which is
the kernel of ${\cal N}$.
(iv) The multiplicative group $K^*$ is a subgroup of $A^*(f)$ and the
restriction of ${\cal N}$
to $K^*$ sends $x \in K^*$ to $x^d \in K^*$. The
factor-group $A^*(f) / K^*$ is a $(d-1)$-dimensional torus which
is isogeneous to $A^*_1(f)$. }
\label{exam.tori}
\end{exam}
\begin{exam}
{\rm Let $K'$ be a finite separable extension of $K$.
By primitive element theorem, $K' \cong A(f)$ for some
irreducible
polynomial $f(z) \in K \lbrack z \rbrack $. Thus, we come to
a particular case of the previous example. In this case, ${\cal N}$ is
the norm $N_{K'/K}$, the
algebraic torus $A^*(f)$ is usually denoted by
\[ R_{K'/K} ({\bf G}_m) , \]
and the torus $A_1^*(f)$ is usually denoted by
\[ R_{K'/K}^1 ({\bf G}_m) . \]
Since ${\rm Spec}(K')$ is irreducible,
$R_{K'/K}^1 ({\bf G}_m)$ and $R_{K'/K} ({\bf G}_m)/ K^*$ are
examples of anisotropic tori.}
\label{exam.field}
\end{exam}
\subsection{Compactifications of split tori}
We recall standard facts about toric varieties over algebraically
closed fields \cite{danilov,demasur,oda}.
Let $M$ be a free abelian group of rank $d$ and $N = {\rm Hom}(M, {\bf Z})$ the
dual abelian group.
\begin{dfn}
{\rm A finite set $\Sigma$ consisting of convex rational polyhedral
cones in $N_{\bf R} = N \otimes {\bf R}$ is called a {\em complete regular
$d$-dimensional fan} if the following conditions are satisfied:
(i) every cone $\sigma \in \Sigma$ contains $0 \in N_{\bf R}$;
(ii) every face $\sigma'$ of a cone $\sigma \in \Sigma$ belongs to $\Sigma$;
(iii) the intersection of any two cones in $\Sigma$ is a face of
both cones;
(iv) $N_{\bf R}$ is the union of cones from $\Sigma$;
(v) every
cone $\sigma \in \Sigma$ is generated by a part of a ${\bf Z}$-basis of
$N$.\\
We denote by $\Sigma(i)$ the set of all $i$-dimensional cones in
$\Sigma$. For each cone $\sigma \in \Sigma$ we denote by
$N_{{\sigma}, \bf R}$ the minimal linear subspace containing $\sigma$. }
\label{def.fan}
\end{dfn}
\noindent
Every complete regular $d$-dimensional fan defines a smooth equivariant
compactification ${\bf P}_{\Sigma}$
of the split $d$-dimensional algebraic torus $T$. The variety
${\bf P}_{\Sigma}$ has the following two geometric properties:
\begin{prop} The
toric variety ${\bf P}_{\Sigma}$ is the union of split algebraic
tori $T_{\sigma}$ $($${\rm dim}\, T_{\sigma} = d - {\rm dim}\, \sigma$$)$:
\[ {\bf P}_{\Sigma} = \bigcup_{ \sigma \in \Sigma } T_{\sigma}. \]
For each $k$-dimensional cone $\sigma \in \Sigma{(k)}$,
$T_{\sigma}$ is the kernel of a homomorphism $T \rightarrow
({\bf G}_m(\overline{K})^k)$ defined by a ${\bf Z}$-basis of
the sublattice $N \cap N_{{\sigma},{\bf R}} \subset N$.
\end{prop}
\noindent
Let $\check{\sigma}$ denote the cone in $M_{\bf R}$ which is
dual to $\sigma$.
\begin{prop} The toric variety ${\bf P}_{\Sigma}$ has a
$T$-invariant open covering by affine subsets $U_{\sigma}$:
\[ {\bf P}_{\Sigma} = \bigcup_{ \sigma \in \Sigma} U_{\sigma} \]
where $U_{\sigma} = {\rm Spec}(\overline{K}) \lbrack M \cap \check{\sigma}
\rbrack$.
\end{prop}
\begin{dfn}
{\rm A continuous function $\varphi\; : \;
N_{\bf R} \rightarrow {\bf R}$ is called {\em $\Sigma$-piecewise linear}
if the restriction $\varphi_{\sigma}$ of $\varphi$ to every
cone $\sigma \in \Sigma$ is a linear function. It is
called {\em integral} if $\varphi(N) \subset {\bf Z}$.}
\end{dfn}
\begin{dfn}
{\rm For any integral $\Sigma$-piecewise linear function
$\varphi\; : \; N_{\bf R} \rightarrow {\bf R}$ and any cone $\sigma
\in \Sigma(d)$, we denote by
$m_{\sigma, \varphi}$ the restriction of $\varphi$ to $\sigma$
considered as an element in $M$. We put $m_{\sigma', \varphi} =
m_{\sigma, \varphi}$ if $\sigma'$ is a face of a $d$-dimensional cone
$\sigma \in \Sigma$. }
\end{dfn}
\begin{dfn}
{\rm For any integral $\Sigma$-piecewise linear function
$\varphi\; : \; N_{\bf R} \rightarrow {\bf R}$, we define the
invertible sheaf $L(\varphi)$ as the subsheaf of the constant sheaf
of rational functions on ${\bf P}_{\Sigma}$ generated over
$U_{\sigma}$ by the element $- m_{\sigma,\varphi}$ considered as
a character of $T \subset {\bf P}_{\Sigma}$. }
\end{dfn}
\begin{rem}
{\rm The $T$-action on the sheaf of rational functions
restricts to the subsheaf
$ L(\varphi)$ so that we can consider $ L(\varphi)$ as
a $T$-linearized line bundle over ${\bf P}_{\Sigma}$. }
\end{rem}
Denote by $e_1, \ldots, e_n$ the primitive integral generators
of all $1$-dimensional cones in $\Sigma$. Let $T_{i}$ $(i =1, \ldots, n)$
be the $(d-1)$-dimensional torus orbit corresponding to the cone
${\bf R}_{\geq 0}e_i \in \Sigma$ and $D_i$ the Zariski closure of $T_i$ in
${\bf P}_{\Sigma}$. Define ${\bf D}({\Sigma}) \cong {\bf Z}^n$
as the free abelian group of $T$-invariant Weil divisors on ${\bf
P}_{\Sigma}$ with the basis $D_1, \ldots, D_n$.
\begin{prop}
The correspondence $\varphi \rightarrow L(\varphi)$
gives rise to an isomorphism between the group of
$T$-linearized line bundles on ${\bf P}_{\Sigma}$ and
the group $PL(\Sigma)$ of all
$\Sigma$-piecewise linear integral
functions on $N_{\bf R}$. There is the canonical isomorphism
\[ PL(\Sigma) \cong {\bf D}(\Sigma), \;\;
\varphi \mapsto (\varphi(e_1), \ldots, \varphi(e_n)). \]
The Picard group ${\rm Pic}({\bf P}_{\Sigma})$
is isomorphic to $PL(\Sigma)/M$ where elements of $M$
are considered as globally
linear integral functions on $N_{\bf R}$, so that we have the
exact sequence
\begin{equation}
0 \rightarrow M \rightarrow {\bf D}(\Sigma) \rightarrow
{\rm Pic}({\bf P}_{\Sigma}) \rightarrow 0
\end{equation}
\end{prop}
\begin{dfn}
{\rm Let $\Lambda_{\rm eff}(\Sigma)$ be the cone in
${\rm Pic}({\bf P}_{ \Sigma })$ generated by classes of effective divisors
on ${\bf P}_{\Sigma}$. Denote by ${\Lambda}_{\rm eff}^*(\Sigma)$ the
dual to $\Lambda_{\rm eff}(\Sigma)$ cone.}
\end{dfn}
\begin{prop}
$\Lambda_{\rm eff}(\Sigma)$ is generated by the classes
$\lbrack D_1 \rbrack, \ldots, \lbrack D_n \rbrack$.
\label{generators}
\end{prop}
{\em Proof.} Any divisor $D$ on ${\bf P}_{\sigma}$ is
linearly equivalent to an integral
linear combination of $D_1, \ldots, D_n$. Assume that
$D = a_1 D_1 + \cdots + a_n D_n$ is effective. Then there exists a
rational function $f$ on ${\bf P}_{\Sigma}$ having no poles and zeros on
$T$ such that
\begin{equation}
(f) + D \geq 0.
\label{effective}
\end{equation}
We can assume that $f$ is character of $T$ defined by an element $m_f \in M$.
Then the condition (\ref{effective}) is equivalent to
\begin{equation}
b_i = \langle m_f, e_i \rangle + a_i \geq 0,\; i =1, \ldots, n
\end{equation}
Then $D' = b_1 D_1 + \cdots + b_n D_n$ is linearly equivalent to $D$.
So every effective class $\lbrack D \rbrack $ is a
non-negative integral linear combination of
$\lbrack D_1 \rbrack, \ldots, \lbrack D_n \rbrack$. \hfill $\Box$
\begin{prop}
Let $\varphi_{\Sigma}$ be the $\Sigma$-piecewise linear
integral function such that $\varphi(e_1) =
\cdots = \varphi(e_n)= 1$. Then $L(\varphi_{\Sigma})$ is
isomorphic to the $T$-linearized anticanonical line bundle
on ${\bf P}_{\Sigma}$.
\end{prop}
\begin{exam} {\sl Projective spaces}.
{\rm Consider a $d$-dimensional fan $\Sigma$
whose $1$-dimensional cones are generated by $d+1$ elements
$e_1, \ldots , e_{d}, e_{d+1} = -(e_1 + \cdots + e_{d})$,
where $\{ e_1, \ldots , e_{d} \}$
is a ${\bf Z}$-basis of ${d}$-dimensional lattice $N$, and
$k$-dimensional cones in $\Sigma$ are generated by all possible
$k$-element subsets in $\{ e_1,\ldots, e_{d+1} \}$. Then the corresponding
compactification ${\bf P}_{\Sigma}$ of the $d$-dimensional split torus is
${\bf P}^d$.}
\label{proj.space}
\end{exam}
\begin{rem}
{\rm It is easy to see that the combinatorial construction of toric
varieties ${\bf P}_{\Sigma}$
immediatelly extends to arbitrary fields $E$; i.e., using
a rational complete polyhedral fan $\Sigma$, one
can define the toric variety ${\bf P}_{\Sigma,E}$
as the equivariant compactification of the split torus
$({\bf G}_m(E))^d$. }
\end{rem}
\subsection{Compactifications of nonsplit tori}
Let $T$ be a $d$-dimensional algebraic torus over $K$
with a splitting field $E$ and
$G = {\rm Gal}\, (E/K)$. Denote by $M$ the lattice $\hat{T}$ and put
$N ={\rm Hom}\, (M, {\bf Z})$. Let
$\rho^*$ be the integral representation of $G$ in ${\rm GL}(N)$
which is dual to $\rho$.
In order to construct a projective compactification of $T$ over $K$,
we need a complete fan $\Sigma$ of cones
having an additional combinatorial structure: an {\em
action of the Galois group } $G$ \cite{vosk}:
\begin{dfn}
{\rm A complete fan $\Sigma \subset N_{\bf R}$ is called
{\em $G$-invariant} if for any $g \in G$ and for any $\sigma \in \Sigma$, one
has $\rho^*(g) (\sigma) \in \Sigma$. }
\label{opr.invar}
\end{dfn}
\begin{theo}
Let $\Sigma$ be a complete regular $G$-invariant fan in $N_{\bf R}$. Then
there exists a complete algebraic variety ${\bf P}_{\Sigma,K}$ over $K$
such that its base extension ${\bf P}_{\Sigma,K} \otimes_{{\rm Spec}\, K}
{\rm Spec}\, E$ is isomorphic to
the toric variety ${\bf P}_{\Sigma,E}$ defined over $E$ by $\Sigma$.
Let $\Sigma^G$ be the subset of all $G$-invariant cones $\sigma \in \Sigma$.
Then
\[ {\bf P}_{\Sigma}(K) = \bigcup_{\sigma \in \Sigma^G} T_{\sigma}(K), \]
where $T_{\sigma}$ is the $(d - {\rm dim}\, \sigma)$-dimensional
algebraic torus over $K$ corresponding to the restriction of
the integral $G$-representation in ${\rm GL}(M)$ to
the sublattice $(\hat{\sigma} \cap - \hat{\sigma}) \cap M \subset M$.
\label{decompos}
\end{theo}
Taking $G$-invariant elements in the short exact sequence
\[ 0 \rightarrow M \rightarrow {\bf D}(\Sigma) \rightarrow
{\rm Pic}({\bf P}_{\Sigma,E}) \rightarrow 0 \]
we obtain the exact sequence
\begin{equation}
0 \rightarrow M^G \rightarrow
{\bf D}(\Sigma)^G \rightarrow
{\rm Pic}({\bf P}_{\Sigma,E})^G \rightarrow H^1(G,M) \rightarrow 0
\label{short3}
\end{equation}
\begin{prop}
The group ${\rm Pic}({\bf P}_{\Sigma,E})^G$ is canonically
isomorphic to the Picard group ${\rm Pic}({\bf P}_{\Sigma,K})$. Moreover
$H^1(G, M)$ is the Picard group of $T$.
\end{prop}
\begin{coro}
The correspondence $\varphi \rightarrow L(\varphi)$
induces an isomorphism between the group of
$T$-linearized invertible sheaves on ${\bf P}_{\Sigma,K}$ and
the group $PL(\Sigma)^G$ of all
$\Sigma$-piecewise linear integral $G$-invariant
functions on $N_{\bf R}$. An invertible sheaf $ L$ on
${\bf P}_{\Sigma,K}$ admits a $T$-linearization if and only if
the restriction of $L$ on $T$ is trivial. In particular,
some tensor power of $ L$ always admits a $T$-linearization.
\end{coro}
\begin{coro}
Let $\Lambda_{\rm eff}(\Sigma,K)$ be the cone of effective divisors
of ${\bf P}_{\Sigma,K}$. Then $\Lambda_{\rm eff}(\Sigma,K)$ consists
of $G$-invariant elements in $\Lambda_{\rm eff}(\Sigma)$.
\end{coro}
\begin{coro}{\rm \cite{colliot1}}
Let ${\bf P}_{\Sigma,K}$ be a compactification of an
anisotropic torus $T$. Then
all $K$-rational points of ${\bf P}_{\Sigma,K}$ are contained in
$T$ itself.
\label{point.aniso}
\end{coro}
{\em Proof.} By \ref{decompos}, it is sufficient to prove
that for an anisotropic
torus $T$ defined by some Galois representation of $G$ in ${\rm GL}(M)$,
there is no $G$-invariant cone $\sigma$ of
positive dimension in $\Sigma$.
Assume that a $k$-dimensional cone $\sigma$ with the generators $\{ e_{i_1},
\dots , e_{i_k} \}$ is $G$-invariant. Then $ e_{i_1} + \cdots + e_{i_k}$ is a
nonzero $G$-invariant integral vector in the interior of $\sigma$.
Hence the sublattice $N^G$ of $G$-invariant
elements in $N$ has positive rank.
Thus $M^G \cong \hat{T}_K$ also has positive rank. Contradiction.
\hfill $\Box$
\begin{prop}
Let ${\bf P}_{\Sigma,K}$ be a a compactification of an anisotropic torus
$T$. Then the cone of effective divisors
${\Lambda}_{\rm eff}({\Sigma,K})$ is
simplicial. The rank of the Picard group ${\rm Pic}({\bf P}_{\Sigma,K})$
equals to the number of $G$-orbits in $\Sigma(1)$.
\label{simp}
\end{prop}
{\em Proof.} Let $A_1({\bf P}_{\Sigma})$ be the group
of $1$-cycles on ${\bf P}_{\Sigma,E}$
modulo numerical equivalence. We identify $A_1({\bf P}_{\Sigma})$
with the dual to ${\rm Pic}({\bf P}_{\Sigma})$ group. Consider
the dual cone ${\Lambda}^*_{\rm eff}({\Sigma,K})$.
Since ${\Lambda}^*_{\rm eff}({\Sigma,K}) =
{\Lambda}^*_{\rm eff}({\Sigma})^G$, by
\ref{generators}, ${\Lambda}^*_{\rm eff}({\Sigma,K})$
consists of non-negative $G$-invariant
${\bf R}$-linear relations among primitive generators of $\Sigma(1)$.
Let
\[ \Sigma(1) = \Sigma_1(1) \cup \ldots \cup \Sigma_l(1) \]
be the decomposition of $\Sigma(1)$ into a union of $G$-orbits.
Then every $G$-invariant linear relation among the primitive
generators $e_1 , \ldots , e_n$ of the
$1$-dimensional cones has the form
\[ \sum_{1 \leq i \leq l }
\lambda_i \left( \sum_{ \sigma_j \in \Sigma_i(1) }
e_j \right) = 0\;\;\;(\sigma_j = {\bf R}_{\geq 0} e_j). \]
For every $i$ ($1 \leq i \leq l$), the sum
\[ \sum_{ \sigma_j \in \Sigma_i(1) } e_j \]
is a $G$-invariant element of the lattice $N$.
Since $T$ is anisotropic, $N^G = 0$ and all sums
$\sum_{ \sigma_j \in \Sigma_i(1) } e_j $ must be equal to zero. These
integral relations give rise to a ${\bf Z}$-basis
$r_1, \dots , r_l$ of the group of
integral linear relations among $e_1 , \ldots , e_n$.
Thus
$A_1 ({\bf P}_{\Sigma})^G_{\bf R}$ is isomorphic to ${\bf Z}^l$ and the cone
$\Lambda_{\rm eff}^*({\Sigma,K})$
consists of nonnegative linear combinations of
$r_1, \dots , r_l$. So the cone $\Lambda_{\rm eff}({\Sigma,K})$ is also
an $l$-dimensional simplicial cone in
${\rm Pic}({\bf P}_{\Sigma,K}) \otimes {\bf R}$.
\hfill $\Box$
Below we consider several examples of compactifications of anisotropic tori.
\begin{exam}
{\rm Consider a $d$-dimensional fan $\Sigma$ as in \ref{proj.space}.
It has a natural action
of the symmetric group $S_{d+1}$. Let $E$ be
a Galois extension of $K$ such that the Galois group
${\rm Gal}\, (E/K)$ is a subgroup of $S_{d+1}$ (for instance,
$E$ is a simple algebraic extension defined by an
$K$-irreducible polynomial $f$).
Then the action of $G$ on
$\Sigma$ defines a ${d}$-dimensional toric variety
${\bf P}_{\Sigma,K}$
which over $E$ is isomorphic to $d$-dimensional
projective space; i.e. ${\bf P}_{\Sigma,K}$ is a
Severi-Brauer variety. In particular, if $E = K(f)$, then
${\bf P}_{\Sigma,K}$ is a compactification of the ${d}$-dimensional
anisotropic torus $R_{E/K}({\bf G}_m)/K^*$.
Since ${\bf P}_{\Sigma,K}$ contains infintely many $K$-rational points,
${\bf P}_{\Sigma,K}$ is in fact isomorphic to ${\bf P}^{d}$ over $K$. }
\label{exam.pn}
\end{exam}
\begin{exam}
{\rm A complete fan $\Sigma$ is called {\em centrally symmetric} if it is
invariant under the map $-Id$ of $N_{\bf R}$.
Let $\Sigma$ be a centrally symmetric $4$-dimensional
fan and let $E$ be an exension of $K$ of degree 2.
The $d$-dimensional torus $T$
corresponding to the integral representation of
${\rm Gal}(E/K) \cong {\bf Z}/2{\bf Z}$
by $Id$ and $-Id$ is isomorphic to the anisotropic torus
$(R^1_{E/K})^d$. The ${\bf Z}/2{\bf Z}$-invariant fan $\Sigma$
defines the compactification ${\bf P}_{\Sigma,K}$ of $(R^1_{E/K})^d$. }
\label{two.aniso}
\end{exam}
\begin{exam}
{\rm Let $K'$ be a cubic extension of a number field $K$.
We construct a smooth compactification
of the $2$-dimensional anisotropic $K$-torus $R^1_{K'/K}({\bf G}_m)$ as
follows.
Let $Y$ be the cubic surface in ${\bf P}^3$ defined by the equation
\[ N_{K'/K}(z_1,z_2,z_3) = z_0^3 \]
where $ N_{K'/K}(z_1,z_2,z_3)$ is the homogeneous cubic norm-form. Over
the algebraic closure $\overline{K}$ it is
isomorphic to the singular cubic surface $z_1 z_2 z_3 = z_0^3$.
The $3$ quadratic
singular points $p_1, p_2, p_3 \in Y_{\overline{K}}$ are
defined over a splitting field $E$ of $R^1_{K'/K}({\bf G}_m)$ and the Galois
group $G = {\rm Gal}\, K'/K$ acts on $\{ p_1, p_2, p_3 \}$ by permutations.
There exists a minimal simultaneous resolution $\psi\; : Y' \rightarrow Y$ of
singularities which is defined over $K$. By
contraction $\psi' : Y' \rightarrow X$ of the proper pull-back of three
$(-1)$ curves which are preimages of lines passing through the singular points
we obtain a
Del Pezzo surface $X$ of anticanonical degree $6$ which is a smooth
compactification of the anisotropic torus $R^1_{K'/K}({\bf G}_m)$. }
\end{exam}
Let $k$ be a finite
field of characteristic $p$ containing $q = p^n$
elements. Any finite extension $k'$ of $k$ is a cyclic Galois extension
and the group $G = {\rm Gal }(k'/k)$ is generated by the Frobenius
automorphism $\phi \; : \; z \rightarrow z^q$. By ~\ref{represent}, any
$d$-dimensional
algebraic torus $T$ over $k$ splitting over $k'$ is uniquely defined
by the conjugacy class
in ${\rm GL} (d, {\bf Z})$
of the integral matrix
\[ \Phi = \rho ( \phi ) .\]
\noindent
The characteristic polynomial of the matrix $\Phi$ gives the following
formula obtained by T. Ono \cite{ono1} for the number of
$k$-rational points in $T$ :
\begin{theo}
Let $T$ be a $d$-dimensional algebraic torus defined over
a finite field $k$. In the above notations, one has
the following formula for the number of $k$-rational points of $T$:
\[ {\rm Card} \lbrack T(k) \rbrack = (-1)^d{\rm det} ( \Phi - q\cdot Id ). \]
\label{fin.tori}
\end{theo}
\begin{prop}
Let ${\bf P}_{\Sigma}$ be a toric variety over a finite field $k$ defined by
a $\Phi$-invariant fan $\Sigma \subset N_{\bf R}$. For any $\Phi$-invariant
cone $\sigma \in \Sigma^G$, let $M_{{\bf R}, \sigma} =
\check{\sigma} \cap (- \check{\sigma})$ be the maximal linear subspace
in the dual cone $\check{\sigma}
\subset M_{\bf R}$. Let $\Phi_{\sigma}$ be the restriction of
$\Phi$ on $M_{{\bf R},\sigma}$. Then
\[ {\rm Card}\lbrack {\bf P}_{\Sigma}(k)\rbrack = \sum_{\sigma \in \Sigma^G}
(-1)^{{\rm dim}\,\sigma} {\rm det}(\Phi_{\sigma} - q \cdot Id). \]
\label{point.var}
\end{prop}
\medskip
{\em Proof.} By \ref{decompos},
\[ {\bf P}_{\Sigma} (k) = \bigcup_{\sigma \in \Sigma^G} T_{\sigma}(k). \]
Observe that $k'$ is a splitting field for every algebraic torus
$T_{\sigma,k}$ defined by the $\rho$-action of $\Phi_{\sigma}$.
Now the statement follows from \ref{fin.tori}.
\hfill $\Box$
\subsection{Algebraic tori over local and global
fields}
First we fix our notations.
Let ${\rm Val}(K)$ be the set of all valuations of a global field $K$.
For any $v \in {\rm Val}(K)$, we denote by $K_v$ the completion of
$K$ with respect to $v$.
Let $v$ be a non-archimedian absolute valuation of a number field $K$ and
$E$ a
finite Galois extension of $K$.
Let ${\cal V}$ be an extension of $v$ to $E$,
$E_{\cal V}$ the completion
of $E$ with respect to ${\cal V}$. Then
\[ {\rm Gal}(E_{\cal V}/ K_v ) \cong G_v \subset G, \]
where $G_{v}$ is the decomposition subgroup of
$G$ and $ K_v \otimes_K E \cong \prod_{{\cal V} \mid v} E_{\cal V}. $
Let $T$ be an algebraic torus over $K$ with the splitting field $E$.
Denote by
$T_{K_v}=T\otimes K_v$.
\begin{dfn}
{\rm We denote the group of characters $\hat{T}_{K_v} = M^{G_v}$ by
$M_v$ and the dual group ${\rm Hom}(\hat{T}_{K_v} , {\bf Z})
= N^{G_v}$ by $N_v$. }
\end{dfn}
\noindent
Let ${(K_v \otimes_K E)}^*$ and $E^*_{\cal V}$ be the multiplicative
groups of $K_v \otimes_K E$ and $E_{\cal V}$ respectively. One has
\[ T_{K_v} = {\rm Hom}_G (\hat{T}, {(K_v \otimes_K E )}^* ) =
{\rm Hom}_{G_v} (M, E_{\cal V}^* ). \]
Denote by ${\cal O}_{\cal V}$
the maximal compact subgroup in $E^*_{\cal V}$.
There is a short exact sequence
\[ 1 \rightarrow
{\cal O}_{\cal V} \rightarrow E_{\cal V}^* \rightarrow {\bf Z}
\rightarrow 1, \;\; b \rightarrow {\rm ord} \mid b \mid_{\cal V}. \]
\noindent
Denote by $T({\cal O}_v)$ the maximal compact subgroup in $T(K_v)$.
Applying the functor
${\rm Hom}_{G_v}(M_v, * )$ to the short
exact sequence above, we obtain the
short exact sequence
\[ 1 \rightarrow N_v \otimes {\cal O}_{\cal V} \rightarrow
N_v \otimes E_{\cal V}^* \rightarrow
N_v \rightarrow 1 \]
which induces an injective homomorphism
\[ \pi_v\; : \; T(K_v) / T({\cal O}_v)
\hookrightarrow N_v = N^{G_v}. \]
\begin{prop} {\rm \cite{drax1}}
The homomorphism $\pi_v$ has finite cokernel. Moreover, $\pi_v$ is
an isomorphism if $E$ is unramified in $v$.
\label{pi-image}
\end{prop}
\begin{dfn}
{\rm Let $S$ be a finite subset of ${\rm Val}(K)$ containing
all archimedian and ramified non-archimedian valuations of $K$.
We denote by $S_{\infty}$ the set of all archimedian
valuations of $K$ and put $S_0 = S \setminus S_{\infty}$. }
\end{dfn}
\medskip
Now we assume that $v$ is an archimedian absolute valuation, i.e., $K_v$ is
${\bf R}$ or ${\bf C}$. It is known that any torus over
${\bf R}$ is isomorphic to the product of some copies of ${\bf C}^*$,
${\bf R}^*$, or $S^1 = \{ z \in {\bf C} \mid z \overline{z} =1 \}$. The
quotient $T(K_v) / T({\cal O}_v)$ is isomorphic to the ${\bf R}$-linear
space $N_v \otimes {\bf R}$.
The homomorphism $T(K_v) \rightarrow T(K_v) / T({\cal O}_v)$
is simply the logarithmic
mapping onto the Lie algebra of $T(K_v)$. Hence, we obtain:
\begin{prop}
For any archimedian absolute valuation $v$, the
quotient $T(K_v) / T({\cal O}_v)$ can be canonically identified with
the real Lie
algebra of $T(K_v)$ embedded in the $d$-dimensional ${\bf R}$-subspace
$N_{\bf R}$.
\end{prop}
\medskip
\begin{dfn}
{\rm Denote by $T({\bf A}_K)$ the adele group of $T$, i.e., the
restricted topological product
\[ {\prod_{v \in {\rm Val}(K)}} T(K_v) \]
consisting of all elements ${\bf t} = \{ t_v \} \in
\prod_{v \in {\rm Val}(K)}T(K_v)$ such that $t_v \in T({\cal O}_v)$ for
almost
all $v \in {\rm Val}(K)$.
Let
\[T^1({\bf A}_K) = \{ {\bf t} \in T({\bf A}_K) \mid
\prod_{v \in {\rm Val}(K)}
\mid m(t_v) \mid_v = 1, \; {\rm for \; all}\; m \in \hat{T}_K \subset M \}.
\]
We put also
\[ {\bf K}_T = \prod_{v \in {\rm Val}(K)} T({\cal O}_v), \]
}
\end{dfn}
\begin{prop} {\rm \cite{ono1}}
The groups $T({\bf A}_K)$, $T^1({\bf A}_K)$, $T(K)$,
${\bf K}_T$ have the following properties which are generalizations
of the corresponding properties of the adelization of ${\bf G}_m(K)$:
{\rm (i)} $T({\bf A}_K)/T^1({\bf A}_K) \cong {\bf R}^k$, where $k$ is the rank
of $\hat{T}_K$;
{\rm (ii)} $T^1({\bf A}_K)/T(K)$ is compact;
{\rm (iii)} $T^1({\bf A}_K)/ {\bf K}_T \cdot T(K) $
is isomorphic to the direct product of a finite group
${\bf cl}(T_K)$ $($this
is an analog of the idele-classes group ${Cl}(K)$$)$ and
a connected compact abelian topological group which dimension
equals the rank $r'$ of the group of ${\cal O}_K$-units in $T(K)$
$($this rank equals $r_1 + r_2 -1$ for ${\bf G}_m$$)$;
{\rm (iv)} $W(T) = {\bf K}_T \cap T(K)$ is a finite
group of all torsion elements in $T(K)$ $($this is the analog of the group
of roots of unity in ${\bf G}_m(K)$$)$.
\label{subgroups}
\end{prop}
The following theorem of A. Weil plays a fundamental role in the
definition of adelic measures on algebraic varieties.
\begin{theo} {\rm \cite{peyre,weil1} } Let $X$ be an $d$-dimensional
smooth algebraic variety over a global field $K$. Denote by
${\cal K}$ the canonical sheaf on $X$ with a family of local metrics
$\|\cdot \|_v $. Then these local metrics uniquely define
natural $v$-adic measures $\omega_{{\cal K},v}$ on $X(K_v)$.
Let $U \subset X$ be a Zariski open
subset of $X$. Then for almost
all $v \in {\rm Val}(K)$ one has
\[ \int_{U({\cal O}_v)} \omega_{{\cal K},v} =
\frac{{\rm Card} \lbrack U(k_v) \rbrack}{q^d_v}, \]
where $k_v$ is the residue field of $K_v$ and $q_v = {\rm Card}
\lbrack k_v \rbrack$.
\end{theo}
\begin{rem}
{\rm We notice that the structure sheaf ${\cal O}_X$ of any
algebraic variety $X$ has a natural metrization defined by
$v$-adic valuations of the field $K$.
If $X = {\cal G}$ is an algebraic group,
then there exists a natural
way to define a metrization of the canonical sheaf ${\cal K}$ on
${\cal G}$ by choosing a ${\cal G}$-invariant algebraic differential $d$-form
$\Omega$. Such a form defines an isomorphism of ${\cal K}$ with
the structure sheaf ${\cal O}_{\cal G}$. We denote the corresponding
local measure on ${\cal G}(K_v)$ by $\omega_{\Omega,v}$.}
\label{l.measures}
\end{rem}
Let $T$ be a $d$-dimensional torus over $K$ with a splitting field $E$.
Take a $T$-invariant
differential $d$-form $\Omega$ on $T$ (it is unique up to a constant
from $K$). According to A. Weil (\ref{l.measures}),
we obtain a family of local measures $\omega_{\Omega,v}$ on $T(K_v)$.
\begin{dfn} {\rm \cite{ono1} Let
\[ L_S(s, T;E/K) = \prod_{v \not\in S} L_v(s, T ;E/K) \]
be the Artin $L$-function corresponding to the representation
\[ \rho \; :\; G= {\rm Gal}(E/K) \rightarrow {\rm GL}(M) \]
and a finite set $S \subset {\rm Val}(K)$ containing all
archimedian valuations and all non-archimedian valuations of $K$ which are
ramified in $E$.
By definition, $L_v(s,T;E/K) \equiv 1$ if $v \in S$. The numbers
\[ c_v = L_v(1, T; E/K) = \frac{1}{{\rm det}(Id - q^{-1}_v \Phi_v)},
\; v \not\in S \]
are called {\em canonical correcting factors} for measures
$\omega_{\Omega,v}$
($\Phi_v$ is the $\rho$-image of a local Frobenius element in
$G$).
}
\end{dfn}
By \ref{fin.tori}, one has
\[ c_v^{-1} = \int_{T({\cal O}_v)} \omega_{\Omega,v} =
\frac{{\rm Card} \lbrack T(k_v)\rbrack}{q^d_v}, \; \; v \not\in S. \]
Let $d\mu_v = c_v \omega_{\Omega,v}$. We put $c_v =1$ for $v \in S$. Since
\[ \int_{T({\cal O}_v)} d\mu_v = 1 \]
for $v \not\in S$, the $\{ c_v \}$ defines
the canonical measure
\[ \omega_{\Omega,S} = \prod_{v \in {\rm Val}(K)} d\mu_v \]
on the adele group $T({\bf A}_K)$. By the product formula,
$\omega_{\Omega,S}$ does
not depend on the choice of $\Omega$.
Let $dx$ be the standard Lebesgue measure on $T({\bf A}_K)/T^1({\bf A}_K)
\cong {\bf R}^k = M_{\bf R}^G$. There exists a unique Haar measure
$\omega^1_{\Omega,S}$ on $T^1({\bf A}_K)$ such that $\omega^1_{\Omega,S}dx =
\omega_{\Omega,S}$.
\begin{dfn}
{\rm The {\em Tamagawa number of } $T_K$ is defined as
\[ \tau(T_K) = \frac{b_S(T_K)}{l_S(T_K)} \]
where
\[ b_S(T_K) = \int_{T^1({\bf A}_K)/T(K)} \omega^1_{\Omega,S} , \]
\[ l_S(T_K) = \lim_{s \rightarrow 1} (s-1)^k L_S(s, T; E/K). \]}
\label{tamagawa1}
\end{dfn}
\begin{rem}
{\rm Although the numbers $b_S(T_K)$ and $l_S(T_K)$ do depend on the choice
of the finite subset $S \subset {\rm Val}(K)$, the Tamagawa number
$\tau(T_K)$ does not depend on $S$.}
\end{rem}
\begin{theo} {\rm \cite{ono1,ono2} }
The Tamagawa number $\tau(T)$ of $T$ does not depend on the
choice of a splitting field $E$. It satisfies the following properties:
{\rm (i)} $\tau({\bf G}_m(K)) =1;$
{\rm (ii)} $\tau(T \times T') = \tau(T) \cdot \tau(T') $
where $T'$ and $T$ are tori over $K$;
{\rm (iii)} $\tau_K (R_{K'/K}(T)) = \tau_{K'}(T)$ for any
torus $T$ over $K'/K$.
Moreover, $\tau(T)$ is the ratio of two positive
integers
\[ h(T_K) = {\rm Card} \lbrack H^1(G, M) \rbrack \]
and $i(T_K) = {\rm Card}\lbrack {\rm III}(T) \rbrack$ where
\[ {\rm III}(T) = {\rm Ker}\, \lbrack
H^1(G, T(K)) \rightarrow \prod_{v} H^1(G_v, T(K_v)) \rbrack; \]
in particular, $\tau(T_K)$ is a rational number.
\label{tamagawa2}
\end{theo}
\begin{dfn}
{\rm
Let $\overline{T(K)}$ be the closure of $T(K)$ in $\prod_vT(K_v) $ in the
direct product topology. Define the {\em
obstruction group to weak approximation} as
\[ A(T)= \prod_v T(K_v)/\overline{T(K)}. \] }
\end{dfn}
\begin{theo} {\rm \cite{sansuc}}
Let ${\bf P}_{ \Sigma }$ be a complete smooth toric variety over $K$.
There is an exact sequence:
\[ 0\rightarrow A(T) \rightarrow Hom (H^1(G,{\rm Pic}({\bf P}_{ \Sigma ,E})),{\bf Q }/{\bf Z })\rightarrow {\rm
III}(T)\rightarrow
0.
\]
The group $H^1(G,{\rm Pic}({\bf P}_{ \Sigma ,E}))$ is canonically isomorphic to
${\rm Br}({\bf P}_{\Sigma,K})/{\rm Br}(K)$, where
${\rm Br}({\bf P}_{\Sigma,K}) = H^2_{\rm et}({\bf P}_{\Sigma,K}, {\bf G}_m)$.
\label{weak}
\end{theo}
\begin{coro}
Denote by $\beta({\bf P}_{\Sigma})$ the cardinality of
$H^1(G,{\rm Pic}({\bf P}_{ \Sigma ,E}))$. Then
\[ {\rm Card} \lbrack A(T) \rbrack =
\frac{\beta({\bf P}_{\Sigma})}{i(T_K)}. \]
\label{weak1}
\end{coro}
\section{Heights and their Fourier transforms}
\subsection{Complexified local Weil functions and heights}
A theory of heights on an algebraic variety $X$ defined over a number field $K$
is
the unique functorial homomorphism from ${\rm Pic}(X) $ to equivalence
classes of functions
$X(K)\rightarrow {\bf R}_{\ge 0}$
which on metrized line bundles ${\cal L}$ is given by the formula
$$
H_{{\cal L}}(x) = \prod_v \| f(x)\|^{-1}_v
$$
where $f$ is a rational section of $L$ not vanishing in $x\in X(K)$.
Two functions are equivalent if
they differ by a bounded on $X(K)$ function.
For our purposes it will be convenient to extend these
notions to the complexified
Picard group ${\rm Pic}(X)\otimes {\bf C}$.
Let ${\bf P}_{\Sigma}$
be a compact toric variety over a global field $K$.
We define a canonical
compact covering of ${\bf P}_{\Sigma}(K_v)$
by compact subsets ${\bf C}_{\sigma,v} \subset U_{\sigma}(K_v)$.
For this purpose we identify lattice elements $m \in M$ with characters
of $T$ and define the compact subset ${\bf C}_{\sigma,v} \subset
U_{\sigma}(K_v)$ as follows
\[ {\bf C}_{\sigma,v} = \{ x_v \in U_{\sigma}(K_v) \mid \| m(x_v) \|_v \leq 1\;
{\rm for}\; {\rm all}\; m \in M^{G_v}
\cap \check{\sigma} \}. \]
\begin{prop}
The compact subsets ${\bf C}_{\sigma,v}$ $(\sigma \in \Sigma)$ form a compact
covering of ${\bf P}_{\Sigma}(K_v)$ such that for any two cones
$\sigma, \sigma' \in \Sigma$ one has
\[ {\bf C}_{\sigma,v} \cap {\bf C}_{\sigma',v} = {\bf C}_{\sigma \cap
\sigma',v}. \]
\end{prop}
{\em Proof.} The last property of the compact subsets ${\bf C}_{\sigma,v}$
follows
immediatelly from their definition. Since the $T(K_v)$-orbit
of maximal dimension is dense in ${\bf P}_{\Sigma}(K)$, it is sufficient
to prove
that the compacts ${\bf C}_{\sigma,v}$ cover $T(K_v)$.
Let $x_v \in T(K_v)$. Denote by $\overline{x_v}$ the image of
$x_v$
in $T(K_v) / T({\cal O}_v) \subset N_{\bf R}$.
By completeness of the fan $\Sigma$, the point $-\overline{x_v}$ is
contained in some cone $\sigma \in \Sigma$.
Hence $x_v \in {\bf C}_{\sigma,v}$.
\hfill $\Box$
\bigskip
Now we define canonical metrizations of $T(K_v)$-linearized
line bundles on ${\bf P}_{\Sigma}(K_v)$.
\medskip
Let $ L(\varphi)$ be a line bundle on ${\bf P}_{\Sigma}( K_v)$
corresponding to a $\Sigma$-piecewise linear integral $G_v$-invariant
function $\varphi$ on $N_{\bf R}$.
\begin{prop}
Let $f$ be a rational section of $ L(\varphi)$. We define the $v$-norm
of $f$ at a point $x_v \in {\bf P}_{\Sigma}(K_v)$ as
\[ \| f(x_v) \|_v = \mid \frac{f(x_v)}{m_{\sigma,\varphi}(x_v)} \mid_v \]
where $\sigma$ is a cone in $\Sigma$ such that $x_v \in {\bf C}_{\sigma,v}$
and $m_{\sigma,\varphi} \in M$ is the restriction of $\varphi$ on $\sigma$.
Then this $v$-norm defines a $T({\cal O}_v)$-invariant
$v$-adic metric on $ L(\varphi)$.
\end{prop}
{\em Proof.} The statement follows from the fact that
\[ \mid m_{\sigma,\varphi}(x_v) \mid_v = \mid m_{\sigma',\varphi}(x_v) \mid_v
\]
if $x_v \in {\bf C}_{\sigma,v} \cap {\bf C}_{\sigma',v}$.
\hfill $\Box$
A family of local metrics on all
$T$-linearized line bundles on ${\bf P}_{\Sigma}$ corresponding
to $\Sigma$-piecewise linear $G$-invariant
functions $\varphi \in PL(\Sigma)^G$ uniquely determines
a family of local Weil functions on $({\bf P}_{\Sigma})$
corresponding to $T$-invariant divisors
\[ D_{\varphi} = \varphi(e_1)D_1 + \cdots + \varphi(e_n)D_n. \]
We extend these local Weil functions to
the group of $T$-invariant Cartier divisors with {\em complex
coefficients} as follows.
\begin{dfn}
{\rm
A $T$-invariant ${\bf C}$-Cartier divisor is a formal linear combination
$D_s = s_1 D_1 + \cdots + s_n D_n$, with $s=(s_1,\ldots,s_n)\in {\bf C}^n$ or
equivalently
a complex piecewise linear function
$\varphi$ in $PL(\Sigma)^G_{\bf C}$ having the property
$\varphi(e_i) = s_i$ $(i = 1, \ldots ,n)$. }
\end{dfn}
\begin{dfn}
{\rm Let $\varphi \in PL(\Sigma)^G_{\bf C}$.
For any point $x_v \in T(K_v) \subset {\bf P}_{\Sigma}(K_v)$,
denote by $\overline{x_v}$ the image of
$x_v$ in $N_v$ (resp. $N_v\otimes {\bf R}$ for archimedian valuations), where
$N_v$ is considered
as a canonical lattice in the real space $N_{\bf R}$.
Define the {\em complexified
local Weil function}
$H_{ \Sigma ,v}(x_v, \varphi)$ by the formula
\[H_{ \Sigma ,v}(x_v,\varphi) = e^{\varphi(\overline{x_v})\log q_v }\]
where $q_v$ is the cardinality of the residue field
$k_v$ of $K_v$ if $v$ is non-archimedian and $\log q_v = 1$ if
$v$ is archimedian. }
\end{dfn}
\begin{prop}
The complexified local Weil function $H_{ \Sigma ,v}(x_v, \varphi)$ satisfies the
following properties:
{\rm (i)} If $s_i = \varphi(e_i) \in {\bf Z}^n$ $( i =1, \ldots, n)$,
then $H_{ \Sigma ,v}(x_v,\varphi)$ is a classical local Weil
function $H_{L(\varphi),v}(x_v)$ corresponding to
a $T$-invariant Cartier divisor
\[ D_s = s_1 D_1 + \cdots + s_n D_n \]
on ${\bf P}_{\Sigma}$.
{\rm (ii)} $H_{ \Sigma ,v}(x_v,\varphi)$ is $T({\cal O}_v)$-invariant.
{\rm (iii)} $H_{ \Sigma ,v}(x_v, \varphi + \varphi') =
H_{ \Sigma ,v}(x_v,\varphi)H_{ \Sigma ,v}(x_v,\varphi')$.
\label{local.f}
\end{prop}
\begin{dfn}
{\rm Let $\varphi \in PL(\Sigma)^G_{\bf C}$. We define the {\em complexified
height function on ${\bf P}_{\Sigma,K}$} by
\[ H_{\Sigma}(x,\varphi) = \prod_{v \in {\rm Val}(K)} H_{ \Sigma ,v}(x,\varphi). \]}
\end{dfn}
\begin{rem}
{\rm Although all local factors $H_{ \Sigma ,v}(x,\varphi)$ of
$H_{\Sigma}(x,\varphi)$ are functions on
$PL(\Sigma)_{\bf C}^G$, by the product formula,
the global complex height function $H_{\Sigma}(x,\varphi)$
depends only on the class of
$\varphi \in PL(\Sigma)_{\bf C}^G$ modulo
complex global linear $G$-invariant functions on $N_{\bf C}$,
i.e., $H_{\Sigma}(x,\varphi)$ depends only
on the class of $\varphi$ in ${\rm Pic}({\bf P}_{\Sigma,K})\otimes {\bf C}$. }
\end{rem}
\begin{dfn}
{\rm We define the zeta-function of the complex
height-function $H_{\Sigma}(x, \varphi)$ as
\[ Z_{\Sigma}(\varphi) = \sum_{x \in T(K)} H_{\Sigma}(x,-\varphi). \]}
\end{dfn}
\begin{rem}
{\rm One can see that the series $Z_{\Sigma}(\varphi)$
converges absolutely and uniformly in the domain
${\rm Re}(\varphi(e_j)) \gg 0$ for all $j$.
Since $H_{\Sigma}(x,\varphi)$ is the product of the local complex
Weil functions $H_{ \Sigma ,v}(x,\varphi)$ and
$H_{ \Sigma ,v}(x,\varphi) = 1$ for almost all $v$ ($x \in T(K)$),
we can immediately
extend $H_{\Sigma}(x,\varphi)$ to a function on the adelic group
$T({\bf A}_K)$.}
\end{rem}
\subsection{Fourier transforms of non-archimedian
heights}
\begin{dfn}
{\rm Let $\Sigma$ be a complete regular fan of cones in $N$
whose 1-dimensional cones are generated by
$e_1, \ldots, e_n$. We establish a
one-to-one correspondence between $e_1, \ldots, e_n$ and $n$
independent variables $z_1, \ldots, z_n$.
The {\em Stanley-Reisner ring} $R(\Sigma)$ is defined as the factor of the
polynomial ring $A[z]= {\bf C} [z_1, \ldots, z_n ]$ by the ideal
$I(\Sigma)$ generated by all monomials
$z_{i_1} \ldots z_{i_k}$ such that
$e_{i_1}, \ldots, e_{i_k}$ are not generators of a $k$-dimensional
cone in $\Sigma$.}
\label{opr.stenley}
\end{dfn}
\begin{prop}
There is a natural identification between the the elements of
the lattice $N$ and the monomial {\bf C}-basis of the ring
$R(\Sigma)$.
\label{prop.basis}
\end{prop}
{\em Proof.} Every integral point $x \in N$ belongs to the interior of
a unique cone $\sigma \in \Sigma$.
Let $e_{i_1}, \ldots, e_{i_k}$ be an integral basis of $\sigma$. Then
there exist positive integers $a_1, \ldots, a_k$ such that
\[ \overline{x} = a_1e_{i_1} + \cdots a_k e_{i_k}. \]
Therefore, $\overline{x}$ defines the
monomial $m(\overline{x}) = z_{i_1}^{a_1} \cdots z_{i_k}^{a_k}$.
By definition, $m(\overline{x}) \notin I(\Sigma)$.
It is clear that $I(\Sigma)$ has
a monomial {\bf C}-basis. Hence, we have
constructed a mapping $\overline{x} \rightarrow m(\overline{x})$
from $N$ to
the monomial basis of $R(\Sigma)$.
It is easy to see that this mapping is bijective.
\hfill $\Box$
\bigskip
Now choose a valuation $v \not\in S$. Then we obtain
a cyclic subgroup $G_v = \langle \Phi_v \rangle \subset G$
generated by a lattice automorphism
$\Phi_v \,:\, N \rightarrow N$ representing
the local Frobenius element at place $v$.
Then $\Sigma(1)$ splits into a disjoint union of $G_v$-orbits
\[ \Sigma(1) = \Sigma_1(1) \cup \cdots \cup \Sigma_l(1). \]
Let $d_j$ be the length of the $G_v$-orbit $ \Sigma _j(1)$.
One has
\[ \sum_{ i =1}^{l} d_j = n. \]
\begin{dfn}
{\rm Define the ${\bf Z}^l{\geq 0}$-grading of the
polynomial ring $A[z] = {\bf C} \lbrack z_1, \ldots, z_n \rbrack$
and the Stanley-Reisner ring $R(\Sigma)$
by the decomposition of the set of variables $\{z_1, \ldots, z_n\}$
into the disjoint union of $l$ sets $Z_1 \cup \cdots \cup Z_l$
which is induced by the decomposition of
$ \Sigma (1)$ into $G_v$-orbits. The standard
${\bf Z}_{\geq 0}$-grading of the
polynomial ring $A[z] = {\bf C} \lbrack z_1, \ldots, z_n \rbrack$
and the Stanley-Reisner ring $R(\Sigma)$ will be
called the {\em total grading}.
}
\end{dfn}
\begin{dfn}
{\rm We define the power series
$P(\Sigma, \Phi_v; t_1, \ldots, t_l)$ by the formula
\[ P(\Sigma,\Phi_v; t_1, \ldots, t_l) = \sum_{(i_1,\ldots, i_l) \in
{\bf Z}_{{\geq 0}}^l} ({\rm Tr}\, \Phi_v^{i_1, \ldots, i_l})
t_1^{i_1} \cdots t_l^{i_l}, \]
where $\Phi_v^{i_1, \ldots, i_l}$ is the linear operator
induced by $\Phi_v$
on the
homogeneous $(i_1, \ldots, i_l)$-component of $R(\Sigma)$. }
\end{dfn}
\begin{prop}
One has
\[ P(\Sigma,\Phi_v; t_1, \ldots, t_l) = \frac{Q_{ \Sigma }(t_1^{d_1}, \ldots,
t_l^{d_l})}
{(1- t_1^{d_1}) \cdots (1 - t_l^{d_l}) } \]
where
$Q_{ \Sigma }(t_1^{d_1}, \ldots, t_l^{d_l})$ is a
polynomial in $t_1^{d_1}, \ldots, t_l^{d_l}$
having the total degree $n$ such its all
nonconstant monomials have the total degree
at least $2$.
\label{p-function}
\end{prop}
{\em Proof.}
Since ${\rm dim} A \lbrack z \rbrack - {\rm dim} R(\Sigma) = n-d$,
there exists the minimal
${\bf Z}^l_{\geq 0}$-graded free resolution
\[ 0 \rightarrow F^{n-d} \rightarrow \cdots \rightarrow F^1 \rightarrow F^0 = A \lbrack z \rbrack \rightarrow
R(\Sigma) \rightarrow 0 \]
of the Stanley-Reisner ring $R(\Sigma)$ considered as a module over the
polynomial ring $A \lbrack z \rbrack$.
Let $\overline{\Phi}_v^{\,i_1, \ldots, i_l}$ be the linear operator
on the homogeneous $(i_1, \ldots, i_l)$-component of $A\lbrack z
\rbrack$ induced by the action of $\Phi_v$ on $z_1, \ldots, z_n$. Then
\[ \sum_{(i_1,\ldots, i_l) \in
{\bf Z}_{{\geq 0}}^l} ({\rm Tr}\, \overline{\Phi}_v^{\,i_1, \ldots, i_l})
t_1^{i_1} \cdots t_r^{i_l} = \frac{1}{\prod_{j = 1}^l (1 - t_j^{d_j})}. \]
We notice that ${\rm Tr}\, \overline{\Phi}_v^{\,i_1, \ldots, i_l}$ and
${\rm Tr}\, {\Phi}_v^{\,i_1, \ldots, i_l}$ can be nonzero
only if the length $d_k$ of the $G_v$-orbit $ \Sigma _k(1)$
divides $i_k$ $( k =1, \ldots, l)$.
Therefore the polynomial $Q(t_1^{d_1}, \ldots, t_l^{d_l})$ is defined by
ranks and ${\bf Z}_{\geq 0}^l$-degrees of generators of
the free $A\lbrack z \rbrack$-modules $F^i$.
Notice that every monomial in $I(\Sigma)$ has the total degree at least $2$,
because every element $e_i \in \{ e_1, \ldots, e_n \}$ generates
a $1$-dimensional cone of $\Sigma$. So all
generators of $F^i$ $(i \geq 1)$ have the total degree at least $2$.
Therefore, the polynomial
$Q_{ \Sigma }$ has only monomials of the total degree at least $2$.
Since $R(\Sigma)$ is a Gorenstein ring, $F^{n-d}$ is a free $A\lbrack z
\rbrack$-module of rank $1$ with a generator of the
degree $(d_1, \ldots, d_l)$. Therefore, $Q$ has the total degree
$n = d_1 + \cdots + d_l$. \hfill $\Box$
\bigskip
Let $\chi$ be a topological character of $T({\bf A}_K)$ such that
its $v$-component $\chi_v\, : \, T(K_v) \rightarrow S^1 \subset
{\bf C}^*$ is trivial on $T({\cal O}_v)$.
For each $ j \in \{ 1, \ldots, l\}$, we denote by
$n_j$ the sum of $d_j$ generators of all $1$-dimensional
cones of the $G_v$-orbit $ \Sigma _j(1)$. Then $n_j$ is a $G_v$-invariant
element of $N$. By \ref{pi-image}, $n_j$ represents an element
of $T(K_v)$ modulo $T({\cal O}_v)$. Therefore, $\chi_v(n_j)$ is
well defined.
\begin{prop}
Let $v \not\in S$. Denote by $q_v$ the cardinality of the
finite residue field $k_v$ of $K_v$. Then
for any local topological character
$\chi_v$ of $T(K_v)$, one has
\[ \hat{H}_{\Sigma,v} (\chi, -\varphi) =
\int_{T(K_v)} H_{ \Sigma ,v}(x_v,-\varphi) \chi_v(x_v)
d\mu_v = \]
\[ \frac{Q_{ \Sigma }\left( \frac{\chi_{v}(n_1)}{{q_v}^{\varphi(n_1)}},
\ldots, \frac{\chi_{v}(n_l)}{{q_v}^{\varphi(n_l)}} \right)}
{(1- \frac{\chi_{v}(n_1)}{{q_v}^{\varphi(n_1)}} )
\cdots (1 - \frac{\chi_{v}(n_l)}{{q_v}^{\varphi(n_l)}} ) } \]
if $\chi_v$ is trivial on $T({\cal O}_v)$, and
\[ \int_{T(K_v)} H_{ \Sigma ,v}(x_v,-\varphi) \chi_v(x_v)
d\mu_v = 0 \]
otherwise.
\label{integral.1}
\end{prop}
{\em Proof.} Since the local Haar measure $\mu_v$ is
$T({\cal O}_v)$-invariant, one has
\[ \int_{T(K_v)} H_{ \Sigma ,v}(x_v, -\varphi) \chi_v(x_v) d \mu_v = \]
\[ = \sum_{\overline{x}_v
\in T(K_v)/T({\cal O}_v)} H_{ \Sigma ,v}(\overline{x}_v, -\varphi)
\chi_v(\overline{x}_v) \int_{T({\cal O}_v)} \chi_v d\mu_v \]
where $\overline{x}_v$ denotes the image of $x_v$ in $T(K_v)/T({\cal O}_v)
= N_v$. Notice that $\int_{T({\cal O}_v)} \chi_v d\mu_v = 0$
if $\chi_v$ has nontrivial restriction on $T({\cal O}_v)$.
By \ref{prop.basis},
there exists a natural identification between $G_v$-invariant
elements of $N$ and $G_v$-invariant monomials in $R(\Sigma)$.
Since $\Phi_v$ acts by permutations on monomials in the homogeneous
$(i_1, \ldots, i_l)$-component of $R(\Sigma)$, the number
of $G_v$-invariant monomials in $R^{i_1, \ldots, i_l}(\Sigma)$ equals
${\rm Tr}\,\Phi_v^{i_1, \ldots, i_l}$.
Take a $G_v$-invariant element $\overline{x}_v \in N$ such that
$m(\overline{x}_v) \in R^{i_1, \ldots, i_r}(\Sigma)$.
Put $i_k = d_k b_k$ $( k =1,
\ldots, l)$.
Then
\[ \varphi(\overline{x}_v) = b_1\varphi(n_1) + \cdots + b_l \varphi(n_l) \]
and
\[ \chi_v(\overline{x}_v) = \chi_{v}^{b_1}(n_1) \cdots \chi_{v}^{b_l}(n_l).
\]
This implies the claimed formula.
\hfill $\Box$
\bigskip
Let $A^*({\bf P}_{\Sigma}) = \bigoplus_{i =0}^d
A^i({\bf P}_{\Sigma})$ be the Chow ring
of ${\bf P}_{\Sigma,E_{\cal V}}$.
The groups $A^i({\bf P}_{\Sigma})$ have natural
$G_v$-action. Denote by $\Phi_v(i)$ the operator on $A^i({\bf P}_{\Sigma})$
induced by $\Phi_v$.
\begin{prop}
Denote by ${\bf 1}_v$ the trivial topological character of $T(K_v)$. Then
the restriction of
\[ \int_{T(K_v)} H_{ \Sigma ,v}(x_v,-\varphi) {\bf 1}_v(x_v) d \mu_v \]
to the line $s_1 = \cdots = s_r = s$ is equal to
\[ L_v( s,T; E/K) \left( \sum_{k =0}^d
\frac{{\rm Tr}\, \Phi_v(i)}{q_v^{ks}} \right). \]
\label{loc-integ}
\end{prop}
{\em Proof.} The Chow ring is the quotient of $R(\Sigma)$ by
a regular sequence \cite{danilov}. This gives the ${\bf Z}_{\geq 0}$-graded
(by the total degree) Koszul resolution having a $\Phi_v$-action:
\[ 0 \rightarrow \Lambda^d M \otimes R(\Sigma) \rightarrow
\cdots \rightarrow \Lambda^1 M \otimes R(\Sigma) \rightarrow
R(\Sigma) \rightarrow A^*({\bf P}_{\Sigma}) \rightarrow 0. \]
We apply the trace operator to the $k$-homogeneous component
of the Koszul complex, then we multiply the result by $1/q_v^{ks}$ and
take the sum over $k \geq 0$. By \ref{integral.1},
we have
\[ L_v^{-1}(s,T;E/K) \cdot
\int_{T(K_v)} H_{ \Sigma ,v}(x_v,-\varphi) {\bf 1}_v(x_v) d \mu_v =
\sum_{k =0}^d \frac{{\rm Tr}\, \Phi_v(i)}{q_v^{ks}} \]
because
\[ \sum_{k =0}^d \frac{(-1)^k}{q_v^{ks}}
{\rm Tr}( \Lambda^k {\Phi_v}) =
{\rm det}(Id - q_v^{-s} \Phi_v) = L_v^{-1}(s,T;E/K). \]
\hfill $\Box$
\subsection{Fourier transforms of archimedian
heights}
\begin{prop}
Let $\chi_v(y) = e^{-2\pi i \langle x,y \rangle}$ be a topological character
of $T(K_v)$ which is trivial on $T({\cal O}_v)$. Then
the Fourier transform $\hat{H}_{ \Sigma ,v}(\chi_v,-\varphi)$ of a
local archimedian Weil function
$H_{ \Sigma ,v} (x,-\varphi)$ is a rational function in $s_j = \varphi(e_j)$ for
${\rm Re}(s_j) > 0$.
\end{prop}
{\em Proof.} First we consider the case $K_v = {\bf C}$. Then
$T(K_v)/T({\cal O}_v) = N_{\bf R}$ and
\[ \frac{\hat{H}_{ \Sigma ,v}(\chi_v(y),-\varphi)}{\int_{T({\cal O}_v)} d\mu_v}
= \int_{N_{\bf R}} e^{-\varphi(x) -
2\pi i \langle x,y \rangle} dx =
\sum_{\sigma \in \Sigma(d)} \int_{\sigma} e^{-\varphi(x) -
2\pi i \langle x,y \rangle} dx \]
where $dx=d\overline{x}_v$ is the standard measure
on $N_v\otimes {\bf R }\simeq {\bf R }^d $. On the other hand,
\[ \int_{\sigma} e^{-\varphi(x) -
2\pi i \langle x,y \rangle} dx = \frac{1}{\prod_{e_j \in \sigma}
(s_j + 2\pi i \langle x,y \rangle)}. \]
If $K_v = {\bf R}$, then the following simple
statements allows to repeat the arguments:
\begin{lem}
Let $\Sigma \subset N_{\bf R}$ be a complete regular $G$-invariant
fan of cones. Denote by $\Sigma^G \subset N_{\bf R}^G$
the fan consisting of $\overline{\sigma} = \sigma \cap N_{\bf R}^G$,
$\sigma \in \Sigma$. Then $\Sigma^G$ is again a complete
regular fan.
\end{lem}
\hfill $\Box$
\medskip
\newline
The proof of the following proposition was suggested to us by W. Hoffmann.
\begin{prop}
Let ${\bf K} \subset {\bf C}^r$ be a compact such that ${\rm Re}( s_j) >
\delta$ for
all $(s_1, \ldots, s_r) \in {\bf K}$. Then there exists a constant
$c({\bf K},\Sigma)$ such that
\[ \mid \hat{H}_{ \Sigma ,v} (y,-\varphi) \mid \leq c({\bf K}, \Sigma ) \sum_{\sigma \in
\Sigma(d)}
\frac{1}{\prod_{e_k \in \sigma}
(1 + \mid \langle y, e_k \rangle \mid)^{1 + 1/d}}. \]
\label{l-estimation}
\end{prop}
{\em Proof.}
Let $f_1, \ldots, f_d$ be a basis of $M$. Put $x_i = \langle x, f_i \rangle$.
We denote by $y_1, \ldots, y_d$ the coordinates of $y$ in the basis
$f_1, \ldots, f_d$.
Let $\varphi_i(x) = \frac{\partial}{\partial x_i}\varphi (x)$.
$\varphi_i(x)$ has a constant value $\varphi_{i,\sigma}$
in the interior of a cone $\sigma \in \Sigma(d)$.
\[ \hat{H}_{ \Sigma ,v} (y,-\varphi) = \int_{N_{\bf R}} e^{-\varphi(x)-2\pi
i<y,x>}dx =
\frac{1}{2\pi iy_j} \int_{N_{\bf R}}
\frac{\partial}{\partial x_j} (e^{-\varphi(x)})
e^{-2\pi i<y,x>}dx \]
\[ = - \frac{1}{2\pi iy_j}
\int_{N_{\bf R}} \varphi_j(x) e^{-\varphi(x)-2\pi i<y,x>}dx \]
\[
= \frac{i}{2\pi y_j} \sum_{\sigma \in\Sigma(d)}
\frac{\varphi_{j,\sigma}}{\prod_{e_k \in \sigma} (s_k+2\pi i<y,e_k>)} \]
Notice that $M_{\bf R}$ is covered by $d$ domains:
\[ V_j = \{ y = \sum_i y_i f_i \in M_{\bf R} \mid\;
\; \mid y_j \mid = \max_i \mid y_i \mid \}. \]
Let $\| y \|^2 = \sum_{i} y_i^2$. Then $\| y \| \leq \sqrt{d} \mid y_j \mid$
for $ y \in V_j$. Then
\[ | \hat{H}_{ \Sigma ,v} (y,-\varphi) |\leq
\frac{\sqrt{d}}{\|y\|}\sum_{\sigma \in \Sigma (d)}\frac{1}{
\prod_{e_k\in \sigma } |s_k+2\pi i<y,e_k>|} \]
for $y\in V_j$.
Furthermore, we obtain
\[ | \hat{H}_{ \Sigma ,v} (y,-\varphi) |\leq
\frac{C'(\delta)}{1 + \|y\|} \sum_{\sigma \in \Sigma (d)}\frac{1}{
\prod_{e_k\in \sigma } (1 + \mid <y,e_k>\mid } \]
using the following obvious statement:
\begin{lem}
Assume that ${\rm Re}(s) > \delta > 0$. Then there exists
a positive constant $C(\delta)$ such that for all $t$ one has
$ C(\delta) (\mid s + 2\pi i t \mid) \geq 1 + \mid t \mid$.
\end{lem}
Since $\mid \langle y, e_k \rangle \mid \leq \| y \| \|e_k \|$, it follows
that there exist constants $c_{\sigma}$ such that
\[ c_{\sigma} (1+|<y,e_k>|)^d \geq \prod_{e_k\in \sigma }(1+|<y,e_k>|).
\]
Finally, we obtain
\[ \mid \hat{H}_{ \Sigma ,v} (y,-\varphi) \mid \leq c_j(\delta, \Sigma)
\cdot \sum_{\sigma \in \Sigma(d)}
\frac{1}{\prod_{e_k \in \sigma}
(1 + \mid \langle y, e_k \rangle \mid)^{1 + 1/d}} \]
for all $y \in V_j$.
It remains to put $c({\bf K}, \Sigma ) = \max_j c_j(\delta, \Sigma)$. \hfill $\Box$
\begin{coro}
Let $g \, : \, M_{\bf C} \rightarrow {\bf C}$ be a continious function
such that
\[ \mid g(iy) \mid \leq \|y\|^{\varepsilon}, \; \varepsilon < 1, \;\;
y \in M_{\bf R}. \]
Then
\[ \sum_{y \in O} g(iy) \hat{H}_{ \Sigma ,v}(y, -\varphi) \]
is absolutely and uniformly convergent on ${\bf K}$ for any function $g(iy)$
for any lattice $O \subset M_{\bf R}$.
\label{lconver}
\end{coro}
\section{Characteristic functions of convex cones}
Let $V$ be an $r$-dimensional real vector space, $V_{\bf C}$ its
complex scalar extension, $\Lambda \subset V$
a convex $r$-dimensional cone such that $\Lambda \cap - \Lambda = 0
\in V$. Denote by $ \Lambda ^{\circ}$ the interior of $ \Lambda $,
${ \Lambda }_{\bf C}^{\circ} =
{ \Lambda }^{\circ} + iV$
the complex tube domain over ${ \Lambda }^{\circ}$, by
$V^*$ the
dual space, by ${ \Lambda }^* \subset V^*$ the dual to ${ \Lambda }$ cone and by
$dy$ a Haar measure on $V^*$.
\begin{dfn}
{\rm The {\em characteristic function of} ${ \Lambda }$ is defined as
the integral
\[ {\cal X}_{ \Lambda }(dy,u) =
\int_{{ \Lambda }^*} e^{- \langle u, y \rangle} dy, \]
where $u \in { \Lambda }_{\bf C}$. }
\end{dfn}
\begin{rem}
{\rm Characteristic functions of convex cones have been investigated
in the theory of homogeneous cones by M. K\"ocher, O.S. Rothaus, and
E.B. Vinberg \cite{koecher,vinberg,rothaus}.}
\end{rem}
\begin{rem}
{\rm We will be interested in characteristic functions of convex cones
${ \Lambda }$ in real spaces $V$ which have natural lattices $L \subset V$
of the maximal rank $r$. Let $L^*$ be the dual lattice in $V^*$, then
we can normalize the Haar measure $dy$ on $V^*$ so
that the volume of the fundamental domain
$V^*/L^*$ equals $1$. In this case the
corresponding characteristic function will be denoted simply by
${\cal X}_{ \Lambda }(u)$. }
\end{rem}
\begin{prop} {\rm \cite{vinberg} }
Let $u \in { \Lambda }^{\circ} \subset V$ be an interior point
of ${\Lambda}$. Denote by ${ \Lambda }^*_u(t)$ the convex $(r-1)$-dimensional
compact
\[ \{ y \in { \Lambda }^* \mid \langle u, y \rangle = t \} \]
We define the $(r-1)$-dimensional measure $dy_t'$ on ${ \Lambda }^*_u(t)$
in such a way that for any function $f\; : \; V \rightarrow
{\bf R}$ with compact support one has
\[ \int_{{V}^*} f(y) dy = \int_{-\infty}^{+\infty}dt
\left(\int_{\langle u,y \rangle=t} f(y)dy_t'\right). \]
Then
\[ {\cal X}_{ \Lambda }(u) = (r-1)!\int_{{ \Lambda }^*_u(1)} dy_1'. \]
\end{prop}
The characteristic function ${\cal X}_{ \Lambda }(u)$ has the following
properties \cite{rothaus,vinberg}:
\begin{prop}
{\rm (i)} If ${\cal A}$ is any invertible
linear operator on ${V}$, then
\[ {\cal X}_{ \Lambda } ({\cal A}u) = \frac{{\cal X}_{ \Lambda }(u)}
{{\rm det}{\cal A}}; \]
{\rm (ii)} If ${ \Lambda }^{\circ} = {\bf R}^r_{\geq 0}$, $L = {\bf Z}^r \subset
{\bf R}^r$, then
\[ {\cal X}_{ \Lambda }(u) = (u_1 \cdots u_r)^{-1}, \;{\rm for }
\;{\rm Re}(u_i) > 0 ; \]
{\rm (iii)} If $z \in { \Lambda }^{\circ}$, then
\[ \lim_{z \rightarrow \partial { \Lambda }} {\cal X}_{ \Lambda }(z) = \infty; \]
{\rm (iv)} ${\cal X}_{ \Lambda }(u) \neq 0$ for all
$u \in { \Lambda }_{\bf C}^{\circ}$.
\label{zeta.cone}
\end{prop}
\begin{prop}
If ${ \Lambda }$ is an $r$-dimensional finitely generated polyhedral cone,
then ${\cal X}_{ \Lambda }(u)$ is a rational function of degree $-r$.
In particular, ${\cal X}_{ \Lambda }(u)$ has
a meromorphic extension to the whole complex space $V_{\bf C}$.
\end{prop}
{\em Proof.} It follows from Proposition \ref{zeta.cone}(i) that
${\cal X}_{ \Lambda }(\lambda u) =
{\lambda}^{-r} {\cal X}_{ \Lambda }(u)$. Hence ${\cal X}_{ \Lambda }(u)$ has
degree $-r$. In order to calculate ${\cal X}_{ \Lambda }(u)$,
we subdivide the dual cone ${ \Lambda }^*$ into
a union of simplicial subcones
\[ { \Lambda }^* = \bigcup_{j} { \Lambda }^*_j .\]
Then ${ \Lambda }$ is the intersection
\[ { \Lambda } = \bigcap_j { \Lambda }_j. \]
For ${\rm Re}(u) \in \bigcap_j { \Lambda }_{j}^{\circ}$, one has
\[ {\cal X}_{ \Lambda }(u) = \sum_j {\cal X}_{{ \Lambda }_j}(u). \]
By Proposition \ref{zeta.cone}(i),(ii), every
function ${\cal X}_{{ \Lambda }_j}(u)$ is rational. \hfill $\Box$
\begin{dfn}
{\rm Let $X$ be a smooth proper algebraic variety. Denote by
$ \Lambda _{\rm eff} \subset {\rm Pic}(X)_{\bf R}$ the cone
generated by classes of effective divisors on $X$.
Assume that the anticanonical class $ \lbrack {\cal K}^{-1}
\rbrack \in {\rm Pic}(X)_{\bf R}$
is contained in the interior of $ \Lambda _{\rm eff}$. We define
the constant $\alpha(X)$ by
\[ \alpha(X) = {\cal X}_{ \Lambda _{\rm eff}}( \lbrack {\cal K}^{-1}
\rbrack). \]
}
\end{dfn}
\begin{coro}
If ${ \Lambda }_{\rm eff}$ is a finitely generated polyhedral cone,
then $\alpha(X)$ is a rational number.
\end{coro}
\begin{exam}
{\rm Let ${\bf P}_{\Sigma,K}$ be a smooth compactification of
an anisotropic torus $T_K$.
By \ref{simp}, $ \Lambda _{\rm eff} \subset
{\rm Pic}({\bf P}_{\Sigma,K}) \otimes {\bf R}$ is a
simplicial cone. Using \ref{zeta.cone} and the exact sequence
\[ 0 \rightarrow PL(\Sigma)^G \rightarrow {\rm Pic}({\bf P}_{\Sigma,K})
\rightarrow H^1(G,M) \rightarrow 0 \]
we obtain
\[ {\cal X}_{ \Lambda _{\rm eff}}(u) = \frac{1}{h(T_K) u_1
\cdots u_r}, \]
where $u = \varphi$, $\varphi(e_j) = u_j$ $(j =1, \ldots l)$.
In particular,
\[ \alpha({\bf P}_{ \Sigma }) = \frac{1}{h(T_K)}. \]
}
\end{exam}
\begin{exam}
{\rm Consider an example of a non-simplicial cone of Mori $ \Lambda _{\rm eff}$
in $V = {\rm Pic}(X)_{\bf R}$ where $X$ is a Del Pezzo
surface of anticanonical degree 6.
The cone ${ \Lambda }$ has 6 generators corresponding to exceptional curves
of the first kind on $X$. We can construct $X$ as the blow up of
3 points $p_1, p_2, p_3$ in general position on ${\bf P}^2$.
The exceptional curves are $C_1, C_2, C_3,
C_{12}, C_{13}, C_{23}$, where $C_{ij}$ is the proper pullback of the line
joining $p_i$ and $p_j$.
If $u = u_1 [C_1] + u_2 [C_2] + u_3 [C_3] + u_{12}[C_{12}] +
u_{13}[C_{13}] + u_{23} [C_{23}] \in \Lambda _{\rm eff}^{\circ}$, then
\[ {\cal X}_{ \Lambda _{\rm eff}}(u) =
\frac{ u_1 + u_2 + u_3 + u_{12} + u_{13} + u_{23} }
{(u_1 + u_{23}) (u_2 + u_{13})(u_3 + u_{12})(u_1 + u_2 + u_3 )
(u_{12} + u_{13} + u_{23})} \]
and
\[ \alpha(X) = 1/12. \]}
\end{exam}
\begin{prop}
Assume that ${ \Lambda }$ is a finitely generated
polyhedral cone and ${ \Lambda } \cap -{ \Lambda } = 0$.
Let $p_0$ and $p_1$ be two points in ${E}$ such that $p_1 \not\in { \Lambda }$
and $p_0 \in { \Lambda }^{\circ}$. Let $t_0$ be a positive real number such
that $t_0p_0 + p_1 \in \partial { \Lambda }$. We define a meromorphic function
in one complex variable $t$ as
\[ Z(p_0,p_1, t) = {\cal X}_{ \Lambda }(tp_0 + p_1 ). \]
Let $k$ be
the codimension of the minimal face of ${ \Lambda }$ containing
$t_0p_0 + p_1$. Then
the rational function $Z(p_0,p_1, t)$ is analytic
for ${\rm Re}(t) > t_0$ and
it has a pole of order $k$ at $t = t_0$.
\end{prop}
{\em Proof.} As in the proof of the previous statement, we can subdivide the
dual cone ${ \Lambda }^*$ into simplicial subcones ${ \Lambda }^*_j$ such
that $t_0p_0 + p_1 \in \partial { \Lambda }_1$ and
$t_0p_0 + p_1 \not\in \partial { \Lambda }_j$
$(j >1)$. It suffices now to apply Proposition \ref{zeta.cone}(i),(ii) to
${ \Lambda }_1$. \hfill $\Box$
\begin{coro}
Assume that ${ \Lambda }$ is only locally polyhedral at the
point $t_0p_0 + p_1$ and $k$ is
the codimension of a minimal polyhedral face of ${ \Lambda }$ containing
$t_0p_0 + p_1$. Then
{\rm (i)} $Z(p_0,p_1, t)$ is an analytical function for ${\rm Re}(t) > t_0$.
{\rm (ii)} $Z(p_0,p_1, t)$ has meromorphic continuation to some
neigbourhood of $t_0$.
{\rm (iii)} $Z(p_0,p_1, t)$ has a pole of order $k$ at $t = t_0$.
\end{coro}
\section{Distribution of rational points}
\subsection{The method of Draxl}
Let $\Sigma$ be a $G$-invariant regular fan,
$\Sigma(1) = \Sigma_1(1) \cup \cdots \cup \Sigma_r(1)$ be
the decomposition of $\Sigma(1)$ into $G$-orbits. We choose a
representative $\sigma_j$ in each $\Sigma_j(1)$ $( j =1, \ldots, r)$.
Let $e_j$ be the primitive integral generator of $\sigma_j$,
$G_j \subset G$ be the stabilizer of $e_j$. Denote by $k_j$ the
length of $G$-orbit of $e_j$, and by $K_j \subset E$ the subfield of
$G_j$-fixed elements. Then $k_j = \lbrack K_j : K \rbrack$
$(j =1, \ldots, r)$.
Consider the $n$-dimensional torus
\[ T' := \prod_{j =1}^r R_{K_j/K}({\bf G}_m). \]
Notice that the group ${\bf D}(\Sigma)$ can be identified with the $G$-module
$\hat{T}'_K$. The homomorphism of $G$-modules
$ M \rightarrow {\bf D}(\Sigma)$
induces the homomorphism $T' \rightarrow T$
and a map
$$
\gamma:\, \prod_{j=1}^r
{\bf G}_{m}({\bf A}_{K_j})/ {\bf G}_{m}(K_j) \rightarrow T({\bf A}_{K})/T(K)
$$
We get a map of characters
$$
\gamma^*:\, (T({\bf A}_{K})/T(K))^*\rightarrow \prod_{j =1}^r
({\bf G}_{m}({\bf A}_{K_j})/{\rm G}_m(K_j))^*.
$$
\begin{rem}{\rm
The kernel of $\gamma^*$ is dual to the obstruction group to weak approximation
$A(T) $ defined above. }
\label{obstr}
\end{rem}
Let
\[ \chi \;: \; T({\bf A}_K) \rightarrow S^1 \subset {\bf C}^* \]
be a topological character which is trivial on
$T(K)$.
Then $\chi \circ \gamma$ defines Hecke characters of the
idele groups
\[ \chi_j \; :\;
{\bf G}_m({\bf A}_{K_j}) \rightarrow S^1 \subset {\bf C}^*. \]
If $\chi$ is trivial on ${\bf K}_T$, then all characters $\chi_j$
$(j =1, \ldots, r)$ are trivial on the maximal
compact subgroups in ${\bf G}_m({\bf A}_{K_j})$.
We denote by $L_{K_j}(s,\chi_j)$ the Hecke $L$-function corresponding to the
character $\chi_j$.
The following statement is well-known:
\begin{theo}
The function $L_{K_j}(s,\chi_j)$ is holomorphic in the whole plane
unless $\chi_j$ is trivial. In the later case,
$L_{K_j}(s,\chi_j)$ is holomorphic for
${\rm Re}(s) >1$ and has a meromorphic extension to the complex plane with
a pole of order $1$ at $s = 1$.
\end{theo}
We come to the main statement which describes the analytical properties
of the Fourier transform of height functions.
\begin{theo}
Define affine complex coordinates
$\{s_1, \ldots, s_r \}$ on the vector space
$PL(\Sigma)_{\bf C}^G$ by $s_j = \varphi(e_j)$ $( j =1, \ldots, r)$.
Then the Fourier transform
$\hat{H}_{ \Sigma }(\chi, -\varphi)$ of the complex height
function $H_{ \Sigma }(x,-\varphi)$ is always an analytic function for
${\rm Re}(s_j) > 1$ $(1 \leq j \leq l)$, and
\[ \hat{H}_{ \Sigma }(\chi,-\varphi)
\prod_{i=1}^r L^{-1}_{K_j}(s_j,\chi_j) \]
has an analytic extension to the domain ${\rm Re}(s_j) > 1/2$
$(1 \leq j \leq r)$.
\label{dmethod}
\end{theo}
{\em Proof.} The idea of the proof is essentialy due to Draxl
\cite{drax1}. We have the Euler product
\[ \hat{H}_{ \Sigma }(\chi,-\varphi) =
\prod_{v \in {\rm Val}(K)} \hat{H}_{ \Sigma ,v}(\chi_v,-\varphi) \]
In order to prove the above properties of
$\hat{H}_{ \Sigma }(\chi, -\varphi)$, it is
sufficient to investigate the product
\[ \hat{H}_{ \Sigma ,S}(\chi,-\varphi) = \prod_{v \not\in S}
\hat{H}_{ \Sigma ,v}(\chi_v,-\varphi). \]
Choose a valuation $v \not\in S$. Then we obtain
a cyclic subgroup $G_v = \langle \Phi_v \rangle \subset G$
generated by a lattice automorphism
$\Phi_v \,:\, N \rightarrow N$ representing
the local Frobenius element at place $v$.
Let $l$ be the number of $G_v$-orbits in $ \Sigma (1)$.
By \ref{integral.1},
\[ \hat{H}_{ \Sigma ,S}(\chi, -\varphi) = \prod_{v \not\in S}
P\left( \Sigma, \Phi_v; \frac{\chi_{v}(n_1)}{{q_v}^{\varphi(n_1)}},
\ldots, \frac{\chi_{v}(n_l)}{{q_v}^{\varphi(n_l)}} \right). \]
By \ref{p-function}, we have
\[ \hat{H}_{ \Sigma ,v}(\chi, -\varphi) =
\frac{Q_{ \Sigma }\left( \frac{\chi_{v}(n_1)}{{q_v}^{\varphi(n_1)}},
\ldots, \frac{\chi_{v}(n_l)} {{q_v}^{\varphi(n_l)}} \right)}
{(1- \frac{\chi_{v}(n_1)}{{q_v}^{\varphi(n_1)}} )
\cdots (1 - \frac{\chi_{v}(n_l)}{{q_v}^{\varphi(n_l)}} )} . \]
Moreover,
\[ \prod_{v \not\in S}
Q_{ \Sigma }\left( \frac{\chi_{v}(n_1)}{{q_v}^{\varphi(n_1)}},
\ldots, \frac{\chi_{v}(n_l)}{{q_v}^{\varphi(n_r)}} \right) \]
is an absolutely convergent Euler product for
${\rm Re}(s_j) > 1/2$ $( j =1, \ldots, l)$.
It remains to show the relation between
\[ \left( 1 - \frac{\chi_{v}(n_1)}{{q_v}^{\varphi(n_1)}} \right)^{-1} \cdots
\left( 1 - \frac{\chi_{v}(n_l)}{{q_v}^{\varphi(n_l)}} \right)^{-1} \]
and local factors of the product of the Hecke $L$-functions
\[ \prod_{j =1}^r L_{K_j}(s_j, \chi_j). \]
For this purpose, we compare two decompositions of $ \Sigma (1)$ into
the disjoint union of $G_v$-orbits and $G$-orbits.
Notice that for every $j \in \{ 1, \ldots, r \}$, the $G$-orbit
$\Sigma^j(1)$ decomposes into a disjoint union of $G_v$-orbits
\[ \Sigma_j(1) = \Sigma_{j1}(1) \cup \cdots \cup \Sigma_{j l_j}(1). \]
Let $d_{ji}$ be the length of the $G_v$-orbit $ \Sigma ^j_i(1)$; i.e., we put
$\{ d_{ji} \} = \{ d_1, \ldots, d_l \}$.
One has
\[ \sum_{ i =1}^{l_j} d_{ji} = k_j. \]
and
\[ l = \sum_{i =1}^r l_j. \]
On the other hand, $l_j$ is the number of different
valuations ${\cal V}_{j1}, \ldots, {\cal V}_{jl_j} \in {\rm Val}(K_j)$
over of $v \in {\rm Val}(K)$. Let $k_v$ be the residue
field of $v \in {\rm Val}(K)$, $k_{{\cal V}_{ji}}$ the residue field
of ${\cal V}_{ji} \in {\rm Val}(K_j)$. Then
\[ d_{ji} = \lbrack k_{{\cal V}_{ji}} : k_v \rbrack. \]
We put also $\{ n_1, \ldots, n_l \} =
\{ n_{ji} \}$, where $n_{ji}$ denotes the sum of $d_{ji}$ generators
of all $1$-dimensional cones of the $G_v$-orbit $ \Sigma _{ji}(1)$.
Therefore, $\chi_v(n_{ji})$ is the
${\cal V}_{ji}$-adic component
of the Hecke character $\chi_j$. Hence
\[ \prod_{i =1}^{l_j} \left( 1 - \frac{
\chi_{v}(n_{ji})}{{q_v}^{\varphi(n_{ji})}} \right)^{-1} \]
equals the product of the local factors
\[ \prod_{{\cal V}_{ji}} \left( 1 -
\frac{\chi_{{\cal V}_{ji}}}{q_{{\cal V}_{ji}}^{\varphi(n_{ji})}}
\right)^{-1} \]
of the Hecke $L$-function $L_{K_j}(s_j, \chi_j)$.
\hfill $\Box$
\subsection{The meromorphic extension of
$Z_{\Sigma}(\varphi)$}
\begin{theo}
For any $\varepsilon > 0$ there exists a $\delta >0$ and a
constant $c({\varepsilon})$ such
that
\[ \mid L_K(s,\chi) \mid \leq c(\varepsilon)(\mbox{\rm Im}(s))^{\varepsilon}\;
\; \mbox{\rm for}\; u= \mbox{\rm Re}(s) > 1 - \delta \]
for every Hecke $L$-function $L_K(s,\chi)$ with a
nontrivial nonramified character $\chi$.
\label{estim}
\end{theo}
{\em Proof.} We use the following standard statement
based on the Phragm\'en-Lindel\"of principle:
\begin{lem} {\rm ( \cite{titchmarsh}, p.181)}
Let $f(s) $ be a single valued analytic function in the strip
$u_1 \le {\rm Re}(s)\le u_2$ satisfying the conditions:
{\rm (i)} $|f(u + it )| < A_0 \exp ( e^{C|t|})$ for some
real constants $A_0 >0$ and $ 0 < C < \pi/(u_2 - u_1)$;
{\rm (ii)} $| f(u_1 + it) | \leq A_1 |t|^{a_1}$,
$| f(u_2 + it) | \leq A_2 |t|^{a_2}$ for some
constants $a_1$, $a_2$.
Then for all $u_1 \leq u \leq u_2$, we have the estimate
\[ | f(u + it) | \leq A_3 |t|^{a(u)} \]
where
\[ a(u) =a_1 \frac{u_2-u}{u_2-u_1} + a_2 {\frac{u-u_1}{u_2-u_1}}.\]
\label{FL}
\end{lem}
Choose a sufficiently small $\delta_1$. Then $| L_K(s,\chi) |$ is
bounded by $A_2(\delta_1)=
\zeta_K(1+\delta_1)$ for ${\rm Re}(s) = 1 + \delta_1$.
Consider the functional equation $L_K(s,\chi) =
C(s)L_K(1-s,\overline{\chi})$.
Since $\chi$ (as well as $\overline{\chi}$) is unramified, the function
$C(s)$ depends only on the field $K$.
Using standard estimates for $\Gamma$-factors in $C(s)$, we obtain
$| L_K(s, \chi) | < A_1 ({\rm Im}(s))^{a_1}$ for ${\rm Re}(s) =
- \delta_1$ and some sufficiently large explicit constants $A_1(\delta_1),
a_1(\delta_1)$.
We apply Lemma \ref{FL} to the $L$-function
$L_K(s,\chi)$ where $u_1 = - \delta_1$,
$u_2 = 1 + \delta_1$, and $a_2 = 0$.
Then for $1 - \delta < {\rm Re}(s) < 1 + \delta_1$, one has
\[ | L_K(s,\chi) | \leq A_3(\delta,\delta_1)({\rm Im}
(s))^{a_1\frac{\delta + \delta_1}{1 + \delta + \delta_1}}. \]
It is possible to choose $\delta$ and $\delta_1$ in such a way that
\[ a_1(\delta_1) \frac{\delta + \delta_1}{1 + 2 \delta_1} < \varepsilon. \]
\hfill $\Box$
\begin{theo} Let $s_j = \varphi(e_j)$ $( j =1, \ldots, r)$. Then
the height zeta function $Z_{\Sigma}(\varphi)$ is holomorphic for
${\rm Re}(s_j) > 1$. There exists an analytic continuation
of $Z_{\Sigma}(\varphi)$ to the domain
${\rm Re}(s_j) > 1- \delta$ such that the only
singularities of $Z_{\Sigma}(\varphi)$ in this domain are poles
of order $ \leq 1$ along the hyperplanes $s_j = 1$ $(j =1, \ldots, r)$.
\label{extension.m}
\end{theo}
{\em Proof.} By the Poisson formula,
\[ Z_{\Sigma}(\varphi) =
\frac{1}{{\rm vol}(T^1({\bf A}_K)/T(K)) } \sum_{\chi \in
(T({\bf A}_K)/T(K))^* } \hat{H}_{ \Sigma }(\chi, -\varphi). \]
Since $H_{\Sigma}(x, -\varphi)$ is ${\bf K}_T$-invariant,
we can assume that in the above formula $\chi$ runs
over the elements of the group ${\cal P}$ consisting of
characters of $T({\bf A}_K)$ which are trivial on
${\bf K}_T \cdot T(K)$.
Let $J$ be a subset of $I = \{ 1, \ldots, r \}$.
Denote by ${\cal P}_J$ the
subset of ${\cal P}$ consisting of all characters $\chi \in {\cal P}$
such that the corresponding Hecke character $\chi_j$ is trivial
if and only if $j \in J$. Then
\[ Z_{\Sigma}(\varphi) = \sum_{J \subset I}
Z_{\Sigma,J}(\varphi) \]
where
\[ Z_{\Sigma,J}(\varphi) =
\frac{1}{{\rm vol}(T^1({\bf A}_K)/T(K)) } \sum_{\chi \in {\cal P}_J}
\hat{H}_{ \Sigma }(\chi, -\varphi). \]
Consider the logarithmic space
\[ N_{{\bf R}, \infty} =
\prod_{ v \in {\rm Val}_{\infty}(K)} T(K_v)/T({\cal O}_v) \]
containing the full sublattice $T({\cal O}_K)/W(T)$
of ${\cal O}_K$-integral points of $T(K)$ modulo torsion.
Let $\chi_{\infty}$ be the restriction of a character
$\chi \in {\cal P}$ to $N_{{\bf R}, \infty}$. Then $\chi_{\infty}(x) =
e^{2\pi i < x,y_{\chi} >}$ where $y_{\chi}$ is
an element of the dual logarithmic space
\[ M_{{\bf R}, \infty} =
\prod_{ v \in {\rm Val}_{\infty}(K)} {\rm Hom}\,( T(K_v)/T({\cal O}_v),
{\bf R}). \]
Moreover, $y_{\chi}$ belongs to the dual lattice
$(T({\cal O}_K)/W(T))^*
\subset M_{{\bf R}, \infty}$.
Let $u_1, \ldots, u_{r'}$ be a basis of the lattice
$T({\cal O}_K)/W(T) \subset N_{{\bf R}, \infty}$, $f_1, \ldots, f_{r'}$
the dual basis of the dual lattice $(T({\cal O}_K)/W(T))^*
\subset M_{{\bf R}, \infty}$. We extend
\[ e^{2\pi i < x, f_1>} , \ldots, e^{2 \pi i<x, f_{r'}>} \]
to some adelic characters
$\eta_1, \ldots, \eta_{r'} \in {\cal P}$. Using \ref{subgroups} and
the basis $\eta_1, \ldots, \eta_{r'}$, we
can extend
\[ \chi_{\infty} = \prod_{k = 1}^{r'}
e^{2\pi i a_k < x,f_k >}, \;\; a_k \in {\bf Z} \]
to a character
\[ \tilde{\chi} = \prod_{k =1}^{r'} \eta_i^{a_k} \in {\cal P} \]
such that $\tilde{\chi}_{\infty} = \chi_{\infty}$ and
$\chi \cdot \tilde{\chi}^{-1}$ is a character of the finite group ${\bf
cl}(T)$.
We fix a character of $\chi_c$ of ${\bf cl}(T)$. Denote by
${\cal P}_{J,\chi_c}$ the
set of all characters $\chi \in {\cal P}_J$ such that
$\chi \cdot \tilde{\chi}^{-1} = \chi_c$.
Then a character $\chi \in {\cal P}_{J,\chi_c}$ is uniquely defined by
its archimedian component $\chi_{\infty}$.
By \ref{dmethod},
\[ Q_{ \Sigma } (\chi,-\varphi) = \prod_{ v \not\in S_{\infty}}
\hat{H}_{ \Sigma ,v}(\chi_v, \varphi)
\prod_{i =1}^r L_{K_j}^{-1}(s_j, \chi_j) \]
is absolutely convergent Euler product for ${\rm Re}(s_j) > 1 -
\delta > 1/2$.
By \ref{estim},
\[ \prod_{j \not\in J} L_{K_j}(s_j, \chi_j) < C(\varepsilon)
\prod_{j \not\in J} {\rm Im}\, (s_j)^{\varepsilon} \]
${\rm Re}(s_j) > 1 - \delta$.
We apply \ref{l-estimation} to the archimedian Fourier
transform
\[ \hat{H}_{ \Sigma ,\infty}(\chi, \varphi) =
\prod_{ v \in S_{\infty}}
\hat{H}_{ \Sigma ,v}(\chi_v, \varphi). \]
Then, by \ref{lconver},
\[ \sum_{\chi \in {\cal P}_{J,\chi_c}} \hat{H}(\chi, -\varphi)
\prod_{j \in J} \zeta_{K_j}(s_j)^{-1} \]
is absolutely convergent for ${\rm Re}(s_j) > 1 - \delta$.
Therefore, we have obtained that
\[ Z_{ \Sigma ,J} (\varphi) \prod_{j \in J} \zeta_{K_j}(s_j)^{-1} \]
is a holomorphic function for ${\rm Re}(s_j) > 1 - \delta$
and for any $J \subset I$.
It remains to notice, that in the considered domain
$\prod_{j \in J} \zeta_{K_j}(s_j)$
has only poles of order $1$ along hyperplanes $s_j = 1$.
\hfill $\Box$
\subsection{Rational points of bounded height}
Recall the standard tauberian statement:
\begin{theo}\cite{delange}
Let $X$ be a countable set, $F\, : \, X \rightarrow {\bf R}_{>0}$ a real valued
function. Assume that
\[ Z_F(s) = \sum_{x \in X} F(x)^{-s} \]
is absolutely convergent for ${\rm Re}(s) > a> 0$ and has a representation
$$
Z_F(s)=(s-a)^{-r}g(s) + h(s)
$$
with $g(s)$ and $h(s)$ holomorphic for $Re(s)\ge a$, $g(a)\neq 0$,
$r\in N$.
Then for any $B>0$ there exists only a finite number
$N(F,B)$ of elements $x \in X$ such that $F(x) \leq B$. Moreover,
\[ N(F,B) = \frac{g(a)}{a(b-1)!} B^a (\log B)^{r-1}(1+o(1)). \]
\label{tauberian}
\end{theo}
Let ${\cal L} = {\cal L}(\varphi_0)$ be a metrized invertible sheaf
over a smooth compactification ${\bf P}_{\Sigma}$ of an anisotropic
torus $T_K$ defined by a $G$-invariant fan $\Sigma$. We denote
by $Z_{\Sigma,{\cal L}}(s) = Z_{\Sigma}(s \varphi_0)$
the restriction of $Z_{ \Sigma }(\varphi)$ to the
line $s\lbrack {\cal L} \rbrack \subset {\rm Pic}({\bf P}_{ \Sigma })_{\bf R}$.
Let $a({\cal L})$ be the abscissa of convergence of $Z_{\Sigma,{\cal L}}(s)$
and
$b({\cal L})$ the order of the pole of $Z_{\Sigma,{\cal L}}(s)$ at
$s = a({\cal L})$.
By \ref{extension.m},
\[ a({\cal L}) \leq \min_{j = 1}^r \frac{1}{\varphi(e_j)}. \]
By \ref{tauberian}, we obtain:
\begin{theo}
Assume that $\varphi(e_j) > 0$ for all $j =1, \ldots, r$; i.e.,
the class $\lbrack {\cal L} \rbrack$ is contained in the interior
of the cone of effective divisors $\Lambda_{\rm eff}(\Sigma)$.
Then there exists only finite number $N({\bf P}_{\Sigma}, {\cal L},B)$ of
$K$-rational points $x \in T(K)$ having the
${\cal L}$-height $H_{\cal L}(x) \leq B$.
Moreover,
\[ N({\bf P}_{\Sigma}, {\cal L},B) =B^{a({\cal L})}\cdot (\log B)^{b({\cal
L})-1}(1+o(1)) \;\; ,
B \rightarrow \infty. \]
\end{theo}
The following statement implies Batyrev-Manin conjectures
about the distribution of rational points of bounded ${\cal L}$-height
for smooth compactifications of anisotropic tori:
\begin{theo}
The number $a({\cal L})$ equals
\[ a({\cal L})={\rm inf} \, \{\lambda \mid \lambda \lbrack {\cal L}\rbrack +
\lbrack {\cal K} \rbrack \in \Lambda_{\rm eff}(\Sigma)\}; \]
i.e.,
\[ a({\cal L}) = \min_{j = 1,r} \frac{1}{\varphi(e_j)}. \]
Moreover, $b({\cal L})$ equals the codimension of the minimal face
of $\Lambda_{\rm eff}(\Sigma)$ containing
$a({\cal L})\lbrack {\cal L}\rbrack +
\lbrack {\cal K} \rbrack$.
\end{theo}
{\em Proof. } By \cite{ono1}, we can choose the finite set $S$ such
that the natural homomorphism
\[ \pi_S \, : \, T(K) \rightarrow
\prod_{v \not\in S} T(K_v)/T({\cal O}_v) =
\prod_{v \not\in S} N_v \]
is surjective. Denote by $T({\cal O}_S)$ the kernel of
$\pi_S$ consisting of all $S$-units in $T(K)$.
The group $T({\cal O}_S)/W(T)$ has the natural embedding
in the finite-dimensional space
\[ N_{S,{\bf R}} = \prod_{v \in S} T(K_v)/T({\cal O}_v) \otimes {\bf R} \]
as a full sublattice.
Let $\Delta$ be the fundamental domain of $T({\cal O}_S)/W(T)$ in
$N_{S,{\bf R}}$. For any $x \in T(K)$, denote by $\overline{x}_S$ the image
of $x$ in $N_{S,{\bf R}}$. Define $\phi(x)$ to be the
element of $T({\cal O}_S)$ such that $\overline{x}_S - \phi(x) \in \Delta$.
Thus, we have obtained the mapping
\[ \phi_S \, : \, T(K) \rightarrow T({\cal O}_S). \]
Define the new height function $\tilde{H}_{ \Sigma }(x,\varphi)$ on $T(K)$ by
\[ \tilde{H}_{ \Sigma }(x, \varphi) = H_{ \Sigma }(\varphi, \phi_S(x))
\prod_{v \not\in S} H_{ \Sigma ,v}(x_v, \varphi). \]
\noindent
Notice the following easy statement:
\begin{lem}
Choose a compact subset ${\bf K} \subset {\bf C}^r$ such that
${\rm Re}(s_j) > \delta$ $( j =1, \ldots, r)$ for $\varphi \in {\bf K}$.
Then there exist positive constants $C_1$, $C_2$ such that
\[ 0 < C_1 < \frac{\tilde{H}_{ \Sigma }(x, \varphi)}{H_{ \Sigma }(x, \varphi)} <
C_2, \; \mbox{\rm for}\; \varphi \in {\bf K}, \; x \in T(K). \]
\label{compare}
\end{lem}
Define $\tilde{Z}_{ \Sigma }(\varphi)$ by
\[ \tilde{Z}_{ \Sigma }(\varphi) = \sum_{x \in T(K)} \tilde{H}_{ \Sigma }(x, -\varphi). \]
Then $\tilde{Z}_{ \Sigma }(\varphi)$ splits into the
product
\[ \tilde{Z}_{ \Sigma }(\varphi) =
\prod_{v \not\in S} \left( \sum_{z \in N_v} H_{ \Sigma ,v}(z,-\varphi) \right)
\cdot \left( \sum_{ u \in T({\cal O}) } H_{ \Sigma }(u, -\varphi) \right). \]
By \cite{drax1}, the Euler product
\[ \prod_{j =1}^r \zeta_{K_j}(s_j) \prod_{v \not\in S}
\left( \sum_{z \in N_v} H_{ \Sigma ,v}(z,-\varphi) \right) \]
is a holomorphic function without zeros for ${\rm Re}(s_j) > 1/2$.
On the other hand,
\[ \sum_{ u \in T({\cal O}) } H_{ \Sigma }(u, -\varphi) \]
is an absolutely convergent series nonvanishing for
${\rm Re}(s_j) > 0$.
Therefore, $\tilde{Z}_{ \Sigma }(\varphi)$ has a meromorphic extension
to the domain ${\rm Re}(s_j) > 1/2$ where it has poles of order $1$
along the hyperplanes $s_j = 1$.
By \ref{compare} and \ref{tauberian}, $\tilde{Z}_{ \Sigma }(\varphi)$ and
$Z_{ \Sigma }(\varphi)$ must have the same poles in the domain
${\rm Re}(s_j) > 1 - \delta$. Therefore,
$Z_{ \Sigma }(\varphi)$ has poles of order $1$ along the hyperplanes $s_j = 1$.
By taking the restriction of $Z_{ \Sigma }(\varphi)$ to the line
$\varphi = s \varphi_0$, we obtain
the statement.
\hfill $\Box$
\subsection{The residue at $s_j = 1$}
Recall the definition of the Tamagawa number of
Fano varieties \cite{peyre}.
This definition immediately extends to arbitrary
algebraic varieties $X$ with a metrized
canonical sheaf ${\cal K}$.
Let $x_1, \ldots, x_d$ be local analytic coordinates on $X$. They
define a homeomorphism $f\,: \, U \rightarrow {K_v}^d$
in $v$-adic topology between an
open subset $U \subset X$ and $f(U) \subset {K_v}^d$.
Let $dx_1 \cdots dx_d$ be the Haar measure on $K_v^d$ normalized
by the condition
\[ \int_{{\cal O}_v^d} dx_1 \cdots dx_d = \frac{1}{(\sqrt{\delta_v})^d} \]
where $\delta_v$ is the absolute different of $K_v$.
Denote by $dx_1 \wedge \cdots \wedge dx_d$ the standard
differential form on $K_v^{d}$. Then
$g = f^*(dx_1 \wedge \cdots \wedge dx_d)$ is a local analytic section of
the metrized canonical sheaf ${\cal K}$.
We define the local measure on $U$ by
\[ \omega_{{\cal K},v} =f^*( \| g(f^{-1}(x)) \|_v dx_1 \cdots dx_d) . \]
The adelic Tamagawa measure $\omega_{{\cal K},S}$ is defined by
\[ \omega_{{\cal K},S} = \prod_{v \in {\rm Val}(K)} \lambda_v^{-1}
\omega_{{\cal K},v} \]
where $\lambda_v = L_v(1, {\rm Pic}(X_{\overline{K}});\overline{K}/K))$
if $v \not\in S$,
$\lambda_v = 1$ if $v \in S$.
\begin{dfn}
{\rm \cite{peyre}
Let $\overline{X(K)}$ be the closure of $X(K) \subset X({\bf A}_K)$
in the
direct product topology. Then the {\em Tamagawa number} of $X$ is defined by
\[ \tau_{\cal K}(X) =
\lim_{s \rightarrow 1} (s-1)^r L_S(s, {\rm Pic}(X_{\overline{K}});\overline{K}/K))
\cdot \int_{\overline{X(K)}} \omega_{{\cal K},S}. \]}
\end{dfn}
\begin{prop}
Let ${\cal K} = {\cal L}(-\varphi_{\Sigma})$ be the metrized
canonical sheaf on a toric variety ${\bf P}_{\Sigma}$. Then
the restriction of the $v$-adic
measure $\omega_{{\cal K},v}$ to $T(K_v) \subset
{\bf P}_{\Sigma}(K_v)$ coincides with the measure
\[ H_{ \Sigma ,v}(x, -\varphi_{\Sigma}) \omega_{\Omega,v}. \]
\label{restriction}
\end{prop}
{\em Proof.} The rational differential $d$-form $\Omega$ is a
rational section of ${\cal K}$. By definition of the $v$-adic metric on
${\cal L}(-\varphi_{\Sigma})$, $H_{ \Sigma ,v}(x, -\varphi_{\Sigma})$
equals the norm $\| \Omega \|_v$ of the $T$-invariant
section $\Omega$. This implies the statement.
\hfill $\Box$
\begin{prop}
One has
\[ \int_{\overline{T(K)}} \omega_{{\cal K},S} =
\int_{\overline{{\bf P}_{ \Sigma }(K)}} \omega_{{\cal K},S}. \]
\end{prop}
{\em Proof.} Since $\overline{{\bf P}_{ \Sigma }(K)} \setminus \overline{T(K)}$
is a subset of ${\bf P}_T({\bf A}_K) \setminus T({\bf A}_K)$, it is
sufficient to prove that
\[ \int_{T(K_v)} \omega_{{\cal K},v} =
\int_{{\bf P}_{ \Sigma }(K_v)} \omega_{{\cal K},v}. \]
Since the measure $\omega_{{\cal K},v}$ is $T({\cal O}_v)$-invariant and
the stabilizer in $T({\cal O}_v)$ of any point $x \in {\bf P}_{ \Sigma }(K_v)
\setminus T(K_v)$ is uncountable, the $\omega_{{\cal K},v}$-volume of
${\bf P}_{ \Sigma }(K_v) \setminus T(K_v)$ is zero.
\label{2int}
\hfill $\Box$.
\begin{theo}
Let $\Theta(\Sigma,K)$ be the
the residue of the zeta-function $Z_{\Sigma}(\varphi)$ at
$s_1 = \cdots = s_r = 1$. Then
\[ \Theta(\Sigma,K) = \alpha({\bf P}_{\Sigma})\beta({\bf P}_{\Sigma})
\tau_{\cal K}({\bf P}_{\Sigma}). \]
\end{theo}
{\em Proof.} By the Poisson formula,
\[ Z_{\Sigma}(\varphi) =
\frac{1}{{\rm vol}(T^1({\bf A}_K)/T(K)) } \sum_{\chi \in
(T({\bf A}_K)/T(K))^* } \hat{H}_{ \Sigma }(\chi, -\varphi). \]
By \ref{extension.m},
the residue of $Z_{\Sigma}(\varphi)$ at
$s_1 = \cdots = s_r = 1$ appears from $Z_{ \Sigma ,I}(\varphi)$
containing only the terms $\hat{H}_{ \Sigma }(\chi, -\varphi)$
such that $\chi_1, \ldots , \chi_r$ are trivial characters of
${\bf G}_m({\bf A}_{K_j})/{\bf G}_m(K_j)$ $(j = 1, \ldots, l)$; i.e.,
$\chi$ is a character of the finite group $A(T)$.
We apply again the Poisson formula to the finite
sum
\[ Z_{ \Sigma ,I}(\varphi)
= \frac{1}{{\rm vol}(T^1({\bf A}_K)/T(K)) }
\sum_{\chi \in (A(T))^* } \hat{H}_{ \Sigma }(\chi, -\varphi). \]
Using \ref{weak}, \ref{weak1}, \ref{obstr}, we have
\[ Z_{ \Sigma ,I}(\varphi)
= \frac{\beta({\bf P}_{\Sigma})}{i(T_K)\cdot{\rm vol}(T^1({\bf A}_K)/T(K)) }
\int_{\overline{T(K)}} H_{ \Sigma }(x,-\varphi) \omega^1_{\Omega,S}. \]
Notice that $\omega^1_{\Omega,S} = \omega_{\Omega,S}$ for
anisotropic tori.
Now we assume that $\varphi(e_1) = \ldots = \varphi(e_r) = s$.
Our purpose is to compute the constant
\[ \Theta(\Sigma,K) = \lim_{ s \rightarrow 1} (s-1)^r Z_{ \Sigma ,I}(s\varphi_{\Sigma}).
\]
By \ref{tamagawa1}, \ref{tamagawa2},
\[ \Theta(\Sigma,K) =
\frac{\beta({\bf P}_{\Sigma})}{h(T_K)} L_S^{-1}(1, T; E/K)
\lim_{ s \rightarrow 1} (s-1)^r
\int_{\overline{T(K)}} H_{ \Sigma }(x,-s\varphi_{\Sigma}) \omega_{\Omega,S}. \]
Notice that $\overline{T(K)}$ contains $T(K_v)$ for $v \not\in S$.
Denote by $\overline{T(K)}_S$ the image of $\overline{T(K)}$ in
$\prod_{v \in S} T(K_v)$.
By \ref{loc-integ}, we have
\[ \int_{\overline{T(K)}} H_{ \Sigma ,v}(x,-s\varphi_{\Sigma}) \omega_{\Omega,S} =
\]
\[ = \prod_{v \not\in S} \int_{T(K_v)} H_{ \Sigma ,v}(x,-s\varphi_{\Sigma}) d\mu_v
\cdot
\int_{\overline{T(K)}_S}
\prod_{v \in S} H_{ \Sigma }(x,-s\varphi_{\Sigma}) \omega_{\Omega,v} = \]
\[ = L_S(s, T;E/K) \cdot \prod_{v \not\in S}
\left( \sum_{k =0}^d
\frac{{\rm Tr}\, \Phi_v(i)}{q_v^{ks}} \right) \cdot
\int_{\overline{T(K)}_S}
\prod_{v \in S} H_{ \Sigma ,v}(x,-s\varphi_{\Sigma}) \omega_{\Omega,v}. \]
\[ L_S^{-1}(1, T; E/K) \omega_{\Omega,S} =
\prod_{v \in {\rm Val}(K)} \omega_{\Omega,v}. \]
By \ref{p-function} and \ref{integral.1},
\[ L_S^{-1}(s, {\rm Pic}({\bf P}_{\Sigma,E}); E/K)
\prod_{v \not\in S} \left( \sum_{k =0}^d
\frac{{\rm Tr}\, \Phi_v(i)}{q_v^{ks}} \right) \]
has no singularity at $s =1$.
Moreover, by \ref{restriction},
\[ \prod_{v \not\in S}
L_S^{-1}(1, {\rm Pic}({\bf P}_{\Sigma,E});E/K) \left( \sum_{k =0}^d
\frac{{\rm Tr}\, \Phi_v(i)}{q_v^{k}} \right)
= \prod_{v \not\in S} \int_{T(K_v)} \lambda_v^{-1} \omega_{{\cal K},v}. \]
Therefore
\[ \Theta(\Sigma,K) =
\frac{\beta({\bf P}_{\Sigma})}{h(T_K)}
\lim_{ s \rightarrow 1} (s-1)^r
L_S(s, {\rm Pic}({\bf P}_{\Sigma,E}); E/K)
\int_{\overline{T(K)}} \omega_{{\cal K},S}. \]
It remains to apply \ref{2int}.
\hfill $\Box$
Using \ref{tauberian}, we obtain:
\begin{coro}
Let $T$ be an anisotropic
torus and ${\bf P}_{\Sigma}$ its smooth compactification $($notice that we do
not need to assume
that ${\bf P}_{\Sigma}$ is a Fano variety$)$. Let $r$ be
the rank of ${\rm Pic}({\bf P}_{\Sigma,K})$.
Then the number $N( {\bf P}_{\Sigma},{\cal K}^{-1}, B)$
of $K$-rational points $x \in T(K)$ having the anticanonical
height $H_{{\cal K}^{-1}}(x) \leq B$ has the asymptotic
\[ N({\bf P}_{\Sigma},{\cal K}^{-1}, B) = \frac{\Theta(\Sigma,K)}{(r-1)!}
\cdot B (\log B)^{r-1}(1+o(1)), \hskip 0,3cm B\rightarrow \infty.\]
\end{coro}
|
1996-01-22T01:48:53 | 9411 | alg-geom/9411013 | en | https://arxiv.org/abs/alg-geom/9411013 | [
"alg-geom",
"math.AG"
] | alg-geom/9411013 | Jean-Bruno Erismann | A. Belkasri, M. Hamade | A New Approach to Characterise all the Transitive Orientations for an
Undirected Graph | 20 pages, LaTeX, macro epsf.sty | null | null | CPT-94/P.3069 | null | A new approach to find all the transitive orientations for a comparability
graph (finite or infinite) is presented. This approach is based on the link
between the notion of ``strong'' partitive set and the forcing theory (notions
of simplices and multiplices). A mathematical algorithm is given for the case
of a comparability graph which has only non limit sub-graphs.
| [
{
"version": "v1",
"created": "Fri, 18 Nov 1994 08:16:05 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Belkasri",
"A.",
""
],
[
"Hamade",
"M.",
""
]
] | alg-geom | \section{Introduction}
The problem of the transitive orientation for a comparability graph
was studied by Golumbic
using the forcing theory [1]. The problem was solved for {\it finite}
comparability graphs and an algorithm was given which gives one
transitive orientation for a {\it finite} comparability graph.
The purpose of this paper is to study the transitive orientation for
the case of {\it infinite} comparability graphs. The results for the
finite case could not be extended to the infinite case because of the
finite type like of the approach used in Ref.[1]. We were then
obliged to consider a new approach, but remaining within the forcing
theory. This was possible by introducing the notion of a {\it
``strong'' partitive set}. It happens that this idea permits to solve
the problem of the transitive orientations by inducing a lot of
characteristics of the forcing theory, from one hand and the
undirected graphs, in general, from the other hand.
The paper is organized as follows: the section 2 is devoted to the
definitions and notations used throughout the paper and also to recall
the main results of the forcing theory which still valid for infinite
graphs. In section 3, we establish narrow links between the notion of
a {\it ``strong'' partitive set} and {\it simplices} (and {\it
multiplices}) which are the principal touls of the forcing theory.
In section 4, we used the results of section 3 to face the problem of
transitive orientation of comparability graphs. We proved a theorem
which is in fact a mathematical algorithm which gives all the
transitive orientations for a comparability graph
which has all its sub-graphs non limit.
\section{Preliminairies}
This section is devoted to the definitions and notations which will be
used throughout this article. We also recall some results about the
partitive sets and the implication classes. These results can be found
with more details in Ref.[1].
We consider here any kind of graphs; finite or infinite. In what
follows we denote by $G=(V,E)$ any graph, where $V$ is the set of
vertices, and by $E(\subseteq V^2)$ the set of edges. Directed edge
will be denoted by $(a,b)$ \newline
(for $a,b\in V$) and an undirected one is denoted
by $ab, ab=\left\{(a,b),(b,a)\right\}$. We say that a graph $G=(V,E)$
is {\it empty} if $E=\phi$.
For any $X\subseteq V$, $G(X)=(X,E(X))$ will denote the sub-graph of
$G$ induced on $X$; where
$$
E(X)=\left\{(a,b)\in E\quad ;\quad \{a,b\}\subseteq X\right\}
$$
we define also the set of vertices $\widetilde{A}$ spanned by a set of
edges $A$ as
$$
\widetilde{A} =\left\{a\in V;\ \hbox{there exists}\ b\in V
\hbox{so that}\ (a,b)\in A
\
\hbox{or}\ (b,a)\in A\right\}
$$
\subsection{Partitive set and ``strong'' partitive set}
Let $\cong$ be a binary relation acting on $V^2$ defined by
$$(a,b)\cong (c,d)\Leftrightarrow\left(\{(a,b),(c,d)\}\cap E=\phi\
\hbox{or}\ \{(a,b),(c,d)\}\subseteq E\right)
$$
This means that the edges $(a,b)$ and $(c,d)$ are both belonging to
$E$ or are both out of $E$.
Let $X\subseteq V$ be a sub-set of $V$. $X$ is a partitive set of $G$
(or $V$) if:
$$
\hbox{for every}\ \{a,b\}\subseteq X\ \hbox{and every}\ c\in V-X
$$
$$
\hbox{we have}\ (a,c)\cong (b,c)\ \hbox{and}\ (c,a)\cong (c,b)
$$
It means that the elements of $X$ are related to any external element
in the same manear. The notion of the partitive set is the analogue of
the notion of interval in an ordered set. We will denote by $I(G)$ the
class of partitive sets of the graph $G$. A partitive set is trivial
if it is a singleton or equal to $V$. By $I^{\star}(G)$ we will denote
the class of non-trivial partitive sets.
\noindent
A graph is indecomposable if all its partitive sets are
trivial; otherwise, it is decomposable.
\noindent
A partitive set $X\in I(G)$ is called a ``strong'' partitive set of
$G$(or $V$) if
for every partitive set $Y\in I(G)$ so that $Y\cap X\not =\phi$,
we have either $X\subseteq Y$ or $Y\subseteq X$.
It means that the eventuality that $X$ and $Y$ shear only a common
part is excluded.
\noindent
We will denote by $I_F(G)$ the class of ``strong'' partitive sets of
$G$ and by $I^{\star}_F(G)$ the class of the non-trivial ``strong''
partitive sets. We say that $G$ is {\it limit} if $I_F(G)$ do not
contain any element different from $V$
which is maximal for the inclusion$(\subseteq)$; otherwise it is
{\it non limit}.
\newpage
\noindent {\it Isomorphism}: Two graphs $G=(V,E)$ and $G'=(V',E')$ are
said to be isomorphic if:
\begin{enumerate}
\item[(i)] {there is a bijection $f:V\to V'$;}
\item[(ii)] {$f$ preserves the edges, i.e., for every
$\{a,b\}\subseteq V:$}
\end{enumerate}
$$
(a,b)\in E\Leftrightarrow (f(a),f(b))\in E'\ .
$$
\noindent {\it Quotient graph}: Let $G=(V,E)$ be a graph and $P$ a
partition of $G$ made of partitive sets $(P\subseteq I(G))$. We define
the {\it quotient graph} of $G$ by $P$, denoted by $G/P$, as the
isomorphic graph to $G(f(P))$, where $f$ is a choice function from $P$
to $V$, i.e., $X\in P\Rightarrow f(X)\in X$.
\begin{proposition} \label{prop2.1}
Let $P$ be a partition of partitive sets (respectively of ``strong''
partitive sets) of $G=(V,E)$ and $X$ a sub-set of $P$. We have then\\
$X\in I(G/P)$ (respectively $X\in I_F(G/P)$) if and only if:
$\cup X\in I(G)$
\noindent $(\hbox{respectively}\ \cup X\in I_F(G))\ ,$
\noindent
where $\cup X$ means the union of the vertices constituting the
partitive sets (respectively the ``strong'' partitive sets) of $X$.
\end{proposition}
\subsection{Implication classes and simplices}
{\it Comparability graph} {\it (or transitively orientable graph)}:
Let
$G=(V,E)$ be an undirected graph. $G$ is a comparability graph if
there exists an orientation of the edges of $G$ so it constitutes a
partial order on $V$.
Comparability graphs are also known as transitively orientable graphs
or partially orderable graphs.
\noindent{\it Implication classes}: Let us define the binary relation
$\Gamma$ on the edges of an undirected graph $G=(V,E)$ as follows:
$$
(a,b)\Gamma(a',b')\Leftrightarrow\left\{\begin{array}{llll}
\hbox{either}& a=a'&\hbox{and}&bb'\not\in E\\
\hbox{or}&b=b'&\hbox{and}&aa'\not\in E.\\
\end{array}\right.
$$
We say that $(a,b)$ directly forces $(a',b')$ whenever $(a,b)\Gamma
(a',b')$. In graphical representation $ab$ and $a'b'$ will have only
one common vertex and the orientation is so that arrows are both
pointed to the extremities or are both pointed to the common vertex
(see Figure 2.1).
Notice that $\Gamma$ is not transitive. In Figure~2.1 $(a,d)$ forces
directly $(a,b)$ and $(a,c)$. The edges $(b,c)$ and $(c,b)$ are
not forced by any
other edge.
\begin{center}
\vglue 0,2cm
\vglue 0,4cm
{\bf Figure 2.1}
\end{center}
Let $\Gamma^{\star}$ be the transitive closure of $\Gamma$. It is easy
to verify that $\Gamma^{\star}$ is an equivalence relation on $E$ and
the equivalence classes related to $\Gamma^{\star}$ are what one
call the {\it implication classes} of $G$.
In what follows we will see that it is useful to define what one call
color classes of $G$ (or shortly colors of $G$). If $A$ is an
implication class, the color class associated to $A$ and denoted by
$\widehat A$, is the union of $A$ and $A^{-1}$ \\
$(\widehat A=A\cup
A^{-1}\subseteq E)$; where $A^{-1}=\{(a,b)\in E$ with $(b,a) \in A\}$.
\begin{theorem}\label{th2.2}
({\it Golumbic Ref.[1]}) Let $G=(V,E)$ be a comparability graph and
$A$ an implication class of $G$. If $O=(V,E')$ is a partial order
associated to $G$, we have necessarily either $E'\cap\widehat A=A$ or
$E'\cap
\widehat A=A^{-1}$ and, in either case, $A\cap A^{-1}=\phi$.
\end{theorem}
\begin{lemma}\label{lem2.3} {\it (The Triangle Lemma).} Let
$A,B$ and
$C$ be implication classes of an undirected graph $G=(V,E)$ with $A
\not = B$ and $A\not = C^{-1}$ and having edges $(a,b)\in C$,
$(a,c)\in B$ and
$(b,c)\in A$, we have then
\begin{enumerate}
\item[(i)] $(b',c')\in A\Rightarrow \left((a,b')\in C\right.$ and
$\left.(a,c')\in B\right)$ ;
\item[(ii)] $\left((b',c')\in A\right.$ and $\left.(a',b')\in
C\right)\Rightarrow (a',c')\in B$ ;
\item[(iii)] $a\not\in\widetilde A$.
\end{enumerate}
\end{lemma}
The following results are consequences of the triangle lemma.
\begin{center}
\vglue 0,2cm
\vglue 0,4cm
{\bf Figure 2.2 } {\it Illustration of the triangle lemma.}
\end{center}
\begin{theorem}\label{th2.4}
({\it Golumbic Ref.[1]}) Let $A$ be an implication class of an
undirected graph $G=(V,E)$. Exactly one of the following alternatives
holds:
\begin{enumerate}
\item[(i)] $A=A^{-1}=\widehat A$ and $\widehat A$ is not transitively
orientable ;
\item[(ii)] $A\cap A^{-1}=\phi$ and then $A$ and $A^{-1}$ are two
transitive orientations of $\widehat A$.
\end{enumerate}
\end{theorem}
\begin{proposition}\label{prop2.5}
Let $X$ be a partitive set $(X\in I(G))$ and $\widehat A$ a color
class of an undirected graph $G=(V,E)$ so that $E(X)\cap\widehat
A\not=
\phi$ we have then
$\widehat A\subseteq E(X)$.
\end{proposition}
\begin{proposition}\label{prop2.6}
If $\widehat A$ is a color class of $G=(V,E)$ then $\widetilde A$ is a
partitive set of $G$ $(\widetilde A\in I(G))$.
\end{proposition}
\noindent {\it Simplex} : Let $G=(V,E)$ be an undirected graph. A
$K_{r+1}$ complete sub-graph of $G$, $S=(V_S,E_S)$ on $r+1$ vertices
is called a {\it simplex of rank} $r$ if each undirected edge $ab$ of
$E_S$ is contained in a different color class of $G$. A simplex is
{\it maximal} if it is not properly contained in any larger simplex.
The {\it multiplex} $M$ generated by a simplex $S$ of rank $r$ is
defined to be the part of $E$ constituted of all edges which their
color classes are present in the simplex $S$
($M(S)={{\cup}_{\widehat A\cap E_S \neq\phi}}\widehat A$).
$M$ is said also a
multiplex of rank $r$. The multiplex $M$ is said to be maximal if $S$
is maximal. We will denote by $\widehat M$ the collection of color
classes present in the multiplex $M$.
\section{Connection between multiplices and ``strong'' partitive sets}
In this section we make connection between the notion of ``strong''
partitive set and multiplices. In our knowledge this was never made
before. It happens that this connection allows us to recover results
of Golumbic [1] for finite graphs and generalize them to the infinite
graphs. The results presented in this section will be used in the
following section to state a theorem on the decomposability of
undirected graphs. In what follows $G$ will denote an undirected graph
unless other mention is pointed out.
\begin{proposition}\label{prop3.1}
If $\widehat A$ and $\widehat B$ are two color classes of $G=(V,E)$ we
have then $(\widehat A=\widehat B)\Leftrightarrow (\widetilde
A=\widetilde B)$.
\end{proposition}
{\it Proof.} It is obvious that $\widehat A=\widehat B \Rightarrow
\widetilde A=\widetilde B$. Let us suppose that $\widetilde A=
\widetilde B$ and $\widehat A\not=\widehat B$, we have then for every
$x\in\widetilde A$ there exists $a\in\widetilde A$ and $b\in\widetilde
B$ so that $ax\in \widehat A$ and $bx\in\widehat B$. Since $\widehat
A\not=\widehat B$ we have necessarily $ab\in E$. Let $C$ be the color
class which contains $ab$. We have then two alternatives:
\begin{enumerate}
\item[(i)] $\widehat C\not=\widehat A\not= \widehat B\Rightarrow b\not\in\widetilde A$
(Triangle Lemma) which is absurd since\\
$b\in \widetilde B=\widetilde A$;
\item[(ii)] $\widehat C=\widehat A\not= \widehat B\Rightarrow a\not\in\widetilde B$
(Triangle Lemma) which is also absurd, since $a\in \widetilde A=\widetilde B$.
\end{enumerate}
\begin{proposition}\label{prop3.2} Let $\widehat A$ be a color class of
$G=(V,E)$ and $X$ a partitive set of $G$ so that $X\subset \widetilde A$,
there exists then $a\in\widetilde A-X$ so that for every $x\in X$, $ax\in\widehat
A$.
\end{proposition}
{\it Proof.} Let $x\in X\subset\widetilde A$, there exists $a\in\widetilde A$ so
that $ax \in \widehat A$. Necessarily $a\in \widetilde A-X$ (otherwise $(ax\in\widehat
A\cap E(X))\Rightarrow\widehat A\subseteq E(X)$~[P.\rf{prop2.5}) then for
every
$x\in X$ we have $xa\in E$, since $X\in I(G)$. Let us assume that
there exists $y\in X$ and a color $ \widehat B\not=\widehat A$ so that $ay\in
\widehat B$, then $xy\in E$. Let $\widehat C$ be the color which contains $xy$.
We have
$\widehat C\not=\widehat A$ and $\widehat C\not=\widehat B$ (otherwise: $(\widehat C=\widehat
A\Rightarrow \widetilde A\subseteq X)$ and $(\widehat C=\widehat
B\Rightarrow \widetilde B\subseteq X)$ which contradicts the fact that
$a\in(\widetilde A\cap\widetilde B)-X$).
Then using the triangle lemma we will have $y\not\in\widetilde A$ which is
absurd. Finally we have for every $x\in X,ax\in \widehat A$.
\begin{theorem}\label{th3.3}
Let $\widehat A$ and $\widehat B$ be two color classes of $G=(V,E)$ so that $\widetilde
A-\widetilde B\not=\phi$ and $\widetilde B-\widetilde A\not=\phi$. We have then that $X=\widetilde
A\cap \widetilde B$ is a ``strong'' partitive set of $G$.
\end{theorem}
{\it Proof.} Since $\widetilde A$ and $\widetilde B$ are partitive sets[P.2.6] we
have that
$X=\widetilde A\cap \widetilde B$ is a partitive set. If $X=\phi$ or $X$ is a
singleton $(|X|=1)$ then $X$ is a ``strong'' partitive set.
Let us suppose now that $|X|>1$. Let $Y$ be a partitive set $(Y\in
I(G))$ so that $X\cap Y\not=\phi$ and $Y-X\not=\phi$ and let
$z\in X\cap Y$ and $y\in Y-X$. We have to show
that $X\subset Y$.
Applying~[P.\rf{prop3.2} we have:
$X\subset\widetilde A\Rightarrow$ there exists $a\in\widetilde A-X$ so that for
every
$x\in X,\ ax\in\widehat A$;
$X\subset\widetilde B\Rightarrow$ there exists $b\in\widetilde B-X$ so that for
every
$x\in X,\ bx\in\widehat B$.
Thus $za\in\widehat A$ and $zb\in \widehat B$.
If $a\in Y$ then $az\in E(Y)\cap\widehat A\not=\phi$, thus $X\subseteq\widetilde
A\subseteq Y$~[P.\rf{prop2.5}.
If $b\in Y$ then $bz\in E(Y)\cap\widehat B\not=\phi$, thus $X\subseteq\widetilde
B\subseteq Y$~[P.\rf{prop2.5}.
Let us now suppose that $\{a,b\}\cap Y=\phi$, we have then:
\begin{enumerate}
\item[(i)] if $y\not\in\widetilde A$ then: $(\{y,z\}\subseteq Y\in I(G)$ and
$za\in E)\Rightarrow ya\in E$; \\
$(\widetilde A\in I(G)$ and $ay\in E)\Rightarrow$
for every $x\in X\subset \widetilde A, yx\in E$. \\
Let $\widehat K$ be the color class of $ya$ and $\widehat C$ that one of
$yz$. We have then: $(y\in(\widetilde K\cap\widetilde C)-\widetilde A$, $a\in\widetilde K-Y$ and
$yz\in\widehat C\cap E(Y)\not=\phi)$ implies that $(\widetilde K\not=\widetilde A\not=
\widetilde C)\Leftrightarrow (\widehat K\not=\widehat A\not=\widehat C)$, where at the last
step we have applied~[P.\rf{prop3.1}. \\
Let $v\in X$ (with $v\not= z$) and $\widehat D$ be the color of $yv$. We
have then: \\
$\widehat K\not=\widehat A\not=\widehat D$ (since $\widehat K\not=\widehat A,yv\in\widehat D\cap
E(Y\cup X)\not=\phi$ and\\
$a\in (\widetilde A\cap\widetilde K)-(Y\cup X)$).
The two tricolor triangles $(a,y,z;\widehat K,\widehat A,\widehat C)$ and $(a,y,v;\widehat
K,\widehat A,\widehat D)$ have two common colors, thus $\widehat D=\widehat C$, thus for
every $x\in X, yx\in\widehat C$ which implies that $X\subset\widetilde C\subseteq
Y$ ($\widetilde C\subseteq Y$ because $yz\in \widehat C\cap E(Y)\neq\phi$[P.2.5])
(see Figure~3.1).
\item[(ii)] If $y\in \widetilde A$ then $(y\not\in X=\widetilde A\cap\widetilde B$ and
$y\in\widetilde A)\Rightarrow y\not\in\widetilde B$. By replacing $a$ by $b$ and
$\widehat A$ by $\widehat B$ up here in (i), we get $X\subset \widetilde C \subseteq Y$.
\end{enumerate}
\begin{center}
\vglue 0,2cm
\vglue 0,4cm
{\bf Figure 3.1} {\em We have : $\widehat D=\widehat C$[Triangle Lemma].}
\end{center}
\smallskip
\begin{theorem}\label{th3.4}
Let $X$ and $Y$ be two ``strong'' partitive sets of $G=(V,E)$ so that
$X\cap Y=\phi$. $X$ and $Y$ can be related to each other by only one
color at most.
\end{theorem}
\begin{center}
\vglue 0,2cm
\vglue 0,4cm
{\bf Figure 3.2}
{\em If $X$ and $Y$ are ``strong'' partitive sets we have necessarily
$\widehat A = \widehat B$}.
\end{center}
\medskip
{\it Proof.} Let $\widehat A$ and $\widehat B$ be two colors connecting $X$ and
$Y$. We have then $X\subset \widetilde A\cap\widetilde B$ and $Y\subset \widetilde A\cap
\widetilde B$ (applying [P.2.6] and the definition of the ``strong''
partitive set). Let us suppose that $\widehat A\not=\widehat B$ and define the
following sets
$$I_A=\left\{Z\in I_F(G)\ \hbox{with}\ Z\subset\widetilde A\right\}\
\hbox{and}\ I_B=\left\{Z\in I_F(G)\ \hbox{with}\ Z\subset\widetilde
B\right\}$$ we have $\cup I_A=\widetilde A$ and $\cup I_B=\widetilde B$ since:
$\cup I_A\subseteq \widetilde A$ and
if $x\in \widetilde A$, then $\{x\}\in I_F(G)\cap I_A$, thus $x\in \cup I_A$
and then $\widetilde A\subseteq \cup I_A$. The same situation holds for $I_B$.
Since $\{X,Y\}\subseteq I_A\cap I_B$ we have $\widetilde A\cap \widetilde
B\not=\phi$, $\widehat A\cap E(\widetilde A\cap \widetilde B)\not=\phi$ and $\widehat B\cap
E(\widetilde A\cap \widetilde B)\not= \phi$ we have then only two alternatives:
\begin{enumerate}
\item[(i)] either $\widetilde A=\widetilde B\Rightarrow\widehat A=\widehat B$~[P.\rf{prop3.1}
\item[(ii)] or $\widetilde A\not=\widetilde B\Rightarrow\left(\widetilde A-(\widetilde A\cap \widetilde
B)\neq\phi
\right.$ or $\left.\widetilde B -(\widetilde A\cap \widetilde B)\neq\phi\right)$ which is
absurd since it contradicts with $\widehat A\cap E(\widetilde A\cap\widetilde B)\not=
\phi$ and $\widehat B\cap E(\widetilde A\cap\widetilde B)\not=\phi$~[P.\rf{prop2.5}.
\end{enumerate}
\begin{corollary}\label{cor3.5}
Let $P$ be a partition of ``strong'' partitive sets of $G=(V,E)$ and
$f$ a choice mapping from $P$ to $V$, i.e., $f:X\in P\to f(X)\in
X\subseteq V$. We have then that the isomorphism from $G/P$ to
$G(f(P))$ conserves the color classes.
\end{corollary}
\begin{corollary}\label{cor3.6}
Let $X$ be a ``strong'' partitive set of $G=(V,E)$ with $X\not= V$ and
let $u\in V-X$. We have then:
\begin{enumerate}
\item[(i)] either for every $x\in X, ux\not\in E$;
\item[(ii)] or there exists a color $\widehat A$ of $G$ so that for every
$x\in X, ux\in \widehat A$.
\end{enumerate}
\end{corollary}
{\it Proof.} This is true because for every $u\in V$ the singleton
$\{u\}$ is a ``strong'' partitive set of $G$.
\begin{theorem}\label{th3.7}
Let $X$ be a ``strong'' partitive set of $G=(V,E)$ and $M(S)$ a
multiplex of $G$ generated by $S=(V_S,E_S)$. We have then the
following implication:
$M\cap E(X)\not=\phi\Rightarrow M\subseteq E(X)$.
\end{theorem}
{\it Proof.} If $E_S\subseteq E(X)$ then for every color $\widehat
A\subseteq M$ we have $\widehat A\cap E(X)\not =\phi$ and thus for every
$\widehat A\subseteq M$, $\widehat A\subseteq E(X)$ which implies that
$M\subseteq E(X)$.
In the other hand if $E_S\cap E(X)=\phi$ then $M\cap E(X)=\phi$. Let
us assume now that $E_S-E(X)\not=\phi$ and $E_S\cap E(X)\not=\phi$,
then $V_S-X\not=\phi$. Let $u\in V_S-X$ and $ab\in E_S\cap E(X)$,
then $\{a,b\}\subseteq X\cap V_S$ and $u$ is related to $a$ and $b$ by
two different colors
(applying the definition of a simplex). This is absurd because it
contradicts with the corollary~[\rf{cor3.6}.
This result is the analogue for multiplices and "strong" partitive
sets of [P.2.5] which deals with colors and partitive sets.
\begin{lemma}\label{lem3.8}
Let $G=(V,E)$ be an undirected graph with the number of vertices
greater than 2 $(|V|>2)$. If $G$ is decomposable and has a color $\widehat
A$ so that $\widetilde A= V$, then $G$ has a non-trivial maximal ``strong''
partitive set.
\end{lemma}
{\it Proof.} Let $X$ be a non-trivial partitive set of $G$ $(X\in
I^{\star}(G))$, thus $X\not=\widetilde A$. Let us define on $I(G)$ the
following binary relation $R$ : \newline
$XRY\Leftrightarrow ((X=Y)$ or $(X\cap
Y\not=\phi, X-Y\not=\phi$ and $Y-X\not=\phi))$. And let $R^{\star}$ be
the transitive closure of $R$.
It is easy to verify that $R^{\star}$ is an equivalence relation on
$I(G)$. Let $X^{\star}$ be the equivalence class of $X$ modulo
$R^{\star}$. By the definition itself of $R^{\star}$, $\cup X^{\star}$ is a
``strong'' partitive set of $G$.
In the other hand $X\not=\widetilde A\Rightarrow$ for every $Y\in X^{\star}$,
$E(Y)\cap\widehat A=\phi$ (otherwise $\widetilde A=Y$ and for every $Z\in
X^{\star}$, $Z\subset Y$ which contradicts with the definition of
$R^{\star}$). We have then $\cup X^{\star}\subset V$. There exists
then
$a\in V-\cup X^{\star}$ so that for every $x\in \cup X^{\star}$,
$ax\in\widehat A$[P.3.2]. But $\{ a\}\in I^{\star}_F(G)$ and if $Y\in
I^{\star}_F(G)$ so that $\{ a\}\subset Y$ then $Y\subseteq V-\cup
X^{\star}$ ( otherwise $\widehat A\cap E(Y)\neq\phi\Rightarrow\widetilde A
=V\subseteq Y $[P.2.5]), consequently $a$ is contained in a maximal
"strong" partitive set different from $V$. Thus $\cup X^{\star}$ is
contained in a non trivial maximal "strong" partitive set.
\begin{theorem}\label{th3.9}
Let $M (S)$ be a multiplex of $G = (V,E)$ generated by the simplex $S
= (V_S,E_S)$. $M (S)$ is maximal if and only if the set of vertices
$\widetilde M$ spanned by $M$ is a ``strong'' partitive set.
\end{theorem}
{\it Proof.} Let us suppose that $M (S)$ is maximal, then $S$ is
maximal. $\widetilde M$ is a partitive set of $G$. Let $Y$ be a partitive set
of $G$ so that $Y \cap \widetilde M \not= \phi$ and $Y - \widetilde M \not= \phi$.
Assume that $\widetilde M - Y \not= \phi$ and let us show that it is absurd.
Let $y \in Y - \widetilde M$. $G (\widetilde M)$ is connected and then there exists
$u
\in Y \cap \widetilde M$ and $v \in \widetilde M - Y$ so that $u v \in M$. The
following statements hold:
\arraycolsep2.5pt
$$
\begin{array}{ll}
( Y \in I (G),\ u \in Y\ \mbox{and}\ v \in \widetilde M - Y ) &
\Rightarrow\ \mbox{For any} x\in Y, xv\in E\Rightarrow\ y v \in E\ ;
\\[3mm]
( {\widetilde M} \in I (G),\ v \in \widetilde M\ \mbox{and}\ y \in Y - \widetilde M ) &
\Rightarrow\ \mbox{for any}\ x \in \widetilde M,\ x y \in E
\end{array}
$$(see Figure 3.3).
\begin{center}
\vglue 0,2cm
\vglue 0,4cm
{\bf Figure 3.3}
{\em For every $x\in {\widetilde M},$ $xy\in E$}.
\end{center}
\medskip
The colors connecting $y$ to the summits of $S$ can not be all
different, otherwise, $M$ will not be maximal.
Let us suppose that there exists $\{ a,b \} \in S$ and a color $\widehat A$
so that $\{ ya, yb \} \subseteq \widehat A$. Since $y \in \widetilde A - \widetilde M$
and
$\widetilde M \in I (G)$, then we have $\widehat A \cap M = \phi$[P.2.5]. If
rank$(M)=1$ then there exists a color $\widehat K$ so that $\widehat K = M$. But
$(ab\in \widehat K
\cap E (\widetilde A) \not= \phi ) \Rightarrow \widetilde K \subseteq \widetilde A$ :
$\{ uv,ab\}\subseteq\widehat K \Rightarrow \{ yu,yv\}\in\widehat A$[Triangle
Lemma], then we have
$(y u \in \widehat A \cap E (Y)) \Rightarrow \widetilde M = \widetilde K \subseteq \widetilde
A \subseteq Y$[P.2.5] ( see Figure 3.4) which is
in contradiction with our proposition.
\begin{center}
\vglue 0,2cm
\vglue 0,4cm
{\bf Figure 3.4}
{\em $yu\in\widehat A\cap E(Y)\neq\phi$ and $ab\in\widehat K\cap E(\widetilde A)\neq
\phi$}.
\end{center}
\medskip
Let us assume now that rank$(M) \ge 2$. Let $c \in V_S - \{ a,b \},\
\widehat B$ the color of $y c,\ \widehat C$ the color of $a c$ and $\widehat D$ the
color of $b c$. If $\widehat A \not= \widehat B$ then $S$ contains two tricolor
triangles: $(y,a,c~;\ a y \in \widehat A,\ c y \in \widehat B,\ a c \in \widehat C)$
and $(y,b,c~;\ b y \in \widehat A,\ c y \in \widehat B,\ b c \in \widehat D)$ having
two common colors, then $\widehat C = \widehat D$~[Triangle lemma] which is
absurd because $S$ is a simplex, then $\widehat A = \widehat B$ (see Figure
3.5).
\begin{center}
\vglue 0,2cm
\vglue 0,4cm
{\bf Figure 3.5}
{\em $\widehat B\neq\widehat A\Rightarrow \widehat C=\widehat D$}.
\end{center}
\medskip
Thus for every $x \in V_S,\ y x \in \widehat A$ and by consequence $V_S
\subset \widetilde A$ which implies that for every color $\widehat H \subset M$
we have $ \widehat H \cap E (\widetilde A) \not= \phi$ then $\widetilde M \subset \widetilde A$.
But
$(y u \in \widehat A \cap E(Y) \not= \phi) \Rightarrow \widetilde A \subseteq Y$
hence $\widetilde M \subset \widetilde A \subseteq Y$, which also contradicts the
first assumption $(\widetilde M - Y \not= \phi)$.
Finally we have showed that $\widetilde M \subset Y$ which means
that $\widetilde M$ is a ``strong'' partitive set. Let us now prove the
converse. Let us assume that $\widetilde M$ is a ``strong'' partitive set of
$G$ and let us show that $M$ is a maximal. If the summits of $\widetilde M$
are related to
$y \in V - \widetilde M$, then they are related by the same color which
achieves the proof.
\begin{corollary}\label{cor3.10}
Let $M (S)$ be a multiplex of $G = (V,E)$ generated by the simplex
$S=(V_S,E_S)$ and let $a \in V - \widetilde M$.
The simplex $S$ is extensible to a simplex $S'=(V_{S'},E_{S'})$
(with $S$ a sub-graph of $S'$) so that $V_{S'} = V_S \cup \{ a \}$ if
and only if there exists $\{ b,c \} \subseteq \widetilde M$ and two colors
$\widehat A$ and $\widehat B$ of $E - M$ so that $a b \in \widehat A$ and $a c \in
\widehat B$.
\end{corollary}
{\it Proof.} Use [C.3.6] and [T.3.9].
\section{Transitive orientations of an undirected graph}
\noindent In this section, using the results of the previous section
we prove the existence of a partition of maximal multiplices for the
set of edges of an undirected graph. Therefore, the transitive
orientations of a comparability graph turn up to the transitive
orientations of their multiplices. These orientations are independent
to each other. A theorem of decomposability for a non limit
undirected graph is proved.
\begin{lemma}\label{lem4.1}
Let $G=(V,E)$ be any graph. Let $X$ and $Y$ be two partitive sets of
$G$ so that $X \subseteq Y$. The following statements hold:
\begin{itemize}
\item[(i)] $X \in I_F (G) \Rightarrow X \in I_F (G (Y))\ ;$
\item[(ii)] $Y \in I_F (G) \Rightarrow (X \in I_F (G) \Leftrightarrow
X
\in I_F (G (Y)).$
\end{itemize}
\end{lemma}
{\it Proof.} Let $X$ be a ``strong'' partitive set of $G$ $(X \in I_F
(G))$. We have for every $Z \in I (G (Y)),\ Z \in I (G)$. Hence $X \in
I_F (G (Y))$. Let $X \in I_F (G (Y)),\ Y \in I_F (G)$ and $Z \in I
(G)$ so that $Z \cap X \not= \phi$ and $Z - X \not= \phi$. Then $Z
\cap X \not=
\phi \Rightarrow Z \cap Y \not= \phi$. But $Y \in I_F (G)$, thus
either
$Z \subseteq Y$ or $Y \subseteq Z$. But $(Y \subseteq Z \Rightarrow X
\subseteq Z)$ and $(X \subseteq Y \Rightarrow Z \in I (G (Y))
\Rightarrow X \subseteq Z)$.
\begin{lemma}\label{lem4.2}
Let $G=(V,E)$ be any graph. Let $F$ and $F'$ be two partitions of $G$
constituted of maximal ``strong'' partitive sets. We have then $F =
F'$.
\end{lemma}
{\it Proof.} Let us assume that $F \not= F'$. Since $F$ and $F'$ are
patitions of $V$, if $X \in F$ then there exists $X' \in F'$ so that
$X
\cap X' \not= \phi$. Thus either $X \subseteq X'$ or $X' \subseteq X$.
But
$X$ and $X'$ are maximal ``strong'' partitive sets, thus $X = X'$.
Hence $F = F'$.
\begin{proposition} \label{prop4.3}
Let $M (S)$ a multiplex of $G = (V,E)$ generated by the
simplex $S$ with rank$(M) \ge 2$. We have then for every $x \in \widetilde M$
there exists two colors $\{ \widehat A,\widehat B \} \subseteq \widehat M$ with $\widetilde
A - \widetilde B \not= \phi$ and $\widetilde B - \widetilde A \not= \phi$ so that $x \in
\widetilde A
\cap
\widetilde B$.
\end{proposition}
{\it Proof.} Let $x \in M$, then there exists $\widehat A \subset M$ and $y
\in \widetilde A$ so that $x y \in \widehat A$. Moreover ($\widehat A \subset M$ and
rank $(M) \ge 2$) $\Rightarrow S$ contains one tricolor triangle:
$(a, b, c\ ;\ \widehat A, \widehat B, \widehat C)$ so that $b c \in \widehat A,\ a c\in
\widehat B$ and $a b \in \widehat C$. Thus $a \not\in \widetilde A,\ b \not\in \widetilde B$
and $c
\not\in \widetilde C$~[Triangle lemma]. Hence $\widetilde A - \widetilde B \not= \phi$
and $\widetilde B - \widetilde A \not= \phi$. In the other hand:
$x y \in \widehat A \Rightarrow ((a x \in \widehat B\ \hbox{and}\ a y \in \widehat
C)\
\hbox{or}\ (a x \in \widehat C\ \hbox{and}\ a y \in \widehat B))$~[Triangle
lemma]. If one suppose $a x \in \widehat B$ then $x \in \widetilde A \cap \widetilde B$.
\begin{theorem}\label{th4.4}
Let $M (S)$ be a multiplex of $G = (V,E)$ generated by the simplex
$S=(V_S,E_S)$.
We have then the following statements:
\begin{itemize}
\item[(i)] $G (\widetilde M)$ has a partition of maximal ``strong''
partitive sets $F_M \not= \{ \widetilde M \}$~;
\item[(ii)] If rank$(M) = 1$, we have either $G (\widetilde M) / F_M$ is
isomorphic to $S$ or $G (\widetilde M) / F_M$ is indecomposable and
isomorphic to a sub-graph $G'=(V',E')$ of $G (\widetilde M)$ so that $E'
\subseteq M$~;
\item[(iii)] If rank $(M) \ge 2$, then $G (\widetilde M) / F_M$ is
isomorphic to $S$.
\end{itemize}
\end{theorem}
{\it Proof.} 1) If rank$(M) = 1$ then $M$ contains only one color $\widehat
A = M$. Moreover if $G (\widetilde M)$ is indecomposable then $F_M = \{ \{ x
\}\ ;\ x \in \widetilde M \}$. Hence $G (\widetilde M) / F_M$ is isomorphic to $G
(\widetilde M)$ and thus $G (\widetilde M) / F_M$ is indecomposable.
Let us assume now that $G (\widetilde M)$ is decomposable. Thus $G
(\widetilde M)$ contains a non-trivial partitive set $X$.
After~[L.\rf{lem4.2}
$F_M$ exists. If $| F_M | = 2$ then $G (\widetilde M) / F_M$ is complete and
isomorphic to $S$.
Let us suppose that $| F_M | \ge 3$. If $G (\widetilde M) / F_M$ has a
non-trivial partitive set $X$, $G (\widetilde M) / F_M$ has a non-trivial
maximal ``strong'' partitive set $Y$~[L.\rf{lem4.2} and [C.3.5].
Hence $\cup Y \in I_F^{\star} (G (M))$[P.21] which is absurd because
$F_M$ is already maximal. Thus $G (\widetilde M) / F_M$ is indecomposable.
\vskip 0.8cm
2) Let us now assume that rank$(M) \ge 2$. Using the
proposition~[\rf{prop4.3} one gets that for every $x \in \widetilde M$ there
exists two colors $\{ \widehat A, \widehat B \} \subseteq \widehat M$ with $\widetilde A -
\widetilde B \not= \phi$ and $\widetilde B - \widetilde A \not= \phi$ so that $x \in \widetilde A
\cap \widetilde B$.
Applying~[T.\rf{th3.3} we have that the intersection $\widetilde A \cap \widetilde B
\not= \widetilde M$ is a ``strong'' partitive set of $G$. Thus $\widetilde A \cap \widetilde
B$ is a ``strong'' partitive set of $G (\widetilde M)$ itself since $\widetilde M
\in I (G)$~[L.\rf{lem4.1}. Let $F_M$ be the set of the intersections
two by two of colors of $M$.
Then for every $a \in V_S$ there exists $X \in F_M$ so that $a
\in X$. Let $\{ a, b \} \subset V_S$ so that $a \in X \in F_M$ and $b
\in Y \in F_M$. Let us assume that there exists a ``strong'' partitive
set $Z$ of $G (\widetilde M)$ so that $X \cup Y \subseteq Z$. $S$ will
contain a tricolor triangle $(a, b, c\ ;\ \widehat A, \widehat B, \widehat C)$ with
$b c \in \widehat A$, $a c \in \widehat B$ and $a b \in \widehat C$. This is absurd
since $\{ c \}$ and $Z$ are two ``strong'' partitive sets of $G (\widetilde
M)$ and can be related at most by only one color[T.3.4]. Hence $F_M$
is a maximal partition of "strong" partitive sets and
separates the summits of $S$.
Finally we get that $G
(\widetilde M) / F_M$ is isomorphic to $S$.
\begin{corollary}\label{cor4.5}
The only multiplices $M (S)$ which might be not transitively
orientable are those of
rank $= 1$ and so that $G (\widetilde M) / F_M$ is non isomorphic to $S$.
\end{corollary}
{\it Proof.} Because complete graphs are orientable.
\begin{corollary}\label{cor4.6}
An undirected graph $G = (V, E)$ can have at most one multiplex which
spanned all its summits.
\end{corollary}
{\it Proof.} Let $M (S)$ and $M (S')$ be two multiplices of $G$ so
that $\widetilde M = V$ and $\widetilde M' = V$, we have then $G (\widetilde M) = G (\widetilde
M') = G$. Let $F$ and $F'$ be two partitions of maximal ``strong''
partitive sets of $G$ related to $M$ and $M'$. Thus $G / F$ is
isomorphic to $S$ and $G / F'$ is isomorphic to $S'$. Moreover the two
isomorphismes conserve the colors~[C.\rf{cor3.5}. But
after~[L.\rf{lem4.2} we have $F = F'$. Hence $S = S'$ and $M = M'$.
\begin{corollary}\label{cor4.7}
Let $G = (V,E)$ be an undirected graph and $M (S),\ M' (S')$ two
maximal multiplices of $G$. We have then $M \cap M' \not= \phi
\Rightarrow M = M'$.
\end{corollary}
{\it Proof.} Since $M$ and $M'$ are ``strong'' partitive sets we
have \\
$M \cap M' \not= \phi \Rightarrow \widetilde M\cap \widetilde {M'}\neq \phi
\Rightarrow ( \widetilde M \subseteq \widetilde M'\ \hbox{or}\ \widetilde M' \subseteq \widetilde
M )$.\newline Let us assume that $\widetilde M \subseteq \widetilde M'$ and let $F,
F'$ be two partitions of maximal ``strong'' partitive sets related
respectively to
$G (\widetilde M)$ and $G (\widetilde M')$. If there exists $X \in F'$ so that $M
\cap E (X) \not= \phi$ then $M \subseteq E (X)$~[T.\rf{th3.7} and $M
\cap M' = \phi$ ( since $G (M') / F'$ is isomorphic to $S'$ and $S$
would be a sub-graph of $G (X)$. Hence $E_S\cap E_{S'}=
\phi$).\newline Finaly we have that for every $X \in F',\ E (X) \cap M
= \phi$. Thus
$S$ is isomorphic to a sub-graph of $S'$. But $S$ is maximal. Hence $S
= S'$ and $M = M'$.
\begin{corollary}\label{cor4.8}
Let $G = (V,E)$ be an undirected graph with $E \not= \phi$. $E$ has
then a partition of maximal multiplices.
\end{corollary}
The theorem 4.4 tell us that a multiplex has the same number of
transitive orientations as the simplex which generated
this multiplex. The simplex itself has a number of transitive
orientations equal to the number of the possible permutations of
its summits. Moreover [C.\rf{cor4.5} asserts that the only multiplices
$M(S)$ which might be not transitively orientable are those with rank
1 so that $G(\widetilde M)/F_M$ is not isomorphic to $S$. Thus using
[C.\rf{cor4.8} the problem of transitive orientation for a
comparability graph come down to the transitive orientation of its
multiplices. But the following problem is rised: if we orientate in
any way and at certain step a given multiplex, will this orientation
influence or not the orientations of the other multiplices at the
following steps ? The response is not and we will prove this statement
using a theorem [T.\rf{th4.11} which is known and for which we propose
a new proof outcoming {}from the forcing theory.\newline
Before announcing [T.\rf{th4.11}, we will announce a theorem
[T.\rf{th4.9} which, in fact, is a mathematical algorithm permitting
to find all the transitive orientations for a comparability graph
which has only non limit sub-graphs, {\em e.g.}, case of finite
graphs.
\begin{lemma}\label{lem4.x}
Let $G=(V,E)$ be a connected undirected graph. Then $G$ can not
contain a multiplex $M$ so that both $\widetilde M\neq V$ and $\widetilde M$ is
maximal for the inclusion among the $\widetilde N$, where $N$ is any
multiplex of $G$
\end{lemma}
{\it Proof.} Let us assume that such a multiplex $M$ exists and
show that it is absurd. Since $\widetilde M$ is maximal for the inclusion it
implies that $M$ is maximal. After[T.\rf{th3.9}, $\widetilde M$ is a "strong"
partitive set. Since $\widetilde M\neq V$ and $G$ is connected, we have
:\newline there exists $x\in V-\widetilde M$ and $y\in \widetilde M$ so that $xy\in
E$. Let $\widehat A$ the color containing $xy$. Then
$\widehat A\not\subset M$ and $x\in (\widetilde A \cap \widetilde M )\neq\phi$.
But $\widetilde M$ is a "strong" partitive set of $G$ and $\widetilde A$ is a
partitive set of $G$, thus $\widetilde M\subset \widetilde A$, which is absurd
because $\widehat A$ is a multiplex of rank 1.
\begin{theorem}\label{th4.9}
Let $G = (V,E)$ be an undirected graph having a partition of maximal
``strong'' partitive sets $F_G \not= \{ V \}$. We have then that $G /
F_G$ satisfies one of the following exclusive assertions:
\begin{itemize}
\item[(i)] $G / F_G$ is empty.
\item[(ii)] $G / F_G$ is indecomposable and there exists a maximal
multiplex $M_G$ of $G$ with rank$(M_G) = 1,\ \widetilde M_G = V$ and $G /
F_G$ is isomorphic to a sub-graph $G'=(V',E')$ of $G$
so that $E' \subseteq
M_G$.
\item[(iii)] $G / F_G$ is complete and isomorphic to a maximal simplex
$S$ generating a maximal multiplex $M_G$ so that $\widetilde M_G = V$.
\end{itemize}
\end{theorem}
{\it Proof.} If $G$ is non connected, $F_G$ is the class of the
connected components and $G / F_G$ is empty. In the other hand, it is
obvious that if $G / F_G $ is empty then $G$ is non connected. Hence
$G / F_G$ is empty if and only if $G$ is non connected.
Let us assume now that $G$ is connected. Since $G$ is non limit,
it implies that $G$ has maximal multiplex $M$ so that $\widetilde M$
is maximal for the inclusion. Thus using the [L.\rf{lem4.x} we have
$\widetilde M =V$. Hence $G=G(\widetilde M)$ and using [T.\rf{th4.4} we get the
result.
\begin{corollary}\label{cor4.10}
Let $G = (V,E)$ be a connected and undirected graph. $G$ has then a
partition of maximal ``strong'' partitive sets $F_G \not= \{ V \}$ if
and only if $G$ has a multiplex $M_G$ so that $\widetilde M_G = V$.
\end{corollary}
{\it Proof.} Applying~[T.\rf{th4.9} we have: $F_G$ exists $\Rightarrow
M_G$ exists. In the other hand applying~[T.\rf{th4.4}, we have the
other implication: $M_G$ exists $\Rightarrow F_G$ exists.
\begin{theorem}\label{th4.11}
Let $O = (V,E)$ be a partial order and $G = (V, E)$ be its
comparability graph. $O$ and $G$ have then the same ``strong''
partitive sets $(I_F (G) = I_F (O))$.
\end{theorem}
Before giving the proof of this theorem, we present some preleminary
results which will be used for the proof.
\begin{lemma}\label{lem4.12}
Let $O = (V,E')$ be a partial order and $G = (V, E)$ its
comparability graph. Then every partitive set of $O$ is a partitive
set of $G$\\
$ (I (O) \subseteq I (G))$.
\end{lemma}
{\it Proof.} Let $Y \in I (O)$. If $Y$ is a trivial partitive set, we
have $Y \in I (G)$. Let us suppose that $Y$ is non trivial, then if $a
\in V - Y$ we have one of the following statements:
\begin{itemize}
\item for every $y \in Y,\ ((a,y) \in E')\Rightarrow a y \in E$.
\item or for every $y \in Y,\ ((y,a) \in E')\Rightarrow a y \in E$.
\item or for every $y \in Y,\ (\{ (a,y), (y,a) \} \cap E' = \phi)
\Rightarrow a y \not\in E$.
\end{itemize}
Therefore $Y \in I (G)$.
\begin{proposition}\label{prop4.13}
Let $O = (V,E')$ be a partial order and $G = (V, E)$ its
comparability graph. Then every ``strong'' partitive set of $G$ is a
``strong'' partitive set of $O$ $ (I_F (G) \subseteq I_F (O))$.
\end{proposition}
{\it Proof.} Let $X$ be a ``strong'' partitive set, {\it i.e.,} $X \in I_F
(G)$. If $X$ is trivial then $X \in I_F (O)$. Let us
suppose that $X$ is non trivial. We have then one of the following
statements:
\begin{itemize}
\item for every $x \in X,\ x a \not\in E$, then for every $\{ x, y \}
\subseteq X,\ (x, a) \cong (y,a)$ (cf.~\S2.1).
\item or there exists a color $\widehat A$ of $G$ so that for every $x \in
X,\ x a \in \widehat A$, then for every $\{ x, y \} \subseteq X,\ (x, a)
\cong (y, a)$ (since $G$ is a comparability graph [T.\rf{th2.2}).
\\ Therefore $X \in I (O)$. Let $Y \in I (O)$ so that $Y \cap X \not=
\phi$ and $Y - X \not= \phi$. Then using the previous
lemma~[\rf{lem4.12} we have $Y \in I (G)$ and then $X \subset Y$. Thus
$X \in I_F (O)$.
\end{itemize}
\begin{proposition}\label{prop4.14}
If $X$ is a maximal ``strong'' partitive set of $G$ then $X$ is a
maximal ``strong'' partitive set of $O$.
\end{proposition}
{\it Proof.} If $X \in I_F (G)$ is maximal, it implies that $G$ is non
limit. Thus $G$ has a partition ${ F}$ of maximal ``strong''
partitive sets and $X \in { F} \subseteq I_F (O)$.
After~[T.\rf{th4.9} we have either $G / { F}$ is empty and then $X$
is a connecting class. Thus $X$ is maximal in $I_F (O)$ ; or $G / {
F}$ is complete and then $O / { F}$ is a chain, therefore $G /
{ F}$ do not has a non trivial ``strong'' partitive sets, thus $X$ is
maximal in $I_F (O)$ ; or $G / { F}$ is indecomposable and the
partitive sets of $G / { F}$ are trivial, therefore,
after~[L.\rf{lem4.12} the partitive sets of $O / { F}$ are trivial,
thus $X$ is maximal in $I_F (O)$.
\begin{proposition}\label{prop4.15}
Let $X$ be a non trivial ``strong'' partitive set of $O$. There exists
then a non trivial ``strong'' partitive set $Y$ of $G$ so that
$X \subseteq Y$.
\end{proposition}
{\it Proof.} If $G$ is non limit then it has a partition ${ F}$ of
maximal ``strong'' partitive sets which is also a partition of maximal
``strong'' partitive sets of $O$[P.4.15]. Thus there exists $Y \in {
F}$ so that $X \subseteq Y$. If $G$ is limit then since $X \in I (O)
\Rightarrow X \in I (G)$~[L.\rf{lem4.12}, therefore $Y$ exists.
\vskip 0.8cm
\noindent
In what follows, we present the proof of the theorem~[\rf{th4.11}.
{\it Proof.} If $X \in I_F (G)$ then after~[P.\rf{prop4.14} $X \in I_F
(O)$.
Let $X \in I_F^{\star} (O)$. Then after~[P.\rf{prop4.15} there exists
$Y
\in I_F^{\star} (G)$ so that $X \subseteq Y$. Let us suppose that $X
\not\in I_F (G)$. Thus $X \subset Y$.\newline
$Y \in I_F^{\star} (G) \Rightarrow Y \in I_F^{\star} (O)$. Therefore
after~[L.\rf{lem4.1} $X \in I_F^{\star} (O (Y))$. Thus there exists
$Y_1
\in I_F^{\star} (G (Y))$ so that $X \subset Y_1$. So we have
constructed a strictly decreasing suite of elements of $I_F^{\star}(G)
\subseteq I_F^{\star} (O)$ (after~[P.\rf{prop4.13}).\newline
The intersection $\cap Y_i$ of this suite is a ``strong'' partitive
set of $O$ and it is the smallest element of $I_F^{\star} (O)$ which
contains
$X$. Therefore $X = \cap Y_i$. But $\cap Y_i$ is also a ``strong''
partitive set of $G$. Thus $X \in I_F^{\star} (G)$ and then $I_F (O) =
I_F (G)$.
\section*{Acknowledgments}
One of the authors, M.~H., would like
to thank Prof. P. ILLE for helpful and stimulating discussions
and also Profs. C. Rauzy and G. Fardoux for their encouragements.
|
1994-11-22T06:20:13 | 9411 | alg-geom/9411014 | en | https://arxiv.org/abs/alg-geom/9411014 | [
"alg-geom",
"math.AG"
] | alg-geom/9411014 | Peter Magyar | Peter Magyar | A Borel-Weil Theorem for Schur Modules | 35pp, LaTeX | null | null | null | null | We present a generalization of the classical Schur modules of $GL(N)$
exhibiting the same interplay among algebra, geometry, and combinatorics. A
generalized Young diagram $D$ is an arbitrary finite subset of $\NN \times
\NN$. For each $D$, we define the Schur module $S_D$ of $GL(N)$. We introduce a
projective variety $\FF_D$ and a line bundle $\LL_D$, and describe the Schur
module in terms of sections of $\LL_D$. For diagrams with the ``northeast''
property,
$$(i_1,j_1),\ (i_2, j_2) \in D \to (\min(i_1,i_2),\max(j_1,j_2)) \in D ,$$
which includes the skew diagrams, we resolve the singularities of $\FD$ and
show analogs of Bott's and Kempf's vanishing theorems. Finally, we apply the
Atiyah-Bott Fixed Point Theorem to establish a Weyl-type character formula of
the form:
$$ {\Char}_{S_D}(x) = \sum_t {x^{\wt(t)} \over \prod_{i,j} (1-x_i
x_j^{-1})^{d_{ij}(t)}} \ ,$$
where $t$ runs over certain standard tableaux of $D$.
Our results are valid over fields of arbitrary characteristic.
| [
{
"version": "v1",
"created": "Mon, 21 Nov 1994 22:44:45 GMT"
}
] | 2015-06-30T00:00:00 | [
[
"Magyar",
"Peter",
""
]
] | alg-geom | \section{$GL(N)$ modules}
\label{Schur modules and Weyl modules}
\subsection{Schur modules}
Given a finite set $T$,
we will also use the symbol $T$ to denote
the order $|T|$ when appropriate.
Thus $GL(T) \stackrel{\rm def}{=} GL(|T|)$, etc.
Let $\Sigma_T$ be the symmetric
group permuting the elements of $T$.
For any left $G$-space $X$, $\Sigma_T$ acts on the
right, and $G$ acts on the left, of the cartesian
product $X^T$ by:
$$
g(x_{t_1}, x_{t_2}, \ldots ) \pi =
(g x_{\pi t_1}, g x_{\pi t_2}, \ldots ) .
$$
A {\em diagram} is a finite subset of
${\bf N} \times {\bf N}$.
Its elements $(i,j) \in D$ are called {\em squares}.
We shall often think of $D$ as a sequence
$(C_1,C_2,\ldots,C_r)$ of columns $C_j \subset {\bf N}$.
The Young diagram corresponding to
$ \lambda = ( \lambda _1 \geq \lambda _2 \geq \cdots \geq \lambda _N \geq 0)$ is the
set $\{(i,j) \mid 1\leq j\leq N,\ 1\leq i\leq \lambda _j \}$.
For any diagram $D$, we let
$$
\mathop{\rm Col}(D) = \{\pi \in \Sigma_D \mid \pi(i,j) = (i',j) \ \exists i'\}
$$
be the group permuting the squares of $D$
within each column, and we define $\mathop{\rm Row}(D)$ similarly
for rows.
Let $F$ be a field.
We shall always write $G = GL(N,F)$, $B = $ the subgroup
of upper triangular matrices, $H =$ the subgroup of diagonal
matrices, and $V = F^N$ the
defining representation.
Now let $F$ have characteristic zero.
Define the idempotents
${\alpha}_D$, $\beta_D$ in the group algebra
$F[\Sigma_D]$ by
$$
{\alpha_D} = {1 \over |\mathop{\rm Row} D|} \sum_{\pi \in \mathop{\rm Row} D} \pi, \ \ \ \
{\beta_D} = {1 \over |\mathop{\rm Col} D|} \sum_{\pi \in \mathop{\rm Col} D} \mathop{\rm sgn}(\pi) \pi ,
$$
where $\mathop{\rm sgn}(\pi)$ is the sign of the permutation.
Define the {\em Schur module}
$$
S_D \stackrel{\rm def}{=} V^{\otimes D} {\alpha_D} {\beta_D} \subset V^{\otimes D},
$$
a representation of $G$.
Note that we get
an isomorphic Schur module
if we change the diagram by permuting the rows or the
columns (i.e., for some permutation $\pi: {\bf N} \rightarrow {\bf N}$,
changing $D = \{(i,j)\}$ to
$D' = \{(\pi(i), j) \mid (i,j) \in D\}$, and similarly for columns).
\subsection{Weyl modules}
Let $W = V^*$, the dual of the defining representation of
$G=GL(N,F)$, where $F$ is an infinite field.
Given a diagram $D$,
define the alternating product
{\em with respect to the columns}
$$
{\bigwedge} ^D W = \{ f:V^D \rightarrow F \mid
f \mbox{ multilinear, and } f(v \pi) = \mathop{\rm sgn}(\pi) f(v) \ \forall \pi
\in \mathop{\rm Col}(D) \},
$$
where {\em multilinear} means $f(v_1,\ldots,v_d)$ is $F$-linear
in each of the $d = |D|$ variables.
Consider the multidiagonal
{\em with respect to the rows}
$$
\Delta^D V = \Delta^{R_1} V \times
\Delta^{R_2} V \times \cdots \subset V^{R_1} \times V^{R_2}
\times \cdots = V^D ,
$$
where $R_1, R_2, \ldots $ are the rows of $D$.
Now define the {\em Weyl module}
$$
W_D \stackrel{\rm def}{=} {\bigwedge} ^D W \mid_{ \Delta^D V},
$$
where $\mid_{ \Delta^D V}$ denotes restriction of functions from
$V^D$ to $ \Delta^D V$.
Since $ \Delta^D V$ is stable under the diagonal action of
$G$, $W_D$ is naturally a $G$-module.
\vspace{1em}
\noindent
{\bf Remark.} For $F$ a finite field,
we make the following modification.
Consider $W = W(F) \hookrightarrow W(\bar{F})$, where
$\bar{F}$ is the algebraic closure.
That is, identify
$$
W = \{ f:\bar{F}^N \rightarrow \bar{F} \mid
f \mbox{ is $\bar{F}$-linear, and }
f(F^N) \subset F \}.
$$
Then define
$$
W_D \stackrel{\rm def}{=} {\bigwedge} ^D W \mid_{ \Delta^D V(\bar{F})}.
$$
This keeps the restriction map from killing
nonzero tensors which happen to vanish on the finite set
$ \Delta^D V(F)$.
With this definition, $W_D$ clearly has the base change property
$W_D(L) = W_D(F) \otimes_F L$ for any extension of
fields $F \subset L$.
Now consider $W_D({\bf Z})$. This is a free ${\bf Z}$-module, since it is
a submodule of the ${\bf Z}$-valued functions on $ \Delta^D V$.
Suppose $D$ satisfies a direction condition. Then our vanishing
results of Proposition \ref{config split} (a),
along with the appropriate universal
coefficient theorems, can be used to show that for any field $F$,
$$
W_D(F) = W_D({\bf Z}) \otimes_{{\bf Z}} F .
$$
\begin{prop}
If $F $ has characteristic zero, then $W_D \cong
S_D^*$ as $G$-modules.
\end{prop}
\noindent
{\bf Proof.} $S_D$ is the image of the composite mapping
$$
V^{\otimes D} {\alpha_D} \hookrightarrow
V^{\otimes D} \stackrel{{\beta_D}}{\rightarrow}
V^{\otimes D} {\beta_D} .
$$
For $W = V^*$, write
$$
W^{\otimes D} = \{f: V^D \rightarrow F \mid f \mbox{ multilinear}\},
$$
$$
\mathop{\rm Sym}\mbox{} ^D W =
\{f: V^D \rightarrow F \mid f \mbox{ multilinear, and }
f(v\pi) = f(v) \ \forall \pi \in \mathop{\rm Row}(D)\}.
$$
Now, representations of $F[\Sigma_D]$ are completely reducible,
so $S_D^*$ is the image of
$$
W^{\otimes D} {\beta_D} \hookrightarrow
W^{\otimes D} \stackrel{{\alpha_D}}{\rightarrow}
W^{\otimes D} {\alpha_D} ,
$$
and $ W^{\otimes D} {\beta_D} \cong \bigwedge^D W$,\
$W^{\otimes D} {\alpha_D} \cong \mathop{\rm Sym}\mbox{}^D W$.
Now, let
$$
{\mathop{\rm Poly}\mbox{}}^D W =
\{f: V^l \rightarrow F \mid f \mbox{ homog poly of multidegree }
(R_1, \ldots, R_l) \},
$$
where $l$ is the number of rows of $D$.
Then we have a $G$-equivariant map
$$
\mathop{\rm rest}\mbox{}_{\Delta} : \mathop{\rm Sym}\mbox{} ^D W \rightarrow \mathop{\rm Poly}\mbox{} ^D W
$$
restricting functions from $V^D$ to the row-multidiagonal $ \Delta^D V \cong V^l$.
It is well known that $\mathop{\rm rest}\mbox{}_{\Delta}$ is an isomorphism:
the symmetric part of a tensor algebra is isomorphic
to a polynomial algebra.
Thus we have the commutative diagram
\begin{eqnarray*}
{\bigwedge} ^D W & \hookrightarrow
W^{\otimes D} \stackrel{{\alpha_D}}{\rightarrow}
& \mathop{\rm Sym}\mbox{} ^D W \\
|| & || & \downarrow \mathop{\rm rest}\mbox{}_{\Delta} \\
{\bigwedge} ^D W & \hookrightarrow
W^{\otimes D} \stackrel{{\alpha_D}}{\rightarrow}
& \mathop{\rm Poly}\mbox{} ^D W .
\end{eqnarray*}
Now, the image in the top row is $S_D^*$,
the image in the bottom row
is $W_D$, and all the vertical maps are isomorphisms,
so we have $\mathop{\rm rest}\mbox{}_{\Delta} : S_D^* \tilde{\rightarrow} W_D$
an isomorphism. $\bullet$
If $D = \lambda $ a Young diagram,
then $W_D$ is isomorphic to Carter and Lusztig's
dual Weyl module for $G = GL(N,F)$.
This will follow from Proposition \ref{Young diagram} in the
following section.
\section{Configuration varieties}
\label{configuration varieties}
{\bf N.B.} Although our constructions
remain valid over ${\bf Z}$, for simplicity we will
assume for the remainder of this paper
that
{\em $F$ is an algebraically closed field}.
\subsection{Definitions and examples}
\label{definitions and examples}
Given a finite set C (a column), and $V=F^N$, consider
$ V^C \cong M_{N\times C}(F)$, the $N \times |C|$ matrices,
with a right multiplication of $GL(C)$.
Let
$$
\mathop{\rm St}(C) = \{X \in V^C \mid \mathop{\rm rank} X = |C| \},
$$
the Stiefel manifold, and
$$
\mathop{\rm Gr}(C) = \mathop{\rm St}(C)/GL(C),
$$
the Grassmannian.
Also, let
$$
{\cal L}_C = \mathop{\rm St}(C) \stackrel{GL(C)}{\times} {\det} ^{-1} \rightarrow
\mathop{\rm Gr}(C)
$$
be the Plucker determinant bundle,
whose sections are regular functions $f: \mathop{\rm St}(C) \rightarrow F$ with
$f(XA) = \det(A) f(X) \ \forall A \in GL(C)$.
In fact, such global sections can be extended to polynomial
functions
$f: V^C \rightarrow F$.
For a diagram $D$ with columns $C_1, C_2,\ldots$,
we let
$$
\mathop{\rm St}(D) = \mathop{\rm St}(C_1) \times \mathop{\rm St}(C_2) \times \cdots,
\mbox{ }
\mathop{\rm Gr}(D) = \mathop{\rm Gr}(C_1) \times \mathop{\rm Gr}(C_2) \times \cdots,
\mbox{ }
\LL_D = {\cal L}_{C_1} \Box\hspace{-0.76em}\times {\cal L}_{C_2} \Box\hspace{-0.76em}\times \cdots.
$$
Recall that $ \Delta^D V \subset V^D$ is the {\em row} multidiagonal
(as opposed to the column constructions above).
Let
$$
\FF_D ^o \stackrel{\rm def}{=} \mathop{\rm Im}\left[ \Delta^D V \cap \mathop{\rm St}(D) \rightarrow \mathop{\rm Gr}(D)\right],
$$
and define the {\em configuration variety of $D$} by
$$
\FF_D = \overline{ \FF_D ^o} \subset \mathop{\rm Gr}(D),
$$
the Zariski closure of $ \FF_D ^o$ in $\mathop{\rm Gr}(D)$.
We denote the restriction of $ \LL_D $ from $\mathop{\rm Gr}(D)$
to $ \FF_D $ by the same symbol $ \LL_D $.
Some properties follow immediately from the definitions.
For instance, $ \FF_D $ is an irreducible variety.
Just as for Schur modules
and Weyl modules,
changing the diagram by permuting the rows
or the columns gives an isomorphic configuration variety and
line bundle.
If we add a column $C$ to $D$ which already
appears in $D$, we get an isomorphic configuration
variety, but the line bundle
is twisted to have higher degree.
Since $ \LL_D $ gives the Plucker embedding on $\mathop{\rm Gr}(D)$,
it is very ample on $ \FF_D $.
\vspace{1em}
\noindent
{\bf Examples. } Set $N=4$. Consider the diagrams
$$
D_1 = \begin{array}{ccc}
\Box & & \\
\Box & \Box & \\
& \Box &
\end{array}
\ \ \
D_2 = \begin{array}{ccc}
\Box & & \\
\Box & \Box & \Box \\
& & \Box
\end{array}
\ \ \
D_3 = \begin{array}{ccc}
\Box & \Box & \\
& \Box & \\
& \Box & \Box
\end{array}
$$
Identifying $\mathop{\rm Gr}(k,F^N)$ with $\mathop{\rm Gr}(k-1,{\bf P}^{N-1}_F)$, we may
consider the $ \FF_D $'s as varieties of configurations in ${\bf P}^3$:\\
(1) $ { \cal F } _{D_1}$ is the variety of pairs $(l,l')$, where
$l,l'$ are intersecting lines in ${\bf P}^3$.
It is singular at the locus where the two lines coincide.\\
(2) $ { \cal F } _{D_2}$ is the variety of triples $(l, p, l')$
of two lines and a point which lies on both of them.
The variety is smooth: indeed, it is a fiber bundle over the partial
flag variety of a line containing a point. There is an obvious
map $ { \cal F } _{D_2} \rightarrow { \cal F } _{D_1}$, which is birational,
and is in fact a small resolution of singularities.
(C.f. Proposition \ref{smooth}.)\\
(3) $ { \cal F } _{D_3}$ is the variety of planes with two marked
points (which may coincide). $ { \cal F } _{D_3}^o$ is the locus where the
marked points are distinct. The variety is smooth as in
the previous example. \\
$$
D_4 = \begin{array}{ccc}
\Box & \Box & \\
\Box & & \Box \\
& \Box & \Box
\end{array}
\ \ \
D_5 = \begin{array}{ccc}
\Box & \Box & \Box \\
\Box & & \\
& \Box & \\
& & \Box
\end{array}
D_6 = \begin{array}{cccc}
\Box & & & \\
& \Box & & \\
& & \Box & \\
& & & \Box
\end{array}
\ \ \
$$
(4) $ { \cal F } _{D_4}$ is the variety of triples of coplanar lines.\\
(5) $ { \cal F } _{D_5}$ is the variety of triples of lines with a common point.
This is the
projective dual of the previous variety,
since the diagrams are complementary
within a $4 \times 3$ rectangle (up to permutation of rows and
columns).
(See Theorem \ref{complement thm}.)
The variety of triples of lines which intersect
pairwise cannot be described by a single diagram, but consists of
$ { \cal F } _{D_4} \cup { \cal F } _{D_5}$. (See Section \ref{intersection varieties}.)\\
(6) $ { \cal F } _{D_6} \cong ({\bf P}^{3})^4$ contains the
$GL(N)$-invariant subvariety where all four points
in ${\bf P}^3$ are colinear. Since the cross-ratio
is an invariant of four points on a line,
this subvariety contains infinitely many $GL(N)$ orbits.
$$
D_7 = \begin{array}{cccc}
\Box & \Box & \Box & \\
& \Box & \Box & \Box \\
& & \Box & \\
& & & \Box
\end{array}
\ \ \
D_8 = \begin{array}{ccccc}
\Box & \Box & \Box & & \\
& \Box & \Box & \Box & \Box \\
& & \Box & & \\
& & & \Box &
\end{array}
$$
(7) $ { \cal F } _{D_7} \cong G \stackrel{B}{\times} X_{ \lambda }$, the $G$-orbit version of
the Schubert variety $X_{ \lambda } \subset \mathop{\rm Gr}(2,4)$
associated to the partition $ \lambda = (1,2)$.
This is the smallest example of a singular Schubert variety. \\
(8) $ { \cal F } _{D_8}$ is a smooth variety
which maps birationally
to $ { \cal F } _{D_7}$ by forgetting the point associated to
the last column. In fact, this is essentially the same
resolution as (1) and (2) above.
Such resolutions of singularities can be given
for arbitrary Schubert varieties of $G = GL(N)$,
and generalize Zelevinsky's resolutions
in~\cite{Z}. C.f. Section \ref{resolution}.
$\bullet$
\begin{thm}
\label{weyl mod is image}
If $F$ is an algebraically closed field, then
$$
W_D \cong
\mathop{\rm Im}\left[\mathop{\rm rest}\mbox{}_{\Delta}: H^0(\mathop{\rm Gr}(D), \LL_D ) \rightarrow H^0( \FF_D , \LL_D )\right],
$$
where $\mathop{\rm rest}\mbox{}_{\Delta}$ is the restriction map.
\end{thm}
\noindent
{\bf Proof.} Note that for $GL(D) = GL(C_1) \times GL(C_2) \times \cdots$,
$$
H^0(\mathop{\rm Gr}(D), \LL_D ) =
\{ f:V^D \rightarrow F \mid
f(XA) = \det(A) \, f(X) \ \forall A \in GL(D)\},
$$
and recall
$$
{\bigwedge} ^D W = \{ f:V^D \rightarrow F \mid
f \mbox{ multilinear, and } f(v \pi) = \mathop{\rm sgn}(\pi) \ f(v) \ \forall \pi
\in \mathop{\rm Col}(D) \}.
$$
But in fact these sets are equal, because a multilinear, anti-symmetric
function $g:V^C\rightarrow F$ always satisfies
$g(XA) = \det(A) \ g(X) \, \forall A \in GL(C)$.
Now $W_D$ and $H^0( \FF_D , \LL_D )$ are gotten by restricting
functions in these identical sets to $ \Delta^D V$, so we are done. $\bullet$
\subsection{Diagrams with at most $N$ rows}
We say $D$ has $\leq N$ rows if $(i,j) \in D \Rightarrow 1\leq i\leq N$.
\begin{prop}
\label{bigorbit}
If $D$ has $\leq N$ rows, then $ \FF_D $ has an open dense $GL(N)$-orbit
$ \FF^{\mbox{\rm \tiny gen}} _D$.
\end{prop}
\noindent
{\bf Proof.}
Let $D$ have columns $C_1, C_2,\ldots$.
Consider a sequence of vectors $X =(v_1,\ldots,v_n) \in V^N$.
For $C = \{i_1, i_2, \ldots \} \subset \{1,\ldots,N\}$,
define $X(C) \stackrel{\rm def}{=} {\mathop{\rm Span}} _F(v_{i_1},v_{i_2},\ldots ) \in \mathop{\rm Gr}(C)$
(for $X$ sufficiently general).
Consider an element $g \in GL(N)$ as a sequence of
column vectors $g = (v_1,\ldots,v_n)$.
Then
$$
g(C) = g\cdot {\mathop{\rm Span}} _F(e_{i_1},e_{i_2},\ldots )
= g\cdot (I(C_1), I(C_2), \ldots) ,
$$
where $e_i$ denotes the $i$-th coordinate vector and
$I$ the identity matrix.
Now define the map
$$
\begin{array}{rccc}
\Psi: & V^N & \rightarrow & \Delta^D V \subset V^D \\
& (v_1,\ldots,v_N) & \mapsto & (v_i)_{(i,j)\in D},
\end{array}
$$
where $(u_{ij})_{(i,j)\in D} $ denotes an element of $V^D$.
Then the composite
$$
V^N \stackrel{\Psi}{\rightarrow} \Delta^D V \rightarrow \FF_D ^o
$$
is an onto map taking
$g \mapsto (g(C_1), g(C_2),\ldots) = g\cdot (I(C_1), I(C_2), \ldots)$.
Since $GL(N)$ is dense in $V^N$,
its image is dense in $ \FF_D ^o$, and hence the composite
image $ \FF^{\mbox{\rm \tiny gen}} _D \stackrel{\rm def}{=} G \cdot (I(C_1),I(C_2),\ldots)$
is a dense $G$-orbit in $ \FF_D $.
$\bullet$
\begin{prop}
\label{Young diagram}
If $D$ is the Young diagram associated to a
dominant weight $ \lambda $ of $GL(N)$, then: \\
(a) $ \FF_D \cong G/P$, a quotient of the flag variety $ { \cal F } = G/B$. \\
(b) The Borel-Weil line bundle
$ {\cal L} _{ \lambda } \stackrel{\rm def}{=} G \stackrel{B}{\times} ( \lambda ^{-1}) \rightarrow { \cal F } $ is
the pullback of $ \LL_D $ under the projection $ { \cal F } \rightarrow \FF_D $. \\
(c) $ \mathop{\rm rest}\mbox{}_{\Delta} :
H^0(\mathop{\rm Gr}(D), \LL_D ) \rightarrow H^0( \FF_D , \LL_D )$
is surjective,
and $W_D \cong H^0( \FF_D , \LL_D )$. \\
\end{prop}
\noindent
{\bf Proof.} (a) Let $\mu = (\mu_1 \geq \mu_2 \geq \cdots) = \lambda ^t $,
the transposed diagram, and let $P = \{(x_{ij}) \in GL(N) \mid
x_{ij} = 0 \mbox{ if } \exists k , \ i > \mu_k \geq j > \mu_{k+1} \}$,
a parabolic subgroup of $G$.
Then $G/P$ is the space of partial flags
$V = F^N \supset V_1 \supset V_2 \supset \cdots $
consisting of subspaces $V_j$ with $dim(V_j) = \mu_j$.
Clearly $G/P \cong \FF_D $. \\
(b) Let $\Psi: G \rightarrow \Delta^D V \cap \mathop{\rm St}(D) $ be the map in the proof
of the previous proposition. Then the map
\begin{eqnarray*}
G \stackrel{P}{\times} F_{ \lambda ^{-1}} & \rightarrow &
\LL_D = (\mathop{\rm St}(D) \stackrel{GL(D)}{\times} {\det} _D^{-1}) \mid_{ \FF_D } \\
(g, \alpha) & \mapsto & (\Psi(g), \alpha)
\end{eqnarray*}
is a $G$-equivariant bundle isomorphism. Then (b) follows
by standard arguments. \\
(c) The surjectivity is a special case of
Proposition \ref{config split}
in Section \ref{cohomology}.
(See also~\cite{In}.)
The other statement then follows by Prop \ref{weyl mod is image}.
$\bullet$
\subsection{Complementary diagrams}
\label{complementary diagrams}
\begin{thm}
\label{complement thm}
Suppose the rectangular diagram
Rect $= \{1,\ldots,N\} \times \{1,\ldots,r\}$ is
the disjoint union of two diagrams $D$, $D'$.
Let $W_D$, $W_{D'}$ be the corresponding Weyl modules for $G =GL(N,F)$.
Then: \\
(a) there is an $F$-linear bijection
$\tau:W_D \rightarrow W_{D'} $
such that
$\tau(g w) = \det_{N\times N}^r(g')\ g'\ \tau(w)$,
where $g'$ is the inverse transpose in $GL(N)$ of the matrix $g$; \\
(b) the characters obey the relation
$ {\mathop{\rm char}} \, {W_{D'}}(h) = \det_{N\times N}^r(h)\ {\mathop{\rm char}} \, {W_D} (h^{-1}) ,$
for diagonal matrices $h\in G$ ; \\
(c) if $F$ has characteristic zero, then as $G$-modules
$$
W_{D'} \cong {\det}\mbox{}^{-r} \otimes W_{D}^* \ \ \ \mbox{ and }\ \ \
S_{D'} \cong {\det}\mbox{}^r \otimes S_{D}^* .
$$
\end{thm}
\noindent
{\bf Proof.} (a) Given $C \subset \{1,\ldots,N\} $
(a column set), we considered above the Plucker line bundle
$$
\mathop{\rm St}(C) \stackrel{GL(C)}{\times} \det\mbox{} ^{-1} \rightarrow \mathop{\rm Gr}(D) .
$$
We may equally well write this as
$$
GL(N) \stackrel{P_C}{\times} \det\mbox{} _C^{-1} \rightarrow \mathop{\rm Gr}(D) ,
$$
where $P_C \stackrel{\rm def}{=} \{ (x_{ij}) \in GL(N) \mid
x_{ij} = 0 \mbox{ if } i \not\in C, \ j \in C\}$
is a maximal parabolic subgroup of $GL(N)$ (not necessarily
containing $B$), and
$\det_C : P_C \rightarrow F$ is the multiplicative character
$\det_C(x_{ij})_{N\times N} \stackrel{\rm def}{=}
\det_{C \times C}(x_{ij})_{i,j \in C}$.
Hence, if $C_1, C_2, \ldots, C_r$ are the columns of $D$,
we may write
$$
Gr(D) \cong G ^r /P_D,
$$
and the bundle
$$
\LL_D \cong G ^r \stackrel{P_D}{\times} \det\mbox{} _D^{-1},
$$
where $P_D \stackrel{\rm def}{=} P_{C_1} \times \cdots \times P_{C_r}$
and $\det_D(X_1,\ldots,X_r) \stackrel{\rm def}{=} \det_{C_1}(X_1)\times \cdots
\times \det_{C_r}(X_r)$.
Under this identification,
$$
\FF_D \cong \mbox{ closure} \mathop{\rm Im}\left[\, \Delta G \hookrightarrow G^r
\rightarrow \mathop{\rm Gr}(D) \, \right]
$$
(c.f. Proposition \ref{bigorbit}).
Now let $\tau: G^r \rightarrow G^r$,
$\tau(g_1,\ldots,g_r) = (g_1', \ldots,g_r')$,
where $g' = \ ^t g^{-1}$, the inverse transpose of a matrix $g\in G$.
Then $\tau(P_D) = P_{D'}$, and $\tau$ induces a map
$$
\tau: \mathop{\rm Gr}(D) \rightarrow \mathop{\rm Gr}(D'),
$$
as well as a map of line bundles
$$
\begin{array}{rccc}
\tau:& \LL_D & \rightarrow & {\cal L}_{D'} \\
& \| & & \| \\
& G^r \stackrel{P_D}{\times} \det\mbox{}_D^{-1}
& &
G^r \stackrel{P_{D'}}{\times} \det\mbox{}_{D'}^{-1} \\
&(g_1, \ldots, g_r, \alpha) & \mapsto &
(g_1', \ldots, g_r', \det(g_1',\ldots,g_r') \alpha) .
\end{array}
$$
This map is not $G$-equivariant. Rather, if we have a
section of $ \LL_D $, $f:G^r \rightarrow F$ (with $f(gp) = \det_D(p) f(g)$
for $p \in P_D$), then for $g_0 \in G$,
we have
$\tau(g_0 f ) = g_0' \det(g_0')^r \tau(f)$
(a section of ${\cal L}_{D'}$).
Since $W_D$ is the restriction of such functions
$f$ to $\Delta G \subset G^r$, and $\tau(\Delta G) \subset \Delta G$,
we have an induced map
$$
\tau: W_D \rightarrow W_{D'}
$$
(an isomorphism of $F$ vector spaces),
satisfying $\tau(g_0 w) = g_0' \det(g_0')^r \tau(w)$
for $g_0 \in G$, $w\in W_D$.
This is the map required in (a),
and now (b), (c) follow trivially.
$\bullet$
\section{Resolution of singularities}
\label{resolution}
In this section,
we define the class of northwest direction diagrams,
which includes
(up to a permutation of rows and columns)
the skew, inversion, Rothe,
and column-convex diagrams.
We construct an explicit resolution of singularities
of the associated configuration varieties
by means of ``blowup diagrams''. We also
find defining equations for these varieties.
One should note that the resolutions constructed are not
necessarily geometric blowups, and can sometimes be
small resolutions, as in Example 8 above.
We shall, as usual, think of a diagram $D$ either as
a subset of ${\bf N} \times {\bf N}$, or as a list $(C_1, C_2, \ldots, C_r)$
of columns $C_j \subset {\bf N}$.
We shall examine only configuration varieties, as
opposed to line bundles on them, so we shall assume that
the columns are without multiplicity: $C_j \neq C_{j'}$
for $j \neq j'$.
\subsection{Northwest and lexicographic diagrams}
A diagram $D$ is {\em northwest} if it possesses the following
property:
$$
(i_1,j_1),\ (i_2, j_2) \in D
\Rightarrow (\min(i_1,i_2),\min(j_1,j_2)) \in D.
$$
Given two subsets $C = \{i_1 < i_2 < \ldots < i_l \}, \ \
C' = \{i'_1 < i'_2 < \ldots i'_{l'}\} \subset {\bf N}$,
we say $C$ is {\em lexicographically less than } $C'$
\ ($C \stackrel{\rm lex}{<} C'$) if
$$
l < l\, ' \mbox{ and } i_1 = i'_1,\ \ldots ,\ i_l = i'_{l'},
$$
$$
\mbox{or } \exists\, m : \ i_1 = i'_1,\ \ldots ,\ i_{m-1} = i'_{m-1},\
i_m < i'_m.
$$
In the first case, we say $C$ is an {\em initial subset} of
$C'$ \ ($C \stackrel{\rm init}{\subset} C'$).
A diagram $D = (C_1, C_2, \ldots)$ is {\em lexicographic}
if $C_1 \stackrel{\rm lex}{<} C_2 \stackrel{\rm lex}{<} \cdots$.
Note that any diagram can be made lexicographic
by rearranging the order of columns.
\begin{lem}
If $D$ is northwest, then the lexicographic rearrangement of $D$ is
also northwest.
\end{lem}
\noindent
{\bf Proof.} (a) I claim that if $j < j'$, then either
$C_j \stackrel{\rm lex}{<} C_{j'}$, or $C_{j} \stackrel{\rm init}{\supset} C_{j'}$.
Let $C_j = \{i_1 < i_2 < \ldots \}$,
$ C_{j'} = \{i'_1 < i'_2 < \ldots \}$.
We have assumed $C_j \neq C_{j'}$.
Thus $C_j \stackrel{\rm lex}{<} C_{j'}$ or
$C_j \stackrel{\rm lex}{>} C_{j'}$.
In the second case, $C_j \stackrel{\rm init}{\supset} C_{j'}$ or
there is an $r$ such that $i_1 = i'_1, \ldots i_{r-1} = i'_{r-1},
i_r < i'_r$.
By the northwest property, this last case would mean
$i'_r \in C_j$, with $i_{r-1} = i'_{r-1} < i'_r < i_r$.
But this contradicts the definition of $C_j$.
Thus the only possibilities are those of the claim. \\
(b) It follows immediately from (a) that
if $C_1 \stackrel{\rm lex}{<} C_2 \stackrel{\rm lex}{<} \cdots \stackrel{\rm lex}{<} C_{s-1} \stackrel{\rm lex}{>} C_s$,
then there is a $t < s$ with
$C_{t-1} \stackrel{\rm lex}{<} C_s$, $C_s \stackrel{\rm init}{\subset} C_t$,
$C_s \stackrel{\rm init}{\subset} C_{t+1}$, \ldots,
$C_s \stackrel{\rm init}{\subset} C_{s-1}$. \\
(c) From (b), we see that to rearrange the columns lexicographically
requires only the following operation:
we start with $C_1, C_2, \ldots$, and when we encounter the
first column $C_s$ which violates lexicographic order, we move it
as far left as possible, passing over those columns $C_i$ with
$C_s \stackrel{\rm init}{\subset} C_i$.
This operation does not destroy the northwest
property, as we can easily
check on boxes from each pair of
columns in the new diagram.
By repeating this operation, we get the lexicographic rearrangement,
which is thus northwest.
\subsection{Blowup diagrams}
The combinatorial lemmas of this section will be used to
establish geometric properties of configuration varieties.
Given a northwest diagram $D$ and two of its columns
$C, C' \subset {\bf N}$, the {\em intersection blowup diagram}
$ \widehat{D} _{C,C'}$ is the diagram with the same columns as $D$
except that the new column
$C \cap C'$ is inserted in the proper lexicographic position
(provided $C \cap C' \neq C, C'$).
\begin{lem}
\label{intlem}
Suppose $D$ is lexicographic and northwest, and $C \stackrel{\rm lex}{<} C'$
are two of its columns. Then:
(a) $C \cap C' \stackrel{\rm init}{\subset} C'$, and
(b) if $C \subset C'$, then $C \stackrel{\rm init}{\subset} C'$.
\end{lem}
\noindent
{\bf Proof.} (a) If $i \in C_j \cap C_{j'}$ and $i > i' \in C_{j'}$,
then $i' \in C_j$ by the northwest property. Similarly for (b).
$\bullet$
\begin{lem}
If $D$ is lexicographic and northwest, then $ \widehat{D} _{C,C'}$ is also
lexicographic and northwest.
\end{lem}
\noindent
{\bf Proof.} If $C = C_j, C' = C_{j'}$ with $j < j'$, and
we insert the column $C \cap C' \stackrel{\rm init}{\subset} C'$ immediately before
$C'$, then we easily check that the resulting diagram is again northwest.
Hence $ \widehat{D} _{C,C'}$, which is the lexicographic rearrangement of this,
is also northwest by a previous lemma.
$\bullet$
Consider the columns $C_1, C_2, \ldots \subset {\bf N}$ of a northwest
diagram $D$, and take the smallest collection
$\{ \widehat{C} _1 \stackrel{\rm lex}{<} \widehat{C} _2 \stackrel{\rm lex}{<} \cdots \}$ of subsets of ${\bf N}$
which contains the $C_i$ and is closed under taking intersections.
Then we define a new diagram
$ \widehat{D} = ( \widehat{C} _1, \widehat{C} _2, \ldots )$
which we call
the {\em maximal intersection blowup diagram} of $D$.
Clearly $ \widehat{D} \, ^{\widehat{}} = \widehat{D} $.
Repeated application of the above lemma shows that
if $D$ is lexicographic and northwest, then so is $ \widehat{D} $.
\vspace{1em}
\noindent
{\bf Examples.} For one of the (non-northwest)
diagrams considered previously,
we have:
$$
D_4 = \begin{array}{ccc}
\Box & \Box & \\
\Box & & \Box \\
& \Box & \Box
\end{array} \ \ \ \ \ \
\widehat{D} _4 = \begin{array}{cccccc}
\Box & \Box & \Box & & & \\
& \Box & & \Box & \Box & \\
& & \Box & & \Box & \Box
\end{array}
$$
For the diagrams $D_7$ and $D_8$ in the previous examples,
$D_8 = \widehat{D} _7$.
$\bullet$
Consider the columns $C \subset {\bf N}$
of a diagram $D$ as a partially ordered
set under $\subset$, ordinary inclusion.
Given two distinct columns $C$, $C'$, we say
$C'$ {\em minimally covers} $C$
(or simply $C'$ {\em covers} $C$)
if $C \subset C'$ and there is no column
of $D$ strictly included between $C$ and $C'$.
\begin{lem}
\label{maxmin}
Let $D$ be a lexicographic northwest diagram,
and $C_L$ be the last column of $D$. Then: \\
(a) there is a column $C_l \neq C_L$ such that
$$
( \bigcup_{C \neq C_L} C)
\cap C_L = C_l \cap C_L ;
$$
(b) if $ \widehat{D} = D$, then $C_L$ covers
at most one other column $C_l$
and is covered by
at most one other column $C_u$.
\end{lem}
\noindent
{\bf Proof.}
(a) Now, by Lemma \ref{intlem}, $C \cap C_L \stackrel{\rm init}{\subset} C_L$
for any column $C$.
Hence the sets $C \cap C_L$ for $C \neq C_L$ are linearly
ordered under inclusion, and there is a largest one
$C_l \cap C_L$.
Thus
$$
( \bigcup_{C \neq C_L} C) \cap C_L
= \bigcup_{C \neq C_L} (C \cap C_L)
= C_l \cap C_L .
$$
(b) By Lemma \ref{intlem},
the columns with $C \subset C_L$ satisfy $C \stackrel{\rm init}{\subset} C_L$ and are
linearly ordered, so there is at most one maximal $C_u$.
Now suppose $C_u, C_u' \stackrel{\rm lex}{<} C_L$ are columns of $D$ both
covering $C_L$. Then again by
Lemma \ref{intlem}, we have $C_u \cap C_u' \stackrel{\rm lex}{\leq} C_u$
or $ \stackrel{\rm lex}{\leq} C_u'$,
so that $C_u \cap C_u' \neq C_L$.
But
$C_u \cap C_u'$ is between $C_L$ and $C_u$,
and between $C_L$ and $C_u'$.
Hence
$C_u = C_u \cap C_u' = C_u'$.
$\bullet$
\subsection{Blowup varieties}
Let $D = (C_1, C_2, \ldots)$ be a lexicographic northwest
diagram,
and $ \widehat{D} = ( \widehat{C} _1, \widehat{C} _2, \ldots)$ be its maximal
intersection blowup.
Recall that $ \widehat{D} $ is obtained by adding certain columns to
$D$, so there is a natural projection map
\begin{eqnarray*}
{\mathop{\rm pr}} : \ \mathop{\rm Gr}( \widehat{D} ) & \rightarrow & \mathop{\rm Gr}(D) \\
(V_{ \widehat{C} })_{ \widehat{C} \in \widehat{D} } & \mapsto &
(V_{C})_{C \in D},
\end{eqnarray*}
obtained by forgetting some of the linear subspaces
$V_{ \widehat{C} } \in \mathop{\rm Gr}( \widehat{C} )$.
\begin{prop}
\label{birational}
If $D$ has $\leq N$ rows, then
$$
{\mathop{\rm pr}} : \ \mathop{\rm Gr}( \widehat{D} ) \rightarrow \mathop{\rm Gr}(D)
$$
induces a birational map of algebraic varieties
$$
{\mathop{\rm pr}} : \ { \cal F } _{ \widehat{D} } \rightarrow \FF_D .
$$
\end{prop}
\noindent
{\bf Proof.} Consider the dense open sets
$ \FF^{\mbox{\rm \tiny gen}} _{ \widehat{D} } \subset { \cal F } _{ \widehat{D} }$ and
$ \FF^{\mbox{\rm \tiny gen}} _D \subset \FF_D $ of Proposition \ref{bigorbit},
consisting of subspaces in general position.
If we consider an element $g \in GL(N)$ as a sequence of
column vectors $g = (v_1,\ldots,v_n)$, and
$C = \{i_1, i_2, \ldots \} \subset \{1,\ldots,N\}$,
recall that we define
$g(C) = {\mathop{\rm Span}} _F(v_{i_1},v_{i_2},\ldots ) \in \mathop{\rm Gr}(C)$.
By definition, any element of $ \FF^{\mbox{\rm \tiny gen}} _{ \widehat{D} }$ can
be written as
$(g( \widehat{C} _1), g( \widehat{C} _2), \ldots ) \in \mathop{\rm Gr}(D)$
for some $g \in GL(N)$.
Now, any column of $ \widehat{D} $ can be written as an intersection
of columns of $D$:
$ \widehat{C} = C_{j_1} \cap C_{j_2} \cap \cdots$.
Then we have
$g( \widehat{C} ) = g(C_{j_1}) \cap g(C_{j_2}) \cap \cdots$,
so the projection map
\begin{eqnarray*}
{\mathop{\rm pr}} : \ \FF^{\mbox{\rm \tiny gen}} _{ \widehat{D} } & \rightarrow & \FF^{\mbox{\rm \tiny gen}} _D \\
(g( \widehat{C} ))_{ \widehat{C} \in \widehat{D} } & \mapsto &
(g(C))_{C \in D}
\end{eqnarray*}
can be inverted:
$$
\begin{array}{rrcl}
{\mathop{\rm pr}} ^{-1} : & \FF^{\mbox{\rm \tiny gen}} _D & \rightarrow & \FF^{\mbox{\rm \tiny gen}} _{ \widehat{D} } \\
& (g(C))_{C \in D} & \mapsto & (g( \widehat{C} ) =
g(C_{j_1}) \cap g(C_{j_2}) \cap \cdots)_{ \widehat{C} \in \widehat{D} }.
\end{array}
$$
Hence the map is birational on the configuration varieties as claimed.
$\bullet$
\subsection{Intersection varieties}
\label{intersection varieties}
Now, given a diagram $D$, define the {\em intersection variety}
$ \II_D $ of $D$ by:
$$
\II_D = \{ (V_C)_{C \in D} \in \mathop{\rm Gr}(D)
\mid \forall C,C',\ldots \in D, \ \ \dim( V_C \cap V_{C'} \cap \cdots
)
\geq |C \cap C' \cap \cdots|\}.
$$
Clearly $ \II_D $ is a projective subvariety of $\mathop{\rm Gr}(D)$,
and $ \FF_D \subset \II_D $.
If $ \widehat{D} = D$ (up to rearrangement of column order),
then the intersection conditions reduce to inclusions:
$$
\II_D = \{ (V_C)_{C \in D} \in \mathop{\rm Gr}(D)
\mid C \subset C' \Rightarrow V_C \subset V_{C'}
\}.
$$
\vspace{1em}
\noindent
{\bf Example.} For the diagram $D_4$ of Section
\ref{definitions and examples},
and $N=4$, $ { \cal I } _{D_4}$ has two irreducible components,
$ { \cal F } _{D_4}$ and $ { \cal F } _{D_5}$. That is, as before,
if we have
three lines in ${\bf P}^3$ with non-empty pairwise intersections,
then either they are coplanar, or they all intersect in a point.
$\bullet$
\begin{lem}
\label{union conditions}
Let $D$ be a northwest diagram, and
$ \II_D $ its intersection variety.
Then any configuration $(V_C)_{C \in D} \in \II_D $
satisfies
$$
\dim(V_C + V_{C'} + \cdots) \leq |C \cup C' \cup \cdots|
$$
for any columns $C, C',\ldots$ of $D$.
\end{lem}
\noindent
{\bf Proof.}
Without loss of generality, assume $D$ is lexicographic.
We use induction on the number of columns in $D$.
Now any list $C, C', \ldots$ of columns of $D$
also constitutes a lexicographic northwest diagram,
so to carry through the induction we need only
prove the statement for {\em all} the columns
$C_1, C_2, \ldots, C_L$ of $D$.
Now, by Lemma \ref{maxmin}, there is a column $C_l \neq C_L$
such that
$( \cup_{C\neq C_L} C) \cap C_L = C_l \cap C_L$.
Then we have
\begin{eqnarray*}
\dim(\ ( \sum_{C\neq C_L} V_C ) \cap V_{C_L}\ )
& \geq & \dim( \sum_{C \neq C_L} (V_C \cap V_{C_L})\ ) \\
& \geq & \dim(\, V_{C_l} \cap V_{C_L} ) \\
& \geq & |\, C_l \cap C_L| \ \ \ \mbox{ since } (V_C) \in \II_D \\
& = & | \ ( \bigcup_{C\neq C_L} C\ ) \cap C_L \ | .
\end{eqnarray*}
Thus we may write
\begin{eqnarray*}
\dim( \sum_{C\in D} V_C ) & = &
\dim( \sum_{C \neq C_L} V_C \, ) + \dim(V_{C_L})
-\dim(\, ( \sum_{C\neq C_L} V_C \, ) \cap V_{C_L} \ ) \\
& \leq & |\! \bigcup_{C\neq C_L} C \, | + |\, C_L|
- |\, ( \bigcup_{C\neq C_L} C \ ) \cap C_L |
\ \ \ \mbox{ by induction} \\
& = & |\bigcup_{C\in D} C \ |
\ \ \ \bullet
\end{eqnarray*}
\begin{lem}
If $D$ is a northwest diagram with
$\leq N$ rows and $ \widehat{D} = D$
(up to rearrangement of column order),
then $ \FF_D $ is an irreducible component of $ \II_D $.
\end{lem}
\noindent
{\bf Proof.}
Recall that $ \FF_D $ is always irreducible. Thus it suffices
to show that $ \FF^{\mbox{\rm \tiny gen}} _D$ is an open subset of $ \II_D $.
Consider the set $ \II_D ^{\mbox{\rm \tiny gen}}$ of configurations
$(V_C)_{C \in D}$ satisfying, for every list $C, C',\ldots$ of
columns in $D$,
$$
\dim (V_C + V_{C'} + \cdots)
= |C \cup C' \cup \cdots |
$$
and
$$
\dim (V_C \cap V_{C'} \cup \cdots)
= |C \cap C' \cap \cdots | .
$$
This is an open subset of $ \II_D $ by the previous lemma.
I claim that $ \FF^{\mbox{\rm \tiny gen}} _D = \II_D ^{\mbox{\rm \tiny gen}}$.
To see this equality,
let $(V_C)_{C \in D} \in \II_D $ satisfy
the above rank conditions,
and we will find a basis $g = (v_1,\ldots,v_N)$ of $V=F^N$
such that $V_C = g(C)$ for all $C$. (C.f. the proof of
Proposition \ref{bigorbit}.)
As before, we consider the columns as a poset under ordinary inclusion.
We begin by choosing mutually independent bases for those
$V_C$ where $C$ is a minimal element of the poset.
This is possible because
$
\dim {\mathop{\rm Span}} (V_C \mid C \mbox{ minimal})
= \sum_{C \ \mbox{\tiny minml}} |C|.
$
Now we consider the $V_C$ where $C$ covers a minimal column.
We start with the basis vectors already chosen, and add enough
vectors, all mutually independent, to span each space.
Again, the dimension conditions ensure there will be no conflict
in choosing independent vectors, since the $V_C$ can have no
intersections with each other except those due to the
intersections of columns.
The condition $ \widehat{D} = D$ ensures that all these intersections
are (previously considered) columns.
We continue in this way for the higher layers of the poset.
We will not run out
of independent basis vectors because
all the columns of $D$
are contained in $\{1,\ldots,N\}$.
$\bullet$
\subsection{Smoothness and equations defining varieties}
\begin{prop}
\label{smooth}
Let $D$ be a northwest diagram
with $\leq N$ rows and $ \widehat{D} = D$
(up to rearrangement of column order).
Then $ \FF_D = \II_D $, and $ \FF_D $ is a smooth variety.
\end{prop}
\noindent
{\bf Proof.}
(a) Let $C_L$ be the last column of $D$, and
let $D'$ be $D$ without the last column.
By lemma \ref{maxmin}, $C_L$ is covered by at most one
other column $C_u$, and covers at most one other column
$C_l$.
If these columns do not exist, take
$C_l = \emptyset$, $C_u = \{1,\ldots,N\}$. \\
(b) Now I claim that there is a fiber bundle
$$
\begin{array}{ccc}
\mathop{\rm Gr}(C_l, C_L, C_u) & \rightarrow & Z \\
& & \downarrow \\
& & \mathop{\rm Gr}(D') \ \,
\end{array}
$$
where $\mathop{\rm Gr}(C_l, C_L, C_u)$ denotes the Grassmannian of
$|C_L|$-dimensional linear spaces which contain a
fixed $|C_l|$-dimensional space and are
contained in a fixed
$|C_u|$-dimensional space;
and
$$
Z = \{ (\, (V_{C'})_{C'}, V_L ) \in \mathop{\rm Gr}(D') \times \mathop{\rm Gr}(C_L) \mid
V_{C_l} \subset V_L \subset V_{C_u} \}.
$$
This is clear. See also~\cite{BD1}.\\
(c) Note that $ \II_D = ( { \cal I } _{D'} \times \mathop{\rm Gr}(C_L) ) \cap Z$.
This is because of the uniqueness of $C_l$ and $C_u$.
Thus the above fiber bundle restricts to
\begin{eqnarray*}
\mathop{\rm Gr}(C_l, C_L, C_u) & \rightarrow & \II_D \\
& & \downarrow \\
& & { \cal I } _{D'} \ \ ,
\end{eqnarray*}
which is thus also a fiber bundle. \\
(d) Now apply the above construction repeatedly,
dropping columns of $D$ from the end.
Finally we obtain $ \II_D $ as an
iterated fiber bundle whose fibers at each step are smooth
and connected
(in fact they are Grassmannians).
In particular, $ \II_D $ is smooth
and connected. \\
(e) Since $ \II_D $ is a smooth, connected, projective algebraic
variety, it must be irreducible. But by a previous lemma,
$ \FF_D $ is an irreducible component of $ \II_D $. Therefore
$ \FF_D = \II_D $, a smooth variety.
$\bullet$
\begin{prop}
\label{conn fibers}
Let $D$ be a northwest diagram with
$\leq N$ rows. Then $ \FF_D = \II_D $, and the
birational projection map
$ { \cal F } _{ \widehat{D} } \rightarrow \FF_D $ has connected
fibers.
\end{prop}
\vspace{1em}
\noindent
{\bf Proof.}
(a) I claim the following:
if $ \widehat{C} $ is a column of $ \widehat{D} $ such that
for all $C \in \widehat{D} $ with $C \stackrel{\neq}{\subset} \widehat{C} $ we have
$C \in D$, then the projection map
$ { \cal I } _{D \cup \widehat{C} } \rightarrow \II_D $ is onto, with
connected fibers.
Suppose $(V_C)_{C\in D}$ is a configuration in $ \II_D $.
Let
$$
V_u = \bigcap_{C \in D \atop C \supset \widehat{C} } V_C
\ \ \ \
\mbox{ and } \ \ \ \
V_l = \sum_{C \in D \atop C \subset \widehat{C} } V_C.
$$
Then $\dim(V_u) \geq | \widehat{C} |$ since $(V_C) \in \II_D $,
and $\dim(V_l) \leq | \widehat{C} |$ by Lemma \ref{union conditions}.
Clearly $V_l \subset V_u$.
Now choose an arbitrary $V_{ \widehat{C} }$ between $V_l$ and $V_u$
with $\dim(V_{ \widehat{C} }) = | \widehat{C} |$.
Then for any list of columns $C, C', \ldots \in D$,
we have either: \\
(i) $ \widehat{C} \cap C \cap C'\cdots = \widehat{C} $, and
$$
V_{ \widehat{C} \cap C \cap C'\cdots} \ = \ V_{ \widehat{C} } \ = \ V_{ \widehat{C} } \cap V_u
\ \subset \ V_{ \widehat{C} } \cap V_C \cap V_{C'} \cap \cdots ;
$$
or (ii) $ \widehat{C} \cap C \cap C'\cdots \stackrel{\neq}{\subset} \widehat{C} $, so that
$ \widehat{C} \cap C \cap C'\cdots \in D$ by hypothesis, and
$$
V_{ \widehat{C} \cap C \cap C'\cdots} \ \subset \ V_l \cap V_C \cap V_{C'} \cdots
\ \subset \ V_{ \widehat{C} } \cap V_C \cap V_{C'} \cap \cdots .
$$
In either case $(V_C)_{C \in D \cup \widehat{C} } \in { \cal I } _{D \cup \widehat{C} }$.
Thus $ { \cal I } _{D \cup \widehat{C} } \rightarrow \II_D $ is onto, and the
fibers are the Grassmannians $\mathop{\rm Gr}(V_l, |C|, V_u)$. \\
(b) We now see that $ { \cal I } _{ \widehat{D} } \rightarrow \II_D $ is onto
(with connected fibers)
by repeated application of (a), starting with
$ \widehat{C} $ minimal in the poset of columns of $ \widehat{D} $ and proceeding
upward. \\
(c) By the previous proposition, the projection map takes
$ { \cal I } _{ \widehat{D} } = { \cal F } _{ \widehat{D} } \rightarrow \FF_D $. But $ { \cal I } _{ \widehat{D} } \rightarrow \II_D $
is onto, so $ \FF_D = \II_D $, and we are done.
$ \bullet $
The above proposition shows that for northwest
diagrams, $ \FF_D $ is defined by the rank conditions of $ \II_D $.
In general, we state the
\begin{conj}
For an arbitrary diagram $D$,
$ \FF_D $ is the set of configurations satisfying
\begin{eqnarray*}
\dim(V_C + V_{C'} + \cdots) & \leq & |\ C \cup C' \cup \cdots| \\
\dim( V_C \cap V_{C'} \cap \cdots )
& \geq & |\ C \cap C' \cap \cdots|
\end{eqnarray*}
for every list $C,C',\ldots $ of columns of $D$.
Equivalently(?), the variety defined by these equations is irreducible.
\end{conj}
\section{Cohomology of line bundles}
\label{cohomology}
Using the technique of Frobenius splitting,
we show certain surjectivity and vanishing theorems for
line bundles on configuration varieties.
In particular, we show that for any
northwest diagram $D$, $ \FF_D $ is normal, and
projectively normal with respect to $ \LL_D $
(so that global sections of $L_D$ on $ \FF_D $ extend to $\mathop{\rm Gr}(D)$);
and $ \FF_D $ has rational singularities.
The material of section \ref{frobenius} was shown to me by Wilberd
van der Kallen.
\subsection{Frobenius splittings of flag varieties}
\label{frobenius}
The technique of Frobenius splitting,
introduced by V.B. Mehta, S. Ramanan, and A. Ramanathan
{}~\cite{MR},~\cite{RR},~\cite{R1},~\cite{R2},
is a method for proving
certain surjectivity and vanishing results.
Given two algebraic varieties $Y \subset
X$ defined
over an algebraically closed
field $F$ of characteristic $p > 0$,
with $Y$ a closed subvariety of $X$,
we say that the pair $Y \subset X$ is
{\em compatibly Frobenius split} if: \\
(i) the $p^{th}$ power map $F: {\cal O} _X \rightarrow F_* {\cal O} _X$
has a splitting, i.e. an $ {\cal O} _X$-module morphism
$\phi: F_* {\cal O} _X \rightarrow {\cal O} _X$ such that $\phi F$ is the
identity; and \\
(ii) we have $\phi(F_* I) = I$, where
$I$ is the ideal sheaf of $Y$.
Mehta and Ramanathan prove the following
\begin{thm}
\label{vanishing}
Let $X$ be a projective variety, $Y$ a closed subvariety,
and $L$ an ample line bundle on $X$.
If $Y \subset X$ is compatibly split, then
$H^i(Y,L) = 0$ for all $i >0$,
and the restriction map
$H^0(X,L) \rightarrow H^0(Y,L)$ is surjective.
Furthermore, if $Y$ and $X$ are defined and projective over ${\bf Z}$
(and hence over any field), and they are compatibly split
over any field of positive characteristic,
then the above vanishing and surjectivity
statements also hold for all fields of characteristic zero.
$\bullet$
\end{thm}
Our aim is to show that, for $D$ a northwest diagram,
$ \FF_D \subset \mathop{\rm Gr}(D)$ is compatibly split.
The above theorem and Theorem \ref{weyl mod is image} will then imply that
$S_D^* \cong H^0( \FF_D , \LL_D ) = \sum_i (-1)^i H^i( \FF_D , \LL_D )$,
the Euler characteristic of $ \LL_D $.
We will also need the following result
of Mehta and V. Srinivas~\cite{MS}:
\begin{prop}
\label{normality}
Let $Y$ be a projective variety which is Frobenius split,
and suppose there exists
a smooth irreducible projective variety $Z$
which is mapped onto $Y$ by an algebraic map
with connected fibers.
Then $Y$ is normal.
Furthermore, if $Y$ is defined over ${\bf Z}$, and is normal
over any field of positive characteristic,
then $Y$ is also normal over all fields of characteristic zero.
$\bullet$
\end{prop}
\begin{prop}
\label{pushforward}
Let $f: Z\rightarrow X$ be a separable morphism
with connected fibers,
where $X$ and $Z$ are projective varieties
and $X$ smooth.
If $Y \subset Z$ is compatibly split, then so is $f(Y) \subset X$.
$\bullet$
\end{prop}
We will show our varieties are split by using the above proposition
to push forward a known
splitting due to Ramanathan~\cite{R2} and O. Mathieu~\cite{M}.
For an integer $n$ and $n$ permutations $w, w', \ldots$,
define
$$
X_n = \underbrace{G \stackrel{B}{\times} G \stackrel{B}{\times} \cdots \stackrel{B}{\times} G}_{n \
\mbox{\rm \tiny factors} }/B,
$$
and the twisted multiple Schubert variety
$$
Y_{w, w' \ldots}
= \overline{B w B} \stackrel{B}{\times} \overline{B w' B} \stackrel{B}{\times} \cdots \subset X_n
$$
Note that we have an isomorphism
$$
\begin{array}{ccc}
X_n & \rightarrow & (G/B)^n \\
(g, g', g'', \ldots) & \mapsto & (g, g g', g g' g'',\ldots).
\end{array}
$$
\begin{prop}{(Ramanathan-Mathieu)}
\label{Ram-Math}
Let $G$ be a reductive algebraic group over a
field of positive characteristic with Weyl group $W$ and
Borel subgroup $B$,
and let $w_0, w_1, \ldots w_r \in W$.
Then $Y_{w_0, w_1, \ldots} \subset X_{r+1}$ is compatibly split.
$\bullet$
\end{prop}
Now, for Weyl group elements $u_1, \ldots, u_r$,
define a variety
$ { \cal F } _{u_1,\ldots,u_r} \subset (G/B)^r$
by
$$
{ \cal F } _{u_1,\ldots,u_r} = \overline{G \cdot ( u_1 B,\ldots, u_r B)}.
$$
\begin{prop}{(van der Kallen)}
\label{splitting}
Let $w_1, \ldots, w_r$ be Weyl group elements,
and define $u_1 = w_1, u_2 = w_1 w_2, \ldots u_r = w_1 \cdots w_r$.
Suppose $w_1, \ldots w_r$ satisfy
$\ell(w_1 w_2 \cdots w_r)
= \ell(w_1) + \ell(w_2) + \cdots + \ell(w_r)$,
or equivalently
$\ell(u_j) = \ell(u_{j-1}) + \ell(u_{j-1}^{-1} u_j)$ for all $j$.
Then the pair $ { \cal F } _{u_1,\ldots,u_r} \subset (G/B)^r$
is compatibly split.
\end{prop}
\noindent
{\bf Proof.}
Define
$$
\begin{array}{rccc}
f: & X_{r+1} & \rightarrow & (G/B)^r \\
& (g_0, g_1, \ldots, g_r) & \mapsto &
(g_0 g_1,\, g_0 g_1 g_2,\, \ldots,\, g_0 g_1 \ldots g_r).
\end{array}
$$
We will examine the image under this map
of
$$
Y \stackrel{\rm def}{=} Y_{w_0, w_1, \ldots w_r}
= G \stackrel{B}{\times} \overline{B w_1 B} \stackrel{B}{\times} \cdots
\stackrel{B}{\times} \overline{B w_r B} \subset X_{r+1} ,
$$
where $w_0$ is the longest element of the Weyl group.
It is well known that, under the given hypotheses,
we have
$(B w_1 B) \cdots (B w_r B) = B w_1 \cdots w_r B$,
and that the multiplication map
$$
B w_1 B \stackrel{B}{\times} \cdots \stackrel{B}{\times} B w_r B \rightarrow B w_1 \cdots w_r B
$$
is bijective.
Thus any element $(g, b_1 w_1 b_1',\ldots, b_r w_r b_r' B) $
(for $b_i, b_i' \in B$) can be written as
$(g , b w_1, w_2, \ldots, w_r B) = (g b, w_1, w_2, \ldots, w_r B)$
for some $b\in B$,
and
$$
f(G \stackrel{B}{\times} B w_1 B \stackrel{B}{\times} \cdots \stackrel{B}{\times} B w_n B)
= f(G \stackrel{B}{\times} w_1 \stackrel{B}{\times} \ldots \stackrel{B}{\times} w_r B)
= G (u_1 B, \ldots , u_r B).
$$
Hence
$f(Y)
= { \cal F } _{u_1,\ldots,u_r}$,
since our varieties are projective.
Now, $f$ is a separable map
with connected fibers
between smooth projective varieties,
so the compatible splitting of the previous
proposition pushes forward
by Proposition \ref{pushforward}.
$\bullet$
We will need the following lemmas to show that
our configuration varieties have rational singularities.
\begin{lem}{(Kempf~\cite{K})}
\label{Kempf}
Suppose $f : Z \rightarrow X$ is a separable
morphism with generically connected fibers
between projective algebraic varieties $Z$ and $X$,
with $X$ normal. Let $L$ be an ample line bundle on $X$,
and suppose that $H^i(Z, f^* L^{\otimes n}) = 0$
for all $i > 0$ and all $n >> 0$.
Then $R^i f_* {\cal O} _Z = 0 $ for all $i>0$.
$\bullet$
\end{lem}
Resuming the notation of Prop \ref{Ram-Math},
let $w_1, \ldots w_n$ be arbitrary Weyl group elements,
and let $ \lambda _1, \ldots \lambda _n$ be arbitrary weights of $G$.
Let $X_n$ be as before, and define the line bundle
$ {\cal L} _{ \lambda _1, \ldots \lambda _n}$ on $X_n$
and on $Y_{w_1, \ldots, w_n} \subset X_n$
as the quotient of
$G^n \times F$
by the $B^n$-action
$$
(b_1, \ldots, b_n)\cdot (g_1, g_2, \ldots, g_n, a) \stackrel{\rm def}{=}
(g_1 b_1, b_1^{-1} g_2 b_2, \ldots , b_{n-1}^{-1} g_n b_n,\,
\lambda _1(b_1) \cdots \lambda _n(b_n) a) .
$$
Note that under the identification $X_n \cong (G/B)^n$,
$ {\cal L} _{ \lambda _1, \ldots, \lambda _n}$ is isomorphic
to the Borel-Weil line bundle
$G^n \stackrel{B^n}{\times} ( \lambda _1^{-1},\ldots \lambda _n^{-1})$.
\begin{lem}
\label{nonample}
Assume $ \lambda _1, \ldots \lambda _n$ are dominant weights
(possibly on the wall of the Weyl chamber).
Then $H^i(Y_{w_1, \ldots, w_n},\, {\cal L} _{ \lambda _1, \ldots, \lambda _n}) = 0$
for all $i > 0$.
\end{lem}
\noindent
{\bf Proof} (van der Kallen).
Note that $ {\cal L} _{ \lambda _1,\ldots, \lambda _n}$ is effective, but not
necessarily ample, so we cannot deduce the conclusion
directly from Theorem \ref{vanishing}.
Recall the following facts from $B$-module theory
{}~\cite{P},~\cite{vdK}: \\
(a) An excellent filtration of a $B$-module is one
whose quotients are isomorphic to Demazure modules
$H^0(\overline{BwB}, {\cal L} _{ \lambda })$, for Weyl group elements $w$
and dominant weights $ \lambda $. \\
(b) If $M$ has an excellent filtration,
and $ {\cal E} (M) \stackrel{\rm def}{=} G \stackrel{B}{\times} M$ is the corresponding
vector bundle on $G/B$, then
$H^i(G/B, {\cal E} (M)) = 0 $ for all $i >0$,
and $H^0(G/B, {\cal E} (M))$ has an excellent filtration. \\
(c) Polo's Theorem: If $M$ has an excellent filtration,
then so does $( \lambda ^{-1}) \otimes M$ for any
dominant weight $ \lambda $.
Now consider the fiber bundle
$$
\begin{array}{ccc}
Y_{w_2,\ldots, w_n} & \rightarrow & Y_{w_1, w_2, \ldots, w_n} \\
& & \downarrow \\
& & \overline{Bw_1B}
\end{array}
$$
which leads to the spectral sequence
$$
H^i(\ \overline{Bw_1B}, \
{\cal E} ( \, ( \lambda _1^{-1}) \otimes
H^j(Y_{w_2,\ldots , w_n}, {\cal L} _{ \lambda _2,\ldots, \lambda _n})\, )\ )
\Rightarrow
H^{i+j}(Y_{w_1, w_2, \ldots, w_n}, {\cal L} _{ \lambda _1, \lambda _2,\ldots, \lambda _n}) .
$$
By induction, assume that
$H^j(Y_{w_2, \ldots, w_n}, {\cal L} _{ \lambda _2,\ldots, \lambda _n}) = 0$ for $j>0$,
and that
$H^0(Y_{w_2, \ldots, w_n}, {\cal L} _{ \lambda _2,\ldots, \lambda _n})$
has an excellent filtration.
Then applying (b) and (c), we find
$$
H^i(Y_{w_1, w_2, \ldots, w_n}, {\cal L} _{ \lambda _1, \lambda _2,\ldots, \lambda _n}) =
H^i(\ \overline{Bw_1B},
\, {\cal E} ( \, ( \lambda _1^{-1}) \otimes
H^0(Y_{w_2,\ldots , w_n}, {\cal L} _{ \lambda _2,\ldots, \lambda _n})\, ) \ )
= 0
$$
for $i>0$,
and that
$H^0(Y_{w_1, w_2, \ldots, w_n}, {\cal L} _{ \lambda _1, \lambda _2,\ldots, \lambda _n})$
has an excellent filtration.
$\bullet$
\subsection{Frobenius splitting of Grassmannians}
We would now like to push forward the
Frobenius splittings found
above for flag varieties
to get splittings of configuration varieties.
For this we need a combinatorial prerequisite.
Given a diagram $D = (C_1, C_2,\ldots, C_r)$
with $\leq N$ rows,
consider a sequence of permutations
(Weyl group elements)
$u_1, u_2, \ldots \in \Sigma_N$
such that, for all $j$: \\
($\alpha$) $\ell(u_j) = \ell(u_{j-1})+\ell(u_{j-1}^{-1} u_{j}) $, and \\
($\beta$) $u_j(\ \{1,2,\ldots,|C_j| \ \}) = C_j$.
The first condition says that the sequence is
increasing in the weak order on the Weyl group.
In the next section, we will give an algorithm
which produces such a sequence for
any northwest diagram, so that the following
theorem will apply:
\begin{prop}
\label{config split}
If $D$ a diagram which admits a sequence
of permutations $u_1, u_2, \ldots$ satisfying
($\alpha$) and ($\beta$) above,
then the pair $ \FF_D \subset \mathop{\rm Gr}(D)$ is compatibly split for any
field $F$ of positive characteristic.
\mbox \\
Hence over an algebraically closed field $F$ of
arbitrary characteristic, \\
(a) the cohomology groups $H^i( \FF_D , \LL_D ) = 0$ for $i > 0$; \\
(b) the restriction map $\mathop{\rm rest}\mbox{}_{\Delta} : H^0(\mathop{\rm Gr}(D), \LL_D )
\rightarrow H^0( \FF_D , \LL_D )$ is surjective;\\
(c) $ \FF_D $ is a normal variety.
\end{prop}
\noindent
{\bf Proof.}
By ($\beta$), the maximal parabolic
subgroups $P_C = \{ (x_{ij}) \in GL(N) \mid
x_{ij} = 0 \mbox{ if } i \not\in C , \ j \in C \}$
satisfy
$u_i B u_i^{-1} \subset P_{C_i}$.
Write
$$
\mathop{\rm Gr}(D) = \mathop{\rm Gr}(C_1) \times \cdots \times \mathop{\rm Gr}(C_r)
\cong G/P_{C_1} \times \cdots \times G/P_{C_r},
$$
and consider the $G$-equivariant projection
$$
\begin{array}{rccc}
\phi: & (G/B)^r & \rightarrow & \mathop{\rm Gr}(D) \\
& (g_1 B,\ldots,g_r B) & \mapsto &
(g_1 u_1^{-1} P_{C_1},\ldots,g_r u_r^{-1} P_{C_r})
\end{array}
$$
Then we have $\phi(u_1 B, \ldots, u_r B) = (I \, P_{C_1},\ldots, I \, P_{C_r})$
and $\phi( { \cal F } _{u_1,\ldots,u_r}) = \FF_D $.
Since $\phi$ is a map with connected fibers
between smooth projective varieties,
we can push forward the compatible splitting
for $ { \cal F } _{u_1,\ldots,u_r} \subset (G/B)^r$ found in the
previous section. Applying
Theorem \ref{vanishing} and Propositions \ref{normality} and
\ref{conn fibers},
we have the assertions of the theorem.
$\bullet$
Note that (b) and (c) of the Proposition are equivalent
to the projective normality of $ \FF_D $ with respect to $ \LL_D $.
\begin{conj}
For any diagram $D$, and any Weyl group elements
$u_1, \ldots u_r$, the pairs $ \FF_D \subset \mathop{\rm Gr}(D)$ and
$ { \cal F } _{u_1, \ldots u_r} \subset (G/B)^r$ are compatibly
split.
\end{conj}
In order to prove the character formula in the last
section of this paper, we will need stronger relations
between the singular configuration varieties and their
desingularizations. In particular,
we will show that our varieties have
rational singularities.
\begin{lem}
Let $X$, $Y$ be algebraic varities with an action of an
algebraic group $G$, and $f: X\rightarrow Y$ an equivariant
morphism. Assume that $X$ has an open dense $G$-orbit $G\cdot x_0$,
and take $y_0 = f(x_0)$, $G_0 = \mathop{\rm Stab}_G y_0$.
Then $f^{-1}(y_0) = \overline{G_0 \cdot x_0}$.
In particular, if $G_0$ is connected,
then $f^{-1}(y_0)$ is connected and irreducible.
\end{lem}
\noindent
{\bf Proof.}
For $F = {\bf C}$, this is trivial. Take $x_1 \in f^{-1}(y_0)$, and
consider a path $x(t) \in X$ such that $x(0) = x_1$ and $x(t) \in G\cdot x_0$
for small $t>0$. Then the path $f(x(t))$ lies in $G\cdot y_0$ for
small $t \geq 0$, and we can lift it to a path $g(t) \in G$ such that
$g(0) = \mathop{\rm id}$ and $f(x(t)) = g(t)\cdot y_0$ for small $t \geq 0$.
Then $\tilde{x}(t) \stackrel{\rm def}{=} g(t)^{-1}\cdot x(t)$ satisfies
$\tilde{x}(0) = x_1$, $\tilde{x}(t) \in G_0\cdot x_0$ for small
$t > 0$.
For general $F$, T. Springer
has given the following clever argument.
Assume without loss of generality that $X$ is irreducible and
$G\cdot y_0$ is open dense in $Y$. Since an algebraic map
is generically flat, and $G\cdot y_0$ is open, all the irreducible
components $C$ of $f^{-1}(y_0)$ have the same dimension
$\dim C = \dim X - \dim Y$. Let $Z = \overline{G\cdot C}$
be the closure of one of these components.
Now, the restriction $f: Z \rightarrow Y$ also satisfies our
hypotheses, with $C \subset Z$ again a component of
the fiber of the restricted $f$,
so we again have $\dim C = \dim Z - \dim Y$, and
$\dim Z = \dim X$. Thus $G\cdot C$ is an open subset
of $X$, since $X$ is irreducible.
Now consider the open set $G\cdot C \cap G\cdot x_0 \subset X$.
Choose a point $z$ in this set which does not lie in any other
component $C'$ of our original $f^{-1}(y_0)$. For any other
component $C'$, choose a similar point $z'$. But we have
$g \cdot z \in C$,
$g'\cdot z' = g_0 g\cdot z \in C'$ for some $g, g', g_0 \in G$,
and in fact $g_0 \in G_0$.
Thus $C' = g_0 \cdot C$, and $G_0$ permutes the components transitively.
Hence, $G_0 \cdot x_0 $ has at least
as many irreducible components as the whole $f^{-1}(y_0)$,
and the lemma follows.
$\bullet$
\begin{prop}{(Inamdar-van der Kallen)}
\label{rational sing}
Suppose $D_1$, $D_2$ are diagrams admitting sequences
of permutations with ($\alpha$) and ($\beta$) as above, such that
$D_2$ is obtained by removing some of the columns of $D_1$.
Denote $ { \cal F } _1 = { \cal F } _{D_1}$, $ { \cal F } _2 = { \cal F } _{D_2}$, $ {\cal L} _2 = {\cal L} _{D_2}$,
and consider the projection
$ {\mathop{\rm pr}} : { \cal F } _1 \rightarrow { \cal F } _2$.\\
Then: \\
(a) $H^0( { \cal F } _1, {\mathop{\rm pr}} ^* {\cal L} _2 ) =
H^0( { \cal F } _2, {\cal L} _2 )$, and this $G$-module has a good
filtration (one whose quotients are isomorphic to
$H^0(G/B, {\cal L} _{ \lambda })$ for dominant weights $ \lambda $).
\\
(b) $H^i( { \cal F } _1, {\mathop{\rm pr}} ^* {\cal L} _2 ) =
H^i( { \cal F } _2, {\cal L} _2 ) = 0$ for all $i > 0$.
\\
(c) $R^i {\mathop{\rm pr}} _* {\cal O} _{ { \cal F } _2} = 0$ for all $i >0$.
\\
(d) If $F$ has characteristic zero, then
$ \FF_D $ has regular singularities for any
northwest diagram $D$.
\end{prop}
\noindent
{\bf Proof.}
(i) Consider a sequence of permutations $w_1, w_2, \ldots w_r$
(where $r$ is the number of columns in $D_1$) such that
$u_1 = w_1, u_2 = w_1 w_2, \ldots $ satisfies
($\alpha$) and ($\beta$), and let $Y = Y_{w_0, w_1, \ldots w_r}$,
(where $w_0$ is the longest permutation).
Then we have a commutative
diagram of surjective morphisms
$$
\begin{array}{ccc}
Y & \stackrel{\Phi_1}{\rightarrow} & { \cal F } _1 \\
& \stackrel{\Phi_2}{\searrow} & \downarrow \mbox{\rm \small pr} \\
& & { \cal F } _2
\end{array}
$$
where $\Phi_j = \phi \, \circ f$, where $\phi$ and $f$ are the
maps defined in the proofs of Propositions \ref{splitting} and
\ref{config split}
in the cases $D = D_j$.
All of these spaces have dense $G$-orbits.
Furthermore, the stabilizer of a general point in $ \FF_D $ is an
intersection of parabolic subgroups and is connected.
Thus, by the above lemma, the fibers of $\Phi_1$ are generically
connected. \\
(ii) Now (i) and Lemma \ref{nonample} insure that the hypotheses
of Kempf's lemma (Proposition \ref{Kempf}) are satisfied.
Thus $R^i(\Phi_1)_* {\cal O} _Y = 0$ for $i>0$,
and by the Leray spectral sequence
we have, for all $i \geq 0$,
$$
H^i(Y, \Phi_1^* {\mathop{\rm pr}} ^* {\cal L} _2) =
H^i( { \cal F } _1, (\Phi_1)_* (\Phi_1)^* {\mathop{\rm pr}} ^* {\cal L} _2) .
$$
(iii) Furthermore, $ { \cal F } _1$ is normal by the previous Proposition,
and $\Phi_1$ is separable with connected fibers, so
\begin{eqnarray*}
(\Phi_1)_* (\Phi_1)^* {\mathop{\rm pr}} ^* {\cal L} _2 & \cong &
[ (\Phi_1)_* (\Phi_1)^* {\cal O} _{ { \cal F } _1} ] \otimes {\mathop{\rm pr}} ^* {\cal L} _2 \\
& \cong & {\mathop{\rm pr}} ^* {\cal L} _2.
\end{eqnarray*}
Thus $H^i(Y, \Phi_1^* {\mathop{\rm pr}} ^* {\cal L} _2) = H^i( { \cal F } _1, {\mathop{\rm pr}} ^* {\cal L} _2)$
for all $i \geq 0$. \\
(iv) An exactly similar argument shows that
$H^i(Y, \Phi_2^* {\cal L} _2) = H^i( { \cal F } _2, {\cal L} _2)$ for all $i \geq 0$.
But $\Phi_2^* = \Phi_1^* {\mathop{\rm pr}} ^*$, so for all $i$,
$$
H^i( { \cal F } _1, {\mathop{\rm pr}} ^* {\cal L} _2) = H^i(Y, \Phi_2^* {\cal L} _2) =
H^i( { \cal F } _2, {\cal L} _2) .
$$
But we saw in Lemma \ref{nonample}
that $H^i(Y, \Phi_2^* {\cal L} _2)$ vanishes for $i>0$,
so (b) of the present Proposition follows. \\
(iv) We also saw in the proof of Lemma \ref{nonample}
that $H^0(Y, \Phi_2^* {\cal L} _2)$ has an excellent filtration
as a $B$-module. But this is equivalent to it having
a good filtration as a $G$-module, so (a) follows. \\
(v) Now consider the spectral sequence
$$
R^i {\mathop{\rm pr}} _* R^j(\Phi_1)_* {\cal O} _Y \Rightarrow R^{i+j} (\Phi_2)_* {\cal O} _Y.
$$
For $i>0$, we have $ R^i(\Phi_1)_* {\cal O} _Y = 0$ and
$R^i(\Phi_2)_* {\cal O} _Y =0 $ by (ii) above.
Because of this and the normality of $ { \cal F } _1$,
we have for all $i>0$,
\begin{eqnarray*}
0 & = & R^i(\Phi_2)_* {\cal O} _Y \\
& = & R^i {\mathop{\rm pr}} _* (\Phi_1)_* {\cal O} _Y \\
& = & R^i {\mathop{\rm pr}} _* {\cal O} _{F_1} .
\end{eqnarray*}
this shows (c). \\
(vi) Now take $D_2 = D$ an arbitrary northwest diagram, and
$D_1 = \widehat{D} $ its maximal blowup. Then $ {\mathop{\rm pr}} $ is a resolution of
singularities by Proposition \ref{smooth}.
Assume, as we will show in the next section,
that $D_1$ admits a sequence of permutations as required.
Then (c) holds, and this is precisely the definition of
rational singularities in characteristic zero, so we have (d).
$\bullet$
\subsection{Monotone sequences of permutations}
Let $D = (C_1, C_2,\ldots, C_r)$
be a northwest diagram
with $\leq N$ rows.
In this section, we will construct by a recursive
algorithm a sequence of permutations
$u_1, u_2, \ldots \in \Sigma_N$
satisfying the conditions
of the previous section: for all $j$, \\
($\alpha$) $\ell(u_j) = \ell(u_{j-1})+\ell(u_{j-1}^{-1} u_{j}) $, and \\
($\beta$) $u_j(\ \{1,2,\ldots,|C_j| \ \}) = C_j$.
For each column $C$ of
$D$, define
the integer
$$
{\mathop{\rm gap}} _ N(C) = \left\{
\begin{array}{l}
\max \{ i\mid i\not\in C,\ \exists i'\in C : i < i' \}
\mbox{, if this set is } \neq \emptyset \\
N \mbox{, if the above set is empty.}
\end{array}
\right.
$$
Since $D$ is northwest, there is an integer $J_N \geq 1$ such that
$$
N = {\mathop{\rm gap}} _ N(C_1) = \cdots = {\mathop{\rm gap}} _ N(C_{J_N-1}) >
{\mathop{\rm gap}} _ N(C_{J_N}) = \cdots = {\mathop{\rm gap}} _ N(C_r).
$$
Now define the {\em derived diagram} $D'$ of $D$
as follows. Given a column $C$ of $D$,
there is a corresponding column $C'$ of $D'$:
$$
C' = \{i\mid i \in C, \ i < {\mathop{\rm gap}} _ N(C) \}
\cup \{ i-1 \mid i\in C,\ i > {\mathop{\rm gap}} _ N(C) \}.
$$
That is, we take $C$ and push
all squares below the $ {\mathop{\rm gap}} _ N(C)$-th row upward
by one place.
\begin{lem}
If $D$ is northwest with $\leq N$ rows, then $D'$ is northwest
with $\leq N-1$ rows.
\end{lem}
\noindent
{\bf Proof.}
The only doubtful case in checking the northwest property is
that of two squares $(i_1,j_1)$ and $(i_2,j_2)$ in $D'$
with $j_1 < J_N \leq j_2$ and $i_1 > i_2$.
Since $j_1 < J_N$, we have $C_{j_1} = \{1,2,\ldots,i_1-1,i_1,\ldots\}$,
so that $i_2 \in C_{j_1}$ and $i_2 \in C_{j_1}'$.
Hence $(i_2,j_1) \in D'$
as required.
$\bullet$
Now, consider the following elements of $\Sigma_N$:
$$
\kappa_n^{(N)}(i) = \left\{ \begin{array}{ll}
i & \mbox{ if } i<n \\
i+1 & \mbox{ if } n\leq i <N \\
n & \mbox{ if } i = N
\end{array}
\right.
$$
Then $\kappa_1^{(N)},\ldots,\kappa_N^{(N)}$ are
minimum length coset representatives
of the quotient $\Sigma_N/ \Sigma_{N-1}$,
and for any permutation $\pi \in \Sigma_{N-1}$,
we have $\ell(\kappa_n \pi) = \ell(\kappa_n) + \ell(\pi)$.
Now, starting with $D$, a northwest diagram with $\leq N$ rows,
we can define a sequence of derived diagrams
$D = D^{(N)}, D^{(N-1)}, \ldots, D^{(1)}$,
where $D^{(i)} = (D^{(i+1)})'$ is a northwest diagram
with $\leq i$ rows.
Let the columns of $D^{(i)}$ be $C_1^{(i)},\ldots,C_r^{(i)}$,
and define $ {\mathop{\rm gap}} (i,j) = {\mathop{\rm gap}} _ i(C_j^{(i)})$.
For each $i$, we have
$$
i = {\mathop{\rm gap}} (i,1) = \cdots = {\mathop{\rm gap}} (i,J_i-1)
> {\mathop{\rm gap}} (i,J_i) = \cdots = {\mathop{\rm gap}} (i,r).
$$
Then either $\kappa_{ {\mathop{\rm gap}} (i,j)}^{(i)}(\{1, 2, \ldots,i-1\})
\supset C_j^{(i)}$,
or $C_j^{(i)} = \{1, 2,\ldots, i\}$.
Notice that $J_N \leq J_{N-1} \leq \cdots$.
Finally, for each column $j = 1,\ldots, r$, define
$$
u_j = \kappa_{ {\mathop{\rm gap}} (N,j)}^{(N)}\,
\kappa_{ {\mathop{\rm gap}} (N-1,j)}^{(N-1)}
\cdots \kappa_{ {\mathop{\rm gap}} (1,j)}^{(1)}.
$$
This is a reduced decomposition of $u_j$, in the sense
that $\ell(u_j)$ is the sum of the lengths of the factors.
Since $\kappa_i^{(i)} = \mathop{\rm id}$, and $J_N \leq J_{N-1} \leq \cdots$,
each $u_j$ is an initial string of $u_{j+1}$.
Thus the $u_j$ have the desired monotonicity property ($\alpha$).
It only remains to show property ($\beta$):
\begin{lem}
For each column $C_j$ of $D$,
$u_j(\{1,2,\ldots,|C_j|\}) = C_j$.
\end{lem}
\noindent
{\bf Proof.}
For each $i$, we have a $u_j^{(i)}$ associated to $D^{(i)}$,
with $u_j^{(i+1)} = \kappa_{ {\mathop{\rm gap}} (i+1,j)}^{(i+1)}\, u_j^{(i)}$.
For a given $i<N$, assume that
$u_j^{(i)}(\{1,2,\ldots, |C_j^{(i)}|\}) \subset C_j^{(i)}$.
Then I claim the same is true for $i+1$.
This is clear, because $C_j^{(i+1)}$ is $C_j^{(i)}$ with some
of its squares pushed down, and $\kappa_{ {\mathop{\rm gap}} (i+1,j)}^{(i+1)}$
pushes down these squares to the proper positions.
In the case that $C_j^{(i+1)} = \{1,2,\ldots,l\}$, for some $l$,
we have
$u_j^{(i+1)} = u_j^{(i)} = \mathop{\rm id}$, and the claim is again true.
The lemma now follows by induction on $i$.
$\bullet$
\section{A Weyl character formula}
The results of the last two sections
allow us to apply the Atiyah-Bott Fixed Point Theorem
to compute the characters of the
Schur modules for northwest diagrams.
To apply this theorem, we must examine the points of
$ \FF_D $ fixed under the action of $H$, the group of
diagonal matrices. We must also understand the action
of $H$ on the tangent spaces at the fixed points.
\subsection{Fixed points and tangent spaces}
\label{fixed points}
The following formula is due to Atiyah and Bott~\cite{AB}
in the complex analytic case, and was extended to the
algebraic case by Nielsen~\cite{N},~\cite{Iv}.
\begin{thm}
\label{AB thm}
Let $F$ be an algebraically closed field,
and suppose the torus $H = (F^{\times})^N$
acts on a smooth projective
variety $X$ with isolated fixed points,
and acts equivariantly on a line bundle $L \rightarrow X$.
Then the character of $H$ acting on the cohomology groups of $L$
is given by:
$$
\sum_i (-1)^i \mathop{\rm tr}(h \mid H^i(X,L)) =
\sum_{p \ \mbox{\tiny fixed}} {\mathop{\rm tr}(h \mid L|_p )
\over \det( \mathop{\rm id} -\, h \mid T^*_p X )},
$$
where $p$ runs over the fixed points of $H$, $L|_p$ denotes the
fiber of $L$ above $p$, and $T^*_p X$ is the cotangent space.
$\bullet$
\end{thm}
We will apply the formula for $X = \FF_D $ a smooth configuration variety,
where $D = (C_1, C_2, \ldots)$ is a lexicographic northwest
diagram with $\leq N$ rows and $ \widehat{D} = D$.
\vspace{1em}
\noindent
{\bf Fixed points.}
Assume for now that the columns are all distinct.
Let
$H = \{h=\mathop{\rm diag}(x_1,\ldots,x_N) \in GL(N) \}$
act on $\mathop{\rm Gr}(D)$ and $ \FF_D $ by the restriction of the $GL(N)$ action.
Then by Proposition \ref{smooth}, we have
$ \FF_D = \II_D = \{(V_C)_{C \in D} \in \mathop{\rm Gr}(D) \mid
C \subset C' \Rightarrow V_C \subset V_{C'} \}$,
a smooth variety.
A point in $ \FF_D \subset \mathop{\rm Gr}(D) = \mathop{\rm Gr}(C_1) \times \mathop{\rm Gr}(C_2) \times \cdots$
is fixed by $H$ if and only if each component is fixed.
Now, the fixed points of $H$ in $\mathop{\rm Gr}(l, F^N)$ are
the coordinate planes
$E_{k_1,\ldots,k_l} = {\mathop{\rm Span}} (e_{k_1}, \ldots, e_{k_l})$, where
the $e_k$ are coordinate vectors in $F^N$
(c.f.~\cite{H}).
For instance, the fixed points in ${\bf P}^{N-1}$ are the $N$ coordinate
lines $F e_k$.
We may describe the fixed points in $\mathop{\rm Gr}(C)$ as
$E_S = {\mathop{\rm Span}} (e_k \mid k\in S)$, where
$S \subset \{1,\ldots,N\}$ is any set with $|S| = |C|$.
Hence the fixed points in $ \FF_D $ are as follows:
Take a function $t$ which assigns to any column $C$
a set $t(C)\subset \{1,\ldots,N\}$ with $|\, t(C)| = |C|$,
and $C \subset C' \Rightarrow t(C) \subset t(C')$.
We will call such a $t$ a {\em standard column tabloid}
for $D$. Then the fixed point corresponding to $t$ is
$E_t = (\, E_{t(C)})_{C\in D}$.
\vspace{1em}
\noindent
{\bf Tangent spaces at fixed points.}
We may naturally identify the tangent space
$T_{V_0} Gr(l, F^N) = \mathop{\rm Hom}_F(V_0, F^N/V_0)$.
If $V_0$ is a fixed point (that is, a space stable under $H$),
then $h \in H$ acts
on a tangent vector $\phi \in Hom_F(V_0,F^N/V_0)$
by $(h\cdot \phi)(v) = h (\phi( h^{-1} v))$.
For $(V_C)_{C\in D} \in \mathop{\rm Gr}(D)$, we have
$T_{(V_C)} \mathop{\rm Gr}(D) = \bigoplus_{C\in D} Hom(V_C, F^N/V_C)$.
Furthermore, if $(V_C)_{C\in D} \in \FF_D $, then
$$
\begin{array}{cl}
T_{(V_C)} \FF_D = \{ \phi = (\phi_C)_{C\in D} &
\! \! \in \bigoplus_{C\in D} Hom(V_C, F^N/V_C) \mid \\
& C \subset C' \Rightarrow \phi_{C'}|_{V_C} \equiv
\phi_C \mbox{ mod } V_{C'} \}
\end{array}
$$
(that is, the values of $\phi_{C}$ and $\phi_{C'}$ on
$V_C$ agree up to translation by elements of $V_{C'}$).
See~\cite{H}.
For a fixed point $E_t = (E_{t(C)})$, we will find a basis
for $T_{E_t}$ consisting of eigenvectors of $H$.
Now, the eigenvectors in $T_{E_t} \mathop{\rm Gr}(D) =
\bigoplus_{C\in D} Hom(V_C, F^N/V_C)$ are precisely
$\phi^{ijC_0}= (\phi^{ijC_0}_C)_{C \in D}$,
where $i,j\leq N$, $C_0$ is a fixed column of $D$,
and $\phi^{ijC_0}_C(e_l) \stackrel{\rm def}{=} \delta_{C_0, C} \delta_{il} e_j$
($\delta$ being the Kronecker delta).
The eigenvalue is
$$
h \cdot \phi^{ijC_0} = \mathop{\rm diag}(x_1, \ldots x_N)
\cdot \phi^{ijC_0} = x_i^{-1} x_j \ \phi^{ijC_0}.
$$
To obtain eigenvectors of $T_{E_t} \FF_D $, we must impose the
compatibility conditions. An eigenvector $\phi$ with eigenvalue
$x_i^{-1} x_j$ must be a linear combination
$\phi = \sum_{C \in D} a_C \phi^{ijC}$ with $a_C \in F$.
By the compatibility, we have that
$$
C\subset C',\, i\in t(C),\, j\not\in t(C')\ \ \Rightarrow \ \ a_C = a_{C'} .
$$
We wish to find the number $d_{ij}$
of linearly independent solutions of this condition
for $a_C$.
Given a poset with a relation $\subset$,
define its {\em connected components}
as the equivalence classes generated
by the elementary relations $x \sim y$ for $x \subset y$.
Now for a given $i,j$ consider the poset whose elements are those
columns $C$ of $D$ such that $i\in t(C)$, $j \not\in t(C)$,
with the relation of ordinary inclusion.
Then $d_{ij}$ is the number of components of this poset.
Note that the eigenvectors for all the eigenvalues
span the tangent space.
Thus
$$
\det(\, \mathop{\rm id} - h \ | \ T^*_{E_t}) = \prod_{i\neq j}
(1 - x_i x_j^{-1})^{d_{ij}(t)}.
$$
\vspace{1em}
\noindent
{\bf Bundle fibers above fixed points.}
Finally, let us examine the line bundles $L$
on $ \FF_D $ obtained by giving each column $C$
a multiplicity $m(C) \geq 0$.
If $m(C) > 0$ for all columns $C$ of $D$, then $L \cong {\cal L} _{D'}$
for the diagram $D'$ with the same columns as $D$, each
repeated $m(C)$ times. If some of the $m(C) = 0$,
then $L$ is the {\em pullback} of $ {\cal L} _{D'}$
for the diagram
$D'$ with the same columns as $D$, each taken $m(C)$ times,
where 0 times means deleting the column.
In the second case, $L$ is effective, but not ample.
It follows easily from the definition that
$$
\mathop{\rm tr}( h \mid L|_{E_t} ) = x_1^{- {\mathop{\rm wt}} _1(t)} \cdots x_N^{- {\mathop{\rm wt}} _N(t)},
$$
where
$$
{\mathop{\rm wt}} _ i(t) = \sum_{C \atop i \in t(C)} m(C) .
$$
Hence we obtain:
$$
\sum_i (-1)^i \mathop{\rm tr}(h \mid H^i( \FF_D , \LL_D )) =
\sum_{t} { \prod_i x_i^{- {\mathop{\rm wt}} _ i(t)} \over
\prod_{i\neq j} (1-x_i x_j^{-1})^{d_{ij}(t)} },
$$
where $t$ runs over the standard column tabloids of $D$.
\subsection{The character formula}
\label{character formula}
We summarize in combinatorial language the implications of the
previous section.
We think of a diagram $D$
as a list of columns $C_1, C_2, \ldots \subset {\bf N}$,
possibly with repeated columns.
Given a diagram $D$, the {\em blowup diagram} $ \widehat{D} $ is the
diagram whose columns consist of all the columns of $D$ and all
possible intersections of these columns. We will call the
columns which we add to $D$ to get $ \widehat{D} $ the {\em phantom columns}.
We may define a {\em standard column tabloid} for the diagram $ \widehat{D} $\,
with respect to $GL(N)$,
to be a filling (i.e. labeling)
of the squares of $ \widehat{D} $ by integers in $\{1,\ldots,N\}$,
such that:\\
(i) the integers in each column are strictly increasing, and\\
(ii) if there is an inclusion $C \subset C'$ between two columns,
then all the numbers in the filling of $C$ also appear in the filling of $C'$.
Given a tabloid $t$ for $ \widehat{D} $, define integers $ {\mathop{\rm wt}} _ i(t)$ to be the
number of times $i$ appears in the filling, but
{\em not counting i's which appear in the phantom columns}.
Also define integers $d_{ij}(t)$ to be the number of connected
components of the following graph: the vertices are columns $C$ of
$ \widehat{D} $ such that $i$ appears in the filling of $C$, but $j$ does not;
the edges are $(C,C')$ such that $C \subset C'$ or $C' \subset C$.
(An empty graph has zero components.)
Recall that a diagram $D$ is {\em northwest} if
$ i \in C_j,\ i' \in C_{j'} \ \Rightarrow \min(i,i') \in C_{\min(j,j')} $.
The following theorem applies without change to {\em northeast} diagrams
and any other diagrams obtainable from northwest ones
by rearranging the order of the rows
and the order of the columns.
Also, we can combine it with Theorem \ref{complement thm} to
compute the character for the complement of
a northwest diagram in an $N \times r$ rectangle.
Denote a diagonal matrix by
$h = \mathop{\rm diag}(x_1,\ldots,x_N).$
\begin{thm}
Suppose $D$ is a northwest diagram with $\leq N$ rows, and
$F$ an algebraically closed field. Then:\\
(a) The character of the Weyl module $W_D$ (for $GL(N,F)$) is given by
$$
{\mathop{\rm char}} _ {W_D}(h) = \sum_{t} { \prod_i x_i^{- {\mathop{\rm wt}} _ i(t)}
\over \prod_{i\neq j} (1-x_i x_j^{-1})^{d_{ij}(t)} },
$$
where $t$ runs over the standard tabloids for $ \widehat{D} $.\\
(b) For $F$ of characteristic zero, the character of the Schur module
$S_D$ (for $GL(N,F)$) is given by
$$
{\mathop{\rm char}} _ {S_D}(h) = \sum_{t} { \prod_i x_i^{ {\mathop{\rm wt}} _ i(t)}
\over \prod_{i\neq j} (1-x_i^{-1} x_j)^{d_{ij}(t)} },
$$
where $t$ runs over the standard tabloids for $ \widehat{D} $.
\end{thm}
\vspace{1em}
\noindent
{\bf Example.}
Consider the following diagram
and some of its standard tabloids for $N = 3$:
$$
D = \begin{array}{ccc}
\Box & \Box & \\
& \Box & \Box
\end{array}
\ \ \ \
t_1 = \begin{array}{ccc}
1 & 1 & \\
& 2 & 2
\end{array}
\ \ \ \
t_2 = \begin{array}{ccc}
1 & 1 & \\
& 2 & 1
\end{array}
\ \ \ \
t_3 = \begin{array}{ccc}
3 & 2 & \\
& 3 & 3
\end{array}
$$
The tabloid $t_1$ has $d_{12} = d_{13} = d_{21} = d_{23} = 1$,
$d_{31} = d_{32} = 0$, and $t_2$ has $d_{12} = 2$, $d_{13} = d_{23} = 1$,
$d_{21} = d_{31} = d_{32} = 0$. The other standard tabloids can be
obtained from $t_1$ and $t_2$ by applying a permutation of
$\{1,2,3\}$ to the entries, and rearranging the entries in the middle
column to make them increasing. For instance, $t_3 = \pi t_2$, where
$\pi$ is the transposition $(13)$.
Note that the standard tabloids are standard
tableaux in the usual sense: they are fillings
with the columns strictly increasing, and the rows non-increasing.
This is true in general when $ \widehat{D} $ is a skew diagram with no repeated
columns, though not all the standard tableaux are obtained in this way.
Applying our formula we find that
$ {\mathop{\rm char}} \, S_D = s_{(3,1,0)} + s_{(2,2,0)}$,
where $s_{( \lambda _1, \lambda _2, \lambda _3)}$ is a classical Schur function,
the character of an irreducible Schur module.
Since $D$ is a skew diagram,
we could have obtained this result
using the Littlewood-Richardson Rule.
It should be possible to prove this rule using
the present methods.
$\bullet$
\vspace{1em}
\noindent
{\bf Proof of the Theorem.}
(i) Consider the map $ {\mathop{\rm pr}} : { \cal F } _{ \widehat{D} } \rightarrow \FF_D $, and the
pullback line bundle $ {\mathop{\rm pr}} ^* \LL_D $. This is the bundle on $ { \cal F } _{ \widehat{D} }$
corresponding to giving the phantom columns $C$ of $ \widehat{D} $ multiplicity
$m_C = 0$.
Let RHS denote the right hand side of our formula in (a).
Then by the analysis of Section \ref{fixed points},
RHS is equal to the right hand
side of the Atiyah-Bott formula (Theorem \ref{AB thm}) for
$X = { \cal F } _{ \widehat{D} }$, $L = {\mathop{\rm pr}} ^* \LL_D $.
Thus
$$
\mbox{RHS} = {\mathop{\rm char}} \sum_i (-1)^i H^i( { \cal F } _{ \widehat{D} }, {\mathop{\rm pr}} ^* \LL_D ) .
$$
(ii) By Proposition \ref{rational sing},
we have
$ H^i( { \cal F } _{ \widehat{D} }, {\mathop{\rm pr}} ^* \LL_D ) = 0 $ for $i>0$,
and $ H^0( { \cal F } _{ \widehat{D} }, {\mathop{\rm pr}} ^* \LL_D ) = H^0( \FF_D , \LL_D )$.
Thus RHS = $ {\mathop{\rm char}} H^0( \FF_D , \LL_D )$. \\
(iii) By Proposition \ref{config split},
the restriction of global sections of $ \LL_D $
from $\mathop{\rm Gr}(D)$ to $ \FF_D $ is surjective,
and we have
\begin{eqnarray*}
\mbox{RHS} & = & {\mathop{\rm char}} H^0( \FF_D , \LL_D ) \\
& = &
{\mathop{\rm char}}
\mathop{\rm Im}\left( \mathop{\rm rest}\mbox{} : H^0(\mathop{\rm Gr}(D), \LL_D ) \rightarrow H^0( \FF_D , \LL_D ) \right) \\
& = & {\mathop{\rm char}} W_D.
\end{eqnarray*}
The last equality holds by
Proposition \ref{weyl mod is image}, and we have proved (a).
Then (b) follows because $S_D = (W_D)^*$.
$\bullet$
\subsection{Betti numbers}
In this section, we compute the betti numbers of the
smooth configuration varieties
of Section \ref{fixed points}.
\begin{prop}{(Bialynicki-Birula~\cite{B})}
Let $X$ be a smooth projective variety over an algebraically
closed field $F$, acted on by the one-dimensional torus $F^{\times}$
with isolated fixed points. Then there is a decomposition
$$
X = \coprod_{p \ \mbox{\rm \tiny fixed}} X_p ,
$$
where the $X_p$ are disjoint, locally closed, $H$-invariant
subvarieties, each isomorphic to an affine space
$X_p \cong \mbox{\rm \bf A}^{d^+\! (p)}$.
The dimensions $d^+(p)$ are given as follows.
Let the tangent space $T_p X \cong \bigoplus_{n \in {\bf Z}} a_n(p) F_n$, where
$a_n(p) \in {\bf N}$ and $F_n$ is the one-dimensional representation
of $F^{\times}$ for which the group element $t \in F^{\times}$
acts as the scalar $t^n$.
Then
$$
d^+\!(p) = \sum_{n > 0} a_n(p) .
$$
$\bullet$
\end{prop}
Over ${\bf C}$, the above proposition does not quite give a CW decomposition
for $X$, since the boundaries of the cells need not lie in cells of lower
dimension. Nevertheless, $\dim_{{\bf R}} \partial X_p \leq \dim_{{\bf R}} X_p - 2$,
and this is enough to fix the betti numbers $\beta_i = \dim_{{\bf R}} H^{i}(X,
{\bf R})$:\ \
$\beta_{2i} = \#\{p \mid d^+(p) = i\}$, and $\beta_{2 i + 1} = 0$.
Now, in our case consider the spaces $X = \FF_D $ of Section \ref{fixed points},
acted on by the $N$-dimensional torus $H$. Consider the embedding
$$
\begin{array}{cccc}
\check{\rho} : & F^{\times} & \rightarrow & H \\
& t & \mapsto & \mathop{\rm diag}(t^{N-1}, t^{N-2}, \ldots, t, 1),
\end{array}
$$
corresponding to the
coweight $\check{\rho} = $ half sum of the positive coroots.
Then $\check{\rho}(F^{\times})$ has the same (isolated)
fixed points as $H$, since none of the
eigenvectors of $H$ on $T_p X$ is fixed
by $\check{\rho}(F^{\times})$.
(I.e., $(\alpha, \check{\rho}) \neq 0$ for any root $\alpha$.)
Also, a given eigenvector of weight $x_i x_j^{-1}$ is
of positive $\check{\rho}(F^{\times})$ weight
exactly when $i < j$.
Thus, for a fixed point (standard tabloid) $t$ of $D$, define
$$
d^+\! (t) = \sum_{i < j} d_{ij}(t) .
$$
We then have the
\begin{prop}
Suppose $F = {\bf C}$, and $D$ is a northwest diagram with $\leq N$ rows and
$ \widehat{D} = D$. Then the betti numbers
$$
\beta_{2i} = \#\{t \mid d^+\! (t) = i\}, \ \ \ \beta_{2i+1} = 0 ,
$$
and the Poincare polynomial
$$
P(x, \FF_D ) \stackrel{\rm def}{=} \sum_i \beta_i x^i = \sum_t x^{2 d^+\! (t)} ,
$$
where $t$ runs over the standard tabloids of $D$.
$\bullet$
\end{prop}
In fact, our proof shows the above propostion for a broader class of
spaces.
Suppose $D = (C_1, C_2, \ldots)$ is an arbitrary
diagram such that the variety
$$
\mbox{\rm Inc}_D \stackrel{\rm def}{=} \{ (V_C)_{C \in D} \in \mathop{\rm Gr}(D) \mid C \subset C'
\Rightarrow V_C \subset V_{C'} \}
$$
is smooth. Then the proposition holds with $ \FF_D $ replaced by
$\mbox{\rm Inc}_D$.
|
1995-07-20T18:16:12 | 9411 | alg-geom/9411005 | en | https://arxiv.org/abs/alg-geom/9411005 | [
"alg-geom",
"math.AG"
] | alg-geom/9411005 | Stein Arild Stromme | G. Ellingsrud and S. A. Str{\o}mme | Bott's formula and enumerative geometry | 22 pages, amslatex 1.1 The paper is a considerably expanded version
of our previous eprint alg-geom/9409006 which had the title "Counting twisted
cubics on general complete intersections" | null | null | null | null | We outline a strategy for computing intersection numbers on smooth varieties
with torus actions using a residue formula of Bott. As an example,
Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some
general complete intersection in projective space are computed. The results are
consistent with predictions made from mirror symmetry computations. We also
compute degrees of some loci in the linear system of plane curves of degrees
less than 10, like those corresponding to sums of powers of linear forms, and
curves carrying inscribed polygons.
| [
{
"version": "v1",
"created": "Thu, 10 Nov 1994 13:15:01 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Ellingsrud",
"G.",
""
],
[
"Strømme",
"S. A.",
""
]
] | alg-geom | \section{Introduction}
One way to approach enumerative problems is to find a suitable
complete parameter space for the objects that one wants to count, and
express the locus of objects satisfying given conditions as a certain
zero-cycle on the parameter space. For this method to yield an
explicit numerical answer, one needs in particular to be able to
evaluate the degree of a given zerodimensional cycle class. This is
possible in principle whenever the numerical intersection ring (cycles
modulo numerical equivalence) of the parameter space is known, say in
terms of generators and relations.
Many parameter spaces carry natural actions of algebraic tori, in
particular those coming from projective enumerative problems. In 1967,
Bott \cite{Bott-1} gave a residue formula that allows one to express
the degree of certain zero-cycles on a smooth complete variety with an
action of an algebraic torus in terms of local contributions supported
on the components of the fixpoint set. These components tend to have
much simpler structure than the whole space; indeed, in many interesting
cases, including all the examples of the present paper, the fixpoints
are actually isolated.
We show in this note how Bott's formula can be effectively used to
attack some enumerative problems, even in cases where the rational
cohomology ring structure of the parameter space is not known.
Our first set of applications is the computation of the numbers of
twisted cubic curves (theorems \ref{main} and \ref{main2}) and
elliptic quartic curves (\thmref{main3}) contained in a general
complete intersection and satisfying suitable Schubert conditions.
The parameter spaces in question are suitable components of the
Hilbert scheme parameterizing these curves. These components are
smooth, by the work of Piene and Schlessinger \cite{Pien-Schl} in the
case of cubics, and Avritzer and Vainsencher \cite{Avri-Vain} in the
case of elliptic quartics.
The second set of applications is based on the Hilbert scheme of
zero-dimensional subschemes
of ${\bold P}^2$, which again is smooth by Fogarty's work \cite{Foga-1}. These
applications deal with the degree of the variety of sums of powers of
linear forms in three variables (\thmref{main4}) and Darboux curves
(\thmref{main5}).
\begin{ack}
Part of this work was done at the Max-Planck-Institut f\"ur Mathematik
in Bonn during the authors' stay there in the spring of 1993. We would
like to thank the MPI for this possibility. We would also like to
express our thanks to S. Katz, J. Le Potier, D. Morrison, and A.
Tyurin for raising some of the problems treated here and for many
stimulating discussions.
\end{ack}
\subsection{Main results}
The first theorem deals with the number of twisted cubics on a general
Calabi-Yau threefold which is a complete intersection in some projective
space. There are exactly five types of such threefolds: the quintic in
${\bold P}^4$, the complete intersections $(3,3)$ and $(2,4)$ in ${\bold P}^5$, the
complete intersection $(2,2,3)$ in ${\bold P}^6$ and finally $(2,2,2,2)$ in
${\bold P}^7$.
\begin{thm}\label{main}
For the general complete intersection Calabi-Yau threefolds, the numbers
of twisted cubic curves they contain are given by the following table:
\smallskip
\begin{center}
\begin{tabular}
{|l|c|c|c|c|c|}\hline
Type of complete intersection\vrule height1em width0cm depth0.4em
& $5$ & $4,2$ & $3,3$ & $3,2,2$ &$2,2,2,2$\\ \hline
Number of twisted cubics \vrule height1em width0cm depth0.4em
&$317206375$ &$15655168$ &$6424326$ &$1611504$ &$416256$\\ \hline
\end{tabular}
\end{center}
\end{thm}
In the case of a general quintic in ${\bold P}^4$, the number of rational
curves of any degree was predicted by Candelas et al.\ in
\cite{Cand-Gree-Ossa-Park}, and the
cubic case was verified by the authors in \cite{Elli-Stro-3}. In
\cite{Libg-Teit}
Libgober and Teitelbaum predicted the corresponding numbers for the
other Calabi-Yau complete intersections. Our results are all in
correspondence with their predictions.
Greene, Morrison, and Plesser \cite{Gree-Morr-Ples} have also predicted
certain numbers of rational curves on higher dimensional Calabi Yau
hypersurfaces. Katz \cite{Katz-2} has verified these numbers for lines
and conics for hypersurfaces of dimension up to 10. The methods of the
present paper have allowed us to verify the following numbers. All but
the last one, $N_3^{1,1,1,1}(8)$, have been confirmed by D.~Morrison
(privat communication) to be consistent with \cite{Gree-Morr-Ples}.
\begin{thm}\label{main2}
For a general hypersurface $W$ of degree $n+1$ in ${\bold P}^n$ $(n\le8)$ and
for a partition $\lambda=(\lambda_1\ge\dots\ge\lambda_m>0)$ of $n-4$,
the number $N_3^{\lambda}(n)$ of twisted cubics on $W$ which meet $m$
general linear subspaces of codimensions $\lambda_1+1,\dots,\lambda_m+1$
respectively is given as follows:
\medskip
\begin{center}
\begin{tabular}{|l|l|l||l|l|l|}\hline
\vrule height1.2em width0cm depth0.6em$n$ & $\lambda$ & $N_3^{\lambda}(n)$&
\vrule height1em width0cm depth0.4em$n$ & $\lambda$ & $N_3^{\lambda}(n)$ \\ \hline
\vrule height1em width0cm depth0.4em$4$&{} &$317206375$ &
\vrule height1em width0cm depth0.4em$7$&$1,1,1$ & $12197109744970010814464$ \\ \hline
\vrule height1em width0cm depth0.4em$5$& $1$ & $6255156277440$ & $8$&
\vrule height1em width0cm depth0.4em$4$ & $897560654227562339370036$ \\ \hline
\vrule height1em width0cm depth0.4em$6$& $2$ & $30528671745480104$ &$8$&
\vrule height1em width0cm depth0.4em$3,1$ & $17873898563070361396216980$ \\ \hline
\vrule height1em width0cm depth0.4em$6$& $1,1$ & $222548537108926490$ &
\vrule height1em width0cm depth0.4em$8$&$2,2$ & $33815935806268253433549768$ \\ \hline
\vrule height1em width0cm depth0.4em$7$&$3$ & $154090254047541417984$ &
\vrule height1em width0cm depth0.4em$8$&$2,1,1$ & $174633921378662035929052320$ \\ \hline
\vrule height1em width0cm depth0.4em$7$&$2,1$ & $2000750410187341381632$ &
\vrule height1em width0cm depth0.4em$8$&$1,1,1,1$& $957208127608222375829677128$ \\ \hline
\end{tabular}
\end{center}
\end{thm}
The number $N_3^{1,1,1,1}(8)$ is not related to mirror symmetry as far
as we know; Greene et.al.~ get numbers only for partitions with at most 3
parts. Our methods also yield other numbers not predicted (so far!) by
physics methods: for example, there are 1345851984605831119032336
twisted cubics contained in a general nonic hypersurface in ${\bold P}^7$ (not
a Calabi-Yau manifold).
A similar method can be used to compute the number of elliptic
quartic curves on general Calabi-Yau complete intersections. Here are
the results for some hypersurfaces, which we state without proof:
\begin{thm}\label{main3}
The number of quartic curves of arithmetic
genus 1 on a general hypersurface of degree $n+1$ in ${\bold P}^n$ are for
$4\le n\le13$ given by the following table. These curves are all smooth.
\smallskip
\begin{center}
\begin{tabular}{|l|l|}\hline
$n$\vrule height1.2em width0cm depth0.6em & Smooth elliptic quartics on a general hypersurface of
degree $n+1$ in ${\bold P}^n$\\ \hline
$4$&\vrule height1em width0cm depth0.4em $3718024750$\\ \hline
$5$&\vrule height1em width0cm depth0.4em $387176346729900$\\ \hline
$6$&\vrule height1em width0cm depth0.4em $81545482364153841075$\\ \hline
$7$&\vrule height1em width0cm depth0.4em $26070644171652863075560960$\\ \hline
$8$&\vrule height1em width0cm depth0.4em $12578051423036414381787519707655$\\ \hline
$9$&\vrule height1em width0cm depth0.4em $8760858604226734657834823089352310000$\\ \hline
$10$&\vrule height1em width0cm depth0.4em $8562156492484448592316222733927180351143552$\\ \hline
$11$&\vrule height1em width0cm depth0.4em $11447911791501360069250820471811603020708611018752$\\ \hline
$12$&\vrule height1em width0cm depth0.4em
$20498612221082029813903827233942127541022477928303274152$\\ \hline
$13$&\vrule height1em width0cm depth0.4em
$48249485834889092561505032612701767175955799366431126942036480$\\ \hline
\end{tabular}
\end{center}
\end{thm}
This computation uses the description given in \cite{Avri-Vain} of the
irreducible component of the Hilbert scheme of ${\bold P}^3$ parameterizing
smooth elliptic quartics. This Hilbert scheme component can be
constructed from the Grassmannian of pencils of quadrics by two explicit
blowups with smooth centers, and one may identify the fixpoints for the
natural action of a torus in a manner analogous to what we carry out for
twisted cubics in this paper. For another related construction, see
\cite{Meur-1}, which treats curves in a weighted projective space.
The number of elliptic quartics on the general quintic threefold was
predicted by Bershadsky et.al.\ \cite{Bers-Ceco-Oogu-Vafa}. Their
number, 3721431625, includes singular quartics of geometric genus 1.
These are all plane binodal quartics, and their number is
$1185*2875=3406875$ by \cite{Vain-1}. Thus the count of
\cite{Bers-Ceco-Oogu-Vafa} is compatible with the number above.
Recently, Kontsevich \cite{Kont-1} has developed a technique for
computing numbers of rational curves of {\em any\/} degree, using the
stack of stable maps rather than the Hilbert scheme as a parameter
space. He also uses Bott's formula, but things get more complicated
than in the present paper because of the presence of non-isolated
fixpoints in the stack of stable maps.
The next theorem deals with plane curves of degree $n$ whose equation
can be expressed as a sum of $r$ powers of linear forms. Let $PS(r,n)$
be the corresponding subvariety of ${\bold P}^{n(n+3)/2}$. Then $PS(r,n)$ is
the $r$-th secant variety of the $n$-th Veronese imbedding of ${\bold P}^2$.
Let $p(r,n)$ be the number of ways a form corresponding to a general
element of $PS(r,n)$ can be written as a sum of $r$ $n$-th powers if
this number is finite, and 0 otherwise. The last case occurs if and
only if $\dim(PS(r,n))$ is less than the expected $3r-1$. We don't know
of an example where $p(r,n)>1$ if $PS(r,n)$ is a proper subvariety. If
$p(r,n)=1$, then $p(r,n')=1$ for all $n'\ge n$. It is easy to see that
$p(2,n)=1$ for $n\ge3$.
\begin{thm} \label{main4} Assume that $n\ge r-1$ and $2\le r\le8$.
Then $p(r,n)$ times the degree of $PS(r,n)$ is $s_r(n)$, where
{\allowdisplaybreaks
\begin{align}
2\,s_2(n)= &\,n^4-10\,n^2+15\,n-6, \notag \\
3!\,s_3(n)= &\,n^6-30\,n^4+45\,n^3+206\,n^2-576\,n+384,\notag \\
4!\,s_4(n)= &\,n^8-60\,n^6+90\,n^5+1160\,n^4-3204\,n^3-5349\,n^2+26586
\,n-23760,\notag\\
5!\,s_5(n)= &\,n^{10}-100\,n^8+150\,n^7+3680\,n^6-10260\,n^5-
52985\,n^4+\notag\\*
&\,224130\,n^3+127344\,n^2-1500480\,n + 1664640,\notag\\
6!\,s_6(n)=
&\,n^{12}-150\,n^{10}+225\,n^9+8890\,n^8-25020\,n^7-244995\,n^6
+1013490\,n^5+\notag
\\*&\,2681974\,n^4-17302635\,n^3+1583400\,n^2+101094660\,n
-134190000,\notag\\
7!\,s_7(n)=&\,n^{14}-210\,n^{12}+315\,n^{11}+
18214\,n^{10}-51660\,n^9-802935\,n^8+\notag\\*
&\,3318210\,n^7+17619994\,n^6-102712365\,n^5
-136396680\,n^4+\notag\\*
&\,1498337820\,n^3-872582544\,n^2-7941265920\,n
+12360418560,\notag\\
8!\,s_8(n)=&\,n^{16}-280\,n^{14}+420\,n^{13}+
33376\,n^{12}-95256\,n^{11}
-2134846\,n^{10}+\notag\\*
&\,8858220\,n^9+75709144\,n^8-
427552020\,n^7-1332406600\,n^6+\notag\\*
&\,11132416680\,n^5+5108998089\,n^4
-145109970684\,n^3+\notag\\*
&\,144763373916\,n^2 +713178632880\,n-1286736675840.\notag
\end{align}
}
\end{thm}
For example, $s_5(4)=0$; this corresponds to the classical but
non-obvious fact that not all ternary quartics are sums of five fourth
powers. (Those who are are called Clebsch quartics; they form a
hypersurface of degree 36).
Note in particular that $s_3(3)=4$. It is classically known that
$PS(3,3)$ is indeed a hypersurface of degree 4, its equation is the
so-called $S$-invariant \cite{Salm}. It follows that $p(3,3)=1$, and
hence that $p(3,n)=1$ for $n\ge3$.
Only the first few of these polynomials are reducible:
$s_r(r-1)=0$ for $r\le5$, but
the higher $s_r$ in the table are irreducible over ${\bold Q}$.
Note that the formulas of the theorem are not valid unless $n\ge r-1$.
For example, a general quintic is uniquely expressable as a sum of
seven fifth powers (cfr.~the references in \cite{Muka-1}), while
$s_7(5)$ is negative.
The final application quite similar. A {\em Darboux curve\/} is a
plane curve of degree $n$ circumscribing a complete $(n+1)$-gon (this
terminology extends the one used in \cite{Bart-1}). This means that
there are distinct lines $L_0,\dots,L_n$ such that $C$ contains all
intersection points $L_i\cap L_j$ for $i<j$. Equivalently, there are
linear forms $\ell_0,\dots,\ell_n$ such that the curve is the divisor
of zeroes of the rational section $\sum_{i=0}^n \ell_i^{-1}$ of
$\OP2(-1)$. Let $D(n)$ be the closure in ${\bold P}^{n(n+3)/2}$ of the
locus of Darboux curves. Let $p(n)$ be the number of inscribed
$(n+1)$-gons in a general Darboux curve, if finite, and 0 otherwise.
\begin{thm} \label{main5}
For $n=5,6,7,8,9$, the product of $p(n)$ and the degree of the Darboux locus
$D(n)$ is $2540, 583020, 99951390, 16059395240, 2598958192572$,
respectively.
\end{thm}
We have no guess as to what $p(n)$ is; it might well be 1 for
$n\ge5$. It is always positive for $n\ge5$ by an argument of Barth's
\cite{Bart-1}.
For $n\le4$ it is 0. For $n\le3$, all curves are Darboux. For
$n=4$, Darboux curves are L\"uroth quartics, and form a degree 54
hypersurface \cite{Morl,LePo,Tyur-1}.
\section{Bott's formula}\label{C}
Let $X$ be a smooth complete variety of dimension $n$, and assume that
there is given an algebraic action of the multiplicative group ${\bold C}^*$
on $X$ such that the fixpoint set $F$ is finite. Let ${\cal E}$ be an
equivariant vector bundle of rank $r$ over $X$, and let
$p(c_1,\dots,c_r)$ be a weighted homogenous polynomial of degree $n$
with rational coefficients, where the variable $c_i$ has degree $i$.
Bott's original formula \cite{Bott-1} expressed the degree of the
zero-cycle $p(c_1({\cal E}),\dots,c_r({\cal E}))\in H^{2n}(X,{\bold Q})$ purely in terms
of data given by the representations induced by ${\cal E}$ and the tangent
bundle $T_X$ in the fixpoints of the action.
Later, Atiyah and Bott \cite{Atiy-Bott-1} gave a more general formula,
in the language of equivariant cohomology. Its usefulness in our
context is mainly that it allows the input of Chern classes of several
equivariant bundles at once. Without going into the theory of
equivariant cohomology, we will give here an interpretation of the
formula which is essentially contained in the work of Carrell and
Lieberman
\cite{Carr-Lieb-1,Carr-Lieb-2}.
To explain this, first note that the ${\bold C}^*$ action on $X$ induces, by
differentiation, a global vector field $\xi\in H^0(X,T_X)$, and
furthermore, the fixpoint set $F$ is exactly the zero locus of $\xi$.
Hence the Koszul complex on the map $\xi^{\vee}\:\Omega_X \to {\cal O}_X$ is a
locally free resolution of ${\cal O}_F$. For $i\ge0$, denote by $B_i$ the
cokernel of the Koszul map $\Omega_X^{i+1}\to \Omega_X^{i}$.
It is well known that $H^j(X,\Omega_X^i)$ vanishes for $i\ne j$, see
e.g. \cite{Carr-Lieb-1}.
Hence there are natural exact sequences for all $i$:
$$
0 \to H^i(X,\Omega_X^i) @>p_i>> H^i(X,B_i) @>r_i>> H^{i+1}(X,B_{i+1}) \to0.
$$
In particular, there are natural maps
$q_i=r_{i-1}\circ\dots\circ r_0\:H^0(F,{\cal O}_F)\to H^i(X,B_i)$.
\begin{defn} Let $f\: F \to {\bold C}$ be a function and $c\in
H^i(X,\Omega_X^i)$ a non-zero cohomology class. We say that $f$ {\em
represents\/} $c$ if $q_i(f)=p_i(c)$.
\end{defn}
For each $i\ge-1$, put $A_i=\ker q_{i+1}$. Then
$$
0=A_{-1}\subseteq{\bold C}=A_0\subseteq A_1\subseteq A_2\subseteq \dots \subseteq A_n = H^0(F,{\cal O}_F)
$$
is a filtration by sub-vector spaces
of the ring of complex-valued functions on $F$.
The filtration has the property that
$A_i A_j \subseteq A_{i+j}$, and the associated graded ring
$\operatornamewithlimits{\oplus} A_{i}/A_{i-1}$ is naturally isomorphic to the cohomology ring
$H^*(X,{\bold C})\simeq \operatornamewithlimits{\oplus} H^i(X,\Omega_X^i)$.
(In \cite{Carr-Lieb-2}, the filtration is constructed as coming from one of
the spectral sequences associated to hypercohomology of the Koszul
complex above.)
An interesting aspect of this is that cohomology classes can be
represented as functions on the fixpoint set. The representation is
unique up to addition of functions coming from cohomology classes of
lower degree (i.e., lower codimension). Since the algebra of functions
on a finite set is rather straightforward, this gives an efficient way
to evaluate zero-cycles, provided that 1) we know how to describe a
function representing a given class, and 2) we have an explicit formula
for the composite linear map
$$
\epsilon_X\: H^0({\cal O}_F) @>q_n>> H^{n}(X,\Omega_X^n)
@>\text{res}_X>\simeq> {\bold C}.
$$
These issues are addressed in the theorem below.
Let ${\cal E}$ be an equivariant vector bundle of rank $r$ on $X$. In each
fixpoint $x\in F$ the fiber of ${\cal E}$ splits as a direct sum of
one-dimensional representations of ${\bold C}^*$; let
$\tau_1({\cal E},x),\dots,\tau_r({\cal E},x)$ denote the corresponding
weights, and for all integers $k\ge0$, let $\sigma_k({\cal E},x)\in{\bold Z}$ be the
$k$-th elementary symmetric function in the $\tau_i({\cal E},x)$.
\begin{thm}\label{bott1}
Let the notation and terminology be as above. Then
\begin{enumerate}
\item
The $k$-th Chern class $c_k({\cal E})\in H^k(X,\Omega_X^k)$ of ${\cal E}$ can be
represented by the function $x\mapsto \sigma_k({\cal E},x)$.
\item
For a function $f\: F \to {\bold C}$, we have
$\epsilon_X(f) =
\displaystyle{\sum_{x\in F} \frac{f(x)}{\sigma_n(T_X,x)}}.$
\end{enumerate}
\end{thm}
\begin{pf} See \cite[equation~3.8]{Atiy-Bott-1}, and \cite{Carr-Lieb-2}.
\end{pf}
Note that the function $\sigma_k({\cal E},-)$ depends on the choice of a
${\bold C}^*$-linearisation of the bundle ${\cal E}$, whereas the Chern class
$c_k({\cal E})$ it represents does not.
\section{Twisted cubics} \label{A}
Let $\operatorname{Hilb}^{3t+1}_{{\bold P}^n}$ be the Hilbert scheme parameterizing
subschemes of ${\bold P}^n$ ($n\ge3$) with Hilbert polynomial $3t+1$, and let
$H_n$ denote the irreducible component of $\operatorname{Hilb}^{3t+1}_{{\bold P}^n}$
containing the twisted cubic curves. Recall from \cite{Pien-Schl} that
$H_3$ is smooth and projective of dimension $12$. Any curve
corresponding to a point of $H_n$ spans a unique 3-space, hence $H_n$
admits a fibration \begin{equation} \label{fibration} \Phi\: H_n\to
G(3,n) \end{equation} over the Grassmannian of 3-planes in ${\bold P}^n$, with
fiber $H_3$. It follows that $H_n$ is smooth and projective of
dimension $4n$.
There is a universal subscheme ${\cal C}\subset H_n\times {\bold P}^n$. For a
closed point $x\in H_n$, we denote by $C_x$ the corresponding cubic
curve, i.e., the fiber of ${\cal C}$ over $x$. Also, let ${\cal I}_x\subseteq\OP{n}$ be
its ideal sheaf. By the classification of the curves of $H_n$ (see
below), it is easy to see that
\begin{equation}
\label{H1}
H^1({\bold P}^n,{\cal I}_x(d))=H^1(C_x,{\cal O}_{C_x}(d))=0\quad
\text{for all $d\ge1$ and for all
$x\in H_n$.}
\end{equation}
For a subscheme $W\subseteq {\bold P}^n$, denote by $H_W\subseteq H_n$ the closed
subscheme parameterizing twisted cubics contained in $W$. There is a
natural scheme structure on $H_W$ as the intersection of $H_n$ with
the Hilbert scheme of $W$. If $C_x\subseteq W$ is a Cohen-Macaulay twisted
cubic, then locally at $x\in H_n$, the scheme $H_W$ is simply the
Hilbert scheme of $W$.
Our goal is to compute the cycle class of $H_W$ in $A^*(H_n)$ in the
case that $W$ is a general complete intersection in ${\bold P}^n$. In
particular, we want its cardinality if it is finite, and its
Gromov-Witten invariants (see below) if it has positive dimension.
For each integer $d$ we define a sheaf ${\cal E}_{d}$ on $H_n$ by
\begin{equation}
\label{defE}
{\cal E}_{d}=p_{1*}({\cal O}_{\cal C}\* p_{2}^* \OP{n}(d)),
\end{equation}
where $p_1$ and $p_2$ are the two projections of $H_n\times {\bold P}^n$. If
$d\ge1$, then the vanishing of the first cohomology groups \eqref{H1}
implies by standard base change theory
\cite{AG} that ${\cal E}_{d}$ is locally free of rank $3d+1$, and moreover
that there are surjections $\rho\:H^0(\OP{n}(d))_{H_n}\to{\cal E}_{d}$ of
vector bundles on $H_n$. In particular, for all $x\in H_n$, there is a
natural isomorphism
\begin{equation}
{\cal E}_{d}(x) @>\simeq>> H^0(C_x,{\cal O}_{C_x}(d)).
\label{basechange}
\end{equation}
A homogenous form $F\in H^0(\OP{n}(d))$ induces a global section
$\rho(F)$ of ${\cal E}_{d}$ over $H_n$, and the evaluation of this section at
a point $x$ corresponds under the identification (\ref{basechange}) to
the restriction of $F$ to the curve $C_x$. Hence the zero locus of
$\rho(F)$ corresponds to the set of curves $C_x$ contained in the
hypersurface $V(F)$.
More generally, in the case of an intersection $V(F_1,\dots,F_p)$ in
${\bold P}^n$ of $p$ hypersurfaces, the section $(\rho(F_1),\dots,\rho(F_p))$
of ${\cal E}={\cal E}_{d_1}\oplus\dots\oplus{\cal E}_{d_p}$ vanishes exactly on the points
corresponding to twisted cubics contained in $V(F_1,\dots,F_p)$.
\begin{prop} \label{propA}
Let $W\subseteq{\bold P}^n$ be the complete intersection of $p$ general
hypersurfaces in ${\bold P}^n$ of degrees $d_1,\ldots,d_p$ respectively.
Assume that $\sum_i(3d_i+1)=4n$. Then the number of twisted cubic
curves contained in $W$ is finite and equals
$$\int_{H_n}c_{4n}({\cal E}_{d_1}\oplus\cdots\oplus{\cal E}_{d_p}).$$
These cubics are all smooth.
\end{prop}
\begin{pf}
By the considerations above, the bundle
${\cal E}={\cal E}_{d_1}\oplus\cdots\oplus{\cal E}_{d_p}$ is a quotient bundle of the
trivial bundle $\operatornamewithlimits{\oplus} H^0(\OP{n}(d_i))_{H_n}$. Hence Kleiman's Bertini
theorem \cite{Klei-1} implies that the zero scheme of the section
$(\rho(F_1),\ldots,\rho(F_p))$ is nonsingular and of codimension
$\operatorname{rank}({\cal E})$. Since $\operatorname{rank}({\cal E})=\dim(H_n)$, the number of points is
finite and given by the top Chern class.
\end{pf}
\subsection{Gromov-Witten numbers}\label{B}
More generally, assume that $W$ is as in \propref{propA}, except that we
only assume an inequality $\sum_i(3d_i+1)\le4n$ instead of the equality.
The top Chern class of ${\cal E}$ still represents the locus $H_n(W)$ of
twisted cubics contained in $W$, although there are infinitely many of
them if the inequality is strict. One may assign finite numbers to this
family by imposing Schubert conditions. For this purpose say that a
{\em Schubert condition\/} on a curve is the condition that it intersect
a given linear subspace of ${\bold P}^n$. If the subspace has codimension
$c+1$, then the corresponding Schubert condition is of codimension $c$
(corresponding to the class $\gamma_c$ below).
\begin{defn}
Let $W$ be a general complete intersection in ${\bold P}^n$ of
$p$ hypersurfaces of degrees $d_1,\dots,d_p$ respectively. Let
$\lambda=(\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_m > 0)$ be a
partition of $4n - \sum_{i=1}^p(3d_i+1)$, and let $P_1,\dots,P_m$ be
general linear subspaces such that $\operatorname{codim} P_i = \lambda_i+1$. The
number of twisted cubics on $W$ meeting all the $P_i$ is called the
$\lambda$-th {\em Gromov-Witten number} of the family of twisted
cubic curves on $W$, and is denoted by $N_3^\lambda(W)$.
\end{defn}
\begin{rem}
This is a slight variation on the definition used in \cite{Katz-2}, and
differs from that by a factor of 3 (resp.~9) for partitions with 2
(resp.~1) parts. In \cite{Katz-2} only partitions of length at most three
are considered, as these numbers are the ones that have been predicted by
mirror symmetry computations (when $W$ is Calabi-Yau). We have used the
term Gromov-Witten number rather than Gromov-Witten invariant, as the
latter term is now being used in a more sophisticated sense
\cite{Kont-Mani-1}.
\end{rem}
Let $h$ denote the hyperplane class of ${\bold P}^n$ as well as its pullback to
$H_n\times{\bold P}^n$, and let $[{\cal C}]\in A^*(H_n\times{\bold P}^n)$ be the cycle class of the
universal curve ${\cal C}$. If $P\subseteq{\bold P}^n$ is a linear subspace of codimension
$c+1\ge2$, then ${\cal C}\cap H_n\times P$ projects birationally to its image under
the first projection, which is the locus of curves meeting $P$. Hence the
class of the locus of curves meeting $P$ is ${p_1}_* (h^{c+1}[{\cal C}])$. For
simplicity, we give this class a special notation:
\begin{notation}
For a natural number $c$, let
$\gamma_c = {p_1}_* (h^{c+1}[{\cal C}]) \in A^c(H_n)$.
\end{notation}
\begin{prop} \label{propB}
Let $W\subseteq{\bold P}^n$ be the complete intersection of $p$ general
hypersurfaces in ${\bold P}^n$ of degrees $d_1,\ldots,d_p$ respectively.
Assume that $\sum_i(3d_i+1)\le 4n$, and let $\lambda=(\lambda_1 \ge
\lambda_2 \ge \dots \ge \lambda_m > 0)$ be a partition of $4n -
\sum_{i=1}^p(3d_i+1)$. Then
$$
N_3^\lambda(W)=
\int_{H_n} c_{{\text{top}}}({\cal E}_{d_1}\oplus\cdots\oplus{\cal E}_{d_p})\cdot
\prod_{i=1}^m\gamma_{\lambda_i}.
$$
Furthermore, if $P_1,\dots,P_m$ are general linear subspaces such that
$\operatorname{codim} P_i = \lambda_i+1$, then the $N_3^\lambda(W)$ twisted cubics on
$W$ which meets each $P_i$ are all smooth.
\end{prop}
\begin{pf} Similar to the proof of \propref{propA}.
\end{pf}
For later use, we want to express the classes $\gamma_i$ in terms of
Chern classes of the bundles ${\cal E}_d$.
\begin{prop} \label{gammaformler}
Let $a_i=c_i({\cal E}_1)$, $b_i=c_i({\cal E}_2)$, $c_i=c_i({\cal E}_3)$, and
$d_i=c_i({\cal E}_4)$. Then we have the following formulas for the $\gamma_c$:
{\allowdisplaybreaks
\begin{align}
\gamma_0 =&\, 3\notag \\
\gamma_1 =&\, 5a_1-14b_1+13c_1-4d_1\notag\\
\gamma_2 =&\, 3a_1^2-9a_1b_1+9a_1c_1-3a_1d_1-3b_1^2+9b_1c_1\notag\\* &
-3b_1d_1-6c_1^2+3c_1d_1+a_2-3b_2+3c_2-d_2\notag\\
\gamma_3 =&\,
3a_1^3-9a_1^2b_1+9a_1^2c_1-3a_1^2d_1-3a_1b_1^2+9a_1b_1c_1\notag\\* &
-3a_1b_1d_1-6a_1c_1^2+3a_1c_1d_1-4a_1a_2-3a_1b_2+3a_1c_2-a_1d_2\notag\\* &+
14a_2b_1-13a_2c_1+4a_2d_1+3a_3\notag\\
\gamma_4 =&\, 3a_1^4-9a_1^3b_1+9a_1^3c_1-3a_1^3d_1-3a_1^2b_1^2\notag\\* &
+9a_1^2b_1c_1-3a_1^2b_1d_1-6a_1^2c_1^2+3a_1^2c_1d_1-7a_1^2a_2\notag\\* &
-3a_1^2b_2+3a_1^2c_2-a_1^2d_2+23a_1a_2b_1-22a_1a_2c_1+7a_1a_2d_1\notag\\* &
+3a_2b_1^2-9a_2b_1c_1+3a_2b_1d_1+6a_2c_1^2-3a_2c_1d_1+8a_1a_3\notag\\* &
-a_2^2+3a_2b_2-3a_2c_2+a_2d_2-14a_3b_1+13a_3c_1-4a_3d_1-3a_4\notag
\end{align}
}
\end{prop}
\begin{pf}
Let $\pi\: B={\bold P}({\cal E}_1) \to H_n$. The natural surjection
$\rho\:H^0(\OP{n}(1))_{H_n}\to{\cal E}_{1}$ induces a closed imbedding $B\subseteq
H_n\times{\bold P}^n$ over $H_n$. Over a closed point $x$ of $H_n$, the fiber of
$B$ is just the ${\bold P}^3$ spanned by $C_x$. So the universal curve ${\cal C}$
is actually a codimension 2 subscheme of $B$. It follows by
the projection formula that
$$
\gamma_c = {\pi}_* (\tau^{c+1}[{\cal C}]_B) \in A^c(H_n),
$$
where $[{\cal C}]_B$ denotes the class of ${\cal C}$ in $A^2(B)$, and $\tau\in
A^1(B)$ is the first Chern class of the tautological quotient linebundle
on $B$. The formulas of the proposition now follow by straightforward
computation (for example using \cite{schubert}) from the next lemma.
\end{pf}
\begin{lem} The class of ${\cal C}$ in $B$ is
\begin{equation*}
\begin{aligned}
[{\cal C}]_B=&\,3\tau^2+(-4d_1+2a_1-14b_1+13c_1)\tau\\&
+3c_1d_1+4a_2-3b_2+3c_2-d_2-2a_1^2\\&
+5a_1b_1-4a_1c_1+a_1d_1-3b_1^2+9b_1c_1-3b_1d_1-6c_1^2
\end{aligned}
\end{equation*}
\end{lem}
\begin{pf}
Let $i\: {\cal C}\to B$ be the inclusion. Then $[{\cal C}]_B$ equals
the degree 2 part of the Chern character of the ${\cal O}_B$-module
$i_*{\cal O}_{\cal C}(\ell\tau)$, for any integer $\ell$. For $\ell=4$, there is a
canonical Beilinson type resolution of $i_*{\cal O}_{\cal C}(4\tau)$:
\begin{equation}
0\to\pi^*{\cal E}_1\*\Omega^3_{B/H_n}(3\tau)\to
\pi^*{\cal E}_2\*\Omega^2_{B/H_n}(2\tau)\to
\pi^*{\cal E}_3\*\Omega^1_{B/H_n}(\tau)\to
\pi^*{\cal E}_4
\label{beilinson}
\end{equation}
Using this it is a straightforward exercise (again using
\cite{schubert}) to compute the Chern character of $i_*{\cal O}_{\cal C}$ in terms of
$\tau$ and the Chern classes of the ${\cal E}_n$. \end{pf}
\subsection{Coarse classification of twisted cubics}\label{D}
We divide the curves $C_x$ for $x\in H_3$ into two groups, according to
whether they are Cohen-Macaulay or not.
A locally Cohen-Macaulay twisted cubic curve $C_x$ is also arithmetically
Cohen-Macaulay, and its ideal is given by the vanishing of the
$2\times2$ minors of a $3\x2$ matrix $\alpha$ with linear coefficients.
There is a resolution of ${\cal O}_{C_x}$:
\begin{equation}
K^{\bullet}:\quad 0 @>>> F\*{\cal O}_{{\bold P}^3}(-3) @>\alpha>> E\*{\cal O}_{{\bold P}^3}(-2)
@>\wedge^2\alpha^t>> \OP3,
\end{equation}
where $F$ and $E$ are vector spaces of dimensions 2 and 3 respectively.
Intrinsically,
\begin{align}
E=&H^0({\bold P}^3,{\cal I}_{x}(2))\subseteq H^0(\OP3(2))\label{EE}\\
F=&\operatorname{Ker}(E\* H^0(\OP3(1)) @>\text{mult}>> H^0(\OP3(3))).\label{FF}
\end{align}
\begin{lem} \label{CMlem}
Let $C_x$ be Cohen-Macaulay, and let $E$ and $F$ be as above. Then
there is a functorial exact sequence:
\begin{equation}
0 @>>> {\bold C} \to \operatorname{End}(F)\operatornamewithlimits{\oplus}\operatorname{End}(E) \to \operatorname{Hom}(F,E)\* H^0(\OP3(1))
\to T_{H_3}(x) \to 0
\end{equation}
\end{lem}
\begin{pf}
Recall the canonical identification
$T_{H_3}(x)=\operatorname{Hom}_{{\bold P}^3}({\cal I}_{x},{\cal O}_{C_x})$. The sequence now
follows from consideration of the total complex associated to the double
complex $\operatorname{Hom}_{{\bold P}^3}(K^{\bullet},K^{\bullet})$.
\end{pf}
Next let us consider the curves $C_x$ for $x\in H_3$ which are not
Cohen-Macaulay. By \cite{Pien-Schl}, these are projectively equivalent
to a curve with ideal generated by the net of quadrics
$x_0(x_0,x_1,x_2)$ plus a cubic form $q$, which can be taken to be of
the form $q=Ax_1^2+Bx_1x_2+Cx_2^2$, with $A$, $B$, and $C$ linear forms
in ${\bold C}[x_1,x_2,x_3]$. If we furthermore impose the conditions that $B$
is a scalar multiple of $x_3$, then the cubic $q$ is unique up to
scalar. (See \cite{Elli-Stro-3}).
Let $Y\subseteq H_3$ be the locus of non-Cohen-Macaulay curves, and denote by
$I$ the 5-dimensional incidence correspondence $\{(p,H)\in
{\bold P}^3\times{{\bold P}^3}^* \mid p\in H \}$. By the above, the quadratic part of
${\cal I}_x$ for $x\in Y$ gives rise to a point of $I$. This gives a morphism
\begin{equation}
g\: Y\to I,
\label{def:g}
\end{equation}
and again from the above it is clear that this makes $Y$ a
${\bold P}^6$-bundle over $I$. Hence $Y$ is a divisor on $H_3$, and it is
clear how to compute the tangent spaces $T_Y(x)$. To get hold of
$T_{H_3}(x)$, we need to identify the normal direction of $Y$ in $H_3$.
For this, let $C_x$ be the curve above, and consider the family $C_t$ of
Cohen-Macaulay curves given for $t\ne0$ by the matrix
\begin{equation}
\alpha_t =\begin{pmatrix} 0 & -x_0\\ x_0 & 0\\ -x_1 & x_2 \end{pmatrix}
+ t\begin{pmatrix} C & B\\ 0 & A\\ 0 & 0 \end{pmatrix}
\end{equation}
Then
\begin{equation}
\det\left(\begin{array}{c|c} & x_1\\ \alpha_t & -x_2\\ & 0
\end{array}\right) = t(Ax_1^2+Bx_1x_2+Cx_2^2) = tq,
\end{equation}
which implies that $\lim_{t\to0}C_t=C_x$. The tangent vector $\xi\in
T_{H_3}(x)=\operatorname{Hom}({\cal I}_x,{\cal O}_{C_x})$ corresponding to this one-parameter
family has this effect on the quadratic equations:
\begin{equation} \label{xi}
\xi(x_0^2)=Bx_0,\quad \xi(x_0x_1)=-Bx_1-Cx_2, \quad \xi(x_0x_2)=Ax_1.
\end{equation}
In particular, $\xi\ne0$. (This argument actually shows that the
blowup of the space of determinantal nets of quadrics
along the locus of degenerate nets maps isomorphically onto $H_3$,
cfr.~\cite{Elli-Pien-Stro}).
\subsection{The torus action}\label{E}
Consider the natural action of $\operatorname{GL}(n+1)$ on ${\bold P}^n$. It induces an
action on $H_n$ and on the bundles ${\cal E}_d$ for $d\ge1$. Let
$T\subseteq\operatorname{GL}(n+1)$ be a maximal torus, and let $(x_0,\ldots,x_n)$ be
homogeneous coordinates on ${\bold P}^n$ in which the action of $T$ is
diagonal. A point $x\in H_n$ is fixed by $T$ if and only if the
corresponding curve $C_x$ is invariant under $T$, i.e., $t(C_x)=C_x$ for
any $t\in T$. This is easily seen to be the case if and only if the
graded ideal of $C_x$ is generated by {\em monomials\/} in the $x_i$.
In particular, the fixpoints are isolated.
We will identify all the fixpoints $x\in H_n$, and for each of them we
will compute the representation on the tangent space $T_{H_n}(x)$. The
tangent space of the Hilbert scheme is $\operatorname{Hom}({\cal I}_x,{\cal O}_x)$, but special
care must be taken at the points where $H_n$ meets another component
of the Hilbert scheme. At these points, $T_{H_n}(x)$ is a proper
subspace of $\operatorname{Hom}({\cal I}_x,{\cal O}_x)$.
By the choice of the coordinates $(x_0,\ldots,x_n)$, there are
characters $\lambda_i$ on $T$ such that for any $t\in T$ we have
$t.x_i=\lambda^{}_i(t)x_i$. The characters $\lambda^{}_i$ generate the
representation ring of $T$, i.e., if $W$ is any finite dimensional
representation of $T$ we may, by a slight abuse of notation, write
$W=\sum a_{p_0,\dots,p_n}\lambda_0^{p_0}\lambda_1^{p_1}
\cdots\lambda_n^{p_n}$, where the $p_i$ and $a_{p_0,\dots,p_n}$ are
integers.
Recall \eqref{fibration} the morphism $\Phi:H_n\to G(3,n)$ which maps a
point $x\in H_n$ to the 3-space spanned by the corresponding curve
$C_x$. This morphism clearly is $\operatorname{GL}(n+1)$ equivariant, and its fibers
are all isomorphic to $H_3$. If $C_x$ is invariant under $T$, then so
is its linear span. Hence up to a permutation of the variables, we may
assume that it is given by the equations $x_4=\dots=x_n=0$, so that
$x_0,\dots,x_3$ are coordinates on the ${\bold P}^3\subseteq{\bold P}^n$ corresponding to
$\Phi(x)$. The torus $T$ acts on ${\bold P}^3$ through the four-dimensional
quotient torus $T_3$ of $T$ with character group spanned by
$\lambda_0,\dots,\lambda_3$.
The tangent space of $H_n$ at a fixpoint $x$ decomposes as a direct
sum
\begin{equation}
\label{dsum}
T_{H_n}(x)=T_{H_3}(x)\oplus T_{G(3,n)}(\Phi(x)),
\end{equation}
and it is well
known that
\begin{equation}\label{tangrass}
T_{G(3,n)}(\Phi(x))=
\operatorname{Hom}(H^0({\cal I}_{{\bold P}^3/{\bold P}^n}(1)), H^0(\OP3(1)))=
\sum_{j=0}^3\sum_{i=4}^n
\lambda_j^{}\lambda_i^{-1}.
\end{equation}
Hence we need to study the tangent space of $H_3=\Phi^{-1}\Phi(x)$.
\begin{prop} Any fixpoint of $T_3$ in $H_3$ is projectively equivalent to
one of the following, where the first four are Cohen-Macaulay and the
last four are not:
\begin{alignat}{2}
(1)\quad& (x_0x_1,x_1x_2,x_2x_3) &\qquad \qquad
(5)\quad&(x_0^2,x_0x_1,x_0x_2,x_1x_2x_3) \notag\\
(2)\quad&(x_0x_1,x_1x_2,x_0x_2) &\qquad \qquad
(6)\quad&(x_0^2,x_0x_1,x_0x_2,x_1^{}x_2^2)\notag\\
(3)\quad&(x_0x_1,x_2^2,x_0x_2) &\qquad \qquad
(7)\quad&(x_0^2,x_0x_1,x_0x_2,x_2^2x_3^{})\notag\\
(4)\quad&(x_0^2,x_0x_1,x_1^2) &\qquad \qquad
(8)\quad&(x_0^2,x_0x_1,x_0x_2,x_2^3)\notag
\end{alignat}
\end{prop}
\begin{pf}
The action of $T_3$ on ${\bold P}^3$ has the four coordinate points
as its fixpoints, and the only one-dimensional orbits are the six lines
of the coordinate tetrahedron. Hence any curve invariant under $T_3$
must be supported on these lines. If $C_x\in H_3$ is Cohen-Macaulay and
$T_3$-fixed, it is connected, has no embedded points and is not plane.
Hence there are only four possibilities: (1) the union of three distinct
coordinate lines, two of which are disjoint, (2) the union of three
concurrent coordinate lines, (3) a coordinate line doubled in a
coordinate plane plus a second line intersecting the first but not
contained in the plane, and finally (4) the full first-order
neighborhood of a coordinate line.
If $x\in Y^{T_3}$, then by the description of the curves in $Y$ we may
assume that the quadratic part of the ideal is $(x_0^2,x_0x_1,x_0x_2)$,
meaning that $C_x$ is a cubic plane curve in the plane $x_0=0$ which is
singular in $P=(0,0,0,1)$ plus an embedded point supported at $P$ but
not contained in the plane. For the cubic we have these possibilities:
(5) the three coordinate lines, (6) one double coordinate line through
$P$ plus another simple line passing through $P$, (7) one double
coordinate line through $P$ plus another simple line not passing through
$P$, and (8) one coordinate line through $P$ tripled in the plane.
\end{pf}
\begin{rem}
There are several fixpoints of each isomorphism class, in
fact it is easy to verify by permuting the variables that in a given
${\bold P}^3$ the numbers of fixpoints of the types 1 through 8 are 12, 4, 24,
6, 12, 24, 24, 24, respectively. This is consistent with the
fact that the (even) betti numbers of $H_3$ are 1, 2, 6, 10, 16, 19, 22,
19, 16, 10, 6, 2, 1, so that the Euler characteristic of $H_3$ is 130,
see \cite{Elli-Pien-Stro}.
\end{rem}
\begin{prop}\label{cm-akk}
Let $x$ be one of the fixpoints 1--4. Then the representation on the
tangent space $T_{H_3}(x)$ is given by
\begin{equation*}
T_{H_3}(x) = \operatorname{Hom}(F,E)\*({\lambda}_0+{\lambda}_1+{\lambda}_2+{\lambda}_3)-\operatorname{End}(E)-\operatorname{End}(F)+1,
\end{equation*}
where the representations $E$ and $F$ are given in the following table:
\smallskip
\begin{center}
\begin{tabular} {|c|c|c|c|} \hline
Type\vrule height1.2em width0cm depth0.6em & ${\cal I}_x$ & $E$ & $F$ \\ \hline
(1)\vrule height1em width0cm depth0.4em & $(x_0x_1,x_1x_2,x_2x_3)$ &
$\lambda^{}_0\lambda^{}_1+\lambda^{}_1\lambda^{}_2
+\lambda^{}_2\lambda^{}_3$ &
$\lambda^{}_0\lambda^{}_1\lambda^{}_2+\lambda^{}_1
\lambda^{}_2\lambda^{}_3$ \\ \hline
(2)\vrule height1em width0cm depth0.4em & $(x_0x_1,x_1x_2,x_0x_2)$ &
$\lambda^{}_0\lambda^{}_1+\lambda^{}_1\lambda^{}_2
+\lambda^{}_0\lambda^{}_2$ &
$2\lambda^{}_0\lambda^{}_1\lambda^{}_2$ \\ \hline
(3)\vrule height1em width0cm depth0.4em & $(x_0x_1,x_2^2,x_0x_2)$ &
$\lambda^{}_0\lambda^{}_1+\lambda_2^2+\lambda^{}_0
\lambda^{}_2$ &
$\lambda^{}_0\lambda^{}_1\lambda^{}_2+\lambda^{}_0
\lambda^{2}_2$ \\ \hline
(4)\vrule height1em width0cm depth0.4em & $(x_0^2,x_0x_1,x_1^2)$ &
$\lambda^{2}_0+\lambda^{}_0\lambda^{}_1+\lambda^{2}_1$ &
$\lambda^{}_0\lambda^{2}_1+\lambda^{2}_0\lambda^{}_1$ \\ \hline
\end{tabular}
\end{center}
\end{prop}
\begin{pf} Follows from \lemref{CMlem}, and the fact that $E$ and $F$
are equivariantly given by \eqref{EE} and \eqref{FF}.
\end{pf}
\begin{prop} \label{pr:akk}
Let $x$ be one of the fixpoints 5--8. Let $\mu$ be the character of
the minimal cubic generator, i.e., ${\lambda}_1{\lambda}_2{\lambda}_3$, ${\lambda}_1{\lambda}_2^2$,
${\lambda}_2^2{\lambda}_3$, and ${\lambda}_2^3$, respectively, and let
\begin{align*}
A=&{\lambda}_0^{-1}({\lambda}_1+{\lambda}_2+{\lambda}_3)+{\lambda}_3({\lambda}_1^{-1}+{\lambda}_2^{-1})\\
B=&{\lambda}_1^3+{\lambda}_1^2{\lambda}_2+{\lambda}_1^2{\lambda}_3+{\lambda}_1{\lambda}_2^2+
{\lambda}_1{\lambda}_2{\lambda}_3+{\lambda}_2^3+{\lambda}_2^2{\lambda}_3
\end{align*}
Then the tangent space of $H_3$ at $x$ is given by
\begin{equation*}
T_{H_3}(x) = A + \mu^{-1}(B-\mu) + ({\lambda}_0{\lambda}_1{\lambda}_2)^{-1}\mu
\end{equation*}
\end{prop}
\begin{pf}
Let $\beta=g(x) \in I$ be as in \secref{E}. In fact, all
types 5--8 lie over the same fixpoint $\beta$. The first term, $A$ in
the sum above, is easily seen to be the representation on the tangent
space $T_I(\beta)$. Now $g\:Y\to I$ is a projective bundle, and the
fiber $g^{-1}(\beta)$ is the projective space associated to the vector
space of cubic forms in $(x_1,x_2)^2{\bold C}[x_1,x_2,x_3]$. The
representation on this vector space is $B$, and the second term of the
formula of the proposition is the representation on $T_{g^{-1}\beta}(x)$.
Thus the first two terms make up $T_Y(x)$. The last term,
$({\lambda}_0{\lambda}_1{\lambda}_2)^{-1}\mu$, is the character on $N_{Y/H_3}(x)$. This can be
seen from equations (\ref{xi}): by checking each case, one verifies that
the normal vector $\xi$ is semi-invariant with character
is $({\lambda}_0{\lambda}_1{\lambda}_2)^{-1}$ times the character of the cubic form $q$.
\end{pf}
\subsection{The computation}\label{F}
Let us briefly describe the actual computation, carried out using
``Maple'' \cite{Maple}, of the numbers in the introduction. $H_n$ has a
natural torus action with isolated fixpoints. By what we have done in
the last section, we can construct a list of all the fixpoints of $H_n$;
there are $130\binom{n+1}4 $ of these. For each of them we compute the
corresponding tangent space representation, by \eqref{dsum} and
propositions \ref{cm-akk} and \ref{pr:akk}.
A consequence of the fact that all fixpoints are isolated is that none
of the tangent spaces contain the trivial one-dimensional
representation. Choose a one-parameter subgroup $\psi\:{\bold C}^* \to T$ of
the torus $T$, such that all the induced weights of the tangent space at
each fixpoint are non-zero. This is possible since we only need to
avoid a finite number of hyperplanes in the lattice of one-parameter
subgroups of $T$. For example, we may choose $\psi$ in such a way that
the weights of the homogeneous coordinates $x_0,\dots,x_n$ are
$1,w,w^2,\dots,w^n$ for a sufficiently large integer $w$. In our
computations (for $n\le8$) we used instead weights taken from the sequence
4, 11, 17, 32, 55, 95, 160, 267, 441, but any choice that will not
produce a division by zero will do.
Since all the tangent weights of the ${\bold C}^*$ action on $H_n$ so obtained
are non-trivial, it follows that this action has the same fixpoints as
the action of $T$, hence a finite number.
By \propref{propB}, we need to evaluate the class
$$\delta=c_{{\text{top}}}({\cal E}_{d_1}\oplus\cdots\oplus{\cal E}_{d_p})\cdot
\prod_{i=1}^m\gamma_{\lambda_i}\in A^{4n}(H_n).$$
Note that the isomorphism \eqref{basechange}
is equivariant. Clearly $H^0({\cal O}_{C_x}(d))$ is spanned by all monomials
of degree $d$ not divisible by any monomial generator of $I_x$. Thus we
know all the representations ${\cal E}_d(x)$ for all fixpoints $x\in H_n$.
By \propref{gammaformler}, $\delta$ is a polynomial
$p(\dots,c_k({\cal E}_d),\dots)$ in the Chern classes of the equivariant
vector bundles ${\cal E}_d$. To find a function $f$ on the fixpoint set which
represents $\delta$, simply replace each occurance $c_k({\cal E}_d)$ by the
localized equivariant Chern class $\sigma_k({\cal E}_d,-)$, i.e., put
$f=p(\dots,\sigma_k({\cal E}_d,-),\dots)$. Then the class is evaluated by
the formula in \thmref{bott1} (2).
\section{The Hilbert scheme of points in the plane}\label{hilbkap}
Let $V$ be a three-dimensional vector space over ${\bold C}$ and let
${\bold P}(V)$ be the associated projective plane of rank-1 quotients of $V$.
Denote by $H_r=\operatorname{Hilb}^r_{{\bold P}(V)}$ the Hilbert scheme parameterizing
length-$r$ subschemes of ${\bold P}(V)$. There is a universal subscheme
${\cal Z}\subseteq H_r\times{\bold P}(V)$. We will use similar notational conventions
as in \secref{A}: for example, if $x\in{H_r}$, the corresponding subscheme
of ${\bold P}^2$ is denoted $Z_x$, its ideal sheaf ${\cal I}_x$ etc.
As in \eqref{defE}, let for any integer $n$
\begin{equation}
\label{def2E}
{\cal E}_{n}=p_{1*}({\cal O}_{\cal Z}\* p_{2}^* {\cal O}_{{\bold P}(V)}(n)),
\end{equation}
where $p_1$ and $p_2$ are
the two projections of $H_r\times{\bold P}(V)$.
Since ${\cal Z}$ is finite over ${H_r}$, all basechange maps
\begin{equation}
{\cal E}_{n}(x) @>\simeq>> H^0(Z_x,{\cal O}_{Z_x}(n)).
\label{basechange2}
\end{equation}
are isomorphisms. In particular, ${\cal E}_n$ is a rank-$r$ vector bundle
on ${H_r}$. Denote by ${\cal L}$ the linebundle
\begin{equation}\label{defL}
{\cal L} = \displaystyle{\operatornamewithlimits{\wedge}}^r{\cal E}_0\*\displaystyle{\operatornamewithlimits{\wedge}}^r{\cal E}_{-1}^{\vee}.
\end{equation}
Then ${\cal L}$ corresponds to the divisor on $H_r$ corresponding to
subschemes $Z$ meeting a given line. We are going to compute integrals
of the form
\begin{equation}\label{generalintegral}
\int_{{H_r}} s_{2r}({\cal E}_n\*{\cal L}^{\*m})
\end{equation}
for small values of $r$. Afterwards we will give interpretations of
some of these numbers in terms of degrees of power sum and Darboux loci
in the system ${\bold P}(S_nV)$ of plane curves of degree $n$ in the dual
projective plane ${\bold P}(V^{\vee})$.
As usual, we start by identifing all the fixpoints and tangent space
representations for a suitable torus action. This has been carried out
in more detail in \cite{Elli-Stro-1}, the following simpler presentation
is sufficient for the present purpose.
As in \secref{E}, let $T\subseteq \operatorname{GL}(V)$ be a maximal torus and let
$x_0,x_1,x_2$ be a basis of $V$ diagonalizing $T$ under the natural
linear action. The eigenvalue of $x_i$ is a character ${\lambda}_i$ of $T$.
We identify characters with one-dimensional representations, hence the
representation ring of $T$ with the ring of Laurent polynomials in
${\lambda}_0,{\lambda}_1,{\lambda}_2$. For example, the natural representation on the vector
space $V^{\vee}$ can be written ${\lambda}_0^{-1}+{\lambda}_1^{-1}+{\lambda}_2^{-1}$.
Fixpoints of ${H_r}$ can be described in terms of partitions,
i.e., integer sequences $b=\{b_r\}_{r\ge0}$ weakly decreasing to zero.
Let $|b|=\sum_{r\ge0}b_r$. The {\em diagram\/} of a partition $b$ is
the set $D(b)=\{(r,s)\in{\bold Z}_{\ge0}^2\mid s< b_r\}$ of cardinality $|b|$.
A {\em
tripartition\/} is a triple $B=(b^{(0)},b^{(1)},b^{(2)})$ of
partitions; put $|B|=\sum_i |b^{(i)}|$; the number being partitioned.
The {\em $n$-th diagram\/} $D_n(B)$ of a tripartition
$B=(b^{(0)},b^{(1)},b^{(2)})$ is defined for
$n\ge|B|$ as follows: Letting the index $i$ be counted modulo 3, we put
$$
D_n^i(B)=\{ (n_0,n_1,n_2)\in{\bold Z}^3 \mid n_0+n_1+n_2=n
\text{ and } (n_{i+1},n_{i+2})\in
D(b^{(i)})\}
$$
and $D_n(B) = D_n^0(B)\cup D_n^1(B)\cup D_n^2(B)$. Intuitively, the
diagram $D_n(B)$ lives in an equilateral triangle with corners $(n,0,0)$,
$(0,n,0)$, and $(0,0,n)$, and originating from the $i$-th corner there is a
(slanted) copy of $D(b^{(i)})$. When $n\ge|B|$, these don't overlap. As
$n$ grows, the shape and size of the three parts of $D_n(B)$ stay the same,
whereas the area separating them grows. We may also define $D_n(B)$ for
integers $n<r$ by the same formula as above, but where the union is taken
in the sense of multisets, i.e., some elements might have multiplicities 2
or even 3. For $n<r$ the diagram $D_n(B)$ may also stick out of the
triangle referred to above.
A fixpoint $x\in{H_r}$ corresponds to a length-$r$ subscheme $Z_x\subseteq{\bold P}(V)$
defined by a monomial ideal. Fix an integer $n\ge r$ and consider the
set
$$
D_n(Z_x)=\{(n_0,n_1,n_2)\in {\bold Z}_{\ge0}^3 \mid \sum_i n_i=n \text{ and }
\prod x_i^{n_i} \notin H^0({\bold P}(V),{\cal I}_x(n))\}
$$
This set is the $n$-th diagram of a tripartition $B$ of $r$, the three
constituent partitions corresponding to the parts of $Z_x$ supported in the
three fixpoints of ${\bold P}(V)$. Conversely, starting with a tripartition of
$r$, we may obviously construct a monomial ideal of colength $r$ in
such a way that we get an inverse of the construction above. Hence there
is a natural bijection between $H_r^T$ and the set of tripartitions of
$r$.
In terms of representations, it follows from the above that for a
fixpoint $x$ corresponding to the tripartition
$B=(b^{(0)},b^{(1)},b^{(2)})$, we have
\begin{equation}\label{reprEn}
{\cal E}_n(x)=H^0({\cal O}_{Z_x}(n)) = \sum_{(n_0,n_1,n_2)\in D_n(B)} \prod {\lambda}_i^{n_i}.
\end{equation}
For $n<r$, the summation index needs to be interpreted as running
through the multiset $D_n(B)$. The representation on ${\cal L}$ in the same
fixpoint is
\begin{equation}\label{reprL}
{\cal L}(x) = \prod {\lambda}_i^{|b^{(i)}|}.
\end{equation}
For $n\ge r$, we also have the following formula for
$I_n:=H^0({\bold P}(V),{\cal I}_x(n))$ in the representation ring:
$$
I_n =S_nV - H^0({\cal O}_{Z_x}).
$$
To compute the tangent space representation, we use a trick that is often
useful even in higher dimensions: functorial free resolutions. The tangent
space of ${H_r}$ in $x$ is canonically isomorphic to $\operatorname{Ext}^1({\cal I}_x,{\cal I}_x)$.
Fix an integer $n\ge r+2$. Then there is a canonical resolution of locally
free ${\cal O}_{{\bold P}(V)}$-modules
$$
K_{\bullet}:\quad 0\to K_2 \to K_1 \to K_0
$$
of
${\cal I}_x(n)$, where $K_p=\Omega_{{\bold P}(V)}^p(p)\*I_{n-p}$. As in
\eqref{beilinson}, this is a special case of Beilinson's spectral sequence.
$T$ acts on $K_\bullet$. Let $S^\bullet$ be the total complex
associated to the double complex $\operatorname{Hom}_{{\bold P}^2}(K_\bullet,K_\bullet)$.
Then the $i$-th cohomology group of $S^\bullet$ is
$\operatorname{Ext}^i({\cal I}_x(n),{\cal I}_x(n))=\operatorname{Ext}^i({\cal I}_x,{\cal I}_x)$ for $i=0,1,2$
\cite[Lemma 2.2]{Elli-Stro-4}.
For $i=0$ this is ${\bold C}$ (with trivial action) and for
$i=2$ it is zero. Using the canonical identifications
$\operatorname{Hom}(\Omega^p(p),\Omega^q(q)) = \displaystyle{\operatornamewithlimits{\wedge}}^{p-q}V^{\vee}$, we end up with the
following formula for the tangent space representation in terms of the
data of the tripartition $B$:
{\allowdisplaybreaks
\begin{align}\label{reprT}
T_{{H_r}}(x) = \,& 1-(\sum_{i=0}^2(-1)^i\operatorname{Ext}^i({\cal I}_x,{\cal I}_x))\\
= \,&1 -(S^0-S^1+S^2) \notag\\
=\,&1-(\operatorname{Hom}(I_n,I_{n})+\operatorname{Hom}(I_{n-1},I_{n-1})+\operatorname{Hom}(I_{n-2},I_{n-2}))\notag\\*
&+ ({\lambda}_0^{-1}+{\lambda}_1^{-1}+{\lambda}_2^{-1})
(\operatorname{Hom}(I_{n-1},I_n)+\operatorname{Hom}(I_{n-2},I_{n-1}))\notag\\*
&-({\lambda}_0^{-1}{\lambda}_1^{-1}+{\lambda}_1^{-1}{\lambda}_2^{-1}+{\lambda}_2^{-1}{\lambda}_0^{-1})\operatorname{Hom}(I_{n-2},I_n)\notag
\end{align}
}
Here are the computational results on the Hilbert scheme which will
be used in the following applications:
\begin{prop} \label{computation1} Let ${\cal E}_n$ be as in \eqref{def2E}.
For $2\le r\le 8$, we have
$$
\int_{{H_r}}s_{2r} ({\cal E}_n) = s_r(n),
$$
where $s_r(n)$ are as in \thmref{main4}.
\end{prop}
\begin{prop} \label{computation2} Let ${\cal E}_{-1}$ and ${\cal L}$ be as in
\eqref{def2E} and \eqref{defL}.
For $r=2,3,4,5,6,7,8,9,10$, we have
$$
\int_{{H_r}}s_{2r}( {\cal E}_{-1}\*{\cal L}) = 0,0,0,0,2540, 583020, 99951390,
16059395240, 2598958192572.
$$
\end{prop}
\begin{pf} For both propositions, apply Bott's formula. The one-parameter
subgroup of $T$ such that the ${\lambda}_i$ have weights 0,1,19 will work. The
contribution at each fixpoint is given by
\eqref{reprEn}, \eqref{reprL}, and \eqref{reprT}. Generate all
fixpoints and perform the summation using e.g. \cite{Maple}.
\end{pf}
\begin{rem} There is no difficulty in principle to evaluate
\eqref{generalintegral} directly with symbolic values of both $n$ and
$m$, for given values of $r$. For example, for $r=3$, the result is
$(n^{6}+24\,n^{5}m+252\,n^{4}m^{2}+1344\,n^{3}m^{3}+3780\,
n^{2}m^{4}+5040\,nm^{5}+
2520\,m^{6}-30\,n^{4}-432\,n^{3}m-2520\,n^{2}m
^{2}-6048\,nm^{3}-5040\,m^{4}+45\,n^{3}+ 504\,n^{2}m+2268\,nm^{2}+3024
\,m^{3}+206\,n^{2}+1200\,nm+1512\,m^{2}-576\,n- 1728\,m+384)/6$. However,
with given computer resources, one gets further the fewer variables one
needs. On a midrange workstation, we could do this integral up to $r=5$.
\end{rem}
\begin{rem} Tyurin and Tikhomirov \cite{Tyur-Tikh} and Le Potier have
shown that \propref{computation2} implies that the Donaldson
polynomial $q_{17} ({\bold P}^2) = 2540 $. It may also be deduced from the
proposition that $q_{21}({\bold P}^2) = 233208$, see \cite{Tyur-Tikh} or our
forthcoming joint paper with J. Le Potier.
\end{rem}
\subsection{Power sum varieties of plane curves}
Closed points of ${\bold P}(S_nV)$ correspond naturally to curves of degree
$n$ in the dual projective plane ${\bold P}(V^{\vee})$. In particular, points of
${\bold P}(V)$ correspond to lines in ${\bold P}(V^{\vee})$, so $H_r=\operatorname{Hilb}^r_{{\bold P}(V)}$
is a compactification of the set of unordered $r$-tuples of linear
forms modulo scalars.
Let $r$ and $n$ be given integers. Let $U(r,n)$ be the set of pairs
$(\{L_1,\dots,L_r\},C)$ where the $L_i\subseteq{\bold P}(V^{\vee})$ are lines in
general position, and $C$ is a curve with equation of the form
$\sum_{i=1}^r a_i\ell_i^n \in S_nV$, where $\ell_i\in V^{\vee}$ is an
equation of $L_i$. Then the power sum variety $PS(r,n)$ is the
closure of the image of $U(r,n)$ in ${\bold P}(S_nV)$ under the projection
onto the last factor. To compute the degree of the image times the
degree $p(r,n)$ of the map $U(r,n)\to PS(r,n)$, we need to find a
workable compactification of $U(r,n)$.
Recall from \eqref{def2E} the rank-$r$ vector bundle ${\cal E}_n$ on
${H_r}$. It comes naturally with a morphism $S_nV_{H_r}\to {\cal E}_n$,
which is surjective if $n\ge r-1$. Now consider the projective bundle
over $H_r$:
\begin{equation}
{\bold P}({\cal E}_n) \subseteq {\bold P}(S_nV) \times {H_r}.
\end{equation}
It is easy to verify that ${\bold P}({\cal E}_n)$ contains $U(r,n)$ as an open
subset. It follows that $p(r,n)$ times the degree of $PS(r,n)$ is given
by the self-intersection of the pullback to ${\bold P}({\cal E}_n)$ of
${\cal O}_{{\bold P}(S_nV)}(1)$. This is
$\int_{{\bold P}({\cal E}_n)} c_1({\cal O}_{{\bold P}({\cal E}_n)}(1))^{3r-1}$, and pushing it
forward to the Hilbert scheme, we get almost by definition,
$\int_{{H_r}} s_{2r}({\cal E}_n)$
\cite[Ch.~ 3]{IT}.
Together with \propref{computation1}, this proves \thmref{main4}.
\subsection{Darboux curves} These curves are also defined in terms of
linear forms, and we may take $H_{n+1}$ as a parameter space for the
variety of complete $(n+1)$-gons. For a length-$(n+1)$ subscheme
$Z\subseteq{\bold P}(V)$, put $E=H^0({\cal O}_{Z}(-1))=H^1({\cal I}_Z(-1))$ and $F=H^1({\cal I}_Z)$.
The multiplication map
$
V\* E \to F
$
gives rise to a bundle map over the dual plane ${\bold P}(V^{\vee})$:
$$
m\: E_{{\bold P}(V^{\vee})}(-1) \to F_{{\bold P}(V^{\vee})}.
$$
If $Z$ consists of $n+1$ points in general position,
the degeneration locus $D(Z)$ corresponds to the set of bisecant
lines to $Z$, i.e., the singular locus of the associated $(n+1)$-gon.
The Eagon-Northcott resolution of $D(Z)$ gives the
following short exact sequence:
$$
0 \to F_{{\bold P}(V^{\vee})}^{\vee}(-1) \to E_{{\bold P}(V^{\vee})}^{\vee} \to L\*_{\bold C}{\cal I}_{D(Z)}(n) \to 0,
$$
showing that there is a natural surjection
$$
S_nV^{\vee} \to H^0({\bold P}(V^{\vee}),{\cal I}_{D(Z)}(n))^{\vee} \simeq H^0(Z,{\cal O}_Z(-1))\*L.
$$
Here $L$ is
the onedimensional vector space $\det(F)\*\det(E)^{-1}$.
Globalizing this construction over $H_{n+1}$ gives a natural map
$$
p\: S_nV_{H_{n+1}} \to {\cal E}_{-1}\*{\cal L}
$$
such that the closure of the image of the induced rational map
${\bold P}({\cal E}_{-1}\*{\cal L})\to{\bold P}(S_nV)$ is the Darboux locus $D(n)$.
By the lemma below, this rational map is actually a morphism. Thus
we may argue as in the power sum case and find that $p(n)$ times the
degree of $D(n)$ is $\int_{{\bold P}({\cal E}_{-1}\*{\cal L})} c_1({\cal O}(1))^{3n+2}=
\int_{H_{n+1}} s_{2n+2}({\cal E}_{-1}\*{\cal L})$. This together with
\propref{computation2} implies \thmref{main5}.
\begin{lem}
The bundle map $p\:S_nV_{H_{n+1}} \to {\cal E}_{-1}\*{\cal L}$ over
$H_{n+1}$ is surjective.
\end{lem}
\begin{pf} Assume the contrary. Since the support of the cokernel is
closed and $\operatorname{GL}(V)$-invariant,
there exists a subscheme $Z$ in $\operatorname{Supp}\operatorname{Coker}(p)$ which is supported in
one point. Without loss of generality we may assume that $Z$ is
supported in the point $x_1=x_2=0$.
Let $E$ and $F$ be as above, and let $K\subseteq E$ be a subspace of
codimension 1. For a linear form $\ell\in V$, let $m_{\ell}\:E\to F$
be multiplication by $\ell$. The assumption that $p$ is not
surjective means that $K$ can be chosen such that the determinant of
the restriction of $m_{\ell}$ to $K$ is 0 for all $\ell\in V$.
Multiplication by $x_0$ induces an isomorphism $E=H^0({\cal O}_Z(-1)) \simeq
H^0({\cal O}_Z) \simeq {\bold C}[x,y]/I_Z$, where $x=x_1/x_0$ and $y=x_2/x_0$. Under
these identifications, if $\ell=1-ax-by$ is the image of a general
linear form, the kernel of $m_{\ell}$ is generated by $1/\ell$.
Consider the set $S$ consisting of all such elements $\ell^{-1}$, with
$a,b\in{\bold C}$. The series expansion $\ell^{-1}=1+(ax+by)+(ax+by)^2+\cdots$
shows that $S$ generates ${\bold C}[x,y]/(x,y)^m$ as a vector space for all
$m\ge0$. Indeed, a hyperplane $W_m\subseteq{\bold C}[x,y]/(x,y)^m$ containing the
image of $S$ would, by induction on $m$, dominate
${\bold C}[x,y]/(x,y)^{m-1}$. Hence $W_m$ could not contain
$(x,y)^{m-1}/(x,y)^m$. But the image of $S$ in $(x,y)^{m-1}/(x,y)^m$ is
the cone over a rational normal curve of degree $m-1$, hence spans this
space.
Since $(x,y)^m\subseteq I_Z$ for $m$ large, it follows that $S$ generates
${\cal O}_Z$ and hence $E$ as a vector space. Now for an $\ell$ such that
$\ell^{-1}\notin K$, the restriction of $m_\ell$ to $K$ will be an
isomorphism. This leads to the desired contradiction.
\end{pf}
\section{Discussion} How general is the strategy of using Bott's
formula in enumerative geometry, as outlined in these examples? The
first necessary condition is probably a torus action, although Bott's
formula is valid in a more general situation: a vector field with
zeros and vector bundles acted on by the vector field. It seems to
us, though, that the cases where one stands a chance of analysing the
local behaviour of bundles near all zeroes of such a field are those
where the both the vector field and its action on the vector bundles
are ``natural'' in some sense. If there are parameter spaces with
natural flows on them, not necessarily coming from torus actions,
presumably Bott's formula could be useful.
It is not necessary that the fixpoints be isolated in order for the
method to give results. Kontsevich's work \cite{Kont-1} is a
significant example of this. Another natural candidate for Bott's
formula is the moduli spaces of semistable torsionfree sheaves on
${\bold P}^2$. These admit torus actions, but not all fixpoints are
isolated. One can still control the structure of the fixpoint
components, however. This may hopefully be used for computing
Donaldson polynomials of the projective plane, at least in some cases.
A more serious obstacle to the use of Bott's formula is the presence
of singularities in the parameter space. For example, all components
of the Hilbert scheme of ${\bold P}^n$ admit torus actions with isolated
fixpoints, but they are almost all singular. The main non-trivial
exceptions are actually the ones treated in the present paper. On the
other hand, singularities present inherent problems for most
enumerative approaches, especially if a natural resolution of
singularities is hard to find.
For all examples in this paper, one needs a computer to actually perform
the tedious computations. We mentioned already that the number of
fixpoints in the case of twisted cubics in ${\bold P}^n$ is $130\binom{n+1}4$.
For the Hilbert scheme of length-8 subschemes of ${\bold P}^2$, the
corresponding number is 810. From the point of view of computer
efficiency, there are some advantages to the use of Bott's formula in
contrast to trying to work symbolically with generators and relations in
cohomology, as for example in \cite{Elli-Stro-3} or \cite{schubert}.
First of all, the method works even if we don't know all relations, as
is the case for the Hilbert scheme of the plane, for example. But the
main advantage is perhaps that Bott's formula is not excessively hungry
for computer resources. It is often straightforward to make a loop over
all the fixpoints. The computation for each fixpoint is fairly simple,
and the result to remember is just a rational number. This means that
the computer memory needed does not grow much with the number of
fixpoints, although of course the number of CPU cycles does. For
example, most of the numbers of elliptic quartics were computed on a
modest notebook computer, running for several days.
Finally, Bott's formula has a nice error-detecting feature, which is
an important practical consideration: If your computer program
actually produces an integer rather than just a rational number,
chances are good that the program is correct!
|
1996-03-05T06:13:25 | 9411 | alg-geom/9411004 | en | https://arxiv.org/abs/alg-geom/9411004 | [
"alg-geom",
"hep-th",
"math.AG",
"math.QA",
"q-alg"
] | alg-geom/9411004 | Ezra Getzler | Ezra Getzler | Operads and moduli spaces of genus $0$ Riemann surfaces | 26 pages, latex2e with amslatex 1.2beta (available from
e-math.ams.com by anonymous ftp), no figures | null | null | null | null | We study a pair of dual operads which arise in the study of moduli spaces of
pointed genus 0 curves (this duality is similar to that between commutative and
Lie algebras). These operads are both quadratic, and even Koszul, and arise in
the theory of quantum cohomology.
| [
{
"version": "v1",
"created": "Wed, 9 Nov 1994 18:51:02 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Getzler",
"Ezra",
""
]
] | alg-geom | \section{$\SS$-modules and operads}
In this section, we recall the basic definitions of the theory of
operads. For more details, see \cite{n-algebras} and \cite{GK}.
\subsection{$\SS$-modules}
An $\SS$-module is a collection of chain complexes (all chain complexes in
this paper are over the field $\mathbb{C}$, and have finite dimensional total
homology)
$$
\{ \v(n) \mid n\ge0 \} ,
$$
together with an action of $\SS_n$ on $\v(n)$. This definition generalizes
Joyal's notion of a linear species \cite{Joyal}, which is an ungraded
$\SS$-module.
A chain complex $V$ may be thought of as an $\SS$-module by setting
$$
\v(n) = \begin{cases} V , & n=1 , \\ 0 , & n\ne1 . \end{cases}
$$
\subsection{Schur functors}
Given an $\SS$-module $\v$ and a finite set $S$, we define
$$
\v(S) = \Bigl(
\bigoplus\begin{Sb}\text{bijections}\\\begin{CD}f:\{1,\dots,n\}@>>>S
\end{CD}\end{Sb}
\v(n) \Bigr)_{\SS_n} .
$$
It is clear that if $S=\{1,\dots,n\}$, then $\v(S)$ is naturally
identified with $\v(n)$.
To an $\SS$-module $\v$ is associated an endofunctor of the category of
chain complexes, called the Schur functor of $\v$, by the formula
$$
V \mapsto \mathsf{S}(\v,V) = \bigoplus_{n=0}^\infty \v\o_{\SS_n} V^{\o n} ;
$$
here $V^{\o n}$ is the graded $n$th tensor power of $V$. Introduce a
monoidal structure on the category of $\SS$-modules, with tensor product
$$
(\v\circ\mathcal{W})(n) = \bigoplus_{k=0}^\infty \Bigl( \v(k) \o
\bigoplus_{\begin{CD}f:\{1,\dots,n\}@>>>\{1,\dots,k\}
\end{CD}} \bigotimes_{i=1}^k
\mathcal{W}(f^{-1}(i)) \Bigr)_{\SS_k} ,
$$
and unit the $\SS$-module ${1\!\!1}$:
$$
{1\!\!1}(n) = \begin{cases} \mathbb{C} , & n=1 , \\ 0 , & n\ne1 .
\end{cases}$$
The peculiar formula for $\v\circ\mathcal{W}$ is justified by
$$
\mathsf{S}(\v,\mathsf{S}(\mathcal{W},V)) = \mathsf{S}(\v\circ\mathcal{W},V) .
$$
Note also that $\mathsf{S}({1\!\!1},V)=V$. For more on this formalism, see Chapter 1
of \cite{n-algebras}.
\subsection{Operads}
An operad is a monoid in the category of $\SS$-modules, that is, an
$\SS$-module $\a$ with product $\begin{CD}\rho:\a\circ\a@>>>\a
\end{CD}$ and unit
$\begin{CD}\eta:{1\!\!1}@>>>\a
\end{CD}$ satisfying the axioms of associativity and unit
\cite{Maclane}. We denote the image under the product $\rho$ of
$$
a\o b_1\o\dots\o b_k \in \a(k)\o\a(n_1)\o\dots\o\a(n_k)
$$
by $a(b_1,\dots,b_k)$ and the unit by $1\in\a(1)$.
An operad structure on an $\SS$-module $\a$ such that $\a(n)=0$ for $n\ne1$
is the same thing as an associative algebra structure on $\a(1)$. Whereas
an element of an algebra has only one ``input'' and one ``output,'' an
element of an operad has multiple inputs and one output.
\subsection{The endomorphism operad of a chain complex}
If $V$ is a chain complex, its endomophism operad is the $\SS$-module
$$
\mathcal{E}_V(n) = \operatorname{Hom}(V^{\o n},V) .
$$
This is an operad, whose product is given by composition: if
$a\in\operatorname{Hom}(V^{\o k},V)$ and $b_i\in\operatorname{Hom}(V^{\o n_i},V)$, then
$$
a(b_1,\dots,b_k) = a \* (b_1\o\dots\o b_k) ,
$$
where we think of $b_1\o\dots\o b_k$ as an element of
$\operatorname{Hom}(V^{\o(n_1+\dots+n_k)},V^{\o k})$.
\subsection{Suspension of operads}
If $V$ is a chain complex, denote by $\Sigma V$ the chain complex such that
$(\Sigma V)_i=V_{i-1}$, with differential $-\delta$. If $\v$ is an
$\SS$-module, denote by $\Lambda\v$ the $\SS$-module
$$
(\Lambda\v)(n) = \Sigma^{1-n} \operatorname{sgn}_n \o \v(n) ,
$$
where $\operatorname{sgn}_n$ is the sign character of $\SS_n$. There is a natural
isomorphism
$$
\mathsf{S}(\Lambda\v,V) \cong \Sigma\mathsf{S}(\v,\Sigma^{-1}V) .
$$
It follows that if $\a$ is an operad, then so is $\Lambda\a$, and if $A$
is an $\a$-algebra, then $\Sigma A$ is a $\Lambda\a$-algebra.
\subsection{Algebras over an operad}
An algebra over an operad $\a$ is a chain complex $A$, together with a
morphism of operads $\begin{CD}\a@>>>\mathcal{E}_A
\end{CD}$. Thus, if $A$ is an algebra over an operad
$\a$ and $\rho\in\a(n)$, there is a product $a_1\o\dots
a_n\mapsto\rho(a_1,\dots,a_n)$ from $A^{\o n}$ to $A$. These products are
equivariant, under the actions of $\SS_n$ on $\a(n)$ and $A^{\o n}$,
associative with respect to the product of $\a$, and $1(a)=a$, where
$1\in\a(1)$ is the unit of $\a$.
\subsection{Construction of operads}
Given an algebraic structure defined by a set of multilinear operations
together with a set of multilinear relations among them, one may construct
the operad $\a$ having this presentation, in such a way that an $\a$-algebra
is the same thing as an instance of the original algebraic structure.
Denote by $V\mapsto T(V)$ the free algebra generated by the chain complex
$V$ with respect to this algebraic structure. To define $\a(n)$, we form
the free algebra $T(x_1,\dots,x_n)$ generated by the free vector space
$\mathbb{C}^n$ of rank $n$. The torus $(\mathbb{C}^\times)^n$ acts on this chain complex;
let $\a(n)$ be the $\SS_n$-submodule on which it acts by the character
$(z_1,\dots,z_n)\mapsto z_1\dots z_n$. The group $\SS_n$ acts on
$T(x_1,\dots,x_n)$, and thus on $\a(n)$, by permutation of the letters
$x_i$. It is not difficult to see that the $\SS$-module thus constructed is
an operad: the unit is the word $x_1\in\a(1)\subset T(x_1)$, while the
product is defined by substitution.
Let us give some examples of this construction. In the case of commutative
algebras, we call the resulting operad ${\operatorname{\mathcal{C}\mathit{om}}}^+$; the $\SS_n$-module
${\operatorname{\mathcal{C}\mathit{om}}}^+(n)$ is spanned by the word $x_1\dots x_n$ the free commutative
algebra generated by letters $\{x_1,\dots,x_n\}$, and carries the trivial
action of $\SS_n$. (Here, we are dealing with non-unital commutative
algebras, so that ${\operatorname{\mathcal{C}\mathit{om}}}^+(0)=0$.)
Let ${\operatorname{\mathcal{A}\mathit{ss}}}^+$ be the operad associated to associative algebras; the
$\SS_n$-module ${\operatorname{\mathcal{A}\mathit{ss}}}^+(n)$ is spanned by the words
$$
\{ x_{\sigma(1)}\dots x_{\sigma(n)} \mid \sigma\in\SS_n \} ,
$$
and carries the regular representation of $\SS_n$. (Again, we set
${\operatorname{\mathcal{A}\mathit{ss}}}^+(0)=0$.)
Finally, the operad associated to Lie algebra is denoted ${\operatorname{\mathcal{L}\mathit{ie}}}^+$. The
underlying $\SS$-module may be studied by means of the
Poincar\'e-Birkhoff-Witt theorem, which implies that
$$
{\operatorname{\mathcal{A}\mathit{ss}}}^+ \cong {\operatorname{\mathcal{C}\mathit{om}}}^+ \circ {\operatorname{\mathcal{L}\mathit{ie}}}^+ .
$$
In Proposition \ref{Lie}, we will show how this leads to a formula for the
character of ${\operatorname{\mathcal{L}\mathit{ie}}}^+(n)$, due to Klyachko \cite{Klyachko}.
\subsection{The configuration spaces $\mathbb{C}^n_0$ and the braid operad $\b$}
Let $\mathbb{C}^n_0$ be the configuration space of $n$ labelled points in
$\mathbb{C}$. Define the $\SS$-module $\mathcal{B}\mathit{raid}$ by
$$
\mathcal{B}\mathit{raid}(n) = \begin{cases} H_{\bullet}(\mathbb{C}^n_0) , & n>0 , \\ 0 , & n=0 .
\end{cases}$$
We now construct a natural operad structure on $\mathcal{B}\mathit{raid}$. (We called this the
braid operad in \cite{n-algebras}.)
In the definition of operads, one can replace the category of chain
complexes by the category of topological spaces, and the tensor product by
Cartesian product, obtaining the notion of a topological operad. If $\mathcal{O}$
is a topological operad and $H_{\bullet}(-)$ is a generalized homology theory
with products, $H_{\bullet}(\mathcal{O})$ is an operad in the category of graded vector
spaces; this gives a useful method of constructing operads.
Boardman and Vogt have constructed a topological operad called the little
discs operad \cite{BoardmanVogt}. Let $D$ be the closed unit disc in $\mathbb{C}$,
and let $\mathcal{O}(n)$ be the topological space
$$\textstyle
\mathcal{O}(n) = \bigl\{ \binom{z_1\dots z_n}{r_1\dots r_n}
\in \binom{D^n}{\mathbb{R}^n_+} \big|
\text{ the discs $r_iD+z_i$ are disjoint subsets of $D$} \bigr\} .
$$
The symmetric group $\SS_n$ acts on $\mathbb{CP}(n)$ by permuting the discs:
$$\textstyle
\sigma\binom{z_1\dots z_n}{r_1\dots r_n} =
\binom{z_{\sigma(1)}\dots z_{\sigma(n)}}{r_{\sigma(1)}\dots r_{\sigma(n)}} .
$$
The product in this operad is defined by gluing of disks: if
$a=\binom{z_1\dots z_k}{r_1\dots r_k}$ and
$b_i=\binom{y_{i,1}\dots y_{i,n_i}}{s_{i,1}\dots s_{i,n_i}}$, then
$$\textstyle
a(b_1,\dots,b_k) = \begin{pmatrix}
r_1y_{1,1}+z_1&\dots&r_1y_{1,n_1}+z_1&\dots&
r_ky_{k,1}+z_k&\dots&r_ky_{k,n_k}+z_k \\
r_1s_{1,1}&\dots&r_1s_{1,n_1}&\dots&r_ks_{k,1}&\dots&r_ks_{k,n_k}
\end{pmatrix} .
$$
The map $\begin{CD}\mathcal{O}(n)@>>>\mathbb{C}^n_0
\end{CD}$ defined by
$\binom{z_1\dots z_n}{r_1\dots r_n}\mapsto(z_1,\dots,z_n)$ is a homotopy
equivalence, and thus the homology operad $H_{\bullet}(\mathcal{O})$ of this
topological operad has $\mathcal{B}\mathit{raid}$ as its underlying $\SS$-module.
The operad $\mathcal{B}\mathit{raid}$ has the following presentation (see \cite{Cohen:thesis}
and \cite{n-algebras}): it is generated by two operations, a commutative
product of degree $0$ and a Lie bracket of degree $1$, satisfying the
Poisson relation:
$$
[a,bc] = [a,b]c + (-1)^{(|a|+1)|b|} b[a,c] .
$$
It follows that the $\SS$-module $\mathcal{B}\mathit{raid}$ is isomorphic to
${\operatorname{\mathcal{C}\mathit{om}}}^+\circ\Lambda^{-1}{\operatorname{\mathcal{L}\mathit{ie}}}^+$. In particular, we obtain another
realization of the $\SS_n$-module ${\operatorname{\mathcal{L}\mathit{ie}}}^+(n)$:
$$
{\operatorname{\mathcal{L}\mathit{ie}}}^+(n) \cong \operatorname{sgn}_n \o H_{n-1}(\mathbb{C}^n_0) .
$$
\section{Cyclic operads}
All of the operads which we discuss in this article, notably the
hypercommutative and gravity operads, are cyclic operads
\cite{cyclic} --- that is, there is a notion of invariant inner product on
algebras over these operads. In this section, we recall the basics of the
theory of cyclic operads.
\subsection{Cyclic $\SS$-modules}
A cyclic $\SS$-module is an $\SS$-module $\v$ together with an action of
$\SS_{n+1}$ on $\v(n)$ extending the action of $\SS_n$. The name derives
from the fact that an action of $\SS_{n+1}$ is determined by compatible
actions of $\SS_n$ and the cyclic group $C_{n+1}\subset\SS_{n+1}$ generated
by the cycle $(01\dots n)$. We denote the action of $(01\dots n)$ on
$\v(n)$ by $v\mapsto v^*$, motivated by the fact that a cyclic $\SS$-module
structure on the $\SS$-module associated to a chain complex $V$ is just an
involution $v\mapsto v^*$. It is convenient to write $\v\(n\)$ for
$\v(n+1)$.
If $\v$ is a cyclic $\SS$-module, denote by $\Lambda\v$ the cyclic
$\SS$-module
$$
(\Lambda\v)\(n\) = \Sigma^{2-n} \operatorname{sgn}_n \o \v\(n\) .
$$
Thus, $\Lambda$ applied to the $\SS$-module underlying $\v$ is isomorphic
to the $\SS$-module underlying $\Lambda\v$.
A stable cyclic $\SS$-modules is a cyclic $\SS$-module $\a$ which satisfies
the condition that $\a\(n\)=0$ for $n<3$. (The word ``stable'' comes from
the theory of algebraic curves.)
\subsection{Cyclic operads}
A cyclic operad $\a$ is a cyclic $\SS$-module $\a$ with an operad
structure, such that $1^*=1$ and for all $a\in\a(k)$ and $b\in\a(l)$,
$$
a(1,\dots,1,b)^* = (-1)^{|a|\,|b|} b^*(a^*,1,\dots,1) .
$$
Note that if $\v$ is a cyclic $\SS$-module associated to a chain complex
$V$ with involution, then a cyclic operad structure on $\v$ is the same
thing as a $\ast$-algebra structure on $V$.
\subsection{Stable cyclic operads}
A cyclic operad $\a$ whose underlying $\SS$-module is stable is called a
stable cyclic operad. Because of this condition, $\a\(2\)=0$, so we can no
longer think of $\a$ as having a unit. This requires the introduction of
non-unital operads, following Markl. Observe that all of the products
$a(b_1,\dots,b_k)$ of a (unital) operad may be obtained by iterating the
products
$$
a\circ_i b = a(\underset{\text{$i-1$ times}}{\underbrace{1,\dots,1}},b,
\underset{\text{$k-i$ times}}{\underbrace{1,\dots,1}}) .
$$
The axioms for non-unital operads may be found in \cite{modular}: they are
of two types, equivariance and associativity. Finally, a non-unital operad
is cyclic if $(a\circ_kb)^*=(-1)^{|a|\,|b|}b^*\circ_1a^*$.
\subsection{Invariant inner products and cyclic algebras}
If $V$ is a chain complex with inner product $\<-,-\>$ (which we suppose to
be non-degenerate), the endomorphism operad $\mathcal{E}_V$ is a cyclic operad,
such that if $a\in\a(n)$ and $v_i\in V$, $0\le i\le n$,
$$
\< v_0 , a(v_1,\dots,v_n) \> = \< v_n , a^*(v_0,\dots,v_{n-1}) \> .
$$
A cyclic algebra over a cyclic operad is a chain complex $A$ with inner
product, together with a morphism of cyclic operads $\begin{CD}\a@>>>\mathcal{E}_A
\end{CD}$.
If $A$ is an algebra over the operad underlying a cyclic operad $\a$, and
$\<-,-\>$ is an inner product on $A$, we say that the inner product is
invariant.
Note that if $\a$ is a cyclic operad, then the operad $\Lambda\a$ is not
cyclic, but rather anticyclic:
$(a\circ_kb)^*=-(-1)^{|a|\,|b|}b^*\circ_1a^*$. This reflects the fact that
if $V$ is a chain complex and $\<-,-\>$ is an inner product on $V$, then
there is induced on $\Sigma V$ an antisymmetric non-degenerate bilinear
form $(-1)^{|v|}\<\Sigma v,\Sigma w\>$. However, $\Lambda^2\a$ is again a
cyclic operad, and if $A$ is a cyclic $\a$-algebra, then $\Sigma^2A$ is a
cyclic $\Lambda^2\a$-algebra.
\subsection{Examples of cyclic operads}
Since associative, commutative and Lie algebras all have well-known notions
of invariant inner product, it is not surprising that the corresponding
operads are cyclic. For the operads $\a^+$, where
$\a\in\{{\operatorname{\mathcal{A}\mathit{ss}}},{\operatorname{\mathcal{C}\mathit{om}}},{\operatorname{\mathcal{L}\mathit{ie}}}\}$, define the $\SS$-module $\a$ by setting
$\a(n)=\a^+(n)$ for $n>1$ and $\a(n)=0$ for $n\le1$.
The cyclic structure of ${\operatorname{\mathcal{C}\mathit{om}}}$ is simple to describe, since an inner
product on a commutative algebra is invariant if and only if
\begin{equation} \label{invariant}
\<a,bc\> = \<ab,c\> .
\end{equation}
It follows from this formula that $a=a^*$ for all $a\in{\operatorname{\mathcal{C}\mathit{om}}}\(n\)$.
In a similar way, the action of $\SS_n$ on ${\operatorname{\mathcal{A}\mathit{ss}}}\(n\)$ is determined by the
condition that an inner product on an associative algebra is invariant if
and only if \eqref{invariant} holds. It turns out that the $\SS_n$-module
${\operatorname{\mathcal{A}\mathit{ss}}}\(n\)$ is the induced representation $\operatorname{Ind}_{C_n}^{\SS_n}{1\!\!1}$, where
$C_n$ is the subgroup of $\SS_n$ generated by $\tau_n$, and ${1\!\!1}$ is its
trivial representation. We may think of ${\operatorname{\mathcal{A}\mathit{ss}}}\(n\)$ as being the
$\SS_n$-module spanned by symbols
$$
\<x_{\sigma(1)},x_{\sigma(2)}\dots x_{\sigma(n)}\> , \quad
\sigma\in\SS_n ,
$$
representing the inner product of $x_{\sigma(2)}\dots x_{\sigma(n)}$ with
$x_{\sigma(1)}$, subject to the relations
$$
\<x_{\sigma(1)},x_{\sigma(2)}\dots x_{\sigma(n)}\> \sim
\<x_{\sigma(n)},x_{\sigma(1)}\dots x_{\sigma(n-1)}\> .
$$
The cyclic structure on the Lie operad is associated with the usual notion
of an invariant inner product (or Killing form) on a Lie algebra,
satisfying
$$
\<[a,b],c\> = \<a,[b,c]\> .
$$
The character of the $\SS_{n+1}$-module ${\operatorname{\mathcal{L}\mathit{ie}}}(n)$ is calculated in
\cite{modular}. We will obtain a realization of this representation as a
homology group in the next section.
It is proved in \cite{cyclic} that the braid operad $\mathcal{B}\mathit{raid}$ is not a
cyclic operad.
\section{The moduli spaces $\overline{\mathcal{M}}_{0,n}$}
In this section, we study the combinatorial structure of the compactified
moduli spaces $\overline{\mathcal{M}}_{0,n}$. We then define the gravity and
hypercommutative operads, and introduce the fundamental exact sequences
relating them, which are obtained by considering the spectral sequence
associated by Deligne to the stratified space $\overline{\mathcal{M}}_{0,n}$.
\subsection{Graphs and trees}
The strata of the compactification $\overline{\mathcal{M}}_{0,n}$ are labelled by trees with
$n$ legs, and we recall some definitions from the theory of trees in this
paragraph (see also \cite{GK} and \cite{modular}).
A graph $G=(F,\pi,\tau)$ is a finite set $F=\operatorname{Flag}(G)$, the set of flags of
the graph, together with a partition $\pi$ and an involution $\tau$ of
$F$. (By a partition, we mean a decomposition of $F$ into disjoint subsets,
possibly empty, called its blocks.)
The vertices $\operatorname{Vert}(G)$ of the graph $G$ are the blocks of $\pi$, the edges
$\operatorname{Edge}(G)$ are the $2$-cycles of $\tau$, while the legs $\operatorname{Leg}(G)$ are the
fixed points of $\tau$. To a graph $G$ is associated a cell complex $G$
whose cells have dimension $0$ and $1$, and whose ends correspond to the
legs $\operatorname{Leg}(G)$. A graph is called a tree if this complex is simply
connected. We will have no further use for non-simply connected graphs in
this paper; however, much of the theory we describe has an analogue for
general graphs \cite{modular}.
The legs $\operatorname{Leg}(v)$ of a vertex $v\in\operatorname{Vert}(G)$ are the flags in the
corresponding equivalence class, while the valence $|v|$ of a vertex is the
cardinality of $\operatorname{Leg}(v)$.
If $S$ is a finite set, let $\mathcal{T}\(S\)$ be the set of isomorphism classes of
trees $T$ whose external edges are labelled by the elements of $S$ and such
that each vertex has valence at least three. Note that $\mathcal{T}\(S\)$ is
finite. The set of trees is graded by the number of edges:
$$
\mathcal{T}\(S\) = \bigcup_{i=0}^{|S|-3} \mathcal{T}_i\(S\) .
$$
In particular, $\mathcal{T}_0\(S\)$ has a single element, the tree with one vertex
whose set of flags equals $S$.
Denote by $\det(S)$ the determinant line $\Lambda^{\text{max}}\mathbb{C}^S$, which
is a representation of $\operatorname{Aut}(S)$. (For example, $\det(\{1,\dots,n\})$ is
just the sign representation $\operatorname{sgn}_n$.) If $T$ is a tree, let $\det(T)$ be
the determinant line $\det(\operatorname{Vert}(T))$ of the set of vertices of $T$. There
are natural isomorphisms
\begin{equation} \label{det}
\det(T) \cong \det(\operatorname{Edge}(T))
\cong \det(\operatorname{Leg}(T)) \o \bigotimes_{v\in\operatorname{Vert}(T)} \det(\operatorname{Leg}(v)) .
\end{equation}
\subsection{Stable curves}
A stable curve with $n$ marked points is a projective curve $\Sigma$ whose
only singularities are double points, together with an embedding of
$\{1,\dots,n\}$ in the set of smooth points of $\Sigma$, such that there
are no continuous automorphisms of $\Sigma$ fixing the marked points and
double points. Knudsen has proved that the moduli space $\overline{\mathcal{M}}_{g,n}$ of
stable curves of arithmetic genus $g$ with $n$ marked points is a compact
orbifold, obtained by adjoining to $\mathcal{M}_{g,n}$ a divisor with normal
crossings \cite{Knudsen}. We will only be interested in the genus zero
case, in which case $\overline{\mathcal{M}}_{0,n}$ is actually a projective variety. Another
reference for the genus $0$ case is \cite{Kapranov}.
A stable curve $\Sigma$ determines a graph $\Gamma_\Sigma$, called the dual
graph of $\Sigma$: the vertices of $\Gamma_\Sigma$ are the components of
the subvariety of smooth points of $\Sigma$, the edges are the double
points, and $\operatorname{Leg}(\Gamma_\Sigma)$ is the set of marked points. In
particular, if $\Sigma$ has arithmetic genus $0$ and $n$ marked points,
then $\Gamma_\Sigma\in\mathcal{T}\(n\)$.
The moduli space $\overline{\mathcal{M}}_{0,n}$ is a stratified space: it has one stratum
$\mathcal{M}\(T\)$ for each tree $T\in\mathcal{T}\(n\)$, consisting of all curves
$\Sigma\in\overline{\mathcal{M}}_{0,n}$ such that $\Gamma_\Sigma=T$. The stratum $\mathcal{M}\(T\)$
is isomorphic to the product
$$
\prod_{v\in\operatorname{Vert}(T)} \mathcal{M}_{0,|v|} ,
$$
and has codimension equal to the number of edges of $T$. For example
$\overline{\mathcal{M}}_{0,4}$ has the following four strata:
$$
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If $T\in\mathcal{T}_k(n)$, denote by $\overline{\mathcal{M}}\(T\)$ the closure of the stratum
$\mathcal{M}\(T\)$ of $\overline{\mathcal{M}}_{0,n}$, and by $[\overline{\mathcal{M}}\(T\)]$ the corresponding cycle
in $H_{2(n-k-3)}(\overline{\mathcal{M}}_{0,n})$. The following theorem is due to Keel
\cite{Keel}.
\begin{theorem} \label{Keel:1}
The cycles $[\overline{\mathcal{M}}\(T\)]$, $T\in\mathcal{T}\(n\)$ span $H_{\bullet}(\overline{\mathcal{M}}_{0,n})$.
\end{theorem}
In Section \ref{Keel:proof}, we give a proof of this theorem which differs
from Keel's, and uses mixed Hodge theory.
\subsection{The gravity operad}
Let $\mathcal{G}\mathit{rav}$ be the stable cyclic $\SS$-module
$$
\mathcal{G}\mathit{rav}\(n\) = \begin{cases}
\Sigma^{3-n}\operatorname{sgn}_n\o H_{\bullet}(\mathcal{M}_{0,n}) , & n\ge3 , \\ 0 , & n<3 .
\end{cases}$$
Note that $\mathcal{G}\mathit{rav}\(n\)$ is concentrated in degrees $3-n\le i\le 0$.
There is a natural cyclic operad structure on $\mathcal{G}\mathit{rav}$. To define the
product
$$\begin{CD}
\circ_i : \mathcal{G}\mathit{rav}\(m+1\)\o\mathcal{G}\mathit{rav}\(n+1\) @>>> \mathcal{G}\mathit{rav}\(m+n\) ,
\end{CD}$$
consider the embedding $j$ of
$\mathcal{M}_{0,\{0,\dots,m\}}\times\mathcal{M}_{0,\{0',\dots,n'\}}$ as a stratum of
$\overline{\mathcal{M}}_{0,\{0,\dots,\hat{\imath},\dots,m,1',\dots,n'\}}$, corresponding to
the joining of the point labelled $i$ in the curve
$\Sigma_1\in\mathcal{M}_{0,\{0,\dots,m\}}$ to the point labelled $0'$ in the curve
$\Sigma_2\in\mathcal{M}_{0,\{0',\dots,n'\}}$. Consider the Poincar\'e residue map
associated to this embedding \cite{Deligne}:
$$\begin{CD}
\operatorname{Res} : H^{\bullet}(\mathcal{M}_{0,\{0,\dots,\hat{\imath},\dots,m,1',\dots,n'\}}) @>>>
H^{\bullet}(\mathcal{M}_{0,\{0,\dots,m\}}\times\mathcal{M}_{0,\{0',\dots,n'\}})
\end{CD}$$
Suitably suspending the adjoint of this map, we obtain the product
$\circ_i$ of $\mathcal{G}\mathit{rav}$. This construction makes it quite obvious that $\mathcal{G}\mathit{rav}$
satisfies the equivariance and associativity axioms of a cyclic operad.
Denote by $[x_1,\dots,x_n]$ the element of $\mathcal{G}\mathit{rav}(n)$ of degree $2-n$
corresponding to the standard basis vector of $H_0(\mathcal{M}_{0,n+1})$. This
operation is graded antisymmetric, since $\SS_{n+1}$ acts on it by the sign
representation, and it is proved in \cite{weil} that this sequence of
operations generates $\mathcal{G}\mathit{rav}$, and that all relations are generated by the
quadratic relations of \eqref{gravity}. Note that in that paper, we work with
the operad $\Lambda^{-1}\mathcal{G}\mathit{rav}$, and the generators are all in degree
$1$: the relationship between these two sets of generators at the level of
algebras is
$$
\{x_1,\dots,x_n\} = (-1)^{(n-1)|x_1|+(n-2)|x_2|+\dots+|x_{n-1}|}
\Sigma^{-1} [\Sigma x_1,\dots,\Sigma x_n] .
$$
In that paper, the operad structure is constructed in a different, though
equivalent, way, using $\mathbb{C}^\times$-equivariant homology.
\subsection{$\mathcal{G}\mathit{rav}$ as a mixed Hodge operad}
Operads (and, more specifically, stable cyclic operads) may be defined in
any symmetric monoidal category with colimits. Up to this point, we have
concentrated on the examples of operads in the categories of chain
complexes (differential graded operads) and topological spaces (topological
operads). However, the category of mixed Hodge complexes \cite{BZ} is a
symmetric monoidal category with colimits, with graded tensor product as
the monoidal structure, and operads in this category are called mixed Hodge
operads. In fact, the mixed Hodge operads which most concern us are pure
and have vanishing differential.
The gravity operad is an example of a mixed Hodge operad. This
carries a unique mixed Hodge structure compatible with the Poincar\'e
residue maps which define the products in $\mathcal{G}\mathit{rav}$:
$$
\mathcal{G}\mathit{rav}\(n\) = \Sigma^{3-n} \operatorname{sgn}_n \o H_{\bullet}(\mathcal{M}_{0,n},\mathbb{C}(n-3)) .
$$
Here, $\mathbb{C}(n-3)$ is the Tate Hodge structure, which is a line with Hodge
numbers $(n-3,n-3)$.
\subsection{The hypercommutative operad}
Let $\mathcal{H}\mathit{ycom}$ be the stable cyclic $\SS$-module
$$
\mathcal{H}\mathit{ycom}\(n\) = \begin{cases} H_{\bullet}(\overline{\mathcal{M}}_{0,n}) , & n\ge3 , \\ 0 , & n<3 .
\end{cases}$$
The cyclic $\SS$-space $\overline{\mathcal{M}}\(n\)=\overline{\mathcal{M}}_{0,n}$ is a topological cyclic
operad: the product is given by gluing stable curves together at marked
points. It follows that $\mathcal{H}\mathit{ycom}$ is a cyclic operad. Kontsevich and Manin
found \cite{KM} that algebras over $\mathcal{H}\mathit{ycom}$ are just hypercommutative
algebras in the sense of \eqref{hypercommutative}, where the operation
$(x_1,\dots,x_n)\in\mathcal{H}\mathit{ycom}\(n+1\)$ corresponds to the fundamental class
$[\overline{\mathcal{M}}_{0,n+1}]\in H_{2(n-2)}(\overline{\mathcal{M}}_{0,n+1})$. The fact that the
operations $(x_1,\dots,x_n)$ generate $\mathcal{H}\mathit{ycom}$ is an elegant restatement of
Theorem \ref{Keel:1}. In Proposition \ref{orthogonal}, we will give a new
proof (obtained jointly with Kontsevich) of the relations between the
generators of $\mathcal{H}\mathit{ycom}$, which relies on the duality between hypercommutative
algebras and gravity algebras, together with our explicit presentation
\eqref{gravity} of the gravity operad $\mathcal{G}\mathit{rav}$.
\subsection{A spectral sequence of Deligne}
Let $M$ be a smooth projective variety of complex dimension $n$, and let
$\{D_1,\dots,D_N\}$ be a sequence of smooth divisors with normal crossings;
we denote their union by $D$. The sheaf of logarithmic differential forms
$\mathcal{E}^{\bullet}_M(\log D)$ on $M$ is generated over the sheaf of differential
forms $\mathcal{E}^{\bullet}_M$ by symbols $d(\log f)$, where $f$ is a section of
$\mathcal{O}(D)$, subject to the relations
$$
d(\log fg) = d(\log f) + d(\log g) \quad\text{and}\quad
f\*d(\log f) = df.
$$
The sheaf $\mathcal{E}^{\bullet}_M$ has a differential $d$, characterized by
$d(d(\log f))=0$, and we have the fundamental isomorphism
$$
H^{\bullet}(U) \cong H^{\bullet}(M,\mathcal{E}^{\bullet}_M(\log D)) ,
$$
where $U=M\setminus D$.
The sheaf $\mathcal{E}^{\bullet}_M(\log D)$ is filtered by subsheaves
$$
W_k\mathcal{E}^{\bullet}_M(\log D) = \operatorname{span}_\mathcal{O}\{
d(\log f_1)\.\dots\.d(\log f_i) \mid i\le k \} .
$$
Let $j^k:D^k\hookrightarrow M$ be the embedding of the closed subvariety
$$
D^k = \coprod_{i_1<\dots<i_k} D_{i_1} \cap \dots \cap D_{i_k} ,
$$
and let $\epsilon_k$ be the locally constant line bundle over $D^k$, which
over the component $D_{i_1}\cap\ldots\cap D_{i_k}$ equals the determinant
line $\det(\{i_1,\dots,i_k\})$. There is a canonical quasi-isomorphism
$$
\operatorname{gr}^W_k \mathcal{E}^{\bullet}_M(\log D) \simeq
\Sigma^{-k}j^k_*\mathcal{E}^{\bullet}_{D^k}\o\epsilon_k .
$$
The associated spectral sequence
$$
E_1^{-p,q} = H^{2p+q}(D^p,\epsilon_p) \Rightarrow
E_\infty^{-p,q} = \operatorname{gr}^W_pH^{-p+q}(U)
$$
carries a Hodge filtration $F$, induced by the Hodge filtration of
$\mathcal{E}^{\bullet}_M(\log D)$, and by the principal of two types, $E_2=E_\infty$
(\cite{Deligne}, Section 3.2). The weight filtration induced on $H^n(U)$ by
$W$ defines, up to translation, its mixed Hodge structure:
$\operatorname{gr}^W_pH^{-p+q}(M)$ carries a pure Hodge structure of weight $q$.
If a finite group $\Gamma$ acts on $M$, preserving $U\subset M$, this
spectral sequence carries an action of $\Gamma$ compatible with its action
on $H^{\bullet}(U)$.
\subsection{The cohomology ring of $\mathcal{M}_{0,n}$}
We now describe the cohomology ring of the moduli space $\mathcal{M}_{0,n}$. Our
main tool is Arnold's description of the cohomology ring of the
configuration space $\mathbb{C}^n_0$ \cite{Arnold}.
If $1\le j\ne k\le n$, let $\omega_{jk}$ be the logarithmic differential form
on the configuration space $\mathbb{C}^n_0$ given by the formula
$$
\omega_{jk} = \frac{d\log(z_j-z_k)}{2\pi i} .
$$
Note that the cohomology class of $\omega_{jk}$ is integral.
\begin{proposition} \label{Arnold}
The cohomology ring $H^{\bullet}(\mathbb{C}^n_0,\mathbb{Z})$ is the graded commutative ring
with generators $[\omega_{jk}]$, and relations $\omega_{jk}=\omega_{kj}$ and
$\omega_{ij}\omega_{jk}+\omega_{jk}\omega_{ki}+\omega_{ki}\omega_{ij}=0$. The symmetric group
$\SS_n$ acts on $H^{\bullet}(\mathbb{C}^n_0,\mathbb{Z})$ through its action on the generators
$\sigma\*\omega_{ij}=\omega_{\sigma(i)\sigma(j)}$.
\end{proposition}
\begin{pf}
The Serre spectral sequence for the fibration
$$\begin{CD}
\mathbb{C}\setminus\{z_1,\dots,z_n\} @>>> \mathbb{C}^{n+1}_0 @>>> \mathbb{C}^n_0
\end{CD}$$
defined by projecting $(z_1,\dots,z_{n+1})$ to $(z_1,\dots,z_n)$ collapses
at $E_2$, and the monodromy of $\pi_1(\mathbb{C}^n_0)$ on
$H^{\bullet}(\mathbb{C}\setminus\{z_1,\dots,z_n\})$ is trivial. The proof now
proceeds by induction on $n$.
\end{pf}
\begin{corollary} \label{equivariant}
The cohomology ring $H^{\bullet}(\mathcal{M}_{0,n+1},\mathbb{C})$ may be identified with the
kernel of the differential $\iota$ on $H^{\bullet}(\mathbb{C}^n_0,\mathbb{C})$ whose action
on the generators is $\iota\omega_{jk}=1$.
\end{corollary}
\begin{pf}
The isotropy group of the point $\infty\in\mathbb{CP}^1$ under the action of
$\operatorname{PSL}(2,\mathbb{C})$ is
$$\textstyle
\operatorname{Aff}(\mathbb{C}) = \Bigl\{ \left( \begin{smallmatrix} a & b \\ 0 & a^{-1}
\end{smallmatrix} \right) \mathop{\Big|}
a\in\mathbb{C}^\times , b\in\mathbb{C} \Bigr\} \subset \operatorname{PSL}(2,\mathbb{C}) .
$$
Since $\operatorname{PSL}(2,\mathbb{C})$ acts transitively on $\mathbb{CP}^1$, we see that
$\mathcal{M}_{0,n+1}\cong\mathbb{C}^n_0/\operatorname{Aff}(\mathbb{C})$. But the group $\operatorname{Aff}(\mathbb{C})$ is homotopy
equivalent to the circle group
$$\textstyle
\Bigl\{ \left( \begin{smallmatrix} a & 0 \\ 0 & a^{-1} \end{smallmatrix}
\right) \mathop{\Big|} |a|=1 \Bigr\} ,
$$
giving a homotopy equivalence $\mathbb{C}^n_0\simeq\mathcal{M}_{0,n+1}\times S^1$.
This allows us to identify the cohomology of $\mathcal{M}_{0,n+1}$ with the
$S^1$-equivariant cohomology of $\mathbb{C}^n_0$.
The infinitesimal generator of the circle action on $\mathbb{C}^n_0$ is
the vector field
$$
T = 2\pi i \sum_{k=1}^n \bigl( z_k \partial_k - \bar{z}_k \bar\partial_k \bigr) ,
$$
whose contraction with a generator $\omega_{jk}$ is $\omega_{jk}(T)=1$.
\end{pf}
The above result leads to yet another realization of the $\SS_n$-module
structure on ${\operatorname{\mathcal{L}\mathit{ie}}}\(n\)\cong{\operatorname{\mathcal{L}\mathit{ie}}}(n-1)$:
$$
{\operatorname{\mathcal{L}\mathit{ie}}}\(n\) \cong \operatorname{sgn}_n \o H_{n-3}(\mathcal{M}_{0,n}) .
$$
\subsection{Application to $\mathcal{M}_{0,n}\subset\overline{\mathcal{M}}_{0,n}$} \label{Deligne}
We now apply Deligne's spectral sequence with $U=\mathcal{M}_{0,n}$ and
$M=\overline{\mathcal{M}}_{0,n}$. (This spectral sequence is also discussed in \cite{GK},
Section 3.4.5.) Denote the closure of the open stratum $\mathcal{M}\(T\)$ by
$\overline{\mathcal{M}}\(T\)$. Then the divisors are the closed strata $\mathcal{M}\(T\)$,
$T\in\mathcal{T}_1\(n\)$, while $D^p$ is the union of the closed strata
$\overline{\mathcal{M}}\(T\)$, $T\in\mathcal{T}_p\(n\)$. The restriction of $\epsilon_p$ to
$\overline{\mathcal{M}}\(T\)$ equals $\det(\operatorname{Edge}(T))$; as this is naturally isomorphic to
$\det(T)$, we see that
$$
E_1^{-p,q} \cong \bigoplus_{T\in\mathcal{T}_p\(n\)}
H^{-2p+q}(\overline{\mathcal{M}}\(T\),\det(T)) .
$$
The differential $\begin{CD}d_1:E_1^{-p,q}@>>>E_1^{-p+1,q}
\end{CD}$ is easy to describe: it
is the composition
$$\begin{CD}
\displaystyle
\bigoplus_{T\in\mathcal{T}_p\(n\)} H^{-2p+q}(\overline{\mathcal{M}}\(T\),\det(T)) @.
\displaystyle
\bigoplus_{T\in\mathcal{T}_{p-1}\(n\)} H^{-2p+2+q}(\overline{\mathcal{M}}\(T\),\det(T)) \\
@| @| \\
\displaystyle
\bigoplus_{T\in\mathcal{T}_p\(n\)} H_{2(n-3)-q}(\overline{\mathcal{M}}\(T\),\det(T)) @>>>
\displaystyle
\bigoplus_{T\in\mathcal{T}_{p-1}\(n\)} H_{2(n-3)-q}(\overline{\mathcal{M}}\(T\),\det(T))
\end{CD}$$
where the vertical isomorphisms are induced by Poincar\'e duality, and the
bottom arrow is the map induced on the homology groups by the inclusion of
the $p$-codimensional closed strata of $\overline{\mathcal{M}}_{0,n}$ into the
$(p-1)$-codimensional closed strata.
The key to unlocking this spectral sequence is the following lemma, which
shows that
$$
E_2^{-k,2k} \cong H^k(\mathcal{M}_{0,n}) ,
$$
while $E_2^{pq}=0$ if $2p+q\ne0$.
\begin{lemma} \label{mixed}
The mixed Hodge structure of $H^k(\mathcal{M}_{0,n})$, $n\ge3$, is pure of weight
$2k$.
In degree $i$, $\mathcal{G}\mathit{rav}\(n\)$ equals the $\SS_n$-module $\operatorname{sgn}_n\o
H_{i+n-3}(\mathcal{M}_{0,n})$, with pure Hodge structure of weight $-2i$.
\end{lemma}
\begin{pf}
By Proposition \ref{Arnold}, the cohomology ring of $\mathbb{C}^n_0$ is generated
by the logarithmic differential forms $\omega_{ij}$; it follows that the mixed
Hodge structure of $H^k(\mathbb{C}^n_0)$ is pure of weight $2k$. By Corollary
\ref{equivariant}, there is an injection of the cohomology of $\mathcal{M}_{0,n+1}$
into the cohomology of $\mathbb{C}^n_0$, induced by the quotient map
$\begin{CD}\mathbb{C}^n_0@>>>\mathcal{M}_{0,n+1}
\end{CD}$, and the result follows.
\end{pf}
Thus, for $q$ even, the $E_1$-term of the spectral sequence gives rise to a
resolution of the graded vector space $H^{\bullet}(\mathcal{M}_{0,n})$
\begin{equation} \label{bar:even} \begin{CD}
0 @>>> H^p(\mathcal{M}_{0,n}) @>>>
\bigoplus_{T\in\mathcal{T}_p\(n\)} H^0(\overline{\mathcal{M}}\(T\),\det(T)) @>>>
\bigoplus_{T\in\mathcal{T}_{p-1}\(n\)} H^2(\overline{\mathcal{M}}\(T\),\det(T)) @>>> \dots
\end{CD} \end{equation}
For $q$ odd, we obtain the exact sequence
$$\begin{CD}
\dots @>>> \bigoplus_{T\in\mathcal{T}_1\(n\)} H^{q-2}(\overline{\mathcal{M}}\(T\),\det(T)) @>>>
H^q(\overline{\mathcal{M}}_{0,n}) = \bigoplus_{T\in\mathcal{T}_0\(n\)}
H^q(\overline{\mathcal{M}}\(T\),\det(T)) @>>> 0 .
\end{CD}$$
which shows, by induction on odd $q$, that the cohomology groups
$H^q(\overline{\mathcal{M}}_{0,n})$ vanish if $q$ is odd.
\subsection{Proof of Theorem \ref{Keel:1}} \label{Keel:proof}
It is clear that the result holds for $n=3$, since $\overline{\mathcal{M}}_{0,3}$ is a
point. We now argue by induction on $n$. As a consequence of the
surjectivity of the differential in the exact sequence \eqref{bar:even}
$$\begin{CD}
\bigoplus_{T\in\mathcal{T}_1\(n\)} H^{2p-2}(\overline{\mathcal{M}}\(T\),\det(T)) @>>>
H^{2p}(\overline{\mathcal{M}}_{0,n}) = \bigoplus_{T\in\mathcal{T}_0\(n\)}
H^{2p}(\overline{\mathcal{M}}\(T\),\det(T))
\end{CD}$$
for $p\le n-2$, we see that all homology classes of $\overline{\mathcal{M}}_{0,n}$ except
the fundamental class are supported on the closures of strata of
codimension $1$. Such a closed stratum is isomorphic to
$\overline{\mathcal{M}}_{0,i}\times\overline{\mathcal{M}}_{0,j}$ where $i+j=n+2$ and $i,j\ge3$, allowing us
to apply the induction. }\end{pf
\subsection{The dimension of $H^2(\overline{\mathcal{M}}_{0,n})$}
Another consequence of \eqref{bar:even} is a simple formula for the
dimension $\dim H^2(\overline{\mathcal{M}}_{0,n})$ of the Picard variety of $\overline{\mathcal{M}}_{0,n}$:
\begin{eqnarray*}
\dim H^2(\overline{\mathcal{M}}_{0,n}) &=& \sum_{k=3}^{n-1} \binom{n-1}{k}
= 2^{n-1} - \frac{n^2-n+2}{2} \\
&=& \binom{n}{n-4} + \binom{n}{n-6} + \dots
\end{eqnarray*}
\begin{pf}
When $p=1$, the short exact sequence \eqref{bar:even} becomes
$$\begin{CD}
0 @>>> H^1(\mathcal{M}_{0,n}) @>>>
\bigoplus_{T\in\mathcal{T}_1\(n\)} H^0(\overline{\mathcal{M}}\(T\),\det(T)) @>>>
H^2(\overline{\mathcal{M}}_{0,n}) @>>> 0 .
\end{CD}$$
By Corollary \ref{equivariant}, the dimension of $H^1(\mathcal{M}_{0,n})$ equals
the coefficient of $-t$ in $(1-2t)\dots(1-(n-2)t)$, or
$$
2 + \dots + (n-1) = \binom{n-1}{2} - 1 .
$$
(See \eqref{poincare-moduli} for more details of this calculation: we will
actually show that $H^1(\mathcal{M}_{0,n})$ is isomorphic to the irreducible
$\SS_n$-module $V_{n-2,2}$.) Each tree in $\mathcal{T}_1\(n\)$ contributes a copy
of $\mathbb{C}$ to $\bigoplus_{T\in\mathcal{T}_1\(n\)} H_{2(n-4)}(\overline{\mathcal{M}}\(T\))$. Let $T$ be
such a tree and consider the set $S$ of the external edges attached to one
vertex of $T$. We see that trees with two vertices correspond to subsets
$S\subset\{1,\dots,n\}$ where $2\le|S|\le n-2$, where we identify the trees
corresponding to the subsets $S$ and $S^c$, the complement of $S$. Thus,
$$
|\mathcal{T}_1\(n\)| = \frac12 \sum_{k=2}^{n-2} \binom{n}{k}
= \frac12 (2^n-2n-2) = 2^{n-1} - n - 1 .
$$
The result follows easily.
\end{pf}
This formula may be compared to the dimension of $H^2(\overline{\mathcal{M}}_{g,n})$, $g>2$,
which follows from the work of Arbarello and Cornalba \cite{AC}:
$$
\dim H^2(\overline{\mathcal{M}}_{g,n}) = 2^{n-1} (g+1) + n + 1 .
$$
We see that this formula is correct for $g=0$ up to a polynomial error.
The above dimension formula may be refined, using the realization of
$\overline{\mathcal{M}}_{0,n}$ as an iterated blowup, to show that as an $\SS_n$-module,
$H^2(\overline{\mathcal{M}}_{0,n})$ is the direct sum of the suitable exterior powers of the
permutation representation $\mathbb{C}^n$ of $\SS_n$.
Note that there are the same number of $n$-linear relations \eqref{gravity}
among the brackets $[x_1,\dots,x_k]$ generating the gravity operad as $\dim
H^2(\overline{\mathcal{M}}_{0,n+1})$; as we will see, this is no coincidence.
\section{Koszul operads}
In this section, we prove our main theorem, the duality of the
hypercommutative and gravity operads. To do this, we must generalize
Ginzburg and Kapranov's theory of Koszul operads \cite{GK} so that it
applies to operads which are not necessarily generated by bilinear
operations. First, we recall their cobar construction for operads, an
analogue of Hochschild's bar construction for associative algebras.
The dual of an operad is only defined up to homotopy, and is represented by
the cobar operad. However, there is a class of operads, the Koszul operads,
for which there is a particularly nice dual, whose generators are in
one-to-one correspondence with those of the original operad. A Koszul
operad is quadratic (the relations among its generators are bilinear), as
is its dual, and the relations in the dual operad may be characterized as
the orthogonal complement of those of the original operad.
\subsection{Free operads and trees}
We now recall from \cite{cyclic} the structure of the free cyclic operad
$\mathbb{T}_+\v$ generated by a cyclic $\SS$-module $\v$. There is an analogous
construction for operads, for which we refer to \cite{n-algebras}. From now
on, all cyclic $\SS$-modules which we discuss will be stable.
If $\v$ is a (stable) cyclic $\SS$-module, let $\mathbb{T}_+\v$ be the
(stable) cyclic $\SS$-module defined by
$$
\mathbb{T}_+\v\(n\) = \bigoplus_{T\in\mathcal{T}\(n\)} \v\(T\) ,
$$
where $\v\(T\)=\bigotimes_{v\in\operatorname{Vert}(T)} \v\(\operatorname{Leg}(v)\)$. Note that
$\mathbb{T}_+\v$ is graded by subspaces
$$
\mathbb{T}_i\v\(n\) = \bigoplus_{T\in\mathcal{T}_i\(n\)} \v\(T\) .
$$
Then $\mathbb{T}_+$ is an endofunctor in the category of (stable) cyclic
$\SS$-modules.
There is a natural structure of a triple on the functor $\mathbb{T}_+$:
\begin{enumerate}
\item since $\mathbb{T}_+\mathbb{T}_+$ is a sum over trees, each vertex of which is
itself a tree, the product of the triple is a natural transformation
from $\mathbb{T}_+\mathbb{T}_+$ to $\mathbb{T}_+$ obtained by gluing the trees at the vertices
into the larger tree;
\item the unit of the triple is the natural transformation from the
identity functor to $\mathbb{T}_+$ induced by the inclusion
$\mathcal{T}_0\(n\)\subset\mathcal{T}\(n\)$.
\end{enumerate}
The following theorem is a melding of results from \cite{cyclic} and
\cite{modular}.
\begin{theorem}
A (non-unital, stable) cyclic operad is the same thing as a $\mathbb{T}_+$-algebra
in the category of (stable) cyclic $\SS$-modules.
\end{theorem}
\subsection{The cobar construction for operads}
The cobar construction $\mathsf{B}$, introduced by Ginzburg and Kapranov
\cite{GK}, is a contravariant functor on the category of operads. We study
here the slight variant of this functor which acts on the category of
(non-unital, stable) cyclic operads.
The dual $V^*$ of a chain complex $V$ is defined as follows:
$\begin{CD}V^*_i=(V_{-i})^*$, and $\delta^*:V^*_i@>>>V^*_{i-1}
\end{CD}$ is the adjoint of
$\begin{CD}\delta:V_{-i+1}@>>>V_{-i}\end{CD}$.
If $\v$ is a stable cyclic $\SS$-module, denote by $\v^\vee$ the
stable cyclic $\SS$-module
$$
\v^\vee\(n\) = \Sigma^{n-3}\operatorname{sgn}_n\o\v\(n\)^* .
$$
This functor is an involution on the category of stable cyclic
$\SS$-modules, that is, $(\v^\vee)^\vee$ is naturally isomorphic to
$\v$.
The cobar operad $\mathsf{B}\a$ of a (non-unital, stable) cyclic operad is
obtained by perturbing the differential of the free cyclic operad
$\mathbb{T}_+\a^\vee$ by a differential $\partial$ which reflects the operad structure
of $\a$, and is defined as follows.
If $T\in\mathcal{T}\(n\)$, and $e$ is an edge of $T$, denote by $T/e$ the tree in
which $e$ is contracted to a point: thus, $T/e$ has one fewer vertices, and
one fewer edges, than $T$. There is a natural map of degree $0$
$$\begin{CD}
\partial_{T/e} : \a\(T\) @>>> \a\(T/e\)
\end{CD}$$
induced by composition in the operad $\a$ along the edge $e$. This induces
a map
$$\begin{CD}
\partial^\vee_{T/e} : \a^\vee\(T/e\) @>>> \a^\vee\(T\)
\end{CD}$$
of degree $-1$. We now define the differential $\partial$ to be the operator
whose matrix element from $\a^\vee\(\tilde{T}\)\subset\mathsf{B}\a$ to
$\a^\vee\(T\)\subset\mathsf{B}\a$ is the sum of the operators $\partial^\vee_{T/e}$
over internal edges $e$ such that $T/e$ is isomorphic to $\tilde{T}$.
Paying careful attention to the signs coming from the suspensions, one
shows that the differential $\partial$ satisfies the formulas
$\partial^2=\delta\partial+\partial\delta=0$, and hence that $\delta+\partial$ is a differential on
$\mathbb{T}_+\a^\vee$. It is also not hard to show that $\partial$ is compatible with
the cyclic operad structure of $\mathbb{T}_+\a^\vee$, so that
$\mathsf{B}\a=(\mathbb{T}_+\a^\vee,\delta+\partial)$ is an operad.
The properties of the resulting functor are summarized by the following
theorem.
\begin{theorem} \label{cobar}
\textup{(1)} The cobar construction is a homotopy functor, that is, if
$\begin{CD}f:\a@>>>\b
\end{CD}$ is a homotopy equivalence, then so is $\begin{CD}
\mathsf{B} f:\mathsf{B}\a@>>>\mathsf{B}\b\end{CD}$.
\noindent \textup{(2)} There is a natural transformation from $\mathsf{B}\BB$ to the
identity functor, and the resulting map $\begin{CD}\mathsf{B}\BB\a@>>>\a
\end{CD}$ is a
homotopy equivalence for all $\a$.
\end{theorem}
\begin{pf}
The homotopy invariance of $\mathsf{B}$ is easy to see by a double complex
argument. The natural map from $\mathsf{B}\BB\a$ to $\a$, which is projection onto
the summand $\a\cong(\a^\vee)^\vee\subset\mathbb{T}_+(\mathbb{T}_+\a^\vee)^\vee$, is
shown to be a homotopy equivalence of operads in Theorem 3.2.16 of
\cite{GK}.
\end{pf}
If $\a$ and $\b$ are operads and $\begin{CD}\Phi:\mathsf{B}\a@>>>\b
\end{CD}$ is a morphism of
operads, there is a bar construction $\mathcal{B}_\Phi$ on $\a$-algebras, defined
for an $\a$-algebra $A$ by twisting the differential on the chain complex
$$
A \oplus
\Sigma^{-1} \bigoplus_{n=3}^\infty \operatorname{Hom}_{\SS_n}(\b(n),(\Sigma A)^{\o n})
$$
in such a way as to reflect the $\a$-algebra structure of $A$. (See
\cite{n-algebras} for details.) When $\b=\mathsf{B}\a$ and $\Phi$ is the identity
map, we denote the resulting functor $\mathcal{B}$. Let $QA$ be the complex of
indecomposables, obtained by taking the cokernel of the map
$$\begin{CD}
\bigoplus_{n=3}^\infty \rho_n :
\bigoplus_{n=3}^\infty \a(n)\o_{\SS_n} A^{\o n} @>>> A .
\end{CD}$$
The following theorem is proved in Chapter 3 of \cite{n-algebras}.
\begin{theorem} \label{Bar}
\textup{(1)} There is a natural transformation of functors $\begin{CD}
\mathcal{B}@>>>Q\end{CD}$, such
that if $A$ is a free algebra, the morphism $\begin{CD}\mathcal{B} A@>>>QA
\end{CD}$ is a homotopy
equivalence.
\noindent \textup{(2)} The functors $\mathcal{B}_\Phi$ are homotopy functors: if
$\begin{CD}f:A@>>>B
\end{CD}$ is a homotopy equivalence, then so is $\begin{CD}\mathcal{B}_\Phi f:\mathcal{B}_\Phi
A@>>>\mathcal{B}_\Phi B\end{CD}$.
\end{theorem}
If $\begin{CD}\Phi:\mathsf{B}\a@>>>\b
\end{CD}$ is a homotopy equivalence of operads, the natural
morphism $\begin{CD}\mathcal{B} A@>>>\mathcal{B}_\Phi A
\end{CD}$ is a homotopy equivalence for all
$\a$-algebras $A$. Thus, the functor $\mathcal{B}_\Phi$ is a left derived functor
$\L Q$ of the indecomposable functor $Q$, that is, a homotopy functor
homotopy equivalent to $Q$ on free $\a$-algebras. It is proved in
\cite{GK} that there are natural homotopy equivalences of operads
$$\begin{CD}
\mathsf{B}{\operatorname{\mathcal{A}\mathit{ss}}} @>>> {\operatorname{\mathcal{A}\mathit{ss}}} \quad \mathsf{B}{\operatorname{\mathcal{C}\mathit{om}}} @>>> {\operatorname{\mathcal{L}\mathit{ie}}} \quad \mathsf{B}{\operatorname{\mathcal{L}\mathit{ie}}} @>>> {\operatorname{\mathcal{C}\mathit{om}}} .
\end{CD}$$
The bar construction associated to the first of these homotopy equivalences
is, up to a shift in degree, the Hochschild bar construction on associative
algebras, while the bar constructions associated to the other two homotopy
equivalences are the functors $\L Q$ on commutative and Lie algebras
discussed in the introduction, equal, up to a shift in degree, to the
Harrison and Chevalley-Eilenberg complexes respectively. Thus, the duality
result Theorem \ref{Harrison} is seen to be a special case of Theorem
\ref{Bar}.
In Theorem \ref{Kontsevich}, we will prove that $\mathcal{H}\mathit{ycom}$-algebras are the
same thing as hypercommutative algebras in the sense of
\eqref{hypercommutative}. Thus the duality between hypercommutative and
gravity algebras announced in the introduction follows from Theorem
\ref{Bar} combined with the following generalization of Theorem 4.25 of
\cite{GK}.
\begin{theorem} \label{duality}
There is a natural homotopy equivalence of operads $\begin{CD}
\mathsf{B}\mathcal{H}\mathit{ycom}@>>>\mathcal{G}\mathit{rav}\end{CD}$.
\end{theorem}
\begin{pf}
Let $\v$ be the $\SS$-module obtained by summing the short exact sequences
\eqref{bar:even} (minus the terms $H^p(\mathcal{M}_{0,n})$), placing the summand
$$
\bigoplus_{T\in\mathcal{T}_p\(n\)} H^q(\overline{\mathcal{M}}\(T\),\det(T))
$$
of $\v\(n\)$ in degree $2(n-3)-p-q$. Using Poincar\'e duality, we see that
$$
\v\(n\) \cong \bigoplus_{p=0}^{n-3}
\bigoplus_{T\in\mathcal{T}_p\(n\)} \Sigma^p \det(T) \o \mathcal{H}\mathit{ycom}\(T\) .
$$
Furthermore, there is a natural homotopy equivalence
$\begin{CD}\Sigma^{2(n-3)}H^{\bullet}(\overline{\mathcal{M}}_{0,n})@>>>\v\(n\)
\end{CD}$, which induces a homotopy
equivalence $\begin{CD}\v^\vee@>>>\mathcal{G}\mathit{rav}\end{CD}$.
By the isomorphism \eqref{det}, the $\SS$-module $\v^\vee$ may be rewritten
as
\begin{eqnarray*}
\v^\vee\(n\) &\cong& \bigoplus_{p=0}^{n-3} \Sigma^{n-3-p}
\bigoplus_{T\in\mathcal{T}_p\(n\)} \bigotimes_{v\in\operatorname{Vert}(T)} \det(\operatorname{Leg}(v)) \o
\mathcal{H}\mathit{ycom}^*\(\operatorname{Leg}(v)\) \\
&\cong& \bigoplus_{p=0}^{n-3} \Sigma^{n-3-p}
\bigoplus_{T\in\mathcal{T}_p\(n\)} \bigotimes_{v\in\operatorname{Vert}(T)} \Sigma^{3-|v|}
\mathcal{H}\mathit{ycom}^\vee\(\operatorname{Leg}(v)\) \\
&\cong& \bigoplus_{p=0}^{n-3} \Sigma^{n-3-p+\sum_{v\in\operatorname{Vert}(T)}(3-|v|)}
\bigoplus_{T\in\mathcal{T}_p\(n\)} \mathcal{H}\mathit{ycom}^\vee\(T\) .
\end{eqnarray*}
If $T\in\mathcal{T}_p\(n\)$, we have
$$
\sum_{v\in\operatorname{Vert}(T)} (3-|v|) = 3 - n + p ,
$$
showing that
$$
\v^\vee\(n\) \cong \bigoplus_{p=0}^{n-3} \bigoplus_{T\in\mathcal{T}_p\(n\)}
\mathcal{H}\mathit{ycom}^\vee\(T\) \cong \mathsf{B}\mathcal{H}\mathit{ycom}\(n\) .
$$
A little diagram chasing shows that the differentials of the $\SS$-modules
$\v^\vee$ and $\mathsf{B}\mathcal{H}\mathit{ycom}$ are the same, and that the resulting homotopy
equivalence between $\mathcal{G}\mathit{rav}$ and $\mathsf{B}\mathcal{H}\mathit{ycom}$ is compatible with the operad
structures.
\end{pf}
\subsection{The cobar construction for mixed Hodge operads}
The free operad functor $\v\mapsto\mathbb{T}\v$ and the functor $\begin{CD}
\v@>>>\v^\vee\end{CD}$
have analogues in the category of mixed Hodge $\SS$-modules, defined in
precisely the same way as in the category of $\SS$-modules. (We recall that
the dual $\v^*$ in the category of mixed Hodge complexes reverses the
weight filtration, sending complexes of weight $k$ to complexes of weight
$-k$.) This allows us to extend the cobar construction to the category of
mixed Hodge operads, by the same definition as in the category of dg
operads.
If we follow through the proof of Theorem \ref{duality} paying attention to
the mixed Hodge structures, we see that the homotopy equivalence
$\begin{CD}\mathsf{B}\mathcal{H}\mathit{ycom}@>>>\mathcal{G}\mathit{rav}
\end{CD}$ is compatible with the Hodge structures of $\mathcal{G}\mathit{rav}$ and
$\mathcal{H}\mathit{ycom}$, where $\mathcal{H}\mathit{ycom}$ carries the natural (pure) Hodge structure coming
from its realization as the cohomology of the smooth K\"ahler manifold
$\overline{\mathcal{M}}_{0,n}$. This observation will be essential in our calculation of the
defining relations of the operad $\mathcal{H}\mathit{ycom}$.
\subsection{Quadratic operads}
We now generalize Ginzburg and Kapranov's notion of a Koszul operad to
operads whose generators are not necessarily bilinear operations. Once
more, we restrict attention to stable cyclic operads.
An ideal $\b\subset\a$ of a cyclic operad is a cyclic $\SS$-submodule such
that for all operations $\circ_i$, $a\circ_ib$ is in $\b$ if either $a$ or
$b$ is. The intersection of two ideals is obviously an ideal. An ideal is
generated by a cyclic $\SS$-submodule $\r\subset\b$ if $\b$ is the
intersection of all ideals of $\a$ containing $\r$.
Let $\a$ be an operad, generated by a cyclic $\SS$-submodule $\v$. The pair
$(\a,\v)$ is a cyclic quadratic operad if the ideal $\begin{CD}
\ker(\mathbb{T}\v@>>>\a)\end{CD}$ in
the free cyclic operad $\mathbb{T}_+\v$ is generated by
$$\begin{CD}
\ker \Bigl( \mathbb{T}_1\v=\bigoplus_{T\in\mathcal{T}_1} \v\(T\) @>>> \a \Bigr) .
\end{CD}$$
The word quadratic is used here because $\v\(T\)$ is quadratic in $\v$ if
$T$ has one internal edge and hence two vertices. Thus, $\r$ is itself
quadratic in $\v$.
The cyclic operads $\a={\operatorname{\mathcal{A}\mathit{ss}}}$, ${\operatorname{\mathcal{C}\mathit{om}}}$ and ${\operatorname{\mathcal{L}\mathit{ie}}}$ are all quadratic, with
generating cyclic submodule $\v$, where
$$
\v\(n\) = \begin{cases} \a\(3\) , & n=3, \\ 0 , & n\ne3 . \end{cases}
$$
For example, ${\operatorname{\mathcal{L}\mathit{ie}}}\(3\)$ is one-dimensional, spanned by $[a_1,a_2]$, and
the cyclic $\SS$-module of relations $\r$ is given by the formula
$$
\r\(n\) =
\begin{cases} \span\{ [a_1,[a_2,a_3]] , [a_2,[a_3,a_1]] \} , & n=4 , \\
0 , & n\ne4 . \end{cases}
$$
\subsection{The naive dual of a quadratic operad}
If $\a$ is a cyclic quadratic operad, the naive dual $\a^!$ of $\a$ is the
cokernel of the composition
$$\begin{CD}
\psi : \mathsf{B}\a @>\partial>> \mathsf{B}\a @>>> \mathbb{T}_+\v^\vee ,
\end{CD}$$
where the second arrow is the surjection of cyclic $\SS$-modules
$\begin{CD}\mathsf{B}\a@>>>\mathbb{T}_+\v^\vee
\end{CD}$ induced by the inclusion of cyclic $\SS$-modules
$\v\subset\a$.
\begin{definition}
If $\a$ is a cyclic quadratic operad, there is a natural morphism of
operads $\begin{CD}\Phi:\mathsf{B}\a@>>>\a^!\end{CD}$, induced by the surjection
$\begin{CD}\mathsf{B}\a@>>>\mathbb{T}_+\v^\vee
\end{CD}$. The operad $\a$ is Koszul if the surjection of
operads $\begin{CD}\mathsf{B}\a@>>>\a^!
\end{CD}$ is a homotopy equivalence, or equivalently, if
$\begin{CD}\mathsf{B}\a^!@>>>\a\end{CD}$ is.
\end{definition}
\begin{proposition} \label{orthogonal}
If $\a$ is a cyclic quadratic operad with generators $\v$ and relations
$\r$, let $\r^\perp$ be the kernel of the natural map from
$\mathbb{T}_1\v^\vee\cong(\mathbb{T}_1\v^\vee)^\vee$ to $\r^\vee$. Then $\a^!$ is a
cyclic quadratic operad with generators $\v^\vee$ and relations $\r^\perp$.
\end{proposition}
\begin{pf}
It suffices to show that the image of $\mathsf{B}\a$ in $\mathbb{T}_+\v^\vee$ under the
above composition is the ideal generated by $\r^\perp$. Denote by
$\a_k\subset\a$ the image of $\mathbb{T}_k\v$ in $\a$ under the quotient map
$\begin{CD}\mathbb{T}_+\v@>>>\a
\end{CD}$. Thus, $\a_0=\v$ and $\a_1=\mathbb{T}_1\v/\r$. Observe that
$\a_1^\vee\cong\r^\perp$.
If $T\in\mathcal{T}\(n\)$, the summand $\a^\vee\(T\)$ of $\mathsf{B}\a\(n\)$ may be
thought of as the vector space spanned by decorations of the tree $T$, in
which each vertex of $T$ is assigned an element of $\a^\vee$ of
appropriate valence. The map $\begin{CD}\psi:\mathsf{B}\a@>>>\mathbb{T}_+\v^\vee
\end{CD}$ vanishes on such
a decorated tree unless the vertex decorations lie in $\v$ at all but one
vertex $v$, which is decorated by $a\in\a_1^\vee$. The map $\psi$ applied
to this decorated tree produces a new decorated tree in which the vertex
$v$ is replaced by the tree underlying $a$ (and thus having one additional
edge). Thus, the image of $\psi$ is the ideal generated by $\r^\perp$.
\end{pf}
\begin{corollary}
A cyclic quadratic operad $\a$ is Koszul if and only if $\a^!$ is, and
$(\a^!)^!\cong\a$.
\end{corollary}
As examples of naive duals, we have ${\operatorname{\mathcal{A}\mathit{ss}}}^!\cong{\operatorname{\mathcal{A}\mathit{ss}}}$, ${\operatorname{\mathcal{C}\mathit{om}}}^!\cong{\operatorname{\mathcal{L}\mathit{ie}}}$
and ${\operatorname{\mathcal{L}\mathit{ie}}}^!\cong{\operatorname{\mathcal{C}\mathit{om}}}$. It is proved in \cite{GK} that the operads ${\operatorname{\mathcal{A}\mathit{ss}}}$,
${\operatorname{\mathcal{C}\mathit{om}}}$ and ${\operatorname{\mathcal{L}\mathit{ie}}}$ are Koszul. A non-cyclic example of a Koszul operad is
$\mathcal{B}\mathit{raid}$ \cite{n-algebras}, which satisfies
$\mathcal{B}\mathit{raid}^!\cong\Lambda^{-1}\mathcal{B}\mathit{raid}$.
The proof of the following theorem occupies the remainder of this
section. This theorem is joint work of the author and M. Kontsevich.
\begin{theorem} \label{Kontsevich}
Let $\v\subset\mathcal{H}\mathit{ycom}$ be the cyclic $\SS$-submodule spanned by the
fundamental classes
$$
[\overline{\mathcal{M}}_{0,n}] \in H_{2(n-3)}(\overline{\mathcal{M}}_{0,n}) \subset \mathcal{H}\mathit{ycom}\(n\) .
$$
The operad $\mathcal{H}\mathit{ycom}$ is Koszul, with generators $\v$, and
$\mathcal{H}\mathit{ycom}^!\cong\mathcal{G}\mathit{rav}$.
\end{theorem}
\begin{pf}
This theorem is proved using the duality between the mixed Hodge operads
$\mathcal{G}\mathit{rav}$ and $\mathcal{H}\mathit{ycom}$, and the fact that the operad $\mathcal{G}\mathit{rav}$ is quadratic.
If $\a$ is a mixed Hodge operad, the natural homotopy equivalence
$\begin{CD}\mathsf{B}\BB\a@>>>\a
\end{CD}$ of Theorem \ref{cobar} is a morphism of mixed Hodge
operads. It follows that we have a diagram in the category of mixed Hodge
operads
$$\begin{CD}
\mathsf{B}\BB\mathcal{H}\mathit{ycom} @>>> \mathsf{B}\mathcal{G}\mathit{rav} \\
@VVV @. \\
\mathcal{H}\mathit{ycom} @.
\end{CD}$$
in which both arrows are homotopy equivalences. The homology of the weight
$-2p$ summand of the complex $\mathsf{B}\mathcal{G}\mathit{rav}\(n\)$, which by this argument is
isomorphic to $H_{2p}(\overline{\mathcal{M}}_{0,n})$, must be concentrated in degree $2p$;
from this, we see that this subcomplex is exact except at the last term,
giving a long exact sequence
\begin{equation} \label{bar:Hodge}\begin{CD}
\dots @>>> \bigoplus_{T\in\mathcal{T}_{n-2-p}\(n\)} \mathcal{G}\mathit{rav}^\vee\(T\)_{2p+1}
@>>> \bigoplus_{T\in\mathcal{T}_{n-3-p}\(n\)} \mathcal{G}\mathit{rav}^\vee\(T\)_{2p}
@>>> \mathcal{H}\mathit{ycom}\(n\)_{2p} @>>> 0 .
\end{CD}\end{equation}
Let $\v^\vee\subset\mathcal{G}\mathit{rav}$ be the cyclic $\SS$-module spanned by the generators
of $\mathcal{G}\mathit{rav}$; for each $n\ge3$, there is one generator, of degree $3-n$ and
weight $2(n-3)$, in $\mathcal{G}\mathit{rav}\(n\)$. Thus $\v\(n\)$ is spanned by an
element of degree $2(n-3)$ and weight $2(3-n)$. Taking $p=n-3$ in the long
exact sequence \eqref{bar:Hodge}, we see that $\v\(n\)$ may be identified
with $H_{2(n-3)}(\overline{\mathcal{M}}_{0,n})$.
Let $\r^\vee\subset\mathcal{G}\mathit{rav}$ be the cyclic $\SS$-module spanned by the
elements of degree $4-n$ in $\mathcal{G}\mathit{rav}$. An $\SS_n$-module, $\r^\vee\(n\)\cong
H_1\(\mathcal{M}_{0,n}\)\o\operatorname{sgn}_n$; it is concentrated in degree $4-n$, has weight
$2(n-4)$, and dimension $\binom{n-1}{2}-1$. The case $p=n-4$ of the long
exact sequence \eqref{bar:Hodge} is the short exact sequence
$$\begin{CD}
0 @>>> \r\(n\) @>>> \bigoplus_{T\in\mathcal{T}_1\(n\)} \v\(T\)
@>>> H_{2(n-4)}(\overline{\mathcal{M}}_{0,n}) @>>> 0 ,
\end{CD}$$
showing that the $\SS$-module $\r$ is a subset of the set of
relations for $\mathcal{H}\mathit{ycom}$. Furthermore, there are no further relations, as may
be seen from \eqref{bar:Hodge}: for general $p\le n-3$, there is an exact
sequence
\begin{multline*}\begin{CD}
\dots @>>> \bigoplus_{T\in\mathcal{T}_{n-2-p}\(n\)} \bigoplus_{v\in\operatorname{Vert}(T)}
\r\(\operatorname{Leg}(v)\) \o \bigotimes_{w\in\operatorname{Vert}(T)\setminus\{v\}} \v\(\operatorname{Leg}(w)\) @>>> \\
\bigoplus_{T\in\mathcal{T}_{n-3-p}\(n\)} \v\(T\) @>>> H_{2p}\(n\)_{2p} @>>> 0 .
\end{CD}\end{multline*}
This shows that the operad $\mathcal{H}\mathit{ycom}$ is quadratic, with generators $\v$ and
relations $\r$.
It remains to identify the $\SS$-module $\r$ with the set of relations
\eqref{hypercommutative} which hold in a hypercommutative algebra. This is
done in two parts: we first show that these relations are are in the
orthogonal complement of the relations which define a gravity algebra, and
thus form a subset of $\r$, and then show that they form a subspace of
$\r\(n\)$ of dimension at least $\binom{n-1}{2}-1$. Since $\r\(n\)$ itself
has dimension $\binom{n-1}{2}-1$, this completes the proof.
It is simple to check that the relations \eqref{hypercommutative} are
orthogonal to those which hold in a gravity algebra. Consider the relation
$$
G_0 = \pm [[a_i,a_j],a_1,\dots,\widehat{a_i},\dots,\widehat{a_j},\dots,a_k] .
$$
The inner product of this relation with the relation
$$
H = \sum_{S_1\coprod S_2=\{1,\dots,n\}} \pm ((a,b,x_{S_1}),c,x_{S_2})
- \sum_{S_1\coprod S_2=\{1,\dots,n\}} \pm (a,(b,c,x_{S_1}),x_{S_2})
$$
vanishes, since only the terms with $S_1=\emptyset$ can contribute: they
each contribute a term $1$, but with opposite sign.
Turning now to the relation
$$
G_\ell = \pm
[[a_i,a_j],a_1,\dots,\widehat{a_i},\dots,\widehat{a_j},\dots,a_k]
- [[a_1,\dots,a_k],b_1,\dots,b_\ell] , \quad \ell>0 ,
$$
we see that there are three cases to consider:
\begin{enumerate}
\item none of the letters $a,b,c$ lie in the set $\{b_1,\dots,b_\ell\}$, in
which case the inner product of relation $H$ with the above relation again
vanishes, for the same reason as when $\ell=0$;
\item one of the letters, say $c$, lies in the set $\{b_1,\dots,b_\ell\}$,
in which case the only terms having a non-zero inner product with $H$ are
$[[a,b],c,x_S]$ and $[[a_1,\dots,a_k],b_1,\dots,b_\ell]$, whose
contributions, each equal to $1$, cancel;
\item two or three of the letters $a,b,c$ lie in the set
$\{b_1,\dots,b_\ell\}$, in which case the inner product of the above
relation with each term of $H$ vanishes.
\end{enumerate}
Finally, we check that the space of relations among $n$ letters in a
hypercommutative algebra has dimension at least $\binom{n}{2}-1$. Consider
the projection of these relations into the space $\a(n)$ of all quadratic
words in the generators of the hypercommutative operad of the form
$$
((x_i,x_j),x_1,\dots,\widehat{x_i},\dots,\widehat{x_j},\dots,x_n) .
$$
(Note that this subspace of $\mathbb{T}\v^\vee\(n+1\)$ is not
$\SS_{n+1}$-invariant, but only $\SS_n$-invariant.) The dimension of
$\a(n)$ is $\binom{n}{2}$, and the relations \eqref{hypercommutative} project
in $\a(n)$ to relations
$$
((a,b),c,x_S) = ((a,c),b,x_S) .
$$
Clearly, the quotient of $\a(n)$ by these relations is
one-dimensional. This completes the proof that \eqref{hypercommutative} are
all the relations in the operad $\mathcal{H}\mathit{ycom}$.
\end{pf}
\section{The equivariant Poincar\'e polynomials of $\mathcal{M}_{0,n}$ and
$\overline{\mathcal{M}}_{0,n}$}
In this section, we use the results of \cite{modular} to calculate the
character of the $\SS_n$-modules $H_i(\mathcal{M}_{0,n})$ and
$H_i(\overline{\mathcal{M}}_{0,n})$. By and large, the results of this section are
independent of the rest of this paper.
\subsection{Symmetric functions}
Let $\Lambda$ be the ring of symmetric functions: this is the limit
$$
\Lambda = \varprojlim \mathbb{Z}\[x_1,\dots,x_k\]^{\SS_k} .
$$
Then $\Lambda$ is the ring $\mathbb{Z}\[h_1,h_2,\dots\]$ of power series in the
complete symmetric functions
$$
h_n(x_i) = \sum_{i_1\le\dots\le i_n} x_{i_1}\dots x_{i_n} ,
$$
and $\Lambda_\mathbb{Q}=\Lambda\o\mathbb{Q}$ is a power series ring $\mathbb{Q}\[p_1,p_2,\dots\]$
in the power sums
$$
p_n(x_i) = \sum_i x_i^n .
$$
If $\sigma\in\SS_n$ has cycles of length
$\lambda_1\ge\dots\ge\lambda_\ell$, its cycle index $\psi(\sigma)$ is the
monomial $p_{\lambda_1}\dots p_{\lambda_\ell}$. If $V$ is
an $\SS_n$-module, its characteristic is the symmetric function
$$
\operatorname{ch}_n(V) = \frac{1}{n!} \sum_{\sigma\in\SS_n} \operatorname{Tr}_V(\sigma) \psi(\sigma) .
$$
It may be shown that $\operatorname{ch}_n(V)\in\Lambda\subset\Lambda_\mathbb{Q}$, and that the
characteristics of the irreducible representations of the symmetric groups
$\SS_n$, $n\ge0$, form a basis of $\Lambda$ over $\mathbb{Z}$, called the Schur
functions \cite{Macdonald}. For example, $h_n$ is the characteristic of the
trivial representation of $\SS_n$.
Define the Poincar\'e characteristic of an $\SS$-module to be
$$
\operatorname{ch}_t(\v) = \sum_{n=0}^\infty \sum_{i=0}^\infty (-t)^i \, \operatorname{ch}_n(\v_i(n))
\in \Lambda\(t\) .
$$
Setting $t=1$, we obtain the (Euler-Frobenius) characteristic $\operatorname{ch}(\v)$.
For example,
$$
\operatorname{ch}({\operatorname{\mathcal{C}\mathit{om}}}^+) = \sum_{n=1}^\infty h_n = \exp\Bigl( \sum_{n=1}^\infty
\frac{p_n}{n} \Bigr) - 1 , \quad\text{and}\quad
\operatorname{ch}({\operatorname{\mathcal{A}\mathit{ss}}}^+) = \sum_{n=1}^\infty p_1^n = \frac{p_1}{1-p_1} .
$$
\subsection{Plethysm}
Consider the ring $\Lambda\(t\)$ of power series in a variable $t$ with
coefficients in $\Lambda$. There is an associative product on
$\Lambda\(t\)$, called plethysm and denoted $f\circ g$, characterized by
the formulas
\begin{enumerate}
\item $(f_1+f_2)\circ g=f_1\circ g+f_2\circ g$;
\item $(f_1f_2)\circ g=(f_1\circ g)(f_2\circ g)$;
\item if $f=f(t,p_1,p_2,\dots)$, then $p_n\circ f=f(t^n,p_n,p_{2n},\dots)$,
and $t\circ f=t$.
\end{enumerate}
The following formula generalizes its analogue for ungraded $\SS$-modules,
proved in \cite{Macdonald}:
$$
\operatorname{ch}_t(\v\circ\mathcal{W}) = \operatorname{ch}_t(\v)\circ\operatorname{ch}_t(\mathcal{W}) .
$$
The operation
$$
\operatorname{Exp}(f) = \sum_{n=0}^\infty h_n\circ f
$$
plays the role for symmetric functions that exponentiation does for
power series. The inverse of $\operatorname{Exp}$ is the operation
$$
\operatorname{Log}(f) = \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(p_n\circ f) ,
$$
where $\mu(n)$ is the M\"obius function.
Using this formula and the Poincar\'e-Birkhoff-Witt theorem
${\operatorname{\mathcal{A}\mathit{ss}}}^+={\operatorname{\mathcal{C}\mathit{om}}}^+\circ{\operatorname{\mathcal{L}\mathit{ie}}}^+$, we may calculate $\operatorname{ch}({\operatorname{\mathcal{L}\mathit{ie}}}^+)$.
\begin{proposition} \label{Lie}
$\displaystyle\operatorname{ch}({\operatorname{\mathcal{L}\mathit{ie}}}^+) = - \sum_{n=1}^\infty \frac{\mu(n)}{n}
\log(1-p_n)$
\end{proposition}
\begin{pf}
We know from the Poincar\'e-Birkhoff-Witt theorem that
$\operatorname{ch}({\operatorname{\mathcal{A}\mathit{ss}}}^+)=\operatorname{Exp}(\operatorname{ch}({\operatorname{\mathcal{L}\mathit{ie}}}^+))$; it follows that
$$
\operatorname{ch}({\operatorname{\mathcal{L}\mathit{ie}}}^+) = \operatorname{Log}(1+\operatorname{ch}({\operatorname{\mathcal{A}\mathit{ss}}}^+))
= \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(1+p_n\circ\operatorname{ch}({\operatorname{\mathcal{A}\mathit{ss}}}^+)) .
$$
Since
$$
1+ p_n \circ \operatorname{ch}({\operatorname{\mathcal{A}\mathit{ss}}}^+) = 1 + \frac{p_n}{1-p_n} = \frac{1}{1-p_n} ,
$$
the result follows.
\end{pf}
It follows from this formula that
$$
\operatorname{ch}_n({\operatorname{\mathcal{L}\mathit{ie}}}^+(n)) = \frac{1}{n} \sum_{d|n} \mu(d) p_d^{n/d} .
$$
This is the characteristic of the induced representation
$\operatorname{Ind}_{C_n}^{\SS_n}\chi$, where $\chi$ is a primitive character of the
cyclic group.
We now turn to calculating the characteristic of the braid operad $\mathcal{B}\mathit{raid}$.
First, we need a lemma.
\begin{lemma}
$\operatorname{ch}_t(\Lambda\v)=-t\operatorname{ch}_t(\v)(-t^{-1}p_1,-t^{-2}p_2,-t^{-3}p_3,\dots)$
\end{lemma}
\begin{pf}
Tensoring with $\operatorname{sgn}_n$ has the effect of replacing $p_n$ by
$(-1)^{n-1}p_n$. Applying $\Sigma^{-n}$ to $\v(n)$ then has the effect of
replacing $p_n$ by $(-t)^{-n}p_n$.
\end{pf}
\begin{proposition} \label{configuration}
For each $n\ge1$, let
$$
P_n(t) = \frac{1}{n} \sum_{d|n} \frac{\mu(n/d)}{t^d} .
$$
Then
$$
\operatorname{ch}_t(\mathcal{B}\mathit{raid}) = \prod_{n=1}^\infty \Bigl( 1 + t^np_n \Bigr)^{P_n(t)} - 1 .
$$
\end{proposition}
\begin{pf}
The $\SS$-module $\Lambda^{-1}{\operatorname{\mathcal{L}\mathit{ie}}}^+$ has Poincar\'e characteristic
$$
\operatorname{ch}_t(\Lambda^{-1}{\operatorname{\mathcal{L}\mathit{ie}}}^+)
= \frac{1}{t} \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(1+t^np_n) .
$$
It follows that
\begin{eqnarray*}
\operatorname{ch}_t(\b) &=& \operatorname{Exp} \circ \operatorname{ch}_t(\Lambda^{-1}{\operatorname{\mathcal{L}\mathit{ie}}}^+) - 1
= \exp\Bigl( \sum_{k=1}^\infty \frac{p_k}{k} \Bigr) \circ
\Bigl( t^{-1} \sum_{n=1}^\infty \frac{\mu(n)}{n} \log(1+t^np_n) \Bigr) - 1 \\
&=& \exp\Bigl( \sum_{k=1}^\infty
\sum_{n=1}^\infty t^{-k} \frac{\mu(n)}{kn} \log(1+t^{kn}p_{kn}) \Bigr) - 1
\\
&=& \prod_{n=1}^\infty ( 1 + t^np_n )^{P_n(t)} - 1 .
}\end{pf
\end{eqnarray*}
\def}\end{pf{}\end{pf}
In particular, setting $p_1=x$ and $p_n=0$, $n>1$, we see that the
Poincar\'e polynomial of the space $\mathbb{C}^n_0$ is
\begin{eqnarray}
\label{poincare-config}
\sum_{i=0}^{n-1} (-t)^n \dim H^i(\mathbb{C}^n_0)
&=& \text{coefficient of $x^n/n!$ in $(1+tx)^{t^{-1}} - 1$} \\
&=& \binom{t^{-1}}{n} t^n = \prod_{i=1}^{n-1} (1-it) .
\end{eqnarray}
\subsection{The characteristic of a cyclic $\SS$-module}
If $\v$ is a stable cyclic $\SS$-module, we define $\operatorname{Ch}_t(\v)$ in a similar
way to $\operatorname{ch}_t(\v)$:
$$
\operatorname{Ch}_t(\v) = \sum_{n=3}^\infty \sum_{i=0}^\infty (-t)^i \,
\operatorname{ch}_n(\v_i\(n\)) \in \Lambda\(t\) .
$$
If $\operatorname{ch}_t(\v)$ denotes the Poincar\'e characteristic of the $\SS$-module
underlying $\v$, then we have the formula
$$
\operatorname{ch}_t(\v) = \frac{\partial\operatorname{Ch}_t(\v)}{\partial p_1} .
$$
We now calculate $\operatorname{Ch}_t(\mathbf{m})$, where $\mathbf{m}$ is the cyclic $\SS$-module
$$
\mathbf{m}\(n\) = \begin{cases} H_{\bullet}(\mathcal{M}_{0,n}) , & n\ge3 , \\ 0 , & n<3 .
\end{cases}$$
Note that the Poincar\'e polynomial of $\mathcal{M}_{0,n}$ is much easier to
calculate than the Poincar\'e characteristic: it is obtained by dividing
the Poincar\'e polynomial \eqref{poincare-config} of $\mathbb{C}^{n-1}_0$ by $1-t$:
\begin{equation} \label{poincare-moduli}
\sum_{i=0}^{n-2} (-t)^n \dim H^i(\mathcal{M}_{0,n}) = \prod_{i=2}^{n-2} (1-it).
\end{equation}
\begin{theorem} \label{moduli:characteristic}
$$
\operatorname{Ch}_t(\mathbf{m}) = \frac{1}{1-t^2} \Bigl( (1+tp_1)
\prod_{n=1}^\infty ( 1 + t^np_n )^{P_n(t)}
- 1 - (1+t)h_1 - (h_2 + te_2) \Bigr)
$$
\end{theorem}
\begin{pf}
{}From the $\SS_n$-equivariant homotopy equivalence
$\mathbb{C}^n_0\simeq\mathcal{M}_{0,n+1}\times S^1$, which holds for $n\ge2$, we see that
\begin{equation} \label{differential}
\frac{\partial\operatorname{Ch}_t(\mathbf{m})}{\partial p_1}
= \operatorname{ch}_t(\mathbf{m}) = \frac{t\bigl(\operatorname{ch}_t(\mathcal{B}\mathit{raid})-p_1\bigr)}{t-1} .
\end{equation}
The Serre spectral sequence for the $\SS_n$-equivariant fibration
$$\begin{CD}
\mathbb{C}\setminus\{1,\dots,n\} @>>> \mathcal{M}_{0,n+1} @>>> \mathcal{M}_{0,n}
\end{CD}$$
collapses at $E^2$, so $H_{\bullet}(\mathcal{M}_{0,n+1}) \cong
H_{\bullet}(\mathbb{C}\setminus\{1,\dots,n\}) \o H_{\bullet}(\mathcal{M}_{0,n})$.
Furthermore, this isomorphism is $\SS_n$-equivariant, where $\SS_n$
acts on $H_{\bullet}(\mathbb{C}\setminus\{1,\dots,n\})$ by the monodromy of the
Gauss-Manin connection.
Now, $H_0(\mathbb{C}\setminus\{1,\dots,n\})$ is the trivial $\SS_n$-module, while
$H_1(\mathbb{C}\setminus\{1,\dots,n\})$ is the irreducible representation
$V_{n-1,1}$, which is the kernel of the natural map $\begin{CD}
\mathbb{C}^n@>>>\mathbb{C}\end{CD}$ obtained
by sending $(x_1,\dots,x_n)$ to $x_1+\dots+x_n$. If $\sigma\in\SS_n$ is a
transitive permutation, $\operatorname{Tr}(\sigma|V_{n-1,1})=-1$; this shows that
\begin{equation} \label{initial}
\operatorname{Ch}_t(\mathbf{m})\big|_{p_1=0} = \frac{\operatorname{ch}_t(\mathcal{B}\mathit{raid})}{1+t} \Big|_{p_1=0} .
\end{equation}
The theorem now follows on solving the differential equation
\eqref{differential} with initial condition \eqref{initial}.
\end{pf}
The first few terms of $\operatorname{Ch}_t(\mathbf{m})$ are as follows:
$$\begin{tabular}{|c|l|}
$n$ & $\operatorname{Ch}_t(\mathcal{M}_{0,n})$ \\ \hline
$3$ & $s_3$ \\
$4$ & $s_4-ts_{2^2}$ \\
$5$ & $s_5-ts_{32}+t^2(s_{31^2})$ \\
$6$ & $s_6-ts_{42}+t^2(s_{41^2}+s_{321})-t^3(s_{41^2}+s_{3^2}+s_{2^21^2})$ \\
\end{tabular}$$
The pattern emerging here, that $H_1(\mathcal{M}_{0,n})\cong V_{n-2,2}$, is easily
verified in general using our formula for $\operatorname{Ch}_t(\mathbf{m})$. We have seen that
there is a natural identification between $\operatorname{sgn}_n\o H^1(\mathcal{M}_{0,n})$ and the
space of relations \eqref{hypercommutative} among $n-1$ letters in the
hypercommutative operad: thus, we see that this space of relations is the
irreducible $\SS_n$-module $V_{2^21^{n-4}}$.
Applying l'H\^opital's rule to Theorem \ref{moduli:characteristic}, we see
that the Euler-Frobenius characteristic of $\mathbf{m}$ is given by the formula
$$
\operatorname{Ch}(\mathbf{m}) = \lim_{\begin{CD}t@>>>1\end{CD}} \operatorname{Ch}_t(\mathbf{m})
= \frac12 (1+p_1)^2 \sum_{n=1}^\infty \frac{\varphi(n)}{n}
\log(1+p_n) - \frac{1}{4} (2p_1 + 3p_1^2 + p_2) .
$$
Finally, it follow easily from the formula for $\operatorname{Ch}_t(\mathbf{m})$ that
$$
\operatorname{Ch}_t(\mathcal{G}\mathit{rav}) = - \frac{t^3}{1-t^2}
\Bigl( (1-p_1) \prod_{n=1}^\infty (1-p_n)^{P_n(t)}
- 1 + (1+t^{-1})h_1 - (t^{-1}h_2 + t^{-2}e_2) \Bigr)
$$
\subsection{The Poincar\'e characteristic of $\mathbb{T}_+\v$}
In \cite{modular}, a formula for the Poincar\'e characteristic of $\mathbb{T}_+\v$
in terms of the Poincar\'e characteristic of $\v$ is derived. If
$F=e_2-\operatorname{Ch}_t(\v)$, define the Legendre transform $G=\mathcal{L} F$ of $F$ in the
sense of symmetric functions by the formula
\begin{equation} \label{Legendre}
F\circ \frac{\partial(\mathcal{L} G)}{\partial p_1} + G = p_1 \frac{\partial G}{\partial p_1} .
\end{equation}
Then $G=h_2+\operatorname{Ch}_t(\mathbb{T}_+\v)$.
Note that \eqref{Legendre} implies that $(\partial F/\partial p_1)\circ(\partial G/p p_1)=p_1$,
from which it is straightforward to calculate $\partial G/\partial p_1$. Substituting
$\partial G/\partial p_1$ into both sides of \eqref{Legendre}, we obtain an explicit
formula for $G$.
As an application of \eqref{Legendre}, we now calculate the Poincar\'e
characteristics of the varieties $\overline{\mathcal{M}}_{0,n}$. We use a slight extension
of \eqref{Legendre}, in which $\v=\v_0\oplus\v_1$ has an internal
$\mathbb{Z}/2$-grading, and $\operatorname{Ch}_t(\v)=\operatorname{Ch}_t(\v_0)-\operatorname{Ch}_t(\v_1)$. Let $\v$ and $\mathcal{W}$
be the stable cyclic $\SS$-modules
\begin{eqnarray*}
\v_i\(n\) &=& \begin{cases} 0 , & i=0 , \\ \mathcal{H}\mathit{ycom}\(n\)\o\operatorname{sgn}_n , & i=1 ;
\end{cases} \\
\mathcal{W}_{ij}\(n\) &=& \begin{cases} \operatorname{sgn}_n\o H^p(\mathcal{M}_{0,n}) , &
\text{$i=2(n-p-3)$ and $j\equiv p+1\pmod{2}$,} \\
0 , & \text{otherwise.}
\end{cases}
\end{eqnarray*}
Thus,
\begin{eqnarray*}
\operatorname{Ch}_t(\v) & =& - \operatorname{Ch}_t(\mathcal{H}\mathit{ycom})\big|_{p_n\to(-1)^{n-1}p_n} , \\
\operatorname{Ch}_t(\mathcal{W}) &=& \frac{- t^{-6}\operatorname{Ch}_t(\mathbf{m})\Big|_{t\to-t^{-2}}
{p_n\to(-1)^{n-1}t^{2n}p_n}} .
\end{eqnarray*}
Using \eqref{det}, we may rewrite \eqref{bar:even} in the form
$$\begin{CD}
0 @>>> H^p(\mathcal{M}_{0,n})\o\operatorname{sgn}_n @>>>
\bigoplus_{T\in\mathcal{T}_p\(n\)} \v\(T\)_{2(n-p-3)} @>>>
\bigoplus_{T\in\mathcal{T}_{p-1}\(n\)} \v\(T\)_{2(n-p-3)} @>>> \dots
\end{CD}$$
which shows that $\operatorname{Ch}_t(\mathcal{W})=\operatorname{Ch}_t(\mathbb{T}_+\v)$, and hence that
$G=h_2+\operatorname{Ch}_t(\mathcal{W})$ is the Legendre transform of $F=e_2-\operatorname{Ch}_t(\v)$. In this
way, we have proved the following proposition.
\begin{theorem} \label{F-G}
The symmetric function
$$
F = e_2 + \operatorname{Ch}_t(\mathcal{H}\mathit{ycom})\big|_{p_n\to(-1)^{n-1}p_n}
$$
is the Legendre transform of the symmetric function
$$
G = h_2 - t^{-6} \operatorname{Ch}_t(\mathbf{m})\Big|_{\frac{t\to t^2}{ p_n\to(-1)^{n-1}t^{2n}p_n}} .
$$
\end{theorem}
Here are some sample calculations of $\operatorname{Ch}_t(\overline{\mathcal{M}}_{0,n})$ for small $n$:
$$\begin{tabular}{|c|l|}
$n$ & $\operatorname{Ch}_t(\overline{\mathcal{M}}_{0,n})$ \\ \hline
$3$ & $s_3$ \\
$4$ & $(1+t^2)s_4$ \\
$5$ & $(1+t^4)s_5+t^2(s_5+s_{41})$ \\
$6$ & $(1+t^6)s_6+(t^2+t^4)(2s_6+s_{51}+s_{42})$ \\
\end{tabular}$$
These formulas simplify if we are only interested in the dimensions of
the vector spaces $H_i(\overline{\mathcal{M}}_{0,n})$. We have the formula
\begin{eqnarray*}
g(x,t) = G'\Big|_{\frac{p_1\to x}{ p_n\to0,n>1}}
&=& x - \sum_{n=2}^\infty \frac{x^n}{n!} \sum_{i=0}^{n-2} (-1)^i
t^{2(n-i-2)} \dim H_i(\mathcal{M}_{0,n+1}) \\
&=& x - \frac{(1+x)^{t^2}-(1+t^2x)}{t^2(t^2-1)} .
\end{eqnarray*}
It is a corollary of Theorem \ref{F-G} that
$$
f(x,t) = F'\Big|_{\frac{p_1\to x}{ p_n\to0,n>1}}
= x + \sum_{n=2}^\infty \frac{x^n}{n!}
\sum_{i=0}^{n-2} t^{2i} \dim H_{2i}(\overline{\mathcal{M}}_{0,n+1}) ,
$$
is the inverse of $g$, in the sense that $f(g(x,t),t)=x$.
This is a reformulation of Fulton and MacPherson's calculation of the
Poincar\'e polynomial of $\overline{\mathcal{M}}_{0,n}$. Note that their proof also makes
use of mixed Hodge theory, in the form of the ``fake Poincar\'e
polynomial.'' Our result Theorem \ref{F-G} is an equivariant version of their
calculation.
\makeatletter\renewcommand{\@biblabel}[1]{\hfill[#1]}\makeatother
|
1997-01-22T09:51:28 | 9610 | alg-geom/9610002 | en | https://arxiv.org/abs/alg-geom/9610002 | [
"alg-geom",
"math.AG"
] | alg-geom/9610002 | Yves Laszlo | Yves Laszlo | Linearization of group stack actions and the Picard group of the moduli
of $\SL_r/\mu_s$-bundles on a curve | 13 pages, PlainTex | null | null | null | null | We first study the descent theory of line bundles under a morphism which is
tors or under a group stack and then use this technical result to determine the
exact structure of $\Pic(\M_G)$ where $G=\SL_r/\mu_s$ (we include a minor
modification to explain the genus 0 case).
| [
{
"version": "v1",
"created": "Thu, 3 Oct 1996 07:38:31 GMT"
},
{
"version": "v2",
"created": "Wed, 22 Jan 1997 08:50:18 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Laszlo",
"Yves",
""
]
] | alg-geom | \section{...} et une sous-section avec
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\null
\centerline{{\bf Linearization of group stack actions and the Picard group}}
\centerline{{\bf of the moduli
of $\mathop{\bf SL}\nolimits_r/\mu_s$-bundles on a curve}}
\medskip
\centerline{Yves {\pc LASZLO}
\footnote{\parindent 0.5cm($\dagger$)}{\sevenrm Partially
supported by the European HCM Project ``Algebraic Geometry in Europe"
(AGE).}}
\def\a{{\cal A}}\def\b{{\cal B}}\def\c{{\cal C}}\def\tp{{\tilde p}}\def\X{{\cal X}}
\def{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}{{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}}
\def{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}{{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def{\bf P}{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}}
\def{\cal U}}\def\d{{\cal D}}\def\F{{\cal F}{{\cal U}}\def\d{{\cal D}}\def\F{{\cal F}}
\let\scp=\scriptstyle
{\bf Introduction}
\medskip
Let $G$ be a complex semi-simple group and $\tilde G\rightarrow \kern -3mm\rightarrow G$ the universal covering. Let $\M_G$
(resp. $\M_{\tilde G}$) be the moduli stack of $G$-bundles over a curve $X$ of degree
$1\in\pi_1(G)$ (resp. of ${\tilde G}$-bundles. In [B-L-S], we have studied the link between the
groups $\mathop{\rm Pic}\nolimits(\M_G)$ and $\mathop{\rm Pic}\nolimits(\M_{\tilde G}$), the later being well understood thanks to [L-S].
In particular, it has been possible to give a complete description in the case where $G={\bf
PSL}_r$ but not in the case $\mathop{\bf SL}\nolimits_r/\mu_s,\ s\mid r$, although we were able to give partial
results. The reason was that we did not have at our disposal the technical background to study
the morphism $\M_{\tilde G}{\rightarrow}\M_G$. It turns to be out that it is a torsor under some group
stack, not far from a Galois \'etale cover in the usual schematic picture. Now, the descent
theory of Grothendieck has been adapted to the set-up of fpqc morphisms of stacks in [L-M] and
gives the theorem
\ref{theo} in the particular case of a morphism which
is torsor under a group stack. We then used this technical result to determine the exact structure
of
$\mathop{\rm Pic}\nolimits(\M_G)$ where
$G=\mathop{\bf SL}\nolimits_r/\mu_s$ (theorem
\ref{theo-pic}).
\medskip I would like to thank L. Breen to have taught me both the notion of torsor and
linearization of a vector bundle in the set-up of group-stack action and for his comments on a
preliminary version of this paper.
\medskip
{\bf Notation}
\medskip
Throughout this paper, all the stacks will be implicitely assumed to be algebraic over a fixed base
scheme and the morphisms locally of finite type. We fix once for all a projective, smooth,
connected genus $g$ curve $X$ over an algebraically closed field $\k$ and a closed point $x$ of
$X$. For simplicity, we assume $g>0$ (see remarks \ref{rema} and \ref{rem} for
the case of
${\bf P}^1$). The Picard stack parametrizing families of line bundles of degree
$0$ on $X$ will be denoted by ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ and the jacobian variety of $X$ by $JX$.
If $G$ is an algebraic group over $\k$, the quotient stack ${\rm Spec}(\k)/G$ (where
$G$ acts trivialy on ${\rm Spec}(\k))$ whose category over a
$\k$-scheme $S$ is the category of $G$-torsors (or $G$-bundles) over $S$ will be denoted by $BG$.
If $n$ is an integer and $A={\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X), JX$ or $BG_m$ we denote by $n_A$ the $n^{\rm\scp th}$-power
morphism $a\longmapsto a^n$. We denote by ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_n$ (resp. $J_n$) the $0$-fiber $A\times_A{\rm Spec}(\k)$
of $n_A$ when $A={\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ (resp. $A=JX$), wh
\section{Generalities} Following [Br], for
any diagram $$ A\fhd{h}{}B\matrix{\fhd{g}{}\cr\Uparrow\rlap{$\scp\lambda$}\cr
\scp f\cr\fhd{}{}}C\fhd{l}{}D$$ of $2$-categories, we'll denote by $l*\lambda:\ l\kern 1pt{\scriptstyle\circ}\kern 1pt f\Rightarrow
l\kern 1pt{\scriptstyle\circ}\kern 1pt g$ (resp. $\lambda*h:\ f\kern 1pt{\scriptstyle\circ}\kern 1pt h\Rightarrow g\kern 1pt{\scriptstyle\circ}\kern 1pt h$) the $2$-morphism deduced from
$\lambda$.
\global\def\currenvir{subsection For the convenience of the reader, let us prove a simple formal lemma which will be usefull in the
section \ref{simplicial}. Let $\a,\b,\c$ be three $2$-categories, a $2$-commutative
diagram $$\matrix{
&&\c\cr
&\llap{${}^{\delta_0}$}\nearrow&\fvh{}{d_0}\cr
\a&\fhd{f}{}&\b\cr
&\llap{${}_{\delta_1}$}\searrow&\fvb{}{d_1}\cr
&&\c\cr}\leqno{(\global\def\currenvir{formule)}$$\label{strict-categorie}
and a $2$-morphism $\mu:\ \delta_0\Rightarrow \delta_1$.
\th Lemma
\enonce Assume that $f$ is an equivalence. There exists a unique $2$-morphism $$\mu*f^{-1}:\
d_0\Rightarrow d_1$$ such that $(\mu*f^{-1})*f=\mu$.
\endth\label{lemme-categorie}
{\it Proof}: let $\epsilon_k, k=0,1$ the $2$-morphism $d_k\kern 1pt{\scriptstyle\circ}\kern 1pt f\Rightarrow \delta_k$. Let $b$ be an object of $\b$. Pick an object $a$ of $\a$ and an isomorphism
$\alpha:\ f(a)\buildrel\sim\over{\rightarrow} b$. Let $\varphi_\alpha :\ d_0(b)\buildrel\sim\over{\rightarrow} d_1(b)$ be the unique isomorphism making
the diagram
$$\matrix{
\delta_0(a)&\fhd{\epsilon_0(a)}{}&d_0\kern 1pt{\scriptstyle\circ}\kern 1pt f(a)&\fhd{d_0(\alpha )}{}&d_0(b)\cr
\fvb{\mu_a}{}&&&&\fvb{}{\varphi_\alpha }\cr
\delta_1(a)&\fhd{\epsilon_1(a)}{}&d_1\kern 1pt{\scriptstyle\circ}\kern 1pt f(a)&\fhd{d_1(\alpha )}{}&d_1(b)\cr
}$$ commutative. We have to show that $\varphi_\alpha $ does not depend on $\alpha $ but only
on $b$. Let $\alpha ':\ f(a')\buildrel\sim\over{\rightarrow} b$ be another isomorphism. There exists a unique isomorphism
$\iota:\ a'\buildrel\sim\over{\rightarrow} a$ such that $\alpha\kern 1pt{\scriptstyle\circ}\kern 1pt f(\iota)=\alpha'$. The one has the equality
$\varphi_{\alpha'}=d_1(\alpha )\kern 1pt{\scriptstyle\circ}\kern 1pt\Phi\kern 1pt{\scriptstyle\circ}\kern 1pt d_0(\alpha)^{-1}$ where
$$\Phi=[d_1\kern 1pt{\scriptstyle\circ}\kern 1pt
f(\iota)]\kern 1pt{\scriptstyle\circ}\kern 1pt\epsilon_1(a')\kern 1pt{\scriptstyle\circ}\kern 1pt \mu_{a'}\kern 1pt{\scriptstyle\circ}\kern 1pt\epsilon_0(a')^{-1}\kern 1pt{\scriptstyle\circ}\kern 1pt[d_0\kern 1pt{\scriptstyle\circ}\kern 1pt f(\iota)]^{-1}.$$
The functoriality of $\epsilon_i$ and $\mu$ ensures that one has the equalities
$$d_k\kern 1pt{\scriptstyle\circ}\kern 1pt f(\iota)\kern 1pt{\scriptstyle\circ}\kern 1pt\epsilon_k(a')=\epsilon_k(a)\kern 1pt{\scriptstyle\circ}\kern 1pt\delta_k(\iota)$$ and
$$\mu_{a}=\delta_1(\iota)\kern 1pt{\scriptstyle\circ}\kern 1pt\mu_{a'}\kern 1pt{\scriptstyle\circ}\kern 1pt\delta_0(\iota)^{-1}.$$ This shows that
$$\Phi=\epsilon_1(a)\kern 1pt{\scriptstyle\circ}\kern 1pt\mu_a\kern 1pt{\scriptstyle\circ}\kern 1pt\epsilon_0(a)^{-1}$$ which proves
the equality $\varphi_\alpha =\varphi_{\alpha '}$. We can therefore define $\mu_b$
to be the isomorphism
$\varphi_\alpha $ for one isomorphism $\alpha:\ f(a)\buildrel\sim\over{\rightarrow} b$. One checks that the construction
is functorial in $b$ and the lemma follows.\cqfd
\section{Linearizations of line bundles on stacks} Let us first recall following [Br] the notion
of torsor in the stack context.
\global\def\currenvir{subsection Let $f:\ {\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}{\rightarrow}\E$ be a faithfully flat morphism of stacks. Let us assume that an
algebraic $gr$-stack $\C$ acts on $f$ (the product of $\C$ is denoted by $\m$ and the unit object
by $1$). Following [Br], this means that there exists a 1-morphism of $\E$-stacks $m:\
\C\times{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}{\rightarrow}{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$ and a $2$-morphism $\mu:\ m\kern 1pt{\scriptstyle\circ}\kern 1pt (\m\times \hbox{\rm Id}_{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M})\Rightarrow m\kern 1pt{\scriptstyle\circ}\kern 1pt
(\hbox{\rm Id}_\C\times m)$ such that the obvious associativity condition (see the diagram (6.1.3) of
[Br]) is satisfied and such that there exists a $2$-morphism $\epsilon:\ m\kern 1pt{\scriptstyle\circ}\kern 1pt(1\times
\hbox{\rm Id}_{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M})\Rightarrow \hbox{\rm Id}_{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$ which is compatible to $\mu$ in the obvious sense (see (6.1.4) of
[Br]).
\global\def\currenvir{rem To say that $m$ is a morphism of $\E$-stacks means that the diagram
$$\matrix{\C\times{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}&\fhd{m}{}&{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}\cr\searrow&&\swarrow\cr&\E\cr}$$ is $2$-commutative. In other
words, if we denote for simplicity the image of a pair of objects $m(g,x)$ by $g.x$, this means
that there exists a functorial isomorphism $\iota_{g,x}:\ f(g.x){\rightarrow} f(x)$.\label{Y-morphism}
\global\def\currenvir{subsection\label{equi} Suppose that $\C$ acts on another such $f':{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}'{\rightarrow}\E$. A morphism
$p:\ {\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}'{\rightarrow}{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$ will be said equivariant if there exists a $2$-morphism
$$q:\ m\kern 1pt{\scriptstyle\circ}\kern 1pt(\hbox{\rm Id}\times p)\Rightarrow p\kern 1pt{\scriptstyle\circ}\kern 1pt m'$$ which is compatible to $\mu$ (as in
[Br] (6.1.6)) and $\epsilon$ (which is implicit in [Br]) in the obvious sense.
\th Definition
\enonce With the above notations, we say that $f$ (or ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$) is a $\C$-torsor over
$\E$ if the morphism $pr_2\times m:\ \C\times {\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}{\rightarrow}{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}\times_\E{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$ is an isomorphism (of stacks)
and the geometrical fibers of $f$ are not empty.
\endth
\global\def\currenvir{rem In down to earth terms, this means that if $\iota:\ f(x){\rightarrow} f(x')$ is an isomorphism in
$\E$ ($x,x'$ being objects of ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$), there exist an object $g$ of $\C$ and a unique isomorphism
$(x,g.x)\fhd{\sim}{}(x,x')$ which induces $\iota$ thanks to $\iota_{g,x}$ (cf. \ref{Y-morphism}).
\global\def\currenvir{ex If $\M_X(G_m)$ is the
Picard stack of $X$, the morphism $\M_X(G_m){\rightarrow}\M_X(G_m)$ of multiplication by $n\in{\bf} Z}\def\Q{{\bf} Q}\def\C{{\bf} C}\def\F{{\bf} F$ is a
torsor under $B\mu_n\times J_n(X)$ (cf. (\ref{torseur})).
\global\def\currenvir{subsection Let a $\L$ be a line bundle on ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$. By definition, the data $\L$
is equivalent to te data of a morphism $l:\ {\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}{\rightarrow} BG_m$ (see [L-M],prop. 6.15). If $\L,\L'$ are
$2$ line bundles on $\X$ defined by $l,l'$, we will view an isomorphism
$\L\buildrel\sim\over{\rightarrow}\L'$ as a $2$-morphism $l\Rightarrow l'$.\label{convention-fibre}
\th Definition
\enonce A $\C$-linearization of $\L$ is a $2$-morphism $\lambda:\ l\kern 1pt{\scriptstyle\circ}\kern 1pt
m\Rightarrow l\kern 1pt{\scriptstyle\circ}\kern 1pt pr_2$ such that the two diagrams of $2$-morphisms $$\matrix{ l\kern 1pt{\scriptstyle\circ}\kern 1pt m\kern 1pt{\scriptstyle\circ}\kern 1pt
(\m\times \hbox{\rm Id}_{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M})&\buildrel{l*\mu}\over{\Longrightarrow}&l\kern 1pt{\scriptstyle\circ}\kern 1pt m\kern 1pt{\scriptstyle\circ}\kern 1pt (\hbox{\rm Id}_\C\times m)\cr
\hfill\Big\Downarrow{\scriptstyle\lambda*(\m\times
\hbox{\rm Id}_{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M})}&&\hfill\Big\Downarrow{\scriptstyle\lambda*(\hbox{\rm Id}_\C\times m)}\cr l\kern 1pt{\scriptstyle\circ}\kern 1pt pr_2\kern 1pt{\scriptstyle\circ}\kern 1pt(\m\times
\hbox{\rm Id}_{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M})=l\kern 1pt{\scriptstyle\circ}\kern 1pt pr_2\kern 1pt{\scriptstyle\circ}\kern 1pt pr_{23}&\buildrel \lambda*pr_{23}\over{\Longleftarrow}&l\kern 1pt{\scriptstyle\circ}\kern 1pt
pr_2\kern 1pt{\scriptstyle\circ}\kern 1pt(\hbox{\rm Id}_\C\times m)=l\kern 1pt{\scriptstyle\circ}\kern 1pt m\kern 1pt{\scriptstyle\circ}\kern 1pt pr_{23}\cr }\leqno(\global\def\currenvir{formule)$$\label{lineA} and $$
\matrix{ l\kern 1pt{\scriptstyle\circ}\kern 1pt m\kern 1pt{\scriptstyle\circ}\kern 1pt(1\times \hbox{\rm Id}_{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M})&\buildrel l*\epsilon\over{\Longrightarrow}&l\cr
\FFvb{\lambda*(1\times \hbox{\rm Id}_{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M})}{}&&\parallel\cr l&=&l\cr
}\leqno(\global\def\currenvir{formule)$$\label{lineB}(stritly) commutes.
\endth
\global\def\currenvir{rem\label{rem-line} In $g_1,g_2$ are objects of $\C$ and $d$ of ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$, the commutativity of the
diagram (\ref{lineA}) means that the diagram
$$\matrix{
\L_{(g_1.g_2)x}&\fhd{\sim}{}&\L_{g_1(g_2.x)}\cr
\fvb{}{\wr}&&\fvb{}{\wr}\cr
\L_x&\fhg{\sim}{}&\L_{g_2.x}\cr}$$ is commutative and the commutativity of (\ref{lineB}) that the
two
isomorphisms $\L_{1.x}\simeq \L_x$ defined by the linearization $\lambda$ and $\epsilon$
respectively are the same.
\section{An example} Let me recall that a closed point $x$ of $X$ has been fixed. Let $S$ be a
$\k$-scheme. The $S$-points of the jacobian variety of $X$ are by definition isomorphism classes of
line bundles on $X_S$ together whith a trivialization along $\{x\}\times S$ (such a pair will be
called a rigidified line bundle). For the covenience of the reader, let me state this well known
lemma which can be founf in SGA4, exp. XVIII, (1.5.4)
\th Lemma
\enonce The Picard stack ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ is canonically isomorphic (as a $\k$-group stack) to $JX\times
BG_m$.
\endth\label{torseur}
{\it Proof}: let $f:\ {\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X){\rightarrow} JX\times BG_m$ be the morphim which associates
-to the line bundle $L$ on $X_S$ the pair
$L\otimes L_{\mid \{x\}\times S}^{-1},L_{\mid \{x\}\times S}$ (thought as an object of $JX\times
BG_m$ over $S$);
-to an isomorphism $L\buildrel\sim\over{\rightarrow} L'$ on $X_S$ its restriction to $\{x\}\times S$.
Let $f':\ JX\times BG_m{\rightarrow}{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ be the morphism which associates
-to the pair $(L,V)$ where $L$ is a rigidified bundle on $X_S$ and $V$ a line bundle on $S$
(thought as an object of $JX\times BG_m$ over $S$), the line bundle $L\otimes_{X_R}V$;
-to an isomorphism $(l,v):\ (L,V)\buildrel\sim\over{\rightarrow} (L',V')$ the tensor product $l\otimes_{X_S} v$.
The morphisms $f$ and $f'$ are (quasi)-inverse each other and are morphisms of $\k$-stacks.\cqfd
We will identify from now ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ and $JX\times BG_m$. Let $\L$ (resp. ${\bf P}$ and $\T$) be the
universal bundle on $X\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ (resp. on $X\times JX$ and $BG_m$) and let $\Theta=(\det
R\Gamma{\bf P})^{-1}$ be the theta line bundle on $JX$. The isomorphism $\L\buildrel\sim\over{\rightarrow}{\bf P}\otimes\T$ yields
an isomorphism $$\det
R\Gamma\L^n(m.x)\buildrel\sim\over{\rightarrow}\Theta^{-n^2}\otimes\T^{(m+1-g)}.\leqno({\global\def\currenvir{formule})$$\label{iso-det}
\section{Descent of $\C$-line bundles} \label{simplicial} The object of this section is to prove
the following statement
\th Theorem
\enonce Let $f:{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}{\rightarrow}\E$ a $\C$-torsor as above. Let $\mathop{\rm Pic}\nolimits^\C({\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M})$ be the group of isomorphism
classes of $\C$-linearized line bundles on ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$.
Then, the pull-back morphism $f^*:\
\mathop{\rm Pic}\nolimits(\E)\buildrel\sim\over{\rightarrow}\mathop{\rm Pic}\nolimits^\C({\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M})$ is an isomorphism.
\endth\label{theo}
The descent theory of Grothendieck has been adapted in the case of algebraic $1$-stacks in
[L-M], essentially in the proposition (6.23).Let ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_\bullet{\rightarrow}\E$ be the (augmented) simplicial
complex of stacks coskeleton of $f$ (as defined in [De] (5.1.4) for instance). By proposition
(6.23) of [L-M], one just has to construct a cartesian ${\cal O}_{D_\bullet}$-module $\L_\bullet$ such
that $L_0$ is the ${\cal O}_{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$-module $\L$ to prove the theorem. The $n$-th piece ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_n$ is inductively
defined by ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_0={\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M},\ {\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_n={\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}\times_\E{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_{n-1}$ for $n>0$. Let $p_n:\ {\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_n{\rightarrow}{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$ be the
projection on the first factor. It is the simplicial morphism associated to the map $$\tp_n:\
\left\{\matrix{\Delta_0&{\rightarrow}&\Delta_n\cr 0&\longmapsto&0\cr}\right.$$ We define $\L_n$ by the
morphism $$l_n:\ {\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_n\fhd{p_n}{}{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}\fhd{l}{}BG_m\leqno{(\global\def\currenvir{formule)}.$$
\global\def\currenvir{subsection\label{rel-iso} Let $\delta_i$ (resp. $s_j$) be the
face (resp. degeneracy) operators (see [De] (5.1.1) for instance) (by abuse of notation, we use the
same notation for $\delta_j,s_j$ and their image by ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_\bullet)$.
The category $(\Delta)$ is generated by the
face and degeneracy operators with the following relations (see for instance the proposition
VII.5.2 page 174 of [McL]) $$\matrix{\delta_i\kern 1pt{\scriptstyle\circ}\kern 1pt\delta_j&=&\delta_{j+1}\kern 1pt{\scriptstyle\circ}\kern 1pt\delta_i&\kern
.5cm&i\leq j\cr}\leqno{(\global\def\currenvir{formule)}$$\label{A}
$$\matrix{s_j\kern 1pt{\scriptstyle\circ}\kern 1pt s_i&=&s_i\kern 1pt{\scriptstyle\circ}\kern 1pt s_{j+1}&\kern
.5cm&i\leq j\cr}\leqno{(\global\def\currenvir{formule)}$$\label{B}
$$\left\{\matrix{ s_j\kern 1pt{\scriptstyle\circ}\kern 1pt\delta_i&=&\delta_i\kern 1pt{\scriptstyle\circ}\kern 1pt
s_{j-1}&\kern .5cm&i<j\cr &=&1&\kern .5cm&i=j,i=j+1\cr
&=&\delta_{i-1}\kern 1pt{\scriptstyle\circ}\kern 1pt s_j&\kern .5cm&i>j+1.\cr
}
\right.\leqno{(\global\def\currenvir{formule)}$$\label{C}
Therefore, the data of a cartesian ${\cal O}_{{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_\bullet}$-module $\L_\bullet$ is equivalent to the
data of isomorphisms $\alpha _j:\ \delta_j^*\L_n\buildrel\sim\over{\rightarrow}\L_{n+1},\ j=0,\ldots,n+1$ and
$\beta _j:\ s_j^*\L_{n+1}\buildrel\sim\over{\rightarrow}\L_{n},\ j=0,\ldots,n$ (where $n$ is a non negative integer) which
are compatible with the relations \ref{A}, \ref{B} and \ref{C}.
Let $n$ be a non negative integer.
\global\def\currenvir{subsection We have first to define for $j=0,\ldots,n+1$ an isomorphism $\alpha _j:\
\delta_j^*\L_n\buildrel\sim\over{\rightarrow}\L_{n+1}$. The line bundle $\delta_j^*\L_n$ is defined by the morphism $l\kern 1pt{\scriptstyle\circ}\kern 1pt
p_n\kern 1pt{\scriptstyle\circ}\kern 1pt\delta_j:\ {\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_{n+1}{\rightarrow} BG_m$ and $\tp_n\kern 1pt{\scriptstyle\circ}\kern 1pt\delta_j$ is associated to the map
$$\left\{\matrix{\Delta_0&{\rightarrow}&\Delta_{n+1}\cr 0&\longmapsto&\delta_j(0)\cr}\right.$$ If $j\not=0$,
one has therefore $\tp_n\kern 1pt{\scriptstyle\circ}\kern 1pt\delta_j=\tp_{n+1}$ and $\delta_j^*\L_n=L_{n+1}$. We define $\alpha
_j$ by the identity in this case.
Suppose now that $j=0$. Let $\pi_n:\ {\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_n{\rightarrow}{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_1$ be the projection on the $2$ first factors
(associated to the canonical inclusion $\Delta_1\hookrightarrow\Delta_n$. The commutativity of the
$2$ diagrams
$$\matrix{
{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_{n+1}&\fhd{\delta_0}{}&{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_{n}\cr\fvb{}{\pi_{n+1}}&&\fvb{p_{n}}{}\cr{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_1&\fhd{\delta_0}{}&{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}\cr}
\ {\rm and}\
\matrix{
{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_{n+1}&\fhd{p_{n+1}}{}&{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}\cr\fvb{}{\pi_{n+1}}&&\fvh{\delta_1}{}{}\cr{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_1&=&{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_1\cr}$$ allows to
reduce the problem to the construction of an isomorphism $$\delta_0^*\L\buildrel\sim\over{\rightarrow} \delta_1^*\L\ {\rm where}\
\delta_i,{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_1{\rightarrow}{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}\ i=0,1$$ are the face morphisms or, what is amounts to the same, to the construction
of a $2$-morphism $\nu:\ l\kern 1pt{\scriptstyle\circ}\kern 1pt \delta_0\Rightarrow l\kern 1pt{\scriptstyle\circ}\kern 1pt \delta_1$ (the morphism $\alpha _j$
will be $\alpha _j=\nu*\pi_{n+1}$). Now the diagram $$\matrix{ &&BG_m\cr &\llap{${}^{l\kern 1pt{\scriptstyle\circ}\kern 1pt
m}$}\nearrow&\fvh{}{l\kern 1pt{\scriptstyle\circ}\kern 1pt \delta_0}\cr \C\times\X&\fhd{pr_2\times m}{}&\X\times_\E\X\cr
&\llap{${}_{l\kern 1pt{\scriptstyle\circ}\kern 1pt pr_2}$}\searrow&\fvb{}{l\kern 1pt{\scriptstyle\circ}\kern 1pt \delta_1}\cr
&&BG_m\cr}\leqno{(\global\def\currenvir{formule)}$$ is strict commutative and $pr_2\times m$ is an equivalence by the
definition of a torsor. By the lemma \ref{lemme-categorie}, the $2$-morphism $\lambda$ induces a
canonical $2$-morphism $\lambda*(pr_2\times m)^{-1}:\ l\kern 1pt{\scriptstyle\circ}\kern 1pt \delta_0\Rightarrow l\kern 1pt{\scriptstyle\circ}\kern 1pt \delta_1$
which is the required $2$-morphism $\nu$.
\global\def\currenvir{subsection We have then to define for $j=0,\ldots,n$ an
isomorphism $\beta _j:\ s_j^*\L_{n+1}\buildrel\sim\over{\rightarrow}\L_{n}$. The line bundle $s_j^*\L$ is defined by the
morphism $l\kern 1pt{\scriptstyle\circ}\kern 1pt p_{n+1}\kern 1pt{\scriptstyle\circ}\kern 1pt s_j$ and $p_{n+1}\kern 1pt{\scriptstyle\circ}\kern 1pt s_j$ is associated
to the canonical inclusion $\Delta_0\hookrightarrow{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_n$ which means $p_{n+1}\kern 1pt{\scriptstyle\circ}\kern 1pt s_j=p_n$.
Therefore, $s_j^*\L_{n+1}=\L_n$ and we define $\beta_j$ to be the identity.
\global\def\currenvir{subsection We have to show that the data $\L_\bullet,\alpha _j,\beta_j,\ j\geq 0$ defines a line
bundle on the simplicial stack ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_\bullet$ as explained in (\ref{rel-iso}). Notice that the fact
that the definition of the $\beta_j$'s is compatible with the relations \ref{B} is tautological
($\beta_j$ is the identity on the relevant $\L_n$).
\global\def\currenvir{subsection Relation \ref{A}: in terms of $l$,
this relation means the following. We have the 2 stricltly commutative diagrams $$\alpha
_i\kern 1pt{\scriptstyle\circ}\kern 1pt(\delta_i*\alpha _j):\ l\kern 1pt{\scriptstyle\circ}\kern 1pt p_n\kern 1pt{\scriptstyle\circ}\kern 1pt \delta_j\kern 1pt{\scriptstyle\circ}\kern 1pt \delta_i\FFhd{\delta_i*\alpha
_j}{}l\kern 1pt{\scriptstyle\circ}\kern 1pt p_{n+1}\kern 1pt{\scriptstyle\circ}\kern 1pt\delta_i\FFhd{\alpha _i}{}l\kern 1pt{\scriptstyle\circ}\kern 1pt p_{n+2}$$ diagrams $$\matrix{
{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_{n+2}&\fhd{\delta_i}{}&{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_{n+1}&\fhd{\delta_j}{}&{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_n\cr &\llap{$\scp
p_{n+2}$}\searrow&\fvb{}{\scp p_{n+1}}&\swarrow\rlap{$\scp p_n$}{}\cr &&{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}&\fhd{l}{}BG_m\cr }\
{\rm and}\ \matrix{ {\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_{n+2}&\fhd{\delta_{j+1}}{}&{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_{n+1}&\fhd{\delta_i}{}&{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}_n\cr
&\llap{$\scp p_{n+2}$}\searrow&\fvb{}{\scp p_{n+1}}&\swarrow\rlap{$\scp p_n$}{}\cr
&&{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}&\fhd{l}{}BG_m\cr
}$$
inducing the two
$2$-morphisms
$$ \alpha _i\kern 1pt{\scriptstyle\circ}\kern 1pt(\alpha _j*\delta_i):\ l\kern 1pt{\scriptstyle\circ}\kern 1pt p_n\kern 1pt{\scriptstyle\circ}\kern 1pt \delta_j\kern 1pt{\scriptstyle\circ}\kern 1pt
\delta_i\FFhd{\alpha _j*\delta_i}{}l\kern 1pt{\scriptstyle\circ}\kern 1pt p_{n+1}\kern 1pt{\scriptstyle\circ}\kern 1pt\delta_i\FFhd{\alpha _i}{}l\kern 1pt{\scriptstyle\circ}\kern 1pt p_{n+2}$$
and
$$ \alpha _{j+1}\kern 1pt{\scriptstyle\circ}\kern 1pt(\alpha _i*\delta_{j+1}):\ l\kern 1pt{\scriptstyle\circ}\kern 1pt p_n\kern 1pt{\scriptstyle\circ}\kern 1pt \delta_i\kern 1pt{\scriptstyle\circ}\kern 1pt
\delta_{j+1}\FFhd{\alpha _i*\delta_{j+1}}{}l\kern 1pt{\scriptstyle\circ}\kern 1pt p_{n+1}\kern 1pt{\scriptstyle\circ}\kern 1pt\delta_{j+1}\FFhd{\alpha
_{j+1}}{}l\kern 1pt{\scriptstyle\circ}\kern 1pt p_{n+2}.$$ The relation \ref{A} means exactly the equality
$$\alpha _i\kern 1pt{\scriptstyle\circ}\kern 1pt(\alpha _j*\delta_i)=\alpha _{j+1}\kern 1pt{\scriptstyle\circ}\kern 1pt(\alpha _i*\delta_{j+1}),
\ i\leq j.\leqno(\ref{A}')$$
If $j=0$, the relation \ref{A}' is just by definition of $\alpha _j$ the condition \ref{lineA}
(see remark \ref{rem-line}).
If $j>0$, both the 2 isomorphisms $\alpha_j$ and $\alpha_{j+1}$ are the relevant identity and
the relation \ref{A}' is tautological.
\global\def\currenvir{subsection Relation \ref{C}: the only non tautological relation in (\ref{C}) is the associated
to the equality $s_0\kern 1pt{\scriptstyle\circ}\kern 1pt \delta_0=1$ in $(\Delta)$ which means as before that $\alpha
_0*\delta_0$ is the identity functor of $l\kern 1pt{\scriptstyle\circ}\kern 1pt p_n=l\kern 1pt{\scriptstyle\circ}\kern 1pt p_n\kern 1pt{\scriptstyle\circ}\kern 1pt\delta_0\kern 1pt{\scriptstyle\circ}\kern 1pt s_0$. But, this
is exactly the meaning of the relation \ref{lineB} (see remark \ref{rem-line}).
\section{Application to the Picard groups of some moduli spaces} Let us chose 3 integers
$r,s,d$ such that
$$r\geq 2\ {\rm and}\ s\mid r\mid ds.$$
If $G$ is the group $\mathop{\bf SL}\nolimits_r/\mu_s$ we denote as in [B-L-S] by
$\M_G(d)$ the moduli stack of $G$-bundles on $X$ and by $\M_{\mathop{\bf SL}\nolimits_r}(d)$ the moduli stack of rank
$r$ vector bundles and determinant ${\cal O}(d.x)$. If $r=s$ (i.e. $G={\bf PSL}_r$), the natural morphism
of algebraic stacks $$\pi:\ \M_{\mathop{\bf SL}\nolimits_r}(d){\rightarrow}\M_G(d)$$ is a ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_r$-torsor (see the corollary of
proposition 2 of [Gr] for instance). Let me explain how to deal with the general case.
\global\def\currenvir{subsection Let $E$ be a a rank
$r$ vector bundle on $X_S$ endowed with an isomorphism $\tau;\ D^{r/s}\buildrel\sim\over{\rightarrow}\det(E)$
where $D$ is some line bundle. Let me define the $\mathop{\bf SL}\nolimits_r/\mu_s$-bundle $\pi(E)$ associated to $E$
(more precisely to the pair $(E,\tau)$).
\th Definition
\enonce An $s$-trivialization of $E$ on the
\'etale neighborhood $T{\rightarrow} X_S$ is a triple $(M,\alpha ,\sigma)$ where $\alpha:\ D\buildrel\sim\over{\rightarrow} M^s$ is
an isomorphism ($M$ is a line bundle on $T$); $\sigma:\ M^{\oplus r}\buildrel\sim\over{\rightarrow} E_T$ is an isomorphism;
$\det(\sigma)\kern 1pt{\scriptstyle\circ}\kern 1pt \alpha^{r/s}:\ D^{r/s}\buildrel\sim\over{\rightarrow}\det(E)$ is equal to $\tau$.
Two $s$-trivializations $(M,\alpha ,\sigma)$ and $(M',\alpha' ,\sigma')$ of $E$ will be said
equivalent if there exists an isomorphism $\iota: M\buildrel\sim\over{\rightarrow} M'$ such that $\iota^s\kern 1pt{\scriptstyle\circ}\kern 1pt\alpha
=\alpha'$.
\endth
The principal homogeneous space
$$T\longmapsto \{\hbox{equivalence classes of $s$-trivializations of } E_T\}$$
defines the
$\mathop{\bf SL}\nolimits_r/\mu_s$-bundle $\pi(E)$\footnote{${}^\dagger$}{\sevenrm We see here a $\scp G$-bundle as a
formal homogeneous space under $\scp G$.}. Now, the construction is obviously functorial and
therefore defines the morphsim $\pi:\ \M_{\mathop{\bf SL}\nolimits_r}(d){\rightarrow} \M_G(d)$ (observe that an object $E$ of
$\M_{\mathop{\bf SL}\nolimits_r}(d)$ has determinant ${\cal O}({ds\over r}.x)^{r/s}$). Let $L$ be a line bundle and
$(M,\alpha,\tau)$ an $s$-trivialization of $E_T$. Then, $(M\otimes L,\alpha\otimes
\hbox{\rm Id}_{L^s},\sigma\otimes \hbox{\rm Id}_L)$ is an $s$-trivialization of $E\otimes L$ (which has determinant
($D\otimes L^s)^{r/s}$). This shows that there exists a canonical functorial isomorphism
$$\pi(E)\buildrel\sim\over{\rightarrow}\pi(E\otimes L)\leqno{(\global\def\currenvir{formule)}.$$\label{iso-pro} In particular,
$\pi$ is
${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s$-equivariant.
\th Lemma \enonce The natural morphism of algebraic stacks $\pi:\ \M_{\mathop{\bf SL}\nolimits_r}(d){\rightarrow}\M_G(d)$
is a ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s$-torsor. \endth
\label{pi-tors} {\it Proof}: let $E,E'$ be two rank $r$ vector
bundles on $X_S$ (with determinant equal to ${\cal O}(d.x)$) and let $\iota:\ \p(E)\buildrel\sim\over{\rightarrow}\p(E)'$ an
isomorphism. As in the proof of the lemma 13.4 of [B-L-S], we have the exact sequence of sets
$$1{\rightarrow}\mu_s{\rightarrow} {\rm Isom}(E,E'){\rightarrow}{\rm Isom}(\p(E),\p(E)')\fhd{\pi_{E,E'}}{} H^1_{\rm
\acute et}(X_S,\mu_s).$$ Let $L$ be a $\mu_s$-torsor such that $\pi_{E,E'}(\iota)=[L]$. Then,
$\p(E\otimes L)$ is canonically equal to $\p(E)$ and $\pi_{E\otimes L,E'}=0$ and $\iota$ is
induced by an isomorphism $E\otimes L\buildrel\sim\over{\rightarrow} E'$ well defined up to multiplication by $\mu_s$. The
lemma follows.\cqfd
\global\def\currenvir{subsection Let ${\cal U}}\def\d{{\cal D}}\def\F{{\cal F}$ be the universal bundle
on $X\times \M_{\mathop{\bf SL}\nolimits_r}(d)$. We would like to know which power of the determinant bundle $\d=(\det
R\Gamma{\cal U}}\def\d{{\cal D}}\def\F{{\cal F})^{-1}$
on $\M_{\mathop{\bf SL}\nolimits_r}(d)$ descends to $\M_G(d)$. As in I.3 of [B-L-S], the rank $r$ bundle
$\F=\L^{\oplus(r-1)}\oplus\L^{1-r}(d.x)$ on $X\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ has determinant ${\cal O}(d.x)$ and therefore
defines a morphism $$f:\ {\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)=JX\times BG_m{\rightarrow}\M_{\mathop{\bf SL}\nolimits_r}(d)$$ which is ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s$-equivariant.
The
vector bundle $\F'={\cal O}^{\oplus (r-1)}\oplus\L^{-r/s}(d.x)$ on $X\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ has determinant
$[\L^{-1}({ds\over r}.x)]^{r/s}$. The
$G$-bundle
$\pi(\F')$ on $X\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ defines a morphism $f':\ {\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}{\rightarrow}\M_G(d)$. The relation
$\L\otimes (\hbox{\rm Id}_X\times s_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H})^**(\F')=\F$ and (\ref{iso-pro}) gives an isomorphism
$\pi(\F)=(\hbox{\rm Id}_X\times s_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H})^*\pi(\F')$ which means that the diagram
$$\matrix{
{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)&\fhd{f}{}&\M_{\mathop{\bf SL}\nolimits_r}(d)\cr
\fvb{}{s_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}}&&\fvb{}{\pi}\cr
{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)&\fhd{f'}{}&\M_G(d)\cr}\leqno{(\global\def\currenvir{formule)}$$\label{dia-rest} is $2$-commutative.
Exactly as in I.3 of [B-L-S], let me prove the
\th Lemma
\enonce The line bundle $f^*\d^k$ on ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ descends through $s_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}$ if and only if $k$
multiple of $s/(s,r/s)$.
\endth\label{lem-rest}
{\it Proof}: let
$\chi=r(g-1)-d$ be the opposite of the Euler characteristic of ($\k$-)points of $\M_{\mathop{\bf SL}\nolimits_r}(d)$. By
(\ref{iso-det}), one has an isomorphism
$f^*\d^k\buildrel\sim\over{\rightarrow}\Theta^{kr(r-1)}\otimes\T^{k\chi}.$
The theory of Mumford groups says that $\Theta^{kr(r-1)}$ descends through $s_J$ if and only if $k$
is a multiple of $s/(s,r/s)$. The line bundle $\T^{k\chi}$ on $BG_m$ descends through $s_{BG_m}$ if
and only if $k\chi$ is a multiple of $s$. The lemma follows from the above isomorphism and from the
observation that the condition
$s\mid r\mid ds$ forces $s\chi$ to be a multiple of $s$.\cqfd
\global\def\currenvir{rem\label{rema} If $g=0$, the jacobian $J$ is a point and the condition on
$\Theta$ is empty. The only condition is in this case being $k\chi$ multiple
of
$s$.
Let me recall that $\d$ is the determinant bundle on $\M_{\mathop{\bf SL}\nolimits_r}(d)$ and $G=\mathop{\bf SL}\nolimits_r/\mu_s$.
\th Theorem
\enonce Assume that the characteristic of $\k$ is $0$. The integers $k$ such
that $\d^k$ descends to $\M_G(d)$ are the multiple of $s/(s,r/s)$.
\endth\label{theo-pic}
{Proof}: by lemma \ref{lem-rest} and diagram (\ref{dia-rest}), we just have to proving that $\d^k$
efectively descends where $k=s/(s,r/s)$. By theorem \ref{theo} and lemma \ref{pi-tors}, this means
exactly that $\d^k$ has a ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s$-linearization. We know by lemma \ref{lem-rest} that the pull-back
$f^*\d^k$ has such a linearization.
\th Lemma
\enonce The pull-back morphism $\mathop{\rm Pic}\nolimits({\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times\M_{SL_r}(d)){\rightarrow}\mathop{\rm Pic}\nolimits({\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X))$ is injective.
\endth
{\it Proof}: by lemma \ref{torseur}, one is reduced to prove that the natural morphism
$$\mathop{\rm Pic}\nolimits(B\mu_s\times\M_{SL_r}(d)){\rightarrow}\mathop{\rm Pic}\nolimits(B\mu_s\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X))$$ is injective. Let ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$ be any stack.
The canonical morphism ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}{\rightarrow}{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}\times B\mu_s$ is a $\mu_s$-torsor (with the trivial action of
$\mu_s$ on ${\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}$). By theorem \ref{theo}, one has the equality $$\mathop{\rm Pic}\nolimits({\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}\times
B\mu_s)=\mathop{\rm Pic}\nolimits^{\mu_s}({\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}).$$ Assume further that
$H^0({\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M},{\cal O})=\k$. The later group is then canonically isomorphic to
$$\mathop{\rm Pic}\nolimits({\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M})\times\mathop{\rm Hom}\nolimits(\mu_s,G_m)=\mathop{\rm Pic}\nolimits({\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M})\times\mathop{\rm Pic}\nolimits(B\mu_s).$$ All in all, we get a functorial
isomorphism $$\iota_{\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}:\ \mathop{\rm Pic}\nolimits({\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M}\times
B\mu_s)\buildrel\sim\over{\rightarrow}\mathop{\rm Pic}\nolimits({\cal X}}\def\C{{\cal G}}\def\L{{\cal L}}\def\E{{\cal Y}}\def\m{{m_\C}}\def\M{{\cal M})\times\mathop{\rm Pic}\nolimits(B\mu_s).\leqno(\global\def\currenvir{formule)\label{pic-pro}$$ By [L-S], the Picard group
of $\M_{\mathop{\bf SL}\nolimits_r}(d)$ is the free abelian group ${\bf} Z}\def\Q{{\bf} Q}\def\C{{\bf} C}\def\F{{\bf} F.\d$ and the formula (\ref{iso-det})
proves that $$f^*:\ \mathop{\rm Pic}\nolimits(\M_{\mathop{\bf SL}\nolimits_r}(d)){\rightarrow}\mathop{\rm Pic}\nolimits({\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X))$$ is an injection. The lemma folows from the
commutative diagram
$$\diagram{
\mathop{\rm Pic}\nolimits(\M_{\mathop{\bf SL}\nolimits_r}(d))\times\mathop{\rm Pic}\nolimits(B\mu_s)&\hookrightarrow&\mathop{\rm Pic}\nolimits({\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X))\times\mathop{\rm Pic}\nolimits(B\mu_s)\cr
\fvb{\iota_\M}{\wr}&&\fvb{\iota_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}}{\wr}\cr
\mathop{\rm Pic}\nolimits(\M_{\mathop{\bf SL}\nolimits_r}(d)\times B\mu_s)&{\rightarrow}&\mathop{\rm Pic}\nolimits({\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)\times B\mu_s)\cr
}$$\cqfd
Let $\H$ (resp. $\H_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}$) be the line bundle on ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times\M_{\mathop{\bf SL}\nolimits_r}(d)$ (resp.
${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$
$$\H=\mathop{{\cal H}om}\nolimits(m_\M^*\d^k,pr_2^*\d^k)\ {\rm resp.}\
\H_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}=\mathop{{\cal H}om}\nolimits(m_\M^*f^*\d^k,pr_2^*f^*\d^k).$$
Let us chose a ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s$-linearization $\lambda_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}$ of $f^*\d^k$. It defines a trivialization of the line
bundle $\H_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}$.
The equivariance of $f$ implies (cf. \ref{equi}) that
there exists a (compatible) $2$-morphism
$$q:\ m_\M\kern 1pt{\scriptstyle\circ}\kern 1pt(\hbox{\rm Id}\times f)\Rightarrow f\kern 1pt{\scriptstyle\circ}\kern 1pt m_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}$$
making the diagram
$$ \matrix{ {\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)&\fhd{m_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}}{}&{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)\cr \fvb{{\rm Id}\times
f}{}&&\fvb{f}{}\cr {\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times\M_{SL_r}(d)&\fhd{m_\M}{}&\M_{SL_r}(d)\cr }$$
$2$-commutative. The $2$-morphism $q$ defines an isomorphism from the pull-back $m_\M^*\d^k$ on
${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ to $m_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}^*(f^*\d^k)$. The pull-back of $pr_2^*\d^k$ on ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ is
tautologically isomorphic to $pr_2^*(f^*\d^k)$. The
preceding isomorphisms induce an isomorphism $$(\hbox{\rm Id}\times f)^*\H\buildrel\sim\over{\rightarrow}\H_J.$$ The later line
bundle being trivial, so is $(\hbox{\rm Id}\times f)^*\H$. The lemma above proves therefore that $\H$
istself is {\it trivial} . Each ($\k$-)point $j$ of $J_s$ defines a morphism
$\M_{SL_r}(d){\rightarrow}{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times\M_{SL_r}(d)$ (resp. ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X){\rightarrow}{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$); let me denote by $\H_j$
(resp. $f^*\H_j$) the pull-back of $\H$ (resp. $(\hbox{\rm Id}\times f)^*\H)$ by this morphism. The pull-back
morphism $$H^0({\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times\M_{\mathop{\bf SL}\nolimits_r}(d),\H){\rightarrow} H^0({\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X),(\hbox{\rm Id}\times f)^*\H)$$ can be
identified to the direct sum $$\oplus_{j\in{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s(\k)}H^0(\M_{\mathop{\bf SL}\nolimits_r}(d),\H_j){\rightarrow}
H^0({\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X),f^*\H_j).$$
Because
$$H^0(\M_{\mathop{\bf SL}\nolimits_r}(d),{\cal O})=H^0({\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X),{\cal O})=\k\leqno{(\global\def\currenvir{formule)},\label{cons}$$ this morphism is a direct
sum of non-zero morphisms of vector spaces of dimension $1$ and therefore an isomorphism. In
particular, a linearization $\lambda_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}$ of $f^*\d^k$ defines canonicaly an isomophism $$\lambda_\M:\
m_\M^*\d^k\buildrel\sim\over{\rightarrow} pr_2^*\d^k$$ such that $(\hbox{\rm Id}\times f)^*\lambda_\M=\lambda_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}$.
Explicitely, $\lambda_\M$ is
characterized as follows: let $x$ be an object of $\M_{\mathop{\bf SL}\nolimits_r}(d)$ over a connected scheme $S$ and
$g$ an object of ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s(S)={\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s(\k)$. The preceding dicussion means that the functorial
isomorphisms $$\lambda_\M(g,x):\ \d^k_{g.x}\buildrel\sim\over{\rightarrow}\d^k_x$$ are determined when $x$ lies in the essential
image of $f$. In this case, let us chose an isomorphism $f(x')\buildrel\sim\over{\rightarrow} x$ (inducing an isomorphism
$g.f(x')\buildrel\sim\over{\rightarrow} g.x$).
Then, the diagram of isomorphisms of line bundles on $S$
$$\matrix{
L'_{x'}=L_{f(x')}&{\rightarrow}&L_x\cr
\fvh{\lambda_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(g,x')}{}&&&\llap{$\scp\lambda_\M(g,x)$}\nwarrow&\cr
L'_{g.x'}=L_{f(g.x')}&\fhd{q_{g,x'}}{}&L_{g.f(x')}&{\rightarrow}&L_{g.x}\cr
}$$ is commutative (where $L=\d^k$ and $L'=f^*\d^k$).
Now, the pull-back of $\lambda_\M$ on
${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}_s\times{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$ satisfies conditions \ref{lineA} and \ref{lineB}. Using (\ref{cons}) and the
equivariance of of $f$ as above, this shows that $\lambda_\M$ is a linearization. For instance,
keeping the notation above, let us check the condition \ref{lineB}. We have to check that the
isomorphism $\iota$ of $L$ induced by $\epsilon$ is the identity. As above, it is enough to check
that on ${\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(X)$. By definition, with a slight abuse of notations, the diagrams
$$\matrix{
L'_{x'}=L_{f(x')}&{\rightarrow}&L_x\cr
\fvh{\lambda_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(1,x')}{}&&&\llap{$\scp\lambda_\M(1,x)$}\nwarrow&\cr
L'_{1.x'}=L_{f(1.x')}&\fhd{q_{1,x'}}{}&L_{1.f(x')}&{\rightarrow}&L_{1.x}\cr
}\kern .5cm {\rm and}\kern .5cm
\matrix{
L_x&\fhd{\iota}{}&L_x\cr
&\llap{$\scp\lambda_\M(1,x)$}\nwarrow&\fvh{\epsilon(x)}{}\cr
&&L_{1.x}\cr
}$$
Because $\lambda_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}$ is a linearization, condition \ref{lineB} gives the commutative diagram
$$\matrix{
L'_{x'}&=&L'_{x'}\cr
&\llap{$\scp\lambda_{\cal J}}\def\p{{\pi}}\def\k{{\bf k}}\def\P{{\cal P}}\def\T{{\cal T}}\def\H{{\cal H}(1,x')$}\nwarrow&\fvh{}{\epsilon'(x')}\cr
&&L_{1.x'}\cr
}$$ showing that the equality $\iota=\hbox{\rm Id}$ remains to prove the commutativity of the diagram
$$\matrix{
L_{f(1.x')}&\fhd{\epsilon'}{}&L_{f(x')}\cr
\fvb{q_{1,x'}}{}&&\parallel\cr
L_{1.f(x')}&\fhd{\epsilon}{}&L_{f(x')}\cr
}
$$
But this follows from the commutativity of the diagram
$$\matrix{
f(1.x')&\fhd{\epsilon'}{}&f(x')\cr
\fvb{q_{1,x'}}{}&&\parallel\cr
1.f(x')&\fhd{\epsilon}{}&f(x')\cr
}
$$ which is by definition the fact that $q$ is compatible to $\epsilon$ as required in
(\ref{equi}). One would check condition \ref{lineA} in an analogous way.\cqfd
\global\def\currenvir{rem\label{rem}In the case $g=0$, the condition is an in remark \ref{rema}.
\global\def\currenvir{rem This linearization can be certainly also deduced from a careful
analysis of the first section of [Fa], but the method above seems simpler.
\bigskip
\centerline{\bf References}
\medskip
[B-L-S] A.Beauville, Y. Laszlo, C. Sorger, {\it The Picard group of the moduli of $G$-bundles on
a curve}, preprint alg-geom/9608002.
\medskip
[Br] L. Breen, {\it Bitorseurs et cohomologie non ab\'elienne
}, in The Grothendieck Festschrift I, Progr. Math. {\bf 86},
Birkh\"auser (1990), 401-476.
\medskip
[De] P. Deligne, {\it Th\'eorie de Hodhe III}, Publ. Math.
I.H.E.S. {\bf 44} (1974), 5-78.
\medskip
[Fa] G. Faltings, {\it Stable $G$-bundles and Projective Connections}, JAG {\bf
2} (1993), 507-568.
\medskip
[Gr] A. Grothendieck, {\it G\'eom\'etrie formelle et g\'eom\'etrie
alg\'ebrique}, FGA, S\'em. Bourbaki {\bf 182} (1958/59), 1-25.
\medskip
[L-M] G. Laumon, L. Moret-Bailly, {\it Champs alg\'ebriques}, preprint
Universit\'e Paris-Sud (1992).
\medskip
[L-S] Y. Laszlo, C. Sorger, {\it The line bundles on the moduli of parabolic $G$-bundles over
curves and their sections}, preprint alg-geom/9507002.
\bigskip
[McL] S. Mac Lane, {\it Categories for the working mathematician}, GTM {\bf 5}, Springer-Verlag (1971).
\bigskip
\hfill\hbox to 5cm{\hfill Y. {\pc LASZLO}\hfill}
\smallskip
\hfill\hbox to 5cm{\hfill DMI - \'Ecole Normale Sup\'erieure\hfill}
\smallskip
\hfill\hbox to 5cm{\hfill ( URA 762 du CNRS )\hfill}
\smallskip
\hfill\hbox to 5cm{\hfill 45 rue d'Ulm\hfill}
\smallskip
\hfill\hbox to 5cm{\hfill F-75230 {\pc PARIS} Cedex 05\hfill}
\smallskip
\closeall\bye
|
1996-10-09T20:46:51 | 9610 | alg-geom/9610008 | en | https://arxiv.org/abs/alg-geom/9610008 | [
"alg-geom",
"math.AG"
] | alg-geom/9610008 | Jim Bryan | Jim Bryan and Marc Sanders | The Rank Stable Topology of Instantons on $\cpbar$ | LaTeX2e | null | null | null | null | Let $\M_{k}^{n}$ be the moduli space of based (anti-self-dual) instantons on
$\cpbar$ of charge $k$ and rank $n$. There is a natural inclusion of rank $n$
instantons into rank $n+1$. We show that the direct limit space is homotopy
equivalent to $BU(k)\times BU(k)$. The moduli spaces also have the following
algebro-geometric interpretation: Let $\linf$ be a line in the complex
projective plane and consider the blow-up at a point away from $\linf$. $\M
_{k}^{n}$ can be described as the moduli space of rank $n$ holomorphic bundles
on the blownup projective plane with $c_{1}=0$ and $c_{2}=k$ and with a fixed
holomorphic trivialization on $\linf$.
| [
{
"version": "v1",
"created": "Wed, 9 Oct 1996 18:41:28 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Bryan",
"Jim",
""
],
[
"Sanders",
"Marc",
""
]
] | alg-geom | \section{Introduction}
In his 1989 paper \cite{Ta}, Taubes studied the stable topology of the
based instanton moduli spaces. He showed that if ${\mathcal{M}}_{k}^{n}(X)$ denotes
the moduli space of based $SU(n)$-instantons of charge $k$
on $X$, then there is a map ${\mathcal{M}} _{k}^{n}(X)\to {\mathcal{M}} _{k+1}^{n}(X)$ and,
in the direct limit topology, ${\mathcal{M}} _{\infty }^{n}(X)$ has the homotopy type
of $\operatorname{Map}_{0}(X,BSU(n))$.
There is also a map ${\mathcal{M}} _{k}^{n}(X)\hookrightarrow {\mathcal{M}} _{k}^{n+1}(X)$ given by
the direct sum of a connection with the trivial connection on a trivial
line bundle and one can consider the direct limit ${\mathcal{M}} _{k}^{\infty }(X)$.
For the case of
$X=S^{4}$ with the round metric, it was shown by Kirwan and also by Sanders
(\cite{Kir},\cite{Sa}) that the direct limit has the homotopy type of
$BU(k)$.
In this note we consider the case of $X=\cpbar $ where $\cpbar $ denotes
the complex projective plane with the Fubini-Study metric and the opposite
orientation of the one induced by the complex structure. Our result is:
\begin{theorem}\label{thm: main result}
${\mathcal{M}} _{k}^{\infty }(\cpbar )$ has the homotopy type of $BU(k)\times BU(k)$.
\end{theorem}
The main tool in the proof of the theorem is a construction of the moduli
spaces ${\mathcal{M}} _{k} ^{n}(\cpbar )$ due to King \cite{Ki}. In general,
Buchdahl \cite{Bu} has shown that, for appropriate metrics on the
$N$-fold connected sum $\# _{N}\cpbar $, the moduli spaces ${\mathcal{M}} _{k}^{n}(\#
_{N}\cpbar )$ are diffeomorphic to certain spaces of equivalence classes of
holomorphic bundles on $\cnums \P^{2}$ blown-up at $N$ points. The
universal $U(k)\times U(k) $ bundle that appears giving the homotopy
equivalence of theorem \ref{thm: main result} can be constructed as
higher direct image bundles (see section \ref{section:proof}).
\begin{rem}\label{rem:motivation}
The cofibration $S^2\to \cpbar \to S^4$ gives rise to the fibration
of mapping spaces $\Omega^4BSU(n)\to {\rm Map}_{\ast}(\cpbar , BSU(n))\to
\Omega^2BSU(n)$ which for K-theoretic reasons is a trivial fibration in the
limit over $n$. The total space of this fibration is homotopy equivalent to
the space of based gauge equivalence classes of all connections on
$\cpbar$. Thus,
from Taubes' result, ${\mathcal{M}}_k^\infty $
must have
the property that taking the limit over $k$ gives $BU\times BU$. For $S^4$,
similar remarks
imply that $\lim_{k\to \infty} {\mathcal{M}} _{k} ^{\infty}(S^4)\simeq BU$ and the
inclusion
of ${\mathcal{M}} _k ^{\infty} (S^4)$ into this limit has been shown to be (up to
homotopy) the
natural inclusion $BU(k)\hookrightarrow BU$ (\cite {Sa}).
Theorem 1.1
and these results for $S^4$ suggest a general conjecture which is supported
by the fact that the higher direct image bundle giving our homotopy equivalence
generalizes in an appropriate way.
\end{rem}
\begin{conj}\label{conj:N fold connected sum}
For appropriate metrics on $\# _{N}\cpbar $, ${\mathcal{M}} _{k}^{\infty }(\#
_{N}\cpbar )$ has the homotopy type of a product $BU(k)\times \cdots
\times BU(k)$ with $N+1$ factors.
\end{conj}
\begin{rem}\label{rem:Bott periodicity}
Combining theorem \ref{thm: main result} with Taubes' stabilization result
leads to an alternate proof of Bott periodicity for the unitary group.
There is a natural map from instantons on $S^4$ to those on $\cpbar$ (given by
pull-back) which in the limit as $n\to \infty$ is homotopy equivalent to the
diagonal $BU(k) \to BU(k)\times BU(k)$. Taking the limit as $k\to\infty$ and
applying Taubes' result, the diagonal map appears as the inclusion of fibers
in
the fibration $BU\simeq\Omega_k^4BU\to BU\times BU\to \Omega^2(BSU)$ (see
remark 1.1).
However, the map $BU\rightarrow BU \times BU$ has homotopy fiber $U\times U
\bigg/ U\: \simeq U$,
and therefore, given the above fibration we must have
$U\simeq\Omega^3BSU\simeq\Omega^2 SU\simeq\Omega^2 U$.
Tian (\cite {Ti})
noticed that results for limits of instantons on $S^4$ (see \cite {Kir} or
\cite {Sa})
already imply the four-fold periodicity $\Omega^4BU\simeq Z\times BU$. The
${\cpbar}$ case thus gives the
finer two-fold periodicity $\Omega^2BU\simeq Z\times BU$, as one might expect
due to the nontrivial $S^2\hookrightarrow {\cpbar}$.
In a future paper we will study limits of $Sp(n)$ and $SO(n)$ instantons
on $\cpbar$ and their relationships to those on $S^4$. As an amusing
corollary, we will be able to rederive many of the
Bott periodicity
relationships among
$Sp$, $U$, $SO$, and their homogeneous spaces.
\end{rem}
\section{The construction of ${\mathcal{M}} _{k}^{n}(\cpbar )$}
Let $x_{0}\in \cpbar $ be the base point. Since $\cpbar -\{x_{0} \}$ is
conformally equivalent to $\til{\cnums} ^{2}$, the complex plane
blown-up at the origin, ${\mathcal{M}} _{k}^{n}(\cpbar )$ can
be regarded as instantons on $\til{\cnums} ^{2}$ based ``at infinity''.
Buchdahl \cite{Bu} proved an analogue in this non-compact setting of
Donaldson's theorem relating instantons to holomorphic bundles: Let
$\til{\cnums}^{2}_{N}$ be the complex plane blown-up at $N$ points with a
K\"ahler metric. Then $\til{\cnums}^{2}_{N}$ has a ``conformal
compactification'' to $\#_{N}\cpbar$ and a ``complex compactification'' to
$\til{\cnums \P}^{2}_{N}$ (the projective plane blown-up at $N$ points).
We have added a point
$x_{0}$ in the former case and a complex projective line $\ell_{\infty }$ in the latter.
Define $\msalg{k}{n}(\til{\cnums \P}^{2}_{N})$ to be the moduli space
consisting of pairs $({\mathcal{E}},\tau
)$ where ${\mathcal{E}} $ is a rank $n$ holomorphic bundle on $\til{\cnums \P}
^{2}_{N}$ with $c_{1}({\mathcal{E}} )=0$, $c_{2}({\mathcal{E}} )=k$, and where $\tau :{\mathcal{E}}
|_{\ell_{\infty }}\to \cnums ^{n}\otimes \O _{\ell_{\infty }}$ is a
holomorphic trivialization of ${\mathcal{E}} $ on $\ell_{\infty }$.
There is a natural map
$\Phi:{\mathcal{M}}_{k}^{n}(\# _{N}\cpbar )\longrightarrow\msalg{k}{n}(\til{\cnums
\P}^{2}_{N})$ defined as follows. Let
$p:\til{\cnums \P}^{2}_{N}\to \# _{N}\cpbar $ be the map that collapses
$\ell_{\infty }\mapsto x_{0}$. If
$[A]\in\ms{k}{n}$ then the $\bar\partial$ operator that defines the holomorphic bundle
$\mathcal{V}=\Phi(A)$ is taken to be $(d_{p^{*}(A)})^{(0,1)}$,
the anti-holomorphic part
of the covariant derivative defined by the pullback of the connection. The
anti-self-duality of $A$ implies that the curvature of $p^{*}(A)$ is a
$(1,1)$-form and so $\bar\partial ^{2}=0$.
Buchdahl's theorem is then
\begin{theorem} \label{thm:buchdahl}
The map $\Phi:{\mathcal{M}}_{k}^{n}(\# _{N}\cpbar
)\longrightarrow\msalg{k}{n}(\til{\cnums \P}^{2}_{N})$ is a
diffeomorphism.
\end{theorem}
The case $N=1$ was first proved by King \cite{Ki}. We
now restrict ourselves to that case and simply write
$\ms{k}{n}$ for ${\mathcal{M}}_{k}^{n}(\cpbar )$ and $\msalg{k}{n}(\til{\cnums \P}^{2})$.
King constructed $\ms{k}{n}$ explicitly in terms of linear algebra data.
We recall his construction. Consider configurations of linear maps:
\begin{picture}(100,75)(-135,0)
\put(20,50){$W_0$}
\put(56,15){\vector(-2,3){21}}
\put(55,5){$V_{\infty}$}
\put(36,52){\vector(1,0){47}}
\put(83,56){\vector(-1,0){47}}
\put(84,50){$W_1$}
\put(86,45){\vector(-2,-3){19}}
\put(35,27){$b$}
\put(56,45){$x$}
\put(48,60){$a_1,a_2$}
\put(81,27){$c$}
\end{picture}
\noindent
where $W_0$, $W_1$ and $V_{\infty }$ are complex vector spaces of
dimensions $k$, $k$, and $n$ respectively.
A configuration $(a_1,a_2,b,c,x)$ is called
{\em integrable} if it satisfies the equation
$$a_1xa_2-a_2xa_1+bc=0.$$
A configuration $(a_1,a_2,b,c,x)$ is {\em non-degenerate} if it satisfies
the following conditions:
$$\forall \hspace{4pt}
(\lambda_1,\lambda_2),(\mu_1,\mu_2)\in\cnums^2 \text{ such that
}\lambda_1\mu_1 + \lambda_2\mu_2=0\text{ and }(\mu_1,\mu_2)\neq(0,0),$$
\begin{eqnarray*}
\exists\hspace{-6pt}\raisebox{1.5pt}{/} v\in W_1\text{ such that } &&\left\{
\begin{array}{lr}
xa_1v=\lambda_1v & (\mu_1 a_1 + \mu_2 a_2)v=0 \\
xa_2v=\lambda_2v & cv=0
\end{array}
\right. \\
\text{and }\exists\hspace{-6pt}\raisebox{1.5pt}{/} w\in W_0^\star\text{ such that } &&\left\{
\begin{array}{lr}
x^*a_1^*w=\lambda_1w & (\mu_1 a^*_1 + \mu_2 a^*_2)w=0 \\
x^*a_2^*w=\lambda_2w & b^*w=0
\end{array}
\right.
\end{eqnarray*}
Let $A_{k}^{n}$ be the space of all integrable
non-degenerate configurations. $G=Gl(W_{0})\times Gl(W_{1})$ acts
canonically on $A_k^n$. The action is explicitly given by
$$
(g_{0},g_{1})\cdot (a_{1},a_{2},b,c,x)=
(g_{0}a_{1}g_{1}^{-1},g_{0}a_{2}g_{1}^{-1},
g_{0}b,cg_{1}^{-1},g_{1}xg_{0}^{-1})
$$
\begin{theorem}\label{thm: monad description}
The moduli space $\ms{k}{n}$ is isomorphic to
$A_{k}^{n}/G$.
\end{theorem}
\proof
King uses such configurations
to determine monads that in turn
determine holomorphic bundles. Configurations in the same $G$
orbit determine the same bundle. For the sake of brevity we refer the
reader to \cite{Ki} or \cite{Br} for details. The construction identifies the
vector spaces $W_0$ and $W_{1}$ canonically as $H^1(\mathcal{E}(-\ell_{\infty }))$
and $H^{1}(\mathcal{E}(-\ell_{\infty }+E))$ respectively, where $E\subset \til{\cnums
\P}^{2}$ is the exceptional divisor. The vector space $V_{\infty } $ is
identified with the fiber over $\ell_{\infty }$.
\section{Proof of theorem \ref{thm: main result} }\label{section:proof}
We prove the theorem in two steps: We first show that the space of monad
data $A_{k}^{n}$ forms a principal $G=Gl(k)\times Gl(k)$ bundle over ${\mathcal{M}}
_{k}^{n}$. We then show that the induced $G$-equivariant inclusion
$A_k^n\hookrightarrow A_k^{n+2k}$ is null-homotopic so that we can conclude
that $A_{k}^{\infty }$ is contractible.
\begin{lem}\label{lem:freeness of G action}
$G$ acts freely on the space of monad data $A_{k}^{n}$.
\end{lem}
\begin{proof}
This is essentially proved in \cite{Ki} where it is implicitly shown that
the non-degeneracy conditions are precisely the conditions that guarantee
freeness. We point out that this also follows more conceptually from the
existence of a universal family $\Bbb{E}\to \ms{k}{n}\times \til{\cnums
\P} ^{2}$ and the cohomological
interpretation of $W_{0}$ and $W_{1}$:
First, the existence of a universal family can be shown via the gauge
theoretic construction:
Let $V$ be a smooth hermitian vector bundle on $\til{\cnums \P}^{2}$ with
$c_{1}(V)=0$ and $c_{2}(V)=k$. Let $\mathcal{A}^{1,1}_{0}$ denote
unitary connections on $V$
with curvature of pure type $(1,1)$ and that restrict to the trivial
connection on $\ell_{\infty } $ and let $\mathcal{G}_{0}^{\cnums }$ denote the
complex gauge transformations of $V$ that are the identity restricted to
$\ell_{\infty } $.
Then $\ms{k}{n}=\mathcal{A}^{1,1}_{0}/\mathcal{G}^{\cnums }_{0}$. The
quotient
$$(\mathcal{A}^{1,1}_{0}\times V )/\mathcal{G}^{\cnums }_{0}\to {\mathcal{M}}
_{k}^{n}\times \til{\cnums \P}^{2}$$
will form a universal bundle if the moduli space is smooth and no ${\mathcal{E}} \in
{\mathcal{M}} _{k}^{n}$ has non-trivial automorphisms ({\em c.f.} \cite{Fr-Mo} Chapt. IV):
\begin{lem}\label{lem:E has no finite automorphisms, moduli space is smooth}
${\mathcal{M}} _{k}^{n}$ is smooth and any ${\mathcal{E}} \in {\mathcal{M}} _{k}^{n}$ has no non-trivial
automorphisms preserving $\tau :{\mathcal{E}} |_{\ell_{\infty }} \to \cnums ^{n}\otimes
\mathcal{O}_{\ell_{\infty } }$.
\end{lem}
By Serre duality $H^{2}({\mathcal{E}} \otimes {\mathcal{E}}^{*} )=H^{0}({\mathcal{E}} \otimes {\mathcal{E}} ^{*}\otimes
K)^{*}$. Since ${\mathcal{E}} \otimes {\mathcal{E}} ^{*}$ is trivial on $\ell_{\infty } $, it is
also trivial on nearby lines. Any section of ${\mathcal{E}} \otimes {\mathcal{E}}^{*}
\otimes K$ restricts to a section of $\cnums ^{n^{2}}\otimes
\mathcal{O}_{\ell_{\infty } }(-3)$ and so must
vanish on $\ell_{\infty } $. Likewise, it must vanish on nearby lines and so it is
$0$ on an open set and must be identically $0$. Thus $H^{2}({\mathcal{E}}\otimes {\mathcal{E}}
^{*})=0$ and smoothness follows once we show there are no automorphisms.
Suppose that there exists an automorphism $\phi \in H^{0}({\mathcal{E}} \otimes {\mathcal{E}}
^{*})$ such that $\phi \neq {{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l $ and $\phi $ preserves $\tau $ so that
$\phi |_{\ell_{\infty } }={{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l |_{\ell_{\infty } }$. Then $\phi -{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l $ is a non-zero section of
${\mathcal{E}} \otimes {\mathcal{E}}^{*} $
vanishing on $\ell_{\infty } $. We then get an injection $0\to \O(\ell_{\infty } )\to {\mathcal{E}}
\otimes {\mathcal{E}} ^{*}$. Restricting this sequence to $\ell_{\infty } $ we get an injection
$0\to \O_{\ell_{\infty } }(1)\to \O_{\ell_{\infty } }\otimes \cnums ^{n^{2}}$
which is a contradiction.
Let $\pi :\ms{k}{n}\times \til{\cnums\P }^{2}\to \ms{k}{n}$. The
higher direct image sheaves $R^{1}\pi _{*}(\Bbb{E}(-\ell_{\infty }))$ and $R^{1}\pi
_{*}(\Bbb{E}(-\ell_{\infty }+E))$ are locally free and rank $k$. This follows from
the index theorem and the vanishing of the $H^{0} $ and $H^{2}$
cohomology of ${\mathcal{E}} (-\ell_{\infty } )$ and ${\mathcal{E}} (-\ell_{\infty } +E)$. The $H^{0}$ vanishing follows
by again considering the restriction of a section of the bundles to lines
nearby to $\ell_{\infty } $. Using Serre duality and the same argument, one gets the
vanishing for $H^{2}$.
Thus the vector spaces $W_{0}$ and $W_{1}$ are the fibers of the vector
bundles $R^{1}\pi_{*}(\Bbb{E}(-\ell_{\infty }))$ and
$R^{1}\pi_{*}(\Bbb{E}(-\ell_{\infty }+E))$. The $G$-orbit of a configuration giving a
bundle ${\mathcal{E}}$ can be identified with the group of isomorphisms
$g_0:H^1({\mathcal{E}}(-\ell_{\infty }))\to
\cnums^k$ and $g_1:H^{1}({\mathcal{E}}(-\ell_{\infty }+E))\to \cnums^k$. Thus
$A_{k}^{n}$ is realized precisely as the total space of the
principal $Gl(k)\times Gl(k)$ bundle associated to
$R^{1}\pi_{*}(\Bbb{E}(-\ell_{\infty }))\oplus R^{1}\pi_{*}(\Bbb{E}(-\ell_{\infty }+E))$.
\end{proof}
Recall that the map $\ms{k}{n}\hookrightarrow \ms{k}{n+1}$ is defined
by the direct sum with the trivial connection: $[A]\mapsto [A\oplus \theta]$.
In terms of holomorphic bundles this is ${\mathcal{E}} \mapsto {\mathcal{E}} \oplus \mathcal{O}$.
Tracing through the monad construction, it is easy to see that the
inclusion induces the $G$-equivariant map $A_k^n\hookrightarrow A_k^{n+1}$
given by $(a_{1},a_{2},x,b,c)\mapsto (a_{1},a_{2},x,b',c')$ where
$b^{\prime}$ is $b$ with an extra first column of zeroes and $c^{\prime}$ is
$c$ with an extra first row of zeroes.
Define $A_k^{\infty}$ to be the direct
limit $\lim_{n\to\infty}A_k^n$ so that there is a homeomorphism between ${\mathcal{M}}
_k^{\infty}$ and $A_k^{\infty}/G$
\begin{lem}
$A_k^{\infty}$ is a contractible space.
\end{lem}
\begin{proof}
Since the $A_k^{n}$'s are
algebraic varieties and the maps
$A_k^n\to A_k^{n+1}$ are algebraic, they admit triangulations compatible with
the maps. Thus $A_k^\infty$ inherits the structure of a CW-complex and so it
is sufficient to show that all of its homotopy groups are zero.
To this end we prove that for any
$k$ and $l$ there is an $r>l$ such that the natural inclusion
from $A_k^n\hookrightarrow A_k^r$ is homotopically trivial.
Consider the homotopy
$H_t:A_k^n\to A_k^{2k+n}$
defined as follows:
\[H_t((a_1,a_2,x,b,c))=
( (1-t)a_1,\; (1-t)a_2,\; (1-t)x,\; b_t,\; c_t)\]
where
\[c_t=
\begin{pmatrix}tI_k\\ 0_{k,k}\\ (1-t)c
\end{pmatrix}
\;\;{\rm ,}\;\;
b_t=(0_{k,k},\; tI_k,\; (1-t)^2b),\]
$I_k$ is the $k\times k$ identity matrix and $0_{k,k}$ is the
$k\times k$ zero matrix. To see that
$H_t(v)\in A_k^{n+2k}$ for any $v\in A_k^n$, we
check that the integrability and non-degeneracy conditions
are satisfied for all $0\leq t\leq 1$.
Integrability holds because $b_tc_t=(1-t)^3bc$.
Non-degeneracy is satisfied for all $t\not=0$
because there is a full rank $k\times k$ block, $tI_k$,
in both
$c_t$ and $b_t$. Furthermore, $H_0$ is just the inclusion
$A_k^n\hookrightarrow A_k^{n+2k}$, so
non-degeneracy also holds when $t=0$.
Finally, note that $H_1$ is a constant map.
\end{proof}
These lemmas show that $A_{k}^{\infty }$ is a contractible space acted on
freely by $G=Gl(k)\times Gl(k)$ and $A_{k}^{\infty }/G=\ms{k}{\infty }$.
Thus $\ms{k}{\infty }$ is homotopic to $BG$ which in turn has the homotopy
type of $BU(k)\times BU(k)$. We end by remarking that the proof shows that the
universal $U(k)\times U(k)$ bundle is the bundle that restricts to any of the
finite ${\mathcal{M}}_k^n$'s as $R^{1}\pi_{*}(\Bbb{E}(-\ell_{\infty }))\oplus
R^{1}\pi_{*}(\Bbb{E}(-\ell_{\infty }+E))$.
|
1996-12-16T11:00:28 | 9610 | alg-geom/9610019 | en | https://arxiv.org/abs/alg-geom/9610019 | [
"alg-geom",
"math.AG"
] | alg-geom/9610019 | Alexander G. Kuznetsov | Alexander Kuznetsov | The Laumon's resolution of Drinfeld's compactification is small | Correction of misprints, alternative definition of Drinfeld's
compactification included AMSLaTeX v 1.1 | null | null | null | null | Let $C$ be a smooth projective curve of genus 0. Let $\FF$ be the variety of
complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple
$\alpha$ of positive integers one can consider the space $\MM\alpha$ of
algebraic maps of degree $\alpha$ from $C$ to $\FF$. This space has drawn much
attention recently in connection with Quantum Cohomology. The space $\MM\alpha$
is smooth but not compact. The problem of compactification of $\MM\alpha$
proved very important. One compactification $\MML\alpha$ (the space of {\em
quasiflags}), was constructed in \cite{L}. However, historically the first and
most economical compactification $\MMD\alpha$ (the space of {\em quasimaps})
was constructed by Drinfeld (early 80-s, unpublished). The latter
compactification is singular, while the former one is smooth. Drinfeld has
conjectured that the natural map $\pi:\MML\alpha\to\MMD\alpha$ is a small
resolution of singularities. In the present note we prove this conjecture. As a
byproduct, we compute the stalks of $IC$ sheaf on $\MMD\alpha$ and, moreover,
the Hodge structure in these stalks. Namely, the Hodge structure is a pure Tate
one, and the generating function for the $IC$ stalks is just the Lusztig's
$q$-analogue of Kostant's partition function (see \cite{Lu}).
| [
{
"version": "v1",
"created": "Sun, 27 Oct 1996 10:37:49 GMT"
},
{
"version": "v2",
"created": "Mon, 28 Oct 1996 05:33:31 GMT"
},
{
"version": "v3",
"created": "Mon, 16 Dec 1996 09:46:05 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Kuznetsov",
"Alexander",
""
]
] | alg-geom | \section{The space of maps into flag variety}
\subsection{Notations}
Let $G$ be a complex semisimple simply-connected Lie group,
$H\subset B$ its Cartan and Borel subgroups,
$N$ the unipotent radical of $B$,
$Y$ the lattice of coroots of $G$ (with respect to $H$),
$l$ the rank of $Y$,
$I=\{i_1,i_2,\dots i_l\}$ the set of simple coroots,
$R^+$ the set of positive coroots,
$X$ the lattice of weights,
$X^+$ the cone of dominant weights,
$\Omega=\{\omega_1,\omega_2\dots\omega_l\}$ the set of fundamental weights
($\langle\omega_k,i_l\rangle=\delta_{kl}$),
${\cal B}=G/B$ the flag variety and
$C$ a smooth projective curve of genus ~$0$.
Recall that there are canonical isomorphisms
$$
\oper{H}\nolimits_2({\cal B},{\Bbb Z})\cong Y\qquad\oper{H}\nolimits^2({\cal B},{\Bbb Z})\cong X.
$$
For $\lambda\in X$ let ${\mbox{\bf L}}_\lambda$ denote the corresponding
$G$-equivariant line bundle on ${\cal B}$.
The map $\varphi:C\to{\cal B}$ has degree $\alpha\in{\Bbb N}[I]\subset Y$
if the following equivalent conditions hold:
\begin{enumerate}
\item $\varphi_*([C])=\alpha$;
\item for any $\lambda\in X\quad\text{we have}\quad
\deg(\varphi^*{\mbox{\bf L}}_\lambda)=\langle\lambda,\alpha\rangle$.
\end{enumerate}
We denote by $\MM\alpha$ the space of algebraic maps from $C$
to ${\cal B}$ of degree $\alpha$.
It is known that $\MM\alpha$ is smooth variety and
$\dim\MM\alpha=2|\alpha|+\dim{\cal B}$. In this paper we compare
two natural compactifications of the space $\MM\alpha$, which
we presently describe.
\subsection{Drinfeld's compactification}
The Pl\"ucker embedding of the flag variety ${\cal B}$ gives rise to the
following interpretation of $\MM\alpha$.
For any irreducible representation $V_\lambda$ ($\lambda\in X^+$) of $G$
we consider the trivial vector bundle ${\cal V}_\lambda=V_\lambda\otimes{\cal O}_C$
over $C$.
For any $G$-morphism $\psi:\ V_\lambda\otimes V_\mu\longrightarrow V_\nu$ we denote
by the same letter the induced morphism $\psi:\ {\cal V}_\lambda\otimes {\cal V}_\mu
\longrightarrow {\cal V}_\nu$.
Then $\MM\alpha$ is the space of collections of line subbundles
${\cal L}_\lambda\subset{\cal V}_\lambda,\ \lambda\in X^+$ such that:
a) $\deg{\cal L}_\lambda=-\langle\lambda,\alpha\rangle$;
b) For any nonzero $G$-morphism $\psi:\ V_\lambda\otimes V_\mu\longrightarrow V_\nu$
such that $\nu=\lambda+\mu$ we have $\psi({\cal L}_\lambda\otimes{\cal L}_\mu)=
{\cal L}_\nu$;
c) For any $G$-morphism $\psi:\ V_\lambda\otimes V_\mu\longrightarrow V_\nu$
such that $\nu<\lambda+\mu$ we have $\psi({\cal L}_\lambda\otimes{\cal L}_\mu)=0$.
\begin{remark}\label{rem1}
Certainly, the property b) guarantees that in order to specify such a
collection it suffices to give ${\cal L}_{\omega_k}$ for the set $\Omega$ of
fundamental weights.
\end{remark}
If we replace the curve $C$ by a point, we get the Pl\"ucker description of
the flag variety ${\cal B}$ as the space of collections of lines
$L_\lambda\subset V_\lambda$ satisfying conditions of type (b) and (c)
(thus ${\cal B}$ is embedded into $\prod\limits_{\lambda\in X^+}{\Bbb P}(V_\lambda)$).
Here, a Borel subgroup ${\mbox{\bf B}}$ in ${\cal B}$ corresponds to
a system of lines $(L_\lambda,\lambda\in X^+)$ if lines are the fixed
points of the unipotent radical of ${\mbox{\bf B}}$, $L_\lambda=(V_\lambda)^{\mbox{\bf N}}$,
or equivalently, if ${\mbox{\bf N}}$ is the common stabilizer for all lines
${\mbox{\bf N}}=\bigcap\limits_{\lambda\in X^+}G_{L_\lambda}$.
The following definition in case $G=SL_2$ appeared in \cite{Drinfeld}.
\begin{defn}[V.Drinfeld]\label{MMD}
The space $\MMD\alpha$ of quasimaps of degree $\alpha$
from $C$ to ${\cal B}$ is the space of
collections of invertible subsheaves
${\cal L}_\lambda\subset{\cal V}_\lambda,\ \lambda\in X^+$ such that:
a) $\deg{\cal L}_\lambda=-\langle\lambda,\alpha\rangle$;
b) For any nonzero $G$-morphism $\psi:\ V_\lambda\otimes V_\mu\longrightarrow V_\nu$
such that $\nu=\lambda+\mu$ we have $\psi({\cal L}_\lambda\otimes{\cal L}_\mu)=
{\cal L}_\nu$;
c) For any $G$-morphism $\psi:\ V_\lambda\otimes V_\mu\longrightarrow V_\nu$
such that $\nu<\lambda+\mu$ we have $\psi({\cal L}_\lambda\otimes{\cal L}_\mu)=0$.
\end{defn}
\begin{remark}
Here is another version of the Definition, also due to V.Drinfeld.
The principal affine space ${\cal A}=G/N$ is an $H$-torsor over ${\cal B}$.
We consider its affine closure ${\frak A}$,
that is, the spectrum of the ring of functions on ${\cal A}$.
The action of $H$ extends to ${\frak A}$ but it is not free anymore. Consider the
quotient stack $\tilde{\cal B}={\frak A}/H$. The flag variety ${\cal B}$ is an
open substack in $\tilde{\cal B}$. A map
$\tilde\phi:C\to\tilde{\cal B}$ is nothing else than an $H$-torsor $\Phi$ over $C$ along
with an $H$-equivariant morphism $f:\Phi\to{\frak A}$. The degree of this map
is defined as follows.
Let $\chi_\lambda:H\to{\Bbb C}^*$ be the character of $H$ corresponding
to a weight $\lambda\in X$. Let $H_\lambda\subset H$ be the kernel
of the morphism $\chi_\lambda$. Consider the induced ${\Bbb C}^*$-torsor
$\Phi_\lambda=\Phi/H_\lambda$ over $C$. The map $\tilde\phi$ has
degree $\alpha\in{\Bbb N}[I]$ if
$$
\text{for any }\lambda\in X\quad\text{we have}\quad
\deg(\Phi_\lambda)=\langle\lambda,\alpha\rangle.
$$
\end{remark}
\begin{defn}\label{MMD1}
The space $\MMD\alpha$ is the space of maps $\tilde\phi:C\to\tilde{\cal B}$
of degree $\alpha$ such that the generic point of $C$ maps into
${\cal B}\subset\tilde{\cal B}$.
\end{defn}
The equivalence of \ref{MMD} and \ref{MMD1} follows immediately from
the Pl\"ucker embedding of ${\frak A}$ into
$\prod\limits_{\lambda\in X^+}V_\lambda$.
\begin{pro}
$\MMD\alpha$ is a projective variety.
\end{pro}
\begin{proof}
The space $\MMD\alpha$ is naturally embedded into the space
$$
\prod_{k=1}^l
{\Bbb P}(\oper{Hom}({\cal O}_C(-\langle\omega_k,\alpha\rangle),{\cal V}_{\omega_k}))
$$
and is closed in it.
\end{proof}
\subsection{The stratification of the Drinfeld's compactification}
In this subsection we will introduce the stratification of the space
of quasimaps.
{\bf Configurations of $I$-colored divisors.}
Let us fix $\alpha\in{\Bbb N}[I]\subset Y,\ \alpha=\sum\limits_{k=1}^N a_ki_k$.
Consider the configuration space $C^\alpha$ of colored
effective divisors of multidegree $\alpha$ (the set of colors is $I$).
The dimension of $C^\alpha$ is equal to the length
$|\alpha|=\sum\limits_{k=1}^N a_k$.
Multisubsets of a set $S$ are defined as elements of some
symmetric power $S^{(m)}$ and we denote the image of
$(s_1,\dots,s_m)\in S^m$ by $\{\{s_1,\dots,s_m\}\}$.
We denote by $\Gamma(\alpha)$ the set of all partitions of $\alpha$,
i.e.\ multisubsets $\Gamma=\{\{\gamma_1,\dots,\gamma_m\}\}$ of ${\Bbb N}[I]$
with $\sum\limits_{r=1}^m\gamma_r=\alpha$, $\gamma_r>0$.
For $\Gamma\in\Gamma(\alpha)$ the corresponding stratum $C^\alpha_\Gamma$
is defined as follows. It is formed by configurations which can be
subdivided into $m$ groups of points, the $r$-th group containing $\gamma_r$
points; all the points in one group equal to each other, the different
groups being disjoint. For example, the main diagonal in $C^\alpha$
is the closed stratum given by partition $\alpha=\alpha$, while the complement
to all diagonals in $C^\alpha$ is the open stratum given by partition
$$
\alpha=\sum\limits_{k=1}^N(\underbrace{i_k+i_k+\ldots+i_k}_{a_k\operatorname{ times}})
$$
Evidently, $C^\alpha=\bigsqcup\limits_{\Gamma\in\Gamma(\alpha)}C^\alpha_\Gamma$.
{\bf Normalization and defect of subsheaves.}\nopagebreak
Let $F$ be a vector bundle on the curve $C$ and let $E$ be a subsheaf in $F$.
Let $F/E=T(E)\oplus L$ be the decomposition of the quotient sheaf $F/E$
into the sum of its torsion subsheaf and a locally free sheaf, and let
$\tilde E=\oper{Ker}(F\to L)$ be the kernel of the natural map $F\to L$.
Then $\tilde E$ is a vector subbundle in $F$ which contains $E$ and has
the following universal property:
$$
\text{for any subbundle ${\cal E}'\subset F$ if ${\cal E}'$ contains $E$
then ${\cal E}'$ contains also $\tilde E$}.
$$
Moreover, $\oper{rank}\tilde E=\oper{rank} E$, $\tilde E/E\cong T(E)$ and
$c_1(\tilde E)=c_1(E)+\ell(T(E))$ (for any torsion sheaf on $C$
we denote by $\ell(T)$ its length).
\begin{defn}
We will call $\tilde E$ the {\em normalization}
of $E$ in $F$ and $T(E)$ the {\em defect} of $E$.
\end{defn}
\begin{remark}\label{detnorm}
If $\tilde E$ is the normalization of $E$ in $F$ then $\Lambda^k(\tilde E)$ is the
normalization of $\Lambda^kE$ in $\Lambda^k F$.
\end{remark}
For any $x\in C$ and torsion sheaf $T$ on $C$ we will denote by
$\ell_x(T)$ the length of the localization of $T$ in the point $x$.
\begin{defn}
For any quasimap
$\varphi=({\cal L}_\lambda\subset{\cal V}_\lambda)_{\lambda\in X^+}\in\MMD\alpha$
we define the {\em normalization} of $\varphi$ as follows:
$$
\tilde\varphi=(\tilde{\cal L}_\lambda\subset{\cal V}_\lambda),
$$
and the {\em defect} of $\varphi$ as follows:
$$
\oper{def}(\varphi)=(T({\cal L}_\lambda))
$$
(the defect of $\varphi$ is a collection of torsion sheaves).
\end{defn}
\begin{pro}\label{normalization}
For any $\varphi\in\MMD\alpha$ there exists $\beta\le\alpha\in{\Bbb N}[I]$,
partition $\Gamma=(\gamma_1,\dots,\gamma_m)\in\Gamma(\alpha-\beta)$ and
a divisor $D=\sum\limits_{r=1}^m\gamma_rx_r\in C^{\alpha-\beta}_\Gamma$ such that
$$
\tilde\varphi\in\MM\beta
\qquad
\ell_x(\oper{def}(\varphi)_\lambda)=\begin{cases}
\displaystyle\langle\lambda,\gamma_r\rangle,&\text{if $x=x_r$}\\
\ds0,&\text{otherwise}\end{cases}
$$
\end{pro}
\begin{proof}
Clear.
\end{proof}
\begin{defn}
The pair $(\beta,\Gamma)$ will be called the {\em type of degeneration}
of $\varphi$. We denote by $\DD\beta\Gamma$ the subspace of $\MMD\alpha$
consisting of all quasimaps $\varphi$ with the given type of degeneration.
\end{defn}
\begin{remark}
Note that $\DD0\emptyset=\MM\alpha$.
\end{remark}
We have
\begin{equation}
\MMD\alpha=\bigsqcup\begin{Sb}
\beta\le\alpha\\
\Gamma\in\Gamma(\alpha-\beta)
\end{Sb}\DD\beta\Gamma
\end{equation}
The map $d_{\beta,\Gamma}:
\DD\beta\Gamma\to\MM\beta\times C^{\beta-\alpha}_\Gamma$ which sends
$\varphi$ to $(\tilde\varphi,D)$ (see \ref{normalization}) is an isomorphism.
The inverse map $\sigma_{\beta,\Gamma}$ can be constructed as follows.
Let $\varphi=({\cal L}_\lambda)\in\MM\beta$. Then
$$
\sigma_{\beta,\Gamma}(\varphi,D)\stackrel{\text{\rm def}}{=}({\cal L}'_\lambda)\qquad
{\cal L}'_\lambda\stackrel{\text{\rm def}}{=}
\bigcap_{r=1}^m{\frak m}_{x_r}^{\langle\lambda,\gamma_r\rangle}\cdot{\cal L}_\lambda,
$$
where ${\frak m}_x$ denotes the sheaf of ideals of the point $x\in C$.
\subsection{Laumon's compactification}
Let $V$ be an $n$-dimensional vector space.\linebreak
>From now on we will assume that $G=SL(V)$ (in this case
certainly $l=n-1$). In this case there is the Grassmann
embedding of the flag variety, namely
$$
{\cal B}=\{(U_1,U_2,\dots,U_{n-1})\in
G_1(V)\times G_2(V)\times\dots\times G_{n-1}(V)\ |\
U_1\subset U_2\subset\dots\subset U_{n-1}\},
$$
where $G_k(V)$ is the Grassmann variety of $k$-dimensional subspaces in $V$.
This embedding gives rise to another interpretation of $\MM\alpha$.
We will denote by ${\cal V}$ the trivial vector bundle $V\otimes{\cal O}_C$ over $C$.
Let $\alpha=\sum\limits_{k=1}^{n-1}a_ki_k$, where $i_k$ is the simple coroot dual
to the highest weight $\omega_k$ of representation $G$ in $\Lambda^kV$.
Then $\MM\alpha$ is the space of complete flags of vector subbundles
$$
0\subset{\cal E}_1\subset{\cal E}_2\subset\dots\subset{\cal E}_{n-1}\subset{\cal V}
\quad\text{ such that }\quad c_1({\cal E}_k)=-\langle\omega_k,\alpha\rangle=-a_k.
$$
\begin{defn}[{Laumon, \cite[4.2]{Laumon}}]
The space $\MML\alpha$ of {\em quasiflags} of degree $\alpha$ is the space
of complete flags of locally free subsheaves
$$
0\subset E_1\subset E_2\subset\dots\subset E_{n-1}\subset{\cal V}
\quad\text{ such that }\quad c_1(E_k)=-\langle\omega_k,\alpha\rangle=-a_k.
$$
\end{defn}
It is known that $\MML\alpha$ is a smooth projective variety
of dimension $2|\alpha|+\dim{\cal B}$ (see loc.\ cit., Lemma 4.2.3).
\subsection{The stratification of the Laumon's compactification}
There is a stratification of the space $\MML\alpha$ similar to
the above stratification of $\MMD\alpha$.
\begin{defn}
For any quasiflag $E_\bullet=(E_1,\dots,E_{n-1})$ we define its
{\em normalization} as
$$
\tilde E_\bullet=(\tilde E_1,\dots,\tilde E_{n-1}),\text{ where }\tilde E_k\text{ is the
normalization of $E_k$ in ${\cal V}$}
$$
and {\em defect}
$$
\oper{def}(E_\bullet)=(\tilde E_1/E_1,\dots,\tilde E_{n-1}/E_{n-1})
$$
Thus, the defect of $E_\bullet$ is a collection of torsion sheaves.
\end{defn}
\begin{pro}\label{Lnormalization}
For any $E_\bullet\in\MML\alpha$ there exist $\beta\le\alpha\in{\Bbb N}[I]$,
partition $\Gamma=(\gamma_1,\dots,\gamma_m)\in\Gamma(\alpha-\beta)$ and
a divisor $D=\sum\limits_{r=1}^m\gamma_rx_r\in C^{\alpha-\beta}_\Gamma$ such that
$$
\tilde E_\bullet\in\MM\beta,
\qquad
\ell_x(\oper{def}(E_k))=\begin{cases}
\langle\omega_k,\gamma_r\rangle,&\text{if $x=x_r$}\\
0,&\text{otherwise}\end{cases}
$$
\end{pro}
\begin{defn}
The pair $(\beta,\Gamma)$ will be called the {\em type of degeneration}
of $E_\bullet$. We denote by $\fL\beta\Gamma$ the subspace in $\MML\alpha$
consisting of all quasiflags $E_\bullet$ with the given type of degeneration.
\end{defn}
\begin{remark}
Note that $\fL0\emptyset=\MM\alpha$.
\end{remark}
We have
\begin{equation}
\MML\alpha=\bigsqcup\begin{Sb}
\beta\le\alpha\\
\Gamma\in\Gamma(\alpha-\beta)
\end{Sb}\fL\beta\Gamma
\end{equation}
\subsection{The map from $\MML\alpha$ to $\MMD\alpha$}
Consider the map $\pi:\MML\alpha\to\MMD\alpha$ which
sends a quasiflag of degree $\alpha$ $E_\bullet\in\MML\alpha$
to a quasimap given by the collection $({\cal L}_{\omega_k})_{k=1}^{n-1}$
(see Remark \ref{rem1}) where
${\cal L}_{\omega_k}=\Lambda^k E_k\subset\Lambda^k{\cal V}={\cal V}_{\omega_k}$.
\begin{pro}
Let $E_\bullet$ be a quasiflag of degree $\alpha$ and let $(\beta,\Gamma)$
be its type of degeneration. Then $\pi(E_\bullet)$ is a quasimap of
degree $\alpha$ and its type of degeneration is $(\beta,\Gamma)$.
\end{pro}
\begin{proof}
Obviously we have
$\deg{\cal L}_{\omega_k}=\deg\Lambda^k E_k=c_1(E_k)=-\langle\omega_k,\alpha\rangle$
which means that $\pi(E_\bullet)\in\MMD\alpha$.
According to the Remark \ref{detnorm},
$\tilde{\cal L}_{\omega_k}=\Lambda^k\tilde E_k$ (i.e. $({\cal L}_{\omega_k})\in\MM\beta$), hence
\begin{equation}\label{lx}
\ell_x(\tilde{\cal L}_{\omega_k}/{\cal L}_{\omega_k})=\ell_x(\tilde E_k/E_k).
\end{equation}
This proves the Proposition.
\end{proof}
\begin{remark}
Note that \refeq{lx} implies that
$\pi$ preserves not only $\beta$ and $\Gamma$ but also $D$ (see
\ref{normalization}, \ref{Lnormalization}).
\end{remark}
Recall that a proper birational map $f:{\cal X}\to{\cal Y}$ is called {\em small}
if the following condition holds: let ${\cal Y}_m$ be the set of all points
$y\in{\cal Y}$ such that $\dim f^{-1}(y)\ge m$. Then for $m>0$ we have
\begin{equation}\label{small}
\oper{codim}{\cal Y}_m>2m.
\end{equation}
{\bf Main Theorem. } The map $\pi$ is a small resolution of singularities.
\section{The fibers of $\pi$}
\subsection{}
We fix ${\cal E}_\bullet\in\MM\beta$, a partition $\Gamma\in\Gamma(\alpha-\beta)$,
and a divisor $D\in C_\Gamma^{\alpha-\beta}$. Then
$({\cal E}_\bullet,D)\in\DD\beta\Gamma$. We define $F({\cal E}_\bullet,D)$
as $\pi^{-1}({\cal E}_\bullet,D)$.
Let $D=\sum_{r=1}^m\gamma_rx_r$. We define the space ${\cal F}({\cal E}_\bullet,D)$
of commutative diagrams
$$
\begin{CD}
{\cal E}_1 @>>> {\cal E}_2 @>>> \dots @>>> {\cal E}_{n-1} \\
@V\varepsilon_1VV @V\varepsilon_2VV @. @V\varepsilon_{n-1}VV \\
T_1 @>\tau_1>> T_2 @>\tau_2>> \dots @>\tau_{n-2}>>T_{n-1}
\end{CD}
$$
such that
a) $\varepsilon_k$ is surjective,
b) $T_k$ is torsion,
c) $\displaystyle \ell_x(T_k)=\begin{cases}
\displaystyle\langle\omega_k,\gamma_r\rangle,&\text{if $\displaystyle x=x_r$}\\
\ds0,&\text{otherwise}\end{cases}$
\begin{lem}\label{torsion}
We have an isomorphism
$$
F({\cal E}_\bullet,D)\cong{\cal F}({\cal E}_\bullet,D).
$$
\end{lem}
\begin{proof}
If $E_\bullet\in F({\cal E}_\bullet,D)$ then by the \ref{Lnormalization}
the collection $(T_1,\dots,T_{n-1})=\oper{def}(E_\bullet)$ satisfies
the above conditions.
Vice versa, if the collection $(T_1,\dots,T_k)$ satisfies the
above conditions, then consider
$$
E_k=\oper{Ker}({\cal E}_k @>\varepsilon_k>> T_k).
$$
Since the square
$$
\begin{CD}
{\cal E}_k @>\varepsilon_k>> T_k \\
@VVV @V\tau_kVV \\
{\cal E}_{k+1} @>\varepsilon_{k+1}>> T_{k+1}
\end{CD}
$$
commutes, we can extend it to the commutative diagram
$$
\begin{CD}
0 @>>> E_k @>>> {\cal E}_k @>\varepsilon_k>> T_k @>>> 0 \\
@. @VVV @VVV @V\tau_kVV @.\\
0 @>>> E_{k+1} @>>> {\cal E}_{k+1}@>\varepsilon_{k+1}>> T_{k+1} @>>> 0
\end{CD}
$$
The induced morphism $E_k\to E_{k+1}$ is injective because
${\cal E}_k\to{\cal E}_{k+1}$ is, and
\begin{multline*}
\qquad c_1(E_k)=c_1({\cal E}_k)-\ell(T_k)=
-\langle\omega_k,\beta\rangle-\sum_{x\in C}\ell_x(T_k)=\\=
-\langle\omega_k,\beta\rangle-\sum_{r=1}^m\langle\omega_k,\gamma_r\rangle=
-\langle\omega_k,\beta+(\alpha-\beta)\rangle=-\langle\omega_k,\alpha\rangle\qquad
\end{multline*}
This means that $E_\bullet\in F({\cal E}_\bullet,D)$.
\end{proof}
\begin{pro}\label{gfibre}
If $D=\sum\limits_{r=1}^m\gamma_rx_r$ is a decomposition
into disjoint divisors then
\begin{equation}
F({\cal E}_\bullet,D)\cong\prod_{r=1}^mF({\cal E}_\bullet,\gamma_rx_r).
\end{equation}
\end{pro}
\begin{proof}
Recall that if $T$ is a torsion sheaf on the curve $C$ then
$$
T=\bigoplus_{x\in C}T_x,
$$
where $T_x$ is the localization of $T$ in the point $x$.
This remark together with Lemma \ref{torsion} proves the Proposition.
\end{proof}
The above Proposition implies, that in order to describe general fiber
$F({\cal E}_\bullet,D)$ it is enough to have a description of the fibers
$F({\cal E}_\bullet,\gamma x)$, which we will call {\em simple fibers}.
\subsection{The stratification of a simple fiber}
We will need the following obvious Lemma.
\begin{lem}\label{coin}
Let ${\cal E}$ be a vector bundle on $C$. Let ${\cal E}'\subset{\cal E}$ be a vector
subbundle, and let $E\subset{\cal E}$ be a (necessarily locally free)
subsheaf. Then $E'={\cal E}'\cap E$ is a vector subbundle in $E$.
Moreover, the commutative square
$$
\begin{CD}
E' @>>> E \\
@VVV @VVV \\
{\cal E}' @>>> {\cal E}
\end{CD}
$$
can be extended to the commutative diagram
$$
\begin{CD}
E' @>>> E @>>> E/E' \\
@VVV @VVV @VVV \\
{\cal E}' @>>> {\cal E} @>>> {\cal E}/{\cal E}'\\
@VVV @VVV @VVV \\
{\cal E}'/E' @>>> {\cal E}/E @>>> \dfrac{{\cal E}/E}{{\cal E}'/E'}\cong\dfrac{{\cal E}/{\cal E}'}{E/E'}
\end{CD}
$$
in which both the rows and the columns form the short exact sequences.
\end{lem}
The sheaf in the lower-right corner of the diagram
will be called {\em cointersection} of $E$ and ${\cal E}'$ inside ${\cal E}$
and denoted by $\coin{\cal E} E{{\cal E}'}$.
Let
$$
\gamma=\sum_{k=1}^{n-1}c_ki_k.
$$
For every $E_\bullet\in F({\cal E}_\bullet,\gamma x)$ we define
\begin{equation}
\mu_{pq}(E_\bullet)\stackrel{\text{\rm def}}{=} l\left(\frac{{\cal E}_q}{E_p\cap{\cal E}_q}\right)
\qquad(1\le q\le p\le n-1),
\end{equation}
\begin{equation}\label{nu}
\nu_{pq}(E_\bullet)=\begin{cases}
\mu_{pq}(E_\bullet)-\mu_{p+1,q}(E_\bullet),&\text{if }1\le q\le p<n-1\\
\mu_{pq}(E_\bullet),&\text{if }1\le q\le p=n-1
\end{cases}
\end{equation}
\begin{equation}\label{ka}
\kappa_{pq}(E_\bullet)=\begin{cases}
\nu_{pq}(E_\bullet)-\nu_{p,q-1}(E_\bullet),&\text{if }1<q\le p\le n-1\\
\nu_{pq}(E_\bullet),&\text{if }1=q\le p\le n-1
\end{cases}
\end{equation}
\begin{remark}
The transformations \refeq{nu} and \refeq{ka} are invertible, so the numbers
$\mu_{pq}$ can be uniquely reconstructed from $\nu_{pq}$ or $\kappa_{pq}$.
Namely,
\begin{equation}\label{inv}
\displaystyle\nu_{pq}=\sum_{r=1}^q\kappa_{pr};\qquad
\displaystyle\mu_{pq}=\sum_{s=p}^{n-1}\nu_{sq}=\sum_{r\le q\le p\le s}\kappa_{sr}.
\end{equation}
\end{remark}
\begin{lem}
We have
\begin{equation}\label{pnu}
\nu_{pq}(E_\bullet)=l\left(\dfrac{{\cal E}_q\cap E_{p+1}}{{\cal E}_q\cap E_p}\right).
\end{equation}
\begin{equation}\label{pka}
\kappa_{pq}(E_\bullet)=l\left(
\coin{{\cal E}_q\cap E_{p+1}}{{\cal E}_q\cap E_p}{{\cal E}_{q-1}\cap E_{p+1}}\right).
\end{equation}
\end{lem}
\begin{proof}
The commutative diagram with exact rows
$$
\begin{CD}
0@>>> {\cal E}_q\cap E_p @>>> {\cal E}_q @>>> \dfrac{{\cal E}_q}{{\cal E}_q\cap E_p} @>>>0\\
@. @VVV @| @VVV\\
0@>>> {\cal E}_q\cap E_{p+1} @>>> {\cal E}_q @>>> \dfrac{{\cal E}_q}{{\cal E}_q\cap E_{p+1}} @>>>0
\end{CD}
$$
implies \refeq{pnu}. In order to prove \refeq{pka} note that
$$
{\cal E}_{q-1}\cap E_p=({\cal E}_q\cap E_p)\cap({\cal E}_{q-1}\cap E_{p+1})
$$
and apply Lemma \ref{coin} and \refeq{pnu}.
\end{proof}
\begin{cor}
Numbers $\mu_{pq}$, $\nu_{pq}$ and $\kappa_{pq}$
satisfy the following inequalities:
\begin{eqnarray}
0\le\kappa_{pq}\\
0\le\nu_{p1}\le\nu_{p2}\le\dots\le\nu_{pp}\\
0\le\mu_{n-1,q}\le\mu_{n-2,q}\le\dots\le\mu_{qq}=c_q.\label{muin}
\end{eqnarray}
\end{cor}
\begin{proof}
See \refeq{pka},\refeq{ka},\refeq{nu} and compare the definition of $\mu_{qq}$
with \ref{torsion}.
\end{proof}
We will denote by $[p,q]$ the positive coroot
\begin{equation}\label{pq}
[p,q]\stackrel{\text{\rm def}}{=}\sum_{k=q}^pi_k\in R^+
\end{equation}
\begin{lem}\label{part}
For any $E_\bullet\in F({\cal E}_\bullet,\gamma x)$ we have
$$
\sum_{1\le q\le p\le n-1}\kappa_{pq}(E_\bullet)[p,q]=\gamma.
$$
\end{lem}
\begin{proof}
Applying \refeq{ka}, \refeq{pq} and \refeq{inv} we get
\begin{multline*}
\sum_{1\le q\le p\le n-1}\kappa_{pq}[p,q]=
\sum_{1\le q\le p\le n-1}(\nu_{pq}-\nu_{p,q-1})[p,q]=\\=\!\!\!
\sum_{1\le q\le p\le n-1}\!\!\!\nu_{pq}([p,q]-[p,q+1])=\!\!\!
\sum_{1\le q\le p\le n-1}\!\!\!\nu_{pq}i_q=
\sum_{q=1}^{n-1}\left(\sum_{p=q}^{n-1}\nu_{pq}\right)i_q=
\sum_{q=1}^{n-1}\mu_{qq}i_q.
\end{multline*}
Now Lemma follows from \refeq{muin}.
\end{proof}
Let ${\frak K}(\gamma)$ be the set of all partitions of $\gamma\in{\Bbb N}[I]$ into the
sum of positive coroots: $\gamma=\sum\limits_{s=1}^t\delta_s$, where
$\delta_s\in R^+$ (note that ${\frak K}(\gamma)\ne\Gamma(\gamma)$).
In other words, since every positive coroot for $G=SL(V)$ is
equal to $[p,q]$ for some $p,q$,
$$
{\frak K}(\gamma)=\{ (\kappa_{pq})_{1\le q\le p\le n-1}\ |\
\kappa_{pq}\ge0\quad\text{ and }\sum_{1\le q\le p\le n-1}\kappa_{pq}[p,q]=\gamma\}.
$$
Let ${\frak M}(\gamma)$ denote the set of all collections $(\mu_{pq})$
which can be produced from some $(\kappa_{pq})\in{\frak K}(\gamma)$ as in \refeq{inv}.
The Lemma \ref{part} implies that for any
$E_\bullet\in F({\cal E}_\bullet,\gamma x)$ we have
$(\mu_{pq}(E_\bullet))\in{\frak M}(\gamma)$.
Define the stratum ${\frak S}((\mu_{pq})_\pq{n-1},({\cal E}_k)_{k=1}^{n-1})$ as follows:
$$
{\frak S}((\mu_{pq})_\pq{n-1},({\cal E}_k)_{k=1}^{n-1})=
\{ E_\bullet\in F({\cal E}_\bullet,\gamma x)\ |\ \mu_{pq}(E_\bullet)=\mu_{pq}\}.
$$
To unburden the notations in the cases when it is clear which flag
${\cal E}_\bullet$ is used we will write just ${\frak S}_\mu$.
We have obviously
\begin{equation}\label{stratification}
F({\cal E}_\bullet,\gamma x)=\bigsqcup_{\mu\in{\frak M}(\gamma)}{\frak S}_\mu.
\end{equation}
\begin{remark}\label{shortflag}
We will also use the similar
varieties ${\frak S}((\mu_{pq})_\pq{N},({\cal E}_k)_{k=1}^N)$ that can
be defined in the same way for any short flag $({\cal E}_k)_{k=1}^N$
(that is the flag of subbundles ${\cal E}_1\subset\dots\subset{\cal E}_N\subset{\cal V}$
with $\oper{rank}{\cal E}_k=k$).
\end{remark}
\subsection{The strata ${\frak S}_\mu$}
In order to study ${\frak S}_\mu$ we will introduce some more varieties.
For every $1\le N\le n-1$, a short flag of subbundles $({\cal E}_k)_{k=1}^N$
(see Remark \ref{shortflag}) and a collection of numbers $(\nu_k)_{k=1}^N$
such that $0\le\nu_1\le\dots\le\nu_N$, we define the space
${\frak T}((\nu_k)_{k=1}^N,({\cal E}_k)_{k=1}^N)$ as follows:
\begin{equation}\label{TT}
{\frak T}((\nu_k)_{k=1}^N,({\cal E}_k)_{k=1}^N)=\{E\subset{\cal E}_N\ |\
\oper{rank}(E)=N\quad\text{ and }\quad
l{\left(\frac{{\cal E}_k}{{\cal E}_k\cap E}\right)}=\nu_k\}
\end{equation}
We define {\em pseudoaffine} spaces by induction in dimension.
First, the affine line ${\Bbb A}^1$ is a pseudoaffine space. Now
a space $A$ is called pseudoaffine if it admits a fibration
$A\to B$ with pseudoaffine fibers and pseudoaffine $B$.
\begin{thm}
The space ${\frak T}((\nu_k)_{k=1}^N,\!({\cal E}_k)_{k=1}^N)$ is
pseudoaffine of dimension $\!\sum\limits_{k=1}^{N-1}\!\!\nu_k$.
\end{thm}
\begin{proof}
We use induction in $N$.
The case $N=1$ is trivial. There is only one subsheaf $E$ in the
line bundle ${\cal E}_1$ with $\ell({\cal E}_1/E)=\nu_1$, namely
$E={\frak m}_x^{\nu_1}\cdot{\cal E}_1$. This means that ${\frak T}(\nu_1,{\cal E}_1)$
is a point and the base of induction follows.
If $N>1$ then consider the map
$$
\tau:{\frak T}((\nu_k)_{k=1}^N,({\cal E}_k)_{k=1}^N)\to
{\frak T}((\nu_k)_{k=1}^{N-1},({\cal E}_k)_{k=1}^{N-1}),
$$
which sends $E\in{\frak T}((\nu_k)_{k=1}^N,({\cal E}_k)_{k=1}^N)$ to
$E'=E\cap{\cal E}_{N-1}\in{\frak T}((\nu_k)_{k=1}^{N-1},({\cal E}_k)_{k=1}^{N-1})$.
\begin{lem}\label{fibre}
Let $L={\frak m}_x^{\nu_N-\nu_{N-1}}\cdot\left(\dfrac{{\cal E}_N}{{\cal E}_{N-1}}\right)$.
For any $E'\in{\frak T}((\nu_k)_{k=1}^{N-1},({\cal E}_k)_{k=1}^{N-1})$
there is an isomorphism
$$
\tau^{-1}(E')\cong\oper{Hom}(L,{\cal E}_{N-1}/E')\cong
{\Bbb A}^{\ell({\cal E}_{N-1}/E')}={\Bbb A}^{\nu_{N-1}}.
$$
\end{lem}
Thus, the space ${\frak T}((\nu_k)_{k=1}^N,({\cal E}_k)_{k=1}^N)$ is
affine fibration over a pseudoaffine space, hence it is pseudoaffine and its
dimension is equal to
$$
\dim\left({\frak T}((\nu_k)_{k=1}^{N-1},({\cal E}_k)_{k=1}^{N-1})\right)+\nu_{N-1}=
\sum_{k=1}^{N-2}\nu_k+\nu_{N-1}=\sum_{k=1}^{N-1}\nu_k.
$$
The Theorem is proved.
\end{proof}
\begin{proofof}{Lemma \ref{fibre}}
Let $E\in\tau^{-1}(E')$. Since $E'=E\cap{\cal E}_{N-1}$ we can apply Lemma
\ref{coin} which gives the following commutative diagram:
$$
\begin{CD}
E' @>>> E @>>> L \\
@VVV @VVV @VVV \\
{\cal E}_{N-1} @>j>> {\cal E}_N @>\psi>> {\cal E}_N/{\cal E}_{N-1} \\
@VVV @VVV @VVV \\
T_{N-1} @>>> T_N @>>> T_N/T_{N-1}
\end{CD}
$$
(Note that since ${\cal E}_N/{\cal E}_{N-1}$ is a line bundle and
$\ell(T_N/T_{N-1})=\ell(T_N)-\ell(T_{N-1})=\nu_N-\nu_{N-1}$ the kernel
of natural map ${\cal E}_N/{\cal E}_{N-1}\to T_N/T_{N-1}$ is isomorphic
to $L$.)
Let $\tilde{\cal E}_N=\psi^{-1}(L)$. Then $E$ is contained in $\tilde{\cal E}_N$ and
we have the following commutative diagram:
$$
\begin{CD}
E' @>>> E @>>> L \\
@VVV @VVV @| \\
{\cal E}_{N-1} @>j>> \tilde{\cal E}_N @>\psi>> L \\
@VVV @V\varepsilon VV \\
T_{N-1} @= T_{N-1}
\end{CD}
$$
This means that the points of $\tau^{-1}(E')$ are in one-to-one correspondence
with maps $\varepsilon:\tilde{\cal E}_N\to T_{N-1}$ such that $\varepsilon\cdot j$ is equal
to the canonical projection from ${\cal E}_{N-1}$ to $T_{N-1}$.
Applying the functor $\oper{Hom}(\bullet,T_{N-1})$ to the middle row of the
above diagram we get an exact sequence:
$$
0\to\oper{Hom}(L,T_{N-1})\to\oper{Hom}(\tilde{\cal E}_N,T_{N-1})@>j^*>>\oper{Hom}({\cal E}_{N-1},T_{N-1})\to
\oper{Ext}\nolimits^1(L,T_{N-1}).
$$
The last term in this sequence is zero because $L$ is locally free
and $T_{N-1}$ is torsion.
This means that the space of maps $\varepsilon$ which we need to describe
is a torsor over the group $\oper{Hom}(L,T_{N-1})$. Hence this space can
be identified with the group. Thus, we have proved that
$\tau^{-1}(E')\cong\oper{Hom}(L,T_{N-1})$ is an affine space.
Now,
$$
\dim(\tau^{-1}(E'))=\dim\oper{Hom}(L,T_{N-1})=\dim\oper{H}\nolimits^0(T_{N-1})=\ell(T_{N-1})=
\nu_{N-1}.
$$
The Lemma is proved.
\end{proofof}
\begin{thm}\label{saff}
The space ${\frak S}((\mu_{pq})_\pq{N},({\cal E}_k)_{k=1}^N)$
is a pseudoaffine space of dimension $\mu_{21}+\mu_{32}+\dots+\mu_{N,N-1}$.
\end{thm}
\begin{proof}
We use induction in $N$.
If $N=1$ then ${\frak S}_\mu$ is a point and the base of induction follows.
If $N>1$ consider the map
$$
\sigma:{\frak S}((\mu_{pq})_\pq{N},({\cal E}_k)_{k=1}^N)\to
{\frak T}((\mu_{N,k})_{k=1}^N,({\cal E}_k)_{k=1}^N),
$$
which sends $(E_k)_{k=1}^N$ to $E_N\subset{\cal E}_N$.
\begin{lem}\label{sfibre}
Let $E\in{\frak T}((\nu_{N,k})_{k=1}^N,({\cal E}_k)_{k=1}^N)$.
Consider $\tilde{\cal E}_k={\cal E}_k\cap E\quad(1\le k\le N-1)$ and
set $\tilde\mu_{pq}=\mu_{pq}-\mu_{Nq}\quad(\pq{N-1})$.
Then $(\tilde{\cal E}_k)_{k=1}^{N-1}$ is a short flag of subbundles and
for any $E\in{\frak T}((\mu_{N,k})_{k=1}^N,({\cal E}_k)_{k=1}^N$
\begin{equation}
\sigma^{-1}(E)\cong{\frak S}((\tilde\mu_{pq})_{\pq{N-1}}),(\tilde{\cal E}_k)_{k=1}^{N-1}).
\end{equation}
\end{lem}
Thus ${\frak S}((\mu_{pq})_\pq{N},({\cal E}_k)_{k=1}^N)$ is a fiber space
with pseudoaffine base and fiber, therefore it is pseudoaffine.
Now, the calculation of the dimension
\begin{multline*}
\dim\left({\frak S}((\mu_{pq})_\pq{N},({\cal E}_k)_{k=1}^N)\right)=
\sum_{k=1}^{N-1}\mu_{N,k}+\sum_{k=1}^{N-2}\tilde\mu_{k+1,k}=\\=
\sum_{k=1}^{N-1}\mu_{N,k}+\sum_{k=1}^{N-2}(\mu_{k+1,k}-\mu_{N,k})=
\mu_{N,N-1}+\sum_{k=1}^{N-2}\mu_{k+1,k}=
\sum_{k=1}^{N-1}\mu_{k+1,k},
\end{multline*}
finishes the proof of the Theorem.
\end{proof}
\begin{proofof}{Lemma \ref{sfibre}}
Assume that $(E_k)_{k=1}^N\in{\frak S}_\mu$ and $E_N=E$. The commutative
diagram
$$
\begin{CD}
0 @>>> {\cal E}_q\cap E_p @>>> {\cal E}_q @>>> \dfrac{{\cal E}_q}{{\cal E}_q\cap E_p} @>>> 0\\
@. @VVV @| @VVV @.\\
0 @>>> {\cal E}_q\cap E @>>> {\cal E}_q @>>> \dfrac{{\cal E}_q}{{\cal E}_q\cap E} @>>> 0
\end{CD}
$$
implies that
\begin{equation}\label{lengths}
l\left(\dfrac{{\cal E}_q\cap E}{{\cal E}_q\cap E_p}\right)=
l\left(\dfrac{{\cal E}_q}{{\cal E}_q\cap E_p}\right)-
l\left(\dfrac{{\cal E}_q}{{\cal E}_q\cap E}\right)=
\mu_{pq}-\mu_{Nq}=\tilde\mu_{pq},
\end{equation}
hence $(E_k)_{k=1}^{N-1}\in{\frak S}_{\tilde\mu}$.
Vice versa, if $(E_k)_{k=1}^{N-1}\in{\frak S}_{\tilde\mu}$ then
the above commutative diagram along with \refeq{lengths} implies that
$(E_k)_{k=1}^N\in{\frak S}_\mu$, where we have put $E_N=E$.
\end{proofof}
\subsection{The cohomology of the simple fiber}
Now we will compute the dimension of the strata ${\frak S}_\mu$ in
terms of the partition $\kappa$.
\begin{defn}
A space ${\cal X}$ is called {\em cellular} if it
admits a stratification with pseudoaffine strata.
\end{defn}
Suppose ${\cal X}=\bigsqcup\limits_{\xi\in\Xi} S_\xi$ is a
pseudoaffine stratification of a cellular space ${\cal X}$.
For a positive integer $j$ we define
$\chi(j)\stackrel{\text{\rm def}}{=}\{\xi\in\Xi\ |\ \dim S_\xi=j\}.$
\begin{lem}\label{hodge}
The Hodge structure $\oper{H}\nolimits^\bullet({\cal X},{\Bbb Q})$ is a direct sum of Tate structures,
and ${\Bbb Q}(j)$ appears with multiplicity $\chi(j)$.
In other words,
$$
\oper{H}\nolimits^\bullet({\cal X},{\Bbb Q})=\oplus_{j\in{\Bbb N}}{\Bbb Q}(j)^{\chi(j)}.
$$
\end{lem}
\begin{proof}
Evident.
\end{proof}
Given a Tate structure ${\cal H}=\oplus_{j\in{\Bbb N}}{\Bbb Q}(j)^{\chi(j)}$ we
consider a {\em generating function}
$$
P({\cal H},t)=\sum_{j\in{\Bbb N}}\chi(j)t^j\in{\Bbb N}[t].
$$
For $\kappa\in{\frak K}(\gamma)$ we define $K(\kappa)\stackrel{\text{\rm def}}{=}\displaystyle\sum\limits_{\pq{n-1}}\kappa_{pq}$
as the number of summands in the partition $\kappa$.
For $\gamma\in{\Bbb N}[I]$ the following $q$-analog of the Kostant's
partition function was was introduced in \cite{Lusztig}:
\begin{equation}\label{cke}
{\cal K}_\gamma(t)=t^{|\gamma|}\sum_{\kappa\in{\frak K}(\gamma)}t^{-K(\kappa)}.
\end{equation}
\begin{lem}\label{dims}
Let $\kappa\in{\frak K}(\gamma)$ and $\mu\in{\frak M}(\gamma)$ be defined
as in \refeq{inv}. Then
\begin{equation}
\dim{\frak S}_\mu=\sum_{k=1}^{n-2}\mu_{k+1,k}=|\gamma|-K(\kappa),
\end{equation}
\end{lem}
\begin{proof}
Applying \refeq{inv} we get
\begin{multline*}
\sum_{k=1}^{n-2}\mu_{k+1,k}=
\sum_{k=1}^{n-2}
\left(\sum\begin{Sb}1\le q\le k\\k+1\le p\le n-1\end{Sb}\kappa_{pq}\right)=
\sum_{\pq{n-1}}(p-q)\kappa_{pq}=\\=
\sum_{\pq{n-1}}(|[p,q]|-1)\kappa_{pq}=
|\gamma|-\sum_{\pq{n-1}}\kappa_{pq}=|\gamma|-K(\kappa).
\end{multline*}
{}\end{proof}
\begin{cor}\label{ck}
For any $\gamma\in{\Bbb N}[I]$, $x\in C$, the simple fiber
$F({\cal E}_\bullet,\gamma x)$ is a cellular space, and the generating
function of its cohomology is equal to the Lusztig--Kostant polynomial
$$
P(\oper{H}\nolimits^\bullet(F({\cal E}_\bullet,\gamma x)),t)={\cal K}_\gamma(t).
$$
\end{cor}
\begin{proof}
Apply \refeq{stratification}, \ref{saff}, \ref{hodge} and \ref{dims}.
\end{proof}
\begin{cor}\label{cck}
Let $D=\sum\limits_{r=1}^m\gamma_rx_r\in C^{\alpha-\beta}_\Gamma$.
The fiber $F({\cal E}_\bullet,D)$ is a cellular space and
\begin{equation}\label{ccke}
P(\oper{H}\nolimits^\bullet(F({\cal E}_\bullet,D)),t)={\cal K}_\Gamma
\stackrel{\text{\rm def}}{=}\prod_{r=1}^m{\cal K}_{\gamma_r}(t).
\end{equation}
\end{cor}
\begin{proof}
Apply \ref{gfibre}, \ref{hodge} and \ref{ck}.
\end{proof}
\begin{lem}\label{est}
Let $D=\sum\limits_{r=1}^m\gamma_r x_r$. We have
$$
\dim F({\cal E}_\bullet,D)\le\left|\sum_{r=1}^m\gamma_r\right|-m.
$$
\end{lem}
\begin{proof}
Note that for any $\kappa\in{\frak K}(\gamma_r)$ we have $K(\kappa)\ge1$, hence
$\deg{\cal K}_{\gamma_r}\le|\gamma_r|-1$. Now, the Lemma follows from \ref{cck}.
\end{proof}
\begin{proofof}{Main Theorem}
Consider the stratum $\DD\beta\Gamma$ of $\MMD\alpha$.
Its dimension is $2|\beta|+\dim{\cal B}+m$ and codimension is
$2|\alpha-\beta|-m$. The Lemma \ref{est} implies that the dimension
of the fiber of $\pi$ over the stratum $\DD\beta\Gamma$ is less than or equal
to $|\alpha-\beta|-m$, which is strictly less then the half
codimension of the stratum.
\end{proofof}
\subsection{Applications}
Let $\underline{\Bbb Q}$ denote the smooth constant Hodge irreducible module on $\MML\alpha$
(as a constructible complex it lives in cohomological degree
$-2|\alpha|-\dim{\cal B}$). Let $IC$ denote the minimal extension
of a smooth constant irreducible Hodge module from $\MM\alpha$ to
$\MMD\alpha$. It is well known that the smallness of $\pi$ implies
the following corollary.
\begin{cor}
$$
IC=\pi_*\underline{\Bbb Q}.
$$
\end{cor}
Now we can compute the stalks of $IC$ as
cohomology of fibers of $\pi$:
for $\varphi\in\MMD\alpha$ we have
$$
IC_{(\varphi)}^\bullet=\oper{H}\nolimits^\bullet(\pi^{-1}(\varphi),\underline{\Bbb Q})
$$
as graded Hodge structures.
\begin{cor}[Parity vanishing]
$$
IC_{(\varphi)}^{j}=0\quad\text{if $j-\dim{\cal B}$ is odd.}
$$
\end{cor}
\begin{proof}
Use \ref{cck}.
\end{proof}
\begin{cor}
For $\varphi\in\DD\beta\Gamma$ we have
$$
IC_{(\varphi)}^{-2|\alpha|-\dim{\cal B}+2j}={\Bbb Q}(j)^{{\frak k}_\Gamma(j)},
$$
where ${\frak k}_\Gamma(j)$ is the coefficient of $t^j$ in ${\frak K}_\Gamma(t)$.
\end{cor}
|
1996-10-15T18:02:52 | 9610 | alg-geom/9610017 | en | https://arxiv.org/abs/alg-geom/9610017 | [
"alg-geom",
"math.AG",
"math.QA",
"q-alg"
] | alg-geom/9610017 | Rodriguez Romo Suemi-FESC | V.K. Kharchenko, J. Keller, S. Rodriguez-Romo | Prime rings with PI rings of constants | 20 pages, LaTex2e, to appear in Israel Journal of Mathematics, volume
96, part B, 1996 (357-377) | null | null | null | null | It is shown that if the ring of constants of a restricted differential Lie
algebra with a quasi-Frobenius inner part satisfies a polynomial identity (PI)
then the original prime ring has a generalized polynomial identitiy (GPI). If
additionally the ring of constants is semiprime then the original ring is PI.
The case of a non-quasi-Frobenius inner part is also considered.
| [
{
"version": "v1",
"created": "Tue, 15 Oct 1996 16:59:22 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Kharchenko",
"V. K.",
""
],
[
"Keller",
"J.",
""
],
[
"Rodriguez-Romo",
"S.",
""
]
] | alg-geom | \section{Introduction}
Rings of constants of restricted differential Lie algebras with an outer
action on prime and semiprime rings were investigated in detail in
papers [Kh82], [Po83], [Pi86] (see also [Kh91,Ch.4, Ch6(6.4)]). In the
present paper we are going to consider actions with a nontrivial inner
part. In the papers [Ko91] and [Kh81] it is shown that the minimal restriction
required is that the inner part should be quasi-Frobenius (selfinjective).
We are interested in the structure of a prime ring $R$ provided it is known
that its ring of constants satisfies a polynomial identity. I.V.L'vov's
example [Lv93] shows that in this case the ring $R$ does not need to
be a PI-ring. We will show that in this case $R$ satisfies a generalized
polynomial identity.
The notion of a generalized polynomial identity was introduced by
S.A.Amitsur in [Am65]. In his paper S.A.Amitsur proved a structure
theorem for primitive rings with generalized polynomial identities.
Later W.S.Martindale [Ma69] generalized this result to arbitrary
prime rings. Using this theorem
we will prove that if the ring of constants is a semiprime PI-ring
and the inner part is quasi-Frobenius, then the ring $R$ is a PI-ring.
\section{Preliminaries}
Recall that a {\it derivation} of a ring $R$ is an additive mapping
$d:R \rightarrow R$ satisfying the condition
$(xy)^d=x^dy+xy^d.$
If $d_1, d_2$ are derivations then it is easy to see that the
commutator $[d_1,d_2]=d_1d_2-d_2d_1$ is also a derivation. Therefore
the set $Der R$ of all derivations of $R$ is a Lie subring in the ring
of endomorphisms of the abelian group $(R,+)$. Moreover, if $z$ is a central
element, then the composition of $d$ with the multiplication by $z$
is a derivation
$$(xy)^{dz}=z(xy)^d=(zx^d)y+x(zy^d)$$
In this case the operators of multiplication may not commute with derivations:
$x^{zd}\stackrel{\rm def}{=} (zx)^d=z^dx+zx^d$ or
$$zd=dz+z^d. \eqno(1)$$
Thus the set $Der R$ is a right module over the center $Z.$ The
module structure of $Der R$
is connected with the commutator operation by the formula
$$[dz,d_1]=[d,d_1]z+dz^{d_1}. \eqno(2)$$
Note that $z^{d_1}$ is again a central element:
$[z^{d_1},x]=[z^{d_1},x]+[z,x^{d_1}]=[z,x]^{d_1}$=0.
Finally, if the characteristic $p$ of the ring $R$ is nonzero, $pR=0,$
then the $p$th power of any derivation will be a derivation by the Leibniz
formula
$$(xy)^{d^p}=\sum_{k=0}^{k=p} C_p^k x^{d^k}y^{d^{p-k}}=x^{d^p}y+xy^{d^p}.$$
Now it is natural to formulate the following definition.
{\bf 2.1. Definition.} A set of derivations is called a {\it differential
restricted Lie $Z$-algebra}, or shortly a {\it Lie $\partial $-algebra}, if
it is a right $Z$-submodule of $DerR$ closed with respect to the operations
$[d_1,d_2]=d_1d_2-d_2d_1$ and $d^{[p]}=d^p.$
Note that the notion of a Lie $\partial $-algebra can be formalized
abstractly as a restricted Lie ring with a structure of right
$Z$-module connected with the main operations by formula (2) and the
following formula
$$(dz)^{[p]}=d^{[p]}z^p+d\cdot
(\ldots ((\overbrace{z^dz)^dz)^d\ldots )^d}^{p-1}z \eqno(3)$$
which follows from (1) (see details in [Kh91, pp. 6-11]; for a slightly more
general approach see in [Pa87]).
Now let $R$ be a prime ring. Denote by $R_{\cal F}$ its left Martindale ring
of quotients (see, for example, [Kh91 pp.19-24]), by $Q$ the symmetric Martindale
ring of quotients. Recall that the center $C$ of $R_{\cal F}$ is called the
{\it extended} (or {\it generalized}) centroid of $R$ and it is a field (see [Ma69]).
All derivations of $R$ can be uniquely extended to derivations of $Q$ and
of $R_{\cal F}.$ The extended derivations are characterized in $Der Q$ by the
property $R^d\subseteq R$ but the linear combinations over $C$ of extended
derivations do not satisfy this property. Therefore we have to consider more
general objects.
{\bf 2.2. Definition.} A derivation $d$ of $Q$ is called $R$-{\it continuous}
if there
exists a nonzero two-sided ideal $I$ of $R$ such that $I^d\subseteq R.$
It is easy to see that the set ${\cal D}(R)$ of all $R$-continuous derivations
is a differential restricted Lie $C$-subalgebra of $Der Q.$
In the present paper we consider Lie
$\partial $-algebras of $R$-continuous derivations which are finite
dimensional over $C.$
Let us fix the notations $R, C, Q, R_{\cal F}, {\cal D} (R)$
for a prime ring, its extended centroid, the symmetric Martindale ring of
quotients, the left Martindale ring of quotients
and the Lie $\partial $-algebra of $R$-continuous derivations,
respectively. Throughout the paper $L$ denotes a
restricted differential Lie $C$-algebra of $R$-continuous derivations,
$L\subseteq {\cal D} (R),$ finite dimensional over $C,$
and $R^L=\{ r\in R:\forall \mu \in L \ \
\ r^{\mu }=0\} $ is its ring of constants.
\section{The inner part of a Lie $\partial $-algebra}
If $a$ is an element of $Q$ then the map $a^{-}:x\rightarrow xa-ax$
is an $R$-continuous derivation, i.e. $Q^{-}\subseteq {\cal D}(R).$
{\bf 3.1. Definition.} The space $K(L)$ generated over $C$ by all
$q\in Q$ such that $q^{-}\in L$ is called the {\it inner linear part of $L.$}
It is clear that $C^{-}=0,$ therefore $K(L)$ contains $C$ and in particular
it contains the unit of $Q.$
{\bf 3.2. Lemma.} {\it The space $K(L)$ is a restricted Lie subalgebra of the
adjoint restricted Lie algebra $Q^{(-)}.$}
Recall that $Q^{(-)}$ is a restricted Lie algebra defined on the $C$-space
$Q$ with the operations $[q_1,q_2]=q_1q_2-q_2q_1,\ \ q^{[p]}=q^p.$
For the proof of the lemma it is enough to show that $K(L)$ is closed
with respect to these operations. This fact immediately follows from the
formulae
$$[a,b]^{-}=[a^{-},b^{-}] \eqno(4)$$
$$(a^p)^{-}=(a^{-})^{[p]}. \eqno(5)$$
{\bf 3.3. Lemma.} {\it $K(L)^{-}$ is equal to the subalgebra $L_{int}$
of all inner derivations of $L.$}
The proof is evident.
{\bf 3.4. Definition.} The associative subalgebra ${\cal B}(L)$
generated in $Q$ by $K(L)$ is called the {\it inner associative part of} $L.$
{\bf 3.5. Lemma.} {\it The algebra ${\cal B}(L)$ is of finite dimension
over $C.$}
{\bf Proof.} By the definition of operations in $K(L),$ the identity map
$id$ is a homomorphism of restricted Lie algebras
$id: K(L)\rightarrow {\cal B}(L)^{(-)}.$ Therefore ${\cal B}(L)$ as an associative
envelope of ${\cal B}(L)^{(-)}$ is a homomorphic image of the universal
restricted associative envelope $U_p(K(L)).$ The latter has dimension
$(\dim K(L))^p.$ The lemma is proved.
{\bf 3.6. Lemma.} {\it The algebra ${\cal B}(L)$ is stable under the action of
$L,$ i.e. ${\cal B}(L)^{\mu } \subseteq {\cal B}(L)$ for all} $\mu \in L.$
The proof follows from the formula
$$(q^{\mu })^{-}=[q^{-},\mu ]. \eqno(6)$$
\section{Differential operators}
Denote by $\Phi (L)$ the associative
subring generated in the endomorphism ring
of the abelian group $(Q,+)$ by $L$ and by the operators of left and
right multiplications by elements from ${\cal B}(L).$ By formula
(1) the ring $\Phi (L)$ may not be an algebra over $C.$ Of course
$\Phi (L)$ is an algebra over the subfield of central constants
$$F=C^L\stackrel{\rm def}{=} \{ c\in C:\forall l\in L \ \ c^l=0 \} .$$
Nevertheless $\Phi (L)$ is a left and a right space over $C$
while the subring of left multiplications, ${\cal B}(L)^l,$ and that of
right multiplications, ${\cal B}(L)^r,$ are algebras over $C.$
{\bf 4.1.} Let us fix derivations $\mu _1, \ldots ,\mu _m \in L$ such
that $\mu _1+K(L)^{-}, \ldots ,\mu _m+K(L)^{-}$
form a basis for the right $C$-space $L/K(L)^{-}.$ An operator $\Delta $
is called {\it correct} if it is of the form:
$$\Delta =\mu ^{s_1}_1\mu ^{s_2}_2 \ldots \mu ^{s_m}_m,$$
where $0\leq s_i<p$ and we suppose that $\mu ^0=1$ is the identity
operator.
Let $U$ be a right linear space generated by all correct operators. By
formula (1) this set will be a left space over $C,$ also.
{\bf 4.2. Proposition.} {\it The ring $\Phi (L)$ of differential operators
is isomorphic as a left and a right space over $C$ to a
tensor product over $C:$
$$\Phi (L)\simeq {\cal B}(L)^r\otimes {\cal B}(L)^l\otimes U\simeq
U\otimes {\cal B}(L)^l\otimes {\cal B}(L)^r, \eqno(7)$$
where $U$ is the linear space generated by correct operators over $C$.}
{\bf Proof.} It is enough to show that each differential operator
$d\in \Phi (L)$ has a unique representation in the form
$$d=\sum _{i,j,k} \alpha ^{(k)}_{ij}a^r_{ik}a^l_{jk}\Delta _k \eqno(8)$$
and a unique representation in the form
$$d=\sum _{i,j,k} \Delta _ka^l_{ik}a^r_{jk}\alpha ^{(k)}_{ij}, \eqno(9)$$
where $a_{ik}, a_{jk} \in A$ and $A$ is some fixed basis
of ${\cal B}(L)$ over $C$
(recall that by associativity, $a^r_{ik}a^l_{jk}=a^l_{jk}a^r_{ik}$)
and the $\Delta _k$'s
are correct words in $\{ \mu _1,\ldots ,\mu _m \} .$
The existence of this presentation follows from the relations
$$\mu a^r=a^r\mu -(a^{\mu })^r \eqno(10)$$
$$\mu a^l=a^l\mu -(a^{\mu })^l \eqno(11)$$
$$\mu ^p=\mu _1c_1+ \ldots +\mu _mc_m+b^r-b^l \eqno(12)$$
$$\mu _i\mu _j=\mu _j\mu _i+\mu _1c_1+ \ldots +\mu_mc_m +b^r-b^l, \eqno(13)$$
where in formula (12) \ $\mu _1c_1+ \ldots +\mu _mc_m+b^{-}$ \
is a representation
of $\mu ^p\in L$ as a linear combination of $\mu _i$'s modulo $K(L)^{-}$
and in (13) \ $\mu _1c_1+ \ldots +\mu _mc_m+b^{-}$ \ is the
corresponding representation
of $[\mu _i,\mu _j]\in L.$
The transformations of the left hand sides to the right hand sides
(in the last formula
only if $i>j$) allow us to reduce the operator to the form (8).
If we write formulae (10), (11) in the form
$$a^r\mu =\mu a^r+(a^{\mu })^r \eqno(14)$$
$$a^l\mu = \mu a^l+(a^{\mu })^l \eqno(15)$$
then in the same way the operator is reduced to the form (9).
For the proof of the uniqueness it is possible to use
the following results on differential
identities (see [Kh91, theorem 2.2.2, corollary 2.5.8] or [Kh78]).
{\bf 4.3. Proposition.} {\it If the derivations $\mu _1, \ldots ,\mu _m \in
{\cal D}(R)$ are linearly independent modulo $Q^{-},$ and if
the ring $R$ satisfies an identity of the type
$$\sum ^{p^n}_{k=1} \sum _i a_{ki} x^{\Delta _k}b_{ki}=0,$$
where $\Delta _1, \ldots , \Delta _{p^n}$ --- are all correct operators and the
coefficients $a_{ki},b_{ki}$ belong to $R_{\cal F},$ then
$\sum _ia_{ki} \otimes b_{ki}=0$ in $R_{\cal F} \otimes _C R_{\cal F}$
for all $k, 1\leq k \leq p^n.$ In the same way if the identity
$$\sum ^{p^n}_{k=1} (\sum _i a_{ki}xb_{ki})^{\Delta _k}=0$$
is valid then $\sum _i a_{ki}\otimes b_{ki}=0, 1\leq k\leq p^n.$}
Since ${\cal D}(I)={\cal D}(R)$ and $Q(I)=Q(R)$ for each nonzero ideal $I$
of $R$ (see [Kh91, Lemma 1.8.4]), then proposition 4.3 shows that the restriction of a nonzero
differential operator $d\in \Phi (L)$ to $I$ is nonzero. This note is
important due to the following lemma:
{\bf 4.4. Lemma.} {\it For each differential operator $d\in \Phi (L)$
there exists a nonzero two sided ideal $I$ of $R$ such that
$I^d\subseteq R.$}
The proof is easily obtained by induction from the formula
$(I^2)^{\mu }=I^{\mu }I+II^{\mu }\subseteq I$
which is valid for the ideal $I$ such that $I^{\mu }\subseteq R.$
\section{Quasi-Frobenius algebras}
Recall that a finite dimensional algebra $B$
over a field $C$ is called quasi-Frobenius if
one of the following equivalent conditions is valid (see [CR62]).
(Q1) {\it For each left ideal $\lambda $ and right ideal $\rho $ of $B$
the following equalities hold:
$$l(r(\lambda ))=\lambda ,\ \ \ r(l(\rho ))=\rho ,$$
where $l(A)=\{ b\in B: bA=0\} $ is the left annihilator,
$r(A)=\{ b\in B: Ab=0\} $ is the right annihilator.}
(Q2){\it The left regular module $_BB$ is injective.}
(Q3) {\it Modules $_BB$ and $(B_B)^*=Hom (B,C)$ have the same indecomposable
components.}
Recall that for any left (right) module $M$ the set of all linear
functionals $M^*$ has a structure of right (left) module defined by the
formula $(m^*b)(m)=m^*(bm)$ (respectively $m(bm^*)=(mb)m^*$).
The modules $M$ and $N$ for $N\simeq M^*$ are called
{\it conjugated modules}. If the
module $M$ is of finite dimension then $(M^*)^*\simeq M$ and the
conjugacy of modules (left and right), $M$ and $N,$ can be
characterized by the existence of a nondegenerate associative bilinear form
$(\ ,\ ):N\times M\rightarrow C.$ In this case for every basis
$a_1,\ldots ,a_n$ of $M$ there exists a {\it dual} basis
$a_1^*,\ldots ,a_n^*$ of $N$ which is characterized by the
following properties
$(a_i^*,a_i)=1, (a_i^*,a_j)=0, i\neq j.$
Condition (Q3) implies the following condition which is important for us:
(Q4) {\it The sum of all right ideals $\rho $ of $B$
conjugated to left ideals of $B$ is equal to
$B.$}
It can be proved that this condition is also equivalent to $B$ being
quasi-Frobenius. Moreover, as (Q1) is left-right symmetric then the left
analog of (Q4) is also valid.
(Q5) {\it The sum of all left ideals ${\lambda }$ of $B$ conjugated to right
ideals of $B$ is equal to $B.$}
The most important subclass of the class of quasi-Frobenius algebras is
the class of Frobenius algebras. These algebras are defined by one of the
following equivalent conditions ([CR62]).
(F1) {\it For each left ideal $\lambda $ and right ideal $\rho $ of $B$
the following equalities hold:
$$l(r(\lambda ))=\lambda ,\ \ \dim r(\lambda )+\dim \lambda =\dim B$$
\ $$ r(l(\rho ))=\rho , \ \ \dim l(\rho )+\dim \rho=\dim B.$$ }
(F2) {\it There exists an element
$\varepsilon \in B^*$ whose kernel contains
no nonzero onesided ideals of $B.$}
(F3) {\it There exists a nondegenerate associative bilinear form
$B\times B\rightarrow C.$ }
(F4) {\it The modules $_BB$ and $(B_B)^*$ are isomorphic.}
Classical examples of Frobenius algebras are: group algebras of finite
groups over a field of arbitrary characteristic, universal restricted
enveloping algebras of finite dimensional Lie $p$-algebras, finite dimensional
Hopf algebras, Clifford algebras. Finite dimensional semisimple algebras
evidently satisfy (F1) therefore they are Frobenius.
\section{Universal constants}
Let $\lambda $ and $\rho $ be left and right conjugated ideals of
${\cal B}(L).$ Let us choose a basis $a_1, \ldots ,a_n$ of $\lambda $
and let $a_1^*, \ldots ,a_n^*$ be the dual basis of $\rho .$
It is well-known that the element $c=\sum a_i\otimes a_i^*$ of the
tensor product ${\cal B}\otimes _C{\cal B}$ commutes with the elements of
${\cal B}, \ \ bc=cb$ for all $b\in {\cal B}.$ This implies that the
set of values of the operator $c_{\lambda ,\rho }=\sum a_i^l(a_i^*)^r$
is contained in the centralizer of ${\cal B}.$ In particular for any
$\mu \in K(L)^-$ we have $$c_{\lambda ,\rho }(x)^{\mu }=0. \eqno(16) $$
Let $U(L)$ be the associative subring of $\Phi (L)$ generated by $L$
and by the operators of multiplication by central elements. It is clear that
$U(L)$ is both a left and a right space over $C$ and an algebra over the field
of central constants $F=C^L.$
Consider the right ideal $I=K(L)^-\cdot U(L)$ of $U(L).$
First of all the formula
$\mu a^-=a^-\mu -(a^{\mu })^-$ shows that $I$ is a two sided ideal of $U(L).$
The same formula and formulae (12), (13)
show that the identity operator and operators of the
form $a_1^-a_2^-\ldots a_s^-\Delta ,$
where $\Delta $ is a correct operator, $a_i\in K(L), \ s\geq 0,$
generate $U(L)$ as a left space over $C.$
{\bf 6.1. Proposition.} {\it The factor-algebra $U(L)/I=\overline{U}$
is Frobenius as
an algebra over $F=C^L.$}
{\bf Proof.} By the well-known R.Baer theorem [Ba27] the dimension of $C$
over $F$ is finite and therefore $\overline{U} =U(L)/I$ has a finite dimension
over $F.$ Since $K(L)^-\subseteq I,$ the elements $\bar{\mu }_1=\mu _1+I,
\ldots , \bar{\mu }_m=\mu _m+I$ generate $\overline{U} $ as a ring over $C.$
Moreover the relations $\bar{\mu } _i\bar{\mu }
_j=\overline{[\mu _i,\mu _j]}+\bar{\mu } _j\bar{\mu } _i$ show that the
images of
correct words
$\bar{\Delta } _k$ generate $\overline{U}$ as a left vector space over $C.$
The main note is that the elements $\bar{\Delta } _k$
are linearly independent over $C.$ If
$$\sum _kc_k\Delta _k=\sum _k d_k\Delta _k\in I,$$
where $d_k$ are linear combinations of products of the type
$a_1^-\cdots a_s^-,$ then taking into account that $a^-=a^r-a^l$
and using Proposition 4.3, we have $c^r_k=d_k$ for all $k,$
which is impossible since $c_k^r(1)=c_k,\ d_k(1)=0.$
Thus $\bar{\Delta }_k$ are linearly independent.
Now let us define Berkson's linear map
(see [Be64]) $\varphi :\overline{U} \rightarrow C$ which
corresponds to
the element $\sum c_k\bar{\Delta }_k$
the coefficient of $\bar{\Delta }_{p^m}=\bar{\mu }^{p-1}_1\ldots \bar{\mu
}^{p-1}_m.$ The kernel of this linear map contains neither left nor right
nonzero ideals, since the product
$$(\bar{\mu }_1^{s_1}\ldots \bar{\mu }_m^{s_m})(\bar{\mu }^{p-s_1-1}_1\ldots
\bar{\mu }^{p-s_m-1}_m)$$ written as a linear combination of correct words
contains a unique member $\bar{\Delta }_{p^m}$ with a coefficient
equal to 1.
If $\psi :C\rightarrow F$ is any projection, then the linear functional
$\varepsilon :d\mapsto \psi (\varphi (d))$ satisfies (F2) and therefore
$\overline{U}$ is a Frobenius algebra. The proposition is proved.
Let us consider the right subspace $\hat{U}$ of $\overline{U}$ over $C$
generated by all nonempty words $\bar{\Delta }_k.$ This space does not
contain the unit (the identity operator) and it is a right
(but, possibly, not
a left) ideal because by formula (14) one has
$$\bar{\Delta }c\bar{\mu }=\bar{\Delta }\bar{\mu }c+\bar{\Delta }c^{\mu }.$$
By formula (13), the product $\bar{\Delta }\bar{\mu }$ can be
written as a linear combination $\sum \bar{\Delta }_kc_k,$ where
$\bar{\Delta }_k$ are nonidentity correct operators.
Thus, the left annihilator $A=l(\hat{U})$ in the algebra $\overline{U}$ is not
equal to zero. Moreover, by (F1) its dimension over $F$ is connected with
the dimension of $\hat{U}$ by the formula
$ \dim _F\overline{U}=\dim _F\hat{U}+\dim _FA.$ On the other hand
$\dim _F\overline{U}=\dim _F\hat{U}+\dim _FC$ i.e. the dimensions of $A$
and $C$ over $F$ coincide. It means that $A$ is one dimensional over $C$
i.e. $A=C\bar{f}$ (but possibly $A\neq \bar{f}C$ as $A$ may not be a
right $C$-space), where $\bar{f}=\sum \bar{\Delta }_kc_k=\sum
c_k^{\prime }\bar{\Delta }_k$ is a nonzero element of $\overline{U}.$
Thus, we have obtained that $\bar{f}\bar{\mu }_i=\bar{0}$ in $\overline{U}.$
In the ring of differential operators this means that
$f\mu _i\in K(L)^{-}\cdot U(L).$ We have also that
$fK(L)^{-}\subseteq K(L)^{-}\cdot U(L)$ as $I=K(L)^{-}\cdot U(L)$
is a two sided ideal. Thus
$$fL\subseteq f(\sum (\mu _iC+K(L)^{-}))\subseteq K(L)^{-}\cdot U(L)$$
which, using formula (16), implies
$$((c_{\lambda ,\rho } (x))^f)^{\mu }=0 \eqno(17)$$
for all ${\mu \in L}.$ Let us formulate the obtained result as a lemma
(see also Lemma 4.6, [Kh95]).
{\bf 6.2. Lemma.} {\it There exists a differential operator $f$ of the type
$\sum \Delta _kc_k=\sum c^{\prime }_k\Delta _k,$ such that for each
conjugated left ideal $\lambda $ and right ideal $\rho $ of ${\cal B}$
with dual bases $a_1, \ldots , a_n$ and $a_1^*,\ldots ,a_n^*,$ the
operator
$$u_{\lambda , \rho }=\sum_i a_i^l(a_i^*)^rf \eqno(18)$$
has values only in the ring of constants $Q^L.$ There exists a nonzero
ideal $I$ of $R$ such that
$$0\neq u_{\lambda ,\rho }(I)\subseteq R^L \eqno(19)$$}
{\bf Proof.} The representation of
$f$ in the form $\sum c^{\prime }_k\Delta _k$
follows from (10).
Formula (19) follows from formula (17), proposition 4.3 and lemma 4.4.
\section{PI rings of constants}
In this secton we will prove the theorem about a generalized polynomial
identity and discuss
its generalization to the case when the inner part is not quasi-Frobenius.
{\bf 7.1. Theorem.} {\it Let $L$ be a finite dimensional restricted
differential Lie $C$-algebra of $R$-continuous derivations of a prime
ring $R$ of positive characteristic $p>0.$ Suppose that the inner associative
part ${\cal B}(L)$ of $L$ is quasi-Frobenius. If the ring of constants $R^L$
is PI then $R$ is GPI.}
{\bf Proof.} Let $f(x_1,\ldots ,x_n)=0$ be a multilinear identity
of $R^L.$ Let us choose arbitrary left ideals $\lambda _1,\ldots ,\lambda _n$
of ${\cal B}(L)$ having conjugated right ones $\rho _1, \ldots ,\rho _n.$
By Lemma 6.2 for every $j, 1\leq j\leq n$ there exists an operator
$$u_j=u_{\lambda _j,\rho _j}=\sum_i a^l_{ij}(a_{ij}^*)^rf_j=\sum_{i,k}
a_{ij}^l(a_{ij}^*)^rc_k^{\prime }\Delta _k$$
and a nonzero ideal $I_j$ of $R,$ such that $0\neq u_j(I_j)\subseteq R^L.$
If $I=\cap I_j$ then $u_j(x)\in R^L$ for all $x\in I$
and therefore the following differential identity holds in $I$ \
$$f(u_1(x_1),\ u_2(x_2),\ldots ,\ u_n(x_n))=0.$$
Let us fix some values of $x_2=b_2, \ldots ,x_n=b_n$ in $I.$ We have
$$f(\sum_{i,k} (c_k^{\prime }a_{i1}x_1a_{i1}^*)^{\Delta _k},\ u_2(b_2),\ldots
,\ u_n(b_n))=0. \eqno(20)$$
By Leibnitz formula any expression of the type $(axb)^{\Delta }$
can be written in the form
$$(axb)^{\Delta }=ax^{\Delta }b+\sum_s a_sx^{\Delta _s}b_s,$$
where $\Delta _s$ are subwords of $\Delta .$ In particular
$$(c_k^{\prime }a_{i1}x_1a_{i1}^*)^{\Delta _k}=
c_k^{\prime }a_{i1}x_1^{\Delta _k}a_{i1}^*+\sum_s a_sx_1^{\Delta _s}b_s.
\eqno(21)$$
If $\Delta _{k_0}$ is the greatest operator such that $c^{\prime }_{k_0}$
is not zero, then this formula allows us to represent (20) in the form
$$\sum^{k_0}_{k=1} \sum_i v_{ki}x_1^{\Delta _k}w_{ki}=0,$$
here we suppose that $\Delta _1<\Delta _2<\ldots <\Delta _{p^m}$ is the
lexicographic ordering of all correct operators.
By Proposition 4.3 applied to the prime ring $I$ we have
$$\sum_i v_{k_0i}\otimes w_{k_0i}=0$$
in the tensor product $I_{\cal F}\otimes _{C(I)}I_{\cal F},$ where
$C(I)$ is the generalized centroid of $I$ and $I_{\cal F}$ is the left
Martindale ring of quotients of $I.$ It is well-known and it is easy to
see that $I_{\cal F}=R_{\cal F}$ and $C(I)=C(R).$ Therefore for any
$x_1\in R_{\cal F}$ we have the identity
$$\sum_i v_{k_0i}x_1w_{k_0i}=0.$$
This identity with (21) and (20) implies that the identity
$$c_{k_0}^{\prime }f(\sum_i a_{i1}x_1a_{i1}^*, u_2(b_2),\ldots
,u_n(b_n))=0 \eqno(22)$$
is valid for each $x_1\in R_{\cal F}.$
Since the values $b_2,\ldots ,b_n$ are arbitrary from $I$, we have an identity
of the form
$$f(\sum_i a_{i1}x_1a_{i1}^*,u_2(x_2),\ldots ,u_n(x_n))=0, \eqno(23)$$
where $x_1\in R_{\cal F}, x_2\in I, \ldots ,x_n\in I.$
Now let us fix values
$x_1\in R_{\cal F}, x_3=b_3\in I, \ldots ,x_n=b_n \in I.$
Then in the same way we obtain
$$f(\sum_i a_{i1}x_1a_{i1}^*,\sum_i a_{i2}x_2a_{i2}^*,\ldots
,u_n(x_n))=0,$$
where $x_1,x_2\in R_{\cal F}, x_3,\ldots x_n \in I.$
Continuing this process we will obtain the following identity on $R_{\cal F}:$
$$f(\sum_i a_{i1}x_1a_{i1}^*,\sum_i a_{i2}x_2a_{i2}^*,\ldots
,\sum_i a_{in}x_na_{in}^*)=0, \eqno(24)$$
This is a generalized identity valid in $R_{\cal F}\supseteq R.$ All we need
is to prove that for some $\lambda _1,\ldots, \lambda_n;\rho _1, \ldots
,\rho _n$ this is not a trivial identity. It means that the left hand side
of (24) is not zero in the free product $R_{\cal F} *_CC\langle
x_1,\ldots ,x_n\rangle $
or, in other words, this identity does not follow from the trivial
generalized identities
$xc=cx,$ where $c\in C.$
Otherwise assume all these identities are trivial.
Any application
of a trivial
identity does not change the order of the indeterminates, therefore
all the generalized monomials (i.e. sums of all monomials with fixed order
of sequence of the indeterminates) in the identities (24) should be (trivial)
identities. These generalized monomials have the form
$$\alpha _{\pi }(\sum_i a_{i\pi (1)}x_{\pi (1)}a^*_{i\pi (1)})
(\sum_i a_{i\pi (2)}x_{\pi (2)}a^*_{i\pi (2)})\cdots
(\sum_i a_{i\pi (n)}x_{\pi (n)}a^*_{i\pi (n)}),$$
where $\pi $ is a permutation and
$$f(x_1,\ldots x_n)=\sum_{\pi }\alpha _{\pi}x_{\pi (1)}\cdots x_{\pi (n)}.$$
Since one of the coefficients $\alpha _{\pi }$ is equal to one
(let $\alpha _1=1$),
$$(\sum_i a_{i1}x_1a^*_{i1})(\sum_i a_{i2}x_2a^*_{i2})\cdots
(\sum_i a_{in}x_na^*_{in})=0 \eqno(25)$$
Let us fix some values of $x_2,\ldots ,x_n$ in $R$ and apply Proposition 4.3
to (25), where we suppose $x=x_1,$ and all coefficients $a_{ki}, \ \ k=2,3,\ldots
p^m$ are zero. We have
$$(\sum_ia_{i1}\otimes a_{i1}^*)(\sum_i a_{i2}x_2a^*_{i2})\cdots
(\sum_i a_{in}x_na^*_{in})=0.$$
The set $\{ a_{i1}\} $ is a basis of the ideal $\lambda _1,$
i.e. this is a linearly independent set, therefore
$$a_{i1}^*(\sum_i a_{i2}x_2a^*_{i2})\cdots
(\sum_i a_{in}x_na^*_{in})=0$$
for all $a_{i1}^*$ from the dual basis $\{ a_{i1}^*\} $ of the conjugated
ideal $\rho _1.$ This implies that
$$\rho _1(\sum_i a_{i2}x_2a^*_{i2})\cdots
(\sum_i a_{in}x_na^*_{in})=0.$$
Since the pair $(\lambda _1,\rho _1)$ was chosen in an arbitrary way,
$$(\sum_{\rho ^*\simeq\ a\ left\ ideal\ of\ {\cal B}}\rho )
(\sum_i a_{i2}x_2a^*_{i2})\cdots
(\sum_i a_{in}x_na^*_{in})=0. \eqno(26)$$
By Property (Q5) of quasi-Frobenius algebras $$1\in {\cal B}=
(\sum_{\rho ^*\simeq\ a\ left\ ideal\ of\ {\cal B}}\rho ) $$ and
therefore
$$(\sum_i a_{i2}x_2a^*_{i2})\cdots
(\sum_i a_{in}x_na^*_{in})=0.$$
Now the evident induction works. The theorem is proved.\
The same proof can be applied also for some cases when the inner
part is not quasi-Frobenius but has enough pairs of conjugated
one-sided ideals. Indeed, let us denote by ${\cal B}_r$ the sum
of all right ideals of a finite dimensional algebra ${\cal B}$
conjugated to left ones.
{\bf 7.2. Lemma.} ${\cal B}_r$ {\it is a two-sided ideal of} ${\cal B}.$
{\bf Proof.} Let $\rho $ be a right ideal such that the dual left
module $\rho ^*={\rm Hom}(\rho ,C)$ is isomorphic to a left ideal
$\lambda .$ If $b\in {\cal B}$ then we have an exact sequence of homomorphisms
of right ideals $\rho \rightarrow b\rho \rightarrow 0.$ The conjugated
sequence has the form $\rho ^*\leftarrow (b\rho )^*\leftarrow 0,$
therefore the right ideal $b\rho $ has a conjugated module $(b\rho )^*$
which is isomorphic to a left subideal of $\lambda \simeq \rho ^*.$
Thus $b\rho \subseteq {\cal B}_r$ and ${\cal B}_r$ is a two-sided ideal.
The lemma is proved.
In the same way one can define an ideal ${\cal B}_l$ --- the sum of all
left ideals conjugated to right ones.
{\bf 7.3. Theorem.} {\it Let $L$ be a finite dimensional restricted
differential Lie $C$-algebra of $R$-continuous derivations of a prime
ring $R$ of positive characteristic $p>0.$ If the algebra of
constants $R^L$ satisfies a multilinear polynomial identity of degree
$n$ and ${\cal B}(L)^n_r\neq 0,$ then $R$ is a GPI-ring.}
{\bf Proof.} In the same way as in the proof of Theorem 7.1 we have
identities (24). If all of these identities are trivial then we also have
the identities (26)
which can be written in the form
$${\cal B}(L)_r(\sum a_{i2}x_2a_{i2}^*)\cdots (\sum
a_{in}x_na_{in}^*)=0. \eqno(27)$$
If $b$ is an arbitrary element of ${\cal B}(L),$ then
$b(\sum_i a_{ik}x_ka_{ik}^*)=(\sum _i a_{ik}x_ka_{ik}^*)b.$
Therefore for $b\in {\cal B}(L)_r,$ identity (27) implies
$$(\sum a_{i2}x_2a_{i2}^*)\cdots (\sum
a_{in}x_na_{in}^*)b=0. \eqno(28)$$
By Proposition 4.3 we have
$$(\sum a_{i2}\otimes a_{i2}^*)\cdots (\sum
a_{in}x_na_{in}^*)b=0,$$
as in the proof of Theorem 7.1 we have
$${\cal B}(L)_r(\sum a_{i3}x_3 a_{i3}^*)\cdots (\sum
a_{in}x_na_{in}^*)b=0,$$
thus
$$(\sum a_{i3}x_3 a_{i3}^*)\cdots (\sum
a_{in}x_na_{in}^*){\cal B}(L)_r^2=0.$$
Now the evident induction implies ${\cal B}(L)_r^n=0.$ Hence
one of the GPI's (24) is not trivial. The theorem is proved.
In a symmetrical way one can prove that the condition ${\cal B}(L)^n_l\neq 0$
also implies that one of the identities (24) is not trivial. It can be proved
that ${\cal B}^n_r=0$ iff ${\cal B}^n_l=0:$
{\bf 7.4. Proposition.} {\it Let ${\cal B}$ be a finite dimensional algebra.
Then all $n+1$ conditions ${\cal B}_r^k{\cal B}_l^{n-k}=0, \ \ k=0,\ldots ,n$
are equivalent to each other.}
{\bf Proof.} It is enough to show that the conditions for $k$
and $k+1$ are equivalent. The condition ${\cal B}_r^k{\cal B}_l^{n-k}=0$
is equivalent to
${\cal B}_r^k{\cal B}_l^{n-k-1}\lambda =0$ for
all pairs of conjugated ideals $\rho , \lambda .$ Since the form
$(\ ,\ ):\rho \times \lambda \rightarrow C$ is
nondegenerate the last condition
for given $\lambda , \rho $ is equivalent to
$(\rho ,{\cal B}_r^k{\cal B}_l^{n-k-1}\lambda )=0.$ By
associativity of the form this is equivalent to
$(\rho {\cal B}_r^k{\cal B}_l^{n-k-1},\lambda )=0$ and since
the form is nondegenerate
this is equivalent to $\rho {\cal B}_r^k{\cal B}_l^{n-k-1}=0.$
The last conditions for all pairs of conjugated ideals $\lambda , \rho $
are equivalent to ${\cal B}_r^{k+1}{\cal B}_l^{n-k-1}=0.$ The proposition
is proved.
Now it is a question of interest whether the condition ${\cal B}(L)^n_r=0$
implies that all identities (24) are trivial generalized polynomial
identities. The answer is yes:
{\bf 7.5. Proposition.} {\it If under the conditions of theorem {\rm 7.3}
\ ${\cal B}(L)_r^n=0,$ then all identities {\rm (24)} are trivial.}
{\bf Proof.} It is enough to show that all the generalized monomials (25)
are trivial identities. We will prove by inverse induction on $k$
that for arbitrary
$b_1, \ldots ,b_k\in {\cal B}(L)_r$ the generalized polynomial
$$(\sum _ia_{i\ k+1}x_{k+1}a_{i\ k+1}^*)\cdots
(\sum _ia_{in}x_na_{in}^*)b_kb_{k-1}\cdots b_1=0 \eqno(29)$$
is a trivial generalized identity.
If $k=n$ then (29) has the form $b_nb_{n-1}\cdots b_1=0$ that is a trivial
identity as ${\cal B}(L)^n_r=0.$
Assume that (29) is a trivial identity. The identities
$$b(\sum
_i a_{is}xa_{is}^*)=(\sum
_i a_{is}xa_{is}^*)b, \ \ b\in {\cal B}(L) \eqno(30)$$
are trivial generalized polynomial identities (as well as any
linear generalized identity). Let $b_k=a_{ik}^*,$ then from (29) and (30)
we have the following trivial identity
$$a_{ik}^*(\sum _ia_{i\ k+1}x_{k+1}a_{i\ k+1}^*)\cdots
(\sum _ia_{in}x_na_{in}^*)b_{k-1}\cdots b_1=0.$$
Multiplication of this equality on the left by $a_{ik}x_k$ and summation
over $i$ gives the equality (29) with a smaller $k.$ The proposition is
proved.\
\section{Semiprime PI-rings of constants}
In this secton we will prove under the conditions of Theorem 7.1,
that if
the ring of constants $R^L$ is a semiprime PI-ring, then $R$ is also PI.
{\bf 8.1. Theorem.} {\it Let $L$ be a finite dimensional restricted
differential Lie $C$-algebra of $R$-continuous derivations of a prime
ring $R$ of positive characteristic $p>0.$ Suppose that the inner associative
part ${\cal B}(L)$ of $L$ is quasi-Frobenius. If the ring of constants $R^L$
is a semiprime PI-ring, then $R$ is PI.}
{\bf Proof.} By Theorem 7.1 the ring $R$ satisfies a generalized polynomial
identity. Moreover all generalized polynomial identities (24) hold
in its left Martindale ring of quotients $R_{\cal F}.$ In particular they hold
in the central closure $RC\subseteq R_{\cal F}$ of the ring $R.$
By the Martindale structure theorem
[Ma69] this central closure has an idempotent
$e,$ such that $D=eRCe$ is a skew field of finite dimension
over $C.$ (Note that formally Martindale
theorem can be applied only if the
coefficients of the identity belong to $R.$ In our case they belong to
$R_{\cal F}$ but may not belong to $R.$ Nevetheless Martindale's original
proof is correct for our case too; see, for instance, [Kh91, Theorem 1.13.4]
or the special investigation in [La86].)
Thus, by the Martindale theorem, $RC$ is a primitive ring with a nonzero socle.
The N. Jacobson structure theorem [Ja64] shows that $RC$ is a dense subring in
the finite topology in the complete ring ${\cal E}$ of linear transformations
of the left space $V=eRCe$ over the skew field $D.$
Moreover, the left Martindale quotient ring $(RC)_{\cal F}$ is equal to
${\cal E}$ (see, [Ha82, Lemma 1.1] and [Ha87, Remark 4.9] or
[Kh91, Theorem 1.15.1]). It is easy to see
that $R_{\cal F} \subseteq (RC)_{\cal F}={\cal E}.$ (Indeed, if $q\in R_{\cal
F}$ and $Iq\subseteq R$ for a nonzero ideal $I$ of $R,$ then we can extend
$q$ to the ideal $IC$ of $RC$ by the obvious formula
$(\sum i_{\alpha }c_{\alpha })q=\sum (i_{\alpha }q)c_{\alpha }.$
This is well-defined. Indeed, if $\sum i_{\alpha }c_{\alpha }=0$
and $J$ is a nonzero
ideal of $R$
such that $Jc_{\alpha }\subseteq R$ then $\sum (jc_{\alpha })i_{\alpha }=0$
for all $j\in J.$ Therefore $\sum (jc_{\alpha })(i_{\alpha }q)=0;$ i.e.
$J(\sum c_{\alpha }(i_{\alpha }q))=0$ and $\sum (i_{\alpha }q)c_{\alpha }=0.$)
Now all the coefficients of (24) belong to
${\cal E}$ and since addition and multiplication are continuous in the finite
topology, the identities (24) hold in ${\cal E}.$ (Here one can use
also Corollary 2.3.2 from [Kh91] which
allows us to extend identities from $RC$ to
$(RC)_{\cal F}.$)
Now we are going to prove that the space $V$ is finite dimensional over $D.$
In that case the dimension of ${\cal E}$ over $C$ will also be finite:
$d=\dim _C{\cal E}= (\dim _DV)^2 \cdot \dim _CD$ and ${\cal E}$ (and
therefore $R$),
like any $d$-dimensional algebra, will
satisfy the standard polynomial identity:
$$S_d(x_1, \ldots , x_{d+1})\equiv \sum (-1)^{\pi }x_{\pi (1)}\cdots x_{\pi
(d+1)}=0.$$
On the contrary, suppose that $V$ has infinite dimension $\dim V=\beta .$
Let $M$ be the set of all linear transformations whose rank is less then $\beta .$
(Recall that the {\it rank } of a transformation $l$ is the
dimension over $D$ of
its image.) It is well-known that $M$ is a maximal ideal of ${\cal E}.$
So the factor ring $\bar{\cal E}={\cal E}/M$ is a simple ring with a unit.
{\bf 8.2. Lemma.} { \it The ring $\bar{\cal E}$ is not Artinian.}
{\bf Proof.} Let $\{ e_i, i\in I\} $ be a basis of $V$ over $D.$ and
$$I_1\supset I_2\supset \ldots \supset I_n\supset \ldots $$ be a chain of
subsets such that $|I_k \setminus I_{k+1}|=\beta ,$ and let
$$A_n=\{ l\in {\cal E}:e_il=0 \ \ \forall i \in I\setminus I_n\} .$$
Then
$$ (A_1+M)/M\supset A_2+M/M\supset \ldots \supset A_n+M/M\supset \ldots $$
is an infinite descending chain of right ideals of $\bar{\cal E}.$
Indeed, if $A_n+M=A_{n+1}+M,$ then for the transformation $w,$ defined by
\[ e_iw=\left\{ \begin{array}{ll}
e_i & \mbox{if $i\in I_n\setminus I_{n+1}$} \\
0 & \mbox{otherwise}
\end{array}
\right. , \]
we should get a presentation $w=a+m,$ where $a\in A_{n+1}, \ m\in M.$
Let $V_1$ be a subspace generated by $\{e_i:i\in I_n\setminus I_{n+1} \} .$
Then $V_1=V_1w\subseteq V_1a+V_1m=V_1m.$ However, $\dim _DV_1=\beta ,$
while $\dim _DV_1m\leq \dim Vm<\beta ,$ which is a contradiction. The lemma is
proved.\
{\bf 8.3. Lemma.} {\it The ring $\bar{\cal E}$ does not satisfy a non trivial
generalized polynomial identity.}
{\bf Proof.} Like any simple ring with a unit, the ring $\bar{\cal E}$ is
primitive. If it satisfies a GPI, then by the S.A. Amitsur structure theorem
[Am65] it has a nonzero socle $S,$ which is a two-sided ideal and
therefore $S=\bar{\cal E}.$ In N. Jacobson presentation of $\bar{\cal E}$
as a dense ring of linear transformations, the socle consists of all
transformations of finite rank. This means that the unit has finite
rank and therefore the space has finite dimension. Thus $\bar{\cal E}$
is the ring of all linear transformations of a finite dimensional space
over a skew field. In particular $\bar{\cal E}$ is Artinian; this is a
contradiction
to Lemma 8.2. The lemma is proved.
Let us consider now identities (24). We have seen that all these identities
hold in ${\cal E}.$ If we apply the natural homomorphism
$\varphi :{\cal E}\rightarrow \bar{\cal E}={\cal E}/M$ we obtain
the following identities of the ring $\bar{\cal E}$
$$f(\sum _i\bar{a}_{i1}x_1\bar{a}^*_{i1},\ldots ,\sum
_i\bar{a}_{in}x_n\bar{a}^*_{in})=0, \eqno(31)$$
where $\bar{a}=\varphi (a)=a+M.$
By Lemma 8.3 all we need is to prove that
one of the identities (31) is a nontrivial
GPI of $\bar{\cal E}.$
First of all we have to calculate the generalized centroid of $\bar{\cal E}.$
As $\bar{\cal E}$ is a simple ring with a unit, it equals its left Martindale
quotient ring and therefore the generalized centroid is equal to the center.
{\bf 8.4. Lemma.} {\it The center of
$\bar{\cal E}$ is canonically isomorphic
to $C, \ \ C(\bar{\cal E})=\varphi (C).$}
{\bf Proof.} See [Ro58, Corollary 3.3].
We will need the following result which gives a criterium
for determining when the
ring of constants is semiprime (see Theorem 5.1 [Kh95]).
{\bf 8.5. Theorem.} {\it Under the conditions of theorem 8.1, the ring of
constants is semiprime if and only if ${\cal B}(L)$ is differentially
semisimple, i.e. it has no nonzero differential (with respect to
action of $L$) ideals with zero
multiplication or, equivalently, it is a sum of a finite number of
differentially simple algebras.}
By this theorem we have that in our situation the algebra ${\cal B}(L)$
is differentially semisimple.
{\bf 8.6. Lemma.} {\it The ideal $M$ is a differential ideal with
respect to $L,$ i.e. $M^{\mu }\subseteq M$ for each $\mu \in L.$}
{\bf Proof.} Note that $M$ is a differential ideal with respect to
each derivarion of ${\cal E}.$ Indeed, if $l\in M$ than $l$ is a
transformation of rank less then $\beta $ and the projection
$e:V\rightarrow {\rm im} l$ also has rank less than $\beta ,$
in which case $l=le.$ We have $l^{\mu }=l^{\mu }e+le^{\mu }\in M$
for each derivation $\mu \in {\rm Der}({\cal E}).$
By proposition 1.8.1 [Kh91] any $R$-continuous derivation has a unique
extension to $R_{\cal F}.$ In particular each derivation from $L$
is defined on $RC.$ Again by the same proposition we have that
the elements of $L$ have a unique extensions
to $(RC)_{\cal F}={\cal E}.$ Thus we have
obtained that the ideal $M$ is differential with respect to $L.$
The lemma is proved.
As a consequence we have that the intersection $M_0=M\cap {\cal B}(L)$ is
a differential
ideal of ${\cal B}(L),$ which is not equal to ${\cal B}(L)$
(it does not contain 1). The left
annihilator $l(M_0)$ of $M_0$ in ${\cal B}(L)$ is also a differential
ideal, therefore $l(M_0)\cap M_0 $ is a differential ideal with zero
mulitiplication. By theorem 8.5, $l(M_0)\cap M_0=0.$ In the same way
the left annihilator of the sum $l(M_0)+M_0$ is zero (it is contained in
$l(M_0)$ and, therefore, has a zero multiplication). Now property (Q1)
of quasi-Frobenius algebras implies that
$l(M_0)+M_0=r(l(l(M_0)+M_0))=r(0)={\cal B}(L)$ and, finally
$${\cal B}(L)=l(M_0)\oplus M_0=e{\cal B}(L) \oplus (1-e){\cal B}(L), \eqno(37)$$
where $e$ is a central idempotent defined by the corresponding decomposition
of the unit $1=e\oplus (1-e).$
Let us return to identities (31). Suppose that in these identities
$\{ a_{ij}\} $ and $\{ a_{ij}^* \} $ are bases of conjugated ideals
$\lambda _j, \rho _j$ contained in $l(M_0).$ In that case the sets
$A_j=\{ \bar{a}_{ij}, i=1, \ldots m \} $ are linearly independent over
the center of $\bar{\cal E}$ (see lemma 8.4). Moreover, the $C$-space
generated by all possible $a_{ij}^*$'s contains the
unit $e$ of $l(M_0)$ because for each
conjugated pair of ideals $\lambda ,\rho $
the one-sided ideals $e\lambda ,e\rho $ are also conjugated with
respect to the same
form (note that $e$ is a central idempotent of ${\cal B}(L)$).
This implies that the linear space over the center of $\bar{\cal E}$
generated by all $\bar{a}_{ij}^*$'s contains the unit $\bar{e}$ of
$\bar{\cal E}.$ This fact allows us to prove that one of the identities
(31) is nontrivial in the same manner as it was done in the end of
the proof of Theorem 7.1. By Lemma 8.3, Theorem 8.1 is proved.
In this proof we used the fact that the
inner part ${\cal B}(L)$ is differentially
semisimple and that it has enough pairs of conjugated ideals.
Therefore in the way analogous to Theorem 7.3 we can formulate a
slightly more general result.
{\bf 8.7. Theorem.} {\it Let $L$ be a finite dimensional restricted
differential Lie $C$-algebra of $R$-continuous derivations of a prime
ring $R$ of positive characteristic $p>0.$ Suppose that the inner
part ${\cal B}(L)$ is a direct sum of differentially simple ideals
$${\cal B}(L)=B_1\oplus B_2\oplus \ldots \oplus B_m.$$
If the algebra of constants $R^L$ satisfies a multilinear polynomial
identity of degree $n$ and $(B_i)^n_r\neq 0, i=1,\ldots ,m,$ then
$R$ is a PI-ring.}
The only place where we have used that ${\cal B}(L)$ is quasi-Frobenius
is decomposition (37). Therefore it is enough to show that
each differential ideal of
the direct sum of differentially simple algebras with units
is a direct summand.
If $B=B_1\oplus B_2\oplus \ldots \oplus B_m$ is a direct sum of
differentially simple algebras then for any differential ideal $A$
we have that $A B_i$ is a differential ideal of $B_i.$ This implies
that either $B_i\subseteq A$ or $A B_i=0.$ In the same way
either $B_i\subseteq A$ or $B_iA=0.$
Let $l(A)$ be the left annihilator of $A,$ then $l(A)\cap A$ is
a differential ideal with zero multiplication, so its product
with each $B_i$ is zero. This is possible only if the intersection
is zero. In the same way the left annihilator of the sum $l(A)+A$
has a zero multiplication and therefore it is equal to zero. It means that
$l(A)+A$ contains all the components $B_i$ and $l(A)\oplus A=B.$
\
ACKNOWLEDGMENT
The authors are grateful to Professor Dalit Baum for her help.
\
{\bf REFERENCES}
\
[Am65]. S.A.Amitsur, {\it Generalized polynomial identities and pivotal
monomials}, Trans. Amer. Math. Soc., v.114(1965), 210--216.
[Ba27]. R.Baer, {\it Algebraiche theorie der differentierbaren funktionen
koper}, I. -- Sitzungsberichte. Heidelberger Academia, 1927, 15--32.
[Be64]. A.J.Berkson, {\it The u-algebra of a restricted Lie algebra is
Frobenius}, PAMS, v.15(1964), 14--15.
[CR62]. C.W.Curtis and I.Reiner, {\it Representation Theory of Finite
Groups and Associative Algebras}, Interscience Publishers, New York -- London,
1962.
[Ha82]. M.Hacque, {\it Anneaux fidelement repr\'esent\'es sur leur socle
droit}, Communications in Algebra, v.10, no.10(1982), 1027--1072.
[Ha87]. M.Hacque, {\it Th\'eorie de Galois des anneaux presque-simples},
Journal of Algebra, v.108, no.2(1987), 534--577.
[Ja64]. N.Jacobson, {\it Structure of Rings}, Amer. Math. Soc. Colloquium.
Publ., Providence, 1964.
[Kh78]. V.K.Kharchenko, {\it Differential identities of prime rings},
Algebra i logika, v.17, no.2(1978), 220--238.
[Kh81]. V.K.Kharchenko, {\it On centralizers of finite dimensional algebras},
Algebra i logika, v.20, no.2(1981), 231--247.
[Kh82]. V.K.Kharchenko, {\it Constants of derivations of prime rings},
Math. USSR Izvestija, v.18, no.2(1982), 381--401.
[Kh91]. V.K.Kharchenko, {\it Automorphisms and Derivations of Associative
Rings}, Kluwer Academic Publishers, v.69(1991).
[Kh95]. V.K.Kharchenko, {\it On derivations of prime rings of
positive characteristic}, Algebra i logika, 1995, to appear.
[Ko91]. A.N.Korjukin, { \it To a question of bicentralizers in prime rings},
Sib. Mat. Journal, v.32, no.6(1991), 81--86.
[La86]. C.Lanski, {\it A note on GPIs and their coefficients}, Proc. Amer.
Math. Soc. v.98(1986), 17--19.
[Lv93]. I.V.Lvov, {\it On centralizers of finite dimensional subulgebras
in the algebra of linear transformations}, Third International Algebraic
Conference, Krasnojarsk, 1993, 213--214.
[Ma69]. W.S.Martindale, {\it Prime rings satisfying a generalized polynomial
identity}, Journal of algebra, v.12, no.4(1969), 576--584.
[Pa87]. D.S.Passman, {\it Prime ideals in enveloping rings}, Trans. Amer.
Math. Soc., v.302(1987), 535--560.
[Pi86]. Piers Dos Santos, {\it Derivationes des anneaux semi-premiers I},
Comm. in algebra, v.14, no.8(1986), 1523--1559.
[Po83]. A.Z.Popov, {\it On derivations of prime rings}, Algebra i Logika,
v.22, no.1(1983), 79--92.
[Ro58]. A.Rosenberg, {\it The structure of the infinite general linear
group}, Ann. Math. ser.2, 68(1958), 278--294
\end{document}
|
1996-10-31T13:12:06 | 9610 | alg-geom/9610021 | en | https://arxiv.org/abs/alg-geom/9610021 | [
"alg-geom",
"hep-th",
"math.AG",
"math.QA",
"nlin.SI",
"q-alg",
"solv-int"
] | alg-geom/9610021 | Nakajima Hiraku | Hiraku Nakajima | Jack polynomials and Hilbert schemes of points on surfaces | AMSLaTeXv1.2 + epic.sty + eepic.sty + youngtab.sty, 20pages | null | null | null | null | The Jack symmetric polynomials $P_\lambda^{(\alpha)}$ form a class of
symmetric polynomials which are indexed by a partition $\lambda$ and depend
rationally on a parameter $\alpha$. They reduced to the Schur polynomials when
$\alpha=1$, and to other classical families of symmetric polynomials for
several specific parameters.
It is well-known that Schur polynomials can be realized as certain elements
of homology groups of Grassmann manifolds. The purpose of this paper is to give
a similar geometric realization for Jack polynomials. However, spaces which we
use are totally different. Our spaces are Hilbert schemes of points on a
surface X which is the total space of a line bundle L over the projective line.
The parameter $\alpha$ in Jack polynomials relates to our surface X by $\alpha
= -<C,C>$, where C is the zero section, and <C,C> is the self-intersection
number of C.
| [
{
"version": "v1",
"created": "Thu, 31 Oct 1996 11:25:57 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Nakajima",
"Hiraku",
""
]
] | alg-geom | \section{Introduction}
The Jack (symmetric) polynomials $P_\lambda^{(\alpha)}(x)$ form a class of symmetric
polynomials which are indexed by a partition $\lambda$ and depend
rationally on a parameter $\alpha$. They reduced to the Schur
polynomials when $\alpha = 1$, and to other classical families of
symmetric polynomials for several specific parameters. Recently they
attracts attention from various points of view, for example the
integrable systems and combinatorics. They are simultaneous
eigenfunctions of certain commuting families of differential
operators, appearing in the integrable system called the
Calogero-Sutherland system (see e.g., \cite{AMOS} and the reference
therein). On the other hand, Macdonald studied their combinatorial
properties. In fact, he introduced an even more general class of
symmetric functions (a two parameter family), which have many common
combinatorial features as Jack polynomials (see
\cite[Chapter~6]{Symmetric}).
It is well-known that Schur polynomials can be realized as certain
elements of homology groups of Grassmann manifolds (see e.g.,
\cite[Chapter~14]{Fulton}). The purpose of this paper is to give a
similar geometric realization for Jack polynomials. However, spaces
which we use are totally different. Our spaces are Hilbert schemes of
points on a surface $X$ which is the total space of a line bundle $L$
over the projective line $\operatorname{\C P}^1$. The parameter $\alpha$ in Jack
polynomials relates to our surface $X$ by
\begin{equation}
\alpha = -\langle{C},{C}\rangle = -c_1(L)[{C}],
\label{eq:selfinter}\end{equation}
where $C$ is the zero section, and $\langle{C},{C}\rangle$ is the
self-intersection number of $C$. It seems difficult to realize Jack
polynomials in homology groups of Grassmann manifolds since they
have no parameter.
The Hilbert scheme $\HilbX{n}$ parametrizing $0$-dimensional
subschemes of length $n$ on $X$ has been studied by various peoples
(see \cite{Go-book} and the reference therein). The author and
Grojnowski independently showed that the direct sum of the homology
groups of $\HilbX{n}$ (the summation is over $n$) is a representation
space of the Heisenberg algebra (boson Fock space) \cite{Gr,Na-hilb}.
It is also well-known that the space of symmetric polynomials is the
same representation space. This is the relationship between symmetric
functions and Hilbert schemes.
Now we explain our realization in more detail.
There are two characterizations of Jack polynomials:
\begin{aenume}
\item an orthogonal basis such that the transition matrix to monomial
symmetric functions is strictly upper triangular (see
\thmref{thm:defJack}), or
\item simultaneous eigenfunctions of a family of commuting
differential operators (see above).
\end{aenume}
We use the characterization~(a). We first identify the complexified
ring of symmetric functions with the direct sum of the middle degree
homology groups of Hilbert schemes as above. We then identify the
inner product with the intersection pairing (\thmref{thm:ident}). This
result was essentially proved in
\cite{Gr,Na-hilb} combined with \cite{ES2}.
Then monomial symmetric functions are identified with fundamental
classes of certain middle dimensional subvarieties in the Hilbert
schemes (\thmref{thm:monom}), which were first introduced by
Grojnowski~\cite{Gr}.
Finally, in order to get mutually orthogonal elements, we use the {\it
localization\/}. We define an $S^1$-action on the surface $X$ which
induces an action on the Hilbert scheme. The localization, which goes
back to Bott residue formula \cite{Bott}, enables computations of the
intersection product to be reduced to the fixed point set of the
$S^1$-action. Then cohomology classes which localize to different
fixed point components are mutually orthogonal. This is the mechanism
to construct a orthogonal basis. (In this paper, we shall use the
equivariant cohomology to formulate
the localization following \cite{AtBo}.)
The relation to the subvarieties corresponding to monomial symmetric
functions can be studied as follows. For each fixed point component,
we associate a locally closed submanifold (``stratum'') consisting of
points which converge to the fixed point components when they moved by
the ${\mathbb C}^*$-actions which extends the $S^1$-action (see
\eqref{eq:limit}). The dominance order is identified with the closure
relation of the strata \eqref{eq:closure}. The subvarieties
corresponding monomial symmetric functions are the closures of
stratum. Then it is easy to check that the transition matrix is
strictly upper triangular.
In fact, almost all arguments work for the total space of a line
bundle over any compact Riemann surface. We use the middle degree
homology group of Hilbert schemes on which only $H^2(C)$
contributes. ($H^0(C)$ and $H^1(C)$ contribute only to lower degrees.)
The motivation of this work comes from Wilson's observation
\cite{Wilson} that ``completed phase space'' for the complex
Calogero-Moser system (a cousin of the Calogero-Sutherland system) is
diffeomorphic to the Hilbert scheme of points on the affine plane.
(The author learned this observation from Segal's talk at Warwick,
1996 March.) This made the author to look for the connection between
Hilbert schemes and Jack polynomials. However, the connection found in
this paper does not follow the Wilson's route. Clarifying relation
between these two connections should be an important problem. For
example, it is desirable to have a geometric realization of commuting
differential operators.
It seems natural to conjecture that we get similar geometric
realization of Macdonald polynomials (two parameter family explained
above) if we replace the homology by the equivariant K-theory.
Analogous phenomenon was found for the affine Weyl group and the
affine Hecke algebra. The group ring of the affine Weyl group is
realized on the homology group of the cotangent bundle of the flag
manifold, while the Hecke algebra is realized on the equivariant
K-theory. (See \cite{Gi-book}.) We hope to return back in near
future.
\subsection*{Acknowledgment}
The author would like to thank K.~Hasegawa who told him the relation
between Jack polynomials and the Calogero-Sutherland system just after
Segal's talk at Warwick.
It is also a pleasure to acknowledge discussions with
T.~Gocho,
A.~Matsuo and
H.~Ochiai
during the seminar on Hilbert schemes of points held in 1996 spring.
In particular, Matsuo's talk on Macdonald polynomials was very helpful.
\section{Preliminaries}\label{sec:pre}
In this section we review the theory of symmetric functions and the
equivariant cohomology for later use.
\subsection{Symmetric Functions}\label{subsec:symmetric}
First we briefly recall the theory of symmetric functions. See
\cite{Symmetric} for detail.
A {\it partition\/} $\lambda = (\lambda_1,\lambda_2,\lambda_3,\dots)$
is a nonincreasing sequence of nonnegative integers such that
$\lambda_i = 0$ for all but finitely many $i$. Let $|\lambda| = \sum_i
\lambda_i$. We say $\lambda$ is a partition of $n$ if $|\lambda| =
n$. We also use another presentation $\lambda = (1^{m_1}2^{m_2}\dots)$
where $m_k = \# \{i\mid \lambda_i = k\}$. Number of nonzero entries in
$\lambda$ is called {\it length\/} of $\lambda$ and denoted by
$l(\lambda)$.
If $\lambda$ and $\mu$ are partitions, we define $\lambda\cup\mu$ be
the partition whose entries are those of $\lambda$ and $\mu$ arranged
in the descending order.
For a partition $\lambda = (\lambda_1,\lambda_2,\dots)$, we give a
Young diagram such that the number of boxes in the $i$th column is
$\lambda_i$. Remark that our convention differs from one used in
\cite{Symmetric}. Our diagram is rotated by $\pi/2$ from one used in
\cite{Symmetric}.
The conjugate of a partition $\lambda$ is the partition $\lambda'$
whose diagram is the transpose of the diagram of $\lambda$, i.e.,
\begin{equation}
\label{eq:conj}
\lambda_i' = \# \{j\mid \lambda_j\ge i\}.
\end{equation}
We define $\lambda\ge\mu$ if $|\lambda| = |\mu|$ and
\begin{equation}
\lambda_1+\lambda_2+\cdots+\lambda_i \ge
\mu_1+\mu_2+\cdots+\mu_i \qquad\text{for all $i$}.
\label{eq:dom}\end{equation}
This defines a partial order on the set of partitions and is called
{\it dominance order}. Note that $\lambda\ge \mu$ if and only if
$\mu'\ge \lambda'$.
Let $\Lambda_N$ be the ring of symmetric functions
\begin{equation*}
\Lambda_N = {\mathbb Z}[x_1,\dots,x_N]^{{\mathfrak S}_N},
\end{equation*}
where the symmetric group ${\mathfrak S}_N$ acts by the permutation of
the variables. It is a graded ring:
\begin{equation*}
\Lambda_N = \bigoplus_{n \ge 0} \Lambda_N^n,
\end{equation*}
where $\Lambda_N^n$ consists of the homogeneous symmetric functions
of degree $n$.
It is more relevant for us to consider symmetric functions in
``infinitely many variables'' formulated as follows: let $M\ge N$ and
consider the homomorphism
\begin{equation*}
{\mathbb Z}[x_1,\dots,x_M] \to {\mathbb Z}[x_1,\dots,x_N]
\end{equation*}
which sends $x_{N+1}$, \dots, $x_M$ to $0$.
We have induced homomorphisms
\begin{equation*}
\rho^n_{M,N}\colon \Lambda_M^n \to \Lambda_N^n,
\end{equation*}
which is surjective for any $M\ge N$,
and bijective for $M\ge N\ge n$.
Let
\begin{equation*}
\Lambda^n \overset{\operatorname{\scriptstyle def.}}{=} \varprojlim \Lambda_N^n.
\end{equation*}
Then the ring of symmetric functions in infinitely many variables is
defined by
\begin{equation*}
\Lambda \overset{\operatorname{\scriptstyle def.}}{=} \bigoplus_n \Lambda^n.
\end{equation*}
In the relationship between symmetric functions and Hilbert schemes,
the degree $n$ corresponds the number of points, while the number of
variables $N$ are irrelevant. This is the only reason why we use the
different notation from \cite{Symmetric}.
There are several distinguished classes of symmetric functions. The
first class is the {\it monomial symmetric function\/} $m_\lambda$.
Let $\lambda$ be a partition with $l(\lambda)\le N$. Let
\begin{equation*}
m_\lambda(x_1,\dots,x_N)
\overset{\operatorname{\scriptstyle def.}}{=} \sum_{\alpha\in{\mathfrak S}_N\cdot\lambda}
x_1^{\alpha_1}\cdots x_N^{\alpha_N}
= \frac{1}{\#\{\sigma\in{\mathfrak S}_N \mid \sigma\cdot\lambda = \lambda\}}
\sum_{\sigma\in{\mathfrak S}_N}
x_1^{\lambda_{\sigma(1)}}\cdots x_N^{\lambda_{\sigma(N)}},
\end{equation*}
where $\alpha = (\alpha_1,\dots,\alpha_N)$ runs over all distinct
permutation of $(\lambda_1,\lambda_2, \dots, \lambda_N)$.
If $l(\lambda) \le N$, we have
\begin{equation*}
\rho_{M,N}^n m_\lambda(x_1,\dots,x_M) = m_\lambda(x_1,\dots,x_N).
\end{equation*}
Hence $m_\lambda$ defines an element in $\Lambda$, which is also denoted
by $m_\lambda$.
Then $\{ m_\lambda\}_{\lambda}$ is a basis for $\Lambda$.
The {\it $n$th power sum\/} is
\begin{equation*}
p_n \overset{\operatorname{\scriptstyle def.}}{=} \sum x_i^n = m_{(n)}.
\end{equation*}
For a partition $\lambda = (\lambda_1,\lambda_2,\dots)$, let
$p_\lambda = p_{\lambda_1}p_{\lambda_2}\cdots$.
Then $\{ p_{\lambda} \}_{\lambda}$ is a basis for $\Lambda\otimes{\mathbb Q}$.
(It is {\it not\/} a ${\mathbb Z}$-basis for $\Lambda$.)
For a positive real number $\alpha$, we define an inner product
$\langle\cdot,\cdot\rangle$ on $\Lambda\otimes{\mathbb Q}$ by
\begin{equation*}
\langle p_\lambda, p_\mu\rangle
\overset{\operatorname{\scriptstyle def.}}{=} \alpha^{l(\lambda)} z_\lambda\delta_{\lambda\mu},
\end{equation*}
where $z_\lambda = \prod k^{m_k} m_k!$ for $\lambda =
(1^{m_1}2^{m_2}\cdots)$.
The Jack polynomials are defined by
\begin{Theorem}[\protect{\cite[10.13]{Symmetric}}]
For each partition $\lambda$, there is a unique symmetric polynomial
$P_\lambda^{(\alpha)}$ satisfying
\begin{enumerate}
\def(\theequation.\arabic{enumi}){(\theequation.\arabic{enumi})}
\def\arabic{enumi}{\arabic{enumi}}
\item\label{tri} $P_\lambda^{(\alpha)} = m_\lambda + \sum_{\mu < \lambda}
u_{\lambda\mu}^{(\alpha)}m_\mu$ for suitable coefficients
$u_{\lambda\mu}^{(\alpha)}$,
\item $\langleP_\lambda^{(\alpha)}, P_\mu^{(\alpha)}\rangle = 0$ if $\lambda\ne\mu$.
\end{enumerate}
\label{thm:defJack}\end{Theorem}
The uniqueness is clear since the basis $\{P_\lambda^{(\alpha)}\}_{\lambda}$ is
obtained by the Gram-Schmidt orthogonalization from
$\{m_\lambda\}_{\lambda}$ with respect to {\it any\/} total order
compatible with the dominance order. The nontrivial point lies in
(\ref{thm:defJack}.\ref{tri}) where the summation is over $\mu$ which
is smaller than $\lambda$ with respect to the dominance order.
Our geometric construction gives a new proof of this theorem.
The ``integral form'' $J_\lambda^{(\alpha)}$ of $P_\lambda^{(\alpha)}$ is defined by
the normalization
\begin{equation}
J_\lambda^{(\alpha)} \overset{\operatorname{\scriptstyle def.}}{=} c_\lambda(\alpha)P_\lambda^{(\alpha)}, \qquad
c_\lambda(\alpha) \overset{\operatorname{\scriptstyle def.}}{=} \prod_{s\in\lambda} (\alpha a(s) + l(s)+1),
\label{eq:normal}\end{equation}
where ``$s\in\lambda$'' means that $s$ is a box of the Young diagram
corresponds to $\lambda$, and $a(s)$ (resp.\ $l(s)$) is the {\it
arm\/} (resp.\ {\it leg\/}) length defined by
\begin{equation}
\label{fig:hooklength}
\newcommand{\hfil}{\hfil}
\newcommand{\heartsuit}{\heartsuit}
\newcommand{\spadesuit}{\spadesuit}
\Yvcentermath1
\young(\hfil,\hfil\heartsuit,\hfil\heartsuit\hfil,\hfil s\spadesuit\sps,\hfil\hf\hfil\hf)\qquad\qquad
\begin{matrix}
a(s) &= \text{number of $\heartsuit$} \\
l(s) &= \text{number of $\spadesuit$}
\end{matrix}
\end{equation}
Since our diagram is rotated, there is no reason to call them arm and
leg. But we follow the traditional convention.
The {\it augmented\/} monomial symmetric function is defined by
\begin{equation*}
\tilde{m}_\lambda \overset{\operatorname{\scriptstyle def.}}{=} u_\lambda m_\lambda, \qquad
u_\lambda\overset{\operatorname{\scriptstyle def.}}{=} \prod_k m_k \quad\text{for
$\lambda = (1^{m_1}2^{m_2}\cdots)$}.
\end{equation*}
Macdonald conjectured \cite[10.26?]{Symmetric} that
$J^{(\alpha)}_\lambda$ is expressed as a linear combination of
augmented monomial symmetric function $\tilde{m}_\lambda$ with
coefficients in ${\mathbb Z}_{\ge 0}[\alpha]$. This conjecture was proved
affirmatively by Knop and Sahi \cite{KS} by a combinatorial method.
\subsection{Equivariant Cohomology}
We define the equivariant cohomology using the Borel construction. We
assume the group is $S^1$, though the adaptation to the general
compact Lie group is straightforward. Let $ES^1\to BS^1$ be the
universal $S^1$-bundle which is given by the inductive limit of the
Hopf fibration $S^{2n+1}\to \operatorname{\C P}^n$.
For a topological space $M$ with a circle action, let $M_{S^1} =
ES^1\times_{S^1} M$. We have a projection $M_{S^1} \to BS^1$, which is
a fibration with fiber $M$. Then, the equivariant cohomology
$\HT^*(M)$ is, by definition, the cohomology of $M_{S^1}$. Similarly,
the equivariant cohomology with compact support, denoted by
$\HTc^*(M)$ is defined by the cohomology of $M_{S^1}$ with compact
support in the fiber direction of $M_{S^1}\to BS^1$.
We assume coefficients are complex numbers.
If $M$ is the space $pt$ consisting of a single point with a trivial
$S^1$-action, we have $\HT^*(pt) = H^*(BS^1)$. Since $BS^1$ is the
infinite dimensional projective space $\operatorname{\C P}^\infty$, $H^*(BS^1)$ is the
polynomial ring with a generator $u$ in $H^2(BS^1)$. We normalize $u$
to be the first Chern class of the tautological line bundle, i.e., the
dual of the hyperplane bundle.
By the projection $M_{S^1}\to BS^1$, we have
$\HT^*(pt)\cong{\mathbb C}[u]$-module structures on $\HT^*(M)$ and $\HTc^*(M)$.
In order to relate the equivariant cohomology $\HT^*(M)$ to the
ordinary cohomology $H^*(M)$, we consider the Leray-Serre spectral
sequence associated with the fibration $M_{S^1}\to BS^1$. It is a
spectral sequence converging to $H^*(M_{S^1}) = \HT^*(M)$ with
$E_2$-term $E_2^{p,q} = H^p(BS^1)\otimes H^q(M)$. (Note $\pi_1(BS^1) =
0$.) The (decreasing) filtration is given by
\begin{equation*}
F^p \HT^*(M) = \{ u^p\varphi \mid \varphi\in \HT^*(M) \}.
\end{equation*}
Then Kirwan proved
\begin{Theorem}\cite[5.8]{Kirwan}
Let $M$ be a compact symplectic manifold with a Hamiltonian
$S^1$-action. Then the Leray-Serre spectral sequence associated with
$M_{S^1}\to BS^1$ degenerates at the $E_2$-term. Thus we have
\begin{equation*}
F^p \HT^{p+q}(M)/ F^{p+1}\HT^{p+q}(M) \cong H^p(BS^1)\otimes H^q(M).
\end{equation*}
\label{thm:degenerate}\end{Theorem}
Kirwan's proof works for a noncompact symplectic manifold provided
$f^{-1}((-\infty,c])$ is compact for all $c\in {\mathbb R}$, where $f$ is the
moment map associated with the Hamiltonian $S^1$-action
(see also \cite[Chapter~5]{Lecture}). Unfortunately, we do not have a
natural K\"ahler metric on the Hilbert scheme of points, so we could
not check the moment map satisfies the above condition.
However, we can take another route to save Kirwan's argument. In stead of
the gradient flow of the moment map, we use the ${\mathbb C}^*$-action which
extends the $S^1$-action. (See \cite[Chapter~7]{Lecture}.)
We have another short-cut when we assume $X$ is the total space of a
line bundle over $\operatorname{\C P}^1$. Since the ordinary cohomology groups vanish
in odd degree, it is obvious that the spectral sequence degenerates at
$E_2$-term.
\section{Homology Group of the Hilbert Scheme}\label{sec:homology}
The purpose of this section is to identify the complexified ring of
symmetric functions $\Lambda\otimes{\mathbb C}$ with the homology group of the
Hilbert scheme of $X$. In fact, the result of this section holds when
$X$ is the total space of a line bundle $L$ over {\it any\/}
compact Riemann surface ${C}$.
Let $X$ be as above.
It is a nonsingular $2$-dimensional quasi-projective
surface containing ${C}$ as the $0$-section.
Let $\HilbX{n}$ be the Hilbert scheme parameterizing $0$-dimensional
subschemes of length $n$. By a result of Fogarty~\cite{Fog}, it is a
nonsingular $2n$-dimensional variety.
Let us consider the homology group $H_*(\HilbX{n})$ of the Hilbert
scheme. In \cite{Na-hilb}, we have constructed an action of the
Heisenberg algebra on the direct sum $\bigoplus_n H_*(\HilbX{n})$.
Let us briefly recall the construction.
For each $i\in\mathbb Z\setminus\{0\}$, let
$P_{C}[i]$ be a subvariety consisting
$(\idl_1,\idl_2)\in \coprod_n \HilbX{n-i}\times\HilbX{n}$ such that
\begin{equation}
\begin{cases}
\mathcal J_1\supset \mathcal J_2,\;
\text{$\Supp(\mathcal J_1/\mathcal J_2) = \{ x\}$ for some $x\in {C}$}
& (\text{when $i > 0$}) \\
\mathcal J_1\subset \mathcal J_2,\;
\text{$\Supp(\mathcal J_2/\mathcal J_1) = \{ x\}$ for some $x\in {C}$}
& (\text{when $i < 0$})
\end{cases},
\label{eq:corr}\end{equation}
where we consider points in $\HilbX{n}$ as ideals of $\shfO_X$.
Let $p_a$ be the projection to the $a$th factor in the product
$\HilbX{n-i}\times\HilbX{n}$. Note that the restriction of the
projection $p_2\colon P_{C}[i]\to \HilbX{n}$ is proper.
We define a homomorphism $H_*(\HilbX{n})\to H_*(\HilbX{n-i})$ by
\begin{equation*}
\varphi\longmapsto
p_{1*}\left(p_2^*\varphi\cap \big[P_{C}[i]\big]\right),
\end{equation*}
where $p_2^*\varphi\cap$ means the cap product of the pull-back of the
Poincar\'e dual of $\varphi$ by $p_2^*$.
Since $p_2\colon P_{C}[i]\to \HilbX{n}$ is proper, the support of
$p_2^*\varphi\cap \big[P_{C}[i]\big]$ is compact, and hence
$p_{1*}\left(p_2^*\varphi\cap \big[P_{C}[i]\big]\right)$ can be defined.
Moving $n$, we get an endomorphism on $\bigoplus_n H_*(\HilbX{n})$,
which we denote by the same symbol $P_{C}[i]$ for brevity.
Since $P_{C}[i]\cap (\HilbX{n-i}\times\HilbX{n})$ is a
$(2n-i)$-dimensional subvariety
(see \cite[\S3]{Na-hilb} or \cite[\S8.3]{Lecture} for the proof),
$P_{C}[i]$ maps $H_{2n+k}(\HilbX{n})$ to
$H_{2(n-i)+k}(\HilbX{n-i})$. In particular, the middle degree part
($k=0$) is preserved. Then the main result of \cite{Na-hilb} is
the following commutator relation
\begin{equation}
\big[P_{C}[i], P_{C}[j]\big]
= (-1)^{i-1}i\delta_{i+j,0}\langle {C},{C}\rangle \operatorname{id}.
\label{eq:comrel}\end{equation}
(The factor $(-1)^{i-1}i$, which was not determined in \cite{Na-hilb},
was given by Ellingsrud-Str\o mme~\cite{ES2}. See also
\cite[Chapter~9]{Lecture} for a proof in the spirit of this paper.)
Let $1$ be the generator of $H_0(\HilbX{0}) = {\mathbb C}$. Applying
$P_{C}[i]$ successively on $1$, we get a subspace in
$\bigoplus_n H_{2n}(\HilbX{n})$ which is the irreducible
representation of the Heisenberg algebra.
For each partition $\lambda=(1^{m_1}2^{m_2}\cdots)$ define
\begin{equation*}
P^\lambda{C} \overset{\operatorname{\scriptstyle def.}}{=}
P_{C}[-1]^{m_1} P_{C}[-2]^{m_2}\cdots 1
\in H_{2|\lambda|}(\HilbX{|\lambda|}).
\end{equation*}
Since the representation of the Heisenberg algebra generated by
$P_{C}[i]$'s is irreducible, $P^\lambda{C}$'s are linearly
independent.
On the other hand, by the formula of G\"ottsche~\cite{Got,GS} for the
Poincar\'e polynomial $P_t(\HilbX{n})$ of $\HilbX{n}$
(see also \cite[Chapter~7]{Lecture} for the proof based on Morse
theory) we have
\begin{equation*}
\sum_{n=0}^\infty q^n P_t(\HilbX{n}) =
\prod_{m=1}^\infty
\frac{(1 + t^{2m-1}q^m)^{b_1(X)}}
{(1 - t^{2m-2}q^m)(1 - t^{2m}q^m)}\, .
\end{equation*}
Hence
\begin{equation}
\sum_{n=0}^\infty q^n \mathop{\text{\rm dim}}\nolimits H_{2n}(\HilbX{n}) =
\prod_{m=1}^\infty \frac{1}{1 - q^m}\, .
\label{eq:char}\end{equation}
The right hand side of \eqref{eq:char} is the same as the character of
the irreducible representation of the Heisenberg algebra, hence
$P^\lambda{C}$'s span $\bigoplus_n H_{2n}(\HilbX{n})$. In other words,
$\{ P^\lambda{C}\,\}_{\lambda}$ is a basis for $\bigoplus_n
H_{2n}(\HilbX{n})$.
We identify $\bigoplus_n H_{2n}(\HilbX{n})$ with the polynomial ring
${\mathbb C}[p_1,p_2,\cdots]$ (and hence with the complexified ring of
symmetric functions $\Lambda\otimes{\mathbb C}$) by
\begin{equation}
p_\lambda = p_1^{m_1}p_2^{m_2}\cdots \longmapsto
P^\lambda{C} =
P_{C}[-1]^{m_1} P_{C}[-2]^{m_2}\cdots 1\qquad
\text{for $\lambda = (1^{m_1}2^{m_2}\cdots)$.}
\label{eq:ident}\end{equation}
Then the operator $P_{C}[-i]$ corresponds to the multiplication by
$p_i$ when $i > 0$.
The main result of this section is
\begin{Theorem}
\textup{(1)} The direct sum $\bigoplus_n H_{2n}(\HilbX{n})$ of the
middle degree homology group is isomorphic to the complexified ring
of symmetric functions $\Lambda\otimes{\mathbb C}$ under the
identification~\eqref{eq:ident}.
\textup{(2)} The intersection pairing $\langle\cdot,\cdot\rangle$ on
$H_{2n}(\HilbX{n})$ is given by
\begin{equation*}
\langle P^\lambda{C}, P^\mu{C}\rangle
= (-1)^n \delta_{\lambda\mu}z_\lambda
(-\langle{C},{C}\rangle)^{l(\lambda)},
\end{equation*}
where $\lambda$, $\mu$ is a partition of $n$.
\label{thm:ident}\end{Theorem}
If we define a new inner product by
$\langle\cdot | \cdot\rangle \overset{\operatorname{\scriptstyle def.}}{=} (-1)^n\langle\cdot,\cdot\rangle$
on $H_{2n}(\HilbX{n})$, it is equal to the one used for the
definition of Jack's symmetric functions, where the parameter $\alpha$
is $-\langle{C},{C}\rangle$.
\begin{proof}[Proof of \thmref{thm:ident}]
The only remaining is to prove the statement~(2).
We identify $\bigoplus_n H_{2n}(\HilbX{n})$ with ${\mathbb C}[p_1,p_2,\cdots]$
by (1).
Then
the commutation relation \eqref{eq:comrel} means that the operator
$P_{C}[i]$ for $i > 0$ corresponds to
\begin{equation*}
(-1)^{i-1} i\, \langle{C},{C}\rangle \pd{}{p_i}.
\end{equation*}
Hence for $i, j > 0$, we have
\begin{equation}
\begin{split}
& \big[P_{C}[i]^m, P_{C}[-j]^n\big]
= \left((-1)^{i-1} i\,\langle{C},{C}\rangle
\pd{}{p_i}\right)^m\; p_j^n \\
=\;& \begin{cases}
n(n-1)\cdots (n-m+1)\delta_{ij}
\left\{(-1)^{i-1}i\,\langle{C},{C}\rangle\right\}^m
P_{C}[-j]^{n-m},
&\text{for $n\ge m$}\\
0, &\text{for $n < m$.}
\end{cases}
\end{split}
\label{eq:PPcom}\end{equation}
By construction, $P_{C}[i]$ is the adjoint of $P_{C}[-i]$ with
respect to the intersection form.
For $\lambda = (1^{m_1}2^{m_2}\cdots)$, $\mu = (1^{n_1}2^{n_2}\cdots)$
let ${\bar\lambda} = (2^{m_2}3^{m_3}\cdots)$,
${\bar\mu} = (2^{n_2}3^{n_3}\cdots)$.
Then we have
\begin{equation*}
\begin{split}
\langle P^\lambda{C}, P^\mu{C}\rangle
=& \langle P_{C}[-1]^{m_1} P^{\bar\lambda}{C},
P_{C}[-1]^{n_1} P^{\bar\mu}{C}\rangle \\
=& \langle P_{C}[1]^{n_1}P_{C}[-1]^{m_1}P^{\bar\lambda}{C},
P^{\bar\mu}{C}\rangle.
\end{split}
\end{equation*}
Since $P_{C}[-1]$ commutes with $P_{C}[i]$ for $i\ne -1$, we
have
\begin{equation*}
\begin{split}
&P_{C}[1]^{n_1}P_{C}[-1]^{m_1}P^{\bar\lambda}{C}
= \Big[P_{C}[1]^{n_1}, P_{C}[-1]^{m_1}\Big]
P^{\bar\lambda}{C}\\
=\;& \begin{cases}
m_1!\; \langle{C},{C}\rangle^{m_1}
P^{\bar\lambda}{C} &\text{for $m_1 = n_1$}\\
0, &\text{otherwise,}
\end{cases}
\end{split}
\end{equation*}
where we have used \eqref{eq:PPcom}.
Hence we have
\begin{equation*}
\langle P^\lambda{C}, P^\mu{C}\rangle = \delta_{m_1,n_1}
m_1! \langle{C},{C}\rangle^{m_1}
\langle P^{\bar\lambda}{C}, P^{\bar\mu}{C}\rangle.
\end{equation*}
Inductively, we get
\begin{equation*}
\begin{split}
\langle P^\lambda{C}, P^\mu{C}\rangle
&= \delta_{\lambda\mu}
\prod_i \left((-1)^{i-1}i
\langle{C},{C}\rangle\right)^{m_i}\, m_i!\\
&= \delta_{\lambda\mu} z_\lambda
(-1)^{\sum i m_i} \left(-\langle{C},{C}\rangle\right)^{\sum m_i}.
\end{split}
\end{equation*}
Since $\sum i m_i = n$, $\sum m_i = l(\lambda)$, we get the assertion.
\end{proof}
\section{Certain Subvarieties and Monomial Symmetric
Functions}\label{sec:subvar}
In this section we define certain middle dimensional subvarieties
parametrized by partitions and identify them with monomial symmetric
functions under the isomorphism given in \thmref{thm:ident}. The
subvarieties were first introduced by Grojnowski~\cite{Gr}, and their
identification with monomial symmetric functions was proved in
\cite[Chapter~9]{Lecture}. We reproduce them for the sake of the
reader.
Let $X$ be as in \secref{sec:homology}, i.e., the total space of a
line bundle over a compact Riemann surface $C$. Let us consider
the ${\mathbb C}^*$-action on $X$ given by the multiplication on fibers. It
induces an action on the Hilbert scheme $\HilbX{n}$. Let us consider
a subvariety of the Hilbert scheme $\HilbX{n}$ defined by
\begin{equation}
\{ Z\in\HilbX{n} \mid \Supp(\shfO_Z)\subset{C} \}.
\label{eq:LSigma}\end{equation}
Since the ${\mathbb C}^*$-action retracts $\HilbX{n}$ to a neighborhood of the
above subvariety, $\HilbX{n}$ is homotopically equivalent to the subvariety.
It was pointed out by Grojnowski \cite{Gr} (see
\cite[Chapter~7]{Lecture} for the proof) that its irreducible components
$L^\lambda{C}$ are indexed by a partition $\lambda$ as follows:
first let $\pi\colon\HilbX{n}\to S^n X$ be the Hilbert-Chow morphism
which assigns to a closed subscheme $Z$ of $X$, the $0$-cycle
consisting of the points of $Z$ with multiplicities given by the
length of the local rings on $Z$. Then \eqref{eq:LSigma} is given by
$\pi^{-1}(S^n{C})$, where $S^n{C}$ is considered as a subvariety
of $S^n X$. We have a stratification of $S^n{C}$ given by
\begin{equation*}
S^n {C} = \bigcup_{\lambda} S^n_\lambda {C}, \quad
\text{where }S^n_\lambda {C} \overset{\operatorname{\scriptstyle def.}}{=}
\left\{ \left.\sum_{i=1}^k \lambda_i [x_i] \in S^n {C} \right|
\text{$x_i \neq x_j$ for $i \neq j$} \right\},
\end{equation*}
where $\lambda = (\lambda_1,\lambda_2,\dots)$ runs over
partitions of $n$.
Now consider a locally closed subvariety $\pi^{-1}(S^n_\lambda{C})$ of
\eqref{eq:LSigma}. By a result of Brian\c{c}on~\cite{Bri}, the
Hilbert-Chow morphism $\pi$ is semismall (see also \cite{GS} or
\cite[Chapter~6]{Lecture}), the dimension of the fiber of $\pi$ over a
point in $S^n_\lambda{C}$ is $n-l(\lambda)$. Hence the dimension of
$\pi^{-1}(S^n_\lambda{C})$ is equal to $n$, which is independent of the
partition $\lambda$. Moreover, the fiber of $\pi$ is irreducible again by
a result of Brian\c{c}on~\cite{Bri}. Thus the irreducible components
of \eqref{eq:LSigma} are given by
\begin{equation}
L^\lambda{C} \overset{\operatorname{\scriptstyle def.}}{=} \text{Closure of }\pi^{-1}(S^n_\lambda{C}).
\label{eq:Llambda}\end{equation}
We need another definition of $L^\lambda{C}$. Let us consider a fixed
point $\idl$ of the ${\mathbb C}^*$-action in $\HilbX{n}$. Since the fixed
point in $X$ is the zero section $C$, we have
$\Supp(\shfO_X/\idl)\subset C$. Let us decompose $\idl$ as
$\idl_1\cap\cdots\cap\idl_m$ according to the support, i.e.,
$\Supp(\shfO_X/\idl_k) = \{x_k\}$ and $x_k\ne x_l$ for $k\ne l$. For
each $k$, we take a coordinate system $(z,\xi)$ around $x_k$ where $z$
is a coordinate of ${C}$ around $x_k$, and $\xi$ is a fiber coordinate
of $L_{x_k}$. Then $\idl_k$ is generated by monomials in $z$ and
$\xi$:
\begin{equation*}
\idl_k = (\xi^{\lambda^{(k)}_1}, z\xi^{\lambda^{(k)}_2}, \dots,
z^{N-1}\xi^{\lambda^{(k)}_N}, z^N),
\end{equation*}
for some partition $\lambda^{(k)} =
(\lambda^{(k)}_1,\lambda^{(k)}_2,\dots)$ with $N = l(\lambda^{(k)})$.
For a partition $\lambda$ and a point $x\in C$, let
\begin{equation}
\idl_{\lambda,x} \overset{\operatorname{\scriptstyle def.}}{=} (\xi^{\lambda_1}, z\xi^{\lambda_2}, \dots,
z^{N-1}\xi^{\lambda_N}, z^N),
\label{eq:idlLx}\end{equation}
where $N = l(\lambda)$ and $(z,\xi)$ is the coordinate system around
$x$. (See Figure~\ref{fig:Young} for the visualization of
$\idl_{\lambda,x}$.) Thus the fixed point can be written as $\idl =
\bigcap \idl_{\lambda^{(k)},x_k}$ for some distinct points $x_k$'s in
$C$ and partitions $\lambda^{(k)}$.
If $x_k$ approaches to $x_l$,
$\idl_{\lambda^{(k)},x_k}\cap\idl_{\lambda^{(l)},x_l}$ converges to
$\idl_{\lambda^{(k)}\cup\lambda^{(l)}, x_l}$.
This shows that the fixed point components are parametrized by
partitions $\lambda = \bigcup \lambda^{(k)}$. Let us denote by
$S^\lambda{C}$ the corresponding component. By the above discussion,
we have
\begin{equation*}
S^\lambda{C} \overset{\operatorname{\scriptstyle def.}}{=}
\left\{ \idl = \bigcap \idl_{\lambda^{(k)},x_k} \mid x_i\in {C},\;
x_i \neq x_j \text{ for $i\neq j$},\;
\lambda = \bigcup \lambda^{(k)}
\right\}.
\end{equation*}
As a complex manifold, $S^\lambda{C}$
is isomorphic to a product of symmetric product
\[
S^\lambda{C}\cong S^{m_1}{C}\times S^{m_2}{C}\times\cdots,
\]
where $\lambda = (1^{m_1}2^{m_2}\dots)$.
Now another description of $L^\lambda{C}$ is
\begin{equation}
L^\lambda{C} = \text{Closure of }
\{ \idl\in\HilbX{n} \mid \lim_{t\to\infty} t\cdot\idl \in
S^\lambda{C} \},
\label{eq:limit}\end{equation}
where $t\cdot$ denotes the ${\mathbb C}^*$-action.
In fact, since the Hilbert-Chow morphism $\pi\colon\HilbX{n}\to S^nX$
is ${\mathbb C}^*$-equivariant, $t\cdot\idl$ stays a compact set, or
equivalently converges to a fixed point if and only if
$\Supp(\shfO_X/\idl)\subset{C}$. Moreover,
$\lim_{t\to\infty} t\cdot\idl$ is contained in the open stratum
\[
\left\{ \idl_{(\lambda_1),x_1}\cap \dots \cap
\idl_{(\lambda_N),x_N} \mid x_i\in {C},\;
x_i \neq x_j \text{ for $i\neq j$}\right\}
\]
of $S^\lambda{C}$
if and only if $\idl\in\pi^{-1}(S^n_\lambda{C})$. This shows the
identification \eqref{eq:limit}.
\begin{Remark}
As is presented in \cite[Chapter~7]{Lecture}, we can start from
\eqref{eq:limit} as a definition of $L^\lambda{C}$.
Studying the weight decomposition of the tangent space at a point in
$S^\lambda{C}$, we can prove that \eqref{eq:limit} is $n$-dimensional
(cf.\ discussion for \propref{prop:pullback}). The irreducibility of
\eqref{eq:limit} follows from that of $S^\lambda{C}$. Hence
\eqref{eq:limit} is an irreducible component of \eqref{eq:LSigma}.
\end{Remark}
\begin{Theorem}
Under the isomorphism $\Lambda\otimes{\mathbb C} \cong\bigoplus
H_{2n}(\HilbX{n})$ given in \thmref{thm:ident},
the class $[L^\lambda{C}]$ corresponds to the monomial symmetric
function $m_\lambda$.
\label{thm:monom}\end{Theorem}
\begin{proof}
We shall show
\begin{equation}
P_{C}[-i][L^\lambda{C}] = \sum_\mu a_{\lambda\mu} [L^\mu{C}],
\qquad\text{for any $i \in\mathbb Z_{>0}$},
\label{eq:Ll-ind}\end{equation}
where the summation is over partitions $\mu$ of $|\lambda|+i$ which is
obtained as follows:
\begin{enumerate}
\refstepcounter{equation}\label{eq:coeff}
\def(\theequation.\arabic{enumi}){(\theequation.\arabic{enumi})}
\def\arabic{enumi}{\arabic{enumi}}
\item Add $i$ to a term in $\lambda$, say
$\lambda_k$ \textup(possibly $0$\textup).
\item Then arrange it in descending order.
\item Define the coefficient $a_{\lambda\mu}$ by
$\#\{ l\mid \mu_l = \lambda_k + i\}$.
\end{enumerate}
This is the same as the relation between the power sum and the
monomial symmetric function:
\begin{equation*}
p_i m_\lambda = \sum_\mu a_{\lambda\mu} m_\mu.
\end{equation*}
By induction, we have $[L^\lambda{C}] = m_\lambda$ under the
identification $P_{C}[-i] = p_i$.
The equation~\eqref{eq:Ll-ind} is proved by studying the intersection
product.
Let $p_a\colon \HilbX{n}\times\HilbX{n}\to \HilbX{n}$ be the
projection onto the $a$th factor ($a = 1,2$).
By the definition~\eqref{eq:corr}, we represent $P_{C}[-i]$ as a
subvariety. Then its set theoretical intersection with
$p_2^{-1}(L^\lambda{C})$ is
\begin{equation}
\{ (\idl_1,\idl_2)\mid \idl_2\in L^\lambda{C},\; \idl_1\subset\idl_2,\;
\Supp (\idl_2/\idl_1) = \{x\}
\text{ for some } x\in{C}\}.
\label{eq:intersection}\end{equation}
Let $\mu = (\mu_1,\mu_2,\dots)$ be a partition of $|\lambda|+i$
which does not necessarily satisfy the condition in \eqref{eq:coeff}.
Let $N = l(\mu)$.
Since $\{[L^\mu{C}]\}_{\mu}$ is a basis for
$H_{2|\mu|}(\HilbX{|\mu|})$, the left hand side of \eqref{eq:Ll-ind}
can be written as a linear combination of $[L^\mu{C}]$'s.
In order to determine the coefficients of
$[L^\mu{C}]$ in $P_{{C}}[-i][L^\lambda{C}]$,
it is enough to take arbitrary point $\idl_1$ in
$L^\mu{C}$ and restrict cycles to a neighborhood of $\idl_1$.
We choose the point $\idl_1 =
\idl_{(\mu_1),x_1}\cap\dots\cap\idl_{(\mu_N),x_N}\in L^\mu{C}$ where
$x_k$'s are distinct points in $C$. Here
$\idl_{(\mu_k),x_k} = (\xi^{\mu_k}, z)$
for the coordinate $(z, \xi)$ around $x_k$.
(See \eqref{eq:idlLx} for the definition of $\idl_{\lambda,x}$.)
Suppose this point $\idl_1$ is contained in the image of
\eqref{eq:intersection} under the projection $p_1$, i.e.,
there exists $\idl_2$ such that $(\idl_1,\idl_2)$ is a point in
\eqref{eq:intersection}.
Then the point $x$ must be one of $x_k$'s, and
\begin{equation*}
\idl_2 = \idl_{(\mu_1),x_1}\cap\dots\cap\idl_{(\mu_{k-1}),x_{k-1}}
\cap\idl_{(\mu_k-i), x_k}
\cap\idl_{(\mu_{k+1}),x_{k+1}}\cap\dots\cap\idl_{(\mu_N),x_N},
\end{equation*}
i.e., $\idl_2$ is obtained from $\idl_1$ by replacing
$\idl_{(\mu_k),x_k}$ by $\idl_{(\mu_k-i), x_k}$.
Since $\idl_2$ must be a point in $L^\lambda{C}$,
$\mu$ is obtained by (a) adding $i$
to $\lambda_k$, and then (b) arranging in descending order.
Moreover, if $a_{\lambda\mu}$ is as in \eqref{eq:coeff}, there are
exactly $a_{\lambda\mu}$ choices of $x_k$'s. This explains the coefficient
$a_{\lambda\mu}$ in the formula. Thus the only remaining thing to check is
that each choice of $(\idl_1, \idl_2)$ contributes to
$P_{C}[-i][L^\lambda{C}]$ by $[L^\mu{C}]$.
This will be shown by checking
\begin{enumerate}
\refstepcounter{equation}\label{eq:wanttoshow}
\def(\theequation.\arabic{enumi}){(\theequation.\arabic{enumi})}
\def\arabic{enumi}{\arabic{enumi}}
\item $P_{C}[-i]$ and $p_2^{-1}(L^\lambda{C})$ intersect
transversally,
\item the intersection \eqref{eq:intersection} is isomorphic to
$L^\mu{C}$ under the first projection $p_1$,
\end{enumerate}
in a neighborhood of $(\idl_1,\idl_2)$.
Since $\idl_1$ and $\idl_2$ are isomorphic outside $x_k$, we can
restrict our concern to $\idl_{(\mu_k),x_k}$ and $\idl_{(\mu_k-i),x_k}$.
We take the following coordinate neighborhood around
$(\idl_{(\mu_k),x_k}, \idl_{(\mu_k-i), x_k})$ in
$\HilbX{\mu_k}\times\HilbX{\mu_k-i}$:
\begin{align*}
&\left\{ ( (\xi^{\mu_k}+f_1(\xi), z+ g_1(\xi)),
(\xi^{\mu_k-i} + f_2(\xi), z+g_2(\xi)))
\mid \text{$f_1$, $g_1$, $f_2$, $g_2$ as follows}\right\} \\
&\qquad\qquad
f_1(\xi) = a_1\xi^{\mu_k-1} + a_2\xi^{\mu_k-2} + \dots + a_{\mu_k}, \\
&\qquad\qquad
g_1(\xi) = b_1 + b_2\xi + \dots + b_{\mu_k-i}\xi^{\mu_k-i-1} \\
&\qquad\qquad\qquad\qquad\qquad
+ (b_{\mu_k-i+1} + b_{\mu_k-i+2}\xi + \dots +
b_{\mu_k}\xi^{i-1})(\xi^{\mu_k-i} + f_2(\xi))\\
&\qquad\qquad
f_2(\xi) = a'_1\xi^{\mu_k-i-1}
+ a'_2\xi^{\mu_k-i-2} + \dots + a'_{\mu_k-i}, \\
&\qquad\qquad
g_2(\xi) = b'_1+ b'_2\xi + \dots + b'_{\mu_k-i}\xi^{\mu_k-i-1}
\end{align*}
where $(a_1,\dots, a_{\mu_k}, b_1,\dots, b_{\mu_k})$
(resp.\ $(a'_1,\dots, a'_{\mu_k-i}, b'_1,\dots, b'_{\mu_k-i})$)
is in a neighborhood of $0$ in ${\mathbb C}^{2\mu_k}$ (resp.\ ${\mathbb C}^{2(\mu_k-i)}$).
Then the above ideal is contained in
$P_{C}[-i]$ if and only if the followings hold
\begin{enumerate}
\refstepcounter{equation}
\def(\theequation.\arabic{enumi}){(\theequation.\arabic{enumi})}
\def\arabic{enumi}{\arabic{enumi}}
\item $\xi^{\mu_k} + f_1(\xi) = \xi^i(\xi^{\mu_k-i} + f_2(\xi))$,
\item $g_1(\xi) - g_2(\xi)$ is divisible by $\xi^{\mu_k-i} + f_2(\xi)$.
\end{enumerate}
Namely, the defining equation for $P_{{C}}[-i]$ is
\begin{equation*}
\begin{split}
& a_1 = a'_1, a_2 = a'_2, \dots, a_{\mu_k-i} = a'_{\mu_k-i}, \\
& a_{\mu_k-i+1} = \cdots = a_{\mu_k} = 0, \\
& b_1 = b'_1, b_2 = b'_2, \dots, b_{\mu_k-i} = b'_{\mu_k-i}.
\end{split}
\end{equation*}
On the other hand, the defining equation for
$p_2^{-1}(L^\lambda{C})$ is
\begin{equation*}
a'_1 = a'_2 = \cdots = a'_{\mu_k-i} = 0.
\end{equation*}
Now our assertions \eqref{eq:wanttoshow} are immediate.
\end{proof}
Our next task is to explain a geometric meaning of the dominance
order \eqref{eq:dom}. It is given by modifying the stratification
introduced in \cite{Bri,Iar}.
For $i\ge 0$, let $\shfO_X(-iC)$ be the sheaf of functions vanishing
to order $\ge i$ along the zero section $C$. Let $\idl\subset\shfO_X$
be an ideal of colength $n$ such that the support of $\shfO_X/\idl$ is
contained in $C$. We consider the sequence
$(\lambda'_1,\lambda'_2,\dots)$ of nonnegative integers given by
\begin{equation*}
\lambda'_i(\idl)\overset{\operatorname{\scriptstyle def.}}{=} \operatorname{length}\left(
\frac{\shfO_X(-(i-1)C)}{\idl\cap\shfO_X(-(i-1)C)+\shfO_X(-iC)}
\right).
\end{equation*}
The reason why we put the prime become clear later. The sequence in
\cite{Bri,Iar} was defined by replacing $\shfO_X(-iC)$ by
$\mathfrak m_x^i$ where $\mathfrak m_x$ is the maximal ideal corresponding to
a point $x$. As in \cite[Lemma~1.1]{Iar}, we have
$\idl\supset\shfO_X(-nC)$, hence $\lambda'_i(\idl) = 0$ for $i\ge
n+1$. From the exact sequence
\begin{equation*}
0 \to \frac{\shfO_X(-iC)}{\idl\cap\shfO_X(-iC)} \to
\frac{\shfO_X(-(i-1)C)}{\idl\cap\shfO_X(-(i-1)C)} \to
\frac{\shfO_X(-(i-1)C)}{\idl\cap\shfO_X(-(i-1)C)+\shfO_X(-iC)}
\to 0,
\end{equation*}
we have
\begin{equation*}
\sum_{i=1}^n \lambda'_i(\idl) = n.
\end{equation*}
Let us decompose the ideal $\idl$ by its support, i.e.,
$\idl = \idl_1\cap\dots\cap\idl_N$ such that
$\{ \Supp(\shfO_X/\idl_k) \}_k$ are $N$ distinct points.
By definition, we have
\begin{equation}
\lambda_i'(\idl) = \sum_{k=1}^N \lambda_i'(\idl_k).
\label{eq:sum}\end{equation}
Suppose that $\idl$ satisfies $\Supp(\shfO_X/\idl) = \{x\}$ for some
$x\in C$. We take a coordinate system $(z,\xi)$ around $x$ where $\xi$
is a coordinate for the fiber.
If $\xi^i f_1(z),\dots, \xi^i f_d(z)$ form a basis of
\begin{equation*}
\frac{\shfO_X(-iC)}{\idl\cap\shfO_X(-iC)+\shfO_X(-(i+1)C)},
\end{equation*}
Then $\xi^{i-1} f_1(z),\dots, \xi^{i-1} f_d(z)$ are linearly
independent in
\begin{equation*}
\frac{\shfO_X(-(i-1)C)}{\idl\cap\shfO_X(-(i-1)C)+\shfO_X(-iC)}.
\end{equation*}
Hence we have $\lambda'_i(\idl) \ge \lambda'_{i+1}(\idl)$. Thus
$(\lambda'_1(\idl), \lambda'_2(\idl), \dots)$ is a partition of $n$.
By \eqref{eq:sum}, the same is true for general $\idl$ which do not
necessarily satisfy $\Supp(\shfO_X/\idl)=\{x\}$.
Let us denote the partition by $\lambda'(\idl)$.
For a partition $\lambda' = (\lambda'_1,\lambda'_2,\dots)$ of $n$,
let $W^{\lambda'}$ be the set of ideals $\idl\subset\shfO_X$ with
colength $n = |\lambda'|$ such that $\shfO_X/\idl$ is supported on
$C$ and $\lambda'(\idl) = \lambda'$.
Since
\begin{equation*}
\operatorname{length}(\shfO_X(-iC)/\idl\cap\shfO_X(-iC)) \leq
\sum_{j=i+1}^n \lambda'_j = n - \sum_{j=1}^i \lambda'_j
\end{equation*}
is a closed condition on $\idl$, the union
\begin{equation*}
\bigcup_{\mu'\ge \lambda'} W^{\mu'}
\end{equation*}
is a closed subset of $\{\idl\in \HilbX{n}\mid
\Supp(\shfO_X/\idl)\subset C\}$. Thus we have
\begin{equation}
\label{eq:closure}
\text{Closure of }W^{\lambda'} \subset\bigcup_{\mu'\ge \lambda'}
W^{\mu'}.
\end{equation}
Suppose that $\lambda'$ is the conjugate of $\lambda$ as in
\eqref{eq:conj}.
We get the following third description of $L^\lambda{C}$.
\begin{Proposition}
$L^\lambda{C} = \text{Closure of }W^{\lambda'}$.
\label{prop:third}\end{Proposition}
\begin{proof}
Let us write $\lambda = (\lambda_1,\dots,\lambda_N)$ with
$N = l(\lambda)$.
By \eqref{eq:limit}, a generic point $\idl$ in $L^\lambda{C}$ satisfies
$\lim_{t\to\infty} t\cdot\idl =
\idl_{(\lambda_1),x_1}\cap \dots \cap \idl_{(\lambda_N),x_N}$
such that $x_i$'s are distinct points in $C$.
Since the support of $\idl$ cannot move as $t\to\infty$, we can
decompose
$\idl = \idl_1\cap\dots\cap\idl_N$ such that
$\Supp(\shfO_X/\idl_k) = \{x_k\}$.
Take a coordinate system $(z,\xi)$ around $x_k$ as before. Then
$\idl_{(\lambda_k),x_k}$ was defined by $(z,\xi^{\lambda_k})$.
Since $\lim_{t\to\infty} t\cdot\idl_k = \idl_{(\lambda_k),x_k}$, we have
\begin{equation*}
\idl_k = (\xi^{\lambda_k}, z + a_1 \xi + a_2 \xi^2 + \cdots +
a_{\lambda_k-1}\xi^{\lambda_k-1})
\end{equation*}
for some $a_1,\dots, a_{\lambda_k-1}$
(cf.\ the proof for \thmref{thm:monom}).
Then $\lambda'(\idl_k) = (1^{\lambda_k})$. By \eqref{eq:sum}, we have
$$
\lambda'(\idl) = \lambda'.
$$
This shows $L^{\lambda}{C}\subset\text{Closure of }W^{\lambda'}$.
Conversely, suppose $\idl$ is a point in $W^{\lambda'}$. Let
\begin{equation*}
\idl_i \overset{\operatorname{\scriptstyle def.}}{=}
\frac{\idl\cap\shfO_X(-(i-1)C)}{\idl\cap\shfO_X(-iC)}.
\end{equation*}
Then $\idl' \overset{\operatorname{\scriptstyle def.}}{=} \bigoplus \idl_i$ satisfies $\lim_{t\to\infty}
t\cdot\idl = \idl'$ and $\idl'\in W^{\lambda'}$. This shows
$W^{\lambda'}\subset L^\lambda{C}$.
\end{proof}
\section{Equivariant Cohomology of Hilbert Schemes}
Let $X$ be the total space of a line bundle over $\operatorname{\C P}^1$ and $C$ the
zero section. $X$ is the quotient space of $({\mathbb C}^2\setminus\{0\})\times{\mathbb C}$
by the ${\mathbb C}^*$-action given by
\begin{equation*}
(z_0, z_1, \xi) \mapsto (\lambda z_0, \lambda z_1, \lambda^{-\alpha}\xi)
\qquad \lambda\in{\mathbb C}^*,
\end{equation*}
where $\alpha = -c_1(L)[{C}] = -\langle{C},{C}\rangle$.
We denote by $[(z_0, z_1, \xi)]$ the equivalence class containing
$(z_0, z_1, \xi)$. The projection $X \to \operatorname{\C P}^1$ is given by
$[(z_0, z_1, \xi)]\mapsto [z_0:z_1]$, hence $\xi$ is a coordinate
for the fiber.
We consider the $S^1$-action on $\HilbX{n}$ induced by the action on
$X$ defined by
\begin{equation}
[(z_0, z_1, \xi)] \mapsto [(z_0, t^{-1} z_1, t^{\alpha}\xi)]\qquad
\text{for $t\in S^1$}.
\label{eq:action}\end{equation}
Note that this differs from the $S^1$-action studied in the previous
section given by
$[(z_0, z_1, \xi)] \mapsto [(z_0, z_1, t\xi)]$.
We shall study the equivariant cohomology of $\HilbX{n}$ with respect
to the above $S^1$-action.
The reason for using this $S^1$-action is to identify the
normalization factor~\eqref{eq:normal} with the equivariant Euler
class (see \propref{prop:pullback}).
Let $pt$ be the space consisting of a single point with a trivial
$S^1$-action. Let us denote the obvious $S^1$-equivariant morphism by
$p\colon \HilbX{n}\to pt$.
We have an equivariant push-forward $p_*$.
Define a bilinear form
$\langle \cdot, \cdot\rangle_{S^1}\colon
\HTc^k(\HilbX{n})\otimes \HTc^k(\HilbX{n})\to\HT^{2(k-2n)}(pt)$ by
\begin{equation}
\langle \varphi,\psi\rangle_{S^1} \overset{\operatorname{\scriptstyle def.}}{=} p_*(\varphi\cup \psi),
\label{eq:inner}\end{equation}
where $\cup$ is the cup product
\begin{equation*}
\cup\colon\HTc^k(\HilbX{n})\otimes\HTc^k(\HilbX{n})\to
\HTc^{2k}(\HilbX{n}).
\end{equation*}
This is trivial unless $k$ is even and $k\ge 2n$. We assume this
condition henceforth.
In particular, $\langle\cdot,\cdot\rangle_{S^1}$ is
symmetric.
Let $j^*\colon \HTc^k(\HilbX{n})\to \HT^k(\HilbX{n})$ be the natural
homomorphism.
Then the cup product $\cup$ factor through $j^*$:
\begin{equation*}
\varphi\cup \psi = \varphi\cup j^*\psi,
\end{equation*}
where $\cup$ in the right hand side is the cup product
\begin{equation*}
\cup \colon \HTc^k(\HilbX{n})\otimes\HT^k(\HilbX{n})\to
\HTc^{2k}(\HilbX{n}).
\end{equation*}
In particular, $\langle\cdot,\cdot\rangle_{S^1}$ is well-defined on
\begin{equation*}
\HTc^k(\HilbX{n})/\operatorname{Ker} j^* \cong \operatorname{Im} j^*.
\end{equation*}
Let $\HilbX{n}_{S^1} \to BS^1$ be the fibration used in the definition
of the equivariant cohomology. By \thmref{thm:degenerate} (see also
the discussion after \thmref{thm:degenerate}), the spectral sequence of
the fibration in cohomology degenerates. Let us denote by
$F^*\HT^k(\HilbX{n})$ the corresponding decreasing filtration.
Since
\(H^q(\HilbX{n}) = 0\) for $q > 2n$,
we have
\begin{equation*}
\begin{split}
& F^p \HT^k(\HilbX{n}) = \HT^k(\HilbX{n}) \qquad\text{for $p\le k-2n$}, \\
& \HT^k(\HilbX{n})/F^{k-2n+1}\HT^k(\HilbX{n}) \cong
\HT^{k-2n}(pt)\otimes H^{2n}(\HilbX{n}).
\end{split}
\end{equation*}
If $j^*\psi \in F^{k-2n+1}\HT^k(\HilbX{n})$, then
$\varphi\cup j^* \psi\in F^{2(k-2n)+1}\HTc^{2k}(\HilbX{n})$.
Hence
$p_*(\varphi\cup j^*\psi)\in F^{2(k-2n)+1}\HT^{2(k-2n)}(pt)\cong 0$.
In particular, $\langle\cdot,\cdot\rangle_{S^1}$
is well-defined on
$\operatorname{Im} j^*/\operatorname{Im} j^*\cap F^{k-2n+1}\HT^k(\HilbX{n})$.
Consider the composition
\begin{equation}
\begin{gathered}
\HTc^{k}(\HilbX{n})/\operatorname{Rad}\langle\ ,\ \rangle_{S^1}
\xrightarrow{j^*} \operatorname{Im} j^*/\operatorname{Im} j^*\cap F^{k-2n+1}\HT^k(\HilbX{n})\\
\to
\HT^k(\HilbX{n})/F^{k-2n+1}\HT^k(\HilbX{n}) \cong
\HT^{k-2n}(pt)\otimes H^{2n}(\HilbX{n}),
\end{gathered}
\label{eq:compo}\end{equation}
which is injective by the above discussion.
We have an analogous injective homomorphism between {\it ordinary\/}
cohomology groups
\begin{equation*}
H^{2n}_c(\HilbX{n})/\operatorname{Rad}\langle\ ,\ \rangle
\to H^{2n}(\HilbX{n}).
\end{equation*}
However, the bilinear form $\langle\ ,\ \rangle$ on
$H^{2n}_c(\HilbX{n})$ is nondegenerate by \thmref{thm:ident}(2).
Hence $\operatorname{Rad}\langle\ ,\ \rangle = 0$. Moreover, the
Poincar\'e duality implies
$\mathop{\text{\rm dim}}\nolimits H^{2n}_c(\HilbX{n}) = \mathop{\text{\rm dim}}\nolimits H^{2n}(\HilbX{n})$.
Therefore the natural homomorphism
$ H^{2n}_c(\HilbX{n})\to H^{2n}(\HilbX{n})$ is an isomorphism.
\begin{Theorem}
Suppose $k$ is even and $k\ge 2n$. Then the composition~\eqref{eq:compo}
\begin{equation}
\HTc^{k}(\HilbX{n})/\operatorname{Rad}\langle\ ,\ \rangle_{S^1} \to
\HT^{k-2n}(pt)\otimes H^{2n}(\HilbX{n})
\label{eq:isom}\end{equation}
is an isomorphism.
Moreover, the bilinear form $\langle\cdot ,\cdot \rangle_{S^1}$ on the left
hand side is equal to one on the right hand side
induced by the intersection pairing on $H^{2n}(\HilbX{n})\cong
H^{2n}_c(\HilbX{n})$.
\label{thm:isom}\end{Theorem}
\begin{proof}
The subvariety $L^\lambda{C}$ defined in \eqref{eq:Llambda}
is invariant under the $S^1$-action.
Hence its Poincar\'e dual defines an element in the equivariant cohomology
$\HTc^{2n}(\HilbX{n})$, which we denote by $[L^\lambda{C}]_{S^1}$.
Under \eqref{eq:compo} with $k=2n$,
$[L^\lambda{C}]_{S^1}\bmod\operatorname{Rad}\langle\ ,\ \rangle_{S^1}$
is mapped to $1\otimes [L^\lambda{C}]$.
More generally, if $u$ is the generator of $\HT^*(pt)$ which lives in
$\HT^{2}(pt)$, $u^{k/2 - n}[L^\lambda{C}]_{S^1}$ is mapped to
$u^{k/2 - n}\otimes [L^\lambda{C}]$.
This shows that \eqref{eq:isom} is surjective.
As we have already seen the injectivity, it is an isomorphism.
Next consider the bilinear forms on the both hand sides of
\eqref{eq:isom}. The doubt arises from that
$\HT^k(\HilbX{n})\cong \bigoplus_{p+q=k}\HT^p(pt)\otimes
H^q(\HilbX{n})$ is not an {\it algebra\/} isomorphism, it is only an
isomorphism between {\it vector spaces}.
If we consider the graded space $G\HT^k(\HilbX{n})$ instead of
$\HT^k(\HilbX{n})$, it becomes an algebra isomorphism.
However, as we pointed out above, $F^{k-2n+1}\HT^k(\HilbX{n})$ does
not contribute to $\langle\cdot ,\cdot \rangle_{S^1}$. Hence we have
the assertion.
\end{proof}
\section{Localization}
We assume $X$ is the total space of a line bundle over $\operatorname{\C P}^1$ with
$\alpha = -c_1(L)[\operatorname{\C P}^1] > 0$.
Let us consider the $S^1$-action given in \eqref{eq:action}. The fixed
point set consists of two components
\begin{equation*}
\{ [(1,0,0)] \} \text{ and } \{ [(0, 1,\xi)] \mid \xi\in{\mathbb C} \}.
\end{equation*}
The first is an isolated point and the latter is isomorphic to ${\mathbb C}$.
If $Z\in \HilbX{n}$ is fixed by the induced action, its support must
be contained in the above fixed point set.
There are a family of fixed points $Z_\lambda$ with support $[(1,0,0)]$
indexed by a partition
$\lambda = (\lambda_1,\lambda_2,\dots)$ of $n$ as follows: let
$(z,\xi)$ be a coordinate system around $[(1,0,0)]$ given by $(z,\xi)
\leftrightarrow [(1,z,\xi)]$.
Then the corresponding ideal $\idl_\lambda$ is given by
\begin{equation*}
\idl_\lambda \overset{\operatorname{\scriptstyle def.}}{=} (\xi^{\lambda_1}, z\xi^{\lambda_2}, \dots,
z^{N-1}\xi^{\lambda_N}, z^N),
\end{equation*}
where $N = l(\lambda)$. Namely, $\idl_\lambda =
\idl_{\lambda,[(1,0,0)]}$ in the notation~\eqref{eq:idlLx}. The ideal
can be visualized by the Young diagram as follows. (Remind that our
diagram is rotated by $\pi/2$ from one used in \cite{Symmetric}.) If
we write the monomial $z^{p-1}\xi^{q-1}$ at the position $(p,q)$, the
ideal is generated by monomials outside the Young diagram. For
example, the Young diagram in figure~\ref{fig:Young} corresponds to
the ideal $(\xi^4, z\xi^3, z^2\xi, z^3)$. Although it is not necessary
for our purpose, one can show that these ideals are a complete list of
fixed points in $\HilbX{n}$ whose support are $[(1,0,0)]$ when $\alpha
> 0$.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\setlength{\unitlength}{0.01250000in}
\begin{picture}(73,114)(0,-10)
\path(22,62)(22,82)(2,82)
(2,62)(22,62)
\path(22,42)(22,62)(2,62)
(2,42)(22,42)
\path(42,42)(42,62)(22,62)
(22,42)(42,42)
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(2,22)(22,22)
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(22,22)(42,22)
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(2,2)(22,2)
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\caption{Young diagram and an ideal}
\label{fig:Young}
\end{center}
\end{figure}
Let $i_\lambda\colon Z_\lambda\to \HilbX{n}$ be the inclusion.
We have a homomorphism between equivariant cohomology groups:
\begin{equation*}
\HT^*(Z_\lambda) \ni \varphi
\longmapsto i_{\lambda*}\varphi \in \HTc^*(\HilbX{n}).
\end{equation*}
Let $e(T_{Z_\lambda}\HilbX{n})$ be the $S^1$-equivariant Euler class
of the tangent space at $Z_\lambda$, considered as an element of
$\HT^{4n}(Z_\lambda)$. Then we have
\begin{Theorem}
Let $\langle\cdot,\cdot\rangle_{S^1}$ be the bilinear pairing defined in
\eqref{eq:inner}. Then we have
\begin{equation*}
\langle i_{\lambda*}\varphi, i_{\mu*}\psi\rangle_{S^1}
=
\begin{cases}
0 & \text{if $\lambda\ne\mu$}, \\
p_{\lambda*} \left(\varphi\cup \psi\cup
e(T_{Z_\lambda}\HilbX{n})\right) &\text{if $\lambda=\mu$},
\end{cases}
\end{equation*}
where $p_\lambda\colon Z_\lambda\to pt$ is the obvious map.
\label{thm:push}\end{Theorem}
\begin{proof}
By the projection formula, we have
\begin{equation*}
i_{\lambda*}\varphi\cup i_{\mu*}\psi
= i_{\mu*}\left( i_{\mu}^*i_{\lambda*}\varphi \cup \psi\right).
\end{equation*}
The homomorphism $i_{\mu}^*i_{\lambda*}$ is zero when
$\lambda\neq\mu$, and is given by the multiplication by the
equivariant Euler class $e(T_{Z_\lambda}\HilbX{n})$ if $\lambda=\mu$.
Therefore,
\makeatletter\tagsleft@false\begin{align}
& \langle i_{\lambda*}\varphi, i_{\lambda*}\psi\rangle_{S^1}
= p_*\left(i_{\lambda*}\varphi\cup i_{\lambda*}\psi\right)
= p_*i_{\lambda*}\left(\varphi \cup \psi \cup
e(T_{Z_\lambda}\HilbX{n})\right) \notag \\
=\;& p_{\lambda*}\left(\varphi \cup \psi \cup
e(T_{Z_\lambda}\HilbX{n})\right)
\tag*{\qed}
\end{align}\makeatother
\renewcommand{\qed}{}
\end{proof}
Taking a class from $\HT^*(Z_\lambda)$ for each $\lambda$, we can give
a set of mutually orthogonal elements in $\HTc^*(\HilbX{n})$. These
are candidates for Jack polynomials. Our remaining task is to choose
classes so that two conditions in \thmref{thm:defJack} will be
satisfied. Thus we need to know monomial symmetric functions, or their
geometric counterparts $[L^\lambda{C}]_{S^1}$.
In order to study $[L^\lambda{C}]_{S^1}$, we need to know the
$S^1$-module structure of its normal bundle at $Z_\lambda$.
In fact, the formula will become clearer if we
consider the two dimensional torus action given by
\begin{equation*}
[(z_0, z_1, \xi)] \mapsto [(z_0, t_1 z_1, t_2\xi)]\qquad
\text{for $(t_1, t_2)\in T = S^1\times S^1$}.
\end{equation*}
The $S^1$-action studied in \secref{sec:subvar} is given by the
restriction to $(1,t)\in T$, while the $S^1$-action given by
\eqref{eq:action} is the restriction to $(t^{-1}, t^{\alpha})$.
The fixed point $Z_\lambda$ defined above is also fixed by
this torus action.
The tangent space $T_{Z_\lambda}\HilbX{n}$ at $Z_\lambda$ is a
$T$-module and have a weight decomposition
\begin{equation*}
T_{Z_\lambda}\HilbX{n} = \bigoplus_{p,q\in\mathbb Z} H(p,q),
\end{equation*}
where $H(p,q) = \{ v\in T_{Z_\lambda}\HilbX{n} \mid
(t_1,t_2)\cdot v = t_1^p t_2^q v\;\text{for $(t_1,t_2)\in T$}\}$.
\begin{Lemma}
The character of the tangent space $T_{Z_\lambda}\HilbX{n}$ is given
by
\begin{equation*}
\sum_{p,q} \mathop{\text{\rm dim}}\nolimits_{\mathbb C} H(p,q) T_1^p T_2^q =
\sum_{s\in\lambda} \left(T_1^{l(s)+1}T_2^{-a(s)}
+ T_1^{-l(s)}T_2^{a(s)+1}\right).
\end{equation*}
\label{lem:weight}\end{Lemma}
\begin{proof}
The corresponding formula was proved in \cite{ES} (see also
\cite[2.2.5]{Go-book} or \cite[Proposition~5.5]{Lecture})
when $X$ is replaced by ${\mathbb C}^2$ with the torus action given by
\begin{equation*}
(z_1,z_2) \mapsto (t_1 z_1, t_2 z_2)\qquad
\text{for $(z_1,z_2)\in{\mathbb C}^2$, $(t_1,t_2)\in T$.}
\end{equation*}
(The presentation in \cite{ES} or \cite{Go-book} is different from the
above. After substituting the arm/leg-length to their formula, we get the
above formula.)
Since the exponential map gives a
$T$-equivariant isomorphism between a neighborhood of $0\in{\mathbb C}^2$
and a neighborhood of $[(1,0,0)]\in X$, we get the assertion.
\end{proof}
Since the $S^1$-equivariant Euler class of an $S^1$-module is given by
the product of weights, we get the following.
\begin{Corollary}
The $S^1$-equivariant Euler class of the tangent space at $Z_\lambda$
is given by
\begin{equation*}
e(T_{Z_\lambda}\HilbX{n}) =
u^{2n}\prod_{s\in\lambda} \left(-\alpha a(s) - l(s) - 1 \right)\cdot
\prod_{s\in\lambda} \left(\alpha(a(s) + 1) + l(s)\right).
\end{equation*}
\label{cor:tangent}\end{Corollary}
In particular, $Z_\lambda$ is an isolated fixed point of $S^1$-action
and $e(T_{Z_\lambda}\HilbX{n})$ is nonzero since $\alpha > 0$.
In particular, we have
\begin{Corollary}
Let $\varphi_\lambda$ be a nonzero element in $\HT^k(Z_\lambda)$. Then
$\{ i_{\lambda*}\varphi_\lambda\pmod{\operatorname{Rad}\langle\ ,\
\rangle} \}_{\lambda}$ forms an orthogonal basis of
$\HTc^{4n+k}(\HilbX{n})/\operatorname{Rad}\langle\ ,\ \rangle$.
\label{cor:orth}\end{Corollary}
Let us decompose the tangent space $T_{Z_\lambda}\HilbX{n}$ as
\begin{gather*}
N_{Z_\lambda}^{>0} \overset{\operatorname{\scriptstyle def.}}{=} \bigoplus_{p\in\mathbb Z}
\bigoplus_{q > 0}H(p,q), \qquad
N_{Z_\lambda}^{\le 0} \overset{\operatorname{\scriptstyle def.}}{=} \bigoplus_{p\in\mathbb Z}
\bigoplus_{q \le 0}H(p,q), \\
T_{Z_\lambda}\HilbX{n} = N_{Z_\lambda}^{>0}\oplus N_{Z_\lambda}^{\le
0}
\end{gather*}
\begin{Proposition}
The pull-back of the equivariant cohomology class
$[L^\lambda{C}]_{S^1}$ by the inclusion $i_{\lambda}\colon
Z_\lambda\to \HilbX{n}$ is equal to the $S^1$-equivariant Euler
class of $N_{Z_\lambda}^{>0}$. It is given by
\begin{equation*}
i_\lambda^*[L^\lambda{C}]_{S^1} = e(N_{Z_\lambda}^{>0}) =
u^{n}\prod_{s\in\lambda} \left(\alpha(a(s) + 1) + l(s)\right).
\end{equation*}
\label{prop:pullback}\end{Proposition}
\begin{proof}
By the description~\eqref{eq:limit} of $L^\lambda{C}$,
it is nonsingular at $Z_\lambda$ and its
tangent space is equal to $N_{Z_\lambda}^{\le 0}$.
Hence the normal bundle is $N_{Z_\lambda}^{>0}$.
Thus we have $i_\lambda^*[L^\lambda{C}]_{S^1} =
e(N_{Z_\lambda}^{>0})$. The above formula follows from
\lemref{lem:weight}.
\end{proof}
\begin{Theorem}
For each partition $\lambda$, let $F_\lambda$ be the cohomology class
\begin{equation}
i_{\lambda*}\left(\frac{u^n}{e(N_{Z_\lambda}^{\le 0})}\right) =
i_{\lambda*}\left( \frac{1}{
\prod_{s\in\lambda} \left(-\alpha a(s) - l(s) - 1 \right)}\right)
\label{eq:Flambda}\end{equation}
considered as an element in
\begin{equation*}
\HTc^{4n}(\HilbX{n})/\operatorname{Rad}\langle\ ,\ \rangle \cong
\HT^{2n}(pt)\otimes H^{2n}_c(\HilbX{n})
\xrightarrow[u^{-n}\otimes\text{P.D.}]{\cong}
H_{2n}(\HilbX{n}).
\end{equation*}
Then they satisfy
\begin{align}
& F_{\lambda} = [L^\lambda{C}]
+ \sum_{\mu < \lambda} u_{\lambda\mu}^{(\alpha)}[L^\mu{C}]
\qquad\text{for some $u_{\lambda\mu}^{(\alpha)}$,} \label{eq:FL}\\
&\langle F_{\lambda}, F_{\mu}\rangle = 0
\qquad\text{for $\lambda\neq\mu$}. \label{eq:orth}
\end{align}
\label{thm:Jack}\end{Theorem}
\begin{Corollary}
Under the identification $\bigoplus H_{2n}(\HilbX{n})\cong\Lambda\otimes{\mathbb C}$
given in \thmref{thm:ident}, $F_{\lambda}$ corresponds to the Jack's
symmetric function $P_{\lambda}^{(\alpha)}$, where the parameter
$\alpha$ is given by $-\langle{C},{C}\rangle$ \textup(see
\eqref{eq:selfinter}\textup).
\end{Corollary}
\begin{proof}[Proof of \protect{\thmref{thm:Jack}}]
The equation \eqref{eq:orth} follows from \thmref{thm:push}.
Let us take the generator $\varphi_\lambda\in\HT^0(Z_\lambda)$. By
\corref{cor:orth}, we can write $[L^\lambda{C}]_{S^1}$ as a linear
combination of $i_{\mu*}\varphi_\lambda$ modulo
$\operatorname{Rad}\langle\ ,\ \rangle$. One can check that $Z_\mu$
is contained in $W^{\mu'}$. Hence $L^\lambda{C}=\text{Closure of }
W^{\lambda'}$ (see \propref{prop:third}) contains $Z_\mu$ only if
$\mu\le\lambda$ by \eqref{eq:closure}. In particular, the pull-back
$i_\mu^*[L^\lambda{C}]_{S^1} = 0$ unless $\mu\le\lambda$. Hence we can
write
\begin{equation*}
[L^\lambda{C}]_{S^1} = \sum_{\mu\le\lambda} v_{\lambda\mu}^{(\alpha)}
i_{\mu*}\varphi_\lambda\mod{\operatorname{Rad}\langle\ ,\ \rangle}
\end{equation*}
for some $v_{\lambda\mu}^{(\alpha)}$.
In order to compute $v_{\lambda\lambda}^{(\alpha)}$, we consider the
pull-back by $i_\lambda^*$. We have
\begin{align*}
i_\lambda^*[L^\lambda{C}]_{S^1} &= e(N_{Z_\lambda}^{>0})
&&\text{(by \propref{prop:pullback})} \\
&= \frac{e(T_{Z_\lambda}\HilbX{n})}
{e(N_{Z_\lambda}^{\le 0})}
&&\text{(since $T_{Z_\lambda}\HilbX{n} =
N_{Z_\lambda}^{>0}\oplus N_{Z_\lambda}^{\le 0}$)} \\
&=\frac{1}{u^n}i_\lambda^* i_{\lambda*}\left(
\frac{u^n}{e(N_{Z_\lambda}^{\le 0})}\right)
&&\text{(by the self-intersection formula)}.
\end{align*}
Thus we get the assertion.
\end{proof}
Finally note that the denominator in \eqref{eq:Flambda} coincides with
the normalization factor~\eqref{eq:normal} up to sign. Hence the
integral form $J_\lambda^{(\alpha)}$ is an {\it integral\/} class.
This shows that $J_\lambda^{(\alpha)}$ is expressed as a linear
combination of $m_\lambda$ with coefficients in ${\mathbb Z}$ for any positive
integer $\alpha$. Thus we have $J_\lambda^{(\alpha)} =
\sum_{\mu<\lambda} \tilde{u}_{\lambda\mu}^{(\alpha)} m_\mu$ with
$\tilde{u}_{\lambda\mu}^{(\alpha)}\in{\mathbb Z}[\alpha]$. This proves a part
of Macdonald conjecture \cite[10.26?]{Symmetric} mentioned in
\subsecref{subsec:symmetric}.
|
1996-10-04T08:53:31 | 9610 | alg-geom/9610004 | en | https://arxiv.org/abs/alg-geom/9610004 | [
"alg-geom",
"dg-ga",
"math.AG",
"math.DG"
] | alg-geom/9610004 | Sacha Sardo Infirri | Alexander V Sardo Infirri | Partial Resolutions of Orbifold Singularities via Moduli Spaces of
HYM-type Bundles | LaTex2e, 30 pages with 1 table | null | null | null | null | Let $\Gamma$ be a finite group acting linearly on $\C^n$, freely outside the
origin, and let $N$ be the number of conjugacy classes of $\Gamma$ minus one. A
construction of Kronheimer of moduli spaces $X_\zeta$ of translation-invariant
$\Gamma$-equivariant instantons on $\C^2$ is generalised to $\C^n$. The moduli
spaces $X_\zeta$ depend on a parameter $\zeta\in\Q^N$. The following results
are proved: for $\zeta=0$, $X_0$ is isomorphic to $\C^n/\Gamma$; if $\zeta\neq
0$, the natural maps $X_\zeta\to X_0$ are partial resolutions. The moduli
$X_\zeta$ are furthermore shown to admit K\"ahler metrics which are
Asymptotically Locally Euclidean (ALE). A description of the singularities of
$X_\zeta$ using deformation complexes is given, and is applied in particular to
the case $\Gamma\subset\SU(3)$. It is conjectured that for general $\Gamma$ and
generic $\zeta$ that the singularities of $X_\zeta$ are at most quadratic. When
$\Gamma\subset\SU(3)$ a natural holomorphic 3-form is constructed on the smooth
locus of $X_\zeta$, which is conjectured to be non-vanishing. The morphims
$X_\zeta\to X_0$ are expected to be crepant resolutions and $X_\zeta$ to be
smooth for generic choices of the parameter $\zeta$. Related open problems in
higher-dimensional complex geometry are also mentioned. The paper has a
companion paper which identifies the moduli $X_\zeta$ with representation
moduli of McKay quivers, and describes them completely in the case of abelian
groups.
| [
{
"version": "v1",
"created": "Fri, 4 Oct 1996 06:49:50 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Infirri",
"Alexander V Sardo",
""
]
] | alg-geom | \part{BlackBoard Bold}
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\begin{document}
\title[\runningheadstring]{\titlestring\footnote{Maths Subject Classification \msc}}
\author{Alexander V.\ Sardo Infirri}
\email{[email protected]}
\address{Research Institute for Mathematical Sciences\\ Ky\=oto University\\
Oiwake-ch\protect\=o\\ Kitashirakawa\\ Saky\protect\=o-ku\\ Ky\=oto
606-01\\ Japan}
\date{2 October 1996}
\begin{abstract}
\hyphenation{trans-la-tion}
Let~$\Gamma$ be a finite group acting linearly on~${\mathbb C}^n$, freely
outside the origin, and let $N$ be the number of conjugacy classes of
$\Gamma$ minus one.
A construction of Kronheimer~\cite{kron:ale} of moduli spaces
$X_\zeta$ of translation-invariant $\Gamma$-equivariant instantons on
${\mathbb C}^2$ is generalised to ${\mathbb C}^n$.
The moduli spaces $X_\zeta$ depend on a parameter~$\zeta\in\Q^N$. The
following results are proved: for $\zeta=0$, $X_0$ is isomorphic to
${\mathbb C}^n/\Gamma$; if $\zeta\neq 0$, the natural maps $X_\zeta\to X_0$
are partial resolutions. The moduli $X_\zeta$ are furthermore shown
to admit K\"ahler metrics which are Asymptotically Locally Euclidean
(ALE).
A description of the singularities of $X_\zeta$ using deformation
complexes is given, and is applied in particular to the case
$\Gamma\subset\SU(3)$. It is conjectured that for general $\Gamma$
and generic $\zeta$ that the singularities of $X_\zeta$ are at most
quadratic. When $\Gamma\subset\SU(3)$ a natural holomorphic 3-form is
constructed on the smooth locus of $X_\zeta$, which is conjectured to
be non-vanishing. The morphims $X_\zeta\to X_0$ are expected to be
crepant resolutions and $X_\zeta$ to be smooth for generic choices of
the parameter $\zeta$. Related open problems in higher-dimensional
complex geometry are also mentioned.
The paper has a companion paper~\cite{sacha:flows} which identifies
the moduli $X_\zeta$ with representation moduli of McKay quivers, and
describes them completely in the case of abelian groups.
\end{abstract}
\maketitle
\tableofcontents
\setcounter{section}{-1}
\section{Introduction}
\label{sec:intro}
This paper is concerned with affine {\em orbifold singularities},
namely with singularities of the type $X={\mathbb C}^n/\Gamma$ for $\Gamma$ a
finite group acting linearly on ${\mathbb C}^n$.
More precisely, this paper gives a method for constructing {\em
partial resolutions\/} of $X$, namely birational morphisms $Y\to X$
which are isomorphisms over the smooth locus of $X$.
\subsection{Background}
\label{sec:intro:back}
The method in question was first introduced by
Kronheimer~\cite{kron:ale}. It can be described in various ways,
depending on one's point of view. One description (although maybe not
the most straight-forward) is to construct moduli spaces $X_\zeta$ of
instantons on the trivial bundle ${\mathbb C}^n\times R\to{\mathbb C}^n$. Here $R$
denotes the regular representation space for the group $\Gamma$, and
$\zeta$ is a linearisation of the bundle action. The instantons are
required to satisfy Hermitian-Yang-Mills-type equations, as well as
additional $\Gamma$-equivariance and translation-invariance
properties.
In Kronheimer's case, $\Gamma\subset\SU(2)$, and the moduli spaces
$X_\zeta$ can in fact be viewed as hyper-K\"ahler\ quotients. Kronheimer
shows that $X_0$ is isomorphic to ${\mathbb C}^2/\Gamma$, that the natural maps
$X_\zeta\to X_0$ are partial resolutions for $\zeta\neq 0$, and that
indeed $X_\zeta$ coincides with the minimal resolution of
${\mathbb C}^2/\Gamma$ for generic choices of $\zeta$. Furthermore, $X_\zeta$
inherit natural hyper-K\"ahler\ metrics on their non-singular locus, which
are shown to be Asymptotically Locally Euclidean (ALE): they
asymptotically approximate the Euclidean metric at infinity (up to
terms vanishing with the inverse of the fourth power of the radial
coordinate).
\subsection{Main Results}
\label{sec:intro:main}
In the present case $n$ is any integer greater than or equal to $2$,
and $\Gamma\subset\U(n)$ is assumed to act on ${\mathbb C}^n$ freely outside
the origin for any $n\geq 2$, which means that $X={\mathbb C}^n/\Gamma$ has an
isolated singularity.\footnote{This is for the purpose of simplicity
--- the method would seem to be applicable to the general case with
some modifications} As a result, the moduli $X_\zeta$ are only {\em
K\"ahler \/} rather than hyper-K\"ahler\ quotients (in actual fact they are
more conveniently described in term of geometric invariant theory). The main result is
\begin{nonumberthm}[c.f.\xspace Thms.~\ref{thm:X0free}, \ref{thm:partial_resolutions} and \ref{thm:ale} in the text]
Let $\Gamma$ act linearly on ${\mathbb C}^n$ and freely outside the origin
and let $X_\zeta$ be the moduli spaces constructed in
Section~{\rm \ref{sec:setup:moment}}.
Then $X_0$ is isomorphic to $X={\mathbb C}^n/\Gamma$, and for $\zeta\neq 0$,
the natural morphisms $X_\zeta\to X_0$ are partial resolutions.
Furthermore, the inherited K\"ahler\ metrics on the smooth loci of
$X_\zeta$ are Asymptotically Locally Euclidean in the sense
of~\cite{kron:ale}.
\end{nonumberthm}
\subsection{Two Conjectures}
\label{sec:intro:discussion}
The final sections of the paper discuss and develop two conjectures.
\begin{nonumberconj}[c.f.\xspace Conj. \ref{conj:formal}]
The singularities of $X_\zeta$ (for generic $\zeta$, say) are at
most quadratic algebraic.
\end{nonumberconj}
This is a common occurrence for moduli spaces of this
kind~\cite{nadel:quadratic,gold_mill:flat,gold_mill:invariance}. Its
proof can be reduced to proving the formality of a certain
differential graded Lie algebra (DGLA) by the methods
of~\cite{gold_mill:invariance}. This is done in
Section~\ref{sec:defcplx}, where the singularities of $X_\zeta$ are
described in terms of deformation complexes~\cite{ahs,don_kron:4mfds}.
The concept of formality is explained, and it is suggested that the
complex relevant to $X_\zeta$ may be formal, in a way similar to
Tian~\cite{tian:smoothness} and Todorov's~\cite{todo:weil-peterson}
work. This would imply that the singularities of $X_\zeta$ at most
quadratic algebraic. This conjecture is also checked by computer for
low order abelian groups in~$\U(3)$ using the methods in the companion
paper~\cite{sacha:flows}.
\begin{nonumberconj}[c.f.\xspace Conj.\ \ref{conj:su3} in the main text]
If $\Gamma\subset\SU(3)$, the morphisms $X_\zeta\to X_0$ are
crepant, and if $\zeta$ is generic, $X_\zeta$ is smooth and its
Euler number is equal to the orbifold Euler number of $X_0$ as
defined in~\cite{dhvw:i}.
\end{nonumberconj}
The fact that $X_\zeta$ has at most quadratic singularities has been
verified for the abelian subgroups of order less than~11. The
smoothness of $X_\zeta$ has been verified in the abelian cases $\qsing
1/3(1,1,1)$, $\qsing 1/6(1,2,3)$, $\qsing 1/7(1,2,4)$, $\qsing
1/8(1,2,5)$, $\qsing 1/9(1,2,6)$, $\qsing 1/10(1,2,7)$ and $\qsing
1/11(1,2,8)$. Both these verifications were done by a brute-force
listing of singularities of $X_\zeta$ for all possible $\zeta$, using
the methods given in the companion paper~\cite{sacha:flows}.
The cases $\Gamma\subset\SU(n)$ present a particular interest. The
problem of constructing a crepant resolution of $C^3/\Gamma$ with the
same orbifold Euler number was only recently
completed~\cite{mar_ols_per,roan:mirror_cy,mark:res_168,roan:res_a5,ito:trihedral,roan:calabi-yau}.
For the case ${\mathbb C}^4/\Gamma$, one can obtain some interesting analogous
results if one considers terminalisations rather than
resolutions~\cite{sacha:sl4}.
In Section~\ref{sec:su3}, a natural holomorphic 3-form is constructed
on the smooth locus of $X_\zeta$: this is conjectured to be
non-vanishing. Its norm is shown to be constant if and only if the
induced metric on $X_\zeta$ is Ricci-flat (which does not usually turn
out to be the case, however).
\subsection{Related Questions}
\label{sec:intro:related}
This paper has a companion paper~\cite{sacha:flows} in which the
moduli $X_\zeta$ are identified with representation moduli of McKay
quivers. This allows one to explicitly describe the case of
Abelian~$\Gamma$ in terms of ``flows'' on the McKay quiver. Explicit
computations are carried out for groups of low order, and the
conjectures about the smoothness and the triviality of the canonical
bundle are checked (by brute-force computer calculations) for abelian
subgroups of $\SU(3)$ of order less than or equal to 11.
Many questions are left open by the present work, besides the
conjectures already mentioned. For instance, do the birational models
$X_\zeta$ of ${\mathbb C}^n/\Gamma$ possess any special properties with regards
to their singularities (are they terminal?) Are all terminal models
for a given 3-fold singularity obtained by this construction? What is
the relationship between the different $X_\zeta$? Are they related by
flips/flops? Is it possible, by choosing very special values of
$\zeta$, to produce blowups $X_\zeta\to X_0$ which are interesting
from the point of view of higher dimensional geometry, for instance,
for the construction of flips in dimensions~4 and greater?
\subsection{Methods}
\label{sec:intro:methods}
The methods used include geometric invariant theory, K\"ahler\ quotients, and elementary theory
of the moduli of bundles, the necessary aspects of which are reviewed
in Sections~\ref{sec:git}, \ref{sec:hym} and~\ref{sec:setup}.
Furthermore, the same construction is presented under different angles
with the intention that the reader who is familiar with one of them
(or with Kronheimer's work~\cite{kron:ale}) will be able to follow the
discussion easily.
The later sections devoted to the various conjectures raised by the
main results touch on the theory of deformation complexes, and
concepts of Kuranishi germs, formality, and so on. Some background is
also provided, although not as extensive as to be able to describe it
as ``self-contained'', given the conjectural nature of the material.
\subsection{Outline}
\label{sec:intro:outline}
The outline of this paper is as follows.
Section~\ref{sec:git} reviews material regarding geometric invariant theory\ quotients which
is necessary to define the moduli $X_\zeta$. No essentially new
material is involved, although the formulation of some of the results
may be un-familiar to non-specialists.
Section~\ref{sec:hym} review material concerning moduli of
Hermitian-Yang-Mills connections. This is not essential to the
understanding of the main results, although some familiarity is
desirable for the understanding of Section~\ref{sec:defcplx}.
Section~\ref{sec:setup} deals with the definition and construction of
the moduli $X_\zeta$.
Section~\ref{sec:partial} gives the proof that $X_0$ is isomorphic to
$X$ and that $X_\zeta\to X_0$ are partial resolutions.
Section~\ref{sec:ale} proves that the induced metrics on $X_\zeta$ are
ALE.
Section~\ref{sec:defcplx} contains the discussion of the singularities
of $X_\zeta$ in the language of deformation complexes. This includes
the conjecture that the singularities of $X_\zeta$ are at most
quadratic algebraic.
Section~\ref{sec:su3} deals with the case $\Gamma\subset\SU(3)$,
the construction of the holomorphic three-form on the non-singular
locus of $X_\zeta$ and conjecture~\ref{conj:su3}.
\subsection{Acknowledgments}
\label{sec:intro:ack}
The present paper and its companion paper~\cite{sacha:flows} consist
mostly\footnote{Minor portions have been rewritten to include
references to advances in the field made since then.} of excerpts
of my D.Phil.\ thesis~\cite{sacha:thesis}, and I wish to acknowledge
the University of Oxford and Wolfson College for their hospitality
during its preparation. I am grateful to the Rhodes Trust for
financial support during my first three years, and to Wolfson College
for a loan in my final year. The conversion from thesis to article
format was done while I was a Research Assistant in RIMS, Kyoto.
I also take the opportunity to thank my supervisors Peter Kronheimer
and Sir~Michael Atiyah who provided me with constant advice,
encouragement and support and whose mathematical insight has been an
inspiration. I also wish to thank William Crawley-Boevey, Michel
Brion, Gavin Brown, Jack Evans, Partha Guha, Katrina Hicks, Frances
Kirwan, Alistair Mees, Alvise Munari, Martyn Quick, David Reed, Miles
Reid, Michael Thaddeus, and, last but not least, my parents and
family.
\section{Geometric Invariant Theory}
\label{sec:git}
This section recalls the geometric invariant theory of affine
varieties and proves some results which shall be needed in the sequel.
These results have been included here, because, although they are
well-known to the experts, no elementary treatment
exists.\footnote{For the generalisation of these results to arbitrary
quasi-projective varieties, see~\cite{thaddeus:git_flips,dolg_hu}.}
\subsection{Linearisations and GIT quotients}
\label{sec:git:lin}
Let~$G$ be a reductive group acting linearly on a complex affine
variety~$X$. In this situation,\footnote{When $X$ is only
quasi-projective the definition of a linearisation involves
specifying an ample bundle over $X$ as well as a lift of the action
to the bundle.}\/ a ($G$-)\emph{linearisation} is a lifting of the
$G$-action to the trivial line bundle $L\to X$. Such a linearisation
is determined completely by the action of $G$ on the fibres of $L$,
namely by a character $\zeta\colon G\to {\C^*}$. For every character
$\zeta$, denote by $L_\zeta$ the trivial bundle endowed with the
corresponding linearisation. The space of $G$-invariant sections of
$L_\zeta$ is denoted by~$H^0(L_\zeta)^G$.
The geometric invariant theory\ (GIT) quotient of $X$ by $G$ with respect to $\zeta$ is defined by
$$X\gitquot\zeta G :=\Proj \bigoplus_{k\in{\mathbb N}}H^0(kL_\zeta)^G.$$
\begin{example}
\label{example:zeta_zero}
If $\zeta=0$ is the trivial character, then the corresponding
quotient $X\gitquot 0G$ coincides with the usual affine GIT quotient
$X\gitquot{}G$. In fact, suppose that $X=\Spec R$ for a finitely
generated ring $R$ and let $z_0$ be a coordinate in the fibre of
~$L_0$. Then $\oplus_{k\in{\mathbb N}}H^0(kL_0)^G=R[z_0]^G=R^G[z_0]$, so
taking Proj gives: $$X\gitquot 0 G =\Proj R^G[z_0]=\Spec
R^G=X\gitquot{} G,$$ which is the usual affine GIT quotient.
\end{example}
\subsection{Stability and Extended $G$-equivalence}
\label{sec:git:stab}
The GIT quotient can be obtained by first restricting attention to the
open set $\sst X(\zeta)\subseteq X$ of so called \emph{semi-stable}
points. A point $x$ in $X$ is called \emph{semi-stable} (with respect
to $\zeta$) if there exists a $G$-invariant section of $kL_\zeta$ (for
some $k$ in ${\mathbb N}$) which is non-vanishing at~$x$. As a set,
$X\gitquot\zeta G$ is the quotient of $\sst X(\zeta)$ by the
\emph{extended $G$-equivalence} relation induced by the closure of the
$G$-orbits:
$$x\sim y \iff \overline{Gx}\cap\overline{Gy}\neq \emptyset.$$ Thus the
$G$-invariant quotient map $\sst X(\zeta)\to X\gitquot\zeta G$ for the
equivalence relation can map several $G$-orbits to the same point.
For most of the points, this does not happen, however. This is
because the closure of an open orbit is obtained by adding orbits of
smaller dimension, so since the dimension of the orbit is a lower
semi-continuous function on $X$, it follows that there is an open
subset $\st X(\zeta)\subseteq\sst X(\zeta)$ of points which have
full-dimensional closed $G$-orbits --- the so-called \emph{stable}
points --- and there is a one-one correspondence between the orbits of
$G$ in $\st X(\zeta)$ and their images in the GIT quotient. In other
words, the GIT quotient contains, as an open set, the geometric quotient
$\st X(\zeta)/G$.
\subsection{Quotients for non-trivial linearisations}
\label{sec:git:gen}
Example~\ref{example:zeta_zero} showed that the GIT quotient for the
trivial linearisation coincides with the affine GIT quotient. The
following theorem uses the notion of stability to show that the
quotients for non-trivial $\zeta$ are closely related to the affine
quotient.
\begin{thm}
\label{thm:partial_res}
The GIT quotients admit projective morphisms
$$\rho_\zeta\colon X\gitquot\zeta G\to X\gitquot 0G$$
which are isomorphisms over
$$\rho_\zeta^{-1}(\st X(0))\to\st X(0).$$
\end{thm}
\begin{proof}
Let $X=\Spec R$ for some finitely generated ring $R$, and let
$z_\zeta$ be a complex coordinate in the fibre of~$L_\zeta$. The
GIT quotient $X\gitquot\zeta G$ is given by taking Proj of the
$G$-invariant part of $R[z_\zeta]$, where $R[z_\zeta]$ is to be
considered as an algebra graded by the powers of~$z_\zeta$. The
previous example showed that when $\zeta$ is zero, $\Proj
R^G[z_0]=\Spec R^G$.
For a general non-zero $\zeta$, $R[z_\zeta]^G\neq R^G[z_\zeta]$.
However, the {\em degree-zero\/} part of $R[z_\zeta]^G$ is
always~$R^G$, and this shows~\cite[Example~II.4.8.1 and
Cor.~II.5.16]{hart:ag} that $X\gitquot\chi G$ is projective over
$X\gitquot{}G=\Spec R^G$.
Finally, the map $X\gitquot\zeta G\to X\gitquot 0G$ comes from the
descent of the composition $\sst X(\zeta)\hookrightarrow \sst X(0)
\to X\gitquot 0G$ and this is one-one whenever $\sst X(0) \to
X\gitquot 0G$ is, so the last statement of the theorem follows.
\end{proof}
\begin{rmk}
\label{rmk:partial_res}
If $X$ contains a $0$-stable point, then $\st X(0)$ is open and
non-empty, so dense in~$X$. Its image $\st X(0)\gitquot 0G$ is open
and dense in $X\gitquot 0G$, and therefore, $\rho_\zeta\colon
X\gitquot\zeta G\to X\gitquot 0G$ is an isomorphism on a dense open
subset, i.e.\ $\rho_\zeta$ is birational.
\end{rmk}
\section{Moduli of Hermitian-Yang-Mills\ Connections}
\label{sec:hym}
This section recalls basic background concerning Hermitian-Yang-Mills\ connections and
the construction of their moduli, following~\cite{don_kron:4mfds}.
The construction of $X_\zeta$ will appear as a specialisation of the
material in this section. However, the reader wishing to go straight
to the point can skip this section and find a self-contained
construction of $X_\zeta$ in section~\ref{sec:setup}.
\subsection{Connections over Symplectic Manifolds}
\label{sec:hym:symplectic}
Suppose $X$ is a compact symplectic manifold $(X,\omega)$ of dimension
$2n$ and let $E$ be a complex vector bundle over~$X$. The bundle of
infinitesimal automorphisms of $E$ will be denoted $\gl(E)$ or~$\End
E$.
\subsubsection{Connections}
Consider connections on $E$, namely linear maps $\nabla\colon
\Omega^0_{X}(E) \to \Omega^1_{X}(E)$ which satisfy the Leibniz condition.
Any connection on $ E$ can be expressed in a local neighbourhood\ $U\subset X$
as
\begin{equation}
d_\alpha= d+\alpha,
\label{eq:dalpha}
\end{equation} for $\alpha\in\Omega^1_U(\End E)$. On the other hand, if one considers the difference of two connections, one gets a {\em
global\/} one-form with values in~$\End E$.
\subsubsection{Hermitian Structure}
Let $\bdh$ be a positive definite Hermitian inner product\ on the fibres of $E$ and denote by ${\mathfrak u}(
E)\subset\gl( E)$ the real sub-bundle of unitary automorphisms
determined by~$\bdh$. The connection $\nabla$ is said to be
\emph{compatible} with the Hermitian structure if
$$d \bdh(s,t) =\bdh(\nabla s,t)+\bdh(s,\nabla t).$$ Such a connection
will have local one-form representatives $\alpha\in\Omega_U^1({\mathfrak u}(
E))$; the space ${\mathcal A}$ of all such connections is an infinite-dimensional
affine space modeled on~$\Omega_X^1({\mathfrak u}( E))$.
\begin{rmk}
Why does one fix a Hermitian structure on $E$? One reason is
because fixing a Hermitian structure on $E$ amounts, by Chern-Weil
theory, to fixing the topological invariants of the connections: for
any Hermitian connection $d_\alpha$ on $E$, the Chern polynomials
$c_1(E)$ and $c_2(E)-c_1(E)^2$ are represented respectively by
$\frac{i}{2\pi}\trace F_\alpha$ and $\trace F_\alpha^2$.
\end{rmk}
\subsubsection{Symplectic Structure and Gauge Group}
The space ${\mathcal A}$ has a symplectic structure defined by
$$\boldsymbol{\omega}(a,b):=\int_X\trace(a\wedge b)\wedge\omega^{n-1},$$ for
$a,b\in\Omega_X^1({\mathfrak u}( E))$ tangent vectors to~${\mathcal A}$.
The \emph{gauge group} ${\mathcal G}$ is the group of automorphisms of $ E$ which
respect the Hermitian structure in the fibres and cover the identity
map of $X$. It acts on ${\mathcal A}$ by $$g\cdot\nabla:= g\nabla
g^{-1}$$ (the condition that $g$ is unitary ensures that the new
connection is compatible with $\bdh$) and preserves the symplectic
form~$\boldsymbol{\omega}$.
\subsubsection{Moment Map}
The Lie algebra of the gauge group is~$\Lie
{\mathcal G}=\Omega^0_X({\mathfrak u}( E))$. The moment map for the action of the
gauge group is given by~\cite[Prop.\ 6.5.8]{don_kron:4mfds}
\mapeq\mu{\mathcal A}{\Omega^0_X({\mathfrak u}( E))^*}\alpha{s\mapsto \int_X \trace
s F_\alpha\wedge\omega^{n-1}}{eq:mucA}
Note that the ${\mathcal G}$-equivariance follows because the curvature transforms as a tensor under gauge transformations.
\subsection{Connections over K\"ahler\ manifolds, Holomorphic Structures and the Hermitian-Yang-Mills\ condition}
\label{sec:hym:kahler}
Now suppose that $X$ is in fact a K\"ahler\ manifold.
\subsubsection{Complex Structure and the Decomposition of Curvature}
There is a natural decomposition $\Omega^1_{X}(\End E) =
(\Omega^{1,0}_{X}\oplus \Omega^{0,1}_{X})\otimes \Omega^0(\End E)$,
and if a connection is expressed in a local holomorphic frame
$\{z_i\}$ according to~\eqref{eq:dalpha} it takes the form
$$\alpha = \sum_i \alpha_i dz_i - \alpha_i^* d \bar z_i,$$ for
$\alpha_i$ smooth sections of~$\End E$.
The local connection $d_\alpha$ splits into a sum of $(1,0)$ and
$(0,1)$ parts $\partial_\alpha+\bar{\partial}_\alpha$ given by:
\begin{align}
\partial_\alpha &= \partial + \sum_i \alpha_i dz_i\\
\bar{\partial}_\alpha &= \bar{\partial} - \sum_i \alpha_i^* d{\bar{z}}_i.
\end{align}
The space ${\mathcal A}$ becomes a (flat) K\"ahler\ manifold when tangent vectors
are identified with their $(0,1)$ parts, or in other words, when
connections are represented by their $\bar{\partial}_\alpha$ operators. The
holomorphic tangent space to ${\mathcal A}$ is of course isomorphic to
$\Omega^{0,1}_{X}(\End E)$.
The $(1,1)$ and $(2,0)$ parts of the curvature ${F_\alpha}$ of
$d_\alpha$ are given by
\begin{align}\label{eq:F11}
F_\alpha^{1,1} &= \sum_{i, j} \left(\frac{\partial\alpha_j}{\partial{\bar{z}}_i} - \frac{\partial\alpha_j^*}{\partial z_i}
-[\alpha_i,\alpha^*_j]\right)dz_i\wedge d{\bar{z}}_j,\\ \intertext{and}
\label{eq:F20}
F_\alpha^{2,0} &= \sum_{i,j} \left(-\frac{\partial\alpha_i}{\partial
{z_j}} +\frac{1}{ 2}[\alpha_i,\alpha_j]\right)dz_i\wedge dz_j,
\end{align}
with $F_\alpha^{0,2}$ equal to minus the Hermitian adjoint of
$F_\alpha^{2,0}$.
\subsubsection{The Hermitian-Yang-Mills condition}
A connection on $E$ is called \emph{Hermitian-Yang-Mills} (\emph{HYM}) if the inner
product of its curvature with the K\"ahler\ form $\omega$ is a central
element of $\Omega^1_X(\gu(E))$. This is in fact a moment map
condition: using the identity
$$F_\alpha\wedge\omega^{n-1}=\frac{1}{ n}\ip{F_\alpha}\omega \omega^n =: \frac{1}{ n}(\Lambda F_\alpha)\omega^n,$$
the map in equation~\eqref{eq:mucA} becomes
$$\mu(\alpha)(s)=\frac{\Vol(X)}{(n-1)!} \trace(s\Lambda F_\alpha),$$
so, embedding the Lie algebra of ${\mathcal G}$ its dual in the usual way, we
see that the moment map becomes a constant multiple of
\map{\mu^*}{\mathcal A}{\Omega^0_X(\gu(E))}\alpha{\Lambda F_\alpha.}
The moduli space of HYM connections is the K\"ahler\ quotient
$\mu^{*-1}(0)/{\mathcal G}$.
\subsubsection{Holomorphic Bundles}
Suppose instead that the Hermitian connection $d_\alpha$ is required to
induce a holomorphic structure on $E$. By the Newlander-Nirenberg
theorem, prescribing such a structure is exactly equivalent to
specifying a connection $d_\alpha$ which is
\emph{integrable}, i.e.\xspace whose $(0,1)$-part is such
that $\bar{\partial}_\alpha\circ \bar{\partial}_\alpha=0$. This is equivalent to the
condition that $F_\alpha$ is of type $(1,1)$ and gives a K\"ahler\
subvariety ${{\mathcal A}^{1,1}}\subset{\mathcal A}$. This variety parametrises all
the possible holomorphic structures which can be put on the $C^\infty$
bundle $E\to X$.
The action of ${\mathcal G}$ on ${\mathcal A}$ extends to an action of its complexification
${\mathcal G}^{\mathbb C}$, which can be thought of naturally as the group of all
general linear automorphisms of $E$ covering the identity map on $X$.
Put $\tilde g:=(g^*)^{-1}$ and let
\begin{align}
g\cdot\bar{\partial}_\alpha &:= g\bar{\partial}_\alpha g^{-1},\\
g\cdot\partial_\alpha &:= \tilde g\partial_\alpha \tilde g^{-1}.
\end{align}
This action of ${\mathcal G}^{\mathbb C}$ preserves the space ${\mathcal A}^{1,1}$, and its
orbits are equivalence classes of holomorphic bundles. To get a nice
moduli space (a quasi-projective variety), one must restrict to
the so-called semi-stable bundles, or in other words, consider a GIT
quotient ${\mathcal A}^{1,1}\gitquot{}{\mathcal G}^{\mathbb C}$. A theorem of Uhlenbeck and
Yau~\cite{uhl_yau:hym} states that the moduli space of stable bundles with
the same topological type as $E\to X$ coincides with the moduli space
of Hermitian-Yang-Mills connections on $E$. This is an
infinite-dimensional version of the correspondence between symplectic
and algebro-geometric quotients.
\begin{rmk}
One should really use the quotient of ${\mathcal G}^{\mathbb C}$ by the scalar
automorphisms, since they act trivially on ${\mathcal A}$. The resulting group then
has a trivial centre so there is only one linearisation of the
action; it essentially determined by the degree and rank of $E$.
\end{rmk}
\section{Construction of $X_\zeta$}
\label{sec:setup}
A construction of $X_\zeta$ from scratch will be given in this
section.
Let $Q$ be an $n$-dimensional complex representation of a finite group
\mbox{$\Gamma$}. Average over the group elements to get a positive definite Hermitian inner product\ on $Q$
such that $\Gamma\subset \U(Q)$. Let ${R}$ be the regular
representation of \mbox{$\Gamma$}, i.e.\
the free \mbox{$\Gamma$}-module which is generated over ${\mathbb C}$ by a basis
$\{e_\gamma | \gamma\in \Gamma\}$, and on which \mbox{$\Gamma$}\ acts via the
morphism $\varphi\colon \Gamma\to\Aut_{\mathbb C} R$ defined by: $$\gamma\cdot
e_\delta := \varphi(\gamma) e_\delta := e_{\gamma\delta}.$$
\subsection{Invariant HYM Connections}
\label{sec:setup:hym}
\begin{note}
The reader who has not read Section~\ref{sec:hym} or is not interested
in the ``moduli of bundles'' point of view can jump directly
to~\ref{sec:setup:M}.
\end{note}
The construction of $X_\zeta$ is based on a variation on the
construction in the previous section. It consists, roughly
speaking, in applying the construction to the case where the compact
K\"ahler\ manifold $X$ is replaced by the germ of the singularity
$Q/\Gamma$.
More precisely, start with $Q^*$, the dual vector-space to $Q$, and
$E=Q^*\times R\to Q^*$ the trivial vector bundle with fibre $R$.
Consider the connections on $E$ which are invariant under all
translations in $Q^*$. These connections are determined by their
value at one point, and they form a finite-dimensional vectorspace
which can be identified with $M=Q\otimes_{\mathbb C} \End_{\mathbb C} R$ by choosing an
isomorphism $\Omega^1_{Q^*}\cong Q$. The constructions in the
previous section are now valid, because translation invariance
eliminates any problems one might have with the non-compactness of
base space $Q^*$. Most aspects are indeed a lot simpler: it suffices to set
all the derivatives of the $\alpha_i$ equal to zero, and to ignore any
integrals over the base space and all the formulas remain valid.
An unusual feature of these invariant connections is that there are
{\em several\/} moduli spaces: the usual Hermitian-Yang-Mills\ condition for a
connection states that the contraction of the curvature with the K\"ahler\
form should be a central element of the Lie algebra of the gauge
group. In case of general connections, there is only one
gauge-invariant momentum level set because the centre of the gauge
group ${\mathcal G}$ is trivial. In the case of invariant connections the
gauge group that is relevant is a much smaller group $K^\Gamma$,
consisting of gauge transformations which are invariant with respect
to all translations and the action of $\Gamma$. This group consists of
unitary endomorphisms of $R$ which commute with the action of $\Gamma$,
and has a non-trivial centre (consisting of the traceless
\mbox{$\Gamma$}-endomorphisms $\zeta\colon R\to R$). The non-triviality of the
centre means that there are several gauge invariant momentum level
sets, and hence several possible moduli $X_\zeta$.
For the sake of the readers unfamiliar with the material in
section~\ref{sec:hym}, the construction of $X_\zeta$ is given in detail
without making any reference to the bundle construction.
\subsection{The Vector Space $M^{\Gamma }$}
\label{sec:setup:M}
Let $M=Q\otimes_{\mathbb C} \End_{\mathbb C} R$ and let \mbox{$\Gamma$}\ act on $\End_{\mathbb C} R$ by
conjugation via $\Gamma\stackrel{\varphi}\hookrightarrow \Aut R$:
\begin{equation}\label{eq:gamma_action_endR}
\gamma \cdot T := \varphi(\gamma)\,T\, \varphi(\gamma)^{-1}
\end{equation}
This makes $M$ into a \mbox{$\Gamma$}-module. Its
\mbox{$\Gamma$}-invariant part is denoted ${M^\Gamma}$:
\begin{equation}\label{eq:definition_M}
M^\Gamma := \left( Q\otimes\End R \right)^\Gamma.
\end{equation}
The spaces $M^\Gamma$ and $M$ can be described explicitly by choosing a basis
$\{q_l\}_{l=1}^n$ for $Q$, and defining the \emph{components} of
$\alpha\in M$ by:
\begin{equation}\label{eq:components_alpha}
\alpha=\sum_{l=1}^n q_l\otimes \alpha_{l}.
\end{equation}
In this way the elements $\alpha\in M$ are identified with
$n$-tuples of linear maps $\alpha_{i}:R\to R$. The elements of
$M^\Gamma$
correspond to those $n$-tuples which satisfy the following
equivariance condition
\begin{equation}\label{eq:alpha_equivariance_condition}
\sum_l \gamma_{kl} \alpha_{l} = \varphi(\gamma)\alpha_{k}\varphi(\gamma)^{-1},\rlap{$\qquad \forall k,\gamma$,}
\end{equation}
where $\boldsymbol{\gamma}=(\gamma_{kl})$ is the matrix corresponding to
the
action of the element $\gamma$ on $Q$ with respect to the basis
$\{q_l\}_{l=1}^n$.
\subsection{Symplectic and K\"ahler\ structure.}
\label{sec:setup:symp}
Endow $R$ with a fixed positive definite Hermitian inner product\ $\<\ ,\ >$ which makes the standard basis
$e_\gamma$
orthonormal. The inner product on $R$ also defines a real structure
on the
space $\End_{\mathbb C} R$ of ${\mathbb C}$-linear endomorphisms of $R$ by the
Hermitian adjoint operation in the usual way:
$$\ip{T^*x}y := \ip{x}{Ty}. \qquad x,y\in R. $$
Define a positive definite Hermitian inner product\ ${\bdh}$ on $M$ by
\mapeq{\bdh}{M\times M}{{\mathbb C}}{(\alpha,\beta)}{\sum_i \trace
(\alpha_i\beta^*_i)}{eq:dfn_omega} The definition of $\bdh$ is independent of the
basis of $Q$ up to a unitary transformations, and restricts to an
inner product on $M^\Gamma$.
As usual, $\bdh$ induces two forms on the underlying real
vector-space to $M$:
\begin{itemize}
\item a non-degenerate symmetric bilinear form ${{\mathbf g}=\Re(\bdh)}$
called the \emph{Riemannian metric} associated to $\bdh$
\map{{\mathbf g}}{M\times M}{{\mathbb R}}{(\alpha,\beta)}{\frac{1}{2}\sum_i \trace
(\alpha_i\beta^*_i+\beta_i\alpha^*_i)}
\item a non-degenerate skew-symmetric bilinear form
${\omega=\Im(\bdh)}$ called the \emph{K\"ahler} form
associated to $\bdh$ \map{\omega}{M\times
M}{{\mathbb R}}{(\alpha,\beta)}{\frac{1}{2\sqrt{-1}} \sum_i
\trace(\alpha_i\beta^*_i-\beta_i\alpha^*_i)}
\end{itemize}
This gives a Riemannian metric ${\mathbf g}$ and a K\"ahler form $\omega$ of
type
$(1,1)$ on $M$ and $M^\Gamma$, related as usual by
\begin{equation}
\label{eq:omega_g}
\omega(\alpha, \beta )={\mathbf g}( \alpha , i\beta ).
\end{equation}
This makes $M^\Gamma, M$ and all their complex subvarieties into K\"ahler\
varieties.
The group $\GL (R)$ of automorphisms of $R$ acts on $M$ by conjugation
on $\End R$:
\begin{equation}
\alpha_i\mapsto g\alpha_i g^{-1},\qquad g\in\GL (R).
\label{eq:action_glr}
\end{equation}
In fact, the scalars act trivially, and the action descends to an
action of $G:=\PGL(R):=\GL(R)/\GL(1)$.
The subgroup $\GL^\Gamma\! R$ of endomorphisms which commute with the action
of $\Gamma$ acts on $M^\Gamma$ and, in the same way, there is a free action
of $G^\Gamma=\PGL(R)^\Gamma$ on $M^\Gamma$. A maximal compact subgroup of $G$
(resp.\xspace $G^\Gamma$) is given by $K=\PU(R)$ (resp.\xspace $K^\Gamma:=
\PU^\Gamma(R)$). The compact group $K$ (resp.\xspace $K^\Gamma$) leaves the
K\"ahler\ structure on $M$ (resp.\xspace $M^\Gamma$) invariant.
\subsection{The Variety $\protect\cN^{\Gamma }$ of Commuting Matrices}
\label{sec:setup:cnga}
Define the following natural map:
\map{\psi}{Q\otimes\End R}{\Lambda^2 Q\otimes \End R}{\sum_k q_k
\otimes \alpha_{k}}{\sum_{k,l} q_k\wedge q_l[\alpha_{k},\alpha_{l}]}
This definition is independent of the basis of $Q$ (in terms of
connections, it corresponds to calculating the $(0,2)$ part of the
curvature). Denote the restriction of $\psi$ to $M^\Gamma$ by the same
letter. Define
\begin{equation}\label{eq:definition_cN}
{\mathcal N} :=\psi{}^{-1}(0)\subset M;
\end{equation}
it is a cone (i.e.\ it is invariant under multiplication by non-zero
scalars) which is an intersection of quadrics in $M$ given by the
coordinate functions of $\psi$. In the representation of equation~
\eqref{eq:components_alpha}, its points consist of $n$-tuples of
commuting $r\times r$ matrices:
$${\mathcal N} = \{(\alpha_1,\dots,\alpha_n): \alpha_i\in \Mat_r({\mathbb C}),
[\alpha_i,\alpha_j]=0.\}$$ Its $\Gamma$-invariant part ${{\mathcal N}^\Gamma}$
consists of those commuting matrices satisfying the equivariance
condition~\eqref{eq:alpha_equivariance_condition}.
\subsection{Moment Map}
\label{sec:setup:moment}
\nopagebreak
Consider the vector space $M$ with the hermitian inner product $h$ and
the action of $K$. The Lie algebra of $K$ is isomorphic to $\su R$,
which consists of traceless skew-Hermitian endomorphisms of $R$. Using
the invariant inner product
$$\ip ab := \trace (a b^*)=-\trace ab,$$ identify $\su R$ with
its dual in the usual way. Then the moment map is for the action of
$K$ is
\mapeq{\mu}M{\su
R}{\alpha}{\sum_{k}[\alpha^*_{k},\alpha_{k}].}{eq:form_mu} (In the
language of of connections, the map $\mu$ corresponds to contracting
the curvature of the connection with the K\"ahler\ form $\omega=\sum_i
dq_i\wedge d\bar{q_i}\in\Omega_{Q^*}^{1,1}$.) The moment map for the
action of $K^\Gamma$ on $M^\Gamma$ is obtained simply by restriction.
The K\"ahler\ quotients are defined by
\begin{equation}\label{eq:kaehler_reductions}
X_\zeta := \frac{\mu{}^{-1}(\zeta)\cap {\mathcal N}^\Gamma}{K^\Gamma}, \qquad \zeta\in
\text{Centre}(\su^\Gamma R).
\end{equation}
As was remarked in section~\ref{sec:git},
to make this definition rigorous, one needs to make sense of the K\"ahler\
structure on $X_\zeta$. One way to do this is by restricting $\zeta$
to take on integral values. Then, by the correspondence between K\"ahler\
and GIT quotients, one has
\begin{equation}
X_\zeta \cong {\mathcal N}^\Gamma\gitquot\zeta G^\Gamma,
\label{eq:git_xzeta}
\end{equation}
where $\zeta$ on the right-hand side specifies the linearisation of
the action of $G^\Gamma$ on the trivial line bundle ${\mathcal N}^\Gamma\times{\mathbb C}$.
\begin{rmk}
In the case $\Gamma\subset SU(2)$, one has $\Lambda^2 Q\cong R_0$ ---
the trivial representation. Identifying ${\mathbb C}^2$ with the
quaternions, the vector space $M^\Gamma$ becomes a hyper-K\"ahler
manifold with 3 distinct complex structures $I,J,K$ and
corresponding associated K\"ahler forms $\omega_I, \omega_J,
\omega_K$. The map $\psi$ is then a moment map for the complex
symplectic form $\omega_{\mathbb C}= \omega_J+i\omega_K$ which is itself
holomorphic with respect to $\omega_I$ and the quotients $X_\zeta$
are quotients of $M^\Gamma$ with respect to the hyper-K\"ahler\ moment map
given by $(\mu,\psi)$, where the second (complex) variable is set to
zero. Kronheimer~\cite{kron:thesis,kron:ale} exploits this fact to
show that $X_\zeta$ are the minimal resolutions of ${\mathbb C}^2/\Gamma$ for
generic values of $\zeta$. Furthermore, by varying the level set of
$\psi$, he obtains universal deformations. If $\dim M>2$ however,
$\psi^{-1}(\zeta)$ is not $G^\Gamma$-invariant for non-zero $\zeta$.
\end{rmk}
\begin{rmk}
\label{rmk:GQaction}
In fact, there is a further action on $M$ by $\GL Q$ (acting on $Q$
on the left). If $\rho\in\GL^\Gamma\! Q$, then it is easy to see that this
preserves $M^\Gamma$, and indeed ${\mathcal N}^\Gamma$. Furthermore, the centre
$Z(\Gamma)$ of $\Gamma$ is a subgroup of $\GL^\Gamma\! Q$
which acts trivially because
of~\eqref{eq:alpha_equivariance_condition}, so there is an action
of $G':=\GL^\Gamma\! Q/Z(\Gamma)$ which the quotients $X_\zeta $
inherit. The compact subgroup $K':=\U^\Gamma\!
(Q)/Z(\Gamma)$ acts in a Hamiltonian fashion, and the moment map for
this is given by \mapeq{\mu'}{X_\zeta}{(\Lie \U^{\Gamma}
Q)^*}{[\alpha]}{b\mapsto \sum_{ij}b_{ij}\trace
\alpha_i\alpha^*_j.}{eq:form_muQ} In general this does not provide
one with very much information: for instance, if $Q$ is irreducible,
$G'={\mathbb C}^*$ and $\mu'(\alpha)$ is the identity endomorphism of $Q$
times the sum of the norm squared of the $\alpha_i$'s. In the case
where $\Gamma$ is abelian, however, $G'$ contains an algebraic torus of
dimension $n$ acting freely. The components of $\mu'$ are the norm
squared of the matrices $\alpha_i$. This is exploited in the
companion paper~\cite{sacha:flows} to obtain a complete description
of $X_\zeta$ by the methods of toric geometry.
\end{rmk}
\section{Variation of Quotients and Partial Resolutions}
\label{sec:partial}
The zero momentum quotient $X_0$ is better understood if viewed as a
two-stage construction: first construct a ``universal quotient"
${\mathcal N}_0$ by ignoring the $\Gamma$-equivariance condition (i.e. perform the
same construction with $\Gamma$ replaced by the trivial group) and then
obtain $X_0$ as its \mbox{$\Gamma$}-invariant part.
\subsection{The Universal Quotient}
\label{sec:partial:uni}
\nopagebreak
Consider taking symplectic quotients of ${\mathcal N}$ with respect to the action of
$K$. Since the centre of its Lie algebra is trivial,
there is only {\em one\/} quotient, with momentum zero:
$${\mathcal N}_0= \frac{ {\mathcal N}\cap \mu{}^{-1}(0)}{K}={\mathcal N}\gitquot{} G.$$
\begin{lemma}
\label{lemma:N0}
The reduction ${\mathcal N}_0$ is isomorphic to configuration space of
$r=|\Gamma|$ points in $Q$:
$${\mathcal N}_0 \cong \Sym^r(Q) := {Q^r/ \Sigma_r},$$
where $\Sigma_r$ denotes the permutation group on $r$ letters acting
component-wise on the Cartesian product $Q^r$.
\end{lemma}
\begin{proof} The proof simply adapts Kronheimer's \cite[Lemma
5.2.1]{kron:thesis}. It is shown that the $K$-orbits in
${\mathcal N}\cap\mu^{-1}(0)$ can be identified in a one-one way with the
$\Sigma_r$-orbits in $Q^r$. Let $\alpha\in{\mathcal N}\cap \mu{}^{-1}(0)$ have
components $\alpha_i$ with respect to a basis $q_i$. The conditions
$\psi(\alpha)=\mu(\alpha)=0$ give
\begin{align*}
[\alpha_i,\alpha_j] &=0,\qquad \text{for all }i,j\\
\sum_i [\alpha^*_i,\alpha_i] &=0.
\end{align*}
If one denotes by $A_i$ the operator $\adj(\alpha_i)$, one has,
using the Jacobi identity and the above equations:
\[
\sum_i A^*_i A_i(\alpha^*_j) = \sum_i
[[\alpha^*_i,\alpha_i],\alpha^*_j]+
[[\alpha^*_j,\alpha^*_i],\alpha_i]= 0.
\] The positivity of $A^*_iA_i$ implies $A^*_iA_i(\alpha^*_j)=0$ for
all $i$ and $j$, and hence $A_j(\alpha^*_j)=[\alpha^*_j,\alpha_j]=0$.
Thus ${\mathcal N}_0$ is the variety of $n$-tuples of normal commuting
endomorphisms of $R$ modulo simultaneous conjugation. Any such
$n$-tuple can simultaneously diagonalised by conjugation by a unitary
matrix. This means that the orbit of an $n$-tuple is determined by the
eigenvalues of its components; more precisely, there are orthonormal
vectors $v_\gamma\in R$ indexed by the elements $\gamma\in\Gamma$ and
corresponding eigenvalues $\lambda^i_\gamma\in{\mathbb C}$ such that:
\begin{equation}
\label{eq:alpha_eigenvectors}
\alpha_i(v_\gamma)=\lambda^i_\gamma v_\gamma, \qquad \text{for all } i,\gamma.
\end{equation}
This gives $r$ elements
$$\lambda_\gamma :=\sum_i \lambda^i_\gamma q_i \in Q,$$ which could be
called the {\em eigenvalues\/} of $\alpha$. These are defined up to a
permutation, because one can always conjugate by an elementary matrix
which permutes the rows of the $\alpha_i$'s.
In geometrical language, denote by~$\Delta\subset M$ the
subspace of $n$-tuples of matrices which are diagonal with respect to
the standard basis~$e_\gamma$ of~$R$. The unitary automorphism of $R$
which maps $e_\gamma$ to $v_\gamma$ moves $\alpha$ into $\Delta$. The
slice $\Delta$ can be identified with $Q^r$ by mapping $\alpha\in
\Delta$ to its $r$ eigenvalues (listed in some specified order). In
this way $\Delta$ inherits an action of $\Sigma_r$, and the
$\U(R)$-orbit of $\alpha\in\mu^{-1}(0)\cap {\mathcal N}$ intersects $\Delta$ in
a single $\Sigma_r$-orbit.
\end{proof}
\subsection{The Zero-Momentum Quotient}
\label{sec:partial:zero}
\nopagebreak
\begin{thm}
\label{thm:X0free}
If $\Gamma$ acts freely outside the origin, then $X_0\cong Q/\Gamma$ as
varieties.
\end{thm}
\begin{proof}
The proof consists in showing that the $K^\Gamma$-orbits in
${\mathcal N}^\Gamma\cap\mu^{-1}(0)$ can be identified in a one-one way with the
$\Gamma$-orbits in $Q$. Let $\alpha\in{\mathcal N}^\Gamma\cap\mu^{-1}(0)$ and let
$v_1$ be an eigenvector of $\alpha$ with eigenvalue
$\lambda_1:=\sum_i\lambda^i_1 q_i\in Q$:
$$\alpha_i(v_1)=\lambda^i_1 v_1,\quad i=1,\dots,n.$$ By equivariance
of $\alpha$,
$$\alpha_i (R(\gamma)v)=(Q(\gamma)\lambda_1)^i(R(\gamma)v), \text {
for all } \gamma,\text{ and all }i,$$ so the eigenvectors and
eigenvalues of $\alpha$ are given by
$$\lambda_\gamma:=Q(\gamma)\lambda_1\text{ and
}v_\gamma:=R(\gamma)v_1,\quad\text{for all }\gamma$$ i.e. they lie in orbits of $\Gamma$.
Since $\Gamma$ acts freely outside the origin, the eigenvalues are
either all zero or all distinct and non-zero. In the latter case, the
eigenvectors therefore form a basis of $R$. The unitary
automorphism of $R$ defined by $e_\gamma \mapsto v_\gamma$ commutes with the action of $\Gamma$, so defines an element of $K^\Gamma$.
If $\Delta^\Gamma\subset M^\Gamma$ denotes the $n$-tuples of
endomorphisms of $R$ which are diagonal with respect to the
standard basis $\{e_\gamma\}$ then the automorphism carries $\alpha$ into $\Delta^\Gamma$. The map $\alpha\mapsto \sum_i\lambda^i_1 q_i$
identifies $\Delta^\Gamma$ with $Q$ in a manner that is compatible
with the $\Gamma$-action on both sides.
Furthermore, the $K^\Gamma$-orbit of $\alpha$ intersects
$\Delta^\Gamma$ in precisely one $\Gamma$-orbit. Thus
$X_0\cong \Delta^\Gamma/\Gamma\cong Q/\Gamma$.
\end{proof}
The following lemma will be useful in the section on ALE metrics
(cf.~\cite{kron:thesis}).
\begin{lemma}
\label{lemma:iso_flat}
If $\Gamma$ acts freely outside the origin the map
$$\mu^{-1}(0)\cap{\mathcal N}^\Gamma/K^\Gamma\to \Delta^\Gamma/\Gamma$$ is an isometry
when $\Delta^\Gamma$ is given the metric it inherits as a subspace of
$M^\Gamma$, namely the Euclidean metric. Furthermore, the bundle
$\mu^{-1}(0)\cap{\mathcal N}^\Gamma\to X_0$ is flat.
\end{lemma}
\begin{proof}
The key point is that the subspace $\Delta^\Gamma$ is everywhere
orthogonal to the orbits of $K^\Gamma$: a tangent vector to the orbits
consists of an $n$-tuple of matrices of the form $[\xi,\alpha_i]$
for some $\xi\in\su^\Gamma(R)$, and these matrices are always zero on
the diagonal, so orthogonal to $\Delta^\Gamma$.
This shows that the bundle $\mu^{-1}(0)\cap{\mathcal N}^\Gamma \to X_0$ is flat,
and the definition of the quotient metric on $X_0$ implies that the
map $X_0=\mu^{-1}(0)\cap{\mathcal N}^\Gamma/K^\Gamma\to\Delta^\Gamma/\Gamma$ is an
isometry.
\end{proof}
\subsubsection{Case when $\Gamma$ does not act freely outside the origin}
Let $\alpha\in\cN^{\Gamma }\cap\mu^{-1}(0)$ have an eigenvalue $\lambda\in Q$.
If the stabiliser $\Gamma_\lambda$ of $\lambda$ is trivial, then
$\alpha$ has $r$ distinct eigenvalues, corresponding to the elements
of the orbit $\Gamma\lambda$. This determines the components
$\alpha_i$ completely on the whole of $R$.
On the other hand, if $\lambda$ has a non-trivial stabiliser
$\Gamma_\lambda$, then this determines $\alpha$ on the
sub-representation $W_\lambda := \spn \Gamma\cdot E_\lambda\subset R$, where
$E_\lambda$ is the eigenspace corresponding to $\lambda$. In fact,
$E_\lambda$ is a representation of the stabiliser subgroup
$\Gamma_\lambda$ and $W_\lambda$ is simply the representation
of $\Gamma$ induced by $E_\lambda$:
$$W_\lambda=\Ind_{\Gamma_\lambda}^\Gamma E_\lambda.$$
If $\dim E_\lambda<|\Gamma_\lambda|$, then $W_\lambda\neq R$ and
$\alpha$ restricts to an endomorphism of $W^\perp_\lambda$. Let
$\lambda'$ be an eigenvalue of the restriction; the equivariance
condition then determines $\alpha$ on the factor $W_{\lambda'}$. Continuing
in this way, one obtains a decomposition of $R$:
$$R=W_\lambda\oplus W_{\lambda'}\oplus\dots.$$
From this discussion, one obtains the following description of the quotient $X_0$:
\begin{thm}
\label{thm:X0}
There is an inclusion $Q/\Gamma \hookrightarrow X_0$; this inclusion is
an isomorphism if and only if $\Gamma$ acts freely on $Q$ outside the
origin.
\end{thm}
\begin{proof}
The first statement follows because, for any orbit $\Gamma\lambda$ in
$Q$, one can construct an $n$-tuple $\alpha$ of diagonal matrices
whose $\lambda$-eigenspace has dimension equal to the stabiliser
$\Gamma_\lambda$. The orbit of such an $\alpha$ under $K^\Gamma$ consists of
commuting matrices with eigenvalue $\gamma\lambda$ with multiplicity
$|\Gamma_\lambda|$, for all $\gamma\in\Gamma$.
For the second statement, note that the {\em if\/} direction is
theorem~\ref{thm:X0free}. The {\em only if\/} direction follows
because if $\lambda$ is an eigenvalue of $\alpha$ with non-trivial
stabiliser and with multiplicity one, one can set $\alpha$ to
be zero on $W^\perp_\lambda$ (since $0$ is a fixed point of $\Gamma$,
the equivariance condition~\eqref{eq:alpha_equivariance_condition}
does not imply the existence of other eigenvalues).
In general, $X_0$ corresponds configurations of $r=|\Gamma|$ points of
$Q$ which are unions of orbits of $\Gamma$, and hence give rise to a
decomposition of $R$ into induced representations
$$R=\bigoplus_i \Ind_{\Gamma_{\lambda_i}}^\Gamma E_{\lambda_i},$$
where $E_{\lambda_i}$ denote the $\lambda$-eigenspace of an element $\alpha$.
\end{proof}
\begin{rmk}
When $\Gamma$ doesn't act freely outside the origin, the quotient
$X_0$ may end up containing all sorts of things. For instance, for
the group action $\qsing 1/5(0,1,-1)$, the quotient $X_0$ contains a
copy of ${\mathbb C}^3/\Z_5$, a copy of ${\mathbb C}^5$, eight copies of ${\mathbb C}^2$, etc...
\end{rmk}
\subsection{Non-zero Momentum and Partial Resolutions}
\label{sec:partial:non-z}
By theorem~\ref{sec:git}.\ref{thm:partial_res}, in the case where
$\zeta$ is integral, there are projective morphisms $\rho_\zeta\colon
X_\zeta\to X_0$ which are isomorphisms over the set of points which
have finite $K^\Gamma$-stabilisers.
\begin{prop}
If $\Gamma$ acts freely outside the origin, The stabilisers of $K^\Gamma$
on ${\mathcal N}^\Gamma\cap\mu^{-1}(0)$ are trivial everywhere except at
$\alpha=0$.
\end{prop}
\begin{proof}
An automorphism $T$ of $R$ which fixes $\alpha\in
{\mathcal N}^\Gamma\cap\mu^{-1}(0)$ must preserve the (simultaneous) eigenspaces
of $\alpha$. If $T$ also commutes with the action of $\Gamma$, its
action on an eigenvector $v\in R$ determines its action on the
linear span of the $\Gamma$-orbit of $v$. In the case where $\Gamma$ acts
freely and $\alpha$ is non-zero this means that $T$ is only allowed
to multiply each eigenvector by the same non-zero constant --- and
this constant must be of modulus one if $T$ is unitary. Such a $T$
thus corresponds to the identity element in the quotient group
$K^\Gamma=\PU^\Gamma(R)$.
\end{proof}
Applying the theorem about K\"ahler\ quotients, one gets the following
theorem:
\begin{thm}
\label{thm:partial_resolutions}
If\/ $\Gamma$ acts on $Q$ freely outside the origin, and $\zeta$ is
integral, there are projective morphisms $\rho_\zeta\colon
X_\zeta\to X_0=Q/\Gamma$ which are isomorphisms outside the set
$\rho_\zeta^{-1}(0)$.
\end{thm}
\begin{rmk}
Even in the case that $\Gamma$ does not act freely outside the
origin, it is likely that there are still birational maps from $X_\zeta$ to
the component of $X_0$ which is isomorphic to $Q/\Gamma$ and which are
isomorphisms outside the singular set.
\end{rmk}
\section{ALE Metrics}
\label{sec:ale}
The quotients $X_\zeta$ inherit a metric ${\mathbf g}_\zeta$ from the metric
${\mathbf g}$ on the ambient space $M^\Gamma$. This section shows that these
are ALE metrics.
A metric ${\mathbf g}$ on a real $m$-dimensional Riemannian manifold $X$ is
called \emph{asymptotically locally Euclidean}
(\emph{ALE}) if there exists a compact subset $C\subset
X$ whose complement $X\setminus C$ has a finite covering
$\widetilde{X\setminus C}$ which is diffeomorphic to the complement of
a ball in ${\mathbb R}^m$, and such that, in the pulled-back coordinates
$x_1,\dots,x_m$ on $\widetilde{X\setminus C}$, ${\mathbf g}$ takes the form
\begin{equation}
{\mathbf g}_{ij}=\delta_{ij} + a_{ij},
\label{eq:ale}
\end{equation}
where $|\partial^p a_{ij}|=O(r^{-4-p})$ for $p\geq 0$, where
$r=\sqrt{\sum_i x_i^2}$ denotes the radial distance in ${\mathbb R}^m$ and
$\partial$ denotes the differentiation with respect to the coordinates
$x_1,\dots,x_m$.
\begin{thm}
\label{thm:ale}
The metrics on $X_\zeta$ are ALE: for any $\zeta$, there is an
expansion in powers of $r$
\begin{equation}
{\mathbf g}_\zeta = \delta + \sum_{k\geq 2} h_k(\theta)r^{-2k},
\label{eq:expansion}
\end{equation}
where $(r,\theta)$ denote polar coordinates in ${\mathbb R}^{2n}\cong Q$. This
expansion is analytic and may be differentiated term by term.
\end{thm}
\begin{proof}
Kronheimer's proof~\cite[Prop.5.5.1]{kron:thesis} goes through with
the appropriate modifications. The metric ${\mathbf g}_\zeta$ restricted
to the unit ball $r=1$ is an analytic function of $\zeta$, so admits
an expansion $${\mathbf g}_{\zeta| r=1} = \sum_\nu f_\nu \zeta^\nu$$ where
$\nu$ are multi-indices in the coordinates of $\zeta$. The moment
map being quadratic homogeneous implies that
$${\mathbf g}_\zeta(r,\theta)={\mathbf g}_{r^{-2}\zeta}(1,\theta).$$ Hence the
expansion for ${\mathbf g}_\zeta$ takes the form $$\sum_{k\geq 0}
h_k(\theta)r^{-2k},$$ where the $h_k=\sum_{|\nu|=k}f_\nu\zeta^\nu$
are analytic functions of the radial coordinates.
It remains to show that $h_0=\delta$ and that $h_1=0$. The first
statement is equivalent to showing that the identification $X_0\to
Q/\Gamma$ is an isometry. This was done in
Lemma~\ref{lemma:iso_flat}.
For the second statement, one must show that the variation of
${\mathbf g}_\zeta$ with $\zeta$ is zero at $\zeta=0$ in every direction
$\lambda\in ((\Lie K^\Gamma)^*)^{K^\Gamma}$. The metric ${\mathbf g}_\zeta$ is determined
entirely by the K\"ahler\ form $\omega_\zeta$ and the induced complex
structure $J_\zeta$. Since the latter is the same for all $\zeta$,
it is sufficient to prove that $$\partial_\lambda
\omega_{\zeta|\zeta=0}=0$$ for all $\lambda$. A general formula for
the variation of the induced symplectic form is given by Duistermaat
and Heckman in~\cite{dui_hec:variation}. Away from the
singularities, the projection $\mu^{-1}(\zeta)\cap{\mathcal N}^\Gamma\to
X_\zeta$ is a principal $K^\Gamma$-bundle whose connection is given by
the Levi-Civita connection for the induced metric on $X_\zeta$. If
$\Omega_\zeta$ denotes the curvature, regarded locally as an element
of $\Omega^2_{X_\zeta}\otimes\su^\Gamma(R)$, then the formula for the
variation of $\omega_\zeta$ is given by $$\partial_\lambda
\omega_\zeta = \<\lambda,\Omega_\zeta>.$$ In the present case,
lemma~\ref{lemma:iso_flat} tells us that $\Omega_0=0$, so the
variation is zero for $\zeta=0$, and this concludes the proof.
\end{proof}
\section{Deformation Complexes}
\label{sec:defcplx}
The question of the local geometry of the moduli spaces $X_\zeta$ can
be studied using the tools of deformation complexes.
\subsection{Differential Forms and Graded Lie Algebras}
\label{sec:defcplx:diff}
Define the vectorspaces $$M^{p,q} := \Omega^{p,q}_{Q^*}\otimes \End R,$$ for
$p,q \in{\mathbb N}$, whose typical element $\beta$ is of the form
$$\beta = \beta^I_{\bar J} dq_I\wedge d\bar q^{J},$$ where the
summation convention is used for the multi-indices $I,J$ and where, as
usual, $dq_I= \wedge_{i\in I} dq_i$, and $d\bar q^J=\wedge_{j\in J}
d\bar q^j$. Define the \emph{degree} of $\beta$ to be $\deg
\beta:=|I|+|J|$ and write $(-1)^\beta$ for $(-1)^{\deg\beta}$. The product of two elements $\alpha,\beta$ is
defined to be
$$\alpha\beta:=\alpha^I_{\bar J}\beta^{I'}_{\bar J'}dq_I\wedge d{\bar{q}}^{J}\wedge
dq_{I'}\wedge d{\bar{q}}^{J'},
$$
and the bracket of
any two elements $\alpha\in M^{p,q}$ and $\beta\in M^{p',q'}$ is
defined by
\begin{equation}
[\alpha,\beta] := [\alpha^I_{\bar J},\beta^{I'}_{\bar J'}]dq_I\wedge d\bar q^{J}\wedge dq_{I'}\wedge d\bar q^{J'} = \alpha\beta-(-1)^{\alpha\beta}\beta\alpha.
\label{eq:bracket}
\end{equation}
In the equation above and elsewhere, $(-1)^{\alpha\beta}$ means
$(-1)^{\deg\alpha\deg\beta}$ and not
$(-1)^{\deg\alpha}(-1)^{\deg\beta}$. Writing
$M^r:=\sum_{p+q=r}M^{p,q}$, the algebra $M^*$ inherits the structure of a
graded Lie algebra, namely a graded algebra with a bracket satisfying
$$[M^r,M^s]\subset M^{r+s},$$
(graded) skew-commutativity
$$[\alpha,\beta]=-(-1)^{\alpha\beta}[\beta,\alpha],$$ and the (graded)
Jacobi identity:
$$(-1)^{\alpha\gamma}[\alpha,[\beta,\gamma]]+(-1)^{\beta\alpha}[\beta,[\gamma,\alpha]]+(-1)^{\gamma\beta}[\gamma,[\alpha,\beta]]=0.$$
There are two sub-algebras $M^{*,0}$ and $M^{0,*}$ of $M^*$. Defining
the adjoint of $\alpha=\alpha^I_{\bar J}dq_I\wedge d{\bar{q}}^{J}$ to be
$$\alpha^* := (\alpha^I_{\bar J})^*d{\bar{q}}^I\wedge dq_{J},$$
then
$(\alpha\beta)^* = (-1)^{\alpha\beta}\beta^*\alpha^*$ and
$[\alpha,\beta]^*=(-1)^{\alpha\beta}[\beta^*,\alpha^*]$.
The Jacobi identity implies that if $\alpha$ or $\beta$ has odd degree then
\begin{equation}
[\alpha,[\alpha,\beta]]=\frac{1}{2}[[\alpha,\alpha],\beta].
\label{eq:aab}
\end{equation}
If $\alpha=\alpha_i d{\bar{q}}^i \in M^{0,1}$, and one
defines
\map{\bar{\partial}_\alpha}{M^{p,q}}{M^{p,q+1}}{\beta}{[\alpha,\beta],} then,
using~\eqref{eq:aab}, one sees that the sequences
\begin{equation} \label{eq:Mpx_complex}
M^{p,*}: M^{p,0}\xrightarrow{\bar{\partial}^0_\alpha}M^{p,1}\xrightarrow{\bar{\partial}^1_\alpha}M^{p,2}\to\dots
\end{equation}
of vectorspaces are complexes precisely when $[\alpha,\alpha]=0$,
i.e.\ when $\alpha\in {\mathcal N}$. Write $H^{p,q}_\alpha$ for the cohomology
groups $H^q(M^{p,*},\bar{\partial}_\alpha)$. If one introduces a metric on
$M^{p,q}$ by using the standard inner product on $\End R$ and making
$$\frac{1}{\sqrt{2^{p+q}}}\{dq_I\wedge d{\bar{q}}^{J}\}_{\substack{|I|=p,\\
|J|=q}}$$
orthonormal~\cite[p.80]{gri_har:ag}, one can define the adjoint
operator $\bar{\partial}^*_\alpha$, and the Laplacian
$\bar\Box_\alpha:=\bar{\partial}_\alpha\bar{\partial}^*_\alpha+\bar{\partial}^*_\alpha\bar{\partial}_\alpha$.
Their kernels give harmonic representatives for the cohomology groups
in the usual way $${\mathcal H}^{p,q}_\alpha:= \ker \bar\Box^{p,q}_\alpha
\subset M^{p,q}.$$
If $\Lambda\colon M^{p,q}\to M^{p-1,q-1}$ denotes
the operation of contraction with the K\"ahler\ form $\omega= dq_i\wedge
d{\bar{q}}^i$, then the definition of the adjoint and the invariance of
the trace under cyclic permutations give
\begin{equation}
\bar{\partial}^*_\alpha \beta= -\Lambda[\alpha^*,\beta],
\label{eq:dstar}
\end{equation}
or, in coordinates,
\begin{equation}(\bar{\partial}^*_\alpha \beta)^I_{\bar J} = [\alpha^*_j,\beta^I_{j\bar J}].\label{eq:dstar_coord}
\end{equation}
Writing $\kappa:= dq_1\wedge \dots \wedge dq_n$,
the $n$-th power of $\omega$ is
\begin{equation}
\omega^n=n!(-1)^{(n-1)(n-2)/2}\frac{i^n}{2^n}\kappa\wedge\kappa^*.
\label{eq:omegan}
\end{equation}
\subsection{Local Description of $X_\zeta$}
\label{sec:defcplx:local}
Under the identification $M=M^{0,1}$, the derivative of the
action~\eqref{eq:action_glr} of $\GL(R)$ on $M$ is given by
$$\partial_\alpha^0\colon M^{0,0}\to M^{0,1}.$$ On the other hand, the
derivative of $\alpha\mapsto F^{0,2}_\alpha$ is (twice)
$$\bar{\partial}^1_\alpha \colon M^{0,1}\to M^{0,2},$$ so the Zariski tangent
space to ${\mathcal N}\gitquot{}\GL(R)$ at an element $\alpha$ is given by the first cohomology group of the Atiyah-Hitchin-Singer~\cite{ahs} deformation complex
$$ M^{0,0}\xrightarrow{\bar{\partial}^0_\alpha}M^{0,1}\xrightarrow{\bar{\partial}^1_\alpha}M^{0,2},$$ i.e.\
by $H^{0,1}_\alpha$. From the point of view of the K\"ahler\ quotient,
one can see this as follows: the Zariski tangent space to $X_\zeta$ at
$[\alpha]$ is given by $\ker d\mu(\alpha)\cap\ker d\psi(\alpha)$. By
definition, the derivative of $\mu$ is dual to the action of $K^\Gamma$,
so $d\mu(\alpha)=-2\bar{\partial}^*_\alpha= 2\Lambda[\alpha^*,\ ]$, as can be
verified by remarking that $\mu(\alpha)=\Lambda[\alpha^*,\alpha]$.
Hence $\ker d\mu(\alpha)=\Image \bar{\partial}^*_\alpha$, so the tangent
spaces indeed coincide.
A local model for ${\mathcal N}\gitquot{}\GL(R)$ in a neighbourhood of a point
$[\alpha]$ where $\alpha\in{\mathcal N}$ has trivial stabiliser is given by
solving the equation $F^{0,2}_{a+\beta}=0$ for $\beta$ in the slice
$$\{\beta\in M^{0,1}: \bar{\partial}_\alpha^*\beta=0,\|\beta\|\text{
small}\}.$$ This comes down to solving the system of equations
\begin{align}
&\bar{\partial}^*_\alpha \beta=0\\
&\bar{\partial}_\alpha \beta + \frac{1}{2}[\beta,\beta]=0,
\end{align}
in a neighbourhood of the origin. Kuranishi's argument~\cite{kura:complex,kura:newproof} shows that the
solution set is given by the zero set of a map $\Phi\colon
{\mathcal H}^{0,1}_\alpha\to {\mathcal H}^{0,2}_\alpha$ whose two-jet at the origin is
given by
\map{\Phi_{(2)}}{{\mathcal H}^{0,1}_\alpha}{{\mathcal H}^{0,2}_\alpha}{\beta}{{\mathcal H}_\alpha([\beta,\beta]).}
where
$${\mathcal H}_\alpha \colon M^{0,2} \to {\mathcal H}^{0,2}_\alpha$$ denotes the
orthogonal projection to the harmonic subspace. Similar statements
hold for $X_\zeta$ and $M^{0,*,\Gamma}$.
\subsection{Kuranishi Germs and Formality}
\label{sec:defcplx:kura}
This whole discussion can be phrased in more abstract language of
deformation functors and differential graded Lie algebras. Additional
details and background can be found
in~\cite{gold_mill:invariance,gold_mill:fundamental} and~\cite{dgms}.
The algebra $M^{0,*}$ is actually a \emph{differential graded Lie
algebra} (DGLA) when
endowed with the differential $\bar{\partial}_\alpha$. The metric on $M^{p,q}$
makes it into an \emph{analytic DGLA}, namely a DGLA which possesses
a norm compatible with its differential and bracket, and which induces
what is essentially a Hodge decomposition of its graded pieces with
finite dimensional topological summands ${\mathcal H}^i$ which are the
analogues of the harmonic forms. When $\alpha$ has trivial
stabiliser, $X_\zeta$ is locally analytically isomorphic in the
neighbourhood of $[\alpha]$ to the \emph{Kuranishi germ} ${{\mathbf K}_M}$
associated to $M^{0,*,\Gamma}$. The results so far are stated in the
following theorem.
\begin{thm}
\label{thm:formal}
The sequence of vector spaces $(M^{0,*},\bar{\partial}_\alpha)$ is a complex
(and therefore a differential graded Lie algebra) if and only if
$\alpha\in{\mathcal N}$. Furthermore, if $\alpha\in{\mathcal N}^\Gamma$, the Zariski
tangent space to $X_\zeta$ at $[\alpha]$ is isomorphic to the first
cohomology group of its $\Gamma$-invariant part
$$H^{0,1,\Gamma}_\alpha:=H^1(M^{0,*,\Gamma},\bar{\partial}_\alpha)$$ and if
$\alpha$ has trivial $K^\Gamma$-stabilisers, then $X_\zeta$ is locally
isomorphic to its Kuranishi germ
$${\mathbf K}_{M^{0,*,\Gamma}}=\{\beta\in M^{0,1,\Gamma} |
\bar{\partial}^*_\alpha\beta=\bar{\partial}_\alpha \beta + \frac{1}{2}[\beta,\beta]=0\}.$$
\end{thm}
In general the Kuranishi germ ${\mathbf K}_L$ of an analytic DGLA $(L,d)$ is
(Banach analytically) isomorphic to the germ at $0$ of
$$\{\beta\in (\Image d)^\perp\subset L^1 | d\beta+\frac{1}{2}[\beta,\beta]=0\},$$ where $(\Image d)^\perp$ is a fixed complement
of the image of $d$ in $L^1$. Goldman
and~Millson~\cite{gold_mill:invariance} prove that ${\mathbf K}_L$ is
invariant under \emph{quasi-isomorphisms}, namely chains of
homomorphisms of DGLAs\footnote{An important point is that the intermediate
$L',L'',\dots$ need {\em not\/} have any analytic structure and that the
intermediate arrows need not preserve any splittings.}
$$L\to L' \leftarrow L'' \to \dots \leftarrow L'''$$ which induce
isomorphisms in cohomology. When $L$ is quasi-isomorphic to its
cohomology (which is a DGLA when endowed with the zero differential),
$L$ is called \emph{formal} and it follows that ${\mathbf K}_L$ is analytically
isomorphic to the quadratic cone $${\mathbf Q}_L:=\{\beta\in{\mathcal H}^1 :
[\beta,\beta]=0\}.$$
One way in which this can happen is if
the bracket of two harmonic elements of degree $1$ is harmonic. This
is the case, for instance for the moduli space of flat
Hermitian-Yang-Mills connections over a compact K\"ahler\
manifold~\cite{nadel:quadratic,gold_mill:flat,gold_mill:invariance}.
If in addition, the cup-product on ${\mathcal H}^1$ is zero, then ${\mathbf K}_L\cong
{\mathcal H}^1$ and the deformation space is a smooth manifold (even if
${\mathcal H}^2\neq 0$). This is the case, for instance for the moduli space
of complex structures over a \emph{Calabi-Yau $n$-fold}, namely a
compact K\"ahler\ manifold with a nowhere vanishing holomorphic
$(n,0)$-form~\cite{gold_mill:invariance}. These moduli were studied by F.~Bogomolov. The key fact which implies
the formality of the DGLA and the vanishing of the cup-product in this
case was proved by Tian~\cite{tian:smoothness} and
Todorov~\cite{todo:weil-peterson}.
In the case of the algebra $M^{0,*}$, formula~\eqref{eq:aab} with
$\alpha$ and $\beta$ interchanged shows that the bracket of two
harmonic elements in ${\mathcal H}^{0,1}_\alpha$ is $\bar{\partial}_\alpha$-closed.
However, it does not follow that $\bar{\partial}^*_\alpha([\beta,\beta])=0$;
indeed this is easily seen to be false, since
$[\beta,\beta]=2\beta\beta$. Nevertheless, it does not seem
unreasonable to expect that $M^{0,*,\Gamma}$ can also be proved to be
formal for generic $\zeta$, maybe by imitating Tian and Todorov's
method.
\begin{conj}
\label{conj:formal}
The differential graded Lie algebra $(M^{0,*,\Gamma},\bar{\partial}_\alpha)$ is formal
for all $\alpha\in{\mathcal N}^\Gamma\cap\mu^{-1}(\zeta)$ and generic $\zeta$, and therefore $X_\zeta$ has, for these $\zeta$, at worst
quadratic algebraic singularities.
\end{conj}
Another conjecture is the following:
\begin{conj}
\label{conj:su3}
If $\Gamma\subset\SU(3)$, can one imitate the Tian-Todorov proof and
show that the Kuranishi germ of $(M^{0,*,\Gamma},\bar{\partial}_\alpha)$ is
isomorphic to ${\mathcal H}^{0,1,\Gamma}_\alpha$ for generic $\zeta$, i.e.~that
$X_\zeta$ is smooth?
\end{conj}
The fact that $X_\zeta$ has at most quadratic singularities has been
verified for the abelian subgroups of order less than~11. The
smoothness of $X_\zeta$ has been verified in the abelian cases $\qsing
1/3(1,1,1)$, $\qsing 1/6(1,2,3)$, $\qsing 1/7(1,2,4)$, $\qsing
1/8(1,2,5)$, $\qsing 1/9(1,2,6)$, $\qsing 1/10(1,2,7)$ and $\qsing
1/11(1,2,8)$. Both these verifications were done by exhaustive
listing of singularities of $X_\zeta$ for all possible $\zeta$, using
the methods given in the companion paper~\cite{sacha:flows}.
A different approach is available in the specific case of $\SU(3)$;
this is presented next.
\section{Subgroups of $\protect\SU(3)$ and Cubic Forms}
\label{sec:su3}
Suppose that $\Gamma\subset\SU(3)$. If $\alpha\in\mu^{-1}(\zeta)$ and
$\beta,\delta\in{\mathcal H}^{0,1,\Gamma}_\alpha$, then, as remarked in the
previous section,
$$\bar{\partial}_\alpha[\beta,\delta] =0,$$ but $[\beta,\delta]$ is not in
${\mathcal H}^{0,2,\Gamma}_\alpha$. However, considerations of type show that it differs
from its harmonic projection by a term $\bar{\partial}_\alpha\epsilon$, for some
$\epsilon\in M^{0,1,\Gamma}$. For $\eta\in{\mathcal H}^{0,1,\Gamma}_\alpha$
\begin{align}
\trace(\eta[\beta,\delta])- \trace(\eta{\mathcal H}_\alpha([\beta,\delta])) &= \trace(\eta[\alpha,\epsilon]) \notag\\
&= \trace(\epsilon[\eta,\alpha])\notag\\
&= 0,\qquad \text{since } \eta\in{\mathcal H}^{0,1,\Gamma}_\alpha.
\label{eq:trace_harmonic}
\end{align}
This shows that the tensor
\corresp{H^{0,1,\Gamma}_\alpha\otimes H^{0,1,\Gamma}_\alpha\otimes
H^{0,1,\Gamma}_\alpha}{{\mathbb C}}{(\eta,\beta,\delta)\phantom{{}^{,1,\Gamma}}}{\kappa\trace(\eta{\mathcal H}_\alpha([\beta,\delta])),}
is totally symmetric on $H^{0,1,\Gamma}_\alpha$ (the
isomorphism $\Omega^{3,3}_{Q^*}\cong{\mathbb C}$ has been used). An easy polarisation
argument shows that it is completely determined by the corresponding cubic form
\map{{\mathbf C}}{H^{0,1,\Gamma}_\alpha}{\mathbb C}\beta{\kappa\trace(\beta([\beta,\beta])).}
\begin{prop}
\label{prop:cubic}
The singularity of $X_\zeta$ has no quadratic part if and only if
${\mathbf C}(\beta)=0$ for all $\beta\in{\mathcal H}^{0,1,\Gamma}_\alpha$ and all
$\alpha\in\mu^{-1}(\zeta)$.
\end{prop}
\begin{proof}
Suppose $X_\zeta$ has no quadratic part at $[\alpha]$. Then
$\Phi_{(2)}(\beta)={\mathcal H}_\alpha([\beta,\beta])=0$. But this implies
that ${\mathbf C}(\beta)=0$ by equation~\eqref{eq:trace_harmonic}.
Conversely, if ${\mathbf C}(\beta)=0$ for all $\beta\in{\mathcal H}^{0,1,\Gamma}$ then
the corresponding totally symmetric tensor vanishes on all triples
$(\eta,\beta,\beta)$ for all $\eta,\beta\in {\mathcal H}^{0,1,\Gamma}_\alpha$.
Since this is true for all $\eta$, it must be that
${\mathcal H}_\alpha([\beta,\beta])\in\Image\bar{\partial}_\alpha$, i.e.\
$\Phi_{(2)}(\beta)=0$ in ${\mathcal H}^{0,2,\Gamma}_\alpha$.
\end{proof}
There is a natural $3$-vector ${\Omega}$ whose value on three elements
of $H^{0,1,\Gamma}_\alpha$ of is given by
\begin{equation}
\Omega(\eta,\beta,\delta):=\kappa\trace( \eta\beta\delta).
\label{eq:Omega}
\end{equation}
This is symmetric under cyclic permutations of the entries, so
decomposes into a totally skew-symmetric part $\Omega_{\text{skew}}$
and a totally symmetric part, which is nothing but the totally
symmetric tensor corresponding to ${\mathbf C}$. The proposition above
implies the
\begin{cor}
\label{cor:3form}
If $X_\zeta$ is smooth, then $\Omega$ defines an element of
$\Omega^{3,0}(X_\zeta)$.
\end{cor}
\begin{conj}
The canonical sheaf ${\mathcal O}_{X_\zeta}(K_{X_\zeta})$ is locally free,
and is generated by the non-vanishing $(3,0)$-form $\Omega$ when
$X_\zeta$ is smooth.
\end{conj}
Taking the wedge of $\Omega$ with its complex conjugate gives
\begin{align}
\label{eq:owo1}
\Omega\wedge\Omega^*(\eta,\beta,\delta,\eta^*,\beta^*,\delta^*) &=
(\epsilon_{ijk}\trace
\eta_i\beta_j\delta_k)\overline{(\epsilon_{{\bar{\imath}}{\bar{\jmath}}{\bar{k}}}\trace
\eta_{\bar{\imath}}\beta_{\bar{\jmath}}\delta_{\bar{k}})}\kappa\wedge\kappa^*,\\
\label{eq:owo2}
&=\left|\sum^\circ_{ijk}\trace \eta_i\beta_j\delta_k\right|^2\kappa\wedge\kappa^*,
\end{align}
where $\sum^\circ_{ijk}$ denotes the sum over distinct $i,j$ and $k$.
On the other hand, the symplectic form $\omega_\zeta$ on $X_\zeta$ is
simply the restriction of the symplectic form $\omega$ defined
in~\eqref{eq:dfn_omega}, and equation~\eqref{eq:omegan} gives
\begin{equation}
\omega_\zeta\wedge\omega_\zeta\wedge\omega_\zeta = \frac{3i}{4}\kappa\wedge\kappa^*.
\end{equation}
Suppose that $(\eta,\beta,\delta)$ are an orthonormal triple in
$T^{1,0}_\alpha X_\zeta$. Then the value of the coefficient of
$\kappa\wedge\kappa^*$ in~\eqref{eq:owo2} is equal to $\|\Omega\|^2
\|\kappa\wedge\kappa^*\|^{-2}$. Hence $X_\zeta$ has trivial canonical
bundle if this coefficient is never zero for all
$\alpha\in\mu^{-1}(\zeta)$.
\begin{lemma}
\label{lemma:bochner}
The K\"ahler\ manifold $X_\zeta$ is Ricci-flat if
and only if the coefficient of $\kappa\wedge \kappa^*$
in~\eqref{eq:owo2} is constant for all orthonormal triples
$(\eta,\beta,\delta)$ in $H^{0,1,\Gamma}_\alpha$ and all
$\alpha\in\mu^{-1}(\zeta)\cap{\mathcal N}^\Gamma$.
\end{lemma}
\begin{proof}
This follows because if $X_\zeta$ is Ricci-flat, there exists a
holomorphic $(3,0)$-form $\Omega'$ which is covariant constant on
$X_\zeta$. Hence $\Omega$ will differ from $\Omega'$ by a
holomorphic function $f$. Now Liouville's theorem implies that $f$
is either constant or unbounded. Since $\Omega$ is clearly bounded
(by $6\kappa\wedge\kappa^*$), $f$ must be constant.
\end{proof}
\subsection{Example}
\label{sec:su3:ex}
Let us work out a specific example. Consider the group $\Gamma=\mu_3$
of order $3$ acting on ${\mathbb C}^3$ with weights $(1,1,1)$. The following
configuration of matrices is easily seen to define a point of
$\mu^{-1}(\zeta)\cap{\mathcal N}^\Gamma$, where $\zeta=(-|A|^2,|A|^2-|B|^2,|B|^2)$, $(A,B\in{\mathbb R})$:
\begin{equation}
\label{eq:point1}
\alpha_1=\begin{pmatrix}
0 &A &0 \\
0 &0 &B \\
0 &0 &0
\end{pmatrix}d{\bar{q}}^1, \quad \alpha_2=\alpha_3=0.
\end{equation}
The tangent space is three-dimensional and is generated by the
following orthonormal elements (recall that $\|d{\bar{q}}^i\|^2=2$)
\begin{equation}
\beta_1=\frac{1}{\sqrt 2}\begin{pmatrix}
0 &0 &0 \\
0 &0 &0 \\
1 &0 &0
\end{pmatrix}d{\bar{q}}^1, \quad
\beta_i= \frac{1}{\sqrt{2(A^2+B^2)}}\begin{pmatrix}
0 &A &0 \\
0 &0 &B \\
0 &0 &0
\end{pmatrix}d{\bar{q}}^i,
\end{equation}
for $i=2,3$, so this defines a smooth point of $X_\zeta$. The value
of $\|\Omega\|^2$ at this point is
\begin{equation}
\left|\frac{1.A.B+1.B.A}{2\sqrt{2}(A^2+B^2)}\right|^2\kappa\wedge\kappa^* =
\frac{1}{2}\left(\frac{AB}{A^2+B^2}\right)^2 \kappa\wedge\kappa^*,
\label{eq:norm1}
\end{equation} and so this is non-zero away from $AB=0$ (which correspond to
non-generic values of $\zeta$).
At the point
\begin{equation}\label{eq:point2}
\alpha_1=\begin{pmatrix}
0 &A+C &0 \\
0 &0 &B+C \\
C &0 &0
\end{pmatrix}d{\bar{q}}^1, \quad
\alpha_2= \alpha_3=0,
\end{equation}
in $\mu^{-1}(\zeta)\cap {\mathcal N}^\Gamma$, the tangent space is still
three-dimensional, with orthonormal generators
\begin{equation}
\beta_i = \frac{1}{\sqrt{6}}\begin{pmatrix}
0 &1 &0 \\
0 &0 &1 \\
1 &0 &0
\end{pmatrix}d{\bar{q}}^i, \quad i=1,2,3,
\end{equation}
The value of $\|\Omega\|^2$ however is now
\begin{equation}
\left|6\left(\frac{1}{\sqrt 6}\right)^3\right|^2 \kappa\wedge\kappa^*=\frac{1}{6}\kappa\wedge\kappa^*.
\label{eq:norm2}
\end{equation}
In fact, all the points of $\mu^{-1}(\zeta)\cap{\mathcal N}^\Gamma$ are of the
form~ \eqref{eq:point1} or ~\eqref{eq:point2} (modulo permutations of
the indices $1,2,3$)\footnote{See~\cite{sacha:flows} for an
explanation of why this is so.}. Thus it has been shown, in a
rather laborious way, that away from certain degenerate values of
$\zeta$, $\Omega$ is non-vanishing on $X_\zeta$ and $K_{X_\zeta}$ is
therefore trivial. In fact, $X_\zeta={\mathcal O}_{{\mathbb P}^2}(-3)$.
Since the coefficient of $\kappa\wedge\kappa^*$ in~\eqref{eq:norm1} is
always smaller than ${1/ 8}$, one also deduces that
$\Omega\wedge\Omega^*$ is not a constant multiple of
$\omega_\zeta\wedge\omega_\zeta\wedge\omega_\zeta$ on any of the
quotients $X_\zeta$, and therefore by lemma~\ref{lemma:bochner} that
the induced metric is never Ricci-flat.
\begin{rmk} The space ${\mathcal O}_{{\mathbb P}^2}(-3)$ does have a standard
Ricci-flat metric, as was first noted by Calabi~\cite{calabi}.
\end{rmk}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
|
1996-10-12T01:24:37 | 9610 | alg-geom/9610013 | en | https://arxiv.org/abs/alg-geom/9610013 | [
"alg-geom",
"math.AG"
] | alg-geom/9610013 | Hans Boden | H. U. Boden and K. Yokogawa | Rationality of Moduli Spaces of Parabolic Bundles | latex2e | J. London Math. Soc. 59 (1999) 461-478 | 10.1112/S0024610799007061 | null | null | The moduli space of parabolic bundles with fixed determinant over a smooth
curve of genus greater than one is proved to be rational whenever one of the
multiplicities associated to the quasi-parabolic structure is equal to one. It
follows that if rank and degree are coprime, the moduli space of vector bundles
is stably rational, and the bound obtained on the level is strong enough to
conclude rationality in many cases.
| [
{
"version": "v1",
"created": "Fri, 11 Oct 1996 23:16:19 GMT"
}
] | 2021-09-29T00:00:00 | [
[
"Boden",
"H. U.",
""
],
[
"Yokogawa",
"K.",
""
]
] | alg-geom | \section{Introduction}
Let $X$ be a smooth complex curve of genus $g \geq 2$, $L$ a line bundle of
degree $d$ over $X,$ and
${\cal M}_{r,L}$ the moduli space of semistable bundles $E$ of rank $r$ with
determinant $L$.
\begin{conj} \label{conj}
${\cal M}_{r,L}$ is {\it rational}, i.e. it is birational
to a projective space.
\end{conj}
\noindent
Despite many positive results \cite{newstead1}, this is still an open problem,
even for $(r,d)=1.$
In this paper, we study a closely related problem, namely the birational
classification of moduli
spaces of parabolic bundles over $X.$
These moduli spaces occur naturally as
\begin{enumerate}
\item[(i)] unitary representation spaces of Fuchsian groups
\cite{mehta-seshadri},
\item[(ii)] moduli spaces of Yang-Mills connections on X with an orbifold
metric \cite{boden1}, and
\item[(iii)] moduli spaces of certain semistable bundles over an elliptic
surface \cite{bauer}.
\end{enumerate}
The theory developed in \cite{bh} and extended here shows that their birational
type depends only on the {\em quasi-parabolic} structure (see Proposition
\ref{prop:qp}). The methods of \cite{newstead1} then prove, in many cases, that
these moduli spaces are rational. The weaker result, Theorem \ref{thm:rat1},
uses only
Newstead's theorem, while the stronger one, Theorem \ref{thm:rat2}, requires
an adaptation of his inductive argument.
Using the theory developed in \S 4, it then follows from
Theorem \ref{thm:rat2} that
${\cal M}_{r,L} \times {\Bbb P}^{r-1}$ is rational
if $(r,d)=1$ (see Corollary \ref{cor}).
Stable rationality of the moduli spaces
had been proved in this case by Ballico
\cite{ballico}, and our result is a strengthening of his.
For instance, a consequence is that one can
conclude Conjecture \ref{conj}
under the assumption that $(r,d)=1$ for most values of the
genus\footnote{Choosing $d' \equiv d \mod(r)$ with $0<d'<r,$ the hypothesis is
that either $(d',g)=1$ or $(r-d',g)=1.$}
(see Corollary \ref{cor:rat}).
A number of useful facts are established along the way. One key point is
Proposition \ref{prop:fine},
which gives a simple criterion for the existence of a universal bundle of
stable parabolic bundles.
We also extend the theory developed in \cite{bh} in several important ways
(Theorems \ref{thm:bh1}, \ref{thm:bh2}, and \ref{thm:bh3});
the first two are standard but necessary for our purposes and the third is
completely new.
Its proof requires the idea of shifting a parabolic sheaf (Definition
\ref{defn:shift}), which also provides
a framework for the Hecke correspondence (equation (\ref{eqn:hecke})).
All of these results play a crucial role in the proofs of Theorems
\ref{thm:rat1} and \ref{thm:rat2}.
A brief word about the organization of this paper:
\S 2 introduces the notation used in the following sections,
\S 3 discusses the existence of universal families,
\S 4 summarizes and extends the theory of \cite{bh},
\S 5 describes shifting and the Hecke correspondence,
and \S 6 contains the proofs of the main results and their corollaries.
Before we begin, we would like to acknowledge a certain debt to the work of
Newstead,
upon which a number of our arguments depend, and
without which this paper would be inconceivable.
\section{Notation}
Let $X$ be a smooth curve of genus $g \geq 2$ and $D$ a reduced divisor in
$X.$
If $E$ is a ${\Bbb C}^r$ bundle over $X,$ then a {\it parabolic structure} on
$E$ with respect to
$D$ is just a collection of weighted flags in the fibers of $E$ over each $p
\in D$ of the form
\begin{eqnarray} \label{defn:parbun1}
&E_p = F_1(p) \supset F_2(p) \supset \cdots \supset F_{\kappa_p}(p) \supset
0,&\\
&\, \;\; 0 \leq \alpha_1(p) < \alpha_2(p)< \cdots < \alpha_{\kappa_p}(p) < 1.&
\label{defn:parbun2}
\end{eqnarray}
Holomorphic bundles $E$ with parabolic structures are called {\it parabolic
bundles}, and we use the notation $E_*$ to indicate the bundle (or,
equivalently, locally-free sheaf) $E$ together with a choice of parabolic
structure.
A morphism $\phi:E_* \longrightarrow E'_*$ of parabolic bundles is a bundle map
satisfying
$\phi(F_i(p)) \subset F'_{j+1}(p)$ whenever $\alpha_i(p) > \alpha'_j(p)$ for all $p
\in D.$
We use the tensor product notation $H^0(E_*^\vee \otimes E'_*)$
for these morphisms, where $E^\vee_*$ denotes the dual parabolic bundle (cf.
\cite{yoko}).
A {\it quasi-parabolic} structure on $E$ is what is left after the weights are
forgotten,
it is determined topologically by its flag type $m,$ which specifies {\it
multiplicities}
$m(p) = (m_1(p), \ldots, m_{\kappa_p}(p))$ for each $p \in D$ defined by
$m_i(p) = \dim F_i(p) - \dim F_{i+1}(p).$
A subbundle $E'$ inherits a parabolic structure from one on $E$ in a canonical
way:
The flag in $E'_p$ is gotten by intersecting with the flag in $E_p$ and the
weights are determined by choosing
maximal weights such that the inclusion map from $E'$ to $E$ is parabolic
(p.\ 213, \cite{mehta-seshadri}). Parabolic structures on quotients have a
similar description (loc.\ cit.).
A parabolic bundle $E_*$ is called {\it stable}
if every proper holomorphic subbundle $E'$ satisfies
$\mu (E'_*) < \mu (E_*),$
where
$$\mu (E_*) = \operatorname{pardeg} E_* / r =
\deg E / r + \sum_{p \in D} \sum_{i=1}^{\kappa_p} m_i(p) \alpha_i (p)/r.$$
The parabolic bundle $E_*$ is called {\it semistable} if $\mu (E'_*) \leq \mu
(E_*)$
for each subbundle $E'_*.$
The construction of the moduli space ${\cal M}_\alpha$ of semistable parabolic bundles,
as a normal, projective variety, is given in \cite{mehta-seshadri}. The
subspace ${\cal M}^s_\alpha$
of stable bundles is smooth and Zariski-open, in particular,
if every semistable bundle is stable, then ${\cal M}_\alpha$ is smooth.
Let $\Delta^r = \{ (a_1,\ldots,a_r) \mid 0 \leq a_1 \leq \cdots \leq a_r <1\}$
and define $W = \{ \alpha:D\longrightarrow \Delta^r \}.$
Points in $W$ determine both the weights and the multiplicities. Conversely,
given a weight $\alpha$ in the sense of (\ref{defn:parbun2}), the associated point
in $W$ is
gotten by repeating each $\alpha_i(p)$ according to its multiplicity $m_i(p)$.
We abuse notation slightly by referring to points in $W$ as weights.
This gives an obvious notion of when a weight is {\it compatible}
with a choice of multiplicities, and for a given $m,$ we define
the open face of weights compatible with $m$ to be
$$V_m = \{ \alpha \in W \mid \alpha_{i-1}(p)=\alpha_i(p)
\Leftrightarrow \sum_{k=1}^j m_k(p) < i \leq \sum_{k=1}^{j+1} m_k(p) \}.$$
A weight in the interior of $W$ specifies full flags at each $p \in D.$
For every other choice of $m$,
$V_m$ is contained in the boundary of $W.$
Now $W$ is a simplicial set, and the face relations give a natural ordering on
$\{V_m\}$
and we write $V_m > V_{m'}$ if $V_{m'}$ is a proper face contained in the
closure of $V_m.$
This agrees with the natural ordering on $m$ gotten by successive refinement.
Weights for which ${\cal M}_\alpha$ is not necessarily smooth
satisfy $\mu(E'_*) = \mu(E_*)$ for some proper subbundle $E'.$
Letting $E''$ be the quotient, then the short exact sequence of parabolic
bundles
$E'_* \stackrel{\iota}{\longrightarrow} E_* \stackrel{\pi}{\longrightarrow}
E''_*$
determines a partition of $(d,r,m)$ in the obvious way:
$(d',d''), (r',r'')$ and $(m',m'')$ are the degrees, ranks, and multiplicities
of $(E',E'').$
(We define $m'$ and $m''$ here slightly unconventionally, namely
\begin{eqnarray*}
m'_i(p) &=& \dim (F_i(p) \cap \iota (E'_p)) - \dim (F_{i+1}(p) \cap \iota
(E'_p)),\\
m''_i(p) &=& \dim (\pi (F_i(p)) \cap E''_p) - \dim (\pi (F_{i+1}(p)) \cap
E''_p),
\end{eqnarray*}
for $p \in D$ and $1 \leq i \leq \kappa_p$.)
Notice that $r',r'' > 0$ and $m'_i(p), m''_i(p) \geq 0.$
Write $\xi = (d',r',m').$ For fixed
$\xi,$ the set of weights
compatible with $m$
for which $\mu(E'_*) = \mu(E_*)$ is
the intersection of a hyperplane $H_\xi$ in $W$ with $V_m$ given by the
equation
\begin{equation}\label{eqn:hyper}
\sum_{i=1}^{\kappa_p} m_i(p) \alpha_i(p))
\sum_{p\in D} \sum_{i=1}^{\kappa_p} (r' m_i(p)-r m'_i(p)) \alpha_i(p) = r d' -
r'd.
\end{equation}
There are only finitely many hyperplanes; the above equation puts a bound
on $d'$ and all other quantities are already bounded.
We shall refer to $H_\xi \cap V_m$ as a {\it wall} in $V_m.$
These walls induce a chamber structure on $V_m,$ a {\it chamber} being
a connected component of $V_m \setminus \cup_\xi H_\xi$
(it is possible that $V_m \subset H_\xi$).
Weights $\alpha \in W \setminus \cup H_\xi$ are called {\it generic}, and for
these weights,
${\cal M}_\alpha = {\cal M}^s_\alpha.$
In the next section, we shall see that $V_m$ contains a generic weight
if and only if the degree $d$ and the set of multiplicities $\{m_i(p)\}$
have greatest common divisor equal to one.
\section{Families of parabolic bundles}
In this section, we present Proposition \ref{prop:fine}, which establishes
the existence of a universal family of stable parabolic bundles
parametrized by ${\cal M}_\alpha^s$ whenever $V_m$ contains a generic weight.
Although results of this type are well-known to experts,
the proposition, as well as the proof, are original
(cf. Th\'eor\`eme 32, \cite{seshadri}).
It is important because, in the case of ordinary bundles,
the non-existence of the universal family
(\cite{ramanan}) is the obstruction to
proving Corollary \ref{conj} by induction,
and as shown in \S 6,
the analogous argument works for parabolic
bundles precisely because the necessary conditions
for the vanishing of this obstruction given by Proposition \ref{prop:fine}
are often satisfied.
Given positive integers $m_1,\ldots,m_\kappa$ such that $m_1+\cdots m_\kappa =
r,$
define ${\cal F}_m$ to be the variety of flags of type $m.$
These are simply flags
${\Bbb C}^r=F_1 \supset \cdots \supset F_s \supset 0$ with $\dim F_i -\dim
F_{i+1} = m_i.$
Furthermore, for any bundle
$E \longrightarrow S$ of rank $r,$ let ${\cal F}_m(E) \longrightarrow S$ be the
bundle of flags of type $m.$
Given a bundle $U \rightarrow S \times X,$
we adopt the notation $U_s = U|_{\{s\} \times X}.$ We also use
$\pi_S$ for the projection map $S\times X \rightarrow S.$
\begin{defn} \label{defn:family}
Fix multiplicities $m(p)$ for each $p \in D.$
\begin{enumerate}
\item[(i)]
A family of quasi-parabolic bundles (of type $m$) parametrized by a variety
$S$ is a bundle $U$ over $S \times X$ together with a section $\phi_p$ of
the flag bundle ${\cal F}_{m(p)}(U|_{S \times \{p\}}) \longrightarrow S$ for
each $p \in D.$
\item[(ii)] Two families $(U,\phi)$ and $(U',\phi')$ parametrized by $S$ are
equivalent, written
$(U,\phi) \sim (U',\phi'),$ if there exists a line bundle $L$ over $S$ and an
isomorphism
$U \cong U' \otimes \pi_S^* L$ under which $\phi \mapsto \phi'.$
\end{enumerate}
\end{defn}
Note that the section $\phi_p$ in (i) above is just a choice of
a nested chain of subbundles of $U|_{S \times \{p\}}$ whose relative coranks
are given by the multiplicities $m(p).$
A family of parabolic bundles is gotten by associating a fixed set of
weights to each chain of subbundles.
Let $U_*=(U,\phi,\alpha)$ be the resulting family of parabolic bundles
and $U_{s,*}=(U_s, \phi(s),\alpha)$ be the parabolic bundle above $s \in S.$
Then $U_*$ is called a family of (semi)stable parabolic bundles if
$U_{s,*}$ is (semi)stable for each $s \in S.$
It follows from the construction of Mehta and Seshadri that ${\cal M}_\alpha$ is a
coarse moduli space.
Proposition 1.8 of \cite{newstead2} then gives two conditions which are
necessary and sufficient for a coarse moduli space to be fine, i.e. to admit a
universal family.
The second condition is not difficult to verify
using an argument similar to that given in Lemma 5.10 of \cite{newstead2}.
The first condition requires that we construct a family
${\cal U}^\alpha_*$ parametrized by ${\cal M}^s_\alpha$ with the property that ${\cal U}^\alpha_{e,*}$
is a parabolic stable bundle isomorphic to $E_*$ for all
$[E_*] = e \in {\cal M}^s_\alpha.$
To construct this family, we need to review the construction of
${\cal M}_\alpha$ (\cite{bhosle}, \cite{mehta-seshadri}).
Let $Q$ be the Hilbert scheme of coherent sheaves over $X$ which are quotients
of ${\cal O}_X^{\oplus N}$ with fixed Hilbert polynomial
(that of $E(k)$ for $k \gg g$), where $N= h^0(E).$ Let $U$ be the universal
family on
$Q \times X.$ Define $R$ to be the subscheme of $Q$ of points $r \in Q$ so that
$U_r$ is a locally free sheaf which is generated by its global sections and
$h^1(U_r)=0.$
Let $\widetilde{R}$ be the total space of the universal flag
bundle over $R$ with flag type $\prod_{p\in D} {\cal F}_{m(p)},$
and let $\widetilde{U}$ be the pullback of $U$ to $\widetilde{R}.$
Then $\widetilde{U}$ is canonically a family of parabolic bundles
parametrized by $\widetilde{R}$ by letting, for each $p \in D,$
$\phi_p$ be the tautological section
and $\alpha(p)$ be the fixed weights.
It follows that $\widetilde{R}$ has the local universal property for parabolic
bundles
(p.\ 16, \cite{bhosle}).
The subsets $\widetilde{R}^{s}$ ($\widetilde{R}^{ss}$) corresponding to the
stable (semistable)
parabolic bundles are invariant under the natural action of $\operatorname{GL}(N) =
\operatorname{Aut}({\cal O}_X^{\oplus N}),$
and ${\cal M}_\alpha$ is a good quotient of $\widetilde{R}^{ss}$
(with linearization induced by the weights $\alpha$),
and ${\cal M}^s_\alpha$ is the geometric quotient of
$\widetilde{R}^{s}.$
The center of $\operatorname{GL}(N)$
acts trivially on $R$ and $\widetilde{R},$ but nontrivially on the locally
universal bundle $\widetilde{U}$.
In fact, $\lambda(\hbox{id})$ acts on $\widetilde{U}$ by scalar multiplication by
$\lambda$
in the fibers (this follows from p.\ 138, \cite{newstead2}).
Given a line bundle $L$ over $\widetilde{R}^{s}$ with a natural
lift of the $\operatorname{GL}(N)$ action such that $\lambda(\hbox{id})$ acts by multiplication by
$\lambda,$
then using $\widetilde{U}^{s}$ to denote $\widetilde{U}|_{\widetilde{R}^{s}
\times X},$
the quotient of $\widetilde{U}^{s} \otimes \pi^*_{\widetilde{R}^{s}} L^{-1},$
together with the tautological sections and weights $\{\phi_p, \alpha(p) \mid p
\in D \}$ mentioned above,
gives the desired family.
\begin{prop} \label{prop:fine}
Such a line bundle $L$ exists if either
\begin{enumerate}
\item[(i)] the elements of the set $\{d, m_i(p) \mid p \in D, 1 \leq i \leq
\kappa_p \}$
have greatest common divisor equal to one, or
\item[(ii)] the face $V_m$ containing $\alpha$ contains a generic weight.
\end{enumerate}
Moreover, these two conditions are equivalent, and when they are satisfied,
the moduli space ${\cal M}^s_\alpha$ is fine.
\end{prop}
The idea of the proof is to find line bundles $L_k$ for each $k \in \{d, m_i(p)
\}$
over $\widetilde{R}^{s}$ with natural actions of $\operatorname{GL}(N)$ such that
$\lambda(\hbox{id})$ acts
by scalar multiplication by $\lambda^k.$
Then (i) gives the existence of $k_1, \ldots, k_\ell \in \{d, m_i(p) \}$
and integers $a_1, \ldots, a_\ell$ so that
$a_1 k_1 + \cdots a_\ell k_\ell = 1.$
The required line bundle is then the tensor product
$L = L^{a_1}_{k_1} \otimes \cdots \otimes L^{a_\ell}_{k_\ell}.$
At the end of the proof, we will show that (i) and (ii) are equivalent.
We start with a lemma.
\begin{lem}
Suppose $E_*$ is parabolic semistable or degree $d$ and rank $r$ and $H_*$ is a
parabolic line
bundle of degree $h$, then
\begin{equation} \label{lem:ineq}
h^1(H^\vee_* \otimes E_*) \neq 0 \quad \Rightarrow \quad d \leq r(2g-2+h) + r^2
n.
\end{equation}
\end{lem}
\begin{pf}
Serre duality for parabolic bundles (Proposition 3.7 of \cite{yoko}) implies
that
$$h^1(H^\vee_* \otimes E_*)
\leq h^0(E^\vee_* \otimes {H}_* \otimes K(D)).$$
(If we had used $h^0(E^\vee_* \otimes \widehat{H}_* \otimes K(D)),$
the circumflex over $H_*$ indicating {\it strongly} parabolic morphisms, we
would get the usual statement
of Serre duality with
equality, cf. \cite{yoko,by}.)
Suppose that $\phi : E \longrightarrow H \otimes K(D)$ is a non-zero map and
let $E'$ be the subbundle generated by $\operatorname{Ker} \phi.$ Then
$$\deg E' \geq \deg E - \deg H\otimes K(D) = d - h - (2g -2 +n).$$
Considering $E'_*$ with its canonical parabolic structure as a subbundle of
rank $r-1,$
the inequality (\ref{lem:ineq}) follows easily from this,
semistability of $E_*,$ and
the inequalities $\operatorname{pardeg} E'_* \geq \deg E'$ and
$\operatorname{pardeg} E_* \geq \deg E + rn.$
\end{pf}
\noindent
{\it Proof of Proposition.}
Write the weights $\alpha$ without repetition.
Choose $\ell: D \longrightarrow {\Bbb Z}$
with
$ 1 \leq \ell_p \leq \kappa_p+1$ and set $\beta(p) = \alpha_{\ell_p}(p).$
(Take $\beta(p) > \alpha_{\kappa_p}$ if $\ell_p = \kappa_p+1.$)
For $h \in {\Bbb Z},$ define
$$\chi(\ell,h) = d + r(1-g-h) - \sum_{p \in D} \sum_{i=1}^{\ell_p-1} m_i(p).$$
Let $H_*$ be the parabolic line bundle with $\deg H = h < d/r - r n -(2g-2)$
and with weights
$\beta(p)$ at $p \in D.$ It follows from the lemma that
if $E_*$ is semistable, then $h^1(H^\vee_* \otimes E_*)=0.$ Thus
$h^0(H^\vee_* \otimes E_*) = \chi(\ell,h)$ by Riemann-Roch.
Hence $(R^0 \pi_{\widetilde{R}^{s}}) (\widetilde{U}^{s} \otimes \pi_X^* H_*)$
is a locally free sheaf of rank $\chi(\ell,h)$ over $\widetilde{R}^{s}.$
Let $L(\ell,h)$ be the determinant of the corresponding bundle.
By construction, the $\operatorname{GL}(N)$ action on $\widetilde{U}$ induces one on this
bundle (and hence on $L(\ell,h)$);
$\lambda(\hbox{id})$ acts by scalar multiplication by $\lambda$ on the bundle and by
$\lambda^{\chi(\ell,h)}$ on $L(\ell,h).$
It is now a simple exercise in high school algebra to see that we can choose
$h,h'$ and $\ell,\ell'$ so
that $\lambda(\hbox{id})$ acts on $L(\ell,h) \otimes L(\ell',h')$ by $\lambda^k$ for
any $k \in \{ d, m_i(p) \}.$
This proves the conclusion of the proposition assuming (i), and now we show
that conditions (i) and (ii)
are equivalent. Suppose first that (i) does not hold.
Consider $E_*$ as
a quasi-parabolic bundle without holomorphic structure, which will be specified
later.
Since the set $\{ d, m_i(p)\}$ is not relatively prime,
there exists a prime number $q$
evenly dividing each element of the set. Clearly $q$ also divides
$r.$
Set $d' =d/q, r' = r/q$ and $m'_i(p) =m_i(p)/q.$
Consider the quasi-parabolic bundle $E'_*$
with degree $d'$, rank $r',$ and multiplicities $m'.$
Any choice of weights $\alpha$ on $E_*$ induces (the same!) weights on $E'_*,$ and
it follows that since $g \geq 2,$ there is some holomorphic structure for
which $E'_*$
is semistable.
Define the holomorphic structure on $E_*$ by
$$E_* = E'_* \oplus \stackrel{q}{\cdots} \oplus E'_*.$$
It follows that $E_*$ is semistable but not stable
for {\it any} choice of compatible weights.
This implies that $V_m$ does not contain a generic weight.
Suppose conversely that $V_m$ does not contain a generic weight.
Since $V_m$ is affine,
$V_m \subset H_\xi$ for some $\xi=(r',d',m')$ Using (\ref{eqn:hyper}), we
conclude that
for all $\alpha \in V_m,$
$$ \sum_{p \in D} \sum_{i=1}^{\kappa_p} (r m_i'(p) - r' m_i(p)) \alpha_i(p) =
rd'-r'd .$$
(Here, we are still thinking of $\alpha$ without repetition.)
We can vary each $\alpha_i(p)$ continuously by some small amount,
and it follows that $$rm'_i(p) - r' m_i(p)=0= rd'-r'd$$
for all $i$ and $p.$
Since $r' < r,$ there exists a prime $q$ such that
$q^k$ divides $r$ but not $r'.$ Hence $q$ divides $d$ and each element of
the set $\{m_i(p) \mid p \in D, 1 \leq i \leq \kappa_p\}.$
$\quad \Box$
\section{The variation and degeneration theorems}
In this section, we describe and extend the theory of \cite{bh}. This allows us
to
compare the moduli spaces of parabolic bundles ${\cal M}_\alpha$ and ${\cal M}_\beta$ when
\begin{enumerate}
\item[(i)] $\alpha, \beta \in V_m$ are generic weights in adjacent chambers,
\item[(ii)] $\alpha \in V_\ell$ and $\beta \in V_m$ are generic weights not
separated by any hyperplanes and
$V_\ell > V_m.$
\end{enumerate}
Cases (i) and (ii) correspond to Theorem 3.1 and Proposition 3.4 of \cite{bh}.
We present slightly stronger versions of those results tailored for our
purposes here.
Starting with (i), suppose that $\alpha, \beta \in V_m$ are generic weights
separated by a single hyperplane $H_\xi.$
Choose $\gamma \in H_\xi$ on the straight line connecting $\alpha$ to $\beta.$
Then ${\cal M}_\gamma$
is stratified by the Jordan-H\"older type of the underlying bundle,
and since $\gamma$ lies on only one hyperplane, there are exactly two strata:
the stable bundles ${\cal M}^s_\gamma$ and the strictly semistable bundles
$\Sigma_\gamma.$
Writing $\xi=(r',d',m')$
for the partition, then it is not hard to see that
$\Sigma_\gamma \cong {\cal M}_{\gamma'} \times {\cal M}_{\gamma''},$ with the obvious definitions
for $\gamma'$ and $\gamma''$
coming from the partition $\xi.$
\begin{thm} \label{thm:bh1}
There are natural algebraic maps $\phi_\alpha$ and $\phi_\beta$
$$\begin{array}{rcl}{\cal M}_\alpha& & {\cal M}_\beta\\
\;\; \phi_\alpha \!\! \searrow \!\!\!\!\!\!\! && \!\!\!\!\!\!\! \swarrow \!\!
\phi_\beta\\& {\cal M}_\gamma
\end{array}$$
which are generized blow-downs along projectivizations of vector bundles over
$\Sigma_\gamma,$
where the projective fiber dimensions $e_\alpha$ and $e_\beta$ satisfy $e_\alpha +
e_\beta + 1 = \operatorname{codim} \Sigma_\gamma.$
\end{thm}
\begin{pf}
The proof is the same as in \cite{bh}, the only difference being the actual
computation of the
numbers $e_\alpha$ and $e_\beta,$ which we discuss now.
We assume that $E_* \sim_S E'_* \oplus E''_*,$ where $[E_*] \in \Sigma_\gamma$ and
$\sim_S$ denotes Seshadri equivalence (i.e.\ isomorphic Jordan-H\"older form).
The topological type of the parabolic bundles $E'_*$ and $E''_*$ does not
change as
$[E_*]$ varies within $\Sigma_\gamma.$ We use $(r',r''), (d',d'')$ and $(m',m'')$
to denote
the ranks, degrees, and multiplicities of $(E'_*, E''_*),$
written as in \S 2.
The moduli spaces ${\cal M}_\alpha, {\cal M}_\beta,$ and ${\cal M}_\gamma$ have dimension
$$(g-1) r^2 +1 + \frac{1}{2}\sum_{p \in D} \left(r^2 - \sum_{i=1}^{\kappa_p}
m_i(p)^2\right).$$
Using a similar formula for $\Sigma_\gamma={\cal M}^{\gamma'} \times {\cal M}^{\gamma''},$ we find
that
$$\operatorname{codim} \Sigma_\gamma = r' r''(2g - 1) -1+\sum_{p\in D}\sum_{i=1}^{\kappa_p}
m'_i(p) m''_i(p).$$
Now we claim that
$$h^0({E''_*}^\vee \otimes E'_*)=0=h^0({E'_*}^\vee \otimes E''_*).$$
This is true for any $\alpha' \in V_m,$ as one of these equations is true for
$\alpha,$ the other for $\beta,$
but $H^0$ is constant as the weights are varied within $V_m.$
Let ${\cal U}'$ and ${\cal U}''$ be the families parametrized by $\Sigma_\gamma$ gotten by
pulling back the universal
families ${\cal U}^{\gamma'}$ and ${\cal U}^{\gamma''},$ whose existence follows from Proposition
\ref{prop:fine}.
Then the vector bundles referred to in the theorem are
$$(R^1 \pi_{\Sigma_\gamma})({{\cal U}''}^\vee \otimes {\cal U}') \hbox{ and } (R^1
\pi_{\Sigma_\gamma})({{\cal U}'}^\vee \otimes {\cal U}'').$$
The projectivizations of these bundles have dimensions
\begin{eqnarray}
e_\alpha &=& h^1({E''_*}^\vee \otimes E'_*)-1 = r'' d' - r' d'' + r' r'' (g-1) +
\chi ({\cal Q}) -1,
\label{formula:ealpha}\\
e_\beta &=& h^1({E'_*}^\vee \otimes E''_*)-1 = r' d'' - r'' d' + r' r'' (g-1) +
\chi ({\cal Q}') -1,
\label{formula:ebeta}
\end{eqnarray}
where
${\cal Q}$ and ${\cal Q}'$ are skyscraper sheaves supported on $D$
obtained as the quotients
\begin{eqnarray*}
&\parhom (E''_*,E'_*) \longrightarrow \operatorname{{\frak H}{\frak o}{\frak m}} (E'',E') \longrightarrow {\cal
Q},&\\
&\parhom (E'_*,E''_*) \longrightarrow \operatorname{{\frak H}{\frak o}{\frak m}} (E',E'') \longrightarrow {\cal
Q}'.&
\end{eqnarray*}
It is a nice exercise to see
$$ \chi ({\cal Q}) + \chi ({\cal Q}') = \sum_{p \in D}
\left( r' r'' -\sum_{(i,j) \in S_e(p)} m'_i(p)m''_j(p) \right),$$
where $S_e(p) = \{ (i,j) \mid \gamma'_i(p) = \gamma''_j(p)\}.$
This shows $e_\alpha + e_\beta +1 = \operatorname{codim} \Sigma_\gamma.$
\end{pf}
\begin{thm} \label{thm:bh2}
Suppose that $\alpha \in V_\ell, \, \beta \in V_m, \, V_\ell > V_m,$ and that
$\alpha$ and $\beta$ are generic and are not separated by any hyperplanes.
Then there exists a fibration $\psi:{\cal M}_\alpha \longrightarrow {\cal M}_\beta$
with fiber a (possibly twisted) product of flag varieties and
this fibration is locally trivial in the Zariski topology.
In particular, ${\cal M}_\alpha$ is birational to the product of ${\cal M}_\beta$ with a
product of flag varieties.
\end{thm}
\begin{pf}
The hypothesis $V_\ell > V_m$ just means that the flag
structure degenerates as we pass from $\alpha$ to $\beta.$
By induction, it is enough to prove the above statement
when the degeneration of the flag structure
is taking place at only one parabolic point.
Given
$E_*$ a parabolic bundle with multiplicities $m$ and weights $\alpha,$
let $E'_*$ be the parabolic bundle
with multiplicities $\ell$ and weights $\beta$
resulting from forgetting part of the flag structure and interchanging the
weights.
One easily verifies that if $E_*$ is $\alpha$-stable, then $E_*'$ is $\beta$-stable,
and the existence of the morphism $\psi$ then follows from the coarseness
property of ${\cal M}_\beta.$
The remaining issue is to identify the fiber
and to prove local triviality.
For the first issue, notice that there is an inverse procedure to the forgetful
map described above.
Given a parabolic bundle $E'_*$ with multiplicities $\ell$ and weights $\beta,$
consider all parabolic bundles $E_*$ with weights $\alpha$ obtained from $E'_*$
by
refining the flag stucture to one with multiplicities $m$ and
exchanging the weights.
For a given $E'_*,$ the set of all such possible refinements $E_*$ is
parametrized by a flag variety.
A straightforward numerical verification shows that applying this procedure
to a $\beta$-stable parabolic bundle $E_*'$ yields
an $\alpha$-stable $E_*$ for every possible refinement.
It is not hard to see that the same procedure, when applied to the universal
family ${\cal U}_*^\beta,$
identifies ${\cal M}_\alpha$ with the total space of the flag bundle of
${\cal U}^\beta$ restricted to ${\cal M}_\beta \times \{p\}$ and the map $\psi$
with the bundle projection.
\end{pf}
One might expect from Theorem \ref{thm:bh1} that the birational type of
${\cal M}_\alpha$ depends only on the underlying quasi-parabolic structure.
This is the content of the following proposition.
\begin{prop} \label{prop:qp}
Suppose that $g \geq 2.$
Then the birational type of ${\cal M}_\alpha$ is independent of the
choice of $\alpha \in V_m.$
\end{prop}
\begin{pf}
We prove the proposition by showing that ${\cal M}_\alpha$ and ${\cal M}_\beta$
are birational whenever
$\alpha,\beta \in V_m$
are not separated by any walls
(although one may lie on a wall which does not contain the other).
So assume that
$\alpha \in \cap_{i=1}^n H_{\xi_i}$ and $\beta \in \cap_{i=1}^m H_{\xi_i},$
where $m \geq n.$
By Theorem 4.1 \cite{mehta-seshadri}, ${\cal M}_\alpha$ and ${\cal M}_\beta$ are normal,
projective varieties and
$\dim {\cal M}_\alpha = \dim {\cal M}_\beta,$ hence we only need to construct an
injective morphism $\phi:{\cal M}^s_\beta \longrightarrow {\cal M}^s_\alpha$
to conclude ${\cal M}_\alpha$ is birational to ${\cal M}_\beta.$
One easily verifies that every $\beta$-stable bundle is $\alpha$-stable,
and the existence of $\phi$ follows from the
coarseness of ${\cal M}_\alpha.$
\end{pf}
\section{Shifting and the Hecke correspondence}
In this section, we introduce the notion of a shifted parabolic bundle,
which is the result of changing the weights, multiplicities, and degree of
$E_*$
in a prescribed way.
In some sense, shifting is a symmetry of a larger weight space, one
which includes bundles of different degrees. Two applications of shifting are
discussed at the end.
Shifting is most naturally described in terms of parabolic sheaves.
If ${\cal E}$ is a locally free sheaf on $X,$
then a {\it parabolic structure} on ${\cal E}$ consists of a weighted filtration of
the form
\begin{eqnarray} \label{eqn:filtration1}
&{\cal E}={\cal E}_{\alpha_1} \supset{\cal E}_{\alpha_2} \supset \cdots \supset {\cal E}_{\alpha_l} \supset
{\cal E}_{\alpha_{l+1}} = {\cal E}(-D),&\\
&0 \leq \alpha_1 < \alpha_2 < \cdots < \alpha_l < \alpha_{l+1}=1.\quad \quad&
\label{eqn:filtration2}
\end{eqnarray}
We can define ${\cal E}_x$ for $x \in [0,1]$ by setting ${\cal E}_x = {\cal E}_{\alpha_i}$
if $\alpha_{i-1} < x \leq \alpha_i,$ and then extend to $x \in {\Bbb R}$ by setting
${\cal E}_{x+1} ={\cal E}_{x}(-D).$
We call the resulting filtered sheaf ${\cal E}_*$ a parabolic sheaf and ${\cal E}={\cal E}_0$
the underlying sheaf.
We can define parabolic subsheaves, degree, and stability for
these objects, and there
is a categorical equivalence between locally free parabolic sheaves and
parabolic bundles.
We describe this in case $D = p,$ the general case being quite similar
(\cite{yoko}, \cite{by}).
Suppose that $E_*$ is a parabolic bundle given by flags and weights in the
fibers as in
(\ref{defn:parbun1}) and (\ref{defn:parbun2}).
Define ${\cal E}_*$ by setting $${\cal E}_x = \ker(E\rightarrow E_{p}/F_i),$$ for $
\alpha_{i-1} < x < \alpha_i.$
Thus ${\cal E}_*$ is a parabolic sheaf.
Conversely, given a parabolic sheaf ${\cal E}_*,$ the quotient ${\cal E}_0/{\cal E}_1 =
{\cal E}/{\cal E}(-p)$
is a skyscraper sheaf with support $p$ and fiber that of ${\cal E}.$
Defining a flag in this fiber by setting
$F_i = ({\cal E}_{\alpha_{i}}/{\cal E}_1)_p$ and associating the weight $\alpha_i,$
we obtain a parabolic bundle in the sense of (\ref{defn:parbun1}) and
(\ref{defn:parbun2}).
The category of parabolic sheaves is developed in \cite{yoko}, where one finds
for example
the definitions of tensor products ${\cal E}_* \otimes {\cal E}'_*$ and duals ${\cal E}^\vee_*.$
We use this notation freely in the
calculations of \S 6 involving sheaf cohomology and point out that
$H^i({\cal E}_*) = H^i({\cal E}).$
\begin{defn} \label{defn:shift}
Given a parabolic sheaf ${\cal E}_*$ and $\eta \in {\Bbb R},$ define the shifted
parabolic sheaf
${\cal E}_*[\eta]_*$ by setting ${\cal E}_*[\eta]_x = {\cal E}_{x+\eta}.$
\end{defn}
\noindent
{\it Remark.} The above operation can be refined in case $D = p_1 + \cdots +
p_n.$ If
$\eta=(\eta_1, \ldots, \eta_n),$ then one can shift ${\cal E}_*$ by $\eta_i$ at each
$p_i \in D$
(\cite{yoko}, \cite{bh}).
\medskip \noindent
It is not difficult to verify that ${\cal E}_*[\eta]_*$ is (semi)stable if and only
if ${\cal E}_*$ is (semi)stable,
and it follows that this defines an isomorphism between the associated moduli
spaces of parabolic bundles.
We can easily describe the parabolic structure on
the shifted bundle ${\cal E}'_* = {\cal E}_*[\eta]_*$ in case $0 < \eta \leq 1$ and $D =
p.$
Let $E'_*$ denote the parabolic bundle associated to ${\cal E}'_*.$
If $i$ is the integer with $\alpha_i < \eta \leq \alpha_{i+1},$ then
the weights of $E'_*$ are given by
\begin{equation} \label{eqn:sft_wts}
\alpha'_j = \begin{cases} \alpha_{j+i} - \eta & \hbox{ for } j=1, \ldots, r-i, \\
1+\alpha_{j-r+i}-\eta & \hbox{ for } j = r-i+1,\ldots,
r.
\end{cases}
\end{equation}
The quasi-parabolic structure of $E'_*$ has
multiplicities $m'$ given by a cyclic permutation of $m,$
i.e.\ $m' = (m_{i+1}, \ldots, m_\kappa, m_1, \ldots, m_i).$
Although ${\cal E}'$ is a subsheaf of ${\cal E},$
$E'$ is {\it not} a subbundle of $E,$ so one must appeal to sheaf
theory in order to define the flag in
$E'_p.$ This is a simple exercise in tracing
through the equivalence between locally free parabolic sheafs and
parabolic bundles given above.
\begin{figure}[b]
\begin{picture}(100,210)(-110,-10)
\put(-70,150){${\cal E}_*$}
\put(-70,100){${\Bbb R}$}
\put(-40,110){\line(1,0){290}}
\put(5,110){\line(0,1){70}}
\put(3,100){$0$}
\put(170,110){\line(0,1){20}}
\put(168,100){$1$}
\put(-12,185){${\cal E}={\cal E}_{\alpha_1}$}
\put(-40,180){\line(1,0){78}}
\put(40,180){\circle*{4}}
\put(34,100){$\alpha_1$}
\multiput(40,110)(0,4){12}{\line(0,1){2}}
\multiput(40,178)(0,-4){4}{\line(0,-1){2}}
\put(55,165){${\cal E}_{\alpha_2}$}
\put(42,160){\line(1,0){48}}
\put(40,160){\circle{4}}
\put(90,160){\circle*{4}}
\put(88,100){$\alpha_2$}
\multiput(90,110)(0,4){7}{\line(0,1){2}}
\multiput(90,158)(0,-4){4}{\line(0,-1){2}}
\put(92,140){\line(1,0){48}}
\put(109,145){${\cal E}_{\alpha_3}$}
\put(90,140){\circle{4}}
\put(140,140){\circle*{4}}
\put(137,100){$\alpha_3$}
\multiput(140,110)(0,4){5}{\line(0,1){2}}
\multiput(140,138)(0,-4){2}{\line(0,-1){2}}
\put(142,130){\line(1,0){48}}
\put(140,130){\circle{4}}
\put(190,130){\circle*{4}}
\put(152,135){${\cal E}(-p)$}
\put(185,100){$1\!+\!\alpha_1$}
\multiput(190,110)(0,4){2}{\line(0,1){2}}
\multiput(190,128)(0,-4){2}{\line(0,-1){2}}
\put(192,120){\line(1,0){58}}
\put(190,120){\circle{4}}
\put(200,125){${\cal E}_{\alpha_2}(-p)$}
\put(-80,50){${\cal E}_*[\eta]_*$}
\put(-70,0){${\Bbb R}$}
\put(-40,10){\line(1,0){250}}
\put(5,10){\line(0,1){50}}
\put(3,0){$0$}
\put(170,10){\line(0,1){10}}
\put(168,0){$1$}
\put(-40,85){${\cal E}_{\alpha_1}$}
\put(-40,80){\line(1,0){18}}
\put(-20,80){\circle*{4}}
\put(-40,0){$\alpha_1\!-\!\eta$}
\multiput(-20,10)(0,4){12}{\line(0,1){2}}
\multiput(-20,78)(0,-4){4}{\line(0,-1){2}}
\put(-5,65){${\cal E}_{\alpha_2}$}
\put(-18,60){\line(1,0){48}}
\put(-20,60){\circle{4}}
\put(30,60){\circle*{4}}
\put(20,0){$\alpha_2\!-\!\eta$}
\multiput(30,10)(0,4){7}{\line(0,1){2}}
\multiput(30,58)(0,-4){4}{\line(0,-1){2}}
\put(32,40){\line(1,0){48}}
\put(49,45){${\cal E}_{\alpha_3}$}
\put(30,40){\circle{4}}
\put(80,40){\circle*{4}}
\put(67,0){$\alpha_3\!-\!\eta$}
\multiput(80,10)(0,4){5}{\line(0,1){2}}
\multiput(80,38)(0,-4){2}{\line(0,-1){2}}
\put(82,30){\line(1,0){48}}
\put(80,30){\circle{4}}
\put(130,30){\circle*{4}}
\put(92,35){${\cal E}(-p)$}
\put(110,0){$1\!+\!\alpha_1\!-\!\eta$}
\multiput(130,10)(0,4){2}{\line(0,1){2}}
\multiput(130,28)(0,-4){2}{\line(0,-1){2}}
\put(132,20){\line(1,0){78}}
\put(130,20){\circle{4}}
\put(140,25){${\cal E}_{\alpha_2}(-p)$}
\end{picture}
\caption{The parabolic sheaf ${\cal E}_*$ shifted by $\eta$ with $\alpha_1 < \eta <
\alpha_2.$}
\end{figure}
There are two interesting applications of shifting we discuss now.
The first is the Hecke correspondence. Using ${\cal M}_{r,d}$ to denote the moduli
space of
semistable bundles of rank $r$ and degree $d,$ the Hecke correspondence gives a
means
of comparing ${\cal M}_{r,d}$ and
${\cal M}_{r,d'}$ through
the use of parabolic bundles. For $r=2,$ this was observed in a remark at the
end of \cite{mehta-seshadri}.
To start, define $\epsilon_+(d,r), \epsilon_-(d,r),$ and $\epsilon(d,r)$ for $d,r \in {\Bbb
Z}$ with $r >0$ by
\begin{eqnarray*}
\epsilon_\pm(d,r) &=& \inf \{ \pm({\textstyle \frac{d}{r}-\frac{d'}{r'}}) \mid
d',r' \in {\Bbb Z}, \; 1 \leq r' < r, \hbox{ and } \pm ({\textstyle
\frac{d}{r}-\frac{d'}{r'}}) > 0 \} \\
\epsilon(d,r) \;\; &=& \min \{ \epsilon_\pm(d,k) \mid k=1, \ldots, r \}.
\end{eqnarray*}
It is easy to see that $\epsilon_\pm(d,k) > 0$ for all $k,$ thus $\epsilon(d,r)>0$ as
well.
Suppose that $E$ is a bundle over $X$ of degree $d$ and rank $r$
and suppose further that $E'$ is a proper subbundle.
If $\mu(E') < \mu(E),$ then $\mu(E)-\mu(E') \geq \epsilon_+(d,r).$
Similarly, if $\mu(E') > \mu(E),$ then $\mu(E')-\mu(E) \geq \epsilon_-(d,r).$
\begin{prop} \label{prop:num}
Suppose that $E_*$ satisfies $ {\displaystyle \sum_{p \in D}
\sum_{i=1}^{\kappa_p} m_i(p)} \alpha_i(p) < \epsilon(d,r)/2.$
\begin{enumerate}
\item[(i)] If $E$ is stable as a regular bundle, then $E_*$ is parabolic
stable.
\item[(ii)] If $E_*$ is parabolic stable, then $E$ is semistable as a regular
bundle.
\end{enumerate}
\end{prop}
\begin{pf}
(i)
If $E'_*$ is a proper parabolic subbundle of $E_*$, then
$$\mu(E'_*) \leq \mu(E') + \epsilon(d,r)/2 < \mu(E') + \epsilon_+(d,r) \leq \mu(E) <
\mu(E_*),$$
thus $E_*$ is parabolic stable.
(ii) If $E'$ is a subbundle of $E,$ then
$$\mu(E') \leq \mu(E'_*) < \mu(E_*) < \mu(E) + \epsilon(d,r)/2 < \mu(E) +
\epsilon_-(d,r),$$
hence $\mu(E') \leq \mu(E)$ and $E$ is semistable.
\end{pf}
We thus get a morphism ${\cal M}_\alpha \longrightarrow {\cal M}_{r,d}$ which is the map of
Theorem \ref{thm:bh2} in case $(r,d)=1.$
By choosing the weights and quasi-parabolic structure correctly, we can fit
${\cal M}_{r,d}$
and ${\cal M}_{r,d-1}$ into a chain diagram of maps as follows. Let $D=p$ and
$m=(1,\ldots, 1),$ and choose weights
$\alpha=(\alpha_1,\ldots,\alpha_r)$ with $\alpha_1 + \cdots + \alpha_r < \epsilon(r,d)/2.$ Suppose
$\alpha_1 < \eta < \alpha_2$ and
set $E'_*$ to be the parabolic bundle $E_*$ shifted by $\eta.$
Notice that $E'_*$ has degree $d-1,$ multiplicities
$m'=(1,\ldots,1),$ and weights $\alpha'=(\alpha_2 - \eta, \ldots, \alpha_r-\eta,
1-\eta+\alpha_1).$
If $\beta' \in V_{m'}$ is generic with
$\beta'_1 + \cdots + \beta'_r < \epsilon(r,d)/2,$ then we can connect
$\alpha'$ to $\beta'$ in $V_{m'}$ by a line passing through a finite number of
hyperplanes $H_{\xi^1}, \ldots, H_{\xi^n},$ all of the form to which Theorem
\ref{thm:bh1} applies.
Choose weights $\alpha^i$ in the intermediate chambers and
$\gamma^i \in H_{\xi^i}$ for $i=1,\ldots, n$ with $\alpha^n = \beta'.$
Applying Theorem \ref{thm:bh1} each time we cross a hyperplane,
we get the following diagram:
\begin{equation} \label{eqn:hecke}
\begin{array}{lllllllll}{\cal M}_\alpha \cong {\cal M}_{\alpha'} & &&&{\cal M}_{\alpha^1} & & &&
\!\!\!{\cal M}_{\beta'}\\
\psi \downarrow & \searrow && \swarrow && \searrow && \swarrow &
\downarrow \psi' \!\!\!\! \\
{\cal M}_{r,d} && {\cal M}_{\gamma^1} &&& & \cdots & & \!\!\!{\cal M}_{r,d-1}
\end{array}
\end{equation}
\medskip \noindent
where, by the above proposition, the vertical maps $\psi$ and $\psi'$
have fibers the (full) flag variety over ${\cal M}^s_{r,d}$ and ${\cal M}^s_{r,d-1},$
respectively.
By Theorem \ref{thm:bh2}, $\psi$ is a
fibration which is locally trivial in
the Zariski topology provided $(r,d)=1,$ and the same follows for $\psi'$ if
$(r,d-1)=1.$
The second application of shifting is to extend the results of \cite{bh} to a
case which is
natural from the point of view of representations of Fuchsian groups but less
natural
from the point of view of parabolic bundles.
Assume for simplicity that $\mu(E_*) = 0$ and $D=p.$
Thus, $\deg E = -k$ for some $0 \leq k < r,$ and the relevant weight space
is
$$W_k = \{ (\alpha_1,\ldots,\alpha_r) \in \Delta^r \mid \alpha_1 + \cdots + \alpha_r = k
\}.$$
Consider the union
${\displaystyle \widetilde{W} = \bigcup_{k=0}^{r-1} W_k},$
where we identify
$$\partial_0 W_k = \{ \gamma \in W_k \mid \gamma_1=0 \}$$
with its companion set
$$\partial_1 \overline{W}_{k+1} = \{ \overline{\gamma} \in \overline{W}_{k+1} \mid
\overline{\gamma}_r=1 \}$$
via the identification
\begin{equation} \label{eqn:glue}
\partial_0 W_k \ni \gamma = (0, \gamma_2, \ldots, \gamma_n) \sim (\gamma_2, \ldots, \gamma_n,
1) = \overline{\gamma} \in \partial_1 \overline{W}_{k+1}.
\end{equation}
One can think of this set
$\widetilde{W}$ as the space of all weights
modulo shifting\footnote{Because every bundle can be shifted so that
$\mu(E_*)=0.$},
which in this case is just
the quotient $\operatorname{SU}(r)/Ad$ and which can be naturally
identified with the standard $r-1$ simplex.
{}From this point of view $\partial_0 W_k$ is
an interior hyperplane of $\widetilde{W}$
because it satisfies condition (\ref{eqn:hyper}).
However, Theorem \ref{thm:bh1} does not obviously carry over to this case
because points in $W_k$ and $W_{k+1}$ are weights on
parabolic bundles of different degrees.
Given a parabolic bundle of degree $-k,$
what is needed is a canonical procedure to construct a
parabolic bundle of degree $-(k+1).$
This is precisely what is provided by the shifting operation.
Thought of in terms of $ \widetilde{W},$
the following theorem extends Theorem \ref{thm:bh1} to
the case where $H_\xi= \partial_0 W.$
We use the notation ${\cal M}_\alpha(k,m)$ for the moduli space when
$E_*$ has degree $-k,$ multiplicities $m,$ and weights $\alpha.$
\begin{thm} \label{thm:bh3}
Suppose that $\gamma \in \partial_0 W_k \cap V_m$ does not lie on any other
hyperplanes and that $\alpha \in W_k \cap V_m$ is a generic
weight near to $\gamma.$ Choose $\eta \in {\Bbb R}$ with
$0<\eta < \gamma_{m_1+1}.$ Define $\overline{\gamma}\in \partial_1
\overline{W}_{k+1}$
as in \rom(\ref{eqn:glue}\rom).
Let $E'_*$ be $E_*$ shifted by $\eta,$ and denote the multiplicities of $E'_*$
by $m'.$
Set $k'= - \deg E' = k+m_1.$
Let $\beta \in W_{k'} \cap V_{m'}$ be generic near $\overline{\gamma}.$
Then there are projective algebraic maps $\phi_\alpha$ and $\phi_\beta$
$$\begin{array}{rcl}{\cal M}_\alpha(m,k)& & {\cal M}_\beta(m',k')\\
\;\; \phi_\alpha \!\! \searrow \!\!\!\!\!\! && \!\!\!\!\! \swarrow \!\!
\phi_\beta\\& {\cal M}_\gamma(m,k)
\end{array}$$
satisfying the conclusion of Theorem \ref{thm:bh1}.
\end{thm}
\begin{pf}
By the choice of $\alpha, \beta$ and $\eta,$ we see that $\alpha_{m_1} < \eta <
\alpha_{m_1+1},
\; \eta < \beta_1$ and $\eta < \gamma_{m_1+1}.$
Consequently, the shifting operation defines the following isomorphisms:
\begin{eqnarray*}
{\cal M}_\alpha(m,k) &\cong& {\cal M}_{\alpha'}(m',k'),\\
{\cal M}_\beta(m',k') &\cong& {\cal M}_{\beta'}(m',k'),\\
{\cal M}_\gamma(m,k) &\cong& {\cal M}_{\gamma'}(m',k'),
\end{eqnarray*}
where $\alpha', \beta', \gamma' \in V_{m'}$ are defined from $\alpha, \beta, \gamma$ as in
(\ref{eqn:sft_wts}).
Now Theorem \ref{thm:bh1} applies to the shifted moduli spaces to prove the
theorem.
One can calculate $e_\alpha$ and $e_\beta$ by applying formulas
(\ref{formula:ealpha}) and (\ref{formula:ebeta}) to $ \alpha', \beta'$ and $\gamma'.$
\end{pf}
\medskip
\noindent
{\it Remark.} Theorem \ref{thm:bh3} solves a problem mentioned at the end of
\cite{bh}
and extends the wall-crossing formula for knot invariants introduced in
\cite{boden3}.
\section{Rationality of moduli spaces of parabolic bundles}
Let $L$ be a holomorphic line bundle over a curve $X$ of genus
$g \geq 2.$ Denote by
\begin{enumerate}
\item[(i)] ${\cal M}_{r,L}$ the moduli space of semistable bundles $E$ of rank $r$
with $\det E = L,$ and by
\item[(ii)] ${\cal M}_{\alpha, L}$ the moduli space of parabolic bundles $E_*$ with
weights $\alpha$ and
$\det E = L.$
\end{enumerate}
The main results of \S 4 hold for the moduli spaces with fixed determinant with
no essential difference. In view of
Theorem \ref{thm:bh2},
the goal is therefore to prove rationality with the coarsest possible choice of
flag structure.
At one extreme, we have the trivial flag, whose moduli space is exactly
${\cal M}_{r,L}.$
Proposition 2 of \cite{newstead1} implies that ${\cal M}_{r,L}$ is rational if $\deg
L = \pm 1 \mod (r),$
and then
Theorem \ref{thm:bh2} and Proposition \ref{prop:qp} imply that ${\cal M}_{\alpha,L}$ is
also rational for any $\alpha \in V_m$
provided $\deg L = \pm 1 \mod (r).$
\begin{thm} \label{thm:rat1}
If $m(p)=(1,\ldots,1)$ for some $p \in D,$ then
${\cal M}_{\alpha, L}$ is rational for all $\alpha \in V_m.$
\end{thm}
\begin{pf}
First, use Theorem \ref{thm:bh2} to reduce to the case $D=p$ by forgetting all
the other flag structures. If $E'_*$ denotes the bundle obtained by shifting
$E_*$
by some $\eta$ with $\alpha_1 < \eta < \alpha_2,$
then $\det E' = L' = L(-p).$ It follows that shifting by $\eta$ defines an
isomorphism from
${\cal M}_{\alpha,L}$ to ${\cal M}_{\alpha',L'}.$
Repeated application of shifting puts us in the case $\deg L = 1 \mod(r),$ and
then Newstead's theorem and Theorem \ref{thm:bh2} imply that ${\cal M}_{\alpha,L}$ is
rational.
\end{pf}
The above argument works in slightly more generality.
We can always shift our bundle to be any of the ${\cal E}_x$ appearing in
the filtration (\ref{eqn:filtration1}) and illustrated in Figure 1.
Thus, whenever one of these terms in the filtration is of a degree to which
Newstead's theorem applies, the corresponding moduli space
of parabolic bundles is rational.
The next theorem is a considerable strengthening of the previous one.
\begin{thm} \label{thm:rat2}
If $m_i(p)=1$ for some $p \in D$ and some $1 \leq i \leq \kappa_p,$
then ${\cal M}_{\alpha,L}$ is rational for all $\alpha \in V_m.$
\end{thm}
Before delving into the proof of this theorem, we mention some interesting
consequences.
Recall first the following definition.
\begin{defn} \label{defn:stablerat}
A variety $V$ is stably rational of level $k$ if $V \times {\Bbb P}^k$ is
rational.
The level is the smallest integer $k$ with this property.
\end{defn}
The following result, with a weaker bound on the level, was proved in
\cite{ballico}.
\begin{cor} \label{cor}
For $(r,d)=1, \; {\cal M}_{r,L}$ is stably rational with level $k \leq r-1.$
\end{cor}
\begin{pf} Theorem \ref{thm:rat2} implies that ${\cal M}_{\alpha,L}$ is rational, where
$m(p)=(r-1,1),$ and
Theorem \ref{thm:bh2} shows that
${\cal M}_{\alpha,L}$ is birational to ${\cal M}_{r,L} \times {\Bbb P}^{r-1},$ which proves
the corollary.
\end{pf}
We now apply this last result to Conjecture \ref{conj}.
\begin{cor} \label{cor:rat}
Suppose $(r,d)=1.$ By tensoring with a line bundle, we can assume that $0 < d <
r.$
If either $(g,d) =1$ or $(g,r-d)=1,$ then ${\cal M}_{r,L}$ is rational.
\end{cor}
\begin{pf}
Suppose first that $(g,r-d)=1.$ Let $L$ be a line bundle of degree $r(g-1)+d.$
Then Newstead's construction applies and proves that
${\cal M}_{r,L}$ is birational to ${\cal M}_{r-d,L} \times {\Bbb P}^\chi,$ where $\chi =
(g-1)(r^2-(r-d)^2).$
But the above corollary implies that ${\cal M}_{r-d,L}$ is stably rational with level
$k \leq r-d-1 \leq \chi,$
hence ${\cal M}_{r,L}$ is rational.
The case $(g,d)=1$ follows by the same argument after applying duality, which
interchanges
$(r,d)$ and $(r,r-d).$
\end{pf}
{\it Remark.} Conjecture \ref{conj} was previously known \cite{newstead1} in
the following three cases:
\begin{enumerate}
\item[(i)] $d = \pm 1 \mod (r),$
\item[(ii)] $(r,d)=1$ and $g$ a prime power, and
\item[(iii)] $(r,d)=1$ and the two smallest distinct primes factors of $g$ have
sum
greater than $r.$
\end{enumerate}
Corollary \ref{cor:rat}
applies in each case. More importantly, it applies
in many cases not covered by (i), (ii) or (iii).
In fact, for a given $r$ and $d$ with $(r,d)=1,$ one can easily
list those $g$ for which the conjecture remains open.
For example, if $r= 110$ and $d=43,$ then Corollary \ref{cor:rat} applies
as long as $g$ is not a multiple of $d \cdot (r-d)= 43 \cdot 67=2881.$
\medskip
\noindent
{\it Proof of Theorem.}
Set $d= \deg L.$
The theorem is clearly true for $r=1$ and follows
from Theorem \ref{thm:rat1} for $r=2,$ so assume $r>2.$
Notice that by tensoring with a line bundle, we can suppose $$r(g-1) < d \leq
rg.$$
By Theorem \ref{thm:bh2}, we can again assume that
$D=p,$ and by shifting and another application of Theorem \ref{thm:bh2}, if
necessary,
we can arrange it so that $m(p)=(r-1,1).$
Write $$\alpha = \alpha(p) = (\overbrace{\alpha_1, \cdots, \alpha_1}^{r-1}, \alpha_2).$$
Proposition \ref{prop:fine} implies that $V_m$ contains a generic weight
and that ${\cal M}_{\alpha,L}$ parametrizes a universal family ${\cal U}_*^\alpha.$
By Proposition \ref{prop:qp}, the birational type of ${\cal M}_{\alpha,L}$ is
independent of
choice of compatible weights, so
we can assume that the weights are small enough to satisfy the hypothesis of
Proposition \ref{prop:num}
(this comes up at various technical points in the argument, e.g. the proof of
Claim \ref{claim}).
\medskip\noindent
Consider the following two cases.
\medskip\noindent
{\sc Case I:} \quad $d=rg.$ \quad
Choose $\eta$ with $\alpha_1 < \eta < \alpha_2,$ and let $E'_* = E_*[\eta]_*.$
Denote the weights of $E'_*$ by $\alpha'$ as in (\ref{eqn:sft_wts}).
If $\det E = L,$ then $\det E' = L' = L(-(r-1)p)$ has degree $d'= d-(r-1).$
Since $d' = 1 \mod(r),$ Proposition 2 of \cite{newstead1}
implies that ${\cal M}_{r,L'}$ is rational, and Theorem \ref{thm:bh2} then implies
that
${\cal M}_{\alpha'\!,L'}$ is also rational.
Rationality of ${\cal M}_{\alpha,L}$ now follows from the isomorphism of the
moduli spaces ${\cal M}_{\alpha,L} \cong {\cal M}_{\alpha'\!,L'}$ defined by shifting by $\eta.$
\medskip\noindent
{\sc Case II:} \quad $r(g-1) < d < rg.$ \quad
The idea is to use induction to construct a nonempty, Zariski-open subset ${\cal M}$
of affine space of dimension
$(r^2-1)(g-1)+r-1 \; (= \dim {\cal M}_{\alpha,L})$
and a family of stable parabolic bundles ${\cal U}_*$ parametrized by ${\cal M}$ with $\det
{\cal U}_{\xi,*} = L$ for all
$\xi \in {\cal M}.$
The universal property of ${\cal U}_*^\alpha$ then gives a map $\psi_{{\cal U}_*} : {\cal M}
\longrightarrow {\cal M}_{\alpha,L}.$
If, in addition, we have ${\cal U}_{\xi_1,*} \cong {\cal U}_{\xi_2,*}\; \Leftrightarrow \;
\xi_1 = \xi_2,$ then $\psi_{{\cal U}_*}$ is injective
and rationality of ${\cal M}_{\alpha,L}$ follows from that of ${\cal M}$ and the dimension
condition.
Set $r' = rg-d, \, r'' = r-r'$ and $\alpha'= (\overbrace{\alpha_1, \cdots,
\alpha_1}^{r'-1}, \alpha_2).$
Assume that both $\alpha$ and $\alpha'$ are generic.
Let ${\cal U}_*^{\alpha'}$ be the universal family parametrized by ${\cal M}_{\alpha'\!,L}$
and $I_*= {\cal O}_X[\alpha_1]_*$ be the trivial parabolic line bundle with weight
$\alpha_1.$
If $e' = [E'_*] \in {\cal M}_{\alpha'\!,L},$
then because ${E'_*}^\vee\otimes I_*$ is a stable parabolic bundle of negative
parabolic degree,
$h^0({E'_*}^\vee \otimes I_*)=0$ and
\begin{equation} \label{eq:rank}
n \stackrel{\hbox{\scriptsize\rm def}}{=} h^1({E'_*}^\vee \otimes I_*) =
(2r'+r'')(g-1) + r'' + 1
\end{equation}
is independent of $e'.$
Since ${\cal U}^{\alpha'}_{e',*} \cong E'_*,$ it follows that
$$(R^1 \pi_{{\cal M}_{\alpha'\!,L}}) (({{\cal U}_*^{\alpha'}})^\vee \otimes \pi_X^*(I_*))$$
is locally free. The associated vector bundle $V
\stackrel{\pi}{\longrightarrow} {\cal M}_{\alpha'\!,L}$
has rank $n$ and fiber over $e'$ naturally isomorphic to $H^1({E'_*}^\vee
\otimes I_*).$
Let ${\cal U}_*'=(\pi^{r''} \times 1_X)^* ({\cal U}_*^{\alpha'})$ be the pullback family
and ${\cal I}^{\oplus r''}_* =\pi_X^* I_*^{\oplus r''}$ the trivial family,
where $\pi^{r''}:V^{\oplus r''} \longrightarrow {\cal M}_{al'\!,L}$.
There is an extension
\begin{equation} \label{eqn:universal_extension}
0 \longrightarrow {\cal I}_*^{\oplus r''} \longrightarrow {\cal U}_* \longrightarrow {\cal U}_*'
\longrightarrow 0
\end{equation}
of families over $V^{\oplus r''} \times X,$
such that, for $\xi \in V^{\oplus r''}_{e'},$
${\cal U}_{\xi,*}$ is the parabolic bundle $E^\xi_*$
described as the short exact sequence
\begin{equation} \label{eq:ses}
0 \longrightarrow I^{\oplus r''}_* \longrightarrow E^\xi_* \longrightarrow
E'_* \longrightarrow 0
\end{equation}
corresponding to the extension class $\xi \in H^1({E'_*}^\vee \otimes I^{\oplus
r''}_*).$
Using stability of $E'_*$ and triviality of $I^{\oplus r''}_*,$
it follows that
$$\operatorname{Aut}(E'_*) \times \operatorname{Aut}(I^{\oplus r''}_*) \cong {\Bbb C}^* \times \operatorname{GL}(r'',
{\Bbb C}).$$
This group acts naturally as fiber-preserving maps on the bundle $V^{\oplus
r''}$ since
$$V^{\oplus r''}_{e'} \cong H^1({E'_*}^\vee \otimes I^{\oplus r''}_*) =
H^1({E'_*}^\vee \otimes I_*)^{\oplus r''},$$
and two extension classes $\xi_1$ and $\xi_2$ in the same orbit
have associated bundles $E^{\xi_1}$ and $E^{\xi_2}$ which are isomorphic.
We can ignore the ${\Bbb C}^*$ action here
because $(z,1)\cdot \xi = (1,z) \cdot \xi$ for $z \in {\Bbb C}^*$ and $\xi \in
V^{\oplus r''}.$
Using the inductive
hypothesis and local triviality of $V,$ we can choose
a nonempty Zariski-open subset ${\cal M}'$ of ${\cal M}_{\alpha'\!,L}$ isomorphic to a
Zariski-open subset of affine space of dimension $({r'}^2-1)(g-1)+r'-1$
such that $V|_{{\cal M}'} \cong {\cal M}' \times H^1({E'_*}^\vee \otimes I_*)$ ($E'_*$ is
fixed).
Lemma 2 of \cite{newstead1} applies here and produces a Zariski-open subspace
${\cal M}'\times W$ of
$V^{\oplus r''}|_{{\cal M}'}$ invariant under the group action,
and an affine subspace $A \subset W$
so that every orbit in $W$ intersects $A$ precisely once.
In fact, $A$ can be chosen as a Zariski-open subset of the Grassmannian
$G(r'',n).$
In any case, it should be clear that $A$ has dimension $r''(n-r'').$
Using equation (\ref{eq:rank}) and the fact that $r'+r'' = r,$ we see
that ${\cal M}' \times A$ is a Zariski-open subset of affine space of dimension
\begin{eqnarray*}
\dim {\cal M}' \times A &=& ({r'}^2-1)(g-1)+r'-1 + r''(n - r'')\\
&=& ({r'}^2-1)(g-1)+r'-1 + r''((2r'+r'')(g-1) + 1) \\
&=& (r^2-1)(g-1)+r-1.
\end{eqnarray*}
Let ${\cal M}$ be the subset of $V^{\oplus r''}$ defined by
$${\cal M} = \{ \xi \in {\cal M}' \times A \mid H^1({\cal U}_{\xi,*}) = 0 \},$$
and consider the bundle ${\cal U}_*$ restricted to ${\cal M},$ which we continue to denote
${\cal U}_*.$
For $\xi \in V^{\oplus r''},$ let $E^\xi_* = {\cal U}_{\xi,*}.$
Clearly $E^\xi_*$ is a parabolic bundle with weights $\alpha$ and determinant $L,$
thus ${\cal M}$ parametrizes a family of parabolic bundles.
By the upper semi-continuity theorem, ${\cal M}$ is Zariski-open in ${\cal M}' \times A.$
We claim that ${\cal M}$ is nonempty.
Fix $e'=[E'_*] \in {\cal M}'$ and consider the set
$$N= \{ \xi \in H^1({E'_*}^\vee \otimes I^{\oplus r''}_*) \mid h^1(E^\xi_*)=0
\}.$$
If $N \cap A \neq \emptyset,$ then ${\cal M}$ is nonempty. Clearly, $N$ is invariant
under the
action of $\operatorname{GL}(r'',{\Bbb C}),$ so it is enough to show $N \cap W \neq
\emptyset.$
There is a natural map
$$\delta : H^1({E'_*}^\vee \otimes I^{\oplus r''}_*) \times H^0(E'_*)
\longrightarrow H^1(I^{\oplus r''}_*)$$
with $\delta_\xi = \delta(\xi,\cdot) : H^0(E'_*) \longrightarrow H^1(I^{\oplus
r''}_*)$ the
coboundary map of the long exact sequence in homology of
(\ref{eq:ses}).
Now $H^0(E'_*)=H^0(E'),$ and since $\alpha_1+(r'-1) \alpha_2 < \epsilon(r,d)/2,$ by
Proposition \ref{prop:num},
$E'$ is semistable as a non-parabolic bundle. Serre duality implies that
$h^1(E')=h^0({E'}^\vee \otimes K),$ and we compute
\begin{eqnarray*}
\deg({E'}^\vee \otimes K) &=& -d +r'(1-g) \\
&\leq& (r+r')(1-g) - r'',
\end{eqnarray*}
which is negative since $r'' \geq 1$ and $g \geq 2.$ This implies that
$h^1(E'_*) =0,$
and Riemann-Roch implies that $h^0(E'_*) = r''g.$
Because $h^1(I^{\oplus r''}_*)=r''g,$
we see that
$$\xi \in N \Longleftrightarrow H^1(E^\xi_*) =0 \Longleftrightarrow \delta_\xi
\hbox{ is an isomorphism.}$$
But $\delta$ is obviously onto and
$\dim (\ker \delta) = r''n.$
The set $N$ has complement
$$N^c = \{ \xi \in H^1({E'_*}^\vee \otimes I^{\oplus r''}_*)
\mid \delta(\xi,s)=0 \hbox{ for some }0 \neq s \in H^0(I^{\oplus r''}_*) \}.$$
But $\delta(\xi,s) = 0 \Rightarrow \delta(\xi, zs)=0$ for all $z \in {\Bbb C},$
which shows that
the map $\ker \delta \longrightarrow N^c$ has fibers of dimension $\geq 1.$
Hence $\dim N^c \leq \dim(\ker \delta) -1 < r''n,$ and we see that $N$ is
nonempty and Zariski-open.
Thus $N \cap W \neq \emptyset$ and it follows that ${\cal M}$ is nonempty.
We now prove that ${\cal M}$ parametrizes a family of stable parabolic bundles,
using again the inequality
$(r-1) \alpha_1 + \alpha_2 < \epsilon(r,d)/2$ and Proposition \ref{prop:num}.
\begin{claim} \label{claim}
\begin{enumerate}
\item[(i)] $E^\xi_*$ is stable for all $\xi \in {\cal M}.$
\item[(ii)] $E^{\xi_1}_* \cong E^{\xi_2}_* \Longleftrightarrow
\operatorname{GL}(r'', {\Bbb C}) \cdot \xi_1 = \operatorname{GL}(r'', {\Bbb C}) \cdot \xi_2$ for all
$\xi_1, \xi_2 \in {\cal M}.$
\end{enumerate}
\end{claim}
\begin{pf}
(i) Suppose to the contrary that $E^\xi_*$ is not parabolic stable for some
$\xi \in {\cal M}.$
Let $G_*$ be a rank $s$ parabolic subbundle of $E^\xi_*$ with $\mu(G_*) >
\mu(E^\xi_*).$
Then $\mu(G) \geq \mu(E^\xi),$ since otherwise
$$\mu(G_*) < \mu(G) + \epsilon(d,r)/2 < \mu(E^\xi) < \mu(E^\xi_*).$$
As in the argument of Lemma 6 of Newstead, the map
$G \longrightarrow E'$ has a factorization as
$G \rightarrow G^1 \rightarrow G^2 \rightarrow E'$ and the arguments there
give the following inequalities:
\begin{eqnarray} \label{eq:1}
\deg(G^2) &\geq& \deg(G) \geq \frac{sd}{r}, \\
\label{eq:2}
\operatorname{rank}(G^2) &\leq& \operatorname{rank}(G) - h^0(G) \leq \frac{sr'}{r}.
\end{eqnarray}
These imply that $\mu(G^2) - \mu(E') \geq 0.$
But $E'_*$ is parabolic stable, so by Proposition \ref{prop:num},
$E'$ is semistable and $\mu(G^2)=\mu(E').$
Thus, we must have equalities
in equations (\ref{eq:1}) and (\ref{eq:2}), in particular $\mu(G) =
\mu(E^\xi)$.
But since $\mu(G_*) > \mu(E^\xi_*),$ we see that $G_*$ must inherit the weight
$\alpha_2,$
which implies that $G^2_*$ also inherits $\alpha_2,$ and it now follows that
$$\mu(G^2_*) - \mu(E'_*) = \frac{(s_2 -1) \alpha_1 + \alpha_2}{s_2} - \frac{(r'-1)
\alpha_1 + \alpha_2}{r'} > 0,$$
where $s_2 = \operatorname{rank} G^2 < r'.$ This contradicts the parabolic stability of
$E'_*$ and completes the proof of part (i).
\medskip\noindent
(ii) Since $\Leftarrow$ is true independent of the vanishing of $H^1,$ we only
prove $\Rightarrow.$
Suppose $E^{\xi_1}_* \cong E^{\xi_2}_*$ and set $\pi_X(E^{\xi_i}_*) =
e_i'=[{E^i_*}'] \in {\cal M}_{\alpha'\!,L}.$
Notice that $h^1(E^{\xi_i}_*) = 0,$
and so $h^0(E^{\xi_i}_*) = \chi(E^{\xi_i}_*) = r''.$ It follows that
every holomorphic section of $E^{\xi_i}_*$ has its image contained in
$I^{\oplus r''}_*.$
Hence any isomorphism $\varphi : E^{\xi_1}_* \longrightarrow E^{\xi_2}_*$
defines
a commutative diagram
$$\CD
0 @>>> I^{\oplus r''}_* @>>> E^{\xi_1}_* @>>> {E^1_*}' @>>> 0 \\
&& @VV{\varphi''}V @VV{\varphi}V @VV{\varphi'}V \\
0 @>>> I^{\oplus r''}_* @>>> E^{\xi_2}_* @>>> {E^2_*}' @>>> 0 \\
\endCD $$
where both $\varphi'$ and $\varphi''$ are isomorphisms, and
so $\xi_2 = (\varphi' \times \varphi'') \cdot \xi_1.$
\end{pf}
Part (i) of the claim and the universal property of ${\cal U}_*^\alpha$
gives a map ${\cal M} \stackrel{\psi}{\longrightarrow} {\cal M}_{\alpha,L},$ which is
injective by part (ii).
Since ${\cal M}$ is nonempty,
$\dim {\cal M} = \dim {\cal M}_{\alpha,L},$ so rationality of ${\cal M}_{\alpha,L}$ follows from that
of ${\cal M}.$
This concludes the proof in Case II.
$\quad \Box$
\medskip
\noindent
{\it Remark.}
We had originally hoped to prove rationality of ${\cal M}_{\alpha,L}$ with the weaker
hypothesis that
$\alpha$ is generic,
but the argument does not hold in this generality.
For consider the case $D=p.$ By tensoring with a line bundle and shifting, we
can assume that
$$r(g-1) < d \leq r(g-1) + m_1.$$ Hence, the subbundle split off in the
induction
is again a sum of parabolic line bundles with the same weights.
The difficulty is in proving that
the quotient $E'_*$ has {\it generic} weights $\alpha'.$
Proposition \ref{prop:fine} implies that
$E'_*$ admits a generic weight if and only if the elements of the set $\{d,
m'_i(p) \}$ greatest common divisor equal to one.
The statement
$$(d,m_1,\ldots,m_\kappa)=1 \Rightarrow (d,m'_1,\ldots,m'_\kappa)=1,$$
which is what we would need to prove here, is unfortunately false
(notice that $m'_1=m_1-d+r(g-1)$ and $m'_i=m_i$ otherwise).
\medskip\noindent
{\it Acknowledgements.} Both authors would like to express their gratitude to
the
Max-Planck-Institut f\"ur Mathematik for providing financial support. The first
author
is also grateful to the Institut des Hautes \'Etudies
Scientifique for partial support.
We would also like to thank I.\ Dolgachev and L.\ G\"ottsche
for helpful discussions.
|
1996-10-04T09:04:22 | 9610 | alg-geom/9610005 | en | https://arxiv.org/abs/alg-geom/9610005 | [
"alg-geom",
"dg-ga",
"math.AG",
"math.DG"
] | alg-geom/9610005 | Sacha Sardo Infirri | Alexander V Sardo-Infirri | Resolutions of Orbifold Singularities and Flows on the McKay Quiver | LaTex2e, 57 pages with 1 table and 19 figures | null | null | null | null | Let $\Gamma$ be a finite group acting linearly on $\C^n$, freely outside the
origin. In previous work a generalisation of Kronheimer's construction of
moduli of Hermitian-Yang-Mills bundles with certain invariance properties was
given. This produced varieties $X_\zeta$ (parameterised by $\zeta\in\Q^N$)
which are partial resolutions of $\C^n/\Gamma$. In this article, it is shown
the same $X_\zeta$ can be described as moduli spaces of representations of the
McKay quiver associated to the action of $\Gamma$. It it shown that, for
abelian groups, $X_\zeta$ are toric varieties defined by convex polyhedra which
are the solution sets for a generalisation of the transportation problem on the
McKay quiver. The generalised transportation problem is solved for an arbitrary
quiver to give a description of the extreme points, faces, and tangent cones to
the solution polyhedra in terms of certain distinguished trees in the quiver.
Applied the McKay quiver, this gives an explicit procedure for calculating
$X_\zeta $, its Euler number, and its singularities for any $\zeta$. The
$\zeta$-parameter-space is thus partitioned into a finite disjoint union of
cones inside which the biregular type of $X_\zeta$ remains constant. Finally,
the example $\C^3/\Z_5$ (weights $1,2,3$) is worked out in detail, and figures
of smooth and singular $X_\zeta$ and their corresponding flows are drawn. A
further example of smooth crepant resolution $X_\zeta$ is drawn for the
singularity ${1/11}(1,4,6)$.
| [
{
"version": "v1",
"created": "Fri, 4 Oct 1996 06:59:13 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Sardo-Infirri",
"Alexander V",
""
]
] | alg-geom | \part{BlackBoard Bold}
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\begin{document}
\title[Orbifold Singularities and McKay Flows]{\titlestring\footnote{Maths Subject Classification (1991): 14M25 (Primary) 05C35, 90C08, 32S45 (Secondary)}}
\author{Alexander V.\ Sardo Infirri}
\email{[email protected]}
\address{Research Institute for Mathematical Sciences\\ Ky\=oto University\\
Oiwake-ch\protect\=o\\ Kitashirakawa\\ Saky\protect\=o-ku\\ Ky\=oto
606-01\\ Japan}
\date{2 October 1996}
\begin{abstract}
Let~$\Gamma$ be a finite group acting linearly on~${\mathbb C}^n$, freely
outside the origin, and let $N$ be the number of conjugacy classes of
$\Gamma$ minus one.
In~\cite{sacha:thesis,sacha:ale} a generalisation of Kronheimer's
construction \cite{kron:ale} of moduli of Hermitian-Yang-Mills bundles
with certain invariance properties was given. This produced varieties
$X_\zeta$ (parameterised by $\zeta\in\Q^N$) which are partial
resolutions of ${\mathbb C}^n/\Gamma$.
In this article, it is shown that $X_\zeta$ can be described as moduli
spaces of representations of the McKay quiver associated to the action
of $\Gamma$, subject to certain natural commutation relations.
This allows a complete description of these varieties in the case when
$\Gamma$ is abelian. They are shown to be toric varieties corresponding
to convex polyhedra which are the solution sets for a generalisation
of the transportation problem on the McKay quiver.
The generalised transportation problem is solved for a general quiver
to give a description of the extreme points, faces, and tangent cones
to the solution polyhedra in terms of certain distinguished trees in
the underlying graph to the quiver. Applied to the case of the McKay
quiver, this gives an explicit computational procedure for calculating
$X_\zeta $, its Euler number, and giving a complete list of the
singularities which can occur for all $\zeta$. The
$\zeta$-parameter-space $\Q^N$ is thus partitioned into a finite
disjoint union of cones inside which the biregular type of $X_\zeta$
remains constant. Passing from one cone to the other correponds to a
birational transformation.
The example ${\mathbb C}^3/\Z_5$ (weights $1,2,3$) is worked out in detail:
there are two types of $\zeta$-cones: ones for which $X_\zeta $ is a
smooth resolution, and others where it has a singularity isomorphic to
a cone over a quadric in ${\mathbb C}^4$. This gives some evidence for the
conjecture expressed in~\cite{sacha:ale} acording to which the
singularities of $X_\zeta$ are at most quadratic for a generic
$\zeta$. Computer calculations also show that the cases where
$\Gamma\subset SU(3)$, and $|\Gamma|\leq 11$ yield crepant $X_\zeta$.
For generic $\zeta$, the Euler number of $X_\zeta$ is also equal to
$|\Gamma|$, and the $X_\zeta$ give smooth crepant resolutions fo the
singularity.
A further example of a picture of the polyhedron corresponding to
$X_\zeta$ is drawn for the singularity $\frac{1}{11}(1,4,6)$.
\end{abstract}
\maketitle
\tableofcontents
\setcounter{section}{-1}
\section{Introduction}
\label{sec:intro}
This paper is concerned with affine \emph{orbifold singularities},
namely with singularities of the type $X={\mathbb C}^n/\Gamma$ for $\Gamma$ a
finite group acting linearly on ${\mathbb C}^n$.
In~\cite{sacha:thesis,sacha:ale} a method using moduli of invariant
Hermitian-Yang-Mills bundles was given for constructing partial
resolutions of $X$ carrying natural asymptotically locally Euclidean
(ALE) metrics. In this article, the same construction is described in
terms of representation moduli of quivers. This allows a complete
description of the moduli to be given for the case of abelian groups
using the language of toric varieties. It turns out that the convex
polyhedra describing the moduli are the solutions of a generalisation
of a well-known network optimization problem (the transportation
problem) on the McKay quiver, in which the quiver plays the role of a
network where commodities are transported, and the parameter $\zeta$
specifies the supplies and demands at each vertex.
\subsection{Background}
\label{sec:intro:back}
The resolution of the Kleinian singularities ${\mathbb C}^2/\Gamma$ for $\Gamma$ a
finite subgroup of $\SL(2)$ is a classical subject; their minimal
resolution $\widetilde X$ was first constructed by Du Val, and
Brieskorn~\cite{briesk} showed that the components of the exceptional
divisors form graphs which are dual to the homogeneous Dynkin diagrams
for the Lie algebras $A,D,E$. In 1980, McKay~\cite{mckay:graphs}
remarked that this establishes a correspondence between the extended
homogeneous Dynkin diagrams $\overline A,\overline D,\overline E$ and the
irreducible representations of $\Gamma$. More recently~\cite{reid_ito}
another correspondence was constructed between conjugacy classes ``of
weight 1'' and crepant divisors for any quotient singularity generated
by a finite subgroup of $\SL(n)$.
In the case where $\Gamma\subset\SL(3)$, Dixon, Harvey, Vafa and
Witten proposed a definition of the orbifold Euler
characteristic~\cite{dhvw:i,dhvw:ii} and conjectured the existence of
smooth resolutions with trivial canonical bundle (i.e.\xspace which are
crepant) and whose Euler number is given by the DHVW orbifold Euler
number. The final cases in the proof of this conjecture were only
recently completed
\cite{mar_ols_per,roan:mirror_cy,mark:res_168,roan:res_a5,ito:trihedral,roan:crepant}.
For the case $\Gamma\subset\SL(4)$, the author has obtained some
interesting analogous results if one considers terminalisations rather
than resolutions~\cite{sacha:sl4}.
A promising link between the representation theory of $\Gamma$ and the
construction of resolutions was established in 1986;
Kronheimer~\cite{kron:thesis,kron:ale} constructed a family of
hyper-K\"ahler\ quotients $X_\zeta $ by looking at a dimensional reduction
of the Hermitian Yang-Mills (HYM) equations on a $\Gamma$-equivariant
bundle over ${\mathbb C}^2$. He used the representation theory of $\Gamma$ and
McKay's observation to prove that, for generic $\zeta$, these
quotients are isomorphic to the minimal resolution $\widetilde X$.
The author's thesis~\cite{sacha:thesis} (from which the present paper
and its companion~\cite{sacha:ale} are in large part extracted) grew
out of the ambition to generalise the results as far as possible to
dimensions higher than two.
Despite the fact that there is little hope for the results
for $\GL(n)$ and general $n$ to be as strong as those for $\SL(2)$
groups, some of the nice properties of the $\SL(2)$ case generalise to
the $\GL(n)$ case.\footnote{Some interesting new aspects are also
present for the $\SL(3)$ case~\cite{sacha:thesis}.}
In~\cite{sacha:thesis} several descriptions were given of the
generalised construction. Probably the most concise is the one given
in~\cite{sacha:ale}: construct moduli spaces $X_\zeta$ of instantons
on the trivial bundle ${\mathbb C}^n\times R\to{\mathbb C}^n$. Here $R$ denotes the
regular representation space for the group $\Gamma$, and $\zeta$ is a
linearisation of the bundle action. The instantons are required to
satisfy Hermitian-Yang-Mills-type equations, as well as additional
$\Gamma$-equivariance and translation-invariance properties.
In the present paper, an entirely different view point is adopted,
namely that of considering $X_\zeta$ are representation moduli of the
McKay quiver. More precisely, one considers representations
$\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R})$ of the McKay quiver ${\mathcal Q}$ into the multiplicity
space ${\mathbf R}$ of the regular representation of $\Gamma$, subject to
certain commutation relations ${\mathcal K}$. A reductive group $\PGL({\mathbf R})$ acts on the above space, and one obtains GIT quotients $X_\zeta$
which depend on a rational parameter $\zeta$.
\subsection{Main Results}
\label{sec:intro:main}
This paper assumes that $\Gamma\subset\GL(n)$ acts on ${\mathbb C}^n$ freely
outside the origin for any $n\geq 2$, which means that $X={\mathbb C}^n/\Gamma$
has an isolated singularity.\footnote{This is for the purpose of
simplicity --- the method would seem to be applicable to the general
case with some modifications.} The main results are as follows.
Firstly, $X_\zeta$ are identified with representation moduli of the
McKay quiver. This is fairly straight-forward and involves mostly
translating quiver concepts over to the language used
in~\cite{sacha:ale}. There are two further sets of results.
\subsubsection{Toric description of $X_\zeta$}
The first is a toric description of the representation variety
$\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R})$ and its moduli $X_\zeta$. The convex
polyhedron $C_\zeta$ corresponding to $X_\zeta$ is identified with the
projection to ${\mathbb R}^n$ of the solution polyhedron for the
transportation problem on the McKay quiver. The transportation
problem is a well-known linear network optimization
problem~\cite{kenn_helg,gond_mino:graphs}: given an assignment of real
numbers to the vertices of a quiver\footnote{The term
\emph{network} is usually used in this context instead of quiver.} ${\mathcal Q}$
(thought of as representing the demand and supply of certain
commodities), the aim is to find an assignment of non-negative real
numbers to the arrows (a \emph{flow} on ${\mathcal Q}$), in such a way that the
demands and supplies at each vertex are satisfied (i.e.\xspace the equation
$\partial f=\zeta$ holds). For any given assignment of weights
$\zeta$, the solution set is a convex polyhedron inside ${\mathbb R}^{{\mathcal Q}_1}$
denoted by $F_\zeta$.
The notation in use is as follows:
\begin{notation}
Let $\Gamma$ be the cyclic group of order $r$ acting freely outside the
origin in $Q$ with weights $w_1,\dots,w_n\in\Z_r=\Z/r\Z$ i.e.\xspace via
\map{\rho}{\mu_r\subset{\C^*}}{{\C^*}^n}{\lambda}{\begin{pmatrix}
\lambda^{w^1} \\
& \lambda^{w^2} \\
&\\
& & & \lambda^{w^n}
\end{pmatrix}.}
The McKay
quiver ${\mathcal Q}$ of $\Gamma$ has vertices
${\mathcal Q}_0=\Z_r$, and arrows $a_v^i:=v\to v-w_i \pmod r$ for each $v\in\Z_r$ and
$i=1,\dots,n$.
Let $\pi\colon{\mathcal Q}_1\to\{1,\dots,n\}$ be the map defined by
$\pi(a_v^i):=i$ and let $\pi\colon{\mathbb R}^{{\mathcal Q}_1}\to{\mathbb R}^n$ be the induced
linear map defined by mapping the standard bases. (Here and
elsewhere we use the exponential notation
${\mathbb R}^{{\mathcal Q}_1}:=\Map({\mathcal Q}_1,{\mathbb R})$. The standard basis of ${\mathbb R}^{{\mathcal Q}_1}$ is
given by the ``indicator'' functions $\chi_a$ defined by
$\chi_a(a')=1$ if $a=a'$ and $\chi_a(a')=0$ otherwise). Let $\Pi$
be the sub-lattice of $\Z^n$ of index $r$ defined by
$$\Pi := \ker\hat\rho=\{ x\in \Z^n | \sum x_iw_i\equiv 0\pmod r\}.$$
For each
element $\zeta\in\Z^{{\mathcal Q}_0}$ such that $\sum_{v\in{\mathcal Q}_0}
\zeta(v)=0$, consider the corresponding solution polyhedron
$F_\zeta$ to the transportation problem on ${\mathcal Q}$, and let
$C_\zeta=\pi F_\zeta$ be its projection to ${\mathbb R}^n$.
\end{notation}
Adopting the above notation, one can state the first main theorem.
\begin{nonumberthm}[c.f.\xspace Thms.\ \ref{thm:toric-d-moduli} and \ref{thm:slices}]
The moduli $X_\zeta$ are isomorphic to the toric varieties
$$X_\zeta\cong\protect\overline{T}{}^{\Pi,C_\zeta},$$
where $C_\zeta$ correspond to
the projection to ${\mathbb R}^n$ of the solution polyhedra to the
transportation problem on ${\mathcal Q}$.
\end{nonumberthm}
\subsubsection{Generalised Transportation Problem}
\label{sec:intro:ab:trans}
The problem of determining $C_\zeta$ can be considered as a
generalised transportation problem. The solution to the classical
transportation problem is well known:
\begin{nonumberthm}[Classical Transportation Problem (c.f.\xspace Theorem~\ref{thm:classical})]
The extreme points of $F_\zeta$ are precisely those flows in
$F_\zeta$ whose supports\footnote{The support of a flow is the set
of arrows on which it is non-zero.} are \emph{trees}, i.e.\xspace contain
no cycles.
\end{nonumberthm}
The basic idea behind the proof of this theorem is to associate to
each cycle $c$ a corresponding flow $\tilde\chi_c$ as follows. A
\emph{cycle} in ${\mathcal Q}$ is a sequence $c_1,\dots,c_m$ of arrows such
that they form a cycle when their orientation is disregarded. The
disjoint union of the arrows $c_i$ whose direction agrees (resp.\xspace disagrees)
with the ordering $c_1,\dots,c_k$ is called the \emph{positive\/}
(resp.\xspace \emph{negative\/}) part of $c$ and is denoted it by $c^+$
(resp.\xspace $c^-$).
For each arrow $a\in{\mathcal Q}_1$, let $\chi_a$ denote the flow which takes
the value $1$ on the arrow $a$ and zero elsewhere. To each cycle $c$,
one can define the \emph{basic flow} associated to $c$ by
$$\tilde\chi_c:=\sum_{c_i\in c^+}\chi_{c_i} - \sum_{c_i\in c^-}\chi_{c_i}\in\Z^n.$$
Note that for any cycle $c$, $\partial \tilde\chi_c=0$, i.e.\xspace the
associated flow does not contribute anything at any vertex. If $f\in
F_\zeta$ contains $c$ in its support, then one can add
$\pm\epsilon\tilde\chi_c$ to $f$ for some small $\epsilon$ and still
remain in $F_\zeta$. Hence $f$ cannot be an extreme point.
Vice-versa, if $f$ is not extreme, then one can reconstruct a cycle in
its support.
The generalisation of this theorem to describe the projection
$C_\zeta=\pi(F_\zeta)$ is quite straight-forward. One begins by
defining the \emph{type} of any given cycle as the element
$\pi(\tilde\chi_c)\in\Z^n$. Cycles of type $0\in\Z^n$ play a special
role, much as ordinary cycles do in the classical case.
\begin{dfn}
The \emph{closure} of $S\subset{\mathcal Q}_1$ is defined to be the
smallest over-set $\overline{S}\supseteq S$ such that $$c^-\subseteq
\overline{S}\iff c^+\subseteq\overline{S},$$
for all cycles $c$ of type
zero. Two configurations $S,S'$ will be called \emph{equivalent}
(written $S\sim S'$) if $\overline{S}=\overline{S'}$.
\end{dfn}
Th same methods as in the classical case allows one to prove the
following generalisation.
\begin{nonumberthm}[Extreme Points of $C_\zeta$]
The extreme points of $C_\zeta$ are the images under $\pi$ of
precisely those flows in $F_\zeta$ whose support contains no cycles
of non-zero type.
\end{nonumberthm}
In fact, by taking into account all cycles in the support of a given
flow, it is possible to determine the dimension of the face which
$\pi(f)$ belongs to. Some additional terminology will be convenient
to state the results.
Let ${\mathcal C}$ denote the set of all \emph{configurations}, i.e.\xspace the set of
non-empty subsets of ${\mathcal Q}_1$. For a configuration $S\subset{\mathcal Q}_1$, let
$F_0(S)$ be the cone generated by the flows $\tilde\chi_c$ for the
cycles $c$ whose negative part is included in $S$ and let $Z_0(S)$ be
its maximal vector subspace (i.e.\xspace the subspace generated by the flows
$\tilde\chi_c$ for the cycles $c\subseteq S$). The \emph{rank} of $S$
is defined to be the dimension of $\pi Z_0({\overline S})$. The set of
configurations (resp.\xspace trees) of rank $k$ is denoted ${\mathcal C}^k$ (resp.\xspace
${\mathcal T}^k$). Also, write ${\mathcal C}_\zeta$ (resp.\xspace ${\mathcal T}_\zeta$) for the subset of
configurations (resp.\xspace trees) which are \emph{admissible for
$\zeta$},\/ namely configurations (resp.\xspace trees) $S\in{\mathcal C}$ which
arise as the support of some element in $F_\zeta$.
\begin{nonumberthm}[c.f.\xspace Theorem \ref{thm:faces}]
\label{thm:intro:faces}
For all $\zeta$, the map
\map{\text{Face}_\zeta}{{\mathcal C}_\zeta}{\text{Faces of }\pi
F_\zeta}{S}{\pi F_\zeta \cap(\pi f+\pi Z_0(\overline{S})))} is
independent of the choice of $f\in F_\zeta\cap\supp^{-1}(S)$, and
induces a bijection
$$\text{Face}_\zeta\colon {\mathcal C}^k_\zeta/\!\!\sim
\xrightarrow{\;\cong\;} \text{$k$-faces of }\pi F_\zeta.$$
Furthermore, for all $[S]\in{\mathcal C}^k_\zeta/\!\!\sim$,
$$T_{\text{Face}_\zeta(S)}\pi F_\zeta =\pi F_0(\overline{S}),$$
where
the left-hand side denotes the tangent cone to the polyhedron $\pi
F_\zeta$ at the face $\text{Face}_\zeta(S)$.
In other words, $\pi F_0(\overline{S})$ gives the tangent cone
corresponding to the configuration $S$ (which is independent of the
value of $\zeta$) and $\text{Face}_\zeta(S)$ gives the corresponding
face of $C_\zeta$ (whose direction is also independent of $\zeta$).
\end{nonumberthm}
This theorem has a number of important
corollaries.
\begin{nonumbercor}[c.f.\xspace Cor.\ \ref{cor:iso_C_zeta}]
\label{cor:intro:iso_C_zeta}
If $\zeta$ and $\zeta'$ have the same admissible configurations
(${\mathcal C}_\zeta={\mathcal C}_{\zeta'}$) or even just the same admissible trees
(${\mathcal T}_\zeta={\mathcal T}_{\zeta'}$) then the corresponding polyhedra
$C_{\zeta}$ and $C_{\zeta'}$ are geometrically isomorphic. Two
polyhedra are said to be \emph{geometrically isomorphic} if they are
combinatorially isomorphic and their tangent cones at the
corresponding faces are identical. In particular, their associated
fans and toric varieties must be identical.
\end{nonumbercor}
\begin{nonumbercor}[c.f.\xspace Cor. \ref{cor:number_extreme_pts}]
\label{cor:intro:number_extreme_pts}
Let ${\mathcal T}\subset{\mathcal C}$ denote the configurations which are {\em
trees,\/} and let $\ext
C_\zeta$ denote the extreme points of $C_\zeta$. Then
$$\card{\ext C_\zeta}=\card{{\mathcal C}^0_\zeta/\!\!\sim}=\card{{\mathcal T}^0_\zeta/\!\!\sim}.$$
\end{nonumbercor}
\begin{rmk}
Since any tree admits a unique flow such that $\partial f = \zeta$,
it is very easy to determine ${\mathcal T}_\zeta$. Furthermore, if $\zeta$
is generic, then ${\mathcal C}_\zeta\subset{\mathcal C}_{\text{span}}$, the subset of
\emph{spanning configurations},\/ i.e.\xspace configurations whose arrows
join any two vertices of the quiver (not necessarily in an oriented
way). In this case, $T\in{\mathcal T}^0_\zeta$ if and only if the tree $T$
admits an assignment of elements of $\Z^n$ to the vertices
which satisfies particular properties (a so-called \emph{$n$-weighting}
in the terminology of Section~\ref{sec:2:examples:weight:fixed}). This
condition can again be checked by a very simple algorithm.
To sum up: the above corollary allows one to determine the extreme
points of $C_\zeta$ for any $\zeta$ very easily.
\end{rmk}
\begin{nonumbercor}[c.f.\xspace Cor.\ \ref{cor:fan} and Lemma \ref{lemma:generic_flow}]
\label{cor:intro:fan}
The extreme points of the polyhedron $\pi F_\zeta$ correspond to the
trees $T$ in ${\mathcal T}^0_\zeta$ and the tangent cone to $\pi F_\zeta$ at
the point corresponding to $T$ is $\pi F_0(\clos{T})=\pi F_0(T)$. Thus
the fan associated to the polyhedron $\pi F_\zeta $ is given by the
dual cones $\pi F_0(T)^\vee$ for the trees $T\in{\mathcal T}^0_\zeta$ and all
their faces.
\end{nonumbercor}
The theorem and corollaries above allow a
complete understanding of the extreme points, faces, and tangent cones
to the solution polyhedra of a generalised transportation problem.
Applied to the case of the McKay quiver, they allow one to determine
the Euler number and the singularities of $X_\zeta$ as well as giving
a complete list of the singularities which can occur for all $\zeta$.
Several interesting questions regarding these moduli nevertheless
remain, such as whether they produce smooth (resp.\xspace terminal)
resolutions in the $\SL(3)$ (resp.\xspace $\SL(4)$) case. A conjecture is
given in the companion paper~\cite{sacha:ale} and several examples are
computed in Section~\ref{sec:2:examples} of this paper.
Depending on the point of view, isomorphic objects are denoted by
different notations. For ease of reference, Table~\ref{tab:notation}
gives a correspondence table between the point of view
in~\cite{sacha:ale} and in Parts~\ref{sec:1} and~\ref{sec:2} of this
paper.
\renewcommand{\arraystretch}{1.4}
\begin{table}[htbp]
\begin{center}
\leavevmode
\setlength{\fboxrule}{0pt}
\begin{tabular}{|l|l|l|}
\hline
Moduli of Bundles & Moduli of Representations&
Toric Varieties\\
\cite{sacha:ale} & Part 1 of this paper & Part 2 of this paper\\
\hline
$M^\Gamma $&$ \Rep_{\mathcal Q}({\mathbf R}) $&$ \protect\overline{T}{}^{\Lambda^1_{\mathbb R},\Lambda^1_{{\mathbb R}+}}={\mathbb C}^{{\mathcal Q}_1}$\\
\hline
${\mathcal N}^\Gamma $&$ \Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R}) $&$ \protect\overline{T}{}^{\Lambda,C}$\\
\hline
$G^\Gamma=\PGL^\Gamma(R) $&$ \PGL({\mathbf R}) $&$ \protect\overline{T}{}^{\Lambda^{0,0}}=\protect\overline{T}{}^{0,0}$\\
\hline
$X_\zeta $&$ {\mathcal M}_\zeta $& $\protect\overline{T}{}^{\Pi,C_\zeta}$\\
\hline
\end{tabular}
\vspace{1em}
\caption{Correspondence between the notations in use in this paper and in the
companion paper~\cite{sacha:ale}.}
\label{tab:notation}
\end{center}
\end{table}
\subsection{Methods and Outline}
\label{sec:intro:outline}
The methods used to prove the main results are notationally cumbersome
due to the quiver notation, but on the whole straightforward. A
familiarity with toric geometry will make reading easier, although all
the required facts and notation are explained when needed.
The paper is divided into two parts; the first dealing with general
groups, and the second specializing to the case of abelian ones. Each
part begins with a section summarizing the notation in use.
\subsubsection{Part \ref{sec:1}}
Section~\ref{sec:1:notation} summarises the notation.
Section~\ref{sec:1:quivers} outlines the basic facts and notation
concerning quivers, their representations and their moduli. The
necessary details regarding geometric invariant theory (GIT) quotients are also given.
Section~\ref{sec:1:mckay} explains the case of the McKay
quiver associated to the representation of a finite group. The McKay
correspondence is briefly mentioned, and the ``canonical'' commutation
relations ${\mathcal K}$ are defined.
\subsubsection{Part \ref{sec:2}}
Section~\ref{sec:2:notation} summarises the notation.
Section~\ref{sec:2:abel} specialises further the discussion from
Part~\ref{sec:1} to the case of abelian groups. The McKay quiver and
commutation relations for these can be described quite simply and some
specialised notation is introduced for this purpose. The
representation variety and its moduli are proved to be toric varieties
corresponding to an $(n+\card{\Gamma}-1)$-dimensional convex cone $C$
and to its $n$-dimensional convex polyhedral slices $C_\zeta$
respectively.
From Section~\ref{sec:2:flow} onwards, the focus is on the
transportation problem; its solution polyhedra coincide with $C_\zeta$
in the case of the McKay quiver.
Section~\ref{sec:2:flow} explains the ordinary transportation problem
on a network.
Section~\ref{sec:2:gtp} explains the generalised transportation
problem and states the theorems describing its solution polyhedra.
An example of an application to the McKay quiver and several important
corollaries are given in Section~\ref{sec:2:mckay}.
The proofs of the theorems are given in
Section~\ref{sec:2:flow:proofs}, except for the proof of some
technical lemmas regarding flows which are left until
Section~\ref{sec:2:exact}.
Section~\ref{sec:2:sing} contains a discussion of the singularities of
$X_\zeta$.
The paper concludes in Section~\ref{sec:2:examples} with some
practical examples and computations.
\subsection{Acknowledgments}
\label{sec:intro:ack}
The present paper and its companion paper~\cite{sacha:ale} consist
mostly\footnote{Minor portions have been rewritten to include
references to advances in the field made since then
(notably~\cite{ito:trihedral,roan:crepant,reid_ito}).} of excerpts
of my D.Phil.\ thesis~\cite{sacha:thesis}, and I wish to acknowledge
the University of Oxford and Wolfson College for their hospitality
during its preparation. I am grateful to the Rhodes Trust for
financial support during my first three years, and to Wolfson College
for a loan in my final year. The conversion from thesis to article
format was done while I was a Research Assistant in RIMS, Kyoto.
I also take the opportunity to thank my supervisors Peter Kronheimer
and Sir~Michael Atiyah who provided me with constant advice,
encouragement and support and whose mathematical insight has been an
inspiration. I also wish to thank William Crawley-Boevey, Michel
Brion, Gavin Brown, Jack Evans, Partha Guha, Katrina Hicks, Frances
Kirwan, Alistair Mees, Alvise Munari, Martyn Quick, David Reed, Miles
Reid, Michael Thaddeus, and, last but not least, my parents and
family.
\section{Summary of Notation for Part \ref{sec:1}}
\label{sec:1:notation}
\subsection{General}
\begin{Pentry}
\item[$k$] Algebraically closed field.
\item[$\Z_+$] Non-negative integers.
\item[${\mathbb R}_+$] Non-negative reals.
\end{Pentry}
\subsection{Quivers, Representations}
\begin{Pentry}
\item[$Q$] $n$-dimensional complex vector-space.
\item[$\Gamma$] Finite sub-group of $\SL(Q)$
\item[${\mathcal Q}$] Generic quiver ${\mathcal Q}=({\mathcal Q}_0,{\mathcal Q}_1)$.\\
From Section~\ref{sec:1:mckay} onwards, ${\mathcal Q}={\mathcal Q}_{\Gamma,{\mathcal Q}}$, the
McKay quiver associated to $(\Gamma,Q)$.
\item[${\mathcal Q}_0$] Vertices of ${\mathcal Q}$; in the case of the McKay quiver,
elements of ${\mathcal Q}_0$ index the irreducible representations $R_v$ of
$\Gamma$.
\item[${\mathcal Q}_1$] Arrows of ${\mathcal Q}$.
\item[$R_v$] Irreducible representation of $\Gamma$ corresponding to
vertex $v$ of the McKay quiver.
\item[$R$] Regular representation of $\Gamma$; $R=\oplus_{v\in{\mathcal Q}_0}
{{\mathbf R}_v}\otimes R_v$.
\item[${\mathbf R}_v$] The trivial $\Gamma$-module giving the multiplicity of
$R_v$ in $R$.
\item[${\mathbf R}$] Multiplicity space for $R$; ${\mathbf R}=\oplus_{v\in{\mathcal Q}_0}{{\mathbf R}_v}$
\item[${\mathbf V}$] Generic representation space of ${\mathcal Q}$; ${\mathbf V}=\oplus_{v\in{\mathcal Q}_0}{\mathbf V}_v$.
\item[$R_{\mathbf V}$] The $\Gamma$-module $\oplus_{v\in{\mathcal Q}_0} {\mathbf V}_v\otimes R_v$
corresponding to ${\mathbf V}$.
\item[$\alpha$] $\Gamma$-invariant element of $(Q\otimes\End R)$.
\item[$q_i$] Basis of $Q$ ($i=1,\dots,n$).
\item[$\alpha_i$] Component of $\alpha$ with respect to $\{q_i\}$:
$\alpha=\sum_{i=1}^n q_i\otimes\alpha_i$.
\item[$\tilde\alpha$] Corresponding representation of the McKay quiver
into ${\mathbf R}$.
\item[$a$] Arrow of ${\mathcal Q}$; also written $t(a)\to h(a)$, where $t(a)$
denotes the tail and $h(a)$ the head of $a$.
\item[$\tilde\alpha_a$] Value of $\tilde\alpha$ on the arrow $a$;
$\tilde\alpha_a\colon {\mathbf R}_{t(a)}\to {\mathbf R}_{h(a)}$.
\item[${\mathcal K}$] Relations on ${\mathcal Q}$.
\item[$\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R})$] Space of all representations of ${\mathcal Q}$ into
${\mathbf R}$ satisfying the relations ${\mathcal K}$.
\item[$\PGL({\mathbf R})$] The isomorphism group for representations of ${\mathcal Q}$
into ${\mathbf R}$
$$\PGL({\mathbf R}):=\times_{v\in{\mathcal Q}_0}\GL({{\mathbf R}_v})/{\C^*}.$$
\item[$\mathfrak{k}$] The Lie algebra of $\PU({\mathbf R})$ (a real form of $\PGL({\mathbf R})$).
\item[${\mathcal M}_{{\mathcal Q},{\mathcal K},\zeta}$] Representation moduli of $({\mathcal Q},{\mathcal K})$
$${\mathcal M}_{{\mathcal Q},{\mathcal K},\zeta}:= \Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R})\gitquot{\chi}\PGL({\mathbf R}).$$
The linearisation $\chi$ is related to $\zeta$ by
$\zeta=d\chi(1)_{|\mathfrak{k}}$.
\item[$X_{\zeta}$] The moduli above in the case when ${\mathcal Q}$ is the
McKay quiver, and ${\mathcal K}$ are the commutation relations defined in
Section~\ref{sec:mckay:mckay:ccr:relns}.
\item[$\rho_{\zeta}$] The natural projective morphism $X_{\zeta}\to
X_{0}$ (a partial resolution).
\end{Pentry}
\part{General Groups}
\label{sec:1}
\section{Quivers, Relations and Representation Moduli}
\label{sec:1:quivers}
\subsection{Quivers}
\label{sec:1:quivers:quivers}
A \emph{quiver} is an oriented graph, possibly with multiple arrows
between the same vertices and with loops (arrows which begin and end
at the same vertex). Formally, a quiver ${\mathcal Q}$ consists of a pair of
finite sets ${\mathcal Q}=({\mathcal Q}_0,{\mathcal Q}_1)$ with two maps ${\mathcal Q}_1
\overset{h}{\underset{t}\rightrightarrows} {\mathcal Q}_0$; the elements of
${\mathcal Q}_0$ are called \emph{vertices} and those of ${\mathcal Q}_1$ are called
\emph{arrows}.\/ The elements $t(a)$ and $h(a)$ are called the
\emph{tail\/} and \emph{head\/} of the arrow $a\in{\mathcal Q}_1$ respectively.
The arrow $a$ is also sometimes denoted $t(a)\to h(a)$.
\subsection{Representations}
\label{sec:1:quivers:reps}
Let $k$ be an algebraically closed field.
A \emph{representation of a quiver} is a realization of its diagram of
vertices and arrows in some category: it corresponds to replacing the
vertices by objects and the arrows by morphisms between the objects.
From now on, only the category of $k$-vector-spaces is considered; a
representation ${\mathbf V}$ of a quiver ${\mathcal Q}$ is thus taken to be a collection
of finite dimensional vector-spaces ${\mathbf V}_v$, indexed by the vertices
$v\in {\mathcal Q}_0$, and of linear maps ${\mathbf V}_{v\to v'}\colon {\mathbf V}_{v}\to {\mathbf V}_{v'},$
indexed by the arrows $v\to v' \,\in{\mathcal Q}_1$.
The set of
all representations of ${\mathcal Q}$ into a fixed ${\mathcal Q}_0$-graded vector-space
${\mathbf V}=\oplus_{v\in {\mathcal Q}_0} {\mathbf V}_v$ forms a vector-space, denoted $\Rep_{\mathcal Q}
{\mathbf V}$.
\begin{example}
\label{ex:endo}
Representations of the quiver ${\mathcal Q}$ with one vertex and one loop are just
endomorphisms of a vector-space.
\end{example}
\begin{example}
A \emph{primitive} representation of
${\mathcal Q}$ is an element of
$$\Rep_{{\mathcal Q}}(k^{{\mathcal Q}_0}),$$ where $k^{{\mathcal Q}_0}$ denotes
the free $k$-vector-space on the vertices. A primitive representation
thus corresponds to an assignment of an element of $k$ to each arrow
in ${\mathcal Q}$, i.e.\xspace to an element of the vector-space $k^{{\mathcal Q}_1}$.
\label{ex:primitive}
\end{example}
\subsection{Relations}
\label{sec:1:quivers:relations}
If the morphisms ${\mathbf V}_{v\to v'}$ are required to satisfy relations
between them, then one talks about a \emph{representation of a quiver with
relations}.
More formally, a \emph{relation} is defined as a formal sum of paths in
a quiver. A (non-trivial) \emph{path} $p$ is a sequence
$a_1\dots a_n$ of arrows which \emph{compose}, i.e.\xspace such that
$t(a_i)=h(a_{i+1})$ for $1\leq i < n$:
$$\stackrel{h(p)}\bullet \xleftarrow[{a_1}]{} \bullet \xleftarrow[{a_2}]{} \dots
\xleftarrow[{a_n}]{}\stackrel{t(p)}\bullet.$$ The vertex $h(a_1)$ ($t(a_n)$) is
called the
\emph{head} (\emph{tail}) of the path $p$ and denote it by $t(p)$ ($h(p)$). If ${\mathbf V}$ is a representation
of ${\mathcal Q}$, then there is an induced morphism
$${\mathbf V}(p)={\mathbf V}_{a_1}{\mathbf V}_{a_{2}}\dots {\mathbf V}_{a_n}\colon {\mathbf V}_{t(p)} \to {\mathbf V}_{h(p)}$$
corresponding to each path. The \emph{trivial path} $e_v$
consists of a single vertex $v$ and no arrows; ${\mathbf V}_{e_v}$ is of
course the identity endomorphism of ${\mathbf V}_v$.
A \emph{representation \/} is said to \emph{satisfy}
$r=\sum\lambda_ip^i$ if $\sum\lambda_i{\mathbf V}(p^i)=0$. If ${\mathcal K}$ denotes
a set of relations, then the set of representations of ${\mathcal Q}$ into
${\mathbf V}$ satisfying the relations ${\mathcal K}$ will be denoted
$\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf V})$. It is an affine variety inside
$\Rep_{\mathcal Q}({\mathbf V})$.
\begin{rmk}
There are in general many classes of relations one can consider, but
in this paper, only \emph{commutation relations}
will be considered, namely relations generated by differences of two
paths, both paths having the same tail and head.
\end{rmk}
\subsubsection{Morphisms of Representations}
\label{sec:1:quivers:morph}
\nopagebreak
Let ${\mathbf V},{\mathbf V}'$ be two representations of the same quiver ${\mathcal Q}$ (possibly
with relations). A \emph{morphism} of
representations $\theta\colon {\mathbf V}\to {\mathbf V}'$ is given by morphisms
$\theta_v\colon {\mathbf V}_v\to {\mathbf V}'_v$ for each vertex $v$, which satisfy
${\mathbf V}'_{a}\theta_{t(a)}=\theta_{h(a)}{\mathbf V}_a$ for each arrow $a$. If
${\mathbf V}_v={\mathbf V}_v'$ and the linear maps $\theta_v$ are all isomorphisms, then
$\theta$ is called an \emph{isomorphism of representations}.
\subsubsection{Representation Moduli}
\label{sec:1:quivers:moduli}
\nopagebreak
Given a quiver ${\mathcal Q}$, some relations ${\mathcal K}$ and a representation space
${\mathbf V}$, one is naturally interested in the \emph{moduli space} of representations. Loosely speaking, this is the
set of isomorphism classes of representations of ${\mathcal Q}$ into ${\mathbf V}$. More
precisely, the group
$$\GL({\mathbf V}):=\times_{v\in{\mathcal Q}_0}\GL({\mathbf V}_v)$$
acts on
$\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf V})$ and its orbits consist of equivalence classes
of representations. The scalar subgroup ${\C^*}\subset\GL({\mathbf V})$ acts
trivially, and one is left with a free action of the quotient
$\PGL({\mathbf V}):=\GL({\mathbf V})/{\C^*}$.
Since $\PGL({\mathbf V})$ is not compact, one must resort to geometric invariant theory\ (GIT) in order
to obtain quotients which are well-defined as quasi-projective
varieties. The ``canonical'' quotient is the affine GIT quotient
\begin{equation}
\label{eq:zero-quot}
{\mathcal M}_{{\mathcal Q},{\mathcal K},0}:= \Rep_{{\mathcal Q},{\mathcal K}}({\mathbf V})\gitquot{} \PGL({\mathbf V}).
\end{equation}
There are other possible quotients however and their existence is in
fact the basis of this paper. In general, given a complex
affine variety $X$ acted upon by a reductive group $G$, and given a
character $\chi\colon G\to{\C^*}$, one defines
$$X\gitquot\chi G := \Proj \bigoplus_{r\in{\mathbb N}} {\mathcal O}_X(rL_\chi)^G,$$
where $L_\chi$ is the trivial bundle over $X$, on which $G$ acts via
$\chi$. The reader is refered to~\cite{sacha:ale,sacha:thesis} for a
detailed treatment.
When the complex reductive group $G$ has a real form $K$, rather than
specifying the character $\chi$ of $G$ one can specify instead its
derivative at the identity $\Lie G\to {\mathbb R}$, or even the restriction of
the later to the Lie algebra of the real form, giving an element of
$\zeta=d\chi(1)_{|\Lie K}\in(\Lie K)^*$.
For the moduli of representations of ${\mathcal Q}$ into ${\mathbf V}$, a real form of
$\PGL({\mathbf V})$ is given by $\PU({\mathbf V}):=\times_{v\in{\mathcal Q}_0}\U({\mathbf V}_v)/\U(1)$,
and the dual of its Lie algebra is the subspace
$${\mathfrak{k}}:=\{\zeta\in \times_{v\in{\mathcal Q}_0} \End_{\mathbb R}({\mathbf V}_v)^*|
\sum_{v\in{\mathcal Q}_0}\trace \zeta_v=0\},$$
where $\zeta_v$ denotes the restriction
of $\zeta$ to $\End_{\mathbb R}({\mathbf V}_v)$.
For $\zeta$ an integral element of $\mathfrak{k}$, define
\begin{equation}
\label{eq:zeta-quot}
{\mathcal M}_{{\mathcal Q},{\mathcal K},\zeta_\chi} := \Rep_{{\mathcal Q},{\mathcal K}}({\mathbf V})\gitquot\chi \PGL({\mathbf V}),\quad\text{where }\zeta_\chi=d\chi(1)_{|\Lie K}
\end{equation}
This is seen to coincide with Definition~\ref{eq:zero-quot} in the case
$\zeta_\chi=0$ (i.e.\xspace $\chi=1$).
\begin{rmk}
In the above, ``$\zeta$ integral'' means integral with respect to
the natural lattice in $\mathfrak{k}$ (i.e.\xspace the kernel of the exponential
map). Strictly speaking, one is not restricted to integral values,
any rational value will do, provided one makes sense of ``fractional
linearisations'' in the obvious manner. We do not bother, as
nothing substantially new is gained from this approach (the moduli
for $k\zeta$ are isomorphic to those for $\zeta$).
\end{rmk}
\section{The McKay Quiver}
\label{sec:1:mckay}
The McKay quiver ${\mathcal Q}_{\Gamma,Q}$ is a quiver which is naturally
associated to a representation $Q$ of a finite group $\Gamma$. As
explained below, its vertices are the irreducible representations of
$\Gamma$ and the arrows describe how the tensor product of $Q$ with each
irreducible decomposes into a sum of irreducibles.
It seems natural to expect a relation between ${\mathcal Q}_{\Gamma,Q}$ and the
quotient singularity $X=Q/\Gamma$. In fact, for the case of a finite
subgroup $\Gamma\subset \SU(2)$ there is a remarkable relation between
${\mathcal Q}_{\Gamma,Q}$ and the minimal resolution
$\widetilde{X}\to X$. McKay remarked~\cite{mckay:graphs} that the
quivers ${\mathcal Q}_{\Gamma,{\mathbb C}^2}$ are precisely the extended Dynkin diagrams of
type $\overline A,\overline D$, and $\overline E$. The ordinary Dynkin
diagrams of type $A,D$ and $E$ had previously been shown by
Brieskorn~\cite{briesk} to be the dual graphs to the graphs of rational
curves in the exceptional fibre of $\widetilde{X}\to X$.
Kronheimer~\cite{kron:thesis,kron:crendus,kron:ale} showed that one
can construct the minimal resolution $\widetilde{X}$ by considering
what turn out to be, upon closer inspection, moduli spaces of
representations of ${\mathcal Q}_{\Gamma,{\mathbb C}^2}$ into the regular representation
space of $\Gamma$. The representations are also required to
satisfy some commutation relations, denoted ${\mathcal K}$.
In this section, the generalisation of these commutation relations to
any McKay quiver is described.
\subsection{The McKay Quiver}
\label{sec:mckay:mckay:dfn}
\nopagebreak
Let $\Gamma$ be a finite group and let $\{R_i, i\in I\}$ be the
set of irreducible representations of $\Gamma$. Any
$\Gamma$-module $R_{\mathbf V}$ decomposes into a sum of irreducibles:
\begin{equation}
R_{\mathbf V}= \bigoplus_{i\in I}{\mathbf V}_i\otimes R_i,
\label{eq:irred_decomp}
\end{equation}
and gives an $I$-graded vector-space (trivial as a $\Gamma$-module)
${\mathbf V}=\oplus_{i\in I}{\mathbf V}_i$, called the \emph{multiplicity space} for $R_{\mathbf V}$.
The dimension of ${\mathbf V}_i$ is called the \emph{multiplicity} of $R_i$ in
$R_{\mathbf V}$ and the vector $\dim {\mathbf V}=(\dim {\mathbf V}_i)_{i\in I}$ is called the
\emph{dimension vector} of the $\Gamma$-module $R_{\mathbf V}$. Conversely, given
any $I$-graded vector-space ${\mathbf V}$, one can construct a corresponding $\Gamma$-module
$R_{\mathbf V}$ by formula~\eqref{eq:irred_decomp}.
The \emph{McKay quiver} ${\mathcal Q}={\mathcal Q}_{\Gamma,Q}$ of a
representation $Q$ is constructed as follows. The vertices ${\mathcal Q}_0=I$
are the irreducible representations of $\Gamma$, and there are $a_{ij}$ (possibly
zero) arrows from vertex $i$ to vertex $j$. The non-negative
integers $a_{ij}$ (which form what is called the \emph{adjacency
matrix} of ${\mathcal Q}$) are defined by the following irreducible
decompositions (for each $i\in I$)
\begin{equation}
\label{eq:aijdecomp}
Q\otimes R_i=\bigoplus_j a_{ji}R_j.
\end{equation}
More invariantly, one may write
\begin{align}
Q\otimes R_i &= \bigoplus_j A^*_{ji}\otimes R_j\\
\intertext{where}
A_{ji}:&=\Hom_\Gamma(Q\otimes R_i,R_j),
\end{align}
and think of the arrows from $i$ to $j$ as giving a basis for the
$a_{ij}$-dimensional vector-space $A_{ij}$.
If ${\mathbf V}$ is a ${\mathcal Q}_0$-graded vector-space, then the representations of
${\mathcal Q}$ into ${\mathbf V}$ correspond to the $\Gamma$-module endomorphisms $R_{\mathbf V}\to
Q\otimes R_{\mathbf V}$. In fact,
\begin{equation}
\begin{split}
\Hom_\Gamma(R_{\mathbf V},Q\otimes R_{\mathbf V})
&= \Hom_\Gamma(\oplus_i {\mathbf V}_i\otimes R_i, Q\otimes (\oplus_j{\mathbf V}_j\otimes R_j))\\
&= \bigoplus_{i,j} A_{ij}^* \otimes \Hom({\mathbf V}_i,{\mathbf V}_j),
\end{split}
\end{equation}
so given $\alpha\in\Hom_\Gamma(R_{\mathbf V},Q\otimes R_{\mathbf V})$, one
can pair it with an arrow $a=i\to j$ (considered as a basis element of
$A_{ij}$) and obtain a map ${\mathbf V}_i\to {\mathbf V}_j$. Thus
\begin{equation}
\label{eq:hom-rep}
\Hom_\Gamma(R_{\mathbf V},Q\otimes R_{\mathbf V}) = \Rep_{\mathcal Q}({\mathbf V}).
\end{equation}
The representation of ${\mathcal Q}$ which corresponds to
$\alpha\in\Hom_\Gamma(R_{\mathbf V},Q\otimes R_{\mathbf V})$ will be denoted $\tilde \alpha$.
\begin{rmk}
Note that the orientation of ${\mathcal Q}$ is reversed when the
representation $Q$ is replaced by its dual $Q^*$, so when $Q$ is
self-dual all the arrows $i\to j$ have opposite arrows $j\to i$. The
pair $i\rightleftarrows j$ is usually denoted $i\mbox{---} j$.
\end{rmk}
\begin{example}[The McKay Correspondence]
\label{ex:dynkin}
For a subgroup $\Gamma<\SU(2)$, the standard 2-dimensional
representation $Q$ is always self-dual, since $\Lambda ^2 Q\cong {\mathbb C}$
as $\Gamma$-modules. McKay's observation~\cite{mckay:graphs} is that
the quivers ${\mathcal Q}_{\Gamma,Q}$ coincide with the extended homogeneous\footnote{A
Dynkin diagram is called \emph{homogeneous} if it has
no multiple bonds} Dynkin diagrams $\overline A_k,\overline D_k,
\overline E_6,\overline E_7,\overline E_8$ represented in
Figure~\ref{fig:dynkin}.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 5cm \epsfbox{fig/dynkin.eps}
\end{center}
\caption[The extended homogeneous Dynkin diagrams $\overline
A_k,\overline D_k, \overline E_6,\overline E_7,\overline E_8$]{The
extended homogeneous Dynkin diagrams $\overline A_k,\overline D_k,
\overline E_6,\overline E_7,\overline E_8$ (The extra vertex is
indicated marked $\bullet$.)}
\label{fig:dynkin}
\end{figure}
\end{example}
\subsection{The Commutation Relations}
\label{sec:mckay:mckay:ccr:relns}
\nopagebreak
Recall from Section~\ref{sec:mckay:mckay:dfn} that if $R_{\mathbf V}$ is any
$\Gamma$-module, the $\Gamma$-module homomorphisms $R_{\mathbf V} \to Q\otimes R_{\mathbf V}$
correspond to representations of the McKay quiver into the $I$-graded
vector-space ${\mathbf V}$ which is the multiplicity space for $R_{\mathbf V}$.
One natural $\Gamma$-module to consider as a candidate for $R_{\mathbf V}$ is
the ``canonical one'' given by the regular representation space $R$ of
$\Gamma$. Its multiplicity space will be denoted ${\mathbf R}=\oplus_{v\in {\mathcal Q}_0}
{\mathbf R}_v$. Its components satisfy the well-known equalities $\dim {\mathbf R}_v=\dim R_v$.
The simplest way to define the commutation relations ${\mathcal K}$ is to use
the isomorphism~\eqref{eq:hom-rep}. Let $q_i$ be a basis of $Q$ which
is orthonormal with respect to some positive definite $\Gamma$-invariant
hermitian inner product. Let $\tilde \alpha\in\Rep_{\mathcal Q}({\mathbf R})$ be a
representation of ${\mathcal Q}$ into ${\mathbf R}$, and let $\alpha\in Q\otimes\End_{\mathbb C}
R$ be the $\Gamma$-invariant element which corresponds via the
isomorphism~\eqref{eq:hom-rep}. The group $\Gamma$ acts on $\End_{\mathbb C} R$
in the natural way, i.e.\xspace by conjugation, and the element $\alpha $
decomposes with respect to the basis $q_i$ into endomorphisms $\alpha
_i\in\End_{\mathbb C} R$ for $i=1,\dots, n$ which satisfy the following
equivariance condition
\begin{equation}\label{eq:alpha_equivariance_condition}
\sum_l \gamma_{kl} \alpha_{l} =
\varphi(\gamma)\alpha_{k}\varphi(\gamma)^{-1},\rlap{$\qquad \forall
k,\gamma$,}
\end{equation}
where $\boldsymbol{\gamma}=(\gamma_{kl})$ is the matrix corresponding to
the
action of the element $\gamma$ on $Q$ with respect to the basis
$\{q_l\}_{l=1}^n$.
The commutation relations are defined by the condition
\begin{equation}
\label{eq:alpha_commute}
[\alpha _i,\alpha _j]=0.
\end{equation}
Thus the variety ${\mathcal M}_{{\mathcal Q},{\mathcal K}}({\mathbf R})$ of representations of ${\mathcal Q}$ into
${\mathbf R}$ satisfying these relations coincides with the variety ${\mathcal N}^\Gamma$
defined in~\cite{sacha:ale,sacha:thesis}.
\subsection{Representation Moduli}
\label{sec:1:mckay:moduli}
Given $\zeta\in\mathfrak{k}$, the representation moduli of the McKay
quiver of $(\Gamma,Q)$ in the multiplicity space ${\mathbf R}$ for the regular
representation $R$ of $\Gamma$ are defined by the GIT quotients
$${\mathcal M}_\zeta= \Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R}) \gitquot\zeta \PGL({\mathbf R}).$$
Since the group $\PGL({\mathbf R})$ coincides with the group $\PGL^\Gamma(R)$ (the
projectivization of the group of $\Gamma$-invariant linear
endomorphisms of $R$), one sees that the moduli ${\mathcal M}_\zeta$ coincide
with the moduli $X_\zeta$ of Hermitian-Yang-Mills type bundles defined
and studied in~\cite{sacha:ale}.
\part{Abelian Groups}
\label{sec:2}
In Part~\ref{sec:1} the moduli $X_\zeta$ were shown to coincide with
the moduli of Hermitian-Yang-Mills bundles defined in~\cite{sacha:ale}
and hence ($\Gamma$ acts freely outside the origin) to provide partial
resolutions $\rho_\zeta\colon X_\zeta \to X_0=Q/\Gamma$ of the
isolated quotient singularity.
The focus from now on will be the moduli $X_\zeta$ in the case where
$\Gamma$ is an abelian group (of order $r$) acting linearly on
$Q\cong{\mathbb C}^n$. (Later we shall specialise to the cyclic group of order
$r$.)
When $\Gamma$ is abelian, the singularity $X$ is toric and can be
resolved within the toric category, in general in many ways. In fact,
as shown in the first section below, the space of representations
$\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R})$ and its quotients $X_\zeta$ are also
toric varieties; in toric notation (reviewed in~\ref{sec:2:abel:toric})
$$\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R}) = \protect\overline{T}{}^{\Lambda,C}\text{ and }X_\zeta =
\protect\overline{T}{}^{\Pi,C_\zeta},$$
where $\Lambda,\Pi$ are certain
lattices, and where the $n$-dimensional convex polyhedra $C_\zeta$ are
obtained by `slicing' the $n+r-1$-dimensional cone $C$ by affine
$n$-planes.
The rest of this paper is devoted to describing the slices $C_\zeta $,
and in particular their extreme points and their tangent cones. The
number of extreme points is the Euler characteristic of $X_\zeta $,
and the geometry of the tangent cones of $C_\zeta $ at these points
describes the singularities of $X_\zeta $.
There are two main steps in the study of the slices $C_\zeta $. The
first step is to reduce their description to a network flow problem on
the McKay quiver. The particular problem turns out to be a
generalisation of the classical transportation
problem~\cite{kenn_helg,gond_mino:graphs}. This is done
in Section~\ref{sec:2:abel:toric}.
The second step is to describe the polyhedra which solve the
generalised transportation problem. A solution for a general quiver
is given in Section~\ref{sec:2:flow}. The main result is
Theorem~\ref{thm:general}, which says that the flows which correspond
to the extreme points of $C_\zeta$ are precisely the ones whose
support contains no cycles of non-zero type in its ``closure'' (see
\*2:flow:cycle* for the definition). This is then applied to the case
of the McKay quiver in Sections~\ref{sec:2:exact} and~\ref{sec:2:sing} to
describe the slices $C_\zeta $ for all $\zeta$.
The last section in this part chapter concerns various conjectures, in
particular for singularities of dimension~3. One hope is that, in
general, the quotients $X_\zeta $ provide a class of partial
desingularisations of the isolated quotient singularity $X$, which
will be in some sense ``natural,'' and, in good cases,
non-singular and minimal with respect to this
property. This is motivated by the fact that
Kronheimer~\cite{kron:thesis} has shown that for the case $\Gamma
\subset\SU(2)$, the $X_\zeta $ coincide with the minimal smooth
resolution of the singularity for generic
values of $\zeta$. One candidate for a ``good case'' is the case when
$\Gamma\subset\SU(3)$.
\begin{conj}
If $\Gamma\subset\SU(3)$ is abelian and acts on ${\mathbb C}^3$ freely
outside the origin then $\rho_\zeta\colon X_\zeta \to X_0={\mathbb C}^3/\Gamma$
is crepant and is a smooth resolution for generic values of $\zeta$.
\end{conj}
Using the description of $C_\zeta$, this conjecture is reduced in
Section~\ref{sec:2:crep} to to a statement concerning the existence
and combinatorial properties of certain trees in the McKay quiver.
Brute-force computer calculations show the conjecture to be true for
the singularities $\qsing 1/6(1,2,3)$, $\qsing 1/7(1,2,4)$, $\qsing
1/8(1,2,5)$, $\qsing 1/9(1,2,6)$, $\qsing 1/10(1,2,7)$ and $\qsing
1/11(1,2,8)$, but there is as yet no general proof. Other conjectures
regarding, for example, the Euler number of the resolutions also
translate to graph theoretical and combinatorial statements on the
McKay quiver.
\section{Summary of Notation for Part \ref{sec:2}}
\label{sec:2:notation}
\begin{Pentry}
\item[$\Gamma$] Finite abelian group $\Gamma$ of order $r$ (usually
$\mu_r$).
\item[$\mu_{r}$] Group of $r$-th roots of unity in ${\C^*}$.
\item[$\widehat\Gamma$] Character group $\widehat\Gamma:=\Hom(\Gamma,{\C^*})$.
\item[$\rho$] The morphism giving the action of $\Gamma$ on ${\C^*}^n$:
$\rho\colon\Gamma\to{\C^*}^{n}$.
\item[$w_{i}$] Weights ($w_i\in \Z_{r}; i=1,\dots,n$) for the action of
$\Gamma$ on ${\mathbb C}^{n}$.
\item[$\hat{\rho}$] Dual morphism $\hat\rho\colon\Z^{n}\to\widehat\Gamma$.
\item[$\Pi$] A sub-lattice of $\Z^{n}$ of index
$r$) given by $$\Pi := \ker\hat\rho= \{ x\in \Z^n | \sum x_iw_i\equiv 0\pmod
r\}.$$
\item[$\qsing1/r(w_1,\dots,w_n)$] The quotient singularity
${\mathbb C}^n/\mu_r$, where $\mu_r$ acts on ${\mathbb C}^n$ with weights $(w_1,\dots,w_n)$.
\item[$R_{v}$] One-dimensional representation on which
$\Gamma$ acts by $\lambda\to\lambda^{v}$.
\item[$\nu$] Natural morphism $\nu\colon\Gamma\to\PGL({\mathbf R})$.
\item[$\hat{\nu}$] Dual morphism $\hat\nu\colon\Lambda^{0,0}\to\widehat\Gamma$.
\item[$a_{v}^{i}$] Arrow of type $i$: $a_{v}^{i}:=v\to v-w_{i}$ ($v\in{\mathcal Q}_{0}$,
$i\in\{1,\cdots,n\}$).
\item[$\tilde{\alpha}(a_{v}^{i})$] Value of $\tilde{\alpha}$ on
$a_{v}^{i}$; Equal to the $(v-w_{i},v)$-th entry of the matrix
$\alpha$.
\end{Pentry}
\subsection{Toric Geometry}
\begin{Pentry}
\item[$M$] Generic lattice.
\item[$M_{\Q}$] Associated rational vector-space.
\item[$T^{M}$] Algebraic torus $T^{M}:=\Spec{\mathbb C}[M]$.
\item[$P$] Generic polyhedron in $M_{\Q}$.
\item[$\protect\overline{T}{}^{M,P}$] Quasi-projective toric variety
$\Proj{\mathbb C}[\widetilde{P}\cap\widetilde{M}]$ defined by $M$ and $P$.
\item[$\widetilde{M}$] Product lattice $\widetilde{M}:=\Z\times M$
\item[$\widetilde{P}$] (Closure of) the cone over $P$: $\widetilde{P} :=
\overline{\Q_{\ge 0}(\{1\}\times P)}\subset M_{\Q}$.
\item[$T_{F}P$] Tangent cone of $P$ at face $F$.
\end{Pentry}
\subsection{Flows}
\begin{Pentry}
\item[$k^A$] Set (lattice, vectorspace, cone) of maps $A\to k$. Used for
$k=\Z,\Z_+, {\mathbb R}, {\mathbb R}_+, {\mathbb C}$, and $A={\mathcal Q}_0, {\mathcal Q}_1, S$.
\item[$f$] Generic flow (element of ${\mathbb R}^{{\mathcal Q}_{1}}$).
\item[$f^{\pm}$] Positive/negative part of $f$.
\item[$\Lambda^{1}$] Lattice $\Z^{{\mathcal Q}_{1}}$ of integer-valued flows on
${\mathcal Q}$.
\item[$\Lambda^{1}_{{\mathbb R}}$] Vectorspace of real-valued flows on ${\mathcal Q}$.
\item[$\Lambda^{1}_{{\mathbb R}_{+}}$] First quadrant in $\Lambda^{1}_{{\mathbb R}}$
(non-negative real-valued flows).
\item[$\chi_{a}$] Flow taking the value 1 on $a$ and zero elsewhere.
Basis element of $\Lambda^{1}$.
\item[$\chi_{v}^{i}$] Alternative notation for $\chi_{a_{v}^{i}}$.
\item[$\Lambda^{0}$] Lattice $\Z^{{\mathcal Q}_{0}}$ of assignments of integers
to the vertices of ${\mathcal Q}$; $\Lambda^0_{\mathbb R}$ coincides with $\Lie \GL({\mathbf R})$.
\item[$\Lambda^{0,0}$] Sub-lattice $\{\zeta\in\Z^{{\mathcal Q}_{0}}|
\sum_{v\in{\mathcal Q}_{0}}\zeta(v)=0\}$ of $\Lambda^0$; $\Lambda^{0,0}_{\mathbb R}$
coincides with $\Lie \PGL({\mathbf R})$.
\item[$\zeta$] Element of $\Lambda^{0,0}$.
\item[$\In(v)$] Set of arrows $a$ such that $h(a)=v$.
\item[$\Out(v)$] Set of arrows $a$ such that $t(a)=v$.
\item[$\partial$] Natural map
$\partial\colon\Lambda^{1}\to\Lambda^{0,0}$ which calculates the net
contribution of a flow at each vertex:
$$\partial f(v)=\sum_{a\in\In(v)} f(a) - \sum_{a\in\Out(v)} f(a).$$
\item[$\chi_{v}$] Function taking the value 1 on $v$ and zero
elsewhere. Basis element of $\Lambda^{0}$.
\item[$\Lambda^{2}$] Sub-lattice of $\Lambda^{1}$ generated by the
elements corresponding to the commutation relations
$$\chi_v^i +\chi_{v- w_i}^j-\chi_v^j-\chi_{v- w_j}^i,\qquad\text{for
} i,j=1,\dots, n$$
\item[$\Lambda$] Quotient lattice $\Lambda^{1}/\Lambda^{2}$.
\item[$C$] Image of cone $\Lambda^{1}_{{\mathbb R}_{+}}$ inside the quotient
$\Lambda$. Also written $\Lambda_{{\mathbb R}_{+}}$.
\item[$C_{\zeta}$] Convex polyhedron obtained by slicing $C$ with the
hyper-plane $\partial f=\zeta$. Equal to the image of $F_{\zeta}$
under $\pi$.
\item[$\pi$] Map $\pi\colon{\mathcal Q}_1\to\{1,\dots,n\}$ which assigns to each
arrow of the McKay quiver its type: $\pi(a^i_v) := i$. Induces a
map $\pi\colon\Lambda^{1}_{{\mathbb R}_{+}}\to{\mathbb R}^{n}$ on the flows:
$$\pi(f)=\left(\sum_{v\in{\mathcal Q}_{0}} f(a_{v}^{1}),\dots,\sum_{v\in{\mathcal Q}_{0}}
f(a_{v}^{n})\right).$$
\end{Pentry}
\subsection{Configurations}
\begin{Pentry}
\item[$S$] Configuration of arrows (non-empty subset of ${\mathcal Q}_{1}$).
\item[$\supp f$] Support of the flow $f$ (arrows on which $f$ takes a
non-zero value).
\item[${\mathcal C}$] Set of all configurations of arrows.
\item[${\mathcal T}$] Set of all configurations of arrows which are trees (i.e.\xspace
which contain no cycles).
\item[$F_{\zeta}$] Admissible (i.e.\xspace non-negative) flows which satisfy
$\partial f=\zeta$.
\item[$F_{\zeta}(S)$] Flows satisfying $\partial f=\zeta$ which are
non-negative outside $S$. (A convex cone if $\zeta=0$).
\item[$Z_{\zeta}(S)$] Flows satisfying $\partial f=\zeta$ which are
zero outside $S$. (Equal to the maximal subspace contained in
$F_{0}(S)$ if $\zeta=0$).
\item[$f_{\zeta}$] Map $f_{\zeta}\colon{\mathcal T}\to\Lambda^{1}_{{\mathbb R}}$
assigning to each tree $T$ the unique flow $f=f_{\zeta}(T)$ such
that $\partial f=\zeta$.
\item[${{\Lambda^{0,0}_{{\mathbb R}}}}(T)$] Open convex cone in $\Lambda^{0,0}_{{\mathbb R}}$ of values
of $\zeta$ for which the tree $T$ is the support of some flow in
$F_{\zeta}$.
\item[${\mathcal D}$] Generic subset of ${\mathcal C}$.
\item[${\mathcal D}_{\zeta}$] Elements of ${\mathcal D}$ which are the support of some
$f\in F_{zeta}$.
\item[${\mathcal D}_{\spn}$] Elements of ${\mathcal D}$ which span all vertices of
${\mathcal Q}$.
\item[${\mathcal D}^{k}$] Elements of ${\mathcal D}$ of rank $k$ (i.e.\xspace elements $S$ such that $\rk \pi F_{0}(\overline{S})=k$.
\end{Pentry}
\subsection{Cycles and Paths}
\begin{Pentry}
\item[$p$] Generic path in ${\mathcal Q}$.
\item[$p^{+}$] Positive part of $p$.
\item[$p^{-}$] Negative part of $p$.
\item[$\tilde{\chi}_{p}$] Basic flow associated to the path $p$.
\item[$c$] Generic cycle in ${\mathcal Q}$.
\item[$\{v\}(j_{0},\dots,j_{k})$] Notation for basic flows.
\end{Pentry}
\subsection{Miscellaneous}
\begin{Pentry}
\item[${\mathcal W}_S$] Weighting of the vertices of ${\mathcal Q}_{0}$ (a map
${\mathcal Q}_{0}\to\Z^{n}$ with special properties).
\end{Pentry}
\section{Representations and Relations in the Abelian Case}
\label{sec:2:abel}
\subsection{The McKay Quivers}
\label{sec:2:abel:mckay}
In the abelian case, all the irreducible representations are
one-dimensional and are determined by an element of the dual group
$\widehat\Gamma$, which is a finite abelian group isomorphic to $\Gamma$. Thus
the vertex set of ${\mathcal Q}$ is nothing but $\widehat\Gamma$. The group $\Gamma$ will
always be identified with a subgroup of ${\mathbb C}^{*n}\subset\GL(n)$, and thus
$\widehat\Gamma$ will be thought of as a product of finite groups $\Z_{r_i}$ with the group
operation denoted additively.
\begin{example}
\label{ex:5123}
Let $\Gamma=\mu_5\subset{\C^*}$, the group of $5$-th roots of unity, acting
on ${\mathbb C}^3$ with \emph{weights} $1,2$ and $3$, i.e.\xspace via
$$\lambda\to
\begin{pmatrix}
\lambda^1 & 0 & 0 \\
0 & \lambda^2 & 0 \\
0 & 0 & \lambda^3
\end{pmatrix}.
$$ This action is denoted symbolically by $\frac{1}{ 5}(1,2,3)$. The
character group of $\mu_5$ is $\widehat{\mu_5}=\Z_5:=\Z/5\Z $, so
${\mathcal Q}$ has $5$ vertices. For each $i\in\Z_5$, let $\chi_i\colon
\lambda\mapsto\lambda^i$ be the corresponding character, so that
$\chi_i\chi_j=\chi_{i+j}$. Writing $R_i$ for the irreducible
representations, one has $R_i\otimes R_j = R_{i+j}$. The representation
$Q$ is just $R_1\oplus R_2\oplus R_3$, so that $$Q\otimes
R_i=R_{i+1}\oplus R_{i+2}\oplus R_{i+3}.$$ Thus, the quiver has arrows
from the vertex $i$ to the vertices $i-1, i-2$ and $i-3$ for each
$i$ (see Figure~\ref{fig:all5123} below).\footnote{The additions are modulo 5.}
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 3cm \epsfbox{fig/all5123.eps}
\end{center}
\caption{The McKay quiver for the action $\frac{1}{ 5}(1,2,3)$}
\label{fig:all5123}
\end{figure}
\end{example}
In fact, it is easy to see that the McKay quiver for the action of any
cyclic group has a similar appearance. To see what the arrows are, decompose $Q$ into a sum of
one-dimensional irreducibles
\begin{equation}
\label{eq:qdecomp}
Q=\bigoplus_{i=1}^n R_{w_i},
\end{equation}
where $w_i\in\widehat\Gamma$ for $i=1,\dots, n$ are the \emph{weights} of the action of $\Gamma$ on $Q$. Using
~\eqref{eq:irred_decomp}, one sees easily that
\begin{equation}
Q\otimes R_v = R_{v+w_1}\oplus \dots\oplus R_{v+w_n},
\label{eq:qtensorrv}
\end{equation}
so there is one arrow from $v$ to $v-w_i$ for each vertex $v$ and
each weight $w_i$, giving a total of $nr$ arrows. The arrows
corresponding to the weight $w_i$ are written $$a_v^i :=
v\to v -w_i,\quad \text{for } v\in{\mathcal Q}_0$$ and are said to be of
\emph{type}~$i$.
The McKay quiver for a general abelian group $\Gamma$ can be obtained by
decomposing $\Gamma$ into products of cyclic groups. Define the
\emph{product} of ${\mathcal Q}$ and ${{\mathcal Q}'}$ to be the quiver
with vertices ${\mathcal Q}_0\times{{\mathcal Q}'}_0$ and with arrows
\begin{multline}
\{(v,t(a'))\to (v,h(a')) : v\in{\mathcal Q}_0,a'\in{{\mathcal Q}'}_1\}\\
\cup \{( t(a),v')\to (h(a),v') : v'\in{{\mathcal Q}'}_0,a\in{\mathcal Q}_1\}.
\end{multline}
Then the McKay quiver for $\Gamma$ is given by taking the product of the
quivers for the cyclic factors.
\subsection{Commutation Relations}
\label{sec:2:abel:comm}
Pick a basis $q_i$ of $Q$ such that the action of $\Gamma$ on $Q$
is diagonal, with $q_i$ corresponding to the irreducible component
$R_{w_i}$. Then decomposing $R$ into irreducibles and using Shur's Lemma
\begin{equation}
\begin{split}
\Hom_\Gamma(R,R_{w_i}\otimes R)
&= \bigoplus_{v,v'\in{\mathcal Q}_0}\Hom_\Gamma({{\mathbf R}_v}\otimes R_v,{\mathbf R}_{v'}\otimes
R_{w_i}\otimes R_{v'})\\
&= \bigoplus_{v\in{\mathcal Q}_0\phantom{v'}} \Hom({{\mathbf R}_v},{\mathbf R}_{v-w_i })
\end{split}
\label{eq:homga}
\end{equation}
This means that the scalar map $\tilde \alpha(a_v^i)\colon {\mathbf R}_{v}\to {\mathbf R}_{v-w_i
}$ is multiplication by the $(v-w_i,v)$-th entry of the $i$-th
component matrix $\alpha_i\in\End R$. The condition
$[\alpha_i,\alpha_j]=0$ therefore translates to the commutation relation
\begin{equation}
\label{eq:comrelations2}
\tilde \alpha(a_v^i)\tilde \alpha(a_{v-w_i }^j)-\tilde \alpha(a_v^j)\tilde \alpha(a_{v-w_j }^i)=0
\end{equation}
for all $v\in{\mathcal Q}_0$ and $i,j\in\{1,\dots, n\}$.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 3cm \epsfbox{fig/comm7123.eps}
\end{center}
\caption[A picture of the commutation
relation~\eqref{eq:comrelations2}]{A picture of the commutation
relation~\eqref{eq:comrelations2} for $v=0$, $i=1$ and $j=3$ in the
McKay quiver for $\frac{1}{ 7}(1,2,3)$. (The product of the representation along the continuous arrows must equal the product along the dotted arrows.)}
\label{fig:comm7_123}
\end{figure}
\subsection{Toric Descriptions of the Representation Moduli}
\label{sec:2:abel:toric}
\subsubsection{Toric Varieties}
Let $M$ be an integral lattice isomorphic to $\Z^n$, let
$M_\Q:=M\otimes\Q$ be the associated rational vector
space, and let $C\subset M_\Q$ be a convex $n$-dimensional cone.
Let $T^M:=\Spec{\mathbb C}[M]$ be the algebraic torus whose
character group is $M$. Then
$$\protect\overline{T}{}^{M,C}:=\Spec{\mathbb C}[C\cap M]$$
is a compactification of $T^M$, called
the \emph{toric variety associated to the cone} $C$ with respect to the
lattice $M$. The covariant notation $\protect\overline{T}{}_{M^\vee, C^\vee}$ which uses
the dual objects\footnote{The dual of a cone $C$ is the cone $C^\vee
:= \{n\in M^\vee | n(c) \geq 0, \forall c\in C\}$} is widely in use in the literature, but the
contravariant version is more convenient for the purposes of this
paper.
This definition can be extended from cones to general convex polyhedra
as follows. If $P$ is any convex polyhedron in $M_\Q$, denote by
$\widetilde{P}$ the closure of the cone over $P$, namely
\begin{equation}
\label{eq:Ptw}
\widetilde{P} := \overline{\Q_{\ge 0}(\{1\}\times P)} \subset \widetilde{M}_\Q=\Q\times
M.
\end{equation}
Then one can define:
\begin{equation}
\label{eq:toric-defn}
\protect\overline{T}{}^{M,P}:=\Proj{\mathbb C}[\widetilde{P}\cap\widetilde{M}],
\end{equation}
and this is easily seen to extend the previous definition for cones.
\subsubsection{GIT quotients of Toric Varieties}
Note that there is a natural line bundle $L^{M,P}:={\mathcal O}(1)$ on $\protect\overline{T}{}^{M,P}$
on which $T^M$ acts and whose sections are given simply by evaluation
on the lattice points of $P$. If one translates $P$ by an element
$\zeta\in M$, one gets the same toric variety
($\protect\overline{T}{}^{M,P+\zeta}=\protect\overline{T}{}^{M,P}$) and the same line bundle $L$, but with a
\emph{different \/} linearisation of the action of $T^M$, differing
from the old by the character $\zeta$.
The consequence of this for GIT quotients is that if $M'$ is any
sub-lattice of $M$, and if $\zeta$ an element of $M'$ (and hence a
character of $T^{M'}$) one can consider either the quotient
$$\protect\overline{T}{}^{M,P}\gitquot{\zeta}T^{M'}$$
with $T^{M'}$ taken to act on $L^{M,P}$ via multiplication by the
section $\zeta$ or, equivalently, the quotient
$$T^{M,\zeta+P}\gitquot{} T^{M'}$$ where $T^{M'}$ now acts on
$L^{M,\zeta+P}$ in the ordinary way.
Using this notation, one can state the following proposition which
describes how toric varieties behave under GIT quotients
(c.f.\xspace~\cite{thaddeus:flips}).
\begin{prop}
\label{prop:toric_quotients}
Let $0\to M''\to M\to M'\to 0$ be an exact sequence of lattices and
$P\subset M_\Q$ be a convex polyhedron. Write
$$(M,P)\gitquot{}M' :=
(M'',P\cap M_\Q'').$$ Then
$$\protect\overline{T}{}^{ M,P}\gitquot{} T^{ M'} = \protect\overline{T}{}^{(M,P)\gitquot{}M'}.$$
\end{prop}
\begin{proof}
An element $x^m\in{\mathbb C}[ M\cap P]$ is invariant under $T^{ M'}$ if $m$
belongs to the stabiliser of $T^{ M'}$ in $T^M$, which is nothing
but $M''$. Thus
${\mathbb C}[\widetilde{P}\cap \widetilde{M}]^{T^ {M'}} ={\mathbb C}[\widetilde{P}\cap\widetilde{M''}]= {\mathbb C}[\widetilde{(P\cap M_\Q'')}\cap\widetilde{M''}]$ and the result follows by taking
$\Proj$.
\end{proof}
\subsubsection{The Representation Variety}
The description of the previous sections shows that
$\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R})$ is a subvariety of ${\mathbb C}^{{\mathcal Q}_1}$ defined by the
equations~\eqref{eq:comrelations2}. The goal is now to describe the quotients
by the group $\PGL({\mathbf R})$. It turns out that $\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R})$ is
a toric variety and that $\PGL({\mathbf R})$ is an algebraic sub-torus, so that one is
able to use Proposition~\ref{prop:toric_quotients}.
Consider the lattice $\Lambda ^1:= \Z^{{\mathcal Q}_1}$ and the
first quadrant $\Lambda ^1_{{\mathbb R}+}:={\mathbb R}_+^{{\mathcal Q}_1}$ inside its associated
real vector-space $\Lambda^1_{\mathbb R}$. Recall that the toric variety which
corresponds to ${\mathbb C}^{{\mathcal Q}_1}$ is
$$\protect\overline{T}{}^{\Lambda ^1,\Lambda ^1_{{\mathbb R}+}} := \Spec {\mathbb C}[\Lambda ^1_+],$$
where $\Lambda ^1_+$ is the semi-group $\Z_+^{{\mathcal Q}_1}$ and ${\mathbb C}[\Lambda
^1_+]$ denotes its group algebra. More precisely, the ${\mathbb C}$-points of
the scheme $\Spec {\mathbb C}[\Lambda ^1_+]$ are into one-one correspondence
with the points of ${\mathbb C}^{{\mathcal Q}_1}$ as follows. If $x\colon {\mathbb C}[\Lambda
^1_+]\to{\mathbb C}$ is an algebra homomorphism representing a point of $\Spec
{\mathbb C}[\Lambda ^1_+]$ then, evaluating it on the generators of $\Lambda
^1_+$ gives a map ${\mathcal Q}_1\to {\mathbb C}$, i.e.\xspace a point of ${\mathbb C}^{{\mathcal Q}_1}$.
Conversely, if $\tilde \alpha \in{\mathbb C}^{{\mathcal Q}_1}$, there is an induced morphism
of semi-groups
\corresp{(\Lambda ^1_+,+)}{({\mathbb C},\cdot)}{f}{\prod_a(\tilde \alpha (a))^{f(a)},}
and this induces an algebra morphism ${\mathbb C}[\Lambda ^1_+]\to{\mathbb C}$.
Under this identification, the points of $\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R})$ correspond to the
semi-group morphisms $ \Lambda ^1_+ \to{\mathbb C}$ which are the identity when
restricted to the
sub-semi-group $\Lambda ^2\cap\Lambda ^1_+$ of $\Lambda ^1_+$. Here
$\Lambda ^2$ is the sub-lattice of $\Lambda ^1$ generated by the
elements
$$\chi_v^i +\chi_{v- w_i}^j-\chi_v^j-\chi_{v- w_j}^i,\qquad\text{for }
i,j=1,\dots, n$$
(and $\chi_v^i$ denotes the indicator function
$\chi_{a_v^i}$ for the element $a_v^i$, i.e.\xspace the basis element of
$\Lambda^1$ which takes the value $1$ on the arrow $a_v^i=v\to
v-w_i\in {\mathcal Q}_1$ and zero elsewhere).
\begin{prop}
\label{prop:rep-variety-toric}
The quotient $\Lambda:=\Lambda ^1/\Lambda ^2$ is a lattice (i.e.\xspace
is torsion free). If one denotes by $\Lambda _+$ (resp.\xspace
$C=\Lambda_{{\mathbb R}+}$) the lattice (resp.\xspace the cone) generated by the
image of the elements of $\Lambda ^1_+$ in $\Lambda$, then the
variety of representations of the McKay quiver with
relations into ${\mathbf R}$ is given by
$$\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R}) = \protect\overline{T}{}^{\Lambda,C} = \Spec {\mathbb C}[\Lambda _+].$$
\end{prop}
\begin{proof}
The only fact which has to be proved is that $\Lambda $ is a
lattice. This will be postponed till
Lemmas~\ref{lemma:kerc_lambda2}--\ref{lemma:imagec}.
\end{proof}
The group $\GL({\mathbf R})$ on the other hand coincides with
${\C^*}^{{\mathcal Q}_0}$. If $\Lambda^0$ denotes the lattice
$\Z^{{\mathcal Q}_0}$, then one can write
$\GL({\mathbf R})=T^{\Lambda^0}$ in the notation of Section~\ref{sec:2:abel:toric}.
Similarly, writing $\Lambda^{0,0}$ for the
sub-lattice of $\Lambda^0$ whose elements $\zeta$ satisfy
$\sum_{v\in{\mathcal Q}_0}(v)=0$, one has $\PGL({\mathbf R})= T^{\Lambda^{0,0}}$.
Applying Proposition~\ref{prop:toric_quotients}, one obtains the
following toric description of the moduli $X_\zeta$.
\begin{thm}
\label{thm:toric-d-moduli}
The moduli spaces $X_\zeta$ are quasi-projective toric varieties
$$X_\zeta = \protect\overline{T}{}^{\Pi ,C_\zeta },$$
where
$\Pi:=\ker\hat\rho=\pi\times\partial(\Lambda)/\Lambda^{0,0}$ is a
sub-lattice of $\Z^n$ of index $\card{\Gamma}$ and $C_\zeta$ are convex
polyhedra in $\Pi_{\mathbb R}$ in obtained by slicing the cone $C$ with the
affine planes $\zeta +\Lambda ^{0,0}_{\mathbb R}$.
\end{thm}
\begin{proof}
By definition
$$X_\zeta=\protect\overline{T}{}^{\Lambda,C}\gitquot{\zeta}T^{\Lambda^{0,0}},$$
and this
coincides with $$\protect\overline{T}{}^{\Lambda,\zeta+C}\gitquot{}T^{\Lambda^{0,0}}.$$
If $\Lambda':=\pi\times\partial(\Lambda)/\Lambda^{0,0}$ was a
lattice then applying Proposition~\ref{prop:toric_quotients} would
give $X_\zeta = \protect\overline{T}{}^{\Lambda' ,C_\zeta }$.
It therefore remains to prove that $\Lambda'=\Pi$. This is done in
Section~\ref{sec:2:exact}:
Lemmas~\ref{lemma:kerc_lambda2}--\ref{lemma:imagec} show that there
is an exact sequence of abelian groups
\begin{equation}
\label{eq:exact1}
0\to\Lambda\xrightarrow{\pi\times\partial} \Z^n\times\Lambda^{0,0}
\xrightarrow{\hat\rho-\hat\nu} \Z^n/\Pi \cong\widehat\Gamma \to 0.
\end{equation}
\end{proof}
Note that the morphisms appearing in the exact sequence in the above
proof are:
\begin{itemize}
\item The projection $\pi\colon\Lambda^1\to\Z^n$ (or better, its descent to
$\Lambda\to\Z^n$).
\item The (descent to $\Lambda$) of the morphism of lattices
$\partial\colon\Lambda^1\to\Lambda^{0,0}$ dual to the action of
$\PGL({\mathbf R})=T^{0,0}$ on $\Rep_{{\mathcal Q}}({\mathbf R})={\mathbb C}^{{\mathcal Q}_1}$ given by
\map{\Hat\partial}{T^{0,0}}{T^1}{\lambda}{a\mapsto\lambda_{t(a)}^{-1}\lambda_{h(a)}.}
Here $\lambda_v$ denotes the component of
$\lambda\in T^{0,0}$ corresponding to the vertex $v\in{\mathcal Q}_0$.
\item The morphism of lattices
\map{\hat\nu}{\Lambda^{0,0}}{\widehat\Gamma}{\zeta}{\sum_{v\in{\mathcal Q}_0}\zeta(v)v}
dual to the action of \/$\Gamma$ on $\End R$ by conjugation.
\item The morphism of
lattices\map{\hat\rho}{\Z^n}{\widehat\Gamma}{\zeta}{\sum_{v\in{\mathcal Q}_0}x_iw_i}
dual to the action $\rho\colon \Gamma\to {\C^*}^n$.
\end{itemize}
The fact that $\pi\times\partial$ descends to $\Lambda$ corresponds to
the fact that $\GL^\Gamma(Q)={\C^*}^n$ and $\PGL({\mathbf R})=T^{0,0}$ act on
$\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R})$ rather than simply on
$\Rep_{{\mathcal Q}}({\mathbf R})={\mathbb C}^{{\mathcal Q}_1}$. On the other hand, the fact that their
image maps into $\ker (\hat\rho-\hat\nu)$ corresponds to the fact that
$\Rep_{{\mathcal Q},{\mathcal K}}({\mathbf R})$ can be identified with the $\Gamma$-invariant
part of $\Hom(R,Q\otimes R)$.
The action of $T^\Pi$ arises from the existence of the map
$$\pi\colon{\mathcal Q}_1\to \{1,\dots, n\},$$ which assigns to each arrow its
type.
This induces a $\Z$-linear map $\pi
\colon {\Lambda^1}\to \Z^n$, which is denoted by
the same letter, and one obtains an action of ${\C^*}^n$ on
${\mathbb C}^{{\mathcal Q}_1}$ via the corresponding morphism of algebraic tori:
$\tau\cdot\tilde \alpha:=\Hat\pi (\tau)\tilde \alpha$.
Explicitly, for $\tau\in{\C^*}^n$, $\tilde \alpha \in{\mathbb C}^{{\mathcal Q}_1}$, and
$a\in{\mathcal Q}_1$,
$$ (\tau\cdot \tilde \alpha)(a)
=\tau_{\pi(a)}\tilde \alpha(a),
$$
where $\tau\in {\C^*}^n$ has components $\tau_i:=\tau(e_i)$ with
respect to the standard basis $e_i$ of $\Z^n$, and $\chi _a$ denotes
the basis element of $\Lambda ^1$ which is the indicator function of
the singleton $\{a\}\subset{\mathcal Q}_1$. The action of ${\C^*}^n$ corresponds
to multiplying the value of $\tilde \alpha $ on all arrows $a$ with the
same $\pi(a)$ by the same factor $\tau_{\pi(a)}$. Since $\Gamma$ is
abelian, $\rho(\Gamma)\subset{\C^*}^n$ acts trivially on the
representations, and one has an action of $T^\Pi:={\C^*}^n/\rho(\Gamma)$.
According to the toric formalism, one has
$$T^\Pi = {\C^*}^n/\Image\rho = T^{\ker\hat\rho},$$
and thus $\Pi=\ker\hat\rho$.
Note also that the exact sequence implies that the cone
$C:=\Lambda_{{\mathbb R}+}$ is isomorphic to
$({\pi\times\partial})(\Lambda^1_{{\mathbb R}+})$. Therefore, one has the
following theorem.
\begin{thm}
\label{thm:slices} The slices $C_\zeta$ (which describe
the toric moduli $X_\zeta$ when $\zeta$ is integral) are given by
\begin{align}
\label{eq:czeta}
C_\zeta &\phantom{:}= \pi_{\mathbb R}(F_\zeta ),\\
\intertext{ where }
\label{eq:fzeta}
F_\zeta &:= {\mathbb R}_+^{{\mathcal Q}_1}\cap \partial^{-1}_{\mathbb R}(\zeta ).
\end{align}
\end{thm}
The next section is devoted to the study of the polyhedra $F_\zeta $
and $C_\zeta $.
\section{Flow Polyhedra}
\label{sec:2:flow}
The question of determining the polyhedra $C_\zeta$ is in fact a
generalisation of a basic problem, well-known to network optimization
specialists:
\begin{problem}[The Transportation Problem ({\protect\cite{kenn_helg,gond_mino:graphs}})]
\label{pb:transp}
Given a quiver --- a ``network" in optimization parlance --- ${\mathcal Q}$
and an element $\zeta\in{\mathbb R}^{{\mathcal Q}_0}$ specifying given demands and
supplies of some commodity at the vertices, find
all\footnote{Actually, the interest in the transportation problem
usually centers around finding the extreme flow which minimizes a
certain cost function, not in finding \emph{all\/} extreme flows.
Furthermore, there are usually capacity constraints for the maximal
flow allowed along each arrow.} the non-negative elements
$f\in{\mathbb R}_+^{{\mathcal Q}_1}$, representing flows of commodities along each
arrow, whose net contribution at each vertex balances the demands and supplies specified by $\zeta$.
\end{problem}
Writing $\In(v)$ ($\Out(v)$) for the arrows
which have their head (tail) at a given vertex $v$, then
$\partial\colon{\mathbb R}^{{\mathcal Q}_1}\to{\mathbb R}^{{\mathcal Q}_0}$ is given by
$$\partial f(v)=\sum_{a\in\In(v)} f(a) - \sum_{a\in\Out(v)} f(a).$$
The element $\partial f$ is called the \emph{net contribution} of $f$.
The flow $f\in{\mathbb R}^{{\mathcal Q}_1}$ is said to be a \emph{$\zeta$-flow} if
\begin{equation}
\partial f = \zeta.
\label{eq:demand}
\end{equation}
If $f$ is
non-negative, it is called an
\emph{admissible $\zeta$-flow}.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 3cm \epsfbox{fig/flow7123.eps}
\end{center}
\caption[An admissible flow on the McKay quiver for $\frac{1}{
7}(1,2,3)$ and its net contribution at each vertex]{An admissible
flow on the McKay quiver for $\frac{1}{ 7}(1,2,3)$. (The numbers along
the edges indicate the values of
$f$ and the numbers at the vertices
indicate the values of $\partial f=\zeta$.)}
\label{fig:flow7123}
\end{figure}
The problem is therefore to determine the set $F_\zeta$ of all
admissible flows for a given $\zeta$. Note that the solution set is
empty unless $\zeta$ satisfies $\sum_{v\in{\mathcal Q}_0}\zeta(v)=0$ (i.e.\xspace
supply = demand). Recall that the sub-space for which this holds was
denoted $\Lambda^{0,0}_{\mathbb R}$. The set $F_\zeta$ is the
intersection of the cone ${\mathbb R}_+^{{\mathcal Q}_1}$ (the first quadrant) with an
affine translate of the vector-space $\ker\partial$, so $F_\zeta$ is a convex
polyhedron; the interest lies in determining its extreme points. For
instance, if one is trying to minimize a convex cost function of the
flows, then the minimum will be attained at one of these extreme
points. A solution to this problem is given by the easy:
\begin{thm}\footnote{This theorem is the basis for the ``Simplex on a graph'' algorithm~\cite{luen}. See Proposition~3.16 in \cite{kenn_helg}, noting that trees correspond to what is called a ``basic solution'' or a ``basis'' for the linear programming problem~\cite[\S 2.3, Defn., p.17]{luen}. For the case of the permutation polytope, see~\cite[\S 5, Th.~1.1]{yem_kov_kra}}
\label{thm:classical}
The extreme points of $F_\zeta$ are precisely the admissible
$\zeta$-flows whose support contains no cycles.
\end{thm}
Here, the \emph{support} of a flow is the set of
arrows on which it takes non-zero values. A \emph{cycle} means a
sequence of arrows in ${\mathcal Q}_1$ which form a cycle in the underlying
graph to ${\mathcal Q}$, when their orientation is disregarded.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 3cm \epsfbox{fig/uncycle7123.eps}
\end{center}
\caption{A cycle in the McKay
quiver for $\frac{1}{7}(1,2,3)$}
\label{fig:uncycle7123}
\end{figure}
\section{Generalised Transportation Problem}
\label{sec:2:gtp}
In the present paper,
the interest is in the image $C_\zeta$ of
$F_\zeta$ under the projection $\pi\colon {\mathbb R}^{{\mathcal Q}_1}\to{\mathbb R}^n$, and
one is therefore let to the following slightly more general problem.
\begin{problem}
\label{pb:labeledq}
Given a quiver ${\mathcal Q}$, a natural integer $n$ and a projection
$\pi\colon{\mathbb R}^{{\mathcal Q}_1}\to{\mathbb R}^{n}$, characterize the (supports of the)
flows which, under $\pi$, map to an extreme point of $\pi F_\zeta $
for a given $\zeta$.
\end{problem}
\begin{rmk}
This problem can be thought of as a transportation problem with some
extra structure: instead of associating a cost with \emph{each\/}
arrow, one supposes that there are linear relations between the arrow
costs which are parametrised by $n$ independent variables --- in
other words, suppose that the cost function really is a (convex) function of these $n$
variables, so that the cost-minimizing flows will correspond
to extreme points of $\pi F_\zeta$.
\end{rmk}
If $f$ is any flow with $\pi(f)=x\in \pi F_\zeta $, then the support
$S$ of $f$ will be called a $\zeta$-\emph{configuration (of arrows)} for
$x$, or simply a \emph{configuration} of $x$ (there will in general
be \emph{many \/} $\zeta $-configurations for a given $x$). If $x$ is
an extreme point of $\pi F_\zeta$, then any configuration $S$ of $x$
will be called an \emph{extreme configuration}. The problem is thus to
determine all the possible extreme configurations.
\subsection{Cycle Type and Closure}
\label{sec:2:flow:cycle}
In order to state the solution to the problem, one needs to introduce
some concepts relating to cycles. Suppose that $c=(c_1,\dots, c_k)$ is
a cycle, i.e.\xspace a sequence of arrows $c_i\in{\mathcal Q}_1$ such that they form
a circuit, when suitably oriented. Consecutive arrows $c_i$'s are not allowed
to be the same (although they can join the same
vertices)\footnote{Note that it is necessary to state this explicitly
for quivers, due to the possibility of several arrows joining the
same two vertices.}. An arrow $c_i$ is said to \emph{agree} with the ordering
$c_1,\dots c_k$ if the tail of $c_i$ is an extremity of $c_{i-1}$ and
the head of $c_i$ is an extremity of $c_{i+1}$, and is said to
\emph{disagree} otherwise. Call the disjoint union\footnote{The
disjoint union is
used in case some cycles contain the same arrow twice.} of the
arrows $c_i$ whose direction agrees (disagrees) with the ordering
$c_1,\dots,c_k$ the \emph{positive\/} (\emph{negative\/}) part of $c$
and denote it by $c^+$ ($c^-$).
For each arrow $a\in{\mathcal Q}_1$, let $\chi_a$ denote the basis element of
$\Z^{{\mathcal Q}_1}$ which takes the value $1$ on the arrow $a$ and zero
elsewhere. To each cycle $c$, one can define the \emph{basic flow}
associated to $c$ by
$$\tilde\chi_c:=\sum_{a\in c^+}\chi_a - \sum_{a\in c^-}\chi_a\in\Z^n.$$
The \emph{type} of $c$ is defined to be element $\pi(\tilde\chi_c)\in\Z^n$.
Cycles of
\emph{type zero} are of special interest.
They correspond to cycles having zero total number of arrows of any
given type, the number being counted algebraically, according to
whether the arrow agrees or disagrees with the
orientation of $c$.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 3cm \epsfbox{fig/nullcycle7123.eps}
\end{center}
\caption[A cycle of type zero in the McKay quiver for $\frac{1}{
7}(1,2,3)$]{A cycle of type zero in the McKay quiver for $\frac{1}{
7}(1,2,3)$. (The numbers next to the arrows indicate their type.)}
\label{fig:nullcycle7123}
\end{figure}
The \emph{cycle closure} of a configuration $S$ is defined to be the
smallest over-set $\clos{S}$ of $S$ such that, for any cycle $c$ in
${\mathcal Q}$ of type $0$,
\begin{equation}
c^+\subseteq \clos{S}\iff c^-\subseteq \clos{S}.
\label{eq:cycle_closure}
\end{equation}
The closure of $S$ can be computed by searching for all the
cycles $c$ of type $0$ satisfying $c^+\subseteq S$, adjoining $c^-$ to
$S$, and repeating this procedure until~\eqref{eq:cycle_closure} is
satisfied. Note that, even if $S$ contains no cycles of non-zero type
(for instance, if $S$ is a tree), the arrows one adjoins may create
such cycles in $\clos{S}$.
Two configurations $S,S'$ will be called
\emph{equivalent} (written $S\sim S'$) if
$\overline{S}=\overline{S'}$.
\subsection{Statements of the Generalised Theorems}
\label{sec:2:flow:state}
The generalised version of
Theorem~\ref{thm:classical} can now be stated.
\begin{thm}[Generalised Extreme Flows]
\label{thm:general}
The extreme points of the polyhedron $\pi F_\zeta$ are the images under $\pi$ of the
admissible $\zeta$-flows whose supports have no cycles of non-zero
type in their closures. If $S$ is a such a configuration
and if there is a $\zeta$-flow with support $S$, then the image of
that flow under $\pi$ is an extreme point of $\pi F_\zeta$.
\end{thm}
Note that if $\pi$ is the identity map, then \emph{any\/} cycle is of
non-zero type and one recovers Theorem~\ref{thm:classical}.
The above theorem can be generalised to get a description of the faces
of $\pi F_\zeta $ of all dimensions. Recall that the \emph{tangent
cone\/} of $P$ at one of its faces $F$ is the convex cone $$T_F
P:={\mathbb R}_+(P-F):={\mathbb R}_+(P-f), \text{ for any }f\in \interior F,$$
where $\interior F$ denotes the relative interior of the face $F$.
For any configuration of arrows $S$ define
$$
F_\zeta(S) := \{f\in\partial^{-1}(\zeta) : a\not\in S \implies
f(a) \geq 0\}$$
and
$$Z_\zeta(S) := \{f\in\partial^{-1}(\zeta) : a\not\in S
\implies f(a) = 0\}.$$
One has $F_\zeta(\emptyset)=F_\zeta$ and
$Z_\zeta(S)=F_\zeta\cap\supp^{-1}(S)$. When $\zeta=0$, $F_0(S)$ is a
cone and $Z_0(S)$ is its maximal vector subspace. The cone
$F_0(S)$ (resp.\xspace the vector-space $Z_0(S)$) is generated
over ${\mathbb R}_+$ by the flows $\tilde\chi_c$ for cycles $c$ such that
$c^-\subseteq S$ (resp.\xspace $c\subseteq S$).
Let ${\mathcal C}$ denote the set of all \emph{configurations},\/ i.e.\xspace
the set of non-empty subsets of ${\mathcal Q}_1$. The
\emph{rank} of $S$ is defined to be the rank
of $\pi Z_0(\overline S)$. It is trivial to see that the rank
function determines a partition of ${\mathcal C}$ into non-empty disjoint sets
${\mathcal C}={\mathcal C}^0\amalg{\mathcal C}^1\amalg\dots\amalg{\mathcal C}^n$ which respects the
equivalence relation $\sim$ induced by $S\mapsto\overline{S}$. As a
further piece of notation, for any subset of ${\mathcal D}\subseteq{\mathcal C}$, denote
by ${\mathcal D}^k$ the subset of configurations in ${\mathcal D}$ which
have rank $k$. Also, write
${\mathcal D}_\zeta$ for the subset of
configurations which are
\emph{admissible for $\zeta$},\/ namely
configurations $S\in{\mathcal D}$ which arise as the support of some element in
$F_\zeta$.
Theorem~\ref{thm:general} says that the configurations corresponding
to the extreme points of $F_\zeta$ are precisely those belonging to
${\mathcal C}^0_\zeta$. In general one has the following complete description
of the extreme faces and tangent cones of $C_\zeta$:
\begin{thm}[Extreme Faces and Tangent Cones]
\label{thm:faces}
For all $\zeta$, the map
\map{\text{Face}_\zeta}{{\mathcal C}_\zeta}{\text{Faces of }\pi
F_\zeta}{S}{\pi F_\zeta \cap(\pi f+\pi Z_0(\overline{S})))} is independent of
the choice of $f\in F_\zeta\cap\supp^{-1}(S)$, and induces a bijection
$$\text{Face}_\zeta\colon {\mathcal C}^k_\zeta/\!\!\sim \xrightarrow{\;\cong\;} \text{$k$-faces of
}\pi F_\zeta.$$ Furthermore, for all $[S]\in{\mathcal C}^k_\zeta/\!\!\sim$,
$$T_{\text{Face}_\zeta(S)}\pi F_\zeta =\pi F_0(\overline{S}),$$
where the left-hand side denotes
the tangent cone to the polyhedron $\pi F_\zeta$ at the face
$\text{Face}_\zeta(S)$. In other words, $\pi F_0(\overline{S})$ gives the
tangent cone corresponding to the configuration $S$ (which is
independent of the value of $\zeta$) and $\text{Face}_\zeta$ gives the
corresponding face of $C_\zeta$ (whose direction is also independent
of $\zeta$).
\end{thm}
\begin{rmk}
Note that both the direction of the face corresponding to $S$ and
the tangent cone of $\pi F_\zeta$ at this face are independent of the
value of $\zeta$.
\end{rmk}
In fact, (see Lemma~\ref{lemma:0cycles}) $Z_0(S)$ is generated
by the flows for the cycles supported in $S$, so that
Theorem~\ref{thm:general} corresponds to the case when $S$ has rank zero.
The concepts needed for the proof of Theorem~\ref{thm:faces} are
developed in Section~\ref{sec:2:flow:basic}. The theorem is then proved
in Section~\ref{sec:2:flow:classpf} for the classical case (where
${\mathbb R}^{{\mathcal Q}_1}={\mathbb R}^n$ and $\pi\colon{\mathcal Q}_1\to{\mathcal Q}_1$ is the identity). The general case follows
easily from this and is treated in Section~\ref{sec:2:flow:genpf}.
Before this, an example of an application and several important
corollaries are given.
\section{Application to the McKay Quiver}
\label{sec:2:mckay}
\subsection{Example}
\label{sec:2:flow:examples}
Consider the McKay quiver
for the action $\frac{1}{ 5}(1,2,3)$. Recall that identifying
${\mathcal Q}_0$ with $\Z_5$, the arrows are $a_v^i := v\to v-i$ for $v\in
\Z_5$ and $i\in\{1,2,3\}$. Let $\pi $ be the map ${\mathcal Q}_1 \to\{1,2,3\}$
which assigns to each arrow $a_v^i$ its type $i$. This induces a
projection $\pi\colon {\mathbb R}^{{\mathcal Q}_1}\to{\mathbb R}^3$ which assigns to the basis element
$\chi_v^i := \chi_{a_v^i}$ the basis element $e_i$ of ${\mathbb R}^3$. With
respect to this map, the cycles of type zero are those with total
number of arrows of any given type equal to zero, where the number of
arrows is counted algebraically according to the orientation of the cycle. Consider
the configuration $T\subset{\mathcal Q}_1$ represented in Figure~\ref{fig:tree5123}.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 1in \epsfbox{fig/tree5123.eps}
\end{center}
\caption{A configuration $T\in{\mathcal C}^0$ in the McKay quiver for $\frac{1}{ 5}(1,2,3)$.}
\label{fig:tree5123}
\end{figure}
What is the closure of~$T$? A little thought
shows\footnote{See Section~\ref{sec:2:examples:comm} for more details on this.} that
$\clos{T}$ is simply $T$ itself. Since $\clos{T}$ has no cycles, $Z_0(\clos{T})=0$, so
$T\in{\mathcal C}^0$. If $T$ is admissible for $\zeta $, and $f$ is any $\zeta $-flow with support $T$, then the theorem says
that $\pi(f)$ is a
$0$-face of $\pi F_\zeta $, i.e.\xspace an extreme point. Furthermore, the
tangent cone to $\pi F_\zeta $ at $\pi(f)$ is the cone $\pi F_0(T)$; this is
generated by the types of the cycles whose negative part is contained
in $T$. These are listed in Figure~\ref{fig:cycles5123}.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 7.5cm \epsfbox{fig/cycles5123.eps}
\end{center}
\caption{The different types of cycles which generate the cone $\pi F_0(T)$ for the configuration $T$ in Figure~\ref{fig:tree5123}.}
\label{fig:cycles5123}
\end{figure}
There are four different types\nopagebreak
\begin{align*}
v_1 &= (1,1,-1)\\
v_2 &= (1,-2,1)\\
v_3 &= (-1,0,2)\\
v_4 &= (-1,3,0),
\end{align*}
and they generate $\pi F_0(T)$ as a cone. Any three of these form a basis
for the lattice $\Pi\subset\Z^3$ of index $5$ given by
$$\Pi:=\ker\hat\rho = \{(a,b,c): a+2b+3c \equiv 0\pmod 5\},$$
and
they satisfy the single relation $v_1+v_3 = v_2+v_4$. This
corresponds via the usual toric formalism to a singularity of
$X_\zeta$ of the type $xw=yz \subset{\mathbb C}^4$. The variety $X_\zeta$
therefore has such a singularity whenever the tree $T$ is admissible
for $\zeta$, i.e.\xspace whenever the flow $f_\zeta(T)$ is
positive. Writing this out explicitly, one sees that the admissible
cone for $T$, ${{\Lambda^{0,0}_{{\mathbb R}}}}(T)$
is given by the following inequalities\footnote{See
\ref{sec:2:mckay:flow:trees} for the exact definition of ${{\Lambda^{0,0}_{{\mathbb R}}}}(T)$.}
\begin{align*}
-\zeta_0 &>0\\
-\zeta_2 &>0\\
-\zeta_4 &>0\\
-\zeta_4-\zeta_1 &>0
\end{align*}
\subsection{Corollaries}
\label{sec:2:flow:cor}
\subsubsection{$k$-Adjacency}
The ``simplex on a graph''
algorithm~\cite[Alg.~3.3]{kenn_helg} can be interpreted as moving from
one extreme point of $F_\zeta$ to an adjacent one by varying the flow
along a single cycle in the network\footnote{See~\cite[\S 5.4]{evans}, or, for
the special case of the ``permutation polytope,''~\cite[\S 5,
Th.~1.3]{yem_kov_kra}.}. Geometrically this says that two extreme flows
are joined by an edge of $F_\zeta$ if and only if the union of their
supports contains only \emph{one\/} cycle (up to multiples obtained by
going around the cycle $k\in\Z$ times).
In order to generalize the notion of being joined by an edge, the
following terminology is introduced: two points of a
polyhedron are called \emph{$k$-adjacent} if they are contained in a
$k$-dimensional face, but in no face of smaller dimension. Note that
by convexity, two points are contained in the same face if
and only if their midpoint is also in that face. Since a
configuration of the midpoint is obtained by taking the union of configurations
of the two points, the answer to the question is deduced as a
corollary to Theorem~\ref{thm:faces}:
\begin{cor}[$k$-Adjacency]
\label{cor:k_adjacent}
Two points $x,x'\in\pi F_\zeta$ are $k$-adjacent if and only if for
some (and hence for any) configurations $S$ of $x$ and $S'$ of $x'$,
$${S\cup S'}\in{\mathcal C}^k,$$ i.e.\xspace their union has rank $k$.
\end{cor}
Note that the case~ $k=0$ gives the condition for two configurations
to give rise to the same extreme point. The case $k=1$ says that two
configurations are joined by an edge of~ $\pi F_\zeta $ if and only if
their union only contains cycles whose types are multiples of a fixed
element $v\in{\mathbb R}^n$. For the case when $\pi$ is the identity, this
reduces to the classical statement that the union $S\cup S'$ contains
only one cycle (up to multiples obtained by going around the cycle
$k\in\Z$ times).
\subsubsection{Trees}
\label{sec:2:mckay:flow:trees}
It can be shown easily (see Corollary~\ref{cor:spanning_trees}) that
any extreme point of~ $\pi F_\zeta$ has at least one configuration~
$T$ which is a \emph{tree}, i.e.\xspace it has no cycles. Thus, to know the
extreme points and the tangent cones of $\pi
F_\zeta$ at these points, one need only answer the question of which
trees occur as extreme configurations. Denoting the set of
trees in ${\mathcal C}$ by ${\mathcal T}$, the generalised theorem
implies that the extreme points of $C_\zeta$ are given by the images
of the flows whose supports are members of~ ${\mathcal T}^0_\zeta$. This set
is easy to determine: for a given $\zeta$, there is, on each tree $T$, a
unique flow $f_\zeta(T)\in\Lambda^1$ which meets the demand $\zeta$
and is supported on $T$; $f_\zeta(T)$ is called the \emph{($\zeta$-)flow
admitted by the tree\/}
$T$. (See \ref{sec:2:examples} for an example). This defines a map
$$f_\zeta\colon{\mathcal T}\to\Lambda^1.$$ The set $f_\zeta^{-1}(\Lambda^1_+)$ of
trees $T$ which admit non-negative $\zeta$-flows is of course
the set ${\mathcal T}_\zeta$ of \emph{admissible tree}\emph{s\/} for $\zeta $.
The classical Theorem~\ref{thm:classical} says that
$$\ext F_\zeta = f_\zeta({\mathcal T}_\zeta).$$ Similarly, the
generalised Theorem~\ref{thm:general} gives the following:
\begin{cor}
\label{cor:number_extreme_pts}
Let $f_\zeta$ be the map which assigns to each tree $T$ the unique
$\zeta$-flow with support equal to $T$. Then the extreme points of
$C_\zeta$ are given by
$$\ext \pi F_\zeta = \pi f_\zeta ({\mathcal T}^0_\zeta ),$$ where ${\mathcal T}^0 :=
{\mathcal C}^0\cap {\mathcal T}$ is the set of trees whose closures have type zero. Furthermore, Theorem~\ref{thm:faces} implies that
$$\card{\ext C_\zeta}=\card{{\mathcal C}^0_\zeta/\!\!\sim}=\card{{\mathcal T}^0_\zeta/\!\!\sim},$$
where $\sim$ is the equivalence relation induced by closure.
\end{cor}
Note (see Lemma~\ref{lemma:maximal_config} for a proof) that for the
sets $S\in {\mathcal C}^0$ one has $\pi F_0(\clos{S})=\pi F_0(S)$. This implies the
following corollary:
\begin{cor}
\label{cor:fan}
The extreme points of the polyhedron $\pi F_\zeta$ correspond to the
trees $T$ in ${\mathcal T}^0_\zeta$ and the tangent cone to $\pi F_\zeta$ at
the point corresponding to $T$ is $\pi F_0(\clos{T})=\pi F_0(T)$. Thus the
fan\footnote{Recall that the \emph{fan \/}
associated to a convex polyhedron $P$ is the collection of dual
cones to the tangent cones of $P$ at all its faces.} $\Sigma
_\zeta$ associated to the polyhedron $\pi F_\zeta $ is given by
the dual cones $\pi F_0(T)^\vee$ for the trees $T\in{\mathcal T}^0_\zeta$ and
all their faces. In fact, its $k$-skeleton, i.e.\xspace the set of its
$k$-dimensional cones, is $\Sigma _\zeta^{(k)} :=\{\pi F_0(\clos{S})^\vee
: S\in{\mathcal C}^{n-k}_\zeta \}$.
\end{cor}
\begin{rmk}
This corollary says that the singularities of $C_\zeta$ are
precisely those given (with respect to the lattice $\Pi$) by the
cones $\{\pi F_0(T)| T\in{\mathcal T}^0_\zeta\}$.
\end{rmk}
\subsubsection{Variation of the Flow Polyhedra with $\zeta $}
\label{sec:2:flow:cor:var}
The following corollary of Theorem~\ref{thm:faces}
describes when two different values of $\zeta$ give isomorphic
polyhedra:
\begin{cor}
\label{cor:iso_C_zeta}
If $\zeta$ and $\zeta'$ have the same admissible configurations
(${\mathcal C}_\zeta={\mathcal C}_{\zeta'}$) or even just the same admissible trees
(${\mathcal T}_\zeta={\mathcal T}_{\zeta'}$) then the corresponding polyhedra
$C_{\zeta}$ and $C_{\zeta'}$ are geometrically isomorphic. Two
polyhedra are said to be geometrically isomorphic is they are combinatorially
isomorphic and their tangent cones at the corresponding faces are
identical. In particular, their associated fans are identical, and
their corresponding toric varieties are isomorphic.
\end{cor}
Note that when one multiplies $\zeta$ by a non-zero number, the
polyhedron $C_\zeta$ is simply scaled-up. Fixing a tree $T\in{\mathcal T}^0$ determines
an open convex cone ${{\Lambda^{0,0}_{{\mathbb R}}}}(T)\subseteq \Lambda^{0,0}_{\mathbb R}$ of values of
$\zeta$ for which $T$ is $\zeta$-admissible. The cone ${{\Lambda^{0,0}_{{\mathbb R}}}}(T)$ is
called the
\emph{admissible cone of $T$}.\/ If ${\mathcal E}$ is a fixed subset of
${\mathcal T}^0$, then the condition ${\mathcal T}^0_\zeta={\mathcal E}$ defines an open cone in
the $\zeta$-parameter space $\Lambda^{0,0}_{\mathbb R}$. As ${\mathcal E}$ varies, one
obtains a partition of $\Lambda ^{0,0}_{\mathbb R}$ into a union of open cones
inside which the polyhedra $C_\zeta $ are geometrically isomorphic.
\subsubsection{Degeneracies}
Another fact which will be of interest to us is that, for {\em
generic\/} values of $\zeta$, any $\zeta$-flow has a support which
is a spanning subgraph of the quiver, i.e.\xspace which connects any two
vertices. This is written ${\mathcal C}_\zeta\subset{\mathcal C}_{\spn
}$.
For instance, the faces of ${{\Lambda^{0,0}_{{\mathbb R}}}}(T)$ consist of degenerate
values of $\zeta$
for which some extreme $\zeta$-flows have supports which are strict
subsets of $T$ and so cannot be spanning subsets.
In the next section, basic flows associated to paths and cycles are
studied in more detail. They provide the key to the proofs of the
other results.
\section{Proofs}
\label{sec:2:flow:proofs}
\subsection{Basic Flows}
\label{sec:2:flow:basic}
Many proofs in the context of network flows use the basic technique of
decomposing a flow into certain basic components associated to paths
in ${\mathcal Q}$. A \emph{path} in ${\mathcal Q}$ means a sequence $p=(p_1,\dots,p_k)$
of arrows in ${\mathcal Q}_1$ which form a connected path in the underlying
graph to ${\mathcal Q}$, once their orientation has been disregarded.
Consecutive $p_i$'s are not allowed to be the same (although they can
join the same vertices). As for cycles, the disjoint union of the
arrows of the path $p=(p_1,\dots,p_k)$ which agree\footnote{See
Section~\ref{sec:2:flow:cycle} for the definition of
\emph{agree.\/}} (resp.\xspace disagree) with the sense of traversal specified by
the sequence $p_1,\dots,p_k$ are called the \emph{positive}
(resp.\xspace\emph{negative}) arrows of $p$ and denoted $p^+$ (resp.\xspace $p^-$). A path
will sometimes be confused with its set of arrows, for instance in
statements such as ``a path $p$ is {\em in\/} a set $S\subseteq{\mathcal Q}_1$
(written $p\subseteq S$)'' which means of course that all its arrows
belong to the set $S$.
To each path $p$ the \emph{basic}\footnote{Basic flows are also
termed \emph{simple flows}~\cite{busacker_saaty:graphs}
or \emph{elementary flows} by other authors.}
flow $\tilde\chi_p\colon{\mathcal Q}_1\to\Z$ is
defined by
\begin{equation}
\tilde\chi_p(a)=\sum_{a\in p^+}\chi_a - \sum_{a\in p^-}\chi_a\in\Z^n.
\label{eq:basic_flow}
\end{equation}
Note that if $p$ is a path from $v$ to $v'$, then $\partial
\tilde{\chi}_p=\chi _{v'}-\chi _v$. Conversely, one has the following lemma.
\begin{lemma}
\label{lemma:basicf_path}
If $f$ is an integral flow with $\partial f=\chi _{v'}-\chi _v$ for
two vertices $v,v'$ of ${\mathcal Q}$, then there exists a path $p$ from $v$
to $v'$ such that $p^{\pm}\subseteq\supp f^\pm$.
\end{lemma}
\begin{proof}
By induction on the $1$-norm of $f$:
$\norm{f}_1:=\sum_{a\in{\mathcal Q}_1}\abs{f(a)}$. If $\norm{f}_1=1$ then,
obviously, $f=\chi _a$ or $-\chi_a$, for some arrow $a\in{\mathcal Q}_1$
which is either $v\to v'$ or $v'\to v$ respectively. Either way,
$f=\tilde{\chi }_{p}$ for the corresponding one-arrow path $p$ from
$v$ to $v'$. Now suppose that $\norm{f}_1>1$ and that $\partial
f=\chi _{v'}-\chi _v$. There must be an arrow $a$ such that one of
the following statements holds
\begin{enumerate}
\item[(1)] $t(a)=v$ and $f(a)>0$, or
\item[(2)] $h(a)=v$ and $f(a)<0$.
\end{enumerate}
If (1) holds then the flow $f'=f-\chi _a$ satisfies the induction
hypothesis with $\partial f'=\chi _{v'}-\chi _{h(a)}$, so there exists
a path $p'\subseteq\supp f'$ from $h(a)$ to $v'$. But then $p=ap'$ is
a path from $v$ to $v'$ with $p^+ = p^{\prime +}\cup
\{a\}\subseteq\supp f^{\prime+}\cup\{a\} = \supp f^+$. Case (2)
follows in a similar way setting $f'=f+\chi _a$ and $p=(-a)p$.
\end{proof}
Obviously, if $p$ is actually a cycle, then the basic flow associated
to $p$ is a $0$-flow (it satisfies $\partial \tilde\chi_p = 0$), and
is supported in $p$. Figure~\ref{fig:basicf7123}
gives an example of such a flow.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 3cm \epsfbox{fig/basicf7123.eps}
\end{center}
\caption{A basic $0$-flow associated to the
cycle in Figure~\protect\ref{fig:uncycle7123}}
\label{fig:basicf7123}
\end{figure}
The converse to this statement is given by the following lemma~\cite[Th.~7.2]{busacker_saaty:graphs}, \cite[{\S5.1.6,Th.~3}]{gond_mino:graphs}, .
\begin{lemma}[Decomposition into basic flows]
Any\/ $0$-flow $f$ can be decomposed into a positive linear
combination of basic flows for finitely many cycles $c_i$ such that
$c^\pm_i\subseteq \supp f^\pm$.
\label{lemma:0cycles}
\end{lemma}
\begin{proof}
Let $f\in\ker\partial$ and $S$ be its support.
If $S$ contains a cycle $c$ then let $v$ be a vertex such that $c^+$
has an arrow $a$ with $t(a)=v$. Then $f':=
f-f(a)\tilde\chi_c\in\ker\partial$ and $\supp f' = \supp f \setminus
\{a\}$ is strictly smaller than $\supp f$. Repeating this reasoning
finitely many times, one obtains a flow $f'\in\ker\partial$ such that
$S'=\supp f'$ contains no cycles. Suppose $S'$ is non-empty, and
let $p$ be a path in $S'$ which is \emph{maximal, \/} i.e.\xspace is not
contained in any strictly larger path. Let $p$ go from vertex $v$ to
vertex $v'$ and $a$ denotes the first arrow of $p$ (the one joined
to $v$). Then this is the only arrow in the quiver which is joined to
$v$ and on which $f'$ is non-zero, so $\partial f'(v)$ must be non-zero.
Since this is impossible, $S'$ must be empty, and $f=\sum_i
x_i \tilde\chi_{c_i}$ for some cycles $c_i\subseteq S$ and real
numbers $x_i$. Replacing $c_i$ by $-c_i$ if necessary, one may assume
$x_i>0$. (Note also that if $f$ is integral to start with, the $x_i$
are also integral.)
I claim that these cycles may be chosen in such a way that their
orientations are \emph{conformal}, i.e.\xspace such that any
two cycles $c_i,c_j$ satisfy $c^+_i\cap c^-_j=\emptyset$. Indeed
suppose that $c_i$ and $c_j$ are not conformal, and suppose, say,
$x_i\geq x_j$.
On the subset $U\subseteq{\mathcal Q}_1$ on which their orientations
disagree, the sum of $x_ic_i$ and $x_jc_j$ will cancel each other out, and only $x_i-x_j$ units
of flow will survive. The complement $c_i\cup c_j\setminus U$ will
consist of one or more disjoint cycles $d_1,\dots,d_k$, all
conformal to $c_i$. Thus upon adding the two basic flows
$$x_i\tilde\chi_{c_i}+x_j\tilde\chi_{c_j} =
(x_i-x_j)\tilde\chi_{c_i} + x_j(\tilde{\chi }_{d_1}+\dots+\tilde{\chi
}_{d_k}),$$ giving $k+1$ conformal cycles with all coefficients
positive or zero. Repeating this procedure for
all the pairs of non-conformal cycles gives the desired conformal
decomposition.
Now writing $$f=\sum_i\tilde\chi_{c_i}=\sum_i
(\chi_{c^+_i}-\chi_{c^-_i}),$$ and since $c^+_i\cap
c^-_j=\emptyset$, one has
$$f^\pm = \sum_i\chi_{c^\pm_i},$$
which is the desired result.
\end{proof}
\begin{rmk}
The decomposition lemma above can be viewed as a consequence of the
following homological argument. Regard ${\mathcal Q}$ as a CW-complex, and for
each cycle $p$ in ${\mathcal Q}$, adjoin a 2-cell whose boundary is the element
$\tilde\chi_{p}$. Then the resulting CW-complex $\widetilde{\mathcal Q}$ has
$H^1(\widetilde{\mathcal Q})=0$.
An elementary proof was given because it illustrates the basic
technique of decomposing a flow into basic flows.
\end{rmk}
\subsection{Proof of Theorem~\protect\ref{thm:faces} --- Classical Case}
\label{sec:2:flow:classpf}
The classical analog of Theorem~\ref{thm:faces} (i.e.\xspace the case when
$\pi$ is the identity map) can now be proved. The statement says that
the $k$-dimensional faces of $F_\zeta$ correspond to the $\zeta
$-configurations $S$ such that $\rk Z_0(S)=k$, i.e.\xspace whose cycles
span a lattice of rank $k$. This theorem follows from the following
four facts:
\begin{fact}
\label{fact:1}
If $f\in F_\zeta$ then the cone ${\mathbb R}_+(F_\zeta-f)$ contains a subspace
of dimension $k$ if and only if $f$ is contained in the (relative)
interior of a face of dimension $k$ (and hence in no lower dimensional
face).
\end{fact}
\begin{fact}
\label{fact:2}
For any $f\in F_\zeta$, one has ${\mathbb R}_+(F_\zeta-f)= F_0(\supp f)$.
\end{fact}
\begin{fact}
\label{fact:3}
The maximal vector subspace in the cone $ F_0(S)$ is $Z_0(S)$.
\end{fact}
\begin{fact}
\label{fact:4}
The subspace $Z_0(S)$ is generated by the basic flows for the
cycles supported in $S$.
\end{fact}
Fact~\ref{fact:1} follows from the definition of a $k$-dimensional
face, \ref{fact:3} is trivial and~\ref{fact:4} follows from Lemma~\ref{lemma:0cycles}. Fact~\ref{fact:2} is proved in the
following lemma.
\begin{lemma}
\label{lemma:tangent_fzeta}
If $f\in F_\zeta$, then ${\mathbb R}_+(F_\zeta -f) = F_0(\supp f)$.
\end{lemma}
\begin{proof}
Let $f\in F_\zeta$, so that
$F_\zeta=(f+\ker\partial)\cap{\mathbb R}_+^{{\mathcal Q}_1}$. The non-negative
multiples of elements in $(f+\ker\partial)\cap{\mathbb R}_+^{{\mathcal Q}_1} -f$
belong to $\ker\partial$ and are non-negative outside $\supp f$, so
belong to $ F_0(\supp f)$. Conversely, suppose $m\in F_0(\supp f)$.
Then $\partial(f+\epsilon m)=\zeta $ for all $\epsilon\in{\mathbb R}$, and it
suffices to show that there exists $\epsilon >0$ such that
$f+\epsilon m\in {\mathbb R}_+^{{\mathcal Q}_1}$. But this follows because $f$ is
bounded below by a positive number on $\supp f$, whereas $m\geq 0$
outside $\supp f$.
\end{proof}
\subsection{Proof of Theorem~\protect\ref{thm:faces} --- General Case}
\label{sec:2:flow:genpf}
In order to prove the general case of Theorem~\ref{thm:faces}, a bit
more has to be said about the various configurations which can occur for
a point $x\in\pi F_\zeta$.
\begin{lemma}
\label{lemma:maximal_config}
If $S$ is a $\zeta $-configuration for $x$ then so is $\clos{S}$.
Furthermore, all $\zeta $-configurations for $x$ have the same
closure.
\end{lemma}
\begin{proof}
Let $f\in F_\zeta$ be a flow such that $\pi f=x$ and let $S=\supp
f$. The proof begins by constructing a flow $\overline f\in F_\zeta\cap
\pi^{-1}(x)$ whose support is $\clos{S}$. If $c$ is a cycle of type $0$
in ${\mathcal Q}$ such that $c^-\subseteq S$ and $c^+\not\subseteq S$, one
can add a small positive multiple of $\tilde\chi_c$ to $f$ and
obtain a non-negative flow $f'$. Since $c$ is a cycle,
$\partial\tilde\chi_c=0$, and since $c$ has type $0$, $\pi f=\pi
f'$ and hence $f'\in F_\zeta$. Continuing in
this way until all cycles of type $0$
satisfy~\eqref{eq:cycle_closure}, one obtains the required flow
$\overline f$.
For the second statement of the lemma, note that if $f,f'$ are two
elements of $F_\zeta\cap\pi^{-1}(x) $, then $\partial(f-f')=0$, so by
Lemma~\ref{lemma:0cycles}, has a decomposition into basic flows for
cycles $c_i$: $$f-f' = \sum_i x_i\tilde\chi_{c_i},$$ with
$c_i^+\subseteq f$, $c_i^-\subseteq f'$ and $x_i>0$. Now
$c_i^+\subseteq S\implies c_i^-\subseteq \overline S$, so $$f'=f+\sum_i
\tilde\chi_{c_i}$$ implies that $S'\subseteq \overline S$. By symmetry, one
also has $S\subseteq \overline{S'}$, and so $\overline S = \overline{S'}$.
\end{proof}
A little corollary needed later is
\begin{cor}
\label{cor:spanning_trees}
Any $S\in{\mathcal C}^0$ contains a tree $T$ such that $\clos{T}=\clos{S}$.
\end{cor}
\begin{proof}
Let $S=\supp f$ be a configuration for $x$. Eliminate any cycles in
the support of $f$ by adding basic flows corresponding to those
cycles, as in the proof of Lemma~\ref{lemma:maximal_config}. The
resulting flow $f'$ is supported in a tree $T$ and, since all the
basic flows have type zero it satisfies $\pi(f')=x$.
Lemma~\ref{lemma:maximal_config} shows that $\clos{T}=\clos{S}$.
\end{proof}
\begin{proof}[Proof of the general Theorem~\ref{thm:faces}]
Consider a point $x\in\pi F_\zeta$. The cone ${\mathbb R}_+(\pi F_\zeta-x)$
gives the tangent cone to $\pi F_\zeta$ at the minimal face of
$F_\zeta$ containing $x$. It is obtained by taking the union over
all $f\in F_\zeta\cap\pi^{-1}(x)$ of $\pi {\mathbb R}_+(F_\zeta-f)$. By
Lemma~\ref{lemma:tangent_fzeta} this gives
$$\bigcup\{\pi F_0(S) : S \text{ a $\zeta $-configuration for }x\},$$
which by Lemma~\ref{lemma:maximal_config} is $\sigma(S)=\pi
F_0(\clos{S})$, for any $S=\supp f$ and $f\in F_\zeta\cap\pi^{-1}(x)$.
The minimal face of $\pi F_\zeta$ containing $x$ is given by
intersecting $\pi Z_0(\clos{S})$, the largest subspace in the cone $\pi
F_0(\clos{S})$, with the tangent cone of $\pi F_\zeta $ at $x$ and then
translating by $x$. This gives
$$x+(\pi F_\zeta )_x \cap Z_0(\clos{S}),$$ which is precisely the face
$\text{Face}_\zeta(S)$ mentioned in the theorem and all faces of
$\pi F_\zeta$ are obtained in this way. It is obvious that
equivalent configurations have the same rank and give the same face,
so the partition of ${\mathcal C}$ has the stated properties and
$\text{Face}_\zeta$ is a bijection ${\mathcal C}^k_\zeta/\!\!\sim\to k\text{-faces}$.
\end{proof}
\section{Exactness Results}
\label{sec:2:exact}
The exactness of the sequence~\eqref{eq:exact1} is proven in this
section.
\subsection{Basic Flows: Sequential Notation}
\label{sec:2:exact:sequ}
\nopagebreak
We introduce some notation which is convenient to describe basic
flows. This is used in~ Section~\ref{sec:2:exact:exact} to give an
elementary proof of the exactness of~\eqref{eq:exact1}.
For $v\in{\mathcal Q}_0$ and $j\in\{1,\dots,n\}$, write $\chi_v^j$ for the
basis element $\Lambda ^1$ which is the indicator function of the
singleton $\{a_v^j\}\subset{\mathcal Q}_1$.
Define the following
symbols:
\begin{align}
\{v\}(j) &:= \chi_v^j\\
\{v\}(-j) &:= -\chi_{v+w_j}^j.
\end{align}
For $k>0$, and $j_0,\dots,j_k\in\pm\{1,\dots,n\}$, define
$$ \{v\}(j_0,\dots,j_k):=\{v\}(j_0)+\{v-w_{j_0}\}(j_1,\dots,j_k).$$
Also define $(j)\{v\} := \{v+w_j\}(j)$ and
$$(j_k,\dots,j_0)\{v\}=(j_k,\dots,j_1)\{v+w_{j_0}\} +
(j_0)\{v\}.$$
This sequential notation is designed especially for representing basic
flows and has the advantage of including only the relevant
information. For instance, if $p=(p_1,\dots,p_k)$ is a
path in ${\mathcal Q}$, then the
basic flow associated to $p$ is, in this notation,
$$\tilde\chi_p=\{t(p_1)\}(j_1,\dots,j_k),$$ where
$$j_i:=
\begin{cases}
\phantom{-}\pi(p_i),\qquad & p_i\in p^+\\
-\pi(p_i),\qquad & p_i\in p^-
\end{cases}
$$
For example, the basic flow represented in
Figure~\ref{fig:basicf7123}
can be written as $$\{0\}(1,1,2,1,1,-3,-3)$$ in this notation.
Figure~\ref{fig:bfl11124} shows yet another example.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 3cm \epsfbox{fig/bfl11124.eps}
\end{center}
\caption{The basic flow $\{0\}(1,1,3,-2,1,3,-1,-1)$ in the
McKay quiver for $\frac{1}{ 11}(1,2,4)$.}
\label{fig:bfl11124}
\end{figure}
Often, when there is a need to specify both endpoints, the notation
$\{v\}(j_0,\dots,j_k)=:\{v\}(j_0,\dots,j_k)\{v'\}$ with
$v'=v-\sum w_{j_i}$ will be adopted. The
following identities are easily checked for any $v\in{\mathcal Q}_0$
and
$j,j_i\in\pm\{1,\dots,n\}$:
\begin{gather}
(j)\{v\}+\{v\}(-j) = 0 \label{eq:v-id1},\\
\{v\}(j_0,\dots,j_k,-j_k,\dots,-j_0)=0 \label{eq:v-id2},\\
\{v\}(j_0,\dots,j_k)\{v'\}+\{v'\}(j_{k+1},\dots,j_l)=
\{v\}(j_0,\dots,j_l),
\label{eq:v-id3}
\end{gather}
where the last identity holds of course for $v'=v-\sum_{i=0}^k
w_{j_i}$.
If $\sum_i w_{j_i}=0$, i.e.\xspace the corresponding flow
$\{v\}(j_1,\dots, j_k)$ is a $\partial$-closed flow associated to a cycle,
there is another identity which allows us to cyclically permute
the indices:
\begin{equation}
\label{eq:v-id4}
\{v\}(j_0,\dots,j_k)=\{v-w_{j_0}\}(j_1,\dots,j_k,j_0).
\end{equation}
Recall that $\Lambda^2$ is the sub-lattice of $\Lambda^1$ generated by
the elements corresponding to the commutation relations. These can be
written in various notations: $$r^{ij}_v=\chi_v^i
+\chi_{v-w_i}^j-\chi_v^j-\chi_{v-w_j}^i=
\{v\}(i,j,-i,-j).$$
Putting these identities together gives the following lemma.
\begin{lemma}
For any permutation $\sigma$ of $\{0,\dots,k\}$ one has
$$\{v\}(j_{\sigma(0)},\dots,j_{\sigma(k)}) =
\{v\}(j_0,\dots,j_k)\mod \Lambda^2.$$
\label{lemma:permutation}
\end{lemma}
\vspace*{-\belowdisplayskip}
\vspace*{-\topsep}
\vspace*{-\partopsep}
\begin{proof}
It is enough to show that the equality holds for any
transposition $\sigma=(q,q+1)$ of consecutive elements.
Let $j_q=i$ and $j_{q+1}=l$. By the identities
\eqref{eq:v-id2}--\eqref{eq:v-id3}
one has
$$
\{v\}(j_0,\dots,j_k) =
\{v\}(j_0,\dots,j_{q-1})+\{v_q\}(i,l)+
\{v_{q+2}\}(j_{q+2},\dots,j_k),
$$ where $v_a:=v-\sum_{i=0}^{a-1} w_{j_i}$ for any $a\in\{0,\dots,
k\}$. Using
$\{v_q\}(l,i,-l,-i)+\{v_q\}(i,l)=\{v_q\}(l,i)$, one sees that
$$\{v\}(j_{\sigma(0)},\dots,j_{\sigma(k)})-\{v\}(j_0,\dots,j_k)
=\{v_q\}(l,i,-l,-i).$$ Now if $i,l\in\{1,\dots,n\} $ then the result
follows because $\{v_q\}(l,i,-l,-i)=r^{il}_{v_q}$. If on the other hand
$-i,l\in\{1,\dots,n\} $, then the result is true because
\begin{align*}
\{v_q\}(l,i,-l,-i) &=\{v_q +w_i\}(-i,l,i,-l),\qquad\text{ (by
equation~\ref{eq:v-id4})}\\
&=r^{-i,l}_{v_q +w_i}.
\end{align*}
The other two possibilities ($(i,-l)$ and $(-i,-l)$) also follow in this way.
\end{proof}
Another useful result follows from these identities and
Corollary~\ref{lemma:0cycles}:
\begin{lemma}
Any $\partial$-closed flow can be written as the basic flow
associated to a single cycle in ${\mathcal Q}$ (not necessarily contained in
its support).
\label{lemma:null_single_cycle}
\end{lemma}
\begin{proof}
Note that in the proof of Lemma~\ref{lemma:0cycles},
$f\in\ker\partial$ was decomposed into a sum of basic flows for
cycles $c$. The sum of two basic flows corresponding to cycles can
be written, using identities \eqref{eq:v-id2}--\eqref{eq:v-id4} as
\begin{multline}
\{v\}(j_1,\dots,j_k)+\{v'\}(j'_1,\dots,j'_{k'}) = \\
\{v\}(j_1,\dots,j_k,a_1,\dots,a_l,j'_1,\dots,j'_{k'},-a_l,\dots,-a_1),
\label{eq:v-sum}
\end{multline}
where $a_i\in\pm\{1,\dots,n\}$ are such that $v-\sum_i
w_{a_i}=v'$,
and one has $\sum_i w_{j_i}=\sum_i w_{j'_i}=0$ since the basic
flows on the left-hand side correspond to cycles. This proves the
lemma.
\end{proof}
\subsection{Exactness Results}
\label{sec:2:exact:exact}
\nopagebreak
The promised proof of the exactness of~\eqref{eq:exact1} can now be
given.
Recall the morphism of lattices $\pi\times\partial\colon\Lambda^1\to
\Z^n\times\Lambda^{0,0}$ defined in
section~\ref{sec:2:abel:toric}.
\begin{lemma}
One has $\ker \pi\times\partial =\Lambda^2$, i.e.\xspace
$$\Lambda^2\to\Lambda^1\stackrel{\pi\times\partial}\to\Z^n\times\Lambda^{0,0}$$
is exact.
\label{lemma:kerc_lambda2}
\end{lemma}
\begin{proof} Since $\pi\times\partial(r^{ij}_v)=0$ one only has to prove that
$\ker \pi\times\partial\subseteq\Lambda^2$. Let $f\in\ker \pi\times\partial$, and suppose that $c'$
is a cycle such that $f=\tilde\chi_{c'}$, as
described in Lemma~\ref{lemma:null_single_cycle}. Then $f$ is of the
form
$$f=\{v\}(j_0,\dots,j_k),$$ for some $v\in{\mathcal Q}_0$ and
$j_i\in\pm\{1,\dots,n\}$. Since $\pi f=0$, one has $\#\{i:
j_i=j\}=\#\{i:j_i=-j\}$ for any $j\in\{1,\dots,n\}$. Thus,
up to permutations, $(j_0,\dots,j_k)$ can be rewritten as
$(j_1,-j_1,j_2,-j_2,\dots,j_l,-j_l)$
for some elements $j_k\in\{1,\dots,n\}$. Since
$\{v\}(j_1,-j_1,j_2,-j_2,\dots,j_l,-j_l)=0$, the result
follows by Lemma~\ref{lemma:permutation}.
\end{proof}
From this lemma it follows that~\eqref{eq:exact1} is exact, and
induces an inclusion $\Lambda\hookrightarrow \Z^n\times\Lambda^{0,0}$.
Recall that $\Pi=\ker\hat\rho$, where
\map{\hat\rho}{\Z^n}{\widehat\Gamma}{x}{\sum_i x_iw_i.}
\begin{lemma}
\label{lemma:Lambda_n_pi}
One has $\Pi=\pi(\ker\partial)$.
\end{lemma}
\begin{proof}
Suppose $x\in\Pi $. One wants to find $f\in\ker\partial$
such that, for all $i\in\{1,\dots,n\}$, $f$ satisfies
$$\sum_{a:\pi(a)=i} f(a) = x_i.$$
This is easy: just take the basic flow given by
\begin{equation}
f=\{v\}(\underbrace{ 1,\dots, 1}_{x_1},
\underbrace{ 2,\dots, 2}_{x_2},\dots,
\underbrace{ n,\dots, n}_{x_n}),
\label{eq:cycle_lambda_n}
\end{equation} for any $v\in{\mathcal Q}_0$. (If $x_i$ is negative under any
brace, the notation is taken to mean $-x_i$ copies of $-i$.) Now $\sum_{i=1}^n w_i{x_i} =
0\text{ in }\widehat\Gamma$ is equivalent to the fact that the basic flow
corresponds to a cycle, and so $\partial f=0$. The converse follows
from Lemma~\ref{lemma:null_single_cycle} and the preceding sentence.
\end{proof}
\begin{lemma}
\label{lemma:imagec}
There is an exact sequence of abelian groups
$$0\to\Lambda \xrightarrow{\pi\times\partial}
\Z^n\times\Lambda^{0,0} \xrightarrow{\hat\rho-\hat\nu} \Z^n/\Pi
\cong\widehat\Gamma \to 0,$$
where
\map{\hat\nu}{\Lambda^{0,0}}{\widehat\Gamma}{\zeta}{\sum_{v\in{\mathcal Q}_0}\zeta(v)v}
is the morphism of lattices dual to the action of \/$\Gamma$ on
$\End R$ by conjugation.
\end{lemma}
\begin{proof}
One needs to show that if $(x,\zeta)\in\Z^n\times\Lambda^{0,0}$ satisfies
\begin{equation}
\sum_{i=1}^n x_iw_i - \sum_v \zeta(v)v=0,
\label{eq:zerosum}
\end{equation}
then there exists a flow
$f\in\Lambda^1$ such that $\pi(f)=x$ and $\partial f=\zeta$. One
can construct this flow in two steps. For convenience, suppose that
the weights of the action of $\Gamma$ on $Q$ have been normalised so
that $w_1=1$. First construct a flow $g$ such that $\partial
g=\zeta$ with only arrows of type $1$: start at vertex $0$ and let
$g(1\to 0)=\zeta(0)$. Next, let $g(2\to 1)=\zeta(1)+\zeta(0)$, and
continue in this way, setting
$$g(k+1\to k) = \sum_{j=0}^k\zeta(j),$$ and $g=0$ on all the other
arrows. Then $\partial g =\zeta$ by construction, and
$\pi(g)=X_1e_1$, where $X_1=\sum_j\zeta(j)j$. Now add on any flow of the form
$$g'=\{v\}(\underbrace{ 1,\dots, 1}_{x_1-X_1}, \underbrace{ 2,\dots,
2}_{x_2},\dots, \underbrace{ n,\dots, n}_{x_n}),$$ using the same
conventions as in the previous lemma for the case when the integers
under the braces are negative. Equation~\eqref{eq:zerosum} means
that $g'$ is the basic flow associated to a cycle, and so $\partial
g'=0$. The flow $f=g+g'$ has the required properties.
\end{proof}
\section{Singular Configurations}
\label{sec:2:sing}
In this section some comments are made regarding the singularities of $C_\zeta$.
Recall that the tangent cone to $C_\zeta =\pi F_\zeta $ at a point $x$
which has a configuration $S\in{\mathcal C}^k$, is the cone $\pi F_0(\clos{S})$.
This is singular if its dual $(\pi F_0(\clos{S}))^\vee$ is
generated by a part of a basis of ${\Pi}^*$. In this case,
$S$ is called a \emph{singular configuration}.\/ The set of
singular configurations is denoted ${\mathcal S}$.
\begin{prop}
The following statements are true.
\begin{enumerate}
\item $C_\zeta$ is singular at the faces corresponding to
configurations $S\in {\mathcal S}_\zeta$.
\item $C_\zeta$ is non-singular in co-dimension $k$ if and only if
${\mathcal S}_\zeta^{n-k}=\emptyset$.
\item $C_\zeta$ is generically non-singular in co-dimension $k$ if
and only if
${\mathcal S}_{\spn}^{n-k}=\emptyset$.
\item $C_\zeta$ is generically non-singular if and only if
${\mathcal S}_{\spn}=\emptyset$.
\end{enumerate}
\end{prop}
\begin{proof}
Part (1) of the following proposition follows from
Theorem~\ref{thm:faces} and Lemma~\ref{lemma:generic_flow}. Part~(2) follows because $S\in{\mathcal S}^{n-k} \iff \dim \sigma_S = k$. The last two
statements follow because ${\mathcal S}_{\spn}=\cup_{\zeta\text{ generic }}{\mathcal S}_\zeta$.
\end{proof}
\begin{question}
What is the lowest $k$ for which ${\mathcal S}^k_\zeta$ is empty (i.e.\xspace in
what co-dimension is $C_\zeta$ smooth)? The cone $C_0$ is smooth in
co-dimension $1$ (it has an isolated singularity), so it seems likely
that ${\mathcal S}^k=\emptyset$ for $k\geq1$. Of course, this is equivalent to
the statement that ${\mathcal S}^1=\emptyset$.
\end{question}
The statements about singular trees are translated here for the record.
\begin{conj}
The polyhedra $C_\zeta$ are non-singular in co-dimension $n-1$,
i.e.\xspace ${\mathcal S}^1=\emptyset$. If\/ $\Gamma\subset\SU(3)$, then $C_\zeta$ are
non-singular for generic values of $\zeta$, i.e.\xspace
${\mathcal S}^0_{\text{span}}=\emptyset$. Furthermore, the Euler number of
$C_\zeta$ for generic $\zeta$ is equal to the order of $\Gamma$,
i.e\ $\card{{\mathcal T}^0_\zeta/\!\!\sim}=\card{\Gamma}$.
\end{conj}
Let us look at the case of singular \emph{points.\/} Suppose that
$S\in{\mathcal C}^0$ is a singular configuration. This means that the primitive
generators $\rho(S)=(\rho^1_S,\dots,\rho^k_S)$ of $\pi F_0(S)$ do not
form a basis of $\Pi $. If $k=n$, $\pi F_0(\clos{S})$ corresponds to a
finite abelian quotient singularity, whereas, if $k>n$, the
singularity is determined by the linear relations holding between the
$\rho_S^i$. To determine whether a given set $S\in{\mathcal C}^0$ is singular
or not, one must find all the cycles $c$ such that $c^-\subseteq S$,
calculate their type, and see what primitive vectors of $\Pi$
one obtains. It is sufficient to restrict one's attention to the
cycles which are not decomposable into a union of cycles. Let us look
at some examples.
\section{Examples and Computations}
\label{sec:2:examples}
\subsection{Commutators}
\label{sec:2:examples:comm}
In this section, the commutator of a configuration $S$ is defined;
this is specific to the McKay quiver and its purpose is purely
computational: it gives a necessary criterion for determining when
$S\in{\mathcal C}^0$ which is extremely useful in practical calculations.
Let us begin by working out what the closure of a subset
$S\subseteq{\mathcal Q}_1$ corresponds to in the case of the McKay quiver for
the action $\frac{1}{ r}(w_1,\dots,w_n)$. Let $\pi$ be the map
${\mathcal Q}_1\to\{1,\dots,n\}$ which assigns to each arrow $a_v^i = v\to
v-w_i$ its type $i$. This induces a map $\pi\colon{\mathbb R}^{{\mathcal Q}_1}\to{\mathbb R}^n$
which assigns to the basis element $\chi_v^i=\chi_{a_v^i}$ the basis
element $e_i$ of ${\mathbb R}^n$.
Recall that to calculate the closure of $S$, one must find the
smallest over-set of $S$ which contains the positive part of cycles of
type zero if and only if it contains the negative part. The simplest
cycles which have type zero are cycles with only four arrows: starting
from vertex $v$, go forward along arrow $a_v^i$ to $v-w_i$, forward
again along $a_{v-w_i}^j$ to $v-w_i-w_j$, back along $a_{v-w_j}^i$ to
$v-w_j$ and back along $a_v^j$ to $v$. This cycle is denoted by
$c_v^{ij}$. The basic flow corresponding to this $c^{ij}_v$ is
$\{v\}(i,j,-i,-j)$ in the notation of Section~\ref{sec:2:exact:sequ}. If
$S$ contains $c^{ij+}_v=\{a_v^i,a_{v-w_i}^j\}$ for some $v,i,j$ the
closure of $S$ must contain $c^{ij-}_v=\{a_v^j,a_{v-w_j}^i\}$. The
two pairs of arrows will be called \emph{complementary pairs}.\/
\begin{dfn}
Denote by $p^{ij}_v$ the pair of arrows $\{a_v^i,a_{v-w_i}^j\}$.
If $S$ is a subset of ${\mathcal Q}_1$, the
smallest over-set $S^{\bowtie}\supseteq S$ satisfying the
``commutation condition''
\begin{equation} p^{ij}_v \subseteq
S^{\bowtie} \iff p^{ji}_v\subseteq S^{\bowtie}
\label{eq:S_commute}
\end{equation} is called the \emph{commutator} of $S$.
\end{dfn}
\begin{figure}[htbp]
\begin{center} \leavevmode
\epsfysize= 3cm \epsfbox{fig/commutator.eps}
\end{center}
\caption[The commutator of the set in
Figure~\protect\ref{fig:uncycle7123}]{The commutator of the set in
Figure~\protect\ref{fig:uncycle7123}. Notice that this is not an
invariant set: there is no way to assign elements of $\Z^n$ to the
vertices in such a way that equation~\protect\ref{eq:S_inv} is
satisfied.} \label{fig:commutator}
\end{figure}
The fact that $\clos{S}\supseteq S^{\bowtie}$ is
extremely useful in practical computations, because it gives the
following easily checked necessary condition for $S$ to be an IC-set.
\begin{lemma}
If $S$ is a spanning IC-set then the morphism
${\mathcal W}_S\colon {\mathcal Q}_0 \to\Z^n$ defined by
equations~\ref{eq:lambda_def} satisfies the following
conditions for all $v\in{\mathcal Q}_0$, and $i,j\in\{1,\dots,n\} $:
\begin{equation}
p^{ij}_v \subseteq S \implies
\begin{cases}
{\mathcal W}_S({v-w_j})-{\mathcal W}_S({v})&=e_j \\
{\mathcal W}_S({v-w_j-w_i})-{\mathcal W}_S({v-w_j})&=e_i,
\end{cases}
\label{eq:lambda_commute}
\end{equation}
where $\{e_i\}$ denotes the standard basis of $\Z^n$.
\end{lemma}
It is rather surprising that this condition is in fact \emph{not\/} sufficient. Figure~\ref{fig:notcc} gives an example of a set $S$ for which
$S=S^{\bowtie}\neq \clos{S}$.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 3cm \epsfbox{fig/notcc.eps}
\end{center}
\caption{A configuration $S$ for which $S=S^{\bowtie}\neq \clos{S}$. The dotted arrows indicate the arrows in $\clos{S}\setminus S^{\bowtie}$}
\label{fig:notcc}
\end{figure}
\subsection{Weightings and Stabilisers}
\label{sec:2:examples:weight}
There is another way of understanding the condition $S\in {\mathcal C}^k$ for a
set $S\subseteq{\mathcal Q}_1$ which relates directly to the toric geometry of
$X_\zeta =\protect\overline{T}{}^{\Pi,C_\zeta }$. The basic idea is that a $k$-dimensional
face of $C_\zeta $ corresponds to elements of $X_\zeta $ which are
fixed by a torus of codimension $k$. For instance, the extreme points
correspond to fixed points of $T^\Pi$. Let us begin with this case for
simplicity.
\subsubsection{Fixed Points and $n$-weightings}
\label{sec:2:examples:weight:fixed}
Recall that the lattice $\Z^{{\mathcal Q}_0}$ is denoted by $\Lambda ^0$. The
sub-lattice of co-rank $1$ defined by the equation $\sum_{v\in{\mathcal Q}_0}
\zeta(v) = 0$ is denoted $\Lambda ^{0,0}$.
For any subset of arrows $S\subseteq{\mathcal Q}_1$ one attempts to find
a morphism ${\mathcal W}_S\colon \Lambda^{0,0}\to\Z^n$
satisfying
\begin{equation}
{\mathcal W}_S(\partial \chi_a)=\pi(\chi_a),\quad\text{ for }a\in S.
\label{eq:S_inv}
\end{equation}
If ${\mathcal W}_S$ exists, it satisfies
\begin{equation}
{\mathcal W}_S(\partial\tilde{\chi }_p)=\pi\tilde{\chi }_p
\label{eq:lambda_def}
\end{equation} for any path $p=(p^1,\dots,p^k)$ in $S$. In
particular, this equation implies that $\pi\tilde{\chi }_c=0$ must
hold for all cycles $c\subseteq S$. In other words, if ${\mathcal W}_S$
exists, then $\rk \pi Z_0(S)=0$. Conversely, if all cycles in $S$
have zero type then one can find a morphism ${\mathcal W}_S$
satisfying~\eqref{eq:S_inv}. Note that this is possible if and only
if there is morphism ${\mathcal W}'_S\colon \Lambda^0\to\Pi$ which extends
it. The latter corresponds to a labeling $${\mathcal W}'_S\colon{\mathcal Q}_0\to\Z^n$$ of the vertices of the quiver by elements of
$\Z^n$ in such a way that equation~\ref{eq:S_inv} holds. This is
called an \emph{$n$-weighting} of $S$, and
$S$ is called~\emph{invariant} if it admits such a weighting.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 3cm \epsfbox{fig/lambdas7123.eps}
\end{center}
\caption{Example of an $n$-weighting of a configuration of arrows.}
\label{fig:lambdas7123}
\end{figure}
In summary, $S$ is an extreme configuration if and only if $\clos{S}$ is
invariant, if and only if $\pi Z_0(S)=0$. Such $S$ are called \emph{
invariant closure configurations\/} or \emph{IC-configuration}\emph{s\/}
for short. Particularly important is the set ${\mathcal T}^0$
of \emph{IC-trees}; it is easy to
determine whether $T\in {\mathcal T}$ is an IC-tree: just check whether the
arrows $a\in \clos{T}$ all satisfy ${\mathcal W}_T\partial\chi _a= \pi \chi _a$.
To see the relationship to the toric geometry of $X_\zeta $, recall
that $X_\zeta$ is the GIT quotient
of a certain affine variety in ${\mathbb C}^{{\mathcal Q}_1}$ by $T^{0,0}$. The extreme
points of $C_\zeta $ therefore correspond to elements $\tilde \alpha
\in{\mathbb C}^{{\mathcal Q}_1}$ which are mapped to the same $T^{0,0}$-orbit under the
morphism of algebraic tori $$\widehat{{\mathcal W}_S}\colon {\C^*}^n\to T^1.$$ This
is the case for any $\tilde \alpha $ such that $\supp\tilde \alpha \subseteq S^{\Box}$, where $S^{\Box}$ is the set of all arrows in ${\mathcal Q}_1$
which satisfy~\eqref{eq:S_inv} (this includes of course $\overline S$).
This discussion can be extended to higher dimensional faces, although
it is not as practical for concrete calculations.
\subsubsection{Higher Dimensional Faces}
\label{sec:2:examples:weight:higher}
The sub-lattice of integral flows on ${\mathcal Q}$ which are zero outside $S$
will be denoted $\Z^S\subseteq\Z^{{\mathcal Q}_1}$. Recall that $Z_0(S)$ is the
set of $0$-flows which are zero outside $S$: $Z_0(S)= \Z^S\cap
\ker\partial$. Define, for each $S\subseteq{\mathcal Q}_1$, a morphism of
lattices
$$ {\mathcal W}_S\colon \partial \Z^S\to{\pi \Z^S/\pi Z_0(S)}$$
defined by
\begin{equation}
\label{eq:ws}
{\partial\chi_a}:= \pi Z_0(S) + \pi\chi_a,\quad \rlap{$a\in S$.}
\end{equation}
This fits into a commutative diagram
$$\begin{CD}
\label{CD:ws}
\partial\Z^S & @>{{\mathcal W}_S}>> & {\pi\Z^S/\pi Z_0(S)}\\
@A{\partial}AA & & @AA{\pr}A\\
\Z^S & @>>{\pi_{|\Z^S}}> & \pi\Z^S.
\end{CD}
$$
Taking the image of this under the functor $\ \widehat{\ } = \Hom(\
\cdot\ ,{\mathbb C}^*)$, one obtains a corresponding diagram of algebraic tori:
$$
\begin{CD}
\label{CD:tori}
T^{0,0}&\supseteq T^{\partial\Z^S}
& @<{\widehat {{\mathcal W}_S}}<< & T^{\pi\Z^S/\pi Z_0(S)}&\\
& @V{\widehat\partial}VV & & @VV{\widehat\pr}V &\\
T^1 & \supseteq T^{\Z^S} & @<<{\widehat {\pi_{|\Z^S}}}< & T^{\pi\Z^S}&
\subseteq {\C^*}^n .
\end{CD}
$$
This diagram shows that the action of the sub-torus $ T^{\pi \Z^S/\pi Z_0(S)}$ of
$T^{\pi \Z^S}$ on $\tilde \alpha \in {\mathbb C}^S$ via
$\widehat{\pi_{|\Z^S}}\widehat{\pr}$ leaves $\tilde \alpha $ in the same orbit of
$T^{0,0}$. In fact, since the torus corresponding to $\Z^{S^c}$
(where $S^c$ denotes the complement of $S$ in ${\mathcal Q}_1$) acts trivially
on ${\mathbb C}^S$, one sees that
$$ T^{\pi\Z^{{\mathcal Q}_1}/\pi Z_0(S)}$$
acts on ${\mathbb C}^S$ fixing the $T^{0,0}$-orbits and is a
sub-torus of $T^{\pi \Z^{{\mathcal Q}_1}} = {\C^*}^n$ of codimension $\rk\pi Z_0(S)$.
\subsubsection{Existence and Uniqueness of $n$-weightings}
\label{sec:2:examples:weight:exist}
\begin{dfn}
Two vertices in $v,v'\in {\mathcal Q}$ are said to be \emph{connected by
a subset\/} $S\subseteq{\mathcal Q}_1$ if there is
a path $p$ in $S$ whose endpoints are $\{v,v'\}$. A subset $S\subseteq
{\mathcal Q}_1$ is called a \emph{spanning set} if it connects any
two vertices in ${\mathcal Q}_0$.
\end{dfn}
One sees easily that the condition that $S$ be a spanning subset is
equivalent to the statement $\partial\Z^S = \Lambda ^{0,0}$. Thus if
$S$ is a spanning subset, any $n$-weighting ${\mathcal W}_S$ of $S$ is
completely determined by~\eqref{eq:lambda_def}, and so is unique.
In fact, if $T\subseteq S$ is any spanning tree in $S$, then $T$
determines a unique morphism ${\mathcal W}_T\colon \Lambda ^{0,0}\to\Z^n$, and
one sees that $S$ is invariant if and only
\begin{equation}
\label{eq:S_inv_wt}
{\mathcal W}_T\partial\chi _a = \pi \chi _a, \quad\forall a\in S\setminus T.
\end{equation}
The following lemma shows that, for generic values of $\zeta $, one
can effectively use the condition above to check the invariance of
configurations.
\begin{lemma}
\label{lemma:generic_flow}
If $\zeta$ is generic in $\Lambda ^{0,0}_{\mathbb R} = \Lambda
^{0,0}\otimes_\Z {\mathbb R}$ and $f$ is a $\zeta$-admissible flow then
$\supp(f)$ is a spanning subset.
\end{lemma}
\begin{proof}
If $\supp(f)$ is not a spanning subset then there exists a partition
of ${\mathcal Q}_0$ into disjoint non-empty subsets $S_0$ and $S_1$ such that
no element of $S_0$ is connected to any element of $S_1$. Consider
the restriction of $f$ to the $S_i$; the previous statement implies
that $\partial (f_{|S_i})=(\partial f)_{|S_i}$. Since, for any
$\zeta$-flow $g$, ``the flow is conserved," i.e.\xspace
$\sum_{v\in{\mathcal Q}_0}(\partial g)_v=0$, one has
\begin{align*}
\sum_{v\in S_i} \zeta_v &= \sum_{v\in S_i} (\partial f)_v \\
&= \sum_{v\in{\mathcal Q}_0} \left( \partial (f_{|S_i}) \right)_v \\
&=0,
\end{align*}
but this does not happen for a generic $\zeta$ in ${\mathbb R}^{{\mathcal Q}_0}_0$.
\end{proof}
First, an example which has some singular cones: the action of the
group of fifth roots of unity on ${\mathbb C}^3$ with weights $1$, $2$ and $3$.
In this case, there are whole cones of values of $\zeta$ for which
$X_\zeta$ is smooth, and others where $X_\zeta $ is singular.
\begin{rmk}[About the computations]
The computations were done using several computer programs. A Pascal
program was used to produce a list of all the IC-trees for any action
of a cyclic group. Then, for each value of $\zeta$, another Pascal
program was used to determine which trees were admissible and to work
out the corresponding flows and extreme points. Then a Mathematica
program was run to draw the pictures of the polyhedra and of the
extreme flows.
\end{rmk}
\subsection{Example: the action $\qsing 1/5(1,2,3)$.}
\label{sec:2:ex:5123}
Recall the action $\frac{1}{ 5}(1,2,3)$ of the group $\mu_5$ of fifth
roots of unity on ${\mathbb C}^3$ with weights $1,2$ and $3$ considered in
example~\ref{ex:5123}. The McKay quiver in this case
is the regular oriented graph with $5$ vertices and an arrow between
each vertex $v$ and $v-1$, $v-2$ and $v-3 \pmod 5$.
The set ${\mathcal T}^0$ of IC-trees contains a total of 55 trees, once one has
factored out by the symmetry which consists in permuting the vertices
cyclicly (the action of $\widehat\Gamma$ on itself). To draw $C_\zeta$,
choose values of $\zeta$, calculate the $\zeta$-flows on all the
IC-trees, discard those which are negative, and project the resulting
points to ${\mathbb R}^3$ via the map $\pi$. The resulting convex polyhedra
$C_\zeta$ are the intersection of the positive orthant with a finite
number of half-spaces. The polyhedron $C_\zeta$ for the value
$\zeta=(-1,-1,-1,-1,4)$ is shown in Figure~\ref{fig:poly5sing}. The
trees and flows corresponding to the extreme points appear in
Figure~\ref{fig:tree5sing}.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 9cm \epsfbox{fig/poly5123sing.eps}
\end{center}
\caption[$C_\zeta$ for $\qsing 1/5(1,2,3)$,
$\zeta=(-1,-1,-1,-1,4)$.]{$C_\zeta$ for $\qsing 1/5(1,2,3)$,
$\zeta=(-1,-1,-1,-1,4)$. (The view is from ``behind'', from the
point with coordinates $(-1.3,-1,-1)$. The vertices have been
numbered (where possible) in order of increasing $z$ coordinate.)}
\label{fig:poly5sing}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 4cm \epsfbox{fig/tree5123sing1.eps}
\epsfysize= 4cm \epsfbox{fig/tree5123sing2.eps}
\epsfysize= 4cm \epsfbox{fig/tree5123sing3.eps}
\vspace*{-1.5cm}
\leavevmode
\epsfysize= 4cm \epsfbox{fig/tree5123sing4.eps}
\epsfysize= 4cm \epsfbox{fig/tree5123sing5.eps}
\epsfysize= 4cm \epsfbox{fig/tree5123sing6.eps}
\vspace*{-1.5cm}
\leavevmode
\epsfysize= 4cm \epsfbox{fig/tree5123sing7.eps}
\epsfysize= 4cm \epsfbox{fig/tree5123sing8.eps}
\epsfysize= 4cm \epsfbox{fig/tree5123sing9.eps}
\vspace*{-1.5cm}
\leavevmode
\epsfysize= 4cm \epsfbox{fig/tree5123sing10.eps}
\end{center}
\vspace*{-1cm}
\caption[Extreme flows for $\qsing 1/5(1,2,3)$,
$\zeta=(-1,-1,-1,-1,4)$.]{Extreme flows for $\qsing 1/5(1,2,3)$,
$\zeta=(-1,-1,-1,-1,4)$. (The values of the flows are indicated
to the right of the arrows. The numbers in the top left-hand
corners correspond to the vertex numbers in
Figure~\ref{fig:poly5sing}.)}
\label{fig:tree5sing}
\end{figure}
One sees immediately from the figure that $X_\zeta$ has a singularity
at the point corresponding to
the extreme point $(1,3,1)$: the tangent cone there has four generators. The
other extreme points are the intersection of three faces: in order to
determine whether they are in fact singular or not one must check
whether the primitive generators in $\Pi\subset \Z^3$ of the
tangent cone actually generate $\Pi$. In this case it turns out
that they do, so they are smooth points.
In fact, there are a total of 7 singular non-isomorphic IC-trees.
These have been listed in Figure~\ref{fig:singtree}. The first tree
(rotated by $\frac{2\pi}{ 5}$) corresponds to the singular point
$(1,3,1)$ above. The corresponding tangent cone to $C_\zeta$ was
described in Section~\ref{sec:2:flow:state} --- it has four
generators $v_1,\dots,v_4$, any three of which generate $\Pi$, and
satisfying a single relation of the form $v_1+v_3=v_2+v_4$. The
corresponding singularity is therefore of the type $xw=yz
\subset{\mathbb C}^4$: a cone over a quadric surface.
\begin{figure}[htbp]
\begin{center}
\vspace*{-3cm}
\leavevmode
\epsfbox{fig/pstree5_123.eps}
\end{center}
\vspace*{-3cm}
\caption{Non-isomorphic singular IC-trees in ${\mathcal T}^0$ for the action $\frac{1}{ 5}(1,2,3)$.}
\label{fig:singtree}
\end{figure}
Checking the cases listed in Figure~\ref{fig:singtree}, one sees that
it is possible to find generic values of $\zeta$ for which none of
these trees (nor their rotations by elements of $\hat\Gamma$) are admissible: for instance, $\zeta=(9,8,-3,-2,-12)$ is
such a value; this gives a smooth resolution of ${\mathbb C}^3/\Z_5$ which has
Euler number $9$. The corresponding polyhedron and flows are shown in
Figures~\ref{fig:poly5non} and~\ref{fig:tree5non}.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 9cm \epsfbox{fig/poly5123non.eps}
\end{center}
\caption[$C_\zeta$ for $\qsing 1/5(1,2,3)$,
$\zeta=(9,8,-3,-2,-12)$.]{$C_\zeta$ for $\qsing 1/5(1,2,3)$,
$\zeta=(9,8,-3,-2,-12)$. (The view is from ``behind'', from the
point with coordinates $(-1.3,-1,-1)$. The vertices have been
numbered (where possible) in order of increasing $z$ coordinate.)}
\label{fig:poly5non}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 4cm \epsfbox{fig/tree5123non1.eps}
\epsfysize= 4cm \epsfbox{fig/tree5123non2.eps}
\epsfysize= 4cm \epsfbox{fig/tree5123non3.eps}
\vspace*{-1.5cm}
\leavevmode
\epsfysize= 4cm \epsfbox{fig/tree5123non4.eps}
\epsfysize= 4cm \epsfbox{fig/tree5123non5.eps}
\epsfysize= 4cm \epsfbox{fig/tree5123non6.eps}
\vspace*{-1.5cm}
\leavevmode
\epsfysize= 4cm \epsfbox{fig/tree5123non7.eps}
\epsfysize= 4cm \epsfbox{fig/tree5123non8.eps}
\epsfysize= 4cm \epsfbox{fig/tree5123non9.eps}
\end{center}
\vspace*{-1cm}
\caption[Extreme flows for $\qsing 1/5(1,2,3)$,
$\zeta=(9,8,-3,-2,-12)$.]{Extreme flows for $\qsing 1/5(1,2,3)$,
$\zeta=(9,8,-3,-2,-12)$. (The values of the flows are indicated
to the right of the arrows. The numbers in the top left-hand
corners correspond to the vertex numbers in Figure~\ref{fig:poly5non}.)}
\label{fig:tree5non}
\end{figure}
\begin{rmk}
In fact, all the singular trees in Figure~\ref{fig:singtree}
correspond to the same type of singularity, namely a cone over a
quadric surface.\footnote{Note that this is consistent with the
conjecture in~\cite{sacha:ale} regarding the
quadratic nature of the singularities of $X_\zeta$.} All the
IC-trees whose cones are simplicial correspond to non-singular points.
Thus one can tell whether $X_\zeta$ is non-singular simply by checking
whether all extreme points have three edges emanating from them. This
is also the case for other singularities: for instance, $\qsing
1/6(1,2,4)$, $\qsing 1/7(1,2,5)$, $\qsing 1/8(1,2,6)$, $\qsing 1/9(1,2,7)$; it
is not always the case however: for instance, for $\qsing 1/7(1,2,3)$ and
$\qsing 1/10(1,2,8)$ where there exists elements of ${\mathcal T}^0$ which
correspond to $\Z_2$-quotient singularities.
\end{rmk}
\subsection{Crepant Resolutions}
\label{sec:2:crep}
Let $\Sigma_\zeta $ denote the fan determined by the
polyhedron $C_\zeta$. By Corollary~\ref{cor:fan}, the one-skeleton of
$\Sigma_\zeta$ is given by
$$\Sigma_\zeta^{(1)} = \{ \sigma(S)^\vee: S\in{\mathcal C}^{n-1}_\zeta\}, $$ so $X_\zeta$ has
trivial canonical bundle if the primitive generators $v_S$ of the
cones $\sigma(S)^\vee$ for $S\in{\mathcal C}^{n-1}$ all belong to the hyper-plane
defined by the equation $\sum n_i = 1$ in $\Pi^* =
\Z^{3}+\frac{\Z}{ r}(w_1,w_2,w_3)$.
\begin{example}
\label{ex:3111}
Consider the action $\qsing 1/3(1,1,1)$. This has only three
IC-trees up to isomorphism. They consist of trees with two arrows of
the same type. For generic values of $\zeta$, a little
thought shows that the polyhedron $C_\zeta$ is the positive quadrant
with a small equilateral triangle chopped off. The dual fan is the
barycentric subdivision of the positive quadrant by the ray passing
through the point $v=\frac{1}{3}(1,1,1)\in \Pi^*$. This is a
non-singular fan, and since the point $v$ belongs to the hyper-plane
$\sum n_i = 1$, this has trivial canonical bundle. The
variety $X_\zeta$ is the total space of the bundle ${\mathcal O}(-3)$ over
${\mathbb P}^2$.
\end{example}
To conclude this section, a picture of a more complicated
example is draw. Note that $\Gamma\subset\SU(3)$ and that the variety
$X_{\zeta}$ is a smooth crepant resolution with Euler number 11.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfysize= 9cm \epsfbox{fig/poly11_146.eps}
\end{center}
\caption{An example of $C_\zeta$ for the action $\qsing 1/11(1,4,6)$.}
\label{fig:11_146}
\end{figure}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
|
1996-10-01T04:57:38 | 9610 | alg-geom/9610001 | en | https://arxiv.org/abs/alg-geom/9610001 | [
"alg-geom",
"math.AG"
] | alg-geom/9610001 | Sacha Sardo Infirri | Alexander V. Sardo Infirri | Crepant Terminalisations and Orbifold Euler Numbers for SL(4)
Singularities | LaTex2e, 22 pages with 1 table | null | null | null | null | Let $X$ and $Y$ be two analytic canonical Gorenstein orbifolds. A resolution
of singularities $Y\to X$ is called an Euler resolution if $Y$ and $X$ have the
same orbifold Euler number. If $Y$ is only terminal rather than smooth, it is
called an Euler terminalisation. It is proved that Euler terminalisations exist
for toric varieties in any dimension, for 4-dimensional toroidal varieties, and
for singularities $\C^4/G$ where $G$ belongs to certain classes of $\SL(4)$
subgroups. The method of proof is expected to be applicable to a sizeable
number of finite $\SL(4)$ subgroups and to lead to a generalisation of the
Dixon-Harvey-Vafa-Witten orbifold Euler number conjecture to dimension~4.
| [
{
"version": "v1",
"created": "Tue, 1 Oct 1996 02:48:18 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Infirri",
"Alexander V. Sardo",
""
]
] | alg-geom | \part{Maps}
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\end{array}$$}
\newcommand{\map}{\map}
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\newcommand{\bar\Gamma}{\bar\Gamma}
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\newcommand{\bar G_{\xi}}{\bar G_{\xi}}
\newcommand{\bar G_{\eta\eta'}}{\bar G_{\eta\eta'}}
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\newcommand{\subpart}{\prec}
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\newcommand{{\cGo}s\xspace}{{canonical Gorenstein orbifold\xspace}s\xspace}
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\newtheorem{prop}[thm]{Proposition}
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\begin{document}
\title[\runningheadstring]{\titlestring\footnote{{Maths Subject
Classification (1991): 32S45 (Primary) 14L30 14E30 (Secondary)}}}
\author{Alexander V. Sardo Infirri}
\email{[email protected]}
\address{Research Institute for Mathematical Sciences\\ Ky\=oto University\\
Oiwake-ch\protect\=o\\ Kitashirakawa\\ Saky\protect\=o-ku\\ Ky\=oto
606-01\\ Japan}
\date{1 October 1996}
\begin{abstract}
Let $X$ and $Y$ be two analytic canonical Gorenstein orbifolds. A
resolution of singularities $Y\to X$ is called an \emph{Euler
resolution} if $Y$ and $X$ have the same orbifold Euler number. If
$Y$ is only terminal rather than smooth, it is called an \emph{Euler
terminalisation}.
It is proved that Euler terminalisations exist for toric varieties in
any dimension, for 4-dimensional toroidal varieties, and for
singularities ${\mathbb C}^4/G$ where $G$ belongs to certain classes of
$\operatorname{\rm SL}(4)$ subgroups. The method of proof is expected to be
applicable to a sizeable number of finite $\operatorname{\rm SL}(4)$ subgroups and to
lead to a generalisation of the Dixon-Harvey-Vafa-Witten orbifold
Euler number conjecture to dimension~4.
\end{abstract}
\maketitle
\tableofcontents
\setcounter{section}{-1}
\section{Introduction}
\label{sec:intro}
An analytic $n$-fold $X$ will be called a {\em canonical Gorenstein
orbifold\/} if it has at most canonical Gorenstein singularities and
is such that, for each $x\in X$, there exists a finite group
$\pi_x<\operatorname{\rm SL}(n)$ such that
$$(X,x)\cong ({\mathbb C}^n/\pi_{x}, 0)$$
as germs of analytic spaces.
\subsection{The Orbifold Euler Number}
\label{sec:intro:orbi}
The {\em orbifold Euler number\/} of $X$ is defined as the (finite) sum
\begin{equation}
\label{eq:orb-eul-def}
\chi_{\text{orb}}(X) := \sum_{k\geq 1} k\chi(m^{-1}(k)),
\end{equation}
where $\chi$ is the ordinary Euler number and
\map{m}{X}{\Z}{x}{|\operatorname{\rm Cl}(\pi_x)|} is the upper semi-continuous map
assigning to each point $x$ the number of conjugacy classes of
$\pi_x$. It is easy to show~\cite{roan:calabi-yau} that if $M$ is an
$n$-fold admitting a $G$-action whose non-trivial elements' fixed-point
loci have codimension at least two, and such that $M/G$ has only
Gorenstein singularities, then
\begin{align}
\chi_{\text{orb}}(M/G) &\phantom{:}= \chi_{DHVW}(M;G),\\
&:= \sum_{[g]\in\operatorname{\rm Cl}(G)}\chi(M^g/\operatorname{\rm N}^G_g)
\end{align}
where the right-hand side denotes the Dixon-Harvey-Vafa-Witten Euler
number proposed in~\cite{dhvw:i,dhvw:ii}.
If $X$ and $Y$ are {\cGo}s\xspace and $Y\to X$ is a bi-meromorphic map such
that $\chi_{\text{orb}}(Y)=\chi_{\text{orb}}(X)$, then $Y$ will be called an {\em Euler
blow-up\/} of $X$. If $Y$ is in addition smooth, then $Y$ will be called
an {\em Euler resolution}. Restated in the above terminology, the
Dixon-Harvey-Vafa-Witten Euler conjecture~\cite{dhvw:i} is
\begin{quotation}
Every $3$-dimensional canonical Gorenstein orbifold\xspace has an Euler resolution.
\end{quotation}
It took ten years to give a positive answer to the conjecture
\cite{mar_ols_per,roan:mirror_cy,mark:res_168,roan:res_a5,ito:trihedral,roan:crepant}.
\subsection{Euler Terminalisations}
\label{sec:intro:euler-term}
Right from the start it was recognized that the analogous conjecture
in dimension 4 (Do all 4-dimensional {\cGo}s\xspace have Euler resolutions?) is
trivially false: the simplest non-smooth example ${\mathbb C}^4/\<<-1>>$ is
already terminal.
However, rephrased slightly, the conjecture can be made to look much
more promising. For this, note that the existence of Euler
resolutions in dimension 3 is equivalent to saying that the minimal
models for these singularities are smooth. In other words ``smooth'' is
equivalent to ``terminal'' for 3-dimensional Gorenstein finite
quotient singularities.
\begin{dfn}
A Euler blow-up $Y\to X$ such that $Y$ has only terminal
singularities is called an {\em Euler terminalisation\/} of $X$.
The property $\text{Term}(X)$ is defined to be true if and only if such a
$Y$ exists.
\end{dfn}
\begin{dfn}
The property $\text{Term}(n)$ is defined to be true if and only if
$\text{Term}(X)$ is true for all $n$-dimensional {\cGo}s\xspace $X$.
\end{dfn}
Thus, $\text{Term}(2)$ is true in virtue of classical work and $\text{Term}(3)$ is
true by the recent work mentioned above.
\begin{question}
\label{conj}
Is $\text{Term}(n)$ true?
\end{question}
The next open case is of course $\text{Term}(4)$. As in the case of
dimension~3, the question reduces to the problem of constructing Euler
terminalisations for the local singularities ${\mathbb C}^{4}/G$ for all the
small subgroups $G$ of $\operatorname{\rm SL}(4)$.
We shall use the following terminology. A subgroup $G < \operatorname{\rm SL}(V)$ will
be called \emph{reducible} if $V$ is reducible as a $G$-module. The
{\em type\/} of a group $G<\operatorname{\rm SL}(n)$ denotes the dimensions of the
irreducible representations of $G$ appearing in the chosen special
linear representation ${\mathbb C}^n$. For instance, irreducible groups have
type $(n)$ and abelian groups have type $(1,1,\dots,1)$.
For any $n$, denote by $Z_n$ the cyclic central subgroup of $\operatorname{\rm SL}(n)$.
For any element $g\in G$, $\operatorname{\rm N}^G_g$ denotes the
centralizer of $g$ in $G$, namely $\{h\in G | h^{-1}gh=g\}$
\subsection{Main Results}
\label{sec:intro:results}
\subsubsection{Toric and Toroidal cases}
The first result is that $\text{Term}(X)$ is true in all dimensions for
toric varieties. Further more, it holds also for toroidal varieties
(analytic varieties which are locally isomorphic to toric varieties)
in dimension~4 and in dimension~$n$ if termination of flips can be
proved.
\begin{thm}
\label{thm:toric-toroidal}
All simplicial toric {\cGo}s\xspace have Euler terminalisations.
Furthermore, if flips terminate in dimension $n$, then all
$n$-dimensional \emph{toroidal } {\cGo}s\xspace have Euler terminalisations. In
particular, this is the case in dimension 4.
\end{thm}
The proof of the toric case is straightforward; crepant blow-ups of
$X$ correspond to subdividing the first quadrant in $\R^n$ by rays
whose generators all lie in the same hyper-plane; the orbifold Euler
number of any cone is equal to its volume (meaning the volume of the
simplex spanned by its generators), and since the sum of the volumes
of the cones in the subdivision is equal to the total volume of the
original quadrant, the orbifold Euler number is seen to remain
invariant under crepant blow-ups. The fact that among all the crepant
blow-ups there exists a terminal one is a consequence of the Toric
Minimal Model Program~\cite{reid:toricmmp}.
There is no minimal model program as yet for non-toric varieties in
dimensions 4 and over. However, in the toroidal case, a well-known
technique makes it is possible to construct flips by patching together
local toric flips and using uniqueness. Thus if termination of flips
is also known, (as it is in dimension 4~\cite{kmm:intro_mmp}), the
existence of the terminal model follows and this again must have the
same orbifold Euler number as the original variety.
\subsubsection{$\operatorname{\rm SL}(4)$ subgroups of type $(3,1)$}
The second set of results concerns 4-dimensional non-abelian
singularities created by finite $\operatorname{\rm SL}(4)$ subgroups of type (3,1).
Let $G<\operatorname{\rm SL}(4)$ be a finite subgroup which stabilises a line
$V^2\subset V={\mathbb C}^{4}$ and let $V^1$ be a $G$-submodule such that
$V=V^1\oplus V^2$. Denote by $\eta$ the generic point of the line
$\{0\}\times V^2$, and by $G_\eta$ the stabiliser of $\eta$. Note that
$G_\eta$ is a subgroup of $\operatorname{\rm SL}(V^1)\times\{1\}\cong \operatorname{\rm SL}(3)$.
\begin{thm}
If $G<\operatorname{\rm SL}(4)$ fixes a line and is such that the group $G_\eta$ does not
contain $Z_3$ as a subgroup, then ${\mathbb C}^4/G$ has an Euler
terminalisation with only toric singularities.
\end{thm}
The method of proof essentially consists in using the results of
Roan \cite{roan:calabi-yau} to construct a 3-dimensional Euler
resolution of ${\mathbb C}^3/G_\eta$ which is equivariant under the larger group
$G$.
As mentioned in the next section, the assumption that $G_\eta$ does
not contain the group $Z_3$ is not essential to the method. However,
so far the author has been unable to construct the equivariant
resolutions without it. An attempt is made in Section~\ref{sec:3-1:centre:2}.
\subsubsection{$\operatorname{\rm SL}(4)$ subgroups containing $Z_4$}
The third set of results reduces the question $\text{Term}({\mathbb C}^4/G)$ for the
irreducible $G$ which contain $Z_4$ to a conjecture regarding the
existence of the equivariant resolutions of $\operatorname{\rm SL}(3)$ mentioned above.
The conjecture is proved for irreducible subgroups of $\operatorname{\rm SL}(3)$ which
do not contain $Z_3$, but remains open in the general case.
\begin{rmk}
The material presented here can no doubt be pushed further, but the
author has so far been unable to do so. Nevertheless, it is hoped
that the attentive reader will be able to perceive a direction in
which to proceed. Is is also conceivable that the general strategy
that emerges from this approach may be applicable to an understanding
of quotient singularities in higher dimensions.
\end{rmk}
The idea here is as follows. Suppose that $G<\operatorname{\rm SL}(n)$ acts on ${\mathbb C}^{n}$
and contains the centre $Z_{n}$, and write $\bar G:=G/Z_n$ and $\bar V :=
\text{Bl}_0V/Z_n$. Then Lemma~\ref{lemma:bV} implies that $\bar V/\bar G$
is another canonical Gorenstein orbifold\xspace and $\bar V/\bar G \to V/G$ is an Euler blow-up. With the
aid of another Lemma (The Patching Lemma~\ref{lemma:patching}), the
problem is thus reduced to the construction and patching of Euler
blow-ups of local neighbourhoods of $\bar V/\bar G$. The advantage of these
is that the stabilisers of $\bar G$ in the tangent space to the blowup
$\bar V$ are simpler than those of $G$ (because the stabiliser of $\bar G$
must fix a line in the tangent space to $\bar V$, so its type must be
$(t,1)$, where $t$ is a partition of $n-1$).
This approach is spelt out in Section~\ref{sec:4-centre} for
irreducible subgroups of $\operatorname{\rm SL}(4)$ which contain $Z_{4}$; it reduces
the problem to constructing and patching together equivariant $\operatorname{\rm SL}(3)$
resolutions.
\subsubsection{Discussion}
What if, on the other hand, the group $G$ is irreducible, but doesn't
contain the centre of $\operatorname{\rm SL}(V)$?
A complete answer to Question~\ref{conj} for all finite $\operatorname{\rm SL}(4)$
singularities seems out of reach of the methods suggested here, if only
because the cases when $G$ does not contain $Z_4$ include simple
groups, such as the alternating group $A_5$; if dimension~3 is any
indication~\cite{mark:res_168,roan:res_a5}, it seems that ad-hoc
methods will be necessary to construct a terminalisation.
However, the possibility is open that, as a general rule, the
non-simple finite subgroups $\operatorname{\rm SL}(n)$ not containing $Z_n$ are few in
number and relatively amenable in form. For instance, if $\dim V=2$
no cases occur. For $\dim V=3$, one has --- apart from the simple
groups (H) and (I) which require ad-hoc methods --- ``half'' the
groups of type (C) and ``half'' of those of type (D). It turns out
that under the assumption $Z_3\not < G$ these are semi-direct products
of abelian groups with the alternating group $A_3$ and the symmetric
group $S_3$ respectively. This allows one to construct their Euler
resolutions from toric resolutions --- see~\cite{roan:calabi-yau}.
Extension of this method to the $\operatorname{\rm SL}(4)$ case would seem to be
feasible.
Table~\ref{tab:sl4} outlines the state of the $\text{Term}({\mathbb C}^{4}/G)$
problem.
\begin{table}[htbp]
\begin{center}
\leavevmode
\setlength{\fboxrule}{0pt}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{3}{|c|}{Type of Group} & Method & Status\\
\hline
\multirow{7}{12mm}{Irred.\ \textbf{(4)}} &
\multirow{2}{16mm}{$G\not > Z_{4}$}
& \multirow{1}{22mm}{Simple} & \fbox{Ad hoc method?} &\\
\cline{3-5}
& & \multirow{1}{22mm}{Non-Simple} &
\fbox{\parbox{3.2cm}{Small \# of cases?
Semi-direct products of abelian groups with $A_{4}$ and
$S_{4}$?}} & \\
\cline{2-5}
& \multicolumn{2}{l|}{$G > Z_{4}$} &
\fbox{\parbox{3.2cm}{4-d Euler blow-up of origin. Patch
together equivariant lower-d blowups.}} &
\S\ref{sec:4-centre}\ \ \ ?\\
\hline
\multirow{5}{12mm}{Red.} & \multirow{2}{16mm}{\textbf{(3,1)}}
& \multirow{1}{22mm}{$ G_\eta\not >Z_{3}$} &
\fbox{\parbox{3.2cm}{Construct equivariant 3-d Euler
resolution. Reduces to the toric case.}}
& \S\ref{sec:3-1:nocentre} OK\\
\cline{3-5}
& & \multirow{1}{22mm}{$G_\eta >Z_{3}$} & \fbox{\parbox{3.2cm}{Euler
blow-up of fixed line. Reduce to 2-d equivariant blow-up.}} &
\S\ref{sec:3-1}\ \ \ ?\\
\cline{2-5}
& \multicolumn{2}{l|}{\textbf{(2,2)}} & \fbox{Use 2-d results} &
\S\ref{sec:2-2} ?\\
\cline{2-5}
& \multicolumn{2}{l|}{\textbf{(2,1,1)}}
& \fbox{Use 2\&3-d results} & \S\ref{sec:2-1-1} ?\\
\cline{2-5}
& \multicolumn{2}{l|}{\textbf{(1,1,1,1)} (Abelian)} & \fbox{Toric
MMP} & \S\ref{sec:toric}\ \ \ OK\\
\hline
\end{tabular}
\vspace{1em}
\caption{Constructing Euler Terminalisations of ${\mathbb C}^4/G$
for $G<\operatorname{\rm SL}(4)$. (Bold numbers in parentheses indicate the {\em
type\/} of the group. In the cases $(3,1)$, the group $G_\eta$ denotes
the stabiliser in $G$ of a generic point of the line fixed by $G$. )}
\label{tab:sl4}
\end{center}
\end{table}
The {\em Status\/} column indicates the section of this paper which
makes some contribution to the problem. Questions which are solved in
this paper are indicated by the mention ``OK''. A question mark
indicates that the question is still open and that methods of this
paper are relevant. Finally, where no results are known, nothing
is indicated in the Status column, but some guesses are given as to
the likely situation, based on the present state of knowledge.
\subsection{Open Problems}
\label{sec:open}
Interesting open problems arise in relation to recent work of Ito and
Reid~\cite{reid_ito} which establishes a one-one correspondence
between crepant divisors of $V/G$ and conjugacy classes ``of weight
1'' in (the dual group to) $G$.
One interesting question is how this correspondence behaves under the
Euler blow-ups which are constructed for the $\operatorname{\rm SL}(4)$ singularities
mentioned above. A deeper understanding of this (apart from being of
interest in itself) would also no doubt allow one to say more about
the singularities of the terminalisation.
For instance, for toric and toroidal varieties, the Euler blow-up can
be made projective rather than just analytic. But in general, the
gluing process in the Patching Lemma~\ref{lemma:patching} does not in
itself guarantee that the blow-up $X$ will be projective, because a
divisor which is reducible when considered locally in the
neighbourhood of one of the points $[\xi]$ might have two of its
components identified by the gluing process. Such troublesome cases
might conceivably be ruled out by a deeper understanding of the
correspondence between divisors and conjugacy classes.
\subsection{Outline}
\label{sec:intro:outline}
Section~\ref{sec:toric} deals with the construction of Euler
terminalisations for the toric and toroidal cases.
Section~\ref{sec:3-1} deals with the finite $\operatorname{\rm SL}(4)$ groups of type
(3,1). Section~\ref{sec:blowing-up} presents a blowing-up
construction and some other technical lemmas which are then used in
Section~\ref{sec:4-centre} in dealing with irreducible finite $\operatorname{\rm SL}(4)$
subgroups containing $Z_4$. Finally, Section~\ref{sec:2-} makes some
comments regarding the finite $\operatorname{\rm SL}(4)$ groups of type (2,2) and
(2,1,1).
\subsection{Acknowledgments}
\label{sec:intro:ack}
I wish to acknowledge S.Mori and S-S.Roan for the many ideas and
suggestions they contributed to this paper.
This research was undertaken at the Research Institute for
Mathematical Sciences of Kyoto University thanks to an European
Commission Science and Technology Fellowship; I am grateful to my host
institution and its staff for their hospitality and to the European
Commission for their financial support.
\section{Toric and Toroidal Cases}
\label{sec:toric}
\begin{thm}[Toric Minimal Model Program]
\label{thm:toric-mmp}
Let $Y$ be a toric variety and $X $ be a simplicial toric variety
which admits a projective birational toric morphism $f\colon X \to Y$.
Then there exists a sequence $X \stackrel h\dashrightarrow
Z\xrightarrow{g}Y$ such that
\begin{enumerate}
\item $h$ is a composite of toric divisorial contractions or toric flips.
\item $g$ is a projective morphism and $Z$ is a simplicial toric
variety with terminal singularities such that $K_Z$ is relatively
nef for $g$.
\end{enumerate}
Note that if $Y$ has canonical singularities, then $K_Z=g^*K_Y$ in
the sense of $\Q$-Cartier divisors, i.e.\xspace $g$ is crepant.
\end{thm}
\begin{proof}
See~\cite[Theorem 0.2]{reid:toricmmp}, where the result is proved
under the assumption that $Y$ is projective. As
remarked in~\cite{reid:toricmmp} this assumption is not essential and
the result is valid for non-complete toric varieties also (the
easiest way to see this is to reduce the non-projective case to the
projective one by completing the fan in an appropriate way).
\end{proof}
\begin{cor}
\label{cor:abelian-terminal}
All toric {\cGo}s\xspace admit (toric) Euler terminalisations.
\end{cor}
\begin{proof}
Note that a toric variety has at most orbifold singularities if and
only if it is simplicial. It is therefore sufficient to prove that
any crepant blow-up of a simplicial Gorenstein toric variety must
have the same orbifold Euler number as the original. But this is
true because the orbifold Euler number of a simplicial toric variety
is just the volume\footnote{Also called the {\em multiplicity\/} of
the cone.} of the cone, meaning the volume
of the simplex defined by the cone's generators; a crepant blow-up
corresponds to a fan subdivision by one-dimensional rays whose
primitive generators all belong to the same plane, and therefore the
sum of the volumes of the cones in such a subdivision is equal
to the volume of the original cone.
\end{proof}
The minimal model program for general varieties is at present only
proved in dimension~3. In dimension~4, although termination has been
shown~\cite{kmm:intro_mmp} existence of flips remains a problem.
Nevertheless, a technique well-known to minimal model program
specialists allows one to use the Theorem above and the termination
result to say something about toroidal varieties, i.e.\xspace varieties
which are only locally isomorphic to toric varieties. The argument
can phrased for general $n$, even though at present, termination has
only been proved for $n\leq 4$.
\begin{thm}
\label{thm:toroidal-mmp}
Assume that flips terminate in dimension $n$. Then all
$n$-dimensional toroidal canonical Gorenstein orbifold\xspace admit Euler terminalisations (which are
themselves toroidal). In particular, this is true in dimension~4.
\end{thm}
\begin{proof}
Let $Y$ be a canonical Gorenstein orbifold\xspace locally isomorphic to a toric variety and let
$p\colon X\to Y$ be any resolution obtained by toric blowups.
Suppose that $K_X$ is not $p$-nef, and let $c\colon X \to W$ be an
extremal contraction. If it is a divisorial contraction, then
replace $X$ by $W$. If $c$ is a small contraction, consider its
restriction $C_{U_X}\colon U_X \to U_W$ to the inverse image of a
local toric neighbourhood $U_Y\subset Y$.
Theorem~\ref{thm:toric-mmp} implies the existence of a local flip
$c^+_{U_X}\colon U^+_X \to U_W$ over each neighbourhood $U_W$. A
flip being unique if it exists, the local flips patch together on
the overlaps to form a global flip $c^+\colon X^+\to W$. Thus,
existence of flips is guaranteed in this case. Applying the same
procedure repeatedly (and using the termination hypothesis) results
in a projective morphism $p\colon Z\to Y$ such that $Z$ has
$\Q$-factorial terminal singularities with $K_Z$ being $p$-nef,
which means that $p$ is crepant, since $Y$ is canonical.
Furthermore, any toric terminalisation which is crepant must have
the same orbifold Euler number by the volume argument in
Corollary~\ref{cor:abelian-terminal}.
\end{proof}
Corollary~\ref{cor:abelian-terminal} and
Theorem~\ref{thm:toroidal-mmp} together give
Theorem~\ref{thm:toric-toroidal}.
\section{Groups $G$ of type (3,1)}
\label{sec:3-1}
\subsection{Notation}
\label{sec:3-1:notation}
Let $G<\operatorname{\rm SL}(n+1)$ be a finite subgroup such that $V={\mathbb C}^{n+1}$
decomposes into two irreducible $G$-modules: $V=V^1\oplus V^2$, with
$V^1$ of dimension $n$, and $V^2$ of dimension 1.
Denote by $\eta$ the generic point of the line $\{0\}\times V^2$, and
by $G_\eta$ the stabiliser of $\eta$. Note that $G_\eta$ is a subgroup of
$\operatorname{\rm SL}(n)\times\{1\}\cong \operatorname{\rm SL}(n)$. The quotient $C:= G/G_\eta$ is cyclic,
being a naturally a subgroup of $\operatorname{\rm GL}(V^2)\cong{\C^*}$. The canonical
quotient map is denoted $\pi\colon G\to C$ and the induced map on
conjugacy classes is denoted $\pi_*\colon\operatorname{\rm Cl}(G)\to
\operatorname{\rm Cl}(C)$.
Note that $G$ can be considered as a subgroup of $\operatorname{\rm GL}(V^1)=\operatorname{\rm GL}(n)$ by
forgetting about the last row and column of any matrix element. Let
$h\in G$ be an element such that $G=\<<G_\eta,h>>$ --- i.e.\xspace a
representative for a generator of $C$ --- and denote by $h_1$ the
$n\times n$ sub-matrix consisting of the first $n$ rows and columns of
$h$. Let $\lambda_h$ be a complex $n$-th root of $\det(h_1)^{-1}$.
Then $h':= \lambda_h h_1\in\operatorname{\rm SL}(n)$ and normalises $G_\eta$. Hence $G_\eta$
is a normal subgroup of $G':=\<<G_\eta,h'>> < \operatorname{\rm SL}(n)$ with quotient a
cyclic subgroup $C'$. Note that $G'$ and $C'$ are defined up to a
choice of the $n$-th root $\lambda_h$.
The canonical quotient map is denoted $\pi'\colon G'\to C'$ and the
induced map on conjugacy classes is denoted
$\pi^{\prime}_*\colon\operatorname{\rm Cl}(G')\to \operatorname{\rm Cl}(C')$.
\subsection{Equivariant Resolution}
\label{sec:3-1:resolution}
From now on, let $n=3$, i.e.\xspace consider the case where $G<\operatorname{\rm SL}(4)$ fixes
a line in ${\mathbb C}^4$ (and therefore $G_\eta<\operatorname{\rm SL}(3)$).
\begin{conj}
\label{conj:3-1-equiv-Euler}
Let $G<\operatorname{\rm SL}(4)$ be a finite subgroup which stabilises a line
$V^2\subset V={\mathbb C}^4$ and let $G_\eta$ denote the stabiliser in $G$ of
the generic point of $V^{2}$ and let $G'$ and $C'$ be defined as in
Section~\ref{sec:3-1:notation} above.
Then there exists a $G'$-invariant Euler resolution $W^1\to
V^1/G_\eta$, satisfying
\begin{equation}
\label{eq:chi-pi}
\chi((W^1)^{c'}/{C'})=|\pi^{\prime -1}_*([c'])|,
\end{equation}
for all $c'\in C'$.
\end{conj}
If true, we shall see that this conjecture implies that
$\text{Term}({\mathbb C}^{4}/G)$ is true for groups stabilizing a line (i.e.\xspace types
$(3,1)$ and $(2,1,1)$) and (cf.\ Section~\ref{sec:4-centre}) for all
irreducible $G$ which contain $Z_{4}$.
\begin{prop}
\label{prop:3-1-invariant-resolution-gives-terminalisation}
Suppose that Conjecture~\ref {conj:3-1-equiv-Euler} is true for $G$.
Then $W^1\times V^2/C\to V/G$ is an Euler blowup,
and $V/G$ has an Euler terminalisation with only
toric singularities.
\end{prop}
\begin{proof}
First, note that since $V^1/G_\eta$ has dimension at most three, it has
a minimal model. The fact that there are only a finite number of
distinct minimal models implies that ${\C^*}$ acts on any of them: for
if the action of some element $\lambda\in{\C^*}$ produced a different
minimal model, then, by continuity, one could produce countably many
distinct minimal models by acting with a countable family of
distinct neighbours of $\lambda$.
Second, note that the equality~\eqref{eq:chi-pi} would follow from
the same equality with $C'$ replaced by the group $C$. For if
$\phi$ is any element of $G$, one can find $\lambda\in{\C^*}$ such that
$\phi'=\lambda \phi\in G'$. The invariant sets $(W^1)^\phi$ and
$(W^1)^{\phi'}$ have the same homotopy type, by an easy application
of Bialynicki-Birula's well-known decomposition
theorem~\cite[Thm.~4.1]{bb:algebraic_groups} to the smooth variety
$W^1$. Hence $\chi((W^1)^\phi)=\chi((W^1)^{\phi'})$ and so,
averaging,
\begin{equation}
\label{eq:chi-equality}
\chi((W^1)^\phi/C)=\chi((W^1)^{\phi'}/C').
\end{equation}
On the other hand, if $c$ and $c'$ denote the images in $C$ and $C'$
of $\phi$ and $\phi'$ respectively,
\begin{equation}
\label{eq:pi-equality}
|\pi_*^{-1}([c])|=|\pi^{\prime -1}_*([c'])|,
\end{equation}
where $\pi$ denotes the projection $\pi\colon G\to C$ and
$\pi_*\colon\operatorname{\rm Cl}(G)\to\operatorname{\rm Cl}(C)$ the induced map on
conjugacy classes, and similarly for the primed symbols.
Hence one can assume formula~\eqref{eq:chi-pi} to be valid for the
group $C$. The variety $(W^1\times V^2)/G$ is a blow-up of $V/G$
having only cyclic quotient singularities resulting from the
residual action of $C=G/G_\eta$ on $W^1\times V^2$.
Its orbifold Euler number can be expressed as a sum:
\begin{align}
\chi_{\text{orb}}(W^1\times V^2/C) &=\sum_{[c]\in\operatorname{\rm Cl}(C)}
\chi((W^1\times V^2)^c/C),\quad\text{since $\operatorname{\rm N}^C_c=C$}\notag\\
&= \sum_{[c]\in\operatorname{\rm Cl}(C)} \chi((W^1)^c/C),\\
&\qquad\text{since $(V^2)^c$ is contractible}.
\label{eq:orb-sum}
\end{align}
On the other hand, one has:
\begin{equation}
\label{eq:cclass-sum}
|\operatorname{\rm Cl}(G)|= \sum_{[c]\in \operatorname{\rm Cl}(C)} |\pi_*^{-1}([c])|,
\end{equation}
which agrees with the previous sum term-by-term. Hence, $Y:=(W^1\times
V^2)/G\to (V^1\times V^2)/G$ is an Euler blow-up with only toric
(cyclic) singularities.
Applying the minimal model program (Theorem~\ref{thm:toroidal-mmp})
to $Y$, one obtains a crepant terminalisation $t\colon Z\to Y$ which
satisfies $\chi_{\text{orb}}(Z)=\chi_{\text{orb}}(Y)=\chi_{\text{orb}}(V/G)$, and has only toric
singularities.
\end{proof}
Thus in the case where $G$ fixes a line in $V={\mathbb C}^4$ it suffices to
prove the existence of a $G'$-equivariant Euler resolution $W^1\to
V^1/G_\eta$ which satisfies equation~\eqref{eq:chi-pi}.
\begin{rmk}
The same method as above can be used to deal with the easier case
when $n=2$ and $G<\operatorname{\rm SL}(3)$ and fixes a line in ${\mathbb C}^3$. In 2
dimensions, the minimal model is unique, so there is no need to
check $G'$-stability of the Euler resolution of $V^1/G_\eta$.
\end{rmk}
\subsection{Case where $G_\eta$ doesn't contain $Z_3$}
\label{sec:3-1:nocentre}
In this section, Conjecture~\ref{conj:3-1-equiv-Euler} is proved in
the case where $G_\eta$ is irreducible and does not contain $Z_{3}$.
In order to construct a $G$-equivariant Euler resolution $W^1\to
V^1/G_\eta$, the cases to be considered are first restricted using the
following lemma (which makes use of the classification of small finite
sub-groups of $\operatorname{\rm SL}(3)$ --- see~\cite{yau_yu}, although the notation
adopted here is that of~\cite{roan:calabi-yau}, which is slightly
different).
\begin{lemma}
\label{lemma:3-1-nocentre-exact}
Suppose that
\begin{equation}
\label{eq:3-1-nocentre-exact}
1 \to G_\eta \to G' \xrightarrow{\pi'} C' \to 1,
\end{equation}
is an exact sequence of finite groups of $\operatorname{\rm SL}(3)$
such that $C'$ is cyclic and non-trivial, and $G_\eta$ is non-abelian and
doesn't contain $Z_3=\<<\omega_3>>$.
Then exactly one of the following is true.
\begin{enumerate}
\item $G_\eta$ is of type (B) and $G'$ is of type (B).
\item $G_\eta$ is of type (C), (D), (H) or (I) and $G' =
\<<G_\eta,\omega_3>>$.
\item $G_\eta$ is of type (C) and $G'$ is of type (D), with
$G'=\<<G_\eta,R>>$ or $G'=\<<G_\eta,\omega_3R>>$.
\end{enumerate}
\end{lemma}
\begin{proof}
The groups of type (E), (F), (G), (H${}^*$), (I${}^*$) all contain $Z_3$, so
do not occur as the group $G_\eta$ by assumption. The only finite subgroup of
$\operatorname{\rm SL}(3)$ containing the simple group (H) (resp.\ the simple group
(I)) as a normal subgroup is (H${}^*$) (resp.\
(I${}^*$))~\cite[p.36]{yau_yu}. Thus for these, the result follows
immediately.
If $G_\eta$ has type (B), a simple argument~\cite[\S1.4, p.18]{yau_yu}
shows that $G'$ must also have type (B).
It remains to deal with the case where $G_\eta$ has type (C) or (D).
Throughout the rest of this proof, write
$$T:=
\begin{pmatrix}
0& 1& 0\cr 0& 0& 1\cr 1& 0& 0
\end{pmatrix},
$$ for the element which, together with a diagonal group, generates a
group of type (C). To get a group of type (D), recall that one must
add to a group of type (C) an element of the form
\begin{equation}
\label{eq:phi}
\phi= \begin{pmatrix} a& 0& 0\cr 0& 0& c\cr 0& b& 0\cr
\end{pmatrix} \quad\text{with }abc=-1.
\end{equation}
\begin{claim}
If $G_\eta$ has type (C) or (D), then $G'$ must also be of type (C)
or (D) (though not necessarily the same type as $G_\eta$).
\end{claim}
\begin{proof}
Denote by $x_1, x_2, x_3$ the standard coordinates on ${\mathbb C}^3$. If
$G_\eta$ is of type (C) or (D) and does not contain the centre $Z_3$,
then the monomial $x_1x_2x_3$ is invariant under $G_\eta$ up to
scale~\cite[\S 1.3]{yau_yu}. It follows from an easy argument
that $G'$ must also leave ${\mathbb C} x_1x_2x_3$ invariant. Thus $G'$
cannot be primitive. This means that $G'$ must be of type (C) or
(D).
\end{proof}
The next step is to study the normal diagonal subgroups of $G_\eta$ and
$G'$, which are denoted by $H_\eta$ and $H'$ respectively.
\begin{notation}
The standard toric notation for diagonal matrices will be used:
$$\qsing1/d(r_1, r_2, \dots, r_n):\equiv [\exp({\frac{2\pi i r_1}{ d}}),
\exp({\frac{2\pi i r_2}{d}}), \dots, \exp({\frac{2\pi i r_n}{d}})]$$
\end{notation}
\begin{claim}
\label{claim:order3}
If $H'$ contains an element of order $3$ then that element must be
$\omega_3$ or $\omega_3^2$. As a consequence, all the elements of
$H_\eta$ have orders prime to $3$.
\end{claim}
\begin{proof}
If $x\in H'$ has order $3$ and its does not belong to the centre
$Z_3$ then it can be chosen to be of the form
$x=\qsing1/3(i,i+1,i+2)$ for some $i\in\{0,1,2\}$. But then
\begin{align*}
(TxT^{-1})x^{-1} &= \qsing1/3(i+1,i+2,i) - \qsing1/3(i,i+1,i+2)\\
&= \qsing1/3(1,1,-2)\\
&= \qsing1/3(1,1,1)=\omega_3
\end{align*}
On the other hand, since $x$ normalises $G_\eta$, one has
$xT^{-1}x^{-1}\inG_\eta$ and so $\omega_3=TxT^{-1}x^{-1}\inG_\eta$, which
contradicts the hypothesis of the lemma. Thus the only elements of
order $3$ in $H'$ are $\omega_3$ and $\omega_3^2$. As a consequence,
$H_\eta$ has no elements of order $3$, since $Z_3\not < H_\eta$. The
claim for $H_\eta$ follows immediately from this.
\end{proof}
\begin{claim}
\label{claim:hprime}
$H' < \<< H_\eta,\omega_3 >>$, i.e. $H'$ is either equal to
$H_\eta$ or equal to $\<< H_\eta,\omega_3>>$.
\end{claim}
\begin{proof}
Let $\varphi\in H'$. I begin by showing that $\varphi^3\in H_\eta$.
Since $H'$ is normal in $G'$, the element $T \varphi T^{-1}$ is
diagonal, and so, therefore, is $f:=\varphi^{-1}T \varphi T^{-1}$.
One has $fT = \varphi^{-1} T\varphi\in G_\eta$, since $G_\eta$ is
normal in $G'$, so $f$ belongs to $G_\eta$. Since $f$ is also
diagonal, it follows that $f\inH_\eta$.
Writing $\varphi = \qsing 1/d(a,b,-a-b)$, one has
\begin{align*}
f &=\varphi^{-1}(T \varphi T^{-1})=
\qsing1/d(-a,-b,a+b)+\qsing1/d(b,-a-b,a)\\
&=\qsing1/d(b-a,-a-2b,2a+b)\\ \intertext{and}
T^{-1}f T &= \qsing1/d(2a+b,b-a,-a-2b).
\end{align*}
Dividing the second element by the first gives
$$T^{-1} f T f^{-1} = \qsing1/d(3a,3b,-3(a+b))= \varphi^3,$$
so $\varphi^3\inH_\eta$.
Let $x:=\varphi^{-3}\inH_\eta$. By Claim~\ref{claim:order3}, the
order of $x$ is prime to 3, so there exists an integer $l$ such that
$x^{3l}=x$. Writing $\alpha := x^l=(\varphi^{-3l})\inH_\eta$, one has
$(\alpha\varphi)^3=1$, and so Claim~\ref{claim:order3} again implies
that $\alpha \varphi =\omega_3$ or $\omega_3^2$.
Thus $H'=\<<H_\eta,\omega_3>>$.
\end{proof}
Now one can deal with the groups $G_\eta$ and $G'$ themselves.
To begin with, since the quotient $C'=G'/G_\eta$ is cyclic, $G'=\<<
G_\eta,\phi>>$ for some $\phi\in G'\setminus G_\eta$. Now any
element $\phi$ in a group of type (C) or (D) has associated to it a
permutation $\sigma(\phi)\in S_3$, defined according to how it
permutes the coordinates $x_1$, $x_2$, $x_3$. If $\sigma(\phi)$ is
the identity, then $\phi$ is diagonal, whereas if $\sigma(\phi)$ is
a permutation of order 3 then $\phi T$ or $\phi T^{-1}$ is diagonal.
In these cases, since $T\inG_\eta$, it follows from the claim above
that $G' = \<< G_\eta,\omega_3>>$.
The only remaining possibility is that $\sigma(\phi)$ equals a
transposition or order 2, which can be assumed to be the
transposition $(12)$, by
multiplying $\phi$ by a suitable power of $T$. Thus $\phi$ is of
the form~\eqref{eq:phi}.
\begin{claim}
\label{claim:type_D}
For any $\phi$ of the form~\eqref{eq:phi}, define
$$\tilde\phi:=
T^{-1}\phi^2T\phi=
\begin{pmatrix}
A& 0& 0\cr 0& 0& C\cr 0& B& 0\cr
\end{pmatrix},$$ with $A=-1$, $B=b^2c$ and $C=-B^{-1}$.
Suppose that $Z_3\not<\<<\tilde\phi,T>>$. Then
there exists an element $t\in \<<T,\tilde\phi>>$ such that
$t\tilde\phi=R$, where
$$R:=\begin{pmatrix}-1& 0& 0\cr 0& 0& -1\cr 0& -1& 0
\end{pmatrix}.$$
\end{claim}
\begin{proof}
Define $f:=\tilde\phi
T\tilde\phi^{-1}T=[-B,-B,B^{-2}]$ and
$f':=fT^{-1}fT=[1,-B^3,-B^{-3}]$. If the order of $B$ is a
multiple of $3$, say $m=3k$, then $f^{2k}=\omega_3$ or~$\omega_3^2$,
so $Z_3<\<<\tilde\phi,T>>$. Thus if
$Z_3\not<\<<\tilde\phi,T>>$ then the order of $B$ is prime to
three, and a suitable power of $f'$ gives the required element
$t=[-1,-B^{-1},-B]$, which satisfies $t\tilde\phi=R$.
\end{proof}
Now the element $\phi^2$ is diagonal, so belongs to $\<< H_\eta,
\omega_3>>$, by claim~\ref{claim:hprime}. Note also that
$\phi=\tilde\phi(T\phi^{-2}T^{-1})\in\tilde\phi H'$.
{\bf Case 1: $\phi^2 \in H_\eta$}
In this case, $C'$ has order 2, so $\omega_3\not\in H'$ --- i.e.\xspace
$H'=H_\eta$. By Claim~\ref{claim:type_D}, $t\tilde\phi=R$ for some
element $t\in \<<T,\tilde\phi>> < G_\eta$. Thus
$G'=\<<G_\eta,\phi>>=\<<G_\eta,\tilde\phi,H'>> = \<<G_\eta,R>>$, since $H'=H_\eta$.
{\bf Case 2: $\phi^2\not\inH_\eta$}
Then $\phi^2=\varphi\omega_3^k$ for $k=1$ or~$2$ and
$\varphi\inH_\eta$. Let $\phi':=\omega_3^k\phi$. Then
$(\phi')^2=\varphi^2\inH_\eta$, so by the preceding case, one may
assume $t\tilde\phi'=R$ for some
$t\in\<<T,\tilde\phi'>>< G_\eta$. Now
$\tilde\phi=\widetilde{\phi'}$,
so $G'=\<<G_\eta,\phi>>=\<<G_\eta,\tilde\phi,H'>> = \<<G_\eta, Rt^{-1},H'>>
= \<<G_\eta,R,\omega_3>>$.
This completes the proof of the lemma.
\end{proof}
\begin{prop}
\label{prop:3-1-nocentre-equiv-Euler}
Conjecture~\ref{conj:3-1-equiv-Euler} is true when $G_\eta$ is
irreducible and doesn't contain $Z_{3}$.
\end{prop}
\begin{proof}
Recall from the proof of
Prop.~\ref{prop:3-1-invariant-resolution-gives-terminalisation},
that ${\C^*}$ acts on any Euler resolution. Thus, in all cases where
$G'=\<<G_\eta,\omega_3>>$, any smooth crepant resolution of $V^1/G_\eta$
admits a $G'$-action --- indeed a $G$-action, as remarked in the
proof of Prop.~\ref{prop:3-1-invariant-resolution-gives-terminalisation},
since $G<\<<G',{\C^*}>>$.
Hence, Lemma~\ref{lemma:3-1-nocentre-exact} implies that it suffices
to deal with the cases where $G=\<<G_\eta,\omega_3>>$ or
$G_\eta=\<<H_\eta,T>>$ is of type (C) with $G'$ of type (D), either equal
to $\<<G_\eta,R>>$ or equal to $\<<G_\eta, R,\omega_3>>$.
In these cases (see~\cite{roan:calabi-yau}) an Euler resolution of
${\mathbb C}^3/G_\eta$ is obtained by taking a toric Euler resolution of
$\widetilde{{\mathbb C}^3/H_\eta} \to {\mathbb C}^3/H_\eta$ which is $\<<T>>$-stable and
then resolving the singularities of $\widetilde{{\mathbb C}^3/H_\eta}/\<<T>>$.
The existence of a $T$-stable toric Euler resolution follows from
the fact that $H_\eta$ is normal in $G_\eta$. However, $H_\eta$ is also
normal in $G'$, so $\widetilde{{\mathbb C}^3/H_\eta}$ can also be chosen to be
$R$-stable. The singularities of $\widetilde{{\mathbb C}^3/H_\eta}/\<<T>>$ are
fixed points of $R$, so resolving them gives the desired
$G'$-invariant resolution of ${\mathbb C}^3/G_\eta$.
The proof that $\chi((W^1)^{c'}/C')=|\pi_*^{\prime -1}([c'])|$ is done
by treating case by case the three possibilities for $G'$ given by
Lemma~\ref{lemma:3-1-nocentre-exact}.
{\bf Case 1: $G'=\<<G_\eta,\omega_3>>$.} In this case, since
$\omega_3\in{\C^*}$, the same argument as the second paragraph of the
proof of
Proposition~\ref{prop:3-1-invariant-resolution-gives-terminalisation}
gives $\chi((W^1)^{\omega_3})=\chi(W^1)=|\operatorname{\rm Cl}(G_\eta)|$. On the other
hand, $G'$ is just a direct product of $G_\eta$ and $Z_3$, so $\pi_*'$
is everywhere $|\operatorname{\rm Cl}(G_\eta)|:1$.
{\bf Case 2: $G'=\<<G_\eta,R>>$.} If we denote by $H_\eta$ and $H'$ the
normal diagonal subgroups of $G_\eta$ and $G'$ respectively, then they
are equal. Since they do not contain $\omega_3$, their order is
$d^2$ (for some $d$ prime to~3) and they are a semi-direct factor in
$G'$~\cite[Lemma 10]{roan:crepant}. The inverse image of the
trivial class in $C'$ is the number of $G'$-conjugate elements in
$G_\eta$. Since $G_\eta$ is normal, this is the same as the number of
$G_\eta$-conjugate elements in $G_\eta$, i.e.\xspace equal to
$\operatorname{\rm Cl}(G_\eta)=\chi(W^1)$. For the non-trivial class $[R]$,
\cite[Formula~(32)]{roan:crepant} implies that $\chi((W^1)^R/\<<R>>)=d$ and
the proof of ~\cite[Lemma 10]{roan:crepant} again gives
$\pi_*^{\prime-1}([R])=|Z_R\cap H'|=d$.
{\bf Case 3: $G'=\<<G_\eta, \omega_3 R>>$.} As we remarked in Case~1,
the Euler number does not depend on scalar factors, so
$\chi((W^1)^{\omega_3 R})=\chi((W^1)^R)$. On the other hand, the
discussion in the second paragraph of this proof implies that
$|\pi^{\prime-1}_*([\omega_3 R])|=|\pi^{\prime-1}_*([R])|$. The
result thus follows from Case~2.
\end{proof}
Hence, if $G_\eta$ is irreducible and does not contain $Z_3$, $V/G$ has
an Euler terminalisation with only toric singularities.
\begin{question}
The proof of Propositions~\ref{prop:3-1-nocentre-equiv-Euler}
depends on the classification of $\operatorname{\rm SL}(3)$ groups. Is it possible to
find a proof which doesn't depend on the classification?
\end{question}
\subsection{Case where $G_\eta$ contains $Z_3$}
\label{sec:3-1:centre}
Unfortunately, the author was not able to prove the corresponding
result to Conjecture~\ref{conj:3-1-equiv-Euler} in the case where
$G_\eta$ is irreducible but contains $Z_3$. A method is suggested in
Section~\ref{sec:3-1:centre:2}, but requires further work.
If it the conjecture can be proved in all cases, the work above
implies that $\text{Term}({\mathbb C}^{4}/G)$ is true for all groups of type $(3,1)$
and $(2,1,1)$.
The results of the next section imply that in that case,
$\text{Term}({\mathbb C}^{4}/G)$ is true for all irreducible $G$ which contain
$Z_{4}$.
\section{Blowing up in the presence of the centre $Z_n$}
\label{sec:blowing-up}
\subsection{Invariant sets in projective space}
First, a lemma about the Euler number of a invariant sets in
projective space.
\begin{lemma}
\label{lemma:euler-proj}
Let $H$ be a finite abelian group acting linearly on $\PP^n$. Then
$$\chi((\PP^n)^H) = \chi(\PP^n)=n+1.$$
\end{lemma}
\begin{proof}
Suppose $H$ has order $r$. Diagonalise the action of $H$, and order
the weights of the action so that they form a non-decreasing
sequence of elements of $\{0,\dots,r-1\}$. The sequence will
consist of $d_1$ occurrences of the smallest weight $w_1$, followed
by $d_2$ occurrences of the second smallest weight $w_2$, and so on,
ending with $d_s$ occurrences of the greatest weight $w_s$. Since
there are $n+1$ (not necessarily distinct) weights in the sequence,
the multiplicities $d_i$ sum to $n+1$.
Computing the invariant part of $\PP^n$ with respect to the
$H$-action, one sees that it consists of a disjoint union over all
$i\in\{1,\dots,s\}$ of projective spaces $\PP^{d_i-1}$. Taking the
sum of the Euler numbers of the invariant components, and
using the fact that the Euler number of $\PP^d$ is $d+1$, one
obtains the value $\sum d_i$, which by the previous paragraph indeed
coincides with the Euler number $n+1$ of $\PP^n$.
\end{proof}
\subsection{Blowing up the origin}
\label{sec:blowing-up:blowing-up}
Now let $V= {\mathbb C}^n$ and let $\text{Bl}_0V$ be the blow-up of $V$ at the
origin. This has a natural $G$-action and one has the following
commutative diagram
$$\begin{CD}
& \text{Bl}_0V @>\sigma_0>> & V \\
& \downarrow & & & \downarrow \\
& \text{Bl}_0V/G @>{\sigma'_0}>> & V/G .
\end{CD}$$
A standard discrepancy calculation yields the following result.
\begin{lemma}
The morphism $\text{Bl}_0V/G \to V/G$ is crepant if and
only if $G$ contains $Z_n=\<<\omega_n>>$.
\end{lemma}
The following lemma contains the basic idea to constructing
Euler blow-ups.
\begin{lemma}
\label{lemma:bV}
Assume that $G$ contains $Z_n$ and write $\bar G:=G/Z_n$. Then $\bar V :=
\text{Bl}_0V/Z_n$ is smooth, and $\bar V/\bar G \to V/G$ is a projective
Euler blow-up.
\end{lemma}
\begin{proof}
The only place where singularities of $\bar V$ could arise is on the
image $\bar E=E/Z_n$ of the exceptional divisor $E$ of $\sigma_0$.
Identifying $E$ with $\PP(V)$, a local chart for $\text{Bl}_0V$ at
a point $\xi\in E$ is given in suitable local coordinates by
$$(x_1,\frac{x_2}{x_1},\dots,\frac{x_n}{x_1}),$$
so therefore $\omega_n$ acts there as $(\omega_n, 1,1,1)$, i.e.\xspace as a
pseudo-reflection.
The Euler number computation goes as follows:
\begin{align*}
\chi_{\text{orb}}(\bar V/\bar G)
&=\sum_{[\bar g]\in\operatorname{\rm Cl}(\bar G)}\chi(\PP(V)^{\bar g}/\operatorname{\rm N}^{\bar G}_{\bar g}), \\
& \qquad\text{since $\bar V\sim E\cong\PP(V)$\ \ (homotopy)}\\
&=\sum_{[\bar g]\in\operatorname{\rm Cl}(\bar G)} \chi(\PP(V)), \\
& \qquad\text{averaging and using Lemma~\ref{lemma:euler-proj}},\\
&=|\operatorname{\rm Cl}(\bar G)| n = |\operatorname{\rm Cl}(G)|\\
&=\chi_{\text{orb}}(V/G).
\end{align*}
\end{proof}
Thus if $G$ contains $Z_n$ then $\text{Term}(V/G)\equiv\text{Term}(\bar V/\bar G)$.
\subsection{Patching}
\label{sec:blowing-up:patching}
In many cases, an Euler blow-up is constructed by patching together
local Euler blow-ups. The following lemma summarises the necessary
conditions to carry this out this procedure.
\begin{lemma}[Patching Lemma]
\label{lemma:patching}
Let $G$ be a finite subgroup of $\operatorname{\rm SL}(n)$ which contains $Z_n$, so
that $Y:=\bar V/\bar G\to V/G$ is an Euler blow-up. Let $\bar E:\equiv
E/Z_n$, where $E$ is the exceptional divisor of $\text{Bl}_0 V\to V$
and denote by $p\colon\bar V\to\bar E$ the projection.
There exists a finite collection of points $y\in\bar E/\bar G$ and
corresponding analytic neighbourhoods $\bar E_\xi\subset\bar E/\bar G$ such that
$Y$ is covered by $\{Y_y\}$, where $Y_y:=p^{-1}(\bar E_y)$.
Suppose that for each $y$, there exists an Euler blow-up
$\varphi_y\colon X_y\to Y_y$, such that if $y\neq y'$, one
nevertheless has
\begin{equation}
\label{eq:patching}
X_{y|Y_y\cap Y_{y'}} = X_{y'|Y_y\cap Y_{y'}}.
\end{equation}
Then the analytic canonical Gorenstein orbifold\xspace\ $X$ obtained by gluing together all the
$ X_y$ is an Euler blow-up of $Y$.
\end{lemma}
\begin{proof}
The existence of the finite covering $\{\bar E_y\}$ follows because
$\bar E$ is compact. Since, all the $X_y$ are birational to each other
above the overlaps,
equation~\eqref{eq:patching} implies that $X$ is well-defined.
Furthermore, since no crepant
divisors are introduced during the local blow-ups and since the orbifold
Euler numbers of $X_y$ and $Y_y$ are the same, $X$ is crepant
over $Y$ and has the same orbifold Euler number.
\end{proof}
\section{Irreducible $G$ which contain $Z_4$}
\label{sec:4-centre}
\subsection{Notation}
\label{sec:4-centre:notation}
When $G$ contains $Z_4$, $\bar V/\bar G\to V/G$ is an Euler blow-up.
Let $\bar E$ be the exceptional divisor of $\bar V\to V/Z_4$, and let
$p\colon \bar V\to \bar E$ be the projection. Let $\xi\in \bar E$ be a point
in the base, and consider the tangent space of $\bar V$ at $\xi$. This
decomposes into $\bar G$-modules
$$V_\xi^1\oplus V_\xi^2,$$
where $V^1_\xi$ is the tangent space to
$\bar E$ and $V^2_\xi$ is the line tangent to the fibre of $p$, and
stabilized by $\bar G$. Let $\xi'\in V_\xi^2$ be the generic point, so
that its stabiliser $\bar G_{\xi'}$ is a subgroup of $\operatorname{\rm SL}(3)$.
\subsection{Local Blow-ups}
\label{sec:4-centre:blowups}
Let $\xi\in \bar V$ and let $\bar\xi$ denote its image in $Y=\bar V/\bar G$.
A local analytic neighbourhood of $\bar\xi$ is isomorphic to
$$Y_{\bar\xi} := (V_\xi^1\oplus V_\xi^2)/\bar G_\xi= (V_\xi^1/\bar G_{\xi'}\oplus
V_\xi^2)/C_{\xi},$$ where $C_\xi$ denotes the quotient $C_\xi:=
\bar G_{\xi}/\bar G_{\xi'}$, which is cyclic, being a naturally a subgroup
of $\operatorname{\rm GL}(V_\xi^2)\cong{\C^*}$.
Restricting attention to the ``base'' $\bar E$, one has, corresponding to
each $\bar\xi$ in $\bar E $, a quotient singularity $\bar E_{\bar\xi} =
V_\xi^1/\bar G_{\xi'}$ which is an $SL(3)$-singularity. Since $\bar E/\bar G$
is compact, the choice of a finite number of points $\bar\xi$ is
sufficient for $\bigcup_{\bar\xi} Y_{\bar\xi}$ to cover the whole of
$\bar V/\bar G$.
\subsection{Gluing}
\label{sec:4-centre:gluing}
\begin{lemma}
\label{lemma:4-centre-gluing}
Let $W_\xi^1\to V_\xi^1/\bar G_{\xi'}$, be the
$\bar G_\xi$-equivariant Euler resolution such as that in
Conjecture~\ref{conj:3-1-equiv-Euler}. Define $$
X_\xi := (W_\xi^1\times V_\xi^2)/\bar G_{\xi}.$$
Then $\varphi_\xi\colon X_\xi\to
Y_\xi$ are Euler blowups which have only cyclic quotient
singularities, and for each $\xi,\xi'$, $ X_\xi$ and $ X_{\xi'}$ agree
above the inverse image of $Y_\xi\cap Y_{\xi'}$. Thus by
Lemma~\ref{lemma:patching}, they glue to form a complex analytic Euler
blow-up $Y \to \bar V/\bar G$ which has only cyclic quotient singularities.
\end{lemma}
\begin{proof}
One must check that $\chi_{\text{orb}}(X_\xi)=\chi_{\text{orb}}(Y_\xi)$ and that , on
$\varphi_\xi^{-1}(Y_\xi\cap Y_{\xi'})$ the blow-ups corresponding to
$\xi$ and $\xi'$ agree. The orbifold Euler number equality is
checked in
Proposition~\ref{prop:3-1-invariant-resolution-gives-terminalisation}.
For the agreement of the blow-ups on the overlaps, knowledge of the
$SL(3)$ singularities implies that these overlaps only occur over
curves of 2-dimensional singularities (the components of the
non-isolated singularities are all curves for $\operatorname{\rm SL}(3)$). Over these, the
resolutions which are being glued-in are trivial families of minimal
resolutions: they are therefore unique, and so resolutions coming
from neighbourhoods corresponding to different $\xi$'s will agree.
\end{proof}
This gives a crepant analytic blow-up $Y$ which is locally
analytically isomorphic to a cyclic quotient (and hence locally
analytically $\Q$-factorial).
\subsection{Terminalisation and the Orbifold Euler Number}
\label{sec:4-centre:terminalisation}
Since the orbifold Euler number can be calculated by summing the
contributions of the various analytic neighbourhoods, the equality
$$\chi_{\text{orb}}(X)=\chi_{\text{orb}}(\bar V/\bar G)$$
will follow by showing that the
resolutions glued in above preserve the orbifold Euler number. This
is proved in Proposition~\ref{prop:3-1-nocentre-equiv-Euler}. Thus
$X$ is an Euler blow-up with only cyclic singularities.
Applying the minimal model program to $X$
(Theorem~\ref{thm:toroidal-mmp}), one obtains a crepant terminalisation
$t\colon Z\to X$ which satisfies $\chi_{\text{orb}}(Z)=\chi_{\text{orb}}(X)=\chi_{\text{orb}}(V/G)$,
and has only toric singularities.
\begin{rmk}
By studying which toric flips can occur, one might be able to prove
that the singularities of $T$ are in fact at most cyclic. They
would then have to necessarily be isolated. For if a
4--dimensional Gorenstein cyclic singularity consisted of a curve
of singularities, these would also have to be (3-dimensional)
terminal Gorenstein cyclic quotients. But the classification of
3-dimensional terminal cyclic quotients~\cite{mor_stev:terminal}
shows that they are all of the form $\frac{1}{r}(1,-1,a)$, and so
can only be Gorenstein if they are smooth.
\end{rmk}
\subsection{Case where $G$ is of type $(3,1)$ revisited}
\label{sec:3-1:centre:2}
A method similar to the one above can be applied to the case treated
in Section~\ref{sec:3-1:centre}, namely the case where $G$ is a group
of type (3,1) and the stabiliser group $G_\eta<\operatorname{\rm SL}(3)$ is irreducible and
contains $Z_3$. This goes some way towards a solution of
Conjecture~\ref{conj:3-1-equiv-Euler}.
Since $G_\eta >Z_3$, there exists an Euler blow-up $\bar V^1/\bar\Ge\to
V^1/G_\eta$ and this is equivariant under the $G$ (and hence $G'$)
action, since it is obtained from blowing up the origin of $V^1$,
which is of course fixed by $G$. Its singularities are of the type
$(2,1)$ and $(1,1,1)$ and the singular locus is
invariant under $G$.
An analytic resolution of $\bar V^1/\bar\Ge$ can be constructed by the same
gluing procedure as in Lemma~\ref{lemma:4-centre-gluing}: in any
local analytic neighbourhood of $\bar V^1/\bar\Ge$, construct an Euler
resolution, doing this equivariantly under the $G'$-action. These
glue together, since they can only intersect over smooth points.
This gives a resolution $W^1\to\bar V^1/\bar\Ge$.
It remains to show that it admits a $\bar G$-action and that it
satisfies the Euler number property of equation~\eqref{eq:chi-pi}.
\section{Groups $G$ of type (2,2) and (2,1,1)}
\label{sec:2-}
\subsection{Groups of type $(2,2)$}
\label{sec:2-2}
Let ${\mathbb C}^4=V^1\oplus V^2$ with $V^i$ 2-dimensional irreducible
$G$-modules and denote by $\eta_i$ the generic point of $V^i$. If
$G<\operatorname{\rm SL}(2)\times\operatorname{\rm SL}(2)$ then the following lemma constrains the
stabilisers of $\eta_i$.
\begin{lemma}
\label{lemma:terminal}
Suppose that $V=V^1\oplus V^2$ and that
$G<\operatorname{\rm SL}(V^1)\times\operatorname{\rm SL}(V^2)<\operatorname{\rm SL}(V)$. Denote by $G_{i}$ the stabiliser
of the generic point $\eta_i\in V^i$ for $i=1,2$. If both
stabilisers $G_1$ and $G_2$ are trivial, then $V/G$ must be terminal.
\end{lemma}
\begin{proof}
This can be proved by an easy discrepancy calculation, or
equivalently, by using the concept of ``weights'' for the group
action~\cite{reid_ito} as follows. The number of crepant divisors
of $V/G$ is equal to the ``number of elements of $G(1)$ of weight
one''. Here, $G(1):=\operatorname{\rm Hom}(\mu_r, G)$, where $r$ is the least common
multiple of the orders of the elements of $G$. The {\em weight\/}
$\text{wt}(\hat g)$ of $\hat g\in G(1)$ is defined by evaluating $\hat g$
on a primitive generator $\epsilon$ of $\mu_r$, diagonalising the
resulting matrix $\hat g(\epsilon)$ and expressing the diagonal
elements in terms of powers of $\epsilon$ ranging between $0$ and
$r-1$. Because $G<\operatorname{\rm SL}(V)$, the sum of these powers divided by $r$
is a non-negative integer, called the {\em weight\/} of $\hat g$.
Note that if we simply want to calculate the {\em number\/} of
elements of $G(1)$ of a given weight, we can identify $G$ with
$G(1)$ by fixing a primitive generator of $\mu_r$, and pretend to be
calculating the weights of the elements of $G$.
Suppose $V/G$ is not terminal, so that there exists and element
$g\in G$ of weight one. For each $i=1,2$, denote by $g_i$ the
part of the matrix of $g$ which represents its action on the module
$V^i$. Since $\text{wt}(g)=\text{wt}(g_1)+\text{wt}(g_2)$, one of the $g_j$'s must be
equal to the identity (whereas the other one must be a non-trivial
matrix). But $G_j=1$ and $g_j=1$ for some $j$ would imply that
$g=1$, so one of the two stabilisers must be non-trivial.
\end{proof}
Thus, $G<\operatorname{\rm SL}(2)\times\operatorname{\rm SL}(2)$ then $V/G$ not terminal implies
(Lemma~\ref{lemma:terminal}) that one of $G_{\eta_i}$ must be
non-trivial, say $G_\eta=G_{\eta_2}\neq 1$. Denoting by $W^1\to
V^1/G_\eta$ the minimal resolution, one obtains a crepant blow-up
$(W^1\times V^2)/(G/G_\eta)\to V/G$ with singularities of type
$(2,1,1)$ or $(1,1,1,1)$.
In order to prove that the orbifold Euler number remains unchanged
under the blowup $(W^1\times V^2)/G\to V/G$, one must prove a formula
similar to that of equation~\eqref{eq:chi-pi}, namely:
$$\chi((W^{1})^{d}/Z_d)=|\pi_*^{-1}([d])|,$$
for all $d\in D:=G/G_{\eta}<\operatorname{\rm GL}(V^1)$.
This could presumably be achieved by examining the finite $\operatorname{\rm SL}(2)$ and
$\operatorname{\rm GL}(2)$ subgroups and determining how many fit into an exact sequence
of the form given in equation~\eqref{eq:3-1-nocentre-exact}, with the
cyclic group $C$ replaced by the $\operatorname{\rm GL}(2)$ subgroup $D$.
\begin{question}
What happens if $G$ is not a subgroup of $\operatorname{\rm SL}(2)\times\operatorname{\rm SL}(2)$?
\end{question}
\subsection{Groups of type $(2,1,1)$}
\label{sec:2-1-1}
Suppose the irreducible decomposition of ${\mathbb C}^4$ is $V^1\oplus V^2
\oplus V^3$ with $\dim V^1=2$ and $\dim V^i=2$ for $i=2,3$. Denote by
$\eta_i$ the generic point of $V^i$, by $G_3$ the stabiliser of
$\eta_3$ in $G$ and by $G_{32}$ the stabiliser of $\eta_2$ in $G_3$.
Then we have exact sequences
\begin{alignat}{3}
\label{eq:2-1-1-exact}
1 &\to G_3 &\to G &\xrightarrow{\pi_3} C \to 1,\\
\intertext{and}
\label{eq:2-1-1-exact2}
1 &\to G_{32} &\to G_3 &\xrightarrow{\pi_{32}} D \to 1,
\end{alignat}
with $C$ and $D$ cyclic subgroups, $G_{32}<\operatorname{\rm SL}(2)$ and $G_3<\operatorname{\rm SL}(3)$.
The analytic germ of $V/G$ is isomorphic to
$$(V^1\times V^2 \times V^3)/G \cong \bigl( (V^1/G_{32}\times V^2)/D\times
V^3\bigr )/C.$$
The term $V^1/G_{32}$ has a minimal resolution $Z^1$ which is unique
and therefore admits an action of $G_3$. The germ $(Z^1\times V^2)/D$
is a cyclic $\operatorname{\rm SL}(3)$ singularity and has an Euler resolution $W^1$.
It remains to be shown that $W^1$ admits a $G$-action and that
$\chi_{\text{orb}}((W^1\times V^3)/C)=\chi_{\text{orb}}(V/G)$. As in
Section~\ref{sec:3-1}, the later statement would follow
from the equalities
$$\chi((W^1)^{c'}/C')=|\pi_{3*}^{\prime -1}([c'])|,$$
for $c'\in C'$, where the primed objects are defined similarly to those
in Section~\ref{sec:4-centre:notation}.
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
|
1996-10-09T23:07:16 | 9610 | alg-geom/9610009 | en | https://arxiv.org/abs/alg-geom/9610009 | [
"alg-geom",
"math.AC",
"math.AG"
] | alg-geom/9610009 | Qun Chen | Ragnar-Olaf Buchweitz and Qun Chen | Hilbert-Kunz functions of cubic curves and surfaces | LaTex 2e with Xy-pic v3.2 for commutative diagrams | null | null | null | null | We determine the Hilbert-Kunz function of plane elliptic curves in odd
characteristic, as well as over arbitrary fields the generalized Hilbert-Kunz
functions of nodal cubic curves. Together with results of K. Pardue and P.
Monsky, this completes the list of Hilbert-Kunz functions of plane cubics.
Combining these results with the calculation of the (generalized) Hilbert-Kunz
function of Cayley's cubic surface, it follows that in each degree and over any
field of positive characteristic there are curves resp. surfaces taking on the
minimally possible Hilbert-Kunz multiplicity.
| [
{
"version": "v1",
"created": "Wed, 9 Oct 1996 20:50:48 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Buchweitz",
"Ragnar-Olaf",
""
],
[
"Chen",
"Qun",
""
]
] | alg-geom | \section{Introduction}
Let $S=k[x_0,\cdots,x_n]$ be the standard polynomial ring in
$n+1$ variables over a field $k$ of prime characteristic
$p$. Given a finite graded $S$-module $M$, the Hilbert-Kunz
function of $M$ is defined on powers of the characteristic,
$q=p^n$, $n\in {\Bbb{N}}$, through
\[ HK_M (q):=\dim_k M/m^{[q]}M \]
where $m^{[q]}=(x^q_0,\cdots,x^q_n)$ is the
$q$-th Frobenius power of the maximal homogeneous ideal
$m=(x_0, \cdots, x_n)$.
If $I\subset S$ is a homogeneous ideal, and
$R=S/I$ the homogeneous coordinate ring of the
underlying projective scheme $X\subset {\Bbb{P}}^n_k$,
the function $HK_R(q)$ is also called the
Hilbert-Kunz function of $X$.
Introduced by E. Kunz [5] in 1969,
these functions were first studied in detail by
P. Monsky [6], and he obtained the
asymptotic formula
\[ HK_M (q)=c q^m+ O(q^{m-1}) \]
with $c\geq 1$ some real number and
$m$ the Krull dimension of $M$.
The number $c$ is called the Hilbert-Kunz multiplicity of
$M$ and P. Monsky conjectures it to be rational.
In general, it seems very difficult to determine these
functions explicitly and a conceptual interpretation of the
constant $c$ is missing
(see [4] for some surprising examples).
Here we exhibit the Hilbert-Kunz functions of plane elliptic
curves in odd characteristic and of plane nodal cubics.
Combining this work with results in [7] and [8]
completes the explicit determination of Hilbert-Kunz
functions of plane cubic curves. The Hilbert-Kunz functions
of reducible cubics were already determined by K. Pardue in
[8], and he also predicted the following list for the
irreducible ones on the basis of computer experiments. Note
that the Hilbert-Kunz function is invariant under extensions
of the coefficient field $k$, so that one may assume $k$
algebraically closed.
\newtheorem{theorem}{Theorem}
\begin{theorem}
Let $f$ be the equation of an irreducible cubic curve
$C$ in ${\Bbb{P}}^2_k$ over an algebraically closed field
$k$, and let $HK_R(q)$ be the Hilbert-Kunz function of
the homogeneous coordinate ring $R=S/f$.
\begin{description}
\item[(1) (K. Pardue {[8]})] If $C$ is a cuspidal cubic,
\[ HK_R (q)=\left\{
\begin{array}{ll}
\frac{7}{3} q^2 &\text{for}\quad p=3\; , \\
\frac{7}{3} q^2-\frac{4}{3} & \text{for}\quad p\neq 3\; .
\end{array}
\right. \]
\item[(2) (Thm. \ref{th:nodal} below)] If $C$ is a nodal cubic,
\[ HK_R (q)=\left\{
\begin{array}{ll}
\frac{7}{3}q^2-\frac{1}{3}q-1 &
\text{for}\quad q\not\equiv 2\bmod 3\;, \\
\frac{7}{3}q^2-\frac{1}{3}q-\frac{5}{3} &
\text{for}\quad q\equiv 2\bmod 3\;.
\end{array}
\right. \]
\item[(3) (Thm. \ref{theorem:main} below)]
If $C$ is an elliptic curve and $p\neq 2$,
\[ HK_R(q)=\frac{9}{4} q^2-\frac{5}{4}\quad .\]
\item[(4) (P. Monsky {[7]})] If $C$ is an elliptic curve
and $p=2$,
\[ HK_R (q)=\left\{
\begin{array}{ll}
\frac{9}{4}q^2 & \text{if the $j$-invariant is $0$}\;, \\
\frac{9}{4}q^2-1 &\text{if the $j$-invariant is not $0$}\;.
\end{array}
\right. \]
\end{description}
\label{theorem:major}
\end{theorem}
At this stage, Hilbert-Kunz functions or multiplicities of
plane curves of higher degree remain mysterious.
However, a corollary of our work shows that for any
$d\ge 2$ and for any field $k$ of
prime characteristic there exists a plane curve of
degree $d$ in ${\Bbb{P}}^2_k$ whose
Hilbert-Kunz multiplicity is $\frac{3}{4} d$
--- and this is the minimal possible value
for such curves.
In particular, the minimal Hilbert-Kunz multiplicity in
each degree is rational and
independent of the characteristic.
We then determine explicitly the Hilbert-Kunz function
of Cayley's cubic surface in ${\Bbb{P}}^3_k$,
and the result allows us to conclude as well
that for any $d\ge 2$ and for any field $k$
of prime characteristic there exists a surface
of degree $d$ in ${\Bbb{P}}^3_k$ whose
Hilbert-Kunz multiplicity is $\frac{2}{3} d$ ---
and this is again the minimal possible value,
again rational and independent
of the characteristic.
\section{Minimal Values of Hilbert-Kunz Functions}
Let $I=(f)$ be a principal ideal generated by a
homogeneous form $f$ of degree $d > 0$ in $S$.
The considerations in this section apply to the
values of the {\em generalized\/} Hilbert-Kunz
function of $R=S/I$, introduced by A. Conca in [1], and
defined as
\[ HK_{R,x}(q) =\dim_k S/(f,x_0^{q},\ldots,x_n^{q}) \quad, \]
where $q$ is now {\em any\/} nonnegative integer,
$k$ any field.
Unless $k$ is of positive characteristic $p$,
and $q$ is a power of $p$,
this dimension will generally depend upon the choice
of the coordinate system
$x = (x_0,\ldots,x_n)$.
For each $q\in{\Bbb N}$ and each choice of coordinates $x$,
set $x^{[q]}=(x_0^{q},\ldots,x_n^{q})$ and
consider the following graded $S$-modules of finite length,
\begin{eqnarray*}
\Theta & = & \frac{S}{x^{[q]}}=\bigoplus_i \Theta_i \quad , \\
\theta & = & \frac{S}{f+x^{[q]}}=\bigoplus_i \theta_i \quad , \\
\vartheta & = & \frac{(x^{[q]}:f)}{x^{[q]}}=\bigoplus_i \vartheta_i \quad .
\end{eqnarray*}
They are related by the exact sequence of graded
$S$-modules
\begin{equation}
\begin{array}{cccccccccccc} 0 &
\longrightarrow & \vartheta (-d)& \longrightarrow & \Theta (-d) &
\stackrel{f}{\longrightarrow} & \Theta & \longrightarrow & \theta &
\longrightarrow & 0 &,
\end{array}
\label{eq:exa}
\end{equation}
and $HK_{R,x}(q)=\dim_k \theta$.
Evaluating dimensions yields universal bounds for the
generalized Hilbert-Kunz function of
$R=S/(f)$, when $f$ varies over polynomials of degree
$d$ in $n+1$ variables,
\begin{equation}
q^{n+1}=\sum_i \dim_k \Theta_i \geq HK_{R,x}(q)\geq
\sum_i \max \{ \dim_k \Theta_i-\dim_k \Theta_{i-d}, 0 \} \quad.
\label{eq:ineq}
\end{equation}
The upper bound, $HK_{R,x}(q)=q^{n+1}$, is achieved iff
$f\in x^{[q]}$;
for example if $d > (n+1)(q-1)$,
or if $q=p^n$ is a power of the characteristic,
$d\geq q$ and $f=l^d$ for some linear form $l$.
Here we are more concerned with the lower bound,
that is taken on
if and only if $f$ {\em is of maximal rank at\/} $q$,
meaning that in each degree $i$ the $k$-linear map
$f|\Theta_{i-d}$ is of maximal rank.
Whether a given polynomial $f$ is of maximal rank at $q$,
can be decided by looking at the socle degree of the
artinian ring $\theta$,
\begin{equation}a(q) =\max\{i:\theta_i\neq 0\}\quad,
\label{eq:a}
\end{equation}
and at the initial degree of $\vartheta$,
\begin{equation}
\imath(q)=\min\{i:\vartheta_i \neq 0\}\quad.
\label{eq:i}
\end{equation}
Indeed, as the socle degree of $\Theta$ is $(n+1)(q-1)$,
outside the range $d\leq i\leq (n+1)(q-1)$ source or
target of $f|\Theta_{i-d}$ is zero, whereas for
a degree $i$ inside that range the map is not surjective
iff $i \leq a(q)$, not injective
iff $i-d \geq \imath(q)$.
Accordingly, all the $k$-linear maps induced by $f$
are of maximal rank
iff $a(q) < \imath(q) + d$.
Moreover, the exact sequence (\ref{eq:exa}) is selfdual,
whence it suffices
to know either $a(q)$ or $\imath(q)$:
\newtheorem{lemma}{Lemma}
\begin{lemma}
For each $q\in{\Bbb N}$, and independent of $f$, one has
\begin{equation}
a(q)+\imath(q)=(n+1)(q-1)\quad .
\end{equation}
Given $q$, all $k$-linear maps $f|\Theta_{i-d}$ are
of maximal rank iff
\begin{equation}
a(q) < \frac{(n+1)(q-1)+d}{2} < \imath(q)+d\quad .
\end{equation}
Moreover, each of the inequalities implies the other.
\label{lemma:mr}
\end{lemma}
{\noindent \it Proof:\quad }
The ring $\Theta=S/x^{[q]}$ is a zerodimensional
complete intersection with its
socle in degree $(n+1)(q-1)$.
Thus for any finite graded $\Theta$-module $M$,
we have an
isomorphism of graded $\Theta$-modules
\[ {\rm Hom}_k (M,k) \cong {\rm Hom}_{\Theta}
(M,\omega_{\Theta})\quad , \]
where $\omega_{\Theta}=\Theta ((n+1)(q-1))$ is
the canonical module of $\Theta$,
and ${\rm Hom}_k (M,k)$ is the $\Theta$-module graded
naturally through
\[ ({\rm Hom}_k (M,k))_i = {\rm Hom}_k (M_{-i},k)\quad . \]
As ${\rm Hom}_{\Theta}(\theta,\Theta)
\cong (x^{[q]}:f)/x^{[q]}=\vartheta$,
we get
\[ {\rm Hom}_k (\theta,k)\cong \vartheta ((n+1)(q-1))\quad . \]
For the dimension of the finite dimensional
$k$-vector space $\theta_i$, this yields
\[ \dim_k \theta_i =\dim_k {\rm Hom}_k (\theta_i,k)
= \dim_k ({\rm Hom}_k (\theta,k))_{-i}
= \dim_k \vartheta_{(n+1)(q-1)-i}\quad, \]
and the equality follows from the definition of
$a(q)$ and $\imath(q)$.
As $f$ induces maps of maximal rank iff $a(q) < \imath(q) + d$,
we can eliminate either one of the two invariants to obtain
the last claim.
$\hfill \square$
\[ \]
If $d>(n+1)(q-1)$, the information is already complete:
$f$, inducing the zero map in (\ref{eq:exa}),
is trivially of maximal rank at $q$, and
$HK_{R,x}(q) = q^{n+1}$.
Also, if $f$ is a polynomial of a single variable, $n=0$,
there are no secrets to discover.
If $n>0$, the (usual) Hilbert series
\[ H_{\Theta}(t) = \sum_i (\dim_k \Theta_i)t^i =
(1+t+t^2+\cdots+t^{q-1})^{n+1}\quad,\]
of the artinian $k$-algebra
$\Theta$ is a {\em reciprocal\/} and {\em unimodal\/}
polynomial of degree $l=(n+1)(q-1)$ in $t$, meaning that
its coefficients, $\alpha_i=\dim_k \Theta_i$, satisfy
\begin{eqnarray*}
\alpha_i & = & \alpha_{l-i} \quad {\rm for\ every}\quad i \quad, \\
\alpha_i & < & \alpha_{i+1} \quad {\rm for} \quad
0\leq i <\left\lfloor \frac{l}{2} \right\rfloor\quad .
\end{eqnarray*}
In particular,
$\dim_k\Theta_i - \dim_k\Theta_{i-d} > 0$ iff
$0\leq i \leq m(q)$, where
\[ m(q) =\left\lfloor \frac{(n+1)(q-1)+(d-1)}{2}
\right\rfloor\quad,\]
--- as for $a(q)$, we suppress the dependence upon
$d$ from the notation.
Thus the lower bound, $L(q)$, in inequality (\ref{eq:ineq})
evaluates to
\begin{eqnarray*}
L(q) & := & \sum_i \max\{ \dim_k \Theta_i-\dim_k
\Theta_{i-d}, 0 \} \\
& = & \sum_{m(q)-d+1}^{m(q)} \dim_k \Theta_i
\quad,\quad {\rm as}\ H_{\Theta}(t)\ {\rm is\ unimodal\
and\ reciprocal}\\
& = & {\rm\ coefficient\ of\ \;} t^{m(q)} {\rm\;\ in\ }
\frac{(1-t^d)(1-t^q)^{n+1}}{(1-t)^{n+2}}\\
& = & \frac{1}{2\pi\sqrt{-1}}\int_{|z|=\epsilon}
\frac{(1-z^d)(1-z^q)^{n+1}}{(1-z)^{n+2}z^{m(q)+1}}dz\quad.
\end{eqnarray*}
\noindent
As $m(q)$ is the largest integer smaller than
$\frac{1}{2}((n+1)(q-1)+d)$,
we get the following result.
\begin{theorem}
If $n>0$, and if $f$ is a homogeneous polynomial
of degree $d\leq (n+1)(q-1)$\; in \;$n+1$\; many variables,
then the socle degree of the graded artinian $k$-algebra
$\theta$ satisfies
\begin{equation}
a(q) \geq m(q)\quad .
\label{eq:aq}
\end{equation}
Furthermore, the following statements are equivalent:
\begin{description}
\item[{\rm (i)}]\quad
The polynomial $f$ is of maximal rank at $q$.
\item[{\rm (ii)}]\quad
The socle degree $a(q)$ is minimal, $a(q)=m(q)$.
\item[{\rm (iii)}]\quad
The initial degree $\imath(q)$ is maximal,
$\imath(q)=(n+1)(q-1)-m(q)$.
\item[{\rm (iv)}]\quad
The Hilbert-Kunz function of $f$ at $q$ achieves
the lower bound $L(q)$.
\end{description}
\label{theorem:minimal}
\end{theorem}
{\noindent \it Proof:\quad}
The first statement follows from the exact sequence
(\ref{eq:exa}), as
\[ \dim_k \Theta_{m(q)} > \dim_k \Theta_{m(q)-d}\quad . \]
As $a(q)$ is an integer and $m(q)$ is the largest integer
smaller than $\frac{1}{2}((n+1)(q-1)+d)$, the just
established lower bound for $a(q)$ implies the equivalences
in view of Lemma \ref{lemma:mr}.
$\hfill \square$
\[ \]
{\noindent \it Example:\quad} For $d=2,3$ and $n=2,3$,
we get the following table:
\[ \begin{tabular}{|l|l|l|l|} \hline
\emph{$d$} & \emph{$n$} & \emph{$m(q)$} &
\emph{lower bound for $HK_{R,x} (q)$} \\ \hline
2 & 2 & $\left\lfloor \frac{3q}{2} \right\rfloor -1$ &
$\left\lfloor \frac{3}{2} q^2 \right\rfloor
=\left\{ \begin{array}{ll}
\frac{3}{2} q^2 & \mbox{for $q$ even} \\
\frac{3}{2} q^2-\frac{1}{2} & \mbox{for $q$ odd}
\end{array}
\right.$ \\ \cline{2-4}
& 3 & $2(q-1)$ & $\frac{4}{3} q^3-\frac{1}{3} q$ \\ \hline
3 & 2 & $\left\lfloor \frac{3q-1}{2} \right\rfloor$ &
$\left\lfloor \frac{9}{4} q^2-\frac{5}{4} \right\rfloor
=\left\{ \begin{array}{ll}
\frac{9}{4} q^2-2 & \mbox{for $q$ even} \\
\frac{9}{4} q^2-\frac{5}{4} & \mbox{for $q$ odd}
\end{array}
\right.$ \\ \cline{2-4}
& 3 & $2q-1$ & $2q^3-q$ \\ \hline
\end{tabular}
\]
For $d=2$, it can be extracted from [1] that
the quadric $x_0^2-x_1x_2$ for $n=2$, respectively
the quadric $x_0x_1-x_2x_3$ for $n=3$,
have generalized Hilbert-Kunz functions that
take on the minimum value at each $q$.
\[ \]
{\noindent \it Remark 1:\quad}
P. Monsky noted that expressing the minimal
possible value $L(q)$ of $HK_{R,x}(q)$ as a residue
leads to an intriguing lower bound for
Hilbert-Kunz multiplicities in terms of the
integrals
\[ \beta_{n+1}=\frac{1}{\pi} \int^{+\infty}_{-\infty}
(\frac{\sin \alpha}{\alpha})^{n+1} d\alpha =
\frac{1}{2^{n}n!}\sum_{i=0}^{\left\lfloor n/2 \right\rfloor}
(-1)^i(n+1-2i)^{n}\binom{n+1}{i}\quad , \]
as
\begin{equation}
d\beta_{n+1} = \lim_{q\to\infty}\frac{1}{q^n{2\pi\sqrt{-1}}}
\int_{|z|=\epsilon}
\frac{(1-z^d)(1-z^q)^{n+1}}{(1-z)^{n+2}z^{m(q)+1}}dz\quad.
\label{eq:beta}
\end{equation}
Thus, for a hypersurface of degree $d$ in ${\Bbb{P}}^n_k$
over a field $k$ of positive characteristic,
the Hilbert-Kunz multiplicity satisfies
\begin{equation}
c \geq d\beta_{n+1}\quad.
\label{eq:lower}
\end{equation}
A direct combinatorial proof is as follows. Expanding
$(1-t^d)/(1-t)^{n+2}$ into its Taylor series at $t=0$,
one can write
\[
\frac{1-t^d}{(1-t)^{n+2}} =
\sum_{\nu=0}^{d-n-1} R(\nu)t^{\nu} +
\sum_{\nu \geq 0} P(\nu)t^{\nu}\quad,\]
where $R(\nu)\in{\Bbb Z}$, and
$P(\tau) = \frac{d}{n!}\tau^n + O(\tau^{n-1})$ is the
corresponding (Hilbert) polynomial, univariate over ${\Bbb Q}$
of degree $n$ with leading coefficient $\frac{d}{n!}$.
Now use that $m(q)/q = (n+1)/2 + O(1/q)$, that the
coefficient of $t^{m(q)}$ in
$\frac{1}{q^n}(1-t^q)^{n+1}\sum_{\nu=0}^{d-n-1} R(\nu)t^{\nu}$
tends to zero with $q$, and that
\[
(1-t^q)^{n+1}\sum_{\nu \geq 0} P(\nu)t^{\nu} =
\sum_{\nu\geq 0}
\big(\sum_{i=0}^{\left\lfloor \nu/q \right\rfloor}
(-1)^i P(\nu - iq)\binom{n+1}{i}\big)t^{\nu}\quad,
\]
to get
\begin{eqnarray*}
\lim_{q\to\infty} \frac{L(q)}{q^n} &=&
\lim_{q\to\infty}
\sum_{i=0}^{\left\lfloor m(q)/q\right\rfloor}
(-1)^i\frac{P(m(q)-iq)}{q^n}\binom{n+1}{i}\\
&=&\lim_{q\to\infty}
\sum_{i=0}^{\left\lfloor m(q)/q\right\rfloor}
(-1)^i\frac{d}{n!}(\frac{m(q)}{q}-i)^n\binom{n+1}{i}\\
&=&\frac{d}{2^n n!}
\sum_{i=0}^{\left\lfloor n/2\right\rfloor}
(-1)^i(n+1-2i)^n\binom{n+1}{i}\\
&=&d\beta_{n+1}\quad.
\end{eqnarray*}
That this combinatorial expression equals the indicated integral
can now be checked in any table of integrals, e.g.
[3, $3.836.5^3$, p 458]. It follows that the sequence
$\{ \beta_{n} \}$ of rational numbers decreases to zero.
The first few values are
\[ \beta_1=1, \; \;\beta_2=1, \; \;
\beta_3=\frac{3}{4}, \; \; \beta_4=\frac{2}{3}, \; \;
\beta_5=\frac{115}{192}, \; \;
\beta_6=\frac{11}{20}\quad . \]
{\noindent \it Remark 2:\quad}
The same argument applies to $\liminf_{q\to\infty}
HK_{R,x}(q)/q^n$, but it is not yet known whether a
``generalized'' Hilbert-Kunz multiplicity exists, i.e.
whether
$HK_{R,x}(q)/q^n$ tends to a limit for $q\to\infty$.
\[ \]
For elliptic curves in odd characteristic, we will prove
that the corresponding cubic polynomial is of maximal rank
at any power $q$ of the characteristic,
whereas for the polynomial $x_0x_1x_2x_3(
\frac{1}{x_0}+\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3})$,
representing Cayley's cubic surface, this will be even
established at any $q\in{\Bbb N}$
over any field $k$.
The proof is accomplished by showing that $a(q)$ equals the
minimum value $m(q)$, and the respective Hilbert-Kunz function
can then be read off from the table above.
None of this applies to elliptic curves in characteristic $2$,
(see [7] for details), nor does it hold for singular
irreducible cubic curves in any characteristic.
But in the latter case, the Hilbert-Kunz function can be
determined completely from the rational parametrization
of the curve as we show next.
\section{Singular Irreducible Cubic Curves}
The Hilbert-Kunz function of a cuspidal cubic is known from [8],
see also [1], but our treatment here deals with the nodal
and cuspidal case at the same time.
Let $k$ be an algebraically closed field
--- of any characteristic for now ---,
and denote by $C$ a singular irreducible
plane cubic curve in ${\Bbb{P}}^2_k$.
In suitable coordinates,
$C$ is given by a Weierstra\ss\ equation
$$
f(x,y,z) = z(y^2 + a_1 xy - a_2 x^2) - x^3 = 0 \quad,
$$
so that $o=[0,0,1]\in{\Bbb{P}}^2$ is its unique singular point.
The curve has a node at $o$ iff the tangential quadric
$Q(x,y) = y^2 + a_1 xy - a_2 x^2$ has distinct roots iff
$a^2_1 + 4 a_2 \ne 0$, otherwise it is cuspidal.
The curve $C$ is rational and a rational parametrization
$\nu:{\Bbb{P}}^1\to C\subset {\Bbb{P}}^2$ normalizes the curve,
pulling back ${\cal{O}}_{{\Bbb{P}}^2}(1)$ along
$C\hookrightarrow{\Bbb{P}}^2$ and then $\nu$ to ${\cal{O}}_{{\Bbb{P}}^1}(3)$.
Algebraically, such a parametrization is given by the monomorphism of
$k$-algebras
\[ \alpha(x,y,z)=(sQ(s,t),tQ(s,t),s^3)\quad , \]
$$
\alpha: R ={ k[x,y,z]\over f(x,y,z)} \cong
\bigoplus_{n\in {\Bbb{Z}}}
H^0(C,{\cal{O}}_{C}(n))\hookrightarrow
\bigoplus_{n\in {\Bbb{Z}}}
H^0({\Bbb{P}}^1, {\cal{O}}_{{\Bbb{P}}^1}(3n))
\cong k[s,t]^{(3)} =:{\tilde R}\quad,
$$
where ${\tilde R}=k[s,t]^{(3)}$ is the Veronese subring
of the polynomial ring $k[s,t]$ spanned by all homogeneous
polynomials whose degree is divisible by $3$.
Notice that ${\tilde R}_n$ consists of all homogeneous
polynomials of degree $3n$.
The cokernel of $\alpha$ can be identified as follows.
A section $p(s,t)\in H^0({\Bbb{P}}^1, {\cal{O}}_{{\Bbb{P}}^1}(3n))$
comes via $\alpha$ from a section in
$H^0(C,{\cal{O}}_{C}(n))$ iff $p(s,t)$ takes on the
same value at the two points $Q(s,t)=0$.
Explicitly, write $p(s,t) = e_1(s) + e_2(s) t + e(s,t)Q(s,t)$
with uniquely determined polynomials $e_1,e_2\in k[s]$
and $e\in k[s,t]$.
The component $e_2(s)t$ represents the class of
$p(s,t)$ in $k[s,t]/(k[s] + Q k[s,t])$ and
$(k[s,t]/(k[s] + Q k[s,t]))^{(3)}$ is the cokernel of $\alpha$.
If $p(s,t)\in k[s,t]^{(3)}$, then $e_2(s) = b(s^3) s^2$
for some unique univariate polynomial $b$ that is necessarily
of degree $\frac{1}{3}\deg p -1$.
\begin{lemma}
{\rm (i)}\quad
The map
$$
\beta : {\tilde R} = k[s,t]^{(3)}\to k[z](-1)\quad,
$$
associating to $p(s,t)$ the polynomial $b(z)$,
is a degree preserving epimorphism of $R$-modules,
the $R$-module structure on ${\tilde R}$ given by
$\alpha$, the one on $k[z]$ by the natural projection
$R=k[x,y,z]/f\to k[z]$.
\noindent {\rm (ii)}\quad
The sequence of graded $R$-modules
\begin{equation}
\begin{array}{ccccccccc}
0 & \to & R & \stackrel{\alpha}{\longrightarrow} &
{\tilde R} = k[s,t]^{(3)} &
\stackrel{\beta}{\longrightarrow} & k[z](-1) &
\to & 0
\end{array}
\label{eq:exa2}
\end{equation}
is exact.
\label{lemma:normalization}
\end{lemma}
{\noindent \it Proof:\quad }
If $p(s,t) = e_1(s) + b(s^3)s^2 t + e(s,t)Q(s,t)$ is the unique
representation of $p(s,t)\in {\tilde R}$, then
\begin{eqnarray*}
\alpha(x)p(s,t) &=& 0 + 0\cdot s^2t + (sp(s,t))Q(s,t) \quad, \\
\alpha(y)p(s,t) &=& 0 + 0\cdot s^2t + (tp(s,t))Q(s,t) \quad, \\
\alpha(z)p(s,t) &=& s^3e_1(s) + (s^3b(s^3))s^2 t + (s^3e(s,t))Q(s,t) \quad,
\end{eqnarray*}
are the corresponding unique representations of
$\alpha(x)p(s,t), \alpha(y)p(s,t)$ and $\alpha(z)p(s,t)$
respectively.
This shows that the image of $\beta$ is annihilated by
$x,y$ and that $\beta(\alpha(z)p) = z\beta(p)$.
Furthermore, $\beta(s^2t)$ generates
the image of $\beta$ already as $k[z]$-module,
thus a fortiori as $R$-module.
As $s^2t$ is of degree one with respect to the grading
on ${\tilde R} = k[s,t]^{(3)}$, (i) follows.
For (ii), note first that $\beta\alpha(1) = 0$,
whence $\beta\alpha = 0$.
To prove that the kernel of $\beta$ is precisely the
image of $\alpha$, consider Hilbert functions:
In degree $i\in {\Bbb N}$,
\[ \dim_k R_i =\left\{
\begin{array}{ll}
1 & for \quad i=0 \\
3i & for \quad i>0
\end{array}
\right.
\quad,
\quad {\rm whereas}
\quad \dim_k {\tilde R}_i = 3i + 1\quad. \]
Accordingly, the quotient ${\tilde R}_i/R_i$ is zero
for $i=0$ and onedimensional for $i>0$.
Thus the cokernel of $\alpha$ and $k[z](-1)$ have
the same Hilbert function and (ii) follows.
$\hfill\square$
\[ \]
Multiplication with $x^q,y^q,z^q$ on (\ref{eq:exa2})
results in a commutative diagram of graded $R$-modules
whose exact rows and columns define the modules
$A$ through $G$,
\[
\diagram
& 0 \dto & 0 \dto & 0 \dto \\
0 \rto & A \dto \rto^{\matrix \overline{\alpha} \endmatrix} &
B \dto \rto^{\matrix \overline{\beta} \endmatrix} &
C \dto \rto & D \rto & 0 \\
0 \rto & R^{\oplus 3}(-q) \xto[ddd]_{\left(
\matrix
x^q \\
y^q \\
z^q
\endmatrix
\right)}
\rto^-{\matrix \alpha^{\oplus 3} \endmatrix} &
{\tilde R}^{\oplus 3}(-q) \xto[ddd]_{\left(
\matrix
\alpha(x)^q \\
\alpha(y)^q \\
\alpha(z)^q
\endmatrix
\right) }
\rto^-{\matrix \beta^{\oplus 3} \endmatrix} &
k[z]^{\oplus 3}(-1-q) \xto[ddd]_{\left(
\matrix
0 \\
0 \\
z^q
\endmatrix
\right)} \rto & 0 \\
&&&& \\
&&&& \\
0 \rto & R \dto \rto^{\matrix \alpha \endmatrix} &
{\tilde R} \dto \rto^{\matrix \beta \endmatrix} &
k[z](-1) \dto \rto & 0 \\
& E \dto \rto & F \dto \rto & G \dto \rto & 0 \\
& 0 & 0 & 0 &&\quad.
\enddiagram
\]
In this diagram, $E,F,G$, and then also $D$,
are finite dimensional and one has
\begin{equation}
HK_{R,(x,y,z)}(q) = \dim_kE = \dim_k F - \dim_k G + \dim_k D\quad.
\label{eq:sum}
\end{equation}
The dimension of $G\cong (k[z]/z^q)(-1)$ equals $q$, and
the next Lemma determines the dimension of $F$, that is the value
of the generalized Hilbert-Kunz function for ${\tilde R}$
with respect to $(x,y,z)$ at $q$.
\begin{lemma}
{\rm (i)}\quad
Set $P=k[s,t]$, the polynomial ring in two variables
with its natural grading.
For any $q\in{\Bbb{N}}$, the $P$-module
$M = k[s,t]/(\alpha(x)^q,\alpha(y)^q,\alpha(z)^q)$
has minimal graded resolution
$$ \diagram
0 \rto &
P(-4q)\oplus P(-5q)
\xto[rrrr]^-{\left(
\matrix
t^q & -s^q & 0 \\
s^{2q} & 0 &-Q(s,t)^q
\endmatrix
\right) } & & & &
P(-3q)^{\oplus3}
\xto[rr]^-{\left(
\matrix
s^qQ(s,t)^q \\
t^qQ(s,t)^q \\
s^{3q}
\endmatrix
\right)} & &
P \rto & M \rto & 0\ .
\enddiagram
$$
{\rm (ii)}\quad
The middle column in the diagram above is obtained from that
resolution by applying the functor $(\ \;)^{(3)}$, and in
particular $F=M^{(3)}$.\\
\noindent {\rm (iii)}\quad
\[ HK_{{\tilde R},(x,y,z)}(q) = \dim_k F =
\left\{
\begin{array}{ll}
{7\over 3} q^2 - {1\over 3} & \text{if}\quad q\not\equiv 0\bmod 3\quad, \\
{7\over 3} q^2 & \text{if} \quad q\equiv 0\bmod 3\quad.
\end{array}
\right.
\]
\label{lemma:F}
\end{lemma}
{\noindent \it Proof:\quad }
(i)\quad As $s$ does not divide $Q(s,t)$, the module $M$ is artinian,
and the result follows from the Hilbert-Burch theorem:
the (signed) $(2\times 2)$-minors of the
leftmost matrix are respectively
$s^q Q(s,t)^q = \alpha(x)^q\; ,\; t^q Q(s,t)^q = \alpha(y)^q\;,
\; s^{3q} =\alpha(z)^q\;$.
\noindent (ii) is clear and (iii) follows then easily from
\[ \dim_k F_i = \dim_k M_{3 i} =
\dim_k P_{3 i} - 3 \dim_k P_{3 i - 3 q} + \dim_k P_{3 i - 4 q} +
\dim_k P_{3 i - 5 q}\]
and $\dim_k P_j = \max\{0,j+1\}$ for $j\in\Bbb Z$.
$\hfill\square$
\[ \]
In equation (\ref{eq:sum}) for $HK_{R,(x,y,z)}(q)$,
it remains to determine the dimension of $D$.
To this end, we exhibit the map ${\overline \beta}$ explicitly.
As multiplication by $z^q$ is injective on $k[z]$,
one has $C\cong k[z]^{\oplus 2}(-1-q)$.
Furthermore, as $B \cong \big( k[s,t](-4q)\oplus k[s,t](-5q)\big)^{(3)}$
by Lemma \ref{lemma:F}, a homogeneous element in
$B$ is represented by a pair $(p_1(s,t),p_2(s,t))$
of homogeneous polynomials satisfying
\[ \deg p_1 = \deg p_2 + q \equiv -4q\bmod 3\quad,\]
and such pair is mapped to
\[
p_1(s,t)(t^q,-s^q,0) + p_2(s,t)(s^{2q},0,-Q(s,t)^q)
\]
in ${\tilde R}^{\oplus 3}(-q)$. Thus
\[
{\overline \beta}(p_1,p_2) = (\beta(t^qp_1 + s^{2q}p_2),\beta(-s^q p_1))
\in k[z]^{\oplus 2}(-1-q)\cong C\quad.
\]
As the field $k$ is algebraically closed, the quadric $Q$
factors,
$Q(s,t) = (t-us)(t-vs)\;;\; u,v\in k$;
and the unique representation of $t^q \bmod Q(s,t)$ is
\[
t^q = \tau_1 s^q + \tau_2 s^{q-1}t + \tau(s,t) Q(s,t)\quad,
\]
where
\[
\tau_1 = -uv\sum_{i=0}^{q-2} u^{q-2-i}v^{i} \quad,\quad
\tau_2 = \sum_{i=0}^{q-1} u^{q-1-i}v^{i}\quad.
\]
Writing now
\[
p_1 = b_1(s) + b_2(s)t + b_3(s,t)Q(s,t)\quad,\quad
p_2 = c_1(s) + c_2(s)t + c_3(s,t)Q(s,t)\quad,
\]
for suitable polynomials $b_i,c_i$, it follows that
\[
{\overline \beta} (p_1,p_2) =
\big(\tau_2 b_1 s^{q-3}+ (\tau_1 - a_1 \tau_2) b_2 s^{q-2} +
c_2 s^{2q-2}, -b_2 s^{q-2}\big)\big|_{s^3=z}\quad.
\]
So the image of ${\overline \beta}$ in
$C\cong k[z]^{\oplus 2}(-1-q)$ is generated as a $k[z]$-module
by the three pairs
\[ (\tau_2,0)z^{(q-3+\epsilon)/3}\quad,\quad
(\tau_1 -a_1 \tau_2 ,-1)z^{(q-2+\eta)/3}\quad,\quad
(1,0)z^{(2q-2+\zeta)/3}\quad,
\]
where $\epsilon,\eta,\zeta\in \{0,1,2\}$ are such that
the exponents become integers. Accordingly, the dimension of $D$
is equal to $(2q-5 +\epsilon+\eta)/3$ if $\tau_2\ne 0$,
whereas it equals $(3q-4+\eta+\zeta)/3$ if $\tau_2 = 0$.
Thus, the decisive
factor is whether or not $\tau_2$ vanishes.
\begin{lemma}
\begin{description}
\item[(i)]
If $q$ is a power of the characteristic of $k$,
then $C$ is nodal over $k$ iff $\tau_2\ne 0$.
\item[(ii)] If $\tau_2\ne 0$, then for any $q$
\[
\dim_k D =\left\{
\begin{array}{ll}
2\lfloor q/3\rfloor & \text{for} \quad q\not\equiv 0\bmod 3\;, \\
{2\over 3}q - 1 & \text{for} \quad q\equiv 0\bmod 3\;.
\end{array}
\right.
\]
\end{description}
\end{lemma}
{\noindent \it Proof:\quad }
(i)\quad If $q$ is a power of the characteristic of $k$,
then $\tau_2 = (u-v)^{q-1}$, whence $\tau_2 \ne 0$ iff $u\ne v$ iff $C$
is nodal.
\smallskip\noindent
(ii) just evaluates the formula for $\dim_k D$ found above in terms of
$q\bmod 3$.$\hfill\square$
\[ \]
Putting everything together yields the Hilbert-Kunz
function in the nodal case.
\begin{theorem}
Let $C$ be a nodal cubic over a field $k$ of prime characteristic $p$.
For a power $q$ of $p$, the Hilbert-Kunz function at $q$ is
\[ HK_C(q) =\left\{
\begin{array}{ll}
{7\over 3}q^2 - {1\over 3}q - 1 &
\text{for} \quad q\not\equiv 2\bmod 3\; ,\\
{7\over 3}q^2 - {1\over 3}q - {5\over 3} &
\text{for} \quad q\equiv 2\bmod 3\;.
\end{array}
\right.
\]
\label{th:nodal}
\end{theorem}
If $C$ is a cuspidal cubic, then $\tau_2=0$ for any $q$ and
we get immediately the generalized Hilbert-Kunz function ---
in accordance with [8] and [1]:
\[ HK_{C,(x,y,z)}(q) =\left\{
\begin{array}{ll}
{7\over 3}q^2 & \text{for} \quad q\equiv 0\bmod 3\;, \\
{7\over 3}q^2 - {4\over 3} & \text{for} \quad q\not\equiv 0\bmod 3\;.
\end{array}
\right.
\]
Note however that,
if $q$ is not a power of the characteristic,
this last result will in general depend upon the
choice of the coordinate system made relative to the given
Weierstra\ss\ form.
The case of the generalized Hilbert-Kunz function for a nodal cubic
can be extracted as well ---
and the dependence upon the coordinate system becomes
apparent: If the distinct roots $u,v$ satisfy $u^q-v^q = 0$ for some $q$,
the generalized Hilbert-Kunz function ``jumps up'',
it takes on the value from the cuspidal case.
For any given $q$, we can avoid this situation by
replacing $y$ with $y+\alpha x$, for a general $\alpha\in k$.
The curve $C$ is then still in Weierstra\ss\ form, and
with respect to $(x,y+\alpha x, z)$,
the generalized Hilbert-Kunz function takes on the value predicted
by Theorem \ref{th:nodal}.
Unless the algebraically closed field $k$
is an algebraic closure of a finite field, one
can even find an $\alpha\in k$ that works for all $q$
simultaneously.
\section{Elliptic Curves in Odd Characteristic}
In this section, we prove the announced result for elliptic
curves in odd characteristic and deduce that the Hilbert-Kunz
multiplicity of a generic plane curve equals $\frac{3}{4}d$ when
$d\geq 2$.
\begin{theorem}
Let $f(x,y,z)\in S=k[x,y,z]$ be a cubic polynomial defining
a plane elliptic curve over a field $k$ of odd characteristic
$p$. For any $n\in {\Bbb{N}}$ and $q=p^n$,
the socle degree $a(q)$ of $\theta = S/(f+m^{[q]})$ is minimal,
\[a(q)=\frac {3}{2}q-\frac {1}{2}\quad,\]
and the Hilbert-Kunz function of $R=S/(f)$ at $q$ is given by
\[ HK_R (q)=\frac {9}{4}q^2 - \frac {5}{4}\quad .\]
\label{theorem:main}
\end{theorem}
\newtheorem{corollary}{Corollary}
\begin{corollary}
For any field $k$ of prime characteristic $p$
and any integer $d\geq 2$,
there is a curve $C\subset {\Bbb{P}}^2_k$
of degree $d$ whose Hilbert-Kunz multiplicity
achieves the minimum $\frac{3}{4}d\;$.
\end{corollary}
{\noindent \it Proof:\quad } As shown in [8],
the Hilbert-Kunz multiplicity of the quadric
$g=x^2-yz$ equals $\frac{3}{2}$. For elliptic
curves in any prime characteristic,
Theorem \ref{theorem:major} shows
that their Hilbert-Kunz multiplicities are minimal,
equal to $\frac{9}{4}$.
As any integer $d\geq 2$ can be written
$d=2u+3v$ for some $u,v\in {\Bbb{N}}$,
additivity of the Hilbert-Kunz multiplicity, see [6],
implies that the curve of degree $d$,
defined by $h=g^uf^v$, $f$
a nonsingular cubic, will achieve the minimum.
$\hfill \square$
\[ \]
{\noindent \it Remark 3. }
Semi-continuity of
the Hilbert-Kunz multiplicity yields that the
Hilbert-Kunz multiplicity of a
generic plane curve of degree $d\geq 2$ equals
$\frac{3}{4}d$. Clearly
$c=1$ if the degree $d=1$. So the Hilbert-Kunz
multiplicity of a generic curve is rational and
independent of the (positive) characteristic.
\[ \]
In section 4.1, we recall
a classical result about determinants of Hankel
matrices whose entries are Legendre polynomials, and
in section 4.2,
we use it to determine the invariant
$a(q)$ and to establish Theorem \ref{theorem:main}.
\subsection {Hankel Determinants of Legendre Polynomials}
The Hankel matrices associated
to a sequence $a=\{ a_i \}$ are
\[ H^{(n)}_k (a) = \left(
\begin{array}{llll}
a_n & a_{n+1} & \cdots & a_{n+k-1} \\
a_{n+1} & a_{n+2} & \cdots & a_{n+k} \\
\cdots & \cdots & \cdots & \cdots \\
a_{n+k-1} & \cdots & \cdots & a_{n+2(k-1)}
\end{array}
\right)\quad,
\]
with corresponding Hankel determinants
\[ D^{(n)}_k (a) = \det H^{(n)}_k (a) \quad . \]
The generating function for the Legendre polynomials,
$\{P_n (t)\}_{n\in\Bbb N},$ is
\[ F(t,x)=\frac{1}{\sqrt{1-2tx+x^2}} =
\sum P_n (t) x^n \in{\Bbb Z}[\tfrac{1}{2},t][\![ x]\!] \quad .\]
For each $k$, consider the determinant of the following
Hankel matrix whose entries are Legendre polynomials,
\[ D^{(0)}_k (P_i (t)) =\det \left(
\begin{array}{llll}
P_0 (t) & P_1 (t) & \cdots & P_{k-1} (t) \\
P_1 (t) & P_2 (t) & \cdots & P_k (t) \\
\cdots & \cdots & \cdots & \cdots \\
P_{k-1} (t) & \cdots & \cdots & P_{2k-2} (t)
\end{array}
\right)\quad.
\]
In [2], J. Geronimus gave the following beautiful formula.
\begin{theorem}
\[ D^{(0)}_k (P_i (t)) =
2^{- (k-1)^2} (t^2-1)^{\frac{1}{2} (k-1)k} \quad . \]
\label{theorem:G}
\end{theorem}
P. Monsky communicated a direct proof to us that we now present.
\begin{lemma}
If $t\in {\Bbb{R}}$ and $t>1$, then
\[ P_n (t)=\frac{1}{\pi} \int^{\pi}_0 [t+\sqrt{t^2-1}
\cos\alpha]^n d \alpha . \]
\end{lemma}
{\noindent \it Proof:\quad}
If $x$ is small,
\[ (1-2tx+x^2)^{-\frac{1}{2}}
=[(1-tx)^2-(x\sqrt{t^2-1})^2]^{-\frac{1}{2}} \]
\[=\frac{1}{\pi} \int^{\pi}_0 \frac{d\alpha}{(1-tx)-x\sqrt{t^2-1}
\cos\alpha }
=\frac{1}{\pi} \int^{\pi}_0 \frac{d\alpha}
{1-x[t+\sqrt{t^2-1}\cos\alpha ]}\quad . \]
The expansion of the integrand into a power series of $x$
yields the lemma. $\hfill \square$
\[ \]
{\noindent \it Proof of Theorem \ref{theorem:G} (P. Monsky):\quad}
As both sides of Geronimus' formula are polynomials in $t$,
it suffices to prove it for $t\in {\Bbb{R}}$ and $t>1$.
Let $V$ be the vector space of real
valued continuous functions on $[0,\pi]$. Define a symmetric
bilinear form $(\cdot,\cdot)$ on $V$ through
\[ (f,g)=\frac{1}{\pi} \int^{\pi}_0 f(\alpha)g(\alpha) d\alpha
\quad . \]
If $h_1,\ldots,h_s \in V$, let
$\Delta (h_1,\ldots,h_s)$ be the determinant of the matrix
$((h_i,h_j))$.
Set $g_m=[t+\sqrt{t^2-1} \cos\alpha]^m$. By
the preceding lemma, $(g_i,g_j) = P_{i+j}$.
So the required Hankel determinant
is $\Delta (g_0,\ldots,g_{k-1})$.
Let $V_m\subset V$ be the subspace spanned by
$\{1,\cos\alpha,\ldots,(\cos\alpha)^m\}$.
Then $f_m=(t^2-1)^{\frac{m}{2}} \cos (m\alpha) \in V_m$,
and it is a linear combination of $g_0,\ldots,g_m$.
Furthermore,
modulo $V_{m-1}$, $\cos(m\alpha) \equiv 2^{m-1} (\cos\alpha)^m$,
and consequently $f_m \equiv 2^{m-1} (t^2-1)^{\frac{m}{2}}
(\cos \alpha)^m \equiv 2^{m-1} g_m$. We conclude that
\[ \Delta (f_0,\cdots,f_{k-1})=(\prod^{k-1}_{m=1} 2^{m-1})^2
\Delta (g_0,\cdots,g_{k-1})=2^{(k-2)(k-1)}\Delta (g_0,\cdots,g_{k-1}).\]
But using the orthogonality of the $f_i$ one finds that
\[ \Delta (f_0,\cdots,f_{k-1})=2^{-(k-1)}(t^2-1)^{\frac{(k-1)k}{2}} \]
and Theorem \ref{theorem:G} follows. $\hfill \square$
\[ \]
Now consider
\[ G(t,x)=\sqrt{1-2tx+x^2}=\sum \widetilde{P_n (t)} x^n\quad. \]
As $F(t,x) G(t,x) =1$,
Geronimus' formula yields also the following corollary.
\begin{corollary}
\[ D^{(2)}_k (\widetilde{P_i (t)} )= (-1)^k D^{(0)}_{k+1} (P_i (t))
= (-2)^{-k^2} (t^2-1)^{\frac{1}{2} k(k+1)}
\quad . \]
\end{corollary}
{\noindent \it Remark 4. }
The coefficients of Legendre polynomials are rational numbers
whose denominators are powers of $2$. Thus Geronimus' identity
and the above corollary hold over any ring in which $2$
is a unit, in particular over a field of odd characteristic.
\subsection{The Invariant $a(q)$ }
We first prove the following theorem showing that
in odd characteristic there are no
nontrivial syzygies of low degree between
the equation of an elliptic curve and Frobenius powers of the
variables.
\begin{theorem}
Let $k$ be a field of odd characteristic $p$,
and let $f\in k[x,y]$ be a cubic polynomial defining
an elliptic curve in ${\Bbb{A}}^2_k$.
For any $q=p^n$, with $n\in {\Bbb{N}}$,
if $f|u x^q+v y^q+w$ for $u,v,w\in k[x,y]$ of degree at most
$\frac{1}{2}(q-1)$, then $f$ divides each of $u,v,w$.
\label{theorem:second}
\end{theorem}
{\noindent \it Proof:\quad}
We give the proof for $q\equiv 1\bmod 4$.
The argument in the other case, $q\equiv 3\bmod 4$,
is analogous and left to the reader. Without loss
of generality assume that $k$ is algebraically closed.
Since the result
is invariant under the action of $GL(2, k)$, and
the characteristic of $k$ is odd,
we can put the cubic into the form $f=y^2-x(1-2tx+x^2)$
with $t^2\neq 1$.
If $f | u x^q + v y^q + w $ for some $u , v , w $
of degree at most $\frac{1}{2} (q-1) $, then
\begin{equation}
u x^q + v y^q + w = f h \quad ,
\label{eq:main}
\end{equation}
where $h$ is a polynomial in $x,y$.
Set
\begin{eqnarray*}
l & = & \frac{q-1}{2}\quad, \\
g & = & x(1-2tx+x^2)\quad .
\end{eqnarray*}
We can then write
\[ u=\sum^l_{i=0} a_i y^i
=A_0 + yA_1 + fu_1\quad, \]
with $a_i\in k[x]$ of degree at most $l-i,
u_1 \in k[x,y]$ and
\[ A_0=\sum^{\frac{l}{2}}_{j=0} a_{2j}g^j\quad,\quad
A_1=\sum^{\frac{l}{2}-1}_{j=0} a_{2j+1}g^j\quad, \]
polynomials in $x$. Similarly, we write
\begin{eqnarray*}
v & = & B_0 +yB_1+fv_1 \\
w & = & C_0 +yC_1+fw_1 \\
y^q & = & y^{2l+1}=yg^l+ f\gamma
\end{eqnarray*}
for polynomials $v_1, w_1, \gamma\in k[x,y];
B_0,B_1,C_0,C_1\in k[x]$.
Equation (\ref{eq:main}) then becomes
\[ (x^qA_0+C_0+g^{l+1}B_1)+y(x^qA_1+C_1+g^lB_0)=fh_1 \quad .\]
Viewing both sides as polynomials in $y$, we get $h_1=0$ and
\begin{equation}
x^qA_0+C_0+g^{l+1}B_1 = 0\quad ,
\label{eq:1}
\end{equation}
\begin{equation}
x^qA_1+C_1+g^lB_0 = 0\quad .
\label{eq:2}
\end{equation}
As $\deg C_0\leq \deg x^q A_0 \leq \frac{7}{2} l + 1$,
it follows that
\[ \deg B_1 \leq (\frac{7}{2} l + 1) -
3 (l + 1) = \frac{l}{2} -2\quad , \]
and similarly
$\deg B_0 \leq \frac{l}{2} -1$.
Thus we can write
\begin{eqnarray*}
B_1 & = & \alpha_{\frac{l}{2}-2} x^{\frac{l}{2}-2} +
\cdots + \alpha_0 \quad, \\
B_0 & = & \beta_{\frac{l}{2}-1} x^{\frac{l}{2}-1}
+ \cdots + \beta_0\quad,
\end{eqnarray*}
for tuples
\[ \alpha = ( \alpha_i ) \in k^{\frac{l}{2}-1} \quad ,
\quad \beta= (\beta_i) \in k^{\frac{l}{2}} \quad . \]
Since $\deg C_0 \leq \frac{3}{2} l $ and
${\rm ord}(x^q A_0) \geq q=2l+1 $,
the intermediate powers of $x$ in
$g^{l+1} B_1 $ have zero coefficients, whence we get
a linear system of equations for $\alpha$, say
$E \alpha =0$, where
$E : k^{\frac{l}{2}-1} \longrightarrow k^{\frac{l}{2}}$
is represented by the matrix
\[ E=\left(
\begin{array}{llll}
e_2 & e_3 & \cdots & e_{\frac{l}{2}} \\
e_3 & e_4 & \cdots & e_{\frac{l}{2}+1} \\
e_4 & \cdots & \cdots & e_{\frac{l+2}{2}} \\
\cdots & \cdots & \cdots & \cdots \\
e_{\frac{l}{2}+1} & \cdots & \cdots & e_{l-1}
\end{array}
\right)\quad,
\]
whose entries $e_i$ are the coefficients in the expansion
\[ (1-2tx+x^2)^{\frac{q+1}{2}} = x^{q+1} + e_q x^q +\cdots
+ e_0 \quad . \]
Analogously, the corresponding powers of $x$ in $g^l B_0$
yield a system of equations for $\beta$, say
$H \beta = 0$,
where
$H : k^{\frac{l}{2}} \longrightarrow k^{\frac{l}{2}+2}$ is
represented by the matrix
\[ H=\left(
\begin{array}{llll}
h_0 & h_1 & \cdots & h_{\frac{l}{2}-1} \\
h_1 & h_2 & \cdots & h_{\frac{l}{2}} \\
\cdots & \cdots & \cdots & \cdots \\
h_{\frac{l}{2}+1} & \cdots & \cdots & h_l
\end{array}
\right)\quad,
\]
whose entries $h_i$ are the coefficients in the expansion
\[ (1-2tx+x^2)^{\frac{q-1}{2}}= x^{q-1} + h_{q-2} x^{q-2} +
\cdots + h_0\quad . \]
As $q$ is a power of the characteristic $p$, one has
\[(1-2tx+x^2)^{\frac{q-1}{2}} \equiv
\frac{(1 - (2tx)^q + x^{2q})^{\frac{1}{2}}}
{\sqrt{1-2tx+x^2}} \equiv (1 - (2tx)^q + x^{2q})^{\frac{1}{2}}
\sum P_n (t) x^n \bmod p\quad,\]
whence $h_i \equiv P_i (t)\bmod p$ and, analogously,
$e_i \equiv \widetilde{P_i (t)}\bmod p$ for $i<q$, where
$P_i (t)$ and $\widetilde{P_i (t)}$ are as in section 4.1.
As $t^2\neq 1$ by assumption, Geronimus' Theorem and its Corollary
imply
\[ {\rm rank} \; E = \frac{l}{2} -1 \quad,\quad
{\rm rank} \; H = \frac{l}{2}\quad , \]
whence each $\alpha_i$ or $\beta_j$ equals zero, thus
$B_0=B_1=0$, so that $v=fv_1$.
As $\deg C_i <{\rm ord} (x^qA_i)$; for $i=0,1$;
it follows further from equations (\ref{eq:1}) and (\ref{eq:2})
that $C_i=A_i=0$ and the theorem follows.
$\hfill \square$
\[ \]
Now we can finish the {\it Proof of Theorem \ref{theorem:main}:}
As $d=3<3(q-1)$, for any power $q=p^n; n\in {\Bbb{N}};$
of an odd prime $p$,
Theorem \ref{theorem:minimal} yields the lower bound
$a(q) \geq \frac{3}{2}q-\frac{1}{2}$,
and the upper bound $\imath (q)\leq \frac{3}{2} q-\frac{5}{2}$.
It remains thus to show
$\vartheta_{\frac{3}{2} q-\frac{7}{2} }=0$,
or, equivalently,
if $f | u x^q+v y^q + w z^q $ for
$u ,v ,w \in k[x,y,z]_{\frac{1}{2} (q-1) }$, then
$f | u ,v , w $.
As it suffices to verify the above statement in
the affine part $(z=1)$ of ${\Bbb{P}}^2_k $, the result in
Theorem \ref{theorem:second} finishes the proof.
$\hfill \square$
\section{Cayley's Cubic Surface}
Let $S=k[x,y,z,w]$ be the polynomial ring in four variables over
an arbitrary field $k$ and let $f=xyz+xyw+xzw+yzw$ be the
Cayley cubic.
We consider the generalized Hilbert-Kunz function of $R=S/f$,
given at $q\in\Bbb N$ through
\[ HK_{R,(x,y,z,w)} (q)=\dim_k S/(f,x^q,y^q,z^q,w^q) \quad. \]
\begin{theorem}
The socle degree of the artinian
ring $\theta= S/(f,x^q,y^q,z^q,w^q)$ is
\[ a(q)=\left\{
\begin{array}{ll}
0 & {\text if} \quad q=1\;, \\
2n-1 & {\text if} \quad q>1\;,
\end{array}
\right.
\]
and the value of the generalized Hilbert-Kunz function
of Cayley's cubic at $q\in\Bbb N$ is
\[ HK_{R,(x,y,z,w)} (q)= 2q^3 - q\quad.\]
\label{theorem:third}
\end{theorem}
{\noindent \it Proof:\quad }
If $q=1$, then $\theta \cong k$ and $a(1)=0$.
Now assume $q>1$.
Since $d=3<4(q-1)$, Theorem \ref{theorem:minimal}
yields the lower bound $a(q)\geq 2q-1$.
Thus it remains to show $\theta_{2q}=0$, i.e,
that any monomial $x^iy^jz^kw^l\in \theta_{2q}$
is equivalent to $0$.
We prove this by ``descent'' on the
{\it dominant exponent\/}, $e=\max\{i,j,k,l\}$,
of a monomial $x^iy^jz^kw^l$ in $S_{2q}$.
Any monomial with sufficiently large dominant exponent ---
for example, when it exceeds $q$ --- will be
equivalent to $0$. We may thus assume
that for a fixed integer $e$, any monomial whose
dominant exponent exceeds $e$ is equivalent to $0$,
to consider then monomials with dominant exponent equal to $e$.
Due to symmetry, it suffices to consider monomials $x^iy^jz^kw^l$
with $e=i\geq j \geq k \geq l$.
{\it Case 1: \quad}
If $l>0$, then
\begin{eqnarray*}
x^iy^jz^kw^l & \equiv &
x^iy^{j-l}z^{k-l} \cdot (-1)^l x^l (yz+yw+zw)^l \bmod f\\
& \equiv &
(-1)^lx^{i+l} y^{j-l}z^{k-l}(yz+yw+zw)^l \bmod f\\
& \equiv & 0\bmod (f,x^q,y^q,z^q,w^q)
\end{eqnarray*}
by assumption.
{\it Case 2: \quad}
Now suppose $l=0$.
If $k\leq 1$, then $i+j+1\geq 2q$ with $i\geq j$, so that
$i\geq q$ whence the monomial is obviously equivalent to
$0$. If $k>1$,
consider first the monomial $g=x^{i-k+1}y^jz^kw^{k-1}$
and set $h=\min\{ i-k+1,k-1 \}$. Since $j>i-k+1>0$ and
$j\geq k>k-1>0$, the argument employed above shows that $g$ is
equivalent to a linear combination of monomials whose dominant
exponent is $j+h$.
As $j+h>i$, our assumption insures that
$g$ is equivalent to $0$. On the other hand,
\[ g=x^{i-k+1}y^jz^kw^{k-1}
\equiv (-1)^{k-1} x^i y^{j-k+1} z (yz+yw+zw)^{k-1}
\bmod f\quad ,\]
and in the expansion of the right hand side of this congruence,
each term involving $w$ involves all four variables,
has dominant exponent $e=i$, and is thus equivalent to
$0$ by Case 1.
So
\[ g=x^{i-k+1}y^jz^kw^{k-1}
\equiv (-1)^{k-1} x^i y^{j-k+1}z (yz)^{k-1}
\equiv (-1)^{k-1} x^i y^j z^k\quad,\]
whence $x^iy^jz^k\equiv 0$.
The claimed result for the generalized Hilbert-Kunz function
follows now from Theorem \ref{theorem:minimal}, as $d=3 < 4(q-1)$
whenever $q >1$, and its validity for $q=0,1$ is clear.
$\hfill \square$
\[ \]
\begin{corollary}
For any field $k$ of prime characteristic $p$
and any integer $d\geq 2$,
there is a surface $X\subset {\Bbb{P}}^3_k$
of degree $d$ whose Hilbert-Kunz multiplicity
achieves $\frac{2}{3}d$, the minimum possible for such surfaces.
\end{corollary}
{\noindent \it Proof:\quad}
The Hilbert-Kunz multiplicity
of the quadric surface $g=xy-zw$ equals $\frac{4}{3}$
by [1].
As just established,
the Hilbert-Kunz multiplicity of the
Cayley cubic $f$ is equal to $2$. Since
$d=2u+3v$ for some $u,v\in {\Bbb{N}}$, additivity of the
Hilbert-Kunz
multiplicity implies that the surface defined by $g^uf^v$
has Hilbert-Kunz multiplicity equal to $\frac{2}{3}d$.
$\hfill \square$
\[ \]
{\noindent \it Remark 5. }
For any field $k$ of positive characteristic,
by virtue of the above corollary and semi-continuity,
a generic surface in
${\Bbb{P}}^3_k$ of degree $d\geq 2$ achieves the minimal
Hilbert-Kunz multiplicity $\frac{2}{3}d$.
Also, $c=1$ if $d=1$.
So the Hilbert-Kunz multiplicity of a generic surface
is rational and independent of the characteristic.
Note that the Hilbert-Kunz multiplicity of
Cayley's cubic is minimal although this surface is
singular --- in contrast to the case of cubic curves.
\[ \]
{\noindent \it Acknowledgments.\quad}
The authors would like to thank P. Monsky, whose influence on
this paper should be clear to every reader. We are
especially grateful for his permission to include the
direct proof of Geronimus' theorem and for
modifications that lead to a shorter proof
of Theorem \ref{theorem:second}.
We also thank K.~Pardue, who got
us interested in the subject,
and A.~Conca for useful conversations.
|
1996-10-10T01:44:46 | 9610 | alg-geom/9610010 | en | https://arxiv.org/abs/alg-geom/9610010 | [
"alg-geom",
"math.AG"
] | alg-geom/9610010 | Misha S. Verbitsky | Misha Verbitsky | Deformations of trianalytic subvarieties of hyperk\"ahler manifolds | 51 pages, LaTeX2e | Selecta Math. (N.S.) 4 (1998), no. 3, 447--490. | null | null | null | Let $M$ be a compact complex manifold equipped with a hyperk\"ahler metric,
and $X$ be a closed complex analytic subvariety of $M$. In alg-geom/9403006, we
proved that $X$ is trianalytic, i. e., complex analytic with respect to all
complex structures induced by the hyperk\"ahler structure, provided that $M$ is
generic in its deformation class. Here we study the complex analytic
deformations of trianalytic subvarieties. We prove that all deformations of $X$
are trianalytic and naturally isomorphic to $X$ as complex analytic varieties.
We show that this isomorphism is compatible with the metric induced from $M$.
Also, we prove that the Douady space of complex analytic deformations of $X$ in
$M$ is equipped with a natural hyperk\"ahler structure.
| [
{
"version": "v1",
"created": "Wed, 9 Oct 1996 23:39:04 GMT"
}
] | 2008-02-03T00:00:00 | [
[
"Verbitsky",
"Misha",
""
]
] | alg-geom | \section{Introduction.}
\label{_Intro_Section_}
\subsection{An overview}
This is the third article studying closed complex analytic subvarieties
of compact holomorphically symplectic\footnote{Holomorphically
symplectic means ``equipped with a holomorphic symplectic form''.
See \ref{_holomorphi_symple_Definition_} for details.}
K\"ahler manifolds. In the first article in series
(\cite{Verbitsky:Symplectic_I_}), we proved that, when a holomorphically
symplectic manifold $M$ is generic in its deformation class, all
subvarieties $X\subset M$ are also holomorphically symplectic,
i. e. restriction of holomorphic symplectic form to nonsingular
strata of $X$ is non-degenerate. In the second article
\cite{Verbitsky:Symplectic_II_}, we obtained a more precise result
about the structure of such $X$, which is related to the hyperk\"ahler
structure of $M$.
By Yau's proof of Calabi conjecture, $M$ admits a natural hyperk\"ahler
metric (\ref{_symplectic_=>_hyperkahler_Proposition_};
for a definition of hyperk\"ahler manifold,
see \ref{_hyperkahler_manifold_Definition_}).
A hyperk\"ahler structure (which is
essentially a quaternion action in the tangent bundle to $M$) gives
a rise to a whole family of complex structure on $M$, parametrized
by ${\Bbb C} P^1$. These complex structures are called {\bf complex
structures induced by the hyperk\"ahler structure}
(\ref{_indu_comple_str_Definition_}).
Denote the set of all induced complex structures
by $\c R_M$. A closed subset $X\subset M$ is called
{\bf trianalytic} if $X$ is complex analytic with respect
to all induced complex structures $L \in \c R_M$
(\ref{_trianalytic_Definition_}). For an induced
complex structure $L$, we denote by $(M, L)$ the $M$ considered
as a complex manifold, with complex structure $L$. In
\cite{Verbitsky:Symplectic_II_}, we proved that for all $L\in \c R_M$,
with exception of may be a countable set, all complex analytic subsets
of $(M, L)$ are trianalytic (\ref{_hyperkae_embeddings_Corollary_}).
Unlike the second article \cite{Verbitsky:Symplectic_II_},
which supersedes results of the first \cite{Verbitsky:Symplectic_I_},
the present one (the third) elaborates on the results of the second.
We start where \cite{Verbitsky:Symplectic_II_} left. Consider
a complex subvariety $X \subset (M, L)$ which happens to be
trianalytic. Such subvarieties are
called {\bf complex analytic subvarieties of trianalytic type}. We
describe the deformations of $X$ in $(M, L)$ and
the Douady space of such deformations.\footnote{The Douady space
\cite{_Douady_} is the same as Chow scheme, but in complex analytic
situation.}
In \cite{Verbitsky:Symplectic_II_}, we gave a simple cohomological
criterion of trianaliticity, for arbitrary complex analytic
subvariety $X \subset (M, L)$
(see \ref{_G_M_invariant_implies_trianalytic_Theorem_}
of this article). \ref{_G_M_invariant_implies_trianalytic_Theorem_}
immediately implies that a complex analytic deformation of
a trianalytic subvariety is again trianalytic.
The main result of this article is the following theorem.
\hfill
\theorem \label{_iso_intro:Theorem_}
Let $M$ be a hyperk\"ahler manifold, $L$ an induced complex structure,
and $X, X'\subset (M, L)$ be closed complex analytic subvarieties in
the same deformation class. Assume that $X$ is
trianalytic.\footnote{Then the subvariety $X'$ is also trianalytic,
as implied by \ref{_G_M_invariant_implies_trianalytic_Theorem_}.}
Consider $X, X'$ as K\"ahler subvarieties of $(M, L)$,
with K\"ahler metric induced from $M$. Then
\begin{description}
\item[(i)] $X, X'$ are naturally
isomorphic as complex varieties, and this isomorphism is compatible
with the K\"ahler metric.
\item[(ii)] The embedding $X\stackrel i \hookrightarrow M$
is compatible with metrics given by geodesics.
\item[(iii)] Also, every map
$X\stackrel \phi{\:\longrightarrow\:} (M, L)$
is complex analytic, provided that $\phi$ is a
deformation of $i$ in the space of isometries
$X{\:\longrightarrow\:} M$.
\end{description}
\hfill
{\bf Caution} The varieties $X, X'$ need not to be non-singular.
\hfill
\ref{_iso_intro:Theorem_} (i) follows from
\ref{_triana_subse_isome_Theorem_} and
\ref{_triana_subse_comple_ana_Theorem_}.
\ref{_iso_intro:Theorem_}
(ii) follows from \ref{_hype_embe_comple_geode:Corollary_},
and (iii) from \ref{_isome_embe_Proposition_}.
\hfill
As a corollary, we obtain the following interesting result
(\ref{_Doua_hyperka_Theorem_}).
\hfill
\theorem \label{_defo_intro:Theorem_}
Let $M$ be a compact holomorphically symplectic K\"ahler
manifold, and $X\subset M$ a closed complex analytic subvariety.
Consider the Douady space $D_M(X)$ of deformations of $X$ inside
$M$. Then $D_M(X)$ is compact and is equipped with a natural
hyperk\"ahler structure, in the sense of
\cite{_Verbitsky:Hyperholo_bundles_},\footnote{If $D_M(X)$
is non-singular, this means exactly that $D_M(X)$ is hyperk\"ahler.
In singular case, there is no satisfactory definition of a hyperk\"ahler
structure. The definition of \cite{_Verbitsky:Hyperholo_bundles_}
is a palliative, which is probably much stronger that the correct
definition. See \cite{_Simpson:hyperka-defi_}, \cite{_Deligne:defi_}
for alternative definitions.} (see \ref{_singu_hype_Definition_}).
\hfill
\remark This gives a new set of examples of hyperk\"ahler
varieties, in addition to those produced by
\cite{_Verbitsky:Hyperholo_bundles_}. The varieties obtained
through \cite{_Verbitsky:Hyperholo_bundles_} are usually not
compact; on contrary, our new examples are compact.
\hfill
The proof of \ref{_iso_intro:Theorem_} is based on the
following argument. First, we show that every trianalytic
submanifold $X\subset M$ (not necessarily closed) is
{\bf completely geodesic} in $M$ (\ref{_hype_embe_comple_geode:Corollary_}).
\footnote{{\bf Completely geodesic} means
that geodesics in $X$ are also geodesics in $M$.}
A simple geometric argument shows that a family $\c X$ of completely
geodesic submanifolds admits a natural {\bf connection}
(\ref{_conne_Definition_},
\ref{_conne_in_fam_of_comple_geode_Proposition_}).
This connection is compatible with the holomorphic
structure if the family $\c X$ is holomorphic
(\ref{_conne_in_fam_of_comple_geode_Proposition_}).
\hfill
For $M$ a compact hyperk\"ahler variety,
$I$ an induced complex structure and $X$ a trianalytic subvariety of $M$,
we study the Douady space $D(X,I)$ which classifies complex analytic
deformations of $X$ in $(M,I)$, where $(M, I)$ is $M$
considered as a complex manifold. We prove that
the real analytic subvariety $D(X)$ underlying
$D(X,I)$ does not depend from the choice of induced
complex structure. This endows $D(X)$ with a
2-dimensional sphere of complex structures, which
induce quaternionic action in the Zariski tangent space.
A real analytic variety with such a system of complex
structures is called {\bf hypercomplex}
(\ref{_hypercomplex_Definition_}).
We return to the study of the families of trianalytic subvarieties.
As we have shown, the base of the universal family is
hypercomplex. The fibers are trianalytic, and therefore,
hypercomplex as well. The
natural connection in such a family is compatible with
the quaternionic action, because this connection is
{\bf holomorphic} with respect to each of
induced complex structures. Thus, a curvature of this
connection is ${\Bbb H}^*$-invariant (\ref{_curva_SU(2)_inva_Lemma_}).
On the other hand, the curvature lies in the tensor product
of three representations of ${\Bbb H}^*$ of weight one.
A trivial linear algebra argument shows that such a
tensor product does not contain non-trivial ${\Bbb H}^*$-invariant
vectors. This proves that the natural connection
in the family of trianalytic subvarieties is {\bf flat}
(\ref{_conne_in_triana_flat_Theorem_}).
Let $\c X\stackrel \pi {\:\longrightarrow\:} S$ be a family of closed trianalytic
subvarieties in a compact hyperk\"ahler manifold $M$. Let
$X_s^{ns}$ be the set of non-singular points in
$X_s:= \pi^{-1}(s)$, $s\in S$. The submanifold $X_s^{ns}$
is completely geodesic. Thus, the family $\c X$ is equipped
with a natural connection. For every two points $s_1, s_2 \in S$,
we show that this connection might be integrated to an
isomorphism of hyperk\"ahler manifolds
$\Psi^{s_2}_{s_1}:\; X^{ns}_{s_1} {\:\longrightarrow\:} X^{ns}_{s_2}$
(\ref{_triana_subse_isome_Theorem_}). Since $X_s^{ns}$ is
completely geodesic in $M$, its completion as a metric space
is naturally isomorphic to a closure $X_s$ of $X_s^{ns}$ in $M$.
Since the map $\Psi^{s_2}_{s_1}:\; X^{ns}_{s_1} {\:\longrightarrow\:} X^{ns}_{s_2}$
is an isometry, it is naturally extended to an isomorphism
of metric spaces $\bar \Psi^{s_2}_{s_1}:\; X^{ns}_{s_1} {\:\longrightarrow\:}
X^{ns}_{s_2}$ (\ref{_triana_subse_isome_Theorem_}). From
construction it follows that $\Psi^{s_2}_{s_1}$ is compatible
with each of induced holomorphic structures. We extend
this assertion to $\bar \Psi^{s_2}_{s_1}$. This is done
in two steps. A general type argument shows that
a homeomorphism of complex varieties is bimeromorphic
if it is holomorphic in a dense open set. The leap from
bimeromorphic to holomorphic is done via a convoluted
algebro-geometric argument involving normalization
and finite unramified maps.
We finish this paper with the definition of singular hyperk\"ahler
varieties. We cite the previously known examples of singular hyperk\"ahler
varieties and then show that constructed above maps
$\bar \Psi^{s_2}_{s_1}:\; X^{ns}_{s_1} {\:\longrightarrow\:} X^{ns}_{s_2}$
are isomorphisms of hyperk\"ahler varieties. The Douady space
of trianalytic subvarieties of a compact hyperk\"ahler manifold
is proven to be hyperk\"ahler.
\subsection{Contents}
\begin{itemize}
\item Section \ref{_Intro_Section_} is an introduction,
independent from the rest of this paper.
\item Section \ref{_basics_Section_} is a compendium of results
pertaining to hyperk\"ahler geometry and Yang--Mills theory.
We define hyperk\"ahler manifolds, trianalytic subvarieties,
hyperholomorphic bundles, and cite their most basic properties.
A reader acquainted with a literature may skip this section.
\item Section \ref{_nonsingu_preli_Section_} illustrates our results
with a simplicistic example of deformations of a smooth trianalytic
subvariety. This section is also independent from the rest,
and its results are further superseded by
more general statements. It is safe to skip this section too.
\item Section \ref{_comple_geode_defo_Section_} contains a study
of completely geodesic submanifolds and their deformations.
Using an easy geometric argument,
we show that a family of completely geodesic submanifolds
is equipped with a natural connection
(\ref{_conne_in_fam_of_comple_geode_Proposition_}). We study this
connection and show that, under certain additional assumptions,
this connection induces isometry (metric equivalence)
of the fibers of this family. This section does not use
results of hyperk\"ahler geometry.
\item In Section \ref{_comple_geode_hyperho_Section_},
we prove that hyperk\"ahler
submanifolds are always completely geodesic. We use arguments from
Yang--Mills theory and the theory of hyperholomorphic
bundles (\cite{_Verbitsky:Hyperholo_bundles_}).
\item Section \ref{_triholo_Section_} shows that the deformational results
of Section \ref{_comple_geode_defo_Section_} can be applied to the
deformations of hyperk\"ahler submanifolds, not necessary closed.
Again, we use arguments from the theory of hyperholomorphic
bundles. In Appendix to this section, we prove equivalence of real
analytic structures induced on a hyperk\"ahler manifold by different
induced complex structures. The proof is based on twistor
geometry (\cite{_HKLR_}, \cite{_NHYM_}).
\item Section \ref{_Douady_Section_} studies the Douady deformation
space $D(X)$ of trianalytic subvariety $X$ of a compact hyperk\"ahler
manifold $M$.
A general argument of K\"ahler geometry shows that the Douady
space is compact (\cite{_Lieberman_}). Using this, we prove that
the underlying real analytic variety does not depend on the
complex structure. As an application, we obtain that
the Douady space is {\bf hypercomplex} (\ref{_hypercomplex_Definition_}),
i. e., has an integrable quaternionic action in the tangent space.
In the Appendix to this section, we give an independent
proof of the compactness of the Douady space. We describe
the Douady space in terms of the space of isometric embeddings
from $X$ to $M$. The Appendix is based on Wirtinger's
inequality (\ref{_Wirti_for_Kahle_Theorem_}).
The body of this section depends only on the result
of the Appendix to Section \ref{_triholo_Section_}
(the equivalence of induced real analytic structures
on a given hyperk\"ahler manifold).
\item In Section \ref{_Conne_in_fami_Section_},
we define a curvature of a connection in a family of
manifolds. We consider the natural connection on a family
of trianalytic subvarieties of a compact hyperk\"ahler manifolds.
Using the results of Section \ref{_Douady_Section_}
(hypercomplex structure on the Douady space), we show that this connection
is {\bf flat} i. e. its curvature is zero. This result is
not used anywhere outside of Section \ref{_singu_hype_Section_}
(and even there, we don't really need it). However,
the flatness of the connection shows that the
natural isomorphism $\Psi^{s_1}_{s_2}$ of the fibers
of a family $\c X {\:\longrightarrow\:} S$ of trianalytic subvarieties
is independent from the choice of a path $\gamma:\; [0,1] {\:\longrightarrow\:} S$
in its homotopy class.
\item Section \ref{_isome=>holo_Section_} deals with the map
$\Psi^{s_1}_{s_2}:\; X_{s_1} {\:\longrightarrow\:} X_{s_2}$
of fibers of a family of trianalytic subvarieties of $M$,
obtained by integrating the natural connection. This map is by
construction a homeomorphism and is holomorphic outside of singularities,
with respect to each of induced complex structures. Using general argument,
we show that $\Psi^{s_1}_{s_2}$ is bimeromorphic and induces an isomorphism
of normalizations of $X_{s_1}$, $X_{s_2}$. Then, we apply the knowledge
that, outside of singularities,
$X_{s_i}$ are completely geodesic in $M$. This allows us to
show that $\Psi^{s_1}_{s_2}$ induces an isomorphism on the
Zariski tangent spaces. An algebro-geometric argument shows
that these properties are sufficient to prove that
$\Psi^{s_1}_{s_2}:\; X_{s_1} {\:\longrightarrow\:} X_{s_2}$ is a complex analytic
isomorphism, with respect to each of induced complex structures.
In Section \ref{_isome=>holo_Section_}, we use the results of
Section \ref{_comple_geode_defo_Section_} (structure of deformations
of completely geodesic submanifolds) and Section
\ref{_triholo_Section_} (that these results may be applied to
the deformations of trianalytic submanifolds).
\item In Section \ref{_singu_hype_Section_}, we give a definition
of singular hyperk\"ahler varieties. After a short discussion,
we give examples of hyperk\"ahler varieties, stemming from the
theory of hyperholomorphic bundles (\cite{_Verbitsky:Hyperholo_bundles_}).
All trianalytic subvarieties of hyperk\"ahler manifolds are
hyperk\"ahler varieties, which is clear from the definition.
Also, the map $\Psi^{s_1}_{s_2}:\; X_{s_1} {\:\longrightarrow\:} X_{s_2}$
constructed in Section \ref{_isome=>holo_Section_}, is an
isomorphism of hyperk\"ahler varieties. We show that
the Douady space of deformations of a trianalytic subvariety
is also a hyperk\"ahler variety.
\end{itemize}
\section{Basic definitions and results.}
\label{_basics_Section_}
This section used mainly for reference, and contains a
compilation of results and definitions from literature.
An impatient reader is advised to skip it and proceed to the next
section.
\subsection{Hyperk\"ahler manifolds}
\label{_basics_hyperka_Section_}
This subsection contains a compression of
the basic and most known results
and definitions from hyperk\"ahler geometry, found, for instance, in
\cite{_Besse:Einst_Manifo_} or in \cite{_Beauville_}.
\hfill
\definition \label{_hyperkahler_manifold_Definition_}
(\cite{_Besse:Einst_Manifo_}) A {\bf hyperk\"ahler manifold} is a
Riemannian manifold $M$ endowed with three complex structures $I$, $J$
and $K$, such that the following holds.
\begin{description}
\item[(i)] the metric on $M$ is K\"ahler with respect to these complex
structures and
\item[(ii)] $I$, $J$ and $K$, considered as endomorphisms
of a real tangent bundle, satisfy the relation
$I\circ J=-J\circ I = K$.
\end{description}
\hfill
The notion of a hyperk\"ahler manifold was
introduced by E. Calabi (\cite{_Calabi_}).
\hfill
Clearly, hyperk\"ahler manifold has the natural action of
quaternion algebra ${\Bbb H}$ in its real tangent bundle $TM$.
Therefore its complex dimension is even.
For each quaternion $L\in \Bbb H$, $L^2=-1$,
the corresponding automorphism of $TM$ is an almost complex
structure. It is easy to check that this almost
complex structure is integrable (\cite{_Besse:Einst_Manifo_}).
\hfill
\definition \label{_indu_comple_str_Definition_}
Let $M$ be a hyperk\"ahler manifold, $L$ a quaternion satisfying
$L^2=-1$. The corresponding complex structure on $M$ is called
{\bf an induced complex structure}. The $M$ considered as a complex
manifold is denoted by $(M, L)$.
\hfill
\definition \label{_holomorphi_symple_Definition_}
Let $M$ be a complex manifold and $\Omega$ a closed
holomorphic 2-form over $M$ such that
$\Omega^n=\Omega\wedge\Omega\wedge...$, is
a nowhere degenerate section of a canonical class of $M$
($2n=dim_{\Bbb C}(M)$).
Then $M$ is called {\bf holomorphically symplectic}.
\hfill
Let $M$ be a hyperk\"ahler manifold; denote the
Riemannian form on $M$ by $<\cdot,\cdot>$.
Let the form $\omega_I := <I(\cdot),\cdot>$ be the usual K\"ahler
form which is closed and parallel
(with respect to the Levi-Civitta connection). Analogously defined
forms $\omega_J$ and $\omega_K$ are
also closed and parallel.
A simple linear algebraic
consideration (\cite{_Besse:Einst_Manifo_}) shows that the form
$\Omega:=\omega_J+\sqrt{-1}\omega_K$ is of
type $(2,0)$ and, being closed, this form is also holomorphic.
Also, the form $\Omega$ is nowhere degenerate, as another linear
algebraic argument shows.
It is called {\bf the canonical holomorphic symplectic form
of a manifold M}. Thus, for each hyperk\"ahler manifold $M$,
and induced complex structure $L$, the underlying complex manifold
$(M,L)$ is holomorphically symplectic. The converse assertion
is also true:
\hfill
\proposition \label{_symplectic_=>_hyperkahler_Proposition_
(\cite{_Beauville_}, \cite{_Besse:Einst_Manifo_})
Let $M$ be a compact holomorphically
symplectic K\"ahler manifold with the holomorphic symplectic form
$\Omega$, a K\"ahler class
$[\omega]\in H^{1,1}(M)$ and a complex structure $I$.
There is a unique hyperk\"ahler structure $(I,J,K,(\cdot,\cdot))$
over $M$ such that the cohomology class of the symplectic form
$\omega_I=(\cdot,I\cdot)$ is equal to $[\omega]$ and the
canonical symplectic form $\omega_J+\sqrt{-1}\:\omega_K$ is
equal to $\Omega$.
\hfill
\ref{_symplectic_=>_hyperkahler_Proposition_} immediately
follows from the conjecture of Calabi, pro\-ven by
Yau (\cite{_Yau:Calabi-Yau_}).
\blacksquare
\subsection{Hyperholomorphic bundles}
\label{_hyperholo_Subsection_}
This subsection contains several versions of a
definition of hyperholomorphic connection in a complex
vector bundle over a hyperk\"ahler manifold.
We follow \cite{_Verbitsky:Hyperholo_bundles_}.
\hfill
Let $M$ be a hyperk\"ahler manifold. We identify the group $SU(2)$
with the group of unitary quaternions. This gives a canonical
action of $SU(2)$ on the tangent bundle, and all its tensor
powers. In particular, we obtain a natural action of $SU(2)$
on the bundle of differential forms. Of a special interest
to us are those forms which are $SU(2)$-invariant.
\hfill
\lemma \label{_SU(2)_inva_type_p,p_Lemma_}
Let $\omega$ be a differential form over
a hyperk\"ahler manifold $M$. The form $\omega$ is $SU(2)$-invariant
if and only if it is of Hodge type $(p,p)$ with respect to all
induced complex structures on $M$.
{\bf Proof:} This is \cite{_Verbitsky:Hyperholo_bundles_},
Proposition 1.2. \blacksquare
\hfill
Further in this article, we use the following statement.
\lemma \label{_SU(2)_commu_Laplace_Lemma_}
The action of $SU(2)$ on differential forms commutes
with the Laplacian.
{\bf Proof:} This is Proposition 1.1
of \cite{_Verbitsky:Hyperholo_bundles_}. \blacksquare
Thus, for compact $M$, we may speak of the natural action of
$SU(2)$ in cohomology.
\hfill
Let $B$ be a holomorphic vector bundle over a complex
manifold $M$, $\nabla$ a connection
in $B$ and $\Theta\in\Lambda^2\otimes End(B)$ be its curvature.
This connection
is called {\bf compatible with a holomorphic structure} if
$\nabla_X(\zeta)=0$ for any holomorphic section $\zeta$ and
any antiholomorphic tangent vector $X$. If there exist
a holomorphic structure compatible with the given
Hermitian connection then this connection is called
{\bf integrable}.
\hfill
One can define a {\bf Hodge decomposition} in the space of differential
forms with coefficients in any complex bundle, in particular,
$End(B)$.
\hfill
\theorem \label{_Newle_Nie_for_bu_Theorem_}
Let $\nabla$ be a Hermitian connection in a complex vector
bundle $B$ over a complex manifold. Then $\nabla$ is integrable
if and only if $\Theta\in\Lambda^{1,1}(M, \operatorname{End}(B))$, where
$\Lambda^{1,1}(M, \operatorname{End}(B))$ denotes the forms of Hodge
type (1,1). Also,
the holomorphic structure compatible with $\nabla$ is unique.
{\bf Proof:} This is Proposition 4.17 of \cite{_Kobayashi_},
Chapter I.
$\hbox{\vrule width 4pt height 4pt depth 0pt}$
\hfill
\definition \label{_hyperho_conne_Definition_}
Let $B$ be a Hermitian vector bundle with
a connection $\nabla$ over a hyperk\"ahler manifold
$M$. Then $\nabla$ is called {\bf hyperholomorphic} if
$\nabla$ is
integrable with respect to each of the complex structures induced
by the hyperk\"ahler structure.
As follows from
\ref{_Newle_Nie_for_bu_Theorem_}, $\nabla$ is hyperholomorphic
if and only if its curvature $\Theta$ is of Hodge type (1,1) with
respect to any of complex structures induced by a hyperk\"ahler
structure.
As follows from \ref{_SU(2)_inva_type_p,p_Lemma_},
$\nabla$ is hyperholomorphic
if and only if $\Theta$ is a $SU(2)$-invariant differential form.
\hfill
\example \label{_tangent_hyperholo_Example_}
(Examples of hyperholomorphic bundles)
\begin{description}
\item[(i)]
Let $M$ be a hyperk\"ahler manifold, $TM$ its tangent bundle
equipped with Levi--Civita connection $\nabla$. Then $\nabla$
is integrable with respect to each induced complex structure,
and hence, Yang--Mills.
\item[(ii)] For $B$ a hyperholomorphic bundle, all its tensor powers
are also hyperholomorphic.
\item[(iii)] Thus, the bundles of differential forms on a hyperk\"ahler
manifold are also hyperholomorphic.
\end{description}
\subsection{Stable bundles and Yang--Mills connections.}
This subsection is a compendium of the most
basic results and definitions from the Yang--Mills theory
over K\"ahler manifolds, concluding in the fundamental
theorem of Uhlenbeck--Yau \cite{_Uhle_Yau_}.
\hfill
\definition\label{_degree,slope_destabilising_Definition_}
Let $F$ be a coherent sheaf over
an $n$-dimensional compact K\"ahler manifold $M$. We define
$\deg(F)$ as
\[
\deg(F)=\int_M\frac{ c_1(F)\wedge\omega^{n-1}}{vol(M)}
\]
and $\text{slope}(F)$ as
\[
\text{slope}(F)=\frac{1}{\text{rank}(F)}\cdot \deg(F).
\]
The number $\text{slope}(F)$ depends only on a
cohomology class of $c_1(F)$.
Let $F$ be a coherent sheaf on $M$
and $F'\subset F$ its proper subsheaf. Then $F'$ is
called {\bf destabilizing subsheaf}
if $\text{slope}(F') \geq \text{slope}(F)$
A holomorphic vector bundle $B$ is called {\bf stable}
if it has no destabilizing subsheaves.
\hfill
Later on, we usually consider the bundles $B$ with $deg(B)=0$.
\hfill
Let $M$ be a K\"ahler manifold with a K\"ahler form $\omega$.
For differential forms with coefficients in any vector bundle
there is a Hodge operator $L: \eta{\:\longrightarrow\:}\omega\wedge\eta$.
There is also a fiberwise-adjoint Hodge operator $\Lambda$
(see \cite{_Griffi_Harri_}).
\hfill
\definition \label{Yang-Mills_Definition_}
Let $B$ be a holomorphic bundle over a K\"ahler manifold $M$
with a holomorphic Hermitian connection $\nabla$ and a
curvature $\Theta\in\Lambda^{1,1}\otimes End(B)$.
The Hermitian metric on $B$ and the connection $\nabla$
defined by this metric are called {\bf Yang-Mills} if
\[
\Lambda(\Theta)=constant\cdot \operatorname{Id}\restrict{B},
\]
where $\Lambda$ is a Hodge operator and $\operatorname{Id}\restrict{B}$ is
the identity endomorphism which is a section of $End(B)$.
Further on, we consider only these Yang--Mills connections
for which this constant is zero.
\hfill
A holomorphic bundle is called {\bf indecomposable}
if it cannot be decomposed onto a direct sum
of two or more holomorphic bundles.
\hfill
The following fundamental
theorem provides examples of Yang-\--Mills \linebreak bundles.
\theorem \label{_UY_Theorem_}
(Uhlenbeck-Yau)
Let B be an indecomposable
holomorphic bundle over a compact K\"ahler manifold. Then $B$ admits
a Hermitian Yang-Mills connection if and only if it is stable, and
this connection is unique.
{\bf Proof:} \cite{_Uhle_Yau_}. \blacksquare
\hfill
\proposition \label{_hyperholo_Yang--Mills_Proposition_}
Let $M$ be a hyperk\"ahler manifold, $L$
an induced complex structure and $B$ be a complex vector
bundle over $(M,L)$.
Then every hyperholomorphic connection $\nabla$ in $B$
is Yang-Mills and satisfies $\Lambda(\Theta)=0$
where $\Theta$ is a curvature of $\nabla$.
\hfill
{\bf Proof:} We use the definition of a hyperholomorphic
connection as one with $SU(2)$-invariant curvature.
Then \ref{_hyperholo_Yang--Mills_Proposition_}
follows from the
\hfill
\lemma \label{_Lambda_of_inva_forms_zero_Lemma_}
Let $\Theta\in \Lambda^2(M)$ be a $SU(2)$-invariant
differential 2-form on $M$. Then
$\Lambda_L(\Theta)=0$ for each induced complex structure
$L$.\footnote{By $\Lambda_L$ we understand the Hodge operator
$\Lambda$ associated with the K\"ahler complex structure $L$.}
{\bf Proof:} This is Lemma 2.1 of \cite{_Verbitsky:Hyperholo_bundles_}.
\blacksquare
\hfill
Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced
complex structure.
For any stable holomorphic bundle on $(M, I)$ there exists a unique
Hermitian Yang-Mills connection which, for some bundles,
turns out to be hyperholomorphic. It is possible to tell when
this happens (though in the present paper we never use
this knowledge).
\hfill
\theorem
Let $B$ be a stable holomorphic bundle over
$(M,I)$, where $M$ is a hyperk\"ahler manifold and $I$
is an induced complex structure over $M$. Then
$B$ admits a compatible hyperholomorphic connection if and only
if the first two Chern classes $c_1(B)$ and $c_2(B)$ are
$SU(2)$-invariant.\footnote{We use \ref{_SU(2)_commu_Laplace_Lemma_}
to speak of action of $SU(2)$ in cohomology of $M$.}
{\bf Proof:} This is Theorem 2.5 of
\cite{_Verbitsky:Hyperholo_bundles_}. \blacksquare
\subsection{Generic holomorphically symplectic manifolds}
In this section, we define generic holomorphically symplectic manifolds.
Such manifolds, as seen later (\ref{_hyperkae_embeddings_Corollary_}),
admit a hyperk\"ahler structure $\c H$
such that every closed complex analytic subvariety is trianalytic
with respect to $\c H$
(for the definition of trianalytic subvarieities,
see \ref{_trianalytic_Definition_}).
We follow \cite{Verbitsky:Symplectic_I_}
(see also \cite{Verbitsky:Symplectic_II_}).
\hfill
Let $M$ be a compact holomorphically symplectic K\"ahler manifold.
By \ref{_symplectic_=>_hyperkahler_Proposition_},
$M$ has a unique hyperk\"ahler structure with
the same K\"ahler class and holomorphic symplectic form.
Therefore one can without ambiguity speak
about the action of $SU(2)$ on $H^*(M,{\Bbb R})$ (see
\ref{_SU(2)_commu_Laplace_Lemma_}).
Of course, this action essentially depends on the choice
of K\"ahler class.
\hfill
\definition \label{_generic_manifolds_Definition_}
Let $\omega\in H^{1,1}(M)$ be the K\"ahler class
of a holomorphically symplectic
manifold $M$. We say that $\omega$ {\bf induces the $SU(2)$-action
of general type} when all elements of the group
\[ \bigoplus\limits_p H^{p,p}(M)\cap H^{2p}(M,{\Bbb Z})\subset H^*(M)\]
are $SU(2)$-invariant.
A holomorphically symplectic manifold $M$ is
called {\bf of general type} if there
exists a K\"ahler class on $M$ which induces
an $SU(2)$-action of general type.
\hfill
As \ref{_gene_type_co_div_by2_Remark_} below
implies, holomorphically symplectic manifolds
of general type have no Weil divisors.
In particular, such manifolds are never algebraic.
\hfill
\proposition \label{_generic_are_dense_Proposition_}
Let $M$ be a hyperk\"ahler manifold and $S$
be the set of induced complex structures over $M$. Denote by
$S_0\subset S$ the set of $L\in S$ such that the natural
K\"ahler metric on $(M,L)$ induces the $SU(2)$ action of
general type. Then $S_0$ is dense in $S$.
{\bf Proof:} This is Proposition 2.2 from
\cite{Verbitsky:Symplectic_II_}
\blacksquare
\hfill
One can easily deduce from the results in
\cite{_Todorov:Moduli_I_II_} and
\ref{_generic_are_dense_Proposition_} that the set of points
associated with holomorphically symplectic
manifolds of general type is dense in the classifying space
of holomorphically symplectic manifolds of K\"ahler type.
\subsection{Trianalytic subvarieties in compact hyperk\"ahler
manifolds.}
In this subsection, we give a definition and a few basic properties
of trianalytic subvarieties of hyperk\"ahler manifolds.
We follow \cite{Verbitsky:Symplectic_II_}.
\hfill
Let $M$ be a compact hyperk\"ahler manifold, $\dim_{\Bbb R} M =2m$.
\hfill
\definition\label{_trianalytic_Definition_}
Let $N\subset M$ be a closed subset of $M$. Then $N$ is
called {\bf trianalytic} if $N$ is a complex analytic subset
of $(M,L)$ for any induced complex structure $L$.
\hfill
Throughout this paper, we implicitly assume that our
trianalytic subvarieties are connected. Most results are trivially
generalized to the general case.
\hfill
Let $I$ be an induced complex structure on $M$,
and $N\subset(M,I)$ be
a closed analytic subvariety of $(M,I)$, $dim_{\Bbb C} N= n$.
Denote by $[N]\in H_{2n}(M)$ the homology class
represented by $N$. Let $\inangles N\in H^{2m-2n}(M)$ denote
the Poincare dual cohomology class. Recall that
the hyperk\"ahler structure induces the action of
the group $SU(2)$ on the space $H^{2m-2n}(M)$.
\hfill
\theorem\label{_G_M_invariant_implies_trianalytic_Theorem_}
Assume that $\inangles N\in H^{2m-2n}(M)$ is invariant with respect
to the action of $SU(2)$ on $H^{2m-2n}(M)$. Then $N$ is trianalytic.
{\bf Proof:} This is Theorem 4.1 of
\cite{Verbitsky:Symplectic_II_} (see Subsection
\ref{_SU(2)-inv=>triana_Subsection_} for a sketch of a proof).
\blacksquare
\remark \label{_triana_dim_div_4_Remark_}
Trianalytic subvarieties have an action of quaternion algebra in
the tangent bundle. In particular,
the real dimension of such subvarieties is divisible by 4.
\hfill
\ref{_G_M_invariant_implies_trianalytic_Theorem_} has the following
immediate corollary:
\corollary \label{_hyperkae_embeddings_Corollary_}
Let $M$ be a compact holomorphically symplectic
manifold of general type, $S\subset M$ be its complex analytic
subvariety, and $\omega$ be a K\"ahler class
which induces an $SU(2)$-action of general type.
Consider the hyperk\"ahler structure associated with
$\omega$ by \ref{_symplectic_=>_hyperkahler_Proposition_}.
Then $S$ is trianalytic with respect to $\c H$.
\blacksquare
\remark \label{_gene_type_co_div_by2_Remark_}
{}From \ref{_hyperkae_embeddings_Corollary_} and
\ref{_triana_dim_div_4_Remark_}, it follows that
a holomorphically symplectic manifold of general type
has no closed complex analytic subvarieties of odd dimension;
in particular, such a manifold has no divisors.
\subsection{Wirtinger's inequality and its use in K\"ahler geometry}
Let $M$ be a compact K\"ahler manifold, $X\subset M$ a
closed real analytic subvariety.
In this section, we give criteria for $X$ to be
complex analytic, in terms of certain integrals.
We follow \cite{Verbitsky:Symplectic_II_} and
\cite{_Stolzenberg_}.
\hfill
\definition \label{_volume_Riema_Definition_}
Let $H$ be an ${\Bbb R}$-linear space equipped with a
positively defined scalar product, $\dim H =h$. The exterrior
form $\operatorname{Vol}\in \Lambda^h(H)$ is called {\bf a volume form}
if the the standard hypercube with the side 1
has the volume 1 in the measure defined by $\operatorname{Vol}$.
\hfill
Clearly, the volume form is defined up to a sign.
This sign is determined by the choice of orientation on $H$.
In the same manner
we define the top degree differential form $\operatorname{Vol}$ called
{\bf a volume form} on any oriented
Riemannian manifold.
\hfill
Let $V$ be a Hermitian linear space, $W\subset V$ be a
${\Bbb R}$-linear subspace, $\dim_{\Bbb R} W = 2n$. Consider
space $\Lambda^{2n}(W)$ of volume forms on $M$. Let
$\omega$ be the imaginary part of the Hermitian form
on $V$. Consider the vectors $\omega^n$,
$\operatorname{Vol} \in \Lambda^{2n}(W)$. Since $\operatorname{Vol}$ is non-zero,
and $\Lambda^{2n}(W)$ is one-dimensional, we can speak
of a fraction $\frac{\omega^n}{\operatorname{Vol}}$, which is a real number,
defined up to a sign (the form $\operatorname{Vol}$ is defined up to a sign).
Denote by $\Xi_W$ the number
$\left|\frac{\omega^n}{\operatorname{Vol}}\right|$.
\hfill
\lemma\label{_Wirtinger_Lemma_}
(Wirtinger's inequality)
In these assumptions, $\Xi_W\leq 2^n.$
Moreover, if $\Xi_W=2^n$, then $W$ is a
complex subspace of $V$.
{\bf Proof:} \cite{_Stolzenberg_} page 7. \blacksquare
\hfill
Let $M$ be a K\"ahler manifold, $N\subset (M,I)$ be a closed
real analytic subvariety of even dimension,
$N_{ns}\subset N$ the set of non-singular
points of $N$. For each
$x\in N_{ns}$, consider $T_xN$ as a subspace of $T_xM$.
Denote the corresponding number
$\Xi_{T_xN}$ by $\Xi_L (x)$. The following statement is a direct
consequence of \ref{_Wirtinger_Lemma_}:
\hfill
\proposition\label{_N_is_analytic_if_eta_is_constant_Proposition_}
Let $J$ be an induced complex structure.
The subset $N\subset M$ is complex analytic
if and only if
\[ \forall x\in N_{ns} \;\;\;\; \Xi_L(x)=2^n. \]
\blacksquare
\hfill
Let $\operatorname{Vol} N_{ns}$ be the volume form of $N_{ns}$, taken with respect to the
Riemannian form (see \ref{_volume_Riema_Definition_}).
\hfill
\theorem \label{_Wirti_for_Kahle_Theorem_}
Let $M$ be a K\"ahler manifold, $N\subset M$, $\dim N =2n$ a
closed real analytic subvariety, $N_{ns}\subset M$ the set
of its non-singular points. Assume that the improper integrals
$\int_{N_{ns}} \operatorname{Vol} N_{ns}$, $\int_{N_{ns}} \omega^n$ exist.
Then
\[
2^n\int_{N_{ns}} \operatorname{Vol} N_{ns} \geq \int_{N_{ns}} \omega^n
\]
and the equality is reached if and only if $N$ is complex analytic
in $M$.
{\bf Proof:} This is a direct consequence of
\ref{_N_is_analytic_if_eta_is_constant_Proposition_}
\blacksquare
\hfill
We use the term ``symplectic volume'' for the number
$\frac{1}{2^n}\int_{N_{ns}} \omega^n $ and ``Riemannian volume''
for $\int_{N_ns} \operatorname{Vol} N_{ns}$. Then,
\ref{_Wirti_for_Kahle_Theorem_} might be rephrased in the form
``a real analytic cycle is complex analytic if and only if
its symplectic volume is equal to its Riemannian volume''.
\subsection{$SU(2)$-invariant cycles in cohomology
and trianalytic subvarieties}
\label{_SU(2)-inv=>triana_Subsection_}
This subsection consists of a sketch of a proof of
\ref{_G_M_invariant_implies_trianalytic_Theorem_}.
We follow \cite{Verbitsky:Symplectic_II_}.
\hfill
For a K\"ahler manifold $M$, $m=dim_{\Bbb C} M$
and a form $\alpha\in H^{2p}(M,{\Bbb C})$, define
\[
\deg(\alpha):=\int_M \omega^{m-p}\wedge\alpha,
\]
where $\omega$ is the K\"ahler form.
Let $M$ be a compact hyperk\"ahler manifold, $\alpha$ a
differential form.
We denote by $\deg_L\alpha$ the degree associated with
an induced complex structure $L$.
\hfill
\proposition \label{_G_M_invariant_cycles_over_Proposition_}
Let $M$ be a compact hyperk\"ahler manifold and
$\alpha$ be an $SU(2)$-invariant form of non-zero degree.
Then the dimension of $\alpha$ is divisible by 4. Moreover,
\[
\deg_{I}\alpha = \deg_{I'}\alpha,
\]
for every pair of induced complex structures $I$, $I'$.
{\bf Proof:} This is Proposition 4.5 of
\cite{Verbitsky:Symplectic_II_}.
\blacksquare
\hfill
\ref{_G_M_invariant_implies_trianalytic_Theorem_} follows
immediately from \ref{_G_M_invariant_cycles_over_Proposition_}
and \ref{_Wirti_for_Kahle_Theorem_}. By
\ref{_Wirti_for_Kahle_Theorem_}, a real analytic subvariety
$N \subset M$ is trianalytic if and only if its symplectic volume,
taken with respect to any of induced complex structures, is
defined and equal to its Riemannian volume.
In notation of \ref{_G_M_invariant_implies_trianalytic_Theorem_},
the symplectic volume of $N$ taken with respect to $L$
is equal to $\frac {1}{2^n} \deg_L\inangles N$.
{}From \ref{_G_M_invariant_cycles_over_Proposition_},
we obtain that symplectic volume of $N$ is the same for all
induced complex structures. Since $N$ is complex analytic
with respect to $I$, the symplectic volume of $N$ is equal
to the Riemannian volume of $N$, again by
\ref{_Wirti_for_Kahle_Theorem_}. This proves
\ref{_G_M_invariant_implies_trianalytic_Theorem_}.
\section[Deformations of non-singular trianalytic subvarieties.]
{Deformations of non-singular trianalytic \\subvarieties.}
\label{_nonsingu_preli_Section_}
Let $M$ be a holomorphically symplectic K\"ahler manifold,
$X\subset M$ a closed complex analytic submanifold
of trianalytic type. We study
the deformations of $X$ in $M$, in order to prove
\ref{_iso_intro:Theorem_}. It turns out that
\ref{_iso_intro:Theorem_} is almost trivial in the case
$X$ non-singular. In this section, we give the proof
of \ref{_iso_intro:Theorem_} (i) for non-singular $X$; the general
case is proven independently. We hope that a simple argument
will be insightful, even if we need to produce a separate proof
for the general case. This section is perfectly safe to skip.
\hfill
\remark For a smooth trianalytic submanifold $X\subset M$,
$X$ is obviously hyperk\"ahler, in a natural way. The quaternion
action comes from quaternion action in $TM$, and the metric is
induced from $M$ too.
\hfill
We start from the following simple, but important, lemma.
\hfill
\lemma \label{_TM-restrict_X_hyperholo:Lemma_}
Let $M$ be a complex manifold equipped with a hyperk\"ahler metric,
$X\subset M$ a closed complex analytic submanifld of trianalytic
type. Consider the
restriction $TM\restrict{X}$ of the tangent bundle to $M$ on $X$.
We equip $TM_\restrict{X}$ with a connection $\nabla$ coming
from Levi-Civita connection in $TM$. Then $(TM\restrict{X}, \nabla)$
is {\bf hyperholomorphic}.\footnote{See
Subsection \ref{_hyperholo_Subsection_}
for the definition of {\bf hyperholomorphic}.
By \ref{_hyperholo_Yang--Mills_Proposition_},
a hyperholomorphic connection is Yang-Mills.}
\hfill
{\bf Proof:} Consider the natural action of $SU(2)$ on the
space
\[
\Lambda^2\left(X, \operatorname{End}\left(TM\restrict{X}\right)\right).
\]
We need to show that
the curvature $\Theta$ of $\nabla$ is $SU(2)$-invariant. Let
$\nabla_{LC}$ be the Levi--Civita connection on $TM$, and
$(\nabla_{LC})^2\in \Lambda^2(M, \operatorname{End}(TM))$ be its curvature. Clearly,
$\Theta$ is a pull-back of $(\nabla_{LC})^2$ to $X$. Therefore,
it suffices to show that $(\nabla_{LC})^2$ is $SU(2)$-invariant.
This is \ref{_tangent_hyperholo_Example_}.
\blacksquare
\hfill
As a corollary, we obtain the following proposition.
\hfill
\proposition \label{_TM-restrict-X_decompo_for_compa_Proposition_}
In assumptions of \ref{_TM-restrict_X_hyperholo:Lemma_},
let $M$, $X$ be compact. Then the following statements are true.
\begin{description}
\item[(i)] The bundle $TM\restrict{X}$ is naturally
isomorphic to the direct sum $TM\restrict X \cong TX \oplus NX$,
where $N$ is the normal bundle to $X$ inside $M$. This
isomorphism is compatible with the natural connections and Hermitian
metrics on $TM\restrict X$, $TX$, $NX$.
\item[(ii)] $NX$ is hyperholomorphic.
\item[(iii)] For each section $\gamma$ of $NX$, $\gamma$ is nowhere
degenerate, and there is a natural decomposition
\begin{equation} \label{_NX=O-gamma+rest:Equation_}
NX = {\cal O}_\gamma \oplus N_\gamma X,
\end{equation}
where ${\cal O}_\gamma$ is a trivial sub-bundle of $NX$ generated by
$\gamma$, and $N_\gamma X$ its orthogonal complement. The
decomposition \eqref{_NX=O-gamma+rest:Equation_} is compatible
with connection.
\item[(iv)] In assumptions of (iii),
consider the connection $\nabla_\gamma$ induced from
$NX$ to ${\cal O}_\gamma X$. Then is $\nabla_\gamma$ flat.
\end{description}
\hfill
\remark In fact,
\ref{_TM-restrict-X_decompo_for_compa_Proposition_} (i)-(ii) holds true
in assumptions of \ref{_TM-restrict_X_hyperholo:Lemma_} for
general, not necessarily compact, $X$ and $M$
(see \ref{_NX_splits_for_hype_Proposition_},
\ref{_NX_hyperholo_Proposition_}).
However, the proof is easier in compact case.
\hfill
{\bf Proof of \ref{_TM-restrict-X_decompo_for_compa_Proposition_}.}
Consider the embedding
\[ TX \hookrightarrow TM\restrict X \]
of bundles with connection. By \ref{_tangent_hyperholo_Example_},
$TX$ is hyperholomorphic, and hence Yang--Mills.
By \ref{_TM-restrict_X_hyperholo:Lemma_}, $TM\restrict X$ is
hyperholomorphic as well. We obtain that
$TX$ is a destabilizing subsheaf in $TX$.
By \ref{_UY_Theorem_}, a Yang--Mills bundle
is a direct sum of stable bundles.\footnote{Bundles which are direct
sum of stable are called {\bf polystable}.}
Thus, $TM$ is a direct sum of $TX$ and its orthogonal
complement $NX$ (see \ref{_YM_exact_split_Proposition_}
for details). This proves
\ref{_TM-restrict-X_decompo_for_compa_Proposition_} (i).
Since the curvature
of $TM\restrict X$ is decomposed onto a
direct sum of the curvature of $NX$ and the curvature of
$TX$, the curvature of $NX$ is $SU(2)$-invariant.
This proves \ref{_TM-restrict-X_decompo_for_compa_Proposition_} (ii).
\hfill
The condition (iii) follows from (ii) and \ref{_UY_Theorem_},
since every holomorphic section $\gamma$ of a
Yang--Mills bundle $B$ of zero degree (such as hyperholomorphic bundles)
spans a destabilizing subsheaf ${\cal O}_\gamma$.
Since $B$ is Yang--Mills, it is polystable; thus,
${\cal O}_\gamma$ is its direct summand.
This proves \ref{_TM-restrict-X_decompo_for_compa_Proposition_}
(iii). To prove (iv), we notice that ${\cal O}_\gamma$
is a trivial holomorphic bundle with Yang--Mills connection.
By \ref{_UY_Theorem_}, Yang--Mills connection is unique,
and therefore, ${\cal O}_\gamma$ is flat.
\ref{_TM-restrict-X_decompo_for_compa_Proposition_}
is proven.
\blacksquare
\hfill
We recall the following general results of the
theory of deformations of complex subvarieties.
Let $X\subset M$ be a closed complex analytic subvariety
of a compact complex manifold. The {\bf Douady space}
of deformations of $X$ inside of $M$ is defined
(\cite{_Douady_}). We denote the Douady space by $D_M X$,
or sometimes by $D(X)$. By definition, $D_M X$ is a
finite-dimensional complex analytic variety.
The points of $D_MX$ are identified with the
subvarieties $X'\subset M$, where $X'$ is a deformation
of $X$. Let $\gamma:\; {\Bbb R} {\:\longrightarrow\:} D_m X$ be a real analytic map.
Assume that for $t=t_0$, $\gamma(t)$ is a smooth subvariety of
$M$. In deformation theory, the differential
$\frac{d\gamma}{dt}(t_0)$ is interpreted as a
holomorphic section of the normal bundle $N \gamma (t_0)$.
Thus, the Zariski tangent space $T_{X} D_M (X)$
is naturally embedded to the space of holomorphic
sections of $N X$. The following proposition is an
easy application of the Kodaira--Spencer theory.
\hfill
\proposition \label{_compa_subva_iso_holo:Proposition_}
Let $X \subset M$ be a closed complex analytic submanifold of
a compact complex manifold $M$, and $D_M X$ the corresponding
Douady space. Let $\gamma:\; {\Bbb R} {\:\longrightarrow\:} D_M X$ be a real analytic
map. Assume that for all $t_0\in {\Bbb R}$, the subvariety
$\gamma(t_0)$ is smooth. Assume also that the section
\[
\nu_{t_0} =
\frac{d \gamma}{dt}(t_0) \in \Gamma_{\gamma(t_0)} (N \gamma(t_0))
\]
is nowhere degenerate. Finally, assume that $\nu_{t_0}$
splits from $N \gamma(t_0)$, i. e, there exists a decomposition
of a holomorphic vector bundle
\[
N \gamma (t_0) = {\cal O}_{\nu_{t_0}} \oplus N'
\]
where ${\cal O}_{\nu_{t_0}}$ is the trivial subbundle of
$N \gamma(t_0)$ generated by $\nu_{t_0}$.
Then the subvarieties $\gamma(t)\subset M$ are naturally
isomorphic for all $t$.
{\bf Proof:} Well known; see, for instance,
\cite{_Kodaira_Spencer_} \blacksquare
\hfill
\proposition \label{_compa_subva_iso_metric:Proposition_}
In assumptions of \ref{_compa_subva_iso_holo:Proposition_},
let $M$ be K\"ahler. Consider the induced
Hermitian structure on $N\gamma(t)$, for all $t\in {\Bbb R}$.
Assume that for all $t\in {\Bbb R}$, the section
\[
\nu_{t_0} =
\frac{d \gamma}{dt}(t_0)
\]
has constant length. Assume also that the
orthogonal decomposition \[ TM\restrict X = NX \oplus NX^\bot \]
is compatible with the Levi-Civita connection in
$TM\restrict X$. Consider the isomorphisms
$\psi_{t_1,t_2}:\; \gamma(t_1){\:\longrightarrow\:} \gamma(t_2)$ constructed in
\ref{_compa_subva_iso_holo:Proposition_}. Then
the maps $\psi_{t_1,t_2}$ are compatible with the K\"ahler
metric induced from $M$.
{\bf Proof:} The proof follows from Kodaira--Spencer construction;
for a complete argument, see
\ref{_conne_in_fami_of_comple_geo_Proposition_} (ii).
\blacksquare
\hfill
The following theorem is the main result of this section.
\hfill
\theorem \label{_iso_for_smooth_subva:Theorem_}
Let $M$ be a compact holomorphically symplectic
K\"ahler manifold,
and $X\subset M$ a complex submanifold
of trianalytic type. Let $X'$ be a deformation of $X$
in $M$. Then there exists a complex analytic
isomorphism $\psi:\; X {\:\longrightarrow\:} X'$. Moreover, if
the K\"ahler metric on $M$ is hyperk\"ahler,
then $\psi:\; X {\:\longrightarrow\:} X'$
is compatible with the K\"ahler structure induced from $M$.
\hfill
{\bf Proof:} The deformations of $X$ are infinitesimally classified
by the sections of $NX$. Let $\gamma$ be such a section.
Applying \ref{_TM-restrict-X_decompo_for_compa_Proposition_},
we obtain that assumptions of
\ref{_compa_subva_iso_holo:Proposition_} and
\ref{_compa_subva_iso_metric:Proposition_} hold for the deformations
of $X$ inside $M$. Now, \ref{_iso_for_smooth_subva:Theorem_}
is directly implied by \ref{_compa_subva_iso_holo:Proposition_} and
\ref{_compa_subva_iso_metric:Proposition_}. \blacksquare
\hfill
\remark
\ref{_iso_for_smooth_subva:Theorem_}
gives a proof of \ref{_iso_intro:Theorem_}
(i), for the subvarieties $X\subset M$ which are
smooth. We prove \ref{_iso_for_smooth_subva:Theorem_} for
general subvarieties in \ref{_triana_subse_comple_ana_Theorem_}.
\section[Completely geodesic embeddings of Riemannian manifolds.]
{Completely geodesic embeddings \\of Riemannian manifolds.}
\label{_comple_geode_defo_Section_}
Before we proceed with the proof of \ref{_iso_intro:Theorem_},
we have to prove a serie of preliminary
results from deformation theory. The arguments of
deformation theory are greatly simplified in the
hyperk\"ahler case, because hyperk\"ahler embeddings are
completely geodesic\footnote{For a definition of completely geodesical
embeddings, see \ref{_comple_geode:Definition_}.}
(\ref{_hype_embe_comple_geode:Corollary_}).
We prove a number of simple statements from the deformation
theory of completely geodesical embeddings of K\"ahler manifolds.
\subsection{Completely geodesic submanifolds}
\nopagebreak
\hspace{5mm}
\proposition \label{_comple_geodesi_basi_Proposition_
Let $X \stackrel \phi\hookrightarrow M$ be an embedding of Riemannian
manifolds (not necessarily compact) compatible with the Riemannian
structure.\footnote{Such embeddings are called
{\bf Riemannian embeddings}.}
Then the following conditions are equivalent.
\begin{description}
\item[(i)] For every point $x\in X$, there exist a neighbourhod $U \ni x$
of $x$ in $X$ such that for all
$x' \in U$ there is a geodesic in $M$
going from $\phi(x)$ to $\phi(x')$ which lies
in $\phi(X)\subset M$.
\item[(ii)] Consider the Levi-Civita connection $\nabla$ on $TM$,
and restriction of $\nabla$ to $TM \restrict{X}$. Consider the
orthogonal decomposition
\begin{equation} \label{TM_decompo_Equation_}
TM\restrict{X} = TX \oplus TX^\bot.
\end{equation}
Then, this decomposition is preserved by the connection $\nabla$.
\end{description}
{\bf Proof:} Well known; see, for instance,
\cite{_Besse:Einst_Manifo_}.
\hbox{\vrule width 4pt height 4pt depth 0pt}
\hfill
\definition\label{_comple_geode:Definition_}
Let $X \stackrel i \hookrightarrow M$ be a Riemannian embedding
satisfying either of the conditions of
\ref{_comple_geodesi_basi_Proposition_}. Then $i$ is called
{\bf a completely geodesic embedding}, and the image $i(X)\subset M$
is called {\bf a completely geodesic submanifold}.
\hfill
\lemma \label{_comple_geo_compa_holo_Lemma_}
Let $M$ be a K\"ahler manifold, $X\subset M$ a complex submanifold.
If $X$ is completely geodesic, then the decomposition
\eqref{TM_decompo_Equation_}
is compatible with the holomorphic structure
on $TX$, $TM\restrict X$.
{\bf Proof:} Clear. \blacksquare
\subsection{Deformations of submanifolds}
\label{_defo_subva_conven_Subsection_}
Let $\c X \stackrel \pi {\:\longrightarrow\:} S$ be a family of complex
manifolds equipped with a map $\c X \stackrel \phi {\:\longrightarrow\:} M$.
Denote the pre-image $\pi^{-1}(s)\subset \c X$ by $X_s$.
Assume that for all $s\in S$
the restriction $\phi_s:\; X_s{\:\longrightarrow\:} M$ of
$\c X \stackrel \phi {\:\longrightarrow\:} M$
to $X_s \subset \c X$ is a smooth embedding.
Assume that for $s_0\in S$, the image $\phi_{s_0}(X_{s_0})\subset M$
coinsides with $X$.
\hfill
\definition
The collection of data
\[ \left( \c X \stackrel \pi {\:\longrightarrow\:} S,
\phi, \phi_{s_0}:\; X_{s_0} \oldtilde {\:\longrightarrow\:} X\right )
\]
is called {\bf a family of submanifolds of $M$},
and {\bf a family of deformations of $X$}.
The same definition can be formulated for $M$, $X$, $S$, $\c X$
real analytic; in this case, we speak of {\bf family of real
analytic submanifolds of $M$}. Also, the submanifolds might be
replaced by subvarieties in order to obtain the definition
of a family of subvarieties.
\subsection{Section of a normal bundle arising from a deformation}
\label{_norma_vecto_Subsection_}
Let $M$ be a complex or real analytic manifold and
\[ \left( \c X \stackrel \pi {\:\longrightarrow\:} S,
\phi:\; \c X {\:\longrightarrow\:} M \right)
\]
be a system of submanifolds.
For each tangent vector $t \in T_{s_0} S$, deformation theory
gives a canonical section $\eta$ of the normal bundle $N X_{s_0}$.
This section is holomorphic if we work in complex situation.
We recall how this section is obtained.
For a sufficiently small neighbourhood of $x\in \c X$,
we can always find coordinates in $X_s$ which analytically
depend on $s$. For each point $x\in X_{s_0}$,
denote by $x(s)$ the point of $X_s$ with the same
coordinates as $x$. Consider the vector
$\tilde \eta_x :=\frac{dx(s)}{ds}(t)\in T_xM$, which is a derivative
of $x(s)$ along $t$. The vector $\tilde \eta_x$ obviously depends
on the choice of
coordinates in $X_s$. Let $\eta_x\in NX_{s_0}\restrict x$ be
the image of $\tilde\eta_x$ under the natural map
$T_x M {\:\longrightarrow\:} N X_{s_0}\restrict x$. Clearly, $\eta_x$ is
independent from the choice of coordinates. Gluing
$\eta_x$ together, we obtain the canonical
section $\eta\in NX_{s_0}$.
We need the following technical lemma in Section
\ref{_triholo_Section_}.
\hfill
\lemma \label{_norma_sec_holom_Lemma_}
Let $M$ be a complex analytic manifold and
\[ \left( \c X \stackrel \pi {\:\longrightarrow\:} S,
\phi:\; \c X {\:\longrightarrow\:} M \right)
\]
be a real analytic system of submanifolds. Assume that for
all $s\in S$, the submanifold $\pi^{-1}(s) := X_s\subset M$
is complex analytic.
Then, for all $t\in T_{s_0} S$, the corresponding section
$\eta \in N X_{s_0}$ is holomorphic.
\hfill
{\bf Proof:} Shrinking $\c X$ if necessary, we
can find a system of holomorphic
coordinates in $X_s$, $s\in S$
which depends smoothly upon $s$. Consider the section
$\tilde \eta\in TM\restrict{X_{s_0}}$ obtained by
deriving $x(s)$ along $t$ as above. Since the map
$x{\:\longrightarrow\:} x(s)$ is by construction holomorphic,
the section $\tilde\eta$ is also holomorphic. Then,
$\eta$ is holomorphic by construction. \blacksquare
\subsection{Connections in families of manifolds}
\label{_conne_Subsection_}
\hfill
\definition \label{_conne_Definition_}
Let $\pi:\; \c X {\:\longrightarrow\:} S$ be a family
of real analytic manifolds parametrized by $S$.
Consider the bundles $T\c X$, $N_\pi X$,
where $N_\pi X$ is a fiberwise normal bundle
to the fibers of $\pi$. There is a natural
projection $p:\; T\c X {\:\longrightarrow\:} N_\pi X$.
The {\bf connection} in $\c S$ is
a section of $p$, i. e. such an embedding
$\nabla:\; N_\pi X {\:\longrightarrow\:} T\c X$ that
$p \circ s = \operatorname{Id} \restrict {N_\pi X}$.
This definition
is naturally adopted to the case of complex analytic family
of manifolds. If the
section $\nabla:\; N_\pi X {\:\longrightarrow\:} T\c X$ is holomorphic,
the connection is called {\bf a holomorphic connection}.
This definition makes sense only when the base variety $S$ is smooth.
However, the connection in a family gives a connection on a pullback
of this family, under all maps $s:\; S' {\:\longrightarrow\:} S$. Connections on the
pullback families are naturally compatible. We shall sometimes speak
of {\bf connection} in a family where the base $S$ is not smooth.
This means that for all maps $s:\; S' {\:\longrightarrow\:} S$, where $S'$ is smooth,
the pullback family is equipped with a connection, and these
connections are compatible. All the definitions and results
(which we state and prove in the case of a smooth base) are
naturally adopted to the case when the base $S$ is singular.
\hfill
\proposition \label{_conne_in_fam_of_comple_geode_Proposition_}
Let
\[ \left( \c X \stackrel \pi {\:\longrightarrow\:} S,
\phi:\; \c X {\:\longrightarrow\:} M \right)
\]
be a family of submanifolds of a Riemannian manifold $M$.
Assume that for all $s\in S$, the submanifold
$X_s:= \psi(\pi^{-1}(s)) \subset M$ is completely geodesic. Then
the family $\c X$ is equipped with a natural connection.
Moreover, if $M$ is K\"ahler and the family $\c X$ is complex
analytic, then the connection $\nabla$ is holomorphic.
\hfill
{\bf Proof:} Let $T_\pi X\subset T\c X$ be the bundle of vectors
tangent to the fibers of $\pi$.
To split the exact sequence
\[
0 {\:\longrightarrow\:} T_\pi X \stackrel i{\:\longrightarrow\:} T\c X {\:\longrightarrow\:} T_\pi X {\:\longrightarrow\:} 0,
\]
we have to construct a surjection
\begin{equation} \label{_surje_secti_for_conne_Equation_}
T{\c X} \stackrel p {\:\longrightarrow\:} T_\pi X
\end{equation}
satisfying $i\circ p = \operatorname{Id}_{T_\pi X}$.
Consider the orthogonal decomposition
\begin{equation} \label{_ortho_compo_for_norma_in_fami_Equation_}
\phi^*TM=\phi^*N_M X_t \oplus TX_t,
\end{equation}
where $N_M X_t$ is the normal bundle to $\phi(X_t)$ in $M$.
This gives a natural epimorphism
\[ \phi^* TM \stackrel e{\:\longrightarrow\:} T_\pi X.\] Taking a composition
of $e$ with \[ d\phi:\; T\c X {\:\longrightarrow\:} \phi^* TM,\]
we obtain a surjection $p$ of
\ref{_surje_secti_for_conne_Equation_}.
In K\"ahler case, the decomposition
\eqref{_ortho_compo_for_norma_in_fami_Equation_}
is holomorphic by \ref{_comple_geo_compa_holo_Lemma_}. Thus,
we constructed a connection which is holomorphic.
\blacksquare
\hfill
For each real analytic path $\gamma:\; [0,1] {\:\longrightarrow\:} S$,
$t, t'\in [0,1]$, we may {\bf integrate} the connection
along $\gamma$. This notion is intuitively
clear (and well known). We recall briefly
the definition of the integral maps associated to connection,
in order to fix the notation.
Restricting the family $\c X$ to the image of $\gamma$
in $S$ does not change the result of integration.
To simplify the exposition, we assume that $S$ coinsides with
the image $D \cong [0,1]$ of $\gamma$. In this case, the bundle $N_\pi X$
is naturally isomorphic to a pullback $\pi^* T D$. Denote by
$\nu$ a canonical unit section of $\pi^* TD$ corresponding to
the unit tangent field to $[0,1]$. Then, $\nabla \nu$ is a vector
field in $T\c X$. Integrating $\nu$ to a
diffeomorphism, we obtain a map $\exp(t \nu)$ defined in an open
subset of $\c X$, for $t\in {\Bbb R}$. Clearly, $\exp(t\nu)$ maps the points
of $X_{t_1}$ to the points of $X_{t_1+t}$. The resulting
diffeomorphism we denote by
\[
\Psi^{t_1}_{t_1+t}:\; U_{t_1} {\:\longrightarrow\:} U_{t+t_1},
\]
where $U_{t_1}$ is an intersection of $X_{t_1}$ with the domain
of $\exp(t\nu)$. Clearly, the same definition applies to the
arbitrary families of manifolds with connection. When $\c X$
is a complex analytic family with a holomorphic connection,
the diffeomorphisms $\Psi^t_{t'}$ are complex analytic.
\subsection{Deformations of completely
geodesic submanifolds (the main statement)}
Let
\[ \left( \c X \stackrel \pi {\:\longrightarrow\:} S,
\phi, \phi_{s_0}:\; X_{s_0} \oldtilde {\:\longrightarrow\:} X\right )
\]
be a real analytic deformation of $X\subset M$,
and $\gamma:\; [0,1] {\:\longrightarrow\:} S$ be a real analytic map.
Slightly abusing the notation, we
shall consider the fibers $X_s$ of $\pi$ as subvarieties in $M$.
Assume that $X_{\gamma(t)}$ is a completely geodesic complex
analytic submanifold of $M$, for all
$t$. Let $\eta_t \in \Gamma \left(N X_{\gamma(t)}\right)$ be a section of
the normal bundle to $X_{\gamma(t)}$ corresponding to the vector
$\frac {d \gamma}{dt} \in T_{\gamma(t)} S$.
Using the natural
holomorphic embedding
\begin{equation} \label{_N_embe_to_TM_Equation_}
N X_{\gamma(t)} \hookrightarrow TM\restrict{X_{\gamma(t)}}
\end{equation}
associated with a decomposition
$TM\restrict{X_{\gamma(t)}} = N X_{\gamma(t)}\oplus TX_{\gamma(t)}$,
we may consider $\eta_t$ as a section of
$TM\restrict{X_{\gamma(t)}}$.
\hfill
\proposition \label{_conne_in_fami_of_comple_geo_Proposition_}
Let $t$, $t'\in [0,1]$.
Let $\Psi^t_{t'}:\; U_t {\:\longrightarrow\:} U_{t'}$ be the map of
Subsection \ref{_conne_Subsection_}
integrating the connection $\nabla$ of
\ref{_conne_in_fam_of_comple_geode_Proposition_},
where $U_t$, $U_{t'}$ are open subsets
of $X_{\gamma(t)}$, $X_{\gamma(t')}$ defined in
Subsection \ref{_conne_Subsection_}.
Assume that for all $t$ the section
$\eta_t\in TM\restrict{X_{\gamma(t)}}$ is
parallel with respect to the natural
connection on $TM\restrict{X_{\gamma(t)}}$
obtained from Levi--Civita on $TM$. Assume also that the Riemannian form
on $M$ is real analytic.\footnote{This assumption is extraneous;
we use it to simplify the exposition. For the case we are interested
in ($M$ a hyperk\"ahler manifold) the Riemannian form
is real analytic by \ref{_Riema_on_hype_real_ana_Corollary_}.}
Then $\Psi^t_{t'}$ satisfies the following conditions.
\begin{description}
\item[(i)] $\Psi^t_{t'}$ is compatible
with a Riemannian structure on $U_t$, $U_{t'}$.
\item[(ii)]
Assume also that $M$ admits a smooth Riemannian
compactification $\bar M$ such that the family $\c X$
admits a compactification in $\bar M$ (such is the case
when $M$ and $\c X$ are quasiprojective, and $\phi$ is
algebraic). Then the maps $\Psi^t_{t'}:\; U_t {\:\longrightarrow\:} U_{t'}$ of (i)
can be found in such a way that $U_t$ is the set
of non-singular points of $X_{\gamma(t)}$.
\item[(iii)] In assumptions of (ii), we can extend $\Psi^t_{t'}$
to an isomorphism of metric spaces
\[
\bar \Psi^t_{t'}:\; \bar X_{\gamma(t)} {\:\longrightarrow\:} \bar X_{\gamma(t')},
\]
where $\bar X_{\gamma(t)}$, $\bar X_{\gamma'(t)}$ are
closures of $X_{\gamma(t)}$, $X_{\gamma'(t)}$ in $\bar M$, with induced metrics.
\end{description}
The next subsection is taken by the proof of
\ref{_conne_in_fami_of_comple_geo_Proposition_}.
\subsection{Deformations of completely
geodesic submanifolds (the proofs)}
The statement (i) is sufficient to prove
in a small neighbourhood, by analytic continuation. Thus, we
may pick an open subset $U\in {\c X}$ in such a way that the restriction of
$\phi:\; \c X {\:\longrightarrow\:} M$ to
\[
\c X':= \pi^{-1}(\gamma([0,1])\cap U
\]
is an embedding. Shrinking $\c X$ to $\c X'$ and pulling
from $M$ the Riemannian metric, we see that
\ref{_conne_in_fami_of_comple_geo_Proposition_} (i) is
implied by the following lemma.
\hfill
\lemma \label{_integra_field_Killing_Lemma_}
Let $\c X \stackrel \pi{\:\longrightarrow\:} [0,1]$
be a real analytic family of manifolds, equipped with a Riemannian
metrics. Let $T _\pi X\subset T\c X$
be the relative tangent bundle consisiting
of all vectors tangent to the fibers of $\pi$, and
$N _\pi X= T \c X /T _\pi X$ be the normal bundle
to the fibration $\pi$. Assume that the fibers
$\pi^{-1} (t) \subset \c X$ are completely geodesic
in $\c X$. Consider the connection
$\nabla:\; N _\pi X {\:\longrightarrow\:} T\c X$ of
\ref{_conne_in_fam_of_comple_geode_Proposition_}.
Let $\Psi^t_{t'}:\; U_t {\:\longrightarrow\:} U_{t'}$ be the
maps obtained by integrating the connection as in
Subsection \ref{_conne_Subsection_}
and $\eta_t \in N _\pi X$ be the normal fields arising
from the deformation theory. Assume that $\eta_t$'s
are parallel with respect to the natural connection on $N _\pi X$.
Then the maps $\Psi^t_{t'}$ are compatible with Riemannian metrics.
\hfill
{\bf Proof:} Let $\eta\in T \c X$ be the tangent vector field
obtained by gluing all $\eta_t$ together. Then $\Psi_t$
can be considered as integral map of this vector field. Thus,
to prove that $\Psi_t$ is compatible with the Riemannian
structure, we have to show that
$\eta$ is {\bf a Killing vector field}.\footnote{A Killing field
is a vector field which integrates to a diffeomorphism which is
compatible with a Riemannian metric.} Denote by
$\nabla_x:\; T\c X {\:\longrightarrow\:} T\c X$ the
action of covariant derivative along the vector field $x$.
By \cite{_Besse:Einst_Manifo_},
Theorem 1.81, to prove that $\eta$ is Killing it
suffices to prove that for all fields $a, b\in T\c X$,
we have
\begin{equation}\label{_Killing_Equation_}
(\nabla_a \eta, b) + (\nabla_b \eta,a) =0,
\end{equation}
where $(\cdot,\cdot)$ is the Riemannian form.
Take coordinates $(x_0, x_1,...x_n)$
on $\c X$ in such a way that $x_0$ comes from a
projection $\pi:\; \c X {\:\longrightarrow\:} [0,1]$ and
$x_1, ... x_n$ are coordinates along the fibers.
Let $\frac{d}{d x_i}$ be the corresponding vector fields.
To prove that $\eta$ is Killing it suffices to check
\eqref{_Killing_Equation_}
for $a$, $b$ coordinate vector
fields. Since $\eta$ is parallel along $X_t$, we have
$\nabla_{\frac{d}{d x_i}} \eta=0$ for $i= 1,..., n$.
Thus, it suffices to prove \eqref{_Killing_Equation_}
in case $a = \frac{d}{dx_0}$. For appropriate choice
of coordinates, $\frac{d}{d x_0}=\eta$; thus,
\eqref{_Killing_Equation_} is implied by
\begin{equation}\label{_nabla_eta_coo_Equation_}
\left(\nabla_\eta \eta,\frac{d}{d x_i}\right) =0,
\;\; i= 0,\dots n.
\end{equation}
Since Levi-Civita connection is compatible with the metrics,
we have
\begin{equation}\label{_Levi_Civi_compa_Equation_}
D_\eta \left(\eta, \frac{d}{d x_i}\right) =
\left(\nabla_\eta \eta,\frac{d}{d x_i}\right) +
\left(\eta,\nabla_\eta\frac{d}{d x_i}\right),
\end{equation}
where $D_\eta$ is the usual (directional) derivative along $\eta$.
For $i=0$, this gives
\[ D_\eta (\eta, \eta) =
(\nabla_\eta \eta,\eta) +
(\eta,\nabla_\eta\eta)=0
\]
($\eta$ is parallel, and hence has constant length).
This proves \eqref{_nabla_eta_coo_Equation_} for $i=0$.
Since $\frac{d}{d x_i}$ are coordinate vector fields
they commute, and therefore, $\nabla_\eta\frac{d}{d x_i}=
-\nabla_{\frac{d}{d x_i}}\eta=0$ for $i>0$. Thus,
the equation \eqref{_Levi_Civi_compa_Equation_} for $i>0$ is
reduced to
\[
D_\eta \left(\eta, \frac{d}{d x_i}\right) =
\left(\nabla_\eta \eta,\frac{d}{d x_i}\right).
\]
Since the vector fields
$\frac{d}{d x_i}$, $i>0$ are tangent to the fibration $\pi$, the function
$(\eta, \frac{d}{d x_i})$ is identically zero. This proves
\eqref{_nabla_eta_coo_Equation_} for $i>0$.
\ref{_integra_field_Killing_Lemma_} and consequently
\ref{_conne_in_fami_of_comple_geo_Proposition_}
(i) is proven. \blacksquare
\hfill
To prove
\ref{_conne_in_fami_of_comple_geo_Proposition_}
(ii) we have to show that the connection of
\ref{_conne_in_fam_of_comple_geode_Proposition_}
can be integrated for all smooth points of $\c X$.
We give a sketch of a simple geometric argument.
By (i), the maps $\Psi_t$ are isometries. Thus, the
distance from the given point
to the singular set of $X_t$ is invariant under
the maps $\Psi_t$. To integrate the connection,
we write a tangent vector field which we
subsequently integrate. Since the normal field $\eta_t$
is parallel, this tangent field is uniformly bounded.
On a certain distance from the singular set $Sing(X_t)$, depending
on this uniform bound, the connection might be always
integrated. In more precise terms,
for all $\epsilon>0$, all $x\in X_t$, with the distance
between $x$ and $Sing(X_t)$ no less than $\epsilon$, there exist
$\delta>0$ and a map
$\Psi^t_{t+\delta}:\;X_t {\:\longrightarrow\:} X_{t+\delta}$ defined
in a neighbourhood of $x$ which integrates the connection.
{}From this statement and \ref{_conne_in_fami_of_comple_geo_Proposition_}
(i), \ref{_conne_in_fami_of_comple_geo_Proposition_} (ii)
follows directly.
Finally, to prove \ref{_conne_in_fami_of_comple_geo_Proposition_} (iii)
we notice that $X_t$ is completely geodesic in $M$ for all $t$. Thus,
the completion of $X_t$ as a metric space coinsides with the
closure of $X_t$ in $\bar M$. Every isometry of metric spaces
extends to a completion, and thus, $\Psi^t_{t'}$ extends to a closure of
$X_t$, $X_{t'}$ in $\bar M$.
\blacksquare
\section[Hyperholomorphic bundles and completely geodesical
embeddings.]{Hyperholomorphic bundles \\and completely geodesical
embeddings.}
\label{_comple_geode_hyperho_Section_}
\hfill
In this section, we prove that hyperk\"ahler embeddings are
completely geodesic.
\subsection{Hyperholomorphic structure on the normal bundle}
Let $M$ be a hyperk\"ahler manifold, not necessary compact,
and $B$ a vector bundle.
Recall that in Section \ref{_basics_Section_}, we defined
{\bf hyperholomorphic connections} in $B$
(\ref{_hyperho_conne_Definition_}).
These are connections $\nabla$ such that the curvature
$\Theta:=\nabla^2\in \Lambda^2(M,End(B))$ is an
$SU(2)$-invariant 2-form with respect to the
natural $SU(2)$-action in $\Lambda^2(M)$.
The hyperholomorphic connections are always Yang-Mills
(\ref{_hyperholo_Yang--Mills_Proposition_}). The Levi--Civita connection
in the tangent bundle $TM$ is a prime example of a hyperholomorphic
connection (\ref{_tangent_hyperholo_Example_}).
\hfill
\proposition \label{_NX_hyperholo_Proposition_}
Let $M$ be a hyperk\"ahler manifold, not necessary compact, and
$X\subset M$ a trianalytic submanifold. Consider the normal
bundle $NX$, equipped with a connection $\nabla$
induced from the Levi--Civita
connection on $M$. Then $NX$ is hyperholomorphic.
\hfill
{\bf Proof:} Let $L$ be an induced complex structure $M$.
Consider the manifold $(X, L)$ as a complex submanifold of $(M, L)$. The
normal bundle $NX = N(X,L)$ has a natural holomorphic structure,
which is compatible with the connection $\nabla$. Therefore,
the curvature $\Theta\in \Lambda^2(X, End(NX))$ is of type
$(1,1)$ with respect to the Hodge decomposition
defined by $L$. Thus, $\Theta$ is of type $(1,1)$ with
respect to any of the induced complex structures.
By definition, this means that $\nabla$ is hyperholomorphic.
\blacksquare
\subsection{Hyperk\"ahler embeddings are completely geodesic}
In assumptions of \ref{_NX_hyperholo_Proposition_},
consider the bundle $TM\restrict X$ with metrics and connection
induced from $TM$. There is a natural exact sequence of
holomorphic vector bundles over $(X, L)$:
\[
0{\:\longrightarrow\:} TX {\:\longrightarrow\:} TM\restrict X {\:\longrightarrow\:} NX {\:\longrightarrow\:} 0 .
\]
The natural connection in each of these bundles is hyperholomorphic
(\ref{_NX_hyperholo_Proposition_}).
\hfill
\proposition \label{_NX_splits_for_hype_Proposition_}
Let $X$ be a hyperk\"ahler manifold, not necessary compact, $L$
induced complex structure, and
\begin{equation}\label{_holo_seque_Equation_}
0{\:\longrightarrow\:} E_1{\:\longrightarrow\:} E_2 {\:\longrightarrow\:} E_3{\:\longrightarrow\:} 0
\end{equation}
be an exact sequence
of holomorphic vector bundles. Let $g_2$ be a Hermitian structure
on $E_2$. Consider the metrics $g_1$, $g_2$, $g_3$ and the connections
$\nabla_1$, $\nabla_2$, $\nabla_3$ on $E_1$, $E_2$, $E_3$
induced by $g_2$. Assume that $\nabla_1$, $\nabla_2$, $\nabla_3$
are hyperholomorphic. Then the exact sequence
\eqref{_holo_seque_Equation_} splits, and moreover,
the orthogonal decomposition $E_2 = E_1 \oplus E_1^\bot$ is preserved by
the connection $\nabla_2$.
\hfill
{\bf Proof:} The same statement is well known
for the Yang--Mills connections: every exact sequence of
holomorphic bundles with compatible Yang--Mills metrics
splits (see Appendix to this section).
By \ref{_hyperholo_Yang--Mills_Proposition_},
hyperholomorphic connections
are always Yang--Mills. \blacksquare
\hfill
\definition
Let $N \stackrel i \hookrightarrow M$ be an embedding of
hyperk\"ahler manifolds. We say that $i$ is a {\bf hyperk\"ahler
embedding} if $i$ is compatible with the quaternionic structure
and Riemannian metric.
\hfill
\corollary\label{_hype_embe_comple_geode:Corollary_}
Let $N \stackrel i \hookrightarrow M$ be a hyperk\"ahler embedding.
Then $i$ is completely geodesic.
{\bf Proof:} Follows directly from
\ref{_comple_geodesi_basi_Proposition_} and
\ref{_NX_splits_for_hype_Proposition_} \blacksquare
\hfill
\subsection{Appendix: every exact
sequence of Yang--Mills bundles splits.}
\hspace{6mm}
\proposition \label{_YM_exact_split_Proposition_}
Let $X$ be a K\"ahler manifold, not necessary compact, and
\begin{equation}\label{_holo_seque2_Equation_}
0{\:\longrightarrow\:} E_1{\:\longrightarrow\:} E_2 {\:\longrightarrow\:} E_3{\:\longrightarrow\:} 0
\end{equation}
be an exact sequence
of holomorphic vector bundles. Let $g_2$ be a Hermitian
structure on $E_2$.
Consider the induced metrics $g_1$, $g_2$, $g_3$.
Assume that either $g_1$ or $g_3$ is Yang--Mills.
Then the exact sequence
\eqref{_holo_seque_Equation_} splits, and moreover,
the orthogonal complement $E_1^\bot\subset E_2$ is preserved by
the connection $\nabla_2$.
\hfill
{\bf Proof:} Consider the second fundamental form
\[ A\in \Lambda^{0,1}(X, \operatorname{Hom}(E_1,E_3)) \]
of the exact sequence \eqref{_holo_seque2_Equation_}.
The curvatures $\Theta_i$ of $E_i$ are expressed through
$A$ as follows (\cite{_Griffi_Harri_}):
\begin{equation}\label{_Theta_1_through_seco_Equation_}
\Theta_1 = \Theta_2 \restrict{E_1} + {}^t\bar A\wedge A,
\end{equation}
\begin{equation}\label{_Theta_3_through_seco_Equation_}
\Theta_3 = \Theta_2 \restrict{E_3} - A\wedge{}^t\bar A.
\end{equation}
The 2-forms ${}^t\bar A\wedge A$, $A\wedge{}^t\bar A$ are {\bf positive}
unless $A=0$. Thus, the endomorphisms
$\Lambda \left(A\wedge{}^t\bar A\right)\in End(E_1)$,
$\Lambda \left({}^t\bar A\wedge A\right)\in End(E_3)$ have positive trace
(again, unless $A=0$). By our assumption,
$\Lambda (\Theta_2)=0$ and either $\Lambda(\Theta_1)=0$
or $\Lambda(\Theta_3)=0$. Applying $\Lambda$ to both sides
of \eqref{_Theta_1_through_seco_Equation_} and
\eqref{_Theta_3_through_seco_Equation_}, we obtain that either
$\Lambda \left(A\wedge{}^t\bar A\right)=0$ or
$\Lambda \left({}^t\bar A\wedge A\right)=0$.
Therefore, $A=0$ and the exact sequence
\eqref{_holo_seque2_Equation_} splits.
\blacksquare
\section[Triholomorphic sections of hyperholomorphic bundles
and deformations of trianalytic submanifolds.]
{Triholomorphic sections \\of hyperholomorphic bundles\\
and deformations of trianalytic submanifolds.}
\label{_triholo_Section_}
In the previous section, we proved that hyperk\"ahler embeddings are
completely geodesic. In this section, we show that, furthermore,
the deformational results of Section
\ref{_comple_geode_defo_Section_} are fully applicable
to the deformations of trianalytic submanifolds.
\subsection{Triholomorphic sections of normal bundle}
\hfill
\definition
Let $M$ be a hyperk\"ahler manifold, $B$ a vector bundle
equipped with a hyperholomorphic connection, and $\alpha$ a section
of $B$. Then $\alpha$ is called {\bf triholomorphic}
if for each induced complex structure $L$ on $M$, $\alpha$
is a holomorphic section of $B$ considered as a holomorphic
bundle over $(M, L)$.
\hfill
Let $M$ be a hyperk\"ahler manifold, not necessarily compact, and
$X\subset M$ a trianalytic submanifold, not necessarily closed.
Fix an induced complex structure $I$ on $M$. Let
\[
\left( \pi:\; \c X {\:\longrightarrow\:} S,
\phi:\; \c X {\:\longrightarrow\:} M, \phi(\pi^{-1}(s_0)) = X \right)
\]
be a family of deformations of $(X,I) \subset (M, I)$
(see Section \ref{_comple_geode_defo_Section_} for details).
Let $\gamma:\; [0,1] {\:\longrightarrow\:} S$ be a real analytic path in $S$, such
that $\gamma(0) = s_0$. Assume that for all $t\in [0,1]$, the
submanifold $X_t = \phi(\gamma^{-1}(t))\subset M$ is trianalytic.
Consider the normal bundle $NX$ to $X$, with the metric and
connection induced from $M$. By \ref{_NX_hyperholo_Proposition_},
$NX$ is hyperholomorphic. Let $\eta$ be the section of
$NX$ corresponding to $\frac{d\gamma}{dt}$ as in
Subsection \ref{_norma_vecto_Subsection_}.
\hfill
\proposition \label{_norma_triholo_Proposition_}
In the above assumptions, $\eta$ is a triholomorphic section of $NX$.
\hfill
{\bf Proof:} Let $L$ be an induced complex structure, and $(M, L)$,
$(X, L)$ the manifolds $M$ and $X$ considered as complex
manifolds with the complex structure $L$. The normal bundle
$N(X,L)$ is naturally identified with $NX$ as a real vector
bundle. Therefore $\eta$ can be considered as a section
of $N(X,L)$. By \ref{_real_ana_indu_on_hype_equiva_Proposition_}
(see Appendix to this section), $\c X$ can be considered
as a real analytic deformation of $(X, L)$.
{}From \ref{_norma_sec_holom_Lemma_}
it is clear that $\eta$ is holomorphic
as a section of $N(X,L)$. This proves
\ref{_norma_triholo_Proposition_}. \blacksquare
\subsection{Triholomorphic sections are parallel}
\hfill
\proposition\label{_triholo_parallel_Proposition_}
Let $M$ be a hyperk\"ahler manifold, not necessary compact,
and $B$ a vector bundle with a hyperholomorphic connection $\nabla$.
Let $\nu$ be a trianalytic section of $B$. Then, $\nu$ is parallel:
\begin{equation}\label{_nu_para_Equation_}
\nabla\nu=0
\end{equation}
{\bf Proof:} Let $L$ be an induced complex structure. Since
$\nu$ is triholomorphic, $\bar \partial_L \nu =0$, where
$\bar \partial_L:\; B {\:\longrightarrow\:} B\times \Lambda^{0,1}_L(M)$ is the
$(0,1)$-part of the connection, taken with respect to $L$.
Taking $L=I, -I$, we obtain
\begin{equation} \label{_bar6_I+bar6_-I_Equation_}
\bar \partial_I + \bar\partial_{-I}(\nu)=0.
\end{equation}
On the other hand, $\bar\partial_{-I}= \partial_I$, where $\partial_I$ is the
$(1,0)$-part of $\nabla$ taken with respect to $I$. Thus,
\[ \bar \partial_I + \bar\partial_{-I} =\bar \partial_I + \partial_{I} =\nabla. \]
{}From \eqref{_bar6_I+bar6_-I_Equation_}
we obtain that $\nabla(\nu)=0$. \blacksquare
\hfill
{}From \ref{_norma_triholo_Proposition_}
and \ref{_triholo_parallel_Proposition_},
we obtain that the section $\eta \in NX$
is parallel with respect to the connection. Then,
\ref{_conne_in_fami_of_comple_geo_Proposition_} (i)
can be applied to the following effect.
\hfill
\corollary \label{_norma_sec_para_for_hype_from_it_Corollary_}
Let $M$ be a hyperk\"ahler manifold, not necessarily compact, and
$X\subset M$ a trianalytic submanifold, not necessarily closed,
and
\[
\left( \pi:\; \c X {\:\longrightarrow\:} S,
\phi:\; \c X {\:\longrightarrow\:} M, \phi(\pi^{-1}(s_0)) = X \right)
\]
be a real analytic family of deformations of $X\subset M$
(see Section \ref{_comple_geode_defo_Section_} for details).
Let $\gamma:\; [0,1] {\:\longrightarrow\:} S$ be a real analytic path in $S$, such
that $\gamma(0) = s_0$. Assume that for all $t\in [0,1]$, the
submanifolds $X_t = \phi(\gamma^{-1}(t))\subset M$ are trianalytic.
Fix an induced complex structure $I$ on $M$.
Let $U_t\subset (X_t, I)$, $t\in [0,1]$ be the subsets constructed
in Subsection \ref{_conne_Subsection_}\footnote{By
\ref{_hype_embe_comple_geode:Corollary_},
trianalytic submanifolds are completely geodesic, and thus,
\ref{_conne_in_fam_of_comple_geode_Proposition_} can be applied.}
with the corresponding holomorphic isomorphisms
$\Psi_t:\; U_0{\:\longrightarrow\:} U_t$. Then, assumptions of
\ref{_conne_in_fami_of_comple_geo_Proposition_} (i) hold.
Thus,
for all $t, t'\in [0,1]$,
the maps $\Psi^t_{t'}$ are isometries. Moreover,
$\Psi^t_{t'}$ are compatible with the hyperk\"ahler structure.
\hfill
{\bf Proof:}
Assumptions of \ref{_conne_in_fami_of_comple_geo_Proposition_} (i)
hold by \ref{_norma_triholo_Proposition_}
and \ref{_triholo_parallel_Proposition_}.
The maps $\Psi^t_{t'}$ are isometries by
\ref{_conne_in_fami_of_comple_geo_Proposition_} (i).
The maps $\Psi^t_{t'}$ are compatible with the
hyperk\"ahler structure because the they are obtained by integrating
a certain connection in the family $\c X$. This connection
is constructed from the Hermitian metric, and thus, does
not depend from the choice of induced complex structure. Taking
different induced complex structures,
we obtain the same maps $\Psi^t_{t'}$.
Thus, $\Psi^t_{t'}$ is holomorphic
with respect to each of induced complex structures
(see also \ref{_norma_sec_holom_Lemma_}).
\blacksquare
\subsection{Appendix: real analytic structures on hyperk\"ahler
manifolds}
Consider the real analytic structures on a given hyperk\"ahler
manifold arising from the different induced complex structures.
We prove that these real analytic structures are equivalent.
\hfill
\proposition \label{_real_ana_indu_on_hype_equiva_Proposition_}
Let $M$ be a hyperk\"ahler manifold, $I_1$, $I_2$ induced complex
structures. Let $(M, I_1)$ and $(M, I_2)$ be the corresponding
complex manifolds, and $(M, I_1)_{\Bbb R}$, $(M, I_2)_{\Bbb R}$ be the real
analytic manifolds underlying $(M, I_1)$, $(M, I_2)$. Consider the
tautological map $(M, I_1)_{\Bbb R}\stackrel \phi {\:\longrightarrow\:} (M, I_2)_{\Bbb R}$.
Then $\phi$ is compatible with the real analytic structure.
\hfill
{\bf Proof:} Consider the {\bf twistor space} for $M$ (see
\cite{_Besse:Einst_Manifo_}), $\operatorname{Tw}(M)$, with the natural
holomorphic map $\pi:\; \operatorname{Tw}(M) {\:\longrightarrow\:} {\Bbb C} P^1$. Let $Sec(M)$
be the space of holomorphic sections of the map $\pi$.
Then $Sec(M)$ is identified naturally with an open subspace
of a Douady space for $\operatorname{Tw}(M)$, and thus, has a natural
complex structure. According to \cite{_HKLR_} (see \cite{_NHYM_}
for details), $Sec(M)$ is a complexification of $(M,I)$, in the
sense of Grauert. In other words, the complex valued real analytic functions
on $(M,I)$ are naturally identified with the germs of complex analytic
functions on $Sec(M)$. Since $Sec(M)$ is defined independently
from the choice of an induced complex structure, the
tautological map $\phi$ is an equivalence.
This proves \ref{_real_ana_indu_on_hype_equiva_Proposition_}.
\blacksquare
\hfill
\ref{_real_ana_indu_on_hype_equiva_Proposition_} implies that we
may speak of a real analytic manifold underlying a given hyperk\"ahler
manifold.
\hfill
\corollary \label{_Riema_on_hype_real_ana_Corollary_}
Let $M$ be a hyperk\"ahler manifold. Consider the Riemannian
form $g$ on $M$ as a section of the real analytic bundle of
symmetric 2-forms. Then $g$ is real analytic.
\hfill
{\bf Proof:} Let $I, J, K$ be the induced complex structures
which form the standard basis in quaternions, and
$\omega_I$, $\omega_J$, $\omega_K$ be the corresponding
K\"ahler forms. Then $\Omega:=\omega_J + \sqrt{-1}\:\omega_K$ is the natural
holomorphically symplectic form on $(M,I)$, and as such,
$\Omega$ is real analytic. Then, its real part $\omega_J$ is
also real analytic. Since the complex structure operator $J$
is real analytic, we obtain that the form
$g(\cdot,\cdot):= - \omega(\cdot, J\cdot)$ is also real analytic.
\blacksquare
\section{Douady spaces for trianalytic cycles and real analytic structure}
\label{_Douady_Section_}
\subsection{Real analytic structure on the Douady space.}
In this section, we consider the Douady space $D_M(X)$
for a subvariety $X$
of a compact complex manifold $M$ equipped with
a hyperk\"ahler structure.
We prove that, when $X$ is trianalytic, $D_M(X)$ is {\bf
hypercomplex} (\ref{_hypercomplex_Definition_}).
We also show that the real analytic
variety underlying $D_M(X)$ does not change if we replace
a complex structure on $M$ by another induced complex structure.
These results are technical and we use them mainly to
simplify the exposition.
\hfill
Let $M$ be a compact hyperk\"ahler manifold, and $X\subset M$ a
closed trianalytic subvariety. For each induced complex
structure $L$, consider $(X, L)$ as a complex subvariety of $(M, L)$.
Let $D_L(X)$ be the Douady deformation space of $(X, L)$ in
$(M, L)$. The points $[X']$ of $D_L(X)$ correspond to the
subvarieties $X'\subset (M, L)$ which are deformation of $X$.
By \ref{_G_M_invariant_implies_trianalytic_Theorem_} (see also
\cite{Verbitsky:Symplectic_II_}), every such $X'$ is trianalytic.
Let $I_1$, $I_2$ be induced complex structures on $M$.
The following proposition shows that a trianalytic subvariety
$X'$ can be obtained as a deformation of $(X, I_1)$ in $(M, I_1)$
if and only if $X'$ can be obtained
as a deformation of $(X, I_2)$ in $(M, I_2)$.
\hfill
\proposition \label{_X_in_Douady_indep_from_I_1_Proposition_}
Let $I_1$, $I_2$ be induced complex
structures on a compact hyperk\"ahler manifold
$M$, and $X\subset M$ a closed trianalytic subvariety.
Consider the corresponding Douady spaces $D_{I_1}(X)$,
$D_{I_2}(X)$. Then, for a trianalytic subvariety $X'\subset M$,
$[X'] \in D_{I_1}(X)$ if and only if $[X']\in D_{I_2}(X)$.
\hfill
{\bf Proof:} Let us recall the notion of {\bf degree} of
a trianalytic subvariety.
Let $M_1$ be a K\"ahler manifold.
By {\bf degree} of a subvariety $X \subset M_1$ we understand a number
\[ \deg X := \int_X \omega^{\dim_{\Bbb C} X}, \]
where $\omega$ is the K\"ahler form. If $X\subset M$ is
a trianalytic subvariety of a hyperk\"ahler manifold, we can
associate a number $\deg X$ to each of induced complex structures.
In \cite{Verbitsky:Symplectic_I_}, we prove that $\deg X$
is in fact independent from the choice of induced complex structure.
This enables us to speak of {\bf degree} of a closed trianalytic
subvariety of a compact complex manifold.
We return to the proof of
\ref{_X_in_Douady_indep_from_I_1_Proposition_}.
Let $D_1$ be the union of all components of the
Douady space for $(M,I_1)$. By \cite{_Lieberman_},
\cite{_Fukjiki_Kahler_}, $D_1$ is compact;
in particular, $D_1$ has a finite number of connected components.
Let $D_1 = \cup D^i_1$, $i\in \Upsilon$
be the decomposition of $D$ unto a union of its
connected components.
Let $X'\subset M$ be a closed trianalytic subvariety of $M$
such that $[X'] \in D_{I_2}(X)$. There exists a real analytic
path $\gamma:\; [0,1] {\:\longrightarrow\:} D_{I_2}(X)$ joining $[X]$ and $[X']$.
For all $t\in [0,1]$, the point $\gamma(t)$ lies in $D_1^i$ for some
$i\in \Upsilon$. Let $S^i\subset [0,1]$ be the set of all
$t\in [0,1]$ such that $\gamma(t)\in D_1^i$. We are going to
show that $S^i$ are compact for all $i$. This will clearly imply
that for all $i$ except one, $S^i$ is empty, thus proving that
$[X]$ and $[X']$ lie in the same component of $D_1$.
Let $D_{\Bbb R}$ be the set of all real analytic deformations of $X$ in $M$
with natural topology. The forgetful map $\psi:\; D_1 {\:\longrightarrow\:} D_{\Bbb R}$
is continous and injective. Since $D_1$ is compact, $\psi$
is a closed embedding. The composition
$\gamma\circ \psi:\; [0,1] {\:\longrightarrow\:} D_{\Bbb R}$ is obviously continous,
and thus, $\gamma:\; [0,1] {\:\longrightarrow\:} D_1$ is also continous.
This implies that all $S_i$, $i\in \Upsilon$ are
closed, thus proving that all $S_i$ except one are empty.
\ref{_X_in_Douady_indep_from_I_1_Proposition_} is proven.
\blacksquare
\hfill
Let $D_{I_1}(X)_{\Bbb R}$, $D_{I_2}(X)_{\Bbb R}$ be the real analytic
varieties underlying $D_{I_1}(X)$, $D_{I_2}(X)$.
\ref{_X_in_Douady_indep_from_I_1_Proposition_} gives a
tautological bijection \[ \psi:\; D_{I_1}(X)_{\Bbb R}{\:\longrightarrow\:} D_{I_2}(X)_{\Bbb R}.\]
\hfill
\proposition\label{_Douady_real_ana_inde_from_indu_Proposition_}
The map $\psi:\; D_{I_1}(X)_{\Bbb R}{\:\longrightarrow\:} D_{I_2}(X)_{\Bbb R}$
is an isomorphism of real analytic varieties.
\hfill
{\bf Proof:} Let $I$ be an induced complex structure on $M$.
For a real analytic function $f$ on $M$, consider the function
$\hat f:\; D_I(X) {\:\longrightarrow\:} {\Bbb R}$,
\[ [X'] \stackrel{\hat f}{\:\longrightarrow\:} \int_{X'} f \cdot \operatorname{Vol} X', \]
with $\operatorname{Vol} X'$ the volume form on $X'$. For
all $f$, the function $\hat f$ is real analytic. From the definition of
Douady spaces it might be seen that converging
power series of different $\hat f$ generate the sheaf
$\c A_I$ of real analytic functions on $D_I(X)$. For an induced
complex structure $I'$, the same set $\{ \hat f\}$
generates the sheaf $\c A_{I'}$
of real analytic functions on $D_{I'}(X)$
(\ref{_real_ana_indu_on_hype_equiva_Proposition_}). Thus,
the sheaves $\c A_I$ and $\c A_{I'}$ coinside.
\blacksquare
\hfill
\ref{_Douady_real_ana_inde_from_indu_Proposition_}
shows that we may speak of the real analytic variety underlying
$D_M(X)$ without specifying an induced complex structure $I$.
\hfill
\definition \label{_hypercomplex_Definition_}
Let $Y$ be a real analytic variety equipped with
three complex structures $I$, $J$ and $K$. Assume that
for every point $y\in Y$, the action of $I$, $J$, $K$ on the
Zariski tangent space $T_yY$ satisfies $I\circ J = - J\circ I =K$.
Then $Y$ is called {\bf a hypercomplex variety}.
\hfill
\remark \label{_Douady_hyperc_Remark_}
\ref{_Douady_real_ana_inde_from_indu_Proposition_}
immediately implies that the induced complex structures on $M$
equip the Douady space $D_M(X)$ with a hypercomplex structure.
\subsection{Appendix: isometric embeddings are hyperk\"ahler.}
Let $M$ be a compact hyperk\"ahler manifold, $X\subset M$
a closed trianalytic subvariety.
There is an alternative way to describe the Douady space
$D(X)$. The \ref{_isome_embe_Proposition_}
below describes $D(X)$ in terms of of the space of isometric embeddings
$\nu:\; X {\:\longrightarrow\:} M$.
\hfill
Let $M$ be a compact K\"ahler manifold, and $X\subset M$
be a closed analytic subvariety. One can make sense of the
{\bf fundamental class} of $X$, $[X] \in H^{\dim_{\Bbb R}(X)}(M)$,
lying in homology of $M$.
Consider $X$ with an induced structure of a metric space,
and let $X \stackrel \nu {\:\longrightarrow\:} M'$ be an isometric embedding,
where $M'$ is a compact Riemannian manifold. The same argument which
allows us to define the fundamental class of $X$, allows
us to define the fundamental class of $\nu(X)$.
\hfill
\proposition \label{_isome_embe_Proposition_}
Let $M$ be a closed analytic subvariety, $X\subset M$ a closed
trianalytic subvariety, and $\nu:\; X {\:\longrightarrow\:} M$ an isometric embedding.
Assume that the fundamental class of $\nu(X)$ is equal to the fundamental
class of $X$. Then $\nu(X)$ is a trianalytic subvariety of $M$.
\hfill
{\bf Proof:} We use notation
introduced in Subsection \ref{_SU(2)-inv=>triana_Subsection_}.
For each induced complex structure $I$,
we have
\begin{equation}\label{_deg_same_appl_nu_Equation_}
\deg_I (X) - \deg_I (\nu(X)),
\end{equation}
because $\nu(X)$ has the same fundamental class as $X$.
Since $X$ is trianalytic,
applying Wirtinger's inequality (\ref{_Wirti_for_Kahle_Theorem_}),
we obtain that
\begin{equation}
2^n \int_{X_{ns}} \operatorname{Vol} X_{ns} = \int_{X_{ns}} \omega^n,
\end{equation}
where $\omega$ is the K\"ahler class of $(M,I)$, $n=\frac{1}{2}\dim_{\Bbb R}(X)$
and $X_{ns}$ is the nonsingular part of $(X,I)$.
By definition, we have
\begin{equation}
\int_{X_{ns}} \omega^n = \deg_I (X), \ \
\int_{\nu(X_{ns})} \omega^n = \deg_I (\nu(X)).
\end{equation}
Since $\nu$
is an isometry, and $\operatorname{Vol}(X)$ is an invariant of a metric,
we have
\begin{equation} \label{_Vol_same_appl_nu_Equation_}
\int_{X_{ns}} \operatorname{Vol} X_{ns} =
\int_{\nu(X_{ns})} \operatorname{Vol} \left(\nu(X_{ns}) \right).
\end{equation}
Together the equations
\eqref{_deg_same_appl_nu_Equation_}--\eqref{_Vol_same_appl_nu_Equation_}
give
\[
2^n\int_{X_{ns}} \operatorname{Vol} X_{ns} = \int_{\nu(X_{ns})} \omega^n.
\]
Applying Wirtinger's inequality (\ref{_Wirti_for_Kahle_Theorem_}) once
again, we obtain that $\nu(X)$ is complex analytic with respect to $I$.
This proves \ref{_isome_embe_Proposition_}. \blacksquare
\hfill
We just proved \ref{_iso_intro:Theorem_} (iii).
As another application of \ref{_isome_embe_Proposition_},
we give a direct proof that $D(X)$ is compact.
\hfill
\corollary
Let $M$ be a compact hyperk\"ahler manifold,
$X\stackrel {\bar i}\hookrightarrow M$ be a closed trianalytic subvariety
and $D(X)$ its Douady space. Then $D(X)$ is compact.
\hfill
{\bf Proof:} Let $X_{ns}$ be the non-singular part of $X$,
and $i:\; X_{ns} \hookrightarrow M$ the natural embedding.
Then $i$ is an isometry. Consider a deformation $\nu$ of $i$ in class of
isometries. The arument proving \ref{_isome_embe_Proposition_}
shows that the closure of $\nu(X_{ns})$ is trianalytic in $M$.
Consider the space $\underline{Is(X_{ns})}$ of isometries
$X_{ns}\stackrel \nu\hookrightarrow M$. Let $Is(X_{ns})$ be
a connected component of this space, containing $i$. There
is a natural continuous surjection
$p:\; Is(X_{ns}) {\:\longrightarrow\:} D(X)$, which maps $\nu \in Is(X_{ns})$
to a closure of $\nu(X_{ns})$ in $M$. To prove that $D(X)$ is
compact it suffices to show that $Is(X_{ns})$ is compact.
This is an implication of the following general statement, which is clear.
\hfill
\claim
Let $X$, $M$ be Riemannian manifolds, $M$ compact. Let
$Is(X, M)$ be the space of isometries (maps which preserve the
geodesic distance) from $X$ to $M$. Then $Is(X, M)$ is compact.
\blacksquare
\section[Connections in the families of trianalytic subvarieties]
{Connections in the families of trianalytic \\subvarieties}
\label{_Conne_in_fami_Section_}
\subsection{Introduction}
\label{_intro_conne_Subsection_}
Let $M$ be a compact hyperk\"ahler manifold, and
\begin{equation} \label{_fami_Equation_}
\left( \pi:\; {\c X} {\:\longrightarrow\:} S, \phi:\; {\c X} {\:\longrightarrow\:} M
\right )
\end{equation}
a real analytic family of subvarieties, not necessarily closed.
Assume that for all $s\in S$, the fiber $X_s = \phi(\pi^{-1} (s))$
is trianalytic in $M$.
By \ref{_conne_in_fam_of_comple_geode_Proposition_}
and \ref{_hype_embe_comple_geode:Corollary_}, the family
${\c X}$ is then equipped with a connection
$\nabla:\; N_\pi X {\:\longrightarrow\:} T {\c X}$. It is natural
to ask whether this connection is {\bf flat}
(see \ref{_flat_conne_in_fami_Definition_}).
The answer is affirmative,
under certain additional assumptions.
\hfill
\definition \label{_admits_co_Definition_}
We say that the family \eqref{_fami_Equation_} {\bf admits
a compactification} if the following conditions hold.
\begin{description}
\item[(i)]
For each $s \in S$, the closure $\bar X_s$ of the fiber
$X_s$ in $M$ is a trianalytic subvariety of $M$.
\item [(ii)] For all $s\in S$, $\bar X_s$ lie in the same
component of the Douady space.
\end{description}
\hfill
We show that in assumptions of
\ref{_admits_co_Definition_}, the connection
\[ \nabla:\; N_\pi X {\:\longrightarrow\:} T {\c X}\] is indeed flat
(\ref{_conne_in_triana_flat_Theorem_}).
However, we don't know whether it is flat
for general families of trianalytic submanifolds.
\hfill
\subsection{Flat connections and curvature.}
\hfill
\definition
(curvature of a connection)
Let $\pi: {\c X} {\:\longrightarrow\:} S$ be a family of manifolds,
$T_\pi X \subset T{\c X}$ be the fiberwise tangent bundle and
$N_\pi X = T{\c X} / T_\pi X$ be a fiberwise normal bundle.
Let $\nabla:\; N_\pi X {\:\longrightarrow\:} T{\c X}$ be a connection in ${\c X}$.
Then the {\bf curvature} of $\nabla$ is the following
tensor
$\Theta\in \operatorname{Hom}_{\Bbb R}\left ( \Lambda^2_{\Bbb R} \left( N_\pi X\right),
T_\pi{\c X}\right)$.
For two sections $a,b \in N_\pi X$, consider the corresponding
vector fields $\nabla a$, $\nabla b\in T{\c X}$. Consider the projection
$T {\c X} \stackrel {pr}{{\:\longrightarrow\:}} T{\c X} / \nabla N_\pi X = T_\pi X$.
Let $\Theta(a,b):= pr([\nabla a, \nabla b])$.
Clearly, $\Theta$ is a tensor.
\hfill
\proposition
Let $\left( \pi:\; {\c X} {\:\longrightarrow\:} S \right)$
be a family of manifolds equipped with a connection $\nabla$.
Then the following conditions are equivalent.
\begin{description}
\item[(i)] The connection $\nabla$ is flat.
\item[(ii)] Let $s_1, s_2\in S$ be a pair of points,
and $\gamma, \gamma':\; [0,1] {\:\longrightarrow\:} S$ real analytic paths.
Let $U_0 \subset X_{s_0}$, $U_1 \subset X_{s_1}$ be the open
subsets such that the connection $\nabla$ might be integrated to a map
$\Psi:\; U_{s_0} {\:\longrightarrow\:} U_{s_1}$ along $\gamma$, and
$U_0' \subset X_{s_0}$, $U_1' \subset X_{s_1}$ be
such that $\nabla$ might be integrated to a map
$\Psi':\; U_{s_0}' {\:\longrightarrow\:} U_{s_1}'$ along $\gamma'$. Then
$\Psi$ coinsides with $\Psi'$ in the intersection
$U_{s_0} \cap U'_{s_0}$.
\end{description}
{\bf Proof:} Well known. \blacksquare
\hfill
\definition \label{_flat_conne_in_fami_Definition_}
Let ${\c X} {\:\longrightarrow\:} S$ be a family of manifolds, equipped with a
connection $\nabla$. Then $\nabla$ is called {\bf flat}
if its curvature is zero.
\subsection{Curvature of holomorphic connections}
Let $\pi:\; {\c X} {\:\longrightarrow\:} S$ be a complex analytic family
of manifolds. Then the vector bundles $T{\c X}$ and
$N_\pi X$ are equipped with a natural holomorphic structure.
The connection $\nabla:\; N_\pi X {\:\longrightarrow\:} T {\c X}$ is called
{\bf holomorphic} if the map $\nabla:\; N_\pi X {\:\longrightarrow\:} T{\c X}$
is holomorphic.
\hfill
\claim \label{_holo_conne_curva_C-line_Claim_}
Let $\pi:\; {\c X} {\:\longrightarrow\:} S$ be a family of
manifolds equipped with with a holomorphic connection
$\nabla$. Consider the curvature
\[ \Theta\in \operatorname{Hom}_{\Bbb R}\left
( \Lambda^2_{\Bbb R} \left(N_\pi X\right), T_\pi{\c X}\right).
\]
Then $\Theta$ is ${\Bbb C}$-linear; in other words, $\Theta$ belongs
to the subspace
\[ \operatorname{Hom}_{\Bbb C}\left ( \Lambda^2_{\Bbb C} \left(N_\pi X\right), T_\pi{\c X}\right)
\subset \operatorname{Hom}_{\Bbb R}\left
( \Lambda^2_{\Bbb R} \left(N_\pi X\right), T_\pi{\c X}\right).
\]
{\bf Proof:} Clear. \blacksquare
\subsection{Flat connections in a family of trianalytic varieties. }
Let $M$ be a compact hyperk\"ahler manifold, $X\subset M$
a closed trianalytic subvariety, $D(X)$ its Douady space
and
\[ \left( \underline \pi: \; \underline{{\c X}} {\:\longrightarrow\:} D(X),
\underline \phi:\; \underline{{\c X}} {\:\longrightarrow\:} M \right )
\]
a universal family of subvarieties corresponding to $D(X)$.
Let ${\c X}$ be the union of all smooth points in all fibers
of $\underline \pi$, and $\pi$, $\phi$ be the restrictions of
$\underline \pi$ and $\underline \phi$ to ${\c X}$.
Consider the corresponding family of manifolds
\[ \left( \pi: {{\c X}} {\:\longrightarrow\:} D(X),
\phi {{\c X}} {\:\longrightarrow\:} M \right ).
\]
We are in the same situation as described in
Subsection \ref{_intro_conne_Subsection_}.
Thus, the family ${\c X}$ is equipped with a natural
connection $\nabla$.
\hfill
\theorem \label{_conne_in_triana_flat_Theorem_}
In these assumptions, $\nabla$ is flat.
\hfill
{\bf Proof:} The proof of \ref{_conne_in_triana_flat_Theorem_}
takes the rest of this section.
\hfill
Clearly, $N_\pi X = \pi^* TS$. By \ref{_Douady_hyperc_Remark_},
$TS$ has a natural quaternionic action, and thus, an action of
$SU(2) \subset {\Bbb H}^*$. The bundle $T_\pi X$ is also equipped
with a natural action of quaternions. This endows the bundle
$\operatorname{Hom}_{\Bbb R}\left ( \Lambda^2_{\Bbb R} \right(N_\pi X\left), T_\pi{\c X}\right)$
with an action of $SU(2)$.
\hfill
\lemma \label{_curva_SU(2)_inva_Lemma_}
Let $\Theta$ be a curvature of $\nabla$,
$\Theta \in\operatorname{Hom}_{\Bbb R}
\left ( \Lambda^2_{\Bbb R} \left( N_\pi X\right), T_\pi{\c X}\right)$.
Then $\Theta$ is $SU(2)$-invariant.
\hfill
{\bf Proof:} Let $U(1) \stackrel {\tilde i}\hookrightarrow SU(2)$ be
an embedding corresponding to an algebra embedding
${\Bbb C} \stackrel i\hookrightarrow {\Bbb H}$. Let $I$ be an
induced complex structure associated with $i$. The
connection $\nabla$ is holomorphic with respect to $I$.
Therefore $\Theta$ is ${\Bbb C}$-linear
with respect to the action of ${\Bbb C}$ given by
${\Bbb C} \stackrel i \hookrightarrow {\Bbb H}$
(\ref{_holo_conne_curva_C-line_Claim_}).
Thus, $\Theta$ is $U(1)$-invariant with respect
to the action of $U(1)$ given by
$\tilde i:\; U(1)\hookrightarrow SU(2)$.
The group $SU(2)$ is generated by the images of $\tilde i$ for all
algebra embeddings $i:\; {\Bbb C} \hookrightarrow {\Bbb H}$. Thus,
$\Theta$ is $SU(2)$-invariant. \blacksquare
\hfill
Consider $N_\pi X$, $T_\pi X$ as representation of $SU(2)$. These representations
are of weight 1. The bundle
\[
\operatorname{Hom}_{\Bbb R}\left ( \Lambda^2_{\Bbb R} \left(N_\pi X\right), T_\pi{\c X}\right)
\]
is
again a representation of $SU(2)$. Since
$N_\pi X$, $T_\pi X$ are of weight one,
$\operatorname{Hom}_{\Bbb R}\left ( \Lambda^2_{\Bbb R} \left(N_\pi X\right), T_\pi{\c X}\right)$ is
a direct sum of representations of weight 3 and 1. Thus,
this bundle cannot have non-zero $SU(2)$-invariant sections.
Therefore, ($\Theta$ is $SU(2)$-invariant)
implies ($\Theta =0$).
\ref{_conne_in_triana_flat_Theorem_} is proven. \blacksquare
\section[Isometries of trianalytic subvarieties
of a compact hy\-per\-k\"ah\-ler manifold]{Isometries
of trianalytic subvarieties \\
of a compact hyperk\"ahler manifold}
\label{_isome=>holo_Section_}
\subsection{Premises}
Let $M$ be a compact hyperk\"ahler manifold, $X\subset M$ a
closed trianalytic subvariety.
{}From \ref{_conne_in_fami_of_comple_geo_Proposition_},
\ref{_conne_in_triana_flat_Theorem_} and
\ref{_norma_sec_para_for_hype_from_it_Corollary_},
we immediately obtain the following theorem.
\hfill
\theorem \label{_triana_subse_isome_Theorem_}
Let $M$ be a compact hyperk\"ahler manifold, $X\subset M$
a closed trianalytic subvariety, $D(X)$ the corresponding
Douady space (\ref{_Douady_hyperc_Remark_}).
Let $\gamma:\; [0,1] {\:\longrightarrow\:} D(X)$
be a real analytic path. Denote by $X_t$
the subvarieties corresponding to points $\gamma(t)\in D(X)$.
Consider $X_t$ as metric spaces with the metric induced
from $M$.
Then, for each $t_1$, $t_2\in [0,1]$,
there exist a natural isometry
$\Psi^{t_1}_{t_2}:\; X_{t_1} {\:\longrightarrow\:} X_{t_2}$, mapping
non-singular points to non-singular points, and
acting compatible with the hyperk\"ahler structure
on $X_{t_1}^{ns}$.\footnote{The non-singular part $X_t^{ns}$
is naturally equipped with a structure of a hyperk\"ahler manifold.}
This isometry depends only on the homotopy class of $\gamma$.
\blacksquare
\hfill
This almost finishes the proof \ref{_iso_intro:Theorem_} (i).
It remains to prove the following
theorem.
\hfill
\theorem \label{_triana_subse_comple_ana_Theorem_}
In assumptions of \ref{_triana_subse_isome_Theorem_},
let $I$ be an induced complex structure on $M$. Then
the isometry $\Psi^{t_1}_{t_2}:\; X_{t_1} {\:\longrightarrow\:} X_{t_2}$
is compatible with the complex analytic structure induced
by $I$.
\hfill
This section is taken fully with the proof of
\ref{_triana_subse_comple_ana_Theorem_}
\subsection{Homeomorphisms of complex varieties and normalization.}
\hfill
\claim \label{_conti_fu_extend_to_holo_Claim_}
(\cite{_Grauert_}) Let $X$ be a normal complex analytic variety, $U$
a dense open subset in $X$ and $f:\; U {\:\longrightarrow\:} {\Bbb C}$ a bounded
holomorphic function. Then $f$ can be extended to a holomorphic function
on $X$.
\blacksquare
\hfill
This statement has an immediate corollary.
\hfill
\corollary \label{_conti_map_norma_holo_Corollary_}
Let $\phi:\; X {\:\longrightarrow\:} Y$ be a continous map of complex analytic
varieties. Assume that $\phi$ is holomorphic in an open
dense subset $U\subset X$. Then, if $X$ is normal, $\phi$
is holomorphic.
\hfill
{\bf Proof:} Let $x\in X$ be an aritrary point, $V$ its neighbourhood,
sufficiently small. Taking
coordinates in a neighbourhood $W$ of $\phi(V)$ and applying
\ref{_conti_fu_extend_to_holo_Claim_}, we find that
$\phi\restrict {U\cap V}$ extends continously to a
holomorphic map $\tilde \phi$ from $U\cap V$ to $W$.
Since $U\cap V$ is dense in $V$, $\tilde \phi$
coinsides with $\phi\restrict V$. Thus, $\phi$ is
holomorphic. \blacksquare
\hfill
Thus, in the situation of \ref{_triana_subse_comple_ana_Theorem_},
were $X_t$ normal, $\Psi^{t}_{t'}$ would have been holomorphic
and \ref{_triana_subse_comple_ana_Theorem_} would have been proven.
Unfortunately, we have no means to show that $X_t$ is normal.
However, from \ref{_conti_map_norma_holo_Corollary_}
we obtain some information about maps of arbitrary
varieties too.
\hfill
\corollary \label{_conti_map_mero_Corollary_}
Let $\phi:\; X {\:\longrightarrow\:} Y$ be a continous map of complex analytic
varieties, which is holomorphic on an open dense subset
$U\subset X$. Then $\phi$ is meromorphic.
\hfill
{\bf Proof:} Take a normalization $\tilde X \stackrel n {\:\longrightarrow\:} X$.
Applying \ref{_conti_map_norma_holo_Corollary_} to the composition
$n\circ \phi:\; \tilde X {\:\longrightarrow\:} Y$, we obtain that
$n\circ \phi$ is holomorphic. Thus, $\phi$ is meromorphic
as a composition of holomorphic
$n\circ \phi$ and a meromorphic map $n^{-1}$. \blacksquare
\hfill
Applying \ref{_conti_map_mero_Corollary_} to the situation
of \ref{_triana_subse_comple_ana_Theorem_}, we obtain the
following corollary.
\hfill
\corollary \label{_Psi_bimero_Corollary_}
In the situation of \ref{_triana_subse_isome_Theorem_},
the map \[ \Psi^{t_1}_{t_2}:\; X_{t_1} {\:\longrightarrow\:} X_{t_2} \]
is bimeromorphic for each induced complex structures.
\blacksquare
\hfill
It remains to make a leap from ``bimeromorphic'' to ``holomorphic''.
This is done in two steps. We prove a number of
algebro-geometric statements about the behaviour of $\Psi^{t_1}_{t_2}$,
concluding with \ref{_pi_i_properties_Proposition_}.
In \ref{_alge_geo_suffi_pi_i_iso_Proposition_},
we show that these statements are strong enough to show that
the $\Psi^{t_1}_{t_2}$ is holomorphic. The premise
of \ref{_alge_geo_suffi_pi_i_iso_Proposition_}
is purely algebro-geometric and its proof is independent
from the rest of this section.
We finish this Subsection with the following statement, which
we use in \ref{_pi_i_properties_Proposition_}.
\hfill
\proposition \label{_Psi_iso_on_norma_Proposition_}
The map $\Psi^{t_1}_{t_2}:\; X_{t_1} {\:\longrightarrow\:} X_{t_2}$ induces
an isomorphism of normalizations
\[ \tilde \Psi^{t_1}_{t_2}:\;\tilde X_{t_1} {\:\longrightarrow\:} \tilde X_{t_2}. \]
{\bf Proof:} From the definition of normalization \cite{_Grauert_}, the
following lemma is evident.
\hfill
\lemma
Let $\Psi:\; X {\:\longrightarrow\:} Y$ be a homeomorphism of complex
varieties. Assume that in an open dense subset $U\subset X$,
$\Psi$ is holomorphic. Then $\Psi$
induces an isomorphism $\tilde \Psi:\; \tilde X {\:\longrightarrow\:} \tilde Y$
of normalizations, if the following statement holds.
\begin{description}
\item[(*)] There exist a Stein covering $\{ U_i\}$ of $X$ such that
$\{ \Psi(U_i)\}$ is a Stein covering for $Y$.
\end{description}
\blacksquare
To prove \ref{_Psi_iso_on_norma_Proposition_}, it remains to show that
the property {\bf (*)} holds for the map
$\Psi^{t_1}_{t_2}:\; X_{t_1} {\:\longrightarrow\:} X_{t_2}$.
For each point $x \in X_{t_1}$, it suffices to
construct a Stein neighbourhood $U$ of $X_{t_1}$, such that
$\Psi^{t_1}_{t_2}(U)$ is Stein. For a K\"ahler variety,
consider an open ball $B$ of radius $r$, taken with respect to the
metric defined by geodesics. Then, for $r$ sufficiently small,
$B$ is Stein (\cite{_Greene_}). Since $\Psi^{t_1}_{t_2}$ is an
isometry (\ref{_triana_subse_isome_Theorem_}),
an image of an open ball of radius $r$ is again
an open ball of radius $r$. This gives a
system of Stein neighbourhoods satisfying
{\bf (*)}. \ref{_Psi_iso_on_norma_Proposition_}
is proven. \blacksquare
\subsection{Homeomorphisms of completely geodesic subvarieties
induce isomorphisms of Zariski tangent spaces.}
\hfill
\proposition\label{_homeo_indu_iso_Zariski_Proposition_}
Let $M_1$, $M_2$ be a K\"ahler manifolds,
$X_1^{ns}\subset M_1$, $X_2^{ns}\subset M_2$
be completely geodesic complex submanifolds, not necessarily
closed, and $X_1$, $X_2$ be the closures of $X_1^{ns}$, $X_2^{ns}$ in
$M_1$, $M_2$. Assume that $X_1$, $X_2$ are complex analytic
subvarieties of $M_1$, $M_2$. Let $\phi:\; X_1{\:\longrightarrow\:} X_2$ be a morphism
of complex varieties, such that $\phi\restrict{X_1^{ns}}$ is an isometry.
Then $\phi$ induces an isomorphism of Zariski tangent spaces.
\hfill
{\bf Proof:}
To prove \ref{_homeo_indu_iso_Zariski_Proposition_}, we interpret
the Zariski tangent space
$T_x X_1$ in terms of the metric structure on $X_i^{ns}$, where $x$
is a point of $X_1$. Let $\gamma:\; [0,1] {\:\longrightarrow\:} X_1$ be a path satisfying
$\gamma\left([0,1] \backslash\{0.5\}\right) \subset X_1^{ns}$,
$\gamma(0.5) =x$.
There is a natural topology on the total space $Tot(T X_1^{ns})$ ,
which comes from the embedding
$Tot(TX_1) \stackrel i\hookrightarrow Tot(TM_1)$.
Since $X_1$ is completely geodesic, $i$ is an isometry.
This topology is compatible with the map
$d\phi:\; TX_1 {\:\longrightarrow\:} TX_2$, because $d\phi$ is also an isometry.
Assume that $\gamma$ is differentiable
outside of $\{0.5\}$ and
\[ \lim\limits_{t\to +0.5} \frac{d\gamma}{dt} =
\lim\limits_{t\to -0.5} \frac{d\gamma}{dt}.
\]
(the limits are taken in the metric completion of $TX_1$,
which might be considered as a subset of $Tot(TM_1)$).
The Zariski tangent space $T_x X_1$ can be identified with
equivalence classes of such paths as follows. Two
paths $\gamma, \gamma'$ are equivalent if
\[
\lim\limits_{t\to 0.5}
\frac{\rho(\gamma_1(t), \gamma_2(t))}{(t-0.5)^2} =0,
\]
where $\rho$ is the distance function in $M_1$. Since the
distance in $M_1$ coinsides with the distance in $X_1$, and
$\phi$ is isometry, this equivalence relation is compatible
with $\phi$. \blacksquare
\subsection{Algebro-geometric properties of the map $\Psi^{s_1}_{s_2}$.}
Let $M$ be a compact hyperk\"ahler manifold, $X\subset M$ a
closed trianalytic subvariety. Fix a choice of induced complex structure.
Let $D(x)$ be the Douady space of $X$, and $s_1, s_2$ points
on $D(X)$, and $\gamma$ a real analytic path in $D(x)$
connecting $s_1$ and $s_2$. Let $X_1 = X_{s_1}, X_2 = X_{s_2}$
be the subvarieties of $M$ corresponding to $s_1$, $s_2$,
and $\Psi:\; X_1 {\:\longrightarrow\:} X_2$ be the bimeromorphic map
of \ref{_Psi_bimero_Corollary_}. Let $X_1^{ns}$, $X_2^{ns}$ be the
non-singular part of $X_1$, and $\Psi^{ns}$ be the restriction of $\Psi$
to $X_1^{ns}$. Let $\Gamma^{ns}\subset M\times M$ be a graph of
$\Phi^{ns}$, and $\Gamma$ its closure. Clearly, $\Gamma$
is a complex analytic subvariety of $M\times M$.
Let $\pi_i:\; \Gamma {\:\longrightarrow\:} X_i$ be the projections of
$\Gamma$ to $X_i$, which are obviously morphisms
of complex varieties.
\hfill
\proposition\label{_pi_i_properties_Proposition_}
The maps $\pi_i$, $i= 1,2$, have the following properties.
\begin{description}
\item[(i)] $\pi_i$ is finite.
\item[(ii)] $\pi_i$ is dominant and induces isomorphism
of normalizations
\item[(iii)] For every point $x\in \Gamma$, the differential
$d \pi_i :\; T_x \Gamma {\:\longrightarrow\:} T_{\pi_i(x)} X_i$ is an
isomorphism.
\end{description}
{\bf Proof:} The statement
(ii) follows from \ref{_Psi_iso_on_norma_Proposition_}
and (iii) from \ref{_homeo_indu_iso_Zariski_Proposition_}.
To prove \ref{_pi_i_properties_Proposition_} (i), consider the
normalization map $n:\; \tilde \Gamma {\:\longrightarrow\:} \Gamma$. On the
level of rings of functions, we have an embedding
\[
{\cal O}(X_i) \hookrightarrow {\cal O}(\Gamma) \hookrightarrow
{\cal O}(\tilde \Gamma).
\]
Since $\tilde \Gamma$ is a normalization of $X_i$ by (ii),
the ring ${\cal O}(\tilde \Gamma)$ is finitely generated
as a ${\cal O}(X_i)$-module. Since ${\cal O}(X_i)$ is Noetherian
(\cite{_Grauert_}), this implies that ${\cal O}(\Gamma)$
is also finitely generated as a ${\cal O}(X_i)$-module.
This proves \ref{_pi_i_properties_Proposition_} (i).
\blacksquare
\hfill
To prove that the map $\Psi:\; X_1 {\:\longrightarrow\:} X_2$, a.k.a.
$\Psi^{s_1}_{s_2}:\; X_{s_1} {\:\longrightarrow\:} X_{s_2}$
is holomorphic, it suffices to show that $\pi_i:\; \Gamma {\:\longrightarrow\:} X_i$
is an isomorphism. In the following Subsection, we show that
conditions (i) -- (iii) of \ref{_pi_i_properties_Proposition_}
are {\it a priori} sufficient to establish that $\pi$ is
an isomorphism. This will finish the proof of
\ref{_triana_subse_comple_ana_Theorem_}.
\subsection{Finite dominant unramified morphisms of complex varieties.}
\hfill
\proposition \label{_alge_geo_suffi_pi_i_iso_Proposition_}
Let $\phi:\; X {\:\longrightarrow\:} Y$ be a map of complex varieties satisfying
conditions (i)--(iii) of \ref{_pi_i_properties_Proposition_}.
Then $\phi$ is an isomorphism.
\hfill
{\bf Proof:} The map $\phi$ is one-to-one in general point (by (ii)).
Thus, to prove that $\phi$ is an isomorphism it suffices to show
that $\phi$ is etale. On the other hand, $\phi$ is unramified by
(iii). By definition of etale morphisms,
to prove that $\phi$ is etale
it remains to show that $\phi$ is flat. To conclude the
proof of \ref{_alge_geo_suffi_pi_i_iso_Proposition_}, we
use the following lemma.
\hfill
\lemma \label{_unrami_domi_flat_Lemma_}
Let $\phi:\; X {\:\longrightarrow\:} Y$ be a dominant morphism of complex varieties.
Assume that for every point $x\in X$, the map $\phi$ induces an
isomorphism $d\phi:\; T_x X {\:\longrightarrow\:} T_{\phi(x)} Y$ of Zariski
tangent spaces. Then $\phi$ is flat.
\hfill
{\bf Proof:} Let $y=\phi (x)$. Conside the associated morphism
of local rings ${\cal O}_y Y \stackrel {\phi_x} \hookrightarrow
{\cal O}_x X$. To prove
that $\phi$ is flat, it suffices to show that $\phi$ is an isomorphism.
Let ${\mathfrak m}_x$, ${\mathfrak m}_y$ be the maximal ideals
in ${\cal O}_x X$, ${\cal O}_y Y$. Then
${\mathfrak m}_x/{\mathfrak m}_x^2$ is generated by $\phi_x({\mathfrak m}_y)$.
Thus, by Nakayama, $\phi_x({\mathfrak m}_y)$ generate ${\mathfrak m}_x$,
and we obtain ${\mathfrak m}_x= \phi_x({\mathfrak m}_y)\otimes {\cal O}_x X$.
Consider ${\cal O}_x X$ as ${\cal O}_y Y$-module. Then
${\cal O}_x X/{\mathfrak m}_y {\cal O}_x X$ is one-dimensional.
Applying Nakayama once more, we obtain that ${\cal O}_x X$
is an ${\cal O}_y Y$-module generated by $1\in{\cal O}_x X$.
Thus, the map $\phi_x$ is surjective. Since $\phi$ is
dominant, $\phi_x$ is also injective. Thus, ${\cal O}_x X$
is a free ${\cal O}_{\phi(x)} Y$-module for every $x\in X$.
By the local criterion of flatness, this implies that
$\phi$ is flat. \ref{_unrami_domi_flat_Lemma_} is proven.
This finishes the proof of \ref{_alge_geo_suffi_pi_i_iso_Proposition_}
and \ref{_triana_subse_comple_ana_Theorem_} \blacksquare
\section{Singular hyperk\"ahler varieties.}
\label{_singu_hype_Section_}
It is an intriguing question, what is the
``correct''\footnote{``Correct'' in Platonic sense: some mathematicians
presume that the unique ``correct'' definition of each and every
significant mathematical object exists in itself and independently
of human perception.}
definition of a singular hyperk\"ahler variety. We don't pretend to
answer this question. Instead, we give an {\it ad hoc} set of axioms
which describe some known examples (deformation spaces of stable bundles
and trianalytic subvarieties). It is likely that this {\it ad hoc}
definition is stronger than the ``correct'' one. A more elegant approach
was suggested by Deligne and Simpson
(\cite{_Deligne:defi_}, \cite{_Simpson:hyperka-defi_}).
\hfill
\definition\label{_singu_hype_Definition_}
(\cite{_Verbitsky:Hyperholo_bundles_}, Definition 6.5)
Let $M$ be a hypercomplex variety (\ref{_hypercomplex_Definition_}).
The following data define a structure of {\bf hyperk\"ahler variety}
on $M$.
\begin{description}
\item[(i)] For every $x\in M$, we have an ${\Bbb R}$-linear
symmetric positively defined
bilinear form $s_x:\; T_x M \times T_x M {\:\longrightarrow\:} {\Bbb R}$
on the corresponding real Zariski tangent space.
\item[(ii)] For each triple of induced complex structures
$I$, $J$, $K$, such that $I\circ J = K$, we have a
holomorphic differential 2-form $\Omega\in \Omega^2(M, I)$.
\item[(iii)]
Fix a triple of induced complex structure
$I$, $J$, $K$, such that $I\circ J = K$. Consider the
corresponding differential 2-form $\Omega$ of (ii).
Let $J:\; T_x M {\:\longrightarrow\:} T_x M$ be an endomorphism of
the real Zariski tangent spaces defined by $J$, and $Re\Omega\restrict x$
the real part of $\Omega$, considered as a bilinear form on $T_x M$.
Let $r_x$ be a bilinear form $r_x:\; T_x M \times T_x M {\:\longrightarrow\:} {\Bbb R}$
defined by $r_x(a,b) = - Re\Omega\restrict x (a, J(b))$.
Then $r_x$ is equal to the form $s_x$ of (i). In particular,
$r_x$ is independent from the choice of $I$, $J$, $K$.
\end{description}
\noindent \remark \nopagebreak
\begin{description}
\item[(a)] It is clear how to define a morphism of hyperk\"ahler varieties.
\item[(b)]
For $M$ non-singular, \ref{_singu_hype_Definition_} is
equivalent to the usual
one (\ref{_hyperkahler_manifold_Definition_}).
If $M$ is non-singular,
the form $s_x$ becomes the usual Riemann form, and
$\Omega$ becomes the standard holomorphically symplectic form.
\item[(c)] It is easy to check the following.
Let $X$ be a hypercomplex subvariety of a hyperk\"ahler
variety $M$. Then, restricting the forms $s_x$ and $\Omega$
to $X$, we obtain a hyperk\"ahler structure on $X$. In particular,
trianalytic subvarieties of hyperk\"ahler manifolds are always
hyperk\"ahler, in the sense of \ref{_singu_hype_Definition_}.
\end{description}
\hfill
{\bf Caution:} Not everything which is seemingly hyperk\"ahler
satisfies the conditions of \ref{_singu_hype_Definition_}.
Take a quotient $M/G$ os a hyperk\"ahler manifold by an action
of finite group $G$, acting in accordance with hyperk\"ahler
structure. Then $M/G$ is, generally speaking, {\it not} hyperk\"ahler
(in fact, $M/G$ is {\it never} hyperk\"ahler). For instance,
take a quotient of a 2-dimensional torus $T$ by $G=\{\pm 1\}$
acting as an involution $t{\:\longrightarrow\:} -t$. This is a beautiful and well
known example of a hyperk\"ahler automorphism; the quotient space
has 16 isolated singular points, which, if blown up, give
a K3 surface. For $x$ a singular point of $T/\{\pm 1\}$,
its Zariski tangent space has real dimension 6.
On the other hand, for a hypercomplex variety,
there is a quaternion action in every Zariski tangent
space, and thus, the dimension real dimension
of Zariski tangent space must be divisible by 4. We obtain that
the space $T/\{\pm 1\}$ is not even hypercomplex. How this happens?
We take a twistor space $\operatorname{Tw}(T)$ of $T$ and take a quotient of
$\operatorname{Tw}(T)$ by the natural action of $G=\{\pm 1\}$, which is
holomorphic. The quotient is a complex variety fibered over ${\Bbb C} P^1$
For $\operatorname{Tw}(T)/\{\pm 1\}$ to be hypercomplex, this fibration must be trivial,
in a real analytic category. But the functor of forgetting the complex
structure does not commute with taking finite quotients! Thus, even
if $Tw(T)$ is (as a real analytic space) trivially fibered over
${\Bbb C} P^1$, there is no way to push down this trivialization
to $\operatorname{Tw}(T)/\{\pm 1\}$.
\hfill
The following theorem, proven in
\cite{_Verbitsky:Hyperholo_bundles_} (Theorem 6.3),
gives a convenient way to construct
examples of hyperk\"ahler varieties.
\hfill
\theorem \label{_hyperho_defo_hyperka_Theorem_}
Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced
complex structure and $B$ a stable holomorphic bundle over $(M, I)$.
Let $D(B)$ be a deformation space of stable holomorphic structures on $B$.
Assume that $c_1(B)$, $c_2(B)$ are $SU(2)$-invariant, with respect
to the standard action of $SU(2)$ on $H^*(M)$. Then $D(B)$ has a
natural structure of a hyperk\"ahler variety.
\nopagebreak
\blacksquare
\hfill
The following theorem is implicit in
\cite{_Verbitsky:Hyperholo_bundles_}.
\hfill
\theorem \label{_hyperholo_functo_Theorem_}
Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced
complex structure and $B_1, B_2, ..., B_n$
stable holomorphic bundles over $(M, I)$.
Let $D(B_i)$ be a deformation space of stable
holomorphic structures on $B_i$.
Assume that $c_1(B_i)$, $c_2(B_i)$, $i=1, 2, ..., n$
are $SU(2)$-invariant, with respect
to the standard action of $SU(2)$ on $H^*(M)$.
Let $\Pi$ be a natural tensor operation on the vector bundles,
such that, e. g.,
\[ B_1, ... B_n {\:\longrightarrow\:} B_1 \otimes B_2 \otimes \Lambda^2 B_3 \otimes S^7
B_4 \otimes ... \otimes B_n ^*.
\]
Assume that $\Pi(B_1, ... B_n)$ cannot be decomposed to a direct sum
of holomorphic bundles. Then $\Pi(B_1, ... B_n)$ is stable,
and the associated map
\[ D(B_1) \times D(B_2) \times ..., \times D(B_n){\:\longrightarrow\:}
D\left(\Pi(B_1, ..., B_n)\right)
\]
(defined in a certain neighbourhood of
$[B_1]\times [B_2]\times ..., \times [B_n] \in
D(B_1) \times D(B_2) \times ..., \times D(B_n)$
is a morphism of hyperk\"ahler varieties.
\blacksquare
\hfill
\ref{_hyperholo_functo_Theorem_} gives a natural way to
construct trianalytic subvarieites of hyperk\"ahler varieties.
\hfill
The following theorem is almost trivial; the reader is advised
to invent his or her own proof instead of reading ours
(which is by necessity sketchy).
\hfill
\theorem \label{_Doua_hyperka_Theorem_}
Let $M$ be a compact hyperk\"ahler manifold, $X\subset M$
a trianalytic subvariety and $D(X)$ its Douady
space. Then $D(X)$ is a compact hyperk\"ahler variety.
\hfill
{\bf Proof:} Clearly, $D(X)$ is hypercomplex, in a natural way
(\ref{_Douady_hyperc_Remark_}). It remains to construct
the forms $s_x$ and $\Omega$. Let $Y$ be a trianalytic
subvariety of $M$ which is a deformation of $X$,
and $U$ be a sufficiently small neighbourhood of $[Y]\in D(X)$.
Let
$\left(\pi:\; {\c X}_U {\:\longrightarrow\:} U,\phi:\; {\c X}_U {\:\longrightarrow\:} M\right)$
be the universal family of subvarieties
of $M$, attached to $U\subset D(X)$. The space ${\c X}_U$
is hypercomplex and the map $\pi$
is compatible with the hypercomplex structure.
Moreover, the map $\phi:\; {\c X}_U {\:\longrightarrow\:} M$ is an
immersion, so ${\c X_U}$ is hyperk\"ahler (hyperk\"ahler structure
is obtained as a pullback from $M$).
\ref{_conne_in_triana_flat_Theorem_}
provides a natural trivialization of
${\c X}_U {\:\longrightarrow\:} U$. Thus, for each $y\in Y$,
there exists a natural section $\sigma_y:\; U\hookrightarrow {\c X}_U$.
By \ref{_triana_subse_comple_ana_Theorem_},
this section is compatible with the hypercomplex structure.
Restricting the hyperk\"ahler structure
from ${\c X_U}$ to $\sigma_y(U)$, we obtain a hyperk\"ahler
structure on $U$. It is easy to check that this hyperk\"ahler
structure is independent from the choice of section
$\sigma_y$. Gluing the hyperk\"ahler structures from
different $U$, we obtain the proof of
\ref{_Doua_hyperka_Theorem_}. \blacksquare
\hfill
{\bf Acknowledgements:} I am indebted to D. Kaledin for enlightening
discussions; he also suggested some of the proofs.
I am thankful to Tony Pantev for the frutiful discussions.
Mohan Ramachandran communicated me the reference
to \cite{_Greene_}. I am grateful to A. Beilinson and J. Bernstein
for the help with algebraic geometry. My gratitude to
M. Finkelberg, D. Kazhdan and S.-T. Yau for their
interest.
\hfill
|
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